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\begin{document}
\title{Incorporating Kinematic Wave Theory into a Deep Learning Method for High-Resolution Traffic Speed Estimation}
\author[1,2]{Bilal~Thonnam~Thodi$^{1,2}$,
~Zaid~Saeed~Khan$^{\dagger1,2}$,
~Saif~Eddin~Jabari$^{1,2}$,
and~M\'onica~Men\'endez
}
\affil[1]{New York University Tandon School of Engineering, Brooklyn NY, U.S.A.} \affil[2]{New York University Abu Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi, U.A.E.} \affil[$^{\dagger}$]{Corresponding author. Email: \href{mailto:zaid.khan@nyu.edu}{zaid.khan@nyu.edu}}
\date{}
\twocolumn[ \begin{@twocolumnfalse}
\maketitle
\begin{abstract}
We propose a kinematic wave-based Deep Convolutional Neural Network (Deep CNN) to estimate high-resolution traffic speed fields from sparse probe vehicle trajectories. We introduce two key approaches that allow us to incorporate kinematic wave theory principles to improve the robustness of existing learning-based estimation methods. First, we propose an anisotropic traffic kernel for the Deep CNN. The anisotropic kernel explicitly accounts for space-time correlations in macroscopic traffic and effectively reduces the number of trainable parameters in the Deep CNN model. Second, we propose to use simulated data for training the Deep CNN. Using a targeted simulated data for training provides an implicit way to impose desirable traffic physical features on the learning model. In the experiments, we highlight the benefits of using anisotropic kernels and evaluate the transferability of the trained model to real-world traffic using the Next Generation Simulation (NGSIM) and the German Highway Drone (HighD) datasets. The results demonstrate that anisotropic kernels significantly reduce model complexity and model over-fitting, and improve the physical correctness of the estimated speed fields. We find that model complexity scales linearly with problem size for anisotropic kernels compared to quadratic scaling for isotropic kernels. Furthermore, evaluation on real-world datasets shows acceptable performance, which establishes that simulation-based training is a viable surrogate to learning from real-world data. Finally, a comparison with standard estimation techniques shows the superior estimation accuracy of the proposed method.
\textbf{\fontfamily{cmss}\selectfont\color{red!40!black} Keywords}:
Traffic state estimation, traffic anisotropy, kinematic wave theory, convolutional neural networks, deep learning. \end{abstract}
\end{@twocolumnfalse} ]
\section{Introduction} \label{sec1}
Traffic management agencies use a variety of monitoring and control tools to ensure the safe and efficient operation of network road traffic. To meet their operational goals, agencies employ tools that identify disturbances and deploy effective control strategies in real time \cite{papageorgiou2004overview}. However, this requires accurate and timely knowledge of traffic conditions over the entire network, which is currently not possible given the limited sensory instrumentation in most (if not all) cities today. Fixed sensors are expensive and tend to be sparsely installed, offering limited spatial coverage. Data from mobile sensors are expected to become more widely available than data from point sensors, but remain extremely limited in practice; their sparsity is temporal \cite{ambhul2016mfd}. To address such data sparsity (spatially or temporally), we need appropriate mechanisms that fill the gaps in the traffic observations. These are known as \emph{traffic state estimation} (TSE) tools \cite{seo2017traffic}. TSE is a critical precursor to a number of real-time traffic control strategies with either conventional vehicles or a mix with connected and autonomous vehicles \cite{papageorgiou2004overview,li2021backpressure}. Such strategies include, but are not limited to, ramp metering, perimeter control, traffic signal control, and vehicle routing \cite{li2019position,yang2017perimeter,lin2021pay}.
Existing TSE approaches can be broadly divided into two categories: model-based and data-driven \cite{seo2017traffic}. The former approach adopts a mathematical model of traffic flow such as the first-order Lighthill-Whitham-Richards (LWR) model \cite{lighthill1955kinematic,richards1956shock} or one of its many higher-order extensions, like the Aw-Rascle-Zhang (ARZ) model \cite{aw2000resurrection,zhang2002non}. These methods assimilate flow model predictions with real-world observations using an exogenous filter (e.g., Ensemble Kalman filter) \cite{vanerp2020relflow,dakic2018smsestimate,fountoulakis2017highway,bekiaris2016highway,hoogen2012lagrang,nantes2016real,jabari2012stochastic,jabari2013gauss}. Traffic flow models ensure that estimates respect basic traffic principles. However, the models are based on simplifying assumptions of traffic physics that can lead to numerical bias when the assumptions are not met. Furthermore, approximation errors can arise from the data assimilation techniques used in TSE. For instance, it is common to linearize a non-linear flow model for the recursive estimation, and the approximations are poor around the capacity region \cite{seo2017traffic,hoogen2012lagrang}. Lastly, model-based methods require additional inputs (e.g., boundary conditions) which are difficult to obtain in real-time.
The other category of TSE approaches include data-driven/learning techniques, which build statistical/machine learning models from large volumes of (historical) traffic data. Some commonly employed tools include (predominantly) deep neural networks \cite{benkraouda2020traffic,yuhan2016dnn-speed}, support vector regression \cite{xiao2018speed}, principal component analysis \cite{li2013efficient}, and matrix factorization methods \cite{li2021nonlinearts}. The estimation results from data-driven methods are often reported to be more accurate than model-based approaches, but these methods also have shortcomings. Being purely data-driven, the models are agnostic to the physics of traffic flow and could lead to infeasible estimation results. These methods are also not often interpretable and lack robustness. More importantly, the generalizability of the models is often weak and depends on the training data distribution.
We aim to develop a methodology that incorporates the desirable features of both categories, namely the combination of domain knowledge with representational power. Such structured learning methods can ensure robust and interpretable estimation results, parsimonious model complexity, and reduced data requirements. Some recent works along these lines include \cite{yuan2021phygp,zhang2020hybrid,jabari2020sparse,jabari2019learning,kaidi2019queueest,jabari2018stochastic}. A common flavor in these approaches is to impose the physical constraints as cost function regularizers (i.e., as soft constraints) or derive the learning model architecture from physical principles. For instance, \cite{jabari2019learning,jabari2018stochastic} combine predictions of stochastic traffic flow model and limited probe vehicle data to infer vehicle trajectory distributions that are consistent with traffic physics. \cite{kaidi2019queueest} estimates queue lengths at signalized intersection as a solution to a convex optimization problem with queue propagation constraints guided by the kinematic wave theory of traffic flow. \cite{yuan2021phygp} uses the dynamical equations of macroscopic flow models to regularize a Gaussian Process regression model, which is efficient in handling sparse and noisy data.
In the context of deep learning, which is a more attractive choice for non-linear modeling, \cite{shi2021pnntse,pmlr2021barreau,aai2021shi,barreau2021physics,huang2020physics} approximate the solution of a macroscopic traffic flow model using deep neural networks and use the governing physical dynamical equations (in the form of PDE/ODE) as a regularizer in the cost function. They demonstrate that these physics informed regularizers reduce the space of feasible solutions and learn solutions that are consistent with the chosen traffic flow models under limited real-world data. However, this requires training deep neural networks for every instance of initial/boundary conditions, which is computationally expensive for real-time implementation.
We propose an addition to this nascent literature on structured learning methods for TSE that incorporates traffic domain knowledge into learning models. Specifically, we propose a methodology to estimate high-resolution macroscopic traffic speed fields from limited probe vehicle measurements. We use a Deep Convolutional Neural Network (Deep CNN) as the learning model for estimation. The Deep CNN model takes as input sparse vehicle trajectory measurements and outputs a high-resolution speed field over a given space-time domain. The model is trained offline and can then be applied for real-time estimation. We incorporate traffic-specific features into the learning model in two ways, which are described below.
First, the na\"ive isotropic kernels in the Deep CNN model are modified to capture the wave propagation characteristics of free-flowing and congested traffic, in accordance with the kinematic wave theory (KWT) of traffic flow \cite{newell1993simplified}. We develop a Deep CNN with \emph{anisotropic kernels} designed to consider space-time inputs that are in the direction of feasible traffic waves, bounded by forward waves in free-flow and backward waves in congested traffic. As a result, we can significantly reduce the effective number of kernel parameters and hence the Deep CNN model complexity. Further, restricting the CNN to consider only the relevant spatio-temporal input points results in feasible and robust estimation of traffic shockwaves.
Second, we train our Deep CNN model using simulated traffic data. Apart from resolving the data availability issue, this approach allows us to take an empirical distribution of any desirable traffic flow model and use it to train the Deep CNN. The empirical distribution is a surrogate representation of the traffic physics underlying the simulation model. This is a broader approach to incorporate the governing physics as it is easier to generate data corresponding to complex traffic behaviors rather than integrating them into the model architecture as in existing physics-informed learning methods. We demonstrate this by training the Deep CNN model with data generated from a microscopic traffic simulator, which consists of behavioral car-following, lane-changing and gap-acceptance models, and then test it with real-world data having similar traffic characteristics. A natural trade-off of this approach is that the learning model does not capture the exact physical traffic dynamics, but can incorporate a wide range of complex traffic behaviors. Similar methods have been explored in the context of automated systems such as robotic controls and object detection, whereby researchers use high-fidelity simulators or synthetic data instead of real-world data to train deep neural network models \cite{abbeel2017domain,jonathan2018domainrand}.
To summarize, the contributions of this paper are: \begin{enumerate}
\item We develop an anisotropic kernel design for CNNs following the wave propagation characteristics of traffic flow. This could be applied to traffic state estimation, prediction, and data imputation. We also suggest an optimization procedure to learn the optimal weights for the anisotropic kernels.
\item We propose to use simulated traffic data for fitting the anisotropic Deep CNN model and test its performance on real-world datasets.
\item We demonstrate the use of the anisotropic Deep CNN model for speed field estimation at fine space-time resolutions ($10$ meters $\times$ $1$ second in our experiments) using limited input vehicle trajectories ($5\%$ probe vehicle penetration rates). We show sample estimations of real-world traffic data from multiple sources.
\item We extend our estimation methodology to handle unknown probe vehicle penetration rates by introducing an ensemble version of our Deep CNN model. \end{enumerate}
The rest of the paper is structured as follows. We present the estimation problem setting, the anisotropic kernel design, and the optimization procedure in Section \ref{sec2}. We then describe the training data generation and the training experiments in Section \ref{sec3}. In Section \ref{sec4}, we present estimation results, compare the anisotropic CNN with the na\"ive isotropic variant, discuss the transferability of the estimation model to real-world freeway traffic, and explore the sensitivity of the results to different probe vehicle penetration levels. Finally, we conclude the study in Section \ref{sec5}.
\section{Estimation Methodology} \label{sec2}
\subsection{Traffic speed field estimation problem} \label{sec2_1}
A space-time domain $\mathcal{D} = \mathcal{X} \times \mathcal{T}$ representing a given road section is discretized into homogeneous segments $x_i \subset \mathcal{X}$ and time intervals $t_i \subset \mathcal{T}$, such that $\cup_i x_i = \mathcal{X}$ and $\cup_i t_i = \mathcal{T}$. Let $V(x,t)$ denote the value of the macroscopic speed field in a cell $(x,t) \in \mathcal{D}$. We use the cell size $|x|$ closer to the length of a vehicle and $|t|$ in the order of seconds (smaller than what existing estimation methods use \cite{seo2017traffic}) to enable high-resolution speed field estimation. Probe vehicles (PVs) provide local speed measurements $\big\{V^{\mathrm{p}}(x_i^\mathrm{p}, t_i^\mathrm{p})\big\}$ for some cells in $\mathcal{D}$; we represent this partial information by the tensor $\mathbf{z}^{\mathrm{p}}$. We assume sparse observation settings, where only a few cells (e.g., 5-10\%) in $\mathbf{z}^{\mathrm{p}}$ have speed information. We denote by $\mathbf{z}^{\mathrm{f}}$ a tensor of estimates of the complete speed field $V(x,t)$ for the entire space-time domain $\mathcal{D}$. The estimation problem can be formally stated as learning a mapping function $g: \mathbf{z}^{\mathrm{p}} \mapsto \mathbf{z}^{\mathrm{f}}$.
The speed field $V(x,t)$ in each cell of the input tensor $\mathbf{z}^\mathrm{p}$ is encoded using a three-dimensional RGB array (with domain $\{1,\hdots,255\}$) instead of a one-dimensional speed value. This is to differentiate cells occupied with a stopped vehicle (i.e., with $V(x,t)=0$) from empty cells. The output tensor $\mathbf{z}^{\mathrm{f}}$ represents the complete macroscopic speed field over the domain $\mathcal{D}$ and can be encoded using the one-dimensional speed values. Thus, we have, $\mathbf{z}^\mathrm{p} \in \{1,\hdots,255\}^{|\mathcal{X}| \times |\mathcal{T}| \times 3}$ and $\mathbf{z}^\mathrm{f} \in \mathbb{R}_{\ge0}^{|\mathcal{X}| \times |\mathcal{T}|}$.
\subsection{Deep Convolutional Neural Network (Deep CNN) model for estimation} \label{sec2_2}
We use a Deep CNN model similar to the one in \cite{benkraouda2020traffic} to represent the mapping function $g$. The model architecture is shown in Fig. \ref{fig1:cnn}. It comprises an encoder $g_\mathrm{enc}$ and a decoder $g_\mathrm{dec}$, each consisting of three CNN layers. Each CNN layer is composed of a 2D convolution operation, a non-linear activation operation called ReLU (Rectified Linear Unit), and a down-sampling operation called max-pooling (up-sampling operation called nearest neighbor in case of $g_\mathrm{dec}$). As shown in Fig.~\ref{fig1:cnn}, the successive CNN layers of $g_\mathrm{enc}$ have reduced spatio-temporal widths and the successive CNN layers of $g_\mathrm{dec}$ have increased spatio-temporal widths. The Deep CNN model takes the input $\mathbf{z}^\mathrm{p}$, passes it through the hierarchical convolution layers, and outputs the estimated speed field $\mathbf{z}^\mathrm{f}$.
\begin{figure}
\caption{The architecture of the Deep CNN (speed reconstruction model).}
\label{fig1:cnn}
\end{figure}
Unlike other neural network architectures, CNNs have proven to be effective in learning spatial data (e.g., images, video, etc.), which is useful for our application since the space-time diagram reflects spatial data. The CNN model has two properties favorable for learning macroscopic traffic features: local connectivity and parameter sharing. The former assumes the traffic speed fields are locally correlated, and the latter implies local traffic features can occur anywhere in the space-time plane, i.e., they are space-time invariant. Furthermore, the specific encoder-decoder structure (bottleneck formation) of the model shown in Fig.~\ref{fig1:cnn} can efficiently handle the sparse nature of the model input \cite{benkraouda2020traffic}.
The discrete convolution using local kernels in the CNN forms the basis of traffic speed field estimation. In a given CNN layer $l$, a convolution operation calculates the activation in a cell $(x, t)$ as a weighted sum of cell activations observed in the previous layer $(l-1)$: \begin{multline}
\mathbf{z}^{(l)} (x, t, \chi) = \mathbf{z}^{(l-1)}(\cdot,\cdot,\chi) \ast \Theta^{(l)}(\cdot,\cdot) \\ = \sum_{(x_j, ~t_j) \in ~I_\mathrm{iso}} \mathbf{z}^{(l-1)}\left( x_j, t_j, \chi \right) \Theta^{(l)} \left( x_j, t_j \right),
\label{eqn1:conv} \end{multline}
where $\mathbf{z}^{(l)} (x, t, \chi)$ is the feature map value in layer $l$ associated with cell $(x,t)$ and color channel $\chi\in\{1,2,3\}$, $\Theta^{(l)}(\cdot,\cdot) \in \mathbb{R}^{|\mathcal{X}| \times |\mathcal{T}|}$ is the kernel (matrix), which is identical for all cells. $\Theta^{(l)} \left( x_j, t_j \right)$ on the right-hand side, an element of the kernel matrix, determines the extent to which neighboring cell $(x_j,t_j) \in I_\mathrm{iso}$ is correlated with the subject cell $(x,t)$. Hereafter, we simply write $\Theta$ to represent the entire kernel, and drop the `$(\cdot,\cdot)$'.
The feature map value in cell $(x,t)$ can be considered as equivalent to (or some function of) the speed field $V(x,t)$ in that cell. Then, operation \eqref{eqn1:conv} simply says: the speed in cell $(x,t)$ is a weighted interpolation of speeds observed in its immediate surrounding cells. The extent of local cell influence $I_\mathrm{iso}$ is depicted visually on the space-time plane in Fig. \ref{fig2:conv}(a). Each kernel in a layer $l$ represents a different weighting function; together, the kernels learn to identify different traffic features.
\subsection{Anisotropic kernel design for Deep CNN} \begin{figure}
\caption{Space-time correlations modeled by the isotropic kernel of the convolution operation, and that in the real traffic (free-flow and congested).}
\label{fig2:conv}
\end{figure}
The isotropic kernel shown in Fig. \ref{fig2:conv}(a) says that the speed in cell $(x,t)$ is correlated with the speeds observed \textit{anywhere} in the shaded rectangular region $I_{\mathrm{iso}}$. This assumes that a speed variation (such as that caused by slowdowns or speed-ups) at $(x,t)$ can propagate at unbounded velocities in the space-time plane. However, this is not true in real traffic. In real traffic, (i) the speed/density variations propagate at finite velocities that are less than or equal to the free-flowing vehicle speed, and (ii) vehicles respond (predominantly) to frontal stimuli with a delay (approximately equal to the reaction time of driver). The former condition is called hyperbolicity and the latter is called anisotropy. Hyperbolicity is a necessary but not sufficient condition for anisotropy in traffic flow models \cite{hoogendorn2013ani,daganzo1995req,treiber2013traffic}. The use of local kernels (i.e., kernel dimensions $\ll$ the dimension of space-time plane) captures the hyperbolicity property, whereas anisotropy can be captured by modifying the kernel shape as discussed below.
The actual propagation velocity of speed variations depends on the traffic state (i.e., speed or density). We assume that traffic at any point in the space-time plane is either in free-flow or is congested in relation to the Fundamental Diagram. There are different traffic conditions associated with free-flow and congestion, respectively. Then, a speed variation in cell $(x,t)$ propagates downstream (i.e., in the direction of traffic) in free-flowing traffic and upstream (i.e., in the opposite direction of traffic) in congested traffic. This is an empirically and theoretically established feature of traffic \cite{newell1993simplified,treiber2002filter,trieber2011filter,daganzo2005variational,daganzo2005var_bott}. Thus, the extent of the space-time plane correlated with cell $(x,t)$ depends on whether the traffic state is free-flow or congested. The respective correlated regions are shown in Fig. \ref{fig2:conv}(b) and Fig. \ref{fig2:conv}(c) as shaded areas $I_{\mathrm{free}}$ and $I_{\mathrm{cong}}$.
The regions $I_{\mathrm{free}}$ and $I_{\mathrm{cong}}$ are bounded by the free flow traffic speed $c_v$ and the backward shockwave speed $c_w$ \cite{hoogendorn2013ani,treiber2013traffic}, respectively. The speed in cell $(x,t)$ influences the region $I_\mathrm{free}$ downstream, and the region $I_\mathrm{cong}$ upstream. Likewise, the regions $I_\mathrm{free}$ upstream and $I_\mathrm{cong}$ downstream influence the speed in cell $(x,t)$. In summary, the speed predicted in cell $(x,t)$ is correlated with the speeds observed anywhere in $I_\mathrm{free} \cup I_\mathrm{cong}$. We use this knowledge of space-time correlations in designing an alternate and causally \textit{correct} kernel (in the traffic sense) for the Deep CNN model in Fig. \ref{fig1:cnn}. We refer to this as the \textit{anisotropic kernel}, and represent it by the tensor $\Theta_{\mathrm{ani}} = [\Theta_{\mathrm{ani}}^{(l)}]_l$. The corresponding convolution operation is slightly modified from \eqref{eqn1:conv} as, \begin{multline}
\mathbf{z}^{(l)} (x, t, \chi) = \mathbf{z}^{(l-1)}(\cdot,\cdot,\chi) \ast \Theta_{\mathrm{ani}}^{(l)} \\ = \sum_{(x_j, ~t_j) \in ~I_\mathrm{ani}} \mathbf{z}^{(l-1)}\left( x_j, t_j, \chi \right) \Theta_{\mathrm{ani}}^{(l)} \left( x_j, t_j \right),
\label{eqn2:conv2} \end{multline} where the effective influence region is defined as $I_\mathrm{ani} := I_\mathrm{free} \cup I_\mathrm{cong}$. This way, we direct the convolution operator to consider only that portion of the space-time plane which is relevant for the speed interpolation according to traffic physics.
In this paper, we propose a specific anisotropic kernel design, whose influence region is further restricted, motivated by empirical observations: (i) congested traffic has a very narrow range of wave propagation velocities (such that they can be regarded as almost constant), and (ii) free-flow traffic wave propagation velocities are limited within the maximum and minimum desired vehicle speeds \cite{newell1993simplified,treiber2002filter,trieber2011filter,treiber2013traffic,hoogendorn2013ani}.
The anisotropic kernel design to replace the isotropic kernel (from Fig. \ref{fig1:cnn}) is illustrated in Fig. \ref{fig3:ker}. We create two kernels, one each for free-flowing and congested traffic. The influence region $I_\mathrm{free}$ contains all the cells passing and bounded between the maximum ($c_v^{\mathrm{max}}$) and minimum ($c_v^{\mathrm{min}}$) desired vehicle speeds. This is relevant for heterogeneous traffic where the desired speed distribution has a wide range. The free-flow traffic kernel is shown in Fig. \ref{fig3:ker}(a). The influence region for congested traffic, $I_\mathrm{cong}$, contains only those cells passing through the backward propagating shockwave speed $c_w$; see Fig. \ref{fig3:ker}(b). The proposed anisotropic kernel is a superposition of the free-flow and congested kernel. This is shown in Fig. \ref{fig3:ker}(c). The corresponding isotropic kernel is shown in Fig. \ref{fig3:ker}(d) for comparison. One can see that the anisotropic design requires $50\%$ fewer parameters than its isotropic variant for a $7 \times 7$ kernel.
\begin{figure}\label{fig3:ker}
\end{figure}
In summary, our proposed anisotropic kernel design takes three input parameters $\{c_v^{\mathrm{max}}, c_v^{\mathrm{min}}, c_w\}$, whose values depend on the traffic characteristics of the road section. The proposed design aims to learn a broad range of forward propagation speeds and a narrow range of backward propagation speeds. Using a wide distribution for propagation speeds can simultaneously handle different road classes, e.g., highways with different speed limits and arterials. The variability in the free-flow speeds, in addition to capturing differences in speed limits, allows our kernels to capture a variety of kinematic wave speeds as combinations of free-flow waves and backward waves.
\begin{figure}
\caption{Number of model parameters for different widths of isotropic and anisotropic kernel.}
\label{fig:kersize}
\end{figure}
A practical benefit of the proposed anisotropic kernel design is the significant reduction in the model complexity of the Deep CNN model. Model complexity here refers to the total number of model parameters, and depends on the widths and depths of CNN kernels. We quantify the parameter requirements for isotropic and anisotropic kernels as a function of kernel widths in Fig. \ref{fig:kersize}. The number of parameters scales \emph{linearly} for anisotropic kernels as opposed to \emph{quadratically} for isotropic kernels. This implies that anisotropic kernels result in a simpler, lower complexity CNN model which is easier to compute and optimize as compared to its isotropic counterpart. This scaling advantage is realized for higher kernel widths which naturally occur for larger problem sizes. We show these benefits experimentally in Section \ref{sec4} as we compare the model complexity requirements for different road network sizes.
We finally note that the proposed anisotropic kernel design is similar to Treiber and Helbing's adaptive smoothing method for speed interpolation \cite{treiber2002filter} except that: (a) we consider a range of wave propagation speeds in free-flowing traffic instead of a constant value, (b) the weights in the kernel are not set apriori as in \cite{treiber2002filter} but learned from data, and (c) the actual speed predicted is a combination of several anisotropic kernels as opposed to a single anisotropic kernel.
\subsection{Learning anisotropic kernels}
We use anisotropic kernels in all layers of the Deep CNN model in Fig. \ref{fig1:cnn}. The optimal weights $\Theta_{\mathrm{ani}}^*$ for the anisotropic kernel are obtained from the following constrained optimization problem: \begin{equation}
\Theta_{\mathrm{ani}}^* := \underset{\Theta \in \mathbb{R}^{|\mathcal{X}| \times |\mathcal{T}|\times L}}{\arg \min} \big\{ \mathcal{L} \left( \mathbf{z}^\mathrm{f}, g\left( \mathbf{z}^\mathrm{p}, \Theta \right) \right): ~ \mathbbm{1}_\mathrm{ani} \odot \Theta = \mathbf{0} \big\},
\label{eqn3:keropt} \end{equation}
where $g(\mathbf{z}^\mathrm{p}, \Theta): \{0,\hdots,255\}^{|\mathcal{X}| \times |\mathcal{T}| \times 3} \rightarrow \mathbb{R}_{\ge 0}^{|\mathcal{X}| \times |\mathcal{T}|}$ is the mapping function (i.e., the Deep CNN) with the kernel parameterization $\Theta$ made explicit (i.e., $g$ performs the mapping $\mathbf{z}^\mathrm{p} \mapsto \mathbf{z}^\mathrm{f}$), $\mathbbm{1}_\mathrm{ani}$ is a binary tensor of the same dimension as $\Theta$ with values of 0 for cells corresponding to the anisotropic influence cell region $I_\mathrm{ani}$, e.g., the shaded cells in Fig.~\ref{fig3:ker}(c), and 1 elsewhere ($\odot$ is the Hadamard product). The loss function $\mathcal{L}$ captures any discrepancies between the estimated and true speed fields, e.g., the squared $\ell_2$ distance (the squared error): \begin{equation}
\mathcal{L} \left( \mathbf{z}^\mathrm{f}, g\left( \mathbf{z}^\mathrm{p}, \Theta \right) \right) = \left\| \mathbf{z}^\mathrm{f} - g\left( \mathbf{z}^\mathrm{p}, \Theta \right) \right\|_2^2. \label{eqn:loss} \end{equation}
The constrained optimization problem \eqref{eqn3:keropt} can be solved using iterative schemes which can handle feasibility constraints, such as the projected gradient descent. In each iteration $i$, the updates are calculated as follows: \begin{equation}
\Theta_{\mathrm{ani}}^{i+1} := \mathsf{P}_{I_{\mathsf{ani}}}\big( \Theta_{\mathrm{ani}}^{i} - \gamma^{i} G(\Theta_{\mathrm{ani}}^{i}) \big), \end{equation} where $\gamma_{i} > 0$ is the step size (or learning rate) in iteration $i$ and $G(\Theta_{\mathrm{ani}}^{i})$ is a gradient tensor (descent direction) at $\Theta_{\mathrm{ani}}^{i}$. The operator $\mathsf{P}_{I_{\mathrm{ani}}}$ assigns zeros to elements of $\Theta_{\mathrm{ani}}^{i} - \gamma^{i} G(\Theta_{\mathrm{ani}}^{i})$ corresponding to cells that lie outside of $I_{\mathrm{ani}}$, thereby ensuring feasibility of the solutions.
\section{Data and Training} \label{sec3}
As mentioned earlier, we use simulated traffic data consisting of different traffic conditions for training the anisotropic Deep CNN model. In the following, we describe the data used for training and evaluating the model.
\subsection{Training data generation}
To generate data for training the CNN model, we simulate a freeway segment using the Vissim microscopic traffic simulator. The simulated segment corresponds to the \textit{E-22 Abu Dhabi-Al Ain road, UAE} (2 miles in length and 3 lanes wide), and includes an entry and exit ramp to a nearby suburban region. The simulation model is calibrated with general traffic behavior, for instance, prioritizing through movements, appropriate yielding gaps for on-ramp vehicles, and minimum gap for lateral movements. A wide distribution of desired vehicle speeds (ranging from $60-100$ kmph) is used to produce different free-flow wave propagation speeds as is the case for heterogeneous traffic.
We simulate three traffic scenarios with different input vehicle demand profiles on the freeway segment: 800-1200 vehs/hr, 2400-3000 vehs/hr, and 4200-5400 vehs/hr. We used these demand profiles to replicate distinct traffic conditions on the simulated freeway, namely free-flowing, slow-moving (moderately congested), and (heavily) congested traffic. We used on-ramp inflows that constitute 15-20\% of the total freeway flows. Each traffic scenario is simulated for 2 hrs and the vehicle trajectory data for an 800 m homogeneous section on the freeway is recorded. The trajectory data corresponding to three traffic scenarios and their traffic dynamics are summarized in Fig.~\ref{fig:sim_data}.
\begin{figure}
\caption{Visualization of the richness or the traffic features contained in the simulated training dataset (300-second snapshot).}
\label{fig:sim_data}
\end{figure}
Fig.~\ref{fig:sim_data}(a) shows a $300$ second snapshot of vehicle trajectories for the three simulated traffic scenarios. One can note the backward and forward propagating waves due to the stop-and-go, slow-moving, and heterogeneous free-flowing traffic (respectively) in Fig.~\ref{fig:sim_data}(a). The anisotropic kernel is designed based on the range of wave propagation speeds seen in these plots. Fig.~\ref{fig:sim_data}(b) is a flow-density scatter-plot of the three scenarios. Together, these figures show the richness of traffic states contained in the training data.
\subsection{Definition of macroscopic speed field}
An important auxiliary task is to define the ``true'' speed field which the Deep CNN model uses as the ``ground truth'' for evaluating the quality of the estimation. This is achieved by \emph{translating} the set of \emph{all} vehicle trajectories (not just PVs) into a speed field $V(x,t)$. The commonly used generalized definition of macroscopic speeds \cite{eddie1963traffic} results in $V(x,t) = 0$ for some cells due to the fine mesh size we use. Therefore, we propose a simple interpolation method for this purpose instead. Our method interpolates the speeds over the road cells at a fixed time according to: \begin{equation}
V(x,t) =
\begin{cases}
V_\mathrm{up} \left( \frac{d_\mathrm{dn}}{d_\mathrm{up}+d_\mathrm{dn}} \right) + V_\mathrm{dn} \left( \frac{d_\mathrm{up}}{d_\mathrm{up}+d_\mathrm{dn}} \right), & \text{if } d_\mathrm{up} < l_\mathrm{up} \\ & \mkern-18mu \text{and } d_\mathrm{dn} < l_\mathrm{dn} \\
V_\mathrm{up} \left( 1- \frac{d_\mathrm{up}}{l_\mathrm{up}} \right) + V_\mathrm{max} \left( \frac{d_\mathrm{up}}{l_\mathrm{up}} \right), & \text{if } d_\mathrm{up} < l_\mathrm{up} \\ & \mkern-18mu \text{and } d_\mathrm{dn} > l_\mathrm{dn} \\
V_\mathrm{dn} \left( 1- \frac{d_\mathrm{dn}}{l_\mathrm{dn}} \right) + V_\mathrm{max} \left( \frac{d_\mathrm{dn}}{l_\mathrm{dn}} \right), & \text{if } d_\mathrm{up} > l_\mathrm{up} \\ & \mkern-18mu \text{and } d_\mathrm{dn} < l_\mathrm{dn} \\
V_{\mathrm{max}}, & \text{otherwise}, \\
\end{cases}
\label{eqn4:speed} \end{equation} where $V_\mathrm{max}$ is the highest free-flow speed (or speed limit of the highway section), $V_\mathrm{dn}$ (resp. $V_\mathrm{up}$) is the speed of the downstream (resp. upstream) vehicle, $d_\mathrm{dn}$ (resp. $d_\mathrm{up}$) is the distance between the cell $(x,t)$ and the cell containing the downstream (resp. upstream) vehicle, and $l_\mathrm{dn}$ (resp. $l_\mathrm{up}$) is the length of spatial interaction downstream (resp. upstream) of $(x,t)$.
Equation \eqref{eqn4:speed} can be understood as follows: the speed field $V(x,t)$ in cell $(x,t)$ is a weighted combination of the speeds upstream and downstream of the cell. The speed $V_\mathrm{dn}$ of the vehicle downstream of $(x,t)$ has an effect only if it is within the downstream interaction range $l_\mathrm{dn}$ from $(x,t)$; otherwise, its value is replaced by the maximum highway speed $V_\mathrm{max}$ (and analogously for the upstream vehicle). The weights of the upstream and downstream components are proportional to the proximity of the respective interactions. The spatial interaction lengths are chosen to satisfy $l_\mathrm{up} < l_\mathrm{dn}$, to reflect the asymmetrically greater influence of frontal interaction.
\subsection{Training procedure}
The simulation output for each scenario is 7200 seconds of trajectory data for each of the three lanes. We first map the trajectories from a single lane onto a space-time plane to form an input and output frame of dimension $80 \times 7200$ (i.e., the mesh size is $10$ m $\times$ $1$ s). The PV trajectories for the input frame are selected at random using a $5\%$ sampling rate. The output frame that forms the ground truth speed field is generated using the interpolation procedure described in eq.~\eqref{eqn4:speed}. We then extract samples of the input ($\mathbf{z}^\mathrm{p}$) and output tensors ($\mathbf{z}^\mathrm{f}$) from the input and output frames respectively, using a $80 \times 60$ sliding window. We generate 6000+ samples for each trajectory dataset using a 2 s spatial gap between sliding windows. We proceed similarly to generate more data with different sets of random input samples for each of the three traffic scenarios using a $5\%$ sampling rate. The final augmented dataset has $64000+$ input-output sample pairs for training the Deep CNN model. Note that the samples extracted from a specific trajectory record form a sequence, which violates the i.i.d assumption (independent and identically distributed) for the neural network training. However, this is rectified during the optimization stage, where only a random subset of the samples is used in each iteration of the CNN training (this is a common trick employed while training reinforcement learning models, for instance, the use of ``replay memory'' in \cite{mnih2015humanlevel}). We use the following additional parameters for training data generation: $|x|=10$ m, $|t|=1$ s, $c_w=18$ kmph, $c_v^{\mathrm{max}}=100$ kmph, $c_v^{\mathrm{min}}=60$ kmph, $V_\mathrm{max}=95$ kmph, $l_\mathrm{up}=80$ m and $l_\mathrm{dn}=40$ m.
We train five instances of the anisotropic and isotropic CNN models, and report the average of their performance results. We use the \textit{TensorFlow} framework \cite{tensorflow} to train all the models. The two major hyper-parameters, namely the CNN kernel width and depth in each layer, are independently optimized using the Hyperband algorithm \cite{Lisha2018hyperband}, which belongs to the class of bandit-based algorithms. Other hyper-parameter choices are: gradient descent batch size: 32 samples, total training epochs: 300, (fixed) learning rate: $1e-3$, and optimizer: Adam \cite{kingma2014adam}. We use a GPU cluster with NVIDIA Tesla V100 32GB for training the models. The run time for a single training experiment is between $120$ and $150$ min. Note that the training can be viewed as an offline procedure.
\subsection{Testing data}
We test our model using three datasets: (i) a hold-out set from the simulated data that is not used for training (from a different lane of the freeway section), (ii) the Next Generation Simulation Program (NGSIM) dataset \cite{ngsim}, and (iii) the German Highway Drone (HighD) dataset \cite{krajewski2018highd}. We choose the US-101 highway trajectory data from NGSIM, which contains the locations and speeds of all vehicles crossing the observed area during a $45$ min time period with a $0.1$ s resolution. The HighD data consists of trajectory data from several German highways, each consisting of a frame-wise recording of all vehicles passing a $400$ m section during a $20$ min duration, with a resolution of $25$ frames/second. The input-output test samples are generated similarly to the training datasets with the respective space-time discretization parameters.
\emph{We emphasize that we do not use the NGSIM or HighD datasets for training the model.} In other words, the model is trained with data from a simulation using a freeway in the United Arab Emirates; and then tested with additional data from that same simulation, as well as real data from a freeway in the United States and several freeways in Germany. This allows us to evaluate the model's transferability to diverse traffic scenarios and dynamics not seen in the training set, and the viability of using simulated data instead of real data for training.
\section{Results and Discussion} \label{sec4}
In this section, we present the anisotropic reconstruction results and compare the isotropic and anisotropic models. We also discuss the transferability of the trained models to real-world traffic conditions and extend the results to handle varying PV penetration rates.
The architecture of the CNN model obtained from the hyper-parameter optimization is shown in Table \ref{tab:model_arch}. We use the same optimized architecture for both the anisotropic and isotropic CNN models.
\begin{table}[!hbt]
\centering
\caption{Model architecture as obtained from the hyper-parameter optimization}
\label{tab:model_arch}
\begin{tabular}{@{}ccc@{}}
\toprule
Layer name & \multicolumn{1}{l}{Kernel widths} & \multicolumn{1}{l}{Kernel depths} \\ \midrule
Conv-1 & ($5 \times 5$) & $40$ \\
Conv-2 & ($7 \times 7$) & $48$ \\
Conv-3 & ($7 \times 7$) & $32$ \\
Conv-4 & ($5 \times 5$) & $48$ \\
Conv-5 & ($5 \times 5$) & $40$ \\
Conv-6 & ($9 \times 9$) & $56$ \\
Output & ($7 \times 7$) & $1$ \\ \bottomrule
\end{tabular} \end{table}
\subsection{Anisotropic CNN model reconstruction}
Fig.~\ref{fig:recon1} shows five sample estimated speed fields from the hold-out simulated test dataset using the anisotropic model. The reconstruction window is $800$ m $\times$ $60$ s with a $10$ m $\times$ $1$~s resolution. The true speed field, PV trajectories, and speed profiles at three time instants ($t = 10, ~30, $ and $50$ s) are also shown for each sample. Three of the samples correspond to congested traffic conditions, one corresponds to slow-moving traffic conditions, and one corresponds to free-flowing traffic conditions.
\begin{figure*}
\caption{Estimated speed field for selected samples in the simulated test data using the anisotropic CNN model. The fourth column shows the speed profile across the road section at $t = 10, ~30$, and $50$ secs (\textit{blue - true speeds, orange - estimated speeds, black vertical dashed line - input PV speed}).}
\label{fig:recon1}
\end{figure*}
There are several points of interest to note about the reconstruction: All the estimated speed fields are feasible in terms of traffic physics and capture the different traffic states well. The model reproduces the existence of free-flow, congested and transition traffic dynamics correctly despite having very limited input information from the PV trajectories. One can observe the accurate prediction of shockwave dynamics in the congested traffic samples (a)-(c). This is also evident from the speed profile comparison. The true speed profile is often noisy, and the reconstruction has a smoothing effect due to the local convolutional operations in the CNN layers.
We have observed that the estimated speeds in slow-moving traffic have a higher root mean squared error (RMSE) than those in congested and free-flowing traffic; see Fig.~\ref{fig:recon1} (d)-(e) and Table~\ref{tab:model_comp}. In slow-moving traffic, heterogeneity (caused by different vehicle characteristics, driving behaviors, etc.) is predominant, and one can see different forward wave propagation velocities in the speed field; see the example in Fig.~\ref{fig:recon1} (d). Therefore, estimation is inherently a challenging problem unless we observe the actual travel speed. This is not the case for congested traffic, where the collective dynamics can be inferred from the trajectory of a single vehicle, or for free-flowing traffic, where the traffic heterogeneity is limited. In short, traffic speed fields with varied forward propagation wave velocities are still difficult to infer.
Interestingly, in all the scenarios, the anisotropic model predicts the average desired vehicle speed in areas where there are no PV trajectories, which is a reasonable conclusion when no vehicles are observed.
We interpret the Deep CNN model as an interpolation function that locally propagates traffic characteristics (forward and backward waves) using the sparse information from input vehicle trajectories. The model ensures sound propagation of traffic information in space and time, resulting in speed field estimates with different traffic states - free-flowing, slow-moving, congested, and their transition states. Introducing anisotropic kernels further limits the propagation speeds of traffic information, in accordance with the Kinematic Wave Theory of traffic flow. This results in speed field estimates with a gradual and physically reasonable transition between the different traffic states. In contrast, traditional Kalman Filter based assimilation techniques only exploit state information from one (or a few) time step(s) when estimating the traffic speeds. This is inefficient in terms of data usage and fails to accurately reconstruct the dynamics.
In addition, we have tuned the Deep CNN model architecture to learn different traffic wave dynamics. Whether to produce a backward or forward wave depends on the traffic regime, which the model infers from the input trajectories. This is confirmed from the latent space projection of the data (i.e., the output from the encoder model), where three distinct clusters were generated, corresponding to free-flowing, slow-moving and congested traffic, respectively. Another way to put this is that a neural network model can solve an under-determined system - a major upside compared to other machine learning models. This is in contrast with traditional estimation methods which require additional information on initial/boundary conditions or traffic demands.
\subsection{Comparison of anisotropic and isotropic models}
We next compare the performance and computational requirements of the anisotropic and isotropic models in Table~\ref{tab:model_comp}. The RMSE calculation shown in the table is the sample average for $4000+$ simulated test samples. Overall, the anisotropic and isotropic models have similar performance in terms of accuracy, but the anisotropic model leads to more physically plausible shockwave dynamics (this is discussed below). In particular, the anisotropic model performs slightly better in estimating the congested and free-flowing traffic in comparison to slow-moving traffic. This is because the slow-moving data samples comprise heterogeneous traffic in the free-flow regime, which might be better observed by an isotropic kernel than a restricted anisotropic kernel. Depending on the desired speed distribution, one can increase the extent of the anisotropic kernel (i.e, values of $c_v^{\rm min}$ and $c_w^{\rm max}$) and rectify this.
\begin{table}[!htb] \centering
\caption{Comparison of anisotropic and isotropic models. Percent change is with respect to isotropic model.}
\label{tab:model_comp}
\resizebox{\columnwidth}{!}{\begin{tabular}{@{}clccc@{}}
\toprule
\multicolumn{2}{c}{Metric} &
\begin{tabular}[c]{@{}c@{}}Isotropic\\ model\end{tabular} &
\begin{tabular}[c]{@{}c@{}}Anisotropic\\ model\end{tabular} &
\begin{tabular}[c]{@{}c@{}}Percent\\ change\end{tabular} \\ \midrule
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}Root mean\\ squared error\\ (\textit{kmph})\end{tabular}} &
Congested &
\begin{tabular}[c]{@{}c@{}}$8.60$\\ $(\pm 3.16)$\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$8.50$\\ $(\pm 3.15)$\end{tabular} &
$-1.2\% $ \\ \cmidrule(l){2-5}
&
Slow-moving &
\begin{tabular}[c]{@{}c@{}}$10.37$\\ $(\pm1.60)$\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$10.53$\\ $(\pm1.70)$\end{tabular} &
$+1.5\% $ \\ \cmidrule(l){2-5}
&
Free-flowing &
\begin{tabular}[c]{@{}c@{}}$7.42$\\ $(\pm 2.23)$\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$7.40$\\ $(\pm2.22)$\end{tabular} &
$-0.3\% $ \\ \cmidrule(l){2-5}
&
Total &
\begin{tabular}[c]{@{}c@{}}$8.71$\\ $(\pm 2.76)$\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$8.76$\\ $(\pm 2.71)$\end{tabular} &
$+0.5\% $ \\ \midrule
\multicolumn{2}{c}{Number of parameters} &
$443193$ &
$215625$ &
$-51.4\% $ \\ \bottomrule
\end{tabular}} \end{table}
Table~\ref{tab:model_comp} also shows that the anisotropic model requires only half as many parameters as the isotropic model, which is a significant improvement in model complexity given that the performance of the two models is very similar. From a computational perspective, this is a substantial advantage, leading to faster model convergence (RMSE reduction per training epoch) and a potential reduction in the number of training samples required. This confirms that exploiting domain knowledge results in simpler and more interpretable learning models.
Although the isotropic and anisotropic models perform comparably in terms of the average error in estimating the speed, there are some examples where they differ in terms of the structure (speed and extent) of the shockwaves they produce. This is illustrated in Fig.~\ref{fig:recon-comp}, which shows certain examples where the anisotropic model clearly reconstructs more physically plausible shockwave dynamics, as mentioned below.
\begin{figure}
\caption{Estimated speed fields for some selected samples in the simulated test data using the anisotropic and the isotropic CNN models. Black lines show the probe vehicle trajectories used for the reconstruction.}
\label{fig:recon-comp}
\end{figure}
In the example in Fig. \ref{fig:recon-comp}(a), the isotropic CNN underestimates the length of the shockwave at the top, whereas the anisotropic CNN correctly predicts that it existed some time prior to the two PV trajectories crossing it. This is because the anisotropic kernel gets more activation along the direction of the shockwave and hence reconstructs the stop-and-go region correctly, whereas the isotropic kernel considers all directions, which possibly results in averaging out all the nearby activations. Similar patterns have been observed in other test instances. The estimation in Fig. \ref{fig:recon-comp}(b) is obtained using a single input trajectory. The anisotropic model gives a plausible reconstruction of the shockwave whereas the isotropic reconstruction shows large dispersion, which is also physically inconsistent with the input data. The design of anisotropic kernels can rule out such inconsistencies arising in the estimation. Fig. \ref{fig:recon-comp}(c) shows a free-flowing traffic estimation. Again the forward wave produced by the isotropic kernel has more dispersion. In summary, one can see that the anisotropic model produces more accurate wave propagation dynamics consistent with traffic physics, even though the RMSEs of the models are similar.
We finally compare the anisotropic and isotropic models from the perspective of model complexity and over-fitting measures. In order to understand how the complexity of CNN models scales with the road network size, we optimize the anisotropic and isotropic CNN model architectures for different road lengths. The optimization is done using the Hyperband algorithm \cite{Lisha2018hyperband}. The results are shown in Fig.~\ref{fig:modelcomplx}(a), where the optimal number of model parameters required for different road lengths are compared (see the scatter plot). We see that the model complexity scales \emph{quadratically} for isotropic kernels, whereas for anisotropic models it scales \emph{linearly} (see the curve plot). Thus, as the problem size becomes large (for e.g., for long road sections, multiple lanes, or network level settings), the optimal CNN model architecture required for learning traffic dynamics becomes significantly large with isotropic kernels. The anisotropic CNN model, on the other hand, scales well to large problem sizes, results in simpler and more manageable models, and is beneficial for practical implementation. This observation is in-line with the scaling results obtained in Fig.~\ref{fig:kersize}.
\begin{figure}
\caption{Comparison of model-complexity and over-fitting measures for isotropic and anisotropic CNN models.}
\label{fig:modelcomplx}
\end{figure}
In order to measure the over-fitting in CNN models, we define an over-fitting metric as, \begin{equation}
O_{\rm fit} = \Bigg| \frac{{\rm RMSE}_{\rm~train} - {\rm RMSE}_{\rm~test}}{{\rm RMSE}_{\rm~test}} \Bigg| \times 100, \end{equation} where ${\rm RMSE}_{\rm~train}$ and ${\rm RMSE}_{\rm~test}$ are the root mean squared error metrics for training and testing data respectively. $O_{\rm fit}$ measures the difference in the model's performance on the training and testing data; higher values for $O_{\rm fit}$ imply more over-fitting. Over-fitting is an undesirable feature, and indicates poor generalization to unseen testing data. Fig.~\ref{fig:modelcomplx}(b) shows $O_{\rm fit}$ for the isotropic and anisotropic CNN models trained with different proportions of the total training data. The simulated training and simulated testing data are used to calculate $O_{\rm fit}$. Note that the hyper-parameters of the CNN models are optimized independently to ensure that $O_{\rm fit}$ is compared for the optimal isotropic and anisotropic models. The trend line in Fig.~\ref{fig:modelcomplx}(b) shows that the isotropic CNN model results in higher over-fitting. Since this observation is consistent at all data levels (and thus independent of model complexity), we conclude that the isotropic model has higher tendency to over-fit than the proposed anisotropic model. This is because the anisotropic CNN model reduces the number of parameters in a principled way, which lowers the model complexity without compromising test accuracy. In other words, the introduction of anisotropic kernels is a natural way to train CNN models that learn traffic speed dynamics with a lowered risk of over-fitting.
\subsection{Transferability to real-world traffic dynamics}
To understand how well the anisotropic CNN model performs in scenarios with different traffic characteristics than those observed in the training dataset, we test it on various real-world freeway sections. Figs. \ref{fig:highd-hw25} to \ref{fig:highd-hw44} show the estimation results for three sample freeway sections from the HighD and NGSIM datasets using data with a PV sampling rate of $5\%$.
\begin{figure}
\caption{Estimated speed field of lane 4 of highway No. 25 in the HighD dataset using $5\%$ probe sampling rate. The road section is $X=400$ m long and the reconstruction period is $T=1140$ s. The RMSE is $6.80$ kmph.}
\label{fig:highd-hw25}
\end{figure}
\begin{figure}
\caption{Estimated speed field of lane 2 of U.S. Highway 101 in the NGSIM dataset using $5\%$ probe sampling rate. The road section is $X=670$ m long and the reconstruction period is $T=2400$ s. The RMSE is $10.50$ kmph.}
\label{fig:ngsim}
\end{figure}
\begin{figure}
\caption{Estimated speed field of lane 6 of Highway No. 44 in the HighD dataset using $5\%$ probe sampling rate. The road section is $X=400$ m long and the reconstruction period is $T=1140$ s. The RMSE is $14.60$ kmph.}
\label{fig:highd-hw44}
\end{figure}
A quick observation shows that all three example reconstructions are plausible, despite having different space-time dimensions from those used in the training dataset. This is possible because of the parameter sharing property of CNNs, whereby the features learned during training (traffic characteristics in this case) are space-time invariant, and hence can be used with any spatio-temporal reconstruction window.
Closer observation reveals variation in the performance across the three examples. The estimated speed field in Fig.~\ref{fig:highd-hw25} has the lowest RMSE ($\approx 6.80$ kmph), and the speed, width and duration of the predicted backward propagating shockwaves are accurate. In the estimation in Fig.~\ref{fig:ngsim}, the shockwave reconstruction and speeds are reasonably correct, and the RMSE is moderate ($\approx 10.50$ kmph). The model correctly predicts the onset of shockwaves $400$ m upstream of the road section during the initial 600 seconds, though it underestimate the shockwave width. Thus, the model can accurately estimates speed fields for a road section that is simultaneously congested and free-flowing during the same period. The estimation result in Fig.~\ref{fig:highd-hw25} also supports this. The third freeway section, shown in Fig.~\ref{fig:highd-hw44}, comprises free-flowing traffic and has the highest RMSE ($\approx 14.60$ kmph). Apart from the inherent difficulties in the estimation of free-flowing traffic speeds, one can also observe that the speed of forward waves predicted by the model is slightly lower than that from the true waves; see the slow-moving band around $400$ secs as an example. This is due to the difference in the traffic characteristics in the training and testing data, which we elaborate on below.
Recall that the CNN model trained on the simulated data encompasses the knowledge of traffic dynamics of a specific freeway section. How well this model transfers to other test scenarios depends on the traffic characteristics of the test segment. One can explain the difference in the RMSE errors in Figs.~\ref{fig:highd-hw25} to \ref{fig:highd-hw44} by comparing the dynamics contained in the simulated data and the test data. One useful tool for this comparison is the flow-density scatter plot, which is shown in Fig.~\ref{fig:data_comp}. The reason for the low RMSE in the first two examples is that the freeways are operating in the congested regime and the shockwave speeds in the simulated and test data are similar. Likewise, the reason for slightly lower prediction of free-flowing speed in the third freeway section (which operates in free-flowing traffic) is evident from Fig.~\ref{fig:data_comp}(c). A similar reasoning is applicable when one discusses the transferability of the simulated section with its own real-world section, as the simulation doesn't capture the complete dynamics. Empirical FD comparisons can be further exploited to calibrate the trained deep learning model to match with the traffic dynamics of the test data. This is beyond the scope of the current work and represents a possible future extension.
\begin{figure*}
\caption{Flow-density scatter plot comparing the traffic characteristics of the real-world datasets with the simulated training data}
\label{fig:data_comp}
\end{figure*}
In short, we observe that the Deep CNN models trained with simulated data from minimally calibrated traffic flow models transfer well to real-world traffic scenarios. Better results are to be expected with simulation models that are calibrated to the testing scenarios. Since the data required for calibration is lower than that needed for training deep neural networks, we can take advantage of well-developed microscopic traffic simulations to fit data-hungry models like CNNs.
We finally compare the anisotropic Deep CNN model performance with two other existing traffic speed estimation techniques in the literature: (a) General Adaptive Smoothing Method (GASM) from \cite{trieber2011filter}, and (b) Velocity-based LWR Ensemble Kalman Filtering technique (LWR-v EnKF) from \cite{daniel2008enkf}. While both techniques directly estimate the speed field over a given time-space plane, the former is a data assimilation technique using a macroscopic traffic flow model and the latter is an informed traffic interpolation procedure. The LWR-v EnKF method additionally requires initial and boundary conditions as inputs. The estimation results for the NGSIM US-101 highway lane 2 using the anisotropic CNN model, GASM and LWR-v EnKF are compared in Table~\ref{tab:est_comp}. The RMSE metric is evaluated for different input PV penetration rates to understand how well the techniques perform in sparse observation setting. Overall, we see that the anisotropic Deep CNN model results in the least estimation error for all PV penetration rates.
\begin{table}[hbt!] \centering \caption{Comparison results to existing estimation technqiues.} \label{tab:est_comp} \resizebox{\columnwidth}{!}{\begin{tabular}{@{}cccc@{}} \toprule \multirow{2}{*}{Estimation techniques} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Root mean squared error (\textit{kmph}) \\ at different PV penetration rates\end{tabular}} \\ \cmidrule(l){2-4}
& $3\%$ & $5\%$ & $10\%$ \\ \midrule \begin{tabular}[c]{@{}c@{}}Aniso CNN model\\ (this paper)\end{tabular} & \begin{tabular}[c]{@{}c@{}}$11.60$\\ $(\pm 1.46)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$10.70$\\ $(\pm 0.59)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$8.88$\\ $(\pm 0.24)$\end{tabular} \\[0.4cm] \begin{tabular}[c]{@{}c@{}}GASM method\\ (Treiber et al. 2011 \cite{trieber2011filter})\end{tabular} & \begin{tabular}[c]{@{}c@{}}$13.49$\\ $(\pm 2.50)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$11.80$\\ $(\pm 1.90)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$9.50$\\ $(\pm 1.06)$\end{tabular} \\[0.4cm] \begin{tabular}[c]{@{}c@{}}LWR-v EnKF method\\ (Daniel et al. 2008 \cite{daniel2008enkf})\end{tabular} & \begin{tabular}[c]{@{}c@{}}$14.93$\\ $(\pm 0.05)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$14.64$\\ $(\pm 0.08)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$13.63$\\ $(\pm 0.20)$\end{tabular} \\ \bottomrule \end{tabular}} \end{table}
We found that the GASM method provides reasonable estimates at higher PV penetration rates (for e.g., $\geq 10\%$). However, at lower PV penetration rates (i.e., when the input only consist of one or two PV trajectories), the GASM method fails to reproduce correct traffic speed waves and results in higher estimation error. We have also noticed that the GASM method produces large dispersion in their estimates, which implies it poorly captures short-term traffic variations and is not suitable for high-resolution estimation. This is because the GASM method only uses two (pre-defined) kernels for interpolation, while our anisotropic CNN model uses an ensemble of (learned) kernels and hence interpolates low-level traffic features well.
Traffic speed estimated using the LWR-v EnKF method results in highest RMSE as shown in Table~\ref{tab:est_comp}. This poor performance could be due to its myopic character $-$ only uses traffic speed inputs at the current time steps whereas the anisotropic CNN model considers inputs from multiple time steps. We also notice that the performance gets worse for longer estimation intervals, since the LWR model predictions deviate significantly from the actual data. Similar to GASM, the LWR-v EnKF also captures macroscopic traffic features and gets better at higher PV penetration rates.
In short, the anisotropic CNN model outperforms the existing traffic speed estimation techniques, especially at lower PV penetration rates.
\subsection{Variable probe vehicle penetration rates}
To conclude our evaluation of the CNN model's performance, we investigate the effect of changing the PV penetration rate. We train six separate models using data consisting of specific PV penetration rates $10\%, ~20\%, ~\dots, ~70\%$ respectively (in addition to the \textit{5\% probe model} discussed so far in this paper). The input-output pairs for training are generated in the same way as explained in Sec.~\ref{sec3} B. Each of these \textit{probe specific models} is evaluated using testing data which has a corresponding PV penetration rate to the respective model. The average test RMSE results are shown in Fig.~\ref{fig:probe_res}(a) (labeled "probe specific model"). As expected, the RMSE decreases with higher PV penetration rates.
However, we find that these probe specific models are not trivially generalizable to handle penetration rates other than what they were trained on, i.e., the models are penetration rate dependent. We demonstrate this by evaluating the performance of the two extreme probe specific models (i.e. the $5\%$ probe model and the $70\%$ probe model) on testing data across the whole range of PV penetration rates. It is clear that these probe specific models perform well only in/near their training domain. This could be due to the unconstrained latent space representation while training the CNN models, and is inevitable in any data driven models unless physical constraints are imposed.
\begin{figure}
\caption{Probe vehicle (PV) penetration rate analysis}
\label{fig:probe_res}
\end{figure}
The actual PV penetration rate depends on the prevailing traffic demand on the freeway, which is hard to measure in practice. We aim for an estimation model that performs well irrespective of the PV penetration rate. In other words, we want an estimation model that doesn't require prior knowledge of the PV penetration rate. Therefore, we test three methods to handle varying PV penetration rates. The first two methods are brute force approaches, which consist of training the CNN model on a dataset containing the whole range of PV penetration rates $5\%, ~10\%, ~20\%, ~\dots, ~70\%$. The third method is to use an ensemble CNN model. The RMSE results from these models are compared in Fig.~\ref{fig:probe_res}(b).
The first model (labeled \textit{generic model-eq}) is trained on a dataset consisting of all PV penetration rates sampled in equal proportion. The second model (labeled \textit{generic model-uneq}) is similar except that we give more importance to lower PV penetration rates which are more difficult to learn. This is achieved by including more data samples for lower PV penetration rates, i.e., training data $\propto$ 1/(PV penetration rate). Both these models, however, perform sub-optimally compared to the probe specific models. The best method is the third model (labeled \textit{ensemble average model}), which takes the average of the predictions of all the probe specific models. This is referred to as “ensemble bagging” in the machine learning literature, and performs better than a single model trained on a wide range of penetration rates. As seen in Fig.~\ref{fig:probe_res}(b), the ensemble CNN performs consistently well across all the PV penetration rates, even outperforming the respective probe specific models in certain cases. In addition to the performance, the individual models in the ensemble CNN (also called weak learners) can be trained in parallel, resulting in significantly lower training time than the other two generic models.
\section{Conclusion} \label{sec5}
Deep learning models have shown success in solving several inverse problems in traffic flow, but they are limited by their lack of robustness and poor model interpretability. In this paper, we overcome these limitations by proposing an anisotropic Deep Convolutional Neural Network (CNN) model for estimating high-resolution traffic speed field using measurements from probe vehicles. The model employs anisotropic traffic kernels which are designed to explicitly capture a broad range of forward and backward propagation speeds in macroscopic traffic. Additionally, the Deep CNN model is trained using simulated traffic data. Since the generalization of Deep CNN performance depends on the distribution of training data, we note that using a targeted simulated data is an alternate method of imposing desirable traffic physics on the estimation model. For instance, we generate data corresponding to different traffic conditions (congested, slow-moving, free-flowing, etc.) so that the Deep CNN can learn traffic wave propagation speeds originating in heterogeneous traffic.
We present estimation results with input PV penetration rates as low as $5\%$ and output resolution as high as $10$m $\times$ $1$s. In the experiments, we primarily focused on the benefits of using anisotropic kernels in the Deep CNN model over the na\'ive isotropic kernels. We found that anisotropic kernels result in parsimonious model complexity and are less prone to model over-fitting, although the estimation error is similar to their isotropic counterparts. The model complexity grows linearly with problem size for anisotropic kernels whereas it grows as quadratic for isotropic kernels. Specific examples are provided to demonstrate that the anisotropic kernels better produce physically correct traffic shockwaves. We further evaluated the anisotropic Deep CNN on real-world traffic datasets and found acceptable transferability performance. This suggests that simulated data is a viable surrogate to real-world data for training Deep CNNs. We also found that the Deep CNN model performance is PV penetration rate dependent and proposed an ensemble model to handle unknown PV penetration rates.
We believe that the optimal way to apply learning techniques to a specific domain such as traffic state estimation is to integrate the fundamental principles of the domain into the framework of the learning model. This paper represents only one possible example of this general approach. In future work, we aim to explore other methods to incorporate traffic flow theory into learning models such as CNNs.
\section*{Acknowledgment} This work was supported by the NYUAD Center for Interacting Urban Networks (CITIES), funded by Tamkeen under the NYUAD Research Institute Award CG001. The views expressed in this article are those of the authors and do not reflect the opinions of CITIES or its funding agencies.
\appendix
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\begin{document} \title{Doubly Robust Inference for Targeted Minimum Loss Based
Estimation in Randomized Trials with Missing Outcome Data}
\begin{abstract}
Missing outcome data is one of the principal threats to the validity
of treatment effect estimates from randomized trials. The outcome
distributions of participants with missing and observed data are
often different, which increases the risk of bias. Causal inference
methods may aid in reducing the bias and improving efficiency by
incorporating baseline variables into the analysis. In particular,
doubly robust estimators incorporate estimates of two nuisance
parameters: the outcome regression and the missingness mechanism
(i.e., the probability of missingness conditional on treatment
assignment and baseline variables), to adjust for differences in the
observed and unobserved groups that can be explained by observed
covariates. To obtain consistent estimators of the treatment effect,
one of these two nuisance parameters mechanism must be consistently
estimated. Such nuisance parameters are traditionally estimated
using parametric models, which generally preclude consistent
estimation, particularly in moderate to high dimensions. Recent
research on missing data has focused on data-adaptive estimation of
the nuisance parameters in order to achieve consistency, but the
large sample properties of such estimators are poorly understood. In
this article we discuss a doubly robust estimator that is consistent
and asymptotically normal (CAN) under data-adaptive consistent
estimation of the outcome regression \textit{or} the missingness
mechanism. We provide a formula for an asymptotically valid
confidence interval under minimal assumptions. We show that our
proposed estimator has smaller finite-sample bias compared to
standard doubly robust estimators. We present a simulation study
demonstrating the enhanced performance of our estimators in terms of
bias, efficiency, and coverage of the confidence intervals. We
present the results of an illustrative example: a randomized,
double-blind phase II/III trial of antiretroviral therapy in
HIV-infected persons, and provide R code implementing our proposed
estimators. \end{abstract}
\section{Introduction}
Missing data are a frequent problem in randomized trials. If the reasons for outcome missingness and the outcome itself are correlated, unadjusted estimators of the treatment effect are biased, thus invalidating the conclusions of the trial. Most methods to mitigate the bias rely on baseline variables to control for the possible common causes of missingness and the outcome, through estimation of certain ``nuisance'' parameters, i.e., parameters that are not of interest in themselves, but that are required to estimate the treatment effect. In addition to aiding in correcting bias, methods that use covariate adjustment often provide more precise estimates \cite[see, e.g.,][]{koch1998issues,Bang05,Zhang2008,moore2009covariate,Colantuoni2015, Diaz2016}. In this article we focus on doubly robust estimators. Doubly robust estimation of treatment effects in randomized trials requires estimation of two possibly high-dimensional nuisance parameters: the outcome expectation within treatment arm conditional on baseline variables (henceforth referred to as outcome regression), and the probability of missingness conditional on baseline variables (henceforth referred to as missingness mechanism).
The large sample properties of doubly robust estimators hinges upon large sample properties of the estimators of the nuisance parameters. In particular: \begin{enumerate*}[label=(\alph*)] \item doubly robust estimators remain consistent if at least one of
the nuisance parameters is estimated consistently, and\label{prop:a} \item the asymptotic distribution of the effect estimator depends on
empirical process conditions on the estimators of the nuisance
parameters.\label{prop:b} \end{enumerate*}
When parametric models are adopted to estimate the nuisance parameters, a straightforward application of the delta method yields the convergence of the doubly robust estimator to a normal random variable at $n^{1/2}$-rate. The nonparametric bootstrap or an influence function based approach yields consistent estimates of the asymptotic variance and confidence intervals. However, the assumptions encoded in parametric models are rarely justified by scientific knowledge. This implies that parametric models are frequently misspecified, which yields an inconsistent effect estimator. In other words, a doubly robust estimators relying on nuisance parametric models makes no use of the double robustness property \ref{prop:a}: it is always inconsistent.
Data-adaptive alternatives to alleviate this shortcoming have been developed over the last decades in the statistics and machine learning literature. These data-adaptive methods offer an opportunity to employ flexible estimators that are more likely to achieve consistency. Methods such as those based on regression trees, regularization, boosting, neural networks, support vector machines, adaptive splines, etc., and ensembles of them offer flexibility in the specification of interactions, non-linear, and higher-order terms, a flexibility that is not available for parametric models. However, the large sample analysis of treatment effects estimates based on machine learning requires hard-to-verify assumptions, and often yield estimators which are not $n^{1/2}$-consistent, and for which no statistical inference (i.e., p-values and confidence intervals) is available. Nonetheless, data-adaptive estimation has been widely used in estimation of causal effects from observational data \citep[a few examples include][]{vanderLaan&Petersen&Joffe05,
Wang&Bembom&vanderLaan06,ridgeway2007, Bembometal08a,
lee2010improving,neugebauer2016case}. Indeed, the statistics field of \textit{targeted learning} \citep[see e.g.,][]{vanderLaan&Rubin06,
vanderLaanRose11, van2014entering} is concerned with the development of optimal ($n^{1/2}$-consistent, asymptotically normal, efficient) estimators of smooth low-dimensional parameters through the use state-of-the art machine learning.
We develop estimators for analyzing data from randomized trials with missing outcomes, when the missingness probabilities and the outcome regression are estimated with data-adaptive methods.
We propose two estimators: an augmented inverse probability weighted estimator (AIPW), and a targeted minimum loss based estimator (TMLE). Our methods are inspired by recent work by \cite{van2014targeted, benkeser2016doubly}, who developed an estimator of the mean of an outcome from incomplete data when data-adaptive estimators are used for the missingness mechanism. In addition to extending their methodology to our problem, our main contribution is to simplify the assumptions of their theorems to two conditions: consistent estimation of at least one of the nuisance parameters, and a condition restricting the class of estimators of the nuisance parameters to Donsker classes (those for which a uniform central limit theorem applies). Though the Donsker condition may be removed through the use of a cross-validated version of our TMLE, the results are straightforward extensions of the work of \cite{zheng2011cross}, and we do not pursue such results here. We show that the doubly robust asymptotic distribution of these novel estimators requires a slightly stronger version of the standard double robustness in which the nuisance parameters converge to their (possibly misspecified) limits at $n^{1/4}$-rate, with at least one of them converging to the correct limit. Specifically, we show that the TMLE is CAN under these empirical process conditions, and provide its influence function. This allows the construction of Wald-type confidence intervals under the assumption that at least one of the nuisance parameters is consistently estimated, though it is not necessary to know which one. We also make connections between the proposed estimators and standard $M$-estimation theory, by noting that our estimators \citep[and those of][]{van2014targeted, benkeser2016doubly} amount to controlling the behavior of the ``drift'' term resulting from the analysis of the estimator's empirical process. Thus, our methods and theory may be used to improve the performance of other $M$-estimators in causal inference and missing data problems. The need to control the behavior of such terms has been previously recognized in the semiparametric estimation literature, for example in Theorem 5.31 of \cite{vanderVaart98} \cite[see also Section 6.6 of][]{bolthausen2002lectures}.
In related work, \cite{vermeulen2015bias, vermeulen2016data} recently proposed estimators that also target minimization of the drift term. However, their methods are not suitable for our application because they rely on parametric working models for the missingness mechanism. Since we do not know the functional form of the missingness mechanism, we must resort to data-adaptive methods to estimate this probability.
The paper is organized as follows. In Section~\ref{sec:applica} we discuss our illustrative application and define the statistical estimation problem.
In Section~ \ref{sec:existing} we present estimators from existing work; in Section~\ref{sec:proposal} we discuss possible ways of repairing the AIPW, and show that such repairs do not help us achieve desirable properties such as asymptotic linearity. In Section~\ref{sec:tmle} we present our proposed TML estimator an show that it is asymptotically normal with known \textit{doubly robust
asymptotic distribution}, where the latter concept means that the distribution is known under consistent estimation of at least one nuisance parameter. Simulation studies are presented in Section~\ref{sec:simula}. These simulation studies demonstrate that our estimators can lead to substantial bias reduction, as well as improved coverage of the Wald-type confidence intervals. Section~\ref{sec:discussion} presents some concluding remarks and directions of future research.
\section{Illustrative Application}\label{sec:applica} We illustrate our methods in the analysis of data from the ACTG 175 study \citep{hammer1996trial}. ACTG 175 was a randomized clinical trial in which 2139 adults infected with the human immunodeficiency virus type I, whose CD4 T-cell counts were between 200 and 500 per cubic millimeter, were randomized to compare four antiretroviral therapies: zidovudine (ZDV) alone, ZDV+didanosine(ddI), ZDV+zalcitabine(ddC), and ddI alone.
One goal of the study was to compare the four treatment arms in terms of the CD4 counts at week 96 after randomization. By week 96, 797 (37.2\%) subjects had dropped out of the study. Dropout rates varied between 35.7-39.6\% across treatment arms. The investigators found dropout to be associated to patient characteristics such as ethnicity and history of injection-drug use, which are also associated with the outcome, therefore causing informative missingness. Other baseline variables collected at the beginning of the study include age, gender, weight, CD4 count, hemophilia, homosexual activity, the Karnofsky score, and prior antiretroviral therapy.
\subsection{Observed Data and Notation} Let $W$ denote a vector of observed baseline variables, let $A$ denote a binary treatment arm indicator (e.g., in our application we have four such indicators). Let $Y$ denote the outcome of interest, observed only when a missingness indicator $M$ is equal to one. Throughout, we assume without loss of generality that $Y$ takes values on $[0, 1]$. We use the word \textit{model} in the classical statistical sense to refer to a set of probability distributions for the observed data $O=(W, A, M, MY)$. We assume that the true distribution of $O$, denoted by $P_0$, is an element of the nonparametric model, denoted by $\cal M$, and defined as the set of all distributions of $O$ dominated by a measure of interest $\nu$. The word \textit{estimator} is used to refer to a particular procedure or method for obtaining estimates of $P_0$ or functionals of it. Assume we observe an i.i.d. sample $O_1,\ldots,O_n$, and denote its empirical distribution by $\mathbb{P}_{n} $. For a general distribution $P$ and a function $f$, we use $Pf$ to denote $\int f(o)dP(o)$. We use $m(w)$ to denote $E(Y\mid M=1,A=1,W=w)$, $g_A(w)$ to denote $P(A=a\mid W=w)$, and $g_M(w)$ to denote $P(M=1\mid A=1,W=w)$. The index naught is added when the expectation and probabilities are computed under $P_0$ (i.e., $m_0$, $g_{A,0}$, and $g_{M,0}$). We define $g(w)= g_A(w)g_M(w)$.
\subsection{Treatment Effect in Terms of Potential Outcomes and
Identification} Define the potential outcome $Y_1$ as the outcome that would have been observed had study arm $A=1$ and missingness $M=1$ been externally set with probability one. The target estimand is defined as $\theta_{\mbox{\footnotesize causal}}=E(Y_1)$. The index ``causal'' denotes a parameter of the distribution of the potential outcome $Y_1$. We show that $\theta_{\mbox{\footnotesize causal}}$ can be equivalently expressed as a parameter $\theta$ of the observed data distribution $P_0(W,A,M,MY)$, under \ref{ass:cons}-4 below. This is useful since the potential outcome is not observed, in contrast to the data vector $(W,A,M,MY)$, which we can make inferences about. Define the following assumptions: \begin{assumptioniden}[Consistency]\label{ass:cons}
$Y=M\{AY_1 + (1-A)Y_0\}$, \end{assumptioniden} \begin{assumptioniden}[Randomization]\label{ass:random}
$A$ is independent of $Y_1$ conditional on $W$, \end{assumptioniden} \begin{assumptioniden}[Missing at random]\label{ass:mar}
$M$ is independent of $Y_1$ conditional on $(A,W)$, \end{assumptioniden} \begin{assumptioniden}[Positivity]\label{ass:pos}
$g(w)>0$ with probability one over draws of $W$. \end{assumptioniden} \ref{ass:cons} connects the potential outcomes to the observed outcome. \ref{ass:random} holds by design in a randomized trial such as our illustrative example. \ref{ass:mar}, which is similar to that in \cite{rubin1987multiple}, means that missingness is random within strata of treatment and baseline variables (which is often abbreviated as ``missing at random'', or MAR). Equivalently, the MAR assumption may be interpreted as the assumption that all common causes of missingness and the outcome are observed and form part of the vector of baseline variables $W$. \ref{ass:pos} guarantees that $m_0$ is well defined.
Under \ref{ass:cons}-4 above, our target estimand $\theta_{\mbox{\footnotesize causal}}$ is identified as $\theta_0 = E_{P_0}\{m_0(W)\}.$ Note that this parameter definition allows us to compute the parameter value at any distribution $P$ in the model $\mathcal M$. According to this observation, we use the notation $\theta(P)=E_{P}\{m(W)\}$, where $\theta_0=\theta(P_0)$.
\subsection{Data Analysis} We present the results of applying our estimators to the ACTG data. To estimate the probability of missingness conditional on baseline variables $g_M$, we fit an ensemble predictor known as super learning \citep{vanderLaan&Polley&Hubbard07, SL} to the missingness indicator in each treatment arm. Super learning builds a convex combination of predictors in a user-given library, where the combination weights are chosen such that the cross-validated prediction risk is minimized. For predicting probabilities, we define the prediction risk as the average of the negative log-likelihood of a Bernoulli variable. The algorithms used in the ensemble along with their weights are presented in Table~\ref{tab:slcoef}. Note that the algorithms that more accurately predict missingness are data-adaptive algorithms with flexible functional forms, or algorithms that incorporate some type of variable selection.
\begin{table}[!htb]
\centering
\begin{tabular}{l|cccc}
\hline
& \multicolumn{4}{c}{Treatment arm} \\
Algorithm & ZVD & ZVD+ddI & ZVD+ddC & ddI \\
\hline
GLM & 0.00 & 0.00 & 0.00 & 0.00 \\
Lasso & 0.02 & 0.21 & 0.00 & 0.85 \\
Bayes GLM & 0.21 & 0.38 & 0.19 & 0.00 \\
GAM & 0.00 & 0.00 & 0.02 & 0.00 \\
MARS & 0.78 & 0.38 & 0.30 & 0.15 \\
Random Forest & 0.00 & 0.03 & 0.49 & 0.00 \\\hline
\end{tabular}
\caption{Coefficients in the super learner convex combination for
predicting 96 week dropout.}
\label{tab:slcoef} \end{table}
We also use the super learner to estimate the expected CD4 T-cell count at 96 weeks after randomization among subjects still in the study, conditional on covariates. The prediction risk in this case is defined as the average of the squared prediction residuals. The results are presented in Table~\ref{tab:slocoef}. For the outcome regression, the best predictive algorithms are also data-adaptive.
\begin{table}[!htb]
\centering
\begin{tabular}{l|cccc}
\hline
& \multicolumn{4}{c}{Treatment arm} \\
Algorithm & ZVD & ZVD+ddI & ZVD+ddC & ddI \\
\hline
GLM & 0.00 & 0.00 & 0.00 & 0.00 \\
Lasso & 1.00 & 0.30 & 0.08 & 0.60 \\
Bayes GLM & 0.00 & 0.02 & 0.00 & 0.00 \\
GAM & 0.00 & 0.00 & 0.60 & 0.34 \\
MARS & 0.00 & 0.00 & 0.00 & 0.06 \\
Random Forest & 0.00 & 0.68 & 0.32 & 0.00 \\\hline
\end{tabular}
\caption{Coefficients in the super learner convex combination for
predicting CD4 T-cell count.}
\label{tab:slocoef} \end{table}
The results in Tables~\ref{tab:slcoef} and \ref{tab:slocoef} highlight the need to use data-adaptive estimators for the nuisance parameters in the construction of a doubly robust estimator for $\theta_0$. As we show below in Section~\ref{sec:existing}, standard doubly robust estimators are not guaranteed to have desirable properties such as $n^{1/2}$-consistency and doubly robust asymptotic linearity when such data-adaptive estimators are used. This motivates the construction of the estimators we propose.
Figure~\ref{fig:estima} shows the estimated CD4 T-cell count for each treatment arm according to several estimators, along with their corresponding 95\% confidence intervals. The targeted maximum likelihood estimator \citep[TMLE][]{vanderLaanRose11} and the augmented inverse-probability weighted estimator (AIPW) are standard doubly robust estimators, whereas DTMLE and DAIPW are the modifications described in Section~\ref{sec:proposal} below.
Unlike the TMLE and AIPW, the confidence intervals of the DTMLE is expected to have correct asymptotic coverage under consistent estimation of at least one nuisance parameter (Theorem~\ref{theo:dr}). Unfortunately, the same claim does not seem to hold for the DAIPW, although we expect this estimator to have similar properties to the DTMLE in finite samples. For reference, we also present the unadjusted estimate obtained by computing the empirical mean of the outcome within each treatment arm among subjects with observed outcomes.
\begin{figure}
\caption{Estimated CD4 T-cell count on week 96 in each treatment arm, according
to several estimators, along with confidence intervals.}
\label{fig:estima}
\end{figure}
The dataset is available in the R package \texttt{speff2trial} \citep{speff2trial}, the super learner predictor was computed using the package \texttt{SuperLearner} \citep{SL}. R code to compute these estimators is given in Appendix~\ref{sec:code}.
\section{Existing Estimators from the Semiparametric Efficiency Literature}\label{sec:existing} We start by presenting the efficient influence function for estimation of $\theta_0$ in model $\cal M$ \citep[see][]{hahn1998role}: \begin{equation} D_{\eta, \theta}(O) = \frac{A\,M}{g(W)}\{Y-m(W)\} + m(W) -
\theta,\label{eq:defD} \end{equation} where we have denoted $\eta=(g, m)$. The efficient influence function $D_{\eta,\theta}$ is a fundamental object for the analysis and construction of estimators of $\theta_0$ in the non-parametric model $\mathcal M$. First, it is a doubly robust estimating function, i.e., for given estimators $\hat m$ and $\hat g$ of $m_0$ and $g_0$, respectively, an estimator that solves for $\theta$ in the following estimating equation is consistent if at least one of $ m_0$ or $g_0$ is estimated consistently \citep[while the other converges to a limit that may be incorrect, see Theorem 5.9 of][]{vanderVaart98}: \begin{equation}
\sum_{i=1}^n\frac{A_i\,M_i}{\hat g(W_i)}\{Y_i-\hat m(W_i)\} + \sum_{i=1}^n\left\{\hat m(W_i) -
\theta\right\}=0.\label{eq:EE} \end{equation} The estimator constructed by directly solving for $\theta$ in the above equation is often referred to as the augmented IPW estimator, and we denote it by $\hat\theta_{\aipww}$. Second, the efficient influence function (\ref{eq:defD}) characterizes the efficiency bound for estimation of $\theta_0$ in the model $\mathcal M$. Specifically, under consistent estimation of $m_0$ and $g_0$ at a fast enough rate (which we define below), an estimator that solves (\ref{eq:EE}) has variance smaller or equal to that of any regular, asymptotically linear estimator of $\theta_0$ in $\mathcal M$. This property is sometimes called \textit{local efficiency}.
The augmented IPW has been criticized because directly solving the estimating equation (\ref{eq:EE}) may drive the estimate out of bounds of the parameter space \citep[see e.g.,][]{Gruber2010t}, which may lead to poor performance in finite samples. Alternatives to repair the AIPW have been discussed by \cite{Kang2007, Robins2007,
tan2010bounded}. One such approach consists in solving the estimating equation (\ref{eq:EE}) with the first term in the left hand side divided by the empirical mean of the weights $A\,M/\hat g(W)$. Alternatively, the targeted minimum loss based estimation (TMLE) approach of \cite{vanderLaan&Rubin06, vanderLaanRose11} provides a more principled method to construct estimators that stay within natural bounds of the parameter space, for any smooth parameter.
The TMLE of $\theta_0$ is defined as a substitution estimator $\hat\theta_{\tmlee}=\theta(\tilde P)$, where $\tilde P$ is an estimate of $P_0$ constructed such that the corresponding $\tilde \eta$ and $\theta(\tilde P)$ solve the estimating equation $\sum_{i=1}^n D_{\tilde \eta, \theta(\tilde P)}(O_i)=0$. The estimator $\tilde P$ is constructed by tilting an initial estimate $\hat P$ towards a solution of the relevant estimating equation, by means of a maximum likelihood estimator in a parametric submodel.
Specifically, a TMLE may be constructed by fitting the logistic regression model \begin{equation} \logit m_\epsilon(w) = \logit \hat m(w) + \epsilon \frac{1}{\hat
g(w)},\label{eq:submodel} \end{equation} among observations with $(A_i,M_i)=(1,1)$. Here $\logit(p)=\log\{p(1-p)^{-1}\}$. In this expression $\epsilon$ is the parameter of the model, $\logit \hat m(w)$ is an offset variable, and the initial estimates $\hat m$ and $\hat g$ are treated as fixed. The parameter $\epsilon$ is estimated using the empirical risk minimizer \[\hat \epsilon = \arg\max_{\epsilon}\sum_{i=1}^n A_iM_i \{Y_i\log
m_\epsilon(W_i) + (1-Y_i)\log(1- m_\epsilon(W_i))\}.\] The tilted estimator of $ m_0(w)$ is defined as $\tilde m(w) = m_{\hat \epsilon}(w) = \expit\{\logit \hat m(w)+ \hat\epsilon / \hat g(w)\}$, where $\expit(x) = \logit^{-1}(x)$, and the TMLE of $\theta_0$ is defined as \[\hat\theta_{\tmlee}=\frac{1}{n}\sum_{i=1}^n \tilde m(W_i).\] Because the empirical risk minimizer of model (\ref{eq:submodel}) solves the score equation \[\sum_{i=1}^n\frac{A_i\,M_i}{\hat g(W_i)}\{Y_i - m_{\hat
\epsilon}(W_i)\}=0,\] it follows that $\sum_{i=1}^n D_{\tilde \eta, \hat\theta_{\tmlee}}(O_i)=0$ with $\tilde \eta = (\tilde g, \tilde m)$. Since this procedure does not update the estimator $\hat g$, we have $\tilde g=\hat g$.
Further discussion on the construction of the above TMLE may be found in \cite{Gruber2010t}. \cite{Porter2011} provides an excellent review of other doubly robust estimators along with a discussion of their strengths and weaknesses. In this article we focus on the estimators $\hat\theta_{\aipww}$ and $\hat\theta_{\tmlee}$ defined above, but our methods can be used to construct enhanced versions of other doubly robust estimators.
\subsection{Analysis of Asymptotic Properties of Doubly Robust Estimators}
The analysis of the asymptotic properties of the AIPW (as well as the TMLE or any other estimator that solves the estimating equation (\ref{eq:EE})) may be based on standard $M$-estimation and empirical process theory. Here we focus on an analysis of the AIPW based on the asymptotic theory presented in Chapter 5 of \cite{vanderVaart98}.
Define the following conditions: \begin{assumption}[Doubly robust consistency]\label{ass:DR1}
Let $||\cdot||$ denote the $L_2(P_0)$ norm defined as
$||f||^2=\int f^2 dP_0$. Assume
\begin{enumerate}[label=(\roman*)] \item There exists $\eta_1=(g_1, m_1)$
with either $g_1 = g_0$ or $ m_1= m_0$ such that
$||\hat m - m_1||=o_P(1)$ and $||\hat g - g_1||=o_P(1)$.
\item For $\eta_1$ as above, $||\hat m - m_1||\,||\hat g -
g_1||=o_P(n^{-1/2})$. \end{enumerate} \end{assumption} \begin{assumption}[Donsker]\label{ass:donsker}
Let $\eta_1$ be as in~\ref{ass:DR1}-(i). Assume the class of
functions $\{\eta=(g, m):|| m- m_1||<\delta, ||g-g_1||<\delta\}$ is
Donsker for some $\delta >0$. \end{assumption} Under \ref{ass:DR1}-(i) and 2, a straightforward application of Theorems 5.9 and 5.31 of \cite{vanderVaart98} \citep[see also example 2.10.10 of][]{vanderVaart&Wellner96} yields \begin{equation}
\hat\theta_{\aipww}-\theta_0= \beta(\hat\eta) +
(\mathbb{P}_{n} - P_0)D_{\eta_1, \theta_0} + o_P\big(n^{-1/2} + |\beta(\hat\eta)|\big),\label{eq:wh} \end{equation} where $\beta(\hat\eta) = P_0 D_{\hat\eta, \theta_0}$. Thus, the probability distribution of doubly robust estimators depends on $\hat \eta$ through the ``drift'' term $\beta(\hat\eta)$. For our parameter $\theta$ the drift term is given by \begin{equation} \beta(\hat\eta) = \int
\frac{1}{\hat g}(\hat g - g_0)(\hat m - m_0)dP_0.\label{eq:defbeta} \end{equation} Note that under \ref{ass:DR1}, $\beta(\hat\eta)$ converges to zero in probability so that $\hat\theta_{\aipww}$ and $\hat\theta_{\tmlee}$ are consistent. Efficiency under $\eta_1=\eta_0$ can be proved as follows. The Cauchy-Schwartz inequality shows that
\[\beta(\hat\eta)\leq C||\hat m - m_0||\,||\hat g - g_0||,\] for some constant $C$. Under \ref{ass:DR1} and $\eta_1=\eta_0$, we get $\beta(\hat\eta) = o_P(n^{-1/2})$ so that (\ref{eq:wh}) yields \begin{equation*}
\hat\theta_{\aipww} - \theta_0=(\mathbb{P}_{n} - P_0)D_{\eta_0, \theta_0} + o_P\big(n^{-1/2}\big).\label{eq:tmleeff} \end{equation*} An identical result holds replacing $\hat\theta_{\aipww}$ by $\hat\theta_{\tmlee}$ in the above display. Asymptotic normality and efficiency follows from the central limit theorem.
In the more common doubly robust scenario in which at most one of $ m_0$ or $g_0$ is consistently estimated, the large sample analysis of doubly robust estimators relies on the assumption that $\beta(\tilde\eta)$ is asymptotically linear \cite[see Appendix 18 of][]{vanderLaanRose11}. If $\hat \eta$ is estimated in a parametric model, the delta method yields the required asymptotic linearity. However, this assumption is hard to verify when $\hat\eta$ uses data-adaptive estimators; in fact there is no reason to expect that it would hold in general.
In the remainder of the paper we construct drift-corrected estimators $\hat\theta_{\daipww}$ and $\hat\theta_{\dtmlee}$ that control the asymptotic behavior through estimation of the drift term in the more plausible doubly robust situation where either $g_1 = g_0$ or $m_1= m_0$, but not necessarily both.
\begin{remark}[Asymptotic bias of the AIPW and TMLE under double
inconsistency]\label{lemma:bias}
Assume $\hat \eta = (\hat g, \hat \eta)$ converges to some
$\eta_1=(g_1, m_1)$. Define $\theta_1=P_0 m_1$, and note that
$D_{\eta_1,\theta_1}=D_{\eta_1,\theta_0}-\theta_1 + \theta_0$. Under
\ref{ass:donsker}, an application of Theorem 5.31 of
\cite{vanderVaart98} yields
\begin{equation*}
\hat\theta_{\aipww}-\theta_1= \beta(\hat \eta) +
(\mathbb{P}_{n} - P_0)D_{\eta_1, \theta_1} + o_P\big(n^{-1/2} + |\beta(\hat\eta)|\big).
\end{equation*}
Substituting $D_{\eta_1,\theta_1}=D_{\eta_1,\theta_0}-\theta_1 +
\theta_0$ yields
\begin{equation*}
\hat\theta_{\aipww}-\theta_0= \beta(\hat \eta) +
(\mathbb{P}_{n} - P_0)D_{\eta_1, \theta_0} + o_P\big(n^{-1/2} + |\beta(\hat\eta)|\big).
\end{equation*}
The above expression also holds for $\hat\theta_{\aipww}$ replaced with $\hat\theta_{\tmlee}$
and $\hat\eta$ replaced with $\tilde\eta$. The empirical process
term $(\mathbb{P}_{n} - P_0)D_{\eta_1, \theta_0}$ has mean zero. Thus,
controlling the magnitude of $\beta(\hat\eta)$ and
$\beta(\tilde\eta)$ is expected to reduce the bias of $\hat\theta_{\aipww}$ and
$\hat\theta_{\tmlee}$, respectively, in the double inconsistency case in which
$m_1\neq m_0$ and $g_1\neq g_0$. \end{remark}
\section{Repairing the AIPW Estimator Through Estimation of
$\beta(\hat\eta)$}\label{sec:proposal}
As seen from the analysis of the previous section, the consistency \ref{ass:DR1} with $\eta_1=\eta_0$ is key in proving the optimality ($n^{1/2}$-consistency, asymptotic normality, efficiency) of doubly robust estimators such as the TMLE and the AIPW. The asymptotic distribution of doubly robust estimators under violations of this condition depends on the behavior of the drift term $\beta(\hat\eta)$.
We propose a method that controls the asymptotic behavior of $\beta(\hat\eta)$. This is achieved through a decomposition into score functions associated to estimation of $m_0$ and $g_0$. In light of Remark~\ref{lemma:bias} controlling the magnitude and variation of $\beta(\hat\eta)$ is also important to reduce the bias of the TMLE when either $g_0$ or $ m_0$ are inconsistently estimated.
We introduce the following strengthened doubly robust consistency condition: \begin{assumption}[Strengthened doubly robust consistency]\label{ass:DR2}
$\hat \eta = (\hat g, \hat \eta)$ converges to some
$\eta_1=(g_1, m_1)$ in the sense that
$||\hat m - m_1||=o_P(n^{-1/4})$ and $||\hat g - g_1||=o_P(n^{-1/4})$ with
either $g_1 = g_0$ or $ m_1= m_0$. \end{assumption}
The following lemma provides an approximation for the drift term in terms of score function in the tangent space of each of the models for $g_0$ and $m_0$. Such approximation is achieved through the definition of the following univariate regression functions: \begin{align}
\gamma_{A,0}(W) &= P_0\big\{A = 1 \mid m_1(W)\big\},\notag\\
\gamma_{M,0}(W) &= P_0\big\{M = 1\mid A = 1, m_1(W)\big\},\notag\\
r_{A,0}(W) &= E_{P_0}\left\{\frac{A - g_{A,1}(W)}{g_{A,1}(W)}\mathrel{\bigg|}
m_1(W)\right\}\label{eq:regs},\\
r_{M,0}(W) &= E_{P_0}\left\{\frac{M - g_{M,1}(W)}{g_1(W)}\mathrel{\bigg|} A=1, m_1(W)\right\}\notag,\\
e_0(W) &= E_{P_0}\big\{Y - m_1(W)\mid A=1,M=1,g_1(W)\big\}\notag. \end{align} Note that the residual regressions $r_{A,0}$, $r_{M,0}$, and $e_0$ are equal to zero if the limits $g_{A,1}$, $g_{M,1}$, and $m_1$ of the nuisance estimators are correct. To see this, it suffices to replace $g_{A,0}$ for $g_{A,1}$ in $r_{A,0}$, and apply the iterated expectation rule conditioning first on $W$.
\begin{theorem}[Asymptotic approximation of the drift term]\label{lemma:betarep}
Denote $\lambda_0=(\gamma_{A,0}, \gamma_{M,0}, r_{A,0}, r_{M,0}, e_0)$, and define the following score functions: \begin{align*}
D_{Y,\hat m, \lambda_0}(O) &= A\,M\left\{\frac{r_{A,0}(W)}{\gamma_{A,0}(W)}
+ \frac{r_{M,0}(W)}{\gamma_0(W)}\right\}\{Y - \hat m(W)\}\\
D_{M, \hat g, \lambda_0}(O) &= \frac{A\,e_0(W)}{\hat
g(W)}\{M - \hat g_M(W)\}\\
D_{A, \hat g, \lambda_0}(O) &= \frac{e_0(W)}{\hat g_A(W)}\{A - \hat g_A(W)\}, \end{align*} where $\gamma_0(w)=\gamma_{A,0}(w)\gamma_{M,0}(w)$. Under \ref{ass:DR2} we have $\beta(\hat\eta)=P_0\{D_{A,\hat g, \lambda_0} + D_{M,\hat g, \lambda_0} + D_{Y,\hat m, \lambda_0}\} + o_P(n^{-1/2})$. \end{theorem}
Unlike expression~\ref{eq:defbeta}, the above approximation of the drift depends only on one-dimensional nuisance parameters which are easily estimable through non-parametric smoothing techniques. These one-dimensional parameters are functions of the possibly misspecified limits of your estimators. However, in what follows this does not prove to be problematic. In particular, $\beta(\hat\eta)$ may be estimated as follows. First, we construct an estimator of $\lambda_0$ component-wise by fitting non-parametric regression estimators. Since all the regression functions in (\ref{eq:regs}) are one-dimensional, they may be estimated by fitting a kernel regression. For instance, for a second-order kernel function $K_h$ with bandwidth $h$ the estimator of $e_0$ is given by \begin{equation}
\hat e(w) = \frac{\sum_{i = 1}^nA_i\,M_i\,K_{\hat h}\{\hat g(W_i) -
\hat g(w)\}\{Y_i - \hat m(W_i)\}}{\sum_{i=1}^nA_i\,M_i\,K_{\hat h}\{\hat g(W_i) -
\hat g(w)\}}.\label{eq:rhat} \end{equation} The bandwidth is chosen as $\hat h= n^{-0.1}\hat h_{\opt}$, where $\hat h_{\opt}$ is the optimal bandwidth chosen using K-fold cross-validation \citep[the optimality of this selector is discussed in][]{vanderVaart&Dudoit&vanderLaan06}. This bandwidth yields a convergence rate that allows application of uniform central limit theorems \citep[see Theorems 4 and 5 of][]{gine2008uniform}.
An estimator of the drift term may be constructed as \begin{multline}\hat \beta(\hat \eta) = \frac{1}{n}\sum_{i=1}^n\left[\frac{\hat e(W_i)}{\hat g_A(W_i)}\{A_i-\hat g_A(W_i)\}
+\frac{A_i\,\hat e(W_i)}{\hat g(W_i)}\{M_i-\hat g_M(W_i)\} +
\right.\\ \left. A_iM_i\left\{\frac{\hat r_A(W_i)}{\hat\gamma(W_i)}
+ \frac{\hat r_M(W_i)}{\hat\gamma_M(W_i)}\right\}\{Y_i - \hat
m(W_i)\}\right].\label{eq:hatbeta} \end{multline} In light of equation~(\ref{eq:wh}), the above estimator may be subtracted from the AIPW (or the TMLE) to obtain a drift-corrected estimator. We denote this estimator by $\hat\theta_{\daipww}=\hat\theta_{\aipww}-\hat\beta(\hat\eta)$.
Though sensible in principle, $\hat\theta_{\daipww}$ suffers from drawbacks similar to the standard AIPW estimator $\hat\theta_{\aipww}$: it may yield an estimator out of bounds of the parameter space and therefore have suboptimal finite sample performance (we illustrate this in our simulation study in Section~\ref{sec:simula}). In addition, a large sample analysis of $\hat\theta_{\daipww}$ suggests that the $n^{1/2}$-consistency of $\hat\theta_{\daipww}$ requires consistent estimation of $\lambda_0$ at the $n^{1/2}$ parametric rate. In particular, under \ref{ass:DR1}-2, equation~(\ref{eq:wh}) yields \begin{equation} \label{eq:wh1} \hat\theta_{\daipww}-\theta_0= \beta(\hat\eta) - \hat\beta(\hat\eta)+
(\mathbb{P}_{n} - P_0)D_{\eta_1, \theta_0} + o_P\big(n^{-1/2} +
|\beta(\hat\eta)|\big). \end{equation} Lemma~\ref{lemma:asbeta} in the appendix shows that, under \ref{ass:DR2}, \begin{equation} \beta(\hat\eta) - \hat\beta(\hat\eta) = -(\mathbb{P}_{n}-P_0)\{D_{A,\hat g, \lambda_0} + D_{M,\hat g, \lambda_0} +
D_{Y,\hat m, \lambda_0}\} + o_P(n^{-1/2}).\label{eq:wh2} \end{equation} Asymptotic linearity of
$\hat\theta_{\daipww}$ would then require that $|\beta(\hat\eta)|=O_P(n^{-1/2})$, so that the last term in the right-hand side of expression~(\ref{eq:wh1}) is $o_P(n^{-1/2})$. This would require $\lambda_0$ to be estimated at rate $n^{1/2}$, which is in general not achievable in the non-parametric model (e.g., the convergence rate of a kernel regression estimator with second order kernel and optimal bandwidth is $n^{2/5}$). It would thus appear that the $\hat\theta_{\daipww}$ estimator will not generally be asymptotically linear if the estimator of $\lambda_0$ converges to zero more slowly than $n^{-1/2}$.
Surprisingly, the large-sample analysis of the $\hat\theta_{\dtmlee}$ counterpart presented in Section~\ref{sec:tmle} below requires slower convergence rates for the estimator of $\lambda_0$, such that a Kernel regression estimator provides a sufficiently fast rate. This fact has been previously noticed in the context of estimation of a counterfactual mean by \cite{benkeser2016doubly}. We note that the optimal bandwidth $\hat h_{\text{opt}}$ in estimation of $\lambda_0$ yields estimators for which uniform central limit theorems do not apply. Therefore we propose to undersmooth using the bandwidth $\hat h$.
\section{Targeted Maximum Likelihood Estimation with
Doubly Robust Inference}\label{sec:tmle}
As transpires from the developments of the previous section, it is necessary to construct estimators $\hat\eta$ such that $\beta(\hat\eta)$ is $O_P(n^{-1/2})$. In light of expression~(\ref{eq:wh2}), this can be achieved through the construction of an estimator $\tilde\eta$ that satisfies $\hat\beta(\tilde\eta)=0$. This construction is based on the fact that $D_{Y,\hat m, \lambda_0}$, $D_{M,\hat g, \lambda_0}$, and $D_{M,\hat g, \lambda_0}$ are score equations in the model for $m_0$, $g_{M,0}$, and $g_{A,0}$, respectively. As a result, adding the corresponding covariates to a logistic tilting model will tilt an initial estimator $\hat \eta=(\hat g,\hat m)$ towards a solution $\tilde \eta$ of the bias-reducing estimating equations $\hat \beta(\tilde \eta)=0$, in a similar way to the logistic tilting submodel (\ref{eq:submodel}).
The proposed drift-corrected TMLE is defined by the following algorithm: \begin{enumerate}[label = Step~\arabic*., align=left, leftmargin=*] \item \textit{Initial estimators.} Obtain initial estimators
$\hat g_A$, $\hat g_M$, and $\hat m$ of $g_{A,0}$, $g_{M,0}$, and
$ m_0$. These estimators may be based on data-adaptive predictive
methods that allow flexibility in the specification of the
corresponding functional forms. Construct estimators $\hat\gamma_A$,
$\hat\gamma_M$, $\hat\mu$ of $\gamma_{A,0}$, $\gamma_{M,0}$,
$\mu_0$, respectively, by fitting kernel regression estimators as
described in the previous subsection. \item \textit{Compute auxiliary covariates.} For each
subject, compute the auxiliary covariates \[W_1(w)=\frac{1}{\hat g(w)},\,
W_2(w)=\frac{\hat r_A(w)}{\hat \gamma(w)} + \frac{\hat
r_M(w)}{\hat \gamma_M(w)},\,
Z_A(w)=\frac{\hat e(w)}{\hat g_A(w)},\, Z_M(w)=\frac{\hat e(w)}{\hat g(w)} \] \item \textit{Solve estimating equations.} Estimate the parameter
$\epsilon = (\epsilon_A, \epsilon_M, \epsilon_{Y,1}, \epsilon_{Y,2})$ in the logistic tilting models
\begin{align}
\logit m_\epsilon(w) &= \logit \hat m(w) + \epsilon_{Y,1} W_1(w) +
\epsilon_{Y,2} W_2(w),\label{eq:submodelY}\\
\logit g_{M,\epsilon}(w) &= \logit \hat g_M(w) + \epsilon_M
Z_M(w).\label{eq:submodelT}\\
\logit g_{A,\epsilon}(w) &= \logit \hat g_A(w) + \epsilon_A
Z_A(w)\label{eq:submodelT}
\end{align}
Here, $\logit \hat m(w)$, $\logit \hat g_A(w)$, and
$\logit \hat g_M(w)$ are offset variables (i.e., variables with
known parameter equal to one). The above parameters may be estimated
by fitting standard logistic regression models. For example,
$(\epsilon_{Y,1}, \epsilon_{Y,2})$ may be estimated through a
logistic regression model of $Y$ on $(W_1,W_2)$, with no intercept
and with offset $\logit \hat m(W)$ among observations with
$(A,M)=(1,1)$. Likewise, $\epsilon_M$ is estimated through a
logistic regression model of $M$ on $Z_M$ with no intercept and an
offset term equal to $\logit \hat g_M(W)$ among observations with
$A=1$. Lastly, $\epsilon_A$ may be estimated by fitting a logistic
regression model of $A$ on $Z_A$ with no intercept and an offset
term equal to $\logit \hat g_A(W)$ using all observations. Let
$\hat\epsilon$ denote these estimates. \item \textit{Update estimators and iterate.} Define the updated
estimators as $\hat m = m_{\hat \epsilon}$,
$\hat g_M = g_{M,\hat\epsilon}$, and $\hat g_A=g_{A,\hat
\epsilon}$. Repeat steps 2-4 until convergence. In practice, we
stop the iteration once
$\max\{|\hat\epsilon_A|, |\hat\epsilon_M|, |\hat\epsilon_{Y,1}|,
|\hat\epsilon_{Y,2}|\}< 10^{-4}n^{-3/5}$. \item \textit{Compute TMLE.} Denote the estimators in the last step of
the iteration with $\tilde m$, $\tilde g_M$, and $\tilde g_M$. The
drift-corrected TMLE of $\theta_0$ is defined as \[\hat\theta_{\dtmlee} = \frac{1}{n}\sum_{i=1}^n \tilde m(W_i).\] \end{enumerate}
The large sample distribution of the above TMLE is given in the following theorem:
\begin{theorem}[Doubly Robust Asymptotic Distribution of $\hat\theta_{\dtmlee}$]\label{theo:dr} Assume \ref{ass:donsker} and
\ref{ass:DR2} hold for $\tilde\eta$, and denote the limit of
$\tilde \eta$ with $\eta_1$. Then
\[n^{1/2}(\hat\theta_{\tmlee} - \theta_0)\to N(0, \sigma^2),\]where
$\sigma^2 = \var\{D_{\dr}(O)\}$ and
$D_{\dr}(O)=D_{\eta_1, \theta_0}(O) - D_{Y,m_1,\lambda_0}(O) -
D_{M,g_1,\lambda_0}(O) - D_{A,g_1,\lambda_0}(O)$. \end{theorem}
Note that, in an abuse of notation, we have denoted the limit of $\tilde\eta$ with $\eta_1$, though this limit need not be equal to the limit of the initial estimator $\hat\eta$.
\ref{ass:DR2}, assumed in the previous theorem, is stronger than the standard double robustness \ref{ass:DR1}. Under \ref{ass:DR1}, $\tilde m$ or $\tilde g$ may converge to their misspecified limits arbitrarily slowly as long as the product of their $L_2(P_0)$ norms converges at rate $n^{1/2}$. Under \ref{ass:DR2} each estimator is required to converge to its misspecified limit at rate $n^{1/4}$. This is a mildly stronger condition that we conjecture is satisfied by many data-adaptive prediction algorithms. In particular, it is satisfied by empirical risk minimizers (minimizing squared error loss or quasi log-likelihood loss) over Donsker classes. An example of a data-adaptive estimator that satisfies \ref{ass:DR2} is the highly adaptive lasso (HAL) proposed by \cite{van2015generally}. \ref{ass:DR2} is necessary to control the convergence rate of the estimator $\hat\lambda$. The reader interested in the technical details is encouraged to consult the proof of the theorem in the Supplementary Materials.
In light of Theorem~\ref{theo:dr}, the Wald-type confidence interval $\hat\theta_{\dtmlee} \pm z_{\alpha} \hat\sigma/\sqrt{n}$, where $\hat\sigma^2$ is the empirical variance of $\hat D_{\dr}(O)=D_{\tilde \eta, \hat\theta_{\dtmlee}}(O) - D_{Y,\tilde m,\hat
\lambda}(O) - D_{M,\tilde g, \hat \lambda}(O) - D_{A,\tilde g, \hat
\lambda}(O)$ has correct asymptotic coverage $(1-\alpha)100\%$, whenever at least one of $\tilde g$ and $\tilde m$ converges to its true value at the stated rate. However, computation of the confidence interval does not require one to know which of these nuisance parameters is consistently estimated.
\section{Simulation Studies}\label{sec:simula}
We compare the performance of our proposed enhanced estimators $\hat\theta_{\dtmlee}$ and $\hat\theta_{\daipww}$ with their standard versions $\hat\theta_{\tmlee}$ and $\hat\theta_{\aipww}$, using the following data distribution: \begin{align*}
\logit g_{M,0}(a,w)=&\,2 -w_1+4w_2-2w_4+3w_2w_6 + 3w_1w_5w_6 -\\ &\,a(1.5-4w_1+4w_2+2w_3-7w_1w_2-3w_2w_4w_5)\\ \logit m_0(a,w)=&\,-0.5 -w_1-w_2+w_4+2w_2w_6 + 2w_1w_5w_6 -\\
&\,a(2-w_1+3w_2+w_3-6w_1w_2-4w_2w_4w_5). \end{align*} For exogenous variables $\varepsilon_1,\ldots,\varepsilon_6$ distributed independently as uniform variables in the interval $(0,1)$, $W_1,\ldots,W_6$ were generated as \begin{align*}
W_1 &= \log(\varepsilon_1 + 1)\\
W_2 &= \varepsilon_2 / (1 + \varepsilon_1^2)\\
W_3 &= \varepsilon_1 + 1 / (\varepsilon_3 + 1)\\
W_4 &= \sqrt{\varepsilon_2 + \varepsilon_4}\\
W_5 &= \varepsilon_5\varepsilon_4\\
W_6 &= 1 / (\varepsilon_2 + \varepsilon_6 + 1). \end{align*} The treatment probabilities were set to $g_{A,0}(w)=0.5$, corresponding with a randomized trial with equal allocation, and the outcome was generated as $Y\mid \{A=a,W=w\}\sim \text{Bernoulli}(m_0(a,w))$. For this data generating mechanism we have a treatment effect of $\theta_0\approx 0.2328$, and $E(Y\mid A=1,M=1) - E(Y\mid A=0,M=1)\approx 0.3258$, indicating a strong selection bias due to informative missingness.
For each sample size $n$ in the grid $\{200,800,1800,3200,5000,7200,9800\}$, we generate 1000 datasets with the above distribution, and test four different scenarios for estimation of $g_{M,0}$ and $m_0$: \begin{enumerate*}[label=(\alph*)] \item consistent estimation of both $g_{M,0}$ and $m_0$, \item consistent estimation of $m_0$ and inconsistent estimation of
$g_{M,0}$, \item consistent estimation of $g_{M,0}$ and inconsistent estimation
of $m_0$, and \item inconsistent estimation of both $g_{M,0}$ and $m_0$. \end{enumerate*}
Consistent estimators of $g_{M,0}$ and $m_0$ are constructed by first creating a model matrix containing all possible interactions of $W$ up to fourth order, and then running $L_1$ regularized logistic regression. Inconsistent estimation follows the standard practice of fitting logistic regression models on main terms only. The use of $L_1$ regularization provides an example in which the asymptotic linearity of the drift term is not guaranteed. Since we do not assume we know which interactions are present, the use of data-adaptive estimators is the only possible way to obtain consistent estimators, as it is in most real data applications.
In all scenarios, the treatment mechanism is consistently estimated by fitting a logistic regression of $A$ on $W$ including main terms only, even though $g_{A,0}$ is known by design. Intuitively, the purpose of this model fit is to capture chance imbalances of the baseline variables $W$ between study arms for a given data set; these imbalances can then be adjusted to improve efficiency. The general theory underlying efficiency improvements through estimation of known nuisance parameters such as $g_A$ is presented, e.g., by \cite{Robins&Rotnitzky&Zhao94} and \cite{vanderLaan2003}.
We compare the performance of the four estimators in terms of four metrics:
\begin{enumerate}[label=(\roman*)] \item Coverage probability of a confidence interval based on the
central limit theorem, with variance estimated
as \[\hat\sigma^2=\frac{1}{n}\sum_{i=1}^n\text{IF}^2(O_i),\] where
IF is the estimated influence function of the corresponding
estimator. For $\hat\theta_{\aipww}$ and $\hat\theta_{\tmlee}$, the influence function used is
the efficient influence function $D_{\eta,\theta}$. For $\hat\theta_{\daipww}$ and
$\hat\theta_{\dtmlee}$, the influence function $D_{\dr}$ given in
Theorem~\ref{theo:dr}.
Confidence intervals for $\hat\theta_{\aipww}$ and $\hat\theta_{\tmlee}$ are expected to have
correct coverage in scenario (a), incorrect coverage in scenario
(b), and conservative coverage in scenario (c). In light of
Theorem~\ref{theo:dr}, the confidence interval based on $\hat\theta_{\dtmlee}$ is
expected to have correct coverage in scenarios (a)-(c). The behavior
of the confidence interval based on $\hat\theta_{\daipww}$ is conjectured to have
similar performance to the $\hat\theta_{\dtmlee}$, but our theory does not show
this in general. \item The absolute value of the bias scaled by $\sqrt{n}$. This value
is expected to converge to zero in scenarios (a)-(c) for all
estimators, and to diverge in scenario (d). For scenario (d), in
light of Remark~\ref{lemma:bias}, we conjecture that $\hat\theta_{\daipww}$ and
$\hat\theta_{\dtmlee}$ have generally smaller bias than $\hat\theta_{\aipww}$ and $\hat\theta_{\tmlee}$,
respectively. \item The squared root of the relative MSE (RMSE), scaled by
$\sqrt{n}$. The RMSE is defined as the MSE divided by the efficiency
bound $\var\{D_{\eta_0,\theta_0}(O)\}$. This metric is expected to
converge to one for all estimators in scenario (a) (i.e., all
estimators are efficient), it is expected
to converge to some other value in scenarios (b)-(c), and it is
expected to diverge in scenario (d). \item The average of the estimated standard deviations $\hat\sigma$
across 1000 datasets divided by the standard deviation of the
estimates $\hat\theta$. This metric is expected to converge to one
for all estimators in scenario (a), and for estimators $\hat\theta_{\daipww}$ and
$\hat\theta_{\dtmlee}$ in scenarios (b)-(c). \end{enumerate}
\begin{figure}
\caption{Results of the simulation study.}
\label{fig:res}
\end{figure} The results of the simulation are presented in Figure~\ref{fig:res}. In addition to corroborating the expected attributes of the estimators outlined in (i)-(iv) above, the following characteristics deserve further observation:
\begin{itemize} \item $\hat\theta_{\daipww}$ has a much higher variance compared to all other
estimators in scenario (a) for small samples ($n=200$) . This is
possibly a consequence of inverse weighting by small probabilities
in the definition of the correction factor $\hat\beta(\tilde\eta)$
(see equation \ref{eq:hatbeta}). This also affects $\hat\theta_{\dtmlee}$, but to
a lesser extent. \item $\hat\theta_{\daipww}$ and $\hat\theta_{\dtmlee}$ have considerably better performance than
$\hat\theta_{\aipww}$ and $\hat\theta_{\tmlee}$ in scenario (b): they achieve the asymptotic
efficiency bound and have significantly smaller bias. \item $\hat\theta_{\daipww}$ has smaller bias than all competitors under
scenario (d). \end{itemize}
\section{Concluding Remarks}\label{sec:discussion} We present estimators of the effect of treatment in randomized trials with missing outcomes, where the outcomes are missing at random. One of our proposed estimators, the DTMLE, is CAN under data-adaptive estimation of the missingness probabilities and the outcome regression, under consistency of at least one of these estimators. We present the doubly-robust influence function of the estimator, which can be used to construct asymptotically valid Wald-type confidence intervals. We show that the implied asymptotic distribution is valid under a smaller set of assumptions, compared to existing estimators.
As an anonymous referee pointed out, the method of \cite{benkeser2016doubly} could be applied to our problem by defining $T=AM$ and estimating $E\{E(Y\mid T=1, W)\}$. We find this approach unsatisfactory because it ignores intrinsic properties of the variables $A$ and $M$, which are more appropriately exploited when modeled independently. For example, $P(A=1\mid W)$ is known in a randomized trial, and a logistic regression model with at least an intercept term provides a consistent estimator. Furthermore, covariate adjustment through such logistic model is known to improve the precision of the resulting estimator. Optimally using auxiliary information of this type involves positing separate models for the conditional distributions of $A$ and $M$.
Our proposed methods share connections with the balancing score theory for causal inference \citep{Rosenbaum&Rubin83}. In particular, note that the score equations $\mathbb{P}_{n} D_{A,\tilde g, \hat \lambda}=0$ and $\mathbb{P}_{n} D_{M,\tilde g, \hat \lambda}=0$ are balancing equations that ensure that the empirical mean of $\hat e(W)$ is equal to its re-weighted mean when using weights $A_i/\tilde g_A(W_i)$ and $A_iM_i/\tilde g(W_i)$. Covariate balanced estimators have been traditionally used to reduce bias in observational studies and missing data models \citep[e.g.,][]{ hainmueller2011entropy,
imai2014covariate, zubizarreta2015stable}, but covariate selection for balancing remains an open problem. We conjecture that our theory may help to solve this problem by shedding light on key transformations of the covariates that require balance, such as $\hat e(W)$.
We also note that the methods presented may be readily extended to estimation of other parameters in observational data or randomized trials. In particular, the estimators for the causal effect of treatment on the quantile of an outcome presented in \cite{diaz2015efficient} are amenable to the correction presented here.
Finally, Donsker \ref{ass:donsker}, which may be restrictive in some settings, may be removed through the use of a cross-validated version of our TMLE. Such development would follow from trivial extensions of the work of \cite{zheng2011cross}, and would be achieved by constructing a cross-validated version of the MLE in step 2 of the TMLE algorithm presented in Section~\ref{sec:tmle}. \begin{appendices}
\section{Proofs}
\subsection{Theorem~\ref{lemma:betarep}} The drift term $\beta(\hat\eta)$ may be decomposed as \begin{align}
\beta(\hat\eta)=&\int \frac{1}{\hat g}\{g_0-\hat
g\}\{m_0-m_1\}dP_0 +\label{b1}\\
&\int \frac{1}{g_1}\{g_0- g_1\}\{m_0-\hat m\}dP_0+\label{b2}\\
&\int \frac{1}{\hat g}\{g_1-\hat
g\}\{m_1-\hat m\}dP_0 +\label{t1}\\
&\int \left\{\frac{1}{\hat
g}-\frac{1}{g_1}\right\}\{g_0- g_1\}\{m_1-\hat m\}dP_0+\label{t2}\\
&\int \frac{1}{g_1}\{g_0- g_1\}\{m_1-m_0\}dP_0\label{t3} \end{align} Under \ref{ass:DR2} we have $(\ref{t1})+(\ref{t2})=o_P(n^{-1/2})$, and $(\ref{t3})=0$. Denote (\ref{b1}) and (\ref{b2}) with $\beta_g(\hat g)$ and $\beta_m(\hat m)$, respectively. Then \begin{equation}\beta(\hat\eta) = \beta_g(\hat g) +
\beta_m(\hat m) + o_P(n^{-1/2}),\label{eq:biasdec} \end{equation}
Define \begin{align*}
\hat\gamma_{A,0}(W) &= P_0\big\{A = 1 \mid m_1(W), \hat m(W)\big\},\\
\hat\gamma_{M,0}(W) &= P_0\big\{M = 1\mid A = 1, m_1(W), \hat m(W)\big\},\\
\hat r_{A,0}(W) &= E_{P_0}\left\{\frac{A - g_{A,1}(W)}{g_{A,1}(W)}\mathrel{\bigg|}
m_1(W), \hat m(W)\right\},\\
\hat r_{M,0}(W) &= E_{P_0}\left\{\frac{M - g_{M,1}(W)}{g_1(W)}\mathrel{\bigg|} A=1, m_1(W), \hat m(W)\right\},\\
\hat e_0(W) &= E_{P_0}\big\{Y - m_1(W)\mid A=1,M=1,g_1(W),\hat g(W)\big\}. \end{align*} First, assume $g_1=g_0$, so that $\beta(\hat\eta) = \beta_g(\hat g) + o_P(n^{-1/2})$. We have \begin{align}
\beta_g(\hat g) =& \int \frac{1}{\hat g(w)}\{g_0(w)-\hat
g(w)\}\{ m_0(w)- m_1(w)\}dP_0(w)\notag\\
=& \int \frac{a\,m}{\hat g(w)g_0(w)}\{g_0(w)-\hat
g(w)\}\{y- m_1(w)\}dP_0(y,m,a,w)\notag\\
=& \int\left[\int \frac{a\,m}{\hat g(w)g_0(w)}\{y-m_1(w)\}\{g_0(w)-\hat
g(w)\}dP_0(y \mid a,m,w,g_0(w),\hat g(w))\right]\,dP_0(m,a,w)\notag\\
=& \int \frac{a\,m\,\hat e_0(w)}{\hat g(w)g_0(w)}\{g_0(w)-\hat
g(w)\}dP_0(m,a,w)\notag\\
=& \int \frac{\hat e_0(w)}{\hat g(w)}\{g_0(w)-\hat
g(w)\}dP_0(w)\notag\\
=& \int \frac{\hat e_0(w)}{\hat g(w)}\{a\,m-\hat
g(w)\}dP_0(m,a, w)\notag\\
=& \int\left[\frac{a\,\hat e_0(w)}{\hat g(w)}\{m - \hat
g_M(w)\} + \frac{\hat e_0(w)}{\hat g_A(w)}\{a-\hat
g_A(w)\}\right]dP_0(m,a,w)\notag\\
=& \int\left[\frac{a\,e_0(w)}{\hat g(w)}\{m - \hat
g_M(w)\} + \frac{ e_0(w)}{\hat g_A(w)}\{a-\hat
g_A(w)\}\right]dP_0(m,a,w)\label{DAMs}\\
&+ \int\left[\frac{a\,\{\hat e_0(w)-e_0(w)\}}{\hat g(w)}\{m - \hat
g_M(w)\} + \frac{\hat e_0(w)-e_0(w)}{\hat g_A(w)}\{a-\hat
g_A(w)\}\right]dP_0(m,a,w)\label{OPAM}. \end{align} Here $P_0(g_0(w),\hat g(w))$ is the distribution of the transformation $W\to (g_0(W), \hat g(W))$, where $\hat g$ is fixed. The third equality follows by the law of iterated expectation and is obtained by first conditioning on the joint distribution of $(M, A)$ and the transformations $g_0(W)$ and $\hat g(W)$.
The term (\ref{DAMs}) is $P_0\{D_{M,\hat g,\lambda_0} +
D_{A,\hat g, \lambda_0}\}$, whereas (\ref{OPAM}) is $O_P\left(||\hat g -
g_0||^2\right)$. Under \ref{ass:DR2} with $g_1=g_0$ the latter term is $o_P(n^{-1/2})$, so that \[\beta_g(\hat g) = P_0\{D_{M,\hat g,\lambda_0} +
D_{A,\hat g, \lambda_0}\} + o_P(n^{-1/2}).\] The result follows because, under $g_1=g_0$ we have $e_0(w)=0$, and thus $D_{Y, \hat\mu, \lambda_0}=0$.
Now assume $m_1=m_0$, we have $\beta(\hat\eta) = \beta_m(\hat m) + o_P(n^{-1/2})$. We have \begin{align*}
\beta_m(\hat m) =& \int\frac{1}{g_1(w)}\{g_0(w) -
g_1(w)\}\{ m_0(w) - \hat m(w)\}dP_0(w)\\
=& \int\left\{\frac{g_{A,0}}{g_1(w)}\{g_{M,0}(w) -
g_{M,1}(w)\} +\frac{1}{g_{A,1}}\{g_{A,0}-g_{A,1}\}\right\}\{ m_0(w) -
\hat m(w)\}dP_0(w)\\
=& \int\left\{\frac{a}{g_1(w)}\{m -
g_{M,1}(w)\} +\frac{1}{g_{A,1}}\{a-g_{A,1}\}\right\}\{ m_0(w) - \hat m(w)\}dP_0(m,a,w)\\
=& \int\left[a\hat r_{M,0}(w) + \hat
r_{A,0}(W)\right]\{ m_0(w) - \hat
m(w)\}dP_0(m,a,w)\\
=& \int\left[\hat \gamma_A(w)\hat r_{M,0}(w) + \hat
r_{A,0}(W)\right]\{ m_0(w) - \hat m(w)\}dP_0(m,a,w)\\
=& \int\frac{a\,m\,}{\hat\gamma_{A,0}(w)\hat\gamma_{M,0}(w)}\left[\hat \gamma_A(w)\hat r_{M,0}(w) + \hat
r_{A,0}(w)\right]\{y - \hat m(w)\}dP_0(m,a,w)\\
=& \int a\,m\left[\frac{r_{M,0}(w)}{\gamma_{M,0}(w)} +
\frac{r_{A,0}(w)}{\gamma_0(w)}\right]\{y - \hat m(w)\}dP_0(m,a,w) +
O_P(||\hat m - m_0||^2) \end{align*}
Under \ref{ass:DR2} with $m_1=m_0$ we have $||\tilde m -
m_0||^2=o_P(n^{-1/2})$ and $r_{A,0}(w)=r_{M,0}(w)=0$. Thus $D_{M,\tilde g, \lambda_0} = D_{A,\tilde g, \lambda_0}=0$. This completes the proof of the theorem.
\subsection{Theorem~\ref{theo:dr}} Arguing as in equation (\ref{eq:wh}) we get \begin{equation*}
\hat\theta_{\dtmlee}-\theta_0 = \beta(\tilde \eta) +
(\mathbb{P}_{n} - P_0)D_{\eta_1, \theta_0} + o_P\big(n^{-1/2} +
|\beta(\tilde\eta)|\big) \end{equation*} Note that, by construction (see Section~\ref{sec:tmle}), $\hat\beta(\tilde\eta)=0$, so that Lemma~\ref{lemma:asbeta} below gives us the asymptotic expression for $\beta(\tilde \eta)$. Substituting this expression we get \[\hat\theta_{\tmlee}-\theta_0 = (\mathbb{P}_{n} - P_0)(D_{\eta_1, \theta_0} -
D_{M,g_1,\lambda_0} - D_{A,g_1,\lambda_0} - D_{Y,m_1,\lambda_0}) + o_P\big(n^{-1/2} + O_P(n^{-1/2})\big).\] The last term is $o_P(n^{-1/2})$. This, together with the central limit theorem concludes the proof. \begin{lemma}[Asymptotic Linearity of $\beta(\hat\eta)$]\label{lemma:asbeta}
Assume \ref{ass:donsker} and \ref{ass:DR2}. Then
\begin{equation*}
\beta(\hat\eta) -\hat\beta(\hat\eta)= -(\mathbb{P}_{n} - P_0)\{D_{M,g_1,\lambda_0}+D_{A,g_1,\lambda_0} + D_{Y,m_1,\lambda_0}\}
+ o_P(n^{-1/2}).
\end{equation*} \end{lemma} \begin{proof}
From Theorem~\ref{lemma:betarep}, we have \[\beta(\hat\eta)=P_0\{D_{A,\hat g,\lambda_0} +
D_{M,\hat m,\lambda_0} + D_{Y,\hat m,\lambda_0}\} + o_P(n^{-1/2})\] Next, we show that $P_0D_{Y,\hat m,\lambda_0} -\mathbb{P}_{n} D_{Y,\hat m,\hat\lambda} = -(\mathbb{P}_{n}-P_0)D_{Y,m_1,\lambda_0} + o_P(n^{-1/2}).$ The result for the other terms follow an analogous analysis.
If $g_1(w)=g_0(w)$ we have $r_{A,0}(w)=r_{M,0}(w)=0$, which implies $D_{Y,\hat m,\lambda_0}(o) = D_{Y,m_1,\lambda_0}(o)=0$, and the result follows trivially. If $m_1=m_0$, we have \[P_0D_{Y,\hat m,\lambda_0}-\mathbb{P}_{n} D_{Y,\hat m,\hat\lambda} = -(\mathbb{P}_{n}-P_0)D_{Y,\hat m,\hat\lambda} +
P_0(D_{Y,\hat m,\lambda_0} - D_{Y,\hat m,\hat\lambda}),\] where we added and subtracted $P_0D_{Y,\hat m,\hat\lambda}$. We have \[ P_0(D_{Y,\hat m,\lambda_0} - D_{Y,\hat m,\hat\lambda})
=\int g_0\left\{\frac{r_{M,0}}{\gamma_{M,0}} - \frac{\hat
r_M}{\hat\gamma_M} + \frac{r_{A,0}}{\gamma_0} - \frac{\hat
r_A}{\hat\gamma}\right\}\{m_0 - \hat m\}dP_0 \] Using the Cauchy-Schwartz and triangle inequalities, we obtain
\[ P_0(D_{Y,\hat m,\lambda_0} - D_{Y,\hat m,\hat\lambda})=O_P\big(||\hat m -
m_0||\{||\hat r_A - r_{A,0}|| + ||\hat r_M - r_{M,0}|| +
||\hat\gamma_A - \gamma_{A,0}|| + ||\hat\gamma_M - \gamma_{M,0}||\}\big)\] In light of Lemma~\ref{lemma:rs} below we get
\[ P_0(D_{Y,\hat m,r_0} - D_{Y,\hat m,\hat r})=O_P\big(||\hat m -
m_0||\{||\hat g - g_1|| + ||\hat m -
m_0|| + n^{-7/20}\}\big).\] By \ref{ass:DR2} this term is $o_P(n^{-1/2})$.
Under \ref{ass:donsker} and \ref{ass:DR2}, $D_{Y,\hat m,\hat\lambda}$ an application of Theorem 4 of \cite{gine2008uniform} and example 2.10.10 of \cite{vanderVaart&Wellner96} yields that $D_{Y,\hat
m,\hat\lambda}$ is in a Donsker class. Thus, according to theorem 19.24 of \cite{vanderVaart98}: $P_0D_{Y,\hat\eta,\lambda_0}-\mathbb{P}_{n} D_{Y,\hat m,\hat\lambda} = -(\mathbb{P}_{n}-P_0)D_{Y,\eta_1,\gamma_0} + o_P(n^{-1/2})$. \end{proof} \begin{lemma}\label{lemma:rs} Assume $\hat\gamma_A$, $\hat \gamma_M$, and $\hat\mu$ use the
bandwidth $\hat h = n^{-0.1}\hat h_{\opt}$ and $K_h$ is a second order kernel. Then \begin{align*}
||\hat \gamma_A - \gamma_{A,0}|| &= O_P\big(||\hat g - g_1|| + ||\hat m -
m_1|| + n^{-7/20}\big)\\
||\hat \gamma_M - \gamma_{M,0}|| &= O_P\big(||\hat g - g_1|| + ||\hat m -
m_1|| + n^{-7/20}\big)\\
||\hat r_A - r_{A,0}|| &= O_P\big(||\hat g - g_1|| + ||\hat m -
m_1|| + n^{-7/20}\big)\\
||\hat r_M - r_{M,0}|| &= O_P\big(||\hat g - g_1|| + ||\hat m -
m_1|| + n^{-7/20}\big)\\
||\hat e - e_0|| &= O_P\big(||\hat g - g_1|| + ||\hat m -
m_1|| + n^{-7/20}\big) \end{align*} \end{lemma} \begin{proof} We prove the result for $\hat e$. The proofs for the other components of $\hat \lambda$ follow symmetric arguments. Let \[\hat e_0(w) = \frac{\sum_{i = 1}^n A_iM_iK_{\hat h}\{g_1(W_i) -
g_1(w)\}\{Y_i - m_1(W_i)\}}{\sum_{i=1}^nA_iM_iK_{\hat h}\{g_1(W_i) -
g_1(w)\}}\] denote the kernel regression estimator that would be computed if $m_1$ and $g_1$ were known. The triangle inequality yields
\[||\hat e - e_0||\leq ||\hat e - \hat e_0||+||\hat e_0 - e_0||\] Under the conditions of the lemma, since
$\hat h=n^{-0.1}\hat h_{\opt}$ is an undersmoothing bandwidth, the leading term of $||\hat e_0 - e_0||^2$ is the variance of a kernel estimator, which is of order $n^{-1}\hat h^{-1}=O_P(n^{-7/10})$, which yields $||\hat e_0 - e_0||=O_P(n^{-7/20})$. The first term concerns estimation of $\mu_1$ and $g_1$ and may be analyzed as follows. To simplify notation, for a given $g$, let \[K^\star_{g, i}(x) = \frac{K_{\hat h}\{g(X_i) -
g(x)\}}{\sum_{i=1}^nK_{\hat h}\{g(X_i) - g(x)\}}.\] Thus \begin{align*}
\hat e(x) - \hat e_0(x) = &\sum_{i=1}^nA_iM_iK^\star_{\hat g,
i}(x)\{Y_i-\hat m(X_i)\} - \sum_{i=1}^nA_iM_iK^\star_{g_1,
i}(x)\{Y_i-m_1(X_i)\} \\
= &\sum_{i=1}^nA_iM_i\{K^\star_{\hat g, i}(x) - K^\star_{g_1,
i}(x)\}\{Y_i-\hat m(X_i)\} \\ &+ \sum_{i=1}^nA_iM_iK^\star_{g_1,
i}(x)\{m_1(X_i)-\hat m(X_i)\}. \end{align*}
Taking $||\cdot||$ on both sides along with the triangle inequality yields the result in the lemma. \end{proof}
\section{R code}\label{sec:code}
\end{appendices}
\end{document} | arXiv | {
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\begin{document}
\title {
Bakry-\'Emery curvature and diameter bounds on graphs }
\author[Liu]{Shiping Liu} \address{S. Liu, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui Province, China} \email{spliu@ustc.edu.cn}
\author[M\"unch]{Florentin M\"unch} \address{F. M\"unch, Institut f\"ur Mathematik\\Universit{\"a}t Potsdam \\14476 Potsdam, Germany }\email{chmuench@uni-potsdam.de}
\author[Peyerimhoff]{Norbert Peyerimhoff} \address{N. Peyerimhoff, Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom} \email{norbert.peyerimhoff@durham.ac.uk}
\begin{abstract}
We prove diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry-\'Emery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from \cite{Fathi2015} and \cite{Horn2014} and solve a conjecture from \cite{Cushing2016}. \end{abstract} \maketitle \section{Introduction}
The classical Bonnet-Myers theorem states that for a complete, connected Riemannian manifold with Ricci-curvature bounded from below by $K>0$, the diameter is bounded by the diameter of the sphere with the same dimension and Ricci-curvature $K$ (see \cite{Myers1941}). Moreover by Cheng's Rigidy theorem (see \cite{Cheng1975}), sharpness is obtained if and only if the manifold is a sphere. Bakry and Ledoux \cite{BL1996} successfully established this theorem for abstract Markov generators which are diffusion and satisfy Bakry-\'Emery-Ricci curvature \cite{BE1985} conditions.
Our aim is to give a simple proof of this theorem in a discrete setting. Indeed, discrete space Markov generators are not diffusion and therefore, the theory of Bakry and Ledoux is not applicable. However, for many discrete curvature notions there is already a Bonnet-Myers-type result established (for sectional curvature on planar graphs, see \cite{DeVos2007,Higuchi2001,Keller2014,Stome1976}, and for Ollivier-Ricci-curvature, see \cite{Ollivier2009}, and for Forman's discrete Ricci curvature, see \cite{Forman2003}).
In this article, we focus on Bakry-\'Emery-Ricci-curvature \cite{BE1985,Schmueckenschlaeger1999,LY2010,Klartag2015}.
Under the assumption of finite measure and bounded vertex degree, a discrete Bonnet-Myers theorem has been proven for Bakry-\'Emery-curvature on discrete Markov-chains \cite{Fathi2015}. Furthermore, a discrete Bonnet-Myers type theorem was established under the $CDE'$-condition in \cite{Horn2014}, whereby $CDE'$ is stronger than the $CD$ condition \cite{Muench2015}. We will prove diameter bounds under $CD$ conditions which give sharp results and improve diameter bounds from \cite{Fathi2015} and \cite{Horn2014}. Moreover, our results solve Conjecture~8.1 from \cite{Cushing2016}.
In contrast to manifolds, we can upper bound the Laplacian by the gradient on graphs, that is, $(\Delta f)^2 \leq C \Gamma f$ for all functions $f$ on the vertices and a constant $C > 0$ depending only on the maximal vertex degree. This property will give us diameter bounds using the vertex degree instead of the dimension parameter in the curvature-dimension condition.
\subsection{Organization of the paper and main results} In Subsection~1.2, we define Bakry-\'Emery-Ricci-curvature and discuss different distance and diameter notions.
In Section~2, we give two versions of diameter bounds.
The first result (Corollary~\ref{cor:Bonnet Myers CD(K,infty)}) is using $CD(K,\infty)$ and bounded vertex degree $\operatorname{Deg}_{\max}$ and gives the sharp estimate $$ \operatorname{diam}_d(G) \leq \frac {2\operatorname{Deg}_{\max}}{K}. $$ This solves Conjecture~8.1 from \cite{Cushing2016} claiming that for every graph satisfying $CD(K,\infty)$ with $K>0$, there should exist an upper diameter bound of the graph only depending on $K$ and $\operatorname{Deg}_{\max}$.
The second result (Theorem~\ref{thm:Bonnet-Myers CD(K,n)}) works in a more general setting, in particular, no boundedness of the vertex degree is needed anymore. But instead, we will assume $CD(K,n)$ with finite $n$ to prove $$ \operatorname{diam}_\rho \leq \pi \sqrt{\frac {n}K} $$ where $\rho$ is the resistance metric (see Definition~\ref{def:resistance metric}).
Finally in Subsection~2.3, we compare these diameter bounds to diameter bounds from \cite{Fathi2015} and \cite{Horn2014}. In comparison to \cite{Horn2014}, we need a weaker curvature assumption and obtain stronger estimates (see Remark~\ref{rem:Horn}). In comparison to \cite{Fathi2015}, we have an improvement by a factor of $2$ under the same curvature assumptions (see Remark~\ref{rem:fathi}).
\subsection{Setup and notations}
A triple $G=(V,w,m)$ is called a \emph{(weighted) graph} if $V$ is a countable set, if $w:V^2 \to [0,\infty)$ is symmetric and zero on the diagonal and if $m:V \to (0,\infty)$. We call $V$ the \emph{vertex set}, $w$ the \emph{edge weight} and $m$ the \emph{vertex measure}. We define the \emph{graph Laplacian} as a map ${\mathbb{R}}^V \to {\mathbb{R}}^V$ via $\Delta f(x) := \frac 1 {m(x)} \sum_y w(x,y)(f(y) - f(x))$. In the following, we only consider \emph{locally finite} graphs, i.e., for every $x \in V$ there are only finitely many $y \in V$ with $w(x,y) >0$. We write $\operatorname{Deg}(x) := \frac{\sum_y w(x,y)}{m(x)}$ and $\operatorname{Deg}_{\max} := \sup_x \operatorname{Deg}(x)$.
\begin{defn} [Bakry-\'Emery-curvature] The \emph{Bakry-\'Emery-operators} are defined via $$ 2\Gamma(f,g) := \Delta (fg) - f\Delta g - g\Delta f $$ and $$ 2\Gamma_2(f,g) := \Delta \Gamma(f,g) - \Gamma(f, \Delta g) - \Gamma(g,\Delta f). $$ We write $\Gamma(f):= \Gamma(f,f)$ and $\Gamma_2(f):=\Gamma_2(f,f)$.
A graph $G$ is said to satisfy the \emph{curvature dimension inequality} $CD(K,n)$ for some $K\in {\mathbb{R}}$ and $n\in (0,\infty]$ if for all $f$, $$ \Gamma_2(f) \geq \frac 1 n (\Delta f)^2 + K \Gamma f. $$ \end{defn}
Next, we define combinatorial and resistance metrics and diameters.
\begin{defn}[Combinatorial metric]\label{def:combinatorial metric}
Let $G=(V,w,m)$ be a locally finite graph.
We define the \emph{combinatorial metric} $d:V^2 \to [0,\infty)$ via
$$
d(x,y) := \min \{n : \mbox{ there exist } x=x_0,\ldots,x_n=y \mbox{ s.t. } w(x_i,x_{i-1})>0 \,\mbox{for all } i=1\ldots n\}
$$
and the \emph{combinatorial diameter} as
$\operatorname{diam}_d(G) := \sup_{x,y \in V} d(x,y)$. \end{defn}
\begin{defn}[Resistance metric]\label{def:resistance metric}
Let $G=(V,w,m)$ be a locally finite graph.
We define the \emph{resistance metric} $\rho:V^2 \to [0,\infty)$ via
$$
\rho(x,y) := \sup \{f(y) - f(x) : \left\| \Gamma f \right\|_\infty \leq 1\}
$$
and the \emph{resistance diameter} as
$\operatorname{diam}_\rho(G) := \sup_{x,y \in V} \rho(x,y)$. \end{defn}
In the case of bounded degree, there is a standard estimate between combinatorial and resistance metric.
\begin{lemma}(Combinatorial and resistance metric)\label{lem:d, rho}
Let $G=(V,w,m)$ be a locally finite graph with $\operatorname{Deg}_{\max} < \infty$. Then for all $x_0,y_0 \in V$,
$$
d(x_0,y_0) \leq \sqrt{\frac{\operatorname{Deg}_{\max}}{2}}\rho(x_0,y_0).
$$ \end{lemma}
\begin{proof}
Let $f:V \to {\mathbb{R}}$ be defined by $f(x) := d(x,x_0)\sqrt{\frac{2} {\operatorname{Deg}_{\max}}}$. Then for all $x \in V$, \begin{align*}
0\leq \Gamma f(x) &= \frac 1 {2 m(x)} \sum_y w(x,y)(f(y) - f(x))^2 \\
&\leq \frac 1 {2 m(x)} \sum_y w(x,y) \frac 2{\operatorname{Deg}_{\max}}\\
&=\frac{\operatorname{Deg}(x)}{\operatorname{Deg}_{\max}}\\
&\leq 1. \end{align*} Hence, \begin{align*} \rho(x_0,y_0) \geq f(y_0) - f(x_0) = d(y_0,x_0)\sqrt{\frac{2} {\operatorname{Deg}_{\max}}}. \end{align*} This directly implies the claim. \end{proof}
\section{Bonnet-Myers via the Bakry-\'Emery curvature-dimension condition}
In the first subsection, we obtain sharp diameter bounds for $CD(K,\infty)$. In the second subsection, we obtain diameter bounds for unbounded Laplacians for $CD(K,n)$. In the third subsection we show that our results improve the diameter bounds from \cite[Theorem~7.10]{Horn2014} and from \cite[Corollary~6.4]{Fathi2015}.
\subsection{Diameter bounds and $CD(K,\infty)$} The key to prove diameter bounds from $CD(K,\infty)$ is the semigroup characterization of $CD(K,\infty)$ which is equivalent to $$ \Gamma P_t f \leq e^{-2Kt} P_t \Gamma f. $$ {Here, $P_t$ denotes the heat semigroup operator.} For details, see e.g. \cite{Gong2015,Lin2015}.
\begin{theorem}[Distance bounds under $CD(K,\infty)$]\label{thm:distance bounds CD(K,infty)}
Let $(V,w,m)$ be a connected graph satisfying $CD(K,\infty)$ and $\operatorname{Deg}_{\max}<\infty$.
Then for all $x_0,y_0 \in V$,
$$
\rho(x_0,y_0) \leq \frac {\sqrt{2\operatorname{Deg}(x_0)} + \sqrt{2\operatorname{Deg}(y_0)}}K.
$$ \end{theorem}
\begin{proof}
By \cite[Theorem~3.1]{Lin2015} and $\operatorname{Deg}_{\max}<\infty$, we have that $CD(K,\infty)$ is equivalent to
\begin{equation} \label{eq:semigrpchar}
\Gamma P_t f \leq e^{-2Kt} P_t \Gamma f
\end{equation}
for all bounded functions $f:V\to {\mathbb{R}}$.
Due to Cauchy-Schwarz, $(\Delta g(x))^2 \leq 2 {\operatorname{Deg}(x)} (\Gamma g)(x)$ for all $x\in V$ and all $g:V\to {\mathbb{R}}$.
We fix $x_0 , y_0 \in V$ and $\varepsilon>0$.
Then by definition of $\rho$, there is a function $f:V\to {\mathbb{R}}$ s.t.
$f(y_0)-f(x_0) > \rho(x_0,y_0) - \varepsilon$ and $\Gamma f \leq 1$.
W.l.o.g., we can assume that $f$ is bounded. Putting everything together yields for all $x \in V$,
\begin{align*}
\left|\partial_t P_t f(x) \right|^2
= \left|\Delta P_t f(x) \right|^2
\leq 2 {\operatorname{Deg}(x)} \Gamma P_t f(x)
\leq 2 {\operatorname{Deg}(x)} e^{-2Kt} P_t \Gamma f(x)
\leq 2 \operatorname{Deg}(x) e^{-2Kt}.
\end{align*}
By taking the square root and integrating from $t=0$ to $\infty$,
we obtain
\begin{align*}
|P_T f(x) - f(x)|
\leq \int_0^{\infty} \left|\partial_t P_t f(x) \right| dt
\leq \int_0^{\infty} \sqrt{2\operatorname{Deg}(x)} e^{-Kt} dt
= \frac {\sqrt{2\operatorname{Deg}(x)}} K{}
\end{align*}
for all $T>0$ and $x \in V$.
Now, the triangle inequality yields
\begin{align*}
\rho(x_0,y_0) - \varepsilon &\leq |f(x_0) - f(y_0)| \\
&\leq \left|P_t f(x_0) - f(x_0) \right|
+ \left|P_t f(x_0) - P_t f(y_0) \right|
+ \left|P_t f(y_0) - f(y_0) \right| \\
& \leq \frac {\sqrt{2\operatorname{Deg}(x_0)} + \sqrt{2\operatorname{Deg}(y_0)}}K + \left|P_t f(x_0) - P_t f(y_0) \right| \\
&\stackrel{t \to \infty}{\longrightarrow} \frac {\sqrt{2\operatorname{Deg}(x_0)} + \sqrt{2\operatorname{Deg}(y_0)}}K
\end{align*}
where $\left|P_t f(x_0) - P_t f(y_0) \right| \rightarrow 0$, since the graph is connected and since $\Gamma P_t f \rightarrow 0$ as $t \to \infty$ because of \eqref{eq:semigrpchar}.
Taking the limit $\varepsilon \to 0$ finishes the proof. \end{proof} We now use the distance bound to obtain a bound on the combinatorial diameter. \begin{corollary}[Diameter bounds under $CD(K,\infty)$]\label{cor:Bonnet Myers CD(K,infty)}
Let $(V,w,m)$ be a connected graph satisfying $CD(K,\infty)$.
Then,
$$
\operatorname{diam}_d(G) \leq \frac {2 \operatorname{Deg}_{\max}}K.
$$ \end{corollary} \begin{proof}
Due to Lemma~\ref{lem:d, rho} and the above theorem, we have for all $x_0, y_0$,
\begin{align*}
d(x_0,y_0) \leq \sqrt{\frac{\operatorname{Deg}_{\max}}{2}} \rho(x_0,y_0)
\leq \sqrt{\frac{\operatorname{Deg}_{\max}}{2}} \frac {\sqrt{2\operatorname{Deg}(x_0)} + \sqrt{2\operatorname{Deg}(y_0)}}K \leq \frac{2\operatorname{Deg}_{\max}}{K}.
\end{align*}
Thus, $
\operatorname{diam}_d(G) \leq \frac {2 \operatorname{Deg}_{\max}}K$ as claimed. \end{proof}
Indeed, this diameter bound is sharp for the $n$-dimensional hypercube which has diameter $n$, curvature bound $K=2$ and vertex degree $\operatorname{Deg}_{\max} =n$.
An interesting question is whether an analog of Cheng's rigidy theorem (see \cite{Cheng1975}) holds true. In particular, we ask whether hypercubes are the only graphs for which the above diameter bound is sharp.
\subsection{Diameter bounds and $CD(K,n)$}
We can also give diameter bounds for unbounded Laplacians. We need two ingredients to do so. First, we have to replace the combinatorial metric by a resistance metric. Second, we have to assume a finite dimension bound.
Furthermore, we will need completeness of the graph and non-degenerate vertex measure to obtain the semigroup characterization of $CD(K,n)$ (see \cite[Theorem~3.3]{Gong2015}, \cite[Theorem~1.1]{Hua2015}). For definitions of completeness of graphs and non-degenerate vertex measure, see Sections 1 and 2.1 of \cite{Hua2015} or \cite[Definition~2.9, Definition~2.13]{Gong2015}.
We start with an easy but useful consequence of Gong and Lin's semigroup characterization of $CD(K,\infty)$, see \cite[Theorem~3.3]{Gong2015}.
\begin{lemma}[Semigroup property of $CD(K,n)$]\label{lem:semigroup characterization CD(K,n)}
Let $G=(V,w,m)$ be a complete graph with non-degenerate vertex measure.
Suppose $G$ satisfies $CD(K,n)$. Then for all bounded $f:V \to {\mathbb{R}}$ with bounded $\Gamma f$, \begin{align}\label{eqn:CD(K,n) semigroup characterization}
\Gamma P_t f \leq e^{-2Kt} P_t \Gamma f - \frac{1-e^{-2Kt}}{Kn}(\Delta P_t f)^2. \end{align} \end{lemma}
\begin{proof} We first assume that $f$ is compactly supported.
By \cite[Theorem~3.3]{Gong2015}, we have
\begin{align}\label{eq:Char CD(K,n) Gong Lin}
\Gamma P_t f \leq e^{-2Kt} P_t \Gamma f - \frac 2 n \int_0^t e^{-2Ks} P_s (\Delta P_{t-s} f)^2 ds.
\end{align}
Jensen's inequality yields $P_s g^2 \geq (P_s g)^2$ for all $g$ and thus by $g:=\Delta P_{t-s} f$, \begin{align}\label{eq:Jensen semigroup} \frac 2 n \int_0^t e^{-2Ks} P_s (\Delta P_{t-s} f)^2 ds \geq \frac 2 n \int_0^t e^{-2Ks} (P_s \Delta P_{t-s} f)^2 ds =\frac{1-e^{-2Kt}}{Kn}(\Delta P_t f)^2. \end{align}
Putting (\ref{eq:Char CD(K,n) Gong Lin}) and (\ref{eq:Jensen semigroup}) together yields the claim for compactly supported $f$.
We now prove the claim for all bounded $f$ with bounded $\Gamma(f)$ by completeness and a density argument.
Completeness implies that there are compactly supported $\left(\eta_k\right)_{k \in {\mathbb{N}}}$ s.t.
$\eta_k \to 1$ from below and $\Gamma \eta_k \leq 1$.
Due to compact support, (\ref{eqn:CD(K,n) semigroup characterization}) holds for $\eta_k f$. Obviously since $\eta_k \to 1$ from below, we have
$\Gamma P_t(\eta_k f) \to \Gamma P_t f$ and $ \Delta P_t(\eta_k f) \to \Delta P_t f$, pointwise for $k \to \infty$.
It remains to show
$$
P_t \Gamma (\eta_k f) \to P_t \Gamma f,
$$
pointwise for $k \to \infty$.
We observe \begin{align*}
\left[ (\eta_k f)(y) - (\eta_k f)(x)\right]^2 &= \left[\eta_k(y)(f(y)-f(x)) + f(x)(\eta_k(y) - \eta_k(x)) \right]^2 \\
&\leq 2\left[\eta_k(y)(f(y)-f(x)) \right]^2 + 2 \left[ f(x)(\eta_k(y) - \eta_k(x)) \right]^2\\
&\leq 2\left\| \eta_k \right\|_\infty^2 (f(y)-f(x))^2 + 2\|f\|_\infty^2 (\eta_k(y) - \eta_k(x))^2 \end{align*}
and thus,
$$
\Gamma(\eta_k f) \leq 2\left\| \eta_k \right\|_\infty^2 \Gamma f + 2\|f\|_\infty^2 \Gamma \eta_k.
$$
This implies that $\Gamma(\eta_k f)$ is uniformly bounded in $k$ and since $\eta_k f \to f$ pointwise, we obtain
$P_t \Gamma (\eta_k f) \to P_t \Gamma f$ as desired. \end{proof}
With this semigroup property in hands, we now can prove diameter bounds. We will use similar methods as in the proof of Theorem~\ref{thm:distance bounds CD(K,infty)}.
\begin{theorem}[Diameter bounds under $CD(K,n)$]\label{thm:Bonnet-Myers CD(K,n)}
Let $G=(V,w,m)$ be a connected, complete graph with non-degenerate vertex measure.
Suppose $G$ satisfies $CD(K,n)$ for some $K>0$ and $n<\infty$. Then,
$$
\operatorname{diam}_\rho(G) \leq \pi \sqrt{\frac n K}.
$$ \end{theorem}
\begin{proof}
Suppose the opposite. Then, there are $x,y \in V$ s.t. $\rho(x,y)> \pi \sqrt{\frac n K}$, and there is a function $f:V \to {\mathbb{R}}$ s.t. $\Gamma f \leq 1$ and $f(y) - f(x) > \pi \sqrt{\frac n K}$. W.l.o.g., we can assume that $f$ is bounded.
By the semigroup property of $CD(K,n)$ from Lemma~\ref{lem:semigroup characterization CD(K,n)} and by ignoring the non-negative term $\Gamma P_t f$, we have
\begin{align*}
\frac{1-e^{-2Kt}}{Kn}\left( \Delta P_t f \right)^2 \leq e^{-2Kt} P_t \Gamma f.
\end{align*}
Taking square root and applying $\Gamma f \leq 1$ yields
$$
|\partial_t P_t f| = |\Delta P_t f| \leq \sqrt{Kn} \sqrt{\frac{e^{-2Kt}}{1-e^{-2Kt}}} = \sqrt{Kn} \sqrt{\frac{1}{e^{2Kt}-1}}.
$$
Integrating from $t=0$ to $\infty$ yields for all $T$,
\begin{align*}
|P_T f - f|
\leq \sqrt {Kn} \int_{0}^{\infty} \sqrt{\frac{1}{e^{2Kt}-1}} dt
= \sqrt {Kn} \frac{ \arctan{\sqrt{ e^{2Kt} - 1 }}}{K} \bigg|_{t=0}^{\infty} = \frac {\pi}{2} \sqrt{\frac{n}{K}}
\end{align*}
Due to $CD(K,\infty)$ which implies $\| \Gamma P_t f \|_\infty \stackrel{t\to \infty}{\longrightarrow} 0$ and since $G$ is connected, we infer
$\left|P_t f(x) - P_t f(y) \right| \stackrel{t\to \infty}{\longrightarrow} 0$.
We now apply the triangle inequality and obtain
\begin{align*}
\pi \sqrt{\frac n K}
&< |f(y) - f(x)| \\
&\leq |P_t f(y) - f(y)| + |P_t f(y) - P_t f(x)| +|P_t f(x) - f(x)| \\
&\leq \pi \sqrt{\frac n K} + |P_t f(y) - P_t f(x)| \\
&\stackrel{t\to \infty}{\longrightarrow} \pi \sqrt{\frac n K}.
\end{align*}
This is a contradiction and thus, $\operatorname{diam}_\rho(G) \leq \pi \sqrt{\frac n K}$ as claimed. \end{proof}
In contrast to Corollary~\ref{cor:Bonnet Myers CD(K,infty)}, we cannot have sharpness in Theorem~\ref{thm:Bonnet-Myers CD(K,n)} since in the proof, we have thrown away $\Gamma P_t f$ which is strictly positive for $t>0$.
\subsection{Comparison with other discrete diameter bounds}
In \cite[Theorem 7.10]{Horn2014}, Horn, Lin, Liu and Yau have proven \[ CDE'(K,n) \quad \Longrightarrow \quad \operatorname{diam}_d(G) \leq 2\pi \sqrt{\frac{6 \operatorname{Deg}_{\max} n}{K} }. \]
Indeed, due to \cite[Corollary~3.3]{Muench2015} and Theorem~\ref{thm:Bonnet-Myers CD(K,n)}, this result can be improved to
\[ CDE'(K,n) \quad \Longrightarrow \quad CD(K,n) \Longrightarrow \quad \operatorname{diam}_\rho(G) \leq \pi \sqrt{\frac{n}{K}}. \] In case of $\operatorname{Deg}_{\max} < \infty$, we have $$ \operatorname{diam}_d(G) \leq \sqrt{\frac{ \operatorname{Deg}_{\max}}{2}} \operatorname{diam}_\rho(G) \leq \pi \sqrt{\frac{\operatorname{Deg}_{\max} n}{ 2K}}, $$ where the first estimate is due to Lemma~\ref{lem:d, rho}.
\begin{rem}\label{rem:Horn}
Summarizing, we can say that our approach improves \cite[Theorem 7.10]{Horn2014} by a factor of $4\sqrt 3$ and by having weaker curvature assumptions. \end{rem}
Let us also compare our results to the results on Markov-chains in \cite{Fathi2015}. Translated into the graph setting, we have the following result in \cite{Fathi2015}.
\begin{theorem}(see \cite[Corollary~6.4]{Fathi2015})\label{thm:Fathi} Let $G=(V,w,m)$ be a graph with $\operatorname{Deg}_{\max} < \infty$ and $m(V) := \sum_x m(x) < \infty$. Suppose $G$ satisfies $CD(K,\infty)$ for some $K>0$.
Then for all $x,y \in V$, \begin{align*} \rho(x,y) \leq 2\sqrt 2 \frac{ \left(\sqrt{\operatorname{Deg}(x)} + \sqrt{\operatorname{Deg}(y)}\right)}{K}. \end{align*} \end{theorem}
\begin{proof}
Indeed, the theorem is just a reformulation of \cite[Corollary~6.4]{Fathi2015}. \begin{comment}
There is the following translation between the Markov-chains and graphs.
\begin{enumerate}
\item $\Delta f(x) = \sum_y (f(y) - f(x))K(x,y) = \frac 1 m(x) \sum_y w(x,y) (f(y)-f(x)) $
\item The Markov-kernel $K(x,y)$ equals $w(x,y)/m(x)$ for $x\neq y$
\item by $\sum_y K(x,y) = 1$, we have $\operatorname{Deg}_{\max} \leq 1$
\item Due to $K(x,y)\mu(x) = K(y,x)\mu(y)$ and due to symmetry of $w$ and due to $\mu(V)=1$, we have $\mu(x) = m(x)/m(V)$ for all $x\in V$.
\item The vertex laziness rate $J(x)$ equals the vertex degree $\operatorname{Deg}(x)$, thus $\operatorname{Deg}(x) \leq 1$.
\end{enumerate} \end{comment} Let $G=(V,w,m)$ be a graph with $m(V) < \infty$ and $\operatorname{Deg}_{\max} < \infty$. We define the corresponding Markov kernel as $$K(x,y) := \begin{cases} \frac{w(x,y)}{m(x)\operatorname{Deg}_{\max}} &: x\neq y\\ 1 - \frac{\operatorname{Deg}(x)}{\operatorname{Deg}_{\max}}&: x=y \end{cases} $$ and the corresponding measure $\mu(x) := m(x)/m(V)$ for all $x\in V$ according to \cite{Fathi2015}. Then, $Lf(x) := \sum_y (f(y) - f(x)) K(x,y) = \frac {\Delta f(x)}{\operatorname{Deg}_{\max}}$.
Since $\Delta$ satisfies $CD(K,\infty)$, we have that $L$ satisfies $CD(\frac K {\operatorname{Deg}_{\max}} , \infty)$. As in \cite{Fathi2015}, we set $$ J(x) := 1- K(x,x) = \frac{\operatorname{Deg}(x)}{\operatorname{Deg}_{\max}}. $$ For all $x,y\in V$, we have \begin{align*} d_\Gamma(x,y) := \sup\left\{f(y)-f(x): \frac 1 2 L(f^2) - fLf \leq 1\right\} = \sqrt{\operatorname{Deg}_{\max}} \rho(x,y). \end{align*}
Now, \cite[Corollary~6.4]{Fathi2015} yields for all $x,y \in V$, \begin{align*}
\rho(x,y) \frac{K}{\sqrt{\operatorname{Deg}_{\max}}} =d_\Gamma(x,y)\frac{K}{\operatorname{Deg}_{\max}} &\leq 2\sqrt 2 \left(\sqrt{J(x)} + \sqrt{J(y)} \right) \\ &= \frac{2 \sqrt{2} \left(\sqrt{\operatorname{Deg}(x)} + \sqrt{\operatorname{Deg}(y)} \right)} {\sqrt{\operatorname{Deg}_{\max}}}. \end{align*} Multiplying with $\frac{\sqrt{\operatorname{Deg}_{\max}}}{K}$ finishes the proof. \end{proof}
\begin{rem}\label{rem:fathi} We observe that Theorem~\ref{thm:distance bounds CD(K,infty)} improves Fathi's and Shu's result \cite[Corollary~6.4]{Fathi2015}, see Theorem~\ref{thm:Fathi}, by a factor of $2$ and by the fact, that we allow $m(V) = \infty$ which corresponds to an infinite reversible invariant measure in the Markov-chain setting. \end{rem}
{\bf Acknowledgement:} We gratefully acknowledge partial support by the EPSRC Grant EP/K016687/1. FM wants to thank the German Research Foundation (DFG) for financial support.
\end{document} | arXiv | {
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\begin{document}
\fancyhead[LO]{Curvature of minimal graphs} \fancyhead[RE]{ D. Kalaj} \fancyhead[RO,LE]{\thepage}
\thispagestyle{empty}
\vspace*{1cm} \begin{center} {\bf\LARGE Curvature of minimal graphs}
\vspace*{0.5cm}
{\large\bf David Kalaj} \end{center}
\vspace*{1cm}
\begin{quote} {\small \noindent {\bf Abstract}\hspace*{0.1cm} We consider the Gaussian curvature conjecture of a minimal graph $S$ over the unit disk. First of all we reduce the general conjecture to the estimating the Gaussian curvature of some Scherk's type minimal surfaces  over a quadrilateral inscribed in the unit disk containing the origin inside. As an application, we obtain the best estimates of the Gaussian curvature so far at the point above the center of the unit disk. Further we obtain an optimal estimate of the Gaussian curvature at the point $\mathbf{w}$ over the center of the disk, provided $\mathbf{w}$ satisfies certain "symmetric" conditions. The result extends a classical result of Finn and Osserman in 1964. In order to do so, we construct a certain family $S^t$, $t\in[t_\circ, \pi/2]$ of Scherk's type minimal graphs over the isosceles trapezoids inscribed in the unit disk. Then we compare the Gaussian curvature of the graph $S$ with that of $S^t$ at the point $\mathbf{w}$ over the center of the disk.
\vspace*{0.2cm}
\noindent{\bf Keywords}\hspace*{0.1cm} conformal minimal surface, minimal graph, curvature
\vspace*{0.1cm}
\noindent{\bf MSC (2010):}\hspace*{0.1cm} 53A10, 32B15, 32E30, 32H02}
\vspace*{0.1cm} \noindent{\bf Date: \today} \end{quote}
\section{Introduction} \label{sec:intro}
Let $M\subset \R^3=\C\times \R$ be a minimal graph lying over the unit disc $\D\subset \C$. Let $w=(w_1,w_2,w_3):\D\to M$ be a conformal harmonic parameterization of $M$ with $w(0)=0$. Its projection $(w_1,w_2):\D\to \D$ is a harmonic diffeomorphism of the disc which may be assumed to preserve the orientation. Let $z$ be the complex variable in $\D$, and write $w_1+\imath w_2 = f$ in the complex notation. We denote by $f_z=\di f/\di z$ and $f_{\bar z}=\di f/\di \bar z$ the Wirtinger derivatives of $f$. The function $\omega$ defined by \begin{equation}\label{eq:omega}
\overline{f_{\bar z}} = \omega f_z \end{equation} is called the {\em second Beltrami coefficient} of $f$, and the above is the {\em second Beltrami coefficient} with $f$ as a solution. Observe that $\bar{f}_z=\overline{f_{\bar z}}$ and this notation will be used in the sequel.
Orientability of $f$ is equivalent to $\mathrm{Jac}(f)=|f_z|^2-|f_{\bar z}|^2>0$, hence to
$|\omega|<1$ on $\D$. Furthermore, the function $\omega$ is holomorphic whenever $f$ is harmonic and orientation preserving. (In general, it is meromorphic when $f$ is harmonic.) To see this, let \begin{equation}\label{eq:fhg}
u+\imath v = f = h+\overline g \end{equation} be the canonical decomposition of the harmonic map $f:\D\to\D$, where $h$ and $g$ are holomorphic functions on the disc. Then, \begin{equation}\label{eq:omega2}
f_z=h',\quad\ f_{\bar z}=\overline g_{\bar z}= \overline{g'}, \quad\
\omega = \overline{f_{\bar z}}/f_z = g'/h'. \end{equation} In particular, the second Beltrami coefficient $\omega$ equals the meromorphic function $g'/h'$
on $\D$. In our case we have $|\omega|<1$, so it is holomorphic map $\omega:\D\to\D$.
We now consider the Enneper--Weierstrass representation of the minimal graph $\varpi=(u,v,T):\D \to M\subset \D\times \R$ over $f$, following Duren \cite[p.\ 183]{Duren2004}. We have \begin{eqnarray*}
u(z) &=& \Re f(z) = \Re \int_0^z \phi_1(\zeta)d\zeta \\
v(z) &=& \Im f(z) = \Re \int_0^z \phi_2(\zeta)d\zeta \\
T(z) &=& \Re \int_0^z \phi_3(\zeta)d\zeta \end{eqnarray*} where \begin{eqnarray*}
\phi_1 &=& 2(u)_z = 2(\Re f)_z = (h+\bar g + \bar h + g)_z = h'+g', \\
\phi_2 &=& 2(v)_z = 2(\Im f)_z = \imath(\bar h+g - h -\bar g)_z = \imath(g'-h'), \\
\phi_3 &=& 2(T)_z = \sqrt{-\phi_1^2-\phi_2^2} = \pm 2\imath \sqrt{h'g'}. \end{eqnarray*} The last equation follows from the identity $\phi_1^2+\phi_2^2+\phi_3^2=0$ which is satisfied by the Enneper--Weierstrass datum $\phi=(\phi_1,\phi_2,\phi_3)=2\di w$ of any conformal minimal (equivalently, conformal harmonic) immersion $w:D\to\R^3$ from a conformal surface $D$. Let us introduce the notation $p=f_z$. We have that \begin{equation}\label{eq:p}
p = f_z = (\Re f)_z + \imath (\Im f)_z = \frac12(h'+g' + h'-g') = h'. \end{equation} By using also $\omega = \overline{f_{\bar z}}/f_z = g'/h'$ (see \eqref{eq:omega2}), it follows that \[
\phi_1 = h'+g'=p(1+\omega),\quad \phi_2 = -\imath(h'-g')=-\imath p(1-\omega),\quad
\phi_3 = \pm 2\imath p \sqrt{\omega}. \] From the formula for $\phi_3$ we infer that $\omega$ has a well-defined holomorphic square root: \begin{equation}\label{eq:q}
\omega = q^2,\qquad q:\D\to \D\ \ \text{holomorphic}. \end{equation}
In terms of the Enepper--Weierstrass parameters $(p,q)$ given by \eqref{eq:p} and \eqref{eq:q} we obtain \begin{equation}\label{eq:EW}
\phi_1 = p(1+q^2),\quad \phi_2 = -\imath p(1-q^2),\quad
\phi_3 = -2\imath p q. \end{equation} (The choice of sign in $\phi_3$ is a matter of convenience; since we have two choices of sign for $q$ in \eqref{eq:q}, this does not cause any loss of generality.) Hence, \[
\varpi(z) = \left(\Re f(z), \Im f(z), \Im \int_0^z 2 p(\varsigma) q(\varsigma) dt \right),\quad z\in\D. \]
The curvature $\mathcal{K}$ of the minimal graph $M$ is expressed in terms of $(h,g,\omega)$ \eqref{eq:omega2}, and in terms of the Enneper--Weierstrass parameters $(p,q)$, by \begin{equation}\label{eq:curvatureformula}
\mathcal{K} = - \frac{|\omega'|^2}{|h'g'|(1 + |\omega|)^4} = - \frac{4|q'|^2}{|p|^2(1 + |q|^2)^4}, \end{equation} where $p=f_z$ and $\omega=q^2=\overline{f_{\bar z}}/f_z$. (See Duren \cite[p.\ 184]{Duren2004}.)
\subsection{Non-parametric minimal surface equation}
Assume that $S=\{(u,v, \mathbf{f}(u,v)):(u,v)\in\D\}$ is a minimal surface, where $\D$ is the unit disk. Then we call such a surface minimal surface above the unit disk. The minimal surface equation is $$f_{uu}(1+f_v^2)-2 f_u f_v f_{uv}+f_{vv}(1+f_u^2)=0.$$
\section{The Heinz-Hopf-Finn-Osserman problem} We are interested in the following problem.
\begin{problem}\label{problem}
What is the supremum of $|\mathcal{K}(\mathbf{w})|$ over all minimal graphs lying over $\D$? Is \begin{equation}\label{eq:FinnOsserman}
|\mathcal{K}(\mathbf{w})|< \frac{\pi^2}{2} \end{equation} the precise upper bound? Here $\mathbf{w}$ is the point above the center of the unit disk and we call it \emph{centre}. \end{problem} The previous conjecture has been also formulated by Duren in his monograph \cite[Conjecture~2.~p.~185]{Duren2004}.
The first result on this topic has been given by E. Heinz on 1952 in \cite{zbMATH03075392} who introduced the constant $c_0$ which is the best constant in the inequality $|\mathcal{K}(\mathbf{w})|\le c_0$, for all minimal graphs over the unit disk with the centre $\mathbf{w}$. Further this result has been improved by E. Hopf in 1953 in \cite{zbMATH03081064}, who introduced the constant $c_1$ which is the best constant in the inequality $$W^2|\mathcal{K}(\mathbf{w})|\le c_1, $$ where $W=\sqrt{1+\mathbf{f}_u^2+\mathbf{f}_v^2}$. So a similar problem to be consider is the following \begin{problem}\label{problem2}
What is the supremum of $W^2|\mathcal{K}(\mathbf{w})|$ over all minimal graphs lying over $\D$? Is \begin{equation}\label{eq:FinnOsserman3}
W^2|\mathcal{K}(\mathbf{w})|< \frac{\pi^2}{2} \end{equation} the precise upper bound? Here $\mathbf{w}$ is the \emph{centre} of minimal surface. \end{problem}
It was shown by Finn and Osserman \cite{FinnOsserman1964} in 1964 that the upper bound in \eqref{eq:FinnOsserman} is indeed sharp if $q(0)=0$, which means that the tangent plane $T_0 M=\C\times\{0\}$ being horizontal (and hence $f$ is conformal at $0$). Although there is no minimal graph lying over the whole unit disc $\D$ whose centre curvature equals $\frac{\pi^2}{2}$, there is a sequence of minimal graphs whose centre curvatures converge to $\frac{\pi^2}{2}$, and the graphs converge to the Scherk's surface lying over square inscribed into the unit disc. The associated Beltrami coefficient of the Scherk's surface is $\omega(z)=z^2$, with $q(z)=z$. We refer to Duren \cite[p.\ 185]{Duren2004} for a survey of this subject. We also refer to the monograph by J. C. C. Nitsche \cite{Nitsche1965} for earlier results.
Let us recall a path to obtain a weaker upper bound on $|\mathcal{K}|$ which holds for every value
$|q(0)|<1$. This is explained in \cite[pp.\ 184--185]{Duren2004}.
Hall proved in \cite{Hall1982} (1982) the following estimate \begin{equation}\label{eq:Hall}
|f_z(0)|^2+|f_{\bar z}(0)|^2\ge \frac{27}{4\pi^2} \end{equation} for any harmonic diffeomorphism $f:\D\to \D$ with $f(0)=0$. This estimate is sharp in general, but is not sharp if the second Beltrami coefficient $\omega$ is the square of a holomorphic function on $\D$. Applying Hall's estimate and noting that \[
|f_z(0)|^2+|f_{\bar z}(0)|^2 = |f_z(0)|^2(1+|q(0)|^4) \] gives \[
|f_z(0)|^2 \ge \frac{27}{4\pi^2} \frac{1}{(1+|q(0)|^4)}. \]
By using also the Pick-Schwarz inequality $|q'(0)|<1-|q(0)|^2$, we obtain \begin{equation}\label{eq:weakestK}
|\mathcal{K}| = \frac{4|q'(0)|^2}{|f_z(0)|^2(1+|q(0)|^2)^4} \le
\frac{16\pi^2}{27} \frac{\bigl(1-|q(0)|^2\bigr)^2 \bigl(1+|q(0)|^4\bigr)}{(1+|q(0)|^2)^4}. \end{equation} So we have the following inequality \begin{equation}\label{halljda}
|\mathcal{K}| \le \frac{16\pi^2}{27}\approx 5.84865. \end{equation} The above constant is better than the constant $5.98$ obtained by Finn and Osserman in \cite{FinnOsserman1964}.
Further if the minimal surface has its non-parametric parameterization $z=\mathbf{f}(u,v)$, and denoting $$W=\sqrt{1+\mathbf{f}_u^2+\mathbf{f}_v^2},$$ then \eqref{eq:weakestK}, in view of \eqref{firsti} and \eqref{secondi} below implies that \begin{equation}\label{eq:weakestK2}
|\mathcal{K}|\cdot W^2 \le
\frac{16\pi^2}{27} \frac{ \bigl(1+|q(0)|^4\bigr)}{(1+|q(0)|^2)^2}\le \frac{16\pi^2}{27}. \end{equation} It follows from \eqref{eq:weakestK}, that the Heinz constant $c_0<\frac{16\pi^2}{27}$, while \eqref{eq:weakestK2} implies that the Hopf constant $c_1<\frac{16\pi^2}{27}$.
We will give better estimate of both constants in Corollary~\ref{coro}.
The estimate \eqref{halljda} is not sharp as it has been proved by R. Hall in \cite{Hall1998} by obtaining a very small improvement of about $10^{-5}$. As said before, the sharp estimate \eqref{eq:FinnOsserman} in the case $q(0)=0$ was given by Finn and Osserman \cite{FinnOsserman1964} (see also \cite{Nitsche1965}).
\section{The main results} We first formulate the following general result \begin{theorem}\label{prejprej}
For every $w\in\D$, there exist four different points $a_0, a_1,a_2,a_3\in\mathbf{T}$ and then there is a harmonic mapping $f$ of the unit disk onto the quadrilateral $Q(a_0,a_1,a_2,a_3)$ that solves the Beltrami equation \begin{equation}\label{beleq}\bar f_z(z) = \left(\frac{w+\frac{\imath \left(1-w^4\right) z}{\left|1-w^4\right|}}{1+\frac{\imath\overline{w} \left(1-w^4\right) z}{\left|1-w^4\right|}}\right)^2 f_z(z),\end{equation} $|z|<1$ and satisfies the initial condition $f(0)=0$, $f_z(0)>0$. It also defines a Scherk's type minimal surface $S^\diamond: \zeta=\mathbf{f}^\diamond(u,v)$ over the quadrilateral $Q(a_0,a_1,a_2,a_3)$, with the centre $\mathbf{w}=(0,0,0)$ so that its Gaussian normal is $$\mathbf{n}^\diamond_{\mathbf{w}}=-\frac{1}{1+|w|^2}(2\Im w, 2\Re w, -1+|w|^2),$$ and $D_{uv}\mathbf{f}^\diamond(0,0)=0$. Moreover, every other non-parametric minimal surface $S:$ $z=\mathbf{f}(u,v)$ over the unit disk, with a centre $\mathbf{w}$, with $\mathbf{n}_{\mathbf{w}}=\mathbf{n}^\diamond_{\mathbf{w}}$ and $D_{uv}\mathbf{f}(0,0)=0$ satisfies the sharp inequality $$|\mathcal{K}_{S}(\mathbf{w})|<|\mathcal{K}_{S^\diamond}(\mathbf{w})|,$$ or what is the same
$$W^2_{S}|\mathcal{K}_{S}(\mathbf{w})|<W^2_{S^\diamond}|\mathcal{K}_{S^\diamond}(\mathbf{w})|.$$
Further we have \begin{equation}\label{finoser}\mathcal{K}_{S^\diamond}(\mathbf{w})=-\frac{4 \left(1-|w|^2\right)^2}{\left(1+|w|^2\right)^4 |f_z(0)|^2}.\end{equation} \end{theorem} \begin{remark}
It follows from the result of Jenkins and Serrin that such a minimal surface described in Theorem~\ref{prejprej} is unique \cite{Jenkins1Serrin1968}, so $Q=Q(w)$ depends only on $w$ and also $f=f^w$ depends only on $w$. It also follows from Theorem~\ref{prejprej} (i.e. from \eqref{finoser}) that the Heinz and the Hopf constants can be defined as
\begin{equation} \label{heinz}c_0 = \sup_{w} \frac{4 \left(1-|w|^2\right)^2}{\left(1+|w|^2\right)^4 |f^w_z(0)|^2}\end{equation}
\begin{equation} \label{heinz}c_1 = \sup_{w} \frac{4 }{\left(1+|w|^2\right)^2 |f^w_z(0)|^2}.\end{equation}
In particular, when $w$ from Theorem~\ref{prejprej} is an imaginary number (in view of Remark~\ref{remica}), then we precisely describe the quadrilaterals, which appear to be isosceles trapezoids (Section~\ref{sectio2}, Proposition~\ref{defshre}). In this case we give the precise bound of the curvature.
Further, if we consider the mapping $$\tilde f(z) = f\left(\frac{\imath\left|1-w^4\right|}{1-w^4}\frac{ (w-z) }{ (1-z \overline{w})}\right),$$ then $\tilde f$ satisfies the Beltrami equation $\overline{\tilde f}_z=z^2 \tilde f_z$ with the initial conditions $\tilde f(w)=0$ and $\imath (1-w^4)\tilde f_z(w)>0$. \end{remark}
In order to formulate our next results, which are extensions of the Finn-Osserman results, we give the following definition. \begin{definition} We call $\zeta\in D$ a symmetric point of a double differentiable real function $\mathbf{f}:D\to \mathbf{R}$ if there is some vector $h\in\mathbf{T}=\partial\D$ so that the equalities hold \begin{equation}\label{symm}\nabla^2_{h, \imath h}\mathbf{f}(\zeta) = \nabla _h \mathbf{f}(\zeta)=0.\end{equation} We call also that point $\zeta$, $h-$symmetric. A point $\mathbf{w}=(\zeta, \mathbf{f}(\zeta))$ on the graph of a function $\mathbf{f}$ is symmetric if $\zeta$ is symmetric for $\mathbf{f}$.
\end{definition} \begin{remark} The motivation for this definition comes from the following observation. Assume that $\mathbf{f}$ is a symmetric real function w.r.t. imaginary axis, i.e. assume that $\mathbf{f}(-u,v)=\mathbf{f}(u,v)$. Then $D_u \mathbf{f}(-u,v)=-D_u \mathbf{f}(u,v)$. So $D_u\mathbf{f}(0,v)=0$. Further $D_{uv} \mathbf{f}(0,v)=0$ for every $v$. This implies that $\nabla^2_{e_1,e_2} \mathbf{f}(0,0)=0$. By using the translation and rotation of the coordinate system, we get a similar fact for functions that are symmetric at some point w.r.t to an arbitrary line, or more general w.r.t. a small segment.
\end{remark} An example of a symmetric point is any stationary point of the function. \begin{example}\label{forward} Prove that if $\nabla \mathbf{f}(0,0)=0$, then $z=(0,0)$ is a symmetric point of $\mathbf{f}$. Namely if $h=e^{ic}$ and $\mathbf{f}^c(z) = \mathbf{f}(e^{ic}z)$, then $$\mathbf{f}^c_u(0,0)=\cos c \,\mathbf{f}_u(0,0)+\sin c \,\mathbf{f}_v(0,0)=\nabla_h \mathbf{f}(0,0).$$ Further $$\mathbf{f}^c_{uv}(0,0)=\cos(2 c) \mathbf{f}_{uv}(0,0)+\cos (c) \sin(c) \left(-\mathbf{f}_{vv}(0,0)+\mathbf{f}_{uu}(0,0)\right)=\nabla^2_{h,\imath h} \mathbf{f}(0,0).$$ Since $\mathbf{f}^{\pi/2}_{uv}(0,0)=-\mathbf{f}_{uv}(0,0)$, there is $c$ so that $\mathbf{f}^c_{uv}(0,0)=0$. \end{example} In the sequel we give two additional examples of symmetric points of classical minimal surfaces and one counterexample. \begin{example}
a) Assume that $w=\cosh^{-1}\sqrt{u^2+v^2}$, $|w|=\sqrt{u^2+v^2}>1$. Then this function defines the catenoid. Moreover $$w_u=\frac{u}{\sqrt{u^2+v^2} \sqrt{-1+\sqrt{u^2+v^2}} \sqrt{1+\sqrt{u^2+v^2}}}$$ and $$w_{uv}=\frac{u v \left(1-2 u^2-2 v^2\right)}{\left(u^2+v^2\right)^{3/2} \left(-1+\sqrt{u^2+v^2}\right)^{3/2} \left(1+\sqrt{u^2+v^2}\right)^{3/2}}.$$ So every point $(u,0)$ and $(0,v)$ is a symmetric point of this surface. Since it is rotation invariant, it follows that every point of this surface is symmetric.
b) Assume that $w=\log\frac{\cos v}{\cos u}$. Then $w_u=\tan u$ and $w_{uv}=0$. So every point $w=\imath v=(0,v)$ is a symmetric point of Scherk's saddle surface.
c) Assume that $w=\tan^{-1}\frac{v}{u}$, where $u\neq 0$. Then this function defines the helicoid. Then $w_u=\frac{v}{u^2+v^2}$ and $w_{uv}=\frac{(u-v) (u+v)}{\left(u^2+v^2\right)^2} $. It follows that this surface has not any symmetric point. \end{example} We give a partial solution of Problem~\ref{problem} and extend Finn-Osserman result by proving the following theorem. \begin{theorem}\label{th:theor} Assume that $S$ is a non-parametric minimal surface above the unit disk and assume that the point $\mathbf{w}$ over the center of the disk is symmetric. Then the Gaussian curvature $\mathcal{K}(\mathbf{w})$ satisfies the sharp inequalities \begin{equation}\label{eq:FinnOsserman1}
|\mathcal{K}(\mathbf{w})|< \frac{\pi^2}{2}. \end{equation} and \begin{equation}\label{eq:FinnOsserman2}
W^2|\mathcal{K}(\mathbf{w})|< \frac{\pi^2}{2}. \end{equation} \end{theorem} \begin{remark} After we wrote this paper we realized that the statement of Theorem~\ref{th:theor} is not new for symmetric minimal surfaces. An approach different from our approach has been given by Nitsche in \cite{zbMATH03431423}. \end{remark} Further we prove the following theorem \begin{theorem}\label{th:theor2}
There is a decreasing diffeomorphism $\Psi:[0,\pi/2]\to [0,\pi^2/2]$ with the following property. Assume that $S$ is a non-parametric minimal surface above the unit disk with a $h-$symmetric point $\mathbf{w}$ above $0$. Assume that $\theta$ is the angle of the tangent plane $TS_\mathbf{w}$ at $\mathbf{w}$ with $h$. Then the Gaussian curvature $|\mathcal{K}|$ at $\mathbf{w}$ satisfies the sharp inequality \begin{equation}\label{eq:FinnOsserman2}
|\mathcal{K}(\mathbf{w})|< \Psi(\theta)(<\frac{\pi^2}{2}), \end{equation} and for every $0\le \phi<\Psi(\theta)$ there is a non-parametric minimal surface $S_\phi$ above the unit disk, whose point above the center of the unit disk is $h-$simmetric and whose tangent plane at $\mathbf{w}$ makes the angle $\theta$ with $h$ so that $$\mathcal{K}_{S_\phi}(\mathbf{w})=\phi.$$ \end{theorem}
As a corollary of our results, we obtain the following improvement of the Hall upper bound of Gaussian curvature (i.e. of Heinz and Hopf constants) without any condition on the centre. \begin{corollary}\label{coro} Let $S: z=\mathbf{f}(u,v)$ be a minimal surface over the unit disk and assume that $\mathbf{w}$ is its centre. Then the Gaussian curvature $$\mathcal{K}(\mathbf{w})< 5.7.$$ Moreover if $W=\sqrt{1+\mathbf{f}_u^2+\mathbf{f}_v^2}$, then $$\mathcal{K}(\mathbf{w})< \frac{5.8}{W^2}.$$ \end{corollary} \section{Proof of main results} This section contains the proof of our results. At the begging we describe the family of Scherk's type minimal surfaces over isosceles trapezoids inscribed in the unit disk. On account of Theorem~\ref{prejprej} we know that a similar family depending on two parameters exists and such a family would solve the general conjecture, provided it can be explicitly expressed.
\subsection{Scherk's type minimal surfaces with 4 sides and auxiliary results}\label{sectio2} We are going to find a harmonic mapping of the unit disk onto a quadrilateral
inscribed in the unit disk that produces a minimal surface. Let $a_1 =1$, $a_2=e^{\imath t}$, $a_3=e^{\imath s}$, $a_4=e^{\imath(t+s)}$, where $s=\arccos \frac{3\cos t-1}{1+\cos t}.$ Let $$F(\sigma)=\begin{array}{ll}
\Bigg\{ & \begin{array}{ll}
1 & \sigma\in[0,\pi/2] \\
e^{\imath t} & \sigma\in[\pi/2,\pi] \\
e^{\imath s} & \sigma\in [\pi, 3\pi/2] \\
e^{\imath (t+s)} & \sigma\in [3\pi/2,2\pi]. \end{array} \end{array}$$ Let $$f_1(z) =P[F](z) = \frac{1}{2\pi}\int_0^{2\pi}\frac{1-r^2}{1+r^2-2 r \cos(\varsigma-\sigma)} F(\sigma)d \sigma, z=re^{\imath\varsigma}.$$
Then $f_1$ maps the unit disk onto the trapezoid $\mathcal{T}$ with the vertices $a_1,a_2, a_3, a_4$, $a_k=a_1$. Moreover $$f_1(0) = \frac{1}{4} \left(1+e^{\imath t}\right) \left(1+e^{\imath \cos^{-1}\left[\frac{-1+3 \cos t}{1+\cos t}\right]}\right).$$ Further, by \cite[p.~63]{Duren2004}, $$f_1(z) = g(z) +\overline{h(z)},$$ where $$g'(z) =\frac{1}{2\pi \imath} \sum_{k=1}^4 \frac{(a_k-a_{k+1})}{z-\imath ^k} $$ and $$h'(z) =-\frac{1}{2\pi \imath} \sum_{k=1}^4 \frac{(\overline{a_k-a_{k+1}})}{z-\imath ^k}.$$ Thus $$g'(z) =-\frac{(1+\imath) \left(\imath+e^{\imath t}\right) \left(-1+\cos t+2 \imath \sqrt{\cos t} \sin \left[\frac{t}{2}\right]\right) \left(1+\frac{z \sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2}{\pi \left(z^4-1\right) (1+\cos t)},$$ and $$h'(z) = \frac{(1+\imath) \left(1-\cos t+2 \imath \sqrt{\cos t} \sin \left[\frac{t}{2}\right]\right) (e^{-\imath t} \left(\imath+e^{\imath t}\right))\left(z+\frac{\sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2 }{\pi \left(z^4-1\right) (1+\cos t)}.$$ Thus we get $$\omega_1=\frac{h'(z)}{g'(z)}=e^{-\imath \left(t+s-\pi\right)}\frac{ \left(z+\frac{\sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2}{\left(1+\frac{z \sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2},$$ where $s=\cos^{-1}\frac{3\cos t - 1}{1+\cos t}$. So $\omega_1=q_1^2$, where $$q_1(z)=e^{\imath\mu}\frac{z+a(t)}{1+z\overline{a(t)}}.$$ Here $$a(t) = \frac{\sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]},$$ and $$\mu = -\frac{1}{2}\left(t+s-\pi \right).$$
Let $\tau = \frac{1}{2} \left(-\pi +t+s\right)$ and define \begin{equation}\label{after}f: \D\to \mathcal{T}, \ \ f(z):=e^{-\imath\tau} f_1(z). \end{equation} Then $f$ maps the unit disk onto the isosceles trapezoid, whose bases are parallel to the real axis. See figure~3.1. \begin{figure}
\caption{An isosceles trapezoid inscribed in the unit disk. Here $t=\pi/2-0.1$}
\label{f1}
\end{figure} Then \begin{equation}\label{be}f(0) = e^{-\imath\tau}\frac{1}{4}(1+e^{\imath t}+e^{\imath s}+e^{\imath (t+s)})=\imath \sqrt{\cos t}.\end{equation}
Further let $$p=e^{-\imath\tau} g'(z), \tilde p = e^{\imath\tau} h'(z).$$
Then \begin{equation}\label{afterv}p= \frac{-2 \imath \left(\cos \left[\frac{t}{2}\right]+\sin\left[\frac{t}{2}\right]\right) \sin t \left(1+\frac{z \sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2}{\pi \left(1-z^4\right) (1+\cos t)},\end{equation} and
$$\tilde p= \frac{-2 \imath \left(\cos \left[\frac{t}{2}\right]+\sin\left[\frac{t}{2}\right]\right) \sin t \left(z+\frac{ \sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2}{\pi \left(1-z^4\right) (1+\cos t)},$$
and $$\frac{\tilde p}{p}=\frac{ \left(z+\frac{\sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2}{\left(1+\frac{z \sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}\right)^2}.$$
Thus \begin{equation}\label{qqq}q=\frac{z+\frac{\sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}}{1+\frac{z \sqrt{\cos t}}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]}},\end{equation}
\begin{equation}\label{pmtp}p-\tilde p = -\frac{8 \imath \csc t \sin \left[\frac{t}{2}\right]^3}{\pi \left(1+z^2\right)}\end{equation}
and
\begin{equation}\label{pptp}p+\tilde p = \frac{4 \imath \left(\left(1+z^2\right) \cos \left[\frac{t}{2}\right] \sin \left[\frac{s}{2}\right]+2 z \cos \left[\frac{s}{2}\right] \sin \left[\frac{t}{2}\right]\right)}{\pi(z^4-1)}.\end{equation}
We also have $ f(z) = u + \imath v+f(0)$, where $$u=\Re\int_0^z (p+\tilde p)dz,$$ and $$v=\Im \int_0^z (p-\tilde p)dz.$$ Thus we obtain that $$u(z) = -\frac{\Im\left[\log\left[\frac{1+z^2}{(1+z)^2}\right] \sin \left[\frac{s-t}{2}\right]+\log\left[\frac{(1-z)^2}{1+z^2}\right] \sin \left[\frac{s+t}{2}\right]\right]}{\pi },$$ and $$v(z) = -\frac{4 \Re(\tan^{-1} z) \sin \left[\frac{s}{2}\right] \sin \left[\frac{t}{2}\right]}{\pi}.$$ So the equation $$ v(r)+\sqrt{\cos t}=0$$ has only one solution $$r = \tan \left[\frac{1}{8} \pi \sqrt{\cos t} \csc\left[\frac{t}{2}\right]^3 \sin t\right].$$ So $ f(z_\circ)=0$ if \begin{equation}\label{zezero}z_\circ = \tan \left[\frac{1}{8} \pi \sqrt{\cos t} \csc\left[\frac{t}{2}\right]^3 \sin t\right].\end{equation}
The Gaussian curvature of the minimal surface at the point $\mathbf{w}$ over the point $0=f(z_\circ)$ is $$K= -\frac{4|q'(z_\circ)|^2}{|p(z_\circ)|^2(1+|q(z_\circ)|^2)^4}$$ which can be written as $K=-\kappa^2(t)$, where
$\kappa$ is a positive function defined by $$\kappa(t) = \frac{2|q'(z_\circ)|}{|p(z_\circ)|(1+|q(z_\circ)|^2)^2}.$$
\subsubsection{Show that $\kappa$ is increasing and $\kappa(t) \le \kappa(\pi/2)=\frac{\pi}{\sqrt{2}}$}\label{subsub1} For $r=z_\circ\in(0,1)$, by direct computations we get \begin{equation}\label{kappa}\kappa(t) = \frac{\pi \left(1-r^4\right) \cos \left[\frac{t}{2}\right]}{2 \left(\left(1+r^2\right) \cos \left[\frac{t}{2}\right]+2 r \sqrt{\cos t}\right)^2}.\end{equation} Notice that for $t_\circ =2 \tan^{-1}\sqrt{1/2 (-1 + \sqrt{5})}\approx 1.33248$ and $t\in(t_\circ,\pi/2)$, $z_\circ=z_\circ(t)\in\D$. For $t=t_\circ$, $z_\circ=1$ and for $t<t_\circ$, $z_\circ$ is outside of the unit disk. Let's choose the substitution $u = \tan\frac{t}{2}$, then $u\in[\sqrt{1/2 (-1 + \sqrt{5})},1]$ and $$\kappa =\Phi(u):= \frac{\pi \sqrt{1+u^2} \cos \left[\frac{\pi \sqrt{1-u^2}}{2 u^2}\right]}{2 \left(1+\sqrt{1-u^2} \sin \left[\frac{\pi \sqrt{1-u^2}}{2 u^2}\right]\right)^2}.$$ Further we have \[\begin{split}\Phi'(u) &= \pi \Bigg(3 \pi \sqrt{1-u^2} \left(2+u^2-u^4\right)+4 u^4 \sqrt{1-u^2} \cos \left[\frac{\pi \sqrt{1-u^2}}{2 u^2}\right]\\& +\pi(2+u^2-u^4)\left(\sqrt{1-u^2}\cos \left[\frac{\pi \sqrt{1-u^2}}{u^2}\right]+2 \sin \left[\frac{\pi \sqrt{1-u^2}}{2u^2}\right]\right)\Bigg)\\&\Bigg/\left(8 \sqrt{1-u^4} \left(u+u \sqrt{1-u^2} \sin \left[\frac{\pi \sqrt{1-u^2}}{2 u^2}\right]\right)^3\right).\end{split}\] In order to show that $\Phi'(u)>0$ we only need to prove that $$\gamma(u):= \pi(2+u^2-u^4)\left(\sqrt{1-u^2}\cos \left[\frac{\pi \sqrt{1-u^2}}{u^2}\right]+2 \sin \left[\frac{\pi \sqrt{1-u^2}}{2u^2}\right]\right)\ge 0$$ because the other terms in the sum are positive due to the fact that $\frac{\pi \sqrt{1-u^2}}{2u^2} \in(0,\pi/2)$.
Further we have \[\begin{split}\gamma(u) &\ge \pi(2+u^2-u^4)\left(-\sqrt{1-u^2}+2 \sin \left[\frac{\pi \sqrt{1-u^2}}{2u^2}\right]\right)\\&\ge \pi(2+u^2-u^4)\left(-\sqrt{1-u^2}+2\frac{2}{\pi}\left[\frac{\pi \sqrt{1-u^2}}{2u^2}\right]\right) \\&=\pi(2+u^2-u^4)\left(-\sqrt{1-u^2}+\frac{2 \sqrt{1-u^2}}{u^2}\right)>0.\end{split}\] This implies that $\kappa$ is an increasing function for $t\in[t_\circ, \pi/2]$ so that $\kappa(t_\circ)=0\le \kappa(t)\le \kappa(\pi/2)=\pi/\sqrt{2}$.
Let $W=\frac{1+|q|^2}{1-|q|^2}.$ Then define $$\phi(t):=W \kappa(t) = \frac{\pi \left(1+z^2\right) \cot \left[\frac{t}{2}\right]}{2 \left(1+z^2\right) \cos \left[\frac{t}{2}\right]+4 z \sqrt{\cos t}}.$$ \subsubsection{Show that $\phi(t)\le \pi/\sqrt{2}$ for $t\in(t_\circ, \pi/2)$}\label{subsub2}
Straightforward calculations give $$\phi(t)=\frac{\pi \cot \left[\frac{t}{2}\right]}{2 \cos \left[\frac{t}{2}\right]+2 \sqrt{\cos t} \sin \left[\frac{1}{4} \pi \sqrt{\cos t} \csc \left[\frac{t}{2}\right]^3 \sin t\right]}$$
or what is the same
$$\phi(t)=\frac{\pi }{2 \sin \left[\frac{t}{2}\right]+2 \sqrt{\cos t} \sin \left[\frac{1}{4} \pi \sqrt{\cos t} \csc \left[\frac{t}{2}\right]^3 \sin t\right] \tan \left[\frac{t}{2}\right]},$$ and we need to show that
$$\psi(t):={ \sin \left[\frac{t}{2}\right]+ \sqrt{\cos t} \sin \left[\frac{1}{4} \pi \sqrt{\cos t} \csc \left[\frac{t}{2}\right]^3 \sin t\right] \tan \left[\frac{t}{2}\right]}\ge\frac{ \sqrt{2}}{2}.$$
Since $t\in(t_\circ, \pi/2)$, we have $$u= \frac{1}{4} \pi \sqrt{\cos t} \csc \left[\frac{t}{2}\right]^3 \sin t\in[0,\pi/2]$$ and thus $$\sin u\ge \frac{2}{\pi}u.$$
So $$\psi(t)\ge \sin \left[\frac{t}{2}\right]+\frac{1}{2} \cos t \csc \left[\frac{t}{2}\right]^2 \sec\left[\frac{t}{2}\right] \sin t$$ or what is the same \begin{equation}\label{rhs}\psi(t)\ge \vartheta(t):=\cos \left[\frac{t}{2}\right] \cot \left[\frac{t}{2}\right].\end{equation} Now the derivative of $\vartheta(t)$ is $\frac{1}{4} (-3+\cos t) \cot \left[\frac{t}{2}\right] \csc \left[\frac{t}{2}\right]$, so $\vartheta(t)$ is decreasing. Thus $\psi(t)\ge \vartheta(\pi/2)=\sqrt{2}/2$ for $t\in(t_\circ, \pi/2)$. This implies the claimed inequality.
Observe that $s>t$ and for $t\in(t_\circ,\pi/2]$ the (isosceles) trapezoid
$R$ contains $0$. For $t=t_\circ$, $\mathcal{T}$ is a certain isosceles trapezoid with the base consisted of the diameter $[-1,1]$.
Let $$S^t =\{(\Re f(z), \Im f(z), T(z)): z\in\D\}.$$ Then $S^t$ is a Scherk's type minimal graph.
The third coordinate of the Enneper-Weierstrass parametrization is given by $$T(z) = \pm \Re \int_0^z \sqrt{p\tilde p}dz.$$ So $$ T(z)=\pm 2\Re\int_0^z\frac{2 \left(z \sqrt{2-2 \cos t}+\left(1+z^2\right) \sqrt{\cos t} \tan \left[\frac{t}{2}\right]\right)}{\pi \left(-1+z^4\right)}dz.$$ Thus we get \begin{equation}\label{TT}T(z)=\pm \Re \frac{\sin \frac{t}{2} \log[\frac{1-z^2}{1+z^2}]-\log\frac{1+z}{1-z} \sqrt{\cos t} \tan \left[\frac{t}{2}\right]}{\pi }. \end{equation}
Then $$T(z)=\pm \frac{\sin \frac{t}{2} \log[\frac{|1-z^2|}{|1+z^2}|]-\log\frac{|1+z|}{|1-z|} \sqrt{\cos t} \tan \left[\frac{t}{2}\right]}{\pi },$$ so $T(z) \to\pm\infty$ when $z\to \pm 1$ or $z\to \pm \imath$. Moreover its noparametric parametrization $(u,v, \mathbf{f}^{t}(u,v))$, $(u,v)\in\mathcal{T}$ satisfies the relation $\mathbf{f}^{t}(u,v))\to \pm \infty$ when $z=(u,v)\to \zeta \in \partial \mathcal{T}$. An example of a Scherk's type minimal graph is shown in the figure~3.2 below.
\begin{figure}
\caption{A generalized Scherk's surface. Here $t=\pi/2-0.1$}
\label{f2}
\end{figure}
Since $S^t$ is symmetric with respect to the plane $xOz$, it follows that it is a graph of a function $\mathbf{f}^t$ defined in the unit disk which is symmetric with respect to the $u-$axis. This implies that $\mathbf{f}^t(-u,v)=\mathbf{f}^t(u,v)$. So $$D_u \mathbf{f}^{t}(-u,v)=-D_u \mathbf{f}^{t}(u,v). $$ and so
$$D_u \mathbf{f}^t(0,v)=0.$$ Thus \begin{equation}\label{xyzero}D_{uv}\mathbf{f}^t(0,v)=0 \text{ for every $v$.}\end{equation}
Thus we have proved the following proposition.
\begin{proposition}\label{defshre}
For any $t\in(0,\pi/2]$ there is an isosceles trapezoid $$\mathcal{T}^t=\mathcal{T}(e^{\imath\alpha(t)},e^{\imath\beta(t)}, e^{\imath\gamma(t)}, e^{\imath\delta(t)})$$ with the vertices at the unit circle, with bases parallel to the $u-$axis and a Scherk's type minimal surface $$S^t=\{(u,v, \mathbf{f}^t(u,v)): (u,v)\in \mathcal{T}^t\}$$ so that
$$\mathbf{f}^t(z)\to \left\{
\begin{array}{ll}
+\infty, & \hbox{if $z\to \zeta$ when $\zeta\in (e^{\imath\alpha(t)}, e^{\imath\beta(t)})\cup (e^{\imath\gamma(t)}, e^{\imath\delta(t)})$;} \\
-\infty, & \hbox{if $z\to \zeta$ when $\zeta\in (e^{\imath\beta(t)}, e^{\imath\gamma(t)})\cup (e^{\imath\delta (t)}, e^{\imath\alpha(t)})$.}
\end{array}
\right.$$
Moreover $D_{uv} \mathbf{f}^t(0,0)=D_u\mathbf{f}^t (0,0)=0$.
Further for $t\in\left(t_\circ,\frac{\pi}{2}\right]$, where $t_\circ = 2\tan^{-1}\sqrt{\frac{1}{2}(\sqrt{5}-1)}$, the trapezoid $\mathcal{T}$ contains zero and the Gaussian curvature of $S_t$ at the point $\mathbf{w} $ above $0$ is equal to $\mathcal{K}(\mathbf{w})=-\kappa^2(t)$, where $\kappa(t)$ is defined in \eqref{kappa}. Furthermore, $\kappa^2(t)\le \frac{\pi^2}{2}$ for every $t$ and $\lim_{t\to t_\circ}=0$ and $\kappa^2(t)$ is an increasing diffeomorphism of $(t_\circ, \pi/2]$ onto $(0,\pi^2/2]$.
For $t=\pi/2$ the obtained surface is the standard Scherk's minimal graph surface over the square.
\end{proposition}
\begin{proof}[Proof of Theorem~\ref{prejprej}] In order to prove Theorem~\ref{prejprej}, we will derive a useful formula for $\mathbf{f}_{uv}$, of a non-parametric minimal surface $w=\mathbf{f}(u,v)$. Namely we will express $\mathbf{f}_{uv}$ as a function of Enneper-Weisstrass parameters. Assume that $q(z) = a(z) + \imath b(z)=\sqrt{\omega(z)}$ and $p$ are Enneper-Weisstrass parameters of a minimal disk $S=\{(u(z), v(z), T(z)), z\in \D\}=\{(u,v,\mathbf{f}(u,v)): (u,v)\in \D\}$ over the unit disk. Here $f=u+iv$ and $\bar f_z=\omega(z) f_z$.
Then the unit normal at $\mathbf{w}\in S$, in view of \cite[p.~169]{Duren2004} is given by $$\mathbf{n}_{\mathbf{w}}=-\frac{1}{1+|q(z)|^2}(2\Im q(z), 2\Re q(z), -1+|q(z)|^2).$$ It is also given by the formula $$\mathbf{n}_{\mathbf{w}}=\frac{1}{\sqrt{1+\mathbf{f}_u^2+\mathbf{f}_v^2}}\left(-\mathbf{f}_u,-\mathbf{f}_v,1\right).$$ Then we have the relations
\begin{equation}\label{firsti}\mathbf{f}_v (u(x,y),v(x,y))=\frac{2 a(x,y)}{-1+a(x,y)^2+b(x,y)^2}
\end{equation}
\begin{equation}\label{secondi}\mathbf{f}_u(u(x,y),v(x,y))=\frac{2 b(x,y)}{-1+a(x,y)^2+b(x,y)^2}.\end{equation} By differentiating \eqref{firsti} and \eqref{secondi} w.r.t. $x$ we obtain the equations
\begin{equation}\label{firsti1}v_x \mathbf{f}_{uv}(u,v)+u_x \mathbf{f}_{uu}(u,v)=-\frac{4 a b a_x+2 \left(1-a^2+b^2\right) b_x}{\left(-1+a^2+b^2\right)^2}\end{equation}
\begin{equation}\label{secondi1}v_x\mathbf{f}_{vv}(u,v) +u_x \mathbf{f}_{uv}(u,v)=-\frac{4 a b b_x+2 \left(1-a^2+b^2\right) a_x}{\left(-1+a^2+b^2\right)^2}.\end{equation} Now recall the minimal surface equation
\begin{equation}\label{mse}\left(1+\mathbf{f}^2_u(u,v)^2\right)\mathbf{f}_{vv}(u,v)+\left(1+\mathbf{f}^2_v(u,v)^2\right) \mathbf{f}_{uu}(u,v)=2 \mathbf{f}_v(u,v) \mathbf{f}_u(u,v) \mathbf{f}_{uv}(u,v) \end{equation}
From \eqref{firsti}, \eqref{secondi}, \eqref{firsti1},\eqref{secondi1} and \eqref{mse} we get $$\mathbf{f}_{uv}=\frac{M}{N}$$ where \[\begin{split}M&=-2 (a^4+2 a^2 (-1+b^2)+(1+b^2)^2) ((1+a^2-b^2) a_x+2 a b b_x) u_x\\&-2((1+a^2)^2+2 (-1+a^2) b^2+b^4) (2 a b a_x+(1-a^2+b^2) b_x) v_x\end{split}\] and \[\begin{split}N&=(1-a^2-b^2)^2 \\&\times ((a^4-2 a^2 (1-b^2)+(1+b^2)^2) u_x^2+8 a b u_x v_x+((1+a^2)^2-2 (1-a^2) b^2+b^4) v_x^2).\end{split}\] Let $q(z)=a+\imath b=r e^{it}$, $q'(z) = a_x+\imath b_x=R e^{is}$ and $p=Pe^{im}$. Because $u_x=\Re (p(1+q^2))$, and $v_x=-\Re (\imath p(1-q^2))$, after straightforward calculation we get $$\mathbf{f}_{uv}=-\frac{2 R \left(\cos[m-s]-r^4 \cos[m-s+4 t]\right)}{P \left(1-r^2\right)^3 \left(1+r^2\right)}$$ which can be written as
\begin{equation}\label{fexpli}\mathbf{f}_{uv}=-\frac{2\Re \left[p(1-q^4)\overline{q'}\right]}{|p|^2(1-|q|^2)^3(1+|q|^2)}.\end{equation}
Now we continue to prove Theorem~\ref{prejprej}. The solution of \eqref{beleq} with such initial conditions exists and is unique \cite[Theorem~A\&~Theorem~1]{zbMATH05159460} and maps the unit disk onto a quadrilateral $Q(a_0,a_1,a_2,a_3)$ whose vertices $a_0,a_1,a_2,a_3$, $a_4=a_0$ belongs to the unit circle. Moreover by \cite[Theorem~B]{zbMATH05159460}, there are four points $b_k=e^{\imath\alpha_k}, \ k=0,1,2,3$, $b_4=b_0, b_5=b_1$, $$F(e^{it})=\sum_{k=1}^4 a_k I_{(\alpha_k, \alpha_{k+1})}(t).$$ Here $F$ is the boundary function of $f$. Therefore (\cite[p.~63]{Duren2004}) we can conclude that $$f_z(z) = \sum_{k=1}^4 \frac{d_k}{z-b_k},$$ and that $$\bar{f}_z(z) = -\sum_{k=1}^4 \frac{\overline{d_k}}{z-b_k},$$ where $$d_k = \frac{a_k -a_{k+1}}{2\pi \imath}.$$
Therefore the third coordinate of conformal parameterisation is
$$T(z) =\pm 2\Re \imath \int_0^z \sqrt{f_z\bar f_z}dz$$ thus when $z$ is close to $b_k$, then $$T(z) =\pm|d_k|^2\log|1-z/b_k|+O(z-b_k).$$
Thus when $z\to b_k$, $T(z)\to \pm \infty$. This implies that $\mathbf{f}(z)\to \pm\infty$ if $z\to a\in(a_k, a_{k+1})$. Since $$q(z) =\frac{w+\frac{\imath \left(1-w^4\right) z}{\left|1-w^4\right|}}{1+\frac{\imath\overline{w} \left(1-w^4\right) z}{\left|1-w^4\right|}},$$ we get $$q(0) = w \ \ \text{and}\ \ q'(0)=\frac{\imath \left(1-w^4\right) \left(1-|w|^2\right) }{\left|1-w^4\right|}.$$
Now \eqref{finoser} follow from \eqref{eq:curvatureformula}.
Further
$$p(0)(1-q(0)^4)\overline{q'(0)}=-\imath f_z(0)|1-w^4|(1-|w|^2).$$ So in view of the formula \eqref{fexpli} we conclude $\mathbf{f}^\diamond_{uv}=0.$
Now we assert that \begin{equation}\label{needed}|\mathcal{K}_{S}(\mathbf{w})|< |\mathcal{K}_{S^{\diamond}}(\mathbf{w})|,\end{equation}
and what is the same \begin{equation}\label{needed1}W_S^2|\mathcal{K}_{S}(\mathbf{w})|< W_{S^\diamond}^2|\mathcal{K}_{S^{\diamond}}(\mathbf{w})|.\end{equation}
Assume the converse $|\mathcal{K}_{S}(\mathbf{w})|\ge |\mathcal{K}_{S^{\diamond}}(\mathbf{w})|$ and argue by a contradiction. Then as in \cite{FinnOsserman1964}, by using the dilatation $L(\zeta) = \lambda \zeta$ for some $\lambda\ge 1$ we get the surface
$$S_1=L(S)=\{(u,v,\lambda \mathbf{f}\left(\frac{u}{\lambda}, \frac{v}{\lambda}\right): |u+\imath v|<{\lambda}\},$$ whose Gaussian curvature $$\mathcal{K}_1(\mathbf{w})=\frac{\frac{1}{\lambda^2} \left(\mathbf{f}_{uu}(0,0)\mathbf{f}_{vv}(0,0)-\mathbf{f}_{uv}(0,0)^2\right)}{(1+\mathbf{f}_u(0,0)^2+\mathbf{f}_v(0,0)^2)^2}.$$ Observe that such transformation does not change the unit normal at $\mathbf{w}$.
Then there is $\lambda_\ast\ge 1$ so that $\mathcal{K}_1(\mathbf{w})=\mathcal{K}_{S^\diamond}(\mathbf{w})$. Let $$\mathbf{f}^\ast (u,v) =\lambda_\ast \mathbf{f}\left(\frac{u}{\lambda_\ast}, \frac{v}{\lambda_\ast}\right).$$ From $\mathbf{n}_\diamond=\mathbf{n}_\ast$ we get \begin{equation}\label{nowafter}\mathbf{f}^\diamond_{u}(0,0)=\mathbf{f}^\ast_{u}(0,0), \ \mathbf{f}^\diamond_{v}(0,0)=\mathbf{f}^\ast_{v}(0,0).\ \end{equation}
Further we have $$(1+(\mathbf{f}^\ast_{u}(0,0))^2)\mathbf{f}^\ast_{vv} (0,0)-2 \mathbf{f}^\ast_{u}(0,0)\mathbf{f}^\ast_{v}(0,0)\mathbf{f}^\ast_{uv} (0,0)+(1+(\mathbf{f}^\ast_{v}(0,0))^2)\mathbf{f}^\ast_{uu} (0,0)=0,$$ $$(1+(\mathbf{f}^\diamond_{u}(0,0))^2)\mathbf{f}^\diamond_{vv} (0,0)-2 \mathbf{f}^\diamond_{u}(0,0)\mathbf{f}^\diamond_{v}(0,0)\mathbf{f}^\diamond_{uv} (0,0)+(1+(\mathbf{f}^\diamond_{v}(0,0))^2)\mathbf{f}^\diamond_{uu} (0,0)=0,$$ $$ \mathbf{f}^\diamond_{uv}(0,0)=\mathbf{f}^\ast_{uv}(0,0)$$ and the equation $$\frac{ \left(\mathbf{f}^\ast _{uu}(0,0)\mathbf{f}^\ast_{vv}(0,0)-\mathbf{f}^\ast_{uv}(0,0)^2\right)}{(1+\mathbf{f}^\ast_u(0,0)^2+\mathbf{f}^\ast_v(0,0)^2)^2}= \frac{ \left(\mathbf{f}^\diamond _{uu}(0,0)\mathbf{f}^\diamond_{vv}(0,0)-\mathbf{f}^\diamond_{uv}(0,0)^2\right)}{(1+\mathbf{f}^\diamond_u(0,0)^2+\mathbf{f}^\diamond_v(0,0)^2)^2}.$$
We can also w.l.g. assume that $\mathbf{f}^\ast_{uu} $ and $\mathbf{f}^\diamond_{uu} $ as well as $\mathbf{f}^\ast_{vv} $ and $\mathbf{f}^\diamond_{vv} $ have the same sign. If not, then we choose $\lambda_\ast\le -1$ and repeat the previous procedure with $$S_1=L(S)=\{(u,v,\lambda \mathbf{f}\left(\frac{u}{\lambda}, \frac{v}{\lambda}\right): |u+\imath v|<{|\lambda|}\}.$$ Thus the function $F(u,v) = \mathbf{f}^\ast(u,v)- \mathbf{f}^\diamond(u,v)$ has all derivatives up to the order $2$ equal to zero in the point $w=0$.
To continue the proof we use the following lemma \begin{lemma}\label{leci} Assume that the quadrilateral $Q=Q(a,b,c,d)$ is inscribed in the unit disk, and assume that $\zeta=\mathbf{f}(u,v)$ is a Scherk's type minimal surface $S$ above $Q$. i.e. assume that $\mathbf{f}(u,v)\to +\infty$ when $\zeta=u+iv \to w\in (a,b)\cup (c,d)$ and $\mathbf{f}(u,v)\to -\infty$ when $\zeta=u+iv \to w\in (b,c)\cup (a,d)$. Then there is not any other bounded minimal graph $\zeta=\mathbf{f}_1(u,v)$ over a domain $\Omega$ that contains $Q$ which has the same Gaussian curvature, the same Gaussian normal, and the same mixed derivative at the same point $\mathbf{w}\in Q$ as the given surface $S$. \end{lemma} \begin{proof}[Proof of Lemma~\ref{leci}] We observe that \cite[Proof of Proposition~1]{FinnOsserman1964} works for every Scherk's type minimal surface, so if we would have a bounded minimal surface having the all derivatives ap to the order 2 equal to zero, then such non-parametric parameterizations $\mathbf{f}$ and $\mathbf{f}_1$, in view of \cite[Lemma~1]{FinnOsserman1964} will satisfy the relation $F(z)=\mathbf{f}(z)-\mathbf{f}_1(z)=O(\zeta^N(z))$, $N\ge 3$, where $\zeta$ is a certain homeomorphism between two open sets containing $0$. Then by following the proof of \cite[Proof of Proposition~1]{FinnOsserman1964} (second part) we get that this is not possible, because Sherk's type surface has four "sides" but the number $2N$ is bigger or equal to $6$ which is not possible. \end{proof}
This leads to the contradiction so \eqref{needed} is true. To finish the proof of Theorem~\ref{prejprej} we need to prove the sharpness. It is similar to the proof of sharpness of Theorem~\ref{th:theor2} below so we omit it. \end{proof} \begin{proof}[Proof of Theorem~\ref{th:theor}]
Assume that $S=\{(u,v,\mathbf{f}(u,v)): (u,v)\in \D\}$ is any surface above the unit disk and assume that $\mathbf{f}(0,0)=0$. Assume also that we have rotated the unit disk so that $\mathbf{f}_{uv}(0,0)=0$ and $\mathbf{f}_u(0,0)=0$. Namely if $h=e^{ic}$ and $\mathbf{f}^c(z) = \mathbf{f}(e^{ic}z)$. Then as in Example~\ref{forward} $$\mathbf{f}^c_u(0,0)=\nabla_h \mathbf{f}(0,0)=0.$$ Further $$\mathbf{f}^c_{uv}(0,0)=\nabla^2_{h,\imath h} \mathbf{f}(0,0)=0.$$
Let $\mathbf{f}_v(0)=V$ and assume w.l.g that $V>0$. Then the Gaussian normal of $\mathbf{w}\in S$ is \begin{equation}\label{secondn}\mathbf{n}=\frac{1}{\sqrt{1+V^2}}(0,-V, 1).\end{equation}
The Gauss map of $S_t$ above $0=f(z_\circ)$ can be expressed as (see \cite[p.~169]{Duren2004}) $$\mathbf{N}_t=-\frac{1}{1+|a(t)|^2}(2\Im a(t), 2\Re a(t), -1+|a(t)|^2),$$ where $a(t) = q(z_\circ)$. By \eqref{qqq} and \eqref{zezero} we have
\begin{equation}\label{inve}a(t)=\frac{\sqrt{\cos t}+\left(\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]\right) \tan \left[\frac{1}{8} \pi \sqrt{\cos t} \csc\left[\frac{t}{2}\right]^3 \sin t\right]}{\cos \left[\frac{t}{2}\right]+\sin \left[\frac{t}{2}\right]+\sqrt{\cos t} \tan \left[\frac{1}{8} \pi \sqrt{\cos t} \csc\left[\frac{t}{2}\right]^3 \sin t\right]} .\end{equation} We need to find $t $ so $\mathbf{N}_t=\mathbf{n}$, where $\mathbf{n}$ is the unit normal at the second minimal surface above $0$ defined in \eqref{secondn}.
Since the function $a(t)$ is continuous for $t\in[t_\circ,\pi/2]$ and $a(\pi/2)=0$ and $$a(t_\circ) = a\left(2 \tan^{-1}\left[\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)}\right]\right)=1,$$ there is $t_0\in(t_\circ , \pi/2)$ so that $$a(t_0)=\frac{-1+\sqrt{1+V^2}}{V}.$$ In this case $\mathbf{N}_{t_0}=\mathbf{n}$.
Assume now that $$S^{\diamond}=\{((u,v), \mathbf{f}^\diamond(u,v)): (u,v)\in \D\},$$ is the Scherk 's type surface above the trapezoid $\mathcal{T}=\mathcal{T}^{t_0}$ so that $\mathbf{f}^\diamond(0,0)=\mathbf{f}(0,0)=0$.
Let $\mathbf{w}=(0,0,0)$. Then instead of \eqref{nowafter} we have \begin{equation}\label{nowafter}\mathbf{f}^\diamond_{u}(0,0)=\mathbf{f}^\ast_{u}(0,0)=0, \ \mathbf{f}^\diamond_{v}(0,0)=\mathbf{f}^\ast_{v}(0,0)=V.\ \end{equation} Then as in the proof of Theorem~\ref{prejprej} we obtain that \begin{equation}|\mathcal{K}_{S}(\mathbf{w})< |\mathcal{K}_{S^{\diamond}}(\mathbf{w})|,\end{equation} and
\begin{equation}W_{S}^2|\mathcal{K}_{S}(\mathbf{w})< W_{\diamond}^2|\mathcal{K}_{S^{\diamond}}(\mathbf{w})|.\end{equation}
Lemma~\ref{leci} works also for the trapezoid instead of the square. The only important thing is that the mapping $T(z)$ defined in \eqref{TT} tends to $\pm \infty$ as $z\to \pm 1$ of $z\to \pm \imath$. This implies that $\mathbf{f}^\diamond(z)\to \pm\infty$ if $z\to \zeta$, where $\zeta$ belongs to an open side of the trapezoid.
Now subsections~\ref{subsub1} and \ref{subsub2} imply that $$|\mathcal{K}_{S}(\mathbf{w})\le W_{S}^2|\mathcal{K}_{S}(\mathbf{w})\le \pi^2/2$$ what we needed to prove. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th:theor2}]
Assume that $S^t$ is as in Proposition~\ref{defshre}. Since $\kappa:[t_\circ, \pi/2]\to [0,\pi/\sqrt{2}]$ is increasing (see subsection~\ref{subsub1}), the function $a(t)=|q(z_\circ)|: [t_\circ, \pi/2]\to [0,1]$ is decreasing. Further the angle $\theta=\arccos\frac{1-|q(0)|^2}{1+|q(0)|^2}$ of the unit normal is uniquely determined by $|q(z_\circ)|$. It follows that there is a bijective correspondence between the curvature at $\mathbf{w}\in S^t$ and the angle that tangent plane $TS^t_\mathbf{w}$ forms with the $v-$axis. In this way it is determined a continuous decreasing function $\Psi(\theta)=|\mathcal{K}(\mathbf{w})|:[0,\pi/2]\to [0,\pi^2/2]$. The proof of the first part is the same as the proof of Theorem~\ref{th:theor}.
Prove the second part. A similar statement for the case that the tangent plane is horizontal has been proved in \cite[Proposition~3]{FinnOsserman1964}. However that proof does not work in this case. Assume that $\omega= (q(z))^2$ where $q$ is defined in \eqref{qqq}. Also assume that $t\in(t_\circ, \pi/2]$. Let $f$ be as in \eqref{after}. Then $f$ is a solution of Beltrami equation $\overline{f}_z=\omega f_z$ satisfying the initial conditions $$f_z(0)=p=\frac{\imath \sec\left[\frac{t}{2}\right] (-1+\cos t-\sin t)}{\pi }$$ (in view of \eqref{pmtp} and \eqref{pptp}) and $f(0)=\imath \sqrt{\cos t}$ (because of \eqref{be}). Further $f$ maps the unit disk onto the convex trapezoid $\mathcal{T}$. This implies that $\tilde f=\imath f$ maps the unit disk onto the trapezoid $\imath\mathcal{T}$ and satisfies the equation $\overline{ f}_z=-\omega f_z$ with the initial condition $\tilde f(0)=- \sqrt{\cos t}$ and $\tilde f_z(0)>0$. Recall also that $f(z_\circ)=0$, where $z_\circ$ is defined in \eqref{zezero}.
Further, for $0<k<1$ assume that $\omega_k=k^2 e^{-\imath\pi/2}\omega.$ Then solve the second Beltrami equation $\overline{f}_z=\omega_k f_z$ that map the unit disk $\D$ onto itself satisfying the initial condition $f(0) = -\sqrt{\cos t}$ and $f_z(0)>0$ \cite{HengartnerSchober1986}. This mapping exists and is unique \cite[p.~134]{Duren2004}. Then this mapping produces a minimal surface $S_k^t$ over the unit disk. Moreover for $k=n/(n+1),$ the sequence $f_n$ converges (up to some subsequence) in compacts of the unit disk, to a mapping $f^\circ$ that maps the unit disk into the unit disk. By using again the uniqueness theorems \cite[Theorem~B\&~Theorem~1]{zbMATH05159460}, because $f^\circ(0)=\tilde f(0)=-\sqrt{\cos t}$ and $f^\circ_z(0)>0$, it follows that $f^\circ \equiv \tilde f$. Let $\mathbf{w}_n$ be the point above $0$ of minimal surface $S^t_n$. Let $z_n\in \D$, so that $f_n(z_n)=0$. Then $\mathbf{w}_n$ converges to $\mathbf{w}$. Moreover the Gaussian curvatures $\mathcal{K}_n(\mathbf{w}_n)$ of $S_n^t$, in view of the formula \eqref{eq:curvatureformula}, is equal to $$- \frac{4|q_n'(z_n)|^2}{|p_n(z_n)|^2(1 + |q_n(z_n)|^2)^4}$$ and converges to the Gaussian curvature $\mathcal{K}(\mathbf{w})=-\kappa^2(t).$ Namely $z_n=f_n^{-1}(0)$, and therefore $\lim_{n\to \infty}z_n=\lim_{n\to\infty}f_n^{-1}(0)=f^{-1}(0)=z_\circ$, because $f_n^{-1}$ and also $f^{-1}$ are quasiconformal in a disk around $0$ and the family is normal. Also $q_n$ and $p_n$ and $q_n'$ converges in compacts to the corresponding $q$, $p$ and $q'$. We proved that for a fixed $\theta$ the inequality \eqref{eq:FinnOsserman2} cannot be improved. In a similar way we can prove the rest of the theorem. \end{proof} \begin{remark}\label{remica} It follows from Section~\ref{sectio2}, see \eqref{afterv}, that the mapping $f$ satisfies the conditions $f(z_\circ)=0$ and $\imath f_z(z_\circ)>0$. So the mapping $\hat f$ defined by $\hat f(z) =\imath f\left(\frac{z+z_\circ}{1+zz_0}\right)$ satisfies the conditions $\hat f(0)=0$ and $\hat f_z(0)>0$. Moreover it satisfies the Beltrami equation \eqref{beleq} with $w=\imath a(t)$, where $a(t)$ is defined in \eqref{inve}. In this case the given trapezoid is symmetric w.r.t real axis. \end{remark} Now \eqref{finoser} and Theorem~\ref{th:theor} (or the result of Finn and Osserman), implies the following lemma. \begin{lemma}
Assume that $f$ solves the equation $$\bar f_z(z) = z^2 f_z(z),$$ with the initial conditions $f(0)=0$ and $f_z(0)>0$. Assume also that $f$ is a limit of harmonic diffeomorphisms $f_n:\D\to D_n \supseteq \D$, whose second dilatations are squares of holomorphic functions, with initial conditions $f_n(0)=(\bar{f}_{n})_z(0)=0$. Then the sharp inequality $$|f_z(0)|\ge \frac{2\sqrt{2}}{\pi}$$ holds. \end{lemma} \begin{proof} The only important think is that $f_n$ can be lifted to a minimal surface, whose projection contains the unit disk with $f_n(0)=(\bar{f}_{n})_z(0)=0$, so the result follows from \eqref{finoser} and the result of Finn and Osserman (or Theorem~\ref{th:theor}). \end{proof} To prove Corollary~\ref{coro} we also need the following lemma. \begin{lemma} Assume that $f$ is a limit of harmonic diffeomorphisms $f_n$ of the unit disk onto $D_n \supseteq \D$ with squared second holomorphic dilatations, that solve the equation $$\bar f_z(z) = \left(\frac{w+e^{is}z}{1+e^{is}\overline{w}z}\right)^2 f_z(z),$$ with the initial conditions $f(0)=0$ and $f_z(0)>0$. Then we have the inequality
\begin{equation}\label{improv}|f_z(0)|\ge \frac{2\sqrt{2}}{\pi}\frac{(1-|f(-we^{-is})|)}{1-|w|^2}.\end{equation} \end{lemma} \begin{proof}
Let $$f^1(z) = \frac{1}{1-|f(-we^{-is})|}\left(f\left(\frac{e^{-\imath s} (w-z)}{-1+z \overline{w}}\right)-f(-we^{-is})\right).$$ Then $f^1$ solves the Beltrami equation $$\overline{f}^1_z(z)=z^2 f^1_z(z)$$ and $f^1(0)=0$, $f^1_{\bar z}(0)>0$. Let $f_n$ be a mapping defined by
$$f_n^1(z) = \frac{1}{1-|f_n(-we^{-is})|}\left(f_n\left(\frac{e^{-\imath s} (w-z)}{-1+z \overline{w}}\right)-f_n(-we^{-is})\right).$$
Then the second dilatation of $f^1_n$ is the square of an analytic function and it satisfies the initial conditions $f^1_n(0)=(\bar{f^1}_{n})_z(0)=0$.
Therefore by Lemma~\ref{leci}, in view of \eqref{finoser} we get $|f^1_z(0)|\ge \frac{\pi}{2\sqrt{2}}$, and this implies the claimed inequality. \end{proof} \begin{proof}[Proof of Corollary~\ref{coro}]
Let $S: \zeta=\mathbf{f}(u,v)$ be a non-parametric minimal surface over the unit disk and assume that $$\mathbf{n}_{\mathbf{w}}=-\frac{1}{1+|w|^2}(2\Im w, 2\Re w, -1+|w|^2),$$ is its Gaussian normal at the \emph{centre}. Let $f=f_w$ be the solution that is provided to us by Theorem~\ref{prejprej} that produces the Scherk type minimal surface $S^\diamond$. Let $\mathcal{K}=\mathcal{K}_{S^\diamond}(\mathbf{w})$. In view of Theorem~\ref{prejprej}, we only need to estimate the curvature $\mathcal{K}$. Now we have the estimate \begin{equation}\label{eq:weakestK1}
\begin{split}|\mathcal{K}| & =\frac{4(1-|w|^2)}{|f_z(0)|^2(1+|w|^2)^4}\\& \le
\frac{16\pi^2}{27} \frac{\bigl(1-|w|^2\bigr)^2 \bigl(1+|w|^4\bigr)}{(1+|w|^2)^4}.\end{split} \end{equation}
Write $|w|=r$ and consider the function \begin{equation}\label{eq:h}
G(r)= \frac{16\pi^2}{27}\frac{\bigl(1-r^2\bigr)^2 \bigl(1+r^4\bigr)}{(1+r^2)^4}, \quad 0\le r\le 1. \end{equation} Note that \eqref{eq:weakestK1} can be written in the form \begin{equation}\label{eq:estK2}
|\mathcal{K}| \le G(r). \end{equation}
Now the proof of Theorem~\ref{prejprej} implies that, $f$ is a limit of a sequence $f_n$ satisfying Corollary~\ref{coro}. In view of \eqref{finoser} and \eqref{improv} and harmonic Schwarz lemma: $|f(w)|\le \frac{4}{\pi}\tan^{-1}(|w|)$, we get
\[\begin{split}|\mathcal{K}(\mathbf{w})|&=\frac{4 \left(1-|w|^2\right)^2}{\left(1+|w|^2\right)^4 |f_z(0)|^2}
\\& \le \frac{4 \left(1-|w|^2\right)^2}{\left(1+|w|^2\right)^4 |\frac{2\sqrt{2}}{\pi}\frac{(1-|f(-w)|)}{1-|w|^2}|^2}
\\& \le\frac{\pi^2}{2} \frac{ \left(1-|w|^2\right)^4}{\left(1+|w|^2\right)^4 (1-|f(-w)|)^2}
\\& \le \frac{\pi^2}{2} \frac{ \left(1-|w|^2\right)^4}{\left(1+|w|^2\right)^4 (1-\frac{4}{\pi}\tan^{-1}(|w|))^2} :=H(|w|). \end{split}\] From the previous relations and \eqref{eq:estK2} we conclude that $$\mathcal{K}(\mathbf{w})\le \max_{r\in[0,1]}\min\{G(r), H(r)\}.$$ Let $r_\diamond \approx 0.067344733$ be the solution of the equation $$ G(r)=H(r), \ \ r\in(0,1),$$ where $G$ is defined in \eqref{eq:h}. It can be easily proved that $H$ increases in $r\in(0,r_\diamond)$ and $G$ decreases in $(0,1)$. Therefore, $\mathcal{K}(\mathbf{w})<G(r_\diamond) \approx 5.6918$.
Further, by using \eqref{firsti} and \eqref{secondi} we get $$W^2=1+\mathbf{f}^2_u+\mathbf{f}^2_v=\frac{\left(1+|w|^2\right)^2}{\left(1-|w|^2\right)^2},$$ where $w=|q(0)|$. Therefore we get that $$\mathcal{K}(\mathbf{w})W^2 \le \frac{16 \pi ^2 \left(1+r_\diamond^4\right)}{27 \left(1+r_\diamond^2\right)^2}\approx 5.79608.$$ \end{proof}
}
\noindent David Kalaj
\noindent University of Montenegro, Faculty of Natural Sciences and Mathematics, 81000, Podgorica, Montenegro
\noindent e-mail: {\tt davidk@ucg.ac.me}
\end{document} | arXiv | {
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\begin{document}
\title{A Direct Method for Solving Optimal Switching Problems of One-Dimensional Diffusions} \begin{abstract}\noindent In this paper, we propose a direct solution method for optimal switching problems of one-dimensional diffusions. This method is free from conjectures about the form of the value function and switching strategies, or does not require the proof of optimality through quasi-variational inequalities. The direct method uses a general theory of optimal stopping problems for one-dimensional diffusions and characterizes the value function as sets of the smallest linear majorants in their respective transformed spaces.
\end{abstract} \section{Introduction} Stochastic \emph{optimal switching} problems (or \emph{starting and stopping} problems) are important subjects both in mathematics and economics. Since there are numerous articles about real options in the economic and financial literature in recent years, the importance and applicability of control problems including optimal switching problems cannot be exaggerated.
A typical optimal switching problem is described as follows: The controller monitors the price of natural resources for optimizing (in some sense) the operation of an extraction facility. She can choose when to start extracting this resource and when to temporarily stop doing so, based upon price fluctuations she observes. The problem is concerned with finding an optimal switching policy and the corresponding value function. A number of papers on this topic are well worth mentioning : Brennan and Schwarz \citeyearpar{BS1985} in conjunction with convenience yield in the energy market, Dixit \citeyearpar{D1989} for production facility problems, Brekke and {\O}ksendal \citeyearpar{BO1994} for resource extraction problems, Yushkevich \citeyearpar{Y2001} for positive recurrent countable Markov chain, and Duckworth and Zervos \citeyearpar{DZ2001} for reversible investment problems. Hamdad\`{e}ne and Jeanblanc \citeyearpar{HJ2004} analyze a general adapted process for finite time horizon using reflected stochastic backward differential equations. Carmona and Ludkovski \citeyearpar{CL2005} apply to energy tolling agreement in a finite time horizon using Monte-Carlo regressions.
A basic analytical tool for solving switching problems is quasi-variational inequalities. This method is indirect in the sense that one first conjectures the form of the value function and the switching policy and next verifies the optimality of the candidate function by proving that the candidate satisfies the variational inequalities. In finding the specific form of the candidate function, appropriate boundary conditions including the smooth-fit principle are employed. This formation shall lead to a system of non-linear equations that are often hard to solve and the existence of the solution to the system is also difficult to prove. Moreover, this indirect solution method is specific to the underlying process and reward/cost structure of the problem. Hence a slight change in the original problem often causes a complete overhaul in the highly technical solution procedures.
Our solution method is direct in the sense that we first show a new mathematical characterization of the value functions and, based on the characterization, we shall \emph{directly} find the value function and optimal switching policy. Therefore, it is free from any guesswork and applicable to a larger set of problems (where the underlying process is one-dimensional diffusions) than the conventional methods. Our approach here is similar to Dayanik and Karatzas \citeyearpar{DK2003} and Dayanik and Egami \citeyearpar{DE2005} that propose direct methods of solving optimal stopping problems and stochastic impulse control problems, respectively.
The paper is organized in the following way. In the next section, after we introduce our setup of one dimensional optimal switching problems, in section \ref{subsec:recursive}, we characterize the optimal switching times as exit times from certain intervals through sequential optimal stopping problems equivalent to the original switching problem. In section \ref{subsec:value-function}, we shall provide a new characterization of the value function, which leads to a direct solution method described in \ref{subsec:method}. We shall illustrate this method through examples in section \ref{sec:example}, one of which is a new optimal switching problem. Section \ref{sec:last-section} concludes with comments on an extension to a further general problem.
\section{Optimal Switching Problems} We consider the following optimal switching problems for one dimensional diffusions. Let $(\Omega, \F, \p)$ be a complete probability space with a standard Brownian motion $W=\{W_t; t\geq 0\}$. Let $Z_t$ be the indicator vector at time $t$, $Z_t\in \{z_1, z_2,..., z_m\}\triangleq \mathcal{Z}$ where each vector $z_i=(a_1, a_2,..., a_k)$ with $a$ is either $0$ (closed) or $1$ (open), so that $m=2^k$. In this section, we consider the case of $k=1$. That is, $Z_t$ takes either $0$ or $1$. The admissible switching strategy is \begin{equation*} w=(\theta_0, \theta_1, \theta_2,..., \theta_k,...; \zeta_0, \zeta_1, \zeta_2,..., \zeta_k,...) \end{equation*} with $\theta_0=0$ where where where $0\leq \theta_1<\theta_2<....$ are an increasing sequence of $\mathcal{F}_t$-stopping times and $\zeta_1$, $\zeta_2...$ are $\mathcal{F}_{\theta_i}$-measurable random variables representing the new value of $Z_t$ at the corresponding switching times $\theta_i$ (in this section, $\zeta_i=1$ or $0$). The state process at time $t$ is denoted by $(X_t)_{t\ge 0}$ with state space $\mathcal{I}=(c, d)\subseteq \R$ and $X_0=x \in \mathcal{I}$, and with the following dynamics:
If $\zeta_0=1$ (starting in open state), we have, for $m=0, 1, 2,.....$, \begin{align}\label{eq:sde-open} dX_t=\begin{cases} dX_t^1=\mu_1(X^1)dt + \sigma_1(X^1)dW_t, &\theta_{2m}\le t<\theta_{2m+1},\\ dX_t^0=\mu_0(X^0)dt + \sigma_0(X^0)dW_t, &\theta_{2m+1}\le t < \theta_{2m+2}, \end{cases} \end{align} and if $\zeta_0=0$ (starting in closed state), \begin{align} \label{eq:sde-closed} dX_t=\begin{cases} dX_t^0=\mu_0(X^1)dt + \sigma_0(X^0)dW_t, &\theta_{2m}\le t<\theta_{2m+1},\\ dX_t^1=\mu_1(X^1)dt + \sigma_1(X^1)dW_t, &\theta_{2m+1}\le t < \theta_{2m+2}. \end{cases} \end{align} We assume that $\mu_i:\R\rightarrow \R$ and $\sigma_i:\R\rightarrow \R$ are some Borel functions that ensure the existence and uniqueness of the solution of (\ref{eq:sde-open}) for $i=1$ and (\ref{eq:sde-closed}) for $i=0$.
Our performance measure, corresponding to starting state $i=0, 1$, is \begin{equation}\label{eq:problem} J^w_i(x)=\ME\left[\int_0^\infty e^{-\alpha s}f(X_s)ds-\sum_{j=1}^\infty e^{-\alpha \theta_j}H(X_{\theta_{j-}}, \zeta_j)\right] \end{equation} where $H: \mathbb{R}\times \mathcal{Z}\rightarrow \mathbb{R}_+$ is the switching cost function and $f: \R\rightarrow \R$ is a continuous function that satisfies \begin{equation} \label{eq:f-condition}
\ME\left[\int_0^\infty e^{-\alpha s}|f(X_s)|ds\right]<\infty. \end{equation} In this section, the cost functions are of the form: \begin{align*} H(X_{\theta-}, \zeta)= \begin{cases}
H(X_{\theta-}, 1) &\text{opening cost},\\
H(X_{\theta-}, 0) &\text{closing cost}.
\end{cases} \end{align*} The optimal switching problem is to optimize the performance measure for $i=0$ (start in closed state) and $1$ (start in open state). That is to find, for both $i=1$ and $i=0$, \begin{eqnarray}\label{eq:problem-2} v_i(x)\triangleq\sup_{w\in W}J^w(x) \quad\text{with}\quad X_0=x \end{eqnarray} where $W$ is the set of all the admissible strategies. \subsection{Characterization of switching times}\label{subsec:recursive} For the remaining part of section $2$, we assume that the state space $X$ is $\mathcal{I}=(c, d)$ where both $c$ and $d$ are natural boundaries of $X$. But our characterization of the value function does not rely on this assumption. In fact, it is easily applied to other types of boundaries, for example, absorbing boundary.
The first task is to characterize the optimal switching times as exit times from intervals in $\R$. For this purpose, we define two functions $g_0$ and $g_1: \R_+\rightarrow \R$ with \begin{equation}\label{eq:g-1-g-0} g_1(x)\triangleq \sup_{w\in W_0}J^w_1(x) \quad\text{and}\quad g_0(x)\triangleq\sup_{w\in W_0}J^w_0(x).
\end{equation} where $W_0\triangleq \{w\in W: w=(\theta_0, \zeta_0, \theta_1=+\infty )\}$. In other words, $g_1(\cdot)$ is the discounted expected revenue by starting with $\zeta_0=1$ and making no switches. Similarly, $g_0(\cdot)$ is the discounted expected revenue by staring with $\zeta_0=0$ and making no switches.
We set $w_0\triangleq g_1$ and $y_0\triangleq g_0$. We consider the following simultaneous sequential optimal stopping problems with $w_n : \R_+ \rightarrow \R$ and $y_n: \R_+\rightarrow \R$ for $n=1, 2,....$: \begin{equation}\label{eq:w-function} w_{n}(x)\triangleq\sup_{\tau\in \mathcal{S}}\ME \left[\int_0^\tau e^{-\alpha s}f(X_s)ds + e^{-\alpha\tau} (y_{n-1}(X_{\tau})-H(X_{\tau-}, 1-Z_{\tau-}))\right], \end{equation} and \begin{equation}\label{eq:y-function} y_{n}(x)\triangleq\sup_{\tau\in \mathcal{S}}\ME\left[\int_0^\tau e^{-\alpha s}f(X_s)ds + e^{-\alpha\tau} (w_{n-1}(X_{\tau})-H(X_{\tau-}, 1-Z_{\tau-}))\right], \end{equation} where $\mathcal{S}$ is a set of $\mathcal{F}_t$ stopping times. Note that for each $n$, the sequential problem \ref{eq:w-function} (resp. (\ref{eq:y-function})) starts in open (resp. closed) state.
On the other hand, we define $n$-time switching problems for $\zeta_0=1$: \begin{equation}\label{eq:q-function} q^{(n)}(x)\triangleq\sup_{w\in W_n}J_1^w(x),
\end{equation} where \begin{equation*}
W_n\triangleq \{w\in W; w=(\theta_1, \theta_2,... \theta_{n+1}; \zeta_1,
\zeta_2,...\zeta_n); \theta_{n+1}=+\infty\}. \end{equation*} In other words, we start with $\zeta_0=1$ (open) and are allowed to make at most $n$ switches. Similarly, we define another $n$-time switching problems corresponding to $\zeta_0=0$: \begin{equation}\label{eq:z-function} p^{(n)}(x)\triangleq\sup_{w\in W_n}J_0^w(x).
\end{equation} We investigate the relationship of these four problems: \begin{lemma} For any $x \in \R$, $w_{n}(x)=q^{(n)}(x)$ and $y_{n}(x)=p^{(n)}(x)$. \end{lemma} \begin{proof} We shall prove only the first assertion since the proof of the second is similar. We have set $y_0(x)=g_0(x)$. Now we consider $w_1$ by using the strong Markov property of $X$: \begin{align*} w_1(x)&=\sup_{\tau\in \mathcal{S}}\ME\left[\int_0^\tau e^{-\alpha s}f(X_s)ds + e^{-\alpha\tau} (g_{0}(X_{\tau})-H(X_{\tau-}, 0))\right]\\ &=\sup_{\tau\in \mathcal{S}}\ME\left[\int_0^\infty e^{-\alpha s}f(X_s)ds -\int_{\tau}^\infty e^{-\alpha s}f(X_s)ds-e^{-\alpha \tau}(g_0(X_{\tau})-H(X_{\tau-}, 0)) \right]\\ &=\sup_{\tau\in \mathcal{S}}\ME\left[e^{-\alpha \tau}(g_0(X_{\tau})-g_1(X_{\tau})-H(X_{\tau-}, 0))\right]+g_1(x). \end{align*} On the other hand, \begin{align*} q^{(1)}(x)&=\sup_{w\in W_1}\ME\left[\int_{0}^{\infty}e^{-\alpha s}f(X_s)ds -e^{-\alpha \theta_1}H(X_{\theta_{1-}}, \zeta_1)\right]\\ &=\sup_{w\in W_1}\ME\left[\int_{0}^{\theta_1}e^{-\alpha s}f(X_s)ds+\int_{\theta_1}^{\infty}e^{-\alpha s}f(X_s)ds -e^{-\alpha \theta_1}H(X_{\theta_{1-}}, 0)\right]\\ &=\sup_{w\in W_1}\ME\left[(g_1(x)-e^{-\alpha\theta_1}g_1(X_{\theta_1}))-e^{-\alpha \theta_1}(g_0(X_{\theta_1})-H(X_{\theta_{1-}}, 0))\right]\\ &=\sup_{w\in W_1}\ME\left[e^{-\alpha\theta_1}(g_0(X_{\theta_1})-g_1(X_{\theta_1})-H(X_{\theta_{1-}}, 0))\right]+g_1(x). \end{align*} Since both $\tau$ and $\theta_1$ are $\F_t$ stopping times, we have $w_1(x)=q^{(1)}(x)$ for all $x\in \R$. Moreover, by the theory of the optimal stopping (see Appendix \ref{appx:ost}, especially Proposition \ref{prop:A4}), $\tau$ and hence $\theta_1$ are characterized as an exit time from an interval. Similarly, we can prove $y_1(x)=p^{(1)}(x)$. Now we consider $q^{(2)}(x)$ which is the value if we start in open state and make at most $2$ switches (open $\rightarrow$ close $\rightarrow$ open).
For this purpose, we consider the performance measure $\bar{q}^{(2)}$ that starts in an open state and is allowed two switches: For arbitrary switching times $\theta_1, \theta_2>\theta_1 \in \mathcal{S}$, we have \begin{align*} \bar{q}^{(2)}(x)&\triangleq\ME\left[\int_{0}^{\infty}e^{-\alpha s}f(X_s)ds -\sum_{j=1}^2e^{-\alpha \theta_j}H(X_{\theta_{j-}}, \zeta_j)\right]\\ &=\ME\Bigg[\int_{0}^{\theta_1}e^{-\alpha s}f(X_s)ds+\int_{\theta_1}^{\theta_2}e^{-\alpha s} f(X_s)ds+\int_{\theta_2}^{\infty}e^{-\alpha s} f(X_s)ds\\ &\hspace{3cm}-e^{-\alpha\theta_1}H(X_{\theta_1-}, 0)-e^{-\alpha\theta_2}H(X_{\theta_2-}, 1)\Bigg]\\ &=\left(g_1(x)-\ME[e^{-\alpha\theta_1}g_1(X_{\theta_1})]\right) +\left(\ME[e^{-\alpha\theta_1}g_0(X_{\theta_1})-e^{-\alpha\theta_2}g_0(X_{\theta_2})]\right) +\ME[e^{-\alpha\theta_2}g_1(X_{\theta_2})]\\ &\hspace{3cm}-\ME[e^{-\alpha\theta_1}H(X_{\theta_1-}, 0)+e^{-\alpha\theta_2}H(X_{\theta_2-}, 1)]. \end{align*} Hence we have the following multiple optimal stopping problems: \begin{align*} \bar{q}^{(2)}(x)&=\sup_{(\theta_1, \theta_2)\in \mathcal{S}^2}\ME\left[e^{-\alpha\theta_1}\Big((g_0-g_1)(X_{\theta_1})-H(X_{\theta_1-}, 0)\Big)+e^{-\alpha\theta_2}\Big((g_1-g_0)(X_{\theta_2})-H(X_{\theta_2-}, 1)\Big)\right]\\ &\hspace{3cm}+g_1(x) \end{align*} where $\mathcal{S}^2\triangleq \{(\theta_1, \theta_2); \theta_1\in\mathcal{S}; \theta_2\in\mathcal{S}_{\theta_1}\}$ and $\mathcal{S}_{\sigma}=\{\tau\in\mathcal{S}; \tau\ge \sigma\}$ for every $\sigma\in\mathcal{S}$. Let us denote $h_1(x)\triangleq g_1(x)-g_0(x)-H(x, 0)$,
$h_2(x)\triangleq g_0(x)-g_1(x)-H(x, 1)$, \begin{align*} V_1(x)\triangleq\sup_{\tau\in\mathcal{S}}\ME\left[e^{-\alpha\tau}h_1(X_{\tau})\right]\quad\text{and}\quad V_2(x)\triangleq\sup_{\tau\in\mathcal{S}}\ME\left[e^{-\alpha\tau}(h_2(X_\tau)+V_1(X_{\tau}))\right]. \end{align*} We also define \begin{equation*}
\Gamma_1\triangleq \{x\in \mathcal{I}:
V_1(x)=h_1(x)\}\quad\text{and}\quad\Gamma_2\triangleq
\{x\in\mathcal{I}: V_2(x)=h_2(x)+V_1(x)\} \end{equation*} with $\sigma_n\triangleq \inf\{t\ge 0: X_t\in\Gamma_n\}$. By using Proposition 5.4. in Carmona and Dayanik \citeyearpar{CD2003}, we conclude that $\theta_1=\sigma_1$ and $\theta_2=\theta_1+\sigma_2\circ s(\theta_1)$ is optimal strategy where $s(\cdot)$ is the shift operator. Hence we only consider the maximization over the set of admissible strategy $W_2^*$ where \begin{equation*} W^{*}_2\triangleq\{w\in W_2: \theta_1, \theta_2 \quad\text{are exit imes from an interval in $\mathcal{I}$}\}, \end{equation*} and can use the relation $\theta_2-\theta_1=\theta \circ s(\theta_1)$ with some exit time $\theta\in\mathcal{S}$. \begin{align*} q^{(2)}(x)&=\sup_{w\in W_2^*}\ME\left[\int_{0}^{\infty}e^{-\alpha s}f(X_s)ds -\sum_{j=1}e^{-\alpha \theta_j}H(X_{\theta_{j-}}, \zeta_j)\right]\\ &=\sup_{w\in W_2^*}\ME\Big[\int_{0}^{\theta_1}e^{-\alpha s}f(X_s)ds+\int_{\theta_1}^{\theta_2}e^{-\alpha s} f(X_s)ds+\int_{\theta_2}^{\infty}e^{-\alpha s} f(X_s)ds\\ &\hspace{3cm}-e^{-\alpha\theta_1}(H(X_{\theta_{1-}}, 0)+e^{-\alpha(\theta_2-\theta_1)} H(X_{\theta_{2-}}, 1))\Big]\\ &=\sup_{w\in W_2^*}\ME\Big[\int_{0}^{\theta_1}e^{-\alpha s}f(X_s)ds+e^{-\alpha\theta_1}\E^{X_{\theta_1}}\left[\left(\int_0^{\theta} +\int_\theta^{\infty}\right)e^{-\alpha s}f(X_s)ds-e^{-\alpha\theta}H(X_{\theta-}, 1)\right]\\ &\hspace{3cm}-e^{-\alpha\theta_1}H(X_{\theta_{1-}}, 0)\Big]. \end{align*} Now by using the result for $p^{(1)}$, we can conclude \begin{align*} q^{(2)}(x)&=\sup_{w\in W_2^*}\ME\left[\int_{0}^{\theta_1}e^{-\alpha s}f(X_s)ds+e^{-\alpha\theta_1}\left(p^{(1)}(X_{\theta_1})-H(X_{\theta_{1-}}, 0)\right)\right]\\ &=\sup_{\theta_1\in \mathcal{S}}\ME\left[\int_{0}^{\theta_1}e^{-\alpha s}f(X_s)ds+e^{-\alpha\theta_1}\left(y_1(X_{\theta_1})-H( X_{\theta_{1-}}, 0)\right)\right] =w_2(x) \end{align*} Similarly, we can prove $y_2(x)=p^{(2)}(x)$ and we can continue this process inductively to conclude that $w_{n}(x)=q^{(n)}(x)$ and $y_{n}(x)=p^{(n)}(x)$ for all $x$ and $n$. \end{proof} \begin{lemma} For all $x \in \R$, $\lim_{n\rightarrow \infty}q^{(n)}(x)=v_1(x)$ and $\lim_{n\rightarrow \infty}p^{(n)}(x)=v_0(x)$. \end{lemma} \begin{proof} Let us define $q(x)\triangleq \lim_{n\rightarrow \infty}q^{(n)}(x)$. Since $W_n\subset W$, $q^{(n)}(x)\le v_1(x)$ and hence $q(x)\le v_1(x)$. To show the reverse inequality, we define $W^+$ to be a set of admissible strategies such that \begin{equation*}
W^+=\{w\in W: J_1^w(x)< \infty \quad\text{for all}\quad x\in \R\}. \end{equation*} Let us assume that $v_1(x)<+\infty$ and consider a strategy $w^+\in W^+$ and another strategy $w_n$ that coincides with $w^+$ up to and including time $\theta_n$ and then takes no further interventions. \begin{equation}\label{eq:difference}
J^{w^+}_1(x)-J^w_1(x)=\ME\left[\int_{\theta_n}^\infty e^{-\alpha
s}(f(X_s)-f(X_{s-{\theta_n}}))
-\sum_{i\ge n+1}e^{-\alpha \theta_i}H(X_{\theta_i-},
\zeta_i)\right], \end{equation} which implies \begin{equation*}
|J^{w^+}_1(x)-J^w_1(x)|\le \ME\left[\frac{2\|f\|}{\alpha}e^{-\alpha\theta_n}-\sum_{i\ge n+1}e^{-\alpha \theta_i}H(X_{\theta_i-},
\zeta_i)\right]. \end{equation*} As $n\rightarrow +\infty$, the right hand side goes to zero by the dominated convergence theorem. Hence it is shown \begin{equation*}
v_1(x)=\sup_{w\in W^+}J^w_1(x)=\sup_{w\in {\cup}_n W_n}J^w_1(x) \end{equation*} so that $v_1(x)\le q(x)$. Next we consider $v_1(x)=+\infty$. Then we have some $m\in \mathbb{N}$ such that $w_m(x)=q^{(m)}(x)=\infty$. Hence $q^{(n)}(x)=\infty$ for all $n\ge m$. The second assertion is proved similarly. \end{proof} We define an operator $\LL: \mathcal{H}\rightarrow \mathcal{H}$ where $\mathcal{H}$ is a set of Borel functions \begin{align*} \LL u(x)\triangleq \sup_{\tau\in\mathcal{S}}\ME\left[\int_0^\tau e^{-\alpha s}f(X_s)ds +e^{-\alpha\tau}\left(u(X_{\tau})-H(X_{\tau-}, 1-Z_{\tau-})\right)\right]. \end{align*} \begin{lemma}\label{lem:fixed-point} The function $w(x)\triangleq\lim_{n\rightarrow \infty}w_n(x)$ is the smallest solution, that majorizes $g_1(x)$, of the function equation $w=\LL w$. \end{lemma} \begin{proof} We renumber the sequence $(w_0, y_1, w_2, y_3...)$ as $(u_0, u_1, u_2, u_3....)$. Since $u_n$ is monotone increasing, the limit $u(x)$ exists. We have $u_{n+1}(x)=\LL u_{n}(x)$ and apply the monotone convergence theorem by taking $n\rightarrow \infty$, we have $u(x)=\LL u(x)$. We assume that $u'(x)$ satisfies $u'=\LL u'$ and majorizes $g_1(x)=u_0(x)$. Then $u'=\LL u'\geq \LL u_0=u_1$. Let us assume, for induction argument that $u'\geq u_n$, then \begin{align*}
u'=\LL u'\geq \LL u_n=u_{n+1}. \end{align*} Hence we have $u'\geq u_n$ for all $n$, leading to $u'\geq \lim_{n\rightarrow \infty}u_n=u$. Now we take the subsequence in $(w_0, y_1, w_2, y_3....)$ to complete the proof. \end{proof}
\begin{proposition} For each $x\in \R$, $\lim_{n\rightarrow \infty}w_n(x)=v_1(x)$ and $\lim_{n\rightarrow \infty}y_n(x)=v_0(x)$. Moreover, the optimal switching times, $\theta^*_i$ are exit times from an interval. \end{proposition} \begin{proof}
We can prove the first assertion by combining the first two lemmas above. Now we concentrate on the sequence of $w_n(x)$. For each $n$, finding $w_{n}(x)$ by solving (\ref{eq:w-function}) is an optimal stopping problem. By Proposition \ref{prop:A4}, the optimal stopping times are characterized as an exit time of $X$ from an interval for all $n$. This is also true in the limit: Indeed, by Lemma \ref{lem:fixed-point}, in the limit, the value function of optimal switching problem $v_1(x)=w(x)$ satisfies $w=\LL w$, implying that $v_1(x)$ is the solution of an optimal stopping problem. Hence the optimal switching times are characterized as exit time from an interval. \end{proof} \subsection{Characterization of the value functions}\label{subsec:value-function} We go back to the original problem (\ref{eq:problem}) to characterize the value function of the optimal switching problems.
By the exit time characterization of the optimal switching times, $\theta_i^*$ are given by \begin{align}\label{eq:optimal-switching-time} \theta_i^*=\begin{cases}\inf\{t>\theta_{i-1}; X_t^1\in \Gamma_1\}\\ \inf\{t>\theta_{i-1}; X_t^0\in \Gamma_0\} \end{cases} \end{align}
where $\Gamma_1=\R\setminus \mathbf{C}_1$ and $\Gamma_0=\R\setminus \mathbf{C}_0$. We define here $\mathbf{C}_i$ and $\Gamma_i$ to be continuation and stopping region for $X_t^i$, respectively. We can simplify the performance measure $J^w$ considerably. For $\zeta_0=1$, we have \begin{align*} J^w_1(x)&=\ME\left[\int_0^\infty e^{-\alpha s}f(X_s)ds-\sum_{j=1}^\infty e^{-\alpha \theta_j}H(X_{\theta_{j-}}, \zeta_j)\right]\\ &=\ME\Bigg[\int_0^{\theta_1}e^{-\alpha s}f(X_s)ds + \int_{\theta_1}^\infty e^{-\alpha s}f(X_s)ds\\ &\hspace{3cm}-e^{-\alpha\theta_1}\left(H(X_{\theta_1-}, 0)+\sum_{j=2}e^{-\alpha(\theta_i-\theta_1)}H(X_{\theta_{j-}}, \zeta_j)\right) \Bigg]\\ &=\ME\Bigg[\int_0^{\theta_1}e^{-\alpha s}f(X_s)ds+e^{-\alpha\theta_1}\E^{X_{\theta_1}}\left[\int_0^{\infty}e^{-\alpha s}f(X_s)ds -\sum_{j=1}e^{-\alpha\theta_j}H(X_{\theta_{j-}}, \zeta_j)\right]\\ &\hspace{3cm}-e^{-\alpha\theta_1}H(X_{\theta_1-}, 0)\Bigg] \end{align*} We notice that in the time interval $(0, \theta_1)$, the process $X$ is not intervened. The inner expectation is just $J_0^w(X_{\theta_1})$. Hence we further simplify \begin{align*} J_1^w(x)&=\ME\left[\int_0^{\theta_1}e^{-\alpha s}f(X_s)ds+e^{-\alpha\theta_1}(J_0^w(X_{\theta_1})-H(X_{\theta_1-}, 0))\right]\\ &=\ME\left[-e^{-\alpha\theta_1}g_1(X_{\theta_1})+e^{-\alpha\theta_1}(J_0^w(X_{\theta_1})-H(X_{\theta_1-}, 0))\right]+g_1(x)\\ &=\ME\left[-e^{-\alpha\theta_1}g_1(X_{\theta_1})+e^{-\alpha\theta_1}J_1^w(X_{\theta_1})\right]+g_1(x). \end{align*} The third equality is a critical observation. Finally, we define $u_1\triangleq J_1-g_1$ and obtain \begin{equation}\label{eq:u-1} u_1(x)=J^w_1(x)-g_1(x)=\ME\left[e^{-\alpha\theta_1}u_1(X_{\theta_1})\right]. \end{equation} Since the switching time $\theta_1$ is characterized as a hitting time of a certain point in the state space, we can represent $\theta_1=\tau_a\triangleq\inf\{t\geq 0: X_t=a\}$ for some $a\in \R$. Hence equation (\ref{eq:u-1}) is an optimal stopping problem that maximizes \begin{equation} u_1(x)=J^w_1(x)-g_1(x)=\ME\left[e^{-\alpha\tau_a}u_1(X_{\tau_a})\right]. \end{equation} among all the $\tau_a\in \mathcal{S}$. When $\theta_1=0$ (i.e., $x=X_{\theta_1}$), \begin{align*} J_1^w(x)&=\ME\left[-g_1(x)+J_0^w(x)-H(x, 0)\right]+g_1(x) \end{align*} and hence \begin{align*} u_1(x)&=J_0^w(x)-H(x, 0)-g_1(x). \end{align*} In other words, we make a switch from open to closed immediately by paying the switching cost. Similarly, for $\zeta_0=0$, we can simplify the performance measure $J_0^w(\cdot)$ to obtain \begin{equation*} J_0^w(x)=\ME\left[-e^{-\alpha\theta_1}g_0(X_{\theta_1})+e^{-\alpha\theta_1}J_0^w(X_{\theta_1})\right]+g_0(x). \end{equation*} By defining $u_0\triangleq J_0^w-g_0$, we have \begin{equation*} u_0(x)=J^w_0(x)-g_0(x)=\ME\left[e^{-\alpha\theta_1}u_0(X_{\theta_1})\right]. \end{equation*} Again, by using the characterization of switching times, we replace $\theta_1$ with $\tau_b$, \begin{equation}\label{eq:u-0} u_0(x)=J^w_0(x)-g_0(x)=\ME\left[e^{-\alpha\tau_b}u_0(X_{\tau_b})\right]. \end{equation} In summary, we have \begin{align}\label{eq:ost-1} u_1(x)&=\begin{cases} u_0(x)+g_0(x)-H(x, 0)-g_1(x), &x\in \Gamma_1,\\ \ME\left[e^{-\alpha\tau_a}u_1(X_{\tau_a})\right]=\ME\left[e^{-\alpha\tau_a}(u_0(X_{\tau_a})+g_0(X_{\tau_a})- g_1(X_{\tau_a})-H(X_{\tau_a}, 0))\right], &x\in \mathbf{C}_1, \end{cases} \end{align} and \begin{align} u_0(x)&=\begin{cases}\label{eq:ost-0} \ME\left[e^{-\alpha\tau_b}u_0(X_{\tau_b})\right]=\ME\left[e^{-\alpha\tau_b}(u_1(X_{\tau_b})+g_1(X_{\tau_b})-g_0(X_{\tau_b})-H(X_{\tau_b}, 1))\right], &x\in \mathbf{C}_0,\\ u_1(x)+g_1(x)-H(x, 1)-g_0(x), &x\in \Gamma_0. \end{cases} \end{align} Hence we should solve the following optimal stopping problems simultaneously: \begin{align}\label{eq:system} \begin{cases} \bar{v}_1(x)\triangleq\sup_{\tau\in\mathcal{S}}\ME\left[e^{-\alpha\tau}(u_1(X_{\tau})\right]\\ \bar{v}_0(x)\triangleq \sup_{\sigma\in\mathcal{S}}\ME\left[e^{-\alpha\sigma}(u_0(X_{\sigma})\right] \end{cases} \end{align}
Now we let the infinitesimal generators of $X^1$ and $X^0$ be $\A_1$ and $A_0$, respectively. We consider $(\A_i-\alpha)v(x)=0$ for $i=0, 1$. This ODE has two fundamental solutions, $\psi_i(\cdot)$ and $\varphi_i(\cdot)$. We set $\psi_i(\cdot)$ is an increasing and $\varphi_i(\cdot)$ is a decreasing function. Note that $\psi_i(c+)=0, \varphi_i(c+)=\infty$ and $\psi_i(d-)=\infty, \varphi_i(d-)=0$. We define \begin{equation*} F_i(x)\triangleq\frac{\psi_i(x)}{\varphi_i(x)}\quad \text{and}\quad G_i(x)\triangleq -\frac{\varphi_i(x)}{\psi_i(x)}\quad \text{for $i=0, 1$}. \end{equation*} \noindent By referring to Dayanik and Karatzas~\citeyearpar{DK2003}, we have the following representation \begin{equation*} \ME[e^{-\alpha\tau_r}1_{\{\tau_r<\tau_l\}}]=\frac{\psi(l)\varphi(x)-\psi(x)\varphi(l)} {\psi(l)\varphi(r)-\psi(r)\varphi(l)},\quad \ME[e^{-\alpha\tau_r}1_{\{\tau_l<\tau_r\}}]=\frac{\psi(x)\varphi(r)-\psi(r)\varphi(x)} {\psi(l)\varphi(r)-\psi(r)\varphi(l)}, \end{equation*} for $x\in[l,r]$ where $\tau_l\triangleq\inf\{t>0; X_t=l\}$ and $\tau_r\triangleq\inf\{t>0; X_t=r\}$.
By defining \begin{equation*}
W_1=(u_1/\psi_1)\circ G^{-1}_1 \quad \text{and}\quad W_0=(u_0/\varphi_0)\circ F^{-1}_0, \end{equation*}
the second equation in (\ref{eq:ost-1}) and the first equation in (\ref{eq:ost-0}) become \begin{align}\label{eq:W1} W_1(G_1(x))&=W_1(G_1(a))\frac{G_1(d)-G_1(x)}{G_1(d)-G_1(a)}+ W_1(G_1(d))\frac{G_1(x)-G_1(a)}{G_1(d)-G_1(a)} \quad x\in[a, d), \end{align} and \begin{align}\label{eq:W0} W_0(F_0(x))&=W_0(F_0(c))\frac{F_0(b)-F_0(x)}{F_0(b)-F_0(c)}+W_0(F_0(b))\frac{F_0(x)-F_0(c)}{F_0(b)-F_0(c)}, \quad x\in(c, b], \end{align} respectively. We should understand that $F_0(c)\triangleq F_0(c+)=\psi_0(c+)/\varphi_0(c+)=0$ and that $G_1(d)\triangleq G_1(d-)=-\varphi_1(d-)/\psi_1(d-)=0$. In the next subsection, we shall explain $W_1(G_1(d-))$ and $W_0(F_0(c+))$ in details. Both $W_1$ and $W_0$ are a \emph{linear function} in their respective transformed spaces. Hence under the appropriate transformations, the two value functions are linear functions in the continuation region.
\subsection{Direct Method for a Solution}\label{subsec:method} We have established a mathematical characterization of the value functions of optimal switching problems. We shall investigate, by using the characterization, a direct solution method that does not require the recursive optimal stopping schemes described in section \ref{subsec:recursive}. Since the two optimal stopping problems (\ref{eq:system}) have to be solved simultaneously, finding $u_0$ in $x\in \C_0$, for example, requires that we find the smallest $F_0$-concave majorant of $(u_1(x)+g_1(x)-g_0(x)-H(x, 1))/\varphi_0(x)$ as in (\ref{eq:ost-0}) that involves $u_1$.
There are two cases, depending on whether $x \in \mathbf{C}_1\cap \C_0$ or $x\in \Gamma_1\cap \C_0$, as to what $u_1(\cdot)$ represents. In the region $x\in \Gamma_1 \cap \C_0$, $u_1(\cdot)$ that shows up in the equation of $u_0(x)$ is of the form $u_1(x)=u_0(x)+g_0(x)-H(x, 1, 0)-g_1(x)$. In this case, the ``obstacle" that should be majorized is in the form \begin{align}\label{eq:on-(c, a)} &u_1(x)+g_1(x)-g_0(x)-H(x, 1)\nonumber\\ &=(u_0(x)+g_0(x)-H(x, 0)-g_1(x))+g_1(x)-g_0(x)-H(x, 1)\nonumber\\ &=u_0(x)-H(x, 0)-H(x, 1)<u_0(x). \end{align} This implies that in $x\in \Gamma_1 \cap \C_0$, the $u_0(x)$ function always majorizes the obstacle. Similarly, in $x\in \Gamma_0\cap \C_1$, the $u_1(x)$ function always majorizes the obstacle.
Next, we consider the region $x\in \mathbf{C}_0 \cap \C_1$.
The $u_0(\cdot)$ term in (\ref{eq:ost-1}) is represented, due to its linear characterization, as \begin{equation*} W_0(F_0(x))=\beta_0(F_0(x))+ d_0 \end{equation*} with some $\beta_0\in \R$ and $d_0 \in \R_+$ in the transformed space. (The nonnegativity of $d_0$ will be shown.) In the original space, it has the form of $\varphi_0(x)(\beta_0F_0(x)+d_0)$. Hence by the transformation $(u_1/\psi_1)\circ G^{-1}$, $W_1(G_1(x))$ is the smallest linear majorant of \begin{equation*} \frac{K_1(x)+\varphi_0(x)(\beta_0F_0(x)+d_0)}{\psi_1(x)}=\frac{K_1(x)+\beta_0\psi_0(x)+d_0\varphi_0(x)}{\psi_1(x)} \end{equation*} on $(G_1(d-), G_1(a^*))$ where \begin{equation} K_1(x)\triangleq g_0(x)-g_1(x)-H(x, 0). \end{equation}
This linear function passes a point $(G_1(d-), l_d)$ where $G_1(d-)=0$ and \begin{equation*} l_d=\limsup_{x\uparrow d}\frac{(K_1(x)+\beta_0\psi_0(x)+d_0\varphi_0(x))^+}{\psi_1(x)}. \end{equation*} Let us consider further the quantity $l_d \geq 0$. By noting \begin{align*} \limsup_{x\uparrow d}\frac{(K_1(x)+\beta_0\psi_0(x))^+}{\psi_1(x)} &\leq \limsup_{x\uparrow d}\frac{(K_1(x)+\beta_0\psi_0(x)+d_0\varphi_0(x))^+}{\psi_1(x)}\\ &\leq \limsup_{x\uparrow d}\frac{(K_1(x)+\beta_0\psi_0(x))^+}{\psi_1(x)}+\limsup_{x\uparrow d}\frac{d_0\varphi_0(x)}{\psi_1(x)} \end{align*} and $\limsup_{x\uparrow d}\frac{d_0\varphi_0(x)}{\psi_1(x)}=0$, we can redefine $l_d$ by \begin{align}\label{eq:ld} l_d\triangleq \limsup_{x\uparrow d}\frac{(K_1(x)+\beta_0\psi_0(x))^+}{\psi_1(x)} \end{align} to determine the finiteness of the value function of the optimal switching problem, $v_1(x)$, based upon Proposition \ref{prop:A5}-\ref{prop:A7}.
Let us concentrate on the case $l_d=0$.
Similar analysis applies to (\ref{eq:ost-0}). $u_1(x)$ in (\ref{eq:ost-0}) is represented as \begin{equation*} W_1(G_1(x))=\beta_1G_1(x)+d_1 \end{equation*} with some $\beta_1\in \R$ and $d_1\in\R_+$. Note that $d_1=l_d\geq 0$. In the original space, it has the form of $\psi_1(x)(\beta_1G_1(x)+d_1)$. Hence by the transformation $(u_0/\varphi_0(x))\circ F^{-1}$, $W_0(F_0(x))$ is the smallest linear majorant of \begin{equation*} \frac{K_0(x)+\psi_1(x)(\beta_1G_1(x)+d_1)}{\varphi_0(x)}=\frac{K_0(x)-\beta_1\varphi_1(x)+d_1\psi_1(x)}{\varphi_0(x)} \end{equation*} on $(F_0(c+), F_0(b^*))$ where \begin{equation} K_0(x)\triangleq g_1(x)-g_0(x)-H(x, 1). \end{equation} This linear function passes a point $(F_0(c+), l_c)$ where $F_0(c+)=0$ and \begin{equation*} l_c=\limsup_{x\downarrow c}\frac{(K_0(x)-\beta_1\varphi_1(x)+d_1\psi_1(x))^+}{\varphi_0(x)}. \end{equation*} Hence we have $l_c=d_0\geq 0$. By the same argument as for $l_d$, we can redefine \begin{align}\label{eq:lc} l_c\triangleq \limsup_{x\downarrow c}\frac{(K_0(x)-\beta_1\varphi_1(x))^+}{\varphi_0(x)}. \end{align}
\begin{remark}\normalfont \begin{itemize}\label{rem:lc-ld} \item[(a)] Evaluation of $l_d$ or $l_c$ does not require knowledge of $\beta_0$ or $\beta_1$, respectively unless the orders of $\max(K_1(x), \psi_1(x))$ and $\psi_0(x)$ are equal, for example. (For this event, see Proposition \ref{prop:odd-case}.) Otherwise, we just compare the order of the positive leading terms of the numerator in (\ref{eq:ld}) and (\ref{eq:lc}) with that of the denominator.
\item[(b)] \emph{A sufficient condition for $l_d=l_c=0$}: since we have \begin{equation*} 0\leq l_d\leq \limsup_{x\uparrow d}\frac{(K_1(x))^+}{\psi_1(x)}+\limsup_{x\uparrow d}\frac{(\beta_0\psi_0(x))^+}{\psi_1(x)}. \end{equation*} a sufficient condition for $l_d=0$ is \begin{equation}\label{eq:sufficent-d} \limsup_{x\uparrow d}\frac{(K_1(x))^+}{\psi_1(x)}=0 \quad \text{and}\quad \limsup_{x\uparrow d}\frac{\psi_0(x)}{\psi_1(x)}=0. \end{equation}
Similarly, \begin{equation*} 0\leq l_c\leq \limsup_{x\downarrow c}\frac{(K_0(x))^+}{\varphi_0(x)}+\limsup_{x\downarrow c}\frac{(-\beta_1\varphi_1(x))^+}{\varphi_0(x)} \end{equation*} Hence a sufficient condition for $l_c=0$ is \begin{equation}\label{eq:sufficient-c} \limsup_{x\downarrow c}\frac{(K_0(x))^+}{\varphi_0(x)}=0\quad\text {and}\quad \limsup_{x\downarrow c}\frac{\varphi_1(x)}{\varphi_0(x)}=0. \end{equation} Moreover, it is obvious $\beta_1<0$ and $\beta_0>0$ since the linear majorant passes the origin of each transformed space. Recall a points in the interval $(c, d)\in\R_+$ will be transformed by $G(\cdot)$ to $(G(c), G(d-))\in \R_-$. \end{itemize} \end{remark} \noindent We summarize the case of $l_c=l_d=0$: \begin{proposition}\label{prop:2} Suppose that $l_d=l_c=0$, the quantities being defined by (\ref{eq:ld}) and by (\ref{eq:lc}), respectively. The value functions in the transformed space are the smallest linear majorants of \begin{equation*} R_1(\cdot)\triangleq \frac{r_1(G_1^{-1}(\cdot))}{\psi_1(G_1^{-1}(\cdot))}\quad\text{and}\quad R_0(\cdot)\triangleq \frac{r_0(F_0^{-1}(\cdot))}{\varphi_0(F_0^{-1}(\cdot))} \end{equation*} where \begin{equation*} r_1(x)\triangleq g_0(x)-g_1(x)+\beta_0\psi_0(x)-H(x, 0) \end{equation*} and \begin{equation*} r_0(x)\triangleq g_1(x)-g_0(x)-\beta_1\varphi_1(x)-H(x, 1) \end{equation*} for \begin{equation} \beta_0>0\quad\text{and}\quad \beta_1<0. \end{equation}
Furthermore, $\Gamma_1$ and $\Gamma_0$ in (\ref{eq:ost-1}) and (\ref{eq:ost-0}) are given by \begin{equation*} \Gamma_1\triangleq \{x\in (c, d): W_1(G_1(x))=R_1(G_1(x))\},
\quad\text{and}\quad
\Gamma_0\triangleq \{x\in (c, d): W_0(F_0(x))=R_0(F_0(x))\}.
\end{equation*} \end{proposition}
\begin{coro}\normalfont If either of the boundary points $c$ or $d$ is \emph{absorbing}, then $(F_0(c), W_0(F_0(c))$ or $(G_1(d), W_1(G_1(d)))$ is obtained directly. We can entirely omit the analysis of $l_c$ or $l_d$. The characterization of the value function (\ref{eq:W1}) and (\ref{eq:W0}) remains exactly the same. \end{coro}
\begin{remark}\normalfont An algorithm to find $(a^*, b^*, \beta_0^*, \beta_1^*)$ can be described as follows: \begin{enumerate} \item Start with some $\beta_1'\in \R$. \item Calculate $r_0$ and then $R_0$ by the transformation $R_0(\cdot)= \frac{r_0(F_0^{-1}(\cdot))}{\varphi_0(F_0^{-1}(\cdot))}$. \item Find the linear majorant of $R_0$ passing the origin of the transformed space. Call the slope of the linear majorant, $\beta_0$ and the point, $F_0(b)$, where $R_0$ and the linear majorant meet . \item Plug $b$ and $\beta_0$ in the equation for $r_1$ and calculate $R_1$ by the transformation $R_1(\cdot)= \frac{r_1(G_1^{-1}(\cdot))}{\psi_1(G_1^{-1}(\cdot))}$. \item Find the linear majorant of $R_1$ passing the origin of the transformed space. Call the slope of the linear majorant, $\beta_1$ and the point, $G_1(a)$, where $R_1$ and the linear majorant meet. \item Iterate step 1 to 5 until $\beta_1=\beta_1'$. \end{enumerate} If both $R_1$ and $R_0$ are differentiable functions with their respective arguments, we can find $(a^*, b^*)$ analytically. Namely, we solve the following system for $a$ and $b$: \begin{eqnarray}\label{eq:differential-system} \begin{cases}
\left.\frac{dR_0(y)}{dy}\right|_{y=F_0(b)}(F_0(b)-F_0(c))=R_0(F_0(b))\\
\left.\frac{dR_1(y)}{dy}\right|_{y=G_1(a)}(G_1(a)-G_1(d))=R_1(G_1(a)) \end{cases} \end{eqnarray}
where $\left.\frac{dR_0(y)}{dy}\right|_{y=F_0(b^*)}=\beta_0^*$ and
$\left.\frac{dR_1(y)}{dy}\right|_{y=G_1(a^*)}=\beta_1^*$. \end{remark}
Once we find $W_1(\cdot)$ and $W_0(\cdot)$, then we convert to the original space and add back $g_1(x)$ and $g_0(x)$ respectively so that $v_1(x)=\psi_1(x)W_1(G_1(x))+g_1(x)$ and $v_0(x)=\varphi_0(x)W_0(F_0(x))+g_0(x)$. Therefore, by (\ref{eq:ost-1}) and (\ref{eq:ost-0}), the value functions $v_1(\cdot)$ and $v_0(\cdot)$ are given by: \begin{proposition}\label{prop:a<b} If the optimal continuation regions for both of the value functions are connected and if $l_c=l_d=0$, then the pair of the value functions $v_1(x)$ and $v_0(x)$ are represented as \begin{align*} v_1(x)=\begin{cases} \hat{v}_0(x)-H(x, 0), &x\leq a^*,\\ \hat{v}_1(x)\triangleq \psi_1(x)W_1(G_1(x))+g_1(x), &a^*<x, \end{cases} \end{align*} \noindent and \begin{align*} v_0(x)=\begin{cases} \hat{v}_0(x)\triangleq \varphi_0(x)W_0(F_0(x))+g_0(x) &x <b^*,\\ \hat{v}_1(x)-H(x, 1), &b^*\leq x, \end{cases} \end{align*} for some $a^*, b^*\in \R$ with $a^*<b^*$. \end{proposition} \begin{proof} If the optimal continuation regions for both of the value functions are connected and if $l_d=l_c=0$, then the optimal intervention times (\ref{eq:optimal-switching-time}) have the following form: \begin{align}\label{eq:optimal-switching-time} \theta_i^*=\begin{cases}\inf\{t>\theta_{i-1}; X_t\notin (a^*, d)\}, &Z=1,\\ \inf\{t>\theta_{i-1}; X_t\notin (c, \hspace{0.2cm} b^*)\}, &Z=0. \end{cases} \end{align} Indeed, since we have $l_c=l_d=0$, the linear majorants $W_1(\cdot)$ and $W_0(\cdot)$ pass the origins in their respective transformed coordinates. Hence the continuation regions shall necessarily of the form of (\ref{eq:optimal-switching-time}).
By our construction, both $v_1(x)$ and $v_0(x)$ are continuous in $x\in \R$. Suppose we have $a^*> b^*$. In this case, by the form of the value functions, $v_0(b-)-H(b, 1, 0)=v_1(b)$. Since the cost function $H(\cdot)>0 $ and continuous, it follows $v_0(b-)>v_1(b)$. On the other hand, $v_0(b+)=v_1(b)-H(b, 0, 1)$ implying $v_0(b+)<v_1(b)$. This contradicts the continuity of $v_0(x)$. Also, $a^*=b^*$ will lead to $v_1(x)=v_1(x)-H(x, 1, 0)$ which is impossible. Hence if the value functions exist, then we must necessarily have $a^*<b^*$. \end{proof}
In relation to Proposition \ref{prop:a<b}, we have the following observations: \begin{remark}\normalfont \begin{itemize} \item[(a)] It is obvious that \begin{equation*} v_0(x)=\hat{v}_0(x)>\hat{v}_0(x)-H(x, 0)=v_1(x), \quad x\in (c, a^*), \end{equation*} and \begin{equation*} v_1(x)=\hat{v}_1(x)>\hat{v}_1(x)-H(x, 1)=v_0(x), \quad x\in (b^*, d). \end{equation*} \item[(b)] Since $u_1(x)$ is continuous in $(c, d)$, the ``obstacle" $u_1(x)+g_1(x)-g_0(x)-H(x, 1)$ to be majorized by $u_0(x)$ on $x\in \C_0=(c, b^*)$ is also continuous, in particular at $x=a^*$. We proved that $u_0(x)$ always majorizes the obstacle on $(c, a^*)$. Hence $F(a^*) \in \{y: W_0(y)>R_0(y)\}$ if there exists a linear majorant of $R_0(y)$ in an interval of the form $(F_0(q), F_0(d))$ with some $q\in (c, d)$: otherwise, the continuity of $u_1(x)+g_1(x)-g_0(x)-H(x, 1)$ does not hold. Similarly, we have $F(b^*) \in \{y:W_1(y)>R_1(y)\}$ if there exists a linear majorant of $R_0(y)$ in an interval of the form $(G_1(c), G_1(q))$. \end{itemize} \end{remark}
\noindent Finally, we summarize other cases than $l_c=l_d=0$: \begin{proposition}\label{prop:odd-case} \item[(a)] If either $l_d=+\infty$ or $l_c=+\infty$, then $v_1(x)=v_0(x)\equiv +\infty$. \item[(b)] If both $l_d$ and $l_c$ are finite, then $l_d=l_c=0$. \end{proposition} \begin{proof} (a) The proof is immediate by invoking Proposition \ref{prop:A5}. (b) When $l_c$ is finite, we know by Proposition \ref{prop:A5} that the value function $v_0(x)$ is finite. On $x\in (c, a^*)$, $u_1(x)+g_1(x)-g_0(x)-H(x, 1)<u_0(x)<+\infty$ is finite (see (\ref{eq:on-(c, a)})) and thereby \begin{equation*} l_c=\limsup_{x\downarrow c}\frac{u_1(x)+g_1(x)-g_0(x)-H(x, 1)}{\varphi_0(x)}=0. \end{equation*} The same argument for $l_d=0$. \end{proof} \noindent Therefore, we can conclude that $l_d=0$ for the situation where the orders of $\max(K_1(x), \psi_1(x))$ and $\psi_0(x)$ are equal ($\Rightarrow l_d$ is finite) as described in Remark \ref{rem:lc-ld} (a). \section{Examples}\label{sec:example} We recall some useful observations. If $h(\cdot)$ is twice-differentiable at $x\in \mathcal{I}$ and $y\triangleq F(x)$, then we define $H(y)\triangleq h(F^{-1}(y))/\varphi(F^{-1}(y))$ and we obtain $H^{'}(y)=m(x)$ and $H^{''}(y)=m^{'}(x)/F^{'}(x)$ with \begin{equation}\label{eq:devH} m(x)= \frac{1}{F^{'}(x)}\left(\frac{h}{\varphi}\right)^{'}(x), \quad \text{and} \quad H^{''}(y) (\mathcal{A}-\alpha)h(x)\geq 0, \quad y=F(x) \end{equation} with strict inequality if $H^{''}(y)\neq 0$. These identities are of practical use in identifying the concavities of $H(\cdot)$ when it is hard to calculate its derivatives explicitly. Using these representations, we can modify (\ref{eq:differential-system}) to \begin{eqnarray}\label{eq:modified-system} \begin{cases} \frac{1}{F_0'(b)}\left(\frac{r_0}{\varphi_0}\right)'(b)(F_0(b)-F_0(c))=\frac{r_0(b)}{\varphi_0(b)}\\ \frac{1}{G_1'(a)}\left(\frac{r_1}{\psi_1}\right)'(a)(G_1(a)-G_1(d))=\frac{r_1(a)}{\psi_1(a)} \end{cases} \end{eqnarray} \begin{example}\normalfont \textbf{Brekke and {\O}ksendal \citeyearpar{BO1994}}:\normalfont \hspace{0.2cm} We first illustrate our solution method by using a resource extraction problem solved by Brekke and {\O}ksendal~\citeyearpar{BO1994}. The price $P_t$ at time $t$ per unit of the resource follows a geometric Brownian motion. $Q_t$ denotes the stock of remaining resources in the field that decays exponentially. Hence we have \begin{equation*} dP_t=\alpha P_tdt + \beta P_tdW_t\quad\text{and}\quad dQ_t=-\lambda Q_tdt \end{equation*} where $\alpha, \beta,$ and $\lambda>0$ (extraction rate) are constants. The objective of the problem is to find the optimal switching times of resource extraction: \begin{equation*} v(x)=\sup_{w\in W}J^w(x)=\sup_{w\in W}\ME\left[\int_0^\infty e^{-\rho t}(\lambda P_tQ_t-K)Z_tdt - \sum_{i}e^{-\rho\theta_i}H(X_{\theta_i-}, Z_{\theta_i})\right] \end{equation*} where $rho\in\R_+$ is a discount factor with $\rho>\alpha$, $K\in\R_+$ is the operating cost and $H(x, 0)=C\in\R_+$ and $H(x, 1)=L\in\R_+$ are constant closing and opening costs. Since $P$ and $Q$ always show up in the form of $PQ$, we reduce the dimension by defining $X_t=P_tQ_t$ with the dynamics: \begin{equation*} dX_t=(\alpha - \lambda Z_t)X_tdt + \beta X_tdW_t. \end{equation*} \textbf{Solution}: (1) We shall calculate all the necessary functions. For $Z_t=1$ (open state), we solve $(\A_1-\rho)v(x)=0$ where $\A_1=(\alpha-\lambda)xv'(x)+\frac{1}{2}\beta^2 x^2v''(x)$ to obtain $\psi_1(x)=x^{\nu_+}$ and $\varphi_1(x)=x^{\nu_-}$ where $\nu_{+, -}=\beta^{-2}\left(-\alpha+\lambda + \frac{1}{2}\beta^2\pm\sqrt{(\alpha-\lambda-\frac{1}{2}\beta^2)^2+2\rho\beta^2}\right)$. Similarly, for $Z_t=0$ (closed state), we solve $(\A_0-\rho)v(x)=0$ where $\A_0=\alpha xv'(x)+\frac{1}{2}\beta^2 x^2 v''(x)$ to obtain $\psi_0(x)=x^{\mu_+}$ and $\varphi_0(x)=x^{\mu_-}$ where $\mu_{+, -}=\beta^{-2}\left(-\alpha + \frac{1}{2}\beta^2\pm\sqrt{(\alpha-\frac{1}{2}\beta^2)^2+2\rho\beta^2}\right)$. Note that under the assumption $\rho>\alpha$, we have $\nu_+, \mu_+>1$ and $\nu_-, \nu_- <0$.
By setting $\Delta_1=\sqrt{(\alpha-\lambda-\frac{1}{2}\beta^2)^2+2\rho\beta^2}$ and $\Delta_0=\sqrt{(\alpha-\frac{1}{2}\beta^2)^2+2\rho\beta^2}$, we have $G_1(x)=-\varphi_1(x)/\psi_1(x)=-x^{-2\Delta_1/\beta^2}$ and $F_0(x)=\psi_0(x)/\varphi_0(x)=x^{2\Delta_0/\beta^2}$. It follows that $G_1^{-1}(y)=(-y)^{-\beta^2/2\Delta_1}$ and $F_0^{-1}(y)=y^{\beta^2/2\Delta_0}$. In this problem, we can calculate $g_1(x), g_0(x)$ explicitly: \begin{equation*} g_1(x)=\ME\left[\int_0^\infty e^{-\rho s}(\lambda X_s -K)ds\right]=\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho} \end{equation*} and $g(x)=0$. Lastly, $K_1(x)=g_0(x)-g_1(x)-H(x, 0)=-\left(\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho}\right)-C$ and $K_0(x)=g_1(x)-g_0(x)-H(x, 1)=\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho}-L$.\\
\noindent (2) The state space of $X$ is $(c, d)=(0, \infty)$ and we evaluate $l_c$ and $l_d$. Let us first note that $\Delta_0-\Delta_1+\lambda>0$. Since $\lim_{x\downarrow 0}\frac{\varphi_1(x)}{\varphi_0(x)}=\lim_{x\downarrow 0}x^{\frac{\Delta_0-\Delta_1+\lambda}{\beta^2}}=0$ and $\lim_{x\downarrow 0}(K_0(x))^+/\varphi_0(x)=0$, we have $l_c=l_0=0$ by (\ref{eq:sufficient-c}). Similarly, by noting $\lim_{x\uparrow +\infty}\frac{\psi_0(x)}{\psi_1(x)}=\lim_{x\uparrow +\infty}x^{\frac{-(\Delta_0-\Delta_1+\lambda)}{\beta^2}}=0$ and $\lim_{x\uparrow +\infty}(K_1(x))^+/\varphi_0(x)=0$, we have $l_d=l_{+\infty}=0$ by (\ref{eq:sufficent-d}).\\
\noindent (3) To find the value functions together with continuation regions, we set \begin{align*} r_1(x)=-\left(\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho}\right)-C+\beta_0\psi_0(x)\quad\text{and}\quad r_0(x)=\left(\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho}\right)-L-\beta_1\varphi_1(x) \end{align*} and make transformations $R_1(y)=r_1(F^{-1}(y))/\psi_1(F^{-1}(y))$ and $R_0(y)=r_0(F^{-1}(y))/\varphi_0(F^{-1}(y))$, respectively. We examine the shape and behavior of the two functions $R_1(\cdot)$ and $R_0(\cdot)$ with an aid of (\ref{eq:devH}). By calculating $(r_0/\varphi_0)'(x)$ explicitly to examine the derivative of $R_0(y)$, we can find a critical point $x=q$, at which $R_0(F(x))$ attains a local minimum and from which $R_0(F(x))$ is increasing monotonically on $(F_0(q), \infty)$. Moreover, we can confirm that $\lim_{y\rightarrow \infty}R_0'(y)=\lim_{x\rightarrow \infty}\frac{(r_0/\varphi_0)'(x)}{F_0'(x)}=0$, which shows that there exists a finite linear majorant of $R_0(y)$. We define \begin{equation*} p(x)=\beta_1 \omega x^{\nu_-}-(\rho-\alpha)\left(\frac{x}{\rho+\lambda-\alpha}\right)+(K+\rho L) \end{equation*} such that $(\A_0-\rho)r_0(x)=p(x)$ where $\omega\triangleq \left(\rho-\frac{1}{2}\beta^2\nu_-(\nu_{-}-1)-\alpha\nu_-\right) =\frac{1}{2\beta^2}(\Delta_0-\Delta_1+\lambda)(\Delta_0+\Delta_1-\lambda)>0$. By the second identity in (\ref{eq:devH}), the sign of the second derivative $R_0''(y)$ is the same as the sign of $p(x)$. It is easy to see that $p(x)$ has only one critical point. For any $\beta_1<0$, the first term is dominant as $x\rightarrow 0$, so that
$\lim_{x\downarrow 0} p(x)<0$. As $x$ gets larger, for $|\beta_1|$ sufficiently small, $p(x)$ can take positive values, providing two positive roots, say $x=k_1, k_2$ with $k_1<k_2$. We also have $\lim_{x\rightarrow +\infty}p(x)=-\infty$. In this case, $R_0(y)$ is concave on $(0, F(k_1)\cup (F(k_2), +\infty)$ and convex on $(F(k_1), F(k_2))$. Since we know that $R_0(y)$ attains a local minimum at $y=F(q)$, we have $q<k_2$, and it implies that there is one and only on tangency point of the linear majorant $W(y)$ and $R_0(y)$ on $(F(q), \infty)$, so that the continuation region is of the form $(0, b^*)$.
>From this analysis of the derivatives of $R_0(y)$, there is only one tangency point of the linear majorant $W_0(y)$ and $R_0(y)$. (See Figure \ref{fig:1}-(a)). A similar analysis shows that there is only one tangency point of the linear majorant $W_1(y)$ and $R_1(y)$. (See Figure \ref{fig:1}-(b)). \begin{figure}
\caption{\small A numerical example of resource extraction problem. with parameters $(\alpha, \beta, \lambda, \rho, K, L, C)=(0.01, 0.25, 0.01, 0.05, 0.4, 2, 2)$(a) The smallest linear majorant $W_0(F_0(x))$ and $R_0(F_0(x))$ with $b^*=1.15042$ and $\beta_0^*=10.8125$. (b)The smallest linear majorant $W_1(G_1(x))$ and $R_1(G_1(x))$ with $a^*=0.18300$ and $\beta_1^*=-0.695324$. (c) The value function $v_0(x)$. (d) The value function $v_1(x)$.}
\label{fig:1}
\end{figure}\\ \noindent (4) By solving the system of equations (\ref{eq:differential-system}), we can find $(a^*, b^*, \beta_0^*, \beta_1^*)$. We transform back to the original space to find \begin{align*} \hat{v}_1(x)&=\psi_1(x)W_1(G_1(x))+g_1(x)=\psi_1(x)\beta_1^*G_1(x)+g_1(x)\\ &=-\beta_1^*\varphi_1(x)+g_1(x)=-\beta_1^* x^{\nu_-}+\left(\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho}\right), \end{align*} and \begin{align*} \hat{v}_0(x)&=\varphi_0(x)W_0(F_0(x))+g_0(x)=\varphi_0(x)\beta_0^*F_0(x)+g_0(x)=\beta_0^*\psi_0(x)+g_0(x)=\beta_0^*x^{\mu_{+}}. \end{align*} Hence the solution is \begin{align} \nonumber \begin{aligned}
v_1(x) &= \begin{cases}
\beta_0^*x^{\mu_+}-C, & x\leq a^*, \\
-\beta_1^* x^{\nu_-}+\left(\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho}\right),
&x>a^*,\end{cases} \hspace{0.4cm}
v_0(x) &= \begin{cases}
\beta_0^*x^{\mu_{+}}, & x\leq b^*, \\
-\beta_1^* x^{\nu_-}+\left(\frac{x}{\rho+\lambda-\alpha}-\frac{K}{\rho}\right)-L,
&x>b^*,
\end{cases} \end{aligned} \end{align} which agrees with Brekke and {\O}kesendal \citeyearpar{BO1994}. \end{example} \begin{example}\normalfont \textbf{Ornstein-Uhrenbeck process}: \hspace{0.2cm} We shall consider a new problem involving an Ornstein-Uhrenbeck process. Consider a firm whose revenue solely depends on the price of one product. Due to its cyclical nature of the prices, the firm does not want to have a large production facilty and decides to rent additional production facility when the price is favorable. The revenue process to the firm is \begin{equation*} dX_t=\delta(m-X_t-\lambda Z_t)dt+\sigma dW_t, \end{equation*} where $\lambda=r/\delta$ with $r$ being a rent per unit of time. The firm's objective is to maximize the \emph{incremental} revenue generated by renting the facility until the time $\tau_0$ when the price is at an intolerably low level. Without loss of generality, we set $\tau_0=\inf\{t>0: X_t=0\}$. We keep assuming constant operating cost $K$, opening cost, $L$ and closing cost $C$. Now the value function is defined as \begin{equation*} v(x)=\sup_{w\in W}J^w(x)=\sup_{w\in W}\ME\left[\int_0^{\tau_0} e^{-\alpha t}(X_t-K)Z_tdt - \sum_{\theta_i<\tau_0}e^{-\alpha\theta_i}H(X_{\theta_i-}, Z_{\theta_i})\right]. \end{equation*} \textbf{Solution}: (1) We denote, by $\tilde{\psi}(\cdot)$ and $\tilde{\varphi}(\cdot)$, the functions of the fundamental solutions for the auxiliary process $P_t\triangleq(X_t-m+\lambda)/\sigma, t\geq 0$, which satisfies $dP_t=-\delta P_t dt+dW_t$. For every $x\in \mathbb{R}$, \begin{equation*} \tilde{\psi}(x)=e^{\delta x^2/2}\mathcal{D}_{-\alpha/\delta}(-x\sqrt{2\delta})\quad\text{and}\quad \tilde{\varphi}(x)=e^{\delta x^2/2}\mathcal{D}_{-\alpha/\delta}(x\sqrt{2\delta}), \end{equation*} which leads to $\psi_1(x)=\tilde{\psi}((x-m+\lambda)/\sigma)$, $\varphi_1(x)=\tilde{\varphi}((x-m+\lambda)/\sigma)$, $\psi_0(x)=\tilde{\psi}((x-m)/\sigma)$, and $\varphi_0(x)=\tilde{\varphi}((x-m)/\sigma)$ where $\mathcal{D}_\nu(\cdot)$ is the parabolic cylinder function; (see Borodin and Salminen (2002, Appendices 1.24 and 2.9) and Carmona and Dayanik (2003, Section 6.3)). By using the relation \begin{equation} \label{eq:Dft} \mathcal{D}_\nu(z)=2^{-\nu/2}e^{-z^2/4}\mathcal{H}_\nu(z/\sqrt{2}), \quad z\in\mathbb{R} \end{equation} in terms of the Hermite function $\mathcal{H}_\nu$ of degree $\nu$ and its integral representation \begin{equation}\label{eq:Hermite} \mathcal{H}_\nu(z)=\frac{1}{\Gamma(-\nu)}\int_0^\infty e^{-t^2-2tz}t^{-\nu-1}dt, \quad \text{Re}(\nu)<0, \end{equation} (see for example, Lebedev(1972, pp 284, 290)). Since $\ME[X_t]=e^{-\delta t}x+(1-e^{-\delta t})(m-\lambda)$, we have $g_0(x)=0$ and $g_1(x)=\frac{x-(m-\lambda)}{\delta+\alpha}+\frac{m-\lambda-K}{\alpha}$.\\
\noindent(2) The state space of $X$ is $(c, d)=(0, +\infty)$. Since the left boundary $0$ is the absorbing, the linear majorant passes $(0, F_0(0))$. Since $\lim_{x\rightarrow +\infty}\psi_0(x)/\psi_1(x)=0$, we have $l_d=0$.\\
\noindent (3) We formulate \begin{equation*} r_1(x)=-\left(\frac{x-(m-\lambda)}{\delta+\alpha}+\frac{m-\lambda-K}{\alpha}\right)-C+\beta_0\psi_0(x) \end{equation*} and \begin{equation*} r_0(x)=\left(\frac{x-(m-\lambda)}{\delta+\alpha}+\frac{m-\lambda-K}{\alpha}\right)-L-\beta_1\varphi_1(x) \end{equation*} and make transformations: $R_1(y)=r_1(F^{-1}(y))/\psi_1(F^{-1}(y))$ and $R_0(y)=r_0(F^{-1}(y))/\varphi_0(F^{-1}(y))$, respectively. We examine the shape and behavior of the two functions $R_1(\cdot)$ and $R_0(\cdot)$ with an aid of (\ref{eq:devH}). First we check the sign of $R_0'(y)$ and find a critical point $x=q$, at which $R_0(F(x))$ attains a local minimum and from which $R_0(F(x))$ is increasing monotonically on $(F_0(q), \infty)$. It can be shown that $R_0^{'}(+\infty)=0$ by using (\ref{eq:Dft}) and (\ref{eq:Hermite}) and the identity $\mathcal{H}'_\nu(z)=2\nu\mathcal{H}_{\nu-1}(z), z\in\mathbb{R}$ (see Lebedev (1972, p.289), for example.) This shows that there must exist a (finite) linear majorant of $R_0(y)$ on $(F(q), \infty)$. To check convexity of $R_0(y)$, we define \begin{equation*} p(x)=-\frac{\sigma^2\beta_1}{2}\varphi_1''(x)+\delta(m-x-\lambda)\left(\frac{1}{\delta+\alpha}-\beta_1\varphi_1'(x)\right) -\alpha r_0(x) \end{equation*} such that $(\A_0-\alpha)r_0(x)=p(x)$. We can show easily $\lim_{x\rightarrow +\infty}p(x)=-\infty$ since $\varphi_1(+\infty)=\varphi_1'(+\infty)=\varphi_1''(+\infty)=0$. Due to the monotonicity of $\varphi_1(x)$ and its derivatives, $p(x)$ can have at most one critical point and $p(x)=0$ can have one or two positive roots depending on the value of $\beta_1$. In either case, let us call the largest positive root $x=k_2$. We also have $\lim_{x\rightarrow +\infty}p(x)=-\infty$.
Since we know that $R_0(y)$ attains a local minimum at $y=F(q)$ and is increasing thereafter, we have $q<k_2$. It follows that there is one and only on tangency point of the linear majorant $W(y)$ and $R_0(y)$ on $(F(q), \infty)$, so that the continuation region is of the form $(0, b^*)$. A similar analysis shows that there is only one tangency point of the linear majorant $W_1(y)$ and $R_1(y)$.\\
\noindent (4) Solving (\ref{eq:modified-system}), we we can find $(a^*, b^*, \beta_0^*, \beta_1^*)$. We transform back to the original space to find \begin{align*} \hat{v}_1(x)&=\psi_1(x)W_1(G_1(x))+g_1(x)=\psi_1(x)\beta_1^*G_1(x)+g_1(x)=-\beta_1^*\varphi_1(x)+g_1(x)\\ &=-\beta_1^* e^{\frac{\delta(x-m+\lambda)^2}{2\sigma^2}}\mathcal{D}_{-\alpha/\delta}\left(\frac{(x-m+\lambda)\sqrt{2\delta}}{\sigma}\right) +\frac{x-(m-\lambda)}{\delta+\alpha}+\frac{m-\lambda}{\alpha} \end{align*} and \begin{align*} \hat{v}_0(x)&=\varphi_0(x)W_0(F_0(x))+g_0(x)=\varphi_0(x)\beta_0^*(F_0(x)-F_0(0))+g_0(x)\\ &=\beta_0^*\{\psi_0(x)-F_0(0)\varphi_0(x)\}+g_0(x)\\ &=\beta_0^*e^{\frac{\delta}{2}\frac{(x-m+\lambda)^2}{\sigma^2}}\left\{\mathcal{D}_{-\alpha/\delta}\left(-\left(\frac{x-m+\lambda}{\sigma}\right)\sqrt{2\delta}\right) -F(0)\mathcal{D}_{-\alpha/\delta}\left(\left(\frac{x-m}{\sigma}\right)\sqrt{2\delta}\right)\right\}. \end{align*} Hence the solution is, using the above functions, \begin{align} \nonumber \begin{aligned}
v_1(x) &= \begin{cases}
\hat{v}_0(x)-C, & x\leq a^*, \\
\hat{v}_1(x),
&x>a^*,\end{cases} \hspace{0.4cm}
v_0(x) &= \begin{cases}
\hat{v}_0(x), & x\leq b^*, \\
\hat{v}_1(x)-L,
&x>b^*.
\end{cases} \end{aligned} \end{align} See Figure \ref{fig:2} for a numerical example. \begin{figure}
\caption{\small A numerical example of leasing production facility problem with parameters $(m, \alpha, \sigma, \delta, \lambda, K, L, C)=(5, 0.105, 0.35, 0.05, 4, 0.4, 0.2, 0.2)$: (a) The value function $v_0(x)$ with $b^*=1.66182$ and $\beta_0^*=144.313$. (b)The value function $v_1(x)$ with $a^*=0.781797$ and $\beta_1^*=-2.16941$.}
\label{fig:2}
\end{figure} \end{example} \section{Extensions and conclusions}\label{sec:last-section} \subsection{An extension to the case of $k\geq 2$}
It is not difficult to extend to a general case of $k\geq 2$ where more than one switching opportunities are available. But we put a condition that $z\in\mathcal{Z}$ is of the form $z=(a_1, a_2,...., a_k)$ where only one element of this vector is $1$ with the rest being zero, i.e., $z=(0, 0, 0,...., 1, 0, 0)$ for example.
We should introduce the switching operator $\mathcal{M}_0$ on $h\in \mathcal{H}$, \begin{equation}\label{eq:switch-operator} \mathcal{M}_0h(u,z)=\max_{\zeta\in \mathcal{Z}\setminus \{z\}}\left\{h(u, \zeta)-H(u, z; \zeta)\right\}. \end{equation} In words, this operator would calculate which production mode should be chosen by moving from the current production mode $z$. Now the recursive optimal stopping (\ref{eq:w-function}) becomes \begin{equation*} w_{n+1}(x)\triangleq\sup_{\tau\in \mathcal{S}}\ME\left[\int_0^\tau e^{-\alpha s}f(X_s)ds + e^{-\alpha\tau} \mathcal{M}w_{n}(X_{\tau})\right]. \end{equation*} Accordingly, the optimization procedure will become two-stage. To illustrate this, we suppose $k=2$ so that $i=0, 1,$ and $2$. By eliminating the integral in (\ref{eq:switch-operator}), we redefine the switching operator, \begin{equation} \M h_z(x)\triangleq \max_{\zeta\in\mathcal{Z}\setminus \{z\}}\left\{h_\zeta(x)+g_{\zeta}(x)-g_z(x)-H(x, z, \zeta)\right\}, \end{equation} where \begin{equation*} g_z(x)\triangleq \sup_{w\in W_0}J_z^w(x)=\ME\left[\int_0^\infty e^{-\alpha s}f(X_s)ds \right]. \end{equation*} Hence (\ref{eq:u-1}) will be modified to
$u_z(x)=\ME[e^{-\alpha\tau}\M u_z(X_\tau)].$
It follows that our system of equations (\ref{eq:system}) is now \begin{align} \begin{cases}\label{eq:3-system} \bar{v}_2(x)\triangleq \sup_{\tau\in\mathcal{S}}\ME\left[e^{-\alpha\tau} \M\bar{v_2}(X_{\tau})\right]\\ \bar{v}_1(x)\triangleq \sup_{\tau\in\mathcal{S}}\ME\left[e^{-\alpha\tau} \M\bar{v_1}(X_{\tau})\right]\\ \bar{v}_0(x)\triangleq \sup_{\tau\in\mathcal{S}}\ME\left[e^{-\alpha\tau} \M\bar{v_0}(X_{\tau})\right]\\ \end{cases} \end{align} The first stage is optimal stopping problem. One possibility of switching production modes is $(0\rightarrow 1, 1\rightarrow 2, 2\rightarrow 0)$. First, we fix this switching scheme, say $c$, and solve the system of equations (\ref{eq:3-system}) as three optimal stopping problems. All the arguments in Section 2.3 hold. This first-stage optimization will give $(x_0^*(c), x_1^*(c), x_2^*(c), \beta_0^*(c), \beta_1^*(c), \beta_2^*(c))$, where $x_i$'s are switching boundaries, depending on this switching scheme $c$.
Now we move to another switching scheme $c'$ and solve the system of optimal stopping problems until we find the optimal scheme.
\subsection{Conclusions} We have studied optimal switching problems for one-dimensional diffusions. We characterize the value function as linear functions in their respective spaces, and provide a direct method to find the value functions and the opening and switching boundaries at the same time. Using the techniques we developed here as well as the ones in Dayanik and Karazas \citeyearpar{DK2003} and Dayanik and Egami \citeyearpar{DE2005}, we solved two specific problems, one of which involves a mean-reverting process. This problem might be hard to solve with just the HJB equation and the related quasi-variational inequalities. Finally, an extension to more general cases is suggested. We believe that this direct method and the new characterization will expand the coverage of solvable problems in the financial engineering and economic analysis.
\begin{appendix} \section{Summary of Optimal Stopping Theory}\label{appx:ost} Let $(\Omega, \F, \p)$ be a complete probability space with a standard Brownian motion $W=\{W_t; t\geq 0\}$ and consider the diffusion process $X^0$ with state pace $\mathcal{I}\subseteq \mathbb{R}$ and dynamics \begin{equation}\label{eq:process} dX^0_t=\mu(X^0_t)dt + \sigma(X^0_t)dW_t \end{equation} for some Borel functions $\mu :\mathcal{I}\rightarrow \mathbb{R}$ and $\sigma :\mathcal{I}\rightarrow (0, \infty)$. We emphasize here that $X^0$ is an uncontrolled process. We assume that $\mathcal{I}$ is an interval with endpoints $-\infty\leq a < b \leq+\infty$, and that $X^0$ is regular in $(a, b)$; in other words, $X^0$ reaches $y$ with positive probability starting at $x$ for every $x$ and $y$ in $(a,b)$. We shall denote by $\mathbb{F}=\{\mathcal{F}_t\}$ the natural filtration generated by $X^0$.
Let $\alpha \geq 0$ be a real constant and $h(\cdot)$ a Borel function such that $\ME[e^{-\alpha \tau}h(X^0_{\tau})]$ is well-defined for every $\mathbb{F}$-stopping time $\tau$ and $x\in \mathcal{I}$. Let $\tau_y$ be the first hitting time of $y\in \mathcal{I}$ by $X^0$, and let $c\in\mathcal{I}$ be a fixed point of the state space. We set: \begin{align} \nonumber \begin{aligned}
\psi(x) &= \begin{cases}
\ME[e^{-\alpha\tau_c}1_{\{\tau_c<\infty\}}], & x\leq c, \\
1/\E^{c}[e^{-\alpha\tau_x}1_{\{\tau_x<\infty\}}],
&x>c,\end{cases} \hspace{0.4cm}
\varphi(x) &= \begin{cases}
1/\E^{c}\left[e^{-\alpha\tau_x}1_{\{\tau_x<\infty\}}\right], & x\leq c, \\
\ME[e^{-\alpha\tau_c}1_{\{\tau_c<\infty\}}],
&x>c,
\end{cases} \end{aligned} \end{align} and \begin{align} \label{eq:F} F(x)&\triangleq\frac{\psi(x)}{\varphi(x)}, \hspace{0.5cm} x\in \mathcal{I}. \end{align} Then $F(\cdot)$ is continuous and strictly increasing. It should be noted that $\psi(\cdot)$ and $\varphi(\cdot)$ consist of an increasing and a decreasing solution of the second-order differential equation $(\mathcal{A}-\alpha)u=0$ in $\mathcal{I}$ where $\mathcal{A}$ is the infinitesimal generator of $X^0$. They are linearly independent positive solutions and uniquely determined up to multiplication. For the complete characterization of $\psi(\cdot)$ and $\varphi(\cdot)$ corresponding to various types of boundary behavior, refer to It\^{o} and McKean \citeyearpar{IM1974}.
Let $F :[c, d]\rightarrow\mathbb{R}$ be a strictly increasing function. A real valued function $u$ is called \emph{$F$-concave} on $[c, d]$ if, for every $a\leq l<r\leq b$ and $x\in[l, r]$, \begin{equation*} u(x)\geq u(l)\frac{F(r)-F(x)}{F(r)-F(l)}+u(r)\frac{F(x)-F(l)}{F(r)-F(l)}. \end{equation*} We denote by \begin{equation}\label{eq:value} V(x)\triangleq \sup_{\tau\in\mathcal{S}}\ME[\D h(X^0_\tau)], \hspace{0.5cm} x\in[c, d] \end{equation} the value function of the optimal stopping problem with the reward function $h(\cdot)$ where the supremum is taken over the class $\mathcal{S}$ of all $\mathbb{F}$-stopping times. Then we have the following results, the proofs of which we refer to Dayanik and Karatzas~\citeyearpar{DK2003}. \begin{proposition}\normalfont\label{prop:A1} For a given function $U$: $[c, d]\rightarrow[0,+\infty)$ the quotient $U(\cdot)/\varphi(\cdot)$ is an $F$-concave function if and only if $U(\cdot)$ is $\alpha$-excessive, i.e., \begin{align} U(x)\geq \ME[e^{-\alpha\tau}U(X^0_\tau)], &\forall \tau\in\mathcal{S}, \forall x\in[c,d]. \end{align} \end{proposition} \begin{proposition}\normalfont \label{prop:A2} The value function $V(\cdot)$ of (\ref{eq:value}) is the smallest nonnegative majorant of $h(\cdot)$ such that $V(\cdot)/\varphi(\cdot)$ is $F$-concave on $[c,d]$. \end{proposition} \begin{proposition}\normalfont \label{prop:A3} Let $W(\cdot)$ be the smallest nonnegative concave majorant of $H\triangleq (h/\varphi)\circ F^{-1}$ on $[F(c), F(d)]$, where $F^{-1}(\cdot)$ is the inverse of the strictly increasing function $F(\cdot)$ in (\ref{eq:F}). Then $V(x)=\varphi(x)W(F(x))$ for every $x\in[c, d]$. \end{proposition} \begin{proposition}\normalfont \label{prop:A4} Define \begin{equation}\label{eq:opt} S\triangleq\{x\in[c, d]: V(x)=h(x)\}, \hspace{0.5cm}\text{and} \hspace{0.5cm} \tau^{*}\triangleq \inf\{t\geqq0: X^0_t\in S\}. \end{equation} If $h(\cdot)$ is continuous on $[c, d]$, then $\tau^{*}$ is an optimal stopping rule. \end{proposition} When both boundaries are natural, we have the following results: \begin{proposition} \normalfont \label{prop:A5} We have either $V\equiv 0$ in $(c, d)$ or $V(x)<+\infty$ for all $(c, d)$. Moreover, $V(x)<+\infty$ for every $x\in (c, d)$ if and only if \begin{equation}\label{eq:test} l_c\triangleq \limsup_{x\downarrow c}\frac{h^+(x)}{\varphi(x)}\quad \text{and}\quad l_d\triangleq \limsup_{x\uparrow d}\frac{h^+(x)}{\psi(x)} \end{equation} are both finite. \end{proposition} In the finite case, furthermore, \begin{proposition}\normalfont \label{prop:A6} The value function $V(\cdot)$ is continuous on $(c, d)$. If $h: (c, d)\rightarrow \R$ is continuous and $l_c=l_d=0$, then $\tau^*$ of (\ref{eq:opt}) is an optimal stopping time. \end{proposition} \begin{proposition}\normalfont\label{prop:A7} Suppose that $l_c$ and $l_d$ are finite and one of them is strictly positive, and $h(\cdot)$ is continuous. Define the continuation region $C\triangleq (c, d)\setminus\Gamma$. Then $\tau^*$ of (\ref{eq:opt}) is an optimal stopping time, if and only if \begin{align*} &\text{there is no $r\in (c, d)$ such that $(c, r)\subset C$ if $l_c>0$}\quad \text{and}\\ &\text{there is no $l\in (c, d)$ such that $(l, d)\subset C$ if $l_d>0$}. \end{align*} \end{proposition}
\end{appendix}
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\advance\headheight by 2pt
\title[Quantum group of type $A$ and queer Lie superalgebra] {Quantum group of type $A$ and representations of queer Lie superalgebra}
\author[Chen]{Chih-Whi Chen} \address{Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan} \email{d00221002@ntu.edu.tw}
\author[Cheng]{Shun-Jen Cheng$^\dagger$} \thanks{$^\dagger$Partially supported by a MOST and an Academia Sinica Investigator grant} \address{Institute of Mathematics, Academia Sinica, Taipei, Taiwan 10617} \email{chengsj@math.sinica.edu.tw}
\begin{abstract} We establish a maximal parabolic version of the Kazhdan-Lusztig conjecture \cite[Conjecture 5.10]{CKW} for the BGG category $\mc{O}_{k,\zeta}$ of $\mf{q}(n)$-modules of ``$\pm \zeta$-weights'', where $k\leq n$ and $\zeta\in\C\setminus\hf\Z$. As a consequence, the irreducible characters of these $\mf q(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $\mf q(n)$-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained. \end{abstract}
\subjclass[2010]{17B67}
\maketitle
\section{Introduction}
Characters for certain classes of finite-dimensional irreducible modules over the queer Lie superalgebra $\mf{q}(n)$ were obtained in the classical works \cite{Pe, Sv2}. The character problem of an arbitrary finite-dimensional irreducible $\mf{q}(n)$-module was then first solved by Penkov and Serganova \cite{PS1, PS2}. They provided an algorithm for computing the coefficient $a_{\la\mu}$ of the character of the irreducible $\mf{q}(n)$-module $L(\mu)$ in the expansion of the character of the associated Euler characteristic $E(\la)$ for given dominant weights $\la, \mu$.
In \cite{Br2} Brundan developed a rather different approach to computing the coefficient $a_{\la\mu}$ for integer dominant weights $\la,\mu$. Let $\mathbb{F}^n$ be the $\emph{n}$th exterior power of the natural representation of the type B quantum group of infinite rank (cf. \cite{JMO}). It was proved that the transition matrix $(a_{\la\mu})$, for $\la$ and $\mu$ dominant integer weights, is given by the transpose of the transition matrix between canonical and the natural monomial bases of $\mathbb{F}^n$ at $q=1$. This gives all irreducible characters of finite-dimensional integer weight modules in terms of a combinatorial algorithm for computing the canonical basis of $\mathbb F^n$. A new interpretation of the irreducible characters of finite-dimensional half-integer weight modules in the same spirit of Lusztig canonical basis of quantum groups was given in \cite{CK} and \cite{CKW} as well.
While finite-dimensional representations of the queer Lie superalgebra $\mf q(n)$ are now fairly well understood, their infinite-dimensional analogues have not been studied much in the literature. Except for $n=2$ and some special cases, e.g.~\cite{FM, Ch}, irreducible characters of infinite-dimensional modules in the BGG category remain largely unknown (see, e.g., the survey article \cite{GG}).
The Brundan-Kazhdan-Lusztig conjecture \cite[Conjecture 4.32]{Br1} for the BGG category of integer weight $\mf{gl}(m|n)$-modules was proved by Lam, Wang and the second author in \cite{CLW} (see also \cite{BLW}). In fact, in \cite{CLW, BLW} irreducible character problem for arbitrary Borel subalgebras was settled; see also \cite{CL} for algorithms. Furthermore, in \cite{CMW}, by means of twisting functors and parabolic induction functors, Mazorchuk, Wang and the second author reduced the irreducible character problem for $\gl(m|n)$ of an arbitrary highest weight to that of an integer highest weight, for which the Brundan-Kazhdan-Lusztig conjecture is then applicable. This gives a complete solution of the irreducible character problem for the full BGG category.
A similar reduction is established for $\mf{q}(n)$ by the first author in \cite{Ch}. As a consequence, the problem of computing the characters of the irreducible modules of arbitrary weights in the BGG category $\mc{O}_{n}$ for $\mf{q}(n)$ is reduced to the irreducible character problem in the following three categories: (i) the BGG category $\mc{O}_{n,\mathbb{Z}}$ of $\mf{q}(n)$-modules of integer weights, see, e.g., \cite{Br2}. (ii) the BGG category $\mc{O}_{n,\frac{1}{2}+\mathbb{Z}}$ of $\mf{q}(n)$-modules of half-integer weights, see, e.g., \cite{CK}, \cite{CKW}. (iii) ($\zeta\notin \mathbb{Z}/2$ and $k\in \{0,1,\ldots ,n\}$) the BGG category $\mc{O}_{n,{\zeta}^{k}}$ of $\mf{q}(n)$-modules of "$\pm \zeta$-weights", see, \cite{CKW} or Section \ref{sec:par:cat}. {\em In the main body of the present paper, we shall use the notation $\mc{O}_{k,\zeta}$ to denote the category $\mc{O}_{n,{\zeta}^{k}}$, as $n$ will be fixed throughout}.
Kwon, Wang and the second author formulated a Kazhdan-Lusztig type conjecture for the BGG category in (iii) (\cite[Conjecture 5.10]{CKW}) above, analogous to Brundan's conjecture for the category (i) in \cite[Section 4.8]{Br2}. In the same paper, the authors also establish some connections between the canonical bases of types A,B,C. Their investigation seems to indicate connections between certain modules over $\mf{q}(n)$ and modules over the general linear Lie superalgebra $\mf{gl}(k|n-k)$ for various $k\leq n$.
In particular, for each $k\leq n$, one has a bijection between the highest weights of the irreducible objects in the category $\mc{O}_{n,{\zeta}^{k}}$ and those of the BGG category
of integer-weight modules for $\mf{gl}(k|n-k)$, that is compatible with the linkage in both categories (see, e.g., \cite{Ch}). In fact, in \cite{Ch} it was proved that blocks of atypicality degree one of a certain maximal parabolic subcategory $\mc F_{k,\zeta}$ of $\mc{O}_{n,{\zeta}^{k}}$ are equivalent to blocks of atypicality degree one of the category of finite-dimensional modules over $\mf{gl}(k|n-k)$.
In the present paper, we study the Kazhdan-Lusztig conjecture for the BGG category $\mc{O}_{n,{\zeta}^{k}}$, formulated in \cite[Conjecture 5.10]{CKW}, which states that the irreducible characters for modules in $\mc{O}_{n,{\zeta}^{k}}$ are determined by the very same Brundan-Kazhdan-Lusztig polynomials as those for the BGG category of the general linear Lie superalgebra $\gl(k|n-k)$ of \cite{Br1}. The main result of the present paper is to (formulate and) prove a parabolic version of that conjecture for the maximal parabolic subcategory $\mc F_{k,\zeta}$ (see Section \ref{FormulationOfPKL}). We wish to point out that the irreducible $\mf q(n)$-modules in $\mc F_{k,\zeta}$ are almost always infinite-dimensional and the character formulas we have obtained in this paper are new.
The paper is organized as follows. In Section \ref{SectionQuantumgroups}, we recall the quantum group of type A and the construction of the Fock space $\mc{E}^{m|n}$. We review the canonical and the dual canonical bases in (a topological completion of) $\mc{E}^{m|n}$, along with Brundan's algorithm for computing canonical basis. Section \ref{SectionRepnOfLieSuperalg} is devoted to the study of representations of the queer Lie superalgebra $\mf{q}(n)$. Certain parabolic subcategories of $\mc O_{k,\zeta}$ of $\mf{q}(n)$-modules are introduced and characterized. \iffalse \red{We also give a $\mathbb{Z}$-gradation of $\mf{q}(n)$ and the corresponding graded BGG category $\mc{O}_{\text{gr}}$ developed in \cite{Br3} in which the theory of tilting modules was generalized to the case of Lie superalgebras. We will identify $\mc O^\rl_{k,\zeta}$ as a full subcategory of the non-graded version of $\mc{O}_{\text{gr}}$ such that whole the tilting theory can work in the subsequent sections.} \fi In Section \ref{SectionTiltingModules} we study in detail the tilting modules in these parabolic subcategory $\mc O_{k,\zeta}$. The parabolic version of the Kazhdan-Lusztig conjecture for the maximal parabolic subcategory $\mc F_{k,\zeta}$ is then formulated precisely in Section \ref{FormulationOfPKL}. We establish a ``queer'' version of Serganova's fundamental lemma \cite[Theorem 5.5]{Ser} in Section \ref{SectionSerganovasFunLem}. This lemma is then used to prove the Kazhdan-Lusztig conjecture for $\mc F_{k,\zeta}$ in Section \ref{ProofOfMainThm}. Our proof here follows the idea of and is inspired by the proof of the main theorem in \cite{Br1}. Finally, we establish a closed Kac-Wakimoto type character formula for a class of $\mf q(n)$-modules in $\mc F_{k,\zeta}$ resembling ``Kostant modules'' for the general linear Lie superalgebra. For those $\mf{q}(n)$-modules resembling polynomial representations of the general linear Lie superalgebra we obtain a Sergeev-Pragacz type character formula as well. This is accomplished in Section \ref{sec:KW:formula}.
\subsection*{Acknowledgments} The results of the present paper were announced by the second author in the conference {\em Categorical Representation Theory and Combinatorics} held in KIAS in December 2015. In the same conference Brundan announced that he and Davidson can establish \cite[Conjecture 5.10]{CKW} in its full generality based on uniqueness of tensor product categorification in the same spirit as \cite{BLW}.
We have been informed by Shunsuke Tsuchioka that his computer calculations show that the conjectures for the irreducible characters of integer and half-integer weights in the full BGG category formulated respectively in \cite{Br2} and \cite{CKW} require corrections. We are indebted to him for kindly sharing his computations with us.
\subsection*{Notation} \label{SectionNotations} We use $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Z}_{\geq 0}$ to denote the sets of natural numbers, integers, and non-negative integers, respectively. Here and below we let $m,n \in \mathbb{Z}_{\geq 0}$ and set \begin{align*} I(m|n):= \{ -m, -m+1, \ldots ,-1\} \cup \{1, 2,\ldots ,n \}.\end{align*} Let $\mathbb{Z}^{m|n}$ be the set of all functions $f: I(m|n) \rightarrow \mathbb{Z}$.
For $p\in\N$, the symmetric group on $p$ letters is denoted by $\mf{S}_{p}$. Let $\mf{S}_{m|n}: = \mf{S}_{m} \times \mf{S}_n$. Note that $\mf{S}_{m|n}$ acts on the right of $\mathbb{Z}^{m|n}$ by composition of functions.
Throughout the paper, we fix a complex number $\zeta \not\in\hf\Z$ which will be used from Section \ref{SubsectionZgradations} on.
\section{Quantum groups and combinatorial preliminaries} \label{SectionQuantumgroups}
In this section we recall the quantum group of type A of infinite rank. We refer to \cite[Section 2-c]{Br1} or \cite[Section 2]{CLW} for more details.
\subsection{Quantum group of type A} Let $\bold U := \bold U_{q}(\mf{gl}_{\infty})$ be the quantum group of type A of infinite rank. This is the $\mathbb{Q}(q)$-algebra generated by $\{E_a, F_a, K_a, K_a^{-1}\}_{a\in \mathbb{Z}}$, subject to the relations \[K_aK_a^{-1} = K_a^{-1}K_a =1,\ \ K_aK_b = K_bK_a,\] \[K_aE_bK_a^{-1} = q^{\delta_{a,b} - \delta_{a,b+1} }E_b, \ \ K_aF_bK_a^{-1} = q^{\delta_{a,b+1} - \delta_{a,b} }F_b,\] \[E_aF_b -F_bE_a = \delta_{a,b} \frac{K_{a,a+1}- K_{a+1,a}}{q-q^{-1}},\]
\[E_a E_b = E_bE_a, \hskip 106pt \text{ if $|a-b|>1$}, \]
\[E_a^2 E_b + E_bE_a^2 =(q+q^{-1})E_aE_bE_a, \hskip 10pt \text{ if $|a-b|=1$},\]
\[F_a F_b = F_bF_a, \hskip 111pt \text{ if $|a-b|>1$},\]
\[F_a^2 F_b + F_bF_a^2 =(q+q^{-1})F_aF_bF_a, \hskip 17pt \text{ if $|a-b|=1$}.\]
Here and below $K_{a,b} := K_a K_b^{-1}$ for $a\neq b\in \mathbb{Z}$.
$\bold U$ is a Hopf algebra with commultiplication $\Delta: \bold U \rightarrow \bold U \otimes \bold U$ defined by \[\Delta(E_a) = 1\otimes E_a + E_a \otimes K_{a+1,a},\] \[\Delta(F_a) = F_a \otimes 1 +K_{a,a+1}\otimes F_a,\] \[\Delta(K_a) = K_a\otimes K_a,\] for $a\in \mathbb{Z}$.
\subsection{Fock space $\mc{E}^{m|n}$} \label{SectionTheFockSpace} Let $\mathbb V$ be the natural $\bold U$-module with basis $\{v_{a}\}_{a\in \mathbb{Z}}$ and let $\mathbb W$ be its restricted dual with basis $\{w_a\}_{a\in \mathbb{Z}}$ normalized by $\langle w_a,v_b\rangle = (-q)^{-a}\delta_{a,b}$, for $a,b \in\ \mathbb{Z}$. The actions of $\bold U$ on $\mathbb V$ and $\mathbb W$ are respectively given by \begin{align*} K_a v_b =q^{\delta_{a,b}}v_b, \ \ E_a v_b =\delta_{a+1,b} v_a, \ \ F_a v_b =\delta_{a,b}v_{a+1},\\ K_a w_b =q^{-\delta_{a,b}}w_b, \ \ E_a w_b =\delta_{a,b}w_{a+1}, \ \ F_a w_b =\delta_{a+1,b} w_a. \end{align*}
For $m,n\in \mathbb{Z}_{\geq 0}$, the tensor space $ \TT^{m|n}:= \mathbb V^{\otimes m}\otimes \mathbb W^{\otimes n}$ can be viewed as a module over $\bold U$ via the comultiplication $\Delta$. For $f\in \mathbb{Z}^{m|n}$, we let \[M_f:= v_{f(-m)}\otimes v_{f(-m+1)} \otimes \cdots \otimes v_{f(-1)} \otimes w_{f(1)} \otimes w_{f(2)} \otimes w_{f(n)} \in \TT^{m|n}.\] The set $\{M_f\}_{f\in \mathbb{Z}^{m|n}}$ is referred to as the {\em standard monomial basis} for $\TT^{m|n}$.
Let $\mf{S}_{m}$ be the symmetric group on the letters in $I(m|0)$ with the set of generators $\{s_i:=(i \ \ i+1)|-m \leq i\leq -2\} \subseteq \mf{S}_{m}$. Recall that the {\em Iwahori-Hecke algebra} $\mathcal{H}_{m}$ associated to $\mf{S}_{m}$ is the associative $\mathbb{Q}(q)$-algebra generated by $H_i$, $-m\le i\le -2$, subject to the relations
\[(H_i -q^{-1})(H_i + q) =0,\]
\[H_iH_{i+1}H_i = H_{i+1}H_iH_{i+1},\] \[H_iH_j=H_jH_i, \text{ for } |i-j|>1. \] Denote the longest element in $\mf{S}_{m}$ by $\omega_{0}^{(m)}$. For each $\sigma\in \mathfrak{S}_{m}$, we have the corresponding element $H_{\sigma}: = H_{i_1}H_{i_2} \cdots H_{i_r}$ for any reduced expression $\sigma = s_{i_1}s_{i_2}\cdots s_{i_r}$. Recall that there is a unique antilinear ($q\rightarrow q^{-1}$) automorphism $\bar{\,}: \mc{H}_{m} \rightarrow \mc{H}_{m}$ such that $\overline{H_{\sigma}} = H^{-1}_{\sigma^{-1}}$, for all $\sigma \in \mf{S}_m$ (see, e.g., \cite[Section 2-e]{Br1}).
We denote by $\preceq_{\mf{a_m}}$ the classical Bruhat ordering on the weight lattice $\mathbb{Z}^m$ of $\mf{gl}(m)$. For $i\in I(m|n)$, let $d_i \in \mathbb{Z}^{m|n}$ be the function $j\mapsto -\text{sgn}(i)\delta_{ij}$. Recall the {\em super Bruhat ordering} $\preceq$ on $\mathbb{Z}^{m|n}$ for Lie superalgebra $\mf{gl}(m|n)$ defined in \cite[Section 2-b]{Br1} as follows.
Let $P$ be the free abelian group with basis $\{\epsilon_a\}_{a\in \mathbb{Z}}$. Let $\leq$ denote the partial ordering of weights on $P$, i.e., $f\leq g$ if and only if $f-g \in \sum_i \mathbb{Z}_{\geq 0}(\epsilon_i - \epsilon_{i+1})$. Let $\text{wt}_r^{\epsilon}: \mathbb{Z}^{m|n} \rightarrow P$ be the $\epsilon$-weight function defined by \begin{align}
\text{wt}_r^{\epsilon}(f) := \sum_{r\leq i}-\text{sgn}(i)\epsilon_{f(i)}, \text{ for } f\in \mathbb{Z}^{m|n},\ \ r\in I(m|n). \end{align}
The super Bruhat ordering $\preceq$ on $\mathbb{Z}^{m|n}$ is defined by $f\preceq g$, if $\text{wt}^\epsilon_r f \leq \text{wt}^\epsilon_r g$, for all $r\in I(m|n)$, and $\text{wt}^\epsilon_{-m}f =\text{wt}^\epsilon_{-m} g$ (\cite[Section 2-b]{Br1}).
For $f\in \mathbb{Z}^{m|n}$, the {\em degree of atypicality} of $f$ is denoted by $\sharp f$ (see, e.g., \cite[(2.3)]{Br1}). We say $f$ is {\em typical} if $\sharp f =0$; otherwise $f$ is {\em atypical}. For $f,g \in \mathbb{Z}^{m|n}$, we have that $f \succeq g$ implies $\sharp f =\sharp g$.
Recall that $\TT^{m|0}= \mathbb V^{\otimes m}$ admits a $\bold U$-$\mathcal{H}_{m}$-bimodule structure \cite{Jim}. Namely, on $\TT^{m|0}$ the Hecke algebra $\mc H_m$ acts as follows: \begin{align} \label{HeckeAlgebaAction} M_f H_i = \left\{ \begin{array}{lll} M_{fs_i}, \text{ if } f \prec_{\mf{a_m}} fs_i,\\ q^{-1}M_{f}, \text{ if } f = fs_i,\\ M_{fs_i} -(q-q^{-1})M_f, \text{ if } f \succ_{\mf{a_m}} fs_i, \end{array} \right. \end{align}
for all $-m\leq i\leq -2$ and $ f\in \Z^{m|0}$. Similarly, we can define a $\bold U$-$\mathcal{H}_{n}$-bimodule structure on $\TT^{0|n} = \mathbb W^{\otimes n}$.
For $p\in \N$, let \begin{align*}
H_0(p) : = \sum_{\sigma\in \mf{S}_{p}} (-q)^{\ell(\sigma) - \ell(\omega_0^{(p)})} H_{\sigma}. \end{align*} Then $H_0(p)$ is a bar-invariant element in $\mc H_p$ (see, e.g., \cite[Lemma 3.2]{Br1}).
It is proved in \cite[Propositions 1.1 and 1.2]{KMS} that $\TT^{m|0} =\text{Ker}H_0(m)|_{\TT^{m|0}} \oplus \TT^{m|0}H_0(m)|_{\TT^{m|0}}$
and $\text{Ker}H_0(m)|_{\TT^{m|0}} = \sum_{i=-m}^{-2} \text{Ker}(H_i-q^{-1})|_{\TT^{m|0}}$. Similarly, $\TT^{0|n}$ and $\text{Ker}H_0(n)|_{\TT^{0|n}}$ have analogous decompositions.
As a conclusion, $\TT^{m|n}$ admits a $\bold U$-$(\mathcal{H}_{m}\otimes \mathcal{H}_{n})$ bimodule structure
(see, e.g., \cite[Section 2-e]{Br1}) with $\TT^{m|n}= \text{Ker}H_0 \oplus \TT^{m|n} H_0$ and $\text{Ker}H_0 = \sum_i \text{Ker}(H_i-q^{-1})$, where the summation is over $i\in I(m|n)\setminus\{-1,n\}$ and $H_0:= H_{0}(m) H_{0}(n) \in \mathcal{H}_{m}\otimes \mathcal{H}_{n}$. We define the {\em Fock space} $\mc{E}^{m|n}:= \TT^{m|n} H_0$.
We can identify $\mc{E}^{m|n}$ with the {\em $q$-wedge space} $\wedge^m\mathbb V\otimes\wedge^{n}\mathbb W$ (see, e.g., \cite[Section 2.4]{CLW}). Let \begin{align*}
\mathbb{Z}^{m|n}_+:= \{f\in \mathbb{Z}^{m|n}| f(-m)>f(-m+1)> \cdots > f(-1),\ \ f(1)<f(2)< \cdots < f(n) \}.\end{align*}
From \eqref{HeckeAlgebaAction}, it follows that $\{M_fH_0\}_{f\in \mathbb{Z}_+^{m|n}}$ forms a $\mathbb{Q}(q)$-basis for $\mc{E}^{m|n}$ and the following bijection from $\mc{E}^{m|n}$ to $\wedge^m\mathbb V\otimes\wedge^{n}\mathbb W$ \begin{align*}
M_fH_0 \mapsto v_{f(-m)} \wedge \ldots \wedge v_{f(-1)}\otimes w_{f(1)} \wedge w_{f(2)} \wedge \ldots \wedge w_{f(n)}, \text{ for }f\in \mathbb{Z}^{m|n}, \end{align*}
gives an isomorphism of $\bold U$-modules. For each $f\in \mathbb{Z}_+^{m|n}$, we define $K_f : = M_fH_0 \in \mc{E}^{m|n}$. We call $\{K_f\}_{\mathbb{Z}_+^{m|n}}$ the {\em standard monomial basis} for $\mc{E}^{m|n}$.
\subsection{Canonical and dual canonical bases of $\mc{E}^{m|n}$} \label{SectionBases}
We let $\widehat{\TT}^{m|n}$ and $\widehat{\mc{E}}^{m|n}$ denote certain topological completions of $\TT^{m|n}$ and ${\mc{E}}^{m|n}$, respectively, see \cite[Section 2-d]{Br1} for precise definition.
According to \cite[Theorem 2.14]{Br1} $\widehat{\TT}^{m|n}$ admits a continuous, anti-linear bar-involution $\bar{\,}: \widehat{\TT}^{m|n} \rightarrow \widehat{\TT}^{m|n}$ such that $\overline{XuH} =\overline{X}\overline{u}\overline{H}$, for all $X\in \bold U$, $u\in \widehat{\TT}^{m|n}$, $H\in \mathcal{H}_{m}\otimes \mathcal{H}_{n}$, and furthermore $\overline{M_f} = M_f$, for $f\in \mathbb{Z}^{m|n}$ with $f(-m)\leq \cdots \leq f(-1)$, $f(1) \geq \cdots \geq f(n)$, $f(i) \neq f(j)$, for all $-m\leq i < 0 < j \leq n$.
\begin{thm} \label{KL-Lemma} \emph{(} \cite[Theorem 3.6]{Br1} \emph{)} There exist unique bar-invariant topological bases $\{U_f\}_{f\in \mathbb{Z}_+^{m|n}}, \{L_f\}_{f\in \mathbb{Z}_+^{m|n}}$ for $\widehat{\mc{E}}^{m|n}$ such that \begin{align*} U_f = K_f +\sum_{g\prec f} u_{g,f}(q)K_g, \ \ L_f = K_f +\sum_{g\prec f} \ell_{g,f}(q)K_g, \end{align*}
where summation is over $g \in \mathbb{Z}^{m|n}_+$, $u_{g,f}(q)\in q\mathbb{Z}[q]$, and $\ell_{g,f}(q)\in q^{-1}\mathbb{Z}[q^{-1}]$. \end{thm}
The topological bases $\{U_f\}_{f\in \mathbb{Z}_+^{m|n}}$ and $\{L_f\}_{f\in \mathbb{Z}_+^{m|n}}$ are respectively referred to as {\em canonical basis} and {\em dual canonical basis} of $\widehat{\mc{E}}^{m|n}$ (see, e.g., \cite[Section 3-b]{Br1}). The polynomials $u_{g,f}(q)$, $\ell_{g,f}(q)$ can be computed combinatorially \cite[Corollary 3.39]{Br1}.
\subsection{Procedure for canonical basis} \label{CombinatorialSetup} We conclude this section with a review of \cite[Procedure 3.20]{Br1} for constructing canonical basis elements $U_f$, which indeed lie in $\mc E^{m|n}$. This will be used for construction of certain tilting modules of $\mf{q}(n)$ in Section \ref{SectionTiltingModules}.
\begin{proc} \label{Br1Procedure} \emph{(}\cite[Procedure 3.20]{Br1}\emph{)} \emph{ Assume that $f\in \mathbb{Z}_+^{m|n}$ with $\sharp f >0$. Define $h\in \mathbb{Z}_+^{m|n}$, $a\in \mathbb{Z}$ and $\widehat{X}_a, \widehat{Y}_a\in \{E_a,F_a\}$, by the following instructions below starting with step (I).}\end{proc}
{\bf Step (I)} Let $-m\leq i\leq -1$ be the largest number such that $f(i) = f(j)$ for some $1\leq j \leq n$. Go to step (II).
{\bf Step (II)} If $i\neq -1$ and $f(i)-1 =f(i+1)$, replace $i$ by $i+1$ and repeat step (II). Otherwise, go to step (III).
{\bf Step (III)} If $f(i)-1 = f(j)$ for some $1\leq j\leq n$, go to step (II*). Otherwise, set $\widehat{X}_a := F_{f(i)- 1}$, $\widehat{Y}_a:=E_{f(i)-1}$ and $h:= f- d_i$.
{\bf Step (II*)} If $j\neq 1$ and $f(j) -1 =f(j-1)$, replace $j$ by $j-1$ and repeat step (II*). Otherwise, go to step (III*).
{\bf Step (III*)} If $f(j)-1 = f(i)$ for some $-m\leq i\leq -1$, go to step (II). Otherwise, set $\widehat{X}_a := E_{f(j)-1}$, $\widehat{Y}_a:=F_{f(j)-1}$ and $h:= f+ d_j$.
\vskip 8pt
After finitely many steps, the procedure reduces $f$ to a typical $g\in \mathbb{Z}_+^{m|n}$, namely, $U_f=\widehat{X}_{b_1}\widehat{X}_{b_2}\cdots \widehat{X}_{b_{\ell}}U_g$ for some $b_1,b_2,\ldots ,b_{\ell} \in \mathbb{Z}$ depending on $g$. Since $U_g = K_g$, we have the following lemma (cf. \cite[Lemma 3.19, 3.21]{Br1}).
\begin{lem} \label{LemmaForProcedure} Let $f$, $h$, $a\in \mathbb{Z}$, $\widehat{X}_a$ and $\widehat{Y}_a$ be given as above. If $\sharp f =\sharp h$ then $\widehat{X}_a U_h =U_f, \widehat{Y}_a\widehat{X}_aU_h = U_h$ and $\widehat{X}_aK_h = K_f$. If $\sharp f =\sharp h +1$ then we have $\widehat{X}_a U_h =U_f,\ \ \widehat{Y}_a\widehat{X}_aU_h = (q+q^{-1})U_h$ and $\widehat{X}_aK_h = K_f +qK_{f-d_i+d_j}$, for some $-m\leq i<0 < j\leq n$, $f(i) = f(j)$ such that $f- d_i+d_j \in \mathbb{Z}^{m|n}_+$. \end{lem}
\vskip 12pt
\section{Representations of the Lie superalgebra $\mf{q}(n)$} \label{SectionRepnOfLieSuperalg}
\subsection{Queer Lie superalgebra} \label{SectionQueerLieSuperalgebra}
Let $\mathbb{C}^{m|n}$ be the complex superspace of dimension $(m|n)$. The {\em general linear Lie superalgebra} $\mathfrak{gl}(m|n)$ may be realized as $(m+n) \times (m+n)$ complex matrices: \begin{align} \label{glrealization}
\left( \begin{array}{cc} A & B\\ C & D\\ \end{array} \right), \end{align}
where $A,B,C$ and $D$ are respectively $m\times m, m\times n, n\times m, n\times n$ matrices. Let $E_{a,b}$ be the elementary matrix in $\mathfrak{gl}(m|n)$ with $(a,b)$-entry $1$ and other entries 0, for $a,b \in I(m|n)$ and let $\h'=\h'_{m|n}$ be the standard Cartan subalgebra of $\gl(m|n)$ spanned by the basis elements $\{E_{aa}\}$ and dual basis elements $\{\delta'_a\}$, for $a\in I(m|n)$. Denote by $\Phi'^+$ the set of positive roots in the standard Borel subalgebra.
For $m=n$, the subspace \begin{align} \label{qnrealization}
\mf{g}:= \mathfrak{q}(n)= \left\{ \left( \begin{array}{cc} A & B\\ B & A\\ \end{array} \right) \middle\vert\ A, B: \ \ n\times n \text{ matrices} \right\} \end{align}
forms a subalgebra of $\mathfrak{gl}(n|n)$ called the {\em queer Lie superalgebra}.
The set $\{e_{ij}, \bar{e}_{ij}|1\leq i,j \leq n\}$ is a linear basis for $\mathfrak{g}$, where $e_{ij}= E_{-n-1+i,-n-1+j}+E_{i,j}$ and $\bar{e}_{ij}= E_{-n-1+i,j}+E_{i,-n-1+j}$. Note that the even subalgebra $\mathfrak{g}_{\bar{0}}$ is spanned by $\{e_{ij}|1\leq i,j\leq n\}$, which is isomorphic to the general linear Lie algebra $\mathfrak{gl}(n)$.
Let $\mathfrak{h} = \mathfrak{h}_{\bar{0}}\oplus \mathfrak{h}_{\bar{1}}$ be the standard Cartan subalgebra of $\mathfrak{g}$, with linear bases $\{h_i:= e_{ii}| 1\leq i \leq n\}$ and $\{\bar{h}_{i}:= \bar{e}_{ii}|1\leq i \leq n\}$ of $\mathfrak{h}_{\bar{0}}$ and $ \mathfrak{h}_{\bar{1}}$, respectively. Let $\{\delta_i| 1\leq i\leq n\}$ be the basis of $\mathfrak{h}_{\bar{0}}^{*}$ dual to $\{h_i|1\leq i\leq n\}$. We define a symmetric bilinear form $( ,) $ on $\mathfrak{h}_{\bar{0}}^{*}$ by $( \delta_i,\delta_j) = \delta_{ij}$, for $1\leq i,j\leq n$.
We denote by $\Phi,\Phi_{\bar{0}},\Phi_{\bar{1}}$ the sets of all roots, even roots and odd roots of $\mf{g}$, respectively. Let $\Phi^+=\Phi^+_\even\sqcup\Phi^+_\odd$ be the set of positive roots in its standard Borel subalgebra $\mf b=\mf b_\even\oplus\mf b_\odd$, which consists of matrices of the form \eqref{qnrealization} with $A$ and $B$ upper triangular. Ignoring parity we have $ \Phi_\even=\Phi_\odd = \{\delta_i - \delta_j| 1\leq i,j \leq n\}$ and $\Phi^+ = \{\delta_i- \delta_j| 1\leq i< j \leq n\}$. We denote by $\leq$ the usual partial order on the weights $\mf{h}_{\bar{0}}^*$ defined by using $\Phi^+$. The Weyl group $W$ of $\mathfrak{g}$ is defined to be the Weyl group of the reductive Lie algebra $\mathfrak{g}_{\bar{0}}$ and hence acts naturally on $\mathfrak{h}_{\bar{0}}^*$ by permutation. For a given root $\alpha = \delta_i - \delta_j \in \Phi$, let $\bar{\alpha} :=\delta_i + \delta_j $.
In the this paper, $\mf{g}$-modules are always supposed to have compatible $\mathbb{Z}_2$-gradations with $\mf{g}$-actions, and $\mf{g}$-homomorphisms are not necessarily even. For a $\mathfrak{g}$-module $M$ and $\mu \in \mathfrak{h}_{\bar{0}}^*$, let $M_{\mu}:=\{m\in M| h\cdot m = \mu(h)m, \text{ for } h\in \mathfrak{h}_{\bar{0}}\}$ denote its $\mu$-weight space. If $M$ has a weight space decomposition $M = \oplus_{\mu\in \mathfrak{h}_{\bar{0}}^*}M_{\mu}$, its character is given as usual by $\text{ch}M = \sum_{\mu \in \mathfrak{h}_{\bar{0}}^*}\text{dim}M_{\mu}e^{\mu} $, where $e$ is an indeterminate. In particular, we have the root space decomposition $\mathfrak{g} = \mf{h}\oplus(\oplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha})$ with respect to the adjoint representation of $\mathfrak{g}$.
Let $\la = \sum_{i=1}^{n} \la_i \delta_i \in \mathfrak{h}_{\bar{0}}^*$, and consider the symmetric bilinear form on $\mathfrak{h}_{\bar{1}}^*$ defined by $\langle\cdot,\cdot\rangle_{\la} : = \la([\cdot,\cdot] )$. Let $\ell(\la)$ be the number of $i$'s with $\la_i \neq 0$. Let $1\leq i_1<i_2< \cdots < i_{\ell(\la)}\leq n$ such that $\la_{i_1}, \la_{i_2}, \ldots , \la_{i_{\ell(\la)}}$ are non-zero. Denote by $\lceil \cdot \rceil$ the ceiling function. Let $\mathfrak{h}'_{\bar{1}}$ be a maximal isotropic subspace of $\mathfrak{h}_{\bar{1}}$ associated to $\langle\cdot,\cdot\rangle_{\la} $. Put $\mathfrak{h}' = \mathfrak{h}_{\bar{0}} \oplus \mathfrak{h}'_{\bar{1}}$. Let $\mathbb{C}v_{\la}$ be the one-dimensional $\mathfrak{h}'$-module with $h\cdot v_{\la} = \la(h)v_{\la}$ and $h' \cdot v_{\la} =0 $ for $h\in \mathfrak{h}_{\bar{0}}$, $h' \in \mathfrak{h}'_{\bar{1}}$. Then $I_{\la} : = \text{Ind}_{\mathfrak{h}'}^{\mathfrak{h}}\mathbb{C}v_{\la}$ is an irreducible $\mathfrak{h}$-module of dimension $2^{\lceil \ell(\la)/2 \rceil}$ (see, e.g., \cite[Section 1.5.4]{CW}). We let $\Delta(\la): = \text{Ind}_{\mathfrak{b}}^{\mathfrak{g}}I_{\la}$ be the {\em Verma module}, where $I_{\la}$ is extended to a $\mathfrak{b}$-module in a trivial way, and define $L(\la)$ to be the unique irreducible quotient of $\Delta(\la)$. Note that $\Delta(\la)$ and $L(\la)$ are unique up to $\mf{g}$-isomorphisms.
For a weight $\la \in \mf{h}_{\bar{0}}^*$, we let $\sharp \la$ to be the {\em atypicality degree} of $\la$ (see, e.g., \cite[Definition 2.49]{CW}).
We say $\la$ is {\em typical} if $\sharp \la =0$; otherwise $\la$ is called {\em atypical}.
\subsection{$\La_{k,\zeta}$-weights, $\mathbb{Z}$-gradations, and categories $\mc{HC}_{k,\zeta}(\mf{l}^{\rl})$} \label{SubsectionZgradations} Let $k,n\in \mathbb{Z}_{\geq 0}$ with $k\le n$ and $\zeta \in \mathbb{C}\backslash \frac{\mathbb{Z}}{2}$. We let \begin{align*}
&\La_{k,\zeta}:=\{\la=\sum_{r=1}^n\la_i\ep_i\in\h_\even^*|\la_i\in \zeta+\Z\text{ and }\la_j\in-\zeta+\Z,1\le i\le k < j\le n\}, \end{align*}
Let $s,t\in\N$ such that $k=r_1+r_2+\ldots+r_s$ and $n-k=l_1+l_2+\ldots+l_t$, where $r_i,l_j\in\N$. Let $\texttt{r}=(r_1,\ldots,r_s)$ and $\texttt{l}=(l_1,\ldots,l_t)$, and put $\texttt{r}^c=\sum_{i=1}^cr_i$, and $\texttt{l}^d=k+\sum_{i=1}^d l_i$, for $c=0,\ldots,s$ and $d=0,\ldots,t$. We define ($c\not=s$ and $d\not=t$) \begin{align*}
&\La_{k,\zeta}^{\texttt{r,l}}:=\{\la\in\La|\la_i-\la_{i+1}\in\N,\text{ for }\texttt{r}^c< i< \texttt{r}^{c+1}\text{ or }\texttt{l}^d< i< \texttt{l}^{d+1}\}. \end{align*}
In the case $\texttt{r}=\underbrace{(1,1,\ldots,1)}_k$ and $\underbrace{\texttt{l}=(1,1,\ldots,1)}_{n-k}$ we have $\La^{(1,\ldots,1),(1,\ldots,1)}=\La_{k,\zeta}$. In the case $\texttt{r}=(k)$ and $\texttt{l}=(n-k)$ we shall use the notation $\La^{+}_{k,\zeta}$ for $\La^{(k),(n-k)}_{k,\zeta}$, i.e., \begin{align*}
&\La_{k,\zeta}^{+}:=\{\la\in\La_{k,\zeta}|\la_i-\la_{i+1}\in\N, \text{ for }0< i< k \text{ or }k< i<n\}. \end{align*}
Associated to $\La_{k,\zeta}^\texttt{r,l}$ we have a Levi subalgebra \begin{align*} \mf l^\rl=\bigoplus_{i=1}^s \mf{q}(r_i) \oplus \bigoplus_{j=1}^t \mf{q}(l_i)\subseteq \mf{q}(n), \end{align*} with corresponding parabolic subalgebra $\mf p^\rl$, nilradical $\mf u^\rl$, and opposite nilradical $\mf u^{\rl,-}$. Denote the roots in $\mf u^\rl$ by $\Phi^+(\mf u^\rl)$. If $\Pi$ denotes the set of simple roots of even positive roots $\Phi_\even$, we let $\Pi^\rl\subseteq\Pi$ denote the subset such that the even and odd root spaces $\G_\alpha\subseteq\mf l^\rl$ if and only if $\alpha\in\Pi^\rl$. Associated to $\La^\rl_{k,\zeta}$ we thus have a $\Z$-gradation of $\G=\bigoplus_{j\in\Z}\G_j$ uniquely determined by \begin{align}\label{Z-gradation} \deg\h=0,\quad \deg\G_{\pm\alpha}=0,\quad\deg\G_{\pm\beta}=\pm1,\quad\text{for } \alpha\in\Pi^\rl,\beta\in\Pi\setminus\Pi^\rl. \end{align} Note that this grading is also given by the formula \begin{align} \label{GradingOperators} [D,X] = jX, \ \ \text{ for }X\in \mf{g}_j, j\in \mathbb{Z}, \end{align} where $D$ is grading operator $\sum_{c=0}^{s-1} (n-c)\sum_{p=\texttt{r}^c+1}^{\texttt{r}^{c+1}} e_{pp} + \sum_{d=0}^{t-1} (n-s-d)\sum_{q=\texttt{l}^d+1}^{\texttt{l}^{d+1}} e_{qq}\in \mf{h}_{\bar{0}}$. Of course we have $\G_0 = \mf l^\rl$.
Let $W^\rl$ denote the Weyl group of $\mf l^\rl$, so that we have $W^\rl\cong\mf S_{r_1}\times\cdots\times\mf S_{r_s}\times\mf S_{l_1}\times\cdots\times\mf S_{l_t}$. Let $w^{\rl}_0$ be the longest element in $W^{\rl}$ so that, for $\la\in\La^{\rl}_{k,\zeta}$, we have $-w^{\rl}_0\la\in\La^{\rl}_{k,-\zeta}$. In the case $\texttt{r}=\underbrace{(1,1,\ldots,1)}_k$ and $\underbrace{\texttt{l}=(1,1,\ldots,1)}_{n-k}$ we shall write $w_0$ for $w_0^{\rl}$, while in the case $\texttt{r}=(k)$ and $\texttt{l}=(n-k)$ we shall write $w^+_0$ for $w_0^\rl$.
For given Levi subalgebra $\mf{s}$ of $\mf{g}$ containing $\h$, denote by $\mc{HC}_{k,\zeta}(\mf{s})$ the category of $\mf{s}$-modules that are direct sums of finite-dimensional simple $\mf{s}_{\bar{0}}$-modules with highest weights in $\La^{\rl}_{k,\zeta}$.
Let $\mf{b}^{\rl}$ be the standard Borel subalgebra of $\mf{ l^\rl}$, namely, $\mf{b}^{\rl}$ is generated by $\mf{h} \oplus (\oplus_{\alpha\in \Pi^\rl} \mf{g}_{\alpha})$. For given $\la\in\La^\rl_{k,\zeta}$, denote by $\Delta^0(\la):= \text{Ind}_{\mf{b}^{\rl}}^{\mf l^\rl} I_{\la}$ the $\mf l^\rl$-Verma module of highest weight $\la$. Let $L^0(\la)$ be its unique irreducible quotient with highest weight $\la$. Note that $L^0(\la)$ is a typical $\mf l^\rl$-module and is furthermore finite dimensional.
\begin{lem} \label{CharacterizationOfLocFinModules}
$\mc{HC}_{k,\zeta}({\mf l^{\emph{\rl}}})$ is a semisimple category with irreducible objects $\{L^0(\la)|\la\in\La^{\emph{\rl}}_{k,\zeta}\}$. \end{lem}
\begin{proof}
It is enough to show that the full subcategory of $\mc{HC}_{k,\zeta}({\mf l^{{\rl}}})$ consisting of objects with composition factors lying in $\{L^0(\la)|\la\in\La^{{\rl}}_{k,\zeta}\}$ is a semisimple category.
Observe that $L^0(\la)$ and $L^0(\mu)$ have different central characters for $\la, \mu\in\La^{{\rl}}_{k,\zeta}$ with $\la \neq \mu$ (see, e.g., \cite[Theorem 2.48]{CW}), and so there are no nontrivial extensions between these two irreducibles. Therefore, it suffices to show that $L^0(\la)$ has no self-extension in $\mc{HC}_{k,\zeta}({\mf l^{{\rl}}})$, for every $\la\in\La^\rl_{k,\zeta}$. Suppose we have a short exact sequence of the form \begin{align} \label{SESInLevi} 0\rightarrow L^0(\la) \rightarrow E \xrightarrow{f} L^0(\la) \rightarrow 0, \end{align} in $\mc{HC}_{k,\zeta}({\mf l^{{\rl}}})$. Since $\mc{HC}_{k,\zeta}({\mf h})$ is a semisimple category (see, e.g., \cite[Lemma 1]{Fr}), \eqref{SESInLevi} implies that as $\h$-modules we have $E_\la=I_\la\oplus I_\la$. To distinguish these two copies let us write $E_\la=I_\la^{(1)}\oplus I^{(2)}_\la$, where we let $I^{(1)}_\la$ be highest weight space of the submodule $L^0(\la)$ in \eqref{SESInLevi}. Now consider the submodule $W=U(\mf l^\rl)I^{(2)}_\la\subseteq E$. Since $U(\mf l^\rl)I^{(1)}_\la=L^0(\la)$ is irreducible and $W_\la=I^{(2)}_\la$, we have $U(\mf l^\rl)I^{(2)}_\la\cap U(\mf l^\rl)I^{(1)}_\la=0$ and hence $E=W\oplus L^0(\la)$. It follows that $W\cong L^0(\la)$, and so \eqref{SESInLevi} is split. \end{proof}
\subsection{Characters of irreducible $\mf l^\rl$-modules of $\La^\rl_{k,\zeta}$-highest weights} \label{Subsection::IrrChOfL0} For $1\le c\le s$ and $1\le d\le t$, let \begin{align*} ^{(c)}\la=(\la_{\texttt{r}^{c-1}+1},\ldots,\la_{\texttt{r}^{c}}) \quad \text{and}\quad \la^{(d)}=(\la_{\texttt{l}^{d-1}+1},\ldots,\la_{\texttt{l}^{d}}), \end{align*} regarded as weights in the even parts of the Cartan subalgebras of the corresponding queer Lie superalgebras $\mf{q}(r_c)$ and $\mf{q}(l_d)$, respectively. Then we have by Penkov's finite-dimensional typical character formula \cite[Theorem 2]{Pe} \begin{align*} \text{ch}L(\mf{q}(r_c),{^{(c)}\la})=2^{\lceil r_c/2\rceil}\prod_{\texttt{r}^{c-1}+1\le i<j\le \texttt{r}^c}\frac{({1+e^{{-\delta_i+\delta_j}}})}{({1-e^{{-\delta_i+\delta_j}}})}\sum_{w\in\mf S_{r_c}}(-1)^{\ell(w)}w(e^{{^{(c)}\la}}),\\ \text{ch}L(\mf{q}(l_d),\la^{(d)})=2^{\lceil l_d/2\rceil}\prod_{\texttt{l}^{d-1}+1\le s<t\le \texttt{l}^d}\frac{({1+e^{{-\delta_s+\delta_t}}})}{({1-e^{{-\delta_s+\delta_t}}})}\sum_{\sigma\in\mf S_{l_d}}(-1)^{\ell(\sigma)}\sigma(e^{\la^{(d)}}). \end{align*} Therefore we obtain the following character formulas. \begin{prop} \emph{(cf}. \cite[Section 3.1.3]{CW}\emph{)} \label{IrrChForLevi} \begin{align*} \label{ChOfL0} \emph{ch}L^0(\la)=&2^{\lceil n/2\rceil} \prod_{c=1}^s\prod_{\texttt{\emph{r}}^{c-1}+1\le i<j\le \texttt{\emph{r}}^c}\frac{({1+e^{{-\delta_i+\delta_j}}})}{({1-e^{{-\delta_i+\delta_j}}})}\sum_{w\in\mf S_{r_c}}(-1)^{\ell(w)}w(e^{{^{(c)}\la}})\\ &\prod_{d=1}^t\prod_{\texttt{\emph{l}}^{d-1}+1\le s<t\le \texttt{\emph{l}}^d}\frac{({1+e^{{-\delta_s+\delta_t}}})}{({1-e^{{-\delta_s+\delta_t}}})}\sum_{\sigma\in\mf S_{l_d}}(-1)^{\ell(\sigma)}\sigma(e^{\la^{(d)}}). \end{align*} \end{prop}
Recall the Levi subalgebra $\mf l^\rl$ with corresponding parabolic subalgebra $\mf{p}^\rl$, nilradicals $\mf u^\rl$, and opposite nilradical $\mf u^{\rl,-}$. Observe that as an $\mf l^\rl$-module, we have \begin{align*}
\mf u^{\rl,-}\cong& \bigoplus_{1\le i<j\le s}\hf\left[\C^{r_i|r_i*}\otimes\C^{r_j|r_j}\right]\oplus \bigoplus_{i,j}\hf\left[\C^{r_i|r_i*}\otimes\C^{l_j|l_j}\right]\oplus\\
& \bigoplus_{1\le i<j\le t}\hf\left[\C^{l_i|l_i*}\otimes\C^{l_j|l_j}\right]. \end{align*}
Above the factor $\hf$ is explained as follows: For given $p,q \in \mathbb{N}$, both $\C^{p|p*}$ and $\C^{q|q}$ are so-called type $\texttt{Q}$ supermodules, and it is known that their tensor product is isomorphic to a direct sum of two copies of the same irreducible $\mf q(p)\oplus\mf{q}(q)$-module. The factor $\hf$ means that we take one copy of it, see, e.g., \cite[Section 3.1.3]{CW}.
\subsection{Parabolic BGG categories}\label{sec:par:cat}
Let $\mc{O}_n$ denote the BGG category of finitely generated $\mf{q}(n)$-modules which are locally finite over $\mf{b}$ and semisimple over $\mf{h}_{\bar{0}}$. In $\mc{O}_n$, we allow arbitrary (not necessarily even) $\mf{g}$-morphisms. It is well-known that $\{L(\la)|\la \in \mf{h}^*_{\bar{0}}\}$ is a complete set of irreducible objects in $\mc{O}_n$, up to isomorphism. Let $\mc O^\rl_{k,\zeta}$ denote the full subcategory of $\mc{O}_n$ consisting of objects whose composition factors lie in $\{L(\la)| \la\in\La^\rl_{k,\zeta}\}$. We shall use the following notations for the two extreme cases: \begin{align*} \mc O_{k,\zeta}:=\mc O^{(1,\ldots,1),(1,\ldots,1)}_{k,\zeta}, \quad \mc F_{k,\zeta}:=\mc O^{(k),(n-k)}_{k,\zeta}. \end{align*}
Recall that $L^0(\la)$ denotes the finite-dimensional irreducible $\mf{l}^\rl$-module of highest weight $\la$ in Section \ref{Subsection::IrrChOfL0}. Note $L^0(\la)$ can be extended to a $\mf p^\rl$-module by letting $\mf u^\rl$ act trivially. Denote the corresponding {\em parabolic Verma module} by \begin{align*} \Delta^\rl(\la)=\text{Ind}_{\mf p^\rl}^\G L^0(\la). \end{align*}
The following proposition is a characterization of the category $\mc O_{k,\zeta}^\rl$.
\begin{prop} {\em $\mc O_{k,\zeta}^\rl$} is the full subcategory of $\mc O_n$ of {\em $\mf p^{\rl}$}-locally finite, completely reducible {\em $\mf l^\rl$}-modules of $\La^{\emph{\rl}}_{k,\zeta}$-highest weights.
\end{prop}
\begin{proof} Let $\la\in\La^\rl_{k,\zeta}$. Note that $\Delta^\rl(\la)\cong \mc S\left(\mf u^{\rl,-}\right)\otimes L^0(\la)$ as an $\mf l^\rl$-module, where $\mc S\left(\mf u^{\rl,-}\right)$ denotes the supersymmetric tensor of $\mf u^{\rl,-}$. Since all the weights in $\mc S\left(\mf u^{\rl,-}\right)$ are integer weight, we see that all the $\mf l^\rl$-weights of $\Delta^\rl(\la)$ are $\mf l^\rl$-typical, and so $\Delta^\rl(\la)$ is a completely reducible $\mf l^\rl$-module by Lemma \ref{CharacterizationOfLocFinModules}. Therefore $\Delta^\rl(\la)$ is $\mf p^\rl$-locally finite and completely reducible over $\mf l^\rl$. Since $L(\la)$ is a quotient of $\Delta^\rl(\la)$, it follows that $L(\la)$ is also $\mf p^\rl$-locally finite and completely reducible as a $\mf l^\rl$-module. This completes the proof. \end{proof}
In the case $\texttt{r}=\underbrace{(1,1,\ldots,1)}_k$ and $\underbrace{\texttt{l}=(1,1,\ldots,1)}_{n-k}$ we shall write $\Delta(\la)$ for $\Delta^\rl(\la)$, which is consistent with earlier notation, while in the case $\texttt{r}=(k)$ and $\texttt{l}=(n-k)$ we shall write $K(\la)$ for $\Delta^\rl(\la)$.
\begin{rem} The $\mf q(n)$-module $L(\la)$, for $\la\in\La^\rl_{k,\zeta}$, is almost always infinite dimensional. Indeed, it follows from \cite[Theorem 4]{Pe} (see also \cite[Theorem 2.18]{CW}) that $L(\la)$ is finite dimensional if and only if $\la\in\La^+_{k,\zeta}$ and $k\in\{0,n\}$. \end{rem}
\begin{rem} \label{ParabolicSubcategoryRem} Basic features of parabolic subcategory for semisimple Lie algebras are well-known, see e.g., \cite[Chapter 9]{Hum08}. In the case of Lie superalgebras, we refer to \cite{Mar14} in which the parabolic subcategory $\widetilde{\mc{O}}^{\mf p^{\rl}}$ corresponding to $\mf p^{\rl}$ is defined to be the full subcategory of $\mc{O}_{n,\bar{0}}$ consisting of $\mf{p}^{\rl}$-locally finite, and $\mf{l}^{\rl}_{\bar{0}}$-semisimple $\mf{q}(n)$-modules, where $\mc{O}_{n,\bar{0}}$ is the underlying even category of $\mc{O}_{n}$. Note that the underlying even category of $\mc O_{k,\zeta}^\rl$ is precisely the full subcategory of $\widetilde{\mc{O}}^{\mf p^{\rl}}$ consisting of $\mf{q}(n)$-modules of $\La_{k,\zeta}$-weights since each weight in $\La_{k,\zeta}$ is $\mf{l}^{\rl}$-typical. \end{rem}
\section{Tilting modules in parabolic categories} \label{SectionTiltingModules}
Let $k,n\in\Z_{\ge 0}$ with $k\leq n$ and $\zeta \in \mathbb{C}\backslash \hf\Z$ as before. In this section, we study tilting modules in $\mc F_{k,\zeta}$, and formulate the BGG reciprocity in terms of tilting modules by means of the Arkhipov-Soergel duality (see, e.g., \cite[Corollary 5.8]{Br3}).
For a given $\la \in \La^\rl_{k,\zeta}$, we recall the definition and existence of tilting modules $T^\rl(\la)$ in $\mc O^\rl_{k,\zeta}$, provided by \cite[Theorem 6.3]{Br3} (also see \cite[Section 4.3]{Mar14}). In the case $\texttt{r}=\underbrace{(1,1,\ldots,1)}_k$ and $\underbrace{\texttt{l}=(1,1,\ldots,1)}_{n-k}$ (respectively, $\texttt{r}=(k)$ and $\texttt{l}=(n-k)$), i.e., $\la\in\La_{k,\zeta}$ (respectively, $\la\in\La^+_{k,\zeta}$), we denote the tilting module by $T(\la)$ (respectively, $U(\la)$).
For given $m\in\N$, recall that $w_0^{(m)}$ denotes the longest element in $\mf S_m$. The following lemma is well-known.
\begin{lem}\label{lem:dual:hwt} Let $m\in \mathbb{N}$. If $L(\la)$ be a finite-dimensional $\mf{q}(m)$-module then $L(\la)^*\cong L(-w_0^{(m)} \la )$. \end{lem} \begin{proof} Since $L(\la)$ is finite-dimensional, $L(\la)$ is a direct sum of irreducible $\gl(m)$-modules with dominant highest weights $\mu$ such that $\la-\mu\in\sum_{\alpha\in\Phi^+}\Z_{\ge 0}\alpha$. Thus, the lowest $\gl(m)$-weight in $L(\la)$ is $w_0^{(m)}\la$, and hence $L(\la)^*$ has highest weight $-w_0^{(m)}\la$. \end{proof}
Recall the supertrace $\text{str}_{V}(f)$ of an endomorphism $f= f_{\bar{0}}+ f_{\bar{1}}$ ($f_{\bar{0}}$ and $f_{\bar{1}}$ are respectively even and odd) on a superspace $V$ is defined by $\text{str}_{V}(f):= \text{tr}_{V_{\bar{0}}}f_{\bar{0}} -\text{tr}_{V_{\bar{1}}}f_{\bar{0}}$. We consider $\G=\bigoplus_{j\in\Z}\G_j$ with the $\Z$-gradation induced from \eqref{Z-gradation}. Recall that a Lie superalgebra homomorphism $\gamma: \G_{0} \rightarrow \mathbb{C}$ is called a semi-infinite character, if $\gamma([X,Y]) = \text{str}_{\G_{0}}(\text{ad}(X)\circ\text{ad}(Y))$, for $X\in \G_{1}, Y\in \G_{-1}$ (cf. \cite[Definition 1.1]{So} and \cite[Section 5]{Br3}). The proof of the following lemma is inspired by the proof of \cite[Lemma 7.4]{So}.
\begin{lem}\label{lem:0:semi} The trivial character $0:\G_0\rightarrow \C$ is a semi-infinite character for the $\Z$-gradation \eqref{Z-gradation} for $\G$. \end{lem}
\begin{proof}Let $X= X_{\bar{0}}+ X_{\bar{1}}$ and $Y= Y_{\bar{0}}+ Y_{\bar{1}}$ with $X_{\bar i}\in (\mf{g}_{1})_{\bar{i}},Y_{\bar i} \in (\mf{g}_{-1})_{\bar{i}}$ for $i =0,1$. We first note that $\text{str}_{\mf{g}_{0}}(\text{ad}X \circ \text{ad}Y) = \text{str}_{\mf{g}_{0}}(\text{ad}X_{\bar{0}} \circ \text{ad}Y_{\bar{0}}) + \text{str}_{\mf{g}_{0}}(\text{ad}X_{\bar{1}} \circ \text{ad}Y_{\bar{1}}) =\text{str}_{\mf{g}_{0}}(\text{ad}X_{\bar{1}} \circ \text{ad}Y_{\bar{1}})$, since $\mf{g}_{\bar{0}}$ and $\mf{g}_{\bar{1}}$ are isomorphic as $\mf{g}_{\bar{0}}$-modules. Thus, we may assume that $X\in (\G_1)_{\bar{1}}$, $Y \in (\G_{-1})_{\bar{1}}$.
Next, observe that, for each $A\in (\G_{0})_{\bar{0}}$, we have
\[
\text{str}_{\G_0}(\text{ad}[A,X]\circ \text{ad}Y) = \text{str}_{\G_0}(\text{ad}A\circ \text{ad}X\circ \text{ad}Y - \text{ad}X\circ \text{ad}A\circ \text{ad}Y) \]
\[= \text{str}_{\G_0}(\text{ad}X\circ \text{ad}Y\circ \text{ad}A - \text{ad}X\circ \text{ad}A\circ \text{ad}Y)
= \text{str}_{\G_0}(\text{ad}X\circ \text{ad}[Y,A]).\]
Furthermore, since $\G_{1}$ is a semisimple $\text{ad}(\G_{0})_{\bar{0}}$-module generated by root vectors of simple roots, it suffices to show that
\begin{align} \label{EqStrIsZero}
\text{str}_{\G_0}(\text{ad}X_{\alpha}\circ \text{ad}Y_{\beta}) =0,
\end{align}
for all $X_{\alpha} \in \G_{\alpha} \cap (\G_1)_\odd, Y_{\beta}\in \G_{\beta}\cap (\G_{-1})_\odd$ with $\alpha\in \Pi \setminus \Pi^{\rl}, \beta\in \Phi$.
Note that if $\alpha+\beta \neq 0$ then $(\text{ad}X_{\alpha}\circ \text{ad}Y_{\beta})(\G_{\gamma}) \subseteq \G_{\alpha+\beta+\gamma} \neq \G_{\gamma}$ and so \eqref{EqStrIsZero} holds. Therefore we may assume that $\beta =- \alpha$.
Consider the triangular decomposition $\G_0=\mf{n}_{0}^+ \oplus \mf{h} \oplus \mf{n}_{0}^-$ of $\G_0$, with $\mf{n}_{0}^+:= \oplus_{\eta\in \Phi^+}(\G_0)_{\eta}$ and $\mf{n}_{0}^-:= \oplus_{\eta\in \Phi \setminus \Phi^+}(\G_0)_{\eta}$. Let $\Phi(\mf{n}_{0}^+)$ and $\Phi(\mf{n}_{0}^-)$ be the sets of roots of $\mf{n}_{0}^+$ and $\mf{n}_{0}^-$, respectively. Note that $\mf{n}_{0}^+$, $\mf{h}$ and $\mf{n}_{0}^-$ are stable under $\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}$. Furthermore,
\begin{align*}
\text{ad}X_{\alpha}(\mf{n}_{0}^-) \subset \G_{\alpha+\Phi(\mf{n}_{0}^-)}=0, \ \
\text{ad}Y_{-\alpha}(\mf{n}_{0}^+) \subset \G_{-\alpha+\Phi(\mf{n}_{0}^+)}=0. \end{align*} Therefore we have \begin{align*}
\text{str}_{\G_0}(\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}) = \text{str}_{\mf{h}}(\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}) + \text{str}_{\mf{n}_{0}^-}(\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}) \\
= \text{tr}_{\mf{h}_{\bar{0}}}(\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}) - \text{tr}_{\mf{h}_{\bar{1}}}(\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}) + \text{str}_{\mf{n}_{0}^-}(\text{ad}[X_{\alpha},Y_{-\alpha}]).
\end{align*}
Note that $[X_{\alpha},Y_{-\alpha}]\in \mf{h}_{\bar{0}}$ and so $\text{str}_{\mf{n}_{0}^-}(\text{ad}[X_{\alpha},Y_{-\alpha}]) =0$ since there is a natural isomorphisms between $(\mf{n}_{0}^-)_{\bar{0}}$ and $(\mf{n}_{0}^-)_{\bar{1}}$ as $\mf{h}_{\bar{0}}$-modules.
Let $\pi: \mf{h}_{\bar{0}} \rightarrow \mf{h}_{\bar{1}}$ be the linear isomorphism defined by $\pi(e_{ii}) =\overline{e}_{ii}$, for $1\leq i \leq n$. Note that \begin{align*}\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}(h_{\bar{0}}) = \alpha(h_{\bar{0}}) [X_{\alpha}, Y_{-\alpha}], \ \ \text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}(h_{\bar{1}}) = \overline{\alpha}(\pi(h_{\bar{1}})) [X_{\alpha}, Y_{-\alpha}],\end{align*} for $i\in \{\bar{0},\bar{1}\}$ and $h_{i} \in \mf{h}_{i}$. It follows that $\text{tr}_{\mf{h}_{\bar{0}}}(\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha})= \text{tr}_{\mf{h}_{\bar{1}}}(\text{ad}X_{\alpha}\circ \text{ad}Y_{-\alpha}) = 0$. This completes the proof. \end{proof}
Lemma \ref{lem:0:semi}, together with \cite[Theorem 6.4]{Br3} (c.f.~\cite[Theorem 5.12]{So}) and Lemma \ref{lem:dual:hwt}, implies the following tilting module version of the BGG reciprocity. \begin{cor}\label{cor:tilting} For $\la ,\mu \in \La_{k,\zeta}^\rl$, we have {\em \begin{align*} \left(U(\la):K(\mu)\right) = [K(-w^\rl_0\mu):L(-w^\rl_0\la)]. \end{align*}} \end{cor}
\section{Formulation of the Kazhdan-Lusztig conjecture in $\mc F_{k,\zeta}$} \label{FormulationOfPKL}
Let $k,n\in\Z_{\ge 0}$ with $k\leq n$ and $\zeta \in \mathbb{C}\backslash \hf\Z$ as before. In \cite[Conjecture 5.10]{CKW} a Kazhdan-Lusztig type conjecture for $\mc{O}_{k,\zeta}$ was formulated in terms of canonical basis of $\TT^{m|n}$. In this section we formulate a parabolic version of the conjecture for $\mc{F}_{k, \zeta}$ in terms of canonical basis of $\mc E^{k|n-k}$.
We identify $\La_{k,\zeta}$ with $\mathbb{Z}^{k|n-k}$ as follows:
For $\la \in \La_{k,\zeta}$, we define $f_{\la} \in \mathbb{Z}^{k|n-k}$ by
\begin{align}\label{aux:fns} f_{\la}(i) =
\left\{ \begin{array}{ll} \la_{i+k+1}- \zeta, \text{ if } -k \leq i \leq -1, \\
-(\la_{i+k}+ \zeta), \text{ if } 1 \leq i \leq n-k.
\end{array} \right.
\end{align}
This gives a bijection between $\La_{k,\zeta}$ and $\mathbb{Z}^{k|n-k}$, and furthermore under this bijection various definitions correspond, e.g., $\sharp f_{\la} = \sharp \la$. Also, for a given $\mu \in \La_{k,\zeta}$, we let $\la \preceq \mu$ if $f_{\la} \preceq f_{\mu}$. Note that $\la \preceq \mu$ implies $\la \leq \mu$, for all $\la,\mu \in\La_{k,\zeta}$. Under this bijection the set $\La_{k,\zeta}^+$ is sent to $\mathbb{Z}^{k|n-k}_+$ so that we can identity these two sets.
Recall the canonical and dual canonical bases in Section \ref{SectionBases}. For $\la, \mu \in \La^+_{k,\zeta}$, we define $\ell_{\la,\mu}(q): =\ell_{f_{\la}, g_{\mu}}(q)$ and $u_{\la,\mu}(q): =u_{f_{\la}, g_{\mu}}(q)$, where $\ell_{g,f}(q)$ and $u_{g,f}(q)$ are as in Theorem \ref{KL-Lemma}. We have the following parabolic version of \cite[Conjecture 5.10]{CKW} for $\mc F_{k,\zeta}$, whose proof will be given in Section \ref{ProofOfMainThm}.
\begin{thm} \label{PKLConjecture} For $\la \in \La_{k,\zeta}^+$, we have \begin{align*} [U(\la)] = \sum_{\mu \preceq \la, \mu\in \La_{k,\zeta}^+} u_{\mu\la}(1)[K(\mu)],\\ [L(\la)] = \sum_{\mu \preceq \la,\mu\in \La_{k,\zeta}^+} \ell_{\mu\la}(1)[K(\mu)]. \end{align*} \end{thm}
\section{Serganova's fundamental lemma for $\mc F_{k,\zeta}$} \label{SectionSerganovasFunLem}
Let $k,n\in\Z_{\ge 0}$ with $k\leq n$ and $\zeta \in \mathbb{C}\backslash \hf\Z$ as before. In this section we shall prove the queer Lie superalgebra version of Serganova's fundamental lemma \cite[Theorem 5.5]{Ser}. Such a ``queer'' version for the category $\mc F_{k,\zeta}$ is needed for the purpose of adapting Brundan's proof of his finite-dimensional irreducible character formula for the general linear Lie superalgebra \cite[Theorem 4.37]{Br1} to our setting of queer Lie superalgebra.
Recall that $\ov{\alpha}:=\ep_i+\ep_j$, for a given $\alpha=\ep_i-\ep_j\in\Phi^+$ (Section \ref{SectionQueerLieSuperalgebra}). We first recall the following lemma of Penkov and Serganova: \begin{lem}\label{lem:verma:hom} \cite[Proposition 2.1]{PS2} Let $\alpha\in\Phi^+$ and suppose that $(\la,\ov{\alpha})=0$. Then $$\rm{Hom}_{\G}(\Delta(\la-\alpha),\Delta(\la))\not=0.$$ \end{lem}
The following theorem and its proof are inspired by \cite[Theorem 5.5]{Ser}.
\begin{thm}\label{thm:fund:fact} Let $\la\in\La^+_{k,\zeta}$. Suppose that $\alpha\in\Phi^+$ such that $(\la,\ov{\alpha})=0$ and $\la-\alpha\in\La^+_{k,\zeta}$. Then \begin{align*} \rm{Hom}_\G\left(K(\la-\alpha),K(\la)\right)\not=0. \end{align*} In particular, $[K(\la):L(\la-\alpha)]\not=0$. \end{thm}
\begin{proof} In this proof we shall respectively denote $\mf p^\rl$, $\mf l^\rl$ and $\mf u^\rl$ by $\mf p$, $\mf l$ and $\mf u$.
First we have an exact sequence of $\mf l$-modules \begin{align*} 0\longrightarrow I^0(\la)\longrightarrow \Delta^0(\la)\longrightarrow L^0(\la)\longrightarrow 0, \end{align*} where $\Delta^0(\la)$ denotes the $\mf l$-Verma module of highest weight $\la$ (Section \ref{SubsectionZgradations}). This exact sequence trivially extends to an exact sequence of $\mf p$-module by letting $\mf u$ act trivially, and thus we have an exact sequence of $\G$-modules by parabolic induction \begin{align*} 0\longrightarrow \text{Ind}^\G_{\mf p}I^0(\la)\longrightarrow \Delta(\la)\longrightarrow K(\la)\longrightarrow 0. \end{align*} By Lemma \ref{lem:verma:hom} we have \begin{align*} \text{Hom}_{\G}(\Delta(\la-\alpha),\Delta(\la))\not=0, \end{align*} and thus there exists a non-zero $\mf b$-singular vector $v_{\la-\alpha}\in\Delta(\la)$. It suffices to show that $v_{\la-\alpha}\not\in \text{Ind}^\G_{\mf p}I^0(\la)$.
Suppose on the contrary that $v_{\la-\alpha}\in \text{Ind}^\G_{\mf p}I^0(\la)$. Now $v_{\la-\alpha}$ is of course $\mf b_\even$-singular. We observe that if $\mu\in\h^*$ is the highest weight of a composition factor in $I^0(\la)$, then \begin{align*} \mu=w(\la), \end{align*} for some $w\in \mf S_k\times\mf S_{n-k}$. This is a direct consequence of \cite[Theorem 1]{FM}, according to which we have an equivalence of categories between strongly typical blocks of $\mf{q}(k)\oplus\mf{q}(n-k)$-modules and the corresponding blocks of $\mf{gl}(k)\oplus\mf{gl}(n-k)$-modules.
Thus, any weight $\mu$ of a $\mf b_\even$-singular vector in $\text{Ind}^\G_{\mf p}I^0(\la)$ is of the form \begin{align*} \mu=w(\la)-\gamma, \end{align*} where $\gamma$ is a linear $\Z_{\geq 0}$-combination roots in $\Phi^+(\mf{u})$. Thus, we have \begin{align*} \mu=\la-\eta-\gamma, \end{align*} where $\eta$ is a $\Z_{\geq 0}$-linear combination of positive roots of $\mf l$. Thus, by assumption we have $\la-\alpha=\la-\eta-\gamma$ and so \begin{align}\label{aux100} \alpha=\eta+\gamma. \end{align} Now, $\alpha$ is a root in $\mf u$, and so \eqref{aux100} implies that $\gamma\in\Phi^+(\mf u)$, and there are three possibilities for $\eta$: \begin{align*} \eta=\begin{cases}\delta_i-\delta_s+\delta_t-\delta_j,\quad 1\le i<s\le k,k+1\le t<j\le n,\cr \delta_i-\delta_s,\quad 1\le i<s\le k,\cr \delta_t-\delta_j,\quad k+1\le t<j\le n. \end{cases} \end{align*}
Let us first consider the case $\eta=\delta_i-\delta_s$, with $1\le i<s\le k$. Thus, we have $w(\la)=\la-\delta_i+\delta_s$. Now we have $w\in\mf S_k\times\mf S_{n-k}$, and also all the $\la_i$s are distinct, for $1\le i\le k$. Thus, we must have \begin{align*} \la_i-1=\la_s. \end{align*} Therefore, we have $(\la,\eta)=\la_i-\la_s=1$ and $(\alpha,\eta)=1$, so that we have $(\la-\alpha,\delta_i-\delta_s)=0$. But then $\la-\alpha\not\in\La^+_{k,\zeta}$, which is a contradiction.
By a similar argument, the case $\eta=\delta_t-\delta_j$ with $k+1\le i<s\le n$ leads to a contradiction as well.
Finally, we assume that $\eta = \delta_i-\delta_s+\delta_t-\delta_j$, for some $1\le i<s\le k$ and $ k+1\le t<j\le n$. In this case, we have $\gamma = \delta_s - \delta_t$. Similarly, since each component of $\la$ are distinct, it follows from $w\in \mf S_k\times\mf S_{n-k}$ that $\la_ i -1 = \la_s$ and $\la_t -1 = \la_j$. Therefore, $ (\la,\eta)= \la_i- \la_s+\la_t-\la_j =2$ and $(\alpha,\eta)=2$. Now $(\la-\alpha,\delta_i-\delta_s) +(\la-\alpha,\delta_t-\delta_j) = (\la-\alpha,\eta)=0$, which also leads to $\la-\alpha\not\in\La^+_{k,\zeta}$. \end{proof}
\section{Proof of the main theorem} \label{ProofOfMainThm}
Let $k,n\in\Z_{\ge 0}$ with $k\leq n$. Recall that $\zeta \in \mathbb{C}\backslash \hf\Z$ is fixed in Section \ref{SectionNotations}, and the free abelian group $P = \oplus_{a\in \mathbb{Z}}\mathbb{Z}\epsilon_a$ is defined in Section \ref{SectionTheFockSpace}. We let $\mc{F}:=\mc{F}_{k,\zeta}$ in this section.
Let $\text{wt}: \Lambda^+_{k,\zeta} \rightarrow P$ be the weight function defined by (c.f. \cite[Section 2-c]{Br2}) \begin{align*} \text{wt}(\la) := \sum_{i=1}^{k}\epsilon_{\la_{i}-\zeta} - \sum_{i=k+1}^{n}\epsilon_{-(\la_{i}+\zeta)}. \end{align*} It is well-known that $\chi_{\la} = \chi_{\mu}$ if and only if $\text{wt}(\la) = \text{wt}(\mu)$ (see, e.g., \cite[Theorem 2.48]{CW}). We have decomposition $\mc F = \oplus_{\la\in \mf{h}_{\bar{0}}^*} \mc{F}_{\chi_{\la}} = \oplus_{\gamma \in P} \mc{F}_{\gamma}$ according to central characters $\chi_{\la}$ with $\text{wt}(\la) = \gamma$.
Let $\mathbb{C}^{n|n}$ and $(\mathbb{C}^{n|n})^*$ be the standard representation and its dual, respectively. Denote the projection functor from $\mc{F}$ to $\mc{F}_{\gamma}$ by $\text{pr}_{\gamma}$. We define the {\em translation functors} $\text{E}_{a}, \text{F}_{a}: \mc{F} \rightarrow \mc{F}$ as follows \begin{align}
\text{E}_{a} (M):= \text{pr}_{\gamma+(\epsilon_a-\epsilon_{a+1})}(M\otimes (\mathbb{C}^{n|n})^*), \ \
\text{F}_{a} (M):= \text{pr}_{\gamma-(\epsilon_a-\epsilon_{a+1})}(M\otimes \mathbb{C}^{n|n}), \end{align} for all $M\in \mc{F}_{\gamma}$, $\gamma \in P$ , $a\in \mathbb{Z}$. For each $a\in \mathbb{Z}$ , it is not hard to see that both $\text{E}_{a}$ and $\text{F}_{a}$ are exact and bi-adjoint to each other.
We write $\la \rightarrow_{a} \mu$ if $\la , \mu \in \Lambda^+_{k,\zeta}$ and there exists $1\leq i \leq k$ such that $\la_i = \mu_i-1 = a+\zeta$ or there exists $k+1 \leq i'\leq n$ such that $\la_{i'} = \mu_{i'}-1 = -a -1 -\zeta $, and in addition, $\la_j = \mu _j$ for all $ j\neq i$ in the former case, for all $j\neq i'$ in the later case. Let $\mc{K}(\mc{F})$ be the Grothedieck group of $\mc{F}$ and denote the element corresponding to $M\in \mc{F}$ by $[M]$.
We have the following lemma \cite[Lemma 4.2]{Ch}.
\begin{lem} \label{chOfKInTranslationFunctor} Let $\la\in \Lambda^+_{k,\zeta}$. Then both $\emph{E}_{a} K(\la)$ and $\emph{F}_{a} K(\la)$ have flags of parabolic Verma modules and we have the following formula: \begin{align*} [\emph{E}_{a} K(\la)] = 2\sum_{\mu \rightarrow_{a} \la} [K(\mu)], \quad [\emph{F}_{a} K(\la)] = 2\sum_{\la \rightarrow_{a} \mu} [K(\mu)]. \end{align*} \end{lem}
We defined the $\mathbb{Z}$-form $\mc{E}_{\mathbb{Z}}^{k|n-k}$ of $\mc{E}^{k|n-k}$, namely, $\mc{E}_{\mathbb{Z}}^{k|n-k}: = \mathbb{Z} \otimes_{\mathbb{Z}[q,q^{-1}]}\mc{E}_{\mathbb{Z}[q,q^{-1}]}^{k|n-k}$ by letting $q=1$, where $\mc{E}_{\mathbb{Z}[q,q^{-1}]}^{k|n-k}$ is the $\mathbb{Z}[q,q^{-1}]$-lattice spanned by $\{K_f\}_{f\in \mathbb{Z}^{k|n-k}_+}$, and for given $f\in \La^+_{k,\zeta}$ we let $K_{f}(1):= 1\otimes K_{f}, U_f(1):=1\otimes U_f\in \mc{E}_{\mathbb{Z}}^{k|n-k}$.
Let $\mc{A}_{k|n-k}^{\Delta}$ be the full subcategory of finite-dimensional modules over the general linear Lie superalgebra $\mf{gl}(k|n-k)$ consisting of objects that have a flag of Kac modules, see \cite[Sections 4-a,b]{Br1}. Recall that $\mc{A}_{k|n-k}^{\Delta}$ is also equipped with translation functors (see e.g., \cite[Section 4-b]{Br1} and \cite[Sections 3.4 and 5.1]{CW08}). Let $ \mc{F}^{\Delta} $ be the full subcategory of $\mc{F}$ of all modules which have a flag of $K(\la)$ with $\la \in \La^+_{k,\zeta}$. Let $\mc K(\mc F^\Delta)$ be the Grothendieck group of $\mc F^\Delta$. Now Lemma \ref{chOfKInTranslationFunctor}, together with \cite[Corollary 4.26 and Theorem 4.28]{Br1}, implies the following proposition that says that the translation functors for $\mc{F}^{\Delta}$ is the same as the translation functors on $\mc{A}_{k|n-k}^{\Delta}$ on the level of Grothendieck groups up to a $2$-factor.
\begin{prop} \label{PropIsoBetweenKandFockSpace}
Let $j:\mc K(\mc F^\Delta)\rightarrow \mc E^{k|n-k}_\Z$ be the $\Z$-isomorphism defined by \begin{align} \label{IsoBetweenKandFockSpace} j([\text{K}(\la)]) = \text{K}_{f_{\la}}(1) , \ \ \text{ for } \la\in \Lambda^+_{k,\zeta}.
\end{align} Then the representation theoretically defined functors $\emph{F}_a$ and $\emph{E}_{a}$ on $\mc{F}$ decategorify to the Chevalley generators $2F_a$ and $2E_a$ of $\bold U_{q}(\mf{gl}_{\infty})|_{q=1}$ on $\mc{E}_{\mathbb{Z}}^{k|n-k}$. \end{prop}
\begin{prop} \label{TypicalTilting} Let $\la\in\La^+_{k,\zeta}$. If $\la$ is typical, then $K(\la)=L(\la)=U(\la)$. \end{prop} \begin{proof} We have a surjection $K(\la)\rightarrow L(\la)$ that sends the highest weight space to the highest weight space. Now, if $K(\la)$ has a singular vector, then its weight $\mu$ lies $\La^+_{k,\zeta}$ and furthermore we have identical central character $\chi_\la=\chi_\mu$. Since $\la$ is typical, we must have $\la=\mu$. Thus, $K(\la)=L(\la)$ is irreducible.
Note that $\la\in\La^+_{k,\zeta}$ is typical if and only if $-w_0^+\la\in\La^+_{k,-\zeta}$ is typical. Thus, we have $K(-w_0^+\la)=L(-w_0^+\la)$, and hence by Corollary \ref{cor:tilting}, we have $U(\la)=K(\la)$. \end{proof}
Let $\la\in \La^+_{k,\zeta}$ and $a\in \mathbb{Z}$. It is known that both $\text{E}_aU(\la)$ and $\text{F}_aU(\la)$ are direct sums of tilting modules (see, e.g., \cite[Corollary 4.27]{Br1}). Furthermore, we have the following lemma \cite[Lemma 4.3]{Ch}.
\begin{lem} \label{TransFunOfTiltings} Let $\la\in \La^+_{k,\zeta}$. Then the multiplicity of each non-zero tilting module summand of $\emph{E}_aU(\la)$ and $\emph{F}_aU(\la)$ is even. \end{lem}
The following lemma follows from Procedure \ref{Br1Procedure}.
\begin{lem} \label{ExpressionOfUf} For every $f\in \mathbb{Z}^{k|n-k}_{+}$, we have $U_f(1) \in K_f(1) + \sum_{g \prec f}\mathbb{Z}_{\geq 0} K_g(1)$. \end{lem}
We have now all the ingredients to adapt Method two of the proof of \cite[Theorem 4.37]{Br1} to prove that
Procedure \ref{Br1Procedure} specialized at $q=1$ gives the construction of the tilting modules in $\mc F$. \begin{thm} \label{ConstructionOfTiltings} Let $\la\in\La^+_{k,\zeta}$. Then $[U(\la)]$ is mapped to $U_{f_{\la}}(1)$ under the isomorphism $j$ in \eqref{IsoBetweenKandFockSpace}. \end{thm}
\begin{proof} We shall proceed by induction on the degree of atypcality $\sharp\la$ of $\la$. If $\sharp\la=0$, then $K(\la)=L(\la)=U(\la)$ by Lemma \ref{TypicalTilting}. Assume that $\sharp \la >0$ and furthermore $j([U(\nu)]) = U_{h}(1)$, where $\nu \in \La^+_{k,\zeta}$ satisfies $h = f_{\nu}$. Let $\widehat{X}_a \in \{E_a, F_a\}_{a\in \mathbb{Z}}$ be the operators given in Procedure \ref{Br1Procedure}. For each tilting module $U\in \mc{F}$ we define $X_aU$ to be a direct summand of the direct sum of two isomorphic copies of $\widehat{X}_aU$ (see Lemma \ref{TransFunOfTiltings}).
First note that $j([X_aU(\nu)]) = \widehat{X}_aU_{h}(1) = U_{f_{\la}}(1)$. Therefore, we may conclude that $U(\la)$ is a direct summand of $X_aU(\nu)$ by Lemma \ref{ExpressionOfUf}. We shall prove that $U(\la)=X_a U(\nu)$ by proving that $X_a U(\nu)$ is indecomposable.
Suppose $X_a U(\nu)$ is decomposable. Let $ X_a U(\nu)=T_1\oplus T_2$ with $T_1=U(\la)$. It follows from Lemma \ref{LemmaForProcedure} that
\begin{align*} j([Y_aX_aU(\nu)]) = \widehat{Y}_a\widehat{X}_aU_h(1)= \left\{ \begin{array}{ll} U_h(1), \text{ if } \sharp \la = \sharp \nu,\\ 2U_h(1), \text{ if } \sharp \la -1 = \sharp \nu.
\end{array} \right.
\end{align*}
Since $\widehat{X}_a,\widehat{Y}_a$ are bi-adjoint to each other, as in the proof of \cite[Theorem 4.37]{Br1}, we have $(Y_aT_i:U(\nu))\neq 0$ for $i=1,2$. This means that $j([Y_aX_aU(\nu)]) =2U_h(1) $. Therefore, \begin{align*} Y_aX_a U(\nu)=U(\nu)\oplus U(\nu), \end{align*} and so $Y_aT_1=Y_aT_2=U(\nu)$. We obtain $[Y_aU(\la):L(\nu)]=1$. We will show that $[Y_a U(\la):L(\nu)]\ge 2$ and so get a contradiction.
By Lemma \ref{LemmaForProcedure} again, there is $\mu=\la-\alpha\in\La^+_{k,\zeta}$ with $\alpha\in\Phi^+(\mf u)$, $(\la,\ov{\alpha})=0$ such that $\widehat{X}_aK_h(1) = K_f(1)+K_{f_{\mu}}(1)$. By Corollary \ref{cor:tilting} we have \begin{align*} \left(U(\la):K(\mu)\right)=[K(-w_0^+\mu):L(-w_0^+\la)]=[K(-w_0^+\la+w_0^+\alpha):L(-w_0^+\la)].\end{align*} Note that \begin{align*} (-w_0^+\la+w_0^+\alpha ,\ov{w_0^+\alpha})=-(w_0^+\la,w_0^+\ov{\alpha})=-(\la,\ov{\alpha})=0. \end{align*} Consequently, by Theorem \ref{thm:fund:fact} we have $[K(-w_0^+\la+w_0^+\alpha):L(-w_0^+\la)]\ge 1$ and hence $\left(U(\la):K(\mu)\right)\ge 1$.
Since $\left(U(\la):K(\la)\right)= 1$ and $[K(\la):L(\mu)]\ge 1$ by Theorem \ref{thm:fund:fact}, we conclude that \begin{align}\label{aux101} \left[U(\la):L(\mu)\right]\ge 2. \end{align}
Furthermore, since $X_aK(\nu)$ has a filtration with $K(\mu)$ on the top, by the adjunction between $\widehat{X}_a, \widehat{Y}_a$ again we have \begin{align*} \text{Hom}_\G\left(K(\nu),\widehat{Y}_a L(\mu)\right)= \text{Hom}_\G\left(\widehat{X}_a K(\nu), L(\mu)\right)\not=0, \end{align*} which implies that $[Y_aL(\mu):L(\nu)]\ge 1$. Finally, combining this with \eqref{aux101} gives $[Y_a U(\la):L(\nu)]\ge 2$. \end{proof}
We are now ready to prove Theorem \ref{PKLConjecture}.
\begin{proof}[Proof of Theorem \ref{PKLConjecture}]
By Theorem \ref{ConstructionOfTiltings} and Corollary \ref{cor:tilting} we have the multiplicity formula $u_{\mu,\la}(1) = (U(\la):K(\mu))$ and $u_{-w_0^+\la,-w_0^+\mu}(1) = (K(\la):L(\mu))$. Namely, we have the character formulas
\begin{align*}
&\text{ch}U(\la) = \sum_{\mu\preceq \la} u_{\mu,\la}(1)\text{ch}K(\mu),\\
&\text{ch}K(\la) = \sum_{\mu\preceq \la} u_{-\omega_0^+ \la, -\omega_0^+ \mu}(1) \text{ch}L(\mu).
\end{align*}
Let $\textsf{1}_{k|n-k}: = \sum_{1\leq i\leq k}\delta_i -\sum_{k+1\leq i\leq n}\delta_i$. From \cite[Corollary 3.14 and (4.17)]{Br1}), we have that the following transition matrix
\begin{align*}
\left(u_{-\omega_0^+ \la, -\omega_0^+ \mu}(1)\right)_{\la,\mu \in \La^+_{k,\zeta}}
\end{align*}
has inverse matrix
\begin{align*}
\left(\ell_{ \mu +(n+1) \textsf{1}_{k|n-k}, \la+(n+1) \textsf{1}_{k|n-k}}(1)\right)_{\la,\mu \in \La^+_{k,\zeta}} =\left(\ell_{ \mu, \la}(1)\right)_{\la,\mu \in \La^+_{k,\zeta}}.
\end{align*}
The completes the proof. \end{proof}
\section{Kac-Wakimoto and Sergeev-Pragacz type character formulas}\label{sec:KW:formula}
In this section we apply Theorem \ref{PKLConjecture} to obtain closed character formula for analogues of Kostant and polynomial modules of $\mf{q}(n)$. We first recall the notation of $\h'_{m|n}$, $\delta'_a$ ($a\in I(m|n)$), and $\Phi'^+$ from Section \ref{SectionQueerLieSuperalgebra}. Furthermore, given a partition $\mu=(\mu_1,\mu_2,\ldots)$, we let $\mu^t$ denote its conjugate partition. Finally, recall that a partition $\mu$ is called a {$(k|n-k)$-hook partition} if $\mu_{k+1}\le n-k$.
Let $0\le k\le n$ and let $\la\in\La_{k,\zeta}$. Define $\rho=\sum_{i=1}^k (k-i+1-\frac{n+1}{2})\delta_i+\sum_{j=k+1}^n (k-j+\frac{n+1}{2})\delta_j$. Define $\la'=\sum_{i=1}^n\la'_i\delta_i$ by $$\la':=\sum_{i=1}^k(\la_i-\zeta-k+i-1+\frac{n+1}{2})\delta_i+\sum_{j=k+1}^n(\la_j+\zeta+j-k-\frac{n+1}{2})\delta_j.$$
Identifying $\delta_i$ with $\delta'_{-k-1+i}$ and $\delta_j$ with $\delta'_{j-k}$, for $1\le i\le k$ and $k+1\le j\le n$, we may regard $\la'$ and $\rho$ as elements in $\h'^*_{k|n-k}$ and thus as weights for $\gl(k|n-k)$. This gives a bijection between the set $\La_{k,\zeta}$ and the set of integral weights for $\gl(k|n-k)$. In this section we shall freely use this identification and thus identify $\h_\even^*$ with $\h_{k|n-k}'^*$.
Recall that the Borel subalgebras of a general linear Lie superalgebra $\gl(k|n-k)$ are in general not conjugate under its Weyl group $\mf S_{k|n-k}=\mf S_k\times \mf S_{n-k}$. However, it is well-known \cite{LSS} that any two non-conjugate Borel subalgebras with identical even subalgebra can be transformed to each other by a sequence of odd reflections. For a Borel subalgebra ${\mf b'}$ of $\gl(k|n-k)$ let us denote the set of positive and simple roots of ${\mf b'}$ by $\Phi'^+_{{\mf b'}}$ and $\Pi'_{{\mf b'}}$, respectively. Recall that the set of positive roots of the standard Borel subalgebra is denoted by $\Phi'^+$.
Let us denote the highest weight irreducible $\gl(k|n-k)$-module of highest weight $\nu$ with respect to the Borel subalgebra ${\mf b'}$ by $L'_{{\mf b'}}(\nu)$. Let $\rho_{{\mf b'}}$ denote the signed half sum of the positive roots in ${\mf b'}$. Above, the notation $\rho$ stands for the Weyl vector with respect to the standard Borel.
Recall the notion of a $\gl(k|n-k)$-Kostant module from \cite{BS}. In the language of \cite{SZ1} a finite-dimensional irreducible $\gl(k|n-k)$-module of highest weight (with respect to the standard Borel subalgebra) $\la$ is a Kostant module, if $\la$ is {\em totally connected}. By \cite{CHR} it follows that a finite-dimensional irreducible module $L'$ is a Kostant module if and only if there exists a weight $\nu$ and a Borel subalgebra ${\mf b'}$ with a distinguished subset $S\subseteq\Pi'_{{\mf b'}}$ consisting of mutually orthogonal roots such that (i) $L'\cong L'_{{\mf b'}}(\nu)$, (ii) $\sharp\nu=|S|$, and (iii) $S$ is orthogonal to $\nu+\rho_{\mf b'}$. Furthermore, the character for such a module is given by the so-called Kac-Wakimoto character formula which was conjectured in \cite{KW} and established (in the type $A$ case) in \cite{CHR}:
\begin{align}\label{KW:gl}
\text{ch} L'_{{\mf b'}}(\nu)=\frac{1}{\sharp\nu!}\frac{\prod_{\beta\in\Phi'^+_{{\mf b'},\bar{1}}}e^{\beta/2}+e^{-\beta/2}}{\prod_{\alpha\in\Phi'^+_{{\mf b'},\bar{0}}}e^{\alpha/2}-e^{-\alpha/2}} \sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w\left(\frac{e^{\nu+\rho_{{\mf b'}}}}{\prod_{\gamma\in S}1+e^{-\gamma}}\right). \end{align}
\begin{lem}\label{KW:KL:iden}
Let $\la\in\La^+_{k,\zeta}$ such that $L'(\la')$ is a $\gl(k|n-k)$-Kostant module. Suppose that $L'(\la')\cong L'_{{\mf b'}}(\la'_{{\mf b'}})$ such that $S\subseteq\Pi'_{\mf b'}$ is a distinguished subset consisting of mutually orthogonal roots with $\sharp\la'=|S|$ and orthogonal to $\la'_{\mf b'}+\rho_{\mf b'}$. Then we have the following identity in $\h_\even^*$: \begin{align*}
\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w(e^{\mu'+\rho}) = \frac{1}{\sharp\la!} \sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w\left(\frac{e^{\la'_{{\mf b'}}+\rho_{{\mf b'}}}}{\prod_{\gamma\in S}1+e^{-\gamma}}\right). \end{align*} \end{lem}
\begin{proof}
Let $K'(\la')$ denote the Kac module of $\gl(k|n-k)$ of highest weight $\la'$ with respect to the standard Borel subalgebra. By \cite[Theorem 4.37]{Br1} we have \begin{align*} \text{ch}L'(\la')=\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\text{ch}K'(\mu'). \end{align*} Combining this with \eqref{KW:gl} we have the identity: \begin{align*}
\sum_{\mu\preceq\la}&\ell_{\mu\la}(1) \frac{\prod_{\beta\in\Phi'^+_\odd}e^{\beta/2}+e^{-\beta/2}} {\prod_{\alpha\in\Phi'^+_\even}e^{\alpha/2}-e^{-\alpha/2}}\sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w(e^{\mu'+\rho})=\\
&\frac{1}{\sharp\la'!}\frac{\prod_{\beta\in\Phi'^+_{{\mf b'},\bar{1}}}e^{\beta/2}+e^{-\beta/2}}{\prod_{\alpha\in\Phi'^+_{{\mf b'},\bar{0}}}e^{\alpha/2}-e^{-\alpha/2}} \sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w\left(\frac{e^{\la_{{\mf b'}}'+\rho_{{\mf b'}}}}{\prod_{\gamma\in S}1+e^{-\gamma}}\right). \end{align*} Since the even subalgebra of ${\mf b'}$ and that of the standard Borel subalgebra coincide, we have \begin{align*} \frac{\prod_{\beta\in\Phi'^+_\odd}e^{\beta/2}+e^{-\beta/2}} {\prod_{\alpha\in\Phi'^+_\even}e^{\alpha/2}-e^{-\alpha/2}} = \frac{\prod_{\beta\in\Phi'^+_{{\mf b'},\bar{1}}}e^{\beta/2}+e^{-\beta/2}}{\prod_{\alpha\in\Phi'^+_{{\mf b'},\bar{0}}}e^{\alpha/2}-e^{-\alpha/2}}. \end{align*} From this the lemma follows. \end{proof}
Note that corresponding to the Borel subalgebra ${\mf b'}$ for $\gl(k|n-k)$ we have a Borel subalgebra of $\G=\mf{q}(n)$, which is obtained in a similar way as for $\gl(k|n-k)$ with the sequence of odd reflections replaced by the corresponding sequence of twisting functors \cite{Ch}.
For $\la\in\La^+_{k,\zeta}$ we call an irreducible $\mf q(n)$-module $L(\la)$ a {\em Kostant module}, if $L'(\la')$ is a Kostant module of $\gl(k|n-k)$. We can now prove the following Kac-Wakimoto type character formula for Kostant modules of $\mf q(n)$.
\begin{thm}\label{thm:KW:formula}
Let $\la\in\La^+_{k,\zeta}$ such that $L(\la)$ is a Kostant module. Let ${\mf b'}$ be the corresponding Borel subalgebra of $\gl(k|n-k)$ with a distinguished set $S\subseteq\Pi'_{{\mf b'}}$ consisting of mutually orthogonal roots and $\sharp\la'=\sharp\la=|S|$ and orthogonal to $\la'_{\mf b'}+\rho_{\mf b'}$. Let $\la_{{\mf b'}}=\la'_{{\mf b'}}+\rho_{{\mf b'}}+\zeta 1_{k|n-k}$. Then we have \begin{align*}
\text{ch}L(\la)=\frac{2^{\lceil{n/2}\rceil}}{\sharp\la!} \prod_{\alpha\in\Phi^+}\frac{1+e^{-\alpha}}{1-e^{-\alpha}}\sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w\left(\frac{e^{\la_{{\mf b'}}}}{\prod_{\gamma\in S}1+e^{-\gamma}}\right). \end{align*} \end{thm}
\begin{proof} By Theorem \ref{PKLConjecture} we have $\text{ch}L(\la)=\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\text{ch}K(\mu)$. Thus, we compute \begin{align*} \text{ch}L(\la)
&=2^{\lceil{n/2}\rceil}\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\prod_{\beta\in\Phi(\mf u^+)}\frac{1+e^{-\beta}}{1-e^{-\beta}}\sum_{w\in \mf S_{k|n-k}}(-1)^{\ell(w)}w\left(e^\mu\right)\prod_{\beta\in\Phi^+(\mf l)}\frac{1+e^{-\beta}}{1-e^{-\beta}}\\
&=2^{\lceil{n/2}\rceil}\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\prod_{\beta\in\Phi^+}\frac{1+e^{-\beta}}{1-e^{-\beta}}\sum_{w\in \mf S_{k|n-k}}(-1)^{\ell(w)}w\left(e^\mu\right)\\
&=2^{\lceil{n/2}\rceil}\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\prod_{\beta\in\Phi^+}\frac{1+e^{-\beta}}{1-e^{-\beta}}\sum_{w\in \mf S_{k|n-k}}(-1)^{\ell(w)}w\left(e^{\mu'+\rho+\zeta 1_{k|n-k}}\right)\\
&=2^{\lceil{n/2}\rceil}\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\prod_{\beta\in\Phi^+}\frac{1+e^{-\beta}}{1-e^{-\beta}}\sum_{w\in \mf S_{k|n-k}}(-1)^{\ell(w)}w\left(e^{\mu'+\rho}\right)e^{\zeta 1_{k|n-k}}\\
&= \frac{2^{\lceil{n/2}\rceil}}{\sharp\la!} \prod_{\beta\in\Phi^+}\frac{1+e^{-\beta}}{1-e^{-\beta}} \sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w\left(\frac{e^{\la'_{{\mf b'}}+\rho_{{\mf b'}}}}{\prod_{\gamma\in S}1+e^{-\gamma}}\right) e^{\zeta1_{k|n-k}}, \end{align*} where in the last identity we have used Lemma \ref{KW:KL:iden}. The theorem now follows. \end{proof}
\begin{example}
Consider $\mf q(4)$ and $\la=(\zeta+2)\delta_1+(\zeta+1)\delta_2+(-\zeta-1)\delta_3+(-\zeta-2)\delta_4$ so that $k=2$ and $\sharp\la=2$. Furthermore, $\Phi^+=\{\delta_i-\delta_j|1\le i<j\le 4\}$ and the integral Weyl group here is $\mf S_2\times\mf S_2$, consisting of permutations on the letters $\{1,2\}$ and $\{3,4\}$. Then $\la_{{\mf b'}}=(\zeta+2)\delta_1+(\zeta+2)\delta_2+(-\zeta-2)\delta_3+(-\zeta-2)\delta_4$ and $S=\{\delta_1-\delta_3,\delta_2-\delta_4\}$. We have \begin{align*}
\text{ch}L(\la)=2 \prod_{1\le i<j\le 4}\frac{1+e^{-\delta_i+\delta_j}}{1-e^{-\delta_i+\delta_j}}\sum_{w\in\mf S_2\times\mf S_{2}}(-1)^{\ell(w)}w\left(\frac{e^{(\zeta+2) 1_{2|2}}}{(1+e^{-\delta_1+\delta_3})(1+e^{-\delta_2+\delta_4})}\right). \end{align*} \end{example}
\begin{rem}
Theorem \ref{thm:KW:formula} suggests that the Kostant modules for $\mf{q}(n)$ have BGG type resolutions in terms of the parabolic Verma modules $K(\mu)$ in analogy to the resolution of $\gl(k|n-k)$-Kostant modules by Kac modules \cite{CKL, BS}. \end{rem}
We recall that every irreducible polynomial module of $\gl(k|n-k)$, i.e., every irreducible submodule of a tensor power of the standard module $\C^{k|n-k}$, is a Kostant module. For such modules, recall that one has another closed classical character formula, called the Sergeev-Pragacz formula (see, e.g., \cite[Page 60]{Mac} or \cite[\S12.2]{Mu}). Below, we shall derive an analogue of this formula for $\mf q(n)$-Kostant modules that correspond to polynomial modules for the general linear Lie superalgebra.
It is well-known that the isomorphism classes of irreducible polynomial modules of the Lie superalgebra $\gl(k|n-k)$ are in bijection with the so-called $(k|n-k)$-hook partitions. To be more precise, let $\nu=\sum_{i=1}^n\nu_i\delta'_i\in\h_{k|n-k}'^*$. A necessary and sufficient condition for $\nu$ to the highest weight (with respect to the standard Borel subalgebra) of an irreducible polynomial representation is that $\nu^{-}=(\nu_1,\ldots,\nu_k)$ and $\nu^+=(\nu_{k+1},\ldots,\nu_{n})$ are both partitions, and in addition $(\nu^{-},(\nu^{+})^t)$ is a $(k|n-k)$-hook partition.
Let $L'(\nu)$ be a polynomial module of $\gl(k|n-k)$. Then we can visualize the corresponding hook partition diagrammatically as follows: \begin{center} \hskip 0cm \setlength{\unitlength}{0.25in} \begin{picture}(7.5,6.5) \put(0,0){\line(1,0){1}} \put(1,0){\line(0,1){2}} \put(1,2){\line(1,0){1}} \put(2,2){\line(0,1){1}} \put(2,3){\line(1,0){1}} \put(3,3){\line(0,1){1}} \put(3,4){\line(1,0){1}} \put(4,4){\line(0,1){1}} \put(4,5){\line(1,0){3}} \put(7,5){\line(0,1){1}} \multiput(-.4,3)(0.4,0){13}{\line(1,0){0.2}} \put(2.5,4.5){\makebox(0,0)[c]{\Large$\nu^-$}} \put(0.6,2.3){\makebox(0,0)[c]{$(\nu^+)^t$}} \put(7,6){\line(-1,0){7}} \put(0,6){\line(0,-1){6}} \put(-.1,3){\line(1,0){0.2}} \put(-.7,3){\makebox(0,0)[c]{$k$}} \put(4,6.1){\line(0,-1){0.2}} \put(4,6.5){\makebox(0,0)[c]{$n-k$}} \put(4,3){\linethickness{1pt}\line(0,-1){3}} \put(4,3){\linethickness{1pt}\line(1,0){3}} \end{picture} \end{center} We can associate to the corresponding hook partition $\nu$ three partitions $M_\nu$, $r_\nu$, and $b_\nu=\nu^+$ as follows: \begin{center} \hskip 0cm \setlength{\unitlength}{0.25in} \begin{picture}(7.5,6.5) \put(0,0){\line(1,0){1}} \put(1,0){\line(0,1){2}} \put(1,2){\line(1,0){1}} \put(2,2){\line(0,1){1}} \put(2,3){\line(1,0){1}} \put(3,3){\line(0,1){1}} \put(3,4){\line(1,0){1}} \put(4,4){\line(0,1){1}} \put(4,5){\line(1,0){3}} \put(7,5){\line(0,1){1}} \put(7,6){\line(-1,0){7}} \put(0,6){\line(0,-1){6}} \multiput(-.4,3)(0.4,0){13}{\line(1,0){0.2}} \multiput(4,1.5)(0,0.4){13}{\line(0,-1){0.2}} \put(1.9,4.5){\makebox(0,0)[c]{\Large$M_\nu$}} \put(5.6,5.4){\makebox(0,0)[c]{\Large$r_\nu$}} \put(0.6,2){\makebox(0,0)[c]{$b_\nu^t$}} \put(-.1,3){\line(1,0){0.2}} \put(-.7,3){\makebox(0,0)[c]{$k$}} \put(4,6.1){\line(0,-1){0.2}} \put(4,6.5){\makebox(0,0)[c]{$n-k$}} \put(4,3){\linethickness{1pt}\line(0,-1){3}} \put(4,3){\linethickness{1pt}\line(1,0){3}} \end{picture} \end{center}
Let $x_i=e^{\delta_i}$, $i=1,\ldots,k$ and $y_j=e^{\delta_{k+j}}$, $j=1,\ldots, n-k$. We have the following Sergeev-Pragacz character formula for $L'(\nu)$: \begin{align}\label{sergeev:pragacz}
\text{ch}L'(\nu)=\sum_{w\in\mf S_{k|n-k}} w\left( \frac{g_\nu(x,y)x^{r_\nu} y^{b_\nu}\prod_{i=1}^k x_i^{k-i}\prod_{j=1}^{n-k}y_j^{n-k-j}}{\Delta(x)\Delta(y)}\right), \end{align} where $g_\nu(x,y)=\prod_{(i,j)\in M_\nu}(x_i+y_j)$, $\Delta(x)=\prod_{i<j}(x_i-x_j)$, and $\Delta(y)=\prod_{p<q}(y_p-y_q)$. Here $x^{r_\nu}:=\prod_{i=1}^k x_i^{(r_\nu)_i}$ and $y^{b_\nu}:=\prod_{j=1}^{n-k} y_j^{(b_\nu)_j}$. (Also here we have used the identification between $\delta_i$s and $\delta'_j$ as explained above)
Let $C_\nu$ be the complement of $M_\nu$ in the $k\times (n-k)$ box, i.e., the Young diagram $\underbrace{(n-k,n-k,\ldots,n-k)}_k$.
\begin{lem}\label{SP:KL:iden}
Let $\la\in\La^+_{k,\zeta}$ such that $\la'$ is the highest weight of an irreducible polynomial module for $\gl(k|n-k)$. Then we have the following identity in $\h_\even^*$: \begin{align*}
\sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}&w\left( \frac{x^{\la'^-} y^{{\la'^+}}\prod_{i=1}^k x_i^{k-i}\prod_{j=1}^{n-k}y_j^{n-k-j}}{\prod_{(i,j)\in C_{\la'}}1+x_i^{-1}y_j}\right) =\\
&\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\sum_{w\in\mf S_{k|n-k}}(-1)^{\ell(w)}w\left(x^{\mu'^-} y^{{\mu'^+}}\prod_{i=1}^k x_i^{k-i}\prod_{j=1}^{n-k}y_j^{n-k-j}\right). \end{align*} \end{lem}
\begin{proof} To simplify notation let us write $x^{\rho_x}:=\prod_{i=1}^k x_i^{k-i}$ and $y^{\rho_y}:=\prod_{j=1}^{n-k}y_j^{n-k-j}$. For an integer $l$ we write $x^l:=\prod_{i=1}^kx_i^l$ and similarly for $y^l$.
We have by \eqref{sergeev:pragacz} \begin{align*}
\text{ch}L'(\la')=&\sum_{w\in \mf S_{k|n-k}} w\left( \frac{\prod_{(i,j)\in M_{\la'}}(x_i+y_j)x^{r_{\la'}}y^{b_{\la'}}x^{\rho_x}y^{\rho_y}}{\Delta(x)\Delta(y)} \right)\\
=&\sum_{w\in \mf S_{k|n-k}} (-1)^{\ell(w)}\frac{1}{\Delta(x)\Delta(y)}w\left( \frac{\prod_{i,j}(x_i+y_j)x^{r_{\la'}}y^{b_{\la'}}x^{\rho_x}y^{\rho_y}}{\prod_{(i,j)\in C_{\la'}}(x_i+y_j)} \right)\\
=&\sum_{w\in \mf S_{k|n-k}} (-1)^{\ell(w)}\frac{\prod_{i,j}(x_i+y_j)}{\Delta(x)\Delta(y)}w\left( \frac{x^{r_{\la'}}y^{b_{\la'}}x^{\rho_x}y^{\rho_y}}{\prod_{(i,j)\in C_{\la'}}(x_i+y_j)} \right)\\
=&\sum_{w\in \mf S_{k|n-k}} (-1)^{\ell(w)}\frac{\prod_{i,j}(x_i+y_j)}{\Delta(x)\Delta(y)}w\left( \frac{x^{r_{\la'}}y^{b_{\la'}}x^{\rho_x}y^{\rho_y}}{x^{C_{\la'}}\prod_{(i,j)\in C_{\la'}}(1+x^{-1}_iy_j)} \right) \\
=&\sum_{w\in \mf S_{k|n-k}} (-1)^{\ell(w)}\frac{\prod_{i,j}(x_i+y_j)}{\Delta(x)\Delta(y)}x^{-n+k}w\left( \frac{x^{{\la'^-}}y^{b_{\la'}}x^{\rho_x}y^{\rho_y}}{\prod_{(i,j)\in C_{\la'}}(1+x^{-1}_iy_j)} \right). \end{align*}
Also by \cite[Theorem 4.37]{Br1} we have \begin{align*}
\text{ch}L'(\la')=\sum_{ \mu\preceq\la}\ell_{\mu\la}(1)\prod_{i,j}(x_i+y_j)\frac{x^{-n+k}}{\Delta(x)\Delta(y)}\sum_{w\in \mf S_{k|n-k}}(-1)^{\ell(w)}\left( x^{\mu'^-}y^{\mu'^+}x^{\rho_x}y^{\rho_y} \right). \end{align*} Comparing these two expressions the lemma follows. \end{proof}
\begin{thm} Let $\la\in\La^+_{k,\zeta}$ such that $\la'$ is the highest weight of an irreducible polynomial module for $\gl(k|n-k)$. Then we have \begin{align*} \text{ch}L(\la) = &\frac{2^{\lceil n/2\rceil}\prod_{i<j}(x_i+x_j)\prod_{p<q}(y_p+y_q)}{\prod_{i,j}(x_i-y_j)} x^{\zeta+\frac{n+1}{2}-k} y^{-\zeta-\frac{n-1}{2}+k}\\
&\quad\times\sum_{w\in\mf S_{k|n-k}} w\left(\frac{g_{\la'}(x,y)x^{r_{\la'}} y^{b_{\la'}}\prod_{i=1}^k x_i^{k-i}\prod_{j=1}^{n-k}y_j^{n-k-j}}{\Delta(x)\Delta(y)} \right). \end{align*} \end{thm}
\begin{proof} We define \begin{align*} \kappa:=\sum_{i=1}^k(\zeta-\frac{n-1}{2})\delta_i +\sum_{j=k+1}^n(k-\frac{n-1}{2}-\zeta)\delta_j. \end{align*} so that we have $\la=\la'+\rho_x+\rho_y+\kappa$. By Theorem \ref{PKLConjecture} and Lemma \ref{SP:KL:iden} we have the following expression for $\text{ch}L(\la)$: \begin{align*} &=2^{\lceil n/2\rceil}\sum_{\mu\preceq\la}\ell_{\mu\la}(1)\prod_{i,j}\frac{x_i+y_j}{x_i-y_j} \frac{\prod_{i<j,p<q}(x_i+x_j)(y_p+y_q)}{\Delta(x)\Delta(y)}\\ &\qquad\qquad\qquad\qquad \qquad\qquad\times e^{\kappa}\sum_{w}(-1)^{\ell(w)}w\left(x^{\mu'^-}y^{\mu'^+}x^{\rho_x}y^{\rho_y}\right)\\ &=2^{\lceil n/2\rceil}e^{\kappa}\prod_{i,j}\frac{x_i+y_j}{x_i-y_j} \frac{\prod_{i<j,p<q}(x_i+x_j)(y_p+y_q)}{\Delta(x)\Delta(y)}\\ &\qquad\qquad\qquad\qquad \qquad\qquad\times \sum_{w}(-1)^{\ell(w)}w\left( \frac{x^{\la'^-} y^{{\la'^+}}x^{\rho_x}y^{\rho_y}}{\prod_{(i,j)\in C_{\la'}}1+x_i^{-1}y_j}\right)\\ &=2^{\lceil n/2\rceil}e^{\kappa} \frac{\prod_{i<j,p<q}(x_i+x_j)(y_p+y_q)}{\prod_{i,j}(x_i-y_j)}\sum_{w}w\left( \frac{\prod_{i,j}(x_i+y_j)x^{\la'^-} y^{{\la'^+}}x^{\rho_x}y^{\rho_y}x^{C_{\la'}}}{\Delta(x)\Delta(y)\prod_{(i,j)\in C_{\la'}}x_i+y_j}\right)\\ &=2^{\lceil n/2\rceil}x^{n-k}e^{\kappa} \frac{\prod_{i<j,p<q}(x_i+x_j)(y_p+y_q)}{\prod_{i,j}(x_i-y_j)}\\
&\qquad\qquad\qquad\qquad \qquad\qquad\times\sum_{w\in\mf S_{k|n-k}} w\left( \frac{\prod_{(i,j)\in M_{\la'}}(x_i+y_j)x^{r_{\la'}} y^{{b_{\la'}}}x^{\rho_x}y^{\rho_y}}{\Delta(x)\Delta(y)}\right). \end{align*} Recalling the definitions of $\kappa$ and $g_{\la'}(x,y)$ gives the theorem. \end{proof}
\begin{rem} Consider the full subcategory of $\mc O_{n,\hf+\Z}$ consisting of objects with composition factors isomorphic to $L(\la)$ with $\la=\sum_{i=1}^n\la_i\delta_i\in\h_\even^*$ of the form $\la_i\in\hf\Z$ and $\la_{k+1}>\la_{k+2}>\cdots>\la_n>0>\la_1>\la_2>\cdots>\la_k$. According to \cite[Proposition 4.1 and Corollary 4.2]{CKW} the canonical basis on the corresponding subspace of the Fock space of type $C$ can be identified naturally with canonical basis of type $A$. Now, a verbatim repetition of the arguments given above can be used to obtain an irreducible character formula for $L(\la)$ in analogy to Theorem \ref{PKLConjecture}. Here, we use $\hf$ for $\zeta$ in the expression \eqref{aux:fns} to define the corresponding Kazhdan-Lusztig polyomials $\ell_{\la\mu}(q)$. This establishes a parabolic version of a special case of the conjecture on the irreducible characters for the half-integer weights in \cite{CKW}. Also, the formula for Kostant modules and analogues of polynomial modules in this section have analogues in this setting as well. We leave the details to the reader.
We expect that the characters of $L(\la)$ in the case when $\la$ satisfies the more general condition of $\la_j>0>\la_i$, for $i=1,\ldots,k$ and $j=k+1,\ldots,n$, and either $\la_l\in\hf\Z$ or $\la_l\in\Z$, for all $l$, are determind by canonical basis of type $A$ quantum groups. This is predicted by \cite{CKW} and one should be able to establish this following the approach in \cite{BLW}. \end{rem}
\frenchspacing
\end{document} | arXiv | {
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\begin{document}
\begin{center} Topology and its Applications, 155 (2008) 965--971. \end{center} \title{$\mathbb{Z}_2$ actions on complexes with three non-trivial cells}
\author{Mahender Singh}
\address{School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, INDIA}
\email{msingh@mri.ernet.in}
\subjclass[2000]{Primary 55S17; Secondary 55R20}
\keywords{Cohomology ring, fibration, group action, join, totally non-homologous to zero, wedge sum}
\begin{abstract} In this paper, we study $\mathbb{Z}_2 $ actions on a cell complex $X$ having its cohomology ring isomorphic to that of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We determine the possible fixed point sets depending on whether or not $X$ is totally non-homologous to zero in $X_{\mathbb{Z}_2}$ and give examples realizing the possible cases. \end{abstract}
\maketitle
\end{document} | arXiv | {
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} | arXiv/math_arXiv_v0.2.jsonl | null | null |
\begin{document}
\title{ extbf{Towards computing high-order p-harmonic\ descent directions and their limits\ in shape optimization}
\begin{abstract}
\noindent We present an extension of an algorithm for the classical scalar $p$-Laplace Dirichlet problem to the vector-valued $p$-Laplacian with mixed boundary conditions in order to solve problems occurring in shape optimization using a $p$-harmonic approach.
The main advantage of the proposed method is that no iteration over the order $p$ is required and thus allow the efficient computation of solutions for higher orders.
We show that the required number of Newton iterations remains polynomial with respect to the number of grid points and validate the results by numerical experiments considering the deformation of shapes.
Further, we discuss challenges arising when considering the limit of these problems from an analytical and numerical perspective, especially with respect to a change of sign in the source term. \end{abstract}
\section{Introduction} \label{sec::Introduction}
Shape optimization constrained to partial differential equations is a vivid field of research with high relevance for industrial grade applications. Mathematically, we consider the problem \begin{equation*}
\begin{aligned}
\min_{\Omega \in \mathcal{A}}\quad & J(v_s,\Omega)\\
\text{s.t.}\quad & \mathcal{E}(v_s,\Omega) = 0
\end{aligned}
\label{eq:ShapeOptProblem} \end{equation*} where $J$ denotes the objective or shape functional depending on a state variable $v_s$ and a Lipschitz domain $\Omega \subset \mathbb{R}^d$, which is to minimize over the set of admissible shapes $\mathcal{A}$. Further, the state and the domain have to fulfill a PDE constraint $\mathcal{E}$. A common solution technique for this type of problem is to formulate it as a sequence of deformations to the initial shape \cite{sokolowski1992}. Each of these transformations has the form \begin{equation}
\tilde{\Omega} = (\mathrm{id}+tv)(\Omega)
\label{eq:PerturbationOfIdentity} \end{equation} for a step-size $0 < t \leq 1$ and a descent vector field $v:\mathbb{R}^d \rightarrow \mathbb{R}^d$ in the sense $J^\prime(\Omega)\, v < 0$ with shape derivative denoted by $J^\prime$. Hence, the set $\mathcal{A}$ is implicitly defined by all shapes reachable via such transformations to the initial shape. From the analytical derivation of this procedure it is required that the $v$ are at least $W^{1,\infty}(\mathbb{R}^d,\mathbb{R}^d)$ inherently ensuring that all admissible shapes remain Lipschitz. However, in practice this condition is often neglected and so-called Hilbert space methods \cite{allaire2021} are used. The most prominent example of those consists of finding $v \in W^{k,2}(\Omega,\mathbb{R}^d)$. This results in smooth shapes, which are relevant for some applications, but not necessarily optimal. Further, the mesh quality often deteriorates and it would require a high differentiability order $k$ to obtain a Lipschitz transformation from the Sobolev embedding.\\
\noindent Recent development suggests using a $p$-harmonic approach \cite{deckelnick2021} to determine descent directions. This technique can be obtained by considering the steepest descent in the space of Lipschitz transformations \begin{equation}
\underset{\substack{v\in W^{1,\infty}(\Omega,\mathbb{R}^d)\\ \lVert \nabla v \rVert_{L^{\infty}} \leq 1}}{\mathrm{arg\;min}}\; J^\prime(\Omega)\, v.
\label{eq:LipschitzSteepestDescent} \end{equation} and then potentially relax the problem. The descent field is obtained by the solution of a minimization problem for $2 \leq p < \infty$ reading \begin{equation}
\underset{v \in W^{1,p}(\Omega,\mathbb{R}^d)}{\mathrm{arg\;min}}\; \frac{1}{p} \int_{\Omega} \lVert \nabla v \rVert_2^p \;\mathrm{d} x + J^\prime(\Omega)\, v
\label{eq:shapeDescentMinimization} \end{equation} Since the classical Hilbert space setting is recovered for the linear case $p=2$, this approach can also be understood as a regularization of such methods. It is demonstrated that this approach is superior in terms of representation of sharp corners as well as the overall mesh quality and yields improving results for increasing $p$ \cite{mueller2021}. On the downside, the stated numerical examples also made clear that it is challenging to compute solutions for $p>5$ due to serious difficulties in numerical accuracy and the need to iterate over increasing $p$ with the presented solution technique.\\
\noindent We now assume that the shape derivative $J^\prime(\Omega)$ can be expressed via integrals over the domain and the boundary. From the perspective of shape calculus \cite{delfour2011, sokolowski1992} this is a reasonable choice to cover relevant applications. An exemplary computation including geometric constraints can be found in \cite{schulz2016}. For the finite setting we recover the general formulation of a problem for the $p$-Laplacian $\Delta_p v = \nabla \cdot (\lVert \nabla v \rVert _2^{p-2} \nabla v)$ reading \begin{equation}
\underset{u \in \mathcal{U}^p}{\mathrm{arg\;min}}\; J_p(u) = \frac{1}{p} \underbrace{\int_{\Omega} \lVert \nabla (u+g) \rVert_2^p}_{ =:\, \lVert u+g \rVert_{X^p(\Omega)}^p} \;\mathrm{d} x - \int_{\Gamma} h u \;\mathrm{d}\Gamma - \int_{\Omega} f u \;\mathrm{d} x \quad
\label{eq:pLaplaceZeroTraceMinimization} \end{equation} over the set $\mathcal{U}^p := \{ u \in W^{1,p}(\Omega,\mathbb{R}^d):\, u=0 \text{ a.e. on } \partial\Omega\setminus\Gamma\}$ with $v := u + g$ for an arbitrary prolongation of $g$ to the whole domain in $\{ v \in W^{1,p}(\Omega,\mathbb{R}^d):\, v=g \text{ a.e. on } \partial\Omega\setminus\Gamma\}$. The corresponding Euler-Lagrange equation is given by \begin{equation*}
\left.\begin{array}{ll}
-\Delta_p v = f &\text{in } \Omega,\\
\lVert\nabla v \rVert_2^{p-2} \partial_{\eta} v = h & \text{on } \Gamma,\\
v=g &\text{on } \partial\Omega\setminus\Gamma
\end{array}\right\}.
\label{eq:pLaplaceEulerLagrange} \end{equation*}
\noindent Consequently, in order to obtain an efficient shape optimization algorithm, it is necessary to find a solid routine to solve this problem for a preferably high order $p$. On the other hand, it is desirable to directly solve the non-relaxed limit case for $p=\infty$ to obtain analytically valid and possibly superior results.\\
\noindent The remainder is structured as follows: In section \ref{sec:HighOrderDescent} we construct the vector-valued extension of the algorithm and integrate support for Neumann boundary conditions. We present numerical results obtained with this algorithm in section \ref{sec:NumericalResults}. After that we discuss we discuss a version of the presented algorithm for $p=\infty$ before we draw conclusions in section \ref{sec:Conclusion}. \section{High-order \texorpdfstring{$p$}{p}-harmonic descent} \label{sec:HighOrderDescent}
\noindent In \cite{loisel2020} an algorithm to solve scalar $p$-Laplace problems with Dirichlet boundary conditions is presented in order to show it is solvable in polynomial time. The approach relies on interior-point methods and the theory of self-concordant barriers, including estimates in terms of the barrier parameter given by Nesterov \cite{nesterov2004}. Besides the computational complexity, one of the main advantages is that the solution to the linear Laplacian is a sufficient initial guess for any $p$ and no iteration over $p$ is required. We construct an extension to the algorithm for vector-valued functions featuring mixed Dirichlet and Neumann boundary conditions in order to apply it to shape deformation problems. However, we will only introduce minor changes to the original proof and show that the polynomial estimate holds with mixed boundary conditions in the scalar setting.\\
\noindent First, we recall some basic notation for finite elements. Let $h_\Omega$ be a parameter and $T_{h_\Omega}$ a triangulation of $\Omega$ with $n$ nodes, $m$ elements and quasi-uniformity parameter $\rho_{\Omega}$. Further let $V_{h_\Omega}$ denote the space of piece-wise linear Lagrange elements over $T_{h_\Omega}$. The vector-valued finite element coefficient vector $u \in \mathbb{R}^{nd^\prime}$ is given by a $d'$-block for each node $k$ associated with a basis function $\Phi_k$. This allows us to extend the notion of discrete derivative matrices to vector-valued functions by $D^{(j,r)} \in \mathbb{R}^{m\times nd^\prime}$ with entries $D_{i,d^\prime(k-1)+r}^{(j,r)} = \frac{\partial}{\partial x_j}\Phi_k(x^{(i)})$ for elements $i=1,\ldots,m$ and nodes $k=1,\ldots,n$ being non-zero if $k \in \mathrm{spt}\; i$. Meaning the multiplication to a coefficient vector returns the discrete derivative in direction $j$ of the $r$-th image dimension on each element midpoint $x^{(i)}$. The vector of weights is given by $\omega^{(l)}$, where $l$ denotes the number of local quadrature points. This also means for triangular elements using the mid-point rule $\omega^{(1)} = \omega$ is the vector of element volumes. The discretization of the $p$-Laplacian term from the problem (\ref{eq:pLaplaceZeroTraceMinimization}) is then given by \begin{equation*}
\lVert u+g \rVert_{X^p(\Omega)}^p =
\sum_{i=1}^m \omega_i \left( \sum_{j=1}^d \sum_{r=1}^{d^\prime} [D^{(j,r)}(u+g)]_i^2 \right)^\frac{p}{2}. \end{equation*}
\noindent Further, we define basis matrices $E \in \mathbb{R}^{mld^\prime \times n d^\prime}$ returning the function value for all image dimensions on the $l$ local quadrature points of all elements on multiplication to a coefficient vector. This discrete operator will later simplify the proof by allowing us apply similar techniques as those for $D^{(j,r)}$ also for the occurring mass matrices $M=[E^{(l)}]^\intercal W^{(l)} E^{(l)}$ where $W^{(l)} = \mathrm{diag}(\omega^{(l)})$.\\
\noindent Note that all definition hold similarly for boundary elements and will be denoted by a bar, e.g. $\bar{M} = [\bar{E}^{(l)}]^\intercal \bar{W}^{(\bar{l})} \bar{E}^{(l)} \in \mathbb{R}^{nd^\prime \times nd^\prime}$.\\
\noindent With these connections established, we can now state the discretized and reformulated problem for the vector-valued $p$-Laplacian with mixed Dirichlet and Neumann boundary data in the following lemma.
\begin{lemma}
The problem (\ref{eq:pLaplaceZeroTraceMinimization}) of minimizing $J_p(u)$ over the given finite element space $V_h$ with $1 \leq p < \infty$ satisfying the additional upper bound $\omega_i \lVert \nabla(u+g)\vert_{K_i} \rVert_2^p \leq R$ is equivalent to the classical convex problem
\begin{equation}
\min_{x \in \mathcal{Q}_p} \langle c,x \rangle \text{ with } c =
\begin{bmatrix}
-Mf-\bar{M}h\\
\frac{\omega}{p}
\end{bmatrix}
\label{eq:ReformulatedFiniteMinimization}
\end{equation}
with the constrained search set given by
\begin{equation}
\mathcal{Q}_p = \left\{ (u,s) \in \mathbb{R}^n \times \mathbb{R}^m \, : \, s_i \geq \left(\sum_{j=1}^{d} \sum_{r=1}^{d'} [D^{(j,r)}(u+g)]_i^2\right)^{\frac{p}{2}} \land \omega_i s_i \leq R \right\}.
\label{eq:FiniteConstrainedSet}
\end{equation} \end{lemma}
\noindent The obtained problem is now a classical convex optimization problem by the minimization of a scalar product over a constrained set. An algorithm is obtained by constructing a self-concordant barrier for $\mathcal{Q}^p$, computing its first and second derivative and then applying an interior-point method \cite{nesterov2004}.
\begin{lemma}
\label{lem:FiniteBarrierFunction}
A $4m$-self-concordant barrier for $\mathcal{Q}_p$ is given by the function
\begin{equation*}
\begin{gathered}
F(u,s) = -\sum_i \log z_i - \sum_i \log \tau_i \quad\text{where}\\
z_i = s_i^{2/p} - \sum_{j=1}^{d}\sum_{r=1}^{d^\prime}[(\underbrace{D^{(j,r)} u + D^{(j,r)} g}_{=: y^{(j,r)}})_i]^2 \quad\text{and}\quad \tau_i = R-\omega_i s_i.
\end{gathered}
\end{equation*} \end{lemma}
\begin{remark}
By construction the barrier function $F$ is twice differentiable with the first derivative reading
\begin{equation*}
\begin{aligned}
F' &= \begin{bmatrix}
F_u \\
F_s
\end{bmatrix}
\text{ where }\\
F_{u} &= 2 \sum_{j=1}^{d} \sum_{r=1}^{d'} [D^{(j,r)}]^\intercal \frac{y^{(j,r)}}{z}
\quad\text{and}\quad
F_{s} = -\frac{2}{p}\frac{1}{z}s^{2/p-1} + \frac{\omega}{\tau}.
\end{aligned}
\end{equation*}
The second derivative is given by
\begin{equation*}
\begin{aligned}
F'' &= \begin{bmatrix} F_{uu} & F_{us} \\ F_{us}^\intercal & F_{ss} \end{bmatrix} \text{ where}\\
F_{uu} &= 2 \sum_{j=1}^{d} \sum_{r=1}^{d'} [D^{(j,r)}]^\intercal Z^{-1} D^{(j,r)}\\
&\quad+ 4 \sum_{j_1=1}^{d} \sum_{r_1=1}^{d'}\sum_{j_2=1}^{d} \sum_{r_2=1}^{d'} (Y^{(j_1,r_1)}D^{(j_1,r_1)})^\intercal Z^{-1} D^{(j)} (Y^{(j_2,r_2)}D^{(j_2,r_2)}),\\
F_{us} &= -\frac{4}{p}\sum_{j=1}^{d} \sum_{r=1}^{d'} (Y^{(j,r)}D^{(j,r)})^\intercal Z^{-2} S^{2/p-1},\\
F_{ss} &= -\frac{2}{p} \left(\frac{2}{p} - 1 \right) Z^{-1} S^{2/p-2} + \frac{4}{p^2} Z^{-2} S^{4/p-2} + W^2 T^{-2},\\
S &= \diag(s),\, W = \diag(\omega),\, Y = \diag(y),\, Z = \diag(z),\, T = \diag(\tau).
\end{aligned}
\end{equation*} \end{remark} \begin{remark}
Note that in the construction the $\frac{p}{2}$-th power of the norm has been moved.
Thus the additional constrained $s_i \geq 0$ would be required, as stated in the original version.
However, $z_i \rightarrow \infty$ as $s_i \searrow \lVert \nabla(u+g)\vert_{K_i} \rVert_2^2$ or $s_i \nearrow -\lVert \nabla(u+g)\vert_{K_i} \rVert_2^2$ and thus leave us with a correct barrier on the intended set as well additional separated set. Therefore, we drop that condition here and in practice we ensure it by the choice of the initial value with $s_i \geq 0$ and keep track via the line-search in the adaptive path-following. This not only simplifies the notation, but also reduces the computational effort. \end{remark}
\noindent For completeness, we state the proof for the computational complexity in the scalar case with additional Neumann boundary conditions. Note that the obtained bound on the iterations differs from the one in the original version \cite[Theo. 1]{loisel2020}. This stems from a change we have to introduce to the Hessian and subsequent computations as well as using a different estimate in terms of the barrier parameter, which features $\lVert c \rVert^*_{x^*_F}$ instead of $\lVert \hat{x} \rVert^*_{x^*_F}$. However, the estimate on the required iterations for a naive interior-point method is only of theoretical interest. Even though the bound is not sharp, the required iterations are not reasonable for practical applications. Therefore, variations with modified step length \cite{nesterov2001} or an adaptive step size control \cite{loisel2020} are used. Although estimates are worse in this setting, we will see for the latter in section \ref{sec:NumericalResults} a significantly improved performance. Consequently, we will not show theoretical results for the vector-valued setting.\\
\begin{theorem}
Let $1 \leq p <\infty$. Assume that $\Omega \subset \mathbb{R}^{d}$ is a Lipschitz polytype of width L and that $T_{h_\Omega}$ is a quasi-uniform triangulation of $\Omega$, parametrized by $0 < h_\Omega < 1$ and with quasi-uniformity parameter $1 \leq \rho_\Omega < \infty$.
Further assume $g \in W^{1,p}(\Omega)$, $f \in \mathrm{L}^q(\Omega)$ and $h \in \mathrm{L}^q(\Gamma)$ with conjugated exponents $\frac{1}{p} + \frac{1}{q} = 1$ are piece-wise linear on $T_{h_\Omega}$ and let $V_{h_{\Omega}} \subset W_{0}^{1,p}(\Omega)$ be the piece-wise linear finite element space on $T_{h_{\Omega}}$ whose trace vanishes.
Fix a quadrature $Q$ with positive weights such that the integration is exact, $R \geq R^* := 2(1 + \lVert g \rVert^p_{X^p(\Omega)})$ sufficiently large and let $\varepsilon > 0$ be an accuracy.\\
\noindent In exact arithmetic, a naive interior-point method consisting of auxiliary and main path-following using the barrier function from lemma \ref{lem:FiniteBarrierFunction} to minimize $J_p(u)$ over $u \in V_{h}$, starting from $\hat{x} = (0,\hat{s})$ with $\hat{s}_i = 1 + (\sum_j \sum_r [D^{(j,r)}g]_i^2)^{p/2}$, converges to the global minimizer in $V_h$ in at most
\begin{equation*}
\begin{aligned}
N &\leq 14.4 \sqrt{|\Omega|d! h^{-d}} [K^* + \log( \varepsilon^{-1} h_{\Omega}^{-1-7.5d} R^5 (1 + \lVert g \rVert_{X^p(\Omega)})(\lVert f \rVert_{\mathrm{L}^q(\Omega)} + \lVert h \rVert_{\mathrm{L}^q(\Gamma)} + 1))].
\end{aligned}
\end{equation*}
iterations.
This results in a computational complexity denoted by $\mathcal{O}(\sqrt{n} \log(n))$.
The constant $K^{*} = K^{*}(\Omega,\rho_{\Omega}, Q)$ depends on the domain $\Omega$, the quasi-uniformity parameter $\rho_{\Omega}$ of the triangulation and the quadrature $Q$.
At convergence, $u$ satisfies
\begin{equation*}
J_p(u) \leq \min_{\substack{v\in V_{h_\Omega}\\ \frac{1}{p}\lVert v+g \rVert^p_{X^p} \leq R}} J_p(v) + \varepsilon.
\end{equation*} \end{theorem}
\begin{proof}
\noindent In order to compute the final estimate, we need to find a bound for $\lVert c \rVert_2$.
Let $f \in \mathrm{L}^q(\Omega)$ and $h \in \mathrm{L}^q(\Gamma)$ piece-wise linear such that the quadrature on $l$ points is exact.
We can obtain a bound on a similar way as the bound for $\lVert F'(\hat{x}) \rVert_2$ in the original paper.
Start with
\begin{equation*}
\lVert c \rVert_2 = \left\lVert \begin{bmatrix} -Mf-\bar{M}h \\ \frac{\omega}{p} \end{bmatrix} \right\rVert_2
\leq \lVert Mf \rVert_2 + \lVert \bar{M}h \rVert_2 + \frac{1}{p}\lVert \omega \rVert_2.
\end{equation*}
\noindent In accordance with the original proof we can bound the first term by
\begin{equation*}
\lVert Mf \rVert_2 = \lVert [E^{(l)}]^\intercal W^{(l)} E^{(l)} f \rVert_2 \leq \lVert [E^{(l)}]^\intercal W^{(l)} \rVert_2 \lVert E^{(l)}f \rVert_2
\end{equation*}
with the same idea of using $\Vert [E^{(l)}]^\intercal W^{(l)} \Vert_2^2 \leq \omega^{(l)}_{\max}\rho ([E^{(l)}]^\intercal W^{(l)} E^{(l)})$ and then bounding the spectral radius $\rho$.
This is even easier here, because
\begin{equation*}
w^\intercal E^\intercal W E w = \int_{\Omega} w^2 dx = \lVert w \rVert^2_{\mathrm{L}^2(\Omega)} \leq [K_{\Omega}']^2 h_{\Omega}^{2d} \lVert w \rVert_2^2
\end{equation*}
can be estimated without further inequalities.
\noindent The remainder can now be bound correspondingly with the equivalence of $p$-norms in finite dimensions
\begin{equation*}
\begin{aligned}
\lVert Ef \rVert_2 &\leq [\omega^{(l)}_{min}]^{-1/2} \left( \sum_{i=1}^{ml} \omega^{(l)}_i [E^{(l)}f]_i^2 \right)^{\frac{1}{2}}\\
&\leq [\omega^{(l)}_{min}]^{-1/2} (ml)^{1/2} \left( \sum_{i=1}^{ml} \omega^{(l)}_i [E^{(l)}f]_i^q \right)^{\frac{1}{q}}\\
&\leq (\frac{\omega_{min}}{C_Q})^{-1/2} (ml)^{1/2} \lVert f \rVert_{\mathrm{L}^q(\Omega)}
\end{aligned}
\end{equation*}
where we used that the weights on the reference elements are fixed positive and thus there exists a constant $C_Q > 0$ that only depends on the quadrature $Q$ such that $C_Q^{-1} \omega_{\min} \leq \omega^{(l)}_{\min} \leq \omega_{\min}$.
Combing the results, we get
\begin{equation*}
\begin{aligned}
\lVert Mf \rVert_2 &\leq \left(K_{\Omega}' h_{\Omega}^d \right) \left(\omega_{min}^{-1/2} C_Q^{1/2} l^{1/2} m^{1/2} \lVert f \rVert_{\mathrm{L}^q(\Omega)} \right)\\
&\leq K_Q K_{\Omega}'' \lVert f \rVert_{\mathrm{L}^q(\Omega)}.
\end{aligned}
\end{equation*}
\noindent A bound for the middle term can be obtained in the same way, just in $(d-1)$ dimensions. Using
\begin{equation*}
\vert \Gamma \vert = \sum_{i=1}^{\bar{m}}\bar{\omega}_i \geq \bar{m}\bar{\omega}_{\min} \geq \bar{m}\frac{h_{\Omega}^{d-1}}{(d-1)!} \Leftrightarrow \bar{m} \leq \vert \Gamma \vert (d-1)! h_{\Omega}^{d-1}
\end{equation*}
this reads
\begin{equation*}
\begin{aligned}
\lVert \bar{M}h \rVert_2 &\leq \left(K_{\Omega}' h_{\Omega}^{d-1} \right) \left(\bar{\omega}_{min}^{-1/2} (C_{Q}\bar{m}\bar{l})^{1/2} \lVert h \rVert_{\mathrm{L}^q(\Gamma)} \right)\\
&\leq K_{Q} K_{\Omega}'' \lVert h \rVert_{\mathrm{L}^q(\Gamma)}.
\end{aligned}
\end{equation*}
\noindent Finally, using $\lVert \omega \rVert_2 = \left( \sum_{i=1}^m \omega_i^2 \right)^{1/2} \leq \sqrt{m} \omega_{\max} \leq \sqrt{m}\frac{(\rho h)^d}{d!}$ and $p^{-1} \leq 1$ we are able to obtain a bound in the form
\begin{equation*}
\lVert c \rVert_2 \leq C^{\ast}_{\Omega} C^{\ast}_{Q}(\lVert f \rVert_{\mathrm{L}^q(\Omega)} + \lVert h \rVert_{\mathrm{L}^q(\Gamma)} + 1).
\end{equation*}
\noindent Together with the previous results from the original proof we can now determine the relevant bounds as:
\begin{equation*}
\begin{aligned}
\lVert c \rVert^*_{x_F^*} &\leq \lambda_{\min}^{-1} \lVert c \rVert_2 \leq \left( c_{\Omega}' R^{-2} h_{\Omega}^{2d} \right)^{-1} \left( C^{\ast}_{\Omega} C^{\ast}_{Q}(\lVert f \rVert_{\mathrm{L}^q(\Omega)} + \lVert h \rVert_{\mathrm{L}^q(\Gamma)} + 1) \right)\\
&= [c_{\Omega}']^{-1} C^{\ast}_{\Omega} C^{\ast}_{Q} R^2 h_{\Omega}^{-2d} (\lVert f \rVert_{\mathrm{L}^q(\Omega)} + \lVert h \rVert_{\mathrm{L}^q(\Gamma)} + 1)\\
\lVert F'(\hat{x}) \rVert^*_{x_F^*} &\leq \lambda_{\min}^{-1} \lVert F'(\hat{x}) \rVert_2 \leq \left( c_{\Omega}' R^{-2} h_{\Omega}^{2d} \right)^{-1} \left( C^*_{\Omega} h_{\Omega}^{-1-1.5d} R (1+\lVert g \rVert_{X^p(\Omega)})\right)\\
&= [c_{\Omega}']^{-1} C_{\Omega}^* R^3 h_{\Omega}^{-1-3.5d} (1 + \lVert g \rVert_{X^p(\Omega)})
\end{aligned}
\end{equation*}
Plugging these into the actual bound by Nesterov \cite[Sec. 4.2.5]{nesterov2004}
\begin{equation*}
N \leq 7.2 \sqrt{\nu}\, [2\, \ln(\nu) + \ln(\lVert F'(\hat{x}) \rVert^*_{x_F^*}) + \ln(\lVert c \rVert^*_{x_F^*}) + \ln(1/\varepsilon)],
\end{equation*}
we obtain the bound in terms of number of grid points as
\begin{equation*}
\begin{aligned}
N &\leq 7.2 \sqrt{4m}[2\log(4m) + \log(1/\varepsilon)\\
&\quad + \log([c_{\Omega}']^{-1} C_{\Omega}^{\ast} R^3 h_{\Omega}^{-1-3.5d} (1 + \lVert g \rVert_{X^p(\Omega)}))\\
&\quad + \log( [c_{\Omega}']^{-1} C^{\ast}_{\Omega} C^{\ast}_{Q} R^2 h_{\Omega}^{-2d} (\lVert f \rVert_{\mathrm{L}^q(\Omega)} + \lVert h \rVert_{\mathrm{L}^q(\Gamma)} + 1))]\\
&\leq 14.4 \sqrt{|\Omega|d! h_{\Omega}^{-d}} [\log([c_{\Omega}']^{-2} [C_{\Omega}^{\ast}]^2 C_{Q}^{\ast} 16 |\Omega|^2 (d!)^2)]\\
&\quad + \log(\varepsilon^{-1} h_{\Omega}^{-1-7.5d} R^5 (1 + \lVert g \rVert_{X^p(\Omega)})(\lVert f \rVert_{\mathrm{L}^q(\Omega)} + \lVert h \rVert_{\mathrm{L}^q(\Gamma)} + 1) )]\\
&= 14.4 \sqrt{|\Omega|d! h_{\Omega}^{-d}} [K^*(\Omega, \rho_{\Omega}, Q)\\
&\quad + \log(\varepsilon^{-1} h_{\Omega}^{-1-7.5d} R^5 (1 + \lVert g \rVert_{X^p(\Omega)})(\lVert f \rVert_{\mathrm{L}^q(\Omega)} + \lVert h \rVert_{\mathrm{L}^q(\Gamma)} + 1) )].
\end{aligned}
\end{equation*} \end{proof}
\begin{remark}
To find the global minimum, $R$ has to be chosen sufficiently large, so that the solution is contained in the search set $\mathcal{Q}$. For $2 \leq p < \infty$ such an upper bound exists and is given by
\begin{equation*}
R = 2 + 4 \lVert g \rVert_{X^p(\Omega)}^p + 8 (p-1) \left[ L^q \lVert f \rVert_{\mathrm{L}^q(\Omega)}^q + C_T^q (L^q + 1) \lVert h \rVert_{\mathrm{L}^q(\Gamma)}^q\right]
\end{equation*}
where $C_T$ denotes the Sobolev trace constant \cite{evans2015}, which only depends on $p$ and $\Omega$. While there even exists an uniform upper bound for it\cite{bonder2003}, usually neither of them is computable. Therefore, in the actual implementation we choose a heuristic approach. Start by dropping the term resulting from the Neumann boundary condition and if the values during the iteration come close to the bound, restart with an increased version. \end{remark} \begin{proof}
We follow the original proof for the pure Dirichlet case, showing an upper bound for a minimizing sequence $u_k$ using Hölder's inequality, the modified Friedrichs inequality for $\lVert \cdot \rVert_{X^p(\Omega)}$ and the Sobolev trace theorem for the new term.\\
\noindent Start by assuming $\lVert u \rVert_{X^p(\Omega)} \geq \lVert g \rVert_{X^p(\Omega)}$, since otherwise the bound is trivial, and compute
\begin{equation*}
\begin{aligned}
J(u) &= \frac{1}{p}\lVert u+g \rVert_{X^p(\Omega)}^p - \int_\Omega fu \;\mathrm{d}x - \int_\Gamma hu \;\mathrm{d}\Gamma\\
&\geq \frac{1}{p} (\lVert u \rVert_{X^p(\Omega)} - \lVert g \rVert_{X^p(\Omega)})^p - \lVert f \rVert_{\mathrm{L}^q(\Omega)}\lVert u \rVert_{\mathrm{L}^p(\Omega)} - \lVert h \rVert_{\mathrm{L}^q(\Gamma)}\lVert u \rVert_{\mathrm{L}^p(\Gamma)}\\
&\geq \frac{1}{p} \lVert u \rVert_{X^p(\Omega)}^p - \frac{1}{p} \lVert g \rVert_{X^p(\Omega)}^p - \lVert f \rVert_{\mathrm{L}^q(\Omega)} L p^{-\frac{1}{p}}\lVert u \rVert_{X^p(\Omega)}^p\\
&\quad- \lVert h \rVert_{\mathrm{L}^q(\Gamma)} C_T (L p^{-\frac{1}{p}}+1) \lVert u \rVert_{X^p(\Omega)}^p
\end{aligned}
\end{equation*}
\noindent Now we can modify the application of Young's inequality for two forcing terms to $a_1 = 4^{\frac{1}{p}} L p^{-\frac{1}{p}} \lVert f \rVert_{\mathrm{L}^q(\Omega)} $, $a_2 = 4^{\frac{1}{p}} C_T (L p^{-\frac{1}{p}}+1) \lVert h \rVert_{\mathrm{L}^q(\Gamma)} $ and $b_{1,2} = 4^{-\frac{1}{p}} \lVert u \rVert_{X^p(\Omega)}$ and thus get
\begin{equation*}
\begin{aligned}
J(u) - J(0) &\geq \frac{1}{2p}\lVert u \rVert_{X^p(\Omega)}^p - \frac{2}{p} \lVert g \rVert_{X^p(\Omega)}^p -\frac{1}{q} \underbrace{4^{\frac{q}{p}}}_{\underset{\text{for}\, p\geq2}{\leq 4}} \underbrace{p^{-\frac{q}{p}}}_{\leq 1} L^q \lVert f \rVert_{\mathrm{L}^q(\Omega)}^q\\
&\quad- \frac{1}{q} 4^{\frac{q}{p}} C_T^q \underbrace{(L p^{-\frac{1}{p}}+1)}_{\leq (L^q p^{-\frac{q}{p}}+1)} \lVert h \rVert_{\mathrm{L}^q(\Gamma)}^q.
\end{aligned}
\end{equation*}
\noindent Therefore, if
\begin{equation*}
\lVert u \rVert_{X^p(\Omega)}^p > 4 \lVert g \rVert_{X^p(\Omega)}^p + 8 (p-1) \left[ L^q \lVert f \rVert_{\mathrm{L}^q(\Omega)}^q + C_T^q (L^q + 1) \lVert h \rVert_{\mathrm{L}^q(\Gamma)}^q\right] =: \tilde{R},
\end{equation*}
then $J(u) - J(0) \geq 0$.
By contradiction any minimizing sequence $u_k$ must fulfill $\lVert u_k \rVert_{X^p(\Omega)}^p \leq \tilde{R}$ for some $k$ large enough.
Thus, it is in the set $\mathcal{Q}$ and the final bound is obtained by $R := 2(\tilde{R}+1)$ to ensure the construction condition for the initial value. \end{proof} \section{Numerical results} \label{sec:NumericalResults}
In this section, we present results for numerical experiments with the scheme presented in section \ref{sec:HighOrderDescent}. The data was calculated with an implementation in julia based on the finite element library MinFEM \cite{minfem2020} and visualized with Paraview. We choose julia since it allows straightforward computations using matrix or vector based operations, easy adaptation of the code and provides great accessibility to accuracy parameters. Therefore, it is highly suitable for the analysis of algorithms and the experiments performed here.\\
\begin{figure}
\caption{$\lVert v \rVert_2$}
\label{fig:ValidationSolMagnitude}
\caption{$\lVert v-v^* \rVert_2$}
\label{fig:ValidationErrorMagnitude}
\caption{$v_r - v_r^*$}
\label{fig:ValidationErrorX}
\caption{Solution and error for validation by method of manufactured solutions for $v^* = \frac{1}{2}\lVert x \rVert_2 \cdot [1,1]^\intercal$ on $\Omega=[0,1]^2$.}
\label{fig:Validation}
\end{figure}
\noindent We start by validating the algorithm and the implementation. For that purpose, we use the method of manufactured solutions \cite{salari2000} to approximate the analytical solution $v^* = \frac{1}{2}\lVert x \rVert_2 \cdot [1,1]^\intercal$ given for the problem \begin{equation*}
\left.\begin{array}{ll}
-\Delta_p v = -p\, 2^{\frac{p-2}{2}} \lVert x \rVert_2^{p-2} \cdot \begin{bmatrix}1\\ 1\end{bmatrix} & \text{in } \Omega\\
v = \frac{1}{2}\lVert x \rVert_2^2 \cdot \begin{bmatrix}1\\ 1\end{bmatrix} & \text{on } \partial\Omega
\end{array}\right\}. \end{equation*}
\noindent We only test the vector-valued setting, since it contains the scalar case, which has not been validated so far, per component. Figure \ref{fig:Validation} shows the obtained solution and the error on the unit square discretized by a regular mesh with $40000$ nodes. The error in the two components is identical and in the order of the intended accuracy $\varepsilon = 10^{-6}$. The largest errors naturally occurs in the bottom left corner, where a function value close to $0$ has to be approximated. Thus we consider the results obtained by our implementation as valid.\\
\begin{figure}
\caption{Sketch of domain for exemplary problem.}
\label{fig:DomainSketch}
\end{figure}
\noindent From now on, we focus on the additional Neumann boundary conditions. Therefore, we consider the domain $\Omega$ as the unit square, where left and upper boundary are free for deformation and denoted by $\Gamma$. A sketch of this domain can be found in figure \ref{fig:DomainSketch}. On the free boundary, we will work with combinations of the function $\hat{h}(x) = \sin(2\pi x_1) - \sin(2\pi x_2)$. This is a sine wave cycle on each part of the boundary, where the one on the upper boundary is inverted such that the two positive parts are next to the upper left corner. As we will see later, this construction leads towards a distance function on the boundary of the limit solution instead of multiple hats.\\
\begin{table}[!hbp]
\centering
\begin{tabular}{ c || c | c | c || c | c | c || c | c | c ||}
$h=\phantom{22}$ & \multicolumn{3}{c||}{$\hat{h}(x)$} & \multicolumn{3}{c||}{$\hat{h}(x) \cdot \eta$} & \multicolumn{3}{c||}{$\hat{h}(x) \cdot [1,1]^\intercal$}\\
$n=\phantom{22}$ & \phantom{~}2500 & 10000 & 40000 & \phantom{~}2500 & 10000 & 40000 & \phantom{~}2500 & 10000 & 40000\\
\hline
$p=\phantom{2}2$ & 95 & 102 & 116 & 94 & 101 & 115 & 95 & 104 & 116\\
$p=\phantom{2}3$ & 95 & 104 & 114 & 93 & 101 & 114 & 96 & 107 & 115\\
$p=\phantom{2}5$ & 92 & 103 & 113 & 92 & 102 & 107 & 98 & 129 & 116\\
$p=\phantom{2}8$ & 118 & 213 & 198 & 86 & 99 & 107 & 186 & 204 & 213\\
$p=15$ & 191 & 253 & 296 & 111 & 126 & 148 & 228 & 278 & 326\\
$p=25$ & 233 & 308 & 464 & 152 & 204 & 181 & 280 & 361 & 481\\
\end{tabular}
\captionsetup{format=hang}
\caption{Required Newton iterations for solving problems for different boundary source terms with $\hat{h}(x) = \sin(2\pi x_1) - \sin(2\pi x_2)$, number of grid points and PDE parameters $p$.}
\label{tab:FiniteEffort} \end{table}
\noindent For this setting, we can observe the number of required Newton iterations in the path-following with adaptive step-size control for various PDE parameter $p$ and refinements the grid. Table \ref{tab:FiniteEffort} shows these values for the scalar setting and two different prolongations of $\hat{h}(x)$ to a vector-valued setting. Note that the vector-valued problems can be a significantly different problem. For example applying $\hat{h}(x)$ to the outer normal vector results in a problem, where per component only one of the free edges features a sine wave and the other one is homogeneous. This is component wise a simpler problem than the regular scalar one, which is despite the connection of the components visible in the required iterations. The first observation is that much higher orders $p$ are possible when the boundary features a free part. For the pure Dirichlet setting the numerical maximal value was around $p=5$, here the source term reduces the stiffness of the problem such that even $p=25$ is possible. In general, we see for all settings the expected behaviour of increasing iterations for increasing $p$ and $n$ with some exceptions due to the adaptive stepping. Further, the overall iterations stay comparably small to the number of grid points and thus significantly better than the theoretical estimate. In the scalar Dirichlet case this behaviour for the adaptive stepping was already known, but it was unclear if it transfers to the vector-valued setting, especially since the analytical problem is inherently more difficult due to nature of the Frobenius norm in the operator. For the intended application to shape optimization this is a crucial observation, since the computation many deformation fields is required and thus determines the overall runtime.\\
\begin{figure}
\caption{Initial mesh}
\caption{$p=2$}
\caption{$p=5$}
\caption{$p=15$}
\caption{Deformation of a square domain with $h(x) = (\sin(2\pi x_1) - \sin(2\pi x_2)) \cdot \eta$ prescribed on the left and upper boundary for increasing $p$. For comparison the solution for the smaller $p$ is given in blue and the higher order one in red.}
\label{fig:FiniteDeformations}
\end{figure}
\noindent Now we will use results obtained for $\hat{h}(x) \cdot \eta$ to calculate the deformation of the square by perturbation of identity (\ref{eq:PerturbationOfIdentity}). A selected sequence of transformed domains are shown in figure \ref{fig:FiniteDeformations} with a reduced number of grid points for improved visibility. For comparison we take the domain for $p=2$. This features strong bends near the two fixed endpoints of the free boundary, where the mesh is highly deteriorated. Increasing the order to $p=5$, the magnitude of the deformation increases significantly and the bends reduce, however the mesh quality is still poor. When changing to $p=15$, the magnitude changes only slightly as well as the bend. However, the mesh at the bends is not deteriorated anymore and the elements in the interior are deformed more uniformly.
\section{Descent in \texorpdfstring{$W^{1,\infty}$}{W 1,inf}} \label{sec:InfiniteDescent}
\noindent In this section, we consider directly solving the steepest descent problem (\ref{eq:LipschitzSteepestDescent}), commonly associated with the variational problem for the $\infty$-Laplacian and discuss the challenges arising. The first important property of the $\infty$-Laplacian is that the solutions are in general non-unique. However, by the approach of regularization with the $p$-Laplacian, we want the limit of those so-called $p$-Extensions, which is known as the \textit{absolutely minimizing Lipschitz extension (AMLE)} due to early work by Aronsson \cite{aronsson2004}. Here, the term ``absolutely minimizing'' denotes functions $v \in C(\Omega)$ with Lipschitz constant $L_v(V) = L_v(\partial V)$ for all $V \Subset \Omega$.\\
\begin{figure}
\caption{Convergence of numerical solutions for the $p$-Laplacian with increasing order and the analytical limit $p=\infty$ in $[0,1]$ for $g=0$ and different source terms $f$.}
\label{fig:1DPlot}
\end{figure}
\noindent While the minimization formulation for the problem is common, there is no variational formulation with test functions under the integral \cite{lindqvist2019}. Further, even only for zero forcing a reformulation to an Euler-Lagrange equation is possible. With this approach it was shown that unique solutions in the above sense exist for boundary extension problems \cite{jensen1993}. This remains true for homogeneous Dirichlet problems with non-negative source terms \cite{ishii2005}. Here, the unique limit solution can be split into two regions. In the support of the source term, the solution is given by the distance to the boundary $\mathrm{dist}(x,\partial\Omega)$. The other region is then given by the unique solution of the $\infty$-Laplacian without forcing and the distance as Dirichlet boundary condition.\\
\noindent In one dimension, the situation is simplified and analytical solutions can be computed even for source terms with changing sign. Figure \ref{fig:1DPlot} shows such solutions and corresponding solutions of $p$-harmonic relaxations for different source terms. We can observe that the limit solution always has slope 1. Its magnitude does not depend on the magnitude of the source term, but only on the length of the interval between two sign changes. However, the $p$-harmonic solutions depend on it, meaning that the approximation quality depends on the magnitude of the source term as well. Another interesting observation can be made in the lower right plot. The solutions are able to eliminate certain areas with different signs. While the the limit solution is identical to the one with constant source term, the $p$-harmonic approximations are different. Especially one can still an impact of the sign change close to the boundary and the kink at tip occurs later. Further, this one dimensional source term is similar to a normalized version of the Neumann boundary condition or boundary source term used in figure \ref{fig:FiniteDeformations}. Thus, we would expect a solution with slope 1 everywhere on the boundary for the limit deformation.\\
\begin{figure}
\caption{Expected limit.}
\label{fig:InfSolArtificial}
\caption{Inner norm $\lVert \cdot \rVert_2$.}
\label{fig:InfSolLoisel}
\caption{Inner norm $\lVert \cdot \rVert_\infty$.}
\label{fig:InfSolSupSup}
\caption{Deformation of a square domain with $h(x) = (\sin(2\pi x_1) - \sin(2\pi x_2)) \cdot \eta$ prescribed on the left and upper boundary for different different algorithmic realizations of the $\infty$-Laplacian.}
\label{fig:DifferentInfSols}
\end{figure}
\noindent Additionally to the algorithm for the finite setting, which we modified in section \ref{sec:HighOrderDescent}, an algorithm for the limit problem is proposed in \cite{loisel2020}. There, the limit of problem (\ref{eq:pLaplaceZeroTraceMinimization}) is formulated as \begin{equation}
\underset{u\in \mathcal{U}^{\infty}}{\mathrm{arg\;min}}\; J_\infty(u) = \underbrace{\sup_{x \in \Omega} \lVert \nabla (u+g) \rVert}_{=: \lVert u+g \rVert_{X^\infty(\Omega)}} - \int_{\Gamma} h u \;\mathrm{d}\Gamma - \int_{\Omega} f u \;\mathrm{d} x.
\label{eq:InfLaplaceMinimization} \end{equation} First, note that we leave the interpretation of the interior norm in the first term open for now. While there is no actual derivation of the formulation given, we can state some arguments to consider it. Interpreting the primary term in equation (\ref{eq:pLaplaceZeroTraceMinimization}) as $\lVert u+g \rVert_{X^p(\Omega)}^p$ one would obtain this primary term as $\lVert u+g \rVert_{X^\infty(\Omega)}$ in the same way classical limits for $\mathrm{L}^p$-norms are constructed. By the theory of Lipschitz extensions \cite{jensen1993}, one can understand the $\infty$-Laplacian as the minimization of the sup norm of the gradient, given further justification to the idea.\\
\noindent We will not give the construction of the algorithm here again, since it is similar to the finite setting and the extensions are done in the same manner. However, it is interesting that the reformulated discrete problem (\ref{eq:ReformulatedFiniteMinimization}) essentially becomes a problem for the $1$-Laplacian over the subspace of constant $s$ meaning that the Lipschitz constant on each element is bounded uniformly and the bound is minimized. On the Neumann boundary this approach would result in $v\rvert_{\Gamma} = 0$ and thus we add the constraint $s \geq 1$ in order to obtain the desired slope on the boundary.\\
\noindent From the sequence of solutions for finite $p$ and the observations in 1D combined with the theoretical definition of the AMLE, we expect the limit artificially shown in figure \ref{fig:InfSolArtificial}. Figure \ref{fig:InfSolLoisel} shows the result for the choice of the inner norm $\lVert \cdot \rVert_2$ in equation (\ref{eq:InfLaplaceMinimization}) as proposed in \cite{loisel2020}. It overshoots the intended tip slightly, yielding an improvement to the solution for $p=15$ from figure \ref{fig:FiniteDeformations}. But it is not able to resolve the intended free boundaries, leading to worse results than the finite setting. However, the grid quality at those boundaries remains good and instead the center becomes significantly worse. By the method of manufactured solutions, we can also show that it is not able to yield the unique solution for a reference problems with the well-known viscosity solution $v=x_1^{4/3}-x_2^{4/3}$ \cite{lindqvist2016}. Another approach is choosing the supremum norm also for the inner norm, which we computed in figure \ref{fig:InfSolSupSup}. While this choice yields the correct outer boundary, it still does not deform the interior mesh uniformly.\\ \section{Conclusion} \label{sec:Conclusion}
We added support for Neumann boundary conditions to a present algorithm for the scalar $p$-Laplacian and proved that the theoretical estimate on the required Newton steps remains polynomial. Further, we constructed the extension of the algorithm to vector-valued problems and performed numerical experiments including validation and shape deformations. The results demonstrate that the extension is indeed applicable to problems occurring in $p$-harmonic shape optimization and yields solutions for higher-order $p>5$ without iterating over $p$. Those provide further improvements in terms of preserved mesh quality and obtained boundary shape. However, we saw that results obtained for the $\infty$-Laplacian are significantly different and even high-order solutions do not yield a sufficient approximation, especially since they depend on the magnitude of the source term. First experiments with a modified algorithm to solve the $\infty$-Laplacian problem directly did not yield the desired results, but small changes could already achieve improvements, allowing the idea to be considered further.\\
\noindent For future research it remains to construct a proper algorithm for the limit problem. On the other hand, the results for finite $p$ may be applied in shape optimization problems potentially including 3D settings. This includes modifications to the implementation for high-performance computer architecture and analysis considering scalability.
\section*{Acknowledgment} The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ``Modeling, Simulation and Optimization of Fluid Dynamic Applications''.
\printbibliography
\end{document} | arXiv | {
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\begin{document}
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\newcommand{ Verification of a many-ion simulator of the Dicke model through slow quenches across a phase transition}{ Verification of a many-ion simulator of the Dicke model through slow quenches across a phase transition}
\title{ Verification of a many-ion simulator of the Dicke model through slow quenches across a phase transition} \date{\today}
\author{ A. Safavi-Naini} \thanks{These two authors contributed equally} \affiliation{JILA, NIST and University of Colorado, 440 UCB, Boulder, CO 80309, USA} \affiliation{Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA} \author{ R. J. Lewis-Swan} \thanks{These two authors contributed equally} \affiliation{JILA, NIST and University of Colorado, 440 UCB, Boulder, CO 80309, USA} \affiliation{Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA} \author{J. G. Bohnet} \affiliation{NIST, Boulder, CO 80305, USA} \author{M. G\"{a}rttner} \affiliation{JILA, NIST and University of Colorado, 440 UCB, Boulder, CO 80309, USA} \affiliation{Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA} \affiliation{Kirchhoff-Institut f\"{u}r Physik, Universit\"{a}t Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany} \author{K. A. Gilmore} \affiliation{NIST, Boulder, CO 80305, USA} \author{J. E. Jordan} \affiliation{NIST, Boulder, CO 80305, USA} \author{J. Cohn} \affiliation{Department of Physics, Georgetown University, Washington, DC 20057, USA} \author{J. K. Freericks} \affiliation{Department of Physics, Georgetown University, Washington, DC 20057, USA} \author{A. M. Rey} \affiliation{JILA, NIST and University of Colorado, 440 UCB, Boulder, CO 80309, USA} \affiliation{Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA} \author{J. J. Bollinger} \affiliation{NIST, Boulder, CO 80305, USA}
\begin{abstract}
We use a self-assembled two-dimensional Coulomb crystal of $\sim 70$ ions in the presence of an external transverse field to engineer a simulator of the Dicke Hamiltonian, an iconic model in quantum optics which features a quantum phase transition between a superradiant/ferromagnetic and a normal/paramagnetic phase. We experimentally implement slow quenches across the quantum critical point and benchmark the dynamics and the performance of the simulator through extensive theory-experiment comparisons which show excellent agreement. The implementation of the Dicke model in fully controllable trapped ion arrays can open a path for the generation of highly entangled states useful for enhanced metrology and the observation of scrambling and quantum chaos in a many-body system. \end{abstract}
\maketitle
\noindent{\it Introduction. } Quantum many-body systems featuring controllable coupled spin and bosonic degrees of freedom are becoming a powerful platform for the realization of quantum simulators with easily tunable parameters. These include for example cavity QED systems \cite{Leroux2010c,Hosten_2016,Ritsch2013,BGB10,BDR07,Baumann2010,Baumann2011,Klinder2015} and trapped-ion arrays \cite{Porras2004,Kim2009}. Most often, these systems have been operated in the far detuned regime where the bosons do not play an active role in the many-body dynamics and instead are used to mediate spin-spin coupling between particles. Great progress has been realized in this effective spin-model regime including the implementation of long range Ising models with and without an external transverse field and the exploration of rich physics with them such as entanglement dynamics \cite{Leroux2010c,Hosten_2016,Blatt2012a,Jurcevic2014,Senko2015,Richerme2014,Islam2011,Bohnet2016,Garttner2016}, many-body localization \cite{Smith2015}, time crystals\cite{Zhanga2017} and dynamical phase transitions ~\cite{Jurcevic2017,Zhang2017}. \begin{figure}
\caption{Implementation and dynamical protocol. (a) The Dicke model is engineered with a Penning trap ion crystal of $N \sim 70$ ions by applying an optical dipole force, resonant only with the center of mass mode (which generates spin-phonon interactions) and resonant microwaves (which generates the transverse field). The system is initially prepared in the normal phase where all the spins point along the transverse field and are decoupled from the phonons. (b) As the transverse field is slowly turned off [using linear or exponential ramp (shown here) profiles with ramp time $\tau_{\rm ramp}$] the system enters the superradiant phase after crossing the quantum critical point at $B(t_{\rm crit})=B_c$ where the gap closes. The superradiant phase with macroscopic phonon population, ferromagnetically aligned spins and large spin-phonon entanglement is described by the order parameter $\langle(\hat{a}+\hat{a}^{\dagger})\hat{S}_z\rangle$, which is tracked closely by the re-scaled spin observable
$|\alpha_0|\langle|\hat{S}_z|\rangle$. (c) In the perfectly adiabatic regime the ground state evolves from a separable spin-paramagnetic and vacuum photon Fock state into a macroscopic spin-phonon cat state: a superposition of two opposite spin aligned and displaced-coherent phonon states (with the sign of the superposition dictated by a parity symmetry, see SM).}
\label{fig:schem}
\end{figure} On the other hand, excluding few particle implementations \cite{Pedernales2015,Lv2017,Lv2018rabi,Johnson2017,Kienzler2016,Monroe1996,Toyoda2015,Debnath2017,Raimond2001,Lamata2018}, the regime where the bosonic degrees of freedom actively participate in the many-body dynamics has remained largely unexplored.
In this work, we focus on this regime and report the implementation of a simulator of the Dicke model, an iconic model in cavity QED which describes the coupling of a (large) spin and an oscillator, in a self-assembled two-dimensional (2D) crystal of ions. The Dicke model is of broad interest as it exhibits rich physics including quantum phase transitions and non-ergodic behaviour \cite{Altland2012}. More recently it has gained renewed attention due to the implementation of the closely related Tavis-Cummings model in circuit QED \cite{Fink2009} and its realization in CQED experiments with ultracold bosonic atoms \cite{Baumann2010,Baumann2011,Klinder2015}. In the latter the Dicke model emerged as an effective Hamiltonian when one encodes a two-level system in two different momentum states of a Bose-Einstein condensate (BEC) coupled by the cavity field. Within this framework the normal to superradiant transition maps to a transition between a standard zero momentum BEC and a quantum phase with macroscopic occupation of the higher-order momentum mode and the cavity mode.
While CQED experiments have used the intracavity light intensity and time of flight images to monitor the phase transition, here we instead probe the two distinct quantum phases of the Dicke model, by using various controlled ramping protocols of a transverse field across the critical point (see Fig. \ref{fig:schem}). We benchmark the dynamics by experimentally measuring full distribution functions of the spin degrees of freedom and then comparing them with theoretical calculations. The spin observables also allow us to infer the development of spin-phonon correlations.
Our implementation of the Dicke model and corresponding observation of the phase transition in a trapped ion setup represents a complementary work with respect to the CQED platform and illustrates the power and universal nature of quantum simulation. It also opens a path for using the high level control and tunability of trapped ions experiments for the generation of highly entangled states suitable to quantum metrology in the near term future, and for the exploration of regimes currently intractable to theory.
\noindent{\it Spin-Boson System. } Our experimental system is comprised of a 2D single-plane array of laser-cooled $^{9}$Be$^{+}$ ions in a Penning trap. The internal states forming the spin-1/2 system are the valence electron spin states in the Be$^+$ ion ground state which,
in the $4.46$~T magnetic field, are split by $124$~GHz \cite{Garttner2016,Bohnet2016,Sawyer2012,Biercuk2009}. The interplay of the Coulomb repulsion and the electromagnetic confining potentials supports a set of normal vibrational modes of the crystal \cite{Wang2013}, which we couple to the spin degrees of freedom via a spin-dependent optical dipole force (ODF), generated by the interference of a pair of lasers with beatnote frequency $\omega_R$ \cite{Sawyer2012}. The frequency $\omega_R$ is detuned from the center-of-mass mode (COM) frequency, $\omega_{\rm COM}$, by $\delta\equiv\omega_R-\omega_{\rm COM}$ (Fig. 1). The detuning is chosen to predominantly excite the COM mode which uniformly couples all the ions in the crystal \cite{Bohnet2016}.
In the presence of an additional transverse field, generated by resonant microwaves, we implement the Dicke Hamiltonian \cite{Dicke1954,Garraway2011,Wall2017} :
\begin{eqnarray}
\hat{H}^{\rm Dicke}/\hbar = -\frac{g_0}{\sqrt{N}} \left(\hat{a}+\hat{a}^{\dagger}\right)\hat{S}_z + B(t) \hat{S}_x - \delta \hat{a}^{\dagger}\hat{a}. \label{eq:HI} \end{eqnarray} in the frame rotating with $\omega_R$. The operator $\hat{a} (\hat a^\dagger)$ is the bosonic annihilation (creation) operator for the COM mode, $B(t)$ is the time-varying strength of the applied transverse field, and $g_0$ represents the homogeneous coupling between each ion and the COM mode. Here, $\delta < 0$.
We have introduced the collective spin operators $\hat{S}_{\alpha} = (1/2)\sum_j \hat{\sigma}^{\alpha}_j$ where $\hat\sigma^{\alpha}_j$ is the corresponding Pauli matrix for $\alpha = x,y,z$ which acts on the $j$th ion.
The Dicke Hamiltonian exhibits a quantum phase-transition at $B_c=g_0^2/|\delta|$ in the thermodynamic limit, i.e. $N\to\infty$, \cite{Emary2003_PRL,Emary2003_PRE,Porras2013}, separating the normal ($B > B_c$ ) and superradiant ($B < B_c$ ) phases. The Hamiltonian remains unchanged under the simultaneous transformations $\hat S_z\to -\hat S_z$, $\hat{S}_y \to -\hat{S}_y$ and $\hat a \to -\hat a$. These are generated by the the parity operator $\hat{\Pi}=e^{i\pi(\hat{a}^{\dagger}\hat{a} + \hat{S}_x + \frac{N}{2})}$.
In the strong-field regime of the normal phase, $B \gg B_c$, the spins and phonons decouple into a product state. When $|B|>|\delta|$ the corresponding ground state, $\vert \psi_{0,N/2}^{\rm{Nor}} \rangle$, and low lying excitations, $\vert \psi_{n=1,2,\dots}^{\rm{Nor}} \rangle$, are $\vert \psi_{n,N/2}^{\rm{Nor}} \rangle = \vert n\rangle \otimes \vert -N/2 \rangle_x$. We use $\vert n \rangle$ to denote Fock states and $\vert M\rangle_{\alpha=\{x,y,z\}}$ to denote the fully symmetric ($S=N/2$) eigenstates of $\hat{S}_{\alpha}\vert M\rangle_{\alpha}=M\vert M\rangle_{\alpha}$ with $-N/2\leq M\leq N/2$.
\begin{figure*}
\caption{Benchmarking the simulator: Column (a) shows the experimentally measured distribution function along $z$, and (b) the corresponding theoretical simulations neglecting decoherence.
Column (c) show the corresponding mean values of the magnetization $\langle |\hat{S}_z| \rangle$, (d) spin-projection $\langle \hat{S}_x \rangle$, and
(e) $\mathcal C_{\rm sp-ph}\equiv \left \langle \left(\hat a + \hat a^\dagger\right) \hat S_y \right\rangle $.
The filled circles are experimental measurements (statistical error is on the order of marker size), the colored solid and black dashed lines are the theory results without and with
dephasing [the latter curve is absent in panel (c) as the $z$-magnetization is less sensitive to this dominant source of decoherence] and the
colored dotted lines are the Lipkin model results. We indicate the time at which $B_c$ is reached in
each ramp by a vertical line. The initial field is $B(t=0)/(2\pi) \approx 7.1$~kHz, $g_0/(2\pi) \approx 1.32$~kHz, $\delta/(2\pi) = -1$~kHz and $J/(2\pi)=1.75$~kHz.
Respective ion numbers are $N = 68$ [EXP -- row (i)] and $N = 69$ [LIN -- row (ii)].}
\label{fig:ExpResults_ExpecValues}
\end{figure*}
In the weak-field limit, $B \ll B_c$, of the superradiant phase, the spin and phonon degrees of freedom are entangled and the ground state is nearly degenerate in the thermodynamic limit. For a finite system it approaches $\vert \psi_{0, N/2}^{S} \rangle = \frac{1}{\sqrt{2}}\Big(\vert\alpha_0,0\rangle \otimes \vert N/2 \rangle_z \pm \vert-\alpha_0,0\rangle \otimes \vert -N/2 \rangle_z\Big)$ as $B \to 0$, where we have introduced the displaced Fock states $\vert\alpha,n\rangle \equiv \hat{D}(\alpha)\vert n\rangle$ with $\hat{D}(\alpha) = e^{\alpha\hat{a}^{\dagger} - \alpha^*\hat{a}}$ the associated displacement operator \cite{Wunsche1991}. Here, the sign of the superposition is dictated by the parity symmetry: for even $N$ the ground-state will be the symmetric superposition with $\langle e^{i\pi(\hat{a}^{\dagger}\hat{a} + \hat{S}_x + \frac{N}{2})} \rangle = 1$, while for odd $N$ the ground-state is the anti-symmetric superposition with $\langle e^{i\pi(\hat{a}^{\dagger}\hat{a} + \hat{S}_x + \frac{N}{2})} \rangle = -1$. In this weak-field regime the spins exhibit ferromagnetic order, characterized by the non-zero value of the order parameter $\vert \hat S_z \vert$, while the phonon mode acquires a macroscopic occupation $\vert\alpha_0\vert^2$, where $\alpha_0 = g_0\sqrt{N}/(2\delta)$.
The low-lying excitations correspond to displaced Fock states, $\vert \psi_{n>0, N/2}^{S} \rangle$, if $\delta^2 < g_0^2$ and to spin-flips along $\hat z$, $\vert \psi_{0, M<N/2}^{S} \rangle $, if $\delta^2>g_0^2$.
{\it Slow quench dynamics. }
At the start of the experimental sequence (see Fig.~\ref{fig:schem}) we prepare the initial spin state $\left|-N/2\right\rangle_{x}$ with the aid of a resonant microwave pulse. Doppler-limited cooling of the phonon degree of freedom leads to an initial phonon thermal state with mean occupation $\bar{n} \sim 6$. For these parameters the system starts in the normal phase close to the ground-state.
The transverse field is then quenched to zero (whilst the spin-phonon coupling and detuning are held constant) according to two different profiles: (i) Linear (LIN): $B(t)=B_0(1-t/\tau_{\mathrm{ramp}})$, and (ii) Exponential (EXP): $B(t) = B_0e^{-t/\tau}$. We set $\tau_{\mathrm{ramp}}=2$ms and $\tau \approx 600$~$\mu$s.
To characterize the performance of the simulator and the entrance into the superradiant phase, we experimentally measure the full spin distribution along the
$z$ direction (Fig.~\ref{fig:ExpResults_ExpecValues}) by determining the global ion fluorescence scattered from the Doppler cooling laser on the cycling transition for ions in $\left|\uparrow\right\rangle _{z}$ \cite{Martin2017_OTOC,Bohnet2016,Sawyer2012modeAndTempSpec,Biercuk2009}. For repeated experimental trials we infer the state populations, $N_{\uparrow}$ and $N_{\downarrow}$ and calculate the spin-projection $M_z \equiv N_{\uparrow}-N/2$ for each experimental shot by counting the total number of photons collected on a photomultiplier tube in a detection period, typically $5$~ms. Off-resonant light scattering from the ODF lasers is our main source of decoherence dominated by single-particle dephasing at a rate $\Gamma_{el}$ \cite{Uys2010}.
As noted above, the experimental implementation and corresponding numerical simulations were carried out with $N\approx70$ atoms. However, a well-defined cross-over between the normal and superradiant phases, signaled by a well-defined minimum in the energy gap between the ground and excited states of the same parity sector [see Fig. 1 (b)], appears for crystals larger than $N\gtrsim5$ (see SM).
Our theory-experiment comparisons are based on numerical solutions of the Dicke model dynamics combined with thermal averaging. If decoherence is neglected the spin degree of freedom is constrained to the $S=N/2$ manifold. In this reduced Hilbert space we can exactly treat the quantum dynamics. Whilst for the non-negligible thermal phonon occupation in this experiment a classical treatment of the dynamics is sufficient to reproduce the measured observables, a complete formulation of the quantum dynamics becomes necessary for colder conditions, when thermal fluctuations are insufficient to drive dynamics and instead quantum correlations must be properly accounted for.
We observe good qualitative agreement between the experimental spin probability distribution and the theoretically computed unitary dynamics as shown in Figs. \ref{fig:ExpResults_ExpecValues}(a) and (b). In particular, both show a clear transition to a bimodal structure as the field strength is ramped down through $B_c$ (indicated by the black vertical line in each plot), with some ``smearing" due to the thermal occupation of the phonons.
To quantitatively determine the performance of the simulator, we plot the evolution of the effective order parameter $\langle \vert \hat{S}_z \vert \rangle/N$ (experimental values are extracted from the measured distribution) in Fig.~\ref{fig:ExpResults_ExpecValues}~(c), which clearly builds up as one crosses $B_c$. The transition is not abrupt and instead exhibits small amplitude oscillations, most clearly evident in the theoretical calculations, which reflect the active role of the phonons given our initial finite thermal phonon occupation. In particular, our numerical simulations show a dependence of the oscillation amplitude on the initial phonon occupation (see SM). However, the frequency of the phonon oscillations is difficult to determine and interpret, as it depends on the complex interplay between the magnitude of the initial phonon occupation and the changing transverse field. We contrast this behavior with the case when the phonons can be adiabatically eliminated and realize an effective spin Lipkin model, $\hat{H}^{\rm LM}/\hbar = (J/N)\hat{S}_z^2 + B(t)\hat{S}_x$ where $J = g_0^2/\delta$. The Lipkin model dynamics features a sharper increase in magnetization after the critical point, and significant disagreement with the experimental observations.
To further benchmark the simulator we carry out similar measurements of the spin distribution along the $x$ direction, extracted by applying a global $\pi/2$ pulse before the fluorescence measurement. Fig.~\ref{fig:ExpResults_ExpecValues}~(d) shows the mean-value of the spin-projection $\langle\hat{S}_x\rangle$. We observe $x$-depolarization as the system exits the normal phase. The Lipkin model dynamics also exhibits a sharper depolarization across $B_c$ than the one seen in the experiment. In this case, however, we do observe deviations between the experiment and the ideal theory. The reason is that unlike the $z$-magnetization, this observable is strongly affected by dephasing. Since treating the full spin-boson system in the presence of decoherence is computationally challenging, we model the effect of dephasing as $\langle \hat{S}_x\rangle \rightarrow \langle \hat{S}_x \rangle e^{-\Gamma t}$ and $\langle \hat{S}_z\rangle\to \langle \hat{S}_z \rangle$ where $\Gamma=\Gamma_{el}/2$, which is asymptotically valid in the $B \gg B_c$ and $B \ll B_c$ limits \cite{Huelga1997}. We can determine $\Gamma_{el}$ experimentally when $B=0$, and we find $\Gamma_{el} \approx 120$~s$^{-1}$. However, at large $B$, most clearly evidenced in the LIN protocol, the demagnetization is faster than this estimate, and is consistent with $\Gamma_{el} = 280$~s$^{-1}$\footnote{This dephasing could be a result of the experimental system going beyond the Lamb-Dicke regime, which is implicitly assumed in the derivation of the Dicke Hamiltonian Eq.~(1).}. For both ramps we observe excellent agreement to the experiment when dephasing is accounted for.
Although measuring the phonon population might be possible following the protocol reported in Ref.~\cite{Gilmore2017}, we instead infer the build-up of spin-phonon correlations from the time evolution of the spin observable $\langle \hat{S}_x \rangle$. Specifically, we assume the dynamics of the system are captured by the Lindblad master equation for the density matrix of the spin-phonon system $\hat{\rho}$, \begin{equation}
\frac{d\hat{\rho}}{dt} = -\frac{i}{\hbar} \left[ \hat{H}^{\mathrm{Dicke}}, \hat{\rho} \right] +\frac{\Gamma_{el}}{2}\sum_{i=1}^N \left( \hat{\sigma}^z_i \hat{\rho} \hat{\sigma}^z_i - \hat{\rho} \right) , \end{equation} where single-particle dephasing is taken to be the dominant decoherence mechanism. From the master equation we derive the equation of motion $\frac {d}{dt}\langle \hat{S}_x \rangle$, and rearrange to obtain the relation (see SM) \begin{equation}
\mathcal C_{\rm sp-ph}\equiv \langle \left(\hat a + \hat a^\dagger\right) \hat S_y \rangle \equiv \frac{\sqrt{N}}{g_0} \left( \Gamma_{el} \langle \hat S_x \rangle + \frac {d}{dt}\langle \hat{S}_x \rangle \right) . \end{equation} We extract the spin-phonon correlation from the experimental data by evaluating the RHS of the above expression, and calculating the time-derivative numerically with a one-sided derivative. The results are plotted in Fig.~\ref{fig:ExpResults_ExpecValues}(e). We use the same value of $\Gamma_{el}$ as in Fig.~\ref{fig:ExpResults_ExpecValues}(d). The results are compared with a theoretical calculation of $C_{\mathrm{sp-ph}}$ [again modelling dephasing using $\langle \hat{S}_x \rangle_{\Gamma} \equiv \langle \hat{S}_x \rangle_{\Gamma=0}e^{-\Gamma t}$]. In principle, the correlator vanishes when evaluated for the ground-state at any field strength. However, for these slow quenches it acquires a finite value, which in particular grows in the superradiant phase, due to population of excited states. This is attributable due to diabatic excitations created during the ramping protocol or the initial thermal phonon ensemble. Thus, while the correlation $\mathcal C_{\rm sp-ph}$ shows similar dynamical features observed in the other observables, it gives an alternative insight into the excitations created during the ramp.
While we have used the two ramp profiles to benchmark the experiment, we note that the EXP ramp has more utility in preparing a final state close to the expected ground-state $\vert \psi_{0,N/2}^{S} \rangle$ in the superradiant phase. For instance, the EXP ramp produces a clearer bimodal structure in the spin probability distribution along $z$, and associated larger mean absolute spin projection $\langle \vert \hat{S}_z \vert \rangle$. Future experiments could improve assesment of the adiabaticity of the quench protocols by measuring any coherences present between the different spin components, as discussed below.
Accounting for spin-phonon entanglement will be key to properly diagnose the generated many-body quantum state. For example, tracing out the phonons from $\vert \psi_{0,N/2}^{S} \rangle$ will exponentially suppress the coherence between the spin states $\vert \pm N/2 \rangle_z$ (see SM). To benchmark the performance of the adiabatic dynamics it is then highly desirable to first perform a protocol to disentangle the spins and phonons and only after it characterize the state by independently measuring the spins and the phonons without information loss.
To disentangle spin and phonons we propose to instantaneously quench the detuning
$\delta \rightarrow \delta^{\prime} = 2\delta$ at the end of the LAA ramp ($B \rightarrow 0$) and then let the system evolve for a time $t_d=\pi/\delta^\prime$. At $t_d$ the phonons are coherently displaced by $ - g_0\sqrt{N}/(2|\delta|) \langle S_z\rangle$ back to the origin, while the spins only acquire an irrelevant global phase \cite{Wall2017}.
The resulting disentangled state ideally becomes $(1/\sqrt{2})[\vert+\alpha_0,0\rangle\vert+ N/2\rangle_z + \vert-\alpha_0,0\rangle\vert- N/2\rangle_z] \rightarrow (1/\sqrt{2})\vert0\rangle\otimes[\vert+ N/2\rangle_z + \vert- N/2\rangle_z]$ which has maximal spin coherence.
{\it Summary and discussion.} We have reported the experimental realization of a simulator of the Dicke model with a 2D ion crystal of $\sim 70$ ions and verified its dynamics through extensive theory-experiment comparisons. Our trapped-ion simulator provides a complementary approach to related realizations in cold atoms \cite{Baumann2010,Baumann2011,Klinder2015}, which is a key step in benchmarking quantum simulators which go beyond the capacity of classical computation.
Our realization of a many-ion simulator of the Dicke model also paves the way for future investigation of dynamical phase transitions \cite{Jurcevic2017,Zhang2017}, quantum chaos and fast scrambling via out-of-time order correlation measurements \cite{Shenker2014,Kitaev2015,Maldacena2016,swingle2016measuring,Garttner2016}. Moreover, the tunability of the trapped-ion setup opens the possibility of investigating more general spin-boson models \cite{Athreya2017sync}, in particular by operating beyond the uniform coupling regime or the preparation of states outside the fully symmetric Dicke manifold.
The slow quench protocols demonstrated above present a path to generate highly entangled states useful for quantum enhanced metrology \cite{Feldmann2018}. Cat-states are a useful metrological resource as they are composed of a coherent superposition of states that are macroscopically displaced in phase-space, leading to quantum-enhanced phase-sensitivity up to the Heisenberg limit \cite{Bollinger1996,Toscano2006}. In particular, the spin-boson cat-state $\vert \psi_{0,N/2}^{S} \rangle$ would be a metrological resource for sensing collective spin rotations \cite{Bollinger1996}, motional rotation \cite{Campbell2017,Johnson2017}, and coherent displacements for force sensing applications \cite{Penasa2016}.
This could be achieved by using smaller systems (e.g., $N \sim 20$), reducing the initial thermal population of the phonon mode, and shifting the detuning $\delta$ away from $B_c$, which increases the minimum energy gap at the critical point, and consequently the characteristic time-scale to remain adiabatic (see SM). We expect this regime will be accessible in the near term future in part due to the successful implementation of electromagnetic induced transparency cooling \cite{futurePaper}.
\begin{acknowledgments} The authors acknowledge fruitful discussions with J. Marino, M. Holland and K. Lehnert. A.~M.~R acknowledges support from Defense Advanced Research Projects Agency (DARPA) and Army Research Office grant W911NF-16-1-0576, NSF grant PHY1521080, JILA-NSF grant PFC-173400, and the Air Force Office of Scientific Research and its Multidisciplinary University Research Initiative grant FA9550-13-1-0086. M.G. acknowledges support from the DFG Collaborative Research Center SFB1225 (ISOQUANT). J.~E.~J. also acknowledges support from Leopoldina Fellowship Programme. JKF and JC acknowledge support from NSF grant PHYS-1620555. In addition, JKF acknowledges support from the McDevitt bequest at Georgetown. Financial support from NIST is also acknowledged. \end{acknowledgments}
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\begin{center} \textbf{\large Supplemental Material: Verification of a many-ion simulator of the Dicke model through slow quenches across a phase transition} \end{center}
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\section{Finite size effects in observing the phase transition} The quantum phase transition of the Dicke model only truly emerges in the thermodynamic limit $N\to\infty$ \cite{Emary2003_PRL,Emary2003_PRE}. It is thus important to consider the relevance of finite size effects, specifically pertaining to the number of ions $N$ and thus the collective spin length $S=N/2$.
In this spirit, we plot the order parameter $\langle ( \hat{a} + \hat{a}^{\dagger} )\hat{S}_z \rangle$ and energy gap $\Delta$ between the ground-state and excited state in the same parity sector, for various ion numbers in Fig.~\ref{fig:NPlot} and as a function of transverse field strength $B$. A minimum in the energy gap, as a function of $B$, emerges for $N\gtrsim5$. This minimum is associated with the crossover between the normal and superradiant phase, and thus we predict that features of the crossover should be observable for $N\gtrsim 5$. This is consistent with the increasingly sharp transition observable in the order parameter for $N\gtrsim5$. Similarly, calculation of the spin observables $\vert S_z \vert$ and $S_x$ from dynamical ramps [plotted as a function of $B(t)$, parameters taken as per Fig.~2b of the manuscript], indicate that the crossover between normal and superradiant phases is evident for $N\gtrsim5$, which is easily satisfied by the experimentally considered crystal of $N\sim70$.
\begin{figure*}
\caption{Key quantities and observables of Dicke model as a function of atom number $N$ and transverse field $B$. Energy gap $\Delta$ is between the ground-state and
excited state in the same parity sector. Magnetization $\vert S_z \vert$ and mean spin projection $S_x$ are computed for a LIN ramp with $t=2$ms and other parameters taken as per Fig.~2b of the manuscript.
}
\label{fig:NPlot}
\end{figure*}
\section{Effect of the resonance on the energy gap \label{app:gap}}
As discussed in the main text, the Dicke Hamiltonian features a spin-boson resonance at $B=\vert \delta \vert $. At this field strength, the states $\vert m \rangle\vert -N/2 \rangle_x $ and $\vert m-k\rangle\vert -N/2 + k \rangle_x$, with $k$ a positive integer, become nearly degenerate and can be resonantly coupled. The location of this resonance, relative to the critical field strength $B_c$, can greatly affect the energy spectrum of the Dicke model and in particular the magnitude of the energy gap $\Delta$ between the ground-state and excited states in the same parity sector. In this context, we can separate the effects of the resonance into two cases, defined by the relative position of the resonance to the critical field strength:
\begin{itemize}
\item {Case (i): $|\delta| \gg B_c$.} In this regime the resonance $B\simeq |\delta|$ is well separated from the critical point. The ground-state $\vert \psi^{\rm Nor}_{0, N/2} \rangle = \vert 0 \rangle\vert -N/2 \rangle_x$ is decoupled from other states at resonance. Thus, the dynamics can be affected by resonant couplings to other states (as above) only if excited states have become occupied throughout the quench.
\item{Case (ii): $|\delta| \sim B_c$.} If the resonance is in the proximity of the quantum critical point then the low-lying excitations near the critical point of the Dicke Hamiltonian are non-trivial superpositions of spin and phonon excitations. A radical consequence of this complex interplay is the relative reduction of the energy gap between the ground and the first excited states of the same parity at the critical point. We illustrate this in Fig.~\ref{fig:norm_gap} as a function of detuning $\delta$, with the spin-phonon coupling $g_0$ scaled such that the critical field strength $B_c = g_0^2/\delta$ is held fixed. \end{itemize}
\begin{figure}
\caption{The size of the gap as a function of the detuning from the COM for $N=40$. As the size of the detuning increases, the resonant region of the Dicke model moves away from the
quantum critical point separating the normal and superradiant phases. The energy gap at the critical point eventually saturates to a maximum value $\Delta E_{\rm gap}^{\rm max}$. }
\label{fig:norm_gap}
\end{figure}
\section{Additional sequence to disentangle the spin cat-state \label{app:cat}} In the main text, we briefly outline a procedure to disentangle the pure spin-cat state from adiabatic preparation of the ground-state of the Dicke Hamiltonian. Here, we expand upon this discussion and give the appropriate details to verify this step.
In the weak-field limit, $B \ll B_c$, the ground-state of the Dicke Hamiltonian is the spin-phonon cat-state: \begin{equation} \vert \psi^{S}_{0,N/2} \rangle = \frac{1}{\sqrt{2}}\Big(\vert\alpha_0,0\rangle\vert N/2 \rangle_z \pm \vert-\alpha_0,0\rangle\vert -N/2 \rangle_z\Big) , \label{eqn:Supp_SpinPhCat} \end{equation} where $\alpha_0 = g_0\sqrt{N}/(2\delta)$. Without loss of generality we fix the sign of the superposition due to conservation of the spin-phonon parity symmetry, which dictates that the positive superposition is prepared by an adiabatic quench from the strong-field ground-state $\vert \psi^{\mathrm{Nor}}_{0,N/2} \rangle$.
The choice of the sign in the superposition state Eq.~(\ref{eqn:Supp_SpinPhCat}) is dictated by the spin-phonon parity symmetry of the Dicke Hamiltonian. Specifically, $\hat{H}$ is preserved under the simultaneous transformation of $\hat S_z\to -\hat S_z$, $\hat{S}_y \to -\hat{S}_y$ and $\hat a \to -\hat a$, and the associated conserved quantity of the Hamiltonian is the generator of the symmetry $\hat{\Pi} \equiv e^{i\pi(\hat{a}^{\dagger}\hat{a} + \hat{S}_x + \frac{N}{2})}$. This symmetry dictates that when ramping from high to low field, the state $\vert \psi_{0, N/2}^{Nor} \rangle$ will adiabatically connect to the superposition $\vert \psi_{0,N/2}^{S} \rangle$, to conserve the parity $\langle \hat{\Pi} \rangle = e^{i\pi N}$. Specifically, for even $N$ the ground-state will be the symmetric superposition with $\langle \hat{\Pi} \rangle = 1$, whilst for odd $N$ the ground-state is the anti-symmetric superposition with $\langle \hat{\Pi} \rangle = -1$. Without loss of generality, we assume for the following that $N$ is even and thus we fix the sign of the superposition to be positive.
Since the spin and phonon degrees of freedom are entangled in the ground-state [Eq.~(\ref{eqn:Supp_SpinPhCat})], the state obtained by tracing over the phonon degree of freedom is characterized by the reduced density operator \begin{multline}
\hat{\rho}_s = \frac{1}{2}\Big[ |N/2\rangle_z\langle N/2|_z + |-N/2\rangle_z\langle -N/2|_z \Big] \notag \\
+ \frac{e^{-|\alpha_0|^2}}{2} \Big[|-N/2\rangle_z\langle N/2|_z + |N/2\rangle_z\langle -N/2|_z \Big] . \end{multline}
As the displacement amplitude $|\alpha_0|$ is increased, the reduced density matrix exponentially loses any information about the coherences which are exhibited in the spin-phonon superposition state. As a concrete example, the ground-states of the main text typically have a mean phonon occupation $|\alpha_0|^2 \sim 2$--$30$ depending on the chosen parameters (i.e., detuning and spin-phonon coupling), leading to $e^{-|\alpha_0|^2}\lesssim 0.1$. To fully probe the available coherences via only the spin degree of freedom, we must first transform Eq.~(\ref{eqn:Supp_SpinPhCat}) to a spin and phonon product state, \begin{equation} \vert \psi_{{\rm SB}} \rangle = \vert \phi\rangle \otimes \frac{1}{\sqrt{2}} \Big(\vert N/2 \rangle_z + \vert -N/2 \rangle_z\Big) , \label{eqn:Supp_SpinPhProduct} \end{equation}
where $|\phi\rangle$ is some arbitrary state characterizing the phonon degree of freedom.
A possible procedure to achieve this decomposition is the following: At the conclusion of the ramp protocol, we fix the transverse field at $B=0$ and quench the detuning $\delta \rightarrow \delta^{\prime} = 2\delta$. The spin-phonon state is then allowed to evolve for a duration $t_{d} = \pi/\delta^{\prime}$. In the interaction picture, the initial spin-phonon superposition state evolves as \begin{eqnarray}
\vert \psi_{{\rm SB}} \rangle = \hat{U}(t) \vert \psi_{0,N/2}^{S} \rangle , \end{eqnarray} where \begin{eqnarray}
\hat{U}(t) & = & \hat{U}_{SB}(t)\hat{U}_{SS}(t) , \\
\hat{U}_{\rm SS}(t) & = & \exp\left( -i \frac{J}{N} \hat S_z^2 t\right) , \\
\hat{U}_{\rm SB}(t)& = & \hat D( \beta(t,\delta') S_z) .
\end{eqnarray} Here, $\hat U(t)$ is the propagator corresponding to the Dicke Hamiltonian with $B=0$ [Eq.~1 of the main text]. The propagator is comprised of two parts, the spin-spin propagator $\hat U_{\rm SS}(t)$ and the spin-phonon propagator $\hat U_{\rm SP}(t)$ where $\beta(t,\delta)= -g_0(1-e^{-i \delta t})/(2\delta\sqrt{N})$ (see \cite{Wall2017} for a more detailed discussion).
If at the end of the ramp we quench the detuning to $\delta^{\prime} = 2\delta$ and apply $\hat U(t)$ for $t_d=\pi/\delta^{\prime}$, such that $\beta(t_d,\delta')=-g N/(2\delta)$, it is then clear that $\hat{U}_{SB}$ will displace the phonon coherent states (in a direction dependent on the sign of the $S_z$ component) back to vacuum, $|\pm\alpha_0,0\rangle \rightarrow |0\rangle$. We illustrate this displacement in Fig.~\ref{fig:Supp_disentangle}. Note that the action of $\hat{U}_{SS}$ on the spin component of the ground-state imprints an irrelevant global phase $\varphi = JNt_d/2$ on the decoupled state Eq.~(\ref{eqn:Supp_SpinPhProduct}).
\begin{figure}
\caption{Schematic of the disentangling protocol to extract a pure spin cat-state from the spin-phonon
ground-state $\vert \psi_{0,N/2}^{S} \rangle$. At the end of the ramp, we quench the detuning $\delta \rightarrow 2\delta$ and evolve the system for an additional duration $t_d = \pi/|2\delta|$ at fixed $B=0$.
The phonon states start at opposing coherent amplitudes and undergo a spin-dependent coherent displacement which maps them to the phonon vacuum state.}
\label{fig:Supp_disentangle}
\end{figure}
An alternative, but closely related, procedure to disentangle the spin-phonon state is to drive the spin-phonon coupling on resonance, $\delta \to \delta^{\prime} = 0$. In this case, one must shift the phase of the drive by $\pi/2$ such that the spin-phonon coupling transforms as $\frac{g_0}{\sqrt{N}}(\hat{a} + \hat{a}^{\dagger})\hat{S}_z \to \frac{ig_0}{\sqrt{N}}(\hat{a} - \hat{a}^{\dagger})\hat{S}_z$, and subsequently evolve the system for a duration
$t_d = 1/|\delta|$. Following this procedure results in a spin-dependent coherent displacement of the phonon state back to vacuum, $|\pm\alpha_0,0\rangle \rightarrow |0\rangle$, in a manner similar to the previously discussed protocol.
We make one further point regarding the disentangling protocols. In the experimental system we generally characterize the initial state of the phonons as a thermal ensemble $\hat{\rho}_{\bar n}$ while the spin-degree of freedom is prepared in a pure state, such that the initial spin-phonon state is $\hat{\rho}_{SB}(0) = \hat{\rho}_{\bar n} \otimes \vert -N/2 \rangle_x \langle -N/2 \vert_x $. If the protocol is adiabatic and there is no coupling between the excited energy levels, then not only is the ground-state component of this initial ensemble mapped to the weak-field ground-state of the Dicke Hamiltonian, but the excited fraction due to the thermal distribution is also mapped identically. This implies that the final state at the end of the ramp protocol will be a mixture of the true ground-state and the low-lying excitations, which, if $\delta^2 < g^2 N$, can be characterised as displaced Fock states $\vert \pm\alpha_0, n \rangle$ where $n$ corresponds to the number of phonon excitations above the true ground-state.
The action of this protocol on these states is to identically displace the phonon state such that $\vert \pm\alpha_0, n \rangle \rightarrow |n\rangle$. This maps the spin-phonon excited states to the form of a product state identical to Eq.~(\ref{eqn:Supp_SpinPhProduct}). Hence, tracing the phonons out of these excited states also recovers the spin cat-state.
\section{Qualitative effects of initial phonon occupation} In the main text we comment that the oscillations in the spin observable $\langle \vert \hat{S}_z \vert \rangle$ at short times is an indication of a non-negligible initial thermal occupation of the phonon mode (Fig.~2 of main text). Here, we support this conclusion by comparing results of theoretical calculations with different initial phonon occupation. Taking relevant parameters as per Fig.~2 of the main text and considering only the EXP ramp for simplicity, we plot the theoretical results for evolution of $\langle \vert \hat{S}_z \vert \rangle$ in Fig.~\ref{fig:PhononOsc}. We observe that if the phonons are taken to be initially in a vacuum state, the short time dynamics displays only extremely weak signs of oscillations. In contrast, when the phonons are taken to be initially described by a thermal ensemble with mean occupation $\bar{n} = 3$-$9$ there are signficant oscillations at short-times, consistent with the observed experimental data. Moreover, the final magnetization at the conclusion of the ramp protocol is much larger than that predicted from the vacuum case. The various values of $\bar{n}$ plotted give relatively similar agreement with the experimental data. However, $\bar{n} = 6$ is chosen in the main text as this is consistent with the estimated limit from Doppler cooling in the experiment.
\begin{figure}
\caption{Comparison of magnetization $\langle |\hat{S}_z|\rangle$ from experimental data and theoretical calculations for different initial thermal occupation $\langle \hat{a}^{\dagger}\hat{a}\rangle = \bar{n}$ of the phonon mode.
The amplitude of the oscillations at $t \lesssim 1$ clearly increase with $\bar{n}$, whilst the frequency appears to remain comparitively fixed. Data is for an EXP ramp, with all other parameters taken as per Fig.~2b of the
manuscript. }
\label{fig:PhononOsc}
\end{figure}
\section{Inference of spin-phonon correlations \label{app:spinphonon}} As detailed in the main text, we infer the presence of spin-phonon correlations from the time evolution of the spin observable $\langle \hat{S}_x \rangle$. Specifically, starting from the Lindblad master equation for the density matrix of the spin-phonon system $\hat{\rho}$, \begin{equation}
\frac{d\hat{\rho}}{dt} = -\frac{i}{\hbar} \left[ \hat{H}^{\mathrm{Dicke}}, \hat{\rho} \right] +\frac{\Gamma_{el}}{2}\sum_{i=1}^N \left( \hat{\sigma}^z_i \hat{\rho} \hat{\sigma}^z_i - \hat{\rho} \right) , \end{equation} wherein we have assumed single-particle dephasing is the dominant decoherence mechanism, it then follows that \begin{equation}
\frac{d\langle \hat{S}_x \rangle}{dt} = \frac{g_0}{\sqrt{N}}\langle \left( \hat{a} + \hat{a}^{\dagger} \right) \hat{S}_y \rangle - \Gamma_{el}\langle \hat{S}_x \rangle . \end{equation} From here it is straightforward to rearrange for the relation between the spin-phonon correlation and the evolution of $\langle \hat{S}_x \rangle$: \begin{equation}
\mathcal{C}_{\mathrm{sp-ph}} \equiv \langle \left( \hat{a} + \hat{a}^{\dagger} \right) \hat{S}_y \rangle = \frac{\sqrt{N}}{g_0} \left( \Gamma_{el}\langle \hat{S}_x \rangle + \frac{d\langle \hat{S}_x \rangle}{dt} \right) . \label{eqn:SpinPhCorr} \end{equation}
We emphasize that evaluation of this spin-phonon correlation directly from either ground-state $\vert \psi^{\mathrm{Nor}}_{0,N/2} \rangle$ $\vert \psi^S_{0,N/2} \rangle$ yields $\mathcal{C}_{\mathrm{sp-ph}} = 0$, and this result has been confirmed numerically for all transverse field strengths $B$ for the systems considered in the main text. This directly implies that the finite value reported in the main text is due to contributions from excited states. Such contributions may come from diabatic excitations created throughout the ramping protocol or from the initial thermal phonon ensemble.
In the main text, we extract the spin-phonon correlation from the experimental data using the RHS of Eq.~(\ref{eqn:SpinPhCorr}) and evaluating the time-derivative numerically with a one-sided derivative. We model dephasing using $\langle \hat{S}_x \rangle_{\Gamma} = \langle \hat{S}_x \rangle_{\Gamma = 0} e^{-\Gamma t}$ in our theoretical calculations, and extract the theoretically predicted spin-phonon correlation in an identical manner.
\section{Experimental Optimisation of ramp protocols \label{app:RampOpt}}
To experimentally optimize the ramp protocols demonstrated in this work, we chose to optimize with respect to the total magnetization $\langle |\hat{S}_z| \rangle$ at the end of the ramp. For the EXP ramp, we compared approximately $20$ different ramp profiles that utilized different exponential decay rates. Specifically, we would perform an experiment where the effective transverse field was ramped from the initial field $B(t=0)$ at a fixed decay rate to $B\approx0$, where we then measured the spin-projection $M_z^{\mathrm{exp}}$ along the $\hat{z}$-axis. This experiment was repeated, typically $500-700$ times, to gather statistics on the resulting distribution and obtain a measurement of $\langle|\hat{S}_z|\rangle$ from the histogram of $M_z^{\mathrm{exp}}$ measurements. We then picked a ramp profile with a different exponential decay rate, and repeated this procedure. After identifying the exponential decay rate that optimized the final magnetization $\langle|\hat{S}_z|\rangle$, we performed experiments that measured the magnetization distribution $P(M_z^{\mathrm{exp}})$ when stopping the ramp at different times, as discussed in the main text.
\begin{figure}\label{fig:Supp_Exp}
\end{figure}
When performing these ramp sequences and observing the distributions of $M_z^{\mathrm{exp}}$, in some cases the distributions would be biased to positive or negative spin-projection. This can be observed in the distribution of Fig.~\ref{fig:Supp_Exp}(a) at zero offset frequency. Such an effect can be explained by a small longitudinal magnetic field that breaks the symmetry of the ground state.
The small longitudinal field was likely due to imperfect nulling of the Stark shift from the off-resonant laser beams that generate the spin-dependent force \cite{Bohnet2016}. We would observe that this effect varies day to day. To compensate for this effect, during the ramp we would apply a small frequency offset to the microwaves that provided the effective transverse field. For each frequency offset, we would measure the distribution of measurements $M_z^{\mathrm{exp}}$ at the end of the transverse field ramp as shown in Fig.~\ref{fig:Supp_Exp}(a). For the appropriate offset, the distribution would be balanced, with large, separated peaks at positive and negative values of $M_z^{\mathrm{exp}}$. To choose the optimum, we plot $\langle\hat{S}_z\rangle$ as a function of the frequency offset and extract the zero crossing, as shown in Fig.~\ref{fig:Supp_Exp}(b).
\end{document} | arXiv | {
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\begin{document}
\title{Realizing topological relativistic dynamics with slow light polaritons at room temperature}
\author{Mehdi Namazi} \author{Bertus Jordaan} \affiliation{Department of Physics and Astronomy, Stony Brook University, New York 11794-3800, USA} \author{Changsuk Noh} \affiliation{Korea Institute for Advanced Study, 85 Hoegiro, Seoul 1307} \author{Dimitris G. Angelakis} \email{dimitris.angelakis@gmail.com} \affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 1175} \affiliation{School of Electrical and Computer Engineering, Technical University of Crete, Chania, 73100, Greece} \author{Eden Figueroa} \email{eden.figueroa@stonybrook.edu} \affiliation{Department of Physics and Astronomy, Stony Brook University, New York 11794-3800, USA}
\begin{abstract} Here we use a slow light quantum light-matter interface at room temperature to implement an analog simulator of complex relativistic and topological physics. We have realized the famous Jackiw-Rebbi model (JR), the celebrated first example where relativity meets topology. Our system is based upon interacting dark state polaritons (DSP's) created by storing light in a rubidium vapor using a dual-tripod atomic system. The DSP's temporal evolution emulates the physics of Dirac spinors and is engineered to follow the JR regime by using a linear magnetic field gradient. We also probe the obtained topologically protected zero-energy mode by analyzing the time correlations between the spinor components. Our implementation paves the way towards quantum simulation of more complex phenomena involving many quantum relativistic particles. \end{abstract}
\maketitle
\begin{figure*}
\caption{\textbf{Experimental setup for creation of polariton relativistic dynamics at room temperature } (a) The scheme used for creating an EIT tripod system (solid lines) and a dual tripod configuration (solid and dashed lines). (b) In order to experimentally create an EIT tripod system, an electric field $E(z,t)$ (red arrow) enters the medium, simultaneously with two strong co-propagating fields ($\Omega_{d}^{+}$ and $\Omega_{u}^{+})$. For the creation of the dual tripod configuration, the original field $E(z,t)$ is converted into two counter-propagating fields ($E_{-}(z,t)$ and $E_{+}(z,t)$) by introducing two pairs of counter-propagating control fields ($\Omega_{d}^{+}$, $\Omega_{u}^{+}$, $\Omega_{d}^{-}$ and $\Omega_{u}^{-})$. $m_{eff}(z)$ in the JR model is created using a spatially varying magnetic field gradient (dashed black lines).}
\end{figure*}
Over the last decade, a variety of exotic physical phenomena have been realized in artificially created quantum systems near zero temperature \cite{Cirac2012}, including ultra-cold atoms \cite{Bloch2012}, trapped-ions \cite{Blatt2012} and superconducting qubits \cite{Houck2012}. Photonic setups have also been used to emulate relativistic and topological models \cite{Aspuru2012,Rechtsman2013}. An unexplored direction for analog simulation, allowing for operation at room temperature conditions, is the use of atoms interfaced with light in the form of collective excitations known as dark state polaritons (DSPs) \cite{FleischhauerLukin2000}. DSPs have formed the basis of many quantum technology applications including quantum memories \cite{Lvovsky2009,Bussieres2013} and quantum non-linear frequency convertors \cite{radnaev_quantum_2010}. In this work we use DSPs to experimentally demonstrate an analog simulation of the Jackiw-Rebbi (JR) model, the celebrated first example where relativity met topology \cite{JackiwRebbi1976}. Our DSPs are created by storing light in Rb atoms prepared in a dual tripod configuration using counter propagating laser fields. We first show how to realize relativistic Dirac spinor dynamics, as initially suggested in \cite{Unanyan2010}, and then create a static soliton background field as required in the JR model, via a spatially varying atom-photon detuning. We observe signatures of JR's famous zero-energy mode using a temporal analysis of the retrieved light pulses.
DSP's are created by storing and manipulating light in atomic media using electromagnetically-induced-transparency (EIT) \cite{Liu_Nature_2001,phillips_storage_2001,FleischhauerLukin2000,EITReview2005}. In a DSP-based analog simulator, the DSPs are made to follow the dynamics outlined by a light-matter Hamiltonian, prepared by addressing specific atomic transitions using control light fields \cite{Chang2017}. The simulation results are then obtained by measuring the output photon wave functions. Along these lines, a spinor-like object consisting of two DSPs has been experimentally implemented in a double tripod configuration \cite{lee_experimental_2014}.
Moreover DSP-based quantum simulators have been proposed to realize Dirac models \cite{Unanyan2010}, and interacting relativistic systems \cite{Angelakis2013}. Among those, the JR model is of paramount relevance \cite{JackiwRebbi1976} as it predicts charge fractionalization \cite{Niemi1986}. This important aspect has been addressed in proposals using optical lattice setups \cite{RuostekoskiJavanainen2002,JavanainenRuostekoski2003}. Additionally, the JR model has gained further attention due to the topological nature \cite{Witthaut2011,Salger2011,Leder2016,Lamata2011,Grossert2016,Casanova2010,Gerritsma2011,Wilczek2016,Zhang2016} of its zero-energy solution \cite{Mugel2016,Tan_PhotTop2014}. This protected mode can be understood as a precursor to topological insulators \cite{Iadecola2016}, a hotly pursued topic nowadays \cite{TIreview1,TIreview2}. Recently, a soliton following a similar model has been observed experimentally in a fermionic superfluid \cite{YefsahZwierlein2013}.
Here we report the experimental realization of relativistic dynamics exhibiting topological aspects as originally conceived in the JR model \cite{JackiwRebbi1976}. We use a room temperature atomic vapour addressed by laser fields in an EIT dual tripod configuration in order to produce slow light Dirac spinor dynamics. We then tune the system to the JR regime by using a linear magnetic field gradient, inducing a kink profile in the corresponding mass term \cite{Angelakis2014}. The topological aspects of this novel EIT light-matter JR system are explored by analyzing the time correlations between the retrieved spinor components.
The structure of the article is as follows: firstly, we review the basics of coherent light propagation in a dual tripod EIT system. We then show the necessary conditions to connect this framework to Dirac and JR dynamics. Lastly, we present the experimental road map to prepare, evolve and benchmark this EIT light-matter JR system in a room temperature atomic interface. \section{Theoretical Background} \subsection{Tripod based dark state polaritons.} We start by describing the dynamics of an EIT tripod system (defined as tripod-type linkage pattern in \cite{Ruseckas_2011}) formed by two control fields ($\Omega_{d}$ and $\Omega_{u}$) and a probe field $E(z,t)$ (see Fig. 1a solid lines for definition of atomic levels). Based on the usual EIT assumptions, the following equation describes the propagation of the probe $E(z,t)$ under $\Omega_{d}$: \begin{equation*} (\partial_{t}+v_{g}\partial_{z})E(z,t)=+i\frac{g_{\varepsilon}^{2}}{\Omega_{d}^{2}}N \frac{\delta}{2} E(z,t) \end{equation*} where $v_{g}=\frac{c}{1+\frac{g_{\varepsilon}^{2}}{\Omega_{d}^{2}}N}$ is the group velocity of the input field in the atomic medium, N is the number of atoms along the beam path, $\delta$ is the two-photon detuning and $g_{\varepsilon}$ is the light-matter coupling constant for $E(z,t)$. In the perturbative and adiabatic limit a similar equation for the atomic operator $\sigma_{gd}$ can be found. The solution to the combined system of equations is a superposition of $E(z,t)$ and $\sigma_{gd}(z,t)$ and is called a dark state polariton, $\Psi_{d}(z,t)$ \cite{FleischhauerLukin2000}.
In the tripod configuration, the second control field $\Omega_{u}$ creates an additional DSP, $\Psi_{u}(z,t)$. The response of the system is then given by a linear combination of the two DSPs also known as tripod DSPs,$\Psi_{T} =\alpha\Psi_{d}+\beta\Psi_{u}$, with $\Psi_{d(u)}(z,t) =\cos\theta_{d(u)} E(z,t)-\sin\theta_{d(u)}\sigma_{gd(u)}(z,t)$ \cite{karpa_resonance_2008}.
\subsection{JR dynamics with DSPs} Assuming two pairs of counterpropagating control fields, see Fig. 1a solid and dashed lines, the evolution of the two probe fields $ E^{+}(z,t)$ and $ E^{-}(z,t)$ can be derived as \cite{Ruseckas_2011}:
\begin{equation*}
\label{eqSSL_Omega}
(\partial_{t}-v_{g}\sigma_{z}\partial_{z})\left(\begin{array}{c}
E^{+}(z,t)\\
E^{-}(z,t)
\end{array}\right)=i\frac{g_{\varepsilon}^{2}}{\bf{\Omega^{2}}}N\sigma_{z}\frac{\delta}{2} \left(\begin{array}{c}
E^{+}(z,t)\\
E^{-}(z,t)
\end{array}\right)
\end{equation*}
where $\mathbf{{\Omega}}=\left(\begin{array}{cc} \Omega_{d}^{+} & \Omega_{u}^{+}\\ \Omega_{d}^{-} & \Omega_{u}^{-} \end{array}\right)$. By individually manipulating the parameters of the control fields, $\mathbf{\Omega}=\Omega\left(\begin{array}{cc} 1 & i\\ i & 1 \end{array}\right)=\Omega(1+i\sigma_{x})$ which results in \begin{equation} \label{eqMatrixSSL} i\hbar(\partial_{t}-v_{g}\sigma_{z}\partial_{z})\left(\begin{array}{c} E^{+}(z,t)\\ E^{-}(z,t) \end{array}\right)=\hbar\frac{g_{\varepsilon}^{2}}{2\Omega^{2}}N\sigma_{y}\frac{\delta}{2} \left(\begin{array}{c} E^{+}(z,t)\\ E^{-}(z,t) \end{array}\right). \end{equation}
It is possible to derive a similar equation for the atomic operators $\sigma^{\pm}(z,t)=\frac{1}{\sqrt{2}}(\sigma_{gu}\pm i\sigma_{gd})$, thus constructing an equation for spinor of slow light (SSL) object $\mathbf{\Psi}=\binom{\Psi^{+}}{\Psi^{-}}$ as:
\begin{equation}
\label{eqDirac}
i\hbar \partial_t \mathbf{\Psi} = \left( i\hbar v_g\sigma_z\partial_z + m_{eff_{0}}v_g^2 \sigma_y \right)\mathbf{\Psi}
\end{equation}
with $\Psi^{\pm}(z,t)=\cos\theta E^{\pm}(z,t)-\sin\theta\sigma^{\pm}(z,t)$ and $\theta = \arctan(\sqrt{\frac{g^2N}{\Omega^2}})$ \cite{Unanyan2010}. Equation \ref{eqDirac} resembles a 1+1 Dirac equation and describes the evolution of two coupled SSL components with an effective mass $m_{eff_{0}}=\hbar \frac{\delta}{2} \frac{1}{v_g^2} \sin^2(\theta)$.\\
The JR model, can be realized by adding an extra coupling of this Dirac SSL to a background bosonic field
\begin{equation}
i\hbar \partial_t \mathbf{\Psi} = \left( i\hbar v_g\sigma_z\partial_z + m_{eff}(z) v_g^2 \sigma_y \right)\mathbf{\Psi}
\end{equation}
where $m_{eff}(z)$ obeys the Klein-Gordon equation. In the case when $m_{eff}(z)$ has a kink profile ($\propto \tanh(z)$), JR showed that it is possible to find a zero-mode solution which is topologically protected: \begin{equation} \psi_{zero} = \exp\left(-v_g \int_0^z ds \,m_{eff}(s) \right) \chi, \end{equation} where $-\sigma_z \sigma_y \chi=-i\chi $. By denoting $\phi(z)=\exp\left(-v_g \int_0^z ds \,m_{eff}(s) \right) $ and $\chi\propto(1,-1)$, we can write $\psi_{zero}(z) = \phi(z)\begin{pmatrix}1 \\ -1 \end{pmatrix}$ (with suitable normalization). Any particle initialised in this mode will not evolve and stay localized whereas in the normal Dirac case, it will spread \cite{Angelakis2014}.
\begin{figure}
\caption{\textbf{Tripod DSP setup.} (a) An original EIT line (dashed black line) is separated into three lines via applying a magnetic field B. Placing the control laser with a proper two-photon detuning ($\pm \delta /2$ from Zeeman states m=0 and m=-1) creates two isolated EIT systems. (b) Pulsing sequence for the creation of dual-tripod dynamics. (c) The SSL $\mathbf{\Psi}$ is created using $\Omega_{u}^{\pm}$ and $\Omega_{d}^{\pm}$. (d) $\mathbf{\Psi}$ is stored for a time $\tau$ after which it is mapped onto $E^{\pm '}$ using $\Omega_{R}^{\pm}$}
\label{FigSpinor}
\end{figure}
\section{Experimental Realization.} \subsection{Creation of tripod DSP.}
We create the tripod DSP $\Psi_T$ in an atomic ensemble using EIT in the following way. Firstly, three separated EIT systems are created by breaking the degeneracy of the Rb atoms Zeeman sub-levels through applying a DC magnetic field B (see Fig. 2a). Secondly, we isolate two of the EIT systems by using a single control laser that is symmetrically detuned ($\pm \delta/2= \pm g_{d}\mu_{B}B/2$) from the transitions $|u\rangle \rightarrow |e\rangle$ and $|d\rangle \rightarrow |e\rangle$, effectively forming two control fields $\Omega_{u}$ and $\Omega_{d}$ (see Fig. 2b). Lastly, we send a pulse of light ($E(z,t)$) undergoing tripod DSP dynamics due to $\Omega_{u}$ and $\Omega_{d}$, thus creating the components of $\Psi_T$ ($\Psi_u$ and $\Psi_d$).
All the transitions used in the experiment are within the $^{87}$Rb $D_{1}$ line. The storage is based on EIT in warm $^{87}$Rb vapor. The probe is stabilized using top-of-fringe locking to saturation spectroscopy of a Rb vapor cell and the control field is stabilized by an OPLL to the probe field. The probe pulses $E(z,t)$ with a width of 400 $\mu$s is tuned to $5S_{1/2} |F, m_{F}= 1,0\rangle$ $\rightarrow$ $5P_{1/2} |F', m_{F'}= 1,0\rangle$ ($|g\rangle$ $\rightarrow$ $|e\rangle$) (with detuning $\Delta=250MHz$). The writing control fields ($\Omega_{u}$ and $\Omega_{d}$) are tuned at $|F, m_{F}= 2,0\rangle$ $\rightarrow$ $|F', m_{F'}= 1,0\rangle$ ($|u\rangle$ $\rightarrow$ $|e\rangle$ with detuning $-\delta/2$) and $|F, m_{F}= 2,-1\rangle$ $\rightarrow$ $|F', m_{F'}= 1,0\rangle$ ($|d\rangle$ $\rightarrow$ $|e\rangle$ with detuning $+\delta/2$)(see Fig. 2a). The EIT lines have an average FWHM of 1.2 MHz \cite{Namazi2017}. A constant magnetic field induces the two-photon detuning with a $\delta/B$ ratio of 1.09 MHz/G.
The time sequence of the creation of $\Psi_T$ and the readout of $\Psi_{T}^{'}$ is shown in fig. 2b. $\Psi_T$ is stored for $2\mu s$ after which it is mapped onto $E_{d}^{'}$ and $E_{u}^{'}$ using $\Omega_R$. $\Omega_{R}$ is tuned to $|F, m_{F}= 1,0\rangle$ $\rightarrow$ $|F', m_{F'}= 1,0\rangle$ ($|g\rangle$ $\rightarrow$ $|e\rangle$). We calibrate the coherence of this tripod scheme by storing $\Psi_T$ and retrieving it using a co-propagating control field ($\Omega_{R}$) coupled to the $|g\rangle \rightarrow |e\rangle$ transition. Polarization elements supply 42 dB of control field attenuation (80\% probe transmission). The retrieved tripod DSP ($\Psi_{T}^{'}$) has two components, $\Psi_{u}^{'}$ and $\Psi_{d}^{'}$ with a frequency difference $\delta= \omega_{ue} -\omega_{de}$. We find a suitable $\delta$ by choosing a magnetic field B that maximizes the beat note in the retrieved mode.
\subsection{Measurement of 1+1 Dirac Dynamics.} Once suitable atom-light detunings are chosen, we proceed to create the SSL $\bf{\Psi}$. We use two control fields ($\Omega_{u}^+$ and $\Omega_{d}^+$) co-propagating with the probe $E(z,t)$ and two additional counter-propagating control fields $\Omega_{u}^-$ and $\Omega_{d}^-$ (see Fig. 2c). The created SSL components $\Psi^{\pm}$ are then stored. During storage, the temporal interaction of $\Psi^{+}$ and $\Psi^{-}$ follows the Dirac dynamics outlined by equation \ref{eqDirac}. After storage, these dynamics are mapped onto the SSL components $\Psi^{'\pm}$ by applying the counter-propagating control fields $\Omega_{R}^+$ and $\Omega_{R}^-$ (see Fig. 2d). We detect the optical form of $\Psi^{'\pm}$ ($E^{'\pm}(z,t)$) simultaneously in independent photo-detectors.
We vary the storage time for fixed two-photon detuning, thus changing the interaction time between the SSL components. Each pair of correlated experimental points are obtained by measuring the respective storage of light signals, integrating its total energy for varying storage time. We observe coupled oscillations for the intensities retrieved in each direction, $|E^{'+}(z,t)|^2$ (blue dots) and $|E^{'-}(z,t)|^2$ (red dots) in Fig 3, as expected from the usual Dirac dynamics coupling the two components of the spinor. Most importantly, the frequency of the oscillation is changed by varying the two-photon detuning, which testifies to the coherent nature of the process (see Fig. 3a and 3b). We note that similar oscillations between SSL frequency components have been shown in previous studies \cite{Lee2014}. However, in our implementation the two spinor components correspond to different propagation directions, which is the key design element for engineering Dirac dynamics.
\begin{figure}
\caption{\textbf{ Dirac dynamics using SSL.} Evaluating $|E^{\pm '}|^2$ for each $\tau$ results in an out of phase oscillation between the forward (blue dots) and backward (red dots) components of the SSLs. We plot the experimental data for $\delta$ = 350kHz (c) and 700kHz (d). Solid lines in (c) and (d) corresponds to numerical solutions of the SSL Dirac equation (eq. 2).}
\end{figure}
We benchmarked the aforementioned results against numerical solutions of the 1+1 dimensional Dirac equation of the form (including a coherence decay rate $\gamma$ to account for losses in the real experiment): \begin{equation*} i\partial_t \mathbf{\Psi} = \left( iv_g\sigma_z\partial_z + m_{eff_{0}}v_g^2 \sigma_y -\gamma \right)\mathbf{\Psi}, \end{equation*} with the initial condition $\mathbf{\Psi_0}=(\Psi_{0}^{+}, \Psi_{0}^{-})$ extracted from the shape of the original SSL right after storage. The solid lines in Fig. 3c and 3d represent the numerical simulation with fixed $v_g=1.0 cm/\mu s$, $m_{eff_{0}}v_g^2=3.3*\delta$ and $\gamma = 0.3$.
As storage time is increased, the SSL components lose their mutual coherence and thus the experimental data begins to deviate from the theoretical prediction. Nonetheless, these measurements provide strong evidence that the SSL dynamics follows that of relativistic particles.
\subsection{Relativistic dynamics with topological behavior.} Having built an analog Dirac simulator, we now move to mimic JR's topological predictions. In order to engineer $m_{eff}(z)$, we use a spatially varying magnetic field changing the two-photon detuning along the propagation axis of the light. In the original proposal, the bosonic field varies from a negative value to a positive value following the hyberbolic tangent function \cite{JackiwRebbi1976}. Due to its topological nature however, other profiles exhibiting similar behaviour at the boundary work too, as discussed in detail in \cite{Angelakis2014}. In this work, we choose a linearly-varying magnetic field, $B(z)=B_{0}z$, and perform experiments akin to our previous section. Figure 4a shows the obtained results. We observe the suppression of the coupled oscillation dynamics of the bare Dirac model by increasing the strength of the gradient, as predicted in \cite{JackiwRebbi1976,Angelakis2014}.
We again benchmarked this result against a numerical solution of the Jackiw-Rebbi equation. The procedure is similar to the one presented in the previous section, reconstructing the initial SSL components $\mathbf{\Psi_0}=(\Psi_{0}^{+}, \Psi_{0}^{-})$ (red and blue solid line in fig. 4b) and using $\gamma=\left(\begin{array}{cc} \gamma_{1} & 0\\ 0 & \gamma_{2} \end{array}\right)$ and $m_{eff}(z)v_g^2\propto \delta =(0.745\frac{MHz}{cm}(z-2.5cm) + 0.35MHz)$. In general each of the wave-functions can be written as $\psi^{\pm}(x) = e^{i\Phi^{\pm}(x)}|\psi^{\pm}(x)|$. Assuming $|\Phi^{\pm}(x)|$ to be constant, we define a global phase between the SSL components that is represented as $\Phi$ (a free parameter in the numerical fit). The numerical solutions ($\int dz |\Psi^+(z)|^2$ and $\int dz |\Psi^-(z)|^2$) are obtained numerically by fixing the group velocity, effective mass and loss rate to experimental estimations. The free parameters $\Phi$ and initial relative intensity are then fitted to the data. The solid lines in Fig. 4a (red and blue) represent the numerical solution with $\Phi=\pi$.
To probe the creation of the predicted JR zero energy mode, we first construct $\phi(z)$ (see Fig. 4b, green line) using the effective mass provided by the magnetic field gradient (purple line in Fig. 4b). We then calculate the overlap of the experimentally extracted $\mathbf{\Psi_0}=(\Psi_{0}^{+}, \Psi_{0}^{-})$ (red and blue lines in Fig. 4b, measured at 1.5 $\mu s$ storage time) with the zero-mode spinor: $\int dz \mathbf{\Psi_{0}^{\dagger}} \psi_{zero}$. In Fig. 4c, we plot this overlap for different values of the global phase between the SSL components, $\Phi$. Noticeably, the best overlap of $\sim 80 \%$ is also obtained for $\Phi=\pi$, which strongly hints that $\mathbf{\Psi_0}$ was prepared in the zero-mode.
In our experimental results, we have two decay rates for $\Psi^{+ '}$ and $\Psi^{- '}$ as they couple to a magnetic field insensitive and a magnetic field sensitive EIT line, respectively (see Fig. 2a) \cite{Maynard_Side2015}. Our interpretation of the high initial overlap at 1.5 $\mu s$ storage time is that we have created a zero-energy mode within the medium during this initial time interval.
\section{Discussion}
We have experimentally demonstrated the realization of a controllable coupling between two counter-propagating SSL components, simulating the dynamics of a relativistic massive fermion in a 1+1 Dirac equation. By adding a static background bosonic field (via the use of a magnetic field gradient), we have also simulated the celebrated Jackiw-Rebbi model. We have benchmarked our work with theoretical simulations by carefully reconstructing the initial SSL wave functions and using them in a numerical solution of the corresponding Dirac and JR differential equations. These values are then compared with the experimental data achieved by varying the storage time, showing a very good correlation within the coherence time of the atoms. Lastly, we have also measured signatures of the JR zero-mode by observing the inhibition of the oscillation between the spinor components as predicted in the theory.
We consider our experiment to be an important first step towards more complex quantum simulations with many quantum relativistic particles. Possible extensions include the study of the Klein paradox \cite{Klein1929} or the MIT bag model \cite{Chodos1974_MITbag} by using coupled light-matter SSLs. Moreover, as slow light polaritons can be made to interact strongly, our work provides a pathway towards analog simulators of complex phenomena described by interacting quantum field theories. Possibilities include the simulation of: charge fractionalisation in bosons \cite{Semenoff1982}, the interacting random Dirac model \cite{Keil2013Random} and the renormalization of mass due to interacting fermions \cite{Shankar1994}. Furthermore, interacting relativistic models such as the famous Thirring model \cite{Thirring_1958} are now within experimental reach. As many of these important QFT predictions are only addressable using high energy experiments, this new breed of light-matter room temperature simulators will be an exciting tool to reach unexplored realms of physics.
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\begin{document}
\title{Possible volumes of $\Ts{t}
\begin{abstract} The concept of $t$-$(v,k)$~trades of block designs previously has been studied in detail. See for example~A. S. Hedayat (1990) and Billington (2003). Also Latin trades have been studied in detail under various names, see~A. D. Keedwell (2004) for a survey. Recently Khanban, Mahdian and Mahmoodian have extended the concept of Latin trades and introduced $\Ts{t}{v}{k}$. Here we study the spectrum of possible volumes of these trades, $S(t,k)$. Firstly, similarly to
trades of block designs we consider $(t+2)$ numbers $s_i=2^{t+1}-2^{(t+1)-i}$, $0\leq i\leq t+1$, as critical points and then we show that $s_i\in S(t,k)$, for any $0\leq i\leq t+1$, \ and if $s\in (s_i,s_{i+1}),0\leq i\leq t$, then $s\notin S(t,t+1)$. As an example, we determine $S(3,4)$ precisely. \end{abstract}
{\bf Keywords}: $t$-Latin trade, Spectrum, Latin bitrade
\title{Possible volumes of $\Ts{t}
\section{Introduction and Preliminaries}
\noindent Let $V := \{1, 2, \ldots, v\}$ and $V^k$ be the set of all ordered $k$-tuples of the elements of $V$, i.e.\ $V^k := \{(x_1, \ldots, x_k) \mid x_i\in V, i=1, \ldots, k\}$. Also, let $V_I^t :=\{(u_1,\ldots,u_t)_I\mid u_i\in V, i=1, \ldots, t\}$, where $I$ is a $t$-subset of $\{1,\ldots,k\}$. For a pair of elements of $V^k$ and $V_I^t$, where $I= \{i_1,\ldots,i_t\}$ and $i_1 < \cdots < i_t$, we define \[ (u_1,\ldots,u_t)_I\in (x_1,\ldots, x_k) \quad \Longleftrightarrow \quad u_j=x_{i_j},\qquad j=1,\ldots,t. \]
Next we define {\sf $t$-inclusion matrix} $M = \M{t}{v}{k}$, as in~\cite{KhanbanMahdianMahmoodian}. The columns of this matrix correspond to the elements of $V^k$ (in lexicographic order) and its rows correspond to the elements of $\cup_I V_I^t$, where the union is over all $t$-subsets of $\{1,\ldots,k\}$. The entries of this matrix are 0 or 1, and are defined as follows. \[ {\rm M}_{(u_1,\ldots,u_t)_I, (x_1,\ldots,x_k)} = 1 \quad \Longleftrightarrow \quad (u_1,\ldots,u_t)_I\in (x_1,\ldots, x_k). \]
A \ $\SFT{t}{v}{k}$ $T=(T_1,T_2)$ of {\sf volume} $s$ consists of two disjoint collections $T_1$ and $T_2$, each of $s$ elements from $V^k$, such that for each $t$-set $I \subseteq \{1,\ldots,k\}$, and for every element $(u_1,\ldots ,u_t)_I$ of $V_I^t$, the number of elements of $T_1$ and $T_2$ that contain $(u_1,\ldots ,u_t)_I$ is the same. Note that in checking the containment of an element $(u_1,\ldots ,u_t)_I$, elements of $I$ are arranged in increasing order. The volume of a Latin trade $T$ is denoted by ${\rm vol}(T)$. It is clear from the definition above, that for any $t' \le t$, every \ $\T{t}{v}{k}$ \ is also a $\T{t'}{v}{k}$. For simplicity, the notation of $t$-Latin trade is commonly used for this combinatorial object. The {\sf spectrum} of $\T{t}{v}{k}$s, $S(t,k)$ is the set of all integers $s$, such that for each $s$ there exists a $\T{t}{v}{k}$ of volume $s$. A $\T{t}{v}{k}$ of volume $0$ is considered always to exist, that is a trade with $T_1=T_2=\emptyset$ which will be called {\sf trivial trade}. In a $\T{t}{v}{k}$ $T=(T_1,T_2)$ both collections $T_1$ and $T_2$ cover the same elements. This set of elements is called the {\sf foundation} of
$T$ and is denoted by ${\rm found}(T)$. Note that $v$ can be any integer such that $v$ is at least the size of the foundation of $T$.
\begin{example}\label{3-(3,4)l.t} In the following a $\ITT{3}{3}{4}$ $T=(T_1,T_2)$ of volume $15$ and with ${\rm found}(T)=\{1,2,3\}$ is given. \end{example}
\begin{center}
\begin{tabular}{|c|c|} \hline $T_1$ & \begin{tabular}{ccccccccccccccc}
3&3&2&2&2&1&1&2&2&1&1&3&3&2&2 \\
3&2&3&2&1&2&1&2&1&2&1&3&2&3&2 \\
3&3&3&3&3&3&3&2&2&2&2&1&1&1&1 \\
2&3&3&1&2&2&1&2&1&1&2&3&2&2&3 \end{tabular} \\ \hline
\end{tabular} \\[.5cm]
\begin{tabular}{|c|c|} \hline
$T_2$ & \begin{tabular}{ccccccccccccccc}
3&3&2&2&2&1&1&2&2&1&1&3&3&2&2 \\
3&2&3&2&1&2&1&2&1&2&1&3&2&3&2 \\
3&3&3&3&3&3&3&2&2&2&2&1&1&1&1 \\
3&2&2&3&1&1&2&1&2&2&1&2&3&3&2 \end{tabular} \\ \hline \end{tabular} \end{center}
As it is noted in~\cite{KhanbanMahdianMahmoodian}, the set of all $\Ts{t}{v}{k}$ is a subset of the null space of the $t$-inclusion matrix $M = \M{t}{v}{k}$. Also $t$-Latin trades have a close relation with orthogonal arrays. For example, the intersection problem of two orthogonal arrays may be studied as a problem in $t$-Latin trades.
One of the important questions is: \begin{ques} \label{spectrumquestion1} What is the spectrum of $\ITTs{t}{v}{k}$? \end{ques} Similar question about the spectrum of trades of block designs was raised in~\cite{MR1196125}, and two basic conjectures were stated. Since then many results on this subject are published. For a survey see \cite{MR1056530} and~\cite{MR2041871}.
The special case of $\Ts{2}{v}{3}$ is previously studied in detail and is referred with different names such as ``disjoint and mutually balanced'' (DMB) partial Latin squares by Fu and Fu (see for example~\cite{MR1125351}), as an ``exchangeable partial groupoids'' by Dr{\'a}pal and Kepka~\cite{MR733686}
as a ``critical partial Latin square'' (CPLS) by Keedwell~(\cite{MR96a:05027} and \cite{MR1393712}), and as a ``Latin interchange'' by Diane Donovan et~al.~\cite{MR98b:05019}, and recently as a ``Latin bitrade'' by~Dr{\'a}pal et~al.~(see\cite{Drapal}, \cite{MR2338087}, and~\cite{Hamalainen}). See~\cite{CavenaghMathSlovac} for a recent survey.
Following~\cite{MR2338087} we will refer them
as Latin bitrades. Let $L_1$ and $L_2$ be two Latin squares of the same order $n$. A {\sf Latin bitrade} $T=(P,Q)$ consists of two partial Latin squares $P$ and $Q$ obtained from $L_1$ and $L_2$, respectively, by deleting their common entries.
Note that $\Ts{2}{v}{3}$ are more general than Latin bitrades: in the former, repeated blocks and multiple symbols in rows, columns and cells are allowed.
\begin{example}\label{2-(3,3)l.t} The following is a Latin bitrade of volume $7$. (It should be noted that one empty row and one empty column are deleted.) \end{example}
\begin{center}
\begin{tabular}{
|@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}}
|} \hline \m12 & \m21 & \xx. \\ \hline \m21 & \m13 & \m32 \\ \hline \xx. & \m32 & \m23 \\ \hline \end{tabular} \end{center}
A result in~\cite{fu} answers the Question~\ref{spectrumquestion1} in the special case of Latin bitrades. Here we state several theorems about existence and nonexistence of $\Ts{t}{v}{k}$ of specified volumes, and we determine the spectrum of $\Ts{t}{v}{t+1}$ for $t=1,2,$ and $3$.
\section{Possible volumes of $t$-Latin trades}
Most of the concepts and definitions about $\Ts{t}{v}{k}$ are borrowed from $t$-$(v,k)$~trades of block designs. For example: volume, spectrum, $t$-inclusion matrix, frequency vector, etc. Specially, we show that there are close relations between the spectrum of these two combinatorial objects. But, in spite of all the similarities, some differences are observed between them, both in properties and in the methods of proof of lemmas and theorems.
By the following lemma all existence results of $\Ts{t}{v}{t+1}$ can be extended to $\Ts{t}{v}{k}$.
\begin{lemma}\label{t+1tok} For any $k \ge t+1$, we have $S(t,t+1)\subseteq S(t,k)$. \end{lemma} \begin{proof}{ Let $T=(T_1,T_2)$ be a $\T{t}{v}{t+1}$ of volume $s$. For each ordered $(t+1)$-tuple in $T_1$ and $T_2$, we add $(k-t-1)$ fixed elements $x$ of $V$ as $(t+2)^{\rm nd \/}$ to $k^ {\rm th \/}$ coordinates. Then we obtain two collections $T_1^*$ and $T_2^*$ containing of ordered $k$-tuples. Clearly $T^*=(T_1^*,T_2^*)$ is a $\T{t}{v}{k}$ of volume $s$.~}\end{proof}
\begin{lemma}\label{2s} By using any $\ITT{t}{v}{k}$ of volume $s$, we can obtain a \linebreak $\ITT{(t+1)}{v}{k+1}$ of volume $2s$. \end{lemma}
\begin{proof}{ Let $T=(T_1,T_2)$ be a $\T{t}{v}{k}$ of volume $s$. Choose two distinct elements $x$ and $ y \in V$. The following construction (see Figure~1) produces a $\T{(t+1)}{v}{k+1}$ $T^*=(T_1^*,T_2^*)$ of volume $2s$. That is, for constructing $T^*$ we adjoin two new distinct symbols $x$ and $y$ (respectively) to the first component of each element of
$T_1$ and $T_2$ (respectively), to obtain $T_1^*$ and $T_2^*$
(respectively). \begin{center} \hspace*{-4.8cm} $T^*_1$ \\[-.55cm] \hspace*{5.1cm} $T^*_2$ \\ [.3cm]
\begin{tabular}{|c|ccccccc|} \hline
x & & & & & & & \\
x & & & & & & & \\
. & & & & & & & \\
. & & & & $T_1$ & & & \\
. & & & & & & & \\
x & & & & & & & \\ \hline
y & & & & & & & \\
y & & & & & & & \\
. & & & & & & & \\
. & & & & $T_2$ & & & \\
. & & & & & & & \\
y & & & & & & & \\ \hline
\end{tabular} \qquad
\begin{tabular}{|c|ccccccc|} \hline
x & & & & & & & \\
x & & & & & & & \\
. & & & & & & & \\
. & & & & $T_2$ & & & \\
. & & & & & & & \\
x & & & & & & & \\ \hline
y & & & & & & & \\
y & & & & & & & \\
. & & & & & & & \\
. & & & & $T_1$ & & & \\
. & & & & & & & \\
y & & & & & & & \\ \hline \end{tabular} \\[1cm] {\bf Figure \ 1} \\[1.5cm] \end{center} \vspace*{-2cm} }\end{proof}
\begin{remark}\label{union}
Assume we have two $\ITTs{t}{v}{k}$, $T=(T_1,T_2)$ and \ $R=(R_1,R_2)$. Then \ $T+R=(T_1\cup R_1,T_2\cup R_2)$ \
and \ $T-R=(T_1\cup R_2,T_2\cup R_1)$ \ are two $\ITTs{t}{v}{k}$.
Note that the elements which appear in both sides are omitted. So
$T+R$ and $T-R$ are of
volumes
$|T_1|+|R_1|-|T_1\cap R_2|-|T_2\cap R_1|$ and
$|T_1|+|R_2|-|T_1\cap R_1|-|T_2\cap R_2|$, respectively. \end{remark}
\begin{remark}\label{corresponding} If we look at each ordered $k$-tuple in $T_i$ and $R_i$ , $i=1$, $2$, as a variable (each element of $T_1$ and $R_1$ with positive sign and each element of $T_2$ and $R_2$ with negative sign), then the two operations above coincide with the concept of two algebraic
$+$ and $-$ operations. For this reason sometimes we denote a
$\ITT{t}{v}{k}$ $T=(T_1,T_2)$ as $T=(T_1-T_2)$. \end{remark}
To apply linear algebra, we correspond to each $\T{t}{v}{k}$ $T=(T_1,T_2)$, a frequency vector {\bf T}, where the components of {\bf T} are corresponded with all elements of $V^k$ (in lexicographic order). For each $x\in V^k$, {\bf T}$(x)$ is defined as in the following: \\ [.4cm] ${\bf T}(x)=\left \{\begin{array}{ll} \hspace*{3.4mm} p & \hspace{3cm} {\rm if} \ x\in T_1 \ (p \ {\rm times}), \\ -q & \hspace{3cm} {\rm if} \ x\in T_2 \ (q \ {\rm times}), \\ \hspace*{3.4mm} 0 & \hspace{3cm} {\rm otherwise.} \end{array} \right. $ \\[.1cm]
Let {\rm M} be the $t$-inclusion matrix $M = \M{t}{v}{k}$. Then it is an easy exercise to prove that {\rm M}{\bf T}=$\overline{\bf 0}$, where $\overline{\bf 0}$ is the zero vector. And conversely if {\bf T}, with integer components, is a vector in the null space of {\rm M} then it determines a $\T{t}{v}{k}$ $T=(T_1,T_2)$. $T_1$ is obtained from the positive components and $T_2$ is obtained from the negative components of vector {\bf T}. In other words, there is a one-to-one correspondence between the null space of {\rm M} over the ring $\mathbb Z$ and the set of all $\Ts{t}{v}{k}$. The following lemma is the fact mentioned in Remark~\ref{union}, but in a linear algebraic approach.
\begin{lemma}\label{T+R} Consider two $\ITTs{t}{v}{k}$, $T=(T_1-T_2)$ and $R=(R_1-R_2)$. Then $T+R=(T_1+R_1)-(T_2+R_2)$ is also a $\ITT{t}{v}{k}$. \end{lemma} \begin{proof}{ Let {\bf T} and {\bf R} be the frequency vectors of $T$ and $R$, respectively, and {\rm M} be the $t$-inclusion matrix $M = \M{t}{v}{k}$. We have {\rm M}{\bf T}= $\overline{\bf 0}$ and {\rm M}{\bf R}= $\overline{\bf 0}$. Thus {\rm M}{\bf (T+R)}= $\overline{\bf 0}$, i.e. {\bf (T+R)} belongs to the null space of {\bf M}. Therefore $T+R$ is a $\T{t}{v}{k}$. }\end{proof}
\begin{remark} In the previous lemma if \ $T_1\cap R_2=R_1\cap T_2=\emptyset$, then $\rm vol (T+R)= \rm vol (T)+ \rm vol (R)$. \end{remark} In~\cite{KhanbanMahdianMahmoodian}, a $\T{t}{v}{k}$ is represented by a homogeneous polynomial of order $k$ as follows. Let $P=P(x_1,x_2,\ldots ,x_{v})$ be a homogeneous polynomial of order $k$ whose terms are ordered multiplicatively (meaning that for example for $i_1 \neq i_2$ the term $x_{i_1}x_{i_2}x_{i_3}\cdots x_{i_k}$ is different from $x_{i_2}x_{i_1}x_{i_3}\cdots x_{i_k}$, etc.) Now we correspond a frequency vector {\bf T}, with $v^k$ components (in lexicographic order) to polynomial $P$ as in the following:
For $x=(i_1,i_2,\ldots ,i_k)\in V^k$ we let {\bf T}($x$) be the coefficient of $x_{i_1}x_{i_2}x_{i_3}\cdots x_{i_k}$ in $P$. So, if the resulting vector {\bf T} satisfies the equation {\rm M}{\bf T}= $\overline{\bf 0}$, then we refer to polynomial $P$ as a $\T{t}{v}{k}$. It is easy to show that this definition is equivalent to the previous definition of $\T{t}{v}{k}$. This representation
helps us in constructing $\Ts{t}{v}{k}$ of desired volumes.
The following theorem is proved by using polynomial representation of \linebreak $\Ts{t}{v}{k}$.
\begin{theorem}\label{existencesi} For each \ $s_i=2^{t+1}-2^{(t+1)-i}$, \ $0\leq i\leq t+1$, \ there exists a $\ITT{t}{v}{k}$ of volume $s_i$ with $k\geq t+1$. \end{theorem} \begin{proof}{ For \ $i=0$ \ the trivial trade is the answer. For each \ $i$, $1\leq i\leq t+1$, let $T=(T_1-T_2)$ and $R=(R_1-R_2)$ be two $\Ts{t}{v}{k}$ defined as follows:
$T=T_1-T_2$ \\ \hspace*{.36cm}$=(x_1-x_2)\cdots(x_{2t-2i+1}-x_{2t-2i+2})(x_{2t-2i+3}-x_{2t-2i+4})\cdots$
\hspace*{.73cm}$(x_{2t+1}-x_{2t+2})x_{2t+3}\cdots x_{k+t+1}$, \quad and \vspace*{6mm} \\ $R=R_1-R_2$ \\ \hspace*{.36cm}$=-(x_1-x_2)\cdots (x_{2t-2i+1}-x_{2t-2i+2})(y_{2t-2i+3}-x_{2t-2i+4})\cdots$ \\
\hspace*{.93cm} $(y_{2t+1}-x_{2t+2})x_{2t+3}\cdots x_{k+t+1}$, \\
where inside each parenthesis variables are different from each other, and also for each $j$, \ $y_{j} \neq x_{j}$. Now \ $T+R$ \ is a $\T{t}{v}{k}$, by Lemma~\ref{T+R}. $T$~and $R$ are the same in $((t+1)-i)$ parentheses. So, in $T+R$, the following terms are cancelled out with their negatives: \\ $(x_1-x_2)\cdots(x_{2t-2i+1}-x_{2t-2i+2})x_{2(t-i+2)}\cdots x_{2(t+1)}x_{2t+3}\cdots x_{k+t+1}$. Thus $T+R$ is a $\T{t}{v}{k}$ of volume $s_i=2^{t+1}-2^{(t+1)-i}$. } \end{proof}
To continue our discussion we need to define levels of a trade. We may decompose a $t$-Latin trade $T$ and obtain other $(t-1)$-Latin trades. Let $T=(T_1,T_2)$ be a $\T{t}{v}{k}$ and let $j \in
\{1,\ldots,k\}$ and $x \in V$. Take \ $T'_i=\{(x_1, \ldots ,x_{k}) | (x_1, \ldots ,x_{k})\in T_i \ {\rm and} \ x_j=x\}$, for $i=1$, $2$. Delete $x$ from the $j^{\rm th \/}$ coordinate in all elements of $T'_1$ and $T'_2$ to obtain $T''_1$ and $T''_2$, respectively. Now $T''=(T''_1,T''_2)$ is a
$\T{(t-1)}{v'}{k-1}$, which is called a {\sf level trade} of $T$ in the direction of~$j$.
\begin{example}\label{3level trade} In Example \ref{3-(3,4)l.t} for $j=3$, there exist three level trades. For example, for $x=3$ the level trade in the direction of~$j=3$ is as follows. \end{example}
\begin{center}
\begin{tabular}{|c|ccccccc|} \hline
\ &3&3&2&2&2&1&1 \\ $T''_1$ &3&2&3&2&1&2&1 \\
\ &2&3&3&1&2&2&1 \\ \hline
\end{tabular} \\[.5cm]
\begin{tabular}{|c|ccccccc|} \hline
&3&3&2&2&2&1&1 \\ $T''_2$ &3&2&3&2&1&2&1 \\
&3&2&2&3&1&1&2 \\ \hline
\end{tabular} \end{center}
Note that the level trade above is a Latin bitrade, which also can be represented as in Example~\ref{2-(3,3)l.t}. \begin{lemma}\label{level} Let $T=(T_1,T_2)$ be a $\ITT{t}{v}{t+1}$ of volume $s$ with only two non-trivial level trades in some direction $j$. Then the volume of these level trades are equal, say to $a$, and so $s=2a$. \end{lemma} \begin{proof}{ Without loss of generality assume $j=1$. It is easy to see that the structure of $T=(T_1,T_2)$ is the same as structure of $T^*$ in Figure~1, where $k=t+1$. So the two level trades of $T$ in the direction of $j=1$ have the same volume $a$. Moreover, if $T'=(T_1,T_2)$ is one of these level trades, then the other level trade is $T''=(T_2,T_1)$. }\end{proof}
Now we investigate the spectrum of $\Ts{t}{v}{t+1}$. \begin{proposition}\label{S(1,2)} $S(1,2)=\mathbb N_0\backslash \{1\}$. \end{proposition} \begin{proof}{ It is clear that a $\T{1}{v}{2}$ of volume $1$ does not exist. Suppose $s\geq 2$, the following array form a $\T{1}{v}{2}$ of volume $s$.
\begin{center}
\begin{tabular}{|c|c|} \hline $T_1$ & \begin{tabular}{cccccc}
1&2&3&$\cdots$ &$s-1$&$s$ \\
1&2&3&$\cdots$ &$s-1$&$s$ \end{tabular} \\ \hline \end{tabular} \\[.5cm]
\begin{tabular}{|c|c|} \hline
$T_2$ & \begin{tabular}{cccccc}
1&2&3&$\cdots$ &$s-1$&$s$ \\
2&3&4&$\cdots$ &$s$&1 \end{tabular} \\ \hline \end{tabular} \end{center} \vspace*{-6.5mm} }\end{proof}
The following result of H-L. Fu. is an instrument in building an induction base. \begin{proposition}\label{Fu}~{\rm \cite{fu}} A Latin bitrade $T=(P,Q)$ of volume $s$ exists if and only if $s\in \mathbb N_0\backslash \{1,2,3,5\}$. \end{proposition}
\begin{proposition}\label{S(2,3)} $S(2,3)=\mathbb N_0\backslash \{1,2,3,5\}.$ \end{proposition} \begin{proof}{ Obviously, there exist no $\Ts{2}{v}{3}$ of volumes $1$ and $2$. Assume that $T$ is a $\T{2}{v}{3}$ of volume $3$ (or $5$). Then by Lemma~\ref{level}, each of these two numbers must decompose into at least three positive numbers from the set $S(1,2)=\{0,2,3,4,\ldots \}$ which is impossible. \\ Each Latin bitrade is a $\T{2}{v}{3}$, so $\mathbb N_0\backslash \{1,2,3,5\}\subseteq S(2,3).$ } \end{proof}
\begin{theorem}\label{nonexistence<s0} There exists no $\ITT{t}{v}{t+1}$ of volume $s$, for any \linebreak $s_0=0<s<2^t=s_1$. \end{theorem} \begin{proof}{ We proceed by induction on $t$. The statement obviously holds for the case $t=1$. Assume, by induction hypotheses, the statement holds for all values less than $t$, i.e. if $a$ is the volume of a $t'$-Latin trade ($t'<t$), then $a \geq 2^{t'}$. We show that theorem holds for $t$ also. Suppose the statement is not true for $t$, and let $T$ be a $\T{t}{v}{t+1}$ of volume $s$ with $0<s<2^t$. $T$ has at least two non-trivial level trades in each direction. Suppose in some direction $j$, \ $T$ has $l$ level trades of volumes $ a_1, a_2, \ldots, a_l$, where $l \geq 2$ and $s=a_1+\cdots +a_l$. By induction hypotheses $a_i \geq 2^{t-1}$, for each $i$. Therefore $s \geq l\cdot 2^{t-1} \geq 2\cdot 2^{t-1} =2^{t}$, which is a contradiction. }\end{proof}
\begin{theorem}\label{nonexistencesi} For any \ $s\in (2^{t+1}-2^{(t+1)-i} ,2^{t+1}-2^{(t+1)-(i+1)}), \linebreak 1\leq i\leq t$, there does not exist any \ $\ITT{t}{v}{t+1}$ of volume~$s.$ \end{theorem} \begin{proof}{ We proceed by induction on $t$. For case $t=1$, there is nothing to be proved. For $t=2$, statement follows from Proposition~\ref{S(2,3)}. Assume, by induction hypotheses, that statement holds for all values less than~$t$ ($t>2$), i.e. if $t'<t$ then there exists no $\T{t'}{v'}{t'+1}$ of volume $s'$, where $s'_i=2^{t'+1}-2^{(t'+1)-i}<s'<2^{t'+1}-2^{(t'+1)-(i+1)}=s'_{i+1}, 1\leq i\leq t'$.
We show that it holds for $t$ also. Suppose in contrary for some $i$ and some $s$, where $ s_i<s<s_{i+1}$, there exists a $\T{t}{v}{t+1}$ of volume $s$. We show a contradiction.
There are three cases to consider:
{\bf Case $1.$} In some direction $T$ has only two non-trivial level trades. So by Lemma~\ref{level} we have $s=2s'$, where $s'$ is the volume of some $\T{(t-1)}{v'}{t}$. Therefore we have $\frac {s_i}{2}<s'=\frac {s}{2} < \frac {s_{i+1}}{2}$, or $$2^{(t-1)+1}-2^{[(t-1)+1]-i} <s'<2^{(t-1)+1}-2^{[(t-1)+1]-(i+1)}$$ which is a contradiction.
{\bf Case $2.$} In each direction $T$ has more than two non-trivial level trades, and in some direction it has only three non-trivial level trades. So $s=a+b+c$, where for each value of $a$, $b$ and $c$ there exist $\Ts{(t-1)}{v'}{t}$ of these volumes. Note that by Theorem~\ref{nonexistence<s0} we have $a, b, c\geq 2^{t-1}.$ We claim that at least two of values $a$, $b$ and $c$ are equal to $2^{t-1}$. \\ Proof of claim: We know that the critical points in the case $t-1$, in increasing order, are $$s'_o=0,s'_1=2^{t-1},s'_2=3\cdot 2^{t-2},s'_3=7\cdot 2^{t-3},\ldots, s'_{t}=2^t-1.$$ If $a=2^{t-1}$ and $b$, $c\geq 3\cdot 2^{t-2}$, then $$s=a+b+c\geq 2^{t-1}+2\cdot 3\cdot 2^{t-2}=2^{t-1}(1+3)=2^{t+1},$$ which is impossible, because, $s<s_{t+1}=2^{t+1}-1$. So we have either \begin{itemize} \item[a)] $a=b=2^{t-1}$ and $c=2^t-2^{t-j}$, for some $j$, \ $1\leq j\leq t$ \ \ or \item[b)] $a=b=2^{t-1}$ and $c>2^{t}-1$. \end{itemize} In $(a)$ we have $s=a+b+c=2\cdot 2^{t-1}+2^t-2^{t-j}=2^{t+1}-2^{(t+1)-(j+1)}.$ \linebreak
This means that $s$ is a critical point of case $t$, which is a contradiction. In $(b)$ we have \ $s=a+b+c>2\cdot 2^{t-1}+2^{t}-1=2^{t+1}-1$, which is also impossible.
{\bf Case $3.$} In all directions $T$ has at least four non-trivial level trades. This means that $s=\sum_{i=1}^{l}a_i$, where $l\geq4$ and for each $a_i$ there exists a $\T{(t-1)}{v'}{t}$ of volume $a_i$. But then we have $s\geq 4\cdot 2^{t-1}=2^{t+1}$, which is impossible. }\end{proof}
\section{Spectrum of $\Ts{3}{v}{4}$}
For two integers $a$ and $b$ with $a<b$ we denote $[a,b]=\{a,a+1,\ldots ,b\}$. We prove the following theorem.
\begin{theorem}\label{S(3,4)} $S(3,4)=\mathbb N_0\backslash ([1,7]\cup [9,11]\cup \{13\}).$ \end{theorem} \begin{proof}{ By Lemma~\ref{2s} and Proposition~\ref{S(2,3)}, for each even number \linebreak $s \in \mathbb N_0\backslash ([1,7]\cup [9,11]\cup \{13\})$ we can construct a $\T{3}{v}{4}$ of volume $s$. A $\T{3}{v}{4}$ of volume 15 is given in Example~\ref{3-(3,4)l.t} and $\Ts{3}{v}{4}$ of volumes 17, 19, and 21 are given in the Appendix. $\Ts{3}{v}{4}$ of volumes 23 and 25 may be constructed by combination of $\Ts{3}{v}{4}$ of volumes (8 and 15) and (8 and 17), respectively (Lemma~\ref{T+R}). So, up to this point we know that for each $s(k)=2k+1$, where $ 7 \leq k \leq 12$, there exists a $\T{3}{v}{4}$ of volume $s(k)$. For $k \geq 13$ we write $s(k)=s(k-4)+8$, and then, by induction and by Lemma~\ref{T+R}, for each $k \geq 13$ we can obtain a $\T{3}{v}{4}$ of volume $s(k)$. Now the proof is complete by Theorems~\ref{nonexistence<s0} and~\ref{nonexistencesi}. }\end{proof}
\section{Future Research} The study of $\Ts{t}{v}{k}$ when $k=t+1$, is of special interest. For example similar to Latin bitrades, some $\Ts{3}{v}{4}$ may also be denoted by $T=(M,N)$, where $M$ and $N$ are two partial Latin cubes obtained from some Latin cubes $C_1$ and $C_2$ by deleting their common entries. This geometrical view will shed a light to studying questions and conjectures about $\Ts{3}{v}{4}$.
\begin{ques} \label{spectrumquestion2} What are the implications in geometrical interpretation of \linebreak $\ITTs{3}{v}{4}$? \end{ques}
A Latin bitrade is called {\sf $k$-homogeneous} if each row and each column contains exactly $k$ elements, and each element appears exactly $k$ times (see for example~\cite{MR2220235}, for more information). We may define a $k$-homogeneous $\T{t}{v}{t+1}$ and seek for their existence.
\begin{ques} \label{spectrumquestion3} What are the possible spectrums of $k$-homogeneous \linebreak $\ITTs{t}{v}{t+1}$? \end{ques}
\section{Appendix}
\begin{center} A \ $\T{3}{4}{4}$ of volume 17: \\[.1cm]
\begin{tabular}{|c|c|} \hline
$T_1$ & \begin{tabular} {c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c}
2&2&2&1&1&1&3&3&1&1&3&3&2&2&2&1&1 \\
3&2&1&3&2&1&3&2&3&2&3&2&3&2&1&2&1 \\
3&3&3&3&3&3&2&2&2&2&1&1&1&1&1&1&1 \\
2&3&1&3&1&2&3&2&2&3&2&3&3&1&2&2&1 \\ \end{tabular} \\ \hline \end{tabular} \\[.4cm]
\begin{tabular}{|c|c|} \hline
$T_2$ & \begin{tabular} {c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c}
2&2&2&1&1&1&3&3&1&1&3&3&2&2&2&1&1 \\
3&2&1&3&2&1&3&2&3&2&3&2&3&2&1&2&1 \\
3&3&3&3&3&3&2&2&2&2&1&1&1&1&1&1&1 \\
3&1&2&2&3&1&2&3&3&2&3&2&2&3&1&1&2 \\ \end{tabular} \\ \hline \end{tabular} \end{center}
\vspace*{.1cm} \begin{center} A \ $\T{3}{4}{4}$ of volume 19: \\[.1cm]
\begin{tabular}{|c|c|} \hline
$T_1$ & \begin{tabular} {c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c}
3&3&2&2&2&1&1&1&2&2&1&1&3&3&2&2&2&1&1 \\
3&2&4&3&1&4&2&1&4&2&4&2&3&2&3&2&1&2&1 \\
3&3&3&3&3&3&3&3&2&2&2&2&1&1&1&1&1&1&1 \\
3&2&3&2&1&1&3&2&1&3&3&1&2&3&3&1&2&2&1 \\ \end{tabular} \\ \hline \end{tabular} \\[.4cm]
\begin{tabular}{|c|c|} \hline
$T_2$ & \begin{tabular} {c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c}
3&3&2&2&2&1&1&1&2&2&1&1&3&3&2&2&2&1&1 \\
3&2&4&3&1&4&2&1&4&2&4&2&3&2&3&2&1&2&1 \\
3&3&3&3&3&3&3&3&2&2&2&2&1&1&1&1&1&1&1 \\
2&3&1&3&2&3&2&1&3&1&1&3&3&2&2&3&1&1&2 \\ \end{tabular} \\ \hline \end{tabular} \end{center}
\vspace*{.1cm} \begin{center} A \ $\T{3}{3}{4}$ of volume 21: \\[.1cm]
\begin{tabular}{|c|c|} \hline
$T_1$ & \begin{tabular} {c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c}
3&3&2&2&2&1&1&3&3&2&2&2&1&1&3&3&2&2&2&1&1 \\
3&2&3&2&1&2&1&3&2&3&2&1&2&1&3&2&3&2&1&2&1 \\
3&3&3&3&3&3&3&2&2&2&2&2&2&2&1&1&1&1&1&1&1 \\
1&3&3&2&1&1&2&2&1&1&3&2&2&3&3&2&2&1&3&3&1 \\ \end{tabular} \\ \hline \end{tabular} \\[.4cm]
\begin{tabular}{|c|c|} \hline
$T_2$ & \begin{tabular} {c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@{\hspace{1.9mm}}c@ {\hspace{1.9mm}}c@{\hspace{1.9mm}}c}
3&3&2&2&2&1&1&3&3&2&2&2&1&1&3&3&2&2&2&1&1 \\
3&2&3&2&1&2&1&3&2&3&2&1&2&1&3&2&3&2&1&2&1 \\
3&3&3&3&3&3&3&2&2&2&2&2&2&2&1&1&1&1&1&1&1 \\
3&1&1&3&2&2&1&1&2&2&1&3&3&2&2&3&3&2&1&1&3 \\ \end{tabular} \\ \hline \end{tabular} \end{center} \noindent {\bf Acknowledgement.} This research was in part supported by a grant from IPM (\#86050213).
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\begin{document}
\twocolumn[ \icmltitle{\textsc{\textit{AutoCoreset}}: An Automatic Practical Coreset Construction Framework}
\icmlsetsymbol{equal}{*}
\begin{icmlauthorlist} \icmlauthor{\href{https://scholar.google.com/citations?user=6r72e-MAAAAJ&hl=en}{Alaa Maalouf$^1$}}{equal} \icmlauthor{\href{https://scholar.google.com/citations?user=721xaz0AAAAJ&hl=en}{Murad Tukan$^2$}}{equal} \icmlauthor{\href{https://scholar.google.com/citations?user=DTthB48AAAAJ&hl=en}{Vladimir Braverman$^3$}}{} \icmlauthor{\href{https://scholar.google.com/citations?user=910z20QAAAAJ&hl=en}{Daniela Rus$^1$}}{}\\ \color{magenta}{$^{1}$Computer Science and Artificial Intelligence Lab (CSAIL), Massachusetts Institute of Technology (MIT)}\\ {$^{2}$DataHeroes}\\ $^{3}$Department of Computer science, Rice University
\end{icmlauthorlist}
\icmlcorrespondingauthor{Alaa Maalouf}{alaam@mit.edu}
\icmlkeywords{Machine Learning, ICML}
\vskip 0.3in ]
\printAffiliationsAndNotice{\icmlEqualContribution}
\begin{abstract} A coreset is a tiny weighted subset of an input set, that closely resembles the loss function, with respect to a certain set of queries. Coresets became prevalent in machine learning as they have shown to be advantageous for many applications. While coreset research is an active research area, unfortunately, coresets are constructed in a problem-dependent manner, where for each problem, a new coreset construction algorithm is usually suggested, a process that may take time or may be hard for new researchers in the field. Even the generic frameworks require additional (problem-dependent) computations or proofs to be done by the user. Besides, many problems do not have (provable) small coresets, limiting their applicability. To this end, we suggest an automatic practical framework for constructing coresets, which requires (only) the input data and the desired cost function from the user, without the need for any other task-related computation to be done by the user. To do so, we reduce the problem of approximating a loss function to an instance of vector summation approximation, where the vectors we aim to sum are loss vectors of a specific subset of the queries, such that we aim to approximate the image of the function on this subset. We show that while this set is limited, the coreset is quite general. An extensive experimental study on various machine learning applications is also conducted. Finally, we provide a ``plug and play" style implementation, proposing a user-friendly system that can be easily used to apply coresets for many problems. Full open source code can be found at \href{https://github.com/alaamaalouf/AutoCoreset}{\text{https://github.com/alaamaalouf/AutoCoreset}}. We believe that these contributions enable future research and easier use and applications of coresets. \end{abstract}
\begin{figure}
\caption{Evaluation of AutoCoreset against other problem dependant coreset construction algorithms for SVM and Logistic regression (on the Dataset~\ref{dataset:credit}). AutoCoreset achieves a much smaller approximation error and a higher test accuracy for the same coreset size while being an automatic and problem-independent framework. Sensitivity-based coreset for 1-mean, Median of means-based coreset, and Caratheordory coreset are all variants of AutoCoreset.}
\label{fig:credit_logistic_regression}
\label{fig:credit_svm}
\label{fig:credit}
\end{figure}
\begin{figure*}
\caption{A flowchart illustrating our automatic coreset construction framework. Note that \textit{VSAlg()} can be any algorithm from Table~\ref{table:ourContrib}. }
\label{fig:flow}
\end{figure*} \section{Introduction and Motivation} In many machine learning (ML) problems, the input is usually a set $P=\br{p_1,\cdots, p_n}$ of $n$ items, a (probably infinite) set of candidate solutions $\mathcal{X}$ called query set, and a loss function $f:P \times \mathcal{X} \to [0, \infty])$. The goal is to find a query (model, classifier) $x^*$ that minimizes the sum $\sum_{i=1}^n f(p_i,x)$ over every query $x\in \mathcal{X}$. Notably, many of these optimization/learning tasks are typically challenging to approximate when the input is very large. Furthermore, in the era of big data, we usually aim towards maintaining a solution for streaming and/or distributed input data, while consuming small memory. Finally, even well-known problems with a close optimal solution, such as ridge regression and other classes of convex optimization involving Cross-validation methods or hyper-parameter tuning methods, must analyze under many restrictions several queries for various subsets of data, leading to a drastic increase in the running time.
\textbf{Coresets.} A common approach to solve such issues is to use data summarization techniques, namely \emph{Coresets}, which got increasing attention over recent years~\cite{bachem2018scalable,pmlr-v84-bachem18a,buadoiu2008optimal,maalouf2022fast,balcan2013distributed, pmlr-v97-braverman19a,tukan2023provable,curtain2019coresets,jubran2020sets,feldman2014coresets,karnin2019discrepancy,tukan2022pruning,maalouf2021coresets,tukan2022new,tukan2022obstacle,tukan2023efficient}; see surveys in~\cite{feldman2020core,munteanu2018coresets,phillips2016coresets}, and introductions in~\cite{maalouf2021introduction,jubran2019introduction}. A coreset, informally, is a tiny weighted subset of the input set $P$, roughly approximating the loss of $P$ for every possible query $x\in \mathcal{X}$, up to a bound of $1\pm \varepsilon$ factor ($0\leq\varepsilon<1$). The size of the coreset is often independent or close to logarithmic in the amount of the input points $n$, but polynomial in $1/\varepsilon$. Coresets are useful in ML as they significantly increase the efficiency of ML solvers. Specifically, employing conventional methods on the constructed coresets should approximate the optimal solution on the entire dataset, in orders of magnitude less expensive time and memory. Furthermore, by repeatedly running existing heuristics on the coreset in the time it takes to run them once on the original (large) dataset, heuristics that are already quick can be more accurate. Additionally, coresets can be maintained for distributed and streaming data.
\textbf{So what's the problem?} Obtaining non-trivial theoretical guarantees is frequently impossible in many contemporary machine learning problems due to either the target model being highly complex or since every input element $p\in P$ is significant in the sense of high sensitivity; see~\cite{NearConvex}. Hence, generating a coreset becomes a highly challenging process, and the corresponding theoretical analysis occasionally falls short of recommending such approximations. As a result, designing a new coreset and demonstrating its accuracy for a new ML problem might take years, even for simple ones.
Another crucial issue with current theoretical frameworks is their lack of universality. Even the most general frameworks (e.g.,~\cite{feldman2011unified, Langberg2010universal} replace the problem of generating a coreset for an input set $P$ of $n$ points with $n$ new optimization problems (one problem for each of the $n$ input points $p\in P$) known as sensitivity bounding. Solving these may be more difficult than solving the original problem, where for every $p\in P$ we are required to bound its own sensitivity defined as $s(p)=\sup_{x\in \mathcal{X}}\frac{f(p,x)}{\sum_{q\in P}f(q,x)}$. As a result, distinct approximation strategies are often adapted to each task. Hence, the main disadvantage of such frameworks is that researchers provide papers solely for bounding the sensitivities with respect to a certain problem or a family of functions~\cite{NearConvex,maalouf2020tight}, limiting the spread of coresets, as non-expert won't be able to suggest coresets for their desired task. These problems raise the following questions:
\textbf{Is it possible to design an automatic and practically robust coreset construction framework (for any desired cost function and input dataset) that does not need sensitivity calculation or any other problem-dependent computation by the user? Can we provide some provable guarantees with respect to this framework?} \subsection{Vision}
\paragraph{Goal.} Our goal is to provide a single algorithm that only receives the loss function we wish to compute a coreset for and the input dataset, then, it practically outputs a good coreset for the input dataset with respect to the given loss. This algorithm should be generic, efficient, and work practically well for many problems.
\paragraph{motivation.} The main motivation behind this goal is (1) to increase the spread and use of coreset to a larger community that is not limited to coreset researchers or pioneers. (2) Additionally, to ease the use of coresets for many other applications that may be out of the scope of the coreset literature, and finally, to (3) easily apply coresets for new problems that do not have provable coresets. Theoretically speaking, it is indeed very hard to provide a ``theoretical strong coreset’’ to any problem – for example, there exist lower bounds on the coreset sizes for different problems~\cite{munteanu2018logistic,tukan2020coresets}. Thus we aimed at a practical result while providing weaker theoretical guarantees, with an extensive experimental study.
\subsection{Our contribution} In this paper, we provide a coreset construction mechanism that answers both questions. Specifically: \begin{enumerate}[label=(\roman*)]
\item The first automatic practical coreset construction system that only needs to receive the loss function associated with the problem. Our coreset does not require any computation to be done by the user, not mathematical nor technical (without the need for sensitivities or any other task-related computation by the user). To the best of our knowledge, this is the first paper to suggest a \textbf{plug-and-play} style framework/compiler for coreset construction. We also provide a theoretical justification for using our framework.
\item An extensive empirical study on real-world datasets for various ML solvers of Scikit-Learn~\cite{scikit-learn}, including k-means, logistic regression, linear regression, and support vector machines (SVM), showing the effectiveness of our proposed system.
\item \textbf{AutoCoreset:} An open-source code implementation of our algorithm for reproducing our results and future research. For simplicity and ease of access, to obtain a coreset, the user only needs to plug in his desired loss function and the data into our system. We believe this system will popularize and expose the use of coresets to other scientific fields. \end{enumerate}
\section{Setup Details} \label{sec:setup_details}
Given a set $P = \br{p_1\cdots,p_n} \subseteq \ensuremath{\mathbb{R}}^d$ of $n$ points\footnote{if $P$ is a set of labeled items, then $P = \br{p_i=\term{p'_i, y_i} \middle| p'_i \in \ensuremath{\mathbb{R}}^{d-1}, y_i \in\ensuremath{\mathbb{R}}}_{i=1}^n$} and a loss function $f : P \times \mathcal{X} \to [0, \infty)$ where $\mathcal{X}$ is a (possibly infinite) set of queries. In this paper, we develop an automatic coreset construction framework for any problem involving cost functions of the form $\sum_{p \in P} f(p, x)$, here $x\in \mathcal{X}$. Formally, we wish to find a small subset $\mathcal{I} \subseteq [n]$ and a weight function $v: \mathcal{I} \to [0,\infty)$ such that $ \max_{x \in \mathcal{X}}\frac{\sum\limits_{j \in \mathcal{I}} v(j) f\term{p_j,x}}{\sum\limits_{i = 1}^n f\term{p_i,x}} \in 1 + O(\varepsilon), $ for some small $\varepsilon \geq 0$.
\begin{figure}
\caption{Illustration of a vector summarization coreset for an input matrix of $7$ rows and $3$ columns which represent the loss function concerning a set $P$ of $7$ input points, and set of queries $x_1,x_2,x_3$.}
\label{fig:vector_sum}
\end{figure}
\subsection{Preliminaries} We now give our notations and used Definition.
\textbf{Notations.} For a pair of integers $n,m>0$, we denote by $[n]$ the set $\br{1,\cdots,n}$, and by $\ensuremath{\mathbb{R}}^{n\times m}$ the set of every possible $n \times m$ real matrix. For a matrix $M\in \ensuremath{\mathbb{R}}^{n\times m}$ and a pair of integers $i\in [n],j\in [m]$, we use $M_{i,\ast}$ to denote its $i$th row (vector), $M_{\ast,j}$ to denote its $j$th column, and $M_{i,j}$ to denote the entry in the $i$th row and $j$th columns.
In what follows, we define a crucial component on which our system relies, namely, \emph{vector summarization coreset}.
\begin{definition}[Vector summarization coreset] \label{def:VSCoreset} Let $M \in \ensuremath{\mathbb{R}}^{n\times m}$, $\mathcal{I} \subseteq [n]$, $v: \mathcal{I} \to [0, \infty)$ be a weight function, and let $\varepsilon > 0$. The tuple $\term{\mathcal{I},v}$ is an vector summarization $\varepsilon$-coreset for $M$ if $ \norm{\sum_{i \in [n]} M_{i,\ast} - \sum_{j \in \mathcal{I}} v(j) M_{j,\ast}}_2^2 \leq \varepsilon.$ \end{definition}
Many papers suggested different algorithms for computing such coresets; in Table~\ref{table:ourContrib} summarizes some of these results, as we can use them all of them in our method.
\section{\emph{AutoCoreset}}
A coreset aims to approximate the probability distribution induced upon the input data by the cost function. Hence, in order to approximate a given cost function, the coreset must contain points that can result in an approximated distribution to that of the full data.
\textbf{Key idea. } Loosely speaking, assume that for a given cost function $f$ and a set $P=\br{p_1,\cdots,p_n} \subset\ensuremath{\mathbb{R}}^d$, we access an infinite matrix $\mathcal{M}^\ast(P,f)$ where the rows correspond to the $n$ points of $P$, and each column corresponds to a query point from the infinite set of queries $\mathcal{X}$. Specifically, each row $i\in [n]$ is of infinite length representing the loss of each point $p_i \in P$ with respect to the infinite set of queries $\mathcal{X}$. A coreset in this context means finding a subset of the rows $\mathcal{I} \subseteq [n]$, and a weight function $v: \mathcal{I} \to [0, \infty]$, that satisfies the vector summarization coreset guarantee (see Definition~\ref{def:VSCoreset}), i.e., \begin{equation} \label{eq:dream} \norm{\sum_{i \in [n]} {\mathcal{M}^\ast(P,f)}_{i,\ast} - \sum_{j \in \mathcal{I}} v(j){\mathcal{M}^\ast(P,f)}_{j,\ast}}_2^2 \leq \varepsilon. \end{equation}
From such a coreset $\mathcal{I} \subseteq [n]$, the cost function can be approximated, since for every query $x\in \mathcal{X}$ (column in the matrix $\mathcal{M}^\ast(P,f)$), the weighted sum of losses over the coreset $\mathcal{I}$ approximate the original sum of losses of the whole data. While such a concept is admirable, having an access to such immense data is rather imaginative.
Recently~\cite{maaloufsine2022coresets} showed that for an input set of points $P$, and query space $X$ that is defined as a family of sine wave functions, a coreset can be constructed. Specifically, it was shown that if the coreset approximates the loss of every query in a smaller set of queries on the input data, then it will also approximate the losses of the whole set of queries (sine waves). Thus, indeed, the sine wave that fits best the coreset approximates the sine wave that best fits the entire data. Inspired by such a result, we aim towards constructing a sub-matrix $\Tilde{\mathcal{M}}(P,f)$ of $\mathcal{M}^\ast(P,f)$ ($\Tilde{\mathcal{M}}(P,f)$ contain a subset of the columns of $\mathcal{M}^\ast(P,f)$; see Figure~\ref{fig:vector_sum} for illustration) such that constructing the coreset on $\Tilde{\mathcal{M}}(P,f)$ (a weighted subset of the rows of $\Tilde{\mathcal{M}}(P,f)$) will also yield a similar coreset to that of~\eqref{eq:dream} on the $\mathcal{M}^\ast(P,f)$. But, how to build the sub-matrix $\Tilde{\mathcal{M}}(P,f)$? how to choose the query set corresponding to the columns of $\Tilde{\mathcal{M}}(P,f)$?
\begin{table}[ht] \centering \caption{{Summary of known vector summarization coresets and their properties.} } \label{table:ourContrib} \begin{adjustbox}{width=1\linewidth } \small
\begin{tabular}{ | c | c | c | c | c |} \hline \textbf{Method} &
\textbf{\makecell{Probability\\of failure}} & \textbf{\makecell{Approximation\\error}} & \textbf{Coreset size $|\mathcal{I}|$} & \textbf{\makecell{Construction time}} \\ \hline \makecell{Caratheodory\\\cite{maalouf2019fast}\\\cite{caratheodory1907variabilitatsbereich}} & 0 & 0 & $m+1$ & $O(\min\{nm+\log^4(m),m^2n^2,nm^3\})$ \\ \hline \makecell{Frank-Wolfe\\ \cite{feldman2017coresets}\\\cite{clarkson2010coresets}} & 0 & $\varepsilon$ & $O(1/\varepsilon)$ & $O(\min\{nd/\varepsilon\})$ \\ \hline \makecell{Median of means \\ tournament\\\cite{minsker2015geometric} }& $\delta$ & $\varepsilon$ & $O(1/\varepsilon)$ & $O(m\log^2(1/\delta) + m\log(1/\delta)/\varepsilon)$ \\ \hline \makecell{Sensitivity sampling\\ \cite{feldman2011unified}} & $\delta$ & $\varepsilon$ & $O(\frac{1}{\varepsilon}(m+\log{\frac{1}{\delta}}))$ & $O(nm)$ \\ \hline Uniform sampling & $\delta$ & $\varepsilon$ & $O(\frac{1}{\varepsilon\delta})$ & $O(1)$ \\ \hline
\end{tabular} \end{adjustbox} \end{table}
\begin{algorithm}[!t]
\caption{$\textsc{AutoCoreset}\term{P, f, \tau, m, \zeta}$}
\label{alg:main} \begin{algorithmic}[1]
\INPUT set of $n$ points $P$, a loss function $f$, a coreset size $\tau$, number of initial models $m$, and an stopping criterion $\zeta$
\OUTPUT A coreset $\term{\mathcal{I},v}$ such that
\STATE $\Tilde{\mathcal{M}}(P,f) \gets \overset{\to}{0}_{n \times m}$ \alglinelabel{line:settingM}
\FOR{each $i \in [m]$ \alglinelabel{line:pre_for_init_M}}
\STATE $x_i \gets $ a randomized approximated solution involving $P$ and $f$\alglinelabel{line:RandomSols}
\FOR{every $j \in [n]$ \alglinelabel{line:for_init_M}}
\STATE ${\Tilde{\mathcal{M}}(P,f)}_{j, i} \gets f\term{p_j, x_i}$ \alglinelabel{line:init_M}
\ENDFOR \alglinelabel{line:ending_for_init_M}
\ENDFOR \alglinelabel{line:ending_pre_for_init_M}
\REPEAT \alglinelabel{line:repeat}
\STATE $\term{\mathcal{I}, v} \gets$ coreset of $m$ indices for vector summarization problem involving $\Tilde{\mathcal{M}}(P,f)$ \COMMENT{See Definition~\ref{def:VSCoreset}} \alglinelabel{line:generate_VSCoreset}
\STATE $x^\ast \gets \argmin_{x \in \mathcal{X}} \sum\limits_{i \in \mathcal{I}} v\term{i} f\term{p_i, x}$ \alglinelabel{line:solve_on_coreset}
\STATE $\Tilde{\mathcal{M}}(P,f) \gets \left[\Tilde{\mathcal{M}}(P,f) \middle| \overset{\to}{0}_{n}\right]$ \alglinelabel{line:concat_zero_column}
\FOR{every $i \in [n]$}
\STATE ${\Tilde{\mathcal{M}}(P,f)}_{i, m + 1} \gets f\term{p_i, x_C}$ \alglinelabel{line:add_column_to_M}
\ENDFOR\alglinelabel{line:end_for_concat_new_column_M}
\STATE $m \gets m + 1$ \alglinelabel{line:add_column_counter}
\UNTIL{$\zeta$ is satisfied}
\item[\algorithmicreturn] $\term{\mathcal{I}, v}$ \alglinelabel{line:return_Coreset} \end{algorithmic} \end{algorithm}
\subsection{A deeper look into \emph{AutoCoreset}} \label{sec:deeper}
We now give and explain our algorithm $\textsc{AutoCoreset}$ (see Algorithm~\ref{alg:main}), which aims to provide a parasitical coreset with similar guarantees.
\textbf{Into the forging of our coresets.} Let $m > 1$ be an integer. First, a matrix $\Tilde{\mathcal{M}}(P,f)$ is generated to contain $n \times m$ zero entries, followed by generating a set $\mathcal{X}^\prime=\br{{x}_1,\ldots,{x}_m }$ of $m$ approximated solutions with respect to $\min\limits_{x \in \mathcal{X}} \sum\limits_{i=1}^n f(p_i,x)$ as depicted at Lines~\ref{line:settingM}--\ref{line:ending_pre_for_init_M}. If no such approximated solution exists, then the initialization may be also completely random. The (sub)matrix $\Tilde{\mathcal{M}}(P,f)$ is now initialized, where for every $i\in [n]$, and $j\in [m]$, the entry $\Tilde{\mathcal{M}}(P,f)_{i,j}$ in the $i$th row and $j$th column is equal to $f\term{p_i,{x}_j}$. While the properties associated with generated solutions at Line~\ref{line:RandomSols} hold with some probability, our framework is always guaranteed practically to generate a good coreset. This is due to the fact that these solutions are merely used as an initialization mechanism.
From this point on, a loop is invoked. First, using the current state of $\Tilde{\mathcal{M}}(P,f)$, a vector summarization coreset $\term{\mathcal{I},v}$ (see Definition~\ref{def:VSCoreset}) is generated with respect to the rows of $\Tilde{\mathcal{M}}(P,f)$.
A coherent claim of our system is that any vector summarization coreset $\mathcal{I}$ for the rows of $\Tilde{\mathcal{M}}(P,f)$, is directly mapped to coreset for $P$ (using the same set of indexes and the same weight function) with respect to the query set $\mathcal{X}^\prime \subset \mathcal{X}$ and the function $f$, where $\mathcal{X}^\prime$ is the set of all queries that brought about the columns of $\Tilde{\mathcal{M}}(P,f)$. More preciously, $ \max_{x \in \mathcal{X}^\prime}\frac{\sum\limits_{j \in \mathcal{I}} v(j) f\term{p_j,x}}{\sum\limits_{i = 1}^n f\term{p_i,x}} \in 1 + O(\varepsilon); $ see Lemma~\ref{lem:VSC2FC}.
Since the computed vector summarization coreset $\mathcal{I}$ is also a coreset with respect to $f, P$, and $\mathcal{X}^\prime$, we can optimize $f$ over the small coreset $\mathcal{I}$ to obtain a new query $x^*\in \mathcal{X}$ that gives an approximated solution to the full data (see Line~\ref{line:solve_on_coreset}). We then apply the loss $f$ function and the new solution $x^*$ on $p_1,\cdots, p_n$ to obtain the vector of losses $l = (f(p_1,x^*),\cdots,f(p_n,x^*))^T$, and concatenate such vector of loss values to $\Tilde{\mathcal{M}}(P,f)$ as its last column. This aids in expanding the exposure of generated coreset to a wider spectrum of queries, leading towards a \emph{strong coreset}. Observe that in the next iteration, when we compute a new coreset for the given set of queries, the coreset will approximate all of the previous ones (set of queries) and the new computed query/solution $x^*$.
This procedure is repeated until some stopping criterion $\zeta$ is invoked -- we provide more details on the used $\zeta$ in Section~\ref{sec:exp}. We refer the reader to Lines~\ref{line:solve_on_coreset}--\ref{line:add_column_counter}. Note that if we were able to run the above procedure infinitely while ensuring that at each iteration a new solution is computed, $\mathcal{M}^\ast(P,f)$ would have been generated, resulting in the \say{strong coreset} this system is leaning towards. To better grasp the idea of the framework, we provide a flowchart illustration at Figure~\ref{fig:flow}.
\textbf{The parameters $\tau, m, \zeta$.} Our Algorithm initializes its matrix $\Tilde{\mathcal{M}}(P,f)$ with respect to the losses of $m>1$ different queries, and outputs a coreset of size $\tau>1$, hence, the larger the $m$ and $\tau$ the better the approximation, but the slower the time; See section~\ref{sec:exp} for more details. Regarding $\zeta$, it is the used stopping criterion, we provide full details regarding the used $\zeta$ in Section~\ref{sec:exp}.
\subsection{Weaker coresets are fine too} Our \emph{AutoCoreset} system, while ambitiously aims towards holding a grasp over $\mathcal{M}^\ast(P,f)$, it finds a weaker version of the \say{strong coresets}. Specifically speaking, it finds a coreset that attains approximation guarantees with respect to a subset of the query set $\mathcal{X}$. Theoretically speaking, the following lemma summarizes one aspect of the theoretical properties guaranteed by \emph{AutoCoreset}.
\begin{lemma}[Vector summarization coreset $\to$ \say{a weak coreset for any loss}] \label{lem:VSC2FC} Let $P=\br{p_1,\cdots,p_n}\subseteq \ensuremath{\mathbb{R}}^{d}$ be a set of $n$ points as defined in Section~\ref{sec:deeper}, $\mathcal{X}^\prime \subset \mathcal{X}$ be a set of queries, $f : P \times \mathcal{X} \to [0, \infty)$ be a loss function, and let $\Tilde{\mathcal{M}}(P,f) \in \ensuremath{\mathbb{R}}^{n \times \abs{\mathcal{X}^\prime}}$ be the loss matrix defined with respect to $P, f, \mathcal{X}^\prime$ as in Algorithm~\ref{alg:main}. Let $\tau \geq 1$ be an integer, and let $\term{\mathcal{I}, v}$ be a $\varepsilon$-vector summarization coreset concerning $\Tilde{\mathcal{M}}(P,f)$ of size $\abs{\mathcal{I}} = \tau$.
Then, for every $x \in \mathcal{X}^\prime$, \[ \abs{\sum_{i \in [n]} f\term{p_i,x} - \sum_{j \in \mathcal{I}} v(j) f\term{p_j,x}}^2 \leq \varepsilon. \] \end{lemma}
\textbf{Implications of Lemma~\ref{lem:VSC2FC}.} \emph{AutoCoreset} guarantees theoretically that for a finite set of queries $\mathcal{X}^\prime$, a coreset can be constructed supporting $\mathcal{X}^\prime$. A key advantage here would be the ability to represent any query $x$ such that its loss vector $(f(p_1,x),\cdots,f(p_n,x))$ lies inside the \say{convex hull} of the loss vectors of the query set $\mathcal{X}^\prime$. Luckily, such a trait is supported by our system. Specifically speaking, for any query such that its corresponding loss vector $\ell$ with respect to $f$ and $P$ can be formulated as a convex combination of the columns of $\Tilde{\mathcal{M}}(P,f)$, then a vector summarization coreset for the rows of $\Tilde{\mathcal{M}}(P,f)$ is also a vector summarization to the rows of concatenating $\Tilde{\mathcal{M}}(P,f)$ and the column vector $\ell$. In what follows, we give the theoretical justification for the above claim.
\begin{claim}[Weak Coreset with hidden abilities] \label{clm:hidden_talent} Let $P=\br{p_1,\cdots,p_n}\subseteq \ensuremath{\mathbb{R}}^d $ be a set of $n$ points as in Section~\ref{sec:deeper}, $f$ be a loss function supported by \emph{AutoCoreset}, and let $m, \tau, \zeta$ be the defined number of initial solutions, sample size, and stopping criterion, respectively. Let $z \geq m$, $\term{\mathcal{I},v}$ be the output of a call to $\textsc{AutoCoreset}\term{P, f, \tau, m, \zeta}$, and let $\Tilde{\mathcal{M}}(P,f) \in \ensuremath{\mathbb{R}}^{d \times z}$ be the matrix of losses that was constructed throughout the running time of $\textsc{AutoCoreset}$; see Lines~\ref{line:settingM},~\ref{line:init_M},~\ref{line:concat_zero_column}~\ref{line:add_column_to_M} at Algorithm~\ref{alg:main}. Then for any weight function $\alpha : [z] \to [0,1]$ where $\sum\limits_{i =1}^z \alpha\term{i} = 1$, and any $x \in \mathcal{X}$ satisfying that for every $i \in [n]$, $f\term{p_i,x} = \sum\limits_{k=1}^z \alpha\term{k} \Tilde{\mathcal{M}}(P,f)_{i, k}$ , we have \[ \abs{\sum\limits_{i =1}^n f\term{p_i, x} - \sum\limits_{j \in \mathcal{I}} v(j) f\term{p_j, x}}^2 \leq \varepsilon, \] where $\varepsilon \geq 0$ is the approximation factor associated with generating a vector summarization coreset of $m$ points. \end{claim}
\textbf{The best of both worlds.} Claim~\ref{clm:hidden_talent} states that even if it seems that our generated coreset only supports a handful of queries from $\mathcal{X}$, our coreset basically supports many more queries. The highlight of such a claim is that if the optimal solution for the objective function involves $f$ and $P$, then our coreset becomes stronger in the sense of ensuring better quality even during the training/optimization process which involves both $f$ and $P$. Such a claim is usually targeted via \say{Strong coresets} and mainly by \say{Weak coresets}. \emph{AutoCoreset} ensures a coreset that resides on the spectrum involving these coresets at its ends, i.e., generating a coreset from the best of both worlds -- a coreset supporting the optimal solution that the user is aiming to solve using accelerated training via coresets while maintaining the provable approximation guarantees of strong coresets to some extent.
\section{Size, Space, and Time Analysis}
\textbf{Time complexity. } Let $\texttt{VAlg}$ be the vector summarization algorithm used at Line 9 of Algorithm~\ref{alg:main} (pick one from Table~\ref{table:ourContrib}). Let $\varepsilon, \delta \in (0,1)$ be the desired vector summarization approximation error, and probability of failure, respectively. Now denote by \begin{itemize}
\item $T(n, i, \varepsilon, \delta )$: the running time of $\texttt{VAlg}$ on a matrix of $n$ rows and $i$ columns with respect to $\varepsilon$ and $\delta$.
\item $S(n, i, \varepsilon, \delta )$: the size of the coreset computed by VAlg on a matrix of $n$ rows and $i$ columns with respect to $\varepsilon$ and $\delta$.
\item $T_{sol}(n,d)$: the time required to compute a solution vector $x^*$ for $n$ points in the $d$ dimensional space with respect to the problem at hand (e.g., the time required to compute the solution of linear regression is $O(nd^2)$).
\item $T_{cost}(n,d)$: the time required to calculate the cost for $n$ points in the $d$ dimensional space on a single query with respect to the problem at hand (e.g., the time required to compute the cost of linear regression for $n$ points in the $d$ dimensional space given a solution vector $x$ is $O(nd)$.
“t”: be the number of iterations of the algorithm. \end{itemize}
At each iteration “i”, Algorithm~\ref{alg:main} \begin{enumerate}
\item applies $\texttt{VAlg}$ on a matrix of $n$ rows and $i$ columns to obtain a coreset of size $S(n, i, \varepsilon, \delta )$. This step requires $T(n, i, \varepsilon, \delta )$ time.
\item Solves the problem on the coreset to obtain a new solution $x^*$. Requiring $T_{sol}(S(n, i, \varepsilon, \delta ) , d)$ time.
\item Calculates the cost of the $n$ points with respect to $x^*$. Requiring $T_{cost}(n,d)$ time \end{enumerate} Thus, for a single step $i$ the running time is
$T(n, i, \varepsilon, \delta ) + T_{sol}(S(n, i, \varepsilon, \delta ), d) + T_{cost}(n,d)$. Summing for $t$ iterations:
$$\sum_{i=1}^t (T(n, i, \varepsilon, \delta ) + T_{sol}(S(n, i, \varepsilon, \delta ) , d)) +t T_{cost}(n,d).$$
For example, in Linear regression and using the Sensitivity sampling as \textit{VAlg}, an immediate bound for the running time is $O(t (nt + (t/\varepsilon+log(1/\delta)/\varepsilon) d^2 + nd))$.
\textbf{Space complexity. } First, note that the input data and the matrix of losses take $O(n(d+t))$ where $t$ here denotes the number of iterations our coreset generation has taken. Recall the definitions of $\textit{VAlg}, \varepsilon, \delta$ and $S(n, i, \varepsilon, \delta )$. We now denote by \begin{itemize}
\item $Mem(\textit{VAlg}, \varepsilon, \delta, i)$ the amount of space needed by Valg to generate an $\varepsilon$-coreset with a success probability of at least $ 1 - \delta$.
\item $Mem_{sol}(n,d)$ the space required to compute a solution vector $x^*$ for $n$ points in the $d$ dimensional space with respect to the problem at hand (e.g., the space required to compute the solution of SVM is $O(n^2 + d)$. \end{itemize}
The total space complexity is thus bounded by $O(n(d+t) + \max_{i \in [t]} (Mem(\textit{VAlg}, \varepsilon, \delta, i) + Mem_{sol}(S(n, i, \varepsilon, \delta )),d).$
For example for SVM and using the Sensitivity sampling vector summarization, an immediate bound for the space complexity is $O(n(d+t) + (1/\varepsilon(t + \log(1/\delta)))^2)$.
\textbf{Coreset size.} The size of the constructed coreset is equal to the used vector summarization coreset size (See Table~\ref{table:ourContrib}), and it depends on the approximation error $\varepsilon$, the probability of failure $\delta$ we wish to have, and the final number of approximated queries – columns of the query matrix.
In short – let $\varepsilon$ be the desired approximation error and let $\delta$ be the probability of failure. Let $t$ be the number of iterations required Algorithm~\ref{alg:main}. Denote by $S(n, i, \varepsilon, \delta )$ the size of the set computed by the used vector summarization algorithm on a matrix of $n$ rows and $i$ columns with respect to $\varepsilon$ and $\delta$ (see Table~\ref{table:ourContrib} for examples). Then, the size of the coreset is $S(n, t, \varepsilon, \delta )$.
For example, using the Sensitivity sampling method (as the used vector summarization coreset), to approximate the currently given $t$ queries after $t$ iterations, with $\varepsilon$ approximation error, and $\delta$ probability of failure, we get a coreset of size $O(t/\varepsilon + log(1/\delta)/\varepsilon)$.
\textbf{From additive to multiplicative approximation error. } Algorithm~\ref{alg:main} can immediately be modified to compute a coreset that yields a multiplicative approximation as follows. Given the set $P$, the current set of queries $\mathcal{X}^\prime$, and the loss $f$, define a new function $g(p,x):= \frac{f(p,x)}{\sqrt{\sum_{p\in P}f(p,x)}}$ for every pair of a query $x \in \mathcal{X}^\prime$ and input data $p\in P$.
Now build the corresponding matrix $\tilde{\mathcal{M}}(P,g)$ (as done in Algorithm~\ref{alg:main} for $f(p,x)$) instead of $\tilde{\mathcal{M}}(P,f)$, and run the exact same vector summarization coreset algorithm on it. Then, by Lemma~\ref{lem:VSC2FC}, for every $x \in \mathcal{X}^\prime$, $ \abs{\sum_{i \in [n]} g\term{p_i,x} - \sum_{j \in \mathcal{I}} v(j) g\term{p_j,x}}^2 \leq \varepsilon,$
and by the definition of $g$ we get that the result is a multiplicative coreset for the given set of queries as for every $x \in \mathcal{X}^\prime$ \[ \abs{\sum_{i \in [n]} f\term{p_i,x} - \sum_{j \in \mathcal{I}} v(j) f\term{p_j,x}}^2 \leq \varepsilon \sum_{p\in P}f(p,x). \]
\section{Experimental Study} \label{sec:exp}
\begin{figure*}
\caption{Evaluation of our coresets against other competing methods on the Dataset~\ref{dataset:cod_rna}.}
\label{fig:cod-rna_logistic_regression}
\label{fig:cod-rna_svm}
\label{fig:cod}
\end{figure*}
\begin{figure*}
\caption{Evaluation of our coresets against other competing methods on the Dataset~\ref{dataset:HTRU}.}
\label{fig:HTRU_2_logistic_regression}
\label{fig:HTRU_2t_svm}
\label{fig:HTRU}
\end{figure*}
In what follows, we first discuss the choices of different vector summarization coresets, and the used parameters in our experiments, followed by evaluating our coreset on real-world datasets, against other famous competing methods: Near Convex Coreset~\cite{NearConvex}, Lewis weights~\cite{munteanu2018logistic} and leverage scores~\cite{munteanu2018logistic} for logistic regression, Near Convex Coreset~\cite{NearConvex} and optimization based coreset~\cite{tukan2020coresets} for support vector machines (SVM), SVD-based coreset~\cite{maalouf2020tight} for linear regression, Bi criteria coreset~\cite{braverman2021efficient} for $k$-means, and uniform sampling in all of the experiments. We note that each experiment was conducted for $16$ trials, we report both the mean and std for all of the presented metrics.
\textbf{Software/Hardware. } Our algorithms were implemented in Python 3.9~\cite{10.5555/1593511} using \say{Numpy}~\cite{oliphant2006guide}, \say{Scipy}~\cite{2020SciPy-NMeth} and \say{Scikit-learn}~\cite{scikit-learn}. Tests were performed on $2.59$GHz i$7$-$6500$U ($2$ cores total) machine with $16$GB RAM.
\subsection{\emph{AutoCoreset} parameters} \textbf{Vector summarization coresets.} There are many methods for computing such coresets, some of them are deterministic, i.e., with no probability of failure, and others work with some probability $1-\delta$. On the other hand, some are accurate, i.e., $\varepsilon=0$, and others yield an approximation error $\varepsilon>0$. In Table~\ref{table:ourContrib} we summarize some of the common methods for computing such coresets, and their properties, such as size, running time, approximation error, and probability of failure. In our system, we implemented all of the given methods and compared them via extensive experiments.
\textbf{Setting the number of initial solutions $m$.} Throughout our experiments, we have set the number of initial solutions to $10$. The idea behind this is to expose $\textsc{AutoCoreset}$ to a number of solutions that is not too high nor too low. Hence, we ensure that the coreset is not too weak nor too dependent on the initial solutions.
\begin{figure*}
\caption{SVM confusion matrices with respect to our coresets against Uniform sampling and the entire data of Dataset~\ref{dataset:cod_rna}.}
\label{fig:logistic_cod_rna_svm}
\end{figure*}
\begin{figure*}
\caption{Logistic regression confusion matrices with respect to our coresets against Uniform sampling and the entire data of Dataset~\ref{dataset:cod_rna}.}
\label{fig:confusion_cod_rna_logistic}
\end{figure*}
\textbf{Choosing a stopping criterion $\zeta$.} Inspired by the early-stopping mechanism of~\cite{prechelt1998early}, we adopt a similar idea. We make use of a parameter, namely \say{patience}, which was set to $7$, to attempt an indication of the occurrence of saturation with respect to the exposure of our coreset paradigm to new queries; see more details at Section~\ref{sec:moredetails}. To correctly use this parameter, we use additional two parameters, one of which is a counter, while the other holds the optimal coreset that resulted in the smallest sum of the entries of the concatenated columns (see Line~\ref{line:add_column_to_M} at Algorithm~\ref{alg:main}). The counter will be reset to $0$ once a new column is added such that its sum is lower than the smallest sum so far, and the optimal coreset will be updated. Otherwise, the counter will be increased. $\textsc{AutoCoreset}$ will keep running until the above counter reaches the \say{patience} parameter. In our experiments, we returned the optimal coreset since it led to better results. For completeness, we refer the reader to the appendix where we conduct an ablation study and check our results without taking the optimal coreset, i.e., in those results, we take the last coreset. Note that, in both sets of experiments, we outperform the competing methods.
\textbf{Datasets.} The following datasets were used throughout our experimentation. These datasets were taken from~\cite{Dua:2019} and~\cite{CC01a}: \begin{enumerate*}[label=(\roman*)]
\item Credit card dataset~\cite{yeh2009comparisons} composed of $30000$ points with $24$ features representing customers' default payments in Taiwan, \label{dataset:credit}
\item Cod-RNA dataset~\cite{uzilov2006detection}: dataset containing $59535$ points with $8$ features, \label{dataset:cod_rna}
\item HTRU dataset~\cite{lyon2016fifty}: Pulsar candidates collected during the HTRU survey containing $17898$ each with $9$ features, \label{dataset:HTRU}
\item $3D$ Road Network~\cite{guo2012ecomark}: $3D$ road network with highly accurate elevation information from Denmark containing $434874$ points each with $4$ features, \label{dataset:3D}
\item Accelerometer dataset~\cite{ScalabriniSampaio2019}: an accelerometer data from vibrations of a cooler fan with weights on its blades containing $153000$ points consisting each of $5$ features, \label{dataset:accemolator} and
\item Energy efficiency Data Set~\cite{tsanas2012accurate}: a dataset containing $768$ points each of $8$ features. \label{dataset:EB} \end{enumerate*}
\textbf{ML models.} Throughout our entire set of experiments, we have relied on \say{Scikit-Learn} ML models.
\textbf{Reported results.} First, for each coreset $(\mathcal{I},v)$ of an input data $P$ and a loss function $f$, we compute the optimal solution on the coreset $x^*_{\mathcal{I}}\in\argmin_{X\in \mathcal{X}}\sum_{i\in \mathcal{I}}v(i)f(p_i,x) $, and on the real data $x^*_{P}\in\argmin_{x\in \mathcal{X}}\sum_{i\in [n]}f(p_i,x) $, and we report the optimal solution approximation error $\varepsilon=\abs{ \sum_{i\in [n]} f(p_i,x^*_{\mathcal{I}}) - \sum_{i\in [n]} f(p_i,x^*_{P})}$. Secondly, we show for classification problems the test accuracy obtained when training on the coreset, while on regression problems we show an estimate of the coefficient of determination of the prediction $R^2$~\cite{ozer1985correlation}. Additional measures are reported for some problems; we discuss them in the following sections. The bars in our graphs reflect the standard deviation.
\subsection{Traditional ML classification problems}
\begin{figure*}
\caption{Evaluation of our coresets against other competitors concerning the linear regression problem.}
\label{fig:3D_spatial_network_linear_regression}
\label{fig:linear_regression_problem}
\end{figure*}
In what follows, we show our results when setting $f$ to be the loss function of either the \emph{Logistic} regression problem or the \emph{SVMs} problem. In both experiments, since, some of the datasets were unbalanced, each sample coreset size has been split -- small classes get a slightly larger portion of the sample size than simply taking $\eta \times $ sample size where $\eta$ represents the class size percentage with respect to the total number of points, while larger classes get a portion of the sample size smaller than $\eta \times $ sample size.
\textbf{Logistic regression.} We have set the maximal number of iterations to $1000$ (for the Scikit-Learn solver) while setting the regularization parameter to $1$. Our system's approximation error was smaller by orders of magnitude, and the accuracy associated with the models trained using our coreset was better than the model trained on the competing methods; see Figure~\ref{fig:credit_logistic_regression} and Figure~\ref{fig:HTRU_2_logistic_regression}. On the other hand, Figure~\ref{fig:cod-rna_logistic_regression} depicts a multiplicative gap of $30$ with respect to the approximation error in comparison to the competing methods while simultaneously acceding by $5\%$ accuracy gap over them. In addition, we present the confusion matrix for each of our coresets using \emph{AutoCoreset}, and compare it to the confusion matrices with respect to the entire data and the uniform sampling coreset; See Figure~\ref{fig:confusion_cod_rna_logistic}. The confusion matrices aim towards explaining our advantage as our system outputs coresets that approximately maintain the structural properties of the confusion matrix of the entire data better than simply using uniform sampling, as our recall and accuracy are closer to their corresponding values when using the entire dataset.
\begin{figure}
\caption{Evaluation of our coresets against other competitors concerning the $k$-means problem.}
\label{fig:k_means_HTRU}
\label{fig:Kmeans}
\end{figure}
\textbf{SVMs.} As for SVMs, we mainly focused on the linear kernel, while setting the regularization parameter to $1$. Similarly to logistic regression, we outperform the competing methods both in accuracy and approximation error; see Figure~\ref{fig:credit_svm} and Figure~\ref{fig:HTRU_2t_svm}.
\textbf{Discussion}
These results show that general frameworks that aim to handle a large family of functions without embedding some crucial information concerning the properties of the problem, usually tend to lose through the race towards smaller coresets sizes with small approximation errors. We thus show that while \emph{AutoCoreset} is general in the reach of its applications, it also embeds the functional properties of the problem into higher consideration than that of~\cite{NearConvex}, and practically achieves robust results (smaller std).
\subsection{Linear regression and $k$-means clustering.} In our experiments for linear regression, we observe a clear gap between each of our vector summarization coresets and the competing methods, leading towards outperforming the competing coreset for the task of fitting linear regression. In addition, we observe that the determination coefficient $R^2$ for our method is much closer to the determination coefficient $R^2$ when using the entire data. This indicates that our coresets lead to better learning and correlation between the input data and the corresponding outputs of the regression problem; see Figure~\ref{fig:linear_regression_problem}. In addition, for $k$-means, our coresets outperform the competitors (see Figure~\ref{fig:Kmeans}), justifying their robustness across a wide range of applications.
\section{Conclusions and Future Work} In this work, we proposed an automatic practical coreset construction framework that requires only two parameters: the input data and the loss function. Our system, namely \emph{AutoCoreset}, results in small coresets with multiplicative approximation errors significantly smaller than traditional coreset constructions for various machine learning problems, as well as showing that the model learned on our coresets gained more information than the other coresets. While \emph{AutoCoreset} is practical, we also show some desirable theoretical guarantees. We believe that \emph{AutoCoreset} can be further enhanced and tuned to work in the context of Deep learning, e.g., subset selection for boosting training of deep neural networks. We leave this as future work.
Finally, we hope \emph{AutoCoreset} will lay the foundation of practical frameworks for coresets, and hope it reaches the vast scientific community, aiding to achieve faster training with provable guarantees due to training on our coresets.
\appendix \onecolumn \section{More details} \label{sec:moredetails}
\textbf{More on the initialization technique. } Initialization using different approximated solutions is a technique commonly used in optimization algorithms, including kmeans~\cite{arthur2007k}. The idea behind this technique is to start the optimization process from different starting points, or initializations, and to use the resulting approximate solutions to improve the overall optimization performance. This is because different initializations may result in different local optima, and by considering multiple initializations, the optimization algorithm may be able to find a better overall solution. In our method, we do not optimize the given approximate solution, but approximating several approximated solutions using our coreset practically moves the coresets towards approximating ``good’’ various regions of the query set, where each of these regions contains a good solution on the dataset. While there is a possibility that the solutions found may be very similar, in practice, the technique tends to provide benefits in terms of improved optimization performance. Practically, we saw that uniform sampling is also sufficient to achieve very good coresets which approximate the optimal solution very well.
\textbf{More on the stopping criteria. } First of all, the intuition behind setting stopping criteria is derived from the theory of training models in deep learning. Specifically speaking, the early stopping technique in deep learning. While we could have set the number of iterations to a hard-coded scalar (e.g., $400$), we would have either made a very weak coreset that has been exposed to not enough queries, or we would have extended the running time of the algorithm beyond the limits of being practical. The idea that we have used in the paper is to put a threshold on the number of times the minimal cost so far has not changed thus implying some sort of convergence. Notably and most importantly, the usage of such criteria is intensively justified practically in many experimental papers (see for example,~\cite{prechelt2002early,zhou2020bert,gu2018recent}) in deep/machine learning.
We also note that the user can use any stopping criterion and of course, the results will change depending on such a choice.
\textbf{The construction of the query set. } We aimed to obtain a coreset that supports a query set that can span a meaningful part of the entire query space. Intuitively speaking, we aim to have a coreset that approximates the loss of a query set containing (i) the optimal solution of the entire data or some fine approximation to it (see next paragraph for an intuitive explanation of how this should intuitively hold) and (ii) the optimal solution on this computed coreset, given a desired problem (e.g., logistic regression). With this in mind, solving the desired problem on our generated coreset will yield a coreset approximating the solution of the entire data up to $O(\varepsilon)$.
Hence, in the $i$th iteration of our algorithm, we add the solution optimizing the current coreset to the supported set of queries (e.g., optimal logistic regression solution for the current coreset).
Since the coreset is biased towards this solution, we have evaluated the quality of such a solution on the entire data and concatenated such a vector of losses to our matrix of losses (denoted by the matrix $\tilde{\mathcal{M}}$).
This, in turn, means that each time a new query is added to the supported set of queries, the coreset in the next iteration will be adapted to approximate every query in the query set and it will become more generalized, or in a sense a “stronger coreset”.
With this in mind, we can initialize our support query set with approximated solutions to the problem (e.g. $\varepsilon$-approximations), so as to ensure a good initial coreset.
\section{Proof of Our Theoretical Results} \subsection{Proof of Lemma~\ref{lem:VSC2FC}} \begin{proof} First, observe that by construction of $\Tilde{\mathcal{M}}(P,f)$, it holds that for every $x \in \mathcal{X}^\prime$, and $j\in [n]$, there exists an integer $i \in [\abs{\mathcal{X}^\prime}]$ such that
\begin{equation} \label{eq:property_M_to_f} \Tilde{\mathcal{M}}(P,f)_{j,i} = f\term{p_j,x}. \end{equation}
By Definition~\ref{def:VSCoreset}, the pair $\term{\mathcal{I}, v}$ satisfies that \begin{equation} \label{eq:VSCCoresetOnM} \norm{\sum_{j= 1}^n \Tilde{\mathcal{M}}(P,f)_{j,\ast} - \sum_{\ell \in \mathcal{I}} v\term{\ell} \Tilde{\mathcal{M}}(P,f)_{\ell,\ast}}_2^2 \leq \varepsilon. \end{equation}
Note that~\eqref{eq:VSCCoresetOnM} dictates that for every $k \in \left[\abs{X^\prime}\right]$, it holds that \begin{equation} \label{eq:VSCCoresetOnM_per_columns} \abs{\sum_{j \in [n]} \Tilde{\mathcal{M}}(P,f)_{j,k} -\sum_{\ell \in \mathcal{I}} v\term{\ell} \Tilde{\mathcal{M}}(P,f)_{\ell,k} }^2 \leq \varepsilon. \end{equation}
Finally, combining~\eqref{eq:property_M_to_f} and \eqref{eq:VSCCoresetOnM_per_columns} yields Lemma~\ref{lem:VSC2FC}.
\end{proof}
\subsection{Proof of Claim~\ref{clm:hidden_talent}}
\begin{proof} For every $k\in [z]$, denote by $x_k$ the query which corresponds to the $k$th column of $\Tilde{\mathcal{M}}(P,f)$.
The claim holds by the following derivations:
\begin{align*} &\abs{\sum_{i =1}^n f\term{p_i, x} - \sum_{j \in \mathcal{I}} v(j) f\term{p_j, x}}^2 \\
&= \abs{\sum_{i =1}^n \sum_{k=1}^z \alpha\term{k} f(p_i,x_k) - \sum_{j \in \mathcal{I}} v(j) \sum_{k=1}^z \alpha\term{k} f(p_i,x_k)}^2
\\&=
\abs{\sum_{k=1}^z \alpha\term{k}\sum_{i =1}^n f(p_i,x_k) - \sum_{k=1}^z \alpha\term{k}\sum_{j \in \mathcal{I}} v(j) f(p_i,x_k)}^2
\\&=
\abs{\sum_{k=1}^z \alpha\term{k}( \sum_{i =1}^n f(p_i,x_k) - \sum_{j \in \mathcal{I}} v(j) f(p_i,x_k))}^2
\\&\leq
\abs{\sum_{k=1}^z \alpha\term{k} \sqrt{\varepsilon}}^2 = \abs{\sqrt{\varepsilon}}^2 \leq \varepsilon , \end{align*}
where the first equality hold by the definition of $x$, the second and thirds are simple rearrangements, the first inequality holds by Claim~\ref{clm:hidden_talent}. \end{proof}
\section{Experimental Results} \label{sec:appendix_res} In this section, we dive into exploring the effect of the actions/parameters used in \emph{AutoCore}.
\subsection{Taking the last coreset} In what follows, we show the results of using the last coresets \emph{AutoCore} has devised, i.e., as Algorithm~\ref{alg:main} suggests.
\begin{figure*}
\caption{Evaluation of our coresets against Uniform sampling on the Dataset~\ref{dataset:credit}.}
\label{fig:credit_logistic_regression_normal}
\label{fig:credit_svm_normal}
\label{fig:credit_normal}
\end{figure*}
\begin{figure*}
\caption{Evaluation of our coresets against Uniform sampling on the Dataset~\ref{dataset:cod_rna}.}
\label{fig:cod-rna_logistic_regression_normal}
\label{fig:cod-rna_svm_normal}
\label{fig:cod_normal}
\end{figure*}
\begin{figure*}
\caption{Evaluation of our coresets against Uniform sampling on the Dataset~\ref{dataset:HTRU}.}
\label{fig:HTRU_2_logistic_regression_normal}
\label{fig:HTRU_2t_svm_normal}
\label{fig:HTRU_normal}
\end{figure*}
As depicted throughout Figures~\ref{fig:credit_normal}--\ref{fig:HTRU_normal}, we observe that~\emph{AutoCore} output coresets that outperform the competing methods almost in all of our experiments. In some, we observe that the desired behavior of our coreset gets delayed (takes 2 to 3 samples to outperform the rest of the competitors). This is due to the fact that taking such coresets means that the coreset is becoming more general, thus requiring a larger sample size to guarantee better approximation, one needs to sample more. Such behavior does not appear in our \say{optimal coresets} where we have taken the coreset with the optimal cost; see Figures~\ref{fig:credit}--~\ref{fig:HTRU}. The reason for this is that the optimal coreset has been exposed to fewer models/queries than the coreset that would be output by the plain \emph{AutoCore}, and thus the need for a larger sample size for smaller approximation error becomes less demanding.
\subsection{Exploration of different algorithms for choosing queries}
In what follows, we show the effect of different methods for choosing the next query for our practical coreset paradigm with respect to the logistic regression problem.
\begin{figure*}
\caption{Evaluation of our coresets with different algorithms for choosing the next query.}
\label{fig:HTRU_logistic_init_caratheodoy}
\label{fig:HTRU_logistic_init_median_of_means}
\label{fig:HTRU_logistic_init_median_of_means}
\label{fig:credit_normal}
\label{fig:differnt_choices}
\end{figure*}
\subsection{Experimenting with Cifar10 and TinyImageNet}
In what follows, we run our coreset paradigm on Cifar10 and TinyImageNet. For TinyImageNet data, we had to use the JL-lemma to reduce the dimensionality of the data.
As seen from Figure~\ref{fig:deep_data_results}, our coreset construction technique yields better coresets than uniform sampling even for large-scale datasets, where our coreset can be better than uniform sampling by at max $\approx 1.5$ times in terms of relative approximation error.
\begin{figure*}
\caption{Evaluation of our coreset on large-scale datasets.}
\label{fig:deep_data_results}
\end{figure*}
\end{document} | arXiv | {
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\begin{document}
\title{$F$-injectivity and Buchsbaum singularities} \author{Linquan Ma} \address{Department of Mathematics\\ University of Michigan\\ Ann Arbor\\ Michigan 48109} \email{lquanma@umich.edu} \maketitle \begin{abstract} Let $(R,\mathfrak{m}, K)$ be a local ring that contains a field. We show that, when $R$ has equal characteristic $p>0$ and when $H_\mathfrak{m}^i(R)$ has finite length for all $i<\dim R$, then $R$ is $F$-injective if and only if every ideal generated by a system of parameters is Frobenius closed. As a corollary, we show that such an $R$ is in fact a Buchsbaum ring. This answers positively a question of S. Takagi that $F$-injective singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. We also study the characteristic $0$ analogue of this question and we show that Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum in the graded case. \end{abstract}
\section{Introduction}
The concept of $F$-injective rings was first introduced in \cite{FedderFPureRationalsingularity} in the early 1980s. This class of rings naturally arises when one studies the Frobenius actions on the local cohomology supported at the maximal ideal, and is a natural generalization of $F$-pure rings. There is a notion of Frobenius closure for ideals (see Section 2 for detailed definition) that has close connections with $F$-pure and $F$-injective singularities. In general, the Frobenius closure is very small and is always contained in the tight closure (we refer to \cite{HochsterHunekeTC1} for basic tight closure theory). And, quite similar to the tight closure characterizations of $F$-regularity and $F$-rationality, it is well known that, under mild conditions on $R$, the $F$-purity of $R$ is the same as the condition that every ideal be Frobenius closed. If we assume $R$ is Cohen-Macaulay, then the $F$-injectivity of $R$ is equivalent to the condition that every ideal generated by a system of parameters be Frobenius closed, and also equivalent to the condition that a single ideal generated by a system of parameters be Frobenius closed.
However, when $R$ is not assumed to be Cohen-Macaulay, the relation between $F$-injectivity and Frobenius closure is not clear. The first goal of this paper is to explore the connections between these conditions without the Cohen-Macaulay assumption. Our first main theorem is the following: \begin{theorem} \label{main theorem} Let $(R,\mathfrak{m})$ be a local ring of equal characteristic $p>0$. Suppose $H_\mathfrak{m}^i(R)$ has finite length for each $i<\dim R$. Then the following are equivalent: \begin{enumerate} \item $R$ is $F$-injective. \item Every ideal generated by a system of parameters is Frobenius closed. \end{enumerate} \end{theorem}
In general, $F$-injective singularities are not necessarily Cohen-Macaulay, but in many cases, they are Buchsbaum. This is a natural weakening of Cohen-Macaulayness and, in a precise sense, the closest condition to being Cohen-Macaulay (see Section 2 for details). In \cite{SchenzelApplicationsOfDualizingComplexes}, Schenzel proved some homological criteria for Bushsbaum singularities. And, utilizing the results of Hochster and Roberts on the purity of the Frobenius map in \cite{HochsterRobertsFrobeniusLocalCohomology}, Schenzel also obtained some sufficient conditions for Buchsbaum rings in the graded case. In fact, results in \cite{SchenzelApplicationsOfDualizingComplexes} indicate that for $(R,\mathfrak{m})$ an $F$-injective graded ring, if $H_\mathfrak{m}^i(R)$ has finite length for each $i<\dim R$, then $R$ is Buchsbaum. We note that, under some mild conditions on $R$, $H_\mathfrak{m}^i(R)$ has finite length for each $i<\dim R$ if and only if $R$ is Cohen-Macaulay on the punctured spectrum (we give a detailed explanation of this in Section 2). So Schenzel's result is basically saying that $F$-injective singularities with isolated non-Cohen-Macaulay locus are Buchsbaum in the graded case. Takagi asked whether the same conclusion holds when $(R,\mathfrak{m})$ is a local ring: \begin{question}[{\it cf.} Open Problem A.3 in \cite{KovacsandSchwedesurveyonlogcanonicalandDuBoissingulaities}] \label{Takagi's question} Suppose $(R,\mathfrak{m})$ is $F$-injective and $H_\mathfrak{m}^i(R)$ has finite length for each $i<\dim R$. Then is $R$ a Buchsbaum ring? \end{question}
This is supported by results of Goto and Ogawa in \cite{GotoOgawaAnoteonringswithFLC} when $R$ is $F$-pure. Using Theorem \ref{main theorem}, we provide a positive answer to this question. \begin{corollary} \label{main corollary} Let $(R,\mathfrak{m})$ be a local ring of equal characteristic $p>0$. Suppose $R$ is $F$-injective and $H_\mathfrak{m}^i(R)$ has finite length for each $i<\dim R$. Then $R$ is a Buchsbaum ring. \end{corollary}
It is known that $F$-injective singularities in characteristic $p>0$ have close connections with Du Bois singularities in characteristic $0$. This connection was studied intensively by Schwede in \cite{SchwedeFInjectiveAreDuBois}, where it was proved that in characteristic $0$, singularities of dense $F$-injective type are Du Bois and it was conjectured that the converse is also true. Based on this connection, it is quite natural to consider the characteristic $0$ analogue of Question \ref{Takagi's question}: \begin{question}[{\it cf.} Open Problem A.4 in \cite{KovacsandSchwedesurveyonlogcanonicalandDuBoissingulaities}] \label{Takagi's question in char 0} Suppose $(R,\mathfrak{m})$ is Du Bois and $H_\mathfrak{m}^i(R)$ has finite length for each $i<\dim R$. Then is $R$ a Buchsbaum ring? \end{question}
Note that by results of Ishida in \cite{IshidaIsolatedDuBoissingularities}, Question \ref{Takagi's question in char 0} has a positive answer if $(R,\mathfrak{m})$ is a normal isolated singularity. Using Schenzel's criterion for Bushsbaum singularities in \cite{SchenzelApplicationsOfDualizingComplexes} and Schwede's simple characterization of Du Bois singularities in \cite{SchwedeEasyCharacterization}, we provide a positive answer to Question \ref{Takagi's question in char 0} in the normal standard graded case. \begin{theorem} \label{main theorem in characteristic 0} Let $(R,\mathfrak{m})$ be a normal standard graded $K$-algebra (i.e., generated over $K$ by 1-forms). If $R$ is Du Bois and $H_\mathfrak{m}^i(R)$ has finite length for each $i<\dim R$, then $R$ is a Buchsbaum ring. \end{theorem}
Throughout this paper we will use $(R,\mathfrak{m})$ to denote either a Noetherian local ring with unique maximal ideal $\mathfrak{m}$ or an $\mathbb{N}$-graded ring finitely generated over $K$ with unique homogeneous maximal ideal $\mathfrak{m}$. Rings are always assumed to be equal characteristic (i.e., contain a field). In Section 2 we recall and review some basic definitions and properties about $F$-pure, $F$-injective and Buchsbaum rings. In Section 3 we work in equal characteristic $p>0$. We prove Theorem \ref{main theorem} and Corollary \ref{main corollary}. In Section 4 we work in equal characteristic $0$. We provide a criterion for Du Bois singularities for section rings of normal projective varieties and we prove Theorem \ref{main theorem in characteristic 0}, and we also remark that the conjecture that Du Bois singularities have dense $F$-injective type implies a positive answer to Question \ref{Takagi's question in char 0}.
\section{Preliminaries}
Let $(R,\mathfrak{m})$ be a local ring that contains a field. If $R$ has equal characteristic $p>0$, then there is a natural action of the Frobenius endomorphism of $R$ on each of its local cohomology modules $H_{\mathfrak{m}}^i(R)$. Recall that a map of $R$-modules $N\rightarrow N'$ is {\it pure} if for every $R$-module $M$ the map $N\otimes_RM\rightarrow N'\otimes_RM$ is injective. $R$ is called {\it $F$-pure} if the Frobenius endomorphism $R\xrightarrow{F} R$ is pure. $R$ is called {\it $F$-injective} if the Frobenius acts injectively on $H_\mathfrak{m}^i(R)$ for every $i$. We point out that $F$-pure always implies $F$-injective (see \cite{HochsterRobertsFrobeniusLocalCohomology}).
When $R$ has equal characteristic $p>0$, for every ideal $I\subseteq R$, we define \[I^F=\{x\in R|\exists e, x^{p^e}\in I^{[p^e]} \}\] to be the {\it Frobenius closure} of $I$. $I$ is called {\it Frobenius closed} if $I^F=I$. It is well known that under mild conditions on $R$, $R$ is $F$-pure if and only if every ideal is Frobenius closed. Moreover, if $R$ is Cohen-Macaulay, then $R$ is $F$-injective if and only if every ideal generated by a system of parameters is Frobenius closed, and also if and only if one single ideal generated by a system of parameters is Frobenius closed.
We say that a local ring $(R,\mathfrak{m})$ has {\it finite local cohomology} if $H_\mathfrak{m}^i(R)$ has finite length for every $i<\dim R$. It is well known that, under mild conditions on $R$, $R$ has finite local cohomology if and only if $R$ is equidimensional and Cohen-Macaulay on the punctured spectrum (see \cite{SchenzelTrungVerallgemeinerteCohenMacaulayModuln}). We will need the following important result characterizing rings with finite local cohomology. This result and its equivalent form appeared in \cite{SchenzelTrungVerallgemeinerteCohenMacaulayModuln}, \cite{Schenzelstandardsystemsofparemeters} and \cite{GotoOgawaAnoteonringswithFLC}. We recall that a sequence of elements $x_1,\dots,x_r$ in a local ring $R$ is called a {\it $d$-sequence} if $(x_1,\dots,x_{i-1}):x_ix_j=(x_1,\dots,x_{i-1}):x_j$ for every $1\leq i\leq j\leq r$.
\begin{theorem}[{\it cf.} Proposition 2.1 and 2.2 in \cite{Schenzelstandardsystemsofparemeters} and the main Theorem in \cite{GotoOgawaAnoteonringswithFLC}] \label{FLCandd-sequence} Let $(R,\mathfrak{m})$ be a local ring of dimension $n$, then the following are equivalent: \begin{enumerate} \item $H_{\mathfrak{m}}^i(R)$ has finite length for all $i\neq n$. \item There exists an integer $N$ such that for every system of parameters $x_1,\dots, x_n$ contained in $\mathfrak{m}^N$, we have \[(x_1,\dots,x_{i-1}):x_i=(x_1,\dots,x_{i-1}):\mathfrak{m}^N.\] \item There exists an integer $N$ such that every system of parameters contained in $\mathfrak{m}^N$ is a $d$-sequence. \item There exists an integer $N$ and a constant $C$ such that for every system of parameters $x_1,\dots,x_n$ of $R$, we have \[l(R/(\underline{x}))-e(\underline{x}, R)\leq C\] with equality when $(x_1,\dots,x_n)\subseteq \mathfrak{m}^N$. \end{enumerate} Moreover, when the equivalent conditions hold, we can let \[C=\sum_{i=1}^{n-1}\binom{n-1}{i}l(H_\mathfrak{m}^i(R)).\] \end{theorem}
Now we give the definition of Buchsbaum rings. It turns out that there are many different ways to define them. We also note that the definition of Buchsbaum rings is characteristic free (in fact it makes sense in mixed characteristic also, but we will not use this). \begin{definition}\label{definition of Buchsbaum ring} The following conditions on a local ring $(R,\mathfrak{m}, K)$ are equivalent: \begin{enumerate} \item For every system of parameters $x_1,\dots,x_n$, we have \[(x_1,\dots,x_{i-1}):x_i=(x_1,\dots,x_{i-1}):\mathfrak{m}\]for every $i$. \item Every system of parameters is a $d$-sequence. \item The difference $l_R(R/J)-e(J,R)$, where $J$ is an ideal generated by a system of parameters, is an invariant of $R$ (i.e., it is independent of $J$). \item There is a system of parameters $\underline{x}=x_1,\dots,x_n$ such that $\tau^nC^{\bullet}(\underline{x}, R)$ is quasi-isomorphic to a complex of $K$-vector spaces, where $\tau^nC^{\bullet}(\underline{x}, R)$ is the \v{C}ech complex truncated from above at the $n$-th place. \end{enumerate} When $R$ satisfies one of these equivalent conditions, it is called a {\it Buchsbaum} ring. \end{definition}
We refer to \cite{HunekeTheoryofdSequence} for $(1)\Leftrightarrow(2)$, \cite{StuckradandVogelEineVerallgemeinerungderCohenMacaulayRinge} for $(1)\Leftrightarrow(3)$ and \cite{SchenzelApplicationsOfDualizingComplexes} for $(1)\Leftrightarrow(4)$. Next we summarize some basic facts about Buchsbaum rings. \begin{remark} \begin{enumerate} \item Cohen-Macaulay rings are obviously Buchsbaum. Moreover, by Definition \ref{definition of Buchsbaum ring}, it is clear that $R$ is Buchsbaum if and only if we can take $N=1$ in (2)-(4) in Theorem \ref{FLCandd-sequence}. So among rings with finite local cohomology, Buchsbaum rings are the closest to Cohen-Macaulay rings. \item By $(4)$ in Definition \ref{definition of Buchsbaum ring}, $R$ is Buchsbaum implies $H_\mathfrak{m}^i(R)$ are $K$-vector spaces for all $i<n=\dim R$. However, there exist local rings such that $H_\mathfrak{m}^i(R)$ are $K$-vector spaces for all $i<n=\dim R$ but $R$ is {\it not} Buchsbaum (see \cite{GotonoteonquasiBuchsbaumrings}). \end{enumerate} \end{remark}
We also mention the notion of Buchsbaum modules introduced in \cite{StuckradandVogelTowardatheoryofBuchsbaumsingularities}. A finitely generated module $M$ over a local ring $(R,\mathfrak{m})$ is called a {\it Buchsbaum module} of dimension $d$ if \[(x_1,\dots,x_{i-1})M:_Mx_i=(x_1,\dots,x_{i-1})M:_M\mathfrak{m}\] for every system of parameters $(x_1,\dots,x_d)$ of $M$ and every $1\leq i\leq d$. So $R$ is Buchsbaum if and only if $R$ is a Buchsbaum module over $R$ of maximal dimension $=\dim R$. We will need the following powerful criterion, the so-called surjectivity criterion, of Buchsbaum modules: \begin{theorem}[{\it cf.} Theorem 1 in \cite{StuckradandVogelTowardatheoryofBuchsbaumsingularities} and Satz 2 in \cite{StuckradsurjectivitycriterionofBuchsbaum}] \label{surjectivity criterion for Buchsbaum modules} Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m}, K)$. If the canonical maps $\Ext_R^i(K, M)\to H_\mathfrak{m}^i(M)$ are surjective for all $i\neq d=\dim M$ then $M$ is a Buchsbaum module. Moreover, if $R$ is regular, then the converse also holds. \end{theorem}
In \cite{SchenzelApplicationsOfDualizingComplexes}, Schenzel observed the following criterion for rings to be Buchsbaum in the graded case which turns out to be very useful. In fact, this follows easily from (4) in Definition \ref{definition of Buchsbaum ring}. \begin{theorem}[{\it cf.} Theorem 3.1 in \cite{SchenzelApplicationsOfDualizingComplexes}] \label{sufficient condision for Buchsbaum ring in the graded case} Let $(R,\mathfrak{m})$ be a special graded $K$ algebra (meaning that $R$ is non-negatively graded of finite type over $K$). If there exists an integer $t$ such that $[H_\mathfrak{m}^i(R)]_s=0$ for all $s\neq t$ and for every $i<\dim R$, then $R$ is a Buchsbaum ring. \end{theorem}
When $(R,\mathfrak{m})$ is special graded, $F$-injective and has finite local cohomology, it is easy to see that for every $i<\dim R$, $H_\mathfrak{m}^i(R)=[H_\mathfrak{m}^i(R)]_0$. Hence in this case, we can apply Theorem \ref{sufficient condision for Buchsbaum ring in the graded case} with $t=0$. Therefore if $(R,\mathfrak{m})$ is special graded, $F$-injective and $R$ has finite local cohomology, then $R$ is Buchsbaum. This fact is known to experts. In fact, this is exactly Proposition 4.1 in \cite{SchenzelApplicationsOfDualizingComplexes}. Although Schenzel requires that $R$ be $F$-pure, exactly the same argument works when $R$ is $F$-injective.
\section{Characteristic $p>0$ results} In this section we prove our main results in characteristic $p>0$, Theorem \ref{main theorem} and Corollary \ref{main corollary}. Throughout this section all rings are of equal characteristic $p>0$ (although we will repeat this sometimes). We start by proving two simple lemmas that we will use. \begin{lemma} \label{lemma on ideals generated by part of a system of parameters} If every ideal generated by a full system of parameters is Frobenius closed, then so is every ideal generated by part of a system of parameters. \end{lemma} \begin{proof} Suppose $(x_1,\dots,x_t)$ is part of a system of parameters, contained in $(x_1,\dots,x_t,x_{t+1},\dots,x_n)$. If $y\in (x_1,\dots,x_t)^F$, then $y\in (x_1,\dots,x_t,x_{t+1}^s,\dots,x_n^s)^F=(x_1,\dots,x_t,x_{t+1}^s,\dots,x_n^s)$ for every $s>0$. So \[y\in \bigcap_s(x_1,\dots,x_t,x_{t+1}^s,\dots,x_n^s)=(x_1,\dots,x_t).\] \end{proof}
\begin{lemma} \label{lemma on Frobenius action on top local cohomology} Let $(R,\mathfrak{m})$ be a local ring such that every ideal generated by a system of parameters is Frobenius closed. Then the Frobenius acts injectively on $H_\mathfrak{m}^j(R)$ for $j=\dim R$ and $j=\depth R$. \end{lemma} \begin{proof} First notice that if $J^{[q]}=(x_1^q,\dots,x_j^q)$ is Frobenius closed for every $q=p^e$, then Frobenius acts injectively on $H_J^j(R)$. This is because we have a direct limit system \[ \xymatrix{
R/J \ar[r] \ar[d]^{F} & R/J^{[p]} \ar[r] \ar[d]^{F} & R/J^{[p^2]} \ar[r]\ar[d]^{F} & \cdots \\
R/J^{[p]}\ar[r] & R/J^{[p^2]} \ar[r] & R/J^{[p^3]} \ar[r] & \cdots } \] where the vertical maps are the Frobenius and the horizontal maps are multiplications by $(x_1\cdots x_j)^{p^e-p^{e-1}}$ at the corresponding spots. The direct limit of both lines are $H_J^j(R)$ and the vertical map is exactly the Frobenius action on $H_J^j(R)$. Since $J^{[q]}$ is Frobenius closed, we know that each vertical map is injective, hence so is the induced map on the direct limit. Therefore Frobenius acts injectively on $H_J^j(R)$. From this it follows immediately that if every ideal generated by a system of parameters is Frobenius closed, then Frobenius acts injectively on $H_\mathfrak{m}^n(R)$ for $n=\dim R$.
Now we let $I=(x_1,\dots,x_n)$ be a system of parameters with $(x_1,\dots,x_r)$ a maximal regular sequence in $R$ (i.e., $r=\depth R$). By Lemma \ref{lemma on ideals generated by part of a system of parameters}, we know that $(x_1^q,\dots,x_r^q)$ is Frobenius closed for every $q=p^e$. We have the local cohomology spectral sequence: \[E_2^{p,q}=H_{I}^p(H_J^q(R))\Rightarrow H_{I+J}^{p+q}(R).\] We apply this spectral sequence to $I=\mathfrak{m}$ and $J=(x_1,\dots,x_r)$. Since $(x_1,\dots,x_r)$ is a regular sequence, $H_J^q(R)$ vanishes unless $q=r$. So this spectral sequence degenerates. So we know that \[H_\mathfrak{m}^0(H_J^r(R))\cong H_\mathfrak{m}^r(R).\] But since $J^{[q]}=(x_1^q,\dots,x_r^q)$ is Frobenius closed for every $q=p^e$, the above argument shows that Frobenius acts injectively on $H_J^r(R)$, hence it also acts injectively on $H_\mathfrak{m}^0(H_J^r(R))\cong H_\mathfrak{m}^r(R)$. \end{proof}
The next proposition is essentially taken from \cite{GotoOgawaAnoteonringswithFLC}, where the authors show that for rings of finite local cohomology, $F$-purity implies Buchsbaumness. But in fact, the argument in \cite{GotoOgawaAnoteonringswithFLC} only uses that every ideal generated by a system of parameters is Frobenius closed. We give a short proof of this proposition for completeness. \begin{proposition}[{\it cf.} main Corollary in \cite{GotoOgawaAnoteonringswithFLC}] \label{s.o.p Frobenius closed implies Buchsbaum} Let $(R,\mathfrak{m})$ be a local ring of equal characteristic $p>0$. Suppose $R$ has finite local cohomology and every ideal generated by a system of parameters is Frobenius closed. Then $R$ is Buchsbaum. \end{proposition} \begin{proof} We claim that every system of parameters is a $d$-sequence. Since $H_\mathfrak{m}^i(R)$ has finite local cohomology, by Theorem \ref{FLCandd-sequence} $(1)\Rightarrow(2)$, there exists $N$ such that every system of parameters contained in $\mathfrak{m}^N$ is a $d$-sequence. Let $x_1,\dots,x_n$ be an arbitrary system of parameters. Note that by Lemma \ref{lemma on ideals generated by part of a system of parameters}, $(x_1,\dots,x_i)$ is Frobenius closed for every $i$. We want to show $(x_1,\dots,x_{i-1}):x_ix_j=(x_1,\dots,x_{i-1}):x_j$. One containment is obvious. For the other one, let $y\in(x_1,\dots,x_{i-1}):x_ix_j$, for $q\geq N$, we have \begin{eqnarray*} &&yx_ix_j\in(x_1,\dots,x_{i-1})\\ &\Rightarrow&y^qx_i^qx_j^q\in(x_1^q,\dots,x_{i-1}^q)\\ &\Rightarrow&y^qx_j^q\in(x_1^q,\dots,x_{i-1}^q)\\ &\Rightarrow&yx_j\in(x_1,\dots,x_{i-1})\\ &\Rightarrow&y\in(x_1,\dots,x_{i-1}):x_j \end{eqnarray*} where the third line we use $(x_1^q,\dots,x_n^q)$ is a $d$-sequence (because $q\geq N$), the fourth line we use that $(x_1,\dots,x_{i-1})$ is Frobenius closed. \end{proof}
We briefly review the $\Gamma$-construction introduced in \cite{HochsterHunekeFRegularityTestElementsBaseChange}. Let $K$ be a field of positive characteristic $p>0$ with a $p$-base $\Lambda$. Let $\Gamma$ be a fixed cofinite subset of $\Lambda$. For $e\in \mathbb{N}$ we denote by $K^{\Gamma,e}$ the purely inseparable field extension of $K$ that is the result of adjoining $p^e$-th roots of all elements in $\Gamma$ to $K$. Now for $(R,\mathfrak{m})$ a complete local ring with $K\subseteq R$ a coefficient field, let $x_1,\dots,x_n$ be a system of parameters for $R$. We know that $R$ is module-finite over $A=K[[x_1,\dots,x_n]]\subseteq R$. Let $A^\Gamma$ denote $\bigcup_{e\in\mathbb{N}}K^{\Gamma,e}[[x_1,\dots,x_n]]$, which is a regular local ring that is faithfully flat and purely inseparable over $A$. The maximal ideal of $A$ expands to that of $A^{\Gamma}$. Let $R^{\Gamma}=A^{\Gamma}\otimes_AR$, which is module-finite over the regular ring $A^{\Gamma}$ and is faithfully flat and purely inseparable over $R$. The maximal ideal of $R$ expands to the maximal ideal of $R^{\Gamma}$ and the residue field of $R^{\Gamma}$ is $K^{\Gamma}=\bigcup_{e\in\mathbb{N}}K^{\Gamma,e}$.
We will use the important fact that $R^\Gamma$ is $F$-finite (see \cite{HochsterHunekeFRegularityTestElementsBaseChange} for details). Moreover, we can preserve many good properties of $R$ when $\Gamma$ is sufficiently small. For example, if $(R,\mathfrak{m})$ is complete and $F$-injective, then $R^\Gamma$ is still $F$-injective for any sufficiently small choice of cofinite $\Gamma$ by Lemma 2.9 in \cite{EnescuHochsterTheFrobeniusStructureOfLocalCohomology}.
Now we state and prove our main result in characteristic $p>0$.
\begin{theorem} \label{main theorem 2} Let $(R,\mathfrak{m})$ be a local ring of equal characteristic $p>0$ and dimension $n$. Suppose $R$ has finite local cohomology. Then the following are equivalent: \begin{enumerate} \item $R$ is $F$-injective. \item Every ideal generated by a system of parameters is Frobenius closed. \end{enumerate} \end{theorem} \begin{proof} We first prove $(1)\Rightarrow(2)$. Since $R$ is $F$-injective, so is $\widehat{R}$. We apply the $\Gamma$-construction to $\widehat{R}$: $\widehat{R}^\Gamma$ is $F$-finite and $F$-injective for any sufficiently small choice of cofinite $\Gamma$. Now we consider $S=\widehat{\widehat{R}^\Gamma}$, we know that $S$ is complete, $F$-finite, $F$-injective and faithfully flat over $R$. If we can show that every ideal generated by a system of parameters in $S$ is Frobenius closed, then the same follows for $R$ because for every $I\subseteq R$ generated by a system of parameters, we have $I^F\subseteq (IS)^F\bigcap R=IS\bigcap R=I$. Therefore, to prove $(1)\Rightarrow(2)$, we may replace $R$ by $S$ and hence assume that $R$ is complete and $F$-finite. Note that in this case since $R$ is $F$-injective, we know $R$ is reduced (for example, see Remark 2.6 in \cite{SchwedeandZhangBertinitheoremsforFsingularities}), hence we may also assume without loss of generality that $R$ is reduced.
Let $R^{1/q}$ denote the ring obtained by adjoining all $q$-th roots of elements of $R$ where $q=p^e$. Since $R$ is reduced, we have a short exact sequence \[0\rightarrow R\rightarrow R^{1/q}\rightarrow R^{1/q}/R\rightarrow 0\] which induces a long exact sequence of local cohomology \begin{equation} \label{long exact sequence of local cohomology} \cdots\rightarrow H_\mathfrak{m}^{i-1}(R^{1/q}/R)\xrightarrow{\phi_i} H_\mathfrak{m}^i(R)\rightarrow H_\mathfrak{m}^i(R^{1/q})\rightarrow H_\mathfrak{m}^i(R^{1/q}/R)\rightarrow\cdots. \end{equation} Because $R$ is $F$-injective, each $H_\mathfrak{m}^i(R)\rightarrow H_\mathfrak{m}^i(R^{1/q})$ is injective. This means each connecting map $\phi_i$ is the zero map. So (\ref{long exact sequence of local cohomology}) actually gives us $n-1$ short exact sequences: \begin{equation} \label{short exact sequence of local cohomology} 0 \rightarrow H_\mathfrak{m}^i(R)\rightarrow H_\mathfrak{m}^i(R^{1/q})\rightarrow H_\mathfrak{m}^i(R^{1/q}/R)\rightarrow0 \end{equation} for every $0\leq i\leq n-1$.
Let $x_1,\dots,x_n$ be any system of parameters, we want to show that $(x_1,\dots,x_n)$ is Frobenius closed. Since $R$ is complete, we may pick a coefficient field $K\cong R/\mathfrak{m}$ of $R$, and by Cohen's structure theorem, $R$ is module finite over $A=K[[x_1,\dots,x_n]]$. Hence $R^{1/q}$ is also module finite over $A$ for every $q=p^e$ since $R$ is $F$-finite. By Theorem \ref{FLCandd-sequence} $(1)\Rightarrow(2)$, if $q>N$, every system of parameters in $R$ satisfies \[(x_1^q,\dots,x_{i-1}^q):_R\mathfrak{m}^{[q]}\subseteq(x_1^q,\dots,x_{i-1}^q):_Rx_i^q=(x_1^q,\dots,x_{i-1}^q):_R\mathfrak{m}^N\subseteq(x_1^q,\dots,x_{i-1}^q):_R\mathfrak{m}^{[q]}.\] So we must have equalities. But after taking $q$-th roots, this implies \[(x_1,\dots,x_{i-1})R^{1/q}:_{R^{1/q}}x_i=(x_1,\dots,x_{i-1})R^{1/q}:_{R^{1/q}}\mathfrak{m}.\] In particular, this implies that when $q>N$, $R^{1/q}$ is a (finitely generated) Buchsbaum $R$-module of dimension $n$, and hence also a Buchsbaum $A$-module of dimension $n$.
Now we claim that for every $q>N$, $R^{1/q}/R$ is also a Buchsbaum module of dimension $n$ over $A$. We prove this using the surjectivity criterion of Buchsbaum modules (Theorem \ref{surjectivity criterion for Buchsbaum modules}). We have the following commutative diagram, which are the long exact sequences of $\Ext_A^i(K,-)$ and $H_\mathfrak{m}^i(-)$ induced by $0\to R\to R^{1/q}\to R^{1/q}/R\to 0$: \[ \xymatrix{
\cdots \ar[r] & \Ext_A^i(K, R) \ar[r] \ar[d] & \Ext_A^i(K, R^{1/q}) \ar[r] \ar[d]^{\alpha_i} & \Ext_A^i(K, R^{1/q}/R) \ar[r] \ar[d]^{\beta_i} & \cdots \\
0\ar[r] & H_\mathfrak{m}^i(R) \ar[r] & H_\mathfrak{m}^i(R^{1/q}) \ar[r] & H_\mathfrak{m}^i(R^{1/q}/R) \ar[r] & 0 } \] where the bottom sequence is exact by (\ref{short exact sequence of local cohomology}). Since $R^{1/q}$ is a Buchsbaum $A$-module of dimension $n$ and $A$ is a regular local ring, we know that for each $0\leq i\leq n-1$, $\alpha_i$ is surjective. So by the commutativity of the above diagram, each $\beta_i$ is also surjective. Hence $R^{1/q}/R$ is a Buchsbaum $A$-module of dimension $n$ for every $q>N$.
Now we apply Theorem \ref{FLCandd-sequence} $(1)\Rightarrow(4)$ for $(\underline{x})=(x_1,\dots,x_n)$, we have \begin{equation} \label{3.5.3} l(R/(\underline{x}))-e(\underline{x}, R)\leq \sum_{i=1}^{n-1}\binom{n-1}{i}l(H_\mathfrak{m}^i(R)) \end{equation} Since $R^{1/q}$ and $R^{1/q}/R$ are Buchsbaum modules over $A$, we know from Bemerkung (4.2) in \cite{SchenzelTrungVerallgemeinerteCohenMacaulayModuln} that \begin{equation} \label{3.5.4} l(R^{1/q}/(\underline{x})R^{1/q})-e(\underline{x}, R^{1/q})=\sum_{i=1}^{n-1}\binom{n-1}{i}l(H_\mathfrak{m}^i(R^{1/q})) \end{equation} and \begin{equation} \label{3.5.5} l(\displaystyle\frac{R^{1/q}/R}{(\underline{x})(R^{1/q}/R)})-e(\underline{x}, R^{1/q}/R)=\sum_{i=1}^{n-1}\binom{n-1}{i}l(H_\mathfrak{m}^i(R^{1/q}/R)) \end{equation} where the length $l(-)$ and multiplicity $e(-)$ are considered as length and multiplicity computed over $A$, which are the same as the length and multiplicity computed over $R$ since $R$ is module finite over $A$ with the same residue field. Now we consider (\ref{3.5.3})-(\ref{3.5.4})+(\ref{3.5.5}). The left hand side is just \[l(R/(\underline{x}))-l(R^{1/q}/(\underline{x})R^{1/q})+l(\displaystyle\frac{R^{1/q}/R}{(\underline{x})(R^{1/q}/R)})\] because the multiplicities cancel. The right hand side is zero because of (\ref{short exact sequence of local cohomology}). Hence we know that \begin{equation} \label{3.5.6} l(R/(\underline{x}))+l(\displaystyle\frac{R^{1/q}/R}{(\underline{x})(R^{1/q}/R)})\leq l(R^{1/q}/(\underline{x})R^{1/q}). \end{equation}
On the other hand, we can also apply $\otimes_RR/(\underline{x})$ to the short exact sequence \[0\to R\to R^{1/q}\to R^{1/q}/R\to 0,\] and we get \[\Tor_1^R(R/(\underline{x}), R^{1/q}/R)\xrightarrow{\varphi} R/(\underline{x})\rightarrow R^{1/q}/(\underline{x})R^{1/q}\rightarrow \displaystyle\frac{R^{1/q}/R}{(\underline{x})(R^{1/q}/R)}\rightarrow 0.\] So (\ref{3.5.6}) implies that $\varphi$ must be the zero map. Hence for every $q>N$, we have an injection $0\to R/(\underline{x})\rightarrow R^{1/q}/(\underline{x})R^{1/q}$. But this map is the same as the Frobenius map: $0\to R/(\underline{x})\rightarrow R/(\underline{x^q})$. Now if $y\in (x_1,\dots,x_n)^F$, then $y^q\in (x_1^q,\dots,x_n^q)$ for some $q>N$, so $\overline{y}$ maps to $0$ under $0\to R/(\underline{x})\rightarrow R/(\underline{x^q})$, hence $y\in(x_1,\dots,x_n)$. This proves that every ideal generated by a system of parameters is Frobenius closed.
Now we prove $(2)\Rightarrow(1)$. Since every ideal generated by a system of parameters is Frobenius closed, by Proposition \ref{s.o.p Frobenius closed implies Buchsbaum} we know that $R$ is Buchsbaum. But since $R$ is Buchsbaum, we know that every system of parameters is a {\it standard} system of parameters in the sense of Schenzel \cite{Schenzelstandardsystemsofparemeters} (see Corollary 3.6 in \cite{Schenzelstandardsystemsofparemeters}). Hence by Proposition 3.3 in \cite{Schenzelstandardsystemsofparemeters}, we know that for every ideal $I=(x_1,\dots,x_n)$ generated by a system of parameters in $R$ and every $0\leq i\leq n-1$, there are natural isomorphisms: \begin{equation} \label{3.6.7} H_\mathfrak{m}^i(R)\cong \displaystyle\frac{(x_1,\dots,x_i):I}{(x_1,\dots,x_i)+\sum_{j=1}^i(x_1,\dots,\widehat{x_j},\dots,x_i):I} \end{equation} Now we observe that one can view the Frobenius map on $H_\mathfrak{m}^i(R)$ as the natural map $H_\mathfrak{m}^i(R)\to H_\mathfrak{m}^i(R^{1/p})$ and then identify $R^{1/p}$ with $R$. It is straightforward to check that under (\ref{3.6.7}), the Frobenius action on $H_\mathfrak{m}^i(R)$ is the same as the map \[\frac{(x_1,\dots,x_i):I}{(x_1,\dots,x_i)+\sum_{j=1}^i(x_1,\dots,\widehat{x_j},\dots,x_i):I}\to \frac{(x_1^p,\dots,x_i^p):I^{[p]}}{(x_1^p,\dots,x_i^p)+\sum_{j=1}^i(x_1^p,\dots,\widehat{x_j^p},\dots,x_i^p):I^{[p]}}\] sending $\overline{y}$ to $\overline{y^p}$.
So in order to show that Frobenius acts injectively on $H_\mathfrak{m}^i(R)$ for $1\leq i\leq n-1$, it suffices to show that if $y^p\in (x_1^p,\dots,x_i^p)+\sum_{j=1}^i(x_1^p,\dots,\widehat{x_j^p},\dots,x_i^p):I^{[p]}$, then $y\in (x_1,\dots,x_i)+\sum_{j=1}^i(x_1,\dots,\widehat{x_j},\dots,x_i):I$. But since $R$ is Buchsbaum, we know that $(x_1,\dots,x_i):I=(x_1,\dots,x_i):\mathfrak{m}$ is the unmixed component of $(x_1,\dots,x_i)$ in its primary decomposition (we refer to \cite{GotoontheassociatedgradedringsofparameteridealsinBuchsbaumrings}, page 502-503 for a more detailed explanation of this). Now by Theorem 4.7 in \cite{GotoontheassociatedgradedringsofparameteridealsinBuchsbaumrings}, for every $1\leq i\leq n-1$ and every fixed $k\geq 2$, we have \begin{equation} \label{3.6.8} (x_1,\dots,x_i)+\sum_{j=1}^i(x_1,\dots,\widehat{x_j},\dots,x_i):I=(x_1^k,\dots,x_i^k):(x_1x_2\cdots x_i)^{k-1}. \end{equation} So for every $1\leq i\leq n-1$, we have \begin{eqnarray*} &&y^p\in (x_1^p,\dots,x_i^p)+\sum_{j=1}^i(x_1^p,\dots,\widehat{x_j^p},\dots,x_i^p):I^{[p]}\\ &\Rightarrow&y^p(x_1^px_2^p\cdots x_i^p)^{k-1}\in(x_1^{pk},\dots,x_i^{pk})\\ &\Rightarrow&y(x_1x_2\cdots x_i)^{k-1}\in(x_1^k,\dots,x_i^k)\\ &\Rightarrow&y\in (x_1^k,\dots,x_i^k):(x_1x_2\cdots x_i)^{k-1}\\ &\Rightarrow&y\in (x_1,\dots,x_i)+\sum_{j=1}^i(x_1,\dots,\widehat{x_j},\dots,x_i):I\\ \end{eqnarray*} where the last implication is by (\ref{3.6.8}), and the second implication we use the fact that $(x_1^k,\dots,x_i^k)$ is Frobenius closed (by Lemma \ref{lemma on ideals generated by part of a system of parameters}). Hence considering (\ref{3.6.7}), we have already showed that Frobenius acts injectively on each $H_\mathfrak{m}^i(R)$ when $1\leq i\leq n-1$.
It remains to show that the Frobenius acts injectively on $H_\mathfrak{m}^0(R)$ and $H_\mathfrak{m}^n(R)$. But since every ideal generated by a system of parameters is Frobenius closed, we know that $R$ is reduced by the same argument as in the proof of Lemma \ref{lemma on ideals generated by part of a system of parameters}. So we know that $\depth R\geq 1$ and hence $H_\mathfrak{m}^0(R)=0$. Furthermore Frobenius acts injectively on $H_\mathfrak{m}^n(R)$ by Lemma \ref{lemma on Frobenius action on top local cohomology}. This completes the proof of $(2)\Rightarrow(1)$. \end{proof}
Finally we can give a positive answer to Question \ref{Takagi's question}.
\begin{corollary} \label{main corollary 2} Let $(R,\mathfrak{m})$ be a local ring of equal characteristic $p>0$. Suppose $R$ is $F$-injective and $R$ has finite local cohomology. Then $R$ is Buchsbaum. \end{corollary} \begin{proof} This follows immediately from Proposition \ref{s.o.p Frobenius closed implies Buchsbaum} and Theorem \ref{main theorem 2}. \end{proof}
\begin{remark} It is quite natural to ask whether $F$-injectivity is always equivalent to the assertion that every ideal generated by a system of parameters is Frobenius closed. We don't have a counter example yet. \end{remark}
\section{Characteristic $0$ results} In this section we study Question \ref{Takagi's question in char 0}, and we provide a positive answer when $R$ is a section ring of a normal projective variety, hence in particular we answer this question when $R$ is normal and standard graded (we say $R$ is standard graded if $R_0=K$ and $R$ is generated over $R_0$ by $R_1$). Throughout this section all rings and schemes are of finite type over a field $K$ of characteristic $0$. All schemes are separated. We first recall the definition of (strong) log resolutions. Let $X$ be a closed subscheme of $Y$ with ideal sheaf $\mathscr{I}$. A morphism $\pi$: $\widetilde{Y}\rightarrow Y$ is called a {\it log resolution} of the pair $(Y, X)$ if \begin{enumerate} \item $\pi$ is proper and birational with $\widetilde{Y}$ smooth \item $\mathscr{I} O_{\widetilde{Y}}=O_{\widetilde{Y}}(-G)$ is an invertible sheaf corresponding to a divisor $-G$ \item $\Supp(G)\bigcup E$ has simple normal crossings where $E$ is the exceptional set of $\pi$. \end{enumerate} When $X=\emptyset$, we simply say $\pi$ is a log resolution of $Y$. We say $\pi$ is a {\it strong log resolution} of the pair $(Y, X)$ if moreover $\pi$ is an isomorphism outside of $X$. We note that when the characteristic of $K$ is $0$, log resolutions always exist and strong log resolutions exist if $Y$ is smooth (see \cite{HironakaResolution}).
We now recall Schwede's characterization of Du Bois singularities, which was shown to be equivalent to the classical definition using Hodge theoretic methods in \cite{SchwedeEasyCharacterization}. \begin{definition}[{\it cf.} Theorem 4.6 in \cite{SchwedeEasyCharacterization}] \label{Schwede's simple characterization of Du Bois} Let $X\hookrightarrow Y$ be a reduced closed subscheme of a smooth scheme $Y$. Let $\pi$: $\widetilde{Y}\rightarrow Y$ be a strong log resolution of $(Y, X)$ and let $E$ be the reduced pre-image of $X$ in $\widetilde{Y}$. Then $X$ has {\it Du Bois} singularities if and only if the natural map $O_X\rightarrow \mathbf{R}\pi_{*}O_E$ is a quasi-isomorphism. \end{definition}
Moreover, when $X$ is a Cohen-Macaulay normal scheme, there is another simple criterion for Du Bois singularities proved in \cite{KovacsSchwedeSmithLCImpliesDuBois} which is also useful. \begin{theorem}[{\it cf.} Theorem 3.1 in \cite{KovacsSchwedeSmithLCImpliesDuBois}] \label{canonicl sheaf of Du Bois singularity} Suppose that $X$ is normal and Cohen-Macaulay. Let $\pi$: $X'\rightarrow X$ be any log resolution of $X$, and denote the reduced exceptional divisor of $\pi$ by $G$. Then $X$ has Du Bois singularities if and only if $\pi_*\omega_{X'}(G)\cong \omega_X$. \end{theorem}
Now we are ready to prove our main result in characteristic $0$. It follows from some more general results. The first one is a Kodaira vanishing result for normal Cohen-Macaulay Du Bois singularities. This is well known to experts, and follows from a more general (and harder) result of Patakfalvi (Theorem 1.3 of \cite{PatakfalviSeminegativityDuBois}). But we also give a short, different proof for completeness. In fact the result follows easily from one of Fujino's vanishing theorems. \begin{theorem} \label{Kodaira vanishing for CM Du Bois} Let $X$ be a normal projective scheme which is Cohen-Macaulay and Du Bois. Then $H^i(X, \omega_X\otimes\mathscr{L})=0$ for every ample line bundle $\mathscr{L}$ and every $i>0$. \end{theorem} \begin{proof} One of Fujino's vanishing results (for example, see Theorem 1.1 in \cite{FujinoInjectivityvanishingandtorsionfreetheorems}) says that if $f$: $Y\rightarrow Z$ is projective with $Y$ smooth, and $B$ is an effective $\mathbb{Q}$-divisor with coefficients $\leq 1$ with simple normal crossing support, then we have \[H^p(Z, R^qf_*O_Y(K_Y+B+H))=0\] for every $p>0$, $q\geq 0$ if $H=f^*H'$ for some ample $H'$.
Take a log resolution $\pi$: $X'\rightarrow X$ of $X$ with reduced exceptional divisor $G$. In particular $G$ has coefficient $\leq 1$ with simple normal crossing support. Now we apply the above vanishing result to $Y=X'$, $Z=X$, $B=G$, $H=\pi^*\mathscr{L}$, $p=i>0$ and $q=0$. We get \begin{equation}\label{3} H^i(X, \pi_*(O_{X'}(K_{X'}+G)\otimes \pi^*\mathscr{L}))=0. \end{equation} Since we know that $\pi_*\omega_{X'}(G)\cong\omega_X$ when $X$ is normal Cohen-Macaulay and Du Bois by Theorem \ref{canonicl sheaf of Du Bois singularity}. Applying the projection formula to (\ref{3}) we get $H^i(X, \omega_X\otimes\mathscr{L})=0.$ \end{proof}
Another key ingredient is the following result which gives a characterization of Du Bois singularities for section rings of ample line bundles. The proof makes use of the ``natural" construction from \cite{EGA}. \begin{theorem} \label{characterization of Du Bois in graded case} Let $Z$ be a normal projective variety over $K$ and let $\mathscr{L}$ be an ample line bundle on $Z$. Let $R=\oplus_{i\in\mathbb{N}} H^0(Z,\mathscr{L}^i)$ be the section ring of $Z$ with respect to $\mathscr{L}$ and $\mathfrak{m}$ be the irrelevant maximal ideal of $R$. Then the following are equivalent: \begin{enumerate} \item $R$ is Du Bois. \item $Z$ has Du Bois singularities and $[H_\mathfrak{m}^i(R)]_{>0}=0$ for every $i\geq0$. \end{enumerate} \end{theorem} \begin{proof} If $R$ is Du Bois, so is $R_P$ for all homogeneous primes $P$. So both $(1)$ and $(2)$ imply $Z$ has Du Bois singularities. So without loss of generality we assume $Z$ is Du Bois. Since $\mathscr{L}$ is ample, we know that $R$ is a finitely generated $K$-algebra. We pick homogeneous elements $x_1,\dots,x_m$ in $R$ that form a set of algebra generators of $R$ over $K$. Let $x_j$ have degree $d_j>0$. We have a natural degree-preserving map $S=K[x_1,\dots,x_m]\twoheadrightarrow R$ where $S$ is the polynomial ring with a possibly non-standard grading. Let $X=\Spec R$, $Y=\Spec S$, we have $X\hookrightarrow Y\cong\mathbb{A}^m$.
Now we use $R^\natural$ and $S^\natural$ to denote the Rees algebra of $R$ and $S$ with respect to the natural filtration $R_{\geq t}$ and $S_{\geq t}$. That is, \[R^\natural=R\oplus R_{\geq 1}\oplus R_{\geq 2}\oplus\cdots,\] \[S^{\natural}=S\oplus S_{\geq 1}\oplus S_{\geq 2}\oplus\cdots.\] Let $\widetilde{X}=\Proj R^\natural$ and $\widetilde{Y}=\Proj S^\natural$. We have natural maps $\widetilde{X}\to X$ and $\widetilde{Y}\to Y$ induced by the inclusion $R\hookrightarrow R^\natural$ and $S\hookrightarrow S^\natural$. We note that when $x_1,\dots,x_m$ all have degree 1, $\widetilde{X}$ and $\widetilde{Y}$ are just the blow ups of $X$ and $Y$ at the homogeneous maximal ideals.
It is straightforward to check that the reduced pre-image of $X$ in $\widetilde{Y}$ is $\overline{X}=\widetilde{X}\bigcup E$ where $E\cong\Proj(S/S_{\geq 1}\oplus S_{\geq 1}/S_{\geq2}\oplus\cdots)\cong \Proj S$ is a weighted projective space. It is also clear that $\widetilde{X}\bigcap E\cong \Proj R=Z$.
We further let $Y_0\xrightarrow{f}\widetilde{Y}$ be a strong log resolution of the pair $(\widetilde{Y}, \overline{X})$ and we use $X_0$ to denote the reduced pre-image of $\overline{X}$ in $Y_0$. We summarize all the above information in the following diagram: \[ \xymatrix{
X_0 \ar@{^{(}->}[r] \ar[d]_{f} & Y_0 \ar[d]_{f} \ar@/^1pc/[dd]^{\pi} \\
\widetilde{X}\bigcup E=\overline{X}\ar@{^{(}->}[r] \ar[d]_g & \widetilde{Y} \ar[d]_g \\
X\ar@{^{(}->}[r] & Y } \]
First notice that we can also interpret $\widetilde{X}=\Proj R^\natural$ as the total space of the tautological line bundle $\mathscr{L}^{-1}$ on $Z$ (see Section 8.7.3 in \cite{EGA}). Hence $\widetilde{X}$ has Du Bois singularities since $Z$ has Du Bois singularities by assumption. Also notice that $E$ is isomorphic to a weighted projective space, so it has rational singularities and hence is Du Bois. Now since $\widetilde{X}$, $E$, and $\widetilde{X}\bigcap E\cong Z$ are all Du Bois, it follows that $\overline{X}$ is Du Bois because Du Bois singularities glue well (for example, see 3.8, 4.10 in \cite{DuBoisMain} or Lemma 3.4 in \cite{SchwedeEasyCharacterization}).
Second notice that we have a spectral sequence \[R^pg_*R^qf_*O_{X_0}\Rightarrow R^{p+q}\pi_*O_{X_0}.\] Since $Y_0\rightarrow \widetilde{Y}$ is a strong log resolution of $(\widetilde{Y}, \overline{X})$ and $\overline{X}$ is Du Bois, we know that $f_*O_{X_0}=O_{\overline{X}}$ and $R^{q}f_*O_{X_0}=0$ for $q>0$ by Definition \ref{Schwede's simple characterization of Du Bois}. So the above spectral sequence degenerates. We have \begin{equation}\label{1} R^i\pi_*O_{X_0}\cong R^ig_*O_{\overline{X}}. \end{equation}
Now we compute $R^ig_*O_{\overline{X}}$. Since $X$ is affine, this is just $H^i(\overline{X}, O_{\overline{X}})$. Since $\overline{X}=\widetilde{X}\bigcup E$, $Z\cong \widetilde{X}\bigcap E$, we have an exact sequence \[0\rightarrow O_{\overline{X}}\rightarrow O_{\widetilde{X}}\oplus O_E\rightarrow O_Z\rightarrow 0.\] This induces a long exact sequence on cohomology: \begin{equation} \label{7}\cdots \rightarrow H^i(\overline{X}, O_{\overline{X}})\rightarrow H^i(\widetilde{X}, O_{\widetilde{X}})\oplus H^i(E, O_E)\rightarrow H^i(Z,O_Z)\rightarrow \cdots. \end{equation} We know that $H^i(E, O_E)=0$ for every $i\geq 1$ because $E$ is a weighted projective space. We also know $H^i(Z, O_Z)\cong [H^{i+1}_\mathfrak{m}(R)]_0$ for every $i\geq 1$ because $Z\cong \Proj R$. Now we use \v{C}ech complex to understand the map $H^i(\widetilde{X}, O_{\widetilde{X}})\rightarrow H^i(Z,O_Z)$. Recall that $x_1,\dots,x_m$ are homogeneous algebra generators of $R$ over $K$ of degree $d_1,\dots,d_m$. The natural map $O_{\widetilde{X}}\rightarrow O_Z$ induces a map between the $s$-th spot of the \v{C}ech complexes of $O_{\widetilde{X}}$ and $O_Z$ with respect to the affine cover $\{D_+(x_i)\}_{1\leq i\leq m}$. This induced map on \v{C}ech complexes can be explicitly described as follows (all the direct sum in the following diagram is taking over all $s$-tuples $1\leq i_1<\cdots <i_s\leq m$): \[ \xymatrix{ \oplus O_{\widetilde{X}}(D_+(x_{i_1}x_{i_2}\cdots x_{i_s})) \ar[r] \ar[d]^{\cong} & \oplus O_Z(D_+(x_{i_1}x_{i_2}\cdots x_{i_s}))\ar[d]^{\cong} \\
\displaystyle\oplus\{\frac{y}{(x_{i_1}x_{i_2}\cdots x_{i_s})^n}| n\geq 0, y\in R_{\geq nd}, d=d_{i_1}+\cdots+ d_{i_s}\} \ar[r] \ar[d]^{\cong} & \oplus[R_{x_{i_1}x_{i_2}\cdots x_{i_s}}]_0 \ar[d]^\cong \\ \oplus[R_{x_{i_1}x_{i_2}\cdots x_{i_s}}]_{\geq 0} \ar[r]^\phi & \oplus[R_{x_{i_1}x_{i_2}\cdots x_{i_s}}]_0 } \]
It is straightforward to check that the induced map on the second line takes the element $\displaystyle\frac{y}{(x_{i_1}x_{i_2}\cdots x_{i_s})^n}$ to $\displaystyle\frac{\overline{y}}{(x_{i_1}x_{i_2}\cdots x_{i_s})^n}$ where $\overline{y}$ denotes the image of $y$ in $R_{\geq nd}/R_{\geq nd+1}$. Hence the same map $\phi$ on the third line is exactly ``taking the degree $0$ part". By the \v{C}ech complex computation of sheaf cohomology, we know that $H^i(\widetilde{X}, O_{\widetilde{X}})\cong [H^{i+1}_\mathfrak{m}(R)]_{\geq 0}$ and the map $H^i(\widetilde{X}, O_{\widetilde{X}})\rightarrow H^i(Z, O_Z)$ is exactly the natural map $[H^{i+1}_\mathfrak{m}(R)]_{\geq 0}\rightarrow [H^{i+1}_\mathfrak{m}(R)]_{0}$ for every $i\geq 1$. Hence the above long exact sequence (\ref{7}) gives (for every $i\geq 1$) \begin{equation} \label{2} H^i(\overline{X}, O_{\overline{X}})=[H_\mathfrak{m}^{i+1}(R)]_{>0}. \end{equation} From (\ref{1}) and (\ref{2}) it is clear that, for every $i\geq 1$, we have \begin{equation}\label{4} R^i\pi_*O_{X_0}\cong[H_\mathfrak{m}^{i+1}(R)]_{>0}. \end{equation}
However, it is straightforward to check that $Y_0\xrightarrow{\pi}Y$ is also a strong log resolution of the pair $(Y, X)$ and $X_0$ is the reduced pre-image of $X$ in $Y_0$. Hence by Definition \ref{Schwede's simple characterization of Du Bois}, we know that $X$ has Du Bois singularities if and only if $\pi_*O_{X_0}=O_X$ and $R^i\pi_*O_{X_0}=0$ for $i\geq 1$. But since $R$ is a section ring of a normal variety, $R$ is normal. So $\pi_*O_{X_0}=O_X$ is always true by Corollary 5.7 in \cite{SchwedeFInjectiveAreDuBois}. So by (\ref{4}), $X=\Spec R$ has Du Bois singularities if and only if $[H_\mathfrak{m}^i(R)]_{>0}=0$ for $i\geq 0$ (note that $H_\mathfrak{m}^0(R)$ and $H_\mathfrak{m}^1(R)$ vanish because $R$ is normal). \end{proof}
We state and prove our main theorem in characteristic $0$.
\begin{theorem} Let $Z$ be a normal projective variety over $K$ and let $\mathscr{L}$ be an ample line bundle on $Z$. Let $R=\oplus H^0(Z,\mathscr{L}^i)$ be the section ring of $Z$ with respect to $\mathscr{L}$ and $\mathfrak{m}$ be the irrelevant maximal ideal of $R$. If $Z$ is Cohen-Macaulay and $R$ has Du Bois singularities, then $R$ is Buchsbaum.
In particular, when $(R,\mathfrak{m})$ is a normal standard graded $K$-algebra, if $R$ is Du Bois and $R$ has finite local cohomology, then $R$ is Buchsbaum. \end{theorem} \begin{proof} The last assertion is clear because every normal standard graded $K$-algebra is the section ring of a normal projective variety $Z\cong\Proj R$ with respect to some very ample line bundle $\mathscr{L}$, and the fact that $R$ has finite local cohomology implies $R$ is Cohen-Macaulay on the punctured spectrum, so it implies $Z$ is Cohen-Macaulay. Therefore it suffices to prove the first assertion.
Following Theorem \ref{sufficient condision for Buchsbaum ring in the graded case}, we will show that $[H_\mathfrak{m}^i(R)]_s=0$ for every $s\neq 0$ and $i<n=\dim R$. Since $R$ is normal, $H_\mathfrak{m}^i(R)=0$ for $i=0,1$ so there's nothing prove in these cases. Now for $2\leq i<n$, we will show $[H_\mathfrak{m}^i(R)]_s=0$ for $s>0$ and $s<0$.
For $s>0$, this follows immediately from Theorem \ref{characterization of Du Bois in graded case} since $R$ has Du Bois singularities. For $s<0$, we want to show $[H_\mathfrak{m}^{i+1}(R)]_s=H^i(Z,\mathscr{L}^s)=0$ for every $i<n-1$. Since $Z=\Proj R$ is Cohen-Macaulay and Du Bois of dimension $n-1$, by Serre duality, \[H^i(Z, \mathscr{L}^s)\cong H^{n-1-i}(Z, \omega_Z\otimes\mathscr{L}^{-s})=0\] where the last equality is by Theorem \ref{Kodaira vanishing for CM Du Bois} since $n-1-i>0$. \end{proof}
In \cite{SchwedeFInjectiveAreDuBois}, the following conjecture was made: \begin{conjecture}[{\it cf.} Question 8.1 in \cite{SchwedeFInjectiveAreDuBois} or Conjecture 4.1 in \cite{BhattSchwedeTakagiweakordinaryconjectureandFsingularity}] \label{conjecture DB=F-injectivetype} Let $X$ be a reduced scheme of finite type over an algebraically closed field of characteristic 0. Then $X$ has Du Bois singularities if and only if $X$ is of dense $F$-injective type. \end{conjecture}
We note that the ``if" direction of the above conjecture is true by the main result in \cite{SchwedeFInjectiveAreDuBois}. And by a recent result of Bhatt, Schwede and Takagi \cite{BhattSchwedeTakagiweakordinaryconjectureandFsingularity}, this conjecture is equivalent to the weak ordinarity conjecture of Musta\c{t}\u{a} and Srinivas \cite{MustataSrinivasweakordinaryconjecture}. The point we want to make here is that under mild conditions on $R$, a positive answer to Conjecture \ref{conjecture DB=F-injectivetype} will give a positive answer to Question \ref{Takagi's question in char 0} by the standard method of reduction mod $p$.
\begin{proposition} Suppose Conjecture \ref{conjecture DB=F-injectivetype} holds. Let $R$ be a ring finitely generated over an algebraically closed field $K$ of characteristic $0$. Suppose $R$ is equidimensional and $\mathfrak{m}$ is the only non-Cohen-Macaulay (closed) point of $\Spec R$. If $R$ is Du Bois, then $R_\mathfrak{m}$ is Buchsbaum. \end{proposition} \begin{proof} Let $R$ be generated over $K$ by $z_1,\dots,z_k$, i.e., $R=K[z_1,\dots,z_k]/J$. Since $R$ is an affine algebra over an algebraically closed field $K$, without loss of generality we may assume $\mathfrak{m}=(z_1,\dots,z_k)$ is the isolated non-Cohen-Macaulay point. If $R_\mathfrak{m}$ is not Buchsbaum, then there exists system of parameters $\underline{x}=x_1,\dots,x_n$, $\underline{y}=y_1,\dots,y_n$ of $R_\mathfrak{m}$ such that \[l(R_\mathfrak{m}/(\underline{x}))-e(\underline{x}, R_\mathfrak{m})\neq l(R_\mathfrak{m}/(\underline{y}))-e(\underline{x}, R_\mathfrak{m}).\] And we can certainly assume $\underline{x}$, $\underline{y}$ are actually elements of $R$. We note that $l(R_\mathfrak{m}/(\underline{x}))-e(\underline{x}, R_\mathfrak{m})=\chi_1(\underline{x}, R_\mathfrak{m})=\sum_{i=1}^n (-1)^il(H_i(\underline{x}, R)_\mathfrak{m})$. Obviously, the support of all $H_i(\underline{x}, R)$, $H_i(\underline{y}, R)$ consist only of finitely many maximal ideals $\mathfrak{m}=\mathfrak{m}_0, \mathfrak{m}_1,\dots,\mathfrak{m}_t$ of $R$. Now we pick $f\notin \mathfrak{m}$ but $f\in \mathfrak{m}_i$ for every $i\geq 1$ and we localize at $f$. Notice that all the hypothesis on $R$ are unchanged. But now each $H_i(\underline{x}, R)$, $H_i(\underline{y}, R)$ is only supported at $\mathfrak{m}$. In particular we know that \begin{equation*} \sum_{i=1}^n(-1)^i l(H_i(\underline{x}, R))=l(R_\mathfrak{m}/(\underline{x}))-e(\underline{x}, R_\mathfrak{m})\neq l(R_\mathfrak{m}/(\underline{y}))-e(\underline{x}, R_\mathfrak{m})= \sum_{i=1}^n (-1)^il(H_i(\underline{y}, R)). \end{equation*} Now we take a finitely generated $\mathbb{Z}$-algebra $A\subseteq K$ such that $R$ is well-defined over $A$ and let $R_A$ be the corresponding subring of $R$ so that $R=R_A\otimes_AK$. By generic freeness, we can shrink $A$ and make all the kernels, cokernels and homologies in the Koszul complex \[0\rightarrow R_A\to R_A^n\to\cdots\to R_A^n\xrightarrow{\underline{x}} R_A\to 0\] to be free as $A$-modules (and similar to $\underline{y}$). In particular, we have $H_i(\underline{x}, R)=H_i(\underline{x}, R_A)\otimes_AK$, $H_i(\underline{y}, R)=H_i(\underline{y}, R_A)\otimes_AK$ and all $H_i(\underline{x}, R_A)$, $H_i(\underline{y}, R_A)$ are free $A$-modules. In particular, we know that $l(H_i(\underline{x}, R))=\rank_A(H_i(\underline{x}, R_A))$ and $l(H_i(\underline{y}, R))=\rank_A(H_i(\underline{y}, R_A))$. Hence we have \begin{eqnarray*} &&\sum_{i=1}^n(-1)^i\rank_A(H_i(\underline{x}, R_A))=\sum_{i=1}^n(-1)^i l(H_i(\underline{x}, R))\\ &\neq& \sum_{i=1}^n(-1)^i l(H_i(\underline{y}, R))=\sum_{i=1}^n(-1)^i\rank_A(H_i(\underline{y}, R_A)). \end{eqnarray*} Now we pass to closed point of $A$, i.e., tensoring with $A/s$ for $s$ a maximal ideal of $A$. Let $R_s=R_A\otimes_AA/s$ and let $\mathfrak{m}_s$ be the maximal ideal of $R_s$ corresponds to $\mathfrak{m}$. Notice that when we pass to $R_s$, the above will give us \[\sum_{i=1}^n(-1)^i l(H_i(\underline{x}, R_s))\neq \sum_{i=1}^n(-1)^i l(H_i(\underline{y}, R_s))\] because $H_i(\underline{x}, R_A)$ and $H_i(\underline{y}, R_A)$ are free over $A$. In particular, this is saying that $(R_s)_{\mathfrak{m}_s}$ is {\it not} Buchsbaum.
On the other hand, we can shrink $A$ to make $A$ regular and each $R_A[\frac{1}{z_i}]$ free over $A$ by generic freeness. Let $L$ be the fraction field of $A$. We know that \[L\otimes_AR_A[\frac{1}{z_i}]\to K\otimes_AR_A[\frac{1}{z_i}]=R[\frac{1}{z_i}]\] is faithfully flat with $0$-dimensional fiber. Since $(z_1,\dots,z_k)$ is the isolated non-Cohen-Macaulay point, each $R[\frac{1}{z_i}]$ is Cohen-Macaulay. It follows that each $L\otimes_AR_A[\frac{1}{z_i}]$ is Cohen-Macaulay. So we can further shrink $A$ such that each $R_A[\frac{1}{z_i}]$ is Cohen-Macaulay. Now since $A$ is regular and $A\to R_A[\frac{1}{z_i}]$ is faithfully flat, it is easy to see that after tensoring with $A/s$ for each maximal ideal $s$ of $A$, the resulting $R_s[\frac{1}{z_i}]$ is still Cohen-Macaulay for each $i$. Hence we may assume that after we pass to each closed point of $A$ (i.e., after we do the mod $p$ reduction), each $R_s$ still has an isolated non-Cohen-Macaulay point $\mathfrak{m}_s$. And a similar argument shows that we may also assume each $R_s$ is equidimensional. Now if Conjecture \ref{conjecture DB=F-injectivetype} is true, then there should be a dense set of $s\in \Spec A$ such that $R_s$ is $F$-injective, equidimensional with an isolated non-Cohen-Macaulay point $\mathfrak{m}_s$. So by Corollary \ref{main corollary 2}, $(R_s)_{\mathfrak{m}_s}$ is Buchsbaum for a dense set of $s$, this is a contradiction. \end{proof}
\end{document} | arXiv | {
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\begin{document}
\markboth{Nilanjan De, Sk. Md. Abu Nayeem, Anita Pal} {F-index of some graph operations}
\title{F-Index of Some Graph Operations}
\author{Nilanjan De}
\address{Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata, India.} \email{de.nilanjan@rediffmail.com}
\author{Sk. Md. Abu Nayeem}
\address{Department of Mathematics, Aliah University, IIA/27, New Town, Kolkata - 700 156, India.} \email{nayeem.math@aliah.ac.in}
\author{Anita Pal}
\address{Department of Mathematics, National Institute of Technology, Durgapur, India.} \email{anita.buie@gmail.com}
\maketitle
\begin{abstract} The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total $\pi$-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [B. Furtula, I. Gutman, A forgotten topological index, \textit{J. Math. Chem.}, \textbf{53(4)}(2015) 1184--1190.] reinvestigated the index and named it ``forgotten topological index" or ``F-index". In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nano-structures.\\[5pt] \textit{Keywords}: Topological index; vertex degree; first and second Zagreb indices; F index; graph operations.\\[5pt] \textit{Mathematics Subject Classification}: Primary: 05C35; Secondary: 05C07, 05C40 \end{abstract}
\section{Introduction}
Suppose $G$ is a simple connected graph and $V(G)$ and $E(G)$ denote the vertex set and edge set of $G$, respectively. For any vertex ${v}\in V(G)$, let ${{d}_{G}}(v)$ denote its degree, that is the number of neighbors of $v$ and $N(v)$ denote the set of vertices which are the neighbors of the vertex $v$, so that $|N(v)|={{d}_{G}}(v)$. In chemistry, biochemistry and nanotechnology different topological indices are found to be useful in isomer discrimination, structure-property relationship, structure-activity relationship and pharmaceutical drug design. The first and second Zagreb indices of a graph $G$, denoted by $M_1(G)$ and $M_2(G)$, are among the oldest, most popular and extremely studied vertex-degree based topological indices and are respectively defined as \[{{M}_{1}}(G)=\sum\limits_{v\in V(G)}{{{d}_{G}}{{(v)}^{2}}}=\sum\limits_{uv\in E(G)}{[{{d}_{G}}(u)+{{d}_{G}}(v)]}\] and \[{{M}_{2}}(G)=\sum\limits_{uv\in E(G)}{{{d}_{G}}(u){{d}_{G}}(v)}.\]
These indices were introduced in a paper in 1972 \cite{gutm72} to study the structure-dependency of the total $\pi$-electron energy ($\varepsilon$). It was found that the $\varepsilon$ depends on $M_1(G)$ and thus provides a measure of carbon skeleton of the underlying molecules. In the same paper, another topological index, defined as sum of cubes of degrees of the vertices of the graph was also shown to influence $\varepsilon$. However this index was not further studied till then, except in a recent article by Furtula and Gutman \cite{fur15} where they reinvestigated this index and studied some basic properties of this index. They showed that the predictive ability of this index is almost similar to that of first Zagreb index and for the entropy and acentric factor, both of them yield correlation coefficients greater than 0.95. They named this index as ``forgotten topological index" or ``F-index". Throughout the present paper we name this index as F-index and denote it by $F(G)$, so that \[F(G)=\sum\limits_{v\in V(G)}{{{d}_{G}}{{(v)}^{3}}}=\sum\limits_{uv\in E(G)}{[{{d}_{G}}{{(u)}^{2}}+{{d}_{G}}{{(v)}^{2}}]}.\]
As we know that some chemically interesting graphs can be obtained by different graph operations on some general or particular graphs, it is important to study such graph operations in order to understand how it is related to the corresponding topological indices of the original graphs. In \cite{kha09}, Khalifeh et al. derived some exact formulae for computing first and second Zagreb indices under some graph operations. In \cite{das13}, Das et al. derived some upper bounds for multiplicative Zagreb indices for different graph operations. In \cite{aza14}, Azari presented some lower bounds for Narumi-Katayama index under several graph operations. In \cite{de14}, the present authors computed some bounds and exact formulae of the connective eccentric index under different graph operations. There are several other results regarding various topological indices under different graph operations are available in the literature. In \cite{aza13}, Azari and Iranmanesh presented explicit formulas for computing the eccentric-distance sum of different graph operations. Interested readers are referred to \cite{Ash10,kha08,tava14,veyl15,esk13,aza13a,aza15,nd15} in this regard.
In this paper, we present some exact expressions for the F-index of different graph operations such as union, join, Cartesian product, composition, tensor product, strong product, corona product, generalized hierarchical product, disjunction, symmetric difference, splice and link of graphs. Also we apply our results to compute the F-index for some important classes of molecular graphs and nano-structures.
\section{Main Results and Discussions} In this section, we study F-index of various graph operations like union, join, Cartesian product, composition, tensor product, strong product, corona product, generalized hierarchical product, disjunction, symmetric difference, link and splice of graphs. These operations are binary and all the graphs are connected, finite and simple. In the following, if not indicated otherwise, we use the notation $V(G_i)$ for the vertex set, ${{E}(G_i)}$ for the edge set, ${{n}_{i}}$ for the number of vertices and ${{m}_{i}}$ for the number of edges of the graph ${{G}_{i}}$, $i\in \left\{ 1,2,\ldots,k \right\}$, respectively. Throughout the paper, we use the familiar notations $P_n$, $C_n$ and $K_n$ to denote a path graph, cycle graph and complete graph with $n$ number of vertices, respectively.
\subsection{Union} Let $G_1$, $G_2$,...,$G_k$ be $k$ graphs with disjoint vertex sets. Then their union ${{G}_{1}}\cup {{G}_{2}}\cup ...\cup {{G}_{k}}$ is the graph with vertex set ${V({G}_{1})}\cup {V({G}_{2})}\cup ...\cup {V({G}_{k})}$ and the edge set ${E({G}_{1})}\cup {E({G}_{2})}\cup ...\cup {E({G}_{k})}$. The degree of a vertex $v$ of ${{G}_{1}}\cup {{G}_{2}}\cup ...\cup {{G}_{k}}$ is equal to the degree of the vertex $v$ in the component $G_i$, $i=1,2,...,k$, that contains it. In the following theorem we obtain the F-index of the union of $k$ number of graphs. \begin{thm}The F-index of ${{G}_{1}}\cup {{G}_{2}}\cup ...\cup {{G}_{k}}$ is given by \[F({{G}_{1}}\cup {{G}_{2}}\cup ...\cup {{G}_{k}})=F({{G}_{1}})+F({{G}_{2}})+...+F({{G}_{k}}).\] \end{thm} \noindent\textit{Proof.} By definition of F-index we have \begin{eqnarray*} F({{G}_{1}}\cup {{G}_{2}}\cup ...\cup {{G}_{k}})&=&\sum\limits_{v\in V({{G}_{1}}\cup {{G}_{2}}\cup ...\cup {{G}_{k}})}{({{d}_{{{G}_{1}}\cup {{G}_{2}}\cup ...\cup {{G}_{k}}}}(v)})^3\\ &=&\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}{{(v)}^{3}}}+\sum\limits_{v\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}{{(v)}^{3}}}+...+\sum\limits_{v\in V({{G}_{k}})}{{{d}_{{{G}_{k}}}}{{(v)}^{3}}}\\ &=&F({{G}_{1}})+F({{G}_{2}})+...+F({{G}_{k}}), \end{eqnarray*} which completes the proof. \qed
\subsection{Join} The join ${{G}_{1}}+{{G}_{2}}$ of two graphs $G_1$ and $G_2$ is the union ${{G}_{1}}\cup{{G}_{2}}$ together with all the edges joining ${V({G}_{1})}$ and ${V({G}_{2})}$. The degree of a vertex $v$ of ${{G}_{1}}+{{G}_{2}}$ is \[{{d}_{{{G}_{1}}+{{G}_{2}}}}(v) = \left\{ \begin{array}{ll} {{d}_{{{G}_{1}}}}(v)+{{n}_{2}},v\in V({{G}_{1}})\\[2mm] {{d}_{{{G}_{2}}}}(v)+{{n}_{1}},v\in V({{G}_{2}}). \end{array}\right.\] In general, for $k$ graphs $G_1$, $G_2$,...,$G_k$, the degree of a vertex $v$ in ${{G}_{1}}+{{G}_{2}}+...+{{G}_{k}}$ is given by ${d}_{{{G}_{1}}+{{G}_{2}}+...+{G}_{k}}(v)$ $= {{d}_{{{G}_{i}}}}(v)+{n}-{{n}_{i}}$, where $v$ is originally a vertex of the graph $G_i$ and $n={{n}_{1}}+{{n}_{2}}+...+{{n}_{k}}$.
In the following theorem we compute the F-index of the join of $k$ number of graphs. \begin{thm} The F-index of ${{G}_{1}}+{{G}_{2}}+...+{{G}_{k}}$ is given by \[F({{G}_{1}}+{{G}_{2}}+...+{{G}_{k}})=\sum\limits_{i=1}^{k}{F({{G}_{i}})}+3\sum\limits_{i=1}^{k}{{{{\bar{n}}}_{i}}{{M}_{1}}({{G}_{i}})}+6\sum\limits_{i=1}^{k}{{{{\bar{n}}}_{i}}^{2}{{m}_{i}}}+\sum\limits_{i=1}^{k}{{{n}_{i}}{{{\bar{n}}}_{i}}^{3}},\] where ${{\bar{n}}_{i}}=n-{{n}_{i}}, i=1,2,...,k$ and $n={{n}_{1}}+{{n}_{2}}+...+{{n}_{k}}$. \end{thm}
\noindent\textit{Proof.} We have \begin{eqnarray*} F({{G}_{1}}+{{G}_{2}}+...+{{G}_{k}})&=&\sum\limits_{i=1}^{k}{\sum\limits_{v\in V({{G}_{i}})}{{{({{d}_{{{G}_{i}}}}(v)+{{{\bar{n}}}_{i}})}^{3}}}}\\
&=&\sum\limits_{i=1}^{k}{\sum\limits_{v\in V({{G}_{i}})}{({{d}_{{{G}_{i}}}}{{(v)}^{3}}+3{{{\bar{n}}}_{i}}{{d}_{{{G}_{i}}}}{{(v)}^{2}}+3{{{\bar{n}}}_{i}}^{2}{{d}_{{{G}_{i}}}}(v)}+{{{\bar{n}}}_{i}}^{3}})\\
&=&\sum\limits_{i=1}^{k}{\sum\limits_{v\in V({{G}_{i}})}{{{d}_{{{G}_{i}}}}{{(v)}^{3}}}}+3\sum\limits_{i=1}^{k}{{{{\bar{n}}}_{i}}\sum\limits_{v\in V({{G}_{i}})}{{{d}_{{{G}_{i}}}}{{(v)}^{2}}}}\\
&&+3\sum\limits_{i=1}^{k}{{{{\bar{n}}}_{i}}^{2}\sum\limits_{v\in V({{G}_{i}})}{{{d}_{{{G}_{i}}}}(v)}}+\sum\limits_{i=1}^{k}{{{n}_{i}}{{{\bar{n}}}_{i}}^{3}}, \end{eqnarray*} which completes the proof. \qed
Let ${{G}_{1}}={{G}_{2}}=...={{G}_{p}}=G$ and $pG$ denote the join of $p$ copies of $G$. Then the following corollaries follow as direct consequence of the previous theorem. \begin{cor} Let, $n$ and $m$ be the number of vertices and edges of $G$, respectively. Then \[F(pG)=pF(G)+3np(p-1){{M}_{1}}(G)+6{{n}^{2}}mp{{(p-1)}^{2}}+{{n}^{4}}p{{(p-1)}^{3}}.\] \end{cor}
\begin{cor} The F-index of ${{G}_{1}}+{{G}_{2}}$ is given by \[F({{G}_{1}}+{{G}_{2}})=F({{G}_{1}})+F({{G}_{2}})+3{{n}_{2}}{{M}_{1}}({{G}_{1}})+3{{n}_{1}}{{M}_{1}}({{G}_{2}})+6{{n}_{2}}^{2}{{m}_{1}}+6{{n}_{1}}^{2}{{m}_{2}}+{{n}_{1}}{{n}_{2}}^{3}+{{n}_{2}}{{n}_{1}}^{3}.\] \end{cor} The suspension of a graph $G$ is defined as ${{K}_{1}}+G$. So from the Corollary 2 the following result follows. \begin{cor} Let, $n$ and $m$ be the number of vertices and edges of $G$, respectively. Then the F-index of suspension of $G$ is given by \[F({{K}_{1}}+G)=F(G)+3{{M}_{1}}(G)+{{n}^{3}}+6m+n.\] \end{cor}
\begin{ex} The complete $n$-partite graph ${{K}_{{{m}_{1}},{{m}_{2}},...,{{m}_{n}}}}$ (Fig.1) on ${{m}_{1}}+{{m}_{2}}+...+{{m}_{n}}$ vertices can be considered as ${{\bar{K}}_{{{m}_{1}}}}+{{\bar{K}}_{{{m}_{2}}}}+...+{{\bar{K}}_{{{m}_{n}}}}$. Then the F-index of ${{K}_{{{m}_{1}},{{m}_{2}},...,{{m}_{n}}}}$ is given by \[F({{K}_{{{m}_{1}},{{m}_{2}},...,{{m}_{n}}}})=\sum\limits_{i=1}^{n}{{{m}_{i}}{{{\bar{m}}}_{i}}^{3}},\] where ${{\bar{m}}_{i}}=({{m}_{1}}+{{m}_{2}}+...+{{m}_{n}})-{{m}_{i}}, i=1,2,...,n$. \end{ex}
\begin{figure}
\caption{The complete $n$-partite graph.}
\label{f3}
\end{figure}
\begin{ex} The wheel graph ${{W}_{n}}$ on $(n+1)$ vertices is the suspension of ${{C}_{n}}$ and the fan graph ${{F}_{n}}$ on $(n+1)$ vertices is the suspension of ${{P}_{n}}$. So their F-indices are given by
(i) $F({{W}_{n}})={{n}^{3}}+27n$,
(ii) $F({{F}_{n}})={{n}^{3}}+27n-38$. \end{ex}
\begin{ex} The dutch windmill graph or flower graph is the suspension of $m$ copies of $K_2$, denoted by $m{K_2}$. So its F-index is given by $F({K_1}+m{K_2})=8m^3+16m$. \end{ex} \begin{ex} The cone graph ${C}_{m,n}$ is defined as ${C_m}+\bar{K}_n$. So its F-index is calculated as $F({C}_{m,n})=m{n^3}+{m^3}n+6m{n^2}+12mn+8m$. \end{ex}
\subsection{Cartesian Product}
The Cartesian product of $G_1$ and $G_2$, denoted by $G_1\otimes G_2$, is the graph with vertex set $V(G_1)\times V(G_2)$ and any two vertices $({{u}_{p}},{{v}_{r}})$ and $({{u}_{q}},{{v}_{s}})$ are adjacent if and only if [${{u}_{p}}={{u}_{q}}$ and ${{v}_{r}}{{v}_{s}}\in E(G_2)$] or [${{v}_{r}}={{v}_{s}}$ and ${{u}_{p}}{{u}_{q}}\in E(G_1)$]. In the following theorem first we find the first Zagreb index of the Cartesian product of $k$ number of graphs.
\begin{thm}Let $n$ be the total number of vertices in $\bigotimes_{i=1}^{k} G_i$, then the first Zagreb index of ${{G}_{1}}\otimes {{G}_{2}}\otimes ...\otimes {{G}_{k}}$ is given by \[{M_1}\left(\bigotimes_{i=1}^k G_i\right)= {n}\sum\limits_{i=1}^{k}{\frac{{{M}_{1}}({{G}_{i}})}{{{n}_{i}}}+}4{n}\sum\limits_{\substack{i,j=1\\i\ne j}}^{k}{\frac{{{m}_{i}}}{{{n}_{i}}}\cdot\frac{{{m}_{j}}}{{{n}_{j}}}}.\] \end{thm} \noindent\textit{Proof.} For the proof of this theorem we refer to Theorem 1 of \cite{kha09}.\qed
In the following theorem we obtain the F-index of the Cartesian product of $k$ number of graphs. \begin{thm} The F-index of ${{G}_{1}}\otimes {{G}_{2}}\otimes ...\otimes {{G}_{k}}$ is given by \[F({{G}_{1}}\otimes {{G}_{2}}\otimes ...\otimes {{G}_{k}})=n\sum\limits_{i=1}^{k}{\frac{F({{G}_{i}})}{{{n}_{i}}}}+6n\sum\limits_{\substack{i,j=1\\i\ne j}}^{k}{\frac{{{M}_{1}}({{G}_{i}})}{{{n}_{i}}}\cdot\frac{{{m}_{j}}}{{{n}_{j}}}}+8n\sum\limits_{\substack{p,q,r=1\\p\ne q\ne r}}^{k}{\frac{{{m}_{p}}}{{{n}_{p}}}\cdot\frac{{{m}_{q}}}{{{n}_{q}}}}\cdot\frac{{{m}_{r}}}{{{n}_{r}}}.\] \end{thm}
\noindent\textit{Proof.} First we prove the result for $k=2$.
We have, ${{d}_{{{G}_{1}}\otimes {{G}_{2}}}}(a,b)={{d}_{{{G}_{1}}}}(a)+{{d}_{{{G}_{2}}}}(b)$. So, from definition of F-index we have \begin{eqnarray*} F({{G}_{1}}\otimes {{G}_{2}})&=&{{\sum\limits_{a\in V({{G}_{1}})}{\sum\limits_{b\in V({{G}_{2}})}{({{d}_{{{G}_{1}}}}(a)+{{d}_{{{G}_{2}}}}(b))}}}^{3}}\\ &=&\sum\limits_{b\in V({{G}_{2}})}{\sum\limits_{a\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}{{(a)}^{3}}}}+\sum\limits_{a\in V({{G}_{1}})}{\sum\limits_{b\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}{{(b)}^{3}}}}\\ &&+3\sum\limits_{a\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}{{(a)}^{2}}\sum\limits_{b\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}(b)+}}3\sum\limits_{a\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}(a)\sum\limits_{b\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}{{(b)}^{2}}}}\\ &=&{{n}_{2}}F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+6{{m}_{2}}{{M}_{1}}({{G}_{1}})+6{{m}_{1}}{{M}_{1}}({{G}_{2}}). \end{eqnarray*}
Let $n'$ be the total number of vertices in $\bigotimes_{i=1}^{k-1} G_i$. Then by an inductive argument, using Theorem 3, we have \begin{eqnarray*} \displaystyle F\left(\bigotimes_{i=1}^k G_i\right)&=&F\left(\bigotimes_{i=1}^{k-1} G_i\bigotimes G_k\right)\\ &=&n_kF\left(\bigotimes_{i=1}^{k-1} G_i\right)+n'F(G_k)+6m_kM_1\left(\bigotimes_{i=1}^{k-1} G_i\right)+6n'\sum_{i=1}^{k-1}\frac{m_i}{n_i}M_1(G_k)\\ &=&{{n}_{k}}\left[ {n}'\sum\limits_{i=1}^{k-1}{\frac{F({{G}_{i}})}{{{n}_{i}}}+}6{n}'\sum\limits_{\substack{i,j=1\\i\ne j}}^{k-1}{\frac{{{M}_{1}}({{G}_{i}})}{{{n}_{i}}}\cdot\frac{{{m}_{j}}}{{{n}_{j}}}+8{n}'\sum\limits_{\substack{p,q,r=1\\p\ne q\ne r}}^{k-1}{\frac{{{m}_{p}}}{{{n}_{p}}}\cdot\frac{{{m}_{q}}}{{{n}_{q}}}\cdot\frac{{{m}_{r}}}{{{n}_{r}}}}} \right]\\ &&+{n}'F({{G}_{k}})+6{{m}_{k}}\left[ {n}'\sum\limits_{i=1}^{k-1}{\frac{{{M}_{1}}({{G}_{i}})}{{{n}_{i}}}+}4{n}'\sum\limits_{\substack{i,j=1\\i\ne j}}^{k-1}{\frac{{{m}_{i}}}{{{n}_{i}}}\cdot\frac{{{m}_{j}}}{{{n}_{j}}}} \right]+6{n}'{{M}_{1}}({{G}_{k}})\sum\limits_{i=1}^{k-1}{\frac{{{m}_{i}}}{{{n}_{i}}}}\\ &=&\left[ n\sum\limits_{i=1}^{k-1}{\frac{F({{G}_{i}})}{{{n}_{i}}}+}n\frac{F({{G}_{k}})}{{{n}_{k}}} \right]+\left[ 6n\sum\limits_{\substack{i,j=1\\i\ne j}}^{k-1}\frac{{{M}_{1}}({{G}_{i}})}{{{n}_{i}}}\cdot\frac{{{m}_{j}}}{{{n}_{j}}}+6n\frac{{{m}_{k}}}{{{n}_{k}}}\sum\limits_{i=1}^{k-1}{\frac{{{M}_{1}}({{G}_{i}})}{{{n}_{i}}}}\right.\\ &&\left.+6n\frac{{{M}_{1}}({{G}_{k}})}{{{n}_{k}}}\sum\limits_{i=1}^{k-1}{\frac{{{m}_{i}}}{{{n}_{i}}}} \right]+\left[ 8n\sum\limits_{\substack{p,q,r=1\\p\ne q\ne r}}^{k-1}{\frac{{{m}_{p}}}{{{n}_{p}}}\cdot\frac{{{m}_{q}}}{{{n}_{q}}}\cdot\frac{{{m}_{r}}}{{{n}_{r}}}}+24n\frac{{{m}_{k}}}{{{n}_{k}}}\sum\limits_{\substack{i,j=1\\i\ne j}}^{k-1}{\frac{{{m}_{i}}}{{{n}_{i}}}\cdot\frac{{{m}_{j}}}{{{n}_{j}}}} \right], \end{eqnarray*}
from where the desired result follows. \qed
\begin{cor} The F-index of of the torus ${{C}_{{{n}_{1}}}}\otimes {{C}_{{{n}_{2}}}}\otimes...\otimes {{C}_{{{n}_{k}}}}$ is given by
$F({{C}_{{{n}_{1}}}}\otimes {{C}_{{{n}_{2}}}}\otimes ...\otimes {{C}_{{{n}_{k}}}})=8{k^3}{{n}_{1}}{{n}_{2}}\ldots{{n}_{k}}.$
\end{cor}
\begin{cor} The F-index of the Hamming graph ${{K}_{{{n}_{1}}}}\otimes {{K}_{{{n}_{2}}}}\otimes ...\otimes {{K}_{{{n}_{k}}}}$ is given by \begin{eqnarray*} F({{K}_{{{n}_{1}}}}\otimes {{K}_{{{n}_{2}}}}\otimes ...\otimes {{K}_{{{n}_{k}}}})&=&{{n}_{1}}{{n}_{2}}\ldots{{n}_{k}}(\sum\limits_{i=1}^{k}{{{({{n}_{i}}-1)}^{3}}}+3\sum\limits_{\substack{i,j=1\\i\ne j}}^{k}{{{({{n}_{i}}-1)}^{2}}({{n}_{j}}-1)}\\ &&+\sum\limits_{\substack{p,q,r=1\\p\ne q\ne r}}^{k}{({{n}_{p}}-1)({{n}_{q}}-1)({{n}_{r}}-1)})\\ &=&{({n_1}+{n_2}+...+{n_k}-k)^3}{n_1}{n_2}...{n_k}. \end{eqnarray*}
\end{cor} \begin{ex} For $k$-dimensional hypercube ${{Q}_{k}}={{K}_{2}}\otimes {{K}_{2}}\otimes ...\otimes {{K}_{2}}$ (k times), by our calculation, we have $F({{Q}_{k}})={{2}^{k}}{{k}^{3}}$. \end{ex} \begin{ex} The F-index of ${{K}_{{{n}_{1}}}}\otimes {{K}_{{{n}_{2}}}}$ torus is given by \[F({{K}_{{{n}_{1}}}}\otimes {{K}_{{{n}_{2}}}})={{n}_{1}}{{n}_{2}}{({n_1}+{n_2}-2 )^3}\] \end{ex}
\begin{ex} Let $R$ and $S$ denote a $C_4$ nanotube and nanotorus, respectively. Then $R\cong{P_n}\otimes{C_m}$ and $S\cong{C_n}\otimes{C_m}$, for some integers $n$ and $m$. Then by our calculation, $F(R)=64{m}{n}-74{m}$ and $F(S)=64{m}{n}$. Also if, $T\cong{P_n}\otimes{P_m}$, then $F(T)=64{m}{n}-74{m}-74{n}+72$. \end{ex} \subsection{Composition}
The composition or lexicographic product of two graphs ${{G}_{1}}$ and ${{G}_{2}}$ is denoted by ${{G}_{1}}[{{G}_{2}}]$. The vertex set of ${{G}_{1}}[{{G}_{2}}]$ is $V({{G}_{1}})\times V({{G}_{2}})$ and the degree of a vertex $(a,b)$ of ${{G}_{1}}[{{G}_{2}}]$ is given by ${{d}_{{{G}_{1}}[{{G}_{2}}]}}(a,b)={{n}_{2}}{{d}_{{{G}_{1}}}}(a)+{{d}_{{{G}_{2}}}}(b)$ and any two vertices $({{u}_{1}},{{u}_{2}})$ and $({{v}_{1}},{{v}_{2}})$ are adjacent if and only if ${{u}_{1}}{{v}_{1}}\in E({{G}_{1}})$ or [${{u}_{1}}={{v}_{1}}$ and ${{u}_{2}}{{v}_{2}}\in E({{G}_{2}})$]. In the following theorem we compute the F-index of the composition of two graphs. \begin{thm} The F-index of ${{G}_{1}}[{{G}_{2}}]$ is given by \begin{eqnarray*} F({{G}_{1}}[{{G}_{2}}])={{n}_{2}}^{4}F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+6{{n}_{2}}^{2}{{m}_{2}}{{M}_{1}}({{G}_{1}})+6{{n}_{2}}{{m}_{1}}{{M}_{1}}({{G}_{2}}). \end{eqnarray*} \end{thm}
\noindent\textit{Proof.} From definition of F-index, we have \begin{eqnarray*} F({{G}_{1}}[{{G}_{2}}])&=&\sum\limits_{(a,b)\in V({{G}_{1}}[{{G}_{2}}])}{{{d}_{{{G}_{1}}[{{G}_{2}}]}}}{{(a,b)}^{3}}\\
&=&\sum\limits_{a\in V({{G}_{1}})}{\sum\limits_{b\in V({{G}_{2}})}{{{[{{n}_{2}}{{d}_{{{G}_{1}}}}(a)+{{d}_{{{G}_{2}}}}(b)]}^{3}}}}\\
&=&{{n}_{2}}^{3}\sum\limits_{b\in V({{G}_{2}})}{\sum\limits_{a\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}{{(a)}^{3}}}} +\sum\limits_{a\in V({{G}_{1}})}{\sum\limits_{b\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}{{(b)}^{3}}}}\\
&&+3{{n}_{2}}^{2}\sum\limits_{a\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}{{(a)}^{2}}\sum\limits_{b\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}(b)+}}3{{n}_{2}}\sum\limits_{a\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}(a)\sum\limits_{b\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}{{(b)}^{2}}}}\\
&=&{{n}_{2}}^{4}F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+6{{n}_{2}}^{2}{{m}_{2}}{{M}_{1}}({{G}_{1}})+6{{n}_{2}}{{m}_{1}}{{M}_{1}}({{G}_{2}}),
\end{eqnarray*} which completes proof. \qed
\begin{ex} The fence graph is the composition of ${{P}_{n}}$ and ${{P}_{2}}$ and the closed fence graph is the composition of ${{C}_{n}}$ and ${{P}_{2}}$. So from the previous theorem, we have
(i) $F({{P}_{n}}[{{P}_{2}}])=250n-392$,
(ii) $F({{C}_{n}}[{{P}_{2}}])=250n.$ \end{ex} \subsection{Tensor Product}
The tensor product or Kronecker product of two graphs ${{G}_{1}}$ and ${{G}_{2}}$ is denoted by ${{G}_{1}}\times {{G}_{2}}$ and any two vertices $({{u}_{1}},{{v}_{1}})$ and $({{u}_{2}},{{v}_{2}})$ are adjacent if and only if ${{u}_{1}}{{u}_{2}}\in E({{G}_{1}})$ and ${{v}_{1}}{{v}_{2}}\in E({{G}_{2}})$. The degree of a vertex $(a,b)$ of ${{G}_{1}}\times {{G}_{2}}$ is given by ${{d}_{{{G}_{1}}\times {{G}_{2}}}}(a,b)={{d}_{{{G}_{1}}}}(a){{d}_{{{G}_{2}}}}(b)$. In the following theorem, the F-index of the tensor product of two graphs is computed. \begin{thm}
The F-index of ${{G}_{1}}\times {{G}_{2}}$ is given by $F({{G}_{1}}\times {{G}_{2}})=F({{G}_{1}})F({{G}_{2}}).$ \end{thm}
\noindent\textit{Proof.} From definition of F-index, we have
\[F({{G}_{1}}\times {{G}_{2}})=\sum\limits_{(a,b)\in V({{G}_{1}}\times {{G}_{2}})}{{{d}_{{{G}_{1}}\times {{G}_{2}}}}}{{(a,b)}^{3}}=\sum\limits_{a\in V({{G}_{1}})}{\sum\limits_{b\in V({{G}_{2}})}{{{[{{d}_{{{G}_{1}}}}(a){{d}_{{{G}_{2}}}}(b)]}^{3}}}}=F({{G}_{1}})F({{G}_{2}}).\] \qed
\begin{ex} (i) $F({{P}_{n}}\times {{P}_{m}})=(8n-14)(8m-14)$,
(ii) $F({{C}_{n}}\times {{C}_{m}})=64nm$,
(iii) $F({{K}_{n}}\times {{K}_{m}})=nm{{(n-1)}^{3}}{{(m-1)}^{3}}$,
(iv) $F({{P}_{n}}\times {{C}_{m}})=8m(8n-14)$,
(v) $F({{P}_{n}}\times {{K}_{m}})=m(8n-14){{(m-1)}^{3}}$,
(vi) $F({{C}_{n}}\times {{K}_{m}})=8nm{{(m-1)}^{3}}$. \end{ex}
\subsection{Strong Product} The strong product of two graphs $G_1$ and $G_2$ is denoted by ${{G}_{1}}\boxtimes {{G}_{2}}$. It has the vertex set $V(G_1)\times V(G_2)$ and any two vertices $({{u}_{p}},{{v}_{r}})$ and $({{u}_{q}},{{v}_{s}})$ are adjacent if and only if [${{u}_{p}}={{u}_{q}}$ and ${{v}_{r}}{{v}_{s}}\in E(G_2)$] or [${{v}_{r}}={{v}_{s}}$ and ${{u}_{p}}{{u}_{q}}\in E(G_1)$] or [${{u}_{p}}{{u}_{q}}\in E(G_1)$ and ${{v}_{r}}{{v}_{s}}\in E(G_2)$]. Note that if both $G_1$ and $G_2$ are connected then ${{G}_{1}}\boxtimes {{G}_{2}}$ is also connected. The degree of a vertex $(a,b)$ of ${{G}_{1}}\boxtimes {{G}_{2}}$ is given by \[{{d}_{{{G}_{1}}\boxtimes {{G}_{2}}}}(a,b)={{d}_{{{G}_{1}}}}(a)+{{d}_{{{G}_{2}}}}(b)+{{d}_{{{G}_{1}}}}(a){{d}_{{{G}_{2}}}}(b).\] In the following theorem we compute the F-index of the strong product of two graphs. \begin{thm}
The F-index of ${{G}_{1}}\boxtimes {{G}_{2}}$ is given by
\[F({{G}_{1}}\boxtimes {{G}_{2}})={{n}_{2}}F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+F({{G}_{1}})F({{G}_{2}})+6{{m}_{2}}{{M}_{1}}({{G}_{1}})+6{{m}_{1}}{{M}_{1}}({{G}_{2}})+6{{m}_{2}}F({{G}_{1}})\]
\[+6{{m}_{1}}F({{G}_{2}})+3F({{G}_{2}}){{M}_{1}}({{G}_{1}})+3F({{G}_{1}}){{M}_{1}}({{G}_{2}})+6{{M}_{1}}({{G}_{1}}){{M}_{1}}({{G}_{2}}).\] \end{thm} \noindent\textit{Proof.} From definition of F-index, we have \begin{eqnarray*} F({{G}_{1}}\boxtimes {{G}_{2}})&=&\sum\limits_{({{v}_{1}},{{v}_{2}})\in V({{G}_{1}}\otimes {{G}_{2}})}{{{d}_{{{G}_{1}}\otimes {{G}_{2}}}}}{{({{v}_{1}},{{v}_{2}})}^{3}}\\
&=&\sum\limits_{v_1\in V(G_1)}\sum\limits_{v_2\in V(G_2)}[d_{G_1}(v_1)+d_{G_2}(v_2)+d_{G_1}(v_1)d_{G_2}(v_2)]^3\\
&=&\sum\limits_{v_1\in V(G_1)}\sum\limits_{v_2\in V(G_2)} [d_{G_1}(v_1)^3+d_{G_2} (v_2)^3+d_{G_1}(v_1)^3d_{G_2}(v_2)^3+3d_{G_1}(v_1)^2d_{G_2}(v_2)\\
&&+3d_{G_1}(v_1)d_{G_2}(v_2)^2+3d_{G_1}(v_1)^3d_{G_2}(v_2)+3d_{G_1}(v_1)^3d_{G_2}(v_2)^2\\
&&+3d_{G_2}(v_2)^3d_{G_1}(v_1)+3d_{G_1}(v_1)^2d_{G_2}(v_2)^3+6d_{G_1}(v_1)^2d_{G_2}(v_2)^2].
\end{eqnarray*} On simplification, we obtain the desired result.\qed
\subsection{Corona Product} The corona product ${{G}_{1}}\odot {{G}_{2}}$ of two graphs $G_1$ and $G_2$ is obtained by taking one copy of ${{G}_{1}}$ and ${{n}_{1}}$ copies of ${{G}_{2}}$ and by joining each vertex of the $i$-th copy of ${{G}_{2}}$ to the $i$-th vertex of ${{G}_{1}}$, where $1\le i\le {{n}_{1}}$. The corona product of ${{G}_{1}}$ and ${{G}_{2}}$ has total $({{n}_{1}}{{n}_{2}}+{{n}_{1}})$ number of vertices and $({{m}_{1}}+{{n}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}})$ number of edges. Clearly, the corona product of two graphs is not commutative. Different topological indices under the corona product of graphs have already been studied by some researchers \cite{pat14,yar12}. It is easy to see that the degree of a vertex $v$ of ${{G}_{1}}\odot {{G}_{2}}$ is given by \[{{d}_{{{G}_{1}}\odot{{G}_{2}}}}(v) = \left\{ \begin{array}{ll} {{d}_{{{G}_{1}}}}(v)+{{n}_{2}},v\in V({{G}_{1}})\\[2mm] {{d}_{{{G}_{2}}}}(v)+1,v\in V({{G}_{2,i}}), i=1,2,\ldots,{n_1}. \end{array}\right.\] where, ${{G}_{2,i}}$ is the i-th copy of the graph $G_2$. In the following theorem, the F-index of the corona Product of two graphs is computed. \begin{thm}
The F-index of ${{G}_{1}}\odot {{G}_{2}}$ is given by \[F({{G}_{1}}\odot {{G}_{2}})=F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+3{{n}_{2}}{{M}_{1}}({{G}_{1}})+3{{n}_{1}}{{M}_{1}}({{G}_{2}})+6{{n}_{2}}^{2}{{m}_{1}}+6{{n}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}({{n}_{2}}^{2}+1).\] \end{thm} \noindent\textit{Proof.} From definition of F-index, we have \begin{eqnarray*} F({{G}_{1}}\odot {{G}_{2}})&=&\sum\limits_{v\in V({{G}_{1}})}{{{({{d}_{{{G}_{1}}}}(v)+{{n}_{2}})}^{3}}}+{{n}_{1}}\sum\limits_{v\in V({{G}_{2}})}{{{({{d}_{{{G}_{2}}}}(v)+1)}^{3}}}\\ &=&\sum\limits_{v\in V({{G}_{1}})}{({{d}_{{{G}_{1}}}}{{(v)}^{3}}+3{{n}_{2}}{{d}_{{{G}_{1}}}}{{(v)}^{2}}+3{{n}_{2}}^{2}{{d}_{{{G}_{1}}}}(v)+{{n}_{2}}^{3})}\\ &&+{{n}_{1}}\sum\limits_{v\in V({{G}_{2}})}{({{d}_{{{G}_{2}}}}{{(v)}^{3}}+3{{d}_{{{G}_{2}}}}{{(v)}^{2}}+3{{d}_{{{G}_{2}}}}(v)+1)}\\ &=&\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}{{(v)}^{3}}}+3{{n}_{2}}\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}{{(v)}^{2}}}+3{{n}_{2}}^{2}\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}(v)}+{{n}_{1}}{{{n}_{2}}^3}\\ &&+{{n}_{1}}\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{2}}}}{{(v)}^{3}}}+3{{n}_{1}}\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{2}}}}{{(v)}^{2}}}+3{{n}_{1}}\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{2}}}}(v)}+{{n}_{1}}{{n}_{2}}\\ &=&F({{G}_{1}})+3{{n}_{2}}{{M}_{1}}({{G}_{1}})+6{{n}_{2}}^{2}{{m}_{1}}+{{n}_{1}}{{n}_{2}}^{3}+{{n}_{1}}F({{G}_{2}})+3{{n}_{1}}{{M}_{1}}({{G}_{2}})\\ &&+6{{n}_{1}}{{m}_{2}}+{{n}_{1}}{{{n}_{2}}^3}+{{n}_{1}}{{n}_{2}}. \end{eqnarray*}
\qed
The $t$-thorny graph $G^t$ of a given graph $G$ is obtained by joining $t$-number of thorns (pendent edges) to each vertex of $G$. A variety of topological indices of thorn graphs have been studied by a number of researchers \cite{ali14,de12,nd12,de3}.It is well known that, the $t$-thorny graph of $G$ is defined as the corona product of $G$ and complement of complete graph with $t$ vertices $\bar{K_t}$. Thus from the previous theorem the following corollary follows. \begin{cor} The F-index of the $t$-thorny graph is given by \[F({{G}^{t}})=F(G)+3t{{M}_{1}}(G)+6m{{t}^{2}}+n{{t}^{3}}+nt\] where, $n$ and $m$ are number of vertices and edges of $G$, respectively. \end{cor}
\begin{ex} The F-index of $t$-thorny graph of ${{C}_{n}}$ and ${{P}_{n}}$ are given by
(i) $F({{C}_{n}}^{t})=n{{t}^{3}}+6n{{t}^{2}}+13nt+8n$,
(ii) $F({{P}_{n}}^{t})=n{{t}^{3}}+6n{{t}^{2}}-6{{t}^{2}}+13nt-18t+8n-14$. \end{ex}
\begin{ex} One of the hydrogen suppressed molecular graph is the bottleneck graph $(B)$ of a graph $G$, which is defined as the corona product of ${K}_{2}$ and $G$, where $G$ is a given graph. The F-index of bottleneck graph of $G$ is given by $F(B)=2F(G)+6{{M}_{1}}(G)+2n^3+6n^2+8n+12m+2$, where $n$ and $m$ are the number of vertices and edges of $G$, respectively. \end{ex}
Next, as an application of corona product of graphs, we find the F index of some particular bridge graphs. Let $G_{1}, G_{2},...,G_{n}$ be a set of finite pairwise disjoint graphs. The bridge graph with respect to the vertices $v_{1}, v_{2},...,v_{n}$, denoted by $B(G_{1}, G_{2},...,G_{n};v_{1}, v_{2},...,v_{n})$ is the graph obtained by connecting the vertex $v_{i}$ of $G_{i}$ and the vertex $v_{i+1}$ of $G_{i+1}$ by an edge for all $i=1,2,...,n-1$. If $G_{1}\cong G_{2}\cong...\cong G_{n}$ and $v_{1}= v_{2}=...= v_{n}=v$, we define $G_{n}(G,v)=B(G, G,...,G;v, v,...,v)$. In particular, let $B_{n}=G_{n}(P_{3},v)$, where $v$ is the degree 2 vertex of $P_{3}$ and $T_{n,k}=G_{n}(C_{k},u)$ be special bridge graphs. Then from definition of corona product of graphs, $B_{n}\cong P_{n}\odot \overline{K}_{2}$ and $T_{n,3}\cong P_{n}\odot K_{2}$. Using the previous theorem the F index of these bridge graphs are calculated as follows.
\begin{ex}
(i) ${F(B_{m})}=66m-74$, for $m\geq2$
(ii) ${F(T_{m,3})}=80m-74$, for $m\geq2$
(iii) ${F(J_{n,m+1})}={n^3}m+6{n^2}m-6{n^2}+39nm+8m-18n-14$ , for $n\geq3$ and $m\geq2$. \end{ex}
\subsection{Generalized Hierarchical Product}
As an extension of Cartesian product of graphs, Barriere et. al. introduced the generalized hierarchical product of graphs in 2009 \cite{barr09}. Several results on different topological indices under generalized hierarchical product of graphs are already studied \cite{elia13,arez13,luo14,nd14a}. Let $G_1$ and $G_2$ be two connected graphs and $\phi \ne U\subseteq V(G_2)$. Then the hierarchical product of $G_1$ and $G_2$ denoted by $G_1\Pi G_2(U)$, is the graph with vertex set $V(G_1)\times V(G_2)$ and any two vertices $(u,v)$ and $({u}',{v}')$ of $G_1(U)\Pi G_2$ are adjacent if and only if [$u={u}'\in V(G_1)$ and $v{v}'\in E(G_2)$] or [$v={v}'\in U$ and $u{u}'\in E(G_1)$].
Clearly, the degree of a vertex $(u_1,u_2)$ of ${{G}_{1}}\Pi {{G}_{2}(U)}$ is given by \[{{d}_{G_1\Pi G_2(U)}}(u) = \left\{ \begin{array}{ll} {{d}_{{{G}_{1}}}}(u_1)+{{d}_{{{G}_{2}}}}(u_2),{u_2}\in U\\[2mm] {{d}_{{{G}_{2}}}}(u_2),{u_2}\in V({{G}_{2}})-U. \end{array}\right.\] In the following theorem we compute the F-index of the generalized hierarchical product of two graphs. \begin{thm} The F index of ${{G}_{1}}\Pi {{G}_{2}}(U)$ is given by
\[F({{G}_{1}}\Pi {{G}_{2}}(U))=|U|F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+3{{M}_{1}}({{G}_{1}})\sum\limits_{v\in U}{{{d}_{{{G}_{2}}}}(v)}+6{{m}_{1}}\sum\limits_{v\in U}{{{d}_{{{G}_{2}}}}{{(v)}^{2}}}.\] \end{thm}
\noindent\textit{Proof.} We have, from definition of F index \begin{eqnarray*} F({{G}_{1}}\Pi {{G}_{2}}(U))&=&\sum\limits_{({{v}_{1}},{{v}_{2}})\in V({{G}_{1}}\Pi {{G}_{2}}(U))}{{{d}_{{{G}_{1}}\Pi {{G}_{2}}(U)}}}{{({{v}_{1}},{{v}_{2}})}^{3}}\\ &=&\sum\limits_{{{v}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{v}_{2}}\in U}{{{[{{d}_{{{G}_{1}}}}({{v}_{1}})+{{d}_{{{G}_{2}}}}({{v}_{2}})]}^{3}}}}+\sum\limits_{{{v}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{v}_{2}}\in V({{G}_{2}})-U}{{{d}_{{{G}_{2}}}}{{({{v}_{2}})}^{3}}}}\\ &=&\sum\limits_{{{v}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{v}_{2}}\in U}{[{{d}_{{{G}_{1}}}}{{({{v}_{1}})}^{3}}+{{d}_{{{G}_{2}}}}{{({{v}_{2}})}^{3}}+3{{d}_{{{G}_{1}}}}{{({{v}_{1}})}^{2}}{{d}_{{{G}_{2}}}}({{v}_{2}})+3{{d}_{{{G}_{1}}}}({{v}_{1}}){{d}_{{{G}_{2}}}}{{({{v}_{2}})}^{2}}]}}\\ &&+\sum\limits_{{{v}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{v}_{2}}\in V({{G}_{2}})-U}{{{d}_{{{G}_{2}}}}{{({{v}_{2}})}^{3}}}}\\
&=&|U|F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+3{{M}_{1}}({{G}_{1}})\sum\limits_{v\in U}{{{d}_{{{G}_{2}}}}(v)}+6{{m}_{1}}\sum\limits_{v\in U}{{{d}_{{{G}_{2}}}}{{(v)}^{2}}},\end{eqnarray*} which completes the proof. \qed
From definition of the Cartesian product of graphs, it is clear that the Cartesian product of graphs is a special case of generalized hierarchical product of graphs, that is, if $U=V(G_2)$, then $G_1\Pi G_2(U)\cong G_1\otimes G_2$. So from the previous theorem we can also obtain the Theorem 3, for $k=2$.
The cluster product of two graphs $G_1$ and $G_2$, denoted by $G_1\left\{ G_2 \right\}$, is obtained by taking one copy of $G_1$ and $n_1$ copies of a rooted graph $G_2$ and by identifying the root of the $i$-th copy of $G_2$ with the $i$-th vertex of $G_1$, $i=1,2,...,n_1$. From definition of cluster product of graphs, $|V(G_1\left\{ G_2 \right\})|=n_1{n_2}$ and $|E(G_1\left\{ G_2 \right\})|=(m_2+n_1{m_2})$. Suppose the root vertex of $G_2$ is denoted by $x$. Note that, if $U=\left\{ x \right\}\subset V(G_1)$ then $G_1\left\{ G_2 \right\}\cong G_1\Pi G_2(U)\cong G_1\Pi G_2(\left\{ x \right\})$. Then from the previous theorem the following results follow.
\begin{cor} Let ${{G}_{1}}$ and ${{G}_{2}}$ be two connected graphs and $x$ be the root vertex of ${{G}_{2}}$, then \[F({{G}_{1}}\{{{G}_{2}}\})=F({{G}_{1}})+{{n}_{1}}F({{G}_{2}})+3{{M}_{1}}({{G}_{1}}){{d}_{{{G}_{2}}}}(x)+6{{m}_{1}}{{d}_{{{G}_{2}}}}{{(x)}^{2}}.\] \end{cor}
Note that, if $U=\left\{ x \right\}$, $x$ is the root vertex of the graph $G_2$, then ${{G}_{1}}\Pi {{G}_{2}}(U)={{G}_{1}}\Pi {{G}_{2}}$, the (standard) hierarchical product of two graphs. It is easy to see that ${{G}_{1}}\Pi {{G}_{2}}\cong{{G}_{1}}\{{{G}_{2}}\}.$ \begin{ex} The square comb lattice $Cq(N)$ with $N={{n}^{2}}$ vertices can be represented as the cluster product ${{P}_{n}}\{{{P}_{n}}\}$, where the root of ${{P}_{n}}$ is one of its vertices of degree 1. Then from the previous corollary, the F-index of $Cq(N)$ is given by $F(Cq(N))=8{{n}^{2}}+12n-38.$ \end{ex} \begin{ex} The sun graph is defined as ${Sun_{(m,n)}}={{C}_{m}}\{{{P}_{n+1}}\}$, such that ${P_{n+1}}$ is rooted at a vertex of degree one. Then the F-index of ${Sun_{(m,n)}}$ is given by ${Sun_{(m,n)}}=4m(2n+5)$. \end{ex}
\subsection{Disjunction} The disjunction of two graphs ${{G}_{1}}$ and ${{G}_{2}}$, denoted by ${{G}_{1}}\wedge{{G}_{2}}$, consists of the vertex set $V(G_1)\times V(G_2)$ and two vertices $(u_1,v_1)$ and $(u_2,v_2)$ are adjacent whenever ${u_1}{u_2}\in{E(G_1)}$ or ${v_1}{v_2}\in{E(G_2)}$. Clearly, the degree of a vertex $(u_1,u_2)$ of ${{G}_{1}}\wedge {{G}_{2}}$ is given by \[{{d}_{{{G}_{1}}\wedge {{G}_{2}}}}(u_1,u_2)={n_1}{{d}_{{{G}_{1}}}}(u_1)+{n_2}{{d}_{{{G}_{2}}}}(u_2)-{{d}_{{{G}_{1}}}}(u_1){{d}_{{{G}_{2}}}}(u_2).\] In the following theorem we obtain the F-index of the disjunction of two graphs.
\begin{thm} The F-index of ${{G}_{1}}\wedge {{G}_{2}}$ is given by \begin{eqnarray*} F({{G}_{1}}\wedge {{G}_{2}})&=&{{n}_{2}}^{4}F({{G}_{1}})+{{n}_{1}}^{4}F({{G}_{2}})-F({{G}_{1}})F({{G}_{2}})+6{{n}_{1}}{{n}_{2}}^{2}{{m}_{2}}{{M}_{1}}({{G}_{1}})+6{{n}_{1}}^{2}{{n}_{2}}{{m}_{1}}{{M}_{1}}({{G}_{2}})\\ &&+3{{n}_{2}}F({{G}_{1}}){{M}_{1}}({{G}_{2}})+3{{n}_{1}}F({{G}_{2}}){{M}_{1}}({{G}_{1}})-6{{n}_{2}}^{2}{{m}_{2}}F({{G}_{1}})-6{{n}_{1}}^{2}{{m}_{1}}F({{G}_{2}})\\ &&-6{{n}_{1}}{{n}_{2}}{{M}_{1}}({{G}_{1}}){{M}_{1}}({{G}_{2}}).\\ \end{eqnarray*} \end{thm}
\noindent\textit{Proof.} We have, from definition of F-index \begin{eqnarray*} F({{G}_{1}}\wedge {{G}_{2}})&=&\sum\limits_{({{u}_{1}},{{u}_{2}})\in V({{G}_{1}}\wedge {{G}_{2}})}{{{d}_{{{G}_{1}}\wedge{{G}_{2}}}}{{({{u}_{1}},{{u}_{2}})}^{3}}}\\ &=&\sum\limits_{{{u}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{u}_{2}}\in V({{G}_{2}})}{({{n}_{2}}{{d}_{{{G}_{1}}}}({{u}_{1}})+}}{{n}_{1}}{{d}_{{{G}_{2}}}}({{u}_{2}})-{{d}_{{{G}_{1}}}}({{u}_{1}}){{d}_{{{G}_{2}}}}({{u}_{2}}){{)}^{3}}\\ &=&\sum\limits_{{{u}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{u}_{2}}\in V({{G}_{2}})}{({{n}_{2}}^{3}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}+}}{{n}_{1}}^{3}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}-{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}\\ &&+3{{n}_{1}}{{n}_{2}}^{2}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{2}}{{d}_{{{G}_{2}}}}({{u}_{2}})+3{{n}_{1}}^{2}{{n}_{2}}{{d}_{{{G}_{1}}}}({{u}_{1}}){{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{2}}-3{{n}_{2}}^{2}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}{{d}_{{{G}_{2}}}}({{u}_{2}})\\ &&-3{{n}_{1}}^{2}{{d}_{{{G}_{1}}}}({{u}_{1}}){{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}+3{{n}_{2}}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{2}}+3{{n}_{1}}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{2}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}\\ &&-6{{n}_{1}}{{n}_{2}}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{2}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{2}}). \end{eqnarray*}
After simple calculations, we obtain the desired result.\qed
\subsection{Symmetric Difference}
The symmetric difference, of two graphs ${{G}_{1}}$ and ${{G}_{2}}$ is denoted by
${{G}_{1}}\oplus {{G}_{2}}$, so that $|V({{G}_{1}}\oplus {{G}_{2}})|=|V({{G}_{1}})|\times |V({{G}_{2}})|$ and
\[E({{G}_{1}}\oplus {{G}_{2}})=\{({{u}_{1}},{{u}_{2}})({{v}_{1}},{{v}_{2}}):{{u}_{1}}{{v}_{1}}\in E({{G}_{1}})\,\mbox{ or } {{u}_{2}}{{v}_{2}}\in E({{G}_{2}})\mbox{ but not both}\}.\] From definition of symmetric difference it is clear that \[{{d}_{{{G}_{1}}\oplus {{G}_{2}}}}({{v}_{1}},{{v}_{2}})={{n}_{2}}{{d}_{{{G}_{1}}}}({{v}_{1}})+{{n}_{1}}{{d}_{{{G}_{2}}}}({{v}_{2}})-2{{d}_{{{G}_{1}}}}({{v}_{1}}){{d}_{{{G}_{2}}}}({{v}_{2}}).\] In the following theorem we obtain the F-index of the symmetric difference of two graphs. \begin{thm} The F-index of ${{G}_{1}}\oplus {{G}_{2}}$ is given by \begin{eqnarray*} F({{G}_{1}}\oplus {{G}_{2}})&=&{{n}_{2}}^{4}F({{G}_{1}})+{{n}_{1}}^{4}F({{G}_{2}})-8F({{G}_{1}})F({{G}_{2}})+6{{n}_{1}}{{n}_{2}}^{2}{{m}_{2}}{{M}_{1}}({{G}_{1}})+6{{n}_{1}}^{2}{{n}_{2}}{{m}_{1}}{{M}_{1}}({{G}_{2}})\\ &&+12{{n}_{2}}F({{G}_{1}}){{M}_{1}}({{G}_{2}})+12{{n}_{1}}F({{G}_{2}}){{M}_{1}}({{G}_{1}})-12{{n}_{2}}^{2}{{m}_{2}}F({{G}_{1}})-12{{n}_{1}}^{2}{{m}_{1}}F({{G}_{2}})\\ &&-12{{n}_{1}}{{n}_{2}}{{M}_{1}}({{G}_{1}}){{M}_{1}}({{G}_{2}}). \end{eqnarray*} \end{thm} \noindent\textit{Proof.} By definition of F-index, we have \begin{eqnarray*} F({{G}_{1}}\oplus {{G}_{2}})&=&\sum\limits_{({{u}_{1}},{{u}_{2}})\in V({{G}_{1}}\wedge {{G}_{2}})}{{{d}_{{{G}_{1}}\oplus {{G}_{2}}}}{{({{u}_{1}},{{u}_{2}})}^{3}}}\\ &=&\sum\limits_{{{u}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{u}_{2}}\in V({{G}_{2}})}{({{n}_{2}}{{d}_{{{G}_{1}}}}({{u}_{1}})+}}{{n}_{1}}{{d}_{{{G}_{2}}}}({{u}_{2}})-2{{d}_{{{G}_{1}}}}({{u}_{1}}){{d}_{{{G}_{2}}}}({{u}_{2}}){{)}^{3}}\\ &=&\sum\limits_{{{u}_{1}}\in V({{G}_{1}})}{\sum\limits_{{{u}_{2}}\in V({{G}_{2}})}{({{n}_{2}}^{3}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}+}}{{n}_{1}}^{3}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}-8{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}\\ &&+3{{n}_{1}}{{n}_{2}}^{2}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{2}}{{d}_{{{G}_{2}}}}({{u}_{2}})+3{{n}_{1}}^{2}{{n}_{2}}{{d}_{{{G}_{1}}}}({{u}_{1}}){{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{2}}-6{{n}_{2}}^{2}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}{{d}_{{{G}_{2}}}}({{u}_{2}})\\ &&-6{{n}_{1}}^{2}{{d}_{{{G}_{1}}}}({{u}_{1}}){{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}+12{{n}_{2}}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{3}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{2}}+12{{n}_{1}}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{2}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{3}}\\ &&-12{{n}_{1}}{{n}_{2}}{{d}_{{{G}_{1}}}}{{({{u}_{1}})}^{2}}{{d}_{{{G}_{2}}}}{{({{u}_{2}})}^{2}}), \end{eqnarray*} from where we obtain the desired result.\qed
\subsection{Splice and Link of Graphs}
Let ${{v}_{1}}\in {{V}({G}_{1})}$ and ${{v}_{2}}\in {{V}({G}_{2})}$ be two given vertices of ${{G}_{1}}$ and ${{G}_{2}}$, respectively. A splice of ${{G}_{1}}$ and ${{G}_{2}}$ at the vertices ${{v}_{1}}$ and ${{v}_{2}}$, denoted by $({{G}_{1}}\bullet {{G}_{2}})({{v}_{1}},{{v}_{2}})$, was introduced by Dosli\'{c} \cite{dosl05}, and is obtained by identifying the vertices ${{v}_{1}}$ and ${{v}_{2}}$ in the union of ${{G}_{1}}$ and ${{G}_{2}}$. The vertex set of $({{G}_{1}}\bullet {{G}_{2}})({{v}_{1}},{{v}_{2}})$ is given by $V(({{G}_{1}}\bullet {{G}_{2}})({{v}_{1}},{{v}_{2}}))=[V({{G}_{1}})\backslash \{{{v}_{1}}\}]\cup [V({{G}_{2}})\backslash \{{{v}_{2}}\}]\cup \{{{v}_{12}}\}$, where we denote the vertex obtained by identifying ${{v}_{1}}$ and ${{v}_{2}}$ by ${{v}_{12}}$. From the construction of splice graphs it is clear that \[ {d_{({{G_1} \bullet {G_2}})({v_1},{v_2})}(v)} = \left\{ \begin{array}{ll} {{d_{{G_i}}}(v)},&\mbox{for } v \in V({G_i})\mbox{ and } v \ne {v_i},\\ {{d_{{G_1}}}({v_1}) + {d_{{G_2}}}({v_2})},&\mbox{for } v = {v_{12}}. \end{array} \right. \]
Similarly a link of ${{G}_{1}}$ and ${{G}_{2}}$ at the vertices ${{v}_{1}}$ and ${{v}_{2}}$ is denoted by $({{G}_{1}}\sim {{G}_{2}})({{v}_{1}},{{v}_{2}})$ and is obtained by joining the vertices ${{v}_{1}}$ and ${{v}_{2}}$ in the union of ${{G}_{1}}$ and ${{G}_{2}}$. From the construction of link graphs it is clear that \[d_{(G_1\sim G_2){(v_1,v_2)}}(v)=\left\{\begin{array}{ll}d_{G_i}(v),& v\in V(G_i)\mbox{ and } v\neq v_i, i=1,2,\\ d_{G_i}(v)+1,& v = v_i, i=1,2.\end{array}\right.\] In the following, we find the F-index of splice and link of two graphs ${{G}_{1}}$ and ${{G}_{2}}$ at the vertices $v_1$ and $z$.
\begin{thm}
The F-index of $({{G}_{1}}\bullet {{G}_{2}})(v_1,v_2)$ is given by \[F(({{G}_{1}}\bullet {\ }{{G}_{2}})(v_1,v_2))=F({{G}_{1}})+F({{G}_{2}})+3{{d}_{{{G}_{1}}}}(v_1){{d}_{{{G}_{2}}}}(v_2)({{d}_{{{G}_{1}}}}(v_1)+{{d}_{{{G}_{2}}}}(v_2)).\] \end{thm} \noindent\textit{Proof.} From the definition of F-index we have \begin{eqnarray*} F(({{G}_{1}}\bullet {{G}_{2}})(v_1,v_2))&=&{{\sum\limits_{v\in V({{G}_{1}}),v\ne v_1}{{{d}_{{{G}_{1}}}}(v)}}^{3}}+{{\sum\limits_{v\in V({{G}_{2}}),v\ne v_2}{{{d}_{{{G}_{2}}}}(v)}}^{3}}+{{({{d}_{{{G}_{1}}}}(v_1)+{{d}_{{{G}_{2}}}}(v_2))}^{3}}\\ &=&{{\sum\limits_{v\in V({{G}_{1}})}{{{d}_{{{G}_{1}}}}(v)}}^{3}}+{{\sum\limits_{v\in V({{G}_{2}})}{{{d}_{{{G}_{2}}}}(v)}}^{3}}+3{{d}_{{{G}_{1}}}}{{(v_1)}^{2}}{{d}_{{{G}_{2}}}}(v_2)+3{{d}_{{{G}_{1}}}}(v_1){{d}_{{{G}_{2}}}}{{(v_2)}^{2}}. \end{eqnarray*} From above we obtain the desired result after simple calculations.\qed
\begin{thm} The F-index of $({{G}_{1}}\sim{\ }{{G}_{2}})(v_1,v_2)$ is given by
\[F(({{G}_{1}}\sim{\ }{{G}_{2}})(v_1,v_2))= F({{G}_{1}})+F({{G}_{2}})+3({{d}_{{{G}_{1}}}}(v_1)+{{d}_{{{G}_{2}}}}(v_2))+3({{d}_{{{G}_{1}}}}{{(v_1)}^{2}}+{{d}_{{{G}_{2}}}}{{(v_2)}^{2}})+2.\] \end{thm} \noindent\textit{Proof.} From the definition of F-index, we have \begin{eqnarray*} F(({{G}_{1}}\sim {{G}_{2}})(v_1,v_2))&=&{{\sum\limits_{v\in V({{G}_{1}}),v\ne v_1}{{{d}_{{{G}_{1}}}}(v)}}^{3}}+{{\sum\limits_{v\in V({{G}_{2}}),v\ne v_2}{{{d}_{{{G}_{2}}}}(v)}}^{3}}+{{({{d}_{{{G}_{1}}}}(v_1)+1)}^{3}}+{{({{d}_{{{G}_{2}}}}(v_2)+1)}^{3}}\\ &=&F({{G}_{1}})+F({{G}_{2}})+3{{d}_{{{G}_{1}}}}(v_1)+3{{d}_{{{G}_{1}}}}{{(v_1)}^{2}}+1+3{{d}_{{{G}_{2}}}}(v_2)+3{{d}_{{{G}_{2}}}}{{(v_2)}^{2}}+1, \end{eqnarray*} from where the desired result follows. \qed
\section{Conclusion} In this paper, we derive some explicit expression of the F-index of different graph operations such as union, join, Cartesian product, composition, tensor product, strong product, corona product, generalized hierarchical product, disjunction, symmetric difference, splice and link of graphs. Also we apply our results to compute the F-index for some important classes of molecular graphs and nano-structures. For further study, F-index of some other graph operations and for different composite graphs can be computed.
\end{document} | arXiv | {
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\begin{document}
\title{On the normalized Shannon capacity of a union}
\begin{abstract} Let $G_1 \times G_2$ denote the strong product of graphs $G_1$ and $G_2$, i.e.\ the graph on $V(G_1) \times V(G_2)$ in which $(u_1,u_2)$ and $(v_1,v_2)$
are adjacent if for each $i=1,2$ we have $u_i=v_i$ or $u_iv_i \in E(G_i)$. The Shannon capacity of $G$ is $c(G) = \lim_{n\to \infty} \alpha (G^n)^{1/n}$, where $G^n$ denotes the $n$-fold strong power of $G$, and $\alpha (H)$ denotes the independence number of a graph $H$. The normalized Shannon capacity of $G$ is $C(G) = \frac {\log c(G)}{\log |V(G)|}$. Alon \cite{alon} asked whether for every $\epsilon > 0$ there are graphs $G$ and $G'$ satisfying $C(G), C(G') < \epsilon$ but with $C(G + G') > 1 - \epsilon $. We show that the answer is no. \end{abstract}
Despite much impressive work (e.g.\ \cite{alon}, \cite{alon lubetzky}, \cite{alon orlitsky}, \cite{haemers}, \cite{lovasz}) since the introduction of the Shannon capacity in \cite{shannon}, many natural questions regarding this parameter remain widely open (see \cite{graph powers}, \cite{korner orlitsky} for surveys). Let $G_1 + G_2$ denote the disjoint union of the graphs $G_1$ and $G_2$. It is easy to see that $c(G_1 + G_2) \geq c(G_1) + c(G_2)$. Shannon \cite{shannon} conjectured that $c(G_1 + G_2) = c(G_1) + c(G_2)$, but this was disproved in a strong form by Alon \cite{alon} who showed that there are $n$-vertex graphs $G_1,G_2$ with $c(G_i) < e^{c\sqrt{\log n \log\log n}}$ but $c(G_1 + G_2) \geq \sqrt n$. In terms of the normalized Shannon capacity, this implies that for any $\epsilon >0$, there exist graphs $G_1$, $G_2$ with $C(G_i) < \epsilon $ but $C(G_1 + G_2) > 1/2 -\epsilon$. Alon \cite{alon} asked whether `$1/2$' can be changed to `$1$' here. In this short note we will give a negative answer to this question. In fact, the following result implies that `$1/2$' is tight.
\begin{thm} \label{normalized shannon capacity} If $C(G_1) \le \epsilon$ and $C(G_2) \le \epsilon$
then $C(G_1 + G_2) \leq \frac{1 + \epsilon }{2} + \frac{1-\epsilon}{2\log _2 (|V(G_1)| + |V(G_2)|)}$. \end{thm}
\noindent \textbf{Proof.}
Let $N_i = |V(G_i)|$ for $i=1,2$. Fix a maximum size independent set $I$ in $(G_1+G_2)^n$ for some $n\in {\mathbb N}$. We write $|I|=\sum_{S \subset [n]} |I_S|$, where $I_S = \{ x=(x_1,\ldots,x_n) \in I: x_i \in V(G_1) \Leftrightarrow i \in S\}$.
To bound $|I_S|$, we may suppose that $S=[m]$ for some $0\leq m \leq n$. Then $I_S$ is an independent set in $G_1^{m} \times G_2^{n-m}$. As $C(G_1) \leq \epsilon $, by supermultiplicativity $\alpha (G_1^{m}) \leq N_1^{\epsilon m}$; similarly, $\alpha (G_2^{n-m}) \leq N_2^{\epsilon (n-m)}$. For any $x \in V(G_1)^m$, the set of $y \in V(G_2)^{n-m}$
such that $(x,y) \in I_S$ is independent in $G_2^{n-m}$, so $|I_S| \le N_1^m N_2^{\epsilon (n-m)}$. Similarly, $|I_S| \le N_1^{\epsilon m}N_2^{n-m}$.
We multiply these bounds:
$|I_S|^2 \le (N_1^m N_2^{n-m})^{1+\epsilon}$. Writing $\gamma = \frac{N_1}{N_1 + N_2}$, we have \begin{eqnarray*}
\alpha ((G_1 + G_2)^n) = |I| & = & \sum _{S \subset [n]} |I_S| \leq \sum _{m=0}^n \binom {n}{m} \left(N_1^{(1 +\epsilon )/2}\right)^m \left(N_2^{(1 +\epsilon )/2}\right)^{n-m}\\ & = & (N_1^{(1 +\epsilon )/2} + N_2^{(1 +\epsilon )/2})^{n} \\ & = & (\gamma ^{(1 +\epsilon )/2} + (1-\gamma )^{(1 +\epsilon )/2})^n(N_1 + N_2)^{(1+\epsilon)n/2}\\ & \leq & 2^{(1-\epsilon )n/2} (N_1 + N_2)^{(1+\epsilon)n/2}, \end{eqnarray*} as $\gamma ^{b} + (1-\gamma )^{b}$ is maximized at $\gamma = 1/2$ for $0<b<1$ and $0 \leq \gamma \leq 1$. Therefore \begin{equation*} C(G_1 + G_2) = \lim _{n\to \infty} \frac{\log \alpha ((G_1 + G_2 )^{n})}{n \log {(N_1 + N_2)}} \leq \frac{1 + \epsilon }{2} + \frac{1 - \epsilon }{2\log _2(N_1 + N_2)}. \end{equation*}
\end{document} | arXiv | {
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\begin{document}
\author{Yuri A Rylov} \title{Crisis in the geometry development and its social consequences} \date{Institute for Problems in Mechanics, Russian Academy of Sciences,\\ 101-1, Vernadskii Ave., Moscow, 119526, Russia.\\ e-mail: rylov@ipmnet.ru\\ Web site: {$http://rsfq1.physics.sunysb.edu/\symbol{126}rylov/yrylov.htm$}\\ or mirror Web site: {$http://gasdyn-ipm.ipmnet.ru/\symbol{126} rylov/yrylov.htm$}} \maketitle
\begin{abstract} The reasons of the crisis in the contemporary (Riemannian) geometry are discussed. The conventional method of the generalized geometries construction, based on a use of the topology, leads to a overdetermination of the Riemannian geometry. In other words, at the Riemannian geometry construction one uses the needless information (topology), which disagrees with other original axioms. The crisis manifests in the fact, that the mathematical community cannot see and does not want to see the overdetermination of the Riemannian geometry. The most geometers-topologists deny the alternative method of the generalized geometry construction, which does not uses the topology, because it does not contain theorems. Most geometers see the geometry presentation as a set of definitions and theorems. They cannot imagine the geometry presentation without customary theorems. As a result the most clever topologists, which have acknowledged the negligible role of the topology in the geometry construction and inconsistency of the conventional method of the generalized geometry construction, appear in the difficult situation (conflict with the mathematical community). \end{abstract}
\section{Introduction}
I wrote on crisis in geometry \cite{R2005}. The fact is that the contemporary (Riemannian) geometry is overdetermined and hence it is inconsistent. Although the character of inconsistency is known long ago, the mathematical community as a whole does not want to acknowledge these inconsistencies and bypasses them. The content of the crisis lies in the fact that the mistakes in geometry (more exact in the construction of generalized geometries) are not acknowledged and not corrected, but not in the existence of mistakes in themselves. As a result the geometry is developed in the direction, which leads in the blind alley. Why does such a situation take place and which are consequences of the crisis? The presented paper is devoted to this question.
Let us note that it the second crisis in the geometry. The first crisis took place in the second half of the 19th century, when the mathematical community did not want to acknowledge the reality of the non-Euclidean geometries. This crisis had exhausted itself somehow, when the non-Euclidean (Riemannian) geometry begun to be applied in the general relativity. But the analysis of the crisis reasons has not been produced (at any rate, I know nothing about such an analysis).
At first, I connected an appearance of the crisis with the strongly crusted preconception, that the straight is a one-dimensional line in any generalized geometry. It followed from this statement that the one-dimensional line is the most important object of any generalized geometry. In its turn this fact leads to the conclusion that the topology, founded on the concept of one-dimensional curve, lies in foundation of a geometry, and it should be used in construction of a generalized geometry. All conventional methods of the geometry construction are founded on the essential employment of the topology. It means that at first one constructs a topological space, and the geometry was constructed on the basis of the topological space. Overcoming the preconception on the one-dimensionality of the straight, one can construct a more general and simpler method of the generalized geometries construction. This alternative method is founded on the deformation principle \cite{R2003}. This method admits, that the straight may be a surface (tube), and it does not need the topology \cite {R2002} for construction of a generalized geometry. The substance of the alternative method may be expressed by the words: "Any generalized geometry may be obtained as a deformation of the proper Euclidean geometry. The alternative method, applied for the Riemannian geometry construction, is free of the well known defects of the conventional method, i.e. it does not contain such properties of the Riemannian geometry as absence of the absolute parallelism and the convexity problem. All scientists understood that these properties of the Riemannian geometry were its defects, but they could not overcome them without a use of the alternative method.
The fact, that the alternative method gives another results than the conventional one, means that at least one of methods is erroneous. To use the conventional method, one needs many different constraints, restricting a possibility of its application. One needs the dimensionality of the space, its continuity and a continuous coordinate system in it. For application of the alternative method one does not need neither dimensionality, nor the space continuity. The coordinate system is not needed also. One needs only the distance function $\rho \left( P,Q\right) $, or the world function $ \sigma \left( P,Q\right) =\frac{1}{2}\rho ^{2}\left( P,Q\right) $, which satisfies the only condition \begin{equation} \sigma \left( P,P\right) =0 \label{a1.1} \end{equation} Even the symmetry condition \begin{equation} \sigma \left( P,Q\right) =\sigma \left( Q,P\right) \label{a1.2} \end{equation} is not an obligatory condition \cite{R2002b}. The alternative method does not contain any logical reasonings, whereas at application of the conventional method of the generalized geometry construction one needs to provide consistency of all used suppositions. It needs the significant efforts. Consistency of all constraints may be provided not always. For the Riemannian geometry the consistency is violated because of the supposition, that the straight is an one-dimensional curve (or because of the application of the topology at the geometry construction, what is essentially the same). The fact is that at construction by means of the right (alternative) method the geodesic, passing through the point $P$, in parallel to the vector at the point $P$, is one-dimensional. However, in the case, when the straight (geodesic) is passed through the point $P$, in parallel to the vector at the other point $P_{1}$, it is not one-dimensional, in general \cite{R2006}. To avoid the non-one-dimensionality of the straight, the absolute parallelism is forbidden in the Riemannian geometry. Thus, it is evident that the alternative method is true, whereas the conventional one is questionable. In the case, when they lead to different results, one should to prefer the alternative method.
I believed that the construction of generalized geometries by means of the alternative method, founded on the deformation principle, admits one to perceive the inadmissibility of the topology application at the generalized geometry construction. But I appeared to be not right in the relation, that I considered the one-dimensionality as an only preconception preventing from the correct construction of the generalized geometry. It appears, that there is once more preconception. I did not guess about this preconception, because I did not meet it at the construction of the tubular geometry (T-geometry), i.e. the geometry constructed on the basis of the deformation principle. I met the preconception on the one-dimensionality of straight at construction of the T-geometry and expended thirty years to overcome it \cite {R2005}. One can obtain a representation on these preconceptions from consideration of following two syllogisms:
1. According to Euclid the straight is one-dimensional in the Euclidean geometry, hence, the straight is one-dimensional in any generalized geometry.
2. Euclid constructed his geometry, formulating and proving theorems, hence, any generalized geometry should be constructed, formulating and proving theorems.
From the viewpoint of logic the considered syllogisms are not valid, especially if one takes into account the opinion of the ancient Egyptians, supposed that all revers flow northward. Their viewpoint may be considered as a corollary of the third syllogism:
3. The great river Nile flows northward, hence all revers flow northward.
The structure of all three syllogisms is the same. They transform a partial case into the general one without a sufficient foundations. Such a generalization is connected with a narrowness of our experience. There is only one (Euclidean) geometry, and the straight in this geometry is one-dimensional. There is the only geometry, and it constructed by means of formulation and proof of theorems. Finally, the ancient Egyptians knew only one river Nile.
The logical inconsistency of syllogisms did not prevent mathematicians from application of them at construction of any generalized geometry, because in the given case the mathematicians used in their practical activity the associations, but not a logic. In all considered cases we deal with the only object. In this case it is very difficult to distinguish, which properties belong to the object in itself and which properties belong to the method of the object description. When the method of description is considered to be a property of the object in itself, we obtain a preconception. The first syllogism is a reason of the preconception on one-dimensionality of the straight. (The method of the straight description, used by Euclid, was acknowledged as the property of the straight in itself). I knew about this preconception, because I was forced to overcome it. The second syllogism became to be a foundation of the preconception, that all activity of a geometers consists in a formulation and a proof of theorems. (Euclid described the geometry by means of theorems, hence description in terms of theorems is a property of the geometry. If there are no theorems, then there is no geometry.) I had met the second preconception at the following circumstances.
I have submitted my report \cite{R2002} to the seminar of one of geometric-topological chairs of the Moscow Lomonosov University. It took place in 2001, and my paper \cite{R2002} was not yet published. I came to the secretary of the seminar, told him about my wish to read a report and gave him the text of the paper. Skimming my paper the secretary of the seminar said thoughtfully: "How strange geometry! No theorems! Only definitions! I believe that it will be not interesting for us." One of leading geometers, which was forced duty bound to read my paper (the paper was, submitted to the International conference on geometry, which took place in Saint-Petersburg in the summer of 2002) said me in private conversation, that he had understood nothing in my paper, because it contained neither axioms, nor theorems. In that time I did not understand his objections. It was quite unclear for me, how one cannot understand such a simple conception as the T-geometry, which did not contain any logical reasonings. Only essentially later I had understand, that here we deal with a preconception. Mathematicians distinguish from other peoples in the relation, that they perceive the geometry exclusively via its formalism. Further I shall try to show formally, how theorems are transformed into definitions, and the demand of the geometry representation in the form of a set of theorems is not more, than a preconception.
Speaking about social consequences of the crisis, I keep in mind as follows. Now the topology is considered to be the most perspective direction of the geometry development. All best geometers are concentrated in the development of this direction. Having overcame the preconception on the one-dimensionality of the straight and discovering the alternative method of the geometry construction, the topology appears to be a cul-de-sac direction in the geometry development. Many papers on verification of the geometry on the basis of topology appear to be depreciated. However, these papers are considered to be important and perspective as long as the mathematicians do not want to take into account the alternative method and do not acknowledge inconsistencies in the Riemannian geometry.
Let us imagine the following situation. A young talented topologist N has solved a very difficult topological problem. A prestige international prize is declared for solution of this problem. Solving the problem and publishing the results, the topologist N discovers suddenly, that the problem is set incorrectly, because the topology is founded on concepts of the Riemannian geometry, but the Riemannian geometry is overdetermined (i.e. it is inconsistent). The topologist N is awarded this prize quite deservedly, because one did not find mistakes in his solution. But, if the problem is stated incorrectly, it may have no correct solution at all, because different methods of solution may lead to different results. The mathematical community considered that the prize has been awarded correctly, but the topologist N disagree with the opinion of the mathematical community, as far as he understand, that his work may be considered as outstanding one, as long as the mathematical community did not discover the weakness of the ground of the solved problem. What does the topologists N do? If he is a conformist, he may affect ignorance, that he does not discover any incorrectness in the statement of the problem and get quietly the awarded deserved prize. (Whilom the inconsistency of the Riemannian geometry will be discovered!). It is impossible to prove, that he has known about inconsistency of the Riemannian geometry. However, the following problem remain before the topologist N. Further some of his previous papers will be depreciated. How and what for to work in the region of the topological verification of the geometry, if this direction leads to the blind alley?
However, if the topologist N is a scrupulous scientists, but not a conformist, he has only one possibility: to keep away from the opinion of the mathematical community and to abandon from the awarded prize. Should he declare the reason of his decision? It is not a simple question. Declaring the reason of his decision, the topologist N conflicts with the mathematical community, which does not acknowledge the Riemannian inconsistency at this moment. In his time the great Gauss did not risk to conflict with the mathematical community on the question of the non-Euclidean geometry existence. (According to information of the Felix Klein \cite{K37} one discovered unpublished manuscripts of Gauss on the non-Euclidean geometry among his papers). However, Gauss worked in many different branches mathematics, he may admit himself to neglect his works in the non-Euclidean geometry. The contemporary topologists works, as a rule, only in topology. Acknowledgment of inconsistencies in the Riemannian geometry is a drama for a topologist. At the same time such an acknowledgment is a very fearless act.
I suspect, that even if the topologists N discussed inconsistency of the Riemannian geometry with his colleagues, he may meet an incomprehension. I have the following reason for such a suspicion. In the summer 2002 the international conference, devoted to the anniversary of the known Russian geometer A.D. Alexandrov, took place in Saint-Petersburg. Submitting the corresponding report \cite{R2002b}, I arrived at the conference. However, the main goal of my participation in the work of the conference was not my report, which appeared to be interesting for nobody. I wanted to discuss with leading geometers of Russia a possibility of the alternative method of the Riemannian geometry construction, which concerns the geometry foundation. It was important for me, because I am not a mathematician, but a physicist - theorist. I succeeded to discuss this question with some leading geometers, but nobody did not understood me (More exactly, only one person had understood me, but he was not a geometer. He was simply a collaborator of the Saint-Petersburg branch of Mathematical institute, where the conference took place). The leading geometers told me very polite, that the question of the geometry foundation had been solved many years ago by Hilbert and other great geometers. In the present time the problem of the geometry foundation is interesting for nobody. In this discussion only the question on the alternative method of the geometry construction was considered. The question of the Riemannian geometry inconsistency was not discussed.
In other time I wanted to discuss the paper \cite{R2002} at the seminar of the well known mathematician, who had usually a preliminary talk with the potential speaker. During this talk I mentioned that the result of application of the alternative method of the Riemannian geometry construction disagree at some points with the results of application of the conventional method. According to my representation, this fact must intrigue the leader of the seminar in discussion of my report. However, I was told immediately, that the Riemannian geometry cannot be inconsistent. Our talk was finished after this declaration.
Summarizing, I should note that the more talented the topologist, the earlier he faces the inconsistency of the Riemannian geometry, lying in the foundation of the topology and makes the necessary conclusions. A failure in his scientific career may be a result of this inconsistency. Let me note, that the Riemannian geometry inconsistencies may be manifested only on the sufficiently high level of the geometry (topology) development, when new contradictions (other than the convexity problem and the fernparallelism problem) appear. This does not concern the standard formalism of the Riemannian geometry (metric tensor, covariant derivatives, curvature, etc.).
At first site the drama in the scientific career of the topologist N seem to be unreal. However, it has taken place already, puzzling the mathematical community, because nobody cannot imagine that the reason lies not in the character of the topologist N, but in the crisis, which the talented topologist has acknowledged earlier, than other scientists. His behavior in the given situation was very dignified, although it was not clear for his colleagues. I shall not denominate the name of this topologist, although persons close to mathematics and topology may easily determine it.
Farther the pure mathematical questions will be considered, which explain my statement on the overdetermination of the Riemannian geometry. The second section is devoted to comparison of the conventional method of the generalized geometry construction with the alternative one. In the third section the question on the Riemannian geometry inconsistency is considered. In the fourth section one considers why the mathematicians discard the T-geometry, constructed on the basis of the deformation principle.
\section{Comparison of the conventional and alternative methods of the geometry construction}
The conventional method of the Riemannian geometry is as follows. One considers $m$-dimensional surfaces in the $n$-dimensional Euclidean space ($ m<n$). Those properties of $m$-dimensional surfaces, which do not depend on the dimension $n$ of the accommodating Euclidean space, declared to be an internal geometry of the $m$-dimensional surface. It is the Riemannian geometry of the $m$-dimensional space. The Riemannian geometries are restricted by the constraint, that the Riemannian space can be embedded isometrically into the Euclidean space of sufficiently large dimension. The geometry which cannot be embedded isometrically in the Euclidean space, cannot be constructed by this method. Besides the Riemannian geometries are continuous, that is connected with the application of the continuous coordinate system at the construction of the Riemannian geometry. A use of the conventional method of the generalized geometry construction, i.e. the geometry more general, than the Riemannian one, contains a series of resricitions on the generalized geometry. In particular, such a constraint is the embeddability of the space with the generalized geometry into the Euclidean space of the sufficiently large dimension. Besides, such a generalized geometry contains such a characteristic of the geometry as the dimension, which is a natural number $n$. Necessity of some dimension $n$ at the generalized geometry seems to be something as a matter-of-course, although, in reality the dimension is a corollary of the applied method of the generalized geometry construction, when one uses the concept of a manifold, which is a coordinate system, consisting $n$ independent coordinates. The fact, that the dimension is not a necessary property of the generalized geometry (it is rather the means of description), follows from the fact that there is an alternative method of the generalized geometry description, where the concept of dimension may be not introduced.
The alternative method of the generalized geometry description is founded on the deformation principle, which states that any generalized geometry can be obtained as a result of a deformation of the proper Euclidean geometry. Any deformation means a change of distance between the points of the space. Any deformation of the proper Euclidean space generates some generalized geometry. It is produced as follows. One proves the theorem, that the proper Euclidean geometry may be formulated in terms and only in terms of the world function \cite{R2002}. (The world function is a half of the squared distance between the two points of the space). It follows from the theorem, that any geometrical object $\mathcal{O}_{\mathrm{E}}$ and any statement $\mathcal{R} _{\mathrm{E}}$ of the proper Euclidean geometry $\mathcal{G}_{\mathrm{E}}$ can be expressed in terms of the world function $\sigma _{\mathrm{E}}$ of the Euclidean geometry $\mathcal{G}_{\mathrm{E}}$ in the form $\mathcal{O}_{ \mathrm{E}}\left( \sigma _{\mathrm{E}}\right) $ and $\mathcal{R}_{\mathrm{E} }\left( \sigma _{\mathrm{E}}\right) $ respectively. \textit{The set of all geometrical objects $\mathcal{O}_{\mathrm{E}}\left( \sigma _{\mathrm{E} }\right) $ and relations $\mathcal{R}_{\mathrm{E}}\left( \sigma _{\mathrm{E} }\right) $ between them forms the Euclidean geometry $\mathcal{G}_{\mathrm{E} }$.} To obtain the corresponding relations of the generalized geometry $ \mathcal{G}$ it is sufficient to replace the world function $\sigma _{ \mathrm{E}}$ of the proper Euclidean geometry with the world function $ \sigma $ of the generalized geometry $\mathcal{G}$: \[ \mathcal{O}_{\mathrm{E}}\left( \sigma _{\mathrm{E}}\right) \rightarrow \mathcal{O}_{\mathrm{E}}\left( \sigma \right) ,\qquad \mathcal{R}_{\mathrm{E} }\left( \sigma _{\mathrm{E}}\right) \rightarrow \mathcal{R}_{\mathrm{E} }\left( \sigma \right) \] \textit{Then the set of all geometrical objects $\mathcal{O}_{\mathrm{E} }\left( \sigma \right) $ and relations $\mathcal{R}_{\mathrm{E}}\left( \sigma \right) $ between them forms the generalized geometry $\mathcal{G}$.}
The alternative method of the geometry construction supposes, that the generalized geometry $\mathcal{G}$ is determined completely by its world function, and any statement of the generalized geometry $\mathcal{G}$ can be obtained from the corresponding statement of the Euclidean geometry. In this case for construction of the generalized geometry $\mathcal{G}$ one does not need a coordinate system. One does not need either its dimension, or any another information on topology of the generalized geometry $\mathcal{G}$. The dimension and topology (if they exist) can be obtained from the world function $\sigma $ of the generalized geometry $\mathcal{G}$. The method of determination of the dimension is a such one, that the dimension may be different at different points of the space, or the dimension may be not exist at all. It means that the definition of the topology and of the dimension independently of the world function is inconsistent, in general. Thus, the conventional method of the generalized geometry construction is overdetermined. It contains too many axioms, which are not independent. The suggested alternative method is insensitive to the continuity or discreteness of the geometry, as far as it uses nowhere the limiting transition or continuous coordinate system. A use of the conventional method of the geometry construction, when one postulates some axiom system, which determines the generalized geometry, appears to be ineffective, because it is very difficult to provide compatibility of original axioms. The demand of their compatibility imposes unnecessary constraints upon the obtained generalized geometries.
For instance, a construction of the Riemannian geometry may be realized by two methods. Using the conventional method on the basis of the metric tensor, giving on the $n$-dimensional manifold, one obtains the Riemannian geometry $\mathcal{G}_{\mathrm{R}}$. Using the alternative method based on the deformation principle, one obtains $\sigma $-Riemannian geometry $ \mathcal{G}_{\sigma \mathrm{R}}$. Prefix $\sigma $ means, that the generalized geometry $\mathcal{G}_{\sigma \mathrm{R}}$ has the property of $ \sigma $-immanence ($\sigma $-immanence is the property of the geometry to be described completely by the world function $\sigma $). If the world function $\sigma $ is the same in geometries $\mathcal{G}_{\mathrm{R}}$ and $ \mathcal{G}_{\sigma \mathrm{R}}$, the obtained generalized geometries $ \mathcal{G}_{\mathrm{R}}$ and $\mathcal{G}_{\sigma \mathrm{R}}$ are very close, but they distinguish in some details. For instance, in the $\sigma $ -Riemannian geometry $\mathcal{G}_{\sigma \mathrm{R}}$ there is the absolute parallelism (fernparallelism), whereas in the Riemannian geometry $\mathcal{G }_{\mathrm{R}}$ one fails to introduce the absolute parallelism. The fact is that the concept of parallelism of two vectors in the Riemannian geometry $ \mathcal{G}_{\mathrm{R}}$ is transitive according to the geometry construction \cite{R2002a,R2003}. In the $\sigma $-Riemannian geometry $ \mathcal{G}_{\sigma \mathrm{R}}$ there is an absolute parallelism, which is intransitive, but the parallelism of two vectors at the same point is transitive. Thus, if one introduces the absolute parallelism in the Riemannian geometry, it will be intransitive, in general, and incompatible with the original statement on the transitivity of the parallelism in Riemannian geometry. To avoid inconsistency, one declares that in the framework of the Riemannian geometry one cannot introduce the absolute parallelism of vectors, placed at different points of the space.
The considered example shows that the conventional method of the generalized geometry construction is overdetermined. Furthermore, if the proper Euclidean geometry is considered to be as a special case of the Riemannian geometry, and one constructs it in the same way, as one constructs the Riemannian geometry, the proper Euclidean geometry acquires an absurd property. If some region of the space is nonconvex, the Riemannian geometry, constructed in this region with the Euclidean metric tensor, is not the Euclidean geometry, in general, because some distances in such a geometry are determined not along the straight lines of the Euclidean geometry, but along lines belonging partly to the boundary of the region. The obtained Riemannian space cannot be embedded, in general, isometrically into the Euclidean space, although the nonconvex region is a part of the Euclidean space. This absurd result shows, that the system of axioms of the conventional method of the generalized geometry construction is overdetermined. An application of the overdetermined method of the generalized geometry construction leads, in general, to inconsistency. The form of these inconsistencies depends essentially on the method, how these axioms are used.
\section{On inconsistency of the Riemannian geometry}
Some defects of the Riemannian geometry, were known years ago, but somehow they are not perceived as inconsistency of the axiom system, which determines the conventional construction of the Riemannian geometry. (Apparently, it is conditioned with the absence of the alternative method of the generalized geometry construction). For instance, if one constructs the Riemannian geometry as a special case of the Riemannian one, using the conventional method of the Riemannian geometry, the convexity problem appears, consisting in the fact, that a nonconvex region of the Euclidean plane cannot be embedded isometrically, in general, in the Euclidean plane, from which it is cut. The result is evidently absurd. However, for the convex region such an embedding is possible, mathematicians bypass this defect, considering geometries only on convex manifolds. (For instance, the book of A.D.Alexandrov "Internal geometry of convex surfaces"). Another defect is the problem of the fernparallelism, i.e. absence of definition of the parallelism of remote vectors in the Riemannian geometry. This fact is not considered as a defect of the Riemannian geometry also. In reality these defects are corollaries of the overdetermination of the Riemannian geometry, i.e. at the Riemannian geometry construction one uses more axioms, than it is necessary for the geometry construction. Some axioms are incompatible between themselves, or they are compatible only at some constraints, imposed on the geometry. In principle, this overdetermination may lead to another contradictions, which are not known now. The overdetermination of the conventional method of the Riemannian geometry construction has been discovered after appearance of the alternative method, using essentially less amount of information, which is necessary for the geometry construction. The alternative method of the geometry construction does not contain overdetermination, the convexity problem, the problem of fernparallelism and other defects, which are corollaries of this overdetermination.
The main preconception, preventing the contemporary geometry from its development, is the statement that the straight is a one-dimensional line in any generalized geometry (the straight has no width) has been traced to Euclid. It is true in the Riemannian geometry for the straight, passing through the point $P$, in parallel to a vector at the point $P$. However, in the case, when the straight (geodesic) passes through the point $P$ in parallel to a vector at another point $P_{1}$, it is not one-dimensional, in general, \cite{R2006}. To avoid the non-one-dimensionality of the straight, the fernparallelism (i.e. the concept of parallelism of two remote vectors) is forbidden in the Riemannian geometry. The one-dimensionality of the straight may not be included in axiomatics of a generalized geometry. The one-dimensionality of the straight may not be used at the generalized geometry construction, because it may appear to be incompatible with other axioms. (Forbidding the fernparallelism, one did not think on a possible non-one-dimensionality of the straight. Simply an ambiguity in the definition of the parallelism contradicted to the axiomatics and seems to be unacceptable). Forbidding fernparallelism and the geometry on non-convex manifolds, one can avoid a manifestation of the Riemannian geometry inconsistency. However, it does not avoid the inconsistency in itself, because it may appear in other form. In the case of arbitrary generalized geometry the character of the straight (one-dimensional line, or multidimensional surface) is determined by the form of the world function, and one does need to make suppositions on the one-dimensionality, or on non-one-dimensionality of the straight. Furthermore, one cannot demand the one-dimensionality of the straight, because it leads to an overdetermination of conditions of the generalized geometry construction, and as a corollary to their inconsistency, or to a restriction of the class of possible geometries. Application of the topology at the generalized geometry construction supposes that a curve (and its kind a straight) is one-dimensional. It is a reason, why the topology may not be used at the generalized geometry construction. Overcoming of this preconception was very difficult. I personally needed almost thirty years for overcoming this preconception (a description of the path of this overcoming can be found in \cite{R2005}), although I have an experience of successful overcoming of like preconception in other branches of physics.
Although the overcoming of the preconception on the one-dimensionality of the straight was difficult, I hoped, that, being elucidated it will be perceived and overcame by the mathematical community. It appeared, that I was wrong in this relation. The overcoming of the perception was very difficult ( by the way, it was difficult also in the case of the preconception on the statistical description). Apparently, overcoming of preconceptions is always difficult. Besides, it appeared, that besides of the main preconception on the one-dimensionality of the straight there are another preconceptions. For instance, one supposes, that any presentation of a geometry is a set of axioms and theorems and that the geometer's activity is a formulation and a proof of theorems.
\section{Why mathematicians do not acknowledge \newline T-geometry constructed on the basis of the \newline deformation principle?}
The relation of the most mathematicians to the alternative method of the geometry construction, which is based on the deformation principle is negative as a rule, although they cannot contradict anything against this method. Indeed, it as very difficult to contradict anything, because of the simplicity of the alternative method, which does not contain any suppositions except for the sufficiently evident deformation principle. Nevertheless, the refereed mathematical journals reject papers on the generalized geometry construction founded on basis of the deformation principle. In the paper \cite{R2005} it is described in detail, how such a rejection is produced. My attempts of reporting these papers at the seminars of topologists - geometers are rejected also. However, my papers were reported and discussed at the seminar on the geometry as a whole in the Moscow Lomonosov University. This seminar was founded by N.V. Efimov.
In the present time the topological approach to the problem of the generalized geometry construction dominates. One supposes, that it is necessary at first to construct a proper topological space. In the topological space one introduces a metric and constructs a generalized geometry. Construction of the generalized geometries on the basis of the deformation principle depreciates essentially papers on geometry, and the mathematicians, constructing generalized geometries by the conventional method, were not enthused over the prospect. As far as they cannot suggest any essential objection against the deformation principle because of its simplicity and effectiveness, they use another methods of resistance, generating doubts in the scientific scrupulosity of advocates of the conventional method. One of them is described in \cite{R2005} in detail.
Of course, I have known, how strongly we believe, that the straight line is always an one-dimensional geometrical object, because this preconception retarded my discovery of the T-geometry almost with thirty years. However, discovery of the non-one-dimensionality of the straight and incomprehension of such a possibility, when the non-one-dimensionality has been already discovered, seemed to be different things for me. It was appeared that a perception of the deformation principle is the more difficult, the better the person knows the geometry in its contemporary formal presentation. The fact is that the mathematicians perceive all things via the formalism. The associative perception like a reference to the deformation principle conveys little to them.
The strangers and the mathematicians perceive the geometry presentation as a formulation and proof of different geometric theorems. Replacement of theorems with definitions seems to them as anything quite obscure. However, in reality, the formulation and the proof of theorems is only one of the methods of work with different geometric statements (axioms and theorems). The proof of theorems is the most laborious part of work. As a result one gets an impression, that the geometry presentation is a proof of theorems (if there are no theorems, there is no geometry). But there are another methods of work with the geometric statements, which does not distinguish between the axioms and theorems.(There is no necessity to repeat the work of Euclid, one should use results of his work.) At such a method of work the theorems are replaced by definitions, and a necessity of their proof falls off.
I shall explain this in the example of the cosine theorem, which states \begin{eqnarray} \left\vert \mathbf{BC}\right\vert ^{2} &=&\left\vert \mathbf{AB}\right\vert ^{2}+\left\vert \mathbf{AC}\right\vert ^{2}-2\left( \mathbf{AB.AC}\right) \nonumber \\ &=&\left\vert \mathbf{AB}\right\vert ^{2}+\left\vert \mathbf{AC}\right\vert ^{2}-2\left\vert \mathbf{AB}\right\vert \left\vert \mathbf{AC}\right\vert \cos \alpha \label{a3.1b} \end{eqnarray} where the points $A,B,C$ are vertices of a triangle, $\left\vert \mathbf{BC} \right\vert $, $\left\vert \mathbf{AB}\right\vert $, $\left\vert \mathbf{AC} \right\vert $ are lengths of the triangle sides and $\alpha $ is the angle $ \measuredangle BAC$. The relation (\ref{a3.1b}) is the cosine theorem which is proved on the basis of the axioms of the proper Euclidean geometry.
Using expression of the length of the triangle side $\mathbf{AB}$ via the world function $\sigma $ \begin{equation} \left\vert \mathbf{AB}\right\vert =\sqrt{2\sigma \left( A,B\right) } \label{a3.2} \end{equation} we may rewrite the relation (\ref{a3.1b}) in the form \begin{equation} \left( \mathbf{AB.AC}\right) =\sigma \left( A,B\right) +\sigma \left( A,C\right) -\sigma \left( B,C\right) \label{a3.3} \end{equation} The relation (\ref{a3.3}) is a definition of the scalar product $\left( \mathbf{AB.AC}\right) $ of two vectors $\mathbf{AB}$\textbf{\ }and $\mathbf{ AC}$ in the T-geometry. Thus, the theorem is replaced by the definition of a new concept (the scalar product), which is not connected directly with the concept of the linear space.
Another example the Pythagorean theorem for the rectangular triangle $ABC$ with the right angle $\measuredangle BAC$, which is written in the form \begin{equation} \left\vert \mathbf{BC}\right\vert ^{2}=\left\vert \mathbf{AB}\right\vert ^{2}+\left\vert \mathbf{AC}\right\vert ^{2} \label{a3.4} \end{equation} In T-geometry instead of the theorem (\ref{a3.4}) we have a definition of the right angle$\measuredangle BAC$. In terms of the world function this definition has the form. The angle $\measuredangle BAC$ is right, if the relation \begin{equation} \sigma \left( A,B\right) +\sigma \left( A,C\right) -\sigma \left( B,C\right) =0 \label{a3.5} \end{equation} takes place
Thus, we see that theorems of the proper Euclidean geometry are replaced by definitions of T-geometry.
From the formal viewpoint the difference between the conventional method of description and the alternative method may be described as follows. The conventional method of the proper Euclidean geometry may be described as a set $\mathcal{S}_{\mathrm{E}}\left( \mathcal{A}\left[ \mathcal{R}_{\mathrm{E} }\right] \right) $ of algorithms $\mathcal{A}\left[ \mathcal{R}_{\mathrm{E}} \right] $, acting on operands $\mathcal{R}_{\mathrm{E}}$, where operands $ \mathcal{R}_{\mathrm{E}}$ are geometrical objects or relations of the Euclidean geometry. The operands $\mathcal{R}_{\mathrm{E}}$ depend on parameters $\mathcal{P}^{n}\equiv \left\{ P_{0},P_{0},..,P_{n}\right\} $, where $P_{0},P_{0},..,P_{n}$ are points of the space. \begin{equation} \mathcal{R}_{\mathrm{E}}=\mathcal{R}_{\mathrm{E}}\left( \mathcal{P} ^{n}\right) \label{a3.6} \end{equation} As far as all geometrical objects and relations $\mathcal{R}_{\mathrm{E}}$ can be expressed via the world function $\sigma _{\mathrm{E}}$ of the Euclidean geometry $\mathcal{G}_{\mathrm{E}}$, one may rewrite the relation ( \ref{a3.6}) in the form \begin{equation} \mathcal{R}_{\mathrm{E}}=\mathcal{R}_{\mathrm{E}}\left( \mathcal{P} ^{n}\right) =\tilde{\mathcal{R}}_{\mathrm{E}}\left[ \mathcal{\sigma }_{ \mathrm{E}}\left( \mathcal{P}^{n}\right) \right] \label{a3.7} \end{equation} where $\mathcal{\sigma }_{\mathrm{E}}\left( \mathcal{P}^{n}\right) $ is a set of world functions $\sigma _{\mathrm{E}}\left( P_{i},P_{k}\right) ,$ $ P_{i},P_{k}\in \mathcal{P}^{n}$.
Taking into account the relation (\ref{a3.7}), one can represent the set $ \mathcal{S}_{\mathrm{E}}\left( \mathcal{A}\left[ \mathcal{R}_{\mathrm{E}} \right] \right) $ of all algorithms $\mathcal{A}\left[ \mathcal{R}_{\mathrm{E }}\left( \mathcal{P}^{n}\right) \right] $ in the form of the set $\mathcal{S} _{\mathrm{E}}\left( \mathcal{A}\left[ \mathcal{R}_{\mathrm{E}}\left( \mathcal{P}^{n}\right) \right] \right) =\mathcal{S}_{\mathrm{E}}\left( \mathcal{A}\left[ \tilde{\mathcal{R}}_{\mathrm{E}}\left[ \mathcal{\sigma }_{ \mathrm{E}}\left( \mathcal{P}^{n}\right) \right] \right] \right) =\mathcal{S} _{\mathrm{E}}\left( \mathcal{A}\left[ \tilde{\mathcal{R}}_{\mathrm{E,} \mathcal{P}^{n}}^{\prime }\left[ \mathcal{\sigma }_{\mathrm{E}}\right] \right] \right) $ of all algorithms $\widetilde{\mathcal{A}}\left[ \mathcal{ \sigma }_{\mathrm{E}}\right] =\mathcal{A}\left[ \tilde{\mathcal{R}}_{\mathrm{ E,}\mathcal{P}^{n}}^{\prime }\left[ \mathcal{\sigma }_{\mathrm{E}}\right] \right] $. In the Euclidean geometry the set of all algorithms $\mathcal{S}_{ \mathrm{E}}\left( \mathcal{A}\left[ \mathcal{R}_{\mathrm{E}}\left( \mathcal{P }^{n}\right) \right] \right) $ takes the form $\mathcal{S}_{\mathrm{E} }\left( \widetilde{\mathcal{A}}\left[ \tilde{\mathcal{R}}_{\mathrm{E,} \mathcal{P}^{n}}^{\prime }\left[ \mathcal{\sigma }_{\mathrm{E}}\right] \right] \right) $. In the generalized geometry $\mathcal{G}$, described by the world function $\mathcal{\sigma }$, the set of all algorithms takes the form $\mathcal{S}_{\mathrm{E}}\left( \widetilde{\mathcal{A}}\left[ \tilde{ \mathcal{R}}_{\mathrm{E,}\mathcal{P}^{n}}^{\prime }\left[ \mathcal{\sigma } \right] \right] \right) $. It means that the set of all algorithms $\mathcal{ S}_{\mathrm{E}}\left( \widetilde{\mathcal{A}}\left[ \tilde{\mathcal{R}}_{ \mathrm{E,}\mathcal{P}^{n}}^{\prime }\left[ \mathcal{\sigma }\right] \right] \right) $ for any generalized geometry is obtained from the set $\mathcal{S} _{\mathrm{E}}\left( \widetilde{\mathcal{A}}\left[ \tilde{\mathcal{R}}_{ \mathrm{E,}\mathcal{P}^{n}}^{\prime }\left[ \mathcal{\sigma }_{\mathrm{E}} \right] \right] \right) $ of all algorithms for the Euclidean geometry by means of replacement of the operand of algorithms $\sigma _{\mathrm{E} }\rightarrow \sigma $. The form of all algorithms is not changed. This is the formal description of the action of the deformation principle. For construction of the generalized geometry one needs no theorems and no logical reasonings. The main part of the work of construction of the generalized geometry consists in transformation of known Euclidean algorithms $\mathcal{A}\left[ \mathcal{R}_{\mathrm{E}}\left( \mathcal{P} ^{n}\right) \right] $ to the form $\widetilde{\mathcal{A}}\left[ \tilde{ \mathcal{R}}_{\mathrm{E,}\mathcal{P}^{n}}^{\prime }\left[ \mathcal{\sigma }_{ \mathrm{E}}\right] \right] $, where all algorithms are represented in terms of the Euclidean world function.
In the T-geometry a construction of new generalized geometry is produced by the same algorithms, as in the proper Euclidean geometry. One replaces only the world function, i.e. operand of algorithms $\sigma _{\mathrm{E} }\rightarrow \sigma $. One does not need theorems which connect different objects of geometry. One uses the definition of the geometrical object or the \textrm{\ }relation of the Euclidean geometry, expressed directly via world function. For instance, the scalar product of two vectors is defined by the relation (\ref{a3.3}) in all T-geometries. The right angle $\measuredangle BAC$ is defined by the relation (\ref{a3.5}) in all T-geometries. There are no necessity to introduce couplings (theorems) between different geometrical concepts, as far as they are expressed via world function.
In T-geometry the main problem is an obtaining of expressions for different geometrical concepts and objects via the world function. As far as these expressions are the same for all T-geometries, it is sufficient to obtain these expressions in the framework of the proper Euclidean geometry. Thus, instead of formulation of different theorems for different generalized geometries, the geometer must express all concepts and objects of the proper Euclidean geometry via the world function.
The formalism of T-geometry distinguishes essentially from the conventional formalism of the generalized geometry construction by the object of consideration. In the conventional formalism the geometrical objects and primarily the straight are objects of consideration. The conventional construction of the generalized geometry is a repetition of the Euclid's construction, which is produced on the basis of other axioms. The true choice of axioms is the main problem of the conventional method of the geometry construction. In the alternative method, based on the deformation principle, the object of consideration is the world function (but not geometrical objects), i.e. a function of two points of the space. It is essentially simpler object of consideration. However, the rules of work (logic of investigation \cite{R2005a}) with the world function are very complicated. It is very complicated to guess these rules, because they are determined by the Euclidean geometry. Primarily one guesses the rules of work with the most simple object: the straight line. As a matter of fact, they were not guessed, they have been taken from the Euclidean geometry. But, to take them, the Euclidean geometry is to be presented in terms of the world function. It means that it was necessary to construct the formalism on the basis of the world function. Construction of the formalism, founded on the world function, was difficult. It needed about forty years. Thirty years of these forty years were lost for the overcoming the preconception on the one-dimensionality of straight. Different stages of this formalism construction are described in \cite{R2005}.
Apparently, the most difficult and uncustomary for geometers - professionals are the facts, that a new object of consideration (world function) appears and the conventional objects of consideration (geometrical objects) turn into the logic (algorithm) of investigation. This transition from one method of investigation to another is difficult for perception of the geometers - professionals. However, it is essentially easier for non- professionals, who do not know, or know skin-deep the conventional method of the geometry construction. They do not need to compare both methods, overcoming complex of concepts of the conventional method and the formalism, connected with this method.
Constructing geometry on the basis of the deformation principle, it seems that the formalism is absent, in general, as far as the theorems are absent. (The geometers became accustomed that the geometric formalism appears only in the form of theorem.) In reality the formalism appears in the implicit form as a reference to the Euclidean geometry with its formalism. The formalism of the Euclidian geometry is modified at the replacement of the world function of the Euclidean space (Algorithms retain, only operand of algorithms is changed). Besides, for a realization of such a modification the Euclidean geometry is to be presented in the terms of the world function. Mathematicians know slightly this presentation, based on the world function formalism. Despite the evidence of the deformation principle, I obtained it, when the generalized geometry (T-geometry), based on its application, has been already constructed. Furthermore, appearance of the world function formalism (description of the Riemannian geometry in terms of the world function) takes priority of the T-geometry construction.
\section{Application of T-geometry physical problems}
Besides, the overdetermination, appearing at a use of the conventional (topological) method of the geometry construction, this methods is not enough general. The method of the geometry construction based on the deformation principle, admits one to construct the generalized geometry, using deformation transforming the one-dimensional line into a surface. Although the everyday experience does not provide an example of such deformations, such deformations exist. Such a deformation may be described as follows. Let us imagine a beam of straight thin elastic wires of the same length. Being collected in a beam they represent a segment of the straight line. Let us grasp the ends of wires by two hands and move hands one to another. At such a deformation the wires diverge, conserving their lengths, and form an inswept surface. In other words, although deformations turning a one-dimensional line into a surface are possible, the Riemannian geometry and the metrical one ignore them, preferring to remain in the framework of deformations, conserving one-dimensionality of the straight. A use of the Euclidean concept of the straight, as a second base concept of the geometry, and consideration of the one-dimensionality of the straight do not admit one to prove out the deformation principle. The way to the deformation principle lies through the construction of the mathematical formalism, based on the application of the world function.
Applying the described above deformation to the construction of the space-time geometry \cite{R2006}, one succeeded to construct such a geometry, where the motion of free particles is primordially stochastic, and intensity of the stochasticity depends on the particle mass. After proper choice of the parameters of the space-time deformation (it depends on the quantum constant) the statistical description of free stochastic particles appears to be equivalent to the quantum description. \cite{R98}. It is important, that the quantum principles are not used. In other words, the quantum properties are descried by means of the correctly chosen space-time geometry. It is impossible in the framework of the Riemannian geometry as well as in the framework of any generalized geometry, constructed by means of the conventional method of the generalized geometry construction.
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\begin{document}
\title{Spectral radius and $k$-factor-critical graphs
} \author{\small Sizhong Zhou$^{1}$\footnote{E-mail address: zsz\_cumt@163.com (S. Zhou)}, Zhiren Sun$^{2}$\footnote{E-mail address: 05119@njnu.edu.cn (Z. Sun)}, Yuli Zhang$^{3}$\footnote{Corresponding author. E-mail address: zhangyuli\_djtu@126.com (Y. Zhang)}\\ \small $1$. School of Science, Jiangsu University of Science and Technology,\\ \small Zhenjiang, Jiangsu 212100, China\\ \small $2$. School of Mathematical Sciences, Nanjing Normal University,\\ \small Nanjing, Jiangsu 210023, China\\ \small $3$. School of Science, Dalian Jiaotong University,\\ \small Dalian, Liaoning 116028, China\\ }
\maketitle \begin{abstract} \noindent For a nonnegative integer $k$, a graph $G$ is said to be $k$-factor-critical if $G-Q$ admits a perfect matching for any $Q\subseteq V(G)$
with $|Q|=k$. In this article, we prove spectral radius conditions for the existence of $k$-factor-critical graphs. Our result generalises one previous result on perfect matchings of graphs. Furthermore, we claim that the bounds on spectral radius in Theorem 3.1 are sharp. \\ \begin{flushleft} {\em Keywords:} graph; spectral radius; perfect matching; $k$-factor-critical graph.
(2020) Mathematics Subject Classification: 05C50, 05C70 \end{flushleft} \end{abstract}
\section{Introduction}
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$ which has neither multiple edges nor loops. We denote by $n=|V(G)|$ the order of $G$. The number of odd components in $G$ is denoted by $o(G)$. The number of connected components in $G$ is denoted by $\omega(G)$. For any $v\in V(G)$, we denote by $d_G(v)$ the degree of $v$ in $G$. For any $D\subseteq V(G)$, we denote by $G[D]$ the subgraph of $G$ induced by $D$, and by $G-D$ the graph formed from $G$ by removing the vertices in $D$ and their incident edges. A vertex subset $X$ of $G$ is called an independent set if any two members of $X$ are not adjacent in $G$. For a graph $G$ of order $n$, the adjacency matrix $A(G)$ of $G$ is the $n$-by-$n$ matrix in which entry $a_{ij}$ is 1 or 0 according to whether $v_i$ and $v_j$ are adjacent or not, where $V(G)=\{v_1,v_2,\cdots,v_n\}$. The eigenvalues of the adjacency matrix $A(G)$ are also called the eigenvalues of $G$. The largest eigenvalue of $G$, denoted by $\rho(G)$, is called the spectral radius of $G$. For an integer $k\geq2$, the sequential join $G_1\vee G_2\vee\cdots\vee G_k$ of graphs $G_1,G_2,\cdots,G_k$ is the graph with vertex set $V(G_1\vee G_2\vee\cdots\vee G_k)=V(G_1)\cup V(G_2)\cup\cdots\cup V(G_k)$ and edge set $E(G_1\vee G_2\vee\cdots\vee G_k)=\{e: e\in E(G_i) \ \mbox{for some} \ 1\leq i\leq k \ \mbox{or an unordered pair between} \ V(G_i) \ \mbox{and} \ V(G_{i+1}) \ \mbox{for some} \ 1\leq i\leq k-1\}$. Let $K_n$ and $P_n$ denote the complete graph and the path of order $n$, respectively.
Let $a$ and $b$ be two integers with $0\leq a\leq b$. Then a spanning subgraph $F$ of a graph $G$ is called an $[a,b]$-factor if
$a\leq d_F(v)\leq b$ for all $v\in V(G)$. When $a=b=1$, an $[a,b]$-factor is simply called a 1-factor (or a perfect matching). For a nonnegative integer $k$, a graph $G$ is said to be $k$-factor-critical if $G-Q$ admits a 1-factor for any $Q\subseteq V(G)$ with $|Q|=k$.
Egawa and Furuya \cite{EF} studied the existence of perfect matchings in star-free graphs. Brouwer and Haemers \cite{BH}, O \cite{Os} presented eigenvalue conditions for graphs to possess perfect matchings. Johansson \cite{J} verified an El-Zah\'ar type condition for the existence of $[1,2]$-factors in graphs. Zhou, Bian and Pan \cite{ZBP}, Zhou, Sun and Liu \cite{ZSLo}, Zhou, Wu and Bian \cite{ZWB}, Zhou \cite{Zs}, Zhou \cite{Zd}, Zhou and Bian \cite{ZB}, Gao and Wang \cite{GW}, Gao, Chen and Wang \cite{GCW}, Liu \cite{L}, Wang and Zhang \cite{WZi}, Wu \cite{Wp}, Li and Miao \cite{LM} demonstrated some theorems on $[1,2]$-factors in graphs. Wang and Zhang \cite{WZr} derived a sufficient condition for the existence of factor-critical graphs. Zhou \cite{Za1}, Zhou, Wu and Liu \cite{ZWL}, Zhou, Liu and Xu \cite{ZLX} posed some sufficient conditions for graphs to be factor-critical graphs. Gu and Liu \cite{GL} established a relationship between eigenvalues and factor-critical graphs. Ananchuen and Plummer \cite{AP} showed a result for the existence of 3-factor-critical graphs. Enomoto, Plummer and Saito \cite{EPS} investigated a relationship between neighborhoods of independent sets and $k$-factor-critical graphs. Plummer and Saito \cite{PS} showed some characterizations for graphs to be $k$-factor-critical graphs. Wang and Yu
\cite{WY} proved that a $k$-connected 3-$\gamma$-edge-critical claw-free graph $G$ with minimum degree at least $k+1$ and $k\equiv|V(G)|$ (mod 2) is a $k$-factor-critical graph. Zhai, Wei and Zhang \cite{ZWZ} presented some characterizations for the existence of $k$-factor-critical graphs. More results on graph factors were obtained by Wang and Zhang \cite{WZo}, Zhou et al \cite{Zr,ZL,Za2,Zb}, Gao, Wang and Chen \cite{GWC}.
In this article, we also investigate the problem on the existence of $k$-factor-critical graphs, and characterize $k$-factor-critical graphs with respect to the spectral radius. Our main result will be given in Sections 3.
\section{Preliminary Lemmas}
For a nonnegative integer $k$, a graph $G$ is said to be $k$-factor-critical if $G-Q$ admits a 1-factor for any $Q\subseteq V(G)$ with $|Q|=k$.
\noindent{\textbf{Lemma 2.1}} (\cite{F}). Let $k$ be a nonnegative integer, and let $G$ be a graph of order $n$ with $n\equiv k$ (mod 2). Then $G$ is $k$-factor-critical if and only if $$
o(G-D)\leq|D|-k $$
for any $D\subseteq V(G)$ with $|D|\geq k$.
The following two results are very useful for the proof of the main theorem.
\noindent{\textbf{Lemma 2.2}} (\cite{B}). Let $G$ be a connected graph, and let $H$ be a proper subgraph of $G$. Then $\rho(G)>\rho(H)$.
Let $M$ be a real symmetric matrix whose rows and columns are indexed by $V=\{1,2,\cdots,n\}$. Suppose that $M$ can be written as \begin{align*} M=\left(
\begin{array}{ccc}
M_{11} & \cdots & M_{1s}\\
\vdots & \ddots & \vdots\\
M_{s1} & \cdots & M_{ss}\\
\end{array} \right) \end{align*} by means of partition $\pi: V=V_1\cup V_2\cup\cdots\cup V_s$ , wherein $M_{ij}$ is the submatrix (block) of $M$ derived by rows in $V_i$ and the columns in $V_j$. We denote by $q_{ij}$ the average row sum of $M_{ij}$. Then matrix $M_{\pi}=(q_{ij})$ is said to be the quotient matrix of $M$. If the row sum of every block $M_{ij}$ is a constant, then the partition is equitable.
\noindent{\textbf{Lemma 2.3}} (\cite{YYSX}). Let $M$ be a real matrix with an equitable partition $\pi$, and let $M_{\pi}$ be the corresponding quotient matrix. Then every eigenvalue of $M_{\pi}$ is an eigenvalue of $M$. Furthermore, if $M$ is a nonnegative, then the largest eigenvalues of $M$ and $M_{\pi}$ are equal.
\section{The main theorem and its proof}
In this section, we establish a relationship between spectral radius and $k$-factor-critical graphs.
\noindent{\textbf{Theorem 3.1.}} Let $k$ be a nonnegative integer, and let $G$ be a $(k+1)$-connected graph of order $n$ with $n\equiv k$ (mod 2).\\ (\romannumeral1) For $n\geq k+4$ and $n\notin\{k+6,k+8\}$, or $(k,n)=(0,8)$, if $\rho(G)>\theta(n,k)$, then $G$ is $k$-factor-critical, where $\theta(n,k)$ is the largest root of $x^{3}-(n-4)x^{2}-(n+2k-1)x+2(k+1)(n-k-4)=0$.\\ (\romannumeral2) For $n=k+6$, if $\rho(G)>\frac{k+1+\sqrt{k^{2}+18k+33}}{2}$, then $G$ is $k$-factor-critical.\\ (\romannumeral3) For $k\geq1$ and $n=k+8$, if $\rho(G)>\frac{k+2+\sqrt{k^{2}+24k+64}}{2}$, then $G$ is $k$-factor-critical.
\noindent{\it Proof.} Let $\varphi(x)=x^{3}-(n-4)x^{2}-(n+2k-1)x+2(k+1)(n-k-4)$ and let $\theta(n,k)$ be the largest root of $\varphi(x)=0$. Suppose, to the contrary, that $G$ is not $k$-factor-critical. According to Lemma 2.1, there exists a subset $D\subseteq V(G)$ with $|D|\geq k$
such that $o(G-D)\geq|D|-k+1$. By parity, $o(G-D)\geq|D|-k+2\geq2$. Let $o(G-D)=\beta$ and $|D|=d$. Then we admit $\beta\geq d-k+2\geq2$. Select a $(k+1)$-connected graph $G$ with $n$ vertices such that its spectral radius is as large as possible. In terms of Lemma 2.2 and the choice of $G$, the induced subgraph $G[D]$ and all connected components of $G-D$ are complete graphs. Furthermore, $G=G[D]\vee(G-D)$.
\noindent{\bf Claim 1.} $d\geq k+1$.
\noindent{\it Proof.} Let $d=k$. Then $\omega(G-D)\geq o(G-D)\geq|D|-k+2=d-k+2=2$, which contradicts that $G$ is $(k+1)$-connected. This completes the proof of Claim 1.
$\Box$
\noindent{\bf Claim 2.} $G-D$ does not admit even components.
\noindent{\it Proof.} Assume that there exists an even component $G_e$ in $G-D$. Then we create a new graph $G^{(1)}$ by by joining $G_e$ and $G_o$ (that is, $G_e\vee G_o$), where $G_o$ is an odd component of $G-D$. Clearly, $G_e\vee G_o$ is an odd component of $G^{(1)}-D$ and
$o(G^{(1)}-D)=o(G-D)\geq|D|-k+2$. Furthermore, $G$ is a proper subgraph of $G^{(1)}$. In view of Lemma 2.2, $\rho(G)<\rho(G^{(1)})$, which contradicts the choice of $G$. This completes the proof of Claim 2.
$\Box$
Let $G_1,G_2,\cdots,G_{\beta}$ be the odd components in $G-D$ with $|V(G_1)|=n_1\geq|V(G_2)|=n_2\geq\cdots\geq|V(G_{\beta})|=n_{\beta}$. By virtue of Claim 2, there exists the partition $\{D,V(G_1),V(G_2),\cdots,V(G_{\beta})\}$ of $G$. Thus, the quotient matrix of the partition $\{D,V(G_1),V(G_2),\cdots,V(G_{\beta})\}$ of $G$ equals \begin{align*} \left(
\begin{array}{ccccc}
d-1 & n_1 & n_2 & \cdots & n_{\beta}\\
d & n_1-1 & 0 & \cdots & 0\\
d & 0 & n_2-1 & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
d & 0 & 0 & \cdots & n_{\beta}-1\\
\end{array} \right). \end{align*} Then the characteristic polynomial of the matrix is equal to \begin{align*} f_1(x)&=(x-d+1)(x-n_1+1)\cdots(x-n_{\beta}+1)-dn_1(x-n_2+1)\cdots(x-n_{\beta}+1)\\ &+dn_2(x-n_1+1)(x-n_3+1)\cdots(x-n_{\beta}+1)+\cdots\\ &+(-1)^{i}dn_i(x-n_1+1)\cdots(x-n_{i-1}+1)(x-n_{i+1}+1)\cdots(x-n_{\beta}+1)+\cdots\\ &+(-1)^{\beta}dn_{\beta}(x-n_1+1)\cdots(x-n_{\beta-1}+1). \end{align*}
Since the partition $\{D,V(G_1),V(G_2),\cdots,V(G_{\beta})\}$ is equitable, it follows from Lemma 2.3 that the largest root, say $\rho_1$, of $f_1(x)=0$ equals the spectral radius of $G$. Thus, we possess $\rho(G)=\rho_1$. Since $K_{d+n_1}$ is a proper subgraph of $G$, it follows from Lemma 2.2 that $\rho_1=\rho(G)>\rho(K_{d+n_1})=n_1+d-1$.
\noindent{\bf Claim 3.} $n_2=n_3=\cdots=n_{\beta}=1$.
\noindent{\it Proof.} We first verify $n_{\beta}=1$. Assume that $n_{\beta}\geq3$. If $n_1=1$, then we deduce $1=n_1\geq n_2\geq\cdots\geq n_{\beta}\geq3$, which is a contradiction. Next, we deal with $n_1\geq3$.
We create a new graph $G^{(2)}$ by deleting two vertices in $G_{\beta}$ and adding two vertices to $G_1$ by joining the two vertices to the vertices in $V(G_1)\cup D$. Obviously, $G[D]$ and all connected components in $G^{(2)}-D$ are complete graphs. In what follows, we prove $\rho(G)<\rho(G^{(2)})$.
Assume that $V(G_1')$ and $V(G_{\beta}')$ are the vertex sets obtained from $V(G_1)$ by adding the two vertices and from $V(G_{\beta})$ by deleting the two vertices, respectively. The quotient matrix of the partition $\{D,V(G_1'),V(G_2),\cdots,V(G_{\beta-1}),V(G_{\beta}')\}$ of $G^{(2)}$ admits the characteristic polynomial $f_2(x)$ obtained from $f_1(x)$ by replacing $n_1$ and $n_{\beta}$ by $n_1+2$ and $n_{\beta}-2$, respectively. Note that the partition $\{D,V(G_1'),V(G_2),\cdots,V(G_{\beta-1}),V(G_{\beta}')\}$ is equitable. It follows from Lemma 2.3 that the largest root, say $\rho_2$, of $f_2(x)=0$ equals the spectral radius of $G^{(2)}$. Thus, we derive $\rho(G^{(2)})=\rho_2$. According to $f_1(\rho_1)=0$, $n_1\geq n_{\beta}$ and $\rho_1>n_1+d-1$, we infer \begin{align*} f_2(\rho_1)&=2(\rho_1-d+1)(\rho_1-n_2+1)\cdots(\rho_1-n_{\beta-1}+1)(n_\beta-n_1-2)\\ &-2d(\rho_1-n_2+1)\cdots(\rho_1-n_{\beta-1}+1)(\rho_1+n_1-n_{\beta}+2)\\ &+2dn_2(\rho_1-n_3+1)\cdots(\rho_1-n_{\beta-1}+1)(n_{\beta}-n_1-2)\\ &-2dn_3(\rho_1-n_2+1)(\rho_1-n_4+1)\cdots(\rho_1-n_{\beta-1}+1)(n_{\beta}-n_1-2)+\cdots\\ &+(-1)^{\beta}2d(\rho_1-n_2+1)\cdots(\rho_1-n_{\beta-1}+1)(n_{\beta}-n_1-2)<0, \end{align*} which yields $\rho(G)=\rho_1<\rho_2=\rho(G^{(2)})$, which contradicts the choice of $G$. Hence, we infer $n_{\beta}=1$. Similarly, we may verify $n_2=n_3=\cdots=n_{\beta-1}=1$. This completes the proof of Claim 3.
$\Box$
\noindent{\bf Claim 4.} $\beta=d-k+2$.
\noindent{\it Proof.} Assume that $\beta\geq d-k+4$. Then we establish a new graph $G^{(3)}$ obtained from $G$ by adding an edge to join $G_{\beta-1}$ and $G_{\beta}$ ($G_{\beta-1}$ and $G_{\beta}$ are two vertices by Claim 3). It is obvious that $o(G^{(3)}-D)\geq d-k+2$ and $G$ is a proper subgraph of $G^{(3)}$. Then it follows from Lemma 2.2 that $\rho(G)<\rho(G^{(3)})$, which contradicts the choice of $G$. Hence, we deduce $\beta\leq d-k+2$ by parity. On the other hand, $\beta\geq d-k+2$. Thus, we infer $\beta=d-k+2$. This completes the proof of Claim 4.
$\Box$
In what follows, we consider two cases by the value of $n_1$.
\noindent{\bf Case 1.} $n_1=1$.
In this case, we possess $G=K_d\vee(n-d)K_1=K_d\vee(d-k+2)K_1$ and $n=d+\beta=d+d-k+2=2d-k+2$ by Claims 2--4. Note that the quotient matrix of the adjacency matrix of $G$ with the partition $\{V(K_d),V((n-d)K_1)\}$ is equal to \begin{align*} \left(
\begin{array}{cc}
d-1 & n-d\\
d & 0\\
\end{array} \right). \end{align*} Then the characteristic polynomial of the matrix is equal to \begin{align*} f_3(x)&=x(x-d+1)-d(n-d)\\ &=x^{2}-(d-1)x-d(n-d). \end{align*} Note that the partition $\{V(K_d),V((n-d)K_1)\}$ is equitable. By virtue of Lemma 2.3, the largest root, say $\rho_3$, of $f_3(x)=0$ is equal to the spectral radius of $G$. Thus, we possess \begin{align}\label{eq:3.1} \rho(G)=\rho(K_d\vee(n-d)K_1)=\rho_3=\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}. \end{align}
Recall that $n=2d-k+2$. If $d=k+1$, then $n=k+4$ and $\rho(G)=\rho_3=\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}=\frac{k+\sqrt{k^{2}+12k+12}}{2}=\theta(k+4,k)$, which contradicts $\rho(G)>\theta(n,k)$ for $n=k+4$. If $d=k+2$, then $n=k+6$ and $\rho(G)=\rho_3=\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}=\frac{k+1+\sqrt{k^{2}+18k+33}}{2}$, which contradicts $\rho(G)>\frac{k+1+\sqrt{k^{2}+18k+33}}{2}$. Next, we deal with $d\geq k+3$.
In light of \eqref{eq:3.1}, $f_3(\rho_3)=0$ and $n=2d-k+2$, we deduce \begin{align}\label{eq:3.2} \varphi(\rho_3)=&\varphi(\rho_3)-\rho_3f_3(\rho_3)\nonumber\\ =&\rho_3^{3}-(n-4)\rho_3^{2}-(n+2k-1)\rho_3+2(k+1)(n-k-4)\nonumber\\ &-\rho_3^{3}+(d-1)\rho_3^{2}+d(n-d)\rho_3\nonumber\\ =&-(n-d-3)\rho_3^{2}+((d-1)n-d^{2}-2k+1)\rho_3+2(k+1)(n-k-4)\nonumber\\ =&-(d-k-1)\rho_3^{2}+(d+1)(d-k-1)\rho_3+4(k+1)(d-k-1)\nonumber\\ =&(d-k-1)(-\rho_3^{2}+(d+1)\rho_3+4k+4). \end{align} Let $g_3(\rho_3)=-\rho_3^{2}+(d+1)\rho_3+4k+4$. Together with \eqref{eq:3.1} and $n=2d-k+2$, we have \begin{align}\label{eq:3.3} g_3(\rho_3)=&-\rho_3^{2}+(d+1)\rho_3+4k+4\nonumber\\ =&-\left(\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}\right)^{2}\nonumber\\ &+(d+1)\cdot\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}+4k+4\nonumber\\ =&-d^{2}-d+kd+4k+3+\sqrt{(d-1)^{2}+4d(d-k+2)}. \end{align}
If $d=k+3$, then it follows from \eqref{eq:3.3} that \begin{align*} g_3(\rho_3)=&-d^{2}-d+kd+4k+3+\sqrt{(d-1)^{2}+4d(d-k+2)}\\ =&-9+\sqrt{k^{2}+24k+64}. \end{align*} For $d=k+3$ and $k=0$, we deduce $n=8$ and $g_3(\rho_3)<0$. Combining this with \eqref{eq:3.2}, we admit $\varphi(\rho_3)=(d-k-1)g_3(\rho_3)<0$, which yields $\rho(G)=\rho_3<\theta(n,k)$ for $(n,k)=(8,0)$. For $d=k+3$ and $k\geq1$, we have $n=k+8$ and $\rho(G)=\rho_3=\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}=\frac{k+2+\sqrt{k^{2}+24k+64}}{2}$, which contradicts $\rho(G)>\frac{k+2+\sqrt{k^{2}+24k+64}}{2}$.
\noindent{\bf Claim 5.} If $d\geq k+4$, then $d^{2}+d-kd-4k-3>\sqrt{(d-1)^{2}+4d(d-k+2)}$.
\noindent{\it Proof.} By a direct computation, we obtain \begin{align}\label{eq:3.4} (d^{2}+d-&kd-4k-3)^{2}-((d-1)^{2}+4d(d-k+2))\nonumber\\ =&d^{4}+(2-2k)d^{3}+(k^{2}-10k-10)d^{2}+(8k^{2}+2k-12)d+16k^{2}+24k+8\nonumber\\ :=&h_3(d), \end{align} where $h_3(d)=d^{4}+(2-2k)d^{3}+(k^{2}-10k-10)d^{2}+(8k^{2}+2k-12)d+16k^{2}+24k+8$. Let $h_3(x)=x^{4}+(2-2k)x^{3}+(k^{2}-10k-10)x^{2}+(8k^{2}+2k-12)x+16k^{2}+24k+8$ be a real function in $x$ with $x\in[k+4,+\infty)$. The derivative function of $h_3(x)$ is $$ h_3'(x)=4x^{3}+3(2-2k)x^{2}+2(k^{2}-10k-10)x+8k^{2}+2k-12. $$ Furthermore, we possess $$ h_3''(x)=12x^{2}+6(2-2k)x+2(k^{2}-10k-10). $$ Note that $$ -\frac{6(2-2k)}{24}=\frac{k-1}{2}<k+4\leq d. $$ Hence, $h_3''(x)$ is increasing in the interval $[k+4,+\infty)$. Thus $h_3''(x)\geq h_3''(k+4)=2k^{2}+40k+220>0$, which implies that $h_3'(x)$ is increasing in the interval $[k+4,+\infty)$ and so $h_3'(x)\geq h_3'(k+4)=2k^{2}+46k+260>0$. Thus, we infer that $h_3(x)$ is increasing in the interval $[k+4,+\infty)$. Together with $d\geq k+4$, we derive $$ h_3(d)\geq h_3(k+4)=4k+184>0. $$ Combining this with \eqref{eq:3.4}, we deduce $(d^{2}+d-kd-4k-3)^{2}>(d-1)^{2}+4d(d-k+2)$, that is, $d^{2}+d-kd-4k-3>\sqrt{(d-1)^{2}+4d(d-k+2)}$. This completes the proof of Claim 5.
$\Box$
If $d\geq k+4$, then it follows from \eqref{eq:3.2}, \eqref{eq:3.3} and Claim 5 that \begin{align*} \varphi(\rho_3)=&(d-k-1)g_3(\rho_3)\\ =&(d-k-1)(-d^{2}-d+kd+4k+3+\sqrt{(d-1)^{2}+4d(d-k+2)})\\ <&0, \end{align*} which leads to $\rho(G)=\rho_3<\theta(n,k)$, which contradicts $\rho(G)>\theta(n,k)$.
\noindent{\bf Case 2.} $n_1\geq3$.
In this case, we infer $G=K_d\vee(K_{n_1}\cup(\beta-1)K_1)=K_d\vee(K_{n_1}\cup(d-k+1)K_1)$ and $n=d+n_1+d-k+1=2d+n_1-k+1$ by Claims 2--4. In terms of the partition $\{V(K_d),V(K_{n_1}),V((d-k+1)K_1)\}$, the quotient matrix of the adjacency matrix of $G$ is equal to \begin{align*} \left(
\begin{array}{ccc}
d-1 & n-2d+k-1 & d-k+1\\
d & n-2d+k-2 & 0\\
d & 0 & 0\\
\end{array} \right). \end{align*} Then the characteristic polynomial of the matrix equals \begin{align*} f_4(x)=x^{3}-(n-d+k-3)x^{2}-(n+d^{2}-kd+k-2)x+d(d-k+1)(n-2d+k-2). \end{align*} Note that the partition $\{V(K_d),V(K_{n_1}),V((d-k+1)K_1)\}$ is equitable. According to Lemma 2.3, the largest root, say $\rho_4$, of $f_4(x)=0$ equals the spectral radius of $G$. Thus, we have $\rho(G)=\rho_4$ and $f_4(\rho_4)=0$.
Next we are to verify $\varphi(\rho_4)<0$. By plugging the value $\rho_4$ into $x$ of $\varphi(x)-f_4(x)$, we possess \begin{align}\label{eq:3.5} \varphi(\rho_4)=&\varphi(\rho_4)-f_4(\rho_4)\nonumber\\ =&(d-k-1)(-\rho_4^{2}+(d+1)\rho_4-(d+2)n+2d^{2}-(k-6)d+2k+8)\nonumber\\ =&(d-k-1)g_4(\rho_4), \end{align} where $g_4(\rho_4)=-\rho_4^{2}+(d+1)\rho_4-(d+2)n+2d^{2}-(k-6)d+2k+8$. Note that $d\geq k+1$ and $n=2d+n_1-k+1\geq2d-k+4$. Then we obtain \begin{align}\label{eq:3.6} \frac{d+1}{2}<\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}. \end{align}
Since $K_d\vee(n-d)K_1$ is a proper subgraph of $G$, we infer \begin{align}\label{eq:3.7} \rho_4=\rho(G)>\rho(K_d\vee(n-d)K_1)=\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2} \end{align} by \eqref{eq:3.1} and Lemma 2.2. Recall that $g_4(\rho_4)=-\rho_4^{2}+(d+1)\rho_4-(d+2)n+2d^{2}-(k-6)d+2k+8$. In terms of \eqref{eq:3.6} and \eqref{eq:3.7}, we get $$ \frac{d+1}{2}<\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}<\rho(G)=\rho_4. $$ Hence, we infer \begin{align}\label{eq:3.8} g_4(\rho_4)<&g_4\left(\frac{d-1+\sqrt{(d-1)^{2}+4d(n-d)}}{2}\right)\nonumber\\ =&-(2d+2)n+3d^{2}+(7-k)d+2k+7+\sqrt{(d-1)^{2}+4d(n-d)}. \end{align}
\noindent{\bf Claim 6.} If $d\geq k+1$ and $n\geq2d-k+4$, then $(2d+2)n-3d^{2}-(7-k)d-2k-7>\sqrt{(d-1)^{2}+4d(n-d)}$.
\noindent{\it Proof.} By a direct computation, we possess \begin{align}\label{eq:3.9} ((2d+2)n&-3d^{2}-(7-k)d-2k-7)^{2}-((d-1)^{2}+4d(n-d))\nonumber\\ =&(2d+2)^{2}n^{2}-(12d^{3}+(40-4k)d^{2}+(60+4k)d+8k+28)n\nonumber\\ &+9d^{4}+3(14-2k)d^{3}+(k^{2}-2k+94)d^{2}\nonumber\\ &+(-4k^{2}+14k+100)d+4k^{2}+28k+48\nonumber\\ :=&h_4(n), \end{align} where $h_4(n)=(2d+2)^{2}n^{2}-(12d^{3}+(40-4k)d^{2}+(60+4k)d+8k+28)n+9d^{4}+3(14-2k)d^{3}+(k^{2}-2k+94)d^{2}+(-4k^{2}+14k+100)d+4k^{2}+28k+48$. Recall that $n\geq2d-k+4$. Then $$ \frac{12d^{3}+(40-4k)d^{2}+(60+4k)d+8k+28}{2(2d+2)^{2}}<2d-k+4\leq n. $$ Hence, we deduce \begin{align}\label{eq:3.10} h_4(n)\geq&h_4(2d-k+4)\nonumber\\ =&d^{4}+(10-2k)d^{3}+(k^{2}-18k+22)d^{2}+(8k^{2}-38k-4)d+16k^{2}-8k\nonumber\\ :=&l_4(d) \end{align} Let $l_4(x)=x^{4}+(10-2k)x^{3}+(k^{2}-18k+22)x^{2}+(8k^{2}-38k-4)x+16k^{2}-8k$ be a real function in $x$ with $x\in[k+1,+\infty)$. The derivative function of $l_4(x)$ is $$ l_4'(x)=4x^{3}+3(10-2k)x^{2}+2(k^{2}-18k+22)x+8k^{2}-38k-4. $$ Furthermore, we have $$ l_4''(x)=12x^{2}+6(10-2k)x+2(k^{2}-18k+22). $$ Note that $$ -\frac{6(10-2k)}{24}=\frac{k-5}{2}<k+1\leq d. $$ Hence, $l_4''(x)$ is increasing in the interval $[k+1,+\infty)$. Thus $l_4''(x)\geq l_4''(k+1)=2k^{2}+36k+116>0$, which implies that $l_4'(x)$ is increasing in the interval $[k+1,+\infty)$ and so $l_4'(x)\geq l_4'(k+1)=4k^{2}+36k+74>0$. Thus, we infer that $l_4(x)$ is increasing in the interval $[k+1,+\infty)$. Together with $d\geq k+1$, we obtain $$ l_4(d)\geq l_4(k+1)=3k^{2}+8k+29>0. $$ Combining this with \eqref{eq:3.9} and \eqref{eq:3.10}, we deduce $((2d+2)n-3d^{2}-(7-k)d-2k-7)^{2}>(d-1)^{2}+4d(n-d)$, that is, $(2d+2)n-3d^{2}-(7-k)d-2k-7>\sqrt{(d-1)^{2}+4d(n-d)}$. This completes the proof of Claim 6.
$\Box$
If $d\geq k+1$ and $n\geq2d-k+4$, then it follows from \eqref{eq:3.5}, \eqref{eq:3.8} and Claim 6 that \begin{align*} \varphi(\rho_4)=&(d-k-1)g_4(\rho_4)\\ \leq&(d-k-1)(-(2d+2)n+3d^{2}+(7-k)d+2k+7+\sqrt{(d-1)^{2}+4d(n-d)})\\ \leq&0, \end{align*} which yields $\rho(G)=\rho_4\leq\theta(n,k)$, which contradicts $\rho(G)>\theta(n,k)$ for $n\geq k+4$ and $n\notin\{k+6,k+8\}$, or $(k,n)=(0,8)$.
As for $n=k+6$, one has $\varphi(x)=x^{3}-(k+2)x^{2}-(3k+5)x+4(k+1)$ and $\varphi'(x)=3x^{2}-2(k+2)x-3k-5$. By a direct computation, we have $\varphi\Big(\frac{k+1+\sqrt{k^{2}+18k+33}}{2}\Big)=k-3+\sqrt{k^{2}+18k+33}>0$ and $\varphi'\Big(\frac{k+1+\sqrt{k^{2}+18k+33}}{2}\Big)=\frac{k^{2}+18k+37+(k-1)\sqrt{k^{2}+18k+33}}{2}>0$, and so $\rho(G)=\rho_4\leq\theta(n,k)<\frac{k+1+\sqrt{k^{2}+18k+33}}{2}$, which is a contradiction to $\rho(G)>\frac{k+1+\sqrt{k^{2}+18k+33}}{2}$ for $n=k+6$.
As for $k\geq1$ and $n=k+8$, one has $\varphi(x)=x^{3}-(k+4)x^{2}-(3k+7)x+8(k+1)$ and $\varphi'(x)=3x^{2}-2(k+4)x-3k-7$. By a direct calculation, we derive $\varphi\Big(\frac{k+2+\sqrt{k^{2}+24k+64}}{2}\Big)=-18+2\sqrt{k^{2}+24k+64}>0$ and $\varphi'\Big(\frac{k+2+\sqrt{k^{2}+24k+64}}{2}\Big)=\frac{k^{2}+24k+72+(k-2)\sqrt{k^{2}+24k+64}}{2}>0$, and so $\rho(G)=\rho_4\leq\theta(n,k)<\frac{k+2+\sqrt{k^{2}+24k+64}}{2}$, which is a contradiction to $\rho(G)>\frac{k+2+\sqrt{k^{2}+24k+64}}{2}$ for $k\geq1$ and $n=k+8$. This completes the proof of Theorem 3.1.
$\Box$
\section{Extremal graphs}
In this section, we claim that the spectral radius conditions in Theorem 3.1 are sharp.
\noindent{\textbf{Theorem 4.1.}} Let $k$ and $n$ be two nonnegative integers with $n\equiv k$ (mod 2), and let $\theta(n,k)$ be the largest root of $x^{3}-(n-4)x^{2}-(n+2k-1)x+2(k+1)(n-k-4)=0$. For $n\geq k+4$ and $n\notin\{k+6,k+8\}$, or $(k,n)=(0,8)$, we possess $\rho(K_{n-k-3}\vee K_{k+1}\vee(2K_1))=\theta(n,k)$ and $K_{n-k-3}\vee K_{k+1}\vee(2K_1)$ is not a $k$-factor-critical graph. For $n=k+6$, we admit $\rho(K_{k+2}\vee(4K_1))=\frac{k+1+\sqrt{k^{2}+18k+33}}{2}$ and $K_{k+2}\vee(4K_1)$ is not a $k$-factor-critical graph. For $n=k+8$, we have $\rho(K_{k+3}\vee(5K_1))=\frac{k+2+\sqrt{k^{2}+24k+64}}{2}$ and $K_{k+3}\vee(5K_1)$ is not a $k$-factor-critical graph.
\noindent{\it Proof.} Consider the vertex partition $\{V(K_{n-k-3}),V(K_{k+1}),V(2K_1)\}$ of $K_{n-k-3}\vee K_{k+1}\vee(2K_1)$. The quotient matrix of the partition $\{V(K_{n-k-3}),V(K_{k+1}),V(2K_1)\}$ of $K_{n-k-3}\vee K_{k+1}\vee(2K_1)$ is equal to \begin{align*} \left(
\begin{array}{ccc}
n-k-4 & k+1 & 0\\
n-k-3 & k & 2\\
0 & k+1 & 0\\
\end{array} \right). \end{align*} Then the characteristic polynomial of the matrix equals $x^{3}-(n-4)x^{2}-(n+2k-1)x+2(k+1)(n-k-4)$. Note that the partition is equitable. Then it follows from Lemma 2.3 that the largest root of $x^{3}-(n-4)x^{2}-(n+2k-1)x+2(k+1)(n-k-4)=0$ equals the spectral radius of the graph $K_{n-k-3}\vee K_{k+1}\vee(2K_1)$. That is, $\rho(K_{n-k-3}\vee K_{k+1}\vee(2K_1))=\theta(n,k)$. Set $D=V(K_{k+1})$. Then
$o(K_{n-k-3}\vee K_{k+1}\vee(2K_1)-D)=3>1=(k+1)-k=|D|-k$. According to Lemma 2.1, $K_{n-k-3}\vee K_{k+1}\vee(2K_1)$ is not a $k$-factor-critical graph.
Consider the vertex partition $\{V(K_{k+2}),V(4K_1)\}$ of $K_{k+2}\vee(4K_1)$. The quotient matrix of the partition $\{V(K_{k+2}),V(4K_1)\}$ of $K_{k+2}\vee(4K_1)$ equals \begin{align*} \left(
\begin{array}{cc}
k+1 & 4\\
k+2 & 0\\
\end{array} \right). \end{align*} Then the characteristic polynomial of the matrix is $x^{2}-(k+1)x-4(k+2)$. Note that the partition is equitable. In view of Lemma 2.3, the largest root of $x^{2}-(k+1)x-4(k+2)=0$ is equal to the spectral radius of the graph $K_{k+2}\vee(4K_1)$. Thus, we derive
$\rho(K_{k+2}\vee(4K_1))=\frac{k+1+\sqrt{k^{2}+18k+33}}{2}$. Set $D=V(K_{k+2})$. Then $o(K_{k+2}\vee(4K_1)-D)=4>2=(k+2)-k=|D|-k$. By means of Lemma 2.1, $K_{k+2}\vee(4K_1)$ is not a $k$-factor-critical graph.
Consider the vertex partition $\{V(K_{k+3}),V(5K_1)\}$ of $K_{k+3}\vee(5K_1)$. The quotient matrix of the partition $\{V(K_{k+3}),V(5K_1)\}$ of $K_{k+3}\vee(5K_1)$ is equal to \begin{align*} \left(
\begin{array}{cc}
k+2 & 5\\
k+3 & 0\\
\end{array} \right). \end{align*} Then the characteristic polynomial of the matrix equals $x^{2}-(k+2)x-5(k+3)$. Note that the partition is equitable. According to Lemma 2.3, the largest root of $x^{2}-(k+2)x-5(k+3)=0$ equals the spectral radius of the graph $K_{k+3}\vee(5K_1)$. Thus, we get
$\rho(K_{k+3}\vee(5K_1))=\frac{k+2+\sqrt{k^{2}+24k+64}}{2}$. Set $D=V(K_{k+3})$. Then $o(K_{k+3}\vee(5K_1)-D)=5>3=(k+3)-k=|D|-k$. In terms of Lemma 2.1, $K_{k+3}\vee(5K_1)$ is not a $k$-factor-critical graph.
$\Box$
\section*{Data availability statement}
My manuscript has no associated data.
\section*{Declaration of competing interest}
The authors declare that they have no conflicts of interest to this work.
\end{document} | arXiv | {
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\begin{document}
\title{An odd $[1,b]$-factor in regular graphs from eigenvalues} \author{Sungeun Kim\thanks{Incheon Academy of Science and Arts, Korea, Incheon, 22009, tjddms9282@gmail.com}\,
Suil O\thanks{Department of Applied Mathematics and Statistics, The State University of New York, Korea, Incheon, 21985, suil.o@sunykorea.ac.kr. Research supported by NRF-2017R1D1A1B03031758 and by NRF-2018K2A9A2A06020345}\,
Jihwan Park\thanks{Incheon Academy of Science and Arts, Korea, Incheon, 22009, bjihwan37@gmail.com}\,
and Hyo Ree\thanks{Incheon Academy of Science and Arts, Korea, Incheon, 22009, reehyo2234@naver.com} }
\maketitle
\begin{abstract} An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $d_H(v)$ is odd and $1\le d_H(v) \le b$. Let $\lambda_3(G)$ be the third largest eigenvalue of the adjacency matrix of $G$. For positive integers $r \ge 3$ and even $n$, Lu, Wu, and Yang~\cite{LWY} proved a lower bound for $\lambda_3(G)$ in an $n$-vertex $r$-regular graph $G$ to gurantee the existence of an odd $[1,b]$-factor in $G$. In this paper, we improve the bound; it is sharp for every $r$.\\
\noindent \textbf{Keywords:} Odd $[1,b]$-factor, eigenvalues\\
\noindent \textbf{AMS subject classification 2010:} 05C50, 05C70 \end{abstract}
\section{Introduction} In this paper we deal only with finite and undirected graphs without loops or multiple edges. The {\it adjacency matrix} $A(G)$ of $G$ is the $n$-by-$n$ matrix in which entry $a_{i,j}$ is 1 or 0 according to whether $v_i$ and $v_j$ are adjacent or not, where $V(G) = \{v_1,\ldots, v_n\}$. The {\it eigenvalues} of $G$ are the eigenvalues of its adjacency matrix $A(G)$. Let $\lambda_1(G),\ldots, \lambda_n(G)$ be its eigenvalues in nonincreasing order. Note that the spectral radius of $G$, written $\rho(G)$ equals $\lambda_1(G)$.
The degree of a vertex $v$ in $V(G)$, written $d_G(v)$, is the number of vertices adjacent to $v$. An {\it odd} (or {\it even) $[a,b]$-factor} of a graph $G$ is a spanning subgraph $H$ of $G$ such that for each vertex $v \in V(G)$, $d_H(v)$ is odd (or even) and $a \le d_H(v) \le b$; an $[a,a]$-factor is called the {\it $a$-factor}. For a positive integer $r$, a graph is {\it $r$-regular} if every vertex has the same degree $r$. Note that $\lambda_1(G)=r$ if $G$ is $r$-regular. Many researchers proved the conditions for a graph to have an $a$-factor, or (even or odd) $[a,b]$-factor. (See \cite{AKNT, K, M, N}) Brouwer and Haemers started to investiage the relations between eigenvalues and the existence of $1$-factor.
In fact, they~\cite{BH2} proved that if $G$ is an $r$-regular graph without an 1-factor, then $$\lambda_3(G) > \begin{cases} r-1 + \frac 3{r+1} ~~\text{ if } r \text{ is even, }\\ r-1 + \frac 3{r+2} ~~\text{ if } r \text{ is odd} \end{cases}$$ by using Tutte’s 1-Factor Theorem~\cite{T}, which is a special case of Berge-Tutte Formula~\cite{B}.
Cioab{\v a}, Gregory, and Haemers~\cite{CGH} improved their bound and in fact proved that if $G$ is an $r$-regular graph without an 1-factor, then $$\lambda_3(G) \ge \begin{cases} \theta = 2.85577... & \text{if } r=3, \\ \frac 12 (r-2 + \sqrt{r^2+12}) & \text{if } r\ge 4 \text{ is } \text{ even}, \\ \frac 12 (r-3 + \sqrt{(r+1)^2+16}) & \text{if } r\ge 5 \text{ is } \text{ odd}, \end{cases}$$ where $\theta$ is the largest root of $x^3 - x^2 - 6x +2=0$. More generally, O and Cioab{\v a}~\cite{CO} determined connections between the eigenvalues of a $t$-edge connected $r$-regular graph and its matching number when $1 \le t \le r - 2$. In 2010, Lu, Wu, and Yang~\cite{LWY} proved that if an $r$-regular graph $G$ with even number of vertices has no odd $[1,b]$-factor, then $$\lambda_3(G) > \begin{cases} r - \frac{\lceil \frac rb \rceil -2}{r+1}+\frac 1{(r+1)(r+2)}~~\text{ if } r \text{ is even and } \lceil \frac rb \rceil \text{ is even,}\\ r - \frac{\lceil \frac rb \rceil -1}{r+1}+\frac 1{(r+1)(r+2)}~~\text{ if } r \text{ is even and } \lceil \frac rb \rceil \text{ is odd,}\\ r - \frac{\lceil \frac rb \rceil -1}{r+1}+\frac 1{(r+2)^2}~~~~~~\text{ if } r \text{ is odd and } \lceil \frac rb \rceil \text{ is even,}\\
r - \frac{\lceil \frac rb \rceil -2}{r+1}+\frac 1{(r+2)^2}~~~~~~\text{ if } r \text{ is odd and } \lceil \frac rb \rceil \text{ is odd.} \end{cases}$$
To prove the above bounds in the paper~\cite{LWY}, they used Amahashi's result.
\begin{thm}{\rm \cite{A}}\label{A}
Let $G$ be a graph and let $b$ be a positive odd integer. Then $G$ contains an odd $[1,b]$-factor
if and only if for every subset $S\subseteq V(G)$, $o(G-S)\le b|S|$, where $o(H)$ is the number of odd components in a graph $H$. \end{thm}
Thoerem~\ref{A} guarantees that if there is no odd $[1,b]$-factor in an $r$-regular graph, then there exists a subset $S \in V(G)$ such that $o(G-S) > b|S|$. By counting the number of edges between $S$ and $G-S$, we can show that $G-S$ has at least three odd components $Q_1, Q_2, Q_3$ such that $|[V(Q_i),S]| \le r-1$ (see the proof of Theorem~\cite{LWY} or Theorem~\ref{main}). Then they found lower bounds for the largest eigenvalue in a graph in the family ${\cal F}_{r,b}$, where ${\cal F}_{r,b}$ is a family of such a possible component depending on $r$ and $b$, and those bounds are appeared above.
In this paper, we improve their bound and in fact prove that if $G$ is an $n$-vertex $r$-regular graph without an odd $[1,b]$-factor, then $$\lambda_3(G) \ge \rho(r,b),$$ where $$\rho(r,b)=\begin{cases}
\frac{r-2+\sqrt{(r+2)^2-4(\lceil \frac rb\rceil-2)}}{2} & \text{ if both } r \text{ and } \lceil \frac rb\rceil \text{ are even, }\\
\frac{r-2+\sqrt{(r+2)^2-4(\lceil \frac rb\rceil-1)}}{2} & \text{ if } r \text{ is even and } \lceil \frac rb \rceil \text{ is odd, }\\
\frac{r-3+\sqrt{(r+3)^2-4(\lceil \frac rb\rceil-2)}}{2} & \text{ if both } r \text{ and } \lceil \frac rb\rceil \text{ are odd,} \\
\frac{r-3+\sqrt{(r+3)^2-4(\lceil \frac rb\rceil-1)}}{2} & \text{ if } r \text{ is odd and } \lceil \frac rb\rceil \text{ is even.}
\end{cases}$$
The bounds that we found are sharp in a sense that there exists a graph $H$ in ${\cal F}_{r,b}$ such that $\lambda_1(H)=\rho(r,b)$.
For undefined terms, see West~\cite{W} or Godsil and Royle~\cite{GR}.
\section{Construction}
Suppose that $\varepsilon=\begin{cases} 2 & \text{if } r \text{ and } \lceil \frac{r}{b} \rceil \text{ has same parity} \\ 1 & \text{otherwise} \end{cases}$ and $\eta= \lceil \frac{r}{b} \rceil - \varepsilon$. In this section, we provide graphs $H_{r,\eta}$ such that $\lambda_1(H_{r,\eta})=\rho(r,b)$. These graphs show that the bounds in Theorem~\ref{main} are sharp.
Now, we define the graph $H_{r,\eta}$ as follows: $$H_{r,\eta}=\begin{cases} \mathrm{K}_{r+1-\eta} \vee \overline{\frac{\eta}{2}\mathrm{K}_2} &\text{if } r \text{ is even, }\\ \overline{\mathrm{C}_\eta} \vee \overline{\frac{r+2-\eta}{2}\mathrm{K}_2} & \text{if } r \text{ is odd. } \end{cases}$$
To compute the spectral radius of $H_{r,\eta}$, the notion of “equitable partition” of a vertex set in a graph is used. Consider a partition $V(G) = V_1 \cup \cdots \cup V_s$ of the vertex set of a graph $G$ into $s$ non-empty subsets. For $1 \le i, j \le s$, let $q_{i,j}$ denote the average number of neighbours in $V_j$ of the vertices in $V_i$. The quotient matrix of this partition is the $s \times s$ matrix whose $(i, j)$-th entry equals $q_{i,j}$. The eigenvalues of the quotient matrix interlace the eigenvalues of $G$. This partition is {\it equitable} if for each $1 \le i, j \le s$, any vertex $v \in V_i$ has exactly $q_{i,j}$ neighbours in $V_j$. In this case, the eigenvalues of the quotient matrix are eigenvalues of $G$ and the spectral radius of the quotient matrix equals the spectral radius of $G$ (see \cite{BH},\cite{GR} for more details).
\begin{theorem} For $r \ge 3$ and $b \ge 1$, we have $\lambda_{1}(H_{r,\eta})=\rho(r,b)$. \end{theorem} \begin{proof} We prove this theorem only in the case when $r$ is odd because the proof of the other case is similar.
Consider the vertex partition $\{V(\overline{\mathrm{C}_\eta}),V(\overline{\frac{r+2-\eta}{2}\mathrm{K}_2})\}$ of $H_{r,\eta}$. The quotient matrix of the vertex partitions equals
$$Q=\begin{pmatrix}
\eta-3 & r+2-\eta \\
\eta & r-\eta
\end{pmatrix}
$$
The characteristic polynomail of $Q$ is
$$ p(x)=(x-\eta+3)(x-r+\eta)-(r+2-\eta)\eta.$$
Since the vertex partition is equitable, the largest root of the graph $H_{r,\eta}$ equals the largest root of the polynomial, which is $\lambda_1(Q) = \frac{ r-3 + \sqrt{ (r+3)^2 -4\eta } }2$.
\end{proof}
\section{Main results}
In this section, we prove an upper bound for $\lambda_3(G)$ in an $r$-regular graph $G$ with even number of vertices to guarantee the existence of an odd $[1,b]$-factor by using Theorem~\ref{A} and Theorem~\ref{ind}.
\begin{thm}{\rm \cite{BH,GR}}\label{ind}
If $H$ is an induced subgraph of a graph $G$, then $\lambda_i(H) \le \lambda_i(G)$ for all $i \in \{1,\ldots,|V(H)|\}$. \end{thm}
\begin{thm}\label{main}
Let $r \ge 3$, and $b$ be a positive odd integer less than $r$. If $\lambda_3(G)$ of an $r$-regular graph $G$ with even number of vertices is smaller than $\rho(r,b)$, then $G$ has an odd $[1,b]$-factor. \end{thm} \begin{proof} We prove the contrapositive. Assume that an $r$-regular graph $G$ with even number of vertices has no odd $[1,b]$-factor. By Theorem~\ref{A}, there exists a vertex subset $S\subseteq V(G)$ such that
$o(G-S) > b|S|$. Note that since $|V(G)|$ is even, $b$ is odd, and $o(G-S)\equiv |S|~(\!\!\!\mod 2)$, we have $o(G-S)\ge b|S|+2$. Let $G_1,\ldots,G_q$ be the odd components of $G-S$, where $q=o(G-S)$.\\
\noindent
{\it Claim 1. There are at least three odd components, say $G_1,G_2,G_3$, such that $|[V(G_i),S]|< \lceil\frac rb\rceil$ for all $i\in\{1,2,3\}$.}
Assume to the contrary that there are at most two such odd components in $G-S$. Since $G$ is $r$-regular, we have
$$r|S| \ge \sum_{i=1}^q |[V(G_i),S]| \ge \lceil\frac rb\rceil(q-2)+2 \ge \lceil\frac rb\rceil b|S|+2 \ge r|S|+2,$$ which is a contradiction.
By Theorem~\ref{ind}, we have \begin{equation} \lambda_3(G) \ge \lambda_3(G_1\cup G_2 \cup G_3) \ge \min_{i \in \{1,2,3\}}\lambda_1(G_i). \end{equation}
Now, we prove that if $H$ is an odd component of $G-S$ such that $|[V(H),S]| < \lceil\frac rb\rceil$, then $\lambda_1(H) \ge \rho(r,b)$.\\
\noindent
{\it Claim 2. If $H$ is an odd components of $G-S$ such that $|[V(H),S]| < \lceil\frac rb\rceil$ and if $\lambda_1(H)\le \lambda_1(H')$ for all odd components $H'$ in $G-S$ such that $|[V(H'),S]| < \lceil\frac rb\rceil$, then we have $|V(H)|=\begin{cases} r+2 \text{ if } r \text{ is odd,}\\ r+1 \text{ if } r \text{ is even}
\end{cases}$, and ~$2|E(H)|=\begin{cases}
r(r+2)-\eta \text{ if } r \text{ is odd,} \\
r(r+1)-\eta \text{ if } r \text{ is even}. \end{cases}$}
Let $x=\begin{cases} 1 \text{ if } r \text{ is odd,}\\ 0 \text{ if } r \text{ is even}. \end{cases}$
Since $|[V(H),S]| < \lceil\frac rb\rceil < r$ and $G$ is $r$-regular, we have $|V(H)|\ge r+1+x$ since $H$ has an odd number of vertices. If $|V(H)| > r+1+x$, then we have $|V(H)| \ge r+3+x$ since $H$ has an odd number of vertices.
Thus it suffices to show $\rho(r,b) < \lambda_1(H)$ if $|V(H)| \ge r+3+x$. By using the fact that $\lambda_1(G) \ge \frac{2|E(G)|}{|V(G)|}$ for any graph $G$, we have
$$\lambda_1(H) > \frac{r|V(H)|-\eta}{|V(H)|} \ge \frac{r(r+3+x)-\eta}{r+3+x} > \frac{r-2-x+\sqrt{(r+2+x)^2-4\eta}}{2}.$$\\
Now, we prove this theorem by considering two cases depending on the parity of $r$.
{\it Case 1. $r$ is even.} By Claim 2, assume that $H$ is an odd component of $G-S$ such that $|[V(H),S]| < \lceil\frac rb\rceil$, $|V(H)|=r+1$, and $2|E(H)|=r(r+1)-\eta$. Then there are at least $r+1-\eta$ vertices of degree $r$. Let $V_1$ be a set of vertices with degree $r$ such that $|V_1|=r+1-\eta$, and let $V_2$ be the remaining vertices in $V(H)$. Then the quotient matrix of the vertex partition $\{V_1,V_2\}$ of $H$ equals $$\begin{pmatrix} r-\eta & \eta\\ r+1-\eta & \eta-2 \end{pmatrix}$$\\ whose characteristic polynomial is $p(x)=(x-r+\eta)(x-\eta+2)-\eta(r+1-\eta)$. Since the largest root of $p(x)$ equals $\rho(r,b)$, we have $\lambda_1(H) \ge \rho(r,b).$\\
{\it Case 2. $r$ is odd.} By Claim 2, assume that $H$ is an odd component of $G-S$ such that $|[V(H),S]| < \lceil\frac rb\rceil$, $|V(H)|=r+2$, and $2|E(H)|=r(r+2)-\eta$. Then there are at least $r+2-\eta$ vertices of degree $r$. Let $V_1$ be a set of vertices with degree $r$ such that $|V_1|=r+2-\eta$, and let $V_2$ be the remaining vertices in $V(H)$. Suppose that there are $m_{12}$ edges between $V_1$ and $V_2$. Note that $(r+2-\eta)(\eta-1) \le m_{12} \le (r+2-\eta)\eta$. Then the quotient matrix of the vertex partion $\{V_1,V_2\}$ of $H$ equals
$$
\begin{pmatrix}
r- \frac{m_{12}}{r+2-\eta} & \frac{m_{12}}{r+2-\eta} \\
\frac{m_{12}}{\eta} & r-1-\frac{m_{12}}{\eta}
\end{pmatrix}
$$
whose characteristic polynomial is $q(x)=(x-r+ \frac{m_{12}}{r+2-\eta})(x-r+1+\frac{m_{12}}{\eta})-\frac{m_{12}^2}{(r+2-\eta)\eta}$.
Note that since $(r+2-\eta)(\eta-1) \le m_{12} \le (r+2-\eta)\eta$, $m_{12}$ can be expressed $m_{12}=(r+2-\eta)\eta -t$, where $0 \le t \le r+2-\eta$. Thus we have
$$q(x)=x^2-(r-3+\frac{t(r+2)}{(r+2-\eta)\eta})x-3r+\eta-\frac{t}{r+2-\eta}+\frac{tr(r+2)}{(r+2-\eta)\eta}$$$$=x^2-(r-3)x-3r+\eta-\frac{t(r+2)}{(r+2-\eta)\eta}x-\frac{t}{r+2-\eta}+\frac{tr(r+2)}{(r+2-\eta)\eta}.$$
Note that $q(\rho(r,b)) =-\frac{t(r+2)}{(r+2-\eta)\eta} (\rho(r,b)+\frac{\eta}{r+2}-r) \le 0$, since $\eta \ge 1$ and $0 \le t \le r+2-\eta$.
\end{proof}
\end{document} | arXiv | {
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\begin{document}
\title{\bf The Minimum Number of Dependent \\ Arcs and a Related Parameter of \\ Generalized Mycielski Graphs}
\author{Hsin-Hao Lai\\ \normalsize Department of Mathematics\\ \normalsize National Kaohsiung Normal University \\ \normalsize Yanchao, Kaohsiung 824, Taiwan\\ \normalsize {\tt Email:hsinhaolai@nknucc.nknu.edu.tw}\\ \and Ko-Wei Lih\\ \normalsize Institute of Mathematics\\ \normalsize Academia Sinica\\ \normalsize Nankang, Taipei 115, Taiwan\\ \normalsize {\tt Email:makwlih@sinica.edu.tw}}
\date{\small January 21, 2010} \maketitle
\newcommand{\cqfd}{
\rule{8pt}{9pt}} \newcommand{\langle}{\langle} \newcommand{\rangle}{\rangle} \newtheorem{define}{Definition} \newtheorem{proposition}[define]{Proposition} \newtheorem{theorem}[define]{Theorem} \newtheorem{lemma}[define]{Lemma} \newtheorem{remark}[define]{Remark} \newtheorem{corollary}[define]{Corollary} \newtheorem{problem}[define]{Problem} \newtheorem{conjecture}[define]{Conjecture}
\newenvironment{proof}{ \par \noindent {\bf Proof.}\rm} {\mbox{}
\rule{0.5em}{0.809em}\par}
\baselineskip=16pt \parindent=0.5cm
\begin{abstract} \noindent Let $D$ be an acyclic orientation of the graph $G$. An arc of $D$ is {\em dependent} if its reversal creates a directed cycle. Let $d_{\min}(G)$ denote the minimum number of dependent arcs over all acyclic orientations of $G$. Let $G(V_0, E_0)$ be a graph with vertex set $V_0 = \{\langle 0,0 \rangle, \langle 0,1 \rangle, \ldots , \langle 0,n-1 \rangle\}$ and edge set $E_0$. The {\em generalized Mycielski graph} ${\sf M}_m(G)$ of $G$, $m > 0$, has vertex set $V=V_0 \cup (\cup_{i=1}^{m} V_i) \cup \{u\}$, where $V_i=\{\langle i,j \rangle \mid 0 \leqslant j \leqslant n-1\}$ for $1 \leqslant i \leqslant m$, and edge set $E=E_0 \cup (\cup_{i=1}^{m} E_i) \cup \{\langle m,j \rangle u \mid 0 \leqslant j \leqslant n-1\}$, where $E_i= \{\langle i-1,j \rangle \langle i,k \rangle \mid \langle 0,j \rangle \langle 0,k \rangle \in E_0\}$ for $1 \leqslant i \leqslant m$. We generalize results concerning $d_{\min}({\sf M}_1(G))$ in K. L. Collins, K. Tysdal, {\em J. Graph Theory} 46 (2004), 285-296, to $d_{\min}({\sf M}_m(G))$. The underlying graph of a Hasse diagram is called a {\em cover graph}. Let $c(G)$ denote the the minimum number of edges to be deleted from a graph $G$ to get a cover graph. Analogue results about $c(G)$ are also obtained.
\noindent {\em Keyword}.\ acyclic orientation, dependent arc, source-reversal, cover graph, generalized Mycielski graph
\noindent {\em MSC 2000}:\ 05C99 \end{abstract}
\section{Introduction}
Graphs considered in this paper are finite, without loops, or multiple edges. We use $|G|$ and $\|G\|$, respectively, to denote the cardinalities of vertex set $V$ and edge set $E$ of a graph $G(V,E)$. The degree of a vertex $v$ of $G$ is denoted $d_G(v)$. An orientation $D$ of $G$ is obtained by assigning an arbitrary direction, either $x \rightarrow y$ or $y \rightarrow x$, on every edge $xy$ of $G$. The original undirected graph is called the {\em underlying graph} of any such orientation. {\em Sources} (or {\em sinks}) are vertices with no ingoing (or outgoing) arcs. An orientation $D$ is called {\em acyclic} if there does not exist any directed cycle.
Suppose that $D$ is an acyclic orientation of $G$. An arc $u \rightarrow v$ of $D$, or its underlying edge, is called {\em dependent} (in $D$) if the new orientation $D'=(D-(u \rightarrow v)) \cup (v \rightarrow u)$ contains a directed cycle. Note that $u \rightarrow v$ is a dependent arc if and only if there exists a directed walk of length at least 2 from $u$ to $v$. Let $d(D)$ denote the number of dependent arcs in $D$. Let $d_{\min}(G)$ and $d_{\max}(G)$, respectively, denote the minimum and maximum values of $d(D)$ over all acyclic orientations $D$ of $G$. It is known
(\cite{fflw}) that $d_{\max}(G)=\|G\|-|G|+ k$ for a graph $G$ having $k$ components.
Let $\chi(G)$ denote the {\em chromatic number} of $G$, i.e., the least number of colors to color the vertices of $G$ so that adjacent vertices receive distinct colors. Let $g(G)$ denote the {\em girth} of $G$, i.e., the length of a shortest cycle of $G$ if there is any, and $\infty$ if $G$ possesses no cycles. Fisher et al. \cite{fflw} showed that $d_{\min}(G)=0$ when $\chi(G) < g(G)$. The Hasse diagram of a finite partially ordered set depicts the covering relation of elements; its underlying graph is called a {\em cover graph}. Pretzel \cite{pret} proved that $d_{\min}(G)=0$ is equivalent to $G$ being a cover graph.
A convenient tool for us is the source-reversal operation first introduced by Mosesian in the context of finite posets and extensively used by Pretzel dealing with cover graphs. We will summarize the main properties of this operation in Section 2. In Section 3, we will introduce another parameter $c(G)$ which lower bounds $d_{\min}(G)$ and show that $c(G)=1$ if and only if $d_{\min}(G)=1$. In Section 4, we will characterize the case $d_{\min}({\sf M}_m(G)) \geqslant 1$. We give generalizations of results established by Collins and Tysdal in Section 5. In the final Section, we derive upper bounds for $c({\sf M}_m(G))$.
\section{Source-reversal}
Let $u$ be a source of the acyclic orientation $D$. A {\em source-reversal} operation applied to $u$ reverses the direction of all outgoing arcs from $u$ so that $u$ becomes a sink. The new orientation remains acyclic. Note that, if there are no dependent arcs in $D$, neither will there be any after a source-reversal.
\begin{theorem}\label{mosesian} Let $D$ be an acyclic orientation of a connected graph $G$. For any vertex $u$ of $G$, there exists an orientation $D'$ of $G$ obtained from $D$ by a sequence of source-reversals so that $u$ becomes the unique source of $D'$. \end{theorem}
The above result originally appeared in Mosesian \cite{mo}. It was put to good use by Pretzel in a series of papers (for example, \cite{pret}, \cite{pret86}, \cite{pret03}, and \cite{py}).
Let $D$ be an acyclic orientation of the graph $G$. For an undirected cycle $C$ of $G$, we choose one of the two traversals of $C$ as the positive direction. An arc is said to be {\em forward} if its orientation under $D$ is along the positive direction of $C$, otherwise it is said to be {\em backward}. We use $(C,D)^+$ (or $(C,D)^-$) to denote the set of all forward (or backward) arcs of $C$ with respect to $D$. The {\em flow difference} of $C$ with respect to $D$, denoted $f_D(C)$, is defined to
be $|(C,D)^+|-|(C,D)^-|$. The cycle $C$ is called {\em $k$-good} if
$|f_D(C)|\leqslant |C|-2k$, i.e., $C$ has at least $k$ forward arcs and $k$ backward arcs. An orientation $D$ is called {\em $k$-good} if all undirected cycles of its underlying graph $G$ are $k$-good. The set of acyclic orientations coincides with the set of 1-good orientations. A graph $G$ has a 2-good orientation if and only if $d_{\min}(G)=0$.
The {\em flow difference} of an orientation $D$ is the mapping $f$ from all cycles of $G$ to integers such that $f(C)=f_D(C)$ for every cycle $C$. Let $D$ and $D'$ be two orientations of the graph $G$. We say that $D$ is an {\em inversion} of $D'$, and vice versa, if $D$ and $D'$ possess the same flow difference.
The following appeared in Pretzel \cite{pret86}.
\begin{theorem}\label{equivalent flow difference} If $D$ and $D'$ are two acyclic orientations of the graph $G$, then the following statements are equivalent. \begin{enumerate} \item $D'$ is an inversion of $D$. \item $D'$ can be obtained from $D$ by a sequence of source-reversals. \end{enumerate} \end{theorem}
\section{The case for $d_{\min} = 1$}
We denote by $c(G)$ the the minimum number of edges to be deleted from $G$ so that the remaining graph is a cover graph, i.e., $$
c(G)= \min\{|F| \mid F \subseteq E(G) \mbox{ and $G-F$ is a cover graph}\}. $$ Bollob\'{a}s et al. \cite{bbn} first introduced and studied this parameter. Their results were extended in R\"{o}dl and Thoma \cite{rt}. It was also one of the four parameters that give lower bounds to $d_{\min}(G)$ investigated in Lai and Lih \cite{ll}. It is straightforward to observe the following.
\noindent {\em Fact 1.}\ \ $c(G) \leqslant d_{\min}(G)$.
\noindent {\em Fact 2.}\ \ A sufficient and necessary condition for $c(G)=0$ is $d_{\min}(G)$ $=0$.
\begin{theorem}\label{c=min=1} A sufficient and necessary condition for $c(G)=1$ is $d_{\min}(G)=1$. \end{theorem}
\begin{proof} It follows from Facts 1 and 2 that $d_{\min}(G)=1$ implies $c(G)=1$.
Now let us assume that $c(G)=1$. Then there exists an edge $e=xy$ such that $G' = G - e$ has a 2-good orientation $D'$. We may assume that there is no directed path from $y$ to $x$ and extend $D'$ to an acyclic orientation $D$ of $G$ by adding the arc $x \rightarrow y$.
Since $G$ has no 2-good orientations, $D$ must have at least one dependent arc. If $D$ has only one dependent arc, then we are done. If $D$ has at least two dependent arcs, then each of them must belong to a cycle containing $e$.
We claim that $x \rightarrow y$ can not be dependent in $D$. Suppose on the contrary that there exists a directed path $x,v_1,v_2,\ldots ,v_s, y,$ $s \geqslant 1$, from $x$ to $y$ in $D$. Since $D$ has at least two dependent arcs, there is a dependent arc $e'$ in $D$ distinct from $x \rightarrow y$, and there exists a cycle $y, u_1, u_2, \ldots ,u_t, x, y$ in $G$ such that $e'$ is the only backward arc in this cycle. Consider the closed walk $W=x,v_1,v_2,\ldots$, $v_s, y, u_1, u_2$, $\ldots ,u_t, x$. Reversing $e'$ converts $W$ into a closed directed walk. Hence, $e'$ is a dependent arc in $D'$ which contradicts the 2-goodness of $D'$. Therefore, $x \rightarrow y$ is not dependent in $D$.
By Theorems \ref{mosesian} and \ref{equivalent flow difference}, we can find an inversion $D^\ast$ of $D$ such that $D^\ast$ and $D$ have the same flow difference and $y$ is a source in $D^\ast$.
Let $e^\ast$ be an arbitrary dependent arc in $D^\ast$ and $C^\ast$ be a cycle of $G$ such that $(C^\ast,D^\ast)^-=\{e^\ast\}$. Then $C^\ast$ must
pass through the arc $y \rightarrow x$. Otherwise, $|(C^\ast,D^\ast)^-|=
|(C^\ast,D)^-|=|(C^\ast,D')^-|=1$ implies that $e^\ast$ is a dependent arc in $D'$, contradicting the 2-goodness of $D'$.
Suppose that $e^\ast$ is different from the arc $y \rightarrow x$. Hence, $y \rightarrow x$ belongs to $(C^\ast,D^\ast)^+$. Then the arc $x \rightarrow y$ belongs to $(C^\ast,D)^-$. Since $x \rightarrow y$ is not
dependent in $D$, we have $|(C^\ast,D)^-| \geqslant 2$. By Theorem
\ref{equivalent flow difference}, $2=2|(C^\ast,D^\ast)^-|=|C^\ast|-
|(C^\ast,D)^+|+|(C^\ast,D)^-|>2$, a contradiction. We conclude that $e^\ast$ must be the arc $y \rightarrow x$ in $D^\ast$. Therefore, $d_{\min}(G)=d(D^\ast)=1$. \end{proof}
An immediate consequence of the above Theorem is the following.
\begin{corollary} If $d_{\min}(G)=2$, then $c(G)=2$. \end{corollary}
\section{Non-cover Mycielski graphs}
Let $G(V_0, E_0)$ be a graph with vertex set $V_0 = \{\langle 0,0 \rangle,$ $\langle 0,1 \rangle,$ $\ldots ,$ $\langle 0,n-1 \rangle\}$ and edge set $E_0$. For $m > 0$, the {\em generalized Mycielski} graph ${\sf M}_m(G)$ of $G$ has vertex set $V=V_0 \cup (\cup_{i=1}^{m} V_i) \cup \{u\}$, where $V_i=\{\langle i,j \rangle \mid 0 \leqslant j \leqslant n-1\}$ for $1 \leqslant i \leqslant m$, and edge set $E=E_0 \cup (\cup_{i=1}^{m} E_i) \cup \{\langle m,j \rangle u \mid 0 \leqslant j \leqslant n-1\}$, where $E_i= \{\langle i-1,j \rangle \langle i,k \rangle \mid \langle 0,j \rangle \langle 0,k \rangle \in E_0\}$ for $1 \leqslant i \leqslant m$. We note that ${\sf M}_1(G)$ is commonly known as the {\em Mycielskian} $M(G)$ of $G$. It is easy to see that if $H$ is a subgraph of $G$, then ${\sf M}_m(H)$ is a subgraph of ${\sf M}_m(G)$. The following was proved in Lih et al. \cite{tower}.
\begin{theorem}\label{iffeven} Let $n \geqslant 3$. Then ${\sf M}_m(C_n)$ is a cover graph if and only if $n$ is even. \end{theorem}
This can be generalized as follows.
\begin{theorem}\label{lai} $d_{\min}({\sf M}_m(G)) \geqslant 1$ if and only if $G$ is not bipartite. \end{theorem}
\begin{proof} If $G$ has no edge, then obviously ${\sf M}_m(G)$ is a cover graph. Let $G$ be a bipartite graph with at least one edge. Then $\chi({\sf M}_m(G)) = 3 < g({\sf M}_m(G))$. Hence ${\sf M}_m(G)$ is a cover graph. If $G$ is not bipartite, then $G$ contains an odd cycle $C$ of length at least 3. By Theorem \ref{iffeven}, ${\sf M}_m(C)$ is not a cover graph. Since ${\sf M}_m(G)$ is a supergraph of ${\sf M}_m(C)$, it is not a cover graph. \end{proof}
\begin{corollary} $c({\sf M}_m(G)) \geqslant 1$ if and only if $G$ is not bipartite. \end{corollary}
We are going to construct examples to show that equality can hold in Theorem \ref{lai}.
\begin{theorem}\label{mycielski d_min=1} Let $G(V_0, E_0)$ be a triangle-free graph that is not bipartite. Suppose that there exists some vertex $\langle 0,v\rangle$ of $G$ such that $G-\langle 0,v\rangle$ is a bipartite graph whose two parts are denoted by $X$ and $Y$. If $\langle 0,v\rangle$ has precisely one neighbor in $X$ and at least one neighbor in $Y$, then $d_{\min}({\sf M}_m(G))=1$. \end{theorem}
\begin{proof} By Theorem \ref{lai}, we know $d_{\min}({\sf M}_m(G))\geqslant 1$. It suffices to construct an acyclic orientation of ${\sf M}_m(G)$ possessing a unique dependent arc.
{\bf Step 1}.\ Define an orientation $D_1$ of $G$ as follows.
(1)\ If $xy$ is an edge in $G-\langle 0,v\rangle$, $x \in X$ and $y \in Y$, then let $x\rightarrow y$.
(2)\ If $\langle 0,v'\rangle$ is the unique neighbor of $\langle 0,v\rangle$ in $X$, then let $\langle 0,v'\rangle\rightarrow \langle 0,v\rangle$.
(3)\ If $\langle 0,v''\rangle$ is any neighbor of $\langle 0,v\rangle$ in $Y$, then let $\langle 0,v\rangle\rightarrow \langle 0,v''\rangle$.
Obviously, each vertex in $X$ is a source, each vertex in $Y$ is a sink, and $\langle 0,v\rangle$ is neither a source nor a sink. It follows that $D_1$ is an acyclic orientation. Moreover, if $P$ is a directed path of length at least 2 in $D_1$, then $\langle 0,v'\rangle$ must be the initial vertex of $P$ and the length of $P$ is precisely 2. Since $G$ is triangle-free, $D_1$ has no dependent arc.
{\bf Step 2}.\ Let $D_2$ be the extension of $D_1$ into ${\sf M}_m(G)-u$ by defining $\langle i,w_1 \rangle \rightarrow \langle i-1,w_2 \rangle$ and $\langle i-1,w_1 \rangle \rightarrow \langle i,w_2 \rangle$ if $\langle 0,w_1\rangle \rightarrow \langle 0,w_2\rangle$ in $D_1$ and $1\leqslant i\leqslant m$.
If $\langle i_1,v_1 \rangle,\langle i_2,v_2 \rangle,\ldots , \langle i_t,v_t \rangle, \langle i_1,v_1 \rangle$ is a directed cycle in $D_2$, then $\langle 0,v_1 \rangle$, $\langle 0,v_2 \rangle$, $\ldots , \langle 0,v_t \rangle, \langle 0,v_1 \rangle$ is a directed closed walk in $D_1$, contradicting the acyclicity of $D_1$. Similarly, $D_2$ has no dependent arc since $D_1$ has none.
{\bf Step 3}.\ Let $D_3$ be the extension of $D_2$ into ${\sf M}_m(G)$ by defining $\langle m,w\rangle \rightarrow u$ for every $\langle 0,w \rangle$.
Since $D_2$ is acyclic and $u$ is a sink in $D_3$, $D_3$ is acyclic. If $e$ is a dependent arc in $D_3$, then $e$ must be some $\langle m,w\rangle \rightarrow u$. If $\langle 0,w \rangle \ne \langle 0,v' \rangle$, then there is a directed path $P'$ from $\langle m,w\rangle$ to a certain $\langle m,w'\rangle$ in $D_3$. Since there is no edge between $\langle m,w\rangle$ and $\langle m,w'\rangle$ in ${\sf M}_m(G)$, $P'$ must have length at least 2. Hence, we can find a directed path of length at least 2 in $D_1$ and $\langle 0,v' \rangle$ is not the initial vertex of that path. This is a contradiction.
Let us consider the arc $\langle m,v'\rangle \rightarrow u$. Let $\langle 0,v'' \rangle$ be a neighbor of $\langle 0,v \rangle$ in $Y$. The cycle $u, \langle m,v'\rangle, \langle m-1,v\rangle, \langle m,v''\rangle, u$ shows that $\langle m,v'\rangle \rightarrow u$ is a unique dependent arc in $D_3$. \end{proof}
A graph $G$ satisfying Theorem \ref{mycielski d_min=1} can be constructed as follows. Let $v$ be a fixed vertex. Let $X$ be a set of $p \geqslant 2$ vertices and $Y$ be a set of $q \geqslant 2$ vertices. Choose a vertex $v'$ in $X$ and a nonempty proper subset $Y'$ of $Y$. Add edges $vv'$ and $vv''$ for all $v'' \in Y'$. Add a path of length at least 3 from $v'$ to some vertex $z$ in $Y'$ which alternately uses vertices in $X$ and $Y$ and uses no vertex in $Y'$ except the terminal vertex $z$.
However, the problem of characterizing graphs $G$ that satisfy $d_{\min}({\sf M}_m(G))=1$ remains open.
\section{Generalizing a theorem of Collins and Tysdal}
The following appeared in Collins and Tysdal \cite{ct}.
\begin{theorem}\label{basic_mycielski} Let $G$ be a triangle-free graph. Then the following statements hold. \begin{enumerate} \item If $d_{\min}(G)\geqslant 1$, then $d_{\min}(M(G))\geqslant 3$.
\item If $d_{\min}(G)\geqslant 2$, then $d_{\min}(M(G))\geqslant 4$.
\item If $d_{\min}(G)\geqslant 3$, then $d_{\min}(M(G))\geqslant 6$. \end{enumerate} \end{theorem}
Let $S$ be a set of vertices of the graph $G(V_0, E_0)$. We use $S'$ to denote the set of vertices $\{ \langle 1, j \rangle \mid \langle 0, j \rangle \in S\}$ and $G-S+S'$ to denote the subgraph of $M(G)$ induced by the set of vertices $(V_0 \setminus S) \cup S'$ in $M(G)$.
\begin{lemma}\label{replace} If $S$ is an independent set of $G$, then the subgraph $G-S+S'$ of $M(G)$ is isomorphic to $G$. \end{lemma}
\begin{proof} The mapping $\sigma : V(G)\rightarrow V(G-S+S')$ defined below is an isomorphism. $\sigma(\langle 0,i \rangle)=\langle 1,i \rangle$ if $\langle 0,i \rangle \in S$ and $\sigma(\langle 0,i \rangle)=\langle 0,i \rangle$ if $\langle 0,i \rangle \notin S$. \end{proof}
Proofs of Lemmas \ref{2edges} and \ref{3edges} are modeled after ideas used in Collins and Tysdal \cite{ct}.
\begin{lemma}\label{2edges} Let $G(V_0, E_0)$ be a triangle-free graph with at least two edges. For any two edges $e_1, e_2$ in $M(G)-u$, $M(G)-u-\{e_1,e_2\}$ contains a subgraph isomorphic to $G$. \end{lemma}
\begin{proof} If none of $e_1$ and $e_2$ is an edge in $E_0$, we are done. Hence, we assume that $e_1=\langle 0,x_1\rangle \langle 0,y_1\rangle \in E_0$ and consider the subgraph $G'$ of $M(G)$ induced by $(V_0 \setminus \{\langle 0,x_1\rangle\}) \cup\{\langle 1,x_1\rangle\}$. The graph $G'$ is isomorphic to $G$. If $e_2$ is not an edge in $G'$, we are done. Assume that $e_2$ is an edge in $G'$.
{\bf Case 1.} The edge $e_2$ is not incident to $\langle 1,x_1\rangle$. Since $G$ is triangle-free, $\langle 0,x_1\rangle$ can not be adjacent to both endpoints of $e_2$. Suppose that $\langle 0,x_2\rangle$ is an endpoint of $e_2$ and not adjacent to $\langle 0,x_1\rangle$. Let $S=\{\langle 0,x_1\rangle, \langle 0,x_2\rangle\}$.
{\bf Case 2.} The vertex $\langle 1,x_1\rangle$ is an endpoint of $e_2$. Let $S= \{\langle 0,y_1\rangle\}$.
In each case, $S$ is an independent set. By Lemma \ref{replace}, $G-S+S'$ is a subgraph of $M(G)-u-\{e_1,e_2\}$ that is isomorphic to $G$. \end{proof}
\begin{theorem} If a graph $G$ is triangle-free with at least two edges and $d_{\min}(G) \geqslant 1$, then $d_{\min}({\sf M}_m(G))\geqslant d_{\min}(G)+2$. \end{theorem}
\begin{proof} By assumption, $d_{\min}(M(G)-u) \geqslant d_{\min}(G) \geqslant 1$. Let $F$ be the set of dependent arcs of an acyclic orientation $D$ of $M(G)-u$
that satisfies $d(D)=d_{\min}(M(G)-u)$, hence $|F| \geqslant 1$. Pick an edge $e_1$ from $F$ and another edge $e_2 \ne e_1$ of $M(G)-u$. By Lemma \ref{2edges}, $M(G)-u-\{e_1,e_2\}$ contains a subgraph isomorphic to $G$. Thus $d_{\min}(M(G)-u) \geqslant d_{\min}(G)+1\geqslant 2$, and hence we can find two distinct edges $e_1'$ and $e_2'$ from $F$. By Lemma \ref{2edges} again, $M(G)-u-\{e_1',e_2'\}$ contains a subgraph isomorphic to $G$. It follows that $d_{\min}(M(G)-u) \geqslant d_{\min}(G)+2$. Finally, $d_{\min}({\sf M}_m(G)) \geqslant d_{\min}(M(G)-u) \geqslant d_{\min}(G)+2$. \end{proof}
If we replace the set $F$ in the above proof by a set $F'$ of edges of
$M(G)-u$ such that $M(G)-u-F'$ is a cover graph and $|F'|=c(M(G)-u)$, then we can use the same argument to get the following.
\begin{corollary} If a graph $G$ is triangle-free and $c(G)\geqslant 1$, then $c({\sf M}_m(G)) \geqslant c(G)+2$. \end{corollary}
\begin{lemma}\label{3edges}
Let $G(V_0, E_0)$ be a triangle-free graph with $\|G\|\geqslant 3$. For any three edges $e_1, e_2, e_3$ in $E_0$, $M(G)-u-\{e_1,e_2,e_3\}$ contains a subgraph isomorphic to $G$. \end{lemma}
\begin{proof} Let $G'$ be the subgraph of $G$ induced by $\{e_1,e_2,e_3\}$.
{\bf Case 1.}\ If $G'$ is a star, then let $x$ be the vertex of degree 3 and let $S=\{x\}$.
{\bf Case 2.}\ If $G'$ is a path $v_0v_1v_2v_3$ of length 3, then let $S=\{v_0,v_2\}$. Since $G$ is triangle-free, $v_0$ and $v_2$ are not adjacent.
{\bf Case 3.}\ If $G'$ consists of the disjoint union of a path $P_3$ of length 2 and an edge $P_2$, then one endpoint $y$ of $P_2$ is not adjacent to the center vertex $x$ of $P_3$ because $G$ is triangle-free. Let $S=\{x,y\}$.
{\bf Case 4.}\ Let the three edges $e_1=x_1y_1$, $e_2=x_2y_2$, and $e_3 =x_3y_3$ be mutually non-incident. Since $G$ is triangle-free, at least one endpoint of $e_2$, say $x_2$, is not adjacent to $x_1$. Similarly, at least one endpoint of $e_3$, say $x_3$, is not adjacent to $x_1$.
If the vertices $x_2$ and $x_3$ are not adjacent, then let $S=\{x_1,$ $x_2,$ $x_3\}$.
If $x_2$ and $x_3$ are adjacent and $x_1$ is not adjacent to $y_i$, $i = 2$ or $3$, then $y_i$ and $x_{5-i}$ are not adjacent. Let $S=\{ x_1,y_i,x_{5-i}\}$.
If $x_2$ and $x_3$ are adjacent and both $y_2$ and $y_3$ are adjacent to $x_1$, then $\{y_1, y_2, y_3\}$ is an independent set. Let $S=\{y_1, y_2,y_3\}$.
In all cases, $S$ so defined is an independent set. By Lemma \ref{replace}, $G-S+S'$ is a subgraph of $M(G)-u-\{e_1,e_2,e_3\}$ isomorphic to $G$. \end{proof}
\begin{theorem} If a graph $G$ is triangle-free with at least three edges and $d_{\min}(G) \geqslant 3$, then $d_{\min}({\sf M}_m(G))\geqslant d_{\min}(G)+3$. \end{theorem}
\begin{proof} Let $F$ be the set of dependent arcs of an acyclic orientation $D$ of $G$
that satisfies $d(D)=d_{\min}(G)$, hence $|F| \geqslant 3$. Pick three edges $e_1, e_2, e_3$ from $F$. By Lemma \ref{3edges}, $M(G)-u-\{e_1,e_2, e_3\}$ contains a subgraph isomorphic to $G$. It follows that $d_{\min}({\sf M}_m(G))\geqslant d_{\min}(M(G)-u) \geqslant d_{\min}(G)+3$. \end{proof}
\begin{corollary} If a graph $G$ is triangle-free with at least three edges and $c(G)\geqslant 3$, then $c({\sf M}_m(G))\geqslant c(G)+3$. \end{corollary}
\section{Upper bounds of $c({\sf M}_m(G))$}
In this section, we derive upper bounds for $c({\sf M}_m(G))$. Since $\chi(G)<g(G)$ implies that $G$ is a cover graph, we have the following inequality.
$$c(G)\leqslant \min \{\|G\|-\|H\| \mid \mbox{$H$ is a subgraph of $G$ and } \chi(H)<g(H)\}.$$
Let $e_k(G)$ be the maximum number of edges in a $k$-colorable subgraph of $G$. Since the girth of a subgraph is never smaller than that of the given graph, the above inequality implies the following. \begin{equation}\label{girth and e_k}
c(G) \leqslant \|G\| - e_{k-1}(G) \mbox{ if } g(G) \geqslant k. \end{equation}
Let $G$ be a triangle-free graph. If $H=(X,Y)$ is a bipartite subgraph of $G$, then the following inequality holds by the above inequality. \begin{equation}\label{girth>=4 pi_E}
c(G) \leqslant \|G\|- \|H\| -e_2(G[X]). \end{equation}
\begin{proof} Let $X'=(X_1,X_2)$ be a bipartite subgraph of $G[X]$ with $e_2(G[X])$
edges. Consider the subgraph $G'=(V(H),E(H)\cup E(X'))$ of $G$. Obviously, $G'$ is 3-colorable. Hence, $e_3(G)\geq \|H\|+\|X'\|=\|H\|+ e_2(G[X])$. By inequality (\ref{girth and e_k}), we are done. \end{proof}
\begin{theorem} If $G$ is a graph and $m$ is a positive integer, then $c({\sf M}_m(G))
\leqslant \|G\|$. Moreover, if $G$ is a triangle-free graph, then
$c({\sf M}_m(G)) \leqslant \|G\|-e_2(G)$. \end{theorem}
\begin{proof} Obviously, ${\sf M}_m(G)-E(G)$ is bipartite. We have $e_2({\sf M}_m(G))$
$\geqslant$ $\|{\sf M}_m(G)\|-\|G\|$. By inequality (\ref{girth and e_k}),
$c({\sf M}_m(G))\leq \|G\|$. If $G$ is triangle-free, so is ${\sf M}_m(G)$. It is easy to see that ${\sf M}_m(G)-E(G)$ is a bipartite graph with bipartition $(X,Y)$ such that the vertices of $G$ belong to the same partite set, say $X$. Since $G$ has a bipartite subgraph with $e_2(G)$ edges, $e_2({\sf M}_m(G)[X])\geqslant e_2(G)$. By inequality (\ref{girth>=4
pi_E}), $c({\sf M}_m(G)) \leqslant \|{\sf M}_m(G)\|- \|{\sf M}_m(G)-E(G)\|
-e_2({\sf M}_m(G)[X])\leqslant \|G\|-e_2(G)$. \end{proof}
{\bf Acknowledgment}.\ The authors are indebted to Dr. Fei-Huang Chang for useful discussions on Theorem \ref{mycielski d_min=1}.
\end{document} | arXiv | {
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\begin{document}
\title{Optimizing classical communication in remote preparation of a general pure qubit}
\author{Congyi Hua}
\affiliation{Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China} \author{Yi-Xin Chen}
\email{yxchen@zimp.zju.edu.cn}
\affiliation{Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China}
\begin{abstract} How to uses shared entanglement and forward classical communication to remotely prepare an arbitrary (mixed or pure) state has been fascinating quantum information scientists. A constructive scheme has been given by Berry for remotely preparing a general pure state with a pure entangled state and finite classical communication. Based on this scheme, for high-dimensional systems it is possible to use a coding of the target state to optimize the classical communication cost. Unfortunately, for low-dimensional systems such as a pure qubit the coding method is inapplicable. Because qubit plays a central role in quantum information theory, we propose an optimization procedure which can be used to minimize the classical communication cost in the remote preparation of a general pure qubit. Interestingly, our optimization procedure is linked to the uniform arrangement of $N$ points on the Bloch sphere, which provides a geometric description. \end{abstract} \pacs{03.65.Yz, 03.67.-a, 03.65.Ta, 85.25.Cp} \maketitle
\section{Introduction}
In the field of quantum information processing, remote state preparation (RSP) is a kind of protocols that transmit a quantum state from a sender (``Alice") to a receiver (``Bob") using preshared entanglement and forward classical communication~\cite{Lo,Pati,Bennett}. Unlike the celebrated teleportation protocols~\cite{BennettTeleportation}, the sender does not possess a copy of the target state, but has complete classical knowledge of the state, which she chooses from a given ensemble. The RSP protocols can be divided into two different categories: exactly (non-asymptotically) faithful and asymptotically faithful. The exactly faithful RSP produces the desired states one at a time, while the asymptotically faithful RSP only has an asymptotic efficiency. We are concerned with exactly faithful RSP in the present paper.
In the simple case where the target ensemble consists of a great circle on the Bloch sphere, the RSP can be done by using one maximally entangled state (ebit) and one classical bit (cbit) communication~\cite{Pati}. The constraint on the ensemble can be extended to the entire Bloch sphere by allowing more classical communication. Lo~\cite{Lo}, Leung and Shor~\cite{Leung} showed that two cbit communication is necessary and sufficient for the RSP of an arbitrary pure qubit with one ebit preshared. These investigations are based on Alice and Bob shared a maximally entangled state, however, the non-maximally entangled cases may occur due to the imperfect devices in the real world. In these cases, the required resource can be traded off between the cbits and ebits. Ye, \textit{et al.} proposed a protocol for remote preparation of an arbitrary pure state, by using finite cbits and non-maximally entangled pure state~\cite{Ye}. Soon a constructive scheme for this RSP protocol was given by Berry~\cite{Berry}.
Just as Berry showed, the classical communication cost increases drastically as the entanglement goes down. Although this is an inevitable consequence of the trade off between the two kind of resources, unnecessary classical communication cost should be minimized. For a large system dimension, Berry employed a coding of the target state to optimize the classical communication cost for the scheme of this type. However, for preparing low-dimensional target states such as a pure qubit, as the coding method is inapplicable, the scheme still suffers from unsatisfying classical communication cost.
Since qubit is one of the central objects of study in quantum information theory, here we propose an optimization method which can be used in the remote preparation of a pure qubit. Our method comes from a rethink of the preliminary of Berry's scheme. We find the preliminary, which Berry described as an approximate scheme, is actually an algorithm for arranging points on the Bloch sphere. And changing the algorithm to one that can construct points distributed uniformly on the Bloch sphere will minimize the classical communication cost.
This paper is organized as follows. In Sec.~\ref{secii}, we restate Berry's scheme for preparing a general qubit as a four-step process. We show how a uniform distribution of points on the Bloch sphere minimizes the classical communication cost for RSP scheme of this type. For clear demonstration of the optimization procedure, in Sec.~\ref{seciii} we introduce an algorithm called spiral points~\cite{Saff}, which can be used for easy construction of considerably uniformly distributed points on a sphere. Then we replace the original distributions in the scheme with the spiral points and compute the cbits versus ebits trade off. By comparing our results with those in Ref.~\cite{Berry}, we show that the cbits versus ebits trade off computed from the spiral points is very near a lower bound. Finally, in Sec.~\ref{seciv}, we summarize our results and draw some conclusions.
\section{Remote preparation of a general pure qubit} \label{secii} Berry's scheme aims at remote preparation of a pure state using any entangled pure state. Here we restate this scheme for preparing a general qubit.
Assume Alice and Bob share an entangled state, which has the form
\begin{equation} \vert A\rangle =\sum _{k=0}^1 \alpha _k\vert k\rangle \vert k\rangle, \label{eqsource} \end{equation} $ \alpha _k>0$, $\sum _{k=0}^1 \alpha _k^2=1$. Any two-qubit pure entangled state can be brought to this form via local unitary operations at Alice's location. The state Alice wants to prepare at Bob's side is denoted by $\vert \beta\rangle$, which is known to Alice but unknown to Bob.
Before we outline the procedure for Alice to remotely prepare $\vert \beta\rangle$, one important result from Ref.~\cite{Berry} need to be stated. By Allowing Alice and Bob to perform local operations and communicate 2 bits of classical information, the possession of an entangled state in Eq.~(\ref{eqsource}) guarantees Alice the ability to remotely prepare an arbitrary qubit of the form \begin{equation} \vert \psi\rangle =\sum _{k=0}^1 \psi _ke^{i \varphi _k}\vert k\rangle, \label{eqcap} \end{equation}
where $\psi _0\geq 1-r^2$, $r=\min\{\alpha_i\}$.
On the Bloch sphere, the ensemble of states that satisfy Eq.~(\ref{eqcap}) is represented by a $\vert 0\rangle$-centered spherical cap, denoted by $c_0$. According to the entanglement for pure qubits~\cite{Horodecki}, one can know that the less entanglement $\vert A\rangle$ has, the smaller spherical cap Alice can prepare.
Now let's outline the procedure for the preparation by four steps. To avoid unnecessary elaboration, we treat step 1 and 2 as briefly as possible. For more details, we refer the readers to Ref.~\cite{Berry}.
Step 0. Construct a distribution of $N$ points (or states) $\vert\beta _i'\rangle$, $i=1, 2, \text{...}, N$, on the Bloch sphere. $N$ should be large enough to make the set of spherical caps $C=\left\{c_1, c_2, \text{...} , c_N\right\}$, where $c_i=\{\vert e\rangle\ \vert\ |\langle \beta _i'\vert e\rangle |^2\geq 1-r^2\}$, a cover of the Bloch sphere. Further, define $N$ unitary transformations $U_i$'s, each transforms $c_0$ into $c_i$.
Step 1-2. Alice prepares at Bob's location a state $\vert\varphi _0\rangle$ in $c_0$ such that Bob can bring $\vert\varphi _0\rangle$ to the desired state $\vert \beta\rangle$ in $c_i$ by some unitary transformation $U_i$. This can be done by an entanglement transformation followed by a disentangling measurement, and costs 2 bits of classical information.
Step 3. Alice send Bob $\log N$ bits classical information to indicate him which $U_i$ should be used to bring $\vert\varphi _0\rangle$ to $\vert \beta\rangle$.
Step 0 is actually the preliminary for the preparation in Berry's original scheme. We treat the preliminary as Step 0 to facilitate the explanation of the optimization procedure. In order to successfully perform the RSP with resource states having different entanglement, an algorithm should be given for constructing distributions of any number $N$ of points. It can be easily seen that the $\log N$ cbits cost in step 3 depends on the point distribution given in step 0. If we have an algorithm can distribute the centers of the spherical caps more uniformly, then less number of spherical caps will be needed to cover the Bloch sphere, and more classical communication will be saved. To minimize the classical communication cost, what one need is an algorithm that constructs uniformly distributed points on the Bloch sphere.
Below is how Berry's algorithm locates $N$ spherical caps. Assume the state locating at the center of a spherical cap is expressed as \begin{equation*} \vert\tilde{\beta '}\rangle =\sum _{k=0}^1 \beta _k\vert k\rangle, \end{equation*} where $\beta_0$ is real, and $\beta_1$ is complex. The state $\vert\tilde{\beta '}\rangle$ is not necessarily normalized and the corresponding normalized state will be denote by $\vert\beta '\rangle$. Berry begins with finding on the interval $\left[0,1\right]$ $D$ uniformly distributed numbers \begin{equation*} (2n-1)/D-1, \end{equation*} $n=1,2, \text{...} , D$. By picking 3 such numbers (repetition is allowed) as $\beta_0$, the real and imaginary parts of $\beta_1$, a spherical cap can be located. It's obvious that the total number of spherical caps constructed by this algorithm satisfies $N=D^3$.
In the above algorithm, although the spherical caps are represented by $N$ points that are uniformly distributed in the unit box, the distribution of these points on the Bloch sphere are nonuniform. Worse, as two or more different $\vert\tilde{\beta '}\rangle$'s may correspond to one the same $\vert\beta '\rangle$, lots of points coincide with each other. Fig.~\ref{fig1} illustrates the case of $N=4^3$.
We already know that the optimization procedure is equivalent to finding an algorithm for constructions of uniformly distributed points on the Bloch sphere. However except for some special cases such as the arrangements of 4, 8, 6, 12, 20 points on a sphere, in which cases we can use the vertices of the Platonic solids due to their perfect symmetry, finding an algorithm that can uniformly arrange an arbitrary number of points on a sphere is still an open question. Fortunately, there're still a variety of algorithms that can construct quite uniform point distribution on a sphere. A simple to describe and compute algorithm is spiral points, which we will use to demonstrate the optimization procedure in the Sec.~\ref{seciii}.
\begin{figure}
\caption{The points distribution given the Berry's algorithm in Ref.~\cite{Berry} in the case of $N=4^3$. Since some points coincide with each other, only 28 (instead of 64) points are distingushable. (View along the negative z direction.)}
\label{fig1}
\end{figure}
\section{Demonstrating the optimization procedure via spiral points}
\label{seciii} The problem of how to uniformly distribute points on a sphere has long been receiving attention by scientists in their work, such as searching for large stable carbon molecules and locating identical charged particles so that they are in equilibrium according to Coulomb's law, etc. Spiral points is an algorithm proposed for the explicit construction of considerably uniformly distributed points on the sphere. It has the advantage of being simple to describe and compute, thus suitable for the demonstration of the optimization procedure in remote preparation of a pure qubit.
Just like the algorithm's name, the construction of the spiral points is like to draw a spiral path along the surface of the unit ball. One begins from setting the first spiral point at the south pole of the sphere. To obtain the next spiral point, one proceeds upward from the current point along a meridian to the height that is $2/(n-1)$ higher and travels counterclockwise along a latitude for a fixed distance of $3.6/\sqrt{N}$ to arrive at the next point. The entire path will end up at the north pole. Using spherical coordinates, the $i$th spiral points $p_i$ may be given as below: \begin{equation*} \theta _i=\arccos \left(z_i\right), \end{equation*} \begin{equation*} z_i=-1+\frac{2(i-1)}{N-1} \text{, } 1\leq i\leq N \text{, } \end{equation*} \begin{equation*} \phi _1=\phi _N=0 \text{, } \end{equation*} \begin{equation*} \phi _i=\left(\phi _{i-1}+\frac{3.6}{\sqrt{N}}\frac{1}{\sqrt{1-z_i^2}}\right)(\text{mod} 2\pi)\text{, } 2\leq i\leq N \text{.} \end{equation*} In Fig.~\ref{fig2}, one can see how uniform the distribution of 64 spiral points looks.
Now let's calculate the cbits versus ebits trade off for the scheme using spiral points. But before we can calculate the trade of, we must introduce the concept of Voronoi diagram~\cite{Aurenhammer}. A Voronoi diagram is a way of dividing space into numbers of regions. In the context of Voronoi diagram the spiral points $p_i$'s are called sites. For each site, there will be a corresponding polygon-shaped region consisting of all points closer to this site than to any other. These regions are called Voronoi cells, whose edges are equidistant from two sites, and vertices equidistant from three or more sites. (Fig.~\ref{fig2} gives an illustration of Voronoi cells corresponding to the 64 spiral points.) Let's denote by $v_{\{i,j\}}$ the $j$th vertex of the Voronoi cell corresponding to $p_i$.
\begin{figure}
\caption{Spiral points for $N=64$. The mesh on the sphere shows the Voronoi cells corresponding to spiral points. (View along the negative z direction.)}
\label{fig2}
\end{figure}
Since every spherical cap $c_i$ is centered at the spiral point $p_i$, $C$ will not become a cover of the Bloch sphere until every $c_i$ covers the Voronoi cell corresponding to $p_i$. One can measure the size of $c_i$ by the fidelity radius $r_F$, which is defined by 1 minus the fidelity between the central state and a boundary state, i.e., $r_F=r^2$. Similarly, one can measure the size of the hardest to cover Voronoi cell by \begin{equation*}
\begin{split}&\rho _F(N)\\=&\max\left\{1-\left|\left\langle v_{\{i,j\}}\vert p_i\right\rangle \right|^2 \vert \text{ for } \text{any } p_i \text{ and } \text{related } v_{\{i,j\}}\text{}\right\}. \end{split} \end{equation*} In order to make $C$ a cover of the Bloch sphere, $N$ should be large enough to ensure $\rho _F(N)\leq r_F$. To compute $\rho _F(N)$, we need to obtain the coordinates of all $v_{\{i,j\}}$'s.
In the problems of generating a Voronoi diagram from a given set of points, except for some special points distributions, it is generally hard to find analytic solutions. One different approach that is commonly seen is to adopt a numerical solution. There're several algorithms developed for computing the spherical Voronoi diagram~\cite{Aurenhammer,Na,Zheng}. We implement in our program the popularly used sweep line algorithm for computing the Voronoi diagram in $O(N \log N)$ time~\cite{Zheng}. By taking the spiral points as input, the program is executed for the input size $N$ from 3 to 1024. We list part of the result (for $N=2^n$, $n=1, 2, ..., 10$) in the table below:
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill} } c|c|c|c|c|c }
\hline
$N$ & 2 & 4 & 8 & 16 & 32\\
\hline
$\rho_F$ & 0.5 & 0.5 & 0.259739 & 0.120679 & 0.054644\\
\hline
$N$ & 64 & 128 & 256 & 512 & 1024\\
\hline
$\rho_F$ & 0.026443 & 0.013054 & 0.006607 & 0.003326 & 0.001669\\
\hline \end{tabular*}
Based on the obtained values of $\rho _F(N)$, we can compute the classical bits cost versus entanglement of the resource state. The smallest integer $N$ which satisfies the inequation $r_F\geq \rho _F(N)$ is used to calculate the classical bits cost $\log N$ and the entanglement is calculated by $-r^2 \log r^2-(1-r^2)\log(1-r^2)$. We show the result in Fig.~\ref{fig3}. Comparing with that of the original scheme proposed in Ref.~\cite{Berry}, we can see that the classical bits cost after using spiral points is significantly reduced. Actually the classical bits cost is reduced to a level very close to the limit for RSP scheme of this type, because it is between an upper bound and a lower bound of this limit (refer respectively to Eq. (23) and Eq. (24) of Ref.~\cite{Berry}).
\begin{figure}
\caption{The cbits cost versus ebits for RSP of pure qubits states using partially entangled state. The dotted curve is that based on the original scheme given in Ref.~\cite{Berry}, and the solid curve is the result obtained when spiral points are used. The dashed-dotted curve and the dashed curve are an upper bound and a lower bound on the classical communication for RSP scheme of this type. The black dots are drawn from the cases which are presented in table.}
\label{fig3}
\end{figure}
It must be emphasized that Fig.~\ref{fig3} only plot the classical bits cost in step 3. The total classical bits cost for RSP schemes of this type should count the 2 bits in step 2 of the scheme.
\section{Conclusions} \label{seciv} We have reanalyzed a RSP scheme for remotely preparing a general pure state, and related the optimization of the scheme to an algorithm which can construct uniformly distributed points on the Bloch sphere. Since the original algorithm does not provide uniform point distributions on the Bloch sphere, we replace it with spiral points, an algorithm that gives a quite uniform point distribution with considerable simplicity.
Using a uniform point distribution algorithm like spiral points in the scheme has two main advantages.
(1) The classical bits cost of this type RSP scheme is reduced to a level near optimal, which the original scheme cannot achieve if the state to be prepared is in low dimension.
(2) Once an appropriate algorithm is determined, the scheme can be constructed easily. There is no need of a coding method to optimize the classical bits cost.
There may be some orther algorithms to choose, we use spiral points partly because its simplicity of describing and computing gives a good demonstration for the optimization procedure. To generalize our method to higher dimensions is possible although further work might be required.
\section{Acknowledgments} This work is supported by the NNSF of China, Grant No. 11375150.
\end{document} | arXiv | {
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\begin{document}
\title{Contraction of matchgate tensor networks on non-planar graphs} \begin{abstract} A tensor network is a product of tensors associated with vertices of some graph $G$ such that every edge of $G$ represents a summation (contraction) over a matching pair of indexes.
It was shown recently by Valiant, Cai, and Choudhary that tensor networks can be efficiently contracted on planar graphs if components of every tensor obey a system of quadratic equations known as matchgate identities. Such tensors are referred to as {\it matchgate tensors}. The present paper provides an alternative approach to contraction of matchgate tensor networks that easily extends to non-planar graphs. Specifically, it is shown that a matchgate tensor network on a graph $G$ of genus $g$ with $n$ vertices can be contracted in time $T=poly(n) + O(m^3)\, 2^{2g}$ where $m$ is the minimum number of edges one has to remove from $G$ in order to make it planar. Our approach makes use of anticommuting (Grassmann) variables and Gaussian integrals. \end{abstract}
\tableofcontents
\section{Introduction and summary of results}
Contraction of tensor networks is a computational problem having a variety of applications ranging from simulation of classical and quantum spin systems~\cite{MarkovShi05,Verstraete04,Vidal05,Vidal07,Levin06} to computing capacity of data storage devices~\cite{Schwartz07}. Given the tremendous amount of applications it is important to identify special classes of tensor networks that can be contracted efficiently. For example, Markov and Shi found a linear time algorithm for contraction of tensor networks on trees and graphs with a bounded treewidth~\cite{MarkovShi05}. An important class of graphs that do not fall into this category are planar graphs.
Although contraction of an arbitrary tensor network on a planar graph is a hard problem, it has been known for a long time that the generating function of perfect matchings known as the {\it matching sum} can be computed efficiently on planar graphs for arbitrary (complex) weights using the Fisher-Kasteleyn-Temperley (FKT) method, see~\cite{Fisher61,Kasteleyn61,Temperley61}. It is based on the observation that the matching sum can be related to Pfaffian of a weighted adjacency matrix (known as the Tutte matrix). The FKT method also yields an efficient algorithm for computing the partition function of spin models reducible to the matching sum, most notably, the Ising model on a planar graph~\cite{Barahona81}. Recently the FKT method has been generalized to the matching sum of non-planar graphs with a bounded genus~\cite{Gallucio99,Zecchina01,Reshetikhin06}.
Computing the matching sum can be regarded as a special case of a tensor network contraction. It is therefore desirable to characterize precisely the class of tensor networks that can be contracted efficiently using the FKT method. This problem has been solved by Valiant~\cite{Valiant02a,Valiant07} and in the subsequent works by Cai and Choudhary~\cite{Cai2006a,Cai2006b,Cai2006c}. Unfortunately, it turned out that the matching sum of planar graphs essentially provides the most general tensor network in this class, see~\cite{Cai2006a,Cai2006c}. Following~\cite{Cai2006a} we shall call such networks {\it matchgate tensor networks}, or simply matchgate networks. A surprising discovery made in~\cite{Cai2006b} is that matchgate tensors can be characterized by a simple system of quadratic equations known as {\it matchgate identities} which does not make references to any graph theoretical concepts. Specifically, given a tensor $T$ of rank $n$ with complex-valued components $T(x)=T_{x_1,\, x_2,\ldots,\, x_n}$ labeled by $n$-bit strings $x\in \{0,1\}^n$ one calls $T$ a {\it matchgate tensor}, or simply a matchgate, if \begin{equation} \label{matchgate_identities} \sum_{a\, : \, x_a\ne y_a} T({x\oplus e^a}) \, T({y\oplus e^a}) \, (-1)^{x_{1} + \ldots + x_{a-1} + y_{1} + \ldots +y_{a-1}} =0 \quad \mbox{for all} \quad x,y\in \{0,1\}^n. \end{equation} Here $e^a$ denotes a string in which the $a$-th bit is $1$ and all other bits are $0$. The symbol $\oplus$ stands for a bit-wise XOR of binary strings. For example, a simple algebra shows that a tensor of rank $n=1,2,3$ is a matchgate iff it is either even or odd\footnote{A tensor $T$ is called even (odd) if $T(x)=0$ for all strings $x$ with odd (even) Hamming weight.}. Furthermore, an even tensor of rank $4$ is a matchgate iff \begin{equation} \label{Grank4} -T(0000)\, T(1111) + T(1100) \, T(0011) - T(1010)\, T(0101) + T(1001)\, T(0110) = 0. \end{equation} A matchgate network is a tensor network in which every tensor is a matchgate.
The purpose of the present paper is two-fold. Firstly, we develop a formalism that allows one to perform {\it partial contractions} of matchgate networks, for example, contraction of a single edge combining its endpoints into a single vertex. More generally, the formalism allows one to contract any connected planar subgraph $G$ of the network into a single vertex $u(G)$ by "integrating out" all internal edges of $G$. The number of parameters describing the contracted tensor assigned to $u(G)$ is independent of the size of $G$. It depends only on the number of "external" edges connecting $G$ to the rest of the network. This is the main distinction of our formalism compared to the original matchgate formalism of Valiant~\cite{Valiant02a}. The ability to implement partial contractions may be useful for designing efficient parallel contraction algorithms. More importantly, we show that it yields a faster contraction algorithm for matchgate networks on non-planar graphs.
Our formalism makes use of anticommuting (Grassmann) variables such that a tensor of rank $n$ is represented by a generating function of $n$ Grassmann variables. A matchgate tensor is shown to have a Gaussian generating function that depends on $O(n^2)$ parameters. The matchgate identities Eq.~(\ref{matchgate_identities}) can be described by a first-order differential equation making manifest their underlying symmetry. Contraction of tensors is equivalent to convolution of their generating functions. Contraction of matchgate tensors can be performed efficiently using the standard Gaussian integration technique. We use the formalism to prove that a tensor satisfies matchgate identities if and only if it can be represented by the matching sum on some planar graph. It reproduces the result obtained earlier by Cai and Choudhary~\cite{Cai2006b,Cai2006c}. Our approach also reveals that the notion of a matchgate tensor is equivalent to the one of a Gaussian operator introduced in~\cite{Bravyi05} in the context of quantum computation.
Secondly, we describe an improved algorithm for contraction of matchgate networks on non-planar graphs. Let $\Sigma$ be a standard oriented closed surface of genus $g$, i.e., a sphere with $g$ handles. \begin{dfn} \label{dfn:planar_cut} Given a graph $G=(V,E)$ embedded into a surface $\Sigma$ we shall say that $G$ is contractible if there exists a region $D\subset \Sigma$ with topology of a disk containing all vertices and all edges of $G$. A subset of edges $M\subseteq E$ is called a planar cut of $G$ if a graph $G_M=(V,E\backslash M)$ is contractible. \end{dfn} A {\it contraction value} $c({\cal T })$ of a tensor network ${\cal T }$ is a complex number obtained by contracting all tensors of ${\cal T }$. Our main result is as follows. \begin{theorem} \label{thm:main} Let ${\cal T }$ be a matchgate tensor network on a graph $G=(V,E)$ with $n$ vertices embedded into a surface of genus $g$. Assume we are given a planar cut of $G$ with $m$ edges. Then the contraction value $c({\cal T })$ can be computed in time $T=O((n+m)^6)+ O(m^3)\, 2^{2g}$. If $G$ has a bounded vertex degree, one can compute $c({\cal T })$ in time $T=O((n+m)^3)+ O(m^3)\, 2^{2g}$. \end{theorem} If a network has a small planar cut, $m\ll n$, the theorem provides a speedup for computing the matching sum and the partition function of the Ising model compared to the FKT method. For example, computing the matching sum of a graph $G$ as above by the FKT method would require time $T=O(n^3)\, 2^{2g}$ since the matching sum is expressed as a linear combination of $2^{2g}$ Pfaffians where each Pfaffian involves a matrix of size $n\times n$, see~\cite{Gallucio99,Zecchina01,Reshetikhin06}, and since Pfaffian of an $n\times n$ matrix can be computed in time $O(n^3)$, see Remark~2 below. In contrast to the FKT method, our algorithm is divided into two stages. At the first stage that requires time $O((n+m)^6)$ one performs a partial contraction of the planar subgraph $G_M$ determined by the given planar cut $M$, see Def.~\ref{dfn:planar_cut}.
The contraction reduces the number of edges in a network down to $m$ without changing the genus\footnote{If the initial network represents a matchings sum, the first stage of the algorithm would require only time $O((n+m)^3)$.}. The first stage of the algorithm yields a new network ${\cal T }'$ with a single vertex and $m$ self-loops such that $c({\cal T }')=c({\cal T })$. At the second stage one contracts the network ${\cal T }'$ by expressing the contraction value $c({\cal T }')$ as a linear combination of $2^{2g}$ Pfaffians similar to the FKT method. However each Pfaffian involves a matrix of size only $O(m)\times O(m)$.
\noindent {\it Remark 1:} The statement of the theorem assumes that all tensors are specified by their generating functions. Thus a matchgate tensor of rank $d$ can be specified by $O(d^2)$ parameters, see Section~\ref{sec:matchgates} for details. The ordering of indexes in any tensor must be consistent with the orientation of a surface. See Section~\ref{subs:tensor_networks} for a formal definition of tensor networks.
\noindent {\it Remark 2:} Recall that Pfaffian of an $n\times n$ antisymmetric matrix $A$ is defined as \[ \mathop{\mathrm{Pf}}\nolimits{(A)}=\left\{\begin{array}{rcl} 0 &\mbox{if}& \mbox{$n$ is odd},\\ \frac1{2^n\, n!} \sum_{\sigma\in S_n} \mathrm{sgn}(\sigma) \, A_{\sigma(1),\sigma(2)}\, A_{\sigma(3),\sigma(4)}\cdots A_{\sigma(n-1),\sigma(n)} &\mbox{if} & \mbox{$n$ is even}.\\ \end{array}\right. \] where $S_n$ is the symmetric group and $\mathrm{sgn}(\sigma)=\pm 1$ is the parity of a permutation $\sigma$. One can efficiently compute Pfaffian up to a sign using an identity $\mathop{\mathrm{Pf}}\nolimits{(A)}^2=\det{(A)}$. However, in order to compute a linear combination of several Pfaffians one needs to know the sign exactly. One can directly compute $\mathop{\mathrm{Pf}}\nolimits{(A)}$ using the combinatorial algorithm by Mahajan et al~\cite{Mahajan99} in time $O(n^4)$. Alternatively, one can use Gaussian elimination to find an invertible matrix $U$ such that $U^T\, A\, U$ is block-diagonal with all blocks of size $2\times 2$. It requires time $O(n^3)$. Then $\mathop{\mathrm{Pf}}\nolimits{(A)}$ can be computed using an identity $\mathop{\mathrm{Pf}}\nolimits{(U\, A\, U^T)}=\det{(U)}\, \mathop{\mathrm{Pf}}\nolimits{(A)}$. This method yields $O(n^3)$ algorithm although it is less computationally stable compared to the combinatorial algorithm of~\cite{Mahajan99}.
\section{Some definitions and notations} \subsection{Tensor networks} \label{subs:tensor_networks}
Throughout this paper a tensor of rank $d$ is a $d$-dimensional complex array $T$ in which the indexes take values $0$ and $1$. Given a binary string of indexes $x=(x_1 x_2 \ldots x_d)$ we shall denote the corresponding component $T_{x_1 x_2 \ldots x_d}$ as $T(x)$.
A tensor network is a product of tensors whose indexes are pairwise contracted. More specifically, each tensor is represented by a vertex of some graph $G=(V,E)$, where $V$ is a set of vertices and $E$ is a set of edges. The graph may have self-loops and multiple edges.
For every edge $e\in E$ one defines a variable $x(e)$ taking values $0$ and $1$. A bit string $x$ that assigns a particular value to every variable $x(e)$ is called an {\it index string}. A set of all possible index strings will be denoted ${\cal X }(E)$. In order to define a tensor network on $G$ one has to order edges incident to every vertex. We shall assume that $G$ is specified by its {\it incidence list}, i.e., for every vertex $u$ one specifies an ordered list of edges incident to $u$ which will be denoted $E(u)$. Thus
$E(u)=\{e_1^u,\ldots,e_{d(u)}^u\}$ where $e^u_j\in E$ for all $j$. Here $d(u)=|E(u)|$ is the degree of $u$. If a vertex $u$ has one or several self-loops, we assume that every self-loop appears in the list $E(u)$ twice (because it will represent contraction of two indexes). For example, a vertex with one self-loop and no other incident edges has degree $2$.
A tensor network on $G$ is a collection of tensors ${\cal T }=\{T_u\}_{u\in V}$ labeled by vertices of $G$ such that a tensor $T_u$ has rank $d(u)$. A {\it contraction value} of a network ${\cal T }$ is defined as \begin{equation} c({\cal T })=\sum_{x\in {\cal X }(E)} \prod_{u\in V} \,T_u(x(e^u_1)\ldots x(e^u_{d(u)})). \end{equation} Thus the contraction value can be computed by taking a tensor product of all tensors $\{T_u\}$ and then contracting those pairs of indexes that correspond to the same edge of the graph. By definition, $c({\cal T })$ is a complex number (tensor of rank $0$).
It will be implicitly assumed throughout this paper that a tensor network is defined on a graph $G$ embedded into a closed oriented surface $\Sigma$. We require that the order of edges incident to any vertex $u$ must agree with the order in which the edges appear if one circumnavigates $u$ counterclockwise. Thus the order on any set $E(u)$ is completely specified by the choice of the first edge $e^u_1\in E(u)$. If the surface $\Sigma$ has genus $g$ we shall say that $G$ has genus $g$ (it may or may not be the minimal genus for which the embedding of $G$ into $\Sigma$ is possible).
\subsection{Anticommuting variables} In this section we introduce notations pertaining to the Grassmann algebra and anticommuting variables (see the textbook~\cite{ItzDr} for more details). Consider a set of formal variables $\theta=(\theta_1,\ldots,\theta_n)$ subject to multiplication rules \begin{equation} \label{grassmann} \theta_a^2=0, \quad \theta_a \theta_b+\theta_b\theta_a=0 \quad \mbox{for all} \quad a,b. \end{equation} The Grassmann algebra ${\cal G }(\theta)$ is the algebra of complex polynomials in variables $\theta_1,\ldots,\theta_n$ factorized over the ideal generated by Eq.~(\ref{grassmann}). Equivalently, ${\cal G }(\theta)$ is the exterior algebra of the vector space $\mathbb{C}^n$, where each variable $\theta_a$ is regarded as a basis vector of $\mathbb{C}^n$. More generally, the variables $\theta_a$ may be labeled by elements of an arbitrary finite set $X$ (in our case the variables will be associated with edges or vertices of a graph). A linear basis of ${\cal G }(\theta)$ is spanned by $2^n$ monomials in variables $\theta_a$. Namely, for any subset $M\subseteq \{1,\ldots,n\}$ define a {\it normally ordered} monomial \begin{equation} \label{normal_order} \theta(M)=\prod_{a\in M} \theta_a \end{equation} where the indexes increase from the left to the right. If the variables are labeled by elements of some set $X$, one can define the normally ordered monomials $\theta(M)$, $M\subseteq X$ by choosing some order on $X$. Let us agree that $\theta(\emptyset)=I$. Then an arbitrary element $f\in {\cal G }(\theta)$ can be written as \begin{equation} \label{linear_basis} f=\sum_{M\subseteq \{1,\ldots,n\}} f(M)\, \theta(M), \quad f(M)\in \mathbb{C}. \end{equation}
We shall use notations $f$ and $f(\theta)$ interchangeably meaning that $f$ can be regarded as a function of anticommuting variables $\theta=(\theta_1,\ldots,\theta_n)$. Accordingly, elements of the Grassmann algebra will be referred to as functions. In particular, $I$ is regarded as a constant function. A function $f(\theta)$ is called even (odd) if it is a linear combination of monomials $\theta(M)$
with even (odd) degree.
Even functions span the central subalgebra of ${\cal G }(\theta)$.
We shall often consider several species of Grassmann variables, for example, $\theta=(\theta_1,\ldots,\theta_n)$ and $\eta=(\eta_1,\ldots,\eta_k)$. It is always understood that different variables anticommute. For example, a function $f(\theta,\eta)$ must be regarded as an element of the Grassmann algebra ${\cal G }(\theta,\eta)$, that is, a linear combination of monomials in $\theta_1,\ldots,\theta_n$ and $\eta_1,\ldots,\eta_k$.
A partial derivative over a variable $\theta_a$ is a linear map $\partial_a\, : \, {\cal G }(\theta) \to {\cal G }(\theta)$ defined by requirement $\partial_a\cdot I=0$ and the Leibniz rule \[ \partial_a\cdot (\theta_b \, f) = \delta_{a,b}\, f - \theta_b (\partial_a\cdot f). \] More explicitly, given any function $f\in {\cal G }(\theta)$, represent it as $f(\theta)=f_0+\theta_a\, f_1$, where $f_0, f_1\in {\cal G }(\theta)$ do not depend on $\theta_a$. Then $\partial_a\, f=f_1$. It follows that $\partial_a\cdot \theta_a=I$, $\partial_a \theta_b=-\theta_b \partial_a$,
$\partial_a \partial_b=-\partial_b\partial_a$ for $a\ne b$ and $\partial_a^2=0$.
A linear change of variables $\theta_a=\sum_{b=1}^n U_{a,b}\, \tilde{\theta}_b$ with invertible matrix $U$ induces an automorphism of the algebra ${\cal G }(\theta)$ such that $f(\theta)\to f(\tilde{\theta})$. The corresponding transformation of partial derivatives is \begin{equation} \label{derivative_change} \partial_a= \sum_{b=1}^n (U^{-1})_{b,a}\, \tilde{\partial}_b. \end{equation}
\subsection{Gaussian integrals} \label{subs:Gintegral} Let $\theta=(\theta_1,\ldots,\theta_n)$ be a set of Grassmann variables. An integral over a variable $\theta_a$ denoted by $\int d\theta_a$ is a linear map from ${\cal G }(\theta_1,\ldots,\theta_n)$ to ${\cal G }(\theta_1,\ldots,\hat{\theta}_a,\ldots,\theta_n)$, where $\hat{\theta}_a$ means that the variable $\theta_a$ is omitted.
To define an integral $\int d\theta_a \, f(\theta)$, represent the function $f$ as $f=f_0+\theta_a\, f_1$, where $f_0,f_1\in {\cal G }(\theta_1,\ldots,\hat{\theta}_a,\ldots,\theta_n)$. Then $\int d\theta_a f(\theta)=f_1$. Thus one can compute the integral $\int d\theta_a f(\theta)$ by first computing the derivative $\partial_a\cdot f(\theta)$ and then excluding the variable $\theta_a$ from the list of variables of $f$.
Given an ordered set of Grassmann variables $\theta=(\theta_1,\ldots,\theta_n)$ we shall use a shorthand notation \[ \int D\theta = \int d\theta_n \cdots \int d\theta_2 \int d\theta_1. \] Thus $\int D\theta$ can be regarded as a linear functional on ${\cal G }(\theta)$, or as a linear map from ${\cal G }(\theta,\eta)$ to ${\cal G }(\eta)$, and so on. The action of $\int D\theta$ on the normally ordered monomials is as follows
\begin{equation} \label{I1} \int D\theta \, \theta(M) = \left\{ \begin{array}{rcl} 1 &\mbox{if}& M=\{1,2,\ldots,n\},\\ 0 && \mbox{otherwise}.\\ \end{array}\right. \end{equation} Similarly, if one regards $\int D\theta$ as a linear map from ${\cal G }(\theta,\eta)$ to ${\cal G }(\eta)$ then \[ \int D\theta \, \theta(M)\, \eta(K) = \left\{ \begin{array}{rcl} \eta(K) &\mbox{if}& M=\{1,2,\ldots,n\},\\ 0 && \mbox{otherwise}.\\ \end{array}\right. \] Although this definition assumes that both variables $\theta$, $\eta$ have a normal ordering, the integral $\int D\theta$ depends only on the ordering of $\theta$.
One can easily check that integrals over different variables anticommute, $\int d\theta_a \int d\theta_b = -\int d\theta_b \int d\theta_a$ for $a\ne b$. More generally, if $\theta=(\theta_1,\ldots,\theta_n)$ and $\eta=(\eta_1,\ldots,\eta_k)$ then \begin{equation} \label{int_com} \int D\theta \, \int D\eta =(-1)^{nk}\, \int D\eta \, \int D\theta. \end{equation} Under a linear change of variables $\theta_a=\sum_{b=1}^n U_{a,b}\, \eta_b$ the integral transforms as \begin{equation} \label{int_change} \int D\theta=\det{(U)}\, \int D\eta. \end{equation} In the rest of the section we consider two species of Grassmann variables
$\theta=(\theta_1,\ldots,\theta_n)$ and $\eta=(\eta_1,\ldots,\eta_k)$. Given an antisymmetric $n\times n$ matrix $A$ and any $n\times k$ matrix $B$, define quadratic forms \[ \theta^T\, A\, \theta = \sum_{a,b=1}^n A_{a,b} \, \theta_a\, \theta_b, \quad \theta^T\, B\, \eta=\sum_{a=1}^n\sum_{b=1}^k B_{a,b}\, \theta_a\, \eta_b. \] Gaussian integrals over Grassmann variables are defined as follows. \begin{equation} \label{Gaussian_Integrals} I(A)\eqdef \int D\theta \, \exp{\left( \frac12 \, \theta^T\, A \, \theta\right)} \quad \mbox{and} \quad I(A,B)\eqdef \int D\theta \, \exp{\left( \frac12 \, \theta^T\, A \, \theta + \theta^T\, B\, \eta\right)}. \end{equation} Thus $I(A)$ is just a complex number while $I(A,B)$ is an element of ${\cal G }(\eta)$. Below we present the standard formulas for the Gaussian integrals. Firstly, \begin{equation} \label{GI1} I(A)=\mathop{\mathrm{Pf}}\nolimits{(A)}. \end{equation} Secondly, if $A$ is an invertible matrix then \begin{equation} \label{GI2} I(A,B)=\mathop{\mathrm{Pf}}\nolimits{(A)}\, \exp{\left( \frac12 \, \eta^T\, B^T A^{-1} B\, \eta\right)}. \end{equation} Assume now that $A$ has rank $m$ for some even\footnote{Note that antisymmetric matrices always have even rank.} integer $0\le m\le n$. Choose any invertible matrix $U$ such that $AU$ has zero columns $m+1,\ldots,n$. (This is equivalent to finding a basis of $\mathbb{C}^n$ such that the last $n-m$ basis vectors belong to the zero subspace of $A$.) Then \[ U^T\, A\, U = \left[ \begin{array}{cc} A_{11} & 0 \\ 0 & 0 \\ \end{array} \right], \] for some invertible $m\times m$ matrix $A_{11}$. Introduce also matrices $B_1$, $B_2$ of size $m\times k$ and $(n-m)\times k$ respectively such that \[ U^T\, B = \left[ \begin{array}{c} B_1 \\ B_2 \\ \end{array} \right]. \] Performing a change of variables $\theta=U\tilde{\theta}$ in Eq.~(\ref{Gaussian_Integrals}) and introducing variables $\tau=(\tau_1,\ldots,\tau_m)$ and $\mu=(\mu_1,\ldots,\mu_{n-m})$ such that $\tilde{\theta}=(\tau,\mu)$ one gets \[ I(A,B)=\det{(U)} \int D\tau \exp{\left( \frac12 \, \tau^T \, A_{11} \, \tau + \tau^T\, B_1\, \eta\right)} \; \int D\mu \exp{\left( \mu^T\, B_2\, \eta\right)}. \] Here we have taken into account Eqs.~(\ref{int_com},\ref{int_change}). Applying Eq.~(\ref{GI2}) to the first integral one gets \begin{equation} \label{GI3} I(A,B)= \mathop{\mathrm{Pf}}\nolimits{(A_{11})}\,\det{(U)}\, \exp{\left( \frac12 \, \eta^T\, B_1^T (A_{11})^{-1} B_1\, \eta\right)}\; \int D\mu \exp{\left( \mu^T\, B_2\, \eta\right)}. \end{equation} One can easily check that $\int D\mu \exp{\left( \mu^T\, B_2\, \eta\right)}=0$ if the rank of $B_2$ is smaller than the number of variables in $\mu$, that is, $n-m$. Since $B_2$ has only $k$ columns we conclude that \[ I(A,B)=0 \quad \mbox{unless} \quad m\ge n-k. \] Therefore in the non-trivial case $I(A,B)\ne 0$ the matrices $B_1^T (A_{11})^{-1} B_1$ and $B_2$ specifying $I(A,B)$ have size $k\times k$ and $k'\times k$ for some $k'\le k$. It means that $I(A,B)$ can be specified by $O(k^2)$ bits. One can compute $I(A,B)$ in time $O(n^3+n^2 k)$. Indeed, one can use Gaussian elimination to find $U$, compute $\det{(U)}$ and $\mathop{\mathrm{Pf}}\nolimits{(A_{11})}$ in time $O(n^3)$. The matrix $A_{1,1}^{-1}$ can be computed in time $O(n^3)$. Computing the matrices $B_1,B_2$ requires time $O(n^2 k)$.
The formula Eq.~(\ref{GI3}) will be our main tool for contraction of matchgate tensor networks.
\section{Matchgate tensors} \label{sec:matchgates}
\subsection{Basic properties of matchgate tensors} Although the definition of a matchgate tensor in terms of the matchgate identities Eq.~(\ref{matchgate_identities}) is very simple, it is neither very insightful nor very useful. Two equivalent but more operational definitions will be given in Sections~\ref{subs:matchgate=gaussian}, \ref{subs:matchgate=matchsum}. Here we list some basic properties of matchgate tensors that can be derived directly from Eq.~(\ref{matchgate_identities}). In particular, following the approach of~\cite{Cai2006b}, we prove that a matchgate tensor of rank $n$ can be specified by a {\it mean vector} $z\in \{0,1\}^n$ and a {\it covariance matrix} $A$ of size $n\times n$. \begin{prop} Let $T$ be a matchgate tensor of rank $n$. For any $z\in \{0,1\}^n$ a tensor $T'$ with components $T'(x)=T(x\oplus z)$ is a matchgate tensor. \end{prop} \begin{proof} Indeed, make a change of variables $x\to x\oplus z$, $y\to y\oplus z$ in the matchgate identities \end{proof} Let $T$ be a non-zero matchgate tensor of rank $n$. Choose any string $z$ such that $T(z)\ne 0$ and define a new tensor $T'$ with components \[ T'(x)=\frac{T(x\oplus z)}{T(z)}, \quad x\in \{0,1\}^n, \] such that $T'$ is a matchgate and $T'(0^n)=1$. Introduce an antisymmetric $n\times n$ matrix $A$ such that \[ A_{a,b}=\left\{ \begin{array}{rcl} T'(e^a\oplus a^b) &\mbox{if}& a<b, \\ -T'(e^a\oplus a^b) &\mbox{if} & a>b,\\ 0 &\mbox{if} & a=b.\\ \end{array} \right. \] \begin{prop} \label{prop:zA} For any $x\in \{0,1\}^n$ \[ T'(x)=\left\{ \begin{array}{ccl} \mathop{\mathrm{Pf}}\nolimits{(A(x))} &\mbox{if} & \mbox{$x$ has even weight}\\ 0 &\mbox{if} & \mbox{$x$ has odd weight}\\ \end{array}\right., \] where $A(x)$ is a matrix obtained from $A$ by removing all rows and columns $a$ such that $x_a=0$. \end{prop} \begin{proof} Let us prove the proposition by induction in the weight of $x$. Choosing $x=0^n$ and $y=e^a$ in the matchgate identities Eq.~(\ref{matchgate_identities}) one gets $T'(e^a)=0$ for all $a$. Similarly, choosing $x=e^b$ and $y=e^a$ with $a< b$
one gets $T'(e^a\oplus e^b)=A_{a,b}=\mathop{\mathrm{Pf}}\nolimits{(A(e^a \oplus e^b))}$. Thus the proposition is true for $|x|=1,2$. Assume it is true for all strings $x$ of weight $\le k$. For any string $x$ of weight $k+1$ and any $a$ such that $x_a=0$ apply the matchgate identities Eq.~(\ref{matchgate_identities}) with $x$ and $y=e^a$. After simple algebra one gets \[ T'(x\oplus e^a)=\sum_{b\, :\, x_b=1} A_{a,b}\, T'(x\oplus e^b) \, (-1)^{\eta(a,b)}, \quad \eta(a,b)=\sum_{j=a}^{b-1} x_j. \] Noting that $x\oplus e^b$ has weight $k$ and applying the induction hypothesis one gets \[ T'(x\oplus e^a)=\sum_{b\, :\, x_b=1} A_{a,b}\, \mathop{\mathrm{Pf}}\nolimits{(A(x\oplus e^b))} \, (-1)^{\eta(a,b)} \] for even $k$ and $T'(x\oplus e^a)=0$ for odd $k$. Thus $T'(y)=0$ for all odd strings of weight $k+2$. Furthermore, let non-zero bits of $x\oplus e^b$ be located at positions $j_1<j_2<\ldots<j_k$. Note that the sign of $A_{a,b}\, (-1)^{\eta(a,b)}$ coincides with the parity of a permutation that orders elements in a set $[a,b,j_1,j_2,\ldots,j_k]$. Therefore, by definition of Pfaffian one gets $T'(x\oplus e^a)=\mathop{\mathrm{Pf}}\nolimits{(A(x\oplus e^a)})$. \end{proof} Thus one can regard the vector $z$ and the matrix $A$ above as analogues of a mean vector and a covariance matrix for Gaussian states of fermionic modes, see for instance~\cite{Bravyi05}. Although Proposition~\ref{prop:zA} provides a concise description of a matchgate tensor, it is not very convenient for contracting matchgate networks because the mean vector $z$ and the covariance matrix $A$ are not uniquely defined. \begin{cor} \label{cor:parity} Any matchgate tensor is either even or odd. \end{cor} \begin{proof} Indeed, the proposition above implies that if a matchgate tensor $T$ has even (odd) mean vector it is an even (odd) tensor. \end{proof}
\subsection{Describing a tensor by a generating function} \label{subs:generating} Let $\theta=(\theta_1,\ldots,\theta_n)$ be an ordered set of $n$ Grassmann variables. For any tensor $T$ of rank $n$ define a {\it generating function} $T\in {\cal G }(\theta)$ according to \[ T(\theta)=\sum_{x\in \{0,1\}^n} T(x)\, \theta(x). \] Here $\theta(x)=\theta_1^{x_1} \cdots \theta_n^{x_n}$ is the normally ordered monomial corresponding to the subset of indexes $x=\{ a\, : \, x_a=1\}$. Let us introduce a linear differential operator $\Lambda$ acting on the tensor product of two Grassmann algebras ${\cal G }(\theta)\otimes {\cal G }(\theta)$ such that \begin{equation} \label{Lambda_ad} \Lambda = \sum_{a=1}^n \theta_a \otimes \partial_a + \partial_a \otimes \theta_a. \end{equation} \begin{lemma} \label{lemma:matchgate2} A tensor $T$ of rank $n$ is a matchgate iff \begin{equation} \label{matchgate_identities2} \Lambda \cdot T\otimes T=0. \end{equation} \end{lemma} \begin{proof} For any strings $x,y\in \{0,1\}^n$ one has the following identity: \[ (\theta_a \otimes \partial_a + \partial_a \otimes \theta_a)\cdot \theta(x)\otimes \theta(y) = \left\{ \begin{array}{rcl} 0 &\mbox{if} & x_a=y_a,\\
(-1)^{x_1+\ldots + x_{a-1} + y_1+\ldots + y_{a-1}}\, \theta(x\oplus e_a)\otimes \theta(y\oplus e_a) &\mbox{if} & x_a\ne y_a.\\ \end{array}\right. \] Expanding both factors $T$ in Eq.~(\ref{matchgate_identities2}) in the monomials $\theta(x)$, $\theta(y)$, using the above identity, and performing a change of variable $x\to x\oplus e_a$ and $y\to y\oplus e_a$ for every $a$ one gets a linear combination of monomials $\theta(x)\otimes \theta(y)$ with the coefficients given by the right hand side of Eq.~(\ref{matchgate_identities}). Therefore Eq.~(\ref{matchgate_identities2}) is equivalent to Eq.~(\ref{matchgate_identities}). \end{proof} Lemma~\ref{lemma:matchgate2} provides an alternative definition of a matchgate tensor which is much more useful than the original definition Eq.~(\ref{matchgate_identities}). For example, it is shown below that the operator $\Lambda$ has a lot of symmetries which can be translated into a group of transformations preserving the subset of matchagate tensors. \begin{lemma} \label{lemma:inv} The operator $\Lambda$ is invariant under linear reversible changes of variables $\theta_a=\sum_{b=1}^n U_{a,b}\, \tilde{\theta_b}$. \end{lemma} \begin{proof} Indeed, let $\tilde{\partial}_a$ be the partial derivative over $\tilde{\theta}_a$. Using Eq.~(\ref{derivative_change}) one gets \[ \sum_{a=1}^n \theta_a \otimes \partial_a + {\partial}_a\otimes {\theta}_a = \sum_{a,b,c=1}^n U_{a,b}\, (U^{-1})_{c,a} \, (\tilde{\theta}_b \otimes \tilde{\partial}_c +
\tilde{\partial}_c \otimes \tilde{\theta}_b) = \sum_{b}^n \, (\tilde{\theta}_b \otimes \tilde{\partial}_b + \tilde{\partial}_b \otimes \tilde{\theta}_b). \] \end{proof} Lemmas~\ref{lemma:matchgate2},\ref{lemma:inv} imply that linear reversible change of variables $T(\theta)\to T(\tilde{\theta})$, where $\theta_a=\sum_{b=1}^n U_{a,b}\, \tilde{\theta_b}$ map matchgates to matchgates. \begin{cor} \label{cor:inv} Let $T$ be a matchgate tensor of rank $n$. Then a tensor $T'$ defined by any of the following transformations is also matchgate.\\ ({\it Cyclic shift}): $T'(x_1,x_2,\ldots,x_n)=T(x_2,\ldots,x_n,x_1)$,\\ ({\it Reflection}): $T'(x_1,x_2,\ldots,x_n)=T(x_n,\ldots,x_2,x_1)$,\\ ({\it Phase shift}): $T'(x)=(-1)^{x\cdot z}\, T(x)$, where $z\in \{0,1\}^n$.\\ \end{cor} \begin{proof} Let $\epsilon=0$ if $T$ is an even tensor and $\epsilon=1$ if $T$ is an odd tensor, see Corollary~\ref{cor:parity}. The transformations listed above are generated by the following linear changes of variables: \begin{eqnarray} \mbox{\it Phase shift} &\mbox{:} & \theta_a \to (-1)^{z_a}\, \theta_a, \quad a=1,\ldots,n.\nonumber \\ \mbox{\it Cyclic shift} &\mbox{:} & \theta_a\to \theta_{a-1} \quad a=2,\ldots,n,
\quad \mbox{and} \quad \theta_1\to (-1)^{\epsilon+1}\, \theta_n. \nonumber \\
\mbox{\it Reflection} &\mbox{:} & \theta_a \to i\, \theta_{n-a}.\nonumber \end{eqnarray} Indeed, let $\theta(x)$ be the normally ordered monomial where $x=(x_1,x_2,\ldots,x_n)$. Let $x'=(x_2,\ldots,x_n,x_1)$ for the cyclic shift and $x'=(x_n,\ldots,x_2,x_1)$ for the reflection. Then the linear changes of variables stated above map $\theta(x)$ to $(-1)^{z\cdot x}\, \theta(x)$ for the phase shift, to $\theta(x')$ for the cyclic shift, and to $i^\epsilon\, \theta(x')$ for the reflection. Therefore, in all three cases $T'$ is a matchgate tensor. \end{proof}
\subsection{Matchgate tensors have Gaussian generating function} \label{subs:matchgate=gaussian} A memory size required to store a tensor of rank $n$ typically grows exponentially with $n$. However the following theorem shows that for matchgate tensors the situation is much better. \begin{theorem} \label{thm:canonical} A tensor $T$ of rank $n$ is a matchgate iff there exist an integer $0\le k\le n$, complex matrices $A$, $B$ of size $n\times n$ and $k\times n$ respectively, and a complex number $C$ such that $T$ has generating function \begin{equation} \label{canonical} T(\theta)=C\exp{\left( \frac12\, \theta^T\, A \, \theta \right)}\int D\mu\, \exp{\left( \mu^T \, B\, \theta \right)}, \end{equation} where $\mu=(\mu_1,\ldots,\mu_k)$ is a set of $k$ Grassmann variables. Furthermore, one can always choose the matrices $A$ and $B$ such that $A^T=-A$ and $BA=0$. \end{theorem} Thus the triple $(A,B,C)$ provides a concise description of a matchgate tensor that requires a memory size only $O(n^2)$. In addition, it will be shown that contraction of matchgate tensors can be efficiently implemented using the representation Eq.~(\ref{canonical}) and the Gaussian integral formulas of Section~\ref{subs:Gintegral}. We shall refer to the generating function Eq.~(\ref{canonical}) as a {\it canonical generating function} for a matchgate tensor $T$. \begin{cor} \label{cor:GI=matchgate} For any matrices $A$ and $B$ the
Gaussian integral $I(A,B)$ defined in Eq.~(\ref{Gaussian_Integrals}) is a matchgate.
\end{cor}
\begin{proof}
Indeed, use Eq.~(\ref{GI3}) and Theorem~\ref{thm:canonical}.
\end{proof}
In the rest of the section we shall prove Theorem~\ref{thm:canonical}. \begin{proof}[Proof of Theorem~\ref{thm:canonical}.] Let us first verify that the tensor defined in Eq.~(\ref{canonical}) is a matchgate, i.e., $\Lambda\cdot T\otimes T=0$, see Lemma~\ref{lemma:matchgate2}. Without loss of generality $A$ is an antisymmetric matrix and $C=1$. Write $T$ as \[ T=T_2\, T_1, \quad \mbox{where} \quad T_2=\exp{\left( \frac12\, \theta^T\, A \, \theta \right)}, \quad T_1=\int D\mu\, \exp{\left( \mu^T \, B\, \theta \right)}. \] Noting that $T_2$ is an even function and $\partial_a\, \theta(x) =\partial_a \cdot \theta(x) + \theta(x)\, \partial_a$ for any even string $x$ one concludes that \begin{equation} \label{T1T2} \Lambda\cdot T\otimes T=\left( \Lambda\cdot T_2\otimes T_2\right)\, T_1\otimes T_1+ T_2\otimes T_2\, \left( \Lambda\cdot T_1\otimes T_1\right). \end{equation} Therefore it suffices to prove that $\Lambda\cdot T_2\otimes T_2=0$ and $\Lambda\cdot T_1\otimes T_1=0$. The first identity follows from $\partial_a\cdot T_2=\sum_{b=1}^n A_{a,b}\, \theta_b \, T_2$ and $A^T=-A$ which implies \[ \Lambda\cdot T_2\otimes T_2=\sum_{a,b=1}^n A_{a,b} \, (\theta_a\otimes\theta_b+ \theta_b\otimes \theta_a)\, T_2\otimes T_2 =0. \] To prove the second identity consider the singular value decomposition $B=L^T\tilde{B} R$, where $L\in SU(k)$ and $R\in SU(n)$ are unitary operators, while $\tilde{B}$ is a $k\times n$ matrix with all non-zero elements located on the main diagonal, $\tilde{B}=\mbox{diag}{(B_1,\ldots,B_k)}$. Introducing new variables $\tilde{\theta}=R\, \theta$ and $\tilde{\mu}=L\, \mu$ one gets \[ T_1=\int D\tilde{\mu}\, \exp{\left( \sum_{a=1}^k B_a\, \tilde{\mu}_a \, \tilde{\theta_a}\right)}= B_1\cdots B_k\, \tilde{\theta}_1\cdots \tilde{\theta}_k. \] Here we have used identity $\int D\tilde{\mu}=\det{(L)}\, \int D\mu =\int D\mu$, see Eq.~(\ref{int_change}). Since $\Lambda$ is invariant under linear reversible changes of variables, see Lemma~\ref{lemma:inv}, and since $\Lambda\cdot \theta(x)\otimes\theta(x)=0$ for any monomial $\theta(x)$ one gets $\Lambda\cdot T_1\otimes T_1=0$. We proved that $\Lambda\cdot T\otimes T=0$, that is, $T$ is a matchgate tensor.
Let us now show that any matchgate tensor $T$ of rank $n$ can be written as in Eq.~(\ref{canonical}). Define a linear subspace ${\cal Z }\subseteq \mathbb{C}^n$ such that \[ {\cal Z }=\{ \xi \in \mathbb{C}^n \, : \, \sum_{a=1}^n \xi_a \theta_a T=0\}. \] Let $\dim{({\cal Z })}=k$. Make a change of variables $\eta= U\, \theta$ where $U$ is any invertible matrix such that the last $k$ rows of $U$ span ${\cal Z }$. Then $\eta_a\, T=0$ for all $a=n-k+1,\ldots,n$. It follows that $T$ can be represented as \begin{equation} \label{T=} T=\eta_{n-k+1} \cdots \eta_{n} \, S \end{equation} for some function $S=S(\eta)$ that depends only on variables $\eta_1,\ldots,\eta_{n-k}$. Equivalently, \[ S=\partial_{n}\cdots \partial_{n-k+1} \cdot T, \] where the partial derivatives are taken with respect to the variables $\eta$. Since $\Lambda$ is invariant under reversible linear changes of variables, see Lemma~\ref{lemma:inv}, and since $\Lambda\, \partial_a\otimes \partial_a = \partial_a\otimes \partial_a \, \Lambda$, we get \begin{equation} \label{LambdaSS} \Lambda \cdot S\otimes S=\sum_{a=1}^{n-k} \eta_a S \otimes \partial_a\cdot S + \partial_a \cdot S \otimes \eta_a S=0. \end{equation} By definition of the subspace ${\cal Z }$ the functions $\eta_1 S,\ldots,\eta_{n-k} S$ are linearly independent. Therefore there exist linear functionals $F_a \, : \, {\cal G }(\eta) \to \mathbb{C}$, $a=1,\ldots,n-k$, such that $F_a(\eta_b S)=\delta_{a,b}$. Applying $F_a$ to the first factor in Eq.~(\ref{LambdaSS}) we get \begin{equation} \label{difur1} \partial_a \cdot S =\sum_{b=1}^{n-k} M_{a,b}\, \eta_b\, S, \quad \mbox{where} \quad M_{a,b}=-F_a(\partial_b \cdot S)\in \mathbb{C}, \end{equation} for all $a=1,\ldots,n-k$. Let $k_{min}$ the lowest degree of monomials in $S$. Let us show that $k_{min}=0$, that is, $S(\eta)$ contains $I$ with a non-zero coefficient. Indeed, let $S_{min}$ be a function obtained from $S$ by retaining only monomials of degree $k_{min}$. Since any monomial in the r.h.s. of Eq.~(\ref{difur1}) has degree at least $k_{min}+1$, we conclude that $\partial_a \cdot S_{min}=0$ for all $a$. It means that $S_{min}=C\, I$ for some complex number $C\ne 0$ and thus $k_{min}=0$.
Applying the partial derivative $\partial_b$ to Eq.~(\ref{difur1})
we get $M_{a,b} =C^{-1} (\partial_b\, \partial_a \cdot \left. S)\right|_{\eta=0}$, where the substitution $\eta=0$ means that the term proportional to the identity is taken. Since the partial derivatives over different variables anticommute, $M$ is an antisymmetric matrix.
Using Gaussian elimination any antisymmetric matrix $M$ can be brought into a block-diagonal form with $2\times 2$ blocks on the diagonal by a transformation $M\to M'=W^T\, X \, W$, where $W$ is an invertible matrix (in fact, one can always choose unitary $W$, see~\cite{Zumino62}). Since our change of variables $\eta=U\theta$ allows arbitrary transformations in the subspace of $\eta_1,\ldots,\eta_{n-k}$ we can assume that $M$ is already bock-diagonal, \[ M=\bigoplus_{a=1}^m \left( \begin{array}{cc} 0 & \lambda_a \\ -\lambda_a & 0 \\ \end{array} \right), \quad \lambda_1,\ldots,\lambda_m\in \mathbb{C}, \] where only non-zero blocks are represented, so that $2m\le n-k$.
Applying Eq.~(\ref{difur1}) for $a=1,2$ we get \begin{equation} \label{difur2} \partial_1 \cdot S = \lambda_1 \eta_2 S, \quad \partial_2\cdot S = -\lambda_1 \eta_1 S. \end{equation} Note that $S$ can be written as \begin{equation} \label{expand} S=\sum_x (\alpha_x \eta_1 + \beta_x \eta_2)\eta(x) + \sum_y (\gamma_y I + \delta_y \eta_1\eta_2) \eta(y), \end{equation} where the sums over $x$ and $y$ run over all odd and even monomials in $\eta_3,\ldots,\eta_{n-k}$ respectively. Substituting Eq.~(\ref{expand}) into Eq.~(\ref{difur2}) one gets $\alpha_x=\beta_x=0$ and $\delta_x=\lambda_1 \gamma_x$, that is \[ S=(I+\lambda_1 \eta_1\eta_2) S', \] where $S'$ depends only on variables $\eta_3,\ldots,\eta_{n-k}$. Repeating this argument inductively, we arrive to the representation \[ S=C\, \prod_{a=1}^m (I+\lambda_a \eta_{2a-1}\eta_{2a}) = C\, \exp{\left( \frac12 \eta^T\, M \, \eta\right)}. \] Here we extended the matrix $M$ such that its last $k$ columns and rows are zero. Combining it with Eq.~(\ref{T=}) one gets \[ T=C\, \eta_{n-k+1} \cdots \eta_n \, \exp{\left( \frac12 \eta^T\, M \, \eta\right)}=C\, \exp{\left( \frac12 \eta^T\, M \, \eta\right)}\, \int D\mu\, \exp{\left( \mu^T \, \tilde{B}\, \eta \right)}, \] where $\mu$ is a vector of $k$ Grassmann variables and $\tilde{B}$ is a $k\times n$ matrix with $0$,$1$ entries such that \[ \mu^T \, \tilde{B}\, \eta = \sum_{a=1}^k \mu_a\, \eta_{n-k+a}. \] Recalling that $\eta=U\, \theta$, we conclude that $T$ has a representation Eq.~(\ref{canonical}) with $A=U^T\, M\, U$ and $B=\tilde{B}\, U$. As a byproduct we also proved that the matrices $A$, $B$ in Eq.~(\ref{canonical}) can always be chosen such that $BA=0$ since $BA=\tilde{B}\, M\, U$ and all non-zero entries of $\tilde{B}$ are in the last $k$ rows. \end{proof}
\subsection{Graph theoretic definition of matchgate tensors} \label{subs:matchgate=matchsum}
Let $G=(V,E,W)$ be an arbitrary weighted graph with a set of vertices $V$, set of edges $E$ and a weight function $W$ that assigns a complex weight $W(e)$ to every edge $e\in E$. \begin{dfn} Let $G=(V,E)$ be a graph and $S\subseteq V$ be a subset of vertices. A subset of edges $M\subseteq E$ is called an $S$-imperfect matching iff every vertex from $S$ has no incident edges from $M$ while every vertex from $V\backslash S$ has exactly one incident edge from $M$. A set of all $S$-imperfect matchings in a graph $G$ will be denoted
${\cal M }(G,S)$. \end{dfn} Note that a perfect matching corresponds to an $\emptyset$-imperfect matching. Occasionally we shall denote a set of perfect matching by ${\cal M }(G)\equiv {\cal M }(G,\emptyset)$. For any subset of vertices $S\subseteq V$ define a {\it matching sum} \begin{equation} \label{Zcycles1} \perfectm{G}{S}=\sum_{M\in {\cal M }(G,S)} \; \prod_{e\in M} W(e). \end{equation} (A matching sum can be identified with a planar matchgate of~\cite{Valiant07}.) In this section we outline an isomorphism between matchgate tensors and matching sums of planar graphs discovered earlier in~\cite{Cai2006c}. For the sake of completeness we provide a proof of this result below. Although the main idea of the proof is the same as in~\cite{Cai2006c} some technical details are different. In particular, we use much simpler crossing gadget.
Specifically, we shall consider planar weighted graphs $G=(V,E,W)$ embedded into a disk such that some subset of $n$ {\it external} vertices $V_{ext}\subseteq V$ belongs to the boundary of disk while all other {\it internal} vertices $V\backslash V_{ext}$ belong to the interior of $D$. Let $V_{ext}=\{u_1,\ldots,u_n\}$ be an ordered list of external vertices corresponding to circumnavigating anticlockwise the boundary of the disk. Then any binary string $x\in \{0,1\}^n$ can be identified with a subset $x\subseteq V_{ext}$ that includes all external vertices $u_j$ such that $x_j=1$. Now we are ready to state the main result of this section. \begin{theorem} \label{thm:cycles} For any matchgate tensor $T$ of rank $n$ there exists a planar weighted graph $G=(V,E,W)$ with $O(n^2)$ vertices, $O(n^2)$ edges and a subset of $n$ vertices $V_{ext}\subseteq V$ such that \begin{equation} \label{T=Z} T(x)=\perfectm{G}{x} \quad \mbox{for all} \quad x\subseteq V_{ext}. \end{equation} Furthermore, suppose $T$ is specified by its generating function, $T=C\exp{\left( \frac12\, \theta^T\, A \, \theta \right)}\int D\mu\, \exp{\left( \mu^T \, B\, \theta \right)}$. Then the graph $G$ can be constructed in time $O(n^2)$ and the weights $W(e)$ are linear functionals of $A$, $B$, and $C$. \end{theorem} The key step in proving the theorem is to show that Pfaffian of any $n\times n$ antisymmetric matrix can be expressed as a matching sum on some planar graph with $O(n^2)$ vertices. This step can be regarded as a reversal of the FKT method that allows one to represent the matching sum of a planar graph as Pfaffian of the Tutte matrix. \begin{lemma} \label{lemma:Pf=cycles} For any complex antisymmetric matrix $A$ of size $n\times n$ there exists a planar weighted graph $G=(V,E,W)$
with $O(n^2)$ vertices, $O(n^2)$ edges such that
the weights $W(e)$ are linear functionals of $A$ and
\begin{equation}
\label{Pf=cycles}
\mathop{\mathrm{Pf}}\nolimits{(A)}=\perfectm{G}{\emptyset}.
\end{equation}
The graph $G$ can be constructed in time $O(n^2)$.
\end{lemma} {\it Remark:} It should be emphasized that we regard both sides of Eq.~(\ref{Pf=cycles}) as polynomial functions of matrix elements of $A$, and the lemma states that the two polynomials coincide. However, even if one treats both sides of Eq.~(\ref{Pf=cycles}) just as complex numbers, the statement of the lemma is still non-trivial, since one can not compute $\mathop{\mathrm{Pf}}\nolimits{(A)}$ in time $O(n^2)$ and thus one has to construct the graph $G$ without access to the value of $\mathop{\mathrm{Pf}}\nolimits{(A)}$. \begin{proof} Let us assume that $n$ is even (otherwise the statement is trivial). Let $D$ be a disk with $n$ marked points $v_1,\ldots,v_n$ on the boundary such that their order corresponds to anticlockwise circumnavigating the boundary of $D$. Let $C_n$ be the complete graph with vertices $v_1,\ldots,v_n$ embedded into $D$. We assume that the embedding is chosen such that all edges of $C_n$ lie inside the disk and there are only double edge crossing points, see Fig.~1. \begin{figure}
\caption{Left: a complete graph $C_6$ embedded into a disk. Right: a perfect matching on $C_6$ with two self-intersections.}
\label{fig:complete}
\end{figure} Let ${\cal M }(C_n)$ be a set of perfect matchings on $C_n$. For any perfect matching $M\in {\cal M }(C_n)$ let $N_c(M)$ be the number of self-intersections in $M$, i.e., the number of edge crossing points in the planar embedding of $C_n$ in which both crossing edges are occupied by $M$. For example, given a planar embedding of $C_6$ shown on Fig.~1, a perfect matching $M=(1,3),(2,5),(4,6)$ has two self-intersections. We claim that \begin{equation} \label{pf-crossing} \mathop{\mathrm{Pf}}\nolimits{(A)}=\sum_{M\in {\cal M }(C_n)} (-1)^{N_c(M)}\, \prod_{(u,v)\in M,\; u<v} A_{u,v}. \end{equation} Indeed, by definition of Pfaffian \begin{equation} \label{pf-standard} \mathop{\mathrm{Pf}}\nolimits{(A)}=\sum_{\sigma} \mathrm{sgn}(\sigma) \, A_{\sigma(1),\sigma(2)} \cdots A_{\sigma(n-1),\sigma(n)}, \end{equation} where the sum is over all permutations of $n$ elements $\sigma$ such that $\sigma(2j-1)<\sigma(2j)$ for all $j$ and $\sigma(1)<\sigma(3)<\ldots <\sigma(n-1)$. Clearly, there exists a one-to-one correspondence between such permutations and perfect matchings in $C_n$. If $M$ is the perfect matching corresponding to the identity permutation, $M=(1,2),\ldots,(n-1,n)$, one has $N_c(M)=0$ and the signs in Eqs.~(\ref{pf-crossing},\ref{pf-standard}) coincide. Furthermore, changing $M$ by any transposition $j\leftrightarrow j+1$ either does not change $M$ or changes the parity of $N_c(M)$, so the signs in Eqs.~(\ref{pf-crossing},\ref{pf-standard}) coincide for all perfect matchings.
In order to represent the sum over perfect matchings in Eq.~(\ref{pf-crossing}) as a sum over perfect matchings in a planar graph we shall replace each edge crossing point of $C_n$ by a {\it crossing gadget}, see Fig.~2. A {\it crossing gadget} is a planar simulator for an edge crossing point. It allows one to establish a correspondence between subsets of edges in the non-planar graph and subsets of edges in a planar graph. In addition, a crossing gadget will take care of the extra sign\footnote{One can gain some intuition about the extra sign factor in Eq.~(\ref{pf-crossing}) if one thinks about the set of edges occupied by a perfect matching $y$ as a family of "world lines" of fermionic particles. The contribution from $y$ to $\mathop{\mathrm{Pf}}\nolimits{(A)}$ can be thought of as a quantum amplitude assigned to this family of world lines. Whenever two particles are exchanged the amplitude acquires an extra factor $-1$.} factor in Eq.~(\ref{pf-crossing}).
\noindent {\it Crossing gadget.} Consider a weighted graph $G_{cross}$ shown on Fig.~2. It has $6$ vertices and $7$ edges. The edge $(5,6)$ carries weight $-1$ and all other edges carry weight $+1$. We fix the embedding of $G_{cross}$ into a disk such that $G_{cross}$ has four external vertices $\{1,2,3,4\}$ on the boundary of the disk. One can easily check that the matching sum of $G_{cross}$ satisfies the following identities: \begin{eqnarray} \perfectm{G_{cross}}{\emptyset}&=&1,\nonumber \\ \perfectm{G_{cross}}{\{1,3\}}=\perfectm{G_{cross}}{\{2,4\}}&=&1, \nonumber \\ \perfectm{G_{cross}}{\{1,2,3,4\}}&=&-1, \nonumber \\ \perfectm{G_{cross}}{\{1,2\}}=\perfectm{G_{cross}}{\{3,4\}}&=&0, \nonumber \\ \perfectm{G_{cross}}{\{1,4\}}=\perfectm{G_{cross}}{\{2,3\}}&=&0. \nonumber
\end{eqnarray} These identities are illustrated in Fig.~3. In addition, $\perfectm{G_{cross}}{S}=0$ whenever $|S|$ is odd. Thus the four boundary conditions for which the matching sum is non-zero represents the four possible configurations (empty/occupied) of a pair of crossing edges if they were attached to the vertices $\{1,2,3,4\}$. For every edge crossing point of $C_n$ one has to cut out a small disk centered at the crossing point and replace the interior of the disk by the gadget $G_{cross}$ such that the four vertices $\{1,2,3,4\}$ are attached to the four external edges, see Fig.~2. Let $\tilde{C}_n$ be the resulting graph. By construction, $\tilde{C}_n$ is planar. It remains to assign weights to edges of $\tilde{C}_n$ such that \begin{equation} \label{correct_weights} \perfectm{\tilde{C}_n}{\emptyset}=\mathop{\mathrm{Pf}}\nolimits{(A)}. \end{equation}
\begin{figure}
\caption{Each edge crossing point in the planar embedding of the complete graph $C_n$ is replaced by the crossing gadget $G_{cross}$. Edges labeled by $\pm $ carry a weight $\pm 1$.}
\label{fig:crossing}
\end{figure}
\begin{figure}
\caption{Matching sums of the graph $G_{cross}$ corresponding to various boundary conditions.}
\label{fig:crossing1}
\end{figure}
Any edge of $\tilde{C}_n$ falls into one of the four categories: (i) edge of $C_n$; (ii) a section of some edge of $C_n$ between two crossing gadgets; (iii) a section of some edge of $C_n$ between a vertex of $C_n$ and some crossing gadget; (iv) an edge that belongs to some crossing gadget. Note that the edges of type (iv) have been already assigned a weight, whereas any edge of type (i),(ii), and (iii) has a unique ancestor edge $e=(u,v)$ in $C_n$.
Let us agree that for every edge $e=(u,v)$, $u<v$ of $C_n$ we choose one of its descendants $\tilde{e}$ in $\tilde{C}_n$ and assign $\tilde{e}$ the weight $A_{u,v}$, while all other descendants of $e$ are assigned the weight $1$. Since all descendants of $e$ appear or do not appear in any perfect matching $M\in {\cal M }(\tilde{C}_n)$ simultaneously, we arrive to Eq.~(\ref{correct_weights}), that is, $\tilde{C}_n$ is the desired graph $G$. It remains to count the number of vertices in $\tilde{C}_n$. There are $O(n^2)$ crossing gadgets each having $O(1)$ vertices. Thus $\tilde{C}_n$ has $O(n^2)$ vertices. Since $\tilde{C}_n$ is a planar graph it has $O(n^2)$ edges, see~\cite{Diestel}. \end{proof}
Let $\tilde{C}_n$ be a planar graph constructed above. Consider a matching sum $\perfectm{\tilde{C}_n}{S}$ for some subset $S\subseteq \{1,2,\ldots,n\}$ of vertices lying on the boundary of the disk. By repeating the arguments used in the proof of Lemma~\ref{lemma:Pf=cycles} one concludes that \begin{equation} \label{Pf=cycles1} \perfectm{\tilde{C}_n}{S} = \mathop{\mathrm{Pf}}\nolimits{(A[S])} \quad \mbox{for all} \quad S\subseteq \{1,2,\ldots,n\}, \end{equation} where $A[S]$ is a matrix obtained from $A$ by removing all rows and columns $a\in S$. Theorem~\ref{thm:cycles} follows from Eq.~(\ref{Pf=cycles1}) and the following simple observation. \begin{lemma} \label{lemma:T=pfaffian} Let $T$ be a matchgate tensor of rank $n$ with a parity $\epsilon(T)$ specified by its generating function \begin{equation} \label{T=GIntegral} T=C\exp{\left( \frac12\, \theta^T\, F\, \theta \right)}\int D\mu\, \exp{\left( \mu^T \, G\, \theta \right)}. \end{equation} Then \begin{equation} \label{T=pfaffian} T(x)=C \epsilon(T)\, \mathop{\mathrm{Pf}}\nolimits{\left( A(x\, 1^{k-n})\right)} \quad \mbox{for all $x\in \{0,1\}^n$}, \quad \mbox{where} \quad A=\left[ \begin{array}{cc} F & -G^T \\ G & 0 \\ \end{array} \right]. \end{equation} The matrix $A$ has size $k\times k$ with $n\le k\le 2n$. \end{lemma} \noindent {\it Remark:} As usual, $A(y)$ denotes a matrix obtained from $A$ by removing all columns and rows $a$ such that $y_a=0$. We assume that $\epsilon(T)=1$ ($\epsilon(T)=-1$) for even (odd) tensors. \begin{proof} Theorem~\ref{thm:canonical} asserts that $T$ always has a generating function Eq.~(\ref{T=GIntegral}) where $G$ has size $m\times n$ for some $m\le n$. Thus $k=n+m\le 2n$. Introducing a set of $k$ Grassmann variables $\eta=(\theta_1,\ldots,\theta_n,\mu_1,\ldots,\mu_m)$ one can rewrite $T$ as \[ T(\theta)=C\, \int D\mu \exp{\left( \frac12\, \eta^T\, A\, \eta \right)}. \] Expanding the exponent one gets \[ \exp{\left( \frac12\, \eta^T\, A\, \eta \right)}=\sum_{z\in \{0,1\}^k} \mathop{\mathrm{Pf}}\nolimits{(A(z)})\, \eta(z). \] Note that \[ \int \, D\mu \, \eta(z)=\left\{ \begin{array}{rcl} (-1)^{m(z_1+\cdots+z_n)} &\mbox{if} & z_{n+1}=\ldots=z_k=1,\\ 0 &&\mbox{otherwise}.\\ \end{array} \right. \] Taking into account that $m$ is even (odd) for even (odd) tensors and so is $z_1+\cdots +z_n$ we conclude that \begin{equation} \label{T=sum} T(\theta)=C \epsilon(T)\, \sum_{x\in \{0,1\}^n} \mathop{\mathrm{Pf}}\nolimits{(A(x1^{k-n})})\, \theta(x), \end{equation} that is $T(x)=C \epsilon(T)\, \mathop{\mathrm{Pf}}\nolimits{(A(x1^{k-n}))}$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:cycles}] Let $A$ be the $k\times k$ matrix constructed in Lemma~\ref{lemma:T=pfaffian} and $\tilde{C}_k$ be the weighted planar graph constructed in Lemma~\ref{lemma:Pf=cycles} such that Eq.~(\ref{Pf=cycles1}) holds for all $S\subseteq \{1,2,\ldots,k\}$. Therefore, \begin{equation} \label{aux314} T(x)=C\epsilon(T)\, \perfectm{\tilde{C}_k}{\bar{x}0^{k-n}} \quad \mbox{for all} \quad x\in \{0,1\}^n \end{equation} where $\bar{x}$ is obtained from $x$ by flipping every bit. In order to transform Eq.~(\ref{aux314}) into Eq.~(\ref{T=Z}) one can incorporate the factor $C\epsilon(T)$ into the matching sum by introducing an extra edge with a weight $C\epsilon(T)$ and adding one extra edge with weight $1$ to every vertex $1,2,\ldots,n$ of the graph $\tilde{C}_k$ in order to flip bits of $x$. \end{proof}
Although it is not necessary, let us mention that the reverse of Theorem~\ref{thm:cycles} is also true, namely, a tensor $T$ defined by Eq.~(\ref{T=Z}) is always a matchgate. The easiest way to prove it is to represent the matching sum $\perfectm{G}{x}$ in Eq.~(\ref{T=Z}) as a contraction of an open matchgate tensor network, see Section~\ref{subs:1shot}, in which every tensor has a linear generating function (thus simulating the perfect matching condition). Then one can use Corollary~\ref{cor:matchgate} to prove that $T$ is a matchgate.
\section{Contraction of matchgate tensor networks}
\subsection{Edge contractions} Consider a tensor network ${\cal T }$ defined on a graph $G=(V,E)$ embedded to a surface $\Sigma$. Suppose one can find a region $D\subset \Sigma$ with topology of a disk such that $D$ contains exactly two vertices $u,v\in V$ and several edges connecting $u$ and $v$ as shown on Fig.~4. We shall define a new tensor network ${\cal T }'$ such that: (i) ${\cal T }'$ coincides with ${\cal T }$ outside $D$; (ii) ${\cal T }'$ contains only one vertex inside $D$; (iii) contraction values of ${\cal T }$ and ${\cal T }'$ are the same. The operation of replacing ${\cal T }$ by ${\cal T }'$ will be referred to as an {\it edge contraction}. The new vertex obtained by contracting all edges connecting $u$ and $v$ inside $D$ will be denoted $u\star v$.
\begin{figure}
\caption{The ordering of edges before and after contraction of $u$ and $v$.}
\label{fig:edge_contraction}
\end{figure}
Suppose there are $b$ edges connecting $u$ and $v$ that lie inside the disk. Applying, if necessary, a cyclic shift of components to the tensors $T_u$ and/or $T_v$ we can assume that these edges correspond to the last $b$ components of the tensor $T_u$ and the first $b$ components of $T_v$, see Fig.~4. Note that if the tensors under consideration are matchgates, the tensors obtained after the cyclic shift are also matchgates, see Corollary~\ref{cor:inv}. In the new network ${\cal T }'$ a pair of vertices $u,v$ is replaced by a single vertex $u\star v$ with degree $d(u\star v)=d(u)+d(v)-2b$. We define a new tensor $T_{u\star v}$ as \begin{equation} \label{edge_contraction} T_{u\star v}(x,y)=\sum_{z_1,\ldots,z_b=0,1} T_u(x,z_b,z_{b-1},\ldots,z_1)\, T_v(z_1,\ldots,z_{b-1},z_b,y), \end{equation} where $x$ and $y$ can be arbitrary bit strings of length $d(u)-b$ and $d(v)-b$ respectively. By definition of the contraction value, $c({\cal T })=c({\cal T }')$.
We shall also define a {\it self-loop contraction} as a special case of edge contraction. Namely, suppose one can find a region $D\subset \Sigma$ with topology of a disk such that $D$ contains exactly one vertex $u\in V$ and several self-loops as shown on Fig.~5. We shall define a new tensor network ${\cal T }'$ such that: (i) ${\cal T }'$ coincides with ${\cal T }$ outside $D$; (ii) ${\cal T }'$ contains one vertex without self-loops inside $D$; (iii) contraction values of ${\cal T }$ and ${\cal T }'$ are the same. The operation of replacing ${\cal T }$ by ${\cal T }'$ will be referred to as a {\it self-loop contraction}. To define this operation, choose the most inner self-loop $\gamma \in E(u)$ introduce a dummy vertex $v$ near the median of $\gamma$ and assign a tensor $T_v(x_1,x_2)=\delta_{x_1,x_2}$ to this vertex. Clearly it does not change a contraction value of a network. Secondly, apply the edge contraction described above to the two edges connecting $u$ and $v$. This reduces the number of self-loops by one. Repeat these two steps until all self-loops inside $D$ are contracted.
\begin{figure}
\caption{Contraction of self-loops can be reduced to edge contraction by adding dummy vertices.}
\label{fig:loop_contraction}
\end{figure}
It should be mentioned that self-loops $\gamma\in E(u)$ can be identified with elements of the fundamental group $[\gamma]\in \pi_1(\Sigma,u)$ of the surface $\Sigma$ with a base point $u$. We do not allow to contract self-loops representing non-trivial homotopy classes (because it cannot be done efficiently for matchgate tensor networks).
\subsection{Edge contraction as a convolution of generating functions} \label{subs:econtraction} Let ${\cal T }=\{T_u\}_{u\in V}$ be a tensor network considered in the previous section. In order to describe each tensor $T_u$ by a generating function $T_u(\theta)$ we shall introduce Grassmann variables $\theta_{u,1},\ldots,\theta_{u,d(u)}$ associated with the edges $e^u_1,\ldots,e^u_{d(u)}\in E(u)$ incident to $u$ such that \begin{equation} \label{T_u(theta)} T_u(\theta)=\sum_{x\in \{0,1\}^n} T(x)\, (\theta_{u,1})^{x_1} (\theta_{u,2})^{x_2} \cdots (\theta_{u,n})^{x_n}, \quad n\equiv d(u). \end{equation} Similarly one can describe the contracted tensor $T_{u\star v}$ in Eq.~(\ref{edge_contraction}) by a generating function \begin{equation} \label{Tu*v_def} T_{u\star v}(\theta)=\sum_{x\in \{0,1\}^p} \sum_{y\in \{0,1\}^q} T_{u\star v}(x,y) \, (\theta_{u,1})^{x_1} \cdots (\theta_{u,p})^{x_p} (\theta_{v,b+1})^{y_1} \cdots (\theta_{v,b+q})^{y_q}, \end{equation} where $p\equiv d(u)-b$ and $q\equiv d(v)-b$. The goal of this section is to represent the function $T_{u\star v}(\theta)$ as an integral of $T_{u}(\theta) T_{v}(\theta)$ in which all variables associated with the edges to be contracted are integrated out.
Let $E(u,v)$ be a set of edges connecting $u$ and $v$. For any edge $e\in E(u,v)$ such that $e$ is labeled as $e_j^u\in E(u)$ and as $e_k^v\in E(v)$ denote \[ \theta(e)=\theta_{u,j}\, \theta_{v,k}, \quad \int\, D\theta(e)=\int\, d\theta_{v,k}\,\int \, d\theta_{u,j}, \quad \mbox{and} \quad \int_{e\in E(u,v)} D\theta(e) =\prod_{e\in E(u,v)} \int\, D\theta(e). \] Note that these definitions make sense only $(u,v)$ is regarded as an ordered pair of vertices. Also note that the integrals $\int\, D\theta(e)$ over different edges commute, see Eq.~(\ref{int_com}), and thus one can take the integrals in an arbitrary order. \begin{lemma} \label{lemma:convolution} Suppose the edges connecting $u$ and $v$ are ordered as shown on Fig.~4, i.e., these are the last $b$ edges incident to $u$ and the first $b$ edges incident to $v$. Then \begin{equation} \label{TuTv} T_{u\star v}=\int_{e\in E(u,v)} D\theta(e)\; T_u\, T_v\, \exp{\left( \sum_{e\in E(u,v)} \theta(e)\right)}. \end{equation} \end{lemma} \begin{proof} By linearity it is enough to prove Eq.~(\ref{TuTv}) for the case when $T_u$ and $T_v$ are monomials in the Grassmann variables, i.e., \[ T_u= (\theta_{u,1})^{x_1} \cdots (\theta_{u,p})^{x_p} (\theta_{u,p+1})^{z_1'} \cdots (\theta_{u,p+b})^{z_b'}, \quad T_v= (\theta_{v,1})^{z_1} \cdots (\theta_{v,b})^{z_b} (\theta_{v,b+1})^{y_1} \cdots (\theta_{v,q+b})^{y_q}, \] where $p\equiv d(u)-b$ and $q\equiv d(v)-b$. By expanding the exponent one gets a sum of all possible monomials in which the two variables associated with any edge $e\in E(u,v)$ are either both present or both absent. Therefore the integral in Eq.~(\ref{TuTv}) is zero unless $z_j=z_{b+1-j}'$ for all $j=1,\ldots,b$. Suppose this is the case. Then one gets after some rearrangement of variables \[ T_u\, T_v= (\theta_{u,1})^{x_1} \cdots (\theta_{u,p})^{x_p} \left( \prod_{e\in S(z)} \theta(e) \right) (\theta_{v,b+1})^{y_1} \cdots (\theta_{v,d(v)})^{y_q}, \] where $S(z)\subseteq E(u,v)$ denotes a set of edges $e$ such that $e$ is labeled as $e_k^v\in E(v)$ and $z_k=1$. Substituting it into the integral Eq.~(\ref{TuTv}), taking into account that $\theta(e)$ is a central element and that $\int D\theta(e) \, \theta(e)=1$ one gets \[ T_{u\star v}=(\theta_{u,1})^{x_1} \cdots (\theta_{u,p})^{x_p} \, (\theta_{v,b+1})^{y_1} \cdots (\theta_{v,b+q})^{y_q} \] which coincides with the desired expression Eq.~(\ref{Tu*v_def}).
\end{proof}
\begin{cor} \label{cor:matchgate} Suppose $T_u$ and $T_v$ are matchgates. Then the contracted tensor $T_{u\star v}$ is also a matchgate. \end{cor} \begin{proof} Since cyclic shifts of indexes map matchgates to matchgates, see Corollary~\ref{cor:inv} in Section~\ref{subs:generating}, we can assume that the edges of $T_u$ and $T_v$ are already ordered as required in
Lemma~\ref{lemma:convolution}. Represent $T_u$, $T_v$ by their canonical
generating functions, see Theorem~\ref{thm:canonical}. Using Eq.~(\ref{TuTv}) one concludes that $T_{u\star v}(\theta)$ is a Gaussian integral $I(A,B)$ for some matrices $A$ and $B$, see Eq.~(\ref{Gaussian_Integrals}). Therefore, $T_{u\star v}$ is a matchgate, see Corollary~\ref{cor:GI=matchgate} in Section~\ref{subs:matchgate=gaussian}. \end{proof}
{\it Remark:} Given the canonical generating functions for $T_u$ and $T_v$, the canonical generating function for the contracted tensor $T_{u\star v}$ can be obtained straightforwardly using Eq.~(\ref{TuTv}) and computing the resulting Gaussian integral $I(A,B)$ using Eq.~(\ref{GI3}). The details can be found in Appendix~A.
\subsection{Contraction of a planar subgraph in one shot} \label{subs:1shot} Suppose a planar connected graph $G=(V,E)$ is a part of a larger non-planar tensor network such that $G$ is connected to the rest of the network by a subset of {\it external edges} $E_{ext}\subseteq E$. The remaining {\it internal edges} $E_{int}=E\backslash E_{ext}$ are the edges that can be contracted "locally" without touching the rest of the network. By abuse of definitions, we shall assume that the external edges have only one endpoint (the other endpoint belongs to the rest of the network) which belongs to the outer face of $G$, see~Fig.~6. For convenience let us also assume that the graph $G$ is embedded into a disk such that the external edges stick out from the disk as shown on Fig.~6. A network that consists of such a graph $G=(V,E)$ and a collection of tensors $\{T_u\}_{u\in V}$ will be referred to as an {\it open tensor network}. Throughout this section we shall consider only open tensor networks in which every tensor is a matchgate. Contraction of an open tensor network amounts to finding a tensor
$T_V$ of rank $|E_{ext}|$ obtained by contracting all internal edges of $G$. It follows from Corollary~\ref{cor:matchgate}, Section~\ref{subs:econtraction} that $T_V$ is a matchgate. The goal of the present section is to represent the generating function for the contracted tensor $T_V$ as a convolution integral similar to Eq.~(\ref{TuTv}) where the integration is taken over all internal edges.
\begin{figure}
\caption{An open tensor network with $7$ external edges equipped with a Kasteleyn orientation. }
\label{fig:planar_open}
\end{figure}
An alternative strategy for computing $T_V$ is to apply the edge contraction described in the previous section sequentially until all internal edges of $G$ are contracted. Although it yields a polynomial-time
algorithm this strategy is not very robust.
An obvious drawback is that every edge contraction involves computing the Gaussian integral Eq.~(\ref{GI3}) which requires a matrix inversion. Contracting sequentially $O(n)$ edges would require $O(n)$ nested matrix inversions which may be difficult or impossible to do if the matrix elements are specified with a finite precision. In order to reduce the number of nested matrix inversions one could organize the edge contractions into a sequence of rounds such that each round involves contractions of pairwise disjoint edges. The contractions involved in every round can be performed in parallel. The number of the rounds can be made $O(\log{n})$ using the techniques developed by F\"urer and Raghavachari~\cite{Furer}. We shall not pursue this strategy though because the approach described below allows one to compute $T_V$ using only one matrix inversion.
The main result of this section is the following theorem. \begin{theorem} \label{thm:partial_contraction} Consider an open matchgate tensor network on a planar graph $G=(V,E)$ with $n$ vertices and $m$ external edges. Assume that the tensors $T_1,\ldots,T_n$ are specified by their canonical generating function, \[ T_j(\theta)=C_j\exp{\left( \frac12\, \theta^T\, A_j \, \theta \right)}\int D\mu\, \exp{\left( \mu^T \, B_j\, \theta \right)}. \] Then the tensor $T_V$ obtained by contracting all internal edges of $G$ can be represented as a Gaussian integral \begin{equation} \label{integral_main} T_V(\eta)= \prod_{j=1}^n C_j \epsilon(T_j)\, \int D\theta \, \exp{\left( \frac12 \, \theta^T\, A \, \theta + \theta^T\, B\, \eta\right)}. \end{equation} Here $A$, $B$ are matrices of size $k\times k$ and $k\times m$ for some $k=O((n+m)^2)$.
Matrix elements of $A$ and $B$ are linear functionals of $A_1,\ldots,A_n$ and $B_1,\ldots,B_n$. One can compute $A$ and $B$ in time $O(k)$. Furthermore, if $G$ has bounded vertex degree then the same statement holds for $k=O(m+n)$. \end{theorem}
Before going into technical details let us explain what is the main difficulty in representing the contracted tensor $T_V$ by a single Gaussian integral. The point is that the convolution formula Eq.~(\ref{TuTv}) holds only if the edges incident to the vertices $u,v$ are ordered in a consistent way as shown on Fig.~4. If the orderings are not consistent, an extra sign may appear while commuting the variables living on the contracted edges towards each other. Assume one wants to contract the combined vertex $u\star v$ with some third vertex $w$. If the ordering of edges at the combined vertex $u\star v$ is not consistent with the ordering at $w$, one has to perform a cyclic shift of indexes in the tensor $T_{u\star v}$ and/or $T_w$ before one can directly apply the formula Eq.~(\ref{TuTv}) to $T_{u\star v}$ and $T_{w}$. Therefore, in general one can not represent the tensor $T_{u\star v\star w}$ obtained by contracting $u,v,w$ as a single Gaussian integral.
In order to avoid the problem with inconsistent edge orderings we shall contract an open matchgate tensor network in two stages. At the first stage one simulates each tensor $T_u$ by a matching sum of some planar graph as explained in Section~\ref{subs:matchgate=gaussian}. It yields an open tensor network in which every tensor has a linear generating function (since every vertex must have exactly one incident edge). At the second stage one represents the contraction of such a network by a single convolution integral analogous to Eq.~(\ref{TuTv}). The problem with inconsistent edge ordering will be addressed by choosing a proper orientation on every edge (which affects the definition of monomials $\theta(e)$ in Eq.~(\ref{TuTv})). One can regard this approach as a generalization of the original Kasteleyn's method~\cite{Kasteleyn61} to the case of a matching sum with "boundary conditions".
\begin{dfn} A tensor $T$ is called linear if it has a linear generating function, $T=\sum_{a=1}^n w_a\, \theta_a$. \end{dfn} Clearly, any linear tensor $T$ can be mapped to $T(\theta)=\theta_1$ by a linear change of variables. Lemma~\ref{lemma:matchgate2} implies that $T(\theta)=\theta_1$ is a matchgate. Therefore any linear tensor is a matchgate, see Lemma~\ref{lemma:inv}. \begin{dfn}
Orientation of a graph $G=(V,E)$ is an antisymmetric matrix $A$ of size $|V|\times |V|$ such that \[ A_{u,v}=\left\{ \begin{array}{rcl} \pm 1 &\mbox{if} & (u,v)\in E,\\ 0 && \mbox{otherwise}.\\ \end{array} \right. \] An edge $(u,v)\in E$ is oriented from $u$ to $v$ iff $A_{u,v}=1$. \end{dfn} Recall that we represent each tensor $T_u$ by a generating function $T_u(\theta)$ that depends on Grassmann variables $(\theta_{u,1},\ldots,\theta_{u,d(u)})$ associated with the edges incident to $u$, see Eq.~(\ref{T_u(theta)}). Given an orientation $A$ of the graph $G$ and an edge $e=(u,v)\in E$ with the labels $e_j^u\in E(u)$ and $e_k^v\in E(v)$, define \begin{equation} \label{thetatheta} \theta(e)=A_{u,v}\, \theta_{u,j}\, \theta_{v,k}, \quad \int D\theta(e) = A_{u,v}\, \int d\theta_{v,k} \, \int \, d\theta_{u,j}, \quad \mbox{and} \quad \int_{e\in E_{int}} D\theta(e)=\prod_{e\in E_{int}} \int D\theta(e). \end{equation} Note that $\theta(e)$ and $\int D\theta(e)$ are symmetric under the transposition of $u$ and $v$. \begin{lemma} \label{lemma:open} Let $T_V$ be a tensor obtained by contraction of an open tensor network on a graph $G=(V,E)$. Assume that all tensors in the network are linear. Then there exists an orientation $A$ and an ordering of the vertices $V=\{v_1,v_2,\ldots,v_n\}$ such that \begin{equation} \label{TuTv_open} T_V=\int_{e\in E_{int}} D\theta(e)\; T_{v_1} T_{v_2} \cdots T_{v_n}\, \exp{\left( \sum_{e\in E_{int}} \theta(e)\right)}. \end{equation} The orientation and the ordering can be found in time $O(n)$. \end{lemma} \noindent {\it Remark 1:} The generating function of $T_V$ is defined for the ordering of the external edges in which they appear as one circumnavigates the boundary of the disk anticlockwise. The order of variables in $T_V$ corresponds to the counterclockwise order of the external edges.
\begin{proof} Without loss of generality $G$ is a $2$-connected graph\footnote{If $G$ has a cut-vertex $u$ one can always add an extra edge to some pair of nearest neighbors of $u$ in order to make $G$ $2$-connected. The new edge must be assigned a zero weight in the two tensors it belongs to. Since the new edge does not contribute to $T_V$ it can be safely removed at the end of the analysis.}. Then the boundary of the outer face of $G$ is a closed loop without self-intersections. Let us denote it $\Gamma_{out}$. Mark some vertex in $\Gamma_{out}$ that has at least one incident external edge (if there are no external edges, mark an arbitrary vertex). Let $\Gamma_{out}=\{1,2,\ldots,m\}$ be an ordered list of all vertices on the outer face of $G$ corresponding to circumnavigating $\Gamma_{out}$ anticlockwise starting from the marked vertex. Extend the ordering of vertices to the rest of $V$ in an arbitrary way, so that $V=\{1,2,\ldots,n\}$ and the first $m$ vertices belong to $\Gamma_{out}$. \begin{dfn} Let $G$ be a planar graph with the vertices ordered as described above. A Kasteleyn orientation (KO) of $G$ is an orientation $A$ such that \\ (1) The number of c.c.w. oriented edges in the boundary of any face of $G$ is odd (except for the outer face).\\ (2) $A_{1,2}=A_{2,3}=\cdots=A_{m-1,m}=1$. \end{dfn} \noindent {\it Remark:} The standard definition of a KO requires that (1) holds for all faces of $G$ including the outer face and does not require (2), see for example~\cite{Reshetikhin06}. By abuse of definitions we shall apply the term KO to orientations satisfying (1),(2). The standard definition is not suitable for our purposes because $G$ may have odd number of vertices while the standard KO exists only on graphs with even number of vertices. The condition~(2) is needed to ensure consistency between different "boundary conditions". Example of a KO is shown on Fig.~6. \begin{prop} \label{prop:Kasteleyn} Any planar graph has a KO. It can be found in a linear time. \end{prop} We postpone the proof of the proposition until the end of the section. Let us choose the orientation $A$ in Eq.~(\ref{thetatheta}) as a KO of the graph obtained from $G$ by removing all external edges. Let us verify that the contracted tensor $T_V$ can be written as in Eq.~(\ref{TuTv_open}).
Indeed, let $S\subseteq E_{ext}$ be a subset of external edges such that any vertex in $\{1,\ldots,m\}$ has at most one incident edge from $S$. (Below we shall consider only such sets $S$ without explicitly mentioning it.) Let $\partial S$ be a set of vertices that have an incident edge from $S$ (clearly all such vertices belong to the outer face). For any $S$ as above and any $\partial S$-imperfect matching $M\in {\cal M }(G,\partial S)$ define a subset of Grassmann variables \[ \Omega(S,M)=\{ (u,j)\, : \, u\in V, \quad \mbox{and} \quad e^u_j \in S\cup M\}. \] In other words, $(u,j)\in \Omega(S,M)$ iff $\theta_{u,j}$ is a Grassmann variable that live on some edge of $S\cup M$. Note that there are two Grassmann variables living on any internal edge and one variable living on any external edge. Thus for any $S$ and $M$ the set $\Omega(S,M)$ contains $n$ variables. Define a normally ordered monomial \begin{equation} \label{order1} \prod_{(u,j)\in \Omega(S,M)} \theta_{u,j} \end{equation} as a product of all variables in $\Omega(S,M)$ ordered according to \begin{equation} \label{corder} (\theta_{1,1},\ldots,\theta_{1,d(1)},\theta_{2,1},\ldots,\theta_{2,d(2)},\ldots,\theta_{n,1},\ldots,\theta_{n,d(n)}). \end{equation} Define also $M$-ordered monomial \begin{equation} \label{order2} \prod_{(u,j)\, : \, e^u_j\in S} \theta_{u,j}\, \prod_{e\in M} \theta(e), \end{equation} where the order in the first product must agree with the chosen ordering of edges in $E_{ext}$, see Fig.~6. Clearly the two products Eqs.~(\ref{order1},\ref{order2}) coincide up to a sign that we shall denote $\mathrm{sgn}(M)$. In order to prove Lemma~\ref{lemma:open} it suffices to show that \begin{equation} \label{sgn(M)=1} \mathrm{sgn}(M)=1 \quad \mbox{for all $\partial S$-imperfect matchings $M$, for all $S\subseteq E_{ext}$}. \end{equation} Indeed, denoting $T_u=\sum_{j=1}^{d(u)} w^u_j\, \theta_{u,j}$ one can rewrite Eq.~(\ref{TuTv_open}) as \begin{eqnarray} T_V&=&\sum_{S\subseteq E_{ext}} \sum_{M\in {\cal M }(G,\partial S)}\, \int_{e\in E_{int}} \, D\theta(e)\, \prod_{(u,j)\in \Omega(S,M)}w^u_j \theta_{u,j} \prod_{e\notin M} \theta(e)\nonumber \\ &=& \sum_{S\subseteq E_{ext}} \prod_{(u,j)\, : \, e^u_j\in S} \theta_{u,j}\, \sum_{M\in {\cal M }(G,\partial S)}\, \mathrm{sgn}(M)\, \prod_{(u,j)\, : \, e^u_j\in M} w^u_j. \end{eqnarray} Assuming $\mathrm{sgn}(M)\equiv 1$ one can identify the sum over $M\in {\cal M }(G,\partial S)$ with the component of the contracted tensor $T_V$ in which the subset $S$ of external edges carries index $1$.
Note that for any $S\subseteq E_{ext}$ and any $\partial S$-imperfect matching $M$ each vertex $u\in V$ contributes exactly one variable to $\Omega(S,M)$. Indeed, at every vertex $u\in V$ there is either one incident edge from $M$ or one incident external edge. All other edges incident to $u$ and the variables living on these edges can be ignored as far as computation of $\mathrm{sgn}(M)$ is concerned. Therefore one can compute the sign $\mathrm{sgn}(M)$ by introducing auxiliary Grassmann variables $\eta=(\eta_1,\ldots,\eta_n)$ associated with vertices of $G$ and comparing the normal ordering of $\eta$ ( the one in which the indexes increase from the left to the right) with the $M$-ordering of $\eta$, namely \[ \prod_{u\in \partial S} \eta_u\, \prod_{e\in M} \eta(e) = \mathrm{sgn}(M)\, \eta_1\eta_2 \cdots \eta_n, \quad \mbox{where} \quad \eta(e)=A_{u,v}\, \eta_u \eta_v \quad \mbox{if} \quad e=(u,v). \] Here the ordering in the first product is normal while the ordering in the second product may be arbitrary since $\eta(e)$ is a central element. Consider any subsets $S,S'\subseteq E_{ext}$. Given any $\partial S$-imperfect matching $M$ and $\partial S'$-imperfect matching $M'$ define a relative sign \begin{equation} \label{rel_sign} \mathrm{sgn}(M,M')\eqdef \mathrm{sgn}(M)\, \mathrm{sgn}(M'), \end{equation} such that \begin{equation} \label{rel_sign1} \prod_{u\in \partial S} \eta_u\, \prod_{e\in M} \eta(e) =\mathrm{sgn}(M,M')\, \prod_{u\in \partial S'} \eta_u\, \prod_{e\in M'} \eta(e). \end{equation} In order to compute $\mathrm{sgn}(M,M')$ consider the symmetric difference $M\oplus M'$. It consists of a disjoint union of even-length cycles $C_1,\ldots,C_p$ and open paths $\Gamma_1,\ldots,\Gamma_q$ such that every path $\Gamma_j$ has both its endpoints in the symmetric difference $\partial S\oplus \partial S'$. Given a path $\Gamma_j$ with endpoints $s,t\in \partial S\oplus \partial S'$, $s<t$ let us orient $\Gamma_j$ from $s$ to $t$. Now one can compute the relative sign as follows. \begin{prop} \label{prop:relative_sign} Consider any subsets $S,S'\subseteq E_{ext}$. Let $C_1,\ldots,C_p$ and $\Gamma_1,\ldots,\Gamma_q$ be the cycles and the paths formed by $M\oplus M'$ for some $\partial S$-imperfect matching $M$ and some $\partial S'$-imperfect matching $M'$. For a path $\Gamma_j$ connecting vertices $s,t\in \partial S\oplus \partial S'$ on the outer face such that $s<t$ let $\omega(\Gamma_j)=1$ if the interval $(s,t)$ contains odd number of vertices from $\partial S$ and $\omega(\Gamma_j)=0$ if this number is even. Then \begin{equation} \label{sigma(M,M')} \mathrm{sgn}(M,M')=(-1)^p\, \prod_{j=1}^p \Phi(C_j) \; \prod_{k=1}^q (-1)^{\omega(\Gamma_k)}\, \Phi(\Gamma_k), \end{equation} where \[ \Phi(C_j)=\prod_{(u,v)\in C_j} A_{u,v} \quad \mbox{and} \quad \Phi(\Gamma_k)=\prod_{(u,v)\in \Gamma_k} A_{u,v}. \]
\end{prop}
\noindent
{\it Remark 1:} The definition of $\omega(\Gamma_j)$ is symmetric under exchange of $S$ and $S'$. Indeed, the overall number of vertices from $\partial S\oplus \partial S'$ contained in the interval $(s,t)$
is even since these vertices are pairwise connected by $\Gamma$'s. The remaining vertices
of $(s,t)$ either belong to both sets $S,S'$ or belong to neither of them. \\ {\it Remark 2:} The product $\prod_{(u,v)\in \Gamma_k} A_{u,v}$ gives the parity of the number of edges in $\Gamma_k$ whose orientation determined by $A$ disagrees with the chosen orientation of $\Gamma_k$. The product $\Phi(C_j)$ does not depend on how one chooses orientation of $C_j$ since every cycle $C_j$ has even length. \begin{proof} Indeed, one can easily check that for every cycle $C_j$ one has \begin{equation} \label{flux3prop} \prod_{e\in C_j\cap M} \eta(e)=-\Phi(C_j)\, \prod_{e\in C_j \cap M'} \eta(e). \end{equation} Therefore changing the $M$-ordering to the $M'$-ordering in a cycle $C_j$ contributes a factor $-\Phi(C_j)$ to the relative sign $\mathrm{sgn}(M,M')$. Consider now a path $\Gamma_j$ connecting vertices $s,t\in \partial S\oplus \partial S'$ where $s<t$. Let us argue that changing the $M$-ordering to the $M'$-ordering on the path $\Gamma_j$ contributes a factor $(-1)^{\omega(\Gamma_j)}\, \Phi(\Gamma_j)$ to the relative sign $\mathrm{sgn}(M,M')$. Indeed, one can easily check the following identities: \begin{eqnarray} s,t\in S &:& \eta_s\eta_t \prod_{e\in \Gamma_j\cap M} \eta(e) = \Phi(\Gamma_j)\, \prod_{e\in \Gamma_j\cap M'} \eta(e),\nonumber \\ s,t\in S' &:& \mbox{the same as above up to $M\leftrightarrow M'$}, \nonumber \\ s\in S, t\in S' &:& \eta_s \prod_{e\in \Gamma_j\cap M} \eta(e) =\Phi(\Gamma_j)\, \eta_t \, \prod_{e\in \Gamma_j\cap M'} \eta(e),\nonumber \\ s\in S', t\in S &:&\mbox{the same as above up to $M\leftrightarrow M'$}. \nonumber \end{eqnarray} Consider as example the case $s,t\in S$. Bringing the variables $\eta_s$ and $\eta_t$ together in the monomial $\prod_{u\in \partial S} \eta_u$ introduces an extra sign $(-1)^{\omega(\Gamma_j)}$. Taking into account that $\eta(e)$ are central elements and using the first identity above one concludes that \[ \prod_{u\in \partial S} \eta_u \, \prod_{e\in \Gamma_j\cap M}\eta(e) = (-1)^{\omega(\Gamma_j)}\, \Phi(\Gamma_j)\, \prod_{u\in \partial S\backslash \{s,t\}} \eta_u \prod_{e\in \Gamma_j\cap M'} \eta(e). \] Other three cases can be considered analogously using Remark~1 above. Combing it with Eq.~(\ref{flux3prop}) one arrives to Eq.~(\ref{sigma(M,M')}). \end{proof} Let us proceed with the proof of Lemma~\ref{lemma:open}. The first condition in the definition of KO implies\footnote{This is the well-known property of a Kasteleyn orientation which we prove below for the sake of completeness.}
that $\Phi(C_j)=-1$ for all cycles $C_j$. Indeed, consider any particular cycle $C_j$ and let $N_0,N_1,N_2$ be the number of vertices, edges, and faces in the subgraph bounded by $C_j$. The Euler formula implies that $N_0+N_2-N_1=1$. Denote also $N_1^{int}$ the number of {\it internal } edges, i.e., edges having at least one endpoint in the interior of $C_j$. Since $C_j$ has even length, $N_1^{int}$ has the same parity as $N_1$. Furthermore, since all vertices of the subgraph bounded by $C_j$ are paired by $M$ (and by $M'$), $N_0$ is even. Since $\Phi(C_j)$ can be regarded as a parity of c.c.w. oriented edges in $C_j$ and each internal edge is c.c.w. oriented with respect to one of the adjacent faces the property~(1) of KO yields \begin{equation} \label{flux1} \Phi(C_j)=(-1)^{N_2+N_1^{int}}=(-1)^{N_2+N_1}=(-1)^{1+N_0}=-1. \end{equation} Therefore Proposition~\ref{prop:relative_sign} implies \begin{equation} \label{flux1'} \mathrm{sgn}(M,M') =\prod_{k=1}^q (-1)^{\omega(\Gamma_k)}\, \Phi(\Gamma_k). \end{equation} Let us now show that \begin{equation} \label{flux2} (-1)^{\omega(\Gamma_k)}\, \Phi(\Gamma_k)=1 \end{equation} for all paths $\Gamma_k$. Indeed, let $s,t\in S\oplus S'$ be the starting and the ending vertices of $\Gamma_k$. Consider a path $\Gamma_k^*$ obtained by passing from $t$ to $s$ along the boundary of the outer face $\Gamma_{out}$ in the clockwise direction. Let $N_0,N_1,N_2$ be the number of vertices, edges, and faces in the subgraph bounded by a cycle $\Gamma_k \cup \Gamma_k^*$. Denote also $N_1^{int}$ the number of edges that have at least one endpoint in the interior of $\Gamma_k \cup \Gamma_k^*$. The Euler formula implies that $N_0+N_2-N_1=1$. Note that $\Phi(\Gamma_k)$ can be regarded as the parity of the number of edges in $\Gamma_k$ whose orientation determined by $A$ corresponds to c.c.w. orientation of the cycle $\Gamma_k \cup \Gamma_k^*$. Repeating the arguments leading to Eq.~(\ref{flux1}) and noting that all edges of the cycle $\Gamma_k \cup \Gamma_k^*$ belonging to $\Gamma_k^*$ are oriented c.c.w. one gets \begin{equation} \label{flux3}
\Phi(\Gamma_k)=(-1)^{|\Gamma_k^*|+N_2+N_1^{int}}=(-1)^{|\Gamma_k|+N_2+N_1}=(-1)^{|\Gamma_k|+N_0+1}.
\end{equation} Here $|\Gamma_k|$ and $|\Gamma_k^*|$ are the numbers of edges in the two paths. Consider three possibility:\\ {\it Case~1:} $s,t\in \partial S$. Then
$|\Gamma_k|$ is odd and thus $\Phi(\Gamma_k)=(-1)^{N_0}$. All $N_0$ vertices of the graph bounded by $\Gamma_k \cup \Gamma_k^*$ are paired by the matching $M$ except for $s,t$ and those belonging to $\partial S$ and lying on the interval $(s,t)$. Therefore the parity of $N_0$ coincides with $\omega(\Gamma_k)$ and we arrive to Eq.~(\ref{flux2}).\\ {\it Case~2:} $s,t\in \partial S'$. The same as Case~1 (see Remark~1 after Proposition~\ref{prop:relative_sign}).\\
{\it Case~3:} $s\in \partial S$, $t\in \partial S'$ (or vice verse).Then $|\Gamma_k|$ is even and thus $\Phi(\Gamma_k)=(-1)^{N_0+1}$. All $N_0$ vertices of the graph bounded by $\Gamma_k \cup \Gamma_k^*$ are paired by the matching $M$ except for $s$ (or except for $t$) and those belonging to $\partial S$ and lying on the interval $(s,t)$. Therefore the parity of $N_0$ coincides with $\omega(\Gamma_k)+1$ and we arrive to Eq.~(\ref{flux2}).
Combining Eqs.~(\ref{flux1},\ref{flux2}) and Proposition~\ref{prop:relative_sign} we conclude that $\mathrm{sgn}(M,M')=1$ for all $M$ and $M'$. Thus either $\mathrm{sgn}(M)=1$ for all $M$ or $\mathrm{sgn}(M)=-1$ for all $M$. One can always exclude the latter possibility by applying a {\it gauge transformation} to the orientation $A$. A gauge transformation at a vertex $u\in V$ reverses orientation of all edges incident to $u$. Let us say that a vertex $u\in V$ is {\it internal} if does not belong to the outer face of $G$.
Clearly a gauge transformation at any internal vertex $u$ maps a KO to a KO and flips the sign $\mathrm{sgn}(M)$ for all $M$. Thus it suffices to consider the case when $G$ does not have internal vertices (i.e. $G$ is an outerplanar graph). If $m=n$ is even, a matching $M=\{(1,2),(3,4),\ldots,(m-1,m)\}$ has sign $\mathrm{sgn}(M)=1$ due to property (1) of a KO and thus all matchings have sign $+1$. If $m=n$ is odd one can apply the same argument using a matching $M=\{(2,3),(4,5),\ldots,(m-1,m)\}$ (recall that the vertex $1$ has at least one external edge and thus it can be omitted in $M$). \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:partial_contraction}]
Let $n_e$ be the number of internal edges in the graph $G$, so that $|E|=m+n_e$. Since $G$ is a planar graph, $n_e=O(n)$, see for example~\cite{Diestel}, and thus $|E|=O(n+m)$. Denote degree of a vertex $u\in V$ by $d(u)$ (it includes both internal and external edges). Applying Theorem~\ref{thm:cycles} one can simulate the tensor $T_u$ at any vertex $u\in V$ by a matching sum of some planar graph $G_u$ with $O(d(u)^2)$ vertices. Combining the graphs $G_u$ together one gets an open tensor network $G'=(V',E')$ in which all tensors are linear. The network $G'$ has $m$ external edges. The number of vertices $n'$ in the network $G'$ can be bounded as
$n'=\sum_{u\in V} O(d(u)^2) = O((\sum_{u\in V} d(u))^2) =O(|E|^2)=O((m+n)^2)$. If $G$ has bounded degree one gets $n'=\sum_{u\in V} O(d(u)^2)=O(n)$. Thus in both cases $n'=O(k)$, where $k$ is defined in the statement of the theorem. It follows from Theorem~\ref{thm:cycles} that the edge weights in the matching sums are linear functions of the matrix elements of $A_1,\ldots,A_n$ and $B_1,\ldots,B_n$. Let $n_e'$ be the number of internal edges in $G'$. Since $G'$ is a planar graph, $n_e'=O(n')=O(k)$. Thus the total number of edges in $G'$
is $|E'|=n_e'+m=O(k)$. Invoking Lemma~\ref{lemma:open} we need to introduce a pair of Grassmann variables for every internal edge of $G'$ and one variable for every external edge. Thus the total number of Grassmann variables is $O(k)$. It determines the number of variables in the vector $\theta$ in Eq.~(\ref{integral_main}). Representing linear tensors $T_j$ as Gaussian integrals, namely \[ T_j = \int d\mu \exp{(\mu\, T_j)}, \] one can combine the multiple integrals in Eq.~(\ref{TuTv_open}) into a single Gaussian integral Eq.~(\ref{integral_main}) with the matrix $A$ having a dimension $O(k)\times O(k)$ and $B$ having a dimension $O(k)\times m$. Thus $A$ and $B$ have the desired properties. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:Kasteleyn}]
Let $G=(V,E)$ be a planar graph with $n$ vertices such that the outer face of $G$ is a simple loop. An orientation $A$ satisfying (1) can be constructed using the algorithm of~\cite{Reshetikhin06}. For the sake of completeness we outline it below. Let $G^*=(V^*,E)$ be the dual graph such that each face of $G$ contributes one vertex to $G^*$ (including the outer face). Let $T$ be a spanning tree of $G^*$ such that the root of $T$ is the outer face of $G$. One can find $T$ in time $O(|V|+|E|)=O(n)$ since for planar graphs $|E|=O(|V|)$. Assign an arbitrary orientation to those edges of $G$ that do not belong to $T$. By moving from the leaves of $T$ to the root assign the orientation to all edges of $T$. Note that for every vertex $u$ of $T$ which is not the root
the orientation of an edge $e$ connecting $u$ to its ancestor is uniquely determined by (1). We obtained an orientation of all edges of $G$ satisfying (1).
In order to satisfy (2) one can apply a series of {\it gauge transformations}. A gauge transformation at a vertex $u\in V$ reverses orientation of all edges incident to $u$. Clearly it preserves the property (1). Applying if necessary a gauge transformation at the vertices $\{1,2,\ldots,m-1\}$ one can satisfy~(2).
\end{proof}
\subsection{Contraction of matchgate networks with a single vertex} \label{subs:1vertex} In this section we explain how to contract a matchgate tensor network ${\cal T }$ that consists of a single vertex $u$ with $m$ self-loops embedded into a surface $\Sigma$ of genus $g$ without self-intersections. Example of such a network with $m=3$ and $g=1$ is shown on Fig.~7. Let $T$ be a tensor of rank $2m$ associated with $u$. Clearly the contraction value $c({\cal T })$ depends only on the pairing pattern indicating what indexes of $T$ are contracted with each other. It will be convenient to represent the pairing pattern by a {\it pairing graph} $P=(V,E)$ with a set of vertices $V=\{1,2,\ldots,2m\}$ such that a pair of vertices $(a,b)$ is connected by an edge iff the indexes $a,b$ of the tensor $T$ are contracted with each other (connected by a self-loop). By definition $P$ consists of $m$ disjoint edges. Let us embed $P$ into a disk such that all the vertices of $P$ lie on the boundary of the disk and their order corresponds to circumnavigating the boundary anticlockwise. The edges of $P$ are represented by arcs lying inside the disk, see Fig.~7. One can always draw the arcs such that there are only pairwise intersection points.
Introduce an auxiliary tensor $R$ of rank $2m$ such that \[ R(x)=\left\{ \begin{array}{rcl} 1 &\mbox{if} & x_a=x_b \quad \mbox{for all $(a,b)\in E$}, \\ 0 & \mbox{if} & x_a\ne x_b \; \mbox{for some $(a,b)\in E$}.\\ \end{array}\right. \]
\begin{figure}
\caption{Left: a tensor network with a single vertex embedded into a torus. Right: the pairing graph $P$. }
\label{fig:pairing}
\end{figure}
The contraction value of ${\cal T }$ can be represented as \begin{equation} \label{TR} c({\cal T })=\sum_{x\in \{0,1\}^{2m}} T(x)\, R(x). \end{equation} Let $\theta=(\theta_1,\ldots,\theta_{2m})$ and $\eta=(\eta_1,\ldots,\eta_{2m})$ be Grassmann variables and $T(\theta)$, $R(\eta)$ be the generating functions of $T$ and $R$. \begin{prop} Let $\epsilon(T)=0,1 $ for even and odd tensors $T$ respectively . Then \begin{equation} \label{TR1} c({\cal T })=i^{\epsilon(T)}\, \int D\theta \int D\eta \, T(\theta)\, R(\eta)\, \exp{(i\, \theta^T\eta)}. \end{equation} \end{prop} \begin{proof} A non-zero contribution to the integral comes from the terms in which $T(\theta)$ contributes monomial $T(x)\, \theta(x)$ and $R(\eta)$ contributes monomial $R(x)\, \eta(x)$ for some $x\in \{0,1\}^{2m}$. A simple algebra shows that for any $x\in \{0,1\}^{2m}$ one has the following identity \[ \theta(x)\, \eta(x)\, \prod_{a\, : \, x_a=0} i \theta_a \eta_a =
i^{-|x|}\, (-1)^{|x|\, (|x|-1)/2}\, \theta(1^{2m})\, \eta(1^{2m}), \]
where $|x|$ is the Hamming weight of $x$.
Taking into account that $T(x)=0$ unless $|x|$ has parity $\epsilon(T)$ one gets \[
i^{-|x|}\, (-1)^{|x|\, (|x|-1)/2}=i^{-\epsilon(T)}. \] Since $\int \, D\theta \int \, D\eta\, \theta(1^{2m})\, \eta(1^{2m})=1$, one gets Eq.~(\ref{TR1}). \end{proof} In general $R$ is not a matchgate tensor because the chosen planar embedding of the pairing graph may have edge crossing points. For example, assume that $P$ has $4$ vertices $\{1,2,3,4\}$ and two edges $(1,3)$, $(2,4)$ (which can be realized on a torus). Then the non-zero components of $R$ are $R(0000)=R(1010)=R(0101)=R(1111)=1$. Substituting them into the matchgate identities Eq.~(\ref{Grank4}) for even rank-$4$ tensors one concludes that $R$ is not a matchgate.
Let us order the edges of $P$ in an arbitrary way, say, $E=\{e_1,e_2,\ldots,e_m\}$. For any edges $e_p,e_q\in E$ let $N_{p,q}$ be the the number of self-intersections of $e_p,e_q$ in the planar embedding shown on Fig.~7. Since we assumed that all intersections are pairwise,
$N_{p,q}$ takes only values $0,1$, i.e., $N$ is a symmetric binary matrix. Let us also agree that $N_{p,p}=0$. We shall see later that the tensor $R$ can be represented as a linear combination of $2^r$ matchgate tensors, where $r$ is a binary rank of the matrix $N$. It is crucial that the rank of $N$ can be bounded by the genus $g$ of the surface $\Sigma$. \begin{lemma} \label{lemma:rank} The matrix $N$ has binary rank at most $2g$. \end{lemma} \begin{proof} Let us cut a small disk $D$ centered at the vertex $u$ from the surface $\Sigma$, embed the pairing graph $P$ into the disk $D$ as shown on Fig.~7 and glue the disk back to the surface $\Sigma$. Thus given any self-loop $\alpha$ connecting indexes $a$ and $b$ of the tensor $T$, a small section of $\alpha$ lying inside $D$ is replaced by an edge $e=(a,b)\in E$ of the pairing graph.
We get a family of $m$ closed loops embedded into $\Sigma$. The loops may have pairwise intersection points inside the disk $D$. Let $\alpha_p$ be a loop that contains an edge $e_p\in E$. To every loop $\alpha_p$ one can assign its homological class $[\alpha_p]\in H_1(\Sigma,\mathbb{Z}_2)$ in the first homological group of $\Sigma$ with binary coefficients. Since all intersection points between the loops are contained in the disk $D$, we get \[ N_{p,q}=\omega([\alpha_p],[\alpha_q]), \] where $\omega\, : \, H_1(\Sigma,\mathbb{Z}_2)\times H_1(\Sigma,\mathbb{Z}_2) \to \{0,1\}$ is the intersection form. It is well known that the intersection form defined on a surface $\Sigma$ of genus $g$ has rank $2g$. Therefore, $N$ has rank at most $2g$. \end{proof} Given any edge $e\in E$, let $l(e),r(e)\in V$ be the two endpoints of $e$ such that $l(e)<r(e)$. Denote $\eta(e)=\eta_{l(e)}\, \eta_{r(e)}$. The generating function for the tensor $R$ can be written as \begin{equation} \label{R(eta)} R(\eta) = \sum_{y\in \{0,1\}^m}
(-1)^{\frac12\, y^T\, N \, y} \, \prod_{e\in y} \eta(e), \quad \mbox{where} \quad \eta(e)=\eta_{l(e)}\, \eta_{r(e)}. \end{equation} Here we identified a binary string $y\in \{0,1\}^m$ with the subset of edges $e_a\in E$ such that $y_a=1$. Indeed, for any $x\in \{0,1\}^{2m}$ such that $R(x)=1$ one has to regroup the factors in $\eta(x)$ to bring together variables corresponding to the same edge. It yields an extra minus sign for every pair of intersecting edges in $y$. Since every pair of edges $e_a,e_b$ contributes a sign $(-1)^{N_{a,b} \, y_a y_b}$, we arrive to Eq.~(\ref{R(eta)}).
Consider binary Fourier transform of the function $(-1)^{\frac12\, y^T\, N \, y}$, \begin{equation} \label{Fourier} f(z)\eqdef\frac1{2^m} \sum_{y\in \{0,1\}^m} (-1)^{\frac12\, y^T\, N \, y + z\cdot y}, \quad z\in \{0,1\}^m. \end{equation} Clearly $f(z)=0$ unless $z\in \mathrm{Ker}(N)^\perp$, where $\mathrm{Ker}(N)=\{ y\in \{0,1\}^m \, : \, Ny=0\}$ is the zero subspace of $N$. If $N$ has rank $r$, the zero subspace of $N$ has dimension $m-r$ and thus $\mathrm{Ker}(N)^\perp$ has dimension $r$. Let us order all the vectors of $\mathrm{Ker}(N)^\perp$ in an arbitrary way \[ \mathrm{Ker}(N)^\perp=\{z^1,\ldots,z^{2^r}\}. \] Applying the reverse Fourier transform one gets \begin{equation} \label{short_sum} (-1)^{\frac12\, y^T\, N \, y}= \sum_{a=1}^{2^r} f(z^a)\, (-1)^{y\cdot z^a}. \end{equation} By Lemma~\ref{lemma:rank} the number of terms in the sum above is bounded by $2^{2g}$. Substituting Eq.~(\ref{short_sum}) into Eq.~(\ref{R(eta)}) we arrive to \begin{equation} \label{GaussianR} R(\eta)= \sum_{a=1}^{2^r} f(z^a)\, \exp{\left( \sum_{e\in E} (-1)^{(z^a)_e} \, \eta(e) \right)}, \end{equation} where $(z^a)_e$ is the component of the vector $z^a$ corresponding to an edge $e$. It shows that $R$ is indeed a linear combination of $2^r$ matchgate tensors with $r\le 2g$.
In order to get an explicit formula for the contraction value Eq.~(\ref{TR}) let us introduce an auxiliary $2m\times 2m$ matrix
\[ A_{j,k}=\left\{ \begin{array}{rcl} +1 &\mbox{if} & \mbox{$j=l(e)$, $k=r(e)$ for some $e\in E$,}\\ -1 &\mbox{if} &\mbox{$j=r(e)$, $k=l(e)$ for some $e\in E$}\\ 0 &&\mbox{otherwise}\\ \end{array} \right. \] Introduce also auxiliary diagonal $2m\times 2m$ matrices $D^a$, $a=1,\ldots,2^r$ such that \[ (D^a)_{j,j}= \left\{ \begin{array}{rcl} (-1)^{(z^a)_e} &\mbox{if}& \mbox{$j=l(e)$ for some $e\in E$},\\ 1 &&\mbox{otherwise}.\\ \end{array}\right. \] Then Eq.~(\ref{GaussianR}) can be rewritten as \begin{equation} R(\eta)= \sum_{a=1}^{2^r} f(z^a)\, \exp{\left( \frac12\, \eta^T\, D^a\, A\, D^a\, \eta\right)}. \end{equation} Theorem~\ref{thm:canonical} implies that $T$ can be described by a generating function \[ T(\theta)=C\exp{\left( \frac12\, \theta^T\, F \, \theta \right)}\int D\mu\, \exp{\left( \mu^T \, G\, \theta \right)}, \] where $F$ and $G$ have size $2m\times 2m$ and $k\times 2m$ for some even integer $0\le k\le 2m$. Using Eq.~(\ref{TR1}) one can express the contraction value $c({\cal T })$ as a linear combination of $2^r$ Gaussian integrals \begin{equation} c({\cal T })=C\, \sum_{a=1}^{2^r} f(z^a)\, \int D\theta\, D\eta\, D\mu \, \exp{\left( \frac12 \, \theta^T\, F\, \theta + \frac12 \, \eta^T \, D^a \, A \, D^a \, \eta + \mu^T\, G\, \theta + i\, \theta^T\, \eta \right)}. \end{equation} Introducing a $(4m+k)\times (4m+k)$ matrix \[ M^a=\left[ \begin{array}{ccc} F & iI & -G^T \\ -iI & -D^a\, A\, D^a & 0 \\ G & 0 & 0 \\ \end{array} \right] \] one finally gets \begin{equation} \label{cv=5} c({\cal T })=C\, \sum_{a=1}^{2^r} f(z^a)\, \mathop{\mathrm{Pf}}\nolimits{(M^a)}. \end{equation} Computing $\mathop{\mathrm{Pf}}\nolimits{(M^a)}$ requires time $O(m^3)$. Lemma~\ref{lemma:rank} implies that the number of terms in the sum is at most $2^{2g}$. Finally, as we show below one can compute $f(z^a)$ in time $O(m^3)$. Thus $c({\cal T })$ can be computed in time $O(m^3)\, 2^{2g}$.
\begin{prop} The function $f(z)$ in Eq.~(\ref{Fourier}) can be represented as \begin{equation} \label{f(z)} f(z)=\frac1{2^{r/2}}\, (-1)^{\frac12\, z^T\, M\, z} \end{equation} for some matrix $M$ computable in time $O(m^3)$. \end{prop} \begin{proof} Using Gaussian elimination any symmetric binary matrix $N$ with zero diagonal can be represented as $N=U^T\, \tilde{N}\, U$, where $U$ is a binary invertible matrix and $\tilde{N}$ is a block diagonal matrix with $2\times 2$ blocks, \[ \tilde{N}=\bigoplus_{j=1}^{r/2} \left( \begin{array}{cc} 0 &1 \\ 1 & 0 \\ \end{array}\right). \] In particular, the rank of $N$ is always even. The matrix $U$ can be found in time $O(m^3)$. Performing a change of variable $y\to Uy$ in Eq.~(\ref{Fourier}) one gets \begin{equation} \label{Fourier1} f(z)=\frac1{2^m} \sum_{y\in \{0,1\}^m} (-1)^{\sum_{j=1}^{r/2} y_{2j-1} y_{2j} + \tilde{z}\cdot y}, \quad \tilde{z}=(U^{-1})^T\,z. \end{equation} It follows that $f(z)=0$ unless $\tilde{z}_{r+1}=\ldots=\tilde{z}_m=0$. Using an identity \[ (-1)^{x_1\cdot x_2} =\frac12 \sum_{y_1,y_2=0,1} (-1)^{y_1\cdot y_2 + y_1 \cdot x_1 + y_2 \cdot x_2} \] one can rewrite Eq.~(\ref{Fourier1}) as \[ f(z)=\frac{1}{2^{r/2}} (-1)^{\sum_{j=1}^{r/2}\, \tilde{z}_{2j-1}\, \tilde{z}_{2j}}= \frac{1}{2^{r/2}} (-1)^{\frac12\, z^T\, U^{-1} \, \tilde{N}\, (U^{-1})^T\, z}. \] We get the desired expression Eq.~(\ref{f(z)}) with $M=U^{-1} \, \tilde{N}\, (U^{-1})^T$. \end{proof}
\subsection{The main theorem} Theorem~\ref{thm:main} can be obtained straightforwardly from Theorem~\ref{thm:partial_contraction} and the contraction algorithm for a network with a single vertex, see Section~\ref{subs:1vertex}. Indeed, let $M$ be a planar cut of $G$ with $m$ edges and $G_M$ be a subgraph obtained from $G$ by removing all edges of $M$. By definition $G_M$ is contained in some region $D$ with topology of a disk. Without loss of generality $D$ contains no edges from $M$ (otherwise one can remove these edges from $M$ getting a planar cut with a smaller number of edges). Thus one can regard $G_M$ as an open tensor network with $2m$ external edges. Since $G_M$ contains all vertices of $G$, the network obtained by contraction of $G_M$ consists of a single vertex and $m$ self-loops. As explained in the previous section, one can compute the contraction value of such a network in time $O(m^3)\, 2^{2g}$.
In order to contract $G_M$ one has to compute the Gaussian integral Eq.~(\ref{integral_main}). Theorem~\ref{thm:partial_contraction} guarantees that this integral involves matrices of size $k$, where $k=O((n+m)^2)$ or $k=O(n+m)$ depending on whether the graph $G$ has bounded vertex degree. As explained in Section~\ref{subs:Gintegral} the Gaussian integral with matrices of size $k$ can be computed in time $O(k^3)$. Combining the two parts together one gets Theorem~\ref{thm:main}.
\section*{Appendix A}
Suppose $T_u$ and $T_v$ are matchgate tensors specified by their canonical generating functions as in Eq.~(\ref{canonical}), that is \[ T_\alpha = C_\alpha \, \exp{\left( \frac12\, \theta^T_\alpha \, A_\alpha \,
\theta_\alpha \right)}\int D\mu_\alpha \, \exp{\left( \mu^T_\alpha \, B_\alpha\, \theta_\alpha \right)},
\quad \mbox{where} \quad \alpha=u,v. \] Here $\theta_u=(\theta_{u,1},\ldots,\theta_{u,d(u)})$ and $\theta_v=(\theta_{v,1},\ldots,\theta_{v,d(v)})$ are the two sets of Grassmann variables associated with the vertices $u$ and $v$. Denote also $\epsilon(T)$ the parity of a matchgate tensor $T$, that is, $\epsilon(T)=0$ ($\epsilon(T)=1$) for even (odd) tensor $T$. In the remainder of this section we explain how to express the canonical generating function for the contracted tensor $T_{u\star v}$, see Eqs.~(\ref{Tu*v_def},\ref{TuTv}),
in terms of the matrices $A_\alpha$, $B_\alpha$.
Applying Eq.~(\ref{TuTv}) one gets \begin{equation} \label{misc1} T_{u\star v}=C_uC_v\, \int_{e\in E(u,v)} \, D\theta(e) \, \int D\mu_u \, \int D\mu_v \, \exp{[ f(\theta_u,\theta_v,\mu_u,\mu_v)]}, \end{equation} where \[
f(\theta_u,\theta_v,\mu_u,\mu_v)=
\sum_{\alpha=u,v} \, \frac12 \, \theta_\alpha^T A_\alpha\, \theta_\alpha + \mu_\alpha^T\, B_\alpha\, \theta_\alpha +
\sum_{e\in E(u,v)} \theta(e). \] Let us split the vectors of Grassmann variables $\theta_u$, $\theta_v$ into external and internal parts, \[ \theta_u=(\theta_u^e,\theta_u^i) \quad \mbox{and} \quad \theta_v=(\theta_v^i,\theta_v^e), \] such that all internal variables are integrated out in $T_{u\star v}$. Then one can rewrite the expression Eq.~(\ref{misc1}) as a product of a Gaussian exponent and the standard Gaussian integral $I(K,L)$, see Eqs.~(\ref{GI2},\ref{GI3}), for some matrices $K,L$ defined below, \begin{equation} \label{misc2} T_{u\star v}(\tau)=C_u C_v\,(-1)^{\frac{b(b-1)}2 + \epsilon(T_u) \epsilon(T_v)}\, \exp{\left( \frac12\, \tau^T\, H \, \tau\right)} \, \int D\eta \exp{\left( \frac12 \, \eta^T\, K\, \eta + \eta^T\, L \, \tau\right)}. \end{equation} Here we introduced auxiliary vectors of Grassmann variables $\tau=(\theta_u^e,\theta_v^e)$, $\eta=(\theta_u^i,\theta_v^i,\mu_u,\mu_v)$. The matrices $H,K,L$ above will be defined using a partition of matrices $A_\alpha$, $B_\alpha$ into "internal" and "external" blocks as follows: \[ A_u=\left[ \begin{array}{cc} A_u^{ee} & A_u^{ei} \\ A_u^{ie} & A_u^{ii} \\ \end{array} \right], \quad A_v=\left[ \begin{array}{cc} A_v^{ii} & A_v^{ie} \\ A_v^{ei} & A_v^{ee} \\ \end{array} \right], \quad B_u=\left[ \begin{array}{cc} B_u^e & B_u^i \\ \end{array} \right], \quad B_v=\left[ \begin{array}{cc} B_v^i & B_v^e \\ \end{array} \right]. \] Introduce also a square matrix $\bar{I}$ that has ones on the diagonal perpendicular to the main diagonal and zeroes everywhere else. Then the matrices $H,K,L$ in Eq.~(\ref{misc2}) are defined as \[ H=\left[ \begin{array}{cc} A_u^{ee} & 0 \\ 0& A_v^{ee} \\ \end{array} \right], \quad K=\left[ \begin{array}{cccc} A_u^{ii} & \bar{I} & -(B_u^i)^T & 0 \\ & A_v^{ii} & 0 & -(B_v^i)^T \\ & & 0 & 0 \\ & & & 0\\ \end{array}\right], \quad L=\left[ \begin{array}{cc} A_u^{ie} & 0 \\ 0 & A_v^{ie} \\ B_u^e & 0\\ 0 & B_v^e \\ \end{array} \right]. \] Finally, the extra sign in Eq.~(\ref{misc2}) takes into account the difference between the order of integrations in Eqs.~(\ref{misc1},\ref{misc2}). Summarizing, Eq.~(\ref{misc2}) together with the Gaussian integration formulas Eqs.~(\ref{GI2},\ref{GI3}) allow one to write down the canonical generating function for the contracted tensor $T_{u\star v}$.
\end{document} | arXiv | {
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\begin{document}
\title[Generalized Class of $\mathcal{P}\mathcal{R}$-warped product submanifolds] {On the Generalized Class of $\mathcal{P}\mathcal{R}$-warped product submanifolds in para-K\"{a}hler Manifolds} \author{A. Sharma} \address{Department of Mathematics, Lovely Professional University, Jalandhar - Delhi G.T. Road, Phagwara, Punjab-144411, INDIA} \email{\textcolor[rgb]{0.00,0.00,0.84}{anilsharma3091991@gmail.com}} \author{S. K. Srivastava} \address{Department of Mathematics, Central University of Himachal Pradesh, Dharamshala-176215, Himachal Pradesh, INDIA} \email{\textcolor[rgb]{0.00,0.00,0.84}{sachin@cuhimachal.ac.in}} \begin{abstract} In this paper, we study a new generalized class of $\mathcal{P}\mathcal{R}$-warped product submanifolds under the name $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifolds in para-K\"{a}hler manifolds $\bar{M}$. The results of existence and non-existence for $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifolds with proper slant factor in $\bar{M}$ are shown. In addition to these results, we give an elementary illustration of such warped product submanifold in $\bar{M}$. \end{abstract} \subjclass[2010]{53B25, 53B30, 53C15, 53C42} \keywords{Para-K\"{a}hler manifold, Warped product, Slant submanifold, Pseudo-Riemannian manifold} \maketitle
\section{Introduction} \noindent The concept of warped product manifolds (more generally warped bundle) was introduced by Bishop and O'Neill for constructing manifolds of nonpositive curvature, as one of the most effective generalization of Riemannian product manifold \cite{Bishop1969}. Later, this notion has been generalized by O'Neill, B. \cite{Neill1983} for semi-Riemannian manifolds. However, the theory attain momentum after Chen came into action with a new class of CR-submanifolds called CR-warped products in Kaehlerian manifolds and give some existence and non existence results \cite{Chen2001}. Analogous to that Hasegawa-Mihai \cite{Hasegawa2003} and Munteanu \cite{Munteanu1999} continued the theory for Sasakian ambient that can be viewed as an odd-dimensional counterpart of K\"{a}hler manifold. Since then geometry of warped product submanifolds in K\"{a}hler-Riemannian and Lorentzian metric structures gain impluse and has been effectively employed to solve problems in mathematics and physics particularly in the field of general theory of relativity and black holes (c.f., \cite{Carot1993, Chen_survey2013, Hong2005, Katanaev1999, Sri-Sharma2017}). The exertions were focused essentially on understanding the parallel Einstein equations, which is responsible for broad consequences in both fields. The metrics of neutral signature also viewed in several geometric and physics problems but they have attained less attention until Davidov et.al \cite{Davidov} presented the analogies and differences between the structures having neutral metric and the Riemannian metric. Earlier \cite{Davidov}, the remarkable works by Ooguri-Vafa \cite{Ooguri1990}, Dunajsky-West \cite{Dunajsky} and Petean \cite{Petean1997}, has perceived various significant role of metric structures having $(m, m)$ signature in mathematical physics and have been fruitfully applied to supersymmetric field theories and string theory. In light of the physical applications of the such neutral metrics structures, the important question of {\it existence or non-existence} of such metric with its warped structure arises naturally. Recently, Chen (\cite{Chen2011}: Chapter 10) initiated the geometry of pseudo-Riemannian warped products submanifolds (called by him PR-warped product) in neutral metric manifold called para-K\"{a}hler manifolds. Motivated by the work of (\cite{Chen2011}: Chapter 10), in the present work we introduce the more general class of \cite{Chen-Munteanu2012} with the name $\mathcal{P}\mathcal{R}$-pseudo-slant submanifolds equipped with warped aspect and investigate the existence and nonexistence of such warped product submanifolds in para-K\"{a}hler manifolds $\bar{M}$.
The organization of article is as follows. In Sect. \ref{pre}, we recall some basic informations about para-K\"{a}hler manifold, slant submanifold and present definition for slant submanifold along with some important results. Sect. \ref{prps} deals with the construction of $\mathcal{P}\mathcal{R}$-pseudo-slant submaniolds, and the conditions of integrablity and totally geodesic foliation for the distributions allied to the characterization of a such submanifolds. In Sect. \ref{prwps}, we first prove the nonexistence of $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifolds of the form $M_{\bot}\times_{f}M_{\lambda}$ and then obtain a necessary and sufficient condition for a submanifold $M$ to be locally a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product and simply product submanifolds of the form $M=M_{\lambda}\times_{f}M_{\bot}$ in $\bar{M}$. Finally in Sect. \ref{exa}, we present a numerical example illustrating warped product submanifold of the form $M_{\lambda}\times_{f}M_{\bot}$ in a para-K\"{a}hler manifold $\bar{M}$.
\section{Preliminaries}\label{pre} \subsection{An almost para-Hermitian manifold}\label{aphm} \noindent A smooth manifold $\bar{M}$ of dimension $2m$ is said to have an almost product structure if \begin{align}\label{phstruct}
\mathcal{P}^2=I, \end{align} where $\mathcal{P}$ is a tensor field of type $(1,1)$ and $I$ is the identity transformation on $\bar{M}^{2m}$. For this, the pair $(\bar{M}^{2m}, \mathcal{P})$ is called almost product manifold. An almost para-complex manifold is an almost product manifold $(\bar{M}^{2m}, \mathcal{P})$ such that the two eigenbundles $T^{\pm}\bar{M}^{2m}$ corresponding to the eigenvalues $\pm1$ of $\mathcal{P}$ have the equal dimension. An almost para-Hermitian manifold $(\bar{M}^{2m}, \mathcal{P}, \bar{g})$ \cite{Libermann1954} is a smooth manifold associated with an almost product structure $\mathcal{P}$ and a pseudo-Riemannian metric $\bar{g}$ satisfying \begin{align} \bar{g}(\mathcal{P}X, \mathcal{P}Y)+\bar{g}(X,Y)=0,\label{phmetric} \end{align}
Clearly, signature of $\bar{g}$ is necessarily $(m,m)$ for any vector fields $X,Y$ tangent to $\bar{M}^{2m}$. Also, Eq. \eqref{phmetric} implies that \begin{align}\label{phantisym}
\bar{g}(\mathcal{P}X,Y)+\bar{g}(X,\mathcal{P}Y)=0, \end{align} for any $X,Y \in \Gamma(T\bar{M})$; $\Gamma(T\bar{M})$ being Lie algebra of vector fields of $\bar{M}^{2m}$. The fundamental $2$-form $\omega$ of $\bar{M}^{2m}$ is defined by \begin{align}\label{phomega} \omega(X,Y) = \bar{g}(X, \mathcal{P}Y ), \quad \forall X,Y \in \Gamma(T\bar{M}). \end{align}
\begin{definition} An almost para-Hermitian manifold $\bar{M}^{2m}$ is called a para-K\"{a}hler manifold if $\mathcal{P}$ is parallel with respect to $\bar{\nabla}$ {\emph i.e.}, \begin{align}\label{pkahlerdef} (\bar{\nabla}_{X} {\mathcal{P}})Y=0, \quad \forall X,Y \in \Gamma(T\bar{M}) \end{align} where $\bar{\nabla}$ is the Levi-Civita connection on $\bar{M}^{2m}$ with respect to $\bar{g}$. \end{definition} \subsection{Geometry of slant submanifolds}\label{gss} \noindent Let us consider that $M$ is an isometrically immersed submanifold of a para-K\"{a}hler manifold in the sense of O'Neill \cite{Neill1983} and Chen \cite{Chen2011} and $g$ denote the induced metric on $M$ such that $g=\bar{g}\vert_M$ having constant signature and rank \cite{Etayo1999}. Let $\Gamma (TM^{\bot })$ indicate the set of vector fields normal to $M$ and $\Gamma(TM)$ the sections of tangent bundle $TM$ of $M$ then the Gauss-Weingarten formulas are given by, respectively, \begin{align} \bar{\nabla }_{X} Y&=\nabla _{X} Y+h(X,Y), \label{gauss}\\ \bar{\nabla }_{X} \zeta &=-A_{\zeta} X+\nabla _{X}^{\bot }\zeta,\label{weingarten} \end{align} for all $X,Y\in \Gamma (TM)$ and $\zeta \in \Gamma (TM^{\bot })$, where $\nabla$ is the induced connection, $\nabla ^{\bot }$ is the normal connection on the normal bundle $TM^{\bot }$, $h$ is the second fundamental form, and the Weingarten operator $A_{\zeta}$ associated with the normal section $\zeta$ (see also, \cite{Chen1973}) is given by \begin{align}\label{shp2form} g\left(A_{\zeta} X,Y\right)=\bar{g}\left(h(X,Y),\zeta\right). \end{align} For any $ \tau \in \Gamma (TM)$ and $\zeta \in \Gamma(TM^{\bot })$, if we write \begin{align} \mathcal{P}{\tau} &=t{\tau}+n{\tau}, \label{jtau} \\ \mathcal{P}{\zeta} &=t'{\zeta}+n'{\zeta}, \label{jzeta} \end{align} where $t{\tau}$ (resp., $n{\tau}$) is tangential (resp., normal) part of $\mathcal{P}{\tau}$ and $t'{\zeta}$ (resp., $n'{\zeta}$) is tangential (resp., normal) part of $\mathcal{P}{\zeta}$, then for any $X,Y\in \Gamma (TM)$ we can easily obtain from Eqs. \eqref{phstruct} and \eqref{jtau} that \begin{align}\label{phantixty} \bar{g}(X, tY)=-\bar{g}(tX, Y). \end{align} A pseudo-Riemannian submanifold $M$ is said to be \cite{Chen2011} \begin{itemize} \item[$\bullet$] {\it totally geodesic} if its second fundamental form vanishes identically. \item[$\bullet$] {\it umbilical} in the direction of a normal vector field $\zeta$ on $M$, if $A_{\zeta} = \eta Id$, for certain function $\eta$ on $M$; here $\zeta$ is called a umbilical section. \item[$\bullet$] {\it totally umbilical} if $M$ is umbilical w.r. t. any local normal vector field. \end{itemize} Here, motivated to \cite{Alegre2017}, we first derive the definition of slant submanifolds in para-K\"{a}hler manifolds $\bar{M}^{2m}$, and then continue the study by introducing a generalized class of $\mathcal{P}\mathcal{R}$-submanifolds \cite{Chen-Munteanu2012} called $\mathcal{P}\mathcal{R}$-pseudo-slant submanifolds in $\bar{M}^{2m}$.
\noindent Let $M$ be any non-degenerate submanifolds of para-K\"{a}hler manifolds $\bar{M}^{2m}$ such that $t^2X=\lambda X=\lambda \mathcal{P}^{2}X, \ g(tX, Y)=-g(X, tY)$ for any $X, Y \in \Gamma(TM)$, where $\lambda$ is a coefficient then with the help of Eq. \eqref{phantixty}, we have \begin{align}\label{theta1} \frac{g(\mathcal{P} X, tY)}{\vert\mathcal{P} X\vert\vert tY\vert}=-\frac{g(X, \mathcal{P} tY)}{\vert\mathcal{P} X\vert\vert tY\vert}=-\frac{g(X, t^{2}Y)}{\vert\mathcal{P} X\vert\vert tY\vert}=-\lambda\frac{g(X, \mathcal{P}^{2}Y)}{\vert\mathcal{P} X\vert\vert tY\vert}=\lambda\frac{g(\mathcal{P} X, \mathcal{P} Y)}{\vert\mathcal{P} X\vert\vert tY\vert}. \end{align} On the other hand, \begin{align}\label{theta2} \frac{g(\mathcal{P} X, tY)}{\vert\mathcal{P} X\vert\vert tY\vert}=\frac{g(tX, tY)}{\vert\mathcal{P} X\vert\vert tY\vert}. \end{align} In particular, from Eqs. \eqref{theta1} and \eqref{theta2}, we obtain for $X=Y$ that $\frac{g(\mathcal{P} X, tX)}{\vert\mathcal{P} X\vert\vert tX\vert}=\sqrt{\lambda}$. Here we call $\lambda$ a slant coefficient and consequently $M$ a slant submanifold. Conversely, assume that $M$ is a slant submanifold then $\lambda\frac{\vert\mathcal{P} X\vert}{\vert tX\vert}=\frac{\vert tX\vert}{\vert\mathcal{P} X\vert}$, where $X$ is a non light like vector field. We obtain by the consequence of previous equation for any non- light like vector fields $X, Y \in \Gamma(TM)$ that $-\lambda\frac{g(X, \mathcal{P}^{2}Y)}{\vert\mathcal{P} X\vert\vert tY\vert}=\frac{g(\mathcal{P} X, tY)}{\vert\mathcal{P} X\vert\vert tY\vert}$, which yields $g(X, t^{2}Y)=\lambda g(X, \mathcal{P}^{2}Y), \ g(tX, Y)=-g(X, tY)$. Hence, $t^2=\lambda I, \quad g(tX, Y)=-g(X, tY)$ by virtue of the fact that structure is para-K\"{a}hler and $X$ is any non-lightlike vector fields. \begin{remark} The slant coefficient $\lambda$ is sometimes $\cos^{2}\theta$ or $\cosh^{2}\theta$ or $-\sinh^{2}\theta$ for all vector fields tangent to $M$, where $\theta$ is a slant angle \cite{Alegre2017, Aydin2013}. \end{remark} \noindent Now by the consequence of above characterization here, we can easily state the definition of slant submanifolds in an almost para-K\"{a}hler manifold and obtain slant submanifold defined for K\"{a}hler-Riemannian Case as one of its important remark: \begin{definition}\label{phslantdfn} Let $M$ be an isometrically immersed submanifold of an almost para-K\"{a}hler manifold $\bar{M}^{2m}$ and $\mathfrak{D_{\lambda}}$ be the distribution on $M$. Then $\mathfrak{D_{\lambda}}$ is said to be \textit{slant distribution} on $M$, accordingly $M$ \textit{slant submanifold}, if there exist a real valued constant $\lambda$ such that \begin{align*}
t^{2} =\lambda I, \quad g(tX, Y)=-g(X, tY), \end{align*} for any non-degenerate tangent vectors $X,Y \in \mathfrak{D_{\lambda}}$ on $M$. Here, we call $\lambda$ \textit{slant coefficient} which is globally constant, i.e., $\lambda$ is independent of the choice of the point on $M$ in $\bar{M}^{2m}$. \end{definition} \begin{remark} Since, our manifold $M$ is non-degenerate ({\it i.e.}, $M$ includes either spacelike vector fields or timelike vector fields), thus our definition of slant submanifolds can be considered as the generalization of definitions given in \cite{Chen1990} which covers only the spacelike vector fields which implies $\lambda=\cos^{2}\theta$, where $\theta$ is slant angle.
\end{remark}
\begin{remark} Here, it is important to note that the {\it invariant} and {\it anti-invariant} submanifolds are improper slant submanifolds with slant coefficients $\lambda=1$ and $\lambda=0$, respectively. Thus, a \textit{proper slant} submanifold is a slant submanifold which is neither invariant nor anti-invariant \cite{Dillen1993}.
\end{remark}
\noindent Further we derive an important result for slant submanifold $M$ of $\bar{M}^{2m}$; \begin{theorem}\label{pointthm2} Let $M$ be any isometrically immersed proper slant submanifold in an almost para-Hermitian manifold $\bar{M}^{2m}$. Then for any $X, Y, Z \in \Gamma(TM)$, $2$-form $\omega$ is closed. \end{theorem} \begin{proof} We have by virtue of exterior differentiation that \begin{align}\label{extdef}
d\omega(X, Y, Z) =& \frac{1}{3} \{ X\omega (Y, Z) + Y\omega (Z, X) + Z\omega (X, Y) \\
\nonumber & - \omega ([X, Y], Z) - \omega ([Y, Z], X) - \omega ([Z, X], Y) \}. \end{align} By the use of Eqs. \eqref{phomega} and \eqref{jtau}, equation \eqref{extdef} simplified to \begin{align}\label{simpdiff} d\omega(X, Y, Z) = &\frac{1}{3} \{ Xg(Y, tZ) + Yg(Z, tX) + Zg(X, tY) \\
\nonumber & - g([X, Y], tZ) - g([Y, Z], tX) - g([Z, X], tY) \}. \end{align} Furthermore, using covariant differentiation and definition of Lie-bracket in Eq. \eqref{simpdiff} we obtain that \begin{align}\label{covdiff} d\omega(X, Y, Z) = &\frac{1}{3} \{ g(Y, \nabla_{X}(tZ)) + g(Z, \nabla_{Y}(tX)) + g(X, \nabla_{Z}(tY)) \\
\nonumber & +g(\nabla_{Y}X, tZ) + g(\nabla_{Z}Y, tX) +g(\nabla_{X}Z, tY) \}. \nonumber \end{align} Above equation using Eqs. \eqref{phantixty}, \eqref{shp2form} and covariant differentiation of $t$ reduced to \begin{align}\label{result} d\omega(X, Y, Z) = 0 \quad \forall X, Y, Z \in \Gamma(TM). \end{align} Thus, Eq. \eqref{result} signifies that the $2$-form $\omega$ is closed. This completes the proof of the theorem. \end{proof}
\section{$\mathcal{P}\mathcal{R}$-pseudo-slant submanifolds}\label{prps} In this section, we introduce $\mathcal{P}\mathcal{R}$-pseudo-slant submanifolds in $\bar{M}^{2m}$ which generalizes the $\mathcal{P}\mathcal{R}$-submanifolds defined by Chen in \cite{Chen-Munteanu2012}. Moreover, since the submanifold $M$ is non-degenerate submanifold so this new class of submanifolds can also be viewed as the generalization of submanifolds defined in Riemannnian settings (see; \cite{Carriazo2002, Chen1981}). \begin{definition} Let $M$ be an isometrically immersed submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then we say that $M$ is a {\it $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold} if it is furnished with the pair of non-degenerate orthogonal distribution $(\mathfrak{D}^{\bot},\mathfrak{D}_{\lambda})$ satisfying the conditions: \begin{itemize} \item[(i)] $TM = \mathfrak{D}^{\bot}\oplus \mathfrak{D}_{\lambda}$, \item[(ii)] the distribution $\mathfrak{D}{^\bot}$ is anti-invariant distribution under $\mathcal{P}$, i.e., $\mathcal{P}(\mathfrak{D}{^\bot})\subseteq \Gamma(TM){^\bot}$ and \item[(iii)] the distribution $\mathfrak{D}_{\lambda}$ is slant distribution with slant coefficient $\lambda.$ \end{itemize} \end{definition} \noindent In consequence of above definition we give following results as a remarks; \begin{remark} Let us denote by $d_1$ and $d_2$ the dimension of $\mathfrak{D}^{\bot}$ and $\mathfrak{D}_{\lambda}$, respectively then we conclude that a $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold $M$ of $\bar{M}^{2m}$ is {\it invariant submanfold} if $d_1=0$, $d_2 \neq 0$ with slant coefficient $\lambda = 1$, {\it anti-invariant submanfold} if $d_1\neq 0$ and $d_2=0$ and {\it $\mathcal{P}\mathcal{R}$-submanifold} if $d_1, d_2 \neq 0$ with slant coefficient $\lambda = 1$ \cite{Chen-Munteanu2012}.
\end{remark} \begin{remark} If $d_1.d_2 \neq 0$ and $\mathfrak{D}_{\lambda}$ is proper slant distribution then we call $M$ a {\it proper} $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of $\bar{M}^{2m}$.
\end{remark} \noindent Furthermore, if we represent the projections on the distributions $\mathfrak{D}^{\bot}$ and $\mathfrak{D}_{\lambda}$ by $P^{\bot}$ and $P_{\lambda}$, respectively. Then we can write \begin{equation} X= P^{\bot} X + P_{\lambda} X \label{pseudo-proj} \end{equation} for any $X \in \Gamma(TM)$. Applying $\mathcal{P}$ to Eq. \eqref{pseudo-proj} and using Eq. \eqref{jtau}, we have \begin{equation} \mathcal{P}X =nP^{\bot} X +tP_{\lambda} X +nP_{\lambda} X. \label{pseudo-jxproj} \end{equation} From above equation we obtain that \begin{align} & nP^{\bot}X \in \Gamma(\mathfrak{D}^{\bot}),\quad tP^{\bot}X=0, \label{pseudo-fpbot}\\ & tP_{\lambda} X \in \Gamma(\mathfrak{D}_{\lambda}),\quad nP_{\lambda} X \in \Gamma(TM^{\bot}).\label{pseudo-tptheta} \end{align} Employing Eq. \eqref{jtau} in \eqref{pseudo-jxproj}, we get \begin{equation} tX = tP_{\lambda}X, \quad nX = nP^{\bot}X+nP_{\lambda} X \label{pseudo-txproj} \end{equation} for $X \in \Gamma(TM)$. Since, $\mathfrak{D}_{\lambda}$ is slant distribution, using definition \ref{phslantdfn}, we conclude that \begin{equation} t^2X = \lambda X \label{pseudo-tsqr} \end{equation} for any $X \in \mathfrak{D}_{\lambda}$ and real-valued constant coefficient $\lambda$ defined on $M$.
\noindent Now, we present the following results for the characterization of proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold: \begin{theorem}\label{pseudo-thm1} Let $M$ be a submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then $M$ is a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold if and only if there exists a real valued constant coefficient $\lambda$ and a distribution $D$ on $M$ such that \begin{enumerate}
\item [$(i)$] $D=\{ X \in \Gamma(TM) \mid (t_{D})^2 X = \lambda X \}$,
\item [$(ii)$] For any $X \in \Gamma(TM)$, orthogonal to $D$, $tX = 0$, \end{enumerate} where $\lambda $ denotes the slant coefficient of $M$. \end{theorem} \begin{proof} Let $M$ be a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of $\bar{M}^{2m}$. Using Eqs. \eqref{pseudo-fpbot}, \eqref{pseudo-tptheta} and \eqref{pseudo-tsqr} we have that $D=\mathfrak{D}_{\lambda} $, which follows $(i)$ and $(ii)$. Conversely $(i)$ and $(ii)$ implies that $TM=\mathfrak{D}_{\lambda} \oplus \mathfrak{D}^{\bot}$. From (ii) , we received that $\mathcal{P}(\mathfrak{D}^{\bot})=\mathfrak{D}^{\bot}$. This completes the proof. \end{proof} \noindent From theorem \ref{pseudo-thm1} we have the following corollary : \begin{corollary}\label{pseudo-cor1} Let $M$ be a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then for all $X, Y \in \Gamma(\mathfrak{D}_{\lambda})$, we have \begin{align} g(tX,tY)&=\lambda \, g(X,Y) \label{pseudo-gtcos}\\ g(nX,nY)&= g(X,Y) - \lambda \, g(X,Y) \label{pseudo-gfsin}. \end{align} \end{corollary} \begin{proof} From Eqs. \eqref{phantisym} and \eqref{jtau} we have $g(tX,tY)=g(\mathcal{P}X-nX,tY)$. Hence $g(tX,tY)=-g(X,\mathcal{P}tY)$. Using Theorem \ref{pseudo-thm1} (i), we obtain formula \eqref{pseudo-gtcos}. Finally, formula \eqref{pseudo-gfsin} directly follows from \eqref{pseudo-gtcos}. \end{proof} \noindent Further, we prove an important lemma for later use: \begin{lemma}\label{lemmain} Let $M$ be a $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then for any $X, Y \in \Gamma(\mathfrak{D}_{\lambda})$, we have \begin{align*} (i)\ t'nX=X-\lambda X\ {and}\ (ii)\ n'nX=-ntX. \end{align*} \end{lemma} \begin{proof} We have from Eq. \eqref{jzeta} that $\mathcal{P}nX=t'fX+n'nX$. Then, taking inner product with $Y$ and applying Eqs. \eqref{phstruct}, \eqref{phantisym} and \eqref{pseudo-gfsin} we derive the formula-$(i)$. For formula-$(ii)$, replacing $X$ by $tX$ and employing Eq. \eqref{pseudo-tsqr} in Eq. \eqref{jtau} we obtain that \begin{align}\label{pseudomain-1}
\mathcal{P}tX=t^{2}X+ntX=\lambda X + ntX. \end{align} On the other hand, using Eq. \eqref{jzeta} and formula-$(i)$ we achieve that \begin{align}\label{pseudomain-2}
\mathcal{P}nX=t'nX+n'nX=(1-\lambda)X + n'nX. \end{align} From Eqs. \eqref{pseudomain-1} and \eqref{pseudomain-2} we get $\mathcal{P}tX+\mathcal{P}nX=X+ntX+n'nX$. Formula-$(ii)$ can be deduced by employing Eqs. \eqref{phstruct} and \eqref{jtau} in previous expression. This completes the proof of the lemma. \end{proof} \noindent Here, we examine the conditions for distributions associated with the definition of proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold being integrable and defines totally geodesic foliation. \begin{lemma}\label{Dbotint} Let $M$ be a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold $\bar{M}$. Then the anti-invariant distribution $\mathfrak{D}^{\bot}$ is integrable if and only if \begin{align} g(A_{\mathcal{P}Y}X, tZ)=g(A_{\mathcal{P}X}Y, tZ) \end{align} for any $X, Y \in \Gamma(\mathfrak{D}^{\bot})$ and $Z \in \Gamma(\mathfrak{D}_{\lambda})$. \end{lemma} \begin{proof} We have from the fact that structure is para-Hermition, Gauss-Weingarten formula that \begin{align}\label{pseudo-thm2_0} g([X,Y],Z) = -\bar{g}(\mathcal{P}\bar\nabla_{X}Y, \mathcal{P}Z)+\bar{g}(\mathcal{P}\bar\nabla_{Y}X ,\mathcal{P}Z).
\end{align}
Furthermore, we can write from Eqs. \eqref{pkahlerdef}, \eqref{jtau} and \eqref{jzeta} that \begin{align}\label{pseudo-thm2_01} -\bar{g}(\mathcal{P}\bar\nabla_{X}Y, \mathcal{P}Z) = -\bar{g}(\bar\nabla_{X}\mathcal{P}Y, tZ)+\bar{g}(\bar\nabla_{X}Y, t'nZ)+\bar{g}(\bar\nabla_{X}Y, n'nZ). \end{align} Using the fact that $h$ is symmetric, lemma \ref{lemmain} and Eqs. \eqref{weingarten}, \eqref{pseudo-tsqr} in equation \eqref{pseudo-thm2_01}, we obtain that \begin{align}\label{pseudo-thm2_02} -\bar{g}(\mathcal{P}\bar\nabla_{X}Y,\mathcal{P}Z) = g(A_{\mathcal{P}Y}X, tZ) + (1-\lambda) \bar{g}(\bar\nabla_{X}Y, Z)-\bar{g}(\bar\nabla_{X}Y, ntZ). \end{align} On the other hand with effect from Eqs. \eqref{pkahlerdef}, \eqref{jtau} and \eqref{jzeta}, we arrive at \begin{align}\label{pseudo-thm2_03} \bar{g}(\mathcal{P}\bar\nabla_{Y}X,\mathcal{P}Z) = -g(A_{\mathcal{P}X}Y, tZ)-\bar{g}(\bar\nabla_{Y}X, t'nZ)-\bar{g}(\bar\nabla_{Y}X, n'nZ) \end{align} Therefore, Eq. \eqref{pseudo-thm2_03} by the use of lemma \ref{lemmain} reduced to \begin{align}\label{pseudo-thm2_04} \bar{g}(\mathcal{P}\bar\nabla_{Y}X,\mathcal{P}Z) = -g(A_{\mathcal{P}X}Y, tZ)-(1-\lambda)\,\bar{g}(\bar\nabla_{Y}X, Z)+ \bar{g}(\bar\nabla_{Y}X, ntZ). \end{align} Employing Eqs. \eqref{pseudo-thm2_02} and \eqref{pseudo-thm2_04} in \eqref{pseudo-thm2_0}, we conclude that \begin{align}\label{pseudo-thm2_05} \lambda g([X,Y],Z)= g(A_{\mathcal{P}Y}X, tZ)-g(A_{\mathcal{P}X}Y, tZ).
\end{align} This completes the proof. \end{proof}
\begin{lemma}\label{Dlambdaint} Let $M$ be a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then the distribution $\mathfrak{D}_{\lambda}$ is integrable if and only if $g(A_{ntZ}X-A_{\mathcal{P}X}tZ, W)=g(A_{ntW}X-A_{\mathcal{P}X}tW, Z)$, for all $X \in \Gamma(\mathfrak{D}^{\bot})$ and $Z, W \in \Gamma(\mathfrak{D}_{\lambda})$. \end{lemma} \begin{proof} The proof of this lemma can be achieved by following same steps as the proof of lemma \ref{Dbotint}. \end{proof}
\begin{lemma}\label{Dbotgeodsic} Let $M$ be a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then the anti-invariant distribution $\mathfrak{D}^{\bot}$ defines a totally geodesic foliation if and only if \begin{align} g(A_{\mathcal{P}Y}X, tZ)=g(A_{ntZ}X, Y) \end{align} for any $X, Y \in \Gamma(\mathfrak{D}^{\bot})$ and $Z \in \Gamma(\mathfrak{D}_{\lambda})$. \end{lemma} \begin{proof} \noindent By the virtue of Eqs. \eqref{phstruct},\eqref{phantisym}-\eqref{jzeta}, we obtain that \begin{align}\label{pseudo-thm2_2} g(\nabla_{X}Y,Z) = -\bar{g}(\bar{\nabla}_{X}\mathcal{P}Y, tZ) + \bar{g}(\bar{\nabla}_{X}Y, t'nZ+n'nZ). \end{align} Using Eq. \eqref{weingarten} and lemma \ref{lemmain} in equation \eqref{pseudo-thm2_2}, we get \begin{align}\label{pseudo-thm2_3} g(\nabla_{X}Y, Z)&= g(A_{\mathcal{P}Y}X, tZ) + \bar{g}(\bar{\nabla}_{X}Y, Z) -\lambda \bar{g}(\bar{\nabla}_{X}Y, Z)-g(A_{ntZ}X, Y). \end{align} Now, employing Eq.\eqref{gauss} in \eqref{pseudo-thm2_3}, we arrive at \begin{align*} \lambda g(\nabla_{X}Y, Z)= g(A_{\mathcal{P}Y}X, tZ)-g(A_{ntZ}X, Y). \end{align*} This completes the proof of the lemma. \end{proof}
\begin{lemma}\label{Dlambdageo} Let $M$ be a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then the distribution $\mathfrak{D}_{\lambda}$ defines a totally geodesic foliation if and only if $g(A_{ntW}Z, X)=g(A_{\mathcal{P}X}Z, tW)$, for all $X \in \Gamma(\mathfrak{D}^{\bot})$ and $Z, W \in \Gamma(\mathfrak{D}_{\lambda})$. \end{lemma} \begin{proof}
To proof this lemma, we follow same steps as in the proof of lemma \ref{Dbotgeodsic}. \end{proof}
\section{$\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifolds}\label{prwps} \noindent In this section, we investigate the existence or nonexistence of non-trivial $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifolds of the form $M_{\bot} \times_fM_{\lambda}$, $M_{\lambda} \times_f M_{\bot} $ a para-K\"{a}hler manifold $\bar{M}^{2m}$, where $M_{\bot}$ and $M_{\lambda}$ are anti-invariant and proper slant submanifolds of $\bar M^{2m}$, respectively. We first recall some basic information about warped product pseudo-Riemannian manifolds;
\noindent Let $\left(B,g_{B} \right)$ and $\left(F ,g_{F} \right)$ be two pseudo-Riemannian manifolds and ${f}$ be a positive smooth function on $B$. Consider the product manifold $B\times F$ with canonical projections \begin{align}\label{cp} \pi:B \times F\to B\quad{\rm and}\quad \sigma:B \times F\to F. \end{align} Then the manifold $M=B \times_{f}F$ is said to be \textit{warped product} (see \cite{Neill1983, Sri-Sharma2017}) if it is endowed with the following warped metric \begin{align}\label{wmetric} g(X,Y)=g_{B}\left(\pi_{\ast}(X),\pi_{\ast}(Y)\right) +(f\circ\pi)^{2}g_{F}\left(\sigma_{\ast}(X),\sigma_{\ast}(Y)\right) \end{align} for all $X,Y\in \Gamma(TM)$ and `$\ast$' stands for derivation map, or equivalently, \begin{align} g=g_{B} +_{f^{2}}g_{F}. \end{align} The function $f$ is called {\it the warping function} and a warped product manifold $M$ is said to be {\it trivial } if $f$ is constant. In view of simplicity, we will determine a vector field $X$ on $B$ with its lift $\bar X$ and a vector field $Z$ on $F$ with its lift $\bar Z$ on $M=B \times_{f} F $ \cite{Bishop1969}.
\begin{proposition}\label{propwp}\cite{Bishop1969} If $X, Y \in \Gamma(TB)$ and $Z, W \in \Gamma(TF)$, then we have for warped product submanifold $M=B \times_{f}F$ that \begin{itemize} \item[(i)] $\nabla _{X}Y \in \Gamma(TB),$ \item[(ii)] $\nabla _{X}Z =\nabla _{Z}X=\left(\frac{X{f}}{f} \right)Z,$ \item[(iii)] $\nabla _{Z}W =\frac{-g(Z, W)}{f} \nabla f,$ \end{itemize} where $\nabla$ denotes the Levi-Civita connection on $M$ and $\nabla f$ is the gradient of $f$ defined by $g(\nabla f, X)=Xf$. \end{proposition} \noindent For a warped product $M=B \times_{f}F$; $B$ is totally geodesic and $F$ is totally umbilical in $M$ \cite{Bishop1969}. \noindent Here, analagous to \cite{Chen-Munteanu2012} we define $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold in a para-K\"{a}hler manifold $\bar{M}^{2m}$: \begin{definition} A proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold is said to be a $\mathcal{P}\mathcal{R}$-{\it pseudo-slant warped product} if it is a warped product of the form : $M_{\lambda} \times_fM_{\bot}$ or $M_{\lambda} \times_fM_{\bot}$ or both, where $M_{\lambda}$ and $M_{\bot}$ are proper slant and anti-invariant submanifolds of $\bar{M}^{2m}$, respectively with $f$ is a non-constant positive smooth function on the first factor. If the warping function $f$ is constant then a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold is said to be a $\mathcal{P}\mathcal{R}$-{\it pseudo-slant product} or {\it trivial product}. \end{definition} \noindent Now, we initiate our study by examining the existence or non-existence of a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of the form $M=M_{\bot} \times_fM_{\lambda}$ in a para-K\"{a}hler manifold $\bar{M}^{2m}$. \begin{theorem}\label{PRpseudothrm1} There doesn't exist any $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of the form $M=M_{\bot} \times_fM_{\lambda}$ in a para-K\"{a}hler manifold $\bar{M}^{2m}$. \end{theorem} \begin{proof} Let us suppose that $M=M_{\bot} \times_fM_{\lambda}$ be any $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then by applying Proposition \ref{propwp}, Eqs. \eqref{phmetric}, \eqref{jtau} and Gauss-Weingarten formulas, we obtain that \begin{align}\label{PRpseudothrm1-1} g(A_{nX}Z, tZ)=X(\ln f)g(tZ, tZ)+g(A_{nZ}X, tZ), \end{align} for any non-degenerate vector field $X \in \Gamma(TM_{\bot})$ and $Z\in \Gamma(TM_{\lambda})$. Employing Corollary \ref{pseudo-cor1} in \eqref{PRpseudothrm1-1}, we deduce that \begin{align}\label{PRpseudothrm1-2} g(A_{nX}Z, tZ)=-\lambda X(\ln f)g(Z, Z)+g(A_{nZ}X, tZ). \end{align} Now, interchanging $Z$ by $tZ$ in above equation, we achieve that \begin{align}\label{PRpseudothrm1-3} g(A_{nX}tZ, t^{2}Z)=-\lambda X(\ln f)g(tZ, tZ)+g(A_{ntZ}X, t^{2}Z). \end{align} Using definition of slant submanifold and Eq. \eqref{pseudo-gtcos} in \eqref{PRpseudothrm1-3}, we get \begin{align}\label{PRpseudothrm1-4} g(A_{nX}tZ, Z)=\lambda X(\ln f)g(Z, Z)+ g(A_{ntZ}X, Z). \end{align} On the other hand, we have from Eqs. \eqref{phantisym} and \eqref{pkahlerdef}-\eqref{jtau} that
\begin{align}\label{PRpseudothrm1-5} g(A_{nZ}X, tZ)=g(\nabla_X{Z}, t^{2}Z)+g(A_{ntZ}X, Z)+g(\nabla_X{tZ}, tZ).
\end{align} Again using definition of slant submanifold and Proposition \ref{propwp} in \eqref{PRpseudothrm1-5}, we deduce that \begin{align}\label{PRpseudothrm1-6} g(A_{nZ}X, tZ)=\lambda X(\ln f) g(Z, Z)+g(A_{ntZ}X,Z)+X(\ln f)g(tZ, tZ). \end{align} Above equation in view of Corollary \ref{pseudo-cor1}, yields \begin{align}\label{PRpseudothrm1-7} g(A_{nZ}X, tZ)=g(A_{ntZ}X, Z). \end{align} From Eqs. \eqref{PRpseudothrm1-2} \eqref{PRpseudothrm1-4} and \eqref{PRpseudothrm1-7}, we conclude that \begin{align}\label{PRpseudothrm1-8} g(A_{nX}Z, tZ)+\lambda X(\ln f)g(Z, Z)=g(A_{nX}Z, tZ)-\lambda X(\ln f)g(Z, Z). \end{align} Thus, we get \begin{align}\label{PRpseudothrm1-9} 2\lambda X(\ln f)g(Z, Z)=0, \end{align} by virtue of Eq. \eqref{shp2form} and the symmetry of $h$. Now, Eq. \eqref{PRpseudothrm1-9} implies that either $\lambda=0$ or $f$ is constant function on $M_{\bot}$, for any non-degenerate vector field $X \in \Gamma(TM_{\bot})$ and $Z\in \Gamma(TM_{\lambda})$. Since $M_{\lambda}$ is a proper slant submanifold so there is only possibility that $f$ must be constant, and this contradicts our supposition. Thus completes the proof of the theorem. \end{proof}
\noindent Next, we first prove an important lemma an then investigate the existence of a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of the form $M=M_{\lambda} \times_fM_{\bot}$ in a para-K\"{a}hler manifold $\bar{M}^{2m}$:
\begin{lemma}\label{pseudo-lem3} If $M=M_{\lambda}\times_{f}M_{\bot}$ be a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of a para-K\"{a}hler manifold $\bar{M}^{2m}$, then we have for any $Z \in \Gamma(TM_{\lambda})$ and $X, Y \in \Gamma(TM_{\bot})$ that \begin{align*} g(A_{\mathcal{P}Y}tZ, X)=-\lambda (Z\ln f) g(X, Y)+g(A_{ntZ}X, Y). \end{align*} \end{lemma} \begin{proof} We know the fact that the structure is para-K\"{a}hler, then by the use of Eqs. \eqref{pkahlerdef}, \eqref{phantisym}, \eqref{shp2form}, \eqref{jtau} and Gauss formula, we achieve that \begin{align}\label{pseudo-lem3-1} g(A_{\mathcal{P}Y}tZ, X)=-g(\nabla_{X}Z, Y)+\bar{g}(\bar\nabla_{X}\mathcal{P}nZ, Y). \end{align} Employing Lemmas \ref{lemmain} and Eq. \eqref{jzeta} in Eq. \eqref{pseudo-lem3-1}, we obtain that \begin{align}\label{pseudo-lem3-2} g(A_{\mathcal{P}Y}tZ, X)=-g(\nabla_{X}Z, Y)+\bar{g}(\bar\nabla_{X}(1-\lambda)Z), Y)+\bar{g}(\bar\nabla_{X}(-ntZ), Y). \end{align} Then by virtue of Gauss-Weingarten formulas, Eq. \eqref{shp2form} and Proposition \ref{propwp} in Eq. \eqref{pseudo-lem3-2}, we completes the proof of lemma. \end{proof}
\begin{theorem}\label{pseudowpthm-2}
Let $M \to \bar{M}^{2m}$ be an isometric immersion of a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold $M$ into a para-K\"{a}hler manifold $\bar{M}^{2m}$. Then $M$ is locally a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of the form $M_{\lambda}\times_{f}M_{\bot}$ if and only if the Weingarten operator of $M$ satisfies \begin{align}\label{pseudo-shpcond} A_{\mathcal{P}Y}tZ - A_{ntZ}Y=-\lambda(Z\ln f)Y, \,\forall \, Y \in \Gamma(\mathfrak{D}^{\bot}), Z, W \in \Gamma(\mathfrak{D}_{\lambda}), \end{align} for some function $\mu$ on $M$ such that $X(\mu)=0$, $X \in \Gamma(\mathfrak{D}^{\bot})$. \end{theorem} \begin{proof} \noindent If $M$ is a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of $\bar{M}^{2m}$. Then from Lemma \ref{pseudo-lem3}, we derive Eq. \eqref{pseudo-shpcond}. Since $f$ is a function on $M_{\lambda}$, setting $\mu = \ln{f}$ implies that $X(\mu)=0$. Conversely, consider that $M$ is proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold of $\bar{M}^{2m}$ such that Eq. \eqref{pseudo-shpcond} holds. Taking inner product of Eq. \eqref{pseudo-shpcond} with $W$ and from lemma \ref{Dlambdageo}, we conclude that the integral manifold $M_{\lambda}$ of $\mathfrak{D}_{\lambda}$ defines totally geodesic foliation in $M$. Then by lemma \ref{Dbotint}, we have that the distribution $\mathfrak{D}_{\bot}$ is integrable if and only if \begin{align}\label{pseudowp-thm1_0} g(A_{\mathcal{P}Y}X, tZ)=g(A_{\mathcal{P}X}Y, tZ), \end{align} for any $X, Y\in \mathfrak{D}^{\bot}$ and $Z \in \mathfrak{D}_{\lambda}$. From the fact that $h$ is symmetric and by employing Eq. \eqref{shp2form} in left hand side of Eq. \eqref{pseudowp-thm1_0}, we arrive at \begin{align}\label{pseudowp-thm1_1} g(A_{\mathcal{P}Y}tZ, X)=g(A_{\mathcal{P}X}Y, tZ). \end{align} Now, by applying the definition of para-K\"{a}hler manifolds, Eqs. \eqref{phstruct}, \eqref{phantisym}, Gauss-Weingarten formulas and Lemma \ref{lemmain} in left hand side of Eq. \eqref{pseudowp-thm1_1}, we derive that \begin{align}\label{pseudowp-thm1_10} g(A_{\mathcal{P}X}Y, tZ)=-\lambda g(\nabla_{X}Z, Y)+g(A_{ntZ}X, Y). \end{align} Now, again taking inner product of Eq. \eqref{pseudo-shpcond} with $X$, we obtain that \begin{align}\label{pseudowp-thm1_11} g(A_{\mathcal{P}Y}tZ , X)=-\lambda g(Z(\mu)Y, X)+g(A_{ntZ}Y, X). \end{align} Therefore from Eqs. \eqref{pseudowp-thm1_0}, \eqref{pseudowp-thm1_10},\eqref{pseudowp-thm1_11} and property of covariant differentiation formula, we deduce that \begin{align*} g(h_{\bot}(X, Y), Z) =-g(Z(\mu)X, Y)=-g(X, Y )g(\nabla{\mu}, Z). \end{align*} This implies $h_{\bot}(X, Y)=-g(X, Y )\nabla{\mu}$, where $h_{\bot}$ is a second fumdamental form of $\mathfrak{D}_{\bot}$ in $M$ and $\nabla\mu$ is gradient of $\mu = \ln{f}$. Hence, the integrable manifold of $\mathfrak{D}_{\bot}$ is totally umbilical submanifold in $M$ and its mean curvature is non-zero and parallel and $X(\mu)=0$ for all $X \in \Gamma(\mathfrak{D}_{\bot})$. Thus, by \cite{Hiepko1979} we achieve that $M$ is a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of $\bar{M}^{2m}$. This completes the proof of the theorem. \end{proof}
\noindent Here, we derive a condition for a $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of the form $M_{\lambda}\times_{f}M_{\bot}$ to be a $\mathcal{P}\mathcal{R}$-pseudo-slant product:
\begin{proposition}\label{pseudoprp-1} A $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of the form $M=M_{\lambda}\times_{f}M_{\bot}$ of a para-K\"{a}hler manifold $\bar{M}^{2m}$ is locally a $\mathcal{P}\mathcal{R}$-pseudo-slant product if and only if the expression $g(h(X,Y), ntZ)$ vanishes, for any $X, Y \in \Gamma(\mathfrak{D}^{\bot})$ and $Z \in \Gamma(\mathfrak{D}_{\lambda})$. \end{proposition} \begin{proof} We have from Eq. \eqref{gauss} that $g(\nabla_{X}Z,Y)=\bar{g}(\bar\nabla_{X}Z, Y)$. Employing Eqs. \eqref{phstruct},\eqref{phantisym}, \eqref{jtau}, \eqref{jzeta} and \eqref{shp2form} in previous expression, we obtain that \begin{align*} g(\nabla_{X}Z,Y)=& -\bar{g}(\mathcal{P}\bar\nabla_{X}Z, \mathcal{P}Y)=-\bar{g}(\bar\nabla_{X}(tZ), \mathcal{P}Y)-\bar{g}(\bar\nabla_{X}(nZ), \mathcal{P}Y) \\ &=\bar{g}(\bar\nabla_{X}(t^{2}Z), Y)+\bar{g}(\bar\nabla_{X}(ntZ), Y)-\bar{g}(\bar\nabla_{X}nZ, \mathcal{P}Y). \end{align*} Using Eq. \eqref{weingarten}, \eqref{pseudo-tsqr} and the fact $g(Y, Z)=0$, we have from above last equation that \begin{align}\label{1prp-1} g(\nabla_{X}Z,Y)&= \lambda \bar{g}(\bar\nabla_{X}Z,Y)-g(h(X,Y), ntZ)+g(\nabla^{\bot}_{X}nZ,\mathcal{P}Y). \end{align} Thus, from Eq. \eqref{1prp-1} and Proposition \eqref{propwp}, we conclude that \begin{align}\label{2prp-1} (1-\lambda)(Z \ln f)g(X, Y)= -g(h(X,Y), ntZ)+g(\nabla^{\bot}_{X}nZ, \mathcal{P}Y). \end{align} Interchanging $X$ and $Y$ in Eq. \eqref{2prp-1} and then subtracting from \eqref{2prp-1}, we obtain that \begin{align}\label{3prp-1} g(\nabla^{\bot}_{X}nZ,\mathcal{P}Y) = g(\nabla^{\bot}_{Y}nZ,\mathcal{P}X). \end{align} Furthermore, from Eqs. \eqref{phstruct}, \eqref{phantisym}, \eqref{jtau} and Gauss-Weingarten formulas, we derive that \begin{align}\label{4prp-1} g(\nabla^{\bot}_{X}nZ,\mathcal{P}Y) &=-(Z\ln f)g(X, Y)-\bar{g}(\bar\nabla_{X}tZ, \mathcal{P}Y). \end{align} Now again by interchanging $X$ with $Y$ in Eq. \eqref{4prp-1} we conclude that Eq. \eqref{3prp-1} hold if and only if \begin{align}\label{5prp-1} \bar{g}(\bar\nabla_{X}tZ, \mathcal{P}Y)= -\bar{g}(\bar\nabla_{X}\mathcal{P}Y, tZ)=0 \end{align} Using Gauss-Weingarten formulas and Eqs. \eqref{phstruct},\eqref{phantisym}, \eqref{jtau}, \eqref{pseudo-tsqr} in Eq. \eqref{5prp-1}, we get \begin{align}\label{6prp-1}
\lambda (Z\ln f) g(X, Y) + g(h(X,Y), ntZ)=0 \end{align} Thus from Eq. \eqref{6prp-1} we can say that $f$ is constant if and only if $g(h(X,Y), ntZ)=0$. Since, $M_{\lambda}$ is proper slant submanifold and $Z$ is non-null vector field. This completes the proof. \end{proof} \section{Example}\label{exa} In addition to results in Sect. \ref{prwps}, here we present an numerical example illustrating $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of the form $M=M_{\lambda}\times_{f}M_{\bot}$ in a para-K\"{a}hler manifold.
\noindent Let $\bar M=\mathbb{R}^6$ be a $6$-dimensional manifold with the standard Cartesian coordinates $(\bar{x}_{1}, \bar{x}_{2}, \bar{x}_{3}, \bar{x}_{4}, \bar{x}_{5}, \bar{x}_{6})$. Define a structure $(\mathcal{P},\bar{g})$ on $\bar M$ by
\begin{align}\label{phstrcex}
\mathcal{P} e_{1}=e_{4},\ \mathcal{P} e_{2}=e_{5},\ \mathcal{P} e_{3}=e_{6},\ \mathcal{P} e_{4}=e_{1},\ \mathcal{P} e_{5}=e_{2}, \mathcal{P} e_{6}= e_{3},
\end{align} \begin{align}\label{phmetricex}
\bar{g}=\sum_{i=1}^{3}(d \bar x_{i})^{2}-\sum_{j=4}^{6}(d \bar x_{j})^{2}, \end{align} where $e_{1} = \dfrac{\partial}{\partial \bar x_{1}},\,e_{2} = \dfrac{\partial}{\partial \bar x_{2}},\,e_{3} = \dfrac{\partial}{\partial \bar x_{3}},\,e_{4} = \dfrac{\partial}{\partial \bar x_{4}}, e_{5} = \dfrac{\partial}{\partial \bar x_{5}}$ and $e_{6} = \dfrac{\partial}{\partial \bar x_{6}}$. By straightforward calculations, one verifies that the structure is an almost para-Hermitian manifold. For Levi-Civita connection $\bar{\nabla}$ with respect to $\bar{g}$, we readily conclude that the manifold $(\bar M, \mathcal{P},\bar{g})$ is a para-K\"{a}hler manifold. \noindent Now, let $M$ is an isometrically immersed smooth submanifold in $\mathcal{R}^{6}$ defined by
\begin{align}\label{phstrucsub} \Omega(x_{1}, x_{2}, x_{3})=\left(x_{1}, x_{1}\cos(x_{2}), x_{1}\sin(x_{2}), x_{3}, k_{1}, k_{2}\right)
\end{align}
where $k_{1}, k_{2}$ are constants and $x_{1} \in \mathbb{R}_{+},\ x_{2}\in(0,\pi/2)$ and $x_{3}$ is non zero. Then the $TM$ spanned by the vectors
\begin{align} \label{phtanbundl} Z_{x_{1}}&=\frac{\partial}{\partial \bar x_{1}}+\cos(x_{2})\frac{\partial}{\partial \bar x_{2}}+\sin(x_{2})\frac{\partial}{\partial \bar x_{3}}, \nonumber\\ Z_{x_{2}}&=-x_{1}\sin(x_{2})\frac{\partial}{\partial \bar x_{2}}+x_{1}\cos(x_{2})\frac{\partial}{\partial \bar x_{3}}, \\ Z_{x_{3}}&=\frac{\partial}{\partial \bar x_{4}}, \nonumber
\end{align}
where $Z_{x_{1}}, Z_{x_{2}}, Z_{x_{3}} \in \Gamma(TM)$.
Using Eq. \eqref{phstrcex}, we obtain that \begin{align}\label{phphibundl} \mathcal{P}(Z_{x_{1}})&=\frac{\partial}{\partial \bar x_{4}}+\cos(x_{2})\frac{\partial}{\partial \bar x_{5}}+\sin(x_{2})\frac{\partial}{\partial \bar x_{6}}, \nonumber \\ \mathcal{P}(Z_{x_{2}})&=-x_{1}\sin(x_{2})\frac{\partial}{\partial \bar x_{5}}+x_{1}\cos(x_{2})\frac{\partial}{\partial \bar x_{6}}, \\ \mathcal{P}(Z_{x_{3}})&=\frac{\partial}{\partial \bar x_{1}}.\nonumber \end{align} From Eqs. \eqref{phtanbundl} and \eqref{phphibundl} we can find that $\mathfrak{D_{\lambda}}$ is a proper slant distribution defined by span\{$ Z_{x_{1}}, Z_{x_{3}}$\} with slant coefficient $\lambda=\frac{1}{\sqrt{2}}$ and $\mathfrak{D}^{\bot}$ is an anti-invariant distribution defined by span\{$Z_{x_{2}}$\} with dimension not equal to zero. So, $M$ turn into a proper $\mathcal{P}\mathcal{R}$-pseudo-slant submanifold. Here, the induced pseudo-Riemannian non-degenerate metric tensor $g$ of $M$ is given by \begin{align*} [g_{ij}]=\begin{bmatrix}
2&0&0\\
0&{x}_{1}^{2}&0\\
0&0&-1 \end{bmatrix}, \end{align*} that is, $g=2dx_{1}^{2}-dx_{3}^{2}+x_{1}^2\{dx_{2}^2\}= g_{M_{\lambda}}+_{f^{2}}g_{M_{\bot}}$. Thus, $M$ is a $3$-dimensional $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifold of $\mathcal{R}^{6}$ with wrapping function $f=x_{1}$.
\end{document} | arXiv | {
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\begin{document}
\begin{abstract} We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction under the same regularity assumptions on the initial data required for the integration of the corresponding nonlinear Schr\"odinger limit system. In contrast to previous works we do not employ any asymptotic/multiscale expansion of the solution. This allows us to derive uniform convergent schemes under far weaker regularity assumptions on the exact solution. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie, respectively, Strang splitting in the nonlinear Schr\"odinger limit. \end{abstract}
\subjclass{35C20 \and 65M12 \and 35L05} \keywords{}
\maketitle
\section{Introduction} Cubic Klein-Gordon equations \begin{equation} \begin{aligned}\label{eq:kgr} & c^{-2} \partial_{tt} z - \Delta z + c^2 z = \vert z\vert^{2} z, \quad z(0,x) = z_0(x),\quad \partial_t z(0,x) = c^2 z'_0(x)\\ \end{aligned} \end{equation} are extensively studied numerically in the relativistic regime $c=1$, see \cite{Gau15,StVaz78} and the references therein. In contrast, the so-called ``non-relativistic regime'' $c\gg 1$ is numerically much more involved due to the highly-oscillatory behavior of the solution. We refer to \cite{EFHI09,HLW} and the references therein for an introduction and overview on highly-oscillatory problems.\\
Analytically, the non-relativistic limit regime $c\to \infty$ is well understood nowadays: The exact solution $z$ of \eqref{eq:kgr} allows (for sufficiently smooth initial data) the expansion \[ z(t,x) = \frac{1}{2}\left( \mathrm{e}^{ic^2 t} u_{\ast,\infty}(t,x) + \mathrm{e}^{-ic^2t} \overline{v}_{\ast,\infty}(t,x) \right) + \mathcal{O}(c^{-2}) \]
on a time-interval uniform in $c$, where $(u_{\ast,\infty}, v_{\ast,\infty})$ satisfy the cubic Schr\"odinger limit system \begin{equation}\label{NLSlimit} \begin{aligned} i \partial_t u_{\ast, \infty} &=& \frac{1}{2} \Delta u_{\ast,\infty} + \frac{1}{8}\big(\left \vert u_{\ast,\infty}\right\vert^2 + 2\left\vert v_{\ast,\infty}\right\vert^2\big) u_{\ast,\infty}\qquad u_{\ast,\infty}(0) = \varphi - i \gamma \\ i \partial_t v_{\ast, \infty} &=& \frac{1}{2} \Delta v_{\ast,\infty} + \frac{1}{8}\big(\left \vert v_{\ast,\infty}\right\vert^2 + 2\left\vert u_{\ast,\infty}\right\vert^2\big) v_{\ast,\infty},\qquad v_{\ast,\infty}(0) = \overline{\varphi} - i \overline{\gamma} \end{aligned} \end{equation} with initial values \begin{align*} z(0,x) \stackrel{c\to\infty}{\longrightarrow} \gamma(x) \quad \text{and}\quad c^{-1}\left(c^2-\Delta\right)^{-1/2} \partial_t z(0,x) \stackrel{c\to\infty}{\longrightarrow} \varphi(x), \end{align*} see \cite[Formula (1.3)]{MaNak02} and for the periodic setting \cite[Formula (37)]{FS13}.\\
Also numerically, the non-relativistic limit regime $c \gg 1$ has recently gained a lot of attention: Gautschi-type methods (see \cite{HoLu99}) are analyzed in \cite{BG}. However, due to the difficult structure of the problem they suffer from a severe time-step restriction as they introduce a global error of order $c^4 \tau^2$ which requires the CFL-type condition $c^2 \tau <1$. To overcome this difficulty so-called limit integrators which reduce the highly-oscillatory problem to the corresponding non-oscillatory limit system (i.e., $c\to \infty$ in \eqref{eq:kgr}) as well as uniformly accurate schemes based on multiscale expansions were introduced in \cite{FS13} and \cite{BaoZ,ChC}. In the following we give a comparison of these methods focusing on their convergence rates and regularity assumptions:
\emph{Limit integrators:} Based on the modulated Fourier expansion of the exact solution (see \cite{CoHaLu03,HLW}) numerical schemes for the Klein-Gordon equation in the strongly non-relativistic limit regime $c \gg 1$ were introduced in \cite{FS13}. The benefit of this ansatz is that it allows us to reduce the highly-oscillatory problem \eqref{eq:kgr} to the integration of the corresponding \emph{non-oscillatory limit Schr\"odinger equation} \eqref{NLSlimit}. The latter can be carried out very efficiently without imposing any $c-$dependent step-size restriction. However, as this approach is based on the asymptotic expansion of the solution with respect to $c^{-2}$, it only allows error bounds of order $$\mathcal{O}(c^{-2} + \tau^2)$$ when integrating the limit system with a second-order method. Henceforth, the limit integration method only yields an accurate approximation of the exact solution for sufficiently large values of $c$.
\emph{Uniformly accurate schemes based on multiscale expansions:} Uniformly accurate schemes, i.e., schemes that work well for small as well as for large values of $c$ were recently introduced for Klein-Gordon equations in \cite{BaoZ,ChC}. The idea is thereby based on a multiscale expansion of the exact solution. We also refer to \cite{BDZ14} for the construction and analysis in the case of highly-oscillatory second-order ordinary differential equations. The multiscale time integrator (MTI) pseudospectral method derived in \cite{BaoZ} allows two independent error bounds at order $$ \mathcal{O}(\tau^2 + c^{-2})\quad \text{and} \quad \mathcal{O}(\tau^2 c^2) $$ for sufficiently smooth solutions. These error bounds immediately imply that the MTI method converges uniformly in time with linear convergence rate at $\mathcal{O}(\tau)$ for all $c \geq 1$ thanks to the observation that $ \mathrm{min}(c^{-2}, \tau^2c^2) \leq \tau $. However, the optimal quadratic convergence rate at $\mathcal{O}(\tau^2)$ is only achieved in the regimes when either $0 < c = \mathcal{O}(1)$ (i.e., the relativistic regime) or $ \frac{1}{\tau} \leq c $ (i.e., the strongly non-relativistic regime). In the context of ordinary differential equations similar error estimates were established for MTI methods in \cite{BDZ14}. The first-order uniform convergence of the MTI-FP method \cite{BaoZ} holds for sufficiently smooth solutions: First-order convergence in time holds in $H^2$ uniformly in $c$ for solutions in $H^7$ with $\sup_{0\leq t \leq T} \Vert z(t)\Vert_{H^{7}} + c^{-2}\Vert \partial_t z(t) \Vert_{H^6} \leq 1$ (see \cite[Theorem 4.1]{BaoZ}). First-order uniform convergence also holds in $H^1$ under weaker regularity assumptions, namely for solutions in $H^6$ satisfying $ \sup_{0\leq t \leq T} \Vert z(t)\Vert_{H^{6}} + c^{-2}\Vert \partial_t z(t) \Vert_{H^5} \leq 1$ if an additional CFL-type condition is imposed in space dimensions $d=2,3$ (see \cite[Theorem 4.9]{BaoZ}).
A second-order uniformly accurate scheme based on the \emph{Chapman-Enskog expansion} was derived in \cite{ChC} for the Klein-Gordon equation. Thereby, to control the remainders in the expansion, second-order uniform convergence in $H^r$ ($r>d/2$) requires sufficiently smooth solutions with in particular $z(0) \in H^{r+10}$. Also, due to the expansion, the \emph{problem needs to be considered in $d+1$ dimensions}.\\
We establish exponential-type integrators which converge with \emph{second-order accuracy in time uniformly in all $c>0$}. In comparison, the multiscale time integrators (MTI) derived in \cite{BaoZ,BDZ14} only converge with first-order accuracy uniformly in all $c \geq 1$. This is due to the fact that the MTI methods are based on the multiscale decomposition $$ z(t,x) = \mathrm{e}^{it c^2} z_{+}^n(t,x) + \mathrm{e}^{-i tc^2} \overline{z^n_{-}}(t,x) + r^n(t,x) $$ which leads to a coupled \emph{second-order system in time} in the $c^{2}$-frequency waves $z_{\pm}^n$ and the rest frequency waves $r^n$ (cf. \cite[System (2.4)]{BaoZ}) and only allows numerical approximations at order $\mathcal{O}(\tau^2 + c^{-2})$ and $\mathcal{O}(\tau^2 c^2)$.
In contrast to \cite{BaoZ,ChC,FS13} \emph{we do not employ any asymptotic/multiscale expansion} of the solution, but construct exponential-type integrators based on the following strategy: \begin{itemize} \item[1.] In a first step we reformulate the Klein-Gordon equation \eqref{eq:kgr} as a coupled \emph{first-order system in time} via the transformations \[ u = z - i \big(c\sqrt{-\Delta +c^2}\big)^{-1} \partial_t z, \quad v = \overline{z}-i \big(c\sqrt{-\Delta +c^2}\big)^{-1} \partial_t \overline{z}. \] \item[2.] In a second step we rescale the coupled first-order system in time by looking at the so-called ``twisted variables'' \[ u_\ast(t) = \mathrm{e}^{ic^2 t} u(t), \qquad v_\ast(t) = \mathrm{e}^{-ic^2t}v(t). \] This essential step will later on allow us to treat the highly-oscillatory phases $\mathrm{e}^{\pm ic^2 t}$ and their interaction explicitly. \item[3.] Finally, we iterate Duhamel's formula in $(u_\ast(t),v_\ast(t))$ and integrate the interactions of the highly-oscillatory phases exactly by approximating only the slowly varying parts. \end{itemize} This strategy in particular allows us to construct uniformly accurate exponential-type integrators up to order two which in addition asymptotically converge to the classical splitting approximation of the corresponding nonlinear Schr\"odinger limit system \eqref{NLSlimit} given in \cite{FS13}. More precisely, the second-order exponential-type integrator converges for $c \to \infty$ to the classical Strang splitting scheme
\begin{equation}\label{limitscheme2a} \begin{aligned} u_{\ast,\infty}^{n+1} &= \mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2} } \mathrm{e}^{-i \tau \frac{3}{8}\vert\mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2} } v_{\ast, \infty}^n\vert^2} \mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2} } u_{\ast,\infty}^n,\qquad u_{\ast,\infty}^0 = \varphi - i \gamma \end{aligned} \end{equation} associated to the nonlinear Schr\"odinger limit system \eqref{NLSlimit} (see also Remark \ref{remarkStrangOk19}) where for simplicity we assumed that $z$ is real-valued such that $u_\ast = v_\ast$. A similar result holds for the asymptotic convergence of the first-order exponential-type integration scheme towards the classical Lie splitting approximation (see also Remark \ref{rem:limit1}).
In \cite{FS13} the Strang splitting \eqref{limitscheme2a} is precisely proposed for the numerical approximation of non-relativistic Klein-Gordon solutions. However, in contrast to the uniformly accurate exponential-type integrators derived here, the scheme in \cite{FS13} only yields second-order convergence in the strongly non-relativistic regime $c > \frac{1}{\tau}$ due to its error bound at order $\mathcal{O}(\tau^2+c^{-2})$.\\
The main novelty in this work thus lies in the development and analysis of efficient and robust exponential-type integrators for the cubic Klein-Gordon equation \eqref{eq:kgr} which \begin{itemize}
\item[$\circ$] allow second-order convergence uniformly in all $c>0$ without adding an extra dimension to the problem.
\item[$\circ$] resolve the solution $z$ in the relativistic regime $c = 1$ as well as in the non-relativistic regime $c \to \infty$ without any $c-$dependent step-size restriction under the same regularity assumptions as needed for the integration of the corresponding limit system.
\item[$\circ$] in addition to converging uniformly in $c$, converge asymptotically to the classical Lie, respectively, Strang splitting for the corresponding nonlinear Schr\"odinger limit system \eqref{NLSlimit} in the non-relativistic limit $c \to \infty$.\\ \end{itemize}
Our strategy also applies to general polynomial nonlinearities $f(z) = \vert z\vert^{2p}z$ with $p \in \mathbb{N}$. However, for notational simplicity, we will focus only on the cubic case $p=1$. Furthermore, for practical implementation issues we impose periodic boundary conditions, i.e., $x \in \mathbb{T}^d$.\\
We commence in Section \ref{sec:scale} with rescaling the Klein-Gordon equation \eqref{eq:kgr} which then allows us to construct first-, and second-order schemes that converge uniformly in $c$, see Section \ref{sec:scheme1} and \ref{sec:scheme2}, respectively.
\section{Scaling for uniformly accurate schemes}\label{sec:scale} In a first step we reformulate the Klein-Gordon equation \eqref{eq:kgr} as a first-order system in time which allows us to resolve the limit-behavior of the solution, i.e., its behavior for $c\to \infty$ (see also \cite{MaNak02,FS13}).
For a given $c > 0$, we define the operator \begin{align}\label{nab} \langle \nabla \rangle_c = \sqrt{- \Delta + c^2}. \end{align} With this notation, equation \eqref{eq:kgr} can be written as \begin{equation}\label{eq:kgrr} \partial_{tt} z + c^2 \langle \nabla \rangle_c^2 z = c^2 f(z) \end{equation} with the nonlinearity \[
f(z) = \vert z \vert^2 z.
\] In order to rewrite the above equation as a first-order system in time, we set \begin{equation}\label{eq:uv} u = z - i c^{-1}\langle \nabla \rangle_c^{-1} \partial_t z , \qquad v = \overline z - i c^{-1}\langle \nabla \rangle_c^{-1} \partial_t \overline z \end{equation} such that in particular \begin{equation}\label{eq:zuv} z = \frac12 (u + \overline{v}). \end{equation} \begin{rem}\label{rem:realz} If $z$ is real, then $u \equiv v$. \end{rem} A short calculation shows that in terms of the variables $u$ and $v$ equation \eqref{eq:kgrr} reads \begin{equation} \label{eq:NLSc} \begin{array}{rcl} i \partial_t u &=& -c\langle \nabla \rangle_c u + c\langle \nabla \rangle_c^{-1} f(\textstyle \frac12 (u + \overline v)), \\[2ex] i \partial_t v &=& -c\langle \nabla \rangle_c v + c\langle \nabla \rangle_c^{-1} f(\textstyle \frac12 (\overline u + v)) \end{array} \end{equation} with the initial conditions (see \eqref{eq:kgr}) \begin{equation} \label{eq:BCc} u(0) = z(0) -ic^{-1}\langle \nabla \rangle_c^{-1} z'(0) , \quad \mbox{and}\quad v(0) =\overline{z(0)} -ic^{-1}\langle \nabla \rangle_c^{-1} \overline{z'(0)}. \end{equation} Formally, the definition of $\langle \nabla \rangle_c$ in \eqref{nab} implies that \begin{align}\label{exnab} c\langle \nabla \rangle_c \quad =\quad c^2 \quad + \quad \text{``lower order terms in $c$''}. \end{align} This observation motivates us to look at the so-called ``twisted variables'' by filtering out the highly-oscillatory parts explicitly: More precisely, we set \begin{align}\label{psi}
u_\ast( t) = \mathrm{e}^{-ic^2 t} u(t), \qquad v_\ast(t) = \mathrm{e}^{-ic^2 t} v(t). \end{align} This idea of ``twisting'' the variable is well known in numerical analysis, for instance in the context of the modulated Fourier expansion \cite{CoHaLu03,HLW}, adiabatic integrators \cite{LJL05,HLW} as well as Lawson-type Runge--Kutta methods \cite{Law67}. In the case of ``multiple high frequencies'' it is also widely used in the analysis of partial differential equations in low regularity spaces (see for instance \cite{Bour93}) and has been recently successfully employed numerically for the construction of low-regularity exponential-type integrators for the KdV and Schr\"odinger equation, see \cite{HS16,OS16}. \\
In terms of $(u_\ast,v_\ast)$ system \eqref{eq:NLSc} reads (cf. \cite[Formula (2.1)]{MaNak02}) \begin{equation}\label{eq:ua1} \begin{aligned}
i\partial_t u_\ast &=& - \mathcal{A}_c u_\ast+ c \langle \nabla \rangle_c^{-1} \mathrm{e}^{-ic^2t}f \left( \textstyle \frac12 ( \mathrm{e}^{ic^2t} u_\ast+ \mathrm{e}^{-ic^2t} \overline{v_\ast})\right)\\
i\partial_t v_\ast &=& - \mathcal{A}_c v_\ast+ c \langle \nabla \rangle_c^{-1} \mathrm{e}^{-ic^2t} f \left( \textstyle \frac12 ( \mathrm{e}^{ic^2t} v_\ast+ \mathrm{e}^{-ic^2t} \overline{u_\ast})\right) \end{aligned} \end{equation} with the leading operator \begin{equation}\label{Ac} \mathcal{A}_c : = c\langle \nabla \rangle_c - c^2. \end{equation} \begin{rem}\label{rem:adstar} The advantage of looking numerically at $(u_\ast,v_\ast)$ instead of $(u,v)$ lies in the fact that the leading operator $-c\langle \nabla \rangle_c$ in system \eqref{eq:NLSc} is of order $c^2$ (see \eqref{exnab}) whereas its counterpart $-\mathcal{A}_c$ in system \eqref{eq:ua1} is ``of order one in $c$'' (see Lemma \ref{lem:boundAc} below). \end{rem} In the following we construct integration schemes for \eqref{eq:ua1} based on Duhamel's formula \begin{equation}\label{du0} \begin{aligned}
u_\ast(t_n+\tau) & = \mathrm{e}^{i \tau \mathcal{A}_c} u_\ast(t_n)\\ & - i c \langle \nabla \rangle_c^{-1} \int_0^\tau \mathrm{e}^{i(\tau-s) \mathcal{A}_c}\mathrm{e}^{-ic^2(t_n+s)}f \left( \textstyle \frac12 ( \mathrm{e}^{ic^2(t_n+s)} u_\ast(t_n+s)+ \mathrm{e}^{-ic^2(t_n+s)} \overline{v_\ast}(t_n+s))\right) \mathrm{d}s,\\
v_\ast(t_n+\tau)& = \mathrm{e}^{i \tau \mathcal{A}_c} v_\ast(t_n)\\ & - i c \langle \nabla \rangle_c^{-1} \int_0^\tau \mathrm{e}^{i(\tau-s) \mathcal{A}_c}\mathrm{e}^{-ic^2(t_n+s)}f \left( \textstyle \frac12 ( \mathrm{e}^{ic^2(t_n+s)} v_\ast(t_n+s)+ \mathrm{e}^{-ic^2(t_n+s)} \overline{u_\ast}(t_n+s))\right) \mathrm{d}s. \end{aligned} \end{equation} Thereby, to guarantee \emph{uniform convergence with respect to $c$} we make the following important observations. We define the Sobolev norm on $\mathbb{T}^d$ by the formula $$
\Vert u \Vert_r^2 = \sum_{k \in \mathbb{Z}^d} (1 + |k|^{2})^r | \hat u_kÊ|^2, \quad \mbox{where} \quad \hat u_k = \frac{1}{(2\pi)^d} \int_{\mathbb{T}^d} u(x) e^{i k \cdot x} d x, $$
where for $k = (k_1,\ldots, k_d) \in \mathbb{Z}^d$, we set $k \cdot x = k_1 x_1 + \cdots k_d x_d$ and $|k|^2 = k\cdot k$. Moreover, for a given linear bounded operator $L$ we denote by $\Vert L \Vert_r$ its corresponding induced norm. \begin{lem}[Uniform bound on the operator $\mathcal{A}_c$] \label{lem:boundAc}\label{lem:bAc} For all $c \in \mathbb{R}$ we have that \begin{align}\label{boundAc} \Vert \mathcal{A}_c u \Vert_r \leq \frac12\Vert u \Vert_{r+2}. \end{align} \end{lem} \begin{proof}
The operator $\mathcal{A}_c$ acts a the Fourier multiplier $(\mathcal{A}_c)_k = c^2 - c\sqrt{c^2+|k|^2}$, $k \in \mathbb{Z}^d$. Thus, the assertion follows thanks to the bound \begin{equation*} \begin{aligned}
\Vert \mathcal{A}_c u \Vert_r^2 &= \sum_{k\in \mathbb{Z}^d} (1 + \vert k \vert^{2})^r \left( c\sqrt{c^2+|k|^2} - c^2 \right)^2 \vert \hat u_k\vert^2 \leq \sum_{k\in \mathbb{Z}^d}(1 + \vert k \vert^{2})^r \left( \frac{|k|^2}{2}\right)^2 \vert \hat u_k\vert^2, \end{aligned} \end{equation*} where we have used that $\sqrt{1+x^2} \leq 1+ \frac{1}{2}x^2$ for all $x \in \mathbb{R}$. \end{proof} \begin{lem}\label{lem:expo} For all $t \in \mathbb{R}$ we have that \begin{equation}\label{approx1} \begin{aligned} \Vert \mathrm{e}^{i t \mathcal{A}_c} \Vert_r = 1 \quad \mbox{and}\quad \left\Vert \left (\mathrm{e}^{-i t \mathcal{A}_c} - 1\right) u\right \Vert_r \leq \frac12 \vert t \vert \Vert u \Vert_{r+2}. \end{aligned} \end{equation} \end{lem} \begin{proof} The first assertion is obvious and the second follows thanks to the estimate $\vert (\mathrm{e}^{ix}-1)\vert \leq \vert x\vert$ which holds for all $x \in \mathbb{R}$ together with the essential bound on the operator $\mathcal{A}_c$ given in \eqref{boundAc}. \end{proof} In particular, the time derivatives $(u_\ast'(t),v_\ast'(t))$ can be bounded uniformly in $c$. \begin{lem}[Uniform bounds on the derivatives $(u_\ast'(t),v_\ast'(t))$]\label{lem:upc} Fix $r>d/2$. Solutions of \eqref{eq:ua1} satisfy \begin{equation}\label{approx2} \begin{aligned} \Vert u_\ast(t_n+s) - u_\ast(t_n) \Vert_r & \leq \frac12 \vert s \vert \Vert u_\ast(t_n) \Vert_{r+2} + \frac18 \vert s \vert \sup_{0 \leq \xi \leq s} \big( \Vert u_\ast(t_n+\xi)\Vert_r+ \Vert v_\ast(t_n+\xi)\Vert_r\big)^3,\\ \Vert v_\ast(t_n+s) - v_\ast(t_n) \Vert_r & \leq \frac12 \vert s \vert \Vert v_\ast(t_n) \Vert_{r+2} + \frac18 \vert s \vert \sup_{0 \leq \xi \leq s} \big( \Vert u_\ast(t_n+\xi)\Vert_r+ \Vert v_\ast(t_n+\xi)\Vert_r\big)^3. \end{aligned} \end{equation} \end{lem} \begin{proof} The assertion follows thanks to Lemma \ref{lem:expo} together with the bound \begin{equation}\label{cnabm} \Vert c \langle \nabla \rangle_c^{-1}\Vert_r \leq 1 \end{equation} which implies by Duhamel's perturbation formula \eqref{du0} that \begin{equation*} \begin{aligned} \Vert u_\ast(t_n+s) - u_\ast(t_n) \Vert_r & \leq \vert s \vert \Vert \mathcal{A}_c u_\ast(t_n) \Vert_r + \frac18 \vert s \vert \Vert c \langle \nabla \rangle_c^{-1} \Vert_r \sup_{0 \leq \xi \leq s} \big( \Vert u_\ast(t_n+\xi)\Vert_r+ \Vert v_\ast(t_n+\xi)\Vert_r\big)^3\\ & \leq \frac12 \vert s \vert \Vert u_\ast(t_n) \Vert_{r+2} + \frac18 \vert s \vert \sup_{0 \leq \xi \leq s} \big( \Vert u_\ast(t_n+\xi)\Vert_r+ \Vert v_\ast(t_n+\xi)\Vert_r\big)^3. \end{aligned} \end{equation*} Similarly we can establish the bound on the derivative $v_\ast'(t)$. \end{proof}
We will also employ the so-called ``$\varphi_j$ functions'' given in the following Definition. \begin{defn}[$\varphi_j$ functions \cite{HochOst10}] \label{def:phi} Set \[ \varphi_0(z ) := \mathrm{e}^{z}\qquad\text{and}\qquad \varphi_k(z) := \int_0^1 \mathrm{e}^{(1-\theta)z} \frac{\theta^{k-1}}{(k-1)!}\mathrm{d} \theta, \quad k \geq 1 \] such that in particular \begin{equation*} \begin{aligned} \varphi_0(z ) = \mathrm{e}^{z}, \qquad \varphi_1(z) = \frac{\mathrm{e}^{z} - 1}{z}, \qquad \varphi_2(z) = \frac{\varphi_0(z) - \varphi_1(z)}{z}. \end{aligned} \end{equation*} \end{defn} In the following we assume local-wellposedness (LWP) of \eqref{eq:ua1} in $H^r$. \begin{ass} Fix $r>d/2$ and assume that there exists a $T_r = T>0$ such that the solutions $(u_\ast(t),v_\ast(t))$ of \eqref{eq:ua1} satisfy \begin{align*} \sup_{0 \leq t \leq T} \Vert u_\ast(t) \Vert_{r}+ \Vert v_\ast(t) \Vert_{r} \leq M \end{align*} uniformly in $c$. \end{ass} \begin{rem} The previous assumption holds under the following condition on the initial data $$ \Vert z(0) \Vert_{r} + \Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0)\Vert_{r} \leq M_0$$ where $M_0$ does not depend on $c$ as can be easily proved from the formulation \eqref{du0}. \end{rem}
\section{A first-order uniformly accurate scheme}\label{sec:scheme1} In this section we derive a first-order exponential-type integration scheme for the solutions $(u_\ast,v_\ast)$ of \eqref{eq:ua1} which allows \emph{first-order uniform time-convergence with respect to $c$}. The construction is thereby based on Duhamel's formula \eqref{du0} and the essential estimates in Lemma \ref{lem:bAc}, \ref{lem:expo} and \ref{lem:upc}. For the derivation we will for simplicity assume that $z$ is real, which (by Remark \ref{rem:realz}) implies that $u = v$ such that system \eqref{eq:ua1} reduces to \begin{equation}\label{eq:ua} i\partial_t u_\ast=- \mathcal{A}_c u_\ast+ \frac{1}{8} c \langle \nabla \rangle_c^{-1} \mathrm{e}^{-ic^2t} \left( \mathrm{e}^{ic^2t} u_\ast+ \mathrm{e}^{-ic^2t} \overline{u_\ast}\right)^3 \end{equation} with mild-solutions \begin{equation}\label{du} \begin{aligned} u_\ast(t_n+\tau)& = \mathrm{e}^{i \tau \mathcal{A}_c} u_\ast(t_n)\\ & - \frac{i}{8} c \langle \nabla \rangle_c^{-1} \int_0^\tau \mathrm{e}^{i(\tau-s) \mathcal{A}_c} \mathrm{e}^{-i c^2 (t_n+s)} \left( \mathrm{e}^{ic^2(t_n+s)} u_\ast(t_n+s) + \mathrm{e}^{-ic^2(t_n+s)} \overline{u_\ast}(t_n+s)\right)^3 \mathrm{d}s. \end{aligned} \end{equation}
\subsection{Construction} In order to derive a first-order scheme, we need to impose additional regularity assumptions on the exact solution $u_\ast(t)$ of \eqref{eq:ua}. \begin{ass}\label{ass:reg1} Fix $r>d/2$ and assume that $u_\ast \in \mathcal{C}([0,T];H^{r+2}(\mathbb{T}^d))$ and in particular \begin{align*} \sup_{0 \leq t \leq T} \Vert u_\ast(t) \Vert_{r+2}\leq M_2 \quad \text{uniformly in $c$}. \end{align*} \end{ass}
Applying Lemma \ref{lem:expo} and Lemma \ref{lem:upc} in \eqref{du} allows us the following expansion \begin{equation}\label{du1} \begin{aligned} u_\ast(t_n +\tau) & = \mathrm{e}^{i \tau \mathcal{A}_c} u_\ast(t_n) - \frac{i}{8} c \langle \nabla \rangle_c^{-1} \mathrm{e}^{i \tau \mathcal{A}_c} \int_0^\tau \mathrm{e}^{-i c^2 (t_n+s)} \left(
\mathrm{e}^{ic^2(t_n+s)} u_\ast(t_n)+ \mathrm{e}^{-ic^2(t_n+s)} \overline{u_\ast}(t_n)\right) ^3 \mathrm{d}s
\\&+ \mathcal{R}(\tau,t_n,u_\ast), \end{aligned} \end{equation} where the remainder $ \mathcal{R}(\tau,t_n,u_\ast)$ satisfies thanks to the bounds \eqref{approx1}, \eqref{approx2} and \eqref{cnabm} that \begin{equation}\label{r1p} \Vert \mathcal{R}(\tau,t_n,u_\ast)\Vert_r \leq \tau^2 k_r(M_2), \end{equation} for some constant $k_r(M_2)$ which depends on $M_2$ (see Assumption \ref{ass:reg1}) and $r$, but is independent of $c$. Solving the integral in \eqref{du1} (in particular, integrating the highly-oscillatory phases $\mathrm{exp}(\pm i l c^2 s)$ exactly) furthermore yields by adding and subtracting the term $ \tau \frac{3i}{8} \mathrm{e}^{i \tau \mathcal{A}_c} \vert u_\ast(t_n)\vert^2 u_\ast(t_n)$ (see Remark \ref{rem:limitLie} below for the purpose of this manipulation) that \begin{equation}\label{du2p} \begin{aligned} & u_\ast(t_n+\tau)
= \mathrm{e}^{i \tau \mathcal{A}_c} \Big( 1 - \tau \frac{3i}{8} \vert u_\ast(t_n)\vert^2 \Big)u_\ast(t_n) - \tau \frac{3i}{8} \left(c \langle \nabla \rangle_c^{-1}-1\right) \mathrm{e}^{i \tau \mathcal{A}_c} \vert u_\ast(t_n)\vert^2 u_\ast(t_n) \\&- \tau \frac{i}{8} c \langle \nabla \rangle_c^{-1} \mathrm{e}^{i \tau \mathcal{A}_c} \Big\{ \mathrm{e}^{2ic^2t_n}\varphi_1(2ic^2 \tau) u_\ast^3(t_n)
+\mathrm{e}^{-2ic^2t_n} \varphi_1(-2ic^2\tau) 3\vert u_\ast(t_n)\vert^2 \overline{u_\ast}(t_n) \\& +\mathrm{e}^{-4ic^2t_n} \varphi_1(-4ic^2\tau) \overline{u_\ast}^3(t_n) \Big\} + \mathcal{R}(\tau,t_n,u_\ast) \end{aligned} \end{equation} with $\varphi_1$ given in Definition \ref{def:phi}.
As the operator $\mathrm{e}^{it \mathcal{A}_c}$ is a linear isometry in $H^r$ and by Taylor series expansion $ \vert 1-x - \mathrm{e}^{-x} \vert = \mathcal{O}(x^2) $ we obtain for $r>d/2$ that \begin{equation}\label{exiL} \begin{aligned} \left \Vert \mathrm{e}^{i \tau \mathcal{A}_c} \Big( 1 - \tau \frac{3i}{8} \vert u_\ast(t_n)\vert^2 u_\ast(t_n)\Big)- \mathrm{e}^{i \tau \mathcal{A}_c} \mathrm{e}^{-\tau \frac{3i}{8} \vert u_\ast(t_n)\vert^2 } u_\ast(t_n) \right\Vert_r
\leq k_{r} 3 \tau^2 \Vert u_\ast(t_n)\Vert_r^3 \end{aligned} \end{equation} for some constant $k_r$ independent of $c$.
The bound in \eqref{exiL} allows us to express \eqref{du2p} as follows \begin{equation}\label{du2} \begin{aligned}
u_\ast(t_n+\tau) & = \mathrm{e}^{i \tau \mathcal{A}_c} \mathrm{e}^{- \tau \frac{3i}{8} \vert u_\ast(t_n)\vert^2}u_\ast(t_n) - \tau \frac{3i}{8} \left(c \langle \nabla \rangle_c^{-1}-1\right) \mathrm{e}^{i \tau \mathcal{A}_c}\vert u_\ast(t_n)\vert^2 u_\ast(t_n) \\&- \tau \frac{i}{8} c \langle \nabla \rangle_c^{-1} \mathrm{e}^{i \tau \mathcal{A}_c} \Big\{ \mathrm{e}^{2ic^2t_n}\varphi_1(2ic^2 \tau) u_\ast^3(t_n)
+\mathrm{e}^{-2ic^2t_n} \varphi_1(-2ic^2\tau) 3 \vert u_\ast(t_n)\vert^2 \overline{u_\ast}(t_n) \\& +\mathrm{e}^{-4ic^2t_n} \varphi_1(-4ic^2\tau) \overline{u_\ast}^3(t_n) \Big\} + \mathcal{R}(\tau,t_n,u_\ast), \end{aligned} \end{equation} where the remainder $ \mathcal{R}(\tau,t_n,u_\ast)$ satisfies thanks to \eqref{r1p} and \eqref{exiL} that \begin{equation}\label{rem1} \Vert \mathcal{R}(\tau,t_n,u_\ast)\Vert_r \leq \tau^2 k_r(M_2), \end{equation} for some constant $k_r(M_2)$ which depends on $M_2$ (see Assumption \ref{ass:reg1}) and $r$, but is independent of $c$.
The expansion \eqref{du2} of the exact solution $u_\ast(t)$ builds the basis of our numerical scheme: As a numerical approximation to the exact solution $u_\ast(t)$ at time $t_{n+1} = t_n + \tau$ we choose the exponential-type integration scheme \begin{equation}\label{scheme100} \begin{aligned} u_\ast^{n+1} & =\mathrm{e}^{i \tau \mathcal{A}_c} \mathrm{e}^{- \tau \frac{3i}{8} \vert u_\ast^n\vert^2} u_\ast^n -\tau \frac{3i}{8} \left(c \langle \nabla \rangle_c^{-1}-1\right) \mathrm{e}^{i \tau \mathcal{A}_c} \vert u_\ast^n\vert^2u_\ast^n \\&- \tau \frac{i}{8} c \langle \nabla \rangle_c^{-1}\mathrm{e}^{i \tau \mathcal{A}_c}\Big\{ \mathrm{e}^{2ic^2t_n} \varphi_1(2ic^2\tau)( u_\ast^n)^3 + \mathrm{e}^{-2ic^2t_n} \varphi_1(-2ic^2\tau)3\vert u_\ast^n\vert^2\overline{u_\ast^n}\\&\qquad\qquad\qquad\qquad + \mathrm{e}^{-4ic^2t_n} \varphi_1(-4ic^2\tau) ( \overline{u_\ast^{n}})^3\Big\}\\
u_\ast^0 & = z(0) -ic^{-1}\langle \nabla \rangle_c^{-1} z'(0) \end{aligned} \end{equation} with $\varphi_1$ given in Definition \ref{def:phi}. Note that the definition of the initial value $u_\ast^0$ follows from \eqref{eq:BCc}.
For complex-valued functions $z$ (i.e., for $u\not \equiv v$) we similarly derive the exponential-type integration scheme \begin{equation}\label{scheme1} \begin{aligned}
u_\ast^{n+1} &=\mathrm{e}^{i \tau \mathcal{A}_c} \mathrm{e}^{- \tau \frac{i}{8} \big( \vert u_\ast^n\vert^2+2 \vert v_\ast^n\vert^2\big)} u_\ast^n -\tau \frac{i}{8} \left(c \langle \nabla \rangle_c^{-1}-1\right) \mathrm{e}^{i \tau \mathcal{A}_c} \big( \vert u_\ast^n\vert^2+2 \vert v_\ast^n\vert^2\big)u_\ast^n \\&- \tau \frac{i}{8} c \langle \nabla \rangle_c^{-1}\mathrm{e}^{i \tau \mathcal{A}_c}\Big\{ \mathrm{e}^{2ic^2t_n} \varphi_1(2ic^2\tau)( u_\ast^n)^2 v_\ast^n + \mathrm{e}^{-2ic^2t_n} \varphi_1(-2ic^2\tau) \big(2 \vert u_\ast^n\vert^2+ \vert v_\ast^n\vert^2\big)\overline{v_\ast^n} \\&\qquad\qquad\qquad\qquad+ \mathrm{e}^{-4ic^2t_n} \varphi_1(-4ic^2\tau) ( \overline{v_\ast^{n}})^2\overline{u_\ast^n}\Big\}\\
u_\ast^0 &= z(0) -ic^{-1}\langle \nabla \rangle_c^{-1} z'(0), \end{aligned} \end{equation} where the scheme in $v_\ast^{n+1}$ is obtained by replacing $u_\ast^n \leftrightarrow v_\ast^n$ on the right-hand side of \eqref{scheme1} with initial value $v_\ast^0 = \overline{z(0)} - i c^{-1}\langle \nabla \rangle_c^{-1}\overline{z'(0)}$ (see \eqref{eq:BCc}).
\begin{rem}[Practical implementation] To reduce the computational effort we may express the first-order scheme \eqref{scheme1} in its equivalent form \begin{equation*}\label{scheme1Pra} \begin{aligned} &u_\ast^{n+1} = \mathrm{e}^{i \tau \mathcal{A}_c} \left( \mathrm{e}^{-\tau \frac{i}{8} \big( \vert u_\ast^n\vert^2+2 \vert v_\ast^n\vert^2\big) } u_\ast^n + \tau \frac{i}{8} \big( \vert u_\ast^n\vert^2+2 \vert v_\ast^n\vert^2\big) u_\ast^n\right)- \frac{i\tau}{8} c \langle \nabla \rangle_c^{-1}\mathrm{e}^{i \tau \mathcal{A}_c}\Big\{ \big( \vert u_\ast^n\vert^2+2 \vert v_\ast^n\vert^2\big) u_\ast^n
\\& +\mathrm{e}^{2ic^2t_n} \varphi_1(2ic^2\tau)( u_\ast^n)^2 v_\ast^n + \mathrm{e}^{-2ic^2t_n} \varphi_1(-2ic^2\tau) \big(2 \vert u_\ast^n\vert^2+ \vert v_\ast^n\vert^2\big)\overline{v_\ast^n} + \mathrm{e}^{-4ic^2t_n} \varphi_1(-4ic^2\tau) ( \overline{v_\ast^{n}})^2\overline{u_\ast^n} \Big\}\\ &\quad u_\ast^0 = z(0) -ic^{-1}\langle \nabla \rangle_c^{-1} z'(0) \end{aligned} \end{equation*} which after a Fourier pseudo-spectral space discretization only requires the usage of two Fast Fourier transforms (and its corresponding inverse counter parts) instead of three.
\end{rem}
In Section \ref{sec:con1} below we prove that the exponential-type integration scheme \eqref{scheme1} is first-order convergent uniformly in $c$ for sufficiently smooth solutions. Furthermore, we give a fractional convergence result under weaker regularity assumptions and analyze its behavior in the non-relativistic limit regime $c \to \infty$. In Section \ref{sec:limit1} we give some simplifications in the latter regime.
\subsection{Convergence analysis}\label{sec:con1} The exponential-type integration scheme \eqref{scheme1} converges (by construction) with first-order in time uniformly with respect to $c$, see Theorem \ref{them:con1}. Furthermore, a fractional convergence bound holds true for less regular solutions, see Theorem \ref{them:con1Frac}. In particular, in the limit $c \to \infty$ the scheme converges to the classical Lie splitting applied to the nonlinear Schr\"odinger limit system, see Lemma \ref{rem:limit1}.
\begin{thm}[Convergence bound for the first-order scheme] \label{them:con1} Fix $r>d/2$ and assume that \begin{align}\label{eq:urged1}
\Vert z(0) \Vert_{r+2} + \Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0)\Vert_{r+2} \leq M_2 \end{align} uniformly in $c$. For $(u_\ast^{n},v_\ast^n)$ defined in \eqref{scheme1} we set \[ z^{n} := \frac12\left( \mathrm{e}^{ic^2 t_n} u_\ast^{n} + \mathrm{e}^{-ic^2 t_n} \overline{v_\ast^{n}}\right). \] Then, there exists a $T_r >0$ and $\tau_0>0$ such that for all $\tau \leq \tau_0$ and $t_n \leq T_r$ we have for all $c >0$ that \begin{align*}\label{glob1} \left \Vert z(t_{n}) - z^{n} \right\Vert_r \leq \tau K_{1,r,M_2} \mathrm{e}^{t_n K_{2,r,M}} \leq \tau K^\ast_{r,M,M_2,t_n}, \end{align*} where the constants $K_{1,r,M_2},K_{2,r,M}$ and $K^\ast_{r,M,M_2,t_n}$ can be chosen independently of $c$. \end{thm} \begin{proof} Fix $r>d/2$. First note that the regularity assumption on the initial data in \eqref{eq:urged1} implies the regularity Assumption \ref{ass:reg1} on $(u_\ast,v_\ast)$, i.e., there exists a $T_r>0$ such that \[ \sup_{0 \leq t \leq T_r} \Vert u_\ast(t)\Vert_{r+2} + \Vert v_\ast(t) \Vert_{r+2} \leq k(M_2) \] for some constant $k$ that depends on $M_2$ and $T_r$, but can be chosen independently of $c$.\\
In the following let $(\phi^t_{u_{ \ast}},\phi^t_{v_{ \ast}})$ denote the exact flow of \eqref{eq:ua1} and let $(\Phi^\tau_{u_{\ast}},\Phi^\tau_{v_{\ast}})$ denote the numerical flow defined in \eqref{scheme1}, i.e., \[ u_\ast(t_{n+1}) = \phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)), \qquad u_\ast^{n+1}= \Phi^\tau_{u_\ast}(u_\ast^n,v_\ast^n) \] and a similar formula for the functions $v_\ast(t_n)$ and $v_\ast^n$. This allows us to split the global error as follows \begin{equation}\label{glob0} \begin{aligned} u_\ast(t_{n+1}) - u_\ast^{n+1}& = \phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)) - \Phi^\tau_{u_\ast}(u_\ast^n,v_\ast^n)\\ &= \Phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)) - \Phi^\tau_{u_\ast}(u_\ast^n,v_\ast^n) +
\phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)) - \Phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)). \end{aligned} \end{equation}
\emph{Local error bound:} With the aid of \eqref{rem1} we have by the expansion of the exact solution in \eqref{du2} and the definition of the numerical scheme \eqref{scheme1} that \begin{equation}\label{local1} \Vert \phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)) - \Phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)) \Vert_r = \Vert \mathcal{R}(\tau,t_n,u_\ast,v_\ast)\Vert_r \leq \tau^2 k_r(M_2) \end{equation} for some constant $k_r$ which depends on $M_2$ and $r$, but can be chosen independently of $c$.
\emph{Stability bound:} Note that for all $l \in \mathbb{Z}$ we have that \[ \Vert \varphi_1(i \tau c^2 l) \Vert_r \leq 2. \] Thus, as $\mathrm{e}^{i t \mathcal{A}_c}$ is a linear isometry for all $t \in \mathbb{R}$ we obtain together with the bound \eqref{cnabm} that as long as $\Vert u_\ast^n\Vert_r \leq 2M$ and $\Vert u(t_n)\Vert_r \leq M$ we have that \begin{equation}\label{stab1} \Vert\Phi^\tau_{u_\ast}(u_\ast(t_n),v_\ast(t_n)) - \Phi^\tau_{u_\ast}(u_\ast^n,v_\ast^n)\Vert_r \leq \Vert u_\ast(t_n) - u_\ast^n\Vert_r +\tau K_{r,M} \left(\Vert u_\ast(t_n) - u_\ast^n\Vert_r + \Vert v_\ast(t_n) - v_\ast^n\Vert_r\right), \end{equation} where the constant $K_{r,M}$ depends on $r$ and $M$, but can be chosen independently of $c$.
\emph{Global error bound:} Plugging the stability bound \eqref{stab1} as well as the local error bound \eqref{local1} into \eqref{glob0} yields by a bootstrap argument that \begin{align}\label{conus} \left \Vert u_\ast(t_{n}) - u_\ast^{n} \right\Vert_r \leq \tau K_{1,r,M_2} \mathrm{e}^{t_n K_{2,r,M}}, \end{align} where the constants are uniform in $c$. A similar bound holds for the difference $v_\ast(t_n) -v_\ast^n$. This implies first-order convergence of $(u_\ast^n,v_\ast^n)$ towards $(u_\ast(t_n),v_\ast(t_n))$ uniformly in $c$.
Furthermore, by \eqref{eq:zuv} and \eqref{psi} we have that \begin{align*}
\Vert z(t_n) - z^n\Vert_r & = \textstyle \left \Vert\frac12 \big( u(t_n) + \overline{v(t_n)}\big) -\frac12 \big(\mathrm{e}^{ic^2 t_n} u_\ast^n + \mathrm{e}^{-ic^2t_n} \overline{v_\ast^n}\big)\right\Vert\\ & \leq \Vert \mathrm{e}^{ic^2t_n} (u_\ast(t_n)-u_\ast^n)\Vert_r + \Vert \mathrm{e}^{ic^2t_n} (v_\ast(t_n)-v_\ast^n)\Vert_r \\ & = \Vert u_\ast(t_n)-u_\ast^n\Vert_r + \Vert v_\ast(t_n)-v_\ast^n\Vert_r . \end{align*} Together with the bound in \eqref{conus} this completes the proof. \end{proof}
\begin{rem} Note that the regularity assumption \eqref{eq:urged1} is always satisfied for initial values \[ z(0,x) = \varphi(x),\qquad \partial_t z(0,x) = c^2 \gamma(x) \qquad \text{with} \quad \varphi, \gamma \in H^{r+2} \] as then thanks to \eqref{cnabm} we have \[ \left \Vert c^{-1}\langle \nabla \rangle_c^{-1}z'(0) \right\Vert_r = \left \Vert c\langle \nabla \rangle_c^{-1} \gamma\right\Vert_r \leq \Vert \gamma \Vert_r. \] \end{rem}
Under weaker regularity assumptions on the exact solution we obtain \emph{uniform fractional convergence} of the formally first-order scheme \eqref{scheme1}. \begin{thm}[Fractional convergence bound for the first-order scheme] \label{them:con1Frac} Fix $r>d/2$ and assume that for some $0< \gamma \leq 1$ \begin{align}\label{eq:urged}
\Vert z(0) \Vert_{r+2\gamma} + \Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0)\Vert_{r+2\gamma} \leq M_{2\gamma} \end{align} uniformly in $c$. For $(u_\ast^{n},v_\ast^n)$ defined in \eqref{scheme1} we set \[ z^{n} := \frac12\left( \mathrm{e}^{ic^2 t_n} u_\ast^{n} + \mathrm{e}^{-ic^2 t_n} \overline{v_\ast^{n}}\right). \] Then, there exists a $T_r>0$ and $\tau_0>0$ such that for all $\tau \leq \tau_0$ and $t_n \leq T_r$ we have for all $c >0$ that \begin{equation*} \begin{aligned} \left \Vert z(t_{n}) - z^{n} \right\Vert_r \leq \tau^\gamma K_{1,r,M_{2\gamma}} \mathrm{e}^{t_n K_{2,r,M}} \leq \tau^\gamma K^\ast_{r,M,M_{2\gamma},t_n}, \end{aligned} \end{equation*} where the constants $K_{1,r,M_{2\gamma}},K_{2,r,M}$ and $K^\ast_{r,M,M_{2\gamma},t_n}$ can be chosen independently of $c$. \end{thm} \begin{proof} The proof follows the line of argumentation to the proof of Theorem \ref{them:con1} using ``fractional estimates'' of the operator $\mathcal{A}_c$.
\end{proof}
Next we point out an interesting observation: For sufficiently smooth solutions the exponential-type integration scheme \eqref{scheme1} converges in the limit $c \to \infty$ to the classical Lie splitting of the corresponding nonlinear Schr\"odinger limit \eqref{NLSlimit}.
\begin{rem}[Approximation in the non relativistic limit $c \to \infty$]\label{rem:limit1} The exponential-type integration scheme \eqref{scheme1} corresponds for sufficiently smooth solutions in the limit $(u_\ast^n,v_\ast^n) \stackrel{c\to \infty}{\longrightarrow} (u_{\ast,\infty}^n,v_{\ast,\infty}^n)$, essentially to the Lie Splitting (\cite{Lubich08,Faou12}) \begin{equation}\label{limitLie} \begin{aligned} u_{\ast, \infty}^{n+1} &=\mathrm{e}^{-i \tau \frac{\Delta}{2}} \mathrm{e}^{-i \tau \frac{1}{8} \big( \vert u_{\ast, \infty}^n\vert^2 +2 \vert v_{\ast, \infty}^n\vert^2 \big)}u_{\ast, \infty}^n,\qquad u_{\ast,\infty}^0 = \varphi - i \gamma, \\ v_{\ast, \infty}^{n+1} &=\mathrm{e}^{-i \tau \frac{\Delta}{2}} \mathrm{e}^{-i \tau \frac{1}{8} \big( \vert v_{\ast, \infty}^n\vert^2 +2 \vert u_{\ast, \infty}^n\vert^2 \big)}v_{\ast, \infty}^n,\qquad v_{\ast,\infty}^0 = \overline{\varphi} - i \overline{\gamma} \end{aligned} \end{equation} applied to the cubic nonlinear Schr\"odinger system \eqref{NLSlimit} which is the limit system of the Klein-Gordon equation \eqref{eq:kgr} for $c \to \infty$ with initial values \begin{align*} z(0) \stackrel{c\to\infty}{\longrightarrow} \gamma \quad \text{and}\quad c^{-1}\langle \nabla \rangle_c^{-1} z'(0) \stackrel{c\to\infty}{\longrightarrow} \varphi. \end{align*}
More precisely, the following Lemma holds. \end{rem}
\begin{lem}\label{rem:limit1} Fix $r>d/2$ and let $ 0 < \delta \leq 2$. Assume that \begin{equation}\label{regass:limit1} \Vert z(0) \Vert_{r+2\delta+\varepsilon} +\Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0)\Vert_{r+2\delta+\varepsilon} \leq M_{2\delta+\varepsilon} \end{equation} for some $\varepsilon >0$ uniformly in $c$ and let the initial value approximation (there exist functions $\varphi,\gamma$ such that) \begin{align}\label{limitIn} \Vert z(0)- \gamma\Vert_r + \Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0) - \varphi\Vert_r \leq k_r c^{-\delta} \end{align} hold for some constant $k_r$ independent of $c$.
Then, there exists a $T>0$ and $\tau_0>0$ such that for all $\tau \leq \tau_0$ the difference of the first-order scheme \eqref{scheme1} for system \eqref{eq:ua1} and the Lie splitting \eqref{limitLie} for the limit Schr\"odinger equation \eqref{NLSlimit} satisfies for $ t_n \leq T$ and all $c>0$ with \begin{equation}\label{ctau1} \tau c^{2-\delta} \geq 1 \end{equation} that \begin{align*} \Vert u_\ast^n- u_{\ast, \infty}^{n} \Vert_r + \Vert v_\ast^n- v_{\ast, \infty}^{n} \Vert_r \leq c^{-\delta} k_r(M_{2\delta+\varepsilon},T) \end{align*} for some constant $k_{r}$ that depends on $M_{2\delta+\varepsilon}$ and $T$, but is independent of $c$. \end{lem} \begin{proof} In the following fix $r>d/2$, $ 0 < \delta \leq 2$ and $\varepsilon>0$:
\emph{1. Initial value approximation:} Thanks to \eqref{limitIn} we have by the definition of the initial value $u_\ast^0$ in \eqref{scheme1}, respectively, $u_{\ast,\infty}^0$ in \eqref{limitLie} that \[ \Vert u_\ast^0 - u_{\ast,\infty}^0 \Vert_r = \Vert z(0) -ic^{-1}\langle \nabla \rangle_c^{-1} z'(0) - (\varphi-i\gamma)\Vert_r \leq k_r c^{-\delta} \] for some constant $k_r$ independent of $c$. A similar bound holds for $v_\ast^0 - v_{\ast,\infty}^0$.
\emph{2. Regularity of the numerical solutions $(u_\ast^n,v_\ast^n)$:} Thanks to the regularity assumption \eqref{regass:limit1} we have by Theorem \ref{them:con1Frac} that there exists a $T>0$ and $\tau_0>0$ such that for all $\tau \leq \tau_0$ we have \begin{equation}\label{regNum1} \Vert u_\ast^n\Vert_{r+2\delta} + \Vert v_\ast^n \Vert_{r+2\delta} \leq m_{2\delta} \end{equation} as long as $t_n \leq T$ for some constant $m_{2\delta}$ depending on $M_{2\delta+\varepsilon}$ and $T$, but not on $c$.
\emph{3. Regularity of the numerical solutions $(u_{\ast,\infty}^n,v_{\ast,\infty}^n)$:} Thanks to the regularity assumption \eqref{regass:limit1} we have by \eqref{limitIn} and the global first-order convergence result of the Lie splitting for semilinear Schr\"odinger equations (see for instance \cite{Faou12,Lubich08}) that there exists a $T>0$ and $\tau_0>0$ such that for all $\tau \leq \tau_0$ we have\begin{equation}\label{regNum2} \Vert u_{\ast,\infty}^n\Vert_{r} + \Vert v_{\ast,\infty}^n \Vert_{r} \leq m_{0} \end{equation} as long as $t_n \leq T$ for some constant $m_{0}$ depending on $M_{r}$ and $T$, but not on $c$.
\emph{4. Approximations:} Using the following bounds, $\gamma > 1$ \begin{equation} \label{kater}
\left|Ê\sqrt{1 + x^2} - 1 - \frac12 x^2\right|Ê \leq x^{2\gamma} \quad \mbox{and} \quad \left|Ê\frac{1}{\sqrt{1 + x^2}¿} - 1\right| \leq x^{2\gamma - 2}, \end{equation} together with the Definition of $\varphi_1$ (see Definition \ref{def:phi}) we have for every $f \in H^{r+2+2\delta}$, \begin{align}\label{boundOpc1} \big\Vert \left(\mathcal{A}_c + \textstyle\frac{\Delta}{2}\right) f \big\Vert_r + \big \Vert \left(c \langle \nabla \rangle_c^{-1}-1\right) f \big \Vert_{r+2} + \big \Vert \varphi_1(ilc^2\tau)f \big\Vert_{r+2+\delta} \leq k_{r} c^{-\delta} \Vert f \Vert_{r+2+2\delta} \end{align} for $l = \pm 2, -4$ and for some constant $k_{r}$ independent of $c$, where we used \eqref{ctau1} for the last estimate.
\emph{5. Difference of the numerical solutions: } Thanks to the a priori regularity of the numerical solutions \eqref{regNum1} and \eqref{regNum2} we obtain with the aid of \eqref{boundOpc1} under assumption \eqref{ctau1} for the difference $u_\ast^n- u_{\ast,\infty}^n$ that \begin{equation}\label{scheme1Ex} \begin{aligned} \Vert u_\ast^{n+1} - u_{\ast,\infty}^{n+1} \Vert_r & \leq \big(1+ \tau k(m_{0})\big) \Vert u_\ast^n - u_{\ast,\infty}^n\Vert_r + (c^{-2+\delta}+\tau) c^{-\delta} k(m_{2\delta})\\ & \leq \big(1+ \tau k(m_{0})\big) \Vert u_\ast^n - u_{\ast,\infty}^n\Vert_r + 2 \tau c^{-\delta} k(m_{2\delta}) \end{aligned} \end{equation} and a similar bound on $v_\ast^n- v_{\ast,\infty}^n$. Solving the recursion yields the assertion. \end{proof}
\subsection{Simplifications in the ``weakly to strongly non-relativistic limit regime''}\label{sec:limit1} In the `` strongly non-relativistic limit regime'', i.e., for large values of $c$, we may simplify the first-order scheme \eqref{scheme1} and nevertheless obtain a well suited, first-order approximation to $(u_\ast,v_\ast)$ in \eqref{eq:ua1}. \begin{rem} Note that for $l = \pm 2, -4$ we have (see Definition \ref{def:phi}) \[ \left \Vert \tau \varphi_1(i l c^2 \tau) \right \Vert_r \leq 2 c^{-2} . \] Furthermore, \eqref{boundOpc1} yields that \[ \Vert \left (c\langle \nabla \rangle_c^{-1} - 1\right) u_\ast(t) \Vert_r \leq c^{-2} k_r \Vert u_\ast(t)\Vert_{r+2} \] for some constant $k_r$ independent of $c$.
Thus, for sufficiently large values of $c$, more precisely if \[ \tau c > 1 \] and under the same regularity assumption \eqref{eq:urged} we may take instead of \eqref{scheme1} the scheme \begin{align*} u_{\ast,c>\tau}^{n+1} & =\mathrm{e}^{i \tau \mathcal{A}_c} \mathrm{e}^{-i \tau\frac{1}{8} \big( \vert u_{\ast,c>\tau}^n\vert^2 +2\vert v_{\ast,c>\tau}^n\vert^2 \big)} u_{\ast,c>\tau}^n \\ v_{\ast,c>\tau}^{n+1} & =\mathrm{e}^{i \tau \mathcal{A}_c} \mathrm{e}^{-i\tau\frac{1}{8} \big( \vert v_{\ast,c>\tau}^n\vert^2 +2\vert u_{\ast,c>\tau}^n\vert^2 \big)} v_{\ast,c>\tau}^n \end{align*} as a first-order numerical approximation to $(u_\ast(t_{n+1}),v_\ast(t_{n+1}))$ in \eqref{eq:ua1}. \end{rem}
However, note that in the strongly non-relativistic limit regime (such that in particular $c \tau > 1$) we may immediately take the Lie splitting scheme proposed in \cite{FS13} as a suitable first-order approximation to \eqref{eq:ua1} thanks to the following observation: \begin{rem}[Limit scheme \cite{FS13}]\label{rem:limitLie} For sufficiently large values of $c$ and sufficiently smooth solutions, more precisely, if \[
\Vert z(0) \Vert_{r+2} + \Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0)\Vert_{r+2}\leq M_{2}\quad \text{and}\quad \tau c > 1 \]
the classical Lie splitting (see \cite{Lubich08,Faou12}) for the nonlinear Schr\"odinger limit equation \eqref{NLSlimit}, namely, \begin{equation}\label{limitscheme1} \begin{aligned} u_{\ast,\infty}^{n+1} &=\mathrm{e}^{-i \tau \frac{1}{2}\Delta } \mathrm{e}^{-i \tau \frac{1}{8}\big( \vert u_{\ast,\infty}^n\vert^2 +2\vert v_{\ast,\infty}^n\vert^2\big) } u_{\ast,\infty}^n\\ v_{\ast,\infty}^{n+1} &=\mathrm{e}^{-i \tau \frac{1}{2}\Delta } \mathrm{e}^{-i \tau \frac{1}{8}\big( \vert v_{\ast,\infty}^n\vert^2 +2\vert u_{\ast,\infty}^n\vert^2\big) } v_{\ast,\infty}^n \end{aligned} \end{equation}
yields as a first-order numerical approximation to $(u_\ast(t_{n+1}),v_\ast(t_{n+1}))$ in \eqref{eq:ua1}.
This assertion follows from \cite{FS13} thanks to the approximation \begin{align*} \Vert u_\ast(t_n) - u_{\ast,\infty}^{n} \Vert_r & \leq \Vert u_\ast(t_n) - u_{\ast,\infty}(t_n)\Vert_r + \Vert u_{\ast,\infty}(t_n) - u_{\ast,\infty}^{n} \Vert_r = \mathcal{O}\big( c^{-1}+\tau\big) \end{align*} and the similar bound on $v_\ast(t_n) - v_{\ast,\infty}^{n}$. \end{rem}
\section{A second-order uniformly accurate scheme}\label{sec:scheme2} In this section we derive a second-order exponential-type integration scheme for the solutions $(u_\ast,v_\ast)$ of \eqref{eq:ua1} which allows \emph{second-order uniform time-convergence with respect to $c$}. For notational simplicity we again assume that $z$ is real, which reduces the coupled system \eqref{eq:ua1} to equation \eqref{eq:ua} with mild-solutions \eqref{du} (see also Remark \ref{rem:realz}).
The construction of the second-order scheme is again based on Duhamel's formula \eqref{du} and the essential estimates in Lemma \ref{lem:bAc}, \ref{lem:expo} and \ref{lem:upc}. However, the second-order approximation is much more involved due to the fact that $$
u_\ast'(t) = \mathcal{O}(1), \quad \text{but}\quad u_\ast''(t) = \mathcal{O}(c^2). $$ The latter observation prevents us from simply applying the higher-order Taylor series expansion $$ u_\ast(t_n+s) = u_\ast(t_n) + s u_\ast'(t_n) + \mathcal{O}\big(s^2 u_\ast''(t_n+\xi)\big) $$ in Duhamel's formula \eqref{du} as this would lead to the ``classical'' $c-$dependent error at order $\mathcal{O}(\tau^2 c^2)$. Therefore we need to carry out a much more careful frequency analysis by iterating Duhamel's formula \eqref{du} twice and controlling the appearing highly-oscillatory terms $\mathrm{e}^{\pm i c^2 t}$ and their interactions $\mathrm{e}^{il c^2 t}$ ($l \in \mathbb{Z}$) precisely.
\subsection{Construction of a second-order uniformly accurate scheme} In Section \ref{submerge} we state the necessary regularity assumptions on the solution $u_\ast$ and derive two useful expansions. In Section \ref{sub:PLD} we collect some useful lemmata on highly-oscillatory integrals and their approximations. These approximations will then allow us to construct a uniformly accurate second-order scheme in Section \ref{sec:USDD}. The rigorous convergence analysis is given in Section \ref{sec:convA2}.
\subsubsection{Regularity and expansion of the exact solution}\label{submerge} In order to derive a second-order scheme, we need to impose additional regularity on the exact solution $u_\ast(t)$ of \eqref{eq:ua}. \begin{ass}\label{ass:reg2} Fix $r>d/2$ and assume that $u_\ast \in \mathcal{C}([0,T];H^{r+4}(\mathbb{T}^d))$ and in particular \[ \sup_{0\leq t \leq T} \Vert u_\ast(t) \Vert_{r+4} \leq M_4 \quad \text{uniformly in $c$}. \] \end{ass} In Lemma \ref{lem:doubleInt} below we derive two useful expansions of the exact solution $u_\ast$ of \eqref{eq:ua}. For this purpose we introduce the following definition. \begin{defn}\label{def:psi} For some function $v$ and $t_n,t \in \mathbb{R}$ we set \begin{equation}\label{psidef} \begin{aligned}
\Psi_{c^2}(t_n,t,v)
&:= t \mathrm{e}^{2ic^2 t_n} \varphi_1\left(2ic^2t\right) v^3 + 3 t \mathrm{e}^{-2ic^2t_n} \varphi_1\left(-2ic^2 t\right)\vert v \vert^2 \overline{v}+ t \mathrm{e}^{-4ic^2t_n} \varphi_1\left(-4ic^2t\right) \overline{v}^3.
\end{aligned} \end{equation} \end{defn}
The above defintion allows us the following expansions of the exact solution $u_\ast$. \begin{lem}\label{lem:doubleInt} Fix $r>d/2$. Then the exact solution of \eqref{eq:ua} satisfies the expansions \begin{equation*} \begin{aligned} u_\ast(t_n+s) & = \mathrm{e}^{i s \mathcal{A}_c} u_\ast(t_n)- \frac{3i}{8}c\langle \nabla \rangle_c^{-1} \int_0^s \mathrm{e}^{i(s-\xi)\mathcal{A}_c} \left \vert \mathrm{e}^{i\xi\mathcal{A}_c}u_\ast(t_n) \right\vert^2 \left(\mathrm{e}^{i\xi\mathcal{A}_c}u_\ast(t_n) \right)
\mathrm{d}\xi
\\& - \frac{i}{8}c\langle \nabla \rangle_c^{-1} \Psi_{c^2}(t_n,s,u_\ast(t_n))+ \mathcal{R}_1(t_n,s, u_\ast) \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} u_\ast(t_n+s) & = \mathrm{e}^{i s \mathcal{A}_c} u_\ast(t_n)- \frac{i}{8}c\langle \nabla \rangle_c^{-1} \Big( 3s \left \vertu_\ast(t_n) \right\vert^2 u_\ast(t_n) + \Psi_{c^2}(t_n,s,u_\ast(t_n))\Big)\\&+ \mathcal{R}_2(t_n,s, u_\ast) \end{aligned} \end{equation*} with $\Psi_{c^2}$ defined in \eqref{psidef} and where the remainders satisfy \begin{equation}\label{remBdoubleInt} \begin{aligned} & \Vert \mathcal{R}_{1}(t_n,s, u_\ast)\Vert_r +\Vert \mathcal{R}_{2}(t_n,s, u_\ast)\Vert_r \leq s^2 k_{r}(M_2) \end{aligned} \end{equation} for some constant $k_r(M_2)$ which depends on $M_2$, but is independent of $c$. \end{lem} \begin{proof} Note that by Duhamel's perturbation formula \eqref{du} we have that \begin{equation} \begin{aligned}
u_\ast(t_n+s) =\mathrm{e}^{i s \mathcal{A}_c} u_\ast(t_n)
- \frac{i}{8}c\langle \nabla \rangle_c^{-1} \int_0^s \mathrm{e}^{i(s-\xi)\mathcal{A}_c}\Big(3 \left\vert u_\ast(t_n+\xi)\right\vert^2 u_\ast(t_n+\xi)+ \mathrm{e}^{2ic^2(t_n+\xi)} u_\ast(t_n+\xi)^3 \\ + 3 \mathrm{e}^{-2ic^2(t_n+\xi)} \left\vert u_\ast(t_n+\xi)\right\vert^2 \overline{u_\ast}(t_n+\xi)+ \mathrm{e}^{-4ic^2(t_n+\xi)} \overline{u_\ast}(t_n+\xi)^3 \Big)\mathrm{d}\xi. \end{aligned} \end{equation} Therefore, the bound on $c\langle \nabla \rangle_c^{-1}$ given in \eqref{cnabm} in particular implies that for $\xi \in \mathbb{R}$ \[ \Vert u_\ast(t_n+\xi) - \mathrm{e}^{i \xi \mathcal{A}_c} u_\ast(t_n)\Vert_r \leq \xi k_r (1+M_0)^3 \] for some constant $k_r$ which is independent of $c$. Together with Lemma \ref{lem:expo} and \ref{lem:upc} the assertion then follows by integrating the highly-oscillatory phases $\mathrm{exp}\left( \pm i l c^2 \xi\right)$ exactly. \end{proof}
In the next section we collect some important definitions and useful lemmata on highly-oscillatory integrals. \subsubsection{Preliminary lemmata on highly-oscillatory integrals}\label{sub:PLD}
The construction of a second-order approximation to $u_\ast$ based on the iteration of Duhamel's formula \eqref{du} that holds uniformly in all $c > 0$ leads to interactions of the highly-oscillatory phases $\mathrm{e}^{i c^2 t}$. More precisely, we need to handle highly-oscillatory integrals of type \begin{equation}\label{inti99}
\int_0^\tau \mathrm{e}^{i s(\delta c^2- \mathcal{A}_c)} \left( \mathrm{e}^{i s \mathcal{A}_c} v\right)^l \left( \mathrm{e}^{-i s \mathcal{A}_c} \overline{v}\right)^m \mathrm{d} s , \qquad \delta \in \{-4, -2, 2\}. \end{equation} In order to control these integrals we first need to distinguish the non-resonant case $\delta \in \{-4, -2\}$ where $$ \forall c > 0,\, k \in \mathbb{N}\, : \quad (\delta c^2 - \mathcal{A}_c)_k = \delta c^2 - c\sqrt{c^2+k^2} + c^2 \neq 0 $$ from the resonant case $\delta = 2$ in which the operator $ \delta c^2 - \mathcal{A}_c $
may become singular.
In Lemma \ref{lemI1} we outline how to control the non-resonant case $\delta \in \{-4, -2\}$. Lemma \ref{lemI2} treats the resonant case $\delta = 2$.
\begin{lem}\label{lemI1} Fix $r>d/2$. Then we have for $\delta_1 = -2$ and $\delta_2 = -4$ that for $j = 1,2$ and $l,m \in \mathbb{N}^*$, \begin{equation} \label{merci} \begin{aligned} & \int_0^\tau \mathrm{e}^{i s(\delta_j c^2- \mathcal{A}_c)} \left( \mathrm{e}^{i s \mathcal{A}_c} v\right)^l \left( \mathrm{e}^{-i s \mathcal{A}_c} \overline{v}\right)^m \mathrm{d} s \\&=\tau \varphi_1\left( i\tau(\delta_j c^2-\mathcal{A}_c)\right)v^l\overline{v}^m + i \tau^2 \varphi_2\left( i\tau(\delta_j c^2-\mathcal{A}_c)\right)\left( l v^{l-1} \overline{v}^m \mathcal{A}_c v -m v^l \overline{v}^{m-1} \mathcal{A}_c \overline{v} \right) \\&+ \mathcal{R}(t_n,s, v), \end{aligned} \end{equation} where the remainder satisfies \begin{equation}\label{lemr1} \Vert \mathcal{R}(t_n,s, v)\Vert_{r} \leq k_r \tau^3 \Vert v\Vert_{r+4} \Vert v\Vert_r^{l+m-1} \end{equation} for some constant $k_r$ which is independent of $c$. \end{lem} \begin{proof} By Taylor series expansion of $\mathrm{e}^{i s \mathcal{A}_c}$ and noting \eqref{boundAc} we obtain that \begin{equation}\label{inti1} \begin{aligned} &\int_0^\tau \mathrm{e}^{-i s \mathcal{A}_c}\mathrm{e}^{i \delta_j c^2 s} \left( \mathrm{e}^{i s \mathcal{A}_c} v\right)^l \left( \mathrm{e}^{-i s \mathcal{A}_c} \overline{v}\right)^m\mathrm{d} s \\ &= \int_0^\tau \mathrm{e}^{i s ( \delta_j c^2 - \mathcal{A}_c)} \left( v^{l} \overline{v}^m + i s \left(l v^{l-1} \overline{v}^m \mathcal{A}_c v - mv^l \overline{v}^{m-1} \mathcal{A}_c \overline{v} \right) \right) \mathrm{d} s + \mathcal{R}(t_n,s, v), \end{aligned} \end{equation} where thanks to \eqref{boundAc} we have for $r>d/2$ that \eqref{lemr1} holds for the remainder. The assertion then follows by the definition of the $\varphi_j$ functions given in Definition \ref{def:phi}. \end{proof}
As our numerical scheme will be built on the approximation in \eqref{merci} we need to guarantee that the constructed term $$
\tau^2 \varphi_2\left( i\tau(\delta_j c^2-\mathcal{A}_c)\right)\left( l v^{l-1} \overline{v}^m \mathcal{A}_c v -m v^l \overline{v}^{m-1} \mathcal{A}_c \overline{v} \right) $$ is uniformly bounded with respect to $c$ in $H^r$ for all functions $v \in H^r$. This stability analysis is carried out in Remark \ref{remidemi} below, where we in particular exploit the bilinear estimate \begin{equation}\label{bili} \textstyle \Vert v w \Vert_r \leq k\, \Vert v \Vert_{r_1} \Vert w \Vert_{r_2} \quad \text{for all } r \leq r_1+r_2-\frac{d}{2} \quad \text{with}\quad r_1,r_2,-r \neq \frac{d}{2} \quad \text{and}\quad r_1+r_2 \geq 0. \end{equation} \begin{rem}[Stability in Lemma \ref{lemI1}]\label{remidemi} Note that for $\delta_1 = -2$, respectively, $\delta_2 = -4$ we have that \begin{equation}\label{bbA}
0 \neq \delta_j c^2 - \mathcal{A}_c = \delta_j c^2 - c\langle \nabla \rangle_c + c^2 = \left\{ \begin{array}{ll} - (c^2 + c \langle \nabla \rangle_c) &\mbox{if} \quad j = 1\\ - (3c^2 + c\langle \nabla \rangle_c) & \mbox{if}\quad j = 2 \end{array} \right. . \end{equation} Thanks to \eqref{bbA} which in particular implies that \begin{align*}
& \left(\langle \nabla \rangle_c\right)_k = \sqrt{c^2 + \vert k\vert^2} \leq \sqrt{c^2} + \sqrt{|k|^2} = c + |k| \quad \mbox{and} \quad \\
& \frac{1}{c^2 + c\left(\langle \nabla \rangle_c\right)_k } \leq \mathrm{min}\left\{ |c|^{-2}, |c\sqrt{c^2+k^2}|^{-1}\right\} \leq \mathrm{min}\left\{ |c|^{-2}, (c |k|)^{-1}\right\} \end{align*}
we obtain together with the bilinear estimate \eqref{bili} that for $\delta_j = -2,-4$ \begin{equation}\label{stab21} \begin{aligned} & \left\Vert \tau^2 \varphi_2\left(i \tau(\delta_j c^2-\mathcal{A}_c)\right) \left(v \mathcal{A}_c w\right) \right\Vert_r
= \tau \left \Vert \frac{\varphi_0(i\tau(\delta_j c^2-\mathcal{A}_c)) - \varphi_1(i\tau(\delta_j c^2-\mathcal{A}_c))}{(\delta_j c^2-\mathcal{A}_c)} \left(v \mathcal{A}_c w\right)\right\Vert_r \\&\qquad \leq 2\tau \left \Vert \frac{1}{(c^2 + c \langle \nabla \rangle_c)} \left( v \mathcal{A}_c w \right) \right\Vert_r \leq 2 \tau \left \Vert \frac{1}{(c^2 + c \langle \nabla \rangle_c)} \left( v2 c^2 w \right) \right\Vert_r + 2\tau \left \Vert \frac{1}{(c^2 + c \langle \nabla \rangle_c)} \left( v c \langle \nabla \rangle_0 w \right) \right\Vert_r \\ &\qquad \leq 4k_r\tau \Vert v \Vert_r \Vert w \Vert_r \end{aligned} \end{equation} for all $r>d/2$ and all functions $v$ and $w$ and some constant $k_r>0$. The estimate \eqref{stab21} guarantees stability of our numerical scheme built on the approximation in \eqref{merci}. \end{rem}
A simple manipulation allows us to treat the resonant case, i.e., $\delta = 2$ in \eqref{inti99}, similarly to Lemma \ref{lemI1}. \begin{lem}\label{lemI2} Fix $r>d/2$ and let $c \neq 0$. Then we have that \begin{equation} \begin{aligned} &\int_0^\tau \mathrm{e}^{i s (2c^2-\mathcal{A}_c)} \left( \mathrm{e}^{i s \mathcal{A}_c} v\right)^l \left( \mathrm{e}^{-i s \mathcal{A}_c} \overline{v}\right)^m\mathrm{d} s =\textstyle \tau \varphi_1\left( i \tau ( 2c^2 - \frac{1}{2} \Delta) \right) \left(v^l \overline{v}^m\right) \\&\qquad\textstyle+i \tau^2 \varphi_2\left( i \tau ( 2c^2 - \frac{1}{2} \Delta)\right) \Big[
(\frac12\Delta - \mathcal{A}_c) \left(v^l \overline{v}^m\right) + \left( l v^{l-1} \overline{v}^m \mathcal{A}_c v - m v^l \overline{v}^{m-1} \mathcal{A}_c \overline{v} \right) \Big] \\ & \qquad + \mathcal{R}(t_n,s, v), \end{aligned}\label{uuli} \end{equation} where the remainder satisfies \begin{equation}\label{lemr2} \Vert \mathcal{R}(t_n,s, v)\Vert_{r} \leq k_r \tau^3 \Vert v\Vert_{r+4} \Vert v\Vert_r^{l+m-1} \end{equation} for some constant $k_r$ which is independent of $c$. \end{lem} \begin{proof} Note that as \[ 2c^2 - \mathcal{A}_c =\textstyle 2c^2 - \frac{1}{2} \Delta + \frac{1}{2} \Delta-\mathcal{A}_c \] we obtain \begin{equation} \begin{aligned}\label{int22} &\int_0^\tau \mathrm{e}^{i s (2c^2-\mathcal{A}_c)} \left( \mathrm{e}^{i s \mathcal{A}_c} v\right)^l \left( \mathrm{e}^{-i s \mathcal{A}_c} \overline{v}\right)^m\mathrm{d} s =
\int_0^\tau \mathrm{e}^{ i s ( 2c^2 - \frac{1}{2} \Delta)} \mathrm{e}^{i s (\frac{1}{2}\Delta-\mathcal{A}_c)} \left( \mathrm{e}^{i s \mathcal{A}_c} v\right)^l \left( \mathrm{e}^{-i s \mathcal{A}_c} \overline{v}\right)^m\mathrm{d} s\\
& = \int_0^\tau \mathrm{e}^{ i s ( 2c^2 - \frac{1}{2} \Delta)}
\Big[ \textstyle\big(1+ i s(\frac12 \Delta-\mathcal{A}_c)\big) \left(v^l\overline{v}^m\right) + i s \left( lv^{l-1} \overline{v}^m \mathcal{A}_c v - m v^l \overline{v}^{m-1} \mathcal{A}_c \overline{v} \right)
\Big]\mathrm{d} s+ \mathcal{R}(t_n,s, v), \end{aligned} \end{equation} where thanks to \eqref{boundAc} we have for $r>d/2$ that \eqref{lemr2} holds for the remainder. The assertion then follows by the definition of the $\varphi_j$ functions given in Definition \ref{def:phi}. \end{proof}
Again we need to verify that the constructed term $$ \textstyle \tau^2 \varphi_2\left( i \tau ( 2c^2 - \frac{1}{2} \Delta)\right) \Big[
(\frac12\Delta - \mathcal{A}_c) \left(v^l \overline{v}^m\right) + \left( l v^{l-1} \overline{v}^m \mathcal{A}_c v - m v^l \overline{v}^{m-1} \mathcal{A}_c \overline{v} \right) \Big] $$ in \eqref{uuli} can be bounded uniformly with respect to $c$ in $H^r$ for all functions $v \in H^r$. This is done in the following remark. \begin{rem}[Stability in Lemma \ref{lemI2}] Note that the operator $2c^2-\frac12\Delta $ satisfies the bounds \[
\frac{ c |k| }{\left(2c^2-\frac12\Delta\right)_k} = \frac{c |k|}{2c^2 + \frac12 \vert k \vert^2} \leq 2, \qquad \frac{ c^2 }{\left(2c^2-\frac12\Delta\right)_k} = \frac{c^2}{2c^2 + \frac12 \vert k \vert^2} \leq \frac12 \] and furthermore \[
\left( \mathcal{A}_c \right)_k = c \sqrt{c^2+|k|^2} - c^2 \leq 2c^2+ c |k|. \] The above estimates together with the bilinear estimate \eqref{bili} imply that for $r>d/2$ \begin{equation}\label{stab22} \begin{aligned} & \left \Vert \textstyle \tau^2 \varphi_2\left( i \tau ( 2c^2 - \frac{1}{2} \Delta) \right) \left(v \mathcal{A}_c w \right) \right \Vert_r^2 \leq \tau
\sum_{k} \frac{( 1 + |k|^2)^r}{(2c^2 + \frac12|k|^2)^2} \Big| \sum_{k = k_1 + k_2} v_{k_1} (\mathcal{A}_c)_{k_2} w_{k_2} \Big|^2\\
& \leq \tau m_r\sum_{k} \frac{( 1 + |k|^2)^r c^4}{(2c^2 + \frac12|k|^2)^2} \Big(\sum_{k = k_1 + k_2} |v_{k_1}| |w_{k_2}|\Big)^2+ \tau m_r \sum_{k} \frac{( 1 + |k|^2)^rc^2}{(2c^2 + \frac12|k|^2)^2} \Big(\sum_{k = k_1 + k_2} |v_{k_1}| |k_2| |w_{k_2}|\Big)^2 \\
& \leq \tau m_r\sum_{k}( 1 + |k|^2)^r \Big(\sum_{k = k_1 + k_2} |v_{k_1}| |w_{k_2}|\Big)^2+ \tau m_r \sum_{k} ( 1 + |k|^2)^{r-1} \Big(\sum_{k = k_1 + k_2} |v_{k_1}| |k_2| |w_{k_2}|\Big)^2 \\ & \leq \tau m_r \Vert v \Vert_r ^2\Vert w \Vert_r^2 + \tau k_r \Vert v \Vert_r^2 \Vert \partial_x w\Vert_{r-1}^2 \leq \tau k m_r \Vert v \Vert_r^2 \Vert w \Vert_r^2 \end{aligned} \end{equation} for some constant $m_r>0$ which guarantees stability of the numerical method built on the approximation in Lemma \ref{lemI2}. \end{rem} Next we need to analyze integrals involving the highly-oscillatory function $\Psi_{c^2}$ defined in \eqref{def:psi}. The following lemma yields a uniform approximation. \begin{lem}\label{intpsiP} Fix $r>d/2$. Then for any polynomial $p(v)$ in $v$ and $\overline{v}$ we have that \begin{equation} \begin{aligned} & \int_0^\tau \mathrm{e}^{i(\tau-s)\mathcal{A}_c} p\left(\mathrm{e}^{is\mathcal{A}_c}v\right) c\langle \nabla \rangle_c^{-1} \Psi_{c^2}(t_n,s,v)\mathrm{d} s= \tau^2 p(v)c\langle \nabla \rangle_c^{-1}\vartheta_{c^2}(t_n,\tau,v) + \mathcal{R}(t_n,\tau,v) \end{aligned} \end{equation} with \begin{equation}\label{defpsip} \begin{aligned}
\vartheta_{c^2}(t_n,\tau,v) : & = \mathrm{e}^{2ic^2 t_n} \frac{\varphi_1\left(2ic^2\tau\right)-1}{2i\tau c^2} v^3\\& + 3 \mathrm{e}^{-2ic^2t_n}\frac{ \varphi_1\left(-2ic^2 \tau\right)-1}{-2i\tau c^2} \vert v\vert^2 \overline{v}+\mathrm{e}^{-4ic^2t_n}\frac{ \varphi_1\left(-4ic^2\tau\right)-1}{-4i\tau c^2} \overline{v}^3 \end{aligned} \end{equation} and where the remainder satisfies \begin{equation}\label{Rintpsi} \left \Vert \mathcal{R}(t_n,\tau,v)\right\Vert_r \leq k_r \tau^3 \left(1+ \Vert v\Vert_{r+2}\right)^5 \end{equation} for some constant $k_r$ independent of $c$. \end{lem} \begin{proof} Thanks to the approximation \eqref{approx1} and the fact that $\Psi_{c^2}(t_n,s,u_\ast(t_n))$ is of order one in $s$ uniformly in $c$ we have that \begin{equation*} \begin{aligned} & \int_0^\tau \mathrm{e}^{i(\tau-s)\mathcal{A}_c} p\left(\mathrm{e}^{is\mathcal{A}_c}v\right) c\langle \nabla \rangle_c^{-1} \Psi_{c^2}(t_n,s,v)\mathrm{d} s\\ & =p\left(v\right) c\langle \nabla \rangle_c^{-1} \int_0^\tau\Psi_{c^2}(t_n,s,v)\mathrm{d} s+ \mathcal{R}(t_n,\tau,v), \end{aligned} \end{equation*} where the remainder satisfies for $r>d/2$ the bound \eqref{Rintpsi}. \end{proof}
Finally, we need to handle the interaction of highly-oscillatory phases $\mathrm{e}^{ilc^2 t}$ with the highly-oscillatory function $\Psi_{c^2}$ defined in \eqref{def:psi}. \begin{lem}\label{lem:intPsic} Let $c\neq 0$. Then, we have for $l \in \mathbb{N}$ that \begin{equation}\label{def:om} \begin{aligned} \Omega_{c^2,l}(t_n,\tau,v)& := \frac{1}{\tau^2}\int_0^\tau \mathrm{e}^{i l c^2s} \Psi_{c^2}(t_n,s,v) \mathrm{d} s\\&= \mathrm{e}^{2ic^2 t_n} \frac{\varphi_1\left( (l+2)ic^2\tau\right) -\varphi_1\left(lic^2\tau\right)}{2i\tau c^2} v^3 \\&+ 3 \mathrm{e}^{-2ic^2t_n} \frac{\varphi_1\left((l-2)ic^2\tau\right) -\varphi_1\left(lic^2\tau\right)}{-2i\tau c^2}\vert v \vert^2 \overline{v}\\
& + \mathrm{e}^{-4ic^2t_n} \frac{\varphi_1\left((l-4)ic^2\tau\right)-\varphi_1\left(lic^2\tau\right)}{-4i\tau c^2} \overline{v}^3 \end{aligned} \end{equation} as well as that \begin{equation*} \int_0^\tau \mathrm{e}^{i lc^2s} s \mathrm{d} s = \tau^2 \varphi_2( i l c^2 \tau). \end{equation*} \end{lem} \begin{proof} Note that by Definition \ref{def:psi} we have that \begin{align*}
\Psi_{c^2}(t_n,s,v) & = \mathrm{e}^{2 ic^2 t_n} \frac{\mathrm{e}^{(l+2)ic^2s}-\mathrm{e}^{lic^2s}}{2ic^2} v^3 + 3 \mathrm{e}^{-2 ic^2t_n} \frac{\mathrm{e}^{-2 ic^2s}-\mathrm{e}^{lic^2s}}{-2ic^2}\vert v \vert^2 \overline{v}\\
& + \mathrm{e}^{-4ic^2t_n} \frac{\mathrm{e}^{(l-4)ic^2s}-\mathrm{e}^{lic^2s}}{-4ic^2} \overline{v}^3 \end{align*} which implies the assertion by Definition \ref{def:phi} of $\varphi_1$ and $\varphi_2$. \end{proof}
With the above lemmata at hand we can commence the construction of the second-order uniformly accurate scheme. \subsubsection{Uniform second-order discretization of Duhamel's formula}\label{sec:USDD} Our starting point is again Duhamel's perturbation formula (see \eqref{du}) \begin{align*}
u_\ast(t_n+\tau) & = \mathrm{e}^{i \tau \mathcal{A}_c} u_\ast(t_n)\\ & - \frac{i}{8} c \langle \nabla \rangle_c^{-1} \int_0^\tau \mathrm{e}^{i(\tau-s) \mathcal{A}_c} \mathrm{e}^{-i c^2 (t_n+s)} \left( \mathrm{e}^{ic^2(t_n+s)} u_\ast(t_n+s) + \mathrm{e}^{-ic^2(t_n+s)} \overline{u_\ast}(t_n+s)\right)^3 \mathrm{d}s\\ \end{align*}
which we split into two parts by separating the linear plus classical cubic part $|u_\ast|^2u_\ast$ from the terms involving $u_\ast^3, \overline u_\ast^3$ and $|u_\ast|^2\overline u_\ast$. More precisely, we set \begin{equation} \begin{aligned}\label{duha}
u_\ast(t_n+\tau) & = I_{\ast}(\tau,t_n,u_\ast) - \frac{i}{8} c \langle \nabla \rangle_c^{-1} I_{c^{2}}(\tau,t_n,u_\ast)
\end{aligned}
\end{equation}
with the linear as well as classical cubic part $|u_\ast|^2u_\ast$
\begin{equation}\label{Iast}
\begin{aligned}
I_{\ast}(\tau,t_n,u_\ast):= \mathrm{e}^{i \tau \mathcal{A}_c} u_\ast(t_n) - \frac{3i}{8} c \langle \nabla \rangle_c^{-1} \int_0^\tau \mathrm{e}^{i (\tau-s) \mathcal{A}_c} \vert u_\ast(t_n+s)\vert^2 u_\ast(t_n+s) \mathrm{d}s
\end{aligned} \end{equation}
and the terms involving $u_\ast^3, \overline u_\ast^3$ and $|u_\ast|^2\overline u_\ast$
\begin{equation}\label{Ic2}
\begin{aligned} I_{c^{2}}(\tau,t_n,u_\ast):= & \int_0^\tau \mathrm{e}^{i (\tau-s) \mathcal{A}_c} \Big( \mathrm{e}^{2 ic^2 (t_n+s)} u_\ast^3(t_n+s) \\&\qquad\qquad + 3 \mathrm{e}^{-2ic^2(t_n+s)} \vert u_\ast(t_n+s)\vert^2 \overline{u_\ast}(t_n+s)+ \mathrm{e}^{-4ic^2 (t_n+s)}\overline{u_\ast}^3(t_n+s)\Big) \mathrm{d}s. \end{aligned} \end{equation}
In order to obtain a second-order uniformly accurate scheme based on the decomposition \eqref{duha} we need to carefully analyze the highly-oscillatory phases in $I_{\ast}(\tau,t_n,u_\ast)$ and $I_{c^2}(\tau,t_n,u_\ast)$. We commence with the analysis of $I_{\ast}(\tau,t_n,u_\ast)$. \\
\emph{1.) First term $I_{\ast}(\tau,t_n,u_\ast)$:} By Lemma \ref{lem:doubleInt} we have that \begin{equation}\label{u18O} \begin{aligned} u_\ast(t_n+s) & = \mathrm{e}^{i s \mathcal{A}_c} u_\ast(t_n)- \frac{3i}{8}c\langle \nabla \rangle_c^{-1} \int_0^s \mathrm{e}^{i(s-\xi)\mathcal{A}_c} \left \vert \mathrm{e}^{i\xi\mathcal{A}_c}u_\ast(t_n) \right\vert^2 \left(\mathrm{e}^{i\xi\mathcal{A}_c}u_\ast(t_n) \right)
\mathrm{d}\xi
\\& - \frac{i}{8}c\langle \nabla \rangle_c^{-1} \Psi_{c^2}(t_n,s,u_\ast(t_n))+ \mathcal{R}_1(t_n,s, u_\ast) \end{aligned} \end{equation} with $\Psi_{c^2}$ defined in \eqref{psidef} and where the remainder $\mathcal{R}_1$ is of order $\mathcal{O}(s^2)$ uniformly in $c$. Plugging the approximation \eqref{u18O} into $I_{\ast}(\tau,t_n,u_\ast)$ defined in \eqref{Iast} yields that \begin{equation} \begin{aligned}\label{Iast1}
I_{\ast}(\tau,t_n,u_\ast) & = \mathrm{e}^{i \tau \mathcal{A}_c} u_\ast(t_n) - \frac{3i}{8} c \langle \nabla \rangle_c^{-1} \int_0^\tau \mathrm{e}^{i (\tau-s) \mathcal{A}_c} \vert u_\ast(t_n+s)\vert^2 u_\ast(t_n+s) \mathrm{d}s \\
&= \mathrm{e}^{i \tau \mathcal{A}_c} u_\ast(t_n) - \frac{3i}{8} c \langle \nabla \rangle_c^{-1} I_\ast^{1}(\tau,t_n,u_\ast) \\& + \frac{3i}{8} c \langle \nabla \rangle_c^{-1} \frac{i}{8} \int_0^\tau \mathrm{e}^{i(\tau-s) \mathcal{A}_c} \Big\{ 2 \left \vert \mathrm{e}^{i s \mathcal{A}_c} u_\ast(t_n)\right\vert^2 c \langle \nabla \rangle_c^{-1} \Psi_{c^2}(t_n,s,u_\ast(t_n)) \\&
- \left(\mathrm{e}^{is\mathcal{A}_c} u_\ast(t_n)\right)^2 c \langle \nabla \rangle_c^{-1} \overline{\Psi_{c^2}}(t_n,s,u_\ast(t_n))
\Big \}\mathrm{d} s
\\& +\mathcal{R}(\tau,t_n,u_\ast),
\end{aligned} \end{equation} where we have set \begin{equation*}\label{iast11} \begin{aligned} I_\ast^{1}(\tau,t_n,u_\ast) &:= \int_0^\tau \mathrm{e}^{i(\tau-s) \mathcal{A}_c} \Big\{
\left \vert \mathrm{e}^{i s \mathcal{A}_c} u_\ast(t_n)\right\vert^2 \mathrm{e}^{is \mathcal{A}_c} u_\ast(t_n) \\& \qquad- \frac{3i}{4} \left \vert \mathrm{e}^{i s \mathcal{A}_c} u_\ast(t_n)\right\vert^2 c \langle \nabla \rangle_c^{-1} \int_0^s \mathrm{e}^{i(s-\xi)\mathcal{A}_c} \vert \mathrm{e}^{i \xi \mathcal{A}_c}u_\ast(t_n)\vert^2 \mathrm{e}^{i \xi \mathcal{A}_c}u_\ast(t_n) \mathrm{d} \xi\\&\qquad + \frac{3i}{8}\left( \mathrm{e}^{is\mathcal{A}_c} u_\ast(t_n)\right)^2 c \langle \nabla \rangle_c^{-1} \int_0^s \mathrm{e}^{- i(s-\xi)\mathcal{A}_c} \vert \mathrm{e}^{i \xi \mathcal{A}_c}u_\ast(t_n)\vert^2 \mathrm{e}^{-i \xi \mathcal{A}_c} \overline{u_\ast(t_n)} \mathrm{d} \xi
\Big\} \mathrm{d} s \end{aligned} \end{equation*} and the remainder satisfies \begin{equation}\label{r3} \Vert \mathcal{R}(\tau,t_n,u_\ast)\Vert_r \leq \tau^3 k_{r}(M_4) \end{equation} for some constant $k_r(M_4)$ which depends on $M_4$, but is independent of $c$.
Lemma \ref{intpsiP} allows us to handle the highly-oscillatory integrals involving the function $\Psi_{c^2}$ in \eqref{Iast1}. Thus, in order to obtain a uniform second-order approximation of $ I_{\ast}(\tau,t_n,u_\ast)$ it remains to derive a suitable second-order approximation to $I_\ast^{1}(\tau,t_n,u_\ast)$.
\emph{1.1.) Approximation of $I_\ast^{1}(\tau,t_n,u_\ast)$:} The midpoint rule yields the following approximation \begin{equation}\label{Iastast} \begin{aligned} I_\ast^{1}(\tau,t_n,u_\ast)
& = \tau \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c} \Big\{
\left \vert \mathrm{e}^{i \frac{\tau}{2} \mathcal{A}_c} u_\ast(t_n)\right\vert^2 \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c} u_\ast(t_n)\\&
\qquad - \frac{3i}{4} \left \vert \mathrm{e}^{i \frac{\tau}{2} \mathcal{A}_c} u_\ast(t_n)\right\vert^2 c \langle \nabla \rangle_c^{-1} \int_0^{\tau/2} \mathrm{e}^{i(\frac{\tau}{2}-\xi)\mathcal{A}_c} \vert \mathrm{e}^{i \xi \mathcal{A}_c}u_\ast(t_n)\vert^2 \mathrm{e}^{i \xi \mathcal{A}_c}u_\ast(t_n) \mathrm{d} \xi\\&\qquad + \frac{3i}{8}\left( \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c} u_\ast(t_n)\right)^2 c \langle \nabla \rangle_c^{-1} \int_0^{\tau/2} \mathrm{e}^{- i(\frac{\tau}{2}-\xi)\mathcal{A}_c} \vert \mathrm{e}^{i \xi \mathcal{A}_c}u_\ast(t_n)\vert^2 \mathrm{e}^{-i \xi \mathcal{A}_c} \overline{u_\ast(t_n)} \mathrm{d} \xi
\Big\} \\&\qquad+\mathcal{R}(\tau,t_n,u_\ast(t_n)), \end{aligned} \end{equation} where the remainder satisfies thanks to \eqref{boundAc} and \eqref{cnabm} that \begin{equation}\label{r4} \begin{aligned} \Vert \mathcal{R}(\tau,t_n,u_\ast(t_n))\Vert_r \leq \tau^3 k_{r}(M_4) \end{aligned} \end{equation} with $k_r$ independent of $c$.
Next we approximate the two remaining integrals in \eqref{Iastast} with the right rectangular rule, i.e., \begin{equation}\label{Iast11} \begin{aligned}
\int_0^{\tau/2} \mathrm{e}^{ i(\frac{\tau}{2}-\xi)\mathcal{A}_c} \vert \mathrm{e}^{i \xi \mathcal{A}_c}u_\ast(t_n)\vert^2 \mathrm{e}^{ i \xi \mathcal{A}_c}u_\ast(t_n) \mathrm{d} \xi = \frac{\tau}{2} \vert \mathrm{e}^{i\frac{\tau}{2}\mathcal{A}_c}u_\ast(t_n)\vert^2 \mathrm{e}^{ i \frac{\tau}{2}\mathcal{A}_c}u_\ast(t_n)+ \mathcal{R}(\tau,t_n,u_\ast(t_n)), \end{aligned} \end{equation} where the remainder satisfies again thanks to \eqref{boundAc} that \begin{equation}\label{r5} \begin{aligned} \Vert \mathcal{R}(\tau,t_n,u_\ast(t_n))\Vert_r \leq \tau^2 k_{r}(M_4) \end{aligned} \end{equation} with $k_r$ independent of $c$.
Plugging \eqref{Iast11} into \eqref{Iastast} yields, with the notation \begin{align}\label{Usplit} \mathcal{U}_{\ast}(t_n) = \mathrm{e}^{i \frac{\tau}{2}\mathcal{A}_c} u_\ast(t_n) \end{align} that \begin{equation*} \begin{aligned}
I_\ast^{1}(\tau,t_n,u_\ast) & = \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c} \Big\{\tau
\left \vert \mathcal{U}_{\ast}(t_n) \right\vert^2 \mathcal{U}_{\ast}(t_n)\\&
\qquad - \frac{\tau^2}{2} \frac{3i}{4} \left \vert \mathcal{U}_{\ast}(t_n)\right\vert^2 c \langle \nabla \rangle_c^{-1}\vert \mathcal{U}_{\ast}(t_n)\vert^2 \mathcal{U}_{\ast}(t_n) +\frac{\tau^2}{2} \frac{3i}{8}\mathcal{U}_{\ast}(t_n)^2 c \langle \nabla \rangle_c^{-1} \vert \mathcal{U}_{\ast}(t_n)\vert^2 \overline{\mathcal{U}_{\ast}(t_n)} \Big\} \\&
\qquad+\mathcal{R}(\tau,t_n,u_\ast(t_n)),
\end{aligned}
\end{equation*}
where thanks to \eqref{r3}, \eqref{r4} and \eqref{r5} the remainder satisfies the bound $ \Vert \mathcal{R}(\tau,t_n,u_\ast(t_n))\Vert_r \leq \tau^2 k_{r}(M_4) $ with $k_r$ independent of $c$.
In order to obtain asymptotic convergence to the classical Strang splitting scheme \eqref{limitscheme2a} associated to the nonlinear Schr\"odinger limit \eqref{NLSlimit} we add and subtract the term
$$
\mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c}\frac{\tau^2}{2} \frac{3i}{8} \vert \mathcal{U}_{\ast}(t_n)\vert^4 \mathcal{U}_{\ast}(t_n)
$$
in the above approximation of $I_\ast^{1}(\tau,t_n,u_\ast)$. This yields that
\begin{equation}\label{midpoint} \begin{aligned}
& I_\ast^{1}(\tau,t_n,u_\ast) \\
& = \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c} \Big\{\tau
\left \vert \mathcal{U}_{\ast}(t_n) \right\vert^2 \mathcal{U}_{\ast}(t_n) - \frac{\tau^2}{2} \frac{3i}{8} \vert \mathcal{U}_{\ast}(t_n)\vert^4 \mathcal{U}_{\ast}(t_n)
\\&
\qquad - \frac{\tau^2}{2} \frac{3i}{4} \left \vert \mathcal{U}_{\ast}(t_n)\right\vert^2 \big(c \langle \nabla \rangle_c^{-1}-1\big)\vert \mathcal{U}_{\ast}(t_n)\vert^2 \mathcal{U}_{\ast}(t_n) +\frac{\tau^2}{2} \frac{3i}{8}\mathcal{U}_{\ast}(t_n)^2 \big(c \langle \nabla \rangle_c^{-1}-1\big) \vert \mathcal{U}_{\ast}(t_n)\vert^2 \overline{\mathcal{U}_{\ast}(t_n)} \Big\} \\&
\qquad+\mathcal{R}(\tau,t_n,u_\ast(t_n)). \end{aligned} \end{equation}
The above decomposition allows us a second-order approximation of $I_\ast(\tau,t_n,u_\ast)$ which holds uniformly in all $c$:
\emph{1.2.) Final approximation of $I_\ast(\tau,t_n,u_\ast)$:} Plugging \eqref{midpoint} into \eqref{Iast1} yields with the aid of Lemma \ref{intpsiP} that \begin{equation*}\label{IastFin} \begin{aligned} & I_\ast(\tau,t_n,u_\ast) = \mathrm{e}^{i \frac{\tau}{2}\mathcal{A}_c} \Big\{ \mathcal{U}_{\ast}(t_n) -\frac{3i}{8} \tau
\left \vert \mathcal{U}_{\ast}(t_n) \right\vert^2 \mathcal{U}_{\ast}(t_n) +\left(-\frac{3i}{8}\right)^2 \frac{\tau^2}{2} \vert \mathcal{U}_{\ast}(t_n)\vert^4 \mathcal{U}_{\ast}(t_n)\Big\}\\& - \tau \frac{3i}{8} \Big(c \langle \nabla \rangle_c^{-1}-1\Big) \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c}
\left \vert \mathcal{U}_{\ast}(t_n) \right\vert^2 \mathcal{U}_{\ast}(t_n) +\tau^2 \theta_{c\langle \nabla \rangle_c-1}\left(t_n,\tau,\mathcal{U}_{\ast}(t_n)\right)\\
&- \tau^2 \frac{3}{32} c\langle \nabla \rangle_c^{-1} \left \vert u_\ast(t_n)\right\vert^2c\langle \nabla \rangle_c^{-1} \vartheta_{c^2}(t_n,\tau,u_\ast(t_n)) + \tau^2\frac{3}{64}c\langle \nabla \rangle_c^{-1} \left( u_\ast(t_n)\right)^2 c\langle \nabla \rangle_c^{-1}\overline{\vartheta_{c^2}}(t_n,\tau,u_\ast(t_n))\\&
+ \mathcal{R}(\tau,t_n,u_\ast)
\end{aligned}
\end{equation*}
with a remainder $\mathcal{R}$ of order $\mathcal{O}(\tau^3)$ uniformly in $c$. The Taylor series expansion
$ \big \vert 1+ x + \frac{x^2}{2} - \mathrm{e}^x \big \vert = \mathcal{O}(x^3) $ furthermore allows us the following final representation of $I_\ast$:
\begin{equation}\label{IastFin}
\begin{aligned}
I_\ast(\tau,t_n,u_\ast) & = \mathrm{e}^{i \frac{\tau}{2}\mathcal{A}_c}\mathrm{exp}\left(-\frac{3i}{8} \tau \vert \mathcal{U}_{\ast}(t_n)\vert^2\right)\mathcal{U}_{\ast}(t_n) \\& - \tau \frac{3i}{8} \Big(c \langle \nabla \rangle_c^{-1}-1\Big) \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c}
\left \vert \mathcal{U}_{\ast}(t_n) \right\vert^2 \mathcal{U}_{\ast}(t_n) +\tau^2 \theta_{c\langle \nabla \rangle_c-1}\left(t_n,\tau,\mathcal{U}_{\ast}(t_n)\right)\\
&- \tau^2 \frac{3}{32} c\langle \nabla \rangle_c^{-1} \left \vert u_\ast(t_n)\right\vert^2c\langle \nabla \rangle_c^{-1} \vartheta_{c^2}(t_n,\tau,u_\ast(t_n)) \\&+ \tau^2\frac{3}{64}c\langle \nabla \rangle_c^{-1} \left( u_\ast(t_n)\right)^2 c\langle \nabla \rangle_c^{-1}\overline{\vartheta_{c^2}}(t_n,\tau,u_\ast(t_n))\\&
+ \mathcal{R}(\tau,t_n,u_\ast) \end{aligned} \end{equation} with \begin{equation}\label{def:theta} \begin{aligned} & \theta_{c\langle \nabla \rangle_c-1}(t_n,\tau,v) : = - \frac{1}{2} \frac{9}{64}\mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c} \Big(c \langle \nabla \rangle_c^{-1}-1\Big) \left \vert v\right\vert^4 v \\ &
-\frac{1}{2} \frac{9}{32} c \langle \nabla \rangle_c^{-1} \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c}\left \vert v\right\vert^2 \Big(c \langle \nabla \rangle_c^{-1} - 1 \Big) \vert v\vert^2 v
+\frac{1}{2} \frac{9}{64}c \langle \nabla \rangle_c^{-1} \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c}v^2 \Big(c \langle \nabla \rangle_c^{-1} - 1 \Big) \vert v\vert^2 \overline{v}
\\ \end{aligned} \end{equation} and where $\vartheta_{c^2}$ is defined in \eqref{defpsip} and the remainder $\mathcal{R}(\tau,t_n,u_\ast)$ satisfies \begin{equation}\label{r6} \Vert \mathcal{R}(\tau,t_n,u_\ast(t_n))\Vert_r \leq \tau^3 k_{r}(M_4) \end{equation} with $k_r$ independent of $c$.
The approximation of $I_\ast(\tau,t_n,u_\ast)$ given in \eqref{IastFin} yields the first terms in our numerical scheme. In order to obtain a full approximation to $u_\ast(t_n+\tau)$ in \eqref{duha} we next derive a second-order approximation to $I_{c^2}(\tau,t_n,u_\ast)$.\\
\emph{2.) Second term $I_{c^2}(\tau,t_n,u_\ast)$:} Applying the second approximation in Lemma \ref{lem:doubleInt} yields together with Lemma \ref{lem:expo} and by the definition of $I_{c^2}(\tau,t_n,u_\ast)$ in \eqref{Ic2} that \begin{equation*}\label{du2s} \begin{aligned}
I_{c^2}(\tau,t_n,u_\ast) & = \int_0^\tau \mathrm{e}^{i(\tau-s)\mathcal{A}_c} \Big\{ \mathrm{e}^{2ic^2(t_n+s)} \left(\mathrm{e}^{is\mathcal{A}_c}u_\ast(t_n)\right)^3 + 3\mathrm{e}^{-2ic^2(t_n+s)} \left\vert\mathrm{e}^{is\mathcal{A}_c}u_\ast(t_n)\right\vert^2 \mathrm{e}^{-is\mathcal{A}_c}\overline{u_\ast}(t_n) \\&+ \mathrm{e}^{-4ic^2(t_n+s)} \left(\mathrm{e}^{-is\mathcal{A}_c}\overline{u_\ast}(t_n)\right)^3 \Big\}\mathrm{d} s \\&+ \int_0^\tau \Big\{- \frac{3i}{8} \mathrm{e}^{2ic^2(t_n+s)}\left(u_\ast(t_n)\right)^2 c\langle \nabla \rangle_c^{-1} \Big[ 3 s \vert u_\ast(t_n)\vert^2 u_\ast(t_n) + \Psi_{c^2}(t_n,s,u_\ast(t_n))\Big]\\ & + 3 \mathrm{e}^{-2ic^2(t_n+s)}\Big(- \frac{i}{8} \left( \overline{u_\ast}(t_n)\right)^2 c\langle \nabla \rangle_c^{-1} \Big[ 3 s \vert u_\ast(t_n)\vert^2 u_\ast(t_n) + \Psi_{c^2}(t_n,s,u_\ast(t_n))\Big]\\
& + \frac{2 i}{8} \left\vertu_\ast(t_n)\right\vert^2 c\langle \nabla \rangle_c^{-1} \Big[ 3 s \vert u_\ast(t_n)\vert^2 \overline{u_\ast}(t_n) + \overline{\Psi}_{c^2}(t_n,s,u_\ast(t_n))\Big] \Big)\\ & +\frac{3i}{8}\mathrm{e}^{-4ic^2(t_n+s)} \left( \overline{u_\ast}(t_n)\right)^2 c\langle \nabla \rangle_c^{-1} \Big[ 3 s \vert u_\ast(t_n)\vert^2 \overline{u_\ast}(t_n) + \overline{\Psi}_{c^2}(t_n,s,u_\ast(t_n))\Big]\Big\} \mathrm{d} s\\ & + \mathcal{R}(t_n,\tau,u_\ast) \end{aligned} \end{equation*} with $\Psi_{c^2}$ defined in \eqref{psidef} and where thanks to Lemma \ref{lem:expo}, \ref{lem:doubleInt} and the fact that $\Psi_{c^2}$ is of order one in $s$ uniformly in $c$ the remainder satisfies $ \Vert \mathcal{R}(\tau,t_n,u_\ast(t_n))\Vert_r \leq \tau^3 k_{r}(M_4) $ with $k_r$ independent of $c$.
Lemma \ref{lemI1}, \ref{lemI2} together with Lemma \ref{lem:intPsic} thus allow us the following expansion of $I_{c^2}$: We have \begin{equation}\label{Ic2calc} \begin{aligned} & I_{c^2}(\tau,t_n,u_\ast) = I^1_{c^2}(\tau,t_n,u_\ast) + \mathcal{R}(t_n,\tau,u_\ast) \end{aligned} \end{equation} with the highly-oscillatory term \begin{equation}\label{IOkti} \begin{aligned} & I^1_{c^2}(\tau,t_n,u_\ast): =\textstyle \tau \mathrm{e}^{2 i c^2 t_n} \mathrm{e}^{i \tau \mathcal{A}_c} \varphi_1\left(i \tau ( 2c^2 - \frac{1}{2} \Delta)\right) u_\ast^3(t_n)\\& \textstyle+ i \tau^2 \mathrm{e}^{2 i c^2 t_n} e^{i \tau \mathcal{A}_c} \varphi_2\left(i \tau ( 2c^2 - \frac{1}{2} \Delta)\right) \Big[ (\frac12\Delta-\mathcal{A}_c)u_\ast^3(t_n) + 3 u_\ast^2(t_n) \mathcal{A}_c u_\ast(t_n) \Big]\\ &+ 3\tau \mathrm{e}^{-2ic^2t_n} \mathrm{e}^{i \tau \mathcal{A}_c} \varphi_1(i\tau(-2c^2-\mathcal{A}_c))\left\vertu_\ast(t_n)\right\vert^2 \overline{u_\ast}(t_n) \\&+ 3 i \tau^2\mathrm{e}^{-2ic^2t_n} \mathrm{e}^{i \tau \mathcal{A}_c}\varphi_2(i\tau(-2c^2-\mathcal{A}_c)) \Big[ \overline{u_\ast}^2(t_n)\mathcal{A}_c u_\ast(t_n) - 2 \vert u_\ast(t_n)\vert^2 \mathcal{A}_c \overline{u_\ast}(t_n) \Big]\\ & + \tau \mathrm{e}^{-4ic^2t_n} \mathrm{e}^{i \tau \mathcal{A}_c}\varphi_1(i\tau(-4c^2 - \mathcal{A}_c)) \overline{u_\ast}^3(t_n) - i \tau^2\mathrm{e}^{-4ic^2t_n} \mathrm{e}^{i \tau \mathcal{A}_c}\varphi_2(i\tau(-4c^2-\mathcal{A}_c)) 3 \overline{u_\ast}^2(t_n) \mathcal{A}_c \overline{u_\ast}(t_n)\\ & - \tau^2 \frac{3i}{8} \mathrm{e}^{2ic^2t_n} \left(u_\ast(t_n)\right)^2 c\langle \nabla \rangle_c^{-1} \Big[3\varphi_2(2ic^2\tau)\vert u_\ast(t_n)\vert^2 u_\ast(t_n) + \Omega_{c^2,2,}(t_n,\tau,u_\ast(t_n)) \Big]\\ & - \tau^2 \frac{3i}{8} \mathrm{e}^{-2ic^2t_n} \left( \overline{u_\ast}(t_n)\right)^2 c\langle \nabla \rangle_c^{-1} \Big[3 \varphi_2(-2ic^2\tau) \vert u_\ast(t_n)\vert^2 u_\ast(t_n) + \Omega_{c^2,-2}(t_n,\tau,u_\ast(t_n))\Big] \\
& + \tau^2 \frac{6 i}{8} \mathrm{e}^{-2ic^2t_n} \left\vert u_\ast(t_n)\right\vert^2 c\langle \nabla \rangle_c^{-1} \Big[3 \varphi_2(-2ic^2\tau) \vert u_\ast(t_n)\vert^2 \overline{u_\ast}(t_n)+ \overline{\Omega}_{c^2,-2}(t_n,\tau,u_\ast(t_n))
\Big] \\ & +\tau^2 \frac{ 3 i}{8}\mathrm{e}^{-4ic^2t_n} \left( \overline{u_\ast}(t_n)\right)^2 c\langle \nabla \rangle_c^{-1} \Big[3 \varphi_2(-4ic^2\tau)\vert u_\ast(t_n)\vert^2 \overline{u_\ast}(t_n) + \overline{\Omega}_{c^2,-4}(t_n,\tau,u_\ast(t_n))\Big] \\ & + \mathcal{R}(t_n,\tau,u_\ast),\\ \end{aligned} \end{equation} where $\Omega_{c^2,l}$ is defined in Lemma \ref{lem:intPsic} and the remainder satisfies \begin{equation}\label{R9} \Vert \mathcal{R}(t_n,\tau,u_\ast)\Vert_r \leq \tau^3 k_r(M_4), \end{equation} with $k_r$ independent of $c$.\\
\emph{3.) Final approximation of $u_\ast(t_n+\tau)$:} Plugging \eqref{IastFin} as well as \eqref{Ic2calc} into \eqref{duha} builds the basis of our second-order scheme: As a numerical approximation to the exact solution $u_\ast$ at time $t_{n+1}$ we take the second-order uniform accurate exponential-type integrator: $\mathcal{U}_{\ast}^n = \mathrm{e}^{i \frac{\tau}{2}\mathcal{A}_c} u_\ast^n$ and \begin{equation} \label{scheme2} \begin{aligned}
u_\ast^{n+1} & = \mathrm{e}^{i \frac{\tau}{2} \mathcal{A}_c} \mathrm{e}^{- i \tau \frac{3}{8} \left \vert \mathcal{U}_{\ast}^n \right\vert^2} \mathcal{U}_{\ast}^n \\&
- \tau \frac{3i}{8} \Big(c \langle \nabla \rangle_c^{-1}-1\Big) \mathrm{e}^{i\frac{\tau}{2} \mathcal{A}_c} \left \vert \mathcal{U}_{\ast}^n\right\vert^2 \mathcal{U}_{\ast}^n +\tau^2 \theta_{c\langle \nabla \rangle_c-1}\left(t_n,\tau,\mathcal{U}_{\ast}^n \right)\\&- \tau^2 \frac{3}{64} c\langle \nabla \rangle_c^{-1} \Big[2 \left \vert u_\ast^n\right\vert^2c\langle \nabla \rangle_c^{-1} \vartheta_{c^2}(t_n,\tau,u_\ast^n) - \left( u_\ast^n\right)^2 c\langle \nabla \rangle_c^{-1}\overline{\vartheta_{c^2}}(t_n,\tau,u_\ast^n)\Big]\\ & - \frac{i}{8} c \langle \nabla \rangle_c^{-1} I^1_{c^2}(\tau,t_n,u_\ast^n), \end{aligned} \end{equation} where $ I^1_{c^2}(\tau,t_n,u_\ast^n)$ is defined in \eqref{IOkti} and with $\varphi_1, \varphi_2$ given in Definition \ref{def:phi}, $\theta_{c\langle \nabla \rangle_c-1}$ given in \eqref{def:theta}, $\vartheta_{c^2}$ in \eqref{defpsip} and $\Omega_{c^2,l}$ in \eqref{def:om}.
\subsection{Convergence analysis}\label{sec:convA2} The exponential-type integration scheme \eqref{scheme2} converges (by construction) with second-order in time uniformly with respect to $c$.
\begin{thm}[Convergence bound for the second-order scheme]\label{them:con2} Fix $r>d/2$ and assume that \begin{equation}\label{eq:urged2} \Vert z(0)\Vert_{r+4} + \Vert c^{-1}\langle \nabla \rangle_c^{-1}z'(0)\Vert_{r+4} \leq M_4 \end{equation} uniformly in $c$. For $u_\ast^n$ defined in \eqref{scheme2} we set \[ z^n := \frac{1}{2} \left( \mathrm{e}^{ic^2t_n} u_\ast^n + \mathrm{e}^{-ic^2 t_n} \overline{u_\ast^n}\right). \] Then, there exists a $T_r>0$ and $\tau_0>0$ such that for all $\tau \leq \tau_0$ and $t_n\leq T_r$ we have for all $c >0$ that \[ \Vert z(t_n)-z^n\Vert_r \leq \tau^2 K_{1,r,M_4} \mathrm{e}^{t_n K_{2,r,M}} \leq \tau^2 K^\ast_{r,M,M_4,t_n}, \] where the constants $K_{1,r,M_2},K_{2,r,M}$ and $K^\ast_{r,M,M_4,t_n}$ can be chosen independently of $c$. \end{thm} \begin{proof} First note that the regularity assumption on the initial data in \eqref{eq:urged2} implies the regularity Assumption \ref{ass:reg2} on $u_\ast(t)$, i.e., there exists a $T_r>0$ such that \[ \sup_{0 \leq t \leq T} \Vert u_\ast(t)\Vert_{r+4} \leq k(M_4) \] for some constant $k$ that depends on $M_4$ and $T_r$, but can be chosen independently of $c$.
In the following let $\phi^t$ denote the exact flow of \eqref{eq:ua}, i.e., $u_\ast(t_{n+1}) = \phi^\tau(u_\ast(t_n))$ and let $\Phi^\tau$ denote the numerical flow defined in \eqref{scheme2}, i.e., \[ u_\ast^{n+1} = \Phi^\tau(u_\ast^n). \] Taking the difference of \eqref{du} and \eqref{scheme2} yields that \begin{equation}\label{glob02} \begin{aligned} u_\ast(t_{n+1}) - u_\ast^{n+1}& = \phi^\tau(u_\ast(t_n)) - \Phi^\tau(u_\ast^n)\\ &= \Phi^\tau(u_\ast(t_n)) - \Phi^\tau(u_\ast^n) +
\phi^\tau(u_\ast(t_n)) - \Phi^\tau(u_\ast(t_n)). \end{aligned} \end{equation}
\emph{Local error bound:} With the aid of the expansion \eqref{IastFin} and \eqref{Ic2calc} we obtain by the representation of the exact solution in \eqref{duha} together with the error bounds \eqref{R9} and \eqref{r6} that \begin{equation}\label{local2} \Vert \phi^\tau(u_\ast(t_n)) - \Phi^\tau(u_\ast(t_n)) \Vert_r = \Vert \mathcal{R}(\tau,t_n,u_\ast)\Vert_r \leq \tau^3 k_r(M_4) \end{equation} for some constant $k_r$ which depends on $M_4$ and $r$, but can be chosen independently of $c$.
\emph{Stability bound:} Note that by the definition of $ \varphi_2$ in Definition \ref{def:phi}, $\theta_{c\langle \nabla \rangle_c-1}$ in \eqref{def:theta}, $\vartheta_{c^2}$ in \eqref{defpsip} and $\Omega_{c^2,l}$ in \eqref{def:om} we have for $l=-4,-2,2$ that \begin{equation}\label{opB} \begin{aligned} \tau^2 \Big( \Vert \varphi_2(lic^2 t) (f-g)\Vert_r + \Vert \Omega_{c^2,l}(t_n,\tau,f)- \Omega_{c^2,l}(t_n,\tau,g)\Vert_r + \Vert \vartheta_{c^2}(t_n,\tau,f)-\vartheta_{c^2}(t_n,\tau,g) \Vert_r \Big)\\ \leq \tau k_r\left(\Vert f\Vert_r,\Vert g\Vert_r\right) \Vert f - g\Vert_r \end{aligned} \end{equation} for some constant $k_r$ independent of $c$. Together with the bound \eqref{cnabm}, the definition of $\varphi_1$ in Definition \ref{def:phi} and the stability estimates \eqref{stab21} and \eqref{stab22} we thus obtain as long as $\Vert u_\ast(t_n)\Vert_r \leq M$ and $\Vert u_\ast^n\Vert_R \leq 2M$ that \begin{equation}\label{stab2} \Vert \Phi^\tau(u_\ast(t_n)) - \Phi^\tau(u_\ast^n) \Vert_r \leq \Vert u_\ast(t_n) - u_\ast^n\Vert_r +\tau K_{r,M} \Vert u_\ast(t_n) - u_\ast^n\Vert_r, \end{equation} where the constant $K_{r,M}$ depends on $r$ and $M$, but can be chosen independently of $c$.
\emph{Global error bound:} Plugging the stability bound \eqref{stab2} as well as the local error bound \eqref{local2} into \eqref{glob02} yields by a bootstrap argument that \begin{align}\label{conus2} \left \Vert u_\ast(t_{n}) - u_\ast^{n} \right\Vert_r \leq \tau^2 K_{1,r,M_4} \mathrm{e}^{t_n K_{2,r,M}}, \end{align} where the constants are uniform in $c$. Note that as $u = v$ we have by \eqref{eq:zuv} and \eqref{psi} that \begin{align*} \Vert z(t_n) - z^n\Vert_r & = \textstyle \left \Vert\frac12 \big( u(t_n) + \overline{u(t_n)}\big) -\frac12 \big(\mathrm{e}^{ic^2 t_n} u_\ast^n + \mathrm{e}^{-ic^2t_n} \overline{u_\ast^n}\big)\right\Vert\\ & \leq \Vert \mathrm{e}^{ic^2t_n} (u_\ast(t_n)-u_\ast^n)\Vert_r = \Vert u_\ast(t_n)-u_\ast^n\Vert_r . \end{align*} Together with the bound in \eqref{conus2} this completes the proof. \end{proof} \begin{rem}[Fractional convergence and convergence in $L^2$]\label{remFrac} A fractional convergence result as Theorem \ref{them:con1Frac} for the first-order scheme also holds for the second-order exponential-type integrator \eqref{scheme2}: Fix $r>d/2$ and let $0\leq \gamma \leq 1$. Assume that \[ \Vert z(0)\Vert_{r+2+2\gamma} + \Vert c^{-1}\langle \nabla \rangle_c^{-1}z'(0)\Vert_{r+2+2\gamma} \leq M_{2+2\gamma}. \] Then, the scheme \eqref{scheme2} is convergent of order $\tau^{1+\gamma}$ in $H^r$ uniformly with respect to $c$.
Furthermore, for initial values satisfying \[ \Vert z(0)\Vert_{4} + \Vert c^{-1}\langle \nabla \rangle_c^{-1}z'(0)\Vert_{4} \leq M_{4,0} \] the exponential-type integration scheme \eqref{scheme2} is second-order convergent in $L^2$ uniformly with respect to $c$ by the strategy presented in \cite{Lubich08}. \end{rem}
In analogy to Remark \ref{rem:limit1} we make the following observation: For sufficiently smooth solutions the exponential-type integration scheme \eqref{scheme2} converges in the limit $c \to \infty$ to the classical Strang splitting of the corresponding nonlinear Schr\"odinger limit equation \eqref{NLSlimit}.
\begin{rem}[Approximation in the non relativistic limit $c \to \infty$] The exponential-type integration scheme \eqref{scheme2} corresponds for sufficiently smooth solutions in the limit $u_\ast^n\stackrel{c\to \infty}{\longrightarrow} u_{\ast,\infty}^n$, essentially to the Strang Splitting (\cite{Lubich08,Faou12}) \begin{equation}\label{limitStrang} \begin{aligned} u_{\ast, \infty}^{n+1} &=\mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2}} \mathrm{e}^{-i \tau \frac{3}{8} \vert \mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2}}u_{\ast, \infty}^n\vert^2}\mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2}}u_{\ast, \infty}^n,\qquad u_{\ast,\infty}^0 = \varphi - i \gamma, \end{aligned} \end{equation} for the cubic nonlinear Schr\"odinger limit system \eqref{NLSlimit}.
More precisely, the following Lemma holds. \end{rem}
\begin{lem}\label{rem:limit2} Fix $r>d/2$. Assume that \begin{equation*} \Vert z(0) \Vert_{r+3} +\Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0)\Vert_{r+3} \leq M_{3} \end{equation*} for some $\varepsilon>0$ uniformly in $c$ and let the initial value approximation (there exist functions $\varphi,\gamma$ such that)
\begin{align*} \Vert z(0)- \gamma\Vert_r + \Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0) - \varphi\Vert_r \leq k_r c^{-1} \end{align*} hold for some constant $k_r$ independent of $c$.
Then, there exists a $T>0$ and $\tau_0>0$ such that for all $\tau \leq \tau_0$ the difference of the second-order scheme \eqref{scheme2} for system \eqref{eq:ua} and the Strang splitting \eqref{limitStrang} for the limit Schr\"odinger equation \eqref{NLSlimit} satisfies for $ t_n \leq T$ and all $c >0$ with \begin{equation*} \tau c \geq 1 \end{equation*} that \[ \Vert u_\ast^n- u_{\ast, \infty}^{n} \Vert_r \leq c^{-1} k_r(M_{3},T) \] for some constant $k_{r}$ that depends on $M_{3}$ and $T$, but is independent of $c$. \end{lem} \begin{proof} The proof follows the line of argumentation to the proof of Lemma \ref{rem:limit1} by noting that for $l=-4,-2$ and $n=-4,-2,2$ \[ \tau \Big( \Vert \varphi_j(2i \tau \langle \nabla \rangle_c^2) \Vert_r + \Vert \varphi_j\big( i \tau (lc^2 - \mathcal{A}_c)\big)\Vert_r + \Vert \varphi_j( n i c^2 \tau) \Vert_r \Big) \leq k_r c^{-2} \] for some constant $k_r$ independent of $c$. \end{proof}
\subsection{Simplifications in the ``weakly to strongly non-relativistic limit regime''}\label{sec:limit2} In the ``weakly to strongly non-relativistic limit regime'', i.e., for large values of $c$, we may again (substantially) simplify the second-order scheme \eqref{scheme2} and nevertheless obtain a well suited, second-order approximation to $u_\ast(t_n)$ in \eqref{eq:ua}.
\begin{rem}[Limit scheme \cite{FS13}]\label{remarkStrangOk19} For sufficiently large values of $c$ and sufficiently smooth solutions, more precisely, if \[
\Vert z(0) \Vert_{r+4} + \Vert c^{-1}\langle \nabla \rangle_c^{-1} z'(0)\Vert_{r+4}\leq M_{4}\quad \text{and}\quad \tau c > 1 \] we may take instead of \eqref{scheme2} the classical Strang splitting (see \cite{Lubich08,Faou12}) for the nonlinear Schr\"odinger limit equation \eqref{NLSlimit}, namely, \begin{equation}\label{limitscheme2} \begin{aligned} u_{\ast,\infty}^{n+1} &=\mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2} } \mathrm{e}^{-i \tau \frac{3}{8}\vert \mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2} } u_{\ast,\infty}^n\vert^2 }\mathrm{e}^{-i \frac{\tau}{2} \frac{\Delta}{2} } u_{\ast,\infty}^n\\ \end{aligned} \end{equation} as a second-order numerical approximation to $u_\ast(t_n)$ in \eqref{eq:ua}. The assertion follows from \cite{FS13} thanks to the approximation \begin{align*} \Vert u_\ast(t_n) - u_{\ast,\infty}^{n} \Vert_r & \leq \Vert u_\ast(t_n) - u_{\ast,\infty}(t_n)\Vert_r + \Vert u_{\ast,\infty}(t_n) - u_{\ast,\infty}^{n} \Vert_r = \mathcal{O}\big( c^{-2}+\tau^2\big). \end{align*} \end{rem}
\section{Numerical experiments} In this section we numerically confirm first-, respectively, second-order convergence uniformly in $c$ of the exponential-type integration schemes \eqref{scheme1} and \eqref{scheme2}. In the numerical experiments we use a standard Fourier pseudospectral method for the space discretization with the largest Fourier mode $K = 2^{10}$ (i.e., the spatial mesh size $\Delta x = 0.0061$) and integrate up to $T = 0.1$. In Figure \ref{fig1} we plot (double logarithmic) the time-step size versus the error measured in a discrete $H^1$ norm of the first-order scheme \eqref{scheme1} and the second-order scheme \eqref{scheme2} with initial values \begin{align*}
& z(0,x) = \frac12 \frac{ \mathrm{cos}(3 x)^2 \mathrm{sin}(2x)}{2-\mathrm{cos}(x)},\qquad \partial_t z(0,x) = c^2 \frac12 \frac{\mathrm{sin}(x)\mathrm{cos}(2x)}{2-\mathrm{cos}(x)} \end{align*} for different values $c = 1, 5, 10, 50, 100, 500, 1000, 5000, 10000 .$
\begin{figure}
\caption{Error of the first-, respectively, second-order exponential-type integration scheme \eqref{scheme1} and \eqref{scheme2}. The slope of the dotted and dashed line is one and two, respectively.}
\label{fig1}
\end{figure}
\end{document} | arXiv | {
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\begin{document}
\title[Non-parametric sets over number fields]{Non-parametric sets of regular realizations over number fields}
\author{Joachim K{\"{o}}nig}
\email{koenig.joach@technion.ac.il}
\address{Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel}
\author{Fran\c cois Legrand}
\email{legrandfranc@technion.ac.il}
\address{Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel}
\date{\today}
\maketitle
\begin{abstract} Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular of a certain type, under a conjectural ``uniform Faltings' theorem". \end{abstract}
\section{Introduction}
\subsection{Topic of the paper}
Given a number field $k$, the {\it{Inverse Galois Problem}} (over $k$) asks whether every finite group $G$ occurs as the Galois group of a Galois extension $F/k$. A classical way to obtain such an extension of $k$ with Galois group $G$ consists in introducing an indeterminate $T$ and in producing a Galois extension $E/k(T)$ with Galois group $G$ and $E/k$ {\it{regular}}\footnote{i.e., $E \cap \overline{\mathbb{Q}}=k$. See \S2.1 for basic terminology. For short, we say that $E/k(T)$ is a ``$k$-regular Galois extension with Galois group $G$" and that $G$ is a ``regular Galois group over $k$" if there exists such an extension.}. Then, by Hilbert's irreducibility theorem, there exist infinitely many $t_0 \in k$ such that the {\it{specialization}} $E_{t_0}/k$ of $E/k(T)$ at $t_0$ has Galois group $G$. Many finite groups have been realized as Galois groups over $k$ by this method; see, e.g., \cite{MM99}.
Then, one can ask whether this geometric approach to solve the Inverse Galois Problem is optimal. This is known as the {\it{Beckmann-Black Problem}}: given a finite group $G$, is every Galois extension $F/k$ with Galois group $G$ a specialization of some $k$-regular Galois extension $E_F/k(T)$ with the same Galois group? Despite its naive appearance, results on the Beckmann-Black Problem are sparse. Namely, the answer is known to be {\it{Yes}} for abelian groups, symmetric groups, alternating groups, and some dihedral groups; see, e.g., \cite[Theorem 2.2]{Deb01b} for more details and references. Moreover, no counter-example is known.
Here, we consider the following generalization of the Beckmann-Black Problem. Given a finite group $G$ and a positive integer $n$, we say that a set $S$ of $k$-regular Galois extensions of $k(T)$ with Galois group $G$ is {\it{$n$-parametric over $k$}} if any given $n$ Galois extensions of $k$ with Galois group $G$ occur as specializations of a same extension $E/k(T) \in S$; see Definition \ref{def1}. With this phrasing, the Beckmann-Black Problem asks whether every finite group $G$ has a 1-parametric set over $k$. The aim of this paper consists in proving that there exist many finite groups $G$ that have no $n$-parametric set over $k$ of a certain type.
\subsection{Finite sets}
Studying the parametricity of a given finite set $S$ of $k$-regular Galois extensions of $k(T)$ with Galois group $G$ has been started by the second author, in the particular case where the set $S$ consists of a {\it{single}} extension $E/k(T)$; see \cite{Leg15, Leg16a}. Following the terminology there, we say for short that the extension $E/k(T)$ is {\it{parametric over $k$}} if every Galois extension of $k$ with Galois group $G$ is a specialization of $E/k(T)$. In this context, the above definition of $n$-parametricity does of course not depend on the integer $n$.
Strikingly, no finite group is known not to have a parametric extension over $k$ \footnote{Hence, no finite group is known not to have a finite 1-parametric set over $k$.} while only a few are known to have one. For example, recall that only four finite groups (namely, $\{1\}$, $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$, and $S_3$) are known to have a parametric extension over $\mathbb{Q}$. This follows from the classical fact that these groups $G$ are the only ones to have a {\it{one parameter generic polynomial over $\mathbb{Q}$}} (see \cite[page 194]{JLY02}), i.e., a polynomial $P(T,Y) \in \mathbb{Q}(T)[Y]$ with Galois group $G$ which provides all the Galois extensions with Galois group $G$ of any given field $L$ of characteristic zero (by specializing the indeterminate $T$ properly). Of course, there might exist finite groups besides these four groups with a finite 1-parametric set over $\mathbb{Q}$ (or even with a parametric extension over $\mathbb{Q}$). But there is no such example available in the literature!
Here, we offer a general approach to show that a given finite group $G$ has no finite 1-parametric set over $k$ \footnote{with the possible exception of the empty set; see Remark \ref{rem1}.}. Below, we give a few examples in the specific case $k=\mathbb{Q}$ which illustrate the wide variety of groups we can cover. See Theorems 5.1-3 for our precise results.
\begin{theorem} \label{thm intro} Let $G$ be a non-trivial finite group whose order is neither a prime number nor 4.
\noindent {\rm{(1)}} Assume that one of the following conditions holds:
{\rm{(a)}} $G$ has order prime to 6,
{\rm{(b)}} $G$ is abelian, but none of the following groups: $\mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$,
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$, $(\mathbb{Z}/3\mathbb{Z})^2$, $(\mathbb{Z}/2\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^4$,
{\rm{(c)}} $G$ is the dihedral group $D_n$ (with $2n$ elements), where $n$ is any
positive integer which is neither a prime number nor in $\{4,6,8,9,12\}$,
{\rm{(d)}} $G={\rm{GL}}_n(\mathbb{F}_q)$, where $n \geq 2$ is any positive integer and $q \geq 3$ is
any prime power such that $(n,q) \not=(2,3)$.
\noindent Then, $G$ has no finite 1-parametric set over $\mathbb{Q}$.
\noindent {\rm{(2)}} Assume that one of the following conditions holds:
{\rm{(a)}} $G$ is abelian,
{\rm{(b)}} $G=S_n$ ($n \geq 6$).
\noindent Then, $G$ has no parametric extension over $\mathbb{Q}$. \end{theorem}
\noindent All our examples are obtained as applications of several criteria which ensure that a given (finite) set $S$ of $k$-regular Galois extensions of $k(T)$ with Galois group $G$ is not 1-parametric over $k$; see \S3. These criteria can be applied to various finite groups and, for the sake of simplicity, we have only considered ``nice" families of groups here. The interested reader is then invited to give more examples. The proofs of our criteria and our explicit examples involve a wide variety of tools, such as a finiteness result for specializations (see \S2.3), solving embedding problems (see \S4), various results on regular Galois realizations (see, e.g., Lemma 6.7), as well as non-existence results of rational points on twisted (hyper)elliptic curves (see \S8). However, our method requires the existence of a non-trivial proper normal subgroup of $G$ with several properties. In particular, this cannot be used if $G$ is a simple group\footnote{In a recent paper of Neftin and the two authors, a completely different method is used to show that $A_n$ ($n \geq 4$) has no finite 1-parametric set over $k$; see \cite[Corollary 7.3]{KLN17}. We also mention this weaker result (which can be used with simple or non-simple groups): any given non-trivial regular Galois group $G$ over $k$ is the Galois group of a $k$-regular Galois extension of $k(T)$ that is not parametric over $k$; see \cite[Theorem 1.3]{Leg16a} and \cite[Theorem 2.2]{Koe17}.}.
Theorem 1.1 can be compared with a recent result of D\`ebes \cite{Deb16}. It is shown there that there exist many finite groups $G$ with this property: no $k$-regular Galois extension $E/k(T)$ with Galois group $G$ is ``parametric over $k(U)$" (with $U$ a new indeterminate), i.e., all the specializations of the extension $E(U)/k(U)(T)$ cannot provide all the Galois extensions of $k(U)$ with Galois group $G$. As an immediate consequence of \cite[Remark 2.3]{Deb16}, we obtain that this weaker conclusion of D\`ebes holds for every finite group $G$ which is covered by our method.
\subsection{Infinite sets}
Finding finite groups with no infinite 1-parametric set over $k$ is a much more challenging problem as this would disprove the Beckmann-Black Problem over the number field $k$ and as such sets occur more often than in the finite case. Stronger results can even happen: for example, recall that, as the polynomial $Y^N + T_1 Y^{N-1} + \cdots + T_{N-1}Y + T_N \in \mathbb{Q}(T_1,\dots,T_N)[Y]$ is generic, the set of all $k$-regular Galois extensions of $k(T)$ with Galois group $S_N$ ($N \geq 1$) is $n$-parametric over $k$ for every positive integer $n$; see \cite[Proposition 3.3.10]{JLY02}.
Here, we consider the set of all $k$-regular Galois extensions of $k(T)$ with Galois group $G$ and (at most) $r_0$ {\it{branch points}}, where $r_0$ is any given positive integer. This is motivated by the fact that some of these sets are known to be 1-parametric over $k$. For example,
\noindent - if $G$ is abelian, then the set of all $k$-regular Galois extensions of $k(T)$ with Galois group $G$ and exactly $r_0$ branch points is $1$-parametric over $k$, unless this set is empty; see \cite{Deb99a},
\noindent - if $G=S_n$ ($n \geq 1$) and $k=\mathbb{Q}$, then the above set is $1$-parametric over $k$ as soon as $r_0 \geq 2n-1$; see \cite[Proposition 1.2]{Bec94}.
\noindent We prove that, for many finite groups $G$, the set of all $k$-regular Galois extensions of $k(T)$ with Galois group $G$ and at most $r_0$ branch points is not $n$-parametric over $k$ for large $n$ (depending on $r_0$). This is shown to hold under a conjectural ``uniform Faltings' theorem", asserting that the number of $k$-rational points on a given smooth curve $X$ defined over $k$ with genus $g$ at least 2 is bounded by a constant $B$ depending only on $g$ and $k$ (and not on $X$). See \S2.3, as well as Theorems 5.4-5, for precise statements.
\subsection{Framework of the paper}
This paper is organized as follows. In \S2, we recall some background that is used in the sequel. \S3 contains our criteria to show the non-parametricity of a given set of regular realizations of a finite group $G$ over a number field $k$. In \S4, we solve embedding problems which are part of the assumptions of the criteria of \S3. \S5 is devoted to the statements of Theorems 5.1-5, while proofs are given in \S6. As to \S7 and \S8, they are devoted to auxiliary results that are used throughout the paper.
{\bf{Acknowledgments.}} We wish to thank Danny Neftin for helpful discussions, as well as the anonymous referee for many valuable comments. The first author is supported by the Israel Science Foundation (grant No. 577/15). The second author is supported by the Israel Science Foundation (grants No. 696/13, 40/14, and 577/15).
\section{Basics}
Below, we survey some standard machinery that will be used in the sequel. In \S2.1, we review some standard background on function field extensions. \S2.2 is devoted to basic properties of parametric sets and parametric extensions. Finally, we give finiteness results on the number of specialization points with prescribed specialization in \S2.3.
For this section, let $k$ be a number field, $G$ a finite group, and $T$ an indeterminate over $k$.
\subsection{Background on function field extensions}
Let $E/k(T)$ be a Galois extension with Galois group $G$ and such that $E/k$ is {\it{regular}} (i.e., $E \cap \overline{\mathbb{Q}}=k$). For short, we say that $E/k(T)$ is a {\it{$k$-regular Galois extension with Galois group $G$.}} In the rest of the paper, by ``genus of the extension $E/k(T)$", we mean the genus of the function field $E$.
We propose the following definition:
\begin{definition} \label{def0} We define the {\it{minimal genus of $G$ over $k$}}, denoted by $m_{G,k},$ as the smallest integer $g$ such that there exists at least one $k$-regular Galois extension of $k(T)$ that has Galois group $G$ and genus $g$. If $G$ is not a regular Galois group over $k$ \footnote{i.e., if there exists no $k$-regular Galois extension of $k(T)$ with Galois group $G$.}, we set $m_{G,k}=\infty$. \end{definition}
\subsubsection{Branch points}
Recall that $t_0 \in \mathbb{P}^1(\overline{\mathbb{Q}})$ is {\it{a branch point of $E/k(T)$}} if the prime ideal $(T-t_0) \overline{\mathbb{Q}}[T-t_0]$ \footnote{Replace $T-t_0$ by $1/T$ if $t_0 = \infty$.} ramifies in the integral closure of $\overline{\mathbb{Q}}[T-t_0]$ in the compositum $E\overline{\mathbb{Q}}$ of $E$ and $\overline{\mathbb{Q}}(T)$ (in a fixed algebraic closure of $k(T)$). The extension $E/k(T)$ has only finitely many branch points, denoted by $t_1,\dots,t_r$ \footnote{One has $r=0$ if and only if $G$ is trivial.}.
\subsubsection{Inertia canonical invariant}
For each positive integer $n$, fix a primitive $n$-th root of unity $\zeta_n$. Assume that the system $\{\zeta_n\}_n$ is {\it{coherent}}, i.e., $\zeta_{nm}^n=\zeta_m$ for any positive integers $n$ and $m$.
To $t_i$ can be associated a conjugacy class $C_i$ of $G$, called the {\it{inertia canonical conjugacy class (associated with $t_i$)}}, in the following way. The inertia groups of the prime ideals lying over $(T-t_i) \, \overline{\mathbb{Q}}[T-t_i]$ in the extension ${E}\overline{\mathbb{Q}}/\overline{\mathbb{Q}}(T)$ are cyclic conjugate groups of order equal to the ramification index $e_i$. Furthermore, each of them has a distinguished generator corresponding to the automorphism $(T-t_i)^{1/e_i} \mapsto \zeta_{e_i} (T-t_i)^{1/e_i}$ of $\overline{\mathbb{Q}}(((T-t_i)^{1/e_i}))$. Then, $C_i$ is the conjugacy class of all the distinguished generators of the inertia groups of the prime ideals lying over $(T-t_i) \, \overline{\mathbb{Q}}[T-t_i]$ in the extension ${E}\overline{\mathbb{Q}}/\overline{\mathbb{Q}}(T)$. The unordered $r$-tuple $({C_1},\dots,{C_r})$ is called {\it{the inertia canonical invariant of ${E}/k(T)$}}.
\subsubsection{Specializations}
Given $t_0 \in \mathbb{P}^1(k) \setminus \{t_1,\dots,t_r\}$, the residue extension of $E/k(T)$ at a prime ideal $\mathcal{P}$ lying over $(T-t_0) k[T-t_0]$ is denoted by ${E}_{t_0}/k$ and called {\it{the specialization of ${E}/k(T)$ at $t_0$}}. This does not depend on the prime $\mathcal{P}$ lying over $(T-t_0) k[T-t_0]$ as the extension ${E}/k(T)$ is Galois. The specialization $E_{t_0}/k$ is Galois with Galois group a subgroup of $G$, namely the decomposition group of ${E}/k(T)$ at $\mathcal{P}$.
\subsection{Parametric sets and parametric extensions}
\begin{definition} \label{def1} Let $n$ be a positive integer and $S$ a set of $k$-regular Galois extensions of $k(T)$ with Galois group $G$. We say that $S$ is {\it{an $n$-parametric set over $k$}} if, given $n$ extensions $F_1/k, \dots, F_n/k$ each of which has Galois group $G$, there exists some extension $E/k(T)$ in $S$ such that $F_1/k, \dots, F_n/k$ all are specializations of $E/k(T)$. \end{definition}
\begin{remark} \label{rem1} {\rm{(1)}} If $S$ is an $n$-parametric set over $k$ for a given integer $n \geq 1$, then it is $m$-parametric over $k$ for every integer $1 \leq m \leq n$.
\noindent {\rm{(2)}} Assume that $S$ consists of only one extension $E/k(T)$. Then, given an integer $n \geq 1$, the set $S=\{E/k(T)\}$ is $n$-parametric over $k$ if and only if it is $1$-parametric over $k$. In this case, we will say for short that {\it{the extension $E/k(T)$ is parametric over $k$.}}
\noindent {\rm{(3)}} The set $S$ consisting of no $k$-regular Galois extension of $k(T)$ with group $G$ is 1-parametric over $k$ if and only if $G$ is not a Galois group over $k$ \footnote{i.e., if there exists no Galois extension of $k$ with Galois group $G$.}. In this case, this set is $n$-parametric over $k$ for each $n \geq 1$. \end{remark}
\subsection{Finiteness of the number of specialization points with prescribed specialization}
Proposition \ref{twisting} below, which is already implicitly used in \cite[\S3.3.5]{Deb99a}, will be used on several occasions.
\begin{proposition} \label{twisting} Let $E/k(T)$ be a $k$-regular Galois extension with Galois group $G$ and genus at least 2, $H$ a subgroup of $G$, and $F/k$ a Galois extension with Galois group $H$. Then, there exist only finitely many points $t_0 \in \mathbb{P}^1(k)$ such that $F/k=E_{t_0}/k$. \end{proposition}
\begin{proof}
Denote the genus of $E/k(T)$ by $g$. By the Twisting Lemma 3.2 of \cite{DL12}, there exist a positive integer $n \leq (2 \cdot |G|)^{|G|}$ and $n$ smooth curves $X_1, \dots, X_n$ defined over $k$ that satisfy the following properties:
\noindent (1) $X_1,\dots,X_n$ all have genus $g$,
\noindent (2) two different points $t_0$ and $t'_0$ in $\mathbb{P}^1(k)$ such that $E_{t_0}=F=E_{t'_0}$ give rise to two different $k$-rational points in $X_1(k) \cup \cdots \cup X_n(k)$.
\noindent Assume that there exist infinitely many $t_0 \in \mathbb{P}^1(k)$ such that $F/k=E_{t_0}/k$. Then, by (2), $X_1(k) \cup \cdots \cup X_n(k)$ is infinite, i.e., some $X_{i_0}(k)$ is infinite as well. By Faltings' theorem, the genus of $X_{i_0}$ then is 0 or 1, which cannot happen by (1) and the assumption on $g$. \end{proof}
Now, recall the following standard conjecture which provides Faltings' theorem for families of curves with the same given genus.
\noindent {\bf{Uniformity Conjecture.}} {\it{Let $g \geq 2$ be an integer. Then, there exists a positive integer $B$, which depends only on $g$ and $k$, such that, for every smooth curve $X$ defined over $k$ with genus $g$, the set $X(k)$ of all $k$-rationals points on $X$ has cardinality at most $B$.}}
\noindent By \cite[Theorem 1.1]{CHM97}, this conjecture is true under the Lang Conjecture, which asserts that the set of all $k$-rational points on a given variety of general type defined over $k$ is not Zariski dense.
Finally, combining the Twisting Lemma and the Uniformity Conjecture provides the following conjectural version of Proposition \ref{twisting}. As the proof is almost identical, details are left to the reader.
\begin{proposition} \label{twisting2}
Assume that the Uniformity Conjecture holds. Then, given an integer $g_0 \geq 2$, there exists an integer $B \geq 1$, depending only on $|G|$, $k$, and $g_0$, and satisfying the following. Let $E/k(T)$ be a $k$-regular Galois extension with Galois group $G$ and genus between 2 and $g_0$, $H$ a subgroup of $G$, and $F/k$ a Galois extension with Galois group $H$. Then, there exist at most $B$ points $t_0 \in \mathbb{P}^1(k)$ such that $F/k=E_{t_0}/k$. \end{proposition}
\section{Criteria for non-parametricity}
For this section, let $k$ be a number field, $O_k$ the integral closure of $\mathbb{Z}$ in $k$, and $G$ a finite group.
The aim of this section is to give criteria for the group $G$ to have no parametric set over $k$ with suitable properties.
\subsection{A preliminary result}
Below, we explain how to derive parametric sets with various Galois groups, assuming we know at least one.
\begin{proposition} \label{prelim1} Let $H$ be a normal sugbroup of $G$. Assume that every Galois extension $F/k$ with Galois group $G/H$ embeds into a Galois extension of $k$ with Galois group $G$ \footnote{i.e., there exists a Galois extension $L/k$ with Galois group $G$ such that $F \subset L$.}. Then, the following two conclusions hold.
\noindent {\rm{(1)}} Every $1$-parametric set over $k$ for the group $G$ with positive cardinality at most $s$ gives rise to a $1$-parametric set over $k$ for the group $G/H$ with positive cardinality at most $s \cdot r$, where $r$ denotes the number of normal sugbroups $H'$ of $G$ such that $G/H' \cong G/H$.
\noindent {\rm{(2)}} Furthermore, assume that $G$ has a unique normal subgroup $H'$ such that $G/H' \cong G/H$. Let $n$ be a positive integer. Then, every $n$-parametric set over $k$ for the group $G$ with positive cardinality at most $s$ gives rise to an $n$-parametric set over $k$ for the group $G/H$ with positive cardinality at most $s$. In particular, every parametric extension over $k$ with Galois group $G$ gives rise to a parametric extension over $k$ with Galois group $G/H$. \end{proposition}
We need the following lemma.
\begin{lemma} \label{subextensions} Let $H$ be a normal subgroup of $G$, $E/k(T)$ a $k$-regular Galois extension with Galois group $G$, and $t_0 \in \mathbb{P}^1(k)$, which is not a branch point of $E/k(T)$. Denote the distinct subextensions of $E/k(T)$ that have Galois group $G/H$ by $E_1/k(T), \dots, E_r/k(T)$. Assume that the specialization $E_{t_0}/k$ has Galois group $G$. Then, the extensions $(E_1)_{t_0}/k, \dots, (E_r)_{t_0}/k$ are the distinct subextensions of $E_{t_0}/k$ that have Galois group $G/H$. \end{lemma}
\begin{proof} Let $i$ between 1 and $r$. As $E_{t_0}/k$ has Galois group $G$, the specialized extension $E_{t_0}/(E_i)_{t_0}$ has Galois group ${\rm{Gal}}(E/E_i)$. As $E_1, \dots, E_r$ are distinct, the same is true for ${\rm{Gal}}(E/E_1), \dots, {\rm{Gal}}(E/E_r)$. Then, the extensions $(E_1)_{t_0}/k, \dots, (E_r)_{t_0}/k$ are distinct and they all have Galois group $G/H$. It then remains to notice that $r$ is the number of normal subgroups $H'$ of $G$ such that $G/H' \cong G/H$ to conclude. \end{proof}
\begin{proof}[Proof of Proposition \ref{prelim1}] Given a positive integer $n$, let $E_1/k(T), \dots,$ $E_s/k(T)$ be $k$-regular Galois extensions with Galois group $G$ giving rise to an $n$-parametric set over $k$ ($s \geq 1$). Denote the $r$ normal subgroups $H'$ of $G$ such that $G/H' \cong G/H$ by $H_1, \dots, H_r$. Let $F_1/k, \dots, F_n/k$ be $n$ Galois extensions each of which has Galois group $G/H$. By our embedding assumption, there exist $n$ Galois extensions $L_1/k, \dots, L_n/k$ each of which has Galois group $G$ and such that $F_j \subset L_j$ for each $j \in \{1,\dots,n\}$. As the set $\{E_1/k(T), \dots, E_s/k(T)\}$ is $n$-parametric over $k$, there exist $i \in \{1,\dots,s\}$ and specialization points $t_{0,1}, \dots, t_{0,n} \in \mathbb{P}^1(k)$ such that $L_j/k$ occurs as the specialization of $E_i/k(T)$ at $t_{0,j}$ for each $j \in \{1,\dots,n\}$. Let $j \in \{1,\dots,n\}$. By Lemma \ref{subextensions}, the extensions $(E_i^{H_1})_{t_{0,j}}/k, \dots, (E_i^{H_r})_{t_{0,j}}/k$ are the distinct subextensions of $L_j/k$ with Galois group $G/H$. Hence, $F_j/k = (E_i^{H_l})_{t_{0,j}}/k$ for some $l \in \{1,\dots,r\}$. First, assume that $n=1$. Then, $F_1/k = (E_i^{H_l})_{t_{0,1}}/k$ for some $l \in \{1,\dots,r\}$. In particular, the set $$\{E_1^{H_1}/k(T), \dots, E_1^{H_r}/k(T), \dots, E_s^{H_1}/k(T), \dots, E_s^{H_r}/k(T)\}$$ is $1$-parametric over $k$, as needed for part (1). Now, suppose that $r=1$. Then, $F_j/k = (E_i^{H})_{t_{0,j}}/k$ ($j \in \{1,\dots,n\}$). Hence, $\{E_1^{H}/k(T), \dots,$ $E_s^{H}/k(T)\}$ is $n$-parametric over $k$, as needed for part (2). \end{proof}
\subsection{A criterion in minimal genus $\geq 2$}
Theorem \ref{genus2} below is the easiest and the most useful of our criteria.
\begin{theorem} \label{genus2} Assume that $G$ has a normal subgroup $H$ such that the following two conditions hold:
\noindent {\rm{(1)}} there exists at least one Galois extension of $k$ with Galois group $G/H$ which embeds into infinitely many Galois extensions of $k$ with Galois group $G$,
\noindent {\rm{(2)}} $m_{{G/H},k} \geq 2$.
\noindent Then, the following two conclusions hold.
\noindent {\rm{(1)}} The group $G$ has no finite 1-parametric set over $k$.
\noindent
{\rm{(2)}} Furthermore, assume that the Uniformity Conjecture of \S2.3 holds. Then, given an integer $r_0 \geq 1$, there exists an integer $n \geq 1$ (depending only on $r_0$, $k$, and $|G|$) such that the set which consists of all $k$-regular Galois extensions of $k(T)$ with Galois group $G$ and at most $r_0$ branch points is not $n$-parametric over $k$. \end{theorem}
\noindent In the light of condition (2) above, it is useful to investigate which finite groups can occur as the Galois group of a $k$-regular Galois extension of $k(T)$ with genus $\leq 1$. We refer to \S7 for more on this classical topic. As to the embedding condition (1), it is studied in \S4.
Theorem \ref{genus2} is a straightforward application of Lemma \ref{lemma genus2} below.
\begin{lemma} \label{lemma genus2} Let $r_0$ be a positive integer and $S$ a set of $k$-regular Galois extensions of $k(T)$ with Galois group $G$. Assume that $G$ has a normal subgroup $H$ such that the following two conditions hold:
\noindent {\rm{(1)}} there exists a Galois extension of $k$ with group $G/H$ which embeds into infinitely many Galois extensions of $k$ with group $G$,
\noindent {\rm{(2)}} for each extension $E/k(T) \in S$ and each $j \in \{1,\dots,r\}$, the genus of $E^{H_j}/k(T)$ is at least 2, where $H_1, \dots, H_r$ denote the distinct normal subgroups $H'$ of $G$ such that $G/H' \cong G/H$.
\noindent Then, the following two conclusions hold.
\noindent {\rm{(1)}} Assume that the set $S$ is finite. Then, there exist infinitely many Galois extensions $F/k$ with Galois group $G$ such that, for each $E/k(T) \in S$, the extension $F/k$ is not a specialization of $E/k(T)$.
\noindent
{\rm{(2)}} Assume that each $E/k(T)$ in $S$ has at most $r_0$ branch points and the Uniformity Conjecture holds. Then, there exist an integer $n$ (depending only on $r_0$, $k$, and $|G|$) and $n$ Galois extensions $L_1/k, \dots, L_n/k$ with Galois group $G$ such that, for each $E/k(T) \in S$, at least one of the extensions $L_1/k, \dots, L_n/k$ is not a specialization of $E/k(T)$. \end{lemma}
\begin{proof} First, assume that $S$ is finite. As condition (1) holds, conclusion (1) holds if $S$ is empty. We may then assume that $S$ is not empty. Set $S= \{E_1/k(T), \dots, E_s/k(T)\}$ with $s \geq 1$. Assume that all but finitely many Galois extensions of $k$ with Galois group $G$ are specializations of $E_1/k(T), \dots, E_s/k(T)$. Let $F/k$ be a Galois extension with Galois group $G/H$ which embeds into infinitely many Galois extensions $L/k$ with Galois group $G$ (condition (1)). Let $L/k$ be such an extension. Up to dropping finitely many of them, we may assume that $L/k = (E_i)_{t_0}/k$ for some $i \in \{1,\dots,s\}$ and some $t_0 \in \mathbb{P}^1(k)$. By Lemma \ref{subextensions}, $(E_i^{H_1})_{t_0}/k, \dots, (E_i^{H_r})_{t_0}/k$ are the distinct subextensions of $L/k$ that have Galois group $G/H$. Hence, $F/k=(E_i^{H_j})_{t_0}/k$ for some $j \in \{1,\dots,r\}$. Repeat this trick infinitely many times to get that there exists $(i,j) \in \{1,\dots,s\} \times \{1,\dots,r\}$ and infinitely many $t_0 \in \mathbb{P}^1(k)$ such that $(E_i^{H_j})_{t_0}=F$. By Proposition \ref{twisting}, the genus of $E_i^{H_j}/k(T)$ is 0 or 1, which cannot happen by condition (2). Hence, conclusion (1) holds.
Now, assume that each extension $E/k(T) \in S$ has at most $r_0$ branch points and the Uniformity Conjecture holds. Let $F/k$ be a Galois extension with Galois group $G/H$ and a sequence $(L_n/k)_{n \geq1}$ of distinct Galois extensions with Galois group $G$ containing $F$. Given $n \geq 1$, assume that $L_1/k, \dots, L_n/k$ all are specializations of some extension $E/k(T)$ in $S$. Apply the same argument as before to get that there exists $j \in \{1,\dots,r\}$ and $n/r$ distinct points $t_0 \in \mathbb{P}^1(k)$ such that $F/k=(E^{H_j})_{t_0}/k$. As $E/k(T)$ has at most $r_0$ branch points, the same is true of the subextension $E^{H_j}/k(T)$. Hence, the genus of $E^{H_j}/k(T)$, which is at least 2 (condition (2)), is bounded by some integer $g_0$ which depends only on $|G|$ and $r_0$. One may then apply Proposition \ref{twisting2} to get that there exists a positive integer $B$, which depends only on $|G|$, $r_0$, and $k$, such that the set of points $t_0 \in \mathbb{P}^1(k)$ satisfying $(E^{H_j})_{t_0}/k=F/k$ has cardinality at most $B$. Hence, $n \leq r \cdot B$, thus ending the proof. \end{proof}
\subsection{A criterion in minimal genus $\geq 1$}
Below, we give an analog of Theorem \ref{genus2}, where we relax condition (2) on the minimal genus. However, we have to add some local conditions in the embedding condition (1) and drop the conjectural conclusion (2).
\begin{theorem} \label{genus1} Assume that $G$ has a normal subgroup $H$ such that the following two conditions hold.
\noindent {\rm{(1)}} There exists an infinite set $\mathcal{S}$ of prime ideals of $O_k$ satisfying the following. Given $\mathcal{P} \in \mathcal{S}$, there exists a Galois extension of $k$ with group $G/H$, with ramification index at $\mathcal{P}$ not in $\{1,2,3,4,6\}$, and which embeds into infinitely many Galois extensions of $k$ with group $G$.
\noindent {\rm{(2)}} One has $m_{{G/H},k} \geq 1$.
\noindent Then, $G$ has no finite 1-parametric set over $k$. \end{theorem}
\noindent {\it{Addendum}} \ref{genus1}. In the special case $k=\mathbb{Q}$, one may replace condition (1) by the following weaker variant.
\noindent (1)' There exists an infinite set $\mathcal{S}$ of prime numbers satisfying the following. Given $p \in \mathcal{S}$, there exists a Galois extension of $\mathbb{Q}$ with Galois group $G/H$, with ramification index at $p$ at least 3, and which embeds into infinitely many Galois extensions of $\mathbb{Q}$ with Galois group $G$.
Theorem \ref{genus1} is a straightforward application of Lemma \ref{lemma genus1} below.
\begin{lemma} \label{lemma genus1} Let $S$ be a finite set of $k$-regular Galois extensions of $k(T)$ with Galois group $G$. Assume that $G$ has a normal subgroup $H$ such that the following two conditions hold.
\noindent {\rm{(1)}} There exists an infinite set $\mathcal{S}$ of prime ideals of $O_k$ satisfying the following. Given $\mathcal{P} \in \mathcal{S}$, there exists a Galois extension of $k$ with group $G/H$, with ramification index at $\mathcal{P}$ not in $\{1,2,3,4,6\}$, and which embeds into infinitely many Galois extensions of $k$ with group $G$.
\noindent {\rm{(2)}} For each extension $E/k(T) \in S$ and each $j \in \{1,\dots,r\}$, the genus of $E^{H_j}/k(T)$ is at least 1, where $H_1, \dots, H_r$ denote the distinct normal subgroups $H'$ of $G$ such that $G/H' \cong G/H$.
\noindent Then, there exist infinitely many Galois extensions $F/k$ with Galois group $G$ such that, for each extension $E/k(T) \in S$, the extension $F/k$ is not a specialization of $E/k(T)$. Moreover, in the special case $k=\mathbb{Q}$, one may replace condition {\rm{(1)}} by condition {\rm{(1)}}' of Addendum \ref{genus1}. \end{lemma}
\begin{proof} As in the case $m_{G/H,k} \geq 2$, one may assume $S \not= \emptyset$. Set $S= \{E_1/k(T), \dots, E_s/k(T)\}$ with $s \geq 1$. Assume that all but finitely many Galois extensions of $k$ with group $G$ are specializations of $E_1/k(T), \dots, $ $E_s/k(T)$. Let $\mathcal{P}$ be one of the infinitely many primes of $\mathcal{S}$ of condition (1) which is a {\it{bad prime}}\footnote{See \cite[Definition 2.6]{Leg16c} for the precise definition. Here, we only use the standard fact that a given finite $k$-regular Galois extension of $k(T)$ has only finitely many bad primes.} for none of the subextensions $E_1^{H_1}/k(T), \dots,$ $E_1^{H_r}/k(T), \dots, E_s^{H_1}/k(T), \dots, E_s^{H_r}/k(T).$ Let $F/k$ be a Galois extension with group $G/H$, ramification index at $\mathcal{P}$ not in $\{1,2,3,4,6\}$, and which embeds into infinitely many Galois extensions $L/k$ with group $G$ (condition (1)). As in the case $m_{G/H,k} \geq 2$, one finds $(i,j) \in \{1,\dots,s\} \times \{1,\dots,r\}$ and infinitely many $t_0 \in \mathbb{P}^1(k)$ such that $(E_i^{H_j})_{t_0}=F$. Then, by Proposition \ref{twisting} and condition (2), the genus of $E_i^{H_j}/k(T)$ is 1. Hence, by Proposition \ref{RH genus 1}, the inertia canonical invariant of this extension consists only of conjugacy classes of elements of order 2, 3, 4 or 6. As $\mathcal{P}$ is a good prime for $E_i^{H_j}/k(T)$, we may apply the Specialization Inertia Theorem \cite[\S2.2.3]{Leg16c}\footnote{The assumption requiring the branch points of $E_i^{H_j}/k(T)$ and their inverses to be integral with respect to $\mathcal{P}$ can also be satisfied (up to dropping more primes).} to get that the ramification index of $\mathcal{P}$ in $F/k$ is 1, 2, 3, 4 or 6, a contradiction.
To get the conclusion under condition (1)' in the case $k=\mathbb{Q}$, it suffices to see that, with the same notation as above and by Proposition 7.2, the inertia canonical invariant of $E_i^{H_j}/\mathbb{Q}(T)$ should consist only of conjugacy classes of involutions of $G/H$. \end{proof}
\subsection{Criteria with no explicit assumption on the minimal genus}
In Theorem \ref{genus0} below, we do not require any assumption on the minimal genus of quotients of $G$, at the cost of making the conclusion weaker.
\begin{theorem} \label{genus0} Assume that $G$ has a normal subgroup $H$ such that the following two conditions hold.
\noindent {\rm{(1)}} There exists a Galois extension of $k$ with group $G/H$ which embeds into infinitely many Galois extensions of $k$ with group $G$.
\noindent {\rm{(2)}} For each normal subgroup $H'$ of $G$ such that $G/H' \cong G/H$, there exist 5 conjugacy classes $C_1, \dots, C_5$ of $G$ such that the following holds for each $i \in \{1,\dots,5\}$:
{\rm{(a)}} $C_i$ is not contained in $H'$,
{\rm{(b)}} each element of $C_i$ generates a maximal cyclic subgroup of $G$ \footnote{By this, we mean a cyclic subgroup of $G$ which is contained in no strictly larger cyclic subgroup of $G$.},
{\rm{(c)}} for each $j \in \{1,\dots,5\} \setminus \{i\}$, the class $C_i$ is not a power of $C_j$,
{\rm{(d)}} $C_i$ belongs to the inertia canonical invariant of some $k$-regular
Galois extension of $k(T)$ with Galois group $G$.
\noindent Then, given a $k$-regular Galois extension $E/k(T)$ with Galois group $G$, there exist infinitely many Galois extensions of $k$ with Galois group $G$ each of which is not a specialization of $E/k(T)$. In particular, $G$ has no parametric extension over $k$. \end{theorem}
\begin{proof} Let $E/k(T)$ be a $k$-regular Galois extension with Galois group $G$. Assume that all but finitely many Galois extensions of $k$ with Galois group $G$ are specializations of $E/k(T)$. Let $H'$ be a normal subgroup of $G$ such that $G/H' \cong G/H$. Below, we show that the subextension $E^{H'}/k(T)$ has at least 5 branch points. Then, by the Riemann-Hurwitz formula, the genus of $E^{H'}/k(T)$ is at least 2. We may then combine Lemma \ref{lemma genus2} and condition (1) to get a contradiction.
Let $i \in \{1,\dots,5\}$. By part (d) of condition (2) and our assumption on $E/k(T)$, there exists a conjugacy class $C'_i$ of $G$ belonging to the inertia canonical invariant of $E/k(T)$ and an integer $a_i \geq 1$ such that ${C'_{i}}^{a_i}=C_i$; see \cite[Theorem 4.2]{Leg16c}. Then, by part (b) of condition (2), one has $C'_i=C_i^{b_i}$ for some integer $b_i \geq 1$ which is coprime to the order $c_i$ of the elements of $C_i$. Moreover, by part (c) of condition (2), one has $C'_i \not=C'_j$ for $i \not=j$. Hence, $E/k(T)$ has at least 5 distinct branch points $t_1, \dots, t_5$ whose associated inertia canonical conjugacy classes are $C_1^{b_1}, \dots, C_5^{b_5}$. By part (a) of condition (2) and as $b_i$ is coprime to $c_i$, one has that, for each $i \in \{1,\dots,5\}$, the class $C_i^{b_i}$ is not contained in $H'$. Hence, $t_1,\dots,t_5$ all are branch points of $E^{H'}/k(T)$, as needed. \end{proof}
Viewing Theorems \ref{genus2} and \ref{genus1}, it is natural to apply Theorem \ref{genus0} when $m_{G/H,k}=0$, i.e., when $G/H$ is one of the groups given in Proposition \ref{list genus 0}. We focus below on the case where $G/H$ is cyclic and leave the other cases to the reader. For simplicity, we take $k=\mathbb{Q}$.
\begin{theorem} \label{cyclic} Assume that $G$ has a normal subgroup $H$ such that the following three conditions hold.
\noindent {\rm{(1)}} The quotient $G/H$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$ for some integer $n \geq 3$.
\noindent {\rm{(2)}} There exists a Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z}/n\mathbb{Z}$ which embeds into infinitely many Galois extensions of $\mathbb{Q}$ with group $G$.
\noindent {\rm{(3)}} For each normal subgroup $H'$ of $G$ such that $G/H' \cong \mathbb{Z}/n\mathbb{Z}$, there exist 3 distinct conjugacy classes $C_1$, $C_2$, and $C_3$ of $G$ such that the following holds for each $i \in \{1,2,3\}$:
{\rm{(a)}} $C_i$ is not contained in $H'$,
{\rm{(b)}} each element of $C_i$ generates a maximal cylic subgroup of $G$,
{\rm{(c)}} $C_i$ belongs to the inertia canonical invariant of some $\mathbb{Q}$-regular
Galois extension of $\mathbb{Q}(T)$ with Galois group $G$.
\noindent Then, given a $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ with Galois group $G$, there exist infinitely many Galois extensions of $\mathbb{Q}$ with Galois group $G$ each of which is not a specialization of $E/\mathbb{Q}(T)$. In particular, $G$ has no parametric extension over $\mathbb{Q}$. \end{theorem}
\begin{proof} The proof is almost identical to that of Theorem \ref{genus0}. To avoid confusion, we reproduce it below with the needed adjustments.
Let $E/\mathbb{Q}(T)$ be a $\mathbb{Q}$-regular Galois extension with Galois group $G$. Assume that all but finitely many Galois extensions of $\mathbb{Q}$ with Galois group $G$ are specializations of $E/\mathbb{Q}(T)$. Let $H'$ be a normal subgroup of $G$ such that $G/H' \cong \mathbb{Z}/n\mathbb{Z}$. Below, we show that $E^{H'}/\mathbb{Q}(T)$ has at least 3 branch points. Then, by Propositions 7.1-2 and \ref{list genus leq 1 abelian} (and as $n \geq 3$), the genus of $E^{H'}/\mathbb{Q}(T)$ is at least 2. We may then combine Lemma \ref{lemma genus2} and condition (2) to get a contradiction.
Let $i \in \{1,2,3\}$. By part (c) of condition (3) and our assumption on $E/\mathbb{Q}(T)$, there exists a conjugacy class $C'_i$ of $G$ in the inertia canonical invariant of $E/\mathbb{Q}(T)$ and an integer $a_i \geq 1$ such that ${C'_{i}}^{a_i}=C_i$; see \cite[Theorem 4.2]{Leg16c}. Then, by part (b) of condition (3), one has $C'_i=C_i^{b_i}$ for some integer $b_i \geq 1$ which is coprime to the order of the elements of $C_i$. By the Branch Cycle Lemma (see \cite{Fri77} and \cite[Lemma 2.8]{Vol96}), $C_i$ is in the inertia canonical invariant of $E/\mathbb{Q}(T)$. As $C_1, \dots, C_3$ are distinct and as none of them is contained in $H'$ (part (a) of condition (3)), $E^{H'}/\mathbb{Q}(T)$ has at least 3 branch points, as needed. \end{proof}
The case where $G/H \cong \mathbb{Z}/2\mathbb{Z}$ requires more attention as this group has a regular realization over $\mathbb{Q}$ with genus 1 and 4 branch points.
\begin{theorem} \label{cyclic2} Assume that the following four conditions hold.
\noindent {\rm{(1)}} The group $G$ has a unique index 2 subgroup $H$.
\noindent {\rm{(2)}} Each quadratic extension of $\mathbb{Q}$ embeds into a Galois extension of $\mathbb{Q}$ with Galois group $G$.
\noindent {\rm{(3)}} There exists at least one quadratic extension of $\mathbb{Q}$ which embeds into infinitely many Galois extensions of $\mathbb{Q}$ with Galois group $G$.
\noindent {\rm{(4)}} There exist $s$ distinct conjugacy classes $C_1, \dots,C_s$ of $G$ such that
{\rm{(a)}} either $s \geq 3$ or $s=2$ and $C_2$ is a power of $C_1$,
{\rm{(b)}} for each $i \in \{1,\dots,s\}$, the class $C_i$ is not contained in $H$,
{\rm{(c)}} for each $i \in \{1,\dots,s\}$, each element of $C_i$ generates a maximal
cyclic subgroup of $G$,
{\rm{(d)}} for each $i \in \{1,\dots,s\}$, $C_i$ belongs to the inertia canonical inva-
riant of some $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ with group $G$.
\noindent Then, given a $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ with group $G$, there exist infinitely many Galois extensions of $\mathbb{Q}$ with group $G$ which are not specializations of $E/\mathbb{Q}(T)$. {\hbox{Hence, $G$ has no parametric extension over $\mathbb{Q}$.}} \end{theorem}
\begin{proof} Let $E/\mathbb{Q}(T)$ be a $\mathbb{Q}$-regular Galois extension with Galois group $G$. Assume that all but finitely many Galois extensions of $\mathbb{Q}$ with Galois group $G$ are specializations of $E/\mathbb{Q}(T)$. As $H$ is unique (condition (1)), it suffices, as in the proof of Theorem \ref{genus0}, to show that $E^H/\mathbb{Q}(T)$ has at least 5 branch points. One shows as in the proof of Theorem \ref{cyclic} that $E^H/\mathbb{Q}(T)$ has at least $s$ branch points whose inertia canonical conjugacy classes are $C_1,\dots,C_s$. Moreover, $H$ is unique and condition (2) holds. Then, one shows as in the proof of Proposition \ref{prelim1} that all but finitely many quadratic extensions of $\mathbb{Q}$ are specializations of $E^H/\mathbb{Q}(T)$. Assume that $E^H/\mathbb{Q}(T)$ has at most 4 branch points. Then, by Proposition \ref{r leq 4}, $E^H/\mathbb{Q}(T)$ has exactly 2 branch points and both are $\mathbb{Q}$-rational. As this cannot happen if $s \geq 3$, one has $s=2$ and $C_2$ is a power of $C_1$. Then, by the Branch Cycle Lemma, $E^H/\mathbb{Q}(T)$ has at least 2 branch points that are not $\mathbb{Q}$-rational, which cannot happen. \end{proof}
\section{Solving embedding problems}
In this section, we give sufficient conditions for the embedding conditions of the criteria of \S3 to hold. From now on, we fix a number field $k$, a finite group $G$, and a non-trivial normal subgroup $H$ of $G$.
\subsection{Solving at least one embedding problem infinitely many times}
First, we give sufficient conditions for the following to hold:
\noindent {\rm{($*$)}} {\it{there exists a Galois extension of $k$ with group $G/H$ which embeds into infinitely many Galois extensions of $k$ with group $G$.}}
\noindent Obviously, a necessary condition for condition {\rm{($*$)}} to hold is that $G$ occurs as a Galois group over $k$.
\subsubsection{The case where $H$ is solvable}
In Proposition \ref{solvable kernel} below, we show that the converse holds if $H$ is solvable.
\begin{proposition} \label{solvable kernel} {\rm{(1)}} Assume that $H$ is solvable. Then, condition {\rm{($*$)}} holds if and only if $G$ is a Galois group over $k$.
\noindent {\rm{(2)}} Assume that $H$ is abelian and $H$ is the unique normal subgroup $H'$ of $G$ such that $G/H' \cong G/H$. Then, any given Galois extension of $k$ with Galois group $G/H$ embeds into zero or infinitely many Galois extensions of $k$ with Galois group $G$. \end{proposition}
\begin{remark} \label{odd order} By a classical result of Shafarevich (see, e.g., \cite[Theorem (9.6.1)]{NSW08}), solvable groups are Galois groups over $k$. Then, condition {\rm{($*$)}} holds if $G$ is solvable (in particular, if $G$ is abelian or of odd order; see \cite{FT63}) and $H$ is any non-trivial normal subgroup of $G$. \end{remark}
\begin{proof} First, we prove (1). Up to replacing the solvable group $H$ by the smallest non-trivial term of its derived series (which is an abelian characteristic subgroup of $H$), we may and will assume that $H$ is abelian. Let $\varphi$ be the canonical surjection of $G$ onto $G/H$ and $n$ a positive integer. Consider the finite group $$G_{\varphi}^n = \{(g_1, \dots, g_n) \in G^n \, : \, \varphi(g_1) = \cdots = \varphi(g_n)\}.$$ Set $$N=\{(g_1, \dots, g_n) \in G_{\varphi}^n \, : \, g_1 \in H, \dots, g_n \in H\}.$$ Since $H$ is a normal subgroup of $G$, $N$ is a normal subgroup of $G_{\varphi}^n$. Moreover, for each $i \in \{1,\dots,n\}$, set $$N_i=\{(g_1, \dots, g_n) \in N \, : \, g_i=1\}.$$ Then, $N_1, \dots, N_n$ all are normal subgroups of $G_{\varphi}^n$ contained in $N$.
\begin{lemma} \label{iso} {\rm{(1)}} One has $G_{\varphi}^n / N_i \cong G$ for each $i \in \{1,\dots,n\}$.
\noindent {\rm{(2)}} One has $G_{\varphi}^n / N \cong G/H$. \end{lemma}
\begin{proof} To prove (1), it suffices to notice that, given $i \in \{1,\dots,n\}$, $(g_1, \dots,g_n) \in G^n_ {\varphi} \mapsto g_i \in G$ is an epimorphism with kernel $N_i$. As to (2), note that the surjection $\varphi$ induces an epimorphism $(g_1, \dots,g_n) \in G^n_ {\varphi} \mapsto \varphi(g_1) \in G/H$ with kernel $N$. Hence, the lemma holds. \end{proof}
Let $L/k$ be a Galois extension with Galois group $G$ and $\theta : G \rightarrow {\rm{Gal}}(L/k)$ an isomorphism. Consider the finite {\it{embedding problem}}\footnote{We refer to \cite[Definition 16.4.1]{FJ08} for more on the terminology.} $$\alpha : (g_1,\dots,g_n) \in G_{\varphi}^n \mapsto \theta (g_1) \in {\rm{Gal}}(L/k).$$ Then, $\alpha$ is split and, by the proof of Lemma \ref{iso}, it has kernel $N_1$, which is abelian (as $H$ is). By a result of Ikeda (see, e.g., \cite[Proposition 16.4.5]{FJ08}), every finite split embedding problem with abelian kernel over a Hilbertian field is solvable\footnote{More generally, Proposition \ref{solvable kernel} holds if $k$ is Hilbertian.}. Thus, there exist a Galois extension $M/k$ such that $L \subset M$ and an isomorphism $\beta : {\rm{Gal}}(M/k) \rightarrow G^n_\varphi$ such that $\alpha \circ \beta$ maps every element of ${\rm{Gal}}(M/k)$ to its restriction to $L$.
Consider the Galois subextensions $M^{\beta^{-1}(N_1)}/k, \dots, M^{\beta^{-1}(N_n)}/k.$ They satisfy the following three properties:
\noindent - ${\rm{Gal}}(M^{\beta^{-1}(N_i)}/k) \cong G$ for each $i \in \{1,\dots,n\}$ (Lemma \ref{iso}),
\noindent - $M^{\beta^{-1}(N_1)}/k = L/k$ (this follows from $\alpha \circ \beta$ mapping every element of ${\rm{Gal}}(M/k)$ to its restriction to $L$),
\noindent - $M^{\beta^{-1}(N_i)}/k \not= M^{\beta^{-1}(N_j)}/k$ for $i \not=j$ (as $H$ is not trivial).
\noindent Moreover, all of them contain the Galois extension $M^{\beta^{-1}(N)}/k$ which has Galois group isomorphic to $G/H$ (Lemma \ref{iso}). Hence, there exists a subextension of $L/k$ with Galois group $G/H$ which embeds into $n$ distinct Galois extensions of $k$ with Galois group $G$. As the given extension $L/k$ has only finitely many subextensions, we are done.
Now, we prove (2). Assume that $H$ is abelian and unique. With no loss of generality, we may assume that $G/H$ is a Galois group over $k$. Fix a Galois extension $F/k$ with Galois group $G/H$ which embeds into at least one Galois extension $L/k$ with Galois group $G$. Apply the same argument as above with the extension $L/k$ to get that, for every integer $n \geq 1$, some subextension $F_n/k$ of $L/k$ with Galois group $G/H$ embeds into $n$ distinct Galois extensions of $k$ with Galois group $G$. As $H$ is unique, one has $F_n=F$ for each $n \geq 1$, thus ending the proof. \end{proof}
\subsubsection{The case where $H$ is not solvable.}
To our knowledge, there is no general criterion asserting that condition ($*$) holds if and only if $G$ is a Galois group over $k$ if $H$ is not solvable. However, this is known for some specific non-abelian simple groups $H$. Here are some of them.
\begin{proposition} \label{GAR 1} Assume that the following two conditions hold:
\noindent {\rm{(1)}} $G/H$ is a Galois group over $k$,
\noindent {\rm{(2)}} $H$ is $A_n$ with $n \not \in \{1,2,3,4,6\}$, or ${\rm{PSL}}_2(\mathbb{F}_p)$ with $p$ odd and $p \not = \pm1 \, \, {\rm{mod}} \, \, 24$, or any sporadic group with the possible exception of $M_{23}$.
\noindent Then, condition {\rm{($*$)}} holds. \end{proposition}
\begin{proof} Assume that $H$ is as in condition (2). Then, by \cite[Chapter IV, Theorem 4.3]{MM99} (see also \cite[Remark 16.9.5]{FJ08}), $H$ has a so-called {\it{GAR-realization over $k$}}. Let $F/k$ be a Galois extension with Galois group $G/H$ (condition (1)), given with an isomorphism $\theta: G/H \rightarrow {\rm{Gal}}(F/k).$ Denote the canonical surjection of $G$ onto $G/H$ by $\varphi$. Then, the embedding problem $\theta \circ \varphi : G \rightarrow {\rm{Gal}}(L/k)$ has kernel $H$. A result of Matzat (see, e.g., \cite[Proposition 16.8.6]{FJ08}) states that every embedding problem whose kernel is non-abelian simple and possesses a GAR-realization over $k$ is regularly solvable over $k$. Since $k$ is Hilbertian, the regular solvability conclusion implies that this embedding problem has infinitely many solutions; cf., e.g., \cite[Lemma 16.4.2]{FJ08}. In particular, condition ($*$) holds. \end{proof}
\subsubsection{A criterion for index 2 subgroups}
In Proposition \ref{Debes 92} below, we give a criterion which uses some regular realizations of $G$ over $k$.
\begin{proposition} \label{Debes 92} Assume that the following two conditions hold.
\noindent {\rm{(1)}} The quotient $G/H$ has order 2.
\noindent {\rm{(2)}} There exists a $k$-regular Galois extension $E/k(T)$ with Galois group $G$ that satisfies the following two conditions:
{\rm{(a)}} $E/k(T)$ has genus at least 2,
{\rm{(b)}} the exact number of conjugacy classes of elements of even order
belonging to the inertia canonical invariant of $E/k(T)$ is either 2 or
3, and the associated two or three branch points are all $k$-rational.
\noindent Then, condition {\rm{($*$)}} holds. \end{proposition}
\begin{proof} By condition (1), the subextension $E^{H}/k(T)$ has degree 2. Moreover, by part (b) of condition (2), it has exactly two branch points $t_1$ and $t_2$, and $t_1$ and $t_2$ are $k$-rational. Up to applying a change of variable, we may and will assume that $E^{H}=k(\sqrt{T})$. Let $b$ be any element of $k \setminus \{0\}$ which is not a root of unity. By \cite[Corollary 1.7]{Deb92}, there exists $a \in k \setminus \{0\}$ such that, for each sufficiently large positive integer $m$, the specialization of $E/k(T)$ at $ab^m$ has Galois group $G$. Fix such an element $a$. For every $m$ as before, the specialization of $E^{H}/k(T)$ at $ab^m$ is quadratic and one has $(E^{H})_{ab^m}= k(\sqrt{ab})$ or $(E^{H})_{ab^m} = k(\sqrt{a})$. To conclude, it suffices to show that all the specializations of $E/k(T)$ at $ab^m$, $m \geq 1$, provide infinitely many distinct extensions of $k$. But this claim holds by part (a) of condition (2) and Proposition \ref{twisting}. \end{proof}
\subsection{Solving every embedding problem at least once}
Now, we give sufficient conditions for the following condition to hold:
\noindent {\rm{($**$)}} {\it{Every Galois extension of $k$ with Galois group $G/H$ embeds into at least one Galois extension of $k$ with Galois group $G$.}}
\noindent Condition {\rm{($**$)}} may fail (e.g., if $H=\mathbb{Z}/2\mathbb{Z}$, $G=\mathbb{Z}/4\mathbb{Z}$, and $k=\mathbb{Q}$; see \cite[Theorem 1.2.4]{Ser92}). However, it holds in the following cases.
\begin{proposition} \label{GAR 2} Condition {\rm{($**$)}} holds in the following two situations:
\noindent {\rm{(1)}} $H$ is $A_n$ with $n \not \in \{1,2,3,4,6\}$, or ${\rm{PSL}}_2(\mathbb{F}_p)$ with $p$ odd and $p \not = \pm1 \, \, {\rm{mod}} \, \, 24$, or any sporadic group with the possible exception of $M_{23}$.
\noindent {\rm{(2)}} $H=A_n$ and $G=S_n$ (for an arbitrary positive integer $n$). \end{proposition}
\begin{proof} In case (1), see the proof of Proposition \ref{GAR 1} to get condition {\rm{($**$)}}. In case (2), it is a result of Neumann; see \cite[Theorem 2]{Neu86}. \end{proof}
\section{Examples}
Let $k$ be a number field and $G$ a non-trivial finite group. This section is organized as follows. In \S5.1, we give examples of finite groups with no finite 1-parametric set over $k$; see Theorems 5.1-2. Then, in \S5.2, we give further examples of groups that have no parametric extension over $k$; see Theorem \ref{thm 2}. Finally, in \S5.3, we give conjectural examples of (infinite) sets of regular realizations over $k$ with the same Galois group which are not $n$-parametric over $k$ for large $n$; see Theorems 5.4-5.
\subsection{Finite groups with no finite 1-parametric set over $k$}
First, we give examples over the given number field $k$.
\begin{theorem} \label{thm 1} Assume that one of the following conditions holds:
\noindent {\rm{(1)}} $G=G_1 \times G_2$, where $G_1$ is any finite group with $m_{G_1,k} \geq 2$ and $G_2$ is any non-trivial finite group,
\noindent {\rm{(2)}} the order of $G$ has no prime factor $p$ such that $[k(\zeta_p):k] \leq 2$ (with $\zeta_p$ a primitive $p$-th root of unity)\footnote{See \S6.2.2 for more on these prime numbers.} and $G$ is not cyclic of prime order,
\noindent {\rm{(3)}} $G$ is abelian, but none of the following groups:
{\rm{(a)}} $\mathbb{Z}/n\mathbb{Z}$ ($n \geq2$),
{\rm{(b)}} $(\mathbb{Z}/p\mathbb{Z})^2$ ($p$ prime),
{\rm{(c)}} $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2p\mathbb{Z}$ ($p$ prime),
{\rm{(d)}} $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3p\mathbb{Z}$ ($p$ prime),
{\rm{(e)}} {\hbox{$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$, $(\mathbb{Z}/4\mathbb{Z})^2$, $(\mathbb{Z}/2\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^2 \times \mathbb{Z}/4\mathbb{Z}$, $(\mathbb{Z}/3\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^4$.}}
\noindent {\rm{(4)}} the center $Z(G)$ of $G$ is not trivial and $G/Z(G)$ is neither solvable nor $A_5$ (for example, $G={\rm{GL}}_n(\mathbb{F}_{q})$ where $n$ is any integer and $q$ is any prime power such that $n \geq 2$, $q \geq 3$, and $(n,q) \not \in \{(2,3), (2,4)\}$).
\noindent Then, $G$ has no non-empty finite 1-parametric set over $k$ \footnote{\label{foot1}See Remark \ref{rem1} for the case of the empty set.}. \end{theorem}
Now, we give other examples in the specific case $k=\mathbb{Q}$ (in addition to those already given in Theorem \ref{thm 1}).
\begin{theorem} \label{thm 1.1} Assume that one of the following conditions holds:
\noindent
{\rm{(1)}} $G$ is any non-abelian group such that $2$ does not divide $|G|$, $3$ divides $|G|$, and $G \not= (\mathbb{Z}/p\mathbb{Z})^k \rtimes \mathbb{Z}/3\mathbb{Z}$ ($k \in \{1,2\}$, $p$ prime, $p \not=3$),
\noindent {\rm{(2)}} $G$ is abelian, but none of the following groups:
{\rm{(a)}} $\mathbb{Z}/p\mathbb{Z}$ ($p$ prime),
{\rm{(b)}} $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$, $(\mathbb{Z}/2\mathbb{Z})^2$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$,
$(\mathbb{Z}/3\mathbb{Z})^2$, $(\mathbb{Z}/2\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^4$,
\noindent {\rm{(3)}} $G$ is the dihedral group $D_n$ (with $2n$ elements), where $n$ is any positive integer which is neither a prime nor in $\{1,4,6,8,9,12\}$,
\noindent {\rm{(4)}} the center $Z(G)$ of $G$ is not trivial and $G/Z(G)$ is neither solvable of even order nor of order $\leq 3$ (for example, $G={\rm{GL}}_2(\mathbb{F}_4)$).
\noindent Then, $G$ has no non-empty finite 1-parametric set over $\mathbb{Q}$ \footref{foot1}. \end{theorem}
\subsection{Finite groups with no parametric extension over $k$} Next, we give other examples of finite groups with no parametric extension over $k$ (in addition to those already given in Theorems \ref{thm 1} and \ref{thm 1.1}).
\begin{theorem} \label{thm 2} Assume that one of the following conditions holds:
\noindent {\rm{(1)}} $G$ is one of the following abelian groups: $\mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$, $(\mathbb{Z}/3\mathbb{Z})^2$, $(\mathbb{Z}/2\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^4$,
\noindent {\rm{(2)}} $G=S_n$, where $n$ is any integer $\geq 6$.
\noindent Then, $G$ has no parametric extension over $\mathbb{Q}$. Furthermore, if $G=S_n$ and $n \geq 8$, then $G$ has no parametric extension over $k$. \end{theorem}
\subsection{Conjectural non-parametric sets}
First, we give conjectural examples of (infinite) sets of regular realizations over the number field $k$ with the same Galois group which are not $n$-parametric over $k$ for large integers $n$.
\begin{theorem} \label{thm 3} Assume that the following three conditions hold:
\noindent {\rm{(1)}} the Uniformity Conjecture of \S2.3 holds,
\noindent {\rm{(2)}} $G$ is any group as in Theorem \ref{thm 1},
\noindent {\rm{(3)}} $G$ is a Galois group over $k$ \footnote{\label{foot2}If $G$ is not a Galois group over $k$, then the conclusion of the result fails trivially.}.
\noindent Then, given an integer $r_0 \geq 1$, there exists an integer $n \geq 1$ (depending only on $r_0$, $k$, and $|G|$) such that the set which consists of all the $k$-regular Galois extensions of $k(T)$ with Galois group $G$ and at most $r_0$ branch points is not $n$-parametric over $k$. \end{theorem}
Now, we give other examples in the specific case $k=\mathbb{Q}$ (in addition to those already given in Theorem \ref{thm 3}).
\begin{theorem} \label{thm 3.1} Assume that the following three conditions hold:
\noindent {\rm{(1)}} the Uniformity Conjecture of \S2.3 holds,
\noindent {\rm{(2)}} either $G$ is any group as in conditions {\rm{(1)}}, {\rm{(2)}}, {\rm{(4)}} of Theorem \ref{thm 1.1} or $G$ is the dihedral group $D_n$ (with 2n elements), where $n$ is any integer which is not a prime and which has a prime factor $\geq 11$,
\noindent {\rm{(3)}} $G$ is a Galois group over $\mathbb{Q}$ \footref{foot2}.
\noindent Then, given an integer $r_0 \geq 1$, there exists an integer $n \geq 1$ (depending only on $r_0$ and $|G|$) such that the set which consists of all the $\mathbb{Q}$-regular Galois extensions of $\mathbb{Q}(T)$ with Galois group $G$ and at most $r_0$ branch points is not $n$-parametric over $\mathbb{Q}$. \end{theorem}
\section{Proofs of Theorems 5.1-5}
\subsection{Direct products}
Assume $G=G_1 \times G_2$, where $G_1$ is any finite group with $m_{G_1,k} \geq 2$ and $G_2$ is any non-trivial finite group. Below, we prove that the conclusions of Theorems \ref{thm 1} and \ref{thm 3} hold with $G$.
Obviously, we may and will assume that $G$ is a regular Galois group over $k$. Since $m_{G_1,k} \geq 2$, it suffices to find a Galois extension $F_1/k$ with Galois group $G_1$ which embeds into infinitely many Galois extensions of $k$ with Galois group $G$; see Theorem \ref{genus2}. Let $F_1/k$ be a Galois extension with Galois group $G_1$ and $E_2/k(T)$ a $k$-regular Galois extension with Galois group $G_2$ (such extensions exist as $G$ is a regular Galois group over $k$). Consider the extension $E_2F_1 / F_1(T)$. By Hilbert's irreducibility theorem, there exist infinitely many $t_0 \in k$ such that the specializations of $E_2F_1/F_1(T)$ at $t_0$ are distinct and all have Galois group $G_2$ (as $G_2$ is not trivial). For such a $t_0$, one has $(E_2 F_1)_{t_0} = (E_2)_{t_0} F_1$. Hence, $(E_2)_{t_0}/k$ has Galois group $G_2$ and $(E_2)_{t_0}$, $F_1$ are linearly disjoint over $k$. In particular, $(E_2)_{t_0} F_1/k$ has Galois group $G$, as needed.
\subsection{Groups whose order has only large prime factors}
This section is organized as follows. In \S6.2.1, we prove the conclusions of Theorems \ref{thm 1} and \ref{thm 3} for the group $G$, where $G$ is any group as in condition (2) of Theorem \ref{thm 1}. \S6.2.2 is devoted to more properties on the set of prime numbers that appear in condition (2) of Theorem \ref{thm 1}. In \S6.2.3, we prove the conclusions of Theorems \ref{thm 1.1} and \ref{thm 3.1} for the group $G$, where $G$ is any group as in condition (1) of Theorem \ref{thm 1.1}.
\subsubsection{Over the number field $k$} \label{6.2.1}
Denote the set of all prime numbers $p$ such that $[k(\zeta_p):k] \leq 2$ by $\mathcal{S}$. Assume that the order $|G|$ of $G$ has no prime factor in $\mathcal{S}$ and $|G|$ is neither 1 nor a prime number. As 2 is in $\mathcal{S}$, the group $G$ has odd order and, since it is not cyclic of prime order, it is not simple. Let $H$ be a non-trivial proper normal subgroup of $G$. By Theorem \ref{genus2} and Remark \ref{odd order}, it suffices to prove the inequality $m_{G/H,k} \geq 2$ to get the desired conclusions.
First, assume that there exists a $k$-regular Galois extension of $k(T)$ with Galois group $G/H$ and genus 1. By Proposition \ref{RH genus 1}, $G/H$ contains an element of order $n$ for some $n \in \{2,3,4,6\}$. But this last conclusion cannot happen since $|G/H|$ has no prime factor in $\mathcal{S}$ and $\{2,3\} \subset \mathcal{S}$.
Now, assume that there exists a $k$-regular Galois extension $E/k(T)$ with Galois group $G/H$ and genus 0. Since $|G/H|$ is coprime to 6, one has $G/H \cong \mathbb{Z}/n\mathbb{Z}$ for some integer $n$ which is coprime to 6 and $E/k(T)$ has exactly two branch points; see Proposition \ref{list genus 0}. Given a prime divisor $p$ of $n$, consider the subextension $E'/k(T)$ of $E/k(T)$ that has degree $p$. By the definition of $\mathcal{S}$, one has $[k(\zeta_p):k] \geq 3$. Then, by the Branch Cycle Lemma, $E'/k(T)$ has at least $3$ branch points, which cannot happen. Hence, one has $m_{G/H,k} \geq 2$, thus ending the proof.
\subsubsection{More on the set $\mathcal{S}$}
Below, we make the set $\mathcal{S}$ of \S\ref{6.2.1} more explicit, at the cost of making it larger. Denote the set of all prime numbers by $\mathcal{P}$. Given $p \in \mathcal{P}$, let $\zeta_p$ be a primitive $p$-th root of unity.
\begin{proposition} \label{S1} {\rm{(1)}} One has $\mathcal{S} \subset \{p \in \mathcal{P} \, : \, p \leq 2[k:\mathbb{Q}]+1\}$.
\noindent {\rm{(2)}} One has $\mathcal{S} \subset \{2,3\} \cup \{p \in \mathcal{P} \setminus \{2,3\} \, : \, k \cap \mathbb{Q}(\zeta_p) \not= \mathbb{Q}\}$.
\noindent {\rm{(3)}} One has $\mathcal{S} \subset \{2,3\} \cup \{p \in \mathcal{P} \setminus \{2,3\} \, : \, p \, \, ramifies \,\, in \, \, k/\mathbb{Q}\}$. \end{proposition}
\begin{proof} Combine the definition of $\mathcal{S}$ and the inequality $[k(\zeta_p):k] \geq (p-1)/{[k:\mathbb{Q}]}$ ($p$ prime) to get the first inclusion. For the second one, fix $p \in \mathcal{S} \setminus \{2,3\}$ and suppose $k \cap \mathbb{Q}(\zeta_p) = \mathbb{Q}$. Then, $[k(\zeta_p):k] = p-1$. As $p$ is in $\mathcal{S}$, one gets $p \leq 3$, which cannot happen. As to the third one, let $p \in \mathcal{S} \setminus \{2,3\}$. By (2), one has $k \cap \mathbb{Q}(\zeta_p) \not= \mathbb{Q}$. If $p$ does not ramify in $k/\mathbb{Q}$, then it does not ramify in $(k \cap \mathbb{Q}(\zeta_p)) / \mathbb{Q}$ either. In particular, the ramification index of $p$ in $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ is at most $[\mathbb{Q}(\zeta_p): k \cap \mathbb{Q}(\zeta_p)] \leq (p-1)/2$, which cannot happen. \end{proof}
As already said, one always has $\{2,3\} \subset \mathcal{S}$. In Proposition \ref{S2} below, we give sufficient conditions on $k$ for the converse to hold.
\begin{proposition} \label{S2} One has $\mathcal{S} = \{2,3\}$ in each of the following cases:
\noindent {\rm{(1)}} $k=\mathbb{Q}$,
\noindent {\rm{(2)}} $k/\mathbb{Q}$ ramifies at most at 2 and 3.
\noindent {\rm{(3)}} $k/\mathbb{Q}$ contains no non-trivial cyclic subextension. \end{proposition}
\begin{proof} If either condition (1) or condition (2) holds, we may apply Proposition \ref{S1} to get $\mathcal{S}=\{2,3\}$. Now, assume that $k/\mathbb{Q}$ contains no non-trivial cyclic subextension. We then need the following easy lemma, whose proof is left to the reader.
\begin{lemma} \label{lemma easy} Let $p$ be an odd prime. Then, $[k(\zeta_p):k] \leq 2$ if and only if $k$ contains the unique subfield $F$ of $\mathbb{Q}(\zeta_p)$ such that $[\mathbb{Q}(\zeta_p):F]=2$. \end{lemma}
\noindent Let $p \in \mathcal{S} \setminus \{2\}$. Then, by Lemma \ref{lemma easy}, the field $k$ contains a subfield $F$ such that $F/\mathbb{Q}$ is cyclic of degree $(p-1)/2.$ If $p \geq 5$, then $F/\mathbb{Q}$ has degree at least 2, which cannot happen. Hence, $p = 3$, as needed. \end{proof}
\subsubsection{Over the rationals}
Here, we assume that $G$ is a non-abelian finite group such that 2 does not divide $|G|$, 3 divides $|G|$, and $G \not= (\mathbb{Z}/p\mathbb{Z})^k \rtimes \mathbb{Z}/3\mathbb{Z}$ ($k \in \{1,2\}$, $p$ prime, $p \not=3$). Below, we show that the conclusions of Theorems \ref{thm 1.1} and \ref{thm 3.1} hold for the group $G$.
As the group $G$ has odd order, it suffices to find a non-trivial normal subgroup $H$ of $G$ such that $m_{G/H, \mathbb{Q}} \geq 2$; see Theorem \ref{genus2} and Remark \ref{odd order}. By Proposition \ref{list genus leq 1}, it then suffices to find a non-trivial proper normal subgroup $H$ of $G$ such that $G/H \not \cong \mathbb{Z}/3\mathbb{Z}$. Consider the derived subgroup $D(G)$ of $G$. It is a non-trivial proper characteristic subgroup of $G$ (since $G$ is solvable but not abelian). If $G/D(G) \not \cong \mathbb{Z}/3\mathbb{Z}$, we are done. One may then and will assume that $G/D(G) \cong \mathbb{Z}/3\mathbb{Z}$. If $D(G)$ has a non-trivial proper characteristic subgroup $H_0$, then one has $G/H_0 \not \cong \mathbb{Z}/3\mathbb{Z}$ (since $G/D(G) \cong \mathbb{Z}/3\mathbb{Z}$) and we are done. One may then and will assume that $D(G)$ has no non-trivial proper characteristic subgroup. As $D(G)$ is also solvable (and non-trivial), one has $D(G)= (\mathbb{Z}/p\mathbb{Z})^k$ for some prime number $p$ and some positive integer $k$; see, e.g., \cite[3.3.15]{Rob96}. First, assume that $p=3$. Since $G/D(G)$ has order 3, we get that $G$ is a 3-group. Set $|G|=3^l$, where $l$ is a positive integer. If $l \in \{1,2\}$, then $G$ is abelian, which cannot happen. One then has $l \geq 3$. Pick a normal subgroup $H$ of $G$ with order 3. Then, $G/H \not \cong \mathbb{Z}/3\mathbb{Z}$ (as $l \geq 3$) and we are done. Now, assume that $p \not=3$. As $G/D(G) \cong \mathbb{Z}/3\mathbb{Z}$, one may then apply the Schur-Zassenhaus theorem to get $G = (\mathbb{Z}/p\mathbb{Z})^k \rtimes \mathbb{Z}/3\mathbb{Z}$. Let $f: \mathbb{Z}/3\mathbb{Z} \rightarrow {\rm{Aut}} ((\mathbb{Z}/p\mathbb{Z})^k)$ be the morphism defining the semidirect product. Denote the elements of $\mathbb{Z}/3\mathbb{Z}$ by $\bar{0}$, $\bar{1}$, and $\bar{2}$. As $G$ is not abelian, $f(\bar{1})$ is not the trivial automorphism of $(\mathbb{Z}/p\mathbb{Z})^k$. Hence, $f(\bar{1})$ has order 3. In particular, there exists a non-zero element $y$ of $(\mathbb{Z}/p\mathbb{Z})^k$ such that $y + f(\bar{1})(y)+f(\bar{2})(y)=0$ in $(\mathbb{Z}/p\mathbb{Z})^k$. Then, the subgroup $H$ of $G$ generated by $(y,\bar{0})$ and $(f(\bar{1})(y),\bar{0})$ is a non-trivial proper normal subgroup of $G$ and, as $H$ is abelian, it has order at most $p^2$. From our assumption on $G$, one has $k \geq 3$. Hence, $G/H \not \cong \mathbb{Z}/3\mathbb{Z}$, thus ending the proof.
\subsection{Abelian groups}
This section is organized as follows. In \S6.3.1, we prove the conclusions of Theorems \ref{thm 1} and \ref{thm 3} for the group $G$, where $G$ is any group as in condition (3) of Theorem \ref{thm 1}. In \S6.3.2, we prove the conclusions of Theorems \ref{thm 1.1} and \ref{thm 3.1} for the group $G$, where $G$ is any group as in condition (2) of Theorem \ref{thm 1.1}. In \S6.3, we prove the conclusion of Theorem \ref{thm 2} for the group $G$, where $G$ is any group as in condition (1) of Theorem \ref{thm 2}.
\subsubsection{Over the number field $k$}
Here, we assume that $G$ is any non-trivial finite abelian group, but none of the following groups:
\noindent {\rm{(a)}} $\mathbb{Z}/n\mathbb{Z}$ ($n \geq 2$),
\noindent {\rm{(b)}} $(\mathbb{Z}/p\mathbb{Z})^2$ ($p$ prime),
\noindent {\rm{(c)}} $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2p\mathbb{Z}$ ($p$ prime),
\noindent {\rm{(d)}} $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3p\mathbb{Z}$ ($p$ prime),
\noindent {\rm{(e)}} $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$, $(\mathbb{Z}/4\mathbb{Z})^2$, $(\mathbb{Z}/2\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^2 \times \mathbb{Z}/4\mathbb{Z}$, $(\mathbb{Z}/3\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^4$.
\noindent Below, we show that the conclusions of Theorem \ref{thm 1} and Theorem \ref{thm 3} hold for the abelian group $G$. By Theorem \ref{genus2}, Remark \ref{odd order}, and Proposition \ref{list genus leq 1 abelian}, it suffices to show that $G$ has a non-trivial proper subgroup $H$ such that $G/H$ is none of the following groups: $\mathbb{Z}/n\mathbb{Z} \, (n \geq 2), (\mathbb{Z}/2\mathbb{Z})^2, \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}, (\mathbb{Z}/3\mathbb{Z})^2,(\mathbb{Z}/2\mathbb{Z})^3.$
Set $G=\mathbb{Z}/d_1\mathbb{Z} \times \cdots \times \mathbb{Z}/d_m\mathbb{Z},$ where $m \geq 1$ and $d_1, \dots,d_m$ are integers $\geq 2$ such that $d_{i}$ divides $d_{i+1}$ for each $i \in \{1,\dots,m-1\}$. As $G$ is not cyclic (condition (a)), one has $m \geq 2$. We split the proof into three cases: $m\ge 4$, $m=3$, and $m=2$. For short, we will call a non-trivial proper quotient of $G$ ``suitable" if it is not in the above list.
Assume that $m\geq 4$. Then, $\mathbb{Z}/d_2\mathbb{Z} \times \cdots \times \mathbb{Z}/d_{m}\mathbb{Z}$ is a suitable quotient of $G$, unless $m=4$ and $d_1= \cdots =d_4=2$. But this last case cannot happen by condition (e).
Assume that $m=3$. First, assume that $d_3 > d_1$ and $d_1 \not=2$. Then, $(\mathbb{Z}/d_1\mathbb{Z})^3$ is a suitable quotient of $G$. Second, assume that $d_3>d_1$ and $d_1=2$. Then, $\mathbb{Z}/d_2\mathbb{Z}\times \mathbb{Z}/d_3\mathbb{Z}$ is a non-trivial proper quotient of $G$ that is suitable, unless $d_2=2$ and $d_3=4.$ But this last case cannot happen by condition (e). Third, assume that $d_1=d_2=d_3$. Then, $(\mathbb{Z}/d_3\mathbb{Z})^2$ is a non-trivial proper quotient of $G$ that is suitable, unless $d_3 =2$ or $d_3=3$. But none of these two cases can happen by condition (e).
Assume that $m=2$. First, assume that $d_2>d_1\geq 4$. Then, $(\mathbb{Z}/d_1\mathbb{Z})^2$ is a suitable quotient of $G$. Second, assume that $d_2=d_1 \geq 4$. Then, $d_2$ is not a prime number (condition (b)). Set $d_2=r \cdot s$, where $r$ and $s$ are integers $\geq 2$. By condition (e), one has $d_2 \geq 6$. Then, $\mathbb{Z}/r\mathbb{Z} \times \mathbb{Z}/d_2\mathbb{Z}$ is a suitable quotient of $G$. Third, assume that $d_1 \leq 3$. Then, by conditions (b), (c), and (d), $d_2/d_1$ is neither $1$ nor a prime. Set $d_2=d_1 \cdot r\cdot s$, where $r$ and $s$ are integers $\geq 2$. Then, $\mathbb{Z}/d_1\mathbb{Z} \times \mathbb{Z}/(d_1 \cdot r)\mathbb{Z}$ is a non-trivial proper quotient of $G$ that is suitable, unless $d_1=2$ and $d_1 \cdot r=4$. This last case leads to $r=2$ and, without loss, to $s=2$ as well. Hence, $G=\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$, which cannot happen by condition (e).
\subsubsection{Over $\mathbb{Q}$}
Here, we assume that $G$ is any non-trivial finite abelian group as in condition (2) of Theorem \ref{thm 1.1}. If $G$ is as in condition (3) of Theorem \ref{thm 1}, then the conclusions of Theorems \ref{thm 1.1} and \ref{thm 3.1} hold with $G$. One may then assume that $G$ is one of the following groups:
\noindent {\rm{(a)}} $\mathbb{Z}/n\mathbb{Z}$ ($n$ is neither a prime number nor in $\{4,6,12\}$),
\noindent {\rm{(b)}} $(\mathbb{Z}/p\mathbb{Z})^2$ ($p$ prime, $p \geq5$),
\noindent {\rm{(c)}} $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2p\mathbb{Z}$ ($p$ prime, $p \geq 5$),
\noindent {\rm{(d)}} $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3p\mathbb{Z}$ ($p$ prime),
\noindent {\rm{(e)}} $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$, $(\mathbb{Z}/4\mathbb{Z})^2$, $(\mathbb{Z}/2\mathbb{Z})^2 \times \mathbb{Z}/4\mathbb{Z}$, $(\mathbb{Z}/3\mathbb{Z})^3$.
First, assume that $G$ is none of the following groups:
\noindent {\rm{(i)}} $\mathbb{Z}/2^a\mathbb{Z}$ ($a \geq3$),
\noindent {\rm{(ii)}} $\mathbb{Z}/3^b\mathbb{Z}$ ($b \geq2$),
\noindent {\rm{(iii)}} $\mathbb{Z}/2^a3^b \mathbb{Z}$ ($a \geq 1$, $b \geq 1$, $(a,b) \not \in \{(1,1),(2,1)\}$),
\noindent {\rm{(iv)}} $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$, $(\mathbb{Z}/4\mathbb{Z})^2$, $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z}$, $(\mathbb{Z}/2\mathbb{Z})^2 \times \mathbb{Z}/4\mathbb{Z}$, $(\mathbb{Z}/3\mathbb{Z})^3$.
\noindent Then, there exist a prime $p \geq 5$ and a non-trivial proper subgroup $H$ of $G$ such that $G/H \cong \mathbb{Z}/p\mathbb{Z}$. Hence, $m_{G/H,\mathbb{Q}} \geq 2$ by Proposition \ref{list genus leq 1 abelian}. Theorem \ref{genus2} and Remark \ref{odd order} then provide the desired conclusions.
Now, assume that $G$ is any group as in conditions (i)-(iv) above. Then, $G$ has a non-trivial proper subgroup $H$ such that $G/H$ is $\mathbb{Z}/8\mathbb{Z}$ or $\mathbb{Z}/9\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$ or $ \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z},$ unless $G=\mathbb{Z}/8\mathbb{Z}$ or $G=\mathbb{Z}/9\mathbb{Z}$. Hence, one has $m_{G/H,\mathbb{Q}} \geq 2$; see Proposition \ref{list genus leq 1 abelian}. As above, one shows that the conclusions of Theorems \ref{thm 1.1} and \ref{thm 3.1} hold for the group $G$.
Finally, assume that $G=\mathbb{Z}/8\mathbb{Z}$ (the other case for which $G=\mathbb{Z}/9\mathbb{Z}$ is similar). In this context, we refer to Lemma \ref{lemma genus2}. By Remark \ref{odd order}, it suffices to show that, given a $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ with Galois group $\mathbb{Z}/8\mathbb{Z}$, the subextension $E^{\mathbb{Z}/2\mathbb{Z}}/\mathbb{Q}(T)$ has genus at least 2. Denote the Euler function by $\varphi$. By the Branch Cycle Lemma, $E/\mathbb{Q}(T)$ has at least $\varphi(8)=4$ branch points that are totally ramified. In particular, $E^{\mathbb{Z}/2\mathbb{Z}}/\mathbb{Q}(T)$ has at least 4 branch points. Then, the genus of $E^{\mathbb{Z}/2\mathbb{Z}}/\mathbb{Q}(T)$ is at least 2 by the Riemann-Hurwitz formula.
\subsubsection{Remaining cases}
Below, we assume that $G$ is among $\mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$, $(\mathbb{Z}/3\mathbb{Z})^2$, $(\mathbb{Z}/2\mathbb{Z})^3$, $(\mathbb{Z}/2\mathbb{Z})^4$.
We need the following lemma, which is a standard consequence of the rigidity method; see, e.g., \cite[\S3.2]{Vol96}.
\begin{lemma} \label{general abelian} Let $G_0$ be a non-trivial finite abelian group. Then, there exists a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ with Galois group $G_0$ and whose inertia canonical invariant contains the conjugacy class $\{g_0\}$ for every non-zero element $g_0$ of $G_0$. \end{lemma}
First, assume that $G=(\mathbb{Z}/2\mathbb{Z})^3$ or $G=(\mathbb{Z}/2\mathbb{Z})^4$. Then, $G$ contains at least 6 elements of order 2, which then generate maximal cyclic subgroups of $G$. One may then apply Theorem \ref{genus0} (with $H$ any subgroup of $G$ with order 2), as well as Remark \ref{odd order} and Lemma \ref{general abelian}, to get the conclusion of Theorem \ref{thm 2} for the group $G$.
Second, assume that $G$ is $\mathbb{Z}/12\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ or $(\mathbb{Z}/3\mathbb{Z})^2$. By Theorem \ref{cyclic}, Remark \ref{odd order}, and Lemma \ref{general abelian}, it suffices to find a non-trivial subgroup $H$ of $G$ such that
\noindent - $G/H$ is cyclic of order at least 3,
\noindent
- $n \geq 2+ |H|$, with $n$ the number of elements of $G$ with maximal order.
\noindent But this last claim holds:
\noindent - if $G=\mathbb{Z}/12\mathbb{Z}$, one has $n=4$ and one can take $H=\mathbb{Z}/2\mathbb{Z}$,
\noindent - if $G=\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, one has $n=4$ and one can take $H=\mathbb{Z}/2\mathbb{Z} \times \{0\}$,
\noindent - if $G=\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$, one has $n =4$ and one can take $H=\mathbb{Z}/2\mathbb{Z} \times \{0\}$,
\noindent - if $G=(\mathbb{Z}/3\mathbb{Z})^2$, one has $n=8$ and one can take $H=\mathbb{Z}/3\mathbb{Z} \times \{0\}$.
Third, assume that $G=\mathbb{Z}/6\mathbb{Z}$. Here, we apply Theorem \ref{cyclic2}. First, note that conditions (1), (2), and (3) are trivially satisfied. Moreover, condition (4) holds with the two conjugacy classes of the two elements of $G$ of order 6.
\subsection{Dihedral groups}
Let $n$ be a positive integer. Assume that $G$ is the dihedral group $D_n$ (with $2n$ elements).
\subsubsection{Proof of the conclusion of Theorem \ref{thm 3.1}}
First, assume that $n$ is not a prime number and $n$ has a prime factor $p \geq 11$. By \cite[Theorem 5.1]{DF94} and the Riemann-Hurwitz formula, $D_p \cong D_n/(\mathbb{Z}/(n/p)\mathbb{Z})$ satisfies $m_{D_p,\mathbb{Q}} \geq 2$. It then remains to apply Theorem \ref{genus2} and Remark \ref{odd order} to get the conclusion of Theorem \ref{thm 3.1} for the group $G$ \footnote{This argument also provides the conclusion of Theorem \ref{thm 1.1} for the dihedral group $D_n$, under the extra assumption that $n$ has a prime factor $p \geq 11$.}.
\subsubsection{Proof of the conclusion of Theorem \ref{thm 1.1}}
Now, assume that $n$ is neither a prime number nor in $\{1,4,6,8,9,12\}.$ Let $p$ be the smallest prime factor of $n$. Then, $G$ has a unique normal subgroup $H$ of order $p$ and the quotient $G/H$ is isomorphic to $D_{n/p}$.
We then need the following two lemmas.
\begin{lemma} \label{lemma 2 dihedral} Let $\mathcal{S}$ be a finite set of primes $q$ such that $q \equiv 1 \, \, {\rm{mod}} \, \, n$. Then, there exists a Galois extension $F/\mathbb{Q}$ that satisfies the following two conditions:
\noindent {\rm{(1)}} ${\rm{Gal}}(F/\mathbb{Q})=D_{n}$,
\noindent {\rm{(2)}} the ramification index of $F/\mathbb{Q}$ at $q$ is equal to $n$ for each $q \in \mathcal{S}$. \end{lemma}
\begin{proof} Given a prime number $q \in \mathcal{S}$, let $F_q/\mathbb{Q}_q$ be a totally ramified cyclic extension of degree $n$ \footnote{For example, one can take $F_q/\mathbb{Q}_q= \mathbb{Q}_q(\sqrt[n]{q})/\mathbb{Q}_q$ (as $q \equiv 1 \, \, {\rm{mod}} \, \, n$).}. Then, by \cite[Theorem 1.1]{DLN17}, there exists a Galois extension $F/\mathbb{Q}$ such that ${\rm{Gal}}(F/\mathbb{Q})=D_{n}$ and the completion of $F/\mathbb{Q}$ at $q$ is equal to $F_q/\mathbb{Q}_q$ for each $q \in \mathcal{S}$. Hence, the lemma holds. \end{proof}
\begin{lemma} \label{lemma 1 dihedral} One has $m_{D_{n/p},\mathbb{Q}} \geq 1$. \end{lemma}
\begin{proof} If $m_{D_{n/p},\mathbb{Q}} =0$, then $n/p \in \{1,2,3,4,6\}$ by Proposition \ref{list genus 0}, i.e., $n$ is a prime number or $n \in \{4, 6,8,9,12\}$, which cannot happen. \end{proof}
Given a prime $q$ such that $q \equiv 1 \, \, {\rm{mod}} \, \, n$, let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_n$ and ramification index $n$ at $q$ (Lemma \ref{lemma 2 dihedral}). Consider the subextension $F^H/\mathbb{Q}$ (that has Galois group $D_{n/p}$). Since the ramification index of $q$ in $F/\mathbb{Q}$ is equal to $n$, that in $F^H/\mathbb{Q}$ is at least $n/p$ and, as $n$ is neither a prime number nor equal to 4, one has $n/p \geq 3$. Moreover, as $H$ is the unique normal subgroup of $D_n$ with order $p$, the extension $F^H/\mathbb{Q}$ embeds into infinitely many Galois extensions of $\mathbb{Q}$ with Galois group $D_n$; see Proposition \ref{solvable kernel}. It then remains to use Theorem \ref{genus1} and Lemma \ref{lemma 1 dihedral} to conclude.
\subsection{Symmetric groups over $k$}
Assume that $G=S_n$ for some $n \geq 8$. Below, we show that $G$ has no parametric extension over $k$. To do this, we apply Theorem \ref{genus0} (with $H=A_n$). Recall that a permutation $\sigma \in S_n$ has {\it{type $1^{l_1} \dots n^{l_n}$}} if, for each index $i \in \{1,\dots, n\}$, there are $l_i$ disjoint cycles of length $i$ in the cycle decomposition of $\sigma$ (for example, an $n$-cycle has type $n^1$). Denote the conjugacy class in $S_n$ of elements of type $1^{l_1} \dots n^{l_n}$ by $[1^{l_1} \dots n^{l_n}]$.
\subsubsection{Checking condition {\rm{(1)}}}
Let $E/\mathbb{Q}(T)$ be a $\mathbb{Q}$-regular Galois extension with group $S_n$ and inertia canonical invariant $([1^{n-2}2^1],$ $[1^1(n-1)^1], [n^1])$ (such an extension exists; see, e.g., \cite[\S8.3.1]{Ser92}). By the Branch Cycle Lemma, the associated branch points all are $\mathbb{Q}$-rational and this inertia canonical invariant contains exactly two conjugacy classes of elements of $S_n$ with even order. Moreover, by the Riemann-Hurwitz formula, the genus of $E/\mathbb{Q}(T)$ is at least 2 (as $n \geq 5$). We may then apply Proposition \ref{Debes 92} to get condition (1) of Theorem \ref{genus0}.
\subsubsection{Checking condition {\rm{(2)}}}
First, we need Lemma \ref{symmetric} below, which relates to part (d) of condition (2) of Theorem \ref{genus0}.
\begin{lemma} \label{symmetric} Every conjugacy class of $S_n$ which is not contained in $A_n$ belongs to the inertia canonical invariant of some $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ with Galois group $S_n$. \end{lemma}
\begin{proof} Let $C$ be a conjugacy class of $S_n$ which is not contained in $A_n$. Denote the type of all the elements of $C$ by $e_1^1 \dots e_s^1$. Let $T_1,\dots,T_n,T, Y$ be indeterminates that are algebraically independent over $\mathbb{Q}$. Set $$H(Y)=(Y-1)^{e_1} \cdots (Y-s)^{e_s} \in \mathbb{Q}[Y],$$ $$G(Y)=Y^n + T_1 Y^{n-1} + \cdots + T_{n-1} Y + T_n \in \mathbb{Q}(T_1,\dots,T_n)[Y]$$ and $$F(Y)=G(Y) - T H(Y) \in \mathbb{Q}(T_1,\dots,T_n,T)[Y].$$ Clearly, $F$ has group $S_n$ over $\mathbb{Q}(T_1,\dots,T_n,T)$. By Hilbert's irreducibility theorem, there are infinitely many $(t_1,\dots,t_n) \in \mathbb{Q}^n$ such that $$\overline{F}(Y):=Y^n + t_1 Y^{n-1} + \cdots + t_{n-1} Y + t_n - TH(Y) \in \mathbb{Q}(T)[Y]$$ has Galois group $S_n$ over $\mathbb{Q}(T)$. Pick such a tuple $(t_1,\dots,t_n)$ and denote the splitting field of $\overline{F}(Y)$ over $\mathbb{Q}(T)$ by $E$. By \cite[Lemma 3.1]{Mue02}, $\infty$ is a branch point of $E/\mathbb{Q}(T)$ and the associated inertia canonical conjugacy class is $C$. Then, the Galois group of $E\overline{\mathbb{Q}}/\overline{\mathbb{Q}}(T)$, which is a normal subgroup of $S_n$, contains an element of $S_n \setminus A_n$. Hence, ${\rm{Gal}}(E\overline{\mathbb{Q}}/\overline{\mathbb{Q}}(T))=S_n$, i.e., $E/\mathbb{Q}$ is regular, thus ending the proof. \end{proof}
Now, we prove condition {\rm{(2)}} of Theorem \ref{genus0}. By Lemma \ref{symmetric} and as each conjugacy class of $S_n$ is rational\footnote{Recall that a conjugacy class $C$ of a given finite group $G$ is {\it{rational}} if $C^i=C$ for each integer $i$ which is coprime to the order of the elements of $C$.}, it suffices to find 5 distinct conjugacy classes $C_1, \dots, C_5$ of $S_n$ such that $C_i \not \subset A_n$ and each element of $C_i$ generates a maximal cyclic subgroup of $S_n$ ($i \in \{1,\dots,5\}$). First, assume that $n \geq 11$. Then, for each integer $1 \leq m \leq n-1$ (resp., $1 \leq m \leq n-2$), the conjugacy class $[m^1(n-m)^1]$ (resp., $[1^1m^1(n-m-1)^1]$) has the desired properties if $n$ is odd (resp., if $n$ is even) and one has $(n-1)/2 \geq 5$ (resp., $(n-2)/2 \geq 5$) such conjugacy classes. In the remaining cases, these conjugacy classes have the desired properties:
\noindent - $[10^1], [1^2 8^1], [1^1 2^1 7^1], [1^1 3^1 6^1], [1^1 4^1 5^1]$ (if $n=10$),
\noindent - $[1^1 8^1], [2^1 7^1], [3^1 6^1], [4^1 5^1], [1^1 2^1 3^2]$ (if $n=9$),
\noindent - $[8^1], [1^2 6^1], [1^1 2^1 5^1], [1^1 3^1 4^1], [2^13^2]$ (if $n=8$).
\subsection{Symmetric groups over $\mathbb{Q}$}
Assume that $G=S_n$ for some $n \geq 6$. Below, we show that $G$ has no parametric extension over $\mathbb{Q}$. By \S6.5, one may and will assume that $n \in \{6,7\}$. In this case, we apply Theorem \ref{cyclic2}. First, note that condition (1) clearly holds, condition (2) holds by Proposition \ref{GAR 2}, and condition (3) holds as well (as explained in \S6.5.1). By Lemma \ref{symmetric}, it remains to find 3 distinct conjugacy classes $C_1,C_2,C_3$ of $S_n$ such that $C_i$ is not contained in $A_n$ and each element of $C_i$ generates a maximal cyclic subgroup of $S_n$ ($i \in \{1,2,3\}$). For $n=6$, one can take $\{C_1,C_2,C_3\} = \{[6^1], [1^2 4^1], [1^12^13^1]\}$. For $n=7$, one can take $\{C_1,C_2,C_3\}=\{[1^16^1], [2^1 5^1], [3^1 4^1]\}$, thus ending the proof.
\subsection{Linear groups (and more general groups)}
Here, we assume that the center $Z(G)$ of $G$ is not trivial and the quotient $G/Z(G)$ is neither solvable nor $A_5$. Below, we show that the conclusions of Theorems \ref{thm 1} and \ref{thm 3} hold with $G$. Without loss, we may and will assume that $G$ is a regular Galois group over $k$. By Proposition \ref{solvable kernel}, there exists a Galois extension of $k$ with Galois group $G/Z(G)$ which embeds into infinitely many Galois extensions of $k$ with group $G$. Moreover, one has $m_{G/Z(G),k} \geq 2$ by Proposition \ref{list genus leq 1}. It then remains to apply Theorem \ref{genus2} to conclude. The same argument provides the conclusions of Theorems \ref{thm 1.1} and \ref{thm 3.1} for the group $G$ if $Z(G)$ is not trivial and $G/Z(G)$ is neither solvable of even order nor of order $\leq 3$.
In Proposition \ref{linear groups} below, we give explicit examples of finite groups $G$ as above.
\begin{proposition} \label{linear groups} Let $n \geq 2$ be an integer and $q \geq 3$ a prime power. Set $G={\rm{GL}}_n(\mathbb{F}_q)$. Then, the following three conclusions hold.
\noindent {\rm{(1)}} The center $Z(G)$ of $G$ is not trivial.
\noindent {\rm{(2)}} Assume that $(n,q) \not \in \{(2,3), (2,4)\}$. Then, $G/Z(G)$ is neither solvable nor $A_5$.
\noindent {\rm{(3)}} Assume that $(n,q) \not=(2,3)$. Then, $G/Z(G)$ is neither solvable of even order nor of order $\leq 3$. \end{proposition}
\begin{proof}
Conclusion (1) follows from the assumption $q \geq 3$. For conclusion (2), assume that $(n,q) \not \in \{(2,3), (2,4)\}$. If $G/Z(G)={\rm{PGL}}_n(\mathbb{F}_q$) was solvable, then this would be also true for ${\rm{PSL}}_n(\mathbb{F}_q)$. By, e.g., \cite[Theorem 9.46]{Rot95}, one would get $(n,q) \in \{(2,2), (2,3)\}$, which cannot happen. Moreover, it is easily checked that $|{\rm{PGL}}_n(\mathbb{F}_q)| \not= 60$ (as $(n,q) \not=(2,4)$). Hence, ${\rm{PGL}}_n(\mathbb{F}_q) \not \cong A_5$ and conclusion (2) holds. Now, assume that $(n,q) \not= (2,3)$. Then, by the above, $G/Z(G)$ is not solvable. In particular, conclusion (3) holds. \end{proof}
\section{Appendix A: groups of minimal genus 0 or 1}
For this section, let $k$ be a number field, $G$ a non-trivial finite group, and $E/k(T)$ a $k$-regular Galois extension with Galois group $G$ and inertia canonical invariant $(C_1,\dots,C_r)$. For each $i \in \{1,\dots,r\}$, let $e_i$ be the order of the elements of $C_i$. The unordered $r$-tuple $(e_1,\dots,e_r)$ is called the {\it{ramification type of $E/k(T)$}}.
Below, we collect some well-known conclusions on the group $G$, the number of branch points $r$, and the ramification type $(e_1,\dots,e_r)$, under the assumption that the genus $g$ of $E/k(T)$ is either 0 or 1.
First, we consider the cases $g=0$ and $g=1$ separately. Propositions 7.1-2 below are essentially applications of the Riemann-Hurwitz formula and the Branch Cycle Lemma. See also, e.g., \cite[Chapter I, Theorem 6.2]{MM99}.
\begin{proposition} \label{list genus 0} Assume that $g=0$. Then, the following two conclusions hold.
\noindent {\rm{(1)}} One of the following conditions holds:
{\rm{(a)}} $G=\mathbb{Z}/n\mathbb{Z}$ for some $n \geq 2$, $r=2$, and $(e_1,e_2)=(n,n)$,
{\rm{(b)}} $G=D_n$ for some $n \geq 2$, $r=3$, and $(e_1,e_2,e_3)=(2,2,n)$,
{\rm{(c)}} $G=A_4$, $r=3$, and $(e_1,e_2,e_3)=(2,3,3)$,
{\rm{(d)}} $G=S_4$, $r=3$, and $(e_1,e_2,e_3)=(2,3,4)$,
{\rm{(e)}} $G=A_5$, $r=3$, and $(e_1,e_2,e_3) = (2,3,5)$.
\noindent {\rm{(2)}} Assume that $k=\mathbb{Q}$. Then, cases {\rm{(a)}} and {\rm{(b)}} cannot occur if $n \not \in \{2,3,4,6\}$, and case {\rm{(e)}} does not occur either. \end{proposition}
\begin{proposition} \label{RH genus 1} Assume that $g=1$. Then, the following two conclusions hold.
\noindent {\rm{(1)}} One of the following conditions holds:
{\rm{(a)}} $r=3$ and $(e_1,e_2,e_3)=(2,3,6)$,
{\rm{(b)}} $r=3$ and $(e_1,e_2,e_3)=(2,4,4)$,
{\rm{(c)}} $r=3$ and $(e_1,e_2,e_3)=(3,3,3)$,
{\rm{(d)}} $r=4$ and $(e_1,e_2,e_3,e_4)=(2,2,2,2)$.
\noindent {\rm{(2)}} Assume that $k=\mathbb{Q}$. Then, cases {\rm{(1)}}, {\rm{(2)}}, and {\rm{(3)}} cannot occur. \end{proposition}
Now, for the convenience of the reader, we summarize some important properties that hold in both the cases $g=0$ and $g=1$.
\begin{proposition} \label{list genus leq 1} Assume that $g \leq 1$. Then, the following two conclusions hold.
\noindent {\rm{(1)}} The group $G$ is either solvable or $A_5$.
\noindent {\rm{(2)}} Assume that $k=\mathbb{Q}$. Then, the group $G$ is either solvable of even order or $\mathbb{Z}/3\mathbb{Z}$. \end{proposition}
\begin{proof} First, we show (1). As the desired conclusion is quite clear if $g=0$ (by Proposition \ref{list genus 0}), we may assume that $g=1$. Then, e.g., \cite[Proposition 2.4]{GT90} provides that $G$ is solvable. As for (2), it is a straightforward combination of Propositions 7.1-2 and the above conclusion (1). \end{proof}
Finally, we make the Galois group $G$ totally explicit, under the extra assumption that this group is abelian. As Proposition \ref{list genus leq 1 abelian} below is a straightforward application of Propositions 7.1-2 and the Branch Cycle Lemma, details are left to the reader.
\begin{proposition} \label{list genus leq 1 abelian} Assume that $G$ is abelian and $g \leq 1$. Then, the following two conclusions hold.
\noindent {\rm{(1)}} The group $G$ is one of the following finite groups: $\mathbb{Z}/n\mathbb{Z} \, (n \geq 2), (\mathbb{Z}/2\mathbb{Z})^2, \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}, (\mathbb{Z}/3\mathbb{Z})^2, (\mathbb{Z}/2\mathbb{Z})^3.$
\noindent {\rm{(2)}} Assume that $k=\mathbb{Q}$. Then, $G$ is one of the following finite groups: $\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/3\mathbb{Z}, \mathbb{Z}/4\mathbb{Z}, \mathbb{Z}/6\mathbb{Z}, (\mathbb{Z}/2\mathbb{Z})^2, (\mathbb{Z}/2\mathbb{Z})^3.$ \end{proposition}
\section{Appendix B: on parametric quadratic extensions in low genus}
The aim of this section consists in proving Proposition \ref{r leq 4} on $\mathbb{Q}$-regular quadratic extensions of $\mathbb{Q}(T)$ below.
\begin{proposition} \label{r leq 4} Let $E/\mathbb{Q}(T)$ be a $\mathbb{Q}$-regular quadratic extension with at most 4 branch points. Then, the following conditions are equivalent:
\noindent {\rm{(1)}} $E/\mathbb{Q}(T)$ is parametric over $\mathbb{Q}$,
\noindent {\rm{(2)}} $E/\mathbb{Q}(T)$ has exactly two branch points and both are $\mathbb{Q}$-rational.
\noindent Moreover, if condition {\rm{(2)}} fails, then there exist infinitely many quadratic extensions of $\mathbb{Q}$ each of which is not a specialization of $E/\mathbb{Q}(T)$. \end{proposition}
The proof is organized as follows. In \S8.1, we recall some standard background on hyperelliptic curves and their quadratic twists. In \S8.2, we explain how to translate the problem in terms of rational points on quadratic twists. Finally, we prove Proposition \ref{r leq 4} in \S8.3.
\subsection{Hyperelliptic curves and their quadratic twists}
Let $P(T) \in \mathbb{Z}[T]$ be a separable polynomial with degree $n$. Set $$P(T)=a_0 + a_1 T + \cdots + a_{n-1} T^{n-1} + a_n T^n.$$
\subsubsection{The even case}
First, we assume that $n$ is even. Consider the equivalence relation $\sim$ on $\overline{\mathbb{Q}}^3 \setminus \{(0,0,0)\}$ defined as follows: $$(y_1,t_1,z_1) \sim (y_2,t_2,z_2)$$ if and only if there exists some $\lambda \in \overline{\mathbb{Q}} \setminus \{0\}$ such that $$(y_2,t_2,z_2) = (\lambda^{n/2}y_1,\lambda t_1, \lambda z_1).$$ The quotient space $(\overline{\mathbb{Q}}^3 \setminus \{(0,0,0)\} )/ \sim$ is a weighted projective space that is denoted by $$\mathbb{P}_{n/2,1,1}(\overline{\mathbb{Q}}).$$ Given $(y,t,z) \in \overline{\mathbb{Q}}^3 \setminus \{(0,0,0)\}$, the corresponding point in $\mathbb{P}_{n/2,1,1}(\overline{\mathbb{Q}})$ is denoted by $[y:t:z].$
Set $$P(T,Z)= a_0 Z^n + a_1 Z^{n-1} T + \cdots + a_{n-1} Z T^{n-1} +a_n T^n.$$ The equation $Y^2= P(T,Z)$ in $\mathbb{P}_{n/2,1,1}(\overline{\mathbb{Q}})$ is {\it{the hyperelliptic curve associated with $P(T)$}}, denoted by $C_{P(T)}$. The set of all $\mathbb{Q}$-rational points on $C_{P(T)}$, i.e., the set of all points $[y:t:z] \in \mathbb{P}_{n/2,1,1}(\overline{\mathbb{Q}})$ such that $(y,t,z) \in \mathbb{Q}^3 \setminus \{(0,0,0)\}$ and $y^2= P(t,z)$, is denoted by $C_{P(T)}(\mathbb{Q}).$ A point $[y:t:z] \in C_{P(T)}(\mathbb{Q})$ is {\it{trivial}} if $y=0$ and {\it{non-trivial}} otherwise. Equivalently, $[y:t:z] \in C_{P(T)}(\mathbb{Q})$ is trivial if $z \not=0$ and $P(t/z)=0$.
Let $d$ be a squarefree integer. The hyperelliptic curve $Y^2 = d \cdot P(T,Z)$ associated with the polynomial $d \cdot P(T)$ is called {\it{the $d$-th quadratic twist of $C_{P(T)}$}}; we denote it by $C_{d \cdot P(T)}$.
\subsubsection{The odd case}
The case where $n$ is odd is quite similar. For the convenience of the reader and to avoid confusion, we redetail it below.
Consider the following equivalence relation $\sim$ on $\overline{\mathbb{Q}}^3 \setminus \{(0,0,0)\}$: $$(y_1,t_1,z_1) \sim (y_2,t_2,z_2)$$ if and only if there exists some $\lambda \in \overline{\mathbb{Q}} \setminus \{0\}$ such that $$(y_2,t_2,z_2) = (\lambda^{(n+1)/2}y_1,\lambda t_1, \lambda z_1).$$ The quotient space $(\overline{\mathbb{Q}}^3 \setminus \{(0,0,0)\} )/ \sim$ is a weighted projective space that is denoted by $\mathbb{P}_{(n+1)/2,1,1}(\overline{\mathbb{Q}}).$ Given $(y,t,z) \in \overline{\mathbb{Q}}^3 \setminus \{(0,0,0)\}$, the corresponding point in $\mathbb{P}_{(n+1)/2,1,1}(\overline{\mathbb{Q}})$ is denoted by $[y:t:z].$
Set $$P(T,Z)= a_0 Z^{n+1} + a_1 Z^{n} T + \cdots + a_{n-1} Z^2 T^{n-1} +a_n Z T^n.$$ The equation $Y^2= P(T,Z)$ in $\mathbb{P}_{(n+1)/2,1,1}(\overline{\mathbb{Q}})$ is {\it{the hyperelliptic curve associated with $P(T)$}}; we denote it by $C_{P(T)}$. The set of all $\mathbb{Q}$-rational points on $C_{P(T)}$, i.e., the set of all points $[y:t:z] \in \mathbb{P}_{(n+1)/2,1,1}(\overline{\mathbb{Q}})$ such that $(y,t,z) \in \mathbb{Q}^3 \setminus \{(0,0,0)\}$ and $y^2= P(t,z)$, is denoted by $C_{P(T)}(\mathbb{Q}).$ A point $[y:t:z] \in C_{P(T)}(\mathbb{Q})$ is {\it{trivial}} if $y=0$ and {\it{non-trivial}} otherwise. Equivalently, $[y:t:z] \in C_{P(T)}(\mathbb{Q})$ is trivial if either $z=0$ (this point, which is $[0:1:0]$, is the point at $\infty$) or $z \not=0$ and $t/z$ is a root of $P(T)$.
Given a squarefree integer $d$, the hyperelliptic curve $Y^2 = d \cdot P(T,Z)$ associated with the polynomial $d \cdot P(T)$ is called {\it{the $d$-th quadratic twist of $C_{P(T)}$}}; we denote it by $C_{d \cdot P(T)}$.
\subsection{From specializations of quadratic extensions to rational points on twisted hyperelliptic curves and vice-versa}
Let $P(T) \in \mathbb{Z}[T]$ be a separable polynomial with degree $n$, roots $t_1,\dots,t_n$, and leading coefficient $a_n$. For short, we set $E=\mathbb{Q}(T)(\sqrt{P(T)})$.
First, we make the set of branch points and the specializations of the $\mathbb{Q}$-regular quadratic extension $E/\mathbb{Q}(T)$ explicit.
\begin{lemma} \label{lemma 1} The set of branch points of $E/\mathbb{Q}(T)$ is either the set $\{t_1,\dots,t_n\}$ (if $n$ is even) or the set $\{t_1,\dots,t_n\} \cup \{ \infty \}$ (if $n$ is odd). \end{lemma}
\begin{proof} See, e.g., \cite[Proposition 6.2.3]{Sti09}. \end{proof}
\begin{lemma} \label{lemma 2} {\rm{(1)}} Let $t_0 \in \mathbb{Q} \setminus \{t_1,\dots,t_n\}$. Then, $E_{t_0} = \mathbb{Q}(\sqrt{P(t_0)}).$
\noindent {\rm{(2)}} Assume that $n$ is even. Then, $E_\infty = \mathbb{Q}(\sqrt{a_n})$. \end{lemma}
\begin{proof} First, we prove part (1). As $P(t_0) \not=0$, the polynomial $Y^2-P(t_0)$ is separable. Since $E$ is the splitting field of $Y^2-P(T)$ over $\mathbb{Q}(T)$, $E_{t_0}$ is the splitting field of $Y^2-P(t_0)$ over $\mathbb{Q}$, i.e., $E_{t_0} = \mathbb{Q}(\sqrt{P(t_0)})$.
Now, we prove part (2). Set $P(T)=a_0 + a_1 T + \cdots + a_{n-1}T^{n-1} + a_n T^n.$ One has $$P(T)=(T^{n/2})^{2} \Big(\frac{a_0}{T^n} + \frac{a_1}{T^{n-1}} + \cdots + \frac{a_{n-1}}{T} + a_n \Big).$$ Set $U=1/T$ and $Q(U)= a_n + a_{n-1} U + \cdots + a_1 U^{n-1} + a_0 U^n.$ Since $n$ is even, $E$ is the splitting field of $Y^2-Q(U)$ over $\mathbb{Q}(U)$ and, as 0 is not a root of $Q(U)$, the polynomial $Y^2-Q(0)$ is separable. Hence, the field $E_\infty$ is equal to the splitting field of $Y^2-Q(0)$ over $\mathbb{Q}$, i.e., $E_\infty=\mathbb{Q}(\sqrt{a_n})$, thus ending the proof of the lemma. \end{proof}
Now, we need the following bridge between specializations of the extension $E/\mathbb{Q}(T)$ and rational points on quadratic twists of $C_{P(T)}$.
\begin{lemma} \label{twisting explicit} Let $d$ be a squarefree integer. Then, the following two conditions are equivalent:
\noindent {\rm{(1)}} the quadratic extension $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$ occurs as a specialization of the $\mathbb{Q}$-regular quadratic extension $\mathbb{Q}(T)(\sqrt{P(T)})/\mathbb{Q}(T)$,
\noindent {\rm{(2)}} the $d$-th quadratic twist $C_{d \cdot P(T)}$ has a non-trivial $\mathbb{Q}$-rational point. \end{lemma}
\begin{proof}
First, assume that $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$ occurs as the specialization of $E/\mathbb{Q}(T)$ at $t_0$ for some $t_0 \in \mathbb{Q}\setminus \{t_1,\dots,t_n\}$. By part (1) of Lemma \ref{lemma 2}, one then has $\mathbb{Q}(\sqrt{d}) = \mathbb{Q}(\sqrt{P(t_0)})$. We then get $y^2=d \cdot P(t_0)$ for some $y \in \mathbb{Q} \setminus \{0\}$. Hence, the non-trivial $\mathbb{Q}$-rational point $[y:t_0:1]$ lies on the quadratic twist $C_{d \cdot P(T)}$. Now, assume that $n$ is even and $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$ occurs as the specialization of $E/\mathbb{Q}(T)$ at $\infty$. By part (2) of Lemma \ref{lemma 2}, one has $\mathbb{Q}(\sqrt{d}) = \mathbb{Q}(\sqrt{a_n})$. We then get $y^2=d \cdot a_n$ for some $y \in \mathbb{Q} \setminus \{0\}$, i.e., the non-trivial $\mathbb{Q}$-rational point $[y:0:1]$ lies on the quadratic twist $C_{d \cdot P(T)}$. Hence, implication (1) $\Rightarrow$ (2) holds.
Conversely, assume that $C_{d \cdot P(T)}$ has a $\mathbb{Q}$-rational point $[y:t:z]$ with $y \not=0$. First, assume that $n$ is odd. Then, $z \not=0$ and $t/z$ is not a root of $P(T)$, i.e., $t/z$ is not a branch point of $E/\mathbb{Q}(T)$; see Lemma \ref{lemma 1}. From the equality $y^2=d \cdot z^{n+1} \cdot P(t/z)$, part (1) of Lemma \ref{lemma 2}, and as $n$ is odd, we get $\mathbb{Q}(\sqrt{d}) =E_{t/z}$, as needed for (1). Now, assume that $n$ is even. If $z=0$, then $y^2=d \cdot a_n t^n$. This gives $\mathbb{Q}(\sqrt{d}) = \mathbb{Q}(\sqrt{a_n})$ (as $n$ is even and $t \not=0$) and then $\mathbb{Q}(\sqrt{d}) = E_\infty$ by part (2) of Lemma \ref{lemma 2}. If $z \not=0$, then $t/z$ is not a root of $P(T)$, i.e., $t/z$ is not a branch point of $E/\mathbb{Q}(T)$. From the equality $y^2=d \cdot z^{n} \cdot P(t/z)$, part (1) of Lemma \ref{lemma 2}, and as $n$ is even, we get $\mathbb{Q}(\sqrt{d}) =E_{t/z}$, thus ending the proof of the lemma. \end{proof}
\subsection{Proof of Proposition \ref{r leq 4}}
Set $E=\mathbb{Q}(T)(\sqrt{P(T)})$, where $P(T)$ $\in \mathbb{Z}[T]$ is separable. Since $E/\mathbb{Q}(T)$ has at most 4 branch points, $P(T)$ has degree at most 4 (Lemma \ref{lemma 1}). First, assume that $P(T)$ has degree 2. Then, the desired conclusion follows from \cite[Proposition 3.1 and \S3.1]{Leg15}. Now, assume that $P(T)$ has degree 4 and no root in $\mathbb{Q}$. Then, by \cite[Theorem 3.1 and \S3.4.1]{Leg16b}, there exist infinitely many squarefree integers $d$ such that the $d$-th quadratic twist $C_{d \cdot P(T)}$ has no $\mathbb{Q}$-rational point. It then remains to use Lemma \ref{twisting explicit} to get the desired conclusion. Finally, assume that either $P(T)$ has degree 3 or $P(T)$ has degree 4 and a root in $\mathbb{Q}$. Then, $E/\mathbb{Q}(T)$ has a $\mathbb{Q}$-rational branch point by Lemma \ref{lemma 1}. Up to applying a change of variable, we may assume that this branch point is $\infty$, i.e., we may assume that $P(T)$ has degree 3; see Lemma \ref{lemma 1}. Moreover, we may assume that $P(T)=T^3+bT+c$, where $b$ and $c$ are in $\mathbb{Z}$, i.e., $C_{P(T)}$ is elliptic. The desired conclusion then follows from the following classical two results (and Lemma \ref{twisting explicit}):
\noindent - $C_{P(T)}$ has infinitely many quadratic twists with Mordell-Weil rank 0 (see, e.g., \cite{Dab08} for references),
\noindent - there exist only finitely many squarefree integers $d$ such that $C_{d \cdot P(T)}$ has a non-trivial torsion point; see \cite[Proposition 1]{GM91}.
\begin{remark} (1) As shown in \cite{DD09}, the conclusion of Proposition \ref{r leq 4} fails in general if $\mathbb{Q}$ is replaced by a larger number number field $k$.
\noindent (2) To our knowledge, whether a given $\mathbb{Q}$-regular quadratic extension of $\mathbb{Q}(T)$ with at least 6 branch points is parametric over $\mathbb{Q}$ is an open problem in general. However,
\noindent - \cite[Proposition 5.6]{Leg16b} shows that the proportion of $\mathbb{Q}$-regular quadratic extensions of $\mathbb{Q}(T)$ with given number of branch points, ``height" at most $H$, and that are not parametric tends to 1 as $H$ tends to $\infty$,
\noindent - by \cite[Corollary 1 and Conjecture 1]{Gra07} and Lemma \ref{twisting explicit}, each $\mathbb{Q}$-regular quadratic extension of $\mathbb{Q}(T)$ with 6 branch points or more is not parametric over $\mathbb{Q}$ under conjectures (for example, the {\it{abc}}-conjecture). \end{remark}
\end{document} | arXiv | {
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\begin{document}
\maketitle
\begin{abstract} We prove the derived equivalence of a pair of non-compact Calabi--Yau 7-folds, which are the total spaces of certain rank 2 bundles on $G_2$-Grassmannians.
The proof follows that of the derived equivalence of Calabi--Yau 3-folds in $G_2$-Grassmannians by Kuznetsov \cite{1611.08386} closely. \end{abstract}
\section{Introduction}
The simply-connected simple algebraic group $G$ of type $G_2$ has three homogeneous spaces $\G \coloneqq G/P_1$, $\Q \coloneqq G/P_2$, and $\F \coloneqq G/B$ associated with the crossed Dynkin diagrams \Fone, \Ftwo, and \Fthree respectively.
The Picard group of $\F$ can be identified with the weight lattice of $G$, which in turn can be identified with $\bZ^2$ as $
(a,b) \coloneqq a \omega_1 + b \omega_2, $ where $\omega_1$ and $\omega_2$ are the fundamental weights associated with the long root and the short root respectively. We write the line bundle associated with the weight $(k,l)$ as $\cO_\F(k,l)$.
Let \begin{align}
R
&\coloneqq \bigoplus_{k,l=0}^\infty H^0 \lb \cO_\F(k,l) \rb
\cong \bigoplus_{k,l=0}^\infty \lb V^G_{(k,l)} \rb^\dual
\end{align} be the Cox ring of $\F$, where $\lb V^G_{(k,l)} \rb^\dual$ is the dual of the irreducible representation of $G$ with the highest weight $(k,l)$.
\begin{comment}
Since $\F$ is a Fano variety, $R$ is Gorenstein by \cite[Theorem 1.2]{MR3055212} (cf.~also \cite[Remark 4.8]{MR3275656}). Since the canonical bundle $\omega_{\F}$ is isomorphic to $\cO_{\F}(-2,-2)$, the canonical module $K_R$ is isomorphic to the shift $R(-2, -2)$ of the free module (see e.g.~\cite[Lemma 2.12]{MR2641200}).
\end{comment}
The $\bZ^2$-grading of $R$ defines a $(\bGm)^2$-action on $\Spec R$, which induces an action of $\bGm$ embedded in $(\bGm)^2$ by the anti-diagonal map
$\alpha \mapsto (\alpha,\alpha^{-1})$. We write the geometric invariant theory quotients as \begin{align}
\V_+
\coloneqq \Proj R_+, \quad
\V_-
\coloneqq \Proj R_-, \quad
\V_0
\coloneqq \Spec R_0,
\end{align} where \begin{align}
R_n = \bigoplus_{i \in \bZ} R_{i, n-i}, \quad
R_+ \coloneqq \bigoplus_{n=0}^{\infty} R_n, \quad
R_- \coloneqq \bigoplus_{n=0}^{\infty} R_{-n}. \end{align}
$\V_+$ and $\V_-$ are the total spaces of the dual of the equivariant vector bundles of rank 2
on $\Q$ and $\G$ associated with irreducible representations of $P_1$ and $P_2$ with the highest weight $(1,1)$. The structure morphisms $
\phi_+ \colon \V_+ \to \V_0 $ and $
\phi_- \colon \V_- \to \V_0 $ are crepant resolutions which contract the zero-sections.
The same construction for the simply-connected simple algebraic group $\Sp(2)$ of type $C_2$, which is accidentally isomorphic to the simply-connected simple algebraic group $\Spin(5)$ of type $B_2$, gives the 5-fold flop discussed in \cite{MR3509912}, where it is attributed to Abuaf.
The main result in this paper is the following:
\begin{theorem} \label{th:main} $\V_+$ and $\V_-$ are derived-equivalent. \end{theorem}
\pref{th:main} provides an evidence for the conjecture \cite[Conjecture 4.4]{MR1957019} \cite[Conjecture 1.2]{MR1949787} that birationally equivalent smooth projective varieties are K-equivalent if and only if they are D-equivalent.
The proof of \pref{th:main} closely follows \cite{1611.08386}, where the derived equivalence of Calabi--Yau complete intersections in $\G$ and $\Q$ defined by sections of the equivariant vector bundles dual to $\V_+$ and $\V_-$. The derived equivalence of these Calabi--Yau 3-folds in turn follows from \pref{th:main} using matrix factorizations.
\begin{notations} We work over a field $\bfk$ throughout this paper. All pull-back and push-forward are derived. The complexes underlying $\Ext^\bullet(-,-)$ and $H^\bullet(-)$ will be denoted by $\hom(-,-)$ and $\h(-)$. \end{notations}
\section{The blow-up diagram}
The $G_2$-Grassmannian $\G$ is the zero locus $
s_\lambda^{-1}(0) $ of the section $
s_\lambda $ of the equivariant vector bundle $\cQ^\dual(1)$ of rank 5 on $\Gr(2,V)$, obtained as the tensor product of the dual $\cQ^\dual$ of the universal quotient bundle $\cQ$ and the hyperplane bundle $\cO(1)$. Here $V \coloneqq V^G_{(0,1)}$ is the 7-dimensional fundamental representation of $G_2$, and $s_\lambda$ corresponds to the $G_2$-invariant 3-form on $V$ under the isomorphism $
H^0(\Gr(2,V),\cQ^\dual(1)) \cong \bigwedge^3 V^\dual. $
We write the $G_2$-equivariant vector bundle associated with the irreducible representation of $P_1$ with the highest weight $(a,b)$ as $\cE_{(a,b)}$. The restriction $
\scrU \coloneqq \cS|_\G $ of the universal subbundle $\cS$ of rank 2 on $\Gr(2,V)$ is isomorphic to $\cE_{(-1,1)}$.
\begin{comment} It is clear that $\scrU^\dual$ is a $G_2$-equivariant vector bundle. Since \begin{align}
H^0(\scrU^\dual)
\cong V^\dual
\cong \lb V^G_{(0,1)} \rb^\dual
\cong H^0(\cE_{(0,1)}), \end{align} one has \begin{align}
\scrU^\dual \cong \cE_{(0,1)}. \end{align}
By applying \cite[Theorem 0.2]{MR2238172} and \cite[Corollary 8.11]{MR2238172} to the case where $L=0$, $X_L=\G$ and $Y_L = \emptyset$, one obtains a full exceptional collection \begin{align} \label{eq:Kuz_col}
\lb \cO_{\G}, \scrU^\dual,
\cO_{\G}(1), \scrU^\dual(1),
\cO_{\G}(2), \scrU^\dual(2) \rb \end{align} in $D^b \coh \G$ \cite[Corollary 8.11]{MR2238172}. This collection is \begin{align}
(\cE_{(0,0)}, \cE_{(0,1)}, \cE_{(1,0)}, \cE_{(1,1)}, \cE_{(2,0)}, \cE_{(2,1)}). \end{align} Note that \begin{align}
\cE_{(a,1)}^\vee \cong \cE_{(-a-1,1)} \end{align} and \begin{align}
\omega_{\G} \cong \cE_{(-3,0)}. \end{align} The collection is \begin{align} \begin{array}{ccc}
(0,1) & (1,1) & (2,1) \\
(0,0) & (1,0) & (2,0), \end{array} \end{align} and helix is continued as \begin{align} \begin{array}{cccccc}
\cdots & (0,1) & (1,1) & (2,1) &\cdots \\
\cdots & (0,0) & (1,0) & (2,0) & (3,0) & \cdots. \end{array} \end{align} One can easily see from \begin{align}
\Hom^*(\cE_{(0,1)}, \cE_{(3,0)})
\cong H^*(\cE_{(-1,1)} \otimes \cE_{(3,0)})
\cong H^*(\cE_{(2,1)}) \end{align} and so on that this helix is strong. \end{comment}
The $G_2$-flag variety $\F$ is isomorphic to the total space of the $\bP^1$-bundle $
\varpi_+ \colon \bP(\scrU) \to \G $ associated with $\scrU$ (or any other equivariant vector bundle of rank 2, since all of them are related by a twist by a line bundle). We write the relative hyperplane class of $\varpi_+$ as $h$, so that \begin{align}
(\varpi_+)_* \lb \cO_\F(h) \rb \cong \scrU^\dual. \end{align} The pull-back to $\F$ of the hyperplane class $H$ in $\G$ will be denoted by $H$ again by abuse of notation.
The other $G_2$-Grassmannian $\Q$ is a quadric hypersurface in $\bP(V)$. We write the equivariant vector bundle on $\Q$ associated with the irreducible representation of $P_2$ with highest weight $(a,b)$ as $\cF_{(a,b)}$. The flag variety $\F$ has a structure of a $\bP^1$-bundle $
\varpi_- \colon \F \to \Q, $ whose relative hyperplane class is given by $H$. We define a vector bundle $\scrK$ on $\Q$ by \begin{align}
\scrK \coloneqq \lb (\varpi_-)_* \lb \cO_\F(H) \rb \rb^\dual, \end{align} so that $
\F \cong \bP_\G(\scrK). $ One can show that $\scrK$ is isomorphic to $\cF_{(1,-3)}$. We write the hyperplane class of $\Q$ as $h$ by abuse of notation, since it pulls back to $h$ on $\F$.
\begin{comment} On $\F$, one has \begin{align}
\cL_{(1,1)}^\vee
&\cong \cL_{(-1,-1)}
\cong \cO_{\F}(-h-H). \end{align} On $\Q$, one has \begin{align}
\cF_{(1,1)}^\vee
&\cong \cF_{(1,-4)}
\cong \scrK(-h). \end{align} On $\G$, one has \begin{align}
\cE_{(1,1)}^\vee
&\cong \cE_{(-2,1)}
\cong \scrU(-H). \end{align}
Recall the diagram \begin{align} \label{eq:Kuznetsov_diagram} \begin{gathered} \xymatrix{ & D \ar@{^(->}[r]^i \ar[ddl]_p & M \ar[d] \ar[ddl]_{\pi_M} \ar[ddr]^{\rho_M} & E \ar@{_(->}[l]_-j \ar[ddr]^q \\ && \F \ar[dl]^\pi \ar[dr]_\rho \\ X \ar@{^{(}->}[r] & \Q && \G & Y \ar@{_{(}->}[l] } \end{gathered} \end{align} from \cite{1611.08386}, which summarizes the situation in \cite{1607.07821,1606.04210}. \end{comment}
Let $\V$ be the total space of the line bundle $\cO_\F(-h-H)$ on $\F$. The structure morphism
will be denoted by $
\pi \colon \V \to \F. $
The Cox ring of $\V$ is the $\bN^2$-graded ring \begin{align}
S = \bigoplus_{k,l=0}^\infty S_{k,l} \end{align} given by \begin{align}
S_{k,l}
&\coloneqq H^0 \lb \cO_\V(k,l) \rb \\
&\cong H^0 \lb \pi_* \lb \cO_\V(k,l) \rb \rb \\
&\cong H^0 \lb \pi_* \cO_\V \otimes \cO_\F(k,l) \rb \\
&\cong H^0 \lb \lb \bigoplus_{m=0}^\infty \cO_\F(m,m) \rb \otimes \cO_\F(k,l) \rb \\
&\cong \bigoplus_{m=0}^\infty H^0 \lb \cO_\F(k+m,l+m) \rb \\
&\cong \bigoplus_{m=0}^\infty \lb V^G_{(k+m,l+m)} \rb^\dual,
\end{align} whose multiple Proj recovers $\V$. Similarly, the Cox ring of the total space $\W_+$ of the bundle $
\cE_{(1,1)}^\dual \cong \scrU(-H) $ is given by $
\bigoplus_{k=0}^\infty H^0 \lb \cO_{\W_+}(k H) \rb $ where \begin{align}
H^0 \lb \cO_{\W_+}(k H) \rb
&\cong H^0 \lb \pi_* \lb \cO_{\W_+}(k H) \rb \rb \\
&\cong H^0 \lb \pi_* \cO_{\W_+} \otimes \cO_\G (k H) \rb \\
&\cong \bigoplus_{m=0}^\infty H^0 \lb \lb \Sym^m \cE_{(1,1)} \rb \otimes \cO_\G(k H) \rb \\
&\cong \bigoplus_{m=0}^\infty H^0 \lb \cE_{(m,m)} \otimes \cE_{(k,0)} \rb \\
&\cong \bigoplus_{m=0}^\infty H^0 \lb \cE_{(m+k,m)} \rb. \end{align} This is isomorphic to $R_+$, so that $\W_+$ is isomorphic to $\V_+$, and the affinization morphism \begin{align} \label{eq:V-affinization}
\V
\to \Spec H^0 \lb \cO_\V \rb
\cong \V_0 \end{align} is the composition of the natural projection $
\varphi_+ \colon \V \to \V_+ $ and the affinization morphism $
\phi_+ \colon \V_+ \to \V_0. $ Since $\V_+$ is the total space of $\cE_{(1,1)}^\dual$, the ideal sheaf of the zero-section is the image of the natural morphism from $\pi_+^* \cE_{(1,1)}$ to $\cO_{\V_+}$, and the morphism $\varphi_+$ is the blow-up along it. Similarly, the affinization morphism \pref{eq:V-affinization} also factors into the blow-up $
\varphi_- \colon \V \to \V_- $ and the affinization morphism $
\phi_- \colon \V_- \to \V_0, $ and one obtains the following commutative diagram: \begin{align} \label{eq:blow-up_diagram} \begin{gathered} \xymatrix{
& \ar[dl]_{\varphi_+} \V \ar[dr]^{\varphi_-} \\
\V_+ \ar[dr]^{\phi_+} & & \ar[dl]_{\phi_-} \V_- \\
& \V_0 } \end{gathered} \end{align}
\section{Some extension groups}
The zero-sections and the natural projections fit into the following diagram: \begin{align} \label{eq:zero-sections_diagram} \begin{gathered} \xymatrix{
& \F \ar[dl]_{\varpi_+} \ar@{^(->}[d]^\iota \ar[dr]^{\varpi_-} & \\
\G \ar@{^(->}[d] & \V \ar[dl]_{\phi_+} \ar[dr]^{\phi_-} & \Q \ar@{^(->}[d] \\
\V+ & & \V_- } \end{gathered} \end{align} We write $
\scrU_\F \coloneqq \varpi_+^* \scrU, $ $
\scrS_\F \coloneqq \varpi_-^* \scrS, $ and $
\scrU_\V \coloneqq \pi^* \scrU_\F. $ By abuse of notation, we use the same symbol for an object of $\D(\F)$ and its image in $\D(\V)$ by the push-forward $\iota_*$. Since $\V$ is the total space of $\cO_\V(-h-H)$, one has a locally free resolution \begin{align}
0 \to \cO_\V(h+H) \to \cO_\V \to \cO_\F \to 0 \end{align} of $\cO_\F$ as an $\cO_\V$-module.
By tensoring $\cO_\F(-h)$ to \cite[Equation (5)]{1611.08386}, one obtains an exact sequence \begin{align} \label{eq:Ext4}
0 \to \cO_\F(H-2h) \to \scrU_\F^\dual(-h) \to \cO_\F \to 0. \end{align}
\pref{lm:Ext1} and \pref{pr:S} below are taken from \cite{1611.08386}:
\begin{lemma}[{\cite[Lemma 1]{1611.08386}}] \label{lm:Ext1} \begin{enumerate}[(i)]
\item Line bundles $\cO_\F(th-H)$ and $\cO_\F(tH-h)$ are acyclic for all $t \in \bZ$.
\item Line bundles $\cO_\F(-2H)$ and $\cO_\F(2h-2H)$ are acyclic and \begin{align*}
H^\bullet(\cO_\F(3h-2H)) \cong \bfk[-1]. \end{align*}
\item Vector bundles $\scrU_\F(-2H)$, $\scrU_\F(-H)$, $\scrU_\F(h-H)$ and $\scrU_\F \otimes \scrU_\F(-H)$ are acyclic, and \begin{align*}
H^\bullet(\scrU_\F(h)) \cong \bfk, \quad
H^\bullet(\scrU_\F \otimes \scrU_\F(h)) \cong \bfk[-1]. \end{align*} \end{enumerate} \end{lemma}
\begin{proposition}[{\cite[Proposition 3 and Lemma 4]{1611.08386}}] \label{pr:S} One has an exact sequence \begin{equation} \label{eq:UUExt}
0 \to \scrU_\F \to \scrS_\F \to \scrU_\F^\dual(-h) \to 0. \end{equation} \end{proposition}
\pref{lm:Ext1} immediately implies the following:
\begin{lemma} \label{lm:orth1} $\cO_\F(-H)$ is right orthogonal to both $\scrU_\F^\dual(-h)$ and $\cO_\F(-h)$. \end{lemma}
\begin{proof} We have \begin{align}
\hom_{\cO_\V} \lb \cO_\F(-h), \cO_\F(-H) \rb
&\cong \hom_{\cO_\V} \lb \lc \cO_\V(H) \to \cO_\V(-h) \rc, \cO_\F(-H) \rb \\
&\cong \h \lb \lc \cO_\F(h-H) \to \cO_\F(-2H) \rc \rb \end{align} and \begin{align}
\hom_{\cO_\V} \lb \scrU_\F^\dual(-h), \cO_\F(-H) \rb
&\cong \hom_{\cO_\V} \lb \lc \scrU_\V^\dual(H) \to \scrU_\V^\dual(-h) \rc, \cO_\F(-H) \rb \\
&\cong \h \lb \lc \scrU_\F(h-H) \to \scrU_\F(-2H) \rc \rb, \end{align} both of which vanish by \pref{lm:Ext1}. \end{proof}
\begin{lemma} \label{lm:Ext2} One has \begin{align}
\hom_{\cO_\V} \lb \scrU_\F^\dual(-h), \scrU_\F \rb
\cong \bfk[-1]. \end{align} \end{lemma}
\begin{proof} One has \begin{align}
\hom_{\cO_\V} \lb \scrU_\F^\dual(-h), \scrU_\F \rb
&\cong \hom_{\cO_\V} \lb \lc \scrU_\V^\dual(H) \to \scrU_\V^\dual(-h) \rc, \scrU_\F \rb \\
&\cong \h \lb \lc \scrU_\F \otimes \scrU_\F(h) \to \scrU_\F \otimes \scrU_\F(-H) \rc \rb. \end{align} \pref{lm:Ext1} shows that the first term gives $\bfk[-1]$ and the second term vanishes. \end{proof}
\begin{lemma} \label{lm:Ext3} One has \begin{align}
\hom_{\cO_\V} \lb \scrU_\F^\dual(-h), \cO_\F \rb
\cong \bfk. \end{align} \end{lemma}
\begin{proof} One has \begin{align}
\hom_{\cO_\V} \lb \scrU_\F^\dual(-h), \cO_\F \rb
&\cong \hom_{\cO_\V} \lb \lc \scrU_\V^\dual(H) \to \scrU_\V^\dual(-h) \rc, \cO_\F \rb \\
&\cong \h \lb \lc \scrU_\F(h) \to \scrU_\F(-H) \rc \rb. \end{align} \pref{lm:Ext1} shows that the first term gives $\bfk$ and the second term vanishes. \end{proof}
\begin{lemma} \label{lm:orth2} One has \begin{align}
\hom_{\cO_\V} \lb \cO_\F(H-2h), \cO_\F(h) \rb
\cong 0. \end{align} \end{lemma}
\begin{proof} One has \begin{align}
\hom_{\cO_\V} \lb \cO_\F(H-2h), \cO_\F(h) \rb
&\cong \hom_{\cO_\V} \lb \lc \cO_\V(2H-h) \to \cO_\V(H-2h) \rc, \cO_\F(h) \rb \\
&\cong \h \lb \lc \cO_\V(3h-H) \to \cO_\V(2h-2H) \rc \rb, \end{align} which vanishes by \pref{lm:Ext1}. \end{proof}
\section{Derived equivalence by mutation}
Recall from \cite{1611.08386} that \begin{align} \label{eq:EC1}
\D(\G) = \la \cO_\G(-H), \scrU, \cO_\G, \scrU^\dual, \cO_\G(H), \scrU^\dual(H) \ra \end{align} and \begin{align} \label{eq:EC2}
\D(\Q) = \la \cO_\Q(-3h), \cO_\Q(-2h), \cO_\Q(-h), \scrS, \cO_\Q, \cO_\Q(h) \ra. \end{align} It follows from \cite{MR1208153} that \begin{align} \label{eq:SOD1}
\D(\V) = \la \iota_* \varpi_+^* \D(\G), \Phi_+(\D(\V_+)) \ra \end{align} and \begin{align} \label{eq:SOD2}
\D(\V) = \la \iota_* \varpi_-^* \D(\Q), \Phi_-(\D(\V_-)) \ra, \end{align} where \begin{align}
\Phi_+ \coloneqq \phi_+^*(-) \otimes \cO_\V(h) \colon \D(\V_+) \to \D(\V) \end{align} and \begin{align}
\Phi_- \coloneqq \phi_-^*(-) \otimes \cO_\V(H) \colon \D(\V_-) \to \D(\V). \end{align} \pref{eq:EC1} and \pref{eq:SOD1} gives \begin{align}
\D(\V) = \la \cO_\F(-H), \scrU_\F, \cO_\F, \scrU_\F^\dual, \cO_\F(H), \scrU_\F^\dual(H), \Phi_+(\D(\V_+)) \ra. \end{align}
By mutating $\Phi_+(\D(\V_+))$ two steps to the left, one obtains \begin{align}
\D(\V) = \la \cO_\F(-H), \scrU_\F, \cO_\F, \scrU_\F^\dual, \Phi_1(\D(\V_+)), \cO_\F(H), \scrU_\F^\dual(H) \ra \end{align} where \begin{align}
\Phi_1 \coloneqq \L_{\la \cO_\F(H), \scrU_\F^\dual(H) \ra} \circ \Phi_+. \end{align} By mutating the last two terms to the far left, one obtains \begin{align}
\D(\V) = \la \cO_\F(-h), \scrU_\F^\dual(-h), \cO_\F(-H), \scrU_\F, \cO_\F, \scrU_\F^\dual, \Phi_1(\D(\V_+)) \ra, \end{align} since $\omega_\V \cong \cO_\V(-h-H)$. \pref{lm:orth1} allows one to move $\cO_\F(-H)$ to the far left without affecting other objects: \begin{align}
\D(\V) = \la \cO_\F(-H), \cO_\F(-h), \scrU_\F^\dual(-h), \scrU_\F, \cO_\F, \scrU_\F^\dual, \Phi_1(\D(\V_+)) \ra. \end{align} By mutating $\scrU_\F$ one step to the left and using \pref{pr:S} and \pref{lm:Ext2}, one obtains \begin{align}
\D(\V) = \la \cO_\F(-H), \cO_\F(-h), \scrS_\F, \scrU_\F^\dual(-h), \cO_\F, \scrU_\F^\dual, \Phi_1(\D(\V_+)) \ra. \end{align} By mutating $\cO_\F(-H)$ to the far right, one obtains \begin{align}
\D(\V) = \la \cO_\F(-h), \scrS_\F, \scrU_\F^\dual(-h), \cO_\F, \scrU_\F^\dual, \Phi_1(\D(\V_+)), \cO_\F(h) \ra. \end{align} By mutating $\Phi_1(\D(\V_+))$ to the right, one obtains \begin{align}
\D(\V) = \la \cO_\F(-h), \scrS_\F, \scrU_\F^\dual(-h), \cO_\F, \scrU_\F^\dual, \cO_\F(h), \Phi_2(\D(\V_+)) \ra \end{align} where \begin{align}
\Phi_2 \coloneqq \R_{\cO_\F(h)} \circ \Phi_1. \end{align} By mutating $\scrU_\F^\dual(-h)$ one step to the right and using \pref{lm:Ext3} and \pref{eq:Ext4}, one obtains \begin{align}
\D(\V) = \la \cO_\F(-h), \scrS_\F, \cO_\F, \cO_\F(H-2h), \scrU_\F^\dual, \cO_\F(h), \Phi_2(\D(\V_+)) \ra. \end{align} Similarly, by mutating $\scrU_\F^\dual$ one step to the right, one obtains \begin{align}
\D(\V) = \la \cO_\F(-h), \scrS_\F, \cO_\F, \cO_\F(H-2h), \cO_\F(h), \cO_\F(H-h), \Phi_2(\D(\V_+)) \ra. \end{align} \pref{lm:orth2} allows one to exchange $\cO_\F(H-2h)$ and $\cO_\F(h)$ to obtain \begin{align}
\D(\V) = \la \cO_\F(-h), \scrS_\F, \cO_\F, \cO_\F(h), \cO_\F(H-2h), \cO_\F(H-h), \Phi_2(\D(\V_+)) \ra. \end{align} By mutating $\Phi_2(\D(\V_+))$ two steps to the left, one obtains \begin{align}
\D(\V) = \la \cO_\F(-h), \scrS_\F, \cO_\F, \cO_\F(h), \Phi_3(\D(\V_+)), \cO_\F(H-2h), \cO_\F(H-h) \ra \end{align} where \begin{align}
\Phi_3 \coloneqq \L_{\la \cO_\F(H-2h), \cO_\F(H-h) \ra} \circ \Phi_2. \end{align} By mutating the last two terms to the far left, one obtains \begin{align} \label{eq:SOD3}
\D(\V) = \la \cO_\F(-3h), \cO_\F(-2h), \cO_\F(-h), \scrS_\F, \cO_\F, \cO_\F(h), \Phi_3(\D(\V_+)) \ra. \end{align} By comparing \pref{eq:SOD3} with \begin{align} \label{eq:SOD4}
\D(\V) = \la \cO_\F(-3h), \cO_\F(-2h), \cO_\F(-h), \scrS_\F, \cO_\F, \cO_\F(h), \Phi_-(\D(\V_-)) \ra \end{align} obtained by combining \pref{eq:EC2} and \pref{eq:SOD2}, one obtains a derived equivalence \begin{align} \label{eq:Phi}
\Phi \coloneqq \Phi_-^! \circ \Phi_3 \colon \D(\V_+) \simto \D(\V_-), \end{align} where \begin{align}
\Phi_-^!(-) \coloneqq (\phi_-)_* \lb (-) \otimes \cO_\V(-H) \rb
\colon \D(\V) \to \D(\V_-) \end{align} is the left adjoint functor of $\Phi_-$. Note that the left mutation along an exceptional object $\cE \in \D(\V)$ is an integral functor $
\Phi_\cK(-) \coloneqq (p_2)_* \lb p_1^*(-) \otimes \cK \rb $ along the diagram \begin{align} \begin{gathered} \xymatrix{
& \V \times_{\V_0} \V \ar[dl]_{p_1} \ar[dr]^{p_2} \\
\V && \V } \end{gathered} \end{align} whose kernel $\cK$ is the cone over the evaluation morphism $
\ev \colon \cE^\dual \boxtimes \cE \to \Delta_\V. $ The functors $
\Phi_+ \colon \D(\V_+) \to \D(\V) $ and $
\Phi_-^! \colon \D(\V) \to \D(\V_-) $ are clearly an integral functor, so that the functor \pref{eq:Phi} is also an integral functor, whose kernel is an object of $\D(\V_+ \times_{\V_0} \V_-)$ obtained by convolution.
\begin{comment} Note that everything is linear over $R_0$. Any integral functor along any diagram of the form \begin{align} \begin{gathered} \xymatrix{
& W \ar[dl]_{p_+} \ar[dr]^{p_-} \\
\V_+ && \V_- } \end{gathered} \end{align} can be written as an integral functor along \begin{align} \begin{gathered} \xymatrix{
& \V_+ \times_{\V_0} \V_- \ar[dl]_{q_+} \ar[dr]^{q_-} \\
\V_+ && \V_- } \end{gathered} \end{align} since one has a diagram \begin{align} \begin{gathered} \xymatrix{
& W \ar[ddl]_{p_+} \ar[ddr]^{p_-} \ar[d]^r \\
& \V_+ \times_{\V_0} \V_- \ar[dl]_{q_+} \ar[dr]^{q_-} \\
\V_+ && \V_- } \end{gathered} \end{align} and \begin{align}
(p_-)_* (p_+^*(-) \otimes \cK)
&\cong (q_-)_* r_* \lb r^* q_+^*(-) \otimes \cK \rb \\
&\cong (q_-)_* \lb q_+^*(-) \otimes r_* \cK \rb. \end{align} \end{comment}
\section{Matrix factorizations}
Let $s_+$ be a general section of the equivariant vector bundle $\cE_{(1,1)}$ on $\G$. The zero $X_+$ of $s_+$ is a smooth projective Calabi--Yau 3-fold. Since $\V_+$ is the total space of the dual bundle $\cE_{(1,1)}^\dual$ on $\G$, the space of regular functions on $\V_+$ which are linear along the fiber can naturally be identified with the space of sections of $\cE_{(1,1)}$. We write the regular function on $\V_+$ associated with $s_+ \in H^0 \lb \cE_{(1,1)} \rb$ as $\varsigma_+ \in H^0 \lb \cO_{\V_+} \rb$. The zero $D_+$ of $\varsigma_+$ is the union of a line sub-bundle of $\V_+$ and the inverse image of $X_+$ by the structure morphism $\pi_+ \colon \V_+ \to \G$. The singular locus of $D_+$ is given by $X_+$.
Let $\varsigma_-$ be a regular function on $\V_-$ corresponding to $\varsigma_+$ under the isomorphism $
H^0 \lb \cO_{\V_+} \rb
\cong H^0 \lb \cO_{\V_0} \rb
\cong H^0 \lb \cO_{\V_-} \rb $ given by the diagram in \pref{eq:blow-up_diagram}, and $X_-$ be the zero of the corresponding section $
s_- \in H^0 \lb \cF_{(1,1)} \rb, $ which is a smooth projective Calabi--Yau 3-fold in $\Q$.
The push-forward of the kernel of $\Phi$ on $\V_+ \times_{V_0} \V_-$ to $\V_+ \times_{\bA^1} \V_-$ gives a kernel of $\Phi$ on $\V_+ \times_{\bA^1} \V_-$. By taking the base-change along the inclusion $0 \to \bA^1$ of the origin and applying \cite[Proposition 2.44]{MR2238172}, one obtains an equivalence $
\Phi_0 \colon \D(D_+) \cong \D(D_-) $ of the bounded derived categories of coherent sheaves. By using either of the characterization of perfect complexes as \emph{homologically finite} objects (i.e., objects whose total $\Ext$-groups with any other object are finite-dimensional) or \emph{compact} objects (i.e., objects such that the covariant functors represented by them commute with direct sums), one deduces that $\Phi_D$ preserves perfect complexes, so that it induces an equivalence $
\Phi_0^\sing \colon \Dsg(D_+) \cong \Dsg(D_-) $ of singularity categories (see \cite[Section 7]{MR2563433} and \cite[Theorem 1.1]{MR2593258}).
Recall that $\V_+$, $\V_-$ and $\V_0$ are geometric invariant theory quotient of $\Spec R$ by the anti-diagonal $\bGm$-action. The residual diagonal $\bGm$-action on both $\V_+$ and $\V_-$ are dilation action on the fiber. The equivalences $\Phi$, $\Phi_0$ and $\Phi_0^\sing$ are equivariant with respect to this $\bGm$-action, and induces an equivalence of $\bGm$-equivariant categories \cite[Theorem 1.1]{1506.00177}, which will be denoted by the same symbol by abuse of notation. Now \cite[Theorem 3.6]{MR3071664} gives equivalences \begin{align} \label{eq:Isik1}
\Dsg([D_+/\bGm]) \cong \D(X_+) \end{align} and \begin{align} \label{eq:Isik2}
\Dsg([D_-/\bGm]) \cong \D(X_-) \end{align} between $\bGm$-equivariant singularity categories and derived categories of coherent sheaves (see also \cite{MR2982435} where the case of line bundles is discussed independently and around the same time as \cite{MR3071664}).
By composing these derived equivalences with $\Phi_0^\sing$, one obtains a derived equivalence between $X_+$ and $X_-$. It is an interesting problem to compare this equivalence with the one obtained in \cite{1611.08386}. Another interesting problem is to prove the derived equivalence using variation of geometric invariant theory quotient along the lines of \cite{0803.2045,MR2795327,1203.6643,MR3327537,MR3370126}, and use it to produce autoequivalences of the derived category \cite{MR3223878,MR3552550}.
\end{document} | arXiv | {
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\begin{document}
\allowdisplaybreaks \def\pd#1#2{\frac{\partial#1}{\partial#2}} \let\oldsection\section \renewcommand\section{\setcounter{equation}{0}\oldsection} \renewcommand\thesection{\arabic{section}} \renewcommand\theequation{\thesection.\arabic{equation}} \newtheorem{theorem}{\indent Theorem}[section] \newtheorem{lemma}{\indent Lemma}[section] \newtheorem{proposition}{\indent Proposition}[section] \newtheorem{definition}{\indent Definition}[section] \newtheorem{remark}{\indent Remark}[section] \newtheorem{corollary}{\indent Corollary}[section]
\title{\LARGE Critical Sharp Front for Doubly Nonlinear Degenerate Diffusion Equations with Time Delay} \author{ Tianyuan Xu$^{a}$, Shanming Ji$^{a,}$\thanks{Corresponding author, email:jism@scut.edu.cn}, Ming Mei$^{b,c}$, Jingxue Yin$^d$, \\ \\ { \small \it $^a$School of Mathematics, South China University of Technology} \\ { \small \it Guangzhou, Guangdong, 510641, P.~R.~China} \\ { \small \it $^b$Department of Mathematics, Champlain College Saint-Lambert} \\ { \small \it Quebec, J4P 3P2, Canada, and} \\ { \small \it $^c$Department of Mathematics and Statistics, McGill University} \\ { \small \it Montreal, Quebec, H3A 2K6, Canada} \\ { \small \it $^d$School of Mathematical Sciences, South China Normal University} \\ { \small \it Guangzhou, Guangdong, 510631, P.~R.~China} } \date{}
\maketitle
\begin{abstract}
This paper is concerned with the critical sharp traveling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by $\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x$ with $m>0$ and $p>1$. The doubly nonlinear diffusion equation is proved to admit a unique sharp type traveling wave for the degenerate case $m(p-1)>1$, the so-called slow-diffusion case. This sharp traveling wave associated with the minimal wave speed $c^*(m,p,r)$ is monotonically increasing, where the minimal wave speed satisfies $c^*(m,p,r)<c^*(m,p,0)$ for any time delay $r>0$. The sharp front is $C^1$-smooth for $\frac{1}{p-1}<m< \frac{p}{p-1}$, and piecewise smooth for $m\ge \frac{p}{p-1}$. Our results indicate that time delay slows down the minimal traveling wave speed for the doubly nonlinear degenerate diffusion equations. The approach adopted for proof is the phase transform method combining the variational method. The main technical issue for the proof is to overcome the obstacle caused by the doubly nonlinear degenerate diffusion.
\end{abstract}
{\bf Keywords}: Doubly nonlinearity, Variational approach, Time delay, Degenerate diffusion, Sharp type wave.
\section{Introduction} This is a continuity of our recent study \cite{Non20} on critical traveling waves for time-delayed degenerate diffusion equation. Our purpose in the present paper is to study the existence, uniqueness and regularity of the critical sharp traveling wave for the following doubly nonlinear diffusion equation with time delay \begin{align}\label{eq-main}
\pd u t=\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x-d(u)+b(u(t-r,x)),\quad x\in \mathbb R,~t>0, \end{align} where $p>1$, $m>0$, $u$ is the population density, $b(u(t-r,x))$ is the birth function,
$r\ge0$ is the time delay, and $d(u)$ is the death rate function. The differential operator $\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x$ is called ``doubly nonlinear'' or non-Newtonian polytropic filtration, see \cite{VazquezJDE17,Jin-JDE} for example. We focus on the slow diffusion case $m(p-1)>1$ such that sharp type (semi-compactly supported) traveling wave exists and the initial perturbation propagates at finite speed for the non-delayed case. The functions $d(s)$ and $b(s)$ satisfy the following conditions: \begin{enumerate} \item[(H$_1$)] Two constant equilibria: $u_-=0$ and $u_+>0$ such that $d(0)=b(0)=0$, $d(u_+)=b(u_+)$, $b'(0)>d'(0)\ge 0$, and $d'(u_+)> b'(u_+)\ge0$; \item[(H$_2$)] Monotonicity: $d(\cdot), \ b(\cdot)\in C^2([0,u_+])$, and $b'(s)> 0$, $d'(s)>0$ for $s\in [0,u_+]$. \end{enumerate} The assumptions (H$_1$)-(H$_2$) are summarized from a large number of evolution equations in ecology, such as the classical Fisher-KPP equation \cite{Fisher}; the well-studied Nicholson's blowflies equation \cite{Gurney80} with the death function $d_1(u)=\delta u$ or $d_2(u)=\delta u^2$, the birth function $$b_1(u)=\tilde{p}u\mathrm{e}^{-au^{\tilde{q}}}, \quad \tilde{p}>0,\quad \tilde{q}>0,\quad a>0;$$ and the Mackey-Glass equation \cite{Mackey} with the growth function $$ b_2(u)=\frac{\tilde{p}u}{1+au^{\tilde{q}}}\quad \tilde{p}>0,\quad \tilde{q}>0,\quad a>0. $$
When $p = 2, \ m=1$, we have the standard heat equation with time delay. As far as we know, reaction diffusion equations with time delay has first been studied by Schaaf in \cite{Schaaf}, where he proved the existence of monotone traveling waves. The proof was based on sub and super solutions and phase plane techniques. Since then, the study of traveling wave solutions for reaction diffusion equations with time delay has drawn considerable attention (see, for example, \cite{Mei_LinJDE09,Chern,Faria,Gomez,LLLM} and references therein). Note that, the results mentioned above are all for the case that the diffusion term is classical Laplacian. Choosing $p=2, \ m>1$, we obtain Porous Medium operator, which describes density-dependent dispersal in biological settings. Here, the important feature of degenerate diffusion equation appears: traveling waves exhibit free boundaries. In \cite{Non20}, we found the sharp type traveling wave (partially compactly supported) corresponding to the critical wave speed and obtained the uniqueness of these waves. Further, we proved that the initial perturbation propagates asymptotically at the same speed \cite{ArXiv21} and later sharp-oscillatory non-monotone traveling waves was found in \cite{JDE20}.
The sharp type (partially compactly supported) traveling wave solutions are essential in the analysis of the propagation properties of degenerate diffusion equations. In many cases, the solutions with (partially) compactly supported initial data propagate asymptotically at the same speed of the sharp waves, which also is the minimal admissible traveling wave speed. This phenomenon was observed by Audrito and V\'azquez \cite{VazquezJDE17} for doubly nonlinear diffusion equation \eqref{eq-main} without time delay (i.e., $r=0$), and further the speed was characterized via a variational approach by Benguria and Depassier \cite{Benguria18}.
The existence of traveling waves of \eqref{eq-main} remains to be technically demanding. Our main objective is to investigate the structure of the critical sharp waves and to estimate the corresponding critical speed using the approach of phase transform method with the help of the variational approach developed recently in our studies \cite{JDE18,Non20}. Precisely speaking, we prove that, the doubly nonlinear diffusion equation \eqref{eq-main} possesses a unique sharp type traveling wave $\phi(x+c^*t)$ for the degenerate case $m(p-1)>1$, and such a sharp traveling wave associated with the minimal wave speed $c^*=c^*(m,p,r)$ is monotonically increasing, where the minimal wave speed satisfies $c^*(m,p,r)<c^*(m,p,0)$ for any time delay $r>0$. Furthermore, we show the optimal regularity of the sharp front $\phi(x+c^*t)$. That is, when $\frac{p-1}{m(p-1)-1}$ is integer, then the sharp front $\phi(x+c^*t)$ is $C^{\frac{p-1}{m(p-1)-1}-1}$-smooth with $\phi$ and all its derivatives $\partial^j \phi$ are Lipschitz continuous for $j=1,\cdots, \frac{p-1}{m(p-1)-1}-1$; while, when $\frac{p-1}{m(p-1)-1}$ is non-integer, then the sharp front $\phi(x+c^*t)$ is $C^{[\frac{p-1}{m(p-1)-1}]}$-smooth, where $[\frac{p-1}{m(p-1)-1}]$ denotes the largest integer which is less then $\frac{p-1}{m(p-1)-1}$, in particular, $\phi$ and its all derivatives $\partial^j \phi$ for $j=1,\cdots, [\frac{p-1}{m(p-1)-1}]$ are $C^{\alpha_{m,p}}$ H\"older continuous with the H\"older exponent $\alpha_{m,p}=\frac{p-1}{m(p-1)-1}-[\frac{p-1}{m(p-1)-1}]$. This implies that the sharp front $\phi(x+c^*t)$ is $C^1$-smooth for $\frac{1}{p-1}<m< \frac{p}{p-1}$, and piecewise smooth for $m\ge \frac{p}{p-1}$. On the other hand, we also prove that the time delay $r>0$ slows down the minimal traveling wave speed $c^*=c^*(m,p,r)$ for the doubly nonlinear degenerate diffusion equations. Finally, let us point out a slightly unexpected phenomenon related to the doubly nonlinear operator. The main difficulty lies in the asymptotic behavior of the phase function $\tilde\psi(\phi)$
defined for the sharp type traveling wave $\phi(\xi)$ by regarding $\psi(\xi):=|(\phi^m(\xi))'|^{p-2}(\phi^m(\xi))'$ as a function of $\phi$. Its asymptotic behavior near the positive equilibrium $u_+$ for the degenerate case $p\in(1,2)$ is quite different from the case $p=2$.
The paper is organized as follows. In Section 2, we state our main results. We defer to Section 3 all the detailed proofs. Section 4 is the brief derivation of models we treat.
\section{Main results} We consider the doubly nonlinear degenerate diffusion equation with time delay \eqref{eq-main}. We are looking for the traveling wave solutions of sharp type that connect the two equilibria $u_-=0$ and $u_+=:K$. Under the hypotheses (H$_1$)-(H$_2$), the birth function $b(u)$ is monotonically increasing on $[u_-,u_+]=:[0,K]$. Let $\phi(\xi)$, where $\xi=x+ct$ and $c>0$, be the traveling wave solution of \eqref{eq-main}, we get (we write $\xi$ as $t$ for the sake of simplicity) \begin{align}\label{eq-tw} \begin{cases}
\displaystyle c\phi'(t)=(|(\phi^m)'(t)|^{p-2}(\phi^m)'(t))'-d(\phi(t))+b(\phi(t-cr)),\quad t\in\mathbb R,\\ \phi(-\infty)=0, \quad \phi(+\infty)=K. \end{cases} \end{align}
Since \eqref{eq-tw} has singularity or degeneracy, we employ the following definition of sharp and smooth traveling waves. Here are some notations used throughout this paper:
$$ C_\mathrm{unif}^\mathrm{b}(\mathbb R):= \{\phi\in C(\mathbb R)\cap L^\infty(\mathbb R);\phi \text{~is uniformly continuous on~}\mathbb R\}, $$ and $$W_\mathrm{loc}^{1,p}(\mathbb R):=\{\phi; \phi\in W^{1,p}(\Omega) \text{~for any compact subset~}\Omega\subset\mathbb R\}.$$
\begin{definition} \label{de-semi} A profile function $\phi(t)$ is said to be a traveling wave solution of \eqref{eq-tw} if $\phi\in C_\mathrm{unif}^\mathrm{b}(\mathbb R)$, $0\le \phi(t)\le K:=u_+$, $\phi(-\infty)=0$, $\phi(+\infty)=K$, $\phi^m\in W_{\mathrm{loc}}^{1,p}(\mathbb R)$, $\phi(t)$ satisfies \eqref{eq-tw} in the sense of distributions. The traveling wave $\phi(t)$ is said to be of sharp type if the support of $\phi(t)$ is semi-compact, i.e., $\text{supp}\,\phi=[t_0,+\infty)$ for some $t_0\in\mathbb R$, $\phi(t)>0$ for $t>t_0$. On the contrary, the traveling wave $\phi(t)$ is said to be of smooth type if $\phi(t)>0$ for all $t\in\mathbb R$. \end{definition}
Without loss of generality, we may always shift $t_0$ to $0$ for the sharp type traveling wave. Therefore, a sharp type traveling wave $\phi(t)$ is a special solution such that $\phi(t)=0$ for $t\le0$, and $\phi(t)>0$ for $t>0$.
For any given $m>0$, $p>1$, such that $m(p-1)>1$, and $r\ge0$, we define the critical (or minimal) wave speed $c^*(m,p,r)$ for the degenerate diffusion equation \eqref{eq-tw} as follows \begin{equation} \label{eq-def} c^*(m,p,r) :=\inf\{c>0; \eqref{eq-tw} \text{~admits increasing traveling waves with speed $c$}\}. \end{equation} For the case without time delay and with degenerate diffusion (i.e. $m(p-1)>1$ and $r=0$), it is proved by Benguria and Depassier in \cite{Benguria18} that \begin{equation} \label{eq-cstar0} c^*(m,p,0)=\sup_{g\in \mathscr{D}} \int_0^K \frac{p}{(p-1)^{(p-1)/p}}(-g'(\phi))^\frac{1}{p}(g(\phi))^\frac{p-1}{p} (m\phi^{m-1}(b(\phi)-d(\phi)))^\frac{p-1}{p}\mathrm{d}\phi, \end{equation} where $\mathscr{D}=\{g\in C^1([0,K]);\int_0^K g(s)\mathrm{d}s=1,g(s)>0,g'(s)<0,\forall s\in(0,K)\}$.
In this paper we show that \eqref{eq-tw} admits a unique sharp type traveling wave, and the sharp traveling wave is monotonically increasing and corresponding to the minimal wave speed $c^*(m,p,r)$, and further $c^*(m,p,r)<c^*(m,p,0)$ for any time delay $r>0$. As a consequence, the time delay slows down the minimal traveling wave speed for the doubly nonlinear degenerate diffusion equations.
Our main results are as follows.
\begin{theorem}[Critical Sharp Traveling Wave] \label{th-existence} Assume that $d(s)$ and $b(s)$ satisfy (H$_1$)-(H$_2$), and $m>0$, $p>1$, $r\ge0$, such that $m(p-1)>1$. There exists a unique $c^*=c^*(m,p,r)>0$ defined in \eqref{eq-def} satisfying $c^*(m,p,r)<c^*(m,p,0)$ for any time delay $r>0$, such that \eqref{eq-tw} admits a unique (up to shift) sharp traveling wave $\phi(x+c^*t)$ with speed $c^*$, which is the critical traveling wave of \eqref{eq-tw} and is monotonically increasing. Moreover, any other traveling wave solution must be smooth and correspond to speed $c>c^*(m,p,r)$. \end{theorem}
\begin{theorem}[Regularity of Sharp Wave] \label{th-sharp} Assume that the conditions in Theorem \ref{th-existence} hold. Let $\gamma_{m,p}$ be the largest integer that is smaller than $\frac{p-1}{m(p-1)-1}$, i.e., $$ \gamma_{m,p}:= \begin{cases} \frac{p-1}{m(p-1)-1}-1, \quad& \text{if~} \frac{p-1}{m(p-1)-1} \text{is an integer},\\ [\frac{p-1}{m(p-1)-1}], \quad& \text{if~} \frac{p-1}{m(p-1)-1} \text{is not an integer}, \end{cases} $$ and denote $\alpha_{m,p}:=\frac{p-1}{m(p-1)-1}-\gamma_{m,p}\in(0,1]$. Then the optimal regularity of sharp wave $\phi(\xi)$ is $\phi\in C^{\gamma_{m,p},\alpha_{m,p}}(\overline{\mathbb R})$, where $C^{\gamma_{m,p},\alpha_{m,p}}(\overline{\mathbb R})$ is the function space defined as: if $\frac{p-1}{m(p-1)-1}$ is an integer, then $\alpha_{m,p}=1$, and \begin{eqnarray} C^{\gamma_{m,p},1}(\overline{\mathbb R}):=\Big\{
\phi \in C^{\gamma_{m,p}}(\overline{\mathbb R})&\Big| & \partial^j \phi, \mbox{ for } j=0,1,\cdots, \frac{p-1}{m(p-1)-1}-1, \notag \\ & &\mbox{ are Lipschitz continuous} \Big\}; \end{eqnarray} while if $\frac{p-1}{m(p-1)-1}$ is not an integer, then $0<\alpha_{m,p}<1$, and \begin{eqnarray} C^{\gamma_{m,p},\alpha_{m,p}}(\overline{\mathbb R}):=\Big\{
\phi \in C^{\gamma_{m,p}}(\overline{\mathbb R})&\Big| & \partial^j \phi, \mbox{ for } j=0,1,\cdots, \Big[\frac{p-1}{m(p-1)-1}\Big], \notag \\ & & \mbox{ are } C^{\alpha_{m,p}} \mbox{ H\"older continuous} \Big\}. \end{eqnarray} \end{theorem}
\begin{remark} If $m\ge\frac{p}{p-1}$, then the sharp traveling wave is not $C^1$ smooth; while if $m\in(\frac{1}{p-1},\frac{p}{p-1})$, then the sharp traveling wave is $C^1$ smooth. See Figure \ref{fig-1}. \end{remark}
\begin{figure}
\caption{Traveling waves: ($a$) non-$C^1$ sharp type for $m\ge\frac{p}{p-1}$; ($b$) $C^1$ sharp type for $m\in(\frac{1}{p-1},\frac{p}{p-1})$.}
\label{fig-1}
\end{figure}
\section{Proof of the main results} For any given $m>0$, $p>1$, and $r>0$, such that $m(p-1)>1$, we solve \eqref{eq-tw} locally for any $c>0$ and then we single out a special one that is a sharp traveling wave with critical wave speed. First, noticing that the sharp wave solution $\phi(t)=0$ for $t\le0$ and then $\phi(t-cr)=0$ for $t\in[0,cr)$, \eqref{eq-tw} is locally reduced to \begin{equation} \label{eq-semi-1} \begin{cases}
c\phi'(t)=(|(\phi^m(t))'|^{p-2}(\phi^m(t))')'-d(\phi(t)), \\ \phi(0)=0, \quad (\phi^m)'(0)=0, \quad t\in(0,cr), \end{cases} \end{equation} whose solutions are not unique and we choose the maximal one such that $\phi(t)>0$ for $t\in(0,cr)$ as shown in the following lemma. Here, $(\phi^m)'(0)=0$ is a necessary and sufficient condition such that the zero extension of $\phi(t)$ to the left satisfies \eqref{eq-tw} locally near $0$ in the sense of distributions.
The proof follows from the similar outline as in \cite{Non20}, the difference lies in the asymptotic behavior of $\psi(t):=|(\phi^m(t))'|^{p-2}(\phi^m(t))'$ in the singular phase plane of $(\phi,\psi)$ for the sharp wave solution $\phi(t)$. Here we mainly sketch the proofs that have differences for the sake of simplicity.
\begin{lemma} \label{le-semi-1} For any $c>0$, the degenerate ODE \eqref{eq-semi-1} admits a unique maximal solution $\phi_c^1(t)$ on $(0,cr)$ such that $\phi_c^1(t)>0$ on $(0,cr)$ and \begin{equation} \label{eq-expansion} \phi(t)=\Big(\frac{c^\frac{1}{p-1}(m(p-1)-1)}{m(p-1)}\Big)^\frac{p-1}{m(p-1)-1}\cdot t_+^\frac{p-1}{m(p-1)-1} +o(t_+^\frac{p-1}{m(p-1)-1}), \quad \text{as~} \ t\to0^+. \end{equation} \end{lemma} {\it\bfseries Proof.} A positive function $\phi(t)>0$ on $(0,cr)$ is a solution to the degenerate ODE \eqref{eq-semi-1} satisfies the following singular differential system on $(0,cr)$ \begin{equation} \label{eq-zsys} \begin{cases} \displaystyle \phi'(t)=\frac{\psi^\frac{1}{p-1}(t)}{m\phi^{m-1}(t)}, \\ \displaystyle \psi'(t)=c\frac{\psi^\frac{1}{p-1}(t)}{m\phi^{m-1}(t)}+d(\phi(t)), \end{cases} \end{equation}
with $\psi(t):=|(\phi^m(t))'|^{p-2}(\phi^m(t))'$. We seek for a solution to \eqref{eq-zsys} such that $\phi(t)>0$ and $\psi(t)>0$ for $t\in(0,cr)$, with $\psi(0)=0$ and $\phi(0)=0$. The system \eqref{eq-zsys} has singularity at some points where $\phi(t)=0$. Therefore, we solve \eqref{eq-zsys} with the initial condition $(\phi_\varepsilon(0),\psi_\varepsilon(0))=(\varepsilon^2,\varepsilon)$, whose local existence and uniqueness are certain according the classical phase plane analysis method.
For any trajectory $(\phi(t),\psi(t))$ such that $\phi(t)>0$ and $\psi(t)>0$ in some interval, we can make change of variables such that we take $\phi$ as an independent variable and regard $\psi$ as a function of $\phi$, denoted by $\tilde \psi(\phi)$, in the phase plane of $(\phi,\psi)$ since $\phi'(t)=\frac{\psi^{1/(p-1)}(t)}{m\phi^{m-1}(t)}>0$. The function corresponding to $(\phi_\varepsilon,\psi_\varepsilon)$ is denoted by $\tilde \psi_\varepsilon(\phi)$. Comparison principle or the analysis of the trajectories shows that $\tilde \psi_\varepsilon(\phi)$ is monotone increasing with respect to $\varepsilon>0$. The limiting function $(\phi(t),\psi(t))$ as $\varepsilon\to0^+$ is the maximal solution to \eqref{eq-semi-1}.
Asymptotic analysis shows that (note that $m(p-1)>1$) $$ \tilde \psi(\phi)=c\phi+o(\phi), \quad \text{as~} \ \phi\to0^+, $$ or equivalently, \begin{equation} \label{eq-ztildepsi}
\psi(t)=|(\phi^m(t))'|^{p-2}(\phi^m(t))'=c\phi(t)+o(\phi(t)), \quad \text{as~} \ t\to0^+. \end{equation} Furthermore, $\phi^m(t)>0$ on $(0,cr)$ with $\phi^m(0)=0$ is the maximal solution to the following singular first order differential equation $$ (\phi^m(t))'=c^\frac{1}{p-1}(\phi^m(t))^\frac{1}{m(p-1)}+o((\phi^m(t))^\frac{1}{m(p-1)}), \quad \text{as~} \ t\to0^+. $$ Therefore, $$ \phi^m(t)=\Big(\frac{c^\frac{1}{p-1}(m(p-1)-1)}{m(p-1)}\Big)^\frac{m(p-1)}{m(p-1)-1}\cdot t_+^\frac{m(p-1)}{m(p-1)-1} +o(t_+^\frac{m(p-1)}{m(p-1)-1}), \quad \text{as~} \ t\to0^+. $$ The proof of \eqref{eq-expansion} is completed. $
\Box$
Next, let $\phi_c^2(t)$ be the solution of the following initial value second order ODE problem \begin{equation} \label{eq-semi-2} \begin{cases}
c\phi'(t)=(|(\phi^m(t))'|^{p-2}(\phi^m(t))')'-d(\phi(t))+b(\phi_c^1(t-cr)), \quad t\in(cr,2cr),\\ \phi(cr)=\phi_c^1(cr), \quad \phi'(cr)=(\phi_c^1)'(cr). \end{cases} \end{equation} The problem \eqref{eq-semi-2} is locally solvable and has no singularity near $t=cr$ since $\phi_c^1(cr)>0$. The above steps can be continued unless $\phi_c^k(t)$ blows up or decays to zero in finite time for some $k\in \mathbb N^+$. Let $\phi_c(t)$ be the connecting function of those functions on each step, i.e., \begin{equation} \label{eq-semi} \phi_c(t)= \begin{cases} \phi_c^1(t), \quad &t\in[0,cr),\\ \phi_c^2(t), \quad &t\in[cr,2cr),\\ \vdots\\ \phi_c^k(t), \quad &t\in[(k-1)cr,kcr),\\ \vdots \end{cases} \end{equation} for some finite steps such that $\phi_c(t)$ blows up or decays to zero, or for infinite steps such that $\phi_c(t)$ is defined on $(0,+\infty)$ and zero extended to $(-\infty,0)$ for convenience.
\begin{lemma} \label{le-semi-decay} For any given $r>0$, there exist two numbers $\overline c>\underline c>0$ such that if $0<c\le \underline c$, then $\phi_c(t)$ decays to zero in finite time; if $c\ge \overline c$, then $\phi_c(t)$ grows up to $+\infty$ as $t$ tends to $+\infty$. \end{lemma} {\it\bfseries Proof.} The above assertions for the special case of $m>1$ and $p=2$ are proved in \cite{JDE20}. Here we consider the generalized phase plane corresponding to the following dynamical system \begin{equation} \label{eq-zphase} \begin{cases} \displaystyle \phi'(t)=\frac{\psi^\frac{1}{p-1}(t)}{m\phi^{m-1}(t)}, \\ \displaystyle \psi'(t)=c\frac{\psi^\frac{1}{p-1}(t)}{m\phi^{m-1}(t)}+d(\phi(t))-b(\phi(t-cr)), \end{cases} \end{equation}
with $\psi(t):=|(\phi^m(t))'|^{p-2}(\phi^m(t))'$. For the local solution $\phi_c(t)$ on its strictly monotone increasing subinterval, we take $\phi_c$ as an independent variable and regard $\psi_c$ as a function of $\phi_c$, denoted by \begin{equation} \label{eq-psi} \tilde \psi_c(\phi_c)=\psi_c(t^{-1}(\phi_c)), \text{~such~that~} t^{-1}(\phi_c) \text{~is~the~inverse~function~of~} \phi_c(t). \end{equation} We may drop the subscripts in $\phi_c$, $\psi_c$, $\tilde\psi_c$, and simply write $\phi$, $\psi$, $\tilde\psi$, for a given $c>0$. Define \begin{equation} \label{eq-phicr} \phi_{cr}:=\inf_{\theta\in[0,\phi]}\Big\{\int_\theta^\phi\frac{ms^{m-1}}{\tilde\psi^\frac{1}{p-1}(s)}\mathrm{d}s\le cr\Big\}. \end{equation} Then the function $\tilde\psi(\phi)$ satisfies \begin{equation} \label{eq-zphasepsi} \begin{cases} \displaystyle \frac{\mathrm{d}\tilde\psi}{\mathrm{d}\phi}=c-\frac{m\phi^{m-1}\cdot(b(\phi_{cr})-d(\phi))}{\tilde\psi^\frac{1}{p-1}(\phi)},\\ \tilde\psi(0)=0, \quad \tilde\psi(\phi)>0 \text{~for~} \phi\in(0,\phi^*), \end{cases} \end{equation} where $\phi^*=\phi_c(t^*)$ and $(0,t^*)$ is the maximum interval such that $\phi_c(t)$ is strictly monotone increasing. The rest of the proof follows similarly as the proofs of Lemma 3.2 and Lemma 3.3 in \cite{JDE20}. $
\Box$
The local solution $\phi_c(t)$ may grow beyond the positive equilibrium $K>0$ or decay to zero in finite interval. The sharp traveling wave is the special one (the uniqueness will be proved) such that $\phi_c(t)$ exists globally and is monotone increasing on $(0,+\infty)$, together with the speed $c$ being identical to the critical wave speed $c^*(m,p,r)$. The existence and other properties of the sharp wave for the case of $m>1$ and $p=2$ are proved in \cite{JDE20,Non20}. Specifically, the existence of sharp wave in the above settings is prove in \cite{JDE20} through a continuous argument for general non-monotone birth function $b(s)$; and the uniqueness is prove in \cite{Non20} via monotone dependence of $\phi_c(t)$ with respect to $c$ for monotone birth function.
\begin{lemma} \label{le-dependent} The solution $\phi_c(t)$ is locally continuously dependent on $c$ and is strictly monotonically increasing with respect to $c$ on their joint existence interval. \end{lemma} {\it\bfseries Proof.} The proof is similar to that in Lemma 3.4 and Lemma 3.6 in \cite{Non20}. Here we omit the details. $
\Box$
\begin{lemma} \label{le-cstar} There exists a unique number $c^*=c^*(m,p,r)>0$ such that $\phi_{c^*}(t)$ is strictly increasing on $(0,+\infty)$ with $\phi_{c^*}(+\infty)=K$, and the function $\phi_{c^*}(t)$ is the unique traveling wave solution of sharp type. The speed of any smooth traveling wave is greater than $c^*(m,p,r)$, and no traveling waves $\phi(x+ct)$ exist when $c\le c^*$. Namely, $c^*$ is the minimal admissible traveling wave speed. \end{lemma} {\it\bfseries Proof.} This is proved in a similar way as Lemma 3.7 and Lemma 3.9 in \cite{Non20}. Here we need to note the following asymptotic behavior near zero of the phase function $\tilde \psi(\phi)$, defined as \eqref{eq-psi}
for any traveling wave solution $\phi(t)$ with $\psi(t):=|(\phi^m(t))'|^{p-2}(\phi^m(t))'$:
(i) if $\phi(t)$ is a sharp traveling wave with speed $c$, then $\tilde\psi(\phi)=c\phi+o(\phi)$, as $\phi\to0^+$, according to Lemma \ref{le-semi-1};
(ii) if $\phi(t)$ is a smooth traveling wave with speed $c$, then $$ \tilde\psi(\phi)=\Big(\frac{m(b'(0)e^{-\lambda cr}-d'(0))}{c}\Big)^{p-1}\phi^{m(p-1)}, \quad \text{as~} \ \phi\to0^+, $$ where $\lambda>0$ is the unique root of the equation $c\lambda+d'(0)=b'(0)e^{-\lambda cr}$.
Suppose that $\hat\phi(t)$ is a smooth traveling wave with speed $c>0$. The phase function corresponds to $\hat\phi(t)$ is denoted by $\hat\psi(\phi)$. Let $\phi_c(t)$ be the local solution of sharp type with the same speed and $\tilde\psi(\phi)$ be its phase function. Locally near zero, $$ \hat\psi(\phi)\sim\Big(\frac{m(b'(0)e^{-\lambda cr}-d'(0))}{c}\Big)^{p-1}\phi^{m(p-1)} <c\phi\sim \tilde\psi(\phi), \quad \text{as~} \ \phi\to0^+, $$ since $m(p-1)>1$. Therefore, the monotone dependence (proved in a similar way as Lemma 3.6 in \cite{Non20}) shows that $\tilde\psi(\phi)>\hat\psi(\phi)$ for $\phi\in(0,K]$. That is, the local solution $\phi_c(t)$ grows beyond $K$ in finite interval and there holds $c>c^*(m,p,r)$. $
\Box$
We also need to describe the asymptotic behavior of the phase function $\tilde\psi(\phi)$ near the positive equilibrium $K$, which is important to the variational characterization of critical wave speed.
\begin{lemma} \label{le-asymp} The phase function $\tilde \psi(\phi)$ defined in \eqref{eq-psi} for the unique sharp traveling wave solution $\phi(t)$ satisfies the following asymptotic expansion:
(i) if $p=2$, then $$
\tilde\psi(\phi)=\kappa_2(K-\phi)+o(|K-\phi|), \quad \text{as~} \ \phi\to K^-, $$ where $\lambda>0$ is the unique positive root of $$mK^{m-1}\lambda^2+c\lambda+b'(K)e^{\lambda cr}-d'(K)=0,$$ and $\kappa_2:=mK^{m-1}\lambda$;
(ii) if $p>2$, then $$
\tilde\psi(\phi)=\kappa_p(K-\phi)^{p-1}+o(|K-\phi|^{p-1}), \quad \text{as~} \ \phi\to K^-, $$ where $\lambda>0$ is the unique positive root of $$c\lambda+b'(K)e^{\lambda cr}-d'(K)=0,$$ and $\kappa_p:=(mK^{m-1}\lambda)^{p-1}$ for $p>2$;
(iii) if $1<p<2$, then $$
\tilde\psi(\phi)=\kappa_p(K-\phi)^\frac{2(p-1)}{p}+o(|K-\phi|^\frac{2(p-1)}{p}), \quad \text{as~} \ \phi\to K^-, $$ where $\lambda>0$ is the unique positive root of $$ m^{p-1}K^{(m-1)(p-1)}\frac{2(p-1)}{p}\Big(\frac{p}{2-p}\lambda\Big)^{p}+b'(K)-d'(K)=0. $$ and $\kappa_p:=m^{p-1}K^{(m-1)(p-1)}\Big(\frac{p}{2-p}\lambda\Big)^{p-1}$ for $p\in(1,2)$. \end{lemma} {\it\bfseries Proof.} For $p\ge2$, we utilize the ansatz of expansion $\tilde\psi(\phi)\sim \kappa(K-\phi)^{p-1}$ and $\phi(t)-K\sim -\mu \mathrm{e}^{-\lambda t}$ as $t\to+\infty$ and $\phi\to K^-$ for some positive constants $\kappa$, $\mu$ and $\lambda$. Further, $d(\phi)-d(K)\sim d'(K)(\phi-K)$, and $b(\phi_{cr})-b(K)\sim b'(K)(\phi_{cr}-K)$, as $t\to+\infty$, where $\phi_{cr}=\phi(\cdot-cr)$. Noticing that $b(K)=d(K)$, we see that \begin{align*} b(\phi_{cr})-d(\phi)&\sim b'(K)(\phi_{cr}-K)-d'(K)(\phi-K) \\ &\sim (b'(K)-d'(K))(\phi-K)+b'(K)(\phi_{cr}-\phi) \\ &\sim (b'(K)-d'(K))(\phi-K)+b'(K)(\mathrm{e}^{\lambda cr}-1)(\phi-K) \\ &\sim (b'(K)\mathrm{e}^{\lambda cr}-d'(K))(\phi-K), \end{align*} as $t\to+\infty$ since $\frac{\phi_{cr}-\phi}{\phi-K}\sim \mathrm{e}^{\lambda cr}-1$. According to \eqref{eq-zphase} and \eqref{eq-zphasepsi}, near $K$, $\tilde\psi$ behaves similar as \begin{equation} \label{eq-zphasepsi-K} \begin{cases} \displaystyle \frac{\mathrm{d}\tilde\psi}{\mathrm{d}\phi}\sim c-\frac{mK^{m-1}\cdot(b'(K)\mathrm{e}^{\lambda cr}-d'(K))(\phi-K)}{\tilde\psi^\frac{1}{p-1}(\phi)},\\ \tilde\psi(K)=0, \quad \tilde\psi(\phi)>0 \text{~for~} \phi\in(0,K). \end{cases} \end{equation}
For the special case $p=2$, $\frac{\mathrm{d}\tilde\psi}{\mathrm{d}\phi}\sim -\kappa$, and the singular ODE \eqref{eq-zphasepsi-K} admits a solution satisfying the expansion $\tilde\psi(\phi)\sim \kappa(K-\phi)$ provided that $\kappa>0$ is the unique positive root of the following equation \begin{equation} \label{eq-zkappa} -\kappa=c+\frac{mK^{m-1}\cdot(b'(K)\mathrm{e}^{\lambda cr}-d'(K))}{\kappa}. \end{equation} Additionally, a necessary condition for the characteristic value $\lambda>0$ of the traveling wave of \eqref{eq-tw} satisfying $\phi(t)-K\sim -\mu \mathrm{e}^{-\lambda t}$ as $t\to+\infty$ is \begin{equation} \label{eq-zlambda} mK^{m-1}\lambda^2+c\lambda+b'(K)\mathrm{e}^{\lambda cr}-d'(K)=0. \end{equation} Moreover, according to the asymptotic expansions $\tilde\psi(\phi)\sim \kappa(K-\phi)$ and $\phi(t)-K\sim -\mu \mathrm{e}^{-\lambda t}$, noticing that $\tilde\psi(\phi)=\psi(t)=(\phi^m(t))'\sim mK^{m-1}\phi'(t)$ for $p=2$, there must hold \begin{equation} \label{eq-zkappa2} \kappa=mK^{m-1}\lambda. \end{equation} Since $d'(K)>b'(K)\ge0$, the characteristic equation \eqref{eq-zlambda} admits a unique positive root $\lambda>0$, and then \eqref{eq-zkappa} is equivalent to \eqref{eq-zkappa2}.
For the case $p>2$, $\frac{\mathrm{d}\tilde\psi}{\mathrm{d}\phi}=o(1)$ as $\phi\to K^-$, and in this situation \eqref{eq-zkappa} reads as \begin{equation} \label{eq-zkappa-p} 0=c+\frac{mK^{m-1}\cdot(b'(K)\mathrm{e}^{\lambda cr}-d'(K))}{\kappa^\frac{1}{p-1}}. \end{equation} The characteristic equation \eqref{eq-zlambda} now is \begin{equation} \label{eq-zlambda-p} c\lambda+b'(K)\mathrm{e}^{\lambda cr}-d'(K)=0. \end{equation} Similar to \eqref{eq-zkappa2} the relation between the expansions $\tilde\psi(\phi)\sim \kappa(K-\phi)^{p-1}$ and $\phi(t)-K\sim -\mu \mathrm{e}^{-\lambda t}$ implies \begin{align*}
\tilde\psi(\phi)=\psi(t)=|(\phi^m(t))'|^{p-2}(\phi^m(t))'\sim (mK^{m-1}\phi'(t))^{p-1} \sim \kappa(K-\phi)^{p-1}. \end{align*} That is, \begin{equation} \label{eq-zkappa2-p} \kappa=(mK^{m-1}\lambda)^{p-1}. \end{equation} The characteristic equation \eqref{eq-zlambda-p} has a unique positive root $\lambda>0$ and \eqref{eq-zkappa-p} is equivalent to \eqref{eq-zkappa2-p} in this case.
The case of $1<p<2$ is quite different, we utilize the ansatz $\tilde\psi(\phi)\sim \kappa(K-\phi)^\frac{2(p-1)}{p}$ and $\phi(t)-K\sim -\mu (1+\lambda t)^{-\frac{p}{2-p}}$ as $t\to+\infty$ and $\phi\to K^-$ for some positive constants $\kappa$, $\mu$ and $\lambda$. It should be addressed that the sharp traveling wave approached the positive equilibrium $K$ algebraically instead of exponentially. Note that $\frac{2(p-1)}{p}\in(0,1)$, $\frac{2(p-1)}{p}-1=1-\frac{2}{p}$, and $\frac{p}{2-p}\in(1,+\infty)$ for $p\in(1,2)$. The algebraical decay behaves differently from the exponential decay, such that \begin{align*} b(\phi_{cr})-d(\phi)&\sim b'(K)(\phi_{cr}-K)-d'(K)(\phi-K) \\ &\sim (b'(K)-d'(K))(\phi-K)+b'(K)(\phi_{cr}-\phi) \\ &\sim (b'(K)-d'(K))(\phi-K), \end{align*} as $t\to+\infty$ since $\frac{\phi_{cr}-\phi}{\phi-K}=o(1)$. The singular ODE \eqref{eq-zphasepsi-K} now shows that \begin{equation} \label{eq-zkappa-p2} \frac{2(p-1)}{p}\cdot \kappa=-\frac{mK^{m-1}\cdot(b'(K)-d'(K))}{\kappa^\frac{1}{p-1}}. \end{equation} For the degenerate case such that $\phi(t)-K\sim -\mu (1+\lambda t)^{-\frac{p}{2-p}}$ (assume $\mu=1$ by rescaling), we have \begin{align*}
(|(\phi^m&(t))'|^{p-2}(\phi^m(t))')'=
(m^{p-1}\phi^{(m-1)(p-1)}|\phi'|^{p-2}\phi')' \\
&=m^{p-1}\phi^{(m-1)(p-1)}(p-1)|\phi'|^{p-2}\phi''+m^{p-1}(m-1)(p-1)\phi^{(m-1)(p-1)-1}|\phi'|^{p} \\
&\sim m^{p-1}\phi^{(m-1)(p-1)}(p-1)|\phi'|^{p-2}\phi'', \end{align*}
since $|\phi'|^{p}=o(|\phi'|^{p-2}\phi'')$ according to $\frac{p}{2-p}>1$. Moreover, $\phi'=o(|K-\phi|)=o(b(\phi_{cr})-d(\phi))$, and then the characteristic equation of \eqref{eq-tw} is $$
-m^{p-1}\phi^{(m-1)(p-1)}(p-1)|\phi'|^{p-2}\phi''\sim b(\phi_{cr})-d(\phi)\sim (b'(K)-d'(K))(\phi-K), $$ which means $$ m^{p-1}K^{(m-1)(p-1)}(p-1)\Big(\frac{p}{2-p}\lambda\Big)^{p-2}\frac{p}{2-p}\frac{2}{2-p}\lambda^2+b'(K)-d'(K)=0. $$ That is, \begin{equation} \label{eq-zlambda-p2} m^{p-1}K^{(m-1)(p-1)}\frac{2(p-1)}{p}\Big(\frac{p}{2-p}\lambda\Big)^{p}+b'(K)-d'(K)=0. \end{equation} Moreover, according to the asymptotic expansions $\tilde\psi(\phi)\sim \kappa(K-\phi)^\frac{2(p-1)}{p}$ and $\phi(t)-K\sim -(1+\lambda t)^{-\frac{p}{2-p}}$, we have \begin{align*}
\tilde\psi(\phi)=&|(\phi^m(t))'|^{p-2}(\phi^m(t))'\sim (mK^{m-1}\phi'(t))^{p-1} \\ \sim& m^{p-1}K^{(m-1)(p-1)}\Big(\frac{p}{2-p}(1+\lambda t)^{-\frac{2}{2-p}}\lambda\Big)^{p-1} \\ \sim& \kappa((1+\lambda t)^{-\frac{p}{2-p}})^\frac{2(p-1)}{p}. \end{align*} Therefore, \begin{equation} \label{eq-zkappa2-p2} \kappa=m^{p-1}K^{(m-1)(p-1)}\Big(\frac{p}{2-p}\lambda\Big)^{p-1}. \end{equation} The characteristic equation \eqref{eq-zlambda-p2} has a unique positive root $\lambda$, and then \eqref{eq-zkappa2-p2} is identical to \eqref{eq-zkappa-p2}. $
\Box$
The critical wave speed $c^*(m,p,0)$ for non-delayed case is characterized via a variational approach by Benguria and Depassier \cite{Benguria-Variational,Benguria,Benguria18}. For time-delayed case, we utilize the variational characterization method to show the dependence of the critical wave speed $c^*(m,p,r)$ with respect to the time delay $r$.
\begin{lemma} \label{le-cstarr} The minimal traveling wave speed $c^*(m,p,r)$ for the time delay $r>0$ is strictly smaller than that without time delay, i.e., $c^*(m,p,r)<c^*(m,p,0)$. \end{lemma} {\it\bfseries Proof.} Let $\phi(t)$ be the unique sharp type traveling wave corresponding to the speed $c=c^*(m,p,r)$ according to Lemma \ref{le-cstar}. This kind of special solution is the unique one that is strictly increasing on $(0,+\infty)$, $\phi(+\infty)=K$ and $\phi'(+\infty)=0$, according to the monotone dependence Lemma \ref{le-dependent} and Lemma \ref{le-cstar}. Detailed discussion can be found in the proof of Lemma 3.10 in \cite{Non20}.
In the proof of Lemma \ref{le-semi-decay}, we formulate the generalized phase plane \eqref{eq-zphase} and \eqref{eq-zphasepsi}, where $\phi_{cr}$ is defined by \eqref{eq-phicr}. Moreover, $\tilde\psi(0)=0$, $\tilde\psi(K)=0$ since $\phi'(+\infty)=0$, $\tilde\psi(\phi)>0$ for all $\phi\in(0,K)$. We rewrite \eqref{eq-zphasepsi} into \begin{equation} \label{eq-zpsi-vc} \frac{\mathrm{d}\tilde\psi}{\mathrm{d}\phi}=c -\frac{m\phi^{m-1}(b(\phi)-d(\phi))}{\tilde\psi^\frac{1}{p-1}} +\frac{m\phi^{m-1}(b(\phi)-b(\phi_{cr}))}{\tilde\psi^\frac{1}{p-1}}, ~\phi\in(0,K). \end{equation} For any $g\in\mathscr{D}=\{g\in C^1([0,K]);\int_0^K g(s)\mathrm{d}s=1,g(s)>0,g'(s)<0,\forall s\in(0,K)\}$, we multiply \eqref{eq-zpsi-vc} by $g(s)$ and integrate it over $(0,K)$ to find \begin{align} \nonumber c=&\int_0^K g(\phi)\frac{\mathrm{d}\tilde\psi}{\mathrm{d}\phi}\mathrm{d}\phi +\int_0^K g(\phi)\frac{m\phi^{m-1}(b(\phi)-d(\phi))}{\tilde\psi^\frac{1}{p-1}}\mathrm{d}\phi \\ \nonumber &-\int_0^K g(\phi)\frac{m\phi^{m-1}(b(\phi)-b(\phi_{cr}))}{\tilde\psi^\frac{1}{p-1}}\mathrm{d}\phi \\ \nonumber =&\int_0^K -g'(\phi)\tilde\psi(\phi)\mathrm{d}\phi +\int_0^K g(\phi)\frac{m\phi^{m-1}(b(\phi)-d(\phi))}{\tilde\psi^\frac{1}{p-1}}\mathrm{d}\phi \\ \nonumber
&+\big[g(\phi)\tilde\psi(\phi)\big]\Big|_{\phi=0}^{\phi=K} -\int_0^K g(\phi)\frac{m\phi^{m-1}(b(\phi)-b(\phi_{cr}))}{\tilde\psi^\frac{1}{p-1}}\mathrm{d}\phi \\ \label{eq-zvar} =& F(g,\tilde\psi) -\int_0^K g(\phi)\frac{Dm\phi^{m-1}(b(\phi)-b(\tilde\phi_{cr}(\phi)))}{\tilde\psi}\mathrm{d}\phi, \end{align} according to
$\int_0^K g(s)\mathrm{d}s=1$ and $[g(\phi)\tilde\psi(\phi)]\big|_{\phi=0}^{\phi=K}=0$ as $\tilde\psi(0)=0=\tilde\psi(K)$, where the functional $$ F(g,\tilde\psi):=\int_0^K -g'(\phi)\tilde\psi(\phi)\mathrm{d}\phi +\int_0^K g(\phi)\frac{m\phi^{m-1}(b(\phi)-d(\phi))}{\tilde\psi^\frac{1}{p-1}}\mathrm{d}\phi. $$
Next, we consider the function $F(g,\tilde\psi)$ over all set $\mathscr{D}$. Utilizing Young's Inequality, we see that \begin{equation} \label{eq-functional} F(g,\tilde\psi) \ge \int_0^K \frac{p}{(p-1)^{(p-1)/p}}(-g'(\phi))^\frac{1}{p}(g(\phi))^\frac{p-1}{p} (m\phi^{m-1}(b(\phi)-d(\phi)))^\frac{p-1}{p}\mathrm{d}\phi, \end{equation} see also in \cite{Benguria18} for non-delayed case. The equality in \eqref{eq-functional} is attainable if there exists a function $\hat g\in \mathscr{D}$ such that \begin{equation} \label{eq-hatg} m\phi^{m-1}(b(\phi)-d(\phi))\hat g(\phi)=(p-1)(-\hat g'(\phi))\tilde\psi^\frac{p}{p-1}(\phi), \quad \phi\in(0,K). \end{equation}
The existence of a $\hat g\in\mathscr{D}$ solving \eqref{eq-hatg} relies heavily on the asymptotic behavior of $\tilde\psi(\phi)$ near $K$ and near $0$ as shown in Lemma \ref{le-asymp} and Lemma \ref{le-semi-1}. According to Lemma \ref{le-asymp}, $$ \tilde\psi^\frac{p}{p-1}(\phi) \sim \begin{cases} \kappa (K-\phi)^p, \quad & p\ge2,\\ \kappa (K-\phi)^2, \quad & p\in(1,2), \end{cases} $$ where $\kappa>0$ is a positive number. Therefore, $\hat g(K)=0$. Otherwise, \begin{equation} \label{eq-zhatg} \frac{-\hat g'(\phi)}{\hat g(\phi)}=\frac{m\phi^{m-1}(b(\phi)-d(\phi))}{(p-1)\tilde\psi^\frac{p}{p-1}(\phi)} \sim \frac{mK^{m-1}(d'(K)-b'(K))}{(p-1)\kappa (K-\phi)^{\max\{p-1,1\}}}, \quad \text{as~} \ \phi\to K^-, \end{equation} which is not integrable, a contradiction. Consider the singular ODE \eqref{eq-zhatg} near $K$ with the condition $\hat g(K)=0$, and $\hat g'(\phi)<0$ for $\phi\in(0,K)$, since $\max\{p-1,1\}\ge1$ for all $p>1$, it has infinitely many solutions such that $\hat g(\phi)>0$ for $\phi\in(0,K)$. If $\hat g(0)$ is finite, we can normalize $\hat g$ such that $\int_0^K\hat g(\phi)\mathrm{d}\phi=1$ and then $\hat g\in\mathscr{D}$. According to Lemma \ref{le-semi-1}, $\tilde\psi(\phi)\sim c\phi$ as $\phi\to0^+$. Then \begin{equation} \label{eq-zhatg0} \frac{-\hat g'(\phi)}{\hat g(\phi)}=\frac{m\phi^{m-1}(b(\phi)-d(\phi))}{(p-1)\tilde\psi^\frac{p}{p-1}(\phi)} \sim \frac{m(b'(0)-d'(0))\phi^{m}}{(p-1)c^\frac{p}{p-1}\phi^\frac{p}{p-1}}, \quad \text{as~} \ \phi\to K^-. \end{equation} It follows that $\hat g(0)<+\infty$ since $m-\frac{p}{p-1}>-1$ as $m(p-1)>1$ such that $\phi^{m-\frac{p}{p-1}}$ is integrable near zero.
Finally, for $\hat g\in\mathscr{D}$, we have \begin{align*} c=&F(\hat g,\tilde\psi) -\int_0^K \hat g(\phi)\frac{Dm\phi^{m-1}(b(\phi)-b(\tilde\phi_{cr}(\phi)))}{\tilde\psi}\mathrm{d}\phi \\ <&F(\hat g,\tilde\psi) \\ \le&\sup_{g\in\mathscr{D}}\int_0^K \frac{p}{(p-1)^{(p-1)/p}}(-g'(\phi))^\frac{1}{p}(g(\phi))^\frac{p-1}{p} (m\phi^{m-1}(b(\phi)-d(\phi)))^\frac{p-1}{p}\mathrm{d}\phi \\ =&c^*(m,p,0), \end{align*} where the last equality is the variational characterization of the speed for non-delayed case as proved in \cite{Benguria18}. The proof is completed. $
\Box$
{\it\bfseries Proof of Theorem \ref{th-existence}.} The existence and uniqueness of sharp type traveling wave are proved in Lemma \ref{le-cstar}. According to Lemma \ref{le-cstarr}, we see that time delay slows down the critical wave speed. Any other traveling waves must be positive and the regularity is trivial since \eqref{eq-tw} is non-degenerate at where $\phi(t)>0$. $
\Box$
{\it\bfseries Proof of Theorem \ref{th-sharp}.} This is proved according to the asymptotic behavior near $0$ in Lemma \ref{le-semi-1}. $
\Box$
\section{Model Formulation}
The models with degenerate diffusion but without time-delay were firstly introduced in \cite{Gurney75,Murry}, and the models with time-delay and regular diffusion were widely studied in \cite{Chern,LLLM,Mei_LinJDE09,MLSS,Mei-Ou-Zhao,Schaaf,So-Yang,So-Wu-Zou}. However, the derivation of the models with both effects of degenerate diffusion and time-delay is not officially derived, even if such models had been proposed and studied in our previous research works \cite{HJMY,JDE18,Non20,JDE20} based on the mathematical concerning. In this section, we develop a degenerate diffusion model with time delay that arises in the modelling of age-structured populations. Here, we give a brief derivation of the equations we treat.
The problem is as follows. Let $a$ denote chronological age, $t$ denote time and $x$ denote spatial position, and let $u(a, t, x)$ denote the population density of age $a$ at time $t$ and at position $x$. Here, we concerns species whose life cycles consist of two demographically distinct phases incorporating immature and reproductive periods.
By $r\ge 0$ we denote the maturation time that divides the two phases, so the matured population density at location $x$ and time $t$ is \begin{equation}\label{eq-P} u(x,t)=\int_r^\infty v(t,x,a) da. \end{equation} Also, since only the mature can reproduce, the functional dependence of birth rate $\beta$ is assumed to enter only through dependence on $u$ and so that $\beta=\beta(u)$.
Assuming that the emigration in species due to intraspecific competition in a way that makes the flux of individuals proportional to the gradient of the mature population, in \cite{Busenberg}, the following age-structured population model with degenerate diffusion is derived \begin{align}\label{eq-age-sturcture} \begin{cases} \displaystyle \pd v a+\pd v t=g(t,x,u,\nabla u)\cdot \nabla v+h(t,x,u,\nabla u,D^2 u)v-\mu v,\quad &x\in \mathbb R^n,~t>0, \\[1mm] v(0,t,x)=\int_r^\infty \beta(u)v(a,t,x)da, \\[1mm] v(a,0,x)=v_0(a,x). \end{cases} \end{align} Using \eqref{eq-P} and integrating the partial differential equation \eqref{eq-age-sturcture} from $r$ to $+\infty$, we obtain for $t>0$ and $x\in \mathbb R^n$ \begin{equation}\label{eq-delaymodel} \pd u t=\tilde g(t,x,u,\nabla u)\cdot\nabla u+\tilde h(t,x,u,\nabla u,D^2u)u +\beta (u(t-r,x))u(t-r,x)S-\mu u. \end{equation} In obtaining \eqref{eq-delaymodel} the following biological realistic assumptions are necessary \cite{Sulsky}: (i) $v(a,t,x)\to 0$ as $a\to +\infty$; (ii) the birth rate at time $t$ reduces to $v(0,t,x)=\beta(u) u$; and (iii) $v(r,t,x)=v(0,t-r,x)S$, where $S$ is the fraction of individuals that survives through the first demographic phase. The last point reflects the fact that the individuals of age $r$ is made up of the survivors that were born at time $t-r$.
The spatial diffusion term $\tilde g\cdot\nabla u+\tilde hu$ includes operators of the form $\Delta u^m$ and doubly nonlinear operator in \eqref{eq-main}. When $r=0, p=2, m>1$, \eqref{eq-delaymodel} may reduce to density dependent models of population dynamics of the form \begin{equation} \label{eq-densitynondelay} \pd u t=\Delta u^m+\beta(u)u-\mu u,\quad x \in \mathbb R^n,~t>0, \end{equation} which was considered by means of existing theory (Aronson \cite{Aronson80Density}, Gurtin \& MacCamy \cite{Gurtin77}). Our model includes a large number of evolution equations in biology, for example, the degenerate delayed Fisher-KPP equations, the degenerate Nicholson's blowflies equation, and the Mackey-Glass equation.
\end{document} | arXiv | {
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\begin{document}
\title{Superposing pure quantum states with partial prior information} \author{Shruti Dogra} \email{shrutidogra.iiserm@gmail.com \\ presently at Department of Physics, IIT Madras, Chennai, India.} \affiliation{Optics and Quantum Information Group, The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600113, India} \affiliation{Fakult\"{a}t Physik, Technische Universit\"{a}t Dortmund, D-44221 Dortmund, Germany} \author{George Thomas} \email{georget@imsc.res.in} \affiliation{Optics and Quantum Information Group, The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600113, India} \author{Sibasish Ghosh} \email{sibasish@imsc.res.in} \affiliation{Optics and Quantum Information Group, The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600113, India} \author{Dieter Suter} \email{dieter.suter@tu-dortmund.de} \affiliation{Fakult\"{a}t Physik, Technische Universit\"{a}t Dortmund, D-44221 Dortmund, Germany} \begin{abstract} The principle of superposition is an intriguing feature of quantum mechanics, which is regularly exploited in many different circumstances. A recent work [PRL \textbf{116}, 110403 (2016)] shows that the fundamentals of quantum mechanics restrict the process of superimposing two unknown pure states, even though it is possible to superimpose two quantum states with partial prior knowledge. The prior knowledge imposes geometrical constraints on the choice of input states. We discuss an experimentally feasible protocol to superimpose multiple pure states of a $d$-dimensional quantum system and carry out an explicit experimental realization for two single-qubit pure states with partial prior information on a two-qubit NMR quantum information processor. \end{abstract} \pacs{03.67.Lx, 03.67.Ac} \maketitle
\section{Introduction \label{intro}} According to the postulates of quantum theory, it is generally possible to generate superpositions of arbitrary pairs of pure states of a quantum system, unless there exists a superselection rule~\cite{dirac-book-1930,dass-qph-2013}. However, a recent study showed that there exists no general quantum protocol for creating superpositions of a completely unknown pair of pure quantum states~\cite{alvarez-sr-2015,oszmaniec-prl-2016}. The difficulty of superimposing unknown quantum states was first discussed in Ref.~\cite{alvarez-sr-2015} in the context of quantum adders. Quantum states that are equivalent up to a global phase, represent the same physical state. Therefore the superposition of unknown quantum states that are equivalent, up to their global phases, may result in a relative phase between these states, and thus in different states. However, some partial prior knowledge about the states can be used to achieve the restricted type of superposition as suggested in a recent work~\cite{oszmaniec-prl-2016}. As shown in Ref.~\cite{oszmaniec-prl-2016}, two unknown quantum states
$\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$ can be superposed, if their overlaps with a reference state $\vert \chi \rangle$ are known and nonzero. For the superposition of two $d$-dimensional states, a tripartite system of dimension $2d^2$ is used. The corresponding state is initialized into $(a \vert 0 \rangle + b \vert 1 \rangle) \vert \psi_1 \rangle \vert \psi_2 \rangle$, with arbitrary complex coefficients $a$, $b$. This state is subsequently transformed by a three-party controlled-SWAP gate. Finally, two projection operators are constructed using the reference state $\vert \chi \rangle$ and its overlaps ($\vert \langle \chi \vert \psi_i \rangle \vert$) with the states to be superimposed. The application of these projectors generates a state proportional to $(a \kappa_2 \vert \psi_1 \rangle + b \kappa_1 \vert \psi_2 \rangle)$, where $\kappa_i= \langle \chi \vert \psi_i \rangle/\vert \langle \chi \vert \psi_i \rangle \vert$. \par In general, for the sake of quantum computation, it may be useful to experimentally superpose unknown quantum states~\cite{lamanta-qph-2017}. For the past few decades, there has been a growing interest for more feasible, robust experimental quantum computation models~\cite{nielsen-book-02, suter-book-2004, suter-book-2008,ladd-nature-2010}. Experimental realization of superposition of unknown quantum states is significant, not only as a quantum computational task, but also as a fundamental principle. There exist experimental techniques based on photons~\cite{hu-pra-2016}, nuclear spins~\cite{li-pra-2016}, and super conducting qubits~\cite{unai-qph-2016} that implemented the superposition protocol discussed in~\cite{oszmaniec-prl-2016}. In Ref.~\cite{hu-pra-2016}, the superposition of two photonic states is realized. The controlled-SWAP implementation was a challenge here; therefore, an effective controlled-SWAP operation was implemented which includes post-selection and is a non-unitary operation. Another work~\cite{li-pra-2016} presents the experimental implementation of the superposition protocol~\cite{oszmaniec-prl-2016} using three nuclear spins, where the controlled-SWAP gate was implemented via numerically optimized pulses. This was followed by a three-qubit tomography and, subsequently, tracing out first and third qubits numerically to imitate projective measurements. A transmons-based implementation of Refs.~\cite{alvarez-sr-2015, oszmaniec-prl-2016} was realized on the IBM Quantum Experience~\cite{unai-qph-2016}. This scheme implemented an optimal quantum circuit obtained using genetic algorithm techniques, but its operation is limited to specific input states.
\par The present work experimentally realizes a full protocol to perform the desired superpositions of pure states of a quantum system, addressing all the aspects discussed in Ref.~\cite{oszmaniec-prl-2016}. The experiment friendly superposition protocol discussed here overcomes the experimental inefficiencies reported in Ref.~\cite{li-pra-2016}. Moreover, this is a two-qubit based experimental implementation to superpose two single-qubit states contrary to the existing implementation that used three physical qubits~\cite{li-pra-2016}. The protocol is further generalized to superpose $n$ higher-dimensional quantum states. A detailed comparison between our experimentally implemented protocol with that of existing experimental implementations in terms of the success probabilities is carried out. We also analyze the enhancement in the success probabilities associated with the desired superpositions for different prior information.
\par The material in this paper is arranged as: theoretical development of the experiment-friendly superposition protocol is described in Section~\ref{theory}. Further, experimental implementation using a system of two-nuclear spins is given in Section~\ref{experiment}. The extension of our scheme to superpose $n$ higher-dimensional quantum states is discussed in Section\ref{nqudits}. The comparison of the success probabilities with respect to previously implemented superposition protocol~\cite{li-pra-2016}, and its enhancement subject to prior information is discussed in section~\ref{discussion}. This is followed by the concluding section~\ref{conclusion}.
\section{Theoretical scheme \label{theory}}
Let us consider the superposition of two arbitrary states $|\Psi_1\rangle$, and $|\Psi_2\rangle$, with desired weights of superposition ($a$ and $b$), and whose respective inner products $\langle \chi|\Psi_i \rangle$ with a known referential state $|\chi\rangle$ are given. It is well known that a state $|\Psi\rangle$ and $e^{\iota \gamma}|\Psi\rangle$ represent the same physical states, despite different values of the overall phase `$\gamma$'. However the superposition of these states depend upon the values of the respective overall phases of the constituent states. While the global phase of a state is intangible, it is possible to determine the overall phase of a state with respect to a reference
state. Here we use the partial prior information given in terms of the inner products $\langle \chi|\Psi_i \rangle$
to obtain the overall phase factors,
$e^{\iota \gamma}=\langle \chi|\Psi_i \rangle/|\langle \chi|\Psi_i \rangle|$. The details of the protocol are worked out in the following stanzas.
Thus, for the class of states $|\Psi_i\rangle=e^{\iota \gamma_i}|\psi_i\rangle$, that are equivalent to each other upto an overall phase, $\gamma_i \in [0,2\pi]$, the
desired superimposed state may be written as, $a |\psi_1 \rangle + b |\psi_2\rangle$.
\par Beginning with an explicit analysis for the superposition of two single-qubit pure states, we consider a system of two coupled spin-$1/2$ particles (denoted here as A and X) under the action of a Hamiltonian \begin{equation}
H= -\Omega_A A_{z}\otimes \mathbb{I}_{X} -\Omega_X \mathbb{I}_{A}
\otimes X_{z} + J A_{z} \otimes X_{z}, \label{ham} \end{equation} where $\Omega_A$ ($\Omega_X$) is the resonance frequency and $A_{z}$ ($X_{z}$) is the $z$-component of angular momentum for spin $A$ ($X$).
$J$ represents the scalar coupling constant. $\vert 0 \rangle_A, \vert 1 \rangle_A$ ($\vert 0 \rangle_X, \vert 1 \rangle_X$) are the eigenvectors of $A_{z}$ ($X_{z}$) with eigenvalues $+1/2, -1/2$ respectively. The single-qubit pure states of our system are encoded in the eigenbasis $\{ \vert 00 \rangle, \vert 01 \rangle, \vert 10 \rangle, \vert 11 \rangle\}$ of the Hamiltonian $H$. We use the subspace spanned by $\vert 0 0 \rangle$, $\vert 0 1 \rangle$ of $H$ to store the single-qubit input state $\vert \Psi_1 \rangle = c_{0 0} \vert 0 \rangle + c_{0 1} \vert 1 \rangle$, where $\vert c_{00}\vert^2+\vert c_{01}\vert^2=1$, while the subspace spanned by the two remaining levels is used to store the input state vector $\vert {\Psi}_2 \rangle = c_{1 0} \vert 0 \rangle + c_{1 1} \vert 1 \rangle$, where $\vert c_{10}\vert^2+\vert c_{11}\vert^2=1$. The state of the two-qubit system ($A+X$) is then \begin{equation} \vert \Psi \rangle^{'} = a \vert 0 \rangle \otimes e^{\iota \gamma_1} \vert {\psi}_1 \rangle + b \vert 1 \rangle \otimes e^{\iota \gamma_2} \vert {\psi}_2 \rangle; \quad \vert a \vert^{2} + \vert b \vert^{2}=1, \label{coded} \end{equation} where $a$ and $b$ are the weights of the superposition. In Eq.~(\ref{coded}), the first qubit is the ancilla and the second qubit is the system-qubit. The superposition protocol that we propose here generates the desired superimposed state, irrespective of the values of phase factors (say $e^{\iota \gamma_j}$ with $j^{th}$ input state)\cite{oszmaniec-prl-2016}. Given any fixed state $\vert \chi \rangle$ of the system qubit (such that $\langle \chi \vert {\psi}_i \rangle \neq 0$), prior knowledge of the inner products $\langle \chi \vert {\psi}_1 \rangle$ and $\langle \chi \vert {\psi}_2 \rangle$ is exploited to find the phases $e^{\iota \gamma_j}$. Using this information, we construct a phase gate ($e^{\iota \theta_{z} (A_{z} \otimes \mathbb{I}_{X})}$), that implements a $z-$rotation on the first qubit by an angle $\theta_{z}=\frac{\gamma_1-\gamma_2}{2}$, leading to the state, \begin{equation} \vert \Psi \rangle^{''} \equiv e^{\iota \frac{\gamma_1+\gamma_2}{2}} (a \vert 0 \rangle \vert \psi_1 \rangle + b \vert 1 \rangle \vert \psi_2 \rangle). \label{eq3}
\end{equation} Thus the phases with the individual single-qubit states are modified, and appear as an overall phase of the two-qubit state. In Appendix~\ref{appA} a detailed view of an alternative protocol is given to encode the states $\vert \psi_1 \rangle$, $\vert \psi_2 \rangle$ and to get rid of their phases $e^{\iota \gamma_1}$, $e^{\iota \gamma_2}$ respectively. Further, a Hadamard gate on the first-qubit in Eq.~(\ref{eq3}) leads to the state (ignoring the overall phase $e^{\iota \frac{\gamma_1 + \gamma_2}{2}}$), \begin{equation}
\vert \Psi \rangle^{'''} \equiv \frac{\vert 0 \rangle}{\sqrt{2}} (a \vert \psi_1 \rangle + b \vert \psi_2 \rangle) + \frac{\vert 1 \rangle}{\sqrt{2}} (a \vert \psi_1 \rangle - b \vert \psi_2 \rangle). \label{eq6} \end{equation} Depending upon the state of the first qubit, one can choose between the sum or difference of the single-qubit states $\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$: a measurement on the first qubit in the basis $\{ \vert 0 \rangle , \vert 1 \rangle\}$ gives rise to the state, $a \vert \psi_1 \rangle + b \vert \psi_2 \rangle$ of the second qubit (in case of outcome $\vert 0 \rangle$) which is proportional to the desired superposed state, $N_{\psi}(a \vert \psi_1 \rangle + b \vert \psi_2 \rangle)$ ($N_{\psi}$ being the normalization constant), obtained with a success probability $N_{\psi}^2/2$. Thus, with the help of only one ancillary qubit, we are able to superpose two single-qubit states. Also, `$e^{\iota \gamma_i}$' does not show up in the final superposed state, which implies that the overall phase factors of the constituent states do not alter the resultant superimposed state in this protocol.
\par In the present context, no-go theorems concerning the implementation of unknown quantum operations~\cite{thompson-njp-2018, arjau-njp-2014,nicolai-pra-2014} are circumvented by using the general protocol, that creates ``arbitrary" pairs of input states within the given constraints. It is important to note that no extra information regarding arbitrary pairs of input states is used further in the superposition protocol.
\section{Experimental implementation \label{experiment}} The NMR pulse sequence to carry out weighted superposition of two single-qubit states is shown in Fig.~\ref{pp}, where the first channel corresponds to the ancillary-qubit $A$ and the second channel corresponds to the system qubit $X$ (here labeled as $^{1}\rm{H}$ and $^{13}\rm{C}$ respectively). Pulse sequence is divided into three blocks: initial, encoding and superposition as mentioned in Fig.~\ref{pp}. In the first block, system and ancillary qubits are jointly initialized in state $\vert 00 \rangle$. A single-qubit rotation by an angle $2\delta$ about the $\hat{\overline{y}}-$axis is applied on the ancillary qubit, generating the state $a \vert 00 \rangle + b \vert 10 \rangle$ (with $a=\cos \delta$ and $b=\sin \delta$). Second block, labeled as `encoding', encodes the arbitrary pair of single qubit states. This is achieved by two two-qubit controlled operations, that encode second qubit with state $\vert \psi_1 \rangle$, when first qubit is in state $\vert 0 \rangle$ and with state $\vert \psi_2 \rangle$ when first qubit is in state $\vert 1 \rangle$. Each controlled-operation is achieved by a controlled-rotation of second-qubit by an angle $(\theta_j)_{n_j}$ where state of the first qubit, $\vert j \rangle$ ($j \in \{ 0,1 \}$) is the control. The axis of rotation, $\hat{n}_j=\cos (\phi_j) \hat{y} + \sin (\phi_j)\hat{x}$. At the end of this step (labeled as $(ii)$ in Fig.~\ref{pp}),
joint state of system and ancilla is given by $a \vert 0 \rangle |\psi_1 \rangle + b \vert 1 \rangle |\psi_2 \rangle$, such that the encoded state $|\psi_j \rangle$ is parametrized by $\{\theta_{j-1}, \phi_{j-1}\}$ ($j=1,2$). This encoded two-qubit state is then fed into the block named `superposition', wherein possible overall phases of the arbitrary input states $\psi_1$ and $\psi_2$ are taken care of by applying a $z-$pulse of angle $\Delta=\frac{\gamma_1-\gamma_2}{2}$ on the first qubit, leading to the state given in Eq.~(\ref{eq3}). This is followed by a pseudo-Hadamard gate on the ancillary qubit, which is a $90^0$ pulse about $-y$ direction, leading to the joint state of system and ancilla as given in Eq.~(\ref{eq6}). A partial read out of the system qubit leads to the expected superposed state. In all the experiments, the referential state ($\vert \chi \rangle$) is chosen as $\vert 0 \rangle$.
\begin{figure}
\caption{(Colour online) NMR pulse sequence to obtain a superposition of two single-qubit states
starting with the pseudo-pure state $\vert 00 \rangle$. The two channels show the
operations on ancilla ($^{1}\rm{H}$) and system qubits ($^{13}\rm{C}$) respectively. Pulse sequence is divided into three parts, shown as separate
blocks of different colors.
Also, various steps are numbered from $(i)$-$(v)$. The radio-frequency pulses are shown as rectangles,
with respective angles of rotations mentioned at the top and the axes of rotations
specified at the bottom. The arbitrary rotation axes are
$\hat{l}_0=\cos(\frac{3\pi}{2}+\phi_0)\hat{x}+\sin(\frac{3\pi}{2}+\phi_0)\hat{y}$,
$\hat{l}_0'=\cos(\phi_0)\hat{x}+\sin(\phi_0)\hat{y}$,
$\hat{l}_1=\cos(\pi+\phi_1)\hat{x}+\sin(\pi+\phi_1)\hat{y}$, and
$\hat{l}_1'=\cos(\frac{\pi}{2}+\phi_1)\hat{x}+\sin(\frac{\pi}{2}+\phi_1)\hat{y}$. At the end of the sequence, a single-qubit measurement is performed on the system qubit. }
\label{pp}
\end{figure}
\par As discussed in the theoretical scheme, the measurement consists of a projective measurement on the first qubit ($\vert 0 \rangle \langle 0 \vert \otimes \mathbb{I}_{2X2}$), followed by a partial-trace operation that retains the state of the second qubit. The measurement applies therefore \textit{only} to the subspace spanned by the eigenvectors $\vert 00 \rangle$ and $\vert 01 \rangle$ of $H$. Experimentally, the corresponding information is contained in the coherence between these two states. Thus the final superposed state is recovered from a two-dimensional subspace by partial quantum state tomography. This approach may also be useful in different experiments as a replacement of projective readout. The desired single-qubit density operator is obtained by a set of two operations: (i) direct readout, to obtain the information about the single-quantum coherence between states $\vert 00 \rangle-\vert 01 \rangle$ and (ii) application of a gradient ($G_z$), followed by a $90^0$ pulse about $y-$axis ($(\frac{\pi}{2})_y^2$) on the second qubit, to obtain the relative populations of the energy levels $\vert 00 \rangle$ and $\vert 01 \rangle$. In both cases, we observe the spectral line corresponding to transition $\vert 00 \rangle-\vert 01 \rangle$. The resultant single-qubit density operator is un-normalized in this protocol. The normalization constant for the desired part of the density operator can be obtained experimentally by measuring the sum of the populations of states $\vert 00 \rangle$ and $\vert 01 \rangle$. This is achieved by applying a gradient to dephase the coherences, followed by a spin-selective $90^0$ pulse on the first qubit ($G_z (\frac{\pi}{2})_y^1$). A readout of the resultant NMR spectrum of the first qubit provides the normalization constant for the desired subspace. This normalization factor is then used to completely characterize the final state density operator of the superposed state.
\par The pulse sequence shown in Fig.~\ref{pp} is implemented experimentally on a sample consisting of $^{13}\rm{C}$ labeled Chloroform in deutrated Acetone. The experiments were performed on a 500 MHz Bruker Avance II NMR spectrometer with a QXI probehead. All pulses were high power, short duration RF pulses applied to the $^{1}\rm{H}$ and $^{13}\rm{C}$ spins on resonance. Scalar coupling constant, $J=215$ Hz. The spin-spin relaxation times ($T_2^{*}$) of the $^{1}\rm{H}$ and $^{13}\rm{C}$ spins were 540 ms and 170 ms, respectively. Nuclear spin systems at thermal equilibrium are in a mixed state. The system was thus initialized into a pseudo-pure state, $\vert 00 \rangle$ by spatial averaging~\cite{cory-pd-1998} with a fidelity of $0.999$. Starting from this pseudo-pure state, various pairs of single-qubit states ($\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$) were encoded on a two-qubit system, as described earlier.
\par In order to ensure the accuracy of this experimental implementation, two-qubit density operators were tomographed at the end of step $(ii)$ and $(iv)$ of the pulse sequence (Fig.~\ref{pp}), thus obtaining the state after encoding ($\rho_{exp}^{(ii)}$) and the state before the measurement ($\rho_{exp}^{(iv)}$) respectively. The two-qubit states were completely reconstructed with a set of four operations: $\{\mathbb{II}, \mathbb{IX}, \mathbb{IY}, \mathbb{XX} \}$, where $\mathbb{X(Y)}$ refers to spin-selective $90^0$ pulse along $x(y)$-axis. Single-qubit density operator of the system qubit is obtained through two operations on the system qubit: $\{ \mathbb{I}, \,G_z \mathbb{Y} \}$, where $G_z$ is the non-unitary gradient implementation about $z-$axis. The resultant single-qubit reduced density operator is then normalized as described earlier in this section. The fidelity between the theoretically expected ($\rho_{t}$) and the experimentally obtained ($\rho_{e}$) states were measured using the following expression, \begin{equation} \mathcal{F} = Tr(\rho_{e} \rho_{t})/\sqrt{Tr(\rho_{e}^{2}) Tr(\rho_{t}^{2})}. \end{equation}
\begin{center} \begin{table} \caption{Summary of experimental results. \label{table1}} \begin{center} \begin{tabular}{lllll} \hline \hline S.No.& Input state $\vert \psi_1 \rangle$ & Input state $\vert \psi_2 \rangle$ & $\frac{a}{b}$ & $\quad\mathcal{F}$ \\ \hline 1 & $\vert 0 \rangle$ & $\frac{1}{\sqrt{2}}(\vert 0 \rangle + \vert 1 \rangle)$ & $1$ & 0.996 \\
2 & $\vert 0 \rangle$ &
$\frac{1}{\sqrt{2}}(\vert 0 \rangle + e^{\frac{\iota \pi}{4}}\vert 1 \rangle)$ & $1$ & 0.995 \\
3 & $\vert 0 \rangle$ & $\frac{1}{\sqrt{2}}(\vert 0 \rangle + e^{\frac{\iota \pi}{2}}\vert 1 \rangle)$ & $1$ & 0.997 \\
4 & $\vert 0 \rangle$ & $\frac{1}{\sqrt{2}}(\vert 0 \rangle + e^{\iota \pi}\vert 1 \rangle)$ & $1$ & 0.997 \\
5 & $\frac{1}{2}(\vert 0 \rangle + \sqrt{3}\vert 1 \rangle)$ & $\frac{1}{2}(\sqrt{3} \vert 0 \rangle + \vert 1 \rangle)$ & $1$ & 0.998 \\
6 & $\frac{1}{2}(\vert 0 \rangle + e^{\frac{\iota \pi}{4}} \sqrt{3}\vert 1 \rangle)$ & $ \frac{1}{2}(\sqrt{3} \vert 0 \rangle + e^{\frac{\iota 2\pi}{3}} \vert 1 \rangle)$ & $1$ & 0.974 \\
7 & $\frac{1}{2}(\vert 0 \rangle + \sqrt{3}\vert 1 \rangle)$ & $\frac{1}{2}(\sqrt{3} \vert 0 \rangle + \vert 1 \rangle)$ & $2$ & 0.999 \\
8 & $\frac{1}{2}(\vert 0 \rangle + \sqrt{3}\vert 1 \rangle)$ & $\frac{1}{2}(\sqrt{3} \vert 0 \rangle + \vert 1 \rangle)$ & $3$ & 0.999 \\
9 & $\frac{1}{2}(\vert 0 \rangle + \sqrt{3}\vert 1 \rangle)$ & $\frac{e^{\frac{2\pi\iota}{3}}}{2}(\sqrt{3} \vert 0 \rangle + \vert 1 \rangle)$ & $1$ & 0.999 \\
10 & $\frac{1}{2}(\vert 0 \rangle + e^{\frac{\iota \pi}{4}} \sqrt{3}\vert 1 \rangle)$ & $ \frac{e^{\frac{2\pi\iota}{3}}}{2}(\sqrt{3} \vert 0 \rangle + e^{\frac{\iota 2\pi}{3}} \vert 1 \rangle)$ & $1$ & 0.981 \\
11 & $ \vert 0 \rangle$ & $ \sin \frac{\pi}{36} \vert 0 \rangle + \cos \frac{\pi}{36} \vert 1 \rangle$ & $1$ & 0.988 \\ \hline \hline
\end{tabular} \end{center} \end{table}
\end{center}
Table~\ref{table1} summarizes the results of various experiments, with columns 2 and 3 showing the single-qubit pure states to be superposed, and column 5 contains the fidelity ($\mathcal{F}$) between the experimentally superposed states and the theoretically expected ones. In the datasets numbered 1-4 of Table~\ref{table1}, we have, $\vert \psi_1 \rangle=\vert 0 \rangle$, and $\vert \psi_2 \rangle =\frac{1}{\sqrt{2}}(\vert 0 \rangle + e^{\iota \phi_2} \vert 1 \rangle)$, with $\phi_2 \in \{0, \frac{\pi}{4}, \frac{\pi}{2},\pi \}$. Each of these pairs corresponds to the same two conical sections as per their Bloch sphere representations. Similarly, the datasets numbered 5 and 6 of the Table~\ref{table1} show the superposition between two pairs of states from the same respective conical sections. A detailed tomographic analysis corresponding to dataset 3 (Table~\ref{table1}) is shown in Fig.~\ref{tomo}. We also generated superpositions of the same constituent states with different weights, as given in datasets 5, 7 and 8 of Table~\ref{table1}. For completeness, the experiments were performed with different overall phases of the input states. These phase factors were introduced while encoding the states $\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$, by applying a pulse of angle $2\delta$ about the axis $`\hat{l}'$ which is aligned with $y-$axis at an angle $\pi+\gamma_2$ (Fig.~\ref{pp}). The encoded state is thus of the form, $a \vert 0 \rangle \vert \psi_1 \rangle + e^{\iota \gamma_2} b \vert 1 \rangle \vert \psi_2 \rangle $. Experiments were performed for two pairs of states shown in datasets 9 and 10 in Table~\ref{table1}. In both cases, $\gamma_2=120^0$ and the remaining parameters were same as those of sets 5 and 6 in Table~\ref{table1}. Now compare the datasets 5 with 9 and 6 with 10. As expected, the presence or absence of the overall phase does not affect the final superposed state. The efficacy of this experimental scheme does not directly depend upon the
values of the overlaps ($|\langle \chi \vert \psi_i \rangle|$). This is evidenced by the dataset 11 of Table~\ref{table1}, where $\vert \psi_2 \rangle$ is very close to $\vert \chi^{\bot} \rangle$ (orthogonal to $\vert \chi \rangle$). Table~\ref{table1} shows that even if we choose the pair of input states ($\vert \psi_1 \rangle, \vert \psi_2 \rangle$) outside the set $\{(\vert \psi_1 \rangle, \vert \psi_2 \rangle): \vert \langle \chi \vert \psi_1 \rangle \vert=\textrm{constant,}\; \vert \langle \chi \vert \psi_2 \rangle \vert=\textrm{constant}\}$, our procedure still generates the expected superposition state $a \vert \psi_1 \rangle +b \vert \psi_2 \rangle$ with high accuracy.
\begin{widetext} \begin{figure*}
\caption{(Colour online) (A) and (B) show the theoretical input states from
dataset 3 of Table~\ref{table1}, part (C) contains the two-qubit state after
encoding ($\rho_{exp}^{(ii)}$), (D) represents the state obtained at the
end of step ($iv$) of the pulse sequence ($\rho_{exp}^{(iv)}$), parts (E)
and (F) show the final experimentally obtained ($\rho_{exp}$) corresponding to
step $(v)$ and theoretically
expected ($\rho_{th}$) single-qubit superposed states respectively. }
\label{tomo}
\end{figure*} \end{widetext}
\section{Superposition of multiple qudits \label{nqudits}} Our procedure can be readily extended to the superposition of arbitrary pure states of $n$ qudits ($d$-dimensional quantum system)~\cite{oszmaniec-prl-2016}. Let $a_1,\, a_2,\dots a_{n}$ be the desired coefficients for creating a superposition of $n$ ($d$-dimensional) states $\vert \Psi_1 \rangle_{d}$, $\vert \Psi_2 \rangle_{d}$, $\dots$ $\vert \Psi_{n} \rangle_{d}$. This requires a hybrid $n\times d-$dimensional qu$n$it-qudit system, where the qu$n$it ($n-$dimensional quantum system) acts as an ancilla (as before) and the qudit acts as the system.
For simplicity, we use a vector representative $|\Psi\rangle_j$
to represent the set of states $e^{\iota \gamma_j}|\Psi\rangle_j$, where $\gamma_j \in [0,2\pi]$. Consider now a $d$-dimensional referential state $\vert \chi \rangle_{d}$, whose non-zero overlaps, $\vert \langle \chi \vert \Psi_j \rangle_{d} \vert^2=c_j$, ($j \in \{1,2,\dots n \}$) are known. Following the same protocol as before, every qudit state is encoded in the $n \times d$ basis vectors of the hybrid qu$n$it-qudit system: $\vert j0 \rangle, \, \vert j1 \rangle, \, \vert j2 \rangle, \dots \vert j(d-1) \rangle$ where $j \in \{ 0,1,\dots n-1\}$. The phases of the constituent states are taken care of by using the information of overlaps of respective constituent states with the referential state (see Appendix~\ref{appA}). This is then followed by Fourier transformation of the qu$n$it, which is in fact the generalization of the Hadamard operation to higher-dimensional states \cite{dogra-ijqi-2015}. The resultant state, which is a generalization of the two-qubit state in Eq.~(\ref{eq6}), is \begin{equation} \frac{1}{N\sqrt{n}} \sum_{j=0}^{n-1} \left(
\vert j \rangle_{n} \otimes \sum_{k=1}^{n} \left( f^{j(k-1)} a_{k} \vert \Psi_{k} \rangle_{d} \right) \right),
\label{eq10} \end{equation} where $f= e^{\iota \frac{2\pi}{n}}$, is the $n^{th}$ root of unity and $N$ is the normalization constant. An arbitrary superposition of $n$ pure states of a qudit is then obtained by the projective measurement $\vert 0 \rangle_{n} \langle 0 \vert_{n} \otimes \mathbb{I}_{d \times d}$ subsequently tracing out the qu$n$it. The final state is a superposition of $n$ $d$-dimensional states, which along with the information of overall phase factors of the constituent ($n$-qudits) states is (from Appendix~\ref{appA}), \begin{equation}
\vert \Psi \rangle = \frac{N_{\Psi}}{N \sqrt{n}} \sum_{k=1}^{n} a_k
\left( \prod_{(j\neq k, j=1)}^{n}{\frac{\langle \chi \vert \Psi_j
\rangle_{d}}{\sqrt{c_j}}} \right) \vert \Psi_{k} \rangle_{d}, \label{eq16} \end{equation} where $N_{\Psi}$ is a constant that normalizes the un-normalized state obtained after the projective measurement. The superposed state $\vert \Psi \rangle$ (Eq.~(\ref{eq16})) is obtained with the success probability, \begin{equation}
P=\frac{N_{\Psi}^2}{N^2 n}=\frac{ \prod_{j=1}^{n}c_j}
{\sum_{j=1}^{n} a_j^2 c_j} \frac{ N_{\Psi}^2}{n}. \label{prob} \end{equation}
\section{Discussion \label{discussion}}
As per superposition protocol discussed in Ref.~\cite{oszmaniec-prl-2016}, a projector $| \mu \rangle \langle \mu |$
(where $|\mu\rangle \propto \sqrt{c_1}|0\rangle + \sqrt{c_2}|1\rangle$) is applied on first qubit to obtain the superposed state. It is discussed in~\cite{li-pra-2016}, that precision of the implementation of this operator highly depends upon the values of
$|\langle \chi|\psi_1 \rangle |$ and $|\langle \chi|\psi_2 \rangle |$. Smaller values of these overlaps lead to huge errors. Detailed analysis of this issue is carried out by Li et al (\cite{li-pra-2016}), where it is shown that when any of
the overlap values ($|\langle \chi|\psi_1 \rangle |$, $|\langle \chi|\psi_2 \rangle |$) approaches zero, the protocol unexpectedly results the final states with poor fidelities. It has been clearly stated in Ref.~\cite{li-pra-2016} that the malfunctioning of the protocol, as
$|\langle \chi|\psi_1 \rangle |$ or $|\langle \chi|\psi_2 \rangle | \rightarrow 0$, is mainly due to experimentally unavoidable imprecisions in the implementation of $| \mu \rangle \langle \mu | \otimes I \otimes I$ projection operator. However in the protocol implemented here, no such projector is used. Instead, we implement a Hadamard operator which due to its ease to implement, neatly gives the resultant state. This is also reflected in one of our experimental results (Table~\ref{table1}, dataset no. 11) where, despite very small value of the overlap between the referential state and the constituent state, experimental superimposed state is obtained with good fidelity. Thus the precision of our protocol is actually independent of the values of these overlaps, which makes this protocol more experimentally feasible. \\ A more close analysis of success probabilities obtained in different superposition protocols, and for different amount of prior information is given in following sub-sections.
\subsection{Comparison between general two-qubit and three-qubit based implementations \label{appB}} In this section, we compare the success probabilities obtained in our scheme with that of previously implemented scheme~\cite{oszmaniec-prl-2016, li-pra-2016} to carry out the superposition of two single-qubit states. With the purpose of comparison, we start with same amount of resources. Thus we use the protocol discussed in Section~\ref{theory} to obtain the present two-qubit based scheme from the existing three-qubit based scheme~\cite{oszmaniec-prl-2016} to superimpose two single-qubit pure states. Recalling Eq.\ref{eq12n}, the resultant superposed state is given as, \begin{equation}
\sqrt{\frac{c_1 c_2}{2(c_1\vert a \vert^2+c_2\vert b \vert^2)}} \left( a
\frac{\langle \chi \vert \psi_2 \rangle}{\vert \langle \chi \vert \psi_2 \rangle \vert}
\vert \psi_1 \rangle + b
\frac{\langle \chi \vert \psi_1 \rangle}{\vert \langle \chi \vert \psi_1 \rangle \vert}
\vert \psi_2 \rangle \right) \label{eq13}. \end{equation} The success probability in this case is given as $P_{2}=\frac{c_1 c_2}{2(c_1\vert a \vert^2+c_2\vert b \vert^2)}N_{\psi}^2$. Here $N_{\psi}$ is the normalization factor for state $a \vert \psi_1 \rangle + b \vert \psi_2 \rangle$ (where $\sqrt{\vert a \vert^2+\vert b \vert^2}=1$). Recalling the treatment in a three-qubit based protocol~\cite{oszmaniec-prl-2016, li-pra-2016}, the resultant state in that case is given as, \begin{equation}
\sqrt{\frac{c_1 c_2}{c_1+c_2}} \left( a
\frac{\langle \chi \vert \psi_2 \rangle}{\vert \langle \chi \vert \psi_2 \rangle \vert}
\vert \psi_1 \rangle + b
\frac{\langle \chi \vert \psi_1 \rangle}{\vert \langle \chi \vert \psi_1 \rangle \vert}
\vert \psi_2 \rangle \right) \label{eq14}. \end{equation} The success probability in this case, $P_{3}=\frac{c_1 c_2}{c_1+c_2}N_{\psi}^2$. Comparing the success probabilities resulting from these two protocols, we have, \begin{eqnarray}
r_p = \frac{P_2}{P_3} &=& \frac{c_1+c_2}{2(c_1\vert a \vert^2+c_2\vert b \vert^2)} \nonumber \\
&=& \frac{r_c+1}{2(1+\vert b \vert^2 (r_c-1))}, \end{eqnarray} where $r_c=\frac{c_2}{c_1} \in (0,\infty)$, $\vert a \vert^2, \vert b \vert^2 \in (0,1)$, and $r_p \in (0, \infty)$. Same value of success probabilities ($P_2$ and $P_3$) result, in case the overlaps, $c_1=c_2$ or the superposition is obtained with equal weights, i.e. $\vert a \vert^2=\vert b \vert^2$. Figure~\ref{plot} shows the variation $r_p$ vs $r_c$ at different values of $\vert b \vert^2$. It is interesting to note that our two-qubit based protocol outperforms the three-qubit based protocol (in terms of success probabilities) in the range $ 0.5 < \vert b \vert^2 < 1$ (when $ 0 < r_c < 1$) and in the range $ 0 < \vert b \vert^2 < 0.5$ (when $ 1 < r_c < \infty$). With reference to Table~\ref{table1}, experimental dataset numbered $7$ has $r_c=3$, $\vert b \vert^2=0.2$ and dataset numbered $8$ corresponds to $r_c=3$, $\vert b \vert^2=0.1$, that correspond to $r_p>1$ as per Figure~\ref{plot}. \begin{figure}
\caption{(Colour online) The variation of $r_p=P_2/P_3$ is shown with the ratio
of overlaps, $r_c=c_2/c_1$ corresponding to different values of
$\vert b \vert^2$. Different curves correspond to different
values of $\vert b \vert^2$,
that are specified on the right side of the plot. Two black points
on the curves for $\vert b \vert^2=0.1, \, 0.2$ correspond to
experimental conditions of the datasets numbered $7$ and $8$ of Table~\ref{table1}.
}
\label{plot}
\end{figure}
\subsection{Enhancement in success probability subject to prior information \label{special}} In general, there is an interplay between the success probability with which the desired superposed state is obtained and the amount of prior information regarding constituent states. We impose certain constraints on the constituent states and observe its impact on the success probabilities. Reconsidering the problem of superposition of two single-qubit states having fixed non-zero overlaps, $\vert \langle \chi \vert \psi_1 \rangle \vert^2=c_1$ and $\vert \langle \chi \vert \psi_2 \rangle \vert^2=c_2$ with the referential state $\vert \chi \rangle$~\cite{oszmaniec-prl-2016}, we have, $\vert \langle \chi^{\bot} \vert \psi_1 \rangle \vert^2=c_1^{\bot} =1-c_1$ and $\vert \langle \chi^{\bot} \vert \psi_2 \rangle \vert^2 =c_2^{\bot}=1-c_2$, where $\langle \chi \vert \chi^{\bot} \rangle=0$. In this case, we consider the action of the identity operator $U_1 =I \otimes I \otimes (\vert \chi \rangle \langle \chi \vert + \vert \chi^{\bot} \rangle \langle \chi^{\bot} \vert)$ (instead of $I \otimes I \otimes \vert \chi \rangle \langle \chi \vert$). Using the overlaps of the input states with both $\vert \chi \rangle$ and $\vert \chi^{\bot} \rangle$, we observe an increase in the success probability (see Appendix~\ref{sec9}). Further, we implement the single-qubit unitary operator $ U_{\chi}$ ($U_{\chi^{\bot}}$) on the first qubit, if the third qubit is in state $\vert \chi \rangle$ ($\vert \chi^{\bot} \rangle$)(see Appendix~\ref{sec9} for details). The explicit forms of the operators are
\begin{displaymath} U_{\chi} = \frac{1}{N_1}
\left(\begin{array}{cc} \frac{1}{\sqrt{c_1}} & \frac{1}{\sqrt{c_2}}
\\ \frac{1}{\sqrt{c_2}} & \frac{-1}{\sqrt{c_1}}
\end{array} \right); \hspace*{2mm}
U_{\chi^{\bot}} = \frac{1}{N_2}
\left(\begin{array}{cc} \frac{1}{\sqrt{c_1^{\bot}}} & \frac{1}{\sqrt{c_2^{\bot}}}
\\ \frac{1}{\sqrt{c_2^{\bot}}} & \frac{-1}{\sqrt{c_1^{\bot}}}
\end{array} \right), \end{displaymath} where $N_1=\sqrt{(c_1+c_2)/c_1c_2}$, $N_2=\sqrt{(c_1^{\bot}+c_2^{\bot})/c_1^{\bot}c_2^{\bot}}$. In this formalism, we mainly study two types of constraints, both $\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$ lie in the $(i)$ same longitudinal plane, and $(ii)$ same transverse plane of the Bloch sphere, In case $(i)$, the desired superposed state is obtained with success probability,
\begin{equation}
P^{tot}=
N_{\psi}^2 \left( \frac{c_1c_2}{c_1+c_2} +
\frac{c_1^{\bot}c_2^{\bot}}{c_1^{\bot}+c_2^{\bot}} \right)=
P_3 + N_{\psi}^2 \frac{c_1^{\bot}c_2^{\bot}}{c_1^{\bot}+c_2^{\bot}} \label{eq11}
\end{equation} For $c_1=c_2^{\bot}$, the success probability, $P^{tot}=2P_3$, becomes double to that of the ordinary case. In case $(ii)$, we have $c_1=c_2 =c~\textrm{(say)}$, which implies $c_1^{\bot}=c_2^{\bot}=c^{\bot}~\textrm{(say)}$. Further, assuming both states occupy diametrically opposite positions on the respective spherical sections of the Bloch sphere, the total success probability obtained then is given by:
\begin{equation}
P^{tot}=N_{\psi}^2 \left( \frac{c}{2} + \frac{c^{\bot}}{2} \right)
= \frac{1}{2}N_{\psi}^2, \label{eq12}
\end{equation}
which is again greater than $P_3$. Further, if both states lie in
the equatorial plane, this pair of states becomes orthogonal, and
the success probability reaches $1/2$. Eqs.~\ref{eq11},~\ref{eq12}
give higher success probabilities (for certain $a,b$ values) as compared to
the $a,b$-dependent protocol discussed in Ref.~\cite{oszmaniec-prl-2016}.
Recently, we came across a different approach~\cite{doosti-pra-2017},
analyzing the superposition of arbitrary pair of orthogonal states.
\section{Conclusions \label{conclusion}} We have experimentally created superposition of single-qubit states in the defined framework, covering all possible aspects, i.e. (i) creation of various single-qubit states and obtaining their superposition, (ii) superposition with arbitrary weights, and (iii) superposition of single-qubit states in the presence of assumed overall phases. All the experimental results have been obtained with fidelities over 0.97. This protocol has also been extended for the superposition of multiple states of a qudit. We have also discussed certain special cases where the desired superposed state is obtained with enhanced success probability.
\begin{acknowledgments}
SD acknowledges the financial support by The Institute of Mathematical Sciences Chennai India, Technische Universit\"{a}t Dortmund Germany, and support by the International Collaborative Research Centre TRR 160 ``Coherent manipulation of interacting spin excitations in tailored semiconductors," funded by the Deutsche Forschungsgemeinschaft. SD, GT, and SG would like to thank Somshubhro Bandyopadhyay, Manik Banik, Prathik Cherian J., Guruprasad Kar, Samir Kunkri, and Ramij Rahaman for useful discussions. \end{acknowledgments}
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\appendix \section{Encoding scheme \label{appA}} \par Let us discuss the case of superposition of $n$ number of pure states of a qudit. Considering a $d$-dimensional referential state $\vert \chi \rangle_{d}$, whose overlap (magnitude) with each of the constituent state is known. Therefore, assuming $\vert \langle \chi \vert \Psi_j \rangle_{d} \vert^2=c_j$, where $j \in \{1,2,\dots,n \}$. Let $a_1,\, a_2,\dots a_{n}$ be the desired weights for creating superposition of $d$-dimensional states $\vert \Psi_1 \rangle_{d}$, $\vert \Psi_2 \rangle_{d}$, $\dots$ $\vert \Psi_{n} \rangle_{d}$ respectively.
We begin with the initial state, \begin{equation}
\frac{1}{N}(a_1' \vert 0 \rangle_{n}+a_2' \vert 1 \rangle_{n}+\dots+a_{n}' \vert n-1 \rangle_{n})
\otimes \vert \Psi_1 \rangle_{d} \otimes \dots
\otimes \vert \Psi_{n} \rangle_{d}, \label{eq21} \end{equation} where $N$ is the normalization factor, which is equal to $\sqrt{\sum_{j=1}^{n} a_j'^2}$. This state belongs to a $n\times (d)^{n}$ dimensional Hilbert space, where the primed coefficients are, \begin{equation}
a_k'=\frac{a_k}{\prod_{(j\neq k,j=1)}^{n} {\vert \langle \chi \vert \Psi_j \rangle_{d} \vert}}
=\frac{a_k}{\sqrt{\prod_{(j\neq k,j=1)}^{n} {c_j}}}. \end{equation} This initial state is then made to undergo a series of controlled-swap operations, $\mathcal{C}\mathcal{S}_{2,3}^1~\mathcal{C}\mathcal{S}_{2,4}^1 \dots \mathcal{C}\mathcal{S}_{2,n}^1$ where state of first spin acts as control. In order to describe the action of this operation, let us reconsider the set of bases vectors of the control spin,
($\vert k \rangle_{n}$, $k\in \{0, 1, \dots, n-1 \}$) in $n$-dimensional Hilbert space, whenever the first spin (qu$n$it) is in state $\vert k \rangle_{n}$, states of first qudit (second spin) and the $(k+1)^{th}$ qudit ($k+2^{th}$ spin) get swapped. The resulting state is of the form, \begin{eqnarray}
\frac{1}{N}&& (a_1' \vert 0 \rangle_{n}
\otimes \vert \Psi_1 \rangle_{d} \otimes \vert \Psi_2 \rangle_{d} \otimes \dots
\otimes \vert \Psi_{n} \rangle_{d} \nonumber \\
&&
+a_2' \vert 1 \rangle_{n}
\otimes \vert \Psi_2 \rangle_{d} \otimes \vert \Psi_1 \rangle_{d} \otimes \dots
\otimes \vert \Psi_{n} \rangle_{d} +\dots \nonumber \\
&&
+a_{n}' \vert n-1 \rangle_{n}
\otimes \vert \Psi_{n} \rangle_{d} \otimes \vert \Psi_3 \rangle_{d} \otimes \dots
\otimes \vert \Psi_{1} \rangle_{d}). \nonumber \\ \label{eq22} \end{eqnarray} This is then acted upon by a set of projection operators constructed using the referential state $\vert \chi \rangle_{d}$. Operator performing $n-1$ number of projections on qudits numbered 2 to $n$ (or spins numbered 3 to $n+1$ in the 1-qu$n$it $\otimes$ n-qudit system) is given as, $I_{n \times n} \otimes I_{d \times d} \otimes \bigotimes_{k=2}^{n}(\vert \chi \rangle_{d} \langle \chi \vert_{d})_k$, where $k$ represents the qudit number. This helps to remove the phases that may be occurring with the constituent states ($\vert \Psi \rangle_{d}$'s). The resulting state is given as, \begin{eqnarray}
\frac{1}{N} \sum_{k=1}^{n} \left( a_k \left( \prod_{(j\neq k, j=1)}^{n}
{\frac{\langle \chi \vert \Psi_j \rangle_{d}}{\sqrt{c_j}}} \right) \vert k-1 \rangle_{n} \vert \Psi_{k} \rangle_{d} \right)
\bigotimes_{m=1}^{n-1} \vert \chi \rangle_{d} \nonumber \\ \label{eq15} \end{eqnarray} Tracing out states of qudits numbered 2 to $n$, we are left with a $n \times d$-dimensional state. Also, shedding the overall phases, the state in Eq.~(\ref{eq15}) is written in a simple manner,
\begin{equation}
\frac{1}{N} (a_1 \vert 0 \rangle_{n} \vert \Psi_1 \rangle_{d} + a_2 \vert 1 \rangle_{n} \vert \Psi_2 \rangle_{d} +
\dots + a_{n} \vert n-1 \rangle_{n} \vert \Psi_{n} \rangle_{d}), \label{eq9} \end{equation} where $N=\sqrt{\sum_{i=1}^{n} \vert a'_i \vert^2}$. In case of superposition of two qubits with weights $a_1=a$ and $a_2=b$, above equation is reduced to, \begin{equation}
\frac{1}{N} \left( a \frac{\langle \chi \vert \Psi_2 \rangle}{\vert \langle \chi \vert \Psi_2 \rangle \vert}
\vert 0 \rangle \otimes \vert \Psi_1 \rangle
+ b \frac{\langle \chi \vert \Psi_1 \rangle}{\vert \langle \chi \vert \Psi_1 \rangle \vert}
\vert 1 \rangle \otimes \vert \Psi_2 \rangle \right), \label{eq12}
\end{equation}
This is the two-qubit encoded state, which after Hadamard implementation on first qubit, followed by a projection operator $|0\rangle \langle 0| \otimes I$ gives rise to the expected superposed state given as, \begin{equation}
\frac{1}{\sqrt{2}N} \left( a \frac{\langle \chi \vert \Psi_2 \rangle}{\vert \langle \chi \vert \Psi_2 \rangle \vert}
\vert \Psi_1 \rangle
+ b \frac{\langle \chi \vert \Psi_1 \rangle}{\vert \langle \chi \vert \Psi_1 \rangle \vert}
\vert \Psi_2 \rangle \right), \label{eq12n}
\end{equation} The additional factor $\frac{1}{N}=\sqrt{\frac{c_1 c_2}{c_1\vert a \vert^2+c_2\vert b \vert^2}}$. Thus we reduce the existing three-qubit based protocol described in~\cite{oszmaniec-prl-2016} to the present two-qubit based protocol described in the main text. It is to be noted that the state Eq.~(\ref{eq12}) has already taken care of the overall phases of states ($\vert \Psi_1 \rangle$ and $\vert \Psi_2 \rangle$).
\section{Prior information and success probabilities\label{sec9}} There is an interplay between the amount of prior information needed to superimpose a pair of partially known single-qubit pure states and the success probability with which the resultant superposed state is obtained. In this section, we discuss the superposition protocol for pair of single-qubit pure sates under additional constraints that further leads to enhanced success probability. We re-consider the problem of superposition of two arbitrary single qubit states with known non-zero overlaps, $\vert \langle \chi \vert \psi_1 \rangle \vert^2=c_1$ and $\vert \langle \chi \vert \psi_2 \rangle \vert^2=c_2$ with the referential single-qubit state $\vert \chi \rangle$. Thus one can obtain the overlaps of the constituent states with $\vert \chi^{\bot} \rangle$ (single-qubit state orthogonal to $\vert \chi \rangle$). We have, $\vert \langle \chi^{\bot} \vert \psi_1 \rangle \vert^2=c_1^{\bot}=1-c_1$ and $\vert \langle \chi^{\bot} \vert \psi_2 \rangle \vert^2=c_2^{\bot}=1-c_2$. Let us begin with a three-qubit initial state, similar to the one given in Eq.~(\ref{eq21}), \begin{equation}
(a \vert 0 \rangle + b \vert 1 \rangle) \otimes \vert \psi_1 \rangle \otimes \vert \psi_2 \rangle.
\label{eq31} \end{equation}
This state is then acted upon by the same three-qubit controlled-swap operation
as described in Appendix (A), such that the resulting state is,
\begin{equation}
a \vert 0 \rangle \otimes \vert \psi_1 \rangle \otimes \vert \psi_2 \rangle
+ b \vert 1 \rangle \otimes \vert \psi_2 \rangle \otimes \vert \psi_1 \rangle.
\label{eq32} \end{equation}
Consider the action of the identity operator $U_1=I \otimes I \otimes
(\vert \chi \rangle \langle \chi \vert + \vert \chi^{\bot} \rangle \langle \chi^{\bot} \vert)$
on the three-qubit state given in Eq.~(\ref{eq31}).
The resultant state is given as,
\begin{eqnarray}
&& \left[ a \langle \chi \vert \psi_2 \rangle \vert 0 \rangle \vert \psi_1 \rangle
+ b \langle \chi \vert \psi_1 \rangle \vert 1 \rangle \vert \psi_2 \rangle \right] \otimes \vert \chi \rangle \nonumber \\
&&+ \left[ a \langle \chi^{\bot} \vert \psi_2 \rangle \vert 0 \rangle \vert \psi_1 \rangle
+ b \langle \chi^{\bot} \vert \psi_1 \rangle \vert 1 \rangle \vert \psi_2 \rangle \right] \otimes \vert \chi^{\bot} \rangle. \nonumber \\
\label{eq33}
\end{eqnarray} Another controlled unitary operation is implemented on the first qubit, where state of third qubit acts as control. Subject to the state of the third qubit ($\vert \chi \rangle$ or $\vert \chi^{\bot} \rangle$), the action of this controlled operation is described (on the first qubit) as, \begin{eqnarray}
U_{\vert \chi \rangle}\vert 0 \rangle &\rightarrow& \frac{1}{N_1}
\left(\frac{1}{\sqrt{c_2}} \vert 0 \rangle +
\frac{1}{ \sqrt{c_1}} \vert 1 \rangle \right), \nonumber \\
U_{\vert \chi \rangle}\vert 1 \rangle &\rightarrow& \frac{1}{N_1}
\left(\frac{1}{\sqrt{c_1}} \vert 0 \rangle -
\frac{1}{\sqrt{c_2}} \vert 1 \rangle \right), \nonumber \\
U_{\vert \chi^{\bot} \rangle}\vert 0 \rangle &\rightarrow& \frac{1}{N_2}
\left(\frac{1}{\sqrt{c_2^{\bot}}} \vert 0 \rangle +
\frac{1}{\sqrt{c_1^{\bot}}} \vert 1 \rangle \right), \nonumber \\
U_{\vert \chi^{\bot} \rangle} \vert 1 \rangle &\rightarrow& \frac{1}{N_2}
\left(\frac{1}{\sqrt{c_1^{\bot}}} \vert 0 \rangle -
\frac{1}{\sqrt{c_2^{\bot}}} \vert 1 \rangle \right), \\
\label{eq34}
\textrm{where,} \frac{1}{N_1}&=&\sqrt{\frac{c_1c_2}{c_1+c_2}} \, \textrm{and} \, \frac{1}{N_2}=\sqrt{\frac{c_1^{\bot}c_2^{\bot}}{c_1^{\bot}+c_2^{\bot}}}. \nonumber \end{eqnarray} Eq.~(\ref{eq33}) thus leads to,
\begin{eqnarray}
&& \frac{a}{N_1} \left( \frac{\langle \chi \vert \psi_2 \rangle}{\sqrt{c_2}} \vert 0 \rangle
+ \frac{\langle \chi \vert \psi_2 \rangle}{\sqrt{c_1}} \vert 1 \rangle \right) \vert \psi_1 \rangle \otimes \vert \chi \rangle \nonumber \\
&+& \frac{b}{N_1} \left(\frac{\langle \chi \vert \psi_1 \rangle}{\sqrt{c_1}} \vert 0 \rangle
- \frac{\langle \chi \vert \psi_1 \rangle}{\sqrt{c_2}} \vert 1 \rangle
\right) \vert \psi_2 \rangle \otimes \vert \chi \rangle \nonumber \\
&+& \frac{a}{N_2} \left( \frac{\langle \chi^{\bot} \vert \psi_2 \rangle}{\sqrt{c_2^{\bot}}} \vert 0 \rangle
+ \frac{\langle \chi^{\bot} \vert \psi_2 \rangle}{\sqrt{c_1^{\bot}}} \vert 1 \rangle \right)
\vert \psi_1 \rangle \otimes \vert \chi^{\bot} \rangle \nonumber \\
&+& \frac{b}{N_2} \left(\frac{\langle \chi^{\bot} \vert \psi_1 \rangle}{\sqrt{c_1^{\bot}}} \vert 0 \rangle
- \frac{\langle \chi^{\bot} \vert \psi_1 \rangle}{\sqrt{c_2^{\bot}}} \vert 1 \rangle
\right) \vert \psi_2 \rangle \otimes \vert \chi^{\bot} \rangle.
\nonumber \\
\label{eq35}
\end{eqnarray}
Application of the projection operator, $\vert 0 \rangle \langle 0 \vert \otimes I_{2 \times 2} \otimes I_{2 \times 2}$
then leads to,
\begin{eqnarray}
&& \frac{1}{N_1} \left( a \frac{\langle \chi \vert \psi_2 \rangle}
{\vert \langle \chi \vert \psi_2 \rangle \vert} \vert \psi_1 \rangle
+ b \frac{\langle \chi \vert \psi_1 \rangle}
{\vert \langle \chi \vert \psi_1 \rangle \vert} \vert \psi_2 \rangle \right) \otimes \vert \chi \rangle \nonumber \\
&+& \frac{1}{N_2} \left( a \frac{\langle \chi^{\bot} \vert \psi_2 \rangle}
{\vert \langle \chi^{\bot} \vert \psi_2 \rangle \vert} \vert \psi_1 \rangle
+ b \frac{\langle \chi^{\bot} \vert \psi_1 \rangle}
{\vert \langle \chi^{\bot} \vert \psi_1 \rangle \vert} \vert \psi_2 \rangle \right) \otimes \vert \chi^{\bot} \rangle.
\nonumber \\ \label{eq36}
\end{eqnarray}
Thus we obtain the weighted superpositions of single-qubit states
$\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$. If state of second qubit here is
$\vert \chi \rangle$, the superposed state,
\begin{equation}
\vert \Psi^{(1)} \rangle = \frac{ N_{\psi}^{(1)}}{N_1} \left( a \frac{\langle \chi \vert \psi_2 \rangle}
{\vert \langle \chi \vert \psi_2 \rangle \vert} \vert \psi_1 \rangle
+ b \frac{\langle \chi \vert \psi_1 \rangle}
{\vert \langle \chi \vert \psi_1 \rangle \vert} \vert \psi_2 \rangle \right) \label{eq37}
\end{equation}
is obtained with a
success probability, $P^{(1)}= (N_{\psi}^{(1)})^2 \frac{c_1c_2}{c_1+c_2}$. While
corresponding to second-qubit state $\vert \chi^{\bot} \rangle$, the superposed state,
\begin{equation}
\vert \Psi^{(2)} \rangle = \frac{ N_{\psi}^{(2)}}{N_2} \left( a \frac{\langle \chi^{\bot} \vert \psi_2 \rangle}
{\vert \langle \chi^{\bot} \vert \psi_2 \rangle \vert} \vert \psi_1 \rangle
+ b \frac{\langle \chi^{\bot} \vert \psi_1 \rangle}
{\vert \langle \chi^{\bot} \vert \psi_1 \rangle \vert} \vert \psi_2 \rangle \right) \label{eq38}
\end{equation}
is resulted with a success probability,
$P^{(2)}= (N_{\psi}^{(2)})^2 \frac{c_1^{\bot}c_2^{\bot}}{c_1^{\bot}+c_2^{\bot}}$.
$N_{\psi}^{(1)}$ and $N_{\psi}^{(2)}$ are the normalization factors of the
first qubit state when states of the second qubit are $\vert \chi \rangle$
and $\vert \chi^{\bot} \rangle$ respectively in Eq.~(\ref{eq36}).
States given in Eqs.~(\ref{eq37}) and~(\ref{eq38}) are weighted superpositions
of the same constituent states $\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$.
But they may be different because of their possibly different relative phases.
The situation of our interest arises when $\vert \Psi^{(1)} \rangle$ varies
from $\vert \Psi^{(2)} \rangle$ only upto a global phase.
Following are few special cases discussing such scenarios.
\begin{figure}
\caption{(Colour online) Bloch sphere representation of
$\vert \psi_1 \rangle$, $\vert \psi_2 \rangle$, and $\vert \chi \rangle$,
marked with unfilled red circle, filled blue circle,
and filled black square respectively.
}
\label{sphere}
\label{sphere}
\end{figure}
\subsubsection{Both states belong to same longitudinal plane on the Bloch sphere}
Assume now that both $\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$ lie in the same
longitudinal plane on the Bloch sphere as shown in Fig.~\ref{sphere}.
More explicitly, for $\frac{\langle \chi^{\bot} \vert \psi_j \rangle}
{\vert \langle \chi^{\bot} \vert \psi_j \rangle \vert}=e^{\iota \phi}\frac{\langle \chi \vert \psi_j \rangle}
{\vert \langle \chi \vert \psi_j \rangle \vert}$,
Eq.~(\ref{eq36}) takes the form,
\begin{eqnarray}
&& \left( a \frac{\langle \chi \vert \psi_2 \rangle}
{\vert \langle \chi \vert \psi_2 \rangle \vert} \vert \psi_1 \rangle
+ b \frac{\langle \chi \vert \psi_1 \rangle}
{\vert \langle \chi \vert \psi_1 \rangle \vert} \vert \psi_2 \rangle \right)
\otimes \left( \frac{1}{N_1} \vert \chi \rangle + \frac{e^{\iota \phi}}{N_2} \vert \chi^{\bot} \rangle \right). \nonumber \\ \label{eq39}
\end{eqnarray}
Tracing out the second qubit, we obtain,
\begin{eqnarray}
&& \sqrt{\frac{1}{N_1^2}+\frac{1}{N_2^2}} ~N_{\psi} \left( a \frac{\langle \chi \vert \psi_2 \rangle}
{\vert \langle \chi \vert \psi_2 \rangle \vert} \vert \psi_1 \rangle
+ b \frac{\langle \chi \vert \psi_1 \rangle}
{\vert \langle \chi \vert \psi_1 \rangle \vert} \vert \psi_2 \rangle \right), \nonumber \\ \label{eq392}
\end{eqnarray}
which is the desired superposed state. This superposed state is obtained with success probability,
\begin{equation}
P^{tot}=
N_{\psi}^2 \left( \frac{c_1c_2}{c_1+c_2} + \frac{c_1^{\bot}c_2^{\bot}}{c_1^{\bot}+c_2^{\bot}} \right)=
P_3 + N_{\psi}^2 \frac{c_1^{\bot}c_2^{\bot}}{c_1^{\bot}+c_2^{\bot}}. \label{seq11}
\end{equation}
This can as well be written as, $P^{\rm tot}=P + P^{\bot}$, where
$P=\left( \frac{N_{\psi}}{N_1} \right)^2$ and $P^{\bot}=\left( \frac{N_{\psi}}{N_2} \right)^2$.
Putting another constraint, $c_1=c_2^{\bot}$,
we obtain $N_1=N_2$ which gives rise to the desired superposed state with a success probability, \begin{equation}
P^{\rm tot}= 2 N_{\psi}^2 \frac{c_1c_2}{c_1+c_2} = 2 P. \end{equation}
\subsubsection{Both states belong to same transverse plane on the Bloch sphere} In this case, we have $c_1=c_2=c~\textrm{(say)}$, which implies $c_1^{\bot}=c_2^{\bot}=c^{\bot}~\textrm{(say)}$.
Eq.~(\ref{eq35}) thus leads to,
\begin{eqnarray}
&& \frac{1}{N}\vert 0 \rangle \left( a \frac{\langle \chi \vert \psi_2 \rangle}{\sqrt{c}}
\vert \psi_1 \rangle +b \frac{\langle \chi \vert \psi_1 \rangle}{\sqrt{c}}
\vert \psi_2 \rangle \right) \otimes \vert \chi \rangle \nonumber \\
&+& \frac{1}{N}\vert 1 \rangle \left( a \frac{\langle \chi \vert \psi_2 \rangle}{\sqrt{c}}
\vert \psi_1 \rangle -b \frac{\langle \chi \vert \psi_1 \rangle}{\sqrt{c}}
\vert \psi_2 \rangle \right) \otimes \vert \chi \rangle \nonumber \\
&+& \frac{1}{N}\vert 0 \rangle \left( a \frac{\langle \chi^{\bot} \vert \psi_2 \rangle}{\sqrt{c^{\bot}}}
\vert \psi_1 \rangle + b \frac{\langle \chi^{\bot} \vert \psi_1 \rangle}{\sqrt{c^{\bot}}}
\vert \psi_2 \rangle \right) \otimes \vert \chi^{\bot} \rangle \nonumber \\
&+& \frac{1}{N}\vert 1 \rangle \left( a \frac{\langle \chi^{\bot} \vert \psi_2 \rangle}{\sqrt{c^{\bot}}}
\vert \psi_1 \rangle - b \frac{\langle \chi^{\bot} \vert \psi_1 \rangle}{\sqrt{c^{\bot}}}
\vert \psi_2 \rangle \right) \otimes \vert \chi^{\bot} \rangle. \nonumber \\
\label{eq41}
\end{eqnarray} Further, assuming both states occupy diametrically opposite positions on respective spheric sections of the Bloch sphere, the azimuthal angles of the two states may be considered as $\phi$ and $\pi+\phi$. Under the action of projection operator, $\vert 0 \rangle \langle 0 \vert \otimes I \otimes \vert \chi \rangle \langle \chi \vert$ Eq.~(\ref{eq41}) gives rise to the desired superposed state,
\begin{eqnarray}
\frac{1}{N_1} \left( a \frac{\langle \chi \vert \psi_2 \rangle}{\sqrt{c}}
\vert \psi_1 \rangle +b \frac{\langle \chi \vert \psi_1 \rangle}{\sqrt{c}}
\vert \psi_2 \rangle \right)
\end{eqnarray}
with a success probability, $P=(\frac{N_{\psi}}{N_1})^2$.
Note that with the projection operator, $\vert 1 \rangle \langle 1 \vert \otimes I \otimes \vert \chi^{\bot} \rangle \langle \chi^{\bot} \vert$ Eq.~(\ref{eq41}) gives rise to the desired superposed state,
\begin{eqnarray}
\frac{1}{N_2} \left( a \frac{\langle \chi^{\bot} \vert \psi_2 \rangle}{\sqrt{c^{\bot}}}
\vert \psi_1 \rangle +b \frac{\langle \chi^{\bot} \vert \psi_1 \rangle}{\sqrt{c^{\bot}}}
\vert \psi_2 \rangle \right)
\end{eqnarray}
with a success probability, $P^{\bot}=(\frac{N_{\psi}}{N_2})^2$.
The total success probability obtained in above two instances,
\begin{equation}
P^{\rm tot}=P+P^{\bot}=N_{\psi}^2 \left( \frac{1}{N_1^2} + \frac{1}{N_2^2} \right) = \frac{1}{2}N_{\psi}^2 \label{seq12}
\end{equation}
Further, if both states lie in the equatorial plane,
this pair of states becomes orthogonal, and the success probability reaches $1/2$.
Eqs.~\ref{seq11},~\ref{seq12} give higher success probabilities (for certain $a,b$ values) as compared to
the $a,b$-dependent protocol discussed in the supplemental material of Ref.~\cite{oszmaniec-prl-2016}.
\end{document} | arXiv | {
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\begin{document}
\title{Nested iterative algorithms for convex constrained image recovery problems\thanks{Part of this work appeared in the conference proceedings of EUSIPCO 2008 \cite{Pustelnik_N_2008_peusipco_constrained_fbairp}
\begin{abstract} The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions $f$ and $g$, where $f$ may be non-smooth and $g$ is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of $g$ is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.
\end{abstract}
\markboth{Pustelnik \emph{et al.}: A constrained forward-backward algorithm\ldots}{SIAM Journal on Imaging Sciences}
\section{Introduction}\label{sec:intro}
Wavelet decompositions \cite{Mallat_S_1999_ap_wavelet_awtosp} proved their efficiency in solving many inverse problems. More recently, frame representations such as Bandlets \cite{LePennec_E_2005_tip_spa_girb}, Curvelets \cite{Candes_EJ_2002_as_Recovering_eiipipoocf}, Grouplets \cite{Mallat_S_2008_acha_geometric_g} or dual-trees \cite{Selesnick_I_2005_dsp_dual_tdtcwt,Chaux_C_2006_tip_ima_adtmbwt} have gained much popularity. These linear tools provide geometrical representations of images and they are able to easily incorporate a priori information (e.g. via some simple statistical models) on the data. Variational or Bayesian formulations of inverse problems using such representations often lead to the minimization of convex objective functions including a non-differentiable term having a sparsity promoting role \cite{Chambolle_A_1998_tip_nonlin_wipvpcnrws,Nikolova_M_2000_siam_Local_shoare,Antoniadis_A_2002_sn_Wavelet_tfscongn,Candes_EJ_2006_ip_Sparsity_aiics,Tropp_JA_2006_tit_convex_jrcpmissn,Combettes_PL_2007_siamopt_proximal_tafmoob}.
In restoration problems, the observed data are corrupted by a linear operator and a noise which is not necessarily additive. To solve this problem, one can adopt a variational approach, aiming at minimizing the sum of two functions $f$ and $g$ over a convex set $C$ in the transform domain. Throughout the paper, $f$ and $g$ are assumed to be in the class $\Gamma_0(\ensuremath{{\mathcal H}})$ of lower semicontinuous convex functions taking their values in $]-\infty,+\infty]$ which are proper (i.e. not identically equal to $+\infty$) and defined on a real separable Hilbert space $\ensuremath{{\mathcal H}}$. Then, our objective is to solve the following: \begin{problem}\label{pb:minimisation} Let $C$ be a nonempty closed convex subset of $\ensuremath{{\mathcal H}}$. Let $f$ and $g$ be in $\Gamma_{0}(\ensuremath{{\mathcal H}})$, where $g$ is differentiable on $\ensuremath{{\mathcal H}}$ with a $\beta$-Lipschitz continuous gradient for some $\beta\in\ensuremath{\,\left]0,+\infty\right[}$. \begin{equation*} \text{Find}\qquad \min_{x\in C} f(x)+g(x). \end{equation*} \end{problem} Problem \ref{pb:minimisation} is equivalent to minimizing $f+g+\ensuremath{\mathrm{\iota_C}}$, where $\ensuremath{\mathrm{\iota_C}}$ denotes the indicator function of $C$, i.e. \begin{equation*} \label{e:iota} (\forall x\in\ensuremath{{\mathcal H}})\quad\ensuremath{\mathrm{\iota_C}}(x)= \begin{cases} 0,&\text{if}\;\;x\in C;\\ \ensuremath{+\infty},&\text{otherwise.} \end{cases} \end{equation*} Up to now, many authors devoted their works to the unconstrained case, i.e. $C= \ensuremath{{\mathcal H}}$. So-called thresholded Landweber algorithms belonging to the more general class of forward-backward optimization methods were proposed in \cite{Figueiredo_M_2003_tosp_EM_afwbir,Bect_J_2004_eccv_unified_vfir,Daubechies_I_2004_cpamath_iterative_talipsc,Bredies_K_2007_coa_gene_cgmcism} in order to solve the problem numerically. Daubechies \textit{et al.} \cite{Daubechies_I_2004_cpamath_iterative_talipsc} investigated the convergence of these algorithms in the particular case when $g$ is a quadratic function and $f$ is a weighted $\ell_p$-norm with $p\in\left[1,2\right]$. These approaches were put into a more general convex analysis framework in \cite{Combettes_PL_2005_mms_Signal_rbpfbs} and extended to frame representations in \cite{Chaux_C_2007_ip_variational_ffbip}. Attention was also paid to the improvement of the convergence speed of the forward-backward algorithm in \cite{Bioucas_J_2007_toip_New_ttsistafir}, for some specific choices of $f$ and $g$. In \cite{Vonesch_C_2008_tip_Fast_lafwrmd}, an accelerated method was suggested in the specific case of a deconvolution in a Shannon wavelet basis. Then, a Douglas-Rachford algorithm relaxing the assumption of differentiability of $g$ was introduced in \cite{Combettes_PL_2007_istsp_Douglas_rsatncvsr}. In recent works \cite{Dupe_FX_2008_pisbi_deconv_cmipisr,Dupe_FX_2008_ip_proximal_ifdpniusr}, a variational approach, which is grounded on a judicious use of the Anscombe transform, was developed for the deconvolution of data contaminated by Poisson noise. A modification of the forward-backward algorithm was subsequently proposed in finite dimension in order to solve the associated optimization problem. Additional comments concerning this approach will be given in Sections \ref{sec:icfDR} and \ref{sec:poisson}. A key tool in the study of the aforementioned methods is the proximity operator introduced by Moreau in 1962 \cite{Moreau_JJ_1962_cras_Fonctions_cdeppdueh,Moreau_J_1965_bsmf_Proximite_eddueh}. The proximity operator of $f \in \Gamma_0(\ensuremath{{\mathcal H}})$ is
$\ensuremath{\mathrm{prox}}_f\colon\ensuremath{{\mathcal H}} \to \ensuremath{{\mathcal H}}\colon x \mapsto \arg\min_{y \in \ensuremath{{\mathcal H}}} \Frac12\left\|y-x\right\|^{2} + f(y)$. We thus see that $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}}$ reduces to the projection $P_C$ onto the convex set $C$. The function $f$ in Problem~\ref{pb:minimisation} may be non-smooth and, actually, it is often chosen as an $\ell^1$-norm, in which case its proximity operator reduces to a componentwise soft-thresholding \cite{Combettes_PL_2005_mms_Signal_rbpfbs}. In \cite{Combettes_PL_2007_siamopt_proximal_tafmoob}, the authors derived the concept of proximal thresholding by considering a larger set of non-differentiable convex functions.
The goal of this paper is to propose iterative algorithms allowing us to solve Problem~\ref{pb:minimisation} when $C\neq \ensuremath{{\mathcal H}}$. The relevance of the proposed methods is shown for image recovery problems where convex constraints on the solution need to be satisfied.
In Section \ref{sec:tools}, we start by recalling some properties of the proximity operator. Then, in Section \ref{se:compprox} we briefly describe the forward-backward and Douglas-Rachford methods. As the proximity operator of the sum of the indicator function of a convex set and a function in $\Gamma_0(\ensuremath{{\mathcal H}})$ cannot be easily expressed in general, we propose two iterative methods to compute this operator: the first one is a forward-backward algorithm, whereas the second one is a Douglas-Rachford algorithm. We also investigate the specific convergence properties of these two algorithms. In Section \ref{sec:algorithm}, we derive two iterative methods to solve Problem \ref{pb:minimisation} and their convergence behaviours are studied. Finally, in Section \ref{sec:application}, these algorithms are applied to a class of image recovery problems. In this case, the Lipschitz-continuity property of the gradient of $g$ is not satisfied in the considered maximum a posteriori criterion. To overcome this difficulty, a quadratic extension technique providing a lower approximation of the objective function is introduced. Numerical results concerning deconvolution problems in the presence of signal-dependent Gaussian noise or Poisson noise are then provided.
\section{Some properties of proximity operators}\label{sec:tools} As already mentioned, the proximity operator of $\ensuremath{\mathrm{\iota_C}} + f$ plays a key role in our approach. Some useful results for the calculation of $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + f}$ are first recalled. Subsequently, the domain of a function $f\,:\,\ensuremath{{\mathcal H}} \to ]-\infty,+\infty]$ is denoted by $\ensuremath{\mathrm{dom}\,} f = \{x \in \ensuremath{{\mathcal H}} \;\mid\; f(x) < +\infty\}$. \begin{proposition}{\rm \cite[Proposition 12]{Combettes_PL_2007_istsp_Douglas_rsatncvsr}} \label{prop:jstsp} Let $f\in \Gamma_0(\mathcal{H})$ and let $C$ be a closed convex subset of $\mathcal{H}$ such that $C\,\cap\, \ensuremath{\mathrm{dom}\,} f\, \neq \, \ensuremath{\varnothing}$. Then the following properties hold. \begin{enumerate} \item\label{prop:jstspi} $(\forall x\in \mathcal{H}),\, \ensuremath{\mathrm{prox}}_f x\in C\Rightarrow \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+f}x = \ensuremath{\mathrm{prox}}_{f}x$ \item\label{prop:jstspii} Suppose that $\mathcal{H}=\mathbb{R}$. Then \begin{equation} \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+f} = P_C\circ \ensuremath{\mathrm{prox}}_f. \label{eq:ppcp} \end{equation} \end{enumerate} \end{proposition}
Note that, the second part of this proposition can be generalized, yielding the following result which appears also as an extension of \cite[Proposition~2.10]{Chaux_C_2007_ip_variational_ffbip} when $C\neq\ensuremath{{\mathcal H}}$: \begin{proposition} \label{l:decomp} Let $\ensuremath{\mathbb K}$ be a nonempty subset of \ensuremath{\mathbb N}, $(o_k)_{k\in\ensuremath{\mathbb K}}$ be an orthonormal basis of $\ensuremath{{\mathcal H}}$ and $(\varphi_k)_{k\in\ensuremath{\mathbb K}}$ be functions in $\Gamma_0(\ensuremath{\mathbb R})$. Set \begin{equation} f\colon\ensuremath{{\mathcal H}}\to\ensuremath{\,\left]-\infty,+\infty\right]}\colon x\mapsto \sum_{k\in\ensuremath{\mathbb K}}\varphi_k(\scal{x}{o_k}).\label{eq:fsep} \end{equation} Let \begin{equation} C = \bigcap_{k\in \ensuremath{\mathbb K}} \{x\in \ensuremath{{\mathcal H}}\;\mid\;\scal{x}{o_k} \in C_k\}\label{eq:Csep} \end{equation} where $(C_k)_{k\in \ensuremath{\mathbb K}}$ are closed intervals in $\ensuremath{\mathbb R}$ such that $(\forall k \in \ensuremath{\mathbb K})$ $C_k \cap \ensuremath{\mathrm{dom}\,}\varphi_k \neq \ensuremath{\varnothing}$.\\ Suppose that either $\ensuremath{\mathbb K}$ is finite, or there exists a subset $\ensuremath{\mathbb L}$ of $\ensuremath{\mathbb K}$ such that: \begin{enumerate} \item \label{p:quezoniva} $\ensuremath{\mathbb K}\smallsetminus \ensuremath{\mathbb L}$ is finite; \item \label{p:quezonivb} $(\forall k\in\ensuremath{\mathbb L})$ $\varphi_k\geq \varphi_k(0)=0$ and $0 \in C_k$. \end{enumerate} Then, \begin{equation} (\forall x\in\ensuremath{{\mathcal H}})\quad\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+f}{x}= \sum_{k\in\ensuremath{\mathbb K}}\pi_k o_k \label{eq:ressep} \end{equation} where \begin{equation} \pi_k = \begin{cases} \inf C_k & \mbox{if $\ensuremath{\mathrm{prox}}_{\varphi_k}\scal{x}{o_k} < \inf C_k$}\\ \sup C_k & \mbox{if $\ensuremath{\mathrm{prox}}_{\varphi_k}\scal{x}{o_k} > \sup C_k$}\\ \ensuremath{\mathrm{prox}}_{\varphi_k}\scal{x}{o_k} & \mbox{otherwise.}\\ \end{cases} \label{eq:ressep2} \end{equation} \end{proposition} \begin{proof} Due to the form of $f$ and $C$, one can write, \begin{eqnarray*} (\forall x\in \ensuremath{{\mathcal H}}) \qquad \big(f + \iota_{C}\big)(x) = \sum_{k\in \ensuremath{\mathbb K}} (\varphi_k+\iota_{C_k})(\langle x,o_k \rangle). \end{eqnarray*} For every $k\in \ensuremath{\mathbb K}$, $\varphi_k + \iota_{C_k} \in \Gamma_0(\ensuremath{\mathbb R})$ since $\varphi_k \in \Gamma_0(\ensuremath{\mathbb R})$ and $C_k$ is assumed to be a closed convex set having a nonempty intersection with $\ensuremath{\mathrm{dom}\,} \varphi_k$. If $\ensuremath{\mathbb K}$ is not finite, in view of Assumption~\ref{p:quezonivb}, we have $(\forall k \in \ensuremath{\mathbb L})$ $\varphi_k + \iota_{C_k} \ge (\varphi_k+i_{C_k})(0) = 0$. From {\rm\cite[Remark~3.2(ii) and Proposition~2.10]{Chaux_C_2007_ip_variational_ffbip}}, it can be deduced that \begin{equation} (\forall x\in \ensuremath{{\mathcal H}}) \qquad \ensuremath{\mathrm{prox}}_{f + \iota_{C}} x = \sum_{k\in \ensuremath{\mathbb K}} \big(\ensuremath{\mathrm{prox}}_{\varphi_k+\iota_{C_k}}\langle x,o_k \rangle\big) o_k. \label{eq:a11} \end{equation} On the other hand, since for every $k\in \ensuremath{\mathbb K}$, $C_k$ is a closed interval in $\ensuremath{\mathbb R}$ such that $C_k \cap \ensuremath{\mathrm{dom}\,} \varphi_k \neq \ensuremath{\varnothing}$, it follows from Proposition~\ref{prop:jstsp}\ref{prop:jstspii}, that \begin{align} \ensuremath{\mathrm{prox}}_{\varphi_k + \iota_{C_k}} \langle x,o_k \rangle = & (P_{C_k} \circ \ensuremath{\mathrm{prox}}_{\varphi_k}) (\langle x,o_k \rangle)\nonumber\\
= & \begin{cases} \inf C_k, &\mbox{if $\ensuremath{\mathrm{prox}}_{\varphi_{k}} \langle x,o_k \rangle <\inf C_k$}\\ \ensuremath{\mathrm{prox}}_{\varphi_k} \langle x,o_k \rangle, &\mbox{if $\ensuremath{\mathrm{prox}}_{\varphi_{k}} \langle x,o_k \rangle \in C_k$}\\ \sup C_k, &\mbox{if $\ensuremath{\mathrm{prox}}_{\varphi_{k}} \langle x,o_k \rangle >\sup C_k$}. \end{cases} \label{eq:a12} \end{align} Combining \eqref{eq:a11} and \eqref{eq:a12} yields \eqref{eq:ressep} and \eqref{eq:ressep2}. \end{proof}
A function $f$ (resp. convex $C$) satisfying \eqref{eq:fsep} (resp. \eqref{eq:Csep}) will be said \emph{separable}. Note that \eqref{eq:ressep} and \eqref{eq:ressep2} imply that \eqref{eq:ppcp} holds. However, this relation has been proved under the restrictive assumption that both $f$ \emph{and} $C$ are separable. In general, when either $f$ \emph{or} $C$ is not separable, \eqref{eq:ppcp} is no longer valid. Let us give two simple counterexamples to illustrate this fact.
\begin{example}\label{ex:ce1} Let $\ensuremath{{\mathcal H}} = \ensuremath{\mathbb R}^2$ and $f$ be the function defined by $(\forall x \in \ensuremath{\mathbb R}^2)$ $f(x) = \frac{1}{2}x^\top\Lambda x$ with $\Lambda = \begin{pmatrix} 1&\Lambda_{1,2} \\ \Lambda_{1,2} & \Lambda_{2,2} \end{pmatrix}$
where $\Lambda_{2,2} \ge 0$ and $|\Lambda_{1,2}| \le \Lambda_{2,2}^{1/2}$. Let $C=[-1,1]^2$. This convex set is separable w.r.t. the canonical basis of $\ensuremath{\mathbb R}^2$.\\ Now, set $x = 2(\Lambda_{1,2},1+\Lambda_{2,2})^\top$. After some calculations (see Appendix \ref{ap:exce1}), one obtains: \begin{itemize} \item $P_C(\ensuremath{\mathrm{prox}}_{f}x)=(0,1)^\top$ \item $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+f}x = (\pi,1)^\top$ where \begin{equation} \pi = \begin{cases} \frac{\Lambda_{1,2}}{2} &\mbox{if $\Lambda_{1,2}\in [-2,2]$}\\ 1 &\mbox{if $\Lambda_{1,2} > 2$}\\ -1 & \mbox{if $\Lambda_{1,2} < -2$.} \end{cases} \label{eq:defpi} \end{equation} \end{itemize} We conclude that \eqref{eq:ppcp} is not satisfied as soon as $\Lambda_{1,2}\neq 0$, that is $f$ is not separable. \end{example}
\begin{example}\label{ex:ce2} Let $\ensuremath{{\mathcal H}} = \ensuremath{\mathbb R}^2$. Consider the separable function defined by $(\forall x = (x^{(1)},x^{(2)})^\top \in \ensuremath{\mathbb R}^2)$ $f(x) = (1+\Lambda_{1,2}) (x^{(1)})^2+ (1-\Lambda_{1,2})
(x^{(2)})^2$ where $0 < |\Lambda_{1,2}| \le 1$. Let the nonseparable convex set $C$ be defined by \begin{equation*}
C = \{x = (x^{(1)},x^{(2)})^\top\in \ensuremath{\mathbb R}^2\;\mid\; \max(|x^{(1)}-x^{(2)}|,|x^{(1)}+x^{(2)}|) \le \sqrt{2}\}. \end{equation*} In this case, it is shown in Appendix \ref{ap:exce2} that \eqref{eq:ppcp} does not hold. \end{example}
In summary, for an arbitrary function in $\Gamma_0(\ensuremath{{\mathcal H}})$ and an arbitrary closed convex set, we cannot trust \eqref{eq:ppcp} to determine the proximity operator of the sum of this function and the indicator function of the convex set. In the next section, we will propose efficient approaches to compute the desired proximity operator in a general setting.
Other more classical properties of the proximity operator which will be used in the paper are provided in the sequel. \begin{proposition}\ \label{prop:proxquad} \begin{enumerate}
\item \label{p:proxlin} If $f = h + \kappa \scal{\cdot}{x}$ where $h \in \Gamma_0(\ensuremath{{\mathcal H}})$, $x \in \ensuremath{{\mathcal H}}$ and $\kappa \in \ensuremath{\mathbb R}$, then $\ensuremath{\mathrm{prox}}_f = \ensuremath{\mathrm{prox}}_h(\cdot -\kappa x)$.
\item \label{p:proxquad} If $f = h + \vartheta \|\cdot\|^2/2$ where $h\in \Gamma_0(\ensuremath{{\mathcal H}})$ and $\vartheta \in \ensuremath{\,\left]0,+\infty\right[}$, then \begin{enumerate} \item \label{p:proxquada} $\ensuremath{\mathrm{prox}}_f = \ensuremath{\mathrm{prox}}_{(1+\vartheta)^{-1}h}\big(\cdot/(1+\vartheta)\big)$ \item \label{p:proxquadb} $(\forall (y,z)\in \ensuremath{{\mathcal H}}^2)$ $\scal{\ensuremath{\mathrm{prox}}_f y - \ensuremath{\mathrm{prox}}_f z}{y-z}
\ge (1+\vartheta) \|\ensuremath{\mathrm{prox}}_f y - \ensuremath{\mathrm{prox}}_f z\|^2$ \item \label{p:proxquadc} $\ensuremath{\mathrm{prox}}_f$ is strictly contractive\footnote{An operator is strictly contractive with constant $\beta$ if it is $\beta$-Lipschitz continuous and $\beta \in ]0,1[$.} with constant $(1+\vartheta)^{-1}$. \end{enumerate} \end{enumerate} \end{proposition} \begin{proof} Properties \ref{p:proxlin} and \ref{p:proxquada} result from straightforward calculations \cite[Lemma 2.6]{Combettes_PL_2005_mms_Signal_rbpfbs}. \ref{p:proxquadb} follows from the fact that $\ensuremath{\mathrm{prox}}_{(1+\vartheta)^{-1}h}$ is firmly nonexpansive \cite[Lemma 2.4]{Combettes_PL_2005_mms_Signal_rbpfbs}, i.e. \begin{equation*}
(\forall (y,z)\in \ensuremath{{\mathcal H}}^2)\qquad \scal{\ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}} y - \ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}} z}{y-z} \ge \|\ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}} y - \ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}} z\|^2. \end{equation*} Thus, by using \ref{p:proxquada}, we have \begin{align*} (\forall (y,z)\in \ensuremath{{\mathcal H}}^2)\qquad &\scal{\ensuremath{\mathrm{prox}}_f y - \ensuremath{\mathrm{prox}}_f z}{y-z}\nonumber\\ = &(1+\vartheta) \left\langle\ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}}\Big(\frac{y}{1+\vartheta}\Big) - \ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}}\Big(\frac{z}{1+\vartheta}\Big), \frac{y}{1+\vartheta}-\frac{z}{1+\vartheta}\right\rangle\nonumber\\
\ge &(1+\vartheta) \left\|\ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}}\Big(\frac{y}{1+\vartheta}\Big) - \ensuremath{\mathrm{prox}}_{\frac{h}{1+\vartheta}} \Big(\frac{z}{1+\vartheta}\Big)\right\|^2\nonumber\\
= &(1+\vartheta)\|\ensuremath{\mathrm{prox}}_f y - \ensuremath{\mathrm{prox}}_f z\|^2. \end{align*} Property \ref{p:proxquadc} can then be deduced, by invoking the Cauchy-Schwarz inequality: \begin{align*} (\forall (y,z)\in \ensuremath{{\mathcal H}}^2)\qquad
(1+\vartheta)\|\ensuremath{\mathrm{prox}}_f y - \ensuremath{\mathrm{prox}}_f z\|^2 &\le \scal{\ensuremath{\mathrm{prox}}_f y - \ensuremath{\mathrm{prox}}_f z}{y-z}\nonumber\\
&\le \|\ensuremath{\mathrm{prox}}_f y - \ensuremath{\mathrm{prox}}_f z\|\|y - z\|. \end{align*} \end{proof}
Recall that a function $f\in \Gamma_0(\ensuremath{{\mathcal H}})$ satisfying the assumptions in \ref{p:proxquad} is said to be strongly convex with modulus $\vartheta$.
\begin{proposition} {\rm \cite[Proposition 11]{Combettes_PL_2007_istsp_Douglas_rsatncvsr}} \label{p:linprox} Let $\ensuremath{\mathcal G}$ be a real Hilbert space, let $f\in\Gamma_0(\ensuremath{\mathcal G})$, and let $L\colon\ensuremath{{\mathcal H}}\to\ensuremath{\mathcal G}$ be a bounded linear operator.
Suppose that the composition of $L$ and $L^*$ satisfies $L\circ L^*=\nu\,\ensuremath{\mathrm{Id}}$, for some $\nu\in\ensuremath{\,\left]0,+\infty\right[}$.
Then $f\circ L\in\Gamma_0(\ensuremath{{\mathcal H}})$ and \begin{equation} \label{e:pfL} \ensuremath{\mathrm{prox}}_{f\circ L}=\ensuremath{\mathrm{Id}}+\nu^{-1}L^*\circ(\ensuremath{\mathrm{prox}}_{\nu f}-\ensuremath{\mathrm{Id}})\circ L. \end{equation} \end{proposition}
\section{Iterative solutions to the minimization of a sum of two convex functions} \label{se:compprox} \subsection{Forward-backward approach} \label{subsec:fb} Consider the following optimization problem, which is a specialization of Problem \ref{pb:minimisation}: \begin{problem}\label{pb:minimisation1} Let $f_1$ and $f_2$ be two functions in $\Gamma_0(\ensuremath{{\mathcal H}})$ such that $\operatorname{Argmin} f_1+f_2 \neq \ensuremath{\varnothing}$ and $f_2$ is differentiable on $\ensuremath{{\mathcal H}}$ with a $\beta$-Lipschitz continuous gradient for some $\beta \in \ensuremath{\,\left]0,+\infty\right[}$. \begin{equation*} \text{Find}\quad\min_{x \in \ensuremath{{\mathcal H}}} f_1(x) + f_2(x). \label{eq:forward-backward} \end{equation*} \end{problem}
As mentioned in the introduction, the forward-backward algorithm is an effective method to solve the above problem.
\subsubsection{Algorithm \cite[Eq.(3.6)]{Combettes_PL_2005_mms_Signal_rbpfbs}} \label{se:fwalgo} Let $x_0\in \ensuremath{{\mathcal H}}$ be an initial value. The algorithm constructs a sequence $(x_n)_{n\in \ensuremath{\mathbb N}}$ by setting, for every $n\in\ensuremath{\mathbb N}$, \begin{eqnarray} x_{n+1} = x_n + \lambda_n \big(\ensuremath{\mathrm{prox}}_{\gamma_n f_1}(x_n - \gamma_n \nabla f_2(x_n) +b_n) + a_n -x_n\big) \label{eq:fb_algo} \end{eqnarray} where $\gamma_n >0$ is the algorithm step-size, $\lambda_n > 0$ is a relaxation parameter and $a_n \in \ensuremath{{\mathcal H}}$ (resp. $b_n\in \ensuremath{{\mathcal H}}$) represents an error allowed in the computation of the proximity operator (resp. the gradient).
The weak convergence of $(x_n)_{n\in \ensuremath{\mathbb N}}$ to a solution to Problem \ref{pb:minimisation1} is then guaranteed provided that: \begin{assumption}
\label{a:gl} \begin{enumerate} \item \label{a:gli} $0 < \underline{\gamma} \le \overline{\gamma} < 2\beta^{-1}$ where $\underline{\gamma} =\inf_{n\in \ensuremath{\mathbb N}} \gamma_n$ and $\overline{\gamma} = \sup_{n\in \ensuremath{\mathbb N}} \gamma_n$. \item \label{a:glii} $(\forall n \in \ensuremath{\mathbb N})$ $0<\underline{\lambda}\le \lambda_n \le 1$.
\item \label{a:gliii} $\sum_{n\in \ensuremath{\mathbb N}} \|a_n\| < +\infty$ and
$\sum_{n\in \ensuremath{\mathbb N}} \|b_n\| < +\infty$. \end{enumerate} \end{assumption} More details concerning this algorithm can be found in \cite{Combettes_PL_2005_mms_Signal_rbpfbs,Chaux_C_2007_ip_variational_ffbip} and conditions for the strong convergence of the algorithm are also given in \cite{Combettes_PL_2007_siamopt_proximal_tafmoob}. An additional result which will be useful in this paper is the following: \begin{lemma} \label{le:convlinfb} Suppose that Assumptions \ref{a:gl}\ref{a:gli} and \ref{a:glii} as well as the assumptions of Problem \ref{pb:minimisation1} hold. If $f_1$ is a strongly convex function with modulus $\vartheta$, then the forward-backward algorithm in \eqref{eq:fb_algo} with $a_n \equiv b_n \equiv 0$ converges linearly to the unique solution $\widetilde{x}$ to Problem \ref{pb:minimisation1}. More precisely, we have \begin{equation}
(\forall n \in \ensuremath{\mathbb N})\qquad \|x_n -\widetilde{x}\| \le
\Big(1-\frac{\underline{\lambda}\underline{\gamma}\vartheta}{1+\underline{\gamma}\vartheta}\Big)^n \|x_0 -\widetilde{x}\|. \label{eq:linconv} \end{equation} \end{lemma} \begin{proof} Since $\operatorname{Argmin}f_1+f_2\neq \ensuremath{\varnothing}$ and $f_1$ is strongly (thus strictly) convex, there exists a unique minimizer $\widetilde{x}$ of $f_1+f_2$. Then, $\widetilde{x}$ is a fixed point of the forward-backward algorithm in \eqref{eq:fb_algo} when $a_n \equiv b_n \equiv 0$. Thus, we have, for all $n\in \ensuremath{\mathbb N}$, \begin{equation*} x_{n+1} -\widetilde{x} = (1-\lambda_n) (x_n -\widetilde{x}) + \lambda_n \big(\ensuremath{\mathrm{prox}}_{\gamma_n f_1}(x_n -\gamma_n \nabla f_2(x_n)) - \ensuremath{\mathrm{prox}}_{\gamma_n f_1}(\widetilde{x} -\gamma_n \nabla f_2(\widetilde{x}))\big) \end{equation*} which yields \begin{multline*}
\|x_{n+1} -\widetilde{x}\| \le (1-\lambda_n) \|x_n -\widetilde{x}\|
\\+ \lambda_n \|\ensuremath{\mathrm{prox}}_{\gamma_n f_1}(x_n -\gamma_n \nabla f_2(x_n))
- \ensuremath{\mathrm{prox}}_{\gamma_n f_1}(\widetilde{x} -\gamma_n \nabla f_2(\widetilde{x}))\|. \end{multline*} Since $f_1$ has been assumed strongly convex with modulus $\vartheta$, $\gamma_n f_1$ is strongly convex with modulus $\gamma_n \vartheta$ and, according to Assumption \ref{a:gl}\ref{a:gli}, it is also strongly convex with modulus $\underline{\gamma}\vartheta$. We deduce from Proposition \ref{prop:proxquad}\ref{p:proxquadc} that $\ensuremath{\mathrm{prox}}_{\gamma_n f_1}$ is strictly contractive with constant $(1+\underline{\gamma}\vartheta)^{-1}$. Hence, we have \begin{equation*}
\|x_{n+1} -\widetilde{x}\| \le (1-\lambda_n) \|x_n -\widetilde{x}\|
+ \frac{\lambda_n}{1+\underline{\gamma}\vartheta} \|x_n -\gamma_n \nabla f_2(x_n)
- \widetilde{x} +\gamma_n \nabla f_2(\widetilde{x})\|. \end{equation*}
Recall that an operator $R\;:\;\ensuremath{{\mathcal H}} \to \ensuremath{{\mathcal H}}$ is nonexpansive if $(\forall (y,z)\in \ensuremath{{\mathcal H}}^2)$ $\|R(x)-R(y)\| \le \|x-y\|$. An operator $T\;:\;\ensuremath{{\mathcal H}} \to \ensuremath{{\mathcal H}}$ is $\alpha$-averaged with $\alpha \in]0,1[$ if $T = (1-\alpha) \ensuremath{\mathrm{Id}} + \alpha R$ where $R$ is a nonexpansive operator.
Since $f_2$ is a differentiable convex function having a $\beta$-Lipschitz continuous gradient with $\beta > 0$, we deduce from the Baillon-Haddad theorem \cite{Baillon_JP_1977_jm_Quelques_pdoabencm}, that $\nabla f_2 /\beta$ is 1/2-average. As $\gamma_n \in ]0,2/\beta[$ , by using \cite[Lemma 2.3]{Combettes_PL_2004_o_Solving_mivconao}, $\ensuremath{\mathrm{Id}} - \gamma_n \nabla f_2$ is $\frac{\gamma_n \beta}{2}$-averaged and it is therefore nonexpansive (see \cite[Lemma 2.1(ii)]{Combettes_PL_2004_o_Solving_mivconao}).
This entails that \begin{equation*}
\|x_n -\gamma_n \nabla f_2(x_n)
- \widetilde{x} +\gamma_n \nabla f_2(\widetilde{x})\|
\le \|x_n -\widetilde{x}\| \end{equation*} and, consequently, \begin{equation*}
\|x_{n+1} -\widetilde{x}\| \le \Big(1-\frac{\lambda_n\underline{\gamma}\vartheta}{1+\underline{\gamma}\vartheta}\Big) \|x_n -\widetilde{x}\| \le \Big(1-\frac{\underline{\lambda}\underline{\gamma}\vartheta}{1+\underline{\gamma}\vartheta} \Big)\|x_n -\widetilde{x}\| \end{equation*} which results in \eqref{eq:linconv}. \end{proof}
The linear convergence of the forward-backward algorithm was also proved in \cite{Bredies_K_2007_submitted_Iterative_stcl,Chen_GHG_1997_jopt_Convergence_rifbs} under different assumptions.
\subsubsection{Computation of $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}$} \label{sss:picgfb} Let $\ensuremath{\kappa}>0$ and $g$ be a differentiable function with $\beta$-Lipschitz continuous gradient where $\beta \in \ensuremath{\,\left]0,+\infty\right[}$. Let $C$ be a closed convex set such that $C\neq \ensuremath{\varnothing}$.
Then, for every $x \in \ensuremath{{\mathcal H}}$, the determination of $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x$ can be viewed as a minimization problem of the form of Problem \ref{pb:minimisation1}. Indeed, by using the definition of the proximity operator, we have: \begin{equation*}
(\forall x\in \ensuremath{{\mathcal H}}) \qquad \ensuremath{\mathrm{prox}}_{\kappa g +\ensuremath{\mathrm{\iota_C}}}x = \arg \min_{y\in\ensuremath{{\mathcal H}}} \frac{1}{2}\left\|y-x\right\|^{2} + \kappa g(y) + \ensuremath{\mathrm{\iota_C}}(y). \end{equation*}
Now, we can set $f_1 = \frac{1}{2}\left\|. -x\right\|^{2}+\ensuremath{\mathrm{\iota_C}}$ and $f_2 = \kappa g$. The proximity operator of $\gamma_n f_1$ with $\gamma_n \in \ensuremath{\,\left]0,+\infty\right[}$, is the proximity operator of $\frac{\gamma_n}{2} \|\cdot\|^2 - \gamma_n\scal{\cdot}{x}+\ensuremath{\mathrm{\iota_C}}$, which is straightforwardly deduced from Proposition~\ref{prop:proxquad}\ref{p:proxlin} and \ref{p:proxquada}: \begin{align} (\forall y \in \ensuremath{{\mathcal H}}) \qquad \ensuremath{\mathrm{prox}}_{\gamma_n f_1}y = P_C\Big(\frac{y+\gamma_n x}{1+\gamma_n}\Big). \label{eq:defproxf1fw} \end{align} whereas $f_2$ has a $\kappa \beta$-Lipschitz continuous gradient. In this case, by setting $a_n \equiv b_n \equiv 0$ in Algorithm \eqref{eq:fb_algo}, we get \begin{equation} (\forall n\in \ensuremath{\mathbb N}) \qquad x_{n+1} = x_n +
\lambda_n \left(P_C\Big(\frac{x_n - \gamma_n ( \kappa \nabla g(x_n)-x)}{1+\gamma_n}\Big)-x_n\right)
\label{eq:proxdifferentiable} \end{equation} with \begin{equation} 0 < \underline{\gamma} \le \gamma_n \le \overline{\gamma} < 2\kappa^{-1}\beta^{-1}. \label{eq:condgammaic} \end{equation} The obtained algorithm possesses the following properties: \begin{proposition} \label{p:fwbc} Suppose that Condition \eqref{eq:condgammaic} and Assumption \ref{a:gl}\ref{a:glii} hold. Consider the algorithm in \eqref{eq:proxdifferentiable} where $x \in \mathcal{H}$.
Then, \begin{enumerate} \item\label{c1:pfb} we have: \begin{equation} (\forall n \in \ensuremath{\mathbb N}) \qquad
\|x_n - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x\| \le \rho^n \|x_0 - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x\| \label{eq:lindecfb} \end{equation} where \begin{equation} \rho = 1- \frac{\underline{\lambda}\underline{\gamma}}{1+\underline{\gamma}}\,; \label{eq:defrho} \end{equation} \item \label{c2:pfb} by setting $x_0=\ensuremath{\mathrm{prox}}_{\kappa g}x$, we get: \begin{equation} \ensuremath{\mathrm{prox}}_{\kappa g} x \in C \quad \Rightarrow \quad (\forall n \in \ensuremath{\mathbb N})\;\; x_n = \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x. \label{eq:consfwpjstsp} \end{equation} \end{enumerate} \end{proposition} \begin{proof} \noindent\ref{c1:pfb} : As $f_1$ is strongly convex with modulus 1, \eqref{eq:lindecfb} is obtained by invoking Lemma~\ref{le:convlinfb}.\\ \ref{c2:pfb} : If $x_0 = \ensuremath{\mathrm{prox}}_{\kappa g} x \in C$, then \eqref{eq:lindecfb} leads to \begin{equation} (\forall n \in \ensuremath{\mathbb N}) \qquad
\|x_n - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x\| \le \Big(1- \frac{\underline{\lambda}\underline{\gamma}}{1+\underline{\gamma}}\Big)^n
\| \ensuremath{\mathrm{prox}}_{\kappa g} x - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x\| = 0 \end{equation} where Proposition \ref{prop:jstsp}\ref{prop:jstspi} has been used in the last equality. This shows that \eqref{eq:consfwpjstsp} is satisfied. \end{proof}
\begin{remark}\ \label{r:fw} \begin{enumerate} \item Eq. \eqref{eq:lindecfb} shows that $(x_n)_{n\in\ensuremath{\mathbb N}}$ converges linearly to $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x$. Although this equation provides an upper bound, it suggests to choose $\lambda_n$ and $\gamma_n$ as large as possible (i.e. $\lambda_n \equiv 1$ and $\gamma_n$ close to $2\kappa^{-1}\beta^{-1}$) to optimize the convergence rate. This fact was confirmed by our simulations. \item\label{r:fw1step} Proposition \ref{p:fwbc}\ref{c2:pfb} may appear as a desirable property since Proposition \ref{prop:jstsp}\ref{prop:jstspi} states that, when $\ensuremath{\mathrm{prox}}_{\kappa g} x \in C$, $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x$ takes a trivial form. In this case, the convergence is indeed guaranteed in just one iteration by appropriately initializing the algorithm. Note however that $\ensuremath{\mathrm{prox}}_{\kappa g}x$ may not always be simple to compute, depending on the form of $g$. \item An alternative numerical method for the computation of
$\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}x$ would consist of setting
$f_1 = \ensuremath{\mathrm{\iota_C}}$ and $f_2 = \frac{1}{2}\left\|. -x\right\|^{2}+\kappa g$ in the forward-backward algorithm, so yielding \begin{equation*} (\forall n\in \ensuremath{\mathbb N}) \qquad x_{n+1} = x_n + \lambda_n \big(P_C(x_n - \gamma_n ( \kappa \nabla g(x_n)+x_n-x)) -x_n\big) \end{equation*} with $0 < \underline{\gamma} \le \overline{\gamma} < 2(\kappa\beta+1)^{-1}$. It can be noticed that the forward-backward algorithm then reduces to a projected gradient algorithm {\rm \cite[Chap. 3., Sect. 3.3.2]{Bertsekas_DP_1997_book_Parallel_adcnm}\cite{Alber_YI_1998_mp_projected_otpsmfncoiahs}}, when $\lambda_n \equiv 1$. In our experiments, it was however observed that the convergence of this algorithm is slower than that in \eqref{eq:proxdifferentiable}, probably due to the fact that $\ensuremath{\mathrm{prox}}_{\gamma_n f_1}$ is no longer strictly contractive for the second choice of $f_1$. \end{enumerate} \end{remark}
\subsection{Douglas-Rachford approach} \label{subsec:dr} Let us relax the Lipschitz continuity assumption in Problem~\ref{pb:minimisation1} and turn our attention to the optimization problem: \begin{problem}\label{pb:minimisation2} Let $g_1$ and $g_2$ be functions in $\Gamma_0(\ensuremath{{\mathcal H}})$ such that $\operatorname{Argmin} g_1+g_2 \neq \ensuremath{\varnothing}$. Assume that one of the following three conditions is satisfied: \begin{enumerate} \item \label{p:1ii} $\ensuremath{\mathrm{dom}\,} g_2\cap\ensuremath{\mathrm{int}\,} \ensuremath{\mathrm{dom}\,} g_1 \neq\ensuremath{\varnothing}$.\footnote{The interior (resp. relative interior) of a set $S$ of $\ensuremath{{\mathcal H}}$ is designated by $\ensuremath{\mathrm{int}\,} S$ (resp. $\ensuremath{\mathrm{rint}\,} S$).} \item \label{p:1i} $\ensuremath{\mathrm{dom}\,} g_1\cap\ensuremath{\mathrm{int}\,} \ensuremath{\mathrm{dom}\,} g_2 \neq\ensuremath{\varnothing}$. \item \label{p:1iii} $\ensuremath{{\mathcal H}}$ is finite dimensional and $\ensuremath{\mathrm{rint}\,} \ensuremath{\mathrm{dom}\,} g_1 \cap\ensuremath{\mathrm{rint}\,} \ensuremath{\mathrm{dom}\,} g_2 \neq\ensuremath{\varnothing}$. \end{enumerate} \begin{equation*} \text{Find}\quad \min_{z \in \ensuremath{{\mathcal H}}} g_1(z) + g_2(z). \label{eq:douglas-rachford} \end{equation*} \end{problem} In the statement of the above problem, the notation differs from that used in Problem~\ref{pb:minimisation1} to emphasize the difference in the assumptions which have been adopted and facilitate the presentation of the algorithms subsequently presented in Section \ref{sec:algorithm}.\\ The Douglas-Rachford algorithm, proposed in \cite{Lions_PL_1979_jna_Splitting_aftsotno,Eckstein_J_2003_mp_douglas_otdrsmatppaftmmo}, provides an appealing numerical solution to Problem \ref{pb:minimisation2}, as described next. \vspace*{0.5cm} \subsubsection{Algorithm \cite[Eq.(19)]{Combettes_PL_2007_istsp_Douglas_rsatncvsr}} Set $z_0 \in \ensuremath{{\mathcal H}}$ and compute, for every $m\in \ensuremath{\mathbb N}$, \begin{equation} \begin{cases} z_{m+\frac{1}{2}} = \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2} z_m + b_m\\ z_{m+1} = z_m + \ensuremath{\tau}_m \big(\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(2z_{m+\frac{1}{2}} - z_m) + a_m - z_{m+\frac{1}{2}}\big)\\ \end{cases} \label{eq:algDR} \end{equation} where $\kappa >0$, $\suite[m]{\ensuremath{\tau}_m}$ is a sequence of positive reals, and $\suite[m]{a_m}$ (resp. $\suite[m]{b_m}$) is a sequence of
errors in $\ensuremath{{\mathcal H}}$ allowed in the computation of the proximity operator of $\ensuremath{\kappa} g_1$ (resp. $\ensuremath{\kappa} g_2$).\\ Then, $(z_m)_{m\in \ensuremath{\mathbb N}}$ converges weakly to $z \in \ensuremath{{\mathcal H}}$ \cite[Corollary 5.2]{Combettes_PL_2004_o_Solving_mivconao} such that $\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}z$ is a solution to Problem~\ref{pb:minimisation2}, provided that:
\begin{assumption}
\label{a:gl2} \begin{enumerate}
\item \label{a:gl21} $(\forall m \in \ensuremath{\mathbb N})$ $\ensuremath{\tau}_m\in ]0,2[$ and $\sum_{m\in\ensuremath{\mathbb N}} \ensuremath{\tau}_m(2-\ensuremath{\tau}_m) = +\infty$.
\item \label{a:gl22} $\sum_{m\in\ensuremath{\mathbb N}} \ensuremath{\tau}_m(\norm{a_m}+\norm{b_m})<+\infty$. \end{enumerate} \end{assumption}
\vspace*{0.3cm} An alternate convergence result is the following: \begin{proposition}\label{prop:constrongDR} Suppose that the assumptions of Problem \ref{pb:minimisation2} hold. If $g_2$ is a strongly convex function, then the Douglas-Rachford algorithm in \eqref{eq:algDR} with $\inf_{m \in \ensuremath{\mathbb N}}\tau_m > 0$, $\sup_{m\in\ensuremath{\mathbb N}} \tau_m \le 2$ and $a_m \equiv b_m \equiv 0$ is such that $(z_{m+1/2})_{m\in \ensuremath{\mathbb N}}$ converges strongly to the unique solution to Problem \ref{pb:minimisation2}. \end{proposition} \begin{proof} Let the $\ensuremath{\mathrm{rprox}}$ operator be defined, for every $f\in \Gamma_0(\ensuremath{{\mathcal H}})$, by \begin{equation} \ensuremath{\mathrm{rprox}}_f = 2\ensuremath{\mathrm{prox}}_f - \ensuremath{\mathrm{Id}}. \end{equation} Let us rewrite the Douglas-Rachford iteration in \eqref{eq:algDR} with $a_m \equiv b_m \equiv 0$ as $z_{m+1} = S_m z_m$, where \begin{equation} S_m = \ensuremath{\tau}_m \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2})+ \ensuremath{\mathrm{Id}} - \ensuremath{\tau}_m \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}. \label{eq:S} \end{equation} For all $(\ensuremath{y},\ensuremath{y'})\in\ensuremath{{\mathcal H}}^2$, we have \begin{equation} \begin{split} \normCar{&S_m\ensuremath{y}-S_m\ensuremath{y'}} = \ensuremath{\tau}_m^{2}\normCar{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y}) -\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'})}\\ &+2 \ensuremath{\tau}_m \prodScal{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y})-\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'})}{\ensuremath{y}-\ensuremath{\tau}_m\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} -\ensuremath{y'} + \ensuremath{\tau}_m \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}}\\ & + \normCar{\ensuremath{y}-\ensuremath{\tau}_m\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} -\ensuremath{y'} + \ensuremath{\tau}_m \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}}. \end{split} \label{strgCV1} \end{equation}
Since $\ensuremath{\kappa} g_1 \in \Gamma_0(\ensuremath{{\mathcal H}})$, $\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}$ is firmly nonexpansive \cite[Lemma 2.4]{Combettes_PL_2005_mms_Signal_rbpfbs} and, the expression in \eqref{strgCV1} can be upper bounded as follows \begin{equation*} \begin{split} &\normCar{S_m\ensuremath{y}-S_m\ensuremath{y'}}\\ & \leq \ensuremath{\tau}_m^{2}\prodScal{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y}) -\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'})}{\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} - \ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}}\\
& { +2 \ensuremath{\tau}_m \prodScal{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y})-\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'})}{\ensuremath{y}-\ensuremath{\tau}_m\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} -\ensuremath{y'} + \ensuremath{\tau}_m \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}}}\\ & + \normCar{\ensuremath{y}-\ensuremath{\tau}_m\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} -\ensuremath{y'} + \ensuremath{\tau}_m \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}} \end{split} \label{strgCV2} \end{equation*}
which yields after simplifications: \begin{equation*} \begin{split} \normCar{S_m\ensuremath{y}-S_m\ensuremath{y'}} \leq \ensuremath{\tau}_m(2 - \ensuremath{\tau}_m)&\prodScal{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y}) -\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}(\ensuremath{\mathrm{rprox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'})}{\ensuremath{y}-\ensuremath{y'}}\\ & + \normCar{\ensuremath{y}-\ensuremath{\tau}_m\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} -\ensuremath{y'} + \ensuremath{\tau}_m \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}}. \end{split}\label{strgCV3} \end{equation*} Using the definition of the operator $S_m$ in \eqref{eq:S}, we thus obtain, after some simple calculations, \begin{multline} \normCar{S_m\ensuremath{y}-S_m\ensuremath{y'}} \leq (2-\ensuremath{\tau}_m)\prodScal{S_m\ensuremath{y}-S_m\ensuremath{y'}}{\ensuremath{y}-\ensuremath{y'}} + (\ensuremath{\tau}_m-1)\normCar{\ensuremath{y}-\ensuremath{y'}} \\ - \ensuremath{\tau}_m ^{2}\big(\prodScal{ \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} - \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'} }{\ensuremath{y}-\ensuremath{y'}}- \normCar{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} - \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}}\big). \label{eq:lasteqy1y2str} \end{multline} Let $\theta$ be the modulus of the strongly convex function $g_2$. Then $\ensuremath{\kappa} g_2$ is strongly convex with modulus $\ensuremath{\kappa} \theta$ and Proposition \ref{prop:proxquad}\ref{p:proxquadb} states that the following inequality holds: \begin{equation*}
\prodScal{ \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} - \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'} }{\ensuremath{y}-\ensuremath{y'}} \geq (\ensuremath{\kappa} \theta +1)\normCar{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} - \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}}, \end{equation*} which combined with \eqref{eq:lasteqy1y2str} leads to \begin{equation} \begin{split} \normCar{S_m\ensuremath{y}-S_m\ensuremath{y'}} + &\ensuremath{\kappa} \theta\ensuremath{\tau}_m ^{2}\normCar{\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y} - \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}\ensuremath{y'}} \\ &\leq (2-\ensuremath{\tau}_m)\prodScal{S_m\ensuremath{y}-S_m\ensuremath{y'}}{\ensuremath{y}-\ensuremath{y'}} + (\ensuremath{\tau}_m-1)\normCar{\ensuremath{y}-\ensuremath{y'}}. \end{split} \label{strgCV8} \end{equation}
Now, let $\ensuremath{\tilde{z}}$ be the unique minimizer of $g_1+g_2$. Hence, $\ensuremath{\tilde{z}} = \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2} z$ where $z$ is a fixed point of $S_m$. Consequently, by setting $\ensuremath{y} = z_m$ and $\ensuremath{y'} = z$ in \eqref{strgCV8}, we deduce that \begin{multline} \normCar{z_{m+1}- z} + \ensuremath{\kappa} \theta\ensuremath{\tau}_m ^{2}\normCar{z_{m+\frac{1}{2}} - \ensuremath{\tilde{z}}} \\
\leq (2-\ensuremath{\tau}_m)\prodScal{z_{m+1}- z}{z_m-z} + (\ensuremath{\tau}_m-1)\normCar{z_m-z}. \label{eq:strgCV8bis} \end{multline} Using the fact that \[ 2\prodScal{z_{m+1}-z}{z_m-z} = \normCar{z_{m+1}-z} + \normCar{z_m-z} -\normCar{z_{m+1}-z_m} \] \eqref{eq:strgCV8bis} can be rewritten as \begin{equation} \begin{split} \ensuremath{\tau}_m \normCar{z_{m+1}- z} &+(2-\ensuremath{\tau}_m)\normCar{z_{m+1} - z_m} + 2\ensuremath{\kappa} \theta\ensuremath{\tau}_m ^{2}\normCar{z_{m+\frac{1}{2}} - \ensuremath{\tilde{z}}} \leq \ensuremath{\tau}_m \normCar{z_m-z}. \end{split} \label{strgCV12} \end{equation}
Considering Assumption \ref{a:gl2}, $(2-\ensuremath{\tau}_m)$ $\normCar{z_{m+1} - z_m}$ is nonnegative and the left-hand side term of inequality \eqref{strgCV12} can be lower bounded, so yielding \begin{equation*} \begin{split} \ensuremath{\tau}_m \normCar{z_{m+1}- z} + 2\ensuremath{\kappa} \theta\ensuremath{\tau}_m ^{2}\normCar{z_{m+\frac{1}{2}} - \ensuremath{\tilde{z}}} \leq \ensuremath{\tau}_m \normCar{z_m-z}. \end{split} \label{strgCV13} \end{equation*}
Finally, by using the assumption that $\underline{\ensuremath{\tau}}=\inf_{m\in \ensuremath{\mathbb N}} \ensuremath{\tau}_m > 0$, we obtain \begin{equation} \begin{split} \normCar{z_{m+1}- z} + 2\ensuremath{\kappa} \theta\underline{\ensuremath{\tau}} \normCar{z_{m+\frac{1}{2}} - \ensuremath{\tilde{z}}} \leq \normCar{z_m-z}. \end{split} \label{strgCV14} \end{equation} This entails that $\normCar{z_{m+1}-z}\leq \normCar{z_m - z}$
and, the sequence $\suite{\|z_m-z\|}$ being decreasing, there exists $c \in \ensuremath{\,\left]0,+\infty\right[}$ such that $\lim_{m\rightarrow\ensuremath{+\infty}}\|z_m-z\|=c$. In turn, from \eqref{strgCV14}, we conclude that $\lim_{m\rightarrow\ensuremath{+\infty}}z_{m+\frac{1}{2}} = \ensuremath{\tilde{z}}$, which shows the strong convergence of $(z_{m+1/2})_{m\in\ensuremath{\mathbb N}}$ to the unique minimizer of $g_1+g_2$. \end{proof}
It can be noticed that, although the convergence of the Douglas-Rachford algorithm generally requires that $\tau_m < 2$, the strong convergence is obtained under the above assumptions, when $\tau_m = 2$. The limit case of the Douglas-Rachford corresponding to $\tau_m \equiv 2$ is known as the Peaceman-Rachford algorithm \cite{Peaceman_DW_1955_siam_numerical_tnsopaede,Combettes_PL_2004_o_Solving_mivconao}.
\subsubsection{Computation of $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+ \gamma\lowercase{f}}$} \label{sec:icfDR} Let $C$ be a nonempty closed convex set of $\ensuremath{{\mathcal H}}$. The Douglas-Rachford algorithm can be used to compute $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+ \gamma f}$ where $f \in \Gamma_0(\ensuremath{{\mathcal H}})$ and $\gamma$ is a positive constant, using again the definition of the proximity operator: \begin{equation}
(\forall x \in \ensuremath{{\mathcal H}})\quad \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \gamma f}x = \arg \min_{y\in \ensuremath{{\mathcal H}}}\frac{1}{2}\left\|y-x\right\|^{2} + \ensuremath{\mathrm{\iota_C}}(y)+ \gamma f(y). \label{eq:prox_definition} \end{equation} The above minimization problem appears as a specialization of Problem \ref{pb:minimisation2} by setting
$g_1 = \gamma f$ and $g_2 = \frac{1}{2}\left\|\cdot-x\right\|^{2} + \ensuremath{\mathrm{\iota_C}}$, provided that one of the following three conditions holds: \begin{assumption}
\label{a:ddd} \begin{enumerate} \item \label{a:ddd1}$ C \cap \ensuremath{\mathrm{int}\,} \ensuremath{\mathrm{dom}\,} f \neq \ensuremath{\varnothing}$. \item \label{a:ddd2}$ \ensuremath{\mathrm{dom}\,} f \cap \ensuremath{\mathrm{int}\,} C \neq \ensuremath{\varnothing}$. \item \label{a:ddd3}$\ensuremath{{\mathcal H}}$ is finite dimensional and $\ensuremath{\mathrm{rint}\,} C \cap \ensuremath{\mathrm{rint}\,}\ensuremath{\mathrm{dom}\,} f \neq \ensuremath{\varnothing}$. \end{enumerate} \end{assumption} Subsequently, we propose to use the Douglas-Rachford algorithm in \eqref{eq:algDR} with $a_m \equiv b_m \equiv 0$, to compute the desired proximity operator. Note that both $\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1}$ and $\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}$ with $\ensuremath{\kappa} > 0$, have to be calculated to apply this algorithm. In our case, we have \begin{eqnarray*} \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_1} = \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} \gamma f} \label{eq:pf1} \end{eqnarray*} and, similarly to \eqref{eq:defproxf1fw}, \begin{eqnarray*} (\forall z \in \ensuremath{{\mathcal H}})\qquad \ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} g_2}z = P_C \Big(\frac{z+\ensuremath{\kappa} x}{1+\ensuremath{\kappa}}\Big). \label{eq:pf2} \end{eqnarray*} The resulting Douglas-Rachford iterations read: for every $m\in \ensuremath{\mathbb N}$, \begin{equation} \begin{cases} \displaystyle z_{m+\frac{1}{2}} = P_C \Big(\frac{z_m+\ensuremath{\kappa} x}{1+\ensuremath{\kappa}}\Big)\\ z_{m+1} = z_m + \ensuremath{\tau}_m \big(\ensuremath{\mathrm{prox}}_{\ensuremath{\kappa} \gamma f}(2z_{m+\frac{1}{2}} - z_m) - z_{m+\frac{1}{2}}\big).\label{eq:algDR2} \end{cases} \end{equation} This algorithm enjoys the following properties: \begin{proposition} \label{prop:DRn} Suppose that one of Assumptions \ref{a:ddd}\ref{a:ddd1}, \ref{a:ddd}\ref{a:ddd2} or \ref{a:ddd}\ref{a:ddd3} holds. Consider the algorithm in \eqref{eq:algDR2} where $x\in \ensuremath{{\mathcal H}}$, $\inf_{m \in \ensuremath{\mathbb N}}\tau_m>0$ and $\sup_{m \in \ensuremath{\mathbb N}}\tau_m\le 2$. Then, \begin{enumerate} \item \label{prop:DRni}
$(z_{m+\frac{1}{2}})_{m\in \ensuremath{\mathbb N}}$ converges strongly to $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \gamma f}x$; \item \label{prop:DRni2}by setting $\kappa = 1$ and $z_0 = 2\ensuremath{\mathrm{prox}}_{\gamma f} x-x$, we get: \begin{equation} \ensuremath{\mathrm{prox}}_{\gamma f} x \in C \quad \Rightarrow\quad (\forall m \in \ensuremath{\mathbb N})\;\;z_{m+\frac{1}{2}} = \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} +\gamma f}x. \label{eq:1stepDR} \end{equation} \end{enumerate} \end{proposition} \begin{proof} \noindent\ref{prop:DRni}: As $g_2$ is strongly convex with modulus 1, \ref{prop:DRni} holds by invoking Proposition \ref{prop:constrongDR}.\\ \ref{prop:DRni2}: Set $\ensuremath{\kappa}=1$, $z_0 = 2\ensuremath{\mathrm{prox}}_{\gamma f}x-x$ with $\ensuremath{\mathrm{prox}}_{\gamma f}x\in C$. By considering the first iteration of the Douglas-Rachford algorithm ($m=0$), we have $z_{\frac{1}{2}} = \ensuremath{\mathrm{prox}}_{\gamma f}x$ and $z_1 = z_0$. So, by induction, $(\forall m \in \ensuremath{\mathbb N})$ $ z_{m+\frac{1}{2}} = \ensuremath{\mathrm{prox}}_{\gamma f}x$, which is also equal to $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \gamma f}x$ according to Proposition~\ref{prop:jstsp}\ref{prop:jstspi}. \end{proof} \begin{remark}\ \begin{enumerate} \item As already observed in Remark \ref{r:fw}\ref{r:fw1step}, \eqref{eq:1stepDR} is a desirable property. It shows that the proposed algorithm converges in one iteration when $\ensuremath{\mathrm{prox}}_{\gamma f} x \in C$, which appears quite consistent in the light of Proposition \ref{prop:jstsp}\ref{prop:jstspi}. \item Other choices can be envisaged for $g_1$ and $g_2$, namely \begin{enumerate}
\item $g_1 = \frac{1}{2}\left\|\cdot-x\right\|^{2} + \ensuremath{\mathrm{\iota_C}}$ and $g_2 = \gamma f$
\item $g_1 =\frac{1}{2}\left\|\cdot-x\right\|^{2}+\gamma f$ and $g_2=\ensuremath{\mathrm{\iota_C}}$
\item $g_1 =\ensuremath{\mathrm{\iota_C}}$ and $g_2=\frac{1}{2}\left\|\cdot-x\right\|^{2}+\gamma f$. \end{enumerate} Nevertheless, the strong convergence of $(z_{m+1/2})_{m\in\ensuremath{\mathbb N}}$ in virtue of Proposition~\ref{prop:constrongDR} is only guaranteed in the third case, whereas Property~\eqref{eq:1stepDR} holds only in the first case (when $\kappa = 1$ and $z_0 = x$). The second case was investigated in \cite{Dupe_FX_2008_ip_proximal_ifdpniusr}, where the good numerical behaviour of the resulting algorithm was demonstrated. \end{enumerate} \end{remark}
\subsection{Discussion} Both Algorithms \eqref{eq:proxdifferentiable} and \eqref{eq:algDR2} allow us to determine the proximity operator of the sum of the indicator function of a closed convex set and a function in $\Gamma_0(\ensuremath{{\mathcal H}})$. The main difference between the two methods is that, in the former one, the convex function needs to be differentiable with a Lipschitz-continuous gradient, whereas the latter requires that the proximity operator of the convex function is easy to compute. In addition, the forward-backward algorithm converges linearly, while we were only able to prove the strong convergence of the Douglas-Rachford algorithm. As we have shown also, the two algorithms are consistent with Proposition~\ref{prop:jstsp}\ref{prop:jstspi}.
\section{Proposed algorithms to minimize $f+g+\ensuremath{\mathrm{\iota_C}}$} \label{sec:algorithm} We have presented two approaches to minimize the sum of two functions in $\Gamma_0(\ensuremath{{\mathcal H}})$. We have also seen that these methods can be employed to compute the proximity operator of the sum of the indicator function of a closed convex set $C$ and a function in $\Gamma_0(\ensuremath{{\mathcal H}})$.
We now come back to the more general form of Problem \ref{pb:minimisation}, for which we will propose two solutions. Both of them correspond to a combination of the forward-backward algorithm and the Douglas-Rachford one.
\subsection{First method: insertion of a forward-backward step in the Dou\-glas-Rachford algorithm} \label{sec:DR(FB)} We propose to apply the Douglas-Rachford algorithm as described in Section \ref{subsec:dr}, when $g_1 = f$ and $g_2= \ensuremath{\mathrm{\iota_C}}+ g$. If we refer to \eqref{eq:algDR}, we need to determine $\ensuremath{\mathrm{prox}}_{\kappa g_1} = \ensuremath{\mathrm{prox}}_{\kappa f}$ and $\ensuremath{\mathrm{prox}}_{\kappa g_2} = \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}$, where $\kappa > 0$. The main difficulty lies in the computation of the second proximity operator. As proposed in Section \ref{sss:picgfb}, we can use a forward-backward algorithm to achieve this goal. The resulting algorithm is:
\begin{algorithm}
\label{algo:main1} \begin{itemize} \item[\Pisymbol{pzd}{192}] Set $\underline{\gamma}\in ]0,2\ensuremath{\kappa}^{-1} \beta^{-1}[$, $\underline{\lambda} \in ]0,1]$ and $\kappa \in \ensuremath{\,\left]0,+\infty\right[}$. Choose $(\tau_m)_{m\in \ensuremath{\mathbb N}}$ satisfying Assumption \ref{a:gl2}\ref{a:gl21}. \item[\Pisymbol{pzd}{193}] Set $m=0$, $z_0 = z_{-1/2} \in C$. \item[\Pisymbol{pzd}{194}] Set $x_{m,0} = z_{m-1/2}$. \item[\Pisymbol{pzd}{195}] For $n = 0,\ldots,N_m-1$ \begin{itemize} \item[a)] Choose $\gamma_{m,n}\in [\underline{\gamma},2\ensuremath{\kappa}^{-1} \beta^{-1}[$ and $\lambda_{m,n} \in [\underline{\lambda},1]$. \item[b)] Compute \[
x_{m,n+1} = x_{m,n} + \lambda_{m,n} \left( P_C \Big( \frac{x_{m,n} - \gamma_{m,n} (\ensuremath{\kappa} \nabla g(x_{m,n})-z_m)}{1+\gamma_{m,n}}\Big) - x_{m,n}\right). \] \end{itemize} \item[\Pisymbol{pzd}{196}] Set $z_{m+\frac{1}{2}} = x_{m,N_m}$. \item[\Pisymbol{pzd}{197}] Set $z_{m+1} = z_m + \ensuremath{\tau}_m \big(\ensuremath{\mathrm{prox}}_{\kappa f}(2z_{m +\frac{1}{2}}-z_m) - z_{m+\frac{1}{2}} \big)$. \item[\Pisymbol{pzd}{198}] Increment $m$ $(m \leftarrow m+1)$ and goto \Pisymbol{pzd}{194}. \end{itemize} \end{algorithm}
Step \Pisymbol{pzd}{192} allows us to set the algorithm parameters and Step \Pisymbol{pzd}{193} corresponds to the initialization of the algorithm. At iteration $m\ge 0$, Step \Pisymbol{pzd}{195} consists of $N_m \ge 1$ iterations of the forward-backward part of the algorithm, where possibly varying step-sizes $(\gamma_{m,n})_n$ and relaxation parameters $(\lambda_{m,n})_n$ are used. Finally Steps \Pisymbol{pzd}{196} and \Pisymbol{pzd}{197} correspond to the Douglas-Rachford iteration. Here, the error term $a_m$ in the computation of $\ensuremath{\mathrm{prox}}_{\kappa f}$ is assumed to be equal to zero but, due to the finite number of iterations $N_m$ performed in Step \Pisymbol{pzd}{195}, an error $b_m = z_{m+1/2} - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}z_m$ may be introduced in Step \Pisymbol{pzd}{195}.
It can be noticed that the forward-backward algorithm has not been initialized in Step \Pisymbol{pzd}{194} as suggested by Proposition \ref{p:fwbc}\ref{c2:pfb}. Indeed, as already mentioned, the computation of $\ensuremath{\mathrm{prox}}_{\kappa g} z_m$ would be generally costly. Furthermore, the initialization in Step \Pisymbol{pzd}{194} is useful to guarantee the following properties: \begin{proposition}\label{p:convfwDR} Suppose that Problem \ref{pb:minimisation} has a solution and that one of Assumptions \ref{a:ddd}\ref{a:ddd1}, \ref{a:ddd}\ref{a:ddd2} or \ref{a:ddd}\ref{a:ddd3} holds. \begin{enumerate} \item\label{p:iconvfwDR} Let $\xi > 0$ and $\rho$ be given by \eqref{eq:defrho}. If $\inf g(C) > -\infty$ and, for every $m\in \ensuremath{\mathbb N}$, the positive integer $N_m$ is chosen such that \begin{subequations} \begin{align} &\rho^{N_m} \sqrt{2\kappa} \big(g(z_0)-\inf g(C)\big)^{1/2} \le \xi &\mbox{if $m=0$} \label{eq:condconvfwDR0}\\
&\rho^{N_m-1} \big(1+\xi^{-1}\rho^{1-m} \|z_m -z_{m-1}\|\big) \le 1 &\mbox{if $m > 0$} \label{eq:condconvfwDRm} \end{align} \end{subequations} then, $(z_m)_{m\in \ensuremath{\mathbb N}}$ converges weakly to $z \in \ensuremath{{\mathcal H}}$ such that $\ensuremath{\mathrm{prox}}_{\iota_C+\ensuremath{\kappa} g}z$ is solution to Problem \ref{pb:minimisation}. \item \label{p:iiconvfwDR} For every $m\in \ensuremath{\mathbb N}$, $(x_{m,n})_{0\le n \le N_m}$ (and thus, $z_{m+1/2}$) lies in $C$. \end{enumerate} \end{proposition} \begin{proof} \noindent\ref{p:iconvfwDR}: According to Proposition \ref{p:fwbc}\ref{c1:pfb}, for every $m\in \ensuremath{\mathbb N}$, \begin{equation*} (\forall n \in \{0,\ldots,N_m\})\qquad
\|x_{m,n}- \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_m \|
\le \rho^n \|x_{m,0}- \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_m \| \end{equation*} and, consequently \begin{equation}
\|b_m \| = \|z_{m+1/2} -\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_m\|
\le \rho^{N_m} \|z_{m-1/2} -\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_m\|. \label{eq:upboundbm} \end{equation} Let us next show by induction that Conditions \eqref{eq:condconvfwDR0} and \eqref{eq:condconvfwDRm} allow us to guarantee that \begin{equation}
\|b_m \| \le \rho^m \xi. \label{eq:geobm} \end{equation} \begin{itemize} \item If $m = 0$, we deduce from \eqref{eq:upboundbm} that \begin{equation}
\|b_0 \| \le \rho^{N_0} \|z_0 -\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_0\|. \label{eq:upboundb0} \end{equation} From the definition of the proximity operator, we have \begin{align*} (\forall x \in C)\qquad
&\frac{1}{2} \|z_0-x\|^2 + \kappa\, g(x)\nonumber\\
&\ge \frac{1}{2} \| z_0-\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_0\|^2 + \kappa \,g\big(\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_0\big)\nonumber\\
& \ge \frac{1}{2} \| z_0-\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_0 \|^2 + \kappa \,\inf g(C) \end{align*} and, since $z_0 \in C$, \begin{equation*}
\| z_0 - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_0 \|^2 \le 2\kappa \big(g(z_0)-\inf g(C)\big). \end{equation*} By combining the latter inequality with \eqref{eq:upboundb0}
and \eqref{eq:condconvfwDR0}, we conclude that $\|b_0\| \le \xi$. \item Now, let us show that \eqref{eq:geobm} holds for $m>0$, by assuming that
$\|b_{m-1}\| \le \rho^{m-1} \xi$. Using \eqref{eq:upboundbm}, we have \begin{align*}
\|b_m\|
& \le \rho^{N_m} \big(\|z_{m-1/2} -\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_{m-1}
+\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_{m-1}-\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_m\|\big)\nonumber\\
& \le \rho^{N_m} \big(\|b_{m-1}\| + \|\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_{m-1}-\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}} + \kappa g} z_m\|\big) \nonumber\\
& \le \rho^{N_m} \big(\|b_{m-1}\| + \|z_{m-1}-z_{m}\|\big) \end{align*} where the nonexpansivity of $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\kappa g}$ has been used in the last inequality. From the induction assumption, we deduce that \begin{equation*}
\|b_m\| \le \rho^{N_m} (\rho^{m-1} \xi + \|z_{m-1}-z_{m}\|\big) \end{equation*} which, according to \eqref{eq:condconvfwDRm}, leads to \eqref{eq:geobm}. \end{itemize} Then, \eqref{eq:geobm} allows us to claim that Assumption \eqref{a:gl2}\ref{a:gl22} is satisfied since \begin{equation*}
\sum_{m\in \ensuremath{\mathbb N}} \tau_m (\|a_m\|+ \|b_m\|) \le 2 \xi (1-\rho)^{-1}. \end{equation*} By further noticing that Assumption \ref{a:ddd} is equivalent to \ref{a:ddd1} $( \ensuremath{\mathrm{dom}\,}(\ensuremath{\mathrm{\iota_C}}+g) \cap \ensuremath{\mathrm{int}\,} \ensuremath{\mathrm{dom}\,} f \neq \ensuremath{\varnothing})$, \ref{a:ddd2} $\left(\ensuremath{\mathrm{dom}\,} f \cap \ensuremath{\mathrm{int}\,} \ensuremath{\mathrm{dom}\,}(\ensuremath{\mathrm{\iota_C}}+g) \neq \ensuremath{\varnothing}\right)$ or, \ref{a:ddd3} $\ensuremath{{\mathcal H}}$ is finite dimensional and
$(\ensuremath{\mathrm{rint}\,} \ensuremath{\mathrm{dom}\,} f \cap \ensuremath{\mathrm{rint}\,}\ensuremath{\mathrm{dom}\,}(\ensuremath{\mathrm{\iota_C}}+g) \neq \ensuremath{\varnothing})$, the conditions for the weak convergence of the Douglas-Rachford algorithm are therefore fulfilled.\\ \ref{p:iiconvfwDR}: The property can be proved by induction by noticing that $x_{0,0} = z_{-1/2} \in C$ and that $x_{m,n+1}$ is a convex combination of $x_{m,n}$ and the projection onto $C$ of an element of $\ensuremath{{\mathcal H}}$.\end{proof}
Eqs. \eqref{eq:condconvfwDR0} and \eqref{eq:condconvfwDRm} constitute more a theoretical guaranty for the convergence of the proposed algorithm than a practical guideline for the choice of $N_m$. In our numerical experiments, these conditions were indeed observed to provide overpessimistic values of the number of forward-backward iterations to be applied in Step \Pisymbol{pzd}{195}.
As a consequence of Proposition \ref{p:convfwDR}\ref{p:iiconvfwDR}, in Step \Pisymbol{pzd}{195}b), the gradient of $g$ is only evaluated on $C$. This means that the assumption of Lipschitz-continuity on the gradient of $g$ is only required on $C$ and therefore, the algorithm can be applied to the following more general setting: \begin{problem} \label{prob:minimisationgen} Let $C$ be a nonempty closed convex subset of $\ensuremath{{\mathcal H}}$. Let $f$ and $g$ be in $\Gamma_{0}(\ensuremath{{\mathcal H}})$, where $g$ is differentiable on $C$ with a $\beta$-Lipschitz continuous gradient for some $\beta\in\ensuremath{\,\left]0,+\infty\right[}$.\footnote{That is there exists an open set containing $C$ on which $g$ is differentiable with a $\beta$-Lipschitz continuous gradient.} \begin{equation*} \text{Find}\qquad \min_{x\in C} f(x)+g(x). \end{equation*} \end{problem} Note that, in the latter problem, the function $g$ does need to be finite.
\subsection{Second method: insertion of a Douglas-Rachford step in the for\-ward-backward algorithm} \label{sec:FB(DR)} For this method, a different association between the functions involved in Problem \ref{pb:minimisation} is considered by setting $f_1 = \ensuremath{\mathrm{\iota_C}}+ f$ and $f_2= g$. Since $f_2$ has then a $\beta$-Lipschitz continuous gradient, we can apply the forward-backward algorithm presented in Section \ref{se:fwalgo}. This requires however to compute $\ensuremath{\mathrm{prox}}_{\gamma_n f_1} = \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\gamma_n f}$, which can be performed with Douglas-Rachford iterations.
Let us summarize the complete form of the second algorithm we propose to solve Problem \ref{pb:minimisation}. \begin{algorithm}
\label{algo:main} \begin{itemize} \item[\Pisymbol{pzd}{192}] Choose sequences $(\gamma_n)_{n\in \ensuremath{\mathbb N}}$ and $(\lambda_n)_{n\in\ensuremath{\mathbb N}}$ satisfying Assumptions \ref{a:gl}\ref{a:gli} and \ref{a:glii}. Set $\underline{\tau} \in ]0,2]$. \item[\Pisymbol{pzd}{193}] Set $n=0$, $x_0 \in C$. \item[\Pisymbol{pzd}{194}] Set $x_n' = x_n - \gamma_n \nabla g(x_n)$. \item[\Pisymbol{pzd}{195}] Set $z_{n,0} = 2 \ensuremath{\mathrm{prox}}_{\gamma_n f} x_n'-x_n'$. \item[\Pisymbol{pzd}{196}] For $m = 0,\ldots,M_n-1$ \begin{itemize} \item[a)] Compute $\displaystyle z_{n,m+\frac{1}{2}} = P_C\Big(\frac{z_{n,m} +x_n'}{2}\Big)$. \item[b)] Choose $\tau_{n,m} \in [\underline{\tau},2]$. \item[c)] Compute $z_{n,m+1} = z_{n,m} +\tau_{n,m}\big(\ensuremath{\mathrm{prox}}_{\gamma_n f}(2z_{n,m+\frac{1}{2}} - z_{n,m}) -z_{n,m+\frac{1}{2}}\big)$. \item[d)] If $z_{n,m+1} = z_{n,m}$, then goto \Pisymbol{pzd}{197}. \end{itemize} \item[\Pisymbol{pzd}{197}] Set $x_{n+1} = x_n + \lambda_n \big(z_{n,m+\frac{1}{2}}-x_n\big)$. \item[\Pisymbol{pzd}{198}] Increment $n$ $(n\leftarrow n+1)$ and goto \Pisymbol{pzd}{194}. \end{itemize} \end{algorithm}
We see that Step \Pisymbol{pzd}{196} consists of at most $M_n \ge 1$ iterations of the Douglas-Rachford algorithm described in Section \ref{sec:icfDR}, which is initialized in accordance with Proposition \ref{prop:DRn}\ref{prop:DRni2}. Steps \Pisymbol{pzd}{194} and \Pisymbol{pzd}{197} correspond to a forward-backward iteration. Let $m_n < M_n$ be the iteration number where the Douglas-Rachford algorithm stops. The error terms involved in Step \Pisymbol{pzd}{197} are $a_n = z_{n,m_n+\frac{1}{2}}-\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\gamma_n f} x_n$ and $b_n=0$. The properties of the algorithm are then the following: \begin{proposition}\label{p:convDRfw} Suppose that Problem \ref{pb:minimisation} has a solution and one of the Assumptions \ref{a:ddd}\ref{a:ddd1}, \ref{a:ddd}\ref{a:ddd2} or \ref{a:ddd}\ref{a:ddd3} holds. \begin{enumerate} \item \label{p:convDRfwi} There exists a sequence of positive integers $(\overline{M}_n)_{n\in \ensuremath{\mathbb N}}$ such that, if $(\forall n \in \ensuremath{\mathbb N})$ $M_n \ge \overline{M}_n$ then, $(x_n)_{n\in \ensuremath{\mathbb N}}$ converges weakly to a solution to Problem \ref{pb:minimisation}. \item \label{p:convDRfwii} The sequence $(x_n)_{n\in \ensuremath{\mathbb N}}$ lies in $C$. \end{enumerate} \end{proposition} \begin{proof} \noindent\ref{p:convDRfwi}: Set $\rho \in ]0,1[$. Let $n\in \ensuremath{\mathbb N}$ and $(z_{n,m})_{m\in \ensuremath{\mathbb N}}$ be defined by iterating Steps~\Pisymbol{pzd}{196}a), b) and c). By invoking Proposition \ref{prop:DRn}\ref{prop:DRni}, we know that $(z_{n,m+\frac{1}{2}})_{m\in \ensuremath{\mathbb N}}$ converges strongly to $\ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\gamma_n f} x_n'$. This implies that there exists $\overline{M}_n\ge 1$ such that \begin{equation*} (\forall m \in \ensuremath{\mathbb N})\qquad m \ge \overline{M}_n -1 \quad\Rightarrow\quad
\|z_{n,m+\frac{1}{2}} - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\gamma_n f} x_n'\| \le \rho^n. \end{equation*} If $M_n \ge \overline{M}_n$, we deduce that \begin{equation*}
\|a_n\|=
\|z_{n,m_n+\frac{1}{2}} - \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\gamma_n f} x_n'\| \le \rho^n \end{equation*} since either $m_n = M_n-1$ or the algorithm stops in Step \Pisymbol{pzd}{196}d) (in which case $z_{n,m_n}$ is a fixed point of the recursion in Step \Pisymbol{pzd}{196}c) and
$z_{n,m_n+\frac{1}{2}} = \ensuremath{\mathrm{prox}}_{\ensuremath{\mathrm{\iota_C}}+\gamma_n f} x_n'$). We therefore have $\sum_{n\in \ensuremath{\mathbb N}} \|a_n\| < \ensuremath{+\infty}$ and the conditions for the weak convergence of the forward-backward algorithm are fulfilled.\\ \ref{p:convDRfwii}: We have chosen $x_0$ in $C$. In addition, $(\forall n \in \ensuremath{\mathbb N})$ $(z_{n,m+\frac{1}{2}})_m$ lies in $C$ and $x_{n+1}$ is convex combination of $x_n$ and $z_{n,m+\frac{1}{2}}$. Hence, it is easily shown by induction that $(\forall n \ge 1)$ $x_n \in C$. \end{proof}
Proposition \ref{p:convDRfw}\ref{p:convDRfwi} guarantees that, by choosing $M_n$ large enough, the algorithm allows us to solve Problem~\ref{pb:minimisation}. Although this result may appear somehow imprecise regarding the practical choice of $M_n$, it was observed in our simulations that small values of $M_n$ are sufficient to ensure the convergence.
In addition, as a direct consequence of Proposition \ref{p:convDRfw}\ref{p:convDRfwii}, in Step \Pisymbol{pzd}{194}, the gradient of $g$ is only evaluated on $C$. This means that, similarly to Algorithm \ref{algo:main1}, this algorithm is able to solve Problem \ref{prob:minimisationgen}. In the next section, we will show that a number of image restoration problems can be formulated as Problem \ref{prob:minimisationgen}.
\section{Application to a class of image restoration problems}\label{sec:application}
\subsection{Context} \label{se:applicont} We aim at restoring an image $\overline{y}$ in a real separable Hilbert space $\ensuremath{\mathcal G}$ from a degraded observation $z\in \ensuremath{\mathcal G}$. Here, digital images of size $N_1\times N_2$ are considered and thus $\ensuremath{\mathcal G} = \ensuremath{\mathbb R}^{N}$ with $N = N_1 N_2$. Let $T$ be a linear operator from $\ensuremath{\mathcal G}$ to $\ensuremath{\mathcal G}$ modelling a linear degradation process, e.g. a convolutive blur. The image $\overline{u} = T\overline{y}$ (resp. $z = (z^{(i)})_{1 \le i \le N}$) is a realization of a real-valued random vector $\overline{U}=(\overline{U}^{(i)})_{1\le i \le N}$ (resp. $Z = (Z^{(i)})_{1\le i \le N}$). The image $\overline{U}$ is contaminated by noise. Conditionally to $\overline{U} = (u^{(i)})_{1 \le i \le N} \in \ensuremath{\mathcal G}$, the random vector $Z$ is assumed to have independent components, which are either discrete with conditional probability mass functions $(\mu_{Z_i \mid \overline{U}^{(i)} = u^{(i)}})_{1 \le i \le N}$, or absolutely continuous with conditional probability density functions which are also denoted by $(\mu_{Z_i \mid \overline{U}^{(i)} = u^{(i)}})_{1 \le i \le N}$. In this paper, we are interested in probability distributions such that: \begin{equation} (\forall i \in \{1,\ldots,N\})(\forall \upsilon \in \ensuremath{\mathbb R}) \qquad \mu_{Z^{(i)} \mid \overline{U}^{(i)}=\upsilon}(z^{(i)})\propto \exp\big(-\psi_i(\upsilon)\big) \label{eq:defpsii} \end{equation} where the functions $(\psi_i)_{1\le i \le N}$ take their values in $]\ensuremath{-\infty},\ensuremath{+\infty}]$ and satisfy the following assumption. \begin{assumption}\label{as:psi} There exists a nonempty subset $\mathbb{I}$ of $\{1,\ldots,N\}$ and a constant $\delta \in \ensuremath{\mathbb R}$ such that, for all $i\in \{1,\ldots,N\}$, \begin{enumerate} \item \label{as:psii} $\ensuremath{\mathrm{dom}\,} \psi_i = ]\delta,\ensuremath{+\infty}[$ if $i\in \mathbb{I}$ and, $\ensuremath{\mathrm{dom}\,} \psi_i = [\delta,\ensuremath{+\infty}[$ if $i\not \in \mathbb{I}$; \item \label{as:psiiii} if $i\in \mathbb{I}$, then $\psi_i$ is twice continuously differentiable on $]\delta,\ensuremath{+\infty}[$ such that $\inf_{\upsilon\in ]\delta,\ensuremath{+\infty}[} \psi_i(\upsilon) > \ensuremath{-\infty}$ and \begin{equation*} \lim_{\substack{\upsilon \to \delta\\\upsilon > \delta}} \psi_i(\upsilon) = \ensuremath{+\infty}. \end{equation*} Its second-order derivative $\psi_i''$ is decreasing and satisfies \begin{equation*} \lim_{\upsilon \to \ensuremath{+\infty}} \psi_i''(\upsilon) = 0; \end{equation*} \item \label{as:psiii} if $i \not\in \mathbb{I}$, then there exists $\alpha_i \in \ensuremath{\left[0,+\infty\right[}$ such that $(\forall \upsilon \in [\delta,\ensuremath{+\infty}[)$ $\psi_i(\upsilon) = \alpha_i \upsilon$. \end{enumerate} \end{assumption} From Assumptions \ref{as:psi}\ref{as:psiiii} and \ref{as:psiii}, it is clear that the functions $(\psi_i)_{1 \le i \le N}$ are convex (since $(\forall i \in \mathbb{I})$ $(\forall \upsilon \in ]\delta,\ensuremath{+\infty}[)$ $\psi''_i(\upsilon) \ge 0$) such that \begin{equation} \lim_{\substack{\upsilon \to \delta\\\upsilon > \delta}} \psi_i''(\upsilon) = \ensuremath{+\infty} \label{eq:psiid} \end{equation} and they are lower semicontinuous (since $(\forall i \in \{1,\ldots,N\})$ $\lim \inf_{\upsilon \to \delta} \psi_i(\upsilon) \ge \psi_i(\delta)$). Examples of such functions will be provided in Sections \ref{sec:gaussian} and \ref{sec:poisson}.
In addition, a both simple and efficient prior probabilistic model on the unknown image $\overline{y}$ is adopted by using a representation of this image in a frame \cite{Daubechies_I_1992_book_ten_lw,Han_D_2000_book_frames_bgr}. The frame coefficient space is the Euclidean space $\ensuremath{{\mathcal H}} =\ensuremath{\mathbb R}^K$ ($K \ge N$). We thus use a linear representation of the form: \begin{equation*} \overline{y} = F^* \overline{x} \end{equation*} where $F^*\,:\; \ensuremath{{\mathcal H}} \to \ensuremath{\mathcal G}$ is a frame synthesis operator, i.e. $\underline{\nu}\, \ensuremath{\mathrm{Id}} \le F^* \circ F \le \overline{\nu}\, \ensuremath{\mathrm{Id}}$ with $(\underline{\nu},\overline{\nu}) \in \ensuremath{\,\left]0,+\infty\right[}^2$ (which implies that $F^*$ is surjective).\footnote{The existence of the lower bound implies the existence of the upper bound in finite dimensional case.} We then assume that the vector $\overline{x}$ of frame coefficients is a realization of a random vector $\overline{X}$ with independent components. Each component $\overline{X}^{(k)}$ with $k \in \{1,\ldots,K\}$ of $\overline{X}$, has a probability density $\exp(-\phi_k(\cdot))/\int_{-\infty}^{+\infty}\exp(-\phi_k(\eta))\,d\eta $ where $\phi_k$ is a finite function in $\Gamma_0(\ensuremath{\mathbb R})$.
Finally, we assume that we have prior information on $\overline{x}$ which can be expressed by the fact that $\overline{x}$ belongs to a closed convex set $C$ of $\ensuremath{{\mathcal H}}$. The constraint set $C$ will be assumed to satisfy: \begin{equation}
(TC^*) \cap \ensuremath{\mathrm{dom}\,}\Psi \neq \ensuremath{\varnothing} \label{eq:TCet} \end{equation} where \begin{equation*} C^* = F^*C = \menge{F^*x}{x\in C} \label{eq:defCet} \end{equation*} and \begin{equation*} \left(\forall u = \big(u^{(i)}\big)_{1 \le i \le N} \in \ensuremath{\mathcal G}\right)\qquad \Psi(u) = \sum_{i=1}^{N} \psi_i\big(u^{(i)}\big). \end{equation*}
With these assumptions, it can be shown (see \cite{Chaux_C_2007_ip_variational_ffbip}) that a Maximum A Posteriori (MAP) estimate of the vector of frame coefficients $\overline{x}$ can be obtained from $z=\big(z^{(i)}\big)_{1 \le i \le N}$ by minimizing in the Hilbert space $\ensuremath{{\mathcal H}} $ the function $f + g + \ensuremath{\mathrm{\iota_C}}$ where \begin{equation} \label{eq:deff} \left(\forall x = \big(x^{(k)}\big)_{1 \le k \le K} \in \ensuremath{{\mathcal H}}\right)\qquad f(x) = \sum_{k=1}^K \phi_k\big(x^{(k)}\big) \end{equation} and \begin{equation}\label{eq:defg} g = \Psi \circ T \circ F^*. \end{equation}
We consequently have: \begin{proposition}\label{p:MAP} Let $\ensuremath{{\mathcal H}} = \ensuremath{\mathbb R}^K$ and $\ensuremath{\mathcal G} = \ensuremath{\mathbb R}^N$ with $K\ge N$. Let $f$ and $g$ be defined by \eqref{eq:deff} and \eqref{eq:defg}, respectively, where $T\colon \ensuremath{\mathcal G} \to \ensuremath{\mathcal G}$ is a linear operator. Under Assumption \ref{as:psi} and Condition \eqref{eq:TCet}, then \begin{enumerate} \item \label{p:MAPi} $f$ and $g$ are in $\Gamma_0(\ensuremath{{\mathcal H}})$;
\item \label{p:MAPii} if $f$ is coercive\footnote{This means that $\lim_{\|x\| \to \ensuremath{+\infty}} f(x) = \ensuremath{+\infty}$.} or $\ensuremath{\mathrm{dom}\,} g\,\cap\, C$ is bounded, then the minimization of $f+g+\ensuremath{\mathrm{\iota_C}}$ admits a solution. In addition, if $f$ is strictly convex on $\ensuremath{\mathrm{dom}\,} g \,\cap\, C$, the solution is unique. \end{enumerate} \end{proposition} \begin{proof} \noindent\ref{p:MAPi}: It is clear that $f$ is a finite convex function of $\ensuremath{{\mathcal H}}$. As the functions $(\psi_i)_{1 \le i \le N}$ are in $\Gamma_0(\ensuremath{\mathbb R})$, $\Psi$ belongs to $\Gamma_0(\ensuremath{\mathcal G})$. In addition, by using \eqref{eq:TCet}, we have $\ensuremath{\mathrm{ran}\,}(T\circ F^*) \cap \ensuremath{\mathrm{dom}\,} \Psi \neq \ensuremath{\varnothing}$. This allows us to deduce that $\ensuremath{\mathrm{dom}\,} g \neq \ensuremath{\varnothing}$ and, therefore, $g \in \Gamma_0(\ensuremath{{\mathcal H}})$.\\ \ref{p:MAPii}: We have $\ensuremath{\mathrm{dom}\,} f\, \cap\, \ensuremath{\mathrm{dom}\,} g\, \cap\, C \neq \ensuremath{\varnothing}$ since $\ensuremath{\mathrm{dom}\,} f = \ensuremath{{\mathcal H}}$ and \eqref{eq:TCet} shows that $\ensuremath{\mathrm{dom}\,} g\, \cap\, C = \ensuremath{\mathrm{dom}\,}(\Psi \circ T \circ F^*)\, \cap\, C \neq \ensuremath{\varnothing}$. Since $f$ and $g$ are in $\Gamma_0(\ensuremath{{\mathcal H}})$, we deduce that $f+g+\ensuremath{\mathrm{\iota_C}}$ is in $\Gamma_0(\ensuremath{{\mathcal H}})$. \\ Suppose now that $f$ is coercive. By Assumption \ref{as:psi}\ref{as:psiiii}, $(\forall i\in \mathbb{I})$ $\inf_{\upsilon \in ]\delta,\ensuremath{+\infty}[} \psi_i(\upsilon) > \ensuremath{-\infty}$ whereas, due to Assumption \ref{as:psi}\ref{as:psiii}, $(\forall i\not\in\mathbb{I})$ $\inf_{\upsilon \in [\delta,\ensuremath{+\infty}[} \psi_i(\upsilon) = \alpha_i \delta$. This implies that $\inf \Psi(\ensuremath{\mathcal G}) > \ensuremath{-\infty}$ and, consequently, $\inf g(\ensuremath{{\mathcal H}}) \ge \inf \Psi(\ensuremath{\mathcal G}) > \ensuremath{-\infty}$. As a result, $f+g+\ensuremath{\mathrm{\iota_C}} \ge f+\ensuremath{\mathrm{\iota_C}}+\inf g(\ensuremath{{\mathcal H}})$ is coercive. When $\ensuremath{\mathrm{dom}\,} g \,\cap\, C$ is bounded, $f+g+\ensuremath{\mathrm{\iota_C}}$ also is coercive. The existence of a solution to the minimization problem follows from classical results in convex analysis \cite[Chap. 3, Prop. 1.2]{Ekeland_I_1999_book_Convex_aavp}.\\ When $f$ is strictly convex on $\ensuremath{\mathrm{dom}\,} g\, \cap\, C$, the uniqueness of the solution follows from the fact that $f+g+\ensuremath{\mathrm{\iota_C}}$ is strictly convex \cite[Chap. 3, Prop. 1.2]{Ekeland_I_1999_book_Convex_aavp}. \end{proof} \begin{remark} \label{re:fcoerc} The function $f$ is coercive (resp. strictly convex) if and only if the functions $(\phi_k)_{1 \le k \le N}$ are coercive {\rm \cite[Prop. 3.3(iii)(c)]{Chaux_C_2007_ip_variational_ffbip}} (resp. strictly convex). \end{remark}
\subsection{Quadratic extension}
If we now investigate the Lipschitz-continuity of the gradient of $g$, it turns out that this property may be violated since $\Psi$ is not finite. Due to \eqref{eq:psiid}, the gradient of $g$ is not even guaranteed to be Lipschitz-continuous on $\ensuremath{\mathrm{int}\,} \ensuremath{\mathrm{dom}\,} g$.
To circumvent this problem, it can be noticed that, because of Assumption \ref{as:psi}\ref{as:psiiii} and \eqref{eq:psiid}, for all $i \in \mathbb{I}$, there exists a decreasing function $\upsilon_i\,:\,\ensuremath{\,\left]0,+\infty\right[} \to ]\delta,\ensuremath{+\infty}[$ such that $\lim_{\theta \to \ensuremath{+\infty}} \upsilon_i(\theta) = \delta$ and \begin{equation} (\forall \theta\in \ensuremath{\,\left]0,+\infty\right[}) (\forall \upsilon \in ]\delta,\ensuremath{+\infty}[)\qquad 0 \le \psi_i''(\upsilon) \le \theta \Leftrightarrow \upsilon \ge \upsilon_i(\theta). \label{eq:defuit} \end{equation} Let us now consider the function $g_\theta = \Psi_\theta \circ T \circ F^*$ with $\theta \in \ensuremath{\,\left]0,+\infty\right[}$, where \begin{equation*} \left(\forall u = \big(u^{(i)}\big)_{1 \le i \le N} \in \ensuremath{\mathcal G}\right)\qquad \Psi_\theta(u) = \sum_{i=1}^{N} \psi_{\theta,i}\big(u^{(i)}\big) \end{equation*} and the functions $(\psi_{\theta,i})_{1\le i \le N}$ are chosen such that, \begin{equation} (\forall \upsilon \in \ensuremath{\mathbb R})\qquad \psi_{\theta,i}(\upsilon) = \begin{cases} \displaystyle \frac{\theta}{2} \upsilon^2+\zeta_{i,1}(\theta)\; \upsilon+\zeta_{i,0}(\theta) & \mbox{if $i\in \mathbb{I}$ and $\delta-\epsilon(\theta) \le \upsilon < \upsilon_{i}(\theta)$}\\ \alpha_i \upsilon & \mbox{if $i \not \in \mathbb{I}$ and $\delta-\epsilon(\theta) \le \upsilon < \delta$}\\ \psi_i(\upsilon) & \mbox{otherwise.} \end{cases} \label{eq:defpsiit} \end{equation} Hereabove, $\epsilon\,:\,\ensuremath{\,\left]0,+\infty\right[} \to\ensuremath{\,\left]0,+\infty\right[}$ is a decreasing function and, \begin{align*} (\forall i \in \mathbb{I})\qquad \zeta_{i,0}(\theta) & = \psi_i\big( \upsilon_{i}(\theta)\big)-\upsilon_{i}(\theta) \psi'_{i}\big( \upsilon_{i}(\theta)\big)+\frac{\theta}{2}\big(\upsilon_{i}(\theta)\big)^2 \\ \zeta_{i,1}(\theta) & = \psi'_{i}\big( \upsilon_{i}(\theta)\big)-\theta \upsilon_{i}(\theta). \end{align*} For every $i\in \mathbb{I}$, the constants $\zeta_{i,0}(\theta)$ and $\zeta_{i,1}(\theta)$ have been determined so as to guarantee the continuity of $\psi_{\theta,i}$ and of its first order derivative at $\upsilon_i(\theta)$. Consequently, the following result can be obtained: \begin{proposition}\label{p:gt} Suppose that Assumption \ref{as:psi} and Condition \eqref{eq:TCet} hold. Then, \begin{enumerate} \item\label{p:gt0} $(\forall \theta \in \ensuremath{\,\left]0,+\infty\right[})$ $g_\theta \in \Gamma_0(\ensuremath{{\mathcal H}})$. \item\label{p:gti} $\big(\forall (\theta_1,\theta_2)\in \ensuremath{\,\left]0,+\infty\right[}^2\big)$, $\theta_1 < \theta_2$ $\Rightarrow$ $g_{\theta_1} \le g_{\theta_2} \le g$. \item\label{p:gtii} For every $\theta \in \ensuremath{\,\left]0,+\infty\right[}$, if $TC^* \subset ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N$, then $g_\theta$ has a Lipschitz-continuous gradient over $C$
with constant $\beta_\theta = \theta \|TF^*\|^2 \le \theta\overline{\nu} \|T\|^2$. \item\label{p:gtiii} For every $\theta \in \ensuremath{\,\left]0,+\infty\right[}$, if $f$ is coercive or if $\ensuremath{\mathrm{dom}\,} g_\theta\,\cap\, C$ is bounded, then the minimization of $f+g_\theta+\ensuremath{\mathrm{\iota_C}}$ admits a solution. In addition, if $f$ is strictly convex on $\ensuremath{\mathrm{dom}\,} g_\theta \cap C$, then $f+g_\theta+\ensuremath{\mathrm{\iota_C}}$ has a unique minimizer $\widetilde{x}_\theta$. \item \label{p:gtv} Assume that \begin{enumerate} \item $\lim_{\theta\to \ensuremath{+\infty}} \epsilon(\theta) = 0$, \item \label{as:enplus} $TC^* \subset [\delta,\ensuremath{+\infty}[^N$, \item \label{as:coercb} $f$ is coercive or $C$ is bounded, \item \label{as:strictc} $f$ is strictly convex on $C$. \end{enumerate} Then, there exists $\overline{\theta} \in \ensuremath{\,\left]0,+\infty\right[}$ such that, for every $\theta \in [\overline{\theta},\ensuremath{+\infty}[$, the minimizer $\widetilde{x}_\theta$ of $f+g_\theta+\iota_C$ is the minimizer of $f+g+\iota_C$. \end{enumerate} \end{proposition} \begin{proof} \ref{p:gt0} Since $\Psi_\theta$ is defined and continuous on $[\delta-\epsilon(\theta),\ensuremath{+\infty}[^N$ and, $(\forall i \in \{1,\ldots,N\})$ $(\forall \upsilon \in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[)$ $\psi_{\theta,i}''(\upsilon)\ge 0$, we have $\Psi_\theta \in \Gamma_0(\ensuremath{\mathcal G})$. In addition, $\ensuremath{\mathrm{dom}\,} \Psi_\theta \cap \ensuremath{\mathrm{ran}\,}(T\circ F^*) \supset \ensuremath{\mathrm{dom}\,} \Psi \cap \ensuremath{\mathrm{ran}\,}(T\circ F^*) \neq \ensuremath{\varnothing}$. Thus, $g_\theta \in \Gamma_0(\ensuremath{{\mathcal H}})$.\\ \ref{p:gti} As a consequence of \eqref{eq:defpsiit} and \eqref{eq:defuit}, we have, for every $i\in \mathbb{I}$, \begin{equation*} (\forall \upsilon \in ]\delta,\upsilon_{i}({\theta_2})[)\qquad \psi_{i}''(\upsilon) > \psi_{{\theta_2},i}''(\upsilon) = {\theta_2}. \end{equation*} So $\psi_i'-\psi_{{\theta_2},i}'$ is a strictly increasing function over $]\delta,\upsilon_i({\theta_2})]$ and \begin{equation*} (\forall \upsilon \in ]\delta,\upsilon_i({\theta_2})[)\qquad \psi_{i}'(\upsilon) -\psi_{{\theta_2},i}'(\upsilon) < \psi_{i}'\big(\upsilon_{i}({\theta_2})\big) - \psi_{{\theta_2},i}'\big(\upsilon_{i}({\theta_2})) = 0 \end{equation*} which, in turn, shows that $\psi_{i}- \psi_{{\theta_2},i}$ is strictly decreasing on $]\delta,\upsilon_i({\theta_2})]$ and \begin{equation*} (\forall \upsilon \in ]\delta,\upsilon_i({\theta_2})[)\qquad \psi_{i}(\upsilon) -\psi_{{\theta_2},i}(\upsilon) > \psi_{i}\big(\upsilon_{i}({\theta_2})\big) - \psi_{{\theta_2},i}\big(\upsilon_{i}({\theta_2})) = 0. \end{equation*} In addition, we know that, if $(i\in \mathbb{I}$ and $\upsilon \le \delta)$ or $(i \not\in \mathbb{I}$ and $\upsilon < \delta)$, then $\psi_{i}(\upsilon)=\ensuremath{+\infty}$ and, if $\big(i\in \mathbb{I}$ and $\upsilon \ge \upsilon_i({\theta_2})\big)$ or $(i\not\in \mathbb{I}$ and $\upsilon \ge \delta)$, then $\psi_{i}(\upsilon) = \psi_{{\theta_2},i}(\upsilon)$. We deduce that, for all $i\in \{1,\ldots,N\}$, $\psi_i \ge \psi_{{\theta_2},i}$ and, therefore $g$ is lower bounded by $g_{\theta_2}$.\\ By proceeding similarly, we have, for every $i\in \mathbb{I}$, \begin{alignat*}{2} &(\forall \upsilon \in [\upsilon_i(\theta_1),\ensuremath{+\infty}[)\qquad &&\psi_{\theta_2,i}(\upsilon)= \psi_i(\upsilon) = \psi_{\theta_1,i}(\upsilon)\\ &(\forall \upsilon \in ]\delta-\epsilon(\theta_2),\upsilon_i(\theta_1)[)\qquad &&\psi''_{\theta_2,i}(\upsilon) > \theta_1 = \psi''_{\theta_1,i}(\upsilon)\nonumber\\ &\Rightarrow\;\; (\forall \upsilon \in ]\delta-\epsilon(\theta_2),\upsilon_i(\theta_1)[)\qquad && \psi'_{\theta_2,i}(\upsilon) < \psi'_{\theta_1,i}(\upsilon) \nonumber\\ &\Rightarrow\;\;(\forall \upsilon \in [\delta-\epsilon(\theta_2),\upsilon_i(\theta_1)[)\qquad && \psi_{\theta_2,i}(\upsilon) > \psi_{\theta_1,i}(\upsilon). \end{alignat*} In addition, \begin{equation*} (\forall i \in \{1,\ldots,N\}) (\forall \upsilon \in ]\ensuremath{-\infty},\delta-\epsilon(\theta_2)[)\qquad \psi_{\theta_2,i}(\upsilon) = \ensuremath{+\infty} \ge \psi_{\theta_1,i}(\upsilon) \end{equation*} and \begin{equation*} (\forall i \not\in \mathbb{I}) (\forall \upsilon \in [\delta-\epsilon(\theta_2),\ensuremath{+\infty}[)\qquad \psi_{\theta_2,i}(\upsilon) =\psi_{\theta_1,i}(\upsilon). \end{equation*} This shows that $\Psi_{\theta_2} \ge \Psi_{\theta_1}$ and, consequently, $g_{\theta_2} \ge g_{\theta_1}$.
\noindent\ref{p:gtii}: As already mentioned, $\ensuremath{\mathrm{dom}\,} \Psi_\theta = [\delta-\epsilon(\theta),\ensuremath{+\infty}[^N$. Consider \begin{equation*} O_\theta = (TF^*)^{-1}(]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N) = \menge{x\in\ensuremath{{\mathcal H}}}{TF^*x \in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N}.
\label{eq:defO} \end{equation*}
$O_\theta$ is an open set and, as $TC^* \subset ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N$, we have: $C \subset O_\theta$. In addition, the function $g_\theta$ is differentiable on $O_\theta$ and its gradient is \cite[Chap. 1, Prop. 5.7]{Ekeland_I_1999_book_Convex_aavp} \begin{equation} (\forall x \in O_\theta) \qquad \nabla g_\theta(x) = FT^*\big(\nabla \Psi_\theta(TF^*x)\big) \label{eq:gradcomp} \end{equation} where \begin{equation*} \big(\forall u =(u^{(i)})_{1\le i \le n} \in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N\big) \qquad \nabla\Psi_\theta(u) = \big(\psi'_{\theta,i}(u^{(i)})\big)_{1 \le i \le N}. \end{equation*} We have then \begin{multline*} \big(\forall u =(u^{(i)})_{1\le i \le n} \in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N\big) \big(\forall v =(v^{(i)})_{1\le i \le n} \in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N\big)\\
\|\nabla\Psi_\theta(u)-\nabla\Psi_\theta(v)\| = \Big(\sum_{i=1}^N \big(\psi'_{\theta,i}(u^{(i)})- \psi'_{\theta,i}(v^{(i)})\big)^2\Big)^{1/2} \end{multline*} and, by the mean value theorem, \begin{align*} (\forall i \in \{1,\ldots,N\})\quad
\big|\psi'_{\theta,i}(u^{(i)})- \psi'_{\theta,i}(v^{(i)})\big|
&\le |u^{(i)}-v^{(i)}| \sup_{\xi \in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[}
|\psi''_{\theta,i}(\xi)| \nonumber\\
&\le \theta |u^{(i)}-v^{(i)}|. \end{align*} This yields \begin{equation*} \big(\forall u\in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N\big) \big(\forall v\in ]\delta-\epsilon(\theta),\ensuremath{+\infty}[^N\big)
\qquad\|\nabla\Psi_\theta(u)-\nabla\Psi_\theta(v)\|
\le \theta \|u-v\| \end{equation*} and, we deduce from \eqref{eq:gradcomp} that \begin{equation*} \big(\forall (x,x') \in O_\theta^2\big)
\qquad\|\nabla g_\theta(x)-\nabla g_\theta(x')\| \le
\theta \|T F^*\|^2 \|x-x'\|. \end{equation*}
and $\|T F^*\|^2 \le \|F\|^2\|T\|^2 \le \overline{\nu}\|T\|^2$.\\ \ref{p:gtiii}: The proof is similar to that of Proposition \ref{p:MAP}\ref{p:MAPii}.\\ \ref{p:gtv}: In the following, we use the notation: $h = f+g+\ensuremath{\mathrm{\iota_C}}$ and $(\forall \theta \in \ensuremath{\,\left]0,+\infty\right[})$ $h_\theta = f+g_\theta+\ensuremath{\mathrm{\iota_C}}$.\\ Let $(\theta_\ell)_{\ell \in \ensuremath{\mathbb N}}$ be an increasing sequence of $\ensuremath{\,\left]0,+\infty\right[}$ such that $\lim_{\ell \to \ensuremath{+\infty}} \theta_\ell = \ensuremath{+\infty}$. As a consequence of \ref{p:gt0} and \ref{p:gti}, $(h_{\theta_\ell})_{\ell \in \ensuremath{\mathbb N}}$ is an increasing sequence of functions in $\Gamma_0(\ensuremath{{\mathcal H}})$. We deduce from \cite[Proposition 7.4(d)]{Rockafellar_RT_2004_book_Variational_a} that $(h_{\theta_\ell})_{\ell \in \ensuremath{\mathbb N}}$ epi-converges to its pointwise limit. By using \eqref{eq:defpsiit} in combination with the facts that $(\forall i \in \mathbb{I})$ $\lim_{\theta\to \ensuremath{+\infty}} \upsilon_i(\theta) = \delta$ and $\lim_{\theta \to \ensuremath{+\infty}} \epsilon(\theta) = 0$, we see that the pointwise limit is equal to $h$.\\ Under Assumptions \ref{as:enplus} and \ref{as:coercb}, $(\forall \ell \in \ensuremath{\mathbb N})$ $h_{\theta_\ell}$ is coercive since $C\,\cap\,\ensuremath{\mathrm{dom}\,} g_{\theta_\ell} = C$. Equivalently, its level sets $\lev{\eta} h_{\theta_\ell} =\menge{x \in \ensuremath{{\mathcal H}}}{h_{\theta_\ell}(x) \le \eta}$ with $\eta \in \ensuremath{\mathbb R}$, are bounded.
$(h_{\theta_\ell})_{\ell \in \ensuremath{\mathbb N}}$ being a sequence of increasing functions, $\cup_{\ell\in\ensuremath{\mathbb N}} \lev{\eta}h_{\theta_\ell} = \lev{\eta}h_{\theta_0}$ is bounded. As the functions $h_{\theta_\ell}$ with $\ell \in \ensuremath{\mathbb N}$ and $h$ are lower semicontinuous and proper, \cite[Theorem 7.33]{Rockafellar_RT_2004_book_Variational_a} allows us to claim that the sequence $(\widetilde{x}_{\theta_\ell})_{\ell \in \ensuremath{\mathbb N}}$ converges to the minimizer $\widetilde{x}$ of $h$ (by Assumptions~\ref{as:enplus} and \ref{as:strictc}, both $h_{\theta_\ell}$ with $\ell \in \ensuremath{\mathbb N}$ and $h$ have a unique minimizer due to the strict convexity of $f$ on $(C\,\cap\,\ensuremath{\mathrm{dom}\,} g)\subset (C\,\cap\,\ensuremath{\mathrm{dom}\,} g_{\theta_\ell}) = C$ and, Propositions \ref{p:MAP}\ref{p:MAPii} and \ref{p:gt}\ref{p:gtiii}). As $\widetilde{x} \in \ensuremath{\mathrm{dom}\,} h$, $(\forall i \in \mathbb{I})$ $(TF^*\widetilde{x})^{(i)} \in \ensuremath{\mathrm{dom}\,} \psi_i = ]\delta,\ensuremath{+\infty}[$, where, for every $x\in \ensuremath{{\mathcal H}}$ and $i\in \{1,\ldots,N\}$, $(TF^* x)^{(i)}$ denotes the $i$-th component of vector $TF^* x$. Since $\lim_{\ell \to \ensuremath{+\infty}} \widetilde{x}_{\theta_\ell} = \widetilde{x}$, we have, for every $i\in \mathbb{I}$, \begin{align*} (\forall \eta \in \ensuremath{\,\left]0,+\infty\right[}) (\exists \ell_{\eta,i} \in \ensuremath{\mathbb N}) \text{ such that }\\ \begin{split} (\forall \ell \in \ensuremath{\mathbb N})\quad \ell \ge \ell_{\eta,i} \Rightarrow &
| (TF^*\widetilde{x}_{\theta_\ell})^{(i)} - (TF^*\widetilde{x})^{(i)}| < \eta\\ \Rightarrow & (TF^*\widetilde{x}_{\theta_\ell})^{(i)} > \min_{i\in \mathbb{I}} (TF^*\widetilde{x})^{(i)} - \eta. \end{split} \end{align*} By setting $\displaystyle \eta = \frac{\min_{i\in \mathbb{I}}(TF^*\widetilde{x})^{(i)}-\delta}{2} > 0$ and $\ell_\eta = \max_{i\in\mathbb{I}} \ell_{\eta,i}$, we deduce that \begin{equation} (\forall \ell \in \ensuremath{\mathbb N})\quad \ell \ge \ell_\eta \Rightarrow (TF^*\widetilde{x}_{\theta_\ell})^{(i)} \ge \underline{\upsilon} \label{eq:minlbound} \end{equation} where $\displaystyle \underline{\upsilon} = \frac{\delta+\min_{i\in \mathbb{I}} (TF^*\widetilde{x})^{(i)}}{2}> \delta$. In addition, since $\lim_{\ell \to \ensuremath{+\infty}} \theta_\ell = \ensuremath{+\infty}$ $\Rightarrow$ $\lim_{\ell \to \ensuremath{+\infty}} \max_{i\in \mathbb{I}} \upsilon_{i}(\theta_\ell) = \delta$, there exists $\overline{\ell}\ge \ell_\eta$ such that $(\forall i \in\mathbb{I})$ $\upsilon_{i}(\theta_{\overline{\ell}}) \le \underline{\upsilon}$. By using \eqref{eq:defpsiit}, this implies that $(\forall i \in \mathbb{I})$ $(\forall \upsilon \in [\underline{\upsilon},\ensuremath{+\infty}[)$, $\psi_{\theta_{\overline{\ell}},i}(\upsilon) = \psi_i(\upsilon)$. By defining now \begin{equation*} D = \menge{x \in \ensuremath{\mathrm{dom}\,} g} {(\forall i \in \mathbb{I})\; (TF^* x)^{(i)} \in [\underline{\upsilon},\ensuremath{+\infty}[} \end{equation*} we deduce that $(\forall x \in D)$ $h_{\theta_{\overline{\ell}}}(x) = h(x)$. Moreover, according to Assumption~\ref{as:enplus}, for every $\ell \in \ensuremath{\mathbb N}$, if $i\not\in \mathbb{I}$, \begin{equation} (TF^*\widetilde{x}_{\theta_\ell})^{(i)}\in [\delta,\ensuremath{+\infty}[. \label{eq:audessus} \end{equation} Altogether, \eqref{eq:minlbound} and \eqref{eq:audessus} show that both $\widetilde{x}_{\theta_{\overline{\ell}}}$ and $\widetilde{x}$ belong to $D$. Consequently, as $\widetilde{x}_{\theta_{\overline{\ell}}} = \arg\min_{x\in \ensuremath{{\mathcal H}}} h_{\theta_{\overline{\ell}}}(x)$, we have: $h(\widetilde{x}_{\theta_{\overline{\ell}}})= h_{\theta_{\overline{\ell}}}(\widetilde{x}_{\theta_{\overline{\ell}}}) \le h_{\theta_{\overline{\ell}}}(\widetilde{x}) = h(\widetilde{x})$, which proves that $\widetilde{x}_{\theta_{\overline{\ell}}}=\widetilde{x}$.\\ Considering now $\theta \in [\theta_{\overline{\ell}},\ensuremath{+\infty}[$, from \ref{p:gti} we get: $h_{\theta_{\overline{\ell}}}\le h_\theta \le h$. Thus, $h(\widetilde{x})=h_{\theta_{\overline{\ell}}}(\widetilde{x}) \le h_\theta(\widetilde{x}) \le h(\widetilde{x})$, which results in $h_\theta(\widetilde{x})= h(\widetilde{x})$, while \begin{equation*} (\forall x \in \ensuremath{{\mathcal H}})\qquad h_\theta(x) \ge h_{\theta_{\overline{\ell}}}(x) \ge h_{\theta_{\overline{\ell}}}(\widetilde{x})= h(\widetilde{x}). \end{equation*} This allows us to conclude that $\widetilde{x}_\theta = \widetilde{x}$ as soon as $\theta \ge \theta_{\overline{\ell}} = \overline{\theta}$. \end{proof} \begin{remark} \begin{enumerate} \item A polynomial approximation of the objective function was considered in \cite{Fessler_JA_1995_tip_Hybrid_ppofftirfts} which is different from the proposed quadratic extension technique. \item As expressed by Proposition \ref{p:gt}\ref{p:gti}, $g_\theta$ (resp. $f+g_\theta+\ensuremath{\mathrm{\iota_C}}$) with $\theta > 0$ constitutes a lower approximation of $g$ (resp. $f+g+\ensuremath{\mathrm{\iota_C}}$), which becomes closer as $\theta$ increases. \item As shown by Proposition \ref{p:gt}\ref{p:gtii}, the main role of parameter $\theta$ is to control the Lipschitz constant of the gradient of this approximation of $g$. \item At the same time, Proposition \ref{p:gt}\ref{p:gtv} indicates that this parameter allows us to control the closeness of the approximation to a minimizer of the original MAP criterion. This approximation becomes perfect when $\theta$ becomes greater than some value $\overline{\theta}$. \end{enumerate} \end{remark} Under the assumptions of Proposition \ref{p:gt}\ref{p:gtii}, the minimization of $f+g_\theta+\ensuremath{\mathrm{\iota_C}}$ with $\theta \in \ensuremath{\,\left]0,+\infty\right[}$ is a problem of the type of Problem \ref{prob:minimisationgen}. Therefore, Propositions~\ref{p:convfwDR} and \ref{p:convDRfw} show that, provided that $f$ is coercive or $C$ is bounded, Algorithms~\ref{algo:main1} and \ref{algo:main} can be applied in this context. In addition, Proposition \ref{p:gt}\ref{p:gtv} suggests that, by choosing $\theta$ large enough, a solution to the original MAP criterion can be found. However, according to Proposition~\ref{p:gt}\ref{p:gtii}, a large value of $\theta$ induces a large value of the Lipschitz constant $\beta_\theta$. This means that a small value of the step-size parameter must also to be used in the forward iteration of the algorithms, which is detrimental to the convergence speed. In practice, the choice of $\theta$ results from a trade-off as will be illustrated by the numerical results.
\subsection{First example}\label{sec:gaussian} \subsubsection{Model} We want to restore an image $\overline{y}\in\ensuremath{\left[0,+\infty\right[}^N$ corrupted by a linear operator $T\,:\,\ensuremath{\mathcal G} \to \ensuremath{\mathcal G}$ and an additive noise $w\in \ensuremath{\mathcal G}$, having the observation \begin{equation*} z=T\overline{y}+w=\overline{u}+w. \end{equation*} In addition, the linear operator $T$ is assumed to be nonnegative-valued (in the sense that the matrix associated to $T$ has nonnegative elements) and, $w = (w^{(i)})_{1 \le i \le N}$ is a realization of an independent zero-mean Gaussian noise $W = (W^{(i)})_{1 \le i \le N}$. The variance of each random variable $W^{(i)}$ with $i\in \{1,\ldots,N\}$ is signal-dependent and is equal to $\sigma_i^2(\overline{u}^{(i)})$ where \begin{equation*} (\forall \upsilon \in [0,+\infty[)\qquad \sigma_i^2(\upsilon) = \frac{\upsilon}{2\alpha_i} \end{equation*} with $\alpha_i \in \ensuremath{\,\left]0,+\infty\right[}$. So, the functions $(\psi_i)_{1\le i \le N}$ as defined in \eqref{eq:defpsii} are, when $z^{(i)} \neq 0$, \begin{equation*} (\forall \upsilon \in \ensuremath{\mathbb R})\qquad \psi_i(\upsilon) = \begin{cases} \displaystyle \frac{\alpha_i\big(\upsilon-z^{(i)}\big)^2}{\upsilon} & \mbox{if $\upsilon \in \ensuremath{\,\left]0,+\infty\right[}$}\\ +\infty & \mbox{otherwise} \end{cases} \end{equation*} and, when $z^{(i)} = 0$, \begin{equation*} (\forall \upsilon \in \ensuremath{\mathbb R})\qquad \psi_i(\upsilon) = \begin{cases} \displaystyle \alpha_i\upsilon & \mbox{if $\upsilon \in \ensuremath{\left[0,+\infty\right[}$}\\ +\infty & \mbox{otherwise.} \end{cases} \end{equation*} So, provided that $z\neq 0$, Assumption \ref{as:psi} is satisfied with $\delta = 0$ and $\mathbb{I} =$\linebreak$ \menge{i\in \{1,\ldots,N\}}{z^{(i)} \neq 0}$ since, for all $i\in \mathbb{I}$, \begin{align*} (\forall \upsilon \in \ensuremath{\,\left]0,+\infty\right[})\qquad \psi_i'(\upsilon) &=\alpha_i \frac{\upsilon^2-(z^{(i)})^2}{\upsilon^2}\\ \psi_i''(\upsilon) & = \frac{2\alpha_i(z^{(i)})^2}{\upsilon^3}. \end{align*} We deduce from \eqref{eq:defuit} that, for every $i \in \mathbb{I}$, \begin{equation*} (\forall \theta \in \ensuremath{\,\left]0,+\infty\right[})\qquad \upsilon_i(\theta) = \Big(\frac{2\alpha_i(z^{(i)})^2}{\theta}\Big)^{1/3}. \end{equation*}
\subsubsection{Simulation results} \label{sec:simulSigDep}
Here, $T$ is either a $3\times 3$ or a $7\times 7$ uniform convolutive blur with $\|T\|=1$. The $512 \times 512$ satellite image $\overline{y}$ ($N = 512^2$) shown in Fig.~\ref{fig:marseille_sigdep}(a) has been degraded by $T$ and a signal-dependent additive noise following the model described in the previous section with $\alpha_i \equiv 1$ or $\alpha_i \equiv 5$. The degraded image $z$ displayed in Fig.~\ref{fig:marseille_sigdep}(b) corresponds to a $7\times 7$ uniform blur and $\alpha_i \equiv 1$.
A twice redundant dual-tree tight frame representation \cite{Chaux_C_2006_tip_ima_adtmbwt} ($\underline{\nu}=\overline{\nu}=2$, $K = 2 N$)
using symlet filters of length $6$ \cite{Daubechies_I_1992_book_ten_lw} has been employed in this example. The potential functions $\phi_k$ are taken of the form $\chi_k |\,.\,|+\omega_k |\,.\,|^{p_k}$ where $(\chi_k,\omega_k)\in \ensuremath{\,\left]0,+\infty\right[}^2$ and $p_k \in \{4/3,3/2,2\}$ are subband adaptive. These parameters have been determined by a maximum likelihood approach. The function $f$ as defined by \eqref{eq:deff} is therefore coercive and strictly convex (see Remark~\ref{re:fcoerc}).
A constraint on the solution is introduced to take into account the range of admissible values in the image by choosing \begin{eqnarray} C^* = [0,255]^N. \label{eq:constval} \end{eqnarray} Due to the form of the operator $T$, $TC^* = C^*$ and Condition \eqref{eq:TCet} is therefore satisfied. Proposition \ref{p:MAP} thus guarantees that a unique solution $\widetilde{x}$ to the MAP estimation problem exists. According to Proposition \ref{p:gt}\ref{p:gtiii}, for every $\theta \in \ensuremath{\,\left]0,+\infty\right[}$, a unique minimizer $\widetilde{x}_\theta$ of $f+g_\theta+\iota_C$ also exists which allows us to approximate $\widetilde{x}$ as stated by Proposition \ref{p:gt}\ref{p:gtv}.
Since, for every $\theta \in \ensuremath{\,\left]0,+\infty\right[}$, $TC^* = C^* \subset [-\epsilon(\theta),\ensuremath{+\infty}[^N$, Proposition \ref{p:gt}\ref{p:gtii} shows that $g_\theta$ has a Lipschitz-continuous gradient over $C$ and Algorithms \ref{algo:main1} and \ref{algo:main} can be used to compute $\widetilde{x}_\theta$. The two algorithms are subsequently tested.
On the one hand, when Algorithm \ref{algo:main1} is used, the initialization is performed by setting $z_0 = P_C z$ and we choose $\ensuremath{\kappa} \equiv 60$ and $\ensuremath{\tau}_m \equiv 1$. The projection onto $C=(F^*)^{-1} C^*$ is $P_C = \ensuremath{\mathrm{prox}}_{\iota_{C^*}\circ F^*}$ which can be computed by using Proposition~\ref{p:linprox} with $L = F^*$. The other parameters have been fixed to $\lambda_{m,n} \equiv 1$ and $\gamma_{m,n} \equiv 0.995/(\kappa\theta)$, in compliance with Proposition \ref{p:gt}\ref{p:gtii}. The convergence of the algorithm is secured by Proposition~\ref{p:convfwDR} since Assumption \ref{a:ddd}\ref{a:ddd1} trivially holds. However, to improve the convergence profile, the following empirical rule for choosing the number $N_m$ of forward-backward iterations has been substituted for the necessary Conditions \eqref{eq:condconvfwDR0} and \eqref{eq:condconvfwDRm}: \begin{equation}
N_m = \inf\menge{n\in \ensuremath{\mathbb N}^*}{\|x_{m,n}-x_{m,n-1}\| \le \eta} \label{eq:Nmchoice} \end{equation} with $\eta = 10^{-4}$.
On the other hand, when Algorithm~\ref{algo:main} is used, the parameters have been chosen as follows : $\lambda_n \equiv 1$, $\tau_{n,m} \equiv 1$ and $\gamma_n \equiv 0.995/\theta$. The algorithm has been initialized by setting $x_0 = P_C z$ where the projection onto $C$ is computed as described previously. The convergence of the algorithm is ensured by Proposition~\ref{p:convDRfw}. The number $M_n$ of Douglas-Rachford iterations has been fixed as follows: \begin{equation}
M_n = \inf\menge{m\in \ensuremath{\mathbb N}^*}{\|z_{n,m}-z_{n,m-1}\| \le \eta} \end{equation} with the same value of $\eta$ as for the first algorithm.
The error between an image $y$ and the original image $\overline{y}$ is evaluated by the signal to noise ratio (SNR) defined as
$20 \log_{10}(\|\overline{y}\|/\|y-\overline{y}\|)$.
Three objectives are targeted in our experiments. First, we want to study the performance of the proposed approach, using the redundant dual-tree transform (DTT). The results presented in Tab.~\ref{tab:snrmars_sigdep} have been generated by Algorithm~\ref{algo:main1}, but Algorithm~\ref{algo:main} leads to the same results.
\begin{table}[htbp] \centering
\begin{tabular}{|c||c|c|c|c|c||c|c|c|c|} \cline{3-10}
\multicolumn{2}{c|}{}& \multicolumn{4}{c||}{$3 \times 3$ blur}& \multicolumn{4}{c|}{$7 \times 7$ blur}\\ \hline
& $\theta$ & 0.025 & 0.05 & 5 & 7 & 0.025 & 0.05 & 5 & 7\\
\cline{2-10}
$\alpha_i=1$ & SNR & 13.9 & 16.3 & \textbf{16.8} & 16.8 & 10.9 & 11.9 & \textbf{12.1} & 12.1\\ \hline \hline
& $\theta$ & 0.15 & 0.25 & 10 & 12 & 0.15 & 0.25 & 10 & 12\\
\cline{2-10}
$\alpha_i=5$ & SNR & 15.9 & 18.0 & \textbf{18.8} & 18.8 & 12.6 & 13.3 & \textbf{13.7} & 13.7 \\ \hline \end{tabular} \caption{\textrm{SNR} for the satellite image.\label{tab:snrmars_sigdep}} \end{table}
As suggested by Proposition \ref{p:gt}\ref{p:gtv}, as $\theta$ increases, the image is better restored. The effectiveness of the proposed approach is also demonstrated visually in Fig.~\ref{fig:marseille_sigdep}(c) showing the restored image when $T$ is a $7 \times 7$ uniform blur, $\alpha_i\equiv 1$ and $\theta=0.05$. It can be observed that the algorithm allows us to recover most of the details which were not perceptible due to blur and noise. \begin{figure}
\caption{Results for a satellite image of the city of Marseille. (a) Original image, (b) degraded image, (c) restored using a DTT. }
\label{fig:marseille_sigdep}
\end{figure}
Secondly, we aim at comparing the two proposed algorithms in terms of convergence for a given value of $\theta$. In Fig. \ref{fig:compalgo}, the MAP criterion value is plotted as a function of the computational time for a $7\times 7$ blur, $\alpha_i\equiv 5$ and $\theta=0.25$. For improved readibility, the criterion has been normalized by subtracting the final value and dividing by the initial one. It can be noticed that Algorithm \ref{algo:main} converges faster than Algorithm \ref{algo:main1}. This fact was confirmed by other simulation results performed in various contexts.
\begin{figure}
\caption{Normalized MAP criterion (Algorithm \ref{algo:main1} in red and Algorithm \ref{algo:main} in blue) versus computational time (in seconds) (Intel Xeon 4 Core, 3.00 GHz).}
\label{fig:compalgo}
\end{figure}
Finally, Fig.~\ref{fig:comptheta} illustrates the influence of the choice of the parameter $\theta$ when Algorithm \ref{algo:main} is used for a $7 \times 7$ blur and $\alpha_i\equiv 5$. \begin{figure}
\caption{Normalized MAP criterion (for $\theta=0.25$ in green and $\theta=10$ in magenta) versus computational time (in seconds) (Intel Xeon 4 Core, 3.00 GHz).}
\label{fig:comptheta}
\end{figure} As expected, the larger $\theta$ is, the slower the convergence of the algorithms is. A trade-off has therefore to be made: $\theta$ must be chosen large enough to reach a good restoration quality but it should not be too large in order to get a fast convergence.
\subsection{Second example}\label{sec:poisson} \subsubsection{Model} In this second scenario, we want to restore an image $\overline{y}\in\ensuremath{\left[0,+\infty\right[}^N$ which is corrupted by a linear operator $T\,:\,\ensuremath{\mathcal G} \to \ensuremath{\mathcal G}$, assumed to be nonnegative-valued and, which is embedded in (possibly inhomogeneous) Poisson noise. Thus, the observed image $z = (z^{(i)})_{1 \le i \le N} \in \ensuremath{\mathbb N}^N$ is Poisson distributed, its conditional probability mass function being given by \begin{equation} (\forall i \in \{1,\ldots,N\})(\forall \upsilon \in \ensuremath{\left[0,+\infty\right[}) \qquad \mu_{Z^{(i)} \mid \overline{U}^{(i)}=\upsilon}(z^{(i)}) = \frac{(\alpha_i \upsilon )^{z^{(i)}}}{z^{(i)}!}\exp\big(-\alpha_i\upsilon \big) \label{eq:probaPoisson} \end{equation} where $(\alpha_i)_{1 \le i \le N} \in \ensuremath{\,\left]0,+\infty\right[}^N$ are scaling parameters.
Consequently, using \eqref{eq:defpsii} and \eqref{eq:probaPoisson}, for every $i \in \{1,\ldots,N\}$, we have, when $z^{(i)} > 0$, \begin{equation} (\forall \upsilon \in \ensuremath{\mathbb R})\qquad \psi_i(\upsilon) = \begin{cases} \displaystyle \alpha_i \upsilon-z^{(i)}
+ z^{(i)} \ln \Big(\frac{z^{(i)}}{\alpha_i \upsilon}\Big) & \mbox{if $\upsilon \in \ensuremath{\,\left]0,+\infty\right[}$}\\ +\infty & \mbox{otherwise} \end{cases} \label{eq:KL} \end{equation} and, when $z^{(i)} = 0$, \begin{equation*} (\forall \upsilon \in \ensuremath{\mathbb R})\qquad \psi_i(\upsilon) = \begin{cases} \displaystyle \alpha_i\upsilon & \mbox{if $\upsilon \in \ensuremath{\left[0,+\infty\right[}$}\\ +\infty & \mbox{otherwise.} \end{cases} \end{equation*} As the functions $(\psi_i)_{1\le i \le N}$ are defined up to additive constants, these constants have been chosen in \eqref{eq:KL} so as to obtain the
expression of the classical Kullback-Leibler divergence term \cite{Byrne_CL_1993_tip_iter_irabcem}.\\ In this context, provided that $z\neq 0$, Assumption \ref{as:psi} holds with $\delta = 0$ and $\mathbb{I} =$\linebreak$ \menge{i\in \{1,\ldots,N\}}{z^{(i)} > 0}$ since, for all $i\in \mathbb{I}$, \begin{align*} (\forall \upsilon \in \ensuremath{\,\left]0,+\infty\right[})\qquad \psi_i'(\upsilon) &=\alpha_i - \frac{z^{(i)}}{\upsilon}\\ \psi_i''(\upsilon) & = \frac{z^{(i)}}{\upsilon^{2}} . \end{align*} We deduce from \eqref{eq:defuit} that, for every $i \in \mathbb{I}$, \begin{equation*} (\forall \theta \in \ensuremath{\,\left]0,+\infty\right[})\qquad \upsilon_i(\theta) = \sqrt{\frac{z^{(i)}}{\theta}}. \end{equation*}
\begin{remark} \label{re:anscombe} At this point, it may be interesting to compare the proposed extension with the approach developed in {\rm \cite{Dupe_FX_2008_ip_proximal_ifdpniusr}}. The use of the Anscombe transform {\rm \cite{Anscombe_F_1948_biometrika_trans_pbnbd}}, in {\rm \cite{Dupe_FX_2008_ip_proximal_ifdpniusr}} is actually tantamount to approximating the anti log-likelihood $\psi_i$ of the Poisson distribution by \begin{equation} (\forall \upsilon \in \ensuremath{\mathbb R})\qquad \widetilde{\psi_i}(\upsilon) = \begin{cases} \frac{1}{2} \Big(2 \sqrt{\alpha_i \upsilon+\frac{3}{8}}- 2 \sqrt{z^{(i)}+\frac{3}{8}} \Big)^2 & \mbox{if $\upsilon \in \ensuremath{\left[0,+\infty\right[}$}\\ \ensuremath{+\infty} & \mbox{otherwise.} \end{cases} \end{equation} The proposed quadratic extension is illustrated in Fig.~\ref{fig:extensionPois} where a graphical comparison with the Anscombe approximation is performed. \end{remark} \begin{figure}
\caption{Graph of the function $\psi_i$ (black continuous line) when $\delta = 0$, $\alpha_i = 1$, $z^{(i)} = 100$. Its quadratic extension $\psi_{\theta,i}$ with $\theta= 0.2$ (purple dashed line) and $\theta= 1$ (red dashed line) for $\epsilon(\theta) = 10^{-16}$
and its Anscombe approximation $\widetilde{\psi}_i$ (cyan continuous line). }
\label{fig:extensionPois}
\end{figure}
\subsubsection{Simulation results} Here, $T$ is a $5\times 5$ uniform
blur with $\|T\|=1$. A $256 \times 256$ ($N = 256^2$) medical image $\overline{y}$ shown in Fig. \ref{fig:resPoisson}(a) is degraded by $T$ and corrupted by a Poisson noise following the model described in the previous section for various intensity levels. The degraded image $z$ is displayed in Fig.~\ref{fig:resPoisson}(b) when $\alpha_i\equiv 0.01$.
An orthonormal wavelet basis representation has been adopted using symlets of length $6$ ($\underline{\nu} = \overline{\nu}=1$, $K = N$). The potential functions $\phi_k$ are taken of the same form as in the first example and, the function $f$ is therefore coercive and strictly convex.
The constraint imposed on the solution is given by $C = (F^*)^{-1}C^*$ where $C^*$ is defined by \eqref{eq:constval}. Since $TC^* = C^*$, Proposition \ref{p:gt}\ref{p:gtiii} guarantees that a unique minimizer $\widetilde{x}_\theta$ of $f+g_\theta+\iota_C$ exists, which has been computed with Algorithm \ref{sec:DR(FB)}. The algorithm has been initialized by setting $z_0 = P_C z$ and, we have chosen $\gamma_{m,n} \equiv 1.99/(\ensuremath{\kappa}\theta)$, $\ensuremath{\kappa} = 60$ and $\lambda_{m,n} \equiv \ensuremath{\tau}_m \equiv 1$. The number of forward-backward iterations is given by \eqref{eq:Nmchoice} with $\eta = 10^{-4}$. Note that the convergence rate could be accelerated by using adaptive step-size methods such as the Armijo-Goldstein search \cite{Tseng_P_2000_jco_modified_fbsmfmmm,Dupe_FX_2008_ip_proximal_ifdpniusr}. However, the computational time of the step-size determination should be taken into account.
To evaluate the performance of our algorithm we use the Signal to Noise Ratio defined in Section \ref{sec:simulSigDep}. Tab. \ref{tab:snr_sebal} shows the values of the $\mathrm{SNR}$ obtained for different values of $\alpha_i$ and $\theta$. As predicted by Proposition \ref{p:gt}\ref{p:gtv}, beyond some value of $\theta$, which is dependent of $\alpha_i$,
the optimal value is found. We also compare our results with those provided by two different approaches. The first one is the regularized Expectation Maximization (EM) approach (also sometimes called SMART) \cite{Byrne_CL_1993_tip_iter_irabcem,Lange_K_1987_tmi_theoretical_atsosmlafeatt} where the Poisson anti-likelihood penalized by a term proportional to the Kullback-Leibler divergence between the desired solution and a reference image is minimized. Its weighting factor has been adjusted manually so as to maximize the $\mathrm{SNR}$ and, the reference image is a constant image whose pixel values has been set to the mean value of the degraded image. The other approach is the method based on the Anscombe transform proposed in \cite{Dupe_FX_2008_ip_proximal_ifdpniusr} and discussed in Remark \ref{re:anscombe}. For fair comparisons, the method here employs the same orthonormal wavelet representation, the same functions $(\phi_k)_{1\le k \le K}$ as ours and the same constraint set $C$.
It can be observed that the approach we propose gives good results. However, for high intensity levels ($\alpha_i \ge 0.1$), the method based on the Anscombe transform performs equally well in terms of SNR. The restored images are shown in Fig. \ref{fig:resPoisson}, when $\alpha_i \equiv 0.01$ and $\theta \equiv 0.001$ after 3000 iterations. In spite of an important degradation of the original image, it can be seen that our approach is able to recover the main features in the image. It can also be noticed that the image restored by the two methods exhibit different visual characteristics.
\begin{table}[htbp] \centering
\begin{tabular}{| p{0.5cm}||c|| c||c|c|c|c|c|} \hline
& Regularized &Anscombe & \multicolumn{5}{c|}{Quadratic extension}\\ \cline{4-8} $\alpha_i$&EM & &$\theta=0.001$ & $\theta=0.005$ & $\theta=0.1$ & $\theta=1$ & $\theta=5$\\ \hline \hline $ 0.01$ & 6.47 & 8.24 & \textbf{9.75} & 9.75 & 9.75 & 9.75 & 9.75\\ \hline $0.05$ & 9.01 & 11.5 & 11.7 & \textbf{11.9} & 11.9 & 11.9 & 11.9 \\ \hline $0.1$ & 10.1 & 12.4 & 12.0 & \textbf{12.5} & 12.5 & 12.5 & 12.5\\ \hline $1$ & 13.8 & \textbf{15.1} & 0 & 10.1 & 13.7 & \textbf{15.1} & 15.1\\ \hline \end{tabular} \caption{$\mathrm{SNR}$ for the medical image.} \label{tab:snr_sebal} \end{table}
\begin{figure}
\caption{Results on the medical image. (a) Original, (b) degraded, (c) restored with EM, (d) restored with Anscombe transform and (e) restored with quadratic extension. }
\label{fig:resPoisson}
\end{figure}
\section{Conclusion} Two main problems have been addressed in this paper.
The first one concerns the minimization on a convex set $C$ of a sum of two functions, one of which $g$ being smooth while the other may be nonsmooth. Such a constrained minimization has been performed by combining forward-backward and Douglas-Rachford iterations. Various combinations of these algorithms can be envisaged and the study we made tends to show that Algorithm~\ref{algo:main} is a good choice. It can be noticed that adding a constraint on the solution for a restoration problem was shown to be useful in another work \cite{Pustelnik_N_2008_peusipco_constrained_fbairp}, where it appeared that the visual quality of the restored image can be much improved w.r.t. the unconstrained case, when both restoration approaches are applicable.
The second point concerns the quadratic lower approximation technique we have proposed. This method offers a means of applying the proposed algorithms in cases when $g$ is differentiable on $C$ but the gradient of $g$ is not necessary Lipschitz continuous on $C$. By quadratically extending $g$, the proposed constrained minimization algorithms can be used. This extension depends on a parameter $\theta$ which controls the precision (closeness to the solution of the original minimization problem) and the convergence speed of the algorithm. As illustrated by the simulations, the choice of this parameter should result from a trade-off. The numerical results have also shown the efficiency of the proposed methods in deconvolution problems involving a signal-dependent Gaussian noise or a Poisson noise.
Finally, it may be interesting to note that nested iterative algorithms similar to those developed in this paper can be used to solve $\min_{x\in\ensuremath{{\mathcal H}}} f + g + h$ where $\ensuremath{{\mathcal H}}$ is a real separable Hilbert space, $f$, $g$ and $h$ are functions in $\Gamma_0(\ensuremath{{\mathcal H}})$ and $g$ is $\beta$-Lipschitz differentiable.
\begin{appendix}
\section{Study of Example \ref{ex:ce1}} \label{ap:exce1} Let $p=\ensuremath{\mathrm{prox}}_{f} x$ and $q = \ensuremath{\mathrm{prox}}_{f+\iota_C} x$ where $x \in \ensuremath{{\mathcal H}}$. Let $g$ be the convex function defined by $(\forall y \in \ensuremath{{\mathcal H}})$
$g(y) = \frac{1}{2}\left\|y-x\right\|^2 + \frac{1}{2}y^{\top}\Lambda y$. Consequently, $p=\big(I+\Lambda\big)^{-1}x$ is the minimizer of $g$ on $\ensuremath{{\mathcal H}}$, whereas $q$ is the minimizer of $g$ on $C$. Thus, we can write $(\forall y \in \ensuremath{{\mathcal H}})$ $g(y) = \tilde{g}(y)+h_x$ where $\tilde{g}(y) = \frac{1}{2}(y-p)^{\top}(I+\Lambda)(y-p)$ and $h_x$ is a function of $x$. Then, $q$ also minimizes $\tilde{g}$ on $C$.\\ In the example, we have chosen $x=2(\Lambda_{1,2},1+\Lambda_{2,2})^{\top}$, which yields $ p = (0,2)^{\top}$ and $P_C(p) = (0,1)^{\top}$.\\ Let $\tilde{q} = (\pi,1)^{\top}$. To show that $q = \tilde{q}$, we have check that $\tilde{q}$ minimizes $\tilde{g}$ on $C$. A necessary and sufficient condition for the latter property to be satisfied \cite[p. 293, Theorem 1.1.1]{Hiriart_Urruty_1996_book_convex_amaIf} is that \begin{equation*} (\forall y \in C)\qquad \big(\nabla\tilde{g}(\tilde{q})\big)^\top(y-\tilde{q}) \ge 0 \end{equation*} where $\nabla\tilde{g}(\tilde{q}) = (I+\Lambda)(\tilde{q}-p)$ is the gradient of $\tilde{g}$ at $\tilde{q}$. This is equivalent to prove that \begin{equation} (\forall (y^{(1)},y^{(2)})^\top \in C)\qquad (2\pi-\Lambda_{1,2})(y^{(1)}-\pi) + (\Lambda_{1,2} \pi - \Lambda_{2,2}-1)(y^{(2)}-1) \geq 0. \label{eq:minequac} \end{equation} Three cases must be considered: \begin{itemize} \item when $\Lambda_{1,2}<-2$, $(y^{(1)},y^{(2)})^\top \in C$ $\Rightarrow$ $y^{(1)} \ge -1 = \pi$ and $y^{(2)} \le 1$. In addition, we have $2\pi-\Lambda_{1,2} = -2 -\Lambda_{1,2} > 0$ and $\Lambda_{2,2}-\Lambda_{1,2}^{2}\geq 0$ $\Rightarrow$ $\Lambda_{1,2} \pi - \Lambda_{2,2}-1 \le -\Lambda_{1,2}^{2}-\Lambda_{1,2}-1 < 0$. So, \eqref{eq:minequac} holds. \item When $\Lambda_{1,2}>2$, similar arguments hold. \item When $\Lambda_{1,2}\in[-2,2]$, $2\pi-\Lambda_{1,2} = 0$ and $\Lambda_{1,2} \pi - \Lambda_{2,2}-1 = \frac{\Lambda_{1,2}^2}{2} - \Lambda_{2,2}-1 \le -\frac{\Lambda_{1,2}^2}{2}-1 \le 0$, which shows that \eqref{eq:minequac} is satisfied. \end{itemize} This leads to the conclusion of Example \ref{ex:ce1}.
\section{Study of Example \ref{ex:ce2}}\label{ap:exce2} Let $f$ be the function defined in Example~\ref{ex:ce2}. Defining the rotation matrix $R=\frac{1}{\sqrt 2}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$, this function can be expressed as \begin{equation*} (\forall x\in \ensuremath{\mathbb R}^{2}) \qquad f(x) = \tilde{f}(Rx) \end{equation*} where $\tilde{f}(x) = \frac{1}{2} x^\top \Lambda x$ with \begin{equation*} \Lambda = \begin{pmatrix} 1 & \Lambda_{1,2}\\ \Lambda_{1,2} & 1 \end{pmatrix}. \end{equation*} In addition, \begin{equation*} C = \{x\in \ensuremath{\mathbb R}^{2}\;\mid \;Rx\in [-1,1]^2\} = R^\top [-1,1]^2. \end{equation*} It can be noticed that $[-1,1]^2$ is the separable convex set considered in Example \ref{ex:ce1} whereas $\tilde{f}$ appears as a particular case in the class of quadratic functions considered in this example (by setting $\Lambda_{2,2} = 1$).\\ Thus, the proximity operator of $f$ is \begin{align*} (\forall x \in \ensuremath{{\mathcal H}})\qquad
\ensuremath{\mathrm{prox}}_f x = &\arg\min_{y\in \ensuremath{{\mathcal H}}} \frac{1}{2} \|x-y\|^2 + f(y)\nonumber\\
= & \arg\min_{y\in \ensuremath{{\mathcal H}}} \frac{1}{2} \|Rx-Ry\|^2 + \tilde{f}(Ry) = R^\top \ensuremath{\mathrm{prox}}_{\tilde{f}}(Rx). \end{align*} and $P_C(\ensuremath{\mathrm{prox}}_f x) = R^\top P_{[-1,1]^2}(R\ensuremath{\mathrm{prox}}_f x) = R^\top P_{[-1,1]^2}\big(\ensuremath{\mathrm{prox}}_{\tilde{f}}(R x)\big)$. Similarly, we have \begin{equation*} (\forall x \in \ensuremath{{\mathcal H}})\qquad \ensuremath{\mathrm{prox}}_{f+\iota_C} x = R^\top \ensuremath{\mathrm{prox}}_{\tilde{f}+\iota_{[-1,1]^2}}(Rx). \end{equation*} So, if $x = 2R^\top (\Lambda_{1,2},2)^\top = \sqrt{2}(2+\Lambda_{1,2},2-\Lambda_{1,2})^\top$, we deduce from Example \ref{ex:ce1} that $P_C(\ensuremath{\mathrm{prox}}_f x) = \frac{1}{\sqrt{2}}(1,-1)^\top$ and $\ensuremath{\mathrm{prox}}_{f+\iota_C} x = \frac{1}{\sqrt{2}} (1+\pi+1,1-\pi)^\top$, where the expression of $\pi$ is given by \eqref{eq:defpi}. It can be concluded that $P_C(\ensuremath{\mathrm{prox}}_f x)\neq \ensuremath{\mathrm{prox}}_{f+\iota_C} x$.
\end{appendix}
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\begin{document}
\setlength{\parindent}{0cm}
\title{Medium-assisted van der Waals dispersion interactions involving chiral molecules}
\author{Hassan Safari} \address{Department of Photonics, Graduate University of Advanced Technology, Kerman, Iran}
\author{Pablo Barcellona} \address{Physikalisches Institut, Albert-Ludwigs-Universit\"at
Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany}
\author{Stefan Yoshi Buhmann} \address{Physikalisches Institut, Albert-Ludwigs-Universit\"at Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany} \address{Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universit\"at Freiburg, Albertstr. 19, 79104 Freiburg, Germany}
\author{A.~Salam} \address{Department of Chemistry, Wake Forest University, Winston-Salem, North Carolina 27109, USA} \address{Physikalisches Institut, Albert-Ludwigs-Universit\"at Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany} \address{Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universit\"at Freiburg, Albertstr. 19, 79104 Freiburg, Germany}
\begin{abstract} The van der Waals dispersion interaction between two chiral molecules in the presence of arbirary magnetoelectric media is derived using perturbation theory. To be general, the molecular polarisabilities are assumed to be of electric, paramagnetic and diamagnetic natures and the material environment is considered to possess a chiral electromagnetic response. The derived formulas of electric--chiral, paramagnetic--chiral, diamagnetic--chiral and chiral--chiral interaction potentials when added to the previously obtained contributions in literature, form a complete set of dispersion interaction formulas. We present them in a unified form making use of electric--magnetic duality. As an application, the case of two anisotropic molecules in free space is considered where we drive the retarded and non-retarded limits with respect to intermolecular distance.
\end{abstract} \pacs{41.20.Cv, 03.50.De, 42.50.Nn, 12.20.-m} \submitto{\NJP}
\maketitle
\section{Introduction} \label{intro} The non-superposability of an object on its mirror image classifies it as chiral. A familiar example is provided by left and right hands. Molecules that are chiral lack an improper axis of rotation, exist as enantiomeric pairs, and exhibit optical activity \cite{atkins1968,power1971, caldwell1971, charney1985,Barron2009}, that is, they are able to rotate the plane of polarization of light either to the left or to the right, and are so termed laevorotatory or dextrorotatory. Other manifestations of molecular handedness include differential absorption (circular dichroism) \cite{Power1974} and differential scattering (Rayleigh and Raman) \cite{Barron1971} of circularly polarized light, their nonlinear analogues \cite{fischer2005}, as well as other chiroptical spectroscopies that depend quadratically or on higher powers of the strengths of electromagnetic fields, such as sum-frequency and second-harmonic generation \cite{giordmaine1965, fischer2010, andrews2018-1}. Many of these phenomena have also been predicted to occur when the incident radiation is of the structured type \cite{andrews2004,cameron2014,cameron2017,forbes2018,babiker2018,forbes2019-1, forbes2019-2, ForbesUnpublished}.
Because chiral compounds possess reduced or no elements of symmetry, selection rules normally in operation that are used to determine whether spectroscopic transitions are allowed or forbidden are considerably relaxed, or no longer apply. This enables multipole moments such as the magnetic dipole and electric quadrupole to feature to leading order in chiroptical processes, in addition to the usually dominant electric dipole moment. Accounting from the outset for both the charge and current density distributions of a collection of protons and electrons that form atoms and molecules is therefore necessary. Furthermore, Maxwell's equations are needed to describe intrinsic electromagnetic effects due to the sources, as well as those arising from any applied radiation fields. Due to the microscopic nature of such elementary charged particles, both radiation and matter ought to be treated rigorously subject to the laws of quantum mechanics \cite{dirac1958}. Hence the theory of molecular QED \cite{craig2012,salam2010,andrews2018-2} naturally lends itself as the obvious means by which to investigate fundamentally the interactions between photons and electrons, and any effects due to the handedness of molecules.
Non-relativistic QED theory in the Coulomb gauge has not only been successfully applied to a whole host of linear and nonlinear optical processes, but also permits inter-particle interactions to be studied using the same formalism. Well-known examples include resonant transfer of electronic excitation energy \cite{craig2012, salam2010, craig1992, daniels2003, grinter2016, salam2018}, and dispersion forces between two and three atoms or molecules \cite{craig2012, salam2010, casimir1948, aub1960, power1993, passante1999, salam2016, passante2018}. For interactions between enantiomers, be they chemically identical or distinct species, both energy transfer and the dispersion energy shift are discriminatory and depend upon the handedness of the molecules \cite{mavroyannis1962,craig1971,craig1998-2,jenkins1994-1, jenkins1994-2, salam1996, salam2000, salam2005}. According to QED theory, the interaction between particles is mediated by the exchange of virtual photons -- by definition unobservable but permitted by the time-energy uncertainty principle \cite{andrews2014,salam2015}. In the case of migration of energy, a single virtual photon is responsible for conveying energy from an excited donor moiety to an unexcited acceptor entity. This compares with the dispersion force, which is understood to arise from the exchange of two virtual photons.
In the presence of an environment, the mediating photons are no longer free-space light quanta associated with the vacuum electromagnetic field but become modified by the medium. One approach for dealing with this is to affect a scheme in which the radiation field and the surrounding environment are quantized, resulting instead in medium-assisted photons. This formed the basis for the construction of a macroscopic QED theory in which a magnetoelectric medium is accounted for and described by frequency dependent electric permittivity and magnetic permeability functions \cite{matloob1995,scheel2008, buhmann2013-1,buhmann2013-2,buhmann2012}. As in molecular QED, its macroscopic counterpart has been cast in terms of minimal- and multipolar-coupling frameworks; it has been applied with great advantage to the calculation of Casimir, Casimir--Polder, and van der Waals forces. For the last type of interaction, this has specifically included the evaluation of pair dispersion potentials in a magnetoelectric medium involving atoms or molecules that are either electrically polarizable or paramagnetically and diamagnetically susceptible, and their behavior has been examined as a function of interparticle separation distance in the non-retarded and retarded coupling regimes \cite{safari2006, buhmann2013-3}.
Within the context of interactions involving optically active species, up to now macroscopic QED has only been employed to study Casimir--Polder forces \cite{Butcher2012, Barcellona2016}, i.e.~between a microscopic particle and a macroscopic object, and the van der Waals dispersion energy between an electrically polarizable molecule and a chiral molecule. While it is known that the last mentioned potential vanishes in free space for an isotropic pair of interacting molecules, it was shown recently that a non-zero energy shift arises by specifically choosing the environment to be a chiral plate \cite{Barcellona2017}.
Not only is the energy finite, but its magnitude and sign may even be controlled and its behavior studied for particular configurations of the chiral molecules and the achiral molecule relative to the chiral plate. For instance, there is maximal enhancement of the interaction when the two particles are aligned parallel to the plate but are each separated a large distance from it. This offers a realistic possibility for the separation of enantiomers, complementing a recent proposal exploiting parity violation in the Casimir--Polder potential when a beam of chiral molecules passes through a pair of chiral mirrors \cite{suzuki2019}.
In the present work, we develop a general and complete macroscopic QED theory for the van der Waals dispersion interaction between two molecules
with arbitrary linear dipolar response
that may or may not be chiral and are described by their electric polarisability, para- and diamagnetic susceptibility, and mixed electric--magnetic dipole (chiral) susceptibility in a magnetoelectric medium. Fourth-order diagrammatic perturbation theory is employed to compute the interaction energy, and explicit contributions involving one or two chiral species are extracted. On inserting the free-space form of the Green's tensor, it is found that the results obtained reduce to those previously derived using molecular QED \cite{salam2010}.
The paper is organized as follows: The basic formalism for QED in the presence of chiral magnetoelectric media together with the introduction of molecule-field interaction Hamiltonians are given in Sec.~\ref{formalism}. In Sec.~\ref{potential} the derivation of electric--chiral, paramagnetic--chiral, and chiral--chiral van der Waals dispersion potentials are presented via fourth order perturbation theory, followed by diamagnetic--chiral interaction for which the third order perturbation theory is applied. A unified form of the formulas of various dispersion interaction potentials is derived in
Sec.~\ref{duality} using the electric--magnetic duality. As an application of the obtained formulas, the interaction between two anisotropic molecules in free space is computed in Sec.~\ref{freespace} where also the retarded and non-retarded limits of intermolecular distances are considered. A summary and concluding remarks are provided in Sec.~\ref{conclusion}.
\section{Basic Formalism} \label{formalism} The Hamiltonian of a system comprised of two molecules $A$ and $B$ in the presence of a quantized electromagnetic field is given as \begin{equation} \label{Ham} \hat{H} = \hat{H}_F + \sum_{A'=\mathrm{A},\mathrm{B}}\hat{H}_{A'} + \sum_{A'=\mathrm{A},\mathrm{B}}\hat{H}_{A'F}, \end{equation} where $\hat{H}_F$, $\hat{H}_{A'}$, and $\hat{H}_{A'F}$ denote, respectively, field, molecule, and molecule--field interaction Hamiltonians. In terms of fundamental photonic annihilation and creation operators, $\hat{\vect{f}}_\lambda$ and $\hat{\vect{f}}_\lambda^\dagger$, the field Hamiltonian reads \begin{equation} \label{HF} \hat{H}_F=\sum_{\lambda=e,m}\int\mathrm{d}^3 r\int_0^\infty \mathrm{d}\omega \hbar\omega\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)\cdot \hat{\vect{f}}_\lambda(\vect{r},\omega) \end{equation} with $\lambda$ referring to the electric $(\lambda=e)$ or magnetic $(\lambda=m)$ nature of the noise source. These operators obey the following commutation relations: \begin{eqnarray} &&\label{comm1} \left[\hat{\vect{f}}_\lambda(\vect{r},\omega),\hat{\vect{f}}_{\lambda'}(\vect{r}',\omega')\right] =\left[\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega) ,\hat{\vect{f}}_{\lambda'}^\dagger(\vect{r}',\omega')\right]=\bm{0},\\ && \label{comm2}\left[\hat{\vect{f}}_\lambda(\vect{r},\omega),\hat{\vect{f}}_{\lambda'}^\dagger(\vect{r}',\omega')\right] =\tens{I}\delta_{\lambda\lambda'}\delta(\vect{r}-\vect{r}') \delta(\omega-\omega'), \end{eqnarray}
with $\tens{I}$ being the identity matrix. The ground-state of the field $|\{0\}\rangle$ is defined by \begin{equation}
\hat{\vect{f}}_\lambda(\vect{r},\omega)|\{0\}\rangle=0,\quad \forall\, \vect{r},\omega,\lambda \end{equation} and the excited photonic states are defined by repeated application of the creation operator to the ground-state. For example, for single- and two-photon excited states, \begin{eqnarray} &&\label{eq6}
|\bm{1}_\lambda(\vect{r},\omega)\rangle = \hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)|\{0\}\rangle,\\
&&\label{eq7}|\bm{1}_\lambda(\vect{r},\omega),\bm{1}_{\lambda'}(\vect{r}',\omega')\rangle =
\frac{1}{\sqrt{2}}\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)\hat{\vect{f}}_{\lambda'}^\dagger(\vect{r}',\omega')|\{0\}\rangle. \end{eqnarray} The molecular Hamiltonian may be written in terms of unperturbed molecular eigenenergies and eigenstates as \begin{equation} \label{HA}
\hat{H}_{A'} =\sum_k E_{A'}^k| k_{A'}\rangle\langle k_{A'}|, \quad A'=\mathrm{A},\mathrm{B}. \end{equation} In the multipolar coupling scheme, the molecule--field interaction Hamiltonian takes the form \cite{buhmann2013-3} \begin{equation} \label{HA'} \hat{H}_{A'F} = -\hat{\vect{d}}_{A'}\cdot\hat{\vect{E}}(\vect{r}_{A'}) -\hat{\vect{m}}_{A'}\cdot\hat{\vect{B}}(\vect{r}_{A'}) -\frac{1}{2}\hat{\vect{B}}(\vect{r}_{A'}) \cdot\hat{\bm{\beta}}^D_{A'}\cdot\hat{\vect{B}}(\vect{r}_{A'}), \end{equation} with $\hat{\vect{d}}_{A'}$ and $\hat{\vect{m}}_{A'}$ being, respectively, electric and magnetic dipole moment operators of the molecule $A'$ and $\hat{\vect{\beta}}^D_{A'}$ is its operator-valued diamagnetisability tensor \begin{equation} \label{beta} \hat{\bm{\beta}}^D_{A'} = -\sum_{\alpha\in A'}\frac{q_\alpha^2}{4m_\alpha}(\hat{\bar{r}}_\alpha^2\bm{I}-\hat{\bar{\vect{r}}}_\alpha \hat{\bar{\vect{r}}}_\alpha)\,, \end{equation} where $q_\alpha$ and $m_\alpha$ are, respectively, the electric charge and mass of the particle $\alpha$, and $\hat{\bar{\vect{r}}}_\alpha$ is its position vector relative to the center of mass of the molecule. The electric and magnetic fields can be expressed as linear combinations of the fundamental photonic operators. To this end, we introduce $\bm{\mathcal{R}}_{\lambda\lambda'}(\vect{r},\omega)$ ($\lambda,\lambda' = e,m$) as the $3\times 3$ blocks of the $6\times 6$ tensor $\bm{\mathcal{R}}(\vect{r},\omega)$, \begin{equation} \bm{\mathcal{R}} = \left( \begin{matrix} \bm{\mathcal{R}}_{ee} &
\bm{\mathcal{R}}_{em}\\ \bm{\mathcal{R}}_{me} & \bm{\mathcal{R}}_{mm} \end{matrix} \right), \end{equation} which is the response tensor of the magnetoelectric material environment \cite{Butcher2012} defined, for locally responding media, according to \begin{equation} \label{R} \bm{\mathcal{R}}\cdot \bm{\mathcal{R}}^\top = \left( \begin{matrix} \varepsilon_0\mathrm{Im}\,(\bm{\varepsilon}- \bm{\kappa}^\top\cdot\bm{\mu}^{-1}\cdot\bm{\kappa}) &
-\mathrm{i}\sqrt{\frac{\varepsilon_0}{\mu_0}}\mathrm{Im}\,(\bm{\kappa}^\top\cdot\bm{\mu}^{-1})\\ \mathrm{i}\sqrt{\frac{\varepsilon_0}{\mu_0}}\mathrm{Im}\,(\bm{\mu}^{-1}\cdot\bm{\kappa}) & -\frac{1}{\mu_0}\mathrm{Im}\, \bm{\mu}^{-1} \end{matrix} \right). \end{equation} In Eq.~\eqref{R} $\bm{\varepsilon}=\bm{\varepsilon}(\vect{r},\omega)$ and $\bm{\mu}=\bm{\mu}(\vect{r},\omega)$ are, respectively, the relative electric permittivity and magnetic permeability tensors of the media, and $\bm{\kappa}=\bm{\kappa}(\vect{r},\omega)$ is the chirality tensor responsible for the contribution of an electric effect to a magnetic response and vise versa. The electric and magnetic fields are given as \begin{eqnarray} &&\label{E} \hat{\vect{E}}(\vect{r})=\sum_{\lambda=e,m}\int\mathrm{d}^3 r'\int_0^\infty \mathrm{d}\omega \tens{G}_\lambda(\vect{r},\vect{r}',\omega)\cdot\hat{\vect{f}}_\lambda(\vect{r}',\omega)+ H.c.,\\ &&\label{B} \hat{\vect{B}}(\vect{r})=\sum_{\lambda=e,m}\int\mathrm{d}^3 r'\int_0^\infty \frac{\mathrm{d}\omega}{\mathrm{i}\omega}\nabla\times \tens{G}_\lambda(\vect{r},\vect{r}',\omega)\cdot \hat{\vect{f}}_\lambda(\vect{r}',\omega)+ H.c.\,, \end{eqnarray} where $\tens{G}_{e}$ and $\tens{G}_m$ are the mode tensors introduced in terms of the Green tensor $\tens{G}$, \begin{multline} \label{Ge}
\tens{G}_\lambda(\vect{r},\vect{r}',\omega)\\ =
-\mathrm{i}\mu_0\omega\sqrt{\frac{\hbar}{\pi}}
\left[\mathrm{i}\omega\tens{G}(\vect{r},\vect{r}',\omega)\cdot
\bm{\mathcal{R}}_{\lambda e}(\vect{r}',\omega)
+
\tens{G}(\vect{r},\vect{r}',\omega)\times\overleftarrow{\nabla}'\cdot
\bm{\mathcal{R}}_{\lambda m}(\vect{r}',\omega)\right], \end{multline} with $\overleftarrow{\nabla}'$ denoting differentiation from the left. The Green tensor $\tens{G}(\vect{r},\vect{r}',\omega)$ obeys the differential equation \begin{eqnarray} &\hspace{-.9in}\nabla\times\bm{\mu}^{-1}\cdot\nabla\times\tens{G} +\frac{\omega}{c}\nabla\times\bm{\mu}^{-1}\cdot\bm{\kappa} \cdot\tens{G} +\frac{\omega}{c}\bm{\kappa}^\top\cdot\bm{\mu}^{-1}\cdot \nabla\times\tens{G} \nonumber\\ &\hspace{.6in}-\frac{\omega^2}{c^2}\left(\bm{\varepsilon} -\bm{\kappa}^\top\cdot\bm{\mu}^{-1}\cdot\bm{\kappa}\right)\cdot\tens{G} =\tens{I}\delta(\vect{r}-\vect{r}'), \end{eqnarray} ($\textit{\textsf T}^\top_{ij} ={\textit{\textsf T}}_{ji}$). All geometric and magnetoelectric properties of the environment are taken into account via $\bm{\varepsilon}(\vect{r},\omega)$, $\bm{\mu}(\vect{r},\omega)$, and $\bm{\kappa}(\vect{r},\omega)$. Furthermore, the Green tensor obeys the Schwarz reflection principle \begin{equation} \label{Schwartz} \tens{G}(\vect{r},\vect{r}',-\omega) =\tens{G}^\ast(\vect{r},\vect{r}',\omega^\ast), \end{equation} Onsager reciprocity \begin{equation} \label{Onsager} \tens{G}(\vect{r},\vect{r}',\omega) =\tens{G}^\top(\vect{r}',\vect{r},\omega), \end{equation} and a useful integral relation \cite{dung2003, buhmann2004} \begin{equation} \label{intrel} \sum_{\lambda=e,m}\int\mathrm{d}^3 s \tens{G}_\lambda(\vect{r},\vect{s},\omega)\cdot \tens{G}_\lambda^{\ast\top}(\vect{r}',\vect{s},\omega) = \frac{\hbar\mu_0\omega^2}{\pi}\,\mathrm{Im}\, \tens{G}(\vect{r},\vect{r}',\omega). \end{equation} These properties will be used in the evaluation of matrix elements appearing in perturbation formulas. \section{Interaction Potential} \label{potential}
For ground-state molecules the van der Waals dispersion interaction is mediated by two virtual-photon exchanges between the molecules accompanied by internal transitions. In achiral molecules the molecular eigenstates are also the eigenstates of the parity operator. Hence, the resulting expression for the interaction potential is obtained such that every molecule may be considered as a superposition of an electric, a paramagnetic, and a diamagnetic contribution for which the molecular transitions are, respectively, electric, paramagnetic, or diamagnetic type only (see Ref.~\cite{buhmann2013-3}). For chiral molecules, however, each molecular transition may possess interdependent electric and magnetic dipole moments and additional higher multipole moments. This leads to an additional chiral contribution to the interaction potential, to which the rest of this section is devoted.
In order to calculate the interaction energy between two ground-state molecules $\mathrm{A}$ and $\mathrm{B}$ in the presence of a medium-assisted electromagnetic field we use perturbation theory. To do so, we consider the sum of the field and molecular Hamiltonians as unperturbed Hamiltonian, and the sum of the molecule--field interaction Hamiltonians, $\hat{H}_{\mathrm{AF}} \!+\!\hat{H}_{\mathrm{BF}}\equiv\hat{H}_{\rm{int}}$, as perturbation. The electric or paramagnetic transitions in the molecules depend linearly on the electric and magnetic fields, respectively, and are associated with the absorption or emission of a single virtual photon, while the diamagnetic coupling involves a two-photon transition. Hence, in the calculation of the energy shift, different orders of perturbation theory are required for diamagnetic molecules in comparison to electric and paramagnetic ones.
\subsection{Electric--chiral, paramagnetic--chiral, and chiral--chiral interaction} The lowest order perturbation leading to the interaction potential, involving an electric, paramagnetic, or a chiral molecule that interacts with a second species which
is chiral via two virtual photon exchange is the fourth order \cite{buhmann2013-3, safari2006}, \begin{equation} \label{p4} U(\vect{r}_{\mathrm{A}},\vect{r}_{\mathrm{B}})= -\sum_{I,I\!I,I\!I\!I\neq0} \hspace{-1ex}
\frac{\langle 0|\hat{H}_{\rm{int}}
|I\!I\!I\rangle
\langle I\!I\!I|\hat{H}_{\rm{int}}|I\!I\rangle\langle I\!I|\hat{H}_{\rm{int}}|I\rangle
\langle I|\hat{H}_{\rm{int}}|0\rangle} {(E_{I\!I\!I}-E_0)(E_{I\!I}-E_0)(E_{I}-E_0)}\,, \end{equation}
with the ground state of the total system denoted by $|0\rangle\equiv|0_\mathrm{A}\rangle|0_\mathrm{B}\rangle|\{0\}\rangle$. The numerator is comprised of multiplication of four matrix elements, each corresponding to an event in which one of the molecules undergoes a transition accompanied by emission or absorption of a virtual photon. The photon emitted by one of the molecules has to be absorbed by the other to contribute to the coupling between molecules. In order to take into account a complete set of intermediate states in calculating the right hand side of Eq.~(\ref{p4}), one may use a diagrammatic language depending on the time-ordered sequence of the propagation of two virtual photons, as sketched in Fig.~\ref{fig1}. \begin{figure}
\caption{ Two-virtual photon exchange diagrams contributing to the van der Waals dispersion potential. The various intermediate states required in Eq.~(\ref{p4}) are readily obtained.}
\label{fig1}
\end{figure} In every single diagram shown, each molecule undergoes two transitions, from the ground state to an excited state and back.
The chirality contribution of a molecule to the interaction potential comes from processes in which an electric upward transition in the molecule is followed by a magnetic downward one in the same molecule, or vice versa. To this order of multipole approximation it can be decomposed into three parts: electric--chiral, paramagnetic--chiral, and chiral--chiral, each of which is examined below.
\paragraph{Electric--chiral interaction.} Let us consider an electric molecule $\mathrm{A}$ and a chiral molecule $\mathrm{B}$ located respectively at positions $\vect{r}_\mathrm{A}$ and $\vect{r}_\mathrm{B}$ in the presence of an arbitrary arrangement of magnetoelectric media. The molecule--field interaction Hamiltonians reduce to \begin{align} \label{eq19} \hat{H}_{\mathrm{A} F} &= -\hat{\vect{d}}_{\mathrm{A}}\cdot\hat{\vect{E}}(\vect{r}_{\mathrm{A}}),\\ \label{eq20}\hat{H}_{\mathrm{B} F} &=-\hat{\vect{d}}_{\mathrm{B}}\cdot\hat{\vect{E}}(\vect{r}_{\mathrm{B}}) -\hat{\vect{m}}_{\mathrm{B}}\cdot\hat{\vect{B}}(\vect{r}_{\mathrm{B}}). \end{align} We first calculate the contribution from every single diagram in Fig.~(\ref{fig1}) in the right hand side of Eq.~(\ref{p4}). For example, for diagram (1) in Fig.~\ref{fig1}, the respective intermediate states read as follows \begin{align} \label{eq21}
|I\rangle &=|m_\mathrm{A}\rangle|0_\mathrm{B}\rangle
|1_{\lambda_1 i_1}(\vect{r}_1,\omega_1)\rangle,\nonumber\\
|I\!I\rangle & = |0_\mathrm{A}\rangle|0_\mathrm{B}\rangle
|1_{\lambda_2i_2}(\vect{r}_2,\omega_2),1_{\lambda_3 i_3}(\vect{r}_3,\omega_3)\rangle, \nonumber\\
|I\!I\!I\rangle & = |0_\mathrm{A}\rangle|l_\mathrm{B}\rangle|1_{\lambda_4 i_4}(\vect{r}_4,\omega_4)\rangle, \end{align}
where $|m_\mathrm{A}\rangle$ and $|l_\mathrm{B}\rangle$ denote excited molecular states. The denominator is equal to $\hbar^3(\omega_\mathrm{A}^{m}+\omega_1) (\omega_2+\omega_3)(\omega_\mathrm{B}^{l}+\omega_4)$ with $\omega_{A'}^{i}\equiv(E_{A'}^i-E_{A'}^0)/\hbar$ denoting the transition frequency of molecule $A'$.
Substitution of Eq.~(\ref{eq19}) together with Eq.~(\ref{E}) for the electric field operator, making use of the definitions (\ref{eq6}) and (\ref{eq7}) for single-- and two--photon states of the electromagnetic field, and applying the commutation relations (\ref{comm1}) and (\ref{comm2}), we find \begin{align} \label{eq22}
\langle I|\hat{H}_{\mathrm{A} F}|0\rangle=&- \left[\vect{d}_{\mathrm{A}}^{m0}\cdot\tens{G}^\ast_{\lambda_1}(\vect{r}_\mathrm{A},\vect{r}_1,\omega_1)\right]_{i_1},\\ \label{eq23}
\langle I\!I|\hat{H}_{\mathrm{A} F}|I\rangle =&-\frac{1}{\sqrt{2}}\left[\vect{d}_\mathrm{A}^{0m}\cdot\tens{G}_{\lambda_2}^\ast(\vect{r}_\mathrm{A},\vect{r}_2,\omega_2)\right]_{i_2}\delta^{(31)}\nonumber\\ &-\frac{1}{\sqrt{2}}\left[\vect{d}_\mathrm{A}^{0m}\cdot\tens{G}_{\lambda_3}^\ast(\vect{r}_\mathrm{A},\vect{r}_3,\omega_3)\right]_{i_3}\delta^{(21)}, \end{align}
where $\vect{d}_{\mathrm{A}}^{ij}=\langle i_\mathrm{A}|\hat{\vect{d}}_{\mathrm{A}}|j_\mathrm{A}\rangle$ is the transition electric dipole moment,
and we have introduced the short--hand notation $\delta^{(\alpha\beta)}$ $\!=$ $\!\delta(\vect{r}_\alpha-\vect{r}_\beta)$ $\!\delta(\omega_\alpha-\omega_\beta)$ $\!
\delta_{i_\alpha i_\beta}$ $\!\delta_{\lambda_\alpha \lambda_\beta}$. The two photons present in state $|I\!I\rangle$ are to be absorbed by the chiral molecule $\mathrm{B}$. Diagram (1) refers to two cases of (1)$a$ and (1)$b$ as shown in Fig.~\ref{fig2}, differing in the time--ordering of the electric--dipole and magnetic--dipole transitions in the chiral atom $\mathrm{B}$. \begin{figure}
\caption{ Two processes corresponding to diagram $(\mathrm{1})$ in Fig.~\ref{fig1}, for electric--chiral interaction.
}
\label{fig2}
\end{figure} Restricting our attention to case (1)$a$, similar to Eqs.~(\ref{eq22}) and (\ref{eq23}) we obtain \begin{align} \label{eq24}
\langle I\!I\!I|\hat{H}_{BF}|I\!I\rangle =&-\frac{1}{\sqrt{2}}[\vect{d}_\mathrm{B}^{l0}\cdot\tens{G}_{\lambda_2}(\vect{r}_\mathrm{B},\vect{r}_2,\omega_2)]_{i_2}\delta^{(34)} -\frac{1}{\sqrt{2}}[\vect{d}_\mathrm{B}^{l0}\cdot\tens{G}_{\lambda_3}(\vect{r}_\mathrm{B},\vect{r}_3,\omega_3)]_{i_3}\delta^{(24)}, \\ \label{eq25}
\langle 0|\hat{H}_{\mathrm{B} F}|I\!I\!I\rangle=&- \frac{1}{\mathrm{i} \omega_4}[\vect{m}_{\mathrm{B}}^{0l}\cdot{\nabla}_{\!\mathrm{B}}\times\tens{G}_{\lambda_4}(\vect{r}_\mathrm{B},\vect{r}_4,\omega_4)]_{i_4}. \end{align} Note that $\hat{H}_{\mathrm{B} F}$ is substituted by $-\hat{\vect{d}}_{\mathrm{B}}\cdot\hat{\vect{E}} (\vect{r}_\mathrm{B})$ and
$-\hat{\vect{m}}_{\mathrm{B}}\cdot\hat{\vect{B}}(\vect{r}_\mathrm{B})$, respectively, for the first and second photon absorptions in molecule $\mathrm{B}$.
Next we substitute these matrix elements into Eq.~(\ref{p4}). The summation present in Eq.~(\ref{p4}) runs over discrete variables of molecular states $m$, $l$, electric or magnetic nature of the noise source $\lambda$, spatial components $i$, as well as continuous variables of frequency and position. Making use of the integral relation (\ref{intrel}) leads to the contribution coming from case (1)$a$ to the fourth-order energy shift as \begin{equation} \label{eq26} U^{1a}_{EC}= -\frac{\mu_0^2}{\hbar\pi^2}\sum_{m,l\neq 0} \int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \omega_1^2\omega_2^2\frac{N_{EC}(\omega_1,\omega_2)}{D_{1}(\omega_1,\omega_2)}, \end{equation} where \begin{align} \label{eq27} N_{EC}(\omega_1,\omega_2)&= -\frac{\mathrm{i}}{\omega_2}\operatorname{tr} \left[\vect{d}_\mathrm{A}^{m0}\vect{d}_{\mathrm{A}}^{0m}\cdot\mathrm{Im}\, \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_1)\cdot \vect{d}_{\mathrm{B}}^{l0}\vect{m}_\mathrm{B}^{0l}\cdot{\nabla}_{\!\mathrm{B}}\times\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_2)\right],\\ D_{1}(\omega_1,\omega_2) &= (\omega_\mathrm{A}^{m}+\omega_1) (\omega_1+\omega_2)(\omega_\mathrm{B}^{l}+\omega_2). \end{align}
In writing Eq.~(\ref{eq27}) we have used Lloyd's theorem which allows us to assume that the matrix elements of
the electric dipole operator are real-valued while those of the magnetic dipole operator are taken to be pure imaginary; $\vect{d}_{A'}^{ij}=\vect{d}_{A'}^{ji}$, $\vect{m}_{A'}^{ij}=-\vect{m}_{A'}^{ji}$ \cite{0867}.
The contribution from diagram (1)$b$ can be obtained in a similar manner and the only difference compared with that of (1)$a$ is found to be that $N(\omega_1,\omega_2)$ is replaced by $N(\omega_2,\omega_1)$. That is \begin{align} \label{eq28} U^1_{EC}&= U^{1a}_{EC}+U^{1b}_{EC}=-\frac{\mu_0^2}{\hbar\pi^2}\sum_{m,l\neq 0} \int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \,\omega_1^2\omega_2^2\frac{N_{EC}(\omega_1,\omega_2)+N_{EC}(\omega_2,\omega_1)}{D_{1}(\omega_1,\omega_2)}\nonumber\\ &=-\frac{\mu_0^2}{\hbar\pi^2}\sum_{m,l\neq 0} \int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \omega_1^2\omega_2^2 N_{EC}(\omega_1,\omega_2)\left[\frac{1}{D_1(\omega_1,\omega_2)}+\frac{1}{D_1(\omega_2,\omega_1)}\right]. \end{align} For all other diagrams in Fig.~\ref{fig1} the result is seen to be obtained similarly to diagram (1). Summing over all contributions leads to (see \ref{AppA}) \begin{align} \label{eq33} U_{EC}&(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})= \frac{\mu_0^2\mathrm{i}}{\hbar\pi^2}\sum_{m,l\neq 0}\frac{4\omega_\mathrm{A}^m}{\omega_\mathrm{A}^m+\omega_\mathrm{B}^l} \int_0^\infty\mathrm{d}\omega_1 \int_0^\infty\mathrm{d}\omega_2 \,{\omega_1^2\omega_2}\nonumber\\ &\times\operatorname{tr}\left[\vect{d}_\mathrm{A}^{m0}\vect{d}_{\mathrm{A}}^{0m}\cdot\mathrm{Im}\, \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_1)\cdot \vect{d}_{\mathrm{B}}^{l0}\vect{m}_\mathrm{B}^{0l}\cdot{\nabla}_{\!\mathrm{B}}\times\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_2)\right]\nonumber\\ &\times\bigg[\frac{1}{(\omega_\mathrm{A}^m+\omega_2)(\omega_\mathrm{B}^l+\omega_2)}-\frac{1}{(\omega_\mathrm{A}^m+\omega_1)(\omega_\mathrm{B}^l+\omega_1)}\bigg] \bigg(\frac{1}{\omega_2+\omega_1}-\frac{1}{\omega_2-\omega_1}\bigg). \end{align}
Equation (\ref{eq33}) can be simplified, first, by performing one of the frequency integrals and then rewriting the remaining integral in terms of imaginary frequency $\omega=\mathrm{i}\xi$ (see \ref{AppB}) to yield \begin{align} \label{eq34} U_{EC}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})=&-\frac{4\mu_0^2\mathrm{i}}{\hbar\pi}\int_0^\infty\mathrm{d} \xi\,\xi^3\sum_{m,l\neq 0} \frac{\xi\omega_\mathrm{A}^m}{[(\omega_\mathrm{A}^m)^2+\xi^2][(\omega_\mathrm{B}^l)^2+\xi^2]}\nonumber\\ &\times\operatorname{tr}\left[\vect{d}_\mathrm{A}^{m0}\vect{d}_{\mathrm{A}}^{0m}\cdot \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\mathrm{i} \xi)\cdot \vect{d}_{\mathrm{B}}^{l0}\vect{m}_\mathrm{B}^{0l}\cdot{\nabla}_{\!\mathrm{B}}\times\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi)\right]. \end{align} This formula can be written as \begin{align} \label{eq35} U_{EC}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})= &\frac{\hbar\mu_0^2}{\pi}\int_0^\infty\mathrm{d} \xi\,\xi^3\nonumber\\ &\times \operatorname{tr}\left[ \bm{\alpha}_\mathrm{A}(\mathrm{i} \xi)\cdot \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\,\mathrm{i} \xi)\cdot {\bm\chi}^{em}_\mathrm{B} (\mathrm{i}\xi)\cdot{\nabla}_{\!\mathrm{B}}\times\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi) \right], \end{align}
where $\bm{\alpha}(\omega)$ and $\bm{\chi}^{em}(\omega)$
are, respectively, electric polarisability and chiral polariz-
ability of the molecules, given as \begin{equation} \label{alpha} \bm\alpha_\mathrm{A}(\omega) =\lim_{\epsilon\to 0^+} \frac{2}{\hbar}\sum_{m} \frac{\omega_\mathrm{A}^m\vect{d}_\mathrm{A}^{m0}\vect{d}_\mathrm{A}^{0m}}{(\omega_\mathrm{A}^m)^2-\omega^2-\mathrm{i} \epsilon \omega}\,, \end{equation}
\begin{equation}
\label{chiem}
\bm\chi^{em}_\mathrm{B}(\omega) = \lim_{\epsilon\to 0^+}-\frac{2}{\hbar}\sum_{l}
\frac{\omega\vect{d}_\mathrm{B}^{l0}\vect{m}_\mathrm{B}^{0l}}{(\omega_\mathrm{B}^l)^2-\omega^2-\mathrm{i} \epsilon\omega}\,.
\end{equation}
Equation (\ref{eq35}) agrees with the corresponding result given in
Ref.~\cite{Barcellona2017} where the calculation was based upon second-order perturbation theory by introducing an effective two-photon interaction Hamiltonian.
\paragraph{ Paramagnetic--chiral interaction.} By replacing the electric molecule $\mathrm{A}$ by a paramagnetic one, the interaction Hamiltonian of molecule $\mathrm{A}$ becomes \begin{equation} \label{Eq38} \hat{H}_{\mathrm{A} F} = -\hat{\vect{m}}_{\mathrm{A}}\cdot\hat{\vect{B}}(\vect{r}_{\mathrm{A}}), \end{equation} while that of molecule $\mathrm{B}$ is again given by Eq.~(\ref{eq20}). Following the same procedure as in obtaining the electric--chiral potential, first, we restrict our attention to case $(1)$ in Fig.~\ref{fig1}, which in turn splits into two cases $(1)a$ and $(1)b$ shown in Fig.~\ref{fig3} for the paramagnetic--chiral interaction. \begin{figure}
\caption{ Diagram $(\mathrm{1})$ in Fig.~\ref{fig1}
for the interaction between a\\ paramagnetic molecule $\mathrm{A}$ and a chiral molecule $\mathrm{B}$ being split into two\\ cases, depending on the sequence of the
two transitions in the chiral\\ molecule $\mathrm{B}$.}
\label{fig3}
\end{figure}
For case $(1)a$ in Fig.~\ref{fig3}, it can be seen easily that the intermediate states and the denominator remain unchanged, the matrix elements $\langle I\!I\!I|\hat{H}_{\mathrm{A} F}|I\!I\rangle$ and $\langle 0|\hat{H}_{\mathrm{A} F}|I\!I\!I\rangle$ are the same as in Eqs.~(\ref{eq24}) and (\ref{eq25}), and the only difference is in the following matrix elements: \begin{align} \label{eq39}
\langle I|\hat{H}_{\mathrm{A} F}|0\rangle=& \left[\frac{1}{\mathrm{i} \omega_1}\vect{m}_{\mathrm{A}}^{m0}\cdot{\nabla}_{\!\mathrm{A}}\times \tens{G}^\ast_{\lambda_1}(\vect{r}_\mathrm{A},\vect{r}_1,\omega_1)\right]_{i_1},\\ \label{eq40}
\langle I\!I|\hat{H}_{\mathrm{A} F}|I\rangle =&\frac{1}{\sqrt{2}}\left[\frac{1}{\mathrm{i}\omega_2}\vect{m}_\mathrm{A}^{0m}\cdot{\nabla}_{\!\mathrm{A}}\times\tens{G}_{\lambda_2}^\ast(\vect{r}_\mathrm{A},\vect{r}_2,\omega_2)\right]_{i_2}\delta^{(31)} \nonumber \\ &+\frac{1}{\sqrt{2}}\left[\frac{1}{\mathrm{i}\omega_3}\vect{m}_\mathrm{A}^{0m}\cdot{\nabla}_{\!\mathrm{A}}\times\tens{G}_{\lambda_3}^\ast(\vect{r}_\mathrm{A},\vect{r}_3,\omega_3)\right]_{i_3}\delta^{(21)}. \end{align} Substitution of these into Eq.~(\ref{p4}) and making use of the integral relation (\ref{intrel}) leads for the contribution of diagram $(1)a$ in Fig.~\ref{fig3}, again, to Eq.~(\ref{eq26}) with $N_{EC}(\omega_1,\omega_2)$ being replaced by \begin{align} \label{eq41} N_{PC}&(\omega_1,\omega_2)=-\frac{\mathrm{i}}{\omega_1\omega_2^2}\nonumber\\ & \times\operatorname{tr}\left[\vect{m}_\mathrm{A}^{m0}\vect{m}_{\mathrm{A}}^{0m}\!\cdot\!{\nabla}_{\!\mathrm{A}}\times\mathrm{Im}\, \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_2)\!\times\!\overleftarrow{\nabla}_{\!\mathrm{B}}\!\cdot\! \vect{m}_{\mathrm{B}}^{l0}\vect{d}_\mathrm{B}^{0l}\!\cdot\!\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_1)\!\times\!\overleftarrow{\nabla}_{\!\mathrm{A}}\right]. \end{align} The contribution of diagram $(1)b$ in Fig.~\ref{fig3} also results in Eq.~(\ref{eq41}) with $N_{PC}(\omega_1,\omega_2)$ being replaced by $N_{PC}(\omega_2,\omega_1)$. Taking into account the contribution from all other diagrams in Fig.~\ref{fig1} we obtain (see \ref{AppC}) \begin{align} \label{eq44} U_{PC}& (\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})= \frac{\mu_0^2\mathrm{i}}{\hbar\pi^2}\sum_{m,l\neq 0} \frac{4\omega_\mathrm{A}^m}{\omega_\mathrm{A}^m+\omega_\mathrm{B}^l}\int_0^\infty\mathrm{d}\omega_1 \int_0^\infty\mathrm{d}\omega_2 \,{\omega_1}\nonumber\\ &\times\operatorname{tr}\left[\vect{m}_\mathrm{A}^{m0}\vect{m}_{\mathrm{A}}^{0m}\cdot{\nabla}_{\!\mathrm{A}}\times \mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_2)\times\overleftarrow{\nabla}_{\!\mathrm{B}}\cdot \vect{m}_{\mathrm{B}}^{l0}\vect{d}_\mathrm{B}^{0l}\cdot\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_1)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\right]\nonumber\\ &\times\bigg[\frac{1}{(\omega_\mathrm{A}^m+\omega_2)(\omega_{\mathrm{B}}^l+\omega_2)}-\frac{1}{(\omega_\mathrm{A}^m+\omega_1)(\omega_\mathrm{B}^l+\omega_1)}\bigg] \bigg(\frac{1}{\omega_2+\omega_1}+\frac{1}{\omega_2-\omega_1}\bigg). \end{align} At this stage, in order to present the result in a more convenient form, we may first use the definition (\ref{a3}) and the identity (\ref{a5}) to perform one of the frequency integrals, and then use complex analysis to write the remaining integral in terms of imaginary frequency as outlined in \ref{AppA}. Doing so, we end up with \begin{align} \label{eq45} &U_{PC} (\vect{r}_\mathrm{A},\vect{r}_\mathrm{B}) =\frac{\hbar\mu_0^2}{\pi}\int_0^\infty\mathrm{d} \xi\,\xi\nonumber\\ &\times\operatorname{tr}\left[\bm{\beta}_\mathrm{A}^P(\mathrm{i} \xi)\cdot{\nabla}_{\!\mathrm{A}}\times \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\mathrm{i} \xi)\times\overleftarrow{\nabla}_{\!\mathrm{B}}\cdot {\bm\chi}_\mathrm{B}^{me}(\mathrm{i} \xi)\cdot\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\right], \end{align} where the paramagnetisability tensor $\bm{\beta}^p_\mathrm{A}$ and the chiral polarisability ${\bm\chi}_\mathrm{B}^{me}$ are defined as
\begin{align}
\label{eq46}
\bm\beta_\mathrm{A}^P(\omega) &= \lim_{\epsilon\to 0^+}\frac{2}{\hbar}\sum_{m}
\frac{\omega_\mathrm{A}^{m}\vect{m}_\mathrm{A}^{m0}\vect{m}_\mathrm{A}^{0m}}{(\omega_\mathrm{A}^{m})^2-\omega^2-\mathrm{i} \epsilon\omega}\,,\\
\label{chime}
\bm\chi^{me}_\mathrm{B}(\omega) &= \lim_{\epsilon\to 0^+}-\frac{2}{\hbar}\sum_{l}
\frac{\omega\vect{m}_\mathrm{B}^{l0}\vect{d}_\mathrm{B}^{0l}}{(\omega_\mathrm{B}^l)^2-\omega^2-\mathrm{i} \epsilon\omega}\,.
\end{align}
\paragraph{Chiral--chiral interaction.} In order to calculate the chiral--chiral contribution to the intermolecular potential, the molecule--field interaction Hamiltonians that have to be considered are \begin{align} \label{eq47} \hat{H}_{A'F} &=-\hat{\vect{d}}_{A'}\cdot\hat{\vect{E}}(\vect{r}_{A'}) -\hat{\vect{m}}_{A'}\cdot\hat{\vect{B}}(\vect{r}_{A'}), \qquad A'\in\{\mathrm{A},\mathrm{B}\} \end{align} as in each molecule, one of the two opposite transitions (mentioned before) have to be considered of an electric nature with the other being considered a magnetic transition. Hence, every single diagram in Fig.~\ref{fig1} splits into four additional time-orderings. As an example, Fig.~\ref{fig4} shows the resulting split of diagram (1).
\begin{figure}
\caption{
Diagram $(\mathrm{1})$ in Fig.~\ref{fig1}
for the interaction between two chiral molecules being split into four cases, depending on the sequence of the
electric and magnetic transitions in each molecule}
\label{fig4}
\end{figure}
Let's consider first, diagram $(1)a$ from Fig.~\ref{fig4}. A calculation similar to the one outlined above Eq.~(\ref{eq22}), leads to the matrix elements $\langle I|\hat{H}_{\mathrm{A} F}|0\rangle$,
$\langle I\!I|\hat{H}_{\mathrm{A} F}|I\rangle$, $\langle I\!I\!I|\hat{H}_{\mathrm{A} F}|I\!I\rangle$, and $\langle 0|\hat{H}_{\mathrm{A} F}|I\!I\!I\rangle$ given, respectively, by Eqs.~(\ref{eq22}), (\ref{eq40}), (\ref{eq24}), and (\ref{eq25}). Substitution of these matrix elements in Eq.~(\ref{p4}), leads to \begin{equation} \label{eq48} U^{(1)a}_{CC}= -\frac{\mu_0^2}{\hbar\pi^2}\sum_{m,l\neq 0} \int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \omega_1^2\omega_2^2\frac{N^{1a}_{CC}(\omega_1,\omega_2)}{D_{1}(\omega_1,\omega_2)}\,, \end{equation} where \begin{align} \label{eq49} N^{1a}_{CC}&(\omega_1,\omega_2) \nonumber\\ &= \frac{1}{\omega_2^2}\operatorname{tr}\left[\vect{d}_\mathrm{A}^{m0}\vect{m}_{\mathrm{A}}^{0m}\cdot{\nabla}_{\!\mathrm{A}}\times\mathrm{Im}\, \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_2)\times\overleftarrow{\nabla}_\mathrm{B}\cdot \vect{m}_{\mathrm{B}}^{l0}\vect{d}_{\mathrm{B}}^{0l}\cdot\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_1)\right]. \end{align} The contributions $U^{(1)b}$, $U^{(1)c}$, and $U^{(1)d}$ are found to have the same form as the right hand side of Eq.~(\ref{eq48}), with $N^{1a}_{CC}$ being replaced, respectively, by $N^{1b}_{CC}$, $N^{1c}_{CC}$, and $N^{1d}_{CC}$, where \begin{align} \label{eq50} N^{1b}_{CC}&(\omega_1,\omega_2)\nonumber\\ &=-\frac{1}{\omega_1\omega_2}\operatorname{tr}\left[\vect{d}_\mathrm{A}^{m0} \vect{m}_{\mathrm{A}}^{0m}\cdot{\nabla}_{\!\mathrm{A}}\times \mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_2) \cdot\vect{d}_\mathrm{B}^{l0}\vect{m}_{\mathrm{B}}^{0l}\cdot{\nabla}_{\!\mathrm{B}}\times\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_1) \right],\nonumber\\ N^{1c}_{CC}&(\omega_1,\omega_2) = N^{1b}_{CC}(\omega_2,\omega_1),\qquad N^{1d}_{CC}(\omega_1,\omega_2) = N^{1a}_{CC}(\omega_2,\omega_1). \end{align} Using these, the contribution of diagram (1) in Fig.~\ref{fig1} to the chiral--chiral interaction potential becomes \begin{align} \label{eq51} U^{(1)}_{CC}=&-\frac{\mu_0^2}{\hbar\pi^2} \sum_{m,l\neq 0}\int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \, \omega_1^2\omega_2^2 \nonumber\\ &\times\left[N_{CC}^{1a}(\omega_1,\omega_2)+N_{CC}^{1b}(\omega_1,\omega_2)\right] \left[\frac{1}{D_1(\omega_1,\omega_2)}+\frac{1}{D_1(\omega_2,\omega_1)}\right]. \end{align}
By calculating the contribution of all other diagrams in Fig.~\ref{fig1} in a similar manner and summing them up, we obtain (see \ref{AppD}) the chiral--chiral interaction potential as \begin{eqnarray} \label{eq58} &&\hspace{-1in}U_{CC}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})=-\frac{\hbar\mu_0^2}{\pi}\int_0^\infty \mathrm{d} \xi\,\xi^2\nonumber\\ &&\hspace{-.5in}\times\Big\{\operatorname{tr}\left[\bm{\chi}^{em}_\mathrm{A}(\mathrm{i} \xi)\cdot{\nabla}_{\!\mathrm{A}}\times \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\mathrm{i} \xi)\times\overleftarrow{\nabla}_\mathrm{B} \cdot\bm{\chi}^{me}_\mathrm{B}(\mathrm{i} \xi)\cdot\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi)\right]\nonumber\\ &&\hspace{-.2in}+\operatorname{tr}\left[\bm{\chi}^{em}_\mathrm{A}(\mathrm{i} \xi)\cdot{\nabla}_{\!\mathrm{A}}\times \tens{G}(\vect{r}_\mathrm{A},\vect{r}_{\mathrm{B}},\mathrm{i} \xi) \cdot\bm{\chi}^{em}_\mathrm{B}(\mathrm{i} \xi)\cdot{\nabla}_{\!\mathrm{B}}\times\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi) \right]\Big\}. \end{eqnarray}
\subsection{Diamagnetic--chiral interaction} While obtaining the formula of interaction potentials so far required fourth-order perturbation theory, the diamagnetic--chiral interaction potential is obtained via third-order perturbation theory (cf.~Ref.~\cite{buhmann2013-3}), \begin{equation} \label{p3} U_{DC}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})= \sum_{I,I\!I\neq 0} \hspace{-1ex}
\frac{\langle 0 |\hat{H}_{\mathrm{int}}|I\!I\rangle\langle I\!I|\hat{H}_{\mathrm{int}}|I\rangle
\langle I|\hat{H}_{\mathrm{int}}|0\rangle} {(E_{I\!I}-E_0)(E_{I}-E_0)}\,. \end{equation} The underlying reason is that the diamagnetic molecule--field interaction is a two--photon process. The two photons are exchanged with the chiral molecule to participate in the two--molecule interaction. To perform the calculation, we replace the paramagnetic molecule $\mathrm{A}$ in the preceeding section with a diamagnetic one. That is \begin{equation} \hat{H}_{\mathrm{A} F} = -\frac{1}{2}\hat{\vect{B}}(\vect{r}_\mathrm{A}) \cdot \hat{\bm{\beta}}_\mathrm{A}^d\cdot \hat{\vect{B}}(\vect{r}_\mathrm{A}), \end{equation}
with $\hat{\bm{\beta}}_\mathrm{A}^d$ defined in Eq.~(\ref{beta}), and $\hat{H}_{\mathrm{B} F} $ was given by Eq.~(\ref{eq20}).
The complete set of intermediate states that are involved in the perturbation formula (\ref{p3}) can be taken into account by the diagrams given in Fig.~\ref{fig5} (see Ref.~\cite{buhmann2013-3}).
\begin{figure}
\caption{
Diagrams for obtaining the interaction potential between a diamagnetic molecule $\mathrm{A}$ and a chiral molecule $\mathrm{B}$ via Eq.~(\ref{p3}).}
\label{fig5}
\end{figure}
Each one of the three diagrams in Fig.~\ref{fig5} is split, as shown in Fig.~\ref{fig6}, into two cases depending on whether the first
transition occurring in the chiral molecule $\mathrm{B}$ is of electric type or magnetic type, labeled by $a$ or $b$, respectively, as shown in Fig.~\ref{fig6}. \begin{figure}
\caption{
Diagram (1) from Fig.~\ref{fig5} being split into two cases depending on the sequence of the transitions in the chiral molecule $\mathrm{B}$.}
\label{fig6}
\end{figure}
Let's consider first diagram (1) in Fig.~\ref{fig5}. The corresponding intermediate states are given as follows \begin{align}
|I\rangle &= |0_\mathrm{A}\rangle|0_\mathrm{B}\rangle|1_{\lambda_1 i_1}(\vect{r}_1,\omega_1),1_{\lambda_2 i_2}(\vect{r}_2,\omega_2)\rangle,\nonumber\\
|I\!I\rangle &= |0_\mathrm{A}\rangle|l_\mathrm{B}\rangle|1_{\lambda_3 i_3}(\vect{r}_3,\omega_3)\rangle. \end{align} To detemine the corresponding matrix elements present in the numerator of Eq.~(\ref{p3}), diagram (1)$a$ or (1)$b$ in Fig.~\ref{fig6} have to be distinguished. For the former one finds \begin{align}
\langle I|\hat{H}_{\mathrm{int}}|0\rangle&= -\frac{1}{2}\beta^D_{\alpha\beta}\langle 1_{\lambda_1 i_1}(\vect{r}_1,\omega_1),1_{\lambda_2 i_2}(\vect{r}_2,\omega_2)|\hat{B}_\alpha(\vect r_\mathrm{A})\hat{B}_\beta(\vect r_\mathrm{A})
|\{0\}\rangle \nonumber\\ & = \frac{\beta^D_{\alpha\beta}}{\sqrt{2}\omega_1\omega_2} \big[{\nabla}_{\!\mathrm{A}}\times\tens{G}_{\lambda_1} (\vect{r}_\mathrm{A},\vect{r}_1,\omega_1)\big]_{\alpha i_1}\big[{\nabla}_{\!\mathrm{A}}\times\tens{G}_{\lambda_2} (\vect{r}_\mathrm{A},\vect{r}_2,\omega_2)\big]_{\beta i_2}, \end{align}
\begin{align}
\langle I\!I|\hat{H}_{\mathrm{int}}|I\rangle&=\langle 0_\mathrm{A}|\langle l_\mathrm{B}|
\langle 1_{\lambda_3 i_3}(\vect{r}_3,\omega_3)| -\hat{\vect{d}}_{\mathrm{B}}\cdot\hat{\vect{E}}(\vect{r}_{\mathrm{B}})
|0_\mathrm{A}\rangle|0_\mathrm{B}\rangle|1_{\lambda_1 i_1}(\vect{r}_1,\omega_1),1_{\lambda_2 i_2}(\vect{r}_2,\omega_2)\rangle\nonumber\\ &= -\frac{d_{\mathrm{B}}^{l0}}{\sqrt{2}} \big[\tens{G}_{\lambda_1} (\vect{r}_\mathrm{B},\vect{r}_1,\omega_1)\big]_{\gamma i_1}\delta^{(32)} -\frac{d_{\mathrm{B}\gamma}^{l0}}{\sqrt{2}} \big[\tens{G}_{\lambda_2} (\vect{r}_\mathrm{B},\vect{r}_2,\omega_2)\big]_{\gamma i_2}\delta^{(31)} , \end{align} \begin{align}
\langle 0|\hat{H}_{\mathrm{int}}|I\!I\rangle
&=\langle 0|-\frac{1}{2}\hat{\vect{m}}_\mathrm{B}\cdot\hat{\vect{B}}(\vect{r}_\mathrm{B})|I\!I\rangle =-\frac{m_\eta^{0l}}{\mathrm{i} \omega_3} \big[{\nabla}_{\!\mathrm{B}}\times\tens{G}_{\lambda_3} (\vect{r}_\mathrm{B},\vect{r}_3,\omega_3)\big]_{\eta i_3}, \end{align} with the respective denominator in Eq.~(\ref{p3}) being $\hbar^2(\omega_3$ $\!+\omega_\mathrm{B}^{l})(\omega_1$ $\!+\omega_2)$. After substituting these matrix elements in Eq.~(\ref{p3}), the implicitly included summations over $\lambda_j$ ($j=e,m$) and position integrals can be performed on making use of the integral relation (\ref{intrel}) to lead to the contribution from diagram (1)$a$ in Fig.~\ref{fig6} as \begin{align} \label{eq65} &\hspace{-2ex}U^{1a}_{DC} =\frac{\mu_0^2\mathrm{i}}{\pi^2}\sum_l \int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 {D}_{1a}(\omega_1,\omega_2) \nonumber\\ &\times \operatorname{tr}\big\{ \bm{\beta}^D\cdot{\nabla}_{\!\mathrm{A}}\times\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_2)\times\overleftarrow{\nabla}_{\!\mathrm{B}}\cdot\vect{m}_\mathrm{B}^{l0}\vect{d}_\mathrm{B}^{0l} \cdot\big[\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_1)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\big] \big\}, \end{align} where \begin{equation} {D}_{1a}(\omega_1,\omega_2)= \frac{\omega_1 }{(\omega_1+\omega_2)(\omega_\mathrm{B}^l+\omega_2)}. \end{equation} The contributions from each diagram, $U^{ia(b)}_{DC}$, can be obtained similarly. It is found that the only difference is in ${D}^{ia(b)}(\omega_1,\omega_2)$
as listed below
\begin{equation}
\label{eq67}
{D}_{1b}={D}_{3a}=\frac{-\omega_1}{(\omega_1+\omega_2)(\omega_\mathrm{B}^l+\omega_1)},
\,\,
{D}_{2a}={D}_{2b}= \frac{-\omega_1}{(\omega_\mathrm{B}^l+\omega_1)(\omega_\mathrm{B}^l+\omega_2)},\,\,{D}_{3b}={D}_{1a}.
\end{equation} Using these for $U^{ia(b)}$ and summing up the resulting expressions, as shown in \ref{AppE}, leads to the diamagnetic--chiral interaction potential as \begin{align} \label{eq68} U_{DC}&(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B}) = \frac{\hbar\mu_0^2}{\pi}\int_0^\infty\mathrm{d}\xi\,\xi \nonumber\\ &\times \operatorname{tr}\big\{ \bm{\beta}^D_\mathrm{A}\cdot\big[{\nabla}_{\!\mathrm{A}}\times\tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\mathrm{i} \xi)\times\overleftarrow{\nabla}_{\!\mathrm{B}}\big]\cdot \bm{\chi}^{me}_\mathrm{B}(\mathrm{i} \xi) \cdot\big[\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\big]\big\}. \end{align} It is worth noting that the formula for the diamagnetic--chiral interaction potential and that for the paramagnetic--chiral energy shift, Eqs.~(\ref{eq45}) and (\ref{eq68}), despite being obtained from different orders of perturbative calculations, have exactly the same form. These may be summed in a single formula by substitution of $\bm{\beta}_P(\omega)+\bm{\beta}_D\to \bm{\beta}(\omega)$ to give an overall magnetic--chiral interaction, \begin{align} \label{eq69} U_{MC}&(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B}) = \frac{\hbar\mu_0^2}{\pi}\int_0^\infty\mathrm{d} \xi\, \xi \nonumber\\ &\times \operatorname{tr}\big\{ \bm{\beta}_\mathrm{A}(\mathrm{i} \xi)\cdot\big[{\nabla}_{\!\mathrm{A}}\times\tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\mathrm{i} \xi)\times\overleftarrow{\nabla}_{\!\mathrm{B}}\big]\cdot \bm{\chi}^{me}_\mathrm{B}(\mathrm{i} \xi) \cdot\big[\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\big]\big\}. \end{align}
The chiral polarisabilities of the two enantiomers of a chiral molecule oppose each other in algebraic sign. This results in a discriminatory vdW dispersion interaction for the two enantiomers as it can be seen from the expressions obtained, Eqs.~\eqref{eq35}, \eqref{eq45}, \eqref{eq58}, and \eqref{eq68}, where a sign change emerges as a chiral molecule is replaced by its enantiomer.
\section{Duality Invariance} \label{duality} The formulas for various contributions to the vdW dispersion interaction given so far in this paper with one or both ground-state molecules being chiral, complete the formulas given for non-chiral species in literature (see Refs.~\cite{safari2006, buhmann2013-3, safari2008}). In this section we gather all contributions and discuss their symmetry with respect to the duality of electric and magnetic fields.
As a preparation, we note that all vdW interactions presented in this paper except for the diamagnetic one derive from the bilinear interaction
\begin{equation} \label{HA'2} \hat{H}_{A'F} = -\hat{\vect{d}}_{A'}\cdot\hat{\vect{E}}(\vect{r}_{A'}) -\hat{\vect{m}}_{A'}\cdot\hat{\vect{B}}(\vect{r}_{A'}). \end{equation}
Recall that for $\vect{r}_{A'}$ in free space, the electric and magnetic fields featuring in this interaction obey duality invariance \cite{Buhmann2009b,buhmann2009}: when combining a known solution to the Maxwell equation into a dual pair $\vect{E}_\lambda$ ($\lambda=e,m$) with $\vect{E}_e=\vect{E}$, $\vect{E}_m=c\vect{B}$, then applying a rotation in this two-dimensional duality space
\begin{equation}
\begin{pmatrix}\vect{E}_e^\star\\
\vect{E}_m^\star\end{pmatrix} =\mathcal{D}(\theta) \begin{pmatrix}\vect{E}_e\\ \vect{E}_m\end{pmatrix},
\quad
\mathcal{D}(\theta)
=\begin{pmatrix}\cos\theta&\sin\theta\\
-\sin\theta&\cos\theta\end{pmatrix} \end{equation}
generates a new solution to the Maxwell equations. It follows immediately that the bilinear interaction Hamiltonian above can be written in duality-invariant form
\begin{equation} \label{HA'3} \hat{H}_{A'F} = -\sum_\lambda\hat{\vect{d}}{}_{A'}^\lambda\cdot\hat{\vect{E}}_\lambda(\vect{r}_{A'}) \end{equation}
by introducing a dual pair vector of molecular dipoles $\vect{d}^\lambda$ with $\vect{d}^e=\vect{d}$, $\vect{d}^m=\vect{m}/c$ and positing that such an upper-index dual pair vector transforms with $\mathcal{D}^{-1}(\theta)=\mathcal{D}^\mathsf{T}(\theta)$ under a duality rotation.
If we now calculate the vdW potential arising from this molecule--field interaction using dual-pair notation, then we arrive at a total interaction
\begin{equation} \label{Uu} U(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B}) = \sum_{\lambda_1,\lambda_2,\lambda_3,\lambda_4= e,m} U_{\lambda_1\lambda_2\lambda_3\lambda_4} \end{equation}
with
\begin{multline}
\label{eq82} U_{\lambda_1\lambda_2\lambda_3\lambda_4}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})= -\frac{\hbar}{2\pi\varepsilon_0^2}\int_0^\infty\mathrm{d} \xi \\ \times \operatorname{tr}\left[ \bm{\alpha}^{\lambda_1\lambda_2}_\mathrm{A}(\mathrm{i} \xi )\cdot \tens{G}_{\lambda_2\lambda_3}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\,\mathrm{i} \xi )\cdot {\bm\alpha}_\mathrm{B}^{\lambda_3\lambda_4} (\mathrm{i} \xi )\cdot\tens{G}_{\lambda_4\lambda_1}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi ) \right] \end{multline}
which is duality-invariant given that both atoms are situated in a free-space region. Here, we have introduced Green tensor components in duality space
\begin{align} \label{eq73} \tens{G}_{ee}(\vect{r},\vect{r}',\omega) &\equiv \frac{\mathrm{i} \omega}{c}\tens{G}(\vect{r},\vect{r}',\omega)\biggl(-\frac{\mathrm{i} \omega}{c}\biggr),\\
\label{eq74}
\tens{G}_{em}(\vect{r},\vect{r}',\omega)&\equiv \frac{\mathrm{i} \omega}{c}\tens{G}(\vect{r},\vect{r}',\omega)\times\bigl(-\overleftarrow\nabla'\bigr),\\ \label{eq75}
\tens{G}_{me}(\vect{r},\vect{r}',\omega)&\equiv \nabla\times\tens{G}(\vect{r},\vect{r}',\omega)\biggl(-\frac{\mathrm{i} \omega}{c}\biggr),\\
\label{eq76}
\tens{G}_{mm}(\vect{r},\vect{r}',\omega)&\equiv \nabla\times\tens{G}(\vect{r},\vect{r}',\omega)\times\bigl(-\overleftarrow\nabla'\bigr), \end{align}
which emerge from expectation values $\langle\hat{\vect{E}}_\lambda\hat{\vect{E}}^\dagger_\lambda\rangle$ and transform via $\mathcal{D}(\theta)$ as well as polarisabilities
\begin{align} \label{eq77} \bm\alpha^{ee}(\omega)&\equiv\bm{\alpha}(\omega) ,\\
\label{eq78}
\bm\alpha^{em}(\omega)&\equiv\bm\chi^{em}(\omega)/c,\\
\label{eq79}
\bm\alpha^{me}(\omega)& \equiv\bm\chi^{me}(\omega)/c,\\
\label{eq80}
\bm\alpha^{mm}(\omega)&\equiv\bm{\beta}(\omega)/c^2 \end{align}
which transform via $\mathcal{D}^\mathsf{T}(\theta)$. Equations~(\ref{Uu}) and (\ref{eq82}) give the most general vdW potential of two molecules with dipole polarisabilities, magnetisabilities, and electromagnetic cross-polarisabilities within an arbitrary bi-anisotropic environment. By virtue of $\bm{\beta}(\omega)=\bm{\beta}_P(\omega)+\bm{\beta}_D$, this includes molecules with both paramagnetic and diamagnetic responses. The general potential is duality invariant, provided that both molecules are situated in free-space regions. For molecules in media, duality-invariance can be ensured by using the real-cavity model where the molecules are surrounded by small free-space cavities \cite{Sambale2007}.
The interactions considered in the previous sections emerge by identifying $E=ee$, $M=mm$, $C=em+me$:
\begin{gather} \label{eq81} U_{EE}=U_{eeee},\quad U_{EM}=U_{eemm},\quad U_{ME}=U_{mmee},\quad U_{MM}=U_{mmmm},\\ \label{eq84} U_{EC} =U_{eeem}+U_{eeme},\quad U_{MC} =U_{mmem}+U_{mmme},\\ \label{eq81b} U_{CE} =U_{emee}+U_{meee},\quad U_{CM} =U_{emmm}+U_{memm},\\ \label{eq84b} U_{CC} =U_{emem}+U_{emme}+U_{meem}+U_{meme} \end{gather}
where for chiral responses, one may use Lloyd's theorem $\bm\chi^{me}=-{{\bm\chi}^{em}}^\top$ [compare Eqs.~(\ref{chiem}) and (\ref{chime})] \cite{0867}.
The duality invariance with $\theta=\pi/2$ can be exploited for molecules with electric, chiral, and paramagnetic reponses by replacing $\bm{\alpha} \leftrightarrow \bm{\beta}/c^2$ and $\bm{\chi}^{em} \leftrightarrow -\bm{\chi}^{me}$, thus generating new potentials from previously calculated results. For instance, this explains why $U_{EE}$ and $U_{MM}$ in free space are so strikingly similar and the same holds for $U_{EC}$ and $U_{MC}$. For diamagnetic molecules, this transformation applies only formally, since the diamagnetic magnetisability is negative, in contrast to the electric polarisability from which it is obtained via the transformation.
One may be tempted to generate chiral potentials from electric or magnetic ones by applying a duality transformation $\theta=\pi/2$. However, as noted in Refs.~\cite{buhmann2012,Buhmann2018charge}, such a transformation will generate nonreciprocal cross-polarisabilities which do not obey Lloyd's theorem and hence do not correspond to chiral molecules, but instead charge-parity violating ones. For instance, starting from a purely electric isotropic molecule of polarisability $\alpha$, the transformed molecule will exhibit cross-polarisabilities $\chi^{em}/c=\chi^{me}/c=\alpha/4$ which clearly violate Lloyd's theorem. This is why vdW potentials involving chiral molecules have a distance dependence which is quite distinct from the known dependences for electric and magnetic molecules. We will see this explicitly below when studying free-space examples.
\section{Interactions in Free Space} \label{freespace} As the simplest application of the obtained formulas, we may consider two molecules in free space for which it can easily be verified that the chirality contribution vanishes in the case of isotropic molecules
unless both molecules are chiral. Hence, for the sake of generality we consider anisotropic molecules and treat the case of isotropic molecules as specific examples. For notational convenience we replace, in what follows, $\bm{\chi}^{em}$ with $\bm{\chi}$ and use the fact that $\bm{\chi}^{me}=-\bm{\chi}^{em\,\top} = -\bm{\chi}^{\top}$.
\subsection{Electric--chiral interaction} \label{EC-free} The free space Green's tensor $\tens{G}^{(0)}$ reads \cite{buhmann2013-1} \begin{equation} \label{G0} \tens{G}^{(0)}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\mathrm{i} \xi) = \frac{1}{4\pi} \left(\tens{I} -\frac{c^2}{\xi^2}\nabla_\mathrm{A}\nabla_\mathrm{A}\right)\frac{\mathrm{e}^{- R\xi/c}}{R} = \frac{\mathrm{e}^{-kR}}{4\pi k^2 R^3}\left[ f(kR)\tens{I} - g(kR) \frac{\vect{R}\vect{R}}{R^2} \right], \end{equation}
where $k= \xi/c$, $R=|\vect{R}|$, $\vect{R} = \vect{r}_\mathrm{A} - \vect{r}_\mathrm{B}$, and \begin{align} \label{f&g} f(x)=&1+x+x^2,\nonumber\\ g(x)=&3+3x+x^2 . \end{align} Using this tensor together with the required derivative \begin{equation} \label{curlG} \nabla_\mathrm{B}\times \tens{G}^{(0)}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\mathrm{i} \xi) = \frac{1}{4\pi}\nabla_\mathrm{B}\left(\frac{\mathrm{e}^{- R\xi/c}}{R}\right)\times\tens{I} =\frac{\mathrm{e}^{-R\xi/c}}{4\pi R^3}(1+ R\xi/c)\vect{R}\times\tens{I} \end{equation} in Eq.~(\ref{eq35}) for an electric molecule A and a chiral molecule B, results in \begin{multline} \label{UECfree} U_{EC}(\vect{R})=\frac{\hbar}{16 \pi^3 \varepsilon_0^2} \epsilon_{ipq}\tilde{R}_q \int_0^\infty \mathrm{d} k \,k^6 \mathrm{e}^{-2kR} \alpha_A^{ij}(\mathrm{i} kc) \chi_B^{rp} (\mathrm{i} kc) \\ \times \left[ \frac{\delta_{jr} -\tilde{R}_j \tilde{R}_r}{k^2R^2} + \frac{2 \left(\delta_{jr} -2\tilde{R}_j \tilde{R}_r \right)}{k^3R^3} +\frac{ 2 \left(\delta_{jr} -3 \tilde{R}_j \tilde{R}_r \right)}{k^4R^4} + \frac{\delta_{jr} -3\tilde{R}_j \tilde{R}_r}{k^5R^5} \right] \end{multline} with $\tilde{\vect{R}} = \vect{R}/R$. In the case of isotropic molecules, for which $\bm{\alpha}$ and $\bm{\chi}$ are diagonal matrices, it can be seen easily that the interaction potential vanishes.
In the retarded limit of intermolecular separation ($R\gg c/\omega_{\mathrm{A}}$ with $\omega_{\mathrm{A}}$ being a typical molecular transition frequency), $\bm{\alpha}$ and $\bm{\chi}$ in Eq.~\eqref{UECfree} can be replaced by their static values $\bm{\alpha}(0)$ and $\bm{\chi}(0)$, \begin{equation} \label{alpha0} \bm{\alpha}_{\mathrm{A}}(0) = \frac{2}{\hbar}\sum_m\frac{\vect{d}_A^{m0}\vect{d}_A^{0m}}{\omega_A^{m}}\equiv \bm{\alpha}_{\mathrm{A}}\,, \qquad \lim_{k\to 0}\bm{\chi}_{\mathrm{B}}(\mathrm{i} k c) = \frac{-2\mathrm{i}}{\hbar}kc\sum_l\frac{\vect{d}_B^{l0}\vect{m}_B^{0l}}{\omega_B^{l}} \equiv k\bm{\chi}'_{\mathrm{B}}. \end{equation} Performing the remaining integral leads to \begin{equation} \label{UECfreeRet} U_{EC}^{\text{r}}(\vect{R})=\frac{7 \hbar}{128 \pi^3 \varepsilon_0^2 R^8} \epsilon_{ipq} \tilde{R}_q \left(5\delta_{jr} -9 \tilde{R}_j \tilde{R}_r \right) \alpha_A^{ij} \chi{'}_B^{rp}. \end{equation}
In the opposite limit of non-retarded coupling, $R\ll c/\omega_{\mathrm{A}}$, the exponential factor in Eq.~\eqref{UECfree} tends to unity and the last term in the square brackets gives the main contribution. Using the definitions \eqref{alpha} and \eqref{chiem} into Eq.~\eqref{UECfree} and carrying out the integral we find \begin{equation} \label{UECnonret} U_{EC}^{\mathrm{nr}}(\vect{R})=-\frac{\mathrm{i}\mu_0}{8 \pi^2 \varepsilon_0\hbar R^5} \epsilon_{ipq} \tilde{R}_q \left(\delta_{jr} -3 \tilde{R}_j \tilde{R}_r \right) \sum_{l,m} \left( \vect{d}_A^{m0}\vect{d}_A^{0m} \right)^{ij}
\left( \vect{d}_B^{l0}\vect{m}_B^{0l} \right)^{rp} \frac{\omega_A^m}{\omega_A^m+\omega_B^l}\,. \end{equation}
\subsection{Magnetic--chiral interaction} \label{MC-free} The interaction potential of a magnetic molecule A and a chiral molecule B in free space may be written by direct calculation of the required derivatives of the Green's tensor, \begin{equation} \label{curlGcurl}{\nabla}_{\!\mathrm{A}}\times \tens{G}^{(0)}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega)\times\overleftarrow{\nabla}_{\!\mathrm{B}} =-\frac{\omega^2}{c^2}\tens{G}^{(0)}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega), \end{equation} and \begin{equation} \label{Gcurl}
\tens{G}^{(0)}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega)\times\overleftarrow{\nabla}_{\!\mathrm{A}} = -{\nabla}_{\!\mathrm{B}}\times \tens{G}^{(0)}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A},\omega), \end{equation} together with Eq.~\eqref{curlG}, and using them in Eq.~\eqref{eq69}. Also, it can be written easily on making use of the duality transformation mentioned below Eq.~\eqref{eq84} as \begin{align} \label{UMCfree}
&U_{MC}(\vect{R}) = \frac{\hbar\mu_0^2c^2}{16 \pi^3} \epsilon_{ipq}\tilde{R}_q \int_0^\infty \mathrm{d} k \,k^6 \mathrm{e}^{-2kR} \beta_A^{ij}(\mathrm{i} kc) \chi_B^{pr} (\mathrm{i} kc) \nonumber\\ &\times \left[ \frac{\delta_{jr} -\tilde{R}_j \tilde{R}_r}{k^2R^2} + \frac{2 \left(\delta_{jr} -2\tilde{R}_j \tilde{R}_r \right)}{k^3R^3} +\frac{ 2 \left(\delta_{jr} -3 \tilde{R}_j \tilde{R}_r \right)}{k^4R^4} + \frac{\delta_{jr} -3\tilde{R}_j \tilde{R}_r}{k^5R^5} \right]. \end{align} Similar to the electric--chiral interaction potential in free space, the magnetic--chiral one vanishes for isotropic molecules by orientational averaging.
In the retarded limit, the result is obtained through a similar manner as for the electric--chiral interacton potential, as \begin{equation} \label{UMCfreeRet} U_{MC}^{\text{r}}(\vect{R})=\frac{7 \hbar\mu_0^2c^2}{128 \pi^3 R^8} \epsilon_{ipq} \tilde{R}_q \left(5\delta_{jr} -9 \tilde{R}_j \tilde{R}_r \right) \beta_A^{ij} \chi{'}_B^{pr}\,, \end{equation} where \begin{equation} \bm{\beta}_\mathrm{A} = \bm{\beta}^d_{\mathrm{A}}+\bm{\beta}^p_{\mathrm{A}}(0) = \bm{\beta}^d_{\mathrm{A}}+\frac{2}{\hbar}\sum_m\frac{\vect{m}_A^{m0}\vect{m}_A^{0m}}{\omega_A^{m}} \,. \end{equation}
In the non-retarded limit, the diamagnetic--chiral and paramagnetic--chiral interactions have to be treated separately because of the frequency-independent nature of the diamagnetisability. For the paramagnetic--chiral interaction, through a similar discussion for electric--chiral contribution that led to Eq.~\eqref{UECfreeRet}, we obtain \begin{equation} \label{UPCnonret} U_{PC}^\mathrm{nr}(\vect{R})=-\frac{\mathrm{i}\mu_0^2}{8 \pi^2 \hbar R^5} \epsilon_{ipq} \tilde{R}_q \left(\delta_{jr} -3 \tilde{R}_j \tilde{R}_r \right) \sum_{l,m} \frac{\omega_A^m \left( \vect{m}_A^{m0}\vect{m}_A^{0m} \right)^{ij}
\left( \vect{d}_B^{l0}\vect{m}_B^{0l} \right)^{pr}}{\omega_A^m+\omega_B^l}\,. \end{equation} For the diamagnetic--chiral interaction, on the contrary, the major contribution to the frequency integral comes from frequencies much greater than $c/\omega_{A}$, for which the chiral polarisability, Eq.~\eqref{chiem}, approximates to
\begin{equation}
\label{eq87}
\bm\chi^{em}_\mathrm{B}(\omega) = \frac{2}{\hbar\omega}\sum_{l} \vect{d}_\mathrm{B}^{l0}\vect{m}_\mathrm{B}^{0l}\,.
\end{equation} Using Eq.~(\ref{eq87}) in Eq.~\eqref{UMCfree} together with $\bm{\beta}\rightarrow \bm{\beta}^d$, results in \begin{equation} \label{UDCnonret} U_{DC}^{\mathrm{nr}}(\vect{R})=- \frac{5\mathrm{i}\mu_0^2c}{64\pi^3 R^6} \epsilon_{ipq} \tilde{R}_q \beta^{dij}_{\mathrm{A}}\left(3\delta_{jr} -7 \tilde{R}_j \tilde{R}_r \right) \sum_{m} \left( \vect{d}_A^{m0}\vect{m}_A^{0m} \right)^{pr} \,. \end{equation}
\subsection{Chiral--chiral interaction} The contribution from the chirality of two molecules A and B to the vdW interaction potential in free space can be obtained by making use of Eqs.~\eqref{curlG} and \eqref{curlGcurl} for the required derivatives of the Green tensor in Eq.~\eqref{eq58}, together with Eqs.~\eqref{G0} and \eqref{f&g}. This leads to \begin{eqnarray} \label{CCfree}
&&\hspace{-1in}U_{CC}(\vect{R}) =\frac{\hbar\mu_0^2c^3}{16\pi^3}\int_0^\infty\mathrm{d} k\,k^6\mathrm{e}^{-2kR}\chi_\mathrm{A}^{ij}(\mathrm{i} k c)\chi_\mathrm{B}^{pq}(\mathrm{i} k c)\nonumber\\ &&\hspace{-.9in}\times\bigg\{ \left(\frac{1}{k^6 R^6}+\frac{2}{k^5R^5}\right) \left(\delta_{jq}-3\tilde{R}_j\tilde{R}_q\right) \left(\delta_{ip}-3\tilde{R}_i\tilde{R}_p\right) \nonumber\\ &&\hspace{-.6in}+\frac{1}{k^4 R^4}\left[3\delta_{jq}\delta_{ip}- 7(\delta_{jq}\tilde{R}_i\tilde{R}_p +\delta_{ip}\tilde{R}_j\tilde{R}_q) +15\tilde{R}_i\tilde{R}_j\tilde{R}_p\tilde{R}_q +\epsilon_{jrp}\epsilon_{qsi}\tilde{R}_r\tilde{R}_s\right]\nonumber\\ &&\hspace{-.6in}+\frac{1}{k^3 R^3}\left[2\delta_{jq}\delta_{ip}- 4(\delta_{jq}\tilde{R}_i\tilde{R}_p+\delta_{ip}\tilde{R}_j \tilde{R}_q) +6\tilde{R}_i\tilde{R}_j\tilde{R}_p\tilde{R}_q +2\epsilon_{jrp}\epsilon_{qsi}\tilde{R}_r\tilde{R}_s\right] \nonumber\\ &&\hspace{-.6in}+\frac{1}{k^2 R^2}\left[\delta_{jq}\delta_{ip}- (\delta_{jq}\tilde{R}_i\tilde{R}_p+\delta_{ip}\tilde{R}_j \tilde{R}_q) +\tilde{R}_i\tilde{R}_j\tilde{R}_p\tilde{R}_q +\epsilon_{jrp}\epsilon_{qsi}\tilde{R}_r\tilde{R}_s\right] \bigg\}. \end{eqnarray} It is worth noting that in the case of isotropic molecules, for which $\chi_{\mathrm{A}'}^{ij} \equiv \chi_{\mathrm{A}'}\delta_{ij}$, Eq.~\eqref{CCfree} reduces to \begin{eqnarray} &&\hspace{-1in}U_{CC}(\vect{R}) =\frac{\hbar\mu_0^2c^3}{8\pi^3R^6}\int_0^\infty\mathrm{d} k\,\mathrm{e}^{-2kR}\chi_\mathrm{A}(\mathrm{i} k c)\chi_\mathrm{B}(\mathrm{i} k c) \left(3+6kR+4k^2R^2\right) \end{eqnarray} in agreement with Refs.~\cite{jenkins1994-1, jenkins1994-2, salam1996, craig1999}. As stated above, this interaction cannot be obtained from the well-known electric--electric potential by means of a duality transformation.
In the retarded intermolecular separation, the chiral polarisabilities in Eq.~(\ref{CCfree}) can be replaced by their static limits. Calculation of the remaining integral results in \begin{align} &U^\text{r}_{CC}(\vect{R}) =\frac{\hbar\mu_0^2c^3}{128\pi^3R^9}\chi_\mathrm{A}^{\prime ij}\chi_\mathrm{B}^{\prime pq}\nonumber\\ &\,\,\times \left(101\delta_{ip}\delta_{jq} -171\delta_{jq}\tilde{R}_i\tilde{R}_p- 171\delta_{ip} \tilde{R}_j\tilde{R}_q +297\tilde{R}_i\tilde{R}_j\tilde{R}_p\tilde{R}_q +81\epsilon_{jrp}\epsilon_{qsi}\tilde{R}_r\tilde{R}_s \right), \end{align} where $\chi'_{\mathrm{A}'}$ is defined according to Eq.~\eqref{alpha0}.
In the opposite limit of non-retarded intermolecular distances, we may set the exponential factor in Eq.~\eqref{CCfree} to unity and retain only the term proportional to $\frac{1}{k^6R^6}$ in the curly brackets of Eq.~(\ref{CCfree}). Doing this together with using Eq.~\eqref{chiem} for chiral polarisabilities and performing the remaining integral leads to \begin{multline} U_{CC}^{\text{nr}}(\vect{R})=-\frac{\mu_0^2c^2}{8 \pi^2 \hbar R^6} \left(\delta_{ip}-3\tilde{R}_i\tilde{R}_p\right) \left(\delta_{jq}-3\tilde{R}_j\tilde{R}_q\right)
\sum_{l,m} \frac{\left( \vect{d}_A^{m0}\vect{m}_A^{0m} \right)^{ij} \left( \vect{d}_B^{l0}\vect{m}_B^{0l} \right)^{pq}}{\omega_A^m+\omega_B^l}\,. \end{multline}
\section{Conclusion} \label{conclusion} In this paper a general expression for the van der Waals dispersion interaction potential between two ground-state molecules in the presence of arbitrary magnetoelectric media is derived making use of perturbation theory. The result for the diamagnetic--chiral interaction potential is seen to have the same form as that of parmagnetic--chiral one, despite being obtained from different orders of perturbative calculations. The formulas obtained in this paper, together with those found in earlier studies (e.g.~Refs.~\cite{safari2006, buhmann2013-3, safari2008}) form a complete set of formulas for the van der Waals interaction potentials for molecules with electric, paramagnetic, and diamagnetic polarisabilities, applicable for chiral and anisotropic molecules and for arbitrary ranges of intermolecular separations as well as allowing for a possible chirality of the surrounding media. The formulas of various contributions to the full dispersion interaction have been brought into a unified form by taking advantage of electric--magnetic duality.
As an application of the obtained formulas, the interaction potential for two anisotropic chiral molecules in free space has been evaluated. The retarded and non-retarded limits of intermolecular separations are obtained and their corresponding distance power laws are summarized in Tab.~\ref{powerlaw}. \begin{table}[!htbp] \caption{ \hspace{-3ex}
\hspace{1ex} Distance power laws for the various dispersion potentials. } \label{powerlaw} \centering
\begin{tabular}{|c|c|c|} \hline Interaction component & Retarded limit & non-retarded limit \\ \hline \hline electric--electric \cite{safari2006}& $-R^{-7}$ & $-R^{-6}$ \\ \hline electric--paramagnetic \cite{safari2008}& $+R^{-7}$ & $+R^{-4}$ \\ \hline electric--diamagnetic \cite{buhmann2013-3}& $-R^{-7}$ & $-R^{-4}$ \\ \hline electric--chiral \cite{SalamThesis} & $\pm R^{-8}$ & $\pm R^{-5}$ \\ \hline paramagnetic--paramagnetic \cite{safari2008}& $-R^{-7}$ & $-R^{-6}$ \\ \hline paramagnetic--diamagnetic \cite{buhmann2013-3}& $+R^{-7}$ & $+R^{-6}$ \\ \hline paramagnetic--chiral \cite{SalamThesis}& $\pm R^{-8}$ & $\pm R^{-5}$ \\ \hline diamagnetic--diamagnetic \cite{buhmann2013-3}& $-R^{-7}$ & $-R^{-6}$ \\ \hline diamagnetic--chiral \cite{SalamUnpublished}& $\pm R^{-8}$ & $\pm R^{-6}$ \\ \hline chiral--chiral \cite{Barcellona2017}& $\pm R^{-9}$ & $\pm R^{-6}$ \\ \hline \end{tabular} \end{table}
\ack AS and SYB thank J.~Franz for discussions. SYB is grateful for support by the German Research Council (grant BU 1803/3-1, S.Y.B.) and the Freiburg Institute for Advanced Studies. AS acknowledges the award of a Mercator Fellowship funded by the Deutsche Forschungsgemeinschaft through the IRTG 2079/Cold Controlled Ensembles in Physics and Chemistry at the University of Freiburg. He also thanks the Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg for the award of a Marie S Curie External Senior Fellowship under the EU Horizon 2020 Grant No. 75430.
\begin{appendix} \section{Calculation of Eq.~(\ref{eq33}) for the chiral--electric potential} \label{AppA} Taking into account the contributions from all diagrams in Fig.~\ref{fig1} to Eq.~(\ref{p4}) for the electric--chiral interaction leads to \begin{eqnarray} \label{eq30} &&\hspace{-.9in}U_{EC}=\sum_{i=1}^{12} U^{i}_{EC}\nonumber\\ &&\hspace{-.8in}=-\frac{\mu_0^2}{\hbar\pi^2} \sum_{m,l\neq 0}\int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \, \omega_1^2\omega_2^2 N_{EC}(\omega_1,\omega_2)[D^+_{EC}(\omega_1,\omega_2)+D^-_{EC}(\omega_2,\omega_1)],\nonumber\\ \end{eqnarray} where \begin{align} \label{eq31} D^{\pm}_{EC}(\omega_1,\omega_2)=&\pm\left(\frac{1}{D_1}-\frac{1}{D_2}+\frac{1}{D_3} -\frac{1}{D_9}-\frac{1}{D_{11}}-\frac{1}{D_{12}}\right)+\frac{1}{D_4}-\frac{1}{D_5}+\frac{1}{D_6}\nonumber\\ & +\frac{1}{D_7}+\frac{1}{D_8}+\frac{1}{D_{10}}\end{align} with $D_{i}$ denoting the energy denominator for case ($i$) as listed in Tab.~\ref{tab1}. It is not difficult to show that \begin{align} \label{eq32} &D^+_{EC}(\omega_1,\omega_2)+D^-_{EC}(\omega_2,\omega_1)\nonumber\\ &=\frac{4\omega_\mathrm{A}^m}{\omega_\mathrm{A}^m+\omega_\mathrm{B}^l}\bigg[\frac{1}{(\omega_\mathrm{A}^m+\omega_2)(\omega_\mathrm{B}^l+\omega_2)}-\frac{1}{(\omega_\mathrm{A}^m+\omega_1)(\omega_\mathrm{B}^l+\omega_1)}\bigg]\left(\frac{1}{\omega_2+\omega_1} -\frac{1}{\omega_2-\omega_1}\right) \end{align} (cf.~Ref.~\cite{craig2012}). Making use of Eq.~(\ref{eq32}) in Eq.~(\ref{eq30}) leads to Eq.~(\ref{eq33}). \begin{table} \centering\begin{tabular}{ll} \hline Diagram & \hspace{9ex}Energy denominator $D_{i}$\\ \hline
(1) &\qquad $D_{1}=(\omega_{\mathrm{A}}^{m}+\omega_1)(\omega_1+\omega_2)(\omega_{\mathrm{B}}^{l}+\omega_2)$ \\
(2) & $\qquad D_{2}=(\omega_{\mathrm{A}}^{m}+\omega_1)(\omega_1+\omega_2)(\omega_{\mathrm{B}}^{l}+\omega_1)$\\
(3) & $\qquad D_{3}=(\omega_{\mathrm{A}}^m+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l)(\omega_{\mathrm{B}}^l+\omega_2)$\\
(4) & $\qquad D_{4}=(\omega_{\mathrm{A}}^m+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l+\omega_1+\omega_2)(\omega_{\mathrm{B}}^l+\omega_1)$\\
(5) & $\qquad D_{5}=(\omega_{\mathrm{A}}^m+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l)(\omega_{\mathrm{A}}^m+\omega_2)$\\
(6) & $\qquad D_{6}=(\omega_{\mathrm{A}}^m+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l+\omega_1+\omega_2)(\omega_{\mathrm{A}}^m+\omega_2)$\\
(7) & $\qquad D_{7}=(\omega_{\mathrm{B}}^l+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l)(\omega_{\mathrm{B}}^l+\omega_2)$\\
(8) & $\qquad D_{8}=(\omega_{\mathrm{B}}^l+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l+\omega_1+\omega_2)(\omega_{\mathrm{B}}^l+\omega_2)$\\
(9) & $\qquad D_{9}=(\omega_{\mathrm{B}}^l+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l)(\omega_{\mathrm{A}}^m+\omega_2)$\\
(10) & $\qquad D_{10}=(\omega_{\mathrm{B}}^l+\omega_1)(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l+\omega_1+\omega_2)(\omega_{\mathrm{A}}^m+\omega_1)$\\
(11) & $\qquad D_{11}=(\omega_{\mathrm{B}}^l+\omega_1)(\omega_1+\omega_2)(\omega_{\mathrm{A}}^m+\omega_2)$\\
(12) & $\qquad D_{12}=(\omega_{\mathrm{B}}^l+\omega_1)(\omega_1+\omega_2)(\omega_{\mathrm{A}}^m+\omega_1)$\\ \hline \end{tabular} \caption{ \label{tab1} The energy denominators corresponding to the diagrams in Fig.~\ref{fig1}. } \end{table} \section{Calculation of Eq.~(\ref{eq34}) for the chiral--electric potential} \label{AppB} In Eq.~(\ref{eq33}) we encounter a double integral in the form of \begin{align} \label{a1} \int_0^\infty\mathrm{d}\omega_1 &\int_0^\infty\mathrm{d}\omega_2\, \frac{\omega_1^2\omega_2\,\mathrm{Im}\,{G}_\alpha(\omega_1)\mathrm{Im}\, {G}_\beta(\omega_2)}{(\omega_{\mathrm{A}}^m+\omega_i)(\omega_{\mathrm{B}}^l+\omega_i)} \bigg(\frac{1}{\omega_2+\omega_1}-\frac{1}{\omega_2-\omega_1}\bigg)\equiv I_i,
\quad i=1,2 \end{align} where $G_\alpha$ and $G_\beta$ denote typical matrix elements of the Green tensor. In order to calculate $I_1$, the integral over $\omega_2$, \begin{equation} \label{a2} \int_0^\infty d\omega_2 \omega_2 \mathrm{Im}\, G_\beta(\omega_2)\left(\frac{1}{\omega_2+\omega_1}-\frac{1}{\omega_2-\omega_1}\right) \equiv g_\beta^{(1)}(\omega_1) \end{equation} is to be performed first, for which we introduce its more general form as \begin{equation} \label{a3} \bm{g}^{(n)} (\omega)= \int_0^\infty\mathrm{d}\omega' \,{\omega'}^{n}\mathrm{Im}\, \tens{G}(\omega') \bigg(\frac{1}{\omega'+\omega}+\frac{(-1)^n}{\omega'-\omega}\bigg), \end{equation} for real $\omega$ and non-negative integer $n$. The Schwartz reflection principle, Eq.~(\ref{Schwartz}), implies that for real $\omega$ the imaginary part of the Green tensor is an odd function of $\omega$. Using this, it is easy to show that the integrand in Eq.~(\ref{a3}) is an even function of $\omega'$, and we may write \begin{equation} \label{a4} \bm{g}^{(n)} (\omega)=\frac{1}{2}\mathrm{Im}\, \int_{-\infty}^\infty\mathrm{d}\omega' \,{\omega'}^{n} \tens{G}(\omega') \bigg(\frac{1}{\omega'+\omega}+\frac{(-1)^n}{\omega'-\omega}\bigg). \end{equation} This integral can be evaluated using contour integral techniques by drawing an infinitely large semicircle in the upper half of the complex frequency plane (where the Green's function is analytic as a response function \cite{dung2003}) to the real axis, and making use of the Cauchy formula. This leads to
\begin{equation} \label{a5} \bm{g}^{(n)} (\omega)= \frac{\pi}{2}(-\omega)^{n}[\tens{G}(\omega)+\tens{G}^\ast(\omega)]. \end{equation} Using Eq.~(\ref{a2}) together with Eq.~(\ref{a5}) in Eq.~(\ref{a1}) results in \begin{eqnarray} \label{a6} &&\hspace{-.9in}I_1=\frac{\pi \mathrm{i}}{4}\int_0^\infty \frac{\mathrm{d}\omega\,\omega^3}{(\omega_{\mathrm{A}}^m+\omega)(\omega_{\mathrm{B}}^l+\omega)} \left[G_\alpha(\omega)-G^\ast_\alpha(\omega)\right] \left[G_\beta(\omega)+G^\ast_\beta(\omega)\right]\nonumber\\ &&\hspace{-.7in}=\frac{\pi \mathrm{i}}{4}\int_0^\infty \frac{\mathrm{d}\omega\,\omega^3}{(\omega_{\mathrm{A}}^m+\omega)(\omega_{\mathrm{B}}^l+\omega)} \left[G_\alpha(\omega)G_\beta(\omega)-G^\ast_\alpha(\omega)G^\ast_\beta(\omega)\right]\nonumber\\ &&\hspace{-.5in}+\frac{\pi \mathrm{i}}{4}\int_0^\infty \frac{\mathrm{d}\omega\,\omega^3}{(\omega_{\mathrm{A}}^m+\omega)(\omega_{\mathrm{B}}^l+\omega)} \left[G_\alpha(\omega)G^\ast_\beta(\omega)-G^\ast_\alpha(\omega)G_\beta(\omega)\right]. \end{eqnarray} The double integral $I_2$ defined by Eq.~(\ref{a1}) can be treated similarly to obtain \begin{eqnarray} \label{a7} &&\hspace{-.9in}I_2=-\frac{\pi \mathrm{i}}{4}\int_0^\infty \frac{\mathrm{d}\omega\,\omega^3}{(\omega_{\mathrm{A}}^m+\omega)(\omega_{\mathrm{B}}^l+\omega)} \left[G_\alpha(\omega)G_\beta(\omega)-G^\ast_\alpha(\omega)G^\ast_\beta(\omega)\right]\nonumber\\ &&\hspace{-.6in}+\frac{\pi \mathrm{i}}{4}\int_0^\infty \frac{\mathrm{d}\omega\,\omega^3}{(\omega_{\mathrm{A}}^m+\omega)(\omega_{\mathrm{B}}^l+\omega)} \left[G_\alpha(\omega)G^\ast_\beta(\omega)-G^\ast_\alpha(\omega)G_\beta(\omega)\right], \end{eqnarray} hence, for $I_2-I_1$ appearing in Eq.~(\ref{eq33}), we end up with \begin{equation} \label{a8} I_2-I_1=-\frac{\pi \mathrm{i}}{2}\int_0^\infty \frac{\mathrm{d}\omega\,\omega^3}{(\omega_{\mathrm{A}}^m+\omega)(\omega_{\mathrm{B}}^l+\omega)} \left[G_\alpha(\omega)G_\beta(\omega)-G^\ast_\alpha(\omega)G^\ast_\beta(\omega)\right]. \end{equation} For the second term in the square brackets we may change the variable as $\omega\to-\omega$ and make use of the Schwartz reflection principle to rewrite Eq.~(\ref{a8}) in the form \begin{equation} \label{a9} I_2-I_1=-\frac{\pi \mathrm{i}}{2} \left\{\int_0^\infty
\frac{\mathrm{d}\omega\,\omega^3 G_\alpha(\omega)G_\beta(\omega)} {(\omega_{\mathrm{A}}^m+\omega)(\omega_{\mathrm{B}}^l+\omega)}+ \int_{-\infty}^0
\frac{\mathrm{d}\omega\,\omega^3 G_\alpha(\omega)G_\beta(\omega)}
{(\omega_{\mathrm{A}}^m-\omega)(\omega_{\mathrm{B}}^l-\omega)}
\right\}. \end{equation} Again, using the fact that both of the integrands are analytic in the upper half of the complex frequency plane including the real axis, we may replace the integration path in the first(second) integral by a quarter-circle in the first(second) quadrant together with the positive imaginary axis ($\omega$ $\!=\mathrm{i} \xi$). The contribution from the quarter circle vanishes due to the limiting behaviour of the Green tensor for large frequencies, and we are left with \begin{equation} I_2-I_1 = -\pi(\omega_{\mathrm{A}}^m+\omega_{\mathrm{B}}^l)\int_0^\infty \mathrm{d} \xi\frac{\xi^4 {G}_\alpha(\mathrm{i} \xi){G}_\beta(\mathrm{i} \xi)}{[(\omega_{\mathrm{A}}^m)^2+\xi^2][(\omega_{\mathrm{B}}^l)^2+\xi^2]}. \end{equation} Using this, leads from Eq.~(\ref{eq33}) to Eq.~(\ref{eq34}).
\section{Calculation of Eq.~(\ref{eq44}) for the paramagnetic--chiral potential} \label{AppC} \begin{eqnarray} \label{eq42} &&\hspace{-1in}U_{PC}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B})\nonumber\\ &&\hspace{-1in}=-\frac{\mu_0^2}{\hbar\pi^2} \sum_{m,l\neq 0}\int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \, \omega_1^2\omega_2^2 N_{PC}(\omega_1,\omega_2)[D^+_{PC}(\omega_1,\omega_2)+D^-_{PC}(\omega_2,\omega_1)], \end{eqnarray} where \begin{align} \label{eq43} D^{\pm}_{PC}(\omega_1,\omega_2)=&\pm\left(\frac{1}{D_1}-\frac{1}{D_2}+\frac{1}{D_3}-\frac{1}{D_9}-\frac{1}{D_{11}} -\frac{1}{D_{12}}\right)-\frac{1}{D_4}+\frac{1}{D_5}-\frac{1}{D_{6}}\nonumber\\ &-\frac{1}{D_7}-\frac{1}{D_8}-\frac{1}{D_{10}} \,. \end{align}
By a straightforward algebra it can be shown that \begin{align} \label{eq43-2}&D^+_{PC}(\omega_1,\omega_2)+D^-_{PC}(\omega_2,\omega_1)\nonumber\\ &=\frac{4\omega_\mathrm{A}^m}{\omega_\mathrm{A}^m+\omega_\mathrm{B}^l}\bigg[\frac{1}{(\omega_\mathrm{A}^m+\omega_2)(\omega_\mathrm{B}^l+\omega_2)}-\frac{1}{(\omega_\mathrm{A}^m+\omega_1)(\omega_\mathrm{B}^l+\omega_1)}\bigg]\left(\frac{1}{\omega_2+\omega_1} +\frac{1}{\omega_2-\omega_1}\right). \end{align} Substituting Eqs.~(\ref{eq41}) and (\ref{eq43-2}) into Eq.~(\ref{eq42}) leads to Eq.~(\ref{eq44}). \section{Calculation of Eq.~(\ref{eq58}) for the chiral--chiral potential} \label{AppD} By calculating the contribution of all other diagrams in Fig.~\ref{fig1} similar to that of diagram (1) and summing them up, we obtain \begin{align} \label{eq53} U_{CC}=- \frac{\mu_0^2}{\hbar\pi^2}\sum_{m,l\neq 0} \int_0^\infty\mathrm{d}\omega_1 \int_0^\infty&\mathrm{d}\omega_2\,{\omega_1^2\omega_2^2} \left\{N^{1a}_{CC}(\omega_1,\omega_2)[{D}^-(\omega_1,\omega_2)+{D}^-(\omega_2,\omega_1)]\right.\nonumber\\ & \left. +N^{1b}_{CC}(\omega_1,\omega_2)[{D}^+(\omega_1,\omega_2)+D^+(\omega_2,\omega_1)]\right\} \end{align} with \begin{align} {D}^{\pm}(\omega_1,\omega_2)=&\frac{1}{D_1}-\frac{1}{D_2}+\frac{1}{D_3}+\frac{1}{D_9}+\frac{1}{D_{11}}-\frac{1}{D_{12}} \nonumber\\ & \pm\left(\frac{1}{D_4}-\frac{1}{D_5}+\frac{1}{D_6}-\frac{1}{D_7}+\frac{1}{D_8}+\frac{1}{D_{10}}\right). \end{align} After some straightforward algebra, it can be shown that \begin{equation} {D}^{\pm}(\omega_1,\omega_2)+{D}^{\pm}(\omega_2,\omega_1)=\frac{4}{\omega_\mathrm{A}^m+\omega_\mathrm{B}^l}F_{\pm}(\omega_1,\omega_2), \end{equation}
where $F_{\pm}(\omega_1,\omega_2)=f_{\pm}(\omega_1,\omega_2)+f_{\pm}(\omega_2,\omega_1)$ with
\begin{equation} f_{\pm}(\omega_1,\omega_2)=\frac{\omega_1}{(\omega_\mathrm{A}^m+\omega_1)(\omega_\mathrm{B}^l+\omega_1)}\bigg(\frac{1}{\omega_1+\omega_2} \pm\frac{1}{\omega_1-\omega_2}\bigg). \end{equation} Upon making use of Eqs.~(\ref{eq49}) and (\ref{eq50}), Eq.~(\ref{eq53}) becomes \begin{eqnarray} &&\hspace{-1.in}U_{CC}= -\frac{4\mu_0^2}{\hbar\pi^2}\sum_{m,l\neq 0}\frac{1}{\omega_\mathrm{A}^m+\omega_\mathrm{B}^l} \int_0^\infty\mathrm{d}\omega_1 \int_0^\infty\mathrm{d}\omega_2\,\omega_1 \nonumber\\ &&\hspace{-1.in}\times\bigg\{\omega_1\operatorname{tr}\left[ \vect{d}_\mathrm{A}^{m0}\vect {m}_{\mathrm{A} }^{0m} \cdot{\nabla}_{\!\mathrm{A}}\times\mathrm{Im}\, \tens{G}(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_2)\times\overleftarrow{\nabla} _\mathrm{B}\cdot \vect{m}_{\mathrm{B}}^{l0}\vect{d}_\mathrm{B}^{0l}\cdot\mathrm{Im}\,\tens{G}(\vect{r}_\mathrm{B},\vect{r}_\mathrm{A}, \omega_1)\right]F_- \nonumber\\ &&\hspace{-.8in}-\omega_2\operatorname{tr}\left[\vect{d}_{\mathrm{A}}^{m0}\vect{m}_{\mathrm{A} } ^{0m} \cdot{\nabla}_{\!\mathrm{A}}\times\mathrm{Im}\,\tens{G }(\vect{r}_\mathrm{A},\vect{r}_\mathrm{B},\omega_2) \cdot\vect{d}_\mathrm{B}^{l0}\vect{m}_{\mathrm{B}}^{0l}\cdot{\nabla}_{\!\mathrm{B}}\times\mathrm{Im}\,\tens{G}(\vect { r}_\mathrm{B},\vect{r}_\mathrm{A},\omega_1) \right]F_+\bigg\}.\nonumber\\ \end{eqnarray} Now, in order to write the result in a compact form, we follow a similar procedure as for obtaining Eqs.~(\ref{eq35}) and (\ref{eq45}).
\section{Calculation of Eq.~(\ref{eq68}) for the diamagnetic--chiral potential} \label{AppE} In order to determine the diamagnetic--chiral interaction potential, Eq.~(\ref{eq65}) has to be added to other portions for those $D_{1(a)}$ is replaced by $D_{ia(b)}$ given by Eq.~(\ref{eq67}). Doing this yields \begin{align} &U_{DC}= -\frac{4\mu_0^2\mathrm{i}}{\pi^2}\sum_l \int_0^\infty\mathrm{d}\omega_1\int_0^\infty\mathrm{d}\omega_2 \frac{\omega_1\omega_2}{(\omega_{\mathrm{B}}+\omega_1)(\omega_1+\omega_2)(\omega_{\mathrm{B}}+\omega_2)} \nonumber\\ &\times\operatorname{tr}\big\{ \bm{\beta}_\mathrm{A}\cdot\big[{\nabla}_{\!\mathrm{A}}\times\mathrm{Im}\,\tens{G}(\vect{r}_{\mathrm{A}},\vect{r}_{\mathrm{B}},\omega_2)\times\overleftarrow{\nabla}_{\!\mathrm{B}}\big]\cdot\vect{m}^{l0}_{\mathrm{B}}\vect{d}^{0l}_{\mathrm{B}} \cdot\big[\mathrm{Im}\,\tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\omega_1)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\big] \big\}. \end{align} This may be written as \begin{align} \label{B1} U_{DC}=-\frac{4\mu_0^2\mathrm{i}}{\pi^2}&\sum_l \int_0^\infty\mathrm{d}\omega_1 \frac{\omega_1}{(\omega_{\mathrm{B}}^l+\omega_1)} \nonumber\\ &\times\operatorname{tr}\big\{ \bm{\beta}_\mathrm{A}\cdot{\nabla}_{\!\mathrm{A}}\times \tens{J}_1\times\overleftarrow{\nabla}_{\!\mathrm{B}}\cdot\vect{m}^{l0}_{\mathrm{B}}\vect{d}^{0l}_{\mathrm{B}} \cdot\big[\mathrm{Im}\,\tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\omega_1)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\big]\big\} \end{align} with \begin{eqnarray} \label{B2} &&\tens{J}_1(\omega)=\int_0^\infty\mathrm{d}\omega_2 \frac{\omega_2}{(\omega+\omega_2)(\omega_{\mathrm{B}}^l+\omega_2)} \mathrm{Im}\,\tens{G}(\vect{r}_{\mathrm{A}},\vect{r}_{\mathrm{B}},\omega_2). \end{eqnarray} The Green's tensor being analytic in the upper half of the complex frequency plane, facilitates rewriting $\vect{J}_1(\omega)$ in terms of the imaginary frequency $\omega_2\to\mathrm{i} \xi$ as \begin{align} \label{B3} \tens{J}_1(\omega)&=\mathrm{Im}\,\int_0^\infty\mathrm{d}\omega_2 \frac{\omega_2}{(\omega+\omega_2)(\omega_{\mathrm{B}}^l+\omega_2)} \tens{G}(\vect{r}_{\mathrm{A}},\vect{r}_{\mathrm{B}},\omega_2)&\nonumber\\ &= \int_0^\infty\mathrm{d} \xi \xi^2\frac{(\omega+\omega_{\mathrm{B}}^l)}{(\omega^2+\xi^2)[(\omega_{\mathrm{B}}^l)^2+\xi^2]}\tens{G}(\vect{r}_{\mathrm{A}},\vect{r}_{\mathrm{B}},\mathrm{i} \xi). \end{align} In the second line we have used the fact that the Green tensor is real-valued for imaginary frequency due to the Schwartz reflection principle, Eq.~(\ref{Schwartz}). Using Eq.~(\ref{B3}) in (\ref{B1}) gives
\begin{align} \label{B4} U_{DC} = -\frac{4\mu_0^2\mathrm{i}}{\pi^2}\sum_l &\int_0^\infty\mathrm{d} \xi \xi^2\frac{1}{[(\omega_{\mathrm{B}}^l)^2+\xi^2]}\nonumber\\ &\times \operatorname{tr}\big\{ \bm{\beta}_\mathrm{A}\cdot{\nabla}_{\!\mathrm{A}}\times\tens{G}(\vect{r}_{\mathrm{A}},\vect{r}_{\mathrm{B}},\mathrm{i} \xi) \times\overleftarrow{\nabla}_{\!\mathrm{B}}\cdot\vect{m}^{l0}_{\mathrm{B}}\vect{d}^{0l}_{\mathrm{B}} \cdot\big[\tens{J}_2\times\overleftarrow{\nabla}_{\!\mathrm{A}}\big]\big\} \end{align} with \begin{eqnarray} \label{B5} &&\tens{J}_2=\mathrm{Im}\,\int_0^\infty\mathrm{d}\omega\frac{\omega}{(\omega^2+\xi^2)} \tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\omega). \end{eqnarray} At this stage, noting that the integrand in (\ref{B5}) has a simple pole at $\omega=\mathrm{i} \xi$ in the upper half of the complex frequency plane, we may use contour integral
techniques to replace the integration path with the positive part of the imaginary axis and excluding the pole by applying an infinitesimal halfcircle around it.
Doing so, we find \begin{align} \label{B6} \int_0^\infty\mathrm{d}\omega&\frac{\omega}{(\omega^2+\xi^2)} \tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\omega) \nonumber\\ &=-P\int_0^\infty\mathrm{d} v\frac{v}{(-v^2+\xi^2)} \tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\mathrm{i} v) +\mathrm{i} \frac{\pi }{2}\tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\mathrm{i} \xi). \end{align} Hence, the right hand side of Eq.~(\ref{B5}) reduces to $(\pi/2)\tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\mathrm{i} \xi)$. Using this in Eq.~(\ref{B4}) results in \begin{align} \label{B7} U_{DC} =& -\frac{2\mu_0^2\mathrm{i}}{\pi}\sum_l \int_0^\infty\mathrm{d} \xi\frac{\xi^2}{(\omega_{\mathrm{B}}^l)^2+\xi^2}\nonumber\\ &\times \operatorname{tr}\big\{ \bm{\beta}_\mathrm{A}\cdot{\nabla}_{\!\mathrm{A}}\times\tens{G}(\vect{r}_{\mathrm{A}},\vect{r}_{\mathrm{B}},\mathrm{i} \xi) \times\overleftarrow{\nabla}_{\!\mathrm{B}}\cdot\vect{m}^{l0}_{\mathrm{B}}\vect{d}^{0l}_{\mathrm{B}} \cdot\big[\tens{G}(\vect{r}_{\mathrm{B}},\vect{r}_{\mathrm{A}},\mathrm{i} \xi)\times\overleftarrow{\nabla}_{\!\mathrm{A}}\big]\big\}. \end{align} Finally, the summation over $l$ in the right hand side can be replaced in terms of the chiral polarisability defined by Eq.~(\ref{chime}),
\begin{equation} \sum_l\frac{\vect{m}^{l0}_{\mathrm{B}}\vect{d}^{0l}_{\mathrm{B}}}{(\omega_{\mathrm{B}}^l)^2+\xi^2}=-\frac{\hbar}{2\mathrm{i} \xi}\bm{\chi}_{\mathrm{B}}^{me}(\mathrm{i} \xi), \end{equation} which leads to Eq.~(\ref{eq68}).
\end{appendix}
\section*{References}
\end{document} | arXiv | {
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\begin{document}
\title{Deep Equivariant Hyperspheres}
\begin{abstract} This paper presents an approach to learning $n$D features equivariant under orthogonal transformations for point cloud analysis, utilizing hyperspheres and regular $n$-simplexes. Our main contributions are theoretical and tackle major issues in geometric deep learning such as equivariance and invariance under geometric transformations. Namely, we enrich the recently developed theory of steerable 3D spherical neurons---$\textup{SO}(3)$-equivariant filter banks based on neurons with spherical decision surfaces---by extending said neurons to $n$D, which we call deep equivariant hyperspheres, and enabling their stacking in multiple layers. Using the ModelNet40 benchmark, we experimentally verify our theoretical contributions and show a potential practical configuration of the proposed equivariant hyperspheres. \end{abstract}
\section{Introduction} \label{introduction} \textit{Spheres}\footnote{By \textit{sphere}, we generally refer to an $n$D sphere or a hypersphere; \eg, a circle is thus a 2D sphere.} serve as a foundational concept in Euclidean space while simultaneously embodying the essence of non-Euclidean geometry through their intrinsic curvature and non-linear nature. This motivated their usage as decision surfaces encompassed by spherical neurons \cite{perwass2003spherical, melnyk2020embed}.
Felix Klein's Erlangen program of 1872 \cite{hilbert_cohnvossen_1952} introduced a methodology to unify non-Euclidean geometries, emphasizing the importance of studying geometries through their invariance properties under transformation groups. Similarly, geometric deep learning (GDL) \cite{bronstein2017geometric, bronstein2021geometric} constitutes a unifying framework for various neural architectures. This framework is built from the first principles of geometry---symmetry and scale separation---and enables tractable learning in high dimensions.
Symmetries play a vital role in preserving structural information of geometric data and allow models to adjust to different geometric transformations. This flexibility ensures that models remain robust and accurate, even when the input data undergoes various changes. In this context, spheres exhibit a maximal set of symmetries compared to other geometric entities in Euclidean space. The orthogonal group $\Og(n)$ fully encapsulates the symmetry structure of an $n$D sphere, including both rotational and reflection symmetries.
In this paper, we consider data that lives in Euclidean space (specifically, point clouds) and undergoes rotations and reflections, \ie, transformations of the $\Og(n)$-group. Enriching the theory of steerable 3D spherical neurons \cite{melnyk2022steerable}, we present a method for constructing $\Og(n)$-equivariant filters using regular $n$-simplexes\footnote{We use that a regular $n$-simplex contains $n+1$ equidistant vertices in $n$D.} and $n$D spheres, which we call \texttt{Deep Equivariant Hyperspheres} (see Figure~\ref{fig:main_figure}). The name also captures the fact that the vertices of a regular $n$-simplex lie on an $n$D sphere, and that our main result enables a way to stack the prior work steerable neurons in multiple layers, thereby enabling $deep$ propagation via them.
Even though the concept of spheres is also an essential part of spherical convolutional neural networks (CNNs) and CNNs designed to operate on 360 imagery \cite{Coors2018ECCV, su2017learning, Esteves_2018_ECCV, cohen2018spherical, perraudin2019deepsphere}, our method does not map input data \textit{on} a sphere, $\mathcal{S}^2$, nor does it perform convolution on a sphere. Instead, it embeds input in a higher-dimensional Euclidean space by means of a quadratic function. \begin{figure}
\caption{The central components of \texttt{Deep Equivariant Hyperspheres} (best viewed in color): regular $n$-simplexes with the $n$D spherical decision surfaces located at their vertices and the simplex change-of-basis matrices $\textbf{M}_n$ (displayed for the cases $n=2$ and $n=3$).}
\label{fig:main_figure}
\end{figure} While spheres have already been used as decision surfaces in prior work \cite{perwass2003spherical, melnyk2020embed} and their symmetries have been utilized for constructing equivariant models \cite{melnyk2022steerable, melnyk2022tetrasphere}, our main contributions are novel theoretical results summarized as follows:
\begin{itemize}
\item We propose $\Og(n)$-equivariant neurons, \texttt{Deep Equivariant Hyperspheres}, by extending the theory of steerable 3D spherical neurons \cite{melnyk2022steerable} to $n$D, which we rigorously prove (Sections~\ref{sec:simplex_change_of_basis} and \ref{sec:main_proofs} in their entirety).
\item We enable stacking steerable 3D spherical neurons in multiple layers by deriving the equivariant sequential $n\textup{D} \rightarrow (n+1)\textup{D}$ feature propagation and show how one can successfully train them by learning normalized spheres and using the activation normalization (Sections~\ref{sec:deep_propagation} and \ref{sec:demonstration}).
\item We provide proof-of-concept experiments using the common ModelNet40 \cite{wu20153d} benchmark and show a potential use of the developed theoretical framework (Section~\ref{sec:demonstration}). \end{itemize}
\section{Background} \label{sec:background} In this section, we present a comprehensive background on the theory of spherical neurons and their rotation-equivariant version, as well as on the general geometric concepts used in our work. \subsection{Spherical neurons via non-linear embedding} \label{sec:spherical_neurons} Spherical neurons \cite{perwass2003spherical, melnyk2020embed} are neurons with, as the name suggests, spherical decision surfaces. By virtue of conformal geometric algebra \cite{li2001generalized}, Perwass \etal \cite{perwass2003spherical} proposed to embed the data vector $\textbf{x}\in\mathbb{R}^n$ and represent the sphere with center $\textbf{c} =(c_1, \dots, c_n)\in\mathbb{R}^n$ and radius $r\in\mathbb{R}$ respectively as \begin{equation}
\label{hypersphere_in_r}
\begin{aligned}
\textbf{\textit{X}} = \big(x_1, \dots, x_n, -1, -\frac{1}{2}\lVert\textbf{x}\rVert^2\big)\in\mathbb{R}^{n+2} ~~~\textup{and}~~~
\textbf{\textit{S}} = \big(c_1, \dots, c_n, \frac{1}{2}(\lVert\textbf{c}\rVert^2 - r^2), 1\big)\in\mathbb{R}^{n+2},
\end{aligned} \end{equation} and used their scalar product $\textbf{\textit{X}}^\top \textbf{\textit{S}} = -\frac{1}{2}\norm{\textbf{x}-\textbf{c}}^2 + \frac{1}{2}r^2$ as a classifier, \ie, the spherical neuron: \begin{equation}
\label{eq:spherical_neuron}
f_{S}(\textbf{\textit{X}}; \textbf{\textit{S}}) = \textbf{\textit{X}}^\top \textbf{\textit{S}},
\end{equation} with learnable parameters $\textbf{\textit{S}}\in \mathbb{R}^{n+2}$.
The sign of this scalar product depends on the position of the point $\textbf{x}$ relative to the sphere $(\textbf{c}, r)$: inside the sphere if positive, outside of the sphere if negative, and on the sphere if zero \cite{perwass2003spherical}. Geometrically, the activation of the spherical neuron \eqref{eq:spherical_neuron} determines the cathetus length of the right triangle formed by $\textbf{x}$, $\textbf{c}$, and the corresponding point on the sphere (see Figure~2~in~\cite{melnyk2020embed}).
We note that with respect to the data vector $\textbf{x}\in\mathbb{R}^n$, a spherical neuron represents a non-linear function $f_{S}(\,\cdot\,;\textbf{\textit{S}}): \mathbb{R}^{n+2} \rightarrow \mathbb{R}$, due to the inherent non-linearity of the embedding \eqref{hypersphere_in_r}, and therefore, does not necessarily require an activation function, as observed by Melnyk \etal~\cite{melnyk2020embed}.
The components of $\textbf{\textit{S}}$ in \eqref{hypersphere_in_r} can be treated as \textit{independent} learnable parameters. In this case, a spherical neuron learns a \textit{non-normalized} sphere of the form $\widetilde{\textbf{\textit{S}}} = ({s_1}, \dots, {s_{n+2}}) \in \mathbb{R}^{n+2}$, which represents the same decision surface as its normalized counterpart defined in \eqref{hypersphere_in_r}, thanks to the homogeneity of the embedding \cite{perwass2003spherical, li2001generalized}. This is, however, not necessarily an ideal way of learning spheres when rotational equivariance must be respected, which we will allude to in our experiments in Section~\ref{sec:demonstration}.
\subsection{Equi- and invariance under orthogonal transformations} \label{sec:equivariance} The elements of the orthogonal group $\Og(n)$ can be represented as $n \times n$ matrices $\textbf{\textit{R}}$ with the properties $\textbf{\textit{R}}^\top \textbf{\textit{R}} = \textbf{\textit{R}} \textbf{\textit{R}}^\top = \textbf{I}_n$, where $\textbf{I}_n$ is the identity matrix, and $\det{\textbf{\textit{R}}} = \pm 1$, geometrically characterizing $n$D rotations and reflections. The special orthogonal group $\SO(n)$ is a subgroup of $\Og(n)$ and includes only orthogonal matrices with the positive determinant, representing rotations.
We say that a function $f : \mathcal{X} \rightarrow \mathcal{Y}$ is $\Og(n)$-equivariant if for every $\textbf{\textit{R}} \in \Og(n)$ there exists the transformation representation, $\rho(\textbf{\textit{R}})$, in the function output space, $\mathcal{Y}$, such that \begin{equation} \label{eq:equivariance}
\rho(\textbf{\textit{R}}) \, f(\textbf{\textup{x}}) = f(\textbf{\textit{R}} \textbf{\textup{x}}) \text{\quad for all~} \textbf{\textit{R}} \in \Og(n), \; \textbf{\textup{x}} \in \mathcal{X}. \end{equation} We call a function $f: \mathcal{X} \rightarrow \mathcal{Y}$ $\Og(n)$-invariant if for every $\textbf{\textit{R}} \in \Og(n)$, $\rho(\textbf{\textit{R}}) = \textbf{I}_n$. That is, if \begin{equation} \label{eq:invariance}
f(\textbf{\textup{x}}) = f(\textbf{\textit{R}} \textbf{\textup{x}}) \text{\quad for all~} \textbf{\textit{R}} \in \Og(n), \; \textbf{\textup{x}} \in \mathcal{X}. \end{equation} Following the prior work convention \cite{melnyk2022steerable, melnyk2022tetrasphere} hereinafter, we write $\textbf{\textit{R}}$ to denote the same $n$D rotation/reflection as an $n \times n$ matrix in the Euclidean space $\mathbb{R}^n$, as an $(n+1) \times (n+1)$ matrix in the projective (homogeneous) space $\textit{P}(\mathbb{R}^n) \subset \mathbb{R}^{n+1}$, and as an $(n+2) \times (n+2)$ matrix in $\mathbb{R}^{n+2}$. For the latter two cases, we achieve this by appending ones to the diagonal of the original $n \times n$ matrix without changing the transformation itself \cite{melnyk2020embed}.
\subsection{Steerable 3D spherical neurons and TetraSphere} \label{sec:steerable_3d_neurons} To propose rotation-equivariant steerable 3D spherical neurons, Melnyk~\etal~\cite{melnyk2022steerable} used as a necessary condition the fact that spherical neuron activations are isometries in 3D \cite{melnyk2020embed}, \ie, that the application of the spherical neurons commutes with rigid transformations (rotations and translations). It was proved in Theorem~4.1~\cite{melnyk2022steerable} that the activation of the spherical neuron $f_{S}(\cdot; \textbf{\textit{S}})$ \eqref{eq:spherical_neuron} and the activation of its arbitrarily rotated version $f_{S}(\cdot; \textbf{\textit{R}}\textbf{\textit{S}})$ differ by the term of the form $A \cos(\theta)$, where $A$ is a positive scalar. Crucially, it applies in \emph{any dimension}.
The main focus of Melnyk~\etal~\cite{melnyk2022steerable} was on 3D, and it was shown that a steerable filter \cite{freeman1991design, knutsson1992aframework} using spherical neurons in 3D needs to comprise \textit{minimum} four 3D spheres: one learnable spherical decision surface $\textbf{\textit{S}} \in \mathbb{R}^5$ \eqref{hypersphere_in_r} and its three copies \textit{rotated} into the other three vertices of the regular tetrahedron, using one of the results of Freeman~and~Adelson~\cite{freeman1991design} that the basis functions must be distributed in the space uniformly.
To construct such a filter, \ie, steerable 3D spherical neuron, the main (learned) sphere center $\textbf{c}_0$ needs to be rotated into $\norm{\textbf{c}_0}\,(1,1,1)$ by the corresponding (geodesic) rotation $\textbf{\textit{R}}_O$. The resulting sphere center is then rotated into the other three vertices of the regular tetrahedron. This is followed by rotating all four spheres back to the original coordinate system. One steerable 3D spherical neuron can thus be defined by means of the $4 \times 5$ matrix $B(\textbf{\textit{S}})$ containing the four spheres: \begin{equation}
\label{eq:sphere_filter_bank}
\textup{F}(\textbf{\textit{X}};\textbf{\textit{S}}) = B(\textbf{\textit{S}}) \textbf{\textit{X}}~,\quad
B(\textbf{\textit{S}}) =
\begin{bmatrix}
(\textbf{\textit{R}}_O^{\top}\, \textbf{\textit{R}}_{T_i}\, \textbf{\textit{R}}_O\, \textbf{\textit{S}})^\top\\
\end{bmatrix}_{i={1\ldots4}} ~, \end{equation} where ${\textbf{\textit{X}}} \in \mathbb{R}^{5}$ is the input 3D point embedded using \eqref{hypersphere_in_r}, $\{\textbf{\textit{R}}_{T_i}\}_{i=1}^{4}$ is the $\mathbb{R}^5$ rotation isomorphism corresponding to the rotation from the first vertex, \ie, $(1, 1, 1)$ to the $i$-th vertex of the regular tetrahedron\footnote{Therefore, $\textbf{\textit{R}}_{T_1}=\textbf{I}_5$, \ie, the original $\textbf{\textit{S}}$ remains at $\textbf{c}_0$.}.
In Theorem~4.2~\cite{melnyk2022steerable}, steerable 3D spherical neurons were proved to be $\SO(3)$-equivariant: \begin{equation} \label{eq:filter_bank_equivariance}
V_{\textbf{\textit{R}}} \, B(\textbf{\textit{S}}) \, \textbf{\textit{X}} = B(\textbf{\textit{S}})\,\textbf{\textit{R}}\textit{\textbf{X}},\quad
V_{\textbf{\textit{R}}} = \textup{\textbf{M}}^\top \textbf{\textit{R}}_O\, \textbf{\textit{R}}\, \textbf{\textit{R}}_O^{\top} \textup{\textbf{M}} ~, \end{equation} where $\textbf{\textit{R}}$ is a representation of the 3D rotation in {$\mathbb{R}^5$}, and $V_{\textbf{\textit{R}}} \in G < \SO(4)$ is the 3D rotation representation in the filter bank output space, with $\textup{\textbf{M}} \in \SO(4)$ being a change-of-basis matrix that holds the homogeneous coordinates of the tetrahedron vertices in its columns as \begin{equation}
\label{eq:M3}
\textbf{M} =
\begin{bmatrix}
\textbf{m}_1 & \textbf{m}_2 & \textbf{m}_3 & \textbf{m}_4 \end{bmatrix} = \frac{1}{2}
\begin{bmatrix}
1 & \phantom{-}1 & -1 & -1 \\
1 & -1 & \phantom{-}1 & -1 \\
1 & -1 & -1 & \phantom{-}1 \\
1 & \phantom{-}1 & \phantom{-}1 & \phantom{-}1 \\
\end{bmatrix}. \end{equation} We note that with respect to the input vector $\textbf{x}\in\mathbb{R}^3$, a steerable 3D spherical neuron represents a non-linear rotational-equvivariant function $\textup{F}(\,\cdot\,;\textbf{\textit{S}}): \mathbb{R}^5 \rightarrow \mathbb{R}^4$ with the learnable parameters $\textbf{\textit{S}} \in \mathbb{R}^5$.
\paragraph{TetraSphere} As the first reported attempt to \textit{learn} steerable 3D spherical neurons in an end-to-end approach, Melnyk \etal~\cite{melnyk2022tetrasphere} has presently introduced an approach for $\Og(3)$-invariant point cloud classification based on said neurons and the VN-DGCNN architecture~\cite{deng2021vector}, called TetraSphere.
Given the point-cloud input $\mathcal{X} \in \mathbb{R}^{N \times 3}$, the TetraSphere approach is to learn 4D features of each point by means of the TetraTransform layer $l_{\textup{TT}}(\,\cdot\,;\textbf{S}): \mathbb{R}^{N \times 3} \rightarrow \mathbb{R}^{N \times 4 \times K}$ that consists of $K$ steerable spherical neurons $B(\textbf{\textit{S}}_k)$ \eqref{eq:sphere_filter_bank} that are shared between the points. The obtained feature map is then propagated through the VN-DGCNN network as-is. After the computation of invariant features by means of the inner product of two equivariant feature maps, aggregation over the $K$ dimensions takes place, and the result is propagated through the remainder of the VN-DGCNN.
However, the work of Melnyk \etal~\cite{melnyk2022tetrasphere} does not investigate the question of how to combine the steerable neurons in multiple layers. In the following, we show that one way to achieve this is by extending the theory of steerable 3D spherical neurons to higher dimensions. \subsection{Regular simplexes} \label{sec:regular_simplexes} Geometrically, a regular $n$-simplex represents $n+1$ equidistant points in $n$D \cite{elte2006semiregular}, lying on an $n$D sphere with unit radius. In the 2D case, the regular simplex is an equilateral triangle; in 3D, a regular tetrahedron, and so on.
Following Cevikalp~and~Saribas~\cite{cevikalp2023deep}, we compute the Cartesian coordinates of a regular $n$-simplex as $n+1$ vectors $\textbf{p}_i \in \mathbb{R}^n$: \begin{equation} \label{eq:simplex} \begin{aligned}
\textbf{p}_i &= \begin{cases}
n^{-1/2} \, \textup{\textbf{1}}, & i = 1 \\
\kappa \, \textbf{1} + \mu \, \textbf{e}_{i-1}, & 2 \leq i \leq n+1 ~,
\end{cases}
\quad \kappa = -\frac{1+\sqrt{n+1}}{n^{3/2}}~,~~ \mu = \sqrt{1 + \frac{1}{n}}~, \end{aligned} \end{equation} where $\textbf{1} \in \mathbb{R}^n$ is a vector with all elements equal to 1 and $\textbf{e}_{i}$ is the natural basis vector with the $i$-th element equal to 1.
For the case $n=3$, we identify the following connection between \eqref{eq:M3} and \eqref{eq:simplex}: the columns of $\textbf{M}$, $\textbf{m}_i \in \mathbb{R}^4$, are the coordinates of the regular 3-simplex appended with a constant and normalized to unit length; that is, $\textbf{m}_i = \frac{1}{p} \begin{bmatrix} \textbf{p}_i \\ 1/\sqrt{3} \end{bmatrix}$ with $p = \left\lVert \begin{bmatrix} \textbf{p}_i \\ 1/\sqrt{3} \end{bmatrix} \right\rVert$, $1 \leq i \leq 4$.
\section{Deep Equivariant Hyperspheres} \label{sec:the_meat} \subsection{The simplex change of basis} \label{sec:simplex_change_of_basis} We generalize the change-of-basis matrix \eqref{eq:M3} to $n$D by introducing $\textup{\textbf{M}}_n$, an $(n+1) \times (n+1)$ matrix holding in its columns the coordinates of the regular $n$-simplex appended with a constant and normalized to unit length:\begin{equation}
\label{eq:nd_basis_matrix}
\textbf{M}_n =
\begin{bmatrix}
\textbf{m}_i \end{bmatrix}_{i={1\ldots n+1}},~~~\textbf{m}_i = \frac{1}{p} \begin{bmatrix} \textbf{p}_i \\ n^{-1/2} \end{bmatrix},~~~p= \left\lVert \begin{bmatrix} \textbf{p}_i \\ n^{-1/2} \end{bmatrix} \right\rVert, \end{equation} where the norms $p$ are constant, since $\lVert\textbf{p}_i\rVert = \lVert\textbf{p}_j\rVert$ for all $i$ and $j$ by definition of a regular simplex. \begin{proposition} \label{pr:M_orthogonal}
Let $\textup{\textbf{M}}_n$ be the-change-of-basis matrix defined in \eqref{eq:nd_basis_matrix}. Then $\textup{\textbf{M}}_n$ is an $(n+1)$D rotation or reflection, \ie, $\textup{\textbf{M}}_n \in \Og(n+1)$. \end{proposition}
\begin{proof} We want to show that $\textbf{M}_n^\top \textbf{M}_n = \textbf{I}_{n+1}$, which will prove that $\textbf{M}_n$ is orthogonal. The diagonal elements of $\textbf{M}_n^\top \textbf{M}_n$ are $\textbf{m}_i^\top \textbf{m}_i = \lVert\textbf{m}_i\rVert ^2 = 1$ since $\lVert \textbf{m}_i \rVert = 1$. The off-diagonal elements are found as $\textbf{m}_i^\top \textbf{m}_j = (\textbf{p}_i^\top \textbf{p}_j + n^{-1}) / p^2$ for $i \neq j$, where $p$ is defined in \eqref{eq:nd_basis_matrix}. Note that $\textbf{p}_i^\top \textbf{p}_j$ is the same for all $i$ and $j$ with $i \neq j$ since, by definition of a regular simplex, the vertices $\textbf{p}_i$ are spaced uniformly. Note that $\textbf{p}_i^\top \textbf{p}_j = -n^{-1}$ for all $i \neq j$ by definition \eqref{eq:simplex}. Hence, the off-diagonal elements of $\textbf{M}_n^\top \textbf{M}_n$ are zeros and $\textbf{M}_n^\top \textbf{M}_n = \textbf{I}_{n+1}$. \end{proof} Since $\textup{\textbf{M}}_n \in \Og(n+1)$, the sign of $\det{\textup{\textbf{M}}}_n$ is determined by the number of reflections required to form the transformation. In the case of a regular $n$-simplex, the sign of the determinant depends on the parity of $n$ \textbf{and} the configuration of the simplex vertices. In our case, $\textup{\textbf{M}}_n$ is a rotation for odd $n$, \ie, $\det{\textup{\textbf{M}}}_n = 1$, and a reflection for even $n$. Consider, for example, the case $n=3$. The matrix $\textbf{M}_3$ shown in \eqref{eq:M3} has $\det{\textbf{M}_3} = 1$, thus, is a 4D rotation, as stated in Section~\ref{sec:steerable_3d_neurons}.
\subsection{Equivariant \texorpdfstring{$n$D}{nD} spheres} \label{sec:main_proofs}
In this section, we generalize steerable 3D spherical neurons reviewed in Section~\ref{sec:steerable_3d_neurons} and denote a provisionally equivariant $n$D-sphere neuron (an \textit{equivariant hypersphere}) by means of the $(n+1)\times (n+2)$ matrix $B_n(\textbf{\textit{S}})$ for the spherical decision surface $\textbf{\textit{S}}\in\mathbb{R}^{n+2}$ with center $\textbf{c}_0 \in \mathbb{R}^n$ and an $n$D input $\textbf{x}\in\mathbb{R}^n$ embedded as $\textbf{\textit{X}}\in\mathbb{R}^{n+2}$ as \begin{equation}
\label{eq:sphere_NDfilter_bank}
\textup{\textbf{F}}_n(\textbf{\textit{X}};\textbf{\textit{S}}) = B_n(\textbf{\textit{S}})\, \textbf{\textit{X}}~,\quad
B_n(\textbf{\textit{S}}) =
\begin{bmatrix}
(\textbf{\textit{R}}_O^{\top}\, \textbf{\textit{R}}_{T_i}\, \textbf{\textit{R}}_O\, \textbf{\textit{S}})^\top \\
\end{bmatrix}_{i={1\ldots n+1}} ~, \end{equation} where $\{\textbf{\textit{R}}_{T_i}\}_{i=1}^{n+1}$ is the $\mathbb{R}^{n+2}$ rotation isomorphism corresponding to the rotation from the first vertex to the $i$-th vertex of the regular $n$-simplex, and $\textbf{\textit{R}}_O \in \SO(n)$ is the geodesic rotation from the sphere center $\textbf{c}_0$ to $\norm{\textbf{c}_0} \textbf{p}_1$ (therefore, $\textbf{\textit{R}}_{T_1}=\textbf{I}_{n+2}$).
We now need to prove that $\textbf{\textup{F}}_n(\, \cdot \,;\textbf{\textit{S}})$ is $\Og(n)$-equivariant. \begin{proposition} Let $\textbf{\textup{F}}_n(\, \cdot \,;\textbf{\textit{S}}): \mathbb{R}^{n+2} \rightarrow \mathbb{R}^{n+1}$ be the neuron defined in \eqref{eq:sphere_NDfilter_bank} and $\textbf{R} \in \Og(n)$ be an $n \times n$ rotation or reflection matrix. Then the transformation representation in the filter output space $\mathbb{R}^{n+1}$ is given by the $(n+1) \times (n+1)$ matrix \begin{equation}
\label{eq:V_n}
V_n = \rho \left(\textbf{\textit{R}}\right)= \textup{\textbf{M}}_n^\top \textbf{\textit{R}}_O \, \textbf{\textit{R}}\, \textbf{\textit{R}}_O^{\top} \textup{\textbf{M}}_n ~, \end{equation} where $\textup{\textbf{M}}_n \in \textup{O}(n+1)$ is the-change-of-basis matrix defined in \eqref{eq:nd_basis_matrix} and $1$ is appended to the diagonals of $\textbf{R}_O$ and $\textbf{R}$ to make them $(n+1) \times (n+1)$. Furthermore, $V_n \in G < \Og (n+1)$.
\end{proposition} \begin{proof} Since $\textup{\textbf{M}}_n \in \Og(n+1)$, $\textbf{\textit{R}}_O \in \SO(n)$, and $\textbf{\textit{R}} \in \Og(n)$ are orthogonal matrices, $V_n$ in \eqref{eq:V_n} is an orthogonal change-of-basis transformation that represents $\textbf{\textit{R}} \in \Og(n)$ in the basis constructed by $\textbf{M}_n$ and $\textbf{\textit{R}}_O$. Note that appending one to the diagonal of $\textbf{\textit{R}}\in \Og(n)$ does not affect the sign of the determinant, which makes $V_n$ a reflection representation if $\det{\textbf{\textit{R}}} = -1$, or a rotation representation if $\det{\textbf{\textit{R}}} = +1$. Since $\textbf{\textit{R}} \in \Og(n)$ and $\textbf{\textit{R}}_O \in \Og(n)$, not all elements of $\Og(n+1)$ can be generated by the operation \eqref{eq:V_n}. Thus, we conclude that $V_n$ belongs to a proper subgroup of $\Og(n+1)$, \ie, $G < \Og (n+1)$.
The original transformation is found directly as \begin{equation}
\label{eq:R_to_V}
\textbf{\textit{R}} = \textbf{\textit{R}}_O^{\top} \textbf{M}_n\, V_n \, \textbf{M}_n^{\top} \textbf{\textit{R}}_O~, \end{equation} followed by the retrieval of the upper-left $n \times n$ sub-matrix. \end{proof} Noteworthy, the basis determined by $\textbf{\textit{R}}_O\in \SO(n)$, which depends on the center $\textbf{c}_0$ of the sphere $\textbf{\textit{S}} \in \mathbb{R}^{n+2}$ (see \eqref{eq:sphere_NDfilter_bank}), will be different for different $\textbf{c}_0$. Therefore, the representation $V_n$ will differ as well.
\begin{theorem} \label{th:the_theorem} The neuron $\textup{\textbf{F}}_n(\, \cdot \,;\textbf{\textit{S}}): \mathbb{R}^{n+2} \rightarrow \mathbb{R}^{n+1}$ defined in \eqref{eq:sphere_NDfilter_bank} is $\Og(n)$-equivariant. \end{theorem} \begin{proof} We need to show that \eqref{eq:equivariance} holds for $\textup{\textbf{F}}_n(\, \cdot \,;\textbf{\textit{S}})$. We substitute \eqref{eq:V_n} to the left-hand side and \eqref{eq:sphere_NDfilter_bank} to the right-hand side, and obtain \begin{equation}
\label{eq:nd_filter_equivariance}
V_n \, B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}} = B_n(\textbf{\textit{S}})\,\textbf{\textit{R}}\textit{\textbf{X}}~. \end{equation} (For the detailed proof, see the Appendix.) \end{proof} We note that with respect to the input vector $\textbf{x}\in\mathbb{R}^n$, the equivariant hypersphere $\textbf{\textup{F}}_n(\, \cdot \,;\textbf{\textit{S}}): \mathbb{R}^{n+2} \rightarrow \mathbb{R}^{n+1}$ represents a non-linear $\Og(n)$-equvivariant function. It is also worth mentioning that the \textit{sum} of the output $\textbf{\textit{Y}} = B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}}$ is an $\Og(n)$-invariant scalar, \ie, the DC-component, due to the regular $n$-simplex construction.
This invariant part can be adjusted by adding a scalar \textit{bias} parameter to the output $\textbf{\textit{Y}}$. The concept of bias is imperative for linear classifiers, but for spherical decision surfaces \cite{perwass2003spherical}, it is implicitly modeled by the embedding \eqref{hypersphere_in_r}. We note, however, that adding a scalar bias parameter, $b \in \mathbb{R}$ to the output of an equivariant hypersphere \eqref{eq:sphere_NDfilter_bank} respects $\Og(n)$-equivariance: \begin{proposition} Let $\textbf{Y}\in\mathbb{R}^{n+1}$ be the output of the $\Og(n)$-equivariant hypersphere $\textup{\textbf{F}}_n(\, \cdot \,;\textbf{\textit{S}}): \mathbb{R}^{n+2} \rightarrow \mathbb{R}^{n+1}$ \eqref{eq:sphere_NDfilter_bank} given the input $\textbf{X}\in \mathbb{R}^{n+2}$, and $b \in \mathbb{R}$ be a bias parameter. Then $\textbf{Y}' = \textbf{Y} + b\,\textbf{\textup{1}}$, where $\textbf{\textup{1}}$ is the vector of ones in $\mathbb{R}^{n+1}$, is also $\Og(n)$-equivariant.
\end{proposition} \begin{proof}
We need to show that \eqref{eq:nd_filter_equivariance} also holds when the bias $b$ is added. First, we use $V_n$---the representation of $\textbf{\textit{R}}\in \Og(n)$ \eqref{eq:V_n}---and the fact that $\textbf{\textit{R}}$ and $\textbf{\textit{R}}_O$ are both appended 1 to their main diagonal to make them $(n+1) \times (n+1)$. Then $V_n \, \textbf{1} = \textup{\textbf{M}}_n^\top \textbf{\textit{R}}_O \, \textbf{\textit{R}}\, \textbf{\textit{R}}_O^{\top} \textup{\textbf{M}}_n \textbf{1} = \textup{\textbf{M}}_n^\top \textbf{\textit{R}}_O \, \textbf{\textit{R}}\, \textbf{\textit{R}}_O^{\top} \begin{bmatrix}\textbf{0}\\ p\,\sqrt{n} \end{bmatrix}= \textup{\textbf{M}}_n^\top \begin{bmatrix}\textbf{0}\\ p\,\sqrt{n} \end{bmatrix} = \textbf{1}$, where $p$ is a scalar defined in \eqref{eq:simplex}. Since the bias $b$ is a scalar, we use that $V_n \, b\textbf{1} = b V_n \, \textbf{1} = b \textbf{1}$. We now consider the left-hand side of \eqref{eq:nd_filter_equivariance}: $V_n \, \textbf{\textit{Y}}' = V_n \, (\textbf{\textit{Y}} + b\textbf{1}) = V_n \, B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}} + V_n \, b\textbf{1} = V_n \, B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}} + b\textbf{1}$. Plugging the equality \eqref{eq:nd_filter_equivariance} into the last equation, we complete the proof: $V_n \, B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}} + b\textbf{1} = B_n(\textbf{\textit{S}})\,\textbf{\textit{R}} \textit{\textbf{X}} + b\textbf{1}$. \end{proof} This result allows us to increase the capacity of the equivariant hypersphere by adding the learnable parameter $b \in \mathbb{R}$.
Finally, an important practical consideration in deep learning is feature normalization \cite{ioffe2015batch, ba2016layer}. We show how the activations of the equivariant hypersphere \eqref{eq:sphere_NDfilter_bank} can be normalized maintaining the equivariance: \begin{proposition} \label{prop:normalization} Let $\textbf{Y}\in\mathbb{R}^{n+1}$ be the $\Og(n)$-equivariant output of the hypersphere filter \eqref{eq:sphere_NDfilter_bank}. Then $\textbf{Y} / \lVert \textbf{Y} \rVert$, where $\lVert \textbf{Y} \rVert \in \mathbb{R}$, is also $\Og(n)$-equivariant.
\end{proposition} \begin{proof}
Let $\textbf{\textit{Y}}' = \textbf{\textit{Y}} / \lVert \textbf{\textit{Y}} \rVert$. We need to show that \eqref{eq:nd_filter_equivariance} holds also in the case of the normalization. We start by rewriting the right-hand side of \eqref{eq:nd_filter_equivariance}: $V_n \, \textbf{\textit{\textit{Y}}}' = \lVert \textbf{\textit{Y}} \rVert ^{-1} V_n \, \textbf{\textit{Y}} = \lVert \textbf{\textit{Y}} \rVert ^{-1} V_n \, B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}}$. We then use the originial equality \eqref{eq:nd_filter_equivariance} and rewrite the last equation: $\lVert \textbf{\textit{Y}} \rVert ^{-1} V_n \, B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}} = \lVert \textbf{\textit{Y}} \rVert ^{-1} B_n(\textbf{\textit{S}})\,\textbf{\textit{R}} \textit{\textbf{X}}$, which completes the proof.
\end{proof} Note that in practice, we often consider multiple vectors $\textbf{\textit{Y}}_i \in \mathcal{Y}^{N \times (n+1)}$, $i = 1 \cdots N$, so we could use, \eg, $\max_i {\lVert\textbf{\textit{Y}}_i\rVert}$ to normalize each $\textbf{\textit{Y}}_i$.
\subsection{Extracting deep equivariant features} \label{sec:deep_propagation} We might want to propagate the equivariant output of $\textbf{F}_n$ \eqref{eq:sphere_NDfilter_bank}, $\textbf{\textit{Y}} = B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}}$, through spherical decision surfaces while maintaining the equivariance properties. One way to achieve it is by using $(n+1)$D spheres, \ie, $\textbf{F}_{n+1}$, since the output $\textbf{\textit{Y}}\in\mathbb{R}^{n+1}$. Thus, the results established in the previous section not only allow us to use the equivariant hyperspheres \eqref{eq:sphere_NDfilter_bank} for $n$D inputs but also to cascade them in multiple layers, thus propagating equivariant representations by successively incrementing the feature space dimensionality with a unit step, \ie, $n\textup{D} \rightarrow (n+1)\textup{D}$.
Consider, for example, the point cloud patch $\mathcal{X} = \{ \textbf{x} \} ^N_{i=1}$ consisting of the coordinates of $N$ points $\textbf{x}\in \mathbb{R}^n$ as input signal. Assuming that all operations are point-wise and that $\textbf{F}_n(\, \cdot \,; \textbf{\textit{S}})$ are shared between the points, a \textit{cascaded} $n\textup{D} \rightarrow (n+1)\textup{D}$ feature extraction procedure using equivariant hyperspheres $\textbf{F}_n(\, \cdot \,; \textbf{\textit{S}})$ for the given output dimensionality $d$ (with $d > n$) can be defined as follows (at the first step, $\textbf{\textit{X}} \gets \textbf{x}$): \begin{equation} \label{eq:algorithm} \begin{aligned}
\textbf{\textit{X}} \in\mathbb{R}^n &\rightarrow \texttt{embed}(\texttt{normalize}(\textbf{\textit{X}}+b)) \rightarrow \textbf{F}_n(\textbf{\textit{X}}; \textbf{\textit{S}}) \rightarrow \texttt{embed}(\texttt{normalize}(\textbf{\textit{X}}+b)) \\ &\rightarrow \textbf{F}_{n+1}(\textbf{\textit{X}}; \textbf{\textit{S}}) \rightarrow \ldots \rightarrow \textbf{F}_{d}(\textbf{\textit{X}}; \textbf{\textit{S}}) \rightarrow \texttt{normalize}(\textbf{\textit{X}}+b) \rightarrow \textbf{\textit{X}} \in \mathbb{R}^{d}~, \end{aligned} \end{equation} where $\texttt{embed}$ is the embedding according to \eqref{hypersphere_in_r}, $\texttt{normalize}$ is the optional activation normalization (see Proposition~\ref{prop:normalization}), and $b$ is an optional scalar bias. \begin{proposition} Given that all operations involved in the procedure~\ref{eq:algorithm} are $\Og(n)$-equivariant, its output will also be $\Og(n)$-equivariant.
\end{proposition} \begin{proof} Let $\textbf{R} \in \Og(n)$ be an orthogonal transformation, $\rho_i(\textbf{\textit{R}})$ the representation of $\textbf{\textit{R}}$ in the respective space, \eg, \eqref{eq:V_n} for the equivariant hypersphere output, and $\textbf{\textit{x}}\in \mathbb{R}^n$ be the input to the procedure~\eqref{eq:algorithm}. We denote the output of the procedure~\eqref{eq:algorithm} as $\textbf{F}(\textbf{x})$, where $\textbf{F}$ is the composition of all operations in the procedure~\eqref{eq:algorithm}. Since each operation is equivariant, \eqref{eq:equivariance} holds for each operation $\boldsymbol{\Phi}$, \ie, we have $\boldsymbol{\Phi}_i(\rho_i (\textbf{\textit{R}})\textbf{\textit{X}}) = \rho_{i+1}(\textbf{\textit{R}})\boldsymbol{\Phi}(\textbf{\textit{X}})$. Consider now the output $\textbf{{F}}(\textbf{x})$ and the transformed output $\textbf{F}(\textbf{\textit{R}}\textbf{x})$. Since each operation in $\textbf{F}$ is equivariant, we have:
\noindent $\begin{aligned} \textbf{F}(\textbf{\textit{R}}\textbf{x}) = \boldsymbol{\Phi}_d(\boldsymbol{\Phi}_{d-1}(\ldots\boldsymbol{\Phi}_2(\boldsymbol{\Phi}_1(\textbf{\textit{R}}\textbf{x})))) = \rho_d(\textbf{\textit{R}}) \boldsymbol{\Phi}_d(\boldsymbol{\Phi}_{d-1}(\ldots \boldsymbol{\Phi}_2(\boldsymbol{\Phi}_1(\textbf{x})))) = \rho_d(\textbf{\textit{R}}) \textbf{F}(\textbf{x}). \end{aligned}$
Thus, the output of the procedure~\eqref{eq:algorithm} is equivariant, as desired. \end{proof}
\section{Experimental validation} \label{sec:demonstration}
In this section, we experimentally verify our theoretical results derived in Section~\ref{sec:the_meat} and investigate what techniques should be used when training the proposed deep equivariant hyperspheres. Rather than proposing a novel architectural design, our main intention has been to integrate and validate our findings within an established model architecture. This allows us to focus on the implications of our theoretical contributions, demonstrating their potential when used in well-studied frameworks.
For this, we use a common 3D point cloud classification benchmark and the $\Og(3)$-invariant VN-DGCNN model \cite{deng2021vector} comprised of equivariant layers, as both the backbone for our proposed equivariant hyperspheres and the baseline model. This choice is motivated by the compatibility of VN-DGCNN layers and features extracted by equivariant hyperspheres, as was established in TetraSphere~\cite{melnyk2022tetrasphere}, where steerable 3D spherical neurons were used to learn the 4D projections of 3D input. We implemented our method using PyTorch \cite{paszke2019pytorch}. For the baseline, we used the code from the official repository\footnote{\url{https://github.com/FlyingGiraffe/vnn}}.
\subsection{Dataset}
\textbf{ModelNet40} \cite{wu20153d} is a common benchmark for 3D shape classification \cite{Chen_2022_CVPR, zhao2020quaternion, deng2021vector, melnyk2022tetrasphere, li2021rotation}. We use the dataset version by \cite{qi2017pointnet}, which contains 12,311 CAD models belonging to 40 distinct classes, with 9843 training and 2468 test samples. We perform the common preprocessing: we sample 1024 points from each 3D model at random, center at the origin and normalize them to a unit sphere.
\textbf{$\Og(n)$-augmentation setup} To demonstrate that our proposed equivariant hyperspheres are indeed $\Og(n)$-equivariant, and hence, can be used to produce invariant features, we employ four different train/test transformation setups: I$/$I, I$/z$, I$/\SO(3)$, and I$/\Og(3)$, with the latter two being the more practical ones. In these setups, ``I'' denotes no rotation augmentation, $z$ stands for augmentation by vertical-axis rotation, and $\SO(3)$ and $\Og(3)$ represent arbitrary 3D rotations and reflections, respectively. All the augmentation settings were generated and applied to the input data on the fly.
\subsection{Implementation details} \label{sec:implementation} \begin{figure}
\caption{\textbf{Top}: A high-level architecture of VN-DGCNN with the specified dimensionality of the feature tensors. Edge-convolutional layers are equivariant; see \cite{deng2021vector} for details. \textbf{Bottom}: The architecture used to validate the theory of the proposed \texttt{Deep Equivariant Hyperspheres}. Input to each $n$D sphere is treated as $N\cdot C_i$~~$n$D vectors. The spheres are shared between these $N\cdot C_i$ vectors, much like the shared multilayer perceptrons in PointNet~\cite{qi2017pointnet}. The $N\cdot C_i \times (n+1)$ output is then reshaped to the original form.}
\label{fig:deh_architecture}
\end{figure} The sketch of the baseline $\Og(3)$-invariant VN-DGCNN is displayed at the top of Figure~\ref{fig:deh_architecture}. We refer to the original paper \cite{deng2021vector} for more details. Here, we only note that its layers are $\Og(3)$-equivariant and operate on the latent dimension produced by the edge computation, inherited from the DGCNN architecture \cite{wang2019dgcnn}, before the first convolutional layer. The invariant features are obtained from the equivariant ones by means of their inner product. We keep the original hyperparameters for all the layers, including $C' = 3$, for the computations in the $\Og(3)\,\textup{inv}$ block (see Figure~\ref{fig:deh_architecture}).
We also include in our comparison the recent TetraSphere model \cite{melnyk2022tetrasphere}, which uses steerable 3D spherical neurons as reviewed in Section~\ref{sec:steerable_3d_neurons}. We use the originally proposed setup with $K=1$ \cite{melnyk2022tetrasphere} and select $C' = 3$ for consistency with the baseline.
Finally, we use VN-DGCNN as the backbone and integrate our proposed \texttt{Deep Equivariant Hyperspheres} (see the procedure~\ref{eq:algorithm}) into it, as demonstrated at the bottom Figure~\ref{fig:deh_architecture}. This integration is viable due to the equivariance of the backbone layers: while VN-DGCNN operates on the latent dimensions, $C_i$, keeping the original signal dimensionality of $3$, our equivariant hyperspheres successively embed the $3$D features into $4$D, $5$D and so on. Crucially, to concatenate the features extracted by the different layers of the network and subsequently obtain invariant features from them, we need to embed the lower-dimensional features by sharing the spheres of the respective dimensionality (see the dashed blocks in Figure~\ref{fig:deh_architecture}). This is in line with the remark made in Section~\ref{sec:main_proofs} that the transformation representations \eqref{eq:V_n} are different for hyperspheres with different centers $\textbf{c}_0$. Similar to the baseline, we select $C' = 3$.
The configuration of our model (see Figure~\ref{fig:deh_architecture}) depends on the following: (a) we may learn normalized hyperspheres \eqref{hypersphere_in_r}, \ie, with the $(n+2)$-th parameter set to always be 1, or we may opt for non-normalized, treating an $n$D sphere as $n+2$ learnable parameters (see Section~\ref{sec:spherical_neurons}); (b) for each point and in each hypersphere output $N\times n \times C_i$, we may normalize the sphere activations according to the procedure~\ref{eq:algorithm} by scaling the rows of each $n \times C_i$ matrix by the maximum norm among the columns; (c) finally, we may or may not use bias parameter (procedure~\ref{eq:algorithm}). In the following, we present a corresponding comparative study. See the Appendix for a discussion on the number of learnable parameters.
To ensure a fair comparison, we use the following training setup for all the methods in the experiments: stochastic gradient descent (SGD) with an initial learning rate of $0.1$, the momentum of $0.9$, weight decay of $10^{-4}$, and a cosine annealing strategy for gradually reducing the learning rate of $0.001$. Following the original work \cite{deng2021vector}, we augment the data during training using random translation in the range $[-0.2, 0.2]$ and scaling ranged in $[2/3, 3/2]$. We train the models for 500 epochs with a batch size of 32. We run each experiment three times using three different seeds.
\subsection{Results and discussion}
We first experiment with the configuration of our model (see Figure~\ref{fig:deh_architecture}). We observe that without normalizing the sphere activations, our model cannot converge on the training data. Therefore, we always normalize the sphere activations, as described in Section~\ref{sec:implementation}. We summarize the experiments on the configuration of our model in Table~\ref{tab:configuration_study}. Therein the case of $n=6$ corresponds to Figure~\ref{fig:deh_architecture}, and shallower versions are obtained by discarding higher-dimensional equivariant hyperspheres; if the bias parameters are used, they are not shared between the shared $n$D spheres. From Table~\ref{tab:configuration_study}, we make two observations for the case $n=3$, which is the closest to TetraSphere: 1) normalization does not improve the performance and 2) a single bias parameter adds enough capacity to a single steerable 3D spherical neuron (3D equivariant hypersphere, in the terminology of our work) to match the performance of the deeper cascaded versions without bias.
For the cascaded \texttt{Deep Equivariant Hyperspheres} case ($n\geq4$), the opposite two observations are true: 1) learning normalized spheres is always a better option than non-normalized, especially as the cascading depth increases; 2) using deeper normalized hyperspheres tends to result in higher performance without requiring the bias parameter. Although increasing the depth to $n>6$ is possible, it is not reasonable given the architecture of the backbone and the deep-hypersphere integration scheme we use in this experiment (see Figure~\ref{fig:deh_architecture}). Using the results from Table~\ref{tab:configuration_study}, we select the model with $n=6$, normalized spheres, and no bias, given the consistency between its high mean performance, low standard deviation, and high best-run performance, for further comparison with the baseline.
In Table~\ref{tab:main_results}, we present the comparison of our chosen model with the baseline. We see a clear improvement in the performance in our case (see Table~\ref{tab:main_results}). The performance of TetraSphere is practically the same as our model. This indicates an interesting phenomenon: equivariant hypersphere representations learned at the level of input (TetraSphere) can be learned just as well with the cascaded hyperspheres (our model). In either case, the result is also indicative of the potential advantage of learning equivariant representations of $n$D input in the higher-dimensional spaces learned by equivariant hyperspheres. \begin{table}[ht]
\centering
\footnotesize
\caption{\texttt{Deep Equivariant Hyperspheres} configuration: Classification accuracy (mean and standard deviation over 3 runs with the best out of them presented in parentheses, \%) on the ModelNet40 test shapes under different transformation augmentation settings. The best results are presented in \textbf{bold}.}
\begin{tabular}{cccl}
\hline & \\ [-1.5ex]
\multicolumn{3}{c}{\texttt{Deep Equivariant Hyperspheres}} & \multicolumn{1}{c}{$\textup{I}/\left\{\textup{I}, z, \SO(3), \Og(3)\right\}$} \\ [0.5ex]
\cline{1-3} & \\ [-1.5ex]
$n$ & normalized spheres & bias & \\
\hline & \\ [-1.5ex]
& \ding{55} & \ding{55} & $89.2 \pm 0.2\% ~~~(\underline{89.3})$ \\
$3$ & \ding{51} & \ding{55} & $89.2 \pm 0.4\% ~~~(\underline{89.5})$ \\
& \ding{51} & \ding{51} & $89.6 \pm 0.4\% ~~~(\underline{90.0})$ \\
\hline & \\ [-1.5ex]
& \ding{55} & \ding{55} & $89.2 \pm 0.3\% ~~~(\underline{89.5})$ \\
$4$ & \ding{51} & \ding{55} & $\textbf{89.7 }\pm 0.2\% ~~(\underline{89.9})$ \\
& \ding{51} & \ding{51} & $89.3 \pm 0.1\% ~~~(\underline{89.5})$ \\
\hline & \\ [-1.5ex]
& \ding{55} & \ding{55} & $56.8 \pm 28.2\% ~(\underline{89.3})$ \\
$5$ & \ding{51} & \ding{55} & $89.4 \pm 0.7\% ~~~(\underline{\textbf{90.1}})$ \\
& \ding{51} & \ding{51} & $89.2 \pm 0.4\% ~~~(\underline{89.6})$ \\
\hline & \\ [-1.5ex]
& \ding{55} & \ding{55} & $44.2 \pm 38.5\% ~(\underline{86.9})$ \\
$6$ & \ding{51} & \ding{55} & $\textbf{89.7} \pm 0.3\% ~~~(\underline{90.0})$ \\
& \ding{51} & \ding{51} & $89.4 \pm 0.5\% ~~~(\underline{89.9})$ \\
[1.0ex]
\hline
\end{tabular}
\label{tab:configuration_study}
\end{table} \begin{table}[ht]
\centering
\footnotesize
\caption{Classification accuracy (mean and standard deviation over 3 runs with the best out of them presented in parentheses, \%) on the ModelNet40 test shapes under different test transformation augmentation settings (since the values are identical, we use a single column to report them). The best results are presented in \textbf{bold}.}
\begin{tabular}{ll}
\hline & \\ [-1.5ex]
Methods & \multicolumn{1}{c}{$\textup{I}/\left\{\textup{I}, z, \SO(3), \Og(3)\right\}$} \\ [0.5ex]
\hline & \\ [-1.5ex]
VN-DGCNN \cite{deng2021vector} & $89.3 \pm 0.3\% ~(\underline{89.6})$ \\
TetraSphere \cite{melnyk2022tetrasphere} & $\textbf{89.8} \pm 0.2\% ~(\underline{89.9})$ \\
\texttt{Deep Equivariant Hyperspheres} & $89.7 \pm 0.3\% ~(\underline{\textbf{90.0}})$ \\
[1.0ex]
\hline
\end{tabular}
\label{tab:main_results}
\end{table}
\textbf{Limitations}: The experimental verification of our theoretical contributions is performed in a noiseless setup with a single synthetic dataset with three runs per model configuration. However, the main conclusion from the obtained results, \ie, the validity of the presented theory, stands as independent of these. \textbf{Compute}: We run our experiments on a single NVIDIA A100, with a single run taking about 40 hours on average.
\section{Conclusion} \label{sec:conslusion}
In this manuscript, we presented \texttt{Deep Equivariant Hyperspheres} --- $n$D neurons equivariant under orthogonal transformations, utilizing spheres and regular $n$-simplexes. We generalized the theory of steerable 3D spherical neurons \cite{melnyk2022steerable} to $n$D and rigorously proved it. Owing to this result, we proposed to cascade these $n$D neurons and thereby enabled their stacking in multiple layers. Albeit tailored to point cloud analysis, the established theory has the potential to be applied to images by, \eg, considering the image grid as 2D coordinates, and other types of data structures such as graphs. This forms a clear direction for future work.
\section*{Acknowledgments} {This work was supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP), by the Swedish Research Council through a grant for the project Algebraically Constrained Convolutional Networks for Sparse Image Data (2018-04673), and the strategic research environment ELLIIT.
The computations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS) partially funded by the Swedish Research Council through grant agreement no. 2022-06725.3, and by the Berzelius resource provided by the Knut and Alice Wallenberg Foundation at the National Supercomputer Centre.}
{
}
\begin{center} \textbf{\Large Appendix} \end{center} \section*{A1\quad Numeric instances for $n=\left\{2, 3, 4\right\}$} To facilitate the reader's understanding of the algebraic manipulations in the next section, herein, we present numeric instances of the central components of our theory defined in \eqref{eq:simplex} and \eqref{eq:nd_basis_matrix}, for the cases $n=2$, $n=3$, and $n=4$. For convenience, we write the vertices of the regular simplex \eqref{eq:simplex} as the $n\times (n+1)$ matrix $\textbf{P}_n = \begin{bmatrix}
\textbf{p}_i \end{bmatrix}_{i=1\ldots n+1}$.
\paragraph{$n=2:$} $\textbf{P}_2 = \frac{\sqrt{2}}{2} \begin{bmatrix} 1 & \phantom{-}(\sqrt{3}-1)/2 & -(\sqrt{3}+1)/2 \\ 1 & -(\sqrt{3}+1)/2 & \phantom{-}(\sqrt{3}-1)/2 \\ \end{bmatrix}$, \quad\quad $p=\sqrt{3/2}$,
\paragraph{$\phantom{n=2:}$} $\textbf{M}_2 = \frac{1}{\sqrt{3}} \begin{bmatrix}
1 & \phantom{-}(\sqrt{3}-1)/2 & -(\sqrt{3}+1)/2 \\
1 & -(\sqrt{3}+1)/2 & \phantom{-}(\sqrt{3}-1)/2 \\
1 & 1 & 1 \\ \end{bmatrix}$.
\paragraph{$n=3:$} $\textbf{P}_3 = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & \phantom{-}1 & -1 & -1 \\ 1 & -1 & \phantom{-}1 & -1 \\ 1 & -1 & -1 & \phantom{-}1 \end{bmatrix}$, \quad\quad\quad\quad\quad\quad\quad\quad\,$p=2/\sqrt{3}$,
\paragraph{$\phantom{n=3:}$} $\textbf{M}_3 =\frac{1}{2}
\begin{bmatrix}
1 & \phantom{-}1 & -1 & -1 \\
1 & -1 & \phantom{-}1 & -1 \\
1 & -1 & -1 & \phantom{-}1 \\
1 & \phantom{-}1 & \phantom{-}1 & \phantom{-}1 \\
\end{bmatrix}$.
\paragraph{$n=4:$} $\textbf{P}_4 = \frac{1}{2} \begin{bmatrix} 1 & \phantom{\;}(3\sqrt{5} - 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 \\ 1 & -(\sqrt{5} + 1)/4 & \phantom{\;}(3\sqrt{5} - 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 \\ 1 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & \phantom{\;}(3\sqrt{5} - 1)/4 & -(\sqrt{5} + 1)/4 \\ 1 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & \phantom{\;}(3\sqrt{5} - 1)/4 \end{bmatrix}$,
\paragraph{$\phantom{n=4:}$}$p = \sqrt{5}/2$,
\paragraph{$\phantom{n=4:}$} $\textbf{M}_4 = \frac{1}{\sqrt{5}} \begin{bmatrix} 1 & \phantom{\;}(3\sqrt{5} - 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 \\ 1 & -(\sqrt{5} + 1)/4 & \phantom{\;}(3\sqrt{5} - 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 \\ 1 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & \phantom{\;}(3\sqrt{5} - 1)/4 & -(\sqrt{5} + 1)/4 \\ 1 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & -(\sqrt{5} + 1)/4 & \phantom{\;}(3\sqrt{5} - 1)/4 \\ 1 & 1 & 1 & 1 & 1 \end{bmatrix}$.
\section*{A2\quad Proof of Theorem 3} \label{sec:complete_proof} In this section, we present the proof of Theorem~3 from the main paper.
\begin{lemma} \label{lem:lemma} Let $\textbf{\textup{M}}_n$ be the change-of-basis matrix defined in \eqref{eq:nd_basis_matrix}, and $\textbf{\textup{P}}_n$ an $n \times (n+1)$ matrix holding the regular $n$-simplex vertices, $\textbf{\textup{p}}_i$, in its columns, and $p = \left\lVert \begin{bmatrix} \textbf{\textup{p}}_i \\ n^{-1/2} \end{bmatrix} \right\rVert$, as defined in \eqref{eq:nd_basis_matrix}. Then \begin{equation}
\label{eq:M_times_P^T}
\textbf{\textup{M}}_n \textbf{\textup{P}}_n^\top = p \begin{bmatrix}
\textbf{\textup{I}}_n\\
\textbf{\textup{0}}^\top
\end{bmatrix}. \end{equation} \end{lemma} \begin{proof} We begin by elaborating on \eqref{eq:nd_basis_matrix}: \begin{equation}
\label{eq:M_and_P}
\textbf{M}_n = \frac{1}{p}\begin{bmatrix}
\textbf{P}_n\\ n^{-1/2} \, \textbf{1}^\top
\end{bmatrix}. \end{equation}
We note that the norms of the rows of $\textbf{P}_n$ are also equal to $p$ since $\textbf{M}_n \in \Og(n+1)$ (as per Proposition~\ref{pr:M_orthogonal}). Recall that $\textbf{P}_n$ is centered at the origin, and, therefore, for a constant $a \in \mathbb{R}$ and a vector of ones $\textbf{1} \in \mathbb{R}^{n+1}$, we obtain $a\,\textbf{1}^\top \textbf{P}_n^\top = \textbf{0}^\top$. Remembering that the product $\textbf{M}_n \textbf{P}_n^\top$ is between $\mathbb{R}^{n+1}$ vectors, we plug \eqref{eq:M_and_P} into the LHS of \eqref{eq:M_times_P^T} and obtain \begin{equation}
\textbf{M}_n \textbf{P}_n^\top = \frac{1}{p}\begin{bmatrix}
\textbf{P}_n\\ n^{-1/2} \, \textbf{1}^\top
\end{bmatrix} \, \textbf{P}_n^\top = \frac{p^2}{p} \begin{bmatrix}
\textbf{I}_n\\
\textbf{0}^\top
\end{bmatrix} = p \begin{bmatrix}
\textbf{I}_n\\
\textbf{0}^\top
\end{bmatrix}. \end{equation} \end{proof}
We now restate and prove Theorem~3 from the paper. \begin{theorem*} The neuron $\textup{\textbf{F}}_n(\textbf{\textit{X}};\textbf{\textit{S}}) = B_n(\textbf{\textit{S}})\, \textbf{\textit{X}}\,$ defined in \eqref{eq:sphere_NDfilter_bank} is $\Og(n)$-equivariant. \end{theorem*}
\begin{proof} \label{proof:theorem3} We need to show that \eqref{eq:equivariance} holds for $\textup{\textbf{F}}_n(\, \cdot \,;\textbf{\textit{S}})$. We substitute \eqref{eq:V_n} to the LHS and \eqref{eq:sphere_NDfilter_bank} to the RHS, and obtain \begin{equation}
\label{eq:nd_filter_equivariance_supp}
V_n \, B_n(\textbf{\textit{S}}) \, \textbf{\textit{X}} = B_n(\textbf{\textit{S}})\,\textbf{\textit{R}}\textit{\textbf{X}}~. \end{equation}
Keeping in mind that the $(n+1)$-th and $(n+2)$-th components, $s_{n+1}$ and $s_{n+2}$, of the sphere $\textbf{\textit{S}}\in \mathbb{R}^{n+2}$ with center $\textbf{c}_0 \in \mathbb{R}^n$ \eqref{hypersphere_in_r} are $\Og(n)$-invariant, as well as our convention on writing the rotation matrices (see the last paragraph of Section~\ref{sec:equivariance}), we rewrite the $(n+1) \times (n+2)$ matrix $B_n(\textbf{\textit{S}})$ using its definition \eqref{eq:sphere_NDfilter_bank}: \begin{equation}
\label{eq:B_n}
B_n(\textbf{\textit{S}}) =
\begin{bmatrix}
(\textbf{\textit{R}}_O^{\top}\, \textbf{\textit{R}}_{T_i}\, \textbf{\textit{R}}_O\, \textbf{\textit{S}})^\top\\
\end{bmatrix}_{i={1\ldots n+1}} =
\begin{bmatrix}
\textbf{c}_0^\top\textbf{\textit{R}}_O^\top\, \textbf{\textit{R}}_{T_i}^\top\, \textbf{\textit{R}}_O\
& s_{n+1} & s_{n+2}
\end{bmatrix}_{i={1\ldots n+1}}. \end{equation}
By definition of the rotation $\textbf{\textit{R}}_O$ \eqref{eq:sphere_NDfilter_bank}, we have that $\textbf{\textit{R}}_O\,\textbf{c}_0 = \lVert \textbf{c}_0 \rVert \textbf{p}_1$, where $\textbf{p}_1 \in \mathbb{R}^n$ is the first vertex of the regular simplex \eqref{eq:simplex}. Since $\textbf{\textit{R}}_{T_i}$ rotates $\textbf{p}_1$ into $\textbf{p}_i$, we obtain \begin{equation}
\label{eq:c_and_P}
\textbf{\textit{R}}_{T_i} \textbf{\textit{R}}_O\,\textbf{c}_0 = \lVert \textbf{c}_0 \rVert \cdot \textbf{p}_i~,\quad 1 \leq i \leq n+1~. \end{equation}
Thus, we can write the RHS of \eqref{eq:nd_filter_equivariance_supp} using the sphere definition \eqref{hypersphere_in_r} as \begin{equation}
\label{eq:RHS} B_n(\textbf{\textit{S}})\,\textbf{\textit{R}}\textbf{\textit{X}} =
\begin{bmatrix}
\lVert \textbf{c}_0 \rVert \cdot \textbf{p}_i^\top\, \textbf{\textit{R}}_O\ &
s_{n+1} & s_{n+2}
\end{bmatrix}_{i={1\ldots n+1}} \textbf{\textit{R}}\textbf{\textit{X}} =
\begin{bmatrix}
\lVert \textbf{c}_0 \rVert \, \textbf{P}_n^\top\, \textbf{\textit{R}}_O \textbf{\textit{R}}\ &
s_{n+1}\, \textbf{1} & s_{n+2}\, \textbf{1}
\end{bmatrix}\,\textbf{\textit{X}}. \end{equation} We now use the definition of $V_n$ along with \eqref{eq:M_times_P^T}, \eqref{eq:M_and_P}, and \eqref{eq:c_and_P} to rewrite the LHS of \eqref{eq:nd_filter_equivariance_supp} as \begin{equation}
\label{eq:LHS}
\begin{aligned}
V_n \, B_n(\textbf{\textit{S}}) \textbf{\textit{X}} &= \textbf{M}_n^{\top}\textbf{\textit{R}}_O\,\textbf{\textit{R}}\,\textbf{\textit{R}}_O^\top\,\textbf{M}_n \begin{bmatrix} \lVert \textbf{c}_0 \rVert \cdot \textbf{P}_n^\top\, \textbf{\textit{R}}_O \ & s_{n+1}\, \textbf{1} & s_{n+2}\, \textbf{1}
\end{bmatrix}\,\textbf{\textit{X}} \\
& = \textbf{M}_n^{\top}\textbf{\textit{R}}_O\,\textbf{\textit{R}}\,\textbf{\textit{R}}_O^\top\,\begin{bmatrix} p\,\lVert \textbf{c}_0 \rVert\begin{bmatrix}
\textbf{I}_n\\
\textbf{0}^\top
\end{bmatrix}\, \textbf{\textit{R}}_O \ &
\textbf{0} & \textbf{0} \\
& {p}\sqrt{n}\,s_{n+1} & {p}\sqrt{n}\,s_{n+2}
\end{bmatrix}\,\textbf{\textit{X}}\\
& = \textbf{M}_n^{\top}\textbf{\textit{R}}_O\,\textbf{\textit{R}}\,
\begin{bmatrix} p\,\lVert \textbf{c}_0 \rVert\,\textbf{\textit{R}}_O^\top\,\textbf{\textit{R}}_O& \textbf{0} & \textbf{0} \\
\textbf{0}^\top & {p}\sqrt{n}\,
s_{n+1} &{p}\sqrt{n}\, s_{n+2}
\end{bmatrix}\,\textbf{\textit{X}}\\
& =
\frac{1}{p}\begin{bmatrix}
\textbf{P}_n^\top\, \textbf{\textit{R}}_O\,\textbf{\textit{R}} & n^{-1/2} \cdot \textbf{1}
\end{bmatrix}
\begin{bmatrix} p\,\lVert \textbf{c}_0 \rVert\, \textbf{I}_n& \textbf{0} & \textbf{0} \\
\textbf{0}^\top & {p}\sqrt{n}\,
s_{n+1} &{p}\sqrt{n}\, s_{n+2}
\end{bmatrix}\,\textbf{\textit{X}}\\
& =
\begin{bmatrix}
\,\lVert \textbf{c}_0 \rVert\, \textbf{P}_n^\top\, \textbf{\textit{R}}_O\,\textbf{\textit{R}}
& \frac{\sqrt{n}}{\sqrt{n}}\, s_{n+1}\, \textbf{1} & \frac{\sqrt{n}}{\sqrt{n}}\, s_{n+2}\, \textbf{1}
\end{bmatrix}
\,\textbf{\textit{X}} \quad = \quad B_n(\textbf{\textit{S}})\,\textbf{\textit{R}}\textbf{\textit{X}}.
\end{aligned} \end{equation} \end{proof}
\section*{A3\quad Model complexities} All the models in the comparison in Table~\ref{tab:main_results} have approximately the same number of learnable parameters close to $2.9 \cdot 10^6$.
In particular, the baseline VN-DGCNN contains $2,899,830$ learnable parameters. Compared to the baseline, TetraSphere in the selected configuration has $1~(\textup{3D sphere}) \cdot 5~(\textup{params.}) = \underline{5}$ extra learnable parameters.
Whereas our proposed \texttt{Deep Equivariant Hyperspheres} in the configuration presented in Table~2 (no bias, normalized spheres) contains $1~(\textup{3D sphere}) \cdot 4~(\textup{params.}) + 1~(\textup{4D sphere}) \cdot 5~(\textup{params.}) + 1~(\textup{5D sphere}) \cdot 6~(\textup{params.}) + 1~(\textup{7D sphere}) \cdot 7~(\textup{params.})= \underline{22}$ additional learnable parameters in comparison with the baseline VN-DGCNN.
\end{document} | arXiv | {
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\begin{document}
\begin{center} {\bf CHARACTERIZATION OF LIL BEHAVIOR IN BANACH SPACE}
\vskip 0.3cm
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} UWE EINMAHL$^{a,}$\footnote[1]{Research supported by an FWO Grant} and DELI LI$^{b,}$\footnote[2]{Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada. }
\vskip 0.3cm
$^a$ {\it Departement Wiskunde, Vrije Universiteit Brussel,} \\ {\it Pleinlaan 2, B-1050 Brussel, Belgium;} \\ $^b$ {\it Department of Mathematical Sciences, Lakehead University,}\\ {\it Thunder Bay, Ontario, Canada P7B 5E1} \end{center}
\begin{abstract} \noindent In a recent paper by the authors a general result characterizing two-sided LIL behavior for real valued random variables has been established. In this paper, we look at the corresponding problem in the Banach space setting. We show that there are analogous results in this more general setting. In particularly, we provide a necessary and sufficient condition for LIL behavior with respect to sequences of the form $\sqrt{nh(n)}$, where $h$ is from a suitable subclass of the positive, nondecreasing slowly varying functions. To prove these results we have to use a different method. One of our main tools is an improved Fuk-Nagaev type inequality in Banach space which should be of independent interest. \end{abstract}
\noindent {\it Short title:} LIL behavior in Banach Spaces
\noindent {\it AMS 2000 Subject Classifications:} 60B12, 60F15, 60G50, 60J15.
\noindent {\it Keywords:} Law of the iterated logarithm, LIL behavior, Banach spaces, regularly varying function, sums of i.i.d. random variables, exponential inequalities.
\section{Introduction}
Let $(B, \| \cdot \| )$ be a real separable Banach space with topological dual $B^{*}$. Let $ \{X,~X_{n};~n \geq 1 \}$ be a sequence of independent and identically distributed (i.i.d.) B-valued random variables. As usual, let $S_{n} = \sum_{i=1}^{n} X_{i},~ n \geq 1$ and set $Lt=\log(t\vee e), LLt = L(Lt), t \ge 0.$
One of the classical results of probability is the Hartman-Wintner LIL and the definitive version of this result in Banach space has been proven by Ledoux and Talagrand (1988).
\noindent{\bf Theorem A} {\it A random variable $X: \Omega \to B$ satisfies the bounded LIL, that is
\begin{equation} \limsup_{n \to \infty} \|S_n\|/\sqrt{nLLn} < \infty \mbox{ a.s.}\label{LT0}\end{equation} if and only if the following three conditions are fulfilled:
\begin{equation} \Bbb{E}\|X\|^2/LL\|X\| < \infty, \;\;\Bbb{E} X =0, \label{LT1}\end{equation} \begin{equation} \Bbb{E} f^2(X) < \infty, f \in B^*, \label{LT2}\end{equation} \begin{equation} \{S_n/\sqrt{nLLn}\} \mbox{ is bounded in probability}.\label{LT3}\end{equation}} Furthermore it is known that if one assumes instead of (\ref{LT3}), \begin{equation} S_n/\sqrt{nLLn} \stackrel{\Bbb{P}}{\to} 0 \label{LT4}\end{equation} one has
\begin{equation} \limsup_{n \to \infty} \|S_n\|/\sqrt{2nLLn} = \sigma \mbox{ a.s.} \label{LT5}\end{equation} where $\sigma^2 = \sup_{f \in B_1^*} \Bbb{E} f^2(X)$ and $B_1^*$ is the unit ball of $B^*.$ It is easy to see that $\sigma^2$ is finite under assumption (\ref{LT2}).\\ If $B$ is a type 2 space then (\ref{LT1}) implies (\ref{LT4}) and the bounded LIL holds if and only if conditions (\ref{LT1}) and (\ref{LT2}) are satisfied. Moreover, in this case we also know the exact value of the limsup in (\ref{LT0}). \\
Recall that we call a Banach space type 2 space if we have for any sequence $\{Y_n\}$ of independent mean zero random variables with $\Bbb{E}\|Y_n\|^2 < \infty, n \ge 1:$
$$\Bbb{E} \|\sum_{i=1}^n Y_i\|^2 \le C \sum_{i=1}^n \Bbb{E} \|Y_i\|^2, n \ge 2.$$ where $C > 0$ is a constant. It is well known that finite-dimensional spaces and Hilbert spaces are type 2 spaces.
Finding the precise value of $\limsup_{n \to \infty}\|S_n\|/\sqrt{2nLLn}$ in general seems to be a difficult problem (see, for instance, Problem 5 on page 457 of Ledoux and Talagrand (1991)). If one imposes the stronger assumption $\Bbb{E} \|X\|^2 < \infty$ and $\Bbb{E} X=0$ instead of (\ref{LT1}) and (\ref{LT2}), de Acosta, Kuelbs and Ledoux (1983) proved that with probability one,
\begin{equation} \sigma \vee \beta_0 \le \limsup_{n \to \infty}\|S_n\|/\sqrt{2nLLn} \le \sigma + \beta_0, \label{alt} \end{equation} where $\beta_0 = \limsup_{n \to \infty} \Bbb{E}\|S_n\|/\sqrt{2nLLn}.$ Moreover, they showed that the lower bound $\sigma \vee \beta_0$ is sharp for random variables in $c_0.$ It is still open whether this is the case in other Banach spaces as well. If $S_n/\sqrt{nLLn} \stackrel{\Bbb{P}}{\to} 0$, one has $\beta_0 = 0$ and one can re-obtain result (\ref{LT5}) if $\Bbb{E} \|X\|^2 < \infty$. Also note that in all other cases one misses the ``true'' value of the $\limsup$ at most by a factor 2. So if $\Bbb{E} \|X\|^2 < \infty,$ we have a fairly complete picture and it is natural to ask whether it is possible to establish (\ref{alt}) under conditions (\ref{LT1}) and (\ref{LT2}). This has been shown by de Acosta, Kuelbs and Ledoux (1983) for certain Banach spaces which satisfy a so-called upper Gaussian comparison principle, but the question of whether this is the case for general Banach spaces seems to be still open. As a by-product of our present work we will be able to answer this in the affirmative.\\
There are also extensions of the Hartman-Wintner LIL to real-valued random variables with possibly infinite variance. Feller (1968) obtained an LIL for certain variables in the domain of attraction to the normal distribution and this was further generalized by Klass (1976, 1977). Kuelbs (1985) and Einmahl (1993) found versions of these results in the Banach space setting. In a recent paper Einmahl and Li (2005) looked at the the following problem for real-valued random variables:
\noindent \emph {Given a sequence, $a_n=\sqrt{nh(n)},$ where $h$ is a slowly varying non-decreasing function, when does one have with probability one,}
$0 < \limsup_{n \to \infty}|S_n|/a_n < \infty ? $
Somewhat unexpectedly it turned out that the classical Hartman-Wintner LIL could be generalized to a ``law of the very slowly varying function''. It is the main purpose of the present paper to investigate whether there are also such results in the Banach space setting. In the process we will derive a very general result on almost sure convergence (see Theorem 5, Sect. 3) which specialized to the classical normalizing sequence $\sqrt{2nLLn}$ also gives result (\ref{alt}) under the weakest possible conditions. \section{Statement of main results} Let ${\cal H}$ be the set of all continuous, non-decreasing functions $h: [0,\infty) \to (0,\infty)$, which are slowly varying at infinity. To simplify notation we set $\Psi(x)= \sqrt{xh(x)}, x \ge 0$ for $h \in \mathcal{H}$ and let $a_n = \Psi(n), n \ge 1.$\\ Given a random variable $X: \Omega \to B$ we consider an infinite-dimensional truncated second moment function $H: [0,\infty) \to [0,\infty)$ defined by
$$ H(t) :=\sup_{f \in B_1^*} \Bbb{E} f^2(X)I\{\|X\| \le t\}, t \ge 0.$$
The first theorem gives a characterization for having $\limsup_{n \to \infty}\|S_n\|/a_n < \infty$ a.s. where $a_n$ is a normalizing sequence of the above form.\vskip 0.3cm
\begin{theo} Let $ X $ be a B-valued random variable. Then we have \begin{equation}\label{2.1} \limsup_{n \rightarrow \infty}
\frac{\|S_{n}\|}{a_{n}} < \infty ~~\mbox{a.s.} \end{equation} if and only if \begin{equation}\label{2.2}
\Bbb{E} X = 0, ~~~\Bbb{E} \Psi^{-1}(\|X\|) < \infty, \end{equation} \begin{equation}\label{2.3} \mbox{the sequence} ~~ \{S_{n}/a_{n};~n \geq 1 \} ~\mbox{is bounded in probability,} \end{equation} and there exists $ c \in [0, \infty)$ such that \begin{equation}\label{2.4} \sum_{n=1}^{\infty} \frac{1}{n} \exp \left \{ - \frac{c^{2} h(n)}{2H(a_{n})} \right \} < \infty. \end{equation} \end{theo}
\noindent By strengthening condition (\ref{2.3}) we can find the exact limsup value in (\ref{2.1}). \begin{theo} Assume (\ref{2.2}) holds and (\ref{2.3}) is strengthened to \begin{equation}\label{2.5} S_{n}/a_{n} \stackrel{\Bbb{P}}{\to} 0, \end{equation} then \begin{equation}\label{2.6} \limsup_{n \rightarrow \infty}
\frac{\|S_{n}\|}{a_{n}} = C_{0} ~~\mbox{a.s.}, \end{equation} where \begin{equation}\label{2.7} C_{0} = \inf \left \{c \geq 0:~\sum_{n=1}^{\infty} \frac{1}{n} \exp \left \{ - \frac{c^{2} h(n)}{2H(a_{n})} \right \} < \infty \right \}. \end{equation} \end{theo} As in the classical case (when considering the sequence $a_n = \sqrt{2nLLn}$ ) one can show that in type 2 spaces (\ref{2.2}) implies (\ref{2.5}) so that in this case (\ref{2.1}) holds if and only if conditions (\ref{2.2}) and (\ref{2.4}) are satisfied. Moreover, the value of the $\limsup$ in (\ref{2.1}) is then always equal to $C_0$. \\
In general, it can be difficult to determine this parameter. For this reason we now look at normalizing sequences $a_n=\sqrt{nh(n)}$ for functions $h$ from certain subclasses of $\mathcal{H}.$ Given $0 \le q < 1$, let ${\cal H}_q \subset {\cal H}$ the class which contains all functions $h \in \mathcal{H}$ satisfying the condition, \[ \lim_{t \to \infty} \frac{h(tf_{\tau}(t))}{h(t)} =1, ~~0 < \tau < 1 - q, \] where $f_{\tau} (t) = \exp ((Lt)^{\tau}), ~0 \le \tau\le 1$. Finally let ${\cal H}_1 = {\cal H}$. Clearly ${\cal H}_{q_{1}} \subset {\cal H}_{q_{2}}$ whenever $0 \leq q_{1} < q_{2} \leq 1$. We call the functions in the smallest subclass ${\cal H}_0$ ``very slowly varying''. From the following theorem it follows that under assumption (\ref{2.5}) we have $C_0 \le \lambda$ for {\it any} $h \in \mathcal{H}$ where $\lambda$ is a parameter which can be easily determined via the $H$-function. If we have $h \in \mathcal{H}_q$, then it also follows that $C_0 \ge (1-q)^{1/2}\lambda.$ Thus, if $h \in \mathcal{H}_0,$ we have $C_0=\lambda$ and this way we can extend the classical LIL to a ``law of the very slowly varying function''. Possible choices for very slowly varying functions are for instance $(LLt)^p,\, p \ge 1$ and $(Lt)^r,\, r >0.$ \begin{theo} Let $X$ be a B-valued random variable. Suppose now that $h \in {\cal H}_q$ where $0 \le q \le 1.$ Assume (\ref{2.2}) and (\ref{2.5}) hold. Then \begin{equation}\label{2.8} (1 - q)^{1/2} \lambda \leq \limsup_{n \rightarrow \infty}
\frac{\|S_{n}\|}{a_{n}} \leq \lambda ~~\mbox{a.s.,} \end{equation} where \begin{equation}\label{2.9} \lambda^{2} = \limsup_{x \rightarrow \infty} \frac{2\Psi^{-1}(x LL x)}{x^{2} LLx}H(x). \end{equation} \end{theo}
Note the $\limsup$ in condition ({\ref{2.9}). If this $\limsup$ is actually a limit, then it easily follows from Theorem 2 that $\limsup_{n\to \infty}\|S_n\|/a_n = \lambda$ a.s. for any function $h \in \mathcal{H}$. This condition, however, is not necessary. It is a special feature of the function class $\mathcal{H}_0$ that under condition (\ref{2.5}) the $\limsup$ in (\ref{2.9}) being equal to $\lambda^2$ is necessary and also sufficient in combination with (\ref{2.2}) for having $\limsup_{n\to \infty}\|S_n\|/a_n = \lambda$ a.s . Moreover, we have for $h \in \mathcal{H}_q$ and $0 \le q <1$ that $\limsup_{n\to \infty}\|S_n\|/a_n < \infty$ a.s. if and only if $\lambda < \infty$ and conditions (\ref{2.2}) and (\ref{2.3}) hold.\\
Theorem 3 gives us analogous corollaries as in the real-valued case. We state two of these. The formulation of the other ones, for instance, a law of the logarithm (see, Corollary 2, Einmahl and Li (2005)), should be then obvious. \begin{cor} Let $X$ be a B-valued random variable. Let $p \geq 1$. Then we have \begin{equation}\label{2.10} \limsup_{n \rightarrow \infty}
\frac{\|S_{n}\|}{\sqrt{2n (LLn)^{p}}} < \infty ~~\mbox{a.s.} \end{equation} if and only if \begin{equation}\label{2.11}
\Bbb{E} X = 0, ~~~\Bbb{E} \|X\|^{2}/(LL\|X\|)^{p} < \infty, \end{equation} \begin{equation}\label{2.12} \lambda^{2} = \limsup_{x \rightarrow \infty} (LLx)^{1-p} H(x) < \infty, \end{equation} and \begin{equation}\label{2.13} \mbox{the sequence} ~~ \{S_{n}/\sqrt{2n (LLn)^{p}};~n \geq 1 \} ~\mbox{is bounded in probability.} \end{equation} Furthermore, \begin{equation}\label{2.14} \limsup_{n \rightarrow \infty}
\frac{\|S_{n}\|}{\sqrt{2n (LLn)^{p}}} = \lambda~~\mbox{a.s.} \end{equation} whenever condition (\ref{2.14}) is strengthened to \begin{equation}\label{2.15} S_{n}/\sqrt{2n (LLn)^{p}} \stackrel{\Bbb{P}}{\to} 0. \end{equation} \end{cor} If $p=1$ we re-obtain Theorem A, but the above corollary actually shows that we have for any $p \ge 1$ an LIL. If $\lambda = 0$ in Theorem 3, we obtain the following useful stability result. \begin{cor} Assume that $X: \Omega \to B$ is a random variable satisfying
\begin{equation} \Bbb{E} X = 0, ~~~\Bbb{E} \Psi^{-1}(\|X\|) < \infty, \label{s1}\end{equation} \begin{equation} \lim_{x \rightarrow \infty} \frac{\Psi^{-1}(x LL x)}{x^{2} LLx} H(x) = 0, \label{s2}\end{equation} \begin{equation} S_n/a_n \stackrel{\Bbb{P}}{\to} 0 \label{s3}\end{equation} then we have \begin{equation}\label{s4} \lim_{n \rightarrow \infty} \frac{S_{n}}{a_{n}} = 0 ~~\mbox{a.s.} \end{equation} Conversely, if $q < 1$ then (\ref{s4}) implies (\ref{s1}) - (\ref{s3}). \end{cor}
The remaining part of the paper is organized as follows. In Sect. 3 we state and prove an infinite-dimensional version of the Fuk-Nagaev inequality improving an earlier version of this inequality given as Theorem 5 in Einmahl (1993). Using a recent result of Klein and Rio (2005) who obtained in some sense an optimal version of the classical Bernstein inequality in infinite-dimensional spaces, we can replace the constant 144 in the exponential term of the earlier version by $2+\delta$ for any $\delta >0$. Employing this improved version of the Fuk-Nagaev inequality one can give much more direct proofs for LIL results than in Einmahl (1993). Especially it is no longer necessary to use randomization arguments and Sudakov minoration for obtaining the precise value of $\limsup_{n \to \infty}\|S_n\|/a_n.$ Readers who are mainly interested in inequalites can read this part independently of the other parts of the present paper. In Sect. 4 we then use the improved Fuk-Nagaev inequality to establish the upper bound part of a general result on almost sure convergence for normalized sums $S_n/c_n$ where $\{c_n; n \ge 1\}$ is a sufficiently regular normalizing sequence. This includes all sequences $a_n =\sqrt{nh(n)}$, where $h \in \mathcal{H}.$ For proving the lower bound part we first use an extension of a method employed in the proof of Theorem 2, Einmahl (1993) to get a first lower bound (see Section 4.2). In the classical case $c_n=\sqrt{2nLLn}$ this bound would be equal to $\sigma$. Our method is fairly elementary and one only needs classical results such as a non-uniform bound on the convergence speed for the CLT on the real line. In Sect. 4.3 we obtain a second lower bound which, in the classical case, matches $\beta_0$ defined in (\ref{alt}). Here we use a modification of an argument based on Fatou's lemma which is due to de Acosta, Kuelbs and Ledoux (1983). In Sect. 5 we finally infer the results stated in Sect. 2 from our general almost sure convergence result (Theorem 5).
\section{A Fuk-Nagaev type inequality} As mentioned in Sect. 2 we use an infinite-dimensional version of the Bernstein inequality which essentially goes back to Talagrand (1994). This inequality turned out to be extremely useful in many applications, but there was a shortcoming that there were no explicit numerical constants. Ledoux (1996) found a different and very elegant method for proving such inequalities which is based on a log-Sobolev type argument in combination with a tensorization of the entropy. He was also able to provide concrete numerical constants for these inequalites. His method was subsequently refined by Massart (2000) and Rio (2002) among other authors. Bousquet (2002) obtained optimal constants in the iid case. Finally, Klein and Rio (2005) generalized this result to independent, not necessarily identically distributed random variables. Their results are formulated for empirical processes, but using a standard argument one can easily obtain inequalities for sums of independent $B$-valued variables from the ones for empirical processes.
\noindent We need the following fact which follows from Lemma 3.4 of Klein and Rio (2005). \\ {\bf Fact A } {\it Let $Y_1,\ldots, Y_n$ be independent $B$-valued random variables with mean zero such that
$$\|Y_i\| \le M \mbox{ a.s.},1 \le i \le n. $$
Then we have for $0 < s < 2/(3M)$:
\begin{equation}
\Bbb{E} \exp(s\|\sum_{i=1}^n Y_i\|) \le \exp\left(s\Bbb{E} \|\sum_{i=1}^n Y_i\| + \beta_n s^2/(2 - 3Ms)\right) \label{KR}
\end{equation} where $\beta_n = 2M\Bbb{E}\|\sum_{i=1}^n Y_i\| + \Lambda_n$ with $\Lambda_n = \sup\{\sum_{j=1}^n \Bbb{E} f^2(Y_j) : f \in B_1^*\}$ and $B_1^*$ is equal to the unit ball of} $B^*$.
To prove this inequality we set $Z_i =Y_i/M, 1\le i \le n$. Recall that $B$ is separable so that we have for any $z \in B$, $\|z\| = \sup_{f \in D} f(z),$ where $D$ is a countable subset of $B_1^*.$ Set in Theorem 1.1 of Klein and Rio (2005) $\mathcal{X}=B$ and consider the following countable class of functions from $\mathcal{X}$ to $[-1,1]^n$: $\mathcal{S} = \{(-1\vee(f \wedge 1),\ldots, -1\vee(f \wedge 1)): f \in D\}.$
Then we readily obtain that $\sup_{s \in \mathcal{S}} \{s^1(Z_1) + \ldots +s^n(Z_n)\} =\|Z_1 + \ldots +Z_n\|$ a.s. and we can infer from the afore-mentioned lemma that for $0 < t <2/3,$
$\Bbb{E} \exp(t\|\sum_{i=1}^n Z_i\|) \le \exp\left(t\Bbb{E} \|\sum_{i=1}^n Z_i\| + \gamma_n t^2/(2 - 3t)\right),$
where $\gamma_n = 2\Bbb{E} \|\sum_{i=1}Z_i\| + V_n$ and $V_n = \sup\{\sum_{j=1}^n \Bbb{E} f^2(Z_j) : f \in B_1^*\}.$ Replacing $Z_i$ by $Y_i/M$ and setting $s=t/M$ we obtain (\ref{KR}).
\noindent Using the well known fact that $\exp(s\|\sum_{i=1}^k Y_i\|), 1 \le k \le n$ is a submartingale if $s >0$ (recall that we assume $\Bbb{E} Y_k = 0, 1 \le k \le n$), we can infer from Doob's maximal inequality for submartingales that for any $x >0,$
$$\Bbb{P}\left\{\max_{1 \le k \le n}\|\sum_{i=1}^k Y_i\| \ge \Bbb{E} \|\sum_{i=1}^n Y_i\| + x\right\} \le \exp( \beta_n s^2/(2 - 3Ms) - sx), 0 < s < 2/(3M).$$ Choosing $s = 2x/(2\beta_n + 3Mx)$ we finally obtain that
\begin{equation} \Bbb{P}\left\{\max_{1 \le k \le n}\|\sum_{i=1}^k Y_i\| \ge \Bbb{E} \|\sum_{i=1}^n Y_i\| + x\right\} \le
\exp\left(-\frac{x^2}{2\Lambda_n + (4\Bbb{E}\|\sum_{i=1}^n Y_i\| + 3x)M}\right).\label{KR1}\end{equation} Next note that we trivially have for any $\epsilon >0,$ \begin{eqnarray} \label{triv}
&&\exp\left(-\frac{x^2}{2\Lambda_n + (4\Bbb{E}\|\sum_{i=1}^n Y_i \| +3x)M}\right)\\ &\le& \exp\left(-\frac{x^2}{(2+\epsilon)\Lambda_n} \right) +
\exp\left(-\frac{x^2}{(1+2/\epsilon) (4\Bbb{E}\|\sum_{i=1}^n Y_i \| +3x)M}\right).\nonumber \end{eqnarray}
Combining (\ref{KR1}) and (\ref{triv}) and setting $x= \eta \Bbb{E}\|\sum_{i=1}^n Y_i \| + y,$ where $0 <\eta \le 1$ and $y>0,$ we can conclude that for any $y >0,$ \begin{equation}
\Bbb{P}\left\{\max_{1 \le k \le n}\|\sum_{i=1}^k Y_i \| \ge (1 + \eta)\Bbb{E}\|\sum_{i=1}^n Y_i \| + y\right\} \le \exp\left(-\frac{y^2}{(2+\epsilon)\Lambda_n} \right) + \exp\left(-\frac{y}{D_{\epsilon,\eta}\,M}\right), \label{KRmod} \end{equation} where $D_{\epsilon,\eta} = (1+2/\epsilon)(3 + 4/\eta).$ We are now ready to prove \begin{theo} \label{FNa}
Let $Z_1,\ldots, Z_n$ be independent B-valued random variables with mean zero such that for some $s >2,$ $\Bbb{E}\|Z_i\|^s < \infty, 1 \le i \le n.$ Then we have for $0 < \eta \le 1, \delta >0$ and any $t >0,$ \begin{equation}
\Bbb{P}\left\{\max_{1 \le k \le n}\|\sum_{i=1}^k Z_i \| \ge (1+\eta)\Bbb{E}\|\sum_{i=1}^n Z_i \| + t\right\}
\le \exp\left(-\frac{t^2}{(2+\delta)\Lambda_n} \right) + C\sum_{i=1}^n \Bbb{E}\|Z_i\|^s/t^s, \end{equation} where $\Lambda_n = \sup\{\sum_{j=1}^n \Bbb{E} f^2(Z_j) : f \in B_1^*\}$ and $C$ is a positive constant depending on $\eta,\delta$ and $s$. \end{theo} {\bf Proof.} To simplify notation we set for $y >0$
$$\beta(y)=\beta_s (y) = \sum_{i=1}^n \Bbb{E} \|Z_i\|^s/y^s.$$ Assume that $\beta(y) < 1.$ For $\epsilon > 0$ fixed we consider the following truncated variables
$$Y_i := Z_i I\{ \|Z_i\| \le \rho\epsilon y\}, \,Y'_i = Y_i - \Bbb{E} Y_i, 1 \le i \le n,$$
where $$\rho= \rho(\epsilon,\eta, y)= 1 \wedge \frac{1}{2\epsilon D_{\epsilon,\eta}\,\log(1/\beta(y))} .$$ Applying inequality (\ref{KRmod}) with $M=2\rho \epsilon y$ we find that \begin{equation}
\Bbb{P}\left\{\max_{1 \le k \le n}\|\sum_{i=1}^k Y'_i \| \ge (1+\eta)\Bbb{E}\|\sum_{i=1}^n Y'_i \| + y\right\} \le \exp\left(-\frac{y^2}{(2+\epsilon)\Lambda_n} \right) + \beta(y). \label{in1} \end{equation} Next consider the variables
$$\Delta_i := Z_i I\{\rho\epsilon y < \|Z_i\| \le \epsilon y\}, 1 \le i \le n.$$ Employing the Hoffmann-J\o rgensen inequality (see, for instance, inequality (6.6) in Ledoux and Talagrand (1991)), we can conclude that \begin{equation}
\Bbb{P}\left\{ \max_{1 \le k \le n}\|\sum_{i=1}^k \Delta_i\| \ge 4\epsilon y \right\}
\le \left(\Bbb{P}\left\{\max_{1 \le k \le n} \|\sum_{i=1}^k \Delta_i\| \ge \epsilon y \right\}\right)^2 \label{in2} \end{equation} which in turn is $$\le \left(\sum_{i=1}^n \Bbb{P}\{\Delta_i \ne 0\} \right)^2
\le \left(\sum_{i=1}^n \Bbb{P}\{\|Z_i\| \ge \rho \epsilon y\} \right)^2.$$ Using Markov's inequality and recalling the definition of $\rho$ we see that this last term is bounded above by $$ (2D_{\epsilon,\eta})^{2s}\beta^2(y)(\log(1/\beta(y)))^{2s} \le K_s (2D_{\epsilon,\eta})^{2s}\beta(y), $$ where $K_s >0$ is a constant so that $(\log a)^{2s}\le K_s a , a \ge 1.$ We can conclude that \begin{equation}
\Bbb{P}\left\{\max_{1 \le k \le n} \|\sum_{i=1}^k \Delta_i\| \ge 4\epsilon y \right\} \le C' \beta(y), \label{in3} \end{equation} where $C'=K_s(2D_{\epsilon,\eta})^{2s}.$ \\
Next set $\Delta'_i := Z_i I\{\|Z_i\| > \epsilon y\}, 1 \le i \le n.$ Then we have once more by Markov's inequality \begin{equation}
\Bbb{P}\left\{\max_{1 \le k \le n}\|\sum_{i=1}^k \Delta'_i\| \ne 0\right\}\le \epsilon^{-s}\beta(y). \label{in4} \end{equation} Combining inequalities (\ref{in1}), (\ref{in3}) and (\ref{in4}), we see that if $\beta(y) < 1$ we have $$
\Bbb{P}\left\{\max_{1 \le k \le n} \|\sum_{i=1}^k (Z_i - \Bbb{E} Y_i) \| \ge (1+\eta)\Bbb{E}\|\sum_{i=1}^n Y'_i \| + (1+4\epsilon)y\right\} \le \exp\left(-\frac{y^2}{(2+\epsilon)\Lambda_n} \right) + C''\beta(y), $$ where $C''=1+ C' + \epsilon^{-s}$. A simple application of the triangular inequality gives \begin{equation}
\Bbb{P}\left\{\max_{1 \le k \le n} \|\sum_{i=1}^k Z_i \| \ge b'_n+ (1+4\epsilon)y\right\} \le \exp\left(-\frac{y^2}{(2+\epsilon)\Lambda_n} \right) + C''\beta(y), \label{in5} \end{equation} where
\begin{eqnarray*}
b'_n &=& (1+\eta)\Bbb{E}\|\sum_{i=1}^n Y'_i\| + \max_{1 \le k \le n} \|\sum_{i=1}^k \Bbb{E} Y_i\|\\ & \le&
(1+\eta)\Bbb{E}\|\sum_{i=1}^n Y_i\| + 3\max_{1 \le k \le n} \|\sum_{i=1}^k \Bbb{E} Y_i\|. \end{eqnarray*} Further note that
\begin{eqnarray*}
\Bbb{E}\|\sum_{i=1}^n Y_i \| &\le & \Bbb{E} \|\sum_{i=1}^n Z_i \| + \Bbb{E}\|\sum_{i=1}^n Z_i I\{\|Z_i\| \ge \rho\epsilon y\} \|\\
&\le&\Bbb{E} \|\sum_{i=1}^n Z_i \| + \sum_{i=1}^n \Bbb{E}\|Z_i\|I\{\|Z_i\| \ge \rho \epsilon y\}
=: \Bbb{E} \|\sum_{i=1}^n Z_i \| + \delta_n. \end{eqnarray*}
As we have $\Bbb{E} Z_i =0, 1 \le i \le n$ it also follows that $\max_{1 \le k \le n} \|\sum_{i=1}^k \Bbb{E} Y_i\| \le \delta_n$ and consequently,
\begin{equation} b'_n \le (1+\eta)\Bbb{E}\|\sum_{i=1}^n Z_i\| + 5\delta_n. \label{cent}\end{equation} Furthermore, we have, $$ \delta_n \le y\beta(y)/\{\rho \epsilon\}^{s-1} \le \epsilon y$$ provided that $\beta(y) \le \epsilon^s \rho^{s-1}.$\\ It is easily checked that if $\rho <1,$ we have $ \beta(y)/(\epsilon^s \rho^{s-1}) \le \beta(y)/(\epsilon^s \rho^{s}) \le (C''\beta(y))^{1/2}.$ (We are assuming that $\beta(y) \le 1.$) Consequently, $\delta_n \le \epsilon y$ whenever $C''\beta(y) \le 1$ and $\rho <1$. This is also true if $\rho =1$ as we have $C'' \ge \epsilon^{-s}.$ We thus can conclude if $\beta(y) \le 1/C'' < 1:$ \begin{equation}
\Bbb{P}\left\{\max_{1 \le k \le n}\|\sum_{i=1}^k Z_i \| \ge (1+\eta)\Bbb{E}\|\sum_{i=1}^n Z_i \| + (1+9\epsilon)y\right\} \le \exp\left(-\frac{y^2}{(2+\epsilon)\Lambda_n} \right) + C''\beta(y). \label{in6} \end{equation} The above inequality is of course trivial if $\beta(y) > 1/C''$ and consequently (\ref{in6}) holds for all $y >0.$ Setting $y= t/(1+9\epsilon)$ and choosing $\epsilon$ in (\ref{in6}) so small that $(2+\epsilon)(1+9\epsilon)^2 \le 2+\delta$, we obtain the assertion. $\Box$ \section{A general result on almost sure convergence}
Let $c_n$ be a sequence of real numbers satisfying the following two conditions, \begin{equation} c_n/\sqrt{n} \nearrow \infty \label{RE} \end{equation} and \begin{equation}
\forall\; \epsilon >0\,\exists\, m_\epsilon \ge 1: c_n/c_m \le (1+\epsilon)(n/m), m_\epsilon \le m < n. \label{REG} \end{equation} Note that condition (\ref{REG}) is satisfied for any sequence $a_n$ considered in Section 2. \\ Let $H$ be defined as in Section 1, that is
$$H(t) = \sup_{f \in B_1^*}\Bbb{E} f^2(X) I\{\|X\| \le t\}, t \ge 0.$$ Set $$\alpha_0 = \sup\left\{\alpha \ge 0: \sum_{n=1}^{\infty} n^{-1}\exp\left(-\frac{\alpha^2 c^2_n}{2nH(c_n)}\right) = \infty\right\} .$$ In general $\alpha_0$ can be any number in $[0,\infty].$ If we are assuming that $\Bbb{E} f^2(X) < \infty, f \in B^*$ and we choose $c_n = \sqrt{2nLLn}$, it follows that $\alpha_0^2 = \sigma^2 = \sup_{f \in B^*_1}\Bbb{E} f^2(X).$\\
Our main result in this section is the following generalization of (\ref{alt}), \begin{theo}\label{th1} Let $X, X_1, X_2, \ldots$ be i.i.d. mean zero random variables taking values in a separable Banach space $B$. Assume that \begin{equation}
\sum_{n=1}^{\infty} \Bbb{P}\{\|X\| \ge c_n\} < \infty, \label{MOM} \end{equation} where $c_n$ is a sequence of positive real numbers satisfying conditions (\ref{RE}) and (\ref{REG}).\\ Then we have with probability one, \begin{equation}
\alpha_0 \vee \beta_0 \le \limsup_{n \to \infty}\|S_n\|/c_n \le \alpha_0 + \beta_0,\label{bounds}
\end{equation} where $\beta_0 = \limsup_{n \to \infty} \Bbb{E} \|S_n\|/c_n.$ \end{theo} The following lemma which is more or less known shows that $\beta_0$ is finite whenever $\{S_n/c_n; n \ge 1\}$ is bounded in probability and that $\beta_0 =0$ if $S_n/c_n \stackrel{\Bbb{P}}{\to} 0.$ So in the latter case we see that the $\limsup$ in (\ref{bounds}) is equal to $\alpha_0.$ \begin{lem} Let $X, X_1, X_2, \ldots$ be iid B-valued random variables with mean zero and let $S_n = \sum_{i=1}^n X_i, n \ge 1$. Let $\{c_n\}$ be a sequence of positive real numbers satisfying conditions (\ref{RE}) and (\ref{REG}). Under assumption (\ref{MOM}) we have the following equivalences: \begin{itemize}
\item[(a)] $\{S_n/c_n; n \ge 1\}$ is bounded in probability $\Longleftrightarrow \limsup_{n \to \infty} \Bbb{E}\|S_n\|/c_n < \infty.$
\item[(b)] $S_n/c_n \stackrel{\Bbb{P}}{\to} 0 \Longleftrightarrow \Bbb{E}\|S_n\|/c_n \to 0.$ \end{itemize} \end{lem} {\bf Proof} We only need to prove the implications ``$\Rightarrow$'' and by a standard symmetrization argument it is enough to do that for symmetric random variables. We have for any $\epsilon >0,$ \begin{equation}
\Bbb{E} \|S_n\| \le \Bbb{E}\left\|\sum_{i=1}^n X_i I\{\|X_i\| \le \epsilon c_n\}\right\| +
n \Bbb{E} \|X\|I\{\|X\| > \epsilon c_n\}. \label{last} \end{equation} The last term is of order $o(c_n)$ under assumption (\ref{MOM}) (see Lemma 1, Einmahl and Li (2005)). Using the trivial inequality
$$\Bbb{P}\left \{\|\sum_{i=1}^n X_i I\{\|X_i\| \le \epsilon c_n\}\| \ge x\right\}
\le \Bbb{P}\{\|S_n\| \ge x\} + n \Bbb{P}\{\|X\| \ge \epsilon c_n\},$$
in conjunction with Proposition 6.8 in Ledoux and Talagrand (1991), we find that if $\{S_n/c_n\}$ is bounded in probability, the first term in (\ref{last}) is $\le C(\epsilon) < \infty$ . Consequently, we have in this case, $\Bbb{E}\|S_n\|/c_n < \infty.$ Assuming $S_n/c_n \stackrel{\Bbb{P}}{\to} 0,$ one can choose $C(\epsilon)$ so that $C(\epsilon) \to 0$ as $\epsilon \to 0$. Since we can make $\epsilon$ arbitrarily small, it follows that $\Bbb{E}\|S_n\|/c_n \to 0$ if $S_n/c_n \stackrel{\Bbb{P}}{\to} 0. \;\Box$\\
If $B$ is a type 2 Banach space, assumption (\ref{MOM}) implies that $\Bbb{E} \|S_n\| = o(c_n).$ (See Lemma 6, Einmahl (1993). The proof given there works also under the present conditions on $\{c_n\}.$) Therefore we have in any type 2 space, $\beta_0=0$ and the $\limsup$ in (\ref{bounds}) is equal to $\alpha_0$. Recalling that finite dimensional spaces are type 2 spaces, we see that this result extends Theorem 3 of Einmahl and Li (2005). Also note that the conditions on $\{c_n\}$ are general enough so that one can infer Theorem 3, Einmahl (1993) from the present Theorem 6 as well (without using randomization and Sudakov minoration).
We now turn to the proof of Theorem \ref{th1}. We assume throughout that condition (\ref{MOM}) is satisfied. Using essentially the same argument as in Lemma 3 of Einmahl (2007) one can infer from the definition of $\alpha_0$ that whenever $n_j \nearrow \infty$ is a subsequence satisfying for large enough $j,$ \begin{equation} 1 < a_1 < n_{j+1}/n_j \le a_2 < \infty,\label{sub}\end{equation} we have: \begin{equation} \sum_{j=1}^{\infty} \exp\left(-\frac{\alpha^2 c_{n_j}^2}{2n_j H(c_{n_j})}\right) \begin{cases} =\infty & \mathrm{if}\;\alpha < \alpha_0,\\ <\infty & \mathrm{if}\;\alpha > \alpha_0.\end{cases} \label{geom} \end{equation}
\subsection{The upper bound part} W.l.o.g. we can and do assume in this part that $\alpha_0 + \beta_0 < \infty.$\\ We first note that on account of (\ref{RE}) and assumption (\ref{MOM}) we have for any subsequence $\{n_j\}$ satisfying (\ref{sub}), \begin{equation}
\sum_{j=1}^{\infty} n_j \Bbb{E} \|X\|^3I\{\|X\| \le c_{n_j}\}/c_{n_j}^3 < \infty. \label{fact1} \end{equation} (See, for instance, Lemma 7.1, Pruitt (1981).)\\ Moreover, we have as $n \to \infty,$
\begin{equation} n\Bbb{E} \|X\|I\{\|X\| \ge c_n\} \le \sum_{i=1}^n \Bbb{E}\|X\|I\{\|X\| \ge c_i\} = o(c_n) \label{fact2} \end{equation}
This last fact follows as in the proof of Lemma 10, Einmahl (1993) replacing $\gamma_n$ by $c_n$. \\
Set $X'_n = X_n I\{\|X_n\| \le c_n\}, n \ge 1$ and denote the sum of the first $n$ of these variables by $S'_n, n \ge 1.$ We obviously have $$\sum_{n=1}^{\infty}\Bbb{P}\{X_n \ne X'_n\} < \infty$$ so that with probability one, $X_n = X'_n$ eventually. Due to relation (\ref{fact2}) we have $\Bbb{E} S'_n = o(c_n)$ and consequently it is enough to show, \begin{equation}
\limsup_{n \to \infty}\|S'_n - \Bbb{E} S'_n\|/c_n \le \alpha_0 + \beta_0 \mbox{ a.s.}\label{upper} \end{equation} This follows via Borel-Cantelli once we have proven for any $0 <\delta <1$ \begin{equation}
\sum_{j=1}^{\infty}\Bbb{P}\left\{\max_{1 \le n \le n_{j+1}}\|S'_n -\Bbb{E} S'_n\| \ge \{\alpha_0 +\delta +\beta_0(1 +\delta)\}(1+ 2\delta)c_{n_j}\right\} < \infty, \label{series} \end{equation} where $n_j \sim \rho^j$ for a suitable $\rho >1.$\\ In order to apply Theorem \ref{FNa} we need an upper bound for
$b_n:=\Bbb{E}\|S'_n - \Bbb{E} S'_n\|.$ Using essentially the same argument as in the proof of Theorem \ref{FNa} we find that this quantity is less than or equal to
$$ \Bbb{E} \|S_n\| + 2\sum_{i=1}^n \Bbb{E}\|X\|I\{\|X\| \ge c_i\}.$$ On account of fact (\ref{fact2}) and condition (\ref{REG}) we have for large enough $j$, $$b_{n_{j+1}} \le (1 + \delta)\beta_0 c_{n_{j+1}}\le (1 + 2\delta)\beta_0 c_{n_{j}},$$
provided we have chosen
$\rho < (1+2\delta)/(1+\delta).$\\ From Theorem \ref{FNa} (where we set $\eta =\delta$) and the $c_r$-inequality it now follows for large $j,$ \begin{eqnarray*}
&&\Bbb{P}\left\{\max_{1 \le n \le n_{j+1}}\|S'_n - \Bbb{E} S'_n\| \ge \{\alpha_0 +\delta +\beta_0(1+\delta)\}(1+ 2\delta)c_{n_j}\right\}\\
&\le& \exp\left(-\frac{(\alpha_0+\delta)^2(1+2\delta)^2 c_{n_j}^2}{(2+\delta)n_{j+1}H(c_{n_{j+1}})}\right) + 8C(\alpha_0+\delta)^{-3} (1+2\delta)^{-3}n_{j+1}\Bbb{E}\|X\|^3I\{\|X\|\le c_{n_{j+1}}\}/c_{n_j}^3\\
&\le& \exp\left(-\frac{(\alpha_0+\delta)^2 c_{n_{j+1}}^2}{2n_{j+1}H(c_{n_{j+1}})}\right) + 8C(\alpha_0+\delta)^{-3}(1 +\delta)^{-3}n_{j+1}\Bbb{E}\|X\|^3I\{\|X\|\le c_{n_{j+1}}\}/c_{n_{j+1}}^3. \end{eqnarray*} Recalling relations (\ref{geom}) and (\ref{fact1}) it is easy now to see that (\ref{series}) holds and the proof of the upper bound is complete. \subsection{The first lower bound} We now can assume that $\alpha_0 >0,$ but it is possible that $\alpha_0 = \infty.$\\ It is obviously enough to show that we have for any $0 < \alpha < \alpha_0$ with probability one \begin{equation}
\limsup_{n \to \infty}\|S_n\|/c_n \ge \alpha. \label{lower} \end{equation} W. l. o. g. we assume that \begin{equation}
\limsup_{n \to \infty} \Bbb{P}\{\|S_n\| \ge \alpha c_n\} \le 1/2. \label{stoch}
\end{equation} Otherwise, we would have $\Bbb{P}\{\limsup_{n \to \infty}\|S_n\|/c_n \ge \alpha\} \ge \limsup_{n \to \infty} \Bbb{P}\{\|S_n\| \ge \alpha c_n\} > 1/2$ which implies (\ref{lower}) via the 0-1 law of Hewitt-Savage. \\ We first prove that under the assumptions (\ref{MOM}) and (\ref{stoch}) we have for any sequence $\{n_j\}$ satisfying condition (\ref{sub}), \begin{equation}
\sum_{j=1}^{\infty}\Bbb{P}\{\|S_{n_j}\| \ge \alpha c_{n_j}\} = \infty. \label{div} \end{equation} To that end we choose for any $j$ a functional $f_j \in B_1^*$ so that
$$ \Bbb{E} f^2_j(X)I\{\|X\| \le c_{n_j}\} \ge (1-\epsilon) H(c_{n_j}),$$ where $0 < \epsilon < 1$ will be specified later on.\\ Set for $j,k \ge 1,$ \begin{eqnarray*}
\xi_{j, k} &:=& f_j(X_k)I\{\|X_k\| \le c_{n_j}\},\\ \xi'_{j, k}&:=& \xi_{j, k} - \Bbb{E} \xi_{j, k}. \end{eqnarray*} Then it is easy to see that \begin{equation}
\Bbb{P}\{\|S_{n_j}\| \ge \alpha c_{n_j}\} \ge \Bbb{P}\{\sum_{k=1}^{n_j} \xi_{j,k} \ge
\alpha c_{n_j}\} - n_j \Bbb{P}\{\|X\| \ge c_{n_j}\}. \end{equation} From assumption (\ref{MOM}) it immediately follows that
$$\sum_{j=1}^{\infty} n_j \Bbb{P}\{\|X\| \ge c_{n_j}\} < \infty.$$
Moreover, we have $|\Bbb{E} \xi_{j, k}| \le \Bbb{E} \|X\| I\{\|X\| \ge c_{n_j}\}$ which is in view of fact (\ref{fact2}) of order $o(c_{n_j}/n_j).$ Consequently, in order to prove (\ref{div}) it is enough to show that for a suitable $0 < \epsilon < 1,$ \begin{equation} \sum_{j=1}^{\infty} \Bbb{P}\left\{\sum_{k=1}^{n_j} \xi'_{j,k} \ge (1+\epsilon)\alpha c_{n_j}\right\} = \infty. \label{f1} \end{equation} To estimate these probabilities we employ a non-uniform bound on the rate of convergence in the central limit theorem (see, e.g., Theorem 5.17 on page 168 of Petrov (1995)). We can conclude that \begin{equation} \Bbb{P}\left\{\sum_{k=1}^{n_j} \xi'_{j,k} \ge
(1+\epsilon)\alpha c_{n_j}\right\} \ge \Bbb{P}\{\sigma_j \zeta \ge (1+\epsilon)\alpha c_{n_j}/\sqrt{n_j}\} - A\alpha^{-3}(1+\epsilon)^{-3} n_j \Bbb{E}|\xi'_{j,1}|^3 c_{n_j}^{-3}, \end{equation} where $\zeta$ is a standard normal variable, $\sigma_j^2 = \mathrm{Var}(\xi_{j,1})$ and $A$ is an absolute constant.\\
Noting that $\Bbb{E}|\xi'_{j,1}|^3 \le 8 \Bbb{E}|\xi_{j,1}|^3 \le 8\Bbb{E}\|X\|^3I\{\|X\| \le c_{n_j}\}$, we can infer from fact (\ref{fact1}) that
$$\sum_{j=1}^{\infty} n_j \Bbb{E}|\xi'_{j,1}|^3 c_{n_j}^{-3} < \infty.$$ Therefore, relation (\ref{f1}) and consequently (\ref{div}) follow if we can show that \begin{equation} \sum_{j=1}^{\infty} \Bbb{P}\{\sigma_j \zeta \ge (1+\epsilon)\alpha c_{n_j}/\sqrt{n_j}\} = \infty. \label{f2} \end{equation} Let $\mathbb{N}_0 = \{j \ge 1: H(c_{n_j}) \le c^2_{n_j}/n_j^2\}.$ Then it is easily checked that for any $\eta >0,$ \begin{equation} \sum_{j \in \mathbb{N}_0}\exp\left(-\frac{\eta c_{n_j}^2}{2n_j H(c_{n_j})}\right) < \infty. \label{f3} \end{equation} Furthermore, we have for large $j \not \in \mathbb{N}_0,$ \begin{eqnarray*}
\sigma^2_j &=& \Bbb{E} f_j^2(X)I\{\|X\| \le c_{n_j}\} - (\Bbb{E} f_j(X)I\{\|X\| \le c_{n_j}\})^2\\
&=&\Bbb{E} f_j^2(X)I\{\|X\| \le c_{n_j}\} - (\Bbb{E} f_j(X)I\{\|X\| > c_{n_j}\})^2\\
&\ge&(1-\epsilon) H(c_{n_j}) - (\Bbb{E}\|X\|I\{\|X\| \ge c_{n_j}\})^2 \\ &\ge& (1-2\epsilon) H(c_{n_j}). \end{eqnarray*}
Here we have used that for large $j$, $\Bbb{E}\|X\|I\{\|X\| \ge c_{n_j}\} \le \sqrt{\epsilon} c_{n_j}/n_j$ (see fact (\ref{fact2})).\\ Employing a standard lower bound for the tail probabilites of normal random variables, we can conclude that for large $j \not \in \mathbb{N}_0,$ $$ \Bbb{P}\{\sigma_j \zeta \ge (1+\epsilon)\alpha c_{n_j}/\sqrt{n_j}\} \ge \exp\left(-\frac{(1+\epsilon)^2\alpha^2 c^2_{n_j}}{2n_j (1-3\epsilon)H(c_{n_j})}\right). $$ Choosing $\epsilon$ so small that $\alpha (1+\epsilon)/\sqrt{1-3\epsilon}) < \alpha_0$ we obtain (\ref{f2}) from relations (\ref{geom}) and (\ref{f3}). This implies relation (\ref{div}).\\ We are now ready to finish the proof by a standard argument.\\ Set $m_k = \sum_{j=1}^k n_j, n \ge 1, $ where $n_j = [(1+\delta^{-2})^j]$ with $0 < \delta < 1/2.$ \\ Note that we then have $n_{j+1}/n_j \ge \delta^{-2}$ and consequently by (\ref{RE}),
\begin{equation} c_{n_{j+1}} \ge \delta^{-1} c_{n_j}, j \ge 1.\label{f4}\end{equation} Likewise it follows that $$m_k \le n_k \left(\sum_{i=0}^{k-1} \delta^{2i} \right) \le n_k/(1-\delta^2).$$ Ik $k$ is large enough we can conclude from (\ref{REG}) that \begin{equation} c_{m_k}/c_{n_k} \le (1+\delta)m_k/n_k \le (1-\delta)^{-1}. \label{f5} \end{equation} Define for $k \ge 1,$ \begin{eqnarray*}
F_k&:=& \left\{\|S_{m_k} - S_{m_{k-1}}\| \ge \alpha c_{n_k}\right\},\\
G_k&:=& \left\{\|S_{m_{k-1}}\| \le 2\alpha \delta c_{n_k}\right\}. \end{eqnarray*} Note that on account of relations (\ref{f4}) and (\ref{f5}) we have for large $k,$
$$\Bbb{P}(G_k) \ge \Bbb{P} \left\{\|S_{m_{k-1}}\| \le 2\alpha c_{n_{k-1}}\right\}
\ge \Bbb{P} \left\{\|S_{m_{k-1}}\| \le 2(1-\delta)\alpha c_{m_{k-1}} \right\}.$$ Thus (recall (\ref{stoch})) $\Bbb{P}(G_k) \ge 1/2$ for large $k$. In view of (\ref{div}) we have $\sum_{k=1}^{\infty} \Bbb{P}(F_k) = \infty.$ The events $F_k$ and $G_k$ are independent. Thus we can conclude via Lemma 3.4. of Pruitt (1981) that $$\Bbb{P}(F_k \cap G_k \mbox{ infinitely often}) =1.$$ We clearly have,
$$ F_k \cap G_k \subset \left\{\|S_{m_k}\| \ge \alpha (1-2\delta) c_{n_k}\right\}$$ which is due to relation (\ref{f5})
$$ \subset \left\{\|S_{m_k}\| \ge \alpha (1-2\delta)(1-\delta) c_{m_k}\right\}$$ provided that $k$ is large enough.\\ It follows that with probability one,
$$\limsup_{k \to \infty} \|S_{m_k}\|/c_{m_k} \ge \alpha (1-2\delta)(1-\delta).$$ Since we can choose $\delta$ arbitrarily small, this implies statement (\ref{lower}). \subsection{The second lower bound} We now assume that $\alpha_0 + \beta_0 < \infty$. If $\alpha_0 = \infty$ the lower bounds follows from part 4.2 and if $\beta_0=\infty$ we can obtain it from Lemma 1.\\
We use essentially the same argument as in Theorem 7 of de Acosta, Kuelbs and Ledoux (1983). There is a small complication: we cannot show for all sequences $\{c_n\}$ satisfying the above conditions that $\Bbb{E} [\sup_n \|S_n\|/c_n] < \infty.$\\ Therefore, we prove this first for $S'_n = \sum_{i=1}^n X'_i, n \ge 1,$ where the random variables $X'_n$ are defined as in part 4.1, that is
$$X'_n = X_n I\{\|X_n\| \le c_n\}, n \ge 1.$$ From the upper bound part (see 4.1) it follows that we have with probability one,
\begin{equation} \limsup_{n \to \infty} \|S'_n\| /c_n\le \alpha_0 + \beta_0 < \infty. \label{fa1}\end{equation} Since $\sup_n \|X'_n\|/c_n \le 1$ we obtain from Corollary 6.12 in Ledoux and Talagrand (1991) that
\begin{equation} \Bbb{E}\left[\sup_n \|S'_n\|/c_n\right] < \infty. \label{fa2}\end{equation} Using the fact that with probability one, $\limsup_{n \to \infty}\|S_n\|/c_n$ is constant, Fatou's lemma implies that with probability one,
\begin{equation} \limsup_{n \to \infty}\|S_n\|/c_n=\limsup_{n \to \infty}\|S'_n\|/c_n\ge \limsup_{n \to \infty} \Bbb{E} \|S'_n\|/c_n. \label{fa3}\end{equation} In view of (\ref{fact2}) we have as $n \to \infty$
$$|\Bbb{E}\|S'_n\| - \Bbb{E}\|S_n\| | \le \sum_{i=1}^n\Bbb{E}\|X\|I\{\|X\| \ge c_i\}=o(c_n)$$ and we find that with probability one,
$$ \limsup_{n \to \infty} \|S_n\|/c_n \ge \beta_0.$$ This completes the proof of Theorem 5.
\section{Proof of Theorem 3} We only prove Theorem 3. The proofs of the corollaries are exactly as in the real-valued case and they are omitted. Also Theorems 1 and 2 follow directly from Theorem 5. \\ In view of Theorem 5 and Lemma 1 we only need to show that \begin{equation} \alpha_0 \le \lambda \label{le} \end{equation} and \begin{equation} \alpha_0 \ge (1-q)^{1/2}\lambda, \; h \in \mathcal{H}_q \label{ge}.\end{equation} As in the real-valued case we can infer from (\ref{2.9}) that $$\limsup_{n \to \infty} (LLn)H(a_n/LLn)/h(n)=\lambda^2/2.$$ The proof of (\ref{ge}) then goes exactly as in the real-valued case and thus it can be omitted as well. In order to prove the corresponding upper bound in the real-valued case, we used another result, namely Theorem 4 in our previous paper, Einmahl and Li (2005). It is possible to extend this result to Banach space valued random variables as well, but there is also a more direct argument for deriving the upper bound (\ref{le}) which we shall give below.
\noindent{\bf Proof of (\ref{le}).} If $\lambda = \infty$ the upper bound is trivial. Thus we can assume that $\lambda \in [0,\infty).$\\We have to show for any $\alpha > \lambda,$ \begin{equation} \sum_{n=1}^{\infty} n^{-1}\exp\left(-\frac{\alpha^2 h(n)}{2 H(a_n)}\right) < \infty. \label{u1}\end{equation} Set $\delta = (\alpha - \lambda)/3.$ Then we clearly have for large enough $n$, $$H(a_n/LLn) \le \frac{(\lambda + \delta)^2}{2} \frac{h(n)}{LLn}.$$ Setting $\mathbb{N}_0 =\{n: H(a_n) - H(a_n/LLn) \le \delta \lambda h(n)/LLn\}$ we get for large $n \in \mathbb{N}_0,$ $$ H(a_n) \le \frac{(\lambda + 2\delta)^2}{2} \frac{h(n)}{LLn}$$ and consequently \begin{equation} \sum_{n \in \mathbb{N}_0} n^{-1}\exp\left(-\frac{\alpha^2 h(n)}{2 H(a_n)}\right) < \infty.\label{u2}\end{equation} Further note that we trivially have, \begin{equation*} \sum_{n=1}^{\infty} \frac{H(a_n) -H(a_n/LLn)}{a_n^2 LLn}
\le \sum_{n=1}^{\infty} \frac{\Bbb{E} \|X\|^3I\{\|X\| \le a_n\}}{a_n^3} < \infty.
\end{equation*}
The latter series is finite because we are assuming $\Bbb{E} \Psi^{-1}(\|X\|) < \infty.$ (See, for instance, Lemma 5(a), Einmahl (1993).) It follows that \begin{equation} \sum_{n \not\in \mathbb{N}_0} \frac{1}{n(LLn)^2} < \infty. \label{u3}\end{equation} Condition (\ref{2.9}) implies that for large enough $n,$ and $0 < \epsilon <1,$ $$H(a_n) \le (\lambda^2 +1)\frac{a_n^2 LL{a_n}}{\Psi^{-1}(a_n LL{a_n})} \le C_{\epsilon} \frac{a_n^2 LL{a_n}}{\Psi^{-1}(a_n) (LL{a_n})^{2-\epsilon}} \le C'_{\epsilon} \frac{h(n)}{(LLn)^{1-\epsilon}},$$ where we have used the fact that $\Psi^{-1}$ is regularly varying at infinity with index 2. ($C_{\epsilon}$ and $C'_{\epsilon}$ are positive constants.)\\ We now can infer from (\ref{u3}) that \begin{equation} \sum_{n \not \in \mathbb{N}_0} n^{-1}\exp\left(-\frac{\alpha^2 h(n)}{2 H(a_n)}\right) \le \sum_{n \not \in \mathbb{N}_0}n^{-1}\exp( -\alpha^2(2C'_{\epsilon})^{-1}(LLn)^{1-\epsilon}) <\infty. \label{u4} \end{equation} Combining (\ref{u2}) and (\ref{u4}) we see that the series in (\ref{u1}) is finite and our proof of (\ref{le}) is complete. $\Box$\\
\begin{center} {\bf REFERENCES} \end{center} \begin{description} \item {\sc Acosta, A. de, Kuelbs, J. and Ledoux, M.} (1983) An inequality for the law of the iterated logarithm. In: Probability in Banach spaces 4. {\it Lecture Notes in Mathematics} {\bf 990} Springer, Berlin Heidelberg, 1--29.
\item {\sc Bousquet, O.} (2002) A Bennett concentration inequality and its application to suprema of empirical processes. {\em C. R. Math. Acad. Sci. Paris} {\bf 334}, no. 6, 495--500.
\item {\sc Einmahl, U.} (1993) Toward a general law of the iterated logarithm in Banach space. {\em Ann. Probab.} {\bf 21}, 2012-2045. \item {\sc Einmahl, U.} (2007) A generalization of Strassen's functional LIL. {\em J. Theoret. Probab.}, to appear. \item {\sc Einmahl, U. and Li, D.} (2005) Some results on two-sided LIL behavior. \emph {Ann. Probab.} {\bf 33}, 1601--1624. \item {\sc Feller, W.} (1968). An extension of the law of the iterated logarithm to variables without variance. {\em J. Math. Mech.} {\bf 18}, 343-355. \item {\sc Klass, M.} (1976) Toward a universal law of the iterated logarithm I. {\em Z. Wahrsch. Verw. Gebiete} {\bf 36}, 165-178. \item {\sc Klass, M.} (1977) Toward a universal law of the iterated logarithm II. {\em Z. Wahrsch. Verw. Gebiete} {\bf 39}, 151-165. \item {\sc Klein, T. and Rio, E.} (2005) Concentration around the mean for maxima of empirical processes. {\em Ann. Probab.} {\bf 33}, 1060--1077. \item {\sc Kuelbs, J.} (1985) The LIL when $X$ is in the domain of attraction of a Gaussian law. {\em Ann. Probab.} {\bf 13}, 825--859. \item {\sc Ledoux, M.} (1996) On Talagrand's deviation inequality for product measures. {\em ESAIM Probab. Statist.} {\bf 1}, 63--87. \item {\sc Ledoux, M. and Talagrand, M.} (1988) Characterization of the law of the iterated logarithm in Banach space. {\em Ann. Probab.} {\bf 16}, 1242--1264. \item {\sc Ledoux, M. and Talagrand, M.}(1991) {\em Probability in Banach Spaces.} Springer, Berlin (1991). \item {\sc Massart, P.} (2000) About the constants in Talagrand's concentration inequalities for empirical processes. {\em Ann. Probab.} {\bf 28}, 863--884. \item {\sc Petrov, V. V.} (1995) {\em Limit Theorems of Probability Theory: Sequences of Independent Random Variables.}~ Clarendon Press, Oxford. \item {\sc Pruitt, W.} (1981) General one-sided laws of the iterated logarithm. {\em Ann. Probab.} {\bf 9}, 1--48. \item {\sc Rio, E.:} (2002) Une in\'egalit\'e de Bennet pour les maxima de processus empiriques. {\em Ann. Inst. H. Poincar\'e Probab. Statist.} {\bf 38}, 1053--1057. \item {\sc Talagrand, M.:} (1994) Sharper bounds for Gaussian and empirical processes. {\em Ann. Probab.} {\bf 22}, 28--76. \end{description}
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\begin{document}
\title{Universum Prescription: Regularization Using Unlabeled Data} \author{ Xiang Zhang \qquad Yann LeCun \\ Courant Institute of Mathematical Sciences, New York University \\ 719 Broadway, 12th Floor, New York, NY 10003 \\ \texttt{\{xiang, yann\}@cs.nyu.edu} \\ } \maketitle
\begin{abstract} This paper shows that simply prescribing ``none of the above'' labels to unlabeled data has a beneficial regularization effect to supervised learning. We call it universum prescription by the fact that the prescribed labels cannot be one of the supervised labels. In spite of its simplicity, universum prescription obtained competitive results in training deep convolutional networks for CIFAR-10, CIFAR-100, STL-10 and ImageNet datasets. A qualitative justification of these approaches using Rademacher complexity is presented. The effect of a regularization parameter -- probability of sampling from unlabeled data -- is also studied empirically. \end{abstract}
\section{Introduction}
The idea of exploiting the wide abundance of unlabeled data to improve the accuracy of supervised learning tasks is a very natural one. In this paper, we study what is perhaps the simplest way to exploit unlabeled data in the context of deep learning. We assume that the unlabeled samples do not belong to any of the categories of the supervised task, and we force the classifier to produce a ``none of the above'' output for these samples. This is by no means a new idea, but we show empirically and theoretically that doing so has a regularization effect on supervised task and reduces the generalization gap, the expected difference between test and training errors. We study three different ways to prescribe ``none of the above'' outputs, dubbed uniform prescription, dustbin class, and background class and show that they improve the test error of convolutional networks trained on CIFAR-10, CIFAR-100 \cite{K09}, STL-10 \cite{ANL11}, and ImageNet \cite{RDSKSMHKKBBF15}. The method is theoretically justified using Radamacher complexity \cite{BM03}.
Here we briefly describe our three universum prescription methods. Uniform prescription forces a discrete uniform distribution for unlabeled samples. Dustbin class simply adds an extra class and prescribe all unlabeled data to this class. Background class also adds an extra class, but it uses a constant threshold to avoid parameterization.
Our work is a direct extension to learning in the presence of universum \cite{WCSBV06} \cite{CASS07}, originated from \cite{V98} and \cite{V06}. The definition of universum is a set of unlabeled data that are known not to belong to any of the classes but in the same domain. We extended the idea of using universum from support vector machines to deep learning.
Most deep learning approaches utilizing unlabeled data belong to the scope of representation learning (reviewed by \cite{BCV13} and \cite{BL07}) and transfer learning \cite{TP98}. They include ideas like pretraining \cite{EBCMVB10} \cite{HOT06} \cite{RPCL06} and semi-supervised training \cite{RBHVR15} \cite{ZMGL15}. Universum prescription incoporates unlabeled data without imposing priors such as sparsity or reconstruction.
Regularization -- techniques for the control of generalization gap -- has been studied extensively. Most approaches implement a secondary optimization objective, such as an \(L_2\) norm. Some other methods such as dropout \cite{SHKSS14} cheaply simulate model averaging to control the model variance. As part of general statistical learning theory \cite{V95}, \cite{V98}, the justification for regularization is well-developed. We qualitatively justify the methods using Radamacher complexity \cite{BM03}, similar to \cite{WZZLF13}.
\section{Universum Prescription} \label{sec:pres}
In this section we attempt to formalize the trick of prescribing ``none of the above'' labels -- universum prescription. Consider the problem of exclusive \(k\)-way classification. In learning we hope to find a hypothesis function \(h \in \mathcal{H}\) mapping to \(\mathbb{R}^k\) so that the label is determined by \(y = \mathrm{argmin}_i ~ h_i(x)\). The following assumptions are made.
\begin{enumerate} \item (Loss assumption) The loss used as the optimization objective is negative log-likelihood:
\begin{equation}
L(h, x, y) = h_y (x) + \log \left[ \sum_{i = 1}^{k} \exp(-h_i (x)) \right].
\end{equation} \item (Universum assumption) The proportion of unlabeled samples belonging to a supervised class is negligible. \end{enumerate}
The loss assumption assumes that the probability of class \(y\) given an input \(x\) can be thought of as \begin{equation}
\Pr[Y = y | x, h] = \frac{\exp(-h_y(x))}{\sum_{i = 1}^{k} \exp(-h_i(x))}, \end{equation} where \((X, Y) \sim \mathbf{D}\) and \(\mathbf{D}\) is the distribution where labeled data are sampled. We use lowercase letters for values, uppercase letters for random variables and bold uppercase letters for distribution. The loss assumption is simply a necessary detail rather than a limitation, in the sense that one can change the type of loss and use the same principles to derive different universum learning techniques.
The universum assumption implicates that labeled classes are a negligible subset. In many practical cases we only care about a small number of classes, either by problem design or due to high cost in the labeling process. At the same time, a very large amount of unlabeled data is easily obtained. Put in mathematics, assuming we draw unlabeled data from distribution \(\mathbf{U}\), the assumption states that \begin{equation}
\label{eq:jtds}
\Pr_{(X,Y) \sim \mathbf{U}}[X, Y \in \{1, 2, \dots, k\}] \approx 0. \end{equation}
The universum assumption is opposite to the assumptions of information regularization \cite{CJ06} and transduction learning \cite{CSZ06T} \cite{GVV98}. It has similarities with \cite{ZZ10} that encourages diversity of outputs for ensemble methods. All our methods discussed below prescribe agnostic targets to the unlabeled data. During learning, we randomly present an unlabeled sample to the optimization procedure with probability \(p\).
\subsection{Uniform Prescription} \label{sect:unif}
It is known that negative log-likelihood is simply a reduced form of cross-entropy \begin{equation}
L(h, x, y ) = -\sum_{i = 1}^{k} Q[Y = i | x] \log \Pr[Y = i | x, h] \end{equation}
in which the target probability \(Q[Y = y | x] = 1\) and \(Q[Y = i | x] = 0\) for \(i \neq y\). Under the universum assumption, if we are presented with an unlabeled sample \(x\), we would hope to prescribe some \(Q\) so that every class has some equally minimal probability. \(Q\) also has to satisfy \(\sum_{i = 1}^{k} Q[Y=i|x] = 1\) by the probability axioms. The only possible choice for \(Q\) is then \(Q[Y | x] = 1/k\). The learning algorithm then uses the cross-entropy loss instead of negative log-likelihood.
It is worth noting that uniform output has the maximum entropy among all possible choices. If \(h\) is parameterized as a deep neural network, uniform output is achieved when these parameters are constantly zero. Therefore, uniform prescription may have the effect of reducing the magnitude of parameters, similar to norm-based regularization.
\subsection{Dustbin Class} \label{sect:dust}
Another way of prescribing agnostic target is to append a ``dustbin'' class to the supervised task. This requires some changes to the hypothesis function \(h\) such that it outputs \(k+1\) targets. For deep learning models one can simply extend the last parameterized layer. All unlabeled data are prescribed to this extra ``dustbin'' class.
The effect of dustbin class is clearly seen in the loss function of an unlabeled sample \((x, k+1)\) \begin{equation} L(h, x, k+1) = h_{k+1} (x) + \log \left[ \sum_{i = 1}^{k + 1} \exp(-h_i (x)) \right]. \end{equation} The second term is a ``soft'' maximum for all dimensions of \(-h\). With an unlabeled sample, the algorithm attempts to introduce smoothness by minimizing probability spikes.
\subsection{Background Class} \label{sect:bgnd}
We could further simplify dustbin class by removing parameters for class \(k + 1\). For some given threshold constant \(\tau\), we could change the probability of a labeled sample to \begin{equation}
\Pr[Y = y | x, h] = \frac{\exp(-h_y(x))}{\exp(-\tau) + \sum_{i = 1}^{k} \exp(-h_i(x))}, \end{equation} and an unlabeled sample \begin{equation}
\Pr[Y = k + 1 | x, h] = \frac{\exp(-\tau)}{\exp(-\tau) + \sum_{i = 1}^{k} \exp(-h_i(x))}. \end{equation}
This will result in changes to the loss function of a labeled sample \((x, y)\) as \begin{equation} L(h, x, y) = h_y (x) + \log \left[ \exp(-\tau) + \sum_{i = 1}^{k} \exp(-h_i (x)) \right], \end{equation} and an unlabeled sample \begin{equation} L(h, x, k + 1) = \tau + \log \left[ \exp(-\tau) + \sum_{i = 1}^{k} \exp(-h_i (x)) \right]. \end{equation}
\begin{table}[h]
\caption{The 21-layer network}
\label{tab:expi}
\begin{center}
\begin{tabular}{ll}
\multicolumn{1}{c}{\bf LAYERS} &\multicolumn{1}{c}{\bf DESCRIPTION}
\\ \hline \\
1-3 &Conv 256x3x3 \\
4 &Pool 2x2 \\
5-8 &Conv 512x3x3 \\
9 &Pool 2x2 \\
10-13 &Conv 1024x3x3 \\
14 &Pool 2x2 \\
15-18 &Conv 1024x3x3 \\
19 &Pool 2x2 \\
20-23 &Conv 2048x3x3 \\
24 &Pool 2x2 \\
25-26 &Full 2048 \\
\end{tabular}
\end{center} \end{table}
We call this method background class and \(\tau\) background constant. Similar to dustbin class, the algorithm attempts to minimize the spikes of outputs, but limited to a certain extent by the inclusion of \(\exp(-\tau)\) in the partition function. In our experiments \(\tau\) is always set to 0.
\section{Theoretical Justification} \label{sec:thry}
In this part, we derive a qualitative justification for universum prescription using probably approximately correct (PAC) learning \cite{V84}. By being ``qualitative'', the justification is in contrast with numerical bounds such as Vapnik-Chervonenkis dimension \cite{VC71} (VC-dim) and others. Our theory is based on Rademacher complexity \cite{BM03}, similar to \cite{WZZLF13} where both dropout \cite{SHKSS14} and dropconnect \cite{WZZLF13} are justified. VC-dim is an upper-bound of Rademacher complexity, suggesting that the latter is more accurate. Previous results on unlabeled data \cite{OAGR11} \cite{OGRA15} assume the same distribution for labeled and unlabeled data, which is impossible under the universum assumption.
\begin{definition}[Empirical Rademacher complexity]
Let \(\mathcal{F}\) be a family of functions mapping from \( \mathcal{X} \) to \(\mathbb{R}\), and \(S = (x_1, x_2, \dots, x_m)\) a fixed sample of size \(m\) with elements in \(\mathcal{X}\). Then, the empirical Rademacher complexity of \(F\) with respect to the sample \(S\) is defined as:
\begin{equation}
\hat{\mathfrak{R}}_S(\mathcal{F}) = \underset{\boldsymbol{\eta}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \frac{1}{m} \sum_{i=1}^{m} \eta_i f(x_i) \right]
\end{equation}
where \(\boldsymbol{\eta} = (\eta_1, \dots, \eta_m)^T\), with \(\eta_i\)'s independent random variables uniformly distributed on \(\{-1, 1\}\). \end{definition}
\begin{definition}[Rademacher complexity]
Let \(\mathbf{D}\) denote the distribution from which the samples were drawn. For any integer \(m \geq 1\), the Rademacher complexity of \(\mathcal{F}\) is the expectation of the empirical Rademacher complexity over all samples of size \(m\) drawn according to \(\mathbf{D}\):
\begin{equation}
\mathfrak{R}_m (\mathcal{F}, \mathbf{D}) = \underset{S \sim \mathbf{D}^m}{\mathrm{E}} [\hat{\mathfrak{R}}_S(F)]
\end{equation} \end{definition}
Two qualitative properties of Rademacher complexity is worth noting here. First of all, Rademacher complexity is always non-negative by the convexity of supremum \begin{equation} \begin{aligned}
\hat{\mathfrak{R}}_S(\mathcal{F}) & = \underset{\boldsymbol{\eta}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \frac{1}{m} \sum_{i=1}^{m} \eta_i f(x_i) \right] \\
& \geq \sup_{f \in \mathcal{F}} \frac{1}{m} \sum_{i=1}^{m} \underset{\eta_i}{\mathrm{E}} [\eta_i] f(x_i) = 0. \end{aligned} \end{equation} Secondly, if for a fixed input all functions in \(\mathcal{F}\) output the same value, then its Rademacher complexity is 0. Assume for any \(f \in \mathcal{F}\) we have \(f(x) = f_0(x)\), then \begin{equation} \begin{aligned}
\hat{\mathfrak{R}}_S(\mathcal{F}) & = \underset{\boldsymbol{\eta}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \frac{1}{m} \sum_{i=1}^{m} \eta_i f(x_i) \right] \\
& = \underset{\boldsymbol{\eta}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \frac{1}{m} \sum_{i=1}^{m} \eta_i f_0(x) \right] \\
& = \frac{1}{m} \sum_{i=1}^{m} \underset{\eta_i}{\mathrm{E}} [\eta_i] f_0(x) = 0. \end{aligned} \end{equation}
Therefore, one way to minimize Rademacher complexity is to regularize functions in \(\mathcal{F}\) such that all functions tend to have the same output for a given input. Universum prescription precisely does that -- the prescribed outputs for unlabeled data are all constantly the same.
The principal PAC-learning result is a bound for functions that are finite in outputs. We use the formulation by \cite{Z13}, but anterior results exist \cite{BBL02} \cite{BM03} \cite{K01} \cite{KP00}. \begin{theorem} [Approximation bound with finite bound on output]
For an energy function \cite{LCHRH06} \(\mathcal{E}(h,x,y)\) over hypothesis class \(\mathcal{H}\), input set \(\mathcal{X}\) and output set \(\mathcal{Y}\), if it has lower bound 0 and upper bound \(M > 0\), then with probability at least \(1-\delta\), the following holds for all \(h \in \mathcal{H} \):
\begin{equation}
\label{eq:pcbd}
\begin{aligned}
& \underset{(x,y) \sim \mathbf{D}}{\mathrm{E}}[\mathcal{E}(h,x,y)] \leq \\
& \frac{1}{m} \sum_{(x,y) \in S} \mathcal{E}(h,x,y) + 2 \mathfrak{R}_m(\mathcal{F}, \mathbf{D}) + M\sqrt{\frac{\log{\frac{2}{\delta}}}{2m}},
\end{aligned}
\end{equation}
where the function family \(\mathcal{F}\) is defined as
\begin{equation}
\mathcal{F} = \left\{ \mathcal{E}(h,x,y) | h \in \mathcal{H} \right\}.
\end{equation}
\(\mathbf{D}\) is the distribution for \((x,y)\), and \(S\) is a sample set of size \(m\) drawn indentically and independently from \(\mathbf{D}\). \end{theorem}
The meaning of the theorem is two-fold. When applying the theorem to the joint problem of training using both labeled and unlabeled data, the third term on the right hand of inequality \ref{eq:pcbd} is reduced by the augmentation of the extra data. The joint Rademacher complexity is written as \(\mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U})\), which is reduced when we prescribe constant outputs to unlabeled data.
The second fold is that when the theorem applies to the supervised distribution \(\mathbf{D}\), we would hope that \(\mathfrak{R}_n(\mathcal{F}, \mathbf{D})\) can be bounded by \(\mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U})\), where \(n\) is the number of supervised samples randomly chosen by the joint problem. Note that the number \(n\) follows a binomial distribution with mean \((1-p)m\). Such a bound can be achieved in a probable and approximate sense.
\begin{theorem} [Rademacher complexity bound on distribution mixture]
\label{thm:rbdm}
Assume we have a joint problem where \(p \leq 0.5\) and there are \(m\) random training samples from the joint distribution \((1-p)\mathbf{D} + p \mathbf{U}\). With probability at least \(1-\delta\), the following holds
\begin{equation}
\label{eq:rbdm}
\begin{aligned}
& \mathfrak{R}_n(\mathcal{F}, \mathbf{D}) \leq \\
& \frac{2-p}{(1-p)\left(1-p - \sqrt{\frac{\log(1/\delta)}{2m}}\right)} \mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U}),
\end{aligned}
\end{equation}
where \(n\) is a random number indicating the number of supervised samples in the total joint samples, and \(m\) is large enough such that
\begin{equation}
1-p-\sqrt{\frac{\log(1/\delta)}{2m}} > 0.
\end{equation} \end{theorem}
We present the proof of theorem \ref{thm:rbdm} in the supplemental material, which utilizes Hoeffding's inequality \cite{H63} \cite{S74}. The theorem tells us that the Rademacher complexity of the supervised problem can be bounded by that of the joint problem. The universum prescription algorithm attempts to make the Rademacher complexity of the joint problem small. Therefore, universum prescription improves generalization by incorporating unlabeled data.
However, theorem \ref{thm:rbdm} has a requirement that \( p \leq 0.5\), otherwise the bound is not achievable. Also, the value of \((2-p)/(1-p)^2\) -- the asymptotic constant factor in inequality \ref{eq:rbdm} when \(m\) is large -- is monitonally increasing with respect to \(p\) with a range of \([2, 6]\) when \(p \leq 0.5\). These facts indicate that we need to keep \(p\) small. The following sections show that there is improvement if \(p\) is small, but training and testing errors became worse when \(p\) is large.
\begin{table}[h]
\caption{The 17-layer network}
\label{tab:expn}
\begin{center}
\begin{tabular}{ll}
\multicolumn{1}{c}{\bf LAYERS} &\multicolumn{1}{c}{\bf DESCRIPTION}
\\ \hline \\
1-3 &Conv 128x3x3 \\
4 &Pool 2x2 \\
5-7 &Conv 256x3x3 \\
8 &Pool 2x2 \\
9-11 &Conv 512x3x3 \\
12 &Pool 2x2 \\
13-15 &Conv 1024x3x3 \\
16 &Pool 2x2 \\
17-19 &Conv 2048x3x3 \\
20 &Pool 2x2 \\
21-22 &Full 4096 \\
\end{tabular}
\end{center} \end{table}
Finally, in terms of numerical asymptotics, theorem \ref{thm:rbdm} suggests that \(\mathfrak{R}_n(\mathcal{F}, \mathbf{D}) \leq \mathbf{O}(1/\sqrt{m})\), instead of the commonly known result \(\mathfrak{R}_n(\mathcal{F}, \mathbf{D}) \leq \mathbf{O}(1/\sqrt{n})\). This bounds the supervised problem with a tighter asymptotical factor because there are more joint samples than supervised samples.
\section{Experiments on Image Classification} \label{sec:expi}
In this section we test the methods on some image classification tasks. Three series of datasets -- CIFAR-10/100 \cite{K09}, STL-10 \cite{ANL11} and ImageNet \cite{RDSKSMHKKBBF15} -- are chosen due to the availability of unlabeled data. For CIFAR-10/100 and STL-10 datasets, we used a 21-layer convolutional network (ConvNet) \cite{LBDHHHJ89} \cite{LBBH98}, in which the inputs are 32-by-32 images and all convolutional layers are 3-by-3 and fully padded. For ImageNet, the model is a 17-layer ConvNet with 64-by-64 images as inputs. These models are inspired by \cite{SZ14}, in which all pooling layers are max-pooling, and ReLUs \cite{NH10} are used as the non-linearity. Two dropout \cite{SHKSS14} layers of probability 0.5 are inserted before the final two linear layers.
\begin{table}[h]
\caption{Result for baseline and uniform prescription. The numbers are percentages.}
\label{tab:expd}
\begin{center}
\addtolength{\tabcolsep}{-3pt}
\begin{tabular}{lrrrrrr}
\textbf{DATASET} & \multicolumn{3}{c}{\textbf{BASELINE}} & \multicolumn{3}{c}{\textbf{UNIFORM}} \\
& \multicolumn{1}{c}{\small\textbf{Train}} & \multicolumn{1}{c}{\small\textbf{Test}} & \multicolumn{1}{c}{\small\textbf{Gap}} & \multicolumn{1}{c}{\small\textbf{Train}} & \multicolumn{1}{c}{\small\textbf{Test}} & \multicolumn{1}{c}{\small\textbf{Gap}} \\
\hline \\
CIFAR-10 & \textbf{0.00} & 7.02 & 7.02 & 0.72 & 7.59 & 6.87 \\
CIFAR-100 F. & \textbf{0.09} & 37.58 & 37.49 & 4.91 & 36.23 & 31.32 \\
CIFAR-100 C. & \textbf{0.04} & 22.74 & 22.70 & 0.67 & 23.42 & 22.45 \\
STL-10 & \textbf{0.00} & \textbf{31.16} & 31.16 & 2.02 & 36.54 & 34.52 \\
STL-10 Tiny & \textbf{0.00} & 31.16 & 31.16 & 0.62 & 30.15 & 29.47 \\
ImageNet-1 & \textbf{10.19} & 34.39 & 24.20 & 13.84 & 34.61 & 20.77 \\
ImageNet-5 & \textbf{1.62} & 13.68 & 12.06 & 3.02 & 13.70 & 10.68 \\
\end{tabular}
\addtolength{\tabcolsep}{4pt}
\end{center} \end{table}
\begin{table}[h]
\caption{Result for dustbin class and background class. Continuation of table \ref{tab:expd}}
\label{tab:expr}
\begin{center}
\addtolength{\tabcolsep}{-3pt}
\begin{tabular}{lrrrrrr}
\textbf{DATASET} & \multicolumn{3}{c}{\textbf{DUSTBIN}} & \multicolumn{3}{c}{\textbf{BACKGROUND}} \\
& \multicolumn{1}{c}{\small\textbf{Train}} & \multicolumn{1}{c}{\small\textbf{Test}} & \multicolumn{1}{c}{\small\textbf{Gap}} & \multicolumn{1}{c}{\small\textbf{Train}} & \multicolumn{1}{c}{\small\textbf{Test}} & \multicolumn{1}{c}{\small\textbf{Gap}} \\
\hline \\
CIFAR-10 & 0.07 & \textbf{6.66} & \textbf{6.59} & 1.35 & 8.38 & 7.03 \\
CIFAR-100 F. & 2.52 & \textbf{32.84} & \textbf{30.32} & 8.56 & 40.57 & 42.01 \\
CIFAR-100 C. & 0.40 & \textbf{20.45} & \textbf{20.05} & 3.73 & 24.97 & 21.24 \\
STL-10 & 3.03 & 36.58 & 33.55 & 14.89 & 38.95 & \textbf{24.06} \\
STL-10 Tiny & 0.00 & \textbf{27.96} & \textbf{27.96} & 0.11 & 30.38 & 30.27 \\
ImageNet-1 & 13.80 & \textbf{33.67} & \textbf{19.87} & 13.43 & 34.69 & 21.26 \\
ImageNet-5 & 2.83 & \textbf{13.35} & \textbf{10.52} & 2.74 & 13.84 & 11.10 \\
\end{tabular}
\addtolength{\tabcolsep}{4pt}
\end{center} \end{table}
The algorithm used is stochastic gradient descent with momentum \cite{P64} \cite{SMDH13} 0.9 and a minibatch size of 32. The initial learning rate is 0.005 which is halved every 60,000 minibatch steps for CIFAR-10/100 and every 600,000 minibatch steps for ImageNet. The training stops at 400,000 steps for CIFAR-10/100 and STL10, and 2,500,000 steps for ImageNet. Table \ref{tab:expi} and \ref{tab:expn} summarize the configurations. The weights are initialized in the same way as \cite{HZRS15}. The following data augmentation steps are used.
\begin{enumerate}
\setlength{\itemsep}{2pt} \item (Horizontal flip.) Flip the image horizontally with probability 0.5. \item (Scale.) Randomly scale the image between \(1/1.2\) and \(1.2\) times of its height and width. \item (Crop.) Randomly crop a 32-by-32 (or 64-by-64 for ImageNet) region in the scaled image. \end{enumerate}
\subsection{CIFAR-10 and CIFAR-100}
The samples of CIFAR-10 and CIFAR-100 datasets \cite{K09} are from the 80 million tiny images dataset \cite{TFF08}. Each dataset contains 60,000 samples, consitituting a very small portion of 80 million. This is an ideal case for our methods, in which we can use the entire 80 million images as the unlabeled data. The CIFAR-10 dataset has 10 classes, and CIFAR-100 has 20 (coarse) or 100 (fine-grained) classes.
Table \ref{tab:expd} and \ref{tab:expr} contain the results. The three numbers in each tabular indicate training error, testing error and generalization gap. Bold numbers are the best ones for each case. The generalization gap is approximated by the difference between testing and training errors. All the models use unlabeled data with probability \(p=0.2\).
We compared other single-model results on CIFAR-10 and CIFAR-100 (fine-grained case) in table \ref{tab:cfcp}. It shows that our network is competitive to the state of the art. Although \cite{G14} has the best results, we believe that by applying out universum prescription methods to their model design could also improve the results further.
\begin{table}[h]
\caption{Comparison of single-model CIFAR-10 and CIFAR-100 results, in second and third columns. The fourth column indicates whether data augmentation is used for CIFAR-10. The numbers are percentages.}
\label{tab:cfcp}
\begin{center}
\begin{tabular}{lccc}
\textbf{REF.} & \textbf{10} & \textbf{100} & \textbf{AUG.}
\\ \hline \\
\cite{G14} & 6.28 & 24.30 & YES \\
(ours) & 6.66 & 32.84 & YES \\
\cite{LXPZT15} & 7.97 & 34.57 & YES \\
\cite{LCY13} & 8.81 & 35.68 & YES \\
\cite{GWMCB13} & 9.38 & 38.57 & YES \\
\cite{WZZLF13} & 11.10 & N/A & NO \\
\cite{MR13} & 15.13 & 42.51 & NO \\
\end{tabular}
\end{center} \end{table}
\subsection{STL-10}
The STL-10 dataset \cite{ANL11} has size 96-by-96 for each image. We downsampled them to 32-by-32 in order to use the same model. The dataset contains a very small number of training samples -- 5000. The accompanying unlabeled data contain 100,000 samples. There is no guarantee that these unlabeled samples do not blong to the supervised classes \cite{ANL11}, therefore universum prescription failed. To verify that the extra data is the problem, an experiment using the 80 million tiny images as the unlabeled dataset is shown in table \ref{tab:expd} and \ref{tab:expr}. In this case the improvement is observed. Due to long training times of our models, we did not perform 10-fold training in the original paper \cite{ANL11}.
One interesting observation is that the results on STL-10 became better with the use of 80 million tiny images instead of the original extra data. It indicates that dataset size and whether universum assumption is satisfied are affecting factors for the effectiveness of universum prescription.
\subsection{ImageNet}
The ImageNet dataset \cite{RDSKSMHKKBBF15} for classification task has in total 1,281,167 training images and 50,000 validation images. The reported testing errors are evaluated on this validation dataset. During training, we resize images to minimum dimension 64, and then feed a random 64-by-64 crop to the network. Same test-time augmentation technique as in \cite{SLJSRAEVR15} are applied, with size variants \{64, 72, 80, 88\}, where each image is viewed in 144 crops.
The extra data comes from the large ImageNet 2011 release\footnote{\texttt{http://www.image-net.org/releases}}, for which we only keep the classes whomself and whose children do not belong to the supervised classes. This is enabled by the super-subordinate (is-a) relation information provided with the WordNet distribution \cite{M95} because all ImageNet classes are nouns of WordNet. Both top-1 and top-5 results are reported in tables \ref{tab:expd} and \ref{tab:expr}.
In all experiments dustbin class provides best results. We believe that it is because the extra class is parameterized, which makes it adapt better on the unlabeled samples.
\begin{table}[h]
\caption{ConvNet for the study of \(p\)}
\label{tab:para}
\begin{center}
\begin{tabular}{ll}
\multicolumn{1}{c}{\bf LAYERS} &\multicolumn{1}{c}{\bf DESCRIPTION}
\\ \hline \\
1 &Conv 1024x5x5 \\
2 &Pool 2x2 \\
3 &Conv 1024x5x5 \\
4-7 &Conv 1024x3x3 \\
8 &Pool 2x2 \\
9-11 &Full 2048 \\
\end{tabular}
\end{center} \end{table}
\section{Effect of the Regularization Parameter} \label{sec:para}
It is natural to ask how would the change of the probability \(p\) of sampling from unlabeled data affect the results. In this section we show the experiments. To prevent an exhaustive search on the regularization parameter from overfitting our models on the testing data, we use a different model for this section. It is described in table \ref{tab:para}, which has 9 parameterized layers in total. The design is inspired by \cite{SEZMFL13}. For each choice of \(p\) we conducted 6 experiments combining universum prescription models and dropout. The dropout layers are two ones added in between the fully-connected layers with dropout probability 0.5. Figure \ref{fig:para} shows the results.
From figure \ref{fig:para} we can conclude that increasing \(p\) will descrease generalization gap. However, we cannot make \(p\) too large since after a certain point the training collapses and both training and testing errors become worse. This confirms the assumptions and conclusions from theorem \ref{thm:rbdm}.
\begin{figure}
\caption{Experiments on regularization parameter. The four rows are CIFAR-10, CIFAR-100 fine-grained, CIFAR-100 coarse and STL-10 respectively.}
\label{fig:para}
\end{figure}
Comparing between CIFAR-10/100 and STL-10, one conclusion is that that the model variance is affected by the combined size of labeled and unlabeled datasets. The variance on training and testing errors are extremely small on CIFAR-10/100 datasets because the extra data we used is almost unlimited (in total 80 million), but on STL-10 the variance seems to be large with much smaller combined size of training and extra datasets. This suggests that using universum prescription with a large abundance of extra data could improve the stability of supervised learning algorithms.
Finally, the comparison between using and not using dropout does not show a difference. This suggests that the regularization effect of universum prescription alone is comparable to that of dropout.
\section{Conclusion and Outlook}
This article shows that universum prescription can be used to regularize a multi-class classification problem using extra unlabeled data. Two assumptions are made. One is that loss used is negative log-likelihood and the other is negligible probability of a supervised sample existing in the unlabeled data. The loss assumption is a necessary detail rather than a limitation. The three universum prescription methods are uniform prescription, dustbin class and background class.
We further provided a theoretical justification. Theorem \ref{thm:rbdm} suggests that asymptotically the generalization ability of the supervised problem could be bounded by the joint problem, which has more samples due to the addition of unlabeled data. Experiments are done using CIFAR-10, CIFAR-100, STL-10 and ImageNet datasets. The effect of the regularization parameter is also studied empirically.
These experiments show that all three universum prescrition methods provide certain improvement over the generalization gap, whereas dustbin class constantly performs the best because the parameterized extra class can adapt better to the unlabeled samples. Further conclusions include that additional unlabeled data can improve the variance of models during training, and that the results are comparable to data-agnostic regularization using dropout.
In the future, we hope to apply these methods to a broader range of problems.
\section*{Acknowledgments}
We gratefully acknowledge NVIDIA Corporation with the donation of 2 Tesla K40 GPUs used for this research. Sainbayar Sukhbaatar offered many useful comments. Aditya Ramesh and Junbo Zhao helped cross-checking the proofs.
\pagebreak
\section*{Supplemental: proof of theorem \ref{thm:rbdm}}
This supplemental material shares the bibliography of the main paper. As an outline of our proof, we first establish a relation between \(\mathfrak{R}_m (\mathcal{F}, \mathbf{D})\) and \(\mathfrak{R}_m (\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U})\), and then another relation between \(\mathfrak{R}_n (\mathcal{F}, \mathbf{D})\) and \(\mathfrak{R}_m (\mathcal{F}, \mathbf{D})\). The first part requires the following lemmas.
\begin{lemma}[Separation of dataset on empirical Rademacher complexity]
\label{lem:sepr}
Let \(S\) be a dataset of size \(m\). If \(S_1\) and \(S_2\) are two non-overlap subset of \(S\) such that \(|S_1| = m - i\), \(|S_2| = i\) and \(S_1 \cup S_2 = S\), then the following two inequalities hold
\begin{equation}
\label{eq:sepr}
\hat{\mathfrak{R}}_S (\mathcal{F}) \leq \frac{m - i}{m} \hat{\mathfrak{R}}_{S_1} (\mathcal{F}) + \frac{i}{m} \hat{\mathfrak{R}}_{S_2} (\mathcal{F}).
\end{equation} \end{lemma} \begin{proof}
Let \((x_j, y_j) \in S_1\) for \(j = 1, 2, \dots, m - i\) and \((x_j, y_j) \in S_2\) for \(i = m - j + 1, m - j + 2, \dots, m\). Denote \(\mathbf{N}\) as the discrete uniform distribution on \(\{1, -1\}\). We can derive by the convexity of supremum and symmetry of \(\mathbf{N}\)
\[
\begin{aligned}
& \hat{\mathfrak{R}}_{S} (\mathcal{F}) = \underset{\boldsymbol{\eta} \sim \mathbf{N}^{m}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \frac{1}{m} \sum_{j=1}^{m} \eta_j f(x_j) \right] \\
= &\frac{2}{m} \underset{\boldsymbol{\eta} \sim \mathbf{N}^{m}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \left( \frac{1}{2} \sum_{j=1}^{m - i} \eta_j f(x_j) + \frac{1}{2} \sum_{j=m-i+1}^{m} \eta_j f(x_j) \right) \right] \\
\leq & \frac{2}{m} \underset{\boldsymbol{\eta} \sim \mathbf{N}^{m}}\mathrm{E} \left[ \frac{1}{2} \sup_{f \in \mathcal{F}} \left( \sum_{j=1}^{m - i} \eta_j f(x_j) \right) + \frac{1}{2} \sup_{f \in \mathcal{F}} \left( \sum_{j=m-i+1}^{m} \eta_j f(x_j) \right) \right] \\
= & \frac{m - i}{m} \underset{\boldsymbol{\eta} \sim \mathbf{N}^{m - i}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \frac{1}{m-i} \sum_{j=1}^{m - i} \eta_j f(x_j) \right] + \\
& \quad \frac{i}{m} \underset{\boldsymbol{\eta} \sim \mathbf{N}^{i}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \frac{1}{i} \sum_{j=m-i+1}^{m} \eta_j f(x_j) \right] \\
= &\frac{m - i}{m} \hat{\mathfrak{R}}_{S_1} (\mathcal{F}) + \frac{i}{m} \hat{\mathfrak{R}}_{S_2} (\mathcal{F}).
\end{aligned}
\] \end{proof}
\begin{lemma}[Sample size inequality for Rademacher complexity]
\label{lem:sinr}
Assume \(0 \leq n \leq m\). If \(|S_n| = n\), \(|S_m| = m\) and \(S_m = S_n \cup \{x_{n+1}, x_{n+2}, \dots, x_{m}\}\), then
\begin{equation}
\label{eq:sine}
n \hat{\mathfrak{R}}_{S_n} (\mathcal{F}) \leq m \hat{\mathfrak{R}}_{S_m} (\mathcal{F}),
\end{equation}
and
\begin{equation}
\label{eq:sint}
n \mathfrak{R}_n (\mathcal{F}, \mathbf{D}) \leq m \mathfrak{R}_m (\mathcal{F}, \mathbf{D}).
\end{equation} \end{lemma} \begin{proof}
First of all, it is obvious that inequality \ref{eq:sint} can be established using mathematical induction if we have \(m \mathfrak{R}_m (\mathcal{F}, \mathbf{D}) \leq (m + 1) \mathfrak{R}_{m + 1} (\mathcal{F}, \mathbf{D})\) for all \(m \geq 0\). To prove this, we first establish that if \(S_m = \{x_1, x_2, \dots, x_m\}\) and \(S_{m + 1} = \{x_1, x_2, \dots, x_m, x_{m+1}\}\) (i.e., \(S_{m+1} = S_m \cup \{x_{m + 1}\}\)), then \(m \hat{\mathfrak{R}}_{S_m} (\mathcal{F}) \leq (m+1) \hat{\mathfrak{R}}_{S_{m+1}} (\mathcal{F})\), which can also establish inequality \ref{eq:sine}.
For any \( \boldsymbol{\eta}_m =\{\eta_1, \eta_2, \dots, \eta_m\} \) and \( \boldsymbol{\eta}_{m + 1} =\{\eta_1, \eta_2, \dots, \eta_m, \eta_{m+1}\}\), that is, \( \boldsymbol{\eta}_{m + 1} = \boldsymbol{\eta}_m \cup \{\eta_{m + 1}\}\), let \( f_0 = \mathrm{argmax}_{f \in \mathcal{F}} \sum_{i = 1}^{m} \eta_i f(x_i)\). By definition of supremum, we have
\[
\begin{aligned}
\sup_{f \in \mathcal{F}} \sum_{i = 1}^{m + 1} \eta_i f(x_i) & \geq \sum_{i = 1}^{m + 1} \eta_i f_0 (x_i) \\
& = \sum_{i = 1}^{m} \eta_i f_0 (x_i) + \eta_{m+1}f_0 (x_{m+1}) \\
& = \sup_{f \in \mathcal{F}} \sum_{i = 1}^{m} \eta_i f(x_i) + \eta_{m+1} f_0(x_{m+1}).
\end{aligned}
\]
Taking espectation over \(\boldsymbol{\eta}_{m+1}\), by the symmetry of distribution \(\mathbf{N}\), we obtain
\[
\begin{aligned}
& \underset{\boldsymbol{\eta}_{m+1} \sim \mathbf{N}^{m+1}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \sum_{i = 1}^{m + 1} \eta_i f(x_i) \right] \\
& \geq \underset{\boldsymbol{\eta}_{m+1} \sim \mathbf{N}^{m}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \sum_{i = 1}^{m} \eta_i f(x_i) + \eta_{m+1} f_0(x_{m+1}) \right] \\
& = \underset{\boldsymbol{\eta}_{m} \sim \mathbf{N}^{m}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \sum_{i = 1}^{m} \eta_i f(x_i) \right] + \underset{\eta_{m + 1} \sim \mathbf{N}}\mathrm{E} \left[ \eta_{m+1} \right] f_0(x_{m+1}) \\
& = \underset{\boldsymbol{\eta}_{m} \sim \mathbf{N}^{m}}\mathrm{E} \left[ \sup_{f \in \mathcal{F}} \sum_{i = 1}^{m} \eta_i f(x_i) \right].
\end{aligned}
\]
By the definition of \(\hat{\mathfrak{R}}_{S_m} (\mathcal{F})\), the inequality above implies \(m \hat{\mathfrak{R}}_{S_m} (\mathcal{F}) \leq (m+1) \hat{\mathfrak{R}}_{S_{m+1}} (\mathcal{F})\). Then, by taking espectation over \(S_{m+1}\) we can obtain
\[
\begin{aligned}
(m + 1) \mathfrak{R}_{m + 1} (\mathcal{F}, \mathbf{D}) & = \underset{S_{m+1} \sim \mathbf{D}^{m+1}} \mathrm{E} \left[ (m+1) \hat{\mathfrak{R}}_{S_{m+1}} \right] \\
& \geq \underset{S_{m} \sim \mathbf{D}^{m}} \mathrm{E} \left[ m \hat{\mathfrak{R}}_{S_{m}} \right] = m \mathfrak{R}_{m} (\mathcal{F}, \mathbf{D}).
\end{aligned}
\]
The lemma can therefore be easily established by mathematical induction. \end{proof}
Using the lemmas above, the relation between \(\mathfrak{R}_m (\mathcal{F}, \mathbf{D})\) and \(\mathfrak{R}_m (\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U})\) can be established as the following theorem, by assuming \(p \leq 0.5\).
\begin{theorem}[Relation of Rademacher complexities in distribution mixture]
\label{thm:rerm}
If \(p \leq 0.5\), then
\begin{equation}
\label{eq:rerm}
\mathfrak{R}_m (\mathcal{F}, \mathbf{D}) \leq \frac{2 - p}{1 - p} \mathfrak{R}_m (\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U}).
\end{equation} \end{theorem}
\begin{proof}
For any function space \(\mathcal{F}\) and distribution \(\mathbf{D}\), denote \(\mathfrak{R}_0(\mathcal{F}, \mathbf{D}) = 0\) and \(\hat{\mathfrak{R}}_{\emptyset}(\mathcal{F}) = 0\). By definition of Rademacher complexity and lemma \ref{lem:sepr}, we get
\[
\begin{aligned}
& \mathfrak{R}_m(\mathcal{F}, \mathbf{D}) = \mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{D}) \\
& = \underset{S \sim ((1-p)\mathbf{D} + p \mathbf{D})^m}{\mathrm{E}} \left [\hat{\mathfrak{R}}_S(\mathcal{F}) \right] \\
& = \sum_{i=0}^m \binom{m}{i} (1-p)^i p^{m-i} \underset{S_1 \sim \mathbf{D}^{i}}{\mathrm{E}} \left[ \underset{S_2 \sim \mathbf{D}^{m-i}}{\mathrm{E}} \left[\hat{\mathfrak{R}}_{S_1 \cup S_2}(\mathcal{F}) \right] \right] \\
& \leq \sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m-i} \\
& \quad \cdot \underset{S_1 \sim \mathbf{D}^{i}}{\mathrm{E}} \left[ \underset{S_2 \sim \mathbf{D}^{m-i}}{\mathrm{E}} \left [ \frac{i}{m} \hat{\mathfrak{R}}_{S_1} (\mathcal{F}) + \frac{m-i}{m} \hat{\mathfrak{R}}_{S_2} (\mathcal{F}) \right] \right] \\
& = \sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m-i} \\
& \quad \cdot \left( \underset{S_1 \sim \mathbf{D}^{i}}{\mathrm{E}} \left[ \frac{i}{m} \hat{\mathfrak{R}}_{S_1} (\mathcal{F}) \right] + \underset{S_2 \sim \mathbf{D}^{m-i}}{\mathrm{E}} \left [ \frac{m-i}{m} \hat{\mathfrak{R}}_{S_2} (\mathcal{F}) \right] \right) \\
& = \sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m-i} \left[ \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) + \frac{m-i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D})\right] \\
& = \left[\sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \right] \\
& \quad + \left[ \sum_{i=0}^m \binom{m}{i} (1-p)^{m-i} p^i \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D})\right] \\
& = \left[\sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \right] \\
& \quad + \left[ \sum_{i=0}^{\lfloor m / 2 \rfloor} \binom{m}{i} (1-p)^{m-i} p^i \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D})\right] \\
& \quad + \left[ \sum_{i=\lfloor m / 2 \rfloor + 1}^{m} \binom{m}{i} (1-p)^{m-i} p^i \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D})\right]. \\
\end{aligned}
\]
The proof proceeds by handling the three parts on the right-hand side of the inequality above separately.
For the first part, using lemma \ref{lem:sinr}, we can get
\[
\begin{aligned}
& \mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U}) \\
& = \sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m-i} \underset{S_1 \sim \mathbf{D}^{i}}{\mathrm{E}} \left[ \underset{S_2 \sim \mathbf{U}^{m-i}}{\mathrm{E}} \left[ \hat{\mathfrak{R}}_{S_1 \cup S_2}(\mathcal{F}) \right ] \right] \\
& \geq \sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m - i} \underset{S_1 \sim \mathbf{D}^{i}}{\mathrm{E}} \left[ \underset{S_2 \sim \mathbf{U}^{m-i}}{\mathrm{E}} \left [\frac{i}{m}\hat{\mathfrak{R}}_{S_1}(\mathcal{F}) \right] \right] \\
& = \sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m - i} \underset{S_1 \sim \mathbf{D}^{i}}{\mathrm{E}} \left[\frac{i}{m}\hat{\mathfrak{R}}_{S_1}(\mathcal{F}) \right] \\
& = \sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m - i} \frac{i}{m} \mathfrak{R}_{i}(\mathcal{F}, \mathbf{D}). \\
\end{aligned}
\]
The second part can also proceed using lemma \ref{lem:sinr}. It is essentially upper-bounded by the first part. By the fact that \(i \leq m - i \) for \(0 \leq i \leq \lfloor m / 2 \rfloor\), we obtain
\[
\begin{aligned}
& \sum_{i=0}^{\lfloor m / 2 \rfloor} \binom{m}{i} (1-p)^{m-i} p^i \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \\
& \leq \sum_{i=0}^{\lfloor m / 2 \rfloor} \binom{m}{i} (1-p)^{m-i} p^i \frac{m - i}{m} \mathfrak{R}_{m-i} (\mathcal{F}, \mathbf{D}) \\
& = \sum_{i=m-\lfloor m / 2 \rfloor}^{m} \binom{m}{i} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \\
& \leq \sum_{i=0}^{m} \binom{m}{i} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \\
& \leq \mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U}) \\
\end{aligned}
\]
The third part takes advantage of the assumption that \( p \leq 0.5\). We know that for \(\lfloor m / 2 \rfloor + 1 \leq i \leq m\), the assumption \(p \leq 0.5\) implies
\[
(1-p)^{m-i} p^i \leq \frac{p}{1-p} (1-p)^{i} p^{m-i}.
\]
Therefore, using the first part, we achieve
\[
\begin{aligned}
& \sum_{i=\lfloor m / 2 \rfloor + 1}^{m} \binom{m}{i} (1-p)^{m-i} p^i \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \\
& \leq \sum_{i=\lfloor m / 2 \rfloor + 1}^{m} \binom{m}{i} \frac{p}{1-p} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \\
& = \frac{p}{1-p} \sum_{i=\lfloor m / 2 \rfloor + 1}^{m} \binom{m}{i} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \\
& \leq \frac{p}{1-p} \sum_{i=0}^{m} \binom{m}{i} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \\
& \leq \frac{p}{1-p} \mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U}).\\
\end{aligned}
\]
By combining all the three parts above, we establish
\[
\begin{aligned}
& \mathfrak{R}_m(\mathcal{F}, \mathbf{D}) \\
& \leq \left[\sum_{i=0}^m \binom{m}{i} (1-p)^{i} p^{m-i} \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D}) \right] \\
& \quad + \left[ \sum_{i=0}^{\lfloor m / 2 \rfloor} \binom{m}{i} (1-p)^{m-i} p^i \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D})\right] \\
& \quad + \left[ \sum_{i=\lfloor m / 2 \rfloor + 1}^{m} \binom{m}{i} (1-p)^{m-i} p^i \frac{i}{m} \mathfrak{R}_{i} (\mathcal{F}, \mathbf{D})\right] \\
& \leq \left( 1 + 1 + \frac{p}{1-p} \right) \mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U})\\
& = \frac{2 - p}{1-p} \mathfrak{R}_m(\mathcal{F}, (1-p)\mathbf{D} + p \mathbf{U}).\\
\end{aligned}
\]
The proof for theorem \ref{thm:rerm} is therefore concluded. \end{proof}
The relation between \(\mathfrak{R}_n (\mathcal{F}, \mathbf{D})\) and \(\mathfrak{R}_m (\mathcal{F}, \mathbf{D})\) is achieved by the following theorem.
\begin{theorem}[Concentration inequality of subset Rademacher complexity]
\label{thm:subr}
Assume in solving the joint problem we obtained \(m\) idependently and identically distributed samples. Let the random number \(n\) represent the number of supervised sample obtained among these \(m\) joint samples with a proprtion probability of \(1-p\). Then, with probability at least \(1-\delta\), the following holds
\begin{equation}
\label{eq:subr}
\mathfrak{R}_n (\mathcal{F}, \mathbf{D}) \leq \frac{\mathfrak{R}_m (\mathcal{F}, \mathbf{D})}{1-p-\sqrt{\frac{\log(1/\delta)}{2m}}},
\end{equation}
for large enough \(m\) such that
\begin{equation}
1-p-\sqrt{\frac{\log(1/\delta)}{2m}} > 0.
\end{equation} \end{theorem}
\begin{proof}
Using lemma \ref{lem:sinr}, we only need to prove an upper bound for \(m/n\). Since we know that \(n\) follows a binomial distribution with mean \((1-p)m\), using Hoeffding's inequality \cite{H63} \cite{S74}, we can obtain
\[
\Pr\left[n \leq (1-p-\epsilon) m\right] \leq \exp(-2 \epsilon^2 m),
\]
or put differently,
\[
\Pr\left[\frac{m}{n} \leq \frac{1}{1-p-\epsilon} \right] \geq 1 - \exp(-2 \epsilon^2 m).
\]
The inequality is obtained by setting \(\delta = \exp(-2 \epsilon^2 m)\). The proof assumes that \(m\) is large enough such that
\[
1-p-\sqrt{\frac{\log(1/\delta)}{2m}} > 0.
\] \end{proof}
As a result, theorem \ref{thm:rbdm} can be obtained by directly combining theorem \ref{thm:rerm} and theorem \ref{thm:subr}.
\end{document} | arXiv | {
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\begin{document}
\newcommand{\Addresses}{{
\footnotesize
\textsc{Laboratory of Algebraic Geometry, Faculty of Mathematics \\ National Research University Higher School of Economics \\
119048 Moscow, Usacheva str., 6}\\
\textit{E-mail:} \texttt{kostyaloginov@gmail.com}
}} \title{Standard models of degree $1$ del Pezzo fibrations}
\begin{abstract} We construct a standard birational model (a model that has Gorenstein canonical singularities) for the three-dimensional del Pezzo fibrations ${ \pi \colon X \longrightarrow C }$ of degree $1$ and relative Picard number $1$. We also embed the standard model into the relative weighted projective space $\mathbb{P}_C (1,1,2,3)$. Our construction works in the $G$-equivariant category where~$G$ is a finite group. \end{abstract}
\section*{Introduction} The Minimal Model Program (the MMP for short, see \cite{Matsuki2002}, \cite{KMM-1987}) is a powerful tool that helps to understand the birational properties of algebraic varieties. The minimal category in which it works is the category $\mathcal C$ of the projective varieties with at worst terminal $\mathbb Q$-factorial singularities. The result of applying this program to a projective variety is either a minimal model, that is a variety $X\in \mathcal C$ whose canonical divisor class $K_X$ is nef, or a Fano\textendash Mori fibration, that is a variety $X\in \mathcal C$ admitting a contraction morphism $\pi \colon X \longrightarrow B$ whose fibers are of positive dimension, the anti-canonical divisor class $-K_X$ is relatively ample, and $\rho (X/B) = 1$.
We will focus on the three-dimensional case. In this case, the base $B$ of the Fano\textendash Mori fibration $\pi \colon X \longrightarrow B$ can be of dimension $0$, $1$ or $2$. If $\dim B=0$ then $X$ is a (possibly singular) Fano variety. It is known that Fano varieties lie in a finite number of algebraic families. However, they are classified only in the smooth case. In the singular case there are partial classificational results (see e.g. \cite{Prokhorov-2016-2}).
If $\dim B=2$ then a general fiber of $\pi$ is a non-singular rational curve. In this case, $\pi \colon X \longrightarrow B$ is called a \textit{$\mathbb Q$-conic fibration}. It is known that in this case there exists a standard model, that is a $\mathbb Q$-conic fibration $\pi' \colon X' \longrightarrow B'$ where $X'$ and $B'$ are non-singular, $X'$ is fiberwise birationally equivalent to $X$, and $\rho (X'/B') = 1$ (see \cite{Sarkisov}).
Finally, if $\dim B=1$ then a general fiber of $\pi$ is a non-singular del Pezzo surface. In this situation the fibration $\pi \colon X \longrightarrow B$ is called a \textit{$\mathbb{Q}$-del Pezzo fibration}. Standard models of such fibrations were considered in the work of A.~Corti \cite{Corti-1996}, see also \cite{Ko-1997}.
For the applications to the problem of classification of the finite subgroups in the Cremona group (see e. g. \cite{Prokhorov-Shramov}), as well as for birational classification of varieties over algebraically non-closed fields one should change the category $\mathcal C$. We will work with the varieties defined over an arbitrary field of characteristic $0$ that admit an action of a finite group $G$. In this case, we can apply the $G$-equivariant Minimal Model Program (the $G$-MMP, see \cite[2.18, 2.19]{KM-1998}, \cite[0.3.14]{Mori-1988}). Again, a final product of applying this program can be either a $G$-minimal model, or a $G$-Fano\textendash Mori fibration. For a three-dimensional $G$-Fano\textendash Mori fibration $\pi \colon X \longrightarrow B$, as in the ``classical'' situation, we have three possibilities: \begin{itemize} \item $G\mathbb{Q}$-Fano varieties. In general, this class is poorly understood. Partial results can be found in the works \cite{Prokhorov-2015}, \cite{Prokhorov-2016}, \cite{Prokhorov-2016-2}, \cite{Prokhorov-Shramov}. \item $G\mathbb{Q}$-conic fibrations. In this case existence of the standard model is proven in the work \cite{Avilov-conic-r}. \item $G\mathbb{Q}$-del Pezzo fibrations (see Definition \ref{defin-1}). In the present work, we construct standard models of $G\mathbb{Q}$-del Pezzo fibrations of degree $1$. \end{itemize}
The following theorems are the main results of this paper (the necessary definitions are given in \S \ref{section-1}).
\begin{thmb} \label{thma-A} Let $X$ be a projective three-dimensional $G$-variety and $C$ be a $G$-curve. Let $\pi \colon X \longrightarrow C$ be a proper $G$-morphism whose generic fiber is a non-singular degree $1$ del Pezzo surface $X_\eta$, and $\mathrm{Pic}^G (X/C)$ is generated by $-K_X$ and $G$-components of fibers of $\pi$. Then there exists a \emph{Gorenstein model}, that is a generalised $G$-del Pezzo fibration $\sigma \colon Y \longrightarrow C'$ such that \begin{enumerate} \item the following diagram is commutative \[ \xymatrix{ X \ar@{-->}[r]^\chi\ar@{->}[d]^{\pi} & Y\ar[d]_{\sigma} \\ C\ar@{-->}[r]&C' } \] where $\chi$ is a birational $G$-equivariant map, \item $Y$ has only $\mathbb{Q}$-factorial canonical Gorenstein singularities, \item $C'$ is non-singular and projective, \item $\chi$ induces an isomorphism between $X_\eta$ and $Y_{\eta'}$ where $Y_{\eta'}$ is the generic fiber of $\sigma$, \item any fiber of $\sigma$ is reduced and irreducible. \end{enumerate}
\end{thmb}
\begin{cor} \label{thma-B} If $\pi: X \longrightarrow C$ is a $G\mathbb{Q}$-del Pezzo fibration of degree $1$ then it has a model with at worst $\mathbb{Q}$-factorial canonical Gorenstein singularities, with irreducible fibers and with the same generic fiber as $\pi$. \end{cor}
\begin{thmb} \label{thma-C} Let $\sigma \colon Y \longrightarrow C $ be a generalised $G$-del Pezzo fibration of degree~$1$, and let Y have only Gorenstein canonical singularities. Then $Y$ admits an embedding over~$C$ into the relative weighted projective space $$Y \hookrightarrow \mathbb{P}_C (1,1,2,3).$$ \end{thmb}
We prove these results in several steps. First, in \S \ref{section-1}, Proposition \ref{claim-4}, we give the main definitions and prove some preliminary results. Second, in \S \ref{section-2} we establish some rigidity properties for del Pezzo surfaces and del Pezzo fibrations of degree $1$. Third, in \S \ref{section-3}, Proposition \ref{claim-3}, starting from a del Pezzo fibration of degree~$1$ as in Theorem \ref{thma-A}, we show that $X$ is $G$-birational over $C$ to a $G\mathbb{Q}$-del Pezzo fibration of degree $1$. After that, in \ref{theorem-7} we construct a canonical model of $X$, that is a fibration $\bar{\pi} \colon \bar{X} \longrightarrow C$ which is $G$-birational to $X$ over $C$ and such that the pair $(\bar{X}, | -K_{\bar{X}}+\bar{\pi}^*\bar{D} |)$ is canonical for some $\bar{D}$. Next, in \S \ref{section-4}, Theorem \ref{theorem-10}, we construct a Gorenstein model which proves Theorem~\ref{thma-A}. After that, in $\S \ref{section-5}$, we recall some facts on the anticanonical algebra of degree $1$ del Pezzo surfaces. Finally, in \S \ref{section-6}, we embed a Gorenstein $G$-fibration $Y$ into $\mathbb{P}_C (1,1,2,3)$ proving Theorem \ref{thma-C}.
\
The author is grateful to his scientific advisor Yu. Prokhorov for posing the problem and constant support in writing the paper, and to A. Kuznetsov and C. Shramov for many helpful discussions.
\section{Preliminaries} \label{section-1}
We work over a field of characteristic $0$, not necessarily algebraically closed. We also fix a finite group $G$. Recall the standard definitions.
\begin{defin} \label{defin-G1} An algebraic variety $X$ is called a \emph{$G$-variety} (or \emph{a variety with an action of the group $G$}) if there exists a homomorphism (not necessarily injective) $${\phi: G \longrightarrow \mathrm{Aut}\ X}.$$
\end{defin}
\begin{defin} \label{defin-G2} A rational map of $G$-varieties $f: X \dashrightarrow Y$ is called a \emph{$G$-map} if $f$ commutes with the action of the group $G$ on $X$ and on $Y$. If the map $f$ is birational we say that $X$ is \emph{$G$-birational} to $Y$. If the map $f$ is a morphism then $f$ is called a \emph{$G$-morphism}. \end{defin}
\begin{defin} \label{defin-G3} A $G$-variety $X$ is called \emph{$G\mathbb{Q}$-factorial} if any $G$-invariant Weil divisor is $\mathbb{Q}$-Cartier. \end{defin}
Let us fix the notation. For a vector space (or a vector bundle) $A$ we denote its $k$-th symmetric power by $S^k A$, and its full symmetric power by $S^\bullet A$. The base locus of a linear system~$\mathscr{L}$ on~$X$ we denote by $\mathrm{Bs} \ \mathscr{L}$. Let $\pi: X \longrightarrow C$ be a proper morphism onto the variety $C$. By $\eta$ we denote the generic point of $C$, and by $X_\eta$ the generic fiber of $\pi$. By a general fiber we mean a fiber over some closed point in an open subset $U \subset C$. We denote by $Z_1(X/C)$ a free abelian group generated by reduced irreducible curves which are mapped to points by $\pi$. There is a natural intersection pairing $$ (\ ,\ ) : \mathrm{Pic} (X) \times Z_1 (X/C) \longrightarrow \mathbb{Z}.$$ We put $\mathrm{Pic} (X/C) = \mathrm{Pic} (X) / \equiv$ where $\equiv$ is the numerical equivalence with respect to the pairing introduced above. We denote by $\rho (X/C)$ the dimension of $\mathrm{Pic} (X/C) $ and by $\rho^G (X/C)$ the dimension of the $G$-invariant subspace $\mathrm{Pic}^G (X/C)$.
Let~$D$ and~$D'$ be divisors on $X$. We write $D \simC D'$ if $D \sim D' + \pi^* E $ for some Cartier divisor $E$ on $C$. If $\mathbb{Q}$-divisors on $X$ are $\mathbb{Q}$-linearly equivalent we write $D \simQ D'$. Finally, we write $D \simQC D'$ if $\mathbb{Q}$-divisors $D$ and $D'$ on $X$ are $\mathbb{Q}$-linearly equivalent over $C$, that is $D \simQ D' + \pi^* E$.
We introduce the main definitions.
\begin{defin} \label{defin-delpezzo} \emph{A del Pezzo surface} $S$ is a (not necessarily normal) projective surface that has at worst Gorenstein singularities and whose anti-canonical divisor class~$-K_S$ is ample. \emph{The degree} of a del Pezzo surface $S$ is the number $(-K_S)^2$. \end{defin}
\begin{defin} \label{defin-1} Let $X$ be a three-dimensional normal projective $G$-variety with at worst terminal singularities and let $C$ be a non-singular $G$-curve. Assume that
\begin{enumerate} \item\label{defin-1a} $X$ is $G\mathbb{Q}$-factorial;
\item \label{defin-1b} there exists a projective $G$-morphism with connected fibers $\pi \colon X \longrightarrow C$;
\item \label{defin-1c} $-K_X$ is $\pi$-ample;
\item \label{defin-1d} $\pi$ is an extremal contraction, that is $\rho^{G} (X / C) = 1$. \end{enumerate}
Then $\pi \colon X \longrightarrow C$ is called a \emph{$G\mathbb{Q}$-del Pezzo fibration}. \emph{The degree of a $G\mathbb{Q}$-del Pezzo fibration} is the degree of its generic fiber $X_\eta$. Since $X$ is terminal $X_\eta$ is a non-singular del Pezzo surface. If $X$ has Gorenstein singularities we call $\pi \colon X \longrightarrow C$ a \emph{$G$-del Pezzo fibration}.
\end{defin}
\begin{defin} \label{defin-2} We call $\pi \colon X \longrightarrow C$ a \emph{generalised $G\mathbb{Q}$-del Pezzo fibration} if $X$ has at worst canonical singularities and the conditions \ref{defin-1b} and \ref{defin-1c} of the definition \ref{defin-1} are satisfied. Notice that in this case the generic fiber $X_\eta$ can be singular. \end{defin}
We will work with the anticanonical linear system on $X$.
\begin{proposition}[{\cite[2.17, 2.19]{Alexeev-1994ge}}] \label{claim-4}
Let $\pi \colon X \longrightarrow C$ be a $G\mathbb{Q}$-del Pezzo fibration. Then there exists a divisor $D$ (which we may assume to be $G$-invariant) on $C$ such that the linear system $\mathscr{H}=| -K_X+\pi^*D |$ on $X$ is non-empty and has no fixed components. \end{proposition}
We will use the language of singularities of the linear systems, introduced in \cite[1.8]{Alexeev-1994ge} and \cite{Corti-1995}. It is easy to see that for a $G\mathbb{Q}$-del Pezzo fibration the restriction of the linear system~$\mathscr{H}$ chosen above to a general fiber of the morphism $\pi$ is surjective. If the degree of a general fiber is $1$ then the linear system $\mathscr{H}$ has one simple base-point on it. Hence it is easy to see that the pair $(X, \mathscr{H})$ is canonical outside a finite number of fibers. Our aim is to construct a canonical model for the pair $(X, \mathscr{H})$. We will need the following lemmas:
\begin{lem} \label{lem-14} Let $\pi \colon X \longrightarrow C$ be a generalised Gorenstein $G$-del Pezzo fibration (that is, $K_X$ is Cartier) of degree $d$ with at worst canonical singularities. Let $F$ be the scheme fiber over a closed point. Write $F=\sum m_i F_i$ where $F_i$ are irreducible components. Then $\sum m_i \le d$. In particular, if $d=1$, then any geometric fiber is reduced and irreducible. \end{lem} \begin{proof}
We may assume the ground field to be algebraically closed. By the adjunction formula $ K_X |_F = K_F $, and we have $$d=(-K_F)^2=(-K_X)^2\cdot F = (-K_X)^2 \cdot \sum m_i F_i =\sum m_i (-K_F |_{F_i})^2 \geq \sum m_i,$$ the last inequality holds since $-K_F$ is an ample Cartier divisor on $F_i$. \end{proof}
\begin{lem}[{\cite[1.22]{Alexeev-1994ge}}] \label{lem-5}
Suppose that the pair $(X, \mathscr{H})$ is terminal where $\mathscr{H}$ is a linear system without fixed components. Then $\mathscr{H}$ has at worst isolated non-singular base-points $P_i$ such that $\mathrm{mult}_{P_i} \mathscr{H} = 1$. \end{lem}
\begin{lem}[{\cite[1.23]{Alexeev-1994ge}}] \label{lem-6}
Suppose that $X$ has only terminal singularities and the pair $(X, \mathscr{H})$ is canonical. Then in the neighbourhood of any base-point $P$ of $\mathscr{H}$ we have $\mathscr{H} \sim -K_X$. \end{lem}
\section{Birational rigidity} \label{section-2}
The following lemma is a variant of birational rigidity of degree $1$ del Pezzo surfaces (cf. \cite[1.6]{Is-1996}).
\begin{lem} \label{lem-15} Let $S$ be a non-singular del Pezzo surface of degree $1$, and let $T$ be either a normal del Pezzo surface of degree $d$ or a conic bundle over a non-singular curve. Let $f: S \dashrightarrow T$ be a birational map between them. Put $$\mathscr{H} = \begin{cases}
| - K_T | ,\ \text{if } d \geq 3, \\
| - 2 K_T | ,\ \text{if } d =2, \\
| - 3 K_T |,\ \text{if } d =1, \\
| nF |, \text{where $F$ is the class of a fiber if T is a conic bundle, $n\geq 1$.} \end{cases}$$
Suppose that $\mathscr{L} : = f^{-1}_* \mathscr{H} \subset | - a K_S | $ for some positive integer $a$. Then $T$ is a degree $1$ del Pezzo surface and $f$ is an isomorphism. \end{lem} \begin{proof} All the properties in the claim can be checked over an algebraically closed field, so we assume that. Consider a resolution of the points of indeterminacy of $f$ \[ \xymatrix{ & Z \ar[dl]_g \ar[dr]^h & \\ S \ar@{-->}[rr]^{f} & & T } \] Consider the case where $T$ is a del Pezzo surface. Since in this case $\mathscr{H}$ is very ample (see Proposition \ref{claim-12}) $h$ is the blow up of the base locus of $\mathscr{L}$. It is easy to check (see \cite[1.3.2]{Is-1996}) that for the strict transform $\widetilde{\mathscr{L}} := g^{-1}_* \mathscr{L} $ we have $$ \widetilde{\mathscr{L}}^2 = \mathscr{L}^2 - \sum r_i ^2$$ $$ K_Z \cdot \widetilde{\mathscr{L}} = K_S \cdot \mathscr{L} + \sum r_i$$ where $r_i$ are the multiplicities of the base points of $\mathscr{L}$ (including infinitely near ones). Notice that $\widetilde{\mathscr{L}} = h^{-1}_* \mathscr{H}$, and since $\mathscr{H}$ is base point free we get $$ \mathscr{H}^2 = \widetilde{\mathscr{L}}^2 = \mathscr{L}^2 - \sum r_i ^2 = a^2 - \sum r_i^2 $$ $$ \mathscr{H} \cdot K_T = K_Z \cdot \widetilde{\mathscr{L}} = - a + \sum r_i $$
Consider the cases:
1) $d\geq 3$. We get $$a^2 = \sum r_i^2 + d, \ \ \ a = \sum r_i + d.$$ It follows that $r_i= 0$ for any $i$ and $d = 1$ which contradicts the assumption.
2) $d = 2$. We get $$a^2 = \sum r_i^2 + 8, \ \ \ a = \sum r_i + 4.$$ These equations easily lead to a contradiction.
3) $d = 1$. Then we get the equations $$a^2 = \sum r_i^2 + 9, \ \ \ a = \sum r_i + 3.$$ We deduce that all the $r_i = 0$, hence the map $f$ is a morphism and $a = 3$. Thus $f$ is a contraction of the exception divisor $E$. Hence $$K_S = f^* K_T + E$$ On the other hand, $aK_S = f^*K_T$. Hence $(1 - a)K_S = E$ which is absurd. We see that there are no contracted curves and $f$ is an isomorphism.
Now consider the case when $T$ is a conic bundle over a non-singular curve. That is there is a morphism $\tau: T \longrightarrow B$ whose general fiber is a non-singular conic.
We have $\mathscr{H}^2 = (nF)^2 = 0$ and $\mathscr{H}\cdot K_T = nF\cdot K_T = -2n$ by adjunction. Considering the resolution of the base points of $\mathscr{L}$ we can write the formulas as above and get $$ a^2 = \sum r_i^2, \ \ \ a = \sum r_i +2n. $$ Again, it is easy to derive a contradiction.
We see that only the case 3) can occur, and the claim follows. \end{proof}
The next proposition gives a generalization of the rigidity property to degree~$1$ del Pezzo fibrations.
\begin{proposition} \label{claim-19} Let $X$ be a projective three-dimensional $G$-variety and $C$ be a non-singular $G$-curve. Let $\pi \colon X \longrightarrow C$ be a $G$-morphism whose generic fiber is a non-singular degree $1$ del Pezzo surface $X_\eta$, and $\mathrm{Pic}^G (X/C)$ is generated by $-K_X$ and $G$-components of fibers. Suppose that $X$ is $G$-birational over $C$ to a generalised $G\mathbb{Q}$-Fano\textendash Mori fibration $\pi': X' \longrightarrow B$ over $C$, that is the following diagram is commutative \[ \xymatrix{ X \ar@{-->}[r]^\chi \ar[d]^{\pi} & X'\ar[d]_{\pi'} \\ C& \ar@{->}[l]_{\psi} B } \] Then $\psi$ is an isomorphism, $X'$ is a $G\mathbb{Q}$-del Pezzo fibration of degree $1$ and $X_\eta \simeq X'_\eta$. Here $X'_\eta$ is the generic fiber of $X'$ over $C$. \end{proposition} \begin{proof} The map $\chi$ induces a birational map $f: X_\eta \dashrightarrow X'_\eta$. Suppose that $B$ is a curve. Since the diagram is commutative, the fibers of $\pi$ and $\pi'$ are connected and $B$ is normal we get that $\psi$ is an isomorphism. Then $X'_\eta$ is a del Pezzo surface over the function field of $C$. We want to apply Lemma \ref{lem-15}.
First consider the case when $X'_\eta$ is a degree $1$ del Pezzo surface. By adjunction we have $-3K_{X'} |_{X'_\eta} = - 3 K_{X'_\eta}$. The class of $-3K_{X'}$ is $G$-invariant in $\mathrm{Pic} (X'/C)$. Since $\chi$ is a $G$-map, the class of $\chi^{-1}_* ( -3K_{X' })$ is also $G$-invariant in $\mathrm{Pic} (X/C)$. Hence it has the form $-a K_X + D$ for some $a$ and some divisor $D$ concentrated in the fibers. Since $(-aK_{X} + D) |_{X_\eta} = - a K_{X_\eta}$ we have $f^{-1}_* ( 3 K_{X'_\eta} ) = - a K_{X_\eta}$, and the conditions of Lemma \ref{lem-15} are satisfied. Thus $f$ is an isomorphism.
If $X'_\eta$ is a del Pezzo surface of degree $d\geq 2$ a similar argument yields a contradiction.
Now suppose that $B$ is a surface in which case $X'_\eta$ as a surface over the function field of $C$ admits a conic bundle structure. Consider the divisor class $F_\eta$ on $X'_\eta$ corresponding to the generic fiber of the map $\pi'|_{X'_\eta}: X'_\eta \longrightarrow B_\eta$. Clearly there is a divisor $F$ on $X'$ such that $F|_{X'_\eta} = F_\eta$. Put $F'=\sum_{g\in G} g. F$. Since $G$ sends a fiber of $\pi'$ to a fiber of $\pi'$ we have $(g.F)|_{X'_\eta} = F_\eta$ for any $g \in G$. Hence $F'$ is $G$-invariant in $\mathrm{Pic}\ X$ and $F'|_{X'_\eta} = n F_\eta$ where $n = |G|$. Since $\chi$ is a $G$-map, $\chi^{-1}_*F'$ is $G$-invariant as well, so it has the form $-aK_X + D$ for some $a$ and some divisor $D$ concentrated in the fibers of $\pi$. Hence $f^{-1}_* (n F_\eta) = -aK_{X_\eta}$, so the conditions of Lemma~\ref{lem-15} are satisfied, and we get a contradiction. \end{proof}
\section{Canonical model} \label{section-3}
First we construct a model that is a $G\mathbb{Q}$-del Pezzo fibration in the sense of Definition \ref{defin-1}.
\begin{proposition} \label{claim-3} Let $X$ be a projective three-dimensional $G$-variety and $C$ be a $G$-curve. Let $\pi \colon X \longrightarrow C$ be a proper $G$-morphism whose generic fiber is a non-singular degree $1$ del Pezzo surface $X_\eta$, and $\mathrm{Pic}^G (X/C)$ is generated by $-K_X$ and $G$-components of fibers. Then there exists a $G\mathbb{Q}$-del Pezzo fibration $\pi' \colon X' \longrightarrow C'$ with the generic fiber $X'_\eta$ isomorphic to $X_\eta$ such that the following diagram is commutative: \[ \xymatrix{ X\ar@{-->}[r]^\chi\ar@{->}[d]^{\pi} & X'\ar[d]^{\pi'} \\ C \ar@{-->}[r] & C' } \] \end{proposition}
\begin{proof} Let $j:~C_1 \longrightarrow C$ be a normalization. Consider the following commutative diagram \[ \xymatrix{ X \ar@{->}[d]^{\pi} \ar@{-->}[dr]^{\pi_1} & \\ C \ar@{<-}[r]^{j} & C_1 } \] where the map $\pi_1:= \pi \circ i^{-1}$ may be not defined over some points of $C_1$.
Let $X_1$ be a $G$-equivariant resolution of singularities (see \cite[Theorem 0.1]{AW-1997}) of $X$ and the indeterminacy points of $\pi_1$: \[ \xymatrix{ X\ar@{<-}[r]^{h} \ar@{->}[d]^{\pi} & X_1 \ar@{->}[d]^{\pi_2} \\ C \ar@{-->}[r] & C_1 } \]
Since a general fiber of $\pi_1$ is non-singular $h$ does not change it. Apply the $G$~-MMP (see \cite[0.3.14]{Mori-1988}) over $C_1$ to the variety $X_1$. We get the following diagram of $G$-maps where the map $g$ is a composition of flips and divisorial contractions \[ \xymatrix{ X_1 \ar@{-->}[r]^g \ar[d]^{\pi_2} & X'\ar[d]_{\pi'} \\ C_1& \ar@{->}[l]_{\psi} B } \]
Put $\chi = g \circ h^{-1}$. The map $\chi$ induces a birational map of the generic fibers $f: X_\eta \dashrightarrow X'_\eta$ where $X'_\eta$ is the generic fiber of the morphism $\psi \circ \pi'$. Then Proposition \ref{claim-19} shows that $f$ and $\psi$ are isomorphisms, and the claim follows. \end{proof}
Notice that the assumption on $\mathrm{Pic}^G (X/C)$ in the above proposition is a relaxation of property \ref{defin-1a} in the Definition \ref{defin-1}. Now we are ready to construct a canonical model.
\begin{thm} \label{theorem-7} Let $\pi \colon X \longrightarrow C$ be a $G\mathbb{Q}$-del Pezzo fibration of degree $1$. Then there exist a $G\mathbb{Q}$-del Pezzo fibration $\bar{\pi} \colon \bar{X} \longrightarrow C$ of degree $1$ with the generic fiber $\bar{X}_\eta \simeq X_\eta$ and a commutative diagram \[ \xymatrix{ X \ar@{-->}[rr]^{h}\ar[dr]^{\pi} && \bar{X} \ar[dl]_{\bar{\pi}} \\ & C & } \]
where the map $h$ is birational and the pair $(\bar{X}, \bar{\mathscr{H}})$ is canonical where ${\bar{\mathscr{H}}=| - K_{\bar{X}}+\bar{\pi}^*\bar{D} |}$ for some ample $G$-invariant divisor $\bar{D}$ on $C$. The pair $(\bar{X}, \bar{\mathscr{H}})$ is called a \emph{canonical model} of $(X, \mathscr{H})$. \end{thm}
\begin{proof}
We put $\mathscr{H}=| - K_X+\pi^*D |$ where $D$ is a $G$-invariant ample divisor on $C$ as in Proposition \ref{claim-4}. We may assume that the following map is surjective $$ \mathrm{H}^0 (X, \mathscr{O}_X (-K_X+\pi^*D)) \twoheadrightarrow \mathrm{H}^0 ( S, \mathscr{O}_S (-K_X+\pi^*D))=\mathrm{H}^0 (S, \mathscr{O}_S (-K_S)).$$ for a general fiber $S$ of the morphism $\pi$. Thus on $S$ the linear system $\mathscr{H}$ has only one simple base-point. There is a corresponding rational section of the morphism $\pi$ whose closure we will denote by $\Gamma$. Thus $\Gamma \subset \operatorname{Bs} \mathscr{H} $.
Let $g_1 \colon X_1 \longrightarrow X$ be the blow-up of the curve $\Gamma$. We denote its exceptional divisor by $E_1$. The linear system $\mathscr{H}_1 = g_*^{-1} \mathscr{H}$ is base-point free on a general fiber of the morphism $\pi_1$.
Let $g_2 \colon \widetilde{X} \longrightarrow X_1$ be a $G$-equivariant resolution of singularities of the pair $(X_1, \mathscr{H}_1)$. We have a commutative diagram \[ \xymatrix{ \widetilde{X} \ar[r]^{g_2} \ar[dr]_{\widetilde{\pi}} & X_1 \ar[d]^{\pi_1} \ar[r]^{g_1}& X \ar[dl]^{\pi} \\ & C & } \] We introduce the notation $$ g = g_2 \circ g_1, \ \ \widetilde{\mathscr{H}} = (g_2)_*^{-1} \mathscr{H}_1, \ \ \widetilde{E}_1 = (g_2)_*^{-1} E_1.$$
On a general fiber $g_2$ is an isomorphism. Hence all the exceptional divisors of the morphism $g_2$ are contained in a finite number of fibers of $\widetilde{\pi}$. Since the linear system~$\widetilde{\mathscr{H}}$ is base-point free, the pair $(\widetilde{X}, \widetilde{\mathscr{H} })$ has the same singularities as $\widetilde{X}$ itself. In particular, $(\widetilde{X}, \widetilde{\mathscr{H} })$ is canonical. Write $$ K_{\widetilde{X}}+\widetilde {\mathscr{H}}+\sum a_i \widetilde{E}_i=g^*(K_X+\mathscr{H}) \simC 0,\qquad (K_X+\mathscr{H}) \simC 0, $$ where $\widetilde{E}_i$ are the exceptional divisors of the morphism $g$. We have $a_i \in \mathbb{Z}$ since $K_X+\mathscr{H}$ is a Cartier divisor on $X$. By construction all the exceptional divisors $\widetilde{E}_i$ are contained in a finite number of fibers of the morphism $\widetilde{\pi} \colon \widetilde{X} \longrightarrow C$ except $\widetilde{E}_1$ for which we have $g_* \widetilde{E}_1=\Gamma$.
We run the $G$-MMP over $C$ for the pair ${( \widetilde{X}, (1 - \epsilon) \widetilde{\mathscr{H}})}$ for $0 < \epsilon \ll 1$.
Let $\widetilde{S}$ be a general fiber of the morphism $\widetilde{\pi}$. It is easy to see that the linear system $$ - (K_{\widetilde {X}}+(1 - \epsilon) \widetilde{\mathscr{H}} )|_{\widetilde{S}} = - \epsilon K_{\widetilde{S}}$$ is nef on $\widetilde{S}$. Hence the result of applying the $G$-MMP is a $G\mathbb{Q}$-Fano\textendash Mori fibration with the base $B$ over $C$. We have the following commutative diagram \[ \xymatrix{ \widetilde{X} \ar@{-->}[r]^{f}\ar[d]_{\widetilde{\pi}} & \bar{X}\ar[d]^{\bar {\pi}}\ar[dl]^{} \\ C & B \ar[l]^{\psi} } \] where $f$ is a composition of divisorial contractions and $K_{\widetilde {X}}+(1 - \epsilon) \widetilde{\mathscr{H}}$-flips. By Proposition \ref{claim-19} we see that $\bar{X}$ is a $G\mathbb{Q}$-del Pezzo fibration of degree $1$ over $B$ (and $B = C$) and $\bar{X}_\eta \simeq X_\eta$.
The pair $(\widetilde{X}, (1 - \epsilon) \widetilde{\mathscr{H}})$ is terminal, therefore the pair $(\bar{X}, (1 - \epsilon) \bar{\mathscr{H}})$ is terminal as well. In particular, $\bar{X}$ has terminal singularities. Since $1$ is not an accumulation point of the set of $3$-dimensional canonical thresholds (see \cite{Prokhorov-2008}) we can choose $\epsilon \ll 1$ such that the pair $(\bar{X}, \bar{\mathscr{H}})$ is canonical.
Since $f^{-1}$ does not contract divisors the following formula holds: \begin{equation} \label{eq:f1} K_{\bar{X}}+\bar{\mathscr{H}}+\sum a_i \bar{E}_i \simC 0, \end{equation} where $f_{*} \widetilde{E}_i = \bar{E}_i$. Thus,
\begin{equation} \label{eq:f2} \bar{\mathscr{H}}+\sum a_i \bar{E}_i \sim - K_{\bar{X}}+\bar{\pi}^*A \end{equation} for a $G$-invariant divisor $A$ on $C$. By construction $\bar{E_i}$ lie in the fibers of $\bar{\pi}$. Since~$\rho^G (\bar{X} / C) = 1$ any fiber of $\widetilde{\pi}$ is $G$-irreducible. Since $\bar{E}_i$ are $G$-irreducible we get that $m_i \bar{E}_i=\bar {\pi}^*(x_i)$ where $x_i \in C$ for some integer $m_i$. The case $m_i > 1$ corresponds to a multiple fiber of the morphism~$\bar{\pi}$. Adding if necessary some number of points $x_i$ to $A$ we may assume that in the equation \eqref{eq:f2} we have $0 \leq a_i < m_i$ for any $a_i$. Suppose that there exists such $i$ that~$a_i > 0$.
By construction we have $f_* \widetilde{E}_1 = \bar{\Gamma}$ where $\bar{\Gamma}$ is a curve that is a section of $\bar{\pi}$ and $\bar{\Gamma} \subset \mathrm{Bs}\ \bar{\mathscr{H}}$. By Lemma \ref{lem-5} in a neighbourhood of any base-point of $\bar{\mathscr{H}}$ we have $ - K_{\bar{X}} \sim \bar{\mathscr{H}}$. If $\bar{\pi}^* ( x_i ) = m_i \bar{E}_i$ then $m_i \bar{E}_i \cdot \bar{\Gamma}=1$ and $\bar{E}_i \cdot \bar{\Gamma}= \frac{1}{m_i}$. Let us consider the base-point $P \in \bar{E}_i \cap \bar{\Gamma}$. From the formula \eqref{eq:f1} it follows that $a_i \bar{E}_i$ is a Cartier divisor in a neighbourhood of $P$. Thus $a_i \bar{E}_i \cdot \bar{\Gamma} \in \mathbb{Z}$. On the other hand, \[ 0 < a_i \bar{E}_i \cdot \bar{\Gamma}=\frac{a_i}{m_i} < 1 \] since $0 < a_i < m_i$. The contradiction shows that \[ K_{\bar{X}}+\bar{\mathscr{H}} \simC 0. \] This completes the proof. \end{proof}
\section{Gorenstein model} \label{section-4} We need the following technical proposition that follows easily from Kodaira's Lemma (see e. g. \cite[0.3.5]{KMM-1987}).
\begin{proposition} \label{claim-8} Let $B$ be a nef big $\mathbb{Q}$-Cartier divisor on a normal projective variety~$X$ with at worst terminal singularities. Then there exists an ample $\mathbb{Q}$-Cartier divisor~$A$ and an effective $\mathbb{Q}$-Cartier divisor $N$ such that $B\simQ A+N$ and the pair~$(X, N)$ is terminal. \end{proposition}
We also use the construction of a terminal modification of a pair.
\begin{proposition}[{\cite[2.8]{Corti-1995}}] \label{claim-9} Let $X$ be a normal projective $G$-variety with at worst terminal singularities and let the pair $(X, \mathscr{H})$ be canonical. Then there exists a $G$-equivariant terminal modification, that is a $G$-variety $\bar{X}$ and a birational $G$-morphism $f \colon \bar{X} \longrightarrow X$ such that the pair $(\bar{X}, \bar{\mathscr{H}})$ is terminal where $\bar {\mathscr{H}} = f^{-1}_* \mathscr{H}$ and the following formula holds $$ K_{\bar{X}}+\bar {\mathscr{H}}=f^*(K_X+\mathscr{H}). $$ \end{proposition}
Now we construct a Gorenstein model.
\begin{thm} \label{theorem-10} Let $\pi \colon X \longrightarrow C$ be a $G\mathbb{Q}$-del Pezzo fibration of degree $1$. Then there exists a Gorenstein model, that is a generalised $G\mathbb{Q}$-del Pezzo fibration $\sigma \colon Y \longrightarrow C$ of degree $1$ such that $Y$ is $G$-birational to $X$ over $C$ and $Y$ has at worst canonical Gorenstein singularities. Moreover, the generic fiber $Y_\eta$ of $\sigma$ is non-singular, $Y_\eta \simeq X_\eta$, and the special fibers are reduced and irreducible. \end{thm}
\begin{proof}
By Theorem \ref{theorem-7} we may assume that the pair $(X, \mathscr{H})$ is canonical where $\mathscr{H} = | -K_X + \pi^*D|$ for some $G$-invariant ample Cartier divisor $D$ on $C$. Let $(\bar{X}, \bar{\mathscr{H}})$ be its $G$-equivariant terminal modification over $C$ (see Proposition \ref{claim-9}). By Lemma \ref{lem-5} the linear system $\bar{\mathscr{H}}$ has at worst isolated non-singular base-points $P_i$ such that $\mathrm{mult}_{P_i} \bar{\mathscr{H}} = 1$. Since $\bar{X}$ is terminal it has at worst isolated singular points. It follows that a general element of $\bar{\mathscr{H}}$ is a Cartier divisor.
The linear system $\bar{\mathscr{H}}$ is nef since it has no curves in its base locus. Restricting $\bar{\mathscr{H}}$ to a general fiber of the morphism ${\bar{\pi} \colon \bar{X} \longrightarrow C}$ we notice that for any $m\geq2$ the image of the map given by the linear system $| m \bar{\mathscr{H}} |$ is three-dimensional. Hence the linear system $ \bar{\mathscr{H}}$ is big. By Proposition \ref{claim-8} for some $\mathbb{Q}$-Cartier divisors $A$ and $N$ we have $$\bar{\mathscr{H}}\simQ A+N,$$ where $A$ is ample, $N$ is effective and the pair $(\bar{X}, N)$ is terminal. By the construction of $\bar{X}$ we have $K_{\bar{X}}+\bar{\mathscr{H}} \simC 0$, hence $$K_{\bar{X}}+N \simQC - A.$$
Thus, for any curve $Z \in \overline{\operatorname{NE}}(X / C)$ with $Z\cdot \bar{\mathscr{H}}=
0$ we have $Z \cdot (K_{\bar{X}}+N) < 0$. By Contraction Theorem \cite[Theorem~3.2.1]{KMM-1987} the linear system $|n \bar{\mathscr{H}}|$ gives a ({$G$-equivariant}) contraction morphism $g \colon \bar{X} \longrightarrow Y$ with connected fibers such that the variety $Y$ is projective over $C$ and the following diagram is commutative \[ \xymatrix{ \bar X \ar[rr]^{g} \ar[dr]^{\bar{\pi}} & & Y \ar[dl]_{\sigma} \\ & C & } \]
It also guarantees the existence of a $\sigma$-ample Cartier divisor $H$ on $Y$ such that $g^*{H}=\bar{\mathscr{H}}$. Since $\bar{\mathscr{H}}$ is ample on an open subset of $\bar{X}$ we see that $g(\bar{X})=Y$ is three-dimensional. Hence $g$ is birational.
Now we prove that $Y$ has Gorenstein canonical singularities. Since by construction of $\bar{X}$ we have $K_{\bar{X}}+\bar{\mathscr{H}}=\bar{\pi}^*D$ for some ample divisor $D$ on $C$ and $ g_* (K_{\bar{X}}+\bar{\mathscr{H}})=K_Y+H$ then from the commutativity of the diagram $$K_Y+H=g_* \bar{\pi}^*D=\sigma^*D.$$ Thus $K_Y$ is a Cartier divisor. We have $$g^*K_Y=g^*(-H+\sigma^*D)=-\bar{\mathscr{H}}+ \bar{\pi}^*D=K_{\bar{X}}.$$ Hence $Y$ has canonical singularities.
The fact that $Y_\eta$ is non-singular and $Y_\eta \simeq X_\eta$ follows from Proposition \ref{claim-19}. The special fibers are reduced and irreducible due to Lemma \ref{lem-14}. It remains to show that $Y$ is $G\mathbb{Q}$-factorial. We show this in the next lemma.
\begin{lem} Let $\sigma \colon Y \longrightarrow C$ be a generalised Gorenstein del Pezzo fibration of arbitrary degree with at worst canonical singularities. Suppose that the fibers of $\sigma$ are reduced and irreducible. Then $Y$ is $\mathbb{Q}$-factorial. \end{lem} \begin{proof} Since $Y$ is canonical it has finitely many non-$\mathbb{Q}$-factorial points by \cite[3.4]{Reid-1985}. Let $F = \sum F_i$ be the union of the fibers of $\sigma$ that contain all the points that are not $\mathbb{Q}$-factorial. Write the excision exact sequence $$ \bigoplus_i \mathbb{Z} F_i \longrightarrow \mathrm{Cl} \ Y \longrightarrow \mathrm{Cl} \ U \longrightarrow 0 $$ where $U = Y \setminus F$. Since $U$ is $\mathbb{Q}$-factorial we have $\mathrm{Cl} \ U \otimes \mathbb{Q} = \mathrm{Pic} \ U \otimes \mathbb{Q}$. We see that any Weil divisor on $Y$ after subtracting some number of fibers $F_i$ and taking multiple comes from $\mathrm{Pic} \ U$. So it is enough to prove that any Cartier divisor on $U$ extends to a Cartier divisor on $Y$. But this follows from the commutative diagram with exact rows (see \cite[21.4.3]{EGA-1967}) \[ \xymatrix{ 0 \ar[r] & \mathrm{Pic} \ C \ar[r]^{\sigma^*} \ar[d] & \mathrm{Pic} \ Y \ar[r] \ar[d] & \mathrm{Pic} \ Y_\eta \ar[d]^{\simeq} \ar[r] & 0 \\
0 \ar[r] & \mathrm{Pic} \ V \ar[r]^{\sigma|_V^*} & \mathrm{Pic} \ U \ar[r] & \mathrm{Pic} \ Y_\eta \ar[r] & 0 } \] where $V = \sigma (U)$. Indeed, the left vertical arrow is clearly surjective, hence by the~Snake Lemma the middle vertical arrow is surjective as well.
\end{proof}
The theorem is proven. \end{proof}
\begin{remark} \label{remark-11} In the proof of the theorem by construction we have $$Y \simeq \operatorname{Proj}_C \bigoplus_{m\geq0} \sigma_* \mathscr{O}_Y (m H).$$ \end{remark}
\
\begin{proof}[Proof of Theorem \ref{thma-A}] Follows immediately from Propositions \ref{claim-4}, \ref{claim-3}, and Theorems \ref{theorem-7}, \ref{theorem-10}. \end{proof}
\section{Anticanonical algebra of a degree $1$ del Pezzo surface} \label{section-5}
We need the following results on the anticanonical algebra $$R=\bigoplus_{m\geq0} \mathrm{H}^0 (S, -mK_S)$$ of a degree $1$ del Pezzo surface $S$.
\begin{proposition} \label{claim-12} The following holds: \begin{enumerate} \item \label{claim-12a} $\dim \mathrm{H}^0 (S, -mK_S)=m(m+1)/2+1;$ \item \label{claim-12b}
the linear system $|-K_S|$ has one simple base-point; \item \label{claim-12c}
the linear system $|-2K_S|$ is generated by global sections; \item \label{claim-12d}
the linear system $|-3K_S|$ is very ample. \end{enumerate} \end{proposition}
\begin{proof} Let $S$ be normal. In this case, the statements \ref{claim-12a}--\ref{claim-12d} are well known if $S$ is non-singular. In the singular case see \cite[\S 4]{HW-1981}.
Let $S$ be non-normal and let $\alpha \colon T \longrightarrow S$ be its normalisation. According to \cite[1.1]{Reid-1994} (see also \cite[1.5]{AF-2003}) we have $$T \simeq \mathbb{P}^2, \quad \alpha^*(-K_S) \simeq \mathscr{O}_{\mathbb{P}^2} (1).$$
It is proven there that $\alpha$ is an isomorphism outside a (possibly singular) conic $Q$ on $\mathbb{P}^2$, and the morphism $\alpha|_Q$ is a $2$ to $1$ covering over a curve on $S$.
According to \cite[2.2]{AF-2003} for any $m\geq1$ the following holds: $$ \dim \mathrm{H}^0 (T, \alpha^*(-mK_S))=\dim \mathrm{H}^0 (S, -mK_S)+ m.$$ Hence we get $$\dim \mathrm{H}^0 (S, -mK_S)=\dim \mathrm{H}^0 (\mathbb{P}^2, \mathscr{O}_{\mathbb{P}^2} (m)) - m=(m+1)(m+2)/2 - m= m(m+1)/2+1.$$
This proves \ref{claim-12a}. The statements \ref{claim-12b} и \ref{claim-12d} are proven in \cite[1.2, 1.5(A)]{AF-2003}, see also \cite[4.10 (ii)]{Reid-1994}.
Since we have $| \sigma^*(- K_S) |=|
\mathscr{O}_{\mathbb{P}^2}(1)|$, all the elements of the linear system $| -K_S |$ are irreducible.
By the adjunction formula they are curves of arithmetic genus~$1$. Since they are rational these curves are either nodal or cuspidal cubic curves. For any $C \in |-3K_S|$ we have the following exact sequence $$ 0 \longrightarrow \mathscr{O}_S (-3K_S-C) \longrightarrow \mathscr{O}_S (-3K_S) \longrightarrow \mathscr{O}_C (-3K_S) \longrightarrow 0. $$ Since any linear system on the curve $C$ of degree $2$ or more does not have base-points and $\mathrm{H}^1(S, \mathscr{O}_S) = 0$, the statement \ref{claim-12c} follows. \end{proof}
\begin{corr} \label{cor-13} \begin{enumerate} \item\label{corr-13a}
The linear system $|-2K_S|$ defines a two-fold covering of the quadratic cone $$\phi=\phi_{|-2K_S|} \colon S \longrightarrow Q \subset \mathbb{P}^3.$$
\item\label{corr-13b} The algebra $R=\bigoplus_{m\geq0} \mathrm{H}^0 (S, -mK_S)$ is isomorphic to $ \mathbb{C} [x, y, z, w] / (f) $ where the degree of generators are $1, 1, 2, 3$ correspondingly and the relation $f$ has degree $6$. Hence, $S$ is isomorphic to the degree $6$ hypersurface in the weighted projective space $\mathbb{P} (1, 1, 2, 3)$. \end{enumerate} \end{corr} \begin{proof} Follows from the previous proposition in full analogy to the non-singular case, see \cite[8.3]{Dolgachev-2012}. \end{proof}
\section{The relative projective space} \label{section-6}
We put $$A_m=\sigma_* (\mathscr{O}_Y (- m K_Y ) ), \quad m\geq 0, $$ $$ A=\bigoplus_{m=0}^{\infty} A_m.$$
By Proposition \ref{claim-12} the sheaf $A_m$ on $C$ is a vector bundle of rank $m(m+1)/2+1$.
\begin{remark} \label{remark-15} We can restrict the sheaf of algebras $A$ on a fiber of the morphism $\sigma$ and apply Corollary \ref{cor-13} to show that $A$ is generated by its components of degree~$\leq 3$. \end{remark}
We will construct the relative weighted projective space $\mathbb{P}_C (1,1,2,3)$, that is a variety which is projective over $C$ and which has $\mathbb{P}(1,1,2,3)$ as a fiber over any point of $C$. We will also construct an embedding over $C$ \[ \xymatrix{ Y \ar[rr]^{i} \ar[dr]^{\sigma} & & \mathbb{P}_C (1,1,2,3) \ar[dl] \\ & C & } \]
We denote the cokernel of the natural inclusion $\alpha \colon S_2 A_1 \longrightarrow A_2$ by $G_2$. We get an exact sequence \begin{equation}\label{equation-2} 0 \longrightarrow S^2 A_1 \longrightarrow A_2 \longrightarrow G_2 \longrightarrow 0. \end{equation}
Consider the multiplication map $\mu \colon A_1 \otimes A_2 \longrightarrow A_3$. Let $V = \operatorname{Ker} \mu$, $G_3 = \operatorname{Coker} \mu$. It is easy to check fiberwise that $V$ и $G_3$ are vector bundles of rank $2$ and $1$ correspondingly. Hence we get an exact sequence \begin{equation}\label{equation-3} 0 \longrightarrow (A_1\otimes A_2)/V \longrightarrow A_3 \longrightarrow G_3 \longrightarrow 0 \end{equation}
Consider a commutative diagram of natural maps \[ \xymatrix{ S^\bullet (A_1 \oplus A_2 \oplus A_3) \ar[r] & A \\ \ar[u] S^\bullet (A_1 \oplus A_2 \oplus ((A_1\otimes A_2)/V)) \ar[ur] & } \]
By Remark \ref{remark-11} we have $Y=\operatorname{Proj}_C A$. We define the relative projective spaces $$\mathbb{P}_C (1^2, 2^4, 3^7) = \mathrm{Proj}_C \ S^\bullet (A_1 \oplus A_2 \oplus A_3 )$$ $$ \mathbb{P}_C (1^2, 2^4, 3^6) = \mathrm{Proj}_C \ S^\bullet (A_1 \oplus A_2 \oplus ((A_1\otimes A_2)/V) ) ) $$
Thus we obtain a commutative diagram of varieties over $C$ \[ \xymatrix{ \mathbb{P}_C (1^2, 2^4, 3^7) \ar@{-->}[d]^{p} & \ar[l]_-{j} Y \ar@{-->}[dl]^{q} \\ \mathbb{P}_C (1^2, 2^4, 3^6) & } \] where $q=p \circ j$. According to Remark \ref{remark-15} the map $j$ is a fiberwise embedding over~$C$.
\begin{proposition} \label{claim-16} The image $ q (Y) $ is a quadratic cone $ \mathbb{P}_C (1,1,2)$ over $C$, that is a projective variety over $C$ such that a fiber over any point $x \in C$ is isomorphic to $ \mathbb{P} (1,1,2)$. Moreover, the following diagram is commutative \[ \xymatrix{ \mathbb{P}_C (1^2, 2^4, 3^7) \ar@{-->}[d]^{p} & \ar[l]_-{j} \ar[d]_{v} Y \ar[dl]_{q} \\ \mathbb{P}_C (1^2, 2^4, 3^6) & \ar[l]_-{u} \mathbb{P}_C (1,1,2) } \] where $u$ is a fiberwise embedding, $v$ is a fiberwise two-fold covering and $p$ is a projection from a point. In particular the map $q$ is a morphism.
\end{proposition}
\begin{proof} On the fiber over any point $x\in C$ the sequences \eqref{equation-2} and \eqref{equation-3} split as the sequences of vector spaces. We have the following non-canonical isomorphisms \begin{eqnarray} \label{equation-4} (A_2)_x &\simeq& (S^2 A_1 \oplus G_2)_x \\ \label{equation-5} (A_3)_x &\simeq& (S^3 A_1 \oplus (A_1 \otimes G_2) \oplus G_3)_x \\ \label{equation-6} ( (A_1 \otimes A_2) /V )_x &\simeq& (S^3 A_1 \oplus (A_1 \otimes G_2) )_x \end{eqnarray}
On the other hand, over a point $x\in C$ we have $$ \operatorname{Proj} S^\bullet (A_1 \oplus G_2)_x \simeq \mathbb{P} (1,1,2)$$
Using the isomorphisms \eqref{equation-4}--\eqref{equation-6} over a point $x\in C$ we construct a natural surjective map $$ \xymatrix@R=10pt{ s \colon S^\bullet (A_1 \oplus (S^2 A_1 \oplus G_2) \oplus (S^3 A_1 \oplus (A_1 \otimes G_2)))_x
\ar[r] \ar@{}[d]|*[@]{\simeq} & S^\bullet (A_1 \oplus G_2)_x,
\\ S^\bullet (A_1 \oplus A_2 \oplus ((A_1 \otimes A_2) /V)))_x & } $$ where $s$ identically maps $$s ( ( S^k A_1 )_x ) = (S^k A_1)_x,\ k\geq 1, \ \ s ( (G_2)_x ) = (G_2)_x$$ and $s$ extends to the corresponding tensor powers in the obvious way. Thus $s$ induces an embedding $u \colon \mathbb{P} (1,1,2) \longrightarrow \mathbb{P} (1^2, 2^4, 3^6)$.
The map $s$ can be extended to a commutative diagram \[ \xymatrix{ S^\bullet (A_1 \oplus (S^2 A_1 \oplus G_2) \oplus (S^3 A_1 \oplus (A_1 \otimes G_2) \oplus G_3))_x \ar[r]^-{s'} & A_x \\ S^\bullet (A_1 \oplus (S^2 A_1 \oplus G_2)_x \oplus (S^3 A_1 \oplus (A_1 \otimes G_2)))_x \ar[r]^-{s} \ar[u]^{z} \ar[ur] & S^\bullet (A_1 \oplus G_2)_x \ar[u]^{w} } \] Here the map $z$ is induced by the natural inclusion of vector spaces, and the map $s'$ is constructed in full analogy with the map $s$: $$ s'(S^k A_1)_x = ( S^k A_1 )_x, \ k\geq1,$$ $$s'(G_2)_x = ( G_2 )_x \subset ( S^2 A_1 \oplus G_2 )_x \simeq (A_2)_x \subset A_x$$ $$s'(G_3)_x = ( G_3 )_x \subset ( S^3 A_1 \oplus(A_1 \otimes G_2)) \oplus G_3 )_x \simeq ( A_3 )_x \subset A_x.$$
It is easy to check that $w$ is a fiberwise two-fold covering of the quadratic cone as in Corollary \ref{cor-13} and that the diagonal map in the diagram induces the map $q$. This proves the claim.\end{proof}
In the notation as above consider a variety $$Z = \overline{p^{-1} (u (\mathbb{P}_C (1,1,2)))} \subset \mathbb{P}_C (1^2, 2^4, 3^7).$$ Clearly it is projective over $C$.
\begin{proposition} \label{claim-17} Each fiber of $Z$ over $C$ is isomorphic to $\mathbb{P} (1,1,2,3)$. Moreover, there exists an embedding $i \colon Y \longrightarrow Z$ over $C$. \end{proposition}
\begin{proof} It is obvious that $Z$ has dimension $3$ over $C$ and that $j(Y) \subset C$. For the fiber $Y_x$ over a point $x\in C$ we construct a commutative diagram \[ \xymatrix{ \mathbb{P} (1^2, 2^4, 3^7) \ar@{-->}[d]^{p} & \ar[l]_-{r} \ar@{-->}[d]^{t} \mathbb{P}(1,1,2,3) \ar@{-->}[dl]^{q} & \ar[l]_-{i} \ar[dl]^-{v} Y_x \\ \mathbb{P} (1^2, 2^4, 3^6) & \ar[l]^-{u} \mathbb{P} (1,1,2) & } \] where $u, v, p$ are as in Proposition \ref{claim-16}, $i$ is an embedding, $t$ is a projection from the point $(0:0:0:1)$, and $q=u \circ t$.
Using the isomorphisms \eqref{equation-4}--\eqref{equation-6} we construct a natural surjective map $$ S^\bullet (A_1 \oplus A_2 \oplus A_3) \simeq S^\bullet (A_1 \oplus (S^2 A_1 \oplus G_2) \oplus (S^3 A_1 \oplus (A_1 \otimes G_2) \oplus G_3)) \longrightarrow S^\bullet (A_1 \oplus G_2 \oplus G_3) $$
This map induces an embedding $r$. We must show that $r(\mathbb{P}(1,1,2,3))$ is contained in $Z_x$ and hence must coincide with it. The variety $Z$ was defined as the closure of the preimage of $u(\mathbb{P}_C (1,1,2))$ under the map $p$. Thus, it is enough to show that there is a fiberwise inclusion $$(p \circ r) (\mathbb{P} (1,1,2,3)) \subset u (\mathbb{P} (1,1,2)).$$ But it follows from the commutativity of the following diagram \[ \xymatrix{ S^\bullet (A_1 \oplus (S^2 A_1 \oplus G_2) \oplus (S^3 A_1 \oplus (A_1 \otimes G_2) \oplus G_3))_x \ar[r] & S^\bullet (A_1 \oplus G_2 \oplus G_3)_x \ar[r] & A_x \\ S^\bullet (A_1 \oplus (S^2 A_1 \oplus G_2) \oplus (S^3 A_1 \oplus (A_1 \otimes G_2)))_x \ar[r] \ar[u] \ar[ur] & S^\bullet (A_1 \oplus G_2)_x \ar[u] \ar[ur] & } \qedhere \] \end{proof}
We denote the variety $Z$ by $\mathbb{P}_C (1,1,2,3)$
\begin{proof}[Proof of Theorem B] Follows immediately from Proposition \ref{claim-17}. \end{proof}
\def$'$} \def\mathbb#1{\mathbf#1{$'$} \def\mathbb#1{\mathbf#1}
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\begin{document}
\title{ Stability of a Nonlocal Traffic Flow Model\\ for Connected Vehicles\thanks{Submitted to the editors DATE. \funding{This work is supported in part by the NSF DMS-2012562, DMS-1937254 and ARO MURI Grant W911NF-15-1-0562. }}}
\headers{Stability of a Nonlocal Traffic Flow Model}{Kuang Huang and Qiang Du}
\author{Kuang Huang\thanks{Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027; {\tt kh2862@columbia.edu}} \and Qiang Du\thanks{Department of Applied Physics and Applied Mathematics, and Data Science Institute, Columbia University, New York, NY 10027; {\tt qd2125@columbia.edu}}}
\maketitle
\begin{center} {\small \color{purple} This work has been published in \emph{SIAM Journal on Applied Mathematics}, 82(1), 221-243. Please refer to the official publication for citation.} \end{center}
\begin{abstract}
The emerging connected and automated vehicle technologies allow vehicles to perceive and process traffic information in a wide spatial range. Modeling nonlocal interactions between connected vehicles and analyzing their impact on traffic flows become important research questions to traffic planners. This paper considers a particular nonlocal LWR model that has been studied in the literature. The model assumes that vehicle velocities are controlled by the traffic density distribution in a nonlocal spatial neighborhood. By conducting stability analysis of the model, we obtain that, under suitable assumptions on how the nonlocal information is utilized, the nonlocal traffic flow is stable around the uniform equilibrium flow and all traffic waves dissipate exponentially. Meanwhile, improper use of the nonlocal information in the vehicle velocity selection could result in persistent traffic waves. Such results
can shed light to the future design of driving algorithms for connected and automated vehicles. \end{abstract}
\begin{keywords} traffic flow, nonlocal LWR, connected vehicles, nonlocal gradient, nonlocal Poincare's inequality, global stability \end{keywords}
\begin{AMS} 35L65, \ 90B20, \ 35R09, \ 47G20 \end{AMS}
\section{Introduction}\label{sec:intro} \setcounter{equation}{0}
In transportation research, one of the central problems is to understand the collective behavior of moving vehicles. With the development of connected vehicle technology, vehicles on the road, such as those traveling on the highway, can be connected through some vehicle-to-vehicle (V2V) or vehicle-to-infrastructure (V2I) communication networks \cite{dey2016vehicle}. As a result, each vehicle perceives nonlocal information on the road. The enhanced access to traffic information brings new opportunities and challenges on many aspects of traffic flows, ranging from traffic management, communication infrastructure and protocols to vehicle design and control. Theoretical studies and modeling efforts are also in great need \cite{huang2020scalable}. On one hand, new models are imperative to study how nonlocal information affects traffic flows and to explore the emergence of new traffic phenomena; On the other hand, car manufacturers will face the problem of designing driving algorithms to guide connected vehicles. This is an interactive and iterative process: a good algorithm is expected to utilize nonlocal information to improve the traffic, and at the same time, a good model can guide the algorithm design.
On the macroscopic level, traffic flows on highways have been modeled via continuum descriptions and hyperbolic conservation laws \cite{lighthill1955kinematic,richards1956shock,payne1971model,aw2000resurrection} similar to models of continuum media. Our present study focuses on such continuum descriptions of the dynamics of vehicle densities on a ring road. The main aim of this work is a mathematical demonstration of how nonlocal information can be utilized to gain desired benefits. By conducting stability analysis of a nonlocal macroscopic traffic flow model, we offer evidence of traffic wave stabilization with proper usage of nonlocal vehicle density information in velocity control.
\subsection{Background mathematical models}\label{sec:background}
A common block of macroscopic traffic flow models is the continuity equation: \begin{align}
\partial_t\rho(x,t)+\partial_x\left(\rho(x,t)u(x,t)\right)=0,\label{eq:continuity} \end{align} that describes the conservation of vehicles, where $\rho(x,t)$ and $u(x,t)$ denote the aggregated traffic density and velocity and $x$ and $t$ are spatial and temporal coordinates. The Lighthill-Whitham-Richards (LWR) model \cite{lighthill1955kinematic,richards1956shock} is the most extensively used macroscopic traffic flow model. It assumes a fundamental relation: \begin{align}
u(x,t)=U\left(\rho(x,t)\right),\label{eq:relation} \end{align} between traffic density and velocity, meaning that the driving speed of a vehicle is determined only by the instantaneous density at the vehicle's current location. The function $U(\cdot)$ is also referred to as the \emph{desired speed function}. The LWR model follows from \eqref{eq:continuity}\eqref{eq:relation} as a scalar conservation law: \begin{align}
\partial_t\rho(x,t)+\partial_x\left(\rho(x,t) U\left(\rho(x,t)\right)\right)=0.\label{eq:lwr} \end{align} The LWR model \eqref{eq:lwr} may produce shock wave solutions even with smooth initial data. Such shock wave solutions qualitatively explain the formation and propagation of traffic jams.
\subsection{Nonlocal LWR model}\label{sec:nonlocal}
The main objective of this paper is to consider the asymptotic stability of a nonlocal extension to the LWR model \eqref{eq:lwr}, proposed by \cite{Blandin2016,goatin2016well}. The basic assumption underneath such a nonlocal model is that each vehicle perceives traffic density information in a road segment of length $\delta>0$ ahead of the vehicle's current location. The driving speed of the vehicle is then based on an weighted average of density within the road segment: \begin{align}
u(x,t)=U\left(\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds\right),\label{eq:nonlocal_relation} \end{align} where the nonlocal kernel $w_\delta(\cdot)$ characterizes the nonlocal effect. \eqref{eq:continuity}\eqref{eq:nonlocal_relation} lead to the following nonlocal LWR model: \begin{align}
\partial_t\rho(x,t)+\partial_x\left(\rho(x,t) U\left(\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds\right)\right)=0.\label{eq:nonlocal_lwr} \end{align}
In the existing studies, some theoretical and numerical results have been developed on the scalar nonlocal conservation law \eqref{eq:nonlocal_lwr}, see Section~\ref{sec:related_work} for a review. However, existing studies on the asymptotic stability of the model are still limited. Under some suitable assumptions, we show that the stability is closely related to the nonlocal kernel $w_\delta(\cdot)$. In particular, {we prove that the solution of the model exponentially converges to a constant density as $t\to\infty$ when the kernel $w_\delta(\cdot)$ is non-increasing and non-constant}. Meanwhile, a constant kernel may lead to traffic waves that persist in time.
To interpret the significance of the mathematical findings made in this work, let us note their connections to issues that are important in real traffic situation. In the traffic research community, it is widely recognized that the presence of traffic waves could result in
elevated risks for traffic safety, an increase in vehicle fuel consumption, as well as a reduction in total traffic throughput \cite{stern2018dissipation}. Thus, the dissipation of traffic waves and the stability of constant density states are features that can offer benefits to both drivers of individual vehicles and the whole traffic ecosystem. The particular mathematical results established here, in plain words, provide further evidence to the following natural principle when designing driving algorithms for connected vehicles: \emph{it can be beneficial to utilize nonlocal interactions between connected vehicles for traffic decisions, and while doing so, suitable forms and ranges of nonlocality should be adopted with nearby information deserving more attention.}
\subsection{Related work}\label{sec:related_work}
The nonlocal LWR model \eqref{eq:nonlocal_lwr} was first proposed in \cite{Blandin2016,goatin2016well}, where the existence, uniqueness and maximum principle of the weak entropy solution were proved using the Lax-Friedrichs numerical approximation. The entropy condition is adopted to ensure the solution uniqueness. In subsequent works, \cite{Chiarello2018} proved the same results for a generalized model of \eqref{eq:nonlocal_lwr}: \begin{align}
\partial_t\rho(x,t)+\partial_x\left(g(\rho(x,t)) U\left(\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds\right)\right)=0,\label{eq:generalize_1} \end{align} and \cite{Chalons2018} developed high-order numerical schemes to solve \eqref{eq:generalize_1}. In a related work, \cite{keimer2017existence} studied a family of nonlocal balance laws: \begin{align*}
\partial_t\rho(x,t)+\partial_x\left(\rho(x,t)U\left(\int_{a(x)}^{b(x)}w_1(x,y,t)\rho(y,t)\,dy\right)\right)=h(x,t), \end{align*} which include \eqref{eq:nonlocal_lwr} as a special case. The existence and uniqueness of the weak solution were proved using the method of characteristics and a fixed-point argument. The latter also leads to solution uniqueness without the use of the entropy condition.
In the existing studies, some analytic properties of the nonlocal LWR model \eqref{eq:nonlocal_lwr} were discussed. In terms of solution regularity, \cite{bressan2019traffic} showed that the solution of \eqref{eq:nonlocal_lwr} has the same regularity as the initial data when: (i) the nonlocal kernel $w_\delta(\cdot)$ is $\mathbf{C}^1$ smooth and non-increasing on $[0,+\infty)$ with the zero extension on $[\delta,+\infty)$; (ii) the desired speed function $U(\cdot)$ is $\mathbf{C}^2$ smooth and $U'\leq-c<0$ for some constant $c$. In contrast, the local LWR model \eqref{eq:lwr} can develop shock solutions from smooth initial data whenever the characteristics impinge each other. In terms of relations between the local and nonlocal models, one fundamental question is whether the solution of the nonlocal model \eqref{eq:nonlocal_lwr} converges to that of the local model \eqref{eq:lwr} when $\delta\to0$, i.e., the vanishing nonlocality limit. In \cite{colombo2019singular}, it was shown that such convergence is in general false with a demonstration based on an example associated with the desired speed function $U(\rho)=\rho$ and discontinuous initial data. Nevertheless, convergence results were given in \cite{keimer2019approximation} when $U(\cdot)$ is a decreasing function and the initial data is monotone. \cite{bressan2019traffic,bressan2020entropy} considered a special case where the nonlocal interaction range is infinite and the nonlocal kernel is exponential, which leads to $$u(x,t)=U\left(\int_0^\infty\delta^{-1} e^{-\frac{s}{\delta}}\rho(x+s,t)\,ds\right).$$ In this case, the nonlocal-to-local convergence as $\delta\to0$ was proved for uniformly positive initial data and any desired speed function $U(\cdot)$ that is $\mathbf{C}^2$ smooth and $U'\leq-c<0$ for some constant $c$. \cite{colombo2018blow} extended the convergence results for exponentially decaying kernels and a family of decreasing desired speed functions, but required the initial data to be uniformly positive and have no negative jumps. It is still an open problem what are sufficient and necessary conditions on the desired speed function, nonlocal kernel and initial data for the vanishing nonlocality limit to be true. In terms of asymptotic behavior, \cite{ridder2019traveling} gave a class of monotone stationary solutions of \eqref{eq:nonlocal_lwr} on an infinitely long road and showed that those solutions are asymptotic local attractors of \eqref{eq:nonlocal_lwr}. \cite{karafyllis2020analysis} studied a generalized model of \eqref{eq:nonlocal_lwr} where a nudging (or ``look behind'') term is added to the nonlocal velocity: \begin{align*}
u(x,t)=U\left(\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds\right)\tilde{U}\left(\int_0^{\tilde{\delta}}\rho(x-s,t)\tilde{w}_{\tilde{\delta}}(s)\,ds\right). \end{align*} Under the assumptions that: (i) the model is solved on a ring road; (ii) $U(\cdot)$ is decreasing and $\tilde{U}(\cdot)$ is increasing; (iii) $w_\delta(s)=1/\delta$; (iv) $\tilde{\delta}$ is the length of the ring road and $\tilde{w}_{\tilde{\delta}}(s)=(\tilde{\delta}-s)/\tilde{\delta}$, the local exponential stability of uniform equilibrium flows as $t\to\infty$ was proved.
Let us also briefly mention other relevant studies. The nonlocal LWR model \eqref{eq:nonlocal_lwr} has been generalized to the case for 1-to-1 junctions \cite{chiarello2019junction} and of multi-class vehicles \cite{chiarello2019non,chiarello2020lagrangian}. There are also nonlocal traffic flow models other than \eqref{eq:nonlocal_lwr}. In \cite{sopasakis2006stochastic}, a model based on Arrhenius ``look-ahead'' dynamics was proposed where the nonlocal velocity: \begin{align*}
u(x,t)=U\left(\rho(x,t)\right)\exp\left(-\int_0^\delta \rho(x+s,t)w_\delta(s)\,ds\right), \end{align*} \cite{Lee2015,Lee2019a,Lee2019b} analyzed shock formation criteria of the model; In \cite{chiarello2020micro}, a nonlocal extension to the traditional Aw-Rascle-Zhang model \cite{aw2000resurrection} was proposed and the micro-macro limit was demonstrated. More broadly, nonlocal models have been drawing increasing attention in our connected world \cite{du2019nonlocal}. Nonlocal conservation laws, in particular, have been studied in many other applications, e.g., pedestrian traffic \cite{Colombo2012,burger2020non}, sedimentation \cite{Betancourt2011} and material flow on conveyor belts \cite{Goettlich2014,rossi2020well}, see \cite{Colombo2016} for a review. \cite{du2012new,du2017nonlocal,du2017numerical} discussed nonlocal conservation laws inspired from discrete descriptions of local conservation laws. Some more analytical and numerical studies on nonlocal conservation laws can be found in \cite{Aggarwal2015,Amorim2015,Colombo2018,Goatin2019,Chiarello2019,Berthelin2019}.
\subsection{Main results}\label{sec:main_results}
Before the rigorous statement of the main results of this paper, let us specify the set-up of the model problem. First of all, we consider the problem on a ring road. Mathematically, we use the spatial domain $x\in[0,1]$ to represent the ring road and assume the periodic boundary condition for the equation \eqref{eq:nonlocal_lwr}. \begin{itemize}
\item[({\bf A1})] {$\rho(0,t)=\rho(1,t)$,\quad $\forall t\geq0$}. \end{itemize}
The periodicity assumption is common in stability analysis of traffic flow models and fits the scenarios in field experiments \cite{sugiyama2008traffic,stern2018dissipation}. The nonlocal LWR model \eqref{eq:nonlocal_lwr} is solved with the periodic boundary condition and the following initial condition: \begin{align}
\rho(x,0)=\rho_0(x),\quad x\in[0,1],\label{eq:ini_data} \end{align} where $\rho_0$ is a nonnegative density distribution in $\mathbf{L}^\infty([0,1])$. We denote: \begin{align}
\bar{\rho}=\int_0^1 \rho_0(x)\,dx,\label{eq:rho_bar} \end{align} the average density of all vehicles on the ring road. Given $\bar{\rho}$, there is a constant solution of \eqref{eq:nonlocal_lwr}: \begin{align}
\rho(x,t)\equiv\bar{\rho}.\label{eq:uniform_sol} \end{align} This constant solution, which is an equilibrium of the dynamics described by the nonlocal LWR model \eqref{eq:nonlocal_lwr}, represents the \emph{uniform flow} in traffic where all vehicles are uniformly distributed and drive at the same speed.
We then make the following assumptions on the desired speed function $U(\cdot)$ and the nonlocal kernel $w_\delta(\cdot)$. \begin{itemize}
\item[({\bf A2})] $U(\rho)=1-\rho$. \end{itemize}
The linear desired speed function $U(\rho)=1-\rho$, usually referred to as the Greenshields speed-density relationship \cite{greenshields1935study}, is widely used in traffic flow modeling. We make the assumption ({\bf A2}) to simplify the problem because in this case \eqref{eq:nonlocal_lwr} can be rewritten as: \begin{align}
\partial_t\rho(x,t)+\partial_x\left(\rho(x,t)\left(1-\rho(x,t)\right)\right)=\nu(\delta)\partial_x\left(\rho(x,t)\mathcal{D}_x^\delta\rho(x,t)\right),\label{eq:nonlocal_diffusion} \end{align} where: \begin{align}
\mathcal{D}_x^\delta\rho(x,t)=\frac{1}{\nu(\delta)}\int_0^\delta \left[\rho(x+s,t)-\rho(x,t)\right]w_\delta(s)\,ds,\quad\text{and}\quad\nu(\delta)=\int_0^\delta sw_\delta(s)\,ds.\label{eq:nonlocal_grad} \end{align}
The equation \eqref{eq:nonlocal_grad} defines the one-sided nonlocal gradient operator $\mathcal{D}_x^\delta$ \cite{dyz17dcdsb,dtty2016cmame}. In \eqref{eq:nonlocal_grad}, the integration is defined with respect to periodicity of the density function. The formulation \eqref{eq:nonlocal_diffusion} reinterprets the nonlocal LWR model \eqref{eq:nonlocal_lwr} as the local one \eqref{eq:lwr} with an additional term that may provide some form of nonlocal diffusion for a suitably chosen kernel $w_\delta(\cdot)$. A sufficient condition is provided in the following assumption.
\begin{itemize}
\item[({\bf A3})] $w_\delta(\cdot)$ is a $\mathbf{C}^1$ function defined on $[0,\delta]$, satisfying
$$w_\delta(s)\geq0, \; \forall s\in[0,\delta]\quad\text{ and }\quad \int_0^\delta w_\delta(s)\,ds=1.$$
In addition, $w_\delta(\cdot)$ is non-increasing and non-constant on $[0,\delta]$. \end{itemize}
The assumption ({\bf A3}) is the key to the main findings of this paper. It is the mathematical reformulation of the natural design principle that density information of nearby vehicles should deserve more attention. Under this assumption, we can deduce that the nonlocal LWR model \eqref{eq:nonlocal_lwr} indeed adds appropriate nonlocal diffusion effect to the local one \eqref{eq:lwr} through a direct spectral analysis, see Section~\ref{sec:spectral_estimate}. More precisely, we will show the following linear stability result.
\begin{thm}\label{thm:linear} Under the assumptions \textup{({\bf A1})\,-\,({\bf A3})}, the uniform flow solution defined by \eqref{eq:uniform_sol} is linearly asymptotically stable for any $\bar{\rho}>0$. \end{thm}
Naturally, for the nonlinear nonlocal system, it is interesting to see if we can extend the linear stability to get global nonlinear stability. For this, we make one additional assumption on the initial data.
\begin{itemize}
\item[({\bf A4})] There exist $0<\rho_{\text{min}}\leq\rho_{\text{max}}\leq1$ such that: \begin{align*}
\rho_{\text{min}}\leq\rho_0(x)\leq\rho_{\text{max}},\quad\forall x\in[0,1]. \end{align*} \end{itemize}
With all above assumptions, we are ready to state the well-posedness of the weak solution as defined below.
\begin{defn}\label{defn:weak_sol}
$\rho\in\mathbf{C}\left([0,\infty);\,\mathbf{L}^1\left([0,1]\right)\right)\cap\mathbf{L}^\infty\left([0,1]\times[0,\infty)\right)$ is a weak solution of \eqref{eq:nonlocal_lwr} with the initial condition \eqref{eq:ini_data} and the periodic boundary condition, if:
\begin{align*}
\int_0^\infty\int_0^1\rho(x,t)\partial_t\phi(x,t)+\rho(x,t)U\left(\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds\right)\partial_x\phi(x,t)\,dxdt+\int_0^1\rho_0(x)\phi(x,0)\,dx=0,
\end{align*}
for all $\phi\in\mathbf{C}^1\left([0,1]\times[0,\infty)\right)$ periodic in space and having compact support. \end{defn}
The well-posedness theorem follows from \cite{keimer2019approximation}. Even though the spatial domain considered in that work is set to be the real line $\mathbb{R}$, the same arguments work with little modifications for the periodic case.
\begin{thm}\label{thm:weak_solution}
Under the assumptions \textup{({\bf A1})\,-\,({\bf A4})}, the nonlocal LWR model \eqref{eq:nonlocal_lwr} admits a unique weak solution in the sense of Definition~\ref{defn:weak_sol}, and the solution satisfies:
\begin{align*}
\rho_{\mathrm{min}}\leq\rho(x,t)\leq\rho_{\mathrm{max}},\quad\forall x\in[0,1],\ t\geq0.
\end{align*} \end{thm}
Although the weak solution always exists, it can be discontinuous. In this paper, the stability analysis is based upon an energy estimate. To do that, we make a regularity assumption on the weak solution of \eqref{eq:nonlocal_lwr}. \begin{itemize}
\item[({\bf A5})] The weak solution $\rho\in\mathbf{C}^1\left([0,1]\times[0,\infty)\right)$. \end{itemize}
The assumption ({\bf A5}) is equivalent to say that $\rho$ is the classical solution of \eqref{eq:nonlocal_lwr}. When the assumptions ({\bf A1})\,-\,({\bf A4}) are true, \cite{bressan2019traffic} proved a sufficient condition for the assumption ({\bf A5}): the initial data $\rho_0$ is $\mathbf{C}^1$ smooth, and the nonlocal kernel $w_\delta(\cdot)$ is $\mathbf{C}^1$ smooth on $[0,+\infty)$ with the zero extension $w_\delta(s)=0$ on $s\in[\delta,+\infty)$.
Now we are in position to state the main results of this paper.
\begin{thm}\label{thm:main}
Under the assumptions \textup{({\bf A1})\,-\,({\bf A5})}, and suppose $\rho(x,t)$ is the solution of the nonlocal LWR model \eqref{eq:nonlocal_lwr}. Then there exists a constant $\lambda>0$ that only depends on $\delta$, $w_\delta(\cdot)$ and $\rho_{\mathrm{min}}$, such that:
\begin{align} \label{eq:main}
\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}\leq e^{-\lambda t}\norm{\rho_0-\bar{\rho}}_{\mathbf{L}^2},\quad\forall t\geq0,
\end{align}
where $\bar{\rho}$ is given by \eqref{eq:rho_bar}.
As a corollary, $\rho(\cdot,t)$ converges to $\bar{\rho}$ in $\mathbf{L}^2\left([0,1]\right)$ as $t\to\infty$. \end{thm}
\begin{rem}
Theorem~\ref{thm:main} says that any classical solution of the nonlocal LWR model \eqref{eq:nonlocal_lwr} {converges exponentially to the uniform flow defined by \eqref{eq:uniform_sol}. In other words, the uniform flow is a globally asymptotically stable equilibrium attracting all initial data}. In such a traffic system, traffic waves will dissipate and all vehicles will quickly adjust their moving positions and driving speeds towards the uniform state from any initial traffic conditions. This conclusion is drawn under the assumptions \textup{({\bf A1})\,-\,({\bf A5})}, which in particular imposes limitations on the nonlocal interactions and other traffic conditions. More discussions in this regard are given in Section~\ref{sec:discussion}, along with additional estimates on the exponent $\lambda$ in \eqref{eq:main} to further illustrate their significance in real traffic scenarios and the design principle for connected vehicles. \end{rem}
\begin{rem}\label{rem:extension} Let us briefly note some possible extensions of Theorem~\ref{thm:main}. First of all, although the regularity assumption ({\bf A5}) is necessary for the energy estimate, it can be removed by considering viscous approximation of the nonlocal LWR model \eqref{eq:nonlocal_lwr} and passing to a vanishing viscosity limit. Secondly, the exponential stability result is also true for the generalized nonlocal LWR model \eqref{eq:generalize_1} with a wide family of functions $g=g(\rho)$. We leave more detailed discussions on these extensions and their interpretations to Section~\ref{sec:discussion}. \end{rem}
The remainder of this paper is organized as follows: Section~\ref{sec:stab_analysis} is devoted to stability analysis of \eqref{eq:nonlocal_lwr} and the proofs of Theorem~\ref{thm:linear} and Theorem~\ref{thm:main}. Section~\ref{sec:numerical_exp} provides numerical experiments to illustrate the results. Conclusions and future research directions follow in Section~\ref{sec:conclusion}.
\section{Stability analysis}\label{sec:stab_analysis} This section aims to establish the main stability results stated earlier in Section~\ref{sec:main_results}. In Section~\ref{sec:spectral_estimate}, we analyze the spectral properties of the nonlocal gradient operator $\mathcal{D}_x^\delta$ and its corresponding nonlocal diffusion operator $\partial_x\mathcal{D}_x^\delta$. The analysis yields the linear stability result and also helps to show the nonlinear stability. The proof of Theorem~\ref{thm:main} builds on an energy estimate that utilizes two ingredients: a nonlocal Poincare inequality and a Hardy-Littlewood rearrangement inequality. Section~\ref{sec:energy_estimate} derives the energy estimate and Section~\ref{sec:proof} completes the proof of Theorem~\ref{thm:main} based on the two inequalities. {Section~\ref{sec:discussion} discusses further extensions of the theorem and compares the local and nonlocal models}. In Section~\ref{sec:counter_example}, a counterexample is shown that convergence to the uniform flow does not hold when the assumption ({\bf A3}) is not satisfied.
\subsection{Spectral analysis and linear stability}\label{sec:spectral_estimate}
Given the assumption ({\bf A1}), we consider the Fourier series expansion of any real-valued periodic function $\rho(x)$ for $x\in[0,1]$: \begin{align*}
\rho(x)=\sum_{k\in\mathbb{Z}} \hat{\rho}(k)e^{2\pi ikx}. \end{align*} For the local gradient operator $\partial_x$ and the nonlocal gradient operator $\mathcal{D}_x^\delta$ defined by \eqref{eq:nonlocal_grad}, a straightforward calculation gives: \begin{align*}
\partial_x\rho(x)=\sum_{k\in\mathbb{Z}}2\pi ik\hat{\rho}(k)e^{2\pi ikx},\quad\text{and}\quad
\mathcal{D}_x^\delta\rho(x)=\sum_{k\in\mathbb{Z}}[ib_\delta(k)+c_\delta(k)]\hat{\rho}(k)e^{2\pi ikx}, \end{align*} where \begin{align}
b_\delta(k)=\frac1{\nu(\delta)}\int_0^\delta \sin(2\pi ks)w_\delta(s)\,ds,\quad \text{and}\quad c_\delta(k)=\frac1{\nu(\delta)}\int_0^\delta [\cos(2\pi ks)-1]w_\delta(s)\,ds.\label{eq:fourier_coeff} \end{align} As a corollary, the spectrum of the nonlocal diffusion operator $\partial_x\mathcal{D}_x^\delta$ is given by the discrete set of eigenvalues $\{-2\pi k b_\delta(k) + 2\pi i k c_\delta(k) \}_{k\in\mathbb{Z}}$. The following lemma gives an estimate on the real parts of the eigenvalues $\{-2\pi k b_\delta(k)\}_{k\in\mathbb{Z}}=\{0\}\cup\{-2\pi k b_\delta(k)\}_{k\geq1}$.
\begin{lemm}\label{lem:spectra}
Under the assumption \textup{({\bf A3})}, we have:
\begin{align}
\alpha\triangleq\inf_{k\geq1}2\pi kb_\delta(k)>0.\label{eq:estimate_alpha}
\end{align} \end{lemm} \begin{proof}
By \cite[Lemma 2]{du2018stability}, the assumption ({\bf A3}) yields that $b_\delta(k)$ is strictly positive for any $k\geq 1$. In fact, since $w_\delta(\cdot)$ is non-increasing and non-constant,
integration by parts gives:
\begin{align}
2\pi kb_\delta(k)&=\frac1{\nu(\delta)}\left[w_\delta(0)-w_\delta(\delta)\cos(2\pi k\delta)+\int_0^\delta \cos(2\pi ks)w'_\delta(s)\,ds\right],\notag\\
&\geq\frac1{\nu(\delta)}\left[w_\delta(0)-w_\delta(\delta)+\int_0^\delta \cos(2\pi ks)w'_\delta(s)\,ds\right]>0,\label{eq:estimate_spectra}
\end{align}
for any $k\geq 1$. Meanwhile, when $k\to\infty$, one can apply the Riemann-Lebesgue Lemma to get:
\begin{align*}
\liminf_{k\to\infty}2\pi kb_\delta(k)\geq \frac1{\nu(\delta)}\left[w_\delta(0)-w_\delta(\delta)\right]>0.
\end{align*}
Combining these facts, we get \eqref{eq:estimate_alpha}. \end{proof}
We now present the proof of the linear stability given in Theorem~\ref{thm:linear}.
\begin{proof}[Proof of Theorem~\ref{thm:linear}] To show the linear stability, we simply need to consider the linearized equation of \eqref{eq:nonlocal_lwr} around the uniform flow $\bar{\rho}$. The equation is given by: \begin{align}
\partial_t\tilde{\rho}(x,t)+(1-2\bar{\rho})\partial_x \tilde{\rho}(x,t)
=\nu(\delta) \bar{\rho} \partial_x \mathcal{D}_x^\delta\tilde{\rho}(x,t).
\label{eq:nonlocal_diffusion_linearize} \end{align} The perturbative solution $\tilde{\rho}(x,t)$ is assumed to have mean zero initially, which remains true for all time. Hence, for the linear stability, we are concerned with the eigenvalues of the nonlocal diffusion operator $\partial_x\mathcal{D}_x^\delta$ except the single zero eigenvalue with a constant eigenfunction. The real parts of those eigenvalues, as shown in Lemma~\ref{lem:spectra}, are uniformly negative. We thus have the linear stability stated in Theorem~\ref{thm:linear}. \end{proof}
\subsection{Energy estimate}\label{sec:energy_estimate}
Suppose $\rho(x,t)$ is any $\mathbf{C}^1$ solution to the nonlocal LWR model \eqref{eq:nonlocal_lwr}. The conservation property gives: \begin{align*}
\int_0^1 \rho(x,t)\,dx=\int_0^1\rho_0(x)\,dx=\bar{\rho},\quad\forall t\geq0. \end{align*}
We define the following \emph{energy function} (aka a Lyapunov functional): \begin{align}
E(t)\triangleq\frac12\int_0^1 \left(\rho(x,t)-\bar{\rho}\right)^2\,dx,\quad\forall t\geq0.\label{eq:energy_fun} \end{align} It is straightforward to get: \begin{align}
\frac{dE(t)}{dt}&=\int_0^1\rho(x,t)\partial_t\rho(x,t)\,dx.\label{eq:tmp_1_1} \end{align} Note that \eqref{eq:nonlocal_diffusion} is equivalent to \eqref{eq:nonlocal_lwr}, substituting \eqref{eq:nonlocal_diffusion} into \eqref{eq:tmp_1_1} yields: \begin{align*}
\frac{dE(t)}{dt}&=-\int_0^1\rho(x,t)\partial_x\left(\rho(x,t)(1-\rho(x,t))\right)\,dx+\nu(\delta)\int_0^1\rho(x,t)\partial_x\left(\rho(x,t)\mathcal{D}_x^\delta\rho(x,t)\right)\,dx. \end{align*} Apply the Newton-Leibniz rule and integration by parts, we obtain: \begin{align}
\frac{dE(t)}{dt}=-\nu(\delta)\int_0^1\rho(x,t)\partial_x\rho(x,t)\mathcal{D}_x^\delta\rho(x,t)\,dx.\label{eq:energy_dt} \end{align} All boundary terms vanish because of the periodic boundary condition.
\subsection{Proof of Theorem~\ref{thm:main}}\label{sec:proof} We present two lemmas to estimate the right hand side of \eqref{eq:energy_dt}. Then the conclusion of the theorem follows from the estimate of the energy function $E(t)$.
\begin{lemm}{[Nonlocal Poincare inequality]}\label{lem: nonlocal_poincare}
Suppose that the nonlocal kernel $w_\delta(\cdot)$ satisfies the assumption \textup{({\bf A3})}. There exists a constant $\alpha>0$ such that for any $\mathbf{C}^1$ periodic function $\rho(x)$ defined on $[0,1]$:
\begin{align}
\int_0^1 \partial_x\rho(x) \mathcal{D}_x^\delta \rho(x)\,dx\geq \alpha\int_0^1 \left(\rho(x)-\bar{\rho}\right)^2\,dx,\label{eq:nonlocal_poincare}
\end{align}
where $\bar{\rho}=\int_0^1\rho(x)\,dx$, $\alpha$ only depends on the nonlocal range $\delta$ and the nonlocal kernel $w_\delta(\cdot)$. \end{lemm}
\begin{proof} We have $b_\delta(-k)=-b_\delta(k)$, $c_\delta(-k)=c_\delta(k)$ and $\hat{\rho}(-k)=\overline{\hat{\rho}(k)}$ for all $k\in\mathbb{Z}$. By Parseval's identity, \begin{align*}
\int_0^1 \partial_x\rho(x) \mathcal{D}_x^\delta \rho(x)\,dx&=\sum_{k\in\mathbb{Z}}-2\pi ik[ib_\delta(k)+c_\delta(k)]|\hat{\rho}(k)|^2=\sum_{k=1}^\infty 4\pi kb_\delta(k)|\hat{\rho}(k)|^2. \end{align*} Meanwhile, \begin{align*}
\int_0^1 \left(\rho(x)-\bar{\rho}\right)^2\,dx=\sum_{k\neq0}|\hat{\rho}(k)|^2=\sum_{k=1}^\infty2|\hat{\rho}(k)|^2. \end{align*} Then the inequality \eqref{eq:nonlocal_poincare} follows from \eqref{eq:estimate_alpha}. \end{proof}
\begin{rem}
The nonlocal Poincare inequality \eqref{eq:nonlocal_poincare} generalizes the classical one:
\begin{align}
\int_0^1 \left(\partial_x\rho(x)\right)^2\,dx\geq \alpha\int_0^1 \left(\rho(x)-\bar{\rho}\right)^2\,dx,\label{eq:classical_poincare}
\end{align}
by introducing the nonlocal gradient operator $\mathcal{D}_x^\delta$. \cite{du2018stability} proposed another generalization of \eqref{eq:classical_poincare}:
\begin{align}
\int_0^1 \left(\mathcal{D}_x^\delta \rho(x)\right)^2\,dx\geq \alpha\int_0^1 \left(\rho(x)-\bar{\rho}\right)^2\,dx,\label{eq:corr_poincare}
\end{align}
to analyze nonlocal Dirichlet integrals. There, $\mathcal{D}_x^\delta$ uses a symmetric difference quotient, so the eigenvalues of $\mathcal{D}_x^\delta$ only have imaginary parts $b_\delta(k)$. In that case, the singularity of the kernel $w_\delta(\cdot)$ at the origin is necessary to bound $b_\delta(k)$ from below when $k\to\infty$, which then implies \eqref{eq:corr_poincare}.
This type of nonlocal Poincare inequality \eqref{eq:corr_poincare}
is further extended in \cite{lee2020nonlocal} where a non-symmetric kernel is used to define the nonlocal gradient, much like the one studied in this work. Then the eigenvalues of $\mathcal{D}_x^\delta$ have both real and imaginary parts. With the kernel $w_\delta(\cdot)$ having no singularity at the origin, the imaginary parts $b_\delta(k)$ decay to zero as $k\to\infty$. However, the real parts $c_\delta(k)$ are bounded from below when $k\to\infty$, thus also leading to \eqref{eq:corr_poincare}.
The inequality \eqref{eq:nonlocal_poincare}, as far as the authors know, has not been presented before. Here, we are estimating the $\mathbf{L}^2$ inner product of $\partial_x\rho$ and $\mathcal{D}_x^\delta\rho$. Although the eigenvalues of $\mathcal{D}_x^\delta$ have both real and imaginary parts because of the non-symmetric kernel, the real parts $c_\delta(k)$ have no contribution to the $\mathbf{L}^2$ inner product. We assume the kernel $w_\delta(\cdot)$ to have no singularity, thus $b_\delta(k)\to0$ when $k\to\infty$. But it does not create any issue since \eqref{eq:nonlocal_poincare} only requires that $2\pi kb_\delta(k)$ is bounded from below. The factor $2\pi k$, which corresponds to the eigenvalues of the local gradient operator $\partial_x$, helps us get the desired result. \end{rem}
\begin{rem}\label{rem:poincareconstant}
Let us mention that in some special cases, the nonlocal Poincare inequality \eqref{eq:nonlocal_poincare} can become an equality. For example, when $\delta=1$ and $w_\delta(s)=2(1-s)$, we have:
\begin{align*}
\mathcal{D}_x^\delta\rho(x)=6\int_0^1 (1-s)[\rho(x+s)-\rho(x)]\,ds.
\end{align*}
A direct calculation gives that $\partial_x\mathcal{D}_x^\delta\rho(x)=-6(\rho(x)-\bar{\rho})-3\partial_x\rho(x)$. That is, the nonlocal diffusion is actually a local term. As a corollary,
\begin{align*}
\int_0^1\partial_x\rho(x)\mathcal{D}_x^\delta\rho(x)\,dx=-\int_0^1(\rho(x)-\bar{\rho})\partial_x\mathcal{D}_x^\delta\rho(x)\,dx=6\int_0^1(\rho(x)-\bar{\rho})^2\,dx,
\end{align*}
which is a key ingredient used in \cite{karafyllis2020analysis} to study the nonlocal LWR model with nudging. For more general choices of the nonlocal range $\delta$ and nonlocal kernel $w_\delta(\cdot)$, Lemma~\ref{lem: nonlocal_poincare} provides a more effective way to derive global asymptotic stability as demonstrated in this work. \end{rem}
A special case of Lemma~\ref{lem: nonlocal_poincare} is when $w_\delta(\cdot)$ is a rescaled kernel: $w_\delta(s)=w_1(s/\delta)/\delta$. That is, the family of kernels $\{w_\delta(\cdot)\}_{\delta\in(0,1]}$ is generated from the kernel $w_1(\cdot)$ defined on $[0,1]$. In this case, it is worthwhile to mention that the constant $\alpha$ in \eqref{eq:nonlocal_poincare} is independent of the nonlocal range $\delta$.
\begin{prop}\label{prop:rescale}
Suppose $w_\delta(s)=w_1(s/\delta)/\delta$ for all $s\in[0,\delta]$, $\delta\in(0,1]$, where $w_1(\cdot)$ satisfies the
assumption ({\bf A3}) for $\delta=1$.
Then there exists a constant $\alpha>0$ only depending on $w_1(\cdot)$ such that for any $\delta\in(0,1]$, \eqref{eq:nonlocal_poincare} holds for the nonlocal kernel $w_\delta(\cdot)$ with the constant $\alpha$. \end{prop}
\begin{proof}
It suffices to show $\alpha\triangleq\inf_{k\geq1,0<\delta\leq1}2\pi kb_\delta(k)>0$ where $b_\delta(k)$ is defined in \eqref{eq:fourier_coeff}.
We denote $a=2\pi k\delta$ and $\nu_1=\int_0^1sw_1(s)\,ds$. Obviously, we have $\nu(\delta)=\delta\nu_1$.
Then we can rewrite \eqref{eq:estimate_spectra} as:
\begin{align*}
2\pi kb_\delta(k)
&\geq\frac1{\nu_1\delta^2}\left[w_1(0)-w_1(1)+\int_0^1 \cos(as)w'_1(s)\,ds\right],\\
&=\frac1{\nu_1\delta^2}\int_0^1 [\cos(as)-1]w'_1(s)\,ds.
\end{align*}
Note that $w_1(1)<w_1(0)$, there exist constants $0<s_1<s_2<1$ and $\eta>0$, which only depend on $w_1(\cdot)$, such that $w'_1(s)\leq-\eta$ when $s\in[s_1,s_2]$. Hence we have:
\begin{align*}
2\pi kb_\delta(k)&\geq\frac\eta{\nu_1\delta^2}\int_{s_1}^{s_2} [1-\cos(as)]\,ds.
\end{align*}
When $0<a<1$, we use the inequality $1-\cos(as)\geq\frac{(as)^2}{2}-\frac{(as)^4}{24}\geq\frac{11}{24}(as)^2$ to get:
\begin{align}
2\pi kb_\delta(k)\geq\frac{\eta a^2}{\nu_1\delta^2}\cdot\frac{11}{72}(s_2^3-s_1^3)=\frac{11\pi^2k^2\eta}{18\nu_1}(s_2^3-s_1^3)=\alpha_1k^2\geq \alpha_1>0,\label{eq:tmp_2_1}
\end{align}
for any $k\geq1$, where the constant $\alpha_1$ only depends on $w_1(\cdot)$.
When $a\geq1$, consider the following integral as a function of $a$:
\begin{align*}
h(a)\triangleq\int_{s_1}^{s_2} [1-\cos(as)]\,ds,\quad a\in[1,+\infty).
\end{align*}
Then $h(a)$ is always positive and $h(a)\to s_2-s_1>0$ when $a\to+\infty$.
Hence $h(a)$ has a lower bound $\alpha_2>0$ for $a\in[1,+\infty)$, and $\alpha_2$ only depends on $w_1(\cdot)$ .
In this case,
\begin{align}
2\pi kb_\delta(k)\geq\frac{\eta\alpha_2}{\nu_1\delta^2}\geq\frac{\eta\alpha_2}{\nu_1}>0.\label{eq:tmp_2_2}
\end{align}
The estimates \eqref{eq:tmp_2_1}\eqref{eq:tmp_2_2} give the conclusion. \end{proof}
Next, we present an inequality similar to \eqref{eq:nonlocal_poincare} to deal with the presence of nonlinearity. Actually, Lemma~\ref{lem:nonlinear_poincare} stated below bridges between linear and nonlinear diffusion in the nonlocal setting. Its proof uses the following Hardy-Littlewood rearrangement inequality on a periodic domain. A similar inequality is used in \cite{bressan2019traffic} to prove that the local limit of nonlocal solutions of \eqref{eq:nonlocal_lwr} satisfies the entropy condition.
\begin{lemm}{[Hardy-Littlewood rearrangement inequality]}\label{lem:hardy-littlewood}
{Suppose $\rho(x)$ is a continuous periodic function defined on $[0,1]$. For any continuous, monotonically increasing function $f(\cdot)$ and $s\in[0,1]$}:
\begin{align}
\int_0^1 f(\rho(x))\rho(x+s)\,dx\leq \int_0^1 f(\rho(x))\rho(x)\,dx.\label{eq:hardy}
\end{align} \end{lemm}
\begin{proof}
We first assume $s\in\mathbb{Q}\cap[0,1]$. Suppose $N$ is a positive integer such that $m=sN$ is a nonnegative integer. Let us consider
the discrete case with
\begin{align*}
0=x_0<x_1<\cdots<x_N=1,\quad x_i=i\Delta x,\quad i=0,\dots,N,
\end{align*}
where $\Delta x=1/N$. Denote:
\begin{align*}
\rho_i=\rho(x_i),\quad f_i=f(\rho(x_i)),\quad i=0,\dots,N, \; \text{ with }\;
\rho_0=\rho_N, \; f_0=f_N.
\end{align*}
Suppose $\sigma(1),\sigma(2),\dots,\sigma(N)$ is a permutation of $1,2,\dots,N$ such that:
\begin{align*}
\rho_{\sigma(1)}\leq\rho_{\sigma(2)}\leq\cdots\leq\rho_{\sigma(N)}.
\end{align*}
The monotonicity of $f(\cdot)$ yields:
\begin{align*}
f_{\sigma(1)}\leq f_{\sigma(2)}\leq\cdots\leq f_{\sigma(N)}.
\end{align*}
Denote $\tau_m$ the shift permutation defined by $\tau_m(i)=i+m$, $i=1,\dots,N$ (use the circular extension when $i+m>N$). The rearrangement inequality gives:
\begin{align*}
\sum_{i=1}^N f_i\rho_{i+m}=\sum_{i=1}^N f_{\sigma(i)}\rho_{\tau_m\circ\sigma(i)}\leq \sum_{i=1}^N f_{\sigma(i)}\rho_{\sigma(i)}=\sum_{i=1}^N f_i\rho_i.
\end{align*}
The inequality \eqref{eq:hardy} can then be derived via a limit process. By the density of $\mathbb{Q}\cap[0,1]$ in $[0,1]$, a further limit process can establish \eqref{eq:hardy} for any $s\in [0,1]$. \end{proof}
\begin{lemm}\label{lem:nonlinear_poincare}
Suppose that the nonlocal kernel $w_\delta(\cdot)$ satisfies the assumption \textup{({\bf A3})}. For any $\mathbf{C}^1$ periodic function $\rho(x)$ defined on $[0,1]$ and satisfying $\rho(x)\geq\rho_{\mathrm{min}}\geq 0$:
\begin{align}
\int_0^1 \rho(x)\partial_x\rho(x) \mathcal{D}_x^\delta \rho(x)\,dx\geq \rho_{\mathrm{min}}\int_0^1 \partial_x\rho(x) \mathcal{D}_x^\delta \rho(x)\,dx.\label{eq:nonlinear_poincare}
\end{align} \end{lemm}
\begin{proof}
Define $f(\rho)=\frac12(\rho-\rho_{\text{min}})^2$. Then \eqref{eq:nonlinear_poincare} can be rewritten as:
\begin{align*}
\int_0^1\partial_x f(\rho(x))\mathcal{D}_x^\delta\rho(x)\,dx\geq0.
\end{align*}
Using integration by parts, it is equivalent to:
\begin{align}
\int_0^1 f(\rho(x))\partial_x\mathcal{D}_x^\delta\rho(x)\,dx\leq0.\label{tmp_3_1}
\end{align}
We only need to show \eqref{tmp_3_1}. A direct calculation gives:
\begin{align}
\partial_x\mathcal{D}_x^\delta\rho(x)&=\frac1{\nu(\delta)}\left[\int_0^\delta \partial_x\rho(x+s)w_\delta(s)\,ds-\partial_x\rho(x)\right],\notag\\
&=\frac1{\nu(\delta)}\left[\int_0^\delta \partial_s\rho(x+s)w_\delta(s)\,ds-\partial_x\rho(x)\right],\notag\\
&=\frac1{\nu(\delta)}\left[\rho(x+\delta)w_\delta(\delta)-\rho(x)w_\delta(0)-\int_0^\delta\rho(x+s)w_\delta'(s)\,ds-\partial_x\rho(x)\right].\label{eq:tmp_3_2}
\end{align}
We multiply both sides of \eqref{eq:tmp_3_2} by $f(\rho(x))$ and integrate them over the domain $[0,1]$.
The Newton-Leibniz rule gives:
\begin{align*}
\int_0^1f(\rho(x))\partial_x\rho(x)\,dx=0.
\end{align*}
Define:
\begin{align*}
I(s)\triangleq\int_0^1f(\rho(x))\rho(x+s)\,dx,\quad s\in[0,\delta].
\end{align*}
Then we have:
\begin{align}
\int_0^1 f(\rho(x))\partial_x\mathcal{D}_x^\delta\rho(x)\,dx&=\frac1{\nu(\delta)}\left[I(\delta)w_\delta(\delta)-I(0)w_\delta(0)-\int_0^\delta I(s)w_\delta'(s)\,ds\right],\nonumber \\
&=\frac1{\nu(\delta)}\left[\left(I(\delta)-I(0)\right)w_\delta(\delta)+\int_0^\delta \left(I(0)-I(s)\right)w_\delta'(s)\,ds\right].\label{eq:tmp_3_3}
\end{align}
When $\rho\geq\rho_{\text{min}}\geq 0$, $f(\rho)$ is monotonically increasing. Lemma~\ref{lem:hardy-littlewood} yields that $I(s)\leq I(0)$ for any $0\leq s\leq\delta$. In addition, the assumption ({\bf A3}) yields that $w_\delta'(s)\leq 0$ for any $0\leq s\leq\delta$. So the both terms on the right hand side of \eqref{eq:tmp_3_3} is non-positive, which gives \eqref{tmp_3_1}. \end{proof}
Naturally, we will be most interested in applying the above lemma to the case where the density $\rho$ satisfies the assumption ({\bf A4}) so that $\rho_{\text{min}}>0$.
Now we can prove our main results. \begin{proof}[Proof of Theorem~\ref{thm:main}]
Define the energy function $E(t)$ by \eqref{eq:energy_fun}. Then the derivative of $E(t)$ is given by \eqref{eq:energy_dt}. By Theorem~\ref{thm:weak_solution}, $\rho(x,t)\geq\rho_{\text{min}}>0$ for all $x\in[0,1]$ and $t\geq0$. Apply Lemma~\ref{lem: nonlocal_poincare} and Lemma~\ref{lem:nonlinear_poincare}, we get the estimate:
\begin{align*}
\frac{dE(t)}{dt}\leq-2\nu(\delta)\alpha\rho_{\text{min}}E(t),\quad \forall t\geq0.
\end{align*}
By the Gronwall's lemma, $E(t)\leq e^{-2\lambda t}E(0)$ where $\lambda=\nu(\delta)\alpha\rho_{\text{min}}$. It immediately yields the conclusion. \end{proof}
\begin{rem}\label{rem:kl_divergence}
An alternative approach to show the exponential stability is to define the Lyapunov functional:
\begin{align*}
V(t)=\int_0^1 \rho(x,t)\ln\frac{\rho(x,t)}{\bar{\rho}}\,dx,
\end{align*}
which is the Kullback–Leibler divergence from the uniform density to $\rho(x,t)/\bar{\rho}$.
A calculation similar to that in Section~\ref{sec:energy_estimate} gives:
\begin{align*}
\frac{dV(t)}{dt}=-\nu(\delta)\int_0^1\partial_x\rho(x,t)\mathcal{D}_x^\delta\rho(x,t)\,dx.
\end{align*}
Apply Lemma~\ref{lem: nonlocal_poincare}, one can get $dV(t)/dt\leq-\nu(\delta)\alpha\int_0^1(\rho(x,t)-\bar{\rho})^2\,dx$.
The Gronwall's lemma together with the inequality (see \cite{karafyllis2020analysis}):
\begin{align*}
\frac1{2\rho_{\mathrm{max}}}\int_0^1(\rho(x,t)-\bar{\rho})^2\,dx\leq V(t)\leq\frac1{2\rho_{\mathrm{min}}}\int_0^1(\rho(x,t)-\bar{\rho})^2\,dx,
\end{align*}
gives the exponential convergence in the Kullback–Leibler divergence:
\begin{align*}
V(t)\leq e^{-2\lambda t}V(0),
\end{align*}
where $\lambda=\nu(\delta)\alpha\rho_{\mathrm{min}}$ and consequently:
\begin{align}
\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}\leq\left(\frac{\rho_{\mathrm{max}}}{\rho_{\mathrm{min}}}\right)^{\frac12}e^{-\lambda t}\norm{\rho_0-\bar{\rho}}_{\mathbf{L}^2}.\label{eq:exp_stab_alt}
\end{align}
The estimate on the Kullback–Leibler divergence was proposed in \cite{karafyllis2020analysis} to prove exponential stability for the nonlocal LWR model with nudging. For the case discussed in this paper, Theorem~\ref{thm:main} provides a sharper result \eqref{eq:main} than \eqref{eq:exp_stab_alt}. \end{rem}
\subsection{Further discussions on the main results}\label{sec:discussion}
The energy estimate presented so far requires the regularity assumption ({\bf A5}) on the solution. A natural question is whether Theorem~\ref{thm:main} holds for the general weak solution. To deal with the weak solution, we consider the following viscous nonlocal LWR model:
\begin{align}
\partial_t\rho(x,t)+\partial_x\left(\rho(x,t) U\left(\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds\right)\right)=\epsilon\partial_x^2\rho(x,t),\label{eq:viscous_nonlocal_lwr}
\end{align}
where $\epsilon>0$ is the viscosity parameter. \cite{colombo2019singular} studied \eqref{eq:viscous_nonlocal_lwr} on the real line and showed the solution well-posedness using a fixed-point theorem and $\mathbf{L}^\infty$ estimates. Based on similar arguments, one can show that \eqref{eq:viscous_nonlocal_lwr} admits a unique weak solution under the conditions in Theorem~\ref{thm:weak_solution}. Further, one can show that the weak solution is $\mathbf{C}^\infty$ smooth by a bootstrap argument. Then we can carry out the energy estimate on \eqref{eq:viscous_nonlocal_lwr} and obtain $E(t)\leq e^{-2(\lambda+c\epsilon) t}E(0)$ where $c>0$ is a constant. Letting $\epsilon\to0$, it can be shown that the solution of \eqref{eq:viscous_nonlocal_lwr} converges to the solution of the original nonlocal LWR model \eqref{eq:nonlocal_lwr} weakly, using similar estimates in \cite{colombo2019singular}. As a corollary, we obtain $E(t)\leq e^{-2\lambda t}E(0)$ since $E(t)$ is a lower semi-continuous functional of $\rho(\cdot,t)$. The conclusion of Theorem~\ref{thm:main} still holds. We only state the extended result, which is the same as Theorem~\ref{thm:main} without the assumption ({\bf A5}). The detailed proof is skipped.
\begin{thm}\label{thm:weak}
Under the assumptions \textup{({\bf A1})-({\bf A4})}, let $\rho(x,t)$ be the weak solution of the nonlocal LWR model \eqref{eq:nonlocal_lwr}. Then there exists a constant $\lambda>0$ that only depends on $\delta$, $w_\delta(\cdot)$ and $\rho_{\mathrm{min}}$, such that the estimate \eqref{eq:main} holds. As a corollary, $\rho(\cdot,t)$ converges to $\bar{\rho}$, which is given by \eqref{eq:rho_bar}, in $\mathbf{L}^2\left([0,1]\right)$ as $t\to\infty$. \end{thm}
We now make a couple of additional remarks.
\begin{rem}\label{rem:speed-function} Let us first have a discussion on the choice of the speed function. The assumption $U(\rho)=1-\rho$ allows us to split the local and nonlocal terms in \eqref{eq:nonlocal_diffusion} and carry out the energy estimate. It is interesting to consider extensions of Theorem~\ref{thm:main} for more general forms of nonlocal velocity selection. For example, we consider the nonlocal velocity in the following form: \begin{align}
u(x,t)=U_0(\rho(x,t))\left(1-\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds\right),\label{eq:reform} \end{align} which leads to the generalized nonlocal LWR model \eqref{eq:generalize_1} with $g(\rho)=\rho U_0(\rho)$. When $U_0\equiv1$, it reduces to the case in Theorem~\ref{thm:main}. Based on \eqref{eq:reform}, the generalized model \eqref{eq:generalize_1} can be rewritten as: \begin{align*}
\partial_t\rho(x,t)+\partial_x\left(g(\rho(x,t))\left(1-\rho(x,t)\right)\right)=\nu(\delta)\partial_x\left(g(\rho(x,t))\mathcal{D}_x^\delta\rho(x,t)\right). \end{align*} \cite{Chiarello2018} proved the same well-posedness results for \eqref{eq:generalize_1} as those for \eqref{eq:nonlocal_lwr} assuming that $g=g(\rho)$ is positive and $\mathbf{C}^1$ smooth. If $g(\rho)$ is bounded away from zero when $\rho\in[\rho_{\mathrm{min}},\rho_{\mathrm{max}}]$, the Hardy-Littlewood rearrangement inequality allows us to remove the nonlinear term $g(\rho)$ and get an estimate similar to \eqref{eq:nonlinear_poincare}. Thus, the conclusion of Theorem~\ref{thm:main} remains true. \end{rem}
\begin{rem}\label{rem:convergence_speed_delta} Let us now discuss the exponent $\lambda$ in the exponential decay estimate \eqref{eq:main}. For a rescaled kernel $w_\delta(s)=w_1(s/\delta)/\delta$ as discussed in Proposition~\ref{prop:rescale}, one can examine how $\lambda$ depends on the kernel $w_1(s)$, the nonlocal range $\delta$ and the initial data. The theoretical analysis in the proof of Theorem~\ref{thm:main} gives a lower bound for the exponent $\lambda$ as $\lambda=\nu(\delta)\alpha \rho_{\mathrm{min}} = \delta \nu_1 \alpha \rho_{\mathrm{min}}$, where $\alpha$ and $\nu_1$ are determined by the kernel $w_1(s)$ and $\rho_{\mathrm{min}}$ is the minimum of initial data. Moreover, a sharper estimate can be derived for some special cases. For example, with the solution (or initial data) sufficiently close to the uniform flow density $\bar{\rho}$, one may replace $\rho_{\mathrm{min}}$ by $\bar{\rho}$. Then, for the linear decreasing kernel $w_1(s)=2(1-s)$, similar to calculations carried out earlier, we can get \begin{align}
\lambda=\frac{2}{\delta}\left(1-\frac{\sin(2\pi \delta)}{2\pi \delta}\right)\bar{\rho},\quad \text{for }\;
\delta\in (0, 1].
\label{decay_rate_precise} \end{align} For sufficiently small $\delta>0$, the above leads to \begin{align}
\lambda=\frac{4\pi^2}{3}\delta\bar{\rho}.\label{decay_rate_asymptotic} \end{align} For numerical validation of these estimates, we refer to Section~\ref{sec:numerical_exp}. Based on both theoretical estimates and numerical observations, one may consider accelerating the convergence to the uniform flow by increasing the nonlocal range $\delta$, at least in a proper range. In traffic terms, this serves to supplement the design principle presented earlier: while nearby information should be given more attention, within a proper nonlocal range, utilizing information gathered over a wider domain could bring more benefits. However, as $\lambda$ may not stay monotonically increasing for all $\delta$, the acceleration might become less effective if $\delta$ gets too large. Thus, one should choose suitably the range of nonlocal information to be utilized. Meanwhile, it should be noted that the value of $\delta$ should also be properly confined in practice to avoid any significant deviation of each vehicle's driving speed from its desired local speed in consideration of driving safety. Likewise, we can also see from the above estimates that the convergence gets faster with larger values of $\bar{\rho}$. We can attribute this property to the nonlinear dependence of the diffusion introduced to the system \eqref{eq:nonlocal_diffusion} on the traffic density. \end{rem}
Finally, let us make some comparisons between the nonlocal LWR model \eqref{eq:nonlocal_lwr} and the local one \eqref{eq:lwr}. In particular, in terms of the practical implication on traffic flows, it is interesting to examine the rate at which the traffic density would get back to the uniform state. On one hand, as shown in Theorem~\ref{thm:main}, the solution of \eqref{eq:nonlocal_lwr} has an exponential convergence towards the uniform flow. On the other hand, \cite{debussche2009long} showed that every solution of \eqref{eq:lwr} with the periodic boundary condition converges to the uniform flow $\bar{\rho}$ given by \eqref{eq:rho_bar} as $t\to\infty$, except the case $\bar{\rho}=0.5$ in which the local flux $f(\rho)=\rho(1-\rho)$ in \eqref{eq:lwr} is degenerate. However, the convergence will be much slower than the exponential convergence of the nonlocal LWR model \eqref{eq:nonlocal_lwr}. We will demonstrate the asymptotic convergence speed of the local LWR model \eqref{eq:lwr} in the following example.
Consider the equation \eqref{eq:lwr} with the following linear initial data: \begin{align}
\rho_0(x)=\beta x,\quad x\in[0,1],\label{eq:linear_ini} \end{align} where $\beta\in[0,1]$ is a constant. In this case, \eqref{eq:lwr} can be solved explicitly and the solution is a piecewise linear function. When $0\leq t\leq\frac1{2\beta}$, there is a moving rarefaction wave and the solution is give by: \begin{align*}
\rho(x,t)=\begin{dcases}
\frac{t-x}{2t},\quad &(1-2\beta)t\leq x<t;\\
\frac{\beta(x-t)}{1-2\beta t},\quad &t\leq x<(1-2\beta)t+1.
\end{dcases} \end{align*} When $t>\frac1{2\beta}$, the solution develops a shock wave. The shock wave moves at a constant speed $1-\beta$ and has a jump from $\rho_l(t)=\frac\beta2-\frac1{4t}$ to $\rho_r(t)=\frac\beta2+\frac1{4t}$. Before the shock formation ($t\leq\frac1{2\beta}$), the $\mathbf{L}^2$ error $\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}$ between the solution and the uniform flow is constant; After the shock formation ($t>\frac1{2\beta}$), a direct calculation gives: \begin{align}
\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}=\frac{1}{2\sqrt{12}t}.\label{eq:local_linear_convergence} \end{align} That is, the solution converges to the uniform flow when $t\to\infty$ with algebraic decay rate, which means that it would take much more time in the local case for the traffic to get to the uniform flow than that predicted in the nonlocal case.
\subsection{A counterexample with the constant kernel}\label{sec:counter_example}
Suppose $\delta=1/m$ where $m$ is a positive integer. We pick the constant kernel: \begin{align}
w_\delta(s)=\frac1\delta,\quad s\in[0,\delta].\label{eq:constant_kernel} \end{align} Suppose that the initial data $\rho_0(x)$ is periodic with period $\delta$, the solution of the nonlocal LWR model \eqref{eq:nonlocal_lwr} can be explicitly given by $\rho(x,t)=\rho_0(x-\bar{u}t)$ where $\bar{u}=1-\bar{\rho}$. Note that, at any time $t$, the density $\rho(\cdot,t)$ is a translation of $\rho_0$ and hence periodic with period $\delta$. Consequently, the velocity: \begin{align*}
u(x,t)=1-\int_0^\delta\rho(x+s,t)w_\delta(s)\,ds=1-\frac1\delta\int_0^\delta\rho(x+s,t)\,ds=1-\bar{\rho}=\bar{u}, \end{align*} is a fixed constant. The nonlocal LWR model \eqref{eq:nonlocal_lwr} then becomes a scalar transport equation: \begin{align*}
\rho_t+\bar{u}\rho_x=0, \end{align*} whose solution is the traveling wave $\rho(x,t)=\rho_0(x-\bar{u}t)$. The traveling wave solution never converges to the uniform flow as $t\to\infty$ unless $\rho_0(x)$ is constant. The same form of counterexample was also proposed in \cite{karafyllis2020analysis} with $\delta\in\mathbb{Q}$ and $\rho_0(x-\bar{u}t)$ being a sine-wave.
This counterexample justifies the key role of the assumption ({\bf A3}). When $\rho_0(x)$ is $\mathbf{C}^1$ smooth and bounded between $\rho_{\text{min}}>0$ and $\rho_{\text{max}}\leq1$, so is the solution. All assumptions in Theorem~\ref{thm:main} are satisfied expect ({\bf A3}). However, the conclusion of the theorem fails to be true because we no longer have the nonlocal Poincare inequality \eqref{eq:nonlocal_poincare} for the constant kernel. To wit, we calculate eigenvalues of the nonlocal gradient operator $\mathcal{D}_x^\delta$ with the constant kernel defined in \eqref{eq:constant_kernel}. For the eigenfunction $e^{2\pi imx}$ with frequency $m=1/\delta$, the real part of the corresponding eigenvalue is: \begin{align*}
b_\delta(m)=\frac1{\delta\nu(\delta)}\int_0^\delta \sin(2\pi ms)\,ds=0, \end{align*} which makes $\alpha=0$ in \eqref{eq:estimate_alpha}. In other words, the properties of the nonlocal kernel $w_\delta(\cdot)$ are essential to guarantee that the nonlocal term in \eqref{eq:nonlocal_diffusion} adds appropriate diffusion effect to dissipate traffic waves.
\section{Numerical experiments}\label{sec:numerical_exp}
In this section, we present results of numerical experiments to further illustrate the established findings and to explore cases not covered by the theoretical results. The following models are considered. \begin{itemize}
\item The local LWR \eqref{eq:lwr};
\item The nonlocal LWR \eqref{eq:nonlocal_lwr} with the linear decreasing kernel $w_\delta(s)=2(\delta-s)/\delta^2$;
\item The nonlocal LWR \eqref{eq:nonlocal_lwr} with the constant kernel $w_\delta(s)=1/\delta$. \end{itemize}
All three models are solved by the Lax-Friedrichs scheme with spatial mesh size $\Delta x=2\times10^{-4}$. For more details about this numerical scheme applying to the nonlocal LWR, see \cite{goatin2016well}. To visualize the evolution of traffic densities solved from the models, we plot their snapshots at selected times, with different colors. Furthermore, we compare asymptotic convergence speeds of the solutions to the uniform flow by plotting the $\mathbf{L}^2$ error $\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}$ as a function of time $t$. We present these convergence speed plots on different time scales, that is, the semi-log plot to represent the cases with an exponential decay in time, i.e., $\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}\propto e^{-\lambda t}$ for some $\lambda>0$, and the log-log plot for cases showing only an algebraic decay in time, in particular, $\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}\propto 1/t$. We also remark that, with the linear decreasing kernel, the exponent $\lambda$ of the exponential decay rate can be estimated theoretically by \eqref{decay_rate_precise} and \eqref{decay_rate_asymptotic}. These theoretical estimates are compared with the value of $\lambda$ estimated from the numerical solutions.
{\bf Experiment 1.} The first experiment aims to validate the quick dissipation of traffic waves established in Theorem~\ref{thm:main}. In this experiment, we choose a bell-shape initial data: \begin{align*}
\rho_0(x)=0.4+0.6\exp\left(-100(x-0.5)^2\right). \end{align*} It represents the scenario that initially vehicles cluster near $x=0.5$ and the traffic is lighter in other places. We compare solutions of three models solved with the initial data. The results are plotted in Figure~\ref{fig:bellshape}.
For the local LWR, the solution first develops a shock wave from the smooth initial data. Then the shock wave dissipates at a speed no faster than the algebraic decay. At time $t=6$, the $\mathbf{L}^2$ error $\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}$ is on the scale of $10^{-2}$ and one can still visually observe a jump in density at the shock.
For the nonlocal LWR, the nonlocal range is set as $\delta=0.2$. With the linear decreasing kernel, the solution remains smooth and the initial high density near $x=0.5$ quickly dissipates. The solution converges to the uniform flow exponentially with the numerically estimated exponent $\lambda=1.26$, which is very close to the theoretical estimate $\lambda=1.23$ given by \eqref{decay_rate_precise}.
At time $t=6$, the whole density profile is nearly uniform with the $\mathbf{L}^2$ error on the scale of $10^{-4}$. With the constant kernel, the solution first has an exponential convergence to the uniform flow, but the $\mathbf{L}^2$ error stagnates on the scale of $10^{-3}$ after $t=2.5$, which means that there are non-dissipative traffic waves with small amplitudes. The contrast between the case using a linear decreasing kernel and that with a constant kernel helps to illustrate the natural design principle concerning the use of nonlocal information, that is, placing more attention on the nonlocal density information of nearby vehicles could result in better traffic conditions.
{\bf Experiment 2.} The second experiment aims to check the case with linear initial data as discussed in Section~\ref{sec:discussion}. In this experiment, we choose the initial data to be a linear function as in \eqref{eq:linear_ini} with $\beta=0.5$. We compare solutions of three models solved with the initial data. The results are plotted in Figure~\ref{fig:linear}.
For the local LWR, the solution is a piecewise linear function as given in Section~\ref{sec:discussion}. At $t=\frac1{2\beta}=1$, a shock wave forms. Then the shock wave dissipates and the solution converges to the uniform flow with an algebraic decay in the $\mathbf{L}^2$ error, i.e., $\norm{\rho(\cdot,t)-\bar{\rho}}_{\mathbf{L}^2}=a/t$ where the estimated value of the coefficient is $a=0.142$. The result validates the analytically derived value $a=\frac{1}{2\sqrt{12}}=0.1443\dots$ based on the estimate in \eqref{eq:local_linear_convergence}.
For the nonlocal LWR, the nonlocal range is set as $\delta=0.2$. With the linear decreasing kernel, the exponential convergence to the uniform flow is observed with the numerically estimated exponent $\lambda=0.66$, also effectively predicted by the theoretical estimate $\lambda=0.61$ given by \eqref{decay_rate_precise}. Meanwhile, we also observe that the traffic density is no longer piecewise linear when $t>0$. A complicated dynamic process is involved in the transition from the linear initial data to the uniform flow. This shows that the nonlocal LWR may have richer transient behaviors than the local LWR. With the constant kernel, similar patterns are observed but the $\mathbf{L}^2$ error stagnates on the scale of $10^{-2}$ after $t=2.5$ because of the existence of non-dissipative traffic waves.
{\bf Experiment 3.} The third experiment aims to validate the counterexample given in Section~\ref{sec:counter_example}. In this experiment, we choose the initial data to be a sine-wave: \begin{align*}
\rho_0(x)=0.5+0.4\sin(4\pi x), \end{align*} which is periodic with a period $0.5$. We compare two solutions with the initial data: one solved from the nonlocal LWR with the linear decreasing kernel, the other solved from that with the constant kernel. For both cases, the nonlocal range is set to be the same as the period of the initial data, i.e., $\delta=0.5$. The results are plotted in Figure~\ref{fig:sinewave}.
With the linear decreasing kernel, the sine wave quickly dissipates and the solution converges to the uniform flow exponentially with the numerically estimated exponent $\lambda=2.02$. In comparison, the theoretical estimate \eqref{decay_rate_precise} gives $\lambda=2.00$ while the asymptotic estimate \eqref{decay_rate_asymptotic} gives $\lambda=3.29$. The result shows that the the actual exponent $\lambda$ may deviate away from the linear relation described by \eqref{decay_rate_asymptotic} when $\delta$ is large but can still be effectively predicted by \eqref{decay_rate_precise}.
With the constant kernel, the solution is a traveling wave moving at the constant speed $\bar{u}=1-\bar{\rho}=0.5$ and the $\mathbf{L}^2$ error stays constant in time and never decays. In this case, vehicles need to repeatedly accelerate and decelerate in accordance with the oscillations in traffic density, resulting in a worse traffic situation even in comparison to that modeled by the local LWR. This again reinforces the advantage of paying more attention to the nearby density information when nonlocal information is utilized.
{\bf Experiment 4.} The fourth experiment aims to examine the impact of the nonlocal range $\delta$. In this experiment, we focus on the nonlocal LWR with the linear decreasing kernel, and choose a piecewise constant initial data: \begin{align*}
\rho_0(x)=\begin{dcases} 0.25,\quad 0\leq x<0.5;\\0.75,\quad 0.5\leq x<1.\end{dcases} \end{align*} For various values of the nonlocal range $\delta$, solutions of the model with the initial data are compared. The results are plotted in Figure~\ref{fig:piece_const}.
In the first row of Figure~\ref{fig:piece_const}, we plot traffic density evolution for the solutions with $\delta=0.1$ and $\delta=0.2$. Although the initial data is discontinuous, the dissipation of traffic waves and the convergence to the uniform flow can still be observed. This result indicates that the regularity assumption in Theorem~\ref{thm:main} might not be necessary, as discussed in Section~\ref{sec:discussion}.
In the bottom left figure of Figure~\ref{fig:piece_const}, we compare the decay rates of convergence for the solutions with $\delta$ ranging from $0.1$ to $0.3$ with a step $0.05$. The result shows that all solutions have exponential convergence to the uniform flow for these values of $\delta$, while the convergence is faster with a larger $\delta$. The exponent $\lambda$ numerically estimated from the solutions with these values of $\delta$ are compared with the theoretical estimates given by \eqref{decay_rate_precise}, as shown in the bottom right figure of Figure~\ref{fig:piece_const}. One can observe an effective match between the theoretical and numerical estimates.
{\bf Experiment 5.} Finally, let us examine how the mean density affects the decay rate of convergence. In this experiment, we focus on the nonlocal LWR with the linear decreasing kernel. The nonlocal range is fixed to be $\delta=0.2$. We choose a family of bell-shape initial data: \begin{align*}
\rho_0(x)=\rho_{\text{min}}+0.6\exp\left(-100(x-0.5)^2\right), \end{align*} with $\rho_{\text{min}}\in[0, 0.4]$. Such a family of initial data have the same variation but different mean densities. We compare the decay rates of convergence for the solutions with $\rho_{\text{min}}$ ranging from $0$ to $0.4$ with a step $0.1$, as shown on the left side of Figure \ref{fig:ini}.
We first observe that even in the case with $\rho_{\text{min}}=0$, meaning that the initial data is supported on a subinterval of the domain and vanishes outside, the solution still converges to the uniform flow exponentially. This example shows the possibility that the established global stability result may still be true for nonnegative initial data with positive mean densities, which presents an interesting problem to be further studied theoretically in the future. Moreover, we observe that the convergence becomes faster as the mean density of the initial data gradually increases, again consistent with the theoretical findings discussed earlier. To better capture the dependence of the exponent $\lambda$ on the mean density $\bar{\rho}$, we do a linear fitting of the numerically estimated values of $\lambda$ with respect to $\bar{\rho}$, see the plot on the right side of Figure~\ref{fig:ini}. The result shows that a linear relation of the form $\lambda=2.53\bar{\rho}$ can effectively describe the dependence of $\lambda$ on $\bar{\rho}$. As a comparison, the theoretical estimate \eqref{decay_rate_precise} gives $\lambda=2.43\bar{\rho}$, which is also shown on the right side of Figure~\ref{fig:ini}. We observe that the theoretical estimate effectively matches numerical observations. In addition, the largest deviation of the numerical estimate of $\lambda$ from both the linear fitting and the theoretical estimate occurs when $\rho_{\text{min}}=0$. This is an interesting phenomenon indicating that there might be a sharper estimate with the existence of vacuum densities in initial data.
\begin{figure}
\caption{Compare solutions from different models with the bell-shape initial data.
}
\label{fig:bellshape}
\end{figure}
\begin{figure}
\caption{Compare solutions from different models with the linear initial data}
\label{fig:linear}
\end{figure}
\begin{figure}
\caption{Compare solutions from different models with the sine-wave initial data}
\label{fig:sinewave}
\end{figure}
\begin{figure}
\caption{Compare solutions with different choices of nonlocal range $\delta$}
\label{fig:piece_const}
\end{figure}
\begin{figure}
\caption{Compare solutions with initial data with different mean densities}
\label{fig:ini}
\end{figure}
\section{Conclusions and future work}\label{sec:conclusion}
This paper studies global stability of a nonlocal traffic flow model, i.e., the nonlocal LWR that assumes vehicles' velocities depend on the nonlocal traffic density. Mathematically, the model is a scalar conservation law with a nonlocal term. Under some assumptions, we prove that the solution of the nonlocal LWR model converges exponentially to the uniform flow as time goes to infinity. The key assumption is that the nonlocal kernel should be non-increasing and non-constant. It reveals a simple but insightful principle for connected vehicle algorithm design that nearby information should deserve more attention. Indeed, equal attention (as associated with the constant kernel) might allow non-uniform traffic patterns to persist in time. Moreover, the analysis on the parameter dependence also shows the importance of choosing suitable ranges of nonlocal information for achieving the best effectiveness in traffic stabilization.
From the mathematical perspective, our proof relies on a couple of assumptions. We believe that the non-increasing and non-constant assumption on the nonlocal kernel plays the key role and other assumptions can be relaxed. For example, we have discussed how the regularity assumption on the solution can be relaxed. Further extensions can also be considered and the same global stability analysis may still be applicable for more general nonlinear desired speed functions as well as variants of other nonlocal macroscopic traffic models. It is also interesting to discuss asymptotic convergence to the uniform flow in other metrics. In particular, it will be interesting to consider in the future the $\mathbf{L}^1$ metric, which is popular for local conservation laws, and more general $\mathbf{L}^p$ metrics for $p\neq 2$.
From the application perspective, it is an interesting question to check the generality of the proposed design principle for connected vehicles in real traffic. For example, we are currently exploring the possible impact on utilizing nonlocal information both in space and time. Moreover, a realistic traffic system can be modeled on different scales using different models. For example, we may consider microscopic traffic models with a given number of discrete vehicles, or nonlocal traffic flow models based on Arrhenius type dynamics or having nudging (``look-behind'') terms. In addition, one may study how to extend the findings presented here to other traffic flow models involving both microscopic and macroscopic scales.
\end{document} | arXiv | {
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\begin{document}
\title{Quantum Network Recovery from Multinode Failure using Network Encoding with GHZ-States on Higher-Order Butterfly Networks}
\begin{abstract} We propose a protocol to transmit three quantum states crossly in a butterfly network with prior entanglement, in the form of GHZ states, between three senders. The proposed protocol requires only one qubit transmission or two classical bits transmission in each channel of the network. We generalize this protocol to higher number of qubits with multiqubit GHZ states towards quantum network operability using network coding with multiqubit GHZ states on higher-order butterfly networks. \end{abstract}
\section{Introduction}
Quantum correlations, particularly prior entanglement across quantum states, can be harnessed for transmitting more classical information through quantum communication links through teleportation and superdense coding schemes \cite{gisin2007quantum,mattle1996dense, pan2001entanglement, harrow2004superdense, vaidman1994teleportation}. The physical realization of quantum networked systems at atomic scale distances using such entangled quantum states is key towards realizing high-throughput quantum communications at such scales. The ideas from classical network coding, such as coding over butterfly networks, can be naturally extended to the quantum case, mindful of the quantum no-go theorems \cite{hayashi2007quantum,ahlswede2000network}. In the butterfly network, two units of information can sent crossly and the channels can transmit only one bit, with the bottleneck being the central channel in the network. It was recently shown that perfect quantum state transmission is impossible in the butterfly network and the bottleneck, in the form of the central channel, cannot be resolved for a quantum network \cite{hayashi2007quantum}. This was subsequently extended to different kinds of networks with quantum network coding \cite{iwama2006quantum}. \textit{Leung et al} proposed network coding using shared entanglement between two parties \cite{leung2010quantum} through quantum teleportation and superdense coding. Hayashi demonstrated the impossibility of transmitting quantum states over the butterfly network between two senders without prior entanglement \cite{hayashi2007prior}. In this \textit{Letter}, we have formulated a non-trivial extension of Hayashi’s results to the case of higher-order butterfly networks using GHZ states. This extension is useful to realize polygon tesselated higher-order butterfly networks for recovery of network operability post detection of erased nodes.
\section{Network Coding with GHZ States on higher-order Butterfly Networks} Our protocol is based on a network with prior entanglement shared between three users. We fundamentally use teleportation for the proposed protocol. \begin{figure}
\caption{Network for Quantum Butterfly Network comprising of three users with three GHZ states and an arbitrary quantum state for each transmission, alongwith four central channels: $E = (X_{1}^{(a)}\oplus X_{2}^{(a)} \oplus X_{3}^{(a)},X_{1}^{(b)}\oplus X_{2}^{(b)} \oplus X_{3}^{(b)})$, $E_1 = (X_{1}^{(a)}\oplus X_{2}^{(a)},X_{1}^{(b)}\oplus X_{2}^{(b)})$, $E_2 = (X_{2}^{(a)}\oplus X_{3}^{(a)},X_{2}^{(b)}\oplus X_{3}^{(b)})$ and $E_3 = (X_{1}^{(a)}\oplus X_{3}^{(a)},X_{1}^{(b)}\oplus X_{3}^{(b)})$.}
\label{Figure 1: Physical Realization}
\end{figure} We assume that the three senders $A_{1}$, $A_{2}$ and $A_{3}$ share three copies of maximally entangled (GHZ) states: $ \vert\phi_{i}\rangle = \frac{1}{\sqrt{2}}(\vert000\rangle + \vert111\rangle)_{(1,i),(2,i),(3,i)}$, where $i = 1, 2, 3$, denotes the $i^{\mathrm{th}}$ GHZ state, while the qubits with the same first index in the subscript belong to the same physical terminal. The senders prepare their states in: $\vert\psi_{j}\rangle = \alpha_{j}\vert0\rangle + \beta_{j}\vert1\rangle$, where $j = 1,2,3$ denotes the $j^{\mathrm{th}}$ user. In the first step, the sender $A_{i}$ performs a Bell state measurement $\{\phi_{+},\phi_{-},\psi_{+},\psi_{-}\}$ on the joint system $A_{i}\otimes A_{i,i}$. We can see the decomposition for $A_{i}\otimes A_{i,i}\otimes A_{(i+1) \mathrm{mod} 3,i}\otimes A_{(i+2) \mathrm{mod} 3,i}, i = 1,2,3$ in \textit{Table 1}. Here $A_{i}$ represents the arbitrary prepared state, while $A_{i,i}, A_{(i+1) \mathrm{mod} 3,i}$ and $A_{(i+2) \mathrm{mod} 3,i}$ are qubits from the three GHZ states at the physical terminal $i$. After the first Bell state measurement, we undertake an additional measurement on the single qubit component of the GHZ state at one of the other two nodes. We denote this measurement as $X_{i}^{(b)}$, with the first measurement being tagged as $X_{i}^{(a)}$. In this measurement, we measure single qubits in the $\vert\pm\rangle= \frac{1}{\sqrt{2}}(\vert0\rangle\pm\vert1\rangle$ basis. \\ \\ The action of Bell state measurement leaves a state $U(X_{3}^{a},X_{2}^{b})^{-1}\vert\psi_{3}\rangle$ or $U(X_{2}^{a},X_{3}^{b})^{-1}\vert\psi_{2}\rangle$ depending on a sequence of measurements at the first qubit. Then the first terminal applies $U(X_{1}^{a},X_{1}^{b})$. The cumulative state can be represented by either of the following two cases \begin{equation}
U(X_{1}^{a},X_{1}^{b})U(X_{3}^{a},X_{2}^{b})^{-1}\vert\psi_{3}\rangle = c_{1132}U((X_{1}^{a},X_{1}^{b})\oplus (X_{3}^{a},X_{2}^{b}))^{-1}\vert\psi_{3}\rangle \end{equation} \begin{equation}
U(X_{1}^{a},X_{1}^{b})U(X_{2}^{a},X_{3}^{b})^{-1}\vert\psi_{2}\rangle=c_{1123}U((X_{1}^{a},X_{1}^{b})\oplus (X_{2}^{a},X_{3}^{b}))^{-1}\vert\psi_{2}\rangle \end{equation} where $\vert c_{1123}\vert = \vert c_{1132}\vert = 1$, depending on whether terminal 2 or 3 measures $(a)$ or $(b)$. If we are to consider the clockwise cyclicity and the indices `wrapping around' (with $X_{4}=X_{1}$), we have the transformation $U((X_{i}^{a},X_{i}^{b})\oplus (X_{i+1}^{a},X_{i+2}^{b}))^{-1}$. To be able to recover the state at the receiving terminals, we formulate a combination of classical bits $X_{1}^{a}, X_{1}^{b}, X_{2}^{a}, X_{2}^{b}, X_{3}^{a}$ and $X_{3}^{b}$ to obtain an inverse unitary transformation for recovering the states. For the scheme with three users and three 3-qubit GHZ states, we use four central channels: $E = (\sum_{\oplus i=1}^{3}X_{i}^{(a)},\sum_{\oplus i=1}^{3}X_{i}^{(b)})$ and $E_{j} = (\sum_{\oplus i=1 \backslash j}^{3}X_{i}^{(a)},\sum_{\oplus i=1 \backslash j}^{3}X_{i}^{(b)})$. Classical information is transmitted using these four central channels to the three users, upon which the inverse transformation can be applied to the quantum state received at the terminal to get the cross-transmitted states. \begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$X_{i}^{(a)}$ & $X_{i}^{(b)}$ & $A_{i}\otimes A_{i,i}$ & $A_{(i+1) \mathrm{mod} 3,i}\otimes A_{(i+2) \mathrm{mod} 3,i}$ & $U$ \\ \hline
(0,0) & 0/1 & $\vert\psi_{+}\rangle$ & $\alpha_{1}\vert00\rangle+\beta_{1}\vert11\rangle$ & $I$/$\sigma_{z}$ \\ \hline
(0,1) & 0/1 & $\vert\psi_{-}\rangle$ & $\alpha_{1}\vert00\rangle-\beta_{1}\vert11\rangle$ & $\sigma_{z}$/$I$\\ \hline
(1,0) & 0/1 & $\vert\phi_{+}\rangle$ & $\alpha_{1}\vert11\rangle+\beta_{1}\vert00\rangle$ & $\sigma_{x}$/$\sigma_{x}\sigma_{z}$ \\ \hline
(1,1) & 0/1 & $\vert\phi_{-}\rangle$ & $\alpha_{1}\vert11\rangle-\beta_{1}\vert00\rangle$ & $\sigma_{x}\sigma_{z}$/$\sigma_{x}$\\ \hline
\end{tabular}
\caption{Decomposition for $ A_{i}\otimes A_{i,i}\otimes A_{(i+1) \mathrm{mod} 3,1}\otimes A_{(i+2) \mathrm{mod} 3,1} \forall i =1,2,3$, and truth values as well as associated unitary transformations for higher-order butterfly network over three nodes}
\label{tab:my_label} \end{table} This proposal can be generalised to the case of $n$-qubits by considering an $n$-qubit GHZ state: $\vert\xi\rangle = \frac{1}{\sqrt{2}}(\vert0^{\otimes n}+\vert1^{\otimes n})$. In this case, there will be $n$+1 channels with $E = (\sum_{\oplus i=1}^{n}X_{i}^{(a)},\sum_{\oplus i=1}^{n}X_{i}^{(b)})$ and $E_{j} = (\sum_{\oplus i=1 \backslash j}^{n}X_{i}^{(a)},\sum_{\oplus i=1 \backslash j}^{n}X_{i}^{(b)})$. As in the case of the two parties, perfect transmission of quantum states is impossible without prior entanglement. Following the analysis of \textit{Hayashi} \cite{hayashi2007prior} for the case of EPR state, we find the bound on fidelity for a generalised case to be \begin{equation}
\sum_{i=1}^{d}f_{i}\leq \frac{2.8512 d}{d+1} \end{equation} where $f_{i}$ is the average entanglement fidelity of the $i^{\mathrm{th}}$ channel and $d$ is the number of nodes in the quantum network. The operation of this higher-order butterfly scheme cannot be undertaken without prior entanglement. \section{Scheme for Quantum Network Recovery from Multinode Failure using Tesselated GHZ-Butterfly Subnetworks}.
The $n$-qubit GHZ states can be used for network recovery after multinode failure for an arbitrary graph state using tesselation of such GHZ-states in partitioned blocks of the graph, as shown in \textit{Figure 2 (a)}. The scheme for undertaking network recovery using the higher-order GHZ state based network coding, formulated in the previous section, comprises of two distinct steps: Checking for operability of nodes and recovery of network operability. We prepare the graph state that constitutes the quantum network by initialising the states at the first terminal/node $\vert\phi\rangle$ and the other nodes with the state $\vert+\rangle=\frac{1}{\sqrt{2}}(\vert0\rangle+\vert1\rangle)$. We then operate with the \textit{CPHASE} operation: $CZ_{ij}=\vert0\rangle\langle0\vert_{i}I_{j}+\vert1\rangle\langle1\vert_{i}Z_{j}$, where $Z$ is the Pauli-z matrix, between adjacent nodes. This gives us the entangled graph state which is the primary resource in the quantum network being studied. \\ \\ For checking for operability of nodes, we use a hybrid classical-quantum approach. Classically, communication of node failure is done using `pings' that are used in the ICMP echo protocol. However, this is not enough to ascertain quantum accessibility and entanglement at the node. Additional to the classical connection, we need an additional layer to enable possibility to assess whether entanglement at the node is accessible and operative. This is done using quantum non-destructive measurements on the nodes of the graph, and any instance of node failure is relayed to the rest of the nodes in the affected subgraph using classical infrastructure.
For optical systems, we can utilise non-destructive detection using an optical resonator containing a single atom prepared in a superposition of two states \cite{reiserer2014quantum}. \\ \begin{figure*}
\caption{Illustrative example for scheme for recovery of network operability from multinode failure: (a) The nodes begin with $\vert\phi\rangle$ at one node and $\vert+\rangle$ at the others, and we apply C-Phase operations on all the adjacent nodes. We then consider the possibility of two nodes failing (marked in grey). The coloured lines represent the GHZ states shared between adjacent nodes. (a)-(c) Mechanism for recovery upon multinode recovery of the network, using GHZ-based network coding in segments of the composite network. Inset is the detector unit comprising of an optical resonator with a three-level atom in a single-sided cavity.}
\label{Figure 1: Physical Realization}
\end{figure*} \\ At each node, we position an optical resonator system with a three-level atom in a single-sided cavity, where one of the two mirrors is perfectly reflecting and there is a small transmission coefficient of the other mirror that allows for in- and outcoupling of light, which is an optical photon. Let the three states of the atom be tagged $\vert1_{a}\rangle$, $\vert2_{a}\rangle$ and $\vert3_{a}\rangle$. The cavity is designed such that it is overcoupled and resonant with the transition between the atomic states $\vert2_{a}\rangle$ and $\vert3_{a}\rangle$. If we prepare the atom initially in the state $\frac{\vert1_{a}\rangle+\vert2_{a}\rangle}{\sqrt{2}}$, an impinging photon makes it transform to $\frac{\vert2_{a}\rangle-\vert1_{a}\rangle}{\sqrt{2}}$ while it remains unchanged in the absence of an impinging photon. This is due to there being no interaction between the atom and photon when the atom is in the state $\vert1_{a}\rangle$ since any transition is far detuned, while when the atom is in the state $\vert2_{a}\rangle$, the strong photon-atom coupling can lead to a normal-mode splitting and the photon undergoes reflection without entering the cavity. We can measure this phase flip using a $\frac{\pi}{2}$ rotation map to map $\frac{\vert1_{a}\rangle+\vert2_{a}\rangle}{\sqrt{2}} \rightarrow \vert1_{a}\rangle$ and $\frac{\vert1_{a}\rangle-\vert2_{a}\rangle}{\sqrt{2}} \rightarrow \vert2_{a}\rangle$, and then cavity-enhanced fluorescence state detection can be used to distinguish between the states $\vert2_{a}\rangle$ and $\vert1_{a}\rangle$ \cite{bochmann2010lossless}. \\ \\ Upon the non-destructive detection of a lost photon, the classical infrastructure used for classical communication in our model for higher-order butterfly networks is used to alert the other nodes in the subnetworks of which the failed node(s) is(are) a part of. This is undertaken using a parity code check. This comprises of $E = (\sum_{\oplus i=1}^{n}X_{i}^{(a)},\sum_{\oplus i=1}^{n}X_{i}^{(b)})$ and $E_{j} = (\sum_{\oplus i=1 \backslash j}^{n}X_{i}^{(a)},\sum_{\oplus i=1 \backslash j}^{n}X_{i}^{(b)})$ for $n$ nodes of a subnetwork. However, unlike in the case of the higher-order butterfly network, we do not send any single- or two-qubit measurements over these channels but rather if the check on each node yields a success (`1') or failure (`0'). In this manner, each node in the subnetwork can exactly know which node(s) is(are) non-operative. Upon ascertaining the same, we operate with a $\sigma_{z}$ operate to remove the particular failed node(s) from the graph state. \\ \\ For recovery of network operability, we can use the network coding formalism for higher-order butterfly networks developed and discussed in the previous section. In this framework, the key point is to tesselate the complex multiqubit network with subnetworks comprising of $n_{i}$ qubit GHZ state for the $i^{\mathrm{th}}$ subnetwork comprising of $n_{i}$ nodes, along with the requisite classical infrastructure, as highlighted in the model formulated for higher-order butterfly networks. As soon as we ascertain which nodes are non-operative, we can use the GHZ-state in the subnetwork to replace the failed node, as shown in \textit{Figure 2 (a)-(c)}. Failure of all nodes that a subnetwork shares with the rest of the network, along with both additional qubits on either side of this set precipitates a point of criticality wherein the network operability cannot be recovered using the formulated scheme. The recovery of rate-optimal coded graph state networks can also be realised using \textit{modified graph-state codes} \cite{priya}. \section{Conclusion} In this \textit{Letter}, we have formulated and constructed a higher-order butterfly network using a multiqubit GHZ states. We studied the fidelity bounds of transmission in such GHZ-based higher-order butterfly networks that scale as $\frac{n}{n+1}$ for the number of terminals, which we use to construct a scheme for quantum network recovery from multinode failure. This is particularly useful for making quantum networks resilient against failures at multiple nodes.
\end{document} | arXiv | {
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\begin{document}
\title{
Colouring bottomless rectangles and arborescences
}
\begin{abstract}
We study problems related to colouring bottomless rectangles. One of our main results shows that for any positive integers $m, k$, there is no semi-online algorithm that can $k$-colour bottomless rectangles with disjoint boundaries in increasing order of their top sides, so that any $m$-fold covered point is covered by at least two colours. This is, surprisingly, a corollary of a stronger result for arborescence colourings. Any semi-online colouring algorithm that colours an arborescence in leaf-to-root order with a bounded number of colours produces arbitrarily long monochromatic paths. This is complemented by optimal upper bounds given by simple online colouring algorithms from other directions.
Our other main results study configurations of bottomless rectangles in an attempt to improve the \textit{polychromatic $k$-colouring number}, $m_k^*$. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, $m_k^*$ is linear in $k$. We also present an improved lower bound for general families: $m_k^* \geq 2k-1$. \end{abstract}
\section{Introduction} The systematic study of polychromatic colourings and cover-decomposition of geometric ranges was initiated by Pach over 30 years ago \cite{P80,P86}. The field has gained popularity in the new millennium, with several breakthrough results; for a (slightly outdated) survey, see \cite{survey}, or see the up-to-date interactive webpage \url{http://coge.elte.hu/cogezoo.html} (maintained by Keszegh and P\'alv\"olgyi).
A family of geometric regions $\ensuremath{\mathcal F}\xspace$ and a point set $P$ in some $\ensuremath{\mathbb R}\xspace^d$ naturally define a primal hypergraph, $H(P, \ensuremath{\mathcal F}\xspace)$. The vertex set of $H$ is the points of $P$, where $\ensuremath{\mathcal F}\xspace' \subset \ensuremath{\mathcal F}\xspace$ is an edge if for some $p \in P$, $p$ is covered by exactly the regions in $\ensuremath{\mathcal F}\xspace'$. We are interested in the dual hypergraph, $H(\ensuremath{\mathcal F}\xspace, P)$, on the vertex set $\ensuremath{\mathcal F}\xspace$, where $p \in P$ is an edge if for some $\ensuremath{\mathcal F}\xspace' \subset \ensuremath{\mathcal F}\xspace$, $p$ is covered by exactly the regions in $\ensuremath{\mathcal F}\xspace'$.
\begin{figure}
\caption{\small{A family of circles, a finite point set, and the corresponding dual hypergraph.}}
\label{fig:circles}
\end{figure}
We can then define the chromatic number $\chi_{\ensuremath{\mathcal F}\xspace}$ of any family $\ensuremath{\mathcal F}\xspace$ of geometric regions. This is the minimum number of colours needed to colour any \textit{finite} point set $P$ so that any region containing at least two points contains at least two colours. The dual chromatic number, $\chi_{\ensuremath{\mathcal F}\xspace}^*$, is defined analogously as the number of colours needed to properly colour any \textit{finite} subfamily of $\ensuremath{\mathcal F}\xspace$. In this paper, we will study the \textit{polychromatic colouring numbers}. \begin{definition} The $k$-th (primal) \textit{polychromatic colouring number} $m_k(\ensuremath{\mathcal F}\xspace)$ is the smallest number needed to $k$-colour any finite point set $P$ so that any region containing at least $m_k(\ensuremath{\mathcal F}\xspace)$ points contains all $k$ colours.
Its dual $m_k^*(\ensuremath{\mathcal F}\xspace)$ is the smallest number needed to $k$-colour any finite subfamily of $\ensuremath{\mathcal F}\xspace$ so that any point covered by $m_k^*$ regions is covered by all $k$ colours. \end{definition} The polychromatic colouring problem is partly motivated by the sensor cover problem; given a set of sensors covering an area, can we decompose them into $k$ sets so that any area covered by $m_k^*$ sensors is covered in each of these sets? When $k=2$, this is called the cover-decomposability problem. In particular, we say a set $P \subset \ensuremath{\mathbb R}\xspace^2$ is \textit{cover-decomposable} if $m_2^*(\ensuremath{\mathcal F}\xspace_P) < \infty$, where $\ensuremath{\mathcal F}\xspace_P$ is the family of all \textit{translates} of $P$. In \cite{P86} it was shown that every centrally-symmetric open convex polygon is cover-decomposable, and this was extended to all open convex polygons in \cite{PT10}. The bound $m_k^*(\ensuremath{\mathcal F}\xspace_P) = O(k)$ was proved for any convex polygon $P$ in \cite{GV09}.
The problem becomes more complicated if we consider \textit{homothets} of a convex polygon. For example, if $\ensuremath{\mathcal F}\xspace_{\square}$
denotes the family of axis-parallel squares in the plane, $m_2(\ensuremath{\mathcal F}\xspace_{\square} ) \leq 215$ \cite{AKV}. On the other hand, $m_2^*(\ensuremath{\mathcal F}\xspace_{\square} ) =\infty$, that is, for any number $m$ there is a family $\ensuremath{\mathcal F}\xspace_m$ of axis-parallel squares such that (1) each point in the plane is covered by at least $m$ squares, but (2) any $2$-colouring of $\ensuremath{\mathcal F}\xspace_m$ produces a point covered by squares of exactly one colour \cite{K13}. Furthermore, if $\ensuremath{\mathcal F}\xspace_{\sqsubset\!\sqsupset}$ denotes the family of axis-parallel rectangles in the plane, $m_2(\ensuremath{\mathcal F}\xspace_{\sqsubset\!\sqsupset}) = \infty$ \cite{CPST}. Consequently, $m_k(\ensuremath{\mathcal F}\xspace_{\sqsubset\!\sqsupset}) = \infty$ for any $k$. The dual $m_k^*(\ensuremath{\mathcal F}\xspace_{\sqsubset\!\sqsupset})$ is infinite as well; there is a constant $C>0$ such that for any numbers $n \geq r \geq 2$, there is a family of $n$ axis-parallel rectangles for which any colouring with at most $C\log n (r\log r)^{-1}$ colours produces a point covered by $r$ monochromatic axis-parallel rectangles\cite{PT08}.
This paper focuses on one particular family: \textit{bottomless rectangles}. \begin{definition} A subset of $\ensuremath{\mathbb R}\xspace^2$ is called a (closed) bottomless rectangle if it is of the form $\{(x,y): l \leq x \leq r, y \leq t \}$. We simply refer to a bottomless rectangle by these paramaters $(l,r,t)$. \end{definition} These range spaces were first defined by Keszegh \cite{Khalf}, who showed $m_2=4$ and $m_2^*=3$. Later Asinowski et al.~\cite{A+13} showed that for any positive integer $k$, any finite set of points in $\ensuremath{\mathbb R}\xspace^2$ can be $k$-coloured such that any bottomless rectangle with at least $3k-2$ points contains all $k$ colors. They also showed that the optimal number that can be written in place of $3k-2$ in the above statement is at least $1.67k$. In our language, if $\ensuremath{\mathcal F}\xspace_{\sqcap}$ denotes the family of all bottomless rectangles in the plane, $1.67 k \leq m_k(\ensuremath{\mathcal F}\xspace_{\sqcap}) \leq 3k-2$.
Our paper studies the dual problem: we would like to determine the optimal $m_k^*(\ensuremath{\mathcal F}\xspace_{\sqcap})$. About this question much less is known; $m_2^* = 3$ \cite{Khalf}, while the best general upper bound is $m_k^*=O(k^{5.09})$, a corollary of a more general result \cite{CKMU} about \emph{octants} (combined with an improvement of the base case \cite{octantnine} that slightly lowered the exponent). The general conjecture, however, is that $m_k^*=O(k)$ for any family for which $m_2^*$ is finite \cite{survey}. It was also proved in \cite{CKMU} that there is no semi-online algorithm ``from above'' for colouring bottomless rectangles. One of our main results is a generalisation of this negative statement.
\begin{restatable}{theorem}{thmrect}\label{thm:rect} For any $k$ and $m$, and any semi-online algorithm that $k$-colours bottomless rectangles from below (resp.\ from above, from the right, or from the left), there is a family of bottomless rectangles such that the algorithm will produce an $m$-fold covered point that is covered by at most one colour. \end{restatable}
Our proof is much more complicated than the one in \cite{CKMU}; while there an Erd\H os-Szekeres \cite{ESz} type incremental argument is used, we need a certain diagonalisation method. In particular, we reduce the semi-online bottomless rectangle colouring problem to a question about semi-online colourings of arborescences, which is interesting in its own right.
\begin{restatable}{theorem}{thmarbone}\label{thm:arb1} For any $k$ and $m$, and any semi-online colouring algorithm that $k$-colours the vertices of an arborescence in a leaf-to-root order, there is an arborescence that has a leaf-to-root order such that the algorithm will produce a directed path of length $m$ that contains at most one colour. \end{restatable}
We apply this theorem to four natural configurations of bottomless rectangles to show that for each configuration, there is a direction from which a semi-online algorithm fails. Furthermore, we show in \autoref{thm:noshallowhittingset} that bottomless rectangles do not admit \textit{shallow hitting sets}, which are another standard tool to bound polychromatic colouring numbers.
These negative results are complemented by optimal upper bounds given by online algorithms from the other directions. We obtain linear bounds for $m^*_k$ for the following families of bottomless rectangles: \begin{itemize}
\item $m^*_k(\ensuremath{\mathcal{F}_{\mathrm{unit}}}\xspace) \leq 2k-1$ for \textit{unit-width} families, \autoref{prop:stepspt},
\item $m^*_k(\ensuremath{\mathcal{F}_{\mathrm{hanging}}}\xspace)\leq 2k-1$ for \textit{hanging} families, \autoref{thm:hanging}, and
\item $m^*_k(\ensuremath{\mathcal{F}_{\mathrm{int}}}\xspace)\leq 3k$ for \textit{intersecting} families. \autoref{thm:intersecting}, \end{itemize}
We also improve the current lower bound for $m^*_k$, by showing $m^*_k \geq 2k-1$ (\autoref{thm:lb}).
In \autoref{sec:configs}, we introduce certain configurations of bottomless rectangles, and define the corresponding colouring problem. We also prove our other main results: that for many families of bottomless rectangles, such as \textit{unit-width}\footnote{See \autoref{sec:unitwidth}} and \textit{intersecting}\footnote{See \autoref{sec:other}} families, $m_k^*$ is linear. In \autoref{sec:arb}, we prove \autoref{thm:arb1} and deduce \autoref{thm:rect} as a corollary of it. In \autoref{sec:further}, we look at other methods to improve the upper bound on $m_k^*$, and improve the lower bound for general families to $m_k^* \geq 2k-1$.
\section{Configurations of bottomless rectangles}\label{sec:configs}
\subsection{\texorpdfstring{Erd{\H o}s}{Erdos}-Szekeres configurations}\label{sec:esconfigs}
We would like to improve the upper bound $m_k^*(\ensuremath{\mathcal{F}_{\sqcap}}\xspace) = O(k^{5.09})$ for general families by classifying some configurations of bottomless rectangles, finding a colouring for each configuration, and combining these to obtain a good colouring for general families. To this end, we will use the classical result of Erd{\H o}s and Szekeres \cite{ESz} that any sequence of length $(k-1)^2+1$ contains a monotone subsequence of length $k$.
Recall that we associated to each rectangle its parameters $(l,r,t)$. We refer to $l$ as its left-coordinate, $r$ its right-coordinate, and $t$ its height. Let $p$ be a point covered by $(m-1)^4+1$ rectangles. Ordering these rectangles by left endpoint, we find a subsequence of length $(m-1)^2+1$ whose right endpoints are monotone. Applying the result of Erd{\H o}s and Szekeres again, we find a (sub)subsequence of length $m$ whose heights are monotone. This proves that any point that is contained in $(m-1)^4 +1$ bottomless rectangles is contained in $m$ bottomless rectangles such that each of the three parameters of these $m$ bottomless rectangles are in increasing or decreasing order. We name these configurations, respectively, \emph{increasing/decreasing steps, towers} and \emph{nested rectangles} (see \autoref{fig:configs}).
We are interested in colouring families with respect to a \textit{fixed} configuration. For example, can we $k$-colour a finite family $\ensuremath{\mathcal F}\xspace$ so that any point covered by an $m$-tower is covered by all $k$ colours? We refer to least such $m$ as $m_k^*$ \textit{for towers}, and analogously for the other configurations. \begin{theorem}\label{thm:configs} $m_k^* = k$ for each fixed configuration. \end{theorem} A result of Berge \cite{Berge} shows that for any family $\ensuremath{\mathcal F}\xspace$ of geometric regions, $m_2^*(\ensuremath{\mathcal F}\xspace) = 2$ if and only if $m_k^*(\ensuremath{\mathcal F}\xspace) = k$ for all $k$. It is not hard to show that $m_2^* = 2$ for each configuration, and then apply the result of Berge. Nevertheless, it is valuable to see that there is a simple online algorithm for each configuration.
\begin{figure}\label{fig:configs}
\end{figure}
\begin{definition} In this paper, we will consider the following two types of colouring algorithms for hypergraphs, which receive the vertices in some order. Both types must colour the vertices \textit{irrevocably} -- they are not allowed to recolour vertices. \begin{enumerate} [(1)]
\item An \emph{online algorithm} must colour each vertex immediately so that at each step, there is no conflict in the partial colouring. \footnote{There is a large literature on online algorithms \cite{GKMZ,HSz,LST}.}
\item A \textit{semi-online algorithm} need not colour each vertex immediately, but must ensure that at each step there is no conflict in the partial colouring. \end{enumerate} \end{definition}
This condition on the partial colouring means that, for example, at every step the algorithm must maintain that a point covered by $m$ rectangles is covered by all $k$ colours, but not all points have to be colored.
\begin{proof}[Proof of \autoref{thm:configs}] We first present a colouring algorithm for towers. We colour the rectangles in increasing order of height, i.e.\ from below, so that at every step the following property holds.
\begin{enumerate} \item[(*)] If a point $p$ is covered by a $j$-tower for $j \leq k$, then $p$ is covered by at least $j$ different colours. \end{enumerate}
At step $1$, colour the rectangle of least height arbitrarily. Suppose the first $t-1$ rectangles have been coloured so that (*) holds. We colour the rectangle $R_t$ as follows. For each $1 \leq i \leq k$, let $y_i$ be the largest number so that if $p \in R_t$ has $y$-coordinate less than $y_i$, then $p$ is covered by colour $i$. (This corresponds to a tallest rectangle $S$ of colour $i$ such that ($S,R_t$) is a tower.) If $y_i$ does not exist for some colour $i$, colour $R_t$ with colour $i$. Otherwise, suppose $y_1 > \dots > y_k$, and colour $R_t$ with colour $k$.
Property (*) holds: if $p$ is covered by a $j$-tower, then either $p$ was already covered by $j$ colours, or we added a new colour to the set of colours covering $p$. We use the same algorithm to colour $k$-nested sets, only we colour the rectangles from above. It is easy to check that with this ordering, the same property holds.
\begin{figure}
\caption{\small{Colouring algorithms for $k$-towers and $k$-nested sets respectively}}
\end{figure}
The algorithm for increasing $k$-steps is only slightly different. We colour the rectangles in decreasing order of right endpoint (from the right). At step $t$, for $1 \leq i \leq k$, let $x_i$ be the least number so that if $p \in R_t$ has $x$-coordinate greater than $x_i$, then $p$ is covered by a rectangle of colour $i$ (corresponding to the leftmost rectangle $S$ of colour $i$ such that ($R_t,S$) form increasing steps). As earlier, if some $x_i$ does not exist, give $R_t$ colour $i$. Otherwise, if $x_1 < \dots < x_k$, give $R_t$ colour $k$.
\begin{figure}
\caption{\small{Colouring algorithms for increasing and decreasing $k$-steps respectively}}
\end{figure} \end{proof} Note that ordering a tower in increasing order of height (from below) is the same as ordering it in decreasing order of right endpoint (from the right), or increasing order of left endpoint (from the left). We may repeat the same algorithm for towers, colouring the rectangles from the left (or right) and we will still obtain a good colouring. Similarly, ordering a nested set from above is the same as ordering it from the left or right. Ordering increasing steps from the right (resp.\ decreasing steps from the left) is the same as ordering them from above. This is all to say that the same algorithm can be used from these ``good'' directions for each fixed configuration.
\begin{center}
\bgroup
\def1.5{1.5}
\begin{tabular}[t]{|c|c|c|c|c|}
\hline
& \small{left ($\rightarrow$)} & \small{right ($\leftarrow$)} & \small{below ($\uparrow$)} & \small{above ($\downarrow$)}\\
\hline
\small{inc.\ steps} & \small{$\infty$} & \small{$=k$} & \small{$\infty$} & \small{$=k$} \\
\hline
\small{dec.\ steps} & \small{$=k$} &\small{$\infty$} &\small{$\infty$}&\small{$=k$}\\
\hline
\small{towers} & \small{$=k$} & \small{$=k$} & \small{$=k$} & \small{$\infty$} \\
\hline
\small{nested} & \small{$=k$} & \small{$=k$} & \small{$\infty$} & \small{$=k$} \\
\hline \end{tabular}\label{tab:configs} \captionof{table}{\small{$m_k^*$ values for each configuration given by semi-online algorithms from different directions.}} \egroup \end{center}
The value $\infty$ indicates the non-existence of semi-online colouring algorithms, which we prove in the next section.
The next natural question to ask is: how can we combine these colourings? Nested rectangles seem to have the ``simplest'' structure of the four configurations. Indeed, ordering nested rectangles from above is the same as ordering them from the left or right. Further, if ($R_1, R_2$) are nested ($R_1$ contains $R_2$), then any point in $R_2$ is neccessarily contained in $R_1$. This shows that we can modify the algorithms for the other configurations to also colour nested rectangles.
\begin{proposition}
$m_k^*=k$ if we $k$-colour any finite family $\mathcal{F}$ with respect to
\begin{enumerate}[(a)]
\item towers and nested sets
\item increasing steps and nested sets
\item decreasing steps and nested sets
\end{enumerate} \end{proposition} \begin{proof} We first present the algorithm for towers and nested sets. The precise statement is that any finite family $\mathcal{F}$ can be $k$-coloured so that any point contained in a $k$-tower or a $k$-nested set is covered by all $k$ colours. We colour the rectangles from the right (this can also be done from the left). We maintain the same property as earlier.
\begin{enumerate}
\item[(*)] If a point $p$ is covered by a $j$-tower or a $j$-nested set for $j \leq k$, then $p$ is covered by at least $j$ different colours.
\end{enumerate}
At step $t$, for $1 \leq i \leq k$, let $y_i$ be the greatest number so that if $p \in R_t$ has $y$-coordinate less than $y_i$, then $p$ is covered by a rectangle of colour $i$.
As earlier, if some $y_i$ does not exist, give $R_t$ colour $i$. Otherwise, suppose $y_1 > \ldots > y_k$, and give $R_t$ colour $k$.
To prove that (*) holds is not as straightforward as in \autoref{thm:configs}.
Let $y$ denote the height of $R_t$.
If $y_k > y$, or $y > y_1$, (*) holds by the same argument as in \autoref{thm:configs}.
If not, we have $y_1 > \dots > y_{l-1} > y > y_l > \dots > y_k$.
Suppose $p \in R_t$ is covered by a $j$-nested set $R_1, \dots, R_{j-1}, R_t$.
Since each $y_i$ is maximal, (*) holds by the same argument as earlier.
The only essentially different case is when $p \in R_t$ is covered by a $j$-tower.
If we did not add a new colour to the set of rectangles containing $p$, this means that $p$ was already covered by a rectangle of colour $k$. However, as $y_k$ was chosen to be maximal, the $y$-coordinate of $p$ must be less than $y_k$, so $p$ is already covered by all $k$ colours.
The algorithms for increasing and decreasing steps are modified in the exact same way. \end{proof}
\subsection{Unit-width rectangles}\label{sec:unitwidth} The next natural pair of Erd{\H o}s-Szekeres configurations to attempt to combine is the increasing and decreasing steps. For this, we consider a different setup; refer to \textit{steps} as the case when we assume that $\ensuremath{\mathcal F}\xspace$ does not contain any towers or nested sets. What is $m_{k, \text{steps}}^*$?
Let \textit{unit bottomless} be the case when all the rectangles in $\ensuremath{\mathcal F}\xspace$ have the same width, or unit width. It is clear that ``\textit{unit bottomless} $\subset$ \textit{steps}'', as any family of unit bottomless rectangles cannot contain towers or nested sets.
\begin{proposition} ``\textit{steps} $=$ \textit{unit bottomless}'', i.e.\ any family of tower- and nested set-free rectangles can be realised as a family of unit width bottomless rectangles so that the corresponding dual hypergraphs are isomorphic. \end{proposition} \begin{proof}
We prove the inclusion steps $\subseteq$ unit bottomless by our favourite method, induction on $|\mathcal{F}|$. Suppose any family $\mathcal{F}$ of $n-1$ rectangles that do not contain towers or nested sets can be realised as a family $\mathcal{F}_{unit}$ of unit bottomless rectangles (with an isomorphic hypergraph), and that this realisation preserves heights and the ordering of left endpoints. That is, the height of a rectangle $R$ in $\mathcal{F}$ is the same as its realisation in $\mathcal{F}_{unit}$.
Let $|\mathcal{F}| = n$, and let $R$ be the leftmost rectangle in $\mathcal{F}$. Take any realisation of $\mathcal{F}\setminus R$ as a family $\mathcal{G}$, and let $R_1, \dots, R_m$ be the rectangles that intersect $R$, and $R_1', \dots, R_m'$ their realisations. Assume without loss of generality that $l(R_1) < \dots < l(R_m)$.
In particular, $l(R_m) < r(R) < r(R_1)$ (as they intersect), so $R_1, \dots, R_m$ also intersect each other. This implies that the interval $[l(R_1'), \dots, l(R_m')]$ has length strictly less than $1$. Thus for $\epsilon$ small enough, if we realise $R$ as a unit width rectangle $R'$ with $r(R') = l(R_m') + \epsilon$ with the same height, then $R'$ will intersect exactly the rectangles $R_1', \dots, R_m'$ (with the same hypergraph structure). \begin{figure}
\caption{\small{We ensure that the realisation of $R$ preserves the hypergraph structure.}}
\end{figure} \end{proof}
So instead of considering colouring points with respect to bottomless rectangles, we may consider colouring steps with respect to points. \begin{proposition} \label{prop:stepspt}
For steps, $m_k^* \leq 2k-1$. \end{proposition} The proof of the proposition will use ABA-free hypergraphs \cite{ABA}. We say a hypergraph $\mathcal{H}$ with an ordering $<$ of its vertex set is ABA-free if there are no hyperedges $A$ and $B$ and vertices $x < y < z$ with $x,z \in A \setminus B$ and $y \in B \setminus A$. For example, interval hypergraphs - where the vertices are points in $\mathbb{R}$ and the hyperedges are the subsets induced by some intervals - are ABA-free. A result of \cite{ABA} tells us that for ABA-free hypergraphs, $m_k \leq 2k-1$.
\begin{proof}[Proof of \autoref{prop:stepspt}] Let $\mathcal{F}$ be a family containing no nested sets or towers and $P$ a finite point set. We claim that by ordering the rectangles by left endpoint, the resulting hypergraph on the vertex set $\mathcal{F}$ with edges induced by $P$ is ABA-free. Suppose for contradiction we have three rectangles with $l(R_1) < l(R_2) < l(R_3)$, and points $p$ and $q$ so that $p \in (R_1 \cap R_3) \setminus R_2$, and $q \in R_2 \setminus (R_1 \cap R_2)$.
Recall that a point $(x,y)$ is in a rectangle $R$ if and only if $x \in [l(R), r(R)]$ and $y < y(R)$. Let $p = (x_p, y_p)$ and $q = (x_q,y_q)$. Then, $p \in R_1, R_3$ but $p \notin R_2$ implies, \[ l(R_1) < l(R_2) < l(R_3) < x_p < r(R_1) < r(R_2) < r(R_3)\text{, and} \] \[ y(R_1), y(R_3) > y_p > y(R_2) \] \begin{figure}
\caption{\small{An arrangement of $p$ and the three rectangles implied by the above conditions.}}
\label{fig:stepsaba}
\end{figure}
As a result, $[l(R_2), r(R_2)]$ is covered by the intervals $[l(R_1), r(R_1)]$ and $[l(R_3), r(R_3)]$, and $R_2$ is below $R_1$ and $R_3$; the ``top side'' of $R_2$ is covered by the top sides of $R_1$ and $R_3$ as in \autoref{fig:stepsaba}. Thus any point in $R_2$ is contained in at least one of $R_1$ and $R_3$, contradicting that $q \in R_2$ but $q \notin R_1$ or $R_3$.
\end{proof}
We end this subsection by extending this to families that do not contain towers.
\begin{theorem}\label{thm:towerfree} For families $\mathcal{F}$ that do not contain towers, $m_k^* \leq 2k-1$. \end{theorem} \begin{proof} As earlier, we want to show that the corresponding hypergraph is ABA-free. Suppose again that we have three rectangles with $l(R_1) < l(R_2) < l(R_3)$, and points $p$ and $q$ so that $p \in R_1, R_3$, $q \notin R_1, R_3$, and $q \in R_2$, $p \notin R_2$.
The previous proposition shows $R_1, R_2, R_3$ must contain at least one nested set. It is also easy to see that not all three of them can form a nested set, so exactly two of them do. Further, the condition $p \in R_1, R_3$ but $q \notin R_1, R_3$ implies that $(R_1,R_3)$ must form a nested set (where $R_1$ contains $R_3$). In this case, it is easy to check that it is not possible to have a rectangle $R_2$ that forms steps with both $R_1$ and $R_3$, and contains $q$ but not $p$. \end{proof}
\subsection{Other families}\label{sec:other} For families that are tower-free, $m_k^*$ is linear. What of families that do contain towers?
We say a family of bottomless rectangles is \textit{hanging} if their left endpoints lie on the line $y=x$. It is clear that we can choose any line with positive slope, as rotating the line $y=x$ and moving the left endpoints along with it preserves the hypergraph structure. So more generally, a \textit{hanging family} is one whose left endpoints all lie on a fixed line with positive slope, which will be $y=x$ for convenience. A tower, for example, can be realised as a hanging arrangement by ``stretching'' the left endpoints.
\begin{proposition}\label{prop:hanging} For \textit{hanging families}, $m_k^* \leq 3k-2$. \end{proposition} This proof relies on a reduction to the primal problem for general families of bottomless rectangles, when we colour points with respect to bottomless rectangles. We want to show that any hanging family $\ensuremath{\mathcal F}\xspace$ and point set $P$ can be realised as a family $\ensuremath{\mathcal F}\xspace(P)$ and point set $P(\ensuremath{\mathcal F}\xspace)$ so that a rectangle $R \in \ensuremath{\mathcal F}\xspace$ covers a point $q \in P$ if and only if the corresponding point $r \in P(\ensuremath{\mathcal F}\xspace)$ is contained in the rectangle $Q \in \ensuremath{\mathcal F}\xspace(P)$.
To each rectangle $R \in \ensuremath{\mathcal F}\xspace$, we associate its right endpoint $r(R)$, and to each point $q=(x,y) \in P$, we associate an infinite hanging rectangle from the point $(x,x)$. Of course, when we say infinite, we simply mean that the right endpoint of the corresponding rectangle in $\ensuremath{\mathcal F}\xspace(P)$ is sufficiently large.
\begin{figure}
\caption{\small{The infinite hanging rectangle from $p$ contains $r(R)$ if and only if $R$ contains $p$.}}
\label{fig:hangingdual}
\end{figure}
Since the best upper bound for the primal problem is $3k-2$, we have the desired bound for hanging families. However, by modifying the proof of this upper bound from \cite{A+13}, we can improve it to $2k-1$. By the duality we observed, it suffices to consider colouring points with respect to infinite hanging rectangles. Order the points in increasing order of $y$-coordinate (from below). Note that if $p$ is to the left of some hanging rectangle $R$, then $R$ does not contain $p$, so once we start colouring points of $R$ we may ``disregard'' $p$. We present the points in increasing order of $y$-coordinate as follows. At step $t$, suppose $q$ is to be presented, and $p$ is the leftmost point at this step. If $q$ is covered by a rectangle to the \textit{right} of $p$, first we ``delete'' $p$, then we present $q$. If not, then we present $q$ without deleting any points. This enables us to only care about the leftmost $2k-1$ points, as we discard the nonessential ones.
More precisely, we frame this as a \textit{dynamic colouring problem} on the line. We wish to colour a dynamically appearing point set $P$ where one of the following kinds of events can occur. \begin{enumerate}[(1)]
\item A new point appears.
\item The leftmost point disappears. \end{enumerate}
\begin{theorem}\label{thm:hanging} For hanging arrangements, $m_k^* \leq 2k-1$. \end{theorem} \begin{proof} Given a dynamically appearing point set $P$ as discussed, we want to $k$-colour it so that at every step, the leftmost $2k-1$ points contain at least one point of each colour. For $i=1, \dots, k$, define an \textit{$i$-gap} to be an inclusion-maximal set of points between two points of colour $i$, and an \textit{$i$-prefix} to be the set of points before the first point of colour $i$. It suffices to maintain the following invariants at each step. \begin{enumerate}[(a)]
\item All $i$-gaps have size at least $k-1$, and
\item all $i$-prefixes have size at most $2k-2$. \end{enumerate} Suppose that these invariants are satisfied at some step, and then an event of type (1) occurs. This can only harm (b) by creating an $i$-prefix of size $2k-1$ for some $i$. At most $k-1$ colours occur in the leftmost $k$ points, and by (a), no colour occurs more than once. So there is some uncoloured point which we can colour with colour $i$; by construction, this is separated from the next point of colour $i$ by at least $k-1$ points, so (a) is preserved.
Now suppose an event of type (2) occurs: the first point disappears. Again, this can only harm (b) by creating an $i$-prefix of size $2k-1$. This means that the leftmost point had colour $i$, so we may once again find an uncoloured point in the leftmost $k$ points of the $i$-prefix, and colour it with colour $i$. \end{proof}
Another type of family we consider is the \textit{intersecting family}. These are families where any two rectangles are pairwise intersecting. In particular, there is a point $v$ contained in the intersection of all the rectangles. We call the vertical line through $v$ the \textit{spine} of the family.
\begin{figure}\label{fig:spine}
\end{figure}
\begin{theorem}\label{thm:intersecting} For intersecting families, $m_k^* \leq 3k$. \end{theorem} Our proof will rely on \textit{shallow hitting sets}. \begin{definition} A subset $S \subset \ensuremath{\mathcal F}\xspace$ is a \textit{$c$-shallow hitting set for depth $d$} if every point that is covered by exactly $d$ rectangles is covered by at least $1$ and at most $c$ rectangles of $S$. \end{definition} Suppose for fixed $c$ and a family $\ensuremath{\mathcal F}\xspace$ of bottomless rectangles, we have a $c$-shallow hitting set for any depth $d$. First we construct a $c$-shallow hitting set $S_1$ for depth $ck$, then remove $S_1$ from $\ensuremath{\mathcal F}\xspace$ and construct a $c$-shallow hitting set $S_2$ for depth $ck-c$, then remove $S_2$ from $\ensuremath{\mathcal F}\xspace$ and so on, yielding some disjoint subfamilies $S_1, \dots, S_k$. It is now easy to check that any point covered by $ck$ rectangles is covered by at least $1$ rectangle from each of $S_1, \dots, S_k$: $m_k^* \leq ck$ for $\ensuremath{\mathcal F}\xspace$. \begin{proof}[Proof of \autoref{thm:intersecting}] Let $d$ be arbitrary. We will construct hitting sets $S_L$ and $S_R$, and show that $S = S_L \cup S_R$ is a $3$-shallow hitting set for depth $d$. Order the points at depth exactly $d$ from below, and we will add rectangles to $S_L$ and $S_R$ in this order. If $p$ is a point at depth $d$ on side $s$ that is not yet covered by any rectangle of $S_s$, add to $S_s$ the rectangle covering $p$ whose extension to the other side is shortest. Once we are finished constructing $S_L$ and $S_R$ in this order, we reduce $S = S_L \cup S_R$ so that it is a minimal hitting set for depth $d$.
To show that it is $3$-shallow, we show that every point $p$ at depth $d$ on side, say $L$, is covered by at most $2$ rectangles from $S_L$ and at most $1$ from $S_R$. This is by minimality; if $p$ is covered by $T_1, T_2, T_3 \in S_L$, removing the one of the lower two whose left endpoint is closer to the spine preserves that $S$ is a hitting set. (This is not true without the fact that $\ensuremath{\mathcal F}\xspace$ is intersecting.) If $p$ is contained by two rectangles $R_1, R_2 \in S_R$, suppose that $y(R_1) > y(R_2)$. Since we chose $R_1$ for $S_R$ from below, there must be a point $q$ on the right side that is covered by $R_1$ but is above $R_2$. As a result, there are $d-1$ other rectangles covering $q$ whose heights are between $R_1$ and $R_2$. By the choice of $R_2$ for $q$ by minimality of its left endpoint, the left endpoints of these $d-1$ rectangles extend beyond the left endpoint of $R_2$, so they cover $p$. This gives $d+1$ rectangles that cover $p$, a contradiction. \end{proof} \begin{figure}
\caption{\small{(left) $T_2$ is not needed to hit points at depth $d$ on the left, and (right) there are $d-1$ rectangles between $R_1$ and $R_2$ that cover $p$.}}
\label{fig:intersecting}
\end{figure} \section{Arborescences}\label{sec:arb}
The goal for this section is to prove \autoref{thm:arb1}, \thmarbone* and to show that \autoref{thm:rect} can be deduced from it. \thmrect* We would like to associate to each family $\ensuremath{\mathcal F}\xspace$ of rectangles a simple graph, and derive a polychromatic colouring of $\ensuremath{\mathcal F}\xspace$ from a suitable colouring of this graph. First we will define the family of graphs that we consider (arborescences), prove \autoref{thm:arb1}, then show how these graphs are obtained from bottomless rectangles. \subsection{The setup}
An arborescence is a directed tree with a distinguished \textit{root} vertex such that all edges are directed away from the root, i.e.\ there is a unique directed path from the root to any vertex. We denote the length of the shortest directed path from $u$ to $v$, if it exists, by $dist(u,v)$. Recall that the length of a path is the number of edges, or one less than the number of vertices. A disjoint union of arborescences is called a \textit{branching}. We say that an ordering of the vertices of a branching is \emph{root-to-leaf} if every vertex is preceded by its in-neighbors and succeeded by its out-neighbors; in particular, from every component first the root is presented and last a leaf.
\begin{claim}
The vertices of any branching can be $k$-coloured by an online algorithm in a root-to-leaf order such that any directed path on $k$ vertices contains all $k$ colours. \end{claim} \begin{proof} If a root is presented, colour it with colour $1$. Every time a new vertex $v$ is presented in the component with root $r$, colour $v$ according to the parity of $dist(r,v) \Mod{k}$ (which can be determined from a root-to-leaf ordering). \end{proof}
We call the reversal of a root-to-leaf ordering a \textit{leaf-to-root} ordering; from each component, first a leaf is presented and last the root. Our main result, \autoref{thm:arb1}, shows that the converse of the above claim fails: any semi-online algorithm will in fact leave arbitrarily long monochromatic paths. In order to apply this result to bottomless rectangles, however, we will need a stronger condition on the leaf-to-root ordering.
For two vertices $u$ and $v$ of a branching, say $u<v$ if they are in the same component and there is a directed path from $u$ to $v$. This forms a partial order where the roots are the minimal elements and the leaves the maximal. A leaf-to-root ordering is a linear extension of this partial order that presents the $<$-maximal element first.
If $u<v$ and there are no other vertices between them, i.e.\ $uv$ is a directed edge, write $u \lessdot v$ and say that $v$ is a \emph{parent} of $u$. (Thus, somewhat contradicting the laws of nature, every vertex can have only one child, but several parents.) When presenting the vertices of a branching in a leaf-to-root order, the newly presented vertex $u$ will always form a root, while its parents were all roots of the branching before $u$ was presented.
\begin{figure}
\caption{\small{A branching with roots $r$ and $r'$. In this example, $u_1 \gtrdot r$, i.e.\ $u_1$ is a parent of $r$, but $u_2$ is not a parent of $r$ even though $u_2 > r$ ($u_2$ is a ``grandparent'' of $r$), and $v'\not>r$. A linear extension of this (or a leaf-to-root ordering) might present the vertices $u', v'$ and $r'$ before $u_3$, so it is not necessary that the roots of the branching are the last vertices presented.}}
\label{fig:branching}
\end{figure}
Denote the roots of the branching before a new vertex $u$ is presented by $v_1,v_2,\ldots$ indexed in the order in which they were presented. We say that a \emph{leaf-to-root} ordering is \emph{geometric} if the parents of $u$ form an interval in this order, i.e., for every $u$, $\{v_i\mid u \lessdot v_i\}=\{v_i \mid l<i<r\}$ for some $l$ and $r$. Intuitively, we do not allow an ordering of the following type.
Now we state a stronger form of \autoref{thm:arb1}. \begin{theorem}\label{thm:arb2} For any numbers $m,k$, there is no semi-online $k$-colouring algorithm that receives the vertices of an arborescence in a geometric leaf-to-root order and maintains that at every stage, all directed paths of length $m$ contain all $2$ colours. \end{theorem} Call a semi-online $k$-colouring algorithm \emph{$m$-proper} if any path on $m$ vertices contains at least two colours. The theorem states there is no $m$-proper semi-online $k$-colouring algorithm for arborescences presented in geometric leaf-to-root order. The idea of the proof is that for any vertex $u$, there are only finitely many possibilities for all directed paths of length $m$ from $u$. However, we can always force the algorithm to produce a new ``type'' of path, leading to a contradiction.
Fix $k$ colours, $C_1, \dots, C_k$, a branching \ensuremath{\mathcal F}\xspace with a geometric leaf-to-root order, a point $p \in V(\ensuremath{\mathcal F}\xspace)$, and the time $t$ at which $p$ appears. To ease future notation, let us get some (many) definitions out of the way. \begin{itemize} \item $p_u$ is a \textit{$u$-parent} of $p$ if there is a directed path $(p,p_1, \dots, p_u)$, i.e.\ $dist(p,p_u)=u$ in the graph. We refer to the subpath $(p_1, \dots, p_u)$ as the \emph{chain} corresponding to $p_u$. \item A $u$-parent $p_u$ of $p$ is \emph{in $C_i$} if $p_u$ is a $u$-parent of $p$ and every point in the chain $(p_1, \dots, p_u)$ is coloured with $C_i$ at time $t$. (Note that $p$ itself need not have colour $C_i$.) \item A $u$-parent $p_u$ of $p$ in $C_i$ is \textit{maximal} if there is no $p_{u+1} \gtrdot p_u$ that is also coloured with $C_i$ at time $t$ (note that this depends \textit{only} on $t$, even if some such $p_{u+1}$ is coloured later). \item Similarly, $p_u$ is an \textit{uncoloured} $u$-parent of $p$ if every point of $(p_1, \dots, p_u)$ is uncoloured, and it is a \textit{maximal} uncoloured $u$-parent if there is no $p_{u+1} \gtrdot p_u$ that is also uncoloured. \item The \textit{type} of $p$, $tp(p)$ is defined as the vector $(t_1, \dots, t_k) \in \mathbb{N}^k$, where \[t_i = \max \{u: p \text{ has a maximal } u\text{-parent in } C_i\}\] \item If two partially coloured trees, $\ensuremath{\mathcal t}\xspace_1$ and $\ensuremath{\mathcal t}\xspace_2$, are isomorphic, we write $\ensuremath{\mathcal t}\xspace_1 \cong \ensuremath{\mathcal t}\xspace_2$. Note that for the isomorphism we require that vertices coloured, say red, must be mapped to red vertices - we do not allow the isomorphism to permute the colours. \end{itemize}
Let $S_t$ be the set of points that have appeared by time $t$ in the same connected component of $\mathcal{F}$ as $p$ (or in the subtree rooted at $p$ at time $t$). We now associate to $p$ a tree $\ensuremath{\mathcal t}\xspace(p)$ by ``trimming'' the induced subgraph $\mathcal{F}[S_t]$ in the following steps. (See \autoref{fig:trimming}.)
\begin{enumerate}
\item If $q$ is uncoloured and $dist(p,q) > m$, delete $q$.
\item If $q_1$ and $q_2$ are both maximal $t_i$-parents in $C_i$ for some remaining $q$, delete $q_2$ and all points that are $>q_2$.
\item For $i=1,\ldots m$, if $q$ is a $(m-i)$-parent of $p$, and $q_1 \gtrdot q$ and $q_2 \gtrdot q$ are such that the subtrees rooted at $q_1$ and $q_2$ are isomorphic, delete $q_2$. \end{enumerate}
\begin{figure}
\caption{\small{Example for trimming with $m=k=2$. In step 1, we delete the uncoloured $3$-parent of $p$, but preserve the red parent of $q_1$. In step 2, we ``trim'' the blue parents of $q'$. In step $3$, the subtrees rooted at $q_1$ and $q_2$ are isomorphic, so we delete $q_2$.}}
\label{fig:trimming}
\end{figure}
The idea of this trimming process is to retain only the ``essential'' information about the colouring when $p$ appears and reduce the number of possible $\ensuremath{\mathcal t}\xspace(p)$ to a bounded number of options. If we assume that the algorithm has produced a $m$-proper colouring until the time that $p$ appears, then we can disregard vertices at distance $>m$ from $p$. If a vertex was not deleted during the trimming, we say that it was \emph{preserved}.
We could modify step $1$ to delete \textit{all} points at distance $> m$ from $p$. However, in the proof we will use the fact that the type of any point at distance $\leq m$ from $p$ is preserved. Of course, if the algorithm is good, then any directed path of length $m$ contains at least $2$ colours, so deleting only the uncoloured points is just a technical condition that simplifies notation. Finally, in step $3$, we ensure that we do not have any ``repetitions''. For example, if all the branches rooted at $p$ are isomorphic, by considering only one of them we do not lose any important information.
We emphasise that $\ensuremath{\mathcal t}\xspace(p)$ depends only on the time at which $p$ appears. For instance, in the above figure, even if $q'$ is coloured blue at a later time, $\ensuremath{\mathcal t}\xspace(p)$ does not change.
\subsection{Proof of the main theorem}
The crucial result of the trimming process is that the following lemma holds. \begin{lemma}\label{lem:trim}
Suppose that a semi-online colouring algorithm as in the statement of the theorem exists. Then the following hold.
\begin{enumerate}
\item The set
$\{\ensuremath{\mathcal t}\xspace(p): \mathcal{F} \text{ is a branching}, p \in V(\mathcal{F})\}$
is finite.
\item If $q \in S_t$ is preserved after the trimming, and $q$ had an $t_i$-parent in $C_i$ in $\mathcal{F}$, then $q$ has an $t_i$-parent in $C_i$ in $\ensuremath{\mathcal t}\xspace(p)$. In particular, the type of $q$ is preserved.
\item Suppose $p' \lessdot p$ is presented, and $q$ was an uncoloured $u$-parent of $p$ in $\ensuremath{\mathcal t}\xspace(p)$ for $u<d$. If none of the points on the chain from $p'$ to $q$ are coloured when $p'$ is presented, then $q$ is preserved in $\ensuremath{\mathcal t}\xspace(p')$.
\end{enumerate} \end{lemma}
\begin{proof}
We show that there are only finitely many possibilities for $\ensuremath{\mathcal t}\xspace(p)$. In step $1$ of the trimming we delete uncoloured points at distance $>m$ from $p$. In step $2$ we preserve only maximal parents in $C_i$ of $p$ for each colour $C_i$. Since the algorithm is $m$-proper, $\ensuremath{\mathcal t}\xspace(p)$ will have depth at most $m$. In step $2$, we also delete ``repetitions'' so there are only finitely many possibilities for each of the branches above $p$. And in step $3$, we delete isomorphic subtrees, so no two of the branches above $p$ are isomorphic. Thus $\ensuremath{\mathcal t}\xspace(p)$ can take only finitely many values.
The second claim follows from our earlier argument.
For the third claim, we only need to consider the case when the algorithm produces an uncoloured $u$-parent $q'$ such that one of $q$ and $q'$ must be trimmed (i.e., the subtrees rooted at $q$ and $q'$ are isomorphic). In this case, we can assume without loss of generality that $q'$ is trimmed so the second property holds. \end{proof}
\begin{lemma}\label{lem:descent}
At any stage of the algorithm, suppose that we have a collection of trees with roots $p_1,\ldots,p_s$ presented in this order such that no two $\ensuremath{\mathcal t}\xspace(p_i)$ and $\ensuremath{\mathcal t}\xspace(p_j)$ are isomorphic.
Then presenting a vertex $p$ with parents $p_1,\ldots,p_s$, will give a tree $\ensuremath{\mathcal t}\xspace(p)$ that is non-isomorphic to any $\ensuremath{\mathcal t}\xspace(p_i)$. \end{lemma}
\begin{proof}
Suppose for contradiction that for some $p_i$, $\ensuremath{\mathcal t}\xspace(p) \cong \ensuremath{\mathcal t}\xspace(p_i)$.
Let $\varphi: \ensuremath{\mathcal t}\xspace(p) \to \ensuremath{\mathcal t}\xspace(p_i)$ be an isomorphism (preserving colourings).
We prove by induction for all $u < m$ that there is a chain $p=r_0 \lessdot r_1 \lessdot \dots \lessdot r_u$ in $\ensuremath{\mathcal t}\xspace(p)$ such that for all $i\le u$ we have $\varphi(r_{i-1}) = r_i$, and $r_i$ is uncoloured.
First suppose $p$ is coloured, say with $C_1$, in $\ensuremath{\mathcal t}\xspace(p)$, and let $t_1$ be maximal such that $p$ has a $t_1$-parent in $C_1$.
$r_1 = p_i= \varphi(p)$ was coloured with $C_1$ in $\ensuremath{\mathcal t}\xspace(r_1)$, and by the isomorphism $r_1$ has a $t_1$-parent in $C_1$.
Since we did not recolour any points, this produces a $(t_1+1)$-parent in $C_1$ of $p$ in $\ensuremath{\mathcal t}\xspace(p)$, contradicting the maximality of $t_1$.
So $p$ must be uncoloured in $\ensuremath{\mathcal t}\xspace(p)$, which implies that $r_1$ was uncoloured in $\ensuremath{\mathcal t}\xspace(r_1)$.
To complete the base case of the induction hypothesis, we need to show that $r_1$ remains uncoloured in $\ensuremath{\mathcal t}\xspace(p)$, i.e., when $p$ appears. Let $t_1$ be as earlier, and suppose again that $r_1$ is coloured with $C_1$ in $\ensuremath{\mathcal t}\xspace(p)$.
Then $p$ has an $(t_1+1)$-parent in $C_1$ in $\ensuremath{\mathcal t}\xspace(p)$, again a contradiction.
Suppose we have produced a chain $p=r_0 \lessdot r_1 \lessdot \dots \lessdot r_{u-1}$ from the induction hypothesis.
If $u-1 = m$, then we are done.
Otherwise, let $r_u = \varphi(r_{u-1})$.
Then $r_u$ is uncoloured in $\ensuremath{\mathcal t}\xspace(r_1)$.
Since $r_{u-1} \gtrdot r_{u-2}$, $r_u \gtrdot \varphi(r_{u-2}) = r_{u-1}$, so $p=r_0 \lessdot r_1 \lessdot \dots \lessdot r_u$ is a chain, and it remains to show that $r_u$ is uncoloured in $\ensuremath{\mathcal t}\xspace(p)$.
Suppose $r_u$ is coloured in $\ensuremath{\mathcal t}\xspace(p)$ with $C_1$.
If $s_1$ is maximal so that $r_{u-1}$ has an $s_1$-parent in $C_1$ in $\ensuremath{\mathcal t}\xspace(p)$, then $r_u$ has an $s_1$-parent in $C_1$ in $\ensuremath{\mathcal t}\xspace(p_1)$, producing an $(s_1+1)$-parent in $C_1$ for $r_{u-1}$ in $\ensuremath{\mathcal t}\xspace(p)$.
This contradicts the maximality of $s_1$.
This eventually produces a chain of $m$ uncoloured points, which contradicts the correctness of the semi-online algorithm. \end{proof} \begin{proof} From here we can finish the proof of \autoref{thm:arb2} with an infinite descent argument as follows. Order the finite sequences of naturals, $\ensuremath{\mathbb N}\xspace^{<\omega}$, such that $(s_1,s_2,\ldots,s_l)>(s_1',s_2',\ldots,s_{l'}')$ if there is some $i$ such that for all $j<i$ we have $s_j=s_j'$ but $s_i>s_i'$, or $l>l'$ and for all $j\le l'$ we have $s_j=s_j'$. For a branching \ensuremath{\mathcal F}\xspace, we define its \emph{associated sequence} as follows. For each root $p_i$ of \ensuremath{\mathcal F}\xspace, consider the sequence of trees $\ensuremath{\mathcal t}\xspace(p_i)$ in the order their roots were presented. Let $i_1$ be the smallest index such that for every $\ensuremath{\mathcal t}\xspace(p_i)$ there is an $i'\le i_1$ such that $\ensuremath{\mathcal t}\xspace(p_i)\cong\ensuremath{\mathcal t}\xspace(p_{i'})$. The number of different trees $\ensuremath{\mathcal t}\xspace(p_i)$ (same as the number of different trees up to $i_1$) is denoted by $s_1$. In general, after $i_{j-1}$ has been defined, let $i_j$ be the smallest index such that for every $\ensuremath{\mathcal t}\xspace(p_i)$ with $i>i_{j-1}$ there is an $i_{j-1}<i'\le i_j$ such that $\ensuremath{\mathcal t}\xspace(p_i)\cong\ensuremath{\mathcal t}\xspace(p_{i'})$. The number of different trees $\ensuremath{\mathcal t}\xspace(p_i)$ for $i_{j-1}<i\le i_{j}$ is denoted by $s_j$. We repeat this for $N$ steps, where $N$ denotes the number of possible different (i.e., non-isomorphic) trees \ensuremath{\mathcal t}\xspace, or until there are no more roots in \ensuremath{\mathcal F}\xspace. The numbers $(s_1,\ldots,s_l)$ are the associated sequence of \ensuremath{\mathcal F}\xspace.
Note that there are finitely many associated sequences, as each $N\ge s_1\ge s_2\ge \dots \ge s_l$, and also $l\le N$. Applying \autoref{lem:descent} to the largest associated sequence that can be attained during the run of the semi-online algorithm, we get a contradiction as follows. Let \ensuremath{\mathcal F}\xspace be a branching whose associated sequence, $(s_1,\ldots,s_l)$, is the largest.
Case 1: If $s_1=N$, then we present a new point $p$ whose parents are the roots of $\ensuremath{\mathcal F}\xspace$, and by \autoref{lem:descent} we produce a new tree, which is not possible.
Case 2: If $N> s_1> \dots> s_l$, then $l<N$. Introduce a new vertex disjoint from all vertices of $\ensuremath{\mathcal F}\xspace$. This will either increase an earlier $s_i$, or give a new $s_{l+1}=1$, but both of these contradict the maximality of $(s_1,\ldots,s_l)$.
Case 3: There is some $j$ for which $s_j=s_{j+1}$. This is only possible if all the trees $\ensuremath{\mathcal t}\xspace(p_i)$ for $i_{j-1}<i\le i_{j}$ have an isomorphic copy $\ensuremath{\mathcal t}\xspace(p_{i'})$ for some $i_{j}<i\le i_{j+1}$. Introduce a new vertex $p$ under all the roots $p_i$ of $\ensuremath{\mathcal F}\xspace$ with index $i>i_{j}$ to obtain a new branching $\ensuremath{\mathcal F}\xspace'$. By \autoref{lem:descent}, the tree $\ensuremath{\mathcal t}\xspace(p)$ is non-isomorphic to any $\ensuremath{\mathcal t}\xspace(p_i)$ with $i_{j-1}<i\le i_{j}$. Therefore, the associated sequence of $\ensuremath{\mathcal F}\xspace'$ will be larger than $(s_1,\ldots,s_l)$, contradicting its maximality.
In summary, it is not possible for a semi-online $k$-colouring algorithm to produce finitely many associated sequences, so it cannot be $m$-proper. \end{proof}
\subsection{Application to bottomless rectangles}
In this section, we apply \autoref{thm:arb2} to semi-online colouring algorithms for Erd{\H o}s-Szekeres configurations. We start with towers.
\begin{corollary}\label{cor:tower}
There is no semi-online colouring algorithm for towers from above, i.e., for any numbers $k$ and $m$, for any semi-online algorithm that $k$-colours bottomless rectangles from above, there is a family of bottomless rectangles such that any two intersecting rectangles form a tower, and the algorithm produces an $m$-fold covered point that is covered by at most one colour. \end{corollary} \begin{proof} In order to apply \autoref{thm:arb2}, we need to show that any branching can be realised as a family of towers so that
\begin{enumerate}
\item ordering the rectangles from above corresponds to a geometric leaf-to-root order of the branching, and
\item a semi-online colouring algorithm for towers from above corresponds to an $m$-proper semi-online $k$-colouring algorithm for branchings in this order.
\end{enumerate}
For any arborescence $\mathcal{F}$ in geometric leaf-to-root order, we show by induction on $|\mathcal{F}|$ that it can be realised as a family of towers with this order. For $|\mathcal{F}| = 1$ this is clear. For the inductive step, we will need to use the fact that the ordering is geometric. For example, suppose we have a non-geometric order and three roots $p,q,r$ that are realised as disjoint rectangles, with $q$ between $p$ and $r$. Then if the next root $s$ is presented with $s<p$ and $s<r$, but $s \nless q$, $s$ cannot be realised as a rectangle.
\begin{figure}
\caption{\small{There is no way to present a new rectangle $s$ that intersects $p$ and $r$ but not $q$.}}
\end{figure}
Now we prove the induction step. Let $|\mathcal{F}| = n$, and $r$ be the last element in the ordering of $V(\mathcal{F})$. Take any realisation of $\mathcal{F} \setminus \{r\}$ as a family of towers. If $r$ is an isolated vertex in $\mathcal{F}$, present $r$ as a disjoint rectangle to the right of the realisation $\mathcal{F} \setminus \{r\}$. Otherwise, since the order is geometric, $r$ will only intersect some geometrically adjacent rectangles of $\mathcal{F}$ (by construction). Hence $r$ can be realised as a minimal rectangle.
\begin{figure}
\caption{\small{By the geometric ordering, we can realise $r$ as a minimal element.}}
\end{figure} The proof for nested rectangles from below is analogous. \end{proof}
\begin{corollary}
There is no semi-online $k$-colouring algorithm from the left or from below for increasing steps. More precisely, for any integers $k$ and $m$, there is no semi-online algorithm to $k$-colour rectangles from the left (or from below) so that at every step, any point covered by $m$-increasing steps is covered by at least $2$ colours.
Similarly, there is no semi-online colouring algorithm for decreasing steps from the right or from below. \end{corollary}
Note that this statement is slightly weaker than \autoref{thm:rect} or \autoref{cor:tower} because we do not exclude the other kind of configurations from the family.
\begin{proof}
We first prove the statement for increasing steps from the left. Again, we will prove the corollary by induction on $|\mathcal{F}|$. However, we also weaken our requirements for the colouring of the steps.
That is, we need not assume that every directed path of length $m$ in the branching corresponds to a point covered by exactly $m$ increasing steps.
It is easy to see that a semi-online colouring algorithm of $\mathcal{F}$ is $m$-proper if and only if when any point $p \in \mathcal{F}$ is presented, any directed path of length $m$ \textit{from} $p$ contains at least $2$ colours.
So it suffices to prove the following by induction.
Any branching $\mathcal{F}$ with a geometric leaf-to-root order can be realised as a family of bottomless rectangles so that
\begin{enumerate}
\item when $p \in \mathcal{F}$ is presented, we realise $p$ as a rectangle so that any directed path of length $m$ from $p$ corresponds to a point covered by exactly $m$ increasing steps, and
\item any two rectangles intersect either as increasing or as decreasing steps.
\end{enumerate}
The second assumption is a technical condition to ensure that $q$ covers the top-right corner of $r$ if and only if $(r,q)$ form increasing steps, so we can choose the top-right corner of an appropriate rectangle as the point satisfying the first induction hypothesis.
The case $|\mathcal{F}| = 1$ is trivial. Let $|\mathcal{F}| = n$, and $r$ be the last element in the ordering of $\mathcal{F}$. Take any realisation of $\mathcal{F} \setminus \{r\}$ satisfying the induction hypotheses. If $r$ is an isolated vertex, let $q$ be the last element presented (thus a root), and realise $r$ as a rectangle so that $(q,r)$ form decreasing $2$-steps (see \autoref{fig:stepsroot}). There are no directed paths of length $m$ from $r$ so both induction hypotheses are satisfied.
\begin{figure}
\caption{\small{The rectangles in decreasing steps correspond to roots of the branching.}}
\label{fig:stepsroot}
\end{figure}
Otherwise, since the ordering is geometric, $r$ will only intersect the rightmost rectangles (by construction), thus can be realised as a rectangle that forms increasing steps with these rightmost roots, and decreasing steps with the other roots (see \autoref{fig:stepsbranching}).
To see that the first hypothesis is satisfied, consider the rectangles corresponding to any directed path of length $m$ from $r$, say $(r_1, \dots, r_{m-1}, r)$. Then the top-right corner of $r_1$ will not be covered by any rectangle other than the ones in this chain - this follows from the induction hypothesis and the fact that $\mathcal{F}$ is a branching, so $r_2$ is the unique child of $r_1$.
\begin{figure}\label{fig:stepsbranching}
\end{figure}
The proof for increasing steps from below follows analogously, except we change the second induction hypothesis to assume that any two rectangles intersect either as increasing steps or as a tower. In this construction, the roots of the branching at any time will correspond to a tower, and the geometric ordering ensures that a new root can be placed in increasing steps with the top-most rectangles of the tower.
The proof for decreasing steps is exactly the same, only interchanging left and right. \end{proof}
\section{Further results for bottomless rectangles}\label{sec:further}
\subsection{No shallow hitting sets}
To prove an upper bound for intersecting families in \autoref{thm:intersecting}, we used the fact that if a family admits a $c$-shallow hitting set for arbitrary depths $d$, then $m_k^* \leq ck$. Unfortunately, general families of bottomless rectangles do not admit such hitting sets. \begin{theorem}\label{thm:noshallowhittingset}
For every integer $c \geq 0$, there exists a real number $r < 1$ and an integer $D \geq 1$ such that for every integer $d \geq D$, there is a family $\ensuremath{\mathcal F}\xspace = \ensuremath{\mathcal F}\xspace(h,d)$ such that the following holds. For any hitting set $H$ of the $d$-cells of $\ensuremath{\mathcal F}\xspace$, there is a vertical line $\ell$ such that $|\ell \cap \ensuremath{\mathcal F}\xspace| \leq \lceil rd \rceil$ and $|\ell \cap H| \geq h$. \end{theorem} In particular, $\ensuremath{\mathcal F}\xspace (h,d)$ witnesses that there are no $(h-1)$-shallow hitting sets for depth $d$. \begin{proof} We construct $\ensuremath{\mathcal F}\xspace(h,d)$ by induction on $h$. For $h=0$, letting $r=0$, $D = 1$, and $\ensuremath{\mathcal F}\xspace(h,d)$ be the empty family, the conditions are satisfied. Let $r_{h-1}$ and $D_{h-1}$ be the values given by the induction hypothesis for $h-1$. Choose $s$ to be a real number such that $0 < s < 1-r_{h-1}$. Set \[ D > \max \Big(\frac{D_{h-1}}{s}, \frac{1}{s(1-r_{h-1})}\Big); \quad r = \max\Big(r_{h-1}+s, 1-s(1-r_{h-1}) - \frac{1}{D}\Big). \] For $d \geq D$, we define $\ensuremath{\mathcal F}\xspace = \ensuremath{\mathcal F}\xspace(h,d)$ as follows. Take $d$ rectangles $(R_1, \dots, R_d)$ in increasing $d$-steps. We will insert families of the form $\ensuremath{\mathcal F}\xspace(h-1, i)$ for some $D_{h-1} \leq i \leq d$. First, set $i_0 = \lfloor sd \rfloor$, so that $i_0 \geq \lfloor sD \rfloor D_{h-1} \geq 1$. Insert a copy of $\ensuremath{\mathcal F}\xspace(h-1,d)$ so that it intersects only the rectangles $R_1, \dots , R_{i_0}$, and the top sides of all the rectangles are above that of $R_{i_0}$. Next, for every $i_0 < i \leq d$, insert a copy of $\ensuremath{\mathcal F}\xspace(h-1,i)$ so that it intersects exactly the rectangles $R_i, \dots, R_d$, and all the top sides lie between those of $R_i$ and $R_{i+1}$ (or just above that of $R_d$ when $i=d$). \begin{figure}\label{fig:nohitting}
\end{figure}
Now, let $H$ be a hitting set for the $d$-cells of $\ensuremath{\mathcal F}\xspace$. Since the point $p$ is in a $d$-cell, $H$ must contain at least one of $R_1, \dots, R_d$. First, suppose that $R_i \in H$ for some $i \leq i_0$. The intersection of $H$ with the leftmost copy of $\ensuremath{\mathcal F}\xspace(h-1,d)$ is a hitting set for the $d$-cells in $\ensuremath{\mathcal F}\xspace(h-1,d)$. By induction, there is a line $\ell$ such that $|\ell \cap \ensuremath{\mathcal F}\xspace(h-1,d)| \leq r_{h-1}d$ and $| \ell \cap H \cap \ensuremath{\mathcal F}\xspace(h-1,d)| \geq h-1$. Then, \[
|\ell \cap \ensuremath{\mathcal F}\xspace| \leq r_{h-1}d + i_0 \leq (r_{h-1}+s)d < rd \] and \[
|\ell \cap H | \geq (h-1) + 1 \geq h. \]
On the other hand, suppose $H$ does not contain any rectangle $R_i$ for $i \leq i_0$. Let $i$ be maximal so that $R_i \in H$. Then, $H \cap \ensuremath{\mathcal F}\xspace(h-1, i)$ is a hitting set for the $i$-cells in $\ensuremath{\mathcal F}\xspace(h-1,i)$. Again, we have a vertical line $\ell$ such that $|\ell \cap \ensuremath{\mathcal F}\xspace(h-1,i)| \leq r_{h-1}i$ and $|\ell \cap H \cap \ensuremath{\mathcal F}\xspace(h-1,i)| \geq h-1$. Similarly, \[
|\ell \cap H| \geq h \] and \begin{align*}
|\ell \cap \ensuremath{\mathcal F}\xspace| &\leq r_{h-1}i + (d-i) + 1\\ &= d+1 - i(1-r_{h-1})\\ &\leq d+1 - sd(1-r_{h-1})\\ &= d(1-s(1-r_{h-1}) + 1/d)\\ &\leq rd. \end{align*} \end{proof}
\subsection{An improved lower bound}
Finally, we present an improved lower bound for general bottomless rectangle families, and a weaker lower bound that can be applied to the steps problem.
\begin{theorem}\label{thm:lb}
$m_k^*(\ensuremath{\mathcal{F}_{\sqcap}}\xspace) \geq 2k-1$ for general families of bottomless rectangles. \end{theorem} \begin{proof}
Our lower bound construction proceeds in two steps.
\begin{enumerate}
\item If $m_k^* < m_{k-1}^*+2$, then every family has a polychromatic $k$-colouring that is proper.
\item There is a family so that no polychromatic $k$-colouring is proper.
\end{enumerate}
This contradiction shows that $m_k^* \geq m_{k-1}^* + 2$, so by induction $m_k^* \geq 2k-1$.
1. Suppose for some family $\mathcal{F}$, no polychromatic $k$-colouring of $\mathcal{F}$ is proper. Let $\mathcal{G}$ be a witness to the sharpness of $m_{k-1}^*$, i.e.\ any $(k-1)$-colouring of $\mathcal{G}$ produces a point covered by $m_{k-1}^*-1$ rectangles but not all $k$ colours. In a small interval around every $2$-covered point in $\mathcal{F}$, we place a thin copy of $\mathcal{G}$ (see \autoref{fig:prop}).
Any polychromatic colouring of this new family $\mathcal{F}'$ must induce a polychromatic colouring of $\mathcal{F}$, so some copy of $\mathcal{G}$ is covered by $2$ rectangles of the same colour, say red.
By hypothesis, any point in this copy of $\mathcal{G}$ covered by at least $m_{k}^*$ rectangles is covered by all $k$ colours.
Since every such point is covered by exactly two red rectangles from $\mathcal{F}$, recolouring every red rectangle in $\mathcal{G}$ blue cannot ruin this property.
However, this induces a $(k-1)$-colouring of $\mathcal{G}$ so that any point in $m_{k-1}^*-1$ rectangles is covered by all $k-1$ colours, a contradiction.
So every family must have a polychromatic colouring that is proper.
\begin{figure}\label{fig:prop}
\end{figure}
2. Consider the family in \autoref{fig:noprop}.
\begin{figure}
\caption{\small{No polychromatic colouring of this family will be proper.}}
\label{fig:noprop}
\end{figure}
We have an $m$-tower (where $m$ may be arbitrarily large), so that each rectangle from the tower meets $R$ in a $2$-covered point. Suppose without loss of generality that $R$ is coloured red in some polychromatic $k$-colouring. For this colouring to be proper, no rectangle of the tower can be red - however the point $p$ will then be covered by $m$ rectangles, none of which are red, so the colouring cannot be polychromatic. This completes our proof. \end{proof}
This lower bound cannot be applied to \textit{unit bottomless}, as this construction relies heavily on towers. For these, we prove the following weaker lower bound.
\begin{proposition}
$m_k^*(\ensuremath{\mathcal{F}_{\mathrm{unit}}}\xspace) \geq 2\lfloor \frac{2k-1}{3}\rfloor + 1$ for \textit{unit bottomless}. \end{proposition} \begin{proof} This is a generalisation of the construction that shows that $m_k^* =3$. Let $\mathcal{F}$ be a family of $2k-1$ rectangles partitioned into $3$ almost equal subfamilies, $\mathcal{F}_1$, $\mathcal{F}_2$ and $\mathcal{F}_3$ as follows.
\begin{figure}
\caption{This construction shows that $m_k \geq 2 \lfloor \frac{2k-1}{3}\rfloor +1$.}
\end{figure}
Consider any $k$-colouring of $\mathcal{F}$. Some colour, say red, is used at most once, so it appears in at most one of $\mathcal{F}_1, \mathcal{F}_2$ and $\mathcal{F}_3$, say $\mathcal{F}_i$. Then the point $p_i$ is covered by the other two subfamilies, and no red rectangle.
Since $\lfloor \frac{2k-1}{3}\rfloor \leq |\mathcal{F}_i| \leq \lceil \frac{2k-1}{3}\rceil$, this proves the lower bound. \end{proof}
Note that the family in the figure does not contain any towers or nested sets. This gives a lower bound to complement \autoref{prop:stepspt}, namely that for steps, $m_k^* \geq 2 \lfloor \frac{2k-1}{3}\rfloor +1$.
\end{document} | arXiv | {
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\begin{document}
\title[Integrable Locally Convex Space Valued Tensor Fields] {Scalarly Essentially Integrable Locally Convex Vector Valued Tensor Fields.\\ Stokes Theorem} \author[B. Silvestri]{Benedetto Silvestri} \date{\today} \keywords{integrable locally convex vector valued tensor fields on manifolds, integration of locally convex vector valued forms on manifolds, Stokes equalities} \subjclass[2010]{46G10, 58C35}
\begin{abstract} This note is propaedeutic to the forthcoming work \cite{sil}; here we develop the terminology and results required by that paper. More specifically we introduce the concept of scalarly essentially integrable locally convex vector-valued tensor fields on a smooth manifold, generalize on them the usual operations, in case the manifold is oriented define the weak integral of scalarly essentially integrable locally convex vector-valued maximal forms and finally establish the extension of Stokes theorem for smooth locally convex vector-valued forms. This approach to the basic theory of scalarly essentially integrable and smooth locally convex vector-valued tensor fields seems to us to be new. Specifically are new (1) the definition of the space of scalarly essentially integrable locally convex vector-valued tensor fields as a $\mathcal{A}(U)$-tensor product, although motivated by a result in the usual smooth and real-valued context; (2) the procedure of $\mathcal{A}(U)$-linearizing $\mathcal{A}(U)$-bilinear maps in order to extend the usual operations especially the wedge product; (3) the exploitation of the uniqueness decomposition of the $\mathcal{A}(U)$-tensor product with a free module in order to define not only (a) the exterior differential of smooth locally convex vector-valued forms, but also (b) the weak integral of scalarly essentially integrable locally convex vector-valued maximal forms; (4) the use of the projective topological tensor product theory to define the wedge product. \end{abstract}
\maketitle
\begin{notation} If $A$ is a ring, then let $A-\mr{mod}$ be the category of $A$-modules and $A$-linear maps. If $E$ is a $A$-module, then let $E^{\ast}$ be its $A$-dual. Let $r,s\in\mathbb{Z}_{+}$ and $E$ be a $A-$module, define $[E,r,s]$ to be such that $[E,0,0]\coloneqq A^{\ast}$, otherwise be the map on $[1,r+s]$ such that \begin{equation*} \begin{aligned} i\in[1,r]\cap\mathbb{Z}\Rightarrow[E,r,s]_{i}&\coloneqq E^{\ast}, \\ j\in[1,s]\cap\mathbb{Z}\Rightarrow[E,r,s]_{r+j}&\coloneqq E. \end{aligned} \end{equation*} Let $\prod[E,0,0]\coloneqq[E,0,0]$ and and let $\prod[E,r,s]$ be the $A$-module product $\prod_{i=1}^{r+s}[E,r,s]_{i}$. If $F$ is a $A$-module, then define $\mf{T}_{s}^{r}(E,F)$ be the $A$-module of $A$-multilinear maps from $\prod[E,r,s]$ into $F$ whose elements are called tensors on $E$ of type $(r,s)$ at values in $F$. Set $\mf{T}_{s}^{r}(E)\coloneqq\mf{T}_{s}^{r}(E,A)$ and identify $\mf{T}_{0}^{0}(E)$ with $A$. Let $\mr{Alt}^{k}(E)$ be the $A$-submodule of the alternating maps in $\mf{T}_{k}^{0}(E)$. \par Let $K\in\{\mathbb{R},\mathbb{C}\}$ and let $G$ be a Hausdorff locally convex space over $\mathbb{K}$. We let $G_{0}$ denote the linear space over $\mathbb{R}$ underlying $G$, while let $G_{\mathbb{R}}$ denote the Hausdorff locally convex space over $\mathbb{R}$ underlying $G$. Let $\mc{L}(G,H)$ be the $\mathbb{K}$-linear space of continuous linear maps from $G$ into $H$ and $G^{\prime}\coloneqq\mc{L}(G,\mathbb{K})$ be the topological dual of $G$, so $(G^{\prime})^{\ast}=\mr{Mor}_{\mathbb{K}-\mr{mod}}(G^{\prime},\mathbb{K})$ is the algebraic dual of $G^{\prime}$. Next if $W$ is an open set of $\mathbb{R}^{n}$ with $n\in\mathbb{N}^{\ast}$ and $k\in\mathbb{Z}_{+}\cup\{+\infty\}$, then we let $\mc{C}^{k}(W,G)$ be the $\mathbb{K}$-linear space of $\mc{C}^{k}$-maps in the sense of Bastiani. For every $(a,v)\in W\times\mathbb{R}^{n}$ we let $D_{v}^{W,G}\vert_{a}f$ denote the derivative of $f$ at $a$ in the direction $v$, and let $D_{v}^{W,G}f:W\ni a\mapsto D_{v}^{W,G}\vert_{a}f\in G$. \par If $X$ is a topological space and $E$ is a Hausdorff locally convex space over $\mathbb{R}$, then we let $\mc{H}(X,E)$ be the $\mathbb{R}$-linear space of compactly supported continous maps defined on $X$ and with values in $E$ provided with the usual locally convex topology, we let $\mc{H}(X)\coloneqq\mc{H}(X,\mathbb{R})$. Let $\mr{Meas}(X,E)$ be the $\mathbb{R}$-linear space of vectorial measures on $X$ with values in $E$, namely the space of $\mathbb{R}$-linear and continuous maps from $\mc{H}(X)$ into $E$ \cite[VI.18 Def. 1]{IntBourb}. Let $\mr{Meas}(X)$ denote $\mr{Meas}(X,\mathbb{R})$ whose elements are called measures on $X$ \cite[Def. 2, $\S1$, $n^{\circ}3$, Ch. $3$]{IntBourb}. A map $g:X\to\mathbb{C}$ is scalarly essentially $\mu$-integrable or simply essentially $\mu$-integrable iff $\mf{R}\circ\imath_{\mathbb{C}}^{\mathbb{C}_{\mathbb{R}}}\circ g$ and $\mf{I}\circ\imath_{\mathbb{C}}^{\mathbb{C}_{\mathbb{R}}}\circ g$ are essentially $\mu$-integrable where $\mf{R}\in\mc{L}(\mathbb{C}_{R},\mathbb{R})$ and $\mf{I}\in\mc{L}(\mathbb{C}_{R},\mathbb{R})$ are the real and imaginary part respectively. Given a Hausdorff locally convex space $G$ over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ and a map $f:X\to G$, we say that $f$ is scalarly essentially $\mu$-integrable iff $\uppsi\circ f$ is essentially $\mu$-integrable for every $\uppsi\in G^{\prime}$. Moreover we say that the integral of $f$ belongs to $G$ iff there exists a necessarily unique $s\in G$ such that $\uppsi(s)=\int\uppsi\circ f$ for every $\uppsi\in G^{\prime}$ in which case we set $\int f\coloneqq s$. \par Let $M$ be a smooth manifold with or without boundary, $N=\mr{dim}\,M$ and $U$ be an open set of $M$. A chart and an atlas of $M$ are understood smooth. Let $\mc{A}(M)$ be the unital algebra of real valued smooth maps on $M$ and let $\mathbf{1}_{M}$ denote its unit. Let $\mc{A}_{c}(M,\mathbb{R})$ be the subalgebra of those $f\in\mc{A}(M)$ whose support is compact, while let $\mc{A}_{c}(M)$ denote the unital subalgebra $\mc{A}_{c}(M,\mathbb{R})\cup\{\mathbf{1}_{M}\}$. If $G$ is a Hausdorff locally convex space over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$, then let $\mc{A}(M,G)$ be the set of maps $f:M\to G$ such that $f\circ\imath_{U}^{M}\circ\phi^{-1}\in\mc{C}^{\infty}(\phi(U),G)$, for every chart $(U,\phi)$ of $M$. A standard argument proves that $f\in\mc{A}(M,G)$ is equivalent to state that for every $x\in M$ there exists a chart $(V,\beta)$ such that $V\ni x$ and $f\circ\imath_{V}^{M}\circ\beta^{-1}\in\mc{C}^{\infty}(\beta(V),G)$. As a result the usual gluing lemma via a covering of charts extends to $\mc{A}(M,G)$. Let $\mc{A}_{c}(U,G)$ be the subset of those maps in $\mc{A}(U,G)$ with compact support, $\mc{A}(U,G)$ and $\mc{A}_{c}(U,G)$ are clearly $\mc{A}(U)$-modules. If $N\neq 0$, then for every chart $(U,\phi)$ of $M$ and $i\in[1,N]\cap\mathbb{Z}$, let $\partial_{i}^{\phi,G}:\mc{A}(U,G)\to\mc{A}(U,G)$ be defined as in the case $G=\mathbb{R}$ with the exception of replacing the operator $D_{e_{i}}$ with $D_{e_{i}}^{\phi(U),G}$, where $\{e_{i}\}_{i=1}^{N}$ is the standard basis of $\mathbb{R}^{N}$. \par Let $TM$ and $T^{\ast}M$ be the tangent and cotangent bundle of $M$ respectively. Let $\mc{V}$ be a smooth vector bundle over $M$, then let $\Gamma_{0}(U,\mc{V})$, $\Gamma^{0}(U,\mc{V})$ and $\Gamma(U,\mc{V})$ be the $\mc{A}(U)$-module of sections, continuous sections and smooth sections respectively of the restriction at $U$ of $\mc{V}$. If $r,s\in\mathbb{Z}_{+}$, let $\mf{T}_{s}^{r}(U,M)\coloneqq\mf{T}_{s}^{r}(\Gamma(U,TM))$ and let $\mf{T}_{s}^{r}(TM)$ be the vector bundle over $M$ whose fiber at $p$ equals $\mf{T}_{s}^{r}(T_{p}M)$; while if $k\in\mathbb{Z}_{+}$, then let $\mr{Alt}^{k}(U,M)\coloneqq\mr{Alt}^{k}(\Gamma(U,M))$ and let $\mr{Alt}^{k}(TM)$ be the vector bundle over $M$ whose fiber at $p$ equals $\mr{Alt}^{k}(T_{p}M)$. Set $\mf{T}_{\bullet}^{\bullet}(U,M)\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\mf{T}_{s}^{r}(U,M)$ and $\mf{T}_{\bullet}^{\bullet}(TM)\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\mf{T}_{s}^{r}(TM)$; while $\mr{Alt}^{\bullet}(U,M)\coloneqq\bigoplus_{k\in\mathbb{Z}_{+}}\mr{Alt}^{k}(U,M)$ and $\mr{Alt}^{\bullet}(TM)\coloneqq\bigoplus_{k\in\mathbb{Z}_{+}}\mr{Alt}^{k}(TM)$. We set $\Omega^{k}(U,M)\coloneqq\Gamma(U,\mr{Alt}^{k}(TM))$ and $\Omega^{\bullet}(U,M)\coloneqq\bigoplus_{k\in\mathbb{Z}_{+}}\Omega^{k}(U,M)$. Clearly $\mr{Alt}^{\bullet}(U,M)$, $\mr{Alt}^{\bullet}(TM)$ and $\Omega^{\bullet}(U,M)$ equal the direct sum over $[1,N]\cap\mathbb{Z}$. \par We shall denote by $\mf{r}_{\mathbb{R}}$ or simply $\mf{r}$ the usual $\mc{A}(U)$-isomorphism from $\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM))$ onto $\mf{T}_{\bullet}^{\bullet}(U,M)$ and by $\mf{t}_{\mathbb{R}}$ or simply $\mf{t}$ the inverse of $\mf{r}$. By abuse of language we let us denote with the same symbol the restriction at $\mr{Alt}^{\bullet}(U,M)$ and at its range of $\mf{t}$ and by $\mf{r}$ its inverse. Given a chart $(U,\phi)$ of $M$, in order to keep the notation as light as possible we convein to let $dx_{i}^{\phi}\in\Gamma(U,T^{\ast}M)$ denote also $\mf{t}(dx_{i}^{\phi})\in\Gamma(U,TM)^{\ast}$. Moreover we let $\{(\otimes(b^{r,s,\phi})^{\ast})_{\alpha}\,\vert\,\alpha\in\Xi(b^{r,s,\phi})\}$ and $\{\mc{E}_{dx^{\phi}}(I)\,\vert\,I\in M(k,N,<)\}$ be the basis of $\mf{T}_{s}^{r}(U,M)$ and $\mr{Alt}^{k}(U,M)$ image via the isomorphism $\mf{r}$ of the basis of $\Gamma(U,\mf{T}_{s}^{r}(TM))$ and $\Gamma(U,\mr{Alt}^{k}(TM))$ associated with the chart $(U,\phi)$ respectively. \par In what follows we let $K\in\{\mathbb{R},\mathbb{C}\}$ and let $G$, $H$, $G_{1}$ and $H_{1}$ be Hausdorff locally convex spaces over $\mathbb{K}$, and let $M$ be a finite dimensional smooth manifold $M$, with or without boudary, such that $N\coloneqq\mr{dim}\,M\neq 0$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^{N}$ and for every open set $A$ of $\mathbb{R}^{N}$ we let $\lambda_{A}$ be the restriction at $A$ of $\lambda$. \end{notation}
\begin{introduction} Let us outline the main ideas underlying this note. We opt to avoid employing the concept of manifold modelled over locally convex spaces via the Bastiani differential calculus. Fortunately this is possible if we generalize to our context the well-known fact that given a finite dimensional vector bundle $\mc{Z}$ on $M$, then $\Gamma(\mc{Z}\otimes\mr{Alt}^{\bullet}(TM))$ is $\mc{A}(M)$-isomorphic to $\Gamma(\mc{Z})\otimes_{\mc{A}(M)}\Gamma(\mr{Alt}^{\bullet}(TM))$. \par Therefore motivated by the above result, given a finite dimensional vector bundle $\mc{V}$ on $M$ and an open set $U$ of $M$, we shall define the space of $G$-valued scalarly essentially $\lambda$-integrable sections of type $\mc{V}$ defined on $U$, as the $\mc{A}(U)$-module \begin{equation} \label{10011652} \mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\Gamma(U,\mc{V}); \end{equation} where $\mf{L}_{c}^{1}(U,G,\lambda)$ is the $\mc{A}(U)$-module of compactly supported scalarly essentially $\lambda$-integrable maps from $U$ at values in $G$ as defined in a natural way in Def. \ref{09281553}. Similar definition is given for $G$-valued smooth sections of type $\mc{V}$ defined on $U$ by replacing $\mf{L}_{c}^{1}(U,G,\lambda)$ with $\mc{A}(U)$. \par The advantages of employing the above definition are the following. \par First it is well-known that for any (possibly noncommutative ring) $A$, any $A$-module $B$ and any \emph{free} $A$-module $C$ we have a unique decomposition of every element of the $\mathbb{Z}$-module $B\otimes_{A}C$ in terms of elements of $B$ and elements of the basis of $C$. In addition when $U$ is the domain of a chart, then $\Gamma(U,\mc{V})$ is a free $\mc{A}(U)$-finite dimensional module. As a result we obtain for instance Cor. \ref{10011800} and Cor. \ref{10011804}. As a result any element of $\mc{A}(U,G)\otimes_{\mc{A}(U)}\Omega^{\bullet}(U,M)$ admits a unique decomposition which among other properties permits to define the exterior differential in a natural way and then to extend it in the usual manner see Def. \ref{08281540} and Thm. \ref{08281401}. Furthermore the unique decomposition applied to any $\mathbb{R}$-valued scalarly essentially $\lambda$-integrable form over an open set of $\mathbb{R}^{N}$, permits to define its integral Def. \ref{09140959a} that is the first step to define the weak integral. \par Second all the standard operations over tensor fields can be extended to the $G$-valued setting just by $\mc{A}(U)$-linearization of $\mc{A}(U)$-bilinears. A paradigmatic example showing this procedure is the wedge product in Def. \ref{09260951} provided a sequence of preliminary results, where an extra care must be implemented since the use in the definition of the projective topological tensor product of two Hausdorff locally convex spaces. \par Third by pushing forward via any continuous functional on $G$ the operation so obtained between $G$-valued sections we obtain the usual corresponding operation between $\mathbb{R}$-valued sections Prp. \ref{09170918} and Prp. \ref{09290547}. \par Fourth and most importantly by pushing forward via any continuous linear map $\uppsi$ from $G$ into $H$ a $G$-valued scalarly essentially $\lambda$-integrable section $\eta$ of type $\mc{V}$ defined on $U$ we obtain a $H$-valued scalarly essentially $\lambda$-integrable section $\uppsi_{\times}(\eta)$ of type $\mc{V}$ defined on $U$ Def. \ref{08281845int}. This permits when $H=\mathbb{K}$ to define in Def. \ref{09281748} the weak integral of a $G$-valued smooth maximal form $\eta$ as the map associating to any continuous functional $\uppsi$ on $G$ the integral of $\uppsi_{\times}(\eta)$, then as a result a vectorial measure on $M$ with values in the real locally convex space $\lr{(G^{\prime})^{\ast}}{\sigma((G^{\prime})^{\ast},G^{\prime})}_{\mathbb{R}}$ is constructed in Thm. \ref{09171005}. Finally the Stokes theorem Thm. \ref{08281926} for a $G$-valued smooth $(N-1)$-form $\theta$ results as a consequence of the usual Stokes theorem applied to $\uppsi_{\times}(\theta)$ for every $\uppsi$ in the topological dual of $G$. \end{introduction}
\section{$G$-Valued Integrable and Smooth Tensor Fields} \label{09172012}
\begin{definition} [\textbf{$G$-Valued Scalarly Essentially Integrable Maps on $M$}] \label{09281553} Define $\mf{L}^{1}(M,G,\lambda)$ to be the set of maps $f:M\to G$ such that $f\circ\imath_{U}^{M}\circ\phi^{-1}$ is scalarly essentially $\lambda_{\phi(U)}$-integrable, for every chart $(U,\phi)$ of $M$. Let $\mf{L}_{c}^{1}(M,G,\lambda)$ be the subset of the maps in $\mf{L}^{1}(M,G,\lambda)$ with compact support. \end{definition}
\begin{remark} \label{09251444int} The theorem of change of variable in multiple integrals along with a standard argument prove that $f\in\mf{L}^{1}(M,G,\lambda)$ is equivalent to state that for every $x\in M$ there exists a chart $(V,\beta)$ such that $V\ni x$ and $f\circ\imath_{V}^{M}\circ\beta^{-1}$ is scalarly essentially $\lambda_{\beta(V)}$-integrable. As a result the usual gluing lemma via a covering of charts extends to $\mf{L}^{1}(M,G,\lambda)$. \end{remark} Recall that $\mc{A}_{c}(M)$ is by definition the unital subalgebra of $\mc{A}(M)$ generated by the unit $\mathbf{1}_{M}$ and by the subalgebra $\mc{A}_{c}(M,\mathbb{R})$ of the maps in $\mc{A}(M)$ with compact support. Thus $\mc{A}_{c}(M)=\mc{A}_{c}(M,\mathbb{R})\cup\{\mathbf{1}_{M}\}$.
\begin{lemma} $\mf{L}^{1}(M,G,\lambda)$ is a $\mc{A}_{c}(M)$-module and $\mf{L}_{c}^{1}(M,G,\lambda)$ is a $\mc{A}(M)$-module. \end{lemma} \begin{proof} $\mf{L}^{1}(M,G,\lambda)$ is a $\mc{A}_{c}(M)$-module since $\mc{A}_{c}(M,\mathbb{R})\subseteq\mc{H}(M)$. Next let $f\in\mf{L}_{c}^{1}(M,G,\lambda)$ and $\psi:M\to\mathbb{R}$ be a smooth bump function for $\mr{supp}(f)$ supported in $M$, then $f=\psi f$ therefore for any $g\in\mc{A}(M)$ we have $gf=g\psi f$, but $g\psi\in\mc{A}_{c}(M)$ and the second sentence of the statement follows by the first sentence of the statement above proven. \end{proof}
Untill the end of this work we let $U$ be an open set of $M$.
\begin{definition} Let $\Gamma(c)(U,TM)$ be the $\mc{A}_{c}(U)$-module $\Gamma(U,TM)$, define $\mf{T}_{s}^{r}(U,M)^{c}=\mf{T}_{s}^{r}(\Gamma(c)(U,TM))$, set $\mf{T}_{s}^{r}(M)^{c}\coloneqq\mf{T}_{s}^{r}(M,M)^{c}$. Moreover define the $\mc{A}(U)$-modules \begin{equation*} \begin{aligned} \Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM))^{c} &\coloneqq \bigl\{f\in\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM))\,\vert\,\mf{r}_{\mathbb{R}}(f)\in\mf{T}_{\bullet}^{\bullet}(U,M)^{c}\bigr\}; \\ \Gamma_{c}(U,\mf{T}_{\bullet}^{\bullet}(TM)) &\coloneqq \bigl\{f\in\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM))\,\vert\,\mr{supp}(f)\in\mr{Cmp}(M)\bigr\}; \\ \mf{T}_{\bullet}^{\bullet}(U,M)_{c} &\coloneqq \bigl\{\zeta\in\mf{T}_{\bullet}^{\bullet}(U,M)\,\vert\,\mr{supp}(\mf{t}_{\mathbb{R}}(\zeta))\in\mr{Cmp}(M)\bigr\}. \end{aligned} \end{equation*} \end{definition} By construction $\mf{T}_{s}^{r}(U,M)^{c}$ is a $\mc{A}_{c}(U)$-module however we have also that
\begin{lemma} \label{09150739} Let $r,s\in\mathbb{Z}_{+}$, thus $\mf{T}_{s}^{r}(U,M)^{c}$ is a $\mc{A}(U)$-module; $\Gamma_{c}(U,\mf{T}_{s}^{r}(TM))$ is a $\mc{A}(U)$-submodule of $\Gamma(U,\mf{T}_{s}^{r}(TM))^{c}$, and then $\mf{T}_{s}^{r}(U,M)_{c}$ is a $\mc{A}(U)$-submodule of $\mf{T}_{s}^{r}(U,M)^{c}$. \end{lemma}
\begin{definition} [\textbf{$G$-Valued Scalarly Essentially Integrable Tensor Fields}] Let $r,s\in\mathbb{Z}_{+}$, define the $\mc{A}(U)$-module of $G$-valued scalarly essentially $\lambda$-integrable tensor fields on $M$ defined on $U$ of type $(r,s)$ to be \begin{equation*} \mf{I}_{s}^{r}(U,M;G,\lambda)\coloneqq\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mf{T}_{s}^{r}(U,M). \end{equation*} Define the $\mc{A}(U)$-modules \begin{equation*} \ms{I}_{s}^{r}(U,M;G,\lambda)\coloneqq\mf{T}_{s}^{r}\bigl(\Gamma(U,TM),\mf{L}_{c}^{1}(U,G,\lambda)\bigr). \end{equation*} and \begin{equation*} \begin{aligned} \mf{I}_{s}^{r}(U,M;G,\lambda)^{c}&\coloneqq\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mf{T}_{s}^{r}(U,M)^{c}; \\ \mf{I}_{s}^{r}(U,M;G,\lambda)_{c}&\coloneqq\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mf{T}_{s}^{r}(U,M)_{c}. \end{aligned} \end{equation*} Finally define \begin{equation*} \begin{aligned} \mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)&\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\mf{I}_{s}^{r}(U,M;G,\lambda), \\ \ms{I}_{\bullet}^{\bullet}(U,M;G,\lambda)&\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\ms{I}_{s}^{r}(U,M;G,\lambda); \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)^{c}&\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\mf{I}_{s}^{r}(U,M;G,\lambda)^{c}; \\ \mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)_{c}&\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\mf{I}_{s}^{r}(U,M;G,\lambda)_{c}. \end{aligned} \end{equation*} \end{definition}
\begin{remark} \label{09161024int} Clearly $\mf{I}_{s}^{r}(U,M;G,\lambda)^{c}$ is $\mc{A}(U)$-isomorphic to a submodule of $\mf{I}_{s}^{r}(U,M;G,\lambda)$ and in what follows we shall identify these two modules. Similarly we identify $\mf{I}_{s}^{r}(U,M;G,\lambda)_{c}$ with a submodule of $\mf{I}_{s}^{r}(U,M;G,\lambda)$, in particular we have $\mf{I}_{s}^{r}(U,M;G,\lambda)_{c}\subseteq\mf{I}_{s}^{r}(U,M;G,\lambda)^{c}$. \end{remark}
\begin{proposition} \label{09161155} $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)=\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)_{c} =\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)^{c}$. \end{proposition} \begin{proof} If $f\in\mf{L}_{c}^{1}(U,G,\lambda)$ and $T\in\mf{T}_{s}^{r}(U,M)$ and $\psi$ is a smooth bump function for $\mr{supp}(f)$ supported in $U$, then $f=\psi f$, so $f\otimes T=(\psi f)\otimes T=f\otimes(\psi T)$. Thus the statement follows since Rmk. \ref{09161024int}. \end{proof}
\begin{lemma} \label{09121429} Assume $\mathbb{K}=\mathbb{C}$, thus $\mf{L}^{1}(U,G,\lambda)=\mf{L}^{1}(U,G_{\mathbb{R}},\lambda)$, in particular $\mf{I}_{\bullet}^{\bullet}(U,M;G_{\mathbb{R}},\lambda)=\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)$. \end{lemma} \begin{proof} $\mf{L}^{1}(U,G_{\mathbb{R}},\lambda)\subseteq\mf{L}^{1}(U,G,\lambda)$ since for every $\uppsi\in\mc{L}(G,\mathbb{C})$ we have $\mf{R}\circ\imath_{\mathbb{C}}^{\mathbb{C}_{\mathbb{R}}}\circ\uppsi\circ\imath_{G_{\mathbb{R}}}^{G}\in\mc{L}(G_{\mathbb{R}},\mathbb{R})$ and $\mf{I}\circ\imath_{\mathbb{C}}^{\mathbb{C}_{\mathbb{R}}}\circ\uppsi\circ\imath_{G_{\mathbb{R}}}^{G}\in\mc{L}(G_{\mathbb{R}},\mathbb{R})$. Next according to what stated immediately after \cite[II.65(1)]{EVT} we have that \begin{equation} \label{09171644} (\forall\phi\in\mf{L}(G_{\mathbb{R}},\mathbb{R}))(\exists\,!\uppsi\in\mf{L}(G,\mathbb{C})) (\phi=\mf{R}\circ\imath_{\mathbb{C}}^{\mathbb{C}_{\mathbb{R}}}\circ\uppsi\circ\imath_{G_{\mathbb{R}}}^{G}); \end{equation} from which we deduce that $\mf{L}^{1}(U,G,\lambda)\subseteq\mf{L}^{1}(U,G_{\mathbb{R}},\lambda)$. \end{proof}
\begin{proposition} \label{09101648int} $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)$ is isomorphic to $\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mf{T}_{\bullet}^{\bullet}(U,M)$; while $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)^{c}$ is isomorphic to $\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mf{T}_{\bullet}^{\bullet}(U,M)^{c}$ as well $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)_{c}$ is isomorphic to $\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mf{T}_{\bullet}^{\bullet}(U,M)_{c}$ in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since \cite[II.61 Prp. 7]{BourA1} there exist (canonical) $\mathbb{Z}$-linear isomorphisms, which are clearly a $\mc{A}(U)-\mr{mod}$ isomorphisms by the definition of the module structure of the tensor product of modules over a commutative ring. \end{proof}
We shall identify the above isomorphic modules.
\begin{proposition} \label{08101431} \begin{multline*} \left(\exists\,!\Upphi\in \mr{Mor}_{\mc{A}(U)-\mr{mod}}\left(\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda),\ms{I}_{\bullet}^{\bullet}(U,M;G,\lambda)\right)\right) (\forall(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}) \\ (\forall f\in\mf{L}_{c}^{1}(U,G,\lambda)) (\forall T\in\mf{T}_{s}^{r}(U,M)) (\forall(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})\in\prod[\Gamma(U,TM),r,s]) \\ \bigl(\Upphi(f\otimes T) (\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})= T(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})\cdot f\bigr). \end{multline*} \end{proposition} \begin{proof} By the universal property of the tensor product over a commutative ring applied to the $\mc{A}(U)$-bilinear map $\ast:\mf{L}_{c}^{1}(U,G,\lambda)\times\mf{T}_{s}^{r}(U,M)\to\ms{I}_{s}^{r}(U,M;G;\lambda)$, $(f,T)\mapsto f\ast T$, defined by $(f\ast T)(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})=T(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})\cdot f$. \end{proof}
\begin{corollary} [\textbf{Unique Decomposition of $G$-Valued Integrable Tensor fields at $M$ defined on a Chart}] \label{10011800} Let $(U,\phi)$ be a chart of $M$, $r,s\in\mathbb{Z}_{+}$ and $\mr{T}\in\mf{I}_{s}^{r}(U,M;G;\lambda)$, thus \begin{equation*} \bigl(\exists\,!f:\Xi(b^{r,s,\phi})\to\mf{L}_{c}^{1}(U,G,\lambda)\bigr) \left(\mr{T}=\sum_{\alpha\in\Xi(b^{r,s,\phi})}f_{\alpha}\otimes(\otimes(b^{r,s,\phi})^{\ast})_{\alpha}\right). \end{equation*} \end{corollary} \begin{proof} $\{(\otimes(b^{r,s,\phi})^{\ast})_{\alpha}\,\vert\,\alpha\in\Xi(b^{r,s,\phi})\}$ is a basis of $\mf{T}_{s}^{r}(U,M)$, thus the statement follows since \cite[II.62 Cor.1]{BourA1}. \end{proof}
\begin{definition} [\textbf{Bar Operators on Integrable Tensor Fields}] \label{09101743} Define the $\mc{A}(U)$-module \begin{equation*} \Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G;\lambda)\coloneqq \mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM)). \end{equation*} Define \begin{equation*} \begin{aligned} \mf{t}_{G}&\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}\bigl(\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda), \Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G,\lambda)\bigr), \\ \mf{t}_{G}&\coloneqq\mr{Id}_{\mf{L}_{c}^{1}(U,G,\lambda)}\otimes\mf{t}_{\mathbb{R}}; \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \mf{r}_{G}&\in \mr{Mor}_{\mc{A}(U)-\mr{mod}} \bigl(\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G,\lambda),\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)\bigr); \\ \mf{r}_{G}&\coloneqq\mr{Id}_{\mf{L}_{c}^{1}(U,G,\lambda)}\otimes\mf{r}_{\mathbb{R}}. \end{aligned} \end{equation*} \end{definition}
\begin{proposition} \label{08281918int} $\mf{t}_{G}$ and $\mf{r}_{G}$ are isomorphisms one the inverse of the other in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since $\mf{t}_{\mathbb{R}}$ and $\mf{r}_{\mathbb{R}}$ are isomorphisms one the inverse of the other in the category $\mc{A}(U)-\mr{mod}$. \end{proof}
\begin{remark} \label{09261218} Since Rmk. \ref{09251444int} the gluing lemma via a covering of charts extends to $\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G,\lambda)$. \end{remark}
\begin{proposition} \begin{equation*} \exists!\,\mf{s}_{G}\in \mr{Mor}_{\mc{A}(U)-\mr{mod}}\bigl(\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G,\lambda), \Gamma_{0}(U,G_{\mathbb{R}}\otimes_{\mathbb{R}}\mf{T}_{\bullet}^{\bullet}(TM))\bigr), \end{equation*} such that \begin{equation*} (\forall f\in\mf{L}_{c}^{1}(U,G,\lambda))(\forall\beta\in\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM))) \bigl(\mf{s}_{G}(f\otimes\beta)=(U\ni p\mapsto f(p)\otimes\beta(p))\bigr). \end{equation*} \end{proposition} \begin{proof} The map $(f,\beta)\mapsto(U\ni p\mapsto f(p)\otimes\beta(p))$ is $\mc{A}(U)$-bilinear thus the statement follows by the universal property of the tensor product of modules over a commutative ring. \end{proof}
Now we are able to define the support as follows \begin{definition} [\textbf{Support}] \label{09161456} Define \begin{equation*} \begin{cases} \mr{supp}:\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)\to\mr{Cmp}(M); \\ \mr{supp}(\theta)\coloneqq\mr{supp}\bigl((\mf{s}_{G}\circ\mf{t}_{G})(\theta)\bigr). \end{cases} \end{equation*} \end{definition}
\begin{convention} \label{09180844} We let $\mf{r}$, $\mf{t}$ and $\mf{s}$ denote $\mf{r}_{G}$, $\mf{t}_{G}$ and $\mf{s}_{G}$ respectively whenever it does not cause confusion. \end{convention}
We will employ the next result in order to construct in Prp. \ref{09171005} a vectorial measure
\begin{proposition} \label{09170948} There exists a unique $\mc{A}(U)$-bilinear map $(g,\theta)\mapsto g\cdot\theta$ from $\mc{H}(U,\mathbb{K})\times\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)$ into $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)$ such that for every $g\in\mc{H}(U,\mathbb{K})$ and every $f\in\mf{L}_{c}^{1}(U,M;G,\lambda)$ and $T\in\mf{T}_{\bullet}^{\bullet}(U,M)$ we have $g\cdot(f\otimes T)=(gf)\otimes T$. \end{proposition} \begin{proof} Let $g\in\mc{H}(U,\mathbb{K})$, thus the map $(f,T)\mapsto(gf)\otimes T$ is $\mc{A}(U)$-bilinear since the $\mc{A}(U)$-module structure of $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)$, then by the universal property there exists a unique $\mc{A}(U)$-linear endomorphism $\mf{k}(g)$ of $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)$ such that $\mf{k}(g)(f\otimes T)=(gf)\otimes T$. Next by the uniqueness characterization present in the universal property we deduce that $\mf{k}$ is a $\mc{A}(U)$-linear map from $\mc{H}(U,\mathbb{K})$ into the $\mc{A}(U)$-module of $\mc{A}(U)$-endomorphisms of $\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda)$. Thus the statement follows since the isomorphism in \cite[II.74 Prp. 1(6)]{BourA1} and by the universal property of the tensor product. \end{proof}
\begin{definition} Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$ be a \textbf{diffeomorphism}. Define \begin{equation*} \begin{aligned} F^{\ast}:\mf{L}_{c}^{1}(U,G,\lambda)&\to\mf{L}_{c}^{1}(W,G,\lambda), \\ f&\mapsto f\circ F; \end{aligned} \end{equation*} well-set since the theorem of change of variable in multiple integrals. \end{definition} Since $F^{\ast}$ is $\mathbb{R}$-linear we can give the following
\begin{definition} [\textbf{Pullback of Integrable Tensors of type $(0,s)$}] Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$ be a \textbf{diffeomorphism}. Define \begin{equation} \label{09180901} \begin{aligned} \overset{\times}{F}&\in\mr{Mor}_{\mathbb{R}-\mr{mod}}( \mf{I}_{\bullet}^{0}(U,M;G,\lambda),\mf{I}_{\bullet}^{0}(W,N;G,\lambda)) \\ \overset{\times}{F}&\coloneqq F^{\ast}\otimes F^{\ast}; \end{aligned} \end{equation} and \begin{equation} \label{09180902} \begin{aligned} \overset{\times}{F}&\in\mr{Mor}_{\mathbb{R}-\mr{mod}}(\Gamma(U,\mf{T}_{\bullet}^{0}(TM);G,\lambda), \Gamma(W,\mf{T}_{\bullet}^{0}(TN);G,\lambda); \\ \overset{\times}{F}&\coloneqq F^{\ast}\otimes F^{\ast}. \end{aligned} \end{equation} \end{definition} Next we prepare for the definition of pushforward.
\begin{definition} Let $\uppsi\in\mc{L}(G,H)$, thus define \begin{equation*} \uppsi_{\ast}:\mf{L}_{c}^{1}(U,G,\lambda)\ni f\mapsto\uppsi\circ f\in\mf{L}_{c}^{1}(U,H,\lambda). \end{equation*} \end{definition} Well-set definition since $\uppsi$ is linear and continuous.
Clearly we have \begin{lemma} \label{08281522int} Let $\uppsi\in\mc{L}(G,H)$, thus $\uppsi_{\ast}\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}(\mf{L}_{c}^{1}(U,G,\lambda),\mf{L}_{c}^{1}(U,H,\lambda))$. \end{lemma} The above result permits to give the following
\begin{definition} [\textbf{Pushforward of $G$-Valued Integrable Tensors}] \label{08281845int} Let $\uppsi\in\mc{L}(G,H)$, define \begin{equation*} \begin{aligned} \uppsi_{\times}&\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}(\mf{I}_{\bullet}^{\bullet}(U,M;G,\lambda),\mf{I}_{\bullet}^{\bullet}(U,M;H,\lambda)); \\ \uppsi_{\times}&\coloneqq\uppsi_{\ast}\otimes\mr{Id}_{\mf{T}_{\bullet}^{\bullet}(U,M)}. \end{aligned} \end{equation*} \end{definition} Then easily we find that
\begin{proposition} [\textbf{Pushforward Commutes with All the Above Operators}] \label{09170918} Let $N$ be a differential manifold, $W$ be an open set of $N$, and $F\in\mc{C}^{\infty}(W,U)$ be a diffeomorphism. If $\uppsi\in\mc{L}(G,H)$, then $\uppsi_{\times}\circ\mf{t}=\mf{t}\circ\uppsi_{\times}$, $\uppsi_{\times}\circ\mf{r}=\mf{r}\circ\uppsi_{\times}$, and $\uppsi_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ\uppsi_{\times}$; \end{proposition} and that
\begin{proposition} \label{09211102} Let $N$ be a differential manifold, $W$ be an open set of $N$, and $F\in\mc{C}^{\infty}(W,U)$ be a diffeomorphism. Thus for every $h\in\mc{A}(U)$ and every $\theta\in\mf{I}_{\bullet}^{0}(U,M;G,\lambda)$ we have $\overset{\times}{F}(h\theta)=(F^{\ast}h)\overset{\times}{F}(\theta)$. \end{proposition}
\begin{corollary} \label{09170919} Assume $\mathbb{K}=\mathbb{C}$. Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$ be a diffeomorphism. If $\{G_{j}\}_{j\in J}$ is a family of \textbf{real} locally convex spaces and $G$ is such that $G_{\mathbb{R}}=\prod_{j\in J}G_{j}$ provided with the product topology. Thus for every $j\in J$ we have that $\Pr^{j}_{\times}\circ\mf{t}=\mf{t}\circ\Pr^{j}_{\times}$, $\Pr^{j}_{\times}\circ\mf{r}=\mf{r}\circ\Pr^{j}_{\times}$, and $\Pr^{j}_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ\Pr^{j}_{\times}$. \end{corollary} \begin{proof} $\Pr^{j}\in\mc{L}(G_{\mathbb{R}},G_{j})$ and the product topology is locally convex as a particular case of what stated in \cite[II.5]{EVT}. Thus the statement is well-set and it follows since Prp. \ref{09170918} applied to $\mathbb{K}=\mathbb{R}$, to $G$ replaced by $G_{\mathbb{R}}$ and to $\uppsi$ replaced by $\Pr^{j}$. \end{proof}
\begin{definition} [\textbf{$G$-Valued Smooth Tensor fields at $M$ defined on $U$}] Let $r,s\in\mathbb{Z}_{+}$, define the $\mc{A}(U)$-module of $G$-valued differential tensor fields at $M$ defined on $U$ of type $(r,s)$ to be \begin{equation*} \mf{T}_{s}^{r}(U,M;G)\coloneqq\mc{A}(U,G)\otimes_{\mc{A}(U)}\mf{T}_{s}^{r}(U,M). \end{equation*} Next we define the $\mc{A}(U)$-module \begin{equation*} \ms{T}_{s}^{r}(U,M;G)\coloneqq\mf{T}_{s}^{r}\bigl(\Gamma(U,TM),\mc{A}(U;G)\bigr). \end{equation*} Finally define the $\mc{A}(U)$-modules \begin{equation*} \begin{aligned} \mf{T}_{\bullet}^{\bullet}(U,M;G)\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\mf{T}_{s}^{r}(U,M;G); \\ \ms{T}_{\bullet}^{\bullet}(U,M;G)\coloneqq\bigoplus_{(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}}\ms{T}_{s}^{r}(U,M;G). \end{aligned} \end{equation*} \end{definition}
\begin{definition} Let $r,s\in\mathbb{Z}_{+}$, define the $\mc{A}(U)$-modules \begin{equation*} \begin{aligned} \mf{T}_{s}^{r}(U,M;G)_{[c]}&\coloneqq\mc{A}_{c}(U,G)\otimes_{\mc{A}(U)}\mf{T}_{s}^{r}(U,M); \\ \mf{T}_{s}^{r}(U,M;G)_{c}&\coloneqq\mc{A}(U,G)\otimes_{\mc{A}(U)}\mf{T}_{s}^{r}(U,M)_{c}; \\ \mf{T}_{s}^{r}(U,M;G)^{c}&\coloneqq\mc{A}(U,G)\otimes_{\mc{A}(U)}\mf{T}_{s}^{r}(U,M)^{c}. \end{aligned} \end{equation*} \end{definition}
\begin{remark} \label{09161024} Clearly $\mf{T}_{s}^{r}(U,M;G)^{c}$ is $\mc{A}(U)$-isomorphic to a submodule of $\mf{T}_{s}^{r}(U,M;G)$ and in what follows we shall identify these two modules. Similarly we identify $\mf{T}_{s}^{r}(U,M;G)_{c}$ (respectively $\mf{T}_{s}^{r}(U,M;G)_{[c]}$) with a submodule of $\mf{T}_{s}^{r}(U,M;G)$, in particular we have $\mf{T}_{s}^{r}(U,M;G)_{c}\subset\mf{T}_{s}^{r}(U,M;G)^{c}$. \end{remark}
\begin{proposition} \label{09161202} Let $r,s\in\mathbb{Z}_{+}$, thus $\mf{T}_{s}^{r}(U,M;G)_{[c]}=\mf{T}_{s}^{r}(U,M;G)_{c}$. \end{proposition} \begin{proof} If $f\in\mc{A}_{c}(U,G)$ and $T\in\mf{T}_{s}^{r}(U,M)$ and $\psi$ is a smooth bump function for $\mr{supp}(f)$ supported in $U$, then $f=\psi f$, so $f\otimes T=(\psi f)\otimes T=f\otimes(\psi T)$. If $g\in\mc{A}(U,G)$ and $S\in\mf{T}_{s}^{r}(U,M)_{c}$ and $\psi$ is a smooth bump function for $\mr{supp}(S)$ supported in $U$, then $S=\psi S$, thus $g\otimes S=g\otimes(\psi S)=(\psi g)\otimes S$. Thus the statement follows since Rmk. \ref{09161024}. \end{proof}
\begin{remark} \label{10041224} $\mf{T}_{s}^{r}(U,M;G)=\mf{T}_{s}^{r}(U,M;G_{\mathbb{R}})$ since $\mc{A}(U,G)=\mc{A}(U,G_{\mathbb{R}})$. \end{remark}
\begin{proposition} \label{09101648} $\mf{T}_{\bullet}^{\bullet}(U,M;G)$ is isomorphic to $\mc{A}(U,G)\otimes_{\mc{A}(U)}\mf{T}_{\bullet}^{\bullet}(U,M)$ in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since \cite[II.61Prp. 7]{BourA1} there exists a (canonical) $\mathbb{Z}$-linear isomorphism, which is clearly a $\mc{A}(U)-\mr{mod}$ isomorphism by the definition of the module structure of the tensor product of modules over a commutative ring. \end{proof}
\begin{proposition} \label{08301601} \begin{multline} \left(\exists\,!\Uppsi\in \mr{Mor}_{\mc{A}(U)-\mr{mod}}\left(\mf{T}_{\bullet}^{\bullet}(U,M;G),\ms{T}_{\bullet}^{\bullet}(U,M;G)\right)\right) (\forall(r,s)\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}) \\ (\forall f\in\mc{A}(U,G)) (\forall T\in\mf{T}_{s}^{r}(U,M)) (\forall(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})\in\prod[\Gamma(U,TM),r,s]) \\ \bigl(\Uppsi(f\otimes T) (\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})= T(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})\cdot f\bigr). \end{multline} \end{proposition} \begin{proof} By the universal property of the tensor product over a commutative ring applied to the $\mc{A}(U)$-bilinear map $\star:\mc{A}(U,G)\times\mf{T}_{s}^{r}(U,M)\to\ms{T}_{s}^{r}(U,M;G)$, $(f,T)\mapsto f\star T$, defined by $(f\star T)(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})=T(\theta_{1},\dots,\theta_{r},X_{1},\dots,X_{s})\cdot f$. \end{proof} The following result justifies the choice of the above definition.
\begin{corollary} [\textbf{Unique Decomposition of $G$-Valued Smooth Tensor fields at $M$ defined on a Chart}] \label{10011804} Let $(U,\phi)$ be a chart of $M$, $r,s\in\mathbb{Z}_{+}$ and $\mr{T}\in\mf{T}_{s}^{r}(U,M;G)$, thus \begin{equation*} \bigl(\exists\,!f:\Xi(b^{r,s,\phi})\to\mc{A}(U,G)\bigr) \left(\mr{T}=\sum_{\alpha\in\Xi(b^{r,s,\phi})}f_{\alpha}\otimes(\otimes(b^{r,s,\phi})^{\ast})_{\alpha}\right). \end{equation*} \end{corollary} \begin{proof} $\{(\otimes(b^{r,s,\phi})^{\ast})_{\alpha}\,\vert\,\alpha\in\Xi(b^{r,s,\phi})\}$ is a basis of $\mf{T}_{s}^{r}(U,M)$, thus the statement follows since \cite[II.62 Cor.1]{BourA1}. \end{proof}
\begin{definition} [\textbf{Bar Operators on Smooth Tensor Fields}] \label{09180944} Define the $\mc{A}(U)$-module \begin{equation*} \Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G)\coloneqq\mc{A}(U,G)\otimes_{\mc{A}(U)}\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM)). \end{equation*} Define with abuse of language the following maps \begin{equation*} \begin{aligned} \mf{t}_{G}&\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}\bigl(\mf{T}_{\bullet}^{\bullet}(U,M;G), \Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G)\bigr), \\ \mf{t}_{G}&\coloneqq\mr{Id}_{\mc{A}(U,G)}\otimes\mf{t}_{\mathbb{R}}; \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \mf{r}_{G}&\in \mr{Mor}_{\mc{A}(U)-\mr{mod}} \bigl(\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G),\mf{T}_{\bullet}^{\bullet}(U,M;G)\bigr); \\ \mf{r}_{G}&\coloneqq\mr{Id}_{\mc{A}(U,G)}\otimes\mf{r}_{\mathbb{R}}. \end{aligned} \end{equation*} \end{definition}
\begin{proposition} \label{08281918} $\mf{t}_{G}$ and $\mf{r}_{G}$ are isomorphisms one the inverse of the other in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since $\mf{t}_{\mathbb{R}}$ and $\mf{r}_{\mathbb{R}}$ are isomorphisms one the inverse of the other in the category $\mc{A}(U)-\mr{mod}$. \end{proof}
\begin{remark} \label{09261218diff} The gluing lemma via a covering of charts extends to $\Gamma(U,\mf{T}_{\bullet}^{\bullet}(TM);G)$, since it extends for maps in $\mc{A}(U,G)$. \end{remark}
We shall use convention \ref{09180844} also for the above defined maps.
\begin{definition} Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$. Define \begin{equation*} \begin{aligned} F^{\ast}:\mc{A}(U,G)&\to\mc{A}(W,G), \\ f&\mapsto f\circ F. \end{aligned} \end{equation*} \end{definition} Since $F^{\ast}$ is $\mathbb{R}$-linear we can give the following
\begin{definition} [\textbf{Pullback of Smooth Tensor of type $(0,s)$}] Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$. Define \begin{equation} \label{09180903} \begin{aligned} \overset{\times}{F}&\in\mr{Mor}_{\mathbb{R}-\mr{mod}}(\mf{T}_{\bullet}^{0}(U,M;G),\mf{T}_{\bullet}^{0}(W,N;G)) \\ \overset{\times}{F}&\coloneqq F^{\ast}\otimes F^{\ast}; \end{aligned} \end{equation} and \begin{equation} \label{09180904} \begin{aligned} \overset{\times}{F}&\in\mr{Mor}_{\mathbb{R}-\mr{mod}}(\Gamma(U,\mf{T}_{\bullet}^{0}(TM);G),\Gamma(W,\mf{T}_{\bullet}^{0}(TN);G); \\ \overset{\times}{F}&\coloneqq F^{\ast}\otimes F^{\ast}. \end{aligned} \end{equation} \end{definition}
\begin{definition} Let $\uppsi\in\mc{L}(G,H)$, thus define by abuse of language \begin{equation*} \uppsi_{\ast}:\mc{A}(U,G)\ni f\mapsto\uppsi\circ f\in\mc{A}(U,H). \end{equation*} \end{definition} Well-set definition since $\uppsi$ is linear and continuous.
Clearly we have \begin{lemma} \label{08281522diff} Let $\uppsi\in\mc{L}(G,H)$, thus $\uppsi_{\ast}\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}(\mc{A}(U,G),\mc{A}(U,H))$. \end{lemma} The above result permits to give the following
\begin{definition} [\textbf{Pushforward of $G$-Valued Smooth Tensors}] \label{08281845diff} Let $\uppsi\in\mc{L}(G,H)$, define \begin{equation*} \begin{aligned} \psi_{\times}&\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}(\mf{T}_{\bullet}^{\bullet}(U,M;G),\mf{T}_{\bullet}^{\bullet}(U,M;H)); \\ \uppsi_{\times}&\coloneqq\uppsi_{\ast}\otimes\mr{Id}_{\mf{T}_{\bullet}^{\bullet}(U,M)}. \end{aligned} \end{equation*} \end{definition} Then easily we find that
\begin{proposition} [\textbf{Pushforward Commutes with All the Above Operators}] \label{09170918diff} Let $N$ be a differential manifold, $W$ be an open set of $N$, and $F\in\mc{C}^{\infty}(W,U)$. If $\uppsi\in\mc{L}(G,H)$, then $\uppsi_{\times}\circ\mf{t}=\mf{t}\circ\uppsi_{\times}$, $\uppsi_{\times}\circ\mf{r}=\mf{r}\circ\uppsi_{\times}$, and $\uppsi_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ\uppsi_{\times}$; \end{proposition} and that
\begin{proposition} \label{09211102diff} Let $N$ be a differential manifold, $W$ be an open set of $N$, and $F\in\mc{C}^{\infty}(W,U)$. Thus for every $h\in\mc{A}(U)$ and every $\theta\in\mf{T}_{\bullet}^{0}(U,M;G)$ we have $\overset{\times}{F}(h\theta)=(F^{\ast}h)\overset{\times}{F}(\theta)$. \end{proposition}
\begin{corollary} \label{09170919diff} Assume $\mathbb{K}=\mathbb{C}$. Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$. If $\{G_{j}\}_{j\in J}$ is a family of \textbf{real} locally convex spaces and $G$ is such that $G_{\mathbb{R}}=\prod_{j\in J}G_{j}$ provided with the product topology. Thus for every $j\in J$ we have that $\Pr^{j}_{\times}\circ\mf{t}=\mf{t}\circ\Pr^{j}_{\times}$, $\Pr^{j}_{\times}\circ\mf{r}=\mf{r}\circ\Pr^{j}_{\times}$, and $\Pr^{j}_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ\Pr^{j}_{\times}$. \end{corollary} \begin{proof} $\Pr^{j}\in\mc{L}(G_{\mathbb{R}},G_{j})$ and the product topology is locally convex as a particular case of what stated in \cite[II.5]{EVT}. Thus the statement is well-set and it follows since Prp. \ref{09170918diff} applied to $\mathbb{K}=\mathbb{R}$, to $G$ replaced by $G_{\mathbb{R}}$ and to $\uppsi$ replaced by $\Pr^{j}$. \end{proof}
\section{$G$-Valued Integrable and Smooth Forms} \label{09172013}
\begin{definition} [\textbf{$G$-valued Scalarly Essentially Integrable Forms at $M$ defined on $U$}] \label{10021357} For every $k\in\mathbb{Z}_{+}$ define the $\mc{A}(U)$-module of $G$-valued scalarly essentially $\lambda$-integrable $k$-forms at $M$ defined on $U$ as follows \begin{equation*} \mr{Alt}^{k}(U,M;G,\lambda)\coloneqq\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mr{Alt}^{k}(U,M). \end{equation*} Next define the $\mc{A}(U)$-module \begin{equation*} \mr{Alt}^{\bullet}(U,M;G,\lambda)\coloneqq\bigoplus_{k\in\mathbb{Z}_{+}}\mr{Alt}^{k}(U,M;G,\lambda). \end{equation*} Set $\mr{Alt}^{k}(M;G,\lambda)\coloneqq\mr{Alt}^{k}(M,M;G,\lambda)$ and $\mr{Alt}^{\bullet}(M;G,\lambda)\coloneqq\mr{Alt}^{\bullet}(M,M;G,\lambda)$. Finally define the $\mc{A}(U)$-modules \begin{equation*} \mr{Alt}_{0}^{\bullet}(U,M;G,\lambda)\coloneqq\mc{H}(U,G_{\mathbb{R}})\otimes_{\mc{A}(U)}\mr{Alt}^{\bullet}(U,M); \end{equation*} and \begin{equation*} \Omega^{\bullet}(U,M;G,\lambda)\coloneqq\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\Omega^{\bullet}(U,M). \end{equation*} \end{definition}
Clearly $\mr{Alt}_{0}^{\bullet}(U,M;G,\lambda)$ is isomorphic to a $\mc{A}(U)$-submodule of $\mr{Alt}^{\bullet}(U,M;G,\lambda)$ and this is isomorphic to a $\mc{A}(U)$-submodule of $\mf{I}_{\bullet}^{0}(U,M;G,\lambda)$. In what follows we shall identify these isomorphic modules.
\begin{proposition} \label{09111218} $\mr{Alt}^{\bullet}(U,M;G,\lambda)$ is isomorphic to $\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mr{Alt}^{\bullet}(U,M)$ in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since \cite[II.61 Prp. 7]{BourA1} there exists a canonical $\mathbb{Z}$-linear isomorphism that is clearly a $\mc{A}(U)-\mr{mod}$ isomorphism by the definition of the module structure of the tensor product of modules over a commutative ring. \end{proof}
\begin{remark} $\mr{Alt}^{\bullet}(U,M;G,\lambda)=\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mr{Alt}_{c}^{\bullet}(U,M)$ since Prp. \ref{09161155} where we employ the convention described in Rmk. \ref{09161024int}. \end{remark}
\begin{corollary} [\textbf{Unique Decomposition of $G$-Valued Scalarly Essentially Integrable Forms}] \label{08262041int} Let $(\phi,U)$ be a chart of $M$ and $\theta\in\mr{Alt}^{k}(U,M;G,\lambda)$, thus \begin{equation*} \bigl(\exists\,!f:M(k,N,<)\to\mf{L}_{c}^{1}(U,G,\lambda)\bigr) \left(\theta=\sum_{I\in M(k,N,<)}f_{I}\otimes\mc{E}_{dx^{\phi}}(I)\right). \end{equation*} \end{corollary} \begin{proof} $\{\mc{E}_{dx^{\phi}}(I)\,\vert\,I\in M(k,N,<)\}$ is a basis of $\mr{Alt}^{k}(U,M)$, thus the statement follows since \cite[II.62 Cor.1]{BourA1}. \end{proof}
\begin{definition} \label{09111121int} Define by abuse of language \begin{equation*} \mf{t}_{G}\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}\bigl(\mr{Alt}^{\bullet}(U,M;G,\lambda),\Omega^{\bullet}(U,M;G,\lambda)\bigr), \end{equation*} be the restriction of $\mf{t}_{G}$ defined in Def. \ref{09101743}, and let \begin{equation*} \mf{r}_{G}\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}\bigl(\Omega^{\bullet}(U,M;G,\lambda),\mr{Alt}^{\bullet}(U,M;G,\lambda)\bigr); \end{equation*} be the restriction of $\mf{r}_{G}$ defined in Def. \ref{09101743}. \end{definition}
\begin{proposition} $\mf{t}_{G}$ and $\mf{r}_{G}$ defined in Def. \ref{09111121int} are isomorphisms one the inverse of the other in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since Prp. \ref{08281918int}. \end{proof}
We shall use convention \ref{09180844} also for the above defined maps.
\begin{definition} Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$ be a diffeomorphism. Define by abuse of language the $\mathbb{R}$-linear map \begin{equation*} \overset{\times}{F}:\mr{Alt}^{\bullet}(U,M;G,\lambda)\to\mr{Alt}^{\bullet}(W,N;G,\lambda); \end{equation*} as the restriction of the map defined in \eqref{09180901}. Similarly define by abuse of language the $\mathbb{R}$-linear map \begin{equation*} \overset{\times}{F}:\Omega^{\bullet}(U,M;G,\lambda)\to\Omega^{\bullet}(W,N;G,\lambda); \end{equation*} as the restriction of the map defined in \eqref{09180902}. \end{definition}
Easily we see that \begin{theorem} [\textbf{Pushforward Commutes with All the Above Operators}] \label{08281542int} Let $N$ be a differential manifold, $W$ be an open set of $N$, $F\in\mc{C}^{\infty}(W,U)$ be a diffeomorphism. If $\uppsi\in\mc{L}(G,H)$, then $\uppsi_{\times}\circ\mf{t}=\mf{t}\circ\uppsi_{\times}$, $\uppsi_{\times}\circ\mf{r}=\mf{r}\circ\uppsi_{\times}$ and $\uppsi_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ\uppsi_{\times}$. \end{theorem}
\begin{definition} [\textbf{$G$-valued Smooth Forms at $M$ defined on $U$}] For every $k\in\mathbb{Z}_{+}$ define the $\mc{A}(U)$-module of $G$-valued differential $k$-forms at $M$ defined on $U$ as follows \begin{equation*} \mr{Alt}^{k}(U,M;G)\coloneqq\mc{A}(U,G)\otimes_{\mc{A}(U)}\mr{Alt}^{k}(U,M); \end{equation*} and define the $\mc{A}(U)$-module of $G$-valued differential forms at $M$ defined on $U$ as follows \begin{equation*} \mr{Alt}^{\bullet}(U,M;G)\coloneqq\bigoplus_{k\in\mathbb{Z}_{+}}\mr{Alt}^{k}(U,M;G), \end{equation*} set $\mr{Alt}^{k}(M;G)\coloneqq\mr{Alt}^{k}(M,M;G)$ and $\mr{Alt}^{\bullet}(M;G)\coloneqq\mr{Alt}^{\bullet}(M,M;G)$. Similarly \begin{equation*} \mr{Alt}_{c}^{k}(U,M;G)\coloneqq\mc{A}_{c}(U,G)\otimes_{\mc{A}(U)}\mr{Alt}^{k}(U,M). \end{equation*} and define the $\mc{A}(U)$-module of $G$-valued differential forms at $M$ defined on $U$ and with compact support as follows \begin{equation*} \mr{Alt}_{c}^{\bullet}(U,M;G)\coloneqq\bigoplus_{k\in\mathbb{Z}_{+}}\mr{Alt}_{c}^{k}(U,M;G), \end{equation*} set $\mr{Alt}_{c}^{k}(M;G)\coloneqq\mr{Alt}_{c}^{k}(M,M;G)$ and $\mr{Alt}_{c}^{\bullet}(M;G)\coloneqq\mr{Alt}_{c}^{\bullet}(M,M;G)$. \end{definition}
\begin{proposition} \label{09101648diff} $\mr{Alt}^{\bullet}(U,M;G)$ is isomorphic to $\mc{A}(U,G)\otimes_{\mc{A}(U)}\mr{Alt}^{\bullet}(U,M)$ and $\mr{Alt}_{c}^{\bullet}(U,M;G)$ is isomorphic to $\mc{A}_{c}(U,G)\otimes_{\mc{A}(U)}\mr{Alt}^{\bullet}(U,M)$ in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since \cite[II.61 Prp. 7]{BourA1} there exists a canonical $\mathbb{Z}$-linear isomorphism that is clearly a $\mc{A}(U)-\mr{mod}$ isomorphism by the definition of the module structure of the tensor product of modules over a commutative ring. \end{proof}
\begin{remark} $\mr{Alt}_{c}^{\bullet}(U,M;G)=\mc{A}(U,G)\otimes_{\mc{A}(U)}\mr{Alt}_{c}^{\bullet}(U,M)$ since Prp. \ref{09161202} where we used the convention described in Rmk. \ref{09161024}. \end{remark}
\begin{corollary} [\textbf{Unique Decomposition of $G$-Valued Smooth Forms}] \label{08262041} Let $(\phi,U)$ be a chart of $M$, $\theta\in\mr{Alt}^{k}(U,M;G)$ and $\eta\in\mr{Alt}_{c}^{k}(U,M;G)$ thus \begin{equation*} \bigl(\exists\,!f:M(k,N,<)\to\mc{A}(U,G)\bigr) \left(\theta=\sum_{I\in M(k,N,<)}f_{I}\otimes\mc{E}_{dx^{\phi}}(I)\right); \end{equation*} and \begin{equation*} \bigl(\exists\,!g:M(k,N,<)\to\mc{A}_{c}(U,G)\bigr) \left(\eta=\sum_{I\in M(k,N,<)}g_{I}\otimes\mc{E}_{dx^{\phi}}(I)\right). \end{equation*} \end{corollary} \begin{proof} $\{\mc{E}_{dx^{\phi}}(I)\,\vert\,I\in M(k,N,<)\}$ is a basis of $\mr{Alt}^{k}(U,M)$, thus the statement follows since \cite[II.62 Cor.1]{BourA1}. \end{proof}
\begin{definition} \label{09111121} By abuse of language define \begin{equation*} \mf{t}_{G}\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}\bigl(\mr{Alt}^{\bullet}(U,M;G),\Omega^{\bullet}(U,M;G)\bigr), \end{equation*} be the restriction of the map $\mf{t}_{G}$ defined in Def. \ref{09180944}. Similarly by abuse of language let \begin{equation*} \mf{r}_{G}\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}\bigl(\Omega^{\bullet}(U,M;G),\mr{Alt}^{\bullet}(U,M;G)\bigr), \end{equation*} be the restriction of the map $\mf{r}_{G}$ defined in Def. \ref{09180944}. \end{definition}
\begin{proposition} \label{09111859} $\mf{t}_{G}$ and $\mf{r}_{G}$ defined in Def. \ref{09111121} are isomorphisms one the inverse of the other in the category $\mc{A}(U)-\mr{mod}$. \end{proposition} \begin{proof} Since Prp. \ref{08281918} \end{proof}
We shall use convention \ref{09180844} also for the above defined maps.
\begin{definition} Let $N$ be a differential manifold, $W$ be an open set of $N$ and $F\in\mc{C}^{\infty}(W,U)$. By abuse of language let \begin{equation*} \overset{\times}{F}:\mr{Alt}^{\bullet}(U,M;G)\to\mr{Alt}^{\bullet}(W,N;G) \end{equation*} be the restriction of the map defined in \eqref{09180903}, and let \begin{equation*} \overset{\times}{F}:\Omega^{\bullet}(U,M;G)\to\Omega^{\bullet}(W,N;G) \end{equation*} be the restriction of the map defined in \eqref{09180904}. \end{definition}
Next we start the sequence of results required to define the wedge product in Def. \ref{09260951}.
\begin{lemma} \label{09251928} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Thus the following \begin{equation*} \begin{aligned} \mf{L}_{c}^{1}(U,G,\lambda)\times\mc{A}(U,H)&\to\mf{L}_{c}^{1}(U,G\widehat{\otimes}H,\lambda), \\ (f,g)&\mapsto(x\mapsto f(x)\otimes g(x)); \end{aligned} \end{equation*} is a well-defined $\mc{A}(U)$-bilinear map. \end{lemma} \begin{proof} Since the topological dual of a Hausdorff topological linear space is $\mathbb{K}$-isomorphic to the topological dual of its completion, we deduce by \cite[Prp.2 pg. 30]{gro} that $(G\widehat{\otimes}H)^{\prime}$ is $\mathbb{K}$-isomorphic via the universal property to the space of bilinear continuous $\mathbb{K}$-forms on $G\times H$. Therefore given any continuous bilinear $\mathbb{K}$-form $b$ on $G\times H$ we have $\widehat{b}\in(G\widehat{\otimes}H)^{\prime}$, where $\widehat{b}$ is the continuous extension at $G\widehat{\otimes}H$ of the linearization of $b$ via the universal property, any element of $(G\widehat{\otimes}H)^{\prime}$ arises uniquely in this way, and finally
there exist $a>0$, $\psi\in X=G^{\prime}$ and $\phi\in Y=H^{\prime}$ such that for every $(u,v)\in G\times H$ we have $|\widehat{b}(u\otimes v)|\leq a|\psi(u)|\,|\phi(v)|$. Then the map in the statement is well-defined since $\psi_{\times}(f)\in\mf{L}_{c}^{1}(U,\mathbb{K},\lambda)$ for every $f\in\mf{L}_{c}^{1}(U,G,\lambda)$, $\phi_{\times}(g)\in\mc{A}(U,\mathbb{K})$ for every $g\in\mc{A}(U,G)$ and by Prp. \ref{09161155} applied to $r=s=0$. The $\mc{A}(U)$-bilinearity is triavially true. \end{proof} Lemma \ref{09251928} permits to give the following
\begin{definition} \label{09260817} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Define \begin{equation*} \uptau\in\mr{Mor}_{\mc{A}(U)-\mr{mod}} \left(\mf{L}_{c}^{1}(U,G,\lambda)\otimes_{\mc{A}(U)}\mc{A}(U,H),\mf{L}_{c}^{1}(U,G\widehat{\otimes}H,\lambda)\right); \end{equation*} such that \begin{equation*} \uptau(f\otimes g)=(x\mapsto f(x)\otimes g(x)). \end{equation*} \end{definition}
\begin{definition} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Let $k,l\in\mathbb{Z}_{+}$, $\omega\in\mr{Alt}^{k}(U,M)$ and $g\in\mc{A}(U,H)$. Define \begin{equation*} \begin{aligned} \wedge_{g,\omega,1}^{l}:\mf{L}_{c}^{1}(U,G,\lambda)\times\mr{Alt}^{l}(U,M)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H,\lambda), \\ (f,\zeta)&\mapsto\uptau(f\otimes g)\otimes(\zeta\wedge\omega), \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \wedge_{g,\omega,2}^{l}:\mf{L}_{c}^{1}(U,G,\lambda)\times\mr{Alt}^{l}(U,M)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H,\lambda), \\ (f,\zeta)&\mapsto\uptau(f\otimes g)\otimes(\omega\wedge\zeta). \end{aligned} \end{equation*} \end{definition}
\begin{proposition} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Let $k,l\in\mathbb{Z}_{+}$, $\omega\in\mr{Alt}^{k}(U,M)$ and $g\in\mc{A}(U,H)$. Thus $\wedge_{g,\omega,2}^{l}=(-1)^{k+l}\wedge_{g,\omega,1}^{l}$ and $\wedge_{g,\omega,i}^{l}$ is $\mc{A}(U)$-bilinear for every $i\in\{1,2\}$. \end{proposition} \begin{proof} The wedge product in $\mr{Alt}^{\bullet}(U,M)$ is $\mc{A}(U)$-bilinear, thus the statement follows since Def. \ref{09260817} and the $\mc{A}(U)$-module structure of $\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H,\lambda)$. \end{proof}
The above result permits the following \begin{definition} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Let $k,l\in\mathbb{Z}_{+}$, $\omega\in\mr{Alt}^{k}(U,M)$ and $g\in\mc{A}(U,H)$. For every $i\in\{1,2\}$ define $\overline{\wedge}_{g,\omega,i}^{l}$ as the unique \begin{equation*} \overline{\wedge}_{g,\omega,i}^{l}\in \mr{Mor}_{\mc{A}(U)-\mr{mod}}\left(\mr{Alt}^{l}(U,M;G,\lambda),\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H,\lambda)\right), \end{equation*} such that \begin{equation*} (\forall f\in\mf{L}_{c}^{1}(U,G,\lambda))(\forall\zeta\in\mr{Alt}^{l}(U,M)) (\overline{\wedge}_{g,\omega,i}^{l}(f\otimes\zeta)=\wedge_{g,\omega,i}^{l}(f,\zeta)). \end{equation*} \end{definition}
Easily we see that \begin{lemma} \label{09260926} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Let $k,l\in\mathbb{Z}_{+}$, Thus the map $(g,\omega)\mapsto\overline{\wedge}_{g,\omega,i}^{l}$ is $\mc{A}(U)$-bilinear. In particular there exists a unique \begin{equation*} \widehat{\wedge}_{i}^{l} \in\mr{Mor}_{\mc{A}(U)-\mr{mod}} \left(\mr{Aut}^{k}(U,M;H), \mr{Mor}_{\mc{A}(U)-\mr{mod}}\left(\mr{Alt}^{l}(U,M;G,\lambda),\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H,\lambda)\right) \right) \end{equation*} such that \begin{equation*} (\forall g\in\mc{A}(U,H))(\forall\omega\in\mr{Aut}^{k}(U,M)) (\widehat{\wedge}_{i}^{l}(g\otimes\omega)=\overline{\wedge}_{g,\omega,i}^{l}). \end{equation*} \end{lemma}
\begin{definition} [\textbf{The Wedge Products of $G$-Valued Integrable Forms}] \label{09260951} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Let $k,l\in\mathbb{Z}_{+}$, define \begin{equation*} \begin{aligned} \wedge_{1}^{k,l}:\mr{Alt}^{l}(U,M;G,\lambda)\times\mr{Alt}^{k}(U,M;H)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H,\lambda); \\ (\theta,\varepsilon)&\mapsto\widehat{\wedge}_{1}^{l}(\varepsilon)(\theta), \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \wedge_{2}^{k,l}:\mr{Alt}^{k}(U,M;H)\times\mr{Alt}^{l}(U,M;G,\lambda)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H,\lambda), \\ (\varepsilon,\theta)&\mapsto\widehat{\wedge}_{2}^{l}(\varepsilon)(\theta). \end{aligned} \end{equation*} Next define \begin{equation*} \begin{aligned} \wedge_{1}:\mr{Alt}^{\bullet}(U,M;G,\lambda)\times\mr{Alt}^{\bullet}(U,M;H)&\to \mr{Alt}^{\bullet}(U,M;G\widehat{\otimes}H,\lambda); \\ (\theta,\varepsilon)&\mapsto\wedge_{1}^{\mr{ord}(\varepsilon),\mr{ord}(\theta)}(\theta,\varepsilon), \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \wedge_{2}:\mr{Alt}^{\bullet}(U,M;H)\times\mr{Alt}^{\bullet}(U,M;G,\lambda)&\to \mr{Alt}^{\bullet}(U,M;G\widehat{\otimes}H,\lambda), \\ (\varepsilon,\theta)&\mapsto\wedge_{2}^{\mr{ord}(\varepsilon),\mr{ord}(\theta)}(\varepsilon,\theta). \end{aligned} \end{equation*} $\wedge_{1}$ will be also denoted by $\wedge$. \end{definition}
\begin{remark} $(f\otimes\zeta)\wedge_{1}(g\otimes\omega)=\uptau(f\otimes g)\otimes(\zeta\wedge\omega)$ and $(g\otimes\omega)\wedge_{2}(f\otimes\zeta)=\uptau(f\otimes g)\otimes(\omega\wedge\zeta)$. \end{remark}
\begin{corollary} [\textbf{The Wedge Products are $\mc{A}(U)$-Bilinear}] \label{09260950} $\wedge_{i}$ in Def. \ref{09260951} is $\mc{A}(U)$-bilinear for every $i\in\{1,2\}$. \end{corollary} \begin{proof} $\wedge_{i}^{k,l}$ is $\mc{A}(U)$-bilinear for every $k,l\in\mathbb{Z}_{+}$ and $i\in\{1,2\}$ as a consequence of Lemma \ref{09260926}, then the statement follows. \end{proof}
\begin{proposition} [\textbf{Pushforward Commutes with Wedge}] \label{09290547} Assume that there exist $\mathbb{K}$-linear subspaces $X$ of $G^{\ast}$ and $Y$ of $H^{\ast}$ such that the topology on $G$ and $H$ are $\sigma(G,X)$ and $\sigma(H,Y)$ respectively. Similarly assume that there exist $\mathbb{K}$-linear subspaces $X_{1}$ of $G_{1}^{\ast}$ and $Y_{1}$ of $H_{1}^{\ast}$ such that the topology on $G_{1}$ and $H_{1}$ are $\sigma(G_{1},X_{1})$ and $\sigma(H_{1},Y_{1})$ respectively. Let $\theta\in\mr{Alt}^{\bullet}(U,M;G,\lambda)$, $\varepsilon\in\mr{Alt}^{\bullet}(U,M;H)$. If $\uppsi\in\mc{L}(G,G_{1})$, and $\upphi\in\mc{L}(H,H_{1})$, then $(\uppsi\otimes\upphi)_{\times}(\theta\wedge\varepsilon)=\uppsi_{\times}(\theta)\wedge\upphi_{\times}(\varepsilon)$. \end{proposition} \begin{proof} The statement is well-set since $\uppsi\otimes\upphi\in\mc{L}(G\widehat{\otimes}H,G_{1}\widehat{\otimes}H_{1})$ by \cite[pg.37]{gro}, then the statement is trivially true. \end{proof}
\begin{corollary} \label{09121123int} Assume $\mathbb{K}=\mathbb{C}$. Let $N$ be a differential manifold, $W$ be an open set of $N$, $F\in\mc{C}^{\infty}(W,U)$ be a diffeomorphism, $\eta\in\mr{Alt}^{\bullet}(U,M;G,\lambda)$ and $\varepsilon\in\mr{Alt}^{\bullet}(U,M;H)$. If $\{G_{j}\}_{j\in J}$ is a family of \textbf{real} locally convex spaces and $G$ is such that $G_{\mathbb{R}}=\prod_{j\in J}G_{j}$ provided with the product topology and if $\{H_{k}\}_{k\in K}$ is a family of real locally convex spaces and $H$ is such that $H_{\mathbb{R}}=\prod_{k\in K}H_{k}$ provided with the product topology; then for every $j\in J$ we have that $(\Pr_{G}^{j})_{\times}\circ\mf{t}_{G}=\mf{t}_{G}\circ(\Pr_{G}^{j})_{\times}$, $(\Pr_{G}^{j})_{\times}\circ\mf{r}_{G}=\mf{r}_{G}\circ(\Pr_{G}^{j})_{\times}$, $(\Pr_{G}^{j})_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ(\Pr_{G}^{j})_{\times}$, moreover for every $k\in K$ we have that \begin{equation*} \big((\Pr_{G}^{j})_{\times}\otimes(\Pr_{H}^{k})_{\times}\bigr) (\eta\wedge\varepsilon)=(\Pr_{G}^{j})_{\times}(\eta)\wedge(\Pr_{H}^{k})_{\times} (\varepsilon). \end{equation*} \end{corollary} \begin{proof} $\eta\in\mr{Alt}^{\bullet}(U,M;G_{\mathbb{R}},\lambda)$ since Lemma \ref{09121429}, while $\Pr_{G}^{j}\in\mc{L}(G_{\mathbb{R}},G_{j})$ and the product topology is locally convex as a particular case of what stated in \cite[II.5]{EVT}. Thus the statement is well-set and it follows since Thm. \ref{08281542int} and Prp. \ref{09290547} applied to $\mathbb{K}=\mathbb{R}$, to $G$ replaced by $G_{\mathbb{R}}$ and to $\uppsi$ replaced by $\Pr_{G}^{j}\in\mc{L}(G_{\mathbb{R}},G_{j})$ and to $H$ replaced by $H_{\mathbb{R}}$ and to $\upphi$ replaced by $\Pr_{H}^{k}\in\mc{L}(H_{\mathbb{R}},H_{k})$. \end{proof}
Next we start to define the wedge product for $G$-valued smooth forms.
\begin{lemma} \label{09251928diff} The following \begin{equation*} \begin{aligned} \mc{A}(U,G)\times\mc{A}(U,H)&\to\mc{A}(U,G\widehat{\otimes}H), \\ (f,g)&\mapsto(x\mapsto f(x)\otimes g(x)); \end{aligned} \end{equation*} is a well-defined $\mc{A}(U)$-bilinear map. \end{lemma} \begin{proof} The bilinear $\otimes:G\times H\to G\widehat{\otimes}H$ is continuous as a result of \cite[Prp.2 pg. 30]{gro}, thus the statement follows since any continuous bilinear map is smooth w.r.t. the Bastiani differential calculus. \end{proof} Lemma \ref{09251928diff} permits to give the following
\begin{definition} \label{09260817diff} Define by abuse of language \begin{equation*} \uptau\in\mr{Mor}_{\mc{A}(U)\mr{mod}}\left(\mc{A}(U,G)\otimes_{\mc{A}(U)}\mc{A}(U,H),\mc{A}(U,G\widehat{\otimes}H)\right); \end{equation*} such that \begin{equation*} \uptau(f\otimes g)=(x\mapsto f(x)\otimes g(x)). \end{equation*} \end{definition}
\begin{definition} Let $k,l\in\mathbb{Z}_{+}$, $\omega\in\mr{Alt}^{k}(U,M)$ and $g\in\mc{A}(U,H)$, define \begin{equation*} \begin{aligned} \wedge_{g,\omega,1}^{l}:\mc{A}(U,G)\times\mr{Alt}^{l}(U,M)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H), \\ (f,\zeta)&\mapsto\uptau(f\otimes g)\otimes(\zeta\wedge\omega), \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \wedge_{g,\omega,2}^{l}:\mc{A}(U,G)\times\mr{Alt}^{l}(U,M)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H), \\ (f,\zeta)&\mapsto\uptau(f\otimes g)\otimes(\omega\wedge\zeta). \end{aligned} \end{equation*} \end{definition}
\begin{proposition} Let $k,l\in\mathbb{Z}_{+}$, $\omega\in\mr{Alt}^{k}(U,M)$ and $g\in\mc{A}(U,H)$. Thus $\wedge_{g,\omega,2}^{l}=(-1)^{k+l}\wedge_{g,\omega,1}^{l}$ and $\wedge_{g,\omega,i}^{l}$ is $\mc{A}(U)$-bilinear for every $i\in\{1,2\}$. \end{proposition} \begin{proof} The wedge product in $\mr{Alt}^{\bullet}(U,M)$ is $\mc{A}(U)$-bilinear, thus the statement follows since Def. \ref{09260817diff} and the $\mc{A}(U)$-module structure of $\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H)$. \end{proof}
The above result permits the following \begin{definition} Let $k,l\in\mathbb{Z}_{+}$, $\omega\in\mr{Alt}^{k}(U,M)$ and $g\in\mc{A}(U,H)$. For every $i\in\{1,2\}$ define $\overline{\wedge}_{g,\omega,i}^{l}$ as the unique \begin{equation*} \overline{\wedge}_{g,\omega,i}^{l}\in \mr{Mor}_{\mc{A}(U)-\mr{mod}}\left(\mr{Alt}^{l}(U,M;G),\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H)\right), \end{equation*} such that \begin{equation*} (\forall f\in\mc{A}(U,G))(\forall\zeta\in\mr{Alt}^{l}(U,M)) (\overline{\wedge}_{g,\omega,i}^{l}(f\otimes\zeta)=\wedge_{g,\omega,i}^{l}(f,\zeta)). \end{equation*} \end{definition}
Easily we see that \begin{lemma} \label{09260926diff} Let $k,l\in\mathbb{Z}_{+}$. Thus the map $(g,\omega)\mapsto\overline{\wedge}_{g,\omega,i}^{l}$ is $\mc{A}(U)$-bilinear. In particular there exists a unique \begin{equation*} \widehat{\wedge}_{i}^{l} \in\mr{Mor}_{\mc{A}(U)-\mr{mod}} \left(\mr{Aut}^{k}(U,M;H), \mr{Mor}_{\mc{A}(U)-\mr{mod}}\left(\mr{Alt}^{l}(U,M;G),\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H)\right), \right) \end{equation*} such that \begin{equation*} (\forall g\in\mc{A}(U,H))(\forall\omega\in\mr{Aut}^{k}(U,M)) (\widehat{\wedge}_{i}^{l}(g\otimes\omega)=\overline{\wedge}_{g,\omega,i}^{l}). \end{equation*} \end{lemma}
\begin{definition} [\textbf{The Wedge Products of $G$-Valued Smooth Forms}] \label{09260951diff} Let $k,l\in\mathbb{Z}_{+}$, define \begin{equation*} \begin{aligned} \wedge_{1}^{k,l}:\mr{Alt}^{l}(U,M;G,\lambda)\times\mr{Alt}^{k}(U,M;H)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H); \\ (\theta,\varepsilon)&\mapsto\widehat{\wedge}_{1}^{l}(\varepsilon)(\theta), \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \wedge_{2}^{k,l}:\mr{Alt}^{k}(U,M;H)\times\mr{Alt}^{l}(U,M;G)&\to\mr{Alt}^{k+l}(U,M;G\widehat{\otimes}H), \\ (\varepsilon,\theta)&\mapsto\widehat{\wedge}_{2}^{l}(\varepsilon)(\theta). \end{aligned} \end{equation*} Next define \begin{equation*} \begin{aligned} \wedge_{1}:\mr{Alt}^{\bullet}(U,M;G)\times\mr{Alt}^{\bullet}(U,M;H)&\to\mr{Alt}^{\bullet}(U,M;G\widehat{\otimes}H); \\ (\theta,\varepsilon)&\mapsto\wedge_{1}^{\mr{ord}(\varepsilon),\mr{ord}(\theta)}(\theta,\varepsilon), \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} \wedge_{2}:\mr{Alt}^{\bullet}(U,M;H)\times\mr{Alt}^{\bullet}(U,M;G)&\to\mr{Alt}^{\bullet}(U,M;G\widehat{\otimes}H), \\ (\varepsilon,\theta)&\mapsto\wedge_{2}^{\mr{ord}(\varepsilon),\mr{ord}(\theta)}(\varepsilon,\theta). \end{aligned} \end{equation*} $\wedge_{1}$ will be also denoted by $\wedge$. \end{definition}
\begin{remark} $(f\otimes\zeta)\wedge_{1}(g\otimes\omega)=\uptau(f\otimes g)\otimes(\zeta\wedge\omega)$ and $(g\otimes\omega)\wedge_{2}(f\otimes\zeta)=\uptau(f\otimes g)\otimes(\omega\wedge\zeta)$. \end{remark}
\begin{corollary} [\textbf{The Wedge Products are $\mc{A}(U)$-Bilinear}] \label{09260950diff} $\wedge_{i}$ in Def. \ref{09260951diff} is $\mc{A}(U)$-bilinear for every $i\in\{1,2\}$. \end{corollary} \begin{proof} $\wedge_{i}^{k,l}$ is $\mc{A}(U)$-bilinear for every $k,l\in\mathbb{Z}_{+}$ and $i\in\{1,2\}$ as a consequence of Lemma \ref{09260926diff}, then the statement follows. \end{proof}
Next we shall use Cor. \ref{08262041} to define the differential of $G$-valued differential forms defined on a chart of $M$.
\begin{definition} [\textbf{Differential of a $G$-valued differential form on a chart}] \label{08281540} Let $(U,\phi)$ be a chart of $M$, define for every $i\in[1,N]\cap\mathbb{Z}$ the following map \begin{equation*} \begin{aligned} \mc{A}(U,G)\times\mr{Alt}^{\bullet}(U,M)&\to\mr{Alt}^{\bullet}(U,M;G) \\ (f,\zeta)&\mapsto\partial_{i}^{\phi,G}(f)\otimes(dx_{i}^{\phi}\wedge\zeta). \end{aligned} \end{equation*} The above map is \textbf{$\mathbb{R}$-bilinear} since the $\mc{A}(U)$-module structure of $\mr{Alt}^{\bullet}(U,M;G)$, since Cor. \ref{09260950diff} and since $\partial_{i}^{\phi,G}$ is $\mathbb{R}$-linear. \footnote{but not $\mc{A}(U)$-linear so the linearization of the above bilinear map is w.r.t. $\mathbb{R}$ rather than $\mc{A}(U)$, however this does not affect the goal for which this map has been introduced, namely to legitimate the definition of $d$ as below.} Therefore by the universal property of the tensor product \begin{equation*} \exists\,!\mf{d}_{i}\in\mr{Mor}_{\mathbb{R}-\mr{mod}}\left(\mr{Alt}^{\bullet}(U,M;G),\mr{Alt}^{\bullet}(U,M;G)\right), \end{equation*} such that \begin{equation*} (\forall f\in\mc{A}(U,G))(\forall\zeta\in\mr{Alt}^{\bullet}(U,M)) (\mf{d}_{i}(f\otimes\zeta)=\partial_{i}^{\phi,G}(f)\otimes(dx_{i}^{\phi}\wedge\zeta)). \end{equation*} Therefore we are legitimate to define $d:\mr{Alt}^{\bullet}(U,M;G)\to\mr{Alt}^{\bullet}(U,M;G)$ such that for every $\theta\in\mr{Alt}^{\bullet}(U,M;G)$ \begin{equation*} d\theta\coloneqq\sum_{I\in M(\mr{ord}\theta,N,<)}\sum_{j=1}^{N}\mf{d}_{j}(f_{I}\otimes\mc{E}_{dx^{\phi}}(I)); \end{equation*} where $f:M(\mr{ord}\theta,N,<)\to\mc{A}(U,G)$ is the unique map in the decomposition of $\theta$ established in Cor. \ref{08262041}. \end{definition}
\begin{theorem} [\textbf{Differential of a $G$-Valued Smooth Form}] \label{08281401} Let $\{U_{\alpha}\}_{\alpha\in D}$ be a collection of domains of charts of $M$ which are subsets of $U$ covering $U$. Thus there exists a unique $d:\mr{Alt}^{\bullet}(U,M;G)\to\mr{Alt}^{\bullet}(U,M;G)$ called the exterior $G$-differentiation such that for all $k\in\mathbb{Z}_{+}$ we have $d:\mr{Alt}^{k}(U,M;G)\to\mr{Alt}^{k+1}(U,M;G)$ and \begin{equation*} (\forall\theta\in\mr{Alt}^{\bullet}(U,M;G)) (\forall\alpha\in D) \bigl(((\imath_{U_{\alpha}}^{U})^{\times}\circ d)\theta = (d\circ(\imath_{U_{\alpha}}^{U})^{\times})\theta \bigr). \end{equation*} \end{theorem} \begin{proof} Since the gluing lemma via charts that is legitimate by Rmk. \ref{09261218diff} where the compatibility is ensured by the uniqueness of the decomposition established in Cor. \ref{08262041}, and by the fact that $(\imath_{U_{\alpha}\cap U_{\beta}}^{M})^{\times}=(\imath_{U_{\alpha}\cap U_{\beta}}^{U_{\alpha}})^{\times}\circ (\imath_{U_{\alpha}}^{M})^{\times}=(\imath_{U_{\alpha}\cap U_{\beta}}^{U_{\beta}})^{\times}\circ(\imath_{U_{\beta}}^{M})^{\times}$. \end{proof}
\begin{remark} \label{09111902} Since Thm. \ref{08281401} and Prp. \ref{09111859} we can define $d$ on $\Omega^{\bullet}(U,M;G)$. \end{remark}
\begin{definition} Let $\uppsi\in\mc{L}(G,H)$, define by abuse of language \begin{equation*} \uppsi_{\ast}:\mc{A}(U,G)\ni f\mapsto\uppsi\circ f\in\mc{A}(U,H). \end{equation*} \end{definition} Well-set definition since $\uppsi$ is linear and continuous.
Clearly we have \begin{lemma} Let $\uppsi\in\mc{L}(G,H)$, thus $\uppsi_{\ast}\in\mr{Mor}_{\mc{A}(U)-\mr{mod}}(\mc{A}(U,G),\mc{A}(U,H))$. If in addition $(U,\phi)$ is a chart of $M$, then \begin{equation} \label{08281526} \partial_{i}^{\phi,H}\circ\uppsi_{\ast}=\uppsi_{\ast}\circ\partial_{i}^{\phi,G}. \end{equation} \end{lemma} The above result permits to give the following
\begin{definition} \label{08281845} Let $\uppsi\in\mc{L}(G,H)$, define by abuse of language \begin{equation*} \begin{aligned} \uppsi_{\times}&\in\mr{Mor}_{\mc{A}(U)-\mr{mod}} (\mr{Alt}^{\bullet}(U,M;G),\mr{Alt}^{\bullet}(U,M;H)); \\ \uppsi_{\times}&\coloneqq\uppsi_{\ast}\otimes\mr{Id}_{\mr{Alt}^{\bullet}(U,M)}, \end{aligned} \end{equation*} and the same symbol denotes also \begin{equation*} \begin{aligned} \uppsi_{\times}&\in\mr{Mor}_{\mc{A}(U)-\mr{mod}} (\Omega^{\bullet}(U,M;G),\Omega^{\bullet}(U,M;H)); \\ \uppsi_{\times}&\coloneqq\uppsi_{\ast}\otimes\mr{Id}_{\Omega^{\bullet}(U,M)}. \end{aligned} \end{equation*} \end{definition}
\begin{theorem} [\textbf{Pushforward Commutes with All the Above Operators}] \label{08281542} Let $N$ be a differential manifold, $W$ be an open set of $N$, $F\in\mc{C}^{\infty}(W,U)$, $\eta\in\mr{Alt}^{\bullet}(U,M;G)$ and $\varepsilon\in\mr{Alt}^{\bullet}(U,M;H)$. If $\uppsi\in\mc{L}(G,G_{1})$, and $\upphi\in\mc{L}(H,H_{1})$, then $\uppsi_{\times}\circ\mf{t}=\mf{t}\circ\uppsi_{\times}$, $\uppsi_{\times}\circ\mf{r}=\mf{r}\circ\uppsi_{\times}$, $\uppsi_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ\uppsi_{\times}$, $\uppsi_{\times}\circ d=d\circ\uppsi_{\times}$ and $(\uppsi\otimes\upphi)_{\times}(\eta\wedge\varepsilon)=\uppsi_{\times}(\eta)\wedge\upphi_{\times}(\varepsilon)$. \end{theorem} \begin{proof} The proof for the operators $\mf{t}$, $\mf{r}$, $\overset{\times}{F}$ and $\wedge$ is trivial, where the statement concerning $\wedge$ is well-set since $\uppsi\otimes\upphi\in\mc{L}(G\widehat{\otimes}H,G_{1}\widehat{\otimes}H_{1})$ by \cite[pg.37]{gro}. The proof for the operator $d$ follows by Def. \ref{08281540}, \eqref{08281526}, by what right now said and by Thm. \ref{08281401}. \end{proof}
\begin{corollary} \label{09121123} Assume $\mathbb{K}=\mathbb{C}$. Let $N$ be a differential manifold, $W$ be an open set of $N$, $F\in\mc{C}^{\infty}(W,U)$, $\eta\in\mr{Alt}^{\bullet}(U,M;G)$ and $\varepsilon\in\mr{Alt}^{\bullet}(U,M;H)$. If $\{G_{j}\}_{j\in J}$ is a family of real locally convex spaces and $G$ is such that $G_{\mathbb{R}}=\prod_{j\in J}G_{j}$ provided with the product topology, and if $\{H_{k}\}_{k\in K}$ is a family of real locally convex spaces and $H$ is such that $H_{\mathbb{R}}=\prod_{k\in K}H_{k}$ provided with the product topology; then for every $j\in J$ we have that $(\Pr_{G}^{j})_{\times}\circ\mf{t}=\mf{t}\circ(\Pr_{G}^{j})_{\times}$, $(\Pr_{G}^{j})_{\times}\circ\mf{r}=\mf{r}\circ(\Pr_{G}^{j})_{\times}$, $(\Pr_{G}^{j})_{\times}\circ\overset{\times}{F}=\overset{\times}{F}\circ(\Pr_{G}^{j})_{\times}$, moreover for every $k\in K$ we have that \begin{equation*} \big((\Pr_{G}^{j})_{\times}\otimes(\Pr_{H}^{k})_{\times}\bigr) (\eta\wedge\varepsilon)=(\Pr_{G}^{j})_{\times}(\eta)\wedge(\Pr_{H}^{k})_{\times} (\varepsilon). \end{equation*} \end{corollary} \begin{proof} Since Thm. \ref{08281542}. \end{proof}
\begin{corollary} [\textbf{Properties of the $G$-differential}] \label{09211057} Let $d$ the operator uniquely determined in Thm. \ref{08281401}. Thus \begin{enumerate} \item $d$ is $\mathbb{R}-$linear; \label{09211057st1} \item For all $\omega\in\mr{Alt}^{\bullet}(U,M)$ and $\eta\in\mr{Alt}^{\bullet}(U,M;G)$ \begin{equation*} d(\omega\wedge\eta) = d\omega\wedge\eta+(-1)^{\mr{ord}(\omega)}\omega\wedge d\eta; \end{equation*} \label{09211057st2} \item $d\circ d=\mathbf{0}$; \label{09211057st3} \item for all $\uppsi\in G^{\prime}$, $f\in\mc{A}(U,G)$ and $X\in\Gamma(U,M)$ we have \begin{equation*} \left((\mf{R}\circ\imath_{\mathbb{K}}^{\mathbb{K}_{\mathbb{R}}}\circ\uppsi\circ\imath_{G_{\mathbb{R}}}^{G})_{\times}\circ d\right)(\imath_{G}^{G_{\mathbb{R}}}\circ f)(X) =X\left((\mf{R}\circ\imath_{\mathbb{K}}^{\mathbb{K}_{\mathbb{R}}}\circ\uppsi\circ f) \right), \end{equation*} where in case $\mathbb{K}=\mathbb{R}$ in the above equality $\mf{R}$ has to be understood $\mr{Id}_{\mathbb{R}}$. \label{09211057st4} \end{enumerate} Moreover let $N$ be a manifold, $U^{\prime}$ be an open set of $N$ and $F\in\mc{C}^{\infty}(U,U^{\prime})$. Thus the following equality of operators defined on $\mr{Alt}^{\bullet}(U^{\prime},N;G)$ holds true \begin{equation*} d\circ\overset{\times}{F}=\overset{\times}{F}\circ d. \end{equation*} \end{corollary} \begin{proof} $(G_{\mathbb{R}})^{\prime}$ separates the points of $G_{\mathbb{R}}$, thus the statement follows by Rmk. \ref{10041224}, by Thm. \ref{08281542} applied for $\mathbb{K}=\mathbb{R}$, $G$ replaced by $G_{\mathbb{R}}$ and for $H=\mathbb{R}$, and by the fact that the statement is true for the special case of real valued smooth forms. \end{proof}
Now the unique decomposition established in Cor. \ref{08262041int} permits to define the integral of a maximal $\mathbb{R}$-valued essentially integrable form defined on an open set of $\mathbb{R}^{N}$ as in the standard case
\begin{definition} \label{09140959a} Let $V$ be an open set of $\mathbb{R}^{N}$, and for every $\omega\in\mr{Alt}^{N}(V,\mathbb{R}^{N};\mathbb{R},\lambda)$ let $f_{\omega}$ be the unique map in $\mf{L}_{c}^{1}(V,\mathbb{R},\lambda)$ such that $\omega=f_{\omega}\otimes\bigwedge_{i=1}^{N}(\imath_{V}^{\mathbb{R}^{N}})^{\ast}(dx_{i})$ via the decomposition established in Cor. \ref{08262041int}. Define the map \begin{equation*} \mr{Alt}^{n}(V,\mathbb{R}^{N};\mathbb{R},\lambda)\ni\omega\mapsto\int f_{\omega}d\lambda_{V}\in\mathbb{R}. \end{equation*} \end{definition}
\begin{definition} Let $M$ be oriented and $(U,\phi)$ be an oriented chart of $M$. Define $\gamma_{\phi}\in\{1,-1\}$ such that $\gamma_{\phi}=1$ if $(U,\phi)$ is positively oriented, otherwise $\gamma_{\phi}=-1$. \end{definition}
Def. \ref{09140959a} and the concept of support as introduced in Def. \ref{09161456} permit to give the following definition as in the standard case
\begin{definition} \label{09140959b} Let $M$ be oriented, $\omega\in\mr{Alt}^{N}(M;\mathbb{R},\lambda)$, $\{(U_{\alpha},\phi_{\alpha})\}_{\alpha\in D}$ be a \emph{finite} family of oriented charts of $M$ such that $\{U_{\alpha}\}_{\alpha\in D}$ is a covering of $\mr{supp}(\omega)$ moreover by setting $D^{\dagger}=D\cup\{\dagger\}$ and $U_{\dagger}=\complement_{M}\mr{supp}(\omega)$, let $\{\psi_{\alpha}\}_{\alpha\in D^{\dagger}}$ be a smooth partition of unity subordinate to $\{U_{\alpha}\}_{\alpha\in D^{\dagger}}$. Define \begin{equation*} \int\omega\coloneqq \sum_{\alpha\in D}\gamma_{\phi_{\alpha}}\int(\imath_{U_{\alpha}}^{M}\circ\phi_{\alpha}^{-1})^{\times}(\psi_{\alpha}\omega). \end{equation*} \end{definition} Standard arguments as for instance \cite[13.1.9]{die2} permit to show that the above definition does not depend by the choice of the covering and of the partition of unity subordinate to it.
Now $\mr{Alt}^{N}(M;\mathbb{K},\lambda)=\mr{Alt}^{N}(M;\mathbb{K}_{\mathbb{R}},\lambda)$ since Lemma \ref{09121429}, while $\mf{R},\mf{I}\in\mf{L}(\mathbb{C}_{\mathbb{R}},\mathbb{R})$ therefore Def. \ref{09140959b} allows us to provide the following
\begin{definition} \label{09121339} Let $M$ be oriented, define \begin{equation} \label{09121103} \begin{aligned} \mr{Alt}^{N}(M;\mathbb{K},\lambda) \ni\beta&\mapsto\int\beta\coloneqq\int\mf{R}_{\times}(\beta)+i\int\mf{I}_{\times}(\beta)\in\mathbb{K}, \text{ if }\mathbb{K}=\mathbb{C}; \\ \mr{Alt}^{N}(M;\mathbb{K},\lambda)\ni\omega&\mapsto\int\omega\in\mathbb{K},\text{ if }\mathbb{K}=\mathbb{R}. \end{aligned} \end{equation} \end{definition}
\begin{definition} [\textbf{Weak Integral of $G$-Valued Scalarly $\lambda$-Integrable Maximal Forms}] \label{09281748} Let $M$ be oriented and $\eta\in\mr{Alt}^{N}(M;G,\lambda)$. Define $\int\eta\in(G^{\prime})^{\ast}$ such that \begin{equation*} \int\eta:G^{\prime}\ni\uppsi\mapsto\int\uppsi_{\times}(\eta)\in\mathbb{K}, \end{equation*} called the weak integral of $\eta$. We say that $\int\eta$ belongs to $G$ or that $\int\eta\in G$ iff there exists a necessarily unique element $s\in G$ such that $\uppsi(s)=\int\uppsi_{\times}(\eta)$ for every $\uppsi\in G^{\prime}$, in such a case and whenever there is no confusion we let $\int\eta$ denote also the element $s$. \end{definition} Clearly $\int$ is a $\mathbb{R}$-linear operator by considering the $\mathbb{R}$-module underlying the $\mc{A}(U)$-module $\mr{Alt}^{N}(M;G,\lambda)$.
By recalling Def. \ref{10021357} a special case is as follows
\begin{proposition} \label{08291022} Let $M$ be oriented and assume that $G$ is quasi-complete and let $\eta\in\mr{Alt}_{0}^{N}(M;G)$, then $\int\eta\in G$ namely \begin{equation*} (\exists\,!b\in G)(\forall\uppsi\in G^{\prime})\left(\uppsi(b)=\int\uppsi_{\times}(\eta)\right). \end{equation*} \end{proposition} \begin{proof} The statement is well set since $\mr{Alt}_{0}^{N}(M;G)$ is isomorphic to a submodule of $\mr{Alt}^{N}(M;G,\lambda)$. The statement follows since Def. \ref{09140959a}, since $\mathbb{R}^{N}$ is locally compact, since the Lebesgue measure on $\mathbb{R}^{N}$ is a measure, and since the weak integral of any compactly supported continuous $G$-valued map against any measure belongs to $G$ as established in \cite[III.38 Cor. 2]{IntBourb}. \end{proof}
\begin{definition} Define $G^{\star}\coloneqq\lr{(G^{\prime})^{\ast}}{\sigma((G^{\prime})^{\ast},G^{\prime})}_{\mathbb{R}}$ \end{definition}
Now we can state the following
\begin{theorem} [\textbf{Vectorial Measure Associated with an Integrable $G$-Valued Form}] \label{09171005} Let $M$ be oriented, thus there exists a unique map \begin{equation*} \mf{m}\in\mr{Mor}_{\mathbb{R}-\mr{mod}}\left(\mr{Alt}^{N}(M;G,\lambda),\mr{Meas}(M,G^{\star})\right); \end{equation*} such that \begin{equation*} (\forall\eta\in\mr{Alt}^{N}(M;G,\lambda))(\forall g\in\mc{H}(M))\left(\mf{m}_{\eta}(g)=\int g\cdot\eta\right); \end{equation*} where $(\cdot)$ is the $\mc{A}(M)$-bilinear map constructed in Prp. \ref{09170948}. \end{theorem} \begin{proof} $\mf{m}$ is $\mathbb{R}$-linear since it is so the weak integral and since $(\cdot)$ is $\mc{A}(M)$-bilinear. Next let $E$ denote $\lr{(G^{\prime})^{\ast}}{\sigma((G^{\prime})^{\ast},G^{\prime})}$ so $G^{\star}=E_{\mathbb{R}}$ and for every $\uppsi\in G^{\prime}$ let $\mr{b}_{\uppsi}:(G^{\prime})^{\ast}\to\mathbb{K}$, $z\mapsto z(\uppsi)$, thus \begin{equation} \label{10021531} E^{\prime}=\{\mr{b}_{\uppsi}\}_{\uppsi\in G^{\prime}}. \end{equation} Let $g\in\mc{H}(M)$ and $\uppsi\in G^{\prime}$ thus $\int\uppsi_{\times}(g\cdot\eta)=\int g\uppsi_{\times}(\eta)$ so \begin{equation} \label{10021534} \mr{b}_{\uppsi}\circ\mf{m}_{\eta}\in\mr{Meas}(M,\mathbb{K}); \end{equation} in particular $\mr{b}_{\uppsi}\circ\mf{m}_{\eta}$ is continuous. Therefore $\mf{m}_{\eta}:\mc{H}(M)\to E$ is continuous by \eqref{10021531}, by \eqref{10021534}, since the definition of weak topologies and since \cite[I.12 Prp. 4]{BourGT}. Hence the statement follows since the topology on $G^{\star}$ is the topology on $E$. \end{proof}
\begin{corollary} \label{09141452} Let $M$ be oriented, $\eta\in\mr{Alt}^{N}(M;G,\lambda)$, $\{(U_{\alpha},\phi_{\alpha})\}_{\alpha\in D}$ be a \emph{finite} family of oriented charts of $M$ such that $\{U_{\alpha}\}_{\alpha\in D}$ is a covering of $\mr{supp}(\eta)$ moreover by setting $D^{\dagger}=D\cup\{\dagger\}$ and $U_{\dagger}=\complement_{M}\mr{supp}(\eta)$, let $\{\psi_{\alpha}\}_{\alpha\in D^{\dagger}}$ be a smooth partition of unity subordinate to $\{U_{\alpha}\}_{\alpha\in D^{\dagger}}$. Thus \begin{equation*} \int\eta=\sum_{\alpha\in D}\int\psi_{\alpha}\cdot\eta. \end{equation*} \end{corollary} \begin{proof} Since $D$ is finite we can define in $\mc{A}(M)$ the map $g=\sum_{\alpha\in D}\psi_{\alpha}$, in particular $g\in\mc{H}(M)$, while $g\circ\imath_{\mr{supp}(\eta)}^{M}=\mathbf{1}_{\mr{supp}(\eta)}$ since $\psi_{\dagger}\circ\imath_{\mr{supp}(\eta)}^{M}=\mathbf{0}_{\mr{supp}(\eta)}$ and since $g+\psi_{\dagger}=\mathbf{1}_{M}$ by definition of partition of unity. Therefore $\eta=g\cdot\eta$, then $\int\eta=\mf{m}_{\eta}(g)=\sum_{\alpha\in D}\mf{m}_{\eta}(\psi_{\alpha})$ where the second equality follows since Thm. \ref{09171005}. \end{proof}
Finally we can establish the following \begin{theorem} [\textbf{Stokes Theorem for $G$-Valued Smooth Forms}] \label{08281926} Let $M$ be oriented and with boundary and $\theta\in\mr{Alt}_{c}^{N-1}(M;G)$, thus \begin{equation*} \int d\theta=\int(\imath_{\partial M}^{M})^{\times}(\theta); \end{equation*} furthermore if $G$ is quasi-complete, then the above integrals belong to $G$. Here if $\partial M=\emptyset$, then the right-hand side of the equality has to be understood equal to $\mathbf{0}$. \end{theorem} \begin{proof} The statement is well set since $\mr{Alt}_{c}^{\bullet}(M;G)$ is isomorphic to a submodule of $\mr{Alt}^{\bullet}(M;G,\lambda)$. Let $\uppsi\in G^{\prime}$, thus $\uppsi_{\times}(d\theta)=d(\uppsi_{\times}(\theta))$ and $\uppsi_{\times}(\imath_{\partial M}^{M})^{\times}\theta=(\imath_{\partial M}^{M})^{\times}\uppsi_{\times}\theta$ since Thm. \ref{08281542}. Henceforth the equality follows by \eqref{09121103}, by Stokes theorem, and in case $\mathbb{K}=\mathbb{C}$ also by $\mr{Alt}_{c}^{\bullet}(M;\mathbb{C},\lambda)=\mr{Alt}_{c}^{\bullet}(M;\mathbb{C}_{\mathbb{R}},\lambda)$ since Rmk. \ref{10041224}, and by Cor. \ref{09121123} applied to the projectors $\mf{R},\mf{I}\in\mf{L}(\mathbb{C}_{\mathbb{R}},\mathbb{R})$. The last sentence of the statement follows since Prp. \ref{08291022}. \end{proof}
\end{document} | arXiv | {
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\begin{document}
\maketitle
\begin{abstract} For projective conifold transitions between Calabi-Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov--Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.
\end{abstract}
\small \tableofcontents \normalsize
\setcounter{section}{-1}
\section{Introduction} \label{s:0}
\subsection{Statements of main results} \label{s:0.1} Let $X$ be a smooth projective 3-fold. A (projective) conifold transition $X \nearrow Y$ is a projective degeneration $\pi: \mathfrak{X} \to \Delta$ of $X$ to a singular variety $\bar X = \mathfrak{X}_0$ with a finite number of ordinary double points (abbreviated as ODPs or nodes) $p_1, \ldots, p_k$, locally analytically defined by the equation $$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0,$$ followed by a projective small resolution $\psi: Y \to \bar X$. In the process of complex degeneration from $X$ to $\bar{X}$, $k$ vanishing spheres $S_i \cong S^3$ with trivial normal bundle collapse to nodes $p_i$. In the process of ``K\"ahler degeneration'' from $Y$ to $\bar{X}$, the exceptional loci of $\psi$ above each $p_i$ is a smooth rational curve $C_i \cong \mathbb{P}^1$ with $N_{C_i/Y} \cong \mathscr{O}_{\mathbb{P}^1}(-1)^{\oplus 2}$. We write $Y \searrow X$ for the reverse process.
Notice that $\psi$ is a crepant resolution and $\pi$ is a finite distance degeneration with respect to the quasi-Hodge metric \cite{cW0, cW1}. A transition of this type (in all dimensions) is called an extremal transition. In contrast to the usual \emph{birational} $K$-equivalence, an extremal transition may be considered as a \emph{generalized} $K$-equivalence in the sense that the small resolution $\psi$ is crepant and the degeneration $\pi$ preserves sections of the canonical bundle. It is generally expected that simply connected Calabi--Yau 3-folds are connected through extremal transitions, of which conifold transitions are the most fundamental. (This has been extensively checked numerically \cite{cyweb}.) It is therefore a natural starting point of investigation.
We study the changes of the so-called $A$ model and $B$ model under a projective conifold transition. In this paper, the $A$ model is the Gromov--Witten (GW) theory of all genera; the $B$ model is the variation of Hodge structures (VHS), which is in a sense only the genus zero part of the quantum $B$ model.
In general, the conditions for the existence of projective conifold transitions is an unsolved problem except in the case of Calabi--Yau 3-folds, for which we have fairly good understanding. For the inverse conifold transition $Y \searrow X$, a celebrated theorem of Friedman \cite{rF} (see also \cite{yK, gT}) states that a small contraction $Y \to \bar{X}$ can be smoothed if and only if there is a totally nontrivial relation between the exceptional curves. That is, there exist constants $a_i \neq 0$ for all $i=1, \ldots, k$ such that $\sum_{i=1}^k a_i [C_i] =0$. These are relations among curves $[C_i]$'s in the kernel of $H_2(Y)_{\mathbb{Z}} \to H_2(X)_{\mathbb{Z}}$. Let $\mu$ be the number of independent relations and let $A \in M_{k \times \mu} (\mathbb{Z})$ be a relation matrix for $C_i$'s, in the sense that the column vectors span all relations.
Conversely, for a conifold transition $X \nearrow Y$, Smith, Thomas and Yau proved a dual statement in \cite{STY}, asserting that the $k$ vanishing 3-spheres $S_i$ must satisfy a totally nontrivial relation $\sum_{i=1}^k b_i [S_i] =0$
in $V_{\mathbb{Z}}:= \ker (H_3(X)_{\mathbb{Z}} \to H_3(\bar{X})_{\mathbb{Z}})$ with $b_i \neq 0$ for all $i$. Let $\rho$ be the number of independent relations and $B \in M_{k \times \rho} (\mathbb{Z})$ be a relation matrix for $S_i$'s.
It turns out that $\mu + \rho =k$ \cite{hC} and the following exact sequence holds.
\begin{theorem}[= Theorem~\ref{t:bes}] \label{t:0.1} Under a conifold transition $X \nearrow Y$ of smooth projective threefolds, we have an exact sequence of weight two Hodge structures: \begin{equation} \label{e:0.1} 0 \to H^2(Y)/H^2(X) \stackrel{B}{\longrightarrow} \mathbb{C}^k \stackrel{A^t}{\longrightarrow} V \to 0. \end{equation} \end{theorem}
We interpret this as a partial exchange of topological information between the \emph{excess $A$ model} of $Y/X$ (in terms of $H^2(Y)/H^2(X)$) and the \emph{excess $B$ model} of $X/Y$ in terms of the space of vanishing cycles $V$.
To study the changes of quantum $A$ and $B$ models under a projective conifold transition of Calabi--Yau 3-folds and its inverse, the first step is to find a $\mathcal{D}$-module version of Theorem~\ref{t:0.1}. We state the result below in a suggestive form and leave the precise statement to Theorem~\ref{p:qbes}:
\begin{theorem}[= Theorem~\ref{p:qbes}] \label{t:0.1.5} Via the exact sequence \eqref{e:0.1}, the trivial logarithmic connection on $(\underline{\mathbb{C}} \oplus \underline{\mathbb{C}}^{\vee})^k \to \mathbb{C}^k$ induces simultaneously the logarithmic part of the Gauss--Manin connection on $V$ and the Dubrovin connection on $H^2(Y)/H^2(X)$. \end{theorem}
Note that the Gauss--Manin connection on $V$ determines the excess $B$ model and Dubrovin connection on $H^2(Y)/H^2(X)$ determines the excess $A$ model in genus zero. The logarithmic part of the connection determines the residue connection and hence the monodromy. One can interpret Theorem~\ref{t:0.1.5} heuristically as "excess $A$ theory $+$ excess $B$ theory $\sim$ trivial''. In other words, the logarithmic parts of two flat connections on excess theories ``glues'' to form a trivial theory. This gives a strong indication towards a unified $A + B$ theory.
``Globalizing'' this result, i.e., going beyond the excess theories, is the next step towards a true $A + B$ theory, which is still beyond immediate reach. Instead we will settle for results on mutual determination in implicit form. Recall that the Kuranishi spaces $\mathcal{M}_X$, $\mathcal{M}_Y$ of Calabi--Yau manifolds are unobstructed (the Bogomolov--Tian--Todorov theorem). For a Calabi--Yau conifold $\bar X$, the unobstructedness of $\mathcal{M}_{\bar X}$ also holds \cite{yK, gT, N}.
\begin{theorem} \label{t:0.2} Let $X \nearrow Y$ be a projective conifold transition of Calabi--Yau threefolds such that $[X]$ is a nearby point of $[\bar{X}]$ in $\mathcal{M}_{\bar{X}}$. Then \begin{enumerate} \item[(1)] $A(X)$ is a sub-theory of $A(Y)$. \item[(2)] $B(Y)$ is a sub-theory of $B(X)$. \item[(3)] $A(Y)$ can be reconstructed from a refined $A$ model of $X^{\circ} := X \setminus \bigcup_{i=1}^k S_i$ ``linked'' by the vanishing spheres in $B(X)$. \item[(4)] $B(X)$ can be reconstructed from a refined $B$ model of $Y^{\circ} := Y \setminus \bigcup_{i=1}^k C_i$ ``linked'' by the exceptional curves in $A(Y)$. \end{enumerate} \end{theorem}
The meaning of these slightly obscure statements will take the entire paper to spell them out. It may be considered as a \emph{categorification} of Clemens' identity $\mu + \rho = k$. Here we give only brief explanations.
(1) is mostly due to Li--Ruan, who in \cite{LR} pioneered the mathematical study of conifold transitions in GW theory. The proof follows from degeneration arguments and existence of flops (cf.~Proposition~\ref{p:1}).
For (2), we note that there are natural identifications of $\mathcal{M}_Y$ with the boundary of $\mathcal{M}_{\bar{X}}$ consisting of equisingular deformations, and $\mathcal{M}_X$ with $\mathcal{M}_{\bar{X}} \setminus \mathfrak{D}$ where the discriminant locus $\mathfrak{D}$ is a \emph{central hyperplane arrangement} with axis $\mathcal{M}_Y$ (cf.~\S\ref{s:4.4}). Therefore, the VHS associated to $Y$ can be considered as a sub-VHS system of VMHS associated to $\bar{X}$ (cf.~Corollary \ref{c:sub-sys}), which is a regular singular extension of the VHS associated to $X$.
With (3), we introduce the ``linking data'' of the holomorphic curves in $X^{\circ}$, which not only records the curve classes in $X$ but also how the curve links with the vanishing spheres $\bigcup_i S_i$. The linking data on $X$ can be identified with the curve classes in $Y$ by $H_2(X^{\circ}) \cong H_2(Y)$ (cf.~Definition~\ref{d:2} and \eqref{e:linking}). We then proceed to show, by the degeneration argument, that the virtual class of moduli spaces of stable maps to $X^\circ$ is naturally a disjoint union of pieces labeled by elements of the linking data (cf.~Proposition \ref{p:4}).
Furthermore, the Gromov--Witten invariants in $Y$ is the same as the numbers produced by the component of the virtual class on $X$ labeled by the corresponding linking data. Thus, the refined $A$ model is really the ``linked $A$ model'' and is equivalent to the (usual) $A$ model of $Y$ (for non-extremal curves classes) in all genera. The vanishing cycles from $B(X)$ plays a key role in reconstructing $A(Y)$.
For (4), the goal is to reconstruct VHS on $\mathcal{M}_X$ from VHS on $\mathcal{M}_Y$ and $A(Y)$. The deformation of $\bar{X}$ is unobstructed. Moreover it is well known that $\operatorname{Def}(\bar{X}) \cong H^1 (Y^{\circ}, T_{Y^{\circ}})$. Even though the deformation of $Y^{\circ}$ is obstructed (in the direction transversal to $\mathcal{M}_Y$), there is a first order deformation parameterized by $H^1 (Y^{\circ}, T_{Y^{\circ}})$ which gives enough initial condition to uniquely determine the degeneration of Hodge bundles on $\mathcal{M}_{\bar{X}}$ near $\mathcal{M}_Y$. A technical result needed in this process is a short exact sequence $$0 \to V \to H^3(X) \to H^3(Y^\circ) \to 0$$ which connects the limiting mixed Hodge structure (MHS) of Schmid on $H^3(X)$ and the canonical MHS of Deligne on $H^3(Y^\circ)$ (cf.~Proposition \ref{p:lift}). Together with the monodromy data associated to the ODPs, which is encoded in the relation matrix $A$ of the extremal rays on $Y$, we will be able to determine the VHS on $\mathcal{M}_X$ near $\mathcal{M}_Y$. In the process, an extension of Schmid's nilpotent orbit theorem \cite{Schmid} to degenerations with certain non-normal crossing discriminant loci is also needed. See Theorem~\ref{p:gnot} for details.
\subsection{Motivation and future plans}
Our work is inspired by the famous Reid's fantasy \cite{mR}, where conifold transitions play a key role in connecting irreducible components of moduli of Calabi--Yau threefolds. Theorems~\ref{t:0.1.5} and \ref{t:0.2} above can be interpreted as the partial exchange of $A$ and $B$ models under a conifold transition. We hope to answer the following intriguing question concerning with ``global symmetries'' on moduli spaces of Calabi--Yau 3-folds in the future: \emph{Would this partial exchange of $A$ and $B$ models lead to ``full exchange'' when one connects a Calabi--Yau threefold to its mirror via a finite steps of extremal transitions? If so, what is the relation between this full exchange and the one induced by ``mirror symmetry''?} To this end, we need to devise a computationally effective way to achieve explicit determination of this partial exchange. One missing piece of ingredients in this direction is a blowup formula in the Gromov--Witten theory for conifolds, which we are working on and have had some partial success \cite{LLW3}. (For smooth blowups with complete intersection centers, we have a fairly good solution in genus zero.)
More speculatively, the mutual determination of $A$ and $B$ models on $X$ and $Y$ leads us to surmise the possibility of a unified ``$A+B$ model'' which will be \emph{invariant} under any extremal transition. For example, the string theory predicts that Calabi--Yau threefolds form an important ingredient of our universe, but it does not specify which Calabi--Yau threefold we live in. Should the $A+B$ model be available and proven invariant under extremal transitions, one would then have no need to make such a choice.
The first step of achieving this goal is to generalize Theorem~\ref{t:0.1.5} to the full local theory, including the non-log part of the connections.
We note that the excess $A$ model on $H^2(Y/X)$ can be extended to the (flat) Dubrovin connection on $Y$ while the excess $B$ model on $H^3(X/Y)$ can be extended to the (flat) Gauss--Manin connection on $X$. We hope to be able to ``glue'' the complete $A$ model on $Y$ and the complete $B$ model on $X$ as flat connections on the unified K\"ahler plus complex moduli.
\section{The basic exact sequence from Hodge theory} \label{s:2} In this section, we recall some standard results on the geometry of projective conifold transitions. Definitions and short proofs are mostly spelled out to fix the notations, even when they are well known. Combined with well-known tools in Hodge theory, we derive the \emph{basic exact sequence}, which is surprisingly absent in the vast literature on the conifold transitions.
\begin{convention} In \S 1-2, all discussions are for projective conifold transitions \emph{without the Calabi--Yau condition}, unless otherwise specified. The Calabi--Yau condition is imposed in \S 3-5. Unless otherwise specified, cohomology groups are over $\mathbb{Q}$ when only topological aspect (including weight filtration) is concerned; they are considered over $\mathbb{C}$ when the (mixed) Hodge-theoretic aspect is involved. All equalities, whenever make sense in the context of mixed Hodge structure (MHS), hold as equalities for MHS. \end{convention}
\subsection{Preliminaries on conifold transitions}
The results here are mostly contained in \cite{hC} and are included here for readers' convenience.
\subsubsection{Local geometry} \label{s:1.2}
Let $X$ be a smooth projective 3-fold and $X \nearrow Y$ a \emph{projective conifold transition} through $\bar X$ with nodes $p_1, \ldots, p_k$ as in \S \ref{s:0.1}. Locally analytically, a node (ODP) is defined by the equation \begin{equation} \label{e:A1}
x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0, \end{equation} or equivalently $uv - ws =0$. The small resolution $\psi$ can be achieved by blowing up the Weil divisor defined by $u=w=0$ or by $u = s = 0$, these two choices differ by a flop.
\begin{lemma} \label{l:1} The exceptional locus of $\psi$ above each $p_i$ is a smooth rational curve $C_i$ with $N_{C_i/Y} \cong \mathscr{O}_{\mathbb{P}^1}(-1)^{\oplus 2}$. Topologically, $N_{C_i/Y}$ is a trivial rank $4$ real bundle. \end{lemma}
\begin{proof} Away from the isolated singular points $p_i$'s, the Weil divisors are Cartier and the blowups do nothing. Locally near $p_i$, the Weil divisor is generated by two functions $u$ and $w$. The blowup $Y \subset \mathbb{A}^4 \times \mathbb{P}^1$ is defined by $z_0 v - z_1 s =0$, in addition to $uv-ws=0$ defining $X$, where $(z_0: z_1)$ are the coordinates of $\mathbb{P}^1$. Namely we have ${u}/{w} = {s}/{v} = {z_0}/{z_1}$. It is now easy to see the exceptional locus near $p_i$ is isomorphic to $\mathbb{P}^1$ and the normal bundle is as described (by the definition of $\mathscr{O}_{\Bbb P^1}(-1)$). Since oriented $\Bbb R^4$-bundles on $\Bbb P^1 \cong S^2$ are classified by the second Stiefel--Whitney class $w_2$ (via $\pi_1({\rm SO}(4)) \cong \Bbb Z/2$), the last assertion follows immediately. \end{proof}
Locally to each node $p = p_i \in \bar X$, the transition $X \nearrow Y$ can be considered as two different ways of ``smoothing'' the singularities in $\bar{X}$: deformation leads to $X_t$ and small resolution leads to $Y$. Topologically, we have seen that the exceptional loci of $\psi$ are $\coprod_{i=1}^k C_i$, a disjoint union of $k$ 2-spheres. For the deformation, the classical results of Picard, Lefschetz and Milnor state that there are $k$ vanishing 3-spheres $S_i \cong S^3$.
\begin{lemma} \label{l:2} The normal bundle $N_{S_i/X_t} \cong T^*_{S_i}$ is a trivial rank $3$ real bundle. \end{lemma}
\begin{proof} From \eqref{e:A1}, after a degree two base change the local equation of the family near an ODP is
$$\sum\nolimits_{j = 1}^4 x_j^2 = t^2 = |t|^2 e^{2\sqrt{-1}\theta}. $$ Let $y_j = e^{\sqrt{-1} \theta} x_j$ for $j = 1, \ldots, 4$, the equation leads to \begin{equation} \label{e:1}
\sum\nolimits_{j = 1}^4 y_j^2 = |t|^2. \end{equation}
Write $y_j$ in terms of real coordinates $y_j = a_j + \sqrt{-1} b_j$, we have $|\vec{a}|^2 = |t|^2 + |\vec{b}|^2$ \text{and} $\vec{a} \cdot \vec{b} =0$, where $\vec{a}$ and $\vec{b}$ are two vectors in $\mathbb{R}^4$. The set of solutions can be identified with
$T^* S_r$ with the bundle structure $T^* S_r \to S_r$ defined by $(\vec{a}, \vec{b}) \mapsto r {\vec{a}}/{|\vec{a}|} \in S_r$
where $S_r$ is the 3-sphere with radius $r =|t|$. The vanishing sphere can be chosen to be the real locus of the equation of \eqref{e:1}. Therefore, $N_{S_r/X_t}$ is naturally identified with the cotangent bundle $T^*{S_r}$, which is a trivial bundle since $S^3 \cong SU(2)$ is a Lie group. \end{proof}
\begin{remark} \label{r:1} The vanishing spheres above are Lagrangian with respect to the natural symplectic structure on $T^* S^3$. A theorem of Seidel and Donaldson \cite{skD} states that this is true globally, namely the vanishing spheres can be chosen to be Lagrangian with respect to the symplectic structure coming from the K\"ahler structure of $X_t$. \end{remark}
By Lemma~\ref{l:2}, the $\delta$ neighborhood of the vanishing 3-sphere $S_r^3$ in $X_t$ is diffeomorphic to the trivial disc bundle $S^3_r \times D^3_\delta$.
By Lemma~\ref{l:1} the $r$ neighborhood of the exceptional 2-sphere $C_i= S^2_{\delta}$ is $D^4_r \times S^2_{\delta}$, where $\delta$ is the radius defined by $4\pi \delta^2 = \int_{C_i} \omega$ for the background K\"ahler metric $\omega$.
\begin{corollary} \cite[Lemma 1.11]{hC} \label{c:surgery} On the topological level one can go between $Y$ and $X_t$ by surgery via $$\partial (S^3_r \times D_\delta^3) = S^3_r \times S^2_\delta = \partial(D^4_r \times S^2_\delta).$$ \end{corollary}
\begin{remark}[Orientations on $S^3$] \label{orient} The two choices of orientations on $S^3_r$ induces two different surgeries. The resulting manifolds $Y$ and $Y'$ are in general not even homotopically equivalent. In the complex analytic setting the induced map $Y \dasharrow Y'$ is known as an ordinary (Atiyah) flop. \end{remark}
\subsubsection{Global topology}
\begin{lemma} \label{l:3} Define $$\mu := \tfrac{1}{2} (h^3(X) - h^3(Y)) \quad \text{and} \quad \rho := h^2(Y) - h^2(X).$$
Then, \begin{equation} \label{e:3}
\mu + \rho = k. \end{equation} \end{lemma}
\begin{proof} The Euler numbers satisfy $$\chi(X) - k\chi(S^3) = \chi(Y) - k \chi(S^2).$$ That is, $$2 - 2h^1(X) + 2h^2(X) - h^3(X) = 2 - 2h^1(Y) + 2h^2(Y) - h^3(Y) - 2k.$$ By the above surgery argument we know that conifold transitions preserve $\pi_1$. Therefore, $\tfrac{1}{2} (h^3(X) - h^3(Y)) + (h^2(Y) - h^2(X)) = k$. \end{proof}
\begin{remark} In the Calabi-Yau case, $\mu = h^{2, 1}(X) - h^{2, 1}(Y) = -\Delta h^{2, 1}$ is the lose of complex moduli, and $\rho = h^{1, 1}(Y) - h^{1, 1}(X) = \Delta h^{1, 1}$ is the gain of K\"ahler moduli. Thus \eqref{e:3} is really $$ \Delta (h^{1, 1} - h^{2, 1}) = k = \tfrac{1}{2}\Delta \chi. $$ \end{remark}
In the following, we study the \emph{Hodge-theoretic meaning} of \eqref{e:3}.
\subsection{Two semistable degenerations} \label{s:2.1} To apply Hodge-theoretic methods on degenerations, we factor the transition $X \nearrow Y$ as a composition of two semistable degenerations $\mathcal{X} \to \Delta$ and $\mathcal{Y} \to \Delta$.
The \emph{complex degeneration} $$ f: \mathcal{X} \to \Delta $$ is the semistable reduction of $\mathfrak{X} \to \Delta$ obtained by a degree two base change $\mathfrak{X}' \to \Delta$ followed by the blow-up $\mathcal{X} \to \mathfrak{X}'$ of all the four dimensional nodes $p_i' \in \mathfrak{X}'$. The special fiber $\mathcal{X}_0 = \bigcup_{j = 0}^k X_j$ is a simple normal crossing divisor with $$\tilde\psi: X_0 \cong \tilde{Y}:= \operatorname{Bl}_{\coprod_{i=1}^k \{p_i\}} \bar{X} \to \bar X$$ being the blow-up at the nodes and with $$X_i = Q_i \cong Q \subset \mathbb{P}^4, \quad i = 1, \ldots, k$$ being quadric threefolds. Let $X^{[j]}$ be the disjoint union of $j + 1$ intersections from $X_i$'s. Then the only nontrivial terms are $X^{[0]} = \tilde Y \coprod_i Q_i$ and $X^{[1]} = \coprod_i E_i$ where $E_i = \tilde Y \cap Q_i \cong \mathbb{P}^1 \times \mathbb{P}^1$ are the $\tilde\psi$ exceptional divisors. The semistable reduction $f$ does not require the existence of a small resolution of $\mathfrak{X}_0$.
The \emph{K\"ahler degeneration} $$ g: \mathcal{Y} \to \Delta $$ is simply the deformations to the normal cone $\mathcal{Y} = {\rm Bl}_{\coprod C_i \times \{0\}} Y \times \Delta \to \Delta$. The special fiber $\mathcal{Y}_0 = \bigcup_{j = 0}^k Y_j$ with $$\phi: Y_0 \cong \tilde Y := {\rm Bl}_{\coprod_{i = 1}^k \{C_i\}}\,Y \to Y$$ being the blow-up along the curves $C_i$'s and $$Y_i = \tilde E_i \cong \tilde E := {P}_{\mathbb{P}^1} (\mathscr{O}(-1)^2 \oplus \mathscr{O}), \quad i = 1, \ldots, k.$$ In this case the only non-trivial terms for $Y^{[j]}$ are $Y^{[0]} = \tilde Y \coprod_i \tilde E_i$ and $Y^{[1]} = \coprod_i E_i$ where $E_i = \tilde Y \cap \tilde E_i$ is now understood as the infinity divisor (or relative hyperplane section) of $\pi_i: \tilde E_i \to C_i \cong \mathbb{P}^1$.
\subsection{Mixed Hodge Structure and the Clemens--Schmid exact sequence} \label{s:2.2}
We now apply the Clemens--Schmid exact sequence \cite{hC2} to the above two semistable degenerations. A general reference is \cite{pG1}. We will mainly be interested in $H^{\leq 3}$. The computation of $H^{> 3}$ is similar.
\subsubsection{} The cohomology of $H^*(\mathcal{X}_0)$, with its canonical mixed Hodge structure, is computed from the spectral sequence $E_0^{p, q}(\mathcal{X}_0) = \Omega^q(X^{[p]})$ with $d_0 = d$, the de Rham differential, and then $$E_1^{p, q}(\mathcal{X}_0) = H^q(X^{[p]})$$ with $d_1 =\delta$ being the combinatorial coboundary operator $$\delta: H^q(X^{[p]}) \to H^q(X^{[p + 1]}).$$ The spectral sequence degenerates at $E_2$ terms.
The weight filtration on $H^*(\mathcal{X}_0)$ is induced from the increasing filtration on the spectral sequence $W_m := \bigoplus_{q \leq m} E^{*, q}.$ Therefore, $$\operatorname{Gr}^W_m(H^j) = E_2^{j-m, m}, \quad \operatorname{Gr}^W_m (H^j) = 0 \quad \text{for $m < 0$ or $m > j$}.$$ Since $X^{[j]} \neq \emptyset$ only when $j = 0, 1$, we have \[
H^0 \cong E_2^{0,0}, \quad H^1 \cong E_2^{1,0} \oplus E_2^{0,1}, \quad
H^2 \cong E_2^{1,1} \oplus E_2^{0,2}, \quad H^3 \cong E_2^{1,2} \oplus E_2^{0,3}. \] The only weight $3$ piece is $E_2^{0,3}$, which can be computed by $$
\delta: E_1^{0,3} = H^3(X^{[0]}) \mathop{\longrightarrow} E_1^{1,3}=H^3(X^{[1]}). $$ Since $Q_i$, $\tilde E_i$ and $E_i$ have no odd cohomologies, $H^3(X^{[1]}) =0$ and $H^3(X^{[1]}) = H^3(\tilde Y)$. We have thus $E_2^{0,3} = H^3(\tilde Y)$.
The weight 2 pieces, which is the most essential part, is computed from \begin{equation} \label{e:4} H^2(X^{[0]}) = H^2(\tilde Y) \oplus \bigoplus\nolimits_{i = 1}^k H^2(Q_i) \mathop{\longrightarrow}\limits^{\delta_2} H^2(X^{[1]}) = \bigoplus\nolimits_{i = 1}^k H^2(E_i). \end{equation} We have $E_2^{1,2} = \operatorname{cok}(\delta_2)$ and $E_2^{0,2} =\ker (\delta_2)$. The weight 1 and weight 0 pieces can be similarly computed. For weight 1 pieces we have $$E_2^{0,1} = H^1(X^{[0]}) = H^1(\tilde Y) \cong H^1(Y) \cong H^1(X),$$ and $E_2^{1,1} = 0$. The weight 0 pieces are computed from $\delta: H^0(X^{[0]}) \to H^0(X^{[1]})$ and we have $E_2^{0,0} = H^0(\tilde Y) \cong H^0(Y) \cong H^0(X)$, and $E_2^{1,0} = 0$. We summarize these calculations as
\begin{lemma} \label{l:4} There are isomorphisms of MHS: \[ \begin{split}
&H^3(\mathcal{X}_0) \cong H^3(\tilde Y) \oplus \operatorname{cok}(\delta_2), \\
&H^2(\mathcal{X}_0) \cong \ker(\delta_2),\\
&H^1(\mathcal{X}_0) \cong H^1(\tilde Y) \cong H^1(Y) \cong H^1(X),\\
&H^0(\mathcal{X}_0) \cong H^0(\tilde Y) \cong H^0(Y) \cong H^0(X). \end{split} \] In particular, $H^j(\mathcal{X}_0)$ is pure of weight $j$ for $j \le 2$. \end{lemma}
\subsubsection{} \label{s:2.2.2} Here we give a dual formulation of \eqref{e:4} which will be useful later. Let $\ell, \ell'$ be the line classes of the two rulings of $E \cong \mathbb{P}^1 \times \mathbb{P}^1$. Then $H^2(Q, \mathbb{Z})$ is generated by $e = [E]$ as a hyperplane class and
$e|_E = \ell + \ell'$. The map $\delta_2$ in \eqref{e:4} is then equivalent to \begin{equation} \label{e:delta2bar} \bar{\delta}_2: H^2(\tilde Y) \longrightarrow \bigoplus\nolimits_{i = 1}^k H^2(E_i)/H^2(Q_i). \end{equation} Since $H^2(\tilde Y) = \phi^* H^2(Y) \oplus \bigoplus_{i = 1}^k \langle [E_i] \rangle$ and
$[E_i]|_{E_i} = -(\ell_i + \ell_i')$, the second component $\bigoplus_{i = 1}^k \langle [E_i] \rangle$ lies in $\ker (\bar{\delta}_2)$ and $\bar{\delta}_2$ factors through \begin{equation} \label{e:5} \phi^* H^2(Y) \to \bigoplus\nolimits_{i = 1}^k H^2(E_i)/H^2(Q_i) \cong \bigoplus\nolimits_{i = 1}^k \langle \ell_i - \ell_i'\rangle \end{equation} (as $\Bbb Q$-spaces). Notice that the quotient is isomorphic to $\bigoplus_{i = 1}^k \langle \ell_i'\rangle$ integrally.
By reordering we may assume that $\phi_* \ell_i = [C_i]$ and $\phi^* [C_i] = \ell_i - \ell_i'$ (cf.~\cite{LLW1}). The dual of \eqref{e:5} then coincides with the fundamental class map $$\vartheta: \bigoplus\nolimits_{i = 1}^k \langle [C_i] \rangle \longrightarrow H_2(Y).$$ In general for a $\mathbb{Q}$-linear map $\vartheta: P \to Z$, we have $\operatorname{im} \vartheta^* \cong (P/\ker \vartheta)^* \cong (\operatorname{im} \vartheta)^*$. Thus \begin{equation} \label{e:6}
\dim_{\mathbb{Q}} \operatorname{cok} (\delta_2) + \dim_{\mathbb{Q}} \operatorname{im} (\vartheta) = k. \end{equation}
We will see in Corollary~\ref{c:1} that $\dim \operatorname{cok} \delta = \mu$ and $\dim \operatorname{im} \vartheta = \rho$. This gives the Hodge theoretic meaning of $\mu + \rho = k$ in Lemma~\ref{l:3}. Further elaboration of this theme will follow in Theorem~\ref{t:bes}.
\subsubsection{} On $\mathcal{Y}_0$, the computation is similar and a lot easier. The weight 3 piece can be computed by the map $H^3(Y^{[0]}) = H^3(\tilde Y) \longrightarrow H^3(Y^{[1]}) = 0$; the weight 2 piece is similarly computed by the map $$ H^2(Y^{[0]}) = H^2(\tilde Y) \oplus \bigoplus\nolimits_{i = 1}^k H^2(\tilde E_i) \mathop{\longrightarrow}\limits^{\delta'_2} H^2(Y^{[1]}) = \bigoplus\nolimits_{i = 1}^k H^2(E_i). $$
Let $h = \pi^* ({\rm pt})$ and $\xi = [E]$ for $\pi: \tilde E \to \mathbb{P}^1$. Then $h|_E = \ell'$ and $\xi|_E = \ell + \ell'$. In particular the restriction map $H^2(\tilde E) \to H^2(E)$ is an isomorphism and hence $\delta'_2$ is surjective. The computation of pieces from weights 1 and 0 is the same as for $\mathcal{X}_0$. We have therefore the following lemma.
\begin{lemma} \label{l:5} There are isomorphisms of MHS: \[ \begin{split}
&H^3(\mathcal{Y}_0) \cong H^3(Y^{[0]}) \cong H^3(\tilde Y) , \\
&H^2(\mathcal{Y}_0) \cong \ker(\delta'_2) \cong H^2(\tilde Y),\\
&H^1(\mathcal{Y}_0) \cong H^1(\tilde Y) \cong H^1(Y) \cong H^1(X),\\
&H^0(\mathcal{Y}_0) \cong H^0(\tilde Y) \cong H^0(Y) \cong H^0(X). \end{split} \] \end{lemma}
\subsubsection{} We denote by $N$ the monodromy operator for both $\mathcal{X}$ and $\mathcal{Y}$ families. The map $N$ induces the unique monodromy weight filtrations $W$ on $H^n(X)$ which, together with the limiting Hodge filtration $F_\infty^\bullet$, leads to Schmid\rq{}s limiting MHS \cite{Schmid, jS}. That is, $$0 \subset W_0 \subset W_1 \subset \cdots \subset W_{2n - 1} \subset W_{2n} = H^n(X)$$ such that $N W_k \subset W_{k - 2}$ and for $\ell \ge 0$, \begin{equation} \label{e:H-sym} N^\ell: G^W_{n + \ell} \cong G^W_{n - \ell} \end{equation} on graded pieces. The induced filtration $F^p_\infty G^W_k := F^p_\infty \cap W_k/F^p_\infty \cap W_{k - 1}$ defines a pure Hodge structure of weight $k$ on $G^W_k$. Similar constructions apply to $H^n(Y)$ as well.
\begin{lemma} \label{l:6} We have the following exact sequences (of MHS) for $H^2$ and $H^3$: \begin{equation*} \begin{split}
0 \to H^3(\mathcal{X}_0) \to & H^3(X) \mathop{\longrightarrow}\limits^{N} H^3(X) \to H_3(\mathcal{X}_0) \to 0,\\
0 \to H^0(X) \to H_6(\mathcal{X}_0) \to H^2(\mathcal{X}_0) \to & H^2(X) \mathop{\longrightarrow}\limits^N 0, \\
0 \to H^3(\mathcal{Y}_0) \to & H^3(Y) \mathop{\longrightarrow}\limits^N 0,\\
0 \to H^0(Y) \to H_6(\mathcal{Y}_0) \to H^2(\mathcal{Y}_0) \to & H^2(Y) \mathop{\longrightarrow}\limits^N 0. \end{split} \end{equation*} \end{lemma}
\begin{proof} These follow from the Clemens--Schmid exact sequence, which is compatible with the MHS. The other terms in the first sequence, namely $H^1(X)\to H_5(\mathcal{X}_0)$ to the left end and $H^5(\mathcal{X}_0) \to H^5(X)$ to the right end, can be ignored since they induce isomorphisms, as can be checked using MHS on $H_5(\mathcal{X}_0)$. Similar comments apply to the third sequence for $H^3(Y)$.
Note that the monodromy is trivial for $\mathcal{Y} \to \Delta$ since the punctured family is trivial. For the second sequence, by Lemma~\ref{l:4}, we know that $H^2(\mathcal{X}_0)$ is pure of weight 2. Hence $N$ on $H^2(X)$ is also trivial and the Hodge structure does not degenerate. Indeed, if $N \ne 0$ then $\ker N$ contains some part of weight $\le 2$ by \eqref{e:H-sym}. \end{proof}
\begin{corollary} \label{c:1} \begin{itemize} \item[(i)] $\rho = \dim \operatorname{im}(\vartheta)$ and $\mu = \dim \operatorname{cok}(\delta_2)$.
\item[(ii)] $H^3(Y) \cong H^3(\mathcal{Y}_0) \cong H^3(Y^{[0]}) \cong H^3(\tilde Y) \cong \operatorname{Gr}^W_3 H^3(X)$.
\item[(iii)] Denote by $K := \ker (N: H^3(X) \to H^3(X))$. Then $H^3(\mathcal{X}_0) \cong K$. More precisely, $\operatorname{Gr}^W_3 (H^3(\mathcal{X}_0)) \cong H^3(Y)$ and $\operatorname{Gr}^W_2 (H^3(\mathcal{X}_0)) \cong \operatorname{cok} (\delta_2)$. \end{itemize} \end{corollary}
\begin{proof} By Lemma~\ref{l:4}, $h^2(\mathcal{X}_0) = \dim \ker (\delta_2)$. It follows from the second and the fourth exact sequences in Lemma~\ref{l:6} that $h^2(X) = \dim \ker (\delta_2) + 1 - (k + 1)$. Rewrite \eqref{e:4} as \begin{equation} \label{e:7} 0 \to \ker (\delta_2) \to H^2(X^{[0]}) \mathop{\longrightarrow}\limits^\delta
H^2(X^{[1]}) \to \operatorname{cok} (\delta_2) \to 0, \end{equation} which implies $\dim \ker(\delta_2) + 2k = \dim \operatorname{cok} (\delta_2) + 2 k + h^2(Y)$.
Combining these two equations with \eqref{e:6}, we have $\rho = h^2(Y) - h^2(X) = k - \dim \operatorname{cok} (\delta_2) = \dim \operatorname{im}(\vartheta)$. This proves the first equation for $\rho$ in (i).
Combining the first equation in Lemma~\ref{l:5} and the third exact sequence in Lemma~\ref{l:6}, we have \begin{equation} \label{e:8} H^3(Y) \cong H^3(\mathcal{Y}_0) \cong H^3(\tilde Y) . \end{equation} This shows (ii) except the last equality.
By Lemmas~\ref{l:6} and \ref{l:4}, $K \cong H^3(\mathcal{X}_0) \cong H^3(\tilde{Y}) \oplus \operatorname{cok}(\delta_2) \cong H^3({Y}) \oplus \operatorname{cok}(\delta_2)$, where the last equality follows from \eqref{e:8}. This proves (iii).
For the remaining parts of (i) and (ii), we investigate the non-trivial terms of the limiting mixed Hodge diamond for $H^n := H^n(X)$: \begin{equation} \label{H^3(Y)} \xymatrix{&&H^{2, 2}_\infty H^3 \ar[dd]^N_\sim\\
H^{3,0}_\infty H^3 &H^{2,1}_\infty H^3 \ar@{-}[ru]&&H^{1,2}_\infty H^3 \ar@{-}[ld]&H^{0,3}_\infty H^3, \\
&&H^{1, 1}_\infty H^3} \end{equation} where $H^{p, q}_\infty H^n = F^p_\infty \operatorname{Gr}^W_{p + q} H^n$. The space $H^{3, 0}(X)$ does not degenerate by \cite{cW1} (which holds for degenerations with canonical singularities, and first proved in \cite{cW0} for the Calabi--Yau case). We conclude that $H^{1,1}_{\infty} H^3 \cong \operatorname{cok}(\delta_2)$ and $\operatorname{Gr}^W_3 H^3(X) \cong H^3(Y)$. By definition $\mu = \frac{1}{2}(h^3(X) - h^3(Y))$, hence $\mu = h^{2, 2}_\infty H^3 = h^{1, 1}_\infty H^3 = \dim \operatorname{cok}(\delta_2)$. \end{proof}
\subsubsection{}
We denote the \emph{vanishing cycle space} $V$ as the $\mathbb{Q}$-vector space generated by vanishing 3-cycles. We first define the abelian group $V_{\mathbb{Z}}$ from \begin{equation} \label{e:9} 0 \to V_{\mathbb{Z}} \to H_3(X, \mathbb{Z}) \to H_3(\bar X, \mathbb{Z}) \to 0, \end{equation} and $V := V_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}$. The sequence \eqref{e:9} arises from the homology Mayer--Vietoris sequence and the surjectivity on the right hand side follows from the fact that $H_2(\coprod^k S^3, \mathbb{Z}) = 0$.
\begin{lemma} \label{l:7} Denote by $H^3 := H^3(X)$. \begin{itemize} \item[(i)] $H^3( \bar{X} ) \cong K \cong H^3(\mathcal{X}_0) \cong W_3\, H^3$.
\item[(ii)] $V^* \cong H^{2, 2}_\infty H^3$ and $V \cong H^{1, 1}_\infty H^3 = \operatorname{cok} (\delta_2)$ via Poincar\'e pairing. \end{itemize} \end{lemma}
\begin{proof} Dualizing \eqref{e:9} over $\mathbb{Q}$, we have $$0 \to H^3(\bar X) \to H^3(X) \to V^* \to 0.$$ The invariant cycle theorem in \cite{BBD} then implies that $H^3(\bar X) \cong \ker N = K \cong H^3(\mathcal{X}_0)$. This proves (i).
Hence we have the canonical isomorphism $$ V^* \cong H^3(X)/H^3(\bar X) = G^W_4 H^3 = F^2_\infty G^W_4 H^3 = H^{2, 2}_\infty H^3. $$ Moreover, the non-degeneracy of the pairing $(\alpha, N\beta)$ on $G_4^W H^3$ implies $$ H^{1, 1}_\infty H^3 = N H^{2, 2}_\infty H^3 \cong (H^{2, 2}_\infty H^3)^* \cong V^{**}_{\mathbb{C}} \cong V_{\mathbb{C}}. $$ This proves (ii). \end{proof}
\begin{remark}[On threefold extremal transitions] \label{r:can} Most results in \S \ref{s:2.2} works for more general geometric contexts. The mixed Hodge diamond \eqref{H^3(Y)} holds for any 3-folds degenerations with at most canonical singularities \cite{cW1}. The identification of vanishing cycle space $V$ via \eqref{e:9} works for 3--folds with only isolated (hypersurface) singularities. Indeed, the exactness on the RHS holds for degenerations $\frak{X} \to \Delta$ such that $\frak{X}$ is smooth and $\frak{X}_0$ has only isolated singularities. This follows from Milnor's theorem that the vanishing cycle has the homotopy type of a bouquet of middle dimensional spheres \cite[Theorem 6.5]{jM}. Hence Lemma \ref{l:7} works for any 3-fold degenerations with isolated hypersurface canonical singularities.
Later on we will impose the Calabi--Yau condition on all the 3-folds involved. If $X \nearrow Y$ is a terminal transition of Calabi--Yau 3-folds, i.e., $\mathfrak{X}_0 = \bar X$ has at most (isolated Gorenstein) terminal singularities, then $\bar X$ has unobstructed deformations \cite{N}. Moreover, the small resolution $Y \to \bar X$ induces an embedding ${\rm Def}(Y) \hookrightarrow {\rm Def}(\bar X)$ which identifies the limiting/ordinary pure Hodge structures $\operatorname{Gr}^W_3 H^3(X) \cong H^3(Y)$ as in Corollary \ref{c:1} (iii).
For conifold transitions all these can be described in explicit terms and more precise structure will be formulated. \end{remark}
\subsection{The basic exact sequence} \label{s:2.3}
We may combine the four Clemens--Schmid exact sequences into one short exact sequence, which we call the \emph{basic exact sequence}, to give the Hodge-theoretic realization ``$\rho + \mu = k$'' in Lemma~\ref{l:3}.
Let $A = (a_{ij}) \in M_{k \times \mu}(\mathbb{Z})$ be a relation matrix for $C_i$'s, i.e., \[
\sum\nolimits_{i=1}^k a_{ij} [C_i] = 0, \qquad j = 1, \ldots, \mu, \] give all relations of the curves classes $[C_i]$'s. Similarly, let $B = (b_{ij}) \in M_{k \times \rho}(\mathbb{Z})$ be a relation matrix for $S_i$'s: \[
\sum\nolimits_{i=1}^k b_{ij} [S_i] = 0, \qquad j = 1, \ldots, \rho. \]
\begin{theorem} [Basic exact sequence] \label{t:bes} The group of 2-cycles generated by exceptional curves $C_i$ (vanishing $S^2$ cycles) on $Y$ and the group of 3-cycles generated by $[S_i]$ (vanishing $S^3$ cycles) on $X$ are linked by the following weight 2 exact sequence $$
0 \to H^2(Y)/H^2(X) \mathop{\longrightarrow}\limits^B
\bigoplus\nolimits_{i = 1}^k H^2(E_i)/H^2(Q_i)
\mathop{\longrightarrow}\limits^{A^t} V
\to 0. $$ In particular $B = \ker A^t$ and $A = \ker B^t$. \end{theorem}
\begin{proof} From \S\ref{s:2.2.2}, $\operatorname{cok}(\delta_2) = \operatorname{cok}(\bar{\delta}_2)$ and \eqref{e:7} can be replaced by \begin{equation} \label{e:10}
0 \to H^2(\tilde{Y})/(\ker \bar{\delta})
\stackrel{D}{\longrightarrow} \bigoplus\nolimits_{i = 1}^k H^2(E_i)/H^2(Q_i)
\stackrel{C}{\longrightarrow} \operatorname{cok}(\delta_2) \to 0. \end{equation} By Lemma~\ref{l:7} (ii), we have $\operatorname{cok}(\delta_2) \cong V$. To prove the theorem, we need to show that $H^2(\tilde Y)/\ker \bar\delta \cong H^2(Y)/H^2(X)$, and $D=B$, $C=A^t$.
By the invariant cycle theorem \cite{BBD}, $H^2(X) \cong H^2(\bar{X})$. Since $H^2 (\bar{X})$ injects to $H^2(Y)$ by pullback, this defines the embedding $$\iota: H^2(X) \hookrightarrow H^2(Y)$$ and the quotient $H^2(Y)/H^2(X)$.
Recast the relation matrix $A$ of the rational curves $C_i$ in $$ 0 \to \mathbb{Q}^\mu \mathop{\longrightarrow}\limits^A \mathbb{Q}^k \cong \bigoplus\nolimits_{i = 1}^k \langle [C_i] \rangle \mathop{\longrightarrow}\limits^S \operatorname{im} (\vartheta) \to 0 $$ where $S = \operatorname{cok}(A) \in M_{\rho \times k}$ is the matrix for $\vartheta$, and $\operatorname{im}(\vartheta)$ has rank $\rho$. The dual sequence reads \begin{equation} \label{e:11}
0 \to (\operatorname{im}\vartheta)^* \cong (\mathbb{Q}^\rho)^*
\mathop{\longrightarrow}\limits^{S^t}
(\mathbb{Q}^k)^* \cong \bigoplus\nolimits_{i = 1}^k H^2(E_i)/H^2(Q_i)
\mathop{\longrightarrow}\limits^{A^t} (\mathbb{Q}^\mu)^* \to 0. \end{equation} Compare \eqref{e:11} with \eqref{e:10}, we see that $(\mathbb{Q}^\mu)^* \cong V$. From the discussion in \S\ref{s:2.2.2}, we have $(\operatorname{im} \vartheta)^* = H^2(Y)/H^2(X)$.
We want to reinterpret the map $A^t: (\mathbb{Q}^k)^* \to V$ in \eqref{e:11}. This is a presentation of $V$ by $k$ generators, denoted by $\sigma_i$, and the relation matrix of which is given by $S^t$. If we show that $\sigma_i$ can be identified with $S_i$, then $(\mathbb{Q}^\mu)^* \cong V$ and $B = S^t = \ker A^t$ is the relation matrix for $S_i$'s.
Consider the following topological construction. For any non-trivial integral relation $\sum_{i = 1}^k a_i [C_i] = 0$, there is a 3-chain $\theta$ in $Y$ with $\partial \theta = \sum\nolimits_{i = 1}^k a_i C_i$. Under $\psi: Y \to \bar X$, $C_i$ collapses to the node $p_i$. Hence it creates a $3$-cycle $\bar \theta := \psi_* \theta \in H_3(\bar X, \mathbb{Z}),$ which deforms (lifts) to $\gamma \in H_3(X, \mathbb{Z})$ in nearby fibers by the surjectivity in \eqref{e:9}. Using the intersection pairing on $H_3(X, \mathbb{Z})$, $\gamma$ then defines an element $\operatorname{PD}(\gamma)$ in $H^3(X, \mathbb{Z})$. Under the restriction $V$, we get $\operatorname{PD}(\gamma) \in V^* $.
It remains to show that $(\gamma.S_i) = a_i$. Let $U_i$ be a small tubular neighborhood of $S_i$ and $\tilde U_i$ be the corresponding tubular neighborhood of $C_i$, then by Corollary \ref{c:surgery}, $$\partial U_i \cong \partial (S_i^3 \times D^3) \cong S^3 \times S^2 \cong \partial(D^4 \times C_i) \cong \partial \tilde U_i.$$ Now $\theta_i := \theta \cap \tilde U_i$ gives a homotopy between $a_i[C_i]$ (in the center of $\tilde U_i$) and $a_i\,{\rm pt}\times [S^2]$ (on $\partial \tilde U_i$). Denote by $\iota: \partial U_i \hookrightarrow X$ and $\tilde \iota: \partial \tilde U_i \hookrightarrow Y$. Then \begin{equation*} \begin{split} (\gamma.S_i)^X &= (\gamma.\iota_*[S^3])^X = (\iota^*\gamma.[S^3])^{\partial U_i} = (\tilde\iota^*\gamma.[S^3])^{\partial \tilde U_i} \\ & = (a_i[S^2], [S^3])^{S^3 \times S^2} = a_i. \end{split} \end{equation*} The proof is complete. \end{proof}
\begin{remark} \label{r:convention} We would like to choose a preferred basis of the vanishing cocycles $V^*$ as well as a basis of divisors dual to the space of extremal curves. These notations will fixed henceforth and will be used in later sections.
During the proof of Theorem~\ref{t:bes}, we establish the correspondence between $A^j = (a_{1j}, \ldots, a_{kj})^t$ and $\operatorname{PD}(\gamma_j) \in V^*$, $1 \le j \le \mu$, characterized by $ a_{ij} = (\gamma_j.S_i)$. The subspace of $H_3(X)$ spanned by $\gamma_j$'s is denoted by $V'$.
Dually, we denote by $T_1, \ldots, T_\rho \in H^2(Y)$ those divisors which form an integral basis of the lattice in $H^2(Y)$ dual (orthogonal) to $H_2(X) \subset H_2(Y)$. In particular they form an integral basis of $H^2(Y)/H^2(X)$. We choose $T_l$'s such that $T_l$ corresponds to the $l$-th column vector of the matrix $B$ via $b_{il} = (C_i.T_l)$. Such a choice is consistent with the basic exact sequence since $$ (A^t B)_{jl} = \sum\nolimits_{i = 1}^k a^t_{ji} b_{il} = \sum\nolimits_{i = 1}^k a_{ij} (C_i.T_l) = \Big(\sum a_{ij} [C_i]\Big).T_l = 0 $$ for all $j, l$. We may also assume that the first $\rho \times \rho$ minor of $B$ has full rank. \end{remark}
\section{Gromov--Witten theory and Dubrovin connections} \label{s:3}
In \S\ref{s:3.1} the $A$ model $A(X)$ is shown to be a sub-theory of $A(Y)$. We then move on to study the genus $0$ excess $A$ model on $Y/X$ associated to the extremal curve classes in \S \ref{s:3.2}. As a consequence the (nilpotent) monodromy is calculated in terms of the relation matrix $B$ at the end of \S \ref{s:dubrovin}.
\subsection{Consequences of the degeneration formula for threefolds} \label{s:3.1} The Gromov--Witten theory on $X$ can be related to that on $Y$ by the degeneration formula through the two semistable degenerations introduced in \S\ref{s:2.1}.
In the previous section, we see that the monodromy acts trivially on $H(X) \setminus H^3(X)$ and we have $$H^3_{inv}(X) = K \cong H^3(Y) \oplus H^{1, 1}_\infty H^3(X) \cong H^3(Y) \oplus V.$$ There we implicitly have a linear map \begin{equation} \label{e:iota}
\iota: H^j_{inv}(X) \to H^j (Y) \end{equation} as follows. For $j=3$, it is the projection $$H^3_{inv}(X) \cong H^3(Y) \oplus V \to H^3 (Y).$$ For $j=2$, it is the embedding defined before and the case $j=4$ is the same as (dual to) the $j=2$ case. For $j=0, 1, 5, 6$, $\iota$ is an isomorphism.
The following is a refinement of a result of Li--Ruan \cite{LR}. (See also \cite{LY}.)
\begin{proposition} \label{p:1} Let $X \nearrow Y$ be a projective conifold transition. Given $\vec{a} \in (H^{\ge 2}_{inv}(X)/V)^{\oplus n}$ and a curve class $\beta \in NE(X) \setminus \{0\}$, we have \begin{equation} \label{e:deg}
\langle \vec a \rangle_{g, n , \beta}^X = \sum\nolimits_{{\psi}_*(\gamma) = \beta} \langle \iota(\vec a)\rangle_{g, n , \gamma}^Y . \end{equation} If some component of $\vec{a}$ lies in $H^0$, then both sides vanish. Furthermore, the RHS is a \emph{finite} sum. \end{proposition}
\begin{proof} A slightly weaker version of \eqref{e:deg} has been proved in \cite{LR, LY}. We review its proof with slight refinements as it will be useful in \S\ref{s:5}.
We follow the setup and argument in \cite[\S 4]{LLW1} closely. By \cite[\S 4.2]{LLW1}, a cohomology class $a \in H^{> 2}_{inv}(X)/V$ can always find a lift to \[
(a_i)_{i=0}^k \in H(\tilde{Y}) \oplus \bigoplus\nolimits_{i=1}^k H(Q_i) \] such that $a_i= 0$ for all $i \neq 0$. We apply J.~Li's algebraic version of degeneration formula \cite{JL2, LY} to the complex degeneration $X \rightsquigarrow \tilde{Y} \cup_E Q$, where $$Q := \coprod\nolimits_{i = 1}^k Q_i $$ is a disjoint union of quadrics $Q_i$'s and $$E := \sum\nolimits_{i = 1}^k E_i .$$ One has $K_{\tilde Y} = \tilde{\psi}^* K_{\bar{X}} + E$. The topological data $(g, n, \beta)$ lifts to two admissible triples $\Gamma_1$ on $(\tilde Y, E)$ and $\Gamma_2$ on $(Q, E)$ such that $\Gamma_1$ has curve class $\tilde{\gamma} \in NE(\tilde Y)$, contact order $\mu = (\tilde \gamma.E)$, and number of contact points $\rho$. Then $$(\tilde{\gamma}. c_1(\tilde Y)) = (\tilde{\psi}_*\tilde{\gamma}.c_1(\bar{X})) - (\tilde \gamma. E) = (\beta.c_1(X)) - \mu.$$ The virtual dimension (without marked points) is given by \begin{equation*} \begin{split} d_{\Gamma_1} = (\tilde \gamma.c_1(\tilde Y)) + (\dim X - 3)(1 - g) + \rho - \mu = d_\beta + \rho - 2\mu \end{split} \end{equation*} where $d_\beta$ is the virtual dimension of the absolute invariant with curve class $\beta$ (without marked points). Since we chose the lifting $(\vec a_i)_{i = 0}^k$ of $\vec a$ to have $\vec a_i = 0$ for all $i \ne 0$, all insertions contribute to $\tilde Y$. If $\rho \ne 0$ then $\rho - 2\mu < 0$. This leads to vanishing relative GW invariant on $(\tilde Y, E)$. Therefore, $\rho$ must be zero.
To summarize, we get \begin{equation} \label{e:i} \langle \vec{a} \rangle^X_{g, n, \beta} = \sum\nolimits_{\tilde\psi_*(\tilde\gamma) = \beta} \langle {\vec{a}_0} \mid \emptyset\rangle^{(\tilde{Y}, E)}_{g, n, \tilde\gamma}, \end{equation} such that \begin{equation} \label{e:lifting} \tilde{\psi}_* \tilde{\gamma} = \beta, \qquad \tilde{\gamma} . E =0,
\qquad \tilde{\gamma}_Q = 0. \end{equation} Formula \eqref{e:i} also holds for $a_i$ a divisor by the divisor axiom.
We use a similar argument to compute $\langle \vec b\rangle^Y_{g, n, \gamma}$ via the K\"ahler degeneration $Y \rightsquigarrow \tilde{Y} \cup \tilde{E}$, where $\tilde{E} $ is a disjoint union of $\tilde{E}_i$ (cf.~\cite[Theorem~4.10]{LLW1}). By the divisor equation we may assume that $\deg b_j \ge 3$ for all $j = 1, \ldots, n$. We choose the lifting $(\vec b)_{i = 0}^k$ of $\vec b$ such that $\vec b_i = 0$ for all $i \ne 0$. In the lifting $\gamma_1$ on $\tilde Y$ and $\gamma_2$ on $\pi: \tilde E = \coprod_i \tilde E_i \to \coprod_i C_i$, we must have $\gamma = \phi_*\gamma_1 + \pi_* \gamma_2$. The contact order is given by $\mu = (\gamma_1.E)$ which has the property that $\mu = 0$ if and only if $\gamma_1 = \phi^*\gamma$ (and hence $\gamma_2 = 0$). If $\rho \ne 0$ we get $d_{\Gamma_1} = d_\gamma + \rho - 2\mu < d_\gamma$ and the invariant is zero. This proves \begin{equation} \label{e:ii} \langle \vec b\rangle^Y_{g, n, \gamma} = \langle \phi^* \vec b \mid \emptyset\rangle^{(\tilde Y, E)}_{g, n, \phi^*\gamma} , \end{equation} with ${\phi}_* \tilde{\gamma} =\gamma$, $\tilde{\gamma} . E =0$, $\tilde{\gamma}_{\tilde{E}} = 0$.
To combine these two degeneration formulas together, we notice that in the K\"ahler degeneration, $\tilde \gamma \in NE(\tilde Y)$ can have contact order $\mu = (\tilde \gamma.E) = 0$ if and only if $\tilde \gamma = \phi^*\gamma$ for some $\gamma \in NE(Y)$ (indeed for $\gamma = \phi_*\tilde \gamma$). Choose $\vec{b} = \iota (\vec{a})$ and \eqref{e:deg} follows. The vanishing statement (of $H^0$ insertion) follows from the fundamental class axiom.
Now we proceed to prove the finiteness of the sum. (This is not stated in \cite{LR}.) For $\phi: \tilde Y \to Y$ being the blow-up along $C_i$'s, the curve class $\gamma \in NE(Y)$ contributes a non-trivial invariant in the sum only if $\phi^* \gamma$ is effective on $\tilde Y$. By combining \eqref{e:5}, \eqref{e:i} and \eqref{e:ii}, the effectivity of $\phi^*\gamma$ forces the sum to be finite. Equivalently, the condition that $\phi^*\gamma$ is effective is equivalent to that $\gamma$ is $\mathscr{F}$-effective under the flop $Y \dasharrow Y'$. (i.e.\ effective in $Y$ and in $Y'$ under the natural correspondence \cite{LLW1}). Recall that under the flop the flopping curve class in $Y$ is mapped to the negative flopping curve in $Y\rq{}$. Therefore, the sum is finite. \end{proof}
\begin{remark} \label{r:4} The phenomena \eqref{e:deg}, including finiteness of the sum, were observed in \cite{HKTY} for Calabi--Yau hypersurfaces in weighted projective spaces from the numerical data obtained from the corresponding $B$ model generating function via mirror symmetry. \end{remark}
\begin{corollary} \label{c:2} Gromov--Witten theory on even cohomology $GW^{ev}(X)$ (of all genera) can be considered as a sub-theory of $GW^{ev}(Y)$. In particular, the big quantum cohomology ring is functorial with respect to $\iota: H^{ev}(X) \to H^{ev}(Y)$ in \eqref{e:iota}. \end{corollary}
\begin{proof} We first note that $\iota$ is an injection on $H^{ev}$. Proposition~\ref{p:1} then implies that all GW invariants of $X$ with even classes can be recovered from invariants of $Y$. The only exception, $H^0$, can be treated by the fundamental class axiom. Therefore, in this sense that $GW^{ev}(X)$ is a sub-theory of $GW^{ev}(Y)$.
In genus zero, this can be rephrased as functoriality. Observe that the degeneration formula also holds for $\beta = 0$. For $g = 0$, this leads to the equality of classical triple intersection $(a, b, c)^X = (\iota(a), \iota(b), \iota(c))^Y.$ Since the Poincar\'e pairing on $H^{ev}(X)$ is also preserved under $\iota$, we see that the classical ring structure on $H^{ev}(X)$ are naturally embedded in $H^{ev}(Y)$.
To see the functoriality of the big quantum ring with respect to $\iota$, we note that $(\iota(a).C_i) = 0$ for any $a \in H^{ev}(X)$ and for any extremal curve $C_i$ in $Y$. Furthermore, for the invariants associated to the extremal rays the insertions must involve only divisors by the virtual dimension count. Hence for generating functions with \emph{at least one insertion} we also have $$ \sum\nolimits_{\beta \in NE(X)} \langle \vec a \rangle^X_\beta q^\beta = \sum\nolimits_{\gamma \in NE(Y)} \langle \iota(\vec a) \rangle^Y_\gamma q^{\psi_*(\gamma)}. $$ Note that the case of $H^0$ is not covered in Proposition~\ref{p:1}, but it can be treated by the fundamental class axiom as above. \end{proof}
\begin{remark} \label{r:5} It is clear that the argument and conclusion hold even if some insertions lie in $H^3_{inv}(X)/V \cong H^3(Y)$ by Proposition~\ref{p:1}. \end{remark}
The full GW theory is built on the full cohomology \emph{superspace} $H = H^{ev} \oplus H^{odd}$. However, the odd part is not as well-studied in the literature as the even one. In some special cases the difficulty does not occur.
\begin{lemma} \label{l:8} Let $X$ be a smooth minimal 3-fold with $H^1(X) = 0$. The non-trivial primary GW invariants are all supported on $H^2(X)$ and hence, by the divisor axiom, reduced to the case without insertion. More generally the conclusion holds for any curve class $\beta \in NE(X)$ with $c_1(X).\beta \le 0$ for any 3-fold $X$ with $H^1(X) = 0$. \end{lemma}
\begin{proof} For $n$-point invariants, the virtual dimension of $\overline{M}_{g, n}(X, \beta)$ is $$\operatorname{vdim} = c_1(X).\beta + (\dim X - 3)(1 - g) + n \le n.$$ Since the appearance of fundamental class in the insertions leads to trivial invariants, we must have the algebraic degree $\deg a_i \ge 1$ for all insertions $a_i$, $i = 1, \ldots, n$. Hence in fact we must have $\deg a_i = 1$ for all $i$ and $c_1(X).\beta = 0$. \end{proof}
\subsection{The even and extremal quantum cohomology} \label{s:3.2} From now on, we restrict to genus zero theory.
Let $s = \sum_\epsilon {s^\epsilon \bar{T}_\epsilon}\in H^2(X)$ where $\bar{T}_\epsilon$'s form a basis of $H^2(X)$. Then the genus zero GW pre-potential on $H^2(X)$ is given by \begin{equation} \label{e:14} F^X_0(s) = \sum_{n = 0}^\infty \sum_{\beta \in NE(X)} \langle s^n \rangle_{0, n, \beta}\, \frac{q^\beta}{n!} = \frac{s^3}{3!} + \sum_{\beta \ne 0} n^X_\beta q^\beta e^{(\beta.s)}, \end{equation} where $n^X_\beta = \langle \rangle_{0, 0, \beta}^X$, and $q^\beta$ the (formal) Novikov variables.
$F^X_0 (s)$ encodes the small quantum cohomology of $X$ (and the big quantum cohomology if $X$ is minimal by Lemma~\ref{l:8}), \emph{except in the topological term $s^3/(3!)$ where we need the full $s \in H^{ev}(X)$}.
Similarly we have $F_0^Y(t)$ on $H^2(Y)$ where \begin{equation} \label{e:split} t = s + u \in H^2(Y) = \iota( H^2(X)) \oplus \bigoplus\nolimits_{l = 1}^\rho \langle T_l \rangle. \end{equation} Namely we identify $s$ with $\iota(s)$ in $H^2(Y)$ and write $u = \sum_{l = 1}^\rho u^l T_l$. $F^Y_0$ can be analytically continued across those boundary faces of the K\"ahler cone corresponding to flopping contractions. In the case of conifold transitions $Y \searrow X$, this boundary face is naturally identified as the K\"ahler cone of $X$.
The following convention of indices on $H^{ev}(Y)$ will be used: \begin{itemize} \item Lowercase Greek alphabets for indices from the subspace $\iota (H^{ev}(X))$; \item lowercase Roman alphabets for indices from the subspace spanned by the divisors $T_l$'s and exceptional curves $C_i$'s; \item uppercase Roman alphabets for variables from $H^{ev}(Y)$. \end{itemize} The generating function associated to an extremal curve $C \cong \mathbb{P}^1$ can be derived from the well-known multiple cover formula $$ E_0^C(t) = \sum\nolimits_{d = 1}^\infty n^N_{d} q^{d[C]} e^{d(C.t)} = \sum\nolimits_{d = 1}^\infty \frac{1}{d^3} \,q^{d[C]} e^{d(C.t)} $$ as $N_{C/Y} = \mathscr{O}_{\mathbb{P}^1}(-1)^{\oplus 2}$. Define $$E_0^Y(t) := \frac{1}{3!} t^3 + \sum\nolimits_{i = 1}^k E^{C_i}_0(t) = E_0^Y(u) + \frac{1}{3!}(t^3 - u^3),$$ where $E_0^{C_i}(t) = E_0^{C_i}(u)$ depends only on $u$. Then the degeneration formula is equivalent to the following restriction $$
F_0^X(s) - \frac{s^3}{3!}= \Big(F_0^Y(s + u) - \frac{(s + u)^3}{3!} - E_0^Y(u) + \frac{u^3}{3!} \Big)\Big|_{q^\gamma \mapsto q^{\psi_*(\gamma)}}, $$ where $q^{[C_i]}$'s are subject to the relations induced from the relations among $[C_i]$'s. More precisely, let $A = (a_{ij})$ be the relation matrix and define $$
\mathbf{r}_j (q) := \prod\nolimits_{a_{ij} > 0} q^{a_{ij} [C_i]} - \prod\nolimits_{a_{ij} < 0} q^{-a_{ij} [C_i]}. $$ Then we have \begin{lemma} \label{l:9} $$ F_0^Y(s + u) = \left[ F_0^X(s) + E_0^Y(u) + \frac{1}{3!}((s + u)^3 - s^3 - u^3) \right]_{\mathbf{r}_j(q) =0, \, 1 \leq j \leq \mu}. $$ \end{lemma}
A splitting of variables of $F_0^Y$ would imply that $QH^{ev}(Y)$ decomposes into two blocks. One piece is identified with $QH^{ev}(X)$, and another piece with contributions from the extremal rays. However, \emph{the classical cup product/topological terms spoil the complete splitting.}
The structural coefficients for $QH^{ev}(Y)$ are $C_{PQR} = \partial^3_{PQR} F^Y_0$. We will determine them according to the partial splitting in Lemma \ref{l:9}.
For $F_0^X(s)$, the structural coefficients of quantum product are given by $$ C_{\epsilon \zeta \iota}(s) := \partial^3_{\epsilon \zeta \iota} F^X_0(s) = (\bar T_\epsilon.\bar T_\zeta.\bar T_\iota) + \sum\nolimits_{\beta \ne 0} (\beta.\bar T_\epsilon)(\beta.\bar T_\zeta)(\beta.\bar T_\iota)\, n_\beta^X\, q^\beta e^{(\beta.s)}. $$
Recall that $B = (b_{ip})$ with $b_{ip} = (C_i.T_p)$ is the relation matrix for the vanishing 3-spheres. For $E_0^Y(u)$, the triple derivatives are \begin{equation} \label{extr-inv} \begin{split} C_{lmn}(u) &:= \partial^3_{lmn} E_0^Y(u) \\ &= (T_l.T_m.T_n) + \sum\nolimits_{i = 1}^k \sum\nolimits_{d = 1}^\infty (C_i.T_l) (C_i.T_m) (C_i.T_n)\, q^{d[C_i]} e^{d(C_i.u)} \\ &= (T_l.T_m.T_n) + \sum\nolimits_{i = 1}^k b_{il} b_{im} b_{in} {\bf f}(q^{[C_i]} \exp {\sum\nolimits_{p = 1}^\rho b_{ip}u^p}). \end{split} \end{equation} Here ${\bf f}(q) = \sum_{d \in \mathbb{N}} q^d = \frac{q}{1 - q} = -1 + \frac{-1}{q - 1}$ is the fundamental rational function with a simple pole at $q = 1$ with residue $-1$ (cf.~\cite{LLW1}). We note that due to the existence of cross terms in Lemma \ref{l:9}, $C_{lmn}$'s do not satisfy the WDVV equations.
Denote by $\bar T^\epsilon \in H^4(X)$ the dual basis of $\bar{T}_\epsilon$'s, and write $T^l$, $1 \le l \le \rho$ the dual basis of $T_l$'s. Also $\bar T_0 = T_0 = {\bf 1}$ with dual $\bar T^0 = T^0$ the point class. Since $H^{ev}(Y) = \iota( H^{ev}(X)) \oplus \big(\bigoplus_{l = 1}^\rho \mathbb{Q} T_l \oplus \bigoplus_{l = 1}^\rho \mathbb{Q} T^l \big)$ is an orthogonal decomposition with respect to the Poincar\'e pairing on $H(Y)$, we have four types of structural coefficients \begin{equation*} C_{\epsilon \zeta}^\iota(s) = C_{\epsilon \zeta \iota}(s), \quad C_{lm}^n(u) = C_{lmn}(u), \quad C_{\epsilon m}^n = C_{\epsilon m n}, \quad C_{mn}^\epsilon = C_{\epsilon mn}, \end{equation*} where the last two are constants. If we consider the topological terms $\frac{1}{2} (s^0)^2 s^{0'} + s^0 \sum_{\epsilon} u^{l} u^{l'}$ where we relabel the indices by $u^{l'} = u_l$ and $s^{0'} = s_0$, then a few more non-trivial constants $C_{000'} = 1$, $C_{mn'0} = \delta_{mn}$ are added.
\subsection{The Dubrovin connection and monodromy} \label{s:dubrovin}
The Dubrovin connection on $T H^{ev}(Y)$ is given by $\nabla^z = d - \frac{1}{z} \sum_{P} dt^P \otimes T_P *$. By Corollary \ref{c:2}, it restricts to the Dubrovin connection on $T H^{ev}(X)$. For the complement with basis $T_l$'s and $T^l$'s, we have \begin{equation} \label{dubrovin} \begin{split} z\nabla^z_{\partial_l} T^m &= -\delta_{lm} T^0, \\ z\nabla^z_{\partial_l} T_m &= -\sum\nolimits_{n = 1}^\rho C_{lmn}(u) T^n - \sum\nolimits_{\epsilon} C_{lm \epsilon} \bar T^\epsilon, \\ z\nabla^z_{\partial_\epsilon} T_m &= -\sum\nolimits_{n = 1}^\rho C_{\epsilon mn} T^n. \end{split} \end{equation}
Along $u = \sum_{l = 1}^\rho u^l T_l$ there is no convergence issue by the explicit expression \eqref{extr-inv}. Thus we drop the Novikov variables henceforth.
From \eqref{extr-inv}, the degeneration loci $\mathfrak{D}$ consists of $k$ hyperplanes in $H^2(Y)$: $$ D_i := \{ v_i := \sum\nolimits_{p = 1}^\rho b_{ip} u^p = 0 \}, \quad 1 \le i \le k, $$ which is the K\"ahler degenerating locus at which $C_i$ shrinks to zero volume. There is a monodromy matrix corresponding to $D_i$, whose main nilpotent block $N_{(i)} = (N_{(i), mn}) \in M_{\rho \times \rho}$ is the residue matrix of the connection in \eqref{dubrovin}. The divisor $\mathfrak{D} = \bigcup_{i = 1}^k D_i$ is \emph{not} normal crossing.
\begin{lemma} \label{l:10} In terms of $\{T_n\}$ and dual basis $\{T^n\}$, the block $N_{(i)}$ is given by $$ N_{(i), mn} = \frac{1}{z} b_{im} b_{in}. $$ \end{lemma}
\begin{proof} Since $dv_i = \sum_{l = 1}^\rho b_{il}\, du^l$, we get from \eqref{dubrovin} and \eqref{extr-inv} that $$ N_{(i), mn} = -\frac{1}{z} b_{im} b_{in} \mathop{{\rm Res}}\limits_{v_i = 0} \frac{-1}{e^{v_i} - 1} $$ which gives the result. \end{proof}
\begin{corollary} \label{c:3} In terms of $\{T_n\}$ and dual basis $\{T^n\}$, the nilpotent monodromy at $u = 0$ along $u^l \to 0$ has its main block given by $N_l = \frac{1}{z} B_l^t B_l$, where $B_l$ is obtained from $B$ by setting those $i$-th rows to $0$ if $b_{il} = 0$. \end{corollary}
\begin{proof} This follows from Lemma \ref{l:10}, which can also be proved directly. To determine $N_{l, mn}$ along $u^l \to 0$ at the locus $u = 0$, we compute \begin{equation*} \begin{split} N_{l, mn} = -\frac{1}{z} \sum_{i = 1}^k b_{il} b_{im} b_{in}\, \mathop{{\rm Res}}\limits_{q = 1} \frac{-1}{e^{b_{il} u^l} - 1} = \frac{1}{z} \sum_{b_{il} \ne 0;\, i = 1}^k b_{im} b_{in} = \frac{1}{z} (B_l^t B_l)_{mn}. \end{split} \end{equation*} This proves the result. \end{proof}
\begin{corollary} The Dubrovin connection on $X$ is the monodromy invariant sub-system on $Y$ at $u = 0$. \end{corollary}
\section{Period integrals and Gauss--Manin connections} \label{s:4}
From this section and on, we assume the Calabi--Yau condition: $$K_X \cong \mathscr{O}_X, \qquad H^1(\mathscr{O}_X) =0.$$ Recall that the Kuranishi space $\mathcal{M}_{\bar{X}}$ is smooth. In \S \ref{s:4.1}, we review well known deformation theory of Calabi--Yau 3-folds with ODPs to derive a local Torelli theorem for $\bar{X}$. Identifying $\mathcal{M}_Y$ with equisingular deformations of $\bar X$ in $\mathcal{M}_{\bar{X}}$, we show that periods of vanishing cycles serve as (analytic) coordinates of $\mathcal{M}_{\bar{X}}$ in the directions transversal to $\mathcal{M}_Y$. To study monodromy, the Bryant--Griffiths formulation is reviewed in \S \ref{s:4.2} and the asymptotics of ($\beta$-)periods near $[\bar X]$ is computed in \S \ref{s:4.3}. The monodromy is determined explicitly in terms of the relation matrix $A$ (Corollary~\ref{c:monodromy}). The technical result (Theorem \ref{p:gnot}) is a version of nilpotent orbit theorem with non-SNC boundary, which is also needed in \S\ref{s:6}. Following these discussions, $B(Y)$ is shown to be a sub-theory of $B(X)$ (Corollary \ref{c:sub-sys}).
\subsection{Deformation theory} \label{s:4.1} The main references for this subsection are \cite{yK, RT}, though we follow the latter more closely. Let $\Omega_{\bar{X}}$ be the sheaf of K\"ahler differential and $\Theta_{\bar{X}} := \mathscr{H}{om} (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})$ be its dual. The deformation of $\bar{X}$ is governed by $Ext^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})$. By local to global spectral sequence, we have \begin{equation} \label{e:15}
\begin{split}
0 \to H^1 (\bar{X}, \Theta_{\bar{X}}) &\stackrel{\lambda}{\to} Ext^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}}) \\
&\to H^0 (\bar{X}, \mathscr{E}xt^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})) \stackrel{\kappa}{\to} H^2 (\bar{X}, \Theta_{\bar{X}}).
\end{split} \end{equation} Since $\mathscr{E}xt^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar X})$ is supported at the ordinary double points $p_i$\rq{}s, we have $H^0 (\bar X, \mathscr{E}xt^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}}) )= \bigoplus_{i=1}^k H^0 ( \mathscr{O}_{p_i})$ by a local computation.
We rephrase the deformation theory on $\bar{X}$ in terms of the log deformation on $\tilde{Y}$. Denote by $E \subset \tilde Y$ the union of the exceptional divisors of $\tilde{\psi}: \tilde{Y} \to \bar{X}$.
\begin{lemma} \label{l:11} We have $R \tilde{\psi}_* K_{\tilde{Y}} = \tilde{\psi}_* K_{\tilde{Y}} = K_{\bar{X}}$ and hence $H^0 (K_{\tilde{Y}}) \cong H^0 (K_{\bar{X}} ) \cong \mathbb{C}$. \end{lemma}
\begin{proof} Apply the Serre duality for the projective morphism $\tilde{\psi}$ and we have $ R \tilde{\psi}_* K_{\tilde{Y}} \cong (\tilde{\psi}_* \mathscr{O}_{\tilde{Y}} \otimes K_{\bar{X}})^{\vee}$. Since $\bar X$ is normal rational Gorenstein, we have $\tilde{\psi}_* \mathscr{O}_{\tilde{Y}} \cong \mathscr{O}_{\bar{X}}$. This proves the first equation, from which the first part of the second equation follows. The second part follows from $K_{\bar{X}} \cong \mathscr{O}_{\bar{X}}$. \end{proof}
\begin{lemma} \label{l:12} There is a canonical isomorphism $$ \Omega^2_{\tilde{Y}} (\log E) \cong K_{\tilde{Y}} \otimes \left( \Omega_{\tilde{Y}} (\log E)(-E) \right)^{\vee}. $$ \end{lemma}
\begin{proof} On $\tilde{Y}$, the isomorphism $\Lambda^3 \Omega_{\tilde{Y}}(\log E) \cong \Omega^3_{\tilde Y}(E)$ leads to the perfect pairing $\Omega_{\tilde{Y}} (\log E) \otimes \Omega^2_{\tilde{Y}} (\log E) \to K_{\tilde{Y}} (E)$. Since $\tilde{Y}$ is nonsingular and $E$ is a disjoint union of nonsingular divisors, all sheaves involved are locally free. Hence the lemma follows. \end{proof}
\begin{lemma}[{\cite[Lemma~2.5]{RT}}] \label{l:13} There are canonical isomorphisms $$ L \tilde{\psi}^* \Omega_{\bar{X}} \cong \tilde{\psi}^* \Omega_{\bar{X}} \cong \Omega_{\tilde{Y}} (\log E) (-E), $$ where $L \tilde{\psi}^*$ is the left-derived functor of the pullback map. \end{lemma}
The first isomorphism follows from the facts that $\bar{X}$ is a local complete intersection and an explicit two-term resolution of $\Omega_{\bar{X}}$ exists. We sketch the argument here and refer to \cite{RT} for details. Locally near a node, defined by \eqref{e:A1}, one has an exact sequence $0 \to \mathscr{O} \stackrel{2 \vec{x}}{\longrightarrow} \mathscr{O}^4 \to \Omega \to 0$. Pulling it back to $\tilde{Y}$, we see that $\tilde{\psi}^*(2 \vec{x}) :\mathscr{O} \to \mathscr{O}^4$ is injective on $Y$ and therefore higher left-derived functors are zero.
The second isomorphism is obtained by a local calculation of the blowing-up of an ordinary double point. If $x_1$ is the local equation of the exceptional divisor $E$, explicit computation in \cite{RT} shows that $ \tilde{\psi}^* \Omega_{\bar{X}}$ is locally generated by $dx_1$ and $x_1 d x_i$ for $i \neq 1$, which is exactly $\Omega_{\tilde{Y}} (\log E) (-E)$.
\begin{lemma}[{\cite[Proposition~2.6]{RT}}] \label{l:sheaf} We have $$ R \mathscr{H}om (\Omega_{\bar{X}}, K_{\bar{X}}) \cong R \tilde{\psi}_* \Omega^2_{\tilde{Y}} (\log E). $$ In particular, $Ext^1 (\Omega_{\bar{X}}, K_{\bar{X}}) \cong H^1 ( \Omega^2_{\tilde{Y}} (\log E))$. \end{lemma}
\begin{proof} By Lemma~\ref{l:12}, $$ R \tilde{\psi}_* \Omega^2_{\tilde{Y}} (\log E) \cong R \tilde{\psi}_* \mathscr{H}om (\Omega_{\tilde{Y}} (\log E)(-E), K_{\tilde{Y}}). $$ By Lemma~\ref{l:13} and the projection formula, the RHS is isomorphic to $$ R \mathscr{H}om (\Omega_{\bar{X}}, R \tilde{\psi}_* K_{\tilde{Y}} )
\cong R \mathscr{H}om (\Omega_{\bar{X}}, K_{\bar{X}}) $$ with the last isomorphism coming from $R \tilde{\psi}_* K_{\tilde{Y}} \cong K_{\bar X}$ in Lemma~\ref{l:11}. \end{proof}
From the general deformation theory, the first term $H^1 (\bar{X}, \Theta_{\bar{X}})$ in \eqref{e:15} parameterizes equisingular deformation of $\bar{X}$. Thanks to the theorem of Koll\'ar and Mori \cite{KM} that this extremal contraction deforms in families, this term parameterizes deformations of $Y$. Therefore, the cokernel of $\lambda$ in (\ref{e:15}), or equivalently the kernel of $\kappa$, corresponds to deformation of the singularities. Since the deformation of $\bar{X}$ is unobstructed \cite{yK}, $\operatorname{Def}(\bar{X})$ has the same dimension as $\operatorname{Def}(X)$, which is $h^{2,1}(X)$. Comparing the Hodge number $h^{2,1}$ of $X$ and $\bar{Y}$ (cf.~\S\ref{s:2}) we have the $\dim \ker (\kappa) = \mu$.
\begin{proposition} \label{p:2} The sequence $$ 0 \to H^1 (\bar{X}, \Theta_{\bar{X}}) \stackrel{\lambda}{\to} Ext^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}}) \to V^* \to 0 $$ is exact. \end{proposition}
\begin{proof} The residue exact sequence on $\tilde{Y}$ is $$ 0 \to \Omega_{\tilde{Y}} \to \Omega_{\tilde{Y}} (\log E) \stackrel{\operatorname{res}}{\longrightarrow} \mathscr{O}_E \to 0. $$ Taking wedge product with $\Omega_{\tilde{Y}}$ we get $$0 \to \Omega^2_{\tilde{Y}} \to \Omega^2_{\tilde{Y}} (\log E) \stackrel{\operatorname{res}}{\longrightarrow} \Omega_E \to 0.$$ Part of the cohomological long exact sequence reads \[
H^0(\Omega_E) \to H^1 (\Omega^2_{\tilde{Y}}) \to H^1 (\Omega^2_{\tilde{Y}} (\log E) ) \to H^1 (\Omega_E) \stackrel{\kappa}{\longrightarrow} H^2 (\Omega^2_{\tilde{Y}}) .
\] Since $H^1(E) =0$, the first term vanishes. By Lemma~\ref{l:sheaf}, the third term is equal to $Ext^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})$. Indeed, it is not hard to see that this exact sequence is equal to that in \eqref{e:15} (cf.\ {\cite[(3.2)]{RT}}).
Using similar arguments as in \S\ref{s:2.2.2}, we have \[
0 \to H^1 (\Omega^2_{\tilde{Y}}) \to H^1 (\Omega^2_{\tilde{Y}} (\log E) )
\to \bigoplus\nolimits_{i=1}^k \langle (\ell_i - \ell_i\rq{} ) \rangle
\stackrel{\bar{\kappa}}{\longrightarrow} \frac{H^2 (\Omega^2_{\tilde{Y}})}{\bigoplus_{i=1}^k \langle (\ell_i + \ell_i\rq{} ) \rangle} .
\] From \eqref{e:delta2bar} and Lemma~\ref{l:7} (ii) we have $$H^2(\tilde Y) \stackrel{\bar{\delta}_2}{\longrightarrow} \bigoplus\nolimits_{i = 1}^k \langle (\ell_i - \ell_i') \rangle \to V \to 0.$$ Now by comparing the dual of the maps $\bar{\delta}_2$ and $\bar{\kappa}$, we see that $\ker(\kappa) = \operatorname{cok} (\bar{\delta}_2)^* = V^*$. The proof is complete. \end{proof}
This proposition shows that the deformation of $Y$ naturally embeds to that of $\bar{X}$, with the transversal direction given by the periods of the vanishing cycles. Moreover, the above discussion also leads to important consequences on the infinitesimal period relations on $\tilde{Y}$ and on $\bar{X}$.
\begin{corollary} \label{c:6} On $\tilde Y$, the natural map \[
H^1( \left( \Omega_{\tilde{Y}} (\log E)(-E) \right)^{\vee})
\otimes H^0 ( K_{\tilde{Y}}) \to H^1 (\Omega^2_{\tilde{Y}} (\log E)) \] is an isomorphism. \end{corollary}
\begin{proof} This follows from Lemma~\ref{l:11} and Lemma~\ref{l:12}. \end{proof}
\begin{corollary} \label{c:5} On $\bar X$, the natural map \[
H^1( R \mathscr{H}om (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}}) )
\otimes H^0 ( K_{\bar{X}}) \to
Ext^1 (\Omega_{\bar{X}}, K_{\bar{X}})
\] is an isomorphism. Indeed, both sides are isomorphic to $Ext^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})$. \end{corollary}
\begin{proof} This is a reformulation of Corollary~\ref{c:6} via Lemma \ref{l:sheaf}. \end{proof}
\begin{remark} \label{r:3.8} Since $\bar{X}$ is rational Gorenstein, $R \mathscr{H}om (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})$ has cohomology only in degrees $0$ and $1$. Indeed, $R^0 \mathscr{H}om (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}}) \cong \Theta_{\bar{X}}$ and $$ R^1 \mathscr{H}om (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})
\cong \mathscr{E}xt^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}})
\cong\bigoplus\nolimits_{i=1}^k \mathscr{O}_{p_i}. $$ By a Leray spectral sequence argument, this gives \eqref{e:15} as well and $$ H^1( R \mathscr{H}om (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}}) ) \cong Ext^1 (\Omega_{\bar{X}}, \mathscr{O}_{\bar{X}}). $$ Interpreting Corollary~\ref{c:5} as a \emph{local Torelli} type theorem, we conclude that the differentiation of any non-zero holomorphic sections of the relative canonical bundle on any deformation parameter of $\bar{X}$ is non-vanishing. \end{remark}
\subsection{Vanishing cycles and the Bryant--Griffiths/Yukawa cubic form} \label{s:4.2}
Recall the Gauss--Manin connection $\nabla^{GM}$ on $$\mathscr{H}^n = R^n f_* \mathbb{C} \otimes \mathscr{O}_S \to S$$ for a smooth family $f: \mathcal{X} \to S$ is a flat connection with its flat sections being identified with the local system $R^n f_* \mathbb{C}$. It contains the integral flat sections $R^n f_* \mathbb{Z}$. Let $\{ \delta_p \in H_n (X, \mathbb{Z})/{\rm (torsions)} \}$ be a homology basis for a fixed reference fiber $X = \mathcal{X}_{s_0}$, with cohomology dual basis $\delta_p^*$'s in $H^n (X, \mathbb{Z})$. Then $\delta_p^*$ can be extended to (multi-valued) flat sections in $R^n f_* \mathbb{Z}$. For $\eta \in \Gamma(S, \mathscr{H}^n)$, we may rewrite it in terms of these flat frames with coefficients being the ``multi-valued'' period integrals ``$\int_{\delta_p} \eta$'' as $\eta = \sum_p \delta_p^* \int_{\delta_p} \eta$. For any local coordinate system $(x_j)$ in $S$, since $\nabla^{GM} \delta_p^* = 0$, we get $$
\nabla^{GM}_{\partial/\partial x_j} \eta =
\sum_p \delta_p^* \frac{\partial}{\partial x_j} \int_{\delta_p} \eta. $$ Thus as far as period integrals are concerned, we may simply regard the Gauss--Manin connection as partial derivatives.
When the family contains singular fibers, by embedded resolution of singularities we may assume that the discriminant loci $\mathfrak{D} \subset S$ is a normal crossing divisor. It is well-known that the Gauss--Manin connection has at worst regular singularities along $\mathfrak{D}$ by the regularity theorem. Namely it admits an extension to the boundary with at worst logarithmic poles.
Let $X \nearrow Y$ be a projective conifold transition, and $V$ the corresonding space of vanishing cycles. Since the vanishing spheres $S_i$ have trivial normal bundles in $X$, we see that $(S_i.S_j) = 0$ for all $i, j$, and hence $V$ is isotropic. Define $V'$ to be the subspace dual to $V$ with respect to the intersection pairing in $H_3(X)$, then $V$ and $V'$ are coisotropic. Furthermore, we have $$H_3(X) \cong H_3(Y) \oplus H_3(Y)^{\perp} \cong H_3(Y) \oplus V \oplus V',$$ from (the proof of) Theorem~\ref{t:bes} and Remark~\ref{r:convention}. Let $\{ \gamma_j \}_{j=1}^{\mu}$ be a basis of $V'$ satisfying $$ \operatorname{PD}(\gamma_j)([S_i]) \equiv (\gamma_j.S_i) = a_{ij}, \qquad 1 \le j \le \mu, $$ where $S_i$'s are the vanishing 3-spheres and $A = (a_{ij})$ is the relation matrix of the exceptional curves $C_i$'s. Additionally, let $\{ \Gamma_j \}_{j=1}^{\mu}$ be the basis of $V$ dual to $\{ \gamma_j \}_{j=1}^{\mu}$ via intersection pairing. Namely $(\Gamma_j.\gamma_l) = \delta_{jl}$.
\begin{lemma} \label{l:15} We may construct a symplectic basis of $H_3(X)$: $$ \alpha_0, \alpha_1, \ldots, \alpha_h, \beta_0, \beta_1, \ldots, \beta_h, \qquad (\alpha_j.\beta_p) = \delta_{jp}, $$ where $h = h^{2, 1}(X)$, with $\alpha_j = \Gamma_j$, $1 \le j \le \mu$. \end{lemma}
\begin{proof} Notice that $V \subset H_3(X, \mathbb{Z})$ is generated by $[S_i^3]$'s, and hence is totally isotropic. Let $W \supset V$ be a maximal isotropic subspace (of dimension $h + 1$). We first select $\alpha_j = \Gamma_j$ for $1 \le j \le \mu$ to form a basis of $V$. We then extend it to $\alpha_1, \ldots, \alpha_h$, and set $\alpha_0 \equiv \alpha_{h + 1}$, to form a basis of $W$.
To construct $\beta_l$, we start with any $\delta_l$ such that $(\alpha_p.\delta_l) = \delta_{pl}$. Such $\delta_l$'s exist by the non-degeneracy of the Poincar\'e pairing. We set $\beta_1 = \delta_1$. By induction on $l$, suppose that $\beta_1, \ldots, \beta_l$ have been constructed. We define $$ \beta_{l + 1} = \delta_{l + 1} - \sum\nolimits_{p = 1}^{l} (\delta_{l + 1}.\beta_p) \alpha_p. $$ Then it is clear that $(\beta_{l + 1}.\beta_p) = 0$ for $p = 1, \ldots, l$. \end{proof}
With a choice of basis of $H_3(X)$, any $\eta \in H^3(X, \mathbb{C}) \cong \mathbb{C}^{2(h + 1)}$ is identified with its ``coordinates'' given by the period integrals $\vec\eta = \big(\int_{\alpha_p} \eta, \int_{\beta_p} \eta\big)$. Alternatively, we denote the cohomology dual basis by $\alpha_p^*$ and $\beta_p^*$ so that $\alpha_j^*(\alpha_p) = \delta_{jp} = \beta_j^*(\beta_p)$. Then we may write $$ \eta = \sum\nolimits_{p = 0}^h \alpha_p^* \int_{\alpha_p} \eta + \beta_p^* \int_{\beta_p} \eta. $$ The symplectic basis property implies that $\alpha_p^*(\Gamma) = (\Gamma.\beta_p) $ and $\beta_p^*(\Gamma) = -(\Gamma.\alpha_p) = (\alpha_p.\Gamma)$. This leads to the following observation.
\begin{lemma} \label{abc} For $1 \le j \le \mu$, we may modify $\gamma_j$ by vanishing cycles to get $\gamma_j = \beta_j$. In particular, $(\gamma_j.\gamma_l) = 0$ for $1 \le j, l \le \mu$ and $\alpha_j^*(S_i) = (S_i.\beta_j) = -a_{ij}$. \end{lemma}
\begin{lemma} \label{l:basis} For all $i = 1, \ldots, k$, $\operatorname{PD}([S_i]) = -\sum_{j = 1}^\mu a_{ij}\,\operatorname{PD}(\Gamma_j)$. \end{lemma}
\begin{proof} Comparing both sides by evaluating at $\alpha_l$'s and $\beta_l$'s for all $l$. \end{proof}
Let $\Omega$ be the non-vanishing holomorphic 3-form on the Calabi--Yau threefold. Bryant--Griffiths \cite{BG} showed that the $\alpha$-periods $x_p = \int_{\alpha_p} \Omega$ form the projective coordinates of the image of the period map inside $\mathbb{P}(H^3) \cong \mathbb{P}^{2h + 1}$ as a Legendre sub-manifold of the standard holomorphic contact structure. It follows that there is a holomorphic \emph{pre-potential} $u(x_0, \ldots, x_h)$, which is homogeneous of weight two, such that $u_j \equiv \frac{\partial u}{\partial x_j} = \int_{\beta_j} \Omega$. In fact, \begin{equation} \label{pp:u} u = \tfrac{1}{2} \sum\nolimits_{p = 0}^h x_p u_p = \tfrac{1}{2} \sum\nolimits_{p = 0}^h x_p \int_{\beta_p} \Omega. \end{equation} Hence $\Omega = \sum_{p = 0}^h (x_p\,\alpha_p^* + u_p\, \beta_p^*)$. In particular, $$ \partial_j \Omega = \alpha_j^* + \sum\nolimits_{p = 0}^h u_{jp}\, \beta_p^*, \qquad \partial^2_{jl} \Omega = \sum\nolimits_{p = 0}^h u_{jlp} \, \beta_p^*. $$ By the Griffiths transversality, $\partial_j \Omega \in F^2$, $\partial^2_{jl} \Omega \in F^1$. Hence we have the \emph{Bryant--Griffiths cubic form}, which is homogeneous of weight $-1$: $$ u_{jlm} = (\partial_m \Omega.\partial^2_{jl} \Omega) = \partial_m (\Omega.\partial^2_{jl}\Omega) - (\Omega.\partial^3_{jlm} \Omega) = -(\Omega.\partial^3_{jlm} \Omega). $$ This is also known as \emph{Yukawa coupling} in the physics literature.
For inhomogeneous coordinates $z_i = x_i/x_0$, the corresponding formulae may be deduced from the homogeneous ones by noticing that $\partial^I u$ is homogeneous of weight $2 - |I|$ for any multi-index $I$.
Under a suitable choice of the holomorphic frames respecting the Hodge filtration, \emph{the Bryant--Griffiths--Yukawa couplings determine the VHS as the structural coefficients of the Gauss--Manin connection}:
\begin{proposition} \label{BGY} Let $\tau_0 = \Omega \in F^3$, $\tau_j = \partial_j \Omega \in F^2$, $\tau^j = \beta_j^* - (x_j/x_0) \beta_0^* \in F^1$ for $1 \le j \le h$, and $\tau^0 = \beta_0^* \in F^0$. Then for $1 \le p, j \le h$, \begin{equation} \begin{split} \nabla_{\partial_p} \tau_0 &= \tau_p, \\ \nabla_{\partial_p} \tau_j &= \sum\nolimits_{m = 1}^h u_{pjm}\, \tau^m, \\ \nabla_{\partial_p} \tau^j &= \delta_{pj}\,\tau^0, \\ \nabla_{\partial_p} \tau^0 &= 0. \end{split} \end{equation} \end{proposition}
\begin{proof} We prove the second formula. Since $u_{pj}$ has weight 0, we have the Euler relation $x_0\, u_{pj0} + \sum_{m = 1}^h x_m\,u_{pjm} = 0$. Hence \begin{equation*} \begin{split} \partial_p \partial_j \Omega &= \sum\nolimits_{m = 1}^h u_{pjm} \,\beta_m^* + u_{pj0} \,\beta_0^* \\ &= \sum\nolimits_{m = 1}^h u_{pjm} \Big(\beta_m^* - \frac{x_m}{x_0} \beta_0^*\Big) = \sum\nolimits_{m = 1}^h u_{pjm} \,\tau^m. \end{split} \end{equation*} It remains to show that $\tau^j \in F^1$. By the first Hodge--Riemann bilinear relations, namely $F^1 = (F^3)^\perp$ and $F^2 = (F^2)^\perp$ in our case, it is equivalent to showing that $\tau^j \in (F^3)^\perp$. This follows from \begin{equation*} \begin{split} (\tau^j, \Omega) = \Big(\beta_j^* - \frac{x_j}{x_0} \beta_0^*, \sum\nolimits_{p = 0}^h (x_p \alpha_p^* + u_p \beta_p^*)\Big) = -x_j + \frac{x_j}{x_0} x_0 = 0. \end{split} \end{equation*} The remaining statements are clear. \end{proof}
\subsection{Degenerations via Picard--Lefschetz and the nilpotent orbit theorem} \label{s:4.3}
Let $\mathcal{X} \to \Delta$ be a one parameter conifold degeneration of threefolds with nonsingular total space $\mathcal{X}$. Let $S_1, \ldots, S_k$ be the vanishing spheres of the degeneration.. The \emph{Picard--Lefschetz} formula (see e.g., \cite[\S 3.B]{eL}) asserts that the monodromy transformation $T: H^3(X) \to H^3(X)$ is given by \begin{equation} \label{PLT} T \sigma = \sigma + \sum\nolimits_{i = 1}^k \sigma([S_i])\, {\rm PD}([S_i]), \end{equation} where $\sigma \in H^3(X)$. It is \emph{unipotent}, with associated \emph{nilpotent} monodromy $$N := \log T = \sum\nolimits_{m = 1}^\infty (T - I)^m/m.$$ We have seen that $(S_i.S_j) = 0$ for all $i, j$. Therefore $T = I + N$ and $N^2 = 0$ (cf.~\S \ref{s:2}). The \emph{main purpose} here is to generalize these to multi-dimensional degenerations, and in particular to the local moduli $\mathcal{M}_{\bar X}$ near $[\bar X]$.
\subsubsection{VHS with simple normal crossing boundaries} \label{SNC} Even though the discriminant loci for the conifold degenerations are in general not simple normal crossing (SNC) divisors, by embedded resolution of singularity they can in principle be modified to become ones. We will begin our discussion in this case for simplicity.
Let $$\mathcal{X} \to \mathbf{\Delta} := \Delta^{\nu} \times \Delta^{\nu'} \ni {\bf t} := (t, s)$$ be a flat family of Calabi--Yau 3-folds such that $X_{\bf t}$ is smooth for $${\bf t} \in \mathbf{\Delta}^* := (\Delta^\times)^{\nu} \times \Delta^{\nu'}.$$ Namely, the discriminant locus is a SNC divisor: $$\mathfrak{D} := \bigcup\nolimits_{j=1}^{\nu} Z( t_j) = \mathbf{\Delta} \setminus \mathbf{\Delta}^*.$$ Around each punctured disk $t_j \in \Delta^\times$, $1\le j \le \nu$, we assume the monodromy $T_j$ is unipotent with nilpotent $N_j$. Note that $N_j N_l = N_l N_j$ since $\pi_1(\mathbf{\Delta}^*) \cong \mathbb{Z}^{\nu}$ is abelian.
If for any ${\bf t} = (t, s)$ we assume that $X_{{\bf t}}$ acquires at most canonical singularities, then $N_j F^3_\infty|_{D_j} = 0$ and $N_j^2 = 0$ for each $j$ (cf.~Remark \ref{r:can}). Different $N_j$ may define different weight filtration $W_j$ and each boundary divisor $Z(t_j)$ corresponds to different set of vanishing cycles. In our case, the structure turns out to be simple. For any $n_j \in \Bbb N$, $1 \le j \le \nu$, the degeneration along the curve $$\gamma(w) := (t (w), s(w)) = (w^{n_1}, \ldots, w^{n_{\nu}}, s_0)$$ has monodromy $$ N_\gamma = \log T_\gamma = \log \prod\nolimits_{j = 1}^\nu T_j^{n_j} = \sum\nolimits_{j = 1}^\nu n_j N_j. $$ Hence $N_\gamma^2 = 0$ for any $(n_1, \ldots, n_\nu) \in \Bbb N^\nu$. That is, $N_{j} N_{l} = 0$ for all $j, l$.
For conifold degenerations, this is clear from the Picard--Lefschetz formula \eqref{PLT}. Indeed $(S_{i_1}.S_{i_2}) = 0$ for all $i_1, i_2$ implies $N_j N_ l = 0$ for all $j, l$.
Let $z_j = \log t_j / 2\pi \sqrt{-1} \in \mathbb{H}$ (the upper half plane), $\mathbf{z} N :=\sum\nolimits_{j = 1}^{\nu} z_j N_j$, and let $\Omega$ denote (the class of) a relative Calabi--Yau 3-form over $\mathbf{\Delta}$, i.e.~a section of $F^3$. By Schmid's nilpotent orbit theorem \cite{Schmid} (cf.~\cite{cW0, cW1}), a natural choice of $\Omega$ takes the form \begin{equation} \label{Omega} \begin{split} \Omega({\bf t}) &= e^{\mathbf{z} N} {\bf a}({\bf t}) = e^{\mathbf{z} N} \Big(a_0(s) + \sum\nolimits_{j = 1}^{\nu} a_j (s) t_j + \cdots \Big) \\ &= {\bf a}({\bf t}) + \mathbf{z} N {\bf a}({\bf t}) \in F^3_{\bf t}, \end{split} \end{equation} where ${\bf a}({\bf t})$ is holomorphic, $N_j a_0(s) = 0$ for all $j$.
In order to extend the theory of Bryant--Griffiths to include the boundary points of the period map, namely to include ODP degenerations in the current case, we need to answer the question if the $\alpha$-periods $ \theta_j({\bf t}) := \int_{\Gamma_j} \Omega({\bf t})$ may be used to replace the degeneration parameters $t_j$ for $1\le j \le \nu$. For this purpose we need to work on the local moduli space $\mathcal{M}_{\bar X}$.
\subsubsection{Extending Yukawa coupling towards non-SNC boundary} \label{s:4.4}
As in \S\ref{s:4.1}, $\bar X$ has unobstructed deformations and $\mathcal{M}_{\bar X} = \operatorname{Def} (\bar{X})$ is smooth. Since $\bar X$ admits a smoothing to $X$, $\dim \mathcal{M}_{\bar X}$ is exactly $h = h^{2, 1}(X)$. The discriminant loci $\mathfrak{D} \subset \mathcal{M}_{\bar X}$ is in general not a SNC divisor. Comparing with the local $A$ model picture on $Y/X$ in \S\ref{s:dubrovin}, the discriminant loci $\mathfrak{D}$ is expected to the union of $k$ hyperplanes. (We intentionally use the same notation $\mathfrak{D}$.)
Recall Friedman's result \cite{rF} on partial smoothing of ODPs. Let $A = [A^1, \ldots, A^\mu]$ be the relation matrix. For any $r \in \mathbb{C}^\mu$, the relation vector $A(r) := \sum_{l = 1}^\mu r_l A^l$ gives rise to a (germ of) partial smoothing of those ODP's $p_i \in \bar X$ with $A(r)_{i} \ne 0$. Thus for $1 \le i \le k$, the linear equation \begin{equation} \label{coor-w} w_i := a_{i1} r_1 + \cdots + a_{i\mu} r_\mu = 0 \end{equation} defines a hyperplane $Z(w_i)$ in $\mathbb{C}^\mu$.
The small resolution $\psi: Y \to \bar X$ leads to an embedding $\mathcal{M}_Y \subset \mathcal{M}_{\bar X}$ of codimension $\mu$. As germs of analytic spaces we thus have $\mathcal{M}_{\bar X} \cong \Delta^{\mu} \times \mathcal{M}_Y \ni (r, s)$. Along each hyperplane $D^i := Z(w_i)_{\Delta^{\mu}} \times \mathcal{M}_Y$, there is a monodromy operator $T^{(i)}$ with associated nilpotent monodromy $N^{(i)} = \log T^{(i)}$. A degeneration from $X$ to $X_i$ with $[X_i] \in D^i$ a general point (not in any $D^{i'}$ with $i' \ne i$) contains only one vanishing cycle $[S^3_i] \mapsto p_i$. We summarize the above discussion in the following lemma.
\begin{lemma} \label{P-L} Geometrically a point $(r,s) \in D^i$ corresponds to a partial smoothing $X_r$ of $\bar X$ for which the $i$-th ordinary double point $p_i$ remains singular. Hence, for $r$ generic, the degeneration from $X$ to $X_r$ has only one vanishing sphere $S^3_i$. Moreover, the Picard--Lefschetz formula \eqref{PLT} says that for any $\sigma \in H^3(X)$, $$ N^{(i)} \sigma = (\sigma([S^3_i])) \operatorname{PD}([S^3_i]). $$ \end{lemma}
Even though the embedded resolution brings he discriminant locus to a SNC divisor, some information might be lost in this process. Therefore we choose to analyze the period map directly by way of the following nilpotent orbit theorem. We call the configuration $\mathfrak{D} = \bigcup_{i = 1}^k D^i \subset \mathcal{M}_{\bar X}$ a \emph{central hyperplane arrangement with axis} $\mathcal{M}_Y$ following the usual convention.
\begin{theorem} \label{p:gnot} Consider a degeneration of Hodge structures over $\Delta^\mu \times M$ with discriminant locus $\mathfrak{D}$ being a central hyperplane arrangement with axis $M$. Let $T^{(i)}$ be the monodromy around the hyperplane $Z(w_i)$ with quasi-unipotency $m_i$, $N^{(i)} := \log ((T^{(i)})^{m_i})/{m_i}$, and suppose that the monodromy group $\Gamma$ generated by $T^{(i)}$'s is \emph{abelian}. Let $\Bbb D$ denote the period domain and $\check{\Bbb D}$ its compact dual. Then the period map $\phi: \Delta^\mu \times M \setminus \mathfrak{D} \to \Bbb D/\Gamma$ takes the following form $$ \phi(r, s) = \exp \left(\sum_{i = 1}^k \frac{m_i\log w_i}{2\pi \sqrt{-1}} N^{(i)}\right) \psi(r, s), $$ where $\psi: \Delta^\mu \times M \to \check{\Bbb D}$ is holomorphic and horizontal. \end{theorem}
\begin{proof} We prove the theorem by induction on $\mu \in \Bbb N$. The case $\mu = 1$ is essentially the one variable case (or SNC case) of the nilpotent orbit theorem. The remaining proof consists of a careful bookkeeping on Schmid's derivation of the multi-variable nilpotent orbit theorem from the one variable case (cf.~\cite[\S 8]{Schmid}, especially Lemma~(8.34) and Corollary~(8.35)).
The essential statement is the holomorphic extension of \begin{equation} \label{hol.ext} \psi(r, s) := \exp \left(-\sum_{i = 1}^k \frac{m_i\log w_i}{2\pi \sqrt{-1}} N^{(i)}\right) \phi(r, s) \in \check{\Bbb D} \end{equation} over the locus $\mathfrak{D}$. For $p \not\in \{0\} \times M$, we can find a neighborhood $U_p$ of $p$ so that the holomorphic extension to $U_p$ is achieved by induction. Notice that the commutativity of $N^{(i)}$'s is needed in order to arrange $\psi(r, s)$ into the form \eqref{hol.ext} with smaller $\mu$. Namely, $$ \psi = \exp \left(-\sum_{w_i(p) = 0} \frac{m_i\log w_i}{2\pi \sqrt{-1}} N^{(i)}\right) \left[\exp \left(-\sum_{w_i(p) \ne 0} \frac{m_i\log w_i}{2\pi \sqrt{-1}} N^{(i)}\right)\phi \right]. $$
Let $R_{\ge 1/2} := \{\,(r, s)\mid |r| \ge \tfrac{1}{2} \,\}$. Then we have a unique holomorphic extension of $\psi$ over $R_{\ge 1/2}$. By the Hartog's extension theorem we get the holomorphic extension to the whole space $\Delta^\mu \times M$. The statement on horizontality follows from the same argument in \cite[\S 8]{Schmid}. \end{proof}
\begin{remark} \label{r:abelian} (i) Let $\mathfrak{D} = \bigcup_{i = 1}^k D^i \subset \Bbb C^\mu$ be a central hyperplane arrangement with axis $0$. Then $\Bbb C^\mu \setminus \mathfrak{D}$ can be realized as $(\Bbb C^\times)^k \cap L$ for $L \subset \Bbb C^k$ being a $\mu$ dimensional subspace. Since $\pi_1((\Bbb C^\times)^k) \cong \Bbb Z^k$, a hyperplane theorem argument shows that $\pi_1(\Bbb C^\mu \setminus \mathfrak{D}) \cong \Bbb Z^k$, hence abelian, if $\mu \ge 3$. However, for $\mu = 2$, $\pi_1(\Bbb C^2 \setminus \mathfrak{D})$ is \emph{not abelian} if $k \ge 3$. Indeed, the natural $\Bbb C^\times$ fibration $\Bbb C^2 \setminus \bigcup_{i = 1}^k D^i \to \Bbb P^1 \setminus \{p_1, \ldots, p_k\}$ leads to $$ 0 \to \pi_1(\Bbb C^\times) \cong \Bbb Z \to \pi_1(\Bbb C^2 \setminus \bigcup D^i) \to \Bbb Z^{*(k - 1)} \to 0, $$ where the RHS is a $k - 1$ free product of $\Bbb Z$.
(ii) Theorem~\ref{p:gnot} is applicable to the conifold transitions since the monodromy representation is abelian and $m_i = 1$ for all $i$. This follows from the Picard--Lefschetz formula \eqref{PLT} and the fact $[S_i].[S_{i'}]=0$ for all vanishing spheres. \end{remark}
\begin{proposition} \label{p:3} There is a holomorphic coordinate system $(r, s) \in \Bbb C^h$ in a neighborhood of $[\bar X] \in \mathcal{M}_{\bar X}$ such that $s \in \Bbb C^{h - \mu}$ is a coordinate system of $\mathcal{M}_Y$ near $[\bar X]$ and $r_j = \int_{\Gamma_j} \Omega$, $1 \le j \le \mu$, are the $\alpha$-periods of the vanishing cycles. Moreover, the section $\Omega(r, s)$ takes the form $$ \Omega = a_0(s) + \sum_{j = 1}^\mu \Gamma_j^* r_j + {\rm h.o.t.} - \sum_{i = 1}^k \frac{w_i \log w_i}{2\pi \sqrt{-1}} \operatorname{PD}([S_i]). $$ Here {h.o.t.}~denotes terms in $V^\perp$ which are at least quadratic in $r_1, \ldots, r_\mu$, and $w_i = a_{i1} r_1 + \cdots + a_{r\mu} r_\mu = \int_{S_i} \Omega$ defines the discriminant locus $D^i$ for $1 \le i \le k$. \end{proposition}
\begin{proof} By Theorem \ref{p:gnot} and the fact $N^{(i_1)} N^{(i_2)} = 0$, we may write \begin{equation} \label{e:Omega} \begin{split} \Omega(r, s) &= \exp \left(\sum_{i = 1}^k \frac{\log w_i}{2\pi \sqrt{-1}} N^{(i)}\right) {\bf a}(r, s) \\ &= {\bf a}(r, s) + \sum_{i = 1}^k \frac{\log w_i}{2\pi \sqrt{-1}} N^{(i)} {\bf a}(r, s) \in F^3_{(r, s)},\end{split} \end{equation} where ${\bf a}(r, s) = a_0(s) + \sum_{j = 1}^\mu a_j(s)\, r_j + O(r^2)$ is holomorphic in $r, s$.
By Lemma \ref{P-L}, all $\alpha$ periods $\theta_l := \int_{\alpha_l} \Omega$ vanish on the logarithmic terms in \eqref{e:Omega}. In particular, $\theta_l(r, s)$'s are single-valued functions. By Corollary \ref{c:5} and Remark~\ref{r:3.8} (the local Torelli property), the $h \times h$ matrix $$ \big(\partial_m \theta_l\big)_{l, m = 1}^h = \Big(\int_{\alpha_l} \partial_m\Omega \Big) $$ is invertible for small $r$. Moreover, along $r = 0$, the off-diagonal block with $1 \le l \le \mu$ (i.e.~with $\alpha_l = \Gamma_l$ being the vanishing cycles) and $\mu + 1 \le m \le h$ (i.e.~with differentiation in the $s$ direction) vanishes. Hence the first $\mu \times \mu$ block $$ \big(\partial_j \theta_l \big)_{l, j = 1}^\mu = \Big(\int_{\Gamma_l} \partial_j \Omega \Big) $$ is also invertible for small $r$. Thus, by the inverse function theorem, $\theta_1, \ldots, \theta_\mu$ and $s$ form a coordinate system near $[\bar X] \in \mathcal{M}_{\bar X}$.
Now we replace $r_j$ by the $\alpha$-period $\theta_j$ for $j = 1, \ldots, \mu$. In order for Theorem~\ref{p:gnot} to be applicable, we need to justify that the discriminant locus $D^i$ is still defined by linear equations in $r_j$'s. This follows from Lemma~\ref{l:basis}: \begin{equation*} \begin{split} \int_{S_i} \Omega = (\Omega, {\rm PD}([S_i])) &= -\sum\nolimits_{j = 1}^\mu a_{ij}(\Omega, {\rm PD}(\Gamma_j)) \\ &= -\sum\nolimits_{j = 1}^\mu a_{ij} r_j =: -w_j. \end{split} \end{equation*}
Denote by ${\rm h.o.t}$ be terms in $V^\perp$ which are at least quadratic in $r_j$'s. The above choice of coordinates implies that $$ \Omega = a_0(s) + \sum_{j = 1}^\mu \Gamma_j^* r_j + {\rm h.o.t.} + \sum_{i = 1}^k \sum_{j = 1}^\mu \frac{\log w_i}{2\pi \sqrt{-1}} N^{(i)} \Gamma_j^* r_j. $$ Then $$ \sum\nolimits_{j = 1}^\mu N^{(i)} \Gamma_j^* r_j = -\sum\nolimits_{j = 1}^\mu a_{ij} r_j \operatorname{PD}([S_i]) = -w_i \operatorname{PD}([S_i]) $$ by Lemma \ref{P-L} and Lemma \ref{abc}. The proof is complete. \end{proof}
Consequently one obtains the asymptotic forms of $\beta$-periods and Bryant--Griffiths form in terms of the above coordinate system $(r, s)$. For $\beta$-periods \[ u_p(r, s) = \int_{\beta_p} \Omega = u_p(s) + {\rm h.o.t.} - \sum_{i = 1}^k \frac{w_i \log w_i}{2\pi \sqrt{-1}} \int_{\beta_p} \operatorname{PD}([S_i]) \] since $\Omega(s) = a_0(s)$. Thus \[ \begin{split}
u_p(r, s) &= u_p(s) +\sum_{i = 1}^k \frac{w_i \log w_i}{2\pi \sqrt{-1}} a_{ip} + {\rm h.o.t.} \quad \text{for $1 \le p \le \mu$} \\
u_p(r, s) &= u_p(s) + {\rm h.o.t.} \quad \text{for } p > \mu. \end{split} \] The Bryant--Griffiths form is then obtained by taking two more derivatives. For $1 \le p, m, n \le \mu$, we get $$ u_{pm} = O(r) + \sum_{i = 1}^k \frac{\log w_i + 1}{2\pi \sqrt{-1}} a_{ip} a_{im} $$ and \begin{equation} \label{Yukawa} u_{pmn} = O(1) + \sum_{i = 1}^k \frac{1}{2\pi \sqrt{-1}} \frac{1}{w_i} a_{ip} a_{im} a_{in}. \end{equation}
\begin{remark} The specific logarithmic function in Proposition \ref{p:3}, which is written in terms of linear combinations of $\alpha$-periods, had appeared in the literature in examples, such as those studied in \cite[p.89]{CGGK} where there are 16 vanishing spheres with a single relation. To our knowledge, it has not been studied in this generality. \end{remark}
\subsubsection{Monodromy calculations} \label{s:monodromy}
As a simple consequence, we determine the monodromy $N(l)$ towards the coordinate hyperplane $Z(r_l)$ at $r = 0$. That is the monodromy associated to the one parameter degeneration $\gamma(r)$ along the $r_l$-coordinate axis ($r_l \in \Delta$ and $r_{j} = 0$ if $j \ne l$). Let $I_l = \{i\mid a_{il} \ne 0\}$ and let $A_l$ be the matrix from $A$ by setting the $i$-th rows with $i \not\in I_l$ to $0$.
\begin{lemma} \label{v-sph} The sphere $S^3_i$ vanishes in $Z(r_l)$ along transversal one parameter degenerations $\gamma$ if and only if $i \in I_l$, i.e., $a_{il} \ne 0$. \end{lemma}
\begin{proof} The curve $\gamma$ lies in $D^i = Z(w_i)$ if and only if $a_{il} = 0$. Thus for those $i \not\in I_l$, the ODP $p_i$ is always present on $X_{\gamma(r)}$ along the curve $\gamma$. In particular the vanishing spheres along $\gamma$ are precisely those $S_i$ with $i \in I_l$. \end{proof}
To calculate the monodromy $N(l)$, recall that (cf.~Lemma \ref{abc}) $\Gamma_j^* \equiv \alpha_j^* = -\operatorname{PD}(\beta_j)$. The Picard--Lefschetz formula (Lemma \ref{P-L}) then says that $$ N(l) \Gamma_j^* = \sum\nolimits_{i \in I_l} (\Gamma_j^*.\operatorname{PD}([S_i]))\operatorname{PD}([S_i]) = -\sum\nolimits_{i \in I_l} a_{ij} \operatorname{PD}([S_i]). $$
\begin{corollary} \label{c:monodromy} For $1 \le p \le \mu$, $$\int_{\beta_p} N(l)\Gamma_j^* = -\sum_{i \in I_l} a_{ij} (S_i.\beta_p) = \sum_{i \in I_l} a_{ij} a_{ip} = (A_l^t A_l)_{jp},$$ while for $p = 0$ or $\mu + 1 \le p \le h$ we have $\int_{\beta_p} N(l) \Gamma_j^* = 0$. \end{corollary}
\begin{corollary} \label{c:sub-sys} The $B(Y)$ is a sub-theory of $B(X)$ by setting $r=0$ and taking the monodromy invariant sub-system. In fact $a_0(s)$ represents the family of Calabi--Yau 3-forms $\Omega(s)$ over $\mathcal{M}_Y$ and the $\alpha$, $\beta$ periods along it gives the VHS on $Y$. \end{corollary}
\subsubsection{On topological logarithmic Gauss--Manin connection} We study the \emph{topological} logarithmic Gauss--Manin connection associated to our conifold degenerations. That is, we seek a topological frame of the bundle $R^3 \pi_* \mathbb{C}$ of a local family $\pi: \mathcal{X} \to \mathcal{M}_{\bar X}$ near the Calabi--Yau conifold $[\bar X]$. By Lemma \ref{l:7} and the Hodge diamond \eqref{H^3(Y)}, part of the frame comes naturally from $H^3(Y)$, while the remaining part is modeled on $V^*$ and $V$. By the same procedure as in the proof of Proposition~\ref{p:3}, the topological frame modeled on $V^* \cong H^{2, 2}_\infty H^3$ can be chosen to be \begin{equation} \label{frame:tau} \begin{split} v_j &:= \exp \left(\sum_{i = 1}^k \frac{\log w_i}{2\pi \sqrt{-1}} N^{(i)} \right) \Gamma_j^* \\ &= \Gamma_J^* + \sum_{i = 1}^k \frac{\log w_i}{2\pi \sqrt{-1}} N^{(i)} \Gamma_j^* = \Gamma_j^* - \sum_{i = 1}^k \frac{\log w_i}{2\pi \sqrt{-1}} a_{ij} \operatorname{PD}([S_i]) \end{split} \end{equation} for $1 \le j \le \mu$. Notice that the correction terms lie in the lower weight piece $H^{1, 1}_\infty H^3$ and $v_j$ is independent of $s$. Moreover, $v_j$ is singular along $D^i$ if and only if $a_{ij} \ne 0$, i.e., $S_i$ vanishes in $Z(r^j)$ by Lemma \ref{v-sph}.
On $V \cong H^{1, 1}_\infty H^3$, we choose the (constant) frame by \begin{equation} \label{frame:tau2} v^j := \exp \left(\sum_{i = 1}^k \frac{\log w_i}{2\pi \sqrt{-1}} N^{(i)} \right) \operatorname{PD}(\Gamma_j) = \operatorname{PD}(\Gamma_j), \qquad 1 \le j \le \mu. \end{equation}
From \eqref{coor-w}, \eqref{frame:tau} and Lemma \ref{l:basis}, it is easy to determine the Gauss--Manin connection on this partial frame in the special directions $\partial/\partial r_p$'s: \begin{equation} \label{GM-P} \begin{split} \nabla^{GM}_{\partial/\partial r_p} v_m &= \frac{1}{2\pi \sqrt{-1}} \sum_{i = 1}^k \frac{a_{ip}}{w_i} \Big( - a_{im} \operatorname{PD}([S_i]) \Big) \\ &= \frac{1}{2\pi \sqrt{-1}} \sum_{i = 1}^k \sum_{n = 1}^\mu \frac{a_{ip} a_{im} a_{in}}{w_i} \,v^n. \end{split} \end{equation}
\begin{proposition} \label{p:constP} Near $[\bar X] \in \mathcal{M}_{\bar X}$, $\nabla^{GM}$ is regular singular along $D^i$'s and smooth elsewhere. The connection matrix $P$ on the block $V^* \oplus V$ takes the form $$ P = \sum_{i = 1}^k \frac{dw_i}{w_i} \otimes P^i = \sum_{i = 1}^k \frac{dw_i}{w_i} \otimes \sum_{m, n = 1}^\mu a_{im} a_{in}\, v^n \otimes (v_m)^* $$ where $P_i$ is a constant matrix in the topological frame $v_m$'s and $v^n$'s. \end{proposition}
Note that there are no higher order terms in $r_j$'s and $\nabla^{GM}$ is block-diagonalized, in contrast to results in \eqref{Yukawa} and the discussions in \S\ref{s:6} where \emph{holomorphic frames} are considered.
\section{Local transitions between $A(Y)$ and $B(X)$} \label{s:local_transitions}
The basic exact sequence in Theorem \ref{t:bes} provides a Hodge theoretic realization of the numerical identity $\mu + \rho = k$.
Now $H^2(Y)/H^2(X) \otimes \mathbb{C} \cong \mathbb{C}^\rho$ is naturally the parameter space of the extremal Gromov--Witten invariants of the K\"ahler degeneration $\psi: Y \to \bar X$, and $V^* \otimes \mathbb{C} \cong \mathbb{C}^\mu$ is naturally the parameter space of periods of vanishing cycles of the complex degeneration from $X$ to $\bar X$. Both of them are equipped with flat connections induced from the Dubrovin and Gauss--Manin connections respectively. Thus it is natural to ask if there is a \emph{$\mathcal{D}$ module lift of the basic exact sequence}.
We rewrite the basic exact sequence in the form \begin{equation*} \xymatrix{H^2_{\mathbb{C}}(Y)/H^2_{\mathbb{C}}(X) \cong \mathbb{C}^\rho \ar[r]^>>>>B & \mathbb{C}^k & V^*_{\mathbb{C}} \cong \mathbb{C}^\mu \ar[l]_<<<<A} \end{equation*} with $A^t B = 0$. This simply means that $\mathbb{C}^k$ is an orthogonal direct sum of the two subspaces ${\rm im}(A)$ and ${\rm im}(B)$. Let $A = [A^1, \ldots, A^\mu]$, $B = [B^1, \ldots, B^\rho]$, and consider the invertible matrix $S = (s^i_{j}) := [A, B] \in M_{k \times k}(\mathbb{Z})$, namely $s^i_j = a_{ij}$ for $1 \le j \le \mu$ and $s^i_{\mu + j} = b_{ij}$ for $1 \le j \le \rho$.
Denote the standard basis of $\mathbb{C}^k$ by $e_1, \ldots, e_k$ with dual coordinates $y_1, \ldots, y_k$. Let $e^1, \ldots, e^k$ be the dual basis on $(\mathbb{C}^k)^\vee$. We consider the standard (trivial) logarithmic connection on the bundle $\underline{\mathbb{C}}^k \oplus (\underline{\mathbb{C}}^k)^\vee$ over $\mathbb{C}^k$ defined by \begin{equation} \label{Ck-log} \nabla = d + \frac{1}{z} \sum_{i = 1}^k \frac{d y_i}{y_i} \otimes (e^i \otimes e_i^*), \end{equation} where $z$ is a parameter. It is a direct sum of $k$ copies of its one dimensional version. We will show that the principal (logarithmic) part of the Dubrovin connection over $\mathbb{C}^\rho$ (cf.~\eqref{extr-inv}) as well as the Gauss--Manin connection on $\mathbb{C}^\mu$ (cf.~\eqref{Yukawa}) are all induced from this standard logarithmic connection through the embeddings defined by $B$ and $A$ respectively.
Recall the basis $T_1, \ldots, T_\rho$ of $\mathbb{C}^\rho$ with coordinates $u^1, \ldots, u^\rho$, and the frame $T_1, \ldots, T_\rho, T^1, \ldots, T^\rho$ on the bundle $\underline{\mathbb{C}}^\rho \oplus (\underline{\mathbb{C}}^\rho)^\vee$ over $\mathbb{C}^\rho$. Notice that $T_j$ corresponds to the column vector $B^j = S^{\mu + j}$, $1 \le j \le \rho$. Let $\hat T_j$ correspond to the column vector $A^j = S^j$ for $1 \le j \le \mu$ with dual $\hat T^j$'s. Then $$ T_j = \sum\nolimits_{i = 1}^k b_{ij}\, e_i = \sum\nolimits_{i = 1}^k s^i_{\mu + j}\, e_i, $$ and dually $$e^i = \sum\nolimits_{j = 1}^\mu s^i_j \,\hat T^j + \sum\nolimits_{j = 1}^{\rho} s^i_{\mu + j}\, T^j = \sum\nolimits_{j = 1}^\mu a_{ij}\, \hat T^j + \sum\nolimits_{j = 1}^{\rho} b_{ij}\, T^j.$$
Denote by $P$ the orthogonal projection $$P: \underline{\mathbb{C}}^k \oplus (\underline{\mathbb{C}}^k)^\vee \to \underline{\mathbb{C}}^\rho \oplus (\underline{\mathbb{C}}^\rho)^\vee .$$ Using \eqref{Ck-log} we compute the induced connection $\nabla^P$ near $\vec 0 \in \mathbb{C}^\rho$: \begin{equation} \label{e:7.3} \begin{split} \nabla^P_{T_l} T_m &= \sum\nolimits_{i,\, i' = 1}^k b_{il} b_{i' m} \big(\nabla_{e_i} e_{i'} \big)^P \\ &= \frac{1}{z} \sum_{i = 1}^k \frac{b_{il} b_{im}}{y_i}\, (e^i)^P = \frac{1}{z} \sum_{n = 1}^\rho \sum_{i = 1}^k \frac{b_{il} b_{im} b_{in}}{y_i}\, T^n. \end{split} \end{equation} We compare it with the one obtained in \eqref{extr-inv} and \eqref{dubrovin}: $$ \nabla^z_{T_l} T_m = -\frac{1}{z} \sum_{n = 1}^\rho \left( (T_l.T_m.T_n) + \sum_{i = 1}^k b_{il} b_{im} b_{in} \frac{q_i}{1 - q_i}\right) T^n, $$ where $$q_i = \exp \sum_{p = 1}^\rho b_{ip} u^p = \exp v_i .$$ The principal part near $u_i = 0$, $1 \le i \le \rho$, gives $$\frac{1}{z} \sum_{n = 1}^\rho \sum_{i = 1}^k \frac{b_{il} b_{im} b_{in}}{v_i}\, T^n ,$$ which coincides with \eqref{e:7.3} by setting $v_i = y_i$ for $1 \le i \le \rho$. We summarize the discussion in the following:
\begin{theorem} \label{p:qbes} Let $X \nearrow Y$ be a projective conifold transition through $\bar X$ with $k$ ordinary double points. Let the bundle $\underline{\mathbb{C}}^k \oplus (\underline{\mathbb{C}}^k)^\vee$ over $\mathbb{C}^k$ be equipped with the standard logarithmic connection defined in \eqref{Ck-log}. Then \begin{itemize} \item[(1)] The connection induced from the embedding $B: \mathbb{C}^\rho \to \mathbb{C}^k$ defined by the relation matrix of vanishing 3 spheres for the degeneration from $X$ to $\bar X$ gives rise to the logarithmic part of the Dubrovin connection on $H^2(Y)/H^2(X)$.
\item[(2)] The connection induced from the embedding $A: \mathbb{C}^\mu \to \mathbb{C}^k$ defined by the relation matrix of extremal rational curves for the small contraction $Y \to \bar X$ gives rise to the logarithmic part of the Gauss--Manin connection on $V^*$, where $V$ is the space of vanishing 3-cycles. \end{itemize} \end{theorem}
Part (1) has just been proved. The proof for (2) is similar (by setting $z = 2\pi \sqrt{-1}$ and $w_i = y_i$, cf.~\eqref{Yukawa}) and is omitted. We remark that the two subspaces $B(\mathbb{C}^\rho)$ and $A(\mathbb{C}^\mu)$ are indeed defined over $\mathbb{Q}$ and orthogonal to each other, hence $A$ and $B$ determine each other up to choice of basis.
\section{From $A(X) + B(X)$ to $A(Y) + B(Y)$} \label{s:5}
In this section we prove Theorem \ref{t:0.2} (3). The main idea is to refine the GW invariants on $X$ to respect the linking data on the vanishing cycles. The GW theory of $Y$ can then be reconstructed from the linked GW theory of $X$.
\subsection{Overview} \label{s:5.1}
\subsubsection{$B(X) \Rightarrow B(Y)$} This is explained in \S\ref{s:4}: The VHS on $Y$ is contained in the logarithmic extension of VHS on $X$ as the monodromy invariant sub-theory along $\mathcal{M}_Y \subset \mathcal{M}_{\bar X}$. This is the easy part.
\subsubsection{$A(X) + B(X)_{classical} \Rightarrow A(Y)$} What we already know about $A(Y)$ consists of the following three pieces of data: \begin{itemize} \item[(1)] $A(X)$, which is given, \item[(2)] the extremal ray invariants on divisors $\{T_l\}_{l = 1}^\rho$ determined by the relation matrix $B$ of the vanishing 3-spheres, and \item[(3)] the cup product on $H^2(Y)$. Since $Y$ comes from surgeries on $X$ along the vanishing spheres, this is determined classically. \end{itemize} The ingredient (2) obviously does not come from $A(X)$ but can be computed explicitly. As discussed in \S\ref{s:3.2} for $g=0$ case, the extremal ray invariants of all genera can be obtained from invariants of $(-1, -1)$ curves by the relation matrix $A$. Therefore, the ingredients needed for (2) is local and independent of the transition. The genus zero case was already discussed. The $g=1$ invariants for $(-1,-1)$ curves was computed in \cite{BCOV} (and justified in \cite{GP}) and $g \geq 2$ invariants in \cite{FP}.
We make a quick comment on reconstruction in genus zero. Using the notations in \eqref{e:split}, (1)--(3) above give the initial conditions on the two coordinates slices $u = 0$ and ``$s = \infty$'' (i.e., $\beta = 0$) respectively. Naively one may wish to reconstruct the genus zero GW theory on the entire cohomology from these two coordinate slices. When $Y$ is Fano, this is often possible by WDVV. However, WDVV gives no information for Calabi--Yau 3-folds. This issue will be resolved by studying the notion of linking data below.
\subsection{Linking data} \label{s:5.2} The homology and cohomology discussed in this subsection are over $\mathbb{Z}$. As a first step, we study the topological information about the holomorphic curves in $X \setminus \bigcup_{i=1}^k S_i$ instead of in $X$. This can be interpreted as the linking data between the curve $C$ and the set of vanishing spheres $\bigcup_{i=1}^k S_i$. We will see that the linking data add extra information to the curve class in $X$ and enable us to recover the missing topological information in the process of transition.
\begin{remark} As mentioned in Remark~\ref{r:1} that the vanishing sphere $S_i$ can be chosen to be Lagrangian with respect to the prescribed K\"ahler form $\omega$ on $X$. When $\omega$ is Ricci flat, it is expected to have special Lagrangian (SL) representatives. A proof to this was recently announced in \cite[Corollary A.2]{HS}. Assuming this, then we have $T_{[S_i]}{\rm Def}(S_i/X) \cong H^1(S_i, \mathbb{R}) = 0$ by McLean's theorem \cite{rcM}. That is, $S_i$ is rigid in the SL category. Thus, given a curve $C$ in $X$ we expect that $C \cap S_i = \emptyset, \forall i$. Furthermore, by a simple virtual dimensional count, this is known to hold for a generic almost complex structure $J$ on $TX$ (cf.\ \cite{kF}). But we shall proceed without these heuristics. \end{remark}
The plan is to assign a \emph{linking data $L$} between $C$ and $S_i$'s so that $L$ represents a refinement of $\beta =[C]$ in $X$ and that $L$ uniquely determines a curve class $\gamma$ in $Y$, such that $n^X_{\beta, L} = n^Y_{\gamma}$. With the choices of lifting $\beta$ in $Y$ being fixed (as above), this is equivalent to saying that $L$ will uniquely determine a curve class $d\ell \in N_1(Y/\bar X)$. Let $B_i = D_\epsilon(N_{S_i/X})$ be the $\epsilon$ open tubular neighborhood of $S_i$ in $X$ with $\epsilon$ small enough such that $C \cap B_i = \emptyset$ for all $i$. Then $\partial B_i = S_\epsilon(N_{S_i/X}) \cong S_i \times S^2_\epsilon \cong S^3 \times S^2$. Let $M := X \setminus \bigcup_{i=1}^k B_i$. Then the pair $(M, \partial M)$ is the common part for both $X$ and $Y$. Indeed let $B^+_i = D_\delta(N_{C_i/Y})$, then $\partial B^+_i = S_\delta (N_{C_i/Y}) \cong S^3_\delta \times C_i \cong S^3 \times S^2.$ This leads to two deformation retracts $$ (Y, \bigcup C_i) \sim (M, \partial M) \sim (X, \bigcup S_i). $$ Consider the sequence induced by the Poincar\'e--Lefschetz duality and excision theorem for $i: \partial M \hookrightarrow M$:
\small \begin{equation} \label{e:5.2} \xymatrix{& H_2(M, \partial M) \ar[r]^<<<<<<\sim & H^4(M) \\H_2(C) \ar[r]^{f_*} & H_2(M) \ar@{->>}[u]_{j_*} \ar[r]^<<<<<<{\sim} & H^4(M, \partial M) \ar@{->>}[u]_{j^*} \\ & \bigoplus_{i} H_2(S^3_i \times S^2_i) \ar[r]^<<<<<\sim \ar[u]_{i_*} & H^3(\partial M) \ar[u]_{\Delta^*}\\ & H_3(M, \partial M) \ar[u]_{\Delta_*} \ar[r]^<<<<<<\sim & H^3(M) \ar[u]_{i^*}.} \end{equation} \normalsize
From the retract $(M, \partial M) \sim (Y, \bigcup C_i)$ and the excision sequence for $(Y, \bigcup C_i)$ we find $H_3(M, \partial M) \to \bigoplus H_2(C_i) \to H_2(Y) \to H_2(M, \partial M) \to 0$. By comparing this with the LHS vertical sequence we conclude by the five lemma that $H_2(M) \cong H_2(Y)$. In particular, the curve class in $Y$ $$ \gamma := f_*[C] \in H_2(M) \cong H_2(Y) $$ is well defined.
\begin{definition} \label{d:2} The linking data $(\beta, L)$ is defined to be $f_*([C]) = \gamma$ above. \end{definition}
From the excision sequence $(X, \bigcup S_i)$, we have $$ 0 \to H^3(M, \partial M) \to H^3(X) \to \bigoplus H^3(S_i) \to H^4(M, \partial M) \to H^4(X) \to 0, $$ where the retract $(M, \partial M) \sim (X, \bigcup S_i)$ is used. Comparing with the right vertical sequence in \eqref{e:5.2}, we find $H^4(M) \cong H^4(X)$ and $h^3(X) = h^3(M) + k - \rho = h^3(M) + \mu$. Since $h^3(X) = h^3(Y) + 2\mu$, this is equivalent to \begin{equation} \label{e:5.3} h^3(M) = h^3(Y) + \mu . \end{equation}
\subsection{Linked GW on $X$ $=$ non-extremal GW on $Y$} \label{s:5.3}
\subsubsection{Analysis of the moduli of stable maps to the degenerating families} \label{s:5.3.1} We recall results in J.~Li's study of degeneration formula \cite{JL1, JL2}: given a projective flat family over a curve $\pi: W \to \mathbb{A}^1$ such that $\pi$ is smooth away from $0 \in B$ and the central fiber $W_0 = Y_1 \cup Y_2$ has only double point singularity with $D := Y_1 \cap Y_2$ a smooth (but not necessarily connected) divisor, Li in \cite{JL1} constructed a moduli stack $\mathfrak{M}(W, \Gamma) \to \mathbb{A}^1$ which has a perfect obstruction theory and hence a virtual fundamental class $[\mathfrak{M}(W, \Gamma)]^{\operatorname{virt}}$ in \cite{JL2}. The following properties will be useful to us. (The notations are slightly changed.) \begin{enumerate} \item[(1)] For every $0 \neq t \in \mathbb{A}^1$, one has \begin{equation*} \label{e:L1}
\mathfrak{M}(W, \Gamma)_t = \overline{M}(X, \beta), \qquad
[\mathfrak{M}(W, \Gamma)]^{\operatorname{virt}}_t = [\overline{M}(X, \beta)]^{\operatorname{virt}} \end{equation*} where $\overline{M}(X, \beta)$ is the corresponding moduli of (absolute) stable maps.
\item[(2)] For the central fiber, the perfect obstruction theory on $\mathfrak{M}(W, \Gamma)$ induces a perfect obstruction theory on $\mathfrak{M}(W_0, \Gamma)$ and \begin{equation*} \label{e:L2}
[\mathfrak{M}(W_0, \Gamma) ]^{\operatorname{virt}}= [\mathfrak{M}(W, \Gamma) ]^{\operatorname{virt}} \cap \pi^{-1} (0) \end{equation*} is a virtual divisor of $[\mathfrak{M}(W, \Gamma) ]^{\operatorname{virt}}$.
\item[(3)] $\mathfrak{M}(W_0, \Gamma)$ and its virtual class are related to the relative moduli and their virtual classes. For each admissible triple (consisting of gluing data) $\epsilon$, there is a ''gluing map'' \begin{equation*} \label{e:L31}
\Phi_{\epsilon} : \mathfrak{M}(Y_1, D; \Gamma_1) \times_{D^{\rho}} \mathfrak{M}(Y_2, D; \Gamma_2)
\to \mathfrak{M}(W_0, \Gamma), \end{equation*} inducing the relation between the virtual cycles \begin{equation*} \label{e:L3}
[\mathfrak{M}(W_0, \Gamma)]^{\operatorname{virt}} = \sum_{\epsilon} m_{\epsilon} {\Phi_{\epsilon}}_* \Delta^{!}
\left([\mathfrak{M}(Y_1, D; \Gamma_1)]^{\operatorname{virt}} \times [\mathfrak{M}(Y_2, D; \Gamma_2)]^{\operatorname{virt}} \right), \end{equation*} where $\Delta : D^{\rho} \to D^{\rho} \times D^{\rho}$ is the diagonal morphism and $m_{\epsilon}$ is a rational number (multiplicity divided by the degree of $\Phi_{\epsilon}$). \end{enumerate}
\subsubsection{Decomposition of $\mathfrak{M}(W_0, \Gamma)$} \label{s:5.3.4}
We study properties of $\mathfrak{M}(W_0, \Gamma)$ and their virtual fundamental classes in the setting of \S\ref{s:3.1}. Namely we specialize the discussions in \S \ref{s:5.3.1} to the two semistable degenerations constructed in \S \ref{s:2.1}.
A comprehensive comparison of the curve classes in $X$, $Y$ and $\tilde{Y}$ is collected in the following diagram. \small $$ \xymatrix{ H_3 (M, \partial M) \ar[r] \ar[d]^= &H_2 ( \bigcup_i E_i) \ar[r] \ar[d]^{\bar{\phi}_*} &H_2(\tilde{Y}) \ar[r] \ar[d]^{\phi_*} &H_2 (M, \partial M) \ar[r] \ar[d]^{=} &0 \ar[d]^{=} \\ H_3 (M, \partial M) \ar[r] \ar[d]^{=} &H_2 ( \bigcup_i C_i) \ar[r] \ar[d]^{\bar{\chi}_*} &H_2({Y}) \ar[r] \ar[d]^{\chi_*} &H_2 (M, \partial M) \ar[r] \ar[d]^{=} &0 \ar[d]^{=} \\ H_3 (M, \partial M) \ar[r] &0 \ar[r] &H_2(X) \ar[r] &H_2 (M, \partial M) \ar[r] &0 } $$ \normalsize
A simple diagram chasing shows that there is a unique lifting $\tilde{\gamma} \in H_2(\tilde Y)$ of $\gamma \in H_2(Y)$ satisfying \eqref{e:lifting}. From this and the degeneration analysis for the K\"ahler degeneration $Y \rightsquigarrow \tilde{Y} \cup_E \tilde{E}$ (now the divisor $D = E = \sum_{i = 1}^k E_i$), we have the following lemma.
\begin{lemma} \label{l:5.2} There is a homotopy equivalence $$ [\overline{M}(Y, \gamma)]^{\operatorname{virt}} \sim [\mathfrak{M}(\tilde{Y}, E; \tilde{\gamma})]^{\operatorname{virt}}. $$ (If $\pi$ can be extended to a family over $\mathbb{P}^1$, then the two cycles are rationally equivalent.) They define the same GW invariants. \end{lemma}
Because of this lemma, we will sometimes \emph{abuse the notation and identify $[\mathfrak{M}(\tilde{Y}, E; \tilde{\gamma})]^{\operatorname{virt}}$ with $[\overline{M}(Y, \gamma)]^{\operatorname{virt}}$}.
\begin{lemma} \label{l:5.3} In the case of complex degeneration $X \rightsquigarrow \tilde{Y} \cup_E Q$ in \S\ref{s:3.1}, images of $\Phi_{\tilde{\gamma}}$ for different $\tilde{\gamma}$ are disjoint from each other. \end{lemma}
\begin{proof} This follows from Li's study on the related moduli stacks. In this special case of $\rho=0$, for any element in $\mathfrak{M}(W_0, \Gamma)$ there is only one way to split it into two "relative maps" (with one of them being empty). We note that this is not true in general, when there are more than one way of splitting of the maps to the central fiber. \end{proof}
Given $\beta \ne 0$, let $\tilde{\gamma}$ and $\tilde{\gamma}'$ be classes appearing in \eqref{e:i}; in particular they are non-exceptional for $\tilde \psi: \tilde Y \to \bar X$. We have $$\tilde{\gamma} - \tilde{\gamma}' = \sum\nolimits_i a_i (\ell_i -\ell'_i) ,$$ where $\ell_i$ and $\ell'_i$ are the $\tilde \psi$ exceptional curve classes (two rulings) in $E_i$, because $\tilde \gamma - \tilde \gamma'$ is $\tilde \psi$ exceptional and $(\tilde \gamma - \tilde \gamma').E_i = 0$. By Proposition~\ref{p:1}, there are only finitely many nonzero $a_i$. For each $\tilde{\gamma}$ above, there is a unique $\gamma = \psi_* \tilde{\gamma}$ in $Y$ which is non-extremal for $\psi: Y \to \bar X$ and satisfies \eqref{e:ii}.
\begin{corollary} \label{c:5.4} Given $\beta \ne 0$ a curve class in $X$, we can associate to it sets of non-$\tilde \psi$-exceptional curve classes $\tilde{\gamma}$ and $\gamma$ discussed above. Then \[
[\overline{M}(X, \beta)]^{\operatorname{virt}}
\sim \sum\nolimits_{\tilde{\gamma}} [\mathfrak{M}(\tilde{Y}, E; \tilde{\gamma})]^{\operatorname{virt}}
\sim \sum\nolimits_{\gamma} [\overline{M}(Y, \gamma)]^{\operatorname{virt}}, \] where $\sim$ stands for the homotopy equivalence and the summations are over the above sets. The conclusion holds for any projective small resolution $Y$ of $\bar X$. \end{corollary}
\begin{proof} This follows from \eqref{e:i}, \eqref{e:ii} and the above discussions. \end{proof}
Recall in \S\ref{s:5.2} we have the identification of the linking data in \begin{equation} \label{e:linking}
H_2(Y^{\circ}) = H_2(Y) = H_2(X^{\circ}) =
H_2 (X \setminus \bigcup\nolimits_i B_i) = H_2(\bar{X} \setminus \bar{X}^{\text{sing}}) \end{equation} where $X \setminus \bigcup_{i=1}^k S_i =: X^{\circ} \sim M \sim Y^{\circ} := Y \setminus \bigcup_{i=1}^k C_i$ and $B_i$ is a tubular neighborhood of the vanishing sphere $S_i$. Therefore, a curve class $\gamma \in H_2(Y)$ can be identified as a "curve class" in $X^{\circ} \sim \bar{X} \setminus \bar{X}^{\text{sing}}$, with the latter a quasi-projective variety, and we can think of $\gamma$ as a curve class in $X^{\circ}$.
\begin{proposition} \label{p:4} For $X_t$ with $t \in \mathbb{A}^1$ very small in the degenerating family $\pi : \mathcal{X} \to \mathbb{A}^1$, we have a decomposition of the virtual class $[\overline{M}(X_t, \beta)]^{\operatorname{virt}}$ into a finite disjoint union of cycles \[
[\overline{M}(X_t, \beta)]^{\operatorname{virt}} = \coprod\nolimits_{\gamma \in H_2(X^{\circ})} [\overline{M}(X_t, \gamma)]^{\operatorname{virt}}, \] where $[\overline{M}(Y, \gamma)]^{\operatorname{virt}} \sim [\overline{M}(X_t, \gamma)]^{\operatorname{virt}} \in A_{\operatorname{vdim}} \left( \overline{M}(X_t, \beta) \right)$ is a cycle class corresponding to the linking data $\gamma$ of $X_t$. \end{proposition}
\begin{proof} By the construction of the virtual class of the family $\pi$, we know that the virtual classes for $X_t$ and for $X_0$ are restrictions of that for $\mathcal{X}$. Lemma \ref{l:5.3} tells us that at $t=0$, the virtual class decomposes into a disjoint union. By semicontinuity of connected components, we conclude that the virtual classes for $X_t$ remain disconnected with (at least) the same number of connected components labeled by $\gamma \in H_2(X^{\circ})$. \end{proof}
We call the numbers defined by $[\overline{M}(X_t, \gamma)]^{\operatorname{virt}}$ the \emph{refined GW numbers} of $X^{\circ}$ with linking data $\gamma$, or simply \emph{linked GW invariants}.
\begin{corollary} \label{c:5.6} The refined GW numbers of $X^{\circ}$ with linking data $\gamma$ are the same as the GW invariants of $Y$ with curve class $\gamma$, where $\gamma$ is interpreted in two ways via \eqref{e:linking}. \end{corollary}
\section{From $A(Y) + B(Y)$ to $A(X) + B(X)$} \label{s:6}
The purpose of this section is to establish part (4) of Theorem \ref{t:0.2}. The main idea is to refine the $B$ model on $Y$ by studying deformations and VHS ``linked'' with the exceptional curves, i.e., on the non-compact $Y \setminus \bigcup_i C_i$. From this, the full VHS of $X$ is then reconstructed via Theorem \ref{p:gnot}.
\subsection{Overview} \label{s:6.1}
\subsubsection{$A(Y) \Rightarrow A(X)$} As is explained in \S\ref{s:3}, $A(X)$ is a sub-theory of $A(Y)$. Indeed, $A(X)$ is obtained from $A(Y)$ by setting all extremal ray invariants to be zero, in addition to ``reducing the linking data'' $\gamma \in NE(Y)$ to $\beta \in NE(X)$.
\subsubsection{$A(Y)_{classical} + B(Y) \Rightarrow B(X)$}
We have seen that $B(Y)$ can be considered as a sub-theory of $B(X)$. In this section, we will show that $B(Y)$, together with the knowledge of extremal curves $\bigcup_i C_I \subset Y$ determines $B(X)$. More precisely, we will show that the ``Hodge filtration'' underlying the variation of MHS of the quasi-projective $Y^{\circ} = Y \setminus \bigcup_i C_i$ on the first jet space of $\mathcal{M}_Y \subset \mathcal{M}_{\bar{X}}$ can be lifted uniquely to the Hodge filtration underlying the degenerating VHS of $X$. Furthermore, the information of the Gauss--Manin connection up to the first jet is sufficient to single out the VHS of $X$.
In the next subsection, we start with a statement of compatibility of MHS which is needed in our discussion. After that we will give a proof showing the unique determination. As in our implication of $B(X) + A(X) \Rightarrow A(Y)$ in \S\ref{s:5}, our $A(Y) + B(Y) \Rightarrow B(X)$ implication is not constructive.
\subsection{Compatibility of the mixed Hodge structures} Recall from \S\ref{s:4.1} that $\mathcal{M}_{\bar{X}}$ is smooth and contains $\mathcal{M}_Y$ in a natural manner. Set $$ U := Y^{\circ} = Y \setminus \bigcup\nolimits_{i = 1}^k C_i \cong \bar{X}^{\circ} = \bar{X} \setminus \bar{X}^{\operatorname{sing}} $$ where $$ \bar{X}^{\text{sing}} = p := \bigcup\nolimits_{i = 1}^k \{p_i\}. $$ To construct the VHS with logarithmic degeneration on $\mathcal{M}_{\bar X}$ near $\mathcal{M}_Y$, we start with the following lifting property.
\begin{proposition} \label{p:lift} There is a short exact sequence of mixed Hodge structures \begin{equation} \label{V-MHS} 0 \to V \to H^3(X) \to H^3(U) \to 0, \end{equation} where $H^3(X)$ is equipped with the limiting MHS of Schmid, $$ V \cong H^{1, 1}_\infty H^3(X), $$ and $H^3(U)$ is equipped with the canonical mixed Hodge structure of Deligne. In particular, $F^3 H^3(X) \cong F^3 H^3(U)$ and $F^2 H^3(X) \cong F^2 H^3(U)$. \end{proposition}
\begin{proof} In the topological level, the short exact sequence \eqref{V-MHS} is equivalent to the defining sequence of the vanishing cycle space \eqref{e:9}. Indeed, since $X$ is nonsingular, $H_3(X) \cong H^3(X)$ by Poincar\'e duality. Also, \begin{equation} \label{U-MHS} H_3(\bar X) = H_3(\bar X, p) \cong H_3(\tilde Y, E) \cong H^3(\tilde Y \backslash E) = H^3(U) \end{equation} by the excision theorem and Lefschetz duality.
Now we consider the mixed Hodge structures. Since $U$ is smooth quasi-projective, it is well know that the canonical mixed Hodge structure on $H^3(U)$ has its Hodge diamond supported on the upper triangular part, i.e., with weights $\ge 3$. Or equivalently, the MHS on $H_3(\bar X)$ has weights $\le 3$ by duality in \eqref{U-MHS}. The crucial point is that Lefschetz duality is compatible with mixed Hodge structures, as stated in Lemma~\ref{L-MHS} below. Hence the short exact sequence \eqref{V-MHS} follows from Lemma~\ref{l:7} which is essentially the invariant cycle theorem.
Notice that $V \cong H^{1, 1}_\infty H^3(X)$ by Lemma \ref{l:7} (ii). In particular, the isomorphisms on $F^i$ for $i = 3, 2$ follows immediately by applying $F^i$ to the sequence \eqref{V-MHS}. \end{proof}
\begin{lemma} \label{L-MHS} Let $Y$ be an $n$ dimensional complex projective variety, $i: Z \hookrightarrow Y$ a closed subvariety with smooth complement $j: U \hookrightarrow Y$ where $U:=Y \backslash Z$. Then the Lefschetz duality $H_i(Y, Z) \cong H^{2n - i}(U)$ is compatible with the canonical mixed Hodge structures. \end{lemma}
This is well known in mixed Hodge theory. For the readers' convenience we include a proof which is communicated to us by M.~de Caltaldo.
\begin{proof} We will make use of the structural theorem of Saito on mixed Hodge modules (MHM) \cite[Theorem 0.1]{mS} which says that there is a correspondence between the derived categories of MHM and that of perverse sheaves (cf.~Axiom A in 14.1.1 of Peters and Steenbrink's book \cite{PS}).
There is a triangle in the derived category of constructible sheaves \[
j_! j^! \mathbb{Q}_Y \to \mathbb{Q}_Y \to i_* i^* \mathbb{Q}_Y . \] This gives maps of MHS $H^i(Y,Z) \to H^i(Y) \to H^i(Z)$ with $H^i(Y,Z)=H^i (Y, j_! j^! \mathbb{Q}_Y)$. In fact, the MHS of $H^i(Y,Z)$ can be defined by the RHS from Saito's theory, since $j_! j^! \mathbb{Q}_Y$ is a complex of MHM.
Dualizing the above setup, we have \begin{equation} \label{e:dualizing}
H_i (Y,Z)=H_i (Y, j_! j^! \mathbb{Q}_Y)^*, \end{equation} where the LHS of \eqref{e:dualizing} having MHS for the same reason as above and compatibly with taking dual as MHS. Furthermore, the RHS of \eqref{e:dualizing} is $H^{-i}_c(Y,j_*j^* \omega_Y)$ by Verdier duality, where $\omega_Y$ is the Verdier dualizing complex. Due to the compactness of $Y$ we have \begin{equation*} \begin{split}
H^{-i}_c(Y,j_*j^* \omega_Y)
&= H^{-i} (Y, j_* j^* \omega_Y)
= H^{-i} (U, \omega_U) \\
&=H^{BM}_i (U)
=H^{2n - i} (U),
\end{split} \end{equation*} where $H^{BM}$ is the Borel--Moore homology. Since all steps are compatible with MHM, the Lefschetz duality is compatible with the MHS. \end{proof}
\subsection{Conclusion of the proof} \label{s:pf}
We now apply the above result to our setting. We have on $\bar X$ (cf.\ \cite{NS}) \[ \cdots H^1_{p} (\Theta_{\bar X}) \to H^1(\Theta_{\bar X}) \to H^1(U, T_U) \to H^2_{p} (\Theta_{\bar X}) \to \cdots. \] Since each $p_i$ is a hypersurface singularity, we have ${\rm depth}\,\mathscr{O}_{p_i} = 3$. Using this fact, Schlessinger \cite{Sch} (see also \cite{rF}) showed that $H^1_p(\Theta_{\bar X}) = 0$ and $H^2_p(\Theta_{\bar X}) \cong \bigoplus_{i = 1}^k \mathbb{C}_{p_i}$. Putting these together, we have \begin{equation} \label{e:6.3.1}
0 \to H^1(\Theta_{\bar X}) \to H^1(U, T_U) \to H^2_{p} (\Theta_{\bar X}) \to \cdots. \end{equation}
Since $\bar X$ is a Calabi--Yau 3-fold with only ODPs, its deformation theory is unobstructed by the $T^1$-lifting property \cite{yK}. Comparing \eqref{e:6.3.1} with \eqref{e:15} we see that $\operatorname{Def} (\bar{X}) \cong H^1(U, T_U)$.
Similarly, on $Y$ we have $$ \cdots H^1_Z(T_Y) \to H^1(T_Y) \to H^1(U, T_U) \to H^2_Z(T_Y) \to H^2(T_Y) \to \cdots , $$ where $Z = Y \setminus U$ is the union of exceptional curves. Since $Y$ is smooth, the depth argument also gives $H^1_Z(T_Y) = 0$ (or by the local duality theorem $H^1_Z (T_Y) \cong H^2(Z, T_Y^\vee \otimes K_Y)^\vee = 0$). Thus $$ \operatorname{Def} (Y) = H^1(T_Y) \subset H^1(U, T_U) \cong \operatorname{Def} (\bar{X}), $$ and $\mathcal{M}_Y$ is naturally a submanifold of $\mathcal{M}_{\bar{X}}$. Write $\mathscr{I} := \mathscr{I}_{\mathcal{M}_{Y}}$ as the ideal sheaf of $\mathcal{M}_Y \subset \mathcal{M}_{\bar{X}}$. Since $H^2(U, T_U) \neq 0$, the deformation of $U$ could be obstructed. Nevertheless, the first-order deformation of ${U}$ exists and is parameterized by $H^1(U, T_U) \supset \operatorname{Def} (Y)$. Therefore, we have the following \emph{smooth family} \[
\pi: \mathfrak{U} \to \mathcal{Z}_1:= Z_{\mathcal{M}_{\bar{X}}}(\mathscr{I}^2) \supset \mathcal{M}_Y, \] where $\mathcal{Z}_1 = Z_{\mathcal{M}_{\bar{X}}}(\mathscr{I}^2)$ stands for the nonreduced subscheme of $\mathcal{M}_{\bar{X}}$ defined by the ideal sheaf $\mathscr{I}^2$. Namely $\mathcal{Z}_1$ is the first jet extension of $\mathcal{M}_Y$ in $\mathcal{M}_{\bar X}$.
Now we may complete the construction of VHS over $\mathcal{M}_{\bar X}$ near the boundary loci $\mathcal{M}_Y \hookrightarrow \mathcal{M}_{\bar X}$. The Gauss--Manin connection for a smooth family over non-reduced base was constructed in \cite{Katz}. For our smooth family $\pi: \mathcal{U} \to \mathcal{Z}_1$, it is defined by the integral lattice $H^3(U, \mathbb{Z}) \subset H^3(U, \mathbb{C})$. Since $U$ is only quasi-projective, the Gauss--Manin connection underlies VMHS instead of VHS. By Proposition \ref{p:lift}, we have $W_i H^3(U) = 0$ for $i \le 2$, $W_3 \subset W_4$ with $\operatorname{Gr}^W_3 H^3(U) \cong H^3(Y)$, and $\operatorname{Gr}^W_4 H^3(U) \cong V^*$.
The Hodge filtration of the local system $F^0 = H^3(U, \mathbb{C})$ has the following structure: $F^\bullet = \{F^3 \subset F^2 \subset F^1 \subset F^0\}$ which satisfies the Griffiths transversality. Since $K_U \cong \mathscr{O}_U$ and $H^0(U, K_U) \cong H^0(Y, K_Y) \cong \Bbb C$, $F^3$ is a line bundle over $\mathcal{Z}_1$ spanned by a nowhere vanishing relative holomorphic 3-form $\Omega \in \Omega^3_{\mathcal{U}/\mathcal{Z}_1}$. Near the moduli point $[Y] \in \mathcal{Z}_1$, $F^2$ is then spanned by $\Omega$ and $v (\Omega)$ where $v$ runs through a basis of $H^1(U, T_U)$. Notice that $v(\Omega) \in W_3$ precisely when $v \in H^1(Y, T_Y)$.
By Proposition~\ref{p:lift}, the partial filtration $F^3 \subset F^2$ on $H^3(U)$ over $\mathcal{Z}_1$ lifts uniquely to a filtration $\tilde F^3 \subset \tilde F^2$ on $H^3(X)$ over $\mathcal{Z}_1$ with $\tilde F^3 \cong F^3$ and $\tilde F^2 \cong F^2$. The complete lifting $\tilde F^\bullet$ is then uniquely determined since $\tilde F^1 = (\tilde F^3)^\perp$ by the first Hodge--Riemann bilinear relation on $H^3(X)$. Alternatively, $\tilde F^1$ is spanned by $\tilde F^2$ and $v (\tilde F^2)$ for $v$ runs through a basis of $H^1(U, T_U)$.
Now $\tilde F^\bullet$ over $\mathcal{Z}_1$ uniquely determines a horizontal map $\mathcal{Z}_1 \to \check{\Bbb D}$. Since it has maximal tangent dimension $h^1(U, T_U) = h^1(X, T_X)$, it determines uniquely the maximal horizontal slice $\psi: \mathcal{M} \to \check{\Bbb D}$ with $\mathcal{M} \cong \mathcal{M}_{\bar X}$ locally near $\mathcal{M}_Y$. The smoothing loci of $\bar X$ in $\mathcal{M}_{\bar X}$ is precisely given by $\mathcal{M}_X$. According to Theorem \ref{p:gnot}, namely an extension of Schmid's nilpotent orbit theorem, under the coordinates ${\bf t} = (r, s)$, the period map $$ \phi: \mathcal{M}_X = \mathcal{M}_{\bar X} \backslash \bigcup\nolimits_{i = 1}^k D^i \to \Bbb D/\Gamma $$ is then given by $$ \phi(r, s) = \exp \left(\sum_{i = 1}^k \frac{\log w_i}{2\pi \sqrt{-1}} N^{(i)}\right) \psi(r, s), $$ where $\Gamma$ is the monodromy group generated by the local monodromy $T^{(i)}= \exp N^{(i)}$ (with $m_i = 1$) around the divisor $D^i$ defined by $w_i = \sum_{j = 1}^\mu a_{ij} r_j = 0$ (cf.~\eqref{coor-w}). Since $N^{(i)}$ is determined by the Picard--Lefschetz formula (Lemma \ref{P-L}), we see that the period map $\phi$ is completely determined by the relation matrix $A$ of the extremal curves $C_i$'s. (The period map gives the desired VHS, with degenerations, over $\mathcal{M}_X$.) This completes the proof that refined $B$ model on $Y \backslash Z = U$ determines the $B$ model on $X$.
\end{document} | arXiv | {
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\begin{document}
\title[Topological entropy on uniform spaces]{Topological entropy of nonautonomous dynamical systems on uniform spaces}
\author{Hua Shao} \address{Department of Mathematics, Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, P. R. China} \address{Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles, Nanjing University of Aeronautics and Astronautics, MIIT, Nanjing 211106, P. R. China} \email{huashao@nuaa.edu.cn} \date{\today}
\maketitle
\begin{abstract} In this paper, we focus on some properties, calculations and estimations of topological entropy for a nonautonomous dynamical system $(X,f_{0,\infty})$ generated by a sequence of continuous self-maps $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ on a compact uniform space $X$. We obtain the relations of topological entropy among $(X, f_{0,\infty})$, its $k$-th product system and its $n$-th iteration system. We confirm that the entropy of $(X, f_{0,\infty})$ equals to that of $f_{0,\infty}$ restricted to its non-wandering set provided that $f_{0,\infty}$ is equi-continuous. We prove that the entropy of $(X, f_{0,\infty})$ is less than or equal to that of its limit system $(X, f)$ when $f_{0,\infty}$ converges uniformly to $f$. We show that two topologically equi-semiconjugate systems have the same entropy if the equi-semiconjugacy is finite-to-one. Finally, we derive the estimations of upper and lower bounds of entropy for an invariant subsystem of a coupled-expanding system associated with a transition matrix. \end{abstract}
{\bf Keywords}: Topological entropy; uniform space; nonautonomous dynamical system; topological conjugacy; coupled-expansion.
{2010 {\bf Mathematics Subject Classification}}: 37B40, 37B55, 54E15.
\section{Introduction} Let $X$ be a compact uniform space equipped with a uniform structure $\mathcal{U}$ and $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ be a sequence of continuous self-maps on $X$. The pair $(X,f_{0,\infty})$ is called a nonautonomous dynamical system. If $f_n=f$ for all $n\geq0$, then $(X, f_{0,\infty})$ becomes the classical dynamical system $(X,f)$. For any $x_0\in X$, the positive orbit $\{x_n\}_{n=0}^{\infty}$ of $(X,f_{0,\infty})$ starting from $x_0$ is defined by $x_n=f_0^n(x_0)$, where $f_0^n=f_{n-1}\circ\cdots\circ f_{0}$ for any $n\geq 1$, and $\{x_n\}_{n=0}^{\infty}$ can be seen as a solution of the nonautonomous difference equation \[x_{n+1}=f_n(x_n),\;n\geq0.\]
Uniform structures were first introduced by Weil in \cite{Weil37}. Recall that a uniform structure on $X$ is a collection $\mathcal{U}$ of subsets of $X\times X$ satisfying that
(i) if $\gamma\in\mathcal{U}$, then $\bigtriangleup_{X}\subset\gamma$;
(ii) if $\eta\in\mathcal{U}$ and $\eta\subset\gamma\subset X\times X$, then $\gamma\in\mathcal{U}$;
(iii) if $\gamma,\eta\in\mathcal{U}$, then $\gamma\cap\eta\in\mathcal{U}$;
(iv) if $\gamma\in\mathcal{U}$, then $\gamma^{-1}\in\mathcal{U}$;
(v) if $\gamma\in\mathcal{U}$, then there exists $\eta\in\mathcal{U}$ such that $\eta\circ\eta\subset\gamma$.
In the above, $\bigtriangleup_{X}=\{(x,x): x\in X\}$ denotes the diagonal in $X\times X$; $\gamma^{-1}=\{(x, y): (y, x)\in\gamma\}$ is the inverse of $\gamma\in\mathcal{U}$, and $\gamma$ is said to be symmetric if $\gamma^{-1}=\gamma$; and $\gamma\circ\eta=\{(x, z) : (x, y)\in\gamma,(y, z)\in\eta \;{\rm for} \;{\rm some}\; y\in X\}$ denotes the composition of $\gamma,\eta\in\mathcal{U}$. Denote $\gamma^n=\underbrace{\gamma\circ\cdots\circ\gamma}_{n}$, and clearly, $\gamma\subset\gamma^n$ for any $n\geq1$. A member of $\mathcal{U}$ is called an index or entourage and the pair $(X,\mathcal{U})$ is called a uniform space. A uniform structure $\mathcal{U}$ is separated if $\bigcap_{\gamma\in\mathcal{U}}\gamma=\bigtriangleup_{X}$. It is known that a uniform space $(X,\mathcal{U})$ is Hausdorff if and only if $\mathcal{U}$ is separated.
The topology induced by the uniformity $\mathcal{U}$ or the uniform topology is the family of all subsets $G$ of $X$ such that for any $x\in G$ there exists an $\gamma\in\mathcal{U}$ such that $\gamma[x]\subset G$, where $\gamma[x]=\{y\in X: (x, y)\in\gamma\}$. Denote by ${\mathcal{U}}^s$, ${\mathcal{U}}^o$ and ${\mathcal{U}}^{s,o}$ be the set of symmetric, open and symmetric open indices in $\mathcal{U}$, respectively. Note that ${\mathcal{U}}^{s,o}$ is a base for $\mathcal{U}$ \cite{Kelley55}. A map $f: X\to X$ is called uniformly continuous if $(f\times f)^{-1}(\gamma)\in\mathcal{U}$ for any $\gamma\in\mathcal{U}$. It is easy to see that $f$ is uniformly continuous if and only if for any $\gamma\in\mathcal{U}$, there exists $\eta\in\mathcal{U}$ such that $(f(x),f(y))\in\gamma$ for any $(x, y)\in\eta$ \cite{Sal21}, if and only if it is continuous relative to the uniform topology \cite{Kelley55}. The dynamics of systems on uniform spaces have been studied by many authors \cite{Arai18,Cecc13,Cecc21,Sal21,Shah20,Yan16,Wang18,Wu19}. For example, Arai recently in \cite{Arai18} showed that for a continuous action of an Abelian group on a second countable Baire Hausdorff uniform space without isolated points, Devaney chaos implies Li-Yorke chaos.
Topological entropy provides a numerical measure for the complexity of dynamical systems. The definition of classical topological entropy was introduced by Adler, Konhelm and McAndrew using open covers for a continuous map on a compact topological space in 1965 \cite{Adler65}. Their definition was directly inspired by the one given by Kolmogorov for measure-theoretic entropy. Later, Bowen gave another definition using separated and spanning sets for a uniformly continuous map on a general metric space \cite{Bowen71}, and this definition is equivalent to Adler's definition when the space is compact. Hood generalized Bowen's definition to a uniformly continuous map on a uniform space, and investigated the relations between the entropy of the original map and the entropy of an induced map on a quotient space \cite{Hood74}. Following Hood's work, Yan and Zeng proved the two definitions of topological entropy using open covers and using separated and spanning sets are equivalent on a compact uniform space, and they investigated the relations between pseudo-orbits and topological entropy \cite{Yan16}. Recently, Ceccherini-Silberstein and Coornaert extended the notion of topological entropy to a uniformly continuous group action on a compact uniform space \cite{Cecc21}.
It is worth mentioning that Kolyada and Snoha extended the concept of topological entropy to a nonautonomous dynamical system generated by a sequence of continuous self-maps on a compact metric space, and they obtained a series of important properties of it. For example, they proved that topological entropy is an invariant under topological equi-conjugacy \cite{Kolyada96}. Note that the majority of complex systems in biology, physics and engineering are driven by sequences of different functions, and thus the study on nonautonomous dynamical systems is of importance in applications. In addition, the behaviors of nonautonomous dynamical systems are much richer and sometimes quite different than what are expected from the classical cases. For example, Balibrea and Oprocha constructed a nonautonomous dynamical system on the interval which has positive topological entropy, but does not exhibit Li-Yorke chaos. For more information on nonautonomous dynamical systems, the readers are referred to \cite{Bali12,Canovas13,Kawan13,Kawan16,Kolyada96,Kolyada99,Liu20,Sal21,Shao16,Shao20,Shao21,Shi09,Xu18,Zhu12} and references therein.
Motivated by the above work, we shall study the topological entropy of a nonautonomous dynamical system generated by a sequence of continuous self-maps on a compact uniform space, concentrating on its properties, calculations and estimations. The rest of the paper is organized as follows. In Section 2, the definitions of topological entropy using open covers and using separated and spanning sets for $(X,f_{0,\infty})$ are introduced, respectively, and these definitions are proved to be equivalent. Several basic properties and calculations of topological entropy of $(X,f_{0,\infty})$ are investigated in Section 3. In Section 4, the relations of topological entropy between two topologically equi-semiconjugate systems are studied, and particularly, they are equivalent if the equi-semiconjugacy is finite-to-one. By establishing the topological equi-semiconjugacy to a subshift of finite type, the estimations of upper and lower bounds of topological entropy for an invariant subsystem of a coupled-expanding system associated with a transition matrix are obtained in Section 5.
\section{Definitions of topological entropy}
In this section, we first recall the definitions of topological entropy using open covers and using separated sets and spanning sets for $(X,f_{0,\infty})$, respectively, and then prove that these definitions are equivalent.
\subsection{Definition with open cover} Let ${\mathscr{A}}_{1},\cdots,{\mathscr{A}}_{n},\; \mathscr{A}$ be open covers of $X$. Denote \[\bigvee_{i=1}^{n}{\mathscr{A}}_{i}=\big\{\bigcap_{i=1}^{n}A_{i}: A_{i}\in{\mathscr{A}}_{i},\ 1\leq i\leq n\big\},\] and \[f_{i}^{-n}(\mathscr{A})=\{f_{i}^{-n}(A): A\in\mathscr{A}\},\; \mathscr{A}_{i}^{n}(f_{0,\infty})=\bigvee_{j=0}^{n-1}f_{i}^{-j}({\mathscr{A}}),\] where \[f_{i}^{n}=f_{i+n-1}\circ\cdots\circ f_{i}, \;f_{i}^{-n}=(f_{i}^{n})^{-1}=f_{i}^{-1}\circ\cdots\circ f_{i+n-1}^{-1},\;i\geq0,\;n\geq1.\] Let $\mathcal{N}(\mathscr{A})$ be the minimal possible cardinality of all subcovers chosen from $\mathscr{A}$. Then the topological entropy of $(X,f_{0,\infty})$ on the cover $\mathscr{A}$ is defined by \begin{align}\label{de} h(f_{0, \infty},\mathscr{A})=\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathcal{N}\big(\mathscr{A}_{0}^{n}(f_{0,\infty})\big), \end{align} and the topological entropy of $(X,f_{0,\infty})$ is defined by \begin{align*} h_{top}(f_{0, \infty})=\sup\{h(f_{0,\infty},\mathscr{A}): \mathscr{A}\; {\rm is\; an\; open\; cover\; of\; X\;}\}. \end{align*} If each element of ${\mathscr{A}}_{2}$ is contained in some member of ${\mathscr{A}}_{1}$, then we say ${\mathscr{A}}_{2}$ is a refinement of ${\mathscr{A}}_{1}$, and denote it as ${\mathscr{A}}_{1}\prec{\mathscr{A}}_{2}$. Clearly, \[{\mathscr{A}}_{1}\prec{\mathscr{A}}_{2}\Rightarrow h(f_{0,\infty},\mathscr{A}_{1})\leq h(f_{0,\infty},\mathscr{A}_{2}).\]
Let $\Lambda$ be any nonempty subset (not necessarily compact or invariant) of $X$. Denote the cover $\{A\cap \Lambda : A\in\mathscr{A}\}$ of the set $\Lambda$ by $\mathscr{A}|_{\Lambda}$. Then the topological entropy of $f_{0, \infty}$ on $\Lambda$ is defined by \begin{align}\label{subset}
h(f_{0,\infty},\Lambda):=\sup\big\{\limsup_{n\to\infty}\frac{1}{n}\log\mathcal{N}\big({\mathscr{A}_{0}^{n}(f_{0,\infty})}|_{\Lambda}\big): \mathscr{A}\; {\rm is\; an\; open\; cover\; of\; X}\big\}. \end{align}
\subsection{Definitions with separated sets and spanning sets} Let $n\geq1$ and $\gamma\in{\mathcal{U}}^{s,o}$. A set $E\subset X$ is called $(n,\gamma)$-separated if for each pair $x, y\in E$ with $x\neq y$, there exists $0\leq j\leq n-1$ such that $(f_0^j(x),f_0^j(y))\notin\gamma$; a set $F\subset X$ is called $(n,\gamma)$-spans another set $K\subset X$ if for each $x\in K$ there exists $y\in F$ such that $(f_0^j(x),f_0^j(y))\in\gamma$ for any $0\leq j\leq n-1$. For a set $\Lambda\subset X$, let $s_n(f_{0,\infty},\Lambda,\gamma)$ be the maximal cardinality of an $(n,\gamma)$-separated set in $\Lambda$, $r_n(f_{0,\infty},\Lambda,\gamma)$ and $r_n^{X}(f_{0,\infty},\Lambda,\gamma)$ be the minimal cardinality of a set in $\Lambda$ and in $X$, respectively, which $(n,\gamma)$-spans $\Lambda$. Clearly, \begin{align}\label{520} r_n^{X}(f_{0,\infty},\Lambda,\gamma)\leq r_n(f_{0,\infty},\Lambda,\gamma). \end{align} Since $X$ is compact, $s_n(f_{0,\infty},\Lambda,\gamma)$, $r_n(f_{0,\infty},\Lambda,\gamma)$ and $r_n^{X}(f_{0,\infty},\Lambda,\gamma)$ are finite.
Note that the cardinality of a set $\Lambda\subset X$ is denoted by $|\Lambda|$. We have the following inequalities.
\begin{proposition}\label{1} Let $\Lambda\subset X$, $\gamma_1,\gamma_2,\gamma,\eta\in\mathcal{U}^{s,o}$ and $n\geq1$.
{\rm(i)}If $\gamma_1\subset\gamma_2$, then \[s_n(f_{0,\infty},\Lambda,\gamma_1)\geq s_n(f_{0,\infty},\Lambda,\gamma_2),\;r_n(f_{0,\infty},\Lambda,\gamma_1)\geq r_n(f_{0,\infty},\Lambda,\gamma_2).\]
{\rm(ii)} If $\eta^2\subset\gamma$, then \[r_n(f_{0,\infty},\Lambda,\gamma)\leq s_n(f_{0,\infty},\Lambda,\gamma)\leq r_n^{X}(f_{0,\infty},\Lambda,\eta)\leq r_n(f_{0,\infty},\Lambda,\eta).\] \end{proposition}
\begin{proof} It is only to prove (ii) since (i) is directly derived by the definitions. Since an $(n,\gamma)$-separated set in $\Lambda$ with the maximal cardinality $(n,\gamma)$-spans $\Lambda$, \begin{align}\label{001} r_n(f_{0,\infty},\Lambda,\gamma)\leq s_n(f_{0,\infty},\Lambda,\gamma). \end{align}
Let $E$ be an $(n,\gamma)$-separated set in $\Lambda$ with $|E|=s_n(f_{0,\infty},\Lambda,\gamma)$ and
$F$ be a subset of $X$ which $(n,\eta)$-spans $\Lambda$ with $|F|=r_n^{X}(f_{0,\infty},\Lambda,\eta)$. For any $x\in E$, there exists $g(x)\in F$ such that $\big(f_0^j(x),f_0^j(g(x))\big)\in\eta$ for all $0\leq j\leq n-1$. This defines a map $g: E\to F$. If $g(x)=g(y)$ for some $x\neq y\in E$, then \[(f_0^j(x),f_0^j(y))\in\eta^2\subset\gamma,\;0\leq j\leq n-1,\] which is a contradiction to the fact that $E$ is an $(n,\gamma)$-separated set. So, $g$ is injective and then
\[|E|=|g(E)|\leq|F|.\] Hence, \begin{align}\label{002} s_n(f_{0,\infty},\Lambda,\gamma)\leq r_n^{X}(f_{0,\infty},\Lambda,\eta). \end{align} Therefore, the inequality follows from (\ref{520})-(\ref{002}). \end{proof}
Let $\Lambda\subset X$ and $\gamma\in{\mathcal{U}}^{s,o}$. Denote \begin{align}\label{12} \bar{s}(f_{0,\infty},\Lambda,\gamma)=\limsup_{n\to\infty}\frac{1}{n}\log s_n(f_{0,\infty},\Lambda,\gamma), \bar{r}(f_{0,\infty},\Lambda,\gamma)=\limsup_{n\to\infty}\frac{1}{n}\log r_n(f_{0,\infty},\Lambda,\gamma). \end{align} Note that ${\mathcal{U}}^{s,o}$ is a base for $\mathcal{U}$, and it is a directed set under set inclusion. Thus, $\bar{s}(f_{0,\infty},\Lambda,\gamma)$ and $\bar{r}(f_{0,\infty},\Lambda,\gamma)$ are nets in $\mathbf{R^{+}}$. So, $\lim_{\gamma\in{\mathcal{U}}^{s,o}}\bar{s}(f_{0,\infty},\Lambda,\gamma)$ and $\lim_{\gamma\in{\mathcal{U}}^{s,o}}\bar{r}(f_{0,\infty},\Lambda,\gamma)$ exist and they are equal by Proposition \ref{1}. Hence, define the topological entropy of $f_{0,\infty}$ on the set $\Lambda$ as \begin{align*} h(f_{0,\infty},\Lambda)=\lim_{\gamma\in{\mathcal{U}}^{s,o}}\bar{r}(f_{0,\infty},\Lambda,\gamma) =\lim_{\gamma\in{\mathcal{U}}^{s,o}}\bar{s}(f_{0,\infty},\Lambda,\gamma). \end{align*} By Proposition \ref{1}, we also have that \[h(f_{0,\infty},\Lambda)=\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log r_n^X(f_{0,\infty},\Lambda,\gamma).\] If $\Lambda=X$, then $h(f_{0,\infty},X)$, briefly write as $h(f_{0,\infty})$, is called the topological entropy of $(X, f_{0,\infty})$.
\subsection{Equivalence of the two definitions} The following basic result is an extension of the Lebesgue covering lemma to uniform spaces.
\begin{lemma}\label{Lebesgue covering lemma}\cite{Kelley55} Let $(X,\mathcal{U})$ be a compact uniform space and $\mathscr{A}$ be an open cover of $X$. Then there exists $\gamma\in{\mathcal{U}}^{s,o}$ such that $\gamma[x]$ is a subset of some member of $\mathscr{A}$ for any $x\in X$. \end{lemma}
Let $\gamma\in{\mathcal{U}}^{s,o}$. Denote $\mathscr{A}_\gamma:=\{\gamma[x]: x\in X\}$. Then $\mathscr{A}_\gamma$ is an open cover of $X$.
\begin{proposition}\label{2} Let $\gamma,\eta\in{\mathcal{U}}^{s,o}$ and $n\geq1$.
{\rm(i)}
\[\mathcal{N}\big((\mathscr{A}_\gamma)_{0}^{n}(f_{0,\infty})|_{\Lambda}\big)\leq r_n(f_{0,\infty},\Lambda,\gamma).\]
{\rm(ii)} If $\eta^2\subset\gamma$, then
\[s_n(f_{0,\infty},\Lambda,\gamma)\leq \mathcal{N}\big((\mathscr{A}_\eta)_{0}^{n}(f_{0,\infty})|_{\Lambda}\big).\]
{\rm(iii)} \[h_{top}(f_{0,\infty},\Lambda)=\lim_{\gamma\in{\mathcal{U}}^{s,o}}
\limsup_{n\to\infty}\frac{1}{n}\log\mathcal{N}\big({({\mathscr{A}_{\gamma}})_{0}^{n}(f_{0,\infty})}|_{\Lambda}\big).\] \end{proposition}
\begin{proof}
(i) Let $F$ be a subset of $\Lambda$ which $(n,\gamma)$-spans $\Lambda$ with $|F|=r_n(f_{0,\infty},\Lambda,\gamma)$. Then for any $y\in \Lambda$, there exists $x\in F$ such that $(f_{0}^{j}(x),f_{0}^{j}(y))\in\gamma$ for any $0\leq j\leq n-1$. Thus, $y\in\bigcap_{j=0}^{n-1}f_{0}^{-j}\big(\gamma[f_{0}^{j}(x)]\big)$. This implies that \[\Lambda\subset\bigcup_{x\in F}\bigcap_{j=0}^{n-1}f_{0}^{-j}\big(\gamma[f_{0}^{j}(x)]\big).\]
So, $\{\big(\cap_{j=0}^{n-1}f_{0}^{-j}(\gamma[f_{0}^{j}(x)])\big)\bigcap\Lambda:x\in F\}$ is an open cover of $\Lambda$, and it is also a subcover of $(\mathscr{A}_\gamma)_{0}^{n}(f_{0,\infty})|_{\Lambda}$. Hence,
\[\mathcal{N}\big((\mathscr{A}_\gamma)_{0}^{n}(f_{0,\infty})|_{\Lambda}\big)\leq |F|=r_n(f_{0,\infty},\Lambda,\gamma).\]
{\rm(ii)} Let $E$ be an $(n,\gamma)$-separated set in $\Lambda$ with $|E|=s_n(f_{0,\infty},\Lambda,\gamma)$. Since every member of $(\mathscr{A}_\eta)_{0}^{n}(f_{0,\infty})|_{\Lambda}$ can contain at most one point of $E$,
\[|E|\leq\mathcal{N}\big((\mathscr{A}_\eta)_{0}^{n}(f_{0,\infty})|_{\Lambda}\big).\] Thus,
\[s_n(f_{0,\infty},\Lambda,\gamma)\leq \mathcal{N}\big((\mathscr{A}_\eta)_{0}^{n}(f_{0,\infty})|_{\Lambda}\big).\]
{\rm(iii)} Let $\mathscr{A}$ be an open cover of $X$. It follows from Lemma \ref{Lebesgue covering lemma} that there exists $\gamma\in{\mathcal{U}}^{s,o}$ such that $\mathscr{A}\prec\mathscr{A}_\gamma$, which yields that
\[\limsup_{n\to\infty}\frac{1}{n}\log\mathcal{N}\big(\mathscr{A}_{0}^{n}(f_{0,\infty})|_{\Lambda}\big)
\leq\limsup_{n\to\infty}\frac{1}{n}\log\mathcal{N}\big((\mathscr{A}_{\gamma})_{0}^{n}(f_{0,\infty})|_{\Lambda}\big).\] Since $\mathscr{A}$ is arbitrary, \[h_{top}(f_{0,\infty},\Lambda)=\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log\mathcal{N}
\big((\mathscr{A}_{\gamma})_{0}^{n}(f_{0,\infty})|_{\Lambda}\big).\] \end{proof}
By Proposition \ref{2}, we have the following result:
\begin{theorem}\label{11} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact uniform space $X$ and $\Lambda$ be a nonempty subset of $X$. Then \[h_{top}(f_{0,\infty},\Lambda)=h(f_{0,\infty},\Lambda).\] \end{theorem}
\begin{remark}\label{1e} {\rm(i)} Below we no longer distinguish between $h_{top}(f_{0,\infty},\Lambda)$ and $h(f_{0,\infty},\Lambda)$, and uniformly use $h(f_{0,\infty},\Lambda)$ to denote the topological entropy of $f_{0,\infty}$ on the set $\Lambda$.
{\rm(ii)} If $f_{n}=f$ for all $n\geq0$, then the limit in {\rm(\ref{subset})} exists by \cite{Adler65}. This, together with Proposition \ref{2}, implies that the $``\limsup"$ in {\rm(\ref{12})} can be replaced by $``\liminf"$. \end{remark}
\section{some properties and calculations of topological entropy}
In this section, some properties and calculations of topological entropy for $(X,f_{0,\infty})$ are investigated.
First, the relations of topological entropy between $(X,f_{0,\infty})$ and its $n$-th iteration system $(X,f_{0,\infty}^{n})$ are studied, where $f_{0,\infty}^{n}=\{f_{kn}^{n}\}_{k=0}^{\infty}$ and $f_{kn}^{n}=f_{kn+n-1}\circ\cdots\circ f_{kn}$ for each $k\geq0$ and $n\geq1$.
\begin{proposition}\label{21} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact uniform space $(X,\mathcal{U})$. Then \[h(f_{0,\infty}^{n})\leq nh(f_{0,\infty}),\;n\geq1.\] Furthermore, if $f_{0,\infty}$ is equi-continuous, then \[h(f_{0,\infty}^{n})=nh(f_{0,\infty}),\;n\geq1.\] \end{proposition}
\begin{proof} Let $n\geq1$. It suffices to prove that $h(f_{0,\infty}^{n})\geq nh(f_{0,\infty})$ when $f_{0,\infty}$ is equi-continuous by Lemma 4.2 in \cite{Kolyada96}. Since $f_{0,\infty}$ is equi-continuous, for any $\gamma\in{\mathcal{U}}^{s,o}$, there exists $\eta\in{\mathcal{U}}^{s,o}$ such that for any $x,y\in X$, \[(x,y)\in \eta\Rightarrow(f_{i}^{k}(x),f_{i}^{k}(y))\in \gamma,\;0\leq k\leq n-1,\;i\geq0.\] This implies that \[(f_{i}^{k}(x),f_{i}^{k}(y))\notin \gamma\; {\rm for}\; {\rm some}\; 0\leq k\leq n-1\;{\rm and}\; i\geq0\Rightarrow(x,y)\notin \eta.\] Thus, if $M$ is an $(mn,\gamma)$-separated set in $X$ under $f_{0,\infty}$ for some $m\geq1$ , then it is also an $(m,\eta)$-separated set in $X$ under $f_{0,\infty}^n$. So, \[s_{mn}(f_{0,\infty},X,\gamma)\leq s_m(f_{0,\infty}^n,X,\eta).\] This, together with the fact that \[s_{(m-1)n+r}(f_{0,\infty},X,\gamma)\leq s_{mn}(f_{0,\infty},X,\gamma),\;r=1,2,\cdots,n,\] implies that \[s_{(m-1)n+r}(f_{0,\infty},X,\gamma)\leq s_m(f_{0,\infty}^n,X,\eta),\;r=1,2,\cdots,n.\] Hence, for any $1\leq r\leq n$, \[\limsup_{m\to\infty}\frac{1}{m}\log s_m(f_{0,\infty}^n,X,\eta)\geq n\limsup_{m\to\infty}\frac{1}{(m-1)n+r} \log s_{(m-1)n+r}(f_{0,\infty},X,\gamma),\] which yields that \[\lim_{\eta\in{\mathcal{U}}^{s,o}}\limsup_{m\to\infty}\frac{1}{m}\log s_m(f_{0,\infty}^n,X,\eta)\geq n\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{m\to\infty}\frac{1}{(m-1)n+r}\log s_{(m-1)n+r}(f_{0,\infty},X,\gamma)\] for any $1\leq r\leq n$, and therefore, $h(f_{0,\infty}^{n})\geq nh(f_{0,\infty})$. \end{proof}
Given another system $(Y,g_{0,\infty})$, where $Y$ is a compact uniform space equipped with a uniform structure $\mathcal{V}$ and $g_{0,\infty}=\{g_{n}\}_{n=0}^{\infty}$ is a sequence of continuous self-maps on $Y$. Let $X\times Y$ be the product space of $X$ and $Y$, and $f_{0,\infty}\times g_{0,\infty}=\{f_{n}\times g_{n}\}_{n=0}^{\infty}$. Then $f_{n}\times g_{n}$ is a continuous self-map on $X\times Y$ for any $n\geq0$. Note that the family of sets of the form $W(\gamma,\eta)$ is a base for the product uniformity of $X\times Y$, where \[W(\gamma,\eta)=\{\big((x,y),(u,v)\big): (x,u)\in \gamma \;{\rm and}\; (y,v)\in \eta\},\;\gamma\in{\mathcal{U}}^{s,o}, \;\eta\in{\mathcal{V}}^{s,o}.\]
\begin{proposition}\label{223} Let $f_{0,\infty}$ and $g_{0,\infty}$ be two sequences of continuous self-maps on compact uniform spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$, respectively. Then \[h(f_{0,\infty}\times g_{0,\infty})\leq h(f_{0,\infty})+h(g_{0,\infty}).\] \end{proposition}
\begin{proof}
Let $\gamma\in{\mathcal{U}}^{s,o}$, $\eta\in{\mathcal{V}}^{s,o}$ and $n\geq1$. Suppose that $F_1$ is a set in $X$ which $(n,\gamma)$-spans $X$ under $f_{0,\infty}$ with $|F_1|=r_n(f_{0,\infty},X,\gamma)$, and $F_2$ is a set in $Y$ which $(n,\eta)$-spans $Y$ under $g_{0,\infty}$ with $|F_2|=r_n(g_{0,\infty},Y,\eta)$. Then $F_1\times F_2$ is a set in $X\times Y$ which $(n,W(\gamma,\eta))$-spans $X\times Y$ under $f_{0,\infty}\times g_{0,\infty}$. In fact, for any $(x,y)\in X\times Y$, there exist $z_1\in F_1$ and $z_2\in F_2$ such that $(f_{0}^{k}(x),f_{0}^{k}(z_1))\in \gamma$ and $(g_{0}^{k}(y),g_{0}^{k}(z_2))\in \eta$ for any $0\leq k\leq n-1$, which yields that \[\big((f_{0}^{k}(x),g_{0}^{k}(y)),(f_{0}^{k}(z_1),g_{0}^{k}(z_2))\big)\in W(\gamma,\eta),\;0\leq k\leq n-1.\] Then \[r_n(f_{0,\infty}\times g_{0,\infty},X\times Y,W(\gamma,\eta))\leq r_n(f_{0,\infty},X,\gamma)r_n(g_{0,\infty},Y,\eta).\] Thus, \begin{align*} &\lim_{\gamma\in{\mathcal{U}}^{s,o},\eta\in{\mathcal{V}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n} \log r_n(f_{0,\infty}\times g_{0,\infty},X\times Y,W(\gamma,\eta))\\ &\leq\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log r_n(f_{0,\infty},X,\gamma) +\lim_{\eta\in{\mathcal{V}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log r_n(g_{0,\infty},Y,\eta). \end{align*} Hence, $h(f_{0,\infty}\times g_{0,\infty})\leq h(f_{0,\infty})+h(g_{0,\infty})$. \end{proof}
If $f_{n}=f$ and $g_{n}=g$ for all $n\geq1$, then ``$\leq$" in Proposition \ref{223} can be replaced by ``$=$".
\begin{Corollary} Let $f$ and $g$ be two continuous self-maps on compact uniform spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$, respectively. Then \[h(f\times g)=h(f)+h(g).\] \end{Corollary}
\begin{proof}
It is only to show $h(f\times g)\geq h(f)+h(g)$ by Proposition \ref{223}. Let $\gamma\in{\mathcal{U}}^{s,o}$, $\eta\in{\mathcal{V}}^{s,o}$ and $n\geq1$. Suppose that $E_1$ is an $(n,\gamma)$-separated set in $X$ under $f$ with $|E_1|=s_n(f,X,\gamma)$, and $E_2$ is an $(n,\eta)$-separated set in $Y$ under $g$ with $|E_2|=s_n(g,Y,\eta)$. It is to show that $E_1\times E_2$ is an $(n,W(\gamma,\eta))$-separated set in $X\times Y$ under $f\times g$. Let $x=(x_1,x_2),y=(y_1,y_2)\in E_1\times E_2$ with $x\neq y$. Without loss of generality, suppose that $x_{1}\neq y_{1}$. Then there exists $0\leq j\leq n-1$ such that $(f^{j}(x_{1}),f^{j}(y_{1}))\notin\gamma$, which implies that \[\big((f^{j}(x_{1}),g^{j}(x_{2})),(f^{j}(y_{1}),g^{j}(y_{2}))\big)\notin W(\gamma,\eta).\] Thus, \[s_n(f\times g,X\times Y,W(\gamma,\eta))\geq s_n(f,X,\gamma)s_n(g,Y,\eta),\] which yields that \begin{align*} &\lim_{\gamma\in{\mathcal{U}}^{s,o},V\in{\mathcal{V}}^{s,o}}\liminf_{n\to\infty}\frac{1}{n}\log s_n(f\times g,X\times Y,W(\gamma,\eta))\\ &\geq \lim_{\gamma\in{\mathcal{U}}^{s,o}}\liminf_{n\to\infty}\frac{1}{n}\log s_n(f,X,\gamma) +\lim_{\gamma\in{\mathcal{U}}^{s,o}}\liminf_{n\to\infty}\frac{1}{n}\log s_n(g,Y,\eta). \end{align*} This, together with Remark \ref{1e} (ii), implies that $h(f\times g)\geq h(f)+h(g)$. \end{proof}
The following result reveals the relations of topological entropy between $(X,f_{0,\infty})$ and its $k$-th product system $(X^k,f_{0,\infty}^{(k)})$, where $X^k=\underbrace{X\times\cdots\times X}_{k}$, $f_{0,\infty}^{(k)}=\{f_{n}^{(k)}\}_{n=0}^{\infty}$ and $f_{n}^{(k)}=\underbrace{f_n\times\cdots\times f_n}_{k}$, $k\geq1$.
\begin{proposition}\label{22} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact uniform space $(X,\mathcal{U})$. Then \[h(f_{0,\infty}^{(k)})=kh(f_{0,\infty}),\;k\geq1.\] \end{proposition}
\begin{proof}
Let $k\geq1$. By Proposition \ref{223}, $h(f_{0,\infty}^{(k)})\leq kh(f_{0,\infty})$. On the other hand, let $\eta\in{\mathcal{U}}^{s,o}$ and $E\subset X$ be an $(n,\eta)$-separated set under $f_{0,\infty}$ with $|E|=s_n(f_{0,\infty},X,\eta)$ for any fixed $n\geq1$. Then $E^k\subset X^k$ is an $(n,W(\underbrace{\eta,\cdots,\eta}_{k}))$-separated set under $f_{0,\infty}^{(k)}$. In fact, for any $x=(x_1,\cdots,x_k),y=(y_1,\cdots,y_k)\in E^k$ with $x\neq y$, there exists $1\leq i_0\leq k$ such that $x_{i_0}\neq y_{i_0}$. Then there exists $0\leq j\leq n-1$ such that $(f_{0}^{j}(x_{i_0}),f_{0}^{j}(y_{i_0}))\notin\eta$, which implies that \[\big((f_{0}^{j}(x_{1}),\cdots,f_{0}^{j}(x_{k})),(f_{0}^{j}(y_{1}),\cdots,f_{0}^{j}(y_{k}))\big)\notin W(\eta,\cdots,\eta).\] Thus, \[s_n\big(f_{0,\infty}^{(k)},X^k,W(\eta,\cdots,\eta)\big)\geq \big(s_n(f_{0,\infty},X^k,\eta)\big)^{k}.\] So, \begin{align*} \lim_{\eta\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log s_n(f_{0,\infty}^{(k)},X^k,W(\eta,\cdots,\eta)) &\geq k\lim_{\eta\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log s_n(f_{0,\infty},X,\eta)\\ &=kh(f_{0,\infty}). \end{align*} Therefore, $h(f_{0,\infty}^{(k)})\geq kh(f_{0,\infty})$. \end{proof}
An open set $U\subset X$ is said to be wandering under $(X,f_{0,\infty})$ if $f_{n}^{k}(U)\cap U=\emptyset$ for all $n\geq0$ and $k\geq1$, and $x\in X$ is called a wandering point if it belongs to some wandering set. Thus, $x\in X$ is called a non-wandering point under $(X,f_{0,\infty})$ if for any open set $U$ containing $x$, there exist $n\geq0$ and $k\geq1$ such that $f_{n}^{k}(U)\cap U\neq\emptyset$. Denote the set of all non-wandering points of $(X,f_{0,\infty})$ by $\Omega(f_{0,\infty})$.
\begin{theorem}\label{h} Let $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ be a sequence of equi-continuous self-maps on a compact uniform space $(X,\mathcal{U})$. Then \[h(f_{0,\infty})= h\big(f_{0,\infty},\Omega(f_{0,\infty})\big).\] \end{theorem}
\begin{proof}
It is only to show that $h(f_{0,\infty})\leq h(f_{0,\infty},\Omega(f_{0,\infty}))$ since ``$\geq$" is trivial. Let $\mathscr{A}$ be an open cover of $X$. Fix $n\geq1$. Let $\mathscr{B}_n$ be a minimal subcover of $\Omega(f_{0,\infty})$ which is chosen from ${\mathscr{A}}_{0}^{n}(f_{0,\infty})$. Denote $K=X\setminus\cup_{B\in\mathscr{B}_n}B$. Then $K$ is compact and all the points in $K$ are wandering. For any $x\in K$, there exists $A_x\in{\mathscr{A}}_{0}^{n}(f_{0,\infty})$ such that $x\in A_x$. Since $x$ is wandering, there exists an open neighborhood $U_{x}$ of $x$ such that $U_{x}$ is a wandering set and $U_{x}\subset A_x$. The open cover $\{U_x\cap K: x\in K\}$ of $K$ has a finite subcover $\{U_{x_i}\cap K: 1\leq i\leq l\}$ for some $l\geq1$. Denote $\mathscr{C}_n=\mathscr{B}_n\cup\{U_{x_1}\cap K,\cdots,U_{x_l}\cap K\}$. Then $\mathscr{C}_n$ is an open cover of $X$ and ${\mathscr{A}}_{0}^{n}(f_{0,\infty})\prec\mathscr{C}_n$. For any $k\geq2$, we consider any nonempty element of $(\mathscr{C}_n)_{0}^{k}(f_{0,\infty}^{n})$, it is of the form \[C_{0}\cap f_{0}^{-n}(C_{1})\cap\cdots\cap f_{0}^{-n}\circ f_{n}^{-n}\circ\cdots\circ f_{(k-2)n}^{-n}(C_{k-1}),\] where $C_i\in\mathscr{C}_n$ for any $0\leq i\leq k-1$. If $C_i=C_j$ for some $i<j$, then \[\emptyset\neq f_{(j-1)n}^{n}\circ\cdots\circ f_{in}^{n}(C_{i})\cap C_i=f_{in}^{(j-i)n}(C_i)\cap C_i,\] which implies that $C_i$ is not wandering, and thus $C_i\in\mathscr{B}_n$. With the same method used in the proof of Lemma 4.1.5 in \cite{Alse93}, one shows that
\[|(\mathscr{C}_n)_{0}^{k}(f_{0,\infty}^{n})|\leq(m+1)!k^m|\mathscr{B}_n|^k,\]
where $|\mathscr{B}_n|$ denotes the number of elements in $\mathscr{B}_n$ and
$m=|\mathscr{C}_n\setminus\mathscr{B}_n|$. Thus, \[h(f_{0,\infty}^n,\mathscr{C}_n) =\limsup_{k\to\infty}\frac{1}{k}\log\mathcal{N}\big((\mathscr{C}_n)_{0}^{k}(f_{0,\infty}^{n})\big)
\leq\limsup_{k\to\infty}\frac{1}{k}\log((m+1)!k^m|\mathscr{B}_n|^k)=\log{\mathscr{B}_n}.\] For any $\epsilon>0$, there exists an open cover $\mathscr{A}$ of $X$ such that \[h(f_{0,\infty}^n)<h(f_{0,\infty}^n,\mathscr{A})+\epsilon.\] This, together with Proposition \ref{21}, implies that \begin{align*} h(f_{0,\infty})&=\frac{1}{n}h(f_{0,\infty}^n)<\frac{1}{n}h(f_{0,\infty}^n,\mathscr{A})+\frac{\epsilon}{n} \leq\frac{1}{n}h\big(f_{0,\infty}^n,{\mathscr{A}}_{0}^{n}(f_{0,\infty})\big)+\frac{\epsilon}{n}\\ &\leq\frac{1}{n}h(f_{0,\infty}^n,\mathscr{C}_n)+\frac{\epsilon}{n}
\leq\frac{1}{n}\log{|\mathscr{B}_n|}+\frac{\epsilon}{n}
=\frac{1}{n}\log\mathcal{N}({\mathscr{A}}_{0}^{n}(f_{0,\infty})|_{\Omega(f_{0,\infty})})+\frac{\epsilon}{n}. \end{align*} Therefore, $h(f_{0,\infty})\leq h\big(f_{0,\infty},\Omega(f_{0,\infty})\big)$. \end{proof}
\begin{lemma}\cite{Kolyada96}\label{131} Let $f_{0,\infty}$ be a sequence of continuous self-maps of a compact topological space $X$. Then for every $1\leq i\leq j$ and every open cover $\mathscr{A}$ of $X$, $h(f_{i,\infty},\mathscr{A})\leq h(f_{j,\infty},\mathscr{A})$ and $h(f_{i,\infty})\leq h(f_{j,\infty})$. \end{lemma}
Based on Lemma \ref{131}, the concept of asymptotical topological entropy is introduced for $f_{0,\infty}$ on a compact topological space in \cite{Kolyada96}. Define \[h^{*}(f_{\infty})=\sup\{h^{*}(f_{\infty},\mathscr{A}): \mathscr{A}\; {\rm is\; an\; open\; cover\; of\;} X\;\}\] as the asymptotical topological entropy of $(X,f_{0,\infty})$, where \[h^{*}(f_{\infty},\mathscr{A})=\lim_{n\to\infty}h(f_{n,\infty},\mathscr{A}).\] It is easy to see that \[h^{*}(f_{\infty})=\lim_{n\to\infty}h(f_{n,\infty}).\]
Recall that a sequence of continuous self-maps $\{f_n\}_{n=0}^{\infty}$ on a compact uniform space $(X,\mathcal{U})$ converges uniformly to $f$ if for any $\gamma\in\mathcal{U}$, there exists $N\geq1$ such that $(f_n(x),f(x))\in\gamma$ for any $x\in X$ and $n\geq N$.
\begin{theorem}\label{s} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact uniform space $(X,\mathcal{U})$. If $f_{0,\infty}$ converges uniformly to $f$, then \[h(f_{0,\infty})\leq \cdots\leq h(f_{n,\infty})\leq\cdots\leq h^{*}(f_{\infty})\leq h(f),\;n\geq1.\] \end{theorem}
\begin{proof} Let $\{x_n\}_{n=0}^{\infty}\subset[0,1]$ be a sequence of mutually different points converging to a point $z$. Define a continuous self-map $F$ on $(\{x_n\}_{n=0}^{\infty}\cup\{z\})\times X$ by \[F(z,y)=(z,f(y)),\;y\in X,\] and \[F(x_n,y)=(x_{n+1},f_n(y)),\;y\in X,\;n\geq0.\] It is easy to see that $\Omega(F)\subset\{z\}\times X$. Then \[h(F)=h(F,\Omega(F))=h(F,\{z\}\times X).\] Thus, \begin{align}\label{113} h(F,\{x_n\}\times X)\leq h(F,\{z\}\times X),\;n\geq0. \end{align} It follows from the definitions of topological entropy that \[h(F,\{x_n\}\times X)=h(f_{n,\infty}),\;n\geq0,\] and \[h(F,\{z\}\times X)=h(f).\] This, together with (\ref{113}), implies that \[h(f_{n,\infty})\leq h(f),\;n\geq0.\] Hence, \[h^{*}(f_{\infty})=\lim_{n\to\infty}h(f_{n,\infty})\leq h(f).\] \end{proof}
\begin{remark} Theorems \ref{h} and \ref{s} are generalizations of Theorems H and E in \cite{Kolyada96} to a uniform space. \end{remark}
Let $\mathscr{K}(X)$ be the hyperspace of all nonempty compact subsets of $X$. Recall that a Borel probability measure $\mu$ on $X$ is $f_{0,\infty}$-homogeneous, if
{\rm (i)} $\mu(K)<\infty$ for all $K\in\mathscr{K}(X)$;
{\rm (ii)} $\mu(K)>0$ for some $K\in\mathscr{K}(X)$;
{\rm (iii)} for any $\gamma\in{\mathcal{U}}^{s,o}$, there exist $\eta\in{\mathcal{U}}^{s,o}$ and $c>0$ such that for any $x,y\in X$ and $n\geq1$, \begin{align}\label{ying} \mu(D_n(f_{0,\infty},y,\eta))\leq c\mu(D_n(f_{0,\infty},x,\gamma)), \end{align} where \[D_n(f_{0,\infty},x,\gamma)=\bigcap_{k=0}^{n-1}f_{0}^{-k}(\gamma[f_{0}^{k}(x)]).\] For such a $\mu$, define \begin{align}\label{ooo} k(f_{0,\infty},\mu)=\lim_{\eta\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}-\frac{1}{n}\log \mu\big(D_n(f_{0,\infty},y,\eta)\big). \end{align} Note that $k(f_{0,\infty},\mu)$ does not depend on $y$ used by (\ref{ying}). Then we have the following calculation of topological entropy.
\begin{theorem}\label{h1l} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact uniform space $(X,\mathcal{U})$. If there exists a Borel probability measure $\mu$ on $X$ which is $f_{0,\infty}$-homogeneous, then \[h(f_{0,\infty})=k(f_{0,\infty},\mu),\] where $k(f_{0,\infty},\mu)$ is specified in {\rm(\ref{ooo})}. \end{theorem}
\begin{proof}
Let $\gamma\in{\mathcal{U}}^{s,o}$, $n\geq1$ and $E\subset X$ be an $(n,\gamma)$-separated set under $f_{0,\infty}$ with $|E|=s_n(f_{0,\infty},X,\gamma)$. Clearly, there exists $\eta\in{\mathcal{U}}^{s,o}$ such that $\eta^2\subset\gamma$. It is easy to verify that \[D_n(f_{0,\infty},x_1,\eta)\cap D_n(f_{0,\infty},x_2,\eta)=\emptyset \;{\rm if}\;x_1\neq x_2\in E.\] Since $\mu$ is $f_{0,\infty}$-homogeneous, there exists $\gamma'\in{\mathcal{U}}^{s,o}$ and $c>0$ such that for all $x,y\in X$, \[\mu(D_n(f_{0,\infty},y,\gamma'))\leq c\mu(D_n(f_{0,\infty},x,\eta)).\] Then \begin{align*} s_n(f_{0,\infty},X,\gamma)\mu(D_n(f_{0,\infty},y,\gamma'))&\leq c\sum_{x\in E}\mu(D_n(f_{0,\infty},x,\eta))\\ &=c\mu\big(\bigcup_{x\in E}D_n(f_{0,\infty},x,\eta)\big)\leq c\mu(X). \end{align*} By the fact that $\mu(X)<\infty$, one gets that \[\limsup_{n\to\infty}\frac{1}{n}\log s_n(f_{0,\infty},X,\gamma)\leq\limsup_{n\to\infty}-\frac{1}{n}\log\mu(D_n(f_{0,\infty},y,\gamma')).\] Then \[\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log s_n(f_{0,\infty},X,\gamma)\leq\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}-\frac{1}{n}\log\mu(D_n(f_{0,\infty},y,\gamma')),\] which implies that $h(f_{0,\infty})\leq k(\mu,f_{0,\infty})$.
On the other hand, let $\gamma\in{\mathcal{U}}^{s,o}$ and $K\in\mathscr{K}(X)$ be the set satisfying that $\mu(K)>0$. Then there exist $\eta\in{\mathcal{U}}^{s,o}$ and $c>0$ such that for all $n\geq1$ and $x,y\in X$, \[\mu(D_n(f_{0,\infty},x,\eta))\leq c\mu(D_n(f_{0,\infty},y,\gamma)).\]
Let $n\geq1$ and $F$ be a subset of $K$ which $(n,\eta)$-spans $K$ with $|F|=r_n(f_{0,\infty},K,\eta)$. Then \[K\subset\bigcup_{x\in F}D_n(f_{0,\infty},x,\eta).\] Thus, \begin{align*} 0<\mu(K)\leq\mu(\bigcup_{x\in F}D_n(f_{0,\infty},x,\eta)) \leq\sum_{x\in F}\mu(D_n(f_{0,\infty},x,\eta)) \leq cr_n(f_{0,\infty},K,\eta)\mu(D_n(f_{0,\infty},y,\gamma)), \end{align*} which implies that \[\limsup_{n\to\infty}\frac{1}{n}\log r_n(f_{0,\infty},K,\eta)\geq\limsup_{n\to\infty}-\frac{1}{n}\log\mu(D_n(f_{0,\infty},y,\gamma)).\] So, \[\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log r_n(f_{0,\infty},\eta,K)\geq\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}-\frac{1}{n}\log\mu(D_n(f_{0,\infty},y,\gamma)).\] Hence, \[h(f_{0,\infty})\geq h(f_{0,\infty},K)\geq k(\mu,f_{0,\infty}).\] \end{proof}
\begin{remark} Theorem \ref{h1l} extends Proposition 7 in \cite{Bowen71} to a nonautonomous dynamical system generated by a sequence of continuous seif-maps on a compact uniform space. \end{remark}
\section{Topological equi-semi-conjugacy}
In this section, the relations of topological entropy between two topologically equi-semicon- jugate nonautonomous dynamical systems are studied.
The sequence $\{\Lambda_n\}_{n=0}^{\infty}$ of subsets of $X$ is called invariant under $(X,f_{0,\infty})$ if $f_n(\Lambda_n)\subset\Lambda_{n+1}$ for all $n\geq0$. Then $(X,f_{0,\infty})$ restricted to $\{\Lambda_n\}_{n=0}^{\infty}$ is called the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$.
\begin{definition}
Let $\{\Lambda_n\}_{n=0}^{\infty}$ and $\{D_n\}_{n=0}^{\infty}$ be invariant under $(X,f_{0,\infty})$ and $(Y,g_{0,\infty})$, respectively. If, for each $n\geq0$, there exists a surjective map $h_n:\Lambda_n\to D_n$ such that $h_{n+1}\circ f_n=g_n\circ h_n$, and the sequence of maps $\{h_n\}_{n=0}^{\infty}$ is equi-continuous, then the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$ is said to be topologically $\{h_n\}_{n=0}^{\infty}$-equi-semiconjugate to the invariant subsystem of $(Y,g_{0,\infty})$ on $\{D_n\}_{n=0}^{\infty}$. Furthermore, if $\{h_n^{-1}\}_{n=0}^{\infty}$ is also equi-continuous, they are said to be topologically $\{h_n\}_{n=0}^{\infty}$-equi-conjugate. In the case that $\Lambda_n=X$ and $D_n=Y$ for each $n\geq0$, $(X,f_{0,\infty})$ is said to be topologically $\{h_n\}_{n=0}^{\infty}$-equi-(semi)conjugate to $(Y,g_{0,\infty})$. If there exists $c>0$ such that $\sup_{y\in Y}|h_{n}^{-1}(y)|\leq c$ for any $n\geq0$, then $\{h_n\}_{n=0}^{\infty}$ is called finite-to-one. \end{definition}
Recall that $\{h_n\}_{n=0}^{\infty}$ is equi-continuous if for any $\gamma\in{\mathcal{V}}^{s,o}$, there exists $\eta\in{\mathcal{U}}^{s,o}$ such that $(h_n(x),h_n(y))\in \gamma$ for each $n\geq0$ and any $x,y\in\Lambda_n$ with $(x,y)\in\eta$.
\begin{remark} It is well known that if two autonomous dynamical systems are topologically conjugate, then their topological properties are all the same. However, this is not necessarily true for nonautonomous dynamical systems, even if two systems are topologically equi-conjugate, see example 3.1 in \cite{Shi09}. Fortunately, topological entropy is an invariant under topological equi-conjugacy. \end{remark}
\begin{theorem}\label{277} Let $f_{0,\infty}$ and $g_{0,\infty}$ be two sequences of continuous self-maps on compact uniform spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$, respectively, $\Lambda_n\subset X$ and $D_n\subset Y$ for each $n\geq0$. Assume that there exists a sequence of maps $h_n: \Lambda_n\to D_n$, $n\geq0$, such that the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$ is topologically $\{h_n\}_{n=0}^{\infty}$-equi-semiconjugate to the invariant subsystem of $(Y,g_{0,\infty})$ on $\{D_n\}_{n=0}^{\infty}$. Then \[h(f_{0,\infty},\Lambda_0)\geq h(g_{0,\infty},D_0).\] Consequently, if the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$ is topologically $\{h_n\}_{n=0}^{\infty}$-equi-conjugate to the invariant subsystem of $(Y,g_{0,\infty})$ on $\{D_n\}_{n=0}^{\infty}$, then \[h(f_{0,\infty},\Lambda_0)=h(g_{0,\infty},D_0).\] \end{theorem}
\begin{proof} Since $\{h_n\}_{n=0}^{\infty}$ is equi-continuous, for any $\gamma\in{\mathcal{V}}^{s,o}$, there exists $\eta\in{\mathcal{U}}^{s,o}$ such that for any $n\geq0$ and $x,y\in\Lambda_n$, \begin{align}\label{18} (x,y)\in \eta\Rightarrow(h_n(x),h_n(y))\in \gamma. \end{align}
Let $n\geq0$ and $E\subset D_0$ be an $(n,\gamma)$-separated set under $g_{0,\infty}$ with $|E|=s_n(g_{0,\infty},D_0,\gamma)$. Fix $x_e\in h_{0}^{-1}(e)\subset\Lambda_0$ for any $e\in E$ and denote $E'=\{x_e: e\in E\}$. Then $|E'|=|E|$. We claim that $E'$ is an $(n,\eta)$-separated set of $\Lambda_0$ under $f_{0,\infty}$. In fact, for any $x_{e_1}\neq x_{e_2}\in E'$, $e_1\neq e_2\in E$, and then there exists $0\leq k\leq n-1$ such that $(g_{0}^{k}(e_1),g_{0}^{k}(e_2))\notin\gamma$. Since \[g_{0}^{k}(e_i)=g_{0}^{k}\circ h_0(x_{e_i})=h_k\circ f_{0}^{k}(x_{e_i}),\;i=1,2,\] $(h_k\circ f_{0}^{k}(x_{e_1}),h_k\circ f_{0}^{k}(x_{e_2}))\notin \gamma$. It follows from (\ref{18}) that $(f_{0}^{k}(x_{e_1}),f_{0}^{k}(x_{e_2}))\notin\eta$. Thus, \[s_n(g_{0,\infty},D_0,\gamma)\leq s_n(f_{0,\infty},\Lambda_0,\eta),\;n\geq0,\;\gamma\in{\mathcal{V}}^{s,o},\] which implies that \begin{align*} \lim_{\gamma\in{\mathcal{V}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log s_n(g_{0,\infty},D_0,\gamma) &\leq\lim_{\gamma\in{\mathcal{V}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log s_n(f_{0,\infty},\Lambda_0,\eta)\\ &\leq\lim_{\eta\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log s_n(f_{0,\infty},\Lambda_0,\eta). \end{align*} Hence, $h(f_{0,\infty},\Lambda_0)\geq h(g_{0,\infty},D_0)$. \end{proof}
In the special case that $\Lambda_n=X$ and $D_n=Y$ for all $n\geq0$, we get the following result.
\begin{Corollary}\label{515} Let $f_{0,\infty}$ and $g_{0,\infty}$ be two sequences of continuous self-maps on compact uniform spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$, respectively. If $(X,f_{0,\infty})$ is topologically equi-semiconjugate to $(Y,g_{0,\infty})$, then \[h(f_{0,\infty})\geq h(g_{0,\infty}).\] Consequently, if $(X,f_{0,\infty})$ is topologically equi-conjugate to $(Y,g_{0,\infty})$, then \[h(f_{0,\infty})=h(g_{0,\infty}).\] \end{Corollary}
Let $R$ be an equivalent relation on $X$ satisfying that for any $\gamma\in\mathcal{U}$, there exists $\eta\in\mathcal{U}$ such that $\eta\circ R\circ\eta\subset R\circ\gamma\circ R$. Then the quotient map $\xi:X\to X/R$ generates a uniformity $\mathcal{U}/R$ on $X/R$ \cite{Cech66,Hood74}, where \[\mathcal{U}/R=\{\gamma'\subset X/R\times X/R: (\xi\times \xi)^{-1}\gamma'\in\mathcal{U}\}=\{\gamma_R=(\xi\times \xi)\gamma: \gamma\in\mathcal{U}\}.\] Note that \begin{align}\label{009} (\xi\times \xi)^{-1}\gamma_R=(\xi\times \xi)^{-1}(\xi\times \xi)\gamma=R\circ\gamma\circ R. \end{align} Let $\mathcal{U}^R$ be the uniformity on $X$ with base $\{R\circ\gamma\circ R: \gamma\in\mathcal{U}\}$.
\begin{lemma}\cite{Hood74}\label{31} Let $f$ be a continuous self-map on a compact uniform spaces $(X,\mathcal{U})$ and $s$ be a self-map on $X/R$ satisfying that $s\circ\xi=\xi\circ f$. Then $f$ is continuous on $(X,\mathcal{U}^R)$ and $s$ is continuous on $(X/R,\mathcal{U}/R)$. \end{lemma}
\begin{theorem}\label{32} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact uniform space $(X,\mathcal{U})$. Assume that there exists a sequence of maps $s_{0,\infty}=\{s_n\}_{n=0}^{\infty}$ on $(X/R,\mathcal{U}/R)$ satisfying that $s_n\circ\xi=\xi\circ f_n$ for any $n\geq0$. Then \[h(f_{0,\infty},\mathcal{U}^{R})\leq h(s_{0,\infty},\mathcal{U}/R)\leq h(f_{0,\infty},\mathcal{U}).\] \end{theorem}
\begin{proof} For any $n\geq0$, $s_n(X/R)\subset X/R$ since $\xi$ is surjective and $s_n\circ\xi=\xi\circ f_n$, and $s_n$ is continuous on $(X/R,\mathcal{U}/R)$ by Lemma \ref{31}. By the fact that $\xi$ is continuous and surjective, $(X,\mathcal{U},f_{0,\infty})$ is topologically equi-semiconjugate to $(X/R,\mathcal{U}/R,s_{0,\infty})$. Thus, $h(s_{0,\infty},\mathcal{U}/R)\leq h(f_{0,\infty},\mathcal{U})$ by Theorem \ref{277}.
Let $\tilde{F}$ be a subset of $X/R$ which $(n,\gamma_R)$-spans $X/R$ under $s_{0,\infty}$ with $|\tilde{F}|=r_n(\gamma_R,X/R,s_{0,\infty})$. For any $x\in X$, there exists $\tilde{y}\in\tilde{F}$ such that $(s_{0}^{j}\circ\xi(x),s_{0}^{j}(\tilde{y}))\in\gamma_R$ for any $0\leq j\leq n-1$. For any $\tilde{z}\in \tilde{F}$, take one point $z\in X$ such that $\xi(z)=\tilde{z}$. Denote
$F=\{z: \tilde{z}\in F\}$. Then $|F|=|\tilde{F}|$ and $\xi(F)=\tilde{F}$. Thus, there exists $y\in F$ such that $\tilde{y}=\xi(y)$. So, \[(\xi\circ f_{0}^{j}(x),\xi\circ f_{0}^{j}(y))=(s_{0}^{j}\circ\xi(x),s_{0}^{j}\circ\xi(y)) =(s_{0}^{j}\circ\xi(x),s_{0}^{j}(\tilde{y}))\in\gamma_R,\;0\leq j\leq n-1.\] This, together with (\ref{009}), implies that \[(f_{0}^{j}(x),f_{0}^{j}(y))\in(\xi\times\xi)^{-1}\gamma_R=R\circ\gamma\circ R,\;0\leq j\leq n-1.\] Hence, $F$ is a subset of $X$ which $(n,R\circ\gamma\circ R)$-spans $X$ under $f_{0,\infty}$ and
\[r_n(f_{0,\infty},X,R\circ\gamma\circ R)\leq|F|=|\tilde{F}|=r_n(s_{0,\infty},X/R,\gamma_R).\] Therefore, $h(s_{0,\infty},\mathcal{U}/R)\geq h(f_{0,\infty},\mathcal{U}^{R})$. \end{proof}
\begin{remark} There may be many different uniformities on the set $X$. Theorem \ref{32} shows that topological entropy depends on the uniformity on $X$. \end{remark}
The topological sup-entropy of $f_{0,\infty}$ on a subset of $X$ is proposed in \cite{Kolyada96}, we extend it to uniform spaces. Suppose that $f_{0,\infty}$ is equi-continuous on $X$. Let $n\geq1$ and $\gamma\in{\mathcal{U}}^{s,o}$. A subset $E^*\subset X$ is called $(n,\gamma)^*$-separated if for each pair $x\neq y\in E^*$, there exist $i\geq0$ and $0\leq j\leq n-1$ such that $(f_i^j(x),f_i^j(y))\notin\gamma$; a subset $F^*\subset X$ is called $(n,\gamma)^*$-spans another set $K\subset X$ if for each $x\in K$ there exists $y\in F^*$ such that $(f_i^j(x),f_i^j(y))\in\gamma$ for any $0\leq j\leq n-1$ and $i\geq0$. For a nonempty subset $\Lambda$ of $X$, let $s_n^*(f_{0,\infty},\Lambda,\gamma)$ be the maximal cardinality of an $(n,\gamma)^*$-separated set in $\Lambda$ under $f_{0,\infty}$ and $r_n^*(f_{0,\infty},\Lambda,\gamma)$ be the minimal cardinality of a set in $\Lambda$ which $(n,\gamma)^*$-spans $\Lambda$ under $f_{0,\infty}$. $s_n^*(f_{0,\infty},\Lambda,\gamma)$ and $r_n^*(f_{0,\infty},\Lambda,\gamma)$ are finite since $X$ is compact and $f_{0,\infty}$ is equi-continuous. With a similar proof to that of Proposition \ref{1}, one easily shows that for any $\gamma_1,\gamma_2\in\mathcal{U}^{s,o}$ with $\gamma_1\subset\gamma_2$, \[s_n^*(f_{0,\infty},\Lambda,\gamma_1)\geq s_n^*(f_{0,\infty},\Lambda,\gamma_2),\;r_n^*(f_{0,\infty},\Lambda,\gamma_1)\geq r_n^*(f_{0,\infty},\Lambda,\gamma_2),\] and for any $\gamma,\eta\in{\mathcal{U}}^{s,o}$ with $\eta^2\subset\gamma$, \[r_n^*(f_{0,\infty},\Lambda,\gamma)\leq s_n^*(f_{0,\infty},\Lambda,\gamma)\leq r_n^*(f_{0,\infty},\Lambda,\eta).\] Define the topological sup-entropy of $f_{0,\infty}$ on the subset $\Lambda$ of $X$ as \[H(f_{0,\infty},\Lambda)=\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log s_n^*(f_{0,\infty},\Lambda,\gamma) =\lim_{\gamma\in{\mathcal{U}}^{s,o}}\limsup_{n\to\infty}\frac{1}{n}\log r_n^*(f_{0,\infty},\Lambda,\gamma).\] Clearly, $h(f_{0,\infty},\Lambda)\leq H(f_{0,\infty},\Lambda)$.
\begin{theorem}\label{h1} Let $f_{0,\infty}$ be a sequence of equi-continuous self-maps on a compact Hausdorff uniform space $(X,\mathcal{U})$ and $g_{0,\infty}$ be a sequence of continuous self-maps on a compact Hausdorff uniform space $(Y,\mathcal{V})$. Assume that $(X,f_{0,\infty})$ is topologically $\{\pi_n\}_{n=0}^{\infty}$-equi-semiconjugate to $(Y,g_{0,\infty})$ and $\{\pi_n\}_{n=0}^{\infty}\subset\{\phi_1,\cdots,\phi_k\}$ for some $k\geq1$. Then \[h(g_{0,\infty})\leq h(f_{0,\infty})\leq h(g_{0,\infty})+\max_{1\leq j\leq k}\sup_{y\in Y}H(f_{0,\infty},\phi_{j}^{-1}(y)).\] Consequently, if $\{\pi_n\}_{n=0}^{\infty}$ is finite-to-one, then \[h(f_{0,\infty})=h(g_{0,\infty}).\] \end{theorem}
\begin{proof} It follows from Corollary \ref{515} that $h(f_{0,\infty})\geq h(g_{0,\infty})$.
Denote $a:=\max_{1\leq j\leq k}\sup_{y\in Y}H(f_{0,\infty},\phi_{j}^{-1}(y))$. It is to show that $h(f_{0,\infty})\leq h(g_{0,\infty})+a$. If $a=\infty$, then we are done. Suppose that $a<\infty$. Let $u\in{\mathcal{U}}^{s,o}$. Then there exists $\gamma\in{\mathcal{U}}^{s,o}$ such that $\gamma^{2}\subset u$. By the definition of $H(f_{0,\infty},\phi_{j}^{-1}(y))$, there exists $m_j(y)\geq1$ such that \[\frac{1}{m_j(y)}\log r_{m_j(y)}^{*}(f_{0,\infty},\phi_{j}^{-1}(y),\gamma)\leq H(f_{0,\infty},\phi_{j}^{-1}(y))\leq a\] for any $y\in Y$ and $1\leq j\leq k$, and let
$F^{*}_y(j)\subset\phi_{j}^{-1}(y)$ which $(m_j(y),\gamma)$-spans $\phi_{j}^{-1}(y)$ with $|F^{*}_y(j)|=r_{m_j(y)}^{*}(f_{0,\infty},\phi_{j}^{-1}(y),\gamma)$, then \begin{align}\label{lo}
\frac{1}{m_j(y)}\log|F^{*}_y(j)|\leq a. \end{align} For any $y\in Y$ and $1\leq j\leq k$, denote \begin{align}\label{lo2} U_{y}^{j}=\bigcup_{z\in F^{*}_y(j)}D^{*}_{m_j(y)}(f_{0,\infty},z,\gamma), \end{align} where \begin{align}\label{lo3} D^{*}_{m_j(y)}(f_{0,\infty},z,\gamma)=\{\omega\in X: (f_{i}^{t}(w),f_{i}^{t}(z))\in\gamma, \;0\leq t\leq m_j(y)-1,\; i\geq0\}. \end{align} Since $f_{0,\infty}$ is equi-continuous, $D^{*}_{m_j(y)}(f_{0,\infty},z,\gamma)$ is open, and thus $U_{y}^{j}$ is open. Clearly, $\phi_{j}^{-1}(y)\subset U_{y}^{j}$, which implies that \[(X\setminus U_{y}^{j})\bigcap(\bigcap_{\gamma\in{\mathcal{U}}^{s,o}}\phi_{j}^{-1}(\bar{\gamma}[y]))=\emptyset.\] Thus, there exists $\gamma_{j,y}\in{\mathcal{U}}^{s,o}$ such that $\phi_{j}^{-1}(\gamma_{j,y}[y])\subset U_{y}^{j}$ since $X$ is compact. Denote $W_y=\bigcap_{1\leq j\leq k}\gamma_{j,y}[y].$ Then \begin{align}\label{ll} \phi_{j}^{-1}(W_{y})\subset U_{y}^{j},\;1\leq j\leq k. \end{align}
Let $\{W_{y_1},\cdots,W_{y_p}\}$, $p\geq1$, be a subcover of the open cover $\{W_{y}: y\in X\}$. By Lemma \ref{Lebesgue covering lemma}, there exists $\eta\in{\mathcal{U}}^{s,o}$ such that $\eta[x]$ is a subset of some member of $\{W_{y_1},\cdots,W_{y_p}\}$ for any $x\in X$. For each $n\geq0$, there exists $1\leq j_n\leq k$ such that \begin{align}\label{5151} \pi_n=\phi_{j_n}. \end{align}
Let $n\geq1$ and $E_n$ be a subset of $Y$ which $(n,\eta)$-spans $Y$ under $g_{0,\infty}$ with $|E_n|=r_n(g_{0,\infty},Y,\eta)$. For any $y\in E_n$, there exists $c_0(y)\in\{y_1,\cdots,y_p\}$ such that $\eta[y]\subset W_{c_0(y)}$, and define $t_0=0$; let $t_1(y)=m_{j_0}(c_0(y))$, and let $c_1(y)\in\{y_1,\cdots,y_p\}$ such that $\eta[g_0^{t_1(y)}(y)]\subset W_{c_1(y)}$. Inductively, assume that $t_0(y),\cdots,t_k(y)$ and $c_0(y),\cdots,c_k(y)$ are already defined, define \[t_{k+1}(y)=t_k(y)+m_{j_{t_{k}}}(c_k(y)),\] and define $c_{k+1}(y)\in\{y_1,\cdots,y_p\}$ satisfying that \begin{align}\label{lll} \eta[g_0^{t_{k+1}(y)}(y)]\subset W_{c_{k+1}(y)}. \end{align} In fact, \begin{align}\label{234} t_{k}(y)=\sum_{s=0}^{k-1}m_{j_{t_{s}}}(c_s(y)),\; k\geq1. \end{align} Then there exists $l\geq0$ such that \begin{align}\label{345} t_l(y)<n\leq t_{l+1}(y). \end{align}
We claim that \[\beta=\{V(y;x_0,\cdots,x_l): y\in E_n, x_s\in F^{*}_{c_s(y)}(j_{t_{s}(y)}),\;0\leq s\leq l\}\] is an open cover of $X$, where \[V(y;x_0,\cdots,x_l)=\{x\in X: (f_{0}^{t+t_s(y)}(x),f_{t_s(y)}^{t}(x_s))\in\gamma,\;0\leq t\leq m_{j_{t_{s}(y)}}(c_s(y))-1, \;0\leq s\leq l\}.\] In fact, let $x\in X$. Since $E_n$ is a set of $Y$ which $(n,\eta)$-spans $Y$ under $g_{0,\infty}$, there exists $y\in E_n$ such that \begin{align}\label{lo1} (g_{0}^{i}(y),g_{0}^{i}(\pi_0(x)))\in\eta,\; 0\leq i\leq n-1. \end{align} For any $0\leq s\leq l$, \[t_{s}(y)\leq t_{l}(y)<n,\] then by (\ref{lo1}), one has that \[\pi_{t_{s}(y)}\circ f_{0}^{t_{s}(y)}(x)=g_{0}^{t_{s}(y)}\circ\pi_{0}(x)\in\eta[g_{0}^{t_{s}(y)}(y)],\] and thus by (\ref{ll})-(\ref{lll}), we have \[f_{0}^{t_{s}(y)}(x)\in\pi_{t_{s}(y)}^{-1}(\eta[g_{0}^{t_{s}(y)}(y)])=\phi_{j_{t_{s}(y)}}^{-1}(\eta[g_{0}^{t_{s}(y)}(y)]) \subset \phi_{j_{t_{s}(y)}}^{-1}(W_{c_{s}(y)})\subset U_{c_{s}(y)}^{j_{t_{s}(y)}}.\] It follows from (\ref{lo2}) and (\ref{lo3}) that there exists $x_s\in F^{*}_{c_{s}(y)}(j_{t_{s}(y)})$ such that \[(f_{i}^{t}(f_{0}^{t_{s}(y)}(x)),f_{i}^{t}(x_s))\in\gamma,\;0\leq t\leq m_{j_{t_{s}(y)}}(c_{s}(y))-1,\;i\geq0.\] Thus, \[(f_{0}^{t+t_{s}(y)}(x),f_{{t_{s}(y)}}^{t}(x_s))=(f_{{t_{s}(y)}}^{t}(f_{0}^{t_{s}(y)}(x)),f_{{t_{s}(y)}}^{t}(x_s))\in\gamma, \;0\leq t\leq m_{j_{t_{s}(y)}}(c_{s}(y))-1,\;0\leq s\leq l.\] So, $x\in V(y;x_0,\cdots,x_l)$. Hence, $\beta$ is an open cover of $X$.
We also claim that any $(n,u)$-separated set intersects each element of $\beta$ at most one point. Otherwise, there exists an $(n,u)$-separated pair $(z,w)$ such that $z,w\in V(y;x_0,\cdots,x_l)$ for some $y\in E_n$ and $x_s\in F^{*}_{c_s(y)}(j_{t_{s}(y)})$, $0\leq s\leq l$. Then \[(f_{0}^{t+t_s(y)}(z),f_{t_s(y)}^{t}(x_s))\in\gamma,\;(f_{0}^{t+t_s(y)}(\omega),f_{t_s(y)}^{t}(x_s))\in\gamma,\;0\leq t\leq m_{j_{t_{s}(y)}}(c_s(y))-1,\;0\leq s\leq l,\] which yields that \begin{align}\label{0091} (f_{0}^{t+t_s(y)}(z),f_{0}^{t+t_s(y)}(\omega))\in \gamma\circ\gamma\subset u,\;0\leq t\leq m_{j_{t_{s}(y)}}(c_s(y))-1,\;0\leq s\leq l. \end{align} This contradicts to the fact that $(z,w)$ is an $(n,u)$-separated pair for $(X,f_{0,\infty})$ since \[\max_{0\leq s\leq l}\big(t+t_s(y)\big)=t_{l+1}(y)-1\geq n-1.\] So, \begin{align}\label{0092}
s_n(f_{0,\infty},X,u)\leq|\beta|=|E_n|\Pi_{s=0}^{l}|F^{*}_{c_s(y)}(j_{t_{s}(y)})|. \end{align} Hence, by (\ref{lo}), (\ref{234}), (\ref{345}) and (\ref{0092}), we have that \begin{align*}
\frac{1}{n}\log s_n(f_{0,\infty}, X,u)&\leq\frac{1}{n}\log|E_n|+\frac{1}{n}\sum_{s=0}^{l}\log|F^{*}_{c_s(y)}(j_{t_{s}(y)})|\\ &\leq\frac{1}{n}\log r_n(g_{0,\infty},Y,\eta)+\frac{a}{n}\sum_{s=0}^{l}\log m_{j_{t_{s}(y)}}(c_s(y))\\ &=\frac{1}{n}\log r_n(g_{0,\infty},Y,\eta)+\frac{a}{n}\big(\sum_{s=0}^{l-1}\log m_{j_{t_{s}(y)}}(c_s(y))+\log m_{j_{t_{l}(y)}}(c_l(y))\big)\\ &\leq\frac{1}{n}\log r_n(g_{0,\infty},Y,\eta)+\frac{a}{n}(n+M), \end{align*} where $M:=\max_{1\leq j\leq k}\{m_{j}(y_1),\cdots,m_{j}(y_p)\}$. This implies that \[h(f_{0,\infty})\leq h(g_{0,\infty})+a.\]
If $\{\pi_n\}_{n=0}^{\infty}$ is finite-to-one, then $a=0$, and thus $h(f_{0,\infty})\leq h(g_{0,\infty}).$ \end{proof}
\begin{remark} {\rm(i)} Theorem \ref{h1} is a generalization of Theorem 17 in \cite{Bowen71} to nonautonomous dynamical systems, and it is also a uniform version of Theorem C in \cite{Kolyada96}.
{\rm(ii)} For two semi-conjugate random dynamical systems on Polish spaces, Liu in \cite{liu05} proved that they have the same entropy if the cardinal number of the pre-image of a point under the semi-conjugacy is finite almost everywhere. \end{remark}
\section{Estimations of topological entropy for $A$-coupled-expanding systems}
In this section, some estimations of upper and lower bounds of topological entropy for an invariant subsystem of an $A$-coupled-expanding system are obtained.
Let us recall the definitions of subshifts of finite type \cite{Rob99}. A matrix $A=(a_{ij})_{N\times N}$ $(N\geq2)$ is said to be a transition matrix if $a_{ij}=0$ or $1$ for all $i, j$; $\sum_{i=1}^{N}a_{ij}\geq1$ for all $j$; and $\sum_{j=1}^{N}a_{ij}\geq1$ for all $i$, $1\leq i,j\leq N$. Given a transition matrix $A=(a_{ij})_{N\times N}$, denote \[\Sigma_{N}^{+}(A)=\{s=(s_0,s_1,\cdots): 1\leq s_j\leq N,\; a_{s_js_{j+1}}=1,\; j\geq0\}.\] Note that $\Sigma_{N}^{+}(A)$ is a compact metric space with the metric \[\rho(\alpha,\beta)=\sum_{i=0}^{\infty}\frac{d(a_i, b_i)}{2^i},\;\alpha=(a_0, a_1,\cdots),\;\beta=(b_0, b_1,\cdots)\in\Sigma_{N}^{+}(A),\] where $d(a_i, b_i)=0$ if $a_i=b_i$, and $d(a_i, b_i)=1$ if $a_i\neq b_i$, $i\geq0$. The map $\sigma_A: \Sigma_{N}^{+}(A)\to\Sigma_{N}^{+}(A)$ with \[\sigma_A((s_0,s_1,s_2,\cdots))=(s_1,s_2,\cdots),\;(s_0,s_1,s_2\cdots)\in\Sigma_{N}^{+}(A),\] is called a subshift of finite type associated with matrix $A$. Its topological entropy is equal to $\log\lambda(A)$, where $\lambda(A)$ is the spectral radius of matrix $A$ and \begin{align}\label{53} \lambda(A)=\lim_{n\to\infty}{\parallel A^n\parallel}^{\frac{1}{n}},\;\parallel A\parallel=\sum_{1\leq i,j\leq N}a_{ij}. \end{align}
\begin{definition} Let $f_{0,\infty}$ be a sequence of self-maps on a uniform space $X$ and $A=(a_{ij})_{N\times N}$ be a transition matrix. If there exist $N$ nonempty subsets $V_1,\cdots,V_N$ of $X$ with pairwise disjoint interiors, such that \[f_n(V_i)\supset\bigcup_{a_{ij}=1}V_j,\;1\leq i\leq N,\;n\geq0,\] then $(X,f_{0,\infty})$ is called $A$-coupled-expanding in $V_i$, $1\leq i\leq N$. Furthermore, $(X,f_{0,\infty})$ is said to be strictly $A$-coupled-expanding in $V_i$, $1\leq i\leq N$, if $\bar{V}_i\cap\bar{V}_j=\emptyset$ for all $1\leq i\neq j\leq N$, where $\bar{V}_i$ denotes the closure of the set $V_i$ with respect to $X$. In the special case that $a_{ij}=1$ for all $1\leq i,j\leq N$, it is briefly called coupled-expanding or strictly coupled-expanding in $V_i$, $1\leq i\leq N$. \end{definition}
An estimation of lower bound of topological entropy for an invariant subsystem of $(X,f_{0,\infty})$ is given in the following result.
\begin{theorem}\label{hh21} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact Hausdorff uniform space $(X,\mathcal{U})$, $A=(a_{ij})_{N\times N}$ be a transition matrix and $V_1,\cdots,V_N$ be nonempty, closed and mutually disjoint subsets of $X$. Assume that \begin{itemize} \item[{\rm(i)}] $(X,f_{0,\infty})$ is $A$-coupled-expanding in $V_i$, $1\leq i\leq N$;
\item[{\rm(ii)}] $f_{0,\infty}$ is equi-continuous in $\bigcup_{i=1}^{N}V_i$. \end{itemize} Then, for each $n\geq0$, there exist a nonempty compact subset $\Lambda_n\subset\bigcup_{i=1}^{N}V_i$ with $f_n(\Lambda_n)=\Lambda_{n+1}$ and a map $h_n:\Lambda_n\to\Sigma_{N}^{+}(A)$ such that the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$ is topologically $\{h_n\}_{n=0}^{\infty}$-equi-semiconjugate to $(\Sigma_{N}^{+}(A), \sigma_A)$. Consequently, \[h(f_{0,\infty},\Lambda_0)\geq\log\lambda(A),\] where $\lambda(A)$ is specified in {\rm(\ref{53})}. \end{theorem}
\begin{proof} Since $(X,f_{0,\infty})$ is $A$-coupled-expanding in $V_i$, $1\leq i\leq N$, for any $m,n\geq0$ and $\alpha=(a_0,a_1,\cdots)\in\Sigma_{N}^{+}(A)$, \begin{align}\label{s1} V_{\alpha}^{m,n}=\bigcap_{k=0}^{m}f_{n}^{-k}(V_{a_k})\neq\emptyset. \end{align} Fix $n\geq0$. Then $V_{\alpha}^{m,n}$ is a nonempty closed subset of $X$ and satisfies that $V_{\alpha}^{m+1,n}\subset V_{\alpha}^{m,n}$ for any $m\geq0$ and $\alpha\in\Sigma_{N}^{+}(A)$. Thus, $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}\neq\emptyset$ for all $\alpha\in\Sigma_{N}^{+}(A)$ by the compactness of $X$. Denote \begin{align}\label{s2} \Lambda_n:=\bigcup_{\alpha\in\Sigma_{N}^{+}(A)}\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}. \end{align} Clearly, $\Lambda_n\neq\emptyset$ and $\Lambda_n\subset\bigcup_{i=1}^{N}V_i$. It is easy to verify that $f_n(x)\in\bigcap_{m=0}^{\infty}V_{\sigma_{A}(\alpha)}^{m,n+1}$. Thus, $f_n(\Lambda_n)\subset\Lambda_{n+1}$. One also easily verify that $f_n(\Lambda_n)\supset\Lambda_{n+1}$ by assumption (i) and the fact that $\sigma_{A}$ is surjective. So, $f_n(\Lambda_n)=\Lambda_{n+1}$.
We claim that $\Lambda_n$ is a compact subset of $X$. Let $\{z_l\}_{l=1}^{\infty}$ be a sequence which converges to a point $z\in X$. Then, there exists $\alpha_l\in\Sigma_{N}^{+}(A)$ such that $z_l\in\bigcap_{m=0}^{\infty}V_{\alpha_l}^{m,n}$ for each $l\geq1$. Since $\Sigma_{N}^{+}(A)$ is compact, $\{\alpha_l\}_{l=1}^{\infty}$ has a convergent subsequence. Without loss of generality, suppose that $\{\alpha_l\}_{l=1}^{\infty}$ converges to $\alpha=(a_0, a_1,\cdots)$. So, for any $m\geq0$, there exists $k_m\geq1$ such that $V_{\alpha_l}^{m,n}=V_{\alpha}^{m,n}$ for all $l\geq k_m$. Hence, $z_l\in V_{\alpha}^{m,n}$ for all $l\geq k_m$, which implies that $z\in V_{\alpha}^{m,n}$. Therefore, $z\in\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$, and consequently, $\Lambda_n$ is closed, and thus compact.
For any $x\in\Lambda_n$, there exists $\alpha\in\Sigma_{N}^{+}(A)$ such that $x\in\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$. Define $h_n(x)=\alpha$. Then the map $h_n:\Lambda_n\to\Sigma_{N}^{+}(A)$ is well defined since $V_i\cap V_j=\emptyset$ for all $1\leq i\neq j\leq N$. Clearly, $h_n$ is surjective. Moreover, \[h_{n+1}\circ f_n(x)=\sigma_{A}(\alpha)=\sigma_{A}\circ h_n(x),\;x\in\Lambda_n.\]
Next, it is to show that $\{h_n\}_{n=0}^{\infty}$ is equi-continuous. Since $V_1,\cdots,V_N$ are mutually disjoint closed subsets of compact Hausdorff space $X$, there exists $\gamma\in{\mathcal{U}}^{s,o}$ such that for any $1\leq i\neq j\leq N$, \begin{align}\label{s3} \gamma\bigcap(V_i\times V_j)=\emptyset. \end{align} For any $\epsilon>0$, there exists $N\geq1$ such that $2^{-N}<\epsilon$. By the equi-continuity of $f_{0,\infty}$ in $\bigcup_{i=1}^{N}V_i$, there exists $\eta\in{\mathcal{U}}^{s,o}$ such that for any $n\geq0$ and any $x,y\in\Lambda_n$ with $h_n(x)=\alpha=(a_0, a_1,\cdots)$ and $h_n(y)=\beta=(b_0, b_1,\cdots)$, \[(x,y)\in\eta\Rightarrow(f_{n}^{j}(x),f_{n}^{j}(y))\in\gamma,\;0\leq j\leq N.\] It follows from (\ref{s1}) and (\ref{s2}) that \[f_{n}^{j}(x)\in V_{a_j},\;f_{n}^{j}(y)\in V_{b_j},\;0\leq j\leq N.\] This, together with (\ref{s3}), implies that \[a_j=b_j,\;0\leq j\leq N.\] Thus, \[\rho(h_n(x),h_n(y))=\rho(\alpha,\beta)\leq2^{-N}<\epsilon.\] Hence, $\{h_n\}_{n=0}^{\infty}$ is equi-continuous in $\{\Lambda_n\}_{n=0}^{\infty}$. Therefore, the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$ is topologically $\{h_n\}_{n=0}^{\infty}$-equi-semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$, and consequently \[h(f_{0,\infty}, \Lambda_0)\geq h(\Sigma_{N}^{+}(A),\sigma_{A})=\log\lambda(A)\] by Theorem \ref{277}. \end{proof}
The next result gives an estimation of lower bound of topological entropy for the full system.
\begin{Corollary}\label{hh212} Let all the assumptions in Theorem \ref{hh21} hold and $\bigcup_{i=1}^{N}V_i=X$ except that assumption {\rm(i)} is replaced by \begin{align}\label{00} f_n(V_i)=\bigcup_{a_{ij}=1}V_j,\;1\leq i\leq N,\;n\geq0. \end{align} Then $(X,f_{0,\infty})$ is topologically equi-semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_A)$. Consequently, \[h(f_{0,\infty})\geq\log\lambda(A).\] \end{Corollary}
\begin{proof} By the proof of Theorem \ref{hh21}, it is only to show that $\Lambda_n=X$ for any $n\geq0$, where $\Lambda_n$ is specified in (\ref{s2}). Let $n\geq0$. By the assumption that $\bigcup_{i=1}^{N}V_i=X$ and (\ref{00}), there exists $\beta\in\Sigma_{N}^{+}(A)$ such that $x\in\bigcap_{m=0}^{\infty}V_{\beta}^{m,n}$ for any $x\in X$, which implies that $X\subset\Lambda_n$. Hence, $\Lambda_n=X$. Then the conclusion follows from Theorem \ref{hh21}. \end{proof}
To proceed, we need the following lemma.
\begin{lemma}\label{hh22} Let $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ be compact uniform spaces, $Y_n\subset Y$ and $\pi_n: X\to Y_n$ be a map for each $n\geq0$. Assume that $\{\pi_n\}_{n=0}^{\infty}$ is equi-continuous at any point of $X$; that is, for any fixed point $x\in X$ and any $\gamma\in{\mathcal{V}}^{s,o}$, there exists $\eta\in{\mathcal{U}}^{s,o}$ such that $\pi_n(y)\in\gamma[\pi_n(x)]$ for any $y\in\eta[x]$ and $n\geq0$. Then $\{\pi_n\}_{n=0}^{\infty}$ is equi-continuous in $X$. \end{lemma}
\begin{proof} Fix $\gamma'\in{\mathcal{V}}^{s,o}$. Then there exists $\gamma\in{\mathcal{V}}^{s,o}$ such that $\gamma^2\subset\gamma'$. By the assumptions, for any $x\in X$, there exists $\eta_x\in{\mathcal{U}}^{s,o}$ such that for any $y\in X$, \begin{align}\label{1011} y\in\eta_x[x]\Rightarrow\pi_n(y)\in\gamma[\pi_n(x)],\;n\geq0. \end{align} For $\eta_x$, there exists $u_x\in{\mathcal{U}}^{s,o}$ such that $u_x^{2}\subset\eta_x$. Since $\{u_x[x]: x\in X\}$ is an open cover of compact space $X$, there exists a finite subcover $\{u_{x_{i}}[x_i]: 1\leq i\leq m\}$ for some $m\geq1$. Denote $u=\bigcap_{i=1}^{m}u_{x_{i}}$. Then $u\in{\mathcal{U}}^{s,o}$. So, for any $y_1,y_2\in X$ with $(y_1,y_2)\in u$, there exists $1\leq i_0\leq m$ such that $y_1\in u_{x_{i_0}}[x_{i_0}]$. Then \[(y_1,x_{i_0})\in u_{x_{i_0}}^2\subset\eta_{x_{i_0}},\; (y_2,x_{i_0})\in u_{x_{i_0}}^2\subset\eta_{x_{i_0}}.\] By (\ref{1011}), we have that \begin{align}\label{1012} (\pi_n(y_1),\pi_n(x_{i_0}))\in\gamma,\;(\pi_n(y_2),\pi_n(x_{i_0}))\in\gamma,\;n\geq0, \end{align} which yields that \[(\pi_n(y_1),\pi_n(y_2))\in\gamma^2\subset\gamma',\;n\geq0.\] Therefore, $\{\pi_n\}_{n=0}^{\infty}$ is equi-continuous in $X$. \end{proof}
An estimation of upper bound of topological entropy for an invariant subsystem of $(X,f_{0,\infty})$ is given in the next result.
\begin{theorem}\label{hh2} Let $f_{0,\infty}$ be a sequence of continuous self-maps on a compact Hausdorff uniform space $(X,\mathcal{U})$, $A=(a_{ij})_{N\times N}$ be a transition matrix and $V_1,\cdots,V_N$ be nonempty and closed subsets of $X$. Assume that \begin{itemize} \item[{\rm(i)}] $(X,f_{0,\infty})$ is $A$-coupled-expanding in $V_i$, $1\leq i\leq N$;
\item[{\rm(ii)}] for any $\alpha\in\Sigma_{N}^{+}(A)$ and $\gamma\in{\mathcal{U}}^{s,o}$, there exists $M\geq1$ such that \begin{align}\label{90} m\geq M\Rightarrow V_{\alpha}^{m,n}\times V_{\alpha}^{m,n}\subset\gamma, \;n\geq0, \end{align} \end{itemize} where $V_{\alpha}^{m,n}$ is specified in {\rm(\ref{s1})}. Then, for each $n\geq0$, there exist a nonempty compact subset $\Lambda_n\subset\bigcup_{i=1}^{N}V_i$ with $f_n(\Lambda_n)=\Lambda_{n+1}$ and a map $\pi_n:\Sigma_{N}^{+}(A)\to\Lambda_n$ such that $(\Sigma_{N}^{+}(A), \sigma_A)$ is topologically $\{\pi_n\}_{n=0}^{\infty}$-equi-semiconjugate to the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$. Consequently, \[h(f_{0,\infty},\Lambda_0)\leq\log\lambda(A).\] \end{theorem}
\begin{proof} For each $n\geq0$, by assumption (i) and the continuity of $f_n$, it is easy to verify that $V_{\alpha}^{m,n}$ is nonempty, closed and satisfies that $V_{\alpha}^{m+1,n}\subset V_{\alpha}^{m,n}$ for any $\alpha\in\Sigma_{N}^{+}(A)$ and $m\geq0$. This, together with assumption (ii) and $\mathcal{U}$ is separated, yields that $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$ is a singleton set for any $\alpha\in\Sigma_{N}^{+}(A)$ and $n\geq0$. Denote \begin{align}\label{111} \bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}=\{x_n(\alpha)\},\; \Lambda_n=\{x_n(\alpha):\alpha\in\Sigma_{N}^{+}(A)\}. \end{align} Clearly, $\Lambda_n\neq\emptyset$, $\Lambda_n\subset\bigcup_{i=1}^{N}V_i$, and \[f_n(x_n(\alpha))=x_{n+1}(\sigma_{A}(\alpha)),\;\alpha\in\Sigma_{N}^{+}(A),\;n\geq0.\] Hence, it follows from the fact that $\sigma_{A}$ is surjective that $f_n(\Lambda_n)=\Lambda_{n+1}$.
Define a map $\pi_n:\Sigma_{N}^{+}(A)\to\Lambda_n$ by $\pi_n(\alpha)=x_n(\alpha)$ for any $\alpha\in\Sigma_{N}^{+}(A)$ and $n\geq0$. Clearly, $\pi_n$ is well defined and surjective. Moreover, we have \[f_n\circ\pi_n(\alpha)=f_n(x_n(\alpha))=x_{n+1}(\sigma_{A}(\alpha))=\pi_{n+1}\circ\sigma_{A}(\alpha),\;\alpha\in\Sigma_{N}^{+}(A),\;n\geq0.\] Next, it is to show that $\{\pi_n\}_{n=0}^{\infty}$ is equi-continuous in $\Sigma_{N}^{+}(A)$. Fix any $\alpha=(a_0,a_1,\cdots)\in\Sigma_{N}^{+}(A)$. By assumption (ii), for any $\gamma\in{\mathcal{U}}^{s,o}$, there exists $M\geq1$ such that (\ref{90}) holds. Set $\delta= 1/2^{M+1}$. For any $\beta=(b_0,b_1,\cdots)\in\Sigma_{N}^{+}(A)$, \[\rho(\alpha,\beta)<\delta\Rightarrow a_j=b_j,\; 0\leq j\leq M+1.\] So, $x_n(\alpha),x_n(\beta)\in V_{\alpha}^{M+1,n}$, and \[(\pi_n(\alpha),\pi_n(\beta))=(x_n(\alpha),x_n(\beta))\in V_{\alpha}^{M+1,n}\times V_{\alpha}^{M+1,n}\subset\gamma,\;n\geq0.\] Thus, $\{\pi_n\}_{n=0}^{\infty}$ is equi-continuous at $\alpha$. Hence, $\{\pi_n\}_{n=0}^{\infty}$ is equi-continuous in $\Sigma_{N}^{+}(A)$ by Lemma \ref{hh22}. Further, for any $n\geq0$, $\Lambda_n$ is compact since $\Sigma_{N}^{+}(A)$ is compact and $\Lambda_n=\pi_n(\Sigma_{N}^{+}(A))$. Hence, $(\Sigma_{N}^{+}(A),\sigma_A)$ is topologically $\{\pi_n\}_{n=0}^{\infty}$-equi-semiconjugate to the invariant subsystem of $(X,f_{0,\infty})$ on $\{\Lambda_n\}_{n=0}^{\infty}$. It follows from Theorem \ref{277} that \[h(f_{0,\infty},\Lambda_0)\leq\log\lambda(A).\] \end{proof}
\begin{remark} Theorems \ref{hh21} and \ref{hh2} are generalizations of Theorem 3 in \cite{Shao20} and Theorem 4.1 in \cite{Shao16}, respectively, to a compact uniform space. \end{remark}
With a similar proof to that of Corollary \ref{hh212}, one has the following result.
\begin{Corollary}\label{hh23} Let all the assumptions in Theorem \ref{hh2} hold and $\bigcup_{i=1}^{N}V_i=X$ except that assumption {\rm(i)} is replaced by (\ref{00}). Then $(\Sigma_{N}^{+}(A),\sigma_A)$ is topologically equi-semiconjugate to $(X,f_{0,\infty})$. Consequently, \[h(f_{0,\infty})\leq\log\lambda(A).\] \end{Corollary}
\end{document} | arXiv | {
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\begin{document}
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\title{\bf \vskip -1.5in Numerical Anisotropy in Finite Differencing} \author{Adrian Sescu\thanks{ Department of Aerospace Engineering, Mississippi State University, 330 Walker at Hardy Rd, Mississippi State, MS 39762; {\it email}: sescu@ae.msstate.edu}\hspace{1.8mm}} \affil{Department of Aerospace Engineering, Mississippi State University, MS 39762}
\date{}
\maketitle
\begin{abstract} Numerical solutions to hyperbolic partial differential equations, involving wave propagations in one direction, are subject to several specific errors, such as numerical dispersion, dissipation or aliasing. In multi-dimensions, where the waves propagate in all directions, there is an additional specific error resulting from the discretization of spatial derivatives along grid lines. Specifically, waves or wave packets in multi-dimensions propagate at different phase or group velocities, respectively, along different directions. A commonly used term for the aforementioned multidimensional discretization error is the numerical anisotropy or isotropy error. In this review, the numerical anisotropy is briefly described in the context of the wave equation in multi-dimensions. Then, several important studies that were focused on optimizations of finite difference schemes with the objective of reducing the numerical anisotropy are discussed.
\end{abstract}
\section{Introduction} \label{intro}
Numerical anisotropy is a discretization error that is specific to numerical approximations of multidimensional hyperbolic partial differential equations (PDE). This error is often neglected, and the focus is directed toward the reduction of other types of discretization errors, such as numerical dissipation, dispersion or aliasing (e.g., Lele \cite{lele}, Tam and Webb \cite{tam1}, Kim and Lee \cite{Kim2}, Zingg et al. \cite{Zingg2}, Mahesh \cite{Mahesh}, Hixon \cite{hixon1}, Ashcroft and Zhang \cite{Ashcroft}, Fauconnier et al. \cite{Fauconnier} or Laizet and Lamballais \cite{Laizet}), or toward improving the accuracy of various time marching schemes (e.g., Hu et al. \cite{Hu}, Stanescu and Habashi \cite{Stanescu}, Mead and Renaut \cite{Mead}, Bogey and Bailly \cite{Bogey} or Berland et al. \cite{Berland}). There are several areas, however, where the numerical anisotropy can significantly affect the numerical solution based on finite difference or finite volume schemes (example include computational acoustics, computational electromagnetics, elasticity or seismology). The numerical anisotropy can be reduced by using, for example, one-dimensional high-resolution discretization schemes, multidimensional optimized difference schemes, or sufficiently fine grids. However, by increasing the number of grid points the computational time may increase considerably, while one-dimensional high-resolution difference schemes may generate spurious waves at the boundaries of the domain. Oftentimes, optimizations of multidimensional difference schemes are more effective.
High-order finite difference schemes that are optimized in one-dimension may not preserve their wavenumber resolution in multi-dimensional problems. These schemes may experience numerical anisotropy, because the dispersion characteristics along grid lines may not be the same as the dispersion characteristics associated with the diagonal directions. Over the years, several attempts to reduce the numerical anisotropy by various techniques were reported. A comprehensive analysis of the numerical anisotropy was performed in the book of Vichnevetsky~\cite{vich} where, among others, the two-dimensional wave equation was solved using two different finite difference schemes for the Laplacian operator. A considerable reduction of the numerical anisotropy was attained by weight averaging the two schemes. A slightly similar approach was previously used by Trefethen~\cite{trefethen} who used the leap frog scheme to solve the wave equation in two dimensions. Zingg and Lomax~\cite{Zingg1} performed optimizations of finite difference schemes applied to regular triangular grids, that give six neighbor points for a given node. They conducted comparisons between the newly derived schemes and conventional schemes that were discretized on square grids, and found that the numerical anisotropy can be significantly reduced by using triangular grids. Tam and Webb ~\cite{tam2} proposed an anisotropy correction to the finite difference representation of the Helmholtz equation. They derived an anisotropy correction factor using asymptotic solutions to the continuous equation and its finite difference approximation.
Jo et al. \cite{Jo}, in the context of solving the acoustic wave equation, proposed a finite difference scheme over a stencil consisting of grid points from more than one direction, by linearly combining two discretizations of the second derivative operator. A notable reduction of the numerical anisotropy was obtined, but the numerical dispersion error was increased. Hustesdt et al. \cite{Hustesdt} proposed a two-staggered-grid finite difference schemes for the acoustic wave propagation in two dimensions, where the first derivative operator was discretized along the grid line and along the diagonal direction. Lin and Sheu~\cite{lin} explored the dispersion-relation-preserving concept of Tam and Webb~\cite{tam1} in two dimensions to optimize the first-order spatial derivative terms of a model equation that resembles the incompressible Navier-Stokes momentum equation. They approximated the derivative using a nine-point grid stencil, resulting in nine unknown coefficients. Eight of them were determined by employing Taylor series expansions, while the ninth one was determined by requiring that the two-dimensional numerical dispersion relation is the same as the exact dispersion relation.
Kumar \cite{Kumar} derived isotropic finite difference schemes for the first and second derivatives in the context of symmetric dendritic solidification, and obtained a notable reduction of the numerical anisotropy. Patra and Kartunnen \cite{Patra} introduced several finite difference stencils for the Laplacian, Bilaplacian, and gradient of Laplacian, with the objective of improving the isotropic characteristics. Their stencils consisted of more grid points than the conventional schemes, but it was shown that the computational cost may decrease with more than 20$\%$ due to some gain in terms of stability. Stegeman et al. \cite{Stegeman} applied spectral analysis to evaluate the error in numerical group velocity (both the magnitude and the direction) of vorticity, entropy and acoustic waves, using the numerical solution to the linearized Euler equations in two-dimensions. They showed that a different measure of the group velocity error must be used to account for the error in the propagation direction of the waves. They also stressed that the numerical group velocity is more important than the numerical phase velocity in analyzing the errors associated with wave propagation. In a series of papers~\cite{sescu1,sescu2,sescu3,sescu4}, Sescu et al. proposed a technique to derive finite difference schemes in multi-dimensions with improved isotropy. The optimization performed in \cite{sescu1,sescu2,sescu3,sescu4} improved the isotropy of the wave propagation and, moreover, the stability restrictions of the multidimensional schemes in combination with either Runge-Kutta or linear multistep time marching methods were found to be more effective. They found that the stability restrictions are more favorable when using multidimensional schemes, even if they involve more grid points in the stencils. However, this was advantageous for low order schemes, such as those of second or fourth order of accuracy, but it was also shown that favorable stability restrictions can be obtained for higher order of accuracy schemes (sixth or eight) by increasing the isotropy corrector factor. The approach was extended to prefactored compact schemes by Sescu and Hixon \cite{sescu6,sescu7}. Beside reducing the numerical anisotropy, the new multidimensional compact schemes are computationally cheaper than the corresponding explicit multidimensional scheme defined on a same stencil.
In computational electromagnetics, there were many attempts to reduce the numerical anisotropy, by applying various techniques. Berini and Wu \cite{Berini} conducted a comprehensive analysis of the numerical dispersion and numerical anisotropy of finite difference schemes applied to transmission-line modeling (TLM) meshes. They found that, under certain circumstances, the time domain nodes introduce anisotropy into the dispersion characteristics of isotropic media, stressing the importance of developing schemes with improved isotropy. Gaitonde and Shang \cite{Gaitonde} proposed a class of high-order compact difference-based finite-volume schemes that minimizes the dispersion and isotropy error functions for the range of wavenumbers of interest. Sun and Trueman~\cite{Sun1} proposed an optimization of two-dimensional finite difference schemes, by considering additional nodes surrounding the point of differencing. They obtained a significant reduction in the numerical anisotropy, dispersion error and the accumulated phase errors over a broad bandwidth. Further optimizations of this scheme were performed in another paper of Sun and Trueman \cite{Sun2}. Koh et al. \cite{Koh} derived a two-dimensional finite-difference time-domain method, discretizing the Maxwell equations, to eliminate the numerical dispersion and anisotropy. They showed that the new algorithm has isotropic dispersion and resemble the exact phase velocity, whose isotropic property is superior to that of other existing schemes. Shen and Cangellaris~\cite{Shen} introduced a new stencil for the spatial discretization of Maxwell's equations. Compared to conventional second-order accurate FDTD scheme, their scheme experienced superior isotropy characteristics of the numerical phase velocity. They also showed that the Courant number cab be increased by using the newly derived schemes. Kim et al. \cite{Kim} derived new three-dimensional isotropic dispersion-finite-difference time-domain schemes (ID-FDTD) based on a linear combination of the traditional central difference equation and a new difference equation using extra sampling points. Among all versions of the proposed finite-difference schemes, three of them showed improved isotropy of the wave propagation compared to the original scheme of the Yee \cite{Yee}. Kong and Chu \cite{Kong} introduced a new unconditionally-stable finite-difference time-domain method with low numerical anisotropy in three-dimensions. Compared with other finite-difference time-domain methods, the normalized numerical phase velocity of their proposed scheme was significantly improved, while the dispersion error and numerical anisotropy have been reduced.
This review will describe and discuss the numerical anisotropy in the framework of wave equation, and will present some of the most important optimizations of finite difference schemes in the context of reducing the numerical anisotropy. In section II, the dispersion error and the numerical anisotropy existing in finite difference discretizations of the wave equation are introduced and discussed. In section III, several approaches to reduce the numerical anisotropy, that were developed over the years by various research groups, are reviewed and discussed. Concluding remarks are included in section IV.
\section{Dispersion Error and Numerical Anisotropy} \label{sec:1}
Let us consider the centered finite difference approximation of the spatial derivative, which contains both the explicit and the implicit (or compact) parts:
\begin{equation}\label{cvb} \sum_{k=1}^{N_c}\alpha_k(u_{j+k}' + u_{j-k}') + u_{j}' = \frac{1}{h} \left( \sum_{k=1}^{N_e}a_k(u_{j+k} - u_{j-k}) \right) + O(h^n), \end{equation} where the gridfunctions are $u_j = u(x_j)$ for $1 \le j \le N$, the derivatives are denoted by a prime, $u_j'$, $h$ is the space step, and $\alpha_k$ and $a_k$ are given coefficients. If $N_{c} = 0$ the scheme is termed explicit, while compact schemes (also known as implicit or Pade schemes), by contrast, have $N_{c} \neq 0$ and require the solution of a matrix equation to determine the derivatives along a grid line. Conventionally, the coefficients $\alpha_k$ and $a_k$ are chosen to provide the largest possible exponent, $n$, in the truncation error, for a given stencil width, but in some instances some of these coefficients are determined to provide improved dispersion characteristics of the scheme. Table~\ref{t1} includes some of these weights for various explicit and compact finite difference schemes: explicit classical second order scheme (E2), explicit classical fourth order scheme (E4), explicit classical sixth order scheme (E6), dispersion-relation-preserving scheme of Tam and Webb~\cite{tam1}, compact classical fourth order scheme (C4), optimized tridiagonal compact scheme of Haras and Ta'asan~\cite{haras} (Haras), optimized pentadiagonal scheme of Lui and Lele~\cite{lui3} (Lui) and spectral-like pentadiagonal compact scheme of Lele~\cite{lele} (Lele). The prefactored compact scheme of Hixon~\cite{hixon1,hixon2} is also included here in the form:
\begin{eqnarray}\label{jjj} a u_{j+1}^{F'} + c u_{j-1}^{F'} + (1-a-c) u_{j}^{F'} = \frac{1}{h} \left[ b u_{j+1} - (2b-1) u_j - (1-b) u_{j-1}) \right], \nonumber \\ c u_{j+1}^{B'} + a u_{j-1}^{B'} + (1-a-c) u_{j}^{B'} = \frac{1}{h} \left[ (1-b) u_{j+1} - (2b-1) u_j - b u_{j-1}) \right], \end{eqnarray} where $F$ and $B$ stand for 'forward' and 'backward', respectively (in a predictor-corrector time marching framework). For sixth order accuracy, $a=1/2-1/(2\sqrt{5})$, $b=1-1/(30 a)$ and $c=0$. The leading order term in the truncation error of a finite difference scheme depends on the choice of the coefficients and the $(n+1)$st derivative of the function $u$.
To study the wavenumber characteristics of finite difference schemes, consider a periodic domain in real space, $x \in [0,L]$, with $N$ uniformly spaced points(the spatial step size is $h=L/N$). The discrete Fourier transform of $u$ is given as $ \hat{u}_m = \frac{1}{N}\sum_{j=1}^{N} u_j e^{-ik_m x_j}$ with $m=-N/2,...,N/2-1, $ where the wavenumber is $k_m = 2\pi m/L$. The $m$th component of the discrete Fourier transform of $u'$ denoted $\hat{u}_m'$ is simply $ik_m \hat{u}_m$. Taking the discrete Fourier transform of equation (\ref{cvb}) implies that
\begin{equation}\label{} (\hat{u}_m')_{num} = iK(k_m h)\hat{u}_m, \end{equation} where the numerical wavenumber is given as
\begin{equation}\label{} K(z) = \frac {\sum_{n=1}^{N_e}2a_n \sin{(nz)}} {1+\sum_{n=1}^{N_c}2\alpha_n \cos{(nz)}}. \end{equation}
Figure \ref{f1} shows the numerical wavenumber for various explicit and compact schemes, corresponding to those given in table \ref{t1}. The numerical wavenumber is compared to the analytical wavenumber which is represented by the straight line in figure \ref{f1}. As one can notice, the compact schemes are superior to the explicit schemes; however, compact schemes are computationally more demanding because large matrices have to be inverted.
\begin{figure}
\caption{Numerical wavenumber compared to the analytical wavenumber.}
\label{f1}
\end{figure}
\begin{figure}
\caption{Numerical wavenumber surfaces compared to the analytical wavenumber surface: a) second order explicit scheme (E2); b) sixth order explicit scheme (E6); c) sixth order prefactored compact scheme (Hixon). The cones represent the exact wavenumber surfaces.}
\label{f2a}
\label{f2b}
\label{f2b}
\label{f2}
\end{figure}
In muldimensions, the numerical wavenumber and the numerical phase and group velocity are also dependent on the direction of propagation. Figure \ref{f2} shows the numerical wavenumber surface for the wave equation in two dimension, corresponding to schemes E2, E6 and Hixon as given in table \ref{t1} and equation (\ref{jjj}), respectively. The cone represents the exact wavenumber surface, obtained by revolving the straight line from figure \ref{f1} around the vertical axis. One can clearly notice the anisotropy in the numerical wavenumber surfaces associated with the finite differencing.
A simple way to reveal the numerical anisotropy is by considering the advection equation in two dimensions,
\begin{equation}\label{hh} \partial_t u =\textbf{c} \nabla u, \end{equation} with the initial condition $u(\textbf{r},0)=u_0(\textbf{r})$, where $\textbf{r}=(x,y)$ is the vector of spatial coordinates, $\textbf{c}=c(\cos\alpha \hspace{2mm} \sin\alpha)$ is the velocity vector ($c$ is a scalar and $\alpha$ the propagation direction angle), $\nabla=(\partial_x \hspace{2mm} \partial_y)^T$ and $u(\textbf{r},t)$ and $u_0(\textbf{r})$ are scalar functions. A simple semi-discretization of equation (\ref{hh}) on a square grid is obtained as
\begin{equation}\label{e2}
d_t u=-\frac{c}{2 h} \big[\cos \alpha (u_{{i+1},j}-u_{{i-1},j})+ \sin \alpha (u_{i,{j+1}}-u_{i,{j-1}})\big], \end{equation} where $h$ is the grid step. Consider the Fourier-Laplace transform:
\begin{equation}\label{e4}
\tilde{u}(\xi,\eta, \omega)=\frac{1}{(2\pi)^{3}}\int_{0}^{\infty} \int \int_{-\infty}^{\infty}u(x,y,t) e^{-i (\xi x +\eta y - \omega t)} dx dy dt \end{equation} where $\xi = K \cos \alpha$ and $\eta = K \sin \alpha$ are the components of the wavenumber and $\omega$ is the frequency ($K$ is the wavenumber magnitude). The application of Fourier-Laplace transform to equation (\ref{hh}) gives the exact dispersion relation:
\begin{equation}\label{e6}
\omega=cK(\cos^2 \alpha+\sin^2 \alpha)=cK. \end{equation} The exact phase velocity is given by $c_{e}=\omega/K=c$. By substituting $\omega$ in equation (\ref{e4}) with (\ref{e6}), $u(\textbf{r},t)$ is obtained as a superposition of sinusoidal solutions in the plane with constant phase lines given by $x\cos\alpha+y\sin\alpha-c_{e}t=const$. As one can notice, the exact phase velocity $c_{e}$ does not depend on the propagation direction $\alpha$, which means that the wave propagates with the same phase velocity in all directions (it is isotropic). Moreover, the exact group velocity defined as $g_{e}=\partial\omega/ \partial K=c$ is the same as the exact phase velocity because the dispersion relation is a linear function of $K$.
We now apply the same Fourier-Laplace transform to the numerical approximation (\ref{e2}) and obtain the numerical dispersion relation in the form
\begin{equation}\label{ii}
\omega=\frac{c}{h}\big[\cos\alpha \sin(Kh\cos\alpha)+\sin\alpha \sin(Kh\cos\alpha)\big] \end{equation} The numerical phase velocity will be given as
\begin{equation}\label{iii}
c_{n}=\frac{\omega}{K}=\frac{c}{Kh}\big[\cos\alpha \sin(Kh\cos\alpha)+\sin\alpha \sin(Kh\cos\alpha)\big]. \end{equation}
The constant phase lines are expressed by the equation $x\cos\alpha+y\sin\alpha-c_{n}t=const$ and move with the phase velocity $c_{n}$. The numerical anisotropy is revealed in equation (\ref{iii}) by the dependence of the numerical phase velocity on the propagation direction angle $\alpha$. In addition, the numerical group velocity is different from the numerical phase velocity (while previously, in the continuous case, they were the same),
\begin{equation}\label{}
g_{n}=\partial_K \omega=c\big[\cos^2 \alpha\cos(Kh\cos\alpha)+\sin^2 \alpha\cos(Kh\sin\alpha
)\big], \end{equation} which is also dependent on the propagation direction. This directional dependence of both phase and group velocities defines the numerical anisotropy. As an illustration, figure \ref{f3} shows polar diagrams for two typical schemes, fourth order explicit E4 and sixth order compact C6 schemes, revealing the numerical anisotropy (the circle of radius $1$ in figure \ref{f3} represents the exact solution).
\begin{figure}
\caption{Polar diagram of normalized phase velocities as a function of points per wavelength (PPW)
and the direction of propagation: a) fourth-order explicit schemes (lowest number of points per wavelength is 4); b) sixth-order compact schemes (lowest number of points per wavelength is 3).}
\label{f3a}
\label{f3b}
\label{f3}
\end{figure}
\section{Reduction of the Numerical Anisotropy} \label{sec:2}
In this section, several attempts to reduce the numerical anisotropy, performed by various research groups over the years, are briefly reviewed. The optimizations of the schemes are grouped according to the mathematical model: wave equation, Helmholtz equations, advection equation, Maxwell equation, and dendritic solidification equations.
\subsection{Wave Equation}
Although the behavior of the numerical anisotropy was often reported in various one-dimensional optimizations of finite difference schemes, one of the first systematic attempts to specifically reduce the numerical anisotropy in finite difference schemes was introduced by Trefethen \cite{trefethen} in the framework of wave equation. To illustrate Trefethen's approach, let us consider the two dimensional wave equation in the form
\begin{equation}\label{sas} \partial_{tt}u = \partial_{xx}u + \partial_{yy}u, \end{equation} defined in $R^2 \times [0,\infty)$, with appropriate initial and boundary conditions. Using the Fourier-Laplace transform, it is ease to find the exact dispersion relation in the form $\omega^2 = \xi^2 + \eta^2$, where $\omega$ is the frequency and $(\xi, \eta)$ is the wavenumber vector. Equation (\ref{sas}) was discretized by Trefethen \cite{trefethen} on a Cartesian grid, using second order accurate schemes for both temporal and spatial derivatives as
\begin{equation}\label{} u_{ij}^{n+1} - u_{ij}^{n} + u_{ij}^{n-1} = \frac{k^2}{h^2} (u_{i+1,j}^{n} + u_{i-1,j}^{n} + u_{i,j+1}^{n} + u_{i,j-1}^{n} - 4u_{i,j}^{n}) \end{equation} which was labeled $LF^2$. Then the same scheme was used to discretize equation (\ref{sas}), except the spatial derivatives were approximated along the diagonal directions with the space step $\sqrt{2}h$; this latter discretization was termed $LF^2$. It was found that the weighted averaging $2/3 LF^2 + 1/3 LF_2$ provided a low numerical anisotropy in the order of $(\sqrt{\xi^2+\eta^2}h)^4$. Slightly the same approach was used by Vichnevetsky \cite{vich} who corrected the numerical isotropy of the wave propagation in two dimensions using either the linear advection equation or the wave equation.
In a series of papers, Sescu et al. \cite{sescu1,sescu2,sescu3} proposed a technique to derive explicit multidimensional finite difference schemes for wave equation and Euler equations. By using the transformation matrix between two orthogonal reference frames, one aligned with the grid line and the other along the diagonal direction, the multidimensional finite difference scheme was obtained as
\begin{equation}\label{} \left( \partial_x u \right)_{i,j} = \frac{1}{h(1+\beta)} \sum_{\nu=-M}^{\nu=M} a_\nu \left( \textbf{E}_{x}^{\nu} + \frac{\beta}{2} \textbf{D}_{x} \right)\cdot u_{i,j} \end{equation} where the multidimensional space shift operator $\textbf{E}_{x}^{\nu}\cdot u_{i,j} = u_{i+\nu,j}$ (see Vichnevetsky and Bowles~\cite{vich} for one dimension) is used. The coefficients $a_n$ are those from the classical centered explicit schemes. The operator $\textbf{D}_{x}^{\nu}\cdot$ was defined as $\textbf{D}_{x}^{\nu}\cdot = \left( \textbf{E}_{x}^{\nu}\textbf{E}_{y}^{\nu} + \textbf{E}_{x}^{-\nu}\textbf{E}_{y}^{\nu} \right)\cdot $ The parameter $\beta$ is called isotropy corrector factor (ICF). The application of the Fourier transform to the multidimensional schemes gives the numerical wavenumber
\begin{equation}\label{}
(\xi h)_{opt}^*=\frac{2}{(1+\beta)} \sum _{n=-N}^M a_n \Big\{e^{nI\xi h}+\frac{\beta}{2}\big[e^{nI(\xi+\eta) h}+e^{nI(\xi-\eta)
h}\big]\Big\}, \end{equation} Then the numerical dispersion relation corresponding to two-dimensional wave equation was considered in the form $
\omega^2-\big[(\xi h)_{opt}^{* \hspace{2 mm} 2} +(\eta h)_{opt}^{* \hspace{2 mm} 2}\big]=0, $ and the ICF was determined by minimizing the integrated error between the phase or group velocities defined along $x$ and $x=y$ directions. Two curves in wavenumber-frequency space were considered: one was the intersection between the numerical dispersion relation surface and $\eta=0$ plane, and the other was the intersection between the numerical dispersion relation surface and the $\xi=\eta$ plane. These two curves were superposed in the $(Kh, \omega)$ plane, where $K h=\big[(\xi h)^2+(\eta h)^2\big]^{\frac{1}{2}}$. Assuming that the equations of the two curves in $(K h, \omega)$ plane are $\omega_1=\omega_1 (Kh,\beta)$ and $\omega_2=\omega_2 (Kh,\beta)$, the integrated error between the phase velocities was then calculated on a specified interval as $
C(\beta)=\int_0^ \eta\big|c_1(Kh,\beta)-c_2(Kh,\beta)\big|^2d(Kh), $ where $c_1(Kh,\beta)$ and $c_2(Kh,\beta)$ are the numerical phase velocities. The minimization was done by equating the first derivative of $C(\beta)$ or $G(\beta)$ with zero, which provided the value of ICF, $\beta$.
Sescu et al. \cite{sescu4,sescu5} conducted a comprehensive stability analysis of the multidimensional schemes combined with either linear-multistep or multi-stage time marching schemes, and obtained several noteworthy results. For the Leap-Frog scheme applied to the advection equations, it was shown that the stability restriction corresponding to multidimensional schemes differs from the corresponding stability restriction via conventional schemes by the factor $(2\beta+2)/(\beta+2)$, where $\beta$ is the isotropy corrector factor. The conclusion was that the stability restrictions corresponding to multidimensional schemes are more convenient compared to the conventional schemes. For an arbitrary direction of the convection velocity with $|c_x| \geq |c_y|$, the stability restriction for conventional stencils was given by $\sigma_x+\sigma_y \leq CFL$, where $\sigma_x=k|c_x|/h$ and $\sigma_y=k|c_y|/h$. For multidimensional stencils the stability restriction was given by $ (1+\beta)\sigma_x+\sigma_y \leq CFL(1+\beta)$ (where, for example, $CFL$ is $1$, $0.72874$ or $0.63052$ corresponding to E2, E4 or E6 scheme, respectively). Adams-Bashforth and Runge-Kutta time marching schemes in combination with conventional and multidimensional schemes were also analyzed, and it was found that the multidimensional schemes provide less restrictive stability limits.
\subsection{Helmholtz Equation}
Tam and Webb \cite{tam2} performed an anisotropy correction of the finite difference representation of the Helmholtz equation,
\begin{equation}\label{zz} \nabla^2 p + \xi^2 p = f \end{equation} where $p$ is the pressure perturbation, $\nabla^2$ is the Laplacian operator, $f$ is the source distribution (e.g., a monopole), $\xi = 2\pi/\lambda$ is the wavenumber, and $\lambda$ is the acoustic wavelength. Tam and Webb \cite{tam2} showed that the finite difference discretization of the Helmholtz equation,
\begin{equation}\label{zx} \frac{p_{i+1,j} - 2p_{i,j} + p_{i-1,j}}{h^2} + \frac{p_{i,j+1} - 2p_{i,j} + p_{i,j-1}}{h^2}
+ \xi^2 p_{i,j} = f_{i,j} \end{equation} with five grid points per wavelength introduces significant numerical anisotropy (equally-spaced grid is assumed in both x- and y-directions, and the spatial step is denoted as before by $h$). They constructed an anisotropy correction factor using asymptotic solutions to the continuous equation (\ref{zz}) and its finite difference approximation (\ref{zx}) as
\begin{equation}\label{} p_a(r,\theta)_{r_{ij}\rightarrow \infty} = \left( \frac{2\pi}{\xi} \right) \frac{\pi}{i r^{1/2}} e^{i(\xi r - \pi/4)} \bar{F}(\bar{\alpha}_s,\bar{\beta}_{+}(\bar{\alpha}_s)) + O(r^{-3/2}) \end{equation} and
\begin{equation}\label{}
p_n(r_{ij},\theta_{ij})_{r_{ij}\rightarrow \infty} = \frac{e^{iK_{ij}r_{ij}}}{r^{1/2}_{ij}} \left[ G_0(\theta_{ij}+\frac{G_1(\theta_{ij}}{r_{ij}}) + O(r^{-5/2}_{ij}) \right] \end{equation} respectively, where $(r_{ij},\theta_{ij})$ are polar coordinates, $K_{ij} = \alpha_s(\theta_{ij})\cos{\theta_{ij}} + \beta_s(\theta_{ij})\sin{\theta_{ij}}$ (with $\alpha_s$ and $\beta_s$ being the wavenumber components from the Fourier transform), and $G_0(\theta_{ij})$ and $G_1(\theta_{ij})$ are functions depending on $\alpha_s$, $\beta_s$, $\theta$ and the Fourier transform $\bar{F}$ of the source term (for more details see equations (19) and (21) in Tam and Webb \cite{tam2}). The anisotropy corrector factor was then defined by the ratio between the absolute values of the two,
\begin{equation}\label{}
D(\theta,\xi h) = \frac{|p_a|}{|p_n|} \end{equation} The correction factor is independent of the distribution of sources, meaning that it can be computed once and for all types of sources. Significant reduction of the anisotropy error was obtained.
\subsection{Advection Equation}
Gaitonde and Shang \cite{Gaitonde} proposed a class of high-order compact difference-based finite-volume schemes which minimized the dispersion and isotropy error functions for the range of wavenumbers of interest. The starting point was the one dimensional advection equation,
\begin{equation}\label{}
\partial_t u + \partial_x f = 0, \hspace{4mm}
f = cu, \hspace{4mm} c > 0 \end{equation} which was discretized using a finite volume approach as
\begin{equation}\label{} d_t \bar{u}_i + \bar{f}_{i+1/2} - \bar{f}_{i-1/2} = 0
\end{equation}
where $\bar{u}$ is the average value of $u$ inside a cell, $\bar{u} = 1/h\int_{x_{i-1/2}}^{x_{i+1/2}}udx$, and $\bar{f}$ is the flux function approximating $f$, which is dependent on the values of $\bar{u}$ from neighbor cells. The reconstruction can be done by considering a primitive function $v = \int_{0}^{x}$ which must be discretized at the cell interface. Gaitonde and Shang \cite{Gaitonde} considered a five-point compact stencil in the form
\begin{equation}\label{ddd}
\alpha v_{i-1/2} + v_{i+1/2} + \alpha v_{i+3/2}
= b\frac{v_{i+5/2} - v_{i-3/2}}{4h} + a\frac{v_{i+3/2} - v_{i-1/2}}{2h}
\end{equation}
where $\alpha$, $a$, and $b$ are constants which determine the order of accuracy of the scheme. Using Taylor series expansions, they sacrificed the order of accuracy of the schemes by writing $a$ and $b$ as functions of $\alpha$,
\begin{equation}\label{}
a = \frac{2(2+\alpha)}{3}, \hspace{4mm} b = \frac{-1+4\alpha}{3}
\end{equation}
The spectral function associated with the scheme (\ref{ddd}) is given as
\begin{equation}\label{}
\hat{A}(w) = \frac{i\left( a\sin(w) + b \sin(2w)/2 \right)}{1 + 2\alpha \cos{w}}
\end{equation}
where $w = 2\pi \xi h/L$ is the scaled wave number. The dispersion error is associated with the imaginary part of the spectral function, $w_d(w) = Im(\hat{A}(w))$. A scaled isotropy wavenumber was defined as
\begin{equation}\label{} w_i(w,\theta) = \cos(\theta) w_d (w \cos(\theta)) + \sin(\theta) w_d (w \sin(\theta)) \end{equation}
where $\theta$ is the angle that the direction of propagation makes with the x-axis. An isotropy error function was defined by Gaitonde and Shang \cite{Gaitonde} in the form
\begin{equation}\label{}
E_i(\alpha,w_{max}) = \int_{0}^{w_{max}}\int_{0}^{\pi/2} |w_i - w| d\theta dw \end{equation} which was minimized to find the value of $\alpha_{opt}$ that gives the lowest numerical anisotropy. Numerical examples confirmed a considerable reduction of the isotropy error.
Sescu and Hixon \cite{sescu6,sescu7} extended the previous optimization performed in \cite{sescu2} to prefactored compact finite difference schemes \cite{hixon1,hixon2} applied to the advection equation. The prefactored compact schemes are defined on a three-point stencil and can return up to eight order of accuracy (see equations (\ref{jjj})). They can be used within a predictor-corrector type time marching scheme framework (MacCormack \cite{MacCormack}), because the numerical derivatives are determined by sweeping from one boundary to the other, in both directions. Following the same analysis as in the case of explicit schemes, the multidimensional prefactored compact schemes were obtained as
\begin{eqnarray}\label{e1} u_{i,j}^{F'} &=& \frac{\alpha}{1+\beta} \left[ u_{i+1,j}^{F'} + \frac{\beta}{2} \left( u_{i+1,j-1}^{F'} + u_{i+1,j+1}^{F'} \right) \right] \\ &+& \frac{1}{h(1+\beta)} \left[ b u_{i+1,j} - e u_{i,j} + \frac{\beta}{2} \left( b u_{i+1,j+1} + b u_{i+1,j-1} - 2e u_{i,j} \right) \right] \nonumber \\ u_{i,j}^{B'} &=& \frac{\alpha}{1+\beta} \left[ u_{i-1,j}^{B'} + \frac{\beta}{2} \left( u_{i-1,j-1}^{B'} + u_{i-1,j+1}^{B'} \right) \right] \\ &+& \frac{1}{h(1+\beta)} \left[ e u_{i,j} - b u_{i-1,j} + \frac{\beta}{2} \left( 2e u_{i,j} - b u_{i-1,j+1} - b u_{i-1,j-1} \right) \right] \nonumber \end{eqnarray} for fourth order of accuracy, and
\begin{eqnarray}\label{e1} u_{i,j}^{F'} &=& \frac{\alpha}{1+\beta} \left[ u_{i+1,j}^{F'} + \frac{\beta}{2} \left( u_{i+1,j-1}^{F'} + u_{i+1,j+1}^{F'} \right) \right] \\ &+& \frac{1}{h(1+\beta)} \left[ b u_{i+1,j} - e u_{i,j} - f u_{i-1,j} + \frac{\beta}{2} \left( b u_{i+1,j+1} - f u_{i-1,j-1} + b u_{i+1,j-1} - f u_{i-1,j+1} - 2e u_{i,j} \right) \right] \nonumber \\ u_{i,j}^{B'} &=& \frac{\alpha}{1+\beta} \left[ u_{i-1,j}^{B'} + \frac{\beta}{2} \left( u_{i-1,j-1}^{B'} + u_{i-1,j+1}^{B'} \right) \right] \\ &+& \frac{1}{h(1+\beta)} \left[ b u_{i+1,j} - e u_{i,j} - b u_{i-1,j} + \frac{\beta}{2} \left( f u_{i+1,j+1} - b u_{i-1,j-1} + f u_{i+1,j-1} - b u_{i-1,j+1} - 2e u_{i,j} \right) \right] \nonumber \end{eqnarray} for sixth order of accuracy. $\beta$ is the isotropy corrector factor (ICF) and its magnitude can be determined by minimizing the dispersion error corresponding to the wave-front propagating along a grid line and the wave-front propagating along a diagonal direction.
Using Fourier analysis, the numerical wavenumbers and the numerical dispersion relation corresponding to the two dimensional wave equation were found. The individual (forward or backward) numerical wavenumber has both real and imaginary parts: the real part of the forward operator is equal to the real part of the backward operator, and the imaginary parts are opposite. As a result, in a MacCormack predictor-corrector scheme the overall imaginary part is zero. The real parts of the numerical wavenumbers corresponding to multidimensional schemes, for derivatives along $x$-direction, were given by:
\begin{eqnarray}\label{e1} Re[(kh)^*_{m}] = \frac{1}{1+\beta} \left\{ f_m(\eta_x) + \frac{\beta}{2} \left[ f_m(\eta_x + \eta_y) + f_m(\eta_x - \eta_y) \right] \right\}, \end{eqnarray} where $m=4$ for fourth and $m=6$ for sixth order of accuracy, $f_4(\eta_x) = 3 \sin{\eta_x}/(2+\cos{\eta_x})$, $f_6(\eta_x) = (28 \sin{\eta_x} + \sin{2\eta_x})/(18+12\cos{\eta_x})$, $\eta_x = \xi h$, $\eta_y = \eta h$ and $\xi$ and $\eta$ are the components of the wavenumber.
In terms of numerical stability, more efficient stability restrictions were obtained as in the case of multidimensional explicit schemes. For example, multidimensional MacCormack schemes were found to provide a stability restriction in the form
\begin{eqnarray}\label{r1} [\sigma_x(1+\beta)]^{2/3} + \sigma_y^{2/3} \leq \frac{(1+\beta)^{2/3}}{\xi_{max}}, \end{eqnarray}
if $|c_x| \geq |c_y|$, and
\begin{equation}\label{r2} \sigma_x^{2/3} + [\sigma_y(1+\beta)]^{2/3} \leq \frac{(1+\beta)^{2/3}}{\xi_{max}}, \end{equation}
if $|c_y| \geq |c_x|$. For diagonal directions, with respect to the grid, ($|c_x|=|c_y|=|c|$) the stability restriction becomes
\begin{equation}\label{s2} \sigma \leq \frac{(1+\beta)}{\xi_{max}^{3/2} \left[ 1+(1+\beta)^{2/3} \right]^{3/2}}. \end{equation} It is obvious that the right hand side of equation (\ref{s2}) is greater than $1/(2\xi_{max})^{3/2}$ when $\beta > 0$, and goes to $1/(\xi_{max})^{3/2}$ when $\beta \rightarrow \infty$. This generated more efficient stability restrictions by using multidimensional compact schemes. Test cases showed that the multidimensional compact schemes were more efficient for both fourth and sixth order accurate schemes.
\subsection{Maxwell Equations}
Sun and Trueman \cite{Sun1} performed an optimization of finite difference schemes applied to Maxwell equations, in terms of reducing the dispersion and isotropy errors. For brevity, we show here the numerical dispersion relations (for finite differencing representations of the Maxwell equations, see equations (1), (2) and (4) in Sun and Trueman \cite{Sun1}):
\begin{equation}\label{q1} \left( \frac{\sin(\omega k/2)}{ck} \right)^2 = \left( w\frac{\sin(\beta_a k/2)}{h} + (1-w)\frac{\sin(3\beta_a k/2)}{3h} \right)^2 \end{equation} corresponding to a grid line, and
\begin{equation}\label{q2} \left( \frac{\sin(\omega k/2)}{ck} \right)^2 = 2\left( w\frac{\sin(\beta_d k/2)}{h} + (1-w)\frac{\sin(3\beta_d k/2)}{3h} \right)^2 \end{equation} corresponding to the diagonal direction, where $w$ is a weighting factor, $\beta_a$ is the numerical phase constant along the grid line, $\beta_d$ is the numerical phase constant along the diagonal direction, $\omega$ is the frequency, and $k$ is the time step (an equally-spaced grid is considered again). The optimization in terms of reducing the numerical anisotropy was done by eliminating the time step terms in equations (\ref{q1}) and (\ref{q2}) to obtain
\begin{equation}\label{} w_i = \frac {\sqrt{2} \sin(3\beta_d k/2)/(3h) - \sin(3\beta_a k/2)/(3h)} {\left[ \sin(\beta_a k/2)/h - \sin(3\beta_a k/2)/(3h) \right] - \sqrt{2} \left[ \sin(\beta_d k/2)/h - \sin(3\beta_d k/2)/(3h) \right]} \end{equation} This optimal weight $w_i$ is a function of mesh density only, and is not dependent of the time step size or the frequency of the signal. This method theoretically provides a uniform phase velocity in all directions. Further optimizations of this scheme were performed in another paper of Sun and Trueman \cite{Sun2}.
Koh et al. \cite{Koh} derived a two dimensional finite-difference time-domain method, discretizing the Maxwell equations, to eliminate the numerical dispersion and anisotropy. The proposed scheme is given as
\begin{eqnarray}\label{ff} d_t^2 H_{x,i,j+1/2}^n = -\frac{k}{\mu h} d_y E_{x,i,j+1/2}^n \nonumber \\ d_t^2 H_{y,i+1/2,j}^n = -\frac{k}{\mu h} d_x E_{y,i+1/2,j}^n \\ d_t^2 E_{z,i,j}^{n+1/2} + \frac{\sigma k}{2\epsilon} [E_{z,i,j}^{n+1} + E_{z,i,j}^{n}]
= \frac{k}{\epsilon h} d_x H_{y,i,j}^{n+1/2} - \frac{k}{\epsilon h} d_y H_{x,i,j}^{n+1/2} \nonumber \end{eqnarray} where $d_t^2$ is the central difference operator with respect to time,
\begin{eqnarray}\label{} d_p f_q = \left( 1-\frac{\alpha}{2} \right) d_p f_q + \frac{\alpha}{4} \left( d^2_p f_{q+1} + d^2_p f_{q-1} \right) \end{eqnarray} with $p$ or $q$ being either $x$ or $y$, and
\begin{eqnarray}\label{} d^2_x f_{i,j} = f_{i+1/2,j} - f_{i-1/2,j}, \hspace{4mm} d^2_y f_{i,j} = f_{i,j+1/2} - f_{i,j-1/2} \end{eqnarray} where $f$ is a generic function. In equation (\ref{ff}), $E$ is the electric field, $H$ is the magnetic field strength, $\sigma$, $\mu$ and $\epsilon$ are the conductivity, permeability and the permittivity, respectively, of the domain, $k$ is the time step, and $h$ is the spatial step in all directions. For a nonconductive media $\sigma = 0$, the numerical dispersion relation of can be obtained as
\begin{eqnarray}\label{mm} \frac{1}{h^2} C_{+} C_{\times} \left( \alpha - \frac{2}{C_{+}} \right)^2 - \frac{1}{h^2} \left( \frac{4C_{\times}}{C_{+}} - C_{+} \right) - \frac{1}{(ck)^2} \sin^2 \left( \frac{\omega k}{2} \right) \end{eqnarray} where $C_{+} = \sin^2 (\xi h/2) + \sin^2 (\eta h/2)$, $C_{\times} = \sin^2 (\xi h/2) \sin^2 (\eta h/2)$, and $\xi$ and $\eta$ are the components of the wavenumber. Equation (\ref{mm}) is a quadratic equation in $\alpha$, and the solution is given as
\begin{eqnarray}\label{mm} \alpha = \frac{2}{C_{+}} \left[ 1 - \sqrt{1 - \frac{h^2 C_{+}}{4C_{\times}} \left( \frac{1}{h^2} C_{+} - \frac{1}{(ck)^2} \sin^2 \left( \frac{\omega k}{2} \right) \right)} \right] \end{eqnarray} An optimal value for $\alpha$, achieving an isotropic numerical phase velocity, can be simply estimated as the mean value of $\alpha$ over the azimuthal angles, and it was found that it remains constant (approximately, $0.167$) for a wide range of grid sizes, and it is insensitive to the value of the Courant number.
Kim et al. \cite{Kim} derived new three-dimensional isotropic dispersion-finite-difference time-domain schemes (ID-FDTD) based on a a linear combination of the traditional central difference equation and a new difference equation based on the extra sampling points. They used the same scaling factors as for the two-dimensional case to attain isotropic dispersion and exact phase velocity. Based on the weighting factors, seven different FDTD schemes were formulated, including the Yee scheme \cite{Yee}. Among the seven proposed FDTD schemes, three showed improved isotropy of the dispersion compared to the dispersion of the Yee scheme. For the sake of brevity, the complete expressions of the schemes are not included here (see Kim et al. \cite{Kim} for more details), and only the numerical dispersion relation is briefly presented. Plane wave solutions were introduced in discretized forms as
\begin{eqnarray}\label{ii1} \textbf{E}^n_{i,j} = \textbf{E}_0 e^{I(n\omega k - \xi i h - \eta j h - \zeta k h)} \end{eqnarray}
\begin{eqnarray}\label{ii2} \textbf{H}^n_{i,j} = \textbf{H}_0 e^{I(n\omega k - \xi i h - \eta j h - \zeta k h)} \end{eqnarray} where $I = \sqrt{-1}$, $\omega$ is the frequency, $(\xi,\eta,\zeta)$ is the numerical wavenumber vector, and $\textbf{E}_0$ and $\textbf{H}_0$ are constant vectors. After inserting (\ref{ii1}) and (\ref{ii2}) into the discretized form of the Maxwell equations (see equation (10) in Kim et al. \cite{Kim}), matrix equations are obtained as $ C \textbf{H}_0 = S_t \epsilon_0 \textbf{E}_0, \hspace{1mm} C \textbf{E}_0 = S_i \mu_0 \textbf{H}_0 $ where
\begin{eqnarray} C = \left[ \begin{array}{ccccc}
0 & -K_z & K_y \\
K_z & 0 & -K_x \\ -K_y & K_x & 0 \end{array} \right] \end{eqnarray} and $K_p = S_p/h [\alpha (P_p - Q_p) - \beta Q_p/2 + 1]$ ($p$ being either $x$, $y$ or $z$), $S_x = \sin(\xi h/2)$, $S_y = \sin(\eta h/2)$, $S_z = \sin(\zeta h/2)$, $P_x = Sy Sz$, $P_y = Sx Sz$, $P_z = Sx Sy$, $Q_x = S_y^2 + S_z^2$, $Q_y = S_x^2 + S_z^2$, $Q_z = S_x^2 + S_y^2$, and $S_t = \sin{\omega k/2}/k$. An eigenvalue equation was obtain as
\begin{eqnarray}\label{} (C^2 + S_t^2 \mu_0 \epsilon_0 I) = 0, \end{eqnarray} and the numerical dispersion relation was obtained by vanishing the associated determinant,
\begin{eqnarray}\label{} \frac{S_t^2}{c_0^2} = K_x^2 + K_y^2+ K_z^2 \end{eqnarray} where $c_0 = 1/\sqrt{\epsilon_0 \mu_0}$. The isotropy correction was performed by defining the values of the weighting factors $\alpha$ and $\beta$, which unlike the two-dimensional case are not unique. Kim et al. \cite{Kim} used the scaling factor from the two-dimensional case, and modified the numerical dispersion relation to estimate the weighting factors.
\subsection{Dendritic Solidification}
Kumar \cite{Kumar} derived isotropic finite difference schemes for the first and second derivatives in the context of symmetric dendritic solidification. The first derivative was discretized as
\begin{eqnarray}\label{vv} (\partial_x u)_{I,i,j} = \frac{1}{2h} \left[ \frac{1}{6}(u_{i+1,j+1} - u_{i-1,j+1}) \right. \nonumber \\ \left.
+ \frac{4}{6}(u_{i+1,j} - u_{i-1,j}) \right. \\ \left.
+ \frac{1}{6}(u_{i+1,j-1} - u_{i-1,j-1}) \right] \nonumber \end{eqnarray} which involves grid points not only along $x$-direction, but also along $y$-direction. The Taylor expansion of the scheme (\ref{vv}) can be written as $(\partial_x u)_{I,i,j} = (1+ h^2/6 \nabla^2)(\partial_x u)_{i,j}$, where the leading order term involves the Laplacian only, implying no directional dependence. The second derivative was discretized as
\begin{eqnarray}\label{} (\partial_{xx} u)_{I,i,j} = \frac{1}{h^2} \left[ \frac{1}{12}(u_{i+1,j+1} - 2u_{i,j+1} u_{i-1,j+1}) \right. \nonumber \\ \left. + \frac{10}{12}(u_{i+1,j} - 2u_{i,j} + u_{i-1,j}) \right. \\ \left. + \frac{1}{12}(u_{i+1,j-1} - 2u_{i,j-1} + u_{i-1,j-1}) \right] \nonumber \end{eqnarray} where the Taylor expansion is given by $(\partial_{xx} u)_{I,i,j} = (1+ h^2/12 \nabla^2)(\partial_{xx} u)_{i,j}$, being again a function of the Laplacian only. The conventional cross derivative $(\partial_{xy} u)_{I,i,j}$ was found to be intrinsically isotropic according to the criterion developed by Kumar \cite{Kumar}. The Laplacian can be obtained by combining the isotropic derivatives along x- and y-directions, $(\nabla^2 u)_{i,j} = (\partial_{xx} u)_{I,i,j} + (\partial_{yy} u)_{I,i,j}$. Significant reduction of the numerical anisotropy was obtained by using these schemes. Shen and Cangellaris \cite{Shen} exploited further this approach to develop new isotropic finite-difference time-domain schemes modeling electromagnetic wave propagation.
\section{Concluding Remarks}
Numerical anisotropy in finite difference discretizations of partial differential equations was discussed and reviewed. In some instances, the numerical anisotropy can be neglected, and the focus is directed toward other types of one-dimensional errors, such as numerical dispersion, dissipation or aliasing. These errors can be analyzed in the context of one dimensional differencec equations, while the extension to multidimensional discretizations is straightforward. By increasing the accuracy of one dimensional schemes or by increasing the number of grid points in the grid, the isotropic characteristics of the waves in multi-dimensions can be improved. These two practices, however, are not always effective since an increase in accuracy may require larger stencils which may introduce spurious waves at the boundaries of the domain, while by increasing of the resolution of the grid may increase the computational time. It is necessary then to analyzed the schemes in multi-dimensions and design specific optimizations with the specific objective of reducing the numerical anisotropy, and at the same time of conserving the dispersion characteristics of the corresponding one dimensional schemes. Various attempts to reduce the numerical anisotropy in finite differencing applied to various model equations were presented and discussed.
Future directions should focus on optimizations of existing compact finite difference schemes in terms of reducing the numerical anisotropy, or derivations of novel compact schemes with low numerical anisotropy. Optimizations and derivations of finite volume schemes (in terms of reducing the numerical anisotropy) applied to either structured or unstructured grids should be also taken into account, especially in the framework of wave propagation problems. Filtering schemes, as applied, for example, in large eddy simulations to separate the small scales from the large scales, may experience numerical anisotropy since they are effective at high wavenumber ranges. Optimizations of such filters in terms of reducing the numerical anisotropy is also another future area of research.
\begin{table}[htpb]
\begin{center}
\caption{Weights of the selected spatial finite difference stencils}
\label{t1}
\begin{tabular}{rrrrrrrrrr} \hline
Stencil & $\alpha_1$ & $\alpha_2$ & $a_1$ & $a_2$ & $a_3$ \\\hline
$E2$ & 0 & 0 & 1/2 & 0 & 0 \\
$E4$ & 0 &0 & 2/3 & -1/12 & 0 \\
$E6$ & 0& 0 & 3/4 & -3/20 & 1/60 \\
$DRP$ & 0 & 0 & 0.770882380 & -0.166705904 & 0.020843142 \\
$C4$ & 1/4& 0 & 3/4 & 0 & 0 \\
$Haras$ & 0.3534620& 0 & 1.5669657/2 & 0.13995831/4 & 0 \\
$Lui$ & 0.5381301& 0.0666331 & 1.36757772/2 & 0.823428170/4 & 0.0185207834/6\\
$Lele$ & 0.5771439& 0.0896406 & 1.3025166/2 & 0.99355/4 & 0.03750245/6 \\ \hline
\end{tabular}
\end{center} \end{table}
\end{document} | arXiv | {
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\begin{document}
\title{Subcomplexes of Certain Free Resolutions}
\begin{abstract} What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this paper, we invoke the Bernstein--Gel\cprime fand--Gel\cprime fand (BGG) correspondence to address this question for free resolutions given by two well-known complexes, the Koszul and the Eagon--Northcott. This novel approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon--Northcott case. \end{abstract}
\section{Introduction}
\begin{question} What are the possible subcomplexes of free resolutions given by the Koszul or Eagon--Northcott complexes? \end{question}
The motivating question of this article is simple to state, but there are no traditional methods in commutative algebra to answer it, or indeed its more general sibling:
\begin{question} Given a minimal free resolution ${\mathbf{G}}$ of a module over a local or graded ring, what are the subcomplexes ${\mathbf{F}}$ that have a split injective map into ${\mathbf{G}}$? \end{question}
We first learned of Question 1.1 for Koszul complexes in relation to ongoing work of Bertram and Ullery on Veronese secants and stable complexes. More broadly, we are also motivated by various uses of subcomplexes in the study of free resolutions. For instance, Question 1.2 is at the heart of recent work on virtual resolutions, where classifying and understanding subcomplexes with specific properties is the key to ~\cite[Theorem 3.1]{besVirtualResFAProdOProjSp} as well as related results such as ~\cite{harada2021virtual}. Even more generally, subcomplexes are fundamental objects of study in the world of syzygies; the linear strand, for example, plays an essential role in many results ~\cite{gKoszulCohomATGeomOProjVar, gKoszulCohomATGeomOProjVarII, ekSLinearSyzConj, gScrollarSyzOGenCanonicalCurvesWGenusle8}.
The goal of this paper is to further understand the structure of subcomplexes from a numerical standpoint. This numerical approach fits in with the well-established broader approach to understanding minimal free resolutions numerically (for instance via the study of Betti tables or Poincaré series). What is more, the numerical realm is the natural place in which to explore our main question, since a change of basis introduces an infinite number of possible subcomplexes.
In order to precisely state the numerical version of our question, we must first introduce some terminology. For a free complex ${\mathbf{F}}$, we define the \emph{rank sequence} of ${\mathbf{F}}$ to be the integer sequence $\rs({\mathbf{F}})~=~(r_0, r_1, \ldots )$ where $r_i$ is the rank of the $i^{th}$ free module in ${\mathbf{F}}$ (we refrain from calling these ``Betti numbers" since our subcomplexes may in general fail to be exact). For a complex ${\mathbf{G}}$, we will denote by $\RS({\mathbf{G}})$ the set of all integer sequences $r$ where $r = \rs({\mathbf{F}})$ for some subcomplex ${\mathbf{F}}$ of ${\mathbf{G}}$. Now we may ask the following:
\begin{question} Given a free resolution ${\mathbf{G}}$, when is an integer sequence $r$ in $\RS({\mathbf{G}})$? \end{question}
One might be tempted to approach the problem directly by trying to explicitly produce subcomplexes of free modules with prescribed ranks, but this raises certain subtleties even in small cases. We find that without some clear strategy for controlling subcomplexes, even this numerical question becomes hard.
For a concrete example, let $S = \mathbbm{k}[x_1, x_2, x_3]$ and ${\mathbf{G}}$ be the minimal free resolution of the residue field which is given by the Koszul complex on $x_1, x_2, x_3$, and consider the question ``Is there a subcomplex ${\mathbf{F}}$ of $ {\mathbf{G}}$ with ranks $r=(1,2,2,0)$?'' That is, ``Is $(1,2,2,0)$ in $\RS({\mathbf{G}})$?'' To answer this question directly requires a linear algebra analysis of the maps required to fill in a diagram like the one below, thus ultimately producing ${\mathbf{F}}$ in its entirety:
\[ \xymatrix{
{\mathbf{F}}: 0 \ar[r]
& 0 \ar[r]\ar[d]
& S(-2)^2 \ar[d]_{\varphi_{2}} \ar[r]^{f_2}
& S(-1)^2 \ar[d]_{\varphi_1} \ar[r]^{f_1}
& S \ar[r] \ar[d]_{\varphi_0}
& 0 \\
{\mathbf{G}}: 0 \ar[r]
& S(-3) \ar[r]
& S(-2)^3 \ar[r]
& S(-1)^3 \ar[r]
& S \ar[r]
& 0
} \]
Thus the restriction of the question to ranks does not give a commensurate improvement in obtaining an answer, and it is clear that a different approach is necessary.
This alternate approach uses the Bernstein--Gel'fand--Gel'fand (BGG) correspondence, which gives an equivalence of categories between graded linear complexes of free modules over a polynomial ring on the one hand and graded modules over an exterior algebra on the other. The question of subcomplexes thus becomes a question of submodules, where the restriction to possible ranks is translated to a question of possible Hilbert functions. Here we make use of existing results in a way that makes broad restrictions possible without constructing entire complexes.
In this paper, we provide some answers to the question of possible subcomplexes of resolutions for two classes of modules over the polynomial ring---complete intersections and quotients by some determinantal ideals---whose minimal free resolutions are given by the Koszul and Eagon--Northcott complexes respectively. The novelty of these theorems is twofold: besides providing constraints on permissible subcomplexes, they demonstrate the efficacy of the correspondence in tackling an otherwise intractable problem and provide insight on how similar results might be obtained for other free resolutions. In fact, since the BGG correspondence is an instance of Koszul duality, these techniques could extend to characterizing subcomplexes of linear complexes over general Koszul algebras, e.g. the Priddy complex.
Our first theorem exactly characterizes the integer sequences that can arise as ranks of subcomplexes of a minimal free resolution of a complete intersection. Let $S = \mathbbm{k}[x_1, \ldots, x_n]$ and ${\mathbf{K}}(f_1, \ldots, f_m)$ be the Koszul complex on $f_1, \ldots, f_m$.
\begin{introtheorem}[Theorem~\ref{thm:generalkoszulranks}]\label{thm:intro1} Let $f_1, \ldots, f_m$ be homogeneous polynomials forming a regular sequence in the polynomial ring $S = \mathbbm{k}[x_1, \ldots, x_n]$ and $r = (r_0, \ldots, r_m)$ be a non-negative integer sequence. Then $r\in \RS({\mathbf{K}}(f_1, \ldots, f_m))$ if and only if it is the zero sequence or $r_0 = 1$ and \[ 0\leq r_{i+1}\leq r_i^{(i)} for 1\leq i\leq n-1, \] where $r_i^{(i)}$ is the shifted Macaulay expansion of $r_i$ as in Definition~\ref{def:macaulayexpansion}. \end{introtheorem}
We are then able to leverage this complete characterization along with the BGG correspondence to obtain meaningful restrictions on the integer sequences that can arise as ranks of subcomplexes of the Eagon--Northcott complex. Before we can state this result we need to introduce some more notation: given complexes ${\mathbf{F}}$ and ${\mathbf{G}}$, we write $\RS({\mathbf{F}}) + \RS({\mathbf{G}})$ for the set of sequences $r$ that can be written as an entry-wise sum $r = r_1 + r_2$ for $r_1\in \RS({\mathbf{F}})$ and $r_2\in \RS({\mathbf{G}})$. For an integer $a$, the set $a\RS({\mathbf{F}})$ is defined to be the $a$-fold sum $\RS({\mathbf{G}}) + \cdots + \RS({\mathbf{G}})$. We denote by ${\mathbf{K}}_{(m)}$ the Koszul complex on $m$ variables
With this notation, we can state our second main result about resolutions of ideals $I$ where the Eagon--Northcott complex gives a minimal free resolution of $S/I$.
\begin{introtheorem}[Theorem~\ref{thm:generalENranks}]\label{thm:intro2} Let $S = \mathbbm{k}[x_1, \ldots, x_n]$ and let $\phi$ be a $p\times q$ matrix with $p\leq q$ such that the maximal minors of $\phi$ generate an ideal $I$ of codimension $q-p+1$ where $S/I$ is Cohen-Macaulay. Let ${\mathbf{F}}$ be the minimal free resolution of $S/I$, and let $r$ be an integer sequence. If $r\in \RS({\mathbf{F}})$, then $r = (0, r_0, r_1,\ldots )$ or $r = (1, r_0, r_1, \ldots)$ where the sequence $(r_0, r_1, \ldots)$ is in \[ \sum_{j=0}^{q-p}\binom{q-j-1}{p-1}\RS({\mathbf{K}}_{(q-p-j)}). \] \end{introtheorem}
\begin{example} Let $S = \mathbbm{k}[x_1,x_2,x_3]$, $\phi = \begin{bmatrix} x_1 & x_2 & x_3 & 0 \\ 0 & x_1 & x_2 & x_3 \end{bmatrix}$, and $I$ be the ideal of $2 \times 2$ minors of $\phi$. The minimal free resolution of $S/I$ is an Eagon--Northcott complex of the form \[ {\mathbf{G}}: \quad 0 \to S(-4)^3 \to S(-3)^8 \to S(-2)^6 \to S, \] with maps as shown in Example~\ref{ex:eagonnorthcottforn3d2}. We can find subcomplexes of ${\mathbf{G}}$ of the form \[ 0 \to S(-4)^2 \xrightarrow{f_3} S(-3)^6 \xrightarrow{f_2} S(-2)^5 \xrightarrow{f_1} S \]
and \[ 0 \rightarrow S(-4)^1 \rightarrow S(-3)^3 \rightarrow S(-2)^3 \rightarrow S, \] but combining Theorem~\ref{thm:intro2} with the characterization of subcomplexes of the Koszul complex given in Theorem ~\ref{thm:intro1} rules out a subcomplex of ${\mathbf{G}}$ of the form \[ 0 \to S(-4)^3 \to S(-3)^5 \to S(-2)^5 \to S. \] This is because $(1,5,5,3)$ is not of the form $(1,r)$, where $r$ is in the set \[R = 3 \RS({\mathbf{K}}_{(2)}) + 2 \RS({\mathbf{K}}_{(1)}) + \RS({\mathbf{K}}_{(0)}).\] By Theorem~\ref{thm:intro1}, $\RS({\mathbf{K}}_{(0)}) = \{(1,0,0), (0,0,0)\}$, $\RS({\mathbf{K}}_{(1)}) = \{(1,1,0), (1,0,0), (0,0,0) \}$, and $\RS({\mathbf{K}}_{(2)}) = \{(1,2,1), (1,2,0), (1,1,0), (1,0,0), (0,0,0)\}$. If a sequence in $R$ has a $3$ in the last spot, we must use the sequence $(1,2,1)$ from $\RS({\mathbf{K}}_{(2)})$ thrice. However, three times $(1,2,1)$ gives a $6$ in the middle position, so any sequence in $R$ with a $3$ in the last spot has at least a $6$ in the middle. Therefore $(5,5,3) \notin R$ and thus $(1,5,5,3) \notin \RS({\mathbf{G}})$. \end{example}
As seen in the example, the common theme throughout our results is that subtle numerics govern whether a given sequence of free modules and maps between them has any hope of being a complex. This situation is reminiscent of numerical conditions that tell us when a given complex can be exact -- a far more well-studied question. Many results are concerned with precisely characterizing exactness, while understanding when a sequence of maps is a complex is taken for granted. To riff on the title of \cite{beWhatMakesAComplexExact}, herein we set out to explore the dual question: ``What makes a complex a complex?''
\subsection*{Acknowledgments} The authors would like to thank Daniel Erman for originally suggesting the motivating question and many helpful discussions throughout all stages of the article's development. We also thank the anonymous referees for their thoughtful comments and suggestions.
\section{Background}\label{section:background}
For the sake of clarity, we settle a formal definition of ``subcomplex.''
\begin{definition} Let ${\mathbf{F}}=(F_i,f_i)$ and ${\mathbf{G}}=(G_i,g_i)$ be two complexes of free modules over the same ring. We say ${\mathbf{F}}$ is a \textit{subcomplex} of ${\mathbf{G}}$ if there are split injective maps $\varphi_i: F_i \rightarrow G_i$ so that $\varphi_i \circ f_{i+1} = g_{i+1} \circ \varphi_{i+1}$, i.e. each square of the following diagram commutes: \[ \xymatrix{
{\mathbf{F}}: \cdots \ar[r] & F_{i+1} \ar[d]_{\varphi_{i+1}} \ar[r]^{f_{i+1}} & F_{i} \ar[d]_{\varphi_{i}} \ar[r]^{f_{i}} & F_{i-1} \ar[d]_{\varphi_{i-1}} \ar[r]^{f_{i-1}} & \cdots \\
{\mathbf{G}}: \cdots \ar[r] & G_{i+1} \ar[r]_{g_{i+1}} & G_{i} \ar[r]_{g_i} & G_{i-1} \ar[r]_{g_{i-1}} & \cdots
} \] \end{definition}
In particular, we exclude injective maps like $\varphi_i: G_i(-1) \xrightarrow{\cdot x} G_i$. In the cases we are interested in, these $\varphi_i$ can be represented by matrices with full column rank and entries from the ground field $\mathbbm{k}$.
Given a free complex ${\mathbf{G}}$, our goal will be to classify the ranks of free modules appearing in subcomplexes of ${\mathbf{G}}$. We introduce some notation that will be used throughout.
\begin{definition} Given a free complex ${\mathbf{F}} = \cdots \to F_1 \to F_0 $, the \emph{rank sequence} of ${\mathbf{F}}$ is \[\rs({\mathbf{F}}) = (r_0, r_1, \ldots)\] where $r_i$ is the rank of the free module $F_i$. For a complex ${\mathbf{G}}$, we use $\RS({\mathbf{G}})$ to denote the set of all possible rank sequences of subcomplexes of ${\mathbf{G}}$. \end{definition}
\begin{notation} Given two sets of rank sequences, say $A = \RS({\mathbf{F}})$ and $B = \RS({\mathbf{G}})$, we will write $A+B$ to refer to the set of sequences that may be expressed as a sum of a sequence in $A$ and a sequence in $B$. Similarly, we will write $nA$ to refer to the set $A+A+\cdots + A$ where the sum has $n$ terms. \end{notation}
\subsection{The BGG Correspondence}
The key tool for our results is the Bernstein-Gel'fand-Gel'fand correspondence \cite{bggAlgVecBunOPnAProbOLinAlg}, which allows us to translate questions about linear free complexes of modules over a symmetric algebra into questions about modules over an exterior algebra. We will cherry-pick what we need of this rich subject; for further detail, see \cite{efsSheafCohomAFreeResOExtAlg} and \cite[Section~7B]{eGeomOSyz}.
Let $\mathbbm{k}$ be a field, $V$ be a $\mathbbm{k}$-vector space with basis $x_1, \ldots, x_n$, and $W$ be the dual vector space of $V$ with basis $e_1, \ldots, e_n$. Let $E = \mathbbm{k}\langle e_1, \ldots, e_n\rangle$ denote the exterior algebra on $W$. Let $S$ denote the symmetric algebra $\Sym(V)$, and identify $S$ with the polynomial ring $\mathbbm{k}[x_1, \ldots, x_n]$. We will assume that the $x_i$ are graded in degree $1$, and the $e_i$ are graded in degree $-1$. Unless otherwise stated, all tensor products are assumed to be over the ground field $\mathbbm{k}$.
We define a pair of functors $\mathbbm{L}$ and $\mathbbm{R}$ as follows: \begin{align*} \mathbbm{L}: \{\text{Graded $E$-modules}\} &\to \{\text{Linear complexes of free $S$-modules}\}\\ N &\longmapsto (\cdots\to S\otimes N_d\xrightarrow[]{\partial_d} S\otimes N_{d-1}\to\cdots) \end{align*} with differential $\partial_d$ defined by linearly extending \[ 1\otimes f\mapsto \sum_{i=1}^n x_i\otimes fe_i \] and \begin{align*} \mathbbm{R}: \{\text{Graded $S$-modules}\} &\to \{\text{Linear complexes of free $E$-modules}\}\\ M &\longmapsto (\cdots\to E\otimes M_d\xrightarrow[]{\partial_d} E\otimes M_{d+1}\to\cdots) \end{align*}
with differential $\partial_d$ defined by linearly extending \[ 1\otimes g\mapsto \sum_{i=1}^n e_i\otimes gx_i. \]
One can check that the functors $\mathbbm{L}$ and $\mathbbm{R}$ preserve exactness.
\begin{example}\label{ex:1441} Consider the module $N = \langle e_1, e_2e_3 \rangle E$ where $E = \mathbbm{k}\langle e_1, e_2, e_3, e_4\rangle$. We will use the following $\mathbbm{k}$-bases for the graded pieces of $N$: \[ \begin{array}{ll}
\operatorname{degree} \ -1 &: e_1 \\
\operatorname{degree} \ -2 &: e_1e_2, \ e_1e_3, \ e_1e_4, \ e_2e_3 \\
\operatorname{degree} \ -3 &: e_1e_2e_3, \ e_1e_2e_4, \ e_1e_3e_4, \ e_2e_3e_4 \\
\operatorname{degree} \ -4 &: e_1e_2e_3e_4 \end{array} \]
Tracing through the definition of $\mathbbm{L}$ we can see, for example, that \[{\partial}_{-2}(1 \otimes e_1e_2) = \sum\limits_{i=1}^4 x_i \otimes e_1e_2e_i = x_3 \otimes e_1e_2e_3 + x_4 \otimes e_1e_2e_4.\]
The entirety of $\mathbbm{L}(N)$ is the complex: \[ 0 \rightarrow S \otimes N_{-1}
\xrightarrow{ \begin{bmatrix} x_2 \\ x_3 \\ x_4 \\ 0 \end{bmatrix} }
S \otimes N_{-2}
\xrightarrow{ \begin{bmatrix} x_3 & -x_2 & 0 & x_1 \\ x_4 & 0 & -x_2 & 0 \\ 0 & x_4 & -x_3 & 0 \\ 0 & 0 & 0 & x_4 \end{bmatrix} }
S \otimes N_{-3}
\xrightarrow{ \begin{bmatrix} x_4 & -x_3 & x_2 & -x_1 \end{bmatrix} }
S \otimes N_{-4} \rightarrow 0 \] \end{example}
The BGG correspondence states that if we consider $\mathbbm{L}$ and $\mathbbm{R}$ as functors on the bounded derived categories then they are adjoint, implying that the derived categories of bounded linear complexes of finitely generated graded $E$-modules and $S$-modules are equivalent. What is more, the functor $\mathbbm{L}$ gives a bijection on objects under which \begin{enumerate}
\item Any linear complex ${\mathbf{F}}$ of $S$-modules may be expressed as $\mathbbm{L}(N)$ for some $E$-module $N$ \cite{efsSheafCohomAFreeResOExtAlg}, and
\item Subcomplexes of ${\mathbf{F}}$ correspond to $E$-submodules of $N$. \end{enumerate}
\begin{remark}\label{rmk:hfrs} For ${\mathbf{F}} = \mathbbm{L}(N)$, we can therefore relate the Hilbert function of $N$ and the rank sequence $\rs({\mathbf{F}})$. Take $r = (r_0, r_1, \ldots, r_n)$ and $h=(h_0, h_1, \ldots, h_n)$ to be two sequences of non-negative integers. Then $r = \rs({\mathbf{F}})$ if and only if $h_i = r_{n-i}$ is the Hilbert function of $N$, that is, if $h_i = \dim_\mathbbm{k}(N_{-i}) = r_{n-i}$. Note that we are still considering the Hilbert function $h$ as a function from $\mathbb{N}$ to $\mathbb{N}$, despite the negative grading on $E$. We will occasionally commit the minor sin of conflating $h$ as a function and an integer sequence, and thus write $h(N)$ for the sequence $(h_0, h_1, \ldots, h_n)$ where $h_i = \dim_\mathbbm{k}(N_{-i})$. \end{remark}
\begin{remark} A quick check reveals that shifting the homological degree of a complex ${\mathbf{F}}$ corresponds with twisting an $E$-module by that same degree, that is, if $\mathbbm{L}(N) = {\mathbf{F}}$, then $\mathbbm{L}(N(i)) = {\mathbf{F}}[i]$, where ${\mathbf{F}}[i]_j = {\mathbf{F}}_{i+j}$. \end{remark}
We also make use of the following relationship between $\mathbbm{L}$ and $\mathbbm{R}$:
\begin{theorem}{(Reciprocity Theorem) \cite[Theorem~3.7]{efsSheafCohomAFreeResOExtAlg}} \label{thm:reciprocitythm} Let $M$ be a graded $S$-module and let $N$ be a graded $E$-module. Then \[ N\to \mathbbm{R}(M) \] is an injective resolution if and only if \[ \mathbbm{L}(N)\to M \] is a free resolution. \end{theorem}
\subsection{Tate Resolutions}
\begin{definition} For any module $N$ over any ring, we can combine a projective resolution ${\mathbf{P}}$ of $N$ and an injective resolution ${\mathbf{I}}$ of $N$ in the following way \[ \begin{tikzcd} \cdots \arrow[r, "{\partial}_2"] & P_1 \arrow[r, "{\partial}_1"] & P_0 \arrow[rd] \arrow[rr, "{\partial}_0"] & & I_0 \arrow[r, "{\partial}_{-1}"] & I_1 \arrow[r, "{\partial}_{-2}"] & \cdots \\
& & & N \arrow[rd] \arrow[ru] & & & \\
& & 0 \arrow[ru] & & 0 & & \end{tikzcd} \] to produce a \textit{Tate resolution}. \end{definition}
More detail about general Tate resolutions can be found in \cite{efsSheafCohomAFreeResOExtAlg}, but we are most interested in Tate resolutions of modules over $E$, where injective and projective modules are both free. In this case, we can take ${\mathbf{P}}$ to be a minimal free resolution of $N$ and ${\mathbf{I}}$ to be the dual of the minimal free resolution of the dual of $N$ to create a unique doubly infinite exact complex of free modules where the image of $P_0$ is isomorphic to $N$. We will call this doubly infinite complex \textit{the} Tate resolution ${\mathbf{T}}(N)$.
\begin{example}\label{ex:tateres} If $N = E/\langle e_1, \ldots, e_n \rangle \cong \mathbbm{k}$, then the Cartan resolution $({\mathbf{C}},{\partial})$ is a projective resolution of $N$ (cf. \cite[Corollary~7.10]{eGeomOSyz}). The dual of ${\mathbf{C}}$ is an injective resolution of $\mathbbm{k}$ (which is its own dual), so stitching the two together yields the Tate resolution of $\mathbbm{k}$. Below is a snippet of ${\mathbf{T}}(\mathbbm{k})$ in the $n=3$ case: \[ \hspace{-1cm} \begin{tikzcd} \cdots \arrow[r] & E^6(2) \arrow[r, "{\partial}_2"] & E^3(1) \arrow[r, "{\partial}_1"] & E \arrow[rd] \arrow[rr, "{\partial}_0"] & & E(-3) \arrow[r, "{\partial}_1^T"] & E^3(-4) \arrow[r, "{\partial}_2^T"] & E^6(-5) \arrow[r] & \cdots \\
& & & & \mathbbm{k} \arrow[rd] \arrow[ru] & & & & \\
& & & 0 \arrow[ru] & & 0 & & & \end{tikzcd} \] where $ {\partial}_0 = \begin{bmatrix} e_1 e_2 e_3 \end{bmatrix}$,
${\partial}_1 = \begin{bmatrix} e_1 & e_2 & e_3 \end{bmatrix}$,
and
${\partial}_2 = \begin{bmatrix} e_1 & e_2 & 0 & e_3 & 0 & 0 \\ 0 & e_1 & e_2 & 0 & e_3 & 0 \\ 0 & 0 & 0 & e_1 & e_2 & e_3 \end{bmatrix}. $
In general, the differential ${\partial}_s$ in the Cartan resolution can be computed by indexing the columns of ${\partial}_s$ with the degree-$s$ monomials in the $x_i$'s and the rows by the degree $s-1$ monomials in the $x_i$'s. Then, if column $i$ is indexed by a monomial $m$ and row $j$ is indexed by a monomial $m'$, the $(i,j)$th entry of ${\partial}_s$ is $e_k$ if $m/m' = x_k$ if $m' \mid m$ and $0$ if $m \nmid m'$. In Example~\ref{ex:tateres}, the indexing monomials for the entries of the ${\partial}_s$ are listed in graded reverse lexicographic order with $x_1 > x_2 > x_3$.
\begin{remark} \label{rem:tateresformoreambient} Because $E$ is free over $\mathbbm{k}\langle e_1, \ldots, e_m \rangle$ for $m \leq n$, extending scalars from $\mathbbm{k}\langle e_1, \ldots, e_m \rangle$ to $E$ is faithfully flat. This means the Tate resolution ${\mathbf{T}}(E/\langle e_1, \ldots, e_m \rangle)$ has the same structure as the Tate resolution ${\mathbf{T}}(\mathbbm{k}\langle e_1, \ldots, e_m\rangle/\langle e_1, \ldots, e_m \rangle)$ as a complex of $\mathbbm{k}\langle e_1, \ldots, e_m\rangle$-modules. That is, the complex of $\mathbbm{k}\langle e_1, \ldots, e_m\rangle$-modules ${\mathbf{T}}(\mathbbm{k})$ and the complex of $E$-modules ${\mathbf{T}}(\mathbbm{k}\langle e_{m+1}, \ldots, e_n \rangle)$ have modules of the same rank and twists, and differentials with the same entries, regardless of the ambient ring. For example, the Tate resolution ${\mathbf{T}}(\mathbbm{k}\langle e_4\rangle)$ over $\mathbbm{k} \langle e_1, \ldots, e_4\rangle$ will ``look'' the same as the one shown in Example~\ref{ex:tateres}, with all $E$'s replaced by $\mathbbm{k}\langle e_1, \ldots, e_4 \rangle$. \end{remark}
\end{example}
\section{Resolutions of ${\mathfrak{m}}^d$}
As before, let $S = \mathbbm{k}[x_1, \ldots, x_n]$, where $\mathbbm{k}$ is a field. Use ${\mathfrak{m}}$ to denote the homogeneous maximal ideal $\langle x_1, \ldots, x_n \rangle$. We begin by exploring the possible rank sequences of subcomplexes of resolutions of ${\mathfrak{m}}^d$, in particular as they are presented by the Koszul complex in the $d=1$ case and the Eagon--Northcott complex in the $d\geq 2$ case.
\subsection{The Koszul Complex}
\begin{definition}\label{def:koszul} The Koszul complex ${\mathbf{K}}(x_1, \ldots, x_m)$ is the graded exact complex \[ {\mathbf{K}}(x_1, \ldots, x_m): 0 \rightarrow S(-m) \xrightarrow{{\partial}_m} \cdots \xrightarrow{{\partial}_3} S(-2)^{\binom{m}{2}} \xrightarrow{{\partial}_2} S(-1)^m \xrightarrow{{\partial}_1} S^1 \rightarrow 0, \] where we index basis elements of $K_d \coloneqq S(-d)^{\binom{m}{d}}$ by size $d$ subsets of $m$. For $T~=~\{i_1, \ldots, i_d\}$, the differential ${\partial}_d$ acts on $e_T$ by ${\partial}_d(e_T) = \sum_{j = 1}^{d} (-1)^{j} x_{i_j} e_{T \backslash i_j}$. \end{definition}
\begin{example}\label{ex:koszul3} The Koszul complex ${\mathbf{K}}(x_1,x_2,x_3)$ is given by \[ {\mathbf{K}}(x_1, x_2, x_3): 0 \rightarrow S(-3) \xrightarrow{{\partial}_3} S(-2)^3 \xrightarrow{{\partial}_2} S(-1)^3 \xrightarrow{{\partial}_1} S \rightarrow 0 \] where \[ {\partial}_1 = \begin{tikzpicture}[anchor=base, baseline]
\matrix [matrix of math nodes,left delimiter={[},right delimiter={]}] (d1)
{x_1 & x_2 & x_3 \\};
\draw [color=red] (d1-1-1.north west) rectangle (d1-1-2.south east); \end{tikzpicture} \quad {\partial}_2 = \begin{tikzpicture}[baseline]
\matrix [matrix of math nodes,left delimiter={[},right delimiter={]}] (d2)
{-x_2 & -x_3 & 0 \\
x_1 & 0 & -x_3 \\
0 & x_1 & x_2 \\};
\draw [color=red] (d2-1-1.north west) rectangle (d2-2-1.south east); \end{tikzpicture} \quad \text{ and } \quad {\partial}_3 = \begin{tikzpicture}[baseline]
\matrix [matrix of math nodes,left delimiter={[},right delimiter={]}] (d3)
{x_3 \\ -x_2 \\ x_1 \\}; \end{tikzpicture}. \] \end{example}
Looking at the above example, we can immediately identify some subcomplexes of the Koszul complex. If $m \leq n$, the complex ${\mathbf{K}}(x_1, \ldots, x_m)$ is a subcomplex of ${\mathbf{K}}(x_1, \ldots, x_n)$ -- one can see the Koszul complex ${\mathbf{K}}(x_1,x_2)$ boxed in red in Example~\ref{ex:koszul3}. One can also simply truncate ${\mathbf{K}}(x_1, x_2, x_3)$ after two modules and omit the $S(-3)$ module at the end, or even omit $S(-3)$ and some summands of the $S(-2)^3$ in the next spot. This observation yields certain sequences that we can be sure must occur as rank sequences of subcomplexes of ${\mathbf{K}}(x_1, \ldots, x_m)$ but to obtain a more complete classification we can peer through the BGG lens and, in particular, use the following key fact.
\begin{fact}[see Example~7.6, \cite{eGeomOSyz}] The linear complex $\mathbbm{L}(E(-n))$ is (isomorphic to) the Koszul complex ${\mathbf{K}}(x_1, \ldots, x_n)$. \end{fact}
The BGG correspondence thus tells us that subcomplexes of the Koszul complex are in correspondence with submodules of the exterior algebra $E$ twisted by $(-n)$, so our question about the possible rank sequences of subcomplexes of ${\mathbf{K}}$ is transformed into a question about the possible Hilbert functions of submodules of $E$ itself (after the appropriate twist). This perspective immediately reveals that subcomplexes of ${\mathbf{K}}$ are less restricted than one might guess from the $n=3$ case. Indeed, Example~\ref{ex:1441} shows that we can obtain a subcomplex of ${\mathbf{K}}(x_1, \ldots, x_4)$ whose rank sequence is $(1,4,4,1,0)$, which is not the rank sequence of a smaller Koszul complex and furthermore cannot be obtained by truncating free summands from the tail of $K$.
This observation also underscores the complexity of the structural question of classifying all subcomplexes in the case of the Koszul complex. Such a task would be equivalent to classifying all ideals in $E$. Though the feasibility of such classification is yet unknown, it is worth noting that the parallel question of classifying ideals in $S$ is impossible by Vakil's Murphy's Law \cite{vMurphysLawIAlgGeoBadBehavedDefSpaces}.
By work of Aramova--Herzog--Hibi \cite[Theorem~4.1]{ahhGotzmannThFExteriorAlgACombinatorics}, possible Hilbert sequences of submodules of the exterior algebra are exactly those corresponding to $f$-vectors of simplicial complexes as described by the Kruskal--Katona Theorem. We can use these results to characterize the possible rank sequences for a subcomplex of the Koszul complex with the following notation.
\begin{definition}\label{def:macaulayexpansion} If $a$ is a positive integer, then, for every positive integer $i$, $a$ has a unique \emph{Macaulay expansion} \[ a = \binom{a_i}{i} + \binom{a_{i-1}}{i-1}+\cdots +\binom{a_j}{j}, \] where $a_i>a_{i-1}>\cdots>a_j\geq j\geq 1$. Define \[ a^{(i)}:= \binom{a_i}{i+1} + \binom{a_{i-1}}{i}+\cdots +\binom{a_j}{j+1}. \] \end{definition}
\begin{theorem}\label{thm:basickoszulranks} A non-negative integer sequence $(r_0, r_1, \ldots, r_n)$ is in $\RS({\mathbf{K}}(x_1, \ldots, x_n))$ if and only if it is the all zeroes sequence or if $r_0=1$ and $r$ satisfies \[ 0\leq r_{i+1}\leq r_i^{(i)} \text{ for } 1 \leq i \leq n-1. \] \end{theorem}
\begin{proof} If $r$ is the sequence of all zeroes, it is the rank sequence of the zero complex, which is a subcomplex of any complex.
Using Corollary~5.3 from \cite{ahhGotzmannThFExteriorAlgACombinatorics} and Remark~\ref{rmk:hfrs}, we see that $h(E/(0:I)) = \rs(\mathbbm{L}(I))$. Because every ideal in $E$ satisfies $0:(0:I) = I$ and can therefore be recognized as an annihilator, classifying Hilbert sequences $h(E/I)$ is equivalent to classifying Hilbert sequences $h(E/(0:I))$. Combining these two sentences, we see that classifying rank sequences $\rs(\mathbbm{L}(I))$ is equivalent to classifying rank sequences $h(E/I)$.
By \cite[Theorem~4.1]{ahhGotzmannThFExteriorAlgACombinatorics}, a non-negative integer sequence $h=(1,h_1, \ldots, h_n)$ is the Hilbert sequence of a module $E/I$ if and only if $0 \leq h_{i+1} \leq h_i^{(i)}$ for all $1 \leq i \leq n-1$. This translates directly to the set $\RS({\mathbf{K}}(x_1, \ldots, x_n))$, and our theorem is proven. \end{proof}
\begin{remark} In an analogous way, Macaulay's Theorem (c.f. \cite[Theorem~4.2.14]{bhCMRings}) characterizes the ranks of subcomplexes of the Cartan resolution of $\mathbbm{k}$ over $E$. \end{remark}
\subsection{The Eagon--Northcott Complex} The Eagon--Northcott complex \cite{enIdDefBMatAACComplexAssocWThem} plays the same role for determinantal ideals that a Koszul complex plays for a sequence of ring elements. We provide a brief presentation here that describes the complex for $S$-modules; more details can be found in \cite[Appendix~A2H]{eGeomOSyz}. Throughout, we will choose bases for our free modules so that we can represent these maps as matrices.
\begin{definition} Let $F = S^f$ and $G=S^g$, with $g \leq f$, and $\alpha: F \rightarrow G$ a map represented by a $g \times f$ matrix $A$ with respect to bases $\{e_1, \ldots, e_f\}$ of $F$ and $\{\varepsilon_1, \ldots, \varepsilon_g\}$ of $G$. Then the Eagon--Northcott complex of the map $\alpha$ is the complex \[ {\mathbf{EN}}(\alpha): 0 \rightarrow EN_{f-g+1} \xrightarrow{d_{f-g+1}} EN_{f-g} \xrightarrow{d_{f-g}} \cdots \xrightarrow{d_3} EN_2 \xrightarrow{d_2} EN_1 \xrightarrow{\Lambda^g \alpha} \Lambda^g G \] where $EN_{k+1} = (\Sym_k G)^* \otimes \Lambda^{g+k} F$ and $d_{k+1}: (\Sym_k G)^* \otimes \Lambda^{g+k} F \rightarrow (\Sym_{k-1} G)^* \otimes \Lambda^{g+k-1} F$ is the map \begin{multline*} (\varepsilon_{1}^{p_1} \cdots \varepsilon_{g}^{p_g})^* \otimes e_{s_1} \wedge \cdots \wedge e_{s_{g+k}} \mapsto \\ \sum\limits_{i=1}^{g+k} (-1)^{i-1} \left[ \sum\limits_{j=1}^{g} A_{j,s_i} (\varepsilon_1^{p_1} \cdots \varepsilon_j^{p_j-1} \cdots \varepsilon_g^{p_g})^* \right] \otimes e_{s_1} \wedge \cdots \wedge \widehat{e_{s_i}} \wedge \cdots \wedge e_{s_{g+k}}. \end{multline*} for $k \geq 1$, where $p_1 + \ldots + p_g = k$ and we adopt the convention that $\varepsilon_j^p = 0$ if $p < 0$. \end{definition}
Note that, if we represent $\alpha$ by the matrix $A$, then $A_{j,s} = \varepsilon_j^*(\alpha(e_s))$, and that using a different basis to express $A$ will give an isomorphic complex.
\begin{example}\label{ex:eagonnorthcottforn3d2} For this example, let $S= \mathbbm{k}[x_1, x_2, x_3]$. Consider the example $\alpha:~S^4~\rightarrow~S^2$ represented by the matrix $A = \begin{bmatrix} x_1 & x_2 & x_3 & 0 \\ 0 & x_1 & x_2 & x_3 \\ \end{bmatrix}.$
Making the appropriate identifications for each module, we can see that ${\mathbf{EN}}(\alpha)$ is the resulting graded complex \[ {\mathbf{EN}}(\alpha): 0 \rightarrow S(-4)^3 \xrightarrow{d_3} S(-3)^8 \xrightarrow{d_2} S(-2)^6 \xrightarrow{d_1} S. \]
Note that the ideal of maximal minors in Example~\ref{ex:eagonnorthcottforn3d2} is the ideal $\langle x_1, x_2, x_3 \rangle^2$. In general, the Eagon--Northcott complex minimally resolves any power of the maximal ideal ${\mathfrak{m}}^d$ by constructing the complex for the $d \times (n+d-1)$ matrix \[ M^{n,d} = \begin{bmatrix} x_1 & x_2 & \ldots & x_n & 0 & \ldots & \ldots & 0 \\ 0 & x_1 & x_2 & \ldots & x_n & 0 & \ldots & 0 \\ \vdots & \ddots & \ddots & \ddots & & \ddots & \ddots & \vdots \\ 0 & \ldots & 0 & x_1 & x_2 & \ldots & x_n & 0 \\ 0 & \ldots & \ldots & 0 & x_1 & x_2 & \ldots & x_n \\ \end{bmatrix}. \]
\end{example}
With this notation, the matrix $A$ in Example~\ref{ex:eagonnorthcottforn3d2} is $M^{3,2}$.
Our eventual goal is to understand, up to rank sequence, the possible subcomplexes of the Eagon--Northcott complex in a similar way as we did with the Koszul complex. We will see that these sequences are much more difficult to classify completely, but that we can still find restrictions that narrow down the possibilities. To do this, we will once again leverage the BGG correspondence in order to understand subcomplexes of Eagon--Northcott complexes. In order to do this, we must restrict ourselves to the linear maps -- and therefore the degree $d$ strand -- of the Eagon--Northcott complex. Note however that with the exception of the first map, the Eagon--Northcott complex is always linear, and any subcomplex of the degree $d$ strand will extend to a subcomplex of the entire Eagon--Northcott complex.
We define the complex $L_{n,d}$ to be the resolution of the ideal $\langle x_1, \ldots, x_n \rangle^d$ as an $S$-module as presented by ${\mathbf{EN}}(M^{n,d})$ and note that $L_{n,d}$ is linear. Indeed, for $d\geq 2$, $L_{n,d}$ is the degree--$d$ strand of ${\mathbf{EN}}(M^{n,d})$ shifted by one homological degree, while for $d=1$ the entire complex is linear, so $L_{n,1}$ is not the entire linear strand since it is missing the first map. This complex corresponds to a specific $E$-module $N_{n,d}$ under the BGG correspondence, that is, $L_{n,d} =\mathbb{L}(N_{n,d})[-n+1]$. Therefore, classifying subcomplexes of $L_{n,d}$ corresponds to understanding the $E$-submodules of $N_{n,d}(-n+1)$.
\begin{remark}\label{rmk:linearstrand} Given an subcomplex ${\mathbf{F}}$ of $L_{n,d}$, we can always extend to a subcomplex of ${\mathbf{EN}}(M^{n,d})$. In fact, since the first term of ${\mathbf{EN}}(M^{n,d})$ is $S^1$, we can extend ${\mathbf{F}}$ by either $0$ or by $S^1$ to obtain a subcomplex of ${\mathbf{EN}}(M^{n,d})$. At the level of rank sequences, this means that $r\in \RS({\mathbf{EN}}(M^{n,d}))$ has the form $(0,r')$ or $(1,r')$ for $r'\in \RS(L_{n,d})$. \end{remark}
\begin{proposition} The module $N_{n,d}$ is the cokernel of ${\partial}_{d-1}^T$ in the Tate resolution ${\mathbf{T}}(\mathbbm{k})$. \end{proposition}
\begin{proof} To obtain a presentation for $N_{n,d}$, we can appeal to the Reciprocity Theorem (Theorem~\ref{thm:reciprocitythm}) and the Tate resolution ${\mathbf{T}}(\mathbbm{k})$. Because $\mathbbm{L}(N_{n,d}) \rightarrow {\mathfrak{m}}^d$ is a free resolution, $N_{n,d} \rightarrow \mathbbm{R}({\mathfrak{m}}^d)$ is an injective resolution, so $N_{n,d}$ is the kernel of $\mathbbm{R}({\mathfrak{m}}^d)$.
The complex $\mathbbm{R}({\mathfrak{m}}^d)$ is known and, in fact, fits nicely into the Tate complex ${\mathbf{T}}(\mathbbm{k})$: if $\mathbbm{k} \rightarrow I_0 \xrightarrow{{\partial}_1^T} I_1 \xrightarrow{{\partial}_2^T} \cdots$ is the injective resolution of $\mathbbm{k}$ as an $E$-module, then $\mathbbm{R}({\mathfrak{m}}^d)$ is the truncation $0 \rightarrow I_d \xrightarrow{{\partial}_{d+1}^T} I_{d+1} \xrightarrow{{\partial}_{d+2}^T} \cdots$. This means that $N_{n,d} = \ker {\partial}_{d+1}^T$, which we can rewrite by the Tate resolution as $\coker {\partial}_{d-1}^T$, since the Tate resolution is the unique exact way to extend $\mathbbm{R}({\mathfrak{m}}^d)$ to the left. \end{proof}
\begin{example} We expound upon the case shown in Example~\ref{ex:tateres}, which considers the matrix $M^{3,2}$ whose minors give the ideal $\langle x_1, x_2, x_3 \rangle^2 \subseteq S = \mathbbm{k}[x_1, x_2, x_3]$. The complex $L_{3,2}$ corresponds to the $E$-module $N_{3,2} = \coker {\partial}_1^T$, where ${\partial}_1^T$ is the map in the Tate resolution given in Example~\ref{ex:tateres}. \end{example}
In this way, our search for possible rank sequences of subcomplexes of $L_{n,d}$ is translated to a search for possible Hilbert functions of submodules of $N_{n,d}$ (again with appropriate twist). We denote this set of the possible Hilbert functions of submodules of $N_{n,d}$ by $\operatorname{HF}(N_{n,d})$ and prove this set satisfies certain constraints. Some persnickety bookkeeping is necessary proceeding.
\begin{remark} The module $N_{m,d}$ has the same presentation matrix when viewed as a $\mathbbm{k} \langle e_1, \ldots, e_m \rangle$-module and as a $\mathbbm{k} \langle e_1, \ldots, e_n \rangle$-module. This follows from combining the argument for the presentation of $N_{n,d}$ and Remark~\ref{rem:tateresformoreambient}. However, the Hilbert function is \textit{not} the same when we consider $N_{m,d}$ as a $\mathbbm{k} \langle e_1, \ldots, e_m \rangle$-module and as a $\mathbbm{k} \langle e_1, \ldots, e_n \rangle$-module. For example, as a $\mathbbm{k}\langle e_1, e_2 \rangle$-module, $N_{1,1} = \coker \begin{bmatrix} e_1 \end{bmatrix}$ has Hilbert function $(1,1,0)$. This differs from the Hilbert function of $\coker \begin{bmatrix} e_1 \end{bmatrix}$ when viewed as a $\mathbbm{k}\langle e_1 \rangle$-module, which is simply $(1,0,0)$. Therefore, in general, the set $\operatorname{HF}(N_{m,d})$ will vary, depending on the ambient exterior algebra, so we must introduce more precise notation. We will continue to let $E = \mathbbm{k}\langle e_1, \ldots, e_n\rangle$ and $N_{n,d}$ for the module where $L_{n,d} = \mathbbm{L}(N_{n,d}(-n+1))$. If we are considering the module $N_{m,d}$ as an $E$-module, we will use the notation $\overline{N_{m,d}}$. Note that $\overline{N_{m,d}} = N_{m,d} \otimes \mathbbm{k}\langle e_{m+1}, \ldots, e_{n} \rangle$. \end{remark}
\begin{theorem}\label{thm:HFNndcontainment} The set of possible Hilbert functions of submodules of $N_{n,d}$ is restricted by the following containment: \[\operatorname{HF}(N_{n,d}) \subseteq \operatorname{HF}(N_{n,d-1}) + \operatorname{HF}(\overline{N_{n-1,d}}). \] \end{theorem}
\begin{proof} First we prove that \[ 0 \rightarrow N_{n,d-1} \rightarrow N_{n,d} \rightarrow \overline{N_{n-1,d}} \rightarrow 0 \] is a short exact sequence. Note that ${\mathfrak{m}}^d$ is the $S$-module $S_{\geq d}$, from which we get the exact sequence of $S$-modules \[ 0 \rightarrow S_{\geq d-1}(-1) \rightarrow S_{\geq d} \rightarrow S'_{\geq d} \rightarrow 0, \] where $S' = \mathbbm{k}[x_1, \ldots, x_{n-1}]$ is an $S$-module in the usual way: $x_n \cdot f = 0$ for any $f \in S'$.
Because $\mathbbm{L}$ preserves exactness, this means that \[ 0 \rightarrow \mathbbm{R}(S_{\geq d-1}(-1)) \rightarrow \mathbbm{R}(S_{\geq d}) \rightarrow \mathbbm{R}(S'_{\geq d}) \rightarrow 0 \] is a short exact complex of linear complexes of $E$-modules. In particular, we know what the kernel of each complex is: exactly the corresponding module $N$. Therefore \[ 0 \rightarrow N_{n,d-1} \rightarrow N_{n,d} \rightarrow \overline{N_{n-1,d}} \rightarrow 0 \] is indeed a short exact sequence of $E$-modules.
Now suppose that $N \subseteq N_{n,d}$ is a submodule with Hilbert function $h(N)$. The image of $N$ in $\overline{N_{n-1,d}}$ is also a submodule, which we will denote $N'$. Let $N''$ be the kernel of the induced map $N\to N'$. It is a submodule of $N_{n,d-1}$, so we have a short exact sequence \[ 0\to N''\to N \to N' \to 0. \]
Because Hilbert functions sum over short exact sequences, we have $h(N) = h(N') + h(N'')$, so the Hilbert function of $N\subseteq N_{n,d}$ is realized as a sum of Hilbert functions of submodules of $\overline{N_{n-1,d}}$ and $N_{n,d-1}$. Thus we see that \[ \operatorname{HF}(N_{n,d})\subset \operatorname{HF}(\overline{N_{n-1,d}}) + \operatorname{HF}(N_{n,d-1}). \] \end{proof}
Note that the containment in Theorem~ \ref{thm:HFNndcontainment} is not an equality. We have shown that any Hilbert function of a submodule of $N_{n,d}$ may be realized as a sum of Hilbert functions of submodules of $\overline{N_{n-1,d}}$ and $N_{n,d-1}$, but there may be submodules of $\overline{N_{n-1,d}}$ and $N_{n, d-1}$ the sum of whose Hilbert functions is not the Hilbert function of a submodule of $N_{n,d}$. Indeed, the following example shows that the containment is strict in even a very small case.
\begin{example}\label{ex:N21} For this example, we will use $E = \mathbbm{k}\langle e_1, e_2 \rangle$ and consider the $E$-module $N_{2,2}$. From above, we have \[ \operatorname{HF}(N_{2,2}) \subseteq \operatorname{HF}(\overline{N_{1,2}}) + \operatorname{HF}(N_{2,1}) \]
Recall that $N_{2,2} = \coker\begin{bmatrix} e_1\\e_2\end{bmatrix}$, $\overline{N_{1,2}} = \coker\begin{bmatrix} e_1\end{bmatrix}$, and $N_{2,1} = \coker\begin{bmatrix} e_1e_2\end{bmatrix}$. The module $\overline{N_{1,2}}$ has Hilbert function $(1,1,0)$, while the submodule $0 \subset N_{2,1}$ has Hilbert function $(0,0,0)$, so we get the sum $(1,1,0)+(0,0,0)$ as a potential Hilbert function for a submodule of $N_{2,2}$. However, there is no submodule of $N_{2,2}$ with Hilbert function $(1,1,0)$ by the following argument.
The module $N_{2,2}$ has $2$ generators in degree $0$, which we will call $\alpha$ and $\beta$. In degree $-1$ we have $e_1\alpha, e_1\beta, $and $e_2\alpha$, with $e_1\alpha = -e_2\beta$. Suppose we have a submodule $N\subset N_{2,1}$ with one generator in degree $0$. If we denote this degree $0$ generator of $N$ by $\zeta$, then we have that $\zeta = a\alpha + b\beta$ for some $a,b\in \mathbbm{k}$. This gives us $2$ elements in degree $-1$: $e_1\zeta = ae_1\alpha + be_2\beta$ and $e_2\zeta = ae_2\alpha+be_2\beta = (ae_2-be_1)\alpha$. We can see that these are linearly independent since the only relation in $N$ is the relation $e_1\alpha = -e_2\beta$, so the Hilbert function of $N$ cannot be $(1,1,0)$.
\end{example}
Our goal now will be to restate Theorem~\ref{thm:HFNndcontainment} as a statement about $\RS(L_{n,d})$ in terms of complexes for which we have a complete characterization of possible rank sequences, namely Koszul complexes. We first introduce some helpful notation: for a nonnegative integer $m$, we will write $E_{(m)} = E/\langle e_{m+1}, \ldots, e_n\rangle$. Note that $E_{(m)} \cong \mathbbm{k}$ when $m=0$. Furthermore, we will write $I_i$ to refer to the ideal generated by the single degree $-i$ monomial $e_1e_2\cdots e_i$ and we will make the convention that $I_0$ is the unit ideal. We will write ${\mathbf{K}}_{(m)}$ to mean the Koszul complex on $m$ variables for $m\leq n$.
\begin{remark}\label{rmk:twist} As a $\mathbbm{k}$-module, $E_{(m)}$ is simply the exterior algebra on $m$ variables, but since we are considering everything over $E$, we have that $\mathbbm{L}(E_{(m)}) = {\mathbf{K}}_{(m)}[-n+m]$, that is to say, the $i{^\text{th}}$ free module in the complex $\mathbbm{L}(E_{(m)})$ is the $(i-n+m){^\text{th}}$ free module in the Koszul complex on $m$ variables. \end{remark}
\begin{lemma}\label{lem:N1jequalsEnminus1} For $1\leq j\leq n$, we have the equality of sets \[ \operatorname{HF}(\overline{N_{1,d}}) = \operatorname{HF}(E_{(n-1)}). \] \end{lemma} \begin{proof} We will show in fact that $\overline{N_{1,d}} \cong E_{(n-1)}$. First observe that $N_{1,d} = \mathbbm{k}\langle e_1 \rangle / \langle e_1 \rangle = \mathbbm{k}$. Therefore $\overline{N_{1,d}} = \mathbbm{k}\otimes \mathbbm{k} \langle e_2, \ldots, e_n \rangle \cong E_{(n-1)}$. \end{proof}
\begin{lemma}\label{lem:hfnnd} For $n,d\geq 2$, the Hilbert functions of submodules of $N_{n,d}$ are restricted by the following containment: \[ \operatorname{HF}(N_{n,d}) \subseteq \sum_{i=1}^n\binom{n+d-2-i}{n-i}\operatorname{HF}(\overline{N_{i,1}}). \] \end{lemma} \begin{proof} Theorem~\ref{thm:HFNndcontainment} gives the containment \[ \operatorname{HF}(N_{n,d}) \subset \operatorname{HF}(\overline{N_{n-1, d}}) + \operatorname{HF}(N_{n,d-1}). \] We can then iterate until we have $\operatorname{HF}(N_{n,d})$ expressed completely in terms of $\operatorname{HF} (\overline{N_{1,j}})$ and $\operatorname{HF} (\overline{N_{i,1}})$ for $2\leq i,j\leq n$. Ultimately, we reduce to \[ \operatorname{HF}(N_{n,d})\subset \sum_{i=2}^n \alpha_{i,1}\operatorname{HF}(\overline{N_{i,1}}) + \sum_{j=2}^d \alpha_{1,j}\operatorname{HF}(\overline{N_{1,j}}) \]
where $\alpha_{i,j}$ counts the number of times that $N_{i,j}$ appears in the sum. This quantity $\alpha_{i,j}$ is the number of times that $(i,j)$ appears as the result of repeatedly subtracting $(1,0)$ and $(0,1)$ from $(n,d)$, with the caveat that, since $(1,i+1)$ is a base case, we never reach $(1,i)$ by subtracting $(0,1)$ from $(1,i+1)$, and similarly for $(1,j)$. One can thus interpret $\alpha_{i,1}$ as the number of integer lattice paths from $(i,2)$ to $(n,d)$ and $\alpha_{1,j}$ as the number of integer lattice paths from $(2,j)$ to $(n,d)$. The number of such lattice paths from $(i,j)$ to $(n,d)$ is given by $\binom{n-i+d-j}{n-i}$. This gives \[ \operatorname{HF}(N_{n,d})\subset \sum_{i=2}^n \binom{n+d-2-i}{n-i}\operatorname{HF}(\overline{N_{i,1}}) + \sum_{j=2}^d \binom{n+d-2-j}{n-2}\operatorname{HF}(\overline{N_{1,j}}). \] Since $\overline{N_{1,j}} = E_{(n-1)}$ regardless of $j$ by the proof of Lemma~\ref{lem:N1jequalsEnminus1}, we can write the sum \[ \sum_{j=2}^d \binom{n+d-2-j}{n-2}\operatorname{HF}(\overline{N_{1,j}}) = \left( \sum_{j=2}^d \binom{n+d-2-j}{n-2}\right)\operatorname{HF}(E_{(n-1)}). \] We can reindex and convert via the hockey stick identity to see that \[ \sum_{j=2}^d \binom{n+d-2-j}{n-2} = \sum_{k=n-2}^{n+d-4}\binom{k}{n-2} = \binom{n+d-3}{n-1} \] so we have \begin{align*} \operatorname{HF}(N_{n,d})&\subset \binom{n+d-3}{n-1}\operatorname{HF}(E_{(n-1)}) + \sum_{i=2}^n \binom{n+d-2-i}{n-i}\operatorname{HF}(\overline{N_{i,1}})\\ &= \sum_{i=1}^n \binom{n+d-2-i}{n-i}\operatorname{HF}(\overline{N_{i,1}}), \end{align*} where the incorporation of the first term into the sum uses the fact that $E_{(n-1)} \cong \overline{N_{1,1}}$
\end{proof}
\begin{lemma}\label{lem:hfni1} For $1\leq i\leq n$, we have the containment of sets \[ \operatorname{HF}(\overline{N_{i,1}}) \subseteq \sum_{j=0}^{i-1} \operatorname{HF}(E_{(n-j-1)}(j)). \] \end{lemma} \begin{proof} When $d = 1$, the module $N_{i,1}$ is easily computable as $N_{i,1} = \coker \begin{bmatrix} e_1 \cdots e_i \end{bmatrix} = \mathbbm{k} \langle e_1, \ldots, e_i \rangle / I_i$, and so $\overline{N_{i,1}} = (E_{(i)}/I_i)\otimes \mathbbm{k} \langle e_{i+1}, \ldots, e_n \rangle = E/I_i$.
Now we can reduce using the short exact sequences of $E/I_i$-modules \[ 0\to \langle e_i\rangle E/I_i \to E/I_i\to E/(\langle e_i\rangle + I_i)\to 0. \] But $\langle e_i\rangle E/I_i \cong E_{(n-1)}/I_{i-1}(1)$ and that $E/(\langle e_i\rangle + I_i) \cong E_{(n-1)}$, so by a similar argument as we have used previously in the proof of Theorem~\ref{thm:HFNndcontainment}, we may now write \[ \operatorname{HF}(E/I_m)\subseteq \operatorname{HF}(E_{(n-1)}/I_{m-1}(1)) + \operatorname{HF}(E_{(n-1)}). \] Now we can split $\operatorname{HF}(E_{(n-1)}/I_{m-1}(1))$ and proceed inductively to get \[ \operatorname{HF}(E/I_i) \subseteq \sum_{j=0}^{i-1}\operatorname{HF}(E_{(n-j-1)}(j)). \] \end{proof}
\begin{theorem}\label{thm:RSnumericsEN} For $n,d\geq 2$, the rank sequence of any subcomplex of $L_{n,d}$ can be written as a positive integral sum of rank sequences of Koszul subcomplexes on fewer than $n$ variables. In particular, \[ \RS(L_{n,d}) \subseteq \sum_{j=0}^{n-1}\binom{n-j+d-2}{d-1}\RS({\mathbf{K}}_{(n-j-1)}). \] \end{theorem}
\begin{proof} Combining Lemmas~\ref{lem:hfnnd}~and~\ref{lem:hfni1}, we can see that \[ \operatorname{HF}(N_{n,d}) \subseteq \sum\limits_{i=1}^{n} \sum\limits_{j=0}^{i-1} \binom{n+d-2-i}{d-2} \operatorname{HF}(E_{(n-j-1)}(j)). \] Twisting each side of this equality by $(-n+1)$ gives \[ \operatorname{HF}(N_{n,d}(-n+1)) \subseteq \sum\limits_{i=1}^{n} \sum\limits_{j=0}^{i-1} \binom{n+d-2-i}{d-2} \operatorname{HF}(E_{(n-j-1)}(-n+j+1)), \] which, fed through the functor $\mathbbm{L}$, yields a containment of sets of rank sequences: \[ \RS(L_{n,d}) \subseteq \sum\limits_{i=1}^{n} \sum\limits_{j=0}^{i-1} \binom{n+d-2-i}{d-2} \RS({\mathbf{K}}_{(n-j-1)}). \] We can switch the order of the double sum, reindex, and apply the hockey stick identity once again to conclude the proof: \[ \begin{aligned} \sum\limits_{i=1}^n \sum\limits_{j=0}^{i-1} \binom{n+d-2-i}{d-2} \RS({\mathbf{K}}_{(n-j-1)}) &\subseteq \sum\limits_{j=0}^{n-1} \sum\limits_{i=j+1}^n \binom{n+d-2-i}{d-2} \RS({\mathbf{K}}_{(n-j-1)}) \\ &= \sum\limits_{j=0}^{n-1} \left( \sum\limits_{i=0}^{n-j-1} \binom{d-2+i}{d-2} \right) \RS({\mathbf{K}}_{(n-j-1)}) \\ &= \sum\limits_{j=0}^{n-1} \binom{n-j+d-2}{d-1} \RS({\mathbf{K}}_{(n-j-1)}). \end{aligned} \] \end{proof}
\begin{example} Let $n=4, d=3$. The complex $L_{4,3}$ resolves the ideal of maximal minors of the matrix \[ \begin{bmatrix} x_1 & x_2 & x_3 & x_4 & 0 & 0\\ 0 & x_1 & x_2 & x_3 & x_4 & 0\\ 0 & 0 & x_1 & x_2 & x_3 & x_4 \end{bmatrix} \] and has the form \[ 0 \to S^{10} \to S^{36} \to S^{45} \to S^{20} \to 0. \] We can use Theorem ~\ref{thm:RSnumericsEN} to rule out some integer sequences as possible rank sequences for subcomplexes of $L_{4,3}$. The containment in Theorem ~\ref{thm:RSnumericsEN} states that \begin{align*} \RS(L_{4,3}) &\subseteq\sum_{j=0}^3\binom{5-j}{2}\RS({\mathbf{K}}_{(3-j)})\\ &= 10\RS({\mathbf{K}}_{(3)}) + 6\RS({\mathbf{K}}_{(2)}) + 3\RS({\mathbf{K}}_{(1)}) + \RS({\mathbf{K}}_{(0)}) \end{align*}
that is, any rank sequence of a subcomplex must be expressible as a sum of 10 rank sequences of subcomplexes of the Koszul complex on 3 variables, 6 rank sequences of subcomplexes of the Koszul complex on 2 variables, 3 rank sequences of subcomplexes of the Koszul complex on 1 variable, and 1 rank sequence of a subcomplex of the Koszul complex on 0 variables.
If we consider the Koszul complex on 3 variables, Theorem ~\ref{thm:basickoszulranks} tells us that any rank sequence $(r_0, r_1, r_2, r_3)$ of a subcomplex of ${\mathbf{K}}_{(3)}$ with $r_3 = 1$ must have $r_2 = 3$. This means that for a rank sequence $(r_0, r_1, r_2, r_3)$ of a subcomplex of $L_{4,3}$, we must have $r_2\geq 3r_3$. For instance, we may say for certain that the sequence $(10,16,20,8)$ is not a possible rank sequence of a subcomplex of $L_{4,3}$, since $20<3\cdot 8$. In this way, we are able to use our complete characterization of rank sequence of Koszul subcomplexes to eliminate certain potential rank sequences from consideration in the Eagon--Northcott case. \end{example}
\section{More General Resolutions}
With our previous results for Koszul and Eagon--Northcott complexes resolving powers of the maximal ideal in hand, we turn now to a more general setting. In particular, we are interested in other ideals $I$ that specialize to powers of the maximal ideal in such a way that $S/I$ is still resolved by the Koszul or Eagon--Northcott complex
\subsection{The general Koszul complex}
The Koszul complex can be defined more generally to give a minimal free resolution of a complete intersection. For $f_1, \ldots, f_m \in S$ a regular sequence of homogeneous elements, we replace the differential in definition ~\ref{def:koszul} by ${\partial}_d(e_T) = \sum_{j = 1}^{d} (-1)^{j} f_{i_j} e_{T - i_j}$, adjusting the twists accordingly.
\begin{theorem}\label{thm:generalkoszulranks} Let $f_1, \ldots, f_m$ be a regular sequence of homogeneous polynomials in $S$. An integer sequence $r = (r_0, \ldots, r_m)$ is in $\RS({\mathbf{K}}(f_1, \ldots, f_m))$ if and only if it satisfies \[ 0 \leq r_{i+1} \leq r_i^{(i)} \text{ for } 1 \leq i \leq m-1. \] \end{theorem}
\begin{proof} We will show that $\RS({\mathbf{K}}(f_1, \ldots, f_m)) = \RS({\mathbf{K}}(x_1, \ldots, x_m))$, then apply Theorem~\ref{thm:basickoszulranks}.
Let ${\mathbf{K}}$ be the Koszul complex on the variables $x_1, \ldots, x_m$. For ${\mathbf{K}}'$ a general Koszul complex ${\mathbf{K}}(f_1, \ldots, f_m)$ over the ring $S' = \mathbbm{k}[y_1, \ldots, y_n]$, there is a map ${\mathbf{K}} \to {\mathbf{K}}'$ induced by the map $S\to S'$ sending $x_i$ to $f_i$.
If ${\mathbf{F}}$ is a subcomplex of ${\mathbf{K}}$, then the image of ${\mathbf{F}}$ under this map is a subcomplex of ${\mathbf{K}}$ with the same rank sequence. So any possible rank sequence of a subcomplex of ${\mathbf{K}}$ must also be possible for a subcomplex of ${\mathbf{K}}'$.
To see that the possible rank sequences for subcomplexes of ${\mathbf{K}}'$ are \emph{exactly} those that are possible for subcomplexes of ${\mathbf{K}}$, we need to check that given a subcomplex ${\mathbf{F}}'$ of ${\mathbf{K}}'$, the differentials of ${\mathbf{F}}'$ are described by matrices over the subalgebra $R = \mathbbm{k}[f_1, \ldots, f_m] \subseteq S'$.
For each $i$, we have \[ \begin{tikzcd} F'_i \arrow[d] \arrow[r, "{\partial}"] & F'_{i-1} \arrow[d] \\ K'_i \arrow[r, "{\partial}'"] & K'_{i-1} \end{tikzcd} \]
where the vertical maps are given by matrices over $\mathbbm{k}$. So the differential ${\partial}$ is a matrix over $R$ if and only if ${\partial}'$ is. But entries of ${\partial}'$ are linear in the $f_i$, so they are defined as matrices over $R$.
Now given a subcomplex ${\mathbf{F}}'$ of ${\mathbf{K}}'$, we need only replace each $F'_i$ by a free $S$-module of the same rank and each $f_i$ in the differential by $x_i$ to obtain a subcomplex ${\mathbf{F}}$ of ${\mathbf{K}}$ with the same rank sequence. \end{proof}
\subsection{More general Eagon--Northcott complexes.} Just as the Koszul complex can be generalized to give a minimal free resolution of a complete intersection, the Eagon--Northcott complex can be generalized to give a minimal free resolution of certain Cohen--Macaulay algebras of the form $S/I$ where $I$ has the maximum possible codimension. We can relate the behavior of subcomplexes of the Eagon--Northcott complex resolving ${\mathfrak{m}}^d$ to the behavior of subcomplexes of a general Eagon--Northcott complex as follows. First, we consider a motivating example.
\begin{example}\label{ex:genericEN} Let $Y = [y_{i,j}]$ be a $p \times q$ generic matrix with $p \leq q$. Then there is a containment of sets \[\RS({\mathbf{EN}}(Y)) \subset \RS({\mathbf{EN}}(M^{q-p+1,p})).\] \end{example}
Let $n = pq$, so our matrix is a map $S^q\to S^p$ for $S = \mathbbm{k}[y_{i,j}] \cong \mathbbm{k}[x_1, \ldots, x_n]$. This specializes to the matrix
\[ M^{q-p+1,p} = \begin{bmatrix} x_1 & x_2 & \cdots &x_{q-p+1} & 0 &\cdots & 0\\ 0 & x_1 & \cdots & x_{q-p} & x_{q-p+1} & \cdots & 0\\ \vdots & & \ddots & & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & x_1 & \cdots & x_{q-p} & x_{q-p+1} \end{bmatrix} \]
under a map that we will call $\varphi$. The maximal minors of this matrix define the ideal $(x_1, \ldots, x_{q-p+1})^p$.
The map $\varphi$ gives us a map of complexes ${\mathbf{EN}}(Y) \to {\mathbf{EN}}(M^{q-p+1,p})$. What is more, under $\varphi$ any subcomplex of ${\mathbf{EN}}(Y)$ gives a subcomplex of ${\mathbf{EN}}(M^{q-p+1,p})$.
This gives us a containment
\begin{align}\label{ineq:specialize} \RS({\mathbf{EN}}(Y)) \subset \RS({\mathbf{EN}}(M^{q-p+1,p})). \end{align}
Note that the generic nature of $Y$ had no bearing on the argument in Example~\ref{ex:genericEN}, so a more general statement relating general Eagon--Northcott complexes to the complex ${\mathbf{EN}}(M^{n,d})$ holds by the same reasoning.
\begin{theorem}\label{thm:generalENranks} Let $Z$ be a $p \times q$ matrix whose maximal minors define an ideal $I$ whose codimension is $q-p+1$ and where $S/I$ is Cohen--Macaulay. Then there is a containment of sets \[\RS({\mathbf{EN}}(Z)) \subset \RS({\mathbf{EN}}(M^{q-p+1,p})).\] \end{theorem} \begin{proof}
With the hypotheses above, ${\mathbf{EN}}(Z)$ gives a minimal free resolution of $S/I$. Furthermore, the artinian reduction of $S/I$ is isomorphic to $S'/{\mathfrak{m}}^p$ for a polynomial ring $S' \cong \mathbbm{k}[x_1, \ldots, x_{q-p+1}]$. This specialization takes any subcomplex of ${\mathbf{EN}}(Z)$ to a subcomplex of ${\mathbf{EN}}(M^{q-p+1,p})$, so (\ref{ineq:specialize}) holds for ${\mathbf{EN}}(Z)$ as it does in Example~\ref{ex:genericEN}.
\end{proof}
While this theorem relates rank sequences of subcomplexes of the entire complexes ${\mathbf{EN}}(Z)$ and ${\mathbf{EN}}(M^{q-p+1,p})$ rather than their degree $d$ strands, Remark~\ref{rmk:linearstrand} tells us that our restrictions on $\RS(L_{q-p+1,p})$, together with the above theorem, still give us valuable information about $\RS({\mathbf{EN}}(Z))$. It should be noted, however, that the result in Theorem~\ref{thm:generalENranks} is a strict containment, as demonstrated in the following example.
\begin{example}\label{ex:specialize} Let $S = \mathbbm{k}[x,y,z,w]$. Consider the Eagon--Northcott complex on the matrix $\begin{bmatrix} 0 & y & z\\ y & z & 0 \end{bmatrix}$, which resolves the square of the maximal ideal in the subalgebra $\mathbbm{k}[y,z]\subset S$. This is a specialization of the Eagon--Northcott complex on $\begin{bmatrix} x & y & z\\ y & z & w \end{bmatrix}$ obtained via the map $\varphi: S\to S$ defined by $x,w\mapsto 0$ and $y,z\mapsto y,z$, so we have the following map of complexes:
\[ \xymatrix{ {\mathbf{F}}': 0 \ar[r] & S(-3)^2 \ar[rr]^{\begin{bsmallmatrix}-z & -w\\ y & z\\ x & -y \end{bsmallmatrix}} \ar[d]_{\varphi^*} & & S(-2)^3 \ar[rrrr]^{\begin{bsmallmatrix} -y^2+xz & -yz+xw & -z^2+yw\end{bsmallmatrix}} \ar[d]_{\varphi^*} & & & & S^1 \ar[d]_{\varphi^*} \\ {\mathbf{F}}: 0 \ar[r] & S(-3)^2 \ar[rr]_{\begin{bsmallmatrix}-z & 0\\ y & z\\ 0 & -y \end{bsmallmatrix}} & & S(-2)^3 \ar[rrrr]_{\begin{bsmallmatrix} -y^2 & -yz & -z^2\end{bsmallmatrix}} & & & & S^1 } \]
Consider the following subcomplex of ${\mathbf{F}}$: \[ {\mathbf{G}}: 0 \longrightarrow S(-3) \xrightarrow{\begin{bsmallmatrix} -z\\ y\end{bsmallmatrix}} S(-2)^2 \xrightarrow{\begin{bsmallmatrix} -y^2 & -yz \end{bsmallmatrix}} S. \]
which has $\rs({\mathbf{G}}) = (1,2,1)$. The subcomplex ${\mathbf{G}}$ is realized as the image of \[ {\mathbf{G}}': 0 \longrightarrow S(-3) \xrightarrow{\begin{bsmallmatrix} -z\\ y\end{bsmallmatrix}} S(-2)^2 \xrightarrow{\begin{bsmallmatrix} -y^2 + xz & -yz +xw \end{bsmallmatrix}} S \] under $\varphi^*$. However, one can check that ${\mathbf{G}}'$ is not a subcomplex of ${\mathbf{F}}'$. Moreover, a straightforward linear algebra computation confirms that there is no subcomplex of ${\mathbf{F}}'$ with rank sequence $(1,2,1)$, so $\RS({\mathbf{F}})\subsetneq\RS({\mathbf{G}})$. \end{example}
\end{document} | arXiv | {
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\begin{document}
\title{Variational quantum simulation of the quantum critical regime} \author{Zhi-Quan Shi} \affiliation{Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China}
\author{Xu-Dan Xie} \affiliation{Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering,
South China Normal University, Guangzhou 510006, China}
\author{Dan-Bo Zhang} \email{dbzhang@m.scnu.edu.cn} \affiliation{Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China} \affiliation{Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China}
\date{\today}
\begin{abstract} The quantum critical regime marks a zone in the phase diagram where quantum fluctuation around the critical point plays a significant role at finite temperatures. While it is of great physical interest, simulation of the quantum critical regime can be difficult on a classical computer due to its intrinsic complexity. In this paper, we propose a variational approach, which minimizes the variational free energy, to simulate and locate the quantum critical regime on a quantum computer. The variational quantum algorithm adopts an ansatz by performing an unitary operator on a product of a single-qubit mixed state, in which the entropy can be analytically obtained from the initial state, and thus the free energy can be accessed conveniently. With numeral simulation, we show, using the one-dimensional Kitaev model as a demonstration, the quantum critical regime can be identified by accurately evaluating the temperature crossover line. Moreover, the dependence of both the correlation length and the phase coherence time with the temperature are evaluated for the thermal states. Our work suggests a practical way as well as a first step for investigating quantum critical systems at finite temperatures on quantum devices with few qubits. \end{abstract}
\maketitle \section{Introduction} The quantum phase transition of quantum many-body systems marks a sharp transition between two phases and plays a central role in physics. Although occurring at zero temperature, the critical quantum fluctuation at the quantum phase transition point has far-reaching effects on the whole phase diagram, especially on a zone of quantum critical regime above the critical point that spans a range of temperatures~\cite{Sachdev}. The quantum critical regime is believed to be key for understanding a broad of physics, such as high-Tc superconductivity~\cite{lee06} and nuclear matter under finite temperature and finite density~\cite{Meyer_RMP_96,Stephanov_PRL_98}. However, the intrinsic complexity of the quantum critical regime, where both quantum fluctuation and thermal fluctuation interplay, makes a simulation of it with classical computers hard due to the sign problem~\cite{troyer2005computational}.
In recent years, rapid advances in quantum technologies, including both quantum hardware and quantum algorithms, enable us to simulate quantum many-body systems. It is natural to exploit current quantum processors to simulate zero-temperature quantum systems, which typically involve pure state, to investigate both static and dynamical properties of quantum systems~\cite{barends_15,bernien_17,kandala_17,zhang_17,yang_observation_2020}. Instead, simulation of finite-temperature quantum systems at equilibrium requires to prepare of a kind of mixed states called thermal states~\cite{terhal_00,poulin_09,temme_11,riera_12,wu_19,verdon_19,liu_ML_21,chowdhury2020variational,wang_PRApp_21,zhu2020generation,zhang2021continuous,xie_PRD_2022}, which can be either obtained as a subsystem of a pure state~\cite{poulin_09,wu_19,zhang2021continuous}, or as a mixture of pure states with a classical probabilistic distribution~\cite{verdon_19,liu_ML_21}.
There are basically two approaches for thermal state preparation on a quantum computer. One is to filter out the thermal state at a temperature from a completely mixed state (infinite-temperature state) by effectively implementing an imaginary time evolution~\cite{poulin_09,mcardle_19,zhang2021continuous}. The other approach refers to variational construction of the thermal state with parameterized quantum circuit~\cite{wu_19,verdon_19,liu_ML_21,chowdhury2020variational,wang_PRApp_21,zhu2020generation}, where the optimization can be obtained by minimizing the variational free energy with a hybrid quantum-classical procedure. The variational method relies on less quantum resource and is suitable for simulating finite-temperature quantum systems on current or near-term quantum processors~\cite{Preskill_18}. However, simulation of the quantum critical regime, which demands for accurately controlling both the quantum fluctuation and thermal fluctuation and their interplay, stands out as an important goal which is less investigated. While Ref.~\cite{zhang2021continuous} has proposed to investigate the quantum critical regime with a continuous-variable assisted quantum algorithm~\cite{lau_17,zhang_20}, it should be run on a hybrid variable quantum computer which still awaits for development. In this regard, a practical way for simulating the quantum critical regime can refer to variational quantum algorithm~\cite{cerezo_variational_2021,Yuan_RMP_2020}.
In this paper, we propose a variational approach to simulate the quantum critical regime, which is demonstrated with the one-dimensional Kitaev model under the periodic condition. The variational quantum algorithm adopts an ansatz that the variational energy can be obtained conveniently, in which the entropy is encoded in the initial state as a product of one-qubit mixed states and the parameterized unitary operator will not change the entropy. By numeral simulation, we show that such a variational quantum algorithm can prepare thermal states faithfully across the phase diagram of the Kitaev model. Remarkably, we reveal that the temperature crossover, which is important as it can locate the quantum critical regime, can be obtained accurately. We also measure both the static and dynamic correlation functions on the optimized variational thermal states, based on which the correlation length and the phase coherence time are fitted. It is shown that both the correlation length and the phase coherence time in the critical regime for a range of intermediate temperatures are proportional to the inverse of temperature. Our work suggests that investigating quantum critical regime with few qubits can be feasible on the current quantum processors.
\section{Quantum critical regime and variational quantum algorithm} \label{sec:model} In this section, we first introduce some backgrounds of the quantum critical regime with the one-dimensional Kitaev model~\cite{kitaev2001unpaired}. Then we introduce a variational quantum computing approach for simulating thermal states of the Kitaev ring~(one-dimensional Kitaev model under periodic boundary condition) as well as locating the quantum critical regime and investigating its properties .
\subsection{Quantum critical regime of the Kitaev model} Quantum phase transition is defined at zero temperature where the phase of state dramatically changes when tuning a parameter of the system across a point. The critical point associated with the quantum phase transition, although occurs at zero temperature, has far-reaching effects for the phases of state at finite temperature~\cite{Sachdev}. By comparing two important energies scales of the system, namely the gap $\Delta$ and the temperature $T$, the phase diagram can be divided into regimes $\Delta\gg T$ and $\Delta\ll T$, as shown in Fig.~\ref{Fig:illustration}. The regime of $\Delta \gg T$ represents low-$T$ regime where dynamics and transport can be described in a semi-classical way. The regime of $\Delta \ll T$, where both quantum fluctuation and classical fluctuation interplay, marks a quantum critical regime, which is separated from the semi-classical regimes with the temperature crossover lines. The quantum critical regime owns universal properties, such as the correlation length and the phase coherence time have scaling behaviors with the temperature~\cite{Sachdev}.
The quantum critical regime plays a key role in understanding a broad of physics such as high-Tc superconductivity and nuclear matter. The difficulty of studying quantum critical systems lies in the intrinsic complexity associated with thermals states that describe phases of state in the quantum critical regime~\cite{troyer2005computational}. Quantum simulation can directly prepare those thermal states and measure the physical properties. In this regard, it provides a bottom-up approach to investigating the quantum critical regime. However, as investigated in our previous work~\cite{zhang2021continuous}, a basic goal of simulating the quantum critical regime by locating the temperature crossover line can be tricky as it is model-dependence. For instance, it typically requires more than a system size of more than $100$ sites for the quantum Ising model, and on the other hand, it demands only a few sites for the Kitaev ring~\cite{zhang2021continuous}. Thus, we chose the Kitaev ring as a model Hamiltonian for simulating the quantum critical regime.
\begin{figure}
\caption{Illustration of the quantum critical regime. At zero temperature $T=0$, there is a phase transition point $\lambda=\lambda_c$. By comparing the energy gap $\Delta$ with the temperature $T$, the whole phase diagram can be divided into the quantum critical regime and the semi-classical regimes, which is separated by the temperature crossover line~(red lines). }
\label{Fig:illustration}
\end{figure}
The Hamiltonian of the Kitaev ring model reads \begin{eqnarray} \label{ham_ks} H_{K}={-J} \sum_{i=1}^{N}[{c }_{i}^{\dagger }{c}_{i+1}+{c }_{i}^{\dagger }{c}_{i+1}^{\dagger }+h.c.]{-u} \sum_{i=1}^{N}{c }_{i}^{\dagger }{c}_{i+1}, \end{eqnarray} where fermion operators ${c}_{N+1}={c}_{1}$ as imposed by the periodic condition. The Kitaev model has a quantum phase transition at $\frac{u}{2J}=1$. To simulate the Kitaev ring model, the fermion operator should be mapped to qubit operator by the Jordan-Wigner transformation, ${c}_{i}=\prod\limits_{j=1}^{i-1}{\sigma}_{j}^{z}{\sigma}_{i}^{-}$,${c}_{i}^{\dagger}={\sigma}_{i}^{\dagger}\prod \limits_{j=1}^{i-1}{\sigma}_{j}^{z}$, ${c}_{i}^{\dagger}{c}_{i}=\frac{1}{2}({\sigma}_{i}^{z}-1)$. Now the spin Hamiltonian reads as~(omitting a constant), \begin{eqnarray} \label{ham_ks} H=-J\sum_{i=1}^{N-1}{\sigma}_{i}^{x}{\sigma}_{i+1}^{x}-J{\sigma}_{1}^{y}P{\sigma}_{N}^{y}{-\lambda} \sum_{i=1}^{N}{\sigma}_{i}^{z}, \end{eqnarray} where $\lambda=\frac{u}{2}$, $P=\prod\limits_{i=2}^{N-1}{\sigma}_{i}^{z}$ is a string operator. Hereafter we set $J=1$. The model in Eq.~\eqref{ham_ks} is in fact close to the transversal field Ising model~(TFIM), except that the boundary term ${\sigma}_{1}^{y}P{\sigma}_{N}^{y}$is different, e.g., the term should be ${\sigma}_{1}^{x}{\sigma}_{N}^{x}$ for TFIM.
We now turn to discuss the Kitaev ring at finite temperature. The quantum system of the Kitaev ring at equilibrium under an inverse temperature $\beta=1/T$ can be described by a thermal state~(also known as Gibbs state)~\cite{kardar_2007}, $\rho(\beta)=e^{-\beta H}/Z(\beta)$, where $Z(\beta)=\text{Tr}e^{-\beta H}$ is the partition function. The free energy is related to the partition function as $F(\beta)=-\beta^{-1}\ln{Z(\beta)}$. The free energy at parameter $\lambda$ is denoted as $F(\beta,\lambda)$.
The thermodynamic properties can be derived from the free energy or by measuring observable the thermal states. One conventional approach to locate the temperature cross lines is to calculate the magnetic susceptibility and identify the temperature cross point $T^*(\lambda)$ for a given $\lambda$ as a temperature where the susceptibility is maximum~\cite{Sachdev,zhang2021continuous}. From statistical physics, it is known that the magnetic susceptibility can be expressed as \begin{equation}\label{eq:chi_def}
\chi(\beta,\lambda)=\frac{\partial^2F(\beta,\lambda)}{\partial^2\lambda}. \end{equation} Thus, the temperature cross point for a given $\lambda$ is, \begin{equation}\label{eq:T_cross_def} T^*=\arg\max_T\chi(1/T,\lambda). \end{equation} It should be pointed out that the quantum critical regime would not include the zone $T>J$, which is dominated by lattice cutoff and the properties are not universal~\cite{Sachdev}.
Another important aspect of the quantum critical regime is the scaling behavior of the correlation length $\xi$ and the phase coherence time $\tau$~\cite{Sachdev}. For the spin chain of $H$, it is known that both $\xi$ and $\tau$ are proportional to the inverse temperature $\beta$ in the quantum critical regime. The correlation length and the phase coherence time should be evaluated from the static correlation function and the dynamical correlation function, respectively. The static correlation function is defined as \begin{equation}\label{eq:R_def}
R(n)=\sum_{i=1}^{N}\text{Tr}[\rho(\beta)\sigma^x_i\sigma^x_{i+n}]. \end{equation} At nonzero temperature, $R(n)$ should be exponentially decreasing with the spatial separation $n$ . The correlation length is defined as a characterization length by $R(n)\propto e^{-n/\xi}$. The dynamical correlation function can be chosen as \begin{equation}\label{eq:C_def}
C(t)=\sum_{i=1}^{N}|\text{Tr}[\rho(\beta)\sigma^x_i(t)\sigma^x_{i}]|, \end{equation} where $\sigma^z_i(t)=e^{iHt}\sigma^z_ie^{-iHt}$. Note that in the summation each dynamical correlation function takes absolute value as it is a complex number. Similarly, $C(t)$ at finite temperature is exponential decreasing with $t$ and the phase coherence time is defined through $C(t)\propto e^{-t/\tau}$.
\subsection{Variational quantum algorithm} We now propose a variational quantum computing approach for simulating the quantum critical regime for the Kitaev ring. This includes two goals: locating the quantum critical regime and investigating the scaling behavior. Reaching those goals relies on preparing thermal states accurately on a quantum computer, based on which physical quantities can be evaluated reliably.
\subsubsection{Complexity of thermal states}\label{subsub:complexity} Let us first illustrate the complexity of preparing thermal states. It is inspiring to decompose the thermal state as, \begin{equation}
\rho(\beta)=\sum_{i=1}^{2^N}p_i(\beta)\ketbra{\psi_i},~~ p_i(\beta)=\frac{e^{-\beta E_i}}{Z(\beta)}, \end{equation} where $H\ket{\psi_i}=E_i\ket{\psi}$. The thermal state thus is a mixture of eigenstates $\{\ket{\psi_i}\}$ with classical probabilities $\{p_i\}$. At low temperatures, only the ground state has a large weighting and the task is almost reduced for preparing the ground state. However, for higher temperatures, such as $T \sim \Delta$, the low-lying eigenstates will have large probabilities and it requires to prepare accurately those low-lying eigenstates $\ket{\psi_i}$ with the corresponding weights $p_i$ at the same time. Such a task is harder than preparing the ground state. However, for very high temperatures the task of preparing the thermal state becomes easy. To see this, we consider the infinite-temperature limit $\beta=0$. The thermal state is a completely mixed state $\rho(\beta=0)=I/2^N$. As $U\rho(\beta=0)U^\dagger=\rho(\beta=0)$, where $U$ is a unitary transformation, $\rho(\beta=0)$ is in fact an equal-mixing of an arbitrary set of complete basis $\{U\ket{\psi}\}$. Another aspect to reveal the simplicity of $\rho(\beta=0)$ is to that it equals to a product of single-qubit mixed state $\otimes_{i=1}^{N} I/2$. In other words, there is no correlation at $\beta=0$ and the temperature is local~\cite{kliesch2014locality}. In fact, it can be proved theoretically that the correlation length scales as $\xi(\beta)=\beta^{\frac{2}{3}}$ for local Hamiltonian at high temperatures~\cite{Kuwahara_PRX_2021}. Since it requires a deeper quantum circuit for preparing states with longer correlation length~\cite{ho2019efficient,Kuwahara_PRX_2021}, such a scaling indicates that the complexity of preparing a thermal state reduces when increasing the temperature at the high-temperature regime.
The above discussion suggests that preparing thermal states at intermediate temperatures will be harder than low-temperature and high-temperature ones. This is just the case for locating the temperature crossover lines which connect the regime of $\Delta\ll T$ and $\Delta\gg T$. In this regard, it can be challenging to simulate the quantum critical regime for quantum computing.
\subsubsection{Variational preparing thermal states} The variational quantum computing approach can meet this challenge by taking the advantage of using a specific ansatz that may require short-depth quantum circuit and thus can be suitable on near-term quantum processors. The variational principle for the quantum system at temperature $T$ is that the free energy should be minimized for the thermal state~\cite{kardar_2007}. One can prepare a variational thermal state $\rho(\boldsymbol{\omega},\beta)$ with a parameter set $\boldsymbol{\omega}$ on a quantum computer. The $\rho(\boldsymbol{\omega};\beta)$ can be generated either as a subsystem of a pure state or as a mixture of pure states. The parameter set $\omega$ should be optimized by minimizing the variational free energy expressed as, \begin{eqnarray} \label{eq:F_E_S}
F(\boldsymbol{\omega};\beta)=E(\boldsymbol{\omega})-TS(\boldsymbol{\omega}), \end{eqnarray} where $E(\boldsymbol{\omega})=\text{Tr}[\rho(\boldsymbol{\omega};\beta)H]$ is the average energy and $S(\boldsymbol{\omega})=-\text{Tr}[\rho(\boldsymbol{\omega};\beta)\log\rho(\boldsymbol{\omega};\beta)]$ is the von Neumann entropy. The energy $E(\boldsymbol{\omega})$ can be evaluated by decomposing the Hamiltonian as a linear combination of local observable and measuring each separately. However, estimating the von Neumann entropy is a difficulty task in general since it does not associate with a Hermitian observable. Indeed some quantum protocols can measure the Reyi entropy which is a function of $\rho^n$~($n$ is a positive integer)~\cite{klich2006measuring,islam2015measuring,brydges2019probing}. However, measuring the von Neumann entropy is more challenging as it involves $\log\rho$ and there is still lack of efficient protocol for generic quantum states, except for some proposals with approximation valid under specific conditions ~\cite{audenaert2007sharp, acharya2019measuring,wang_PRApp_21}.
\begin{figure}
\caption{Illustration of the variational quantum algorithm. The variational Gibbs state is prepared by performing a parameterized unitary operator on an initial state $\otimes_i\rho_i(\theta_i)$. The parameterized circuit consists of $p$ blocks. By measuring the energy $E$ on a quantum computer by calculating the entropy $S$ from $\theta$, the variational free energy $F$ can be obtained. With a hybrid quantum-classical optimization, parameters ($\theta, \alpha,\gamma)$) are iteratively updated to minimize the variational free energy. }
\label{Fig:illustration_circuit}
\end{figure}
One solution is to use some specific ansatz so that the von Neumann entropy can be calculated directly without measurements. This is possible by using an ansatz where the classical probability and the eigenstates are parameterized separately~\cite{martyn2019product,liu2021solving,xie_PRD_2022}, e.g., the variational thermal state can take a formula~(with a parameter set $\boldsymbol{\omega}=(\theta,\phi)$), \begin{equation}\label{eq:PSA_ansatz} \rho(\boldsymbol{\omega};\beta)=\sum_{i=1}^{2^N}p_i(\theta)U(\phi)\ketbra{i}U^\dagger(\phi)=U(\phi)\rho_0(\theta)U^\dagger(\phi), \end{equation} where $\rho_0(\theta)=\sum_{i=1}^{2^N}p_i(\theta)\ketbra{i}$. The unitary evolution $U(\phi)$ will not change the entropy of the initial state $\rho_0(\theta)$. Thus the entropy can be obtained from the classical probability $\{p_i(\theta)\}$, \begin{equation}
S(\theta)=-\sum_{i=1}^{2^N}p_i(\theta)\log p_i(\theta). \end{equation}
Following Refs.~\cite{martyn2019product,xie_PRD_2022}, we chose the initial state as a product state, \begin{eqnarray}
\label{eq:initial}
\rho_0(\theta)&=&\otimes_{i=1}^{N}\rho_i(\theta_i),\nonumber \\ \text{where}~ \rho_i(\theta_i)&=&\sin^2\theta_i\ketbra{0_i}+\cos^2\theta_i\ketbra{1_i}. \end{eqnarray} Such a choice of initial state together with Eq.~\eqref{eq:PSA_ansatz} is known as the product-spectrum ansatz~\cite{martyn2019product}. Calculation of the entropy can be simplified as it is a summation of entropy for each qubit, \begin{equation}
S(\theta)=\sum_{i=1}^{N}[-\sin^2\theta_i\log\sin^2\theta_i-\cos^2\theta_i\log\cos^2\theta_i]. \end{equation} While using only $N$ parameters to characterize the classical probability, it is shown that the product-spectrum ansatz can faithfully represent thermal states of a broad of physical systems~\cite{martyn2019product,xie_PRD_2022}. It is noted that the single-qubit mixed state $\rho_i(\theta_i)$ can be obtained by tracing one qubit of a two-qubit pure state $\sin\theta_i\ket{00}+\cos\theta_i\ket{11}$. This takes an overall $2N$ qubits to prepare the thermal state. One may also take $\rho_0(\theta)$ as a mixture of computational basis $\ket{s_1s_2...s_N}$, where $s_i=0,1$. If $N$ is small, $\rho_0(\theta)$ can be generated by preparing an initial state $\ket{s_1s_2...s_N}$ with a probability $\prod_{i=1}^{N}f_{s_i}(\theta_i)$, where $f_0=\sin$ and $f_1=\cos$. The probabilistic way uses only $N$ qubits but involves an ensemble of quantum circuits whose number increases exponentially with $N$.
The unitary operator $U(\phi)$ can be constructed with different types of parameterized quantum circuits. Here we use the following structure involving $p$-blocks of alternatively Hamiltonian evolution~(with a parameter set $\phi=(\alpha,\eta)$)~\cite{xie_PRD_2022,Chen_CPL_2013}, \begin{eqnarray}
\label{Eq:unitary}
U(\phi)\equiv U(\alpha,\eta)=\prod_{l=1}^{p} e^{-iH_{2}(\eta_{l})}e^{-iH_1({\alpha_{l}})} \end{eqnarray} where $H_{1}(\alpha_{l})=\sum_{i=1}^{N}\alpha_{l,i}\sigma^z_{i}$ and $H_{2}(\eta_{l})=\sum_{i=1}^{N-1}\eta_{l,i}\sigma_{i}^{x}{\sigma}_{i+1}^{x}+\eta_{l,N}{\sigma}_{1}^{y}P{\sigma}_{N}^{y}$. As terms in $H_{1}(\alpha_{l})$ commute to each other, $e^{-iH_1({\alpha_{l}})}$ can be directly decomposed as a series of quantum gates. The same situation applies for $e^{-iH_{2}(\eta_{l})}$. The choice of $U(\phi)$ is physical motivated as $H_{1}(\alpha_{l})$ and $H_{2}(\eta_{l})$ inherit from the Hamiltonian $H$. While similar to the Hamiltonian ansatz~(HVA)~\cite{wiersema-PRXQuantum2020}~(also known as Quantum Alternating Operator Ansatz \cite{Hadfield2017FromTQ}), it allocates each term with a variational parameter, rather than allocating the same parameter to all terms. We may call the unitary in Eq.~\eqref{Eq:unitary} as multi-angle HVA~\cite{Chen_CPL_2013}. The original HVA is proposed to prepare the ground state. A promotion of HVA to the multi-angle HVA is necessary as preparing thermal state is more difficulty and demands more representation power of the ansatz.
With the initial state $\rho_0(\theta)$ in Eq.~\eqref{eq:initial} and the unitary operator $U(\alpha,\eta)$ in Eq.~\eqref{Eq:unitary}, the variational state can be written as $\rho(\theta,\alpha,\eta;\beta)$. To optimize the parameter $(\theta,\alpha,\eta)$, one can minimize the variational free energy $F(\theta,\alpha,\eta;\beta)$ using a hybrid quantum-classical procedure. The parameter for the initial state $\rho(\theta)$ and the parameter $(\alpha,\eta)$ in the quantum circuit are updated respectively after evaluating the free energy in each iteration. An illustration of the variational quantum algorithm for preparing the thermal state as well as the hybrid quantum-classical optimization is given in Fig.~\ref{Fig:illustration_circuit}.
\subsubsection{Evaluation of physical quantities}
With the optimized variational free energy, we now can evaluate the susceptibility in Eq.~\eqref{eq:chi_def} by a second-order difference scheme, \begin{equation}\label{eq:diff_chi}
\chi(\beta,\lambda)\approx \frac{F(\beta,\lambda+\delta\lambda)+F(\beta,\lambda-\delta\lambda)-2F(\beta,\lambda)}{(\delta\lambda)^2}, \end{equation} where $\delta\lambda$ is a small number and variational parameters in $F$ have not been written explicitly. For each $\lambda$, a series of $\chi(\beta,\lambda)$ are calculated, and the temperature crossover point is identified as a temperature that $\chi(\beta,\lambda)$ is maximum as in Eq.~\eqref{eq:T_cross_def}. By going through all $\lambda$, the temperature crossover lines $T(\lambda)$ can be identified.
With the optimized variational thermal state, the spatial and dynamical correlation function can be measured as in Eq.~\eqref{eq:R_def} and Eq.~\eqref{eq:C_def}, respectively. Evaluating the spacial correlation function $R(n)$ relies on a joint-measurement of two qubits. By measuring $R(n)$ at different spacing $n$, the correlation length $\xi$ is obtained by fitting $R(n)\propto e^{-n/\xi}$. Measuring the dynamical correlation function $C(t)$ is also a standard technique~\cite{Pedernales_PRL_14,Li_PRD_2022}. The corresponding quantum circuit is given in Fig.~\ref{Fig:circuit_dynamical}. The value of $C(t)$ is recorded by measuring $\ave{\sigma^x+i\sigma^y}$ on the ancillary qubit. Similarly, by measuring $C(t)$ at different time $t$, the phase coherent time $\tau$ can be obtained by fitting $C(t)\propto e^{-t/\tau}$.
\begin{figure}
\caption{Quantum circuit for computing the dynamical correlation function $\text{Tr}[\rho(\beta)\sigma^x_i(t)\sigma^x_{i}]$. The first qubit initialed as $\ket{0}$ is an ancillary qubit. }
\label{Fig:circuit_dynamical}
\end{figure}
It should be emphasized that studying the scaling behavior should refer to the thermodynamic limit where the system size is infinite. As quantum simulation can only be performed on finite-size systems, one may conduct finite-size scaling~\cite{Fisher_PRL_72}. In this work, however, we only do some primary investigations on the scaling behavior without sophisticated finite-size scaling analysis due to limited simulation capacity.
\section{Results}\label{sec:results} In this section, we represent simulation results. The simulation is performed with the open-source package $\it{projectQ}$~\cite{projectQ} on classical computers. We use BFGS for the optimization, which is a gradient-based method. We also adopt a strategy to boost the optimization by utilizing a continuous relation between optimized variational parameters with the temperature~\cite{zhang_PRA_2020,Yuan_PRA_2021}. Concretely, we first get optimized variational parameters for high-temperature thermal states as they are comparatively easy to solve. The optimized parameters then are set as initial parameters of the thermal state of lower-temperature system. The procedure is repeated when the temperature is reduced to zero temperature. With this strategy, the thermal states of the whole phase diagram can be obtained more efficiently.
\begin{figure}
\caption{Numeral simulation results~(scatters) of free energy are compared with exact ones~(lines) by increasing the number of blocks $p$ in the quantum circuit.}
\label{Fig:F_with_p}
\end{figure}
\begin{figure}
\caption{Numeral simulation results~(dots) of free energy are compared with exact ones~(lines) with increasing temperature. The number of blocks is $p=5$. }
\label{Fig:F_with_T}
\end{figure}
We test the variational algorithm for preparing the thermal states of the Kitaev ring by comparing the optimized free energy with the exact ones. Firstly, the accuracy for solving thermal states at different system sizes is investigated with increasing circuit depth characterized by the number of blocks $p$ in the unitary operator Eq.~\eqref{Eq:unitary}. As seen Fig.~\ref{Fig:F_with_p}, with increasing $p$, the optimized free energy will converge nearly to the exact one. Moreover, a larger-size system will require a larger $p$ for obtaining accurate free energy. The demanding of more quantum resources for simulating larger quantum systems at finite temperatures is expected but the exact scaling of quantum resources with the system size is still an open question. For the transversal field Ising model, it has been argued and numerically verified that the critical point requires a depth $O(N)$ to prepare the ground state with the HVA~\cite{ho2019efficient}. For the multi-angle HVA, the requirement of $p$ with the system size $N$ is still awaited investigation. Secondly, the accuracy of free energy at different temperatures is shown in Fig.~\ref{Fig:F_with_T}. We choose $p=5$ for all cases. For different sizes~($N=3,4,5$) and different $h$~($h=0.9,1.1$), it is shown that the accuracy is good for the whole temperature range.
\begin{figure}
\caption{Locating the quantum critical regime by identifying the temperature crossover line. (a) For a given $\lambda$, the susceptibility $\chi$ is evaluated for varied temperatures~(only several $\lambda$ are shown); (b) The temperature crossover point for each $\lambda$ corresponds to a temperature that $\xi$ is peaked in (a). For all scatters are simulation results and lines are exact results. }
\label{Fig:regime_VQA}
\end{figure}
We then turn to the first goal of simulating the quantum critical regime: identifying the temperature crossover line. We chose the Kitaev model at $N=3$ as a minimal model to demonstrate the temperature crossover. The first step is to calculate the susceptibility $\chi(\beta,\lambda)$ according to the difference scheme in Eq.~\eqref{eq:diff_chi}, where $\delta\lambda=0.001$ is chosen. As shown in Fig.~\ref{Fig:regime_VQA}a, $\chi(\beta,\lambda)$ for a given $\lambda$ at different temperatures are calculated. The $\chi\sim T$ curve for each $\lambda$ is peaked at a temperature, which can be identified as a temperature crossover point. Based on Fig.~\ref{Fig:regime_VQA}a, the temperature crossover line can be obtained. As shown in Fig.~\ref{Fig:regime_VQA}b, the temperature crossover line obtained with VQA~(red dots) fits well with the exact one~(red line).
\begin{figure}
\caption{Dependence of the correlation length and the phase coherence time with temperature. (a) and (c) show the spatial correlation function $R$ with the spatial separation $n$ and the dynamical correlation function $C$ with the time period $t$, respectively. Only results of $T=0.1,0.5,1.5$ are shown; (b) shows the dependence of the correlation length $\chi$ with $T$ and (d) shows the dependence of the phase coherent time with $T$. For all, simulation results marked by scatters are compared with exact results marked by lines. }
\label{Fig:correlation}
\end{figure}
The second goal is to investigate the scaling behavior in the quantum critical regime. At this stage, the simulation is only limited to a very small size. We chose $N=6$. This allows $n$ in the spatial correlation function $R(n)$ can take $n=1,2,3$, which can be used for fitting the function $R(n)=ae^{-\frac{n}{\xi}}$ with three data. For a finite-size system, the largest time period $t$ in the dynamical correlation function $C(t)$ should be chosen so that $C(t)$ will not oscillate. As $t$ can be continuous, the number of data can be large to fit $C(t)=b e^{-t/\tau}$. With those in mind, $R(n)$ and $C(t)$ are measured for different $T$ and $\lambda$, which are shown in Fig.~\ref{Fig:correlation}a and Fig.~\ref{Fig:correlation}c respectively. Then, the correlation length and the phase coherent time are fitted with $R(n)$ and $C(t))$, respectively. In Fig.~\ref{Fig:correlation}b and Fig.~\ref{Fig:correlation}d, the relation $\xi\sim T$ and $\tau\sim T$ are presented. It can be seen that both are proportional to $T^{-1}$ in an intermediate regime of temperature. There are some mismatches at low temperatures. Remarkably at $\lambda=1$, it is expected that the correlation length should be divergent while the numeral simulation shows to be almost flat. This can be due to the finite-size effect. On the other hand, the dramatically increasing~(decreasing) of $\xi$ for $h=0.95$ is qualitatively consistent with theory. This corresponding to the semi-classical regimes, where $\chi$ should be exponentially growing with $T^{-1}$ for $\lambda<1$. Similarly, the phase coherent time ceases to increase with reducing temperature may be due to finite size. It is also observed a deviation of relations of $\xi\propto T^{-1}$ and $\tau\propto T^{-1}$ at high temperature. This can be explained that the high-temperature regime $T>J=1$ is governed by the lattice cutoff and thus no universal behavior can be expected. The above discussions suggest that the scaling behavior may be captured for an intermediate temperature regime.
\section{Conclusions}\label{sec:conclusion} In summary, we have proposed a variational quantum computing approach for simulating the quantum critical regime, using the Kitaev ring as a prototype model, by investigating the temperature crossover and the scaling behavior. The variational quantum algorithm adopts an ansatz that the free energy can be obtained free of the difficulty of measuring the entropy. By numeral simulation, we have shown that the variational quantum algorithm can identify the temperature crossover accurately. Moreover, we have shown that both the correlation length and the phase coherence time are proportional to the inverse temperature in an intermediate regime of temperature. Our work has paved the way for simulating finite-temperature critical systems on a quantum computer.
\end{document} | arXiv | {
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\begin{document}
\title{Hadamard type operations for qubits} \author{Arpita Maitra and Preeti Parashar} \affiliation{Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B T Road, Kolkata 700 108, India, Email: \{arpita\_r, parashar\}@isical.ac.in}
\begin{abstract} We obtain the most general ensemble of qubits, for which it is possible to design a universal Hadamard gate. These states when geometrically represented on the Bloch sphere, give a new trajectory. We further consider some Hadamard `type' of operations and find ensembles of states for which such transformations hold. Unequal superposition of a qubit and its orthogonal complement is also investigated.
\end{abstract} \maketitle \newcommand{\qed}{
\rule{2mm}{2mm}} \newcommand{{\bf Proof : }}{{\bf Proof : }} \newtheorem{definition}{Definition} \newtheorem{algorithm}{Algorithm} \newtheorem{construction}{Construction} \newtheorem{theorem}{Theorem} \newtheorem{question}{Question} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \newtheorem{remark}{Remark} \newtheorem{corollary}{Corollary} \newtheorem{example}{Example} \newcommand{\binom}[2] {\mbox{$\left( { #1 \atop #2 } \right)$}}
{\bf Keywords:} Unitary operations, Hadamard Gate, Bloch Sphere, Qubits.
{\bf PACS:} 03.67.Lx
\section{I. Introduction} Qubits and quantum gates are the two basic building blocks of quantum computers which are believed to be computationally stronger than their classical counterparts. One such important gate is the Hadamard gate which has found wide applications in computer and communication science ~\cite{qNC02}. There are a number of seminal papers in quantum computation and information theory where Hadamard transform has been used~\cite{qDJ92,BV93,BV97,qGR96, DD89}. Shor's fast algorithm for factoring and discrete logarithm~\cite{Sh94} are based on Fourier transform which is a generalization of the Hadamard transform in higher dimensions. Furthermore, the Toffoli and Hadamard gates comprise the simplest quantum universal set of gates \cite{Shi02, DA03}. So, in order to achieve the full power of quantum computation, one needs to add only the Hadamard gate to the classical set. Thus, the role played by the Hadamard gate in quantum algorithms is indeed significant.
Of late, Pati~\cite{PT02} has shown that one can not design a universal Hadamard gate for an arbitrary unknown qubit. Linearity, which is at the heart of quantum mechanics, does not allow linear superposition of an unknown state $|\psi\rangle$ with its orthogonal complement $|\psi_{\perp}\rangle$. However, if one considers qubit states from the polar or equatorial great circles on a Bloch sphere, then it is possible to design Hadamard type of gates. By a Hadamard `type' gate we mean a unitary matrix that is not exactly a Hadamard matrix. However, it still creates an equal superposition (up to a sign or a phase) of a qubit and its complement to produce two orthogonal states. Very recently, Song et. al.~\cite{Son04} have tried to implement the Hadamard gate in a probabilistic manner for any unknown state chosen from a set of linearly independent states.
Motivated by Pati's work, our primary aim in this paper is to construct the most general class of qubit states, for which the Hadamard gate can be designed in a deterministic way. This is achieved in Sec. II, by imposing restrictions ( due to linearity ) on a completely arbitrary unknown quantum state. States from this set are geometrically represented on the three - dimensional unit sphere known as the Bloch sphere. In Sec. III, we show that certain Hadamard `type' transformations are indeed possible for arbitrary states when partial information is available. A Hadamard type gate is obtained for qubits chosen from, not only the polar great circle but also from any polar circle. We also demonstrate with an example, that there is a unique class of states (up to isomorphism) associated with a particular gate, satisfying a fixed transformation. As for the second Hadamard type of transformation, which is related to the states lying on the equatorial great circle, a new ensemble of states is found. In Sec. IV, unequal superposition of a qubit with its orthogonal complement is investigated. This is a generalization of the usual Hadamard transformation when the two amplitudes are not equal. In this context, many new classes of quantum states are found for which the unequal superposition works.Summary and Concluding remarks are made in Sec. V.
\section{II. Hadamard Transform for Special Qubits}
The Hadamard transform $H$, which is a one qubit gate, rotates the two computational basis vectors $|0\rangle$ and
$|1\rangle$
to two other orthogonal vectors $\frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)$
and $\frac{1}{\sqrt{2}} (|0\rangle - |1\rangle)$, respectively. Thus it creates an equal superposition of the amplitudes of the state and its orthogonal. The matrix representation of the Hadamard gate is given by $H = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & 1 \\ 1 & -1 \\ \end{array}\right]$.
The question we ask in this paper is: What is the most general set of qubit states $\{|\psi\rangle, |\psi_{\perp}\rangle\}$, such that the application of the Hadamard gate $H$ takes them to two other orthogonal states
$\frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle)$
and $\frac{1}{\sqrt{2}} (|\psi\rangle - |\psi_{\perp}\rangle)$
respectively? If $|\psi\rangle$ is completely arbitrary and unknown, then such a universal Hadamard gate does not exist \cite{PT02}. So, we shall obtain a special class of qubit states such that \begin{equation} \label{eq1}
H(|\psi\rangle) = \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle), H(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}}
(|\psi\rangle - |\psi_{\perp}\rangle). \end{equation}
\begin{figure*}
\caption{Points on Bloch sphere in reference to Theorem~\ref{th1} and Theorem~\ref{th3}}
\label{fig1}
\end{figure*}
We start by considering a completely arbitrary, unknown qubit state
$|\psi\rangle = a|0\rangle + b |1\rangle$ and its orthogonal complement $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Here $a, b$ are complex numbers obeying the normalization condition
$|a|^2 + |b|^2 = 1$. The operator $H$ acts linearly, i.e.,
$H(|\psi\rangle) = a H(|0\rangle) + b H(|1\rangle)$. In what follows, we show that this restricts the form of $|\psi\rangle$ to
$(\alpha + i\beta) |0\rangle + \alpha |1\rangle$, with $2\alpha^2 + \beta^2 = 1$, where $\alpha$ and $\beta$ are real.
Now we substantiate the assertions made herein above. Ideally, from the Hadamard transformation we obtain \begin{eqnarray}
H(|\psi\rangle) &=& \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle) \nonumber \\ \ &=& \frac{1}{\sqrt{2}}
(a |0\rangle + b |1\rangle + b^* |0\rangle - a^* |1\rangle) \nonumber \\
\ &=& \frac{(a+b^*) |0\rangle - (a^*-b) |1\rangle}{\sqrt{2}}, \nonumber \end{eqnarray} and from linearity we get \begin{eqnarray}
H(|\psi\rangle) &=& a H(|0\rangle) + b H(|1\rangle) \nonumber \\
\ &=& a \frac{|0\rangle + |1\rangle}{\sqrt{2}} +
b \frac{|0\rangle - |1\rangle}{\sqrt{2}} \nonumber \\
\ &=& \frac{(a+b) |0\rangle + (a-b) |1\rangle}{\sqrt{2}}. \nonumber \end{eqnarray} These two expressions should be equal. Hence $a + b^* = a + b$, i.e., $b = b^*$. Thus $b$ is real. Let $b = \alpha$, where $\alpha$ is a real number. Moreover, $-(a^* - b) = (a - b)$, i.e.,
$a+a^* = 2b$. So the real part of $a$ is $\alpha$. Let $a = \alpha + i\beta$, where $\beta$ is a real number. Thus $|\psi\rangle$ is of the form $(\alpha + i\beta) |0\rangle +
\alpha |1\rangle$. Clearly $-\frac{1}{\sqrt{2}} \leq \alpha \leq \frac{1}{\sqrt{2}}$, since
$\beta^2 = 1 - 2 \alpha^2$. Therefore, $|\psi\rangle$ has complex as well as real amplitudes when expressed in the computational basis $\{|0\rangle, |1\rangle \}$; the real parts of which are equal.
It can be easily checked, in a similar fashion, that if we consider
$H(|\psi_{\perp}\rangle)$, we get $b = b^*$ (i.e., $b$ is real) and
$a+a^* = 2b^* = 2b$ (as $b$ is real) leading to the same result. Thus $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$ is restricted to the form $\alpha |0\rangle - (\alpha - i\beta) |1\rangle$.
Our next task is to map the states from this ensemble to points on the Bloch sphere. But before attempting to do this, we give a brief pedagogical description of how to geometrically represent a general qubit state on the Bloch sphere. Consider $|\psi\rangle = a
|0\rangle + b |1\rangle$. Since $a$ and $b$ are complex, assume $a
= r_1 e^{i\gamma}, b = r_2 e^{i(\gamma+\phi)}$. Then $|a| = r_1, |b| = r_2$. Let $r_1 = \cos{\frac{\theta}{2}}, r_2 = \sin{\frac{\theta}{2}}$. Hence, $a = \cos{\frac{\theta}{2}} e^{i\gamma}, b = \sin{\frac{\theta}{2}} e^{i(\gamma+\phi)}$, where $\theta, \phi$ and $\gamma$ are real.
Thus, any qubit $|\psi\rangle = a |0\rangle + b |1\rangle$ can be written as $e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle + e^{i\phi}
\sin{\frac{\theta}{2}} |1\rangle)$. Further, two qubits
$e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle + e^{i\phi}
\sin{\frac{\theta}{2}} |1\rangle)$ and $(\cos{\frac{\theta}{2}}
|0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle)$ are treated on equal footing under measurement, since they differ only by an overall phase factor which has no observable effect.
The qubit $(\cos{\frac{\theta}{2}} |0\rangle +
e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle)$ is mapped to a point $(1, \theta, \phi)$ on the unit Bloch sphere. Here $\theta$ and $\phi$ are the usual polar and azimuthal angles respectively, and they are related to the cartesian coordinates $(x, y, z)$ through the usual relations $x = \cos{\phi} \sin{\theta}$, $y = \sin{\phi} \sin{\theta}$, $z = \cos{\theta}$. If we fix $\phi = 0$, then we obtain states of the form
$\cos{\frac{\theta}{2}} |0\rangle + \sin{\frac{\theta}{2}} |1\rangle$ and
$\cos{\frac{\theta}{2}} |1\rangle - \sin{\frac{\theta}{2}} |0\rangle$ for $0 \leq \theta \leq \pi$. These lie on the polar great circle of the Bloch sphere. On the other hand, for $\theta = \pi /2$, one obtains states of the form
$\frac{1}{\sqrt{2}}(|0\rangle + e^{i\phi} |1\rangle)$ and
$\frac{1}{\sqrt{2}}(|1\rangle - e^{-i\phi} |0\rangle)$ for $0 \leq \phi \leq 2\pi$, which lie on the equatorial great circle.
Now we can conveniently plot the states from our special ensemble
$|\psi\rangle = (\alpha + i\beta) |0\rangle + \alpha |1\rangle$. Bringing it to the desired form, it is clear that $e^{i\gamma} = \frac{\alpha + i\beta}{\sqrt{\alpha^2 + \beta^2}}, \cos{\frac{\theta}{2}} = \sqrt{\alpha^2 + \beta^2}, \sin{\frac{\theta}{2}} = \alpha, e^{i\phi} = \frac{\sqrt{\alpha^2 + \beta^2}}{\alpha + i\beta}$. We thus arrive at the following identification: \begin{eqnarray} \label{al1} x &=& \cos{\phi}\sin{\theta} = 2\alpha^2, \nonumber \\ y &=& \sin{\phi} \sin{\theta} = - 2\alpha \sqrt{1-2\alpha^2}, \nonumber \\ z &=& \cos{\theta} = 1 - 2\alpha^2. \end{eqnarray}
These points are represented by curve 1 on the Bloch sphere.
Next, we consider
$|\psi\rangle = a|0\rangle + b |1\rangle$ and the other orthogonal complement
$|\psi_{\perp}\rangle = - b^* |0\rangle + a^* |1\rangle$, which differs from the first one just by an overall negative sign. This yields that
$|\psi\rangle$ must be of the form
$(\alpha + i\beta) |0\rangle + i\beta |1\rangle$, with $\alpha^2 = 1 - 2\beta^2$ where $-\frac{1}{\sqrt{2}} \leq \beta \leq \frac{1}{\sqrt{2}}$, since
$\alpha$ is real. Therefore, in the computational basis, the qubit state $|\psi\rangle$ has complex and imaginary amplitudes; the imaginary parts of which are equal. As for $|\psi_{\perp}\rangle = -b^* |0\rangle + a^* |1\rangle$, it assumes the form $i\beta |0\rangle + (\alpha - i\beta) |1\rangle$.
For the qubits of the form $(\alpha + i\beta) |0\rangle + i\beta |1\rangle$, we have $e^{i\gamma} = \frac{\alpha + i\beta}{\sqrt{\alpha^2 + \beta^2}}, \cos{\frac{\theta}{2}} = \sqrt{\alpha^2 + \beta^2}, \sin{\frac{\theta}{2}} = \beta, e^{i\phi} = i\frac{\sqrt{\alpha^2 + \beta^2}}{\alpha + i\beta}$. Thus on the Bloch sphere: \begin{eqnarray} \label{al2} x &=& \cos{\phi}\sin{\theta} = 2\beta^2, \nonumber \\ y &=& \sin{\phi} \sin{\theta} = 2\beta \sqrt{1-2\beta^2}, \nonumber \\ z &=& \cos{\theta} = 1 - 2\beta^2. \end{eqnarray}
It is immediately clear that a point represented by Eq(\ref{al1}), for a particular value of $\alpha$, is exactly equal to the one obtained from Eq(\ref{al2}) for the same value of $(- \beta)$. This implies that these two ensembles give the same trajectory on the Bloch sphere. Hence, we shall consider them to be isomorphic to each other. Our result is thus summarized in the following theorem. \begin{theorem} \label{th1} The most general qubit states for which it is possible to design a universal Hadamard gate satisfying Eq(\ref{eq1}) are given by
$\{{|\psi\rangle, |\psi_{\perp}\rangle} || |\psi\rangle = (\alpha + i\beta)
|0\rangle + \alpha |1\rangle ;
|\psi_{\perp}\rangle = \alpha |0\rangle - (\alpha - i\beta) |1\rangle\}$ where $\alpha, \beta$ are real such that $2\alpha^2 + \beta^2 = 1$ and $-\frac{1}{\sqrt{2}} \leq \alpha \leq \frac{1}{\sqrt{2}}$. \end{theorem}
Note that if we choose $\alpha = 0$, then from Eq(\ref{al1}), we obtain the point $(0, 0, 1)$ on the Bloch sphere (i.e., north pole), which can be identified with the computational basis state $|0\rangle$ on curve 1.
In a similar fashion, the trajectory of $|\psi_\perp\rangle$
can be sketched, which would lie on the other side of the Bloch sphere (not visible in the figure). It can be checked that the orthogonal state $|1\rangle$ would be one of its points $(0, 0, -1)$ (i.e., south pole).
We demonstrate that this trajectory has some intersection points with the equatorial great circle also. To this end, for $\theta = \frac{\pi}{2}$,
$z = \cos{\theta} = 0 = 1 - 2\alpha^2$, i.e., $\alpha = \pm \frac{1}{\sqrt{2}}$, and $2\alpha^2 + \beta^2 = 1$, i.e., $\beta = 0$. Substituting these values we get $|\psi\rangle = (\alpha + i \beta) |0\rangle + \alpha |1\rangle
= \pm \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and
$|\psi_\perp\rangle = \alpha |0\rangle + (\alpha - i\beta) |1\rangle
= \pm \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$. For the second case, $\beta = \pm \frac{1}{\sqrt{2}}$ and $\alpha = 0$. This yields
$|\psi\rangle =
= \pm \frac{i}{\sqrt{2}}(|0\rangle + |1\rangle)$ and
$|\psi_\perp\rangle = = \pm \frac{i}{\sqrt{2}}(|0\rangle - |1\rangle)$.
Therefore,if one chooses any qubit $|\psi\rangle$ from curve 1 in the figure, and takes its orthogonal complement, then the Hadamard transformation works perfectly well to generate the superposition. To see it explicitly, take $|\psi\rangle = (\alpha + i\beta)
|0\rangle + \alpha |1\rangle$ and $|\psi_\perp\rangle = \alpha
|0\rangle - (\alpha - i\beta) |1\rangle$. $H$ rotates $|\psi\rangle$ to \begin{eqnarray}
H|\psi\rangle &=& \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & 1 \\ 1 & -1 \\ \end{array}\right] \left[\begin{array}{c} \alpha + i\beta \\ \alpha \\ \end{array}\right] = \frac{1}{\sqrt{2}} \left[\begin{array}{c} \alpha + i\beta + \alpha\\ \alpha + i\beta - \alpha\\ \end{array}\right] \nonumber \\ \ &=& \frac{1}{\sqrt{2}} \left[\begin{array}{c} (\alpha + i\beta) + (\alpha)\\ (\alpha) - (\alpha - i\beta)\\ \end{array}\right]
= \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle), \nonumber \end{eqnarray}
while it acts on $|\psi_{\perp}\rangle$ to give \begin{eqnarray}
H|\psi_{\perp}\rangle &=& \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & 1 \\ 1 & -1 \\ \end{array}\right] \left[\begin{array}{c} \alpha \\ -(\alpha - i\beta) \\ \end{array}\right] \nonumber \\ \ &=& \frac{1}{\sqrt{2}} \left[\begin{array}{c} \alpha - (\alpha - i\beta)\\ \alpha + (\alpha - i\beta)\\ \end{array}\right] \nonumber \\ \ &=& \frac{1}{\sqrt{2}} \left[\begin{array}{c} (\alpha + i\beta) - (\alpha)\\ (\alpha) + (\alpha - i\beta)\\ \end{array}\right]
= \frac{1}{\sqrt{2}} (|\psi\rangle - |\psi_{\perp}\rangle).\nonumber \end{eqnarray}
Alternatively, one can also prove the above theorem by unitarity, as was done in~\cite{PT02} for the general case. Take any two quantum states $\{|\psi^{(k)}\rangle, |\psi^{(l)}\rangle\}$ from this special ensemble, and their complement states
$\{|\psi_{\perp}^{(k)}\rangle, |\psi_{\perp}^{(l)}\rangle\}$. Applying the Hadamard transformation (\ref{eq1}) on them and taking inner product, we get
\begin{eqnarray} \label{eqip1} \langle\psi^{(k)}
|\psi^{(l)}\rangle &=& \frac{1}{2}(\langle\psi^{(k)}
|\psi^{(l)}\rangle + \langle\psi^{(k)}
|\psi_{\perp}^{(l)}\rangle + \langle\psi_{\perp}^{(k)}
|\psi^{(l)}\rangle \nonumber \\
\ &+& \langle\psi_{\perp}^{(k)}|\psi_{\perp}^{(l)}\rangle) \end{eqnarray}
\begin{eqnarray} \label{eqip2} \langle\psi_{\perp}^{(k)}
|\psi_{\perp}^{(l)}\rangle &=& \frac{1}{2}(\langle\psi^{(k)}
|\psi^{(l)}\rangle - \langle\psi^{(k)}
|\psi_{\perp}^{(l)}\rangle - \langle\psi_{\perp}^{(k)}
|\psi^{(l)}\rangle \nonumber \\
\ &+& \langle\psi_{\perp}^{(k)}|\psi_{\perp}^{(l)}\rangle). \end{eqnarray}
Any two qubits from this ensemble obey the conjugation rules \begin{eqnarray} \label{eqip3}
\langle\psi^{(k)}| \psi_{\perp}^{(l)}\rangle &=&
- \langle\psi_{\perp}^{(k)}| \psi^{(l)}{\rangle}^*
= \langle\psi_{\perp}^{(k)}| \psi^{(l)}\rangle, \nonumber \\
\langle\psi^{(k)}| \psi^{(l)}\rangle &=&
\langle\psi_{\perp}^{(k)}| \psi_{\perp}^{(l)}{\rangle}^*. \end{eqnarray}
Substituting these conditions in the above inner product relations, it is straightforward to check that the inner product is preserved. Hence, a universal Hadamard gate exists for any qubit chosen from this special class.
\section{III. Hadamard Type Transforms} In this section, we consider some operations which are not exactly Hadamard transforms, but similar, in the sense that they produce equal superposition of the amplitudes up to a sign or a phase. These have been discussed by Pati, in the context of qubits from the polar and equatorial great circles. Here, we elaborate more on these transformations and present some general results.
\subsection{Polar Type Transformation}
Any two orthogonal vectors on the polar great circle,
$|\psi\rangle =
\cos{\frac{\theta}{2}} |0\rangle + \sin{\frac{\theta}{2}} |1\rangle$ and
$|\psi_{\perp}\rangle =
\cos{\frac{\theta}{2}} |1\rangle - \sin{\frac{\theta}{2}} |0\rangle$, can be shown to transform as \cite{PT02} \begin{equation} \label{eq1a}
U(|\psi\rangle) = \frac{1}{\sqrt{2}} (|\psi\rangle +
|\psi_{\perp}\rangle), U(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}} (|\psi_{\perp}\rangle -
|\psi\rangle). \end{equation} This differs from the usual Hadamard transformation by an overall sign in the second part. The appropriate unitary operator $U$ which does the job is denoted by
$H_P = \sigma_x H = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & -1 \\ 1 & 1 \\ \end{array}\right]$, where $\sigma_x$ is the Pauli flip matrix $\left[\begin{array}{cr} 0 & 1 \\ 1 & 0 \\ \end{array}\right]$.
We now extend this result to vectors from any polar circle. Take $|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle +
e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle$ on any of the polar circles, and its orthogonal complement
$|\psi_{\perp}\rangle = -\sin{\frac{\theta}{2}} |0\rangle
+ e^{i\phi} \cos{\frac{\theta}{2}} |1\rangle$. Then, for any $\phi$, we can construct a unitary operator
$H_G^{\phi} = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & -e^{-i\phi} \\ e^{i\phi} & 1 \\ \end{array}\right]$, such that Eq(\ref{eq1a}) is satisfied, i.e.,
$H_G^{\phi} |\psi\rangle = \frac{1}{\sqrt{2}}
(|\psi\rangle + |\psi_{\perp}\rangle)$ and
$H_G^{\phi} |\psi_\perp\rangle = \frac{1}{\sqrt{2}}
(|\psi_\perp\rangle - |\psi\rangle)$. For $\phi = 0$, $H_G^{\phi}$ reduces to $H_P$, thereby covering the polar great circle case.
\begin{theorem} \label{th2}
For any state $|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle +
e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle$ and its orthogonal state
$|\psi_{\perp}\rangle = -\sin{\frac{\theta}{2}} |0\rangle
+ e^{i\phi} \cos{\frac{\theta}{2}} |1\rangle$, it is possible to design a Hadamard type gate $H_G^{\phi}$ that satisfies the transformation (\ref{eq1a}), once $\phi$ is known. \end{theorem}
It is clear that the unitary operator $U$, satisfying Eq(\ref{eq1a}), depends on the type of states chosen. For instance, we get two different gates above, depending on whether $\{|\psi\rangle, |\psi_{\perp}\rangle\}$ belongs to the polar great circle or some other polar circle.
Therefore, if one fixes the operator $U$, then one can show that there is a unique ensemble of states satisfying the transformation (\ref{eq1a}). For this purpose, let us consider the gate $H_P$, and find the associated class of states for which it works.
We follow our previous procedure of taking
$|\psi\rangle = a|0\rangle + b |1\rangle$ and its orthogonal complement $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Linearity yields, \begin{eqnarray}
H_P(|\psi\rangle) &=& aH_P(|0\rangle) + bH_P(|1\rangle)\\ \ &=&
\frac{a}{\sqrt{2}} (|0\rangle + |1\rangle) +
\frac{b}{\sqrt{2}} (|1\rangle - |0\rangle) \nonumber \\
\ &=& \frac{(a-b) |0\rangle - (a+b) |1\rangle}{\sqrt{2}}. \nonumber \end{eqnarray} On the other hand, \begin{eqnarray}
H(|\psi\rangle) &=& \frac{1}{\sqrt{2}}(
|\psi\rangle+|\psi_\perp\rangle) \nonumber \\
\ &=& \frac{a|0\rangle + b|1\rangle}{\sqrt{2}} +
\frac{b^*|0\rangle - a^*|1\rangle}{\sqrt{2}} \nonumber \\
\ &=& \frac{(a+b^*) |0\rangle + (b-a^*) |1\rangle}{\sqrt{2}} \nonumber \end{eqnarray} Thus, $a - b = a + b^*$, i.e., $b+b^* = 0$. So $b$ is imaginary. Moreover, $a + b = b - a^*$, i.e., $a+a^* = 0$. Therefore, $a$ is also imaginary. Hence, given $\alpha, \beta$ real, we get
$|\psi\rangle = i\alpha |0\rangle + i \beta |1\rangle$ and
$|\psi_{\perp}\rangle = -i\beta |0\rangle + i \alpha |1\rangle$. Rewriting $|\psi\rangle = i(\alpha |0\rangle + \beta |1\rangle)$ and representing it on the Bloch sphere, one can readily check that $e^{i\gamma} = i$, $\cos{\frac{\theta}{2}} = \alpha$, $\sin{\frac{\theta}{2}} = \beta$, $e^{i\phi} = 1$. The resulting trajectory is that of the polar great circle.
However, if we had taken
$|\psi_{\perp}\rangle = -b^* |0\rangle + a^* |1\rangle$, we would have got $|\psi\rangle$ of the form $\alpha |0\rangle +
\beta |1\rangle$, which again are the states on the polar great circle. We thus conclude that, up to isomorphism, this is the only class of qubit states which transforms according to Eq(\ref{eq1a}), under the action of the gate $H_P$. Alternatively, one can also fix the states and determine the corresponding gate uniquely. \subsection{Equatorial Type Transformation}
The second kind of operation discussed in~\cite{PT02} is that of an equal superposition of amplitudes up to a phase such that \begin{equation} \label{eq2}
U(|\psi\rangle) = \frac{1}{\sqrt{2}} (|\psi\rangle + i|\psi_{\perp}\rangle), U(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}}
(i|\psi\rangle + |\psi_{\perp}\rangle). \end{equation}
This alternative universal definition of a Hadamard type gate, has the advantage that it is invariant under the interchange of $|\psi\rangle$ and $|\psi_{\perp}\rangle$. Vectors of the form
$|\psi(\phi)\rangle = H(\cos\frac{\phi}{2} |0\rangle) -
i \sin\frac{\phi}{2} |1\rangle) =
\frac{1}{\sqrt{2}} e^{-i\phi /2} (|0\rangle + e^{i\phi} |1\rangle)$ and the corresponding orthogonal
$|\psi_{\perp}(\phi)\rangle = H(i\sin\frac{\phi}{2} |0\rangle) -
\cos\frac{\phi}{2} |1\rangle) =
\frac{1}{\sqrt{2}} e^{i\phi /2} (|1\rangle - e^{-i\phi} |0\rangle)$ chosen from the equatorial great circle satisfy this transformation provided the unitary matrix is $H_E = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1-i & 0 \\ 0 & 1+i \\ \end{array}\right]$. We wish to clarify here that the matrix $H_E$ presented in \cite{PT02} does not work for the states considered, and the correct form of $H_E$ should essentially be what we have given above.
Our next task is to find the most general class of states satisfying the phase dependent transformation (\ref{eq2}), provided the computational basis vectors $\{|0\rangle, |1\rangle\}$ also transform in the same fashion, i.e., to
$\frac{1}{\sqrt{2}}(|0\rangle + i |1\rangle)$ and
$\frac{1}{\sqrt{2}}(i |0\rangle + |1\rangle)$, respectively.
Thus fixing the unitary operator as $U = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & i \\ i & 1 \\
\end{array}\right]$, we obtain conditions on the form of $|\psi\rangle$ and
$|\psi_{\perp}\rangle$. Following the earlier procedure, we assume that $|\psi\rangle = a|0\rangle + b |1\rangle$ and
$|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Then using linearity of the operation, we find that $|\psi\rangle$ must be of the form
$i\alpha |0\rangle + \beta |1\rangle$. The complement
$|\psi_{\perp}\rangle$ is restricted to
$\beta |0\rangle + i\alpha |1\rangle$. Now $|\psi\rangle$ can be written as
$i\sqrt{\alpha^2+\beta^2} (\frac{\alpha}{\sqrt{\alpha^2+\beta^2}} |0\rangle
- i\frac{\beta}{\sqrt{\alpha^2+\beta^2}} |1\rangle)$, i.e.,
$e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle +
e^{i\frac{3\pi}{2}} \sin{\frac{\theta}{2}} |1\rangle)$. On the Bloch sphere, $x = \cos{\phi}\sin{\theta} = 0, y = \sin{\phi} \sin{\theta} = 2\alpha \sqrt{1-\alpha^2}, z = \cos{\theta} = 2\alpha^2 - 1$, where $-1 \leq \alpha \leq 1$.
Similarly considering the second complement $|\psi_{\perp}\rangle =
-b^* |0\rangle + a^* |1\rangle$, and using linearity of the operation, we get $|\psi\rangle = \alpha |0\rangle + i\beta
|1\rangle$ and $|\psi_{\perp}\rangle = i\beta |0\rangle + \alpha
|1\rangle$. Identifying with the Bloch sphere picture,
$|\psi\rangle$ can be written as $\sqrt{\alpha^2+\beta^2}
(\frac{\alpha}{\sqrt{\alpha^2+\beta^2}} |0\rangle +
i\frac{\beta}{\sqrt{\alpha^2+\beta^2}} |1\rangle)$, i.e.,
$e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle + e^{i\frac{\pi}{2}}
\sin{\frac{\theta}{2}} |1\rangle)$. Therefore, on the Bloch sphere, $x = \cos{\phi}\sin{\theta} = 0, y = \sin{\phi} \sin{\theta} = 2\alpha \sqrt{1-\alpha^2}, z = \cos{\theta} = 2\alpha^2 - 1$, where $-1 \leq \alpha \leq 1$.
Hence $|\psi\rangle$, when expressed in computational basis, is made up of one real and one imaginary amplitude. As expected, the above two ensembles give the same trajectory, represented by curve 2 in the figure. We thus have the following result. \begin{theorem} \label{th3} It is possible to design a universal Hadamard type gate $U$, satisfying the transformation (\ref{eq2}), for any state of the form
$|\psi\rangle =
i\alpha |0\rangle + \beta |1\rangle$ and its orthogonal complement
$|\psi_{\perp}\rangle =
\beta |0\rangle + i\alpha |1\rangle$, where $\alpha, \beta$ are real such that $\alpha^2 + \beta^2 = 1$ and $-1 \leq \alpha \leq 1$. \end{theorem}
One can check explicitly that
$U(|\psi\rangle) = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & i \\ i & 1 \\ \end{array}\right]$ $\left[\begin{array}{c} i\alpha \\ \beta \\ \end{array}\right]$ $= \frac{1}{\sqrt{2}} \left[\begin{array}{c} i(\alpha + \beta)\\ -\alpha + \beta\\ \end{array}\right]
= \frac{1}{\sqrt{2}} (|\psi\rangle + i|\psi_{\perp}\rangle)$. Similarly,
$U(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & i \\ i & 1 \\ \end{array}\right]$ $\left[\begin{array}{c} \beta \\ i\alpha \\ \end{array}\right]$ $= \frac{1}{\sqrt{2}} \left[\begin{array}{c} \beta - \alpha\\ i(\beta + \alpha)\\ \end{array}\right]
= \frac{1}{\sqrt{2}} (i|\psi\rangle + |\psi_{\perp}\rangle)$. Interestingly, this trajectory cuts the equatorial great circle when $z = 0$, i.e., $\alpha = \pm \frac{1}{\sqrt{2}}$. These intersection points imply that there are quantum states (and their orthogonals ) from this ensemble which also belong to the equatorial great circle. Let us find out these states.
For $\theta = \frac{\pi}{2}$, $\alpha = \pm \frac{1}{\sqrt{2}}$, and from normalization condition, $\beta = \pm \frac{1}{\sqrt{2}}$. Substituting these values in $|\psi\rangle$ and $|\psi_\perp\rangle$ of Theorem~\ref{th3}, we get
$|\psi\rangle = \pm \frac{1}{\sqrt{2}} (i |0\rangle \pm |1\rangle)$ and
$|\psi_\perp\rangle = \pm \frac{1}{\sqrt{2}} (|0\rangle \pm i|1\rangle)$. One can similarly find the states corresponding to the second orthogonal complement.
\section{IV. Unequal Superposition} We shall now focus our attention on unequal superposition of the amplitudes of a qubit state. Like the equal superposition case, it is impossible to create unequal superposition of an arbitrary unknown qubit with its complement state \cite{PT02}. Our task therefore, is to obtain special classes of states for which such a superposition would be possible. This can be regarded as a generalized version of the usual Hadamard transformation and is given by \begin{equation} \label{equ1}
U(|\psi\rangle) = p|\psi\rangle + q|\psi_{\perp}\rangle, U(|\psi_{\perp}\rangle) = q^*|\psi\rangle - p^*|\psi_{\perp}\rangle. \end{equation}
Here, $p, q$ are known complex numbers with $|p|^2 + |q|^2 = 1$. We again demand that
$|\psi\rangle$ and $|\psi_{\perp}\rangle$ transform like the computational basis vectors $|0\rangle$ and
$|1\rangle$ respectively. Thus $U$ can be fixed as $\left[\begin{array}{cr} p & q^* \\ q & -p^* \\
\end{array}\right]$. Repeating the linearity procedure, take $|\psi\rangle = a|0\rangle + b |1\rangle$ and
$|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Ideally we should have \begin{eqnarray}
U(|\psi\rangle) &=& p|\psi\rangle + q|\psi_{\perp}\rangle \nonumber \\
\ &=& p(a |0\rangle + b |1\rangle) +
q(b^* |0\rangle - a^* |1\rangle) \nonumber \\
\ &=& (pa+qb^*) |0\rangle + (pb - qa^*) |1\rangle. \nonumber \end{eqnarray} On the other hand, from linearity we get \begin{eqnarray}
U(|\psi\rangle) &=& a U(|0\rangle) + b U(|1\rangle) \nonumber \\
\ &=& a (p|0\rangle + q|1\rangle) +
b (q^*|0\rangle - p^*|1\rangle) \nonumber \\
\ &=& (ap+bq^*) |0\rangle + (aq-bp^*) |1\rangle \nonumber \end{eqnarray} Hence $pa + qb^* = pa + q^*b$, i.e., $qb^* = q^*b = (qb^*)^*$. Thus $qb^*$ is real, which implies that \begin{enumerate} \item both $q, b$ are real, or \item both $q, b$ are imaginary, or \item both $q, b$ are complex, with the constraint $\frac{q_1}{q_2} = \frac{b_1}{b_2}$. (Here, any complex number $z = (a, b, q, p)$ has been written as $z= z_1 + i z_2$). \end{enumerate} Further, $pb - qa^* = qa - p^*b$, i.e., $q(a + a^*) = b(p + p^*)$, so, $q \cdot Re(a) = b \cdot Re(p)$, i.e., $Re(a) = \frac{b}{q} \cdot Re(p)$.
Therefore, $|\psi\rangle$ and $|\psi_{\perp}\rangle$ are restricted to the form
$|\psi\rangle = (\frac{b}{q} \cdot Re(p) + ia_2) |0\rangle + b
|1\rangle$ and
$|\psi_{\perp}\rangle = b^* |0\rangle -
(\frac{b}{q} Re(p) - i a_2) |1\rangle$. Depending on whether $q$ and $b$ are both real, or imaginary or complex ( with $\frac{q_1}{q_2} = \frac{b_1}{b_2}$), we get different classes of states for which the unequal superposition transformation (\ref{equ1}) holds. For the special value of $p = q = \frac{1}{\sqrt{2}}$, this ensemble goes over to the set of states
$|\psi\rangle = (b + ia_2) |0\rangle + b |1\rangle$ (i.e., complex and real amplitudes such that real parts are the same) obtained in Sec. II. Also, the associated Hadamard matrix $H$, satisfying the transformation (\ref{eq1}), can be recovered for these values of $p,q$ from the above $U$. Now, analogous to the previous section, we concentrate below, on two specific unequal superposition transformations.
\subsection{Unequal Polar Type Transformation} According to the prescription outlined in~\cite{PT02}, for the vectors on the polar great circle, one can find a unitary gate $U_P = \left[\begin{array}{cr} p & -q \\ q & p \\
\end{array}\right]$, where $p^2 + q^2 = 1$ and $p, q$ are now real. In this case $|\psi\rangle =
\cos{\frac{\theta}{2}} |0\rangle + \sin{\frac{\theta}{2}} |1\rangle$ and
$|\psi_{\perp}\rangle =
\cos{\frac{\theta}{2}} |1\rangle - \sin{\frac{\theta}{2}} |0\rangle$ and they transform as \begin{equation} \label{equ2}
U_P (|\psi\rangle) = q|\psi_{\perp}\rangle + p|\psi\rangle, U_P (|\psi_{\perp}\rangle) = p|\psi_{\perp}\rangle - q|\psi\rangle. \end{equation} This is almost similar to Eq(\ref{equ1}), up to an overall sign (when $p, q$ are real).
Now we present a generalization of this result. Take a qubit
$|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle +
e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle$ on any polar circle, and the orthogonal qubit
$|\psi_{\perp}\rangle = -\sin{\frac{\theta}{2}} |0\rangle
+ e^{i\phi} \cos{\frac{\theta}{2}} |1\rangle$. Then, for any $\phi$, one can construct a corresponding unitary matrix $U_G^{\phi} = \left[\begin{array}{cr} p & -q e^{-i\phi} \\ q e^{i\phi} & p \\ \end{array}\right]$, such that
$U_G^{\phi} |\psi\rangle =
p |\psi\rangle +
q |\psi_\perp\rangle$ and
$U_G^{\phi} |\psi_\perp\rangle =
p |\psi_\perp\rangle - q |\psi\rangle$. In the limit when $\phi = 0$, we recover the polar great circle case since $U_G^0 = U_P$. Thus if partial information $(\phi)$ is known, given any arbitrary state, it is possible to design a generalized Hadamard type gate for unequal superposition. Note that for $p = q = \frac{1}{\sqrt{2}}$, $U_G^{\phi} = H_G^{\phi}$, thereby yielding the result of Theorem~\ref{th2}.
\subsection{Unequal Equatorial Type Transformation}
The generalized version of the phase dependent Hadamard type of transformation can be written as \begin{equation} \label{equ1x}
U(|\psi\rangle) = p|\psi\rangle + iq|\psi_{\perp}\rangle, U(|\psi_{\perp}\rangle) = iq^*|\psi\rangle + p^*|\psi_{\perp}\rangle. \end{equation}
Here again $p, q$ are known complex numbers with $|p|^2 + |q|^2 =
1$. Under the assumption that $\{|\psi\rangle,
|\psi_{\perp}\rangle\}$ transform in the same way as $\{|0\rangle,
|1\rangle\}$, $U$ is fixed to be $\left[\begin{array}{cr} p & iq^* \\ iq & p^* \\
\end{array}\right]$. In order to obtain classes of states obeying this transformation under the action of $U$, take $|\psi\rangle = a|0\rangle + b |1\rangle$ and
$|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Ideally we should have \begin{eqnarray}
U(|\psi\rangle) &=& p|\psi\rangle + iq|\psi_{\perp}\rangle \nonumber \\
\ &=& p(a |0\rangle + b |1\rangle) +
iq(b^* |0\rangle - a^* |1\rangle) \nonumber \\
\ &=& (ap+iqb^*) |0\rangle + (bp - iqa^*) |1\rangle. \nonumber \end{eqnarray} Then from linearity we get \begin{eqnarray}
U(|\psi\rangle) &=& a U(|0\rangle) + b U(|1\rangle) \nonumber \\
\ &=& a (p|0\rangle + iq|1\rangle) +
b (iq^*|0\rangle + p^*|1\rangle) \nonumber \\
\ &=& (ap+ibq^*) |0\rangle + (iaq+bp^*) |1\rangle \nonumber \end{eqnarray} Hence $pa + iqb^* = pa + iq^*b$, i.e., $qb^* = q^*b = (qb^*)^*$. Thus $qb^*$ is real, i.e., \begin{enumerate} \item both $q, b$ are real or \item both $q, b$ are imaginary or \item both $q, b$ are complex, with the constraint $\frac{b_1}{b_2} = \frac{q_1}{q_2}$. \end{enumerate} Further, $pb - iqa^* = iqa + p^*b$, i.e., $iq(a + a^*) = b(p - p^*)$, so $a_1 = \frac{b}{q} \cdot p_2$.
Hence we get a general class of states,
$|\psi\rangle = (\frac{b}{q} p_2 + i \cdot a_2) |0\rangle
+ b |1\rangle$ and $|\psi_\perp\rangle = b^* |0\rangle -
(\frac{b}{q} \cdot p_2 - i \cdot a_2) |1\rangle)$. Depending on the above three possible solutions, i.e., whether $q, b$ are both real, or imaginary or complex, we get different classes of states for which the unequal superposition transformation (\ref{equ1x}) holds. Again, it is easy to see that for the special case $p = q = \frac{1}{\sqrt{2}}$, this reduces to the class of states obtained in Theorem~\ref{th3}.
\section{V. Summary and Conclusions} In this paper we have found the most general class of qubits (up to isomorphisms) for which the Hadamard gate can be designed. This was achieved using one of the fundamental axioms of quantum mechanics, namely linearity. If expressed in the computational basis, the qubit state assumes a specific form: one complex and one pure real (imaginary) amplitude; the real (imaginary) parts of which are equal. The Hadamard gate is universal for this class of ensemble, i.e., it works for any state belonging to this particular ensemble. When represented on a Bloch sphere, these states give a new trajectory.Interestingly, it has some intersection points with the polar and equatorial great circles.
Equal superposition of $|0\rangle$ and $|1\rangle$ states has played a very crucial role in quantum algorithms to study various problems, e.g., distinguishing between constant and balanced Boolean functions~\cite{qDJ92}, database search~\cite{qGR96} etc. It would be interesting to construct specific computational problems, which one can study by exploiting the superposition of $|\psi\rangle$ and $|\psi_\perp\rangle$ from the class of states obtained in this paper.
We have also considered some Hadamard type transformations which hold for polar and equatorial qubits, and have obtained some new results. The situation becomes more general when the superposition of the two amplitudes is not equal. Many new classes of states have been found and all the results of the equal superposition case can be recovered by letting the parameters to be equal.
The next step would be to generalize these results to higher dimensions,where the analogue of the Hadamard transform would be the discrete Fourier transform. In this direction we have obtained partial results so far and further work is in progress.
\ \\ \noindent{\bf Acknowledgments:} We thank G. Kar and P. Mukhopadhyay for useful discussions. PP acknowledges financial assistance from DST under the SERC Fast Track Proposal scheme for young scientists.
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\title[Pair correlation of lacunary sequences]{The metric theory of the pair correlation function of real-valued lacunary sequences} \author{Ze\'ev Rudnick and Niclas Technau} \date{\today}
\address{School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel}
\email{rudnick@tauex.tau.ac.il} \email{niclast@mail.tau.ac.il} \thanks{This result is part of a project that received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 786758).} \subjclass[2010]{11J54; 11J71}
\keywords{Pair correlation; Poisson statistics; lacunary sequence} \begin{abstract} Let $\{ a(x) \}_{x=1}^{\infty}$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $\alpha a(x)$ is Poissonian for Lebesgue almost every $\alpha\in \mathbb{R}$. By using harmonic analysis, our result --- irrespective of the choice of the real-valued sequence $\{ a(x) \}_{x=1}^{\infty}$ --- can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$ \vert n_1 (a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2)) \vert < 1 $$ in integer six-tuples $(n_1,n_2,x_1,x_2,y_1,y_2)$ located in the box $[-N,N]^6$ with the ``excluded diagonals'', that is $$x_1\neq y_1, \quad x_2 \neq y_2, \quad (n_1,n_2)\neq (0,0),$$ is at most $N^{4-\delta}$ for some fixed $\delta>0$, for all sufficiently large $N$.
\end{abstract} \maketitle
\section{Introduction} A sequence of points $\{\theta_n\}_{n=1}^\infty$ is uniformly distributed modulo one if given any fixed interval $I$ in the unit circle ${\mathbb R}/{\mathbb Z}$,
the proportion of fractional parts $\theta_n \bmod 1$ which lie in $I$ tends to the length of the interval $I$, that is
\begin{equation*}
\#\{n\leq N: \theta_n\bmod 1 \in I\} \sim \operatorname{length}(I) \cdot N, \quad N\to \infty .
\end{equation*}
We study the {\em pair correlation function} $R_2$, defined for every fixed interval $I\subset {\mathbb R}$ by the property that \[
\lim_{N\to \infty} \frac 1N \#\{ 1\leq m\neq n\leq N: |\theta_n-\theta_m| \in \frac 1N I \} = \int_I R_2(x)dx
\] assuming that the limit exists. For a random sequence of $N$ elements, that is $N$ uniform independent random variables in $[0,1)$ (the Poisson model), the limiting pair correlation function is $R_2(x)\equiv 1$.
There are very few positive results on the pair correlation function available for specific sequences, a notable exception being the fractional parts of $\sqrt{n}$ \cite{EMV}; a more tractable problem is to randomize (a ``metric'' theory, in the terminology of uniform distribution theory) by looking at random multiples\footnote{A different notion of randomizing has recently been investigated in \cite{AB}, which studies the pair correlation of the sequence $\alpha^n \bmod 1$ with $\alpha$ random.} $\theta_n=\alpha a(n) \bmod 1$, for almost all $\alpha$.
There is a well-developed metric theory of the pair correlation function for {\em integer} valued sequences $\{a(n)\}_{n=1}^\infty$, initiated in \cite{RudnickSarnak}, where polynomial sequences such as $a(n)=n^d$, $d\geq 2$, are studied,
with several developments in the last few years, see e.g. \cite{ALL, BCC, CLZ, LT, LarcherStockinger, PS, RZ pc, Walker}.
In this note we study the case of real-valued lacunary sequences: Let $a(x)>0$ be a lacunary sequence of positive reals, that is there is some $C>1 $ so that for all integers $x\geq 1$, \[ a(x+1)\geq C a(x). \] For instance, we can take $a(x) = e^x$. It is known that for almost all $\alpha$, the sequence $\alpha a(x)\bmod 1$ is uniformly distributed mod one \cite[Chapter 1, Corollary 4.3]{KN}. Here and throughout this note ``almost all'' is meant with respect to the Lebesgue measure on ${\mathbb R}$.
\begin{thm}\label{thm:lacunary} Assume that $\{a(x)\}_{x=1}^\infty$ is a lacunary sequence of positive reals. Then the pair correlation function of the sequence $\{\alpha a(x)\}_{x=1}^\infty$ is Poissonian for almost all $\alpha$. \end{thm}
When $a(x)$ takes {\em integer} values, \cite{RZ pc} showed that for almost all $\alpha$, the pair correlation function is Poissonian. The case of pair correlation of sequences of {\em rationals} $x_n=a_n/b_n$ with $a_n$ integer-valued and lacunary and $b_n$ integer-valued and (roughly speaking) sufficiently small (e.g. $a_n/b_n=2014^n/[\log\log n]$) was treated in \cite{CLZ}. Here we treat any real-valued sequences.
We will reduce the problem to giving a bound for the number of lattice points satisfying a Diophantine inequality: Let $$M=N^{1+\varepsilon}, \quad K=N^\varepsilon $$
and assume that there is some $\delta>0$ so that for all $\varepsilon>0$ sufficiently small \begin{equation}\label{cond expec} \tag{A}
\#\{1\leq n\leq M, 1\leq x\neq y\leq N: n|a(x)-a(y)|<K \} \ll N^{2-\delta}. \end{equation} Let $\mathcal S(N)$ be the set of integer six-tuples with \[
1\leq y_i\neq x_i\leq N, \quad 1\leq |n_i|\leq M, \quad (i=1,2), \] satisfying \[
| n_1(a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2)) |< K. \] Assume that \begin{equation}\label{lacunary ineq}\tag{B} \#\mathcal S(N) \ll N^{4-\delta}. \end{equation}
\begin{thm}\label{prop:Reduce pc to ineq} Let $\{a(x)\}_{x=1}^\infty$ be a sequence of distinct positive reals. Assume that \eqref{cond expec} and \eqref{lacunary ineq} hold for some $\delta>0$. Then the pair correlation function of $\alpha a(x)$ is Poissonian for Lebesgue almost all $\alpha\in \mathbb{R}$. \end{thm}
In the case of integer-valued sequences, the almost sure convergence of the pair correlation function to the Poisson limit (metric Poisson pair correlation) follows \cite{RudnickSarnak, RZ pc} from a similar bound for the equation $$ n_1 (a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2))=0 $$ See \cite{ALL, BCGW} for a streamlined criterion for metric Poisson pair correlation in terms of the {\em additive energy} of the sequence.
In \S~\ref{sec:lacunary}, we verify that that \eqref{cond expec} and \eqref{lacunary ineq} hold for lacunary sequences, hence obtain Theorem~\ref{thm:lacunary}.
\section{The pair correlation function}\label{sec:pc}
To study the pair correlation function, we use a smooth count cf. \cite{RudnickSarnak}: For $f\in C_c^\infty({\mathbb R})$ or $f$ being an indicator function of a compact interval, set
\[
F_N(x) = \sum_{j\in {\mathbb Z}} f\big(N(x+j)\big)
\]
which is periodic and localized on scale $1/N$.
For a sequence $\{\theta_n\}_{n=1}^\infty\subset {\mathbb R}/{\mathbb Z}$,
we define its pair correlation function by \begin{equation} R_2(f,N)(\{\theta_n\}_{n=1}^\infty)= \frac{1}{N} \sum_{1\leq m\neq n\leq N} F_N(\theta_n-\theta_m). \end{equation} In particular, for a fixed sequence $\{x_n\}_{n=1}^\infty$, we take $\theta_n=\alpha x_n\bmod 1$, and abbreviate the pair correlation function $R_2(f,N)(\{\theta_n\}_{n=1}^\infty)$, having fixed $f$, by $R_2(f,N)(\alpha)=R_2(\alpha)$.
It suffices to restrict $\alpha$ to lie in a fixed finite interval and to consider a smooth average: Let $\rho\in C_c^\infty({\mathbb R})$, $\rho\geq 0$, be a smooth, compactly supported, non-negative weight function, normalized to give a probability density: $ \int_{{\mathbb R}}\rho(\alpha)d\alpha=1$. We define a smooth average
\begin{equation}\label{eq: smooth ave}
\ave{X} = \int_{{\mathbb R}} X(\alpha)\rho(\alpha) \, \mathrm{d}\alpha .
\end{equation}
\subsection{The expected value} \begin{lem}\label{lem:expectation} Let $M=N^{1+\varepsilon}$, $K=N^\varepsilon$ and assume \eqref{cond expec} holds. Then the expected value of $R_2(f,N)(\alpha)$ is \[ \ave{R_2(f,N)} =\int_{-\infty}^\infty f(x)dx +O(N^{-\delta}) \] \end{lem} \begin{proof} Let $f\in C_c^\infty({\mathbb R})$. By using Poisson summation, we have the expansion
\[
F_N(x) = \sum_{j\in {\mathbb Z}} f\big(N(x+j)\big) = \frac 1N\sum_{n\in {\mathbb Z}} \widehat f\Big(\frac nN\Big) e(nx)
\]
with $e(z):=e^{2\pi i z}$, which gives \begin{equation}\label{eq: Fourier expansion of R}
R_2(\alpha) = \frac 1{N^2}\sum_{n\in {\mathbb Z}}
\widehat f\Big(\frac nN\Big) S_{n,N}(\alpha) \end{equation} where $$ S_{n,N}(\alpha) = \sum_{1\leq x \neq y \leq N} e(\alpha n(a(x)-a(y))). $$ Therefore the expected value is \[ \ave{R_2}=\int_{-\infty}^\infty R_2(\alpha)\rho(\alpha)d\alpha = \frac 1{N^2}\sum_{n\in {\mathbb Z}}
\widehat f\Big(\frac nN\Big) \sum_{1\leq x \neq y \leq N}
\^\rho(n(a(x)-a(y))). \] The zero mode $n=0$ gives a contribution of \[ \frac 1{N^2}\^f(0) N(N-1) = \int_{-\infty}^\infty f(x)dx (1+O(1/N)). \]
We split the sum over non-zero modes into two terms: Those with $1\leq |n|\leq M=N^{1+\varepsilon}$, and those with $|n|>M$.
To treat the contribution of modes with $|n|>M=N^{1+\varepsilon}$, we use $|\^f(x)| \ll x^{-A}$ and $|\^\rho|\ll 1$ to bound that term by \[
\frac 1{N^2} \sum_{|n|>M} \Big(\frac nN \Big)^{-A} \sum_{1\leq x\neq y\leq N} 1 = \frac {N^A}{M^{A-1}} \ll \frac{1}{N^{1-\varepsilon}} \] on choosing $A=2/\varepsilon$.
To bound the contribution of modes with $1\leq |n|\leq M$, we separate into a contribution of terms with $|n(a(x)-a(y))|<K$ and the rest.
We use $|\^\rho|, |\^f|\ll 1$ to obtain that the contribution of terms with $|n(a(x)-a(y))|<K$ is \[ \ll \frac 1{N^2}\#\{1\leq n<N^{1+\varepsilon}, 1\leq y\neq x\leq N: n(a(x)-a(y))<N^\varepsilon\}. \] By \eqref{cond expec}, this is $\ll N^{-\delta}$.
The contribution of terms with $|n(a(x)-a(y))|>K$ is bounded using
$$|\^\rho(n(a(x)-a(y)))\ll |n(a(x)-a(y))|^{-A}\leq K^{-A} =N^{-2}$$
and $|\^f|\ll 1$ by \[
\frac 1{N^2}\sum_{\substack{1\leq |n|\leq M\\1\leq x\neq y\leq N\\ |n(a(x)-a(y))|>K}} |\^f(\frac nN)|
\^\rho( n(a(x)-a(y)) |
\ll \frac 1{N^2}\sum_{\substack{1\leq |n|\leq M\\1\leq x\neq y\leq N}} \frac 1{N^2} \leq \frac{M}{N^2}
\] which is $\ll N^{-1+\varepsilon}$. \end{proof}
\subsection{The variance}
\begin{prop} \label{prop:lacunary variance}
Assume that $a(x)$ is a sequence of real numbers such that \eqref{cond expec} and \eqref{lacunary ineq} hold. Then \[
\ave{\Big| R_2(f,N)-\int_{-\infty}^\infty f(x)dx \Big|^2}\ll N^{-\delta}. \]
\end{prop} \begin{proof} By Cauchy-Schwarz, \begin{align*}
\ave{\Big| R_2(f,N)-\int_{-\infty}^\infty f(x)dx \Big|^2} & \leq
2\ave{\Big| R_2(f,N)-\ave{R_2} \Big|^2} \\
& +2 \ave{\Big| \ave{R_2}-\int_{-\infty}^\infty f(x)dx \Big|^2}. \end{align*} By Lemma~\ref{lem:expectation},
$$ \ave{\Big| \ave{R_2}-\int_{-\infty}^\infty f(x)dx \Big|^2}\ll N^{-2\delta}. $$ We now show that \begin{equation}\label{bound on var}
\operatorname{Var} R_2=\ave{\Big| R_2(f,N)-\ave{R_2}\Big|^2} \ll N^{-\delta}
\end{equation} which will prove Proposition~\ref{prop:lacunary variance}.
To prove \eqref{bound on var}, it suffices to show by \eqref{lacunary ineq} that
\begin{equation} \label{lem: variance bounded in terms of counting} \operatorname{Var} R_2 \ll_f \frac{\#\mathcal S(N) }{N^4} \end{equation}
By using the expansion \eqref{eq: Fourier expansion of R}, the variance can be written as
\begin{equation}\label{eq: variance in terms of Fourier coefficients}
\operatorname{Var}(R_{2})=\frac{1}{N^{4}} \sum_{(n_1,n_2) \in \mathbb{Z}^{2}\setminus\{0\}}
\widehat{f}\Big(\frac{n_1}{N}\Big) \widehat{f}\Big(\frac{n_2}{N}\Big)
w(n_{1},n_{2},N).
\end{equation} where for integers $n_1,n_2$, we let $$w(n_{1},n_{2}, N) = \sum_{\substack{1 \leq x_1 \neq x_3 \leq N, \\ 1\leq x_2 \neq x_4 \leq N}} \widehat{\rho}\Big(n_{1} (a(x_3)-a(x_1)) -n_{2}(a(x_4)-a(x_2))\Big)$$ and $\rho$ as in \eqref{eq: smooth ave}.
Due to the rapid decay
of $\widehat{f}$, the contribution from the range in which $|n_{1}|$ or $|n_{2}|$ exceeds $M=N^{1+ \varepsilon}$
is negligible, as we will argue now. We detail only the case $\max\{|n_{1}|,|n_{2}|\} =n_{1} \geq M$, since the other case can be done similarly. We observe the trivial bound $\vert w(n_{1},n_{2},N) \vert \ll N^4$. Moreover, $$n_1=n^{\varepsilon/2}_{1}n^{1-\varepsilon/2}_{1} \geq n^{\varepsilon/2}_{1} N^{1+\varepsilon/2-\varepsilon^{2}/2} $$ which, since $\varepsilon$ is small, yields $n_1 > n^{\varepsilon/2}_{1} N^{1+\varepsilon/3}$. Hence, the contribution to the right hand side of \eqref{eq: variance in terms of Fourier coefficients}
arising from the terms with $\max\{|n_{1}|,|n_{2}|\} =n_{1} \geq M=N^{1+ \varepsilon}$ and $n_2 \neq 0$ is
\begin{align*} \ll \frac 1{N^4} \sum_{\substack{n_{1},n_{2}
\in \mathbb{Z}\setminus\{0\}\\
\vert n_{1} \vert >N^{1+ \varepsilon}}}
\Big(\frac{n_{1}}{N}\Big)^{- 18/\varepsilon}
\sum_{n_2\neq 0} \widehat f\Big(\frac{n_2}{N}\Big)
N^{4}
\ll \frac 1{N^4}.
\end{align*}
Moreover, the terms satisfying $\max\{|n_{1}|,|n_{2}|\} =n_{1} \geq N^{1+ \varepsilon}$, and $n_{2}=0$ are in absolute value
$$
\ll \frac 1{N^4}\sum_{\vert n_{1} \vert \geq N^{1+\varepsilon}}
\widehat{f}\Big(\frac{n_{1}}{N}\Big) N^{4}\ll \frac 1{N^4}.
$$ So, the upshot is that on the right hand side of \eqref{eq: variance in terms of Fourier coefficients}
the sum over all $(n_1,n_2)$ with $\max(|n_1|,|n_2|)>N^{1+\varepsilon}$
contributes $\ll N^{-4}.$ By the rapid decay of $\widehat{\rho}$, we can dispose of the regime where
$ \vert n_1 (a(x_3)-a(x_1))-n_2(a(x_4)-a(x_2)) \vert \geq N^{\varepsilon}.
$
By bounding $\hat{\rho}$ trivially, we find that \[ \operatorname{Var} R_2 \ll \frac{\#\mathcal S(N)}{N^4} + O\Big(\frac 1{N^4}\Big) . \] Since $\#\mathcal S(N)\geq N^3$, we obtain \eqref{lem: variance bounded in terms of counting}. \end{proof}
\section{Almost everywhere convergence: Proof of Theorem~\ref{prop:Reduce pc to ineq}}
We now deduce almost everywhere convergence
from a polynomial variance bound.
\subsection{Preparations}\label{secion: preparations}
We need a general property of the pair correlation function. Recall that for any sequence of points
$\{\theta_n\}_{n=1}^\infty \subset {\mathbb R}/{\mathbb Z}$, we defined
\[
R_2(f,N) = \frac 1N \sum_{1\leq j\neq k\leq N} F_N(\theta_j-\theta_k)
\]
with $ F_N(x) = \sum_{j\in {\mathbb Z}} f(N(x-j))$.
\begin{lem}\label{lem: conv subsequences}
Suppose there is a strictly increasing sequence
$\{ N_m \}_{m=1}^\infty \subseteq \mathbb{Z}_{\geq 1} $, with
\[ \lim_{m\to \infty} \frac{N_{m+1}}{N_m} = 1
\]
so that for all $f\in C_c^\infty({\mathbb R})$,
\begin{equation}\label{convergence on subsequence}
\lim_{m\to \infty} R_2(f,N_m) = \int_{-\infty}^\infty f(x)dx .
\end{equation}
Then we can pass from the sub-sequence to the set of all integers: \begin{equation}\label{convergence on full sequence}
\lim_{N\to \infty} R_2(f,N) = \int_{-\infty}^\infty f(x)dx
\end{equation}
for all $f\in C_c^\infty({\mathbb R})$.
\end{lem}
\begin{proof} We will first deduce that \eqref{convergence on subsequence} holds for the indicator functions
\[
I_s(x) = \begin{cases} 1,& |x|<s/2,\\ 0,&{\rm otherwise}, \end{cases}
\]
by approximating with smooth functions, and show that \eqref{convergence on full sequence} holds for the functions $I_s$, and then deduce by approximating a general even smooth $f\in C_c^\infty({\mathbb R})$ by linear combinations of $I_s$
that \eqref{convergence on full sequence} holds for all such $f$.
Note that for odd smooth $f\in C_c^\infty({\mathbb R})$, we have
$F_N(-x)=-F_N(x)$ which entails $R_2(f,N)=0$,
so the pair correlation function $R_2(f,N)$
converges trivially to the right limit.
From the definition of $R_2(I_s,N)$ we have a monotonicity property: Let $0<\varepsilon<1$. If $ (1-\varepsilon)N'<N<N'$ and $N''<N<(1+\varepsilon)N''$ then
\begin{equation}\label{monotonicity prop}
(1-\varepsilon) R_2(I_{(1-\varepsilon)s} ,N'') \leq R_2(I_s,N) \leq \frac 1{1-\varepsilon} R_2(\ I_{s/(1-\varepsilon)},N') .
\end{equation}
Indeed, using positivity of $ I_s$ (hence of $F_N$)
\[
N\cdot R_2(I_s,N) = \sum_{1\leq j\neq k\leq N} F_N(\theta_j-\theta_k) \leq \sum_{1\leq j\neq k\leq N'} F_N(\theta_j-\theta_k) .
\]
Now if $1>N/N'\geq 1-\varepsilon>0$ then since $I_s$ is even and decreasing on $[0,\infty)$, we have
\[
I_s(Ny) = I_s\Big(N' y\cdot \frac{N}{N'}\Big)
\leq I_s(N'y(1-\varepsilon)) = I_{s/(1-\varepsilon)}(N'y ).
\] So
\[
F_N(x) = \sum_{j\in {\mathbb Z}} I_s(N \cdot(x-j))
\leq \sum_{j\in {\mathbb Z}} I_{s/(1-\varepsilon)} (N'\cdot(x-j))
= \tilde F_{N'}(x) \] where $\tilde F_{N'}(y) = \sum_{j\in {\mathbb Z}} I_{s/(1-\varepsilon)}({N'}(y-j))$. Hence
\[
R_2(I_s,N) \leq \frac{N'}{N} R_2(I_{s/(1-\varepsilon)},N') \leq \frac 1{1-\varepsilon} R_2(I_{s/(1-\varepsilon) },N') \] which proves the upper bound in \eqref{monotonicity prop}. The lower bound of \eqref{monotonicity prop} follows from switching the roles of $N$ and $N''$ and inserting in the upper bound.
Next, fix $\varepsilon\in (0,1)$ small, let $N\gg 1$, and take $m\gg 1$ so that
\[
N_m<N_{m+1}<(1+\varepsilon)N_m
\]
and so if $N_m\leq N<N_{m+1}$ then
\[
(1-\varepsilon)N_{m+1} <N < N_{m+1}, \quad N_{m}\leq N<(1+\varepsilon)N_m .
\]
Then for all $s>0$, we have
\[
(1-\varepsilon) R_2(I_{(1-\varepsilon)s}, N_m) \leq R_2(I_s,N) \leq \frac 1{1-\varepsilon} R_2(I_{s/(1-\varepsilon)},N_{m+1}) .
\]
Taking $m\to \infty$, we find by \eqref{convergence on subsequence}
\[
\limsup_{N\to \infty} R_2(I_s,N) \leq \frac 1{1-\varepsilon} \int_{-\infty}^\infty I_{s/(1-\varepsilon)} dx = \frac{s}{(1-\varepsilon)^2}
\]
and
\[
\liminf_{N\to \infty} R_2(I_s,N) \geq (1-\varepsilon) \int_{-\infty}^\infty I_{ (1-\varepsilon)s} dx = (1-\varepsilon)^2s .
\]
Since $\varepsilon>0$ is arbitrary, we finally obtain
\[
\lim_{N\to \infty} R_2(I_s,N) = s = \int_{-\infty}^\infty I_s(x)dx
\]
so that \eqref{convergence on full sequence} holds for all indicator functions $I_s$.
Therefore \eqref{convergence on full sequence} holds for all test functions $f\in C_c^\infty({\mathbb R})$.
\end{proof}
\subsection{Proof of Theorem~\ref{prop:Reduce pc to ineq}}
It suffices to show that for almost every $\alpha$
in a fixed compact interval $I$ we have
\begin{equation}\label{full conv} \lim_{N\to \infty}R_2(f,N)(\alpha)=\int_{-\infty}^\infty f(x)dx
\end{equation}
for all $f\in C_c^\infty({\mathbb R})$.
Let $\rho\in C^\infty_c({\mathbb R})$ be a non-negative function majorizing the indicator function of the interval $I$: $\mathbf 1_{I}\leq \rho$. Then from the variance bound of Proposition~\ref{prop:lacunary variance}, we find that
for some $ \delta>0$, for all $f\in C_c^\infty({\mathbb R})$,
\[
\int_{I} \Big |R_2(f,N_m)(\alpha)-\int_{-\infty}^\infty f(x)dx \Big|^2 \rho(\alpha)d\alpha \ll_f N^{-\delta} .
\]
Hence for the sequence
\[
N_m = \lfloor m^{2/\delta} \rfloor
\]
we have that for almost all $\alpha \in I$,
\begin{equation}\label{subsequential conv} \lim_{m\to \infty} R_2(f,N_m)(\alpha) = \int_{-\infty}^\infty f(x)dx
\end{equation}
for all $f$. Indeed, for each fixed $f$ set
\[
X_m (\alpha) = \Big|R_2(f,N_m)(\alpha)-\int_{-\infty}^\infty f(x)dx \Big|^2 .
\] Then
\[
\int_{I} X_m(\alpha)d\alpha \leq \int_{-\infty}^{\infty} X_m (\alpha) \rho(\alpha)d\alpha \ll \frac 1{N_m^{\delta}} \ll \frac 1{m^2} .
\]
Therefore
\[
\int_{I} \Big( \sum_{m\geq 1} X_m(\alpha) \Big) d\alpha \leq \sum_{m\geq 1} \int_{-\infty}^\infty X_m(\alpha) d\alpha \ll_f \sum_{m\geq 1} \frac 1{m^2}<\infty
\]
so that $\sum_{m\geq 1} X_m(\alpha)$ converges
for almost all $\alpha\in I$. Thus
$$\lim_{m\to \infty}X_m(\alpha)=0$$
for almost all $\alpha$, i.e. \eqref{subsequential conv} holds for our specific $f$ for almost all $\alpha\in I$.
By a diagonalization argument (see \cite{RudnickSarnak}) there is a set of $\alpha$ whose complement has measure zero so that \eqref{subsequential conv} holds for all $f$. Since $N_{m+1}/N_m\to 1$,
we can use Lemma~\ref{lem: conv subsequences} to deduce \eqref{full conv} holds, proving Theorem~\ref{prop:Reduce pc to ineq}.
\section{Lacunary sequences}\label{sec:lacunary}
From now on, we assume that $\{a(x)\}_{x=1}^\infty$
is a lacunary sequence of (strictly) positive reals,
that is there is some $C>1 $ so that for all integers $x\geq 1$, \[ a(x+1)\geq C a(x) \] for all $x\geq 1$. Consequently, we have for all $x\geq y\geq 1$ that \[ a(x)\geq C^{x-y} a(y) . \] We will show that \eqref{cond expec} and \eqref{lacunary ineq} hold, hence proving that the pair correlation function of $\{\alpha a(x) \;\operatorname{mod} 1\}_{x=1}^\infty$ is Poissonian for almost all $\alpha$, that is Theorem~\ref{thm:lacunary}.
\subsection{The condition \eqref{cond expec}}
\begin{lem} Assume that the sequence $\{a(x)\}_{x=1}^\infty$ is lacunary. Then \eqref{cond expec} holds, in fact with a bound of \[
\#\{1\leq n\leq M, 1\leq x\neq y\leq N: n|a(x)-a(y)|<K \} \ll N^{2\varepsilon}.
\] \end{lem} \begin{proof} Since the sequence is lacunary, we have for $y<x$ that \[ a(x)-a(y) \geq a(x)\Big(1-\frac 1{C^{x-y}}\Big)\geq C^x\Big(1-\frac 1{C^{x-y}}\Big) \] which is $\gg N^\varepsilon$ as soon as $x\geq \epsilon \log_C N$ where $\log_C N=(\log N)/\log C$ . Hence to satisfy the inequality we must have $n<N^\varepsilon$, and $y<x\ll \log N$, so that we have at most $N^{2\varepsilon}$ solutions.
\end{proof}
\subsection{The condition \eqref{lacunary ineq}}
\begin{prop}\label{prop:counting} Assume that $\{a(x)\}_{x=1}^\infty$ is a lacunary sequence of positive real numbers, and that $N^\gamma\ll M\ll N^\Gamma$ for some $0<\gamma<\Gamma<2$. Then \[ \#\mathcal S(N) \ll MN^2(\log M)^2. \] \end{prop} In view of Theorem~\ref{prop:Reduce pc to ineq}, we deduce Theorem~\ref{thm:lacunary}.
\begin{proof} The proof is a modification of \cite[Proposition 2]{RZ pc}: We are given the inequality \begin{equation}\label{lacunary ineq2}
| n_1 (a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2)) | < K. \end{equation} We may assume that $n_i>0$, and $1\leq y_i<x_i\leq N$, $i=1,2$, and that $x_1\geq x_2$. In particular, we may then assume that $x_1\geq 4\log_C M \gg \log N$, because the number of such tuples with $x_1\ll \log N$ is at most $O(M^2(\log N)^4)$, which is admissible (that is, $o(MN^2(\log N)^2)$) if $M=O(N^\Gamma)$ for $\Gamma<2$.
We fix $n_1,x_1,y_1$, and first show that (recall $x_1\geq x_2$) \begin{equation}\label{diff between s_i} x_1 - x_2 \leq 2\log_C M . \end{equation} Indeed, we have a lower bound \[
n_1(a(x_1)-a(y_1)) \geq 1\cdot (a(x_1)-a(x_1-1))
\geq \Big(1-\frac 1C \Big) a(x_1) \] (since $y\leq x_1-1$), and an upper bound \[ n_2(a(x_2)-a(y_2)) \leq M a(x_2) =a(x_1)M \frac{a(x_2)}{a(x_2+(x_1-x_2))} \leq \frac{M}{C^{x_1-x_2}}a(x_1) \] since $a(x+h)\geq C^h a(x)$ for $h\geq 1$. Hence \[ n_1(a(x_1)-a(y_1)) - n_2(a(x_2)-a(y_2)) \geq \Big(1-\frac 1C\Big) a(x_1) - \frac{M}{C^{x_1-x_2}}a(x_1) \] Assuming that $ x_1-x_2>2\log_C M $ gives in particular \[ 1-\frac 1C -\frac{M}{C^{x_1-x_2}} >1-\frac 1C-\frac 1M > \frac 12 \Big(1-\frac 1C\Big)>0 \] for sufficiently large $N$. The condition \eqref{lacunary ineq2} now forces \[ \frac 12(1-\frac 1C)< (1-\frac 1C) - \frac{M}{C^{x_1-x_2}} \leq \frac K{a(x_1)} \ll \frac K{C^{x_1}} \] which forces $x_1 \ll \log_C K\leq \varepsilon \log M$, which we assumed was not the case. Thus we may assume that $x_1-x_2>2\log_C M$, which forces $x_2\geq 2\log_CM$ since $x_1>4\log_C M$.
Now fix $x_2$ as well; then $n_2$ will be determined by $y_2$, because \[ n_2 = \frac{n_1 (a(x_1)-a(y_1))}{a(x_2)-a(y_2)} + O\Big(\frac K{ a(x_2)-a(y_2)}\Big) \] and since $a(y)$ is lacunary, $K/(a(x_2)-a(y_2)) =o(1)$ if $ y_2 \geq \log_C N$, because \[
a(x_2)-a(y_2) \geq a(x_2)-a(x_2-1)\geq a(x_2)(1-\frac 1C) \gg C^{x_2}\geq M^2
\] since $x_2>2\log_CM$.
So we will be done if we show that there is at most one choice of $y_2$ such that $x_2-y_2>2\log_C M$. Indeed, if there are two pairs $(y_2,n_2)$ and $(y_2',n_2')$ for which \eqref{lacunary ineq2} holds (recall all other variables are now fixed), with $x_2-y_2>2\log_C M$, $x_2-y_2'>2\log_C M$, then since \[ a(y_2) \leq \frac{a(x_2)}{C^{x_2-y_2}}\leq \frac{a(x_2)}{M^2} \] we find that \eqref{lacunary ineq2} implies \[ \begin{split} n_1(a(x_1)-a(y_1)) &= n_2(a(x_2)-a(y_2)) +O(K) \\ &= n_2 a(x_2)\Big( 1+ \frac{a(y_2)}{a(x_2)} +O\Big(\frac K{n_2a(x_2)}\Big)\Big) \\ &= n_2 a(x_2)\Big( 1+ O\Big(\frac{K}{M^2}\Big)\Big) \end{split} \] since $n_2a(x_2)\geq a(x_2)\geq C^{x_2} \gg M^2$ if $x_2\geq 2\log_C M$, and $a(x_2)/a(y_2)\geq C^{x_2-y_2}\gg M^2$. If $n_2',y_2'$ is another such pair then we also find \[ n_1(a(x_1)-a(y_1)) = n'_2 a(x_2) \Big( 1+ O \Big(\frac{K}{M^2} \Big)\Big) \] so that \[ n_2 a(x_2)\Big( 1+ O\Big( \frac{K}{M^2} \Big)\Big) = n'_2 a(x_2)\Big( 1+ O\Big( \frac{K}{M^2}\Big)\Big) \] which gives \[ n_2'=n_2 \Big( 1+ O\Big(\frac{K}{M^2}\Big)\Big) = n_2 + O\Big( \frac{K}{M}\Big) \] since $n_2\leq M$. Thus for $M\gg N^\gamma$, while $K\ll N^\varepsilon = o(M)$, we obtain $n_2'=n_2$. \end{proof}
\end{document} | arXiv | {
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\begin{document}
\title{Verified computations for\\ closed hyperbolic 3-manifolds} \author{Matthias Goerner} \email{enischte@gmail.com} \urladdr{http://unhyperbolic.org/}
\subjclass[2010]{57M50, 65G40}
\begin{abstract} Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3-manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann-Zagier and Moser for ideal triangulations upon which HIKMOT is based) showing that there is a redundancy among the edge equations if the edges avoid ``gimbal lock''. We successfully test the algorithm on known examples such as the orientable closed manifolds in the Hodgson-Weeks census and the bundle census by Bell. We also tackle a previously unsolved problem and determine all knots and links with up to 14 crossings that have a hyperbolic branched double cover. \end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
Up to isometry, a finite hyperbolic 3-simplex is determined by its $6$ edge parameters, by which we mean either the edge lengths $l_{ij}$ or the respective entries of its vertex Gram matrix $v_{ij}=-\cosh(l_{ij})$. Thus, an assignment of a parameter to each edge of a finite triangulation $\trig$\!{} of a closed 3-manifold determines a hyperbolic structure for each $3$-simplex of $\trig\!{}$. If certain conditions are fulfilled, the hyperbolic structures on the individual simplices are compatible and form a hyperbolic structure on the manifold (see \cite{heardThesis} and Section~\ref{sec:hypStru}).
Existing software (such as Casson's Geo \cite{casson:geo} and Heard's Orb \cite{orb}) finds a numerical approximation for the edge parameters using Newton's method and reports whether the necessary equations are fulfilled within an error smaller than a certain $\varepsilon$. This suggests but does not prove hyperbolicity. The aim of this paper is to describe how to take such a numerical approximation and rigorously prove hyperbolicity by giving real intervals that are verified to contain a solution to all the necessary equations and inequalities. An algorithm either returning such intervals or (conservatively) reporting failure is described in Section~\ref{sec:algo}. The algorithm is a hyperbolicity verification procedure but not a hyperbolicity decision procedure since its failure just means that the given candidate approximation was not close enough to a hyperbolic structure or needs to be perturbed to avoid ``gimbal lock'' (explained below). An implementation of this algorithm is available at \cite{veriClosedRepo}. Therefore, this paper is achieving for finite triangulations what Hoffman, Ichihara, Kashiwagi, Masai, Oishi, and Takayasu \cite{hikmot} did for ideal triangulations (HIKMOT's functionality has been integrated into SnapPy \cite{SnapPy} by the author since version~2.3).
This is motivated by applications that benefit from using geometric finite triangulations in place of geometric spun triangulation. In particular, such applications no longer need to overcome the incompleteness locus. An example is the generation of cohomology fractals for a closed hyperbolic 3-manifold. As shown in \cite[Figure~8.7]{cohomologyFractals}, the incompleteness locus produces artifacts in the raytraced image of a cohomology fractal which simply disappear when using finite triangulations instead. Another example is the algorithm proposed in \cite{HHGT} to rigorously compute the length spectrum for a hyperbolic 3-manifold. This algorithm requires tiling $\H^3$ with translates of a fundamental domain to cover a ball of specified radius and thus would not work if there is incompleteness locus.
An even more basic motivation is proving hyperbolicity of a closed 3-manifold by finding a triangulation admitting a geometric structure. Restricting ourselves to just spun triangulations introduces a bottleneck. For example, the obvious spun triangulation of a closed census manifold such as \texttt{m135(1,3)} can fail to be geometric. Thus finding a geometric spun triangulation requires drilling and filling (or, in other words, finding a different closed geodesic $\gamma$ such that there is a geometric triangulation spun about $\gamma$). Even worse, some hyperbolic 3-manifolds such as \texttt{m007(3,1)} seem to lack any geometric spun triangulation unless we pass to a cover\footnote{In not yet published work, Maria Trnkova has proven that there is no geometric spun triangulation of \texttt{m007(3,1)} with a small number of tetrahedra.}. Note that a geometric spun triangulation is known for every orientable closed manifold in the SnapPy census (except for \texttt{m007(3,1)} where a 3-fold cover is needed), see \cite{hikmot}. However, the process of finding such (covers admitting) geometric spun triangulations can be tedious and is not known to be possible in general. Furthermore, passing to a cover can complicate applications such as computing the length spectrum.
Potential future work might generalize the techniques of this paper to Heard's work \cite{heardThesis} on 3-orbifolds and Frigerio and Petronio's work \cite{frigPet} on 3-manifolds with geodesic boundary. To find hyperbolic structures on these, Heard's program Orb \cite{orb} uses triangulations with finite as well as ideal and ``hyperinfinite'' vertices. Note that some of the theory in this paper also carries over to spherical and Euclidean geometry and might generalize to yield methods for verifying spherical or Euclidean structures on finite triangulations.
Like \cite{hikmot}, we use interval arithmetic methods such as the interval Newton method or the Krawczyk test. These methods can only show the existence of a solution to a system of equations if the Jacobian matrix is invertible near that solution. If the Jacobian fails to be invertible, these methods can only show the existence of a solution to a subset of the equations. This applies to the edge equations whether we are solving for shapes in the cusped case or for edge lengths in the closed case. Hence, in both cases, we need an additional result showing that there is a redundancy among the edge equations such that solving a suitable subset of them is sufficient. For ideal triangulations, this result is due to Neumann-Zagier \cite{NeumannZagier,NeumannComb} and Moser \cite{moser} (see Appendix). For finite triangulations, we derive such a result in this paper.
Note that while we actually have exactly as many variables as equations in the case of finite triangulations (namely, one per edge), the Jacobian of this system of equations has a kernel at a solution corresponding to a hyperbolic structure. This is because we can move each individual finite vertex of a triangulation in the hyperbolic manifold and obtain a whole family of solutions (see Theorem~\ref{thm:hypStructAreSubmanifold}). Thus, we need to use a two-step strategy to verify a hyperbolic structure: First, we drop some edge equations and fix an equal number of edge parameters such that we can apply interval arithmetic methods to find intervals verified to contain a solution to the subsystem of equations we kept. Next, we show that this solution is also a solution to the equations we dropped earlier and thus that the intervals for the edge parameters contain a point giving a hyperbolic structure. Interval arithmetic can verify that the error of the dropped equations is small and we will show that if the dropped equations are fulfilled approximately, then they are fulfilled exactly provided that a certain condition we call ``gimbal lock'' is avoided.
To define gimbal lock, we will look at the complex of doubly-truncated simplices associated to the triangulation and an assignment of $\myPGL{2}{\C}$-matrices to the edges of the complex computed from the edge parameters (see Section~\ref{sec:cocycles}). The cocycle condition says that the matrices on the edges of a polygon must multiply to the identity. Since a subset of the edge equations is known to be fulfilled, the cocycle condition is known to hold for some polygons but not necessarily for others. The goal is to show that it holds for all polygons so that we get a $\myPGL{2}{\C}$-representation of the fundamental group (see Section~\ref{sec:vertexCocycles} and \ref{sec:extendCocycles} and examples in Section~\ref{section:examples}). Roughly speaking, the idea is that if the product of three small rotations about three axes in generic position is the identity, then each rotation must be the identity. Inspired by the mechanical device called gimbal (see Figure~\ref{fig:gimbal}), we say that we avoid ``gimbal lock'': if a gimbal is not in its locked position, then we can apply any small rotation to the inner-most ring, or equivalently, if we fix the inner-most ring, none of the other rings can be turned.
We describe the resulting algorithm to obtain real intervals in Section~\ref{sec:algo}. The algorithm is effective and able to verify a hyperbolic structure on all 36093 closed orientable manifolds in the Hodgson-Weeks census \cite{hwcensus} and in the census bundle by Bell \cite{bellBundleCensus}, see Section~\ref{sec:results}. Branched double covers of knots or links (or more precisely: double covers of $S^3$ branched over a knot or link) provide a good class of test cases since finding a geometric spun-triangulation of some of them can be challenging. Using finite triangulations instead, we are able to prove the following new result: \begin{theorem} \label{thm:hyperbolicBranchedDoubleCovers} Out of the 313230 knots with up to 15 crossings (not including the unknot), exactly 193839 have a hyperbolic branched double cover.\\ Out of the 120573 links with up to 14 crossings (with at least two components), exactly 37709 have a hyperbolic branched double cover. \end{theorem}
Section~\ref{sec:discussion} concludes with a conjecture that implies that a hyperbolic structure on a finite triangulation can always be perturbed so that the algorithm can verify it.
The appendix in Section~\ref{sec:hikmotGap} points out a gap in the argument (but not the algorithm) of the HIKMOT paper.
\section{Hyperbolic structures on finite triangulations} \label{sec:hypStru}
Consider the isometry class of a positively oriented, finite geodesic simplex $\Delta$ with vertices labeled $0, \dots, 3$ in $\H^3$. We briefly review the relationship of the edge lengths and the angles of $\Delta$ following \cite{heardThesis} with one difference though: we use a slightly simpler definition for the vertex Gram matrix $G$ where all diagonal entries are $-1$ since we are not interested in generalized simplices here. To be consistent with the vertex labels, we $0$-index the rows and columns of a matrix (so $m_{00}$ denotes the top left-most entry).
\begin{figure}
\caption{Angles of simplex.}
\label{fig:anglesInTet}
\caption{A doubly truncated simplex $\overline{\Delta}$, also known as permutahedron.}
\label{fig:DoublyTruncated}
\caption{A prism.}
\label{fig:Prism}
\end{figure}
Let $l_{ij}$ denote the length of the edge between vertex $i$ and $j$. The vertex Gram matrix $G$ associated to the simplex is the symmetric $4\times 4$-matrix with entries $$v_{ij}=-\cosh(l_{ij}).$$ The edge lengths as well as the vertex Gram matrix uniquely determine the isometry class of the simplex. Let $c_{ij}$ denote the respective cofactor of $G$ which is given by $$c_{ij}=(-1)^{i+j} \det(G_{ij}), $$ where $G_{ij}$ is obtained by deleting the $i$-th row and $j$-th column. The dihedral angle between face $i$ and $j$ and the angle at vertex $i$ of the triangle $ijk$ (derived from the law of cosines) are then given by (also see Figure~\ref{fig:anglesInTet}): \begin{equation} \theta_{ij}=\arccos\left(\frac{c_{ij}}{\sqrt{c_{ii}c_{jj}}}\right) \quad\mbox{and}\quad \eta_{i,jk}=\arccos\left(\frac{v_{ij}v_{ik}+v_{jk}}{\sqrt{v_{ij}^2-1}\sqrt{v_{ik}^2-1}}\right). \label{eqn:dihedral} \end{equation}
\begin{definition} Let $G$ be a real symmetric $4\times 4$-matrix with $-1$ on the diagonal. We say that $G$ is realized if $G$ is the vertex Gram matrix of some finite, non-flat simplex. \end{definition}
The following theorem is a special case of \cite[Theorem~1.5]{heardThesis} (also compare to \cite[Theorem~7.2.2]{ratcliffe:hyp}): \begin{lemma} $G$ is realized if and only if \begin{enumerate} \item $G$ has one negative and three positive eigenvalues (which is equivalent to the characteristic polynomial $p_G(x)=\det(x I - G)=x^4+4x^3+a_2x^2+a_1x+a_0$ having coefficients $a_2 < 0$, $a_1>0$, $a_0<0$ by the Budan-Fourier theorem \cite{algRealGeom}), \item $c_{ii}<0$ for all $i$, and\label{item:condNegAdj} \item $c^2_{ij} < c_{ii} c_{jj}$ for all $i$ and $j$,\label{item:wellDefinedCos} \end{enumerate} where $c_{ij}$ denotes the respective cofactor of $G$. \label{lemma:singleGeomSimp} \end{lemma}
Let $\trig$\!{} be an oriented, finite 3-dimensional triangulation (i.e., all vertex links are $2$-spheres) and let $E(\trig)$ denote the set of edges of $\trig$\!{}. Assume we have an assignment of a length $l_e>0$, or equivalently, a parameter $\nu_e < -1$ to each $e\in E(\trig)$ where the two are related by the formula $\nu_e=-\cosh(l_e)$. This induces a symmetric $4\times 4$-matrix $G_\Delta$ for each simplex $\Delta$ of $\trig$\!{} where $v_{ii}=-1$ and $v_{ij}=\nu_e$ if $i\not=j$ and the edge of $\Delta$ from vertex $i$ to $j$ is incident to $e$. Let $\Theta_e$ denote the sum of all dihedral angles $\theta_{ij}$ incident to the edge $e$ of $\trig$\!{}.
We can now use \cite[Lemma~2.4]{heardThesis} to check whether this assignment yields a hyperbolic structure on $\trig$: \begin{theorem} An assignment of a parameter $\nu_e<-1$ to each edge $e$ of an oriented, finite triangulation $\trig$\!{} induces a hyperbolic structure on $\trig$\!{} if \begin{enumerate} \item \label{main:condA} each matrix $G_\Delta$ is realized (i.e., fulfills the conditions of Lemma~\ref{lemma:singleGeomSimp}) and \item \label{main:condB} $\Theta_e=2\pi$ for every $e\in E(\trig)$. \end{enumerate} \label{thm:hypStruct} \end{theorem}
\section{Cocycles} \label{sec:cocycles}
We will describe how to compute a representation\footnote{For consistency on how arrows in a category compose, the loop traversing the loop $a$ first and the loop $b$ second is denoted by $ba$ in $\pi_1(\trig)$.} $\pi_1(\trig)\to\myPGL{2}{\C}$ from an assignment of parameters $\nu_e$ as in Theorem~\ref{thm:hypStruct} using cocycles inspired by \cite[Section~9]{higherGluingEqns} as follows: \begin{definition} Let $\groupG$ be a group and $X$ be a space with a polyhedral decomposition. A $\groupG$-cocycle on $X$ is an assignment of elements in $\groupG$ to the oriented edges of $X$ such that the product around each face is the identity and such that reversing the orientation of an edge replaces the labeling by its inverse. \end{definition}
All cocycles in this section are $\myPGL{2}{\C}$-cocycles.
Given a matrix $G$ fulfilling the conditions in Lemma~\ref{lemma:singleGeomSimp}, we will construct a cocycle on the doubly truncated simplex $\overline{\Delta}$ coming from a simplex $\Delta$, see Figure~\ref{fig:DoublyTruncated}. We index a vertex $v$ of $\overline{\Delta}$ by the permutation $\sigma\in \permS_4$ such that the vertex of $\Delta$ closest to $v$ is $\sigma(0)$, the edge of $\Delta$ closest to $v$ is $\sigma(0)\sigma(1)$, and the face of $\Delta$ closest to $v$ is $\sigma(0)\sigma(1)\sigma(2)$. We label an oriented long, middle, or short edge of $\overline{\Delta}$ by $\alpha^{\sigma(0)\sigma(1)\sigma(2)}, \beta^{\sigma(0)\sigma(1)\sigma(2)},$ respectively $\gamma^{\sigma(0)\sigma(1)\sigma(2)}$ if it starts at the vertex indexed by $\sigma$. Let us introduce the notion of standard position to define the $\myPGL{2}{\C}$-matrices assigned to these edges.
\begin{figure}
\caption{A simplex in different standard positions.}
\label{fig:standardPos}
\end{figure}
We think of the upper half space model of hyperbolic 3-space as a subset $\H^3=\{w = z+t\myJ : t>0\}$ of the quaternions. Recall that $\mathrm{Isom}^+(\H^3)\cong\myPGL{2}{\C} \cong\myPSL{2}{\C}$ where the action of a $\mySL{2}{\C}$-matrix on $\H^3$ is given by $$\left(\begin{array}{cc}a &b\\ c&d\end{array}\right)\mapsto \left( w \mapsto (aw+b)\cdot(cw+d)^{-1} \right).$$
\begin{definition} \label{def:standardPosSimplex} We say that a positively oriented, finite simplex $\Delta$ is in $\sigma$-standard position where $\sigma\in \permS_4$ if vertex \begin{itemize} \item $\sigma(0)$ is at $\myJ$, \item $\sigma(1)$ at $t\myJ$ with $t>1$, \item $\sigma(2)$ at $a+t\myJ$ with $a>0$ and \item $\sigma(3)$ at $z+t\myJ$ with $\myIm(z)>0$ if $\sigma$ is even and $\myIm(z)<0$ otherwise. \end{itemize} \end{definition} An example of this is shown in Figure~\ref{fig:standardPos}. Geometrically, the motivation for this definition is that two faces of two (not necessarily distinct) simplices line up in $\H^3$ if the respective edge lengths match and the two simplices are in the respective standard positions. More precisely, let $f_1\in\{0,1,2,3\}$ be a face of the simplex $\Delta_1$ and $f_2\in\{0,1,2,3\}$ of $\Delta_2$. Let $\sigma\in \permS_4\setminus \permA_4$ with $\sigma(f_1)=f_2$ be a pairing of the two faces. If the edge lengths match under this pairing, then the faces match if each $\Delta_k$ is in $\sigma_k$-standard position where $\sigma_1$ is any permutation with $\sigma_1(3)=f_1$ and $\sigma_2=\sigma\circ \sigma_1$.
This definition also gives us a cocycle as follows (also see Figure~\ref{fig:standardPos}):
\begin{definition} Consider the isometry class of a finite simplex $\Delta$. Given an oriented edge $e$ of $\overline{\Delta}$, let $\sigma$ and $\sigma'$ be the permutations that index the vertex where $e$ starts and, respectively, ends. The natural cocycle on $\overline{\Delta}$ is the cocycle assigning to each edge $e$ the $\myPGL{2}{\C}$-matrix taking $\Delta$ from $\sigma$-standard position to $\sigma'$-standard position. \end{definition}
Note that we can identify Euclidean 3-vector space isometrically with the tangent space of a point in $\H^3$ such that the tangents corresponding to the $x$-, $y$-, and $z$-axis are parallel to the real line, the imaginary line, respectively, the line $t\myJ$ (see Figure~\ref{fig:standardPos}). Thus, we can associate a $\mySO{3}$-matrix to an element in $\Isom^+(\H^3)$ fixing a (finite) point of $\H^3$. Let $$R_\omega = \left(\begin{array}{ccc} \cos\omega & -\sin\omega & 0 \\ \sin\omega & \phantom{-}\cos\omega & 0\\0 &0 & 1\end{array}\right)$$ be the rotation about the $z$-axis by the angle $\omega$.
\begin{lemma} The natural cocycle on $\overline{\Delta}$ can be computed from the vertex Gram matrix $G$ as follows (apply even permutations $\sigma\in \permA_4$ to obtain labels for all edges): $$\alpha^{120}=\alpha^{210}=\alpha^{123}=\alpha^{213}=\left(\begin{array}{cc}0 & \sqrt{v_{12}^2-1} - v_{12}\\1 & 0 \end{array}\right),$$ $$\beta^{123}=\beta^{132}=\left(\begin{array}{cc}-\cos(\eta_{1,32}/2) & \sin(\eta_{1,32}/2)\\ \phantom{-}\sin(\eta_{1,32}/2) & \cos(\eta_{1,32}/2)\end{array}\right),$$ $$\gamma^{123}=\left(\gamma^{120}\right)^{-1}=\gamma^{210}=(\gamma^{213})^{-1}=\left(\begin{array}{cc}\myExp{\myI \theta_{03}} & 0 \\ 0 & 1\end{array}\right).$$ Note that $\beta^{123}$ and $\gamma^{123}$ fix the point $\myJ\in\H^3$ and the associated $\mySO{3}$-matrices are: $$\left(\begin{array}{ccc} -\cos\eta_{1,32} & \phantom{-}0 & \sin \eta_{1,32} \\
0 & -1 & 0 \\
\phantom{-}\sin\eta_{1,32} & \phantom{-}0 & \cos\eta_{1,32} \end{array}\right)\quad\mbox{and}\quad R_{\theta_{03}}. $$ \label{lemma:coycleAssignment} \end{lemma}
\begin{proof} $\alpha^{120}$ is an involution exchanging the points $\myJ$ and $\myExp{l_{12}}\myJ$. The associated $\C P^1$-automorphism is of the form $z\mapsto x/z$ and exchanges $1$ and $\myExp{l_{12}}$. An elementary calculation gives the value for $x$.\\ Consider the hyperbolic plane $\H^2=\{x+t\myJ : x\in \R, t>0\}\subset \H^3$. $\beta^{123}$ is the composition of the rotation of $\H^2$ about $\myJ$ by $\eta_{1,32}$ with the involution fixing the line $t\myJ$ pointwise. We obtain the $\myPSL{2}{\R}$-matrix for the rotation of $\H^2$ by conjugating the rotation of the unit disk by $\eta_{1,32}$ with the matrix taking the upper half plane model to the Poincare disk model of hyperbolic 2-space: $$\left(\begin{array}{cc}\myI & 1\\ 1 & \myI\end{array}\right)^{-1} \cdot \left(\begin{array}{cc} \myExp{\myI \eta_{1,32}/2} & 0 \\ 0 & \myExp{-\myI \eta_{1,32}/2}\end{array}\right)\cdot \left(\begin{array}{cc}\myI & 1\\ 1 & \myI\end{array}\right)$$ \end{proof}
Given an oriented triangulation $\trig$\!{}, let $\overline{\trig}$ be the complex obtained by replacing each simplex $\Delta$ by the double truncated simplex $\overline{\Delta}$. The long and short edges of $\overline{\trig}$ about an edge $e$ of $\trig$\!{} form a prism, see Figure~\ref{fig:Prism}. Given an assignment of edge parameters $\nu_e$ for $\trig$\!{}, this prism about an edge $e\in E(\trig)$ is a cocycle if and only if the short edges compose to the identity, which is equivalent to $\Theta_e$ being a multiple of $2\pi$. Let $\hat\trig$\!{} denote the complex $\overline{\trig}\cup\mbox{Prisms}$.
\begin{theorem} \label{thm:cocycleExtendingToTrig} Consider an assignment of a parameter $\nu_e<-1$ to each edge of an oriented, finite triangulation $\trig$\!{}. \begin{enumerate} \item \label{cocycCondA} If each matrix $G_\Delta$ is realized, we obtain a natural cocycle on $\overline{\trig}$ and, thus, a representation $\pi_1(\overline{\trig})\to\myPGL{2}{\C}$ (up to conjugation, unless we pick a vertex of $\overline{\trig}$ as basepoint). \item \label{cocycCondB} If, furthermore, $\Theta_e$ is a multiple of $2 \pi$ for every $e\in E(\trig)$, the cocycle extends to $\hat{\trig}$\!{} and, thus, yields a representation of $\pi_1(\trig)\to\myPGL{2}{\C}$. \item \label{cocycCondC} If, furthermore, $\Theta_e = 2\pi$ for every $e\in E(\trig)$, the representation is giving a hyperbolic structure on $\trig$\!{}. \label{thm:cocycle} \end{enumerate}
\end{theorem}
\begin{proof} Note that $\alpha$ and $\beta$ in Lemma~\ref{lemma:coycleAssignment} are involutions and only involve the parameters $v_{ij}$ on the edges of the triangle containing the respective $\alpha$ and $\beta$. Hence, the matrices on two big hexagons on two doubly-truncated simplices are compatible and the hexagons can be identified if the edge parameters on the respective triangles of the corresponding tetrahedra match. This proves \eqref{cocycCondA}.\\ \eqref{cocycCondB} follows from the above comment about the prisms being cocycles and the fact that $\hat{\trig}$ differs from $\trig$\!{} only by a set of 3-balls which do not change $\pi_1$.\\ \eqref{cocycCondC} is just restating Theorem~\ref{thm:hypStruct}.
\end{proof}
\section{Extending cocycles on genus 0 surfaces} \label{sec:vertexCocycles}
Let us define a punctured topological polyhedron which we will use as model for a ``vertex link'' of $\hat\trig$\!{} when removing some prisms.
\begin{definition} \label{def:puncTopPolyhed}
Let $L$ be a topological polyhedron, i.e., a decomposition of an oriented $2$-sphere into polygons. Let $P_1, \dots, P_p$ be selected two-cells of $L$ such that
$\partial P_i \cap \partial P_j \not =\emptyset$ for all $i\not=j$. We call $(L,L\setminus \bigcup P_l)$ a punctured topological polyhedron. \end{definition}
The complex $L\setminus \bigcup P_l$ will often be equipped with the following kind of cocycle.
\begin{definition} Let $X$ be a surface with boundary and with a decomposition into polygons. A $(\mySO{3}, \mySO{2})$-cocycle on $X$ is a $\mySO{3}$-cocycle where edges in the boundary $\partial X$ are labeled by rotations $R_\omega\in\mathrm{Im}(\mySO{2}\hookrightarrow\mySO{3})$ about the $z$-axis. \end{definition}
The goal of this section is to give a criterion when such a cocycle on $L\setminus \bigcup P_l$ extends to a $\mySO{3}$-cocycle on $L$.
\begin{definition}
Let $X$ be a $2$-complex. An edge-loop $\myPath$ is a word in the oriented edges of $X$ such that the end of one edge coincides with start of the next edge (when reading the word from right to left and cyclically).
\end{definition}
Note that by parametrizing the oriented edges of $X$ and concatenating them in the order given by $\myPath$, we obtain a geometric realization of $\myPath$ that is a (based) topological loop in $X$. Also note that a $\groupG$-cocycle on $X$ assigns a value in $\groupG$ to an edge-loop $\myPath$ obtained by multiplying the labels of the oriented edges in the respective order. The value assigned to $\myPath$ depends only on the homotopy type of (the geometric realization of) $\myPath$ and, in particular, is trivial if the loop is contractible in $X$.
\begin{definition} \label{def:gimbalLoop} Let $(L,L\setminus \bigcup P_l)$ be a punctured topological polyhedron. A gimbal loop $\myPath$ is a word in the oriented edges of $L$ and the selected two cells $P_1, \dots, P_p$ such that \begin{itemize} \item each $P_1, \dots, P_p$ is contained in $\myPath$ exactly once, \item each $P_l$ in $\myPath$ is preceeded (when reading the word from right to left and cyclically) by an edge $e_j$ such that the endpoint of $e_j$ is a vertex of $\partial P_l$, and \item dropping all $P_1, \dots, P_p$ from $\myPath$ results in an edge loop. This edge loop bounds a disk in $L\setminus\bigcup P_l$. The interior of the disk embeds into $L\setminus\bigcup P_l$ matching the orientation of $L$.
\end{itemize} \end{definition}
\begin{example} \label{example:GimbalLoop} Figure~\ref{fig:gimbalLoop} shows an example of a gimbal loop $\myPath$ for $p=3$. The corresponding word is $e_6 P_2 e_5 e_4 P_1 e_3 e_2 P_3 e_1.$ \end{example}
\begin{figure}
\caption{The edge loop obtained when dropping the $P_l$ from a gimbal loop $\myPath$.}
\label{fig:gimbalLoop}
\caption{The edge loop obtained when replacing $P_l$ by $\partial P_l$ in a gimbal loop $\myPath$.}
\label{fig:splicedGimbalLoop}
\end{figure}
\begin{definition} \label{def:defGimbalFunction} Let $(L,L\setminus \bigcup P_l)$ be a punctured topological polyhedron and $\myPath$ be a gimbal loop. Fix a $(\mySO{3},\mySO{2})$-cocycle on $L\setminus \bigcup P_l$. Assume we are given numbers $(d_1,\dots,d_p)$. Recall that the cocycle assigns a $\mySO{3}$-matrix to every edge in $\myPath$. Assign to each $P_l$ in $\myPath$ the rotation $R_{d_l}\in\mathrm{Im}(\mySO{2}\hookrightarrow\mySO{3})$. We call the product of the matrices assigned to the letters in $\myPath$ the gimbal matrix $m_\myPath(d_1,\dots,d_p)$. Furthermore, we call the function $$g_\myPath:\R^p\to\R^3,\quad (d_1,\dots, d_p)\mapsto\left(m_\myPath(d_1,\dots,d_p)_{01},m_\myPath(d_1,\dots,d_p)_{02},m_\myPath(d_1,\dots,d_p)_{12}\right)$$ assigning the upper triangular entries of the gimbal matrix the gimbal function. \end{definition}
Recall that a $(\mySO{3},\mySO{2})$-cocycle on $L\setminus \bigcup P_l$ assigns a rotation $R_\omega\in\mathrm{Im}(\mySO{2}\hookrightarrow\mySO{3})$ to each edge in $\partial P_l$ (with orientation induced from the orientation of $P_l\subset L$) where we pick $\omega\in(-\pi,\pi]$. Let $\delta_l$ denote the sum of all the angles $\omega$ over the edges of $\partial P_l$. For example, $\delta_2$ in Figure~\ref{fig:gimbalLoop} is the sum of the angles $\omega$ associated to the edges $e^{-1}_{6}, e_{19}, e_{18}, e_{17}, e_{16},$ and $e_{15}.$
\begin{lemma} Using the same setup as in Definition~\ref{def:defGimbalFunction}, we have $g_\myPath(2\pi,\dots,2\pi)=g_\myPath(\delta_1,\dots,\delta_p)=(0,0,0)$. \label{lemma:gimbalMatrixId} \end{lemma}
\begin{proof} Assume each $d_l=2\pi$. Then each $R_{d_l}$ is the identity and thus can be dropped from the gimbal matrix $m_\myPath(2\pi,\dots,2\pi)$. Thus the gimbal matrix is equal to the matrix the cocycle assigns to the edge loop obtained when dropping the $P_i$. This edge loop is contractible. Thus the gimbal matrix is the identity and $g_\myPath(2\pi,\dots,2\pi)=(0,0,0)$.\\ The gimbal loop $\myPath$ can also be turned into an edge loop by replacing each $P_l$ in $\myPath$ by a word in the oriented edges in $\partial P_l$. Here we use the orientation of $\partial P_l$ induced from the $P_l\subset L$ and start traversing $\partial P_l$ at the endpoint of the preceding edge. Figure~\ref{fig:splicedGimbalLoop} shows this edge loop for Example~\ref{example:GimbalLoop}. The corresponding word is: $$e_6(e_6^{-1}e_{19}e_{18}e_{17}e_{16}e_{15})e_5e_4(e_4^{-1}e_{14}e_{13}e_{12}e_{11})e_3e_2(e_2^{-1}e_{10}e_9e_8e_7)e_1.$$ Note that by definition of $\delta_l$, the matrix assigned to this edge loop by the cocycle is equal to the gimbal matrix $m_\myPath(\delta_1,\dots,\delta_p)$. This edge loop is again contractible and thus $g_\myPath(\delta_1,\dots,\delta_p)=(0,0,0)$. To see that the edge loop is contractible, note that a genus 0 surface with $p$ boundary components can be obtained by attaching a 2-cell to a 1-complex consisting of $p$ loops and $p$ arcs connecting the loops to common base point. The edge looped can be homotoped into this form since it bounds a disk with interior embedding into $L\setminus \cup P_l$, see Figure~\ref{fig:gimbalLoopSkeleton}. \begin{figure}
\caption{Homotoping the edge loop obtained from gimbal loop $\myPath$ such that it bounds a disk.}
\label{fig:gimbalLoopSkeleton}
\caption{A gimbal.}
\label{fig:gimbal}
\end{figure} \end{proof}
Let $Dg_\myPath:\R^p\to M(3\times p, \R)$ denote the Jacobian of $g_\myPath$ and $[Dg_\myPath(K)]$ be the interval closure of $Dg_\myPath(K)$, i.e., the smallest (axis-aligned) closed box containing $Dg_\myPath(K)$ when thinking of the matrix space $M(3\times p, \R)$ as $\R^{3p}$. We say that $[Dg_\myPath(K)]$ is invertible if every matrix in $[Dg_\myPath(K)]$ is invertible.
\begin{definition} \label{def:gimbalLockAvoided} Using the same setup as in Definition~\ref{def:defGimbalFunction}, let $K\subset \R^p$ be a box. We say that $K$ avoids gimbal lock with respect to $\myPath$ if $[Dg_\myPath(K)]$ is invertible. \end{definition}
\begin{remark} The term gimbal lock is inspired by the mechanical device called gimbal or Cardan suspension used to achieve an arbitrary rotation in $\mySO{3}$, see Figure~\ref{fig:gimbal}. Letting $d_1, d_2$ and $d_3$ denote the Euler angles at the joints from the grounding boxes to the inner-most ring, the rotation achieved by the gimbal is given by the matrix $$(d_1,d_2,d_3)\mapsto R_{d_1} \left(\begin{smallmatrix} & & -1\\ & 1 & \\ 1 & & \end{smallmatrix}\right) R_{d_2} \left(\begin{smallmatrix} & & 1\\ & 1 & \\ -1 & & \end{smallmatrix}\right) R_{d_3}$$ and a configuration of $(d_1, d_2, d_3)$ where this map does not have full rank is known as gimbal lock. Note the formal similarity of this product to the gimbal matrix $m_\myPath(d_1,\dots, d_p)$. \end{remark}
The following lemma is not useful in the general setting, but illustrates the principle we will use later (also see Example~\ref{example:singleEdgeViolation}):
\begin{lemma} Using the same setup as in Definition~\ref{def:defGimbalFunction}, let $K\subset \R^p$ be a box avoiding gimbal lock with respect to $\myPath$ such that $(2\pi, \dots, 2\pi) \in K$. If $(\delta_1,\dots,\delta_p)\in K$, then all $\delta_l=2\pi$ and the cocycle extends to $L$. \label{lemma:noGimbalImpliesAllEqnsVertexLink} \end{lemma}
\begin{proof} A standard result about the interval Newton method says that $g_\myPath$ is injective on $K$ since $[Dg_\myPath(K)]$ is invertible. Thus, Lemma~\ref{lemma:gimbalMatrixId} implies that $\delta_l=2\pi$. \end{proof}
\section{Extending cocycles on triangulations} \label{sec:extendCocycles}
Let $\trig$ be a triangulation with an assignment of edge parameters $\nu_e<-1$ such that each $G_\Delta$ is realized (i.e., fulfills the conditions of Lemma~\ref{lemma:singleGeomSimp}). Let $E^\sim \cup E^==E(\trig)$ be a partition of the edges into two disjoint sets where $|E^\sim| = 3o$ with $o=|V(\trig)|$ being the number of vertices of $\trig$\!{}. Assume that we know that $\Theta_e=2\pi$ for every edge $e\in E^=$, but we only know that $\Theta_e$ is close to $2\pi$ for $e\in E^\sim$. Thus, the $\myPGL{2}{\C}$-cocycle in Theorem~\ref{thm:cocycleExtendingToTrig} might only extend to $\hat\trig^==\hat\trig\setminus \mathrm{Prisms}(E^\sim)$, the complex where the prisms (including the top and bottom face) about the edges in $E^\sim$ have been removed. The goal of this section is to give a criterion that forces $\Theta_e=2\pi$ for all $e\in E(\trig)$, so that the conclusions of Theorem~\ref{thm:cocycleExtendingToTrig} apply and the edge parameters $\nu_e$ yield a hyperbolic structure for $\trig$\!{}.
Let $V(\trig)=\{v_1,\dots,v_o\}$ be the vertices of $\trig$\!{}. If we drop all all $\alpha$-edges and all $2$- and $3$-cells adjacent to any $\alpha$-edge from $\hat\trig$, the remaining complex has a connected component for each vertex $v_k$. We call such a connected component the ``vertex link'' of $v_k$ and denote it by $L_k$. Starting with $\hat\trig^=$ instead of $\hat\trig$, we similarly obtain a subcomplex $L^=_k\subset L_k$ for each $v_k$. Note that $(L_k,L^=_k)$ is a punctured topological polyhedron as in Definition~\ref{def:puncTopPolyhed} and the $\mySO{3}$-cocycle on $\hat\trig^=$ descends to a $(\mySO{3},\mySO{2})$-cocycle on each $L^=_k$ formed by the $\beta$- and $\gamma$-edges. Let us fix a gimbal loop $\myPath_k$ for each $(L_k,L^=_k)$, giving us gimbal functions $g_1,\dots, g_o$ as in Definition~\ref{def:defGimbalFunction}.
\begin{figure}
\caption{Gimbal loops for two ``vertex links'' (only some part of each is shown). The thick line is the edge in the triangulation.}
\label{fig:gimbalLoopsInTriangulation}
\end{figure}
Let us label the edges in $E^\sim = \{e^\sim_1,\dots,e^\sim_{3o}\}$ and $E^==\{e^=_1,\dots, e^=_{m-3o}\}$. We again want to construct a gimbal function, but this time in one variable $T_j$ per edge $e^\sim_j\in E^\sim$ instead of a variable $d_{k,l}$ per removed polygon as in Section~\ref{sec:vertexCocycles}. Note that each edge $e^\sim_j\in E^\sim$ corresponds to two polygons $P_{k,l}$ and $P_{k',l'}$ that have been removed (from the ``vertex links'' $L_k$ and $L_{k'}$ in $\hat\trig$) to obtain $L^=_k$ and $L^=_{k'}$ (with $k$ and $k'$ not necessarily distinct), see Figure~\ref{fig:gimbalLoopsInTriangulation}. We obtain a gimbal function in $(T_1,\dots, T_{3o})$ for each vertex by setting $d_{k,l}=d_{k',l'}=T_j$ in Definition~\ref{def:defGimbalFunction}. We combine these gimbal functions into a single one:
\begin{definition} \label{def:trigGimbalFunction} Let $\trig$\!{} be a triangulation with an assignment of edge parameters $\nu_e<-1$ such that each $G_\Delta$ fulfills the conditions of Lemma~\ref{lemma:singleGeomSimp}. Let $E^\sim = \{e^\sim_1,\dots,e^\sim_{3o}\}$ and $E^==\{e^=_{1},\dots, e^=_{m-3o}\}$ be a partition of $E(\trig)$ and fix gimbal loops $\myPath_1,\dots,\myPath_o$. The gimbal function is given by $$g: \R^{3o} \to \R^{3 o}, \quad (T_1,\dots, T_{3o}) \mapsto \left(g_1(d_{1,1}, \dots, d_{1, p_1}), \dots, g_o(d_{o,1}, \dots, d_{o,p_o})\right).$$ A box $K\subset \R^q$ avoids gimbal lock if $[Dg(K)]$ is invertible. \end{definition}
Note that the orientation on $(L_k,L^=_k)$ in the above definitions must be chosen such that the boundary of a small hexagon in $(L_k,L^=_k)$ contains a $\gamma^{\sigma(0)\sigma(1)\sigma(2)}$ for $\sigma\in \permA_4$.
\begin{example} Figure~\ref{fig:gimbalLoopsInTriangulation} shows examples of gimbal loops for two ``vertex links'' of $\hat{\trig}^=$. The corresponding gimbal matrices would be given by $$\cdots \gamma^{031}_{\Delta_i} \cdot
\beta^{013}_{\Delta_i} \cdot
R_{T_j} \cdot
\gamma^{012}_{\Delta_i} \cdot
\beta^{021}_{\Delta_i}
\cdots
\quad\mbox{and}\quad
\cdots
\gamma^{103}_{\Delta_i}
\cdot
\gamma^{310}_{\Delta_{i'}}
\cdot
R_{T_j} \cdot \beta^{301}_{\Delta_{i'}}
\cdot
\gamma^{302}_{\Delta_{i'}}\cdots .$$ The gimbal function $g$ for the triangulation is obtained by taking the upper triangular entries of the gimbal matrix for each vertex. \end{example}
\begin{theorem} \label{thm:mainThm} Let $\trig\!{}$, $\nu_e$, $E^\sim$ and $E^=$, $\myPath_1, \dots, \myPath_o$ and $g$ be as in Definition~\ref{def:trigGimbalFunction}. Let $K\subset \R^{3o}$ be a box avoiding gimbal lock such that $(2\pi, \dots, 2\pi)\in K$. If \begin{enumerate} \item $(\Theta_{e^\sim_1},\dots,\Theta_{e^\sim_{3o}})\in K$ and \item $\Theta_{e^=_1}=\dots=\Theta_{e^=_{m-3o}}=2\pi$ \end{enumerate} then $\Theta_e=2\pi$ for all $e\in E(\trig)$, so the edge parameters $\nu_e$ yield a hyperbolic structure on $\trig$\!{}. \end{theorem}
\begin{proof} By applying Lemma~\ref{lemma:gimbalMatrixId} to each vertex $v_1,\dots, v_o$, we see that $$g(2\pi,\dots,2\pi)=g(0,\dots,0)=(0,\dots,0).$$ Hence, the proof from Lemma~\ref{lemma:noGimbalImpliesAllEqnsVertexLink} applies here as well and we have $\Theta_{e^\sim_1}=\dots=\Theta_{e^\sim_{3o}}=2\pi$. Therefore, all conditions of Theorem~\ref{thm:cocycleExtendingToTrig} are fulfilled. \end{proof}
\section{Examples of (non-)gimbal lock} \label{section:examples}
This section is giving examples where the condition $\Theta_e=2\pi$ from Theorem~\ref{thm:hypStruct} is known to be fulfilled for most but not all edges. Let $\trig$ be a finite, orientable triangulation with $o$ vertices and $m$ edges. We partition the edges as
$E^\sim=\{e^\sim_1,\dots,e^\sim_l\}$ and $E^==\{e^=_1,\dots,e^=_{m-l}\}$ depending on whether we know that $\Theta_e$ is close to or, respectively, exactly equal to $2\pi$.
Geometrically, this yields a singular hyperbolic structure with cone singularities along the edges with $\Theta_{e^\sim_i}\not=2\pi$. Looking at a vertex $v$ connected to such an edge, its neighborhood is the hyperbolic cone of its link which is $S^2$ topologically but has a spherical cone structure different from the standard $S^2$. In this section, we extend Definition~\ref{def:trigGimbalFunction} to the case where $l=|E^\sim|$ might not be $3o$ and say that gimbal lock is avoided if $Dg$ has no kernel.
\begin{example} \label{example:singleEdgeViolation} Consider the case $l=1$ with $e^\sim_1$ connecting two distinct vertices $v_1$ and $v_2$. Assume $\Theta_{e^\sim_1}\not=2\pi$. The links of $v_1$ and $v_2$ would have a spherical cone structure with exactly one singularity. These do not exist, so $\Theta_{e'}=2\pi$. This also follows from Lemma~\ref{lemma:noGimbalImpliesAllEqnsVertexLink}. \end{example}
\begin{example}[Gimbal lock]\label{ex:gimbalLockGeodesic} Let us start with a (non-singular) hyperbolic structure on $\trig$\!{} such that the edges $e^\sim_1,\dots, e^\sim_l$ form a simple closed geodesic. Such a hyperbolic structure can often be deformed so that the geodesic becomes a cone singularity\footnote{For example, \texttt{Manifold("m003(-3.3,1.1)")} in SnapPy gives a geometric spun triangulation for the singular hyperbolic structure on the closed Weeks manifold \texttt{m003(-3,1)} with cone angle $2\pi/1.1$.}, i.e., such that $\Theta_{e^\sim_1}=\cdots=\Theta_{e^\sim_l}$ are close but not equal to $2\pi$. Note that the spherical structure of the link of a vertex $v_i$ connected to $e^\sim_i$ and $e^\sim_{i+1}$ gets deformed to have two conical singularities at antipodal points with equal cone angle. In the language of cocycles, pick a gimbal loop $\Gamma_i$ for $v_i$ as in Definition~\ref{def:gimbalLoop} such that the associated gimbal matrix is of the form \begin{equation} \label{eqn:gimbalMatrixAntipodal} m_{\Gamma_i}=R_{\Theta_{e^\sim_i}}\cdot \prod\nolimits_1 \cdot R_{\Theta_{e^\sim_{i+1}}}\cdot \prod\nolimits_1^{-1} = R_{\Theta_{e^\sim_i}} R_{-\Theta_{e^\sim_{i+1}}} \end{equation}
where $\prod_1\in\mySO{3}$ is a rotation by $\pi$ about an axis orthogonal to $z$. Then $m_{\Gamma_i}=\mathrm{Id}$ whenever $\Theta_{e^\sim_i}=\Theta_{e^\sim_{i+1}}$, so $Dg$ has a non-trivial kernel. If we add edges to $E^\sim$ such that $|E^\sim|=3o$, $Dg$ still has a non-trivial kernel, so we have gimbal lock and thus not a contradiction to Theorem~\ref{thm:mainThm}. \end{example}
\begin{example}[Perturbing previous example to avoid gimbal lock] \label{ex:avoidGimbalLockGeodesic} Consider the situation in the previous example but move one vertex, say $v_1$, slightly. Assume $\Theta_{e^\sim_1}\not=2\pi$. The link of $v_1$ would be a spherical cone structure with exactly two cone singularities having distance $<\pi$ (when scaling such that the curvature is $1$). Such a cone structure does not exist \cite[Corollary~3.5]{mondelloPanov}. In the language of cocycles, $\prod_1$ in Equation~\ref{eqn:gimbalMatrixAntipodal} is now a rotation by an angle strictly between $0$ and $\pi$, so $m_{\Gamma_1}$ is the product of two rotations about two different axes and thus only the identity when $\Theta_{e^\sim_l}=\Theta_{e^\sim_{1}}=2\pi$. In fact, this forces $\Theta_{e^\sim_1}=\dots=\Theta_{e^\sim_l}=2\pi$. \end{example}
\begin{example}[A non-hyperbolic manifold fulfilling all but six edge equations] Consider the two tetrahedron triangulation of $S^3$ obtained by identifying the boundary of the first tetrahedron with the boundary of the second tetrahedron via the identity map. Assign the same length to all edges such that each tetrahedron has a hyperbolic structure individually but all of the six edges equations are violated. Perform several 1-4 moves on one or both of the tetrahedra preserving the hyperbolic structure on each tetrahedron. This gives an example of a non-hyperbolic manifold where more than $m-3o$ but not all edge equations are fulfilled. Compare this to the ideal case where Neumann-Zagier (see Appendix) state that all edge equations must be fulfilled whenever $m-o$ are fulfilled. \end{example}
We give a characterization of gimbal lock in 1-vertex triangulations later in Theorem~\ref{thm:singleVertexTriangulationGimbalLockCondition}.
\section{Algorithm} \label{sec:algo}
\subsection{Overview}
Let $\trig$\!{} be an oriented, finite 3-dimensional triangulation. Let $o=V(\trig)$ be the number of vertices and index the edges $E(\trig)=\{e_1,\dots,e_m\}$. Assume we are given floating-point approximations for the edge parameters $\nu_{e_1},\dots,\nu_{e_m}$ such that the $\Theta_{e_1}, \dots, \Theta_{e_m}$ are approximately $2\pi$ (e.g, using the program Orb \cite{orb} based on the methods described in \cite[Section~2.2]{heardThesis}).
We will either fail in one of the steps or obtain intervals $\du{\nu_{e_1}},\dots,\du{\nu_{e_m}}$ guaranteed to contain values for $\nu_{e_1},\dots,\nu_{e_m}$ yielding a hyperbolic structure on $\trig$\!{} as follows: \begin{enumerate} \renewcommand{\arabic{enumi}.}{\Roman{enumi}.} \renewcommand{\arabic{enumi}.}{\Roman{enumi}} \item \label{step:submatrix} Find a subsystem of equations of full rank:\\ Since the given approximations for the edge parameters are close to a hyperbolic structure, evaluating the Jacobian \begin{equation} \label{eqn:jacobian} M=\left(\frac{\partial \Theta_{e_j}}{\partial \nu_{e_i}}\right)_{i=1,\dots,m;j=1,\dots,m} \end{equation} gives a matrix close to a singular matrix that we expect to have rank $m-3o$ (see Conjecture~\ref{conjecture:main}). Pick a suitable set of $m-3o$ rows and columns such that the resulting submatrix of $M$ has full rank.\\ This corresponds to picking two partitions of the edges of $\trig$ \begin{eqnarray*} E(\trig)=E^\sim \cup E^=&\mbox{with}& E^\sim=\{e^\sim_1,\dots,e^\sim_{3o}\},~E^= = \{e^=_1,\dots,e^=_{m-3o}\}\\ E(\trig)=E^\mathrm{fixed} \cup E^\mathrm{var}& \mbox{with}& E^\mathrm{fixed} = \{e^{\mathrm{fixed}}_1,\dots,e^{\mathrm{fixed}}_{3o}\}, ~E^\mathrm{var} = \{e^{\mathrm{var}}_1,\dots,e^{\mathrm{var}}_{m-3o}\} \end{eqnarray*} such that $$M'=\left(\frac{\partial \Theta_{e^=_j}}{\partial \nu_{e^\mathrm{var}_i}}\right)_{i=1,\dots,m-3o;j=1,\dots,m-3o}$$ has full rank near the given values. Keep the values of $\nu_{e^\mathrm{fixed}_i}$ fixed from now on. \item \label{step:intervalNewton} Find an interval solution to the above subsystem of equations:\\ Using interval Newton method or Krawczyk test, find large enough intervals $$\du{\nu_{e^\mathrm{var}_1}},\dots,\du{\nu_{e^\mathrm{var}_{m-3o}}}$$ for the edges in $E^\mathrm{var}$ such that they contain a point where $\Theta_{e^=_j}=2\pi$ for every $e^=_j\in E^=$. To have intervals $\du{\nu_{e_1}},\dots,\du{\nu_{e_m}}$ for all edges of the triangulation, simply use the interval $[\nu_{e^\mathrm{fixed}_i},\nu_{e^\mathrm{fixed}_i}]$ containing only the fixed value $\nu_{e^\mathrm{fixed}_i}$ for each remaining edge $e^\mathrm{fixed}_i\in E^\mathrm{fixed}$. \item \label{step:validGram} Ensure solutions are valid for a simplex:\\ Using the intervals for each edge, use interval arithmetic to verify the conditions of Lemma~\ref{lemma:singleGeomSimp} for each simplex $\Delta$. \item \label{step:approxEdgeEqn} Ensure that the remaining edge equations are ``approximately'' fulfilled:\\ Use these intervals to compute intervals $\du{\Theta_{e^\sim_1}},\dots,\du{\Theta_{e^\sim_{3o}}}$ for the sums of dihedral angles adjacent to an edge in $E^\sim$ to ensure that $2\pi \in \du{\Theta_{e^\sim_j}}$ for every such edge. \item \label{step:avoidGimbal} Ensure that the remaining edge equations are fulfilled exactly:\\ Find a gimbal loop $\myPath_1,\dots,\myPath_o$ for each vertex and consider the resulting gimbal function $g$. Ensure that $K=\du{\Theta_{e^\sim_1}}\times\cdots\times\du{\Theta_{e^\sim_{3o}}}$ avoids gimbal lock for the intervals computed in the previous step using interval arithmetics. \end{enumerate}
\subsection{Computing the Jacobian}
Note that the formulas given in \cite[Lemma~2.5]{heardThesis} for the derivatives $\partial\theta_{ij}/\partial{v_{mn}}$ (where $v_{mn}=v_{nm}$) run into a division by zero when $\theta_{ij}=\pi/2$. To obtain formulas avoiding this problem, we take the total derivative of Equation~\ref{eqn:dihedral} $$\frac{\partial\theta_{ij}}{\partial{v_{mn}}} =\frac{-1}{\sqrt{c_{ii}c_{jj}-c_{ij}^2}} \left(\frac{\partial c_{ij}}{\partial v_{mn}} - \frac{c_{ij}}{2 c_{ii}}\cdot \frac{\partial c_{ii}}{\partial v_{mn}} - \frac{c_{ij}}{2c_{jj}}\cdot \frac{\partial c_{jj}}{\partial v_{mn}}\right)
$$ and note that each $\partial c_{kl}/\partial v_{mn}=(-1)^{k+l} \partial (\det G_{kl}) / \partial v_{mn}$ is, up to sign, the sum of at most two cofactors of $G_{kl}$, namely, the ones corresponding to the $2\times 2$-matrices that can be obtained by deleting from $G_{kl}$ the row and column corresponding to row $m$ or $n$ and column $n$, respectively, $m$ of $G$.
Using this, we can compute the Jacobian $M$ in \eqref{eqn:jacobian} using floating point (in Step~\ref{step:submatrix}), respectively, interval arithmetics (in Step~\ref{step:intervalNewton}) avoiding the need for automatic differentiation.
\subsection{Step~\ref{step:submatrix}: Finding a submatrix of full rank}
This step is necessary since interval methods (e.g., interval Newton method or Krawczyk test) only work for systems with invertible Jacobian. To obtain a subsystem of full rank, we apply the following algorithm to $M$ with $h=m-3o$: \begin{center} \fbox{ \parbox{12.7cm}{ \begin{tabular}{rp{9.8cm}} {\bf Input:} & Square-matrix $M=(m_{r,c})$ with expected rank $h$.\\ {\bf Output:} & Sets $R$ and $S$ of indices of $h$ rows, respectively, $h$ columns. The submatrix $M'$ of $M$ formed by these rows and columns will have full rank.\\ {\bf Algorithm:} \end{tabular} \begin{enumerate} \itemsep0em \renewcommand{\arabic{enumi}.}{\arabic{enumi}.} \renewcommand{\arabic{enumi}.}{\arabic{enumi}.} \renewcommand{\arabic{enumii}.}{\arabic{enumii}.} \renewcommand{\arabic{enumii}}{\arabic{enumii}} \item $R\leftarrow\{\}$. $C\leftarrow\{\}$. \item Repeat $h$ times:\label{step:Repeat} \begin{enumerate} \itemsep0em \item Let $(r,c)$ be the index of the entry $m_{r,c}$ in $M$ with the largest absolute value when ignoring the rows in $R$ and columns in $C$.\label{substep:maxEntry} \item Add multiples of row $r$ to all other rows of $M$ to make all entries except for $m_{r,c}$ in column $c$ zero. \item Add multiples of column $c$ to all other columns of $M$ to make all entries except for $m_{r,c}$ in row $r$ zero. \item $R\leftarrow R\cup\{r\}$. $C\leftarrow C\cup\{c\}$. \end{enumerate} \end{enumerate} } } \end{center} \begin{remark} Note that the algorithm has the following stability properties: \begin{enumerate} \item We obtain the same set of rows and columns of $M$ when permuting the rows or columns of the input matrix $M$, i.e., the result is obtained by applying the same permutation to $R$, respectively, $C$ --- unless there are ties in Step \ref{substep:maxEntry}. \item Transposing $M$ results in interchanging $R$ and $C$. \end{enumerate} \end{remark} \begin{remark} This algorithm is a simplified version of $LDU$-factorization with full pivoting, i.e., a decomposition $M=PLDUP'$ where $P$ and $P'$ are permutation matrices, $L$ and $U$ unit-lower, respectively, unit-upper triangular matrices and $D$ a diagonal matrix.
\end{remark}
\subsection{Step~\ref{step:intervalNewton}}
This step is a straightforward application of the interval Newton method or Krawczyk test to the equations $\Theta_{e^=_j}-2\pi = 0$ in variables $\nu_{e^\mathrm{var}_i}$ (keeping $\nu_{e^\mathrm{fixed}_i}$ fixed). Note that even for high precision solutions, computing the approximate inverse in the Krawczyk test using IEEE754 double-precision floating point numbers is usually sufficient.
If we are interested in increasing the precision of the solution, we can optionally perform the ordinary Newton method to the subsystem from Step~\ref{step:submatrix} before Step~\ref{step:intervalNewton}.
\subsection{Steps~\ref{step:validGram} and \ref{step:approxEdgeEqn}}
These conditions are straightforward to check with interval arithmetics given the comment about the Budan-Fourier theorem in Lemma~\ref{lemma:singleGeomSimp}.
\subsection{Step~\ref{step:avoidGimbal}: Finding gimbal loops}
We need to pick a gimbal loop $\myPath_k$ in each ``vertex link'' $(L_k,L^=_k)$. Note that this complex consists of the small hexagons (with alternating $\beta$- and $\gamma$-edges) of the doubly-truncated simplices (see Figure~\ref{fig:DoublyTruncated}) and polygons coming from the ends of the prisms (see Figure~\ref{fig:Prism} and \ref{fig:gimbalLoopsInTriangulation}). For a vertex $v_k$, pick a hexagon in $L^=_k$, mark it as ``used'' and starting with its boundary (oriented such that it traverses a $\gamma^{\sigma(0)\sigma(1)\sigma(2)}$ with $\sigma\in A_4$), expand this edge-loop until it touches each boundary component of $L^=_k$ as follows: pick a $\beta$-edge of the edge-loop that is adjacent to an unused hexagon $H$ and replace the $\beta$-edge by the five other edges of $H$, marking $H$ as ``used'' (see Figure~\ref{fig:findingAGimbalLoop}). It is faster to exhaust all $\beta$-edges of one hexagon first in breadth-first search manner (vs depth-first search) before moving on to the next hexagon.
\begin{figure}
\caption{Procedure to find a gimbal loop $\myPath$.}
\label{fig:findingAGimbalLoop}
\end{figure}
\subsection{Step~\ref{step:avoidGimbal}: Computing the gimbal function's derivative}
We then need to compute $[Dg(K)]$ which boils down to computing the derivatives $\partial m_{\myPath_k}/\partial T_i$ of the gimbal matrices $m_{\myPath_k}$ associated to the gimbal loops $\myPath_k$. Focusing on one $i$ and $k$, note that $m_{\myPath_k}$ is given by an alternating product the form $$ \prod\nolimits_1 \cdot R_{T_i}\cdot \prod\nolimits_2 \cdot R_{T_i} \cdot \prod\nolimits_3 \cdot R_{T_i} \cdots \prod\nolimits_q $$ where $\prod\nolimits_k$ stands for a product of $\beta$- and $\gamma$-matrices and rotations $R_{T_{i'}}$ with $i'\not=i$ ($q$ is actually at most three since an edge of $\trig$\!{} has two ends which might or might not end in the same vertex). We can then compute the derivative as \begin{eqnarray*} \frac{\partial m_{\myPath_k}}{\partial T_i} &=& \phantom{+} \prod\nolimits_1 \cdot R'_{T_i}\cdot \prod\nolimits_2 \cdot R_{T_i} \cdot \prod\nolimits_3 \cdots R_{T_i} \cdot \prod\nolimits_q\\
& & + ~ \cdots \\
& & +\prod\nolimits_1 \cdot R_{T_i}\cdot \prod\nolimits_2 \cdot R_{T_i} \cdot \prod\nolimits_3 \cdots R'_{T_i} \cdot \prod\nolimits_q \quad \mbox{where}~ R'_\omega = \left(\begin{array}{ccc} -\sin\omega & -\cos\omega & \\ \phantom{-}\cos\omega & -\sin\omega & \\ & & 0\end{array}\right). \end{eqnarray*}
We need to evaluate this using the intervals from Step~\ref{step:intervalNewton} for the $\beta$- and $\gamma$-matrices and the intervals $\du{\Theta_{e^\sim_1}},\dots,\du{\Theta_{e^\sim_{3o}}}$ from Step~\ref{step:approxEdgeEqn} for $R'_{T_i}$ and all the $R_{T_j}$.
\subsection{Step~\ref{step:avoidGimbal}: Verifying invertibility}
To show that a square matrix $m$ with real interval entries is invertible (in the sense used in Definition~\ref{def:gimbalLockAvoided} and \ref{def:trigGimbalFunction}), we can find an approximate inverse $n$ (usually IEEE754-double precision is sufficient) and verify that each entry of $mn-\mathrm{Id}$ has absolute value strictly less than $1/r^2$ where $r$ is the number of rows.
\section{Results} \label{sec:results}
Our implementation of the algorithm in Section~\ref{sec:algo} is available at \cite{veriClosedRepo}. Lists of all knots and links from Theorem~\ref{thm:hyperbolicBranchedDoubleCovers} as well as the isomorphism signatures of the finite triangulations we used are available at \cite{veriClosedData}.
To produce the input to the algorithm, we used SnapPy \cite{SnapPy} to produce a finite triangulation of a manifold and Orb \cite{orb} to find unverified floating point edge parameters. Note that there are finite triangulations that admit hyperbolic structures but Orb is unable to find one. However, for all examples of hyperbolic manifolds considered here, we are always able to make Orb succeed in finding edge parameters by randomizing the finite triangulation in SnapPy several times.
We were able to verify a hyperbolic structure on a finite triangulation of each of the 11031 orientable closed manifolds in the Hodgson-Weeks census \cite{hwcensus} and the 21962 genus 2 surface bundles and 3100 genus 3 surface bundles in the census by Bell \cite{bellBundleCensus}. In particular, we have an independent proof of \cite[Theorem~5.2]{hikmot} that all manifolds in SnapPy's \texttt{OrientableClosedCensus} are hyperbolic.
To prove Theorem~\ref{thm:hyperbolicBranchedDoubleCovers}, we went through all knots and links up to 15, respectively, 14 crossings tabulated by Hoste-Thistlethwaite\footnote{See \texttt{HTLinkExteriors}, \texttt{AlternatingKnotExteriors}, and \texttt{NonalternatingKnotExteriors} in SnapPy.} and used SnapPy to produce the branched double cover. We either used the above method to prove that the resulting manifold is hyperbolic or used Regina \cite{Regina} to prove that it is not hyperbolic by \begin{itemize} \item finding the triangulation in Regina's census or \item recognizing that the triangulation has a structure fitting one of Regina's\\ \texttt{StandardTriangulation}'s making it, e.g., a Seifert fiber space, or \item finding an essential torus using normal surface theory. \end{itemize} Note that the first two methods sometimes require some randomizations and simplifications of the triangulation in SnapPy to work and that the last method can be quite expensive.
The number of tetrahedra in the triangulations used to prove hyperbolicity was between 9 and 46. Most triangulations are single vertex, few have two vertices, and only one had three vertices.
Testing the algorithm on the bundle census and branched double covers was suggested by Nathan Dunfield since the geodesics isotopic to the components of the branching locus tend to be long and spun triangulations spinning about one of these geodesics very often fail to be geometric. By filling and drilling the triangulation, Dunfield found geometric spun triangulations (about a different geodesic) for the branched double covers of 42367 non-alternating knots and links up to 14 crossings, missing 1593.
\begin{remark} \label{remark:experimentEdgePartition}
We also investigate how the occurrence of gimbal lock depends on the edge partition $E^{\sim}\cup E^==E(\trig)$ with $|E^{\sim}|=3o$ where $o$ is number of vertices. For this, we fix a triangulation and a hyperbolic structure and check numerically whether the derivative $Dg$ of the gimbal function has singular values close to zero for different partitions (exhausting all partitions when $o\leq 2$ and sampling otherwise). We did this for several triangulations of orientable closed census manifolds including some with three and four vertices obtained by performing 1-4 moves. This lead to Conjecture~\ref{conjecture:main}. \end{remark}
\section{Discussion} \label{sec:discussion}
Let $\trig$\!{} be a finite, orientable triangulation with $o$ vertices and $m$ edges. We have proven that $\trig$\!{} admits a hyperbolic structure if the checks in each step of the algorithm in Section~\ref{sec:algo} pass. But are there hyperbolic structures which the algorithm cannot verify --- even as we increase the precision and give the algorithm better and better approximations of the edge lengths of the hyperbolic structure as input?
We conjecture that such hyperbolic structures are special and can always be avoided by a random perturbation. We state this in the following conjecture where ``generic'' means that a statement is true except for a closed measure zero set of hyperbolic structures on $\trig$ (we will see later that a natural measure exists on the space of all hyperbolic structures in Theorem~\ref{thm:hypStructAreSubmanifold}):
\begin{conjecture} \label{conjecture:main} Let $\trig$\!{} be a finite, orientable triangulation with $o$ vertices and $m$ edges admitting a hyperbolic structure. Then a generic hyperbolic structure on $\trig$\!{} gives rise to edge lengths $l_{e_1},\dots, l_{e_m}>0$ or equivalently edge parameters $\nu_{e_1},\dots,\nu_{e_m}\leq -1$ such that \begin{enumerate} \item $M$ in \eqref{eqn:jacobian} has rank $m-3o$ and \label{subconjecture:Rank} \item any choice of $m-3o$ linearly independent rows from $M$ avoids gimbal lock. More precisely, for any partition $E(\trig)=E^\sim \cup E^=$ into $3o$ and $m-3o$ edges, we have that\label{subconjecture:gimbalLock} \begin{enumerate} \item[1.] the rows of $M$ corresponding to the edges in $E^=$ are linearly independent \end{enumerate} implies that \begin{enumerate} \item[2.] the derivative $Dg$ of the gimbal function in Definition~\ref{def:trigGimbalFunction} is invertible. \end{enumerate} \end{enumerate} \end{conjecture}
Note that if Part~\eqref{subconjecture:gimbalLock} were false, the choice of the partition $E(\trig)=E^\sim\cup E^=$ made in Step~\ref{step:submatrix} could be such that Step~\ref{step:intervalNewton} passes but the algorithm fails later in Step~\ref{step:avoidGimbal}.
\begin{remark} Numerically, we found that the converse of Part~\eqref{subconjecture:gimbalLock} is not true, i.e., we found examples where the gimbal function is invertible even though the chosen rows of $M$ are linearly dependent. \end{remark}
\begin{remark} Compare the conjecture to the ideal case where we do not know in general whether every cusped, finite volume hyperbolic 3-manifold has a geometric triangulation (see \cite{PetronioWeeks:partiallyFlatTrig,LuoSchleimerTillmann:virtualGeometricTrig,Goerner:DodecahedralGeometricTriangulation}). \end{remark}
\subsection{The space of hyperbolic structures}
Let $\trig$\!{} be a finite, orientable triangulation with $o$ vertices and $m$ edges admitting a hyperbolic structure. We are able to prove that the solution set of the edge equations has dimension $3o$. This is weaker than Part~\eqref{subconjecture:Rank} of Conjecture~\ref{conjecture:main} since it implies that the rank of $M$ is at most $m-3o$ (for example, $x^3=0$ yields a $0$-dimensional submanifold of the 1-dimensional $\R$, yet the Jacobian has rank $0$ at $0$ instead of the expected $1-0=1$).
Earlier, we defined a hyperbolic structure on $\trig$ as a compatible assignment of an isometry class of finite simplices in $\H^3$ to each simpex in $\trig$\!{}. The space of hyperbolic structures on $\trig\!{}$ can be described in the following ways: \begin{enumerate} \item $\edgeEqSol(\trig)\subset\R_{>0}^m$, the set of all tuples $(l_{e_1},\dots,l_{e_m})$ fulfilling the conditions of Theorem~\ref{thm:hypStruct} (since the edge lengths of a hyperbolic structure yield a point in $\edgeEqSol(\trig)$ and determine the hyperbolic structure uniquely). \item The space of geodesic homeomorphisms $\trig\to\mathcal{M}$ in a fixed homotopy class in $[\trig\!{},\mathcal{M}]$ where $\mathcal{M}$ is a hyperbolic 3-manifold homeomorphic to $\trig\!{}$. By Mostow rigidity, this yields all hyperbolic structures. By fixing the homotopy class, a hyperbolic structure gives a unique geodesic homeomorphism, instead of multiple related by the isometries of $\mathcal{M}$. \label{item:geodHomeo} \item The space of all developing homeomorphisms, i.e., $\rho$-equivariant homeomorphisms $d:\tilde{\trig}\to\H^3$ where $p$ is a fixed vertex of $\trig\!{}$ and $\rho:\pi_1(\trig\!{},p)\to\myPSL{2}{\C}$ is a fixed geometric representation. This gives the homeomorphism $\trig\!{}\to\mathcal{M}$ in \eqref{item:geodHomeo} when letting $\mathcal{M}=\H^3/\Gamma$ where $\Gamma=\mathrm{Im}(\rho)$. \end{enumerate}
\begin{theorem} \label{thm:hypStructAreSubmanifold} Let $\trig$\!{} be a finite, orientable triangulation with $o$ vertices and $m$ edges admitting a hyperbolic structure. The space $\edgeEqSol(\trig)$ of hyperbolic structures on $\trig$ is a $3o$-dimensional smooth submanifold of \,$\R_{>0}^m$, is (non-canonically) diffeomorphic to an open subset $U\subset \left(\H^3\right)^o$, and has a canonical measure induced from $\left(\H^3\right)^o$. \end{theorem}
\begin{proof} We call a $\rho$-equivariant geodesic map $d:\tilde{\trig}\to\H^3$ a developing map.\\ {\bf Claim:} Fix a vertex $p$ of $\trig\!{}$ and a geometric representation $\rho:\pi_1(\trig\!{},p)\to\myPSL{2}{\C}$. Then developing maps are in 1-1 correspondence to $(\H^3)^o$.\\ Pick a lift $\tilde{v_1},\dots, \tilde{v_o}$ in $\tilde{\trig}\!{}$ of each vertex of $v_1,\dots,v_o$ of $\trig\!{}$. By $\rho$-equivariance, the images of all vertices of $\tilde{\trig}$ are determined by $(d(\tilde{v_1}), \dots, d(\tilde{v_o}))\in(\H^3)^o$. A geodesic map is determined uniquely by the image of all the vertices.\\ {\bf Claim:} Under the above assumptions, the homeomorphisms among the developing maps form an open subset $U\subset(\H^3)^o$. We have a bijection $l:U\to\edgeEqSol(\trig)$.\\ By invariance of domain, a developing map is a homeomorphism if and only if it is an embedding. A developing map is an embedding if and only if the image of each simplex is positively oriented. The condition on a single simplex in $\tilde{\trig}\!$ to be positively oriented is open in the four vertex positions of the simplex. As a function of $(x_1,\dots,x_o)\in (\H^3)^o$, each of these four vertex positions is given by some $m x_k$ where $m$ is a hyperbolic isometry in $\Gamma=\mathrm{Im}(\rho)$. Thus, the condition is also open in $(\H^3)^o$. By $\rho$-equivariance, it is sufficient to check this for one lift $\tilde{\Delta}$ of each of the finitely many simplices $\Delta$ of $\trig\!{}$, so $U$ is open.\\ {\bf Claim:} The map $L=i\circ l: U\to\R^m_{>0}$ is smooth where $i: \edgeEqSol(\trig)\hookrightarrow \R^m_{>0}$ is the inclusion.\\ The distance function $\H^3\times\H^3\to\R_{\geq 0}$ is smooth for all $(x,y)$ with $x\not= y$.\\ {\bf Claim:} The inverse $l^{-1}:\edgeEqSol(\trig)\to U$ is continuous.\\ Fix a simplex $\tilde{\Delta}$ in $\tilde{\trig}$ and $\sigma\in \permA_4$. Given a point in $\edgeEqSol(\trig)$, let us consider the developing map $d':\tilde{\trig}\to\H^3$ obtained by starting with $\tilde{\Delta}$ in $\sigma$-standard position (see Definition~\ref{def:standardPosSimplex}) and developing simplex by simplex. Note that $d'$ is equivariant with respect to a conjugate but different representation $\rho':\pi_1(\trig\!{}, p)\to\myPGL{2}{\C}$. In other words, there is a unique element $h\in\myPGL{2}{\C}$ such that $\rho(\gamma)=h\circ \rho'(\gamma)\circ h^{-1}$ for all $\gamma\in\pi_1(\trig\!{}, p)$ and $d=h\circ d'$ is $\rho$-equivariant. We can compute $h$ as follows:
fix $\gamma_1,\gamma_2,\gamma_3\in\pi_1(\trig\!{},p)$ such that the attractive fixed points $p_i\in\partial \H^3$ of $\rho(\gamma_i)$ are distinct. Let $p'_i\in\partial\H^3$ be the attractive fixed points of $\rho'(\gamma_i)$. Then $h$ is the unique Moebius transformation with $p_i=h(p'_i)$. Note that the all $d'(\tilde{v_k})$ as well as all $\rho'(\gamma_i)$ and thus $h$ depend continuously on the point in $\edgeEqSol(\trig)$. Thus,
$(d(\tilde{v_1}), \dots, d(\tilde{v_o}))\in U\subset (\H^3)^o$ depends continuously on the point in $\edgeEqSol(\trig)$.\\ {\bf Claim:} The differential $DL$ of $L: U\to\R^m_{>0}$ is injective at every point in $U$.\\
Assume that $DL$ has non-trivial kernel. That is, there is a path $\gamma:(-1,1)\to U$ with $(d\gamma(t)/dt)_{|t=0}\not =0$ but $(d(L\circ\gamma)(t)/dt)_{|t=0}=0$. In other words, we have a 1-parameter family of developing maps $d:\tilde\trig\to \H^3$ where at least one of the points $d(\tilde{v_k})$ is moving with non-zero velocity but the derivatives of all the edge lengths is zero. As we shall see, this cannot happen.\\ From the edge lengths $(L\circ\gamma)(t)\in\edgeEqSol(\trig)$, we can construct a 1-parameter family of developing maps $d'$ as above. Since the derivatives of the lengths is zero, the vertices $d'(\tilde{v_k})$ all have velocity zero. It is not hard to see that the entries of the $h\in\myPGL{2}{\C}$ from above also have zero derivative. Thus, the points $d(\tilde{v_k})$ have zero velocity.\\ {\bf Claim:} The measure on $\edgeEqSol(\trig)$ induced from $\left(\H^3\right)^o$ is independent of the above choices.\\ A different choice of lifts $\tilde{v_1},\dots,\tilde{v_o}$ of vertices corresponds to the action of $(\H^3)^o$ by an element in $\Gamma^o$ and thus does not change the induced measure. A different choice of $\rho':\pi_1(\trig\!{},p)\to\myPGL{2}{\C}$ is conjugate to $\rho$ by an element $h\in\myPGL{2}{\C}$. The measure on $(\H^3)^o$ is invariant under the action of $h$. \end{proof}
\subsection{1-vertex triangulations} \label{sec:singleVertexTrigs} Consider a 1-vertex triangulation $\trig$ admitting a hyperbolic structure. Let $p$ be the vertex of $\trig$ and $\rho:\pi_1(\trig\!{}, p)\to\myPSL{2}{\C}$ be a geometric representation. Note that each edge $e$ of $\trig$ forms a loop in $\pi_1(\trig\!{}, p)$ and denote by $F_e=\{x\in \C P^1 : x = \rho(e)(x)\}$ the set of the corresponding fixed points. \begin{theorem} \label{thm:singleVertexTriangulationGimbalLockCondition} Let $E^\sim = \{e^\sim_1, e^\sim_2, e^\sim_3\}$ and $E^=$ be a partition of the edges of $\trig\!{}$. Gimbal lock occurs for every hyperbolic structure on $\trig$ if there are $i\not= j$ with $F_{e^\sim_i}=F_{e^\sim_j}$. Otherwise, gimbal lock is generically avoided. \end{theorem}
\begin{remark}
It is imaginable that there is a 1-vertex triangulation such that $|\{F_e : e\in E(\trig)\}|=2$ and gimbal lock occurs for each choice of $E^\sim$ giving a counterexample to Conjecture~\ref{conjecture:main}. \end{remark}
\begin{figure}
\caption{Edge in 1-vertex triangulation.}
\label{fig:edgeSingVertTrig}
\end{figure}
\begin{proof} The edges $e^\sim_i$ become geodesics with a potential kink at the image of $p$ in $\mathcal{M}$. Let $\overrightarrow{a}_i^0$ and $\overrightarrow{a}_i^1$ be the directions (of unit length) in which the two ends of $e^\sim_i$ approach $d(p)\in\H^3$ in the developing embedding $d:\tilde{\trig}\to\H^3$. We can identify the tangent space at $d(p)$ with Euclidean 3-space such that the gimbal matrix $m_\Gamma$ is the product of six rotations (the order depending on the choice of gimbal loop $\Gamma$) $$R_{T_1}^{\overrightarrow{a}_1^0},~R_{T_1}^{\overrightarrow{a}_1^1},~R_{T_2}^{\overrightarrow{a}_2^0},~R_{T_2}^{\overrightarrow{a}_2^1},~R_{T_3}^{\overrightarrow{a}_3^0},~\mbox{and}~R_{T_3}^{\overrightarrow{a}_3^1}$$ where $R_\omega^{\overrightarrow{v}}\in\mySO{3}$ denotes the rotation about $\overrightarrow{v}$ by angle $\omega$. Let $\overrightarrow{a}_{i,k}^j$ denote the $k$-th component of $\overrightarrow{a}_i^j$. At the point where all $T_i=2\pi$, we have $$\frac{\partial m_\Gamma}{\partial T_i} = \frac{R_{T_i}^{\overrightarrow{a}_i^0}}{\partial T_i} + \frac{R_{T_i}^{\overrightarrow{a}_i^1}}{\partial T_i}= \left(\begin{array}{ccc} 0 & -\overrightarrow{a}_{i,2}^0 & \overrightarrow{a}_{i,1}^0 \\ \overrightarrow{a}_{i,2}^0 & 0 & -\overrightarrow{a}_{i,0}^0 \\ -\overrightarrow{a}_{i,1}^0 & \overrightarrow{a}_{i,0}^0 & 0 \end{array}\right) + \left(\begin{array}{ccc} 0 & -\overrightarrow{a}_{i,2}^1 & \overrightarrow{a}_{i,1}^1 \\ \overrightarrow{a}_{i,2}^1 & 0 & -\overrightarrow{a}_{i,0}^1 \\ -\overrightarrow{a}_{i,1}^1 & \overrightarrow{a}_{i,0}^1 & 0 \end{array}\right), $$ $$\mbox{so}~\quad \frac{\partial g}{\partial T_i} = \left(-\big(\overrightarrow{a}_{i,2}^0+\overrightarrow{a}_{i,2}^1\big), \big(\overrightarrow{a}_{i,1}^0+\overrightarrow{a}_{i,1}^1\big), -\big(\overrightarrow{a}_{i,0}^0+\overrightarrow{a}_{i,0}^1\big)\right)=\left(-\overline{a}_{i,2}, \overline{a}_{i,1}, -\overline{a}_{i,0}\right)$$
where $\overline{a}_{i}=\overrightarrow{a}_i^0+\overrightarrow{a}_i^1.$ Thus, $Dg=\left(\partial g/\partial T_1, \partial g/\partial T_2, \partial g/\partial T_3\right)$ is invertible if $\overline{a}_1, \overline{a}_2,$ and $\overline{a}_3$ are linearly independent. Figure~\ref{fig:edgeSingVertTrig} shows that $\overline{a}_i$ is pointing from $d(p)$ to the geodesic $g_i$ spanned by $F_{e^\sim_i}$ (respectively pointing to $F_{e^\sim_i}$ if $|F_{e^\sim_i}|=1$) and $\overline{a}_i=0$ if it lies on that geodesic.\\
{\bf Claim:} Given a point $x\in\H^3$, consider the directions $\overline{a}_i$ from $x$ to the point on $g_i$ closest to $x$ (respectively pointing to $F_{e^\sim_i}$ if $|F_{e^\sim_i}|=1$). The set $V$ of $x$ where these directions lie in a plane is closed and has measure zero if all sets $F_{e^\sim_i}$ are distinct. Otherwise $V=\H^3$.\\ Let $f:\H^3\setminus \cup_i g_i\to \R, x\mapsto \omega(\overline{a}_1,\overline{a}_2,\overline{a}_3)$ where $\omega$ is the volume form on $\H^3$. Since $\overline{a}_i$ is the Hodge dual to the gradient of the distance of $x$ to $g_i$, $f$ is analytic. It is not hard to see that there is some $x$ with $f(x)\not = 0$ as long as no two $F_{e^\sim_i}$ and $F_{e^\sim_j}$ are equal. Thus $f^{-1}(0)$ and $V=f^{-1}(0)\cup \cup_i g_i$ have measure zero. \end{proof}
\section{Appendix: The uncited theorem HIKMOT relies on} \label{sec:hikmotGap}
As pointed out in the introduction, the full system of equations for finding a hyperbolic structure on a triangulation fails to have rank equal to the number of variables in both the finite and ideal case. Thus, verification by interval methods in the ideal case also relies on a theorem showing that if a suitable subsystem is fulfilled, all equations are fulfilled. Unfortunately, the paper \cite{hikmot} did not cite this crucial theorem correctly. We want to point out that the algorithm in \cite{hikmot} is correct and its implementation can be trusted since the theorem it relies on has been proven by \cite[Lemma~2.4]{moser}. However the theorem from \cite{NeumannZagier} cited in \cite{hikmot} is insufficient since it assumes hyperbolicity to begin with and its conclusion is too weak. Being the counterpart of the result in this paper in the ideal case, we will briefly state the correct theorem missing from \cite{hikmot}.
Recall, that given an ideal triangulation with $o$ vertices, Thurston \cite[Chapter 4]{ThurstonNotes} gave compatibility equations in one complex variable $z_1,\dots,z_m$ per tetrahedron which are all of the form $$\sum_{j=1}^m a_{r,j} \log\Big(z_j\Big) + b_{r,j} \log\Big(\frac{1}{1-z_j}\Big) + c_{r,j}\log\Big(1-\frac{1}{z_j}\Big) - 2\pi \myI d_r = 0$$ such that a solution with $\myIm(z_j)>0$ (called a geometric solution) yields a complete hyperbolic structure on the manifold obtained by filling in some cusps. There is one such equation with $d_r=1$ for each of the $m$ edges, one with $d_r=1$ for each filled cusped and one (sometimes two are given, but it is easy to see that one is sufficient) with $d_r=0$ for each unfilled cusp. Note that this system of equations consists of $m+o$ equations in $m$ variables and thus is overdetermined. To find a non-overdetermined system, let $\alpha_{r,j}=a_{r,j}-c_{r,j}$ and $\beta_{r,j}=-b_{r,j}+c_{r,j}$ and consider the $m\times m$-matrices $A=(\alpha_{r,j})$ and $B=(\beta_{r,j})$ where $r$ ranges over the edge equations: \begin{theorem}
The matrix $(A|B)$ has rank $m-o$. Pick $m-o$ linearly independent rows of $(A|B)$ and consider the system of equations consisting of the corresponding $m-o$ edge equations and the $o$ cusp equations. \begin{enumerate} \item Any solution to this system fulfills the remaining $o$ edge equations as well.\label{item:moser1} \item Near a geometric solution, the Jacobian of this system of equations is invertible.\label{item:moser2} \end{enumerate} \end{theorem} Note that \eqref{item:moser1} is sufficient for an algorithm to prove a manifold to be hyperbolic. Statement \eqref{item:moser2} ensures that such an algorithm succeeds if given a solution close enough to the geometric one.
\cite{NeumannZagier} states the rank of $(A|B)$ but assumes hyperbolicity. Neumann revisted the result in \cite {NeumannComb} to give a purely combinatorial statement where the rank of $(A|B)$ occurs as rank of the map $\beta$ in a certain chain complex. However, even Theorem~4.1 in \cite{NeumannComb} only implies that the remaining $o$ edge equations in the above theorem are fulfilled modulo $2\pi\myI \Q$ since it does not involve the $d_r$ of the edge equations. For a proof of the above Theorem, see \cite[Section~2.3.1]{moser}.
\end{document} | arXiv | {
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\begin{document}
\title{The homogenous tree as an electric network}
\abstract{Let T be an infinite homogenous tree of homogeneity $q+1$. Attaching to each edge the conductance $1$, the tree will became an electric network. The reversible Markov chain associated to this network is the simple random walk on the homogenous tree. Using results regarding the equivalence between a reversible Markov chain and an electric network, we will express voltages, currents, the Green fuction hitting times, transitions number, probabilities of reaching a set before another, as functions of the distance on the homogenous tree. This connection enables us to give simpler proofs for the properties of the random walk under discussion.}
\section{Introduction}
It is well-known that every reversible Markov chain equivalent to an electric network (see \cite{DS}, \cite{MR0124932}). More precisely, let us consider a reversible Markov chain of state space $\mathbf{G}$ ($\exists \pi : \mathbf{G}\rightarrow ]0,\infty[,\quad \pi(x)p_{xy}=\pi(y)p_{yx}\quad \forall x,y$) and transition matrix $P=(P_{xy})_{x,y\in \mathbf{G}}$. We will associate to this chain an electric network (a weighted graph, the weights being thought as conductances) in the following way: we take the vertex set of the graph to be $\mathbf{G}$, we define $x\sim y$ if $p_{xy}> 0$ and take the weights to be $c(x,y)=\prod(x)p_{xy}$. So $(\mathbf{G},\sim,c)$ will become a weighted graph. Conversely, let's start from a weighted graph $(\mathbf{G},\sim,c)$. We will consider il to be countable and satisfying $\sum c(x,y)<\infty$.\\ The associated reversible Markov chain will be created as follows: we will take $\mathbf{G}$ as a state space, we will define its transition matrix to be $P=(p_{xy})_{x,y\in \mathbf{G}}$, $p_{xy}=\frac{c(x,y)}{\sum_{y\sim x}c(x,y)}$ for $y\sim x$ and $0$ otherwise. We will take the reversibility function $\pi$ to be $\pi(x)=\sum_{y\sim x}c(x,y)$. One can easily check that the matrix $P$ is stochastic and that the function $\pi$ satisfies the condition $\pi(x)p_{xy}=\pi(y)p_{yx}\quad \forall x,y$. So $(\mathbf{G},P,\pi)$ will be a reversible Markov chain.\\ Taking into account this equivalence, we will not make any distinction in what follows between the notions of reversible Markov chain and electric network.\\ Applying the procedure above to the network obtained by attaching unit conductances to the edges of the homogenous tree, we will obtain that its attached Markov chain is the one having the matrix $P=(p_{xy})_{x,y\in T}$, $p_{x,y}=\frac{1}{q+1}$ for $y\sim x$ and $0$ otherwise. This is exactly the simple random walk on the tree (For more details on random walks on trees we refer to \cite{MR1001523} and \cite{MR1743100}).\\ The paper will analyse this process viewing it as an electric network. We will obtain our main results using electric network techniques.
\
\textbf{Acknowledgments}: The author kindly thanks M. Abbassi and F. Baudoin for their kind help.
\
\section{The probabilistic framework} The probabilistic framework of this paper will be the following:\\ Let us consider an irreducible and reversible Markov chain, having the following elements:\\ \textbf{The state space:} $\mathbf{G}$\\ \textbf{The transition matrix:} \[P=(p_{xy})_{x,y\in \mathbf{G}}\] \textbf{The reversibility function:} \[\pi:\mathbf{G}\rightarrow \mathbb{R}^{*}, \pi(x)p_{xy}=\pi(y)p_{xy}\quad \forall x,y\] \textbf{The equivalent states:} \[x\sim y\Leftrightarrow^{def}p_{xy}>0\] \textbf{The hitting vector} of a set $M\subset \mathbf{G}$: \[T_{M}=\inf\{n\geq 1: X_{n}\in M\}.\] When we have $M=\{a\}$, the hitting time will be denoted by $\tau_{a}$.\\ Let us distinguish between $\tau_{M}$ and $T_{M}$, where $T_{M}$ is the following: \[T_{M}=\inf\{n\geq 1: X_{n}\in M\}.\] When we have $M=\{a\}$, $T_{M}$ will be denoted by $T_{a}$.\\ \textbf{The hitting vector} of a set $M\subset \mathbf{G}$: \[\nu^{M}:=\left(P_{x}(\tau_{M}<\infty)\right)_{x\in\mathbf{G}}.\] \textbf{The transience:}\\ The chain is transient $\Leftrightarrow^{def}P_{a}(T_{a}<\infty)<1\quad \forall a$\\ \textbf{The Green function:} \[G(x,y):=\delta_{xy}+p_{xy}+p_{xy}^{2}+p_{xy}^{3}+...,\quad \forall x,y \in \mathbf{G},\] Where $p_{xy}^{n}$ is the $xy$ element of matrix $p^{n}$. The Green function can also be expressed in the following way (see [S], theorem 10.1): \[G(x,y)=E_{x}[\sum_{n\geq 0}1_{x_{n}=y}].\] \textbf{The harmonic functions in a point x:} \[f:\mathbf{G}\rightarrow \mathbb{R},\text{ having } Pf(x)=f(x),\] \[\text{that is }\sum_{y\sim x}p_{xy}f(y)=f(x).\] \textbf{The harmonic functions on a set} $\mathbf{M}\in \mathbf{G}$:\\ \[f:\mathbf{G}\rightarrow\mathbb{R},\text{ f being harmonic in every } x\in M.\] \textbf{The transitions number} from $x$ to $y$ (after one step):\\ \[S_{xy}:=\sum_{n\geq 0}1_{\{x_{n}=x,x_{n+1}=y\}}, \text{ for } x\sim y.\] \section{The electric network framework (first part)} The first part of the electric network framework of this paper will be described by the following definition: \begin{defn} Let us consider an electric network having the following elements:\\ a) $\mathbf{G}$: \textbf{the vertex set.}\\ b) E: \textbf{the edges set} (E contains its edge with its both possible orientations).\\ c) v: \textbf{the voltage function} that appears as a consequence of hooking up a battery between two vertices of the network ($v: \mathbf{G}\rightarrow R$).\\ d) i: \textbf{the electric current intensity function} that appears as a cosequence of hooking up a battery between two vertices of the network ($i: E\rightarrow R$).\\ e) c: \textbf{the conductance function} defined on the set of the oriented edges of the graph ($c: E\rightarrow R$); c is a symmetric function.\\ f) r: \textbf{the resistance function} defined on the set of the oriented edges of the graph ($r: E\rightarrow R$); r is the inverse of the function c, so it is symmetric.\\ g) $C(a\leftrightarrow Z)$ \textbf{the effective conductance from} $\mathbf{a}$ \textbf{to} $\mathbf{Z}$ where $\{a\}$ and $Z$ are two disjoint subsets of $\mathbf{G}$, this notion is defined for a finite graph; it will be the ratio between the sum of the currents that enter the current through the point $a$ and the strictly positive voltage in $a$ when hooking up a battery between $a$ and $Z$ with $v/Z=a$: \[C(a\leftrightarrow Z)=\frac{\sum_{x\sim a}i(a,x)}{v(a)}.\] h) $R(a\leftrightarrow Z)$: \textbf{the efective resistance from} $\mathbf{a}$ \textbf{to} $\mathbf{Z}$, where $\{a\}$ and $Z$ are two disjoint subsets of $\mathbf{G}$, this notion is defined for a finite graph, to be the inverse of the effective conductance from $a$ to $Z$.\\ i) $C (a\leftrightarrow Z)$: \textbf{the effective conductance from} $\mathbf{a}$ \textbf{to infinity}, this notion is defined for an infinite graph G; for defining it, we have to exhaust the graph $\mathbf{G}$ by a sequence $(\mathbf{G}_{n})_{n\geq 0}$ of finite subgraphs. This can be done, for instance, as follows: \[\mathbf{G}_{0}:=\{0\} \text{ (an arbitrarily fixed root)}\] \[\mathbf{G}_{n}:=\mathbf{G}_{n-1}\cup\{y:\exists x\in\mathbf{G}, y\sim x\},\quad \forall n\geq 1.\] One can easily check that $\mathbf{G}_{n-1}\leq \mathbf{G}_{n}\quad \forall n\geq 1$ and that $\cup_{n}\mathbf{G}_{n}=\mathbf{G}$.\\ Let us now denote $\mathbf{G}\backslash\mathbf{G}_{n}$ by $Z_{n}$. For each $n\geq 1$, we identify all the vertices of $C(a\leftrightarrow z_{n})$ the effective conductance from $a$ to $z_{n}$ in $\mathbf{G}^{(n)}$.\\ Finally, we define $C(a\leftrightarrow \infty)$ as follows: \[C(a\leftrightarrow \infty):=\lim_{n}C(a\leftrightarrow z_{n}).\] j) $R(a\leftrightarrow \infty)$: \textbf{the effective resistance from} $\mathbf{a}$ \textbf{to infinity.}\\ We define this notion for an infinite graph; it will be the inverse of the effective conductance from $a$ to infinity. \end{defn} In the above framework, it will be useful for the purpose of this paper to present the following characterisation of transience by means of conductances (see [LP], theorem 2.3). \begin{thm} An infinite electric network $\mathbf{G}$ is \textbf{transient iff the effective conductance from any vertex to infinity is strictly positive.} \end{thm} The idea of proof is the following (for details see \cite{LP}):\\ We start from the following proposition, called the Maximum Principle:\\ THE MAXIMUM PRINCIPLE: Let $\mathbf{G}$ be a finite graph, $\mathbf{H}\subseteq \mathbf{G}$, $\mathbf{H}$ connected and $\overline{\mathbf{H}}:=\{y\in \mathbf{G}:\exists x\in \mathbf{H}\text{ s.t. } y\sim x\}$. Let $f:\mathbf{G}\rightarrow \mathbb{R}$ $f$ harmonic on $\mathbf{H}$ and having $\max_{\mathbf{H}}f=\max_{\mathbf{G}}f$. Then $f/\overline{\mathbf{H}}=\max_{\mathbf{G}}f$. (If $f$ attains its maximum on a set where it is harmonic, then $f$ will be constant on that set).\\ By means of this Maximum Principle, we prove the so - called Uniqueness Principle:\\ THE UNIQUENESS PRINCIPLE: Let $\mathbf{G}$ be a finite graph, $\mathbf{A}\subset\mathbf{G}$ $f$ and $g$ two real functions on $\mathbf{G}$, harmonic on $\mathbf{H}$ and such that $f=g$ on $\mathbf{H}$. Then $f=g$. (A function is perfectly determined by its harmonicity on a certain set and by its values on the complementary of this set).\\ This principle will be used to prove the coincidence of the following two real functions defined on a finite graph $\mathbf{G}$:\\ $\cdot$ the function $F(x)=P_{x}(\tau_{a}<\tau_{z})$, where $\{a\}$ and $Z$ are fixed, disjoint subsets of $\mathbf{G}$.\\ $\cdot$ the voltage function $v$ that will appear on $\mathbf{G}$ as a consequence of hooking up a battery between $a$ and $Z$ with $v(a)=1$ and $v/Z=0$.\\ These functions are both harmonic on $\left(\{a\}\cup Z\right)$ and they take the same values on $\{a\}\cup Z$, so, according to the Uniqueness Principle, they will coincide. Thus we obtain the following probabilistic expression of the voltage function: \[v(x)=P_{x}(\tau_{a}<\tau_{z}).\] This fact will allow us to express the effective conductance from $a$ to $Z$ in a probabilistic way: \[C(a\leftrightarrow Z)=\pi (a)P[a\rightarrow Z]\] The chain is transient iff $P_{a}(T_{a}=\infty)>0 \quad \forall a$, that is iff $\lim_{n}P[a\rightarrow Z_{n}]>0$, where $Z_{n}$ are those in defenition 3.1 i). Now, using (3.1), we will obtain the required characterisation of transience. \section{The electric network framwork (second part)} The second part of the electric network framework of this paper will be describer through the following three definitions: \begin{defn} We consider an (electric) network having the following elements:\\ a)$\mathbf{G}$: \textbf{the vertex set.}\\ b)E: \textbf{the edges set} (E contains each edge with its both possible orientations).\\ c)$\breve{e}$: \textbf{the opposite} of the edge $e-e^{-}e^{+}$; $e^{+}$ is the \textbf{head} of $e$.\\ d)$l^{2}(\mathbf{}G)$: the Hilbert space of the \textbf{real functions on} $\mathbf{G}$ with the property $\sum_{x\in \mathbf{G}}f^{2}(x)<\infty$, with the scaler product \[(f,g)=\sum_{x\in \mathbf{G}}f(x)g(x).\] e)$l^{2}-(E)$: the Hilbert space of the \textbf{real antisymetric functions on E} with the property $\sum_{e\in E}\theta^{2}(e)<\infty$, with the scalar product \[(\theta,\theta^{'})=\frac{1}{2}\sum_{e\in E}\theta (e)\theta^{'}(e).\] f) the operator $d$: \[d: \quad l^{2}(\mathbf{G})\rightarrow l^{2}-(E),\quad dF(e)=F(e^{-})-F(e^{+}).\] g) the operator $d^{*}$: \[d^{*}:\quad l^{2}-(E)\rightarrow l^{2}(\mathbf{G}), \quad d^{*}\theta (x)=\sum_{e^{-}=x}\theta (e).\] \end{defn} \begin{defn} For this definition, we will assume that every function $\theta \in l^{2}-(E)$ represents a certain type of liquid that flows through the network. Let $\mathbf{G}$ be a finite network for the points a),...,f) and an infinite one for the point g).\\ a) \textbf{The quantity of liquid} (of type $\theta$) \textbf{that enters into the network through a vertex} $\mathbf{a}$ is by definition $d^{*}\theta (a)$.\\ b) For $A$ and $Z$ two disjoint, fixed subset of $\mathbf{G}$ (intuitively thought as a source and, respectively, \textbf{exit point} of the flow), we will call a function $\theta \in l^{2}-(E)$ a \textbf{flow from A to Z} if: \[d^{*}\theta > 0 \textbf{ on } A\] \[d^{*}\theta < 0 \textbf{ on } Z\] \[d^{*}\theta = 0 \textbf{ on } (A\cup Z).\] c) If $\theta$ is a flow from $A$ to $Z$. \textbf{the quantity of liquid that enters into the network} is by definition $\sum_{a \in A}d^{*}\theta (a)$.\\ d) If $\theta$ is a flow from $A$ to $Z$. \textbf{the quantity of liquid that flows out of the network} is by definition $\sum_{z \in Z}d^{*}\theta (z)$.\\ e) If $\theta$ is a flow from $A$ to $Z$, we define $\mathbf{strength(\theta)}$ to be $\sum_{a \in A}d^{*}\theta (a)$.\\ f) A \textbf{unit flow} is by definition a flow of strength $1$.\\ g) An antisymmetric function $\theta: \quad E\rightarrow\mathbb{R}$ is called a \textbf{unit flow from} $\mathbf{a}$ \textbf{to infinity} if $d^{*}\theta=1_{\{a\}}$.\\ \end{defn} \begin{defn} a) On the space $l^{2}-(E)$ we define the scalar product $(.;.)_{r}$ in the following way: \[(\theta,\theta^{'})_{r}=\frac{1}{2}\sum_{e\in E}\theta (e)\theta^{'}(e)r(e).\] b) \textbf{The energy} of an antisymmetric function $\theta: \quad E\rightarrow \mathbb{R}$ is defined to be \[\varepsilon (\theta)=\parallel\theta\parallel_{r}^{2}.\] c)We define \textbf{the unit flow along} $\mathbf{e}$ to be the function $\chi^{e}\in l^{2}-(E)$, \[\chi^{e}:=1_{e}-1_{\breve{e}}.\] d)We define the spaces \[\begin{array}{c}
\bigstar = sp\{\sum_{e^{-}=x}c(e)\chi^{e}; x\in \mathbf{G}\}\quad (\textbf{star space})\\
\lozenge=sp\{\sum_{i=1}^{n}\chi^{e_{i}}; e_{1},...,e_{n}\in E \text{ oriented cycle; } n\geq 0\}\quad (\textbf{cycle space})
\end{array} \] \end{defn}
In the above framework, it will be useful for the purpose of this paper to present the following characterisation of transience by means of flows (see [LP], theorem 2.10). \begin{thm} An infinite electric network $\mathbf{G}$ is \textbf{transient iff there is a unit flow on} $\mathbf{G}$ \textbf{of finite energy from any vertex to infinity.} \end{thm} the steps of the proof are the following (for details see \cite{LP}):\\ 1) Proving that the operators $d$ and $d^{*}$ from definition 4.1 f), g) are adjoint in the sense that \[(\theta,dF)=(d^{*}\theta,F),\] for any $\theta \in l^{2}-(E)$ and any $F\in l^{2}(\mathbf{G})$.\\ 2) Writing down Ohm and Kirchhoff Laws by means of the operators $d$ and $d^{*}$: \[\text{Ohm's Law: } dv=i.r\] \[\text{Kirchhoff's Law: } d^{*}i(x)=0,\] if $x$ is not connected to any battery.\\ 3) Proving the two properties of flows:\\ a) $\sum_{a\in A}d^{*}\theta (a)=-\sum_{a\in Z}d^{*}\theta (a)$\\ (The quantity of liquid that enters into the network is equal to the quantity of liquid that flows out of the network).\\ b) $(\theta,dF)=Strength(\theta)[F(A)-F(Z)],$\\ for any real function $F$ that is constant on $A$ and $Z$.\\ 4) Writing down Kirchhoff's Law by mean's of the scalar product $(.;.)_{r}$: \[\text{The node's Law: }\left(\sum_{e^{-}=x}c(e)\chi^{e},i\right)_{r}=0,\] if $x$ is not connected to any battery. \[\text{The cycles's Law: }\left(\sum_{j=1}^{n}\chi^{ej},i\right)_{r}=0,\] for any oriented cycle $e_{1},...,e_{n}$.\\ 5) Proving that the space $l^{2}-(E)$ is the direct sum of the orthogonal subspaces $\bigstar$ and $\lozenge$.\\ 6) Proving Thomson's Principle:\\ THOMSON'S PRINCIPLE: Let $\mathbf{G}$ be a finite network, $A$ and $Z$ two disjoint subsets of $\mathbf{G}$, $\theta$ a unit flow from $A$ to $Z$ and $i$ the electric current flow from $A$ to $Z$ with $d^{*}i=d^{*}\theta$. Then \[\varepsilon (\theta)\geq \varepsilon (i).\] (Of all the flow from $A$ to $Z$ having the same $d^{*}$, the electric current flow is the energy minimizes).\\ 7)Completing the proof of the theorem by means of the previous six steps.
\section{The basic theorem} The two characterisations of the transience of an infinite network presented in sections $3$ and $4$ are the background for the following theorem (see \cite{LP}, proposition 2.11), whose importance for our future purposes the following facts:\\ 1) It extends Ohm's law from the finite to the infinite case.\\ 2) It gives the electric expression of the hitting vector for an infinite network.\\ 3) It gives the electric expression of the Green function for an infinite network. \begin{thm} Let $\mathbf{G}$ be a trasient network and $(\mathbf{G}_{n})_{n\geq 0}$ a sequence of finite subgraphs, containing a vertex a, that exhaust $\mathbf{G}$. We contract to $z_{n}$ the vertices outside $\mathbf{G}$, forming $\mathbf{G}^{(n)}$ (see Figure \ref{fig1}).\\ \end{thm} Let $i_{n}$ be the unit current flow $i_{n} C^{n}$ from $a$ to $z_{n}$. Then $(i_{n})_{n}$ has a point wise limit $i$ on $\mathbf{G}$, that is the unique unit flow on $\mathbf{G}$ from $a$ to infinity of minimum energy.\\ Let $v_{n}$ be the voltages on $\mathbf{G}^{(n)}$ corresponding to $i_{n}$ and with $v_{n}(z_{n})=0$. Then $v:=\lim_{n}v_{n}$ exists on $\mathbf{G}$, is finite and has the properties:\\ 1) $dv=ir$\\ 2) $v(a)=\varepsilon (i)=R(a\leftrightarrow \infty)$\\ 3) $\frac{v(x)}{v(a)}=P_{x}[\tau_{a}<\infty],\quad \forall x$.\\ Let's start in a the random walk on $\mathbf{G}$. For any $x$, the expected number of visits to $x$ is \[\mathbf{G}(a,x)=\pi (x)v(x).\] For any edge $e$, the expected signed number of crossings of $e$ is $i(e)$.\\ For the proof see [LP]. \begin{figure}\label{fig1}
\end{figure}
\section{The homogenous tree as an electric network} \begin{defn} a) A \textbf{tree} is a graph that is locally finite, connected and without loops. Notation: for $x$ and $y$ neighborn, we will write $x\sim y$.\\ b) A \textbf{path} is a finite or infinie sequence of vertices $[v_{0},v_{1},...]$ such that $v_{k}\sim v_{k+1}$ for any $k\geq 0$.\\ c) A \textbf{geodesic path} is a path $[v_{0},v_{1},...]$ such that $v_{k-1}\sim v_{k+1}$ for any $k\geq 0$. \end{defn} The tree we will consider will be \textbf{infinite} (most of the time), homogenous, of degree $q+1$ (each vertex has exactly $q+1$neighbours), $q \geq 2$. It can be viewed as in figures $\ref{fig2}$ or $\ref{fig3}$. We will denote by $0$ an arbitrary vertex, fixed as a root. We will denote by $T$ the vertex set. \begin{figure}\label{fig2}
\end{figure} \begin{figure}\label{fig3}
\end{figure}
\begin{defn} a) We define the distance $\mathbf{d(u,v)}$ between two vertices $u$ and $v$ to be the number of edges of the geodesic path from $u$ to $v$.\\
b)\textbf{The length of a vertex v} is by definition $d(0,v)$; it will be denoted by $|v|$. \end{defn} We will attach to each edge the conductance $1$. So the tree will become an electric network. The relation $\sum_{y \sim x}c(x,y)<\infty$ will be true for any vertex $x$, then, as in the introduction, we will associate a reversible Markov chain to this network. This is the chain of matrix $P=(p_{xy})_{x,y\in T}$, \[p_{xy}=\begin{array}{c}
\frac{1}{q+1},\text{ if } y\sim x \\
0,\text{ otherwise}
\end{array} \]
It represents the simple random walk on the homogenous tree. Writing the relation $c(x,y)=\pi (x)p_{xy}$ for an arbitrary edge $xy$, we obtain the expression of the reversibility function $\pi$: \[\pi:\quad T\rightarrow \mathbb{R}_{+}^{*}, \quad\quad \pi (x)=q+1.\] \begin{rem} This tree has finite resistance to infinity it represents the prototype of network that has finite resistance to infinity. \end{rem} \begin{proof} Let's compute the effective resistance from $0\text{ to } \infty$. According to definition $3.1$ j), we will have:\\ \[R(0\leftrightarrow \infty)=\lim_{n}R(0\leftrightarrow z_{n}),\] where $z_{n}$ is the point which concentrates all the points of level $\geq n$.\\ During this procedure, we throw away the resulting loops, so it is just like concentrating in $z_{n}$ all the vertices of level $n$ (see figure \ref{fig4}). \begin{figure}\label{fig4}
\end{figure} We have to transform this network successively into a simpler one, in order to compute $R(0\leftrightarrow z_{n})$ (see figure \ref{fig5}) \begin{figure}\label{fig5}
\end{figure} We have:\\ a) Unit conductances.\\ b) Unit conductances.\\ c) The conductances are respectively equal to $q+1$, $q(q+1)$, $q^2 (q+1)$,...\\ d) The resistance is equal to $\frac{1}{q+1}+\frac{1}{q(q+1)}+\frac{1}{q^{2}(q+1)}+...$\\ So, \[R(0\leftrightarrow z_{n})=\frac{1}{q+1}.\frac{1-\left(\frac{1}{q}\right)^{2}}{1-\frac{1}{q}}\] Consequently, \[R(0\leftrightarrow z_{n})=\lim_{n}\frac{1}{q+1}.\frac{1-\left(\frac{1}{q}\right)^{2}}{1-\frac{1}{q}}=\frac{q}{q^2-1}<\infty.\] \end{proof} \begin{rem} Having a finite resistance from any vertex to infinity, the homogenous tree will have a strictly positive conductance from any vertex to infinity. Thus, according to theorem 3.1, the associated Markov chain will be transient. \end{rem} \section{New results} We present now the final purpose of this paper: results upon the electric network of the homogenous tree.We first prove a proposition that will express voltages and currents intensities as functions of distance to the root. \begin{prop} Let $T$ be the transient network given by the infinite homogenous tree of degree q+1. We take root $0$ to be the reference point in theorem 5.1. Then:\\ a) The limit function $i$ given by the same theorem is: \[i(x,y)=\begin{array}{c}
\frac{1}{q+1}.\frac{1}{q^{\mid x \mid}}, \quad \forall x,y\in T, \quad y\sim x, \mid y\mid > \mid x\mid\\
-\frac{1}{q+1}.\frac{1}{q^{\mid x\mid -1}}, \quad x,y \in T, \quad y\sim x, \mid y\mid <\mid x\mid
\end{array} \] The limit function $v$ given by the same theorem is: \[v(x)=\frac{q}{q^2-1}.q^{-\mid x\mid},\quad \forall x\in T.\] \end{prop} \begin{proof} a) Let $(T_{n})_{n\geq 0}$ be the following sequence of finite subtrees, containing $0$, that exhaust $T$: \[T_{0},\quad T_{n}=\{x\in T, \mid x\mid \leq n\},\quad \forall n\geq 1.\] We contract to $z_{n}$ the vertices outside $T_{n}$ and denote $T^{(n)}$ the union between $T_{n}$ and $\{z_{n}\}$. (see \ref{fig6}). Let in be the unit current flow in $T^{(n)}$ from $0$ to $z_{n}$. Let $v_{n}$ be the voltage on $T^{(n)}$ corresponding to $i_{n}$ and with $v_{n}(z_{n})=0$. Let $T_{1}^{(n)},...,T_{q+1}^{(n)}$ be the $q+1$ circuits that appear between $0$ and $z_{n}$ when we hook up a battery between $0$ and $z_{n}$, with $v_{n}(0)>0$ and $v_{n}(z_{n})=0$ (see figure \ref{fig7}).\\ The homogenity of the edges resistors will imply: \[R^{T_{1}^{n}}(0\leftrightarrow z_{n})=...=R^{T_{q+1}^{n}}(0\leftrightarrow z_{n}).\] Hence, according to Ohm's Law, written $q+1$ times between $0$ and $z_{n}$, \[i_{n}(0,x_{1})=...=i_{n}(0,x_{q+1}),\] $x_{1},...,x_{q+1}$ being the points of level 1.\\ We know $d^{*}i_{n}=1$, so we will have: \[i_{n}(0,x_{1})=...=i_{n}(0,x_{q+1})=\frac{1}{q+1}.\] Consequently, Ohm's Law written respectively for $0$ and $x_{1},...,0$ and $x_{q+1}$ will imply: \[v_{n}(x_{1})=...=v_{n}(x_{q+1})\] The above procedure for the points $0$ and $z_{n}$ will be now applied, successively, to the points $x_{1}$ and $z_{n},...,x_{q+1}$ and $z_{n}$. We will obtain the intensities of the currents passing between the first two levels: \[i_{n}(x_{1},...,y_{1})=...=i_{n}(x_{q+1},...,y_{q(q+1)})=\frac{1}{q+1}.\frac{1}{q},\] where we have denoted by $y_{1},...,y_{q(q+1)}$ the points of the second level.\\ We obtain in a similar manner the intensities of the currents going between any two consecutive levels: \[i_{n}(x,y)=\frac{1}{q+1}.\frac{1}{q^{\mid x\mid}}, \quad \forall x,y, x\sim y, \mid y\mid=\mid x\mid + 1.\] Let us now consider arbitrary $x$ and $y$, $x\sim y$, $\mid y\mid=\mid x\mid +1$ $(T_{n})_{n}$ exhaust $T$, sowe can find $n_{0}$ such that $x,y\in T_{n_{0}}$. Applying the above procedure to each $T_{n}$, $n\geq n_{0}$, we get: \[i_{n}(x,y)=\frac{1}{q+1}.\frac{1}{q^{\mid x\mid}}, \quad \forall n\geq n_{0},\] hence, by passing to the limit, \[i_{n}(x,y)=\frac{1}{q+1}.\frac{1}{q^{\mid x\mid}}.\] We will also have: \[i_{n}(x,y)=-i_{n}(y,x)=\frac{1}{q+1}.\frac{1}{q^{\mid x\mid}}=-\frac{1}{q+1}.\frac{1}{q^{\mid y\mid-1}}\] so we have proved a). b) The proof will be completed by induction after $\mid x\mid$.\\ I. $\mid x\mid=1$: Ohm's Law between $0$ and $x$ will imply \[v(0)-v(x)=i(0,x).1\] But $v(0)=R(0\leftrightarrow \infty)=\frac{q}{q^{2}-1}$ according to theorem 5.1 and $i(0,x)=\frac{1}{q+1}$ according to a), so we get \[\frac{q}{q^{2}-1}-v(x)=\frac{1}{q+1}.\] Hence \[v(x)=\frac{q}{q^2-1}.\frac{1}{q}\] II. $\mid x\mid \sim \mid y \mid +1$ Ohm's law written between $x$ and $y$, $x\sim y$, $\mid y\mid=\mid x\mid + 1$, will imply: \[v(x)-v(y)=i(x,y).1\] But we have $v(x)=\frac{q}{q^{2}-1}.\frac{1}{q^\mid x\mid}$ and $i(x,y)=\frac{1}{q+1}.\frac{1}{q^{x}}$ according to a), so \[v(y)=\frac{q}{q^{2}-1}.\frac{1}{q^{\mid x\mid}}-\frac{1}{q+1}.\frac{1}{q^{\mid x\mid}}=\frac{q}{q^{2}-1}.\frac{1}{q^{\mid x\mid +1}}.\] I and II will imply \[v(x)=\frac{q}{q^2-1}.\frac{1}{q^{\mid x\mid}}\quad \forall x\] \end{proof} \begin{figure}\label{fig6}
\end{figure} \begin{figure}\label{fig7}
\end{figure} The corollary to come will present the particular form of the Green function for the transient network of the homogenous tree. Thus we arrive to the particular form of this function mentioned in \cite{GS}, Section 2. \begin{cor} Let $T$ be the transient network of the infinite homogenous tree off degree $q+1$ and $\mathbf{G}(.;.)$ the associated Green function. Then $\mathbf{G}(.;.)$ has the following expression: \[\mathbf{G}(a,x)=\frac{q}{q-1}.q^{-d(a,x)},\quad \forall a,x\in T.\] \end{cor} \begin{proof} Theorem 5.1 asserts that \[\mathbf{G}(0,x)=\pi(x)v(x),\] so according to proposition 7.1 b), we will get \[\mathbf{G}(0,x)=(q+1).\frac{q}{q^{2}-1}.\frac{1}{q^{\mid x\mid}}=\frac{q}{q-1}.q^{-\mid x\mid}.\] Writing this in the form \[G(0,x)=\frac{q}{q-1}.q^{-d(0,x)}\] and taking into account the fact that $0$ may be arbitrarily chosen, we get: \[G(a,x)=\frac{q}{q-1}.q^{-d(a,x)}\quad \forall a,x\in T.\] \end{proof} The following proposition will express the hitting times and the transitions number on the homogenous tree network as functions of the distance to the root. \begin{prop} Let $T$ be the transient network of the infinite homogenous tree of degree $q+1$. Then\\ a) The hitting vector of the root is \[\left(q^{-\mid x\mid}\right)_{x\in T}\] b) The hitting vector of an arbitrary point $a\in T$ is \[\left(q^{-d(a,x)}\right)_{x\in T}.\] c) For any starting point $a$, the expected number of transitions of an edge $xy$ is \[E_{a}[S_{xy}]=\frac{q}{q^{2}-1}.q^{-d(a,x)}.\] \end{prop} \begin{proof} a) According to theorem 5.1, the hitting of the root is \[Px(\tau_{a}<\infty)=\frac{v(x)}{v(0)}.\] Using the voltages expressions given by proposition 7.1, we obtain \[Px(\tau_{0}<\infty)=\frac{\frac{q}{q^{2}-1}.q^{-\mid x\mid}}{\frac{q}{q^{2}-1}}=q^{-\mid x\mid}.\] b) Using the similarity property of the vertices (we have no intrinsec way to distinguish a vertex from another) and noticing that $\mid x\mid=d(0,x)$, we generalise a) writing that \[\forall a, \forall x : \quad P_{x}(\tau_{a}<\infty)=q^{-d(a,x)}.\] c) We have: \[\forall a,\quad E_{a}[S_{xy}]=E_{a}[\sum_{n\geq 0}1_{\{x_{n}=x,x_{n+1}=y\}}]=E_{a}[\sum_{n\geq 0}1_{x_{n}=a}]p_{xy}=\mathbf{G}(a,x)p_{xy}\] Using the particular form of the Green function for the homogenous tree and the relation $p_{xy}=\frac{1}{q+1}$, we get: \[E_{a}[S_{xy}]=\frac{q}{q-1}.q^{-d(a,x)}.\frac{1}{q+1}=\frac{q}{q^{2}-1}/q^{-d(a,x)}.\] \end{proof}
\begin{rem} This expected number of transitions is exactly the voltage that appears in $x$ when the unit injection of current is made in $a$. \end{rem} Let $\{a\}$ and $Z$ be two disjoint subsets of the finite homogenous tree $T$. The last proposition will express the probability of reaching $Z$ before $a$ as a function of the degree and of the distance. The finite homogenous tree is obtained from the infinite one by keeping only the first $n$ levels.The edges conductances are still equal to one $P^{'}$, the transition matrix of the associated Markov chain, will obviously be finite. It can be expressed in the following way as function of the matrix $P$ of the infinite network: \[p^{'}_{xy}=\begin{array}{c}
1,\text{ for } x \text{ on level } n \text{ and } y\sim x \\
p_{xy}, \text{ otherwise}.
\end{array} \] On this finite tree, the effective resistance between two points $a$ and $x$ is exactly the distance between them. We can see this as follows: we hook up a battery between $a$ and $x$ and write Kirchhoff's Law for all vertices starting with the level $n$ ones and going up to those on the geodesic path between $a$ and $x$. The result will be that no current goes through any edge expert those on this geodesic path. So we can remove the edges outside it reducing the circuit to the resistors of the geodesic path from $a$ to $x$. The effective resistance from $a$ to $x$ will be the sum of their resistances. This is exactely $d(a,x)$. \begin{prop} Let $T$ be the transient network of the finite homogenous tree of degree $q+1$ and $0$ its root. We will have: a) When $Z$ equals the level $n$ vertex set, \[P[0\rightarrow Z]=q^{n-1}.\frac{q-1}{q^{n}-1}. \text{ see \ref{fig8}}\] b) When the levels number is $n$, $a$ i an arbitrary level $n$ point and $Z$ equals the union of the subtrees rooted in $0$ and not containing $a$, \[P[a\rightarrow Z]=\frac{1}{n}. \text{ see \ref{fig9}}\] c) When the levels number is greater than $n$, $a$ is an arbitrary level $n$ point and $Z$ is the one in b), \[P[a\rightarrow Z]=\frac{1}{n(q+1)}. (\text{see figure \ref{fig10}})\] d)When $a$ is a terminal vertex and $x$ is arbitrary, \[P[a\rightarrow x]=\frac{1}{d(a,x)}. (\text{see figure \ref{fig11}})\] e)When $a$ is not terminal and $x$ is arbitrary, \[P[a\rightarrow x]=\frac{1}{d(a,x).(q+1)}. (\text{see figure \ref{fig12}})\] \begin{figure}\label{fig8}
\end{figure} \begin{figure}\label{fig9}
\end{figure} \begin{figure}\label{fig10}
\end{figure} \begin{figure}\label{fig11}
\end{figure} \begin{figure}\label{fig12}
\end{figure} \end{prop}
\begin{proof} We will have:\\ a) According to 3.1, \[P[0\rightarrow Z]=\frac{C(0\leftrightarrow Z)}{\pi (0)}=\frac{1}{R(0\leftrightarrow Z)\pi (0)}\] $R(0\leftrightarrow Z)$ is given by remark 6.1: \[R(0\leftrightarrow Z)=\frac{1}{q+1}.\frac{1-\left(\frac{1}{q}\right)^{n}}{1-\frac{1}{q}}=\frac{1}{q^{2}-1}.\frac{q^{n}-1}{q^{n-1}}\] $\pi (0)=q+1$, so we get: \[R(0\leftrightarrow Z)=(q^{2}-1).\frac{q^{n-1}}{q^{n}-1}.\frac{1}{q+1}=q^{n-1}.\frac{q-1}{q^{n}-1}.\] b) $P[a\rightarrow Z]=P[a\rightarrow 0]=\frac{C(a\leftrightarrow 0)}{\pi (a)}=\frac{1}{R(a\leftrightarrow 0)\pi (a)}=\frac{1}{n.1}=\frac{1}{n}$\\ c)$P[a\rightarrow Z]=\frac{1}{R(a\leftrightarrow 0)\pi (a)}=\frac{1}{n(q+1)}$\\ d) $P[a\rightarrow x]=\frac{C(a\leftrightarrow x)}{\pi (a)}=\frac{1}{R(a\leftrightarrow x)}.\frac{1}{\pi (a)}=\frac{1}{d(a,x).1}=\frac{1}{d(a,x)}$\\ e) $P[a\rightarrow x]=\frac{1}{R(a\leftrightarrow x)\pi (a)}=\frac{1}{d(a,x)(q+1)}$ \end{proof}
\end{document} | arXiv | {
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\begin{document}
\begin{abstract} The spectral sequence associated to the Arone-Goodwillie tower for the $n$-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod $2$ cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod $2$ Steenrod algebra, together with properties of a generalized class of almost unstable modules which is introduced here.
An essential ingredient of the proof is a non-vanishing result for certain extension groups in the category of unstable modules localized away from nilpotents. \end{abstract}
\maketitle
\section{Introduction} \label{sect:intro}
A fundamental question in algebraic topology is to ask what modules over the mod $p$ Steenrod algebra $\mathscr{A}$ can be realized as the reduced mod $p$ cohomology $H^* (X)$ of a space. First obstructions are provided by the fact that the module must be unstable and that this structure should be compatible with the cup product. For example, Steenrod asked what polynomial algebras can be realized as the cohomology of a space. At the opposite extreme one can ask what unstable algebras with trivial cup square (at the prime two) can be realized; a partial response is provided below in Theorem \ref{THM:K}.
The structure theory of the category $\mathscr{U}$ of unstable modules allows the formulation of precise questions of particular interest in the case where $H^* (X)$ is nilpotent.
Kuhn \cite{Kuhn_annals} proposed a series of highly-influential non-realization conjectures, postulating significant restrictions on the structure of $H^* (X)$ as an unstable module.
Many of these are now theorems \cite{Schwartz,CGPS}. The conjectures are phrased in terms of the nilpotent filtration of the category $\mathscr{U}$; this is a decreasing filtration, where $\mathscr{N}il_n$
is the smallest localizing subcategory of $\mathscr{U}$ containing all $n$-fold suspensions. A general question is the following: what can be said about the structure of $H^* (X)$ as an unstable module
if it belongs to $\mathscr{N}il_n$?
The condition $H^* (X) \in \mathscr{N}il_n$ has topological significance when Lannes' mapping space technology can be applied: it is equivalent to $\mathrm{map} (BV, X)$ being $(n-1)$-connected for all elementary abelian $p$-groups $V$. It is clear that $n$-fold suspensions $\Sigma^n Y$ satisfy the hypothesis and,
since the algebraic suspension restricts to a functor $\Sigma: \mathscr{N}il_n \rightarrow\mathscr{N}il_{n+1}$, it is most interesting to consider the case where $X$ is not a suspension.
The largest submodule of an unstable module $M$ which lies in $\mathscr{N}il_n$ is written $\mathrm{nil}_n M$; if $M$ lies in $\mathscr{N}il_n \backslash \mathscr{N}il_{n+1}$, then there is a short
exact sequence of unstable modules:
\[
0
\rightarrow
\mathrm{nil}_{n+1} M
\rightarrow
M
\rightarrow
\Sigma^n \rho_n M
\rightarrow
0
\] where $\rho_n M$ is a reduced unstable module (ie contains no non-trivial nilpotent submodule) which is non-zero. If in addition $M$ is $n$-connected, then $\rho_n M$ is connected (trivial in degree zero).
It is now known that (at least up to nilpotent unstable modules) there are few restrictions which can be placed on $\rho_n M$ (for $n \geq 1$), following the affirmation of the Lannes and Schwartz Artinian conjecture \cite{2014arXiv1408.3694P,2014arXiv1409.1670S}. Namely, generalizing Kuhn's observation \cite{Kuhn_annals}, examples can be manufactured by considering the topological realization of the beginning of an injective resolution (modulo nilpotents) of a given reduced module and forming the homotopy cofibre: \[ \mathrm{hocofib} \Big\{ \bigvee_i \Sigma^n BV(i)_+ \rightarrow \bigvee_j\Sigma^n
BV(j)_+ \Big\} \] where $\{ V(i)\}$, $\{V(j)\}$ are finite sets of finite rank elementary abelian $p$-groups. The Artinian conjecture ensures that all finitely cogenerated modules admit such finite type presentations
(modulo nilpotents).
Kuhn's non-realization conjectures highlight the interest of the case where $\rho_n M$ is finitely generated over the Steenrod algebra. If $M$ is $n$-connected,
the cases which have already been proved show that $\mathrm{nil}_{n+1}M$ must actually be large; for example, $M$ cannot itself be finitely generated under these hypotheses. These results rely upon
Lannes' $T$-functor technology to reduce to the smallest non-trivial case \cite{CGPS}.
The purpose of this paper is to show how the analysis of the first two non-trivial columns of the spectral sequence associated to the Arone-Goodwillie tower for the functor $ X \mapsto \Sigma^\infty \Omega^n X$
impose conditions on the first two non-trivial layers of the nilpotent filtration of $H^*(X)$. In the case $n=1$, this is an application of the Eilenberg-Moore spectral sequence and the result recovered is a generalization
of the main result of \cite[Section 6]{CGPS}. The case $n >1$ is new and arose as an offshoot of the author's programme to express the non-realization results obtained by Kuhn \cite{Kuhn_nonrealization} (for $p=2$) and
by Büscher, Hebestreit, Röndigs and Stelzer \cite{BHRS} (for odd primes), in terms of obstruction classes in suitable $\mathrm{Ext}$ groups. The key result is provided, on passage to the quotient category $\mathscr{U}/ \mathscr{N}il$, by a
generalization of a theorem of Kuhn (see Theorem \ref{thm:split_mono_gen}) showing that certain $\mathrm{Ext}$ groups in $\mathscr{U}/ \mathscr{N}il$ are non-trivial.
An application of the case $n=1$ is the following:
\begin{THM}
\label{THM:USigmaM}
Let $M$ be a connected unstable module over $\mathbb{F}_2$ of finite type such that $\rho_0 M$ is non-zero and finitely generated.
Then $U (\Sigma M)$, the Massey-Peterson enveloping algebra of the suspension of $M$, is not realizable as the $\mathbb{F}_2$-cohomology of a space.
\end{THM}
The enveloping algebra $U (\Sigma M)$ is isomorphic to the exterior algebra $\Lambda^* (\Sigma M)$, so this is a case of the following Theorem
(for a more refined statement, see Corollary \ref{cor:refined_COR}), in which $QK$ denotes the module of indecomposables:
\begin{THM} \label{THM:K} Let $K$ be a connected unstable algebra of finite type over $\mathbb{F}_2$ such that the cup square $Sq_0$ acts trivially on the augmentation ideal $\overline{K}$. If $QK$ is a $1$-connected unstable module such that $\rho_1 QK$ is non-zero and finitely generated, then $K$ is not realizable as the $\mathbb{F}_2$-cohomology of a space. \end{THM}
Note that the non-realization result of Gaudens and Schwartz \cite{GS12,CGPS} does not apply directly here, since no restriction is placed upon the higher nilpotent filtration of $QK$.
\begin{rem} The hypothesis that $\rho_1 QK$ is a finitely generated unstable module is essential. For example, consider $H^* (X)$ for $X=SU$, so that $\Omega X \simeq
BU$. \end{rem}
The main results of the paper are Theorem \ref{thm:main} (for $n>1$) and Theorem \ref{thm:main_n=1} (for $n=1$). For these, the prime is taken to be two; the modifications necessary in the odd primary case are indicated in Section \ref{sect:podd}.
To give an idea of the flavour of the results, consider the following:
\begin{THM}
For $1 \leq n \in \mathbb{N}$ and $X$ an $n$-connected space such that $H^* (X)$ is of finite type and $\rho_n H^* (X)$ is finitely generated and non-trivial,
$$H^* (X) / \mathrm{nil}_{n+2} H^* (X) $$
is not the $n$-fold suspension of an unstable module.
\end{THM}
At first sight, this result may not appear surprising: the structure theory of unstable algebras (modulo nilpotents) implies that, under the given hypotheses, $\rho_n H^* (X)$ cannot be finitely generated if $X$ is the $n$-fold suspension of a connected space. The theorem shows that this is already exhibited algebraically by the structure of $H^* (X) / \mathrm{nil}_{n+2} H^* (X) $.
This is a fundamental point: in the spectral sequence the columns will not in general be unstable modules. For $n=1$, this is not a serious difficulty, since it is known that the columns of the $E_2$-term of the Eilenberg-Moore spectral sequence are unstable. For the general case, this is no longer true, yet the spectral sequence converges to an unstable module. To allow the nilpotent filtration of unstable modules to be brought to bear, the notion of an {\em almost} unstable module is introduced here, which is shown to be sufficient to cover the cases of interest.
Theorem \ref{thm:main} and Theorem \ref{thm:main_n=1} are much more precise, relating $\rho_n H^* (X)$ and the next layer of the nilpotent filtration, $\rho_{n+1} H^* (X)$. Roughly speaking, for $n >1$ the result states that $\rho_{n+1}H^* (X)$ is at least as large as $\rho_n H^* (X)$; this is exhibited by the injectivity (modulo smaller objects relative to the Krull filtration of $\mathscr{U}$) of the natural transformation \[
\Phi \rho_n H^* (X) \rightarrow \rho_{n+1} H^* (X) \] that arises from the non-exactness of the iterated loop functor $\Omega^n : \mathscr{U} \rightarrow \mathscr{U}$.
For $n=1$, the result is stronger, here the relevant transformation is \[
S^2 (\rho_1 H^* (X)) \rightarrow \rho_2 H^* (X), \] induced by the cup product of $H^* (X)$. The approach is unified here, showing how the two cases are related.
This gives information on the beginning of the nilpotent filtration, whereas the proofs of the known cases of Kuhn's non-realization conjectures give global information. Where applicable, Lannes' mapping space technology can be used to study the higher parts of the nilpotent filtration; the ideas involved will be transparent to the experts and to the readers of \cite{CGPS} and are not developed here.
{\bf Organization of the paper:}
Background is surveyed in Section \ref{sect:alg_prelim}; readers should consult this as and when is necessary.
The technical notion of an almost unstable module is introduced in Section \ref{sect:almost}; this is necessary to be able to control the
image of differentials in the spectral sequence, as is explained in Section \ref{sect:auss}. The spectral sequence derived from
the Arone-Goodwillie tower is reviewed in Section \ref{sect:tower} and the calculational input is provided in Section \ref{sect:cohomEP},
namely the calculation of the $E_1$-term of the spectral sequence via the cohomology of extended powers. In particular, it is shown that, for the case
at hand, the columns of the $E_1$-term are almost unstable. The case of the second extended power admits an explicit algebraic model, as explained in Section \ref{sect:alg_model}, which not only makes the results of the previous section more explicit in this case, but also provides a model for the differential $d_1$. The input from homological algebra is explained in Section \ref{sect:essential},
refining a theorem of Kuhn; this is the key ingredient in the proofs of the
two main theorems, which are given in Section \ref{sect:main}. Section \ref{sect:podd}
sketches the modifications required in the odd primary case.
\section{Algebraic preliminaries} \label{sect:alg_prelim}
This section reviews background, referring to the literature (in particular \cite{schwartz_book} and \cite{K14}) for details. As usual, $\mathscr{U}$ denotes the full subcategory of unstable modules in $\mathscr{M}$, the category of graded modules over the mod $p$ Steenrod algebra $\mathscr{A}$. The inclusion $\mathscr{U} \rightarrow \mathscr{M}$ has left adjoint $\Omega^\infty : \mathscr{M} \rightarrow \mathscr{U}$, the destabilization functor.
The suspension functor $\Sigma : \mathscr{M} \rightarrow \mathscr{M}$ restricts to $\Sigma : \mathscr{U} \rightarrow \mathscr{U}$ and the iterated suspension functor $\Sigma^t : \mathscr{U} \rightarrow \mathscr{U}$ ($t \in \mathbb{N}$)
has left adjoint the iterated loop functor $\Omega^t : \mathscr{U} \rightarrow \mathscr{U}$, which identifies with the composite functor $\Omega^\infty \Sigma^{-t}$ restricted to $\mathscr{U}$.
The category of unstable algebras is denoted $\mathscr{K}$ and the Massey-Peterson enveloping algebra $U : \mathscr{U} \rightarrow \mathscr{K}$ is the left adjoint to the forgetful functor $\mathscr{K} \rightarrow \mathscr{U}$; $U$ takes values in the category $\unstalg_a$ of augmented unstable algebras. The indecomposables functor $Q: \unstalg_a \rightarrow \mathscr{U}$ is given explicitly by $Q K := \overline{K} / \overline{K}^2$, where $\overline{K}$ is the augmentation ideal.
\subsection{The nilpotent filtration}
The category of unstable modules $\mathscr{U}$ has nilpotent filtration: \[
\ldots \subset \mathscr{N}il_{i+1} \subset \mathscr{N}il_i \subset \ldots \subset \mathscr{N}il_1 \subset \mathscr{N}il_0 = \mathscr{U}, \] where $\mathscr{N}il_s$ is the smallest localizing subcategory containing all $s$-fold suspensions \cite{schwartz_book,K14}. In particular $\mathscr{N}il_1$ is the subcategory of nilpotent unstable modules $\mathscr{N}il$.
The inclusion $\mathscr{N}il_s \hookrightarrow \mathscr{U}$ admits a right adjoint $\mathrm{nil}_s : \mathscr{U} \rightarrow \mathscr{N}il_s \subset \mathscr{U}$ so that an unstable module $M$ has a natural, convergent decreasing filtration: \[
\ldots \subset \mathrm{nil}_{s+1} M \subset \mathrm{nil}_s M \subset \ldots \subset \mathrm{nil}_0 M = M \] and, for $s \in \mathbb{N}$, $\mathrm{nil}_s M /\mathrm{nil}_{s+1} M \cong \Sigma^s \rho_sM$, where $\rho_s M$ is a reduced unstable module\footnote{The notation $\rho_s$ is used to avoid possible confusion with the Singer functors.}. (An unstable module is reduced if it contains no non-trivial suspension.)
\subsection{Functors between $\mathbb{F}$-vector spaces} \label{subsect:F}
Functors on vector spaces over a finite field $\mathbb{F}$ arise naturally in the study of unstable modules via Lannes' $T$-functor \cite{schwartz_book}.
\begin{nota}
For $\mathbb{F}$ a finite field, let $\mathscr{F}$ denote the category of functors from finite-dimensional $\mathbb{F}$-vector spaces to $\mathbb{F}$-vector spaces and $\mathscr{F}_\omega \subset \mathscr{F}$ the full subcategory of locally finite (or analytic) functors. \end{nota}
The category $\mathscr{F}$ is tensor abelian with enough projectives and injectives. A functor is finite if it has a finite composition series and is locally finite if it is the colimit of its finite subobjects.
A functor $F$ is polynomial of degree $d$ if $\Delta ^{d+1} F=0$, where $\Delta:\mathscr{F} \rightarrow \mathscr{F}$ is the difference functor defined by $\Delta F(V):= F(V \oplus \mathbb{F})/ F(V)$.
Over the prime field $\mathbb{F}_p$, the quotient category $\mathscr{U} / \mathscr{N}il$ is equivalent to $\mathscr{F}_\omega$ and the localization functor $\mathscr{U} \rightarrow \mathscr{U} /\mathscr{N}il$ gives the exact functor $l : \mathscr{U} \rightarrow \mathscr{F}$ which can be identified in terms of Lannes' $T$-functor as $M \mapsto \{V \mapsto (T_V M )^0 \}$ \cite{schwartz_book}.
\begin{nota} For $d \in \mathbb{N}$, denote by $\mathscr{F}_d$ the full subcategory of $\mathscr{F}$ of functors of polynomial degree $d$. \end{nota}
\subsection{Examples of polynomial functors} \label{subsect:basic_functors}
A number of polynomial functors arise here, which are closely related to the $n$th tensor power functor $T^n : V \mapsto V^{\otimes n}$. The symmetric group $\mathfrak{S}_n$ acts naturally by place permutations on $T^n$, giving: \begin{enumerate}
\item
the $n$th divided power $\Gamma^n := (T^n)^{\mathfrak{S}_n}$;
\item
the $n$th symmetric power $S^n:= (T^n)/\mathfrak{S}_n$. \end{enumerate} These functors are dual under the Kuhn duality functor $D : \mathscr{F}^\mathrm{op} \rightarrow \mathscr{F}$, given by $DF (V):= F(V^*) ^*$ (see \cite{KI}).
Similarly, there is the $n$th exterior power functors $\Lambda ^n$, which is self-dual. The functors $S^1, \Lambda^1 , \Gamma^1$ coincide with $\mathrm{Id} : V \mapsto V$.
The Frobenius $p$th power map induces a natural transformation $S^n \rightarrow S^{np}$. Henceforth taking $p=2$, there is a non-split short exact sequence \begin{eqnarray} \label{eqn:frob_ses}
0 \rightarrow \mathrm{Id} \rightarrow S^2 \rightarrow \Lambda^2 \rightarrow 0, \end{eqnarray} representing a non-zero class $\varphi \in \mathrm{Ext}^1_\mathscr{F} (\Lambda^2 , \mathrm{Id})$.
The composite of $S^2 \twoheadrightarrow \Lambda^2$ with its dual is the norm map $S^2 \rightarrow \Gamma^2$; this occurs in the top row of the following pullback diagram of exact sequences: \begin{eqnarray}
\label{eqn:ext2_pullback_diag}
\xymatrix{
0
\ar[r]
&
\mathrm{Id}
\ar[r]
\ar@{=}[d]
&
S^2
\ar[r]
\ar@{=}[d]
&
\Gamma^2
\ar@{^(->}[d]
\ar[r]
&
\mathrm{Id}
\ar@{^(->}[d]
\ar[r]
&
0\\
0
\ar[r]
&
\mathrm{Id}
\ar[r]
&
S^2
\ar[r] & T^2 \ar[r] & S^2 \ar[r] & 0.
} \end{eqnarray}
The rows represent non-zero classes in $\mathrm{Ext}^2 _\mathscr{F}$: \[
\tilde{e}_1 \in \mathrm{Ext}^2_\mathscr{F} (S^2, \mathrm{Id}) \mapsto e_1 \in \mathrm{Ext}^2 _\mathscr{F}(\mathrm{Id}, \mathrm{Id}). \] These classes appear for example in \cite{FLS}.
\subsection{The Krull filtration}
The category $\mathscr{U}$ of unstable modules has Krull filtration: \[
\mathscr{U}_0 \subset \mathscr{U}_1 \subset \ldots \subset \mathscr{U}_n \subset \ldots \subset \mathscr{U} \] (see \cite{schwartz_book,K14}); $\mathscr{U}_0$ identifies as the full subcategory of locally finite modules. For current purposes, the following is the key result:
\begin{prop} \label{prop:krull_poly} \cite{schwartz_book} For $d \in \mathbb{N}$, the functor $l : \mathscr{U} \rightarrow \mathscr{F}$ restricts to
\[
l : \mathscr{U}_d \rightarrow \mathscr{F}_d.
\] Moreover, if $M$ is a reduced unstable module, then $M \in \mathrm{Ob}\ \mathscr{U}_d$ if and only if $f(M)$ has polynomial degree $d$. \end{prop}
\subsection{Functors on $\mathscr{A}$-modules}
The categories $\mathscr{M}$ and $\mathscr{U}$ are tensor abelian; in particular, for $n \in \mathbb{N}$, the $n$th tensor functor $T^n : \mathscr{M} \rightarrow \mathscr{M}$, $M \mapsto M^{\otimes n}$ is defined, which restricts to $T^n : \mathscr{U} \rightarrow \mathscr{U}$. Again, $\mathfrak{S}_n$ acts naturally by place permutations on $T^n$, giving the $n$th symmetric invariants $\Gamma^n := (T^n)^{\mathfrak{S}_n}$ and the $n$th symmetric coinvariants $S^n:= (T^n)/\mathfrak{S}_n$. The functors $\Gamma^n, S^n : \mathscr{M} \rightarrow \mathscr{M}$ restrict to $\Gamma^n, S^n : \mathscr{U} \rightarrow \mathscr{U}$. Similarly, the exterior power functor $\Lambda ^n : \mathscr{M} \rightarrow \mathscr{M}$ restricts to $\Lambda^n : \mathscr{U} \rightarrow \mathscr{U}$.
For the remainder of the section, the prime $p$ is taken to be $2$. Thus, the Frobenius functor $\Phi : \mathscr{M} \rightarrow \mathscr{M}$ \cite[Section 1.7]{schwartz_book} is the usual degree-doubling functor.
\begin{lem}
\label{lem:Phi}
For $p=2$, the Frobenius functor $\Phi : \mathscr{M} \rightarrow \mathscr{M}$ is exact, commutes with tensor products and
there is a natural isomorphism
\[
\Phi \Sigma \cong \Sigma^2 \Phi.
\] Moreover, $\Phi$ restricts to $\Phi : \mathscr{U} \rightarrow \mathscr{U}$ and, if $M$ is a reduced unstable module, $\Phi M$ is reduced.
For $i \in \mathbb{N}$, $\Phi :\mathscr{U} \rightarrow \mathscr{U}$ restricts to: $
\Phi : \mathscr{N}il_i \rightarrow \mathscr{N}il_{2i}. $ \end{lem}
The Frobenius short exact sequence (\ref{eqn:frob_ses}) and its dual have analogues in $\mathscr{M}$:
\begin{lem} \label{lem:U_ses_Frobenius} For $p=2$ and $M \in \mathrm{Ob}\ \mathscr{M}$, there are natural short exact sequences \begin{eqnarray*} && 0 \rightarrow \Lambda^2 M \rightarrow \Gamma^2 M \rightarrow \Phi M \rightarrow 0 \\ && 0 \rightarrow \Phi M \rightarrow S^2 M \rightarrow \Lambda^2 M\rightarrow 0 . \end{eqnarray*} \end{lem}
\subsection{The Singer functors} (In this section, to simplify presentation, $p$ is taken to be $2$.) The Singer functor $R_1$ was introduced for unstable modules in the form used here by Lannes and Zarati \cite{LZ}. For $M$ an unstable module, $R_1 M$ is the sub $\mathbb{F} [u]$-module of $\mathbb{F} [u] \otimes M$ generated by the image of the total Steenrod square
$\mathrm{St}_1 (x) := \sum_i u^{|x|-i} \otimes Sq^i x \in \mathbb{F} [u] \otimes M$,
for $x \in M$. A key fact is that $R_1 M$ is stable under the action of $\mathscr{A}$ on $\mathbb{F} [u] \otimes M$.
The extension to all $\mathscr{A}$-modules requires $\mathbb{F} [u] \otimes M$ to be replaced by a half-completed tensor product, since the sum in $\mathrm{St}_1 (x)$ is no longer finite in general. This is reviewed in \cite{p_viasm} and details are given (for odd primes) in \cite{p_destab}.
\begin{rem} This Singer functor must not be confused with the stabilized version which occurs in \cite[Chapter II, Section 5]{BMMS}, following ideas of Miller. For the current situation, compare the treatment (in homology) in \cite{KMcC}. \end{rem}
\begin{nota} Write $\mathbb{F} [u]$ for the unstable algebra generated by $u$ of degree $1$ and $\mathbb{F} [u]\mbox{-} \mathscr{M} $ for the category of $\mathbb{F}[u]$-modules in $\mathscr{M}$ (respectively $\mathbb{F}[u]\mbox{-} \mathscr{U}$ for $\mathbb{F}[u]$-modules in $\mathscr{U}$). \end{nota}
\begin{lem} The categories $\mathbb{F}[u]\mbox{-} \mathscr{M}$ and $\mathbb{F} [u]\mbox{-} \mathscr{U}$ are abelian and there are exact forgetful functors $ \mathbb{F}[u]\mbox{-} \mathscr{M} \rightarrow \mathbb{F}[u]\mbox{-} \mathscr{U}$, \ \
$\mathbb{F}[u]\mbox{-} \mathscr{M} \rightarrow \mathscr{M}$ and
$\mathbb{F}[u]\mbox{-} \mathscr{U} \rightarrow \mathscr{U}.$ \end{lem}
\begin{prop} \cite{LZ,p_destab,p_viasm} \begin{enumerate}
\item The functor $ R_1 : \mathscr{M} \rightarrow \mathbb{F} [u] \mbox{-} \mathscr{M}
$ is exact and restricts to an exact functor $R_1 : \mathscr{U} \rightarrow \mathbb{F}[u]\mbox{-} \mathscr{U}$. \item There is a natural surjection $R_1 \twoheadrightarrow \Phi$ which, for $M \in \mathrm{Ob}\ \mathscr{M}$, fits into a short exact sequence:
\[
0 \rightarrow u R_1 M \rightarrow R_1 M \rightarrow \Phi M \rightarrow 0.
\] In particular, the $u$-adic filtration of $R_1M $ has filtration quotients of the form $\Sigma^j \Phi M$. \end{enumerate} \end{prop}
\begin{rem} Forgetting the $\mathbb{F}[u]$-module structure, the Singer functor can be considered as an exact functor $R_1 : \mathscr{M} \rightarrow \mathscr{M}$ which restricts to $R_1 : \mathscr{U} \rightarrow \mathscr{U}$. \end{rem}
\begin{nota} \cite{p_viasm}
For $n \in \mathbb{N}$ and $M \in \mathrm{Ob}\ \mathscr{M}$, write:
$$R_{1/n} M := (R_1 M)\otimes_{\mathbb{F}[u]} \mathbb{F}[u]/(u^n).$$
\end{nota}
\begin{lem} \label{lem:R_truncated}
For $1 \leq n \in \mathbb{N}$, $R_{1/n}$ is an exact functor $R_{1/n} : \mathscr{M} \rightarrow \mathscr{M}$ which restricts to $R_{1/n} : \mathscr{U} \rightarrow \mathscr{U}$.
For $M \in \mathrm{Ob}\ \mathscr{M}$ \begin{enumerate}
\item
for $n \geq 2$, there is a natural short exact sequence
\[
0
\rightarrow
\Sigma^{n-1} \Phi M \rightarrow R_{1/n}M \rightarrow R_{1/n-1} M \rightarrow 0;
\]
\item
the $u$-adic filtration of $R_{1/n}M$ has filtration quotients
$
\Sigma^i \Phi M$ , $0 \leq i < n$;
\item
the natural surjection $R_1 M \twoheadrightarrow \Phi M$ factors across $R_1 M \twoheadrightarrow R_{1/n}M$ as
$$
R_{1/n}M\twoheadrightarrow \Phi M, $$ which is an isomorphism mod $\mathscr{N}il$. \end{enumerate} \end{lem}
\section{Almost unstable modules} \label{sect:almost}
Suppose that $\mathbb{F}=\mathbb{F}_2$. (The results of this section have analogues for odd primes.)
\begin{defn} \label{defn:alunst}
An unstable module $M \in \mathrm{Ob}\ \mathscr{U}$ is almost unstable if it admits a finite filtration with subquotients of the form
$\Sigma^{-i} N$ for some $i \in \mathbb{N}$ and $N \in \mathscr{N}il_i$.
The full subcategory of almost unstable modules in $\mathscr{M}$ is denoted $\widetilde{\unst}$. \end{defn}
\begin{prop} \label{prop:alunst}
There are inclusions of subcategories $\mathscr{U} \subset \widetilde{\unst} \subset \mathscr{M}$ and $\widetilde{\unst}$ is
an abelian Serre subcategory.
Moreover,
\begin{enumerate}
\item
$\widetilde{\unst}$ is closed under $\otimes$ and hence under $\Sigma$;
\item
any almost unstable module is concentrated in non-negative degrees;
\item
\label{item:alunst_bounded}
a module $M \in \mathrm{Ob}\ \mathscr{M}$ concentrated in non-negative degrees and bounded above is almost unstable.
\end{enumerate} \end{prop}
\begin{proof} It is clear that $\widetilde{\unst}$ contains $\mathscr{U}$. To show that $\widetilde{\unst}$ is a Serre subcategory, it suffices to show that it is closed under formation of subobjects and quotients, since closure under extension is clear.
For $M \in \mathrm{Ob}\ \widetilde{\unst}$, write $f_i M $ for an increasing finite filtration that satisfies the defining property of Definition \ref{defn:alunst}. If $K \subset M$ is a submodule, consider the induced filtration $f_i K := K \cap f_i M$. Then, by construction, $f_i K/f_{i-1}K \hookrightarrow f_i M/f_{i-1} M$. By hypothesis the right hand module is of the form $\Sigma^{-t} N$ for some $t \in \mathbb{N}$ and $N \in \mathscr{N}il_t$;
since $\mathscr{N}il_t$ is a Serre subcategory \cite{schwartz_book}, $f_iK $ is a filtration of the required form.
Similarly, for $M \twoheadrightarrow Q$, consider the quotient filtration $f_iQ := \mathrm{image} \{ f_i M \rightarrow Q \}$. Then
$f_i Q /f_{i-1} Q$ is a quotient of $f_i M /f_{i-1} M$, whence the result as before, {\em mutatis mutandis}.
Closure under tensor product is a consequence of the fact that $\otimes$ restricts to $\otimes : \mathscr{N}il_i \times \mathscr{N}il_j \rightarrow \mathscr{N}il_{i+j}$ \cite{schwartz_book,K14}. Closure under $\Sigma$ follows since the suspension functor identifies with $\Sigma \mathbb{F} \otimes - $.
The remaining statements are straightforward. \end{proof}
\begin{exam} \label{exam:almost} As usual, extend the unstable algebra structure of $\mathbb{F} [u]$ to an algebra structure in $\mathscr{M}$ on $\mathbb{F} [u^{\pm 1}]$. Consider the submodule $M:= \mathbb{F}[u^{\pm 1}]_{\geq -1} \subset \mathbb{F} [u^{\pm 1}] \in \mathrm{Ob}\ \mathscr{M}$, so that
$\Sigma M$ occurs in the short exact sequence:
\[
0
\rightarrow
\Sigma \mathbb{F} [u]
\rightarrow
\Sigma M
\rightarrow
\mathbb{F}
\rightarrow
0.
\] This exhibits $\Sigma M$ as an almost unstable module, whereas $M$ is not, since it is non-zero in degree $-1$. Note that $\Sigma^{t} M$ ($t \in \mathbb{Z}$) is never unstable. \end{exam}
\begin{rem}
Proposition \ref{prop:alunst} implies that $\widetilde{\unst}$ is closed under the formation of finite limits and finite colimits.
\begin{enumerate}
\item
That closure under inverse limits fails in general is clear from Proposition \ref{prop:alunst} (\ref{item:alunst_bounded}), since {\em any} $\mathscr{A}$-module $M$
is the inverse limit of its system of truncations $M^{\leq k}$ (the quotient of $M$ by elements of degrees $>k$) as $k \rightarrow \infty$.
\item
Closure under colimits also fails in general, as exhibited by the following example. For $0< t \in \mathbb{N}$, let $N(t)$ denote the subquotient $\Phi^t F(1) / \Phi^{2t} F(1)$ of the free unstable module $F(1)$; this has total dimension $t$, with classes in degrees $2^i$, $t \leq i < 2t$, linked by the operation $Sq_0$.
Consider the $\mathscr{A}$-module:
\[
M:= \bigoplus _{t>1} \Sigma^{t -2^t} N(t).
\] Proposition \ref{prop:alunst} (\ref{item:alunst_bounded}) implies that each $ \Sigma^{t -2^t} N(t)$ is almost unstable (and the choice of the desuspension ensures that $M$ is of finite type).
However, $M$ is not almost unstable; if it were, there would exist $T \in \mathbb{N}$ such that $\Sigma ^T M$ admits a finite filtration (say of length $l \in \mathbb{N}$) such that each subquotient is unstable. Choosing $t \in \mathbb{N}$ such that $t> l$ and $t -2^t + T <0$, consideration of the factor $ \Sigma^{t -2^t} N(t)$ leads to a contradiction. \end{enumerate} \end{rem}
\subsection{The nilpotent filtration of $\widetilde{\unst}$}
The above notions can be refined by introducing an analogue of the nilpotent filtration of $\mathscr{U}$.
\begin{defn}
For $i \in \mathbb{N}$, let $\widetilde{\nil}_i \subset \widetilde{\unst}$ be the full subcategory of objects which admit a finite filtration with subquotients of the form
$\Sigma^{-t} N$ for some $t \in \mathbb{N}$ and $N \in \mathscr{N}il_{i+t}$. \end{defn}
By definition, there is a decreasing filtration: \[
\ldots \subset \widetilde{\nil}_{t+1} \subset \widetilde{\nil}_t \subset \ldots \subset \widetilde{\nil}_0 = \widetilde{\unst}. \]
Proposition \ref{prop:alunst} generalizes to:
\begin{prop} \label{prop:alnil_serre_sigma}
For $s, t \in \mathbb{N}$: \begin{enumerate}
\item
$\widetilde{\nil}_t$ is a Serre subcategory of $\widetilde{\unst}$;
\item
tensor product restricts to $\otimes : \widetilde{\nil}_s \times \widetilde{\nil}_t \rightarrow \widetilde{\nil}_{s+t}$: \item suspension induces $\Sigma : \widetilde{\nil}_s \rightarrow \widetilde{\nil}_{s+1}$ which is an equivalence of categories, with inverse $\Sigma^{-1}$. \end{enumerate}
\end{prop}
\begin{proof} Once established that $\Sigma$ induces an equivalence of categories, the properties follow from the case of $\widetilde{\unst} = \widetilde{\nil}_0$.
To show that $\Sigma^{-1} : \mathscr{M} \rightarrow \mathscr{M}$ induces a functor $\widetilde{\nil}_{s+1} \rightarrow \widetilde{\nil}_s$, since $\Sigma^{-1}$ is exact, it suffices to check on an almost unstable module of the form $\Sigma^{-i} N$ with $i \in \mathbb{N}$ and $N \in \mathscr{N}il_{s+1 +i}$.
Then $\Sigma^{-1} (\Sigma^{-i} N)$ can be written $\Sigma^{-(i+1)} N$ with $N$ considered as lying in $\mathscr{N}il_{s+(i+1)}$. \end{proof}
The category $\widetilde{\unst}$ is not stable under $\Sigma^{-1}$. In combination with Proposition \ref{prop:destab_alnil} below, the above result should be compared with
the fact \cite{schwartz_book} that the loop functor $\Omega : \mathscr{U} \rightarrow \mathscr{U}$ restricts to $\Omega : \mathscr{N}il_{i+1} \rightarrow \mathscr{N}il_i$.
\begin{prop} \label{prop:alunst_stable_Phi_R} For $i \in \mathbb{N}$,
the Frobenius functor $\Phi : \mathscr{M} \rightarrow \mathscr{M}$ restricts to
\[
\Phi : \widetilde{\nil}_i \rightarrow \widetilde{\nil}_{2i}.
\] Hence, for $1 \leq n \in \mathbb{N}$, the truncated Singer functor $R_{1/n}: \mathscr{M} \rightarrow \mathscr{M}$ restricts to:
\[
R_{1/n} : \widetilde{\nil}_i \rightarrow \widetilde{\nil}_{2i}.
\] \end{prop}
\begin{proof}
The functors considered are exact, hence it suffices to consider behaviour on a module of the form $\Sigma^{-t} N$ with $N \in \mathrm{Ob}\ \mathscr{N}il_{i+t}$.
Now $\Phi \Sigma^{-t} N \cong \Sigma^{-2t} \Phi N$ and $\Phi N \in \mathscr{N}il_{2(i+t)}$ by Lemma \ref{lem:Phi}, which implies the first statement. The corresponding statement for
$R_{1/n}$ then follows using the $u$-adic filtration (cf. Lemma \ref{lem:R_truncated}). \end{proof}
\begin{prop} \label{prop:destab_alnil}
For $i \in \mathbb{N}$, the destabilization functor restricts to
\[
\Omega^\infty : \widetilde{\nil}_i \rightarrow \mathscr{N}il_i.
\] \end{prop}
\begin{proof}
The category $\mathscr{N}il_i$ is localizing and $\Omega^\infty$ is right exact, hence it suffices to consider $\Omega^\infty$ applied
to a module of the form $\Sigma^{-t} N$ with $N \in \mathrm{Ob}\ \mathscr{N}il_{i+t}$. By construction, the composite functor $\Omega^\infty \Sigma^{-t}$ restricted
to $\mathscr{U}$ is the iterated loop functor $\Omega^t$. Since $\Omega^t$ restricts to
$\Omega ^t : \mathscr{N}il_{i+t} \rightarrow \mathscr{N}il_i$ \cite{schwartz_book}, this establishes the result. \end{proof}
Recall that $\mathrm{nil}_i : \mathscr{U} \rightarrow \mathscr{N}il_i \subset \mathscr{U}$ denotes the right adjoint to $\mathscr{N}il_i \hookrightarrow \mathscr{U}$.
\begin{cor} \label{cor:nili_adjunction}
For $M \in \mathrm{Ob}\ \widetilde{\nil}_i $ and $N \in \mathrm{Ob}\ \mathscr{U}$, the inclusion $\mathrm{nil}_i N \hookrightarrow N$ induces an isomorphism
\[
\mathrm{Hom}_{\mathscr{M}} (M, \mathrm{nil}_i N) \stackrel{\cong}{\rightarrow} \mathrm{Hom}_{\mathscr{M}} (M, N).
\] \end{cor}
\begin{proof}
Since $N$ and $\mathrm{nil}_i N$ are unstable, the morphism identifies with
\[
\mathrm{Hom}_{\mathscr{M}} (\Omega^\infty M , \mathrm{nil}_i N) {\rightarrow} \mathrm{Hom}_{\mathscr{M}} (\Omega^\infty M, N).
\] Proposition \ref{prop:destab_alnil} shows that $\Omega^\infty M \in \mathrm{Ob}\ \mathscr{N}il_i$, whence the result.
\end{proof}
As a particular case of Corollary \ref{cor:nili_adjunction}, one obtains:
\begin{cor}
\label{cor:alnil_1_red}
For $M \in \mathrm{Ob}\ \widetilde{\nil}_1 $ and $N \in \mathrm{Ob}\ \mathscr{U}$ a reduced unstable module,
\[
\mathrm{Hom}_{\mathscr{M}} (M, N)=0.
\]
\end{cor}
\subsection{Good almost unstable modules}
For $M \in \mathrm{Ob}\ \widetilde{\unst}$, there is a canonical surjection to a reduced unstable module, namely: \[
M \twoheadrightarrow (\Omega^\infty M)/\mathrm{nil}_1(\Omega^\infty M). \] In many cases of interest, the kernel of this map lies in $\widetilde{\nil}_1$. This motivates the following:
\begin{defn} \label{defn:good_alunst}
A module $M \in \mathrm{Ob}\ \mathscr{M}$ is a good almost unstable module if it is almost unstable and the kernel of $M \twoheadrightarrow (\Omega^\infty M)/\mathrm{nil}_1(\Omega^\infty M)$ lies in $\widetilde{\nil}_1$. \end{defn}
\begin{lem}
\label{lem:good_equivalent}
A module $M \in \mathrm{Ob}\ \mathscr{M}$ is a good almost unstable module if and only if the kernel of $M \twoheadrightarrow \Omega^\infty M$ lies in $\widetilde{\nil}_1$. \end{lem}
\begin{proof} By definition $\mathrm{nil}_1(\Omega^\infty M)$ lies in $\mathscr{N}il_1$, whence the result. \end{proof}
\begin{nota} \label{nota:good_alunst}
If $M \in \mathrm{Ob}\ \widetilde{\unst} $ is good,
write the associated short exact sequence:
\[
0
\rightarrow M'
\rightarrow M
\rightarrow
\rho_0 M
\rightarrow 0,
\] where $M' \in \mathrm{Ob}\ \widetilde{\nil}_1$ and $\rho_0 M \in \mathrm{Ob}\ \mathscr{U}$ is reduced. \end{nota}
\begin{exam}
Every unstable module is good when considered as an almost unstable module. More generally, if $M$ is
of the form $\Sigma^{-t}N$ with $N \in \mathscr{N}il_t$, then $M$ is good almost unstable, with associated exact sequence:
\[ 0 \rightarrow \Sigma^{-t} \mathrm{nil}_{t+1} N \rightarrow M \rightarrow \rho_t N \rightarrow 0.
\] In particular, if $t=0$ (so that $M$ is unstable), there is no conflict with the notation $\rho_0 M $. \end{exam}
\begin{prop}
\label{prop:good_alunst}
A subquotient of a good almost unstable module is good almost unstable.
Moreover, if $f : M \twoheadrightarrow Q $ is a surjection from a good almost unstable module, then
$f$ induces a surjection $\rho_0 f : \rho_ 0 M \twoheadrightarrow \rho_0 Q$ which is an isomorphism if and only if
$\ker f$ lies in $\widetilde{\nil}_1$.
\end{prop}
\begin{proof}
Let $M$ be a good almost unstable module with associated short exact sequence as in Notation \ref{nota:good_alunst}.
Consider a submodule $K$; setting $K' := K \cap M'$, one has the morphism of short exact sequences:
\[
\xymatrix{
0
\ar[r]
&
K' \ar[r]
\ar@{^(->}[d]
&
K
\ar[r]
\ar@{^(->}[d]
& K''
\ar[r]
\ar@{^(->}[d]
&
0
\\
0
\ar[r]
&
M'
\ar[r]
&
M
\ar[r]
&
\rho_0M
\ar[r]
&
0,
}
\] which shows that $K'' \subset \rho_0 M $ is a reduced unstable module and $K' \subset M'$ belongs to $\widetilde{\nil}_1$, thus $K$ is good.
Similarly, for a surjection $M \twoheadrightarrow Q$, there is a morphism of short exact sequences:
\[
\xymatrix{
0
\ar[r]
&
M' \ar[r]
\ar@{->>}[d]
&
M
\ar[r]
\ar@{->>}[d]
& \rho_0 M
\ar[r]
\ar@{->>}[d]
&
0
\\
0
\ar[r]
&
\tilde{Q}
\ar[r]
&
Q
\ar[r]
&
Q''
\ar[r]
&
0,
}
\]
where $\tilde{Q}$ is defined by the commutative square on the left, hence belongs to $\widetilde{\nil}_1$ since $M'$ does, and $Q''$ is unstable, as a quotient of $\rho_0 M$. Now $Q''$ is a quotient of $\Omega^\infty Q$, by Lemma \ref{lem:good_equivalent}, thus
$Q$ is good.
By construction there is a surjection $\rho_0 M \twoheadrightarrow \rho_0 Q$. The final statement is clear. \end{proof}
\begin{prop}
\label{prop:good_alunst_stability}
The class of good almost unstable modules is stable under finite direct sums and under $\otimes$.
Moreover, it is preserved by the functors:
\begin{enumerate}
\item
$\Phi : \widetilde{\unst} \rightarrow \widetilde{\unst}$;
\item
$R_{1/n} : \widetilde{\unst} \rightarrow \widetilde{\unst}$, for $1 \leq n \in \mathbb{N}$.
\end{enumerate} If $M$ is good almost unstable, then
\begin{eqnarray*}
\rho_0 (\Phi M) & \cong & \Phi (\rho_0 M) \\
\rho_0 (R_{1/n} M) & \cong & R_{1/n} (\rho_0 M) \ \cong \ \Phi (\rho_0 M).
\end{eqnarray*} \end{prop}
\begin{proof}
Straightforward. The statement for $\Phi$ and $R_{1/n}$ is a generalization of Proposition \ref{prop:alunst_stable_Phi_R}, using
the fact that $\Phi$ preserves reduced unstable modules. \end{proof}
\section{Almost unstable spectral sequences} \label{sect:auss}
The main interest in this section is in spectral sequences which converge to an unstable module and the following natural question: to what extent can the theory of unstable modules be used to understand the structure of the spectral sequence?
\begin{hyp} \label{hyp:ss} Suppose that the spectral sequence $(E_r^{*,*} , d_r)$ satisfies the following conditions: \begin{enumerate}
\item it is second quadrant ($E_r ^{s,t} = 0$ if $s >0$ or $t<0$) and cohomological $d_r : E_r ^{*,*} \rightarrow E_r^{*+r, *+1-r}$; \item $E_1^{0,*}=0$;
\item
each $E_r^{-k, *}$ ($k \in \mathbb{N}$) is an $\mathscr{A}$-module and $d_r$ is $\mathscr{A}$-linear, namely:
\[
d_r : \Sigma (\Sigma^{-k} E_r^{-k,*} )
\rightarrow
(\Sigma^{-k+r} E_r^{-k+r,*} )
\] is a (degree zero) morphism of $\mathscr{M}$;
\item
the spectral sequence converges strongly and $\bigoplus _k \Sigma^{-k} E_\infty^{-k,*}$ is the associated graded of an unstable module, in particular each $\Sigma^{-k} E_\infty^{-k,*}$ is unstable. \end{enumerate} \end{hyp}
\begin{defn}
A spectral sequence $(E_r^{*,*} , d_r)$ satisfying Hypothesis \ref{hyp:ss} is almost unstable (respectively good almost unstable) if
$\Sigma^{-k} E_1 ^{-k, *}$ is almost unstable (resp. good almost unstable) for all $k \in \mathbb{N}$. \end{defn}
\begin{prop} \label{prop:good_alunst_ss} For a spectral sequence $(E_r^{*,*} , d_r)$ satisfying Hypothesis \ref{hyp:ss}, which is good almost unstable, and $k \in \mathbb{N}$, \begin{enumerate}
\item
$\Sigma^{-k} E_r ^{-k, *}$ is a good unstable module for all $1 \leq r \in \mathbb{N}$;
\item
for $1 \leq r \in \mathbb{N}$, $\rho_0 (\Sigma^{-k} E_{r+1}^{-k, *}) \subset \rho_0 (\Sigma^{-k} E_{r}^{-k, *})$
with equality if $r>k$; in particular, $\rho_0 (\Sigma^{-k} E_{\infty}^{-k, *}) = \rho_0 (\Sigma^{-k} E_{r}^{-k, *})$ for $r \geq k$. \end{enumerate} \end{prop}
\begin{proof}
The first statement follows from Proposition \ref{prop:good_alunst}.
For the second, the differential is of the form
\[
d_r : \Sigma (\Sigma^{-k} E_r^{-k,*} )
\rightarrow
(\Sigma^{-k+r} E_r^{-k+r,*} ),
\] where $\Sigma (\Sigma^{-k} E_r^{-k,*} ) \in \mathrm{Ob}\ \widetilde{\nil}_1$ and $ \Sigma^{-k+r} E_r^{-k+r,*} \in \widetilde{\unst}$ is a good unstable module. In particular, by Corollary \ref{cor:alnil_1_red}, the composite map
\[
d_r : \Sigma (\Sigma^{-k} E_r^{-k,*} )
\rightarrow
(\Sigma^{-k+r} E_r^{-k+r,*} )
\twoheadrightarrow
\rho_0 (\Sigma^{-k+r} E_r^{-k+r,*} )
\]
is trivial, thus the image of $d_r$ lies in $(\Sigma^{-k+r} E_r^{-k+r,*} )'$.
It follows from the final statement of Proposition \ref{prop:good_alunst} that $\rho_0 (\Sigma^{-k} E_{r+1}^{-k,*} )$ identifies with $\rho_0 (\ker d_r)$, which is a submodule of $\rho_0 (\Sigma^{-k} E_{r}^{-k, *})$,
by the argument employed in the proof of {\em loc. cit.}.
The second point follows since the spectral sequence is concentrated in the second quadrant with trivial column $E^{0,*}_1$, by hypothesis.
\end{proof}
\begin{cor}
\label{cor:good_auss_d1}
For a spectral sequence $(E_r^{*,*} , d_r)$ satisfying Hypothesis \ref{hyp:ss}, which is good almost unstable,
such that $E_1^{-1,*} = \Sigma (\Sigma^{-t} M) $ for some $t \in \mathbb{N}$ and $M \in \mathscr{N}il_t$, \begin{eqnarray*}
\rho_0 (\Sigma^{-1} E_\infty ^{-1,*})& = & \rho_0 (\Omega^t M)\ \cong \ \rho_t (M) \\
\rho_0 (\Sigma^{-2}E_\infty ^{-2,*})& = & \rho_0 (\Sigma^{-2} E_2 ^{-2,*}) \end{eqnarray*} Moreover, the differential $d_1 : \Sigma (\Sigma^{-2} E_1^{-2,*} )
\rightarrow
\Sigma^{-1} E_1^{-1,*} = \Sigma^{-t} M$ factors across the inclusion
\[
\Sigma^{-t} \mathrm{nil}_{t+1} M \hookrightarrow \Sigma^{-t} M
\] and induces a morphism $d_1 : \rho_0 (\Sigma^{-2} E_1^{-2,*} ) \rightarrow \rho_{t+1} M$ and \[
\rho_0 (\Sigma^{-2}E_\infty ^{-2,*}) \cong \ker \{ \rho_0 (\Sigma^{-2} E_1^{-2,*} ) \rightarrow \rho_{t+1} M\}. \]
\end{cor}
\begin{proof}
The first part follows from Proposition \ref{prop:good_alunst_ss}.
By hypothesis, $(\Sigma^{-2} E_1 ^{-2,*})$ is a good almost unstable module, hence there is a short exact sequence
\[
0
\rightarrow
(\Sigma^{-2} E_1 ^{-2,*})'
\rightarrow
(\Sigma^{-2} E_1 ^{-2,*})
\rightarrow \rho_0 (\Sigma^{-2} E_1 ^{-2,*}) \rightarrow 0 \]
with $(\Sigma^{-2} E_1 ^{-2,*})' \in \mathrm{Ob}\ \widetilde{\nil}_1$. The differential $d_1$ is of the form:
\[
d_1 : \Sigma (\Sigma^{-2} E_1 ^{-2,*})
\rightarrow
\Sigma^{-t} M
\] hence $\Sigma^t d_1 : \Sigma^{t+1} (\Sigma^{-2} E_1 ^{-2,*})
\rightarrow M$. As $(\Sigma^{-2} E_1 ^{-2,*})$ is good almost unstable,
$\Sigma^{t+1} (\Sigma^{-2} E_1 ^{-2,*})$ lies in $\widetilde{\nil}_{t+1}$ and
$\Sigma^{t+1} \big((\Sigma^{-2} E_1 ^{-2,*})'\big)$ in $\widetilde{\nil}_{t+2}$.
Since $M$ is unstable, Proposition \ref{prop:destab_alnil} implies that $\Sigma^t d_1$ maps to $\mathrm{nil}_{t+1} M$ and
its restriction to $\Sigma^{t+1} (\Sigma^{-2} E_1 ^{-2,*})'$ maps to $\mathrm{nil}_{t+2}M$.
The result follows as in the proof of Proposition \ref{prop:good_alunst_ss}.
\end{proof}
\section{The Arone-Goodwillie spectral sequence} \label{sect:tower}
In this section, the presentation of \cite{Kuhn_nonrealization} is followed, since the results of Section \ref{sect:cohomEP} use {\em loc. cit.}.
For $X$ a pointed space (respectively a spectrum), the Arone-Goodwillie tower associated to the functor $X \mapsto \Sigma^\infty \Omega^n X$ for $n \in \mathbb{N}$
has the following form: \[
\xymatrix{
& \ & \ar[d]
\\
&&P^n _3 X
\ar[d]
\\
&&
P^n _2 X
\ar[d] \\
X
\ar[rruu]|{\varepsilon_3}
\ar[rru]|{\varepsilon_2}
\ar[rr]|{\varepsilon_1}
&&
P^n_1 X,
} \] where $P^n _1 X= \Sigma^{-n} X$ for $n <\infty$ (ie the spectrum $\Sigma^{-n} \Sigma^{\infty} X$) and $P^n_0 = 0$.
Ahearn and Kuhn \cite{AK} identify the fibres of the tower in terms of the extended power construction via the cofibre sequence: \[
D_{n,j} \Sigma^{-n} X \rightarrow P^n_j X \rightarrow P^n_{j-1} X \] for $1 \leq j \in \mathbb{N}$, where, for a spectrum $Y$, \[
D_{n,j} Y:= \Big(\mathfrak{C}(n,j)_+\wedge Y^{\wedge j} \Big) _{h \mathfrak{S}_j}, \] $\mathfrak{C}(n,j)$ the Boardman-Vogt space of $j$ little $n$-cubes in an $n$-cube.
For a pointed space $X$, the adjunction unit $X \rightarrow \Omega \Sigma X$ induces an $n$-fold loop map $\Omega^n X \rightarrow \Omega^{n+1} \Sigma X$, for $n \in \mathbb{N}$; by \cite[Corollary 1.2]{AK}, this induces a natural map of towers $P^n_\bullet X \rightarrow P^{n+1}_{\bullet}\Sigma X$ which identifies on the level of the fibres as the natural transformation $D_{n,j} \Sigma^{-n} X \rightarrow D_{n+1,j} \Sigma^{-n}X$ induced by the inclusion $\mathfrak{C} (n,j) \hookrightarrow \mathfrak{C}(n+1,j)$. Similarly, the natural evaluation map $\varepsilon : \Sigma \Sigma^\infty \Omega^{n+1} X \rightarrow \Sigma^\infty \Omega^n X$ induces a map of towers $\Sigma P^{n+1}_\bullet X \rightarrow P^{n}_\bullet X$ and, on fibres, $
\varepsilon : \Sigma D_{n+1,j} X \rightarrow D_{n,j} \Sigma X $
(see \cite{AK}).
If $X$ is $n$-connected for $n \in \mathbb{N}$, the connectivity of the maps $\varepsilon_j$ increases linearly with $j$, hence:
\begin{prop}
\cite{Kuhn_nonrealization} For $X$ an $n$-connected space with $H^* (X)$ of finite type, the spectral sequence associated to the Arone-Goodwillie tower satisfies Hypothesis \ref{hyp:ss} with $$E^{-j, *}_1 = H^* (\Sigma^j D_{n,j} \Sigma^{-n} X)$$ and converges strongly to $H^* (\Omega^n X)$.
The associated filtration of $H^* (\Omega^n X) $ is \[
0 = F_0 H^*(\Omega^n X) \subset F_1 H^*(\Omega^n X)\subset F_2 H^*(\Omega^n X) \subset \ldots \subset H^*(\Omega^n X), \] where $F_j H^* (\Omega^n X) = \mathrm{image} \{ H^* (P^n _j X) \rightarrow H^* (\Omega^n X) \} $. \end{prop}
There is a commutative diagram in $\mathscr{U}$, in which $ Q H^* (X_+)$ denotes the module of indecomposables of the unstable algebra $H^* (X_+)$: \[
\xymatrix{
\Omega^n Q H^* (X_+) \ar[rr]
\ar@{->>}[rd]
&&
H^* (\Omega^n X) .
\\
&
F_1 H^* (\Omega^n X)
\ar@{^(->}[ur]
} \]
The functor $U: \mathscr{U} \rightarrow \mathscr{K}$ induces a morphism of unstable algebras: \begin{eqnarray} \label{eqn:Umap}
U (\Omega^n Q H^* (X_+) ) \rightarrow H^* ((\Omega^n X)_+). \end{eqnarray}
If $M$ is connected, there is a natural inclusion of unstable modules \[
M \rightarrow \overline{UM} \subset UM \] which induces a surjection onto the indecomposables $Q (UM)$; the product of $UM$ induces an increasing filtration of the augmentation ideal $\overline{U M}$: \[
M = F_1 \overline{U M} \subseteq F_2 \overline{UM} \subseteq F_3 \overline{UM} \subseteq \ldots \subseteq F_j \overline{UM} \subseteq \ldots \subseteq \overline{U M}. \]
The results of Ahearn and Kuhn \cite{AK} imply that this filtration is compatible with the filtration $F_j H^* (\Omega^n X)$; namely, for $1 \leq j \in \mathbb{N}$, the morphism (\ref{eqn:Umap}) restricts to a morphism of unstable modules: \[
F_j \overline{U \Omega^n Q H^* (X_+) } \rightarrow F_j H^* (\Omega^n X). \]
At the prime $p=2$, it is the submodule $F_2 \overline{UM}$ which is of interest. The construction of $UM$ implies the following:
\begin{lem} \label{lem:F_2UM} For $M \in \mathrm{Ob}\ \mathscr{U}$ and $\lambda : \Phi M \rightarrow M$ the morphism of unstable modules induced by $Sq_0$, $F_2 \overline{UM}$ occurs in the pushout of short exact sequences: \[
\xymatrix{ \Phi M \ar[d]_\lambda \ar[r] & S^2 M \ar[r] \ar[d] & \Lambda^2 M \ar@{=}[d] \\ M \ar[r] & F_2 \overline{UM} \ar[r] & \Lambda^2 M . } \]
\end{lem}
\begin{rem} \label{rem:cup_product} Taking $M= F_1 H^* (\Omega^n X)$, one obtains the fundamental morphism of short exact sequences: \begin{eqnarray} \label{eqn:filt_ss}
\xymatrix{ \ \ \ & F_1 H^* (\Omega^n X) \ar[r] \ar@{=}[d] & F_2 \overline{U F_1 H^* (\Omega^n X)} \ar[r] \ar[d]^{\cup} & \Lambda^2 ( F_1 H^* (\Omega^n X)) \ar[d]^{\overline{\cup}} \\ & F_1 H^* (\Omega^n X) \ar[r] & F_2 H^* (\Omega^n X) \ar[r] & F_2 H^* (\Omega^n X)/ F_1 H^* (\Omega^n X). } \end{eqnarray}
The identification of $\overline{\cup}$ in terms of the structure of the spectral sequence will be important in Section \ref{sect:main}. \end{rem}
\section{Cohomology of extended powers at $p=2$} \label{sect:cohomEP}
Fix an integer $n \geq 1$ and a spectrum $Y$. In \cite[Section 3]{Kuhn_nonrealization}, Kuhn
describes the mod $2$ cohomology of the extended powers $H^ *(D_{n,j} Y)$; we follow {\em loc. cit.} in considering only spectra with $H^* (Y)$ bounded below and of finite type.
The structure of $H^ *(D_{n,\bullet} Y)$ is determined (see \cite[Theorem 3.14]{Kuhn_nonrealization}) in terms of the following morphisms:
\begin{enumerate}
\item The product \cite[Definition 3.3]{Kuhn_nonrealization} \[
\star : H^* (D_{n,i} Y) \otimes H^*(D_{n,j} Y) \rightarrow H^* (D_{n,i+j} Y), \] which is a morphism of $\mathscr{A}$-modules and induces a commutative (bi)graded algebra structure on $H^ *(D_{n,\bullet} Y)$. \item For $1 \leq j \in \mathbb{N}$, the dual Browder operation \cite[Definition 3.5]{Kuhn_nonrealization} \[
L_{n-1} : H^{*} (Y)^{\otimes j} \rightarrow H^* (\Sigma^{(1-j)(n-1)}D_{n,j} Y), \] which is $\mathscr{A}$-linear. \item The dual Dyer-Lashof operations (for $r \geq 0$) \cite[Definition 3.2]{Kuhn_nonrealization} \[
\mathfrak{Q} _r : H^d (D_{n,j} Y) \rightarrow H^ {2d+r} (D_{n,2j} Y). \] The operation $\mathfrak{Q}_r$ is trivial for $r\geq n$ \cite[Proposition 3.8]{Kuhn_nonrealization}. \end{enumerate}
\begin{rem} The dual Dyer-Lashof operations $\mathfrak{Q}_r$ are not $\mathscr{A}$-linear, but satisfy Nishida relations. Moreover, the operation $\mathfrak{Q}_0$ is not $\mathbb{F}$-linear; the default of linearity is given by the interaction with the $\star$-product: \[
\mathfrak{Q}_0 (x+y) = \mathfrak{Q} _0 (x) + \mathfrak{Q}_0 (y) + x \star y. \] Thus $\mathfrak{Q}_0$ behaves like a divided square operation. \end{rem}
\begin{lem} For $1 \leq j \in \mathbb{N}$, the $\star$-product induces a morphism of $\mathscr{A}$-modules:
\[
\star: \Lambda^2 H^* (D_{n,i}Y) \rightarrow H^* (D_{n,2i} Y).
\] \end{lem}
\begin{proof}
Follows from \cite[Proposition 3.11 (iii)]{Kuhn_nonrealization}. \end{proof}
\begin{lem} \label{lem:images_ddl}
For $1 \leq j \in \mathbb{N}$ and an integer $l>0$, the sub vector space of $H^* (D_{n,2j} Y) $ generated by the images of \[
\mathfrak{Q} _r : H^* (D_{n,j} Y) \rightarrow H^ {2*+r} (D_{n,2j} Y) \] ($r \geq l$) is a sub $\mathscr{A}$-module $\Big(\sum_{r \geq l} \mathrm{image}(\mathfrak{Q}_r)\Big)$ of $H^* (D_{n,2j} Y) $. \end{lem}
\begin{proof}
Follows from the Nishida relation given in \cite[Proposition 3.1(ii)]{Kuhn_nonrealization}. \end{proof}
\begin{rem}
The higher dual Dyer-Lashof operations $\mathfrak{Q}_r$, $r \geq n$ act trivially by \cite[Proposition 3.8]{Kuhn_nonrealization}, hence the sum $\sum_{r \geq l} \mathrm{image}(\mathfrak{Q}_r)$ is finite (and zero for
$l \geq n$). \end{rem}
\begin{prop} \label{prop:filter_ddl}
For $1 \leq j \in \mathbb{N}$, the dual Dyer-Lashof operations induce $\mathscr{A}$-linear maps: \begin{eqnarray*} \overline{\mathfrak{Q}_0} &:& \Phi H^* (D_{n,j} Y) \rightarrow H^* (D_{n,2j} Y)/ \Big ( \Lambda^2 H^* (D_{n,j} Y) + \sum_{r \geq 1} \mathrm{image}(\mathfrak{Q}_r) \Big) \\ \overline{\mathfrak{Q}_l} &:& \Sigma^l \Phi H^* (D_{n,j} Y) \hookrightarrow H^* (D_{n,2j} Y)/ \Big ( \sum_{r \geq l+1} \mathrm{image}(\mathfrak{Q}_r) \Big), \end{eqnarray*} where $l>0$. \end{prop}
\begin{proof}
By \cite[Proposition 3.11(i)]{Kuhn_nonrealization}, the operation $\mathfrak{Q}_0$ becomes $\mathbb{F}$-linear after the passage to the quotient by the submodule $\Lambda^2 H^* (D_{n,j} Y)$. Moreover, the Nishida relation for $\mathfrak{Q}_0$ \cite[Proposition 3.15(i)]{Kuhn_nonrealization} establishes the $\mathscr{A}$-linearity, after passing to the additional quotient by the image of the higher dual Dyer-Lashof operations, which is a sub $\mathscr{A}$-module by Lemma \ref{lem:images_ddl}.
The argument for $\mathfrak{Q}_l$ ($l>0$) is similar, using the Nishida relation \cite[Proposition 3.1(ii)]{Kuhn_nonrealization}. \end{proof}
\begin{thm} \label{thm:cohomEP_good} Suppose that $Y$ is a spectrum such that $H^* (Y) \in \mathrm{Ob}\ \widetilde{\unst}$ is almost unstable and is of finite type. Then for $1 \leq j \in \mathbb{N}$: \begin{enumerate}
\item $H^* (D_{n,j} Y) \in \mathrm{Ob}\ \widetilde{\unst}$ is almost unstable; \item if $H^* (Y)$ is a good almost unstable module, then $H^* (D_{n,j} Y)$ is good and \begin{enumerate} \item $
\rho_0 (H^* (D_{n,j}Y))\cong \Gamma^j \rho_0 (H^* (Y)) $, $n \geq 2$; \item $\rho_0 (H^* (D_{1,j}Y))\cong T^j \rho_0 (H^* (Y)). $ \end{enumerate} \end{enumerate} \end{thm}
\begin{proof} The case $n=1$ is straightforward (compare \cite[Remark 2.1]{Kuhn_nonrealization}), hence suppose that $n \geq 2$.
The proof that the modules are almost unstable is based on \cite[Theorem 3.14]{Kuhn_nonrealization}, which states that, $H^* (D_{n,\bullet} X) $ is generated as a (bi)graded commutative algebra (under the $\star$-product) by elements of the form \[
\mathfrak{Q} _{r_1} \ldots \mathfrak{Q}_{r_t} L_{n-1} (x_1 \otimes \ldots \otimes x_k) \in H^* (D_{n,2^t k}Y), \] subject to the relations given in \cite[Section 3.3]{Kuhn_nonrealization}.
The category $\widetilde{\unst}$ is a Serre subcategory of $\mathscr{M}$ and is stable under $\otimes$, thus an increasing induction upon $j$ implies that it is sufficient to work modulo $\star$-decomposables. Hence one is reduced to considering $\star$-indecomposables of the above form (note that Adem-type relations intervene in considering the words in the dual Dyer-Lashof operations, by \cite[Proposition 3.13]{Kuhn_nonrealization}). Moreover, since the dual Dyer-Lashof operations double the $j$ degree and $\mathfrak{Q}_r$ is trivial for $r\geq n$, in a given $j$-degree, there are only finitely many words $\mathfrak{Q} _{r_1} \ldots \mathfrak{Q}_{r_t}$ which arise.
For $k$ tensor factors, the morphism $L_{n-1}$ is $\mathscr{A}$-linear: \[ L_{n-1} : \Sigma^{(k-1)(n-1)} H^* (Y)^{\otimes k} \rightarrow H^* (D_{n,k} Y). \] The hypothesis implies that $\Sigma^{(k-1)(n-1)} H^* (Y)^{\otimes k}$ lies in $\widetilde{\nil}_{(k-1)(n-1)}$ (in particular is almost unstable), hence so does the image of $L_{n-1}$, by Proposition \ref{prop:alnil_serre_sigma}. For $k>1$, $(k-1) (n-1)>0$, whereas for $k=1$, $L_{n-1}$ is an isomorphism.
A straightforward filtration argument based on Proposition \ref{prop:filter_ddl} allows words in dual Dyer-Lashof operations to be treated. Namely, up to higher terms, the image of an operation $\mathfrak{Q}_r$ is a quotient of the functor $\Sigma^r \Phi$ and, by Proposition \ref{prop:alunst_stable_Phi_R}, the functor $\Sigma^r \Phi$ induces: \[
\Sigma^r \Phi : \widetilde{\nil}_i \rightarrow \widetilde{\nil}_{2i+r}. \] Since $\widetilde{\nil}_{2i+r}$ is a Serre subcategory, up to filtration, this exhibits the image under $\mathfrak{Q}_r$ of an element of $\widetilde{\nil}_i$ as lying in $\widetilde{\nil}_{2i+r}$.
Putting these facts together, one concludes that $H^* (D_{n,j} Y)$ is almost unstable. Moreover, the argument shows that the only possible contributions not in $\widetilde{\nil}_1$ arise from $\star$-products of terms from the image of iterates of $\mathfrak{Q}_0$. If $H^* (Y)$ is good almost unstable, then there is an associated short exact sequence in $\mathscr{M}$: \[
0 \rightarrow (H^* Y)' \rightarrow H^* Y \rightarrow \rho_0 (H^* Y) \rightarrow 0. \] Any terms arising from $(H^* Y)'$ also lie in $\widetilde{\nil}_1$. Hence, to prove the result, it suffices to show that the projection $H^* Y \twoheadrightarrow \rho_0 (H^* Y)$ induces a surjection (recall $n \geq 2$, by hypothesis) \[
H^* (D_{n,j} Y) \twoheadrightarrow \Gamma^j (\rho_0 (H^* Y)), \] since $\Gamma^j (\rho_0 (H^* Y))$ is a reduced unstable module. This is clear as graded vector spaces; to check that the morphism is $\mathscr{A}$-linear, use \cite[Proposition 3.15(i)]{Kuhn_nonrealization}, which shows that the action of the Steenrod squares on $\mathfrak{Q}_0$ is correct modulo the higher dual Dyer-Lashof operations. \end{proof}
\begin{cor} \label{cor:d1_niln} For $1 \leq n \in \mathbb{N}$ and $X$ an $n$-connected space such that $H^*(X) \in \mathscr{N}il_n$ and is of finite type, the spectral sequence calculating $H^* (\Omega^n X)$ associated to the Arone-Goodwillie tower is good almost unstable.
In particular, the morphism $d_1$ from the $(-2)$-column to the $(-1)$-column induces \begin{eqnarray*}
\Gamma^2 (\rho_n H^* (X)) \rightarrow \rho_{n+1} H^* (X) & \ \ & n \geq 2\\ T^2 (\rho_1 H^* (X)) \rightarrow \rho_2 H^* (X) && n=1. \end{eqnarray*} \end{cor}
\begin{proof}
The final statement follows from Corollary \ref{cor:good_auss_d1}. \end{proof}
\section{Algebraic Models} \label{sect:alg_model}
In the case of the second extended power, it is possible to give explicit algebraic models for their cohomology. For current purposes, this is not strictly necessary; it is included since it makes the results of Section \ref{sect:cohomEP} much more explicit.
\subsection{The case of the second extended power}
The calculation of $H^* (D_{\infty,2} Y) $ (see \cite{KMcC}, where homology is used) is a stable version of the calculation of the quadratic construction \cite{Milgram,GLZ,HLS2}. Here $H^* (D_{n,2} Y) $ is considered for finite $n$; this brings the dual Browder operations into the picture.
\begin{lem} \label{lem:nat_incl}
For $Y$ a spectrum with $H^* (Y)$ bounded below and of finite type,
\begin{enumerate}
\item
the $\star$-product induces a monomorphism of $\mathscr{A}$-modules:
\[
\star: \Lambda^2 H^* (Y) \hookrightarrow H^* (D_{n,2} Y);
\] \item the dual Browder operation induces a monomorphism of $\mathscr{A}$-modules: \[ L_{n-1} : \Sigma^{n-1} S^2 H^* (Y) \hookrightarrow H^* (D_{n,2} Y); \] \item the sum of these induces a monomorphism of $\mathscr{A}$-modules: \[
\star \amalg L_{n-1} : \Lambda^2 H^* (Y) \oplus \Sigma^{n-1} S^2 H^* (Y) \hookrightarrow H^* (D_{n,2} Y). \]
\end{enumerate} \end{lem}
\begin{proof}
The morphisms are provided respectively by \cite[Proposition 3.11(iii)]{Kuhn_nonrealization} and \cite[Proposition 3.12]{Kuhn_nonrealization}. The injectivity is a consequence of
\cite[Theorem 3.14]{Kuhn_nonrealization}. \end{proof}
Recall from Section \ref{sect:alg_prelim} that, for $M \in \mathrm{Ob}\ \mathscr{M}$ and $1 \leq n \in \mathbb{N}$, there are natural morphisms: \begin{eqnarray} \label{eqn:pb_po}
\xymatrix{ & & \Gamma^2 M \ar@{->>}[d] \\ \Sigma^{n-1} \Phi M \ar@{^(->}[r] \ar@{^(->}[d] & R_{1/n} M \ar@{->>}[r] & \Phi M \\ \Sigma^{n-1} S^2 M . } \end{eqnarray} Here the middle row is not in general a sequence (for $n=1$ the morphisms are isomorphisms) and not in general exact (for $n \geq 3$).
\begin{defn} For $1 \leq n \in \mathbb{N}$ and $M \in \mathrm{Ob}\ \mathscr{M}$, let $\mathscr{E}_n M$ denote the $\mathscr{A}$-module given by forming the pushout and pullback of diagram (\ref{eqn:pb_po}). \end{defn}
\begin{exam} \label{exam:n=1}
For $n=1$ and $M \in \mathrm{Ob}\ \mathscr{M}$, $\mathscr{E}_1 M$ is naturally isomorphic to $M^{\otimes 2}$. \end{exam}
\begin{prop} \label{prop:functor_E}
For $1 \leq n \in \mathbb{N}$, the above construction defines a functor $
\mathscr{E}_n : \mathscr{M} \rightarrow \mathscr{M} $ which restricts to $\mathscr{E}_n : \mathscr{U} \rightarrow \mathscr{U} $.
For $M \in \mathrm{Ob}\ \mathscr{M}$, there is a natural short exact sequence:
\[
0
\rightarrow
\Lambda^2 M \oplus \Sigma^{n-1} S^2 M \rightarrow \mathscr{E}_n M \rightarrow R_{1/n-1}M \rightarrow 0.
\] \end{prop}
\begin{proof}
Straightforward. \end{proof}
The functor $\mathscr{E}_n$ provides an algebraic model for $H^* (D_{n,2}Y)$:
\begin{prop} \label{prop:alg_model_2col}
For $n \in \mathbb{N} \cup \{ \infty \}$ and $Y$ a spectrum with $H^* (Y)$ bounded below and of finite type, there is a natural isomorphism \[
\mathscr{E}_n H^* (Y) \cong H^* (D_{n, 2} Y) \] which extends the inclusion $\star \amalg L_{n-1} : \Lambda^2 H^* (Y) \oplus \Sigma^{n-1} S^2 H^* (Y) \hookrightarrow H^* (D_{n,2} Y)$ of Lemma \ref{lem:nat_incl}.
\end{prop}
\begin{proof} The result is essentially a restatement of the results of \cite[Section 3]{Kuhn_nonrealization}, using the algebraic functors introduced in Section \ref{sect:alg_prelim}. \end{proof}
\begin{exam} For $n=1$, one recovers from Example \ref{exam:n=1} and Proposition \ref{prop:alg_model_2col} the standard identification $H^* (D_{1,2} X) \cong H^* (X) ^{\otimes 2}$. \end{exam}
\begin{rem}
For $n \geq 1$, there are natural transformations $\mathscr{E}_{n+1} \rightarrow \mathscr{E}_n$, $ \mathscr{E}_n \Sigma \rightarrow \Sigma \mathscr{E}_{n+1}$ that provide algebraic models (via Proposition \ref{prop:alg_model_2col}) for the morphisms in cohomology induced respectively by $D_{n,2} Y \rightarrow D_{n+1,2}Y$ and $ \Sigma D_{n+1,2} Y \rightarrow D_{n,2}\Sigma Y$. \end{rem}
\subsection{The algebraic differential}
There is an algebraic differential which is related to the differential used by Singer (see \cite{p_viasm} for references).
Recall that $\mathbb{F} [u^{\pm 1}]$ has an $\mathscr{A}$-module structure extending that of $\mathbb{F}[u]$; it is a fundamental fact that the residue map $
\mathbb{F} [u^{\pm 1}] \rightarrow \Sigma^{-1} \mathbb{F} $ is $\mathscr{A}$-linear. This gives rise to a natural transformation $
d_M : R_1 M \rightarrow \Sigma^{-1} M $ in $\mathscr{M}$, since $R_1 M $ embeds in the half-completed tensor product $\mathbb{F}[u^{\pm 1}] \underline{\underline{\otimes}} M$. If $M$ is unstable then $d_M$ is trivial.
\begin{lem} \label{lem:diff_R1t} \cite{p_viasm}
For $1 \leq n \in \mathbb{N}$ and $N \in \mathrm{Ob}\ \mathscr{U}$, the differential $d_{\Sigma^{-n}N}$ induces a natural transformation $d_{1/n} : R_{1/n} (\Sigma^{-n}N )\rightarrow \Sigma^{-n-1}N$ which fits into a commutative diagram \[
\xymatrix{ R_1 (\Sigma^{-n} N) \ar[r]^{d_{\Sigma^{-n}N}} \ar@{->>}[d] & \Sigma^{-n-1}N \ar@{=}[d] \\ R_{1/n} (\Sigma^{-n}N ) \ar[r]_{d_{1/n}} & \Sigma^{-n-1}N. } \] The cokernel of $\Sigma d_{1/n} : \Sigma R_{1/n} (\Sigma^{-n}N ) \rightarrow \Sigma^{-n}N $ is $\Omega^n N$. \end{lem}
\begin{proof}
Straightforward. \end{proof}
By the definition of $\mathscr{E}_n M$ (for general $M \in \mathrm{Ob}\ \mathscr{M}$), the quotient $\mathscr{E}_n M/ \Lambda^2 M$ occurs as the pushout of the diagram: \[
\xymatrix{ \Sigma^{n-1} \Phi M \ar@{^(->}[r] \ar@{^(->}[d] & R_{1/n} M \\ \Sigma^{n-1} S^2 M. } \]
\begin{prop} \label{prop:model_d1}
For $1 \leq n \in \mathbb{N}$ and $K$ a connected unstable algebra with augmentation ideal $\overline{K}$, the natural transformation $$ d_{1/n} : R_{1/n} (\Sigma^{-n}\overline{K} ) \rightarrow \Sigma^{-n-1}\overline{K}$$
together with the product $
S^2 (\overline{K}) \rightarrow \overline{K} $ induce a natural transformation in $\mathscr{M}$: \[
d_1 : \mathscr{E}_n (\Sigma^{-n} \overline{K}) \rightarrow \Sigma^{-n-1} \overline{K}. \] \end{prop}
\begin{proof} The subobject $\Sigma^{n-1} S^2(\Sigma^{-n} \overline{K}) $ of $\mathscr{E}_n(\Sigma^{-n} \overline{K}) $ is naturally isomorphic to $\Sigma^{-n-1}S^2 ( \overline{K})$, hence the product induces a natural morphism of $\mathscr{A}$-modules $$ \Sigma^{n-1} S^2(\Sigma^{-n} \overline{K}) \rightarrow \Sigma^{-n-1} \overline{K}. $$
The verification that this is compatible with $d_{1/n}$ given by Lemma \ref{lem:diff_R1t} is straightforward. \end{proof}
\subsection{The spectral sequence differential $d_1$}
Consider the first stages of the Arone-Goodwillie tower for $\Omega^n X$, with $X$ an $n$-connected space. There is a cofibre sequence of spectra \[
D_{n,2} \Sigma^{-n}X \rightarrow P^n_2 X \rightarrow \Sigma^{-n} X \] and the differential $d_1$ from the $-2$-column to the $-1$-column of the spectral sequence is the connecting morphism \[
d_1 : \Sigma H^* ( D_{n,2} \Sigma^{-n} X) \rightarrow H^* (\Sigma^{-n} X). \] This can be identified algebraically in terms of the isomorphism of Proposition
\ref{prop:alg_model_2col}.
\begin{prop} \label{prop:compat_d1}
For $1 \leq n \in \mathbb{N}$ and $X$ a connected space with $H^* (X)$ of finite type, the following diagram commutes: \[
\xymatrix{ \Sigma \mathscr{E}_n (\Sigma^{-n} H^* (X) ) \ar[d]_\cong \ar[r]^(.55){\Sigma d_1} &
\Sigma^{-n} H^* (X) \ar[d]^\cong \\ \Sigma H^* ( D_{n,2} \Sigma^{-n} X) \ar[r]_(.55){d_1} &
H^* (\Sigma^{-n} X) } \] in which the $d_1$ of the top row indicates the algebraic differential of Proposition \ref{prop:model_d1}. \end{prop}
\begin{proof} This result corresponds to \cite[Proposition 4.3]{Kuhn_nonrealization}. \end{proof}
\subsection{Exploiting the nilpotent filtration} \label{subsect:exploit_nil}
\begin{prop} \label{prop:alg_d1_niln}
For $1 \leq n \in \mathbb{N}$ and an unstable module $N \in \mathrm{Ob}\ \mathscr{N}il_n$, the algebraic differential $d_{1/n}:R_{1/n} (\Sigma^{-n} N) \rightarrow \Sigma^{-n-1} N$ factors across $\Sigma ^{-n-1} \mathrm{nil}_{n+1}N$ and the resulting map fits into a natural commutative diagram: \[
\xymatrix{ R_{1/n}(\Sigma^{-n}N) \ar[r] \ar@{->>}[d] & \Sigma ^{-n-1} \mathrm{nil}_{n+1}N \ar@{->>}[d] \\ \Phi \rho_n N \ar[r]_{\delta_n} & \rho_{n+1}N, } \] where the vertical morphisms are induced by the natural projections $R_{1/n} \twoheadrightarrow \Phi$ of Lemma \ref{lem:R_truncated} together with $\Sigma^{-n}N \twoheadrightarrow \rho_n N$.
The natural transformation $\delta_n : \Phi \rho_n N \rightarrow \rho_{n+1}N$ is induced by the linear transformation $x \in \Sigma^{-n} N \mapsto Sq^{|x|+1}(x)$. In particular, if $\delta_n$ is non-trivial, then $N$ is not an $n$-fold suspension. \end{prop}
\begin{proof}
Straightforward, unravelling definitions to identify the morphism $\delta_n$. \end{proof}
\begin{rem} \label{rem:identify_delta}
Using the notation of Proposition \ref{prop:alg_d1_niln}, if $x = \Sigma^{-n} y$ for $y \in N$, then
$Sq^{|x|+1} (\Sigma^{-n} y) = \Sigma^{-n} Sq^{|y|+1 -n}y$. In particular, for $n=1$, the map $\Phi \rho_1 N \rightarrow \rho_2 N$ is
induced by the operation $Sq_0$ on $N$. \end{rem}
\begin{cor} \label{cor:d1_ngeq2}
In the situation of Corollary \ref{cor:good_auss_d1} for $n \geq 2$, the induced morphism factors as \[
\Gamma^2 (\rho_n H^* (X) ) \twoheadrightarrow \Phi (\rho_n H^* (X)) \stackrel{\delta_n}{\rightarrow} \rho_{n+1} H^* (X). \] \end{cor}
There is an alternative viewpoint on the natural transformation $\delta_n: \Phi \rho_n N \rightarrow \rho_{n+1}N$ for $N \in \mathscr{N}il_n$, based on the following result, in which $\Omega^n_1 : \mathscr{U} \rightarrow \mathscr{U}$ denotes the first left derived functor of the iterated loop functor $\Omega^n$.
\begin{prop} \label{prop:Omega_n_1}
For $1 \leq n \in \mathbb{N}$ and $s \geq n$, the functor $\Omega^n_1 : \mathscr{U} \rightarrow \mathscr{U}$ restricts to \[
\Omega^n_1 : \mathscr{N}il_s \rightarrow \mathscr{N}il_{2 (s-n)+1}. \] Moreover, for $N \in \mathrm{Ob}\ \mathscr{N}il_n$ (so that $\Omega^n_1 N \in \mathscr{N}il_1$) \[
\rho_1 (\Omega^n_1 N) \cong \Phi (\rho_n N). \] \end{prop}
\begin{proof}
The proof is by induction on $n$; for $n=1$ this is \cite[Lemma 6.1.3]{schwartz_book} (which also states that $\Omega_1$ takes values in $\mathscr{N}il_1$).
The inductive step uses the short exact sequence $\Omega \Omega^{n-1}_1 \rightarrow \Omega^n_1 \rightarrow \Omega_1 \Omega^{n-1}$ associated to the identification $\Omega^n = \Omega \circ \Omega^{n-1}$ (see \cite{p_viasm} and the references therein). Namely, if $M \in \mathscr{N}il_{s}$ with $s \geq n$, then $\Omega^{n-1}_1 M \in \mathscr{N}il_{2 (s-(n-1))+1} = \mathscr{N}il_{2 (s-n) +3}$, hence
$\Omega \Omega^{n-1}_1 M \in \mathscr{N}il_{2(s-n)+2}$. Similarly, $\Omega^{n-1}M \in \mathscr{N}il_{s-(n-1)} = \mathscr{N}il _{(s-n)+1}$, hence $\Omega_1 \Omega^{n-1} M \in \mathscr{N}il_{2 ((s-n)+1)-1}= \mathscr{N}il_{2 (s-n) +1}$. The result follows, since $\mathscr{N}il_{2 (s-n) +1}$ is closed under extensions. \end{proof}
\begin{cor} \label{cor:les_Omegan}
For $1 \leq n \in \mathbb{N}$ and $N \in \mathrm{Ob}\ \mathscr{N}il_n$, there is a natural exact sequence \[
\Sigma \Phi \rho_{n} M \stackrel{\Sigma \delta_n} {\rightarrow} \Sigma \rho_{n+1} M \rightarrow \Omega^n(M/\mathrm{nil}_{n+2}M) \rightarrow \rho_n M \rightarrow 0. \] \end{cor}
\begin{proof}
This follows by considering the long exact sequence for $\Omega^n_\bullet$ associated to the short exact sequence \[
0 \rightarrow \Sigma^{n+1} \rho_{n+1} M \rightarrow M/\mathrm{nil}_{n+2}M \rightarrow \Sigma^n \rho_n M \rightarrow 0 \] together with the factorization provided by Proposition \ref{prop:Omega_n_1}. \end{proof}
\section{Essential extensions} \label{sect:essential}
Let $\mathbb{F}$ be a finite field and recall that $\mathscr{F}$ is the category of functors from finite-dimensional $\mathbb{F}$-vector spaces to $\mathbb{F}$-vector spaces.
The aim of this section is to give a generalization of the following result:
\begin{thm}
\cite[Theorem 4.8]{KIII}
For $F$ a non-constant finite functor, precomposition with $F$ induces a (naturally split) monomorphism:
\[
\mathrm{Ext}_\mathscr{F}^* (G, H)
\hookrightarrow
\mathrm{Ext}_\mathscr{F}^* (G \circ F, H \circ F).
\] \end{thm}
This is refined by using the observation that the result only depends on the top polynomial degree behaviour of $F$.
\begin{rem} For $f : F_1 \rightarrow F_2$ a morphism between functors taking finite-dimensional values, for any morphism $\alpha : G \rightarrow H$, there is a commutative diagram: \[
\xymatrix{
G \circ F_1
\ar[r]^{\alpha_{F_1}}
\ar[d]_{G f}
&
H \circ F_1
\ar[d]^{Hf}
\\
G \circ F_2
\ar[r]_{\alpha_{F_2}}
&
H \circ F_2
} \] which corresponds to the two (equivalent) ways to define a natural transformation: $\mathrm{Hom}_{\mathscr{F}} (G, H) \rightarrow \mathrm{Hom}_{\mathscr{F}} (G\circ F_1, H\circ F_2)$. By naturality this extends to $\mathrm{Ext}^* _\mathscr{F}$. \end{rem}
Recall that a functor $F$ has polynomial degree exactly $d$ if it is polynomial of degree $d$ but not of degree $d-1$ (ie $F \in \mathscr{F}_d \backslash \mathscr{F}_{d-1}$).
\begin{thm} \label{thm:split_mono_gen}
For $f: F_1\rightarrow F_2$ a morphism between finite functors of polynomial degree exactly $d>0$, if $\mathrm{image}(f)$ has polynomial degree exactly $d$, then precomposition with $F_1$ together with $f$ induce a (naturally split) monomorphism:
\[
\mathrm{Ext}_\mathscr{F}^* (G, H)
\hookrightarrow
\mathrm{Ext}_\mathscr{F}^* (G \circ F_1, H \circ F_2).
\] \end{thm}
\begin{proof}
The proof follows that of \cite[Theorem 4.8]{KIII}, which relies upon the fact that the full subcategory of functors of polynomial degree at most $1$ (which contains the constant and additive functors) is semisimple.
The hypotheses ensure that $\Delta^{d-1}F_1$ and $\Delta^{d-1}F_2$ are non-constant functors of polynomial degree $\leq 1$ and that $\Delta^{d-1}f$ maps to a non-constant functor.
Applying \cite[Lemma 4.12]{KIII}, there exists a finite functor $E$ such that the identity functor $\mathrm{Id}$ is a direct summand of $\mathrm{image}(f) \circ E$. (The proof of the lemma is based on the fact that $\Delta^{d-1} F$ is a natural direct summand of the functor $V \mapsto F (V \oplus \mathbb{F}^{d-1})$, for $F \in \mathrm{Ob}\ \mathscr{F}$.)
By semi-simplicity the splitting factors:
\[
\xymatrix{
\mathrm{Id}
\ar@{^(->}[r]
\ar@/_2pc/[rrrr]^{1}
&
F_1 \circ E
\ar@{->>}[r]
\ar@/^1pc/[rr]^{f_E}
&
\mathrm{image}(f) \circ E
\ar@{^(->}[r]
&
F_2 \circ E
\ar@{->>}[r]
&\mathrm{Id}.
}
\] The proof is completed, {\em mutatis mutandis}, as in \cite[Section 4.3]{KIII}. \end{proof}
\begin{exam} \label{exam:split_mono_ses_frob}
Consider a finite functor $F$ of polynomial degree exactly $d>0$ and let $q_{d-1} F$ be the largest quotient of $F$ of polynomial degree $\leq d-1$, with
associated short exact sequence:
\[
0
\rightarrow
\overline{F}
\rightarrow
F
\rightarrow
q_{d-1} F
\rightarrow
0
\] where $\overline{F} \hookrightarrow F$ satisfies the hypotheses of Theorem \ref{thm:split_mono_gen}. The theorem shows that:
\[
\mathrm{Ext}_\mathscr{F}^* (G, H)
\hookrightarrow
\mathrm{Ext}_\mathscr{F}^* (G \circ \overline{F}, H \circ F).
\]
is a split monomorphism.
Taking $\mathbb{F} = \mathbb{F}_2$, there is a non-zero class $\varphi \in \mathrm{Ext}^1_\mathscr{F} (\Lambda^2 , \mathrm{Id}) $ given by the short exact sequence (\ref{eqn:frob_ses}). Applying the above result gives a pull-back diagram of short exact sequences
\[
\xymatrix{
0\ar[r]
&
F
\ar@{=}[d]
\ar[r]
&
\mathscr{E}
\ar[r]
\ar@{^(->}[d]
&
\Lambda^2 \circ \overline{F}
\ar[r]
\ar@{^(->}[d]
&
0
\\
0\ar[r]
&
F
\ar[r]
&
S^2 \circ F
\ar[r]
&
\Lambda^2 \circ F
\ar[r]
&
0
}
\] in which both short exact sequences are essential, in particular the top row corresponds to a non-zero class of $ \mathrm{Ext}_\mathscr{F}^1 (\Lambda^2 \circ \overline{F}, F)$. \end{exam}
For $\mathbb{F}=\mathbb{F}_2$, recall the extension classes from Section \ref{subsect:basic_functors}, which are related by the Frobenius morphism $\mathrm{Id} \rightarrow S^2$: \[ \tilde{e}_1 \in \mathrm{Ext}^2_\mathscr{F} (S^2, \mathrm{Id}) \mapsto e_1 \in \mathrm{Ext}^2 _\mathscr{F}(\mathrm{Id}, \mathrm{Id}). \]
\begin{cor} \label{cor:classes_S2_non-trivial} For $\mathbb{F}= \mathbb{F}_2$ and $\tilde{F} \subset F$ finite functors of polynomial degree exactly $d>0$, the following classes are non-trivial
\begin{enumerate}
\item
$i^* (e_1 \circ F) \in \mathrm{Ext}^2 _\mathscr{F} (\tilde{F} , F);$
\item
$i^* (\tilde{e}_1 \circ F) \in \mathrm{Ext}^2 _\mathscr{F} (S^2 \circ \tilde{F} , F),$
\end{enumerate} where $i^*$ denotes the pullback induced by the inclusion $i : \tilde{F} \hookrightarrow F$.
Moreover, under the Frobenius morphism $\mathrm{Id} \rightarrow S^2$, these classes are related by \[
i^* (\tilde{e}_1 \circ F) \in \mathrm{Ext}^2 _\mathscr{F} (S^2 \circ \tilde{F} , F) \mapsto i^* (e_1 \circ F) \in \mathrm{Ext}^2 _\mathscr{F} (\tilde{F} , F). \] \end{cor}
\begin{proof}
Follows directly from Theorem \ref{thm:split_mono_gen}, as in Example \ref{exam:split_mono_ses_frob}. \end{proof}
\section{The Main results} \label{sect:main}
This section gives the proofs of the main results of the paper, Theorems \ref{thm:main} and \ref{thm:main_n=1}, based upon the non-triviality result of Section \ref{sect:essential}. Namely an obstruction class $\omega_X $ living in a suitable group $\mathrm{Ext}^2_\mathscr{F} (-,-)$ is introduced, which must vanish. Combined with the non-vanishing result Theorem \ref{thm:split_mono_gen}, this provides restrictions on the structure of $H^*(X)$, in particular on the relationship between the first two non-trivial layers of its nilpotent filtration.
\subsection{Compatibility with the $\star$-product}
In the following, note that \cite[Proposition 4.1]{Kuhn_nonrealization} shows that the spectral sequence associated to the Arone-Goodwillie tower is a spectral sequence of bigraded algebras with respect to the $\star$-product.
\begin{prop} \label{prop:compatibility_star_cup}
For $X$ an $n$-connected space with $H^* (X)$ of finite type, there are identifications via the Arone-Goodwillie spectral sequence calculating
$H^* (\Omega^n X)$:
\begin{eqnarray*}
F_1 H^* (\Omega^n X) &\cong& \Sigma E^{-1,*}_\infty \\
F_2 H^* (\Omega^n X) / F_1 H^* (\Omega^n X) &\cong& \Sigma E^{-2,*}_\infty
\end{eqnarray*} and the morphism \[
\overline{\cup} : \Lambda^2 (F_1 H^* (\Omega^n X)) \rightarrow F_2 H^* (\Omega^n X) / F_1 H^*( \Omega^n X) \] coincides with the morphism induced by the $\star$-product in the spectral sequence. \end{prop}
\begin{proof}
The result follows from the compatibility of the spectral sequence with the multiplicative structure, established by Ahearn and Kuhn \cite{AK},
as used in \cite{Kuhn_nonrealization}.
\end{proof}
\subsection{Working modulo nilpotents}
Throughout this section, the following is supposed:
\begin{hyp} \label{hyp:X_niln} For fixed $1 \leq n \in \mathbb{N}$, $X$ is an $n$-connected space such that $H^* (X)\in \mathscr{N}il_n$ and is of finite type. \end{hyp}
The first condition ensures strong convergence of the spectral sequence calculating $H^* (\Omega^n X)$ and the second specifies the class of spaces of interest here. In particular, there is a surjection $
H^* (X)
\twoheadrightarrow
\Sigma^n \rho_n H^* (X). $
\begin{nota}
For $j \in \{1, 2 \}$, set $
F_j := l\big( F_j H^* (\Omega^n X)\big).
$ \end{nota}
Corollary \ref{cor:good_auss_d1} implies:
\begin{lem} \label{lem:identify_F1}
There is a natural isomorphism
$
F_1 \cong l \big(\rho_n H^* (X)\big).
$ \end{lem}
For clarity of presentation, the case $n=1$ is postponed to Section \ref{subsect:n=1}.
\begin{nota}
Set $K := l \big(\mathrm{ker} \{ \Phi \rho_n H^* (X)\stackrel{\delta_n}{\rightarrow} \rho_{n+1} H^* (X) \}\big)$,
so that $K \subset F_1$
and write $\widetilde{\Gamma^2 \circ F_1}$ for the functor defined by the pullback diagram:
\[
\xymatrix{
0
\ar[r]
&
\Lambda^2\circ F_1
\ar[r]
\ar@{=}[d]
&
\widetilde{\Gamma^2\circ F_1}
\ar@{^(->}[d]
\ar[r]
& K \ar@{^(->}[d]
\ar[r]
&
0
\\
0
\ar[r]
&
\Lambda^2\circ F_1
\ar[r]
&
\Gamma^2\circ F_1
\ar[r]
&
F_1
\ar[r]
&
0
}
\] \end{nota}
\begin{lem} \label{lem:identify:F2/F1}
For $n \geq 2$ there is an isomorphism
$
F_2/ F_1 \cong \widetilde{\Gamma^2 \circ F_1 }
$ and the morphism induced by $\overline{\cup} : \Lambda^2 ( F_1 H^* (\Omega^n X)) \rightarrow F_2 H^* (\Omega^n X) / F_1 H^* (\Omega^n X)$ identifies with the inclusion $
\Lambda^2 \circ F_1 \hookrightarrow \widetilde{\Gamma^2 \circ F_1}. $ \end{lem}
\begin{proof}
The result follows from Corollary \ref{cor:good_auss_d1}, using the identification of the differential $d_1$ modulo nilpotents, which follows from Corollary \ref{cor:d1_ngeq2}. \end{proof}
\begin{nota}
For $n \geq 2$, let $\omega_ X \in \mathrm{Ext}^2 _\mathscr{F} (K, F_1)$ be the Yoneda product of the class $\varphi\circ F_1 \in \mathrm{Ext}^1 _\mathscr{F} (\Lambda^2 \circ F_1 , F_1) $ with the class in $\mathrm{Ext}^1_\mathscr{F} (K , \Lambda^2 \circ F_1 )$ representing $F_2/F_1$. \end{nota}
\begin{lem} \label{lem:omega_X_descriptions} For $n \geq 2$, \begin{enumerate}
\item $\omega_X = i^* (e_1 \circ F_1)$, where $i^* : \mathrm{Ext}^2_\mathscr{F} (F_1, F_1) \rightarrow \mathrm{Ext}^2 _\mathscr{F} (K, F_1)$ is induced by the inclusion $i : K \hookrightarrow F_1$; \item there is a commutative diagram in which the three-term rows and columns are short exact: \[
\xymatrix{
F_1
\ar@{=}[d]
\ar[r]
&
S^2\circ F_1
\ar[r]
\ar[d]
&
\Lambda^2 \circ F_1
\ar[d]
\\
F_1
\ar[r]
&
F_2
\ar[r]
& F_2/F_1
\ar[d]
\\
&&
K
} \] and $S^2 \circ F_1 \rightarrow F_2$ is induced by $\cup : S^2 (F_1 H^* (\Omega^n X) ) \rightarrow F_2 H^* (\Omega^n X)$. \item In particular $\omega_ X = 0 \in \mathrm{Ext}^2 _\mathscr{F} (K, F_1)$. \end{enumerate} \end{lem}
\begin{proof} The first point follows from the identification of $F_2/F_1$ given in Lemma \ref{lem:identify:F2/F1} and the definition of $\widetilde{\Gamma^2 \circ F_1}$.
The second point is a consequence of the compatibility between the cup product and the $\star$-product and follows by combining Proposition \ref{prop:compatibility_star_cup} with the identifications of $F_1$ (Lemma \ref{lem:identify_F1}) and $F_2/F_1$.
The final point follows from homological algebra. \end{proof}
\begin{thm} \label{thm:main}
Suppose that $n \geq 2$ and $X$ is a topological space satisfying Hypothesis \ref{hyp:X_niln}.
If $F_1 = l \big(\rho_n H^* (X)\big)$ is a finite functor of polynomial degree exactly $d >0$, then $K:= l \Big(\mathrm{ker} \{ F_1 \stackrel{\delta_n}{\rightarrow} \rho_{n+1} H^* (X) \}\Big) $ has polynomial degree $<d$.
Equivalently, if $\rho_n H^* (X)$ is finitely generated over $\mathscr{A}$ and lies in $\mathscr{U}_d \backslash \mathscr{U}_{d-1}$ for $d>0$, then
\[
\delta_n : \Phi \rho_n H^* (X)
\rightarrow
\rho_{n+1} H^* (X)
\] has kernel in $\mathscr{U}_{d-1}$, in particular $\delta_n$ is non-trivial.
Hence \begin{enumerate} \item $ \rho_{n+1} H^* (X)\not \in \mathscr{U}_{d-1}$;
\item
$H^*(X) / \mathrm{nil}_{n+2} H^* (X) $ is not an $n$-fold suspension. \end{enumerate} \end{thm}
\begin{proof}
Lemma \ref{lem:omega_X_descriptions} shows that the obstruction class $\omega_X$ is trivial and also that it identifies with $i^* (e_1 \circ F_1)$.
If $F_1$ is a finite functor then, by Theorem \ref{thm:split_mono_gen}, this class is non-trivial if $K$ has exact polynomial degree $d$.
The fact that $H^*(X) / \rho_{n+2} H^* (X) $ is not an $n$-fold suspension follows from the identification of $\delta_n$ in Corollary \ref{cor:les_Omegan}.
\end{proof}
\begin{cor}
\label{cor:main_n>1}
Let $M$ be an unstable module of finite type such that
\begin{enumerate}
\item
$M \in \mathrm{Ob}\ \mathscr{N}il_n$, for $2 \leq n \in \mathbb{N}$;
\item $M$ is $n$-connected;
\item
$\rho_n M$ is finitely generated over $\mathscr{A}$ and lies in $\mathscr{U}_d \backslash \mathscr{U}_{d-1}$ for some $d >0$;
\item
the morphism induced by $Sq_{n-1}$, $\Phi M \rightarrow \Sigma^{1-n}\big( M/ \mathrm{nil}_{n+2} M \big)$ has image in $\mathscr{U}_{d-1}$.
\end{enumerate} Then $M$ cannot be the reduced $\mathbb{F}_2$-cohomology of an $n$-connected space. \end{cor}
\begin{proof}
A consequence of Theorem \ref{thm:main}, using the identification of $\delta_n$ from Section \ref{subsect:exploit_nil} (cf.
Remark \ref{rem:identify_delta}) to express the condition in terms of $Sq_{n-1}$.
\end{proof}
\begin{rem}
The statement has an unavoidably technical nature, due to the identification of the morphism $\delta_n$ in terms of an operation $Sq_i$ with $i >0$.
In the case $n=1$, the analogous result is conceptually simpler, since the corresponding operation is the cup square $Sq_0$ (see Corollary \ref{cor:refined_COR}).
\end{rem}
\subsection{The case $n=1$}
\label{subsect:n=1}
The exceptional case ($n=1$) corresponds to the Eilenberg-Moore spectral sequence. Theorem \ref{thm:cohomEP_good} implies that the cohomology
of $D_{1,2}(\Sigma^{-1} X)$ is the $\mathscr{A}$-module $H^*(\Sigma^{-1} X) ^{\otimes 2}$. The differential $d_1$
is induced by the product $\mu : H^*( X)^{\otimes 2} {\rightarrow} H^*(X)$, which factors by commutativity as
$
S^2 (H^*( X)){\rightarrow} H^*(X).
$ Since $H^* (X)$ is nilpotent, by hypothesis, this induces \[ S^2 ( \rho_1 H^* (X)) \rightarrow \rho_2 H^* (X). \]
\begin{nota}
Write $K$ for the kernel of the corresponding morphism in $\mathscr{F}$:
\[
S^2 \circ F_1 \rightarrow l \big(\rho_2 H^* (X)\big); \] \end{nota}
\begin{lem} \label{lem:n=1_F2/F1}
The quotient $F_2/F_1$ fits into the pullback diagram of short exact sequences:
\[
\xymatrix{
0
\ar[r]
&
\Lambda^2\circ F_1
\ar[r]
\ar@{=}[d]
& F_2/F_1
\ar@{^(->}[d]
\ar[r]
&
K
\ar@{^(->}[d] ^j
\ar[r]
&
0
\\
0
\ar[r]
&
\Lambda^2\circ F_1
\ar[r]
&
T^2\circ F_1
\ar[r]
&
S^2 \circ F_1
\ar[r]
&
0
}
\] \end{lem}
\begin{proof}
Straightforward. \end{proof}
\begin{nota}
For $n = 1 $, let $\omega_ X \in \mathrm{Ext}^2 _\mathscr{F} (K, F_1)$ be the Yoneda product of the class $\varphi\circ F_1 \in \mathrm{Ext}^1 _\mathscr{F} (\Lambda^2 \circ F_1 , F_1) $ with the class in $\mathrm{Ext}^1_\mathscr{F} (K , \Lambda^2 \circ F_1 )$ representing $F_2/F_1$.
\end{nota}
The following result is the analogue for $n=1$ of Lemma \ref{lem:omega_X_descriptions}:
\begin{lem} \label{lem:omega_X_descriptions_n=1} For $n =1$, \begin{enumerate}
\item $\omega_X = j^* (\tilde{e}_1 \circ F_1)$, where $j^* : \mathrm{Ext}^2_\mathscr{F} (S^2\circ F_1, F_1) \rightarrow \mathrm{Ext}^2 _\mathscr{F} (K, F_1)$ is induced by the inclusion $j : K \hookrightarrow S^2 \circ F_1$; \item there is a commutative diagram in which the three-term rows and columns are short exact: \[
\xymatrix{
F_1
\ar@{=}[d]
\ar[r]
&
S^2 \circ F_1
\ar[r]
\ar[d]
&
\Lambda^2 \circ F_1
\ar[d]
\\
F_1
\ar[r]
&
F_2
\ar[r]
&
F_2/F_1
\ar[d]
\\
&&K.
} \] \item In particular $\omega_ X = 0 \in \mathrm{Ext}^2 _\mathscr{F} (K, F_1)$. \end{enumerate} \end{lem}
\begin{proof} Analogous to the proof of Lemma \ref{lem:omega_X_descriptions}, {\em mutatis mutandis}, using Lemma \ref{lem:n=1_F2/F1}. \end{proof}
Before stating the theorem, it is worth resuming the situation; $F_1$ is of polynomial degree exactly $d>0$ and $\tilde{F_1} \stackrel{i}{\hookrightarrow} F_1$ an arbitrary subfunctor. There is a commutative square of inclusions: \[
\xymatrix{
\tilde{F_1}
\ar@{.>}[rd] _\alpha
\ar@{^(->}[rr]^i
\ar@{^(->}[dd]
&&
F_1
\ar@{^(->}[dd]
\\
&
K
\ar@{^(->}[rd]_j
\\
S^2 \circ \tilde{F_1}
\ar@{-->}[ur]_\beta
\ar@{^(->}[rr]_{S^2 i}
&&
S^2 \circ F_1.
} \] Applying $\mathrm{Ext}^2_\mathscr{F} (-, F_1)$ gives a diagram of $\mathrm{Ext}$-groups. Of particular interest are the following observations:
\begin{enumerate}
\item
If the dotted factorization $\alpha$ exists, then the class $\alpha^ * (\omega_X) \in \mathrm{Ext}^2 _\mathscr{F} (\tilde{F_1}, F_1)$ coincides with $i^* (e_1 \circ F_1)$, in the notation of Corollary \ref{cor:classes_S2_non-trivial}. \item If the dashed factorization $\beta$ exists (and hence $\alpha$), then the class $\beta^ * (\omega_X) \in \mathrm{Ext}^2 _\mathscr{F} (S^2 \circ \tilde{F_1}, F_1)$ coincides with $i^* (\tilde{e}_1 \circ F_1)$, in the notation of Corollary \ref{cor:classes_S2_non-trivial}. \end{enumerate}
\begin{thm} \label{thm:main_n=1} Let $X$ be a simply-connected space such that $H^* (X)$ is nilpotent and of finite type and $F_1= l \big(\rho_1 H^* (X)\big)$ is finite of polynomial degree exactly $d >0$. For a subfunctor $\tilde{F_1}$ of $F_1$ of polynomial degree exactly $d$, the following properties hold: \begin{enumerate}
\item
the subfunctor $S^2 \circ \tilde{F_1} \subset S^2 \circ F_1$ is not contained within $K$, equivalently the morphism \[
S^2 \circ \tilde{F_1} \rightarrow l \big(\rho_2 H^* (X)) \] is non trivial; \item the subfunctor $\tilde{F_1} \subset S^2 \circ F_1$ is not contained within $K$, equivalently the composite: \[
\tilde{F_1} \hookrightarrow S^2 \circ \tilde{F_1} \rightarrow l \big(\rho_2 H^* (X)) \] is non-trivial. \end{enumerate} Hence \begin{enumerate}
\item
the image of $S^2 \circ \tilde{F_1} \rightarrow l \big(\rho_2 H^* (X))$ has polynomial degree exactly $2d$, in particular $ \rho_2 H^* (X) \not\in \mathscr{U}_{2d-1}$;
\item
the morphism induced by $Sq_0$:
\[
\Phi \rho_1 H^* (X) \rightarrow \rho_2 H^* (X)
\] has kernel in $\mathscr{U}_{d-1}$, in particular is non-trivial, so that $H^* (X)/ \mathrm{nil}_3 H^* (X)$ is not a suspension. \end{enumerate} \end{thm}
\begin{proof}
The argument follows the proof of Theorem \ref{thm:main}, {\em mutatis mutandis}.
Suppose that $\tilde{F_1} \subset F_1$ is a subfunctor such that $S^2 \circ \tilde{F_1} \subset K$. Then the class $\omega_X \in \mathrm{Ext}^2 _\mathscr{F} (K, F_1)$ pulls back to the class of $\mathrm{Ext}^2_\mathscr{F} (S^2 \circ \tilde{F_1}, F_1)$ which is the pullback of $\tilde{e}_1 \circ F_1$ under the morphism induced by $\tilde{F_1} \subset F_1$. Suppose that $\tilde{F}_1$ has polynomial degree exactly $d$, then this class is non-trivial, by Corollary \ref{cor:classes_S2_non-trivial}; this contradicts Lemma \ref{lem:omega_X_descriptions_n=1}.
The argument using the hypothesis that $ \tilde{F_1} \subset K$ is similar {\em mutatis mutandis}, again using Corollary \ref{cor:classes_S2_non-trivial}.
To show that the polynomial degree of the image of $S^2 \circ \tilde{F_1} \rightarrow l \big(\rho_2 H^* (X))$ is exactly $2d$, without loss of generality we may assume that $\tilde{F_1}$ has no non-trivial quotient of polynomial degree $<d$. In this case, the functor $S^2 \circ \tilde{F_1}$ has no non-trivial quotient of degree $<2d$ (see \cite{CGPS}, for example).
Finally, the composite $\tilde{F_1} \hookrightarrow S^2 \circ \tilde{F_1} \rightarrow l \big(\rho_2 H^* (X))$ is given, upon passage to $\mathscr{U}/ \mathscr{N}il$, by the morphism $\Phi \Sigma \rho_1 H^* (X) \rightarrow \Sigma^2 \rho_2 H^* (X)$
induced by $Sq_0$, noting that the natural isomorphism $\Phi \Sigma \cong \Sigma^2 \Phi$ allows
the suspensions to be removed. Taking $\tilde{F_1}:= K \cap F_1$, the above argument shows that this has polynomial degree $\leq d-1$, whence the result. \end{proof}
\begin{rem}
Theorem \ref{thm:main_n=1} is an improvement upon the main result of \cite[Section 6]{CGPS}, where only the case $\tilde{F_1} = \overline{F_1}$, the kernel of $F_1 \twoheadrightarrow q_{d-1}F_1$ (see Example \ref{lem:omega_X_descriptions_n=1} for this notation) is considered, for the morphism
$S^2 \circ \overline{F_1} \rightarrow l (\rho_2 H^* (X))$. The above theorem
unifies this with the case $n \geq 2$.
The improvement is obtained by using Theorem \ref{thm:split_mono_gen} rather than \cite[Theorem 4.8]{KIII}. \end{rem}
\begin{cor} \label{cor:refined_COR} Let $K$ be a connected unstable algebra of finite type over $\mathbb{F}_2$ such that $\overline{K}$ is $1$-connected and $\rho_1 \overline{K}$ is finitely generated over $\mathscr{A}$ and lies in $\mathscr{U}_d \backslash \mathscr{U}_{d-1}$ for $d>0$ (in particular is non-zero).
If the morphism induced by the cup square, $ Sq_0 : \Phi \overline{K} \rightarrow K / \mathrm{nil}_3 K $, has image in $\mathscr{U}_{d-1}$, then $\overline{K}$ cannot be the reduced $\mathbb{F}_2$-cohomology of a simply-connected space. \end{cor}
\begin{proof} Follows from Theorem \ref{thm:main_n=1} (cf. Corollary \ref{cor:main_n>1}). \end{proof}
\begin{rem} Corollary \ref{cor:refined_COR} applies if the image of $Sq_0 : \Phi \overline{K} \rightarrow K$ lies in $\mathscr{U}_{d-1}$. This occurs, for example, if $Sq_0$ is zero, the case considered in Theorem \ref{THM:K}. Under this hypothesis, $QK$ is a quotient of $\Sigma \Omega \overline{K}$, and it follows that $\rho_1 \overline{K}$ is isomorphic to $\rho_1 QK$. \end{rem}
\section{The case $p$ odd} \label{sect:podd}
This section sketches the modifications necessary for $p$ an odd prime, writing $H^* (-)$ for singular cohomology with $\mathbb{F}_p$-coefficients. Here the difference stems from the fact that the enveloping algebra $UM$ of an unstable module $M$ is the quotient of the free symmetric algebra $S^* (M)$ by the relation $x^p = P_0 x$ for $x$ of even degree; in particular, the relation depends only upon the even degree elements of $M$.
The inclusion of the full subcategory $\mathscr{U} ' \subset \mathscr{U}$ of modules concentrated in even degree admits a right adjoint and induces an equivalence $\mathscr{U} ' / \mathscr{N}il ' \cong \mathscr{U} / \mathscr{N}il$, where $\mathscr{N}il' = \mathscr{U} ' \cap \mathscr{N}il$ (see \cite[Section 5.1]{schwartz_book}). Since the final part of the proof works modulo nilpotents, this allows the avoidance of problems at $p$ odd associated with the action of the Bockstein (cf. \cite{Schwartz} and \cite{BHRS}).
The fundamental exact sequence of functors (cf. Section \ref{subsect:basic_functors}) is now \[
0 \rightarrow \mathrm{Id} \stackrel{x \mapsto x^p}{\longrightarrow} S^p \longrightarrow \overline{S^p} \rightarrow 0, \] where the truncated symmetric power $\overline{S^p}$ is simple and self dual, together with the norm sequence: \begin{eqnarray*}
\xymatrix{
0
\ar[r]
&
\mathrm{Id}
\ar[r]
&
S^p
\ar[r]^N
&
\Gamma^p
\ar[r]
&
\mathrm{Id}
\ar[r]
&
0
} \end{eqnarray*} which represents a non-zero class in $\mathrm{Ext}^2_{\mathscr{F}} (\mathrm{Id}, \mathrm{Id})$ \cite{FLS}.
The $E^1$-page of the Arone-Goodwillie spectral sequence for $X \mapsto \Sigma^\infty \Omega^n X$ at $p$ odd is calculated as in \cite{BHRS}. The arguments again use almost unstable modules (for the odd primary case); although only a very limited part of the structure of the $E^1$-page intervenes, it is necessary to establish that the spectral sequence is good almost unstable (as in Section \ref{sect:cohomEP}).
One considers the $p$th filtration $F_p H^* (\Omega^n X) \subset H^* (\Omega^n X)$; the multiplicative structure induces a morphism of unstable modules \[
\bigoplus _{j=2}^{p-1} S^j (F_1 H^*(\Omega^n X) )
\rightarrow
F_p (H^* (\Omega^n X)) \] which is a monomorphism in $\mathscr{U} / \mathscr{N}il$. The cokernel $\overline{F_p} (H^* (\Omega^n X)) $ lies in a short exact sequence in $\mathscr{U} /\mathscr{N}il$: \[
0
\rightarrow
\Sigma^{-1} E^{-1, *}_\infty
\rightarrow
\overline{F_p} (H^* (\Omega^n X))
\rightarrow
\Sigma^{-p} E^{-p, *}_\infty \rightarrow 0. \] Moreover the cup product induces \[
S^p (F_1 H^*(\Omega^n X))
\rightarrow
\overline{F_p} (H^* (\Omega^n X)) \] which is a monomorphism in $\mathscr{U} / \mathscr{N}il$.
The identification of $ \Sigma^{-p} E^{-p, *}_\infty $ modulo nilpotents relies on the calculation of $d_{p-1} : \Sigma ( \Sigma^{-p} E^{-p, *}_{p-1}) \rightarrow \Sigma^{-1} E^{-1, *}_{p-1}$, since \cite[Proposition 4.1]{BHRS}
shows that lower differentials act trivially on the dual Dyer-Lashof operations.
Hereafter, the argument proceeds as for the case $p=2$, {\em mutatis mutandis}, by analysing $ \overline{F_p} (H^* (\Omega^n X))$ (for $p=2$, this is simply $ F_2 (H^* (\Omega^n X))$).
Since the proof reduces to an argument in $\mathscr{U}/\mathscr{N}il$, the ability to work in $\mathscr{U}'$ provides a useful simplification, making the parallel with the case $p=2$ transparent. For example, for $n>1$, the analogue of Corollary \ref{cor:d1_niln} for $d_{p-1}$ reduces to considering a morphism \[
\Gamma^p (\rho'_n H^* (X)) \rightarrow \rho'_{n+1} H^* (X). \] where, for $M$ an unstable module, $\rho'_k M$ denotes the largest submodule of $\rho_k M$ concentrated in even degrees. Moreover, Lemma \ref{lem:U_ses_Frobenius} has the following analogue: for $M' \in \mathrm{Ob}\ \mathscr{U}'$ there is a short exact sequence: \[
0
\rightarrow
\overline{S^p} M'
\rightarrow
\Gamma^p M'
\rightarrow
\Phi M'
\rightarrow 0
\] where the Frobenius functor $\Phi$ restricted to $\mathscr{U}'$ is directly analogous to that for $p=2$. As in the case $p=2$, the contribution $ \Phi(\rho'_n H^* (X))$ is provided by $\mathfrak{Q}_0$.
The proofs of the analogues of the results of Section \ref{sect:main} follow an identical strategy, depending upon Theorem \ref{thm:split_mono_gen}, which is prime independent.
\begin{rem} A refinement of the results can be obtained by exploiting the weight splitting of $\mathscr{U} / \mathscr{N}il $ associated to the action of the multiplicative group $\mathbb{F}_p^\times$. \end{rem}
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\begin{document}
\begin{center} \begin{large} {\bf Harmonic oscillator chain in noncommutative phase space with rotational symmetry} \end{large} \end{center}
\centerline {Kh. P. Gnatenko \footnote{E-Mail address: khrystyna.gnatenko@gmail.com}}
\centerline {\small \it Ivan Franko National University of Lviv, Department for Theoretical Physics,} \centerline {\small \it 12 Drahomanov St., Lviv, 79005, Ukraine} \centerline {\small \it Laboratory for Statistical Physics of Complex Systems} \centerline {\small \it Institute for Condensed Matter Physics, NAS of Ukraine, Lviv, 79011, Ukraine}
\begin{abstract} We consider a quantum space with rotationally invariant noncommutative algebra of coordinates and momenta. The algebra contains tensors of noncommutativity constructed involving additional coordinates and momenta. In the rotationally invariant noncommutative phase space harmonic oscillator chain is studied. We obtain that noncommutativity affects on the frequencies of the system. In the case of a chain of particles with harmonic oscillator interaction we conclude that because of momentum noncommutativity the spectrum of the center-of-mass of the system is discrete and corresponds to the spectrum of harmonic oscillator.
Key words: harmonic oscillator, composite system, tensors of noncommutativity \end{abstract}
\section{Introduction}
Owing to development of String Theory and Quantum Gravity \cite{Witten,Doplicher} studies of idea that space coordinates may be noncommutative has attracted much attention. Noncommutativity of coordinates leads to existence of minimal length, minimal area \cite{Romero2003,GnatenkoUFG18}, it leads to space quantization. Canonical version of noncommutative phase space is characterized by the following algebra \begin{eqnarray} [X_{i},X_{j}]=i\hbar\theta_{ij},\label{form101}{}\\{} [P_{i},P_{j}]=i\hbar\eta_{ij},\label{form1001}{}\\{} [X_{i},P_{j}]=i\hbar(\delta_{ij}+\gamma_{ij}).\label{form10001}{} \end{eqnarray} where $\theta_{ij}$, $\eta_{ij}$, $\gamma_{ij}$ are elements of constant matrixes. Parameters $\gamma_{ij}$ are considered to be defined as $\gamma_{ij}=\sum_k \theta_{ik}\eta_{jk}/4$ \cite{Bertolami}.
Noncommutative algebra (\ref{form101})-(\ref{form10001}) with $\theta_{ij}$, $\eta_{ij}$, $\gamma_{ij}$ being constants is not rotationally invariant \cite{Chaichian,Balachandran1}. Different generalizations of commutation relations (\ref{form101})-(\ref{form10001}) were considered to solve the problem of rotational symmetry breaking in noncommutative space \cite{Moreno,Galikova,Amorim,GnatenkoPLA14}. Many papers are devoted to studies of position-dependent noncommutativity \cite{Lukierski,Lukierski2009,BorowiecEPL,Borowiec,Borowiec1,Kupriyanov2009,Kupriyanov}, spin noncommutativity \cite{Falomir09,Ferrari13}. The algebras of these types of noncommutativity are rotationally invariant by they are not equivalent to noncommutative algebra of canonical type.
In paper \cite{GnatenkoIJMPA17} a rotationally invariant noncommutative algebra of canonical type was constructed on the basis of idea of generalization of parameters of noncommutativity to a tensors. Introducing additional coordinates and additional momenta $\tilde{a}_i$, $\tilde{b}_i$ $\tilde{p}^a_i$, $\tilde{p}^b_i$, we proposed to define these tensors in the following form \begin{eqnarray} \theta_{ij}=\frac{c_{\theta} l^2_{P}}{\hbar}\sum_k\varepsilon_{ijk}\tilde{a}_{k}, \label{form130}\\ \eta_{ij}=\frac{c_{\eta}\hbar}{l^2_{P}}\sum_k\varepsilon_{ijk}\tilde{p}^b_{k}.\label{for130} \end{eqnarray}
Constants $c_{\theta}$, $c_{\eta}$ are dimensionless, $l_P$ is the Planck length. To preserve the rotational symmetry the coordinates and momenta $\tilde{a}_i$, $\tilde{b}_i$ $\tilde{p}^a_i$, $\tilde{p}^b_i$ are supposed to be governed by a rotationally invariant systems. The systems are considered to be harmonic oscillators
\begin{eqnarray}
H^a_{osc}=\hbar\omega_{osc}\left(\frac{(\tilde{p}^{a})^{2}}{2}+\frac{\tilde{a}^{2}}{2}\right),\label{form104}\\
H^b_{osc}=\hbar\omega_{osc}\left(\frac{(\tilde{p}^{b})^{2}}{2}+\frac{\tilde{b}^{2}}{2}\right),\label{for104}
\end{eqnarray} with $\sqrt{{\hbar}}/\sqrt{{m_{osc}\omega_{osc}}}=l_{P}$ and large frequency $\omega_{osc}$ (the distance between energy levels is very large and oscillators are considered to be in the ground states). The algebra for additional coordinates and additional momenta is the following
\begin{eqnarray} [\tilde{a}_{i},\tilde{a}_{j}]=[\tilde{b}_{i},\tilde{b}_{j}]=[\tilde{a}_{i},\tilde{b}_{j}]=0,\\{} [\tilde{p}^{a}_{i},\tilde{p}^{a}_{j}]=[\tilde{p}^{b}_{i},\tilde{p}^{b}_{j}]=[\tilde{p}^{a}_{i},\tilde{p}^{b}_{j}]=0,\\{} [\tilde{a}_{i},\tilde{p}^{b}_{j}]=[\tilde{b}_{i},\tilde{p}^{a}_{j}]=0,\\{} [\tilde{a}_{i},X_{j}]=[\tilde{a}_{i},P_{j}]=[\tilde{p}^{b}_{i},X_{j}]=[\tilde{p}^{b}_{i},P_{j}]=0,\\{} [\tilde{a}_{i},\tilde{p}^{a}_{j}]=[\tilde{b}_{i},\tilde{p}^{b}_{j}]=i\delta_{ij}.
\end{eqnarray}
Therefore, we have $[\theta_{ij}, X_k]=[\theta_{ij}, P_k]=[\eta_{ij}, X_k]=[\eta_{ij}, P_k]=[\gamma_{ij}, X_k]=[\gamma_{ij}, P_k]=0$ as in the case of canonical noncommutativity (\ref{form101})-(\ref{form10001}) with $\theta_{ij}$, $\eta_{ij}$, $\gamma_{ij}$ being constants.
In the present paper we study influence of noncommutativity of coordinates and noncommutativity of momenta on the spectrum of a harmonic oscillator chain. Studies of a system of harmonic oscillators are important in various fields of physics among them molecular spectroscopy and quantum chemistry \cite{Ikeda,Fillaux,Hong90,Michelot92}, quantum optics \cite{Caves85,Schumaker85,Plenio}, nuclei physics \cite{Isgur,Glozman,Capstick}, quantum information processing \cite{Audenaert,Plenio,Plenio1}.
Harmonic oscillator was intensively studied in the frame of noncommutative algebra \cite{Hatzinikitas,Kijanka,Jing,Smailagic,Smailagic1,Muthukumar,Alvarez,Djemai,Dadic,Giri,Geloun,Abreu,Saha11,Nath,Shyiko}. Recently experiments with micro- and nano-oscillators were implemented for probing minimal length \cite{Bawaj}. In noncommutative space of canonical type two coupled harmonic oscillators were studied in \cite{Jellal,Bing,GnatenkoJPS17}. In \cite{Gnatenko_arx181} a spectrum of a system of $N$ oscillators interacting with each other (symmetric network of coupled harmonic oscillators) has been examined in rotationally invariant noncommutative phase space. In \cite{Daszkiewicz} classical $N$ interacting harmonic oscillators were examined in noncommutative space-time. In \cite{BastosPhysA,Laba} influence of noncommutativity of coordinates and noncommutativity of momenta on the properties of a system of free particles was examined.
The paper is organized as follows. In Section 2 we study energy levels of a harmonic oscillator chain in rotationally invariant noncommutative phase space. Particular case of a chain of particles with harmonic oscillator interaction is examined. Conclusions are presented in Section 3.
\section{Spectrum of harmonic oscillator chain in rotationally invariant noncommutative phase space}
Let us consider a chain of $N$ interacting harmonic oscillators with masses $m$ and frequencies $\omega$ in a space with (\ref{form101})-(\ref{form10001}) and (\ref{form130}), (\ref{for130}) in the case of the closed configuration of the system. So, let us study the following Hamiltonian \begin{eqnarray}
H_s=\sum_{n=1}^N\frac{( {\bf P}^{(n)})^{2}}{2m}+\sum_{n=1}^N\frac{m\omega^2( {\bf X}^{(n)})^{2}}{2}+\nonumber\\+k{\sum_{n=1}^N}({\bf X}^{(n+1)}-{\bf X}^{(n)})^2\label{form777} \end{eqnarray} with periodic boundary conditions ${\bf X}^{(N+1)}={\bf X}^{(1)}$, $k$ is a constant.
In general case coordinates and momenta which correspond to different particles satisfy noncommutative algebra with different tensors of noncommutativity. We have
\begin{eqnarray} [X^{(n)}_{i},X^{(m)}_{j}]=i\hbar\delta_{mn}\theta^{(n)}_{ij},\label{ffor101}\\{} [X^{(n)}_{i},P^{(m)}_{j}]=i\hbar\delta_{mn}\left(\delta_{ij}+\sum_k\frac{\theta^{(n)}_{ik}\eta^{(m)}_{jk}}{4}\right),\label{for1001}\\{} [P^{(n)}_{i},P^{(m)}_{j}]=i\hbar\delta_{mn}\eta^{(n)}_{ij},\label{ffor10001}\\{} \theta^{(n)}_{ij}=\frac{c_{\theta}^{(n)}l_P^2}{\hbar}\sum_k\varepsilon_{ijk}\tilde{a}_{k}, \label{tn}\\ \eta^{(n)}_{ij}=\frac{c_{\eta}^{(n)}\hbar}{l_P^2}\sum_k\varepsilon_{ijk}\tilde{p}^b_{k},\label{etn}
\end{eqnarray} where indexes $m,n=(1...N)$ label the particles \cite{GnatenkoIJMPA18}.
Because of presence of additional coordinates and momenta in (\ref{tn}), (\ref{etn}) we have to study Hamiltonian which include Hamiltonians of harmonic oscillators \begin{eqnarray} H=H_s+H^a_{osc}+H^b_{osc}\label{total} \end{eqnarray} The noncommutative coordinates and noncommutative momenta can be represented as \begin{eqnarray} X^{(n)}_{i}=x^{(n)}_{i}+\frac{1}{2}[{\bm \theta}^{(n)}\times{\bf p}^{(n)}]_i,\label{repx0}\\ P^{(n)}_{i}=p^{(n)}_{i}-\frac{1}{2}[{\bf x}^{(n)}\times{\bm \eta}^{(n)}]_i,\label{repp0} \end{eqnarray} where coordinates and momenta $x^{(n)}_i$, $p^{(n)}_i$ satisfy the ordinary commutation relations
\begin{eqnarray} [x^{(n)}_{i},x^{(m)}_{j}]=[p^{(n)}_{i},p^{(m)}_{j}]=0,\label{orx}\\{} [x^{(n)}_{i},p^{(m)}_{j}]=i\hbar\delta_{mn}.\label{orp}
\end{eqnarray}
and vectors ${\bm \theta}^{(n)}$, ${\bm \eta}^{(n)}$ have the components $\theta^{(n)}_i=\sum_{jk}\varepsilon_{ijk}{\theta^{n}_{jk}}/2,$ $\eta^{(n)}_i=\sum_{jk}\varepsilon_{ijk}{\eta^{(n)}_{jk}}/2.$ In our paper \cite{GnatenkoIJMPA18} we proposed the constants $c^{(n)}_{\theta}$, $c^{(n)}_{\eta}$ in tensors of noncommutativity to be determined by mass as $c^{(n)}_{\theta}m_n=\tilde{\gamma}=const$, $c^{(n)}_{\eta}/{m_n}=\tilde{\alpha}=const$ with $\tilde{\gamma}$, $\tilde{\alpha}$ being the same for different particles. Therefore one has
\begin{eqnarray} \theta^{(n)}_{ij}=\frac{\tilde{\gamma}l_P^2}{m_n\hbar}\sum_k\varepsilon_{ijk}\tilde{a}_{k}, \label{ctn}\\ \eta^{(n)}_{ij}=\frac{\tilde{\alpha} \hbar m_n}{l_P^2}\sum_k\varepsilon_{ijk}\tilde{p}^b_{k}.\label{cetn}
\end{eqnarray} Determination of tensors of noncommutativity in forms (\ref{ctn}), (\ref{cetn}) gives a possibility to consider noncommutative coordinates as kinematic variables \cite{GnatenkoIJMPA18}, to recover the weak equivalence principle \cite{Gnatenko_arxiv}. Taking into account (\ref{ctn}), (\ref{cetn}), in the case when the system consists of oscillators with the same masses one has $\theta^{(n)}_{ij}=\theta_{ij}$, $\eta^{(n)}_{ij}=\eta_{ij}$. Using (\ref{repx0})-(\ref{repp0}) the Hamiltonian $H_s$ reads \begin{eqnarray}
H_s=\sum_{n=1}^N\left(\frac{({\bf p}^{(n)})^{2}}{2m}+\frac{m\omega^2( {\bf x}^{(n)})^{2}}{2}+\right.\nonumber\\ \left.+k({\bf x}^{(n+1)}-{\bf x}^{(n)})^2-\frac{({\bm \eta}\cdot[{\bf x}^{(n)}\times{\bf p}^{(n)}])}{2m}-\right.\nonumber\\ \left.-\frac{m\omega^2({\bm \theta}\cdot[{\bf x}^{(n)}\times{\bf p}^{(n)}])}{2}-\right.\nonumber\\\left.-
{k}({\bm \theta}\cdot[({{\bf x}}^{(n+1)}-{{\bf x}}^{(n)})\times ({\bf p}^{(n+1)}-{\bf p}^{(n)})])+\right.\nonumber\\\left.+\frac{[{\bm \eta}\times{\bf x}^{(n)}]^2}{8m}+\frac{m\omega^2}{8}[{\bm \theta}\times{\bf p}^{(n)}]^2+\right.\nonumber\\\left.+\frac{k}{4}[{\bm \theta}\times ({\bf p}^{(n+1)}-{\bf p}^{(n)})]^2\right).\label{orm777} \end{eqnarray}
In \cite{GnatenkoIJMPA18} we showed that up to the second order in $\Delta H$ defined as \begin{eqnarray} \Delta H=H_s-\langle H_s\rangle_{ab}, \end{eqnarray} Hamiltonian \begin{eqnarray} H_0=\langle H_s\rangle_{ab}+H^a_{osc}+H^b_{osc},\label{2h0} \end{eqnarray} can be studied because up to the second order in the perturbation theory the corrections to spectrum of $H_0$ caused by terms $\Delta H=H-H_0=H_s-\langle H_s\rangle_{ab}$ vanish. Here notation $\langle...\rangle_{ab}$ is used for averaging over the eigenstates of $H^a_{osc}$ $H^b_{osc}$ which are well known
$\langle...\rangle_{ab}=\langle\psi^{a}_{0,0,0}\psi^{b}_{0,0,0}|...|\psi^{a}_{0,0,0}\psi^{b}_{0,0,0}\rangle$. For the harmonic oscillator chain we have \begin{eqnarray} \Delta H=\sum_{n=1}^N\left(\frac{[{\bm \eta}\times{\bf x}^{(n)}]^2}{8m}+\frac{m\omega^2}{8}[{\bm \theta}\times{\bf p}^{(n)}]^2-\right.\nonumber\\\left.-\frac{m\omega^2({\bm \theta}\cdot[{\bf x}^{(n)}\times{\bf p}^{(n)}])}{2}-\frac{({\bm \eta}\cdot[{\bf x}^{(n)}\times{\bf p}^{(n)}])}{2m}-\right.\nonumber\\\left.- {k}{\bm \theta}\cdot[({{\bf x}}^{(n+1)}-{{\bf x}}^{(n)})\times ({\bf p}^{(n+1)}-{\bf p}^{(n+1)})]+\right.\nonumber\\\left.+\frac{k}{4}[{\bm \theta}\times ({\bf p}^{(n+1)}-{\bf p}^{(n)})]^2-\frac{\langle\eta^2\rangle({\bf x}^{(n)})^2}{12m}-\right.\nonumber\\\left.-\frac{\langle\theta^2\rangle m\omega^2({\bf p}^{(n)})^2}{12}-\frac{k}{6}\langle{\theta}^2\rangle ({\bf p}^{(n+1)}-{\bf p}^{(n)})^2\right). \nonumber\\\label{delta}
\end{eqnarray}
here we take into account that $\langle\psi^{a}_{0,0,0}|\tilde{a}_i|\psi^{a}_{0,0,0}\rangle=\langle\psi^{b}_{0,0,0}|\tilde{p}_i|\psi^{b}_{0,0,0}\rangle=0$ and use the following notations \begin{eqnarray}
\langle\theta_i\theta_j\rangle=\nonumber\\=\frac{c_{\theta}^2l_P^4}{\hbar^2}\langle\psi^{a}_{0,0,0}| \tilde{a}_i\tilde{a}_j|\psi^{a}_{0,0,0}\rangle=\frac{c_{\theta}^2l_P^4}{2\hbar^2}\delta_{ij}=\frac{\langle\theta^2\rangle\delta_{ij}}{3},\label{thetar2}\\
\langle\eta_i\eta_j\rangle=\nonumber\\= \frac{\hbar^2 c_{\eta}^2}{l_P^4}\langle\psi^{b}_{0,0,0}| \tilde{p}^{b}_i\tilde{p}^{b}_j|\psi^{b}_{0,0,0}\rangle=\frac{\hbar^2 c_{\eta}^2}{2 l_P^4}\delta_{ij}=\frac{\langle\eta^2\rangle\delta_{ij}}{3}.\label{etar2} \end{eqnarray}
So, analyzing the form of $\Delta H$ (\ref{delta}), we have that up to the second order in the parameters of noncommutativity one can study Hamiltonian $H_0$. This Hamiltonian for convenience can be rewritten as
\begin{eqnarray} H_0=\sum_{n=1}^N\left(\frac{({\bf p}^{(n)})^{2}}{2m_{eff}}+\frac{m_{eff}\omega_{eff}^2( {\bf x}^{(n)})^{2}}{2}+\right.\nonumber\\\left.+k({{\bf x}}^{(n+1)}-{{\bf x}}^{(n)})^2+\right.\nonumber\\ \left.+\frac{k}{6}\langle{\theta}^2\rangle ({\bf p}^{(n+1)}-{\bf p}^{(n)})^2+H^a_{osc}+H^b_{osc} \right),\label{h5}
\end{eqnarray} with
\begin{eqnarray}
m_{eff}={m}\left({1+\frac{m^2\omega^2\langle\theta^2\rangle}{6}}\right)^{-1},\label{meff}\\
\omega_{eff}=\left({\omega^2+\frac{\langle\eta^2\rangle}{6m^2}}\right)^{\frac{1}{2}}\left({1+\frac{m^2\omega^2\langle\theta^2\rangle}{6}}\right)^{\frac{1}{2}}.\label{omegaeff}
\end{eqnarray} The terms $H^a_{osc}+H^b_{osc}$ commute with $H_0$. Coordinates and momenta ${\bf x}^{(n)}$, ${\bf p}^{(n)}$ satisfy (\ref{orx}), (\ref{orp}). Let us rewrite $H_0$ as \begin{eqnarray} H_0=\nonumber\\\frac{\hbar\omega_{eff}}{2}\sum_{n}\left(1+\frac{4km_{eff}\langle{\theta}^2\rangle}{3}\sin^2\frac{\pi n}{N}\right){\tilde {\bf p}}^{(n)}({\tilde{\bf p}^{(n)}})^{\dag}+\nonumber\\+\frac{\hbar\omega^2_{eff}}{2}\sum_{n}\left(1+\frac{8k}{m_{eff}\omega^2_{eff}}\sin^2\frac{\pi n}{N}\right){\tilde {\bf x}^{(n)}}({\tilde {\bf x}^{(n)}})^{\dag},\nonumber\\ \end{eqnarray} using
\begin{eqnarray} {\bf x}^{(n)}=\sqrt{\frac{\hbar}{Nm_{eff}\omega_{eff}}}\sum^{N}_{l=1}\exp\left({\frac{2\pi i nl}{N}}\right)\tilde{{\bf x}}^{(l)},\\ {\bf p}^{(n)}=\sqrt{\frac{\hbar m_{eff}\omega_{eff}}{N}}\sum^{N}_{l=1}\exp\left(-{\frac{2\pi i nl}{N}}\right)\tilde{{\bf p}}^{(l)}
\end{eqnarray} (see, for example, \cite{Plenio}). Introducing operators $a^{(n)}_j$ defined as
\begin{eqnarray} a^{(n)}_j=\frac{1}{\sqrt{2w_n}}\left(w_n\tilde{x}^{(n)}_j+i\tilde{p}^{(n)}_j\right),\\ w_n=\left(1+\frac{8k}{m_{eff}\omega^2_{eff}}\sin^2\frac{\pi n}{N}\right)^{\frac{1}{2}}\times\nonumber\\\times\left(1+\frac{4km_{eff}\langle{\theta}^2\rangle}{3}\sin^2\frac{\pi n}{N}\right)^{-\frac{1}{2}}
\end{eqnarray} we have \begin{eqnarray} H_0=\hbar\omega_{eff}\sum^N_{n=1}\sum^{3}_{j=1}\left(1+\frac{4km_{eff}\langle{\theta}^2\rangle}{3}\sin^2\frac{\pi n}{N}\right)^{\frac{1}{2}}\times\nonumber\\\times\left(1+\frac{8k}{m_{eff}\omega^2_{eff}}\sin^2\frac{\pi n}{N}\right)^{\frac{1}{2}}\left((a^{(n)}_j)^{\dag} a^{(n)}_j+\frac{1}{2}\right).\nonumber\\ \end{eqnarray} The spectrum of $H_0$ reads \begin{eqnarray} E_{\{n_1\},\{n_2\},\{n_3\}}=\hbar\sum^N_{a=1}\left(\omega^2_{eff}+\frac{8k}{m_{eff}}\sin^2\frac{\pi a}{N}\right)^{\frac{1}{2}}\times\nonumber\\\times\left(1+\frac{4km_{eff}\langle{\theta}^2\rangle}{3}\sin^2\frac{\pi a}{N}\right)^{\frac{1}{2}}\left(n^{(a)}_1+n^{(a)}_2+\right.\nonumber\\\left.+n^{(a)}_3+\frac{3}{2}\right)= \sum^N_{a=1}\hbar\omega_a\left(n^{(a)}_1+n^{(a)}_2+n^{(a)}_3+\frac{3}{2}\right),\nonumber\\\label{ennh} \end{eqnarray} where $n^{(a)}_i$ are quantum numbers ($n^{(a)}_i=0,1,2...$). Taking into account (\ref{meff}), (\ref{omegaeff}) the frequencies reads
\begin{eqnarray} \omega^2_a=\left(\omega^2+\frac{\langle\eta^2\rangle}{6m^2}\right)\left(1+\frac{m^2\omega^2\langle\theta^2\rangle}{6}+\right.\nonumber\\\left.+ \frac{4k^2m\langle \theta^2\rangle}{3}\sin^2\frac{\pi a}{N}\right)+\frac{8k}{m}\sin^2\frac{\pi a}{N}+\nonumber\\+\frac{32k^2\langle\theta^2\rangle}{3}\sin^4\frac{\pi a}{N}.\label{omaa} \end{eqnarray}
For a chain of particles with harmonic oscillator interaction, describing by Hamiltonian (\ref{form777}) with $\omega=0$, up to the second order in the parameters of noncommutativity one has
\begin{eqnarray} E_{\{n_1\},\{n_2\},\{n_3\}}=\nonumber\\= \sum^N_{a=1}\hbar\omega_a\left(n^{(a)}_1+n^{(a)}_2+n^{(a)}_3+\frac{3}{2}\right),\label{enh} \end{eqnarray} with \begin{eqnarray} \omega^2_a=\frac{8k}{m}\sin^2\frac{\pi a}{N}+\frac{\langle\eta^2\rangle}{6m^2}+ \frac{32k^2\langle{\theta}^2\rangle}{3}\sin^4\frac{\pi a}{N}.\label{omegaa} \end{eqnarray} It is worth noting that in the case of a space with noncommutative coordinates and commutative momenta (\ref{form101})-(\ref{form10001}) with (\ref{form130}) and $\eta_{ij}=0$ the spectrum of a chain of particles with harmonic oscillator reads (\ref{enh}) with \begin{eqnarray} \omega^2_a=\frac{8k}{m}\sin^2\frac{\pi a}{N}+\frac{32k^2\langle{\theta}^2\rangle}{3}\sin^4\frac{\pi a}{N}.\label{omegat} \end{eqnarray} Note that $\omega^2_N=0$ and corresponds to the spectrum of the center-of-mass of the system. Noncommutativity of momenta leads to discrete spectrum of the center-of-mass of a chain of interacting particles. From (\ref{enh}), (\ref{omegaa}) we have that the spectrum of the center-of-mass of the system corresponds to the spectrum of three dimensional harmonic oscillator with frequency determined as \begin{eqnarray} \omega^2_N=\frac{\langle\eta^2\rangle}{6m^2}.\label{omet} \end{eqnarray}
In the limit $\langle\theta^2\rangle\rightarrow0$, $\langle\eta^2\rangle\rightarrow0$ form (\ref{omaa}) we obtain well known result $\omega^2_a=\omega^2+\frac{8k}{m}\sin^2\frac{\pi a}{N}$.
\section{Conclusions}
Rotationally invariant algebra with noncommutativity of coordinates and noncommutativity of momenta has been considered. The algebra is constructed involving additional coordinates and additional momenta (\ref{form101})-(\ref{form10001}) with (\ref{form130}), (\ref{for130}). We have studied influence of noncommutativity on the spectrum of harmonic oscillator chain with periodic boundary conditions. For this purpose the total Hamiltonian has been examined (\ref{total}) and energy levels of harmonic oscillator chain have been obtained up to the second order in the parameters of noncommutativity. We have found that noncommutativity does not change the form of the chain's spectrum (\ref{ennh}). Noncommutativity of coordinates and noncommutativity of momenta affects on the frequencies of the system (\ref{omaa}).
The case of a chain of particles with harmonic oscillator interaction describing by Hamiltonian (\ref{form777}) with $\omega=0$ has been studied. We have obtained that the spectrum of the center-of-mass of the system is discrete because of noncommutativity of momenta. This spectrum corresponds to the spectrum of harmonic oscillator with frequency (\ref{omet}).
\vskip3mm \textit{Acknowledgement.} The author thanks Prof. V. M. Tkachuk for his advices and support during research studies. The publication contains the results of studies conducted by President's of Ukraine grant for competitive projects (F-75).
\begin{thebibliography}{99}
\bibitem{Witten} N. Seiberg, E. Witten. String theory and noncommutative geometry. {\it J. High Energy Phys.} {\bf 9909}, 032 (1999). \bibitem{Doplicher} S. Doplicher, K. Fredenhagen, J.E. Roberts. Spacetime quantization induced by classical gravity. {\it Phys. Lett. B} {\bf 331}, 39 (1994). doi: 10.1016/0370-2693(94)90940-7 \bibitem{Romero2003} J.M. Romero, J.A. Santiago, J.D. Vergara. Note about the quantum of area in a noncommutative space. {\it Phys. Rev. D} {\bf 68}, 067503 (2003). doi: 10.1103/PhysRevD.68.067503 \bibitem{GnatenkoUFG18} Kh. P. Gnatenko, V. M. Tkachuk. Lenght in a noncommutative phase space. {\it Ukr. J. Phys.} {\bf 63}, 102, (2018) doi: 10.15407/ujpe63.2.102.
\bibitem{Bertolami} O. Bertolami, R. Queiroz. Phase-space noncommutativity and the Dirac equation. {\it Phys. Lett. A} {\bf 375}, 4116 (2011). doi: 10.1016/j.physleta.2011.09.053 \bibitem{Chaichian}M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanu. Hydrogen atom spectrum and the lamb shift in noncommutative QED. {\it Phys. Rev. Lett.} 86, 2716 (2001). doi: 10.1103/PhysRevLett.86.2716
\bibitem{Balachandran1} A. P. Balachandran, P. Padmanabhan. Non-Pauli effects from noncommutative spacetimes. {\it J. High Energy Phys.} {\bf 1012}, 001 (2010).
\bibitem{Moreno} E. F. Moreno. Spherically symmetric monopoles in noncommutative space. {\it Phys. Rev. D} {\bf 72}, 045001 (2005). doi: 10.1103/PhysRevD.72.045001 \bibitem{Galikova} V. G\'alikov\'a, P. Presnajder. Hydrogen atom in fuzzy spaces-Exact solution. {\it J. Phys: Conf. Ser.} {\bf 343}, 012096 (2012). doi:10.1088/1742-6596/343/1/012096 \bibitem{Amorim} R. Amorim. Tensor operators in noncommutative quantum mechanics. {\it Phys. Rev. Lett.} {\bf 101}, 081602 (2008). doi: 10.1103/PhysRevLett.101.081602 \bibitem{GnatenkoPLA14} Kh.P. Gnatenko, V. M. Tkachuk. Hydrogen atom in rotationally invariant noncommutative space. {\it Phys. Lett. A} {\bf 378}, 3509 (2014). doi: 10.1016/j.physleta.2014.10.021 \bibitem{Lukierski} M. Daszkiewicz, J. Lukierski, M. Woronowicz, Towards quantum noncommutative -deformed field theory {\it Phys. Rev. D} {\bf77}, 105007 (2008) doi: 10.1103/PhysRevD.77.105007 \bibitem{Lukierski2009} M. Daszkiewicz, J. Lukierski, M. Woronowicz. $\kappa$-deformed oscillators, the choice of star product and free $\kappa$-deformed quantum fields {\it J. Phys. A: Math. Theor.} {\bf42}, 355201 (2009). doi: 10.1088/1751-8113/42/35/355201 \bibitem{BorowiecEPL} A. Borowiec, Kumar S. Gupta, S. Meljanac, A. Pachol, Constraints on the quantum gravity scale from $\kappa$-Minkowski spacetime {\it EPL } {\bf92}, 20006 (2010). doi: 10.1209/0295-5075/92/20006 \bibitem{Borowiec} A. Borowiec, J. Lukierski, A. Pachol. Twisting and-Poincare. {\it J. Phys. A: Math. Theor.} {\bf47} 405203 (2014). doi: 10.1088/1751-8113/47/40/405203 \bibitem{Borowiec1} A. Borowiec, A. Pachol. $\kappa$ Deformations and Extended $\kappa$-Minkowski Spacetimes {\it SIGMA } {\bf10}, 107 (2014). doi: 10.3842/SIGMA.2014.107 \bibitem{Kupriyanov2009} M. Gomes, V.G. Kupriyanov. Position-dependent noncommutativity in quantum mechanics. {\it Phys. Rev. D} {\bf79}, 125011 (2009). doi: 10.1103/PhysRevD.79.125011 \bibitem{Kupriyanov} V. G. Kupriyanov. A hydrogen atom on curved noncommutative space. {\it J. Phys. A: Math. Theor.} {\bf 46}, 245303 (2013). doi: 10.1088/1751-8113/46/24/245303
\bibitem{Falomir09} H. Falomir, J. Gamboa, J. López-Sarrión, F. Mendez, P.A.G. Pisani. Magnetic-dipole spin effects in noncommutative quantum mechanics {\it Phys. Lett. B} {\bf680} 384 (2009). doi: 10.1016/j.physletb.2009.09.007 \bibitem{Ferrari13} A.F. Ferrari, M. Gomes, V.G. Kupriyanov, C.A. Stechhahn, Dynamics of a Dirac fermion in the presence of spin noncommutativity, {\it Phys. Lett. B} {\bf718}, 1475 (2013). doi: 10.1016/j.physletb.2012.12.010
\bibitem{GnatenkoIJMPA17} Kh. P. Gnatenko, V. M. Tkachuk. Noncommutative phase space with rotational symmetry and hydrogen atom. {\it Int. J. Mod. Phys. A} {\bf32}, 1750161 (2017). doi: 10.1142/S0217751X17501615
\bibitem{Ikeda} S. Ikeda, F. Fillaux. Incoherent elastic-neutron-scattering study of the vibrational dynamics and spin-related symmetry of protons in the $KHCO_3$ crystal. {\it Phys. Rev. B} {\bf 59} 4134 (1999). doi: 10.1103/PhysRevB.59.4134 \bibitem{Fillaux} F. Fillaux. Quantum entanglement and nonlocal proton transfer dynamics in dimers of formic acid and analogues. {\it Chem. Phys. Lett.} {\bf 408} 302 (2005).doi: 10.1016/j.cplett.2005.04.069 \bibitem{Hong90} Fan Hong-yi. Unitary transformation for four Harmonically coupled identical oscillators {\it Phys. Rev. A}, {\bf42}, 4377 (1990). doi: 10.1103/PhysRevA.42.4377 \bibitem{Michelot92} F. Michelot. Solution for an arbitrary number of coupled identical oscillators. {\it Phys. Rev. A} {\bf45}, 4271 (1992). doi: 10.1103/PhysRevA.45.4271
\bibitem{Caves85} C.M. Caves, B.L. Schumaker, New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states {\it Phys. Rev. A} {\bf 31}, 3068 (1985). doi: 10.1103/PhysRevA.31.3068 \bibitem{Schumaker85} B.L. Schumaker, C.M. Caves. New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation. {\it Phys. Rev. A} {\bf 31}, 3093 (1985). doi: 10.1103/PhysRevA.31.3093
\bibitem{Plenio} M.B. Plenio, J. Hartley, J. Eisert, Dynamics and manipulation of entanglement in coupled harmonic systems with many degrees of freedom {\it New J. Phys.} {\bf6}, 36 (2004). doi: 10.1088/1367-2630/6/1/036
\bibitem{Isgur} N. Isgur, G. Karl. P-wave baryons in the quark model. {\it Phys. Rev. D} {\bf 18}, 4187 (1978). doi: 10.1103/PhysRevD.18.4187 \bibitem{Glozman} L. Ya. Glozman, D.O. Riska. The Spectrum of the nucleons and the strange hyperons and chiral dynamics. {\it Phys. Rept.} {\bf 268} 263, (1996) doi: 10.1016/0370-1573(95)00062-3 \bibitem{Capstick} S. Capstick, W. Roberts. Quark models of baryon masses and decays. {\it Prog. Part. Nucl. Phys.} {\bf 45}, 241, (2000). doi: 10.1016/S0146-6410(00)00109-5
\bibitem{Audenaert} K. Audenaert, J. Eisert, M.B. Plenio, R.F. Werner. Entanglement properties of the harmonic chain. {\it Phys. Rev. A} {\bf66}, 042327 (2002). doi: 10.1103/PhysRevA.66.042327 \bibitem{Plenio1} M. B Plenio, F. L Semiao. High efficiency transfer of quantum information and multiparticle entanglement generation in translation-invariant quantum chains. {\it New J. Phys.} {\bf7}, 73 (2005).
\bibitem{Hatzinikitas} A. Hatzinikitas, I. Smyrnakis. The noncommutative harmonic oscillator in more than one dimension. {\it J. Math. Phys.} {\bf43}, 113 (2002). doi: 10.1063/1.1416196 \bibitem{Kijanka} A. Kijanka, P. Kosinski. Noncommutative isotropic harmonic oscillator. {\it Phys. Rev. D} {\bf70}, 127702 (2004). doi: 10.1103/PhysRevD.70.127702 \bibitem{Jing} Jing Jian, Jian-Feng Chen. Non-commutative harmonic oscillator in magnetic field and continuous limit. {\it Eur. Phys. J. C} {\bf60}, 669 (2009). doi: 10.1140/epjc/s10052-009-0950-1 \bibitem{Smailagic} A. Smailagic, E. Spallucci. Isotropic representation of the noncommutative 2D harmonic oscillator. {\it Phys. Rev. D} {\bf65}, 107701 (2002). doi: 10.1103/PhysRevD.65.107701 \bibitem{Smailagic1} A. Smailagic, E. Spallucci. Noncommutative 3D harmonic oscillator. {\it J. Phys. A} {\bf35}, 363 (2002). doi: 10.1088/0305-4470/35/26/103 \bibitem{Muthukumar} B. Muthukumar, P. Mitra. Noncommutative oscillators and the commutative limit. {\it Phys. Rev. D} {\bf66} 027701 (2002). doi: 10.1103/PhysRevD.66.027701 \bibitem{Alvarez} P. D. Alvarez, J. Gomis, K. Kamimura, M. S. Plyushchay. Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton–Hooke symmetry. {\it Phys. Lett. B} {\bf 659} 906 (2008). doi: 10.1016/j.physletb.2007.12.016 \bibitem{Djemai} A. E. F. Djemai, H. Smail. On quantum mechanics on noncommutative quantum phase space. {\it Commun. Theor. Phys.} {\bf41}, 837 (2004). doi: 10.1088/0253-6102/41/6/837 \bibitem{Dadic} I. Dadic, L. Jonke, S. Meljanac. Harmonic oscillator on noncommutative spaces. {\it Acta Phys. Slov.} {\bf 55} 149 (2005). \bibitem{Giri} P. R. Giri, P. Roy. The non-commutative oscillator, symmetry and the Landau problem. {\it Eur. Phys. J. C} {\bf57}, 835 (2008). doi: 10.1140/epjc/s10052-008-0705-4 \bibitem{Geloun} J. Ben Geloun, S. Gangopadhyay, F. G. Scholtz. Harmonic oscillator in a background magnetic field in noncommutative quantum phase-space. {\it EPL} {\bf86}, 51001 (2009). doi: 10.1209/0295-5075/86/51001 \bibitem{Abreu} E. M.C. Abreu, M. V. Marcial, A. C.R. Mendes, W. Oliveira. Analytical and numerical analysis of a rotational invariant D= 2 harmonic oscillator in the light of different noncommutative phase-space configurations. {\it JHEP} 2013, 138 (2013). doi: 10.1007/JHEP11(2013)138 \bibitem{Saha11} A. Saha, S. Gangopadhyay, S. Saha. Noncommutative quantum mechanics of a harmonic oscillator under linearized gravitational waves. {\it Phys. Rev. D} {\bf83}, 025004 (2011). doi: 10.1103/PhysRevD.83.025004 \bibitem{Nath} D. Nath, P. Roy. Noncommutative anisotropic oscillator in a homogeneous magnetic field. {\it Ann. Phys}, {\bf377}, 115 (2017). doi: 10.1016/j.aop.2016.12.028 \bibitem{Shyiko} Kh. P. Gnatenko, O. V. Shyiko. Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space. {\it Mod. Phys. Lett. A} {\bf33}, 1850091 (2018). doi: 10.1142/S0217732318500918
\bibitem{Bawaj} M. Bawaj, et. al. Probing deformed commutators with macroscopic harmonic oscillators. {\it Nature Commun.} {\bf6}, 7503 (2015). doi: 10.1038/ncomms8503
\bibitem{Jellal} A. Jellal, El Hassan El Kinani, M. Schreiber. Two coupled harmonic oscillators on noncommutative plane. {\it Int. J. Mod. Phys. A} {\bf20}, 1515 (2005). doi: 10.1142/S0217751X05020835 \bibitem{Bing} Bing-Sheng Lin, Si-Cong Jing, Tai-Hua Heng. Deformation quantization for coupled harmonic oscillators on a general noncommutative space {\it Mod. Phys. Lett. A} {\bf 23}, 445, (2008). doi: 10.1142/S0217732308023992 \bibitem{GnatenkoJPS17} Kh. P. Gnatenko, V. M. Tkachuk. Two-particle system with harmonic oscillator interaction in noncommutative phase space {\it J. Phys. Stud.} {\bf 21}, 3001 (2017). \bibitem{Gnatenko_arx181} Kh. P. Gnatenko. System of interacting harmonic oscillators in rotationally invariant noncommutative phase space (2018) arXiv:1808.08515
\bibitem{Daszkiewicz} M.~Daszkiewicz, C.J.~Walczyk. Classical Mechanics of Many Particles Defined on Canonically Deformed Nonrelativistic Spacetime. {\it Mod. Phys. Lett} A {\bf26}, 819 (2011). doi: 10.1142/S0217732311035328 \bibitem{BastosPhysA} C. Bastos, A. E. Bernardini, J. F. G. Santos. Probing phase-space noncommutativity through quantum mechanics and thermodynamics of free particles and quantum rotors. {\it Physica A} {\bf438}, 340 (2015). doi: 10.1016/j.physa.2015.07.009 \bibitem{Laba} Kh. P. Gnatenko, H. P. Laba, V. M. Tkachuk. Features of free particles system motion in noncommutative phase space and conservation of the total momentum. {\it Mod. Phys. Lett. A} {\bf33}, 1850131 (2018). doi: 10.1142/S0217732318501316
\bibitem{GnatenkoIJMPA18} Kh. P. Gnatenko, V. M. Tkachuk. Composite system in rotationally invariant noncommutative phase space. {\it Int. J. Mod. Phys. A {\bf 33}}, 1850037 (2018). doi: 10.1142/S0217751X18500379 \bibitem{Gnatenko_arxiv} Kh. P. Gnatenko. Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle. (2018) arXiv:1808.00498.
\bibitem{GnatenkoPLA13} Kh. P. Gnatenko. Composite system in noncommutative space and the equivalence principle {\it Phys. Lett. A} {\bf 377}, 3061 (2013). doi: 10.1016/j.physleta.2013.09.036 \bibitem{GnatenkoPLA17} Kh.P. Gnatenko, V. M. Tkachuk. Weak equivalence principle in noncommutative phase space and the parameters of noncommutativity. {\it Phys. Lett. A} {\bf 381}, 2463 (2017). doi: 10.1016/j.physleta.2017.05.056 \bibitem{GnatenkoMPLA17} Kh. P. Gnatenko. Kinematic variables in noncommutative phase space and parameters of noncommutativity. {\it Mod. Phys. Lett. A} {\bf32}, 1750166 (2017). doi: 10.1142/S0217732317501668
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\begin{document}
\runningheads{M.~V.~Kulikova, J.~V.~Tsyganova}{Constructing stable adaptive Kalman filter-based techniques}
\title{Constructing numerically stable Kalman filter-based algorithms for gradient-based adaptive filtering}
\author{M.~V.~Kulikova\affil{1}\corrauth\, J.~V.~Tsyganova\affil{2}}
\address{\affilnum{1}CEMAT, Instituto Superior T\'{e}cnico, Universidade de Lisboa, Portugal\break \affilnum{2}Department of Mathematics and Information Technologies, Ulyanovsk State University, Russian Federation}
\corraddr{CEMAT, Instituto Superior T\'{e}cnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal. E-mail: maria.kulikova@ist.utl.pt}
\begin{abstract} This paper addresses the numerical aspects of adaptive filtering (AF) techniques for simultaneous state and parameters estimation arising in the design of dynamic positioning systems in many areas of research. The AF schemes consist of a recursive optimization procedure to identify the uncertain system parameters by minimizing an appropriate defined performance index and the application of the Kalman filter (KF) for dynamic positioning purpose. The use of gradient-based optimization methods in the AF computational schemes yields to a set of the filter sensitivity equations and a set of matrix Riccati-type sensitivity equations. The filter sensitivities evaluation is usually done by the conventional KF, which is known to be numerically unstable, and its derivatives with respect to unknown system parameters. Recently, a novel square-root approach for the gradient-based AF by the method of the maximum likelihood has been proposed. In this paper, we show that various square-root AF schemes can be derived from only two main theoretical results. This elegant and simple computational technique replaces the standard methodology based on direct differentiation of the conventional KF equations (with their inherent numerical instability) by advanced square-root filters (and its derivatives as well). As a result, it improves the robustness of the computations against roundoff errors and leads to accurate variants of the gradient-based AFs. Additionally, such methods are ideal for simultaneous state estimation and parameter identification since all values are computed in parallel. The numerical experiments are given. \end{abstract}
\keywords{Linear discrete-time stochastic systems; Kalman filtering; square-root implementation; filter sensitivity computation; maximum likelihood estimation; adaptive filtering.}
\maketitle
\section{Introduction}
The problem of developing the adaptive filtering (AF) techniques for simultaneous state and parameters estimation arising in the design of dynamic positioning systems has received increasing attention in recent years. Any AF method consists of a recursive optimization procedure to identify the uncertain system parameters by minimizing an appropriate defined performance index (e.g. the negative likelihood function) and the application of the Kalman filter (KF) for a dynamic positioning purpose. The gradient-based AF techniques additionally require the performance index (PI) gradient evaluation. It yields to a set of the filter sensitivity equations and a set of matrix Riccati-type sensitivity equations~\cite{Gupta1974,Mehra1974}. The sensitivities evaluation is usually done by the conventional KF and the direct differentiation of its equations (with respect to unknown system parameters); see~\cite{sandell1978maximum,Segal1988,segal1989new,hassani2013,Leander2014} and many others. A serious limitation of this methodology is the numerical instability of the conventional KF (with respect to round off errors) that may destroy the filter and, hence, the PI evaluation with the entire AF computational scheme.
Since 1960s there has been a great practical interest in the design of numerically stable and computationally efficient KF implementation methods. This has resulted in a large number of square-root (SR) filters, UD-based KF implementations and the fast Chandrasekhar-Kailath-Morf-Sidhu KF algorithms~\cite{Dyer1969,KaminskiBryson1971,Morf1974,Bierman1977,Sayed1994,ParkKailath1995}. Any of these advanced KF methods can replace the conventional KF in the AF schemes for a more stable PI evaluation. We may remark that current implementations of the KF are most often expressed in (what is called) an array square-root (ASR) form. They imply utilization of numerically stable orthogonal transformations for each recursion step. This feature enables more efficient parallel implementation and leads to algorithms with better numerical stability and conditioning properties; see \cite[Chapter~12]{KailathSayed2000} for an extended explanation.
Despite the existing diversity of the efficient KF algorithms, the PI \emph{gradient} evaluation (with respect to unknown system parameters) in terms of advanced KF methods is seldom addressed. In this paper we design simple and elegant computational scheme that allows for a natural extension of any ASR KF on the case of the filter sensitivities evaluation. Such methods are ideal for simultaneous state estimation and parameter identification since all values are computed in parallel. Additionally, our approach avoids implementation of the conventional KF (and its derivatives) because of its inherent numerical instability and, hence, improves the robustness of the computations. The first paper on a stable filter sensitivity computation has suggested an extension of the information-type KF~\cite{Bierman1990}. Then, the stable methods in terms of the covariance-type ASR KFs have been investigated in~\cite{2009Kul-IEEE,2009Kul-Matcom,2012TsyKul-AiT}. In this paper, we show that all types of the gradient-based AF schemes within stable ASR-based filters can be derived from two main theoretical results proven here. In contrast to the earlier published works, we do not derive a particular PI gradient evaluation method, but present a general approach that is able to extend any ASR KF (existing or new) on the robust filter derivatives computation. Additionally, the lower triangular scheme for the PI gradient evaluation is designed. This case has never been studied before. The numerical experiments are also given.
\section{State and parameter estimation of state-space models} \label{sec:MLE}
Consider discrete-time linear stochastic system of the form
\begin{align}
x_{k} = & F(\theta) x_{k-1}+B(\theta)u_{k-1} + G(\theta) w_{k-1}, \quad w_{k-1} \sim {\mathcal N}(0, Q(\theta)), \label{eq2.1} \\
z_k = & H(\theta) x_k+v_k, \quad v_{k} \sim {\mathcal N}(0, R(\theta)) \label{eq2.2}
\end{align} where $k$ is a discrete time ($k=1, \ldots, N$), i.e. $x_k$ means $x(t_k)$; vectors $x_k \in \mathbb R^n$ and $z_k \in \mathbb R^m$ are, respectively, the unknown dynamic state and the available measurements; $u_k \in \mathbb R^d$ is the deterministic input signal. The process noise, $\{w_k\}$, and the measurement noise, $\{v_k\}$, are uncorrelated Gaussian white-noise processes, with covariance matrices $Q(\theta) \ge 0$ and $R(\theta) > 0$, respectively. All random variables have known mean values, which we can take without loss of generality to be zero. The initial state $x_0$ is Gaussian random vector with the mean $\bar x_0(\theta)$ and the covariance matrix $\Pi_0(\theta)$, i.e. $x_0 \sim {\mathcal N}(\bar x_0(\theta), \Pi_0(\theta))$. It is independent from $\{w_k\}$ and $\{v_k\}$. Additionally, system~\eqref{eq2.1}, \eqref{eq2.2} is parameterized by a vector of unknown system parameters $\theta \in \mathbb R^p$, which needs to be estimated. This means that the state-space model is known up to certain parameters, i.e. the matrices $F(\theta) \in {\mathbb R}^{n\times n}$, $B(\theta) \in {\mathbb R}^{n\times d}$, $G(\theta) \in {\mathbb R}^{n\times q}$, $Q(\theta) \in {\mathbb R}^{q\times q}$, $H(\theta) \in {\mathbb R}^{m\times n}$ and $R(\theta) \in {\mathbb R}^{m\times m}$ may all depends on $\theta$. We stress that the initials conditions, i.e. $\bar x_0(\theta)$ and $\Pi_0(\theta) \in {\mathbb R}^{n\times n}$ may also depend on the parameters, however, such situation is seldom studied in the literature.
If there is no uncertainties in the system (i.e. $\theta$ is known and, hence, the state-space model is time-invariant), then the KF can be used for estimating the unobservable dynamic state $x_{k}$ from the corrupted measurements $z_1, \ldots, z_k$ as follows~\cite{KailathSayed2000}: \begin{align}
\hat x_{k+1|k} & = F \hat x_{k|k-1}+Bu_k + K_{p,k}e_k, & \hat x_{0|-1} & = \bar x_0, \label{state} && \\
K_{p,k} & = FP_{k|k-1}H^TR_{e,k}^{-1}, & e_k \; \; & = z_k-H\hat x_{k|k-1}, & R_{e,k} & = HP_{k|k-1}H^T+R \label{riccati1} \end{align}
where $K_{p,k}=\E{\hat x_{k+1|k} e_k^T}$ and $e_k \sim {\cal N}(0, R_{e,k})$ are innovations of the discrete-time KF. The matrix $P_{k|k-1}$ appearing in the above formulas is the error covariance matrix, i.e.
$P_{k|k-1}=\E{ (x_{k}-\hat x_{k|k-1})(x_{k}-\hat x_{k|k-1})^{T}}$, and satisfies the difference Riccati equation \begin{align} \label{riccati2}
P_{k+1|k} = & FP_{k|k-1}F^T+GQG^T - K_{p,k}R_{e,k}K_{p,k}^T, & P_{0|-1} = & \Pi_0 > 0. \end{align}
In the next section we consider the problem of parameters estimation by the gradient-based AF techniques.
\subsection{Gradient-based adaptive filtering schemes}
The state-space model~\eqref{eq2.1}, \eqref{eq2.2} under examination is known up to certain parameters, $\theta \in \mathbb R^p$. This means that the associated KF~\eqref{state}~-- \eqref{riccati2} depends on the unknown $\theta$ as well. We stress that both the dynamic state, $x_k$, and system parameters, $\theta$, must be estimated simultaneously from only the observed noisy signal $z_k$. The classical way of solving such a problem is to use \emph{adaptive} KF techniques, where the model parameters are estimated together with the dynamic state~\cite{Sarkka2009}.
To start implementing any AF scheme, one should choose first a PI that reflects the difference between the actual system and the utilized model with associated KF, which needs to be tuned up~\cite{hassani2013}. Then, a particular AF method is to be applied. At present, there are available many commonly used ways for the AF design in practice. Among them are the output-error techniques, the least-squares approach, the maximum-likelihood method, min-max entropy algorithms, {\it etc}~\cite{mehra1972approaches}. An important problem arising in this setting is convergence conditions of the constructed AF, i.e. convergence properties of the unknown parameter estimates, for both linear and nonlinear systems; see a consistency-oriented discussion in~\cite{Luders1974,Ljung1978,Bastin1988,marino1992global} and many others. For instance, \cite[Lemma~3.1]{Ljung1978} proves the main convergence result on this issue. It applies to quite a general situation and can be used as a common framework for the convergence and consistency analysis of many above-cited AF design methods. Throughout the paper we assume that all the assumptions of Lemma 3.1 hold; see details in~\cite[p.~776]{Ljung1978}.
The method of maximum likelihood is a general method for parameter estimation and often used in practice; see, for instance,~\cite{Gupta1974,Mehra1974,Segal1988,Bierman1990} and many others. It requires the maximization of the likelihood function (LF) given as follows~\cite{Schweppe1965}: \begin{equation} {\mathcal L}_{\theta}\left(Z_1^N\right) = -\frac{Nm}{2}\ln(2\pi) - \frac{1}{2} \sum \limits_{k=1}^N \left\{
\ln\left(\det R_{e,k}\right) + e_k^T R_{e,k}^{-1}e_k \right\} \label{LLF} \end{equation} where $Z_1^N=\{z_1,\ldots, z_N\}$ is $N$-step measurement history and $e_k \sim {\cal N}\left(0,R_{e,k}\right)$ are the innovations generated by the discrete-time KF~\eqref{state}~-- \eqref{riccati2}.
Hence, the negative log LF represents the PI for solving the parameters estimation problem by the method of maximum likelihood. Then, a recursive optimization procedure is used to identify the unknown system parameters $\theta$ by minimizing the PI. The optimization is often done by gradient-based or Newton's type methods where the computation of the LF gradient (LG) is necessary. The basic iteration in gradient-type non-linear programming methods has the following form~\cite{Gupta1974}: \begin{equation} \label{optimization}
\theta_{n} = \theta_{n-1} - \gamma \left.\nabla \mu(\theta) \right|_{\theta = \theta_{n-1}}, \; n=1,2, \ldots \end{equation}
where $\theta_n$ is the parameter vector at the $n$-th iteration and $\left.\nabla \mu(\theta) \right|_{\theta = \theta_{n-1}}$ is the gradient of the PI with respect to $\theta$ evaluated at $\theta = \theta_{n-1}$. The $\gamma$ is a scalar step size parameter chosen to ensure that $\left.\mu(\theta) \right|_{\theta_{n}} \le \left. \mu(\theta) \right|_{ \theta_{n-1}} + \epsilon$ where $\epsilon$ is a positive number that can be chosen in a variety of ways; see~\cite{Gupta1974} for more details.
As can be seen, the gradient-based AF approach requires the run of the KF at each iteration step of the optimization method (i.e. for each $\theta_{n-1}$) to generate the $\{ e_k, R_{e,k} \}$, $k = 1, \ldots, N$ for the PI evaluation, $\left.\mu(\theta)\right|_{\theta = \theta_{n-1}}$, corresponding to the current approximation $\theta_{n-1}$. Additionally, it demands the gradient computation, $\left.\nabla \mu(\theta)\right|_{\theta = \theta_{n-1}}$ at each $\theta_{n-1}$. This leads to a set of $p$ vector equations, known as the {\it filter sensitivity equations}, and a set of $p$ matrix equations, known as the {\it Riccati-type sensitivity equations}. The described {\it forward filter method} demands roughly an implementation of $p+1$ equivalent KF's all running in the forward time direction where $p$ is a number of the unknown system parameters.
In this manuscript, we do not discuss the particular optimization method that can be applied in each particular situation, but explain how the PI (the negative log LF) and its gradient can be computed accurately together with the system state. Such methods are ideal for simultaneous state estimation and parameter identification since all values are calculated in parallel.
\subsection{The problem of numerically instability of the conventional KF}
Both parts of the AF scheme, i.e. the chosen optimization method (for finding the optimal $\hat \theta^*$) and the chosen KF algorithm (for computing the PI and estimating $x_k$), play an important role in the computational scheme and affect the accuracy of the recursive adaptive estimator. Most of the previously proposed AF techniques are based on the conventional KF~\eqref{state}~-- \eqref{riccati2} and the direct differentiation of its equations for the PI gradient evaluation~\cite{sandell1978maximum,Segal1988,segal1989new,hassani2013,Leander2014}. The main disadvantage of this approach is numerical instability of the conventional KF while the requirement to compute the filter sensitivities in parallel deteriorates the situation. Here, we improve the accuracy of gradient-based AF methodology by replacing the numerically unstable conventional KF to advanced KF methods and their derivatives with respect to unknown system parameters. More precisely, we are focusing in the techniques developed in the KF community to solve ill conditioned problems. To start the presentation of our main results, we first discuss the ASR filters.
The matrix $P_{k|k-1}$ appearing in~\eqref{state}~-- \eqref{riccati2} has the physical meaning of being the variance of the state prediction error, $x_k-\hat x_{k|k-1}$, and therefore has to be nonnegative-definite. Round off errors may destroy this property leading to a failure of the filter. In contrast to the conventional KF~\eqref{state}~-- \eqref{riccati2}, the ASR methods propagate only square-root factors\footnote{Throughout the paper we use the Cholesky decomposition of the form $A=A^{T/2} A^{1/2}$, where $A^{1/2}$ is an upper triangular matrix with positive diagonal entries.
}
$P_{k|k-1}^{1/2}$ of the covariance matrices $P_{k|k-1}$, $k=1, \ldots, N$. The point is that the product of the computed factors, say $\hat P_{k|k-1}=\hat P_{k|k-1}^{T/2}\hat P_{k|k-1}^{1/2}$, is a symmetric matrix with positive elements on the diagonal and it is almost certainly nonnegative-definite; see~\cite[Chapter~12]{KailathSayed2000} for more details. Furthermore, any ASR filter uses a numerically stable orthogonal rotation at each iteration step. This feature enables more efficient parallel implementation and leads to algorithms with better numerical stability and conditioning properties.
All types of the ASR implementations can be divided into two simple cases. Some of them uses the orthogonal transformation of the form $QA=R$ with $R$ being an upper triangular matrix and others imply the transformation $QA=L$ where $L$ is a lower triangular matrix\footnote{ The left-hand side matrix $A$ is called the pre-array of the ASR filter. The right-hand side matrices $R$ and $L$ are called the post-arrays.}. We illustrate this statement by two ASR KF algorithms designed in~\cite{ParkKailath1995}.
\textsc{The extended square-root covariance filter} (\textbf{eSRCF}). Given the initial values for the filter: $P_{0|-1}^{-T/2} \hat x_{0|-1}=\Pi_{0}^{-T/2} \bar x_{0} $ and $P_{0|-1}^{1/2}=\Pi_0^{1/2}$, recursively update ($k=1, \ldots, N$): \begin{align} Q \left[
\begin{array}{cc|c} R^{1/2} & 0 & -R^{-T/2}z_k \\
P_{k|k-1}^{1/2}H^T & P_{k|k-1}^{1/2}F^T &
P_{k|k-1}^{-T/2} \hat x_{k|k-1} \\ 0 & Q^{1/2}G^T & 0 \end{array} \right]
& =
\left[
\begin{array}{cc|c} R_{e,k}^{1/2} & \bar K_{p,k}^T & -\bar e_k \\
0 & P_{k+1|k}^{1/2} & P_{k+1|k}^{-T/2} \hat x_{k+1|k} \\ 0 & 0 & \gamma_k \end{array} \right],\label{eq:esrcf:1}\\
\hat x_{k+1|k} & = \left( P_{k+1|k}^{T/2}\right) \left(
P_{k+1|k}^{-T/2}\hat x_{k+1|k}\right) \label{eq:esrcf:2} \end{align}
where $Q$ is any orthogonal transformation such that the first two (block) columns of the matrix on the right-hand side of formula~\eqref{eq:esrcf:1} is upper triangular. We introduce a notation for the normalized innovations $\bar e_k = R_{e,k}^{-T/2}e_k$ and the normalized Kalman gain $\bar K_{p,k} = FP_{k|k-1}H^TR_{e,k}^{-1/2}$. The matrix $R_{e,k}^{1/2}$ is a square-root factor of $R_{e,k}$.
\begin{remark} \label{remark:1}
The parentheses in~\eqref{eq:esrcf:2} are used to indicate the quantities that can be directly read off from the post-array in~\eqref{eq:esrcf:1}. Hence, no matrices need to be inverting for finding the state vector estimate $\hat x_{k+1|k}$, $k=1, \ldots, N$. \end{remark}
\textsc{The extended square-root information filter} (\textbf{eSRIF}). Given the initial values for the filter: $P_{0|-1}^{-T/2} \hat x_{0|-1}=\Pi_{0}^{-T/2} \bar x_{0} $ and $P_{0|-1}^{-T/2}=\Pi_0^{-T/2}$, recursively update ($k=1, \ldots, N$): \begin{eqnarray} & Q & \left[
\begin{array}{ccc|c} R^{-T/2} & -R^{-T/2}HF^{-1} & R^{-T/2}HF^{-1}GQ^{T/2} & -R^{-T/2}z_k \\
0 & P_{k|k-1}^{-T/2}F^{-1} & -P_{k|k-1}^{-T/2}F^{-1}GQ^{T/2}
& P_{k|k-1}^{-T/2}\hat x_{k|k-1} \ \\ 0 & 0 & I & 0 \end{array} \right] \nonumber \\ & = & \left[
\begin{array}{ccc|c} R_{e,k}^{-T/2} & 0 & 0 & -\bar e_k \\
-P_{k+1|k}^{-T/2} K_{p,k} &
P_{k+1|k}^{-T/2} & 0 &P_{k+1|k}^{-T/2} \hat x_{k+1|k} \\ * & * & * & * \end{array} \right] \label{eq:esrif:1} \end{eqnarray} where $Q$ is any orthogonal transformation such that the first three (block) columns of the post-array is a lower triangular matrix. The predicted state estimate can be found by solving the triangular system of the following form: \begin{equation}
\left( P_{k+1|k}^{-T/2}\right) \left(
\hat x_{k+1|k}\right)= \left(P_{k+1|k}^{-T/2}\hat x_{k+1|k}\right). \label{eq:esrif:2} \end{equation}
\begin{remark} \label{remark:2} The eSRCF and eSRIF can be verified by ``squaring'' both sides of the $QA=R$ (or $QA=L$), using the fact that $QQ^T=I$, and comparing the entries of both sides of the result. The detailed derivations can be also found in~\cite{KailathSayed2000}. \end{remark}
As mentioned earlier, the maximum likelihood estimation procedure leads to implementation of the KF (and its derivatives with respect to unknown system parameters), which is known to be numerically unstable. It is desirable to avoid the use of the conventional KF in the computational scheme. In other words, we would like to replace the disadvantageous conventional KF by numerically stable ASR filters, e.g. by the eSRCF/eSRIF presented above. The log LF and its gradient can be expressed in terms of the quantities appearing in the ASR filters as follows~\cite{2009Kul-IEEE}: \begin{align} {\mathcal L}_{\theta}\left(Z_1^N\right) = & -\frac{Nm}{2}\ln(2\pi) - \frac{1}{2} \sum \limits_{k=1}^N \left\{2
\ln\left(\det R_{e,k}^{1/2}\right) + \bar e_k^T \bar e_k \right\}, \label{LLF:sqrt} \\ \frac{\partial {\mathcal L}_{\theta}\left(Z_1^N \right)}{\partial \theta_i} = & -\sum \limits_{k=1}^N \left\{
\tr{ R_{e,k}^{-1/2} \cdot \frac{\partial R_{e,k}^{1/2}}{\partial \theta_i}} + \bar e_k^T \frac{ \partial \bar e_k}{\partial \theta_i} \right\}, \quad i=1, \ldots, p \label{grad-esrcf} \end{align} where ${\rm \bf tr}[ \cdot ]$ denotes the trace of matrices.
In the next section, we design a simple and convenient technique for computing derivatives of the ASR filter variables required in equation~\eqref{grad-esrcf}.
\section{ASR Filter Derivatives Computation} \label{sec:main_result}
First, we note that each iteration of ASR filters has the following form: $QA=B$ where $Q$ is any orthogonal transformation such that the post-array $B$ is either a lower triangular or upper triangular matrix. Treating these two cases separately, we prove the following main results.
\begin{lemma} [\textsc{The lower triangular case}] \label{lemma-1}
Let entries of the pre-array $A \in {\mathbb R}^{(s+k)\times (s+l)}$ be known differentiable functions of a parameter $\theta$. Consider the equation of the form $QA=L$ with the following partitioning: \begin{equation} \label{assume-2} \begin{array}{rcc} & \begin{array}{cc} s & \; \; \; \; l
\end{array} & \\
Q & \left[
\begin{array}{c|c}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}
\right]
& \!\!\!\!\!\!\!\! \begin{array}{cc}
k \\
s
\end{array} \end{array}
\begin{array}{rcc}
& \begin{array}{cc}
s & \; \; \; \; l
\end{array} & \\
= & \left[
\begin{array}{c|c}
0 & L_{12}\\
L_{21} & L_{22}
\end{array}
\right] &
\!\!\!\!\!\!\!\! \begin{array}{c}
k \\
s
\end{array} \end{array} \end{equation} where $Q \in {\mathbb R}^{(s+k)\times (s+k)}$ is an orthogonal matrix that lower-triangularizes the first (block) column of the matrix on the left-hand side of~(\ref{assume-2}) and $L_{21} \in {\mathbb R}^{s\times s}$ is lower triangular. Introduce the notation \begin{equation} \label{notation-lemma:1} \begin{array}{rcc} &
\begin{array}{cc}
s & \; \; \;\; l
\end{array} & \\
Q & \left[
\begin{array}{c|c} \left( A_{11}\right)'_{\theta} & \left( A_{12}\right)'_{\theta} \\
\left( A_{21}\right)'_{\theta} & \left( A_{22}\right)'_{\theta}
\end{array}
\right]
& \!\!\!\!\!\!\!\!
\begin{array}{c}
k \\
s
\end{array} \end{array}
\begin{array}{rcc} & \begin{array}{cc}
s & \; \; \; \; l
\end{array} & \\
= & \left[
\begin{array}{c|c}
X & N \\
Y & V
\end{array}
\right] &
\!\!\!\!\!\!\!\!
\begin{array}{c}
k \\
s
\end{array} \end{array}\;. \end{equation}
Then given the derivative of the pre-array $A'_{\theta}$, the following formulas calculate the corresponding derivatives of the post-array blocks: \begin{equation}
\label{lemma2:eq:3}
\left(L_{21}\right)'_{\theta}=(\bar {\cal U}^T + {\cal D} + \bar {\cal L})L_{21}, \end{equation} \begin{equation} \label{lemma2:eq:4} \left(L_{22}\right)'_{\theta} = \left[ \bar {\cal U}^T -\bar {\cal U} \right] L_{22} + L_{21}^{-T}X^T L_{12} + V \end{equation} where $\bar {\cal L}$, ${\cal D}$ and $\bar {\cal U}$ are respectively strictly lower triangular, diagonal and strictly upper triangular parts of the following matrix product $YL_{21}^{-1}$. \end{lemma}
\begin{proof} At first, we show that $Q'_{\theta}Q^T$ is a skew symmetric matrix. For that, we differentiate both sides of the formula $QQ^T=I$ with respect to $\theta$ and arrive at $Q'_{\theta}Q^T+Q\left(Q^T\right)'_{\theta}=0$, or in the equivalent form $Q'_{\theta}Q^T=-\left(Q'_{\theta}Q^T\right)^T$. The latter implies that the matrix $Q'_{\theta}Q^T$ is skew symmetric and can be presented as a difference of two matrices, i.e. $Q'_{\theta}Q^T = \bar {\cal U}^T- \bar {\cal U}$ where $\bar {\cal U}$ is an $(s+k)\times (s+k)$ strictly upper triangular matrix. Thus, the last $(s\times s)$-block located at the main diagonal of $Q'_{\theta}Q^T$ has the same form, i.e. \begin{equation} \label{new:new} \left[Q'_{\theta}Q^T\right]_{s\times s} = \bar {\cal U}^T_{s\times s}- \bar {\cal U}_{s\times s} \end{equation}
where $\bar {\cal U}_{s\times s}$ is a $s \times s$ strictly upper triangular matrix and $\left[Q'_{\theta}Q^T\right]_{s\times s}$ stands for the $(s\times s)$-matrix composed of the entries located at the intersections of the last $s$ rows with the last $s$ columns of the product $Q'_{\theta} Q^T$.
Next, we prove that the above-mentioned matrix $\bar {\cal U}_{s\times s}$ is, in fact, the upper triangular part of the matrix product $YL_{21}^{-1}$. To do this, we differentiate the first equation in formula~(\ref{assume-2}), i.e. $$
\begin{array}{rcc}
& \begin{array}{c}
s
\end{array} & \\
Q & \left[
\begin{array}{c}
A_{11} \\
A_{21}
\end{array}
\right]
& \!\!
\begin{array}{c}
k \\
s
\end{array} \end{array} \begin{array}{rcc}
& \begin{array}{c}
s
\end{array} & \\
= & \left[ \begin{array}{c}
0 \\
L_{21}
\end{array}
\right] &
\!\! \begin{array}{c}
k \\
s
\end{array} \end{array} $$ with respect to $\theta$. Then, taking into account notation~(\ref{notation-lemma:1}) and equality $A=Q^TL$, we obtain \begin{equation} \label{proof2:1} \left[ \begin{array}{c}
0 \\
\left(L_{21}\right)'_{\theta}
\end{array}
\right]
= Q'_{\theta} \left[ \begin{array}{c}
A_{11} \\
A_{21}
\end{array}
\right] +Q \left[ \begin{array}{c} \left( A_{11}\right)'_{\theta} \\
\left(A_{21}\right)'_{\theta}
\end{array}
\right] =Q'_{\theta}Q^T \left[ \begin{array}{c}
0 \\
L_{21}
\end{array}
\right] +\left[\begin{array}{c} X \\ Y
\end{array}\right].
\end{equation}
Further, it is not difficult to see that the pseudoinverse matrix (Moore-Penrose inversion) of
$\left[\; 0 \; | \; L_{21}\right]^T$ is $\left[ \; 0 \; | \; L_{21}^{-1}\right]$. Therefore the right multiplication of both sides of~(\ref{proof2:1}) by the pseudoinverse yields \begin{equation} \label{proof2:2} \left[
\begin{array}{c|c}
0 & 0\\ \hline
0 & \left(L_{21}\right)'_{\theta} L^{-1}_{21}
\end{array}
\right]=
Q'_{\theta}Q^T\left[0_{k\times k} \oplus I_{s\times s} \right]+\! \left[\begin{array}{c} X \\
Y
\end{array}\right]\! \left[ 0 \; L_{21}^{-1}\right]\! \end{equation} where $I_{s\times s}$ is the identity matrix of dimension~$s$ and $0_{k\times k}$ is the zero block of size $k \times k$. The $\left[0_{k\times k} \oplus I_{s\times s}\right]$ means $\mbox{\rm diag}\{0_{k\times k},I_{s\times s}\}$. Now we remark that \begin{equation} \label{proof2:3}
\left[
\begin{array}{c|c} 0 & 0\\[3pt] \hline
0 & \left(L_{21}\right)'_{\theta} L^{-1}_{21}
\end{array}
\right] =
\left[
\begin{array}{c|c} 0 & \left[Q'_{\theta}Q^T\right]_{col:~last~s}^{row:~first~k}\\[3pt]
\hline
0 & \left[Q'_{\theta}Q^T\right]_{s\times s}
\end{array}
\right] + \left[
\begin{array}{c|c} 0 & XL_{21}^{-1} \\[3pt]
\hline
0 & YL_{21}^{-1}
\end{array}
\right] \end{equation} where $\left[Q'_{\theta}Q^T\right]_{col:~last~s}^{row:~first~k}$ stands for the $(k\times s)$-matrix composed of the entries located at the intersection of the first $k$ rows with the last $s$ columns of the matrix $Q'_{\theta}Q^T$.
From the matrix equation~(\ref{proof2:3}), we conclude the following. First, the matrix on the left-hand side of~(\ref{proof2:3}) is block lower triangular. Thus, the strictly upper triangular part of the matrix $\left[Q'_{\theta}Q^T\right]_{s\times s}$ must exactly annihilate the strictly upper triangular part of the corresponding second term on the right-hand side of~(\ref{proof2:3}). In other words, if the matrix product $YL_{21}^{-1}$ is represented as $$
YL_{21}^{-1}=\bar {\cal L}_{s\times s}+{\cal D}_{s\times s}+\bar {\cal U}_{s\times s} $$ where $\bar {\cal L}_{s\times s}$, ${\cal D}_{s\times s}$ and $\bar {\cal U}_{s\times s}$ are respectively the strictly lower triangular, diagonal and strictly upper triangular parts, then the matrix $\bar {\cal U}_{s\times s}$, in fact, satisfies~(\ref{new:new}).
Now formula~(\ref{lemma2:eq:3}) is easily justified. Indeed, from the matrix equation~(\ref{proof2:3}), we obtain \begin{eqnarray}
\left(L_{21}\right)'_{\theta} L^{-1}_{21} & = & \underbrace{\bar {\cal U}^T_{s\times s} - \bar {\cal U}_{s\times s}}_{\left[Q'_{\theta}Q^T\right]_{s\times s}} + \underbrace{\bar {\cal L}_{s\times s} + {\cal D}_{s\times s} + \bar {\cal U}_{s\times s}}_{ YL_{21}^{-1}}, \nonumber \\ \left(L_{21}\right)'_{\theta} & = & (\bar {\cal U}^T_{s\times s} + {\cal D}_{s\times s} + \bar {\cal L}_{s\times s})L_{21}. \nonumber \end{eqnarray} For the sake of simplicity, in equation~(\ref{lemma2:eq:3}) we omit the subscripts of the matrices $\bar {\cal L}$, ${\cal D}$ and $\bar {\cal U}$.
Second, from the matrix equation~(\ref{proof2:3}) we observe that the first (block) row of the left-hand side matrix in~(\ref{proof2:3}) is zero. Thus, the first (block) row of the matrix $Q'_{\theta}Q^T$ must exactly cancel the corresponding block of the second term in~(\ref{proof2:3}), i.e. we arrive at \begin{equation}
\label{part_new:1} \left[Q'_{\theta}Q^T\right]_{col:~last~s}^{row:~first~k} = -XL_{21}^{-1}. \end{equation}
Next, we wish to validate~(\ref{lemma2:eq:4}). By differentiating the last equation in~(\ref{assume-2}) with respect to $\theta$, and then taking into account notation~(\ref{notation-lemma:1}), we derive $$
\left[ \begin{array}{c} \left(L_{12}\right)'_{\theta} \\ \left(L_{22}\right)'_{\theta} \end{array} \right] = Q'_{\theta} \left[ \begin{array}{c} A_{11} \\ A_{21}
\end{array} \right]
+ Q \left[ \begin{array}{c} \left(A_{12}\right)'_{\theta} \\ \left(A_{22}\right)'_{\theta}
\end{array} \right]= Q'_{\theta}Q^T \left[ \begin{array}{c} L_{12} \\ L_{22}
\end{array} \right]
+ \left[ \begin{array}{c} N \\ V
\end{array} \right]. $$
The previous formula implies that \begin{eqnarray} \label{th:proof:3} \left(L_{22}\right)'_{\theta} & = & V + \left[ Q'_{\theta}Q^T \right]_{col:~first~k}^{row:~last~s} L_{12}+\left[ Q'_{\theta}Q^T \right]_{s\times s}L_{22} \nonumber \\ & = & V + \left[ Q'_{\theta}Q^T \right]_{col:~first~k}^{row:~last~s} L_{12} + \left[ \bar {\cal U}_{s\times s}^T -\bar {\cal U}_{s\times s} \right] L_{22} \end{eqnarray} where $\bar {\cal U}_{s\times s}$ is the upper triangular matrix from~(\ref{new:new}) and $\left[ Q'_{\theta} Q^T \right]_{col:~first~k}^{row:~last~s}$ stands for the $(s\times k)$-matrix composed of the entries located at the intersections of the last $s$ rows with the first $k$ columns of the product $Q'_{\theta} Q^T$.
Eventually, formula~(\ref{part_new:1}) and the fact that $Q'_{\theta} Q^T$ is skew symmetric result in
\begin{equation}
\label{th:proof:1}\!\!
\left[Q'_{\theta} Q^T \right]_{col:~first~k}^{row:~last~s} =
- \left[\!\left[ Q'_{\theta} Q^T \!\right]_{col:~last~s}^{row:~first~k}\right]^T\! \!\!= - \left[ -
XL_{21}^{-1}\right]^T = L_{21}^{-T}X^T.
\end{equation}
Thus, the substitution of~(\ref{th:proof:1}) in~(\ref{th:proof:3}) validates~(\ref{lemma2:eq:4}) and completes the proof of Lemma~\ref{lemma-1}. \end{proof}
\begin{lemma} [\textsc{The upper triangular case}] \label{lemma-2} Let entries of the pre-array $A \in {\mathbb R}^{(s+k)\times (s+l)}$ be known differentiable functions of a parameter $\theta$. Consider the equation of the form $QA=R$ with the following partitioning: \begin{equation} \label{assume-2-2} \begin{array}{rcc} & \begin{array}{cc} s & \; \; \; \; l
\end{array} & \\
Q & \left[
\begin{array}{c|c}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}
\right]
& \!\!\!\!\!\! \begin{array}{cc}
s \\
k
\end{array} \end{array}
\begin{array}{rcc}
& \begin{array}{cc}
s & \; \; \; \; l
\end{array} & \\
= & \left[
\begin{array}{c|c}
R_{11} & R_{12} \\
0 & R_{22}
\end{array}
\right] &
\!\!\!\!\!\!\!\! \begin{array}{c}
s \\
k
\end{array} \end{array} \end{equation} where $Q \in {\mathbb R}^{(s+k)\times (s+k)}$ is an orthogonal matrix that produces the block zero entry on the right-hand side of~(\ref{assume-2-2}) and $R_{11} \in {\mathbb R}^{s\times s}$ is upper triangular. Introduce the notation~(\ref{notation-lemma:1}). Then given the derivative of the pre-array $A'_{\theta}$, the following formulas calculate the corresponding derivatives of the post-array: \begin{equation}
\label{lemma2-1}
\left(R_{11}\right)'_{\theta}=(\bar {\cal L}^T + {\cal D} + \bar {\cal U})R_{11}, \end{equation} \begin{equation} \label{lemma2-2} \left(R_{12}\right)'_{\theta} = \left[ \bar {\cal L}^T -\bar {\cal L} \right] R_{12} + R_{11}^{-T}Y^T R_{22}+N \end{equation} where $\bar {\cal L}$, ${\cal D}$ and $\bar {\cal U}$ are respectively strictly lower triangular, diagonal and strictly upper triangular parts of the following matrix product $XR_{11}^{-1}$. \end{lemma}
\begin{proof} Lemma~\ref{lemma-2} can be proved at the same way as Lemma~\ref{lemma-1}. The detail derivation of the formulas above can be also found in~\cite{2009Kul-Matcom}. \end{proof}
\section{Summary of the Computations}
Theoretical results presented in Lemmas~\ref{lemma-1}, \ref{lemma-2} yield a general computational scheme for the filter derivative computations. This new approach is able to replace the conventional KF (and its derivatives with respect to unknown system parameters) by any numerically stable ASR filter in the gradient-based AF techniques. The ASR methodology utilizes the pre-array $A$ of any chosen ASR filter and its derivatives in order to compute the post-array and its derivatives, respectively. Algorithms~\ref{alg:1}, \ref{alg:2} summarize the entire computational schemes in details.
\noindent
\begin{tabular}{p{0.5\textwidth}|p{0.5\textwidth}} \begin{Algorithm} \label{alg:1} {\small (\textsc{The lower triangular case}) }
\underline{Input Data:} The pre-array $A$ and its derivatives $\partial A/ \partial \theta_i$, $i=1, \ldots, p$.
\underline{Process:} Compute the post-array $L$ by~(\ref{assume-2}). Save matrices $\{ Q, L\}$ for future steps. Then, for each component $\theta_i$, $i=1, \ldots, p$: \begin{itemize} \item Find $Q\displaystyle\frac{\partial A}{\partial \theta_i}$ and introduce the notations as in~(\ref{notation-lemma:1}). Save the blocks $\{ X_i, Y_i, N_i, V_i\}$; \item Calculate $Y_iL_{21}^{-1}$. Split it into strictly lower triangular $\bar {\cal L}_i$, diagonal ${\cal D}_i$ and strictly upper triangular $\bar {\cal U}_i$ parts; \item Compute $\displaystyle\frac{\partial L_{21}}{\partial \theta_i}=\bigl(\bar {\cal U}_i^T+{\cal D}_i+\bar {\cal L}_i\bigr)L_{21};$ \item $\displaystyle\frac{\partial L_{22}}{\partial \theta_i} \! =\! \left[ \bar {\cal U}_i^T \!\! -\!\bar {\cal U}_i \right] L_{22}\! +\! L_{21}^{-T}X_i^T L_{12}\! + \! V_i$. \end{itemize} \underline{Output Data:} The post-array $L$ and its derivatives: $\partial L_{21}/ \partial \theta_i$, $\partial L_{22}/ \partial \theta_i$, $i=1, \ldots, p$. \end{Algorithm} &
\begin{Algorithm} \label{alg:2} {\small (\textsc{The upper triangular case})}
\underline{Input Data:} The pre-array $A$
and its derivatives $\partial A/ \partial \theta_i$, $i=1, \ldots, p$.
\underline{Process:} Compute the post-array $R$ by~(\ref{assume-2-2}). Save matrices $\{ Q, R\}$ for future steps. Then, for each component $\theta_i$, $i=1, \ldots, p$: \begin{itemize} \item Find $Q\displaystyle\frac{\partial A}{\partial \theta_i}$ and introduce the notations as in~(\ref{notation-lemma:1}). Save the blocks $\{ X_i, Y_i, N_i, V_i\}$; \item Calculate $X_iR_{11}^{-1}$. Split it into strictly lower triangular $\bar {\cal L}_i$, diagonal ${\cal D}_i$ and strictly upper triangular $\bar {\cal U}_i$ parts; \item Compute $\displaystyle\frac{\partial R_{11}}{\partial \theta_i}=\bigl(\bar {\cal L}^T_i+{\cal D}_i+\bar {\cal U}_i\bigr)R_{11};$ \item $\displaystyle\frac{\partial R_{12}}{\partial \theta_i} \! = \!\left[ \bar {\cal L}_i^T \!\! - \! \bar {\cal L}_i \right] R_{12} \! + \! R_{11}^{-T}Y_i^TR_{22}\!+ \!N_i$. \end{itemize}
\underline{Output Data:} The post-array $R$ and its derivatives: $\partial R_{11}/ \partial \theta_i$ and $\partial R_{12}/ \partial \theta_i$, $i=1, \ldots, p$. \end{Algorithm} \end{tabular}
Having applied Algorithms~\ref{alg:1}, \ref{alg:2} at each iteration of the ASR KF, we obtain the post array of the filter and its derivatives with respect to unknown system parameters for each $k=1, \ldots, N$. These quantities contain the $\{ e_k, R_{e,k} \}$ and $\{ \partial e_k/\partial \theta_i, \partial R_{e,k}/\partial \theta_i \}$, $i=1, \ldots p$, required for the PI and its gradient evaluation; see~\eqref{LLF:sqrt}, \eqref{grad-esrcf}. Hence, the entire gradient-based AF computational scheme can be formulated as follows. Let $\theta_{n-1}$ denotes the value of $\theta$ after $n-1$ iterations of the optimization algorithm~\eqref{optimization}. In this section we explain how the next cycle for computing $\theta_n$ can be obtained by using the chosen gradient-based optimization method, the chosen PI and any ASR filter, e.g. the eSRCF/eSRIF presented above.
\begin{Algorithm} \label{alg:3} (\textsc{Adaptive filtering scheme})
\noindent \underline{Input Data:} A current approximation $\theta_{n-1}$.
\noindent
\underline{Process:} Evaluate the system matrices (and its derivatives) at the current $\theta_{n-1}$: $\hat F(\theta)=F(\theta)\left|_{\theta_{n-1}}\right.$,
$\hat G(\theta) =G(\theta)\left|_{\theta_{n-1}} \right.$ {\it etc}. To improve robustness of the computations, replace the unstable conventional KF~(\ref{state})~-- (\ref{riccati2}) by any ASR filtering algorithm. Use the Cholesky decomposition to find the square-root of the matrices: $\hat \Pi_{0}^{1/2}$ and $\hat R^{1/2}$, $\hat Q^{1/2}$. Set the initial values for the filter and, then, process the measurements $\{ z_1, \ldots, z_N\}$ as follows: \begin{itemize} \item Form the pre-array and its derivatives of the chosen ASR filter. \item Given the pre-array (and its derivatives), find the post-array and its derivatives (with respect to each $\theta_i$, $i=1, \ldots, p$) as follows.
If the post-array has the form of a lower triangular matrix, then apply Algorithm~\ref{alg:1}. If the post-array has the form of an upper triangular matrix, then apply Algorithm~\ref{alg:2}.
\item Extract $\bar e_k$ and $R_{e,k}^{1/2}$ from the post-array. Compute new term in the PI. \item Extract $\partial \bar e_k/ \partial \theta_i$ and $\partial R_{e,k}^{1/2} / \partial \theta_i $, $i=1, \ldots, p$ from the derivatives of the post-array. Compute new term in the PI gradient. \end{itemize}
After processing all measurements $\{ z_1, \ldots, z_N\}$, the PI and its gradient are evaluated. Next, use the chosen gradient-based method in order to find the next approximation $\theta_n$.
\noindent \underline{Output Data:} Next approximation $\theta_{n}$. \end{Algorithm}
Repeat Algorithm~\ref{alg:3} for the next $\theta_{n+1}$ ($n=1, 2, \ldots$) until the stopping criterion is satisfied. The proposed technique simultaneously identifies the uncertain system parameters by minimizing the PI and estimates the unknown state vector of dynamic system.
\section{Illustrative examples: the eSRCF- and eSRIF-based AF methods}\label{sec:method}
The detailed derivation of the eSRCF-based technique for the log LF and its gradient evaluation can be found in~\cite{2009Kul-IEEE}. Here we show how the method can be easily obtained from Lemma~\ref{lemma-2} and Algorithms~\ref{alg:2}. First, we note that the post-array of the eSRCF filter is an upper triangular matrix. Next, the matrix that needs to be triangularized is of size $n+m$. Hence, we apply Lemma~\ref{lemma-2} to the eSRCF pre-array with $s=m+n$, $k=q$, $l=1$ and the following partitioning: $$ Q\; \underbrace{ \begin{bmatrix} \boxed{\begin{matrix} R^{1/2} & 0 \\
P_{k|k-1}^{1/2}H^T & P_{k|k-1}^{1/2} F^T \end{matrix}} & \boxed{\begin{matrix}
-R^{-{\rm T}/2}z_k \\
P_{k|k-1}^{-{\rm T}/2} \hat x_{k|k-1} \end{matrix}} \\
\mbox{\tiny $A_{11} \in {\mathbb R}^{(m+n)\times (m+n)}$} & \mbox{\tiny $A_{12} \in {\mathbb R}^{(m+n)\times 1}$} \\ \boxed{\begin{matrix} \qquad 0 \qquad & Q^{1/2}G^{\rm T} \; \;\; \end{matrix}}
& \boxed{\begin{matrix} \qquad \; 0 \; \qquad \;\; \; \end{matrix}}\\
\mbox{\tiny $A_{21} \in {\mathbb R}^{q\times (m+n)}$} & \mbox{\tiny $A_{22} \in {\mathbb R}^{q\times 1}$} \end{bmatrix} }_{Pre-array \; A} = \underbrace{ \begin{bmatrix} \boxed{ \begin{matrix} R_{e,k}^{1/2} & \bar K_{p,k}^T \\
0 & P_{k+1|k}^{1/2} \end{matrix}} & \boxed{\begin{matrix} -\bar e_k \\
P_{k+1|k}^{-T/2} \hat x_{k+1|k} \end{matrix}} \\
\mbox{\tiny $R_{11} \in {\mathbb R}^{(m+n)\times (m+n)}$} & \mbox{\tiny $R_{12} \in {\mathbb R}^{(m+n)\times 1}$} \\ \boxed{\begin{matrix} \quad 0 \quad & \quad 0 \; \; \;\; \end{matrix}}
& \boxed{ \begin{matrix} \qquad \gamma_k \qquad \; \; \; \end{matrix}} \\
\mbox{\tiny $R_{21} \in {\mathbb R}^{q\times (m+n)}$} & \mbox{\tiny $R_{22} \in {\mathbb R}^{q\times 1}$} \end{bmatrix} }_{Post-array \; R}. $$ The computational scheme of Algorithm~\ref{alg:2} leads to the filter derivative computations and, in particular, to the $\partial R_{e,k}^{1/2}/ \partial \theta_i$ and $\partial \bar e_{k}/ \partial \theta_i$, $i=1, \ldots, p$ evaluation required in the PI and its gradient evaluation.
At the same way the information-type algorithm can be easily obtained from the eSRIF; see also the detailed derivation for the log LF and its gradient evaluation in~\cite{2006KulSem-LN}. We note that the post-array of the eSRIF filter is a lower triangular matrix. Hence, we apply Lemma~\ref{lemma-1} and Algorithm~\ref{alg:1} to the eSRIF with $s=m+n+q$, $k=0$, $l=1$ and the following partitioning: \begin{align*} Q & \underbrace{ \begin{bmatrix} \boxed{\begin{matrix} \phantom{R^{-T/2}} & \phantom{-R^{-T/2}HF^{-1}} & \phantom{R^{-T/2}HF^{-1}GQ^{T/2} } \end{matrix}} & \boxed{\begin{matrix}
\phantom{P_{k|k-1}^{-T/2}\hat x_{k|k-1}} \end{matrix}} \\
\mbox{\tiny $A_{11}$ is empty} & \mbox{\tiny $A_{12}$ is empty} \\ \boxed{\begin{matrix} R^{-T/2} & -R^{-T/2}HF^{-1} & R^{-T/2}HF^{-1}GQ^{T/2} \\
0 & P_{k|k-1}^{-T/2}F^{-1} & -P_{k|k-1}^{-T/2}F^{-1}GQ^{T/2}\\ 0 & 0 & I \end{matrix}}
& \boxed{\begin{matrix} -R^{-T/2}z_k \\
P_{k|k-1}^{-T/2}\hat x_{k|k-1} \\ 0 \end{matrix}}\\
\mbox{\tiny $A_{21} \in {\mathbb R}^{(m+n+q)\times (m+n+q)}$} & \mbox{\tiny $A_{22} \in {\mathbb R}^{(m+n+q)\times 1}$} \end{bmatrix} }_{Pre-array \; A} \\ = & \underbrace{ \begin{bmatrix} \boxed{ \begin{matrix}
\phantom{-P_{k+1|k}^{-T/2} K_{p,k}} & \phantom{P_{k+1|k}^{-T/2}} & \phantom{ 0} \end{matrix}} & \boxed{\begin{matrix}
\phantom{P_{k+1|k}^{-T/2} \hat x_{k+1|k} } \end{matrix}} \\
\mbox{\tiny $L_{11}$ is empty} & \mbox{\tiny $L_{12}$ is empty} \\ \boxed{\begin{matrix} R_{e,k}^{-T/2} & 0 & 0 \\
-P_{k+1|k}^{-T/2} K_{p,k} & P_{k+1|k}^{-T/2} & 0 \\ * & * & * \end{matrix}}
& \boxed{ \begin{matrix} -\bar e_k \\
P_{k+1|k}^{-T/2} \hat x_{k+1|k} \\ * \end{matrix}} \\
\mbox{\tiny $L_{21} \in {\mathbb R}^{(m+n+q)\times (m+n+q)}$} & \mbox{\tiny $L_{22} \in {\mathbb R}^{(m+n+q)\times 1}$} \end{bmatrix} }_{Post-array \; L}. \end{align*}
In summary, the proposed computational schemes naturally extend any ASR filter and allow {\it the filter and the filter sensitivity equations} to be updated in parallel. Hence, such methods are ideal for simultaneous state estimation and parameter identification.
\begin{remark} Some modern ASR KF implementations are based on the $UDU^T$ factorization of the pre-array. Hence, an alternative approach to a problem of numerically stable PI and its gradient evaluation can be found in, the so-called, UD-based filters developed first in~\cite{Bierman1977}. The problem of the UD-based filters' derivative computation (with respect to unknown system parameters) has been formulated by Bierman {\it et al.} in~\cite{Bierman1990} and has been open since 1990s. It was recently solved in~\cite{Tsyganova2013IEEE}. \end{remark}
\section{Numerical Examples\label{experiments}}
First, we wish to check our theoretical derivations presented in Lemma~\ref{lemma-1} and~\ref{lemma-2}. To do so, we consider the following simple test problems.
\begin{example}\label{ex:1:1} (\textsc{Simple test problem: the upper triangular case})
\noindent For the given pre-array $$
A= \left[
\begin{array}{ccc|c} {\theta^5}/20 \; & \; \theta^4/8 & {\theta^3}/{6} & \; \theta^3/3\\ {\theta^4}/8 \; & \; \theta^3/3 & {\theta^2}/{2} & \; {\theta^2}/{2}\\ {\theta^3}/{6} \; & \; {\theta^2}/{2} & \theta & \; 1 \end{array} \right], $$ compute the post-arrays $R$ and its derivative $R'_{\theta}$, say, at $\theta=2$ where the first three (block) columns of the post-array $R$ is an upper triangular matrix. \end{example}
We note, that the unknown parameter $\theta$ is a scalar value, i.e. $p=1$. For simplicity, we assume that $N=1$, i.e. we illustrate the detailed explanation of only one iteration step of the algorithm. Next, we remark that the post-array should be an upper triangular matrix and, hence, Lemma~\ref{lemma-2} and Algorithm~\ref{alg:2} should be applied to solve the stated problem. Then, we pay an attention to the partitioning in~(\ref{assume-2-2}) from Lemma~\ref{lemma-2} and conclude that $s=3$, $l=1$ and $k=0$. Hence, the blocks $A_{21}$, $A_{22}$ of the pre-array $A$ and, respectively, the $R_{21}$, $R_{22}$ of the post-array $R$ are empty. Indeed, according to Example~\ref{ex:1:1} the first three (block) columns of the post-array $R$ is an upper triangular matrix. This means that $s=3$ and, hence, $k=0$, i.e. $A_{21}$, $A_{22}$ are empty. As a result, $l=1$.
Having applied the computational scheme from Algorithm~\ref{alg:2} to the pre-array in Example~\ref{ex:1:1}, we compute the post-array $R$ and its derivative (at the point $\theta=2$). The obtained results are summarized in Table~\ref{tab:1}. All codes were written in MATLAB. To check our derivations, we compute the norm $\left|\left|{(A^T A)}'_{\theta =2}-{(R^T R)}'_{\theta=2}\right|\right|_{\infty}$. Indeed, from equation $QA=R$ we have $A^T A = R^TR$. Thus, the derivatives of both sides of the latter formula must also agree. The obtained value is $1.33\cdot 10^{-14}$. This confirms the correctness of the calculation of Algorithm~\ref{alg:2} and validates the theoretical derivations of Lemma~\ref{lemma-2}.
\begin{table} \caption{Numerical results for Example~\ref{ex:1:1}} \label{tab:1} \centering \tabsize
\begin{tabular}{|ll|} \toprule
\multicolumn{2}{|l|}{ We are given the pre-array $A$ and its derivatives with respect to each $\theta_i$, (in the example $p=1$):} \\
& Pre-array $
A= \left[
\begin{array}{ccc|c} {\theta^5}/20 \; & \; \theta^4/8 & {\theta^3}/{6} & \; \theta^3/3\\ {\theta^4}/8 \; & \; \theta^3/3 & {\theta^2}/{2} & \; {\theta^2}/{2}\\ {\theta^3}/{6} \; & \; {\theta^2}/{2} & \theta & \; 1 \end{array}
\right]$, i.e. $\left.A\right|_{\theta=2}=\left[
\begin{array}{rrr|r}
1.6000 & 2.0000 & 1.3333 & 2.6667 \\
2.0000 & 2.6667 & 2.0000 & 2.0000 \\
1.3333 & 2.0000 & 2.0000 & 1.0000 \end{array} \right]$, \\
& and $A'_{\theta}=\left[
\begin{array}{ccc|c} \theta^4/4 & \theta^3/2 & \theta^2/2 & \theta^2 \\ \theta^3/2 & \theta^2 & \theta & \theta \\ \theta^2/2 & \theta & 1 & 0 \end{array}
\right]$. So, $\left.A'_{\theta}\right|_{\theta=2}=\left[
\begin{array}{ccc|c}
4 & 4 & 2 & 4 \\
4 & 4 & 2 & 2 \\
2 & 2 & 1 & 0 \end{array} \right]$. \\ \midrule
\multicolumn{2}{|l|}{Compute the post-array $R$ using $QR$ algorithm and save matrices $\{ Q, R\}$ for future steps:} \\
& Post-array $R=\left[
\begin{array}{rrr|r}
-2.8875 & -3.8788 & -3.0476 & -3.3247\\
0 & -0.2576 & -0.6954 & 0.8886\\
0 & 0 & 0.0797 & 0.5179 \end{array} \right]$, $Q= \begin{bmatrix}
-0.5541 & -0.6926 & -0.4618 \\
0.5795 & 0.0773 & -0.8113 \\
0.5976 & -0.7171 & 0.3586
\end{bmatrix}$. \\ \midrule
\multicolumn{2}{|l|}{Apply the designed derivative computation method ($p=1$):} \\ $\bullet$ & Compute $QA'_{\theta}$. Denote $X_1=\begin{bmatrix}
-5.9105 & -5.9105 & -2.9552 \\
1.0045 & 1.0045 & 0.5022 \\
0.2390 & 0.2390 & 0.1195 \end{bmatrix}$, $\begin{matrix} Y_1 = [\quad] \\ V_1 = [\quad] \\ \end{matrix} $, $N_1 = \begin{bmatrix}
-3.6017 \\
2.4725 \\
0.9562 \end{bmatrix}$.\\ $\bullet$ & Find $X_1R_{11}^{-1}= \begin{bmatrix}
2.0469 & -7.8778 & -27.5511 \\
-0.3479 & 1.3388 & 4.6822 \\
-0.0828 & 0.3186 & 1.1143 \end{bmatrix}$. Split it into $\bar {\cal L}_1 = \begin{bmatrix}
0 & 0 & 0 \\
-0.3479 & 0 & 0 \\
-0.0828 & 0.3186 & 0 \end{bmatrix}$, \\
& \phantom{Find} ${\cal D}_1= \begin{bmatrix}
2.0469 & 0 & 0 \\
0 & 1.3388 & 0 \\
0 & 0 & 1.1143 \end{bmatrix}$, $\bar {\cal U}_1 = \begin{bmatrix}
0 & -7.8778 & -27.5511 \\
0 & 0 & 4.6822 \\
0 & 0 & 0 \end{bmatrix}$.\\
$\bullet$ & Calculate $\left.R'_{11}\right|_{\theta=2}= \begin{bmatrix}
-5.9105 & -5.8209 & -2.7199 \\
0 & -0.3448 & -0.5325 \\
0 & 0 & 0.0888
\end{bmatrix}$ and $\left.R'_{12}\right|_{\theta=2}= \begin{bmatrix}
-3.9537 \\
1.4810 \\
0.3978 \end{bmatrix}$. \\ \midrule
\multicolumn{2}{|l|}{Hence, the derivative of the post-array is} \\
& $\left.R'_{\theta}\right|_{\theta=2}= \left[
\begin{array}{rrr|r}
-5.9105 & -5.8209 & -2.7199 & -3.9537 \\
0 & -0.3448 & -0.5325 & 1.4810 \\
0 & 0 & 0.0888 & 0.3978 \end{array} \right]$. \\ \bottomrule
\multicolumn{2}{|l|}{Accuracy of the computations: $\left|\left|{(A^T A)}'_{\theta
=2}-{(R^T R)}'_{\theta=2}\right|\right|_{\infty}=1.33\cdot 10^{-14}$} \\ \bottomrule \end{tabular} \end{table}
\begin{example}\label{ex:1:2} (\textsc{Simple test problem: the lower triangular case})
\noindent
For the pre-array $A$ from example~\ref{ex:1:1}, compute the post-arrays $L$ and its derivative $L'_{\theta}$ (at $\theta=2$) where the first three (block) columns of the post-array $L$ is a lower triangular matrix; see equation~(\ref{assume-2}). \end{example}
The lower triangular case can be justified at the same way. We note, that $l=1$, $s=3$, $k=0$ and, hence, we have the partitioning~(\ref{assume-2}) of the pre-array $A$ with the empty blocks $A_{11}$, $A_{12}$. The post-array $L$ is block lower triangular and, hence, we apply the computational scheme presented in Algorithm~\ref{alg:1}. The obtained results are summarized in Table~\ref{tab:matrix:2}. The accuracy of the computation is $\left|\left|{(A^T A)}'_{\theta
=2}-{(L^T L)}'_{\theta=2}\right|\right|_{\infty}=2.57\cdot 10^{-14}$. This confirms the correctness of the calculation of Algorithm~\ref{alg:1} and validates the theoretical derivations of Lemma~\ref{lemma-1}.
\begin{table} \caption{Numerical results for Example~\ref{ex:1:2}} \label{tab:matrix:2} \centering \tabsize
\begin{tabular}{|ll|} \toprule
\multicolumn{2}{|l|}{We are given the pre-array $A$ and its derivatives with respect to each $\theta_i$, (in the example $p=1$):} \\
& Pre-array $
A= \left[
\begin{array}{ccc|c} {\theta^5}/20 \; & \; \theta^4/8 & {\theta^3}/{6} & \; \theta^3/3\\ {\theta^4}/8 \; & \; \theta^3/3 & {\theta^2}/{2} & \; {\theta^2}/{2}\\ {\theta^3}/{6} \; & \; {\theta^2}/{2} & \theta & \; 1 \end{array}
\right]$, i.e. $\left.A\right|_{\theta=2}=\left[
\begin{array}{rrr|r}
1.6000 & 2.0000 & 1.3333 & 2.6667 \\
2.0000 & 2.6667 & 2.0000 & 2.0000 \\
1.3333 & 2.0000 & 2.0000 & 1.0000 \end{array} \right]$, \\
& and $A'_{\theta}=\left[
\begin{array}{ccc|c} \theta^4/4 & \theta^3/2 & \theta^2/2 & \theta^2 \\ \theta^3/2 & \theta^2 & \theta & \theta \\ \theta^2/2 & \theta & 1 & 0 \end{array}
\right]$. So, $\left.A'_{\theta}\right|_{\theta=2}=\left[
\begin{array}{ccc|c}
4 & 4 & 2 & 4 \\
4 & 4 & 2 & 2 \\
2 & 2 & 1 & 0 \end{array} \right]$. \\ \midrule
\multicolumn{2}{|l|}{Compute the post-array $L$ using $QL$ algorithm and save matrices $\{ Q, L\}$ for future steps:} \\
& Post-array $L=\left[
\begin{array}{rrr|r}
-0.0306 & 0 & 0 & -0.6882 \\
-0.6456 & -0.6195 & 0 & -1.5163 \\
-2.8142 & -3.8376 & -3.1269 & -3.0559 \end{array} \right]$, $Q= \begin{bmatrix}
-0.6882 & -0.5869 & -0.4264 \\
0.6882 & -0.3424 & -0.6396 \\
-0.2294 & 0.7337 & -0.6396
\end{bmatrix}$. \\ \midrule
\multicolumn{2}{|l|}{Apply the designed derivative computation method ($p=1$):} \\ $\bullet$ & Compute $QA'_{\theta}$. Denote $\begin{array}{c} X_1 = [\quad] \\ N_1 = [\quad] \\ \end{array} $, $Y_1= \begin{bmatrix}
-0.4588 & -0.4588 & -0.2294 \\
-2.2499 & -2.2499 & -1.1250 \\
-5.5432 & -5.5432 & -2.7716 \end{bmatrix}$, $V_1 = \begin{bmatrix}
-1.3765 \\
-3.0325 \\
-2.9848 \\ \end{bmatrix}$.\\ $\bullet$ & Find $X_1L_{21}^{-1}= \begin{bmatrix}
2.2105 & 0.2861 & 0.0734 \\
10.8396 & 1.4031 & 0.3598 \\
26.7057 & 3.4569 & 0.8864 \end{bmatrix}$. Split it into $\bar {\cal L}_1 = \begin{bmatrix}
0 & 0 & 0 \\
10.8396 & 0 & 0 \\
26.7057 & 3.4569 & 0 \end{bmatrix}$, \\
& \phantom{Find} ${\cal D}_1= \begin{bmatrix}
2.2105 & 0 & 0 \\
0 & 1.4031 & 0 \\
0 & 0 & 0.8864 \end{bmatrix}$, $\bar {\cal U}_1 = \begin{bmatrix}
0 & 0.2861 & 0.0734 \\
0 & 0 & 0.3598 \\
0 & 0 & 0 \end{bmatrix}$.\\
$\bullet$ & Calculate $\left.L'_{21}\right|_{\theta=2}= \begin{bmatrix}
-0.0676 & 0 & 0 \\
-1.2462 & -0.8693 & 0 \\
-5.7777 & -5.7661 & -2.7716
\end{bmatrix}$ and $\left.L'_{22}\right|_{\theta=2}= \begin{bmatrix}
-0.7184 \\
-2.1301 \\
-3.5808 \end{bmatrix}$. \\ \midrule
\multicolumn{2}{|l|}{Hence, the derivative of the post-array is} \\
& $\left.L'_{\theta}\right|_{\theta=2}= \left[
\begin{array}{rrr|r}
-0.0676 & 0 & 0 & -0.7184 \\
-1.2462 & -0.8693 & 0 & -2.1301 \\
-5.7777 & -5.7661 & -2.7716 & -3.5808 \end{array} \right]$. \\ \bottomrule
\multicolumn{2}{|l|}{Accuracy of the computations: $\left|\left|{(A^T A)}'_{\theta
=2}-{(L^T L)}'_{\theta=2}\right|\right|_{\infty}= 2.57\cdot 10^{-14}$} \\ \bottomrule \end{tabular} \end{table}
Next, we wish to discuss the convergence of the parameter $\theta$ to its real value, i.e. to discuss the accuracy of the designed recursive AF estimator presented in Algorithm~\ref{alg:3}. As mentioned earlier, the new AF scheme is developed from the techniques designed in the Kalman filtering community to solve ill conditioned problems. This should improve accuracy and robustness of the computations for a finite-precision computer arithmetics. To check this property, we consider the set of ill-conditioned test problems from~\cite{Tsyganova2013IEEE}.
\begin{example}\label{ex:3} (\textsc{Set of ill-conditioned test problems})
\noindent Consider the state-space model~(\ref{eq2.1})-(\ref{eq2.2}) with $\{F, G, B, H, \Pi_0, Q, R \}$ given by \begin{align*} F= & \begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}, B=\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix},
G=\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix},
Q=\begin{bmatrix}
1
\end{bmatrix}, \quad
R= \begin{bmatrix}
\delta^2\theta^2 & 0 \\
0 & \delta^2\theta^2
\end{bmatrix},
H= \begin{bmatrix}
1 & 1 & 1\\
1 & 1 & 1+\delta
\end{bmatrix} \\
\mbox{ with } & x_0 \sim {\cal N}\left(\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix},
\begin{bmatrix}
\theta^2 & 0 & 0 \\
0 & \theta^2 & 0 \\
0 & 0 & \theta^2
\end{bmatrix}\right) \end{align*} where $\theta$ is an unknown system parameter, that needs to be estimated. To simulate roundoff we assume that $\delta^2<\epsilon_{\text{roundoff}}$, but $\delta>\epsilon_{\text{roundoff}}$ where $\epsilon_{\text{roundoff}}$ denotes the unit roundoff error\footnote{Computer roundoff for floating-point arithmetic is often characterized by a single parameter $\epsilon_{\text{roundoff}}$, defined in different sources as the largest number such that either $1+\epsilon_{\text{roundoff}} = 1$ or $1+\epsilon_{\text{roundoff}}/2 = 1$ in machine precision.}, i.e. the machine precision limit. \end{example}
The set of ill-conditioned problems is constructed as follows. When $\theta = 1$, Example~\ref{ex:3} coincides with well-known test from~\cite{GrewalAndrews2001} that demonstrates how a problem that is well conditioned, as posed, can be made ill-conditioned by the filter. It is often used in the Kalman filtering community for observing the influence of round off errors on various KF implementations. The difficulty is in matrix inversion $R_{e,k}$. After processing only the first measurement $z_1$, the matrix $ R_{e,1}=R+H\Pi_{0}H^T$ becomes singular in machine precision, i.e. as $\delta \to \epsilon_{\text{roundoff}}$. This yields the failure of the conventional KF. To construct a proper test problem for the gradient-based AF estimators, the authors of~\cite{Tsyganova2013IEEE} introduced an unknown system parameter $\theta$, making sure that the same problem is now applied to the matrix $(R_{e,1})'_{\theta}$. In other words, for any fixed value of the parameter $\theta \ne 0$, the matrices $R_{e,1}=R+H\Pi_{0}H^T$ and $(R_{e,1})'_{\theta}$ are ill-conditioned in machine precision, i.e. as $\delta \to \epsilon_{\text{roundoff}}$. As a consequence, both parts of the gradient-based AF techniques (the PI and its gradient evaluation, respectively) fail after processing the first measurement. This destroys the entire AF estimator grounded in the conventional KF implementation. Hence, such test allows for observing the influence of the round off errors on various gradient-based AF schemes.
\begin{figure}
\caption{The computed maximum likelihood estimates of $\theta$ by three gradient-based AF techniques: within conventional KF (marker $\circ$); the eSRCF implementation (marker $\bullet$) and the eSRIF filer (marker $\times$). The initial parameter value, i.e. $\theta^{(0)}=1$, is marked by $*$.}
\label{fig:2}
\end{figure}
We perform the following set of numerical experiments. Given the ``true'' value of the parameter $\theta$, say $\theta^*=5$, the system is simulated for $1000$ samples for various values of $\delta$ while $\delta \to \epsilon_{\text{roundoff}}$. The generated data is then used to solve the inverse problem, i.e. to compute the maximum likelihood estimates by gradient-based AF schemes. We consider the AF recursive estimator based on the conventional KF, on the eSRCF and eSRIF. The designed Algorithms~\ref{alg:1}, \ref{alg:2} are used for the PI and its gradient evaluation within numerically stable ASR filters (the eSRCF and eSRIF). Algorithm~\ref{alg:3} represents the general gradient-based AF scheme where we implemented the standard MATLAB built-in function \verb"fminunc" for optimization purpose. This optimization function utilizes the PI (the negative Log LF) and its gradient that are calculated by the conventional KF approach and the designed ASR methodology. The same initial value of $\theta^{(0)}=1$ is applied in all examined AF estimators. To observe the convergence of the parameter $\theta$ from the initial value $\theta^{(0)}=1$ to its real value $\theta^*=5$, we perform $100$ Monte Carlo simulations and illustrate the obtained results by Fig.~\ref{fig:2}.
From the first two graphs in Fig.~\ref{fig:2} we see that when $\delta=10^{-2}$ and $\delta=10^{-3}$, i.e. when the considered problem is well-posed, all gradient-based AF techniques work equally well. We can observe their perfect convergence from the initial value $\theta^{(0)}=1$ to the real value $\theta^*=5$ in all $100$ Monte Carlo simulations. However, the situation dramatically changes for $\delta=10^{-5}$ when the problem becomes moderately ill-conditioned. The gradient-based AF scheme within conventional KF exhibits perfect performance for $\delta=10^{-2}$ and $\delta=10^{-3}$, but it completely fails for $\delta=10^{-5}$. Indeed, the conventional approach leads to incorrect parameter estimate in most cases among $100$ Monte Carlo simulations when $\delta=10^{-5}$. Meanwhile, the AF techniques based on the numerically stable ASR implementations work well for all examined $\delta$ as $\delta \to \epsilon_{\text{roundoff}}$.
\section{Conclusion}
In this paper, we developed an elegant and simple general computational scheme that extends functionality of any array square-root Kalman filtering algorithm on the filter derivative computations. These values are required in the gradient-based adaptive filtering techniques for simultaneous state and parameter estimation of dynamic positioning systems in many areas of research. The proposed approach yields the improved robustness of the computations against roundoff errors.
\ack The first author thanks the support of Portuguese National Fund ({\it Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia}) within the scope of project SFRH/BPD/64397/2009. The authors also would like to express their gratitude to the anonymous referees for their valuable remarks and comments on the paper.
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\end{document} | arXiv | {
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\begin{document}
\title[Asymptotics of eigenvalues of Hankel operators]{Asymptotic behaviour of eigenvalues of Hankel operators}
\author{Alexander Pushnitski} \address{Department of Mathematics, King's College London, Strand, London, WC2R~2LS, U.K.} \email{alexander.pushnitski@kcl.ac.uk}
\author{Dmitri Yafaev} \address{Department of Mathematics, University of Rennes-1, Campus Beaulieu, 35042, Rennes, France} \email{yafaev@univ-rennes1.fr}
\subjclass[2010]{47B35, 47B06}
\keywords{Hankel and pseudodifferential operators, eigenvalues, Weyl asymptotics}
\begin{abstract} We consider compact Hankel operators realized in $ \ell^2({\mathbb Z}_+)$ as infinite matrices $\Gamma$ with matrix elements $h(j+k)$. Roughly speaking, we show that if $h(j)\sim (b_{1}+ (-1)^j b_{-1}) j^{-1}(\log j )^{-\alpha}$ as $j\to \infty$
for some $\alpha>0$, then the eigenvalues of $\Gamma$ satisfy
$\lambda_{n}^{\pm} (\Gamma)\sim c^{\pm} n^{-\alpha}$ as $n\to \infty$. The asymptotic coefficients $c^{\pm}$ are explicitly expressed in terms of the asymptotic coefficients $b_{1} $ and $b_{-1}$. Similar results are obtained for Hankel operators ${\mathbf{\Gamma}}$ realized in $ L^2({\mathbb R}_+)$ as integral operators with kernels ${\mathbf{h}}(t+s)$. In this case the asymptotics of eigenvalues $\lambda_{n}^{\pm} ({\mathbf{\Gamma}})$ are determined by the behaviour of ${\mathbf{h}}(t)$ as $t\to 0$ and as $t\to \infty$. \end{abstract}
\date{8 December 2014}
\maketitle
\section{Introduction}\label{sec.a}
\subsection{Overview}\label{sec.a1}
For a sequence $\{h(j)\}_{j=0}^\infty$ of complex numbers, a Hankel operator $\Gamma(h)$ in the space $ \ell^2({\mathbb Z}_+)$ is formally defined as the ``infinite matrix'' $\{h(j+k)\}_{j,k=0}^\infty$, that is, \begin{equation} (\Gamma(h) u) (j)= \sum_{k=0}^\infty h(j+k) u (k), \quad u= (u (0), u (1), \ldots). \label{eq:a5} \end{equation} We also consider integral Hankel operators ${\mathbf{\Gamma}}({\mathbf{h}})$ in the space $L^2({\mathbb R}_+)$ (${\mathbb R}_+=(0,\infty)$), formally defined by \begin{equation} ({\mathbf{\Gamma}}({\mathbf{h}}){\mathbf{u}})(t)=\int_0^\infty {\mathbf{h}}(t+s){\mathbf{u}} (s)ds, \label{a5} \end{equation} where ${\mathbf{h}}\in L^1_\mathrm{loc}({\mathbb R}_+)$; this function is called the \emph{kernel} of the Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}})$. Under the assumptions below the operators $\Gamma(h)$ and ${\mathbf{\Gamma}}({\mathbf{h}})$ are bounded. We will refer to the theory of Hankel operators in $ \ell^2({\mathbb Z}_+)$ as to the ``discrete case'' and to the one of the integral Hankel operators in $L^2({\mathbb R}_+)$ as to the ``continuous case''; objects related to the continuous case will be denoted by boldface symbols. Of course the operator $\Gamma(h)$ (resp. ${\mathbf{\Gamma}}({\mathbf{h}})$) is self-adjoint if and only if the sequence $\{h(j)\}$ (resp. the function ${\mathbf{h}}(t)$) is real valued. Background information on the theory of Hankel operators can be found in the book \cite{Peller} by V.~Peller.
In this paper, we are interested in compact self-adjoint Hankel operators. Sharp estimates of eigenvalues of Hankel operators (and, more generally, of singular values in the non-self-adjoint case) are very well known. At the same time, there are practically no results on the asymptotic behaviour of these eigenvalues. The only exceptions known to us are the papers \cite{Widom, Yafaev2}, which will be discussed below. This state of affairs is in a sharp contrast with the case of differential operators, where the Weyl type asymptotics of eigenvalues is established in a large variety of situations. Our goal here is to fill in this gap by describing a class of Hankel operators where the eigenvalue asymptotics (in the power scale) can be found explicitly.
Our approach relies on the following three ingredients: \begin{enumerate}[(i)] \item A result of \cite{Yafaev3} which establishes the unitary equivalence of Hankel operators to pseudodifferential operators ($\Psi$DO) in $L^2 ({\mathbb R})$ of a certain special class. \item Standard Weyl type spectral asymptotics for the corresponding $\Psi$DO, obtained by Birman and Solomyak in \cite{BS4,BS3}. \item Estimates for singular values of Hankel operators from \cite{0} (based on earlier results by Peller). \end{enumerate}
In general, the study of eigenvalue asymptotics for any class of operators involves two steps: construction of an appropriate model problem where the eigenvalue asymptotics can be determined more or less explicitly, and using eigenvalue estimates (or variational methods) to extend the asymptotics to a wider class of operators.
As mentioned above, the relevant estimates in a convenient form were prepared in our previous paper \cite{0}. The most important novel feature of this work is the construction of the appropriate model Hankel operators. In order to construct model Hankel operators, we proceed in two steps. Given a Hankel operator $\Gamma$, first we construct a suitable $\Psi$DO $\Psi_*$ of a negative order such that the spectral asymptotics of $\Psi_*$ can be established (see item (ii) above). Then we use the unitary equivalence (see item (i) above) to map $\Psi_*$ into a Hankel operator $\Gamma_*$ with the same spectrum. For a ``correct" choice of $\Psi_*$, the Hankel operators $\Gamma$ and $\Gamma_*$ are close to each other, and so they have the same leading terms of eigenvalue asymptotics. In more detail, our approach is outlined in Sections~3.1 and 4.1 for the operators \eqref{a5} and \eqref{eq:a5}, respectively.
\subsection{Discrete case}
Let $\{\lambda_n^+(\Gamma)\}_{n=1}^\infty$ be the non-increasing sequence of positive eigenvalues of a compact self-adjoint operator $\Gamma$ (with multiplicities taken into account), and let $\lambda_n^-(\Gamma)=\lambda_n^+(-\Gamma)$. We define also the eigenvalue counting function \begin{equation} n_\pm(\varepsilon;\Gamma) = \# \{n: \lambda_n^\pm(\Gamma)>\varepsilon\}, \quad \varepsilon>0. \label{eq:CF} \end{equation}
We start our discussion from the discrete case. In order to motivate our main result, let us consider the sequence \begin{equation} h(j)=\frac1{(j+1)^\gamma}, \quad \gamma\geq1. \label{a1d} \end{equation} If $\gamma=1$ then the corresponding Hankel operator $\Gamma(h)$, known as the Hilbert matrix, is bounded (but not compact). From here by a simple argument one obtains \begin{align} h(j)=O(j^{-1}), \quad j\to\infty \quad &\Rightarrow \quad \Gamma(h) \text{ is bounded,} \label{a1ff} \\ h(j)=o(j^{-1}), \quad j\to\infty \quad &\Rightarrow \quad \Gamma(h) \text{ is compact.} \label{a1f} \end{align} Roughly speaking, one expects that a faster rate of convergence of the sequence $h(j)$ to zero as $j\to\infty$ results in a faster convergence of the eigenvalues $\lambda^\pm_n(\Gamma(h))$ to zero as $n\to\infty$. Indeed, there is a deep result of H.~Widom who showed in \cite{Widom} that for $\gamma>1$ the Hankel operator corresponding to the sequence \eqref{a1d} is non-negative and its eigenvalues converge to zero \emph{exponentially} fast: $$ \lambda_n^+(\Gamma(h))=\exp(-\pi \sqrt{2\gamma n}+o(\sqrt{n})), \quad n\to\infty. $$
Our goal is to study the case intermediate between $\gamma=1$ and $\gamma> 1$, when $h(j)$ behaves as $j^{-1} ( \log j)^{-\alpha}$ with some $\alpha>0$ for large $j$. To give the flavour of our main result, first we state it in a particular case; the full statement is given in Theorem~\ref{cr.a3} below. We use the notation $x_\pm=\max\{0,\pm x\}$; $B(\cdot,\cdot)$ is the standard Beta function, \begin{equation} B(a,b)=\int_1^\infty (t-1)^{a-1}t^{-a-b}dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \label{a16} \end{equation} (of course, the symbols $\Gamma$ in the r.h.s. of \eqref{a16} stand for the Gamma function rather than for Hankel operators). Put \begin{equation} v(\alpha):=2^{-\alpha} \pi^{1-2\alpha} B(\tfrac1{2\alpha},\tfrac12)^{\alpha}; \label{a4} \end{equation} in particular, $v(1)=2^{-\alpha}$. In what follows, $\log$ denotes the natural (base $e$) logarithm.
\begin{theorem}\label{thm.aa1}
Let $\alpha>0$, $b_1,b_{-1}\in{\mathbb R}$, and let \begin{equation} h(j)=(b_1+(-1)^j b_{-1}) j^{-1}(\log j)^{-\alpha}, \quad j\geq2; \label{a2} \end{equation} the choice of $h(0)$ and $h(1)$ \emph{(\emph{or of any finite number of $h(j)$})} is not important. Then the eigenvalues of the corresponding Hankel operator $\Gamma(h)$ have the asymptotic behaviour \begin{equation} \lambda_n^\pm(\Gamma(h)) = c^\pm n^{-\alpha}+o(n^{-\alpha}), \quad c^\pm=v(\alpha)\bigl((b_1)^{1/\alpha}_\pm+(b_{-1})^{1/\alpha}_\pm\bigr)^\alpha, \label{a1c} \end{equation} as $n\to\infty$. \end{theorem}
Theorem~\ref{thm.aa1} shows that between the cases $\gamma=1$ and $\gamma>1$ in \eqref{a1d} there is a whole scale (the logarithmic scale) of sequences $h (j) $ such that the eigenvalues of $\Gamma(h)$ have power spectral asymptotics.
The full version of this result (which is stated below as Theorem~\ref{cr.a3}) also allows for an error term in \eqref{a2}.
\subsection{Discussion} (1) Theorem~\ref{thm.aa1} is consistent with the Hilbert-Schmidt conditions for $\Gamma(h)$. Indeed, in the self-adjoint case we have \begin{equation}
\sum_{j=0}^\infty |h(j)|^2 (j+1) = \sum_{n=1}^\infty \bigl(\lambda_n^+(\Gamma)^2+\lambda_n^-(\Gamma)^2\bigr), \quad \Gamma=\Gamma(h), \label{eq:HS} \end{equation} and the operator $\Gamma$ belongs to the Hilbert-Schmidt class if and only if the series in the l.h.s. of \eqref{eq:HS} converges. This is true if \begin{equation} h(j) = O(j^{-1} (\log j)^{-\alpha}), \quad j\to\infty, \label{eq:HS1} \end{equation} for some $\alpha> 1/2$. On the other hand, the series in the r.h.s. of \eqref{eq:HS} converges if \begin{equation} \lambda_n^\pm(\Gamma(h)) = O(n^{-\alpha}), \quad n\to\infty, \label{a1cx} \end{equation} with some $\alpha>1/2$. This agrees with Theorem~\ref{thm.aa1}.
(2) It is shown in \cite{0} that, for $0<\alpha<1/2$, condition \eqref{eq:HS1} implies \eqref{a1cx}. This result remains true also for $ \alpha\geq 1/2$ if additionally one imposes some conditions on the iterated differences of the sequence $h(j)$; see Section~\ref{sec.a20} for the precise statement. This is also consistent with Theorem~\ref{thm.aa1}. In particular, Theorem~\ref{thm.aa1} shows that the above result of \cite{0} is sharp.
(3) According to formula \eqref{a1c} the sequences \begin{equation} h_1(j)= j^{-1}(\log j)^{-\alpha} \quad\text{ and }\quad h_{-1}(j)=(-1)^j j^{-1}(\log j)^{-\alpha} \label{a4b} \end{equation} yield the same spectral asymptotics. This fact has a simple explanation:
if $h_1(j)$ and $h_{-1} (j)$ are any two sequences such that $h_{-1}(j)=(-1)^j h_1(j)$, then we have \begin{equation} \Gamma(h_{-1})=F^* \Gamma(h_1) F \quad\text{ where }\quad (F u)(j)=(-1)^j u(j), \quad j\geq0. \label{a4a} \end{equation} Thus, the operators $\Gamma(h_1)$ and $\Gamma(h_{-1})$ are unitarily equivalent and so they have the same eigenvalues.
(4) Let us discuss the structure of the formula \eqref{a1c} for the asymptotic coefficient $c^\pm$.
In terms of the counting function, it can be equivalently rewritten as $$
n_\pm(\varepsilon; \Gamma(h)) =
v(\alpha)^{1/\alpha}\bigl((b_1)^{1/\alpha}_\pm+(b_{-1})^{1/\alpha}_\pm\bigr) \varepsilon^{-1/\alpha}(1+o(1)), \quad \varepsilon\to 0.
$$ It follows that (using notation \eqref{a4b}) \begin{equation} n_\pm(\varepsilon;\Gamma(h)) = n_\pm(\varepsilon; b_1 \Gamma(h_1))+n_\pm(\varepsilon; b_{-1}\Gamma(h_{-1}))+o(\varepsilon^{-1/\alpha}), \quad \varepsilon\to0. \label{a4d} \end{equation} Roughly speaking, this means that the operator $\Gamma(h)$ is in some sense asymptotically equivalent to the orthogonal sum $b_1\Gamma(h_1)\oplus b_{-1}\Gamma(h_{-1})$. The ``asymptotic orthogonality'' of $\Gamma(h_1)$ and $\Gamma(h_{-1})$ may look mysterious here, but it will become clearer in the course of constructing the model operators, see Remark~\ref{rmk.b2}.
\subsection{Continuous case}\label{sec.a1a}
In the discrete case, the spectral asymptotics of $\Gamma(h)$ is determined by the asymptotic behaviour of the sequence $h(j)$ as $j\to\infty$. In the continuous case, the behaviour of the kernel ${\mathbf{h}}(t)$ for $t\to0$ and for $t\to\infty$ as well as the local singularities of ${\mathbf{h}} (t)$ contribute to the asymptotics of the eigenvalues of the Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}})$. In this paper, we consider the kernels without local singularites for $t>0$.
It is well known that the Carleman operator, corresponding to the kernel $$ {\mathbf{h}}(t)=1/t, $$ is bounded, but not compact. From here, similarly to \eqref{a1ff}, \eqref{a1f}, one easily obtains $$ {\mathbf{h}}(t)=O(1/t) \text{ as $t\to0$ and as $t\to\infty$ } \quad \Rightarrow \quad {\mathbf{\Gamma}}({\mathbf{h}}) \text{ is bounded}, $$ $$ {\mathbf{h}}(t)=o(1/t) \text{ as $t\to0$ and as $t\to\infty$ } \quad \Rightarrow \quad {\mathbf{\Gamma}}({\mathbf{h}}) \text{ is compact.} $$ All our kernels will satisfy the latter condition. In the same way as for the ``discrete" Hankel operator, first we give the result in an important particular case; the full statement is given below as Theorem~\ref{thm.d1}.
\begin{theorem}\label{thm.aa2}
Let ${\mathbf{h}}$ be a real valued function in $C^\infty({\mathbb R}_+)$ such that $$ {\mathbf{h}}(t)= \begin{cases} {\mathbf{b}}_0 t^{-1} (\log(1/t))^{-\alpha}, & t<1/a, \\ {\mathbf{b}}_\infty t^{-1} (\log t)^{-\alpha}, & t>a, \end{cases} $$ with some $\alpha>0$, $a>1$ and ${\mathbf{b}}_0,{\mathbf{b}}_\infty\in{\mathbb R}$. Then \begin{equation} \lambda_n^\pm({\mathbf{\Gamma}}({\mathbf{h}})) = {\mathbf{c}}^\pm n^{-\alpha} + o(n^{-\alpha}), \quad {\mathbf{c}}^\pm = v(\alpha)\bigl(({\mathbf{b}}_0)_\pm^{1/\alpha}+({\mathbf{b}}_\infty)_\pm^{1/\alpha}\bigr)^\alpha, \label{d3} \end{equation} as $n\to\infty$, where $v(\alpha)$ is given by \eqref{a4}. \end{theorem}
In order to discuss the continuous case further, it is convenient to introduce some notation. Let us fix two cut-off functions $\chi_0,\chi_\infty\in C^\infty({\mathbb R}_+)$ such that \begin{equation} \chi_0(t)= \begin{cases} 1& \text{for $t\leq1/4$,} \\ 0& \text{for $t\geq1/2$,} \end{cases} \qquad \chi_\infty(t)= \begin{cases} 0& \text{for $t\leq2$,} \\ 1& \text{for $t\geq4$,} \end{cases} \label{a7a} \end{equation} and define the model kernels \begin{equation} {\mathbf{h}}_0(t)=t^{-1}\abs{\log t}^{-\alpha}\chi_0(t), \quad {\mathbf{h}}_\infty(t)=t^{-1}\abs{\log t}^{-\alpha}\chi_\infty(t), \quad t>0. \label{a7b} \end{equation} Observe that the hypothesis of Theorem~\ref{thm.aa2} is equivalent to the representation $$ {\mathbf{h}}={\mathbf{b}}_0{\mathbf{h}}_0+{\mathbf{b}}_\infty {\mathbf{h}}_\infty+{\mathbf{g}}, $$ where ${\mathbf{g}}(t)$ is a smooth function that vanishes both for small and for large $t$. We will see that the contribution of ${\mathbf{\Gamma}}({\mathbf{g}})$ to the eigenvalue asymptotics is negligible. Next, since the singularity of ${\mathbf{h}}_0$ is located at zero and the singularity of ${\mathbf{h}}_\infty$ is located at infinity, it is not surprising that the operators ${\mathbf{\Gamma}}({\mathbf{h}}_0)$ and ${\mathbf{\Gamma}}({\mathbf{h}}_\infty)$ are ``asymptotically orthogonal'', i.e.\ that, similarly to \eqref{a4d}, we have $$ n_\pm(\varepsilon;{\mathbf{\Gamma}}(h)) = n_\pm(\varepsilon; {\mathbf{b}}_0{\mathbf{\Gamma}}({\mathbf{h}}_0))+n_\pm(\varepsilon; {\mathbf{b}}_\infty{\mathbf{\Gamma}}({\mathbf{h}}_\infty))+o(\varepsilon^{-1/\alpha}), \quad \varepsilon\to 0. $$ This explains the structure of formula \eqref{d3} for the asymptotic coefficient ${\mathbf{c}}^\pm$.
Compact Hankel operators ${\mathbf{\Gamma}}({\mathbf{h}})$ with kernels ${\mathbf{h}}(t)$ that have a singularity at a single point $t=t_0>0$ were considered in \cite[Section 3]{GLP} and in \cite[Section 6]{Yafaev2}. In this case the eigenvalues $\lambda_{n}^\pm$ also have
the power asymptotics as $n\to \infty$, but the leading terms of $\lambda_{n}^+$ and $\lambda_{n}^-$ are the same. In the present paper we consider locally regular kernels with a slow decay as $t\to \infty$ and singular at $t=0$. Thus the results as well as the methods of \cite{GLP,Yafaev2} and those of the current paper are independent and complement each other.
The results of this paper in the discrete and continuous cases are not fully independent of each other. There are different ways of relating the operators $\Gamma(h)$ and ${\mathbf{\Gamma}}({\mathbf{h}})$, e.g. via the Laguerre transform, or by linking the corresponding symbols via a conformal change of variable (see e.g. \cite[Section~1.8]{Peller}). Either of these methods shows that the singularity (see \eqref{a7b}) of the kernel ${\mathbf{h}}(t) $ at $t=0$ (resp., at $t=\infty$) corresponds to $h(j)$
with asymptotics $j^{-1} (\log j)^{-\alpha}$ (resp., $(-1)^j j^{-1} (\log j)^{-\alpha}$) as $j\to \infty$. However, technically it turns out to be more convenient to give two independent arguments for the discrete and continuous cases. We also note that some features of the problem are more transparent in the discrete case, while others are in the continuous case.
\subsection{The structure of the paper} As already mentioned, our paper relies on a synthesis of various results. They are collected in Section~\ref{sec.a2}. It is convenient to start the proofs with the continuous case. Thus, in Section~\ref{sec.d} we state and prove our main result in the continuous case, and in Section~\ref{sec.e} we return to the discrete case. Finally, a proof of an assertion for $\Psi$DO in $L^2({{\mathbb R}})$ supplementing \cite{BS4} is given in the Appendix.
\section{Preliminaries}\label{sec.a2}
Here we discuss one by one the three key ingredients of our approach mentioned in Section~1.1.
\subsection{Reduction to $\Psi$DO}\label{sec.a2a}
Let $X$ and $D$ be self-adjoint operators in $L^2({\mathbb R})$ defined by \begin{equation} (Xf)(x)=xf(x), \quad (Df)(x)=-if'(x). \label{a8a} \end{equation} Denote \begin{equation} {\mathfrak{b}}(x):=(\pi/\cosh(\pi x))^{1/2}, \quad x\in{\mathbb R}. \label{eq:Xx} \end{equation} This standard function plays a distinguished role in the theory of Hankel operators.
\begin{theorem}\label{thm.d2}\cite[Theorem 4.3]{Yafaev3}
Let $\sigma\in L^\infty({\mathbb R}_+)$, and let ${\mathbf{h}}$ be the Laplace transform of $\sigma :$ $$ {\mathbf{h}}(t) =\int_0^\infty e^{-\lambda t}\sigma(\lambda)d\lambda = : (\mathcal{L} \sigma)(t), \quad t>0. $$ Then the Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}})$ in $L^2({\mathbb R}_+)$ is unitarily equivalent to the $\Psi$DO \begin{equation} \Psi={\mathfrak{b}}(X){\mathfrak{s}} (D){\mathfrak{b}}(X) \label{a9b} \end{equation} in $L^2({\mathbb R}) $ with \begin{equation} {\mathfrak{s}}(\xi)=\sigma(e^{-\xi}), \quad \xi\in{\mathbb R}. \label{eq:a9b}\end{equation} \end{theorem}
The unitary equivalence of the operators ${\mathbf{\Gamma}}({\mathbf{h}})$ and $\Psi$ is given essentially by the Mellin transform. In \cite{Yafaev2}, the function ${\mathfrak{s}}(\xi)$ is called the sign-function of the kernel ${\mathbf{h}} (t)$ since it determines the sign of the corresponding Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}})$.
In the discrete case, the role of the Laplace transform of $\sigma (\lambda)$ is played by the sequence of moments of a function $\eta(\mu)$ defined on the interval $(-1,1)$. Similarly to Theorem~\ref{thm.d2}, we have
\begin{theorem}\label{thm.e2}\cite[Theorem 7.7]{Yafaev3}
Let $\eta\in L^\infty(-1,1)$, and let $h(j)$ be the sequence of moments of $\eta$: $$ h(j)=\int_{-1}^1 \eta(\mu)\mu^j d\mu, \quad j=0,1,2,\dots. $$ Then the Hankel operator $\Gamma(h)$ in $\ell^2({\mathbb Z}_+)$ is unitarily equivalent to the $\Psi$DO \eqref{a9b} in $L^2({\mathbb R}) $ with $$ {\mathfrak{s}}(\xi)=\eta\left(\frac{2e^{-\xi}-1}{2e^{-\xi}+1}\right), \quad \xi\in{\mathbb R}. $$ \end{theorem}
\begin{remark} A similar statement, but requiring $\eta\geq0$, was proven earlier by Widom in \cite{Widom}. Widom establishes the unitary equivalence of $\Gamma(h)$ to \begin{equation} {\mathfrak{s}}(D)^{1/2}{\mathfrak{b}}(X)^2 {\mathfrak{s}}(D)^{1/2}. \label{d5} \end{equation} Since for any bounded operator $T$, the non-zero parts of the operators $T^*T$ and $TT^*$ are unitarily equivalent, taking $T={\mathfrak{s}}(D)^{1/2}{\mathfrak{b}}(X)$, we see that Widom's result is essentially equivalent to Theorem~\ref{thm.e2}, if ${\mathfrak{s}}\geq0$. We note that the study of spectral asymptotics of $\Gamma(h)$ in \cite{Widom} also relies on the reduction to the $\Psi$DO \eqref{d5}. \end{remark}
\subsection{Weyl asymptotics of $\Psi$DO}\label{sec.a2b}
We need the following result.
\begin{theorem}\label{thm.b2}
Let $a\in C^\infty ({{\mathbb R}})$ be a real-valued function such that \begin{equation} a(\xi)= \begin{cases} A(+\infty)\xi^{-\alpha}(1+o(1)), & \xi\to\infty, \\ A(-\infty)\abs{\xi}^{-\alpha}(1+o(1)), & \xi\to-\infty, \end{cases} \label{b2} \end{equation} for some $\alpha>0$ and some constants $A(+\infty)$ and $A(-\infty)$. Assume that $b(x)=\overline{b(x)}$ and \begin{equation} \abs{b(x)}\leq C\jap{x}^{-\rho}, \quad x\in{\mathbb R}, \label{b1a} \end{equation} for some $\rho>\alpha/2$. Then for the pseudodifferential operator $\Psi=b(X)a(D)b(X)$ in $L^2({\mathbb R})$ one has \begin{equation} \lambda_n^\pm(\Psi) = C^{\pm} n^{-\alpha}+o(n^{-\alpha}), \quad n\to\infty, \label{b4} \end{equation} where the coefficients $C^\pm$ are given by \begin{equation} C^{\pm} = (2\pi)^{-\alpha} \bigl( A(-\infty)_\pm^{1/\alpha}+A(+\infty)_\pm^{1/\alpha} \bigr)^\alpha \biggl( \int_{\mathbb R} \abs{b(x)}^{2/\alpha}dx\biggr)^\alpha. \label{b5} \end{equation} \end{theorem}
\begin{remark} The asymptotic relations \eqref{b4}, \eqref{b5} can be equivalently rewritten in terms of the eigenvalue counting functions as \begin{equation} \lim_{\varepsilon\to0} \varepsilon^{1/\alpha} \#\{n: \lambda_n^\pm(\Psi)>\varepsilon\} = \frac1{2\pi}\lim_{\varepsilon\to0}\varepsilon^{1/\alpha} \meas\{(x,\xi)\in{\mathbb R}^2: \pm a(\xi)b(x)^2>\varepsilon\}, \label{a12a} \end{equation} which is
the Weyl semiclassical formula. \end{remark}
\begin{remark}\label{rmk.b2} Fix some $b$ satisfying the estimate \eqref{b1a}; let $\Psi_+$ correspond to some function $a$ with $A(+\infty)=1$, $A(-\infty)=0$ and let $\Psi_-$ correspond to the case $A(+\infty)=0$, $A(-\infty)=1$. Then, for a general $\Psi$ as in Theorem~\ref{thm.b2}, we can write $$ \Psi=A(+\infty)\Psi_++A(-\infty)\Psi_-+\text{error term}, $$ and \eqref{b4}, \eqref{b5} mean that the operators $\Psi_+$, $\Psi_-$ are ``asymptotically orthogonal'', i.e. $$ n_\pm(\varepsilon;\Psi) = n_\pm(\varepsilon; A(+\infty)\Psi_+)+n_\pm(\varepsilon; A(-\infty)\Psi_-)+o(\varepsilon^{-1/\alpha}), \quad \varepsilon\to+0. $$ In the context of the Weyl formula \eqref{a12a}, this asymptotic orthogonality does not look very surprising as it corresponds to the symbols of the operators $\Psi_+$ and $\Psi_-$ ``living'' in different parts of the phase space. \end{remark}
For compactly supported $b$, Theorem~\ref{thm.b2} was proven in \cite{BS4} where the multi-dimensional case was considered. Extension to arbitrary functions $b$ satisfying \eqref{b1a} is an easy application of Cwikel type estimates for $\Psi$DO of the type $f(X)g(D)$; for completeness we give the proof in the Appendix. We also note that there was an inessential restriction $\alpha\not\in{{\mathbb Z}}_{+}$ in \cite{BS4}. It
appeared only because $\Psi$ was regarded in \cite{BS4} as an integral operator rather than a $\Psi$DO.
Theorem~\ref{thm.b2} concerns a very special class of $\Psi$DO with factorisable amplitudes. For general $\Psi$DO with amplitudes asymptotically homogeneous at infinity, Weyl type formula for the asymptotics of the spectrum was obtained in \cite{BS3}.
\subsection{Spectral estimates}\label{sec.a20}
Let us start with the discrete case when Hankel operators are defined by formula \eqref{eq:a5} in the space ${\ell}^2({{\mathbb Z}}_{+})$. Now we do not assume that the operators $\Gamma$ are self-adjoint. We denote by $\{s_n(\Gamma)\}_{n=1}^\infty$ the non-increasing sequence of singular values of $\Gamma$, i.e. $s_n(\Gamma)=\lambda_n^+(\sqrt{\Gamma^*\Gamma})$.
Here we discuss spectral estimates for Hankel operators corresponding to the sequences $g (j)$ that satisfy $$ g(j)=o(j^{-1}(\log j)^{-\alpha}), \quad j\to\infty, $$ for some $\alpha>0$. We also need some assumptions on iterated differences $g^{(m)} (j)$. These are the sequences defined iteratively by setting $g^{(0)} (j)=g (j)$ and $$ g^{(m)}(j)=g^{(m-1)}(j+1)-g^{(m-1)}(j), \quad j\geq0. $$ Let \begin{equation} M(\alpha)= \begin{cases} [\alpha]+1& \text{ if } \alpha\geq1/2, \\ 0, & \text{ if } \alpha<1/2, \end{cases} \label{c5} \end{equation} where $[\alpha]=\max\{ m\in{\mathbb Z}_{+}: m\leq \alpha\}$ is the integer part of $\alpha$. We impose conditions on $M(\alpha)$ iterated differences of $g(j)$.
\begin{theorem}\label{thm.a1}\cite{0}
Let $\alpha>0$, and let $M=M(\alpha)$ be as in \eqref{c5}. Let $g (j)$ be a sequence of complex numbers that satisfies \begin{equation}
g^{(m)}(j) = o(j^{-1-m}(\log j)^{-\alpha}), \quad j\to\infty, \label{ca7} \end{equation} for all $m=0,\dots,M$. Then $$ s_n(\Gamma(g))=o(n^{-\alpha}), \quad n\to\infty. $$ \end{theorem}
\begin{remark} \begin{enumerate} \item If instead of \eqref{ca7}, we have \begin{equation}
g^{(m)}(j) = O (j^{-1-m}(\log j)^{-\alpha}), \quad j\to\infty, \label{ca7x} \end{equation} then $$ s_n(\Gamma(g))=O(n^{-\alpha}), \quad n\to\infty. $$ \item For the sequence defined by $g(j)=j^{-1}(\log j)^{-\alpha}$, $j\geq2$, condition \eqref{ca7x} is satisfied for all $m$. \item For $\alpha\geq1/2$, our choice of $M(\alpha)$ is probably not optimal, but it is not far from being so. Example~4.7 in \cite{0} shows that for $\alpha \geq 2$ one cannot take $M(\alpha)= [\alpha]-2$ in this theorem. \end{enumerate} \end{remark}
Let us give the analogue of Theorem~\ref{thm.a1} in the continuous case, that is, for the operators ${\mathbf{\Gamma}}={\mathbf{\Gamma}}({\mathbf{g}})$ defined by formula \eqref{a5} in the space $L^2 ({{\mathbb R}}_{+})$. We use the notation $\jap{x}=(\abs{x}^2+1)^{1/2}$. Similarly to the discrete case, for $\alpha<1/2$ we only need an assumption on $\abs{{\mathbf{g}}(t)}$; for $\alpha\geq1/2$ we also need assumptions on the derivatives ${\mathbf{g}}^{(m)}(t)$.
\begin{theorem}\label{thm.a2}\cite{0}
Let $\alpha>0$ and let $M=M(\alpha)$ be the integer given by \eqref{c5}. Let ${\mathbf{g}}$ be a complex valued function, ${\mathbf{g}}\in L^\infty_\mathrm{loc}({\mathbb R}_+)$; if $\alpha\geq1/2$, suppose also that ${\mathbf{g}}\in C^M({\mathbb R}_+)$. Assume that ${\mathbf{g}}$ satisfies $$ {\mathbf{g}}^{(m)}(t)=o(t^{-1-m}\jap{\log t}^{-\alpha}) \quad \text{ as $t\to0$ and as $t\to\infty$.} $$ Then $$ s_n({\mathbf{\Gamma}}({\mathbf{g}}))=o(n^{-\alpha}), \quad n\to\infty. $$ \end{theorem}
As in the discrete case, Theorem~\ref{thm.a2} remains true if $o$ is replaced by $O$.
Theorems~\ref{thm.a1} and \ref{thm.a2} will be used in combination with the following standard result (see e.g. \cite[Section 11.6]{BSbook}) in spectral perturbation theory, which asserts the stability of eigenvalue asymptotics.
\begin{lemma}\label{lma.b1}\cite[Section 11.6]{BSbook}
Let $A$ and $B$ be compact self-adjoint operators and let $\alpha>0$. Suppose that, for both signs $``\pm"$, $$ \lambda^\pm_n(A) = C^\pm n^{-\alpha}+o(n^{-\alpha}), \quad \text{ and }\quad s_n(B) = o(n^{-\alpha}), \quad \quad n\to\infty. $$ Then $$ \lambda^\pm_n(A+B) = C^\pm n^{-\alpha}+o(n^{-\alpha}), \quad n\to\infty. $$ \end{lemma}
\section{Continuous case}\label{sec.d}
\subsection{Statement of the main result} Our main result in the continuous case is
\begin{theorem}\label{thm.d1}
Let $\alpha>0$, ${\mathbf{b}}_0,{\mathbf{b}}_\infty\in{\mathbb R}$, and let $M=M(\alpha)$ be as defined in \eqref{c5}. Let ${\mathbf{h}}$ be a real valued function in $L^\infty_\mathrm{loc}({\mathbb R}_+)$; if $\alpha\geq1/2$, assume also that ${\mathbf{h}}\in C^M({\mathbb R}_+)$. Suppose that \begin{align} \biggl( \frac{d}{dt}\biggr)^m \bigl({\mathbf{h}}(t)-{\mathbf{b}}_0 t^{-1}(\log(1/t))^{-\alpha}\bigr) &= o(t^{-1-m}\jap{\log t}^{-\alpha}), \quad t\to0, \label{d1} \\ \biggl( \frac{d}{dt}\biggr)^m \bigl({\mathbf{h}}(t)-{\mathbf{b}}_\infty t^{-1}(\log t)^{-\alpha} \bigr) &= o(t^{-1-m}\jap{\log t}^{-\alpha}), \quad t\to\infty \label{d2} \end{align} for all $m=0,1,\dots,M$. Then the eigenvalues of the corresponding Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}})$ have the asymptotic behaviour \begin{equation} \lambda_n^\pm({\mathbf{\Gamma}}({\mathbf{h}})) = {\mathbf{c}}^\pm n^{-\alpha} + o(n^{-\alpha}), \quad {\mathbf{c}}^\pm = v(\alpha)\bigl(({\mathbf{b}}_0)_\pm^{1/\alpha}+({\mathbf{b}}_\infty)_\pm^{1/\alpha}\bigr)^\alpha, \label{d3a} \end{equation} as $n\to\infty$, where $v(\alpha)$ is given by \eqref{a4}. \end{theorem}
Of course, this includes Theorem~\ref{thm.aa2} as a particular case.
Let us describe the plan of the proof of Theorem~\ref{thm.d1}. The first and the most important step is to construct a \emph{model operator}. To that end, we introduce an auxiliary explicit function $\sigma_*(\lambda)$ such that its Laplace transform ${\mathbf{h}}_* (t)=(\mathcal{L} \sigma_*) (t)$ has the same asymptotics for $t\to 0$ and $t\to\infty$ as the kernel ${\mathbf{h}} (t)$. To be more precise, we check that the difference ${\mathbf{h}}-{\mathbf{h}}_{*}$ satisfies the assumptions of Theorem~\ref{thm.a2} (singular value estimates). Then we apply the abstract Lemma~\ref{lma.b1} to conclude that the eigenvalues of the Hankel operators ${\mathbf{\Gamma}}({\mathbf{h}})$ and ${\mathbf{\Gamma}}({\mathbf{h}}_{*})$ have the same asymptotic behaviour.
Next, Theorem~\ref{thm.d2} (reduction to $\Psi$DO) implies that the model Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}}_*)$ is unitarily equivalent
to the $\Psi$DO $\Psi_{*} = {\mathfrak{b}}(X) {\mathfrak{s}}_*(D){\mathfrak{b}}(X)$ in $L^2({\mathbb R})$ where ${\mathfrak{b}}(x)$ is the function \eqref{eq:Xx} and ${\mathfrak{s}}_{*}(\xi)=\sigma_*(e^{-\xi})$. Theorem~\ref{thm.b2} (Weyl spectral asymptotics of $\Psi$DO) allows us to find the spectral asymptotics of the operator $\Psi_{*}$ and hence of ${\mathbf{\Gamma}}({\mathbf{h}}_*)$.
\subsection{The model operator}
Let us define the auxiliary function $\sigma_*(\lambda)$ by the formula \begin{equation} \sigma_*(\lambda) = {\mathbf{b}}_\infty\abs{\log \lambda}^{-\alpha}\chi_0(\lambda) + {\mathbf{b}}_0\abs{\log \lambda}^{-\alpha}\chi_\infty(\lambda), \quad \lambda>0, \label{d6} \end{equation} where the smooth cut-off functions $\chi_0$ and $\chi_\infty$ are defined by \eqref{a7a}. Our model operator is the Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}}_*)$ where ${\mathbf{h}}_*=\mathcal{L} \sigma_*$.
\begin{lemma}\label{lma.c0}
The eigenvalues of the model Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}}_*)$ obey the asymptotic relation \begin{equation} \lambda_n^\pm({\mathbf{\Gamma}}({\mathbf{h}}_*)) = {\mathbf{c}}^\pm n^{-\alpha}+o(n^{-\alpha}), \quad n\to\infty, \label{e11} \end{equation} where the coefficients ${\mathbf{c}}^\pm$ are given by \eqref{d3a}. \end{lemma}
\begin{proof} According to Theorem~\ref{thm.d2}, the Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}}_*)$ is unitarily equivalent to the $\Psi$DO $\Psi_{*} = {\mathfrak{b}}(X) {\mathfrak{s}}_*(D){\mathfrak{b}}(X)$ in $L^2({\mathbb R})$ where ${\mathfrak{b}}(x)$ is the standard function \eqref{eq:Xx} and $$ {\mathfrak{s}}_{*}(\xi) = \sigma_*(e^{-\xi}) = {\mathbf{b}}_\infty\abs{\xi}^{-\alpha}\chi_0(e^{-\xi}) + {\mathbf{b}}_0\abs{\xi}^{-\alpha}\chi_\infty(e^{-\xi}), \quad \xi\in{\mathbb R}. $$ In particular, we have \begin{equation} \lambda_n^\pm({\mathbf{\Gamma}}({\mathbf{h}}_*)) = \lambda_n^\pm(\Psi_*), \quad n\in{\mathbb N} . \label{eq:d6} \end{equation} Obviously, the function ${\mathfrak{s}}_{*}$ belongs to $C^\infty ({{\mathbb R}})$ and has the asymptotic behaviour \eqref{b2} with $A(+\infty)= {\mathbf{b}}_{\infty}$ and $A(-\infty)= {\mathbf{b}}_0$. Therefore Theorem~\ref{thm.b2} (Weyl spectral asymptotics of $\Psi$DO) applies to the operator $\Psi_{*}$. This yields the asymptotic formula \begin{equation} \lambda_n^\pm(\Psi_*) = C^\pm n^{-\alpha}+o(n^{-\alpha}), \quad n\to\infty, \label{e11a} \end{equation} where \begin{equation} C^\pm = (2\pi)^{-\alpha} \bigl(({\mathbf{b}}_0)_\pm^{1/\alpha}+({\mathbf{b}}_\infty)_\pm^{1/\alpha}\bigr)^\alpha \biggl(\int_{-\infty}^\infty(\pi(\cosh(\pi x))^{-1})^{1/\alpha} dx\biggr)^\alpha. \label{d10} \end{equation} Using the change of variables $y=(\cosh(\pi x))^2$, the integral representation \eqref{a16} for the Beta function and the definition \eqref{a4} of $v(\alpha)$, we can rewrite the integral in \eqref{d10} as \begin{multline} (2\pi)^{-\alpha} \biggl(\int_{-\infty}^\infty(\pi(\cosh(\pi x))^{-1})^{1/\alpha} dx\biggr)^\alpha = \pi^{1-\alpha} \biggl(\int_{0}^\infty(\cosh(\pi x))^{-1/\alpha} dx\biggr)^\alpha \\ = 2^{-\alpha}\pi^{1-2\alpha} \biggl(\int_1^\infty y^{-\tfrac1{2\alpha}-\tfrac12}(y-1)^{-\tfrac12}dy\biggr)^\alpha = v(\alpha). \label{d10a} \end{multline} Thus, $C^\pm={\mathbf{c}}^\pm$, where ${\mathbf{c}}^\pm$ are the coefficients in \eqref{d3a}. Combining \eqref{eq:d6} with \eqref{e11a}, we obtain \eqref{e11}. \end{proof}
\subsection{Laplace transforms of functions with logarithmic singularities}
Let $\sigma_*(\lambda)$ be given by formula \eqref{d6}, and let ${\mathbf{h}}_*=\mathcal{L} \sigma_*$. Here we find the asymptotics of the function ${\mathbf{h}}_{*} (t)$ as $t\to \infty$ and $t\to 0$. To that end, we need some elementary technical statements about the Laplace transforms of functions with logarithmic singularities at $\lambda=0$ and $\lambda= \infty$. The results below are well known; see, e.g., Lemmas~3 and 4 in \cite{Erdelyi}. However, for completeness we give simple straightforward proofs.
\begin{lemma}\label{L}
Let $$ I_m(t)= \int_{0}^c (-\log \lambda)^{-\alpha}\lambda^{m} e^{-\lambda t}d\lambda, $$ where $\alpha>0$, $m\in{\mathbb Z}_+$ and $c\in (0,1)$. Then \begin{equation}
I_m(t)=m!\, t^{-1-m} |\log t|^{-\alpha} (1+O(\abs{\log t}^{-1})), \label{eq:L2} \end{equation} as $t\to\infty$. \end{lemma} \begin{proof} Let us split $I_m(t)$ into the integrals over $(0,t^{-1/2})$ and over $(t^{-1/2}, c)$. Due to the factor $e^{-\lambda t}$ the second integral can be estimated as \begin{equation} \int_{t^{-1/2}}^{c}
(-\log\lambda)^{-\alpha} \lambda^m e^{-\lambda t}d\lambda = O(t^{-N}), \quad \forall N>0. \label{A3a} \end{equation} Thus, it suffices to consider the integral over $(0,t^{-1/2})$. Making the change of variables $x=\lambda t$, we see that $$ J (t):= \int_0^{t^{-1/2}}
(-\log\lambda)^{-\alpha} \lambda^m e^{-\lambda t}d\lambda = t^{-1-m}(\log t )^{-\alpha}\int_0^{t^{1/2}}
\bigl(1- \tfrac{\log x}{\log t}\bigr)^{-\alpha} x^m e^{-x}d x. $$ Since $u= - \frac{\log x}{\log t} \geq -1/2$ for $x\leq t^{1/2}$, we can use the estimate \begin{equation} \Abs{ (1+u)^{-\alpha} -1} \leq C\abs{u}, \quad u\geq-1/2. \label{eq:L} \end{equation} Thus we see that \begin{equation} J (t)=t^{-1-m} (\log t )^{-\alpha} \int_0^{t^{1/2}} x^m e^{-x}d x+R(t), \label{eq:L4} \end{equation} where the remainder $R(t)$ is estimated by \begin{equation}
t^{-1-m}(\log t )^{-\alpha-1}\int_0^{\infty} |\log x| x^m e^{-x}d x. \label{eq:L5} \end{equation} The integral in \eqref{eq:L4} can be extended to ${{\mathbb R}}_{+}$ and then calculated in terms of the Gamma function. The arising error decays faster than any power of $t^{-1}$ as $t\to\infty$. Putting together \eqref{A3a} and \eqref{eq:L4}, we get \eqref{eq:L2}. \end{proof}
Let us now state the assertion dual to Lemma~\ref{L}. Its proof is almost the same as that of Lemma~\ref{L}.
\begin{lemma}\label{M}
Let
\begin{equation} I_m(t)= \int_{c}^\infty (\log \lambda)^{-\alpha}\lambda^{m} e^{-\lambda t}d\lambda \label{eq:M1} \end{equation} where $\alpha>0$, $m\in{\mathbb Z}_+$ and $c>1$. Then $I_m(t)$ has the asymptotic behaviour \eqref{eq:L2} as $t\to 0$. \end{lemma}
\begin{proof} Now we split \eqref{eq:M1} into the integrals over $(c,t^{-1/2})$ and over $(t^{-1/2}, \infty)$. The first integral can be estimated by $C t^{-(m+1)/2}\abs{\log t}^{-\alpha}$. In the integral over $(t^{-1/2}, \infty)$, we make the change of variables $x=\lambda t$ which yields the integral $$ \int_{t^{-1/2}}^\infty
(\log\lambda)^{-\alpha} \lambda^m e^{-\lambda t}d\lambda =
t^{-1-m}|\log t |^{-\alpha}\int_{t^{1/2}}^\infty
\bigl(1+ \tfrac{\log x}{|\log t|}\bigr)^{-\alpha} x^m e^{-x}d x
=:J (t). $$
Since $ u= \frac{\log x}{|\log t|} \geq -1/2$ for $x\geq t^{1/2}$, we can use \eqref{eq:L} again so that \begin{equation} J (t)=t^{-1-m}\abs{\log t}^{-\alpha} \int_{t^{1/2}}^\infty x^m e^{-y}dy+R(t), \label{eq:MZ2} \end{equation} where the remainder is estimated by \eqref{eq:L5}. The integral in \eqref{eq:MZ2} can be extended to ${\mathbb R}_+$ with an arising error estimated by $C t^{(m+1)/2}$. Putting together the results obtained, we obtain the asymptotics \eqref{eq:L2} as $t\to0$ for the integral \eqref{eq:M1}. \end{proof}
Combining Lemmas~\ref{L} and \ref{M} we easily obtain the following result.
\begin{lemma}\label{lma.d4}
Let the function $\sigma_*$ be given by \eqref{d6}, and let ${\mathbf{h}}_{*}=\mathcal{L}\sigma_*$ be
its Laplace transform. Then $$ {\mathbf{h}}_{*}={\mathbf{b}}_0 {\mathbf{h}}_0+{\mathbf{b}}_\infty {\mathbf{h}}_\infty+\widetilde {\mathbf{g}}, $$ where the model kernels ${\mathbf{h}}_0$, ${\mathbf{h}}_\infty$ are defined by \eqref{a7b} and the error term $\widetilde {\mathbf{g}}\in C^\infty({\mathbb R}_+)$ satisfies the estimates \begin{equation} \abs{\widetilde {\mathbf{g}}^{(m)}(t)}\leq C_m t^{-1-m}\jap{\log t}^{-\alpha-1}, \quad t>0, \label{d8} \end{equation} for all integers $m\geq0$. \end{lemma}
\begin{proof} First consider the case ${\mathbf{b}}_0=0$, ${\mathbf{b}}_\infty=1$. We have \begin{equation} \widetilde {\mathbf{g}}(t) = \int_0^\infty \abs{\log\lambda}^{-\alpha}\chi_0(\lambda)e^{-\lambda t}d\lambda - t^{-1}\abs{\log t}^{-\alpha}\chi_\infty(t), \quad t>0. \label{d8a} \end{equation} It is clear that $\widetilde {\mathbf{g}}\in C^\infty({\mathbb R}_+)$ and that for all $m\in{\mathbb Z}_+$ we have $\widetilde {\mathbf{g}}^{(m)}(t)=O(1)$ as $t\to0$. Thus, it suffices to check the estimates \eqref{d8} for $t\to \infty$. Let us split the integral in \eqref{d8a} into the sum of the integrals over $(0,1/4)$ and $(1/4,1)$. The integral over $(1/4,1)$ (along with all of its derivatives in $t$) decays exponentially fast as $t\to\infty$. Since $\chi_0(\lambda)=1$ for $\lambda\leq 1/4$, we have \begin{multline*} \biggl(\frac{d}{dt}\biggr)^m \biggl( \int_0^{1/4}\abs{\log \lambda}^{-\alpha} \chi_0(\lambda) e^{-\lambda t} d\lambda - t^{-1}(\log t)^{-\alpha} \biggr) \\ = (-1)^m \biggl( \int_0^{1/4}\abs{\log \lambda}^{-\alpha}\lambda^m e^{-\lambda t} d\lambda - m! t^{-1-m}(\log t)^{-\alpha} \biggr)+ O(t^{-1-m}(\log t)^{-\alpha-1}) \end{multline*} as $t\to\infty$. Using Lemma~\ref{L}, we see that the first term in the right-hand side is also $O(t^{-1-m}(\log t)^{-\alpha-1})$.
Similarly, in the case ${\mathbf{b}}_0=1$, ${\mathbf{b}}_\infty=0$ we have \begin{equation} \widetilde {\mathbf{g}}(t) = \int_0^\infty \abs{\log\lambda}^{-\alpha}\chi_\infty(\lambda)e^{-\lambda t}d\lambda - t^{-1}\abs{\log t}^{-\alpha}\chi_0(t), \quad t>0. \label{d8b} \end{equation} Evidently, $\widetilde {\mathbf{g}}\in C^\infty({\mathbb R}_+)$, and this function, along with all of its derivatives in $t$, decays exponentially fast as $t\to\infty$. So one only needs to prove the estimates \eqref{d8} for $t\to0$. We split the integral in \eqref{d8b} into the sum of the integrals over $(0,4)$ and $(4,\infty)$. The integral over $(0,4)$ is a function of $t$ bounded with all its derivatives as $t\to 0$. Differentiating the integral over $(4,\infty)$ and applying Lemma~\ref{M} to it, we complete the proof of \eqref{d8}.
Finally, the general case is a linear combination of the two cases considered above.
\end{proof}
\subsection{Proof of Theorem~\ref{thm.d1}}
By the hypothesis of the theorem, we have the representation $$ {\mathbf{h}}={\mathbf{b}}_0{\mathbf{h}}_0+{\mathbf{b}}_\infty {\mathbf{h}}_\infty+{\mathbf{g}}, $$ where ${\mathbf{g}}$ satisfies the hypothesis of Theorem~\ref{thm.a2} (singular value estimates). As above, let the function $\sigma_*$ be given by \eqref{d6}, and let ${\mathbf{h}}_{*}=\mathcal{L}\sigma_*$ be
its Laplace transform. Using Lemma~\ref{lma.d4}, we obtain that the difference $$ {\mathbf{h}}-{\mathbf{h}}_*={\mathbf{g}}-\widetilde {\mathbf{g}} $$ also satisfies the hypothesis of Theorem~\ref{thm.a2}. Thus, \begin{equation} s_n({\mathbf{\Gamma}}({\mathbf{h}}-{\mathbf{h}}_*))=o(n^{-\alpha}), \quad n\to\infty. \label{eq:We} \end{equation} Let us now apply the abstract Lemma~\ref{lma.b1} to $A={\mathbf{\Gamma}}({\mathbf{h}}_*)$ and $B={\mathbf{\Gamma}}({\mathbf{h}}-{\mathbf{h}}_*)$. Then the desired result for the operator ${\mathbf{\Gamma}}({\mathbf{h}})=A+B$ follows from Lemma~\ref{lma.c0} and from \eqref{eq:We}. \qed
\subsection{Matrix valued kernels}\label{MV} Let $N\in{\mathbb N}$, and let ${\mathbf{h}}$ be an $N\times N$ matrix valued function on $(0,\infty)$. The Hankel operator ${\mathbf{\Gamma}}({\mathbf{h}})$ in the space $L^2({\mathbb R}_+,{\mathbb C}^N)$ is defined by the same formula \eqref{a5} as in the scalar case. Such operators appear, for example, in applications to systems theory, see, e.g. \cite{GLP}.
Theorem~\ref{thm.d1} extends to the matrix case without difficulty. In this case, ${\mathbf{h}}(t)$ is a Hermitian matrix for all $t>0$ (this ensures the self-adjointness of ${\mathbf{\Gamma}}({\mathbf{h}})$), and the coefficients ${\mathbf{b}}_0$, ${\mathbf{b}}_\infty$ in \eqref{d1}, \eqref{d2} are also Hermitian matrices. Formula \eqref{d3a} for the asymptotic coefficients ${\mathbf{c}}^\pm$ becomes $$ {\mathbf{c}}^\pm = v(\alpha)\Bigl(\Tr\bigl(({\mathbf{b}}_0)^{1/\alpha}_\pm\bigr)+\Tr\bigl(({\mathbf{b}}_\infty)_\pm^{1/\alpha}\bigr)\Bigr)^\alpha, $$ where the matrices $({\mathbf{b}}_0)_\pm^{1/\alpha}$, $({\mathbf{b}}_\infty)_\pm^{1/\alpha}$ are defined in the sense of the standard functional calculus for Hermitian matrices.
Let us comment on the proof of this statement. Theorem~\ref{thm.a2} (singular value estimates) extends to the matrix case trivially. Theorem~\ref{thm.d2} (reduction to $\Psi$DO) also extends to the case when $\sigma$ is a matrix valued function; in this case $\Psi$ is a $\Psi$DO acting on vector valued functions. This $\Psi$DO is given by the same formula \eqref{a9b} as in the scalar case with
the matrix valued
function $\mathfrak s(\xi)$ and the standard scalar valued
function $\mathfrak b(x)$ defined by the same formulas \eqref{eq:Xx} and \eqref{eq:a9b} as before. Finally, the extension of Theorem~\ref{thm.b2} (Weyl spectral asymptotics of $\Psi$DO) to the matrix valued case is not quite trivial, but fortunately it has been proven in \cite{BS4} already in the matrix case. Putting together these ingredients in the same way as in the scalar case, one obtains the spectral asymptotics for the matrix valued kernels.
\section{Discrete case}\label{sec.e}
\subsection{Statement of the main result} Below is our main result in the discrete case.
\begin{theorem}\label{cr.a3}
Let $\alpha>0$, $b_1, b_{-1} \in{\mathbb R}$, and let $h$ be a sequence of real numbers given by \begin{equation} h(j) =(b_1+ (-1)^j b_{-1} )j^{-1}(\log j)^{-\alpha}+g_1(j)+(-1)^j g_{-1}(j), \quad j\geq2, \label{a2a} \end{equation} where the error terms ${g}_{\pm 1}$ satisfy conditions \eqref{ca7} for all $m=0,1,\dots,M(\alpha)$ $(M(\alpha)$ is defined in \eqref{c5}$)$. Then the eigenvalues of the corresponding Hankel operator $\Gamma(h)$ have the asymptotic behaviour \begin{equation} \lambda_n^\pm(\Gamma(h)) = c^\pm n^{-\alpha}+o(n^{-\alpha}), \quad c^\pm=v(\alpha)\bigl((b_1)^{1/\alpha}_\pm+(b_{-1})^{1/\alpha}_\pm\bigr)^\alpha, \label{e3} \end{equation} as $n\to\infty$, where $v(\alpha)$ is given by \eqref{a4}. \end{theorem}
Theorem~\ref{thm.aa1} is a particular case of the last theorem corresponding to $g_1=g_{-1}=0$. Observe that if $g_{-1}$ satisfies \eqref{ca7} with some $m>0$, then the sequence $(-1)^j g_{-1}(j)$ does not necessarily satisfy the same condition. Thus, the two term remainder $g_1+(-1)^j g_{-1}$ in \eqref{a2a} in general does not reduce to one term $g_1$.
Just as in the continuous case (see Subsection~\ref{MV}), one can consider Hankel operators in $\ell^2({\mathbb Z}_+,{\mathbb C}^N)$ defined by sequences $\{h(j)\}_{j=0}^\infty$ of Hermitian $N\times N$ matrices. In this case, the asymptotic coefficients $b_{\pm 1}$ in \eqref{a2a} are also Hermitian matrices, and formula \eqref{e3} holds true with the asymptotic coefficient $$ c^\pm = v(\alpha)\Bigl(\Tr\bigl((b_1)^{1/\alpha}_\pm\bigr)+\Tr\bigl((b_{-1})_\pm^{1/\alpha}\bigr)\Bigr)^\alpha. $$
Let us describe the plan of the proof of Theorem~\ref{cr.a3}. We follow the same steps as in Section~\ref{sec.d}, but instead of the Laplace transform ${\mathbf{h}}_*=\mathcal{L} \sigma_*$ of the function $\sigma(\lambda)$, $\lambda>0$, we consider the sequence of moments \begin{equation} h_*(j)=\int_{-1}^1 \eta_*(\mu)\mu^j d\mu, \quad j\geq0 , \label{e19} \end{equation} of some explicit function $ \eta_*(\mu)$ of $\mu\in(-1,1)$. Our model operator is $\Gamma(h_*)$. With our choice of $ \eta_*(\mu)$, the difference $h-h_{*}$ satisfies the assumptions of Theorem~\ref{thm.a1} (singular value estimates). Therefore the eigenvalues of the Hankel operators $\Gamma(h)$ and $\Gamma(h_{*})$ have the same asymptotic behaviour.
Next, Theorem~\ref{thm.e2} implies that the Hankel operator $\Gamma(h_*)$ is unitarily equivalent to the $\Psi$DO $\Psi_{*} = {\mathfrak{b}}(X) {\mathfrak{s}}_*(D){\mathfrak{b}}(X)$ in $L^2({\mathbb R})$ where ${\mathfrak{b}}(x)$ is the standard function \eqref{eq:Xx} and \begin{equation} {\mathfrak{s}}_{*}(\xi)=\eta_*\left(\frac{2e^{-\xi}-1}{2e^{-\xi}+1}\right), \quad \xi\in{\mathbb R}. \label{eq:ss} \end{equation}
Theorem~\ref{thm.b2} (Weyl spectral asympotics for $\Psi$DO)
allows us to find spectral asymptotics of the operators $\Psi_{*}$ and hence of $\Gamma(h_*)$.
\subsection{The model operator}
Let us define the function $\eta_*(\mu)$ by
the following explicit formula: \begin{equation} \eta_*(\mu) = \Abs{\log \frac{1+\mu}{2(1-\mu)}}^{-\alpha} \left( b_1\chi_\infty\Big(\frac{1+\mu}{2(1-\mu)}\Big) + b_{-1}\chi_0\Big(\frac{1+\mu}{2(1-\mu)}\Big) \right), \quad \mu\in(-1,1), \label{e8} \end{equation} where the smooth cut-off functions $\chi_\infty$ and $\chi_0$ are given by equalities \eqref{a7a}. Note that the function $\eta_*$ belongs to the class $C^\infty (-1,1)$ and $\eta_*(\mu)\to 0$ logarithmically as $\mu\to 1$ and as $\mu\to -1$.
\begin{lemma}\label{lma.e0}
Let $h_* (j)$ be the sequence of moments \eqref{e19} of the function \eqref{e8}. Then the eigenvalues of the Hankel operator $\Gamma(h_*)$ obey the asymptotic relation \begin{equation} \lambda_n^\pm(\Gamma(h_*)) = c^\pm n^{-\alpha}+o(n^{-\alpha}), \quad n\to\infty, \label{e18} \end{equation} where the coefficients $c^\pm$ are given by \eqref{e3}. \end{lemma}
\begin{proof} Let us use Theorem~\ref{thm.e2} with $\eta=\eta_*$. We get that the corresponding Hankel operator $\Gamma(h_*)$ is unitarily equivalent to the $\Psi$DO $ \Psi_{*} = {\mathfrak{b}}(X){\mathfrak{s}}_{*}(D){\mathfrak{b}}(X), $
where the functions ${\mathfrak{b}}(x)$ and ${\mathfrak{s}}_{*}(\xi)$ are given by formulas \eqref{eq:Xx} and \eqref{eq:ss}, respectively. By the definition \eqref{e8} of $\eta_*$, we have $$ {\mathfrak{s}}_{*}(\xi)=\abs{\xi}^{-\alpha}(b_1\chi_\infty(e^{-\xi})+b_{-1}\chi_0(e^{-\xi})), \quad \xi\in{\mathbb R}. $$ Applying Theorem~\ref{thm.b2} to the $\Psi$DO $\Psi_{*}$, we obtain $$ \lambda_n^\pm(\Gamma(h_*)) = \lambda_n^\pm(\Psi_*) = C^\pm n^{-\alpha} + o(n^{-\alpha}), \quad n\to\infty, $$ where $$ C^\pm = (2\pi)^{-\alpha} \bigl((b_1)_\pm^{1/\alpha}+(b_{-1})_\pm^{1/\alpha}\bigr)^\alpha \biggl(\int_{-\infty}^\infty(\pi(\cosh(\pi x))^{-1})^{1/\alpha} dx\biggr)^\alpha. $$ It follows from \eqref{d10a} that $C^\pm=c^\pm$, where the numbers $c^\pm$ are given by \eqref{e3}. \end{proof}
\subsection{Moments of functions with logarithmic singularities}
Our goal here is to obtain the asymptotics of the sequence $h_*$ of moments of the function $\eta_*$. We use again Lemma~\ref{L} but in order to replace the continuous parameter $t$ with the discrete one $j$, we need the following simple statement.
\begin{lemma}\label{lma.A2}
Let $m\in{\mathbb Z}_+$, and let $\mathbf{g}\in C^m({\mathbb R}_+)$ be a function that satisfies the estimate $$ \abs{{\mathbf{g}}^{(m)}(t)}\leq C t^{-1-m}(\log t)^{-\alpha}, \quad t\geq 2. $$ Let $\{g(j)\}_{j=2}^\infty$ be a sequence defined by $g(j)={\mathbf{g}}(j)$, $j\geq2$. Then $$ \abs{g^{(m)}(j)} \leq C j^{-1-m}(\log j)^{-\alpha}, \quad j\geq 2. $$ \end{lemma}
\begin{proof} It suffices to use the explicit formula $$ g^{(m)}(j) = \int_0^1 dt_1 \int_0^1 dt_2 \cdots \int_0^1 dt_m \, {\mathbf{g}}^{(m)}(j+t_1+\cdots+t_m), $$ which can be checked by induction in $m$. \end{proof}
The following assertion plays the same role here as Lemma~\ref{lma.d4} played in the previous Section.
\begin{lemma}\label{lma.e3}
Let the sequence $h_*$ be defined by \eqref{e19}, \eqref{e8}. Then $h_* (j)$ has the asymptotics \begin{equation} h_{*}(j) =(b_1+ (-1)^j b_{-1})j^{-1}(\log j)^{-\alpha}+ \widetilde g_{1}(j)+(-1)^j \widetilde g_{-1}(j), \quad j\geq2, \label{eq:a2a} \end{equation} where the error terms $\widetilde g_{\pm 1}(j)$ satisfy the estimates \begin{equation}
\widetilde g_{\pm 1}^{(m)} (j) =O(j^{-1-m}(\log j)^{-\alpha-1}), \quad j\to\infty, \label{e10} \end{equation} for all $m=0,1,2,\dots$. \end{lemma}
\begin{proof} Let the function $\eta_{*} (\mu)$ be defined by \eqref{e8}. If $b_1=1$, $b_{-1}=0$, then $\eta_{*} (\mu)=0$ for $\mu\leq 0$ and setting $\mu=e^{-\lambda}$ in \eqref{e8}, we see that $$ h_*(j) = \int_0^1 \Big(\log \frac{1+\mu}{2(1-\mu)}\Big)^{-\alpha} \chi_\infty \Big(\frac{1+\mu}{2(1-\mu)}\Big) \mu^j d\mu
= (\mathcal{L}\sigma_1)(j) $$ where $$ \sigma_1(\lambda) = \Big(\log \frac{1+e^{-\lambda}}{2(1-e^{-\lambda})}\Big)^{-\alpha} \chi_\infty \Big(\frac{1+e^{-\lambda}}{2(1-e^{-\lambda})}\Big) e^{-\lambda}. $$ If $b_1=0$, $b_{-1}=1$, then $\eta_{*} (\mu)=0$ for $\mu\geq 0$ and setting $\mu=-e^{-\lambda}$ in \eqref{e8}, we see that $h_*(j)=(-1)^j (\mathcal{L}\sigma_{-1})(j)$ where $$ \sigma_{-1}(\lambda) = \Abs{\log \frac{1-e^{-\lambda}}{2(1+e^{-\lambda})}}^{-\alpha} \chi_0 \Big(\frac{1-e^{-\lambda}}{2(1+e^{-\lambda})}\Big) e^{-\lambda}. $$ In both cases we have $$ \sigma_{\pm 1}(\lambda) =(-\log \lambda)^{-\alpha} +O(\abs{\log\lambda}^{-\alpha-1}) $$ as $\lambda\to 0$.
Thus, we can write \begin{equation} (\mathcal{L}\sigma_{\pm 1})(t) = \int_0^{1/2} (-\log \lambda)^{-\alpha} e^{-\lambda t}d\lambda + \int_0^\infty\widetilde{\sigma}_{\pm 1}(\lambda)e^{-\lambda t}d\lambda, \label{eq:LL}\end{equation} where the functions $\widetilde{\sigma}_{\pm 1}(\lambda)$ satisfy the estimate $$ \abs{\widetilde{\sigma}_{\pm 1}(\lambda)}\leq C\jap{\log \lambda}^{-\alpha-1}, \quad \lambda>0, $$ and vanish identically for large $\lambda$. Differentiating \eqref{eq:LL} and using Lemma~\ref{L}, we see that \begin{equation} (\mathcal{L}\sigma_{\pm 1}) (t) = t^{-1}(\log t)^{-\alpha}\chi_\infty(t) + {\mathbf{g}}_{\pm 1}(t), \label{eq:LL1}\end{equation} where the functions ${\mathbf{g}}_{\pm 1} (t)$ satisfy the estimates \eqref{d8} for $t\geq 2$.
For arbitrary $b_1$, $b_{-1}$, we have $$ h_* (j)= b_{1}(\mathcal{L}\sigma_{1})(j) + (-1)^j b_{-1}(\mathcal{L}\sigma_{-1})(j) . $$ According to \eqref{eq:LL1} this yields representation \eqref{eq:a2a} with $\widetilde{g}_{\pm 1} (j)= b_{\pm 1} {\mathbf{g}}_{\pm 1}(j)$. Estimates \eqref{e10} follow from Lemma~\ref{lma.A2}. \end{proof}
\subsection{Proof of Theorem~\ref{cr.a3}}
Let $h$ be as in the hypothesis of Theorem~\ref{cr.a3}, and let, as above, the sequence $h_*$ be defined by \eqref{e19}, \eqref{e8}. Comparing \eqref{a2a} and \eqref{eq:a2a}, we see that \begin{equation} h(j) =h_{*} (j)+ f_1(j)+(-1)^j f_{-1}(j) \label{eq:LL2}\end{equation} where the error terms $f_{\pm 1}(j)= g_{\pm 1}(j)- \widetilde{g}_{\pm 1}(j)$ satisfy the condition $$ f_{\pm 1}^{(m)}(j)= o(j^{-1-m} (\log j)^{-\alpha}), \quad j\to\infty, $$ for all $m=0,1,\dots,M(\alpha)$. According to Theorem~\ref{thm.a1} (singular value estimates) we have $s_n(\Gamma(f_{\pm 1}))=o(n^{-\alpha}), \quad n\to\infty$.
Put $ \widetilde f_{-1}(j)=(-1)^j f_{-1}(j) $; then by the unitary equivalence \eqref{a4a}, one has $s_n(\Gamma(\widetilde f_{-1}))=s_n(\Gamma(f_{-1}))$ and hence \begin{equation} s_n(\Gamma(f_1 + \widetilde f_{-1}))=o(n^{-\alpha}), \quad n\to\infty. \label{e14+} \end{equation} Put $A=\Gamma(h_*)$ and $B=\Gamma(f_1 + \widetilde f_{-1})$ so that, by \eqref{eq:LL2}, $\Gamma(h)=A+B$. In view of \eqref{e18} and \eqref{e14+}, we can apply the abstract Lemma~\ref{lma.b1} to these operators. This
yields the eigenvalue asymptotics \eqref{e3} for $\Gamma(h)$. \qed
\appendix
\section{Theorem~\ref{thm.b2} for $\Psi$DO in $L^2({{\mathbb R}})$ }
Theorem~\ref{thm.b2} was proven in \cite{BS3} for compactly supported $b$. We only have to extend it to general $b$ satisfying the estimate \eqref{b1a}.
\subsection{Schatten class estimates}\label{sec.b1}
For $p>0$, the weak Schatten class $\mathbf{S}_{p,\infty}$ consists of compact operators $A$ such that $$ \norm{A}_{\mathbf{S}_{p,\infty}}:= \sup_{n\geq 1} n^{1/p}s_n(A)<\infty. $$ The functional $\norm{\cdot}_{\mathbf{S}_{p,\infty}}$ is a quasinorm on $\mathbf{S}_{p,\infty}$. Recall that the eigenvalue counting functions of self-adjoint compact operators was defined by formula \eqref{eq:CF}. It is convenient to introduce the following functionals for a self-adjoint operator $A\in \mathbf{S}_{p,\infty}$: $$ \Delta_p^\pm(A)=\limsup_{\varepsilon\to0} \varepsilon^p n_\pm(\varepsilon,A), \quad \delta_p^\pm(A)=\liminf_{\varepsilon\to0} \varepsilon^p n_\pm(\varepsilon,A). $$ In applications, one usually has $\Delta^\pm_p(A)=\delta^\pm_p(A)$, but it is technically convenient to analyse the functionals $\Delta^\pm_p$ and $\delta^\pm_p$ separately. Observe that for $\alpha>0$ and $p=1/\alpha$ the relations $$ \lim_{n\to\infty }n^\alpha\lambda^\pm_n(A) = C^\pm \quad \text{ and }\quad \Delta_{p}^\pm(A)=\delta_{p}^\pm(A)=(C^\pm)^p $$ are equivalent. The functionals $\Delta_{p}^\pm$, $\delta_{p}^\pm$ are continuous in $\mathbf{S}_{p,\infty}$. In fact, one has (see, e.g., \cite{BSbook}, formulas (11.6.10), (11.6.14), (11.6.15)) \begin{align} \abs{(\Delta_{p}^\pm(A_1))^{\frac{1}{1+p}} -(\Delta_{p}^\pm(A_2))^{\frac{1}{1+p}}} \leq \norm{A_1-A_2}^{\frac{1}{1+p}}_{\mathbf{S}_{p,\infty}}, \label{C5} \\ \abs{(\delta_{p}^\pm(A_1))^{\frac{1}{1+p}} -(\delta_{p}^\pm(A_2))^{\frac{1}{1+p}}} \leq \norm{A_1-A_2}^{\frac{1}{1+p}}_{\mathbf{S}_{p,\infty}}. \label{C6} \end{align}
Let $X$, $D$ be the operators in $L^2({\mathbb R})$ defined by \eqref{a8a}. We need an estimate for operators of the form $f(X)g(D)$ in weak Schatten classes. This estimate uses lattice function classes. For $f\in L^2_{\mathrm{loc}}({\mathbb R})$, one writes $$ v_f(n)=\left(\int_{n}^{n+1} \abs{f(x)}^2 dx\right)^{1/2}, \quad n\in{\mathbb Z}, $$ and for $p>0$ one defines the lattice classes \begin{align*} f\in \ell^p(L^2) &\quad \Leftrightarrow \quad v_f\in \ell^p, \\ f\in \ell^{p,\infty}(L^2) &\quad \Leftrightarrow \quad v_f\in \ell^{p,\infty}. \end{align*} Recall that $$ v\in \ell^{p,\infty} \quad \Leftrightarrow \quad \norm{v}_{\ell^{p,\infty}}=\sup_{\varepsilon >0}\varepsilon (\#\{n\in{\mathbb Z}: \abs{v(n)}>\varepsilon\})^{1/p}<\infty. $$
The following theorem is a combination of ideas of Cwikel \cite{C} (who had a version for $p\geq2$) and Birman-Solomyak \cite[Theorem 11.1]{BS5}. The case $1<p\leq2$ appeared in \cite{Simon}. Theorem~\ref{thm.C1} in the form given below and the proof can be found, e.g., in \cite{BKS}.
\begin{theorem}\cite{BKS}\label{thm.C1}
Let $0<p\leq2$, $f\in \ell^p(L^2)$, $g\in \ell^{p,\infty}(L^2)$. Then $f(X)g(D)\in \mathbf{S}_{p,\infty}$ and $$ \norm{f(X)g(D)}_{\mathbf{S}_{p,\infty}} \leq C_p \norm{f}_{\ell^p(L^2)}\norm{g}_{\ell^{p,\infty}(L^2)}. $$ \end{theorem} In fact, we need only a very special case of this theorem.
\subsection{Approximation arguments.} Let us come back to Theorem~\ref{thm.b2}. For $N\in{\mathbb N}$, let $\mathbbm{1}_{(-N,N)}$ be the characteristic function of the interval $(-N,N)$. Set $$ b_N(x)=b(x)\mathbbm{1}_{(-N,N)}(x), \quad \widetilde b_N(x)=b(x)-b_N(x), \quad \Psi_N=b_N(X)a(D)b_N(X). $$
\begin{lemma}\label{appr}
Under the assumptions of Theorem~$\ref{thm.b2}$ we have \begin{equation} \lim_{N\to \infty}\norm{\Psi-\Psi_N}_{\mathbf{S}_{p,\infty}} = 0, \quad p=1/\alpha. \label{b7} \end{equation} \end{lemma}
\begin{proof} Since $$ \Psi-\Psi_N = b(X)a(D)\widetilde b_N(X)+\widetilde b_N(X)a(D)b_N(X), $$ we see that $$ \norm{\Psi-\Psi_N}_{\mathbf{S}_{p,\infty}} \leq 2\norm{b(X)a(D)\widetilde b_N(X)}_{\mathbf{S}_{p,\infty}}. $$ Using the ``Schwarz inequality for classes $\mathbf{S}_{p,\infty}$'' (see e.g. \cite{BSbook}, (11.6.17)), we obtain $$ \norm{b(X)a(D)\widetilde b_N(X)}_{\mathbf{S}_{p,\infty}} \leq \norm{b(X)\abs{a(D)}^{1/2}}_{\mathbf{S}_{2p,\infty}} \norm{\abs{a(D)}^{1/2}\widetilde b_N(X)}_{\mathbf{S}_{2p,\infty}}. $$ Under our assumptions on $a$ and $b$, we have $a\in \ell^{p,\infty}(L^2)$, $b\in \ell^{2p}(L^2)$ and $\norm{\widetilde b_N}_{\ell^{2p}(L^2)}\to0$ as $N\to\infty$. Therefore, by Theorem~\ref{thm.C1}, we have $b(X)\abs{a(D)}^{1/2}\in\mathbf{S}_{2p,\infty}$ and $$ \norm{\abs{a(D)}^{1/2}\widetilde b_N(X)}_{\mathbf{S}_{2p,\infty}} \leq C \norm{a}^{1/2}_{\ell^{p,\infty}(L^2)} \norm{\widetilde b_N}_{\ell^{2p}(L^2)}. $$ This leads to \eqref{b7}. \end{proof}
As already mentioned, Theorem~\ref{thm.b2} was proven in \cite{BS3} for compactly supported $b$. Hence for an arbitrary $b\in L^{2p}_{\mathrm{loc}} ({{\mathbb R}})$ and all $N$ we have $$ \Delta_{p}^\pm(\Psi_N) = \delta_{p}^\pm(\Psi_N) = \frac1{2\pi} \bigl( A(-\infty)_\pm^{p}+A(+\infty)_\pm^{p} \bigr) \int_{-N}^N \abs{b(x)}^{2p}dx. $$ Thus, to extend Theorem~\ref{thm.b2} to general $b$ satisfying estimate \eqref{b1a} we only have to pass here to the limit $N\to\infty$. Lemma~\ref{appr} together with the estimates \eqref{C5}, \eqref{C6} allows us to do this. This yields \eqref{b5}.
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\begin{frontmatter} \title{Sobriety of quantale-valued cotopological spaces}
\author{Dexue Zhang} \address{School of Mathematics, Sichuan University, Chengdu 610064, China}
\begin{abstract} For each commutative and integral quantale, making use of the fuzzy order between closed sets, a theory of sobriety for quantale-valued cotopological spaces is established based on irreducible closed sets. \end{abstract}
\begin{keyword}
Fuzzy topology \sep quantale \sep quantale-valued order \sep quantale-valued cotopological space \sep sobriety \sep irreducible closed set \end{keyword}
\end{frontmatter}
\section{Introduction} A topological space $X$ is sober if each of its irreducible closed subsets is the closure of exactly one point in $X$. Sobriety of topological spaces can be described via the well-known adjunction \[\mathcal{O}\dashv{\rm pt}\] between the category {\sf Top} of topological spaces and the opposite of the category {\sf Frm} of frames \cite{Johnstone}. Precisely, $X$ is sober if $\eta_X\colon X\longrightarrow {\rm pt}(\mathcal{O}(X))$ is a bijection (hence a homeomorphism), where $\eta$ denotes the unit of the adjunction $\mathcal{O}\dashv{\rm pt}$.
In the classical setting, a topological space can be described in terms of open sets as well as closed sets, and we can switch between open sets and closed sets by taking complements. So, it makes no difference whether we choose to work with closed sets or with open sets. In the fuzzy setting, since the table of truth-values is usually a quantale, not a Boolean algebra, there is no natural way to switch between open sets and closed sets. So, it may make a difference whether we postulate topological spaces in terms of open sets or in terms of closed sets. An example in this regard is exhibited in \cite{CZ09,CLZ11}.
The frame approach to sobriety of topological spaces makes use of open sets; while the irreducible-closed-set approach makes use of closed sets. Extending the theory of sober spaces to the fuzzy setting is an interesting topic in fuzzy topology. Most of the existing works focus on the frame approach; that is, to find a fuzzy counterpart of the category {\sf Frm} of frames, then establish an adjunction between the category of fuzzy topological spaces and that of \emph{fuzzy frames}. Works in this regard include Rodabaugh \cite{Rodabaugh}, Zhang and Liu \cite{ZL95}, Kotz\'{e} \cite{Kotze97,Kotze01}, Srivastava and Khastgir \cite{SK98}, Pultr and Rodabaugh \cite{PR03a,PR03b,PR08a,PR08b}, Guti\'{e}rrez Garc\'{i}a, H\"{o}hle and de Prada Vicente \cite{GHP}, and Yao \cite{Yao11,Yao12}, etc. But, the irreducible-closed-set approach to sobriety of fuzzy topological spaces is seldom touched, except in Kotz\'{e} \cite{Kotze97,Kotze01}.
In this paper, making use of the fuzzy inclusion order between closed sets, we establish a theory of sobriety for quantale-valued topological spaces based on irreducible closed sets. Actually, this theory concerns sobriety of \emph{quantale-valued cotopological spaces}. By a quantale-valued cotopological space we mean a ``fuzzy topological space" postulated in terms of closed sets (see Definition \ref{cotopology}). The term \emph{quantale-valued topological space} is reserved for ``fuzzy topological space" postulated in terms of open sets (see Definition \ref{topology}).
It should be noted that in most works on fuzzy frames, the table of truth-values is assumed to be a complete Heyting algebra (or, a frame), even a completely distributive lattice sometimes. But, in this paper, the table of truth-values is only assumed to be a commutative and integral quantale. Complete Heyting algebra, BL-algebras and left continuous t-norms, are important examples of such quantales.
The contents are arranged as follows. Section 2 recalls basic ideas about quantale-valued ordered sets and quantale-valued $\mathcal{Q}$-cotopological spaces. Section 3, making use of the quantale-valued order between closed sets in a $\mathcal{Q}$-cotopological space, establishes a theory of sober $\mathcal{Q}$-cotopological spaces based on irreducible closed sets. In particular, the sobrification of a stratified $\mathcal{Q}$-cotopological space is constructed. The last section, Section 4, presents some interesting examples in the case that $\mathcal{Q}$ is the unit interval $[0,1]$ coupled with a (left) continuous t-norm.
\section{Quantale-valued ordered sets and quantale-valued cotopological spaces} In this paper, $\mathcal{Q}=(Q,\&)$ always denotes a commutative and integral quantale, unless otherwise specified. Precisely, $Q$ is a complete lattice with a bottom element $0$ and a top element $1$, $\&$ is a binary operation on $Q$ such that $(Q,\&, 1)$ is a commutative monoid and $p\&\bigvee_{j\in J}q_j=\bigvee_{j\in J}p\& q_j$ for all $p\in Q$ and $\{q_j\}_{j\in J}\subseteq Q$.
Since the semigroup operation $\&$ distributes over arbitrary joins, it determines a binary operation $\rightarrow$ on $Q$ via the adjoint property \begin{equation*} p\&q\leq r\iff q\leq p\rightarrow r. \end{equation*} The binary operation $\rightarrow$ is called the \emph{implication}, or the \emph{residuation}, corresponding to $\&$.
Some basic properties of the binary operations $\&$ and $\rightarrow$ are collected below, they can be found in many places, e.g. \cite{Belo02,Rosenthal90}.
\begin{prop} Let $\mathcal{Q}$ be a quantale. Then \begin{enumerate}[(1)] \item $1\rightarrow p=p$. \item $p\leq q \iff 1= p\rightarrow q$. \item $p\rightarrow(q\rightarrow r)=(p\& q)\rightarrow r$. \item $p\&(p\rightarrow q)\leq q$. \item $\Big(\bigvee_{j\in J}p_j\Big)\rightarrow q=\bigwedge_{j\in J}(p_j\rightarrow q)$. \item $p\rightarrow\Big(\bigwedge_{j\in J} q_j\Big)=\bigwedge_{j\in J}(p\rightarrow q_j)$. \end{enumerate} \end{prop} We often write $\neg p$ for $p\rightarrow 0$ and call it the \emph{negation} of $p$. Though it is true that $p\leq\neg\neg p$ for all $p\in Q$, the inequality $\neg\neg p\leq p$ does not always hold. A quantale $\mathcal{Q}$ is said to satisfy the {\it law of double negation} if \[(p\rightarrow 0)\rightarrow 0 = p,\] i.e., $\neg\neg p= p$, for all $p\in Q$.
\begin{prop}\cite{Belo02}\label{properties of negation} Suppose that $\mathcal{Q}$ is a quantale that satisfies the law of double negation. Then \begin{enumerate}[(1)] \item $p\rightarrow q = \neg(p\&\neg q)=\neg q\rightarrow \neg p$. \item $p\&q =\neg (q\rightarrow\neg p)=\neg (p\rightarrow\neg q)$. \item $\neg(\bigwedge_{i\in I}p_i) = \bigvee_{i\in I}\neg p_i$. \end{enumerate}\end{prop}
In the class of quantales, the quantales with the unit interval $[0,1]$ as underlying lattice are of particular interest in fuzzy set theory \cite{Ha98,KMP00}. In this case, the semigroup operation $\&$ is called a left continuous t-norm on $[0,1]$ \cite{KMP00}. If a left continuous t-norm $\&$ on $[0,1]$ is a continuous function with respect to the usual topology, then it is called a continuous t-norm.
\begin{exmp} (\cite{KMP00})
Some basic t-norms: \begin{enumerate}[(1)] \item The minimum t-norm: $a\&b=a\wedge b=\min\{a,b\}$. The corresponding implication is given by \[a\rightarrow b=\left\{\begin{array}{ll} 1, & a\leq b;\\ b, & a>b.\end{array}\right.\] \item The product t-norm: $a\&b=a\cdot b$. The corresponding implication is given by $$a\rightarrow b=\left\{\begin{array}{ll} 1, & a\leq b;\\ b/a, & a>b.\end{array}\right.$$ \item The {\L}ukasiewicz t-norm: $a\&b=\max\{a+b-1,0\}$. The corresponding implication is given by $$a\rightarrow b= \min\{1, 1-a+b\}. $$ In this case, $([0,1],\&)$ satisfies the law of double negation. \item The nilpotent minimum t-norm: $$a\& b=\left\{\begin{array}{ll} 0, & a+b\leq 1;\\ \min\{a,b\}, & a+b>1.\end{array}\right.$$ The corresponding implication is given by $$a\rightarrow b=\left\{\begin{array}{ll} 1, & a\leq b;\\ \max\{1-a,b\}, & a>b.\end{array}\right.$$ In this case, $([0,1],\&)$ satisfies the law of double negation. \end{enumerate}\end{exmp}
A $\mathcal{Q}$-order (or an order valued in the quantale $\mathcal{Q}$) \cite{Belo02,Wagner97} on a set $X$ is a reflexive and transitive $\mathcal{Q}$-relation on $X$. Explicitly, a $\mathcal{Q}$-order on $X$ is a map $R\colon X\times X\longrightarrow Q$ such that $1= R(x,x)$ and $R(y,z)\& R(x,y)\leq R(x,z)$ for all $x,y,z\in X$. The pair $(X,R)$ is called a $\mathcal{Q}$-ordered set. As usual, we write $X$ for the pair $(X, R)$ and $X(x,y)$ for $R(x,y)$ if no confusion would arise.
If $R\colon X\times X\longrightarrow Q$ is a $\mathcal{Q}$-order on $X$, then $R^{\rm op}\colon X\times X\longrightarrow Q$, given by $R^{\rm op}(x,y)=R(y,x)$, is also a $\mathcal{Q}$-order on $X$ (by commutativity of $\&$), called the opposite of $R$.
A map $f\colon X\longrightarrow Y$ between $\mathcal{Q}$-ordered sets is $\mathcal{Q}$-order-preserving if $X(x_1,x_2)\leq Y(f(x_1),f(x_2))$ for all $x_1,x_2\in X$. We write \[\mbox{$\mathcal{Q}$-{\sf Ord}}\] for the category of $\mathcal{Q}$-ordered sets and $\mathcal{Q}$-order-preserving maps.
Given a set $X$, a map $A\colon X\longrightarrow Q$ is called a fuzzy set (valued in $\mathcal{Q}$), the value $A(x)$ is interpreted as the membership degree. The map \[{\rm sub}_X\colon Q^X\times Q^X\longrightarrow Q,\] given by \begin{equation*}{\rm sub}_X(A, B)=\bigwedge_{x\in X}A(x)\rightarrow B(x),\end{equation*} defines a $\mathcal{Q}$-order on $Q^X$. The value ${\rm sub}_X(A, B)$ measures the degree that $A$ is a subset of $B$, thus, ${\rm sub}_X$ is called the \emph{fuzzy inclusion order} on $Q^X$ \cite{Belo02}. In particular, if $X$ is a singleton set then the $\mathcal{Q}$-ordered set $(Q^X,{\rm sub}_X)$ reduces to the $\mathcal{Q}$-ordered set $(Q,d_L)$, where \[d_L(p,q)=p\rightarrow q.\]
For any $p\in Q$ and $A\in Q^X$, write $p\&A, p\rightarrow A\in Q^X$ for the fuzzy sets given by $(p\&A)(x)=p\&A(x)$ and $(p\rightarrow A)(x)=p\rightarrow A(x)$, respectively. It is easy to check that \begin{equation*}p\leq {\rm sub}_X(A, B)\iff p\&A\leq B\iff A\leq p\rightarrow B \end{equation*} for all $p\in Q$ and $A, B\in Q^X$. In particular, $A\leq B$ if and only if $1= {\rm sub}_X(A, B)$. Furthermore, \begin{equation*}p\rightarrow {\rm sub}_X(A, B)={\rm sub}_X(p\&A, B)={\rm sub}_X(A,p\rightarrow B)\end{equation*} for all $p\in Q$ and $A, B\in Q^X$.
Given a map $f\colon X\longrightarrow Y$, as usual, define $f^\rightarrow\colon Q^X\longrightarrow Q^Y$ and $f^\leftarrow\colon Q^Y\longrightarrow Q^X$ by \[f^\rightarrow(A)(y)=\bigvee_{f(x)=y}A(x), \quad f^\leftarrow(B)(x)= B\circ f(x).\] The fuzzy set $f^\rightarrow(A)$ is called the image of $A$ under $f$, and $f^\leftarrow(B)$ the preimage of $B$.
The following proposition is a special case of the enriched Kan extension in category theory \cite{Kelly,Lawvere73}. A direct verification is easy and can be found in e.g. \cite{Belo02,LZ07}. \begin{prop}For any map $f\colon X\longrightarrow Y$, \begin{enumerate}[(1)] \item $f^\rightarrow\colon (Q^X,{\rm sub}_X)\longrightarrow (Q^Y,{\rm sub}_Y)$ is $\mathcal{Q}$-order-preserving; \item $f^\leftarrow\colon (Q^Y,{\rm sub}_Y)\longrightarrow (Q^X,{\rm sub}_X)$ is $\mathcal{Q}$-order-preserving; \item $f^\rightarrow$ is left adjoint to $f^\leftarrow$, written $f^\rightarrow\dashv f^\leftarrow$, in the sense that \[{\rm sub}_Y(f^\rightarrow(A),B) = {\rm sub}_X(A,f^\leftarrow(B))\] for all $A\in Q^X$ and $B\in Q^Y$. \end{enumerate} \end{prop}
A fuzzy upper set \cite{LZ06} in a $\mathcal{Q}$-ordered set $X$ is a map $\psi\colon X\longrightarrow Q$ such that \[ X(x,y)\&\psi(x) \leq\psi(y)\] for all $x,y\in X$. It is clear that $\psi\colon X\longrightarrow Q$ is a fuzzy upper set if and only if $\psi\colon X\longrightarrow(Q, d_L)$ is $\mathcal{Q}$-order-preserving.
Dually, a fuzzy lower set in a $\mathcal{Q}$-ordered set $X$ is a map $\phi\colon X\longrightarrow Q$ such that \[\phi(y)\&X(x,y) \leq\phi(x)\] for all $x,y\in X$. Or equivalently, $\phi\colon X^{\rm op}\longrightarrow(Q, d_L)$ is a $\mathcal{Q}$-order-preserving map, where $X^{\rm op}$ is the opposite of the $\mathcal{Q}$-ordered set $X$.
\begin{defn} \label{irreducible} A fuzzy lower set $\phi$ in a $\mathcal{Q}$-ordered set $X$ is irreducible if $\bigvee_{x\in X}\phi(x)=1$ and \[{\rm sub}_X(\phi, \phi_1\vee\phi_2) = {\rm sub}_X(\phi, \phi_1)\vee{\rm sub}_X(\phi, \phi_2)\] for all fuzzy lower sets $\phi_1,\phi_2$ in $X$. \end{defn}
Irreducible fuzzy lower sets are a counterpart of directed lower sets \cite{Gierz2003} in the quantale-valued setting. In particular, the condition $\bigvee_{x\in X}\phi(x)=1$ is a $\mathcal{Q}$-version of the requirement that a directed set should be non-empty.
The following definition is taken from \cite{CZ09, Zhang07}.
\begin{defn}\label{cotopology} A $\mathcal{Q}$-cotopology on a set $X$ is a subset $\tau$ of $Q^X$ subject to the following conditions: \begin{enumerate} \item[(C1)] $p_X\in\tau$ for all $p\in Q$; \item[(C2)] $A\vee B\in\tau$ for all $A, B\in\tau$; \item[(C3)] $\bigwedge_{j\in J} A_j\in\tau$ for each subfamily $\{A_j\}_{j\in J}$ of $\tau$.\end{enumerate}The pair $(X,\tau)$ is called a $\mathcal{Q}$-cotopological space; elements in $\tau$ are called closed sets of $(X,\tau)$. A $\mathcal{Q}$-cotopology $\tau$ is stratified if \begin{enumerate} \item[(C4)] $p\rightarrow A\in\tau$ for all $p\in Q$ and $A\in\tau$. \end{enumerate} A $\mathcal{Q}$-cotopology $\tau$ is co-stratified if \begin{enumerate} \item[(C5)] $p\&A\in\tau$ for all $p\in Q$ and $A\in\tau$. \end{enumerate} A $\mathcal{Q}$-cotopology $\tau$ is strong if it is both stratified and co-stratified. \end{defn}
As usual, we often write $X$, instead of $(X,\tau)$, for a $\mathcal{Q}$-cotopological space.
\begin{rem} If $\mathcal{Q}$ is the quantale obtained by endowing $[0,1]$ with a continuous t-norm $T$, then $\tau\subseteq [0,1]^X$ is a strong $\mathcal{Q}$-cotopology on $X$ if and only if $(X,\tau^*)$ is a fuzzy $T$-neighborhood space in the sense of Morsi \cite{Morsi95a}, where $\tau^*=\{1-A\mid A\in\tau\}$. In particular, if $\mathcal{Q}$ is the quantale $([0,1],\min)$, then $\tau\subseteq [0,1]^X$ is a strong $\mathcal{Q}$-cotopology on $X$ if and only if $(X,\tau^*)$ is a fuzzy neighborhood space in the sense of Lowen \cite{Lowen1982}. \end{rem}
A map $f\colon X\longrightarrow Y$ between $\mathcal{Q}$-cotopological spaces is continuous if $f^\leftarrow(A)=A\circ f$ is closed in $X$ whenever $A$ is closed in $Y$. We write \[\mbox{$\mathcal{Q}$-{\sf CTop}}\] for the category of $\mathcal{Q}$-cotopological spaces and continuous maps; and write \[\mbox{{\sf S}$\mathcal{Q}$-{\sf CTop}}\] for the category of stratified $\mathcal{Q}$-cotopological spaces and continuous maps. It is easily seen that both $\mathcal{Q}$-{\sf CTop} and {\sf S}$\mathcal{Q}$-{\sf CTop} are well-fibred topological categories over {\sf Set} in the sense of \cite{AHS}.
Given a $\mathcal{Q}$-cotopological space $(X,\tau)$, its closure operator $^-\colon Q^X\longrightarrow Q^X$ is defined by \[\overline{A}=\bigwedge\{B \in\tau\mid A\leq B \}\] for all $A\in Q^X$. The closure operator of a $\mathcal{Q}$-cotopological space $(X,\tau)$ satisfies the following conditions: for all $A, B\in Q^X$, \begin{enumerate} \item[\rm (cl1)] $\overline{p_X}=p_X$ for all $p\in Q$; \item[\rm (cl2)] $ \overline{A} \geq A$; \item[\rm (cl3)] $\overline{A\vee B}=\overline{A}\vee\overline{B}$; \item[\rm (cl4)] $\overline{\overline{A}}=\overline{A}$. \end{enumerate}
\begin{prop} Let $X$ be a $\mathcal{Q}$-cotopological space. The following are equivalent: \begin{enumerate}[\rm(1)] \item $X$ is stratified. \item $p\&\overline{A}\leq \overline{p\&A}$ for all $p\in Q$ and $A\in Q^X$. \item The closure operator $^-\colon (Q^X,{\rm sub}_X)\longrightarrow (Q^X,{\rm sub}_X)$ is $\mathcal{Q}$-order-preserving. \end{enumerate} \end{prop} \begin{proof} $(1)\Rightarrow(2)$ Since $p\rightarrow \overline{p\&A}$ is closed and $A\leq p\rightarrow \overline{p\&A}$, it follows that $\overline{A} \leq p\rightarrow \overline{p\&A}$, hence $p\&\overline{A}\leq \overline{p\&A}$.
$(2)\Rightarrow(3)$ For any $A,B\in Q^X$, let $p={\rm sub}_X(A,B)=\bigwedge_{x\in X}(A(x)\rightarrow B(x))$. Since $p\&A\leq B$, then $p\&\overline{A}\leq \overline{p\&A}\leq \overline{B}$, hence ${\rm sub}_X(A,B)=p\leq {\rm sub}_X(\overline{A},\overline{B})$.
$(3)\Rightarrow(1)$ Let $A$ be a closed set and $p\in Q$. In order to see that $p\rightarrow A$ is also closed, it suffices to check that $\overline{p\rightarrow A}\leq p\rightarrow A$, or equivalently, $p\leq {\rm sub}_X(\overline{p\rightarrow A}, A)$. This is easy since $p\leq {\rm sub}_X(p\rightarrow A,A)\leq {\rm sub}_X(\overline{p\rightarrow A},\overline{A}) ={\rm sub}_X(\overline{p\rightarrow A},A)$. \end{proof}
It follows immediately from item (3) in the above proposition that if $X$ is a stratified $\mathcal{Q}$-cotopological space and if $B$ is a closed set in $X$, then for all $A\in Q^X$, \[ {\rm sub}_X(\overline{A},B)\leq {\rm sub}_X(A,B)\leq{\rm sub}_X(\overline{A},\overline{B}) ={\rm sub}_X(\overline{A},B),\] hence \begin{equation*}
{\rm sub}_X(A,B)={\rm sub}_X(\overline{A},B). \end{equation*} This equation will be useful in this paper.
\begin{cor}In a stratified $\mathcal{Q}$-cotopological space $X$, \[\overline{A}=\bigwedge\big\{{\rm sub}_X(A,B)\rightarrow B\mid B~\text{is closed in $X$}\big\}\] for all $A\in Q^X$. \end{cor}
Given a $\mathcal{Q}$-cotopological space $(X,\tau)$, define $\Omega(\tau)\colon X\times X\longrightarrow Q$ by \[\Omega(\tau)(x,y) =\bigwedge_{A\in\tau}(A(y)\rightarrow A(x)).\] Then $\Omega(\tau)$ is a $\mathcal{Q}$-order on $X$, called the \emph{specialization $\mathcal{Q}$-order} of $(X,\tau)$ \cite{LZ06}. It is clear that each closed set in $(X,\tau)$ is a fuzzy lower set in the $\mathcal{Q}$-ordered set $(X,\Omega(\tau))$.
As said before, we often write $X$, instead of $(X,\tau)$, for a $\mathcal{Q}$-cotopological space. Accordingly, we will write $\Omega(X)$ for the $\mathcal{Q}$-ordered set obtained by equipping $X$ with its specialization $\mathcal{Q}$-order.
The correspondence $X\mapsto\Omega(X)$ defines a functor \[\Omega\colon \mathcal{Q}\text{-}{\sf CTop}\longrightarrow\mathcal{Q}\text{-}{\sf Ord}.\] In particular, if $f\colon X\longrightarrow Y$ is a continuous map between $\mathcal{Q}$-cotopological spaces, then $f\colon \Omega(X)\longrightarrow\Omega(Y)$ is $\mathcal{Q}$-order-preserving, i.e., $\Omega(X)(x,y)\leq\Omega(Y)(f(x),f(y))$ for all $x,y\in X$.
Conversely, given a $\mathcal{Q}$-ordered set $(X,R)$, the family $\Gamma(R)$ of fuzzy lower sets in $(X,R)$ forms a strong $\mathcal{Q}$-cotopology on $X$, called the \emph{Alexandroff $\mathcal{Q}$-cotopology} on $(X,R)$. The correspondence $(X,R)\mapsto\Gamma(X,R)=(X,\Gamma(R))$ defines a functor \[\Gamma\colon \mathcal{Q}\text{-}{\sf Ord}\longrightarrow \mathcal{Q}\text{-}{\sf CTop} \] that is left adjoint to the functor $\Omega$ \cite{CZ09,LZ06}.
The following conclusion says that in a stratified $\mathcal{Q}$-cotopological space, the specialization $\mathcal{Q}$-order is determined by closures of singletons, as in the classical case.
\begin{prop}\cite{Qiao} If $X$ is a stratified $\mathcal{Q}$-cotopological space, then $\Omega(X)(x,y) = \overline{1_y}(x)$ for all $x,y\in X$. \end{prop}
\section{Sober $\mathcal{Q}$-cotopological spaces} Let $X$ be a topological space. A closed set $F$ in $X$ is irreducible if it is non-empty and for any closed sets $A,B$ in $X$, $F\subseteq A\cup B$ implies either $F\subseteq A$ or $F\subseteq B$. A topological space is sober if every irreducible closed set in it is the closure of exactly one point. Sobriety is an interesting property in the realm of non-Hausdorff spaces and it plays an important role in domain theory \cite{Gierz2003}. In order to extend the theory of sober spaces to the fuzzy setting, the first step is to postulate irreducible closed sets in a $\mathcal{Q}$-cotopological space. Fortunately, this can be done in a natural way with the help of the fuzzy inclusion order between the closed sets in a $\mathcal{Q}$-cotopological space.
\begin{defn} A closed set $F$ in a $\mathcal{Q}$-cotopological space $X$ is irreducible if $\bigvee_{x\in X}F(x)=1$ and \[{\rm sub}_X(F,A\vee B) = {\rm sub}_X(F, A)\vee{\rm sub}_X(F,B)\] for all closed sets $A,B$ in $X$.\end{defn}
This definition is clearly an extension of that of irreducible closed sets in a topological space. We haste to emphasize that the fuzzy inclusion order, not the pointwise order, between closed sets is used here. The condition $\bigvee_{x\in X}F(x)=1$ is a $\mathcal{Q}$-version of the requirement that $F$ is non-empty.
\begin{exmp}\begin{enumerate}[(1)]\item Let $X$ be a stratified $\mathcal{Q}$-cotopological space. For any $x\in X$, the closure $\overline{1_x}$ of $1_x$ is irreducible. This follows from that ${\rm sub}_X(\overline{1_x},A)= A(x)$ for any closed set $A$ in $X$.
\item A fuzzy lower set $\phi$ in a $\mathcal{Q}$-ordered set $X$ is irreducible in the sense of Definition \ref{irreducible} if and only if $\phi$ is an irreducible closed set in the Alexandroff $\mathcal{Q}$-cotopological space $\Gamma(X)$.\end{enumerate} \end{exmp} Having the notion of irreducible closed sets (in a $\mathcal{Q}$-cotopological space) at hand, we are now able to formulate the central notion of this paper.
\begin{defn}A $\mathcal{Q}$-cotopological space $X$ is sober if it is stratified and each irreducible closed in $X$ is the closure of $1_x$ for a unique $x\in X$. \end{defn}
Write \[\mbox{{\sf Sob}$\mathcal{Q}$-{\sf CTop}}\] for the full subcategory of {\sf S}$\mathcal{Q}$-{\sf CTop} consisting of sober $\mathcal{Q}$-cotopological spaces. This section concerns basic properties of sober $\mathcal{Q}$-cotopological spaces. First, we show that the subcategory {\sf Sob}$\mathcal{Q}$-{\sf CTop} is reflective in {\sf S}$\mathcal{Q}$-{\sf CTop} and that the specialization $\mathcal{Q}$-order of each sober $\mathcal{Q}$-cotopological space $X$ is \emph{directed complete} in the sense that every irreducible fuzzy lower set in the $\mathcal{Q}$-ordered set $\Omega(X)$ has a supremum. Then we will discuss the relationship between \begin{itemize}\setlength{\itemsep}{-2pt} \item sobriety and Hausdorff separation in a stratified $\mathcal{Q}$-cotopological space; \item sober topological spaces and sober $\mathcal{Q}$-cotopological spaces via the Lowen functor $\omega_\mathcal{Q}$; \item sober $\mathcal{Q}$-cotopological spaces and sober $\mathcal{Q}$-topological spaces in the case that $\mathcal{Q}$ satisfies the law of double negation. \end{itemize}
Given a stratified $\mathcal{Q}$-cotopological space $X$, let \[\text{irr}(X)\] denote the set of all irreducible closed sets in $X$. For each closed set $A$ in $X$, define \[s(A)\colon \text{irr}(X)\longrightarrow Q\] by \[s(A)(F) ={\rm sub}_X(F,A).\] \begin{lem}Let $X$ be a stratified $\mathcal{Q}$-cotopological space. \begin{enumerate}[(1)]\item $s(p_X)(F)=p$ for all $p\in Q$ and $F\in {\rm irr}(X)$. \item $s(A)=s(B)\Leftrightarrow A=B$ for all closed sets $A,B$ in $X$. \item $s(A\vee B)=s(A)\vee s(B)$ for all closed sets $A,B$ in $X$. \item $s\big(\bigwedge\limits_{j\in J}A_j\big) = \bigwedge\limits_{j\in J}s(A_j)$ for each family $\{A_j\}_{i\in J}$ of closed sets in $X$. \item $s(p\rightarrow A)= p\rightarrow s(A)$ for all $p\in Q$ and all closed sets $A$ in $X$. \item ${\rm sub}_X(A,B)={\rm sub}_{{\rm irr}(X)}(s(A),s(B))$ for all closed sets $A,B$ in $X$. \end{enumerate}\end{lem} \begin{proof} We check (6) for example. On one hand, \[{\rm sub}_X(A,B) \leq \bigwedge_{F\in \text{irr}(X)}( {\rm sub}_X(F,A) \rightarrow {\rm sub}_X(F,B)) = {\rm sub}_{\text{irr}(X)}(s(A),s(B)).\] On the other hand, \begin{align*} {\rm sub}_{\text{irr}(X)}(s(A),s(B)) & =\bigwedge_{F\in \text{irr}(X)}(s(A)(F)\rightarrow s(B)(F)) \\ & = \bigwedge_{F\in \text{irr}(X)}({\rm sub}_X(F,A)\rightarrow {\rm sub}_X(F,B)) \\ & \leq \bigwedge_{x\in X}({\rm sub}_X(\overline{1_x}, A)\rightarrow {\rm sub}_X(\overline{1_x},B)) \\ & = \bigwedge_{x\in X}(A(x)\rightarrow B(x)) \\ & = {\rm sub}_X(A,B). \end{align*} The proof is finished. \end{proof}
By the above lemma, \[\{s(A)\mid A~\text{is a closed set of $X$}\}\] is a stratified $\mathcal{Q}$-cotopology on $\text{irr}(X)$. We will write $s(X)$, instead of $\text{irr}(X)$, for the resulting $\mathcal{Q}$-cotopological space.
\begin{prop}\label{s(X) is sober} $s(X)$ is sober for each stratified $\mathcal{Q}$-cotopological space $X$. \end{prop} \begin{proof}First of all, we note that for each irreducible closed set $F$ in $X$, the closure of $1_F$ in $s(X)$ is given by $s(F)$. So, it suffices to show that for each closed set $A$ in $X$, if $s(A)$ is irreducible in $s(X)$, then $A$ is irreducible in $X$. We prove the conclusion in two steps.
\textbf{Step 1}. $\bigvee_{x\in X}A(x)=1$. Since $s(A)$ is an irreducible closed set in $s(X)$, it holds that \[\bigvee_{F\in s(X)}s(A)(F)= \bigvee_{F\in s(X)}{\rm sub}_X(F,A)=1.\] Since for each $F\in s(X)$ and $x\in X$, \[F(x)\&{\rm sub}_X(F,A)=F(x)\& \bigwedge_{z\in X}(F(z)\rightarrow A(z))\leq A(x),\] it follows that \begin{align*}\bigvee_{x\in X}A(x)&\geq \bigvee_{x\in X}\bigvee_{F\in s(X)} F(x) \& {\rm sub}_X(F,A) \\ & = \bigvee_{F\in s(X)}\bigvee_{x\in X} F(x)\& {\rm sub}_X(F,A)\\ &= \bigvee_{F\in s(X)}{\rm sub}_X(F,A)\\ &=1.\end{align*}
\textbf{Step 2}. ${\rm sub}_X(A,B\vee C)= ({\rm sub}_X(A,B))\vee({\rm sub}_X(A,C))$ for all closed sets $B, C$ in $X$. Since $s(A)$ is irreducible in $s(X)$, \begin{align*}{\rm sub}_X(A,B\vee C)&= {\rm sub}_{s(X)}(s(A), s(B \vee C)) \\&= {\rm sub}_{s(X)}(s(A),s(B)\vee s(C))\\ & = {\rm sub}_{s(X)}(s(A),s(B))\vee{\rm sub}_{s(X)}(s(A),s(C))\\ &= {\rm sub}_X(A,B)\vee{\rm sub}_X(A,C).\end{align*}
Therefore, $A$ is an irreducible closed set in $X$. \end{proof}
\begin{prop}For a stratified $\mathcal{Q}$-cotopological space $X$, define \[\eta_X\colon X\longrightarrow s(X)\] by $\eta_X(x)=\overline{1_x}$. Then \begin{enumerate}[(1)] \item $\eta_X\colon X\longrightarrow s(X)$ is continuous. \item $X$ is sober if and only if $\eta_X$ is a homeomorphism. \end{enumerate} \end{prop}
\begin{proof}(1) For any closed set $A$ in $X$ and $x\in X$, \[\eta_X^\leftarrow(s(A))(x) = s(A)(\eta_X(x))= A(x),\] hence $\eta_X^\leftarrow(s(A))=A$. This shows that $f$ is continuous.
(2) If $X$ is sober then each irreducible closed set in $X$ is of the form $\overline{1_x}=\eta_X(x)$ for a unique $x\in X$, hence $\eta_X\colon X\longrightarrow s(X)$ is a bijection. Since for each closed set $A$ in $X$ and $x\in X$, \[\eta_X^\rightarrow(A)(\overline{1_x})=\bigvee\big\{A(z)\mid\eta_X(z)= \overline{1_x}\big\}=A(x)= s(A)(\overline{1_x}), \] it follows that $\eta_X^\rightarrow(A)=s(A)$. This shows that $\eta_X$ is a continuous closed bijection, hence a homeomorphism. The converse conclusion is trivial, since $s(X)$ is sober by Proposition \ref{s(X) is sober}. \end{proof}
\begin{thm}\label{soberification} Let $f\colon X\longrightarrow Y$ be a continuous map between stratified $\mathcal{Q}$-cotopological spaces. If $Y$ is sober, there is a unique continuous map $f^*\colon s(X)\longrightarrow Y$ such that ~$f=f^*\circ\eta_X$. \end{thm}
\begin{lem}Let $f\colon X\longrightarrow Y$ be a continuous map between stratified $\mathcal{Q}$-cotopological spaces. Then for each irreducible closed set $F$ of $X$, the closure $\overline{f^\rightarrow(F)}$ of the image of $F$ under $f$ is an irreducible closed set of $Y$. \end{lem} \begin{proof}For any closed sets $A,B$ in $Y$, \begin{align*}{\rm sub}_Y(\overline{f^\rightarrow(F)},A\vee B)& ={\rm sub}_Y(f^\rightarrow(F),A\vee B) & (A\vee B~\text{is closed})\\ &= {\rm sub}_X(F, f^\leftarrow(A\vee B))&(f^\rightarrow\dashv f^\leftarrow)\\ &={\rm sub}_X(F, f^\leftarrow(A))\vee {\rm sub}_X(F, f^\leftarrow(B)) &(F ~\text{is irreducible})\\ & = {\rm sub}_Y(f^\rightarrow(F), A)\vee{\rm sub}_Y(f^\rightarrow(F), B)\\ & = {\rm sub}_Y(\overline{f^\rightarrow(F)}, A)\vee{\rm sub}_Y(\overline{f^\rightarrow(F)},B),\end{align*} hence $\overline{f^\rightarrow(F)}$ is irreducible. \end{proof}
\begin{proof}[Proof of Theorem \ref{soberification}] \textbf{Existence}. For each $F\in s(X)$, since $\overline{f^\rightarrow(F)}$ is an irreducible closed set of $Y$ and $Y$ is sober, there is a unique $y\in Y$ such that $\overline{f^\rightarrow(F)}$ equals the closure of $1_y$. Define $f^*(F)$ to be this $y$. We claim that $f^*\colon s(X)\longrightarrow Y$ satisfies the conditions.
First, we show that $B\circ f^*$ is a closed set in $s(X)$ for any closed set $B$ in $Y$, hence $f^*$ is continuous. Let $A$ be the closed set $f^\leftarrow(B)=B\circ f$ in $X$. Then for any $F\in s(X)$, \begin{align*}s(A)(F) &= {\rm sub}_X(F, f^\leftarrow(B))= {\rm sub}_Y(f^\rightarrow(F), B) \\ &= {\rm sub}_Y(\overline{f^\rightarrow(F)}, B) = {\rm sub}_Y(\overline{1_{f^*(F)}}, B)\\ & = B(f^*(F))=B\circ f^*(F),\end{align*} thus, $B\circ f^*=s(A)$ and is closed in $s(X)$.
Second, for any $x\in X$, since \[1_{f(x)}\leq f^\rightarrow(\overline{1_x})\leq \overline{f^\rightarrow(1_x)} = \overline{1_{f(x)}},\] it follows that \[\overline{1_{f(x)}}=\overline{f^\rightarrow(\overline{1_x})}= \overline{f^\rightarrow(\eta_X(x))},\] hence $f^*(\eta_X(x))= f(x)$, showing that $f^*\circ\eta_X= f$.
Therefore, $f^*\colon s(X)\longrightarrow Y$ satisfies the conditions.
\textbf{Uniqueness}. Since $Y$ is sober, it suffices to show that if $g\colon s(X)\longrightarrow Y$ is a continuous map such that $f=g\circ\eta_X$, then $\overline{1_{g(F)}} = \overline{f^\rightarrow(F)}$ for all $F\in s(X)$.
Since for any $E\in s(X)$, \[\Omega(s(X))(E, F)= \overline{1_F}(E) ={\rm sub}_X(E,F), \] it follows that for any $x\in X$, \begin{align*}F(x) &= {\rm sub}_X(\overline{1_x}, F) \\ &= \Omega(s(X))(\eta_X(x), F)\\ &\leq \Omega(Y)(g(\eta_X(x)), g(F))\\ &=\Omega(Y)(f(x), g(F))\\ &= \overline{1_{g(F)}}(f(x)), \end{align*} showing that $f^\rightarrow(F)\leq \overline{1_{g(F)}}$, hence $\overline{f^\rightarrow(F)}\leq \overline{1_{g(F)}}$.
Conversely, since $\overline{\eta_X^\rightarrow(F)}$ is closed in $s(X)$, there is some closed set $A$ in $X$ such that $\overline{\eta_X^\rightarrow(F)}=s(A)$. For any $x\in X$, since \[F(x)\leq \eta_X^\rightarrow(F)(\eta_X(x))\leq s(A)(\eta_X(x))= A(x),\] it follows that $F\leq A$, then \[g^\rightarrow(s(A)) = g^\rightarrow(\overline{\eta_X^\rightarrow(F)}) \leq \overline{g^\rightarrow\circ\eta_X^\rightarrow(F)} = \overline{f^\rightarrow(F)}, \] hence \[\overline{f^\rightarrow(F)}(g(F))\geq g^\rightarrow(s(A))(g(F))\geq s(A)(F)=1.\] Therefore, $\overline{1_{g(F)}}\leq \overline{f^\rightarrow(F)}$. \end{proof}
Theorem \ref{soberification} shows that the full subcategory of sober $\mathcal{Q}$-cotopological spaces is reflective in {\sf S}$\mathcal{Q}$-{\sf CTop}. For any $\mathcal{Q}$-cotopological space $X$, the sober space $s(X)$ is called the \emph{sobrification} of $X$.
An important property of sober spaces is that the specialization order of a sober space is directed complete \cite{Gierz2003,Johnstone}. The following Proposition \ref{sober is dcpo} says this is also true in the quantale-valued setting if we treat irreducible fuzzy lower sets as ``directed fuzzy lower sets".
\begin{defn}\cite{Wagner97,LZ07} A supremum of a fuzzy lower set $\phi$ in a $\mathcal{Q}$-ordered set $X$ is an element $\sup\phi$ in $X$ such that \[X(\sup\phi, x)=\bigwedge_{z\in X}(\phi(z)\rightarrow X(z,x))={\rm sub}_X(\phi,X(-,x))\] for all $x\in X$. \end{defn}
The notion of supremum of a fuzzy lower set in a $\mathcal{Q}$-ordered set is a special case of that of \emph{weighted colimit} in category theory \cite{Kelly}.
\begin{prop}\label{sober is dcpo} Let $X$ be a sober $\mathcal{Q}$-cotopological space. Then each irreducible fuzzy lower set in the specialization $\mathcal{Q}$-order of $X$ has a supremum. \end{prop}
\begin{proof}Let $\phi$ be an irreducible fuzzy lower set in the $\mathcal{Q}$-ordered set $\Omega(X)$. First, we show that the closure $\overline{\phi}$ of $\phi$ in $X$ is an irreducible closed set. Let $A,B$ be closed sets in $X$. By definition of the specialization $\mathcal{Q}$-order, both $A$ and $B$ are fuzzy lower sets in $\Omega(X)$. Hence \[{\rm sub}_X(\overline{\phi},A\vee B) ={\rm sub}_X(\phi,A\vee B) = {\rm sub}_X(\phi, A)\vee {\rm sub}_X(\phi, B) = {\rm sub}_X(\overline{\phi}, A)\vee {\rm sub}_X(\overline{\phi}, B),\] showing that $\overline{\phi}$ is an irreducible closed set in $X$. Since $X$ is sober, there is a unique $a\in X$ such that $\overline{\phi}=\overline{1_a}$. We claim that $a$ is a supremum of $\phi$ in $\Omega(X)$. That is, for all $x\in X$, \[\Omega(X)(a,x) = \bigwedge_{z\in X}(\phi(z)\rightarrow\Omega(X)(z,x)). \] On one hand, for each $z\in X$, since $\phi(z)\leq \overline{\phi}(z) = \overline{1_a}(z) =\Omega(X)(z,a)$, it follows that \[\phi(z)\rightarrow\Omega(X)(z,x)\geq \Omega(X)(z,a)\rightarrow\Omega(X)(z,x)\geq \Omega(X)(a,x),\] hence \[\Omega(X)(a,x) \leq \bigwedge_{z\in X}(\phi(z)\rightarrow\Omega(X)(z,x)). \] On the other hand, \begin{align*} \bigwedge_{z\in X}(\phi(z)\rightarrow\Omega(X)(z,x))&= {\rm sub}_X(\phi, \overline{1_x}) &(\overline{1_x}(z) = \Omega(X)(z,x)) \\ & = {\rm sub}_X(\overline{\phi},\overline{1_x}) & (\text{$\overline{1_x}$ is closed})\\ &\leq \overline{\phi}(a)\rightarrow\overline{1_x}(a) \\ &= \Omega(X)(a,x). & (\overline{\phi}(a)=1)\end{align*} This completes the proof. \end{proof}
It is well-known that a Hausdorff topological space is always sober \cite{Gierz2003,Johnstone}. The following proposition says this is also true for $\mathcal{Q}$-cotopological spaces if $\mathcal{Q}$ is linearly ordered.
A $\mathcal{Q}$-cotopological space $X$ is Hausdorff if the diagonal $\Delta\colon X\times X\longrightarrow Q$, given by \[\Delta(x,y)=\begin{cases}1, & x=y,\\ 0, & x\not= y, \end{cases}\] is a closed set in the product space $X\times X$. \begin{prop}Let $\mathcal{Q}=(Q,\&)$ be a linearly ordered quantale. Then each stratified Hausdorff $\mathcal{Q}$-cotopological space is sober. \end{prop}
\begin{proof} Let $X$ be a stratified Hausdorff $\mathcal{Q}$-cotopological space. In order to see that $X$ is sober, it suffices to show that if $F$ is an irreducible closed set in $X$, then $F(x)\not=0$ for at most one point $x$ in $X$. Suppose on the contrary that there exist different $x,y$ in $X$ such that $F(x)>0$ and $F(y)>0$. Let $b=\min\{F(x),F(y)\}$. Then $b>0$ by linearity of $\mathcal{Q}$. Since $X$ is Hausdorff, there exist two families of closed sets in $X$, say, $\{A_j\}_{j\in J}$ and $\{B_j\}_{j\in J}$, such that \[\Delta(x,y) = \bigwedge_{j\in J} A_j(x)\vee B_j(y) .\] Since $\Delta(x,y)=0$, there exists some $i\in J$ such that $A_i(x)\vee B_i(y)< b$. Since $A_i(z)\vee B_i(z)=1$ for all $z\in X$, we have either $F\leq A_i$ or $F\leq B_i$, hence either $F(x)<b$ or $F(y)<b$, a contradiction. \end{proof}
\begin{note}\label{T2 is not sober} The assumption that $\mathcal{Q}$ is linearly ordered is indispensable in the above proposition. To see this, let $\mathcal{Q}=\{0,a,b,1\}$ be the Boolean algebra with four elements; let $X$ be the discrete $\mathcal{Q}$-cotopological space with two points $x$ and $y$. It is clear that $X$ is Hausdorff. One can verify by enumerating all possibilities that the map $\lambda$ given by $\lambda(x)=a$ and $\lambda(y)=b$, is an irreducible closed set in $X$, but it is neither the closure of $1_x$ nor that of $1_y$. \end{note}
An element $a$ in a lattice $L$ is a coprime if for all $b,c\in L$, $a\leq b\vee c$ implies that either $a\leq b$ or $a\leq c$ \cite{Johnstone}. A complete lattice $L$ is said to have enough coprimes if every element in $L$ can be written as the join of a family of coprimes. It is clear that every linearly ordered quantale has enough coprimes and the complete lattice of closed sets in a topological space has enough coprimes.
We say that an element in a quantale $\mathcal{Q}=(Q,\&)$ is a coprime if it is a coprime in the underlying lattice $Q$; and $\mathcal{Q}$ has enough coprimes if the complete lattice $Q$ has enough coprimes.
It is easily seen that if $1\in Q$ is a coprime and if $F$ is an irreducible closed set in a $\mathcal{Q}$-cotopological space $X$, then for any closed sets $A,B$ in $X$, $F\leq A\vee B$ implies either $F\leq A$ or $F\leq B$. Said differently, in this case, an irreducible closed set in a $\mathcal{Q}$-cotopological space is a coprime in the lattice of its closed sets.
Let $\mathcal{Q}$ be a quantale and $X$ be a (crisp) topological space. We say that a map $\lambda\colon X\longrightarrow Q$ is upper semicontinuous if for all $p\in Q$, \[\lambda_{[p]}=\{x\in X\mid \lambda(x)\geq p\}\] is a closed set in $X$.
\begin{lem}Let $\mathcal{Q}$ be a quantale with enough coprimes and $X$ be a topological space. \begin{enumerate}[(1)] \item $\lambda\colon X\longrightarrow Q$ is upper semicontinuous if and only if $\lambda_{[p]}$ is a closed set in $X$ for each coprime $p$ in $Q$. \item If both $\lambda, \mu\colon X\longrightarrow Q$ are upper semicontinuous then so is $\lambda\vee\mu$. \item The meet of any family of upper semicontinuous maps is upper semicontinuous. \item If $\lambda\colon X\longrightarrow Q$ is upper semicontinuous then so is $p\rightarrow\lambda$ for all $p\in Q$. \end{enumerate} \end{lem}
\begin{proof} (1) follows from the fact that $\mathcal{Q}$ has enough coprimes and (2) is an immediate consequence of (1). The verification of (3) is straightforward. And (4) follows from \[(p\rightarrow\lambda)_{[q]}=\{x\in X\mid q\leq p\rightarrow\lambda(x)\}= \{x\in X\mid p\&q\leq \lambda(x)\}= \lambda_{[p\&q]}\] for all $p,q\in Q$.
\end{proof}
The above lemma shows that if $\mathcal{Q}$ is a quantale with enough coprimes, then for each topological space $X$, the family of upper semicontinuous maps $X\longrightarrow Q$ forms a stratified $\mathcal{Q}$-cotopology on $X$. We write $\omega_\mathcal{Q}(X)$ for the resulting stratified $\mathcal{Q}$-cotopological space.
For each closed set $K$ in $X$, $1_K\colon X\longrightarrow Q$ is obviously upper semicontinuous, hence every closed set in $X$ is also a closed in $\omega_\mathcal{Q}(X)$. Moreover, for any $A\subseteq X$, the closure of $1_A$ in $\omega_\mathcal{Q}(X)$ equals $1_{\overline{A}}$, where $\overline{A}$ is the closure of $A$ in $X$.
The correspondence $X\mapsto \omega_\mathcal{Q}(X)$ defines an embedding functor \[\omega_\mathcal{Q}\colon {\sf Top}\longrightarrow{\sf S}\mathcal{Q}\text{-}{\sf CTop}.\] This functor is one of the well-known Lowen functors in fuzzy topology \cite{Lowen1976}.
The following conclusion says that for a linearly ordered quantale $\mathcal{Q}$, the notion of sobriety for $\mathcal{Q}$-cotopological spaces is a \emph{good extension} in the sense of Lowen \cite{Lowen1976}.
\begin{prop}\label{good extension} If $\mathcal{Q}$ is a linearly ordered quantale, then a topological space $X$ is sober if and only if the $\mathcal{Q}$-cotopological space $\omega_\mathcal{Q}(X)$ is sober. \end{prop}
\begin{proof}\textbf{Necessity}. Let $\lambda$ be an irreducible closed in $\omega_\mathcal{Q}(X)$. Firstly, we show that for each $x\in X$, the value $\lambda(x)$ is either $0$ or $1$. Suppose on the contrary that there is some $x\in X$ such that $\lambda(x)$ is neither $0$ nor $1$. We proceed with two cases. If there is no element $y$ in $X$ such that $\lambda(y)$ is strictly between $\lambda(x)$ and $1$, let $\phi=1_{\lambda_{[1]}}$ and $\psi$ be the constant map $X\longrightarrow Q$ with value $\lambda(x)$. Then both $\phi$ and $\psi$ are closed in $\omega_\mathcal{Q}(X)$ and $\lambda\leq\phi\vee\psi$, but neither $\lambda\leq\phi$ nor $\lambda\leq\psi$, contradictory to that $\lambda$ is irreducible. If there is some $y\in X$ such that $\lambda(x)<\lambda(y)<1$, let $\phi=1_{\lambda_{[\lambda(y)]}}$ and $\psi$ be the constant map $X\longrightarrow Q$ with value $\lambda(y)$. Then both $\phi$ and $\psi$ are closed in $\omega_\mathcal{Q}(X)$ and $\lambda\leq\phi\vee\psi$, but neither $\lambda\leq\phi$ nor $\lambda\leq\psi$, contradictory to that $\lambda$ is irreducible. Therefore, $\lambda=1_K$ for some closed set $K$ in $X$. Since $\lambda$ is irreducible in $\omega_\mathcal{Q}(X)$, $K$ must be an irreducible closed set in $X$. Thus, in the topological space $X$, $K$ is the closure of $\{x\}$ for a unique $x$, i.e., $K=\overline{\{x\}}$. This shows that in $\omega_\mathcal{Q}(X)$, $\lambda$ is the closure of $1_x$ for a unique $x$. Therefore, $\omega_\mathcal{Q}(X)$ is sober.
\textbf{Sufficiency}. We prove a bit more, that is, if $\mathcal{Q}$ has enough coprimes and $\omega_\mathcal{Q}(X)$ is sober, then $X$ is sober.
Let $K$ be an irreducible closed set in $X$. Firstly, we show that $1_K$ is an irreducible closed set in the $\mathcal{Q}$-cotopological space $\omega_\mathcal{Q}(X)$. That $1_K\colon X\longrightarrow Q$ is upper semicontinuous is trivial. For any closed sets $\lambda,\mu$ in $\omega_\mathcal{Q}(X)$, one has by definition that \[ {\rm sub}_X(1_K,\lambda\vee\mu)= \bigwedge_{x\in K}(\lambda(x)\vee\mu(x)).\] For any coprime $p\leq {\rm sub}_X(1_K,\lambda\vee\mu)$, $K$ is clearly a subset of $(\lambda\vee\mu)_{[p]} = \lambda_{[p]}\cup\mu_{[p]}$, hence either $K\subseteq\lambda_{[p]}$ or $K\subseteq\mu_{[p]}$, and then either $p\leq {\rm sub}_X(1_K, \lambda)$ or $p\leq {\rm sub}_X(1_K, \mu)$. Therefore, \[{\rm sub}_X(1_K,\lambda\vee\mu) \leq {\rm sub}_X(1_K, \lambda)\vee{\rm sub}_X(1_K, \mu).\] The converse inequality \[{\rm sub}_X(1_K,\lambda\vee\mu) \geq {\rm sub}_X(1_K, \lambda)\vee{\rm sub}_X(1_K, \mu)\] is trivial. Thus, $1_K$ is an irreducible closed set in $\omega_\mathcal{Q}(X)$.
Since $\omega_\mathcal{Q}(X)$ is sober, there is a unique $x\in X$ such that $1_K$ is the closure of $1_x$ in $\omega_\mathcal{Q}(X)$. Because the closure of $1_x$ in $\omega_\mathcal{Q}(X)$ equals $1_{\overline{\{x\}}}$, one gets $K=\overline{\{x\}}$.
\end{proof}
\begin{note}The assumption in Proposition \ref{good extension} that $\mathcal{Q}$ is linearly ordered is indispensable. To see this, let $\mathcal{Q}=\{0,a,b,1\}$ be the Boolean algebra with four elements; let $X$ be the discrete (hence sober) topological space with two points $x$ and $y$. It is clear that $\omega_\mathcal{Q}(X)$ is the discrete $\mathcal{Q}$-cotopological space in Note \ref{T2 is not sober}, hence it is not sober. \end{note}
At the end of this section, we discuss the relationship between sober $\mathcal{Q}$-cotopological spaces and sober $\mathcal{Q}$-topological spaces in the case that the quantale $\mathcal{Q}$ satisfies the law of double negation.
\begin{defn} \label{topology} A $\mathcal{Q}$-topology on a set $X$ is a subset $\tau$ of $Q^X$ subject to the following conditions: \begin{enumerate} \item[(O1)] $p_X\in\tau$ for all $p\in Q$; \item[(O2)] $U\wedge V\in\tau$ for all $U,V\in\tau$; \item[(O3)] $\bigvee_{j\in J}U_j\in\tau$ for each subfamily $\{U_j\}_{j\in J}$ of $\tau$.\end{enumerate}The pair $(X,\tau)$ is called a $\mathcal{Q}$-topological space; elements in $\tau$ are called open sets of $(X,\tau)$.\end{defn} A $\mathcal{Q}$-topological space in the above definition is also called a \emph{weakly stratified} $\mathcal{Q}$-topological space in the literature, see e.g. \cite{HS95,HS99}. A $\mathcal{Q}$-topology $\tau$ is stratified \cite{HS99} if \begin{enumerate} \item[(O4)] $p\&U \in\tau$ for all $p\in Q$ and $U\in \tau$.\end{enumerate}
It is clear that if $\mathcal{Q}=(Q,\&)$ is a frame, i.e., if $\&=\wedge$, then every $\mathcal{Q}$-topology is stratified.
Let $\mathcal{Q}$ be a quantale that satisfies the law of double negation. If $\tau$ is a (stratified) $\mathcal{Q}$-cotopology on a set $X$, then \[\neg(\tau)=\{\neg A \mid A\in\tau\} \] is a (stratified) $\mathcal{Q}$-topology on $X$, where $\neg A (x)=\neg(A(x))$ for all $x\in X$. Conversely, if $\tau$ is a (stratified) $\mathcal{Q}$-topology on $X$, then \[\neg(\tau)=\{\neg A \mid A\in\tau\} \] is a (stratified) $\mathcal{Q}$-cotopology on $X$. So, for a quantale $\mathcal{Q}$ that satisfies the law of double negation, we can switch freely between (stratified) $\mathcal{Q}$-topologies and (stratified) $\mathcal{Q}$-cotopologies, hence between open sets and closed sets.
If $\mathcal{Q}=(Q,\&)$ satisfies the law of double negation, then for any $A,B\in Q^X$, \begin{equation*}{\rm sub}_X(A, B) = {\rm sub}_X(\neg B, \neg A) \end{equation*} and \begin{equation*}\bigvee_{x\in X}A(x)\&B(x) =\neg{\rm sub}_X(A, \neg B) =\neg{\rm sub}_X(B,\neg A). \end{equation*} These equations are clearly extensions of the properties listed in Proposition \ref{properties of negation}.
\begin{prop}\label{FR}Let $\mathcal{Q}$ be a quantale that satisfies the law of double negation; and let $(X,\tau)$ be a stratified $\mathcal{Q}$-cotopological space. Then for each irreducible closed set $F$ in $(X,\tau)$, the map \[f_F\colon \neg(\tau)\longrightarrow Q, \quad f_F(U)=\bigvee_{x\in X}F(x)\&U(x) \] satisfies the following conditions: \begin{enumerate}[(Fr1)]\item $f_F(p_X)=p$. \item $f_F(U\wedge V)=f_F(U)\wedge f_F(V)$. \item $f_F\big(\bigvee_{i\in I}U_i\big) = \bigvee_{i\in I}f_F(U_i)$. \item $f_F(p\&U) = p\& f_F(U)$. \end{enumerate} Conversely, if $g\colon \neg(\tau)\longrightarrow Q$ is a map satisfying {\rm (Fr1)--(Fr4)}, then there is a unique irreducible closed set $F$ in $(X,\tau)$ such that $g=f_F$. \end{prop}
\begin{proof} We check (Fr2) for example. \begin{align*}f_F(U\wedge V)&=\neg ({\rm sub}_X(U\wedge V,\neg F))\\ &=\neg ({\rm sub}_X(F,\neg (U\wedge V)))\\ &=\neg ({\rm sub}_X(F,\neg U\vee\neg V)) \\ &=\neg({\rm sub}_X(F,\neg U)\vee{\rm sub}_X(F,\neg V))\\ &=\neg({\rm sub}_X(U,\neg F))\wedge\neg({\rm sub}_X(V,\neg F))\\ & =f_F(U)\wedge f_F(V).\end{align*}
Conversely, suppose $g\colon \neg(\tau)\longrightarrow Q$ is a map that satisfies (Fr1)--(Fr4). Let \[F=\bigwedge\{A\in\tau\mid g(\neg A)=0\}.\] We show that $F$ is an irreducible closed set in $(X,\tau)$ and $g=f_F$.
\textbf{Step 1}. $g(\neg F)=0$. This follows from (Fr3) and Proposition \ref{properties of negation}(3).
\textbf{Step 2}. $g(U) =\bigvee_{x\in X}F(x)\&U(x)$ for all $U\in\neg(\tau)$. On one hand, if we let $p={\rm sub}_X(U,\neg F)$, then $p\&U\leq\neg F$, hence \[p\&g(U)=g(p\&U)\leq g(\neg F)=0.\] Therefore, \[g(U)\leq\neg {\rm sub}_X(U,\neg F)=\bigvee_{x\in X}F(x)\&U(x).\] On the other hand, since $g(\neg(g(U))\&U)= \neg(g(U))\&g(U)=0$, it follows that $\neg(g(U))\&U\leq \neg F$ by definition of $F$. Therefore, $\neg(g(U))\leq {\rm sub}_X(U,\neg F)$, hence \[g(U)\geq\neg {\rm sub}_X(U,\neg F)=\bigvee_{x\in X}F(x)\&U(x).\]
\textbf{Step 3}. $\bigvee_{x\in X}F(x)=1$. Otherwise, let $\bigwedge_{x\in X}\neg F(x)=p$. Then $p\not=0$ and $p_X\leq\neg F$. Therefore, $g(\neg F)\geq g(p_X)=p$, contradictory to that $g(\neg F)=0$.
\textbf{Step 4}. ${\rm sub}_X(F,A\vee B)={\rm sub}_X(F,A)\vee{\rm sub}_X(F,B)$ for all closed sets $A,B$ in $(X,\tau)$. In fact, \begin{align*}{\rm sub}_X(F,A\vee B) &= {\rm sub}_X(\neg A\wedge\neg B, \neg F)\\ & =\neg(g(\neg A\wedge\neg B)) \\ & =\neg(g(\neg A))\vee \neg(g(\neg B))\\ &={\rm sub}_X(\neg A, \neg F)\vee{\rm sub}_X(\neg B, \neg F)\\ &= {\rm sub}_X(F,A)\vee{\rm sub}_X(F,B). \end{align*}
The proof is completed. \end{proof}
For any stratified $\mathcal{Q}$-topological space $(X,\tau)$ and $x\in X$, the map \[f_x\colon \tau\longrightarrow Q, \quad f_x(U)=U(x)\] clearly satisfies (FR1)--(FR4) in Proposition \ref{FR}. This fact leads to the following:
\begin{defn}A stratified $\mathcal{Q}$-topological space $(X,\tau)$ is sober if for each map $f\colon \tau\longrightarrow Q$ satisfying (FR1)--(FR4) in Proposition \ref{FR}, there is a unique $x\in X$ such that $f(U)=U(x)$ for all $U\in \tau$. \end{defn}
We leave it to the reader to check that if $\mathcal{Q}=(Q,\&)$ is a frame, i.e., $\&=\wedge$, then the above definition of sober $\mathcal{Q}$-topological spaces coincides with that in \cite{ZL95}.
\begin{prop} Let $\mathcal{Q}$ be a quantale that satisfies the law of double negation. Then a $\mathcal{Q}$-topological space $(X,\tau)$ is sober if and only if the $\mathcal{Q}$-cotopological space $(X,\neg(\tau))$ is sober. \end{prop}
\begin{proof} \textbf{Necessity}: Let $F$ be an irreducible closed set in the $\mathcal{Q}$-cotopological space $(X,\neg(\tau))$. By Proposition \ref{FR}, the map \[f_F\colon \tau \longrightarrow Q, \quad f_F(U)=\bigvee_{x\in X}F(x)\&U(x) \] satisfies (Fr1)--(Fr4), hence there is a unique $a\in X$ such that $f_F(U)=U(a)$ for all $U\in\tau$. We claim that the closure of $1_a$ in $(X,\neg(\tau))$ is $F$. Since \[\neg F(a)= f_F(\neg F)= \bigvee_{x\in X} F(x) \&(\neg F(x)) =0,\] then $F(a)=1$. Therefore, $\overline{1_a}\leq F$. Conversely, since \[\bigvee_{x\in X}F(x)\&\neg(\overline{1_a})(x)= f_F(\neg(\overline{1_a})) =\neg(\overline{1_a})(a)=0,\] it follows that \[F(x)\leq \neg(\neg(\overline{1_a})(x))= \overline{1_a}(x)\] for all $x\in X$, hence $F\leq \overline{1_a}$.
\textbf{Sufficiency}. Let $f\colon \tau\longrightarrow Q$ be a map satisfying (FR1)--(FR4). By Proposition \ref{FR}, there is an irreducible closed set $F$ in the $\mathcal{Q}$-cotopological space $(X,\neg(\tau))$ such that $f(U) = \bigvee_{x\in X}F(x)\&U(x)$ for all $U\in\tau$. Since $ (X,\neg(\tau))$ is sober, there is a unique $a\in X$ such that $\overline{1_a}=F$, Therefore, \begin{align*}f(U)&=\bigvee_{x\in X}F(x)\&U(x) \\ &= \neg{\rm sub}_X(F,\neg U) \\ &= \neg{\rm sub}_X(1_a, \neg U)&(\overline{1_a}=F, ~\neg U~\text{is closed})\\ &= U(a),\end{align*}
completing the proof. \end{proof}
\section{Examples} This section discusses the sobriety of some natural $\mathcal{Q}$-cotopological spaces in the case that $\mathcal{Q}$ is the unit interval $[0,1]$ endowed with a (left) continuous t-norm. In this section, we will write $\underline{a}$, instead of $a_{[0,1]}$, for the constant map $[0,1]\longrightarrow[0,1]$ with value $a$.
Let $\mathcal{Q}=([0,1],\&)$ with $\&$ being a (left) continuous t-norm on $[0,1]$. We consider three $\mathcal{Q}$-cotopologies on $[0,1]$: \begin{itemize}\setlength{\itemsep}{-2pt} \item $\tau_{C\&}$: the stratified $\mathcal{Q}$-cotopology on $[0,1]$ generated by $\{{\rm id}\}$ as a subbasis; \item $\tau_{S\&}$: the strong $\mathcal{Q}$-cotopology on $[0,1]$ generated by $\{{\rm id}\}$ as a subbasis; \item $\tau_{A\&}$: the Alexandroff $\mathcal{Q}$-cotopology on the $\mathcal{Q}$-ordered set $([0,1], d_R)$, where $d_R(x,y)=y\rightarrow x$. \end{itemize}
First of all, we list some facts about these $\mathcal{Q}$-cotopologies.
\begin{enumerate}[(F1)] \item A closed set in $([0,1],\tau_{A\&})$ is, by definition, a $\mathcal{Q}$-order-preserving map $\phi\colon ([0,1], d_L)\longrightarrow ([0,1], d_L)$, where $d_L(x,y)=x\rightarrow y$. So, it is easy to verify that for all $\phi\in\tau_{A\&}$: \begin{itemize} \setlength{\itemsep}{0pt} \item $\phi$ is increasing, i.e., $\phi(x)\leq\phi(y)$ whenever $x\leq y$; \item $\phi(1)=1\iff \phi\geq{\rm id}$. \end{itemize}
\item For each $x\in[0,1]$, the closure of $1_x$ in $([0,1],\tau_{A\&})$ is $x\rightarrow{\rm id}$, i.e., \[\overline{1_x}= x\rightarrow{\rm id}.\] On one hand, $x\rightarrow{\rm id}$ is a closed set in $([0,1],\tau_{A\&})$ and $(x\rightarrow{\rm id})(x)=1$, hence $\overline{1_x}\leq x\rightarrow{\rm id}$. On the other hand, for any $\phi\in\tau_{A\&}$, if $\phi(x)=1$, then $\phi(t)=\phi(x)\rightarrow\phi(t)\geq x\rightarrow t$ for all $t\leq x$, hence $\phi\geq x\rightarrow{\rm id}$.
\item Since every Alexandroff $\mathcal{Q}$-cotopology is a strong $\mathcal{Q}$-cotopology and ${\rm id}\in \tau_{A\&}$, it follows that
\[\tau_{C\&}\subseteq \tau_{S\&}\subseteq \tau_{A\&}.\] Moreover, since $x\rightarrow{\rm id}\in\tau_{C\&}$ for all $x\in X$, the closure of $1_x$ in both $([0,1],\tau_{C\&})$ and $([0,1],\tau_{S\&})$ is $x\rightarrow{\rm id}$.
\item Since finite joins and arbitrary meets of right continuous maps $[0,1]\longrightarrow[0,1]$ are right continuous, and $x\rightarrow{\rm id}$ is right continuous for all $x\in[0,1]$, it follows that every closed set in $([0,1],\tau_{C\&})$, as a map from $[0,1]$ to itself, is right continuous. \item The space $([0,1],\tau_{C\&})$ is initially dense in the category of stratified $\mathcal{Q}$-cotopological spaces. Indeed, for each stratified $\mathcal{Q}$-cotopological space $(X,\tau)$, \[ \{(X,\tau)\stackrel{A}{\longrightarrow}([0,1],\tau_{C\&})\}_{A\in\tau}\] is an initial source in the topological category {\sf S}$\mathcal{Q}$-{\sf CTop}. \end{enumerate}
\begin{prop}\label{Sierpinski is sober} Let $\mathcal{Q}=([0,1],\&)$ with $\&$ being a left continuous t-norm on $[0,1]$. Then the stratified $\mathcal{Q}$-cotopological space $([0,1],\tau_{C\&})$ is sober. \end{prop}
\begin{proof} It suffices to show that if $\phi$ is an irreducible closed set in $([0,1], \tau_{C\&})$, then $\phi= x\rightarrow{\rm id}=\overline{1_x}$ for some $x\in [0,1]$. Since $\phi$ is increasing and $\bigvee_{t\in[0,1]}\phi(t)=1$, one obtains that $\phi(1)=1$. Let \[x=\inf\{t\in[0,1]\mid \phi(t)=1\}.\] Since $\phi$ is right continuous by (F4), then $\phi(x)=1$, hence $\phi\geq\overline{1_x}=x\rightarrow{\rm id}$. We claim that $\phi=x\rightarrow{\rm id}$. Otherwise, there is some $t<x$ such that $\phi(t)>x\rightarrow t$. Since $x\rightarrow t=\bigwedge_{y<x}(y\rightarrow t)$, there is some $y\in(t,x)$ such that $\phi(t)>y\rightarrow t$. It is clear that both $ y\rightarrow{\rm id}$ and $\underline{\phi(y)}$ belong to $\tau_{C\&}$ and $\phi\leq (y\rightarrow{\rm id})\vee\underline{\phi(y)}$, but neither $\phi\leq (y\rightarrow{\rm id}) $ nor $\phi\leq \underline{\phi(y)}$, contradictory to the assumption that $\phi$ is irreducible. \end{proof}
In the following we consider sobriety of the $\mathcal{Q}$-cotopologies $\tau_{S\&}$ and $\tau_{A\&}$ in the case that $\&$ is the minimum t-norm, the product t-norm and the {\L}ukasiewicz t-norm.
\begin{prop} \label{alexandroff for product t-norm} Let $\mathcal{Q}=([0,1],\&)$ with $\&$ being the product t-norm. Then the Alexandroff $\mathcal{Q}$-cotopology $\tau_{AP}$ on $([0,1], d_R)$ is not sober; but the strong $\mathcal{Q}$-cotopology $\tau_{SP}$ on $[0,1]$ generated by $\{{\rm id}\}$ is sober. \end{prop} \begin{proof} By Example 3.11 in \cite{LZ06}, the Alexandroff $\mathcal{Q}$-cotopology $\tau_{AP}$ consists of maps $\phi\colon [0,1]\longrightarrow[0,1]$ subject to the following conditions: \begin{enumerate}\item[(i)] $\phi$ is increasing; and \item[(ii)] $y/x \leq \phi(y)/{\phi(x)}$ whenever $x> y$, where we agree by convention that $0/0=1$.\end{enumerate}
We note that each $\phi$ in $\tau_{AP}$ is continuous on $(0,1]$, but it may be discontinuous at $0$.
It is easy to verify that the map $\phi\colon [0,1]\longrightarrow[0,1]$, given by $\phi(0)=0$ and $\phi(t)=1$ for all $t>0$, is an irreducible closed set in $([0,1],\tau_{AP})$, but it is not the closure of $1_x$ for any $x\in[0,1]$. So, $([0,1],\tau_{AP})$ is not sober.
In the following we prove in three steps that $\tau_{SP}$ on $[0,1]$ is sober.
\textbf{Step 1}. We show that the stratified $\mathcal{Q}$-cotopology $\tau_{CP}$ on $[0,1]$ generated by $\{{\rm id}\}$ is given by \[\tau_{CP}= \{\phi\wedge \underline{a} \mid a\in[0,1], \phi\in{\cal B}\}, \] where, \[{\cal B}=\{\phi\in\tau_{AP} \mid \phi\geq{\rm id}, \text{$\phi$ is continuous}\}.\]
It is routine to verify that $ \mathcal{C}=\{\phi\wedge \underline{a} \mid a\in[0,1], \phi\in{\cal B}\}$ is a stratified $\mathcal{Q}$-cotopology on $[0,1]$ that contains the identity ${\rm id}\colon [0,1]\longrightarrow[0,1]$, hence $\tau_{CP}\subseteq \mathcal{C}$. To see that $\mathcal{C}$ is contained in $\tau_{CP}$, it suffices to check that ${\cal B}\subseteq\tau_{CP}$. Let $\phi\in{\cal B}$. For each $x\in(0,1]$, define $g_x\colon [0,1]\longrightarrow[0,1]$ by \[g_x=\underline{\phi(x)}\vee((\phi(x)\rightarrow x)\rightarrow t)=\underline{\phi(x)}\vee\Big(\frac{x}{\phi(x)}\rightarrow {\rm id}\Big).\] Then $g_x\in \tau_{CP}$. We leave it to the reader to check that \[\phi= \bigwedge_{x\in(0,1]}g_x= \bigwedge_{x\in(0,1]}\Big(\underline{\phi(x)}\vee \Big(\frac{x}{\phi(x)}\rightarrow {\rm id}\Big)\Big),\] hence $\phi\in\tau_{CP}$. Therefore, $\mathcal{C}\subseteq \tau_{CP}$.
\textbf{Step 2}. We show that \[\tau_{SP}= \{\phi\in\tau_{AP} \mid \text{$\phi$ is continuous}\}.\] It is easily verified that $\mathcal{S}=\{\phi\in\tau_{AP} \mid \text{$\phi$ is continuous}\}$ is a strong $\mathcal{Q}$-cotopology on $[0,1]$ that contains the identity ${\rm id}\colon [0,1]\longrightarrow[0,1]$, hence $\tau_{SP}\subseteq \mathcal{S}$. Conversely, for any $\phi\in\mathcal{S}$ with $\phi(1)>0$, let $\psi=\phi(1)\rightarrow\phi = \phi/\phi(1)$. Then $\psi\in\tau_{AP}$, $\psi(1)=1$, and $\psi$ is continuous. Thus, $\psi\in{\cal B}\subseteq \tau_{SP}$. Since $\tau_{SP}$ is strong and $\phi=\phi(1)\&\psi$, then $\phi\in\tau_{SP}$, therefore $\mathcal{S}\subseteq \tau_{SP}$.
\textbf{Step 3}. $([0,1], \tau_{SP})$ is sober. Suppose $\phi$ is an irreducible closed set in $([0,1], \tau_{SP})$. Since $\phi$ is increasing and $\bigvee_{t\in[0,1]}\phi(t)=1$, one has $\phi(1)=1$, hence $\phi\in{\cal B}\subset\tau_{CP}$. Since $\tau_{CP}$ is coarser than $\tau_{SP}$, then $\phi$ is an irreducible closed set in the sober space $([0,1], \tau_{CP})$, and consequently, $\phi=x\rightarrow {\rm id}$ for a unique $x\in [0,1]$. This shows that $\phi$ is the closure of $1_x$ for a unique $x\in[0,1]$ in $([0,1], \tau_{SP})$, hence $([0,1], \tau_{SP})$ is sober.
\end{proof}
\begin{prop}\label{alexandroff for Luka t-norm} Let $\mathcal{Q}=([0,1],\&)$ with $\&$ being the {\L}ukasiewicz t-norm. Then the strong $\mathcal{Q}$-cotopology $\tau_{SL}$ on $[0,1]$ generated by $\{{\rm id}\}$ is sober and coincides with the Alexandroff $\mathcal{Q}$-cotopology $\tau_{AL}$ on the $\mathcal{Q}$-ordered set $([0,1], d_R)$. \end{prop} \begin{proof} By Example 3.10 in \cite{LZ06}, the Alexandroff $\mathcal{Q}$-cotopology $\tau_{AL}$ on $([0,1], d_R)$ consists of maps $\phi\colon [0,1]\longrightarrow[0,1]$ that satisfy the following conditions: \begin{enumerate}\item[(i)] $\phi$ is increasing; and \item[(ii)] $\phi$ is 1-Lipschitz, i.e., $\phi(x)-\phi(y)\leq x-y$ for all $x\geq y$. \end{enumerate}
Firstly, we show that the stratified $\mathcal{Q}$-cotopology $\tau_{CL}$ on $[0,1]$ generated by $\{{\rm id}\}$ is given by \[\tau_{CL} =\{\phi\wedge \underline{a} \mid a\in[0,1], \phi\in\tau_{AL}, \phi\geq{\rm id}\}. \] It is routine to verify that $\mathcal{C} =\{\phi\wedge \underline{a} \mid a\in[0,1], \phi\in\tau_{AL}, \phi\geq{\rm id}\}$ is a stratified $\mathcal{Q}$-cotopology on $[0,1]$ that contains the identity ${\rm id}\colon [0,1]\longrightarrow[0,1]$, hence $\tau_{CL}\subseteq \mathcal{C}$. To see that $\mathcal{C}$ is contained in $\tau_{CL}$, it suffices to check that for any $\phi\in\tau_{AL}$, if $\phi\geq{\rm id}$ then $\phi\in\tau_{CL}$. For each $x\in[0,1]$, define $g_x\colon [0,1]\longrightarrow[0,1]$ by \[g_x=\underline{\phi(x)}\vee((\phi(x)\rightarrow x)\rightarrow {\rm id}),\] i.e., \[g_x(t)= \max\{\phi(x),\min\{\phi(x)+ t-x, 1\}\}.\] Then $g_x\in \tau_{CL}$. We leave it to the reader to check that \[\phi= \bigwedge_{x\in[0,1]}g_x= \bigwedge_{x\in[0,1]} (\underline{\phi(x)}\vee((\phi(x)\rightarrow x)\rightarrow {\rm id})),\] hence $\phi\in\tau_{CL}$. Therefore, $\mathcal{C}\subseteq \tau_{CL}$.
Secondly, we show that $\tau_{SL}= \tau_{AL}$. For any $\phi\in\tau_{AL}$ with $\phi(1)>0$, it is clear that $\phi(1)\rightarrow\phi \in\tau_{AL}$ and $(\phi(1)\rightarrow\phi)(1)=1$. Thus, $\phi(1)\rightarrow\phi\in\tau_{CL}\subseteq \tau_{SL}$. Since $\tau_{SL}$ is strong and $\phi=\phi(1)\&(\phi(1)\rightarrow\phi)$, it follows that $\phi\in\tau_{SL}$, hence $\tau_{AL}\subseteq \tau_{SL}$. The converse inclusion $\tau_{SL}\subseteq \tau_{AL}$ is trivial.
Finally, we show that $([0,1], \tau_{SL})$ is sober. Let $\phi$ be an irreducible closed set in $([0,1], \tau_{SL})$. Since $\phi$ is increasing and $\bigvee_{t\in[0,1]}\phi(t)=1$, then $\phi(1)=1$, hence $\phi\in \tau_{CL}$. Since $\tau_{CL}$ is coarser than $\tau_{SL}$, it follows that $\phi$ is an irreducible closed set in the sober space $([0,1], \tau_{CL})$, hence $\phi=x\rightarrow {\rm id}$ for a unique $x\in [0,1]$. This shows that $\phi$ is the closure of $1_x$ for a unique $x\in[0,1]$ in $([0,1], \tau_{SL})$. Therefore, $([0,1], \tau_{SL})$ is sober. \end{proof}
\begin{prop}Let $\mathcal{Q}=([0,1],\&)$ with $\&$ being the t-norm $\min$. Then the strong $\mathcal{Q}$-cotopology $\tau_{SM}$ on $[0,1]$ generated by $\{{\rm id}\}$ is sober; but the Alexandroff $\mathcal{Q}$-cotopology $\tau_{AM}$ on the $\mathcal{Q}$-ordered set $([0,1], d_R)$ is not sober.
\end{prop}
\begin{proof} First of all, since $\&=\min$, every stratified $\mathcal{Q}$-cotopological space is obviously a strong $\mathcal{Q}$-cotopological space, then the strong $\mathcal{Q}$-cotopology $\tau_{SM}$ on $[0,1]$ generated by $\{{\rm id}\}$ coincides with the stratified $\mathcal{Q}$-cotopology $\tau_{CM}$ on $[0,1]$ generated by $\{{\rm id}\}$, hence it is sober by Proposition \ref{Sierpinski is sober}.
With the help of Example 3.12 in \cite{LZ06}, it can be verified that \[\tau_{AM}=\{\phi\wedge \underline{a}\mid a\in[0,1], \text{$\phi\colon [0,1]\longrightarrow[0,1]$ is increasing}, \phi\geq{\rm id} \}.\] For any $a\in(0,1)$, the map $\phi\colon [0,1]\longrightarrow[0,1]$ given by \[\phi(t)=\begin{cases}1,& t>a,\\ t, & t\leq a, \end{cases}\] is an irreducible closed set in $([0,1],\tau_{AM})$, and it is not the closure of $1_x$ for any $x\in[0,1]$, so, $([0,1],\tau_{AM})$ is not sober.
\end{proof} We would like to record here that \[\tau_{CM}=\tau_{SM}= \{\phi\in\tau_{AM} \mid \text{$\phi$ is right continuous}\}.\]
It is easily verified that $\mathcal{C}= \{\phi\in\tau_{AM} \mid \text{$\phi$ is right continuous}\}$ is a stratified $\mathcal{Q}$-cotopology on $[0,1]$ that contains the identity ${\rm id}\colon [0,1]\longrightarrow[0,1]$, hence $\tau_{CM}\subseteq \mathcal{C}$. To see the converse inclusion, we show firstly that for all $\phi\in \mathcal{C}$, if $\phi(1)=1$ then $\phi\in\tau_{CM}$. For each $x\in[0,1]$, define $g_x\colon [0,1]\longrightarrow[0,1]$ by $g_x=\underline{\phi(x)}\vee(x\rightarrow {\rm id})$.
Then $g_x\in \tau_{CM}$. Since \[\phi= \bigwedge_{x\in[0,1]}g_x= \bigwedge_{x\in[0,1]}(\underline{\phi(x)}\vee(x\rightarrow {\rm id})),\] then $\phi\in\tau_{CM}$. Secondly, for any $\phi\in \mathcal{C}$, since $\phi(1)\rightarrow\phi\in \mathcal{C}$, $(\phi(1)\rightarrow\phi)(1)=1$, and $\phi=\underline{\phi(1)}\wedge(\phi(1)\rightarrow\phi)$, it follows that $\phi\in \tau_{CM}$. Therefore, $\mathcal{C}\subseteq \tau_{CM}$.
\vskip 6pt
{\bf Acknowledgement} The author thanks Dr. Hongliang Lai for the stimulating discussions during the preparation of this paper.
\end{document} | arXiv | {
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\begin{document}
\begin{abstract} In this paper, we prove the semi-continuity theorem of Diederich-Forn{\ae}ss index and Steinness index under a smooth deformation of pseudoconvex domains in Stein manifolds. \end{abstract}
\maketitle
\section{\bf Introduction}
Let $\Omega$ be a relatively compact, Levi pseudoconvex domain in a complex manifold $X$ with $C^{\infty}$-smooth boundary. Let $\rho : X \rightarrow \mathbb{R}$ be a smooth defining function of $\Omega$, i.e., $\Omega = \{z \in X : \rho(z) < 0 \}$ and $d\rho \neq 0$ on $\partial \Omega$. The Diederich-Forn{\ae}ss index $DF(\Omega)$ and the Steinness index $S(\Omega)$ of $\Omega$ are defined by \begin{align*} DF(\Omega) &:= \sup_{\rho} \left\{ 0< \gamma < 1 : -(-\rho)^{\gamma} \text{ is strictly plurisubharmonic on } \Omega \cap W \right\}, \\ S(\Omega) &:= \inf_{\rho} \left\{ \gamma > 1 : \rho^{\gamma} \text{ is strictly plurisubharmonic on } \overline{\Omega}^{\complement} \cap W \right\}, \end{align*} where the supremum and infimum are taken over all smooth defining function $\rho$, and $W$ is some neighborhood of $\partial\Omega$ that may depend on $\rho$ and $\gamma$. If such $\rho$ and $\gamma$ do not exist, we define $DF(\Omega)=0$ and $S(\Omega)=\infty$. When the ambient space $X$ is a Stein manifold, $DF(\Omega) > 0$ implies the existence of a bounded strictly plurisubharmonic function on $\Omega$, i.e., $\Omega$ is hyperconvex, and $S(\Omega) < \infty$ implies that $\Omega$ admits a Stein neighborhood basis. In 1977, Diederich and Forn{\ae}ss (\cite{diederich-fornaess}) proved the positivity of $DF(\Omega)$ if the boundary is of $C^2$-smoothness. The second named author (\cite{Yum1}) provided a necessary and sufficient condition for $S(\Omega) < \infty$.
In this paper, we shall study the semi-continuity of the both indices of a relatively compact pseudoconvex domain with smooth boundary under a smooth deformation. More precisely, a smooth deformation of a relatively compact domain $\Omega_0$ in a complex manifold $X_0$ is given as follows:
A surjective holomorphic submersion $\pi : (X,\Omega) \rightarrow\D$ from a complex manifold $X$ with a relatively compact domain $\Omega$ in $X$ to the unit disc $\D$ is called a \emph{smooth deformation of $\Omega_0$ in $X_0$ over $\D$} if
\begin{itemize} \item $X_0 = \pi^{-1}(0)$ and $\Omega_0 = \Omega \cap X_0$. \item $\Omega$ admits a defining function $\delta$ such that $\delta\vert_{X_t}$ is a defining function of $\Omega_t:=\Omega \cap \pi^{-1}(t)$ in $X_t:=\pi^{-1}(t)$ for $t\in\D$. \end{itemize} Note that the above conditions guarantee that every fiber $\Omega_t$ is diffeomorphic to $\Omega_0$. (cf, see \cite{Saeki}.)
The main theorem of this paper is as follows:
\begin{thm} \label{thm:main}
If every fiber $X_t$ is a Stein manifold, then
\[
\liminf_{t \rightarrow 0} DF(\Omega_t) \ge DF(\Omega_0) \quad \text{ and } \quad
\limsup_{t \rightarrow 0} S(\Omega_t) \le S(\Omega_0).
\] In other words, $DF(\Omega_t)$ is lower semi-continuous and $S(\Omega_t)$ is upper semi-continuous at $t=0$, respectively. \end{thm}
We would like to emphasize that the Steinness of each fiber $X_t$ is a superfluous condition. In fact, the condition in Theorem \ref{thm:DF,S formulas}, which is our main tool, is sufficient. That is, if each $\Omega_t$ admits a defining function $\rho_t$ such that either $\overline {\partial} \omega_{\rho_t} > 0$ on $\mathcal{N}$ or $\overline {\partial} \omega_{\rho_t} < 0$ on $\mathcal{N}$, then we have the same conclusion. Here, $\omega_{\rho_t}$ is a D'Angelo $(1,0)$-form and $\mathcal{N}$ is the kernel of the Levi form (see Section \ref{sec:Preliminaries}). We refer readers to Corollary 5.4 and Remark 5.6 in \cite{Adachi-Yum} for the details.
\subsection*{\bf Acknowledgment} The authors would like to thank A. Seo for suggesting the problem and M. Adachi for a useful comment on Lemma \ref{lem:null-space}. This work was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea government (No. 2018R1C1B3005963).
\section{\bf Preliminaries} \label{sec:Preliminaries}
In this section, we introduce the D'Angelo $1$-form and the characterizations of Diederich-Forn{\ae}ss and Steinness indices by the D'Angelo $1$-form due to Adachi and the second named author (\cite{Adachi-Yum}).
First, we recall the definition of D'Angelo 1-form, which was introduced by D'Angelo (\cite{D'Angelo0}, \cite{D'Angelo}), and developed by Boas and Straube (\cite{Boas-Straube}). Let $\Omega$ be a relatively compact, Levi pseudoconvex domain in a complex manifold $X$ with $C^{\infty}$-smooth boundary. Let $\rho$ be a smooth defining function of $\Omega$. Denote the kernel of the Levi form by $\mathcal{N} = \bigcup_{p \in M} \mathcal{N}_p \subset T^{1,0}(\partial \Omega)$ where \[ \mathcal{N}_p := \{ L_p \in T^{1,0}_p(\partial \Omega) \mid \mathscr{L}_{\rho}(L_p, L'_p) = 0 \quad \forall L_p' \in T^{1,0}_p(\partial \Omega) \} \] for each $p \in \partial \Omega$, where $\mathscr{L}_{\rho}$ is the Levi-form of $\rho$. Note that the kernel $\mathcal{N}$ does not depend on a defining function $\rho$, and $\mathcal{N}_p = \{ L_p \in T^{1,0}_p(\partial \Omega) \mid \mathscr{L}_{\rho}(L_p, L_p) = 0 \}$ when $\Omega$ is a Levi pseudoconvex domain. Define \[ \eta_{\rho} := \frac{1}{2}\left( \partial \rho - \overline{\partial} \rho \right), \] which is a purely imaginary, non-vanishing 1-form on $\partial \Omega$ that annihilates $T^{1,0}(\partial \Omega) \oplus T^{0,1}(\partial \Omega)$. Let $T_{\rho} \in \Gamma(\mathbb{C} \otimes T(\partial\Omega))$ be a purely imaginary, non-vanishing, smooth vector field on $\partial \Omega$ such that $\eta_{\rho}(T_{\rho}) = 1$. Then $T_{\rho}$ yields a decomposition \[ \mathbb{C} \otimes T(\partial\Omega) = T^{1,0}(\partial\Omega) \oplus T^{0,1}(\partial\Omega) \oplus \mathbb{C} T_{\rho}. \] We call such $T_{\rho}$ a \emph{transversal vector field} normalized with respect to $\eta_{\rho}$. Denote $\eta_{\rho}$ and $T_{\rho}$ by $\eta$ and $T$, respectively, if there is no ambiguity.
\begin{defn} \label{def D'Angelo 1-form}
A {\it D'Angelo 1-form} $\alpha_{\rho}$ of $\rho$ on $\partial \Omega$ is defined by
$$ \alpha_{\rho} := - \mathcal{L}_{T_{\rho}} \eta_{\rho} ,$$
where $\mathcal{L}_{T_{\rho}}$ is the Lie derivative in the direction of $T_\rho$.
A {\it D'Angelo $(1,0)$-form} $\omega_{\rho} := \pi_{1,0} \alpha_{\rho}$ is the projection of $\alpha_{\rho}$ onto its $(1,0)$-component. \end{defn}
\begin{rmk}
In \cite{Adachi-Yum}, they defined the D'Angelo 1-form on a compact abstract CR manifold $M$ of hypersurface type without using a defining function. However, this definition is equivalent to Definition \ref{def D'Angelo 1-form} when $M$ bounds a relatively compact domain in a complex manifold. In this paper, we use Definition \ref{def D'Angelo 1-form} because we deal with only domains. \end{rmk}
Note that although, for a defining function $\rho$, a transversal vector field $T_{\rho}$ normalized with respect to $\eta_{\rho}$ is not unique, $\omega_{\rho}$ and $\overline {\partial} \omega_{\rho}$ are well-defined on $\mathcal{N}$, that is, they are independent of the choice of $T_{\rho}$ (see Lemma 2.5 and 2.6 in \cite{Adachi-Yum}). We regard $\overline {\partial} \omega_{\rho}$ and $\omega_{\rho} \wedge \overline {\omega}_{\rho}$ as quadratic forms on $\mathcal{N}$, i.e., $\overline {\partial} \omega_{\rho} > 0$ on $\mathcal{N}$ means $\overline {\partial} \omega_{\rho}(L, \overline {L}) > 0$ for all $L \in \mathcal{N}$. The main tool we will use in Section \ref{sec:Semi-continuity of two indices} is the following.
\begin{thm}[Adachi, Yum \cite{Adachi-Yum}] \label{thm:DF,S formulas}
Suppose that there exists a defining function $\rho_1$ of $\Omega$ such that $\overline {\partial} \omega_{\rho_1} > 0$ on $\mathcal{N}$ or $\overline {\partial} \omega_{\rho_1} < 0$ on $\mathcal{N}$. Then
\begin{align*}
DF(\Omega) &= \sup_{\rho} \left\{ 0 < \gamma < 1 : \overline{\partial} \omega_{\rho} - \frac{\gamma}{1 - \gamma}(\omega_{\rho} \wedge \overline{\omega}_{\rho}) > 0 \quad \text{ on } \mathcal{N} \right\}, \\
S(\Omega) &= \inf_{\rho} \left\{ \gamma > 1 : -\overline{\partial} \omega_{\rho} - \frac{\gamma}{\gamma - 1}(\omega_{\rho} \wedge \overline{\omega}_{\rho}) > 0 \quad \text{ on } \mathcal{N} \right\}.
\end{align*}
If the supremum or infimum does not exist, then $DF(\Omega)=0$ or $S(\Omega)=\infty$, respectively. \end{thm}
\section{\bf Semi-continuity of two indices} \label{sec:Semi-continuity of two indices}
In this section, we prove the main theorem.
Let $\pi : (X,\Omega) \rightarrow \D$ be a smooth deformation of $\Omega_0$ in $X_0$ over $\D$. Let $\dim_{\mathbb{C}}X = n+1$ and $\dim_{\mathbb{C}}X_t = n$. Let $\delta$ be a defining function of $\Omega \subset X$ such that $\delta_t := \delta|_{X_t}$ is also a defining function of $\Omega_t \subset X_t$ for each $t \in \D$, and $M := \partial \Omega$. For a point $p \in \partial\Omega_t \subset M$, denote by \[ \mathcal{N}_p := \{ L_p \in T^{1,0}_p(\partial \Omega_t) \mid \mathscr{L}_{\delta_t}(L_p, L'_p) = 0 \quad \forall L_p' \in T^{1,0}_p(\partial \Omega_t) \}. \]
\begin{lem} \label{lem:null-space}
For $p \in \partial \Omega_0 \subset M$, suppose that $dim_{\mathbb{C}} (\mathcal{N}_p) = m$ $(1 \le m \le n-1)$. Then there exist a neighborhood $U_p$ of $p$ in $M$ and $m$ linearly independent smooth $(1,0)$ vector fields $L_1, \cdots, L_m$ on $U_p$ satisfying the following conditions:
\begin{itemize}
\item $\mathcal{N}_p = \left< L_1(p), \cdots, L_m(p) \right> $,
\item For each $q \in U_p$, $\mathcal{N}_q \subset \left< L_1(q), \cdots, L_m(q) \right> $.
\end{itemize}
\end{lem} \begin{proof}
Choose a basis $\{ L_{1,p}, \cdots, L_{n-1,p} \}$ for $T^{1,0}_p(\partial \Omega_0)$ such that $\{ L_{1,p}, \cdots, L_{m,p} \}$ forms a basis for $\mathcal{N}_p$.
Then, for each $j=1, \cdots, n-1$, we extend $L_{j,p}$ smoothly to a $(1,0)$ vector field $X_j$ on a neighborhood $U_p$ of $p$ in $M$ such that
$X_j(p) = L_{j,p}$ and $\{ X_1(q), \cdots, X_{n-1}(q) \}$ forms a basis for $T^{1,0}_q(\partial \Omega_t)$ for $q \in U_p$ with $\pi(q)=t$.
This is possible because $\bigcup_{q \in \partial \Omega} T^{1,0}_q(\partial \Omega_t)$ is a vector subbundle of $T^{1,0}(\partial \Omega)$.
Let $M(q)$ be the matrix representation of the Levi-form with respect to a basis $\{ X_j(q) \}^{n-1}_{j=1}$ at $q \in U_p$, i.e.,
$M(q) = [ M_{i\overline{j}}(q) ]_{(n-1)\times(n-1)} := \left[ \mathscr{L}_{\delta_t}(X_i(q), X_j(q)) \right] $. Then $M(q)$ is a hermitian matrix and let \\
\[
M(q) =
\begin{bmatrix}
A(q) & B(q) \\
B(q)^* & C(q)
\end{bmatrix}
\]
with blocks of size $m$ and $(n-m-1)$, where $B(q)^*$ is the conjugate transpose of $B(q)$.
Also, by the construction, $A(p) = B(p) = 0$ and $C(p)$ is invertible.
By the continuity, $C(q)$ is still invertible on $U_p$ (by shrinking $U_p$ if necessary).
Let
$$\Psi(q) :=
\begin{bmatrix}
I_{m \times m} & 0 \\
-C(q)^{-1} B(q)^* & C(q)^{-1}
\end{bmatrix}
,$$
where $I_{m \times m}$ is the identity matrix.
Define vector fields $L_j(q)$ $(1 \le j \le n-1)$ on $U_p$ by
\[
\left[ L_1(q) \cdots L_{n-1}(q) \right] := \left[ X_1(q) \cdots X_{n-1}(q) \right] \times \overline{\Psi}(q),
\]
where $\times$ is the matrix multiplication.
Then $\mathcal{N}_p = \left< L_1(p), \cdots, L_m(p) \right> $ because $L_j(p) = X_j(p) = L_{j,p}$ for $j = 1, \cdots, m$, and
the matrix representation of the Levi-form with respect to a basis $\{ L_j(q) \}^{n-1}_{j=1}$ is
\begin{equation*}
\Psi(q)^* \times M(q) \times \Psi(q) =
\begin{bmatrix}
A(q) - B(q)C(q)^{-1}B(q)^* & 0 \\
0 & C(q)^{-1}
\end{bmatrix}
.
\end{equation*}
Therefore, since $C(q)^{-1}$ is invertible, we conclude that $\mathcal{N}_q \subset \left< L_1(q), \cdots, L_m(q) \right> $. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main}]
Since each fiber $X_t$ is a Stein manifold, one may apply Theorem \ref{thm:DF,S formulas} for each $\Omega_t$ (see Corollary 5.4 in \cite{Adachi-Yum}).
By Theorem \ref{thm:DF,S formulas}, for each $0 < \gamma < DF(\Omega_0)$, there exists a defining function $\rho : X_0 \rightarrow \mathbb{R}$ of $\Omega_0 \subset X_0$ such that
\begin{equation} \label{inq:3.1}
\overline{\partial} \omega_{\rho} - \frac{\gamma}{1 - \gamma}(\omega_{\rho} \wedge \overline{\omega}_{\rho}) > 0 \quad \text{ on } \mathcal{N},
\end{equation}
where $\mathcal{N} := \bigcup_{p \in \partial \Omega_0} \mathcal{N}_p \subset T^{1,0}(\partial \Omega_0)$.
We will extend $\rho$ smoothly to $\widetilde{\rho} : \pi^{-1}(U_0) \rightarrow \mathbb{R}$ such that $\widetilde{\rho}_t := \widetilde{\rho}|_{X_t}$ is a defining function of $\Omega_t$ for each $t \in U_0$ and $\widetilde{\rho}_0 = \rho$, where $U_0$ is a neighborhood of $0 \in \D$. Since $\delta_0$ is also a defining function of $\Omega_0 \subset X_0$, there exists a smooth function $\psi \in C^{\infty}(X_0)$ such that $\rho = e^{\psi} \delta_0$.
Then since $X_0$ is a closed submanifold of $X$, we may extend $\psi$ to $\widetilde{\psi} \in C^{\infty}(\pi^{-1}(U_0))$ (by shrinking $U_0$ if necessary), and $\widetilde{\rho} := e^{\widetilde{\psi}} \delta$ is the desired function. We denote $\widetilde{\rho}$ by $\rho$ again, and $\omega_{\rho_t}$ by $\omega_{t}$.
Now for $p \in \partial \Omega_0$, suppose that $dim_{\mathbb{C}} (\mathcal{N}_p) = m >0 $. Then by Lemma \ref{lem:null-space},
there exist a neighborhood $U_p$ of $p$ in $M$ and $m$ linearly independent smooth $(1,0)$ vector fields $L_1, \cdots, L_m$ on $U_p$ satisfying the following conditions:
\begin{itemize}
\item $\mathcal{N}_p = \left< L_1(p), \cdots, L_m(p) \right> $,
\item For each $q \in U_p$, $\mathcal{N}_q \subset \left< L_1(q), \cdots, L_m(q) \right> $.
\end{itemize}
Then, by the continuity, the inequality (\ref{inq:3.1}) implies that there exists a neighborhood $V_p \subset U_p$ such that
\[
\overline{\partial} \omega_{t} - \frac{\gamma}{1 - \gamma}(\omega_{t} \wedge \overline{\omega}_{t}) > 0 \quad \text{ on } \left< L_1(q), \cdots, L_m(q) \right>
\]
for all $q \in V_p$ with $\pi(q)=t$.
Since $\mathcal{N}_q \subset \left< L_1(q), \cdots, L_m(q) \right> $,
\[
\overline{\partial} \omega_{t} - \frac{\gamma}{1 - \gamma}(\omega_{t} \wedge \overline{\omega}_{t}) > 0 \quad \text{ on } \mathcal{N}_q
\]
for all $q \in V_p$ with $\pi(q)=t$. Therefore, since the set $\{ p \in \partial \Omega_0 : \dim_{\mathbb{C}}(\mathcal{N}_p) > 0 \}$ is compact, we conclude that
$$\liminf_{t \rightarrow 0} DF(\Omega_t) \ge DF(\Omega_0),$$
by applying Theorem \ref{thm:DF,S formulas} again.
$\limsup_{t \rightarrow 0} S(\Omega_t) \le S(\Omega_0)$ follows from the same argument as above. \end{proof}
\section{\bf Example}
In 1977, Diederich and Forn{\ae}ss (\cite{diederich-fornaess2}) constructed a 1-parameter family of bounded, pseudoconvex domains $\Omega_{\beta}$ $(\beta > \frac{\pi}{2})$, called worm domains, in $\mathbb{C}^2$ with $C^{\infty}$-smooth boundaries. They showed that the Diederich-Forn{\ae}ss indices of worm domains are non-trivial, i.e., $0 < DF(\Omega_{\beta}) < 1$, for all $\beta$, and $\Omega_{\beta}$ does not admit a Stein neighborhood basis for some $\beta$. Recently, Liu (\cite{Liu1}) calculated the exact value of the $DF(\Omega_{\beta})$. Also, the second named author (\cite{Yum1}) calculated the exact value of the $S(\Omega_{\beta})$ and found the following relation between two indices for worm domains: \[
\frac{1}{DF(\Omega_{\beta})} + \frac{1}{S(\Omega_{\beta})} = 2, \] whenever $DF(\Omega_{\beta}) > 0$ and $S(\Omega_{\beta}) < \infty$.
In this section, we give an explicit example, by modifying worm domains, which shows the Diederich-Forn{\ae}ss and Steinness indices do not admit upper semi-continuity and lower semi-continuity in general, respectively. We first recall the $\beta$-worm domain.
\begin{defn} \label{worm defn}
The {\it $\beta$-worm domain} $D_{\beta}$ $(\beta > \frac{\pi}{2})$ is defined by
$$ D_{\beta} := \left\lbrace (z,w)\in \mathbb{C}^2 : \left| z - e^{i \log|w|^2} \right|^2 - (1 - \phi_{\beta}(\log|w|^2) ) < 0 \right\rbrace $$
where $\phi_{\beta} : \mathbb{R} \rightarrow \mathbb{R}$ is a fixed smooth function with the following properties :
\begin{itemize} \item[(\romannumeral1)] $\phi_{\beta}(x) \ge 0$, $\phi_{\beta}$ is even and convex. \item[(\romannumeral2)] $\phi^{-1}_{\beta}(0) = I_{\beta - \frac{\pi}{2}} = [-(\beta - \frac{\pi}{2}), \beta - \frac{\pi}{2} ].$ \item[(\romannumeral3)] $\exists$ $a>0$ such that $\phi_{\beta}(x)>1$ if $x<-a$ or $x>a$. \item[(\romannumeral4)] $\phi'_{\beta}(x) \neq 0$ if $\phi_{\beta}(x) = 1$. \end{itemize} \end{defn}
Let $\Omega$ be a domain in $\mathbb{C}^3$ defined by the defining function \[
\rho(z,w,\gamma) := \left| z - e^{i \log|w|^2} \right|^2 - \left(1 - \phi_{\frac{3}{4}\pi}(\log|w|^2) - |\gamma|^2 \right), \] i.e., $\Omega := \{ (z,w,\gamma) \in \mathbb{C}^3 : \rho(z,w,\gamma) < 0 \}$, and $\Omega_{\gamma} := \{ (z,w) \in \mathbb{C}^2 : \rho_{\gamma}(z,w) := \rho(z,w,\gamma) < 0 \}$ for each $\gamma \in \D$. Here, the choice of $\beta = \frac{3}{4}\pi$ is not important in our example.
Now, for the following argument, we refer readers to \cite{diederich-fornaess2} and \cite{Krantz-Peloso}. Let \[
\widetilde{\rho}_{\gamma}(z,w) := \rho_{\gamma}(z,w) e^{2 \arg w}, \] which is a local defining function near any boundary point $p \in \partial \Omega_{\gamma}$. Note that the holomorphic tangent plain $T^{1,0}_p(\partial \Omega_{\gamma})$ is spanned by a vector \[
L_p := -\frac{\partial \rho_{\gamma}}{\partial w}(p) \left.\frac{\partial}{\partial z}\right|_p
+ \frac{\partial \rho_{\gamma}}{\partial z}(p) \left.\frac{\partial}{\partial w}\right|_p. \] Then \begin{align*}
\mathscr{L}_{\widetilde{\rho}_{\gamma}}(L_p, L_p) = \frac{e^{2\arg w}}{4|w|^2}
&\left[
\left| i\overline{z}(z - |w|^{2i}) + \phi'_{\frac{3}{4}\pi}(\log|w|^2) \right|^2 \right. \\
& \left. + \left|z - |w|^{2i} \right|^2 \left( \phi_{\frac{3}{4}\pi}(\log|w|^2) + \phi''_{\frac{3}{4}\pi}(\log|w|^2) \right)
\right]. \end{align*}
Observe that $\left| z - |w|^{2i} \right|$ never vanish on $\partial \Omega_{\gamma}$. If $\log |w|^2 \notin [-(\beta - \frac{\pi}{2}), \beta - \frac{\pi}{2} ]$, then $\phi_{\frac{3}{4}\pi}(\log|w|^2) + \phi''_{\frac{3}{4}\pi}(\log|w|^2) > 0$ by the definition of $\phi_{\frac{3}{4}\pi}$.
If $\log |w|^2 \in [-(\beta - \frac{\pi}{2}), \beta - \frac{\pi}{2} ]$, then $\left| i\overline{z}(z - |w|^{2i}) \right| > 0$ provided that $z \neq 0$. Therefore, if $\gamma \neq 0$ then $z(p) \neq 0$, hence, $\mathscr{L}_{\widetilde{\rho}_{\gamma}}(L, L)(p) > 0$ for all $p \in \partial \Omega_{\gamma}$. This means that $\Omega_{\gamma}$ is strongly pseudoconvex for all $\gamma \neq 0 \in \D$. Since every strongly pseudoconvex domain admits a strictly plurisubharmonic defining function, $DF(\Omega_{\gamma}) = 1$ and $S(\Omega_{\gamma}) = 1$ for all $\gamma \neq 0 \in \D$. Moreover, $\Omega_{0} = D_{\frac{3}{4}\pi}$ implies that $DF(\Omega_{0}) = \frac{2}{3}$ and $S(\Omega_{0})=2$ from the results in \cite{Liu1} and \cite{Yum1}. We conclude that $DF(\Omega_{\gamma})$ and $S(\Omega_{\gamma})$ are not upper semi-continuous and lower semi-continuous at $\gamma = 0$, respectively.
\begin{bibdiv} \begin{biblist}
\bib{Adachi-Yum}{article}{
author={Adachi, Masanori},
author={Yum, Jihun},
title={Diederich--Forn{\ae}ss and Steinness indices for abstract CR manifolds},
journal={},
volume={},
date={2020},
number={},
pages={},
status={Preprint},
eprint={arXiv:2003.01330}, }
\bib{Boas-Straube}{article}{
author={Boas, Harold P.},
author={Straube, Emil J.},
title={de Rham cohomology of manifolds containing the points of infinite
type, and Sobolev estimates for the $\overline\partial$-Neumann problem},
journal={J. Geom. Anal.},
volume={3},
date={1993},
number={3},
pages={225--235}, }
\bib{D'Angelo0}{article}{
author={D'Angelo, John P.},
title={Finite type conditions for real hypersurfaces},
journal={J. Differential Geometry},
volume={14},
date={1979},
number={1},
pages={59--66 (1980)}, } \bib{D'Angelo}{article}{
author={D'Angelo, John P.},
title={Iterated commutators and derivatives of the Levi form},
conference={
title={Complex analysis},
address={University Park, PA.},
date={1986},
},
book={
series={Lecture Notes in Math.},
volume={1268},
publisher={Springer, Berlin},
},
date={1987},
pages={103--110}, }
\bib{diederich-fornaess}{article}{
author={Diederich, Klas},
author={Forn{\ae}ss, John Erik},
title={Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion
functions},
journal={Invent. Math.},
volume={39},
date={1977},
number={2},
pages={129--141}, }
\bib{diederich-fornaess2}{article}{
author={Diederich, Klas},
author={Forn{\ae}ss, John Erik},
title={Pseudoconvex Domains: an example with nontrivial neighborhood},
journal={Math. Ann.},
volume={225},
date={1977},
number={3},
pages={275--292}, }
\bib{Krantz-Peloso}{article}{
author={Krantz, Steven G.},
author={Peloso, Marco M.},
title={Analysis and geometry on worm domains},
journal={J. Geom. Anal.},
volume={18},
date={2008},
number={2},
pages={478--510}, }
\bib{Liu1}{article}{
author={Liu, Bingyuan},
title={The Diederich-Forn\ae ss index I: For domains of non-trivial index},
journal={Adv. Math.},
volume={353},
date={2019},
pages={776--801}, }
\bib{Saeki}{book}{
author={Saeki, Osamu},
title={Topology of Singular Fibers of Differentiable Maps},
series={Lecture notes in Mathematics},
publisher={Springer},
date={2004} }
\bib{Yum1}{article}{
author={Yum, Jihun},
title={On the Steinness Index},
journal={J. Geom. Anal.},
volume={29},
date={2019},
number={2},
pages={1583--1607}, }
\bib{Yum2}{article}{
author={Yum, Jihun},
title={CR-invariance of the Steinness index},
status={Preprint},
eprint={arXiv:1908.01214},
date={2019} }
\end{biblist} \end{bibdiv}
\end{document} | arXiv | {
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\begin{document}
\thispagestyle{empty}
\setcounter{page}{1} \begin{center}
\textbf{\Large {A gradient method in a Hilbert space with an
optimized inner product:} achieving a Newton-like convergence } \end{center}
\begin{center} Arian {\sc Novruzi}\footnote{\revt{Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada, Email: {\tt novruzi@uottawa.ca}; the corresponding author}} and Bartosz {\sc Protas}\footnote{\revt{Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada, Email: {\tt bprotas@mcmaster.ca}}} \end{center}
\begin{abstract}
In this paper we introduce a new gradient method which attains
quadratic convergence in a certain sense. Applicable to
infinite-dimensional {unconstrained minimization problems posed in a
Hilbert space $H$}, the approach consists in finding the energy
gradient $g(\lambda)$ {defined with respect to an optimal} inner
product {selected from an infinite family of equivalent} inner
products $(\cdot,\cdot)_\lambda$ in the space $H$. {The inner
products are} parameterized by a space-dependent weight \revt{function}
$\lambda$.
At each iteration of the method, \revt{where an approximation to the
minimizer is given by an element $u\in H$,} an optimal weight
${\widehat{\lambda}}$ is found as a solution of a nonlinear minimization
problem in the space of weights $\Lambda$. It turns out that the
projection of $\kappa g({\widehat{\lambda}})$, where $0<\kappa \ll 1$ is a
fixed step size, onto a certain finite-dimensional subspace
generated by the method is consistent with Newton's step $h$, in the
sense that $P_u(\kappa g({\widehat{\lambda}}))=P_u(h)$, where $P_u$ is \revt{an
operator describing the projection onto the subspace}. As
demonstrated by rigorous analysis, this property ensures that thus
constructed gradient method attains quadratic convergence for error
components contained in these subspaces, in addition to the linear
convergence typical of the standard gradient method. We propose a
numerical implementation of {this} new approach and analyze its
complexity. Computational results obtained based on a simple model
problem confirm the theoretically established convergence
properties, demonstrating that the proposed approach performs much
better than the standard steepest-descent method based on Sobolev
gradients. The presented results offer an explanation of a number of
earlier empirical observations concerning the convergence of
Sobolev-gradient methods.
\\
\\
{\bf Keywords}: \revt{unconstrained optimization in Hilbert spaces}, variable
inner products, \revt{Sobolev} gradients, Newton's method, gradient method with
quadratic convergence
\\
{\bf MSC (2010)}: primary \revtt{49M05, 49M15, 97N40}; secondary \revtt{65K10, 78M50} \end{abstract}
\tableofcontents
\section{Introduction}
In this investigation we consider
solution of general {unconstrained} optimization problems using the steepest-descent method and focus on {modifying the definition of the
gradient} such that in certain circumstances {this approach} will achieve a quadratic convergence, characteristic of Newton's method. This is accomplished by judiciously exploiting the freedom inherent in the choice of different equivalent norms defining the gradient through the Riesz representation theorem. This freedom can be used to adjust the definition of the inner product, such that the resulting gradient will, in a suitable sense, best resemble the corresponding Newton step. While such ideas can be pursued in both finite-dimensional and infinite-dimensional settings, the formulation is arguably more interesting mathematically and more useful in practice in the latter case. We will thus consider unconstrained optimization problems of the general form \begin{equation} \label{e:min{e(u)}} e({\widehat{u}}) = \min\{e(u),\; u \in H \}, \end{equation} where $e \; : \; H \mapsto \mathbb R$ is the objective functional and $H$ is a suitable function space with Hilbert structure.
Applications which have this form include, for example, minimization of various energy functionals in physics and optimization formulations of inverse problems, where evaluation of the objective functional $e(u)$ may involve solution of a complicated (time-)dependent partial differential equation (PDE). In such applications $H$ is typically taken as a Sobolev space $H^p(\Omega)$, $p \in \mathbb N$, where $\Omega \subset \mathbb R^d$ is the spatial domain assumed to be sufficiently smooth (Lipschitz) and $d \ge 1$ is the dimension \cite{af05}. Therefore, to fix attention, here we will assume $H := H^1_0(\Omega)$ with the inner product and norm in $H$ defined as \begin{equation} \label{e:(u,v)_Ht} (u,v)_H=\int_{\Omega}((\nabla u\cdot\nabla v) + uv)\, dx, \qquad
\|u\|_H=(u,v)_H^{1/2}. \end{equation}
\begin{algorithm} \begin{algorithmic}[1] \STATE set $u = u_0$ \REPEAT \STATE compute $e'(u;\cdot)$ \STATE determine $g\in H$ as the solution of $(g,v)_H=e'(u;v)$, $\forall v\in H$ \STATE set $\widetilde{u} = u - \kappa g$ \STATE set $u = \widetilde{u}$
\UNTIL{ \ $|e(u)|<$ \texttt{err}} \end{algorithmic} \caption{
(Sobolev) Gradient method \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $u_0 \in H$ --- initial guess \newline
\hspace*{0.5cm} $\kappa > 0$ --- step size (sufficiently small) \newline
\hspace*{0.5cm} $\texttt{err} > 0$ --- tolerance \newline
\textbf{Output:} \newline
\hspace*{0.5cm} $\widetilde{u}$ --- approximation to the solution ${\widehat{u}}$ of problem \eqref{e:min{e(u)}}} \label{alg:gradient} \end{algorithm}
\begin{algorithm} \begin{algorithmic}[1] \STATE set $u = u_0$ \REPEAT \STATE {compute $e'(u;\cdot)$} \STATE {compute $e''(u;\cdot,\cdot)$} \STATE determine $h\in H$ as the solution of $e''(u;h,v)=e'(u;v)$, $\forall v\in H$ \STATE set $\widetilde{u} = u - h$ \STATE set $u = \widetilde{u}$
\UNTIL{ \ $|e(u)|<$ \texttt{err}} \end{algorithmic} \caption{
Newton's method \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $u_0 \in H$ --- initial guess \newline
\hspace*{0.5cm} $\texttt{err} > 0$ --- tolerance \newline
\textbf{Output:} \newline
\hspace*{0.5cm} $\widetilde{u}$ --- approximation to the solution ${\widehat{u}}$ of problem \eqref{e:min{e(u)}}} \label{alg:newton} \end{algorithm}
The two most elementary approaches to solve problem \eqref{e:min{e(u)}} are the gradient and Newton's method which, for the sake of completeness, are defined in Algorithms \ref{alg:gradient} and \ref{alg:newton}, respectively. The former approach is sometimes also referred to as the ``Sobolev gradient'' method \cite{n10}. While gradient approaches often involve an adaptive step size selection \cite{nw00}, for simplicity of analysis in Algorithm \ref{alg:gradient} we consider a fixed step size $\kappa = \text{Const}$. Likewise, in order to keep the analysis tractable, we do not consider common modifications of the gradient approach such as the conjugate-gradient method. As regards convergence of the gradient and Newton's method, we have the following classical results, see for example \cite{ciarlet-1}. \begin{theorem} \label{th:gradient} Let ${\widehat{u}}\in H$ be {a} solution of (\ref{e:min{e(u)}}). Assume that $e$ is twice differentiable near ${\widehat{u}}$ and there exist $\delta_0>0$ and $\alpha_0>0$ such that $e''(u;v,v)\geq
\alpha_0\|v\|^2_H$ for all $\|u-{\widehat{u}}\|_H\leq \delta_0$ and $v\in H$. Then, for all $u_0\in B_H({\widehat{u}},\delta_0)$ the gradient {method given by} Algorithm \ref{alg:gradient} converges linearly to ${\widehat{u}}$ in $H$, i.e., \begin{eqnarray*}
\|\widetilde{u}-{\widehat{u}}\|_H &\leq&
\epsilon\|u-{\widehat{u}}\|_H, \end{eqnarray*} with $\epsilon\in(0,1)$ depending only on $e$ and $\kappa$. \end{theorem}
\begin{theorem}\label{th:newton}
Let ${\widehat{u}}\in H$ be {a} solution of (\ref{e:min{e(u)}}). Assume
that $e$ is three times differentiable near ${\widehat{u}}$ and that the map
$v\in H\mapsto e''({\widehat{u}};v,\cdot)\in H'$ is invertible. Then, there
exists $\delta_0>0$ such that for all $u_0\in B_H({\widehat{u}},\delta_0)$
Newton's {method given by} Algorithm \ref{alg:newton} converges
quadratically to ${\widehat{u}}$ in $H$, i.e., \begin{eqnarray*}
\|\widetilde{u}-{\widehat{u}}\|_H
&\leq& C \|u-{\widehat{u}}\|^2_H , \end{eqnarray*} with $C>0$ depending only on $e$. \end{theorem}
We emphasize that in Algorithm \ref{alg:gradient} the gradient $g$
must be computed with respect to the topology of the space $H$ in which the solution {to \eqref{e:min{e(u)}}} is sought \cite{n10}, an aspect of the problem often neglected in numerical investigations. A metric equivalent to $\| \cdot \|_H$ (in the precise sense of norm equivalence) can be obtained by redefining the inner product in \eqref{e:(u,v)_Ht} more generally as follows \begin{equation} \label{e:(u,v)_lam0} (u,v)_{\lambda_0} = \int_{\Omega}(\lambda_0 (\nabla u\cdot\nabla v) + uv)\, dx, \end{equation} where $\lambda_0\in(0,\infty)$ is a fixed constant. While as compared to \eqref{e:(u,v)_Ht} definition \eqref{e:(u,v)_lam0} does not change the analytic structure of the optimization problem \eqref{e:min{e(u)}}, there is abundant computational evidence obtained in the solution of complicated optimization problems \cite{pbh02,p07b,ap15a} that convergence of gradient Algorithm \ref{alg:gradient} may be significantly accelerated by replacing the inner product from \eqref{e:(u,v)_Ht} with the one introduced in \eqref{e:(u,v)_lam0} for some judicious {choices} of the parameter $\lambda_0$. Likewise, a similar acceleration was also observed when the inner product in \eqref{e:(u,v)_Ht} was replaced with another equivalent definition motivated by the structure of the minimization problem and different from \eqref{e:(u,v)_lam0}, cf.~\cite{rssl09,ms10,kd12}. In the absence of an understanding of the mechanism responsible for this acceleration, the parameter $\lambda_0$, or other quantities parameterizing the equivalent inner product, were chosen empirically by trial and error, which is unsatisfactory.
In the present investigation we will consider a more general form of the inner product \eqref{e:(u,v)_lam0} in which the constant $\lambda_0$ is replaced with a space-dependent weight $\lambda = \lambda(x)$. Our goal is to develop a rational and systematic approach allowing one to accelerate the convergence of gradient iterations in Algorithm \ref{alg:gradient} in comparison to the standard case by adaptively adjusting the weight $\lambda(x)$. This will result in a reduction of {the total number of iterations needed to solve
problem \eqref{e:min{e(u)}} to a given accuracy,} but each iteration will be more costly.
Modifications of the inner product with respect to which the gradient is defined may also be interpreted as gradient preconditioning and this perspective is pursued in the monograph \cite{fk02} focused on related problems arising in the solution of nonlinear elliptic equations. The relationship between the gradient and Newton's methods was explored in \cite{kn07} where a variable inner product was considered. In contrast to \revt{the present} approach \revt{in which
the} inner-product \revt{weights are} sought by matching the projections of \revt{the} gradient and Newton's steps \revt{onto} a certain subspace, \revt{in \cite{kn07} optimal inner products were
found by maximizing the descent achieved at a given iteration with
respect to the structure of the corresponding preconditioning
operator.}
The structure of the paper is as follows. In the next section we define the new approach in a general form, whereas in Section \ref{sec:errors} we prove its convergence properties. Then, in Section \ref{sec:optimal} we describe the numerical approach implementing the general method introduced in Section \ref{sec:newgrad} {in two
variants} and analyze its computational complexity. Our model problem and computational results are presented in Section \ref{sec:results}, whereas discussion and conclusions are deferred to Section \ref{sec:final}.
\section{A new gradient method based on an optimal inner product} \label{sec:newgrad}
In this section we introduce our modified version of Algorithm \ref{alg:gradient} for the solution of the minimization problem \eqref{e:min{e(u)}}. We begin by making the following assumptions on the energy $e(u)$: \begin{eqnarray} e&&\mbox{\it is $C^2$ in $H$}, \\
|e''(u;v,w)| &\leq&
M\|u\|_H \|w\|_H,\quad u,v,w \in H, \label{e:e->C0}\\ e''(u;v,v) &\geq&
m\|v\|_H^2, \label{e:e->convex} \end{eqnarray} with certain $M,m>0$.
Let us point out that at each step of both the gradient and Newton's methods the descent direction is defined by the solution of the equation \begin{equation}\label{e:b=e'->1}
b(z,v)=e'(u;v),\quad \forall v\in H, \end{equation} where $b(u,v)$, a symmetric bilinear continuous elliptic form, and $z\in H$ are specific to each method, namely, \begin{itemize} \item $b(u,v)=(u,v)_H$ and $z=g$ in the case of {the} gradient method, and \item $b(u,v)=e''(u;u,v)$ and $z=h$ in the case of Newton's method. \end{itemize} Moreover, we note that the solution $z$ of \eqref{e:b=e'->1} is also the solution of the minimization problem \begin{equation} \label{e:min{b-l}} \min\left\{\frac{1}{2}b(v,v)-e'(u;v),\;\, v\in H\right\}. \end{equation} We emphasize that, in fact, Newton's method may be also viewed as a ``gradient'' method with a particular choice of the inner product at each iteration, namely, the one induced by $e''(u)$. Therefore, the idea for improving the classical gradient method is to make the gradient step $g$ ``close'' to the Newton step $h$ by suitably adapting the inner product in $H$.
We thus propose the following modification of the gradient method from Algorithm \ref{alg:gradient}. We want to consider the gradient $g$ {defined} with respect to {an} inner product in $H$ depending on a function parameter $\lambda$. Typically, $0<\lambda\in C^0(\overline{\Omega})$, {however, to make our method more
attractive from the computational point of view} we will consider $\lambda$ with a finite range. Namely, let $\{\Omega_{i},\, i=1,\ldots,N\}$, be a partition of $\Omega$ into open Lipschitz sets, $\Lambda=\{\lambda:\Omega\mapsto\mathbb R,\,\; \lambda(\Omega_i)=\lambda_i \in\mathbb R\} = \Span \{\ell^i,\, i=1,\dots,N\}\subset L^\infty(\Omega)$, where $\ell^i\in\Lambda$, $\ell^i=\delta_{i,j}$ in $\Omega_j$, $i,j=1,\ldots,N$ with $\delta_{i,j}$ the Kronecker symbol, $\Lambda^+=\Lambda\cap\{0<\lambda<\infty\}$. Sometimes {without
the risk of confusion we will write
$\lambda=[\lambda_1,\ldots,\lambda_N]\in\mathbb R^N$ for
$\lambda\in\Lambda$}, meaning $\lambda(\Omega_i)=\lambda_i$, for all $i=1,\ldots, N$. Then, for $\lambda\in \Lambda^+$, we define the following inner product and norm in $H$ \begin{equation}\label{e:(u,v)_lambda} (v,w)_\lambda = \int_{\Omega} \lambda (\nabla v\cdot \nabla w) + v w \, dx, \quad
\|v\|_\lambda=(v,v)_\lambda^{1/2}, \quad \forall v,w\in H. \end{equation} Clearly, $(\cdot,\cdot)_\lambda$ and $(\cdot,\cdot)_H=(\cdot,\cdot)_1$ are equivalent in $H$ and therefore we can use $(\cdot,\cdot)_\lambda$ instead of $(\cdot,\cdot)_H$ for the gradient method.
The idea is to use the inner product $(\cdot,\cdot)_\lambda$ in the gradient method, with $\lambda$ judiciously chosen. More specifically, for $\lambda\in\Lambda^+$, let $g=g(\lambda)$ be the solution of \begin{equation}\label{e:g(lambda)} (g,v)_\lambda = \int_{\Omega} \lambda (\nabla {g}\cdot \nabla v) + {g} v \, dx = e'(u;v),\quad\forall v\in H. \end{equation}
\begin{remark} \label{r:Riesz}
In the following we will, \revt{in particular}, refer to the
gradient \revtt{$g_1:=g(1)$} which corresponds to \revt{the usual inner
product \eqref{e:(u,v)_Ht} and is also obtained by setting
$\lambda=1$ in \eqref{e:g(lambda)}}, and to \revt{the gradient}
\revtt{$g_0:=g(0)$} which corresponds to $\lambda=0$ \revt{in
\eqref{e:g(lambda)}}. Usually, \revt{$g_1$ and $g_0$ are referred
to as, respectively, the $H^1$ and $L^2$ Riesz representations of
$e'(u)$.}
In general $g_0\notin H^1_0(\Omega)$, but we have \begin{eqnarray} -\Delta g_1+g_1 &=& g_0 \;\, \mbox{\it in ${\cal D'}(\Omega)$}, \label{e:g_1,g_0}\\ -\nabla\cdot(\lambda\nabla g(\lambda))+g(\lambda) &=&g_0\;\, \mbox{\it in ${\cal D'}(\Omega)$}. \label{e:g(l),g_0} \end{eqnarray} \end{remark}
Note that we will use the symbol $g$ to denote the operator $\lambda\in\Lambda\mapsto g=g(\lambda)\in H$, or to denote an element of $H$ --- the meaning will always be clear from the context.
Now assume we are at a certain iteration of the gradient method with $u$ known, which we seek to update to a new value $\widetilde{u}$, cf.~step 5 in Algorithm \ref{alg:gradient}. For this, first we look for a certain ${\widehat{\lambda}} \in \Lambda^+$, defined by\footnote{All along this
paper, the symbol ``$\widehat{\phantom{x}}$'' will be used to denote
the solution of a minimization problem, whereas the symbol
``$\widetilde{\phantom{x}}$'' will be used to represent an updated value
of a variable.} \begin{equation}\label{e:min(lambda)->1} j({\widehat{\lambda}}) := \min\left\{ j(\lambda):=f\circ g(\lambda),\;\, \lambda\in \Lambda^+ \right\}, \quad \text{\it where} \quad f(g):=\frac{\kappa}{2}e''(u;g,g)-e'(u;g). \end{equation} The reason for introducing the step size $\kappa$ in this equality will be clear from Remark \ref{r:name} and also later during the error analysis in Section \ref{sec:errors}. Note that, if problem \eqref{e:min(lambda)->1} has a solution ${\widehat{\lambda}} \in \Lambda^+$, then we will show (see Proposition \ref{p:g'->1}) that ${\widehat{\lambda}}$ solves \begin{equation}\label{e:e''(lambda)=e'} e''(u;\kappa g({\widehat{\lambda}}),g'({\widehat{\lambda}};\ell))=e'(u;g'({\widehat{\lambda}};\ell)),\quad\forall \ell\in \Lambda, \end{equation} where $g'({\widehat{\lambda}};\ell)$ denotes the derivative of $g$ at ${\widehat{\lambda}}$ in the direction $\ell$. Then, the modified gradient approach will consist of Algorithm \ref{alg:gradient} with step 4 amended as follows \begin{eqnarray}
&4.&\mbox{determine $g=g({\widehat{\lambda}})$, where ${\widehat{\lambda}}\in\Lambda^+$ is such that $g({\widehat{\lambda}})\in H$ solves (\ref{e:e''(lambda)=e'})}. \label{alg:gradient+2} \end{eqnarray}
\begin{remark}\label{r:name}
Clearly, our approach is equivalent to the gradient method, but with
the classical inner product $(\cdot,\cdot)_H$ replaced with
$(\cdot,\cdot)_\lambda$.
From equation \eqref{e:e''(lambda)=e'} it follows that
$e''(u;h-\kappa g,g'({\widehat{\lambda}};\ell))=0$, where $h$ is
Newton's step. This means that $P_u(h-\kappa g)=0$,
where $P_u:H\mapsto T_u$ is the projection from $H$ to $T_u$, in
which $T_u= \Span\{g'({\widehat{\lambda}};\ell),\; \ell\in\Lambda\}$ is the
tangent space to the manifold $\{g(\lambda),\,
\lambda\in\Lambda^+\}\subset H$ at $g({\widehat{\lambda}})$, determined with
respect to the inner product $e''(u;\cdot,\cdot)$.
If $T_u=H$, then $\kappa g({\widehat{\lambda}})=h$ and our method reduces to
Newton's method. However, here we have $dim(\Lambda)=N$, so that
{in general} $T_u\neq H$ and $\kappa g$ will be close to $h$ in
the sense that $P_u(h-\kappa g)=0$. This relation will be the key
ingredient to prove {in the demonstration} that our gradient
method, in addition to the linear convergence of a standard gradient
method, has also a quadratic convergence in a certain sense
depending on $T_u$ and the projection $P_u$. This will be explained
in the next sections.
\end{remark}
Our method critically depends on the choice of $\lambda$ and the following proposition offers a first glimpse of what may happen with the solution of problem \eqref{e:min(lambda)->1}. \begin{proposition}\label{p:lambda^k->}
Let $(\lambda^k)$ be a minimizing sequence of $j$ in $\Lambda^+$ and
$(g(\lambda^k))$ be the \revt{corresponding} sequence of gradients
\revt{defined in} \eqref{e:g(lambda)}. Then, up to a subsequence,
$(g(\lambda^k))$ converges weakly in $H^1_0(\Omega)$ and strongly
in $L^2(\Omega)$ to \revt{an element} $g\in H$, while for the
sequence $(\lambda^k)$ one of the following cases may occur.\\
(i)
There
exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]\in\Lambda^+$ and a subsequence of $(\lambda^k)$,
still denoted $(\lambda^k)$, such that $\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all
$i=1,\dots,N$.
In this case $g=g({\widehat{\lambda}})$, i.e. \begin{equation}\label{e:g(hlambda),1} \int_\Omega {\widehat{\lambda}}(\nabla g({\widehat{\lambda}})\cdot\nabla v)+g({\widehat{\lambda}})v \, dx = e'(u;v),\;\, \forall v\in H, \end{equation} and ${\widehat{\lambda}}$ solves \eqref{e:min(lambda)->1}. \\ (ii)
There exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]\in\partial\Lambda^+$,
$I_0\subset I$, $I_\infty\subset I$, with
${\widehat{\lambda}}_i=0$ for all $i\in I_0$,
${\widehat{\lambda}}_i=+\infty$ for all $i\in I_\infty$,
$0<{\widehat{\lambda}}_i<+\infty$ for all $i\in I\backslash(I_0\cup I_\infty)$,
and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$, such that
$\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$, for all $i=1,\dots,N$.
In this case $g\in H^1_0(\Omega;\Omega_\infty^\text{const})$ solves \begin{eqnarray} \hspace*{-8mm} \int_{\Omega\backslash \Omega_0} {\widehat{\lambda}} (\nabla g\cdot\nabla v) + gv \,dx + \int_{\Omega_0} gv \, dx &=& e'(u;v), \quad \forall v\in H^1_0(\Omega;\Omega_\infty^\text{const}), \label{e:g-in-G-G0-Ginf} \end{eqnarray} where \begin{eqnarray} \hspace*{-6mm} \Omega_0&=&\cup\{\Omega_i,\; i\in I_0\}, \\ \hspace*{-6mm} \Omega_\infty&=&\cup\{\Omega_i,\; i\in I_\infty\}, \\ \hspace*{-6mm} H^1_0(\Omega;\Omega_\infty^\text{const}) &=& \{ v\in H^1_0(\Omega), \;\, v=C_i\in\mathbb R\;\, in\;\, \Omega_i,\;\, \forall i\in I_\infty\}. \label{e:H1_0(G;G0,Gi)} \end{eqnarray} Furthermore, if we define $g({\widehat{\lambda}})=g$, with $g$ given by \eqref{e:g-in-G-G0-Ginf}, we have \begin{equation} j({\widehat{\lambda}})\leq\liminf_{k\to\infty}j(\lambda^k). \label{e:j(hl),=liminf} \end{equation}
\end{proposition} {\bf Proof}. Let $(\lambda^k)$ be a minimizing sequence of $j$ in $\Lambda^+$ and $\lambda^k=[\lambda^k_1,\ldots,\lambda^k_N]$. Note that $g(\lambda^k)$ is well defined by \begin{equation}\label{e:g(lambda^k)} \int_{\Omega} \lambda^k(\nabla g(\lambda^k)\cdot\nabla v) + g(\lambda^k)v \, dx = e'(u;v),\quad\forall v\in H. \end{equation} Note also that from the ellipticity of $f$ in $H$, cf.~\eqref{e:e->C0}, \eqref{e:e->convex} and \eqref{e:min(lambda)->1}, it follows that the sequence $g(\lambda^k)$ is bounded in $H^1(\Omega)$. Therefore, up to a subsequence, we may assume that $g(\lambda^k)$ converges weakly in $H$ and strongly in $L^2(\Omega)$ to a certain $g\in H$.
As ${\rm dim}(\Lambda)=N$, there exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_\revt{N}]$, with ${\widehat{\lambda}}_i\in[0,+\infty]$, and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$, such that $\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$. Two cases may occur.\\ 1) ${\widehat{\lambda}}\in\Lambda^+$, i.e., ${\widehat{\lambda}}_i\in(0,\infty)$ for all $i\in I$. From (\ref{e:g(lambda^k)}) we obtain \begin{eqnarray*} \int_{\Omega} {\widehat{\lambda}} (\nabla g\cdot\nabla v) + gv \,dx &=& \int_{\Omega}({\widehat{\lambda}}-\lambda^k) (\nabla g\cdot\nabla v) \,dx \\ &+& \int_{\Omega}\lambda^k (\nabla(g-g(\lambda^k))\cdot\nabla v) + (g-g(\lambda^k))v \,dx \\ &+& \int_{\Omega}\lambda^k (\nabla g(\lambda^k)\cdot \nabla v) + g(\lambda^k)v \,dx \\ &=& \int_{\Omega}({\widehat{\lambda}}-\lambda^k) (\nabla g\cdot\nabla v) \,dx \\ &+& \int_{\Omega}\lambda^k (\nabla(g-g(\lambda^k))\cdot\nabla v) + (g-g(\lambda^k))v \,dx \\ &+& e'(u;v),\quad\forall v\in H. \end{eqnarray*} Then, letting $k$ go to infinity gives \begin{eqnarray*} \int_{\Omega} {\widehat{\lambda}} (\nabla g\cdot \nabla v) + gv \,dx &=& e'(u;v),\quad\forall v\in H, \end{eqnarray*} which proves that $g=g({\widehat{\lambda}})$. Note that it is easy to show that the subsequence $(g(\lambda^k))$ converges to $g$ strongly in $H$, and therefore ${\widehat{\lambda}}$ is the solution of (\ref{e:min(lambda)->1}) because $j$ is continuous in $H$.
\\ 2) ${\widehat{\lambda}}\in\partial\Lambda^+$. \revt{Then,} there exist $I_0\subset I$ \revt{and} $I_\infty\subset I$ such that ${\widehat{\lambda}}_i=0$ for $i\in I_0$, ${\widehat{\lambda}}_i=+\infty$ for $i\in I_\infty$, ${\widehat{\lambda}}_i\in(0,+\infty)$ for $i\in I\backslash(I_0\cup I_\infty)$, and a subsequence of $(\lambda^k)$, still denoted by $(\lambda^k)$, such that $\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$. From (\ref{e:g(lambda^k)}), for each $g(\lambda^k)$ and $v\in H^1_0(\Omega;\Omega_\infty^{const})$ we have \begin{equation*} \int_{\Omega\backslash \Omega_0} \lambda^k (\nabla g(\lambda^k)\cdot \nabla v) + gv \, dx + \int_{\Omega_0} \lambda^k (\nabla g(\lambda^k)\cdot \nabla v) +gv \, dx = e'(u;v). \end{equation*} Then, letting $k$ go to infinity gives \eqref{e:g-in-G-G0-Ginf}, because $\nabla v = 0$ in $\Omega_\infty$.
To show that $g$ is constant on each $\Omega_i$, $i\in I_\infty$, we take $v=g(\lambda^k)$ in \eqref{e:g(lambda^k)}, so that we obtain \begin{eqnarray*} \int_{\Omega_\infty}
\lambda^k |\nabla g(\lambda^k)|^2 + |g|^2 \, dx &=& e'(u;g(\lambda^k))
- \int_{\Omega\backslash\Omega_\infty} \lambda^k |\nabla g(\lambda^k)|^2 + |g|^2 \, dx \\ &\leq& C, \end{eqnarray*} because $(g(\lambda^k))$ is bounded in $H$ and $(\lambda^k)$ is bounded in $L^\infty(\Omega\backslash\Omega_\infty)$. It follows that
$\lim_{k\to\infty}|\nabla g(\lambda^k)|=0$ and $\lim_{k\to\infty}g(\lambda^k)=g$ in $L^2(\Omega_\infty)$. Hence, $g=C_i$ in $\Omega_i$, $C_i\in\mathbb R$. \revt{Thus, $g\in
H^1_0(\Omega;\Omega_\infty^\text{const})$ solves
\eqref{e:g-in-G-G0-Ginf}}.
Finally, \eqref{e:j(hl),=liminf} follows from the fact that $f$ is convex and strongly continuous in $H$, so $f$ is weakly lower semi-continuous, see \cite{brezis-1}.
$\Box$
\begin{remark}\label{r:I0,Iinf}
While analyzing case (ii) we will use the following notation. For
$\lambda=[\lambda_1,\ldots,\lambda_N]\in\partial\Lambda^+$ we write \begin{equation} \left\{ \begin{array}{rcl}
I_{0,\lambda}&=&\cup\{i\in I,\;\, \lambda_i=0\},\\
\Omega_{0,\lambda}&=&\cup\{\Omega_i,\;\; i\in I_{0,\lambda}\}, \end{array} \right. \quad \left\{ \begin{array}{rcl}
I_{\infty,\lambda}&=&\cup\{i\in I,\;\, \lambda_i=+\infty\},\\
\Omega_{\infty,\lambda}&=&\cup\{\Omega_i,\;\; i\in I_{\infty,\lambda}\}. \end{array} \right. \end{equation} For ${\widehat{\lambda}}\in\partial\Lambda^+$ instead we write \begin{equation} \left\{ \begin{array}{rcl}
I_0&=&\cup\{i\in I,\;\, {\widehat{\lambda}}_i=0\},\\
\Omega_0&=&\cup\{\Omega_i,\;\; i\in I_0\}, \end{array} \right. \quad \left\{ \begin{array}{rcl}
I_\infty&=&\cup\{i\in I,\;\, {\widehat{\lambda}}_i=+\infty\},\\
\Omega_\infty&=&\cup\{\Omega_i,\;\; i\in I_\infty\}. \end{array} \right. \end{equation} \end{remark}
\begin{remark}\label{r:hlambda,H10}
In the case when $dim(\Lambda)=1$, i.e., $\Lambda=\mathbb R$, and the space $H$
is equipped with the inner product \begin{equation}
(u,v)_H=\int_\Omega \nabla u \cdot\nabla v \, dx, \label{eq:ipH10} \end{equation} the optimal weight ${\widehat{\lambda}}$ is given explicitly. Indeed, if $\displaystyle (u,v)_\lambda=\int_\Omega \lambda(\nabla u \cdot\nabla v) \, dx$, then $g(\lambda)$ is defined by \begin{equation*}
\int_\Omega\lambda(\nabla g(\lambda)\cdot\nabla v) \, dx = e'(u;v). \end{equation*} This implies $g(\lambda)=\frac{1}{\lambda}g_1$ and then \begin{equation*}
j(\lambda)=\frac{\kappa}{2}\frac{1}{\lambda^2}e''(u;g_1,g_1)-\frac{1}{\lambda}e'(u;g_1). \end{equation*} It follows that the solution ${\widehat{\lambda}}$ of \eqref{e:min(lambda)->1} is given by \begin{equation} \label{e:hlambda,N=1} {\widehat{\lambda}}=\kappa\frac{e''(u;g_1,g_1)}{e'(u;g_1)}. \end{equation} Note that ${\widehat{\lambda}}>0$ because $e'(u;g_1)>0$ and $e''(u;g_1,g_1)>0$. Thus, in the case when $dim(\Lambda)=1$ and the space $H$ is endowed with inner product \eqref{eq:ipH10}, the proposed approach will consist of Algorithm \ref{alg:gradient} with step 4 amended as \begin{eqnarray} &4.& \mbox{determine $g=g({\widehat{\lambda}})$, where ${\widehat{\lambda}}\in\Lambda^+$ is given by \eqref{e:hlambda,N=1}}. \label{alg:gradient+2,N=1} \end{eqnarray} We remark that, interestingly, since ${\widehat{\lambda}}$ is proportional to the step size $\kappa$ and $g({\widehat{\lambda}})$ is proportional to $1/{\widehat{\lambda}}$, in the present case the iterations produced by Algorithm \ref{alg:gradient} will not depend on $\kappa$.
The optimal ${\widehat{\lambda}}$ given in \eqref{e:hlambda,N=1} plays a similar role to the parameter $\alpha$ used in the Barzilai-Borwein version of the gradient method for minimization in $\mathbb \mathbb R^n$ \cite{barzilai-1}. However, here the idea behind the choice of ${\widehat{\lambda}}$ given by \eqref{e:min(lambda)->1} or \eqref{e:hlambda,N=1} is to approximate Newton's step. On the other hand, in \cite{barzilai-1} the optimal $\alpha$ is chosen such that the resulting gradient is a two-point approximation to the secant direction used in the quasi-Newton methods. \end{remark}
\section{Error analysis} \label{sec:errors} In the following, we first present the analysis of case (i) of Proposition \ref{p:lambda^k->}.
\subsection{Error analysis: case ${\widehat{\lambda}}\in\Lambda^+$} \label{s:analysis->1}
\noindent The following proposition gives the differentiability of the map $g$. \begin{proposition} \label{p:g'->1} Let ${\widehat{\lambda}}\in\Lambda^+$ be a solution of \eqref{e:min(lambda)->1}. Then $g\in C^1(\Lambda,H)$ and $j\in C^1(\Lambda)$ near ${\widehat{\lambda}}$. Furthermore, for all $v\in H$ and $\ell\in\Lambda$ we have \begin{eqnarray} \int_{\Omega} {\widehat{\lambda}} (\nabla g'({\widehat{\lambda}};\ell)\cdot\nabla v) + g'({\widehat{\lambda}};\ell)v \, dx &=& - \int_{\Omega} \ell (\nabla g({\widehat{\lambda}})\cdot \nabla v) \, dx, \label{e:g'->1}\\ e''(u;\kappa g({\widehat{\lambda}}),g'({\widehat{\lambda}};\ell))&=&e'(u;g'({\widehat{\lambda}};\ell)), \label{e:e''=e'->1} \end{eqnarray} where $g'({\widehat{\lambda}};\ell)$ is the derivative of $g$ at ${\widehat{\lambda}}$ in the direction $\ell$. \end{proposition} {\bf Proof}. To prove the differentiability of $g$ we consider the map \begin{eqnarray*} F:\Lambda\times H&\mapsto&H' \\ (\lambda,g)&\to& F(\lambda,g), \quad F(\lambda,g)(v) = \int_{\Omega} \lambda (\nabla g\cdot\nabla v) + gv \, dx - e'(u;v),\;\, v\in H. \end{eqnarray*} Note that $F$ is $C^1$ and \begin{equation*} \partial_g F({\widehat{\lambda}},g)(z) = \int_{\Omega} {\widehat{\lambda}} (\nabla z\cdot\nabla v) + zv \, dx,\; z\in H. \end{equation*} It follows from the Lax-Milgram lemma that $\partial_g F({\widehat{\lambda}},g)$ defines an isomorphism from $H$ to $H'$. Then, the differentiability of $g$ is easily deduced by using the implicit function theorem and the fact that the \revt{equation} $F(\lambda,g)=0$ has a unique solution $g\in H$ for any given $\lambda\in\Lambda^+$. In addition, it follows that $\lambda\in\Lambda\mapsto j(\lambda)\in\mathbb R$ is also $C^1$ near ${\widehat{\lambda}}$, because $g\in C^1(\Lambda;H)$ and $f$ is continuous in $H$.
Equalities \eqref{e:g'->1} and \eqref{e:e''=e'->1} are obtained after straightforward computations.
$\Box$
\begin{corollary} \label{c:T->1} Let ${\widehat{\lambda}}\in\Lambda^+$ be a solution of \eqref{e:min(lambda)->1}, $T_u=\Span\{g'({\widehat{\lambda}};\ell),\;\ell\in\Lambda\}$ and $P_u:H\mapsto T_u$ be the projection operator with respect to the inner product $e''(u;\cdot,\cdot)$, i.e., \begin{equation} e''(u;w-T_u w,v) = 0,\quad \forall w\in H,\;\, v\in T_u. \end{equation} Then $P_uh = P_u(\kappa g)$ and $d\leq\dim(T_u)\leq {N}$, where $d=\rank\{i,\; g({\widehat{\lambda}})\neq g_0 \ \textrm{in} \ {\cal D}'(\Omega_i) \}$ (we recall that $g_0$ is the $L^2$ \revt{representation of
$e'(u)$, cf.~Remark \ref{r:Riesz}}). \end{corollary} {\bf Proof}. From \eqref{e:e''=e'->1} and $e''(u;h,v)=e'(u;v)$ for all $v\in H$, it follows $P_u h = P_u(\kappa g)$.
Clearly $\dim(T_u)\leq N$. Now we show that $d\leq \dim(T_u)$. For simplicity and without loss of generality we assume that $g({\widehat{\lambda}})\neq g_0$ in ${\cal D}'(\Omega_i)$ for all $i=1,\ldots,d$.
It is enough to show that $\{g'({\widehat{\lambda}};\ell^i),\; i=1,\ldots,d\}$ are linearly independent. Let $\sum_{i=1,d}\alpha_i g'({\widehat{\lambda}};\ell^i)=0$, $\alpha_i\in\mathbb R$. From \eqref{e:g'->1} we obtain \begin{eqnarray*} 0 &=& \int_{\Omega}{\widehat{\lambda}} \nabla \left(\sum_{i=1,d}\alpha_i g'({\widehat{\lambda}};\ell^i)\right)\cdot\nabla v \, dx + \left(\sum_{i=1,d}\alpha_i g'({\widehat{\lambda}};\ell^i)\right)v \, dx \\ &=& - \int_{\Omega} \left(\sum_{i=1,d}\alpha_i\ell^i\right)\nabla g({\widehat{\lambda}})\cdot\nabla v \, dx. \end{eqnarray*} Then, taking $v\in{\cal D}(\Omega_i)$ gives \[ 0 = \sum_{j=1}^d \alpha_j \int_{\Omega} \ell^j (\nabla g({\widehat{\lambda}})\cdot\nabla v) \, dx = \alpha_i \int_{\Omega_i} \nabla g({\widehat{\lambda}})\cdot \nabla v \, dx = \alpha_i \int_{\Omega_i} \Delta g({\widehat{\lambda}}) v\, dx.
\]
Hence $\Delta g({\widehat{\lambda}})=0$ in ${\cal D}'(\Omega_i)$. Since $-\nabla\cdot({\widehat{\lambda}}\nabla g({\widehat{\lambda}})) +g({\widehat{\lambda}})=g_0$ in ${\cal
D}'(\Omega)$ and since ${\widehat{\lambda}}$ is constant in each $\Omega_i$, it follows \revt{that} $\alpha_i(g({\widehat{\lambda}})-g_0)=0$ in ${\cal
D}'(\Omega_i)$, hence $\alpha_i = 0$.
$\Box$
\begin{remark}
The \revt{estimate of} the dimension of $T_u$ is optimal. In fact,
we can prove that $d={\rm dim}\{i\in I,\;
g'({\widehat{\lambda}};\ell^i)\neq0\}$. Indeed, if $\Delta g({\widehat{\lambda}})=0$ in
$\Omega_i$, we can \revt{show that} $\partial_\nu g({\widehat{\lambda}})=0$ on
$\partial\Omega_i$, \revt{where $\nu$ is the direction of the normal
vector on $\partial\Omega_i$,} and then from (\ref{e:g'->1}) we
get $g'({\widehat{\lambda}};\ell^i)=0$. \end{remark}
\noindent Now we are able to prove the error estimates for our method. \begin{theorem} \label{th:P+newton->1} Assume $e(u)$ satisfies the assumptions of Theorems \ref{th:gradient} and \ref{th:newton} near the solution ${\widehat{u}}$ of (\ref{e:min{e(u)}}). Let $u$ be close to ${\widehat{u}}$ and $\widetilde{u}$ be given by Step 5 of Algorithm \ref{alg:gradient} with $g=g({\widehat{\lambda}})$ and ${\widehat{\lambda}}\in\Lambda^+$ a solution of (\ref{e:min(lambda)->1}). Then we have \begin{eqnarray}
\|\widetilde{u}-{\widehat{u}}\|_{\widehat{\lambda}} &\leq&
\epsilon \|u-{\widehat{u}}\|_{\widehat{\lambda}},
\label{e:|tu-u|->1}\\
\|P_u(\widetilde{u}-{\widehat{u}})\|_H &\leq&
C\|u-{\widehat{u}}\|_H^2,
\label{e:|P(tu-u)|->1} \end{eqnarray} with $\epsilon\in(0,1)$ depending only on $\kappa$ and $e$ and $C>0$ depending only on $e$. \end{theorem}
{\bf Proof}. Estimate \eqref{e:|tu-u|->1} follows from Theorem \ref{th:gradient}, where the norm is changed to
$\|\cdot\|_{\widehat{\lambda}}$, because the gradient is now computed with respect to the inner product $(\cdot,\cdot)_{\widehat{\lambda}}$.
For \eqref{e:|P(tu-u)|->1}, we note that $P_u$ is a linear continuous operator with
$\|P_u\|_H\leq M$. Then \begin{eqnarray*} P_u(\widetilde{u}-{\widehat{u}}) &=& P_u(u-\kappa g({\widehat{\lambda}}) - {\widehat{u}}) \\ &=& P_u(u-{\widehat{u}}) - P_u(\kappa g({\widehat{\lambda}})) \qquad(use\;\ Corollary\;\,\ref{c:T->1}) \\ &=& P_u(u-{\widehat{u}}) - P_u(h) \\&=& P_u(u-h - {\widehat{u}}). \end{eqnarray*} Therefore \begin{eqnarray*}
\|P_u(\widetilde{u}-{\widehat{u}})\|_H & = &
\|P_u(u-h-{\widehat{u}})\|_H \\ &\leq&
M\|u-h-{\widehat{u}}\|_H \qquad(use\; Theorem\; \ref{th:newton}) \\ &\leq&
C\|u-{\widehat{u}}\|_H^2. \end{eqnarray*}
\begin{remark}
\label{r:error->1}
Theorem \ref{th:P+newton->1} states that $\|P_u(\widetilde{u}-{\widehat{u}})\|_H$,
the error of our method at a given step projected onto the tangent plane $T_u$,
decreases at least quadratically in terms of $\|u-{\widehat{u}}\|_H$. \end{remark}
\subsection{Error analysis: case ${\widehat{\lambda}}\in\partial\Lambda^+$} \label{s:analysis->2}
In case (ii) of Proposition \ref{p:lambda^k->} we are led to consider $g({\widehat{\lambda}})$ associated to ${\widehat{\lambda}}\in\partial\Lambda^+$ with ${\widehat{\lambda}}_i=0$ for $i\in I_0$ and ${\widehat{\lambda}}_i=+\infty$ for $i\in I_\infty$, which solves \eqref{e:g(lambda)}. To obtain error estimates similar to the ones given by Theorem \ref{th:P+newton->1}, we would like to have differentiability results similar to the ones given by Proposition \ref{p:g'->1}, which means that we \revt{would} have to compare $g({\widehat{\lambda}})$ with $g(\lambda)$, $\lambda\in\partial\Lambda^+$. \revt{However}, in general, for $\lambda\in\partial\Lambda^+$ equation \eqref{e:g(lambda)} does not provide \revt{an} estimate {in $H$} for $g(\lambda)$ and therefore {the analysis from the previous section cannot be applied directly}.
\revt{On the other hand,} equation \eqref{e:g(lambda)} with ${\widehat{\lambda}}\in\partial\Lambda^+$ implies extra regularity for $e'(u)$, in particular in $\Omega_0$. Assuming that $e''(u)$ {possesses} the same kind of regularity, which comes naturally from the problem, we will prove an error estimate for case (ii) of Proposition \ref{p:lambda^k->} similar to the one already given in Theorem \ref{th:P+newton->1}, but in a weaker norm.
\begin{proposition}\label{p:g=g_0} Let $\Omega_{0,{H^1}}$ be the largest union of $\overline{\Omega}_i$ such that $g_0\in H^1(\Omega_{0,H^1})$ and $I_{0,H^1}=\{i\in I,\; \Omega_i\subset\Omega_{0,H^1}\}$. \\ (i) If $I_{0,H^1}=\emptyset$, then case (ii) of Proposition \ref{p:lambda^k->} does not happen. \\ (ii) If $I_{0,H^1}\neq\emptyset$, then $I_0\subset I_{0,H^1}$, $\Omega_0\subset \Omega_{0,H^1}$ and \begin{alignat}{2}
g({\widehat{\lambda}})&=g_0 & &\; \text{\it in} \ H^1(\Omega_0). \label{e:g-in-G0}
\end{alignat} (iii) Furthermore, $e'(u;\cdot)$ is continuous in $H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap H^1(\Omega_0)$, where \begin{eqnarray} H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap H^1(\Omega_0) &=& \{ v\in H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0),\;\; \mbox{\it $v=0$\;\, on\;\, $\partial\Omega$\;\, and} \nonumber\\ && \hspace*{2mm} v=C_i\;\,\mbox{\it in}\; \Omega_\infty^i,\;\, i\in I_\infty\}. \label{e:N1_0(G-G0;Ginf)} \end{eqnarray} \end{proposition} {\bf Proof}. Indeed, from \eqref{e:g-in-G-G0-Ginf} and \eqref{e:g(l),g_0}, we get $g({\widehat{\lambda}})=g_0$ in ${\cal
D}'(\Omega_0)$. As $g({\widehat{\lambda}})\in H^1_0(\Omega)$, the claims (i) and (ii) follow.
The form of the inner product $(g,v)_{{\widehat{\lambda}}}$ and the fact that ${\widehat{\lambda}}=0$ in $\Omega_0$ \revt{imply} (iii).
$\Box$\\
\noindent Motivated by Proposition \ref{p:lambda^k->}, we are led to the following definition.
\begin{definition}
Let $\lambda=[\lambda_1,\ldots,\lambda_N]\in\partial\Lambda^+$ with
$I_{0,\lambda}\subset I_{0,H^1}$.
We define $g(\lambda)\in
H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda})$ by
\begin{eqnarray}
\int_{\Omega\backslash\Omega_{0,\lambda}} \lambda (\nabla g(\lambda)\cdot\nabla v) + g(\lambda)v \, dx &+& \int_{\Omega_{0,\lambda}}g(\lambda)v \, dx = e'(u;v),\nonumber\\ && \forall v\in H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda}). \label{e:g''} \end{eqnarray} \end{definition}
\begin{proposition} \label{p:g->2;wd} Let $\lambda\in\partial\Lambda^+$ with with $I_{0,\lambda}\subset I_{0,H^1}$. Then \eqref{e:g''} has a unique solution $g(\lambda)\in H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda})$ and $g(\lambda)=g_0$ in $H^1(\Omega_{0,\lambda})$. \end{proposition} {\bf Proof}. The existence and uniqueness of $g(\lambda)$ follows from the Lax-Milgram lemma applied in the space $H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap L^2(\Omega_{0,\lambda})$ equipped with the inner product \begin{equation*} (g,v)_\lambda = \int_{\Omega} \lambda (\nabla g\cdot\nabla v) + gv \, dx = \int_{\Omega\backslash\Omega_{0,\lambda}} \lambda (\nabla u\cdot\nabla v) + gv \, dx + \int_{\Omega_{0,\lambda}}gv \, dx. \end{equation*} Reasoning as in \revt{case (ii) of} Proposition \ref{p:lambda^k->}, we see that $g(\lambda)$ is constant in $\Omega_{\infty,\lambda}^i$, for all $i\in I_{0,\lambda}$. Finally, taking $v\in{\cal D}(\Omega_0)$ we \revt{obtain} $\int_{\Omega_0}(g(\lambda)-g_0)v \,dx=0$, which implies \revt{that} $g(\lambda)=g_0$ in ${\cal D}(\Omega_{0,\lambda})$ and, as $g_0\in H^1(\Omega_{0,H^1})$, completes the proof.
$\Box$ \\
Returning to the minimization problem \eqref{e:min(lambda)->1} and in view of \revt{case (ii) of} Proposition \ref{p:lambda^k->}, we are led to consider the problem \begin{equation} \label{e:min(lambda)->2} \text{find ${\widehat{\lambda}}\in\partial\Lambda^+$ such that }\; j({\widehat{\lambda}}) := \min\{j(\lambda)=(f\circ g)(\lambda),\;\, \lambda\in\partial\Lambda^+\} \end{equation} and \revt{from it eventually obtain} a necessary condition
\revt{analogous to} \eqref{e:e''=e'->1}, which was a key element in proving estimate \eqref{e:|P(tu-u)|->1}.
We would repeat the \revtt{analysis} already applied to problem \eqref{e:min(lambda)->1}, \revtt{as} in Section \ref{s:analysis->1}. However, since $g(\lambda)$ \revt{now} defined via \eqref{e:g''} does not in general belong to $H^1_0(\Omega)$, $j(\lambda)$ may not be well defined.
It appears that there are no general conditions on the data which would ensure that $g(\lambda)\in H^1_0(\Omega)$ when $\lambda\in\partial\Lambda^+$. We will thus proceed with the analysis of this case under the following stronger assumptions on $e'$ and $e''$, which are motivated by the continuity of $e'(u;\cdot)$ in $H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$, see Proposition \ref{p:g=g_0}.
Let us \revt{introduce} the following definitions \begin{eqnarray} {\cal H} &=& \{u\in H^1(\Omega\backslash\Omega_{0,H^1}),\; v\in H^1(\Omega_i),\;\, i\in I_{0,H^1},\;\, u=0\;\; on\;\,\partial\Omega\}, \\ (u,v)_{{\cal H}} &=& \int_{\Omega\backslash\Omega_{0,H^1}}(\nabla u\cdot\nabla v) + uv \, dx + \sum_{i\in I_{0,H^1}}\int_{\Omega_i}(\nabla u\cdot\nabla v) + uv \, dx, \\
\|v\|_{{\cal H}}^2 &=& (v,v)_{{\cal H}}. \end{eqnarray} The set ${\cal H}$ equipped with the inner product $(v,v)_{{\cal H}}$ is a Hilbert space.
In the reminder of this section we will assume
\begin{equation}\label{e:assumption(H10)} \left\{ \begin{array}{l} \mbox{\it $e$, $e'$ and $e''$ satisfy all the conditions of Theorems \ref{th:gradient} and \ref{th:newton} with} \\ \mbox{\it $H$ replaced by ${\cal H}$}. \end{array} \right. \end{equation} Moreover, we will assume \begin{eqnarray}
|e''(u;v,w)|&\leq&M\|v\|_{{\cal H}}\|w\|_{{\cal H}}, \label{e:e->C0,'}\\ e''(u;v,v) &\geq&
m\|v\|_{{\cal H}}^2, \label{e:e->convex,'} \end{eqnarray} with certain $0<m<M<\infty$.
\begin{proposition} \label{p:min(lambda)->2} Assume $e''$ satisfies \eqref{e:assumption(H10)}--\eqref{e:e->convex,'}. For $\lambda\in\partial\Lambda^+$ with $I_{0,\lambda}\subset I_{0,H^1}$ let $g(\lambda)\in H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda})$ be defined by (\ref{e:g''}). Then the problem \eqref{e:min(lambda)->2} has a solution ${\widehat{\lambda}}\in\partial\Lambda^+$. \end{proposition} {\bf Proof}. Let $(\lambda^k)$ be a sequence in $\partial\Lambda^+$ minimizing $j$ in $\partial\Lambda^+$. As $\dim(\Lambda)=N$, without loss of generality, we may assume that there exist $I_0\subset I$, $I_\infty\subset I$ such that $I_{0,\lambda^k}=I_0$, $I_{\infty,\lambda^k}=I_\infty$ for all $k$. It follows {that} $\Omega_{0,\lambda^k}=\Omega_0$, $\Omega_{\infty,\lambda^k}=\Omega_\infty$.
Since $f$ is elliptic in ${\cal H}$ and $g(\lambda^k)\in H^1(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$, for all $k$, necessarily $(g(\lambda^k))$ is bounded in $H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$. Therefore, without loss of generality, we may assume that $(g(\lambda^k))$ converge weakly in $H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$ and strongly in $L^2(\Omega)$ to a certain $g\in H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$. As $f$ is convex, it follows that $f(g)\leq\liminf_{k\to\infty}j(\lambda^k)$, \revt{see
\cite{brezis-1}}.
To conclude that \eqref{e:min(lambda)->2} has a solution, it is enough to show that $g=g({\widehat{\lambda}})$ for a certain ${\widehat{\lambda}}\in\partial\Lambda^+$. For the sequence $\lambda^k$ two cases may occur.\\ (i) There exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]$ with ${\widehat{\lambda}}_i=0$ for $i\in I_0$, ${\widehat{\lambda}}_i=+\infty$ for $i\in I_\infty$, ${\widehat{\lambda}}_i\in(0,+\infty)$ for $i\in I\backslash(I_0\cup I_\infty)$, and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$, such that $\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$. Note that $g_k=g(\lambda^k)\in H^1(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$ satisfies \eqref{e:g''}, i.e.
\begin{eqnarray} \int_{\Omega\backslash\Omega_0} \lambda^k (\nabla g_k\cdot\nabla v) + g_kv \, dx +
\int_{\Omega_0} g_kv \, dx &=& e'(u;v), \label{e:g'',k} \end{eqnarray} for all $v\in H^1(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$. Passing {to the} limit in \eqref{e:g'',k}, we find that $g\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap H^1(\Omega_0)$ solves \eqref{e:g''} so $g=g({\widehat{\lambda}})$. \\ (ii) There exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]$, $i_0\subset I$, $i_\infty\subset I$ with ${\widehat{\lambda}}_i=0$ for $i\in I_0\cup i_0$, ${\widehat{\lambda}}_i=+\infty$ for $i\in I_\infty\cup i_\infty$, ${\widehat{\lambda}}_i\in(0,+\infty)$ for $i\in I\backslash((I_0\cup i_0)\cup (I_\infty\cup i_\infty))$, and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$, such that $\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$.
We take $v\in H^1_0(\Omega\backslash\Omega_0;(\Omega_\infty\cup\omega_\infty)^\text{const})\cap H^1(\Omega_0)$ in \eqref{e:g'',k}, where $\omega_0=\cup\{\Omega_i,\; i\in i_0\}$, $\omega_\infty=\cup\{\Omega_i,\; i\in i_\infty\}$, and we obtain \begin{eqnarray*} \int_{\Omega\backslash(\Omega_0\cup \omega_0)} \lambda^k (\nabla g_k\cdot\nabla v) + g_kv \, dx &+& \int_{\Omega_0} g_kv \, dx + \int_{\omega_0} \lambda^k (\nabla g_k\cdot\nabla v) + g_kv \, dx \nonumber\\ &=& e'(u;v). \label{e:g'',k'} \end{eqnarray*} Letting $k\to\infty$ gives \begin{eqnarray*} \int_{\Omega\backslash(\Omega_0\cup \omega_0)} \hspace*{-5mm} {\widehat{\lambda}} (\nabla g\cdot\nabla v) + gv \, dx + \int_{\Omega_0\cup\omega_0} \hspace*{-3mm} gv \, dx &=& e'(u;v), \end{eqnarray*} for all $v\in H^1_0(\Omega\backslash\Omega_0;(\Omega_\infty\cup\omega_\infty)^\text{const})\cap H^1(\Omega_0)$. Reasoning as in Proposition \ref{p:lambda^k->}, we find that $g=g({\widehat{\lambda}})\in H^1_0(\Omega\backslash\Omega_0;(\Omega_\infty\cup\omega_\infty)^\text{const})\cap H^1(\Omega_0)$ solves \eqref{e:g''} with $\lambda={\widehat{\lambda}}$ and $\Omega_0\cup\omega_0$ (respectively, $\Omega_\infty\cup\omega_\infty$) instead of $\Omega_0$ (respectively, $\Omega_\infty$).
$\Box$
\begin{remark}\label{r:sigma(lambda)}
For ${\widehat{\lambda}}\in\partial\Lambda^+$, in order to control the
variations of ${\widehat{\lambda}}$ in the set
$\Omega\backslash(\Omega_0\cup\Omega_\infty)$ we consider ${\mathbb
1}_{0,\infty}\in\partial\Lambda^+$ defined by \[ {\mathbb 1}_{0,\infty}(x) = \left\{ \begin{array}{ll} 0,&x\in \Omega_0\cup\Omega_\infty,\\ 1,&x\in\Omega\backslash(\Omega_0\cup\Omega_\infty). \end{array} \right. \] Then we perturb ${\widehat{\lambda}}$ with the elements of $\revt{\widehat{\Lambda}}:={\mathbb 1}_{0,\infty}\cdot \Lambda = \{{\mathbb 1}_{0,\infty}\cdot\lambda,\; \lambda\in\Lambda\}$. \end{remark}
\begin{proposition}\label{p:g'->2} Let ${\widehat{\lambda}}\in\partial\Lambda^+$ be the solution of \eqref{e:min(lambda)->2} as given by Proposition \ref{p:min(lambda)->2}. The map $\lambda\in\hat{\Lambda} \mapsto g(\lambda)\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$ is $C^1$ near ${\widehat{\lambda}}$. Furthermore, $g'({\widehat{\lambda}};\ell)\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$, where $g'({\widehat{\lambda}};\ell)$ is the derivative of $g({\widehat{\lambda}})$ at ${\widehat{\lambda}}$ in \revt{the} direction $\ell\in\hat{\Lambda}$, and satisfies \begin{eqnarray} \int_{\Omega\backslash\Omega_0} {\widehat{\lambda}}(\nabla g'({\widehat{\lambda}};\ell)\cdot\nabla v) &+& g'({\widehat{\lambda}};\ell) v \, dx + \int_{\Omega_0} g'({\widehat{\lambda}};\ell) v \, dx = -\int_{\Omega\backslash\Omega_0} \ell (\nabla g({\widehat{\lambda}})\cdot\nabla v) \, dx, \nonumber\\ && \forall v\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0). \label{e:Dg''} \end{eqnarray} In particular, $g'({\widehat{\lambda}};\ell)=0$ in $\Omega_0$ and for every $\ell\in \revt{\widehat{\Lambda}}$ we have \begin{equation}\label{e:e''=e'->2}
e''(u;\kappa g({\widehat{\lambda}}),g'({\widehat{\lambda}};\ell))=e'(u;g'({\widehat{\lambda}};\ell)). \end{equation} \end{proposition} {\bf Proof}. The differentiability of $g$ is deduced from the implicit mapping theorem as follows. Consider the map $F$ \begin{equation*}
\begin{array}{lcll}
F:&\hat{\Lambda}\times H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap L^2(\Omega_0)
&\mapsto &
(H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap L^2(\Omega_0))'
\\
&(\lambda,g)&\to& F(\lambda,g),
\end{array} \end{equation*} with \begin{equation*} F(\lambda,g) = \int_{\Omega\backslash\Omega_0} \lambda(\nabla g(\lambda)\cdot\nabla v) + g(\lambda) v \, dx + \int_{\Omega_0} g(\lambda) v \, dx - e'(u;v). \end{equation*} Clearly, $F$ is $C^1$ near $({\widehat{\lambda}},g({\widehat{\lambda}}))$. Furthermore, we have \begin{eqnarray*} \partial_g F({\widehat{\lambda}},g({\widehat{\lambda}}))(z) &=& \int_{\Omega\backslash\Omega_0} {\widehat{\lambda}}(\nabla z\cdot\nabla v) + z v \, dx + \int_{\Omega_0} zv \, dx, \end{eqnarray*} which defines an isomorphism from $H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap L^2(\Omega_0)$ to its dual. Then, the implicit mapping theorem and the fact that $F(\lambda,g)=0$ has a unique solution $g\in H^1(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap H^1(\Omega_0)$ for any $\lambda\in \revt{\widehat{\Lambda}}$ gives the differentiability of the map $g$. Note that, a priori, \revt{the} implicit mapping theorem \revt{ensures} the differentiability of the map $g$ in $L^2(\Omega_0)$. \revt{Then,} as $g(\lambda)=g_0\in H^1(\Omega_0)$, see Proposition \ref{p:g->2;wd}, the differentiability of the map $g$ in $H^1(\Omega_0)$ \revt{follows as well}.
Next, by direct computations we can easily show \eqref{e:Dg''}. Furthermore, $g'({\widehat{\lambda}};\ell)=0$ in $\Omega_0$ because $g(\lambda)=g_0$ in $H^1(\Omega_0)$.
In regard to \eqref{e:e''=e'->2}, we recall that $e'(u)$ and $e''(u)$ are, respectively, linear and bilinear, and continuous in ${\cal H}$, which together with the identity $g(\lambda)=g_0$ in $\Omega_0$ and the differentiability of $g(\lambda)$ \revt{implies} the differentiability of $\lambda\mapsto e'(u;g(\lambda))$ and $\lambda\mapsto e''(u;g(\lambda),g(\lambda))$. Then, (\ref{e:e''=e'->2}) follows from straightforward computations.
$\Box$ \\
The error estimates are obtained in an analogous way to the corresponding results in Section \ref{s:analysis->1}. First, we have a result similar to Corollary \ref{c:T->1}.
\begin{corollary}\label{c:T->2} Assume $e''(u)$ satisfies conditions \eqref{e:assumption(H10)}--\eqref{e:e->convex,'}. Let ${\widehat{\lambda}}\in\partial\Lambda^+$ be a solution of \eqref{e:min(lambda)->2}, $g=g({\widehat{\lambda}})$, $T_u=\Span\{g'({\widehat{\lambda}};\ell),\;\ell\in\hat{\Lambda}\}$ and $P_u:{\cal H}\mapsto T_u$ be the projection operator with respect to the inner product $e''(u;\cdot,\cdot)$, i.e., \begin{equation} e''(u;w-P_uw,v) = 0,\quad \forall w\in {\cal H},\;\, \forall v\in T_u. \end{equation}
Then, $P_u h = P_u(\kappa g)$ and $d\leq \dim(T_u)\leq N-|I_0|-|I_\infty|$, where $d=\rank\{i,\; g\neq g_0\; \mbox{in}\; {\cal D}'(\Omega_i)\}$.
\end{corollary} {\bf Proof}. From \eqref{e:e''=e'->2} and the relation $e''(u;h,v)=e'(u;v)$ for all $v\in H$, it follows that $P_u h = P_u(\kappa g)$.
Clearly $\dim(T_u)\leq N-|I_0|-|I_\infty|$. Now we show that $\dim(T_u)\geq d$. We assume that $g({\widehat{\lambda}})\neq g_0$ in ${\cal D}'(\Omega_i)$, for all $i=1,\ldots,d$. Let $\ell^i\in\hat{\Lambda}$, $\ell^i(\Omega_j)=\delta_{ij}$. It is enough to demonstrate that $\{g'({\widehat{\lambda}};\ell^i),\; i=1,\ldots,d\}$ are linearly independent. Let $\sum_{i=1,d}\alpha_i g'({\widehat{\lambda}};\ell_i)=0$, $\alpha_i\in\mathbb R$. Then, \begin{eqnarray*} 0 &=& \int_{\Omega\backslash\Omega_0} {\widehat{\lambda}}
\nabla \left(\sum_{i=1,d}\alpha g'({\widehat{\lambda}};\ell^i)\right)\cdot\nabla v + \left(\sum_{i=1,d}\alpha_i \nabla g'({\widehat{\lambda}};\ell^i)\right)v \, dx \\ &+& \int_{\Omega_0} \left(\sum_{i=1,d}\alpha_i \nabla g'({\widehat{\lambda}};\ell^i)\right)v \, dx \\ &=& \int_{\Omega\backslash\Omega_0} \left(\sum_{i=1,d}\alpha_i \ell^i\right) (\nabla g({\widehat{\lambda}})\cdot\nabla v) \, dx. \end{eqnarray*} In the equality above we take $v\in{\cal D}(\Omega_i)$. Then \begin{eqnarray*} 0 &=& \int_{\Omega\backslash\Omega_0} \alpha_i(\nabla g({\widehat{\lambda}})\cdot \nabla v) \, dx = - \alpha_i\int_{\Omega}\Delta g({\widehat{\lambda}}) v \, dx = - \alpha_i\int_{\Omega}(g({\widehat{\lambda}})-g_0)v \, dx, \end{eqnarray*} because $-{\widehat{\lambda}}\Delta g({\widehat{\lambda}})+g({\widehat{\lambda}})=g_0$ in $\Omega_i$, which implies $\alpha_i=0$ and proves the claim.
$\Box$ \\
\noindent Finally, we are able to prove the error estimate for the case ${\widehat{\lambda}}\in\partial\Lambda^+$. \begin{theorem} \label{th:P+newton->2} Assume $e$ satisfies the conditions of Theorems \ref{th:gradient}, \ref{th:newton} with ${\cal H}$ instead of $H$, and $e''(u)$ satisfies conditions \eqref{e:assumption(H10)}--\eqref{e:e->convex,'}. Let ${\widehat{\lambda}}\in\partial\Lambda^+$ be a solution of \eqref{e:min(lambda)->2}, as given by Proposition \ref{p:min(lambda)->2}. If $\widetilde{u}$ and $u$ are given as in Algorithm \ref{alg:gradient} with $g=g({\widehat{\lambda}})$, then \begin{eqnarray}
\|\widetilde{u}-{\widehat{u}})\|_{\widehat{\lambda}} &\leq&
\epsilon \|u-{\widehat{u}}\|_{\widehat{\lambda}},
\label{e:|tu-u|->2}\\
\|P_u(\widetilde{u}-{\widehat{u}})\|_{{\cal H}} &\leq&
C\|u-{\widehat{u}}\|_{{\cal H}}^2,
\label{e:|P(tu-u)|->2} \end{eqnarray} with $\epsilon\in(0,1)$ depending only on $\kappa$ and $e$ and $C>0$ depending only on $e$. \end{theorem} {\bf Proof}. The proof is analogous to the proof of Theorem \ref{th:P+newton->1}.
$\Box$
\begin{remark} \label{r:error->2} The estimates in Theorem \ref{th:P+newton->2} are similar to the ones in Theorem \ref{th:P+newton->1}. However, in general, the quadratic convergence established in Theorem \ref{th:P+newton->2} is slower than the one provided by Theorem \ref{th:P+newton->1}, because in Theorem \ref{th:P+newton->2} the dimension of the space $T_u$ is in general smaller than the dimension of the space $T_u$ in Theorem
\ref{th:P+newton->1}. Finally, estimate \eqref{e:|P(tu-u)|->2} {might
be of no} interest because we may have $\Omega\backslash(\Omega_0\cup\Omega_\infty)=\emptyset$ and so $T_u=\{0\}$. In \revt{such} case $g=g_0$ \revt{would be} the $L^2$ gradient \revt{defined in the entire domain} $\Omega$ and the question of its impact on the \revt{performance} of the gradient method is open.
\revt{Lastly}, Theorem \ref{th:P+newton->2} \revt{provides an estimate
applicable at a single step of the gradient approach, cf.~Algorithm
\ref{alg:gradient},} where a certain $u$ is given and the regularity of $e$ is \revt{determined} in terms of the set where the $L^2$ gradient $g_0=g_0(u)$ is $H^1$. In order to \revt{be able to apply} Theorem \ref{th:P+newton->2} at each step, \revt{one} should rather consider \revt{iterations in the space} ${\cal H}=\{u\in H^1(\Omega_i),\; i\in I,\; u=0\; on\; \partial\Omega\}$ and impose the same assumptions as in Theorem \ref{th:P+newton->2}.
\end{remark}
\section{Determination of optimal weights ${\widehat{\lambda}}$ and the corresponding gradients} \label{sec:optimal}
In this section we describe the computational approach which can be used to determine the optimal form of the inner product \eqref{e:(u,v)_lambda}, encoded in its weight $\lambda$, and the corresponding gradient $g(\lambda)$ at each iteration, cf.~modified step 4 of Algorithm \ref{alg:gradient} given by \eqref{alg:gradient+2}. We will focus on the case when ${\widehat{\lambda}} \in \Lambda^+$, cf.~Section \ref{s:analysis->1}, and in order to ensure non-negativity of the weight, in our approach we will use the representation $\lambda(x) = \eta^2(x)$, $\forall x \in \Omega$, where $\eta \; : \; \Omega \mapsto \mathbb R$ is a function {defined} below. For consistency with the notation introduced in the previous sections and without risk of confusion, hereafter we will use both $\lambda$ and $\eta$. Relation \eqref{e:g(lambda)} can then be expressed in the strong form as \begin{equation} \left\{ \begin{alignedat}{2} g -\nabla\cdot(\eta^2 \nabla g) &= g_0(u) & \quad & \text{\it in} \ \Omega, \\ g & = 0 & \quad & \text{\it on} \ \partial\Omega, \end{alignedat} \right. \label{eq:g(eta)} \end{equation} where here $g=g(\eta)$, whereas the minimization problem \eqref{e:min(lambda)->1} becomes \begin{equation} \label{e:min(eta)->1} j({\widehat{\eta}}) := \min\left\{ j(\eta):=f\circ g(\eta),\;\, \eta^2 \in \Lambda^+ \right\}. \end{equation} We will assume that the function $\eta(x)$ is represented with the ansatz \begin{equation} \eta(x) = \sum_{i=1}^N \eta_i \, \ell^i(x), \label{eq:etaN} \end{equation} where $\{ \ell^i\}_{i=1}^N$ is a set of suitable basis functions. Since in a fixed basis the function $\eta(x)$ is determined by the real coefficients $\{ \eta_i\}_{i=1}^N$, we will also use the notation $\eta = [ \eta_1,\dots,\eta_N ]$. We thus obtain a finite-dimensional minimization problem \begin{equation} \min\{ j(\eta),\;\, \eta=[ \eta_1,\dots,\eta_N ] \in \mathbb R^N \}. \label{eq:minj(eta)} \end{equation} Its minimizers ${\widehat{\eta}} = [ {\widehat{\eta}}_1,\dots,{\widehat{\eta}}_N ]$ satisfy the following optimality conditions, which can be viewed as a discrete form of \eqref{e:e''(lambda)=e'}, \begin{equation} \left[ F_i({\widehat{\eta}}) \right] : = \left[\Dpartial{j}{\eta_i}({\widehat{\eta}})\right] = [e''(u;\kappa g({\widehat{\eta}}),g'_i({\widehat{\eta}}))-e'(u;g'_i({\widehat{\eta}}))] = [0], \qquad i=1,\dots,N, \label{eq:alpha->F=0} \end{equation} where $F=[F_1,\ldots,F_N] \; : \; \mathbb R^N \rightarrow \mathbb R^N$ and $g'_i =g'_i(\eta)= \left[\Dpartial{g}{\eta_i}({\widehat{\eta}})\right]$ satisfy the equations \begin{equation} \left\{ \begin{alignedat}{2} g'_i - \nabla\cdot\left({\widehat{\eta}}^2 \, \nabla g'_i \right) &= 2 \nabla\cdot\left( {\widehat{\eta}} \, \ell_i \nabla g\right) & \qquad & \mbox{\it in } \Omega, \\ g'_i &= 0 & & \mbox{\it on }\partial\Omega. \end{alignedat} \right. \label{eq:g'i} \end{equation} The optimal weight ${\widehat{\eta}}$ can be found either by directly minimizing $j(\eta)$, cf.~\eqref{eq:minj(eta)}, using a version of the gradient-descent method, or by solving the optimality conditions \eqref{eq:alpha->F=0} with a version of Newton's method. The two approaches are described below.
\subsection{Optimal weights via gradient minimization} \label{sec:etagrad}
While in practice one may prefer to use a more efficient minimization approach, such as, e.g., the nonlinear conjugate-gradients method \cite{nw00}, for simplicity of presentation here we focus on the gradient steepest-descent method. The step size $\tau_k$ along the gradient can be determined by solving a line-minimization problem, which can be done efficiently using for example Brent's method \cite{nw00}. Step 4 of Algorithm \ref{alg:gradient}, cf.~\eqref{alg:gradient+2}, is then realized by the operations summarized as Algorithm \ref{alg:etagrad}. In actual computations it may also be beneficial to prevent any of the values $\eta_i$ from becoming too close to zero, which is achieved easily by imposing a suitable bound on the step size $\tau_n$.
Having in mind the complexity analysis presented in Section \ref{sec:complexity}, the termination condition for the main loop in Algorithm \ref{alg:etagrad} is expressed in terms of the maximum number $N_g$ of iterations, although in practice it will be more convenient to base this condition on the relative decrease of $j(\eta)$. \begin{algorithm}[h!] \begin{algorithmic}[1] \STATE evaluate adjoint states $z$ and $z(g)$ (if $e(u)$ depends on a PDE equation) \STATE set $k = 1$ \REPEAT \STATE evaluate $g(\eta)$ by solving \eqref{eq:g(eta)} \STATE evaluate $g'_i(\eta)$, $1=1,\dots,N$, by solving problems \eqref{eq:g'i} \STATE evaluate $e'(u; g'_i(\eta))$, $i=1,\ldots,N$ \STATE evaluate $e''(u; \kappa g(\eta), g'_i(\eta))$, $i=1,\ldots,N$ \STATE evaluate $F(\eta)$, cf.~\eqref{eq:alpha->F=0} \STATE perform line-minimization to determine optimal step size \newline
\hspace*{1.0cm} $\tau_k = \argmin_{\tau>0} j(\eta - \tau F(\eta))$ \qquad (Brent's method \cite{nw00})
\\
\STATE set ${\widehat{\eta}} = \eta - \tau_k F(\eta)$ \STATE set $\eta = {\widehat{\eta}}$ \STATE set $k = k + 1$ \UNTIL{ \ $k = N_g$} \STATE obtain $g({\widehat{\eta}})$ by solving \eqref{eq:g(eta)} with $\eta = {\widehat{\eta}}$ \end{algorithmic} \caption{Determination of optimal weight ${\widehat{\lambda}}$ via gradient minimization \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $N$ --- dimension of the space in which optimal weights are sought \newline
\hspace*{0.5cm} $u \in H$ --- current approximation of minimizer ${\widehat{u}}$ \newline
\hspace*{0.5cm} $\kappa > 0$ --- step size in the outer loop (Algorithm \ref{alg:gradient}) \newline
\hspace*{0.5cm} $\{ \ell^i\}_{i=1}^N$ --- basis function for ansatz \eqref{eq:etaN} \newline
\hspace*{0.5cm} $N_g$ --- maximum number of gradient iterations \newline
\hspace*{0.5cm} $\eta$ --- initial guess for the weight \newline
\textbf{Output:} \newline
\hspace*{0.5cm} ${\widehat{\eta}}$ --- optimal weight \newline
\hspace*{0.5cm} $g({\widehat{\eta}})$ --- corresponding optimal gradient } \label{alg:etagrad} \end{algorithm}
\subsection{Optimal weights via Newton's method} \label{sec:etanewton}
In addition to the gradient of $j(\eta)$ already given in \eqref{eq:alpha->F=0}--\eqref{eq:g'i}, the key additional step required for Newton's method is the evaluation of the Hessian of $j(\eta)$, i.e., \begin{align} \left[ \partial_jF_i(\eta) \right] & = \left[\Dpartialmix{j}{\eta_i}{\eta_j}(\eta)\right] \qquad\qquad \qquad\qquad (i,j=1,\dots,N) \nonumber \\ & = \kappa (e''(u;g'_j(\eta),g'_i(\eta)) + e''(u;g(\eta),g''_{ij}(\eta))) - e'(u;g''_{ij}(\eta)), \label{eq:DF} \end{align} where $g(\eta)$ is given by \eqref{eq:g(eta)}, $g'_i(\eta)$ is given by \eqref{eq:g'i} and $g''_{ij}=g''_{ij}(\eta) = \left[\Dpartialmix{g}{\eta_i}{\eta_j}({\widehat{\eta}})\right]$ satisfy the equations \begin{equation} \begin{alignedat}{2} g''_{ij}(\eta)-\nabla\cdot(\eta^2\nabla g''_{ij}(\eta)) &= 2\left( \nabla\cdot(\ell_j \eta\nabla g'_i(\eta)) + \nabla\cdot(\ell_i \eta\nabla g'_j(\eta))\right) \\ &+ 2 \nabla\cdot(\ell_i \ell_j\nabla g(\eta)) & \quad & \text{\it in} \ \Omega,\\ g''_{ij}(\eta) & = 0 & \quad & \text{\it on} \ \partial\Omega. \end{alignedat} \label{eq:g''} \end{equation} For brevity, Newton's approach is stated in Algorithm
\ref{alg:etanewton} in its simplest form and in practice one would typically use its damped (globalized) version in which the step along Newton's direction $-\left[ DF(\eta) \right]^{-1}\cdot F(\eta)$ may be reduced to ensure the residual $\| F(\eta)\|_2$ of equation \eqref{eq:alpha->F=0} decreases between iterations \cite{k03}. A similar step-size limitation may also be imposed in order to prevent any of the values $\eta_i$ from becoming too close to zero. In addition, in practice, a termination criterion based on the residual
$\| F(\eta)\|_2$ will be more useful. The criterion involving the total number of iterations $N_n$ is used in Algorithm \ref{alg:etanewton} only to simplify the complexity analysis which is presented next. \begin{algorithm}[h!] \begin{algorithmic}[1] \STATE evaluate adjoint states $z$ and $z(g'_i)$, $i=1,\ldots,N$ (if $e(u)$ depends on a PDE equation) \STATE set $k = 1$ \REPEAT \STATE evaluate $g(\eta)$ by solving \eqref{eq:g(eta)} \STATE evaluate $g'_i(\eta)$, $i=1,\dots,N$, by solving \eqref{eq:g'i} \STATE evaluate $g''_{ij}(\eta)$, $i,j=1,\dots,N$, by solving \eqref{eq:g''} \STATE evaluate $e'(u; g'_i(\eta))$, $i=1,\ldots,N$ \STATE evaluate $e'(u; g'_{ij}(\eta))$, $i,j=1,\ldots,N$ \STATE evaluate $e''(u; g'_i(\eta),g'_j(\eta))$, $i,j=1,\ldots,N$ \STATE evaluate $e''(u; g(\eta),g'_{ij}(\eta))$, $i,j=1,\ldots,N$ \STATE evaluate the function $F(\eta)$, cf.~\eqref{eq:alpha->F=0} \STATE evaluate the Hessian $DF(\eta)$, cf.~\eqref{eq:DF} \STATE set ${\widehat{\eta}} = \eta - \left[ DF(\eta) \right]^{-1}\cdot F(\eta)$ \STATE set $\eta = {\widehat{\eta}}$ \STATE set $k = k + 1$ \UNTIL{ \ $k = N_n$} \STATE obtain $g({\widehat{\eta}})$ by solving \eqref{eq:g(eta)} with $\eta = {\widehat{\eta}}$ \end{algorithmic} \caption{Determination of optimal weight ${\widehat{\lambda}}$ using Newton's method \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $N$ --- dimension of the space in which optimal weights are sought \newline
\hspace*{0.5cm} $u \in H$ --- current approximation of minimizer ${\widehat{u}}$ \newline
\hspace*{0.5cm} $\kappa > 0$ --- step size in the outer loop (Algorithm \ref{alg:gradient}) \newline
\hspace*{0.5cm} $\{ \ell^i\}_{i=1}^N$ --- basis function for ansatz \eqref{eq:etaN} \newline
\hspace*{0.5cm} $N_n$ --- maximum number of Newton iterations \newline
\hspace*{0.5cm} $\eta$ --- initial guess for the weight \newline
\textbf{Output:} \newline
\hspace*{0.5cm} ${\widehat{\eta}}$ --- optimal weight \newline
\hspace*{0.5cm} $g({\widehat{\eta}})$ --- corresponding optimal gradient } \label{alg:etanewton} \end{algorithm}
\FloatBarrier
\subsection{Complexity analysis} \label{sec:complexity}
In this section we estimate the computational cost of a single iteration of Algorithms \ref{alg:etagrad} and \ref{alg:etanewton} in which the optimal weight ${\widehat{\lambda}}$ is computed using gradient minimization and Newton's method, respectively, as described in Sections \ref{sec:etagrad} and \ref{sec:etanewton}. This cost will be expressed in terms of: (i) the number $N$ of the degrees of freedom characterizing the dimension of the weight space $\Lambda$, cf.~\eqref{eq:etaN}; (ii) the number $M$ {determining the cost of} the numerical solution of the elliptic boundary-value problems \eqref{eq:g(eta)}, \eqref{eq:g'i}, \eqref{eq:g''}; this latter quantity can be viewed as the number of computational elements used to discretize the domain $\Omega$ (such as finite elements/volumes, grid points or spectral basis functions); (iii) the number $K$ which is the typical number of line-search iterations (line 9 in Algorithm \ref{alg:etagrad}). In the following we will assume that the constants $C_1, C_2, \dots$ are all positive and $\mathcal O(1)$.
Both algorithms require first the evaluation of $e'(u;\ell_i)$, $i=1,\ldots,N$. {In general, the linear form can be expressed as} \begin{equation} e'(u;v) = \int_{\Omega} z v \, dz \label{eq:z} \end{equation} {and, assuming that $v$ is already available, the cost of its
approximation is determined by the cost of evaluating $z$ on
$\Omega$ and the cost of the quadrature which is typically
$\mathcal O(M)$.} If $z$ is a function given explicitly in terms of $u$, then {it can be evaluated on $\Omega$ in terms of $\mathcal O(M)$ operations.
However,} in general, {when the energy depends on $u$ through
some PDE,} which is the case of interest {here,} $z$ will be given in terms of the solution of a suitably-defined adjoint PDE problem. {Then,} for example, if the governing system is an elliptic PDE problem in dimension $(d+1)$ with $u$ acting as the boundary condition, {the numerical solution of the PDE will
require discretization with $\mathcal O(M^q)$, $q=\frac{d+1}{d}$, degrees
of freedom} and, assuming direct solution of the resulting algebraic problems, the cost of evaluating $z$ on $\Omega$ will be $\mathcal O(M^{3q})$. {Thus,} for simplicity, we will restrict our attention to problems in which the cost of approximating $z$ on $M$ points/elements discretizing the domain $\Omega$ will be $C_1(M^{3q}+M)$, $q\in\mathbb N$ ($q=0$ represents the case when the dependence of $e$ on $u$ does not involve a PDE).
A similar argument applies to the evaluation of the second derivative $e''(u;v,w) = \int_{\Omega} z(v) w \, dx$, except that now $z = z(v)$. As the {operator defining the} adjoint PDE is the same for {both} $z$ and $z(v)$, {to determine} $z(v)$ we {only
need} to perform a {back-}substitution at a computation cost $C_2 M^{2q}$, {as explained below}.
We note that the cost of evaluating the gradient $g$ corresponding to a certain $\lambda$ (or equivalently $\eta$) and its derivatives $g'_i$, $g''_{ij}$, see \eqref{eq:g(eta)}, \eqref{eq:g'i}, \eqref{eq:g''}, will primarily depend on $M$. In general, solution of each problem of this type requires $\mathcal O(M^3)$ operations. However, when several such problems need to be solved with the same differential operator, then it is more efficient to perform an LU-type matrix factorization, at the cost $C_3 M^3$, followed by solution of individual problems via back-substitution, each at the cost $C_4 M^2$.
With these estimates in place and assuming $K \ll M$ and $N_g, N_n \ll M$, we are now in the position to characterize the complexity of Algorithms \ref{alg:etagrad} and \ref{alg:etanewton}. The cost of a single iteration of the gradient-minimization approach in Algorithm \ref{alg:etagrad} will be dominated by: \begin{itemize} \item[g.1)] one evaluation of $z$ at the computational cost $C_1
(M^{3q}+M)$,
\item[g.2)] one evaluation of $z(g)$ at the computational cost $C_2
M^{2q}$,
\item[g.3)] the following computations {repeated $N_g$ times:} \begin{itemize} \item[i.1)] $N+K$ elliptic solves (with factorization) for $g$, $g'_i$
and $g(\eta-\tau F(\eta))$ at the cost $C_3 M^3 + C_4 (N+K) M^2$,
\item[i.2)] $N+K$ evaluations of $e'(u;v)$ ($v=g$, $v=g'_i$,
$v=g(\eta-\tau F(\eta))$), and $N+K$ evaluations of $e''(u;g,v)$
($v=g$, $v=g'_i$) at the cost at the cost $C_5 (N+K) M$. \end{itemize} \end{itemize} Thus, finding ${\widehat{\lambda}}$ and $g({\widehat{\lambda}})$ with Algorithm \ref{alg:etagrad} will require \begin{equation} {\cal C}_g = \mathcal O(1) \left( M^{3q} + (M^3 + (N+K)M^2)N_g \right) \approx \mathcal O(1) \left( M^{3q} + M^3N_g \right) \ \mbox{\it flops}. \label{eq:cost(gradient)} \end{equation}
The cost of a single iteration of Newton's approach in Algorithm \ref{alg:etanewton} will be dominated by: \begin{itemize} \item[n.1)] one evaluation of $z$ at the computational cost $C_1 M^{3q}$,
\item[n.2)] $N$ evaluation of $z(g'_i)$ at the computational cost $C_2 N M^{2q}$, \item[n.3)] the following computations {repeated $N_n$ times:} \begin{itemize} \item[i.1)] $\frac{1}{2}N^2$ elliptic solves (with factorization) for
$g''_{ij}$ (noting that $g''_{ij} = g''_{ji}$) at the total cost
proportional to $C_3 M^3 + C_4 N^2 M^2$,
\item[i.2)] $\frac{1}{2}N^2$ evaluations of $e'(u;v)$ (with $v=g''_{ij}$,
$1\leq i\leq j\leq N$), and
$\frac{1}{2}N^2$ evaluations of $e''(u;v,w)$ (with $(v,w)=(g,g''_{ij})$,
$(v,w)=(g'_i,g'_j)$, $i,j=1,\dots,N$),
at the cost $C_6 N^2 M$,
\item[i.3)] one evaluation of $[\partial_j F_i(\eta)]^{-1}\cdot
[F_i(\eta)]$ at the cost $C_7 N^3$. \end{itemize} \end{itemize} Thus, the cost for computing ${\widehat{\lambda}}$ and $g({\widehat{\lambda}})$ with Algorithm \ref{alg:etanewton} would require \begin{eqnarray} {\cal C}_{n} &=& \mathcal O(1) \left( M^{3q}+ N M^{2q} + (M^3 + N^2 M^2 +N^3)N_n \right) \nonumber\\ &\approx& \mathcal O(1) \left( M^{3q}+ N M^{2q} + (M^3 + N^2 M^2)N_n \right)
\quad
\mbox{\it flops}. \label{eq:cost(newton)} \end{eqnarray} Note that the cost of an iteration of a simple gradient algorithm is \begin{equation} {\cal C}_{sg} = \mathcal O(1) \left( M^{3q}+ M^3 \right) \quad \mbox{\it flops}, \label{eq:cost(g)} \end{equation} Then we {obtain} \begin{alignat}{2}
\lim_{N/M\to0}
\frac{{\cal C}_g}{{\cal C}_{sg}}
& =
{\cal O}(1)
\left(1+M^{3(1-q)}N_g\right),
& \quad
\lim_{N/M\to1}
\frac{{\cal C}_g}{{\cal C}_{sg}}
& =
{\cal O}(1)
\left(1+M^{3(1-q)}N_g\right), \label{eq:Cg/Csg} \\
\lim_{N/M\to0}
\frac{{\cal C}_{n}}{{\cal C}_g}
& =
{\cal O}(1)
\frac{1+M^{3(1-q)}N_n}{1+M^{3(1-q)}N_g}, & \quad
\lim_{N/M\to1}
\frac{{\cal C}_{n}}{{\cal C}_g}
& =
{\cal O}(1)
\frac{1+MM^{3(1-q)}N_n}{1+M^{3(1-q)}N_g}. \label{eq:Cn/Cg} \end{alignat} Equations \eqref{eq:Cg/Csg} show that the ratio of the cost of our method using Algorithm \ref{alg:etagrad} {and} the cost of the simple gradient method is of the same order ${\cal
O}(1)\left(1+M^{3(1-q)}N_g\right)$, regardless {of} $N$. Furthermore, the methods tend to {have a comparable cost when $q\geq1$
and $M$ is} large. In view of \eqref{eq:Cn/Cg}, it follows that the same conclusion {also} holds when comparing our method using Algorithm \ref{alg:etagrad} and Algorithm \ref{alg:etanewton} for $N \ll M$. {However,} when $N\approx M$, equations \eqref{eq:Cn/Cg} {indicate} that the cost of our method with Algorithm \ref{alg:etanewton} becomes substantially higher {(by a factor of $M$) as} compared to the cost when Algorithm \ref{alg:etagrad} {is used}. These comments suggest that it {may be more cost efficient} to use Algorithm \ref{alg:etagrad} with {large $N$} (under the assumption $K \ll M$), or Algorithm \ref{alg:etanewton} with $N \ll M$. {In either case,} the cost will depend also on $N_g$ {and} $N_n$, i.e., on how fast Algorithms \ref{alg:etagrad} {and \ref{alg:etanewton} can converge to
${\widehat{\eta}}$}. In conclusion, the relative efficiency of {original
Algorithm \ref{alg:gradient} versus its versions using Algorithms
\ref{alg:etagrad} or \ref{alg:etanewton} to find the optimal
gradients} will depend on the extend to which the increased per-iteration cost {in the latter cases} can be offset by the reduced number of iterations. This trade-off is illustrated based on a simple model in the next section.
\section{A model problem and computational results} \label{sec:results}
In order to illustrate the approach developed in this study, in the present section we consider the following model problem defined on the domain $\Omega = (-1,1)$ \begin{equation}
e({\widehat{u}})
=
\inf\left\{e(u)
:=
\int_\Omega \left( 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right)^{1/2} \, dx, \quad u \in H^1_0(\Omega) \right\}, \label{eq:E} \end{equation} where $a = a(x) = 1 - x^2 / 2$. Clearly, the solution is ${\widehat{u}}=0$ and $e({\widehat{u}}) = 2$. Energy \eqref{eq:E} gives rise to the following expressions for its first and second derivative \begin{align*}
e'(u;v)
& = \int_{-1}^1 \left\{ \frac{ a u } {\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{1/2}} - \frac{d}{dx} \left(\frac{ a \frac{du}{dx} } {\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{1/2}}\right)\right\} v \, dx, \\
e''(u;v,w) & = \int_{-1}^1 \left\{ \frac{ a v w + a \frac{dv}{dx} \frac{dw}{dx} } {\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{1/2}} - \frac{ \left(a u v + a \frac{du}{dx} \frac{dv}{dx}\right) \left(a u w + a \frac{du}{dx} \frac{dw}{dx}\right)} {\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{3/2}} \right\} \, dx. \end{align*} To solve problem \eqref{eq:E} we will use the initial guess $u_0(x) = (1-x^2) \cos(6x) e^x$ chosen such that $u_0\in H^1_0(\Omega)$ and it has a large $H^1$ norm ensuring that $u_0$ is a ``significant distance'' away from the solution ${\widehat{u}}$.
In order to mimic the setting with a refined discretization of the domain $\Omega$, i.e., the case when $M \rightarrow \infty$, in our computations all functions defined on $\Omega$ (i.e, $u$, $\widetilde{u}$, $\lambda$, $g_0(u)$, $g(\lambda)$, $g'_i(\lambda)$ and $g''_{ij}(\lambda)$) will be approximated using {\tt Chebfun} \cite{chebfun}. In this approach all the functions involved are represented in terms of Chebyshev-series expansions truncated adaptively to ensure that the truncation errors do not exceed a prescribed bound (typically related to the machine precision). {\tt
Chebfun} also makes it possible to solve elliptic boundary-value problems such as \eqref{eq:g(eta)},\eqref{eq:g'i} and \eqref{eq:g''} with comparable accuracy. By minimizing the errors related to the discretization in space, this approach allows us to focus on the effect of the main parameter in the problem, namely, the dimension $N$ of the space $\Lambda$ in which the optimal weights are constructed, cf.~\eqref{eq:etaN}. In terms of the basis $\{ \ell^i \}_{i=1}^N$ we take the standard piecewise-linear ``hat'' functions which, unless stated otherwise, are constructed based on an equispaced grid. With such data and choice of the discretization parameters, minimization problem \eqref{eq:E} is already rich enough to reveal the effect of the parameter $N$ on convergence and the differences between different approaches.
We now move on to present computational results obtained solving problem \eqref{eq:E} using the following approaches: \begin{itemize} \item[(a)] steepest-descent method from Algorithm \ref{alg:gradient}
with Sobolev gradients $g(\lambda_0)$ defined through the inner
product \eqref{e:(u,v)_lam0} with {\em constant} weight $\lambda_0 =
10$ (this value of $\lambda_0$ was found by trial and error to
produce fastest convergence),
\item[(b)] steepest descent method from Algorithm \ref{alg:gradient}
with optimal Sobolev gradients $g({\widehat{\lambda}})$ determined using
Algorithm \ref{alg:etagrad} for different values $N$; at every iteration
Algorithm \ref{alg:etagrad} is restarted
with the same initial guess $\lambda(x) = \lambda_0$,
\item[(c)] Newton's method from Algorithm \ref{alg:newton},
\item[(d)] steepest descent method from Algorithm \ref{alg:gradient}
with optimal Sobolev gradients $g({\widehat{\lambda}})$ determined using a
simplified version of Algorithm \ref{alg:etanewton} for different
values $N$ (see below for details); at every iteration simplified
Algorithm \ref{alg:etanewton} is restarted with the same initial
guess $\lambda(x) = \lambda_0$. \end{itemize} Approaches (a), (b) and (d) use the same fixed step size $\kappa = 50$. Approximations of the exact solution $\widehat{u}$ obtained at the $n$th iteration will be denoted $u_n$. In order to prevent the optimal weights ${\widehat{\lambda}}(x)$ from becoming too close to zero for certain $x$, which would complicate the numerical solution of problems \eqref{eq:g(eta)}, \eqref{eq:g'i} and \eqref{eq:g''}, the line-search in Algorithm \ref{alg:etagrad} and the length of Newton's step in Algorithm \ref{alg:etanewton} are restricted such that $\min_{x \in
[-1,1]} {\widehat{\lambda}}(x) > \epsilon_{\tau} \lambda_0$, where we used $\epsilon_{\tau} = 10^{-2}$. In addition, since this will make it possible to objectively compare cases with different values of $N$, here we modify the termination condition in Algorithm
\ref{alg:etagrad}, cf.~line 13, by replacing it with one given in terms of a minimum relative decrease of $j(\eta)$, i.e., $|j({\widehat{\eta}}) -
j(\eta)| / j(\eta) \le \epsilon_{\lambda}$, where $\epsilon_{\lambda}$ is a prescribed tolerance.
We now examine the effect of different parameters on the results obtained with each of the approaches (a)--(d) defined above.
\subsection{Analysis of the effect of the tolerance $\epsilon_\lambda$} The decrease of the (shifted) energy $e(u_n)-e(\hat{u})$ and of the
$H^1$ approximation error $\|u_n - {\widehat{u}}\|_{H^1}$ are shown for approaches (a), (b) and (c) in Figures \ref{fig:eps}a and \ref{fig:eps}b, respectively, where in case (b) we used a single value $N=50$ and three different tolerances $\epsilon_{\lambda} = 10^{-1}, 10^{-2}, 10^{-3}$. In Figure \ref{fig:eps}a we see that minimization with optimal gradients $g({\widehat{\lambda}})$ produces a significantly faster decrease of energy $e(u_n)$ than optimization with ``standard''
Sobolev gradients $g(\lambda_0)$ and analogous trends are also evident in the decrease of the approximation error $\|u_n - {\widehat{u}}\|_{H^1}$, cf.~Figure \ref{fig:eps}b. We add that in order to solve the minimization problem to the same level of accuracy the method based on the ``standard'' Sobolev gradients $g(\lambda_0)$ requires as many as 42 iterations (for clarity, these later stages are not shown in the figures).
In addition, in Figures \ref{fig:eps}a and \ref{fig:eps}b we also observe that convergence of the proposed method systematically accelerates as the tolerance $\epsilon_{\lambda}$ is refined, i.e., as the optimal weights ${\widehat{\lambda}}$ are approximated more accurately. However, we remark that reducing $\epsilon_{\lambda}$ below $10^{-3}$ did not produce further improvement of convergence. Hence, hereafter we will set $\epsilon_{\lambda} = 10^{-3}$.
\begin{figure}\label{fig:eps}
\end{figure}
\subsection{Analysis of the effect of the dimension $N$ of the approximation space $\Lambda$} {The results concerning} the effect of $N$ on the performance of approach (b) are compared with the data for approaches (a) and (c) in Figures \ref{fig:min}a and \ref{fig:min}b for the (shifted) energy
$e(u_n)-e(\widehat{u})$ and the $H^1$ approximation error $\|u_n -
{\widehat{u}}\|_{H^1}$, respectively. We observe that, when optimal gradients
$g({\widehat{\lambda}})$ are used, both $e(u_n)-e({\widehat{u}})$ and $\|u_n - {\widehat{u}}\|_{H^1}$ initially reveal a quadratic convergence, similar to the behavior of these quantities in Newton's method, followed at later iterations by a linear convergence, typical of the standard gradient method. In the light of Theorem \ref{th:P+newton->1}, cf.~estimate
\eqref{e:|P(tu-u)|->1}, this observation can be explained by the fact that at early iterations dominant components of the error $(u_n-{\widehat{u}})$ are contained in the subspaces $T_{u_n}$ where the optimal gradients $g({\widehat{\lambda}})$ are consistent with Newton's steps $h$, cf.~Remark \ref{r:name}. Then, once these error components are eliminated, at later iterations the error $(u_n - {\widehat{u}})$ is dominated by components in directions orthogonal to $T_{u_n}$ where the optimal gradients $g({\widehat{\lambda}})$ do not well reproduce the Newton steps $h$. In Figures \ref{fig:min}a and \ref{fig:min}b we also see that the convergence improves as the dimension $N$ is increased until it saturates for $N$ large enough (here $N \gtrapprox 25)$. This could be explained by the conjecture that increasing $N$ above a certain limit (approximately $25$ in this case) does not increase the ``effective'' dimension of $T_{u_n}$ in $H$ anymore (such a possibility is allowed by the error analysis presented in Section \ref{s:analysis->1}).
In this context it is also interesting to investigate the evolution of the spatial structure of the optimal weights ${\widehat{\lambda}}(x)$ and these results are shown for different values of $N$ at an early ($n=2$) and a later ($n=8$) iteration in Figures \ref{fig:lam}a and \ref{fig:lam}b, respectively. In the first case ($n=2$ corresponding to the quadratic convergence) we see that the optimal weights ${\widehat{\lambda}}(x)$ converge to a well-defined profile as $N$ increases, which features a number of distinct ``spikes''. On the other hand, at later iterations ($n=8$ corresponding to the linear regime) the convergence of the optimal weights ${\widehat{\lambda}}(x)$ with $N$ is less evident and the resulting profiles tend to be more uniform. \begin{figure}\label{fig:min}
\label{fig:lam}
\end{figure}
\noindent We want to {highlight} the case when $N=1$ and space $H$ is endowed with the inner product redefined as in \eqref{eq:ipH10}. As shown in Remark \ref{r:hlambda,H10}, in such circumstances the optimal $\lambda$ can be found analytically, cf.~relation \eqref{e:hlambda,N=1}, at essentially no cost and the iterations produced by Algorithm \ref{alg:gradient} do not depend on the step size $\kappa$. The results obtained with this approach and using the optimal gradients $g({\widehat{\lambda}})$ defined in terms of the inner product \eqref{e:(u,v)_lambda} are compared in Figures \ref{fig:N1}a and \ref{fig:N1}b for the (shifted) energy $e(u_n)-e(\widehat{u})$ and the
$H^1$ approximation error $\|u_n - {\widehat{u}}\|_{H^1}$, respectively. As is evident from these figures, the performance of the approaches corresponding to the two definitions of the inner product, \eqref{e:(u,v)_lambda} and \eqref{eq:ipH10}, is comparable and {in both cases} much better than when a fixed weight $\lambda_0$ is used. We stress that in the case corresponding to the inner product \eqref{eq:ipH10} determination of the optimal $\lambda$ does not require an iterative solution. \begin{figure}\label{fig:N1}
\end{figure}
\subsection{Analysis of the robustness of approach (b) with respect to
{variations of the basis functions defining $\eta$}} This {analysis is performed} by constructing basis functions $\{ \ell^i \}_{i=1}^N$ based on grid points distributed randomly with an uniform probability distribution over the interval $(-1,1)$, except for the leftmost and the rightmost grid points which are always at $x = \pm 1$. The results obtained in several realizations with $N=5$ are compared to the reference case of basis functions constructed based on equispaced grid points as well as with the results obtained with approaches (a) and (c) in Figures \ref{fig:rand}a and \ref{fig:rand}b for the (shifted) energy $e(u_n)-e(\widehat{u})$ and the $H^1$
approximation error $\|u_n - {\widehat{u}}\|_{H^1}$, respectively. One can see in these figures that, expect for one {realization} corresponding to a very special distribution of the grid points, the convergence is little affected by the choice of the basis $\{ \ell^i \}_{i=1}^N$. \begin{figure}\label{fig:rand}
\end{figure}
\subsection{{Analysis of the performance of a simplified version of Algorithm \ref{alg:etanewton}}} Finally, we consider approach (d) where the optimal weights ${\widehat{\lambda}}(x)$ and the corresponding optimal gradients $g({\widehat{\lambda}})$ are determined with Algorithm \ref{alg:etanewton} simplified as follows. The complexity analysis presented in Section \ref{sec:complexity} shows that Algorithm \ref{alg:etanewton} may be quite costly from the computational point of view when $N \gg 1$. To alleviate this difficulty, we consider its simplified version where only one iteration ($N_n = 1$) is performed on system \eqref{eq:alpha->F=0} in which the ``test'' functions $g'_i$, $i=1,\dots,N$, are assumed not to depend on $\lambda$ (or $\eta$). In other words, since instead of $g'_i(\eta)$, $i=1,\dots,N$, the functions $g'_i(\eta_0)$ are used to obtain expressions for $[F(\eta)]_i$, $i=1,\dots,N$, in \eqref{eq:alpha->F=0}, the second derivatives $g''_{ij}$ are eliminated from the Hessian $[DF(\eta)]_{ij}$, $i,j=1,\dots,N$ in \eqref{eq:DF}, which very significantly reduces the computational cost. The results obtained with this simplified approach are shown in Figures \ref{fig:lin}a and \ref{fig:lin}b, respectively, for the decrease of the (shifted) energy $e(u_n)-e({\widehat{u}})$ and for the decrease of the $H^1$ approximation error
$\|u_n - {\widehat{u}}\|_{H^1}$. In these figures we observe general trends qualitatively similar to those evident in Figures \ref{fig:min}a and \ref{fig:min}b, except that the convergence is slower and the transition from the quadratic to linear convergence tends to occur at earlier iterations. \begin{figure}\label{fig:lin}
\end{figure}
\section{Conclusions} \label{sec:final} We have developed a gradient-type numerical approach for {unconstrained} optimization problems in infinite-dimensional Hilbert spaces.
Our method consists in finding an optimal inner product among a family of equivalent inner products parameterized by a space-dependent weight $\lambda$ function. The optimal weight ${\widehat{\lambda}}$ solves a nonlinear optimization problem in a finite dimensional subspace.
Rigorous analysis demonstrates that, in addition to the linear convergence characterizing the standard gradient method, the proposed approach also attains quadratic convergence in the sense that the projection error in a finite-dimensional subspace generated in the process decreases quadratically. Or, equivalently, in this finite dimensional subspace, the optimal gradients and Newton's steps are equivalent. The dimension of these subspaces is determined by the number $N$ of discrete degrees of freedom parameterizing the inner products through the weight $\lambda$.
This analysis is confirmed by numerical experiments, performed based on a simple optimization problem in a setting mimicking high spatial resolution. More specifically, at early iterations both the minimized energy and the error with respect to the exact solution exhibit quadratic convergence followed by the linear convergence at later iterations. The behavior of the proposed method also reveals expected trends with variations of the numerical parameters, namely, the dimension of the space in which the optimal weights ${\widehat{\lambda}}$ are constructed, properties of the basis in this space and the accuracy with which the inner optimization problems are solved. In all cases the convergence of the proposed approach is much faster than obtained using Sobolev gradients with fixed weights. For the ease of analysis and comparisons we focused on a gradient-descent method with a fixed step size $\kappa$, but it can be expected that a similar behavior will also occur when the step size is determined adaptively through suitable line minimization.
The complexity analysis performed in Section \ref{sec:complexity} indicates that the per-iteration cost of the proposed approach and of the standard Sobolev gradient method have the same order if, for example, the energy depends on a elliptic PDE. When the dimension of weight space is $N = 1$ and the inner product does not have the $L^2$ term, cf.~\eqref{eq:ipH10}, then the optimal weight is given explicitly, eliminating the need for its numerical determination. In this particular case the proposed approach has some similarity to the Barzilai-Borwein algorithm \cite{barzilai-1} and produces iterates which do not depend on the step size in the gradient method. The computational cost of the proposed approach is also significantly reduced when Algorithm \ref{alg:etanewton} is used in a simplified form, as described in Section \ref{sec:results}. {We thus
conclude that the gradient-descent method from Algorithm
\ref{alg:gradient} combined with Algorithms \ref{alg:etagrad} and
\ref{alg:etanewton} used to find optimal gradients} are promising approaches suitable for large-scale optimization problems and applications to some such problems will be investigated in the near future.
Our approach based on optimal gradients differs from the family of quasi-Newton methods in that instead of approximating the full Hessian using gradients from earlier iterations, see for example \cite{nw00}, it relies on computing the action of the exact \revt{Hessian and
gradients,} but only on a few judiciously selected directions, and then matching them by appropriately choosing the inner product. Consequently, the resulting algebraic problem is of a much smaller dimension thereby avoiding complications related to poor conditioning and computational cost.
Finally, we believe that the analysis and results presented here explain the acceleration of gradient minimization reported for a range of different problems in \cite{pbh02,p07b,rssl09,r10,ms10,dk10} when Sobolev gradients with suitable (constant) weights were used. Moreover, our work also provides a rational and constructive answer to the open problem of finding an optimal form of the inner product defining the Sobolev gradients.
\end{document} | arXiv | {
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\begin{document}
\thanks{\noindent The authors are partially supported by Departamento de Ciencia, Universidad y Sociedad del Conocimiento del Gobierno de Arag{\'o}n (grant code: E22\_20R: ``{\'A}lgebra y Geometr{\'i}a''), the second author is partially supported by MCIN/AEI/10.13039/501100011033 (grant code: PID2020-114750GB-C31) and the first and third authors are partially supported by the Spanish Government PGC2018-101179-B-100}
\begin{abstract} In this paper we study Sigma-invariants of even Artin groups of $\FC$-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong $n$-link condition for a graph $\Gamma$ and prove that it gives a sufficient condition for a character $\chi:A_\Gamma\to\mathbb{Z}$ to satisfy $[\chi]\in\Sigma^n(A_\Gamma,\mathbb{Z})$. This implies that the kernel $A^\chi_\Gamma=\ker \chi$ is of type $\FP_n$. We prove the homotopical version of this result as well and discuss partial results on the converse. We also provide a general formula for the free part of $H_n(A^\chi_\Gamma;\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $\chi$. This gives a characterization of when $H_n(A^\chi_\Gamma;\mathbb{F})$ is a finite dimensional vector space over~$\mathbb{F}$. \end{abstract}
\subjclass[2020]{Primary 20J06, 20F36; Secondary 57M07, 55P20}
\title{On the Sigma-invariants of even Artin groups of $\FC$-type}
\keywords
\section{Introduction} The Sigma-invariants of a group $G$ are certain sets $\Sigma^n(G,\mathbb{Z})$, $\Sigma^n(G)$ of equivalence classes of characters $\chi:G\to\mathbb{R}$ that provide information about the cohomological --\,in the case of $\Sigma^n(G,\mathbb{Z})$\,-- and homotopical --\,for $\Sigma^n(G)$\,-- finiteness conditions of subgroups lying over the commutator of $G$. The first version of these invariants was defined by Bieri and Strebel in~\cite{Bieri-Strebel-Valuations} and the theory was later developed by Bieri-Neumann-Strebel~\cite{Bieri-Neumann-Strebel-Geometric}, Bieri-Renz~\cite{Bieri-Renz-Valuations}, and Renz~\cite{Renz-Geometrische}. Usually, it is extremely difficult to compute these invariants explicitly but there are some remarkable cases in which a full computation is available.
One of those cases occurs when $G$ is a right-angled Artin group (RAAG for short). These groups are defined from a given finite graph $\Gamma$ which will be assumed here to be simple, i.e., without loops or multiple edges between vertices. Associated to $\Gamma$ one can describe the RAAG, denoted by $A_\Gamma$, as the group generated by the vertices of $\Gamma$ with relators of the form $[v,w]=1$ for any edge $\{v,w\}$ of $\Gamma$. This is a remarkable family of groups that range between finitely generated free abelian groups (corresponding to complete graphs) and finitely generated free groups (associated with graphs with no edges). Many properties of RAAGs can be determined in terms of the combinatorial properties of the graph. This is precisely the case for their Sigma-invariants, which were computed by Meier-Meinert-VanWyk in~\cite{MMVW}. To describe their computation we will need to introduce some terminology.
We recall the concept of link in our context as follows. Fix a simple finite graph $\Gamma$ as before and denote by $V_\Gamma$ (resp. $E_\Gamma$) its set of vertices (resp. edges). If $\Gamma_1\subseteq\Gamma$ is a subgraph and $v\in V_\Gamma$, then the link $\lk_{\Gamma_1}(v)$ of $v$ in $\Gamma_1$ is defined as the full subgraph induced by $V_{\Gamma_1}(v):=\{w\in V_{\Gamma_1}\mid \{v,w\}\in E_\Gamma\}$.
We extend this definition for subsets $\Delta\subseteq\Gamma$ by setting $$\lk_{\Gamma_1}(\Delta)=\cap_{v\in\Delta}\lk_{\Gamma_1}(v).$$ By convention we allow $\Delta$ to be empty, then $\lk_{\Gamma_1}(\Delta)=\Gamma_1.$
We also recall the concept of the \emph{flag complex} associated with $\Gamma$. This the simplicial complex, denoted as $\hat\Gamma$, resulting after attaching a ($k-1$)-simplex to each \emph{$k$-clique}, i.e., to each complete subgraph of $k$ vertices. We use the same notation for arbitrary graphs. Note that, if $\Delta\subseteq\Gamma$ is a clique and $\Gamma_1\subseteq \Gamma$ a subgraph, then $\hat{\lk}_{\Gamma_1}(\Delta)$ is the intersection with $\hat{\Gamma}_1$ of the ordinary simplicial link of the cell $\sigma$ associated to $\Delta$, i.e., the subcomplex of $\hat\Gamma_1$ consisting of those simplices $\tau$ such that $\tau\cup\sigma$ is also a simplex of $\hat\Gamma_1$.
Now, let $\chi:A_\Gamma\to\mathbb{R}$ be a character and $n\geq 0$ an integer. Consider the full subgraph $\mathcal{L}^\chi_0$ induced by the vertices $v$ of $\Gamma$ with $\chi(v)\neq 0$. Following Meier-Meinert-VanWyk~\cite{MMVW}, we call $\mathcal{L}^\chi_0$ the {\sl living} subgraph of $\Gamma$ and say that vertices not in $\mathcal{L}^\chi_0$ are {\sl dead}. Dead vertices are also called {\sl resonant} in~\cite{Blasco-conmar-ji-Homology}. We will say that the character $\chi$ satisfies the \emph{$n$-link condition} if for any clique $\Delta\subseteq\Gamma\setminus\mathcal{L}^\chi_0$,
$$\hat{\lk}_{\mathcal{L}^\chi_0}(\Delta)\text{ is }(n-1-|\Delta|)\textrm{-acyclic}.$$ Then Meier-Meinert-VanWyk proved (see Subsection~\ref{subsec:SigmaRAAGs}) that $\chi\in\Sigma^n(G,\mathbb{Z})$ if and only if $\chi$ satisfies the $n$-link condition. In fact, they also proved the homotopical version of this result that characterizes $\Sigma^n(G)$ in terms of a {\sl homotopical $n$-link condition} (with ``being
$(n-1-|\Delta|)$-connected'' instead of ``being $(n-1-|\Delta|)$-acyclic'').
Here, we want to extend this result for another remarkable family: even Artin groups of $\FC$-type. Given a finite simple graph $\Gamma$ as above, one can consider an even labeling on the edges, that is, for any edge $e=\{u,v\}$, its label $\ell(e)$ is an even number. Any such \emph{even graph} $\Gamma$ defines an \emph{even Artin group} $A_\Gamma$ generated by the vertices of $\Gamma$ and whose relators have the form $(uv)^k=(vu)^k$, where $\ell(e)=2k$. These special Artin groups where first considered in detail in~\cite{Blasco} and~\cite{Blasco-PF}. Note that any subgraph $X\subset \Gamma$ of an even graph $\Gamma$ generates an even Artin group $A_X$. In addition, an even Artin group is said to have $\FC$-type if $A_X$ is of finite type for each clique $X\subset \Gamma$: this means that the \emph{standard parabolic Coxeter group} $W_X$, i.e., the quotient of $A_X$ by the normal subgroup generated by $\langle u^2; u\in V_X\rangle$ is finite.
For a character $\chi$ on an even Artin group we consider a generalization of the living subgraph as follows (see~\cite{M2001}). Denote $m_v=\chi(v)$, $v\in V_\Gamma$ and $m_e=m_v+m_w$, $e=\{v,w\}\in E_\Gamma$. We say that an edge is {\sl dead} if $e$ has label $\ell(e)>2$ and $m_e=0$. We will consider the subgraph $\mathcal{L}^\chi$ obtained from $\Gamma$ after removing all dead vertices and the interior of all dead edges. Note that if all the edges have label precisely 2, i.e., for a RAAG, then $\mathcal{L}^\chi_0=\mathcal{L}^\chi$.
To state our first main result, we also need to introduce the {\sl clique poset}, that is the poset of subgroups of $A_\Gamma$ which are generated by cliques of $\Gamma$: $$\mathcal{P}=\{A_\Delta\mid\Delta\subseteq\Gamma\text{ clique}\}.$$ Note that the poset structure of $\mathcal{P}$ is the poset structure of the poset of cliques of $\Gamma$. Also, we allow $\Delta$ to be empty, in that case $A_\emptyset=\{1\}$. So the geometric realization of the clique poset is the cone of the barycentric subdivision of the flag complex where the vertex of the cone corresponds to the empty clique.
A special role is played by the subset $\mathcal{B}^\chi\subset \mathcal{P}$ of those subgroups $A_\Delta$ where $\Delta\subseteq\Gamma$ is a clique such that for each vertex $v$ in $\Delta$ either $v$ is dead or $v\in e$ for $e$ a dead edge in $\Delta$. Observe that $1=A_\emptyset\in\mathcal{B}^\chi$. We will see that this is equivalent to asking that the center of $A_\Delta$ lies in the kernel of $\chi$, that is, $\chi(Z(A_\Delta))=0$.
\begin{dfn} \label{nlink} Let $\mathcal{B}^\chi\subset \mathcal{P}$ be as above.
Assume that for any $A_\Delta\in\mathcal{B}^\chi$ with $|\Delta|\leq n$ the link $\hat{\lk}_{\mathcal{L}^\chi}(\Delta)$
is $(n-1-|\Delta|)$-acyclic. Then we say that $\chi$ satisfies \emph{the strong $n$-link condition}.
We also define a \emph{homotopical strong $n$-link condition} in a similar way just changing $(n-1-|\Delta|)$-acyclic
by $(n-1-|\Delta|)$-connected. \end{dfn}
Note that the homotopical strong $n$-link condition implies the strong $n$-link condition.
\begin{teo}\label{teo:mainsigma} Let $G=A_\Gamma$ be an even Artin group of $\FC$-type, and $0\neq\chi:G\to\mathbb{R}$ a character such that the strong $n$-link condition holds for $\chi$. Then $[\chi]\in\Sigma^n(G,\mathbb{Z})$. \end{teo}
In the case $n=1$, it is known for several types of Artin groups (see Theorem~\ref{teo:knownArtin}) that $[\chi]\in\Sigma^n(G,\mathbb{Z})$ if and only if $\mathcal{L}^\chi$ is {\sl connected} and {\sl dominant}. This is equivalent to saying that $\chi$ satisfies the strong $n$-link condition (see Subsection~\ref{subsec:SigmaRAAGs}).
We do not know whether the converse of Theorem~\ref{teo:mainsigma} is true in general, but in Section~\ref{sec:free} we prove a partial converse. To do that, we use some of the techniques of \cite{Blasco-conmar-ji-Homology} to perform computations on the homology groups $\mathrm{H}_n(A_\Gamma^\chi,\mathbb{F})$ where $\mathbb{F}$ is a field, $\chi:A_\Gamma\to\mathbb{Z}$ is assumed to be discrete and $A_\Gamma^\chi=\ker\chi$. More precisely, we show that these homology groups are finite dimensional as $\mathbb{F}$-vector spaces if and only if certain $p$-local version of the strong $n$-link condition holds. Recall that a consequence of the well-known properties of the Sigma-invariants (see Section~\ref{sec:Sigma}) is that if for a discrete character $\chi$ we have $[\chi]\in\Sigma^n(G,\mathbb{Z})$, then $A_\Gamma^\chi$ is of type $\FP_n$ and therefore the homology groups with coefficients over any field must be finite dimensional. As a by-product, an explicit computation of independent interest is provided in Theorem~\ref{teo:free} for the free part of $\mathrm{H}_n(A_\Gamma^\chi,\mathbb{F})$ when seen, via $\chi$, as a module over the principal ideal domain~$\mathbb{F}[t^{\pm 1}]$.
Moreover, section~\ref{sec:homotopic} is devoted to stating and proving a partial homotopic analogue of Theorem~\ref{teo:mainsigma} in Theorem~\ref{teo:mainsigmahomotopic}.
\section{Sigma-invariants}\label{sec:Sigma}
Let $G$ be a finitely generated group. In this section we will consider arbitrary non-trivial characters $\chi:G\to\mathbb{R}$. We say that two characters $\chi_1,\chi_2$ are equivalent if one is a positive scalar multiple of the other, i.e., if $\chi_1=t\chi_2$ for some $t>0$. We denote by $[\chi]$ the equivalence class of the character $\chi$ and by $S(G)$ the set of equivalence classes of characters. Note that if $G/G'$ has finite torsion and free rank $r$ then $S(G)$ can be identified with the sphere $S^{r-1}$.
The homological $\Sigma$-invariants of $G$ are certain subsets $$\Sigma^\infty(G,\mathbb{Z})\subseteq\dots\subseteq\Sigma^n(G,\mathbb{Z})\subseteq\dots\subseteq\Sigma^2(G,\mathbb{Z}) \subseteq\Sigma^1(G,\mathbb{Z})\subseteq \Sigma^0(G,\mathbb{Z})=S(G)$$ which are very useful to understand the cohomological finiteness properties of subgroups of $G$ containing the commutator $G'$.
For a formal definition, consider $\chi:G\to\mathbb{R}$ a character and $G_\chi$ the monoid $G_\chi=\{g\in G\mid\chi(g)\geq 0\}$. Then $$\Sigma^n(G,\mathbb{Z})=\{[\chi]\in S(G)\mid \mathbb{Z} G_\chi\text{ is of type }\FP_n\}.$$
There is also a homotopical version $$\Sigma^\infty(G)\subseteq\dots\subseteq\Sigma^n(G)\subseteq\dots\subseteq \Sigma^2(G)\subseteq\Sigma^1(G)\subseteq \Sigma^0(G)=S(G).$$ We can sketch the definition as follows (see ~\cite{MMVW}). Let $G$ be a group of type $\F_n$. We can choose a $CW$-model $X$ for the classifying space for $G$ with a single 0-cell and finite $n$-skeleton. Let $Y$ be the universal cover of $X$. Then we may identify $G$ with a subset of $Y$ and given a character $\chi:G\to\mathbb{R}$, we can extend $\chi$ to a map $\chi:Y\to\mathbb{R}$ that we denote in the same way. To do that, map the vertex labeled by, say, $g$ to $\chi(g)$ and extend linearly to the rest of~$Y$.
For $a\in\mathbb{R}$ denote by $Y_{\chi}^{[a,+\infty)}$ the maximal subcomplex in $Y\cap\chi^{-1}([a,+\infty))$. Assuming $a\leq 0$, the inclusion $Y_{\chi}^{[0,+\infty)}\subseteq Y_{\chi}^{[a,+\infty)}$ induces a map
$$\pi_i(Y_{\chi}^{[0,+\infty)})\to\pi_i(Y_{\chi}^{[a,+\infty)})$$ and we say that $[\chi]\in\Sigma^n(G)$ if there is some $a$ such that this map is trivial for all $i<n$. The reader can find more details about $\Sigma^n(G,\mathbb{Z})$, $\Sigma^n(G)$ in~\cite{MMVW}. We recall now only two well-known properties: both $\Sigma^n(G,\mathbb{Z})$, $\Sigma^n(G)$ are open subsets of $S(G)$ that determine the cohomological and homotopical finiteness conditions of subgroups containing the commutator thanks to the following fundamental Theorem:
\begin{teo}\label{teo:charsigma} Let $G$ be a group of type $\FP_n$ and $G'\leq N\leq G$. Then $N$ is also of type $\FP_n$ if and only if $$\{[\chi]\in S(G)\mid \chi(N)=0\}\subseteq\Sigma^n(G,\mathbb{Z}).$$ Moreover, if $G$ is of type $\F_n$, then $N$ is of type $\F_n$ if and only if $$\{[\chi]\in S(G)\mid\chi(N)=0\}\subseteq\Sigma^n(G).$$ \end{teo}
In particular, if $\chi:G\to\mathbb{Z}$ is discrete, we have that $\ker\chi$ is of type $\FP_n$ if and only if $[\chi], [-\chi]\in\Sigma^n(G,\mathbb{Z})$.
If $R$ is a commutative ring, one can also define $R$-Sigma-invariants $\Sigma^n(G,R)$ by substituting the homology groups in the definition above by homology groups with coefficients in $R$. Theorem~\ref{teo:charsigma} remains true when $\FP_n$ is substituted by $\FP_n$ over $R$. Moreover we have $$\Sigma^n(G)\subseteq\Sigma^n(G,\mathbb{Z})\subseteq\Sigma^n(G,R)$$ for any $G$, $R$ and $n\geq 2$ and $$\Sigma^1(G)=\Sigma^1(G,\mathbb{Z})=\Sigma^1(G,R).$$
We will also need the following useful result.
\begin{lem}\label{lem:center}\cite[Lemma 2.1]{M2001} Let $G$ be any group of type $\F_n$ and $\chi:G\to\mathbb{R}$ a character with $\chi(Z(G))\neq 0$ where $Z(G)$ is the center of $G$. Then $[\chi]\in\Sigma^n(G)\subseteq\Sigma^n(G,\mathbb{Z})$. \end{lem}
Finally, we state here the following result which was proven by Meier-Meinert-VanWyk~\cite{MMVW}. This Theorem will be the main tool in the proof of Theorem~\ref{teo:mainsigma} as it was one of the main tools in their description of the Sigma-invariants for RAAGs.
\begin{teo}\cite[Theorem 3.2]{MMVW}\label{teo:sigmacomplex} Let $G$ be a group acting by cell-permuting homeomorphisms on a $CW$-complex $X$ with finite $n$-skeleton modulo $G$. Let $\chi:G\to\mathbb{R}$ be a character such that for any $0\leq p\leq n$ and any $p$-cell $\sigma$ of $X$ the stabilizer $G_\sigma$ is not inside $\ker\chi$. Then \begin{enumerate}[label*=$\roman*)$]
\item If $X$ is $(n-1)$-connected and $[\chi|_{G_\sigma}]\in\Sigma^{n-p}(G_\sigma)$ for any $p$-cell $\sigma$, $0\leq p\leq n$, then $[\chi]\in\Sigma^n(G)$.
\item If $X$ is $(n-1)$-$R$-acyclic and $[\chi|_{G_\sigma}]\in\Sigma^{n-p}(G_\sigma,R)$ for any $p$-cell $\sigma$, $0\leq p\leq n$, then $[\chi]\in\Sigma^n(G,R)$. \end{enumerate} \end{teo}
\section{Artin groups and their Sigma-invariants}
As we have seen in the introduction, Artin groups can be defined in terms of a labeled graph. Using the symmetry of the standard presentation of an Artin group we can show the following.
\begin{prop}\label{prop:symmetricsigma} Let $G=A_\Gamma$ be an Artin group. Then $-\Sigma^n(G)=\Sigma^n(G)$ and $-\Sigma^n(G,\mathbb{Z})=\Sigma^n(G,\mathbb{Z})$. \end{prop}
\begin{proof}
Due to the symmetry of the relations in $G$, there is a well-defined map $\varphi:G\to G$ given as $\varphi(v):=v^{-1}$ for $v\in V_\Gamma$, which defines an automorphism of $G$. Since $\chi\circ\varphi=-\chi$ and both $\Sigma^n(G)$ and $\Sigma^n(G,\mathbb{Z})$ are invariant under automorphisms of $G$, the result follows. \end{proof}
So we have:
\begin{lem}\label{lem:discrete} Let $G=A_\Gamma$ be an Artin group and $\chi:G\to\mathbb{Z}$ a discrete character. Then $\ker\chi$ is of type $\FP_n$ if and only if $[\chi]\in\Sigma^n(G,\mathbb{Z})$. In particular, if there is some field $\mathbb{F}$ and some $0<i\leq n$ such that $\dim_\mathbb{F}\mathrm{H}^i(\ker\chi,\mathbb{F})$ is infinite, $\chi\not\in\Sigma^n(G,\mathbb{Z})$. \end{lem} \begin{proof} The first statement is a direct consequence of Theorem~\ref{teo:charsigma} and Proposition~\ref{prop:symmetricsigma}. For the second one, recall that if a group is of type $\FP_n$, then after tensoring a finite type resolution of the trivial module by $\mathbb{F}$, one obtains a finite type resolution of projective modules over its group ring and thus it is also $\FP_n$ over~$\mathbb{F}$. \end{proof}
\subsection{Sigma-invariants for RAAGs}\label{subsec:SigmaRAAGs} The explicit computation of the Sigma-invariants for a particular group is usually very difficult. In~\cite{MVW1995}, Meier and VanWyk computed $\Sigma^1(A_\Gamma)$ for $A_\Gamma$ a RAAG:
\begin{teo}[Meier-VanWyk~\cite{MVW1995}] \label{thm:MVW95} Let $G=A_\Gamma$ be a RAAG and $\chi:G\to\mathbb{R}$ a character. Then $$[\chi]\in\Sigma^1(G)\text{ if and only if }\mathcal{L}^\chi_0\text{ is connected and dominating in }\Gamma.$$ \end{teo}
Recall that $\mathcal{L}^\chi_0$ is the subgraph obtained from $\Gamma$ by removing the vertices $v$ with $\chi(v)=0$. As we are assuming that $A_\Gamma$ is a RAAG, $\mathcal{L}^\chi_0=\mathcal{L}^\chi$ is the living subgraph defined above. Also, we say that a subgraph $\Delta\subseteq\Gamma$ is {\sl dominating} if for any $v\in\Gamma\setminus\Delta$ there is some $w\in\Delta$ linked to $v$. In other words, the condition of $\mathcal{L}^\chi_0$ being dominant is equivalent to saying that for every $v\in\Gamma\setminus\mathcal{L}^\chi_0$, $\lk_{\mathcal{L}^\chi_0}(v)\neq\emptyset.$ And therefore the Theorem can be reformulated as follows: $[\chi]\in\Sigma^1(G)$ if and only if \begin{itemize} \item[i)] $\lk_{\mathcal{L}^\chi_0}(\emptyset)$ is 0-connected and,
\item[ii)] for every $v\in\Gamma\setminus\mathcal{L}^\chi_0$, $\lk_{\mathcal{L}^\chi_0}(v)$ is (-1)-connected. \end{itemize}
This can be restated using the 1-link condition defined in the introduction: $$[\chi]\in\Sigma^1(G)\text{ if and only if }\chi\text{ satisfies the homotopical 1-link condition}.$$
Later on, in~\cite{MMVW} Meier-Meinert-VanWyk, extending Theorem~\ref{thm:MVW95}, were able to give a full description of the higher Sigma-invariants of a RAAG in terms of the $n$-link condition.
\begin{teo} Let $G=A_\Gamma$ be a RAAG and $\chi:G\to\mathbb{R}$ a character. Then $[\chi]\in\Sigma^n(G,\mathbb{Z})$ if and only if the $n$-link condition holds for $\chi$. \end{teo}
\subsection{Some partial results for Artin groups}\label{subsec:SigmaArtin} Not much is known about $\Sigma$-invariants of general Artin groups. Let $\chi:A_\Gamma\to\mathbb{R}$, $A_\Gamma$ an Artin group and $Z(S)$ the center of $S\subset A_\Gamma$. We highlight the following partial result.
\begin{teo}[{Meier-Meinert-VanWyk~\cite[Theorem B]{M2001}}] Assume $A_\Gamma$ is of $\FC$-type. If $\hat\Gamma$ is $(n-1)$-acyclic and $\chi(Z(A_\Delta))\neq 0$ for any $\emptyset\neq\Delta\subseteq\Gamma$ clique, then $[\chi]\in\Sigma^n(G,\mathbb{Z})$. \end{teo}
We will see below that the hypothesis $\chi(Z(A_\Delta))\neq 0$ for any $\emptyset\neq\Delta\subseteq\Gamma$ clique means that $\mathcal{B}^\chi$ consists of the trivial subgroup only. Therefore this result is a particular case of our main Theorem~\ref{teo:mainsigma}.
A full characterization is available in few cases only.
\begin{teo}[Meier-Meinert-VanWyk~\cite{M2001}, Almeida~\cite{Almeida18}, Almeida-Kochloukova~\cite{AK15,AK15b}, Kochloukova~\cite{Kochloukova20}] \label{teo:knownArtin} Assume that one of the following conditions holds: \begin{itemize}
\item $\Gamma$ is a connected tree,
\item $\Gamma$ is connected and $\pi_1(\Gamma)$ is free of rank at most 2,
\item $\Gamma$ is even and whenever there is a closed reduced path in $\Gamma$ with all labels bigger than 2, then the length of such path is always odd. \end{itemize} Then $[\chi]\in\Sigma^1(A_\Gamma)\iff\mathcal{L}^\chi$ is connected and dominating.
\end{teo}
Moreover, the class of Artin groups that satisfy the hypothesis in Theorem~\ref{teo:knownArtin} is known to be closed under graph products and, as a consequence, every $\FC$-type Artin group also does (\cite{A21}). Other concrete examples of Artin groups satisfying this hypothesis can be found in \cite{AK15b} and \cite{A17}.
Note that by a similar observation as above, this result can be stated as follows: For $\Gamma$ connected and with $\pi_1(\Gamma)=1$ or free of rank at most 2, then $$[\chi]\in\Sigma^1(A_\Gamma)\iff\chi\text{ satisfies the strong 1-link condition}.$$
Observe also that here we are not assuming that $A_\Gamma$ is even.
\subsection{An easier particular case: direct products of Artin dihedral groups}\label{subsec:productdihedral}
It will be important below to understand the Sigma-invariants of the finite type Artin subgroups $A_\Delta$ of a given even Artin group of $\FC$-type $A_\Gamma$. In general, finite type Artin groups are direct products of finite type irreducible Artin groups and the only possible irreducible finite type Artin groups are those of dihedral type, which are the Artin groups associated to a single edge. In the even case the edge is labeled by an even integer, say $2\ell$ and the associated group is
$$\D_{2\ell}=\langle x,y\mid (xy)^{\ell}=(yx)^{\ell}\rangle.$$
The (homotopical) Sigma-invariants for irreducible Artin groups of finite type have been described in \cite[Section 2]{M2001}. In the particular case of a dihedral Artin group we have the following result.
\begin{lem}\cite[Pg 76]{M2001}\label{lem:dihedral} Let $G=\D_\ell$ be a dihedral Artin group and $n\geq 1$. For any commutative ring $R$ \begin{itemize} \item[i)] If $\ell$ is odd, then $S(G)$ is a 0-sphere and $S(G)=\Sigma^n(G)=\Sigma^n(G,R)$ for any $n$.
\item[ii)] If $\ell=2\tilde{\ell}$ is even $S(G)$ is a 1-sphere. Denoting by $x$, $y$ the standard generators, we have
$\Sigma^n(G)=\Sigma^n(G,R)=S(G)\setminus\{[\chi],[-\chi]\}$ where $\chi(x)=1$, $\chi(y)=-1$.
\end{itemize} \end{lem} \begin{proof} For the homotopical result, see \cite[p.~76]{M2001}. For the homological one note that $\Sigma^1(G)=\Sigma^1(G,R)$ and $$\Sigma^n(G)\subseteq\Sigma^n(G,R)\subseteq\Sigma^1(G,R).$$ \end{proof}
If $A_\Gamma$ is an even Artin group of $\FC$-type and $A_\Delta$ is a finite type subgroup with $\Delta\subseteq\Gamma$, then $\Delta$ must be a clique and $A_\Delta$ is a direct product of even Artin dihedral groups and possibly a factor which is free abelian of finite rank. In this subsection we will give a full description of the Sigma-invariants of such an $A_\Delta$. But we will consider the slightly more general case of a product of {\sl arbitrary} Artin dihedral groups and possibly a free abelian groups of finite rank. Assume $$G=G_1\times\dots \times G_s$$ where each of the $G_i$'s is either $\mathbb{Z}$ or Artin dihedral.
Using \cite[Theorem 1.4]{BG10} and the fact that according to Lemma~\ref{lem:dihedral} the $R$-Sigma-invariants for Artin dihedral groups for $R=\mathbb{Z}$ and $R=\mathbb{Q}$ coincide, we deduce (the upper script $c$ means the complementary of the corresponding subset) $$\Sigma^n(G,\mathbb{Z})^c=\bigcup_{n_1+\ldots+n_s=n,n_i\geq 0}\Sigma^{n_1}(G_1,\mathbb{Z})^c\star\dots\star\Sigma^{n_s}(G_s,\mathbb{Z})^c,$$ where $\star$ is the {\sl join} product in the corresponding spheres (see \cite{BG10}). We have $\Sigma^m(G_i,\mathbb{Z})^c=\emptyset$ unless $m\geq 1$ and $G_i$ is dihedral of even type, in that case $\Sigma^m(G_i,\mathbb{Z})^c=\{[\chi_i],-[\chi_i]\}$ where $\chi_i$ maps the standard generators of $G_i$ to $1$ and $-1$ resp. As a consequence, if we order the factors so that $G_1,\ldots,G_t$ are precisely those which are dihedral of even type, and denote by $v_{t+1},\ldots v_s$ the vertices not involved in dihedral type edges, we have
$$ \Sigma^m(G,\mathbb{Z})^c=\{[\chi]\in S(G)\mid\chi(Z(G)=0\}\text{ if } m\geq t. $$ Note that $\chi(Z(G))=0$ is equivalent to $m_e=0\text{ for } e\in E_\Gamma \text{ if } \ell(e)>2\text{ and }\chi(v_i)=0\text{ for }t+1\leq i\leq s.$ For $m<t$, $\Sigma^m(G,\mathbb{Z})^c$ consists of characters that vanish in all the vertices $v_i$ for $t+1\leq i\leq s$ and in at least $t-m$ of the factors $G_i$ for $1\leq i\leq t$, having zero $m_e$-value in the rest.
\subsection{Coset posets}
In this section we prove some results on coset posets that will be used in the main proofs later on.
\begin{dfn} Let $G$ be a group and $\mathcal{P}$ a poset (ordered by inclusion) of subgroups of $G$. The {\sl coset complex} $C_G(\mathcal{P})$ (or simply $C(\mathcal{P})$ if the group $G$ is clear by the context) is the geometric realization of the poset $G\mathcal{P}$ of cosets $gS$ where $g\in G$ and $S\in\mathcal{P}$. In other words, it is the geometric realization of the simplicial complex whose $k$-simplices are the chains \begin{equation}\label{eq:simplices} g_0S_0\subset g_1S_1\subset\dots\subset g_kS_k=g_0(S_0\subset S_1\subset\dots\subset S_k), \end{equation} where $g_0,\ldots,g_k\in G$ and $S_0,\ldots,S_k\in\mathcal{P}$. Let $\mathcal{P}_\chi$ be the subposet of
$\mathcal{P}$ consisting of those subgroups $S\in\mathcal{P}$ such that $\chi|_S\neq 0$. It yields a subcomplex $C(\mathcal{P})_\chi$ of $C(\mathcal{P})$. We identity $C(\mathcal{P})$ with its geometric realization. \end{dfn}
We will consider posets of subgroups $\mathcal{P}$ having a {\sl height} function $h:\mathcal{P}\to\mathbb{Z}^+\cup\{0\}$ such that whenever $S\subsetneq T$ both sit in $\mathcal{P}$, we have $h(S)<h(T)$. We also assume that there is a bound for the height of the elements of $\mathcal{P}$. We denote that bound by $h(\mathcal{P})$. Now, assume we have a subposet $\mathcal{H}\subseteq\mathcal{P}$. Then $C(\mathcal{H})$ is a subcomplex of $C(\mathcal{P})$. We want to compare the homology of $C(\mathcal{P})$ with the homology of $C(\mathcal{H})$. Let $$\sigma:g(S_0\subset S_1\subset\dots\subset S_k)$$ be a $k$-simplex in $C(\mathcal{P})$. Using the height function $h$ we set $$h_{\mathcal{H}^c}(\sigma)= \begin{cases} -1 & \text{ if }S_i\in\mathcal{H} \text{ for every }0\leq i\leq k,\\ \mathrm{max}\{h(S_j)\mid S_j\not\in\mathcal{H}\} & \text{ in other case}.\\ \end{cases}$$
Note that if $\tau\subseteq\sigma$ is a face, then $h_{\mathcal{H}^c}(\tau)\leq h_{\mathcal{H}^c}(\sigma)$. This implies that we can use this function to define a subcomplex $$D^s:=\{\sigma\in C(\mathcal{P})\mid h_{\mathcal{H}^c}(\sigma)\leq s\}$$ for $-1\leq s\leq h(\mathcal{P})$. So we have a filtration $$C(\mathcal{P}_\chi)=D^0\subseteq D^1\subseteq\dots\subseteq D^{h(\mathcal{P})}=C(\mathcal{P}).$$ For $s\geq 0$, simplices in $D^{s}$ but not in $D^{s-1}$ are of the form $$\sigma:g(S_0\subset S_1\subset\dots\subset S_k)$$ such that there is some $0\leq i\leq k$ with $S_i\in\mathcal{P}\setminus\mathcal{H}$ of height precisely $s$ and $S_j\in\mathcal{H}$ for $j>i$.
Fix an $S\in\mathcal{P}\setminus\mathcal{H}$ of height precisely $s$ and consider the set of all simplices of the form \begin{equation}\label{simplex} \sigma:g(S_0\subseteq S_1\subseteq\dots \subseteq S_i\subseteq S_{i+1}\subseteq\dots\subseteq S_k) \end{equation} with $S_i=S$ and $S_j\in\mathcal{H}$ for each $j>i$. Those simplices lie in $D^s$. The boundary $\partial\sigma$ of such a $k$-simplex $\sigma$ consists of $k-1$-simplices $\tau$ which are of the same form except of the case when $$\tau:g(S_0\subseteq S_1\subseteq\dots\subseteq S_{i-1}\subseteq S_{i+1}\subseteq\dots\subseteq S_k)$$ and then $\tau\in D^{s-1}$. Consider now the complex $Z^S$ which is the geometric realization of $C_S(\mathcal{P}_S)\setminus \{S\}$ for the poset $\mathcal{P}_S=\{S\cap T\mid T\in\mathcal{P}\}$ (note that $Z^S$ could be empty if $S$ is empty). And let $\mathcal{J}^S$ be the poset $$\mathcal{J}^S:=\{T\in\mathcal{H}\mid S\subseteq T\}.$$
The join $Z^S\star|\mathcal{J}^S|$ is in a natural way an $S$-space and we may form the induced $G$-space $G/S\times(Z^S\star|\mathcal{J}^S|)$. Its chain complex is a $G$-complex that consists of induced modules of the form $\mathcal{C_\bullet}(Z^S\star|\mathcal{J}^S|)\uparrow_S^G$. We claim that the quotient complex $D^{s+1}/D^s$ can be decomposed as \begin{equation}\label{complexS}
\bigoplus_{S\in\mathcal{P}\setminus\mathcal{H},h(S)=s}\tilde{\mathcal{C}}_{\bullet +1}(G/S\times(Z^S\star|\mathcal{J}^S|) \end{equation} where $\tilde{\mathcal{C}}_{\bullet +1}$ is the augmented chain complex shifted by one. To see it, consider a simplex $\sigma$ as in~\eqref{simplex} and put $g=xy$ for $y\in S$ so that $$\sigma:x(yS_0\subseteq\dots \subseteq yS_{i-1}\subseteq S\subseteq S_{i+1}\subseteq\dots\subseteq S_k).$$ Then $\sigma$ yields a free direct summand of the chain complex of $D^{s+1}/D^s$ at dimension $k$
that we map onto the summand of $\tilde{\mathcal{C}}_{\bullet +1}(G/S\times(Z^S\star|\mathcal{J}^S|)$ associated to $x\otimes(\sigma_1\star\sigma_2)$ with $$\sigma_1:y(S_0\subseteq\dots\subseteq S_{i-1})$$ and $$\sigma_2: S_{i+1}\subseteq\dots\subseteq S_k.$$
\begin{lem}\label{lem:cosetposets} With the previous notation, assume that for any $S\in\mathcal{P}\setminus\mathcal{H}$ with $h(S)=s$ we have that
$Z^S$ is $(s-2)$-acyclic or empty if $s=0$ and $|\mathcal{J}^S|$ is $(n-1-s)$-acyclic. Then, if $C(\mathcal{P})$ is $(n-1)$-acyclic, so is $C(\mathcal{H})$. \end{lem} \begin{proof}
Using Mayer-Vietoris one can determine the homology groups $\mathrm{H}_r(Z^S\star|\mathcal{J}^S|)$ in terms of $\mathrm{H}_i(Z^S)$ and
$\mathrm{H}_j(|\mathcal{J}^S|)$ for $i+j<r$ so the hypothesis imply that the complex $Z^S\star|\mathcal{J}^S|$ is $(n-1-s+s-2+2)=(n-1)$-acyclic.
By equation~\eqref{complexS},
$$\mathrm{H}_j(D^{s+1}/D^s)=\bigoplus_{S\in\mathcal{P}\setminus\mathcal{H},h(S)=s}\tilde{\mathrm{H}}_{j-1}(Z^S\star|\mathcal{J}^S|)\uparrow_{S}^G=0$$ for $0\leq j\leq n$. And this implies the result: to see it assume by induction that $D^{s+1}$ is $R$-$(n-1)$-acyclic (the beginning of the induction is $D^{h(\mathcal{P})}=C(\mathcal{P})$ which is $(n-1)$-acyclic) and consider the long exact homology sequence of the short exact sequence of complexes $0\to D^s\to D^{s+1}\to D^{s+1}/D^s\to 0$ $$\ldots\to0=\mathrm{H}_{i+1}(D^{s+1})\to\mathrm{H}_{i+1}(D^{s+1}/D^s)\to\mathrm{H}_i(D^s)\to\mathrm{H}_i(D^{s+1})=0\to\ldots$$ Thus also $D^s$ is $(n-1)$-acyclic. Eventually, $C(\mathcal{H})=D^0$ is $(n-1)$-acyclic. \end{proof}
\begin{obs}\label{obs:Morse} As noted by an anonymous referee of this paper, Lemma~\ref{lem:cosetposets} can also be proven using Morse theory,
having $h_{\mathcal{H}^c}$ as the Morse function. The complex $Z^S\star|\mathcal{J}^S|$ is the link of $S$ in $D^{s+1}$ so the hypothesis of Lemma~\ref{lem:cosetposets} is in fact a condition on the acyclicity of the link. The proof presented in this paper exhibits the fact from Morse theory that acyclicity of the links yields isomorphisms between the homology groups of the involved subcomplexes. \end{obs}
\begin{obs}\label{obs:R} In Lemma~\ref{lem:cosetposets} we can substitute the instances of ``being acyclic'' by ``being $R$ acyclic'' for any unital commutative ring $R$. \end{obs}
\section{proof of Theorem~\ref{teo:mainsigma}} As stated above, our proof of Theorem~\ref{teo:mainsigma} is based in Theorem~\ref{teo:sigmacomplex}. To do that we need a suitable complex $X$.
Let $A_\Gamma$ be an Artin group and consider the {\sl clique poset} $$\mathcal{P}=\{A_\Delta\mid\Delta\subseteq\Gamma\text{ clique}\}$$ (recall that a {\sl clique} is a complete subgraph). If the Artin group $A_\Gamma$ is of $\FC$-type, then any clique of $\Gamma$ generates a finite type Artin subgroup so the coset complex $C_G(\mathcal{P})=C(\mathcal{P})$ of $\mathcal{P}$ is the {\sl modified Deligne complex} for $A_\Gamma$ considered by Charney and Davis in \cite{Charney-kpi1}. In \cite{GodelleParis} the modified Deligne complex is used to construct what is called the {\sl clique cube-complex} which is a CAT-(0) cube complex.
For Artin groups of $\FC$-type, the modified Deligne complex was shown to be contractible in \cite{Charney-kpi1} but for completeness, we give a direct easy proof of the fact that the coset complex $C_G(\mathcal{P})$ is contractible in general, i.e., for arbitrary Artin groups $A_\Gamma$ possibly without the $\FC$ condition.
\begin{lem}\label{lem:complex} Let $G=A_\Gamma$ be an Artin group and consider $\mathcal{P}$ the clique poset. The coset complex $C_G(\mathcal{P})=C(\mathcal{P})$ is contractible. \end{lem} \begin{proof} We argue by induction on the number of vertices of $\Gamma$. Observe first that the result is obvious if $\Gamma$ is complete, because then $G$ itself lies in $\mathcal{P}$. If $\Gamma$ is not complete we may decompose $\Gamma=\Gamma_1\cup\Gamma_2$ with $\Gamma_1,\Gamma_2\subsetneq\Gamma$ such that for $\Gamma_0=\Gamma_1\cap\Gamma_2$ no vertex in $\Gamma_1\setminus\Gamma_0$ is linked to any vertex in $\Gamma_2\setminus\Gamma_0$. This decomposition induces a decomposition of $G$ as the free product with amalgamation $$G=A_{\Gamma_1}\star_{A_{\Gamma_0}}A_{\Gamma_2}.$$ Let $C(\mathcal{P}_1)$, $C(\mathcal{P}_2)$ and $C(\mathcal{P}_0)$ be the corresponding coset complexes for the clique posets of $A_{\Gamma_1}$, $A_{\Gamma_2}$ and $A_{\Gamma_0}$. By induction we may assume that they are all contractible. Consider the poset $$\mathcal{BS}=\{gA_{\Gamma_1}\mid g\in G\}\cup\{gA_{\Gamma_2}\mid g\in G\}\cup\{gA_{\Gamma_0}\mid g\in G\}.$$ The geometric realization of this poset is precisely the barycentric subdivision of the Bass-Serre tree associated to the free amalgamated product above. Consider the map $$\begin{aligned}f:C(\mathcal{P})&\to\mathcal{BS}\\ gA_\Delta&\mapsto\left\{ \begin{aligned} &gA_{\Gamma_0}\text{ if }\Delta\subseteq \Gamma_0\\ &gA_{\Gamma_1}\text{ if }\Delta\subseteq \Gamma_1, A_{\Delta}\not\leq A_{\Gamma_0}\\ &gA_{\Gamma_2}\text{ if }\Delta\subseteq \Gamma_2, A_{\Delta}\not\leq A_{\Gamma_0}.\\ \end{aligned}\right. \end{aligned} $$ Observe that each clique of $\Gamma$ is a subgraph either of $\Gamma_1$ or of $\Gamma_2$. Moreover, if $gS\subseteq gT$ with $S$ and $T$ both in $\mathcal{P}$, then $f(gS)\subseteq f(sT)$ so it is a well-defined poset map and for any $gA_{\Gamma_i}\in\mathcal{BS}$, $$\{gS\in C(\mathcal{P})\mid f(gS)\leq gA_{\Gamma_i}\}= gC_{A_{\Gamma_i}}(\mathcal{P}_i)$$ i.e., it is the coset poset of the clique poset of $A_{\Gamma_i}$ shifted by $g$. By induction, the posets $C_{A_{\Gamma_i}}(\mathcal{P}_i)$ have contractible geometric realizations for $i=0,1,2$. So we may apply Quillen poset map Lemma (\cite{Benson}) and deduce that $f$
induces a homotopy equivalence between the geometric realizations. As the geometric realization $|\mathcal{BS}|$
is contractible, we deduce the same for the geometric realization $|C(\mathcal{P})|$. \end{proof}
Note that the Artin group acts on the clique poset so we have a nice action on the geometric realization. However, this is not what we need to apply Theorem~\ref{teo:sigmacomplex} because the stabilizers of this action are not nice enough. In order to construct our suitable $X$, we will also need an auxiliary Lemma.
\begin{lem}\label{lem:spheres} Let $\Delta$ be a (non empty) complete graph with $s$ vertices and with $S:=A_\Delta$ even of $\FC$-type and consider the simplicial complex $Z^S$ with simplices $$hS_0\subseteq hS_1\subseteq\dots \subseteq hS_r$$ for $h\in A_\Delta$ and each $S_j$ a special {\sl proper} subgroup of $A_\Delta$. Then $Z^S$ is $(s-2)$-acyclic. \end{lem}
\begin{proof} We will denote $\Delta_v=\Delta\setminus \{v\}$. For each $g\in S$, $v\in\Delta$ let $gX_{v}$ be the subcomplex of $Z^S$ induced by $\{gA_{\Omega}\mid \Omega\subseteq\Delta_v\}.$ Obviously, the set $\{gX_{v}\mid g\in S,v\in\Delta\}$ is a covering of $Z^S$ as each special proper subgroup of $A_\Delta$ is inside one of the form $gA_{\Delta_v}$.
Observe that for any $\Delta_1\subset\Delta$, if $g_1,g_2\in S$ are such that $g_1A_{\Delta_1}\neq g_2A_{\Delta_1}$, then $g_1A_{\Delta_1}\cap g_2A_{\Delta_1}=\emptyset$. And if we have $\Delta_1,\Delta_2$ and $g_1,g_2\in S$ so that $g_1A_{\Delta_1}\cap g_2A_{\Delta_2}\neq\emptyset$, then for some $x\in A_{\Delta_1}$, $y\in A_{\Delta_2}$, $g_1x=g_2y$ so $$g_1A_{\Delta_1}\cap g_2A_{\Delta_2}=g_1(A_{\Delta_1}\cap xA_{\Delta_2})=g_1A_{\Delta_1\cap\Delta_2}$$ because $g\in A_{\Delta_1}\cap xA_{\Delta_2}$ implies $x^{-1}g\in A_{\Delta_1}\cap A_{\Delta_2}=A_{\Delta_1\cap\Delta_2}$ ($A_{\Delta_1}\cap A_{\Delta_2}=A_{\Delta_1\cap\Delta_2}$ by \cite{Van}). This implies that any set of pairwise distinct $g_0X_{v_0},\ldots,g_kX_{v_k}$ has empty intersection if two of the $v_i$'s are equal and also that whenever such as set has non empty intersection, then the intersection is the subcomplex induced by $\{gA_{\Omega}\mid \Omega\subseteq\Delta_{v_1}\cap\ldots\cap\Delta_{v_k}\}$ which is contractible (it is the geometric realization of a poset with a maximal element). The last property implies that $Z^S$ is homotopy equivalent to the nerve of the covering (see \cite[Chap. VII Theorem 4.4]{BrownBook}) which we denote $N_\Delta$. The nerve has $k$-simplices the sets $\{g_0X_{v_0},\ldots,g_kX_{v_k}\}$ with non empty intersection and the discussion above implies that the cardinality of such as set can be at most $s$ so this nerve is a complex of dimension $s-1$. To deduce that this complex is $(s-2)$-connected one can use \cite[(4.18), Theorem (2.15)]{Deligne-immeubles}, but it is also possible to give a direct argument as follows. As we are assuming that $S$ is even of $\FC$-type, we have a decomposition $$S=S_1\times\ldots\times S_t$$ where each $S_i=A_{\Delta_i}$ is either infinite cyclic or Artin dihedral (of even type). This means that for each $g\in S$ and $v\in\Delta$, $gA_{\Delta_v}=g_iA_{\Delta_v}$ for some $g_i\in S_i$ where $i$ is the index such that $v\in\Delta_i$. From this one deduces that $N_\Delta$ is the join $$N_\Delta=N_{\Delta_1}\star\ldots\star N_{\Delta_t}$$ where each $N_{\Delta_i}$ is the nerve associated to the same construction but for $S_i$. Moreover, if $S_i$ is infinite cyclic, obviously $N_{S_i}$ is a disjoint union of points, thus non empty and if $S_i$ is Artin dihedral, then it is easy to see that $N_{S_i}$ is 0-connected. As $s=a+2b$ where $a$ is the number of $S_i$'s which are infinite cyclic, we get the result.
\end{proof}.
At this point, we are able to construct the desired $X$. Note that Theorem~\ref{teo:sigmacomplex} together with the next result imply Theorem~\ref{teo:mainsigma}.
In the introduction we have defined the subposet $\mathcal{B}^\chi$ of $\mathcal{P}$ as those $A_\Delta$ for $\Delta\subseteq\Gamma$ clique such that for each vertex $v$ in $\Delta$ either $v$ is dead or $v\in e$ for $e$ a dead edge in $\Delta$, this includes the case when $\Delta=\emptyset$. The hypothesis that $A_\Gamma$ is even of $\FC$-type implies that $A_\Delta$ is a direct product of dihedral groups (corresponding to edges with label bigger than 2) and of infinite cyclic groups (generated by dead vertices). Taking into account that the center of the dihedral group generated by, say, $x$ and $y$ is the infinite cyclic group generated by $xy$ we see that for such a $\Delta$ we have $\chi(Z(A_\Delta))=0$. The converse is also obvious. So we have $$\mathcal{B}^\chi=\{A_\Delta\in\mathcal{P}\mid \chi(Z(A_\Delta))=0\}.$$ Denote $$\mathcal{H}^\chi:=\mathcal{P}\setminus\mathcal{B}^\chi,$$
one has the following result on the homotopy of the geometric realization of the coset poset $X:=|C\mathcal{H}^\chi|$.
\begin{prop}
Let $A_{\Gamma}$ be an even Artin group of $\FC$-type. Let $\chi:A_\Gamma\to\mathbb{R}$ be a character. If the strong $n$-link condition holds, then the geometric realization of the coset poset $X:=|C\mathcal{H}^\chi|$ is $(n-1)$-acyclic. \end{prop}
\begin{proof}
Use Lemma~\ref{lem:cosetposets} for $\mathcal{P}$ the clique poset with $h(A_\Delta)=|\Delta|$. Fix $S=A_\Delta\in\mathcal{B}^\chi=\mathcal{P}\setminus\mathcal{H}^\chi$ with $h(S)=s$. The complex denoted $Z^S$ in Lemma~\ref{lem:cosetposets}
is the simplicial complex of Lemma~\ref{lem:spheres} and $\mathcal{J}^S$ is the poset
$$\mathcal{J}^S:=\{T\in\mathcal{P}\mid S\subseteq T,\chi(Z(T))\neq 0\}.$$
Now, consider the poset
$$\mathcal{L}^S:=\{L=A_{\sigma}\in\mathcal{P}\mid \emptyset\neq\sigma\text{ clique of }{\lk}_{\mathcal{L}^\chi}(\Delta)\}.$$ Let $A_\sigma\in\mathcal{L}^S$. Then for $T=A_{(\sigma\star\Delta)}$, we have $S\leq T$ and $T\in\mathcal{P}$. We claim that $T\in\mathcal{J}^S$. To see it, note that as $\sigma\neq\emptyset$ and $\sigma\subseteq\mathcal{L}^\chi$, $Z(A_\sigma)\neq 0$. As $\Gamma$ is even of $\FC$-type, this implies that either there is some $v\in\sigma$, $v\in Z(A_\sigma)$ with $\chi(v)\neq 0$ or there are $v,w\in\sigma$, $(vw)^k\in Z(A_\sigma)$ for some $k$ with $\chi(v)+\chi(w)\neq 0$. Again, the condition that $\Gamma$ is of $\FC$-type implies that in the second case, $(vw)^k\in Z(T)$ so $\chi(Z(T))\neq 0$. In the first case, either $v\in Z(T)$ so again $\chi(Z(T))\neq 0$ or there is some $w\in\Delta$ with $(vw)^k\in Z(T)$ for some $k$. In this case moreover $w\in Z(S)$ thus $\chi(w)=0$. Therefore $\chi((vw)^k)\neq 0$ and again $\chi(Z(T))\neq 0$. The claim follows and therefore we have a well defined poset map
$$ \begin{aligned} f: \mathcal{L}^S&\to\mathcal{J}^S\\
A_\sigma&\mapsto A_{(\sigma\star\Delta)}.\\
\end{aligned}$$
We claim that this map induces a homotopy equivalence between the corresponding geometric realizations. To see it, let $T\in\mathcal{J}^S$ and consider
$$f^{-1}_{\leq T}=\{U\in \mathcal{L}^S\mid f(U)\leq T\}.$$
By Quillen's poset map Theorem (see~\cite{Benson}) it suffices to check that the poset $f^{-1}_{\leq T}$ has contractible geometric realization. Put $T=A_\nu$. Then $\nu$ is a clique of $\Gamma$
such that $\nu=\Delta\star\sigma$ for some $\emptyset\neq\sigma$ clique in $\lk_\Gamma(\Delta)$. We can describe $\sigma$ as a join
$$\sigma=e_1\star\dots\star e_t\star p_1\star\dots p_s$$
where each $e_i$ is a single edge with label $>2$ and each $p_i$ is a single point and all the edges not in some $e_i$ are labeled by 2.
We may order them so that $\chi(e_1),\ldots,\chi(e_l)=0$, $\chi(e_{l+1}),\ldots,\chi(e_t)\neq 0$, $\chi(p_1),\ldots,\chi(p_r)=0$, $\chi_(p_{r+1}),\ldots,\chi(p_s)\neq 0$. Then
$$\sigma\cap\mathcal{L}^\chi=w_1\star\dots\star w_l\star e_{l+1}\star\dots \star e_t\star p_{r+1}\star\dots\star p_s$$
where each $w_i$ is the disconnected set consisting of the two vertices of each $e_i$ has as barycentric subdivision precisely the geometric realization of the poset $f^{-1}_{\leq T}$. As $e_{l+1}\star\dots \star e_t\star p_{r+1}\star\dots\star p_s$ is either contractible or empty, so show that $f^{-1}_{\leq T}$ is contractible we only have to show that $e_{l+1}\star\dots \star e_t\star p_{r+1}\star\dots\star p_s$ is not empty.
As $T\in\mathcal{J}^S$, $\chi(Z(T))\neq 0$. If there is some $v$ vertex of $\nu$ with $v\in Z(T)$ and $\chi(v)\neq 0$ then the fact that $\chi(Z(S))=0$ implies $v\in\sigma$ so $v$ belongs to $\{p_{r+1},\ldots,p_s\}$. So we are left with the case when $\chi(vw)\neq 0$ for $v,w$ vertices of an edge of $\nu$ with label $>2$. If, say, $v$ lies in $\Delta$, then $v\in Z(S)$ so $\chi(v)=0$ and we deduce $\chi(w)\neq 0$. Moreover, in this case the $\FC$-condition implies that $w$ is in the center of $A_\sigma$, i.e., $w$ belongs to $\{p_{r+1},\ldots,p_s\}$. So we may assume that both $v,w$ lie in $\sigma$ so the edge joining them lies in the set $\{e_{l+1},\ldots,e_t\}$. \end{proof}
We finish this section with a couple of example to illustrate how to apply Theorem~\ref{teo:mainsigma}.
\begin{exam} Let $\Gamma$ be the graph and $\chi$ the character $$ \begin{tikzpicture}[scale=1.5, transform shape] \tikzstyle{subj} = [circle, minimum width=3pt, fill, inner sep=0pt] \tikzstyle{obj} = [circle, minimum width=3pt, draw, inner sep=0pt] \node[subj] (n1) at (1,1) {}; \node[subj] (n2) at (2,1) {}; \node[subj] (n3) at (2,2) {}; \node[subj] (n4) at (1,2) {}; \draw (0.3,1.5) node {{\small{$\Gamma$}}}; \draw (1,2) node[left] {\tiny{$a$}} ; \draw (0.75,2) node[left] {\tiny 1} ; \draw (0.75,1) node[left] {\tiny 0} ; \draw (2.25,2) node[right] {\tiny -1} ; \draw (2.25,1) node[right] {\tiny1} ; \draw (2,2) node[right] {\tiny{$b$}}; \draw (1,1) node[left] {\tiny{$c$}} ; \draw (2,1) node[right] {\tiny{$d$}} ; \draw(1.5,2) node[above]{\tiny{4}}; \draw(1.5,1) node[below]{\tiny{4}}; \draw(1,1.5) node[left]{\tiny{2}}; \draw(2,1.5) node[right]{\tiny{2}}; \draw(1.5,1.5) node[right]{\tiny{2}}; \draw (1,1) -- (1,2) -- (2,2) -- (2,1)--(1,1); \draw (1,2)--(2,1); \draw (3.8,1.5) node {{\small{$\mathcal{L}^\chi$}}}; \node[subj] (n2) at (2+3,1) {}; \node[subj] (n3) at (2+3,2) {}; \node[subj] (n4) at (1+3,2) {}; \draw (1+3,2) node[left] {\tiny{$a$}} ; \draw (0.75+3,2) node[left] {\tiny 1} ; \draw (2.25+3,2) node[right] {\tiny -1} ; \draw (2.25+3,1) node[right] {\tiny 1} ; \draw (2+3,2) node[right] {\tiny{$b$}}; \draw (2+3,1) node[right] {\tiny{$d$}} ; \draw(2+3,1.5) node[right]{\tiny{2}}; \draw(1.5+3,1.5) node[right]{\tiny{2}}; \draw (2+3,2) -- (2+3,1); \draw (1+3,2)--(2+3,1); \end{tikzpicture} $$ For $\Delta=(ab)$, $Z(A_\Delta)$ is generated by $ab$ so $\chi(Z(A_\Delta))=0$.
For $\Delta=(a,b,d)$, $Z(A_\Delta)$ is generated by $ab$, $d$ so $\chi(Z(A_\Delta))\neq 0$.
We get: $\mathcal{P}\setminus{\mathcal{H}_\chi}=\{\emptyset,(c),(ab)\}$. The links are:
$$ \begin{tikzpicture}[scale=1.5, transform shape] \tikzstyle{subj} = [circle, minimum width=2pt, fill, inner sep=0pt] \tikzstyle{obj} = [circle, minimum width=2pt, draw, inner sep=0pt] \draw(-2,0) node{\tiny{$\lk_{\mathcal{L}^\chi}(\emptyset)=\mathcal{L}^\chi$}}; \draw(-0.2,0) node[left=-15] {\tiny{$\lk_{\mathcal{L}^\chi}(c)=$}}; \draw(0.5,0.15)--(0.8,-0.15); \node[subj] (n1) at (0.5,0.15) {}; \node[subj] (n2) at (0.8,-0.15){}; \draw(0.8,-0.15) node[right]{\tiny{$d$}}; \draw(0.5,0.15) node[left]{\tiny{$a$}}; \draw(2,0) node{\tiny{$\lk_{\mathcal{L}^\chi}(ab)=d$}}; \end{tikzpicture} $$ All the links are contractible so $\chi\in\Sigma^\infty(A_\Gamma,\mathbb{Z})$. \end{exam}
\begin{exam} Let $\Gamma$ be the graph and $\chi$ the character $$ \begin{tikzpicture}[scale=1.5, transform shape] \tikzstyle{subj} = [circle, minimum width=3pt, fill, inner sep=0pt] \tikzstyle{obj} = [circle, minimum width=3pt, draw, inner sep=0pt] \node[subj] (n1) at (1,1) {}; \node[subj] (n2) at (2,1) {}; \node[subj] (n3) at (2,2) {}; \node[subj] (n4) at (1,2) {}; \draw (0.3,1.5) node {{\small{$\Gamma$}}}; \draw (1,2) node[left] {\tiny{$a$}} ; \draw (0.75,2) node[left] {\tiny 1} ; \draw (0.75,1) node[left] {\tiny 0} ; \draw (2.25,2) node[right] {\tiny -1} ; \draw (2.25,1) node[right] {\tiny1} ; \draw (2,2) node[right] {\tiny{$b$}}; \draw (1,1) node[left] {\tiny{$c$}} ; \draw (2,1) node[right] {\tiny{$d$}} ; \draw(1.5,2) node[above]{\tiny{4}}; \draw(1.5,1) node[below]{\tiny{4}}; \draw(1,1.5) node[left]{\tiny{2}}; \draw(2,1.5) node[right]{\tiny{2}}; \draw (1,1) -- (1,2) -- (2,2) -- (2,1)--(1,1); \draw (3.8,1.5) node[scale=.7,right] {$\mathcal L^\chi$}; \node[subj] (n2) at (2+3,1) {}; \node[subj] (n3) at (2+3,2) {}; \node[subj] (n4) at (1+3,2) {}; \draw (1+3,2) node[left] {\tiny{$a$}} ; \draw (0.75+3,2) node[left] {\tiny 1} ; \draw (2.25+3,2) node[right] {\tiny -1} ; \draw (2.25+3,1) node[right] {\tiny 1} ; \draw (2+3,2) node[right] {\tiny{$b$}}; \draw (2+3,1) node[right] {\tiny{$d$}} ; \draw(2+3,1.5) node[right]{\tiny{2}}; \draw (2+3,2) -- (2+3,1); \end{tikzpicture} $$ As before: $\mathcal{P}\setminus{\mathcal{H}_\chi}=\{\emptyset,(c),(ab)\}$. However, $\lk_{\mathcal{L}^\chi}(\emptyset)=\mathcal{L}^\chi$ is not connected, so $[\chi]$ might not even be in $\Sigma^1(A_\Gamma,\mathbb{Z})$.
Incidentally, in this case, the hypothesis of Corollary~\ref{corol:sigmachar} below is satisfied for $p=2$, hence the converse of Theorem~\ref{teo:mainsigma} also holds true so $$[\chi]\not\in\Sigma^1(A_\Gamma,\mathbb{Z}).$$ \end{exam}
\section{The free part of the homology groups of Artin kernels}\label{sec:free}
Let $A_\Gamma$ be an even Artin group of $\FC$-type and $\chi:A_\Gamma\to\mathbb{Z}$ a discrete character. In this section, we are interested in the homology groups $\mathrm{H}_n(A_\Gamma^\chi,\mathbb{F})$ where $\mathbb{F}$ is a field of characteristic $p$ (either zero or a prime). More precisely, we want to characterize when they are finite $\mathbb{F}$ dimensional and, more generally, to compute their free part when seen as $\mathbb{F}[t^{\pm 1}]$-modules via~$\chi$.
To do that we first develop a $p$-local version of some of the notions that we used above.
We say that an edge $e$ of $\Gamma$ with label $2\tilde{\ell}_e$ is {\sl $p$-dead} if $m_e=\chi(v)+\chi(w)=0$ and $p\mid \tilde{\ell}$. In \cite{Blasco-conmar-ji-Homology}, $p$-dead edges were called $\mathbb{F}$-resonant (recall that $\mathbb{F}$ is a field of characteristic $p$).
We set $\mathcal{L}^\chi_p$ for the subgraph of $\Gamma$ that we get when we remove dead vertices and open $p$-dead edges. This notation is consistent with $\mathcal{L}^\chi_0$ because no edge can be $0$-dead. Note that the set of dead edges is the union of the sets of $p$-dead edges where $p$ runs over all prime numbers and therefore $\mathcal{L}^\chi=\bigcap_{p\textrm{ prime}}\mathcal{L}_p^\chi$.
Let $\mathcal{B}_p^\chi$ be the set of those $A_{\Delta}\in\mathcal{P}$ for a clique $\Delta\subseteq\Gamma$ such that for each vertex $v$ in $\Delta$ either $v$ is dead or $v\in e$ for $e$ a dead edge in $\Delta$ (this includes the case when $\Delta=\emptyset$). Note that as edges which are $p$-dead for some $p$ are dead, this condition implies that~$\chi(Z(A_{\Delta}))=0$.
\begin{dfn} Assume that for any
$A_\Delta\in\mathcal{B}_p^\chi$ with $|\Delta|\leq n$ the link $\hat{\lk}_{\mathcal{L}_p^\chi}(\Delta)$
is $p-(n-1-|\Delta|)$-acyclic, meaning that its homology up to degree $(n-1-|\Delta|)$ with coefficients in a field of characteristic $p$ vanishes. Then we say that $\chi$ satisfies \emph{the strong $p$-$n$-link condition}. \end{dfn}
The homology groups $\mathrm{H}_n(A_\Gamma^\chi,\mathbb{F})$ are precisely the homology groups of the $\mathbb{F}$-chain complex $C_n(\Sal^\chi_\Gamma)$ described in \cite[Section 2]{Blasco-conmar-ji-Homology}. This complex was obtained using the $\chi$-cyclic cover of the Salvetti complex of $A_\Gamma$ (see~\cite{Salvetti-topology,Charney-finite,Paris-lectures}) and has $$C_n(\Sal^\chi_\Gamma)=\mathbb{F}[t^{\pm 1}]\otimes_\mathbb{F} \bar{C}(\hat\Gamma)_{n-1}$$ where $\bar{C}(\hat\Gamma)_{n-1}$ is the augmented chain complex of the flag complex $\hat\Gamma$ shifted by one. The differential of $C_n(\Sal^\chi_\Gamma)$ can be described as follows (see \cite{Blasco-conmar-ji-Homology}, after Remark 2.3). For each edge $e\in E_\Gamma$ let $2\tilde{\ell}_e$ be its label in $\Gamma$ and denote $q_k(x)=(x^k-1)/(x-1)$. Then for any $X\subseteq\Gamma$ complete we have \begin{equation}\label{eq:differential}\partial_n^\chi\sigma^\chi_X=\sum_{v\in X}\langle X_v\mid X\rangle b_{v,X}\sigma^\chi_{X_v}\end{equation} where we are denoting $X_v$ the clique obtained from $X$ by removing $v$ and \begin{equation}\label{eq:coefficient}b_{v,X}:=(t^{m_v}-1) \prod_{{\tiny{\array{c}w\in X_v\\e=\{v,w\}\in E_\Gamma\endarray}}}q_{\tilde{\ell}_e}(t^{m_e}). \end{equation}
In particular, if $\tilde{\ell}_e=1$, then $q_{\tilde{\ell}_e}(t^{m_e})=1$ and if $m_e=0$, $q_{\tilde{\ell}_e}(t^{m_e})=\tilde{\ell}_e$. Recall that an edge $e\in E_\Gamma$ is called $p$-dead if $m_e=0$ and $p\mid \tilde{\ell}_e$; otherwise will be called {\sl $p$-living}. So we see that in~\eqref{eq:differential}, the coefficient $b_{v,X}$ vanishes if either $v$ is dead or belongs to a $p$-dead edge in $X$.
Let $I$ be the augmentation ideal of the ring $R=\mathbb{F}[t^{\pm 1}]$, i.e., the kernel of the augmentation map $R\to \mathbb{F}$ with $t\mapsto 1$, $R$ is a principal ideal domain and $I$ is the ideal generated by $t-1$. Since $I$ is a prime ideal, we can localize and get a new ring $R_I$. We can also localize the complex $C_n(\Sal^\chi_\Gamma)$ and get a new complex $C_n(\Sal^\chi_\Gamma)_I$ with $n$-term $$C_n(\Sal^\chi_\Gamma)_I=R_I\otimes_\mathbb{F} \bar{C}(\hat\Gamma)_{n-1}$$ whose differential we also denote by $\partial_n^\chi$. Since localizing is flat, the $R$-free part of the homology of $(R\otimes_\mathbb{F} \bar{C}(\hat\Gamma)_{n-1},\partial_n^\chi)$ has the same rank as the $R_I$-free part of the homology of $(R_I\otimes_\mathbb{F} \bar{C}(\hat\Gamma)_{n-1},\partial_n^\chi)$. But in this complex we can normalize over the living vertices and $p$-living edges in the following way.
Let $X\subseteq\Gamma$ be a clique and put $$a_X=\prod_{v\in X\textrm{ living}}(t^{m_v}-1)\prod_{e\in E_X\textrm{ $p$-living}}q_{\tilde{\ell}_e(t^{m_e})}.$$ Let $\mu_X$ be the multiplicity of $t-1$ as a factor of $a_X$. Then $$a_X=(t-1)^{\mu_X}h_X$$ where $h_X$ is a unit in our ring $R_I$. Observe that for any $v\in X$ we have \begin{equation}\label{eq:Xv}(t-1)^{\mu_X}h_X=a_X=b_{v,X}a_{X_v}=b_{v,X}(t-1)^{\mu_{X_v}}h_{X_v}.\end{equation}
We can choose an integer $k$ such that $k|X|\geq\mu_X$ for any $X\subseteq\Gamma$ clique. Let $X\subseteq\Gamma$ be a clique and set
$$\tilde{\sigma}_X:=(t-1)^{k|X|-\mu_X}\frac{1}{h_X}\sigma_X.$$ Then
$$\partial_n^\chi(\tilde{\sigma}_X)=(t-1)^{k|X|-\mu_X}\frac{1}{h_X}
\partial_n^\chi(\sigma_X)=(t-1)^{k|X|-\mu_X}\frac{1}{h_X}\sum_{v\in X}\langle X_v\mid X\rangle b_{v,X}\sigma^\chi_{X_v}.$$
Recall that the summand associated to each $v\in X$ vanishes if either $v$ is dead or it belongs to a $p$-dead edge in $X$. Otherwise, using~\eqref{eq:Xv} we see that that summand is, up to a sign,
$$(t-1)^{k|X|-\mu_{X}}\frac{b_{v,X}}{h_X}\sigma^\chi_{X_v}=(t-1)^{k|X|-\mu_{X_v}}\frac{1}{h_{X_v}}
\sigma^\chi_{X_v}=(t-1)^k(t-1)^{k|X_v|-\mu_{X_v}}\frac{1}{h_{X_v}}\sigma^\chi_{X_v}={\tilde\sigma}^\chi_{X_v}.$$
Hence, if we denote by $\mathcal{F}^\chi_p(X)$ the subgraph that we get from $X$ when we remove dead vertices and {\sl closed} $p$-dead edges we have \begin{equation} \label{eq:partial} \partial_n^\chi(\tilde{\sigma}_X)= (t-1)^k\sum_{v\in\mathcal{F}^\chi_p(X)}\langle X_v\mid X\rangle\tilde{\sigma}_{X_v}. \end{equation} Observe that $\mathcal{F}^\chi_p(X)\subseteq X\cap\mathcal{L}_p^\chi$ where $\mathcal{L}^\chi_p$ is the $p$-living subgraph defined in the introduction, i.e., the subgraph of $\Gamma$ that we get when we remove dead vertices and {\sl open} $p$-dead edges.
Now, for each $n$ let $\tilde C_n(\Sal^\chi_\Gamma)_I$ be the sub $R_I$-module of $C_n(\Sal^\chi_\Gamma)_I$ generated by the $\tilde\sigma_X$, $|X|=n$. The computations above imply that $(\tilde C_n(\Sal^\chi_\Gamma)_I,\partial_n^\chi)$ is a subcomplex of $(C_n(\Sal^\chi_\Gamma)_I,\partial_n^\chi)$ and by definition each quotient $C_n(\Sal^\chi_\Gamma)_I/\tilde C_n(\Sal^\chi_\Gamma)_I$ is a $R_I$-torsion module. Using the long exact homology sequence we see that the $R_I$-free part of the homology of $(C_n(\Sal^\chi_\Gamma)_I,\partial_n^\chi)$ equals the $R_I$-free part of the homology of $(\tilde C_n(\Sal^\chi_\Gamma)_I,\partial_n^\chi)$. So from now on we consider this last complex.
We define a new map $d_n^\chi:\tilde C_n(\Sal^\chi_\Gamma)_I\to\tilde C_{n-1}(\Sal^\chi_\Gamma)_I$ by $$d_n^\chi={\frac{1}{(t-1)^k}}\partial_n^\chi.$$
\begin{lem}\label{lem:tecd} With the notation above we have \begin{enumerate}[label*=$\roman*)$] \item \label{lem:tecd-1} $\ker\partial_n^\chi=\ker d_n^\chi$,
\item \label{lem:tecd-2} $\im\partial^\chi_n\subseteq\im d^\chi_n$,
\item \label{lem:tecd-3} $\im d_n^\chi\cap I^k\tilde C_n(\Sal^\chi_\Gamma)_I=\im\partial_n^\chi,$
\item \label{lem:tecd-4} $\dim_\mathbb{F}(\im d_n^\chi/\im\partial_n^\chi)<\infty$. \end{enumerate} \end{lem} \begin{proof} \ref{lem:tecd-1} is obvious. For~\ref{lem:tecd-2}, take $a\in\im\partial_n^\chi$. Then $a=(t-1)a_1$ and $a=\partial^\chi_n(b)$ so $a_1=d_n^\chi(b)\in\im d_n^\chi$ so $a=(t-1)a_1\in\im d_n^\chi$. For~\ref{lem:tecd-3}, the fact that $\im\partial_n^\chi\subseteq\im d_n^\chi\cap I^k\tilde C_n(\Sal^\chi_\Gamma)_I$ is obvious because of~\ref{lem:tecd-2}. Conversely, take $a\in\im d_n^\chi\cap I^k\tilde C_n(\Sal^\chi_\Gamma)_I$. Then $a=d_n^\chi(b)$ and the fact that $a\in I^k\tilde C_n(\Sal^\chi_\Gamma)_I$ together with the definition of $d_n^\chi$ implies that also $b\in I^k\tilde C_n(\Sal^\chi_\Gamma)_I$ so $b=(t-1)^kb_1$ and $\partial_n^\chi(b_1)={\frac{1}{(t-1)^k}}\partial_n^\chi(b)=d_n^\chi(b)=a$ thus $a\in\im\partial_n^\chi$. Finally,~\ref{lem:tecd-4} follows from~\ref{lem:tecd-3}. \end{proof}
\begin{prop} For each $n$, $\dim_\mathbb{F}\mathrm{H}_n(A_\Gamma^\chi,\mathbb{F})<\infty$ if and only if the $n$-th homology of the localized chain complex $I^k\tilde C_n(\Sal^\chi_\Gamma)_I$ respect to $d_\bullet^\chi$ has finite $\mathbb{F}$-dimension, i.e., if and only if $\dim_\mathbb{F}\ker d_n^\chi/\im d_{n+1}^\chi<\infty$. \end{prop} \begin{proof} Note that for each $n$ there is a short exact sequence $$0\to \im d_{n+1}^\chi/\im\partial_{n+1}^\chi\to \ker\partial_n^\chi/\im\partial_{n+1}^\chi\to \ker\partial_n^\chi/\im d_{n+1}^\chi\to 0.$$ Since the left-hand side is of finite $\mathbb{F}$-dimension by Lemma~\ref{lem:tecd}\ref{lem:tecd-4} and $\ker\partial_n^\chi=\ker d_n^\chi$ by Lemma~\ref{lem:tecd}\ref{lem:tecd-1}, the result follows. \end{proof}
\begin{prop}\label{freehomology}
$$\ker d_n^\chi/\im d_n^\chi=R_I\otimes_\mathbb{F}\bigoplus_{A_X\in\mathcal{B}_p^\chi,|X|\leq n}
\overline{\mathrm{H}}_{n-1-|X|}(\hat\lk_{\mathcal{L}^\chi_p}(X)).$$ \end{prop} \begin{proof} From~\eqref{eq:partial} we have $$d_n^\chi(\tilde{\sigma}_X)=\sum_{v\in \mathcal{F}^\chi_p(X)}\langle X_v\mid X\rangle\tilde{\sigma}_{X_v}.$$
Let $\emptyset\neq X\subseteq\Gamma$ be a clique. Let $\mathcal{B}_p^\chi(X)=Y$ be the subgraph of $X$ generated by dead vertices and closed $p$-dead edges and $Z=\mathcal{F}^\chi_p(X)$. Then any vertex of $X$ lies either in $Y$ or in $Z$, in other words, $X$ is the subgraph generated by $Y\cup Z$. Note that $Z\subseteq\lk_{\mathcal{L}^\chi_p}(Y)$ is a clique and $A_Y\in\mathcal{B}_p^\chi$, obviously $Y$ is the biggest subgraph of $X$ satisfying this.
Conversely, given $A_Y\in\mathcal{B}_p^\chi$ and a clique $Z\subseteq\lk_{\mathcal{L}^\chi_p}(Y)$, then the subgraph $X$ of $\Gamma$ generated by $Y\cup Z$ is a clique. We claim that $Y=\mathcal{B}_p^\chi(X)$, obviously $Y\subseteq\mathcal{B}_p^\chi(X)$. If there is some $v\in\mathcal{B}_p^\chi(X)$, $v\not\in Y$, then $v\in Z$ so it can not be dead and there must be some $p$-dead edge $e\in X$ with $e=(v,w)$. As $Z$ is a clique we cannot have $w\in Z$ so $w\in Y$. Then $0=m_e=m_v+m_w$ so $m_w\neq 0$, in other words, $w$ is not a dead vertex and as $A_Y\in\mathcal{B}_p^\chi$, we deduce that there must be some $p$-dead edge $e_1\in Y$ with $e_1=(w,u)$ for some other $u\in Y$. But then observe that the vertices $v,u,w$ from a triangle in $\Gamma$ and the fact that both $e$ and $e_1$ are $p$ dead implies that both have labels bigger than 2 which contradicts the $\FC$-condition. Moreover we also deduce that $Z=\mathcal{F}^\chi_p(X)$.
We will check that for each $A_Y\in\mathcal{B}_p^\chi$ there is a subcomplex $(D_Y)_\bullet$ of $(R_I\otimes_\mathbb{F} \bar{C}(\hat\Gamma)_\bullet,d_\bullet^\chi)$ so that, as complexes, $$\tilde C_\bullet(\Sal^\chi_\Gamma)_I=\bigoplus_{A_Y\in\mathcal{B}_p^\chi}(D_Y)_\bullet.$$
To see it, let $(D_Y)_k=0$ for $0\leq k\leq |Y|-1$ and for $k\geq|Y|$,
$$(D_Y)_k=\oplus\{R_I\tilde\sigma_X\mid |X|=k, X\subseteq\Gamma\text{ clique}, Y= \mathcal{B}_p^\chi(X)\}.$$ The fact that this is a $d_\bullet^\chi$-subcomplex follows from the fact that for $\tilde{\sigma}_X\in (D_Y)_n$, $d_n^\chi(\tilde{\sigma}_X)$ vanishes in all the summands not in $(D_Y)_{n-1}$, more explicitly: $$d_n^\chi(\tilde{\sigma}_X)=\sum_{v\in \mathcal{F}^\chi_p(X)}\langle X_v\mid X\rangle\tilde{\sigma}_{X_v}$$ and as $v\in \mathcal{F}^\chi_p(X)$, $\tilde{\sigma}_{X_v}\in (D_Y)_{n-1}$.
Moreover, the discussion above implies that we can identify
$$(D_Y)_k=R_I\otimes\overline{C}_{k-|Y|-1}(\hat\lk_{\mathcal{L}^\chi_p}(Y))$$ and the fact that each $X$ determines uniquely $Y=\mathcal{B}_p^\chi(X)$ implies that
$$R_I\otimes_\mathbb{F} \bar{C}(\hat\Gamma)_\bullet=\bigoplus_{A_Y\in\mathcal{B}_p^\chi}R_I\otimes\overline{C}_{\bullet+1+|Y|}(\hat\lk_{\mathcal{L}^\chi_p}(Y)).$$ Therefore the result follows. \end{proof}
As a consequence, we obtain the following result.
\begin{teo}\label{teo:free} Let $G=A_\Gamma$ be an even Artin group of $\FC$-type, $\chi:G\to\mathbb{Z}$ a discrete character with kernel $A_\Gamma^\chi$ and $\mathbb{F}$ a field of characteristic $p$. Then the free part of the homology groups $\mathrm{H}_n(A_\Gamma^\chi,\mathbb{F})$ seen as $\mathbb{F}[t^{\pm1}]$-modules has rank
$$\sum_{A_X\in\mathcal{B}_p^\chi,|X|\leq n}
\dim_\mathbb{F}\overline{\mathrm{H}}_{n-1-|X|}(\hat\lk_{\mathcal{L}^\chi_p}(X),\mathbb{F}).$$ \end{teo}
Therefore, \begin{corol} Let $G=A_\Gamma$ be an even Artin group of $\FC$-type, $\chi:G\to\mathbb{R}$ a character and $\mathbb{F}$ a field of characteristic $p$. Then $\dim_\mathbb{F}\mathrm{H}_i(A^\chi_\Gamma,\mathbb{F})<\infty$ for $0\leq i\leq n$ if and only if $\chi$ satisfies the strong $p$-$n$-link condition. \end{corol}
In the particular case $p=0$, note that $\mathcal{B}_0^\chi$ is just the set of those $A_X\in\mathcal{P}$ with $X\subseteq\Gamma\setminus\mathcal{L}^\chi_0$.
We also deduce a partial converse to Theorem~\ref{teo:mainsigma}.
\begin{corol}\label{corol:sigmachar} Let $G=A_\Gamma$ be an even Artin group of $\FC$-type, and $0\neq\chi:G\to\mathbb{R}$ be a character such that $\mathcal{L}^\chi_p=\mathcal{L}^\chi$ for some $p$ either zero or prime. Assume that the strong $p$-$n$-link condition fails for $\chi$. Then $[\chi]\not\in\Sigma^n(G,\mathbb{Z})$. \end{corol} \begin{proof} Let $\chi$ be a character that does not satisfy the strong $p$-$n$-link condition. Assume first that $\chi$ is discrete, i.e., $\chi(G)\subseteq\mathbb{Z}$. Let $\mathbb{F}$ be a field of characteristic $p$. By Proposition~\ref{freehomology} and the discussion above we deduce that some of the homology groups $\mathrm{H}_i(A_\Gamma^\chi,\mathbb{F})$ has infinite dimension as an $\mathbb{F}$-vector space, thus $A_\Gamma^\chi$ is not of type $\FP_n$ thus $[\chi]\not\in\Sigma^n(G,\mathbb{Z})$. For the general case, i.e., when $\chi:G\to\mathbb{R}$ is not necessarily discrete, consider the set $$\{[\varphi]\mid\varphi:G\to\mathbb{Z},\mathcal{L}^\varphi=\mathcal{L}^\chi\}.$$ Observe that $[\chi]$ lies in the closure of this set. The discrete case considered above implies $$\{[\varphi]\mid\varphi:G\to\mathbb{Z},\mathcal{L}^\varphi=\mathcal{L}^\chi_0\}\subseteq\Sigma^c(G,\mathbb{Z})$$ and as $\Sigma^c(G,\mathbb{Z})$ is closed we deduce that also $[\chi]\in\Sigma^c(G,\mathbb{Z})$. \end{proof}
\begin{exam} Let $G=\D_{\ell}$ be the dihedral Artin group associated to a graph $\Gamma$ which consists of a single edge $e$ with vertices $v,w$ and label $\ell=2\tilde{\ell}$ and let $\chi:G\to\mathbb{Z}$ given by $\chi(v)=1$, $\chi(w)=-1$. Let $\mathbb{F}$ be a field of characteristic $p$. In this case the homology groups $\mathrm{H}_n(A_\Gamma^\chi,\mathbb{F})$ vanish for $n>1$ and one can compute directly $\mathrm{H}_1(A_\Gamma^\chi,\mathbb{F})$ for a field $\mathbb{F}$ using the description of the differential~\eqref{eq:differential} above and gets: $$ H_1(A_\Gamma^\chi;\mathbb{F})= \begin{cases}
\mathbb{F}[t^{\pm 1}] & \text{ if } p|\tilde{\ell}\\ \frac{\mathbb{F}[t^{\pm 1}]}{(t-1)} & \text{ otherwise.} \end{cases} $$ This is precisely what Theorem~\ref{teo:free} predicts: if $p\nmid\tilde{\ell}$, there are no $p$-dead edges and no dead vertices which means $\mathcal{B}_p^\chi=\{1\}$ and $\mathcal{L}_p^\chi=\Gamma$. The link $\lk_{\mathcal{L}^\chi_p}(\emptyset)$ is the whole $\Gamma$ so the
associated flag complex is contractible and the associated reduced homology groups vanish. By contrast, if $p|\tilde{\ell}$, the edge $(v,w)$ is $p$-dead so $\mathcal{B}_p^\chi=\{1,e\}$ and $\mathcal{L}_p^\chi$ consists of 2 isolated points. According to Theorem~\ref{teo:free}, the free rank of $\mathrm{H}_1(A_\Gamma^\chi,\mathbb{F})$ is
$$\sum_{A_X\in\mathcal{B}_p^\chi,|X|\leq 1}\dim_\mathbb{F}
\overline{\mathrm{H}}_{0-|X|}(\hat\lk_{\mathcal{L}^\chi_p}(X),\mathbb{F})=\dim_\mathbb{F}\overline{\mathrm{H}}_{0}(\hat\lk_{\mathcal{L}^\chi_p}(\emptyset),\mathbb{F})= \dim_\mathbb{F}\overline{\mathrm{H}}_{0}(\hat\mathcal{L}^\chi_p,\mathbb{F})=1.$$ \end{exam}
\begin{exam} Let $G=\D_4\times \D_6$ where $\D_4$ (resp. $\D_6$) is the dihedral Artin group associated to the edge with label 4 (resp. 6). Then $G=A_\Gamma$, where $\Gamma$ is a full graph with 4 vertices and two disjoint edges labeled with 4 and 6. Denote the standard generators of the factor $\D_4$ by $v$, $w$ and the standard generators of the factor $\D_6$ by $x$, $y$ and consider the character $\chi:G\to\mathbb{Z}$ induced by $\chi(v)=\chi(x)=1$, $\chi(w)=\chi(y)=-1$. Taking into account the computation of the Sigma-invariants for this type of groups that we performed in Subsection~\ref{subsec:productdihedral}, we see that $[\chi]\not\in\Sigma^2(G,\mathbb{Z})$ so its kernel $A_\Gamma^\chi$ is not of type $\FP_2$. In fact $G$ does not satisfy the strong $2$-link condition. To see it, note that $\mathcal{B}^\chi=\{\emptyset,e_1,e_2\}$ where $e_1=(v,w)$ and $e_2=(x,y)$ and $\mathcal{L}^\chi$ is a square with vertices $v,x,w,y$ (that we get when we remove the interior of $e_1$ and $e_2$ from $\Gamma$). For $X=\emptyset\in\mathcal{B}^\chi$ we have $\lk_{\mathcal{L}^\chi}(\emptyset)=\mathcal{L}^\chi$. Then
$$\overline{\mathrm{H}}_{2-1-|X|}(\hat{\lk}_{\mathcal{L}^\chi}(X))=\overline{\mathrm{H}}_{1}(\hat{\mathcal{L}^\chi})=\mathbb{Z}\neq 0.$$ It is easy to see that also the strong $3$-link condition fails: to check it consider for example $X=e_1$, its link in $\mathcal{L}^\chi$ consists of the isolated vertices $x$ and $y$.
We claim however that this $\chi$ does satisfy the strong $p$-$n$-link condition for each $p$ (zero or a prime). As a consequence, for any field $\mathbb{F}$, $$\dim_\mathbb{F}\mathrm{H}_2(A_\Gamma^\chi,\mathbb{F})<\infty.$$
Assume first that $p=2$. Then $\mathcal{B}_2^\chi=\{\emptyset,e_1\}$ and $\mathcal{L}_2^\chi$ is the graph obtained from $\Gamma$ when we remove the open edge $e_1$. Then $\lk_{\mathcal{L}_2^\chi}(\emptyset)=\mathcal{L}_2^\chi$ and $\lk_{\mathcal{L}_2^\chi}(e_1)=e_2$ and both associated flag complexes are contractible.
The argument for $p=3$ is analogous. Finally, if $p\neq 2,3$, $\mathcal{B}_p^\chi=\{\emptyset\}$ and $\mathcal{L}_p^\chi=\Gamma$. Again, the flag complex is contractible.
\end{exam}
\begin{exam} Things are very different if we consider for example $G_1=\D_4\times \D_4$ and $\chi$ as before. Then one easily checks that the strong $2$-$2$-link condition fails so $\dim_\mathbb{F}\mathrm{H}_2(A_\Gamma^\chi,\mathbb{F})$ is infinite. \end{exam}
\section{The homotopic invariants} \label{sec:homotopic} In this section we explain how to modify the statement of Theorem~\ref{teo:mainsigma} to obtain the analogous homotopic result.
Basically, we have to change the hypothesis to the homotopic version. As we have said in Definition~\ref{nlink}, we define the strong homotopic $n$-link condition as follows:
Consider again the set $\mathcal{B}^\chi\subset \mathcal{P}$ of those $A_\Delta$ in the clique poset such that $\chi(Z(A_\Delta))=0$.
Assume that for any $A_\Delta\in\mathcal{B}^\chi$ with $|\Delta|\leq n$ the link $\lk_{\mathcal{L}^\chi}(\sigma)$
is $(n-1-|\sigma|)$-connected. Then we say that $\chi$ satisfies \emph{the strong homotopic $n$-link condition}.
\begin{teo}\label{teo:mainsigmahomotopic} Let $G=A_\Gamma$ be an even Artin group of $\FC$-type, and $0\neq\chi:G\to\mathbb{R}$ a character such that the strong homotopic $n$-link condition holds for $\chi$. Then $[\chi]\in\Sigma^n(G)$. \end{teo}
\begin{proof} The proof follows that of Theorem~\ref{teo:mainsigma} in its homotopic version. The homotopic analogue of Lemma~\ref{lem:cosetposets} can be proved using relative homotopy groups,
and the $(n-1)$-connectedness of $Z^S\star|\mathcal{J}^S|$ (see~\cite[p.57 (2.5)]{Whitehead-homotopy}). \end{proof}
\end{document} | arXiv | {
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\begin{document}
\subjclass[2010]{35K59, 35K90, 35K92} \keywords{Quasilinear parabolic equations; Moving domains; $L^1$-data; Weak solutions; Aubin-Lions lemma}
\begin{abstract}
The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the $p$-Laplace equation as a special case. Weak solutions are shown to be global by obtaining appropriate estimates on the gradient as well as a suitable version of Aubin-Lions lemma in moving domains.
\end{abstract}
\maketitle
\section{Introduction} Problems defined on a domain which changes its shape in time have recently attracted a lot of attention from mathematical community since not only they lead to interesting mathematical questions, but also they arise naturally in physics, biology, chemistry and many other fields. Examples include the studies of pattern formation on evolving surfaces \cite{BEM11,GMMSV11}, of surfactants in two-phase flows \cite{GLS14}, of dealloying by surface dissolution of a binary alloy \cite{EE08}, of a diffusion interface model for linear surface partial differential equations \cite{ES09}, or of modeling and simulation of cell mobility \cite{ESV12}. We refer the interested reader to the extensive review paper \cite{KK15}. In this paper, we study the global existence of a quasilinear parabolic equation in moving domains, i.e. domains with shapes evolving in time. The equation contains a quasilinear diffusion operator, which includes the $p$-Laplacian as a special case, has a nonlinearity depending on the zero and first order terms, and has external force and initial data which are only integrable.
To precisely state the problem under consideration, we consider a bounded domain $\Omega_0\subset \mathbb R^d$, $d\geq 1$, with smooth boundary $\partial\Omega_0$. Let $\mathbf v: \mathbb R^d\times \mathbb R \to \mathbb R^d$ be a smooth and compactly supported vector field and $\zeta: \mathbb R^d \times \mathbb R \to \mathbb R^d$ be its corresponding flow, i.e. $\zeta$ solves \begin{equation*}
\partial_t\zeta(x,t) = \mathbf v(\zeta(x,t),t), \qquad \zeta(x_0,0) = x_0 \end{equation*} for any $x_0 \in \mathbb R^d$. Note that for each fixed $x$, the mapping $t\mapsto \zeta_t(x)$ is an integral curve of $\mathbf v$ and for fixed $t$, the mapping $x_0 \mapsto \zeta_t(x_0)$ is a diffeomorphism. Assuming that $\Omega_0 \subset \mathrm{supp}(\mathbf v)$, we define $\Omega_t = \zeta_t(\Omega_0)$ and the non-cylindrical domain as \begin{equation*}
Q_T:= \bigcup_{t\in (0,T)}\Omega_t\times \{t\} = \bigcup_{t\in (0,T)}\zeta_t(\Omega_0)\times \{t\}, \end{equation*} \begin{equation*}
\Sigma_T:= \bigcup_{t\in (0,T)}\partial\Omega_t\times \{t\} = \bigcup_{t\in (0,T)}\zeta_t(\partial\Omega_0)\times\{t\}. \end{equation*} We choose an open and bounded subset $\widehat{\Omega}$ of $\mathbb R^d$ such that $\cup_{t\in[0,T]}\Omega_t \subset \widehat{\Omega}$ and let $\widehat{Q}:=\widehat{\Omega}\times (0,T)$.
We also need to define time-space spaces in moving domains. Let $\{X(t)\}_{t\in [0,T]}$ be a family of Banach spaces, then we define \begin{equation*}
L^p(0,T;X(t)) = \{f:Q_T\to \mathbb R:\; f(t)\in X(t) \text{ for a.e. } t\in (0,T)\} \end{equation*} with the norm \begin{equation*}
\|f\|_{L^p(0,T;X(t))} = \left(\int_0^T\|f(t)\|_{X(t)}^pdt\right)^{\frac 1p} <+\infty. \end{equation*} Very common in this paper we use $X(t) = L^q(\Omega_t)$ or $X(t) = W_0^{1,q}(\Omega_t)$. In particular, when $p=q$, then we write simply $L^p(Q_T)$ instead of $L^p(0,T;L^p(\Omega_t))$.
The main goal of the present paper is to study the global existence of the following quasilinear problem
\begin{equation}\label{e1}
\begin{cases}
\partial_t u - \text{div} (a(x,t,\nabla u)) + \text{div} (u\mathbf v) + g(x,t,u, \nabla u) = f, &(x,t)\in Q_T,\\
u(x,t) = 0, &(x,t)\in \Sigma_T,\\
u(x,0) = u_0(x), &x\in \Omega_0,
\end{cases}
\end{equation}
with the external force $f\in L^1(Q_T)$ and initial data $u_0 \in L^1(\Omega_0)$. The nonlinear diffusion $a$ is assumed to satisfy
\begin{enumerate}[label=(A\theenumi),ref=A\theenumi]
\item\label{A0} $a: \widehat{Q}\times \mathbb R^d \to \mathbb R^d$ is a Carath\'eodory function;
\item\label{A2} there exists $p>\frac{2d+1}{d+1}$ such that, for $(x,t)\in \widehat{Q}$ and $\xi \in \mathbb R^d$,
\[
|a(x,t,\xi)| \leq \varphi(x,t) + K|\xi|^{p-1}
\]
where $\varphi \in L^{p'}(\widehat{Q})$, $1/p+1/p' = 1$ and $K\geq 0$;
\item\label{A3} there exists $\alpha>0$ such that
\[
a(x,t,\xi)\xi \geq \alpha|\xi|^p,
\]
where $(x,t)\in \widehat{Q}$ and $\xi\in\mathbb R^d$;
\item\label{A1} for almost $(x,t)\in \widehat{Q}$ and all $\xi, \xi' \in \mathbb R^d$,
\[
(a(x,t,\xi) - a(x,t,\xi'))(\xi - \xi') \geq \frac{1}{\Theta(x,t,\xi,\xi')}|\xi-\xi'|^\theta \quad \text{ and } \quad a(x,t,0) = 0;
\]
for $\theta>1$, and $\Theta$ is a nonnegative function which satisfies
\begin{equation}\label{beta}
|\Theta(x,t,\xi,\xi')| \leq C(1+ |\xi| + |\xi'|)^{\vartheta}
\end{equation}
where
\begin{equation}\label{varrho}
\vartheta < (\theta-1)\left(p-\frac{d}{d+1}\right).
\end{equation}
\end{enumerate}
The nonlinearity $g: \widehat{Q}\times \mathbb R\times \mathbb R^d \to \mathbb R$ satisfies: $g$ is continuous with respect to the third and fourth variables, and
\begin{enumerate}[label=(G\theenumi),ref=G\theenumi]
\item\label{G1} it holds
\begin{equation*}
\lambda g(x,t,\lambda, \xi) \geq 0 \quad \text{ for all } \quad \lambda\in \mathbb R, \xi \in \mathbb R^d;
\end{equation*}
\item\label{G2}
$g$ has a {\it subcritical} growth on the gradient, i.e. there exists $0 \leq \sigma < p$ such that
\begin{equation*}
|g(x,t,\lambda,\xi)| \leq h(|\lambda|)(\gamma(x,t) + |\xi|^\sigma)
\end{equation*}
where $\gamma \in L^1(\widehat{Q})$, and $h$ is an increasing function in $\mathbb R_+$.
\end{enumerate}
Let us briefly discuss about the above conditions. The conditions \eqref{A0}--\eqref{A1} of $a$ assure that it contains the important case of $p$-Laplacian, i.e. $$a(x,t,\xi)\equiv a_p(\xi) = |\nabla \xi|^{p-2}\nabla \xi \quad \text{ for } \quad p>\frac{2d+1}{d+1}.$$ Moreover, the technical condition \eqref{A1} is weaker than the usual {\it strong monotonicity} condition
\begin{equation*}\label{strong_monotone}
(a(x,t,\xi) - a(x,t,\xi'))(\xi - \xi') \geq C|\xi - \xi'|^p,
\end{equation*}
for some $C>0$, but still stronger than the mere monotone property condition, i.e.
\begin{equation}\label{mon}
(a(x,t,\xi) - a(x,t,\xi'))(\xi - \xi') \geq 0.
\end{equation}
We also remark that the condition \eqref{G2} allows $g$ to have arbitrary growth in the zero order term, as long as it has the suitable sign stated in \eqref{G1}. A typical example of $g$ is
\begin{equation*}
g(x,t,u,\nabla u) = Cu^{2k+1}(\gamma(x,t) + |\nabla u|^{\sigma})
\end{equation*}
where $k\in \mathbb N$ is arbitrary and $0\leq \sigma < p$.
Elliptic or parabolic equations with irregular data such as $L^1$ or bounded measure appear frequently in applications and therefore are of interest and importance. Concrete examples include elliptic systems modeling electronical devices \cite{GH94}, the Fokker-Planck equation arising from populations dynamics \cite{GS98}, models of turbulent flows in oceanography and climatology \cite{Lew97}, incompressible flows with small Reynolds number \cite{Lio96}, or Keller-Segel or Shigesada-Kawasaki-Teramoto type systems \cite{winkler2019role}. Global existence of weak or renormalized solutions to special cases of \eqref{e1} in fixed domains has been studied extensively in the literature. Let us mention several related works: in \cite{BG89,boccardo1997nonlinear}, the authors considered \eqref{e1} where conditions \eqref{A0}--\eqref{A1} are imposed, but the function $g$ is either zero or does not have the first order term; similar results were shown in \cite{Bla93} assuming \eqref{mon} instead of \eqref{A1}; the case when $g$ depends on the first order term was considered in e.g. \cite{GS01, andreu2002existence}, but the second order term therein is a linear elliptic operator, for instance $\mathrm{div}(a(x,t,\nabla u)) = \Delta u$; when $p>1$ arbitrary, one can show global renormalized solutions \cite{BM97} (see also Remark \ref{remark1}). Related results are also obtained for systems without the first order terms \cite{BS05}.
The global existence of solutions to \eqref{e1} in moving domains with $L^1$-data, up to our knowledge, is not studied therefore is the main motivation of our paper. We would like also to emphasize that, even in the case of a fixed domain, our results extend that of \cite{BG89} and \cite{GS01}. Related results for quasilinear parabolic problems in time-dependent domains can also be found in e.g. \cite{CNO17,bogelein2018existence}.
The main goal of this paper is to prove the global existence of a weak solution to \eqref{e1} under the conditions \eqref{A0}--\eqref{A1}, \eqref{G1}--\eqref{G2} and data $f\in L^1(Q_T)$ and $u_0\in L^1(\Omega_0)$. To state the main result, we first give the precise definition of a weak solution.
\begin{definition}[Weak solutions]\label{def:sol}
Let $T>0$ be arbitrary.
A function
\begin{equation*}
u\in C([0,T];L^1(\Omega_t))\cap \left(\underset{1\leq q < p - \frac{d}{d+1}}{\bigcap}L^q(0,T;W_0^{1,q}(\Omega_t))\right)
\end{equation*}
is called a weak solution to \eqref{e1} on $(0,T)$ if $g(x,t,u,\nabla u)\in L^1(Q_T)$ and for all test functions $\psi\in C([0,T];W^{1,\infty}_0(\Omega_t))\cap C^1((0,T);L^{\infty}(\Omega_t))$, the following weak formulation holds
\begin{equation*}
\begin{aligned}
&\int_{\Omega_T}u(T)\psi(T)dx - \int_0^T\int_{\Omega_t}u\psi_t dxdt\\
&+ \int_0^T\int_{\Omega_t}[a(x,t,\nabla u)\cdot \nabla \psi - u\mathbf v\cdot \nabla \psi + g(x,t,u,\nabla u)\psi]dxdt\\
&= \int_{\Omega_0}u_0\psi(0)dx + \int_0^T\int_{\Omega_t}f\psi dxdt.
\end{aligned}
\end{equation*}
\end{definition}
All the terms above are obviously well-defined except for the term containing $a(x,t,\nabla u)\cdot \nabla \psi$. From the growth assumption \eqref{A2} of $a$, and the fact that $u\in L^q(0,T;W_0^{1,q}(\Omega_t))$ for all $1\leq q < p - d/(d+1)$, it follows that $a(\cdot,\cdot,\nabla u)\in L^s(Q_T)$ for all $1\leq s < 1 + \frac{1}{(p-1)(d+1)}$, and therefore, the integration $\int_{Q_T}a(x,t,\nabla u)\cdot \nabla \psi dxdt$ makes sense since $\nabla \psi\in L^\infty(Q_T)$.
\begin{remark}\label{remark1}
The condition $p > (2d+1)/(d+1)$ is needed to define the weak solution. When $p \leq (2d+1)/(d+1)$, we can only obtain $\nabla u \in L^q(Q_T)^d$ for $q\in (0,1)$. In this case, one can either show the existence of renormalized solutions, see e.g. \cite{BM97}, or weak solutions belonging to $L^r(0,T;W^{1,q}_0(\Omega_t))$ for $r,q$ are different, see e.g. \cite{boccardo1997nonlinear} where \eqref{e1} was studied in a cylindrical domain for all $p>1$ but without the nonlinearity $g$. These two directions go beyond the scope of this paper and therefore are left for upcoming research.
\end{remark}
The main result of this paper is the following theorem.
\begin{theorem}[Global existence of weak solutions]\label{thm:main}
Assume that the vector field $\mathbf v\in C^1(\mathbb R^d,C^0(\mathbb R))$.
Assume the conditions \eqref{A0}--\eqref{A1} and \eqref{G1}--\eqref{G2}. Then for any $u_0 \in L^1(\Omega_0)$ and any $f\in L^1(Q_T)$, there exists a global weak solution $u$ to \eqref{e1} on $(0,T)$ as in Definition \ref{def:sol}.
\end{theorem}
Let us describe the main ideas in proving Theorem \ref{thm:main}. To treat moving domains, one can transform the problem into the case of fixed domains and then study the new equation, with the cost of some additional terms. Usually these additional terms depend significantly on the problem itself, and therefore each problem needs to be treated separately. As an attempt to have a more unified mechanism, a different approach is to derive a mechanism to work on the moving domains directly, that is to establish parallel tools for moving domains corresponding to that of fixed domains. This research direction has been investigated by many authors (see e.g. \cite{AES15,AET18,MB08,Vie14}).
In this paper, we adapt the second approach to prove Theorem \ref{thm:main}, meaning that we treat \eqref{e1} directly on the non-cylindrical domain $Q_T$. More precisely, first, we consider an approximation of \eqref{e1} in which the data is approximated by $f_\varepsilon \in L^\infty(Q_T)$ and by $u_0\in L^\infty(\Omega_0)$. Moreover, we also regularize the nonlinearity $g_\varepsilon = g(1+\varepsilon|g|)^{-1}$ which is bounded for any fixed $\varepsilon>0$. Thanks to this regularization, we can use the method from \cite{CNO17} to obtain the global existence of an approximate solution $u_\eps$. The next goal is to derive estimates of this approximate solution uniformly in $\varepsilon$. In order to do that, due to the low regularity of the data, we refine the analysis in \cite{GS01} to adapt to the case of quasilinear problem \eqref{e1}. Once the uniform estimates for $u_\eps$ are obtained, we would like to pass to the limit as $\varepsilon\to 0$, which consequently requires an Aubin-Lions lemma in the case of moving domains. A similar lemma has been shown in different works (see e.g. \cite{Mou16} or \cite{Fuj70}), but they are not applicable to our situation. Therefore, we prove a new Aubin-Lions lemma in moving domains, which allows us to first obtain the almost everywhere convergence $u_\eps \to u$ and then consequently $\|u_\eps - u\|_{L^1(Q_T)} \to 0$. Due to the dependence of the nonlinearity on $\nabla u$, this convergence is not yet enough. By using the ideas from \cite{GS01}, we utilize the assumptions \eqref{G1} and \eqref{G2} to show that the convergence $\nabla u_\eps \to \nabla u$ holds almost everywhere. This in turn helps to get $g_\varepsilon(x,t,u_\eps,\nabla u_\eps) \to g(x,t,u,\nabla u)$ and $a(x,t,\nablau_\eps) \to a(x,t,\nabla u)$ in appropriate spaces, and eventually to obtain $u$ to be a weak solution to \eqref{e1}.
{\bf The rest of this paper is organized as follows:} In the next Section, we derive uniform a-priori estimates for approximate solutions, which are needed to pass to the limit in Section \ref{proof} to obtain the weak solution of \eqref{e1}. The Appendix \ref{appendix1} and \ref{appendix2} provide the existence of an approximate solution and a proof of the Aubin-Lions lemma in moving domains respectively.
{\bf Notation.} We will use in this paper the following set of notations.
\begin{itemize}
\item Recall that we simply write $L^p(Q_T)$ instead of $\LQ{p}{p}$.
\item The double integration $\int_0^T\int_{\Omega_t}dxdt$ is written using the shorthand notation $\int_{Q_T}dxdt$.
\item We usually write $C = C(\alpha, \beta,\gamma,\ldots)$ to indicate that the constant $C$ depends on the arguments $\alpha, \beta, \gamma$, etc.
\item As we will use it frequently in the paper, for fixed $T>0$ we write
\begin{equation*}
\|\mathbf v\|_\infty:= \|\mathbf v\|_{L^\infty(Q_T)} \quad \text{ and } \quad \|\text{div} \mathbf v\|_{\infty}:= \|\text{div} \mathbf v\|_{L^\infty(Q_T)},
\end{equation*}
which are well-defined when $\mathbf v\in C^1(\mathbb R^d,C^0(\mathbb R))$.
\end{itemize}
\section{Uniform estimates}\label{approximate} In this section, we consider an approximate problem to \eqref{e1} and derive uniform {\it a priori} estimates for the approximate solution. These estimates play a crucial role in passing to the limit to obtain a weak solution to \eqref{e1}. For simplicity we write $g(u,\nabla u)$ instead of $g(x,t,u,\nabla u)$.
Fix an arbitrary time horizon $T>0$. As usual we regularize the initial data $u_0$ and the external term $f$ by more regular data $u_{0,\varepsilon}\in L^\infty(\Omega_0)$ and $f_\varepsilon \in L^{\infty}(Q_T)$ for $\varepsilon>0$, such that \begin{equation}\label{u0f}
\lim_{\varepsilon \to 0}\|u_{0,\varepsilon} - u_0\|_{L^1(\Omega_0)} = 0 \quad \text{ and } \quad \lim_{\varepsilon \to 0}\|f_{\varepsilon} - f\|_{L^1(Q_T)} = 0, \end{equation} and \begin{equation}\label{increasing}
\|u_{0,\varepsilon}\|_{L^1(\Omega_0)} \leq \|u_0\|_{L^1(\Omega_0)} \quad \text{ and } \|f_\eps\|_{L^1(Q_T)} \leq \|f\|_{L^1(Q_T)}. \end{equation} Moreover, we also regularize the nonlinear first order term by a bounded nonlinearity, namely, for $\varepsilon > 0$, \begin{equation*}
g_\eps(w,\nabla w):= \frac{g(w,\nabla w)}{1+\varepsilon|g(w,\nabla w)|}. \end{equation*} Note that for any fixed $\varepsilon>0$, we have \begin{equation*}
|g_\eps(w,\nabla w)| \leq \frac 1\varepsilon \quad \text{ for all } \quad (x,t)\in Q_T\quad \text{ and all } \quad w. \end{equation*} The approximate problem reads as, \begin{equation}\label{approx}
\begin{cases}
\partial_tu_\eps - \text{div} (a(x,t,\nabla u_\eps)) + \text{div}(u_\eps\mathbf v) + g_\varepsilon(u_\eps,\nabla u_\eps) = f_\varepsilon, &(x,t)\in Q_T,\\
u_\eps(x,t) = 0, &(x,t)\in \Sigma_T,\\
u_\eps(x,0) = u_{0,\varepsilon}(x), &x\in \Omega_0.
\end{cases} \end{equation}
\begin{definition}[Weak solutions to \eqref{approx}]
A weak solution to \eqref{approx} on $(0,T)$ is a function $u_\eps\in C([0,T];L^p(\Omega_t))\cap L^p(0,T;W_0^{1,p}(\Omega_t))$ with $\partial_tu_\eps \in L^{p'}(0,T;W^{-1,p'}(\Omega_t))$, where $W^{-1,p'}(\Omega_t) = (W_0^{1,p}(\Omega_t))^*$, such that
\begin{align*}
\int_0^T\langle \partial_t u_\eps, \phi \rangle_{W^{-1,p'}, W_0^{1,p}}dt + \int_0^T\int_{\Omega_t}a(x,t,\nablau_\eps)\cdot \nabla \phi dxdt\\
-\int_0^T\int_{\Omega_t}u_\eps \mathbf v \cdot\nabla\phi dxdt + \int_0^T\int_{\Omega_t}g_\varepsilon(u_\eps,\nablau_\eps)\phi dxdt = \int_0^T\int_{\Omega_t}f_\eps \phi dxdt
\end{align*}
for all test function $\phi \in \LW{p}{1}{p}$. \end{definition}
The global existence of a weak solution to \eqref{approx} can be obtained by the slicing technique in e.g. \cite{CNO17} with suitable, slight modifications. For the sake of completeness, we sketch the main steps of the proof, and postpone it to the Appendix \ref{appendix1} in order to not interrupt the train of thought. \begin{theorem}[Existence of a global solution to the approximate problem]\label{thm:approximate} Fix $T>0$. For any $u_{0,\varepsilon}\in L^\infty(\Omega_0)$ and $f_\eps\in L^\infty(Q_T)$, there exists a weak solution to \eqref{approx} on $(0,T)$. \end{theorem} The focus of this section is therefore to obtain a-priori estimates of solutions to \eqref{approx} which are {\it uniform in $\varepsilon$.} We divide the section further into two subsections, in which the first one shows uniform bounds of approximate solutions in Sobolev spaces, while the second provides uniform bounds of the nonlinearity $g_\eps(u_\eps,\nablau_\eps)$.
\subsection{Uniform bounds of approximate solutions} The following lemma is the main result of this subsection. \begin{lemma}\label{UniformBounds}
There exists a constant $C(T)$ depending on $T, \mathbf v$, $\|u_0\|_{L^1(\Omega_0)}$ and $\|f\|_{L^1(Q_T)}$ but {\normalfont independent of $\varepsilon$} such that
\begin{equation*}
\|u_\eps \|_{\LW{q}{1}{q}} \leq C(T)
\end{equation*}
for all $1 \leq q < p - d/(d+1)$.
\end{lemma} The proof of this lemma is long and technical and is therefore divided into several steps. As a preparation, we need a lemma about Sobolev embeddings in moving domains. \begin{lemma}[Sobolev embeddings]\label{embedding}
Fix $T>0$. Then there exists a constant $C_{\Omega,T}$ depending on $T$ and $\mathbf v$ such that
\begin{equation}\label{desired}
\|u\|_{L^{q^*}(\Omega_t)} \leq C_{\Omega,T}\|\nabla u\|_{L^q(\Omega_t)} \quad \text{ for all } t\in [0,T]\quad \text{ and } \quad u\in W_0^{1,q}(\Omega_t)
\end{equation}
where $q<d$ and
\begin{equation*}
q^* = \frac{dq}{d-q}.
\end{equation*} \end{lemma} \begin{proof}
The classical Sobolev embedding gives
\begin{equation*}
\|u\|_{L^{q^*}(\Omega_0)} \leq C(\Omega_0)\|\nabla u\|_{L^q(\Omega_0)}.
\end{equation*}
Since $\zeta \in C([0,T];C^1(\mathbb R^d))$, there exists $a(T), b(T)$ such that
\begin{equation*}
a(T)\leq |\det(D\zeta_t)(x)| \leq b(T) \quad \text{for all} \quad t\in [0,T].
\end{equation*}
Now, for $t\in [0,T]$, we know that $\Omega_t = \zeta_t(\Omega_0)$. Therefore,
\begin{equation*}
\begin{aligned}
\left(\int_{\Omega_t}|u(x)|^{q^*}dx\right)^{\frac{1}{q^*}} &= \left(\int_{\Omega_0}|u(\zeta_t(y))|^{q^*}|\det(D\zeta_t)|dy\right)^{\frac{1}{q^*}}\\
&\leq b(T)^{\frac{1}{q^*}}C(\Omega_0)\left(\int_{\Omega_0}|\nabla u(\zeta_t (y))|^{q}dy\right)^{\frac 1q}\\
&\leq b(T)^{\frac{1}{q^*}}a(T)^{-\frac 1q}C(\Omega_0)\left(\int_{\Omega_t}|\nabla u(x)|^{q}dx\right)^{\frac 1q}
\end{aligned}
\end{equation*}
which proves the desired estimate \eqref{desired}. \end{proof}
\begin{lemma}\label{lem:inter}
Assume that $u_\eps \in \LW{p}{1}{p}$ satisfies
\begin{equation}\label{cond1}
\sup_{t\in (0,T)}\int_{\Omega_t} |u_\eps| dx \leq \beta,
\end{equation}
and for each $n\in \mathbb N$,
\begin{equation}\label{cond2}
\int_{B_n}|\nabla u_\eps|^pdxdt \leq C_0 + C_1\int_{E_n}|\nabla u_\eps|dxdt
\end{equation}
for some $\beta, C_0, C_1 >0$ {\normalfont independent of $\varepsilon$} where
\begin{equation}\label{def_BnEn}
B_n = \{(x,t)\in Q_T: n\leq |u_\eps(x,t)| \leq n+1 \} \; \text{ and } \; E_n = \{ (x,t)\in Q_T: |u_\eps(x,t)| > n+1 \}.
\end{equation}
Then there exists $C(T,p,q,\beta,C_0,C_1)$ depending on $T,p,q,\beta, C_0$ and $C_1$, but {\normalfont independent of $\varepsilon$}, such that
\begin{equation}\label{e2_1}
\|u_\eps\|_{\LW{q}{1}{q}} \leq C(T,p,q,\beta,C_0,C_1)
\end{equation}
for all $1\leq q < p - \frac{d}{d+1}$.
\end{lemma} \begin{remark}
We remark that since $p>q$, obviously $u_\eps\in \LW{q}{1}{q}$ follows immediately from $u_\eps\in \LW{p}{1}{p}$ and $\|u_\eps\|_{\LW{q}{1}{q}} \leq C(T)\|u_\eps\|_{\LW{p}{1}{p}}$. However, the essential role of \eqref{e2_1} is that the constant $C(T)$ therein is independent of $\varepsilon$, while the norm $ \|u_\eps\|_{\LW{p}{1}{p}}$ might blow up as $\varepsilon\to 0$. \end{remark} \begin{proof}[Proof of Lemma \ref{lem:inter}]
Let $1 \leq q < p$ be arbitrary. From \eqref{cond2}, by using H\"{o}lder's inequality, we have
\begin{equation}\label{e3}
\begin{aligned}
\int_{B_n} |\nabla u_\eps |^pdxdt &\leq C_0+C_1\Big(\int_{E_n}|\nabla u_\eps|^qdxdt\Big)^{1/q}|E_n|^{(q-1)/q}\\
&\leq C_0+C_1\|\nabla u_\eps\|_{L^q(Q_T)}|E_n|^{(q-1)/q}.
\end{aligned}
\end{equation}
Since $q < p$ we can use H\"{o}lder's inequality, inequality \eqref{e3}, and the elementary inequality $(a+b)^{q/p} \leq a^{q/p} + b^{q/p}$ for $a,b\geq 0$, to obtain
\begin{equation}\label{e5}
\begin{aligned}
\int_{B_n} |\nabla u_\eps|^q dxdt &\leq |B_n|^{(p-q)/p}\Big(\int_{B_n}|\nabla u_\eps|^pdxdt\Big)^{q/p}\\
&\leq |B_n|^{(p-q)/p}\Big(C_0^{q/p}+C_1^{q/p}\|\nabla u_\eps\|^{q/p}_{L^q(Q_T)}|E_n|^{(q-1)/p}\Big).
\end{aligned}
\end{equation}
Let $r \geq 0$ be chosen later. We have, by using the definitions of $B_n$ and $E_n$,
\begin{equation}\label{e6}
\begin{cases}
&|B_n| \leq \frac{1}{n^r}\int_{B_n}|u_\eps|^r dxdt,\\
&|E_n| \leq \frac{1}{n^r}\int_{E_n} |u_\eps|^r dxdt \leq \dfrac{1}{n^r}\|u_\eps\|^r_{L^r(Q_T)}.
\end{cases}
\end{equation}
Inserting \eqref{e6} into \eqref{e5} yields
\begin{equation}\label{e7}
\begin{aligned}
\int_{B_n} |\nabla u_\eps|^q dxdt &\leq C_0^{q/p}\Big(\dfrac{1}{n}\Big)^{r(p-q)/p}\Big(\int_{B_n}|u_\eps|^rdxdt\Big)^{(p-q)/p}\\
&+C_1^{q/p}\|\nabla u_\eps\|^{q/p}_{L^q(Q_T)}\|u_\eps\|^{r(q-1)/p}_{L^r(Q_T)}\Big(\frac{1}{n}\Big)^{r(p-1)/p}\Big(\int_{B_n}|u_\eps|^rdxdt \Big)^{(p-q)/p}.
\end{aligned}
\end{equation}
Let $K\in \mathbb N$ be chosen later. We split $\|\nabla u_\eps\|^q_{L^q(Q_T)}$ as follows
\begin{equation}\label{e8}
\|\nablau_\eps\|_{L^q(Q_T)}^q = \int_{Q_T} |\nabla u_\eps|^q dxdt =\sum_{n = 0}^K \int_{B_n} |\nabla u_\eps|^q dxdt + \sum_{n = K+1}^\infty \int_{B_n} |\nabla u_\eps|^q dxdt.
\end{equation}
Since $|B_n| \leq |Q_T|$ and $|E_n| \leq |Q_T|$, we simply evaluate the first term in the right hand side of \eqref{e8} using \eqref{e5} as follows
\begin{equation}\label{e9}
\sum_{n = 0}^K \int_{B_n}|\nabla u_\eps|^q dxdt \leq (K+1)C_2\Big(1+\|\nabla u_\eps\|^{q/p}_{L^q(Q_T)}\Big),
\end{equation}
where $C_2 = \max\{C_0^{q/p}|Q_T|^{(p-q)/p}, C_1^{q/p}|Q_T|^{(q-1)/p}\}$.
Using Young's inequality in \eqref{e8}-\eqref{e9}, we get
\begin{equation}\label{e10}
\|\nabla u_\eps\|^q_{L^q(Q_T)}\leq C(K)+2\sum_{n= K+1}^\infty \int_{B_n}|\nabla u_\eps|^q dxdt,
\end{equation}
where
\begin{equation*}
C(K) = 2\frac{p-1}{p}\cdot((K+1)C_2)^{\frac{p}{p-1}}\left(\frac 2p\right)^{\frac{1}{p-1}} + 2(K+1)C_2.
\end{equation*}
Note that the constant $C(K)$ tends to infinity as $K\to\infty$. It remains to proceed to the study of the series which appears on the right hand side of \eqref{e10}.
Applying H\"{o}lder's inequality on the series with exponents $p/(p-q)$ and $p/q$ and using \eqref{e7}, we have
\begin{equation}\label{e11}
\begin{aligned}
&\sum_{n = K+1}^\infty \int_{B_n} |\nabla u_\eps|^q dxdt\\
&\leq C_0^{q/p}\Big(\sum_{n=K+1}^\infty \dfrac{1}{n^{r(p-q)/q}}\Big)^{q/p}\Big(\sum_{n=K+1}^\infty \int_{B_n} |u_\eps|^r dxdt\Big)^{(p-q)/p}\\
&\quad +C_1^{q/p}\|\nabla u_\eps\|_{L^q(Q_T)}^{q/p}\|u_\eps\|_{L^r(Q_T)}^{r(q-1)/p}\Big(\sum_{n=K+1}^\infty\dfrac{1}{n^{r(p-1)/q}}\Big)^{q/p}\Big(\sum_{K+1}^\infty \int_{B_n}|u_\eps|^r dxdt \Big)^{(p-q)/p}\\
&\leq C_0^{q/p}\Big(\sum_{n=K+1}^\infty \dfrac{1}{n^{r(p-q)/q}}\Big)^{q/p}\|u_\eps\|_{L^r(Q_T)}^{r(p-q)/p}\\
&\quad +C_1^{q/p}\|\nabla u_\eps\|^{q/p}_{L^q(Q_T)}\|u_\eps\|^{r(p-1)/p}_{L^r(Q_T)}\Big(\sum_{n=K+1}^\infty \dfrac{1}{n^{r(p-1)/q}}\Big)^{q/p}.
\end{aligned}
\end{equation}
We choose $r$ so that the remainder of the series above converges to zero as $K \to \infty$, i.e.
\begin{equation}\label{cond_r}
\frac{r(p-q)}{q} > 1.
\end{equation}
Note that due to $q \geq 1$, this already implies $r(p-1)/q > 1$.
It follows from \eqref{e11} that
\begin{equation}\label{e12}
\|\nabla u_\eps\|_{L^q(Q_T)}^q \leq C(K) + \delta(K)\left(\|u_\eps\|_{L^r(Q_T)}^{r(p-q)/p} + \|\nabla u_\eps\|_{L^q(Q_T)}^{q/p}\|u_\eps\|_{L^r(Q_T)}^{r(p-1)/p} \right)
\end{equation}
with
\begin{equation*}
\delta(K) = 2\max\left\{C_0^{q/p}\Big(\sum_{n=K+1}^\infty \dfrac{1}{n^{r(p-q)/q}}\Big)^{q/p}; C_1^{q/p}\Big(\sum_{n=K+1}^\infty \dfrac{1}{n^{r(p-1)/q}}\Big)^{q/p} \right\}
\end{equation*}
with the property $\lim_{K\to\infty}\delta(K) = 0$ thanks to \eqref{cond_r}. From Young's inequality, and recalling that $q/p <q$, we have
\begin{equation*}
\|\nabla u_\eps\|_{L^q(Q_T)}^{q/p}\|u_\eps\|_{L^r(Q_T)}^{r(p-1)/p} \leq \frac 1p\|\nabla u_\eps\|_{L^q(Q_T)}^q + \frac{p-1}{p}\|u_\eps\|_{L^r(Q_T)}^{r}.
\end{equation*}
Therefore, \eqref{e12} implies
\begin{equation}\label{e13}
\begin{aligned}
&\|\nabla u_\eps\|_{L^q(Q_T)}^q\\
&\leq C(K) + \delta(K)\left[\|u_\eps\|_{L^r(Q_T)}^{r(p-q)/p} + \frac{p-1}{p}\|u_\eps\|_{L^r(Q_T)}^r + \frac 1p\|\nablau_\eps\|_{L^q(Q_T)}^q\right]\\
&\leq C(K) + \delta(K)\left[\frac qp + \frac{2p-q-1}{p}\|u_\eps\|_{L^r(Q_T)}^r + \frac 1p\|\nablau_\eps\|_{L^q(Q_T)}^q\right]
\end{aligned}
\end{equation}
where we used $\frac{r(p-q)}{p} = r - \frac{rq}{p} < r$ and the Young inequality $y^{r(p-q)/q} \leq \frac{p-q}{p}y^r + \frac qp$ at the last step. We will show now that by choosing a suitable $r$ (which satisfies \eqref{cond_r}) we can estimate
\begin{equation*}\label{e13_1}
\|u_\eps\|_{L^r(Q_T)}^r \leq C(T,\beta)\|\nablau_\eps\|_{L^q(Q_T)}^q
\end{equation*}
with $\beta$ is in \eqref{cond1}. Indeed, by setting
\begin{equation}\label{chose_r}
r = \frac{q(d+1)}{d},
\end{equation}
we have
\begin{equation*}
\frac{r(p-q)}{q} = \frac{(d+1)(p-q)}{d} > 1 \quad \text{ since } \quad q < p - \frac{d}{d+1},
\end{equation*}
thus \eqref{cond_r} is satisfied. Note that from \eqref{chose_r} we also have $r < q^* = \frac{dq}{d-q}$. Therefore, we can use the interpolation inequality with $\frac{1}{r} = \frac{\eta}{1} + \frac{1-\eta}{q^*}$, and $\sup_{t\in(0,T)}\|u_\eps\|_{L^1(\Omega_t)}\leq \beta$ to estimate
\begin{equation}\label{e16}
\begin{aligned}
\|u_\eps\|_{L^r(Q_T)}^r &= \int_0^T\|u_\eps\|_{L^r(\Omega_t)}^rdt \leq \int_0^T\|u_\eps\|_{L^1(\Omega_t)}^{r\eta}\|u_\eps\|_{L^{q^*}(\Omega_t)}^{r(1-\eta)}dt\\
&\leq \beta^{r\eta}\int_0^T\|u_\eps\|_{L^{q^*}(\Omega_t)}^{r(1-\eta)}dt.
\end{aligned}
\end{equation}
From \eqref{chose_r}, we can easily check that $r(1-\eta)=q$. Therefore, \eqref{e16} yields
\begin{equation}\label{e16_1}
\|u_\eps\|_{L^r(Q_T)}^r\leq \beta^{r\eta}\|u_\eps\|_{\LQ{q}{q^*}}^q.
\end{equation}
By using Lemma \ref{embedding},
\begin{equation}\label{e16_2}
\|u_\eps\|_{\LQ{q}{q^*}}^q \leq C_{\Omega,T}^q\int_0^T\|\nablau_\eps\|_{L^{q}(\Omega_t)}^qdt = C_{\Omega,T}^q\|\nablau_\eps\|_{L^q(Q_T)}^q.
\end{equation}
Combining \eqref{e13}, \eqref{e16_1} and \eqref{e16_2} leads to
\begin{equation}\label{e16_3}
\|\nablau_\eps\|_{L^q(Q_T)}^q \leq C(K)+\delta(K)\left[\frac qp + \left(\frac{2p-q-1}{p}\beta^{r\eta}C_{\Omega,T}^q + \frac 1p \right)\|\nablau_\eps\|_{L^q(Q_T)}^q\right].
\end{equation}
Recalling that $\lim_{K\to\infty} \delta(K) = 0$. We choose $K$ large enough to have
\begin{equation*}
\delta(K)\left(\frac{2p-q-1}{p}\beta^{r\eta}C_{\Omega,T}^q + \frac 1p \right) \leq \frac 12,
\end{equation*}
which, in combination with \eqref{e16_3}, implies
\begin{equation*}
\|\nablau_\eps\|_{L^q(Q_T)}^q \leq 2\left(C(K) + \delta(K)\frac qp\right),
\end{equation*}
which is the desired estimate \eqref{e2_1}. \end{proof} In order to prove Lemma \ref{UniformBounds}, thanks to Lemma \ref{lem:inter}, it is sufficient to prove \eqref{cond1} and \eqref{cond2} for solutions to the approximate problem \eqref{approx}. These will be shown in the next consecutive lemmas. \begin{lemma}\label{prove_cond1}
There exists a constant $\beta = \beta\left(T,\|u_0\|_{L^1(\Omega_0)}, \|f\|_{L^1(Q_T)}\right)$ {\normalfont independent of $\varepsilon$} such that for any solution to \eqref{approx}, the following holds
\begin{equation*}
\|u_\eps\|_{\LQ{\infty}{1}} \leq \beta.
\end{equation*} \end{lemma} \begin{proof}
Let $k \in \mathbb{R}^+$. We define the truncated function
\begin{equation*}\label{T_k}
T_k(z) = \begin{cases}
z, &\text{ if } |z| \leq k,\\
k, &\text{ if } z > k,\\
-k, &\text{ if } z < -k.
\end{cases}
\end{equation*}
It is clear that $T_k$ is a Lipschitz function, and if $u_\eps \in \LW{p}{1}{p}$ then $T_k(u_\eps ) \in \LW{p}{1}{p}$ with
\begin{equation}\label{na-T_k}
\nabla T_k(u_\eps ) = \chi_{\{|u_\eps | \leq k\}}\nabla u_\eps ,
\end{equation}
where $\chi_{\{|u_\eps | \leq k\}}$ is the characteristic function of the set $\{|u_\eps (x, t)|\leq k\}$. Define $S_k(z) = \int_0^zT_k(\tau)d\tau$. We will show the following weak chain rule
\begin{equation}\label{t0}
\int_{Q_t}\partial_su_\eps T_k(u_\eps)dxds = \int_{\Omega_s}S_k(u_\eps)dx\bigg|_{s=0}^{s=t}.
\end{equation}
Indeed, choose a smooth sequence $\{v^m\}\subset C^1([0,T];C^1_c(\Omega_t))$ such that $v^m \xrightarrow{m\to\infty} u_\eps$ in $L^p(0,T;W^{1,p}_0(\Omega_t))\cap C([0,T];L^p(\Omega_t))$ and $\partial_t v^m \xrightarrow{m\to\infty} \partial_t u_\eps$ in $L^{p'}(0,T;W^{-1,p'}(\Omega_t))$. Since $S_k'(z) = T_k(z)$ and $v^m$ is smooth, it holds
\begin{align*}
\int_{Q_t}\partial_sv^m T_k(v^m)dxds = \int_{Q_t}\partial_s(S_k(v^m))dxds &= \int_0^t\left[\frac{d}{ds}\int_{\Omega_s}S_k(v^m)dx - \int_{\partial\Omega_s}S_k(v^m)(\mathbf v\cdot \nu)dS\right]ds\\ &=\int_{\Omega_s}S_k(v^m)dx\bigg|_{s=0}^{s=t}
\end{align*}
where we used $v^m|_{\partial\Omega_s} = 0$ an $S_k(0) = 0$ at the last step. Let $m\to \infty$, thanks to $v^m \to u_\eps$ in $C([0,T];L^p(\Omega_t))$ and $|S_k(z)| \leq C(1+|z|)$, it follows that \begin{equation}\label{t1}\int_{\Omega_s}S_k(v^m)dx\bigg|_{s=0}^{s=t} \to \int_{\Omega_s}S_k(u_\eps)dx\bigg|_{s=0}^{s=t}.
\end{equation}
For the left hand side, we estimate
\begin{equation}\label{t2}
\begin{aligned}
&\left|\int_{Q_t}(\partial_sv^mT_k(v^m) - \partial_su_\eps T_k(u_\eps))dxds \right|\\
&\leq \int_{Q_t}\left|\partial_sv^m - \partial_su_\eps \right||T_k(v^m)|dxds+ \left|\int_{Q_t}(T_k(v^m)-T_k(u_\eps))\partial_su_\eps dxds \right|\\
&=: (I) + (II).
\end{aligned}
\end{equation}
Since
\begin{equation*}
(I)\leq \|\partial_sv^m - \partial_su_\eps\|_{L^{p'}(0,T;W^{-1,p'}(\Omega_t))}\|T_k(v^m)\|_{L^p(0,T;W^{1,p}_0(\Omega_t))}
\end{equation*}
and $\{T_k(v^m)\}_{m\geq 1}$ is bounded in $L^p(0,T;W^{1,p}_0(\Omega_t))$,
we have $\lim_{m\to\infty}(I) = 0$. For $(II)$ it follows from $v^m\to u_\eps$ a.e. in $Q_T$ and $T_k$ is continuous that $T_k(v^m)\to T_k(u_\eps)$ a.e. in $Q_T$.
By combining this with $\{T_k(v^m)\}$ is bounded in $L^p(Q_T)$, we obtain $T_k(v^m) \rightharpoonup T_k(u_\eps)$ weakly in $L^p(Q_T)$ (see e.g. \cite[Lemma 8.3]{robinson2001infinite}). Now since $T_k(u_\eps)\in L^p(0,T;W^{1,p}_0(\Omega_t))$, it yields $T_k(v^m) \rightharpoonup T_k(u_\eps)$ weakly in $L^p(0,T;W^{1,p}_0(\Omega_t)$ and therefore $\lim_{m\to\infty}(II) = 0$. By combining \eqref{t1} and \eqref{t2}, we obtain the desired relation \eqref{t0}.
Choosing $\phi=T_k(u_\eps )$ as test function for \eqref{approx} and using \eqref{t0}, we get, for $0<t\leq T$,
\begin{equation}\label{test-T_k}
\begin{aligned}
\int_{\Omega_s}S_k(u_\eps )dx\biggr|_{s=0}^{s=t} & + \int_{Q_t}a(x,s, \nabla u_\eps )\nabla T_k(u_\eps )dxds \\
&+\int_{Q_t}\text{div}(u_\eps\mathbf v)T_k(u_\eps )dxds +\int_{Q_t} g_\eps(u_\eps , \nabla u_\eps )T_k(u_\eps )dxds\\
&= \int_{Q_t}f_\eps T_k(u_\eps )dxds.
\end{aligned}
\end{equation}
Note that the boundary terms vanish due to the homogeneous Dirichlet boundary condition, which consequently implies that $S(u_\eps) = 0$ on the boundary.
From \eqref{A3} and \eqref{na-T_k}, we have
\begin{equation}\label{a}
\begin{aligned}
\int_{Q_t}a(x,s, \nabla u_\eps )\nabla T_k(u_\eps )dxds &= \int_{Q_t}\chi_{\{|u_\eps | \leq k\}}a(x,s, \nabla u_\eps ) \nabla u_\eps dxds\\
&\geq \alpha \int_{Q_t}\chi_{\{|u_\eps | \leq k\} } |\nabla u_\eps |^p dxds.
\end{aligned}
\end{equation}
Applying intergration by parts for penultimate term on the left hand side of \eqref{test-T_k}, we obtain
\begin{equation}\label{e17}
\begin{aligned}
\int_{Q_t}\text{div} (u_\eps \mathbf v)T_k(u_\eps )dxds = - \int_{Q_t}u_\eps \mathbf v \nabla T_k(u_\eps )dxds= -\int_{Q_t}\chi_{\{|u_\eps | \leq k\}}u_\eps \mathbf v \nabla u_\eps dxds.
\end{aligned}
\end{equation}
Combining \eqref{a}-\eqref{e17} with \eqref{test-T_k}, we get
\begin{equation}\label{e18}
\begin{aligned}
\int_{\Omega_s}S_k(u_\eps )dx\biggr|_{s=0}^{s=t} &+\alpha\int_{Q_t}\chi_{\{|u_\eps | \leq k\}}|\nabla u_\eps |^p dxds +\int_{Q_t}g_\eps(u_\eps,\nablau_\eps)T_k(u_\eps)dxds\\
&\leq \int_{Q_t}|f_\eps T_k(u_\eps )|dxds + \int_{Q_t}\chi_{\{|u_\eps | \leq k\}}u_\eps \mathbf v \nabla u_\eps dxds.
\end{aligned}
\end{equation}
Applying Young's inequality for last term in right-hand side above, we have
\begin{equation}\label{e19}
\begin{aligned}
\int_{Q_t}\chi_{\{|u_\eps | \leq k\}} u_\eps \mathbf v \nabla u_\eps dxds &\leq \|\mathbf v\|_{\infty}\int_{Q_t}|\chi_{\{|u_\eps | \leq k\}} u_\eps||\nablau_\eps|dxds\\ &\leq \frac{\alpha}{2}\int_{Q_t}\chi_{\{|u_\eps | \leq k\}}|\nabla u_\eps |^p dxds + C(\alpha,\|\mathbf v\|_{\infty})\int_{Q_t}\chi_{\{|u_\eps | \leq k\}} |u_\eps |^{p'}dxds\\
&\leq \frac{\alpha}{2} \int_{Q_t}\chi_{\{|u_\eps | \leq k\}}|\nabla u_\eps |^p dxdt +C(\alpha,\|
\mathbf v\|_{\infty})|k|^{p'}|Q_T|\\
&=\frac{\alpha}{2} \int_{Q_t}\chi_{\{|u_\eps | \leq k\}}|\nabla u_\eps |^p dxdt +C(T, \alpha, \|\mathbf v\|_{\infty},k),
\end{aligned}
\end{equation}
where $\frac{1}{p} + \frac{1}{p'} = 1$. Since $|T_k(u_\eps)| \leq k$,
\begin{equation}\label{e20}
\int_{Q_t}|f_\eps T_k(u_\eps)|dxdt \leq k\|f_\eps\|_{L^1(Q_T)}.
\end{equation} We remark that $u_\varepsilon T_k(u_\eps ) \geq 0$, combining with \eqref{G1}, we have $g_\eps(u_\eps,\nablau_\eps)T_k(u_\eps) \geq 0$. Therefore, inserting \eqref{e19} and \eqref{e20} into \eqref{e18} gives
\begin{equation}\label{e23}
\sup_{t\in(0,T)}\|S_k(u_\eps)(t)\|_{L^1(\Omega_t)} \leq \|S_k(u_{\varepsilon,0})\|_{L^1(\Omega_0)} + k\|f_\eps\|_{L^1(Q_T)} + C(T,\alpha,\|\mathbf v\|_\infty,k).
\end{equation}
We set $k= 1$ in \eqref{e23}. Note that $0\leq S_1(z) \leq |z|$ and recall \eqref{increasing}, we get
\begin{equation*}\label{e25}
\begin{aligned}
\sup_{t\in(0,T)}\int_{\Omega_t}S_1(u_\eps )(t)dx\leq \|u_{0}\|_{L^1(\Omega_0)} + k\|f\|_{L^1(Q_T)} + C(T,\alpha,\|\mathbf v\|_\infty,1).
\end{aligned}
\end{equation*}
Therefore, by using $u_\eps = S_1(u_\eps)$ for $|u_\eps| \geq 1$,
\begin{equation*}\label{e26}
\begin{aligned}
\sup_{t\in(0,T)}\|u_\eps\|_{L^1(\Omega_t)} &=\sup_{t\in(0,T)}\int_{\{x\in \Omega_t: \;|u_\eps |\leq 1\}}|u_\eps |dx + \sup_{t\in(0,T)}\int_{\{x\in \Omega_t: \;|u_\eps |\geq 1\}}|u_\eps |dx\\
&\leq \sup_{t\in(0,T)}|\Omega_t| + \sup_{t\in(0,T)}\int_{\Omega_t}|S_1(u_\eps )|dx\\
&\leq \sup_{t\in(0,T)}|\Omega_t|+\|u_{0}\|_{L^1(\Omega_0)} + k\|f\|_{L^1(Q_T)} + TC(T,\alpha,\|\mathbf v\|_\infty,1)\\
&=:\beta.
\end{aligned}
\end{equation*}
This completes the proof of Lemma \ref{prove_cond1}. \end{proof}
\begin{lemma}\label{prove_cond2}
There exist positive constants $C_0, C_1$ {\normalfont independent of $\varepsilon$ and $n\in\mathbb N$} such that the following estimate holds
\begin{equation*}
\int_{B_n}|\nablau_\eps|^pdxdt \leq C_0 + C_1\int_{E_n}|\nablau_\eps|dxdt \quad \text{ for all } \quad \varepsilon>0 \text{ and all } n\in \mathbb N,
\end{equation*}
where $u_\eps$ is a solution to \eqref{approx}. \end{lemma} \begin{proof} For $n\in \mathbb N$, we define the function $\phi_n: \mathbb R\to \mathbb R$ as \begin{equation}\label{phi_n} \phi_n(z) = \begin{cases} 1, &\text{ if } z> n+1,\\ z-n, &\text{ if } n \leq z \leq n+1,\\ 0, &\text{ if } -n < z < n,\\ -z-n, &\text{ if } -n-1 \leq z \leq -n,\\ 1, &\text{ if } z \leq -n-1, \end{cases} \end{equation} and we set $\Psi_n(z) = \int_0^z\phi_n(\tau)d\tau$. We note that $\phi_n$ is a Lipschitz function, and therefore $u_\eps \in \LW{p}{1}{p}$ implies $\phi_n(u_\eps ) \in \LW{p}{1}{p}$ with \begin{equation*}\label{nabla-phi} \nabla \phi_n(u_\eps ) = \chi_{B_n}\nabla u_\eps , \end{equation*}
$\chi_{B_n}$ denoting the characteristic function of the set $B_n = \{(x,t) \in Q_T: n \leq |u_\eps (x,t)| \leq n+1\}$ defined in \eqref{def_BnEn}. We now take $\phi_n(u_\eps ) \in \LW{p}{1}{p}$ as test function for \eqref{approx} to get, by using a weak chain rule similar to \eqref{t0}, \begin{equation}\label{test-phi_n_1} \begin{aligned} \int_{\Omega_T}&\Psi_n(u_\eps)(T)dx +\int_{Q_T}a(x,t, \nabla u_\eps ) \nabla \phi_n(u_\eps )dxdt+\int_{Q_T}\text{div}(u_\eps \mathbf v)\phi_n(u_\eps )dxdt\\
&+\int_{Q_T} g_\eps(u_\eps , \nabla u_\eps )\phi_n(u_\eps )dxdt \leq \int_{\Omega_0}\Psi_n(u_{0,\varepsilon})dx + \int_{Q_T}|f_\eps\phi_n(u_\eps )|dxdt. \end{aligned} \end{equation} From \eqref{A3}, we obtain \begin{equation}\label{e27} \begin{aligned} \int_{Q_T}a(x,t, \nabla u_\eps )\nabla \phi_n(u_\eps )dxdt &= \int_{Q_T}\chi_{B_n}a(x,t, \nabla u_\eps ) \nabla u_\eps dxdt\\
&\geq \alpha \int_{Q_T}\chi_{B_n} |\nabla u_\eps |^p dxdt\\
&= \alpha\int_{B_n}|\nablau_\eps|^pdxdt. \end{aligned} \end{equation} The penultimate term on the left hand side of \eqref{test-phi_n_1} can be rewritten as \begin{equation}\label{e28} \begin{aligned} \int_{Q_T}\text{div} (u_\eps \mathbf v)\phi_n(u_\eps )dxdt = \int_{Q_T}(\nablau_\eps\cdot\mathbf v + u_\eps\text{div}\mathbf v) \phi_n(u_\eps )dxdt. \end{aligned} \end{equation} Combining \eqref{e27}-\eqref{e28} with \eqref{test-phi_n_1}, and the fact that $\Psi_n$ is nonnegative, we get \begin{equation}\label{e29} \begin{aligned}
&\alpha\int_{B_n}|\nabla u_\eps |^p dxdt + \int_{Q_T}g_\eps(u_\eps,\nablau_\eps)\phi_n(u_\eps)dxdt \\
&\leq \int_{\Omega_0}\Psi_n(u_{0,\varepsilon})dx+ \int_{Q_T}|f_\eps||\phi_n(u_\eps )|dxdt\\
&\quad + \int_{Q_T}|\nablau_\eps \cdot \mathbf v||\phi_n(u_\eps)| dxdt + \int_{Q_T}|\text{div}\mathbf v||u_\eps||\phi_n(u_\eps)|dxdt\\
&\leq \int_{\Omega_0}\Psi_n(u_{0,\varepsilon})dx + \|f\|_{L^1(Q_T)} + \|\mathbf v\|_{\infty}\int_{Q_T}|\nablau_\eps||\phi_n(u_\eps)|dxdt + \|\text{div}\mathbf v\|_{\infty}T\beta \end{aligned} \end{equation}
where we used $\sup_{t\in(0,T)}\|u_\eps\|_{L^1(\Omega_t)}\leq \beta$ at the last step. Using $|\Psi_n(z)| \leq |z|$ and $\textrm{supp}(\phi_n) \subset (-\infty, -n] \cup [n,\infty)$, we can estimate \begin{equation*}
\left|\int_{\Omega_0}\Psi_n(u_{0,\varepsilon})dx \right| \leq \|u_{0,\varepsilon}\|_{L^1(\Omega_0)} \leq \|u_0\|_{L^1(\Omega_0)} \end{equation*} and \begin{align*}
\int_{Q_T}|\nablau_\eps||\phi_n(u_\eps)|dxdt &\leq \int_{B_n}|\nablau_\eps|dxdt + \int_{E_n}|\nablau_\eps|dxdt\\
&\leq \frac{\alpha}{2}\int_{B_n}|\nablau_\eps|^pdxdt + C(\alpha)|Q_T| + \int_{E_n}|\nablau_\eps|dxdt, \end{align*} where Young's inequality was applied at the last step. Inserting these estimates into \eqref{e29} yields \begin{equation}\label{est_1} \begin{aligned}
&\alpha \int_{B_n}|\nablau_\eps|^pdxdt + 2\int_{Q_T}g_\eps(u_\eps,\nablau_\eps)\phi_n(u_\eps)dxdt\\
&\leq C\left(\|u_0\|_{L^1(\Omega_0)}, \|f\|_{L^1(Q_T)}, \alpha, \beta, \mathbf v, T\right) + 2\|\mathbf v\|_{\infty}\int_{E_n}|\nablau_\eps|dxdt, \end{aligned} \end{equation} which implies the desired estimate and therefore completes the proof of Lemma \ref{prove_cond2}. \end{proof} \begin{proof}[Proof of Lemma \ref{UniformBounds}]
The proof of Lemma \ref{UniformBounds} is an immediate consequence of Lemmas \ref{lem:inter}, \ref{prove_cond1} and \ref{prove_cond2}. \end{proof} \subsection{Uniform bounds of the nonlinearity} \begin{lemma}\label{bound_non}
Let $u_\eps$ be a solution of \eqref{approx}. Then the following estimate holds
\begin{equation}\label{bound_non_1}
\|g_\eps(u_\eps,\nablau_\eps)\|_{L^1(Q_T)} \leq K
\end{equation}
where $K$ is {\normalfont independent of $\varepsilon$}. \end{lemma} \begin{proof}
To prove \eqref{bound_non_1} we fix $n\in \mathbb N$ and write
\begin{equation*}
\|g_\eps(u_\eps,\nablau_\eps)\|_{L^1(Q_T)} \leq \int\limits_{\{|u_\eps| \leq n+1\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt + \int\limits_{\{|u_\eps|\geq n+1\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt =: G_1 + G_2.
\end{equation*}
Recall the function $\phi_n$ defined in \eqref{phi_n}, we have $\phi_n(z) = 1$ for $z\geq n+1$. Therefore, by using the fact that $\phi_n(u_\eps)g_\eps(u_\eps,\nablau_\eps) \geq 0$ thanks to \eqref{G1},
\begin{align*}
G_2 &= \int_{\{|u_\eps|\geq n+1\}}\phi_n(u_\eps)g_\eps(u_\eps,\nablau_\eps)dxdt\\
&\leq \int_{Q_T}\phi_n(u_\eps)g_\eps(u_\eps,\nablau_\eps)dxdt\\
&\leq C + \|\mathbf v\|_{\infty}\int_{E_n}|\nablau_\eps|dxdt \quad (\text{by using } (\ref{est_1}))\\
&\leq C + \|\mathbf v\|_{\infty}\|\nablau_\eps\|_{L^1(Q_T)}\\
&\leq C \quad (\text{by applying Lemma }\ref{UniformBounds} \text{ for } q= 1).
\end{align*}
Therefore, $G_2$ is bounded uniformly in $\varepsilon$. We now estimate $G_1$ by using the assumption \eqref{G2}
\begin{equation}\label{est_G1}
\begin{aligned}
G_1&\leq \int_{\{|u_\eps| \leq n+1\}}h(u_\eps)(\gamma(x,t) + |\nablau_\eps|^\sigma)dxdt\\
&\leq h(n+1)\|\gamma\|_{L^1(Q_T)} + h(n+1)\int_{\{|u_\eps|\leq n+1 \}}|\nablau_\eps|^\sigma dxdt.
\end{aligned}
\end{equation}
Now, recalling $B_j = \{(x,t): j\leq u_\eps(x,t) \leq j+1 \}$,
\begin{align*}
\int_{\{|u_\eps|\leq n+1\}}|\nablau_\eps|^\sigma dxdt &= \sum_{j=0}^n\int_{B_j}|\nablau_\eps|^\sigma dxdt\\
&\leq \sum_{j=0}^n|B_j|^{\frac{p-\sigma}{p}}\left(\int_{B_j}|\nablau_\eps|^pdxdt \right)^{\frac{\sigma}{p}}\\
&\leq |Q_T|^{\frac{p-\sigma}{p}}\sum_{j=0}^n\left(C + \frac{2\|\mathbf v\|_{\infty}}{\alpha}\int_{E_j}|\nablau_\eps|dxdt \right)^{\frac{\sigma}{p}} \quad (\text{using }(\ref{est_1}))\\
&\leq |Q_T|^{\frac{p-\sigma}{p}}(n+1)\left(C + \frac{2\|\mathbf v\|_{\infty}}{\alpha}\|\nablau_\eps\|_{L^1(Q_T)} \right)^{\frac{\sigma}{p}}\\
&\leq C(n,T)
\end{align*}
where we used Lemma \ref{UniformBounds} with $q = 1$ at the last step. From this and \eqref{est_G1}, it follows that $G_1$ is bounded uniformly in $\varepsilon>0$. Thus \eqref{bound_non_1} is proved. \end{proof}
\section{Proof of Theorem \ref{thm:main}}\label{proof} The uniform bounds in Section \ref{approximate} imply that there exists a subsequence of $\{u_\eps\}_{\varepsilon>0}$ such that \begin{equation*}
u_\eps \rightharpoonup u \quad \text{ weakly in } \quad \LW{q}{1}{q} \quad \text{ for all } \quad 1<q<p - \frac{d}{d+1}. \end{equation*} This limit function $u$ is a candidate for a weak solution to \eqref{e1}, but the weak convergence is far from enough to show that it is the case. We need convergence in stronger topologies, especially to pass to the limit for the nonlinearities. We start with a pointwise and $L^1$-convergence. \begin{lemma}\label{L1convergence}
Let $\{u_\eps\}_{\varepsilon>0}$ be solutions to \eqref{approx}. Then there exists a subsequence of $\{u_\eps\}_{\varepsilon>0}$ (not relabeled) such that
\begin{equation*}
u_\eps \to u \quad \text{ strongly in } L^{s}(Q_T) \quad \text{ for all } \quad 1\leq s < p-\frac{d}{d+1}.
\end{equation*} \end{lemma} To prove Lemma \ref{L1convergence}, we need an Aubin-Lions lemma for the case of moving domains. A similar lemma was recently shown in \cite{Mou16}, but it is not directly applicable to our case. Therefore, a new version is necessary. \begin{lemma}[An Aubin-Lions lemma in moving domains]\label{AL-lemma}
Let $1\leq q <+\infty$ and $\{u_n\}_{n\geq 1}$ be a sequence which is bounded in $L^q(0,T;W_0^{1,q}(\Omega_t))$. Moreover, for any smooth function $\psi\in \mathcal{D}(Q_T)$ it holds
\begin{equation}\label{time_derivative}
\left|\int_{Q_T} u_n\partial_t \psi dxdt \right| \leq C\sup_{t\in (0,T)}\|\psi\|_{H^m(\Omega_t)}
\end{equation}
for some $m\in \mathbb N$. Then $\{u_n\}_{n\geq 1}$ is precompact in $L^1(Q_T)$, and when $q > 1$ then $\{u_n\}_{n\geq 1}$ is precompact in $L^{s}(Q_T)$ for all $1\leq s<q$. \end{lemma} \begin{remark}
In \cite{Mou16}, instead of \eqref{time_derivative}, the following stronger condition was imposed
\begin{equation*}
\left|\int_{Q_T} u_n \partial_t \psi dxdt \right| \leq C\sum_{|\alpha|\leq m}\|\partial_{\alpha}\psi\|_{\LQ{2}{2}}.
\end{equation*}
In our case, due to the fact that the right hand side belongs only $L^1(Q_T)$, it seems that \eqref{time_derivative} is unavoidable. \end{remark} \begin{proof}[Proof of Lemma \ref{AL-lemma}]
Though Lemma \ref{AL-lemma} is an improved version of that in \cite{Mou16}, its proof still follows closely from the ideas therein with some suitable changes. We therefore postpone it and provide the full technical proof in the Appendix \ref{appendix2}. \end{proof}
We can now apply the Aubin-Lions lemma to prove Lemma \ref{L1convergence}.
\begin{proof}[Proof of Lemma \ref{L1convergence}]
Thanks to Lemma \ref{UniformBounds}, we only need to check the condition \eqref{time_derivative}. First, we choose $m\in \mathbb N$ such that $H^{m-1}(\Omega_t)\subset L^\infty(\Omega_t)$ for all $t\in [0,T]$. Moreover, using similar arguments to Lemma \ref{embedding} we deduce that there exists a constant $C = C(\mathbf v,T)$ such that
\begin{equation*}
\|v\|_{L^r(\Omega_t)} \leq C\|v\|_{H^{k}(\Omega_t)} \quad \text{ for all } \quad t\in [0,T]
\end{equation*}
for $k\in \{m-1,m\}$ and for any $r\in[1,\infty]$. Now, we multiply the approximating problem \eqref{approx}
by $\psi\in \mathcal D(Q_T)$ and then integrate on $Q_T$ to get
\begin{equation}\label{est_2}
\begin{aligned}
\int_{Q_T} u_\eps \partial_t\psi dxdt &= -\int_{Q_T} a(x,t,\nablau_\eps)\cdot \nabla\psi dxdt - \int_{Q_T}[\nablau_\eps \cdot \mathbf v + u_\eps\text{div} \mathbf v]\psi dxdt\\
&- \int_{Q_T} g_\eps(u_\eps,\nablau_\eps)\psi dxdt + \int_{Q_T} f_\varepsilon \psi dxdt.
\end{aligned}
\end{equation}
From the assumption \eqref{A2} of $a$ we have
\begin{equation*}
\begin{aligned}
\left|\int_{Q_T} a(x,t,\nablau_\eps)\cdot \nabla\psi dxdt \right| &\leq \int_{Q_T}|\varphi||\nabla\psi|dxdt + K\int_{Q_T}|\nablau_\eps|^{p-1}|\nabla\psi|dxdt\\
&\leq \|\varphi\|_{L^{p'}(Q_T)}\|\nabla \psi\|_{L^p(Q_T)} + K\|\nablau_\eps\|_{L^q(Q_T)}^{p-1}\|\nabla \psi\|_{L^{\frac{q}{q-p+1}}(Q_T)}\\
&\leq C\sup_{t\in (0,T)}\|\psi\|_{H^m(\Omega_t)}.
\end{aligned}
\end{equation*}
Similarly, by using the bounds in Lemmas \ref{UniformBounds} and \ref{bound_non} and $\|f_\varepsilon\|_{L^1(Q_T)} \leq \|f\|_{L^1(Q_T)}$ we get
\begin{align*}
&\left| \int_{Q_T} [\nablau_\eps\cdot\mathbf v + u_\eps\text{div} \mathbf v]\psi dxdt\right|\\
&\leq \|\mathbf v\|_{\infty}\|\nablau_\eps\|_{L^q(Q_T)}\|\psi\|_{L^{q'}(Q_T)} + \|\text{div}\mathbf v\|_{\infty}\|u_\eps\|_{L^q(Q_T)}\|\nabla\psi\|_{L^{q'}(Q_T)}\\
&\leq C(\|\psi\|_{L^{q'}(Q_T)} + \|\nabla\psi\|_{L^{q'}(Q_T)})\\
&\leq C\sup_{t\in(0,T)}\|\psi\|_{H^m(\Omega_t)},
\end{align*}
and
\begin{equation*}
\left|\int_{Q_T} g_\eps(u_\eps,\nablau_\eps)\psi dxdt \right| \leq \|g_\eps(u_\eps,\nablau_\eps)\|_{L^1(Q_T)}\|\psi\|_{L^\infty(Q_T)} \leq C\sup_{t\in (0,T)}\|\psi\|_{H^m(\Omega_t)},
\end{equation*}
and
\begin{equation*}
\left|\int_{Q_T} f_\varepsilon\psi dxdt \right| \leq \|f\|_{L^1(Q_T)}\|\psi\|_{L^\infty(Q_T)} \leq C\sup_{t\in (0,T)}\|\psi\|_{H^m(\Omega_t)}.
\end{equation*}
Putting all these into \eqref{est_2} we get \eqref{time_derivative}, and therefore Lemma \ref{AL-lemma} implies the desired result of Lemma \ref{L1convergence} since $q < p - \frac{d}{d+1}$ is arbitrary.
\end{proof}
Due to the nonlinearities in the gradient in $a(x,t,\nablau_\eps)$ and in $g_\eps(u_\eps,\nablau_\eps)$, we need also stronger convergence of the gradient. \begin{lemma}[Almost everywhere convergence of the gradient]\label{gradients}
Let $\{u_\eps\}_{\varepsilon>0}$ be solutions of the approximate problem \eqref{approx}. Then the sequence $\{\nabla u_\eps \}_{\varepsilon>0}$ converges to $\nabla u$ almost everywhere as $\varepsilon$ goes to zero.
\end{lemma} \begin{proof} We will show that $\{\nabla u_\eps \}_\varepsilon$ is a Cauchy sequence in measure, i.e. for all $\mu > 0$ \begin{equation}\label{gra1}
\mathcal{A}:= \text{meas}\{(x,t)\in Q_T: |\nabla u_{\varepsilon'} - \nabla u_\eps | > \mu\} \to 0, \end{equation} as $\varepsilon', \varepsilon \to 0$. From this, after extracting a subsequence, we have the convergence $\nablau_\eps \to \nabla u$ almost everywhere.
To prove \eqref{gra1}, we let $k>0$ and $\delta>0$ be chosen later and observe that \begin{equation*}
\mathcal{A} \subset \mathcal{A}_1 \cup \mathcal{A}_2 \cup \mathcal{A}_3 \cup \mathcal{A}_4 \end{equation*} where \begin{equation*}\label{split} \begin{aligned}
&\mathcal{A}_1 = \{(x,t)\in Q_T:\; |\nablau_\eps| \geq k \},\\
&\mathcal{A}_2 = \{(x,t)\in Q_T: \; |\nabla u_{\varepsilon'}| \geq k \},\\
&\mathcal{A}_3 = \{(x,t)\in Q_T:\; |u_\eps - u_{\varepsilon'}| \geq \delta \},\\
&\mathcal{A}_4 = \{(x,t)\in Q_T: \; |\nablau_\eps - \nabla u_{\varepsilon'}| \geq \mu, |\nablau_\eps| \leq k, |\nabla u_{\varepsilon'}| \leq k \text{ and } |u_\eps - u_{\varepsilon'}| \leq \delta \}. \end{aligned} \end{equation*} We will estimate $\mathcal{A}_i$, $i=1,\ldots, 4$ separately. Firstly, for $\mathcal{A}_1$, by applying Lemma \ref{UniformBounds} with $q = 1$, we have \begin{equation}\label{est_A1}
|\mathcal{A}_1| = \int_{\mathcal{A}_1}dxdt \leq \frac{1}{k}\int_{\mathcal{A}_1}|\nablau_\eps|dxdt \leq \frac{1}{k}\|\nablau_\eps\|_{L^1(Q_T)} \leq \frac{C}{k} \end{equation} for $C$ independent of $\varepsilon$. Similarly, \begin{equation}\label{est_A2}
|\mathcal{A}_2| \leq \frac{1}{k}\|\nabla u_{\varepsilon'}\|_{L^1(Q_T)} \leq \frac{C}{k}. \end{equation} For $\mathcal{A}_3$, \begin{equation}\label{est_A3}
|\mathcal{A}_3| = \int_{\mathcal{A}_3}dxdt \leq \frac{1}{\delta}\|u_\eps - u_{\varepsilon'}\|_{L^1(Q_T)}. \end{equation}
It remains to estimate $\mathcal{A}_4$. Firstly, by using $T_{\delta}(u_\eps - u_{\varepsilon'}) = u_\eps - u_{\varepsilon'}$ on the set $\{(x,t): \;|u_\eps - u_{\varepsilon'}|\leq \delta\}$, we have \begin{equation}\label{h1} \begin{aligned}
|\mathcal{A}_4| \leq \frac{1}{\mu}\int_{\{|u_\eps - u_{\varepsilon'}| \leq \delta\}}|\nabla(u_\eps - u_{\varepsilon'})|dxdt = \frac{1}{\mu}\int_{Q_T}\chi_{\{|u_\eps - u_{\varepsilon'}| \leq \delta\}}|\nabla (u_\eps - u_{\varepsilon'})|dxdt. \end{aligned} \end{equation} Subtracting the equation \eqref{approx} for $\varepsilon$ and $\varepsilon'$, then taking $\phi = T_\delta(u_\eps - u_{\varepsilon'})$ as a test function, we get \begin{equation}\label{h2} \begin{aligned} &\int_{\Omega_T}S_\delta(u_\eps(x,T) - u_{\varepsilon'}(x,T))dx + \int_{Q_T}(a(x,t,\nablau_\eps) - a(x,t,\nabla u_{\varepsilon'}))\nabla T_{\delta}(u_\eps - u_{\varepsilon'})dxdt\\ &= \int_{\Omega_0}S_\delta(u_{0,\varepsilon} - u_{0,\varepsilon'})dx + \int_{Q_T}(f_\eps - f_{\varepsilon'})T_{\delta}(u_\eps - u_{\varepsilon'})dxdt\\ &- \int_{Q_T}((u_\eps - u_{\varepsilon'})\text{div}\mathbf v + \mathbf v\cdot\nabla(u_\eps - u_{\varepsilon'}))T_{\delta}(u_\eps - u_{\varepsilon'})dxdt\\ &- \int_{Q_T}(g_\eps(u_\eps,\nablau_\eps) - g_{\varepsilon'}(u_{\varepsilon'},\nabla u_{\varepsilon'}))T_{\delta}(u_\eps - u_{\varepsilon'})dxdt. \end{aligned} \end{equation} Since $S_\delta$ is nonnegative and thanks to the assumption \eqref{A1}, the left hand side of \eqref{h2} is bounded below by \begin{equation}\label{h3} \begin{aligned}
&\int_{Q_T}(a(x,t,\nablau_\eps)-a(x,t,\nabla u_{\varepsilon'})(\nabla u_\eps - \nabla u_{\varepsilon'})\chi_{\{|u_\eps - u_{\varepsilon'}| \leq \delta\}}dxdt\\
&\geq C\int_{Q_T}\chi_{\{|u_\eps - u_{\varepsilon'}| \leq \delta\}}\frac{1}{\Theta(x,t,\nablau_\eps,\nabla u_{\varepsilon'})}|\nablau_\eps - \nabla u_{\varepsilon'}|^{\theta}dxdt. \end{aligned} \end{equation}
For the right hand side of \eqref{h2}, we use $|T_\delta(z)| \leq \delta$ and $S_\delta(z) \leq \delta|z|$ to estimate \begin{equation}\label{h4} \begin{aligned}
\text{ Right hand side of } (\ref{h2}) &\leq \delta\|u_{0,\varepsilon} - u_{0,\varepsilon'}\|_{L^1(\Omega_0)} + \delta\|f_\varepsilon - f_{\varepsilon'}\|_{L^1(Q_T)}\\
&\; + \delta\|\text{div} \mathbf v\|_{\infty}\|u_\eps - u_{\varepsilon'}\|_{L^1(Q_T)} + \delta\|\mathbf v\|_{\infty}\|\nabla(u_\eps - u_{\varepsilon'})\|_{L^1(Q_T)}\\
&\; + \delta(\|g_\eps(u_\eps,\nablau_\eps)\|_{L^1(Q_T)} + \|g_{\varepsilon'}(u_{\varepsilon'},\nabla u_{\varepsilon'})\|_{L^1(Q_T)})\\
&\leq C\delta \end{aligned} \end{equation} with $C$ independent of $\varepsilon, \varepsilon'$, and where we used the fact that $\{u_{0,\varepsilon}\}$ is bounded in $L^1(\Omega_0)$, and all $\{f_\eps\}, \{u_\eps\}, \{\nabla u_\eps\}, \{g_\eps(u_\eps,\nablau_\eps)\}$ are bounded in $L^1(Q_T)$. Inserting \eqref{h3} and \eqref{h4} into \eqref{h2} gives \begin{equation*}
\int_{Q_T}\chi_{\{|u_\eps - u_{\varepsilon'}| \leq \delta \}}\frac{1}{\Theta(x,t,\nablau_\eps,\nabla u_{\varepsilon'})}|\nablau_\eps - \nabla u_{\varepsilon'}|^{\theta}dxdt \leq C\delta. \end{equation*} By using H\"older's inequality, we have \begin{align*}
\int_{Q_T}\chi_{\{|u_\eps - u_{\varepsilon'}| \leq \delta \}}|\nablau_\eps - \nablau_{\eps'}|dxdt
&\leq \left(\int_{Q_T}\chi_{\{|u_\eps - u_{\varepsilon'}| \leq \delta \}}\frac{1}{\Theta(x,t,\nablau_\eps,\nablau_{\eps'})}|\nablau_\eps - \nablau_{\eps'}|^{\theta}dxdt\right)^{\frac 1\theta}\\
&\quad \times \left(\int_{Q_T}\chi_{\{|u_\eps - u_{\varepsilon'}| \leq \delta \}}\Theta(x,t,\nablau_\eps,\nablau_{\eps'})^{\frac{1}{\theta-1}}dxdt \right)^{\frac{\theta-1}{\theta}}\\
&\leq C\delta^{\frac 1\theta}\left(\int_{Q_T}\left[1+|\nablau_\eps|^{\frac{\vartheta}{\theta-1}} + |\nablau_{\eps'}|^{\frac{\vartheta}{\theta-1}}\right]dxdt\right)^{\frac{\theta-1}{\theta}} \end{align*} where we used \eqref{beta} at the last step. Thanks to \eqref{varrho}, $\dfrac{\vartheta}{\theta-1} < p - \dfrac{d}{d+1}$. Therefore, Lemma \ref{UniformBounds} implies \begin{equation*}
\int_{Q_T}\left[|\nablau_\eps|^{\frac{\vartheta}{\theta-1}} + |\nablau_{\eps'}|^{\frac{\vartheta}{\theta-1}} \right]dxdt \leq C \end{equation*} and thus \begin{equation*}
\int_{Q_T}\chi_{\{|u_\eps-u_{\eps'}|\leq \delta\}}|\nablau_\eps - \nablau_{\eps'}|dxdt \leq C\delta^{\frac 1\theta}. \end{equation*} Inserting this into \eqref{h1} leads to \begin{equation}\label{est_A4}
|\mathcal{A}_4| \leq \frac{C\delta^{\frac 1\theta}}{\mu} \end{equation} for a constant $C$ independent of $\varepsilon, \varepsilon'$.
Now let $\kappa>0$ be arbitrary. We first choose $k$ to be large enough so that \eqref{est_A1} and \eqref{est_A2} give \begin{equation*}
|\mathcal{A}_1| + |\mathcal{A}_2| \leq \frac{\kappa}{2}. \end{equation*} We next choose $\delta$ to be small enough ($k$ is now fixed) so that \eqref{est_A4} implies \begin{equation*}
|\mathcal{A}_4| \leq \frac{\kappa}{4}. \end{equation*} With $k$ and $\delta$ are fixed, since $\{u_\eps\}_{\varepsilon>0}$ is a Cauchy sequence in $L^1(Q_T)$, thanks to Lemma \ref{L1convergence}, there exists $\varepsilon_0 > 0$ such that, for all $\varepsilon, \varepsilon' \leq \varepsilon_0$, \eqref{est_A3} implies \begin{equation*}
|\mathcal{A}_3| \leq \frac{1}{\delta}\|u_\eps - u_{\varepsilon'}\|_{L^1(Q_T)} \leq \frac{\kappa}{4}. \end{equation*} Therefore, \begin{equation*}
|\mathcal A| \leq |\mathcal{A}_1| + |\mathcal{A}_2| + |\mathcal{A}_3| + |\mathcal{A}_4| \leq \kappa \quad \text{ for all } \quad \varepsilon, \varepsilon' \leq \varepsilon_0. \end{equation*} Thus \eqref{gra1} is proved. \end{proof} We are now ready to get strong convergence for the nonlinear term $a(x,t,\nabla u)$. \begin{lemma}\label{convergence_a} Let $\{u_\eps\}_{\varepsilon>0}$ be solutions to the equation \eqref{approx}. Then, up to a subsequence, \begin{equation*}
a(x,t,\nablau_\eps) \to a(x,t,\nabla u) \quad \text{ strongly in} \quad L^s(Q_T) \quad \text{for all} \quad 1\leq s < 1+\frac{1}{(p-1)(d+1)}. \end{equation*} \end{lemma} \begin{proof}
From Lemma \ref{gradients} and the fact that $a$ is continuous with respect to the third variable, we have
\begin{equation}\label{est_a1}
a(x,t,\nabla u_\eps) \to a(x,t,\nabla u) \quad \text{ almost everywhere } \quad \text{ in } \quad Q_T.
\end{equation}
By using assumption \eqref{A2} and Lemma \ref{UniformBounds} we have for any $1\leq s < 1+\frac{1}{(p-1)(d+1)}$,
\begin{align}\label{est_a2}
\|a(x,t,\nablau_\eps)\|_{L^s(Q_T)}^s \leq C\int_{Q_T}|\varphi|^sdxdt + CK^{r}\int_{Q_T}|\nablau_\eps|^{s(p-1)}dxdt \leq C
\end{align}
thanks to $s(p-1) < p-\frac{d}{d+1}$ and $s < 1 + \frac{1}{(p-1)(d+1)} < \frac{p}{p-1} = p'$. From \eqref{est_a1} and \eqref{est_a2}, the Egorov theorem implies that $\{a(x,t,\nablau_\eps)\}_{\varepsilon>0}$ is precompact in $L^s(Q_T)$ for all $1\leq s < 1+\frac{1}{(p-1)(d+1)}$, which finishes the proof of Lemma \ref{convergence_a}. \end{proof} Due to the subcritial growth of the nonlinearity $g$ in \eqref{G2}, its convergence cannot be obtained in the same way as for $a$ in Lemma \ref{convergence_a}. A different approach should be used, for which we need the following lemma. \begin{lemma}\label{lem:g}
Let $\{u_\eps\}_{\varepsilon>0}$ be solutions to \eqref{approx}. Then
\begin{equation*}\label{bound_non_2}
\lim_{k\to\infty}\sup_{\varepsilon>0}\int_{\{|u_\eps|\geq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt = 0.
\end{equation*} \end{lemma} \begin{proof}
Since $T_k(u_\eps) = k$ for $u_\eps \geq k$, we have
\begin{equation}\label{l1}
\int_{\{|u_\eps|\geq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt \leq \frac{1}{k}\int_{Q_T}g_\eps(u_\eps,\nablau_\eps)T_k(u_\eps)dxdt.
\end{equation}
By integrating \eqref{e18} on $(0,T)$ and using \eqref{e17} we obtain, in particular,
\begin{equation}\label{l2}
\begin{aligned}
&\int_{Q_T}g_\eps(u_\eps,\nablau_\eps)T_k(u_\eps)dxdt\\
&\leq \int_{\Omega_0}S_k(u_{0,\varepsilon})dx + \int_{Q_T}|f_\eps T_k(u_\eps)|dxdt \\
&\quad + \|\mathbf v\|_{\infty}\int_{Q_T}|\nabla u_\eps||T_k(u_\eps)|dxdt + \|\text{div}\mathbf v\|_{\infty}\int_{Q_T}|u_\eps||T_k(u_\eps)|dxdt.
\end{aligned}
\end{equation}
Let $M>0$. We then have the following useful estimates
\begin{equation*}
0 \leq S_k(z) \leq M^2 + k|z|\chi_{\{|z|>M\}} \quad \text{ and } \quad |T_k(z)| \leq M + k\chi_{\{|z|>M\}}.
\end{equation*}
Therefore,
\begin{equation*}
\int_{\Omega_0}S_k(u_{0,\varepsilon})dx \leq M^2|\Omega_0| + k\int_{\{|u_{0,\varepsilon}|>M\}}|u_{0,\varepsilon}|dx,
\end{equation*}
\begin{equation*}
\int_{Q_T}|f_\eps T_k(u_\eps)|dxdt \leq M\|f_\eps\|_{L^1(Q_T)} + k\int_{\{|u_\eps|>M \}}|f_\eps|dxdt,
\end{equation*}
\begin{equation*}
\int_{Q_T}|\nablau_\eps||T_k(u_\eps)|dxdt \leq M\|\nablau_\eps\|_{L^1(Q_T)} + k\int_{\{|u_\eps|>M\}}|\nablau_\eps|dxdt,
\end{equation*}
and
\begin{equation*}
\int_{Q_T}|u_\eps||T_k(u_\eps)|dxdt \leq M\|u_\eps\|_{L^1(Q_T)} + k\int_{\{|u_\eps|>M\}}|u_\eps|dxdt.
\end{equation*}
Using these estimates in \eqref{l1} and \eqref{l2}, we get
\begin{equation}\label{l3}
\begin{aligned}
\int_{\{|u_\eps|\geq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt &\leq \frac{M^2}{k}|\Omega_0| + \frac{CM}{k}\left(\|f_\eps\|_{L^1(Q_T)} + \|u_\eps\|_{\LW{1}{1}{1}}\right)\\
&\quad +\int_{\{|u_{0,\varepsilon}|>M\}}|u_{0,\varepsilon}|dx + \int_{\{|u_\eps|>M \}}|f_\eps|dxdt\\
&\quad + \|\mathbf v\|_\infty\int_{\{|u_\eps|>M\}}|\nablau_\eps|dxdt + \|\text{div} \mathbf v\|_\infty\int_{\{|u_\eps|>M\}}|u_\eps|dxdt.
\end{aligned}
\end{equation}
Due to the uniform bound of $\{\|u_\eps\|_{L^1(Q_T)}\}_{\varepsilon>0}$ we have
\begin{equation*}
\lim_{M\to\infty}\sup_{\varepsilon>0}|\{(x,t)\in Q_T:\; u_\eps(x,t)>M \}| \leq \lim_{M\to\infty}\frac{1}{M}\sup_{\varepsilon>0}\|u_\eps\|_{L^1(Q_T)} = 0.
\end{equation*}
Therefore, from the fact that, as $\varepsilon\to 0$, $\|u_{0,\varepsilon} - u_0\|_{L^1(\Omega_0)} \to 0$, $\|f_\eps-f\|_{L^1(Q_T)} \to 0$ (by the constructions of $u_{0,\varepsilon}$ and $f_\eps$), and $\|u_\eps-u\|_{L^1(Q_T)} \to 0$ (due to Lemma \ref{L1convergence}), and $\|\nablau_\eps - \nabla u\|_{L^1(Q_T)} \to 0$ (due the fact that $\nablau_\eps \to \nabla u$ almost everywhere, and $\|\nablau_\eps\|_{L^q(Q_T)}$ is bounded uniformly in $\varepsilon$ for some $q>1$), we imply that the last four terms on the right hand side of \eqref{l3} become arbitrary small as $M$ tends to infinity.
Let $\kappa>0$ be arbitrary. We first choose $M$ large enough such that the sum of the last four terms on the right hand side of \eqref{l3} is smaller than $\kappa/2$. Then using the boundedness of $\|f_\eps\|_{L^1(Q_T)}$ and $\|u_\eps\|_{\LW{1}{1}{1}}$, there exists $k_0$ large enough, which is independent of $\varepsilon$, such that for all $k\geq k_0$,
\begin{equation*}
\frac{M^2}{k}|\Omega_0| + \frac{CM^2}{k}\left(\|f_\eps\|_{L^1(Q_T)} + \|u_\eps\|_{\LW{1}{1}{1}}\right) \leq \frac{\kappa}{2}.
\end{equation*}
Therefore,
\begin{equation*}
\sup_{\varepsilon>0}\int_{\{|u_\eps|\geq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt \leq \kappa \quad \text{ for all } \quad k\geq k_0,
\end{equation*}
which proves the claim \ref{bound_non_2}. \end{proof} \begin{lemma}[Strong convergence of the first order terms]\label{lemg_varepsilon}
As $\varepsilon \to 0$, there exists a subsequence of $\{g_\varepsilon(u_\eps , \nabla u_\eps )\}$ that converges to $g(u, \nabla u)$ almost everywhere in $Q_T$ and strongly in $L^1(Q_T)$.
\end{lemma} \begin{proof}
From Lemmas \ref{L1convergence} and \ref{gradients}, and the fact that $g$ is continuous with respect to the third and fourth variables, we have
\begin{equation*}
g_\eps(u_\eps,\nablau_\eps) = \frac{g(x,t,u_\eps,\nablau_\eps)}{1+\varepsilon|g(x,t,u_\eps,\nablau_\eps)|} \to g(x,t,u,\nabla u) \; \text{ almost everywhere in } \; Q_T.
\end{equation*}
To show that this convergence is in fact strong in $L^1(Q_T)$-topology, it's sufficient to show that the set $\{g_\eps(u_\eps,\nablau_\eps)\}_{\varepsilon>0}$ is weakly compact in $L^1(Q_T)$, or equivalently to show that
\begin{equation}\label{weak_compact_G}
\lim_{A\in \text{meas}( Q_T), |A|\to 0}\sup_{\varepsilon>0}\int_{A}|g_\eps(u_\eps,\nablau_\eps)|dxdt = 0
\end{equation}
where $A\in \text{meas}(Q_T)$ means that $A\subset Q_T$ is a measurable subset of $Q_T$. Indeed, we have for any $k\in \mathbb N$,
\begin{equation}\label{split_g}
\int_{A}|g_\eps(u_\eps,\nablau_\eps)|dxdt = \int_{A\cap \{|u_\eps|\leq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt + \int_{A\cap \{|u_\eps|\geq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt.
\end{equation}
For the second part, we have
\begin{equation}\label{est_g0}
\int_{A\cap \{|u_\eps|\geq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt \leq \int_{\{|u_\eps|\geq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt
\end{equation}
in which the right-hand side tends to $0$, as $k\to\infty$, uniformly in $\varepsilon$, thanks to Lemma \ref{lem:g}. It remains to estimate the first part in \eqref{split_g}. From the assumption \eqref{G2}, we have
\begin{equation}\label{est_g1}
\begin{aligned}
&\int_{A\cap \{|u_\eps|\leq k\}}|g_\eps(u_\eps,\nablau_\eps)|dxdt\\
&\leq \int_{A\cap \{|u_\eps|\leq k\}}|g(u_\eps,\nablau_\eps)dxdt\\
&\leq h(k)\int_{A\cap \{|u_\eps|\leq k\}}\left(|\gamma(x,t)| + |\nablau_\eps|^{\sigma}\right)dxdt\\
&\leq h(k)\int_{A}|\gamma(x,t)|dxdt + h(k)\left(\int_{A\cap \{|u_\eps|\leq k\}}|\nablau_\eps|^pdxdt\right)^{\frac{\sigma}{p}}|A|^{\frac{p-\sigma}{p}}
\end{aligned}
\end{equation}
where we used H\"older's inequality and the obvious estimate $|A\cap \{u_\eps \leq k\}| \leq |A|$ at the last step. By using H\"older's inequality again we find
\begin{equation}\label{est_g2}
h(k)\int_{A}|\gamma(x,t)|dxdt \leq h(k)\|\gamma\|_{L^{p'}(Q_T)}|A|^{\frac{p'-1}{p'}}
\end{equation}
where we recall that $p' = \frac{p}{p-1}$. From Lemmas \ref{prove_cond2} and \ref{UniformBounds} (with $q = 1$) we can estimate
\begin{equation}\label{est_g3}
\begin{aligned}
\int_{A\cap \{|u_\eps|\leq k\}}|\nablau_\eps|^pdxdt &\leq \sum_{j=0}^{k}\int_{B_j}|\nablau_\eps|^pdxdt\\
&\leq \sum_{j=0}^{k}\left(C_0 + C_1\int_{E_j}|\nablau_\eps|dxdt\right)\\
&\leq \sum_{j=0}^k\left(C_0 + C_1\|\nablau_\eps\|_{L^1(Q_T)}\right)\\
&\leq C(k+1).
\end{aligned}
\end{equation}
Inserting \eqref{est_g2} and \eqref{est_g3} into \eqref{est_g1} gives us
\begin{equation}\label{est_g4}
\int_{A\cap \{|u_\eps|\leq k \} }|g_\eps(u_\eps,\nablau_\eps)|dxdt \leq Ch(k)|A|^{\frac{p'-1}{p'}} + C(k+1)h(k)|A|^{\frac{p-\sigma}{p}}.
\end{equation}
Using \eqref{est_g0} and \eqref{est_g4} yields the desired estimate \eqref{weak_compact_G} which finishes the proof of Lemma \ref{lemg_varepsilon}. \end{proof} The last lemma is about the continuity in time. \begin{lemma}\label{lem:time_continuity}
The sequence $\{u_\eps\}_{\varepsilon>0}$ is a Cauchy sequence in $C([0,T];L^1(\Omega_t))$ as $\varepsilon\to 0$, and therefore $u\in C([0,T];L^1(\Omega_t))$. \end{lemma} \begin{proof}
Let $\varepsilon, \varepsilon'>0$, subtracting the equations for $u_\eps$ and $u_{\eps'}$ and taking $T_1(u_\eps - u_{\eps'})$ as the test function, we have
\begin{equation*}
\begin{aligned}
&\int_{\Omega_t}S_1(u_\eps - u_{\eps'})(t)dx + \int_0^t\int_{\Omega_s}(a(x,s,\nablau_\eps) - a(x,s,\nablau_{\eps'}))(\nablau_\eps - \nablau_{\eps'})\chi_{\{|u_\eps - u_{\eps'}|\leq 1\}}dxds\\
&\leq \int_{\Omega_0}S_1(u_{0,\varepsilon} - u_{0,\varepsilon'})dx - \int_0^t\int_{\Omega_s}(\mathbf v\cdot \nabla(u_\eps - u_{\eps'}) + (u_\eps - u_{\eps'})\text{div}\mathbf v)T_1(u_\eps - u_{\eps'})dxds\\
&-\int_0^t\int_{\Omega_s}(g_\eps(u_\eps,\nablau_\eps) - g_{\varepsilon'}(u_{\eps'},\nablau_{\eps'}))T_1(u_\eps-u_{\eps'})dxds + \int_0^t\int_{\Omega_s}(f_\eps - f_{\varepsilon'})T_1(u_\eps - u_{\eps'})dxds.
\end{aligned}
\end{equation*}
Using the assumption \eqref{A1} and $|T_1(z)| \leq 1$ and $S_1(z) \leq |z|$, we obtain
\begin{equation*}
\begin{aligned}
&\sup_{t\in(0,T)}\int_{\Omega_t}S_1(u_\eps - u_{\eps'})(t)dx\\ &\leq m_{\varepsilon,\varepsilon'}:= \|u_{0,\varepsilon}-u_{0,\varepsilon'}\|_{L^1(\Omega_0)} + \|\mathbf v\|_{\infty}\|\nabla u_\eps - \nabla u_{\eps'}\|_{L^1(Q_T)} + \|\text{div}\mathbf v\|_{\infty}\|u_\eps-u_{\eps'}\|_{L^1(Q_T)}\\
&+\|g_\eps(u_\eps,\nablau_\eps) - g_{\varepsilon'}(u_{\eps'},\nablau_{\eps'})\|_{L^1(Q_T)} + \|f_\eps-f_{\varepsilon'}\|_{L^1(Q_T)}
\end{aligned}
\end{equation*}
where clearly $\lim_{\varepsilon,\varepsilon'\to 0}m_{\varepsilon,\varepsilon'} = 0$. Now by using $|z|\chi_{\{|z|>1\}}/2 \leq S_1(z)\chi_{\{|z|>1\}}$ and $|z|^2\chi_{\{|z|\leq 1\}}/2 \leq S_1(z)\chi_{\{|z| \leq 1\}}$, we can estimate
\begin{equation*}
\begin{aligned}
\|u_\eps(t) - u_{\eps'}(t)\|_{L^1(\Omega_t)} &\leq \int_{\{|u_\eps(t) - u_{\eps'}(t)| \leq 1\}}|u_\eps(t) - u_{\eps'}(t)|dx + \int_{\{|u_\eps(t) - u_{\eps'}(t)| > 1\}}|u_\eps(t) - u_{\eps'}(t)|dx\\
&\leq |\Omega_t|^{1/2}\left(\int_{\{|u_\eps(t) - u_{\eps'}(t)| \leq 1\}}|u_\eps(t) - u_{\eps'}(t)|^2dx\right)^{1/2}+ 2\int_{\Omega_t}S_1(u_\eps - u_{\eps'})(t)dx\\
&\leq |\Omega_t|^{1/2}\left(2\int_{\Omega_t}S_1(u_\eps - u_{\eps'})(t)dx\right)^{1/2} + 2\int_{\Omega_t}S_1(u_\eps - u_{\eps'})(t)dx\\
&\leq \sqrt 2 |\Omega_t|^{1/2}\sqrt{m_{\varepsilon,\varepsilon'}} + 2m_{\varepsilon,\varepsilon'}.
\end{aligned}
\end{equation*}
Hence,
\begin{equation*}
\lim_{\varepsilon,\varepsilon'\to 0}\sup_{t\in(0,T)}\|u_\eps(t) - u_{\eps'}(t)\|_{L^1(\Omega_t)} = 0.
\end{equation*}
Therefore, $\{u_\eps\}_{\varepsilon>0}$ is a Cauchy sequence in $C([0,T];L^1(\Omega_t))$, and thus $u\in C([0,T];L^1(\Omega_t))$. \end{proof} We are now ready to prove the main theorem of this paper. \begin{proof}[Proof of Theorem \ref{thm:main}]
Let $\phi\in C([0,T];W^{1,\infty}_0(\Omega_t))\cap C^1((0,T);L^{\infty}(\Omega_t))$ be the test function to the approximate problem. We have
\begin{align*}
\int_{\Omega_T}u_\eps(T)\phi(T) dx - \int_{Q_T}u_\eps \partial_t \phi dxdt + \int_{Q_T}a(x,t,\nablau_\eps)\cdot\nabla\phi dxdt\\
- \int_{Q_T}u_\eps \mathbf v\cdot \nabla\phi dxdt + \int_{Q_T}g_\eps(u_\eps,\nablau_\eps)\phi dxdt\\
= \int_{\Omega_0}u_{0,\varepsilon}\phi(0)dx + \int_{Q_T}f_\eps \phi dxdt.
\end{align*}
By applying Lemmas \ref{L1convergence}, \ref{convergence_a}, \ref{lemg_varepsilon}, and \ref{lem:time_continuity}, and using \eqref{u0f}, we can pass to the limit as $\varepsilon\to 0$ in all the terms to obtain that
\begin{align*}
\int_{\Omega_T}u(T)\phi(T) dx - \int_{Q_T}u \partial_t \phi dxdt + \int_{Q_T}a(x,t,\nabla u)\cdot\nabla\phi dxdt\\
- \int_{Q_T}u \mathbf v\cdot \nabla\phi dxdt + \int_{Q_T}g(u,\nabla u)\phi dxdt\\
= \int_{\Omega_0}u_{0}\phi(0)dx + \int_{Q_T}f \phi dxdt
\end{align*}
or in other words, $u$ is a weak solution to \eqref{e1} on $(0,T)$. The proof of Theorem \ref{thm:main} is complete. \end{proof} \appendix \section{Existence of approximate solutions}\label{appendix1}
This section is devoted to a proof of the global existence of a weak solution to the approximate system \eqref{approx}. We follow the ideas in \cite{CNO17}.
We divide the time interval $[0;T]$ into $N\in \mathbb{N}$ smaller intervals $(t_j,t_{j+1})$ for $j=0,\ldots, N-1$ and define $\Delta:= \max_j|t_j-t_{j+1}|$. The points $t_j$ are chosen so that \begin{enumerate}
\item $\cup_{j=0,N-1}\Omega_{t_j}\times[t_j,t_{j+1}) \subset \widehat{\Omega}$,
\item $\Omega_{t_j}$ has smooth boundary for all $j\in \{0,\cdots,N-1\}$,
\item \eqref{A0}--\eqref{A1} hold for a.e. $x\in\Omega_{t_j}$ and for all $\xi\in\mathbb R^d$,
\item $t_j$ are Lebesgue points of the map $[0,T]\ni t\mapsto a(\cdot,t,\cdot)\in L^1(\widehat{\Omega} \times B(0,R))$ for any $R>0$, where $B(0,R)\subset \mathbb R^d$ is the open ball centered at zero with radius $R$,
\item $\Delta \to 0$ as $N\to\infty$. \end{enumerate} \newcommand{\widehat}{\widehat}
We define the extended function $\widehat{f}_\varepsilon:\widehat{Q} \to \mathbb R$ as $\widehat{f}_\varepsilon(x,t) = f_\eps(x,t)$ if $(x,t)\in \widehat{Q}$ and $\widehat{f}_\varepsilon(x,t) = 0$ otherwise. Let us denote by $I_j=[t_j,t_{j+1})$. For each $j\in \{0,\ldots, N-1\}$ we consider the following equation \begin{equation}\label{timeapprox} \begin{cases} \partial_t w^{(j)} - \text{div}(a(x,t_j,\nabla w^{(j)})) + \text{div}(w^{(j)}\mathbf v) + g_\eps(w^{(j)}, \nabla w^{(j)}) = \widehat{f}_\varepsilon, &x\in \Omega_{t_j}, \; t\in I_j,\\ w^{(j)}(x,t) = 0, &x\in \partial\Omega_{t_j}, t\in I_j,\\ w^{(j)}(x,t_j) =\begin{cases} \lim_{t\to t_j^{-}}w^{(j-1)}(\zeta_{t_j - t_{j-1}}^{-1}(x), t), &x\in \Omega_{t_j}\cap \Omega_{t_{j-1}},\\ 0&x\in \Omega_{t_j}\setminus \Omega_{t_{j-1}}.\end{cases} \end{cases} \end{equation} If $t_0=0$ then we let $w^{(0)}(x,0)=u_{0,\varepsilon}(x)$. Note that we have the semigroup property $\zeta_{t+s} = \zeta_t\circ \zeta_s$ and the domains $\Omega_{t_j} = \zeta_{t_j - t_{j-1}}^{-1}(\Omega_{t_{j-1}})$ for $j=\overline{0,N-1}$.
For any fixed $j\in \{0,\cdots,N-1\}$, by classical results, see e.g. \cite{Lio69}, one obtains the existence of a solution $w^{(j)}\in L^1(\Omega_{t_j}\times I_j)\cap L^p(I_j;W^{1,p}_0(\Omega_{t_j}))$ with $\partial_tw^{(j)}\in L^{p'}(I_j;W^{-1,p'}(\Omega_{t_j}))$ of \eqref{timeapprox}. Moreover, $w^{(j)}\in C(I_j; L^1(\Omega_{t_j}))$. Denote by $$ \Omega^\Delta:=\{(x,t): x\in\Omega_{t_j}, t\in I_j, j=0,\cdots,N-1\}=\bigcup\limits_{j=0,\cdots,N-1}\Omega_{t_j}\times I_j. $$ From \cite[Lemma 3.4]{CNO17}, we know that as $\Delta\to 0$, $\Omega^\Delta$ converges to $Q_T$ in Hausdorff sense, and as a consequence $\chi_{\Omega^\Delta}$ converges strongly to $\chi_{Q_T}$ in $L^s(\widehat{Q})$ for all $s<\infty$. We now glue the solutions $w^{(j)}(x,t)$ of \eqref{timeapprox} together and define the approximate solutions \begin{equation*}\label{appsol} w^{\Delta}: \widehat{Q}\to \mathbb R\quad \text{with} \quad w^{\Delta}(x,t) = \sum_{j=0}^{N-1}\chi_{I_j}(t)\chi_{\Omega_{t_j}}(x)w^{(j)}(x,t) \end{equation*} for $(x,t)\in \widehat{Q}$. The function $w^{(j)}(x,t)\chi_{\Omega_{t_j}}(x)$ in the formulae above is the function which coincides with $w^{(j)}(x,t)$ in $\Omega_{t_j}$ and is equal to zero outside $\Omega_{t_j}$, that means $w^{\Delta}\equiv 0$ in $\widehat{Q}\backslash \Omega^{\Delta}$.
In the sequel, we prove some {\it a priori} estimates of $w^\Delta$ which are independent of $\Delta$, thus allowing us to pass to the limit $\Delta \to 0$. In conclusion we have $w^\Delta \to v$ where $v$ is a solution to \eqref{approx}. We are ready to give a proof of Theorem \ref{thm:approximate}. \begin{proof}[Proof of Theorem \ref{thm:approximate}]
The proof follows the ideas of \cite{CNO17}, so we only sketch some main steps here. For simplicity we set $$ G_\varepsilon(u,\nabla u)=\mathrm{div}(u\mathbf v)+g_\varepsilon(u,\nabla u). $$
{\bf Step 1: Establishing a priori estimates of $w^{\Delta}$.}
First, we will prove $w^{\Delta}\in L^\infty(\Omega^\Delta)$ for any $t>0$. It is enough to prove the estimate in $\Omega_0\times (0, t_1)$.
Let $k \geq 2$ be arbitrary. By choosing $|w^{\Delta}|^{k-2}w^{\Delta}$ as a test function of \eqref{approx}, we have
\begin{equation}\label{S.1}
\begin{aligned}
\dfrac{d}{dt}\int_{\Omega_0}|w^{\Delta}|^kdx&+k(k-1)\int_{\Omega_0}a(x,0,\nabla w^{\Delta})\cdot\nabla w^{\Delta}|w^{\Delta}|^{k-2}dx\\
&=-k\int_{\Omega_0}G_\varepsilon(w^\Delta,\nabla w^\Delta)|w^\Delta|^{k-2}w^\Delta dx+k\int_{\Omega_0}f_\varepsilon|w^\Delta|^{k-2}w^\Delta dx.
\end{aligned}
\end{equation}
From \eqref{A3}, equation \eqref{S.1} becomes
\begin{equation*}\label{S.2}
\begin{aligned}
\dfrac{d}{dt}\int_{\Omega_0}|w^\Delta|^kdx&+k(k-1)\alpha\left(\dfrac{p}{p+k-2}\right)^p\int_{\Omega_0}|\nabla (w^\Delta)^{\frac{k+p-2}{p}}|^pdx\\
&\leq k\int_{\Omega_0}|G_\varepsilon(w^\Delta,\nabla w^\Delta)||w^\Delta|^{k-1}dx+k\int_{\Omega_0}|f_\varepsilon||w^\Delta|^{k-1}dx.
\end{aligned}
\end{equation*}
By integrating the inequality above from 0 to $t_1$ we have
$$\int_{\Omega_0}|w^\Delta(t)|^{k}dx\leq \int_{\Omega_0}|u_{0,\varepsilon}|^{k}dx+k\int_0^{t_1}\int_{\Omega_0}(|G_\varepsilon|+|f_\varepsilon|)|w^\Delta|^{k-1}dxdt.$$
Fix $\xi>t_1$. By using H\"{o}lder and Young inequalities, we have
\begin{align*}
(1-&t_1\xi^{-\frac{k}{k-1}})\sup\limits_{t\in(0,t_1)}\int_{\Omega_0}|w^\Delta(t)|^kdx+\xi^{-\frac{k}{k-1}}\int_0^{t_1}\int_{\Omega_0}|w^\Delta(t)|^kdxdt\\
&\leq\sup\limits_{t\in(0,t_1)}\int_{\Omega_0}|w^\Delta(t)|^kdx\leq \int_{\Omega_0}|u_{0,\varepsilon}|^{k}dx+k\int_0^{t_1}\int_{\Omega_0}(|G_\varepsilon|+|f_\varepsilon|)|w^\Delta|^{k-1}dxdt.\\
&\leq\int_{\Omega_0}|u_{0,\varepsilon}|^kdx+\xi^k\int_0^{t_1}\int_{\Omega_0}(|G_\varepsilon|+|f_\varepsilon|)^kdxdt+\xi^{-\frac{k}{k-1}}\int_0^{t_1}\int_{\Omega_0}|w^\Delta(t)|^kdxdt.
\end{align*}
Hence
{\small$$ \left(1-t_1\xi^{-\frac{k}{k-1}}\right)^{1/k}\sup\limits_{t\in(0,t_1)}\left(\int\limits_{\Omega_0}|w^\Delta(t)|^kdx\right)^{1/k}\leq \left(\int\limits_{\Omega_0}|u_{0,\varepsilon}|^kdx+\xi^k\int\limits_0^{t_1}\int\limits_{\Omega_0}(|G_\varepsilon|+|f_\varepsilon|)^kdxdt\right)^{1/k}. $$}
Letting $k\to\infty$, we obtain
\begin{equation*}\label{S6}
\|w^\Delta(t)\|_{L^\infty(\Omega_0)}\leq \|u_{0,\varepsilon}\|_{L^\infty(\Omega_0)}+\xi(\|G_\varepsilon\|_{L^\infty(0,t_1;\Omega_0)}+\|f_\varepsilon\|_{L^\infty(0,t_1;\Omega_0)}),\,\forall \xi>t_1.
\end{equation*}
Second, by using the same arguments in \cite[Lemma 3.6 and 3.9]{CNO17}, we obtain two results respectively, for precisely there is some constant $C>0$ depending only on $Q_T$ such that
\begin{equation*}\label{S7}
\sum\limits_{j=0}^{N-1}\int_{t_j}^{t_{j+1}}\int_{\Omega_{t_j}}|\nabla w^\Delta|^p dx\leq C,
\end{equation*}
and let $0<j\leq N$ be fixed, then
\begin{equation}\label{S10}
w^{(j)}_t\chi_{\Omega_{t_j}}\in L^{p'}(I_j; W^{-1,p'}(\Omega_{t_j})).
\end{equation}
{\bf Step 2: Passing to the limits.} From the above estimates, and recalling that $w^{\Delta} \equiv 0$ in $\widehat{Q}\backslash \Omega^{\Delta}$, we will show that there exists a subsequence of $\{w^\Delta\}_N$, also denoted by $\{w^\Delta\}_N$, such that
\begin{itemize}
\item[(i)] $w^\Delta\rightharpoonup v$ in $L^p(0,T;W^{1,p}(\widehat{\Omega}))\cap L^\infty(\widehat{Q})$,
\item[(ii)] $a(x,t,\nabla w^{\Delta})\rightharpoonup a(x,t,\nabla v)$ in $L^{p'}(\widehat{Q})$,
\item[(iii)] $G_\varepsilon(w^\Delta,\nabla w^\Delta)\rightharpoonup G_\varepsilon(v,\nabla v)$ in $L^{p'}(\widehat{Q})$,
\end{itemize}
where $ a(x,t,\nabla w^{\Delta}):=\sum\limits_{j=0}^{N-1}\chi_{[t_j,t_{j+1})}(t)a(x,t_j,\nabla w^{(j)})\chi_{\Omega_{t_j}}(x)$. The limit (i) is straightforward. The limits (ii) and (iii) are proved in the following.
\begin{itemize}
\item \underline{Proof of the limit (ii)}. We only give the main ideas while refer the reader to \cite[Lemma 3.10 and 4.7]{CNO17} for more details.
At first, we show that $\{w^{\Delta}|_{C}\}_{\Delta>0}$ is precompact in $L^1(C)$ where $C = (s_1,s_2)\times K$ is an open cylinder contained in $Q_T$ with $0\leq s_1<s_2 \leq T$ such that $\overline{C}\subset Q_T$. Since $\Omega^\Delta \to Q_T$, we can choose $N$ large enough such that $\overline{C}\subset \Omega^\Delta$. Moreover, if $(s_1,s_2)\cap I_j\ne \emptyset$ then $\overline{K}\subset\Omega_{t_j}$. Therefore,
\begin{equation*}
w^{\Delta}|_C(x,t) = \sum_{j=0}^{N-1}(\chi_{I_j}(t)\chi_{\Omega_{t_j}}(x)w^{(j)}(x,t))|_{C}.
\end{equation*}
It follows that $\{w^{\Delta}|_C\}$ is bounded in $L^p(s_1,s_2;W^{1,p}(K))$. We will now show
\begin{equation}\label{limit}
\int_{s_1}^{s_2-h}\|w^\Delta(t+h)-w^\Delta(t)\|_{W^{-1,p'}(K)}dt\to 0,\quad\text{as }h\to 0^+
\end{equation}
uniformly in $N$. For $z\in W^{1,p}_0(K)$ we can use zero extension to have $z\in W^{1,p}_0(\Omega_{t_j})$ and $\|z\|_{W^{1,p}_0(K)} = \|z\|_{W^{1,p}_0(\Omega_{t_j})}$ thanks to $\overline{K}\subset \Omega_{t_j}$. Thus we can estimate for $t\in I_j^h\cap (s_1,s_2-h)$ with $I_j^h:= [t_j,t_{j+1}-h)$
\begin{equation}\label{t3}
\begin{aligned}
&\int_{I_j^h\cap (s_1,s_2-h)}\left|\int_{K}(w^{\Delta}(t+h) - w^{\Delta}(t))\cdot zdx\right|dt\\
&\leq\int_{I_j^h\cap(s_1,s_2-h)}\int_{t}^{t+h}\int_{K}|w^{(j)}_t\cdot z| dxdsdt\\
&\leq h^{1/p}|I_j|\|w_t^{(j)}\|_{L^{p'}(I_j;W^{-1,p'}(\Omega_{t_j}))}\|z\|_{W^{1,p}_0(K)}.
\end{aligned}
\end{equation}
For $t\in (I_j\backslash I_j^h)\cap (s_1,s_2-h)$,
\begin{equation}\label{t4}
\begin{aligned}
\qquad&\int_{(I_j\backslash I_j^h)\cap (s_1,s_2-h)}\left|\int_{K}(w^{\Delta}(t+h) - w^{\Delta}(t))\cdot zdx\right|dt\\
&\leq\int\limits_{(I_j\backslash I_j^h)\cap(s_1,s_2-h)}\left(\|w^{(j+1)}(t+h)\|_{W^{-1,p'}(\Omega_{t_{j+1}})} + \|w^{(j)}(t)\|_{W^{-1,p'}(\Omega_{t_j})} \right)\|z\|_{W^{1,p}_0(K)}dt\\
&\leq h\left(\|w^{(j+1)}\|_{C(I_j;W^{-1,p'}(\Omega_{t_{j+1}}))}+\|w^{(j)}\|_{C(I_j;W^{-1,p'}(\Omega_{t_j}))}\right)\|z\|_{W^{1,p}_0(K)}.
\end{aligned}
\end{equation}
From \eqref{t3} and \eqref{t4} we can write
\begin{gather*}
\int_{s_1}^{s_2-h}\|w^{\Delta}(t+h)-w^{\Delta}(t)\|_{W^{-1,p'}(K)}dt\\ = \sum_{j=0}^{N-1}\int\limits_{I_j\cap(s_1,s_2-h)}\|w^{\Delta}(t+h)-w^{\Delta}(t)\|_{W^{-1,p'}(K)}dt
\end{gather*}
to obtain the desired limit \eqref{limit}.
By using \cite[Theorem 5]{Si86}, $\{w^\Delta|_{C}\}_{\Delta >0}$ is relatively compact in $L^1(C)$. For any compact subset of $Q_T$, it can be covered by a finite number of open cylinders, then by applying a diagonal procedure, we deduce that the sequence $\{w^\Delta\}_{\Delta>0}$ is relatively compact in $L^1_{\text{loc}}(Q_T)$. Together with the uniform bound of $w^\Delta$ in $L^\infty(\Omega^\Delta)$ and $w^{\Delta}\equiv 0$ in $\widehat{Q}\backslash \Omega^{\Delta}$, we obtain the following lemma.
\begin{lemma}\label{strgcon}
There exists a subsequence of $\{w^\Delta\}_{\Delta>0}$ which converges strongly in $L^1(\widehat{Q})$.
\end{lemma} It's clear that $a(x,t, \nabla w^{\Delta}) \rightharpoonup \overline{a}$ in $L^{p'}(\widehat{Q})$ for some $\overline{a}$. Moreover, we have the following result.
\begin{lemma}
Let $\phi$ be smooth and such that $\text{supp}\phi\subset \Omega^\Delta\cap ([0,T]\times \mathbb R^d)$. Then
\begin{equation*}\label{S0}
\underset{N\to\infty}{\lim\sup}\int_0^T\int_{\Omega^\Delta_t} a(x,t,\nabla w^\Delta)\cdot \nabla w^\Delta\phi dxdt\leq \int_0^T\int_{\Omega_t}\overline{a}\cdot\nabla w\phi dxdt
\end{equation*}
where $\Omega_t^{\Delta} = \Omega_{t_j}$ if $t\in [t_j,t_{j+1})$, $j=0,\ldots, N-1$.
\end{lemma}
Then, we can now use the same arguments as in \cite[Lemma 4.8]{CNO17} to obtain $$\overline{a}(x,t,\nabla v)=a(x,t,\nabla v) \text{ a.e. in } Q_T,$$
hence (ii).
\item \underline{Proof of limit (iii)}. From the boundedness of $G_\varepsilon$, we have
\begin{equation*}
G_\varepsilon(w^\Delta, \nabla w^{\Delta}) \rightharpoonup \overline{g} \quad \text{in}\quad L^{p'}(\widehat{Q}).
\end{equation*}
It remains to prove $\overline{g}=G_\varepsilon(v,\nabla v)$ a.e. in $Q_T$. Since $G_\varepsilon$ is a continuous function with respect to $w$ and $\nabla w$, by classical results (see e.g. \cite{Lio69}), the sequence $G_\varepsilon(w^\Delta,\nabla w^\Delta)\rightharpoonup G_\varepsilon(v,\nabla v)$ in $L^1(\widehat{Q})$ if we show that the sequence $\{\nabla w^\Delta\}$ converges to $\nabla v$ a.e. as $\Delta\to 0$. This property is obtained as we show that $\{\nabla w^\Delta\}$ is a Cauchy sequence in measure, see \cite{RE65}, i.e. for all $\mu>0$
\begin{equation}\label{S19}
\text{meas}\{(x,t)\in \widehat{Q}:|\nabla w^\Delta-\nabla w^{\Delta'}|\geq\mu\}\to 0,\quad \text{as }\Delta, \Delta'\to 0.
\end{equation}
Let us denote by $\mathcal A$ the subset of $\widehat{Q}$ involved in \eqref{S19}. Let $k>0$ and $\eta>0$, we have
\begin{equation*}\label{S20}
\mathcal A\subset \mathcal A_1\cup \mathcal A_2\cup \mathcal A_3\cup\mathcal A_4,
\end{equation*}
where
\begin{align*}
\mathcal A_1&=\{|\nabla w^\Delta|\geq k\},\\
\mathcal A_2&=\{|\nabla w^{\Delta'}|\geq k\},\\
\mathcal A_3&=\{|w^\Delta-w^{\Delta'}|\geq \eta\},\\
\mathcal A_4&=\{|\nabla w^\Delta-\nabla w^{\Delta'}|\geq \mu, |\nabla w^\Delta|\leq k,|\nabla w^{\Delta'}|\leq k,|w^\Delta-w^{\Delta'}|\leq \eta\}.
\end{align*}
By repeating the arguments in Lemma \ref{gradients} we have \eqref{S19}.
\end{itemize}
{\bf Step 3: Recovery initial conditions.}
We refer the reader to \cite[Proposition 4.9]{CNO17} for a proof that $v$ in {\bf Step 2} is a weak solution of problem \eqref{approx} and furthermore, $v(t)\to u_{0,\varepsilon}$ a.e. as $t\to 0$. \end{proof} \section{An Aubin-Lions lemma in moving domains}\label{appendix2} This appendix provides a proof of the Aubin-Lions lemma in Lemma \ref{AL-lemma}. We follow the ideas from \cite{Mou16}. For any $\delta>0$, we write $\Omega_t^\delta = \{x\in \Omega_t: d(x,\partial\Omega_t)> \delta\}$ and \[ Q_T^\delta = \bigcup\limits_{t\in (0,T)}\Omega_t^\delta \times \{t\}. \] Let $\varphi: \mathbb R^d \to \mathbb R$ be a $C^\infty_c$ function such that \begin{itemize}
\item $\varphi$ is radially symmetric;
\item $\mathrm{supp}(\varphi)\subset B(0,1)$;
\item $\int_{\mathbb R^d}\varphi(x)dx = 1$. \end{itemize} \newcommand{\varphi^\varepsilon}{\varphi^\varepsilon} We define the scaled mollifier as $\varphi^\varepsilon(x) = \varepsilon^{-d}\varphi(x/\varepsilon)$ and for any distribution $g\in \mathcal D'(Q_T)$ we have the convolution \begin{equation*} (g(\cdot,t)*\varphi^\varepsilon)(x) = \int_{\Omega_t}g(t,x-y)\varphi^\varepsilon(y)dy = \int_{\mathbb R^d}g(x-y,t)\varphi^\varepsilon(y)dy \end{equation*} defined on $\Omega_t^\varepsilon$, and consequently on $\Omega_t^\delta$ for all $\delta < \varepsilon$ by trivial extension. \begin{lemma}\label{lemma_lim}
Let $\delta > 0$. If $\{\nabla u_n\}$ is bounded in $L^p(Q_T)$ for some $p\geq 1$, then
\begin{equation}\label{lim}
\lim_{\varepsilon\to 0}\sup_{n\geq 1}\|u_n*\varphi^\varepsilon - u_n\|_{L^p(Q_T^\delta)} = 0.
\end{equation} \end{lemma} \begin{proof}
By definition and the fact that $\int_{\mathbb R^d}\varphi^\varepsilon(y)dy = 1$ we have
\begin{equation*}
\begin{aligned}
|(u_n*\varphi^\varepsilon)(x,t) - u_n(x,t)| &\leq \int_{\mathbb R^d} |u_n(x-y,t)-u_n(x,t)||\varphi^\varepsilon(y)|dy\\
&\leq \int_{\mathbb R^d}|y||\nabla u_n(s(x-y) + (1-s)x,t)||\varphi^\varepsilon(y)|dy\\
&= \int_{|y| \leq \varepsilon}|y||\nabla u_n(s(x-y) + (1-s)x,t)||\varphi^\varepsilon(y)|dy.
\end{aligned}
\end{equation*}
Integrating the above inequality over $Q_T^\delta$ and using the fact that $\{\nabla u_n\}$ is bounded in $L^p(Q_T)$, we get
\begin{equation*}
\sup_{n\geq 1}\|u_n *\varphi^\varepsilon - u_n\|_{L^p(Q_T^\delta)} \leq C\int_{Q_T}\int_{|y|\leq \varepsilon}|y||\varphi^\varepsilon(y)|dydxdt,
\end{equation*}
and consequently \eqref{lim} as $\varepsilon\to 0$. \end{proof} \begin{lemma}[A local compactness lemma]\label{local}
Assume all the conditions in Lemma \ref{AL-lemma} are fulfilled. Then there exists $\delta_0>0$ small enough such that for any $\delta\leq \delta_0$, $\{u_n\}$ is precompact in $L^s(Q_T^\delta)$ for all $1\leq s < p$. \end{lemma} \begin{proof}
We first prove that for any fixed $\varepsilon<\delta_0$, the sequence $\{u_n*\varphi^\varepsilon \}_{n}$ is precompact in $L^1(Q_T^\delta)$. Indeed, using the condition \eqref{time_derivative}, and the fact that $\varphi^\varepsilon$ is radially symmetric we have
\begin{equation*}
\begin{aligned}
\left|\int_{Q_T} \psi\partial_t(u_n*\varphi^\varepsilon)dxdt \right| & = \left|\int_{Q_T} (\psi *\varphi^\varepsilon)\partial_t u_n dxdt \right|\\
&\leq C\sup_{t\in (0,T)}\|\psi*\varphi^\varepsilon\|_{H^m(\Omega_t)}\\
&\leq C_\varepsilon\sup_{t\in(0,T)}\|\psi\|_{L^\infty(\Omega_t)}\\
&\leq C_\varepsilon\|\psi\|_{L^\infty(Q_T)}.
\end{aligned}
\end{equation*}
By duality, we get that $\{\partial_t (u_n*\varphi^\varepsilon) \}_{n}$ is bounded in $L^1(Q_T^\delta)$. From the assumption of $u_n$, we obtain that $\{u_n*\varphi^\varepsilon\}_{n}$ and $\{\nabla(u_n*\varphi^\varepsilon) \}_n$ are bounded in $L^1(Q_T^\delta)$. Therefore we have $\{u_n*\varphi^\varepsilon \}_n$ is bounded in $W^{1,1}(Q_T^\delta)$, and thus, by the compact embedding $W^{1,1}(Q_T^\delta) \hookrightarrow L^1(Q_T^\delta)$ we get that $\{u_n*\varphi^\varepsilon\}_n$ is a Cauchy sequence in $L^1(Q_T^\delta)$.
By applying estimate \eqref{lim} in Lemma \ref{lemma_lim} and by writing
\begin{equation*}
\begin{aligned}
&\|u_n - u_m\|_{L^1(Q_T^\delta)}
\\ &\leq \|u_n - u_n*\varphi^\varepsilon\|_{L^1(Q_T^\delta)} + \|u_n*\varphi^\varepsilon - u_m*\varphi^\varepsilon\|_{L^1(Q_T^\delta)} + \|u_m*\varphi^\varepsilon - u_m\|_{L^1(Q_T^\delta)}
\end{aligned}
\end{equation*}
we obtain that $\{u_n\}_n$ is precompact in $L^1(Q_T^\delta)$. Using the boundedness of $\{u_n\}_n$ in $L^p(Q_T^\delta)$ and interpolation we obtain the precompactness of $\{u_n\}_n$ in $L^s(Q_T^\delta)$ for all $1\leq s < p$. \end{proof}
We will also use the following result from \cite{Mou16}. \begin{lemma}\cite[Proposition 8]{Mou16}\label{Mou}
If $\{u_n\}_n$ and $\{\nabla u_n\}_n$ are bounded in $L^p(Q_T)$ and $\{u_n\}_n$ is precompact in $L^p(Q_T^\delta)$ for all $\delta < \delta_0$, then $\{u_n\}_n$ is precompact in $L^p(Q_T)$. \end{lemma}
We have all the ingredients to prove Lemma \ref{AL-lemma}. \begin{proof}[Proof of Lemma \ref{AL-lemma}]
The proof follows directly from Lemmas \ref{local} and \ref{Mou} above. \end{proof}
\par{\bf Acknowledgements:} We would like to thank the referees for their helpful and constructive comments, which help to improve the presentation of this paper.
This research is funded by Thuyloi University Foundation for Science and Technology under grant number TLU.STF.19-04.\\ The third author is supported by the International Research Training Group IGDK 1754 "Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures", funded by the German Research Council (DFG) project number 188264188/GRK1754 and the Austrian Science Fund (FWF) under grant number W 1244-N18. This work is partially supported by NAWI Graz.
\end{document} | arXiv | {
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\begin{document}
\begin{abstract}
Recently, Benedetti et al.~introduced an Ehrhart-like polynomial associated to
a graph. This polynomial is defined as the volume of a certain flow polytope
related to a graph and has the property that the leading coefficient is the
volume of the flow polytope of the original graph with net flow vector
$(1,1,\dots,1)$. Benedetti et al.~conjectured a formula for the Ehrhart-like
polynomial of what they call a caracol graph. In this paper their conjecture
is proved using constant term identities, labeled Dyck paths, and a cyclic
lemma. \end{abstract}
\title{Volumes of flow polytopes related to caracol graphs}
\section{Introduction}
The main objects in this paper are flow polytopes, which are certain polytopes associated to acyclic directed graphs with net flow vectors. Flow polytopes have interesting connections with representation theory, geometry, analysis, and combinatorics. A well known flow polytope is the Chan--Robbins--Yuen polytope, which is the flow polytope of the complete graph $K_{n+1}$ with net flow vector $(1,0,\dots,0,-1)$. Chan, Robbins, and Yuen \cite{CRY2000} conjectured that the volume of this polytope is a product of Catalan numbers. Their conjecture was proved by Zeilberger \cite{Zeilberger1999} using the Morris constant term identity \cite{Morris1982}, which is equivalent to the famous Selberg integral \cite{Selberg1944}.
Since the discovery of the Chan--Robbins--Yuen polytope, researchers have found many flow polytopes whose volumes have nice product formulas, see \cite{Benedetti2019, CKM2017,Meszaros2015, MMR2017,MMS2019, MSW2019, Yip2019} and references therein. In this paper we add another flow polytope to this list by proving a product formula for the volume of flow polytope coming from a caracol graph, which was recently conjectured by Benedetti et al.~\cite{Benedetti2019}. In order to state our results we introduce necessary definitions.
We denote $[n]:=\{1,2,\dots,n\}$. Throughout this paper, we only consider connected directed graphs in which every vertex is an integer and every directed edge is of the form $(i,j)$ with $i<j$.
Let $G$ be a directed graph on vertex set $[n+1]$ with $m$ directed edges. We allow $G$ to have multiple edges but no loops. Let $\vec{a}=(a_1,a_2,\dots ,a_n)\in \ZZ^{n}$. An $m$-tuple $(b_{ij})_{(i,j)\in E}\in \mathbb{R}_{\geq 0}^m$ is called an \emph{$\vec{a}$-flow of $G$} if \[\sum_{(i,j)\in E}b_{ij}(\vec{e}_i-\vec{e}_j)=\left(a_1,\dots,a_n, -\sum_{i=1}^n a_i\right),\] where $\vec{e}_i$ is the standard basis vector in $\mathbb{R}^{n+1}$ with a one in the $i$th entry and zeroes elsewhere. The \emph{flow polytope $\FF_{G}(\vec{a})$ of $G$ with net flow $\vec{a}$} is defined as the set of all $\vec{a}$-flows of $G$.
In this paper we consider the following two graphs, see Figures~\ref{fig:PS} and \ref{fig:Car}: \begin{itemize} \item The \emph{Pitman-Stanley graph} $\PS_{n+1}$ is the graph with vertex set $[n+1]$ and edge set \[ \{(i,i+1): i=1,2,\dots, n\}\cup\{(i,n+1):i=1,2,\dots ,n-1\}. \] \item The \emph{Caracol graph} $\Car_{n+1}$ is the graph with vertex set $[n+1]$ and edge set \[ \{(i,i+1):i=1,2,\dots,n\}\cup\{(1,i):i=3,4,\dots,n\}\cup\{(i,n+1):i=2,3,\dots,n-1\}. \] \end{itemize}
\begin{figure}
\caption{The Pitman-Stanley graph $\PS_{n+1}$.}
\label{fig:PS}
\end{figure}
\begin{figure}
\caption{The caracol graph $\Car_{n+1}$.}
\label{fig:Car}
\end{figure}
We note that the flow polytope $\FF_{\PS_{n+1}}(a_1,\dots,a_n)$ are affinely equivalent to the polytope \[
\Pi_{n-1}(a_1,\dots,a_{n-1}):= \{(x_1,\dots,x_{n-1}): x_i\ge0,
x_1+\dots+x_{i}\le a_1+\dots+a_{i}, 1\le i\le n-1\}, \] considered in \cite{PitmanStanley}. Pitman and Stanley \cite{PitmanStanley} found volume formulas for certain polytopes, which can be restated as normalized volumns of
flow polytopes as follows: \begin{align}
\label{eq:PS11}
\vol\FF_{\PS_{n+1}}(a,b^{n-2},d) &= a(a+(n-1)b)^{n-2}, \\
\label{eq:PS12}
\vol\FF_{\PS_{n+1}}(a,b^{n-3},c,d) &= a(a+(n-1)b)^{n-2} + (n-1)a(c-b)(a+(n-2)b)^{n-3}, \\
\label{eq:PS13}
\vol\FF_{\PS_{n+1}}(a,b^{n-m-2},c,0^{m-1},d) &= a\sum_{j=0}^m \binom nj (c-(m+1-j)b)^j(a+(n-1-j)b)^{n-j-2}, \end{align} where $b^k$ means the sequence $b,b,\dots,b$ of $k$ $b$'s. We note that $\vol\FF_{\PS_{n+1}}(a_1,\dots,a_{n})$ is independent of $a_n$.
In \cite{Benedetti2019}, Benedetti et al.~introduced combinatorial models called gravity diagrams and unified diagrams to compute volumes of flow polytopes. Using these models they showed \begin{align}
\label{eq:5} \vol\FF_{\Car_{n+1}}(a^n) &=C_{n-2}a^n n^{n-2},\\ \vol\FF_{\Car_{n+1}}(a,b^{n-1})&=C_{n-2} a^{n-2}(a+(n-1)b)^{n-2}, \end{align} where $C_k:=\frac{1}{k+1}\binom{2k}{k}$ is the $k$th Catalan number.
For a positive integer $k$ and a directed graph $G$ on $[n+1]$, let $\GG(k)$ be the directed graph obtained from $G$ by adding a new vertex $0$ and $k$ multiple edges $(0,i)$ for each $1\le i\le n+1$. Then we define \begin{equation}
\label{eq:EG} E_G(k) = \vol\mathcal{F}_{\GG(k)}(1,0^{n}). \end{equation}
In \cite{Benedetti2019}, Benedetti et al.~showed that $E_G(k)$ is a polynomial function in $k$. Therefore we can consider the polynomial $E_G(x)$. They also showed that these polynomials $E_G(x)$ have similar properties as Ehrhart polynomials. For example, the leading coefficient of $E_G(x)$ is the normalized volume of $\FF_G(1^n)$. For this reason, they called $E_G(x)$ an Ehrhart-like polynomial. In the same paper they proved the following theorem.
\begin{theorem}\label{thm:PS} We have
\[
E_{\PS_{n+1}}(k)=\frac{1}{kn-1}\binom{(k+1)n-2}{n}.
\] \end{theorem}
Our main result is the following theorem, which was conjectured in \cite{Benedetti2019}. \begin{theorem}\label{thm:main} We have \[E_{\Car_{n+1}}(k)=\frac{1}{kn+n-3}\binom{kn+2n-5}{n-1}\binom{n+k-3}{k-1}.\] \end{theorem}
In this paper we prove both Theorems~\ref{thm:PS} and \ref{thm:main}.
The remainder of this paper is organized as follows.
In Section~\ref{sec:const-term-ident} we use the Lidskii volume formula to interpret $E_{\Car_{n+1}}(k)$ as a Kostant partition function, which is equal to the constant term of a Laurent series. In Section~\ref{sec:labeled-dyck-paths} we introduce labeled Dyck paths and show that the constant term is equal to the number of certain labeled Dyck paths. In Section~\ref{sec:cyclic-lemma} we enumerate these labeled Dyck paths using a cyclic lemma. In Section~\ref{sec:more-prop-label} using our combinatorial models we show the following volume formulas: \begin{align} \label{eq:PS3} \vol\FF_{\PS_{n+1}}(a,b,c^{n-2}) & =(a+b-c)(a+b+(n-2)c)^{n-2}+(-b+c)(b+(n-2)c)^{n-2},\\ \label{eq:PS4}
\vol\FF_{\PS_{n+1}}(a,b,c,d^{n-3})&=(a+b+c-2d)(a+b+c+(n-3)d)^{n-2} \\ \notag & \qquad \qquad -(b+c-2d)(b+c+(n-3)d)^{n-2}\\ \notag&\qquad\qquad -(n-1)a(c-d)(c+(n-3)d)^{n-3} ,\\ \label{eq:conj} \vol\FF_{\Car_{n+1}}(a,b,c^{n-2}) &=C_{n-2}a^{n-1}(a+b(n-1))(a+b+c(n-2))^{n-3}, \end{align} where \eqref{eq:conj} was conjectured by Benedetti et al. in \cite{Benedetti2019}.
\section{Constant term identities} \label{sec:const-term-ident}
In this section we review the Lidskii volume formula and restate Theorems~\ref{thm:PS} and \ref{thm:main} as constant term identities.
Let $G$ be a directed graph on $[n+1]$ and $\vec a\in\ZZ^n$. The \emph{Kostant partition function} $K_G(\vec{a})$ of $G$ at $\vec{a}$ is the number of integer points of $\FF_{G}(\vec{a})$, \emph{i.e.}, if $G$ has $m$ edges, \[
K_G(\vec{a})=|\FF_G(\vec{a}) \cap \ZZ^m|. \]
We denote by $G|_{n}$ the restriction of $G$ to the vertices in $[n]$. Let $\outdeg(i)$ denote the out-degree of vertex $i$ in $G$. The following formula, known as the Lidskii volume formula, allows us to express the (normalized) volume of the flow polytope $\FF_G(\vec{a})$ in terms of Kostant partition functions, see \cite[Theorem~38]{Baldoni2008}.
\begin{theorem}[Lidskii volume formula] \label{thm:lidskii}
Let $G$ be a connected directed graph on $[n+1]$ with $m$ directed edges, where every directed edge is of the form
$(i, j)$ with $i<j$ and let $\vec{a}=(a_1,a_2,\dots ,a_n)\in \ZZ^{n}$. Denoting
$\vec{t}=(t_1,\dots ,t_n):=(\outdeg(1)-1,\dots,\outdeg(n)-1)$, we have \begin{equation*}
\vol\FF_{G}(\vec a) = \sum_{\substack{|\vec s|=m-n\\ \vec{s}\ge \vec t}}\binom{m-n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{n}^{s_n}K_{G|_{n}}(\vec{s}-\vec{t}), \end{equation*} where the sum is over all sequences $\vec{s}=(s_1,\dots ,s_n)$ of nonnegative integers such that
$|\vec s|=s_1+\dots+s_n=m-n$ and $\vec{s}\geq \vec{t}$ in dominance order, \emph{i.e.}, $\sum_{i=1}^{k}s_{i} \geq \sum_{i=1}^{k}t_{i}$ for $k=1,2,\dots,n$. \end{theorem}
Note that if $\vec a = (1,0^n)$ in Theorem~\ref{thm:lidskii} there is only one term in the sum giving the following corollary, see \cite{Baldoni2008}, \cite{PitmanStanley}, or \cite[Corollary~1.4]{VolumesandEhrhart}.
\begin{corollary} For a directed graph $G$ on $[n+1]$, we have \begin{equation}
\label{eq:outdeg} \vol \FF_G(1,0^n) = K_G(p,1-\outdeg(2),1-\outdeg(3),\dots,1-\outdeg(n),0), \end{equation} where $p=\outdeg(2)+\outdeg(3)+\cdots+\outdeg(n)-n+1$. \end{corollary}
For a multivariate rational function $f(x_1,x_2,\dots,x_n)$ we denote by $\CT_{x_i}f$ the constant term of the Laurant series expansion of $f$ with respect to $x_i$ by considering other variables as constants. Since $\CT_{x_1} f$ is a rational function in $x_2,\dots,x_n$, we can apply $\CT_{x_2}$ to it. Repeating in this way the constant term $\CT_{x_{n}}\dots \CT_{x_{1}} f$ is defined. We also define $[x_n^{a_n}\dots x_1^{a_1}]f$ to be the coefficient of the monomial $x_n^{a_n}\dots x_1^{a_1}$ in the Laurent expansion of $f$ when expanded in the variables $x_1, x_2,\dots,x_n$ in this order. Note that we have \begin{equation}
\label{eq:1} [x_n^{a_n}\dots x_1^{a_1}]f = \CT_{x_{n}}\dots \CT_{x_{1}} \left(x_n^{-a_n}\dots x_1^{-a_1} f\right). \end{equation}
Let $G$ be a directed graph on $[n+1]$. Then for $\vec a=(a_1,\dots,a_{n}) \in \ZZ^{n}$ and $a_{n+1}=-(a_1+\cdots+a_n)$, the Kostant partition function $K_G(\vec a)$ can be computed by \begin{equation}
\label{eq:KG} K_G(\vec a) = [x_{n+1}^{a_{n+1}} \cdots x_1^{a_1}] \prod_{(i,j)\in E(G)}\left( 1-\frac{x_i}{x_j}\right)^{-1}. \end{equation}
Now we are ready to express $E_{\PS_{n+1}}(k)$ and $E_{\Car_{n+1}}(k)$ as constant terms of Laurent series. Throughout this paper the factor $(x_j-x_i)^{-1}$, where $i<j$, means the Laurent expansion \[ (x_j-x_i)^{-1} = \frac{1}{x_j} \left( 1-\frac{x_i}{x_j} \right)^{-1} = \frac{1}{x_j}\sum_{l\ge0} \left( \frac{x_i}{x_j}\right)^l. \]
\begin{proposition} \label{prop:car} We have \begin{align} \label{eq:CT1} E_{\PS_{n+1}}(k) &=\CT_{x_{n}}\dots \CT_{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1},\\ \label{eq:CT2} E_{\Car_{n+1}}(k) &= \CT_{x_{n}}\dots \CT_{x_{1}}\frac{1}{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{n}-x_{i})^{-1}\prod _{i=1}^{n-2}(x_{i+1}-x_{i})^{-1}. \end{align}
\end{proposition} \begin{proof} We will only prove \eqref{eq:CT2} since \eqref{eq:CT1} can be proved similarly. Let $G=\Car_{n+1}$ and $H=\GG(k)$. Then $H$ is a graph with vertices $0,1,2,\dots,n+1$, and by \eqref{eq:EG} and \eqref{eq:outdeg}, \[ E_{\Car_{n+1}}(k)=K_H(p,1-\outdeg_H(1),1-\outdeg_H(2),\dots,1-\outdeg_H(n),0), \] where $p=\outdeg_H(1)+\outdeg_H(2)+\cdots+\outdeg_H(n)-n$. Since $\outdeg_{H}(1)=n-1$, $\outdeg_{H}(n)=1$, and $\outdeg_{H}(i)=2$ for $2\le i\le n-1$, we can rewrite the above equation as \[ E_{\Car_{n+1}}(k)=K_H(2n-4,2-n,(-1)^{n-2},0,0). \] Then, by \eqref{eq:KG}, we obtain \begin{equation}
\label{eq:3} E_{\Car_{n+1}}(k)= [x_{n-1}^{-1} \cdots x_2^{-1} x_1^{2-n} x_0^{2n-4}] \prod_{(i,j)\in E(H)}\left( 1-\frac{x_i}{x_j}\right)^{-1}. \end{equation}
Since every term in the expansion of \[ \prod_{(i,j)\in E(H)}\left( 1-\frac{x_i}{x_j}\right)^{-1}= \prod_{i=1}^{n}\left( 1-\frac{x_0}{x_{i}} \right)^{-k} \prod_{i=2}^n\left( 1-\frac{x_1}{x_{i}} \right)^{-1} \left( 1-\frac{x_i}{x_{n+1}} \right)^{-1} \prod_{i=2}^{n-1}\left( 1-\frac{x_i}{x_{i+1}} \right)^{-1} \] is homogeneous of degree 0 in the variables $x_0,x_1,\dots,x_{n+1}$, we can set $x_{0}=1$ in \eqref{eq:3}. Moreover, since every term in the expansion of $(1-x_i/x_{n+1})^{-1}$ has a negative power of $x_{n+1}$ except for the constant term $1$, we can omit the factors involving $x_{n+1}$ in \eqref{eq:3}. Then, by the same argument, we can also omit the factors involving $x_n$ in \eqref{eq:3} to obtain \[ E_{\Car_{n+1}}(k) =[x_1^{2-n}x_2^{-1}\cdots x_{n-1}^{-1}] \prod_{i=1}^{n-1}\left( 1-\frac{1}{x_{i}} \right)^{-k} \prod_{i=2}^{n-1}\left( 1-\frac{x_1}{x_{i}} \right)^{-1} \prod_{i=2}^{n-2}\left( 1-\frac{x_i}{x_{i+1}} \right)^{-1}. \] By replacing $x_i$ by $x_{n-i}^{-1}$ for each $1\le i\le n-1$ we have \[ E_{\Car_{n+1}}(k) = [x_{n-1}^{n-2} x_{n-2} \cdots x_1] \prod_{i=1}^{n-1}\left( 1-x_{i} \right)^{-k} \prod_{i=1}^{n-2}\left( 1-\frac{x_i}{x_{n-1}} \right)^{-1} \prod_{i=1}^{n-3}\left( 1-\frac{x_{i}}{x_{i+1}} \right)^{-1}, \] which is equivalent to \eqref{eq:CT2} by \eqref{eq:1}. \end{proof}
By Proposition \ref{prop:car}, we can restate Theorems~\ref{thm:PS} and \ref{thm:main} as follows.
\begin{theorem} \label{Thm:PS} We have \[
\CT_{x_{n}}\dots \CT_{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1} = \frac{1}{kn-1}\binom{(k+1)n-2}{n}. \] \end{theorem}
\begin{theorem} \label{Thm:Car} We have \begin{multline*}
\CT_{x_{n}}\dots \CT_{x_{1}}\frac{1}{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{n}-x_{i})^{-1}\prod _{i=1}^{n-2}(x_{i+1}-x_{i})^{-1}\\ =\frac{1}{k(n+1)+n-2}\binom{kn+k+2n-3}{n}\binom{n+k-2}{k-1}. \end{multline*} \end{theorem}
\section{Labeled Dyck Paths} \label{sec:labeled-dyck-paths}
In this section we give combinatorial meanings to the constant terms in Theorems~\ref{Thm:PS} and \ref{Thm:Car} using labeled Dyck paths.
A \emph{Dyck path of length $2n$} is a lattice path from $(0,0)$ to $(2n,0)$ consisting of
\emph{up-steps} $(1,1)$ and \emph{down-steps} $(1,-1)$ lying on or above the line $y=0$. The set
of Dyck paths of length $2n$ is denoted by $\Dyck_n$.
Let $k$ be a positive integer. A \emph{$k$-labeled Dyck path} is a Dyck path with a labeling on the
down-steps such that the label of each down-step is an integer $0\le i\le k$ and the labels of any
consecutive down-steps are in weakly decreasing order, see Figure~\ref{fig:LD}. A \emph{doubly
$k$-labeled Dyck path} is a $k$-labeled Dyck path together with an additional labeling on the
down-steps labeled $0$ and the up-steps with integers from $\{1,2,\dots,k\}$ such that the additional labels on these
steps are weakly increasing, see Figure~\ref{fig:DLD}.
We denote by $\LD_n(k)$ (resp.~$\DLD_n(k)$) the set of $k$-labeled Dyck paths (resp.~doubly
$k$-labeled Dyck paths) of length $2n$. We also denote by $\LD_n(k,d)$ the set of $k$-labeled Dyck
paths of length $2n$ with exactly $d$ down-steps labeled $0$.
\begin{figure}
\caption{A $5$-labeled Dyck path of length $20$. The labels of every consecutive down-steps must be in weakly decreasing order.}
\label{fig:LD}
\end{figure}
\begin{figure}
\caption{A doubly $5$-labeled Dyck path of length $20$. The down-steps labeled $0$ and the up-steps are the red steps and their additional labels are written in red. The red labels must be in weakly increasing order.}
\label{fig:DLD}
\end{figure}
A \emph{multiset} is a set with repetitions allowed. Let $\multiset{n}{m}:=\binom{n+m-1}{m}$. Then $\multiset{n}{m}$ is the number of multisets with $m$ elements taken from $[n]$. Equivalently, $\multiset{n}{m}$ is the number of nonnegative integer solutions $(a_{1},a_{2},\dots ,a_{n})$ to $a_{1}+a_{2}+\dots +a_{n}=m$ and also the number of $m$-tuples $(i_{1},i_{2},\dots ,i_{m})$ of nonnegative integers satisfying $1\leq i_{1} \leq i_{2} \leq \dots \leq i_{m} \leq n$.
The following proposition is immediate from the definitions of $\DLD_n(k)$ and $\LD_n(k,d)$.
\begin{proposition} We have \label{prop:DLD&LD} \begin{align*}
|\DLD_n(k)|=\sum_{d=0}^{n}|\LD_n(k,d)| \Multiset{k}{n+d}. \end{align*} \end{proposition}
We now show that the constant terms in Theorems~\ref{Thm:PS} and \ref{Thm:Car} have the following combinatorial interpretations.
\begin{theorem} We have \label{Thm:PS=LD} \begin{align*}
\CT_{x_{n}}\dots \CT_{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1} = |\LD_{n-1}(k,0)|. \end{align*} \end{theorem} \begin{proof} Consider that we choose $x_i^{a_{i1}}x_i^{a_{i2}}\cdots x_i^{a_{ik}}$ in $(1-x_{i})^{-k}=(1+x_{i}+x_{i}^{2}+\cdots)\cdots (1+x_{i}+x_{i}^{2}+\cdots)$ for $i=1,2,\dots ,n$ and we choose $x_{i}^{b_{i}}/x_{i+1}^{b_{i}+1}$ in $(x_{i+1}-x_{i})^{-1}=1/x_{i+1}+x_{i}/x_{i+1}^{2}+x_{i}^{2}/x_{i+1}^{3}+\cdots$ for $i=1,2,\dots,n-1$. Then $\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1}=\sum\prod_{i=1}^{n}(x_{i}^{a_{i1}+\dots+ a_{ik}+b_{i}-b_{i-1}-1})$, where the sum is over all nonnegative integers $a_{ij},b_{i}$ for $1\leq i\leq n$ and $1\leq j \leq k$ with $b_{0}=-1$ and $b_{n}=0$. Hence the left-hand side is the number of the nonnegative integer solutions to the equations $a_{i1}+\dots +a_{ik}=b_{i-1}-b_{i}+1$ for $i=1,2,\dots,n$ with $b_{0}=-1,b_{n}=0$. If we set $r_{i}=b_{i-1}-b_{i}+1$, so that $r_1+r_2+\dots +r_j=j+b_0-b_j\leq j-1$, then the number of solutions is $\sum\prod_{i=1}^{n}\multiset{k}{r_{i}}$, where the sum is over all nonnegative integers $r_{1},\dots,r_{n}$ with $r_{1}+\dots +r_{j}\leq j-1$ for $j=1,2,\dots, n-1$ and $r_{1}+\dots +r_{n}=n-1$. For such an $n$-tuple $(r_1,\dots,r_n)$, let $D$ be the Dyck path of length $2(n-1)$ such that the number of consecutive down-steps after the $i$th up-step is $r_{i+1}$ for $i=1,\dots,n-1$. The map $(r_1,\dots,r_n) \mapsto D$ is a bijection from the set of $n$-tuples satisfying the above conditions to $\Dyck_{n-1}$. Under this correspondence, $\prod_{i=1}^{n}\multiset{k}{r_i}$ is the number of $k$-labeled Dyck paths in $\LD_{n-1}(k,0)$ whose underlying Dyck path is $D$. Therefore we obtain the result. \end{proof}
\begin{theorem} We have \label{Thm:Car=DLD} \begin{align*}
\CT_{x_{n}}\dots \CT_{x_{1}}\frac{1}{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{n}-x_{i})^{-1}\prod _{i=1}^{n-2}(x_{i+1}-x_{i})^{-1} = |\DLD_{n-1}(k)|. \end{align*} \end{theorem} \begin{proof} Similarly to the previous theorem, considering $x_i^{a_{i1}}x_i^{a_{i2}}\cdots x_i^{a_{ik}}$ in $(1-x_{i})^{-k}$ and $x_{i}^{b_{i}}/x_{i+1}^{b_{i}+1}$ in $(x_{i+1}-x_i)^{-1}$ and $x_{i}^{c_{i}}/x_{n}^{c_{i}+1}$ in $(x_n-x_i)^{-1}$, we get that the left-hand side is the number of the nonnegative integer solutions to the equations $a_{i1}+\dots +a_{ik}+b_{i}+c_{i}=1+b_{i-1}$ for $i=1,2,\dots,n-1$ and $a_{n1}+\dots+ a_{nk}+b_{n}+c_{n}=n-1+c_{1}+\dots +c_{n-1}$ with $b_{0}=b_{n-1}=b_{n}=c_{n}=0$. If we set $r_{i}=b_{i-1}-b_{i}+1$ for $i=1,2,\dots,n-1$, then the number of solutions is $\sum_{r_i,c_i}\prod_{i=1}^{n-1}\multiset{k}{r_{i}-c_i}\multiset{k}{n-1+c_{1}+\dots +c_{n-1}}$
where the sum is over all nonnegative integers $r_i,c_i$ for $i=1,2,\dots, n-1$ with $r_{1}+\dots +r_{j}\leq j$ for $j=1,2,\dots, n-2$ and $r_{1}+\dots +r_{n-1}=n-1$. For such an $n$-tuple $(r_1,\dots,r_n)$, let $D$ be the Dyck path of length $2(n-1)$ such that the number of consecutive down-steps after the $i$th up-step is $r_{i}$ for $i=1,\dots,n-1$. The map $(r_1,\dots,r_n) \mapsto D$ is a bijection from the set of $n$-tuples satisfying the above conditions to $\Dyck_{n-1}$. Regard $\multiset{k}{r-c}$ as the number of $r$-tuples $(i_1,\dots,i_r)$ of integers with $k\geq i_1 \geq \dots \geq i_{r-c} \geq 1$ and $ i_{r-c+1} =\dots=i_r= 0$. Then $\sum_{c_i}\prod_{i=1}^{n-1}\multiset{k}{r_{i}-c_i}\multiset{k}{n-1+c_{1}+\dots +c_{n-1}}$, where the sum is over all nonnegative integers $c_i$ for $i=1,2,\dots,n-1$ with $c_i \leq r_i$, is the number of doubly $k$-labeled Dyck paths whose underlying Dyck path is $D$. Therefore we obtain the result. \end{proof}
Note that by Proposition~\ref{prop:DLD&LD}, we can compute the constant terms in Theorems~\ref{Thm:PS=LD} and \ref{Thm:Car=DLD} if we have a formula for the cardinality $|\LD_n(k,d)|$. Therefore our next step is to find this number.
\section{A cyclic lemma} \label{sec:cyclic-lemma}
Let $\LD_{n}(k;a_0,a_1,\dots,a_k)$ denote the set of $k$-labeled Dyck paths of length $2n$ such that the number of down-steps with label $i$ is $a_i$ for $0\le i\le k$. In this section we prove the following theorem using a cyclic lemma.
\begin{theorem} \label{thm:dyck} We have \[
|\LD_n(k;a_0,a_1,\dots,a_k)|=\frac{1}{n+1}\prod_{i=0}^{k}\Multiset{n+1}{a_i}. \] \end{theorem}
\begin{remark} A \emph{parking function} of length $n$ is a tuple $(p_1,p_2,\dots,p_n)\in \ZZ^n_{>0}$ with a condition that $q_i\leq i$ for $i=1,2,\dots,n$ where $(q_1,q_2,\dots,q_n)$ is the rearrangement of $(p_1,p_2,\dots,p_n)$ in weakly increasing order. Let $PF_n$ be the set of parking function of length $n$. There is a well-known bijection between $PF_n$ and $n$-labeled Dyck paths of length $2n$ which the number of each label from $1$ to $n$ equals $1$. Thus, using Theorem \ref{thm:dyck}, we have
\[|PF_n|=|\LD_n(n;0,1,1,\dots,1)|=(n+1)^{n-1}.\] \end{remark}
\begin{remark}
Recently, Yip \cite[Theorem 3.18]{Yip2019} considered a set
$\mathcal{T}_k(n,i)$ of certain labeled Dyck paths and found a simple
formula for its cardinality using a cyclic lemma. Using our notation, this set can be written
\[ \mathcal{T}_k(n,i) = \bigcup_{a_0+\dots+a_{k-1}=n-i}\Dyck_n(k+i-1;a_0,\dots,a_{k-1},1^i).
\] The proof of Theorem~\ref{thm:dyck} in this section is essentially the same as that in \cite[Theorem 3.18]{Yip2019}. \end{remark}
A \emph{$k$-labeled Dyck word} of length $2n$ is a sequence $w = w_1\dots w_{2n}$ of letters in $\{U,D_0,D_1,\dots,D_k\}$ satisfying the following conditions: \begin{itemize} \item The number of $U$'s is equal to $n$. \item For any prefix $w_1\dots w_j$, the number of $U$'s is greater than or equal to the total number of $D_i$'s for
$0\le i\le k$. \item The labels of any consecutive $D_i$'s are in weakly decreasing order, i.e., if $w_i=D_a$ and $w_{i+1}=D_b$, then $a\ge b$. \end{itemize}
Replacing each up step by $U$ and each down step labeled $i$ by $D_i$ is an obvious bijection from $k$-labeled Dyck paths to $k$-labeled Dyck words. For example, the $k$-labeled Dyck word corresponding to the $k$-labeled Dyck path in Figure~\ref{fig:LD} is \begin{equation}
\label{eq:w} UD_0UUUUD_5UD_3D_0UUUD_4D_2D_0D_0UD_1D_1. \end{equation}
From now on, we will identify $k$-labeled Dyck paths with $k$-labeled Dyck words using this bijection. Note that $\LD_n(k;a_0,a_1,\dots,a_k)$ is then the set of $k$-labeled Dyck words of length $2n$ in which the number of $D_i$'s is equal to $a_i$ for $0\le i\le k$. We can count such words by using a well-known cyclic argument. We first need another definition.
An \emph{extended $k$-labeled word} of length $2n+1$ is a sequence $w = w_1\dots w_{2n+1}$ of letters in $\{U,D_0,D_1,\dots,D_k\}$ with $w_1=U$ and exactly $n+1$ $U$'s that satisfies the third condition of a $k$-labeled Dyck word: the labels of any consecutive $D_i$'s are in weakly decreasing order, i.e., if $w_i=D_a$ and $w_{i+1}=D_b$, then $a\ge b$. The set of extended $k$-labeled words of length $2n+1$ is denoted by $\EW_n(k)$.
Let $w = w_1\dots w_{2n+1}\in \EW_n(k)$. We define the integer $\ind(w)$ using the following algorithm. Here, $w = w_1\dots w_{2n+1}$ is cyclically ordered, which means that $w_1$ is followed by $w_2$, $w_2$ is followed by $w_3$, and so on, and $w_{2n+1}$ is followed by $w_1$. \begin{itemize} \item Find a letter $U$ followed by a $D_i$ for some $0\le i\le k$ in cyclic order and delete this pair $U$ and $D_i$
from $w$. Repeat this until there is only one letter left, which must be $U$. \item If the remaining $U$ is the $j$th $U$ in the original word $w$ then define $\ind(w)=j$. \end{itemize} We also define the \emph{shifting operator} $s:\EW_n(k)\to \EW_n(k)$ by \[
s(w): = w_{i}w_{i+1}\cdots w_{2n+1} w_1 \dots w_{i-1}, \] where $i$ is the largest integer with $w_i=U$.
\begin{example} Let $w=UD_1D_0UUD_1U \in \EW_3(1)$. Then by the algorithm, \[ \underline{UD_1}D_0UUD_1U \rightarrow D_0U\underline{UD_1}U \rightarrow \underline{D_0}U\underline{U} \rightarrow U, \] we get $\ind(w)=2$ since the remaining $U$ is the second $U$ in $w$. \end{example}
Since the above algorithm treats the word $w$ cyclically one can easily see that the following lemma holds.
\begin{lemma}\label{lem:cyclic} For any element $w\in \EW_n(k)$, we have
\[
\ind(s(w)) \equiv \ind(w) + 1 \mod n+1.
\] \end{lemma}
Observe that for $w=w_1w_2\dots w_{2n+1}\in \EW_n(k)$ we have $w_1\dots w_{2n}\in\Dyck_n(k)$ if and only if $\ind(w)=n+1$. Therefore, by Lemma~\ref{lem:cyclic}, for each $w\in \EW_n(k)$ there is a unique integer $0\le j\le n$ such that $s^j(w) = w'U$ for some $k$-labeled Dyck word $w'$ of length $2n$. This defines a map $p: \EW_n(k)\to\Dyck_n(k)$ sending $w$ to $p(w)=w'$. Again, by Lemma~\ref{lem:cyclic}, this is a $(n+1)$-to-$1$ map. Note that $w$ and $w'$ have the same the number of steps $D_i$ for each $0\le i\le k$. We have proved the following proposition.
\begin{proposition}\label{prop:n-to-1} There is an $(n+1)$-to-$1$ map $p: \EW_n(k)\to\Dyck_n(k)$ preserving the number of $D_i$'s for $0\le i\le k$. \end{proposition}
We now can prove Theorem~\ref{thm:dyck} easily.
\begin{proof}[Proof of Theorem~\ref{thm:dyck}]
By Proposition~\ref{prop:n-to-1}, $(n+1)|\LD_n(k;a_0,a_1,\dots,a_k)|$ is the number of
elements $w\in \EW_n(k)$ in which $D_i$ appears $a_i$ times for $0\le i\le k$. Since consecutive $D_i$'s are always
ordered according to their subscripts, such elements $w$ are obtained from the sequence $U\dots U$ of $n+1$ $U$'s by
inserting $a_i$ $D_i$'s after $U$'s in $\multiset{n+1}{a_i}$ ways for $0\le i\le k$ independently. Thus we have
\[
(n+1)|\LD_n(k;a_0,a_1,\dots,a_k)|=\prod_{i=0}^{k}\Multiset{n+1}{a_i}, \] which completes the proof. \end{proof}
As corollaries we obtain formulas for $|\LD_{n}(k,d)|$ and $|\DLD_{n-1}(k)|$.
\begin{corollary} \label{cor:LD} We have \[
|\LD_{n}(k,d)|=\frac{1}{n+1}\Multiset{n+1}{d} \Multiset{k(n+1)}{n-d}. \] \end{corollary} \begin{proof}
By Theorem~\ref{thm:dyck},
\[
|\LD_{n}(k,d)|=\frac{1}{n+1} \Multiset{n+1}{d} \sum_{a_1+\dots+a_k=n-d}\prod_{i=1}^{k}\Multiset{n+1}{a_i}.
\]
The above sum is equal to the number of $k$-tuples $(A_1,\dots,A_k)$ of multisets such that
each element $x\in A_i$ satisfies $(n+1)(i-1)+1\le x\le (n+1)i$ and $\sum_{i=1}^k |A_i| = n-d$. Since such a
$k$-tuple is completely determined by $A:=A_1\cup \cdots\cup A_k$, the sum is
equal to $\multiset{k(n+1)}{n-d}$, the number of multisets of size $n-d$ whose elements are in $[k(n+1)]$. Thus we
obtain the formula. \end{proof}
\begin{corollary} \label{cor:DLD} We have \begin{align*}
|\DLD_{n-1}(k)|=\frac{1}{k(n+1)+n-2}\binom{kn+k+2n-3}{n}\binom{n+k-2}{k-1}. \end{align*} \end{corollary} \begin{proof} We will use the standard notation in hypergeometric series, see for example \cite[Chapter 2]{AAR_SP}. By Proposition \ref{prop:DLD&LD} and Corollary~\ref{cor:LD}, \begin{align}
\notag |\DLD_{n-1}(k)|&=\sum_{d=0}^{n-1}|\LD_{n-1}(k,d)|\Multiset{k}{d+n-1}
\\ \notag &=\sum_{d=0}^{n-1}\frac{1}{n}\Multiset{kn}{n-d-1}\Multiset{n}{d}\Multiset{k}{d+n-1}
\\ \label{eq:2} &=\frac{(kn+n-2)!(k+n-2)!}{n!(kn-1)!(n-1)!(k-1)!}\Hyper21{-n+1, k+n-1}{-kn-n+2}{1} . \end{align} By the Vandermonde summation formula \cite[Corollary~2.2.3]{AAR_SP} \[
\Hyper21{-n,b}{c}{1} = \frac{(c-b)_n}{(c)_n}, \] we have \begin{equation}
\label{eq:4} \Hyper21{-n+1, k+n-1}{-kn-n+2}{1} = \frac{(-kn-2n-k+3)_{n-1}}{(-kn-n+2)_{n-1}} =\frac{(kn+2n+k-3)!}{(kn+n+k-2)!}\frac{(kn-1)!}{(kn+n-2)!}. \end{equation} By \eqref{eq:2} and \eqref{eq:4} we obtain the result. \end{proof}
The constant term identities in Theorems~\ref{Thm:PS} and \ref{Thm:Car} follow immediately from Theorems~\ref{Thm:PS=LD}, \ref{Thm:Car=DLD} and Corollary~\ref{cor:DLD}. This completes the proof of Theorems~\ref{thm:PS} and \ref{thm:main} in the introduction.
\section{More Properties of Labeled Dyck Paths} \label{sec:more-prop-label}
In this section we find volumes of flow polytopes of Pitman--Stanley graph $\PS_{n+1}$ and Caracol graph $\Car_{n+1}$ for certain flow vectors using Lidskii's formula and $k$-labeled Dyck prefixes.
A \emph{$k$-labeled Dyck prefix} is the part of a $k$-labeled Dyck path from $(0,0)$ to $(a,b)$ for some point $(a,b)$ in the path. The set of $k$-labeled Dyck prefixes from $(0,0)$ to $(2n-i,i)$ is denoted by $\Dyck_{n,i}$. We also denote by $\Dyck_{n,i}(k;a_0,a_1,\dots,a_k)$ the set of $k$-labeled Dyck prefixes in $\Dyck_{n,i}$ such that the number of down-steps labeled $j$ is $a_j$ for $0\le j\le k$.
Recall that $\Dyck_n(k)$ is in bijection with the set of $k$-Dyck words of length $2n$. Therefore one can consider an element in $\Dyck_{n,i}$ as a $k$-Dyck word of length $2n$ whose last $i$ letters are $D_0$'s. See Figure~\ref{fig:LD2}.
\begin{figure}
\caption{An element of $\Dyck_{20,2}(3;2,3,2,1)$ whose steps are drawn in solid segments. By appending it with the two
dashed down-steps, this element can be considered as an element in $\Dyck_{20}(3)$. This path can be expressed as
$UD_0UUUUD_3UD_1D_1UUUD_2D_2D_0UD_1D_0D_0$, where the last two $D_0$ steps correspond to the dashed down-steps.}
\label{fig:LD2}
\end{figure}
Now we find the cardinality of $\Dyck_{n,i}(k;a_0,a_1,\dots,a_k)$.
\begin{lemma} \label{lem:dyck_n,i} We have \[
|\Dyck_{n,i}(k;a_0,a_1,\dots,a_k)|=\dfrac{i+1}{n+1}\prod_{j=0}^{k}\Multiset{n+1}{a_j}. \] \end{lemma} \begin{proof} Let $\EW_{n,i}(k)$ be the set of words $w=w_1\dots w_{2n-i+1}$ of letters in $\{U,D_0,D_1,\dots,D_k\}$ with $w_1=U$ and exactly $n+1$ $U$'s that satisfies the third condition of a $k$-labeled Dyck word: the labels of any consecutive $D_i$'s are in weakly decreasing order, i.e., if $w_i=D_a$ and $w_{i+1}=D_b$, then $a\ge b$. For $w \in \EW_{n,i}(k)$, an \emph{index candidate} of $w$ is an integer $j$ satisfying the following condition: \begin{itemize} \item Find a letter $U$ followed by a $D_i$ for some $0\le i\le k$ in cyclic order and delete this pair $U$ and $D_i$
from $w$. Repeat this until there are $i+1$ letters left, which must be all $U$'s. Then the $j$th $U$ in the original word $w$ is one of the
remaining $U$'s. \end{itemize} Note that there are $i+1$ index candidates for any $w\in \EW_{n,i}(k)$.
Let $\EW_{n,i}'(k)$ be the set of words obtained from a word $w\in\EW_{n,i}(k)$ by adding $i$ $D_0$'s to the left of the $j$th $U$ in $w$ for an index candidate $j$ of $w$. Note that $\EW_{n,i}'(k)$ is a subset of $\EW_{n}(k)$ which is defined in Section \ref{sec:cyclic-lemma}. Thus every $w'\in \EW_{n,i}'(k)$ has length $2n+1$ and the unique index $\ind(w')$ exists. Then by the map $p$ defined in Proposition \ref{prop:n-to-1}, there is an $(n+1)$-to-$1$ map from $\EW_{n,i}'(k)$ to $\Dyck_{n,i}(k)$. Since there are $(i+1)$ ways to choose an index candidate for $w\in \EW_{n,i}(k)$, we have \[
(i+1)\prod_{j=0}^k\Multiset{n+1}{a_j}=(n+1)|\Dyck_{n,i}(k;a_0,a_1,\dots,a_k)|, \] which completes the proof. \end{proof}
\subsection{Volumes of flow polytopes for the Pitman--Stanley graph.} \label{subsec:PS} Recall that $\vol\FF_{\PS_{n+1}}(a^n)$ and $\vol\FF_{\PS_{n+1}}(a,b^{n-1})$ were computed in \cite{Benedetti2019} and \cite{PitmanStanley}. In this subsection using Lemma~\ref{lem:dyck_n,i} we compute $\vol\FF_{\PS_{n+1}}(a_1,\dots ,a_k,b^{n-k})$ for $0\le k\le 3$. For simplicity, we will consider $\PS_{n+2}$ instead of $\PS_{n+1}$.
Note that $\PS_{n+2}$ has $2n+1$ edges and $\vec{t}:=(\outdeg(1)-1,\dots,\outdeg(n+1)-1)=(1,1,\dots,1,0)$. Since
$\PS_{n+2}|_{n+1}$ is the path graph on $[n+1]$ with edges $(i, i+1)$ for $1\le i\le n$, one can easily see that
$K_{\PS_{n+2}|_{n+1}}(\vec{s}-\vec{t})=1$ for any sequence $\vec{s}\ge \vec t$. Moreover, if $\vec s=(s_1,\dots,s_{n+1})\ge \vec t$, then $s_{n+1}=0$. Thus Lidskii's formula (Theorem~\ref{thm:lidskii}) implies \[
\vol\FF_{\PS_{n+2}}(a_1,\dots ,a_{n+1})=\sum_{\substack{s_1+\dots+s_{n}=n\\ (s_1,\dots,s_{n})\ge
(1^{n})}}\binom{n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{n}^{s_{n}} . \] Thus, we have \begin{align} \notag & \vol\FF_{\PS_{n+2}}(a_1,\dots ,a_k,b^{n-k+1}) \\ \notag &=\sum_{\substack{s_1+\dots+s_{n}=n\\ (s_1,\dots,s_{n})\ge (1^{n})}} \binom{n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{k}^{s_{k}} b^{s_{k+1}+\dots+s_{n}}\\ \notag &=\sum_{m=0}^{n} \sum_{\substack{s_1+\dots+s_{k}=m\\ (s_1,\dots,s_{k})\ge (1^k)}} \sum_{\substack{s_{k+1}+\dots+s_{n}=n-m\\ (m,s_{k+1},\dots,s_{n})\ge (k,1^{n-k})}} \binom{n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{k}^{s_{k}} b^{n-m}\\ \label{eq:PSA} &=\sum_{m=0}^{n} \binom nm b^{n-m} A_{k,m}(a_1,\dots,a_k)B_{n,k,m}, \end{align} where \begin{align*} A_{k,m}(a_1,\dots,a_k) &= \sum_{\substack{s_1+\dots+s_{k}=m\\ (s_1,\dots,s_{k})\ge (1^k)}} \binom{m}{s_{1},s_{2},\dots ,s_{k}}a_{1}^{s_1}\dots a_{k}^{s_{k}}, \\ B_{n,k,m}&= \sum_{\substack{s_{k+1}+\dots+s_{n}=n-m\\ (m,s_{k+1},\dots,s_{n})\ge (k,1^{n-k})}} \binom{n-m}{s_{k+1},\dots ,s_{n}}. \end{align*} The following lemma shows that $B_{n,k,m}$ has a simple formula.
\begin{lemma} We have \label{lem:B}
\[
B_{n,k,m} = (m-k+1)(n-k+1)^{n-m-1}.
\] \end{lemma} \begin{proof}
For a sequence $(s_{k+1},\dots,s_n)$ of nonnegative integers, we have
$s_{k+1}+\dots+s_n=n-m$ and $(m,s_{k+1},\dots,s_n)\ge (k,1^{n-k})$ if and only if
$UD^{s_n}UD^{s_{n-1}}\dots UD^{s_{k+1}}U^kD^m$ is a Dyck path from $(0,0)$ to $(2n,0)$, or equivalently, $UD^{s_n}UD^{s_{n-1}}\dots UD^{s_{k+1}}$ is a Dyck prefix from $(0,0)$ to $(2n-m-k,m-k)$. Moreover, if such a sequence
$(s_{k+1},\dots, s_n)$ is given, $\binom{n-m}{s_{k+1},\dots ,s_{n}}$ is the number of ways to label the down steps of
this Dyck prefix with labels from $\{0,1,\dots,n-m-1\}$ such that there is exactly one down step labeled $j$ for each
$0\le j\le n-m-1$ and the labels of consecutive down steps are in decreasing order. Thus
\[
B_{n,k,m} =|\Dyck_{n-k,m-k}(n-m-1;1^{n-m})|.
\] By Lemma~\ref{lem:dyck_n,i} we obtain the formula. \end{proof}
By \eqref{eq:PSA} and Lemma~\ref{lem:B}, we obtain the following proposition. \begin{proposition}\label{prop:vPS} We have \begin{multline*} \vol\FF_{\PS_{n+2}}(a_1,\dots ,a_k,b^{n-k+1}) \\ =\sum_{m=0}^{n} \binom nm b^{n-m} (m-k+1)(n-k+1)^{n-m-1} A_{k,m}(a_1,\dots,a_k), \end{multline*} where \[ A_{k,m}(a_1,\dots,a_k) = \sum_{\substack{s_1+\dots+s_{k}=m\\ (s_1,\dots,s_{k})\ge (1^k)}} \binom{m}{s_{1},s_{2},\dots ,s_{k}}a_{1}^{s_1}\dots a_{k}^{s_{k}}. \] \end{proposition}
By Proposition~\ref{prop:vPS}, in order to compute $\vol\FF_{\PS_{n+2}}(a_1,\dots ,a_k,b^{n-k+1})$, it is enough to find $A_{k,m}(a_1,\dots,a_k)$. For $k=0,1$, using this method we can easily recover the following formulas in \cite{Benedetti2019,
PitmanStanley}: \begin{align*} \vol\FF_{\PS_{n+2}}(a^{n+1}) & =a^{n}(n+1)^{n-1},\\ \vol\FF_{\PS_{n+2}}(a,b^{n}) & =a(a+nb)^{n-1}. \end{align*}
We now find a formula for this volume for $\vol\FF_{\PS_{n+2}}(a_1,\dots ,a_k,b^{n-k+1})$ for $k=2,3$.
\begin{proposition} For positive integers $a$, $b$, and $c$, we have \[ \vol\FF_{\PS_{n+2}}(a,b,c^{n-1})=(a+b-c)(a+b+(n-1)c)^{n-1}-(b-c)(b+(n-1)c)^{n-1}. \] \end{proposition} \begin{proof} By Proposition~\ref{prop:vPS}, \[
\vol\FF_{\PS_{n+2}}(a,b,c^{n-1}) =
\sum_{m=0}^n \binom{n}{m}c^{n-m}(m-1)(n-1)^{n-m-1} A_{2,m}(a,b), \] where $A_{2,0}(a,b) =A_{2,1}(a,b)=0$ and for $m>2$, \[
A_{2,m}(a,b)=\sum_{\substack{i+j=m\\ (i,j)\ge (1,1)}}\binom{m}{i,j}a^{i}b^j
=(a+b)^m-b^m. \] Thus \begin{align} \notag\vol\FF_{\PS_{n+2}}(a,b,c^{n-1}) &= \sum_{m=2}^n \binom{n}{m}c^{n-m}\left((a+b)^m-b^m\right)(m-1)(n-1)^{n-m-1}\\ \label{eq:vol(a,c,b,)_1} &=\dfrac{1}{n-1}\left(g_n(a+b,c^{n-1})-g_n(b,c^{n-1})-f_n(a+b,c^{n-1})+f_n(b,c^{n-1})\right), \end{align} where \begin{align*} f_n(x,y)&=\sum_{m=0}^n \binom{n}{m}x^my^{n-m} = (x+y)^n,\\ g_n(x,y)&=\sum_{m=0}^nm \binom{n}{m}x^my^{n-m} = nx(x+y)^{n-1}. \end{align*} Simplifying \eqref{eq:vol(a,c,b,)_1} we obtain the result. \end{proof}
In a similar way one can check $A_{3,m}(a,b,c)=(a+b+c)^m-(b+c)^m-ac^{m-1}$ and obtain the following proposition. We omit the details.
\begin{proposition} For positive integers $a$, $b$, $c$, and $d$, we have \begin{multline*}
\vol\FF_{\PS_{n+2}}(a,b,c,d^{n-2})=(a+b+c-2d)(a+b+c+(n-2)d)^{n-1}\\
-(b+c-2d)(b+c+(n-2)d)^{n-1}-na(c-d)(c+(n-2)d)^{n-2}. \end{multline*} \end{proposition}
\subsection{Volumes of flow polytopes for the Caracol graph.} In \cite{Benedetti2019}, Benedetti et al. computed $\vol\FF_{\Car_{n+1}}(a^n)$ and $\vol\FF_{\Car_{n+1}}(a,b^{n-1})$ using unified diagrams and conjectured a formula for $\vol\FF_{\Car_{n+1}}(a,b,c^{n-2})$, see Proposition~\ref{prop:VolCar(a,b,c)} below. In this subsection we prove their conjecture. As before, for simplicity, we consider $\Car_{n+2}$ instead of $\Car_{n+1}$.
The Caracol graph $\Car_{n+2}$ has $3n-1$ edges and $\vec{t}':=(\outdeg(1)-1,\dots,\outdeg(n+1)-1)=(n-1,1,1,\dots,1,0)$. Note that $\vec{s}=(s_1,\dots,s_{n+1}) \geq \vec{t}'$ implies $s_{n+1}=0$. Thus, by Lidskii's formula, \begin{multline*} \vol\FF_{\Car_{n+2}}(a_1,\dots ,a_{n+1}) \\ = \sum_{\substack{s_1+\dots+s_{n}=2n-2\\(s_1,\dots,s_{n})\geq(n-1,1^{n-1})}}
\binom{2n-2}{s_{1},\dots ,s_{n}}a_{1}^{s_1}\dots a_{n}^{s_{n}}K_{\Car_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(n-1,1^{n-1})) . \end{multline*}
Our goal is to find a formula for $X:=\vol\FF_{\Car_{n+2}}(a,b,c^{n-1})$. By the above equation, \begin{multline*} X= \sum_{\substack{s_1+\dots+s_{n}=2n-2\\ (s_1,\dots,s_{n})\ge (n-1,1^{n-1})}} \binom{2n-2}{s_{1},s_{2},\dots ,s_{n}}a^{s_1}b^{s_{2}} c^{s_{3}+\dots+s_{n}}\\
\times K_{\Car_{n+2}|_{n+1}}(s_1-n+1,s_2-1,\dots,s_n-1). \end{multline*} By replacing $s_1$ by $s_1+n-2$, we obtain \begin{multline*} X=\sum_{\substack{s_1+\dots+s_{n}=n\\ (s_1,\dots,s_{n})\ge (1^{n})}} \binom{2n-2}{s_{1}+n-2,s_{2},\dots ,s_{n}}a^{s_1+n-2}b^{s_{2}} c^{s_{3}+\dots+s_{n}}\\
\times K_{\Car_{n+2}|_{n+1}}(s_1-1,\dots,s_n-1). \end{multline*} Considering $p=s_1$, $q=s_2$, and $r=s_3+\dots+s_n$ separately, we can rewrite the above equation as \begin{equation}
\label{eq:8} X = \sum_{\substack{p+q+r=n\\ (p,q)\ge (1,1)}} \binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r A(p,q,r), \end{equation} where \[ A(p,q,r)=\sum_{\substack{s_3+\dots+s_{n}=r\\(p,q,s_3,\dots,s_{n})\geq (1^{n})}}
\binom{r}{s_3,\dots,s_{n}} K_{\Car_{n+2}|_{n+1}}(p-1,q-1,s_3-1,\dots,s_n-1). \]
In the next two lemmas we find a formula for $A(p,q,r)$ using labeled Dyck paths.
Note that every Dyck path of length $2n$ can be expressed uniquely as a sequence $UD^{d_n}UD^{d_{n-1}}\dots UD^{d_1}$ of up steps $U$ and down steps $D$ for some $n$-tuple $(d_1,\dots,d_n)\in\ZZ_{\ge0}^n$ such that $d_1+\dots+d_n=n$ and $(d_1,\dots,d_n)\ge (1^n)$. For nonnegative integers $a_1,\dots,a_n$ whose sum is at most $n$, let \[ D_n(a_1,\dots,a_n) := \{UD^{d_n}UD^{d_{n-1}}\dots UD^{d_1} \in \Dyck_n: d_i\ge a_i\}. \]
\begin{lemma} \label{lem:Car1} Let $(s_1,\dots,s_n)\in \ZZ_{\geq 0}^n$ with $\sum_{i=1}^n s_i=n$ and $(s_1,\dots,s_n)\geq (1^n)$. Then \[
K_{\Car_{n+2}|_{n+1}}(s_1-1,\dots,s_{n}-1) = |D_{n-1}(s_2,\dots,s_n)|. \] \end{lemma} \begin{proof}
Note that $\Car_{n+2}|_{n+1}$ is a directed graph on $[n+1]$ with edges $(1,i)$ for $2\leq i \leq n+1$ and $(j,j+1)$ for $2\leq j \leq n$. By definition of Kostant partition function,
$K_{\Car_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(1^n))$ is the number of nonnegative integer solutions $\{b_{1,i}, b_{j,j+1}: 2\leq i \leq n+1, 2\leq j \leq n \}$ satisfying \begin{align*}
b_{1,2}+b_{1,3}+\dots+b_{1,n+1}&=s_1-1,
\\b_{2,3}-b_{1,2}&=s_2-1,
\\b_{j,j+1}-b_{j-1,j}-b_{1,j}&=s_{j}-1, \qquad (3\le j\le n)
\\-b_{n,n+1}-b_{1,n+1}&=-(s_1+\dots+s_n)+n. \end{align*} Since $\sum_{i=1}s_i = n$, we must have $b_{1,n+1}=b_{n,n+1}=0$ and
the above equations are equivalent to \begin{align*}
b_{1,2}+b_{1,3}+\dots+b_{1,n}&=s_1-1,\\
b_{j,j+1}&=(s_2+\dots+s_{j})+(b_{1,2}+\dots+b_{1,j})-(j-1), \qquad (2\le j\le n). \end{align*} Thus the integers $b_{j,j+1}$ for $2\le j\le n$ are completely determined by the integers $b_{1,i}$ for $2\le i\le n+1$. Moreover, the condition $b_{j,j+1}\ge0$ for $2\le j\le n$ is equivalent to $(s_2,\dots,s_{n})+(b_{12},\dots,b_{1n})\geq (1^{n-1})$
in dominance order. Hence $K_{\Car_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(1^n))$ is the number of $(n-1)$-tuples $(b_{12},b_{13},\dots,b_{1n})\in \ZZ_{\geq
0}^{n-1}$ such that $b_{12}+b_{13}+\dots+b_{1n}=s_1-1$ and $(s_2,\dots,s_{n})+(b_{12},\dots,b_{1n})\geq (1^{n-1})$.
Now let $d_i=s_{i+1}+b_{1,i+1}$ for $1\le i\le n-1$. Then we can reinterprete
$K_{\Car_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(1^n))$ as the number of $(n-1)$-tuples $(d_1,\dots,d_{n-1})\in \ZZ_{\geq 0}^{n-1}$ such that $d_1+\dots+d_{n-1}=n-1$, $(d_1,\dots,d_{n-1})\geq (1^{n-1})$ and $d_i\ge s_{i+1}$ for $1\le i\le n-1$. Since the condition $(d_1,\dots,d_{n-1})\geq (1^{n-1})$ is equivalent to the condition $UD^{d_{n-1}} UD^{d_{n-2}}\cdots UD^{d_{1}}\in\Dyck_{n-1}$, we obtain the desired result. \end{proof}
\begin{lemma} Let $p,q$ and $r$ be fixed nonnegative integers with $p+q+r=n$ and $(p,q)\geq (1,1)$. Then \label{lem:Car2} \[ A(p,q,r) =(p+q-1)\binom{n+p-2}{n-1}(n-1)^{r-1}-\binom{n+p-2}{n}(n-1)^r. \] \end{lemma} \begin{proof}
By Lemma~\ref{lem:Car1},
\[ A(p,q,r)= \sum_{\substack{s_3+\dots+s_{n}=r\\(p,q,s_3,\dots,s_{n})\geq (1^{n})}}
\binom{r}{s_3,\dots,s_{n}} |D_{n-1}(q,s_3,\dots,s_n)|. \]
We will give a combinatorial interpretation of each summand in the above formula using labeled Dyck paths. Let $s_3,\dots,s_n$ be nonnegative integers satisfying $s_3+\dots+s_{n}=r$ and $(p,q,s_3,\dots,s_{n})\geq (1^{n})$. Consider a Dyck path $\pi=UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{1}}\in D_{n-1}(q,s_3,\dots,s_n)$. Then $d_1\ge q$ and $d_i\ge s_{i+1}$ for $2\le i\le n-1$. Now we label the down steps of $\pi$ except the last consecutive down steps $D^{d_1}$ as follows: \begin{itemize} \item Distribute the $r$ labels $1,2,\dots,r$, each label occurring exactly once, to the sequences $D^{d_{n-1}}, D^{d_{n-2}},\dots, D^{d_2}$ consecutive down steps of $\pi$ so that the sequence $D^{d_i}$ gets $s_{i+1}$ labels. There are $\binom{r}{s_3,\dots,s_{n}}$ ways to do this. \item Add $d_i-s_{i+1}$ zero labels to the sequence $D^{d_i}$ and arrange the labels in weakly decreasing order. \end{itemize} Considering the resulting objects of this process we obtain that
$\binom{r}{s_3,\dots,s_{n}} |D_{n-1}(q,s_3,\dots,s_n)|$ is the number of Dyck paths $\pi=UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{1}}$ together with a labeling on the down steps except the last consecutive down steps $D^{d_{1}}$ satisfying the following conditions: \begin{enumerate} \item $d_i\ge s_{i+1}$ for $2\le i\le n-1$. \item $q\le d_1\le n-1-r$. \item The number of down steps labeled $i$ is $1$ for $1\le i\le r$. \item The number of down steps labeled $0$ is $n-1-r-d_1$. \item The labels of any consecutive down steps are weakly decreasing. \end{enumerate} Summing over all possible $s_3,\dots,s_n$ we obtain that $A(p,q,r)$ is the number of Dyck paths $\pi=UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{1}}$ together with a labeling on the down steps of its prefix $UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{2}}$ from $(0,0)$ to $(2n-3-d_1,d_1-1)$ satisfying the above conditions except (1). This implies \[
A(p,q,r)= \sum_{d_1=q}^{n-1-r} |\Dyck_{n-2,d_1-1}(r;n-1-r-d_1,1^{r})|. \] By Lemma~\ref{lem:dyck_n,i}, \begin{align*}
A(p,q,r) & = \sum_{d_1=q}^{n-1-r} \frac{d_1}{n-1} \Multiset{n-1}{n-1-r-d_1} (n-1)^r\\ &= (n-1)^{r-1}\sum_{d_1=q}^{n-1-r} d_1 \binom{2n-3-r-d_1}{n-2}. \end{align*} Replacing $d_1$ by $n-1-r-i$, we have \[
A(p,q,r) = (n-1)^{r-1} \sum_{i=0}^{p-1} (n-1-r-i) \binom{n-2+i}{n-2}. \] Since \begin{align*} (n-1-r-i) \binom{n-2+i}{n-2} &= \left( (2n-2-r) - (n-1+i) \right) \binom{n-2+i}{n-2}\\ &= (2n-2-r) \binom{n-2+i}{n-2} - (n-1) \binom{n-1+i}{n-1}\\ &= (n-1-r) \binom{n-2+i}{n-2} - (n-1) \binom{n-2+i}{n-1}, \end{align*} we have \[
A(p,q,r) = (n-1)^{r-1} \left((p+q-1) \sum_{i=0}^{p-1} \binom{n-2+i}{n-2} -(n-1)\sum_{i=0}^{p-2}\binom{n-1+i}{n-1} \right). \] Finally the identity $\sum_{i=0}^{k}\binom{m+i}{m}=\binom{m+k+1}{m+1}$ finishes the proof. \end{proof}
Now we are ready to compute $X=\vol\FF_{\Car_{n+2}}(a,b,c^{n-1})$.
\begin{proposition}\cite[Conjecture~6.16]{Benedetti2019} \label{prop:VolCar(a,b,c)} For positive integers $a$, $b$, and $c$, we have \[ \vol\FF_{\Car_{n+2}}(a,b,c^{n-1})=C_{n-1}a^{n-1}(a+nb)(a+b+(n-1)c)^{n-2}. \] \end{proposition} \begin{proof}
By \eqref{eq:8} and Lemma~\ref{lem:Car2}, we have
\[
\vol\FF_{\Car_{n+2}}(a,b,c^{n-1}) = X = Y-Z, \] where \[ Y = \sum_{\substack{p+q+r=n\\ (p,q)\ge (1,1)}}
\binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r (p+q-1)\binom{n+p-2}{n-1}(n-1)^{r-1}, \] \[ Z = \sum_{\substack{p+q+r=n\\ (p,q)\ge (1,1)}} \binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r \binom{n+p-2}{n}(n-1)^r. \] Note that in the above two sums, the condition $(p,q)\ge(1,1)$ can be omitted since the summand is zero if $p=0$ or $(p,q)=(1,0)$. Thus \begin{align*} Y &= \frac{a^{n-1}}{n-1}\binom{2n-2}{n-1} \sum_{p+q+r=n} (p+q-1) \binom{n-1}{p-1,q,r}a^{p-1}b^q (c(n-1))^r,\\ Z &= \sum_{p+q+r=n} \binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r \binom{n+p-2}{n}(n-1)^r\\ &=a^{n}\binom{2n-2}{n} \sum_{p+q+r=n} \binom{n-2}{p-2,q,r}a^{p-2}b^q (c(n-1))^r . \end{align*} Using the multinomial theorem \[ \sum_{i+j+k=m} \binom{m}{i,j,k}x^iy^j z^k t^{i+j} = (xt+yt+z)^m, \] and its derivative with respect to $t$, i.e, \[ \sum_{i+j+k=m} (i+j) \binom{m}{i,j,k} x^i y^j z^k t^{i+j-1} = m(x+y)(xt+yt+z)^{m-1}, \] we obtain \begin{align*} Y &= C_{n-1}a^{n-1}n(a+b)(a+b+(n-1)c)^{n-2},\\ Z &=C_{n-1}a^n(n-1)(a+b+(n-1)c)^{n-2}, \end{align*} and the proof follows. \end{proof}
\section*{Acknowledgments} The authors would like to thank Alejandro Morales for informing them that Theorems~\ref{thm:PS} and \ref{thm:main} are equivalent to Theorems~\ref{Thm:PS} and \ref{Thm:Car}. They also thank Nathan Williams for helpful discussion.
\end{document} | arXiv | {
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\begin{document}
\title {
On the skein polynomial for links
} \author {
Boju Jiang
} \author {
Jiajun Wang
} \author {
Hao Zheng
} \address {
Department of Mathematics\\
Peking University\\
Beijing 100871\\
China
} \email {
bjjiang@math.pku.edu.cn
} \email {
wjiajun@math.pku.edu.cn
} \email {
hzheng@math.pku.edu.cn
} \thanks {Partially supported by NSFC grant \#11131008}
\subjclass [2010]{Primary 57M25; Secondary 20F36}
\keywords {the skein polynomial, HOMFLY polynomial, Jones polynomial, Alexander-Conway polynomial, skein relations}
\date{}
\begin{abstract} We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual ``smoothing of a crossing'' move. As by-products we have characterizations of these polynomials for knots, and for links with any given number of components. \end{abstract}
\maketitle
\section {Introduction} \label{sec:Intro}
The skein polynomial (as called in \cite[Chapter 8]{K1}, also known as HOMFLY or HOMFLY-PT polynomial), $P_L(a,z) \in \mathbb Z[a^{\pm1},z^{\pm1}]$, is an invariant for oriented links. Here $\mathbb Z[a^{\pm1},z^{\pm1}]$ is the ring of Laurent polynomials in two variables $a$ and $z$, with integer coefficients. It is defined to be the invariant of oriented links satisfying the axioms \begin{gather*} a^{-1}\cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1a_dd-0} \end{pmatrix} -a\cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1b_dd-0} \end{pmatrix} =z\cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1o_dd-0} \end{pmatrix} ; \tag*{\rm(I)} \\ P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/O-0} \end{pmatrix} =1 . \tag*{\rm(O)} \end{gather*}
The Alexander-Conway polynomial $\Delta_L \in \mathbb Z[t^{\pm\frac12}]$ and the Jones polynomial $V_L \in \mathbb Z[t^{\pm\frac12}]$ are related to the skein polynomial: \[ \Delta_L(t)=P_L(1,t^{\frac12}-t^{-\frac12}), \qquad V_L(t)=P_L(t,t^{\frac12}-t^{-\frac12}). \]
Our main result is
\begin{thm} \label{thm:HOMFLY} The skein polynomial $P_L \in \mathbb Z[a^{\pm1},z^{\pm1}]$ is the invariant of oriented links determined uniquely by the following four axioms. {\allowdisplaybreaks \begin{gather*} a^{-2}\cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1a1a_dd-0} \end{pmatrix} +a^2\cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1b1b_dd-0} \end{pmatrix} =(2+z^2) \cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1oo_dd-0} \end{pmatrix} ; \tag*{\rm(II)} \\ a^{-1}\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0} \end{pmatrix} -a\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0} \end{pmatrix} = a^{-1}\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0} \end{pmatrix} -a\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0} \end{pmatrix} ; \tag*{\rm(III)} \\ P \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/IO-0} \end{pmatrix} =z^{-1}(a^{-1}-a)\cdot P \begin{pmatrix} \;\; \includegraphics[height=1.2cm]{fig_012/I-0} \;\; \end{pmatrix} ; \tag*{\rm(IO)} \\ P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/O-0} \end{pmatrix} =1 . \tag*{\rm(O)} \end{gather*} } \end{thm}
A parallel result is for the Jones polynomial. It is not a direct corollary of the above theorem, because the substitutions $a\mapsto t$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$ do not send $\mathbb Z[a^{\pm1},z^{\pm1}]$ into $Z[t^{\pm\frac12}]$.
\begin{thm} \label{thm:Jones} The Jones polynomial $V_L \in \mathbb Z[t^{\pm\frac12}]$ is the invariant of oriented links determined uniquely by the following four axioms. {\allowdisplaybreaks \begin{gather*} t^{-2}\cdot V \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1a1a_dd-0} \end{pmatrix} +t^2\cdot V \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1b1b_dd-0} \end{pmatrix} =(t+t^{-1}) \cdot V \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1oo_dd-0} \end{pmatrix} ; \tag*{\rm(II$_V$)} \\ t^{-1}\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0} \end{pmatrix} -t\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0} \end{pmatrix} = t^{-1}\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0} \end{pmatrix} -t\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0} \end{pmatrix} ; \tag*{\rm(III$_V$)} \\ V \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/IO-0} \end{pmatrix} =-(t^{\frac12}+t^{-\frac12})\cdot V \begin{pmatrix} \;\; \includegraphics[height=1.2cm]{fig_012/I-0} \;\; \end{pmatrix} ; \tag*{\rm(IO$_V$)} \\ V \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/O-0} \end{pmatrix} =1 . \tag*{\rm(O$_V$)} \end{gather*} } \end{thm}
For the Alexander-Conway polynomial, the result takes a slightly different form. We switch to a ($\Phi$)-type axiom because the (IO)-type one degenerates into a consequence of (II) and (III) (see Corollary~\ref{cor:stabilization_Alexander-Conway}).
\begin{thm} \label{thm:Alexander-Conway} The Alexander-Conway polynomial $\Delta_L \in \mathbb Z[t^{\pm\frac12}]$ is the invariant of oriented links determined uniquely by the following four axioms. {\allowdisplaybreaks \begin{gather*} \Delta \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1a1a_dd-0} \end{pmatrix} + \Delta \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1b1b_dd-0} \end{pmatrix} =(t+t^{-1}) \cdot \Delta \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1oo_dd-0} \end{pmatrix} ; \tag*{\rm(II$_\Delta$)} \\ \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0} \end{pmatrix} - \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0} \end{pmatrix} = \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0} \end{pmatrix} - \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0} \end{pmatrix} ; \tag*{\rm(III$_\Delta$)} \\ \Delta \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/Phi+-0} \end{pmatrix} =(t^{\frac12}-t^{-\frac12})\cdot \Delta \begin{pmatrix} \;\; \includegraphics[height=1.2cm]{fig_012/I-0} \;\; \end{pmatrix} ; \tag*{($\Phi_\Delta$)} \\ \Delta \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/O-0} \end{pmatrix} =1 . \tag*{\rm(O$_\Delta$)} \end{gather*} } \end{thm}
If we restrict our attention to oriented links with a fixed number $\mu>0$ of components, the axiom (IO) becomes irrelevant but we must pick a suitable normalization. Let $U_\mu$ denote the $\mu$-component unlink, and let $C_\mu$ denote the $\mu$-component oriented chain where adjacent rings have linking number $+1$. (In terms of closed braids, $U_\mu$ is the closure of the trivial braid $e\in B_\mu$, and $C_\mu$ is the closure of the braid $\sigma_1^2\sigma_2^2\dots\sigma_{\mu-1}^2\in B_\mu$.) We can use either $U_\mu$ or $C_\mu$ (but $U_\mu$ is preferred) to normalize the skein or Jones polynomial, but for Alexander-Conway polynomial we can only use $C_\mu$.
\begin{thm} \label{thm:HOMFLY_knot} The skein polynomial $P_L$ is the invariant of oriented $\mu$-compo\-nent links determined uniquely by the following three axioms. {\allowdisplaybreaks \begin{gather*} a^{-2}\cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1a1a_dd-0} \end{pmatrix} +a^2\cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1b1b_dd-0} \end{pmatrix} =(2+z^2) \cdot P \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1oo_dd-0} \end{pmatrix} ; \tag*{\rm(II)} \\ a^{-1}\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0} \end{pmatrix} -a\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0} \end{pmatrix} = a^{-1}\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0} \end{pmatrix} -a\cdot P \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0} \end{pmatrix} ; \tag*{\rm(III)} \\[2ex] P(U_\mu) = (z^{-1}(a^{-1}-a))^{\mu-1} . \tag*{\rm(U)} \end{gather*} } \end{thm}
\begin{thm} \label{thm:Jones_knot} The Jones polynomial $V_K$ is the invariant of oriented $\mu$-component links determined uniquely by the following three axioms. {\allowdisplaybreaks \begin{gather*} t^{-2}\cdot V \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1a1a_dd-0} \end{pmatrix} +t^2\cdot V \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1b1b_dd-0} \end{pmatrix} =(t+t^{-1}) \cdot V \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1oo_dd-0} \end{pmatrix} ; \tag*{\rm(II$_V$)} \\ t^{-1}\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0} \end{pmatrix} -t\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0} \end{pmatrix} = t^{-1}\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0} \end{pmatrix} -t\cdot V \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0} \end{pmatrix} ; \tag*{\rm(III$_V$)} \\[2ex] V(U_\mu) = (-(t^{\frac12}+t^{-\frac12}))^{\mu-1} . \tag*{\rm(U$_V$)} \end{gather*} } \end{thm}
\begin{thm} \label{thm:Alexander-Conway_knot} The Alexander-Conway polynomial $\Delta_K$ is the invariant of oriented $\mu$-component links determined uniquely by the following three axioms. {\allowdisplaybreaks \begin{gather*} \Delta \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1a1a_dd-0} \end{pmatrix} + \Delta \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1b1b_dd-0} \end{pmatrix} =(t+t^{-1}) \cdot \Delta \begin{pmatrix} \includegraphics[width=.8cm]{fig_012/1oo_dd-0} \end{pmatrix} ; \tag*{\rm(II$_\Delta$)} \\ \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0} \end{pmatrix} - \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0} \end{pmatrix} = \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0} \end{pmatrix} - \Delta \begin{pmatrix} \includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0} \end{pmatrix} ; \tag*{\rm(III$_\Delta$)} \\ \Delta(C_\mu) = (t^{\frac12}-t^{-\frac12})^{\mu-1} . \tag*{\rm(C$_\Delta$)} \end{gather*} } \end{thm}
Note that the foundational relation (I) cannot appear in Theorems~\ref{thm:HOMFLY_knot}--\ref{thm:Alexander-Conway_knot} because it involves links with different number of components.
Our approach is via closed braids. We explain the language of relators in Section~\ref{sec:relators} and give an algebraic reduction lemma in Section~\ref{sec:key_lem}. This approach is adapted from the corresponding sections of \cite{J1} on Conway's potential function for colored links. The current context of uncolored links makes the reduction argument more transparent. Section~\ref{sec:stabilizations} discusses closed braids with different number of strands. The theorems are proved in the last two sections.
\section{Braids and skein relators} \label{sec:relators}
For braids, we use the following conventions: Braids are drawn from top to bottom. The strands of a braid are numbered at the top of the braid, from left to right. The product $\beta_1\cdot\beta_2$ of two $n$-braids is obtained by drawing $\beta_2$ below $\beta_1$. The set $B_n$ of all $n$-braids forms a group under this multiplication, with standard generators $\sigma_1,\sigma_2,\dots,\sigma_{n-1}$.
It is well known that links can be presented as closed braids. The closure of a braid $\beta\in B_n$ will be denoted $\widehat\beta$. Two braids (possibly with different number of strands) have isotopic closures if and only if they can be related by a finite sequence of two types of moves: \begin{enumerate} \item Conjugacy move: $\beta$ $\leftrightsquigarrow$ $\beta'$ where $\beta,\beta'$ are conjugate in a braid group $B_n$; \item Markov move: $\beta\in B_n$ $\leftrightsquigarrow$ $\beta\sigma_n^{\pm1}\in B_{n+1}$. \end{enumerate}
Let $\Lambda$ be the Laurent polynomial ring $\mathbb Z[a^{\pm1},z^{\pm1}]$. Let $\Lambda B_n$ be the group-algebra on $B_n$ with coefficients in $\Lambda$.
\begin{defn} \label{defn:skein_relation} We say that an element \[ \lambda_1\cdot\beta_1+\dots+\lambda_k\cdot\beta_k \] of $\Lambda B_n$ is a \emph{skein relator}, or equivalently, say that the corresponding formal equation (in which $P_{L_{\beta_h}}$ stands for the $P$ of the link $L_{\beta_h}$) \[ \lambda_1\cdot P_{L_{\beta_1}}+\dots+\lambda_k\cdot P_{L_{\beta_k}}=0 \] is a \emph{skein relation}, if the following condition is satisfied: For any links $L_{\beta_1},\dots,L_{\beta_k}$ that are identical except in a cylinder where they are represented by the braids $\beta_1,\dots,\beta_k$ respectively, the formal equation becomes an equality in $\Lambda$. \end{defn}
\begin{exam} \label{exam:relator_vs_relation} To every element $\lambda_1\cdot\beta_1+\dots+\lambda_k\cdot\beta_k\in \Lambda B_n$, by taking braid closures we have a corresponding element \[ \lambda_1\cdot P_{\widehat\beta_1}+\dots +\lambda_k\cdot P_{\widehat\beta_k} \in \Lambda. \] The latter vanishes if the former is a skein relator. \end{exam}
\begin{exam} The skein relations (I), (II) and (III) in Section~\ref{sec:Intro} correspond to the following relators, respectively: (The symbol $e$ stands for the trivial braid.) {\allowdisplaybreaks \begin{gather*} \text{\rm(I$_\text{B}$)}:= a^{-1}\cdot\sigma_1^2 -a\cdot\sigma_1^{-2} -z\cdot{e} ; \\ \text{\rm(II$_\text{B}$)}:= a^{-2}\cdot\sigma_1^2 +a^2\cdot\sigma_1^{-2} -(2+z^2)\cdot{e} ; \\ \text{\rm(III$_\text{B}$)}:= \begin{aligned}[t] &a^{-1}\cdot{\sigma_1\sigma_2\sigma_1^{-1}} + a\cdot{\sigma_1^{-1}\sigma_2^{-1}\sigma_1} \\ &- a^{-1}\cdot{\sigma_1^{-1}\sigma_2\sigma_1} - a\cdot{\sigma_1\sigma_2^{-1}\sigma_1^{-1}} . \end{aligned} \end{gather*} } \end{exam}
\begin{prop} \label{prop:relator_ideal} Assume that \[ \lambda_1\cdot P_{L_{\beta_1}}+\dots+\lambda_k\cdot P_{L_{\beta_k}}=0
\] is a skein relation. Then for any given braid $\alpha\in B_n$, the following equations are also skein relations: \begin{gather*} \lambda_1\cdot P_{L_{(\beta_1\alpha)}}+\dots+\lambda_k\cdot P_{L_{(\beta_k\alpha)}}=0;
\\ \lambda_1\cdot P_{L_{(\alpha\beta_1)}}+\dots+\lambda_k\cdot P_{L_{(\alpha\beta_k)}}=0.
\end{gather*}
Hence skein relators form a two-sided ideal\/ $\mathfrak R_n$ (called the \emph{relator ideal\/}) in $\Lambda B_n$. \end{prop}
\begin{proof} Look at the cylinder where the links $L_{(\beta_1\alpha)},\dots,L_{(\beta_k\alpha)}$ are represented differently by braids $\beta_1\alpha,\dots,\beta_k\alpha$, respectively. In the upper half cylinder they are represented by braids $\beta_1,\dots,\beta_k$. So the assumption implies the first equality. Similarly for the second equality.
By Definition~\ref{defn:skein_relation}, this means skein relators form a two-sided ideal. \end{proof}
\section{An algebraic reduction lemma} \label{sec:key_lem}
\begin{defn} \label{defn:equivalence} Let $\mathfrak I_n$ be the two-sided ideal in $\Lambda B_n$ generated by $\text{\rm(II$_\text{B}$)}$ and $\text{\rm(III$_{\text{B}}$)}$. (When $n=2$ we ignore $\text{\rm(III$_{\text{B}}$)}$.)
Two elements of the algebra $\Lambda B_n$ are \emph {equivalent modulo $\mathfrak I_n$} (denoted by $\sim$\,) if their difference is in $\mathfrak I_n$. \end{defn}
For example, by conjugation in $B_n$ we have $a^{-2}\cdot \sigma_i^2 +a^2\cdot \sigma_i^{-2} -(2+z^2)\cdot e \sim 0$ and $a^{-1}\cdot\sigma_i\sigma_{i+1}\sigma_i^{-1} +a\cdot \sigma_i^{-1}\sigma_{i+1}^{-1}\sigma_i -a^{-1}\cdot \sigma_i^{-1}\sigma_{i+1}\sigma_i -a\cdot \sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1} \sim 0$, for any $i$.
\begin{lem} \label{lem:reduction} Modulo $\mathfrak I_n$, every braid $\beta\in B_n$ is equivalent to a $\Lambda$-linear combination of braids of the form $\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ and $k\in\{0,\pm1,2\}$. \end{lem}
A braid $\beta\in B_n$ can be written as \[ \beta=\beta_0\sigma_{n-1}^{k_1} \beta_1\sigma_{n-1}^{k_2} \dots \sigma_{n-1}^{k_r}\beta_r \] where $\beta_j\in B_{n-1}$ and $k_j\neq0$. We allow that $\beta_0$ and $\beta_r$ be trivial, but assume other $\beta_j$'s are nontrivial. The number $r$ will be denoted as $r(\beta)$.
The lemma will be proved by an induction on the double index $(n,r)$. Note that the lemma is trivial when $n=2$, or $r(\beta)\leq1$.
It is enough to consider the case $r=2$, because induction on $r$ works beyond $2$. Indeed, if $r(\beta)>2$, let $\beta'=\beta_1\sigma_{n-1}^{k_2} \dots \sigma_{n-1}^{k_r}\beta_r$, then $r(\beta')<r(\beta)$. By inductive hypothesis $\beta'$ is equivalent to a linear combination of elements of the form $\alpha'\sigma_{n-1}^{k'} \gamma'$, hence $\beta$ is equivalent to a linear combination of elements of the form $\beta_0\sigma_{n-1}^{k_1} \alpha'\sigma_{n-1}^{k'} \gamma'$. This brings the problem back to the $r=2$ case. Henceforth we assume $r=2$.
Since the initial and terminal part of $\beta$, namely $\beta_0$ and $\beta_r$, do not affect the conclusion of the lemma, we can drop them. So we assume $\beta=\sigma_{n-1}^{k_1} \beta_1\sigma_{n-1}^{k_2}$, where $\beta_1\in B_{n-1}$.
By the induction hypothesis on $n$, $\beta_1\in B_{n-1}$ is a linear combination of elements of the form $\alpha_1\sigma_{n-2}^\ell \gamma_1$. Note that $\alpha_1,\gamma_1\in B_{n-2}$ commute with $\sigma_{n-1}$. So it suffices to focus on braids of the form $\beta=\sigma_{n-1}^{k}\sigma_{n-2}^{\ell}\sigma_{n-1}^{m}$.
For the sole purpose of controlling the length of displayed formulas, we assume $n=3$ below. The proof for a general $n$ can be obtained by a simple change of subscripts, replacing $\sigma_1,\sigma_2$ with $\sigma_{n-2},\sigma_{n-1}$ and replacing $t_1,t_2,t_3$ with $t_{n-2},t_{n-1},t_{n}$, respectively.
Thus, Lemma~\ref{lem:reduction} has been reduced to the following \begin{lem} \label{lem:key_reduction} Every $\sigma_2^{k}\sigma_1^{\ell}\sigma_2^{m}$ is equivalent \textup{(modulo $\mathfrak I_n$)} to a linear combination of braids of the form $\sigma_1^{k'}\sigma_2^{\ell'}\sigma_1^{m'}$ where $\ell'$ is $0$, $\pm1$ or $2$. \end{lem}
\begin{proof} Modulo $\text{(II$_\text{B}$)}$, we may restrict the exponent $k$ to take values $1$, $2$ and $3$ (we are done if $k$ is $0$). If $k>1$ we can decrease $k$ by looking at $\sigma_2^{k-1}(\sigma_2\sigma_1^{\ell}\sigma_2^m)$, so it suffices to prove the case $k=1$. Again modulo $\text{(II$_\text{B}$)}$, we can restrict the exponents $\ell, m$ to the values $\pm1$ and $2$. There are altogether 9 cases to verify.
{\it 5 trivial cases (braid identities) }: \begin{alignat*}{3} &\sigma_2\sigma_1\sigma_2=\sigma_1\sigma_2\sigma_1, \quad &&\sigma_2\sigma_1\sigma_2^{-1}=\sigma_1^{-1}\sigma_2\sigma_1, \quad &&\sigma_2\sigma_1^{-1}\sigma_2^{-1}=\sigma_1^{-1}\sigma_2^{-1}\sigma_1, \quad \\ &\sigma_2\sigma_1\sigma_2^2=\sigma_1^2\sigma_2\sigma_1, \quad &&\sigma_2\sigma_1^2\sigma_2^{-1}=\sigma_1^{-1}\sigma_2^2\sigma_1. \quad \end{alignat*}
{\it The case $\sigma_2\sigma_1^{-1}\sigma_2$ }: Multiplying $\text{\rm(III$_{\text{B}}$)}$ by $\sigma_2$ on the right and $\sigma_1^{-1}$ on the left, and taking braid identities into account, we get the relation \[ a^{-1}\cdot \sigma_2\sigma_1^{-1}\sigma_2 + a\cdot \sigma_1^{-1}\sigma_2\sigma_1^{-1} - a^{-1}\cdot \sigma_1^{-1}\sigma_2\sigma_1 - a\cdot \sigma_1\sigma_2^{-1}\sigma_1^{-1} \sim 0. \] Then $\sigma_2\sigma_1^{-1}\sigma_2$ is equivalent to a linear combination of braids of the form $\sigma_1^{\pm1}\sigma_2^{\pm1}\sigma_1^{\pm1}$. So the case $\sigma_2\sigma_1^{-1}\sigma_2$ is verified.
{\it The case $\sigma_2\sigma_1^{-1}\sigma_2^2$ }: Multiplying the previous relation by $\sigma_2$ on the right, and taking braid identities into account, we see that \[ a^{-1}\cdot \sigma_2\sigma_1^{-1}\sigma_2^2 + a\cdot \sigma_1^{-1}(\sigma_2\sigma_1^{-1}\sigma_2) - a^{-1}\cdot \sigma_2\sigma_1 - a\cdot \sigma_1^2\sigma_2^{-1}\sigma_1^{-1} \sim 0. \] Similar to the above case, this reduces $\sigma_2\sigma_1^{-1}\sigma_2^2$ to the verified case $\sigma_2\sigma_1^{-1}\sigma_2$.
{\it The case $\sigma_2\sigma_1^2\sigma_2$ }: Multiplying $\text{\rm(III$_{\text{B}}$)}$ on the right by $\sigma_1\sigma_2\sigma_1$, we get \[ a^{-1}\cdot \sigma_1\sigma_2^2\sigma_1 + a\cdot \sigma_2^2 - a^{-1}\cdot \sigma_2\sigma_1^2\sigma_2 - a\cdot \sigma_1^2 \sim 0. \tag*{$\text{\rm(III$'_{\text{B}}$)}$} \] This verifies the case $\sigma_2\sigma_1^2\sigma_2$.
{\it The case $\sigma_2\sigma_1^2\sigma_2^2$ }: Multiplying $\text{\rm(III$'_{\text{B}}$)}$ by $\sigma_2$ on the right, we get \[ a^{-1}\cdot \sigma_1^2\sigma_2\sigma_1^2 + a\cdot \sigma_2^3 - a^{-1}\cdot \sigma_2\sigma_1^2\sigma_2^2 - a\cdot \sigma_1^2\sigma_2 \sim 0. \] The case $\sigma_2\sigma_1^2\sigma_2^2$ is also verified.
We have verified all 9 cases. Modulo $\text{\rm(II$_\text{B}$)}$ we can assume $\ell'\in\{0,\pm1,2\}$. Thus Lemma~\ref{lem:key_reduction} is proved.
The inductive proof of Lemma~\ref{lem:reduction} is now complete. \end{proof}
The resulting $\Lambda$-linear combination of braids of the form $\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ in the Lemma is not unique, but the inductive proof gives us a recursive algorithm to find one.
To compare the ideal $\mathfrak I_n$ with the relator ideal $\mathfrak R_n$ of Section~\ref{sec:relators}, we have
\begin{prop} $\mathfrak I_n\subset \mathfrak R_n$ but $\mathfrak I_n\neq \mathfrak R_n$. \end{prop}
\begin{proof} The inclusion is easy. Indeed, $(\text{I}_\text{B})$ is in the relator ideal $\mathfrak R_n$, and \begin{gather*} \text{\rm(II$_\text{B}$)}= \text{\rm(I$_\text{B}$)}^2 +2z\cdot \text{\rm(I$_\text{B}$)} , \\ \text{\rm(III$_\text{B}$)}= \sigma_2^{-1}\cdot \text{\rm(I$_\text{B}$)} \cdot\sigma_2 -\sigma_2\cdot\text{\rm(I$_\text{B}$)} \cdot\sigma_2^{-1} . \end{gather*} So both $\text{\rm(II$_\text{B}$)}$ and $\text{\rm(III$_\text{B}$)}$ are in $\mathfrak R_n$. Therefore $\mathfrak I_n\subset \mathfrak R_n$.
To show they are not equal, we need the notion of homogeneity. Each $n$-braid $\beta$ has an \emph{underlying permutation} of $\{1,\dots,n\}$, denoted $i\mapsto i^{\beta}$, where $i^{\beta}$ is the position of the $i$-th strand at the bottom of $\beta$. In this way the braid group $B_n$ projects onto the symmetric group $\mathfrak S_n$. An element of $\Lambda B_n$ is called \emph{homogeneous} if all its terms (with nonzero coefficients) have the same underlying permutation. As a $\Lambda$-module, $\Lambda B_n$ splits into a direct sum according to underlying permutations of braids. Under this splitting, every element of $\Lambda B_n$ decomposes into a sum of its \emph{homogeneous components}.
Since $\text{\rm(II$_\text{B}$)}$ and $\text{\rm(III$_\text{B}$)}$ are homogeneous, the ideal $\mathfrak I_n\subset\Lambda B_n$ is generated by homogeneous elements. Then every homogeneous component of any element of $\mathfrak I_n$ is also in $\mathfrak I_n$. Now the relator $\text{\rm(I$_\text{B}$)}\in \mathfrak R_n$ has a homogeneous component $-z\cdot e$ which is not a relator. Hence $\text{\rm(I$_\text{B}$)}$ is not in $\mathfrak I_n$. Thus $\mathfrak I_n$ is strictly smaller than $\mathfrak R_n$. \end{proof}
\section{Stabilizations} \label{sec:stabilizations}
Suppose a braid $\beta\in B_n$ is written as a word in the standard generators $\sigma_1$, $\sigma_2$, \dots, $\sigma_{n-1}$. The same word $\beta$ gives a braid in $B_{n+k}$ for any $k\ge0$. Thus $B_n$ is standardly embedded in $B_{n+k}$.
However, when talking about a closed braid $\widehat\beta$, the number of strands in $\beta$ does matter. We shall use the notation $[\beta]_n$ to emphasize that $\beta$ is regarded as an $n$-braid, and use $[\beta]_n^{\;\widehat{}}$ for its closure. For example, $[\beta]_{n+1}^{\;\widehat{}}$ adds a free circle to $[\beta]_n^{\;\widehat{}}$. The Markov move says $[\beta\sigma_n^{\pm1}]_{n+1}^{\;\widehat{}}$ is isotopic to $[\beta]_n^{\;\widehat{}}$.
For a braid $\beta\in B_n$ and an integer $k\ge0$, we shall use $\beta^{\triangleright k} \in B_{n+k}$ to denote the $k$-th shifted version of $\beta$, i.e., the braid obtained from the word $\beta$ by replacing each generator $\sigma_i$ with $\sigma_{i+k}$. Its closure $[\beta^{\triangleright k}]_{n+k}^{\;\widehat{}}$ is isotopic to $[\beta]_{n+k}^{\;\widehat{}}$.
Suppose $\beta,\beta'\in B_n$ and $\gamma\in B_p$. Observe from the diagram defining braid closure that the closed braid $[\beta\sigma_n^{\pm1} \gamma^{\triangleright n}\beta']_{n+p}^{\;\widehat{}}$ is isotopic to $[\beta\gamma^{\triangleright (n-1)}\beta']_{n+p-1}^{\;\widehat{}}$ (which is in fact a connected sum of oriented links $[\beta\beta']_n^{\;\widehat{}}$ and $[\gamma]_p^{\;\widehat{}}$). By an abuse of language, we will call this a \emph{Markov move}.
If $\beta'$ brings the $n$-th position at its top to the same position at its bottom, then $[\beta\gamma^{\triangleright (n-1)}\beta']_{n+p-1}^{\;\widehat{}}$ is isotopic to $[\beta\beta'\gamma^{\triangleright (n-1)}]_{n+p-1}^{\;\widehat{}}$. We will refer to it as a \emph{slide} move (in the connected sum, sliding $[\gamma]_p^{\;\widehat{}}$ down the last strand of $\beta'$).
\begin{lem} \label{lem:stabilization3} Assume that $P_L \in \mathbb Z[a^{\pm1},z^{\pm1}]$ is an invariant of oriented links that satisfies skein relations \textup{(II)} and \textup{(III)}. Then for $\beta\in B_n$ and $\gamma\in B_p$ we have \[ (1+z^2-a^2)\cdot P\left( [\beta\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) = (a^{-2}-1)\cdot P\left( [\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) . \] \end{lem}
\begin{proof} The braid form of axioms (II) and (III) are the relators (II$_{\text{B}}$) and (III$_{\text{B}}$), respectively. Multiplying (III$_{\text{B}}$) by $\sigma_2\sigma_1^{-1}$ on the right we get another relator \[ a^{-1}\cdot \sigma_2^{-2}\sigma_1^2\sigma_2 + a\cdot \sigma_2\sigma_1^{-2} - a^{-1}\cdot \sigma_2 - a\cdot \sigma_1^2\sigma_2^{-1}\sigma_1^{-2} . \] It gives us an equality between the $P$'s of closed $(n+p+1)$-braids: \begin{align*} &a^{-1}\cdot P\left( [\beta(\sigma_{n+1}^{-2}\sigma_{n}^2\sigma_{n+1})\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} \right) + a\cdot P\left( [\beta(\sigma_{n+1}\sigma_{n}^{-2})\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} \right) \\ & - a^{-1}\cdot P\left( [\beta\sigma_{n+1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} \right) - a\cdot P\left( [\beta(\sigma_{n}^2\sigma_{n+1}^{-1}\sigma_{n}^{-2})\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} \right) =0 . \end{align*} These closed braids can be simplified via isotopy moves (c=conjugacy, M=Markov and s=slide): \begin{align*} [\beta\sigma_{n+1}^{-2}\sigma_{n}^2\sigma_{n+1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} & \overset{\text{c}}\rightsquigarrow [\beta\sigma_{n}^2\sigma_{n+1}\gamma^{\triangleright (n+1)}\sigma_{n+1}^{-2}]_{n+p+1}^{\;\widehat{}} \\ & \overset{\text{s}}{\rightsquigarrow} [\beta\sigma_{n}^2\sigma_{n+1}^{-1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} \overset{\text{M}}{\rightsquigarrow} [\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} ; \\ [\beta\sigma_{n+1}\sigma_{n}^{-2}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} & \overset{\text{M}}{\rightsquigarrow} [\beta\gamma^{\triangleright n}\sigma_{n}^{-2}]_{n+p}^{\;\widehat{}} \overset{\text{s}}\rightsquigarrow [\beta\sigma_{n}^{-2}\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} ; \\ [\beta\sigma_{n+1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} & \overset{\text{M}}{\rightsquigarrow} [\beta\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} ; \\ [\beta\sigma_{n}^2\sigma_{n+1}^{-1}\sigma_{n}^{-2}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\widehat{}} & \overset{\text{M}}{\rightsquigarrow} [\beta\sigma_{n}^2\gamma^{\triangleright n}\sigma_{n}^{-2}]_{n+p}^{\;\widehat{}} \overset{\text{s}}{\rightsquigarrow} [\beta\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} . \end{align*} Since $P_L$ is isotopy invariant, the above equality becomes \begin{gather*} a^{-1}\cdot P\left( [\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) + a\cdot P\left( [\beta\sigma_{n}^{-2}\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) = (a^{-1}+a)\cdot P\left( [\beta\gamma^{\triangleright n}]_{n+p}]_{n+1}^{\;\widehat{}} \right) . \\ \intertext{Comparing it with the equality (from (II$_{\text{B}}$))} a^{-2}\cdot P\left( [\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) + a^2\cdot P\left( [\beta\sigma_{n}^{-2}\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) = (2+z^2)\cdot P\left( [\beta\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) , \end{gather*} we get the desired conclusion. \end{proof}
\begin{cor} \label{cor:stabilization} Under the assumption of the above lemma, the following two relations are equivalent to each other: \begin{gather*} P \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/IO-0} \end{pmatrix} =z^{-1}(a^{-1}-a)\cdot P \begin{pmatrix} \;\; \includegraphics[height=1.2cm]{fig_012/I-0} \;\; \end{pmatrix} ; \tag*{\rm(IO)} \\ P \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/Phi+-0} \end{pmatrix} =az^{-1}(1+z^2-a^2)\cdot P \begin{pmatrix} \;\; \includegraphics[height=1.2cm]{fig_012/I-0} \;\; \end{pmatrix} . \tag*{($\Phi$)} \end{gather*} \end{cor}
\begin{proof} The braid form of these two relations are, respectively, \begin{alignat}{2} P\left( [\beta]_{n+1}^{\;\widehat{}} \right) &= z^{-1}(a^{-1}-a)\cdot P\left( [\beta]_{n}^{\;\widehat{}} \right) &&\quad \text{for any braid } \beta\in B_n; \tag*{(IO$_{\text{B}}$)} \\ P\left( [\beta\sigma_{n}^2]_{n+1}^{\;\widehat{}} \right) &= az^{-1}(1+z^2-a^2)\cdot P\left( [\beta]_{n}^{\;\widehat{}} \right) &&\quad \text{for any braid } \beta\in B_n. \tag*{($\Phi_{\text{B}}$)} \end{alignat} They are equivalent to each other by the above lemma with $[\gamma]_p:=[e]_1$. \end{proof}
There is a parallel statement for Jones polynomial:
\begin{cor} \label{cor:stabilization_Jones} Assume that $V_L \in \mathbb Z[t^{\pm\frac12}]$ is an invariant of oriented links that satisfies skein relations \textup{(II$_V$)} and \textup{(III$_V$)}. Then the following two relations are equivalent to each other: \begin{gather*} V \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/IO-0} \end{pmatrix} =-(t^{\frac12}+t^{-\frac12})\cdot V \begin{pmatrix} \;\; \includegraphics[height=1.2cm]{fig_012/I-0} \;\; \end{pmatrix} ; \tag*{\rm(IO$_V$)} \\ V \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/Phi+-0} \end{pmatrix} =-t^{\frac32}(t+t^{-1})\cdot V \begin{pmatrix} \;\; \includegraphics[height=1.2cm]{fig_012/I-0} \;\; \end{pmatrix} . \tag*{($\Phi_V$)} \end{gather*} \end{cor}
For the Alexander-Conway polynomial, we have:
\begin{cor} \label{cor:stabilization_Alexander-Conway} Assume that $\Delta_L \in \mathbb Z[t^{\pm\frac12}]$ is an invariant of oriented links that satisfies skein relations \textup{(II$_\Delta$)} and \textup{(III$_\Delta$)}. Then $\Delta(L)=0$ for any split link $L$. In particular, the following relation holds true: \[ \Delta \begin{pmatrix} \includegraphics[height=1.2cm]{fig_012/IO-0} \end{pmatrix} =0 . \tag*{\rm(IO$_\Delta$)} \] \end{cor}
\begin{proof} For links $L_1=[\beta]_n^{\;\widehat{}}$ and $L_2=[\gamma]_p^{\;\widehat{}}$, the split link $L=L_1\sqcup L_2=[\beta\gamma^{\triangleright n}]_{n+p}^{\;\widehat{}}$. Then apply Lemma~\ref{lem:stabilization3} with substitutions $a\mapsto 1$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$. \end{proof}
\section{Proof of Theorems \ref{thm:HOMFLY}--\ref{thm:Alexander-Conway}} \label{sec:proof_Theorems}
We shall focus on Theorem~\ref{thm:HOMFLY}, then remark on the other two.
\begin{proof}[Proof of Theorem~\ref{thm:HOMFLY}] Let us forget about the original definition of the skein polynomial, and regard the symbol $P_L$ as a well-defined invariant of oriented links which satisfies the axioms (II), (III), (IO) and (O). By Corollary~\ref{cor:stabilization}, $P_L$ also satisfies axiom ($\Phi$). We shall show that such an invariant $P_L$ is computable, hence uniquely determined.
It suffices to prove the following claim by induction on $n$. \subsection* {Inductive Claim($n$)} For every $n$-braid $\beta\in B_n$, $P \left( [\beta]_n^{\;\widehat{}} \right)$ is computable.
When $n=1$, Claim($1$) is true because there is only one $1$-braid $[e]_1$. Its closure is the trivial knot, whose $P$ must be $1$ by axiom (O).
Now assume inductively that Claim($n-1$) is true, we shall prove that Claim($n$) is also true.
Suppose $\beta$ is an $n$-braid. By Lemma~\ref{lem:reduction}, the braid $\beta\in B_n$ is equivalent to (in a computable way) a $\Lambda$-linear combination of braids of the form $\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ and $k\in\{0,\pm1,2\}$. By Example~\ref{exam:relator_vs_relation} the (mod $\mathfrak I_n$) equivalence preserves the $P$ of closure of braids. So $P \left( [\beta]_n^{\;\widehat{}} \right)$ is a $\Lambda$-linear combination (with computable coefficients) of $P \left( [\alpha\sigma_{n-1}^k \gamma]_n^{\;\widehat{}} \right)$'s. For $k\in\{\pm1,0,2\}$, respectively, we have \begin{alignat*}{2} P \left( [\alpha\sigma_{n-1}^{\pm1} \gamma]_n^{\;\widehat{}} \right) &= P \left( [\alpha \gamma]_{n-1}^{\;\widehat{}} \right) &&\qquad\text{by isotopy}, \\ P \left( [\alpha\sigma_{n-1}^0 \gamma]_n^{\;\widehat{}} \right) &= z^{-1}(a^{-1}-a) \cdot P \left( [\alpha \gamma]_{n-1}^{\;\widehat{}} \right) &&\qquad\text{by (IO)}, \\ P \left( [\alpha\sigma_{n-1}^2 \gamma]_n^{\;\widehat{}} \right) &= az^{-1}(1+z^2-a^2) \cdot P \left( [\alpha \gamma]_{n-1}^{\;\widehat{}} \right) &&\qquad\text{by $(\Phi)$}. \end{alignat*}
Since $P \left( [\alpha \gamma]_{n-1}^{\;\widehat{}} \right)$ is computable by the inductive hypothesis Claim($n-1$), we see $P \left( [\alpha\sigma_{n-1}^k \gamma]_n^{\;\widehat{}} \right)$ is also computable. Thus Claim($n$) is proved.
The induction on $n$ is now complete. Hence $P$ is computable for every closed braid. \end{proof}
\begin{rem} The induction above, together with the reduction argument of Section \ref{sec:key_lem}, provides a recursive algorithm for computing $P \left( [\beta]_n^{\;\widehat{}} \right)$. \end{rem}
\begin{rem} A remarkable feature of this algorithm is that it never increases the number of components of links. In fact, all the reductions in Section~\ref{sec:key_lem} are by axioms (II) and (III) which respect the components, while in this Section, components could get removed but never added, by axioms (IO) and ($\Phi$). So if we start off with a knot, we shall always get knots along the way, the axioms (IO) and ($\Phi$) becoming irrelevant. This observation works even for links with any given number of components, once we set up a suitable normalization. Hence the Theorem~\ref{thm:HOMFLY_knot}. \end{rem}
\begin{rem} For the Jones polynomial, the proof above works well with the substitutions $a\mapsto t$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$. \end{rem}
\begin{rem} The case of Alexander-Conway polynomial is only slightly different. Corollary~\ref{cor:stabilization_Alexander-Conway} says (IO$_\Delta$) is a consequence of axioms (II$_\Delta$) and (III$_\Delta$), and ($\Phi_\Delta$) is taken as an axiom. So the proof above also works through with the substitutions $a\mapsto 1$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$.
Actually, the argument in Section~\ref{sec:stabilizations} can be adapted to work for Conway potential function of colored links, to the effect that in \cite[Main Theorem]{J1}, the relation (IO) is a consequence of axioms (II) and (III) hence can be removed from the list of axioms. \end{rem}
\section{Proof of Theorems \ref{thm:HOMFLY_knot}--\ref{thm:Alexander-Conway_knot}} \label{sec:proof_Theorems2}
Suppose $\mu$ is a given positive integer. Regard $P_L$ as a well-defined invariant of oriented $\mu$-compo\-nent links which satisfies the axioms (II), (III) and (U). We shall temporarily expand the ring $\Lambda:=\mathbb Z[a^{\pm1},z^{\pm1}]$, where the invariant $P_L$ takes value, to $\widetilde\Lambda:=\mathbb Z[a^{\pm1},z^{\pm1},(a^{-2}-1)^{-1}]$, to allow fractions with denominator a power of $(a^{-2}-1)$. We shall show that such an invariant $P_L$ is computable, hence uniquely determined. It is the normalization (U) that brings the value $P_L$ back into the original $\Lambda$.
\begin{lem} \label{lem:reduction_HOMFLY} Suppose $\beta\in B_n$, $p\ge0$, and $[\beta]_{n+p}^{\;\widehat{}}$ has $\mu$ components. If $n>1$, then $P \left( [\beta]_{n+p}^{\;\widehat{}} \right)$ is computable as a $\widetilde\Lambda$-linear combination of terms of the form $P \left( [\beta']_{n'+p'}^{\;\widehat{}} \right)$, each with $\mu$ components, $\beta'\in B_{n'}$, $n'<n$, $p'\ge p$, and the $(a^{-2}-1)^{-1}$-exponent of the corresponding coefficient is at most $p'-p$. \end{lem}
\begin{proof} By Lemma~\ref{lem:reduction}, the braid $\beta\in B_n$ is equivalent to (in a computable way) a $\Lambda$-linear combination of braids of the form $\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ and $k\in\{0,\pm1,2\}$. So $P \left( [\beta]_{n+p}^{\;\widehat{}} \right)$ is a $\widetilde\Lambda$-linear combination (with computable coefficients) of $P \left( [\alpha\sigma_{n-1}^k \gamma]_{n+p}^{\;\widehat{}} \right)$'s. For $k\in\{\pm1,0,2\}$, respectively, we have \begin{alignat*}{2} P \left( [\alpha\sigma_{n-1}^{\pm1} \gamma]_{n+p}^{\;\widehat{}} \right) &= P \left( [\gamma\alpha \sigma_{n-1}^{\pm1}]_{n+p}^{\;\widehat{}} \right) &&\quad\text{by braid conjugation}, \\ &= P \left( [\gamma\alpha ]_{(n-1)+p}^{\;\widehat{}} \right) &&\quad\text{by Markov move}, \\ P \left( [\alpha\sigma_{n-1}^0 \gamma]_{n+p}^{\;\widehat{}} \right) &= P \left( [\alpha \gamma]_{(n-1)+(p+1)}^{\;\widehat{}} \right) &&\quad\text{obvious}, \\ P \left( [\alpha\sigma_{n-1}^2 \gamma]_{n+p}^{\;\widehat{}} \right) &= P \left( [\gamma\alpha\sigma_{n-1}^2]_{n+p}^{\;\widehat{}} \right) &&\quad\text{by braid conjugation} \\ &= \frac{1+z^2-a^2}{a^{-2}-1} \cdot P \left( [\gamma \alpha]_{(n-1)+(p+1)}^{\;\widehat{}} \right) &&\quad\text{by Lemma~\ref{lem:stabilization3}}. \end{alignat*} The $P$'s on the right hand sides satisfy the required conditions. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:HOMFLY_knot}] Suppose a link $L$ with $\mu$ components is presented as $[\beta]_{n}^{\;\widehat{}}=[\beta]_{n+0}^{\;\widehat{}}$. Apply Lemma~\ref{lem:reduction_HOMFLY} repeatedly until no such reduction is possible. Then $P \left( [\beta]_{n+0}^{\;\widehat{}} \right)$ is computed as a $\widetilde\Lambda$-linear combination of terms $P \left( [\beta']_{n'+p'}^{\;\widehat{}} \right)$, each with $n'=1$ hence $\beta'=[e]_1$, such that \begin{itemize} \item[(1)] every $[\beta']_{n'+p'}^{\;\widehat{}}=[e]_{1+p'}^{\;\widehat{}}$ has $\mu$ components, hence $p'=\mu-1$, and $[\beta']_{n'+p'}^{\;\widehat{}} =[e]_\mu^{\;\widehat{}} =U_\mu$; and \item[(2)] the $(a^{-2}-1)^{-1}$-exponent of every coefficient is at most $p'-0=\mu-1$. \end{itemize} Therefore, $P (L)$ is computable and, by axiom (U), every term $P \left( [\beta']_{n'+p'}^{\;\widehat{}} \right) $ has a factor $(a^{-2}-1)^{\mu-1}$ that can cancel the $(a^{-2}-1)^{-1}$-exponent in its coefficient, so $P (L) \in \Lambda$. \end{proof}
Theorem~\ref{thm:Jones_knot} can be proved similarly, but Theorem~\ref{thm:Alexander-Conway_knot} needs modifications. Define $\delta_p:=\sigma_1^2\sigma_2^2\dots\sigma_{p-1}^2 \in B_p$ whose closure $[\delta_p]_p^{\;\widehat{}}$ is the oriented $p$-component chain $C_p$.
\begin{lem} \label{lem:reduction_Alexander-Conway} Suppose $\beta\in B_{n+1}$, $p\ge1$, and $[\beta\delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}}$ has $\mu$ components. If $n>0$, then $\Delta \left( [\beta\delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right)$ is computable as a $\mathbb Z[t^{\pm\frac12}]$-linear combination of terms of the form $\Delta \left( [\beta'\delta_{p'}^{\triangleright n'}]_{n'+p'}^{\;\widehat{}} \right)$, each with $\mu$ components, $\beta'\in B_{n'+1}$, $n'<n$ and $p'\ge p$. \end{lem}
\begin{proof} By Lemma~\ref{lem:reduction} (with substitutions $a\mapsto 1$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$), the braid $\beta\in B_{n+1}$ is equivalent to (in a computable way) a $\mathbb Z[t^{\pm\frac12}]$-linear combination of braids of the form $\alpha\sigma_{n}^k \gamma$ with $\alpha,\gamma\in B_{n}$ and $k\in\{0,\pm1,2\}$. Multiplication by $\delta_p^{\triangleright n}$ makes the braid $\beta\delta_p^{\triangleright n}$ equivalent to a linear combination of braids of the form $\alpha\sigma_{n}^k \gamma\delta_p^{\triangleright n}$, so $\Delta \left( [\beta\delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right)$ is computed as a linear combination of the $\Delta \left( [\alpha\sigma_{n}^k \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right)$'s. For $k\in\{\pm1,0,2\}$, respectively, we have \begin{alignat*}{2} \Delta \left( [\alpha\sigma_{n}^{\pm1} \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) &= \Delta \left( [\gamma\alpha\sigma_{n}^{\pm1} \delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) &&\quad\text{by braid conjugacy}, \\ &= \Delta \left( [\gamma\alpha \delta_p^{\triangleright (n-1)}]_{n+p-1}^{\;\widehat{}} \right) &&\quad\text{by a Markov move}, \\ \Delta \left( [\alpha\sigma_{n}^0 \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) &= 0 &&\quad\text{by Corollary~\ref{cor:stabilization_Alexander-Conway}}, \\ \Delta \left( [\alpha\sigma_{n}^2 \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) &= \Delta \left( [\gamma\alpha\sigma_{n}^2 \delta_p^{\triangleright n}]_{n+p}^{\;\widehat{}} \right) &&\quad\text{by braid conjugacy} \\ &= \Delta \left( [\gamma\alpha \delta_{p+1}^{\triangleright (n-1)}]_{n+p}^{\;\widehat{}} \right) &&\quad\text{by definition}. \end{alignat*} The $\Delta$'s on the right hand sides are in the desired form with $n'=n-1$ and with $\mu$ components. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:Alexander-Conway_knot}] Suppose a link $L$ with $\mu$ components is presented as $[\beta]_{n+1}^{\;\widehat{}}=[\beta\delta_1^{\triangleright n}]_{n+1}^{\;\widehat{}}$. Apply Lemma~\ref{lem:reduction_Alexander-Conway} repeatedly until no such reduction is possible. Then $\Delta \left( [\beta\delta_1^{\triangleright n}]_{n+1}^{\;\widehat{}} \right)$ is computed as a $\mathbb Z[t^{\pm\frac12}]$-linear combination of terms $\Delta \left( [\beta'\delta_{p'}^{\triangleright n'}]_{n'+p'}^{\;\widehat{}} \right)$, with $n'=0$. Hence each $\beta'=[e]_1$, $p'=\mu$ the number of components, and each $[\beta'\delta_{p'}^{\triangleright n'}]_{n'+p'}^{\;\widehat{}} = [\delta_\mu]_\mu^{\;\widehat{}} = C_\mu$. Therefore $\Delta (L)$ is computable and moreover, by axiom (C$_\Delta$), divisible by $\Delta(C_\mu) = (t^{\frac12}-t^{-\frac12})^{\mu-1}$. \end{proof}
\end{document} | arXiv | {
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\begin{document}
\title[Embedding in Euclidean space of coherent configuration of type (2,2;3)]{On the two-distance embedding in real Euclidean space of coherent configuration of type (2,2;3)} \author{Eiichi Bannai, Etsuko Bannai, Chin-Yen Lee, Ziqing Xiang, Wei-Hsuan Yu}
\begin{abstract}
Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lison\v ek in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by Lison\v ek has a distinguished structure of a coherent configuration of type $(2,2;3)$ and is embedded in two concentric spheres in $\mathbb R^8$. In this paper we study whether there exists any other similar embedding of a coherent configuration of type $(2,2;3)$ as a $2$-distance set in $\mathbb R^n$, without assuming any restriction on the size of the set. We prove that there exists no such example other than that of Lison\v ek. The key ideas of our proof are as follows: (i) study the geometry of the embedding of the coherent configuration in Euclidean spaces and to drive diophantine equations coming from this embedding. (ii) solve diophantine equations with certain additional conditions of integrality of some parameters of the combinatorial structure by using the method of auxiliary equations.
\end{abstract} \maketitle \section{Introduction}
Let $X,Y$ be finite subsets of $\mathbb R^d$, we define $A(X,Y)=\{\|x-y\|: x\in X,y\in Y,\text{ and }x\neq y\}$. A finite set $X\subset\mathbb R^d$ is called an \emph{$s$-distance set} if $|A(X,X)|=s$. By assigning a set of linearly independent polynomials to an $s$-distance set, Bannai, Bannai and Stanton \cite{bannai1983upper} and Blokhuis \cite{blokhuis1983few} gave an upper bound for the cardinality of an $s$-distance sets $X$ in $\mathbb R^d$, \begin{align*}
|X|\leq \binom{d+s}{s}. \end{align*}
Sz{\"o}ll{\H{o}}si and {\"O}sterg{\aa}rd \cite{szollHosi2018constructions} had the latest progress for the construction of $s$-distance sets for $s\leq 6, d\leq 8$. They gave an algorithm to exhaust the $s$-distance sets in the space of small dimension and small $s$ at least $3$. Lison\v ek \cite{lisonvek1997new} classified all two-distance sets for the dimension $d=4,5,6$ and $7$. For dimension $8$, he gave an example which attains the maximum cardinality $\binom{8+2}{2}=45$. However the classification problem is still open. The example of Lison\v ek, given below, is the only known $s$-distance set that attain the Bannai-Bannai-Stanton and Blokhuis bound $\binom{d+s}{s}$. In fact, these 45 points are distributed in a very symmetrical manner. It can be divided in two parts : one part is 9 points forming a regular simplex, and the other part is $36$ points coming from the spherical embedding of a Johnson scheme $J(9,2)$.
\begin{example}(Lison\v ek) \label{example} Let $X_1=\{ -e_i+\frac 1 3\sum_{k=1}^9e_k : 1\leq i\leq 9\}$ and $X_2= \{e_i+e_j : 1\leq i<j\leq 9\}$, where $\{e_i:1\leq i\leq 9\}$
are the standard orthonormal basis of $\mathbb R^9$. All the points are on the hyperplane $H=\{x=\sum_{i=1}^9x_ie_i:\sum_{i=1}^9x_i=2\}$, and its affine dimension is $8$. Then, $X_1\cup X_2$ is a maximum two-distance set in $\mathbb R^8$. Notice that the distinct two distances in $X_1 \cup X_2$ are $\sqrt 2$ and $2$. The radius of $X_1$ and $X_2$ are $\frac{2}{\sqrt 3}$ and $\sqrt 2$ respectively. \end{example}
In fact, $X_2\subset \mathbb R^d$ can be interpreted as a block design with the underlying set $X_1$. The union $X_1\cup X_2$ has the elegant structure:
\begin{enumerate} \item $ X_1$ is the regular simplex in $\mathbb R^{d}$; \item $ X_2$ is a scaling of the spherical embedding of a strongly regular graph in $\mathbb R^{d}$; \item distance between point and block only depends on whether the point in block or not;
\item $|A( X_1, X_1)|=1$ and $|A( X_1, X_2)|=|A( X_2, X_2)|=2$. \end{enumerate} Condition (4) is a consequence of (1)-(3), and we list here for convenience.
A coherent configuration of type (2,2;3) is a combinatorial structure and it consists of a point set $V$, a block set $B$ and a set of finite number of relations $\{R_i\}$. The block design in Example \ref{example} is quasi-symmetric, and it is equivalent to the type (2,2;3) coherent configuration. We prove that such coherent configurations can always be embedded into a Euclidean space with the above structure.
\begin{theorem}\label{embcond} Let $(V\cup B, \{R_1,\dots,R_9\})$ be a coherent configuration of type (2,2;3), where $V$ is a finite set. Then there is a map $i:V\cup B\to\mathbb R^{d}$, $d=|V|-1$, such that $X_1=i(V)$ and $X_2=i(B)$ satisfying the conditions (1)-(4) in Example \ref{example}. \end{theorem}
In this paper we will consider the following problem:
When the embedding of type (2,2;3) coherent configuration in Theorem \ref{embcond} gives a two-distance set in Euclidean space? Dropping the condition on sizes (i.e., we don't assume $|X_1\cup X_2|=\binom{d+2}{2}$) makes the problem very broad and difficult. However, to our surprise, we are able to show our main theorem that there is no other example at all.
\begin{theorem}(Main result) \label{main thm} The example given by Lison\v ek is the unique coherent configuration of type (2,2;3) which can be embedded in Euclidean space as a two-distance set satisfying the conditions given in Theorem \ref{embcond}.
\end{theorem}
Nozaki and Shinohara \cite{nozaki2020maximal} study two-distance sets in $\mathbb R^d$ that contain a regular simplex. They solve the case when Larman–Rogers–Seidel (LRS) \cite{larman1977two} ratio is 2. When LRS ratio is 3, they give a partial result by adding some block design structures. So, our present paper is relevant to \cite{nozaki2020maximal}. However, it seems that the situation in our paper is eluded in their consideration.
The outline of our paper is as follows. In section 2, we introduce the notions of type (2,2;3) coherent configurations and quasi-symmetric designs and we will prove the Theorem \ref{embcond}. In section 3, we study when the embedding given by Theorem \ref{embcond} is a two-distance set. We derive the conditions for the embedding of a coherent configuration forming a two-distance set into three Diophantine equations $p_1(S,m,x,y)=p_2(S,m,x,y)=p_2(S,m,x,y)=0 $. In section 4, we determine the complete solutions of the system of Diophantine equations. In section 5, we prove our main result, Theorem \ref{main thm}. Please note that we use computers crucially. The results are rigorously proved by applying the notion of real algebraic geometry. We can see the precedents of the basic idea of the proof in Xiang \cite{xiang2018nonexistence}, and Bannai, Bannai, Xiang, Yu and Zhu \cite{bannai2021classification}.
\section{Preliminaries} We introduce the two well-known concepts, quasi-symmetric designs and coherent configurations of type (2,2;3). They are proved as the equivalent notions in Higman \cite{higman1987coherent}. Quasi-symmetric design is a combinatorial object whose parameters are non-negative integers. The integral conditions between them are crucial for classifying them. \subsection{Quasi-symmetric designs} \begin{definition}\cite{cameron1975graph} A \emph{$t$-design} with parameters $(m,S,\Lambda)$ (or a \emph{$t$-$(m,S,\Lambda)$ design}) is a collection of subsets $B$ (called \emph{blocks}) of a set $V$ of $m$ points such that \begin{enumerate}
\item every member of $B$ contains $S$ points;
\item any set of $t$ points is contained in exactly $\Lambda$ members of $B$. \end{enumerate} \end{definition}
A $2$-design is called a \emph{quasi-symmetric design} if the cardinality of the intersection of two different blocks takes just two distinct values. These two numbers are called the \emph{intersection numbers}. We express these intersection numbers by $\alpha$ and $\beta$, with $\beta<\alpha$. By definition each element $v\in V$ contains in $T$ distinct blocks. For each point $v\in V$ and each block $b\in B$, the following condition is satisfied: \begin{align*}
|\{ w\in B: |b\cap w|=\alpha\text{ and } v\in w \}|=\begin{cases}
N,&\text{ if }v\in b,\\
P,&\text{ if }v\not\in b.
\end{cases} \end{align*}
We denote this number by $N$ when $v\in b$ and by $P$ when $v\not \in b$. The cardinality of quasi-symmetric design is bounded above by ${m\choose 2}$ (see Proposition 3.4 in Cameron and van Lint \cite{cameron1975graph}).
\begin{theorem}\cite[Proposition 3.6]{cameron1975graph} \label{thm3} For a $2$-$(m,S,\Lambda)$ design $\mathcal D$ with $4\leq S\leq m-4$, any two of the following imply the third: \begin{enumerate} \item $\mathcal D$ is quasi-symmetric; \item $\mathcal D$ is a $4$-design;
\item $|B|={m\choose 2}$ . \end{enumerate}
\end{theorem}
The 4-design attaining the lower bound ${m \choose 2}$ is called a \emph{tight} 4-design. Enomoto-Ito-Noda \cite{enomoto1979tight} classified the tight 4-designs.
\begin{theorem}\cite{enomoto1979tight} \label{unique4des}Up to complementation, the $4$-$(23, 7, 1)$ design is the only tight $4$-$(m,S,\Lambda)$ design with $2<S<m-2$. \end{theorem}
By Theorem \ref{thm3}, if $\mathcal D$ is quasi-symmetric with $|B|={m\choose 2}$ ($4\leq S\leq m-4$), then it is the $4$-$(23, 7, 1)$ design by Theorem \ref{unique4des}. This fact will be used in the proof of the Main Theorem.
\subsection{Coherent configurations} Here, we review the notion of coherent configuration given by Higman in \cite{higman1987coherent}. The concept of the coherent configuration is a purely combinatorial axiomatization of permutation groups. \begin{definition}Let $X$ be a finite set and let $\{R_i\}_{i\in I}$ be a set of relations on $X$ such that: \begin{enumerate} \item $\{R_i\}_{i\in I}$ is a partition of $X\times X$;
\item $R_i^t =R_{i^*}$ for some $i^*\in I$ where $R_i^t=\{(y,x): (x,y)\in R_i\}$;
\item there is a subset $\Omega\subset I$ such that $\{(x,x):x\in X\}=\bigcup_{i\in\Omega}R_i$;
\item given $(x,y)\in R_k$, $|\{z: (x,z)\in R_i, (z,y)\in R_j\}|$ is a constant $p_{ij}^k$ which depends only on $i,j,k$. \end{enumerate} Then $(X,\{R_i\}_{i\in I})$ is said to be a \emph{coherent configuration}. \end{definition} By the third condition, $X$ is split into several parts : $X_i=\{x:(x,x)\in R_i\}$ for $i\in\Omega$ and each part is called a \emph{fiber}.
Let $s_{i,j}$ denote the number of relations in $X_i\times X_j$. The \emph{type} of coherent configuration is the matrix with size $|\Omega|\times |\Omega|$ and the $(i,j)$-entry is $s_{i,j}$. When $|\Omega|=1$, it is an \emph{association scheme}. Each fiber carries a structure of association scheme. \begin{example} Let $G=(V,E)$ be a regular graph that is neither complete nor empty. Then $G$ is said to be \emph{strongly regular} with \emph{parameters} $(n,k,\lambda,\mu)$ if it is of order $n$, $k$-regular, every pair of adjacent vertices has $\lambda $ common neighbors, and every pair of distinct nonadjacent vertices has $\mu$ common neighbors. The concept of strongly regular graph and symmetric association scheme with three relations are equivalent. The eigenvalues of $G$ are listed from large to small by $k>r>s$. The eigenvalue $s$ must be a negative number. The reader can find details for strongly regular graphs in \cite{godsil2001algebraic}. \end{example}
\begin{example} Let $\mathcal D$ be a quasi-symmetric design with intersection number $\alpha,\beta$. Define $X=V\cup B$ and \begin{align*}
R_1&=\{(x,x):x\in V\};\\
R_2&=\{(x,x):x\in B\};\\
R_3&=\{(x,y):x,y\in V,x\neq y\};\\
R_4&=\{(x,y)\in B\times B: |x\cap y|=\alpha\};\\
R_5&=\{(x,y)\in B\times B: |x\cap y|=\beta\};\\
R_6&=\{(x,y)\in V\times B: x \in y\};\\
R_7&=\{(x,y)\in V\times B: x \not\in y\};\\
R_8&=R_6^t;\,\,\,\,\,R_9=R_7^t. \end{align*} Then $(X, \{R_1,\dots,R_9\})$ is a coherent configuration of type $\begin{pmatrix} 2&2\\2 &3\end{pmatrix}$ (abbreviated as (2,2;3)). The fibers are $V$ and $B$. \end{example}
\subsection{Connection between type (2,2;3) and quasi-symmetric designs} Higman showed in \cite{higman1987coherent} that a coherent configuration $(X_1\cup X_2,\{R_1,\dots,R_9\})$ of type (2,2;3) is equivalent to a complementary pair of quasi-symmetric design. The relation $R_6$ restricted in $X_1\times X_2$ carries a structure of quasi-symmetric design $\mathcal D$. There is a relation $R$ in $\{R_4,R_5\}$ restricted in $X_2\times X_2$
equivalent to $\{(x,y)\in B\times B: |x\cap y|=\alpha\}$ where $\alpha$ is the largest intersection number of $\mathcal D$. Without loss of generality, we assume $R=R_4$. The $R_4$ restricted in $X_2\times X_2$ gives a structure of strongly regular graph (which is called the \emph{block graph} of $\mathcal D$).
\begin{proposition}\cite[section 9C]{higman1987coherent}\label{9C} Let $\mathcal D$ be a $2$-$(m,S,\Lambda)$ quasi-symmetric design with intersection numbers $\beta<\alpha$. We can express for the parameters of the block graph in terms of the parameters $S,m,\alpha,\beta$ of $\mathcal D$. \begin{align*} \Lambda=&\frac{S(S-1)(S-\alpha)(S-\beta)} {S^4-2S^3-((\alpha+\beta-1)(m-1)-1)S^2+\alpha\beta m(m-1)};\\ T =&\frac{(m-1)\Lambda}{S-1};\,\,\,\,\,N =\frac{\alpha(m-S)(S(S-1)-\beta(m-1))\Lambda }{S(\alpha-\beta)(S-\alpha)(S-1)}; \\ P =& \frac{ (S(S-1)-\beta(m-1)) \Lambda}{(\alpha-\beta)(S-1)};\,\,\,\,\,r =\frac{ 1}{\alpha-\beta}\left( \frac{(m-S)\Lambda}{S-1}-(S-\beta) \right);\\ k =&\frac{ (m-S)(S(S-1)-\beta(m-1) )\Lambda }{ (\alpha-\beta) (S-\alpha)(S-1) };\,\,\,\,\,n =\frac{mT}S ;\,\,\,\, \,s =\frac{\beta-S}{\alpha-\beta} . \end{align*}
\end{proposition}
\begin{remark} The parameters $S,m,\alpha,\beta$ are integers with \begin{align*}
0\leq \beta<\alpha< S<m. \end{align*} By the definition of $2$-design, $\Lambda\in \mathbb Z_{\geq 1}$. By the definition of $N,P$, they belong to $\mathbb Z_{\geq 0}$.
\end{remark}
\subsection{Representation of coherent configurations}
The \emph{adjacency matrices} of a coherent configuration are the $|X|\times |X|$ matrices $A_i$ whose $(x,y)$-entry is $1$ if $(x,y)\in R_i$ and $0$ otherwise. The \emph{adjacency algebra} is $\mathcal A=\text{span}_{\mathbb C}\{A_i:i\in I\}$. Let $\triangle_1,\dots,\triangle_p$ be the non-isomorphic irreducible representations of degree, $e_1,\dots,e_p$ correspondingly. There is a basis $\{\epsilon_{ij}^s\}$ of $\mathcal A$ defined by $\triangle_t(\epsilon_{ij}^s)=\delta_{st}E_{ij}^s$, where $E_{ij}^s$ is the $e_s\times e_s$ matrix with $(i,j)$-entry 1 and all other entries 0. They satisfy the equation $\epsilon_{ij}^s\epsilon_{kl}^t=\delta_{st}\delta_{jk}\epsilon_{il}^s$.
Higman showed in \cite{higman1987coherent} that for a type (2,2;3) coherent configuration, the adjacency algebra $\mathcal A$ has three non-isomorphic irreducible representations. For $x=c_1A_1+\cdots+c_9A_9\in \mathcal A$, the three representations $\triangle _1,\triangle _2:\mathcal A\to \text{Mat}_2(\mathbb C),\triangle_3:\mathcal A\to \mathbb C$ are defined by \begin{align*} \triangle_1(x)&=\begin{pmatrix} c_1+(m-1)c_3 & \alpha_1 c_6+\alpha_2 c_7\\ \alpha_1 c_8+\alpha_2c_9& c_2+kc_4+(n-k-1)c_5 \end{pmatrix},\\ \triangle_2(x)&=\begin{pmatrix} c_1-c_3 & \beta_1c_6+\beta_2c_7\\ \beta_1c_8+\beta_2c_9 & c_2+rc_4-(r+1)c_5 \end{pmatrix},\\ \triangle_3(x)&=(c_2+sc_4-(s+1)c_5), \end{align*}
where $\alpha_1=\sqrt{ST}, \alpha_2= P^{-1}(k-N)\sqrt{ST}, \beta_1= -\beta_2=\sqrt{T-\Lambda}$.
Let $\epsilon_{11}^2=c_1A_1+\cdots+c_9A_9$. Since $\triangle_s(\epsilon_{11}^2)=\delta_{s,2}E^2_{11}$, we have $(c_2+sc_4-(s+1)c_5)=0$, \begin{align*} \begin{pmatrix} c_1+(m-1)c_3 & \alpha_1 c_6+\alpha_2 c_7\\ \alpha_1 c_8+\alpha_2c_9& c_2+kc_4+(n-k-1)c_5 \end{pmatrix}=\begin{pmatrix} 0 & 0\\0&0\end{pmatrix},\\ \begin{pmatrix} c_1-c_3 & \beta_1c_6+\beta_2c_7\\ \beta_1c_8+\beta_2c_9 & c_2+rc_4-(r+1)c_5 \end{pmatrix}=\begin{pmatrix} 1&0\\0&0\end{pmatrix}. \end{align*} Now, we have $9$ variables and $9$ conditions, so we can solve $c_1,\dots,c_9$. Others $\epsilon_{ij}^s$ also have $9$ variables and $9$ conditions, so we can explicitly determine them. We list the solution of $\epsilon_{ij}^2$ below, \begin{align}\label{epsilonbasis} &\epsilon_{11}^2= \frac{(m-1)A_1-A_3}m,\,\,\,\,\, \epsilon_{12}^2= \frac{ \alpha_2A_6-\alpha_1A_7}{\alpha_1\beta_2-\alpha_2\beta_1},\,\,\,\,\,\epsilon_{21}^2 =\frac{\alpha_2A_8-\alpha_1A_9}{\alpha_1\beta_2-\alpha_2\beta_1},\notag\\ &\epsilon_{22}^2= \frac{-((n-k-1)s+ks+k)}{n(r-s)}A_2+\frac{n-k+s}{n(r-s)}A_4+\frac{s-k}{n(r-s)}A_5, \end{align}
where $n,k,r,s$ are the parameters of the strongly regular graph given by $A_4|_{X_2\times X_2}$.
\subsection{Proof of Theorem \ref{embcond}} \begin{proof}[Proof of Theorem \ref{embcond}:] Let $ E:=(\epsilon_{11}^2+\epsilon_{12}^2+\epsilon_{21}^2+\epsilon_{22}^2)/2$ where $\epsilon_{ij}^2$ are given in (\ref{epsilonbasis}). Then $ E^2= E$ and $ E^t= E$. The trace of $ E$ is $m-1$. Namely, $E$ is a symmetric idempotent of rank $m-1$. Let $\{e_x:x\in V\cup B\}$ be the standard basis of $\mathbb R^{V\cup B}$. Observe that $\langle E e_x, Ee_y\rangle= e_x^t E^t E e_y=e_x^t Ee_y$ is the $(x,y)$-entry of $ E$. In other words, \begin{align}\label{emb} \langle E e_x, Ee_y\rangle=\begin{cases} (m-1)/ (2m), &x,y\in V, x=y,\\ -1/(2m), &x,y \in V,x\neq y,\\ \alpha_2/(2\alpha_1\beta_2-2\alpha_2\beta_1),&x\in V,y\in B, x\in y,\\ -\alpha_1/(2\alpha_1\beta_2-2\alpha_2\beta_1),& x\in V,y\in B,x\not\in y,\\ -((n-k-1)s+ks+k)/(2n(r-s)), &x,y\in B,x=y,\\ (n-k+s)/(2n(r-s)), &x,y\in B, x\cap y=a,\\ (s-k)/(2n(r-s)), &x,y\in B,x\cap y=b. \end{cases} \end{align} Define $i:V\cup B\to\mathbb R^{V\cup B}$ by $i(x)=Ee_x$.
Note that $i(V)$ is on one sphere and $i(B)$ is on another sphere. So, $|A(i(V),i(B))|$ is the number of different inner product $\langle i(v),i(b)\rangle$ for $v\in V$ and $b\in B$. From equation (\ref{emb}) we know that it is $2$. And the distance between $i(v)$ and $i(b)$ only depends on $v$ in $b$ or not. Similarly, $i(V)$ is an one-distance set on a sphere which is a regular simplex in $\mathbb R^{m-1}$. Finally, the $F=\epsilon_{22}^2$ is an idempotent of rank $m-1$ on the algebra $\text{span}_{\mathbb C}\{A_2,A_4,A_5\}$. The inner product between points in $\mathbb R^B$ is \begin{align*}
\langle i(x),i(y) \rangle=\frac1 2 e_x^t 2Ee_y=\frac 1 2e_x^tFe_y=\frac 1 2 e_x^t F^tFe_y=\frac 1 2 \langle Fe_x,Fe_y\rangle. \end{align*} So, the $i(B)$ is the scaling of the spherical embedding of a strongly regular graph. Now, $i$ is an embedding satisfying the conditions in the statement of theorem. \end{proof}
\begin{remark} In general, this embedding has at most 5 distinct distances. If we fix the regular simplex and rescale the sphere that containing $i(B)$, it is still an embedding satisfying conditions of Theorem \ref{embcond}. We will need smartly choosing the radius $R_2$ to make the embedding as a two-distance set. How to determine the $R_2$ will be discussed in the next section. \end{remark}
\begin{example} For the Lison\v ek's example, the parameters of $\mathcal D$ are $(S,m,\alpha,\beta,\Lambda,T,N,P)=( 2,9, 1, 0, 1, 8, 7, 2)$ and the parameters of the block graph are $(n,k,\lambda,\mu)=(36,14,7,4)$. Now, \begin{align} \langle E e_x, Ee_y\rangle=\begin{cases} 4/9, &x,y\in V, x=y,\\ -1/18, &x,y \in V,x\neq y,\\ -\sqrt 7 /18,&x\in V,y\in B, x\in y,\\ \sqrt 7/63,& x\in V,y\in B,x\not\in y,\\ 1/9, &x,y\in B,x=y,\\ 5/126, &x,y\in B, x\cap y=a,\\ -2/63, &x,y\in B,x\cap y=b. \end{cases} \end{align} Hence, the radius $R_1$ of the sphere containing $EV$ is $2/3$ and the radius $R_2$ of the sphere containing $EB$ is $1/3$. $A(EV,EV)=\{1\}$, $A(EV,EB)=\left\{ \frac{\sqrt{\sqrt 7+5}}3, \sqrt{\frac 5 9-\frac{2\sqrt7}{63}} \right\}$, $A(EB,EB)=\{\sqrt{1/7}, \sqrt{2/7}\}$. After rescaling $R_1$ with $\sqrt 2$ and $R_2$ with $\sqrt{14}$, $E(V\cup B)$ becomes a two-distance set where $A(E(V\cup B),E(V\cup B))=\{\sqrt{2},2\}$. \end{example}
\section{Embedding as a two-distance set}
Let $\mathcal D=(V,B)$ be a quasi-symmetric design and $i$ be an embedding satisfying conditions in Theorem \ref{embcond}. In this section, we assume that $|A( i(X), i(X) )|=|\{\sqrt{2},\sqrt{\gamma}\}|=2$ where $X=V\cup B$. The following theorem is the main theorem of this section.
\begin{theorem}\label{thm1} Let $(V\cup B, \{R_1,\dots,R_9\})$ be a coherent configuration of type (2,2;3). Let $\mathcal D=(V,B)$ be the corresponding $2$-$(m,S,\Lambda)$ quasi-symmetric design with two intersection numbers $x$ and $y$. Suppose the embedding is a two-distance set and satisfies the conditions of Theorem \ref{embcond}. Then $p_1(S,m,x,y)=p_2(S,m,x,y)=p_3(S,m,x,y)=0$, where \begin{align*} p_1(S,m,x,y)=& S^4 - 2S^2xm + x^2m^2 - 2S^3 + 2S^2x - 2Sxm + 2x^2m + S^2 - 2Sx + x^2;\\ p_2(S,m,x,y)= &S^4 + 2S^2xm - 4S^2ym + x^2m^2 - 4xym^2 + 4y^2m^2 - 2S^3 + 2S^2x \\ &+ 8S^2m - 6Sxm - 2x^2m- 4Sym + 4xym - 4Sm^2 + 4xm^2 + S^2 - 2Sx + x^2;\\ p_3(S,m,x,y)=&S^2x^2 - Smx^2 - 2S^2xy + 2Smxy + S^2y^2 - Smy^2 + S^2m \\ &+ 2S^2x - 2Smx - 2Sx^2 + mx^2 - 2S^2y + 2Sxy + S^2 - 2Sx + x^2. \end{align*}
\end{theorem}
\subsection{Calculation of A(i(X),i(X))} Let $A( i( V), i( V))=\{d_1\}$, $A( i( B), i(B))=\{d_2,d_3\} $ and $A( i( V), i( B))=\{d_4,d_5\}$ with $d_2<d_3$ and $d_4<d_5$.
Let $ i( V)=\{e_1,\dots,e_m\}$ be the standard orthonormal basis. It forms a regular simplex of $\mathbb R^{m-1}$ located on the affine hyperplane $$ H=\left\{ x\in\mathbb R^{m}: \sum_{i=1}^{m}x_i=1\right\}\subset \mathbb R^m.$$ For convenience, we define $f(x)=(x-1)/x$ for all $x\neq 0$. We will determine the elements in $A(i(X),i(X))$ expressed by the parameters $S,m,\alpha$, and $\beta$.
\begin{theorem}\label{thm:A(X,X)} The points set $i(V)$ and $i(B)$ are located on the spheres with radius $R_1$ and $R_2$ respectively. Then \begin{align*} R_1&=\sqrt{f(m)},\\ R_2&=\begin{cases} \sqrt{2-f(m-S)}+\sqrt{f(m)-f(m-S)}, &\gamma>2,\\ \sqrt{2-f(m-S)}-\sqrt{f(m)-f(m-S)}, &\gamma<2, \end{cases}\\
d_1&=\sqrt{2},\,\,\,\,\,d_2=R_2\sqrt{\frac{2(S - \alpha)m}{(m-S)S}},\,\,\,\,\, d_3=R_2\sqrt{\frac{2(S - \beta)m}{(m-S)S }},\\
d_4&=\begin{cases}
\sqrt{ R_2^2-2R_2\sqrt{ f(m)-f(m-S) } +f(m)}, &\text{ if }\gamma>2,\\
\sqrt{ R_2^2-2R_2\sqrt{ f(m)-f(S) } +f(m)},&\text{ if }\gamma<2
\end{cases}\\ d_5&=\begin{cases} \sqrt {R_2^2+2R_2\sqrt{ f(m)-f(S) }+f(m)},&\text{ if }\gamma>2,\\ \sqrt {R_2^2+2R_2\sqrt{ f(m)-f(m-S) }+f(m)},&\text{ if }\gamma<2. \end{cases} \end{align*} \end{theorem}
The Theorem \ref{thm:A(X,X)} is the consequence of Lemma \ref{par1} and Lemma \ref{par2}.
\begin{lemma}\label{par1} The radius $R_1$ and the distances $d_1,d_2,$ and $d_3$ between the points satisfy the formula in Theorem \ref{thm:A(X,X)}. \end{lemma}
\begin{proof} We assume that the vertices of the regular simplex are $e_1,e_2,\dots,e_m$ and the center of the simplex is $(\frac 1 m, \frac 1 m, \cdots, \frac 1 m)$. With easy calculation, we can obtain $R_1=\sqrt{\frac{m-1}{m}}=\sqrt{f(m)}$ and $d_1=\sqrt 2$. We require the $ i( B)$ as the spherical embedding of a strongly regular graph. If it is on the sphere with radius $R_2$, then the elements in $A(i(B),i(B))$ are \begin{align*}
R_2\sqrt{2-2 \frac r k},\,\,\,\,\, R_2\sqrt{2-2\frac{-1-r} {n-k-1}} \end{align*} due to the formula (\ref{emb}) (the block graph is neither complete nor empty, $n-k-1>0$ and $k>0$). If we substitute $r, k$ by $S,m,\alpha,\beta$, assuming $\beta<\alpha$ and $d_2<d_3$, we have $d_2= R_2\sqrt{\frac{2(S - \alpha)m}{(m-S)S}} $ and $d_3=R_2\sqrt{\frac{2(S - \beta)m}{(m-S)S }} $. \end{proof}
The centroid of the regular simplex $i(V)$ is $ o=\frac 1{m} (1,\dots,1)$. Take $v,w\in V$ and $b\in B$ with $v\in b$ and $w\not\in b$. Let $$p=\frac{1}{S}\sum_{\substack{t\in V\\ t\in b}} i(t),\,\,\,\,\, q=\frac 1 {m-S}\sum_{\substack{t\in V\\ t\not\in b}} i(t)$$ be the centroids of points in $b$ and points not in $b$. Direct computation shows that $\langle i( v)- p,p- o\rangle=0=\langle o- q, q- i( w)\rangle$. Geometrically, we may regard that the centroid of $v$ will lie in the perpendicular bisector plane. We have
$ \| i(v)-p\|_2= \| (1,0, \cdots, 0) - (\frac 1 S, \frac 1 S, \dots, \frac 1 S, 0, \dots, 0)\|_2= \sqrt {\frac{S-1}{S}}= \sqrt{f(S)} $. Similarly, we can obtain $\| i(w)-q\|_2=\sqrt{f(m-S)} $. To visualize our calculation, we give the figure \ref{fig4} in the following. \begin{figure}
\caption{length of $\|i(v)-p\|_2,\|p-o\|_2,\|o-q\|_2,\|q-i(w)\|_2$}
\label{fig4}
\end{figure}
We divide $i$ into $4$ cases. If $i$ belongs to case $t$, then use $\iota_t$ denote $i$. First, the embedding of vertices located in a smaller sphere or the other way. Let $\iota_1,\iota_2$ send vertices to the smaller sphere and $\iota_3,\iota_4$ send vertices to the larger sphere. Second, $\iota_1,\iota_4$ send elements not in block located closer than elements in block, and $\iota_2,\iota_3$ sends elements in block located closer than elements not in block. We show the $4$ categories geometrically in the Figure \ref{fig1}. \begin{figure}
\caption{$\iota_1,\iota_2,\iota_3,\iota_4$}
\label{fig1}
\end{figure} \begin{figure}
\caption{$i(b)=q$ }
\label{fig2}
\end{figure}
\begin{remark} It is impossible that $i(b)=q$ (see the Figure \ref{fig2}). We will explain it in the following.
Let $D_1=\|i(v)-i(w)\|_2$, $D_2=\|i(w)-i(b)\|_2$ and $D_3=\|i(v)-i(b)\|_2$. We already knew $D_1=\sqrt 2$ and $D_2=\sqrt{f(m-S)}$. Now, \begin{align*}
D_3=\left\|\left(1,\underbrace{0,\dots,0}_{S-1 \text{ terms}},\underbrace{0,\dots, 0}_{m-S\text{ terms}}\right)-\frac 1{m-S}\left(\underbrace{0,\dots,0}_{S \text{ terms}},\underbrace{1,\dots, 1}_{m-S\text{ terms}}\right)\right\|_2=\sqrt{\frac{m-S+1}{m-S}}>1. \end{align*} Hence, $D_3>D_2$. According to the two-distance set assumption, $D_3=D_1=\sqrt{2}$ implies $m-S=1$ and $D_2=\sqrt{f(m-S)}=0$. Hence, we send every point not in block to the same place which is a contradiction.
\end{remark}
\begin{lemma}\label{par2} The radius $R_2$ and the distances $d_4$ and $d_5$ between the points satisfy the formula in Theorem \ref{thm:A(X,X)}. \end{lemma}
\begin{proof}
According to $\gamma>2$ or $\gamma<2$, there are total $8$ cases need to be considered. We listed them in Table \ref{tab2}.
\begin{table}[h] \centering \begin{tabular}{ccccc}
&$\iota_1$&$\iota_2$&$\iota_3$&$\iota_4$\\ $\gamma>2$ &(A)&(B)&(C) &(D) \\ $\gamma<2$ &(E)&(F)&(G)&(H) \end{tabular} \caption{$8$ cases}\label{tab2} \end{table} Now, we can determine the $d_4$ and $d_5$. Take $v,w\in V$ and $b\in B$ with $v\in b$ and $w\not\in b$. According to Figure \ref{fig4}, for case 1 and case 4, we have
$d_5^2=\|i(v)-i(b)\|^2_2=\|p-i(v)\|_2^2+ (R_2+\|o-p\|_2)^2 =f(S)+(R^2_2+\sqrt{f(m)-f(S)} )^2
$. Also, $d_4^2=\|i(w)-i(b)\|^2_2=\|q-i(w)\|_2^2+ (R_2-\|o-q\|_2)^2 =f(m-S)+(R^2_2+\sqrt{f(m)-f(m-S)} )^2$. Hence, \begin{align*}
&\|i(b)-i(v)\|_2 = \begin{cases} d_4=\sqrt{ R_2^2-2R_2\sqrt{ f(m)-f(m-S) } +f(m)}, &\text{ if }v\not \in b;\\ d_5=\sqrt {R_2^2+2R_2\sqrt{ f(m)-f(S) }+f(m)},&\text{ if } v\in b. \end{cases} \end{align*} For case 2 and case 3, \begin{align*}
&\|i(b)-i(v)\|_2 = \begin{cases} d_4=\sqrt{ R_2^2-2R_2\sqrt{ f(m)-f(S) } +f(m)}, &\text{ if }v\in b;\\ d_5=\sqrt {R_2^2+2R_2\sqrt{ f(m)-f(m-S) }+f(m)},&\text{ if } v\not\in b. \end{cases} \end{align*} Denote the complement of $\mathcal D$ as $\overline{\mathcal D}$. If the embedding of $\mathcal D$ is case 1, the embedding of $\overline{\mathcal D}$ is case 2. Moreover, $\iota_1(\mathcal D)=\iota_2(\overline{\mathcal D})$. The same is true for case 3 and case 4. This reduces to four cases. Without loss of generality, we consider cases (A), (D), (F), (G).
Suppose $\gamma>2$. From the formula for $d_4, d_5$, we find that $R_2$ is a root of the degree two polynomial $x^2+2x\sqrt{f(m)-f(m-S)} +f(m)-d_4^2$. But, since $f(m)<2$, the only positive root is $ \sqrt{ f(m)-f(m-S)} + \sqrt{2- f(m-S) } $. Since this number greater than $R_1$ we conclude that case (D) is impossible.
Suppose $\gamma<2$. From the formula for $d_4, d_5$, we find that $R_2$ is a root of the polynomial $x^2-2x\sqrt{f(m)-f(S)} +f(m)-d_4^2$. The only positive root is $ \sqrt{2-f(m-S)}-\sqrt{f(m)-f(m-S)} $. Since this number is less than $R_1$, we conclude that case (F) is impossible. \end{proof}
\begin{remark} The parameters of conference graphs are
$(n,(n-1)/2,(n-5)/4,(n-1)/4)$. If the block graph is a conference graph, then $d:=m-1$ is the multiplicity of $r$. Hence, $n=2m-1$. Now, the $|X|=m+n>2d+3$ (when $m>2$). By \cite[Theorem 2]{larman1977two} and Theorem \ref{thm:A(X,X)}, $s\in\mathbb Z$ (and hence $r\in\mathbb Z$). \end{remark}
\subsection{Proof of Theorem \ref{thm1}} \begin{proof}[Proof of Theorem \ref{thm1}] Let $A(i(X),i(X))=\{\sqrt 2,\sqrt{\gamma}\}$ and let $(S,m,\alpha,\beta)$ be the parameters of $\mathcal D$.
We will prove the following statement: For $\gamma>2$, $p_1(S,m,\alpha,\beta)=p_2(S,m,\alpha,\beta)=p_3(S,m,\alpha,\beta)=0$ and for $\gamma<2$, $p_1(S,m,\beta,\alpha)=p_2(S,m,\beta,\alpha)=p_3(S,m,\beta,\alpha)=0$.
If $\gamma>2$, then the embedding is of case 1. Since $2=d_2^2=R_2^2 ( 2(S-\alpha)m)/(mS-S^2) $, we have $R_2=\sqrt{\frac{ (m-S)S}{(S-\alpha)m} } $ and $ d_4^2 =2=R_2^2-2R_2\sqrt{f(m)-f(m-S)} +f(m)$ and equivalent to $ 2S \sqrt{S-\alpha} = - (S^2 - \alpha m + S - \alpha)$ which implies $ p_1(S,m,\alpha,\beta)=0$.
Next, $ d_3^2=d_5^2$ if and only if $ R_2^2 (2(S -\beta )m )/(mS -S^2 ) = R_2^2+2R_2\sqrt{f(m )-f(S )} +f(m ) $ and equivalent to $2(m -S ) \sqrt{S -\alpha } =-(S ^2 + \alpha m - 2\beta m + S - \alpha )$ which implies $p_2(S ,m ,\alpha ,\beta )=0$.
By Lemma \ref{par1}, $R_2=\sqrt{2-f(m -S )}+\sqrt{f(m )-f(m -S )}$. By Theorem \ref{thm:A(X,X)}, $ \frac{d_2^2}{d_3^2}=\frac{-s-1}{-s}. $ So, $ \frac{d_2^2}{d_3^2}=f(-s )$ if and only if $$ \frac{2}{ R_2^2+2R_2\sqrt{f(m )-f(S )}+f(m )} =\frac{S -\alpha }{S -\beta }$$ which
implies that $ p_3(S ,m ,\alpha ,\beta )=0$.
If $\gamma<2$, then embedding is of case 3. We apply a similar argument. Now, $R_2=\sqrt{\frac{(m-S)S}{(S-\beta)m}}$. Hence, $d_3^2=d_5^2$ implies $p_1(S,m,\beta,\alpha)=0$ and $d_2^2=d_4^2$ implies $p_2(S,m,\beta,\alpha)=0$. Finally, $R_2=\sqrt{2-f(m-S)}-\sqrt{f(m)-f(m-S)}$ and $ \frac{d_2^2}{d_3^2}=f(-s) =\frac{S-\alpha}{S-\beta}$ implies $ p_3(S,m,\beta,\alpha)=0 $.
\end{proof}
\begin{remark}\label{rem4.2} If $p_1(S ,m ,0,y )=0$, then $S \in\{0,1\}$. Now, $p_2(1,m ,0,y )=4(my + m - 2)m(y - 1)$ implies $m \in\{0,2/(y +1)\}$ or $y =1$. Finally, $p_3(1,\frac 2 {y +1},0,y )=(y -1)(y -3)$. Since $0\leq \beta <\alpha <S <m $, we can't find any feasible solution. \end{remark}
\section{Integer solutions of the polynomials system}
We want to find integer solutions of $ p_i(S_0,m_0,x_0,y_0)=0$ for $i=1,2,3$, in conjunction of that all the parameters $ S_0,m_0,y_0,\Lambda_0,T_0,n_0,k_0,r_0,s_0,l_0,N_0,P_0$ are integers, where \begin{align*} p_1(S,m,x,y)=& S^4 - 2S^2xm + x^2m^2 - 2S^3 + 2S^2x - 2Sxm + 2x^2m + S^2 - 2Sx + x^2;\\ p_2(S,m,x,y)= &S^4 + 2S^2xm - 4S^2ym + x^2m^2 - 4xym^2 + 4y^2m^2 - 2S^3 + 2S^2x \\ &+ 8S^2m - 6Sxm - 2x^2m- 4Sym + 4xym - 4Sm^2 + 4xm^2 + S^2 - 2Sx + x^2;\\ p_3(S,m,x,y)=&S^2x^2 - Smx^2 - 2S^2xy + 2Smxy + S^2y^2 - Smy^2 + S^2m \\ &+ 2S^2x - 2Smx - 2Sx^2 + mx^2 - 2S^2y + 2Sxy + S^2 - 2Sx + x^2. \end{align*}
It would not be an easy task, but we can completely determine the solutions in the following steps. First, by the condition $p_1=p_2=p_3=0$, we could express all parameters $\Lambda, T, n, k, r, s, N, P$ given in Proposition \ref{9C} by the variables $x$ and $z$. Then, we will define a magic auxiliary function $g(x,z)$ such that the function $g(x,z)$ could be bounded values on the unbounded domain of the $zx$ plane. Since $g$ is an integer and bounded, then there are only finitely many cases to be discussed and analysis. Therefore, we can completely determine the solution set.
Notice that the subscript $|_0$ on these variables mean that the variables are evaluated at some particular values. For instance, $x_0$ is the notation for given value of variable $x$. Let \begin{equation} \label{eq:unprovide} z := \frac{x (m + 1) - S (S + 1)}{2 S}. \end{equation}
Lemma \ref{lem:semiportable} allows us to rewrite the parameters in Proposition \ref{9C} as rational functions in just two variables, $x$ and $z$. \begin{lemma} \label{lem:semiportable} Assume that $S, m, x \neq 0$, $p_1(S, m, x, y) = 0$ and $p_2(S, m, x, y) = 0$. Then, \begin{align} \label{eq:baud} S & = x + z^2, \\ \label{eq:bronchioli} m & = (x + z^2 + z)^2 / x, \\ \label{eq:oligandrous} y & = \text{$y_1$ or $y_2$}, \end{align} where \begin{align*} y_1 & = x - z, \\ y_2 & = \frac{x^3+2 x^2 z^2+x^2 z+x z^4+2 x z^3+3 x z^2+z^5+2 z^4+z^3}{\left(x+z^2+z\right)^2}. \end{align*}
Moreover, if $S_0, m_0, x_0, y_0$ are nonzero integers such that $p_1(S_0, m_0, x_0, y_0) = p_2(S_0, m_0, x_0, y_0) = 0$, then $z_0 := z|_{S = S_0, m = m_0, x = x_0, y = y_0}$ is an integer. \end{lemma}
\begin{proof} From (\ref{eq:unprovide}), we get \begin{equation} \label{eq:twiller} m = \frac{S (S + 1 + 2 z)}{x} - 1. \end{equation} Equation (\ref{eq:baud}) follows from substituting (\ref{eq:twiller}) into $p_1 / S^2 = 0$. Substitute equation (\ref{eq:baud}) back into (\ref{eq:twiller}) and we get (\ref{eq:bronchioli}). Substitute both equation (\ref{eq:baud}) and (\ref{eq:bronchioli}) into $p_2 / m = 0$, and we get a quadratic equation in $y$, which has two solutions of the equation (\ref{eq:oligandrous}).
Now let $S_0, m_0, x_0, y_0$ be integers satisfying the assumptions. We see from (\ref{eq:baud}) that $z_0^2 = S_0 - x_0$ is an integer, and from (\ref{eq:baud}) and (\ref{eq:bronchioli}) that $(S_0 + z_0)^2 = m_0 x_0$ is an integer. Therefore, $z_0$ is an integer. \end{proof}
There are two cases that $y = y_1$ and $y = y_2$.
For $y=y_1$ we discuss in Proposition \ref{prop:did} and $y=y_2$ in Proposition \ref{prop:retinker}.
\begin{proposition} \label{prop:did} Treat all parameters $\Lambda, T, n, k, r, s, N, P$ given in Proposition \ref{9C} as elements in $\R(x, z)$ by Lemma \ref{lem:semiportable} and $y = y_1$. Let $x_0$ be a positive integer and $z_0$ an integer such that $\Lambda_0, T_0, n_0, k_0, r_0, s_0, l_0, N_0, P_0$ are all integers. Then, either $x_0 = z_0 (z_0 + 1) / 2$, or $z_0 = 0$, or $z_0 = -1$. \end{proposition}
\begin{proof} Consider the auxiliary polynomial \[
g := 3 + x + 19 z + 16 z^2 + 3 k + 3 \Lambda - m - 4 n - 18 P + 21 z r. \] Since this is a polynomial of parameters with integral coefficients, $g_0$ is an integer. Observe that $$\mathbb R^2=\bigcup_{z,k\in\mathbb R}(z,z(z+1)/2+k).$$ We want to find $(z,x)\in \mathbb Z\times\mathbb Z_{\geq 1}$, so $k\in\mathbb Z$. Use computers to prove that: \begin{itemize} \item(Region1): $g \in (0, 1)$ when $z \in (-\infty, -2] \cup [1, +\infty)$, $x \geq 1$ and $x \geq z (z + 1) / 2 + 1$; \item(Region2): $g \in (0, 2)$ when $x \geq 1$ and $x = z (z + 1) / 2 - 1$; \item(Region3): $g \in (0, 1)$ when $x \geq 1$ and $x \leq z (z + 1) / 2 - 2$. \end{itemize} \begin{figure}
\caption{Regions}
\label{area}
\end{figure} The regions are showed in Figure \ref{area}. By the facts above and that $g_0$ is an integer, if $(x_0, z_0)$ does not satisfy the conclusion of this lemma, then the only possibility is that $x_0 = z_0 (z_0 + 1) / 2 - 1$ and $g_0 = 1$. Solving this system of equations would give us $z_0 = \frac{1}{2}(- 1 \pm \sqrt{41})$, which contradicts the fact that $z_0$ is an integer. \end{proof}
\begin{proposition} \label{prop:retinker} Under the same assumption of Proposition \ref{prop:did} except that $y = y_2$, we assume in addition that $p_3(S_0, m_0, x_0, y_0) = 0$. Then, $z_0 = 0$. \end{proposition}
\begin{proof} Consider the auxiliary polynomial \[
g := -72 \Lambda + 13 m + 13 n + 99 T - 45 x + 32 y - 14 z - 13 m z + 39 n z - 13 T z + 13 x z - 33 z^2 + 13 z^3. \] Since it is an integer coefficient polynomial in parameters, $g_0$ is an integer.
Use computers to prove that: \begin{itemize} \item $g \in (31, 33)$ when $z \in (-\infty, -15] \cup [10, +\infty)$ and $x \in \{1, 2\}$. \item $g \in (-1, 38)$ when $x \geq 3$. \end{itemize}
\noindent Case 1. $x_0 \in \{1, 2\}$.
An enumeration of small pairs $(x_0, z_0)$ shows that there are no small integral pair which gives $g_0 = 32$. For $x_0\in\{1,2\}$ and $z_0\in \{-14,\dots,9\}$, only $ g(1,-1)=136, g(1,0)=0 ,g(2,0)=0$ are integers. But, $\Lambda(1,-1)=-1$.
\noindent Case 2. $x_0\geq3$ and $g_0 \in \{1, \dots, 37\}$.
For each $i \in \{1, \dots, 37\}$, the intersection of the curves $p_3 = 0$ and $g = i$ consists of finitely many points. An explicit calculation shows that the only integral points are $(x_0, z_0) = (0, 1)$ and $(x_0, z_0) = (-9, 3)$. Both violate the assumption that $x_0$ is positive.
\noindent Case 3. $x_0\geq 3$ and $g_0 = 0$.
The intersection of the curves $p_3 = 0$ and $g = 0$ consists of a curve $z = 0$, and $8$ points, all of which are not integral points. \end{proof}
\begin{theorem}\label{solufamily} Consider the system of equations $p_1(S, m, x, y) = p_2(S, m, x, y) = p_3(S, m, x, y) = 0$. The only integral solutions $(S_0, m_0, x_0, y_0)$ such that $S_0, m_0 \neq 0$, $x_0 \geq 1$ and $\Lambda_0,$ $T_0,$ $n_0,$ $k_0,$ $r_0,$ $s_0,$ $l_0,$ $N_0,$ $P_0$ are all integers are on the following parametrized curves. \begin{enumerate}
\item[(i)] $S = \frac{1}{2} z (3 z + 1)$, $m = \frac{9}{2} z (z + 1)$, $x = \frac{1}{2} z (z + 1)$, $y = \frac{1}{2} z (z - 1)$.
\item[(ii)] $S = z$, $m = z$, $x = z$, $y = z$.
\item[(iii)] $S = z + 1$, $m = z$, $x = z$, $y = z + 1$. \end{enumerate}
\end{theorem} \begin{proof} Recall that \begin{align*} S & = x + z^2, \\ m & = (x + z^2 + z)^2 / x, \\ y & = \text{$y_1$ or $y_2$}, \end{align*} where \begin{align*} y_1 & = x - z, \\ y_2 & = \frac{x^3+2 x^2 z^2+x^2 z+x z^4+2 x z^3+3 x z^2+z^5+2 z^4+z^3}{\left(x+z^2+z\right)^2}. \end{align*} For the case $y=y_1$, we have $x_0 = z_0 (z_0 + 1) / 2$, or $z_0 = 0$, or $z_0 = -1$ by Proposition \ref{prop:did}. If $x = z (z + 1) / 2$, then $S = \frac{1}{2} z (3 z + 1)$, $m = \frac{9}{2} z (z + 1)$, $x = \frac{1}{2} z (z + 1)$, $y = \frac{1}{2} z (z - 1)$. If $z=0$, then $S=m=y=x$. If $z=-1$, then $S=y=x+1$ and $m=x$.
For the case $y=y_2$, we have $z_0 = 0$ by Proposition \ref{prop:retinker}. Thus, $S=m=y=x$ holds.\end{proof}
\section{Proof of main Theorem}
\begin{proof}[Proof of the Theorem \ref{main thm}] Let $\mathcal D$ be a $2$-$(m_0,S_0,\Lambda_0)$ quasi-symmetric design satisfying the conditions in Theorem \ref{thm1}. It follows from Theorem \ref{thm1} that $$ p_1(S_0,m_0,x_0,y_0)=p_2(S_0,m_0,x_0,y_0)=p_3(S_0,m_0,x_0,y_0)=0.$$ By Remark \ref{rem4.2}, $x_0\geq 1$. From Theorem \ref{solufamily}, we have three possible solutions need to be discussed. Parametric (ii) and (iii) give $S_0=\alpha_0$ which is a contradiction. It remains to consider the Parametrized solution (i). From Parametrized solution (i): \begin{align*} S = \frac{1}{2} z (3 z + 1),\,\,\, m = \frac{9}{2} z (z + 1),\,\,\, x = \frac{1}{2} z (z + 1),\,\,\, y = \frac{1}{2} z (z - 1)
\end{align*} By Remark \ref{rem4.2} and Lemma \ref{lem:semiportable}, $z_0$ should be an integer.
Suppose $2\leq S_0 \leq 3$.
If $S_0=2$, then $z_0=1$ or $-4/3$. Lemma \ref{lem:semiportable} implies $z_0=1$ which is corresponding to the Lison\v ek's example.
If $S_0=3$, then $z_0=\pm \sqrt{73}/6 - 1/6$ which is impossible.
Suppose $4\leq S_0\leq m_0-4$. Since $n_0={m_0\choose 2}$, by Theorem \ref{thm3}, $\mathcal D$ is the $4$-$(23,7,1)$ design. Now, $S_0=7$ implies $z_0\in \{-7/3,2\}$. If $z_0=2$, then $m_0=27\neq 23$.
Suppose $S_0=m_0-3$, then $z_0=\pm\sqrt{13}/3 - 2/3$ which is impossible.
Suppose $S_0=m_0-2$, then
$z_0=\pm\sqrt{10}/3 - 2/3$ which is impossible.
Suppose $S_0=m_0-1$, then
$z_0=\pm \sqrt 7/3 - 2/3$ which is impossible. \end{proof}
\section{Discussions} We would like to propose some further research problems.
(1) In Nozaki-Shinohara \cite{nozaki2020maximal}, they consider the two-distance sets in $\mathbb R^d$ which contain a regular simplex and a strongly regular graph. We believe that the case where: the regular simplex is of size $d+1$ and the strongly regular graph comes from a natural embedding (with respect to a primitive idempotent of rank $d$) would be the most interesting (and extremal) case. We discussed this problem assuming the additional condition that the two-distance set has the structure of a coherent configuration of type (2,2;3). We wonder whether it is possible to drop this additional condition on the existence of the coherent configuration.
(2) It would be interesting whether there exists any two-distance set in $\mathbb R^d$ coming from the natural embedding of a coherent configuration of type (3,2;3).
(3) It is natural, although it is not so easy, to try to generalize the discussion on two-distance sets to three-distance sets in some way. For example, can we classify three-distance set in $\mathbb R^d$ coming from the natural embedding of a coherent configuration of type (2,2;4). (There are many other possibilities.)
(4) Although it seems to be a difficult problem, it would be interesting to study two-distance set $X$ in $\mathbb R^d$ of the maximum cardinality ${d+ 2 \choose 2},$ whether we can find a structure of a coherent configuration, or some combinatorial structure close to the coherent configuration?
\end{document} | arXiv | {
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\begin{document}
\title[Hessian quotient equations]{The Dirichlet problem for Hessian quotient equations on exterior domains}
\author[T.Y. Jiang]{Tangyu Jiang} \address[T.Y. Jiang]{School of Mathematical Sciences\\ Beijing Normal University \\
100875 Beijing\\
P.R. China} \email{202131081002@mail.bnu.edu.cn}
\author[H.G. Li]{Haigang Li} \address[H.G. Li]{School of Mathematical Sciences\\ Beijing Normal University \\
100875 Beijing\\
P.R. China} \email{hgli@bnu.edu.cn}
\author[X.L. Li]{Xiaoliang Li} \address[X.L. Li]{School of Mathematical Sciences\\ Beijing Normal University \\
100875 Beijing\\
P.R. China} \email{rubiklixiaoliang@163.com}
\subjclass[2010]{35J60, 35J25, 35D40, 35B40}
\keywords{Hessian quotient equations; Exterior Dirichlet problem; Existence and uniqueness; Prescribed asymptotic behavior; Perron's method}
\begin{abstract}
In this paper, we consider the exterior Dirichlet problem for Hessian quotient equations with the right hand side $g$, where $g$ is a positive function and $g=1+O(|x|^{-\beta})$ near infinity, for some $\beta>2$. Under a prescribed generalized symmetric asymptotic behavior at infinity, we establish an existence and uniqueness theorem for viscosity solutions, by using comparison principles and Perron's method. This extends the previous results for Monge--Amp\`ere equations and Hessian equations. \end{abstract}
\maketitle
\section{Introduction}\label{sec:intro} The aim of this paper is to study the Dirichlet problem for Hessian quotient equation \begin{equation}\label{eq:pro} S_{k,l}(D^2u):=\frac{\sigma_k(\lambda(D^2u))}{\sigma_l(\lambda(D^2u))}=g(x) \end{equation} in the exterior domain $\mathbb{R}^n\setminus\overline{D}$, where $D$ is a bounded open set in $\mathbb{R}^n$ $(n\geq 3)$, $0\leq l<k\leq n$, $\lambda(D^2 u)=(\lambda_1,\cdots,\lambda_n)$ denotes the eigenvalue vector of the Hessian matrix $D^2u$, \begin{equation*} \sigma_j(\lambda(D^2u))= \begin{cases} \sum_{1\leq i_1<\cdots<i_j\leq n}\lambda_{i_1}\cdots\lambda_{i_j}, & j=1,\cdots, n,\\ 1,& j=0, \end{cases} \end{equation*} and $g\in C^0(\mathbb{R}^n\setminus D)$ is a positive function satisfying \begin{equation}\label{eq:beta-g}
\inf_{\mathbb{R}^n\setminus D}g>0\quad\text{and}\quad\limsup_{|x|\to\infty}|x|^\beta|g(x)-1|<\infty \end{equation} for some constant $\beta>0$. We are mainly concerned with the existence and uniqueness of viscosity solutions to \eqref{eq:pro}, with prescribed boundary data and asymptotic behavior at infinity.
Note that equation \eqref{eq:pro} embraces several typical cases. When $l=0$, it is Poisson equation \begin{equation}\label{eq:Poisson} \sigma_1(\lambda(D^2u))=\Delta u=g \end{equation} if $k=1$ and fully nonlinear $k$-Hessian equation \begin{equation}\label{eq:k-Hessian-g} \sigma_k(\lambda(D^2u))=g \end{equation} if $2\leq k\leq n$, which in particular corresponds to the famous Monge--Amp\`ere equation \begin{equation}\label{eq:M-A-g} \det(D^2u)=g \end{equation} for $k=n$. It is well-known that, as a special class of nonlinear, second-order elliptic equations of the form $f(\lambda(D^2u))=g(x)$, the classical solvability of the interior Dirichlet problem for equation \eqref{eq:pro} has been extensively studied. We would like to mention the work of Caffarelli--Nirenberg--Spruck \cite{Caffarelli1984,Caffarelli1985} on equations \eqref{eq:k-Hessian-g}--\eqref{eq:M-A-g}, and that of Trudinger \cite{Trudinger1995} on general equation \eqref{eq:pro}; see also Ivochkina \cite{Ivochkina1983,Ivochkina1985}, Urbas \cite{Urbas1990}, Guan \cite{Guan1994}, and the references therein.
In exterior domains, the study of the Dirichlet problem for equation \eqref{eq:pro} also has received increasing attentions in recent years. It goes back to Meyers--Serrin \cite{Serrin1960}, who gave a sufficient condition for the source term $g$ to obtain the existence and uniqueness of solutions to \eqref{eq:Poisson} with an assigned limit at infinity. For nonlinear case of \eqref{eq:pro}, it started by Caffarelli--Li \cite{Caffarelli-Li-2003}, which is an extension of celebrated J\"{o}rgens--Calabi--Pogorelov theorem \cite{Calabi1958,Jorgens1954,Pogorelov1972} stating that any classical convex solution of \begin{equation}\label{eq:M-A} \det(D^2u)=1 \end{equation} in $\mathbb{R}^n$ must be a quadratic polynomial (see also \cite{Caffarelli1995,Cheng-Yau-1986,Jost2001}). They showed that if $u$ is a convex viscosity solution of \eqref{eq:M-A} outside a bounded open convex subset of $\mathbb{R}^n$ ($n\geq3$), then there is a symmetric positive definite matrix $A\in\mathbb{R}^{n\times n}$ with $\det (A)=1$, a vector $b\in\mathbb{R}^n$ and a constant $c\in\mathbb{R}$ such that \begin{equation} \label{eq:C-Li}
\limsup_{|x|\to\infty}\left(|x|^{n-2}\left|u(x)-\left(\frac12x^TAx+b\cdot x+c\right)\right|\right)<\infty. \end{equation} Moreover, by Perron's method they established an existence and uniqueness theorem for exterior solutions of \eqref{eq:M-A} prescribed by \eqref{eq:C-Li}. Then Bao--Li--Zhang \cite{Bao-Li-Zhang-2015} extended that to general equation \eqref{eq:M-A-g}, assuming \eqref{eq:beta-g} with $\beta>2$. Recently, Li--Lu \cite{Li-Lu-2018} further obtained nonexistence results to the exterior problems considered in \cite{Bao-Li-Zhang-2015,Caffarelli-Li-2003}, giving a complete characterization for their solvability.
In spirit of \cite{Caffarelli-Li-2003}, when $g\equiv 1$ the exterior Dirichlet problem for Hessian equations \eqref{eq:k-Hessian-g} and Hessian quotient equations \eqref{eq:pro} was subsequently treated by Bao--Li--Li \cite{Bao-Li-Li-2014} and Li--Li \cite{Li-Li-2018} respectively, under proper prescribed asymptotic condition similar to \eqref{eq:C-Li}. Inspired by \cite{Bao-Li-Zhang-2015}, Cao--Bao \cite{Cao-Bao-2017} further extended the result in \cite{Bao-Li-Li-2014} to equation \eqref{eq:k-Hessian-g} with a general right hand side $g$ satisfying \eqref{eq:beta-g}. However, equation \eqref{eq:pro} with general $g\not\equiv 1$ is much more complicated to deal with than that since whenever $l>0$ elementary symmetric functions $\sigma_k$ and $\sigma_l$, of different order of homogeneities, are involved simultaneously. In particular, in this case we are confronted with the effect of irreducible fraction involved in \eqref{eq:pro}, as will be explained in detail in Remark \ref{rk:thm} (ii) below. More importantly, the comparison principle for viscosity solutions of equation \eqref{eq:pro} have not been found in a general setting of the function $g$, as far as we know, so that ones are unable to carry out the Perron process to construct the solution of the Dirichlet problem as in the Monge--Amp\`ere case and $k$-Hessian case.
Regarding more related studies on exterior domains, we refer the readers to \cite{Bao-Li-2013,Jiang-Li-Li-2020,Li-Bao-2014,Li2019} where other Hessian type equations associated with \eqref{eq:pro}, including the special Lagrangian equations, are considered. Besides, the Liouville type results for equation \eqref{eq:pro} can be found in \cite{Bao2003,CY2010,Li-Li-Yuan-2019,Li-Ren-Wang-2016} and the references therein.
In this paper, we shall prove the solvability of the exterior Dirichlet problem for equation \eqref{eq:pro}, under a prescribed quadratic condition of type \eqref{eq:C-Li}. Inspired by the discussions of Trudinger \cite{Trudinger1990} for the prescribed curvature equations, we investigate the comparison principles in a general setting of Hessian type fully nonlinear elliptic equations which may embrace \eqref{eq:pro} as a special case; see Appendix \ref{sec:comparison}. Then in the spirit of Crandall, Ishii, and Lions \cite{Ishii1992}, we are allowed to establish an existence and uniqueness theorem for viscosity solutions of \eqref{eq:pro} by applying an adapted Perron's method. This extends previously known results on Monge--Amp\`ere equation \eqref{eq:M-A-g} in \cite{Bao-Li-Zhang-2015} and on $k$-Hessian equation \eqref{eq:k-Hessian-g} in \cite{Cao-Bao-2017}, and also extends those obtained in \cite{Li-Li-2018} particularly for the case $g\equiv 1$ to the general $g$ fulfilling condition \eqref{eq:beta-g}. In order to state the result specifically, we introduce below some notations.
Let $$\mathcal{A}_{k,l}=\left\{A\in S^+(n):S_{k,l}(A)=1\right\}$$ where $S^+(n)$ is the space of real $n\times n$ symmetric positive definite matrices. For $A\in S^+(n)$, we define $$H_k(A)=\max_{1\leq i\leq n}\frac{\sigma_{k-1;i}(\lambda(A))\lambda_i(A)}{\sigma_k(\lambda(A))}$$
where $\sigma_{k-1;i}(\lambda):=\sigma_{k-1}(\lambda)|_{\lambda_i=0}$. Then $H_k(A)\in [\frac{k}{n},1)$ for $1\leq k\leq n-1$ and $H_n(A)\equiv 1$. Let $$\mathscr{A}_{k,l}=\left\{A\in\mathcal{A}_{k,l}: H_k(A)<\frac{k-l}{2}\right\}.$$ To avoid $\mathscr{A}_{k,l}$ to be empty, we require the indices $k$ and $l$ to satisfy
\begin{equation}\label{eq:index-k-l} \begin{cases} 0\leq l\leq n-3 &\text{if }k\geq l+2,\\ 0\leq l<\frac{n}{2}-1 &\text{if }k=l+1. \end{cases} \end{equation}
Our main result is the following. \begin{theorem}\label{thm:main}
Let $D$ be a smooth, bounded, strictly convex domain in $\mathbb{R}^n$ ($n\geq 3$) and let $\varphi\in C^2(\partial D)$. Assume that $g\in C^0(\mathbb{R}^n\setminus D)$ satisfies \eqref{eq:beta-g} with $\beta>2$. Then for any $A\in\mathscr{A}_{k,l}$ and $b\in\mathbb{R}^n$, there exists a constant $c_*$ depending only on $n,A,b,D,g$ and $\|\varphi\|_{C^2(\partial D)}$, such that for every $c>c_*$ there exists a unique viscosity solution $u\in C^0(\mathbb{R}^n\setminus D)$ to the problem \begin{equation}\label{eq:asym-thm} \begin{cases} S_{k,l}(D^2u)=g \quad\text{in}\ \mathbb{R}^n\setminus\overline{D},\\ u=\varphi \quad\text{on}\ \partial D,\\
\limsup_{|x|\to\infty}\left(E(x)\Big{|}u(x)-(\frac12x^TAx+b\cdot x+c)\Big{|}\right)<\infty, \end{cases} \end{equation} where \begin{equation}\label{eq:intro-E} E(x):=\left\{ \begin{array}{ll}
|x|^{\min\{\beta,\frac{k-l}{H_k(A)}\}-2} & \text{if }\beta\neq\frac{k-l}{H_k(A)},\\
|x|^{\frac{k-l}{H_k(A)}-2}(\ln |x|)^{-1} & \text{if }\beta=\frac{k-l}{H_k(A)}. \end{array} \right. \end{equation} \end{theorem}
\begin{remark}\label{rk:thm} (i) Notice that when $l=0$ and $2\leq k\leq n$, $\mathscr{A}_{k,0}=\mathcal{A}_{k,0}$ by Proposition \ref{G-pro:A-k-l} and Theorem \ref{thm:main} was proved in \cite{Bao-Li-Zhang-2015} for equation \eqref{eq:M-A-g} and in \cite{Cao-Bao-2017} for equation \eqref{eq:k-Hessian-g}, respectively. We here extend to the general case \eqref{eq:pro} and are able to prove Theorem \ref{thm:main} in a systematic way for all of $(l,k)$ satisfying \eqref{eq:index-k-l}.
(ii) When $l\geq1$ and $g\equiv 1$, Theorem \ref{thm:main} was proved in \cite{Li-Li-2018} by rewriting \eqref{eq:pro} as $$\sigma_k(\lambda(D^2u))=\sigma_l(\lambda(D^2u)),$$ namely, not considering it in a sense of fraction. However, whenever $g\not\equiv 1$ one has to tackle the difficulty caused by the irreducible fraction in \eqref{eq:pro} when seeking subsolutions and supersolutions of \eqref{eq:pro} for carrying out the Perron process. Indeed, different from that adopted in \cite{Li-Li-2018}, here we turn to solve a new ODE to construct the desired subsolutions; see Remark \ref{G-rk:no-h-l} in Subsection \ref{G-sec:sub} for more details. Also, we can obtain the desired supersolutions in a parallel way, since unlike \cite{Li-Li-2018} we are unable to directly pick quadratic polynomials $\frac12x^TAx+b\cdot x+c$ as such ones unless assuming $\inf_{\mathbb{R}^n\setminus D}g=1$. \end{remark} \begin{remark} It is not difficult to observe that Theorem \ref{thm:main} still holds with $\mathscr{A}_{k,l}$ adapted to a slightly larger set $ \mathscr{A}^*_{k,l}:=\{A\in\mathbb{R}^{n\times n}: A^*\in\mathscr{A}_{k,l}\}$ where $A^*=\frac{A+A^T}{2}$. \end{remark}
The proof of Theorem \ref{thm:main} is based on Perron's method, formulated in Theorem \ref{thm:Perron-m} for the general fully nonlinear elliptic equation of form \eqref{app-eq:F-x-D2u} which includes \eqref{eq:pro} as a specific case. We remark that Theorem \ref{thm:Perron-m} is an adaption of arguments as in \cite{Ishii1992,Ishii1989,Ishii-Lions-1990} once the uniqueness and comparison results are well established (see Theorem \ref{thm:comparison-b} and Corollary \ref{thm:comparison-ub}). With the help of that, the key ingredient of our proof lies in finding a family of appropriate subsolutions and supersolutions of \eqref{eq:pro} both with uniformly quadratic asymptotics at infinity. As a realization, we work with the so-called generalized symmetric functions (an extension of radial functions, specified in Definition \ref{G-def:G-Sym}), on which $k$-Hessian operators $S_{k,0}$ can be computed explicitly (see Lemma \ref{G-lem:k-He-G-Sym} below), thus helping us obtain the desired subsolutions and supersolutions by solving two corresponding second-order ODEs (\eqref{G-eq:ode-sub} and \eqref{G-eq:ode-super}). We would like to point out that the reason why we restrict $\beta>2$ and $A\in\mathscr{A}_{k,l}$ in Theorem \ref{thm:main} is to ensure the asymptotically quadratic property of subsolutions and supersolutions of \eqref{eq:pro} to be constructed in Propositions \ref{G-pro:sub} and \ref{G-pro:super}. One can see this from the computations performed in part \textbf{(c)} in Subsection \ref{G-sec:sub}.
The remainder of this paper is organized as follows. Section \ref{sec:G} is devoted to the construction of a family of subsolutions and supersolutions of \eqref{eq:pro} with asymptotically quadratic property. Then we apply Theorem \ref{thm:Perron-m} to prove Theorem \ref{thm:main} in Section \ref{sec:3}. In Appendix \ref{sec:comparison}, in a general setting of fully nonlinear elliptic equations related to the eigenvalues of the Hessian, we employ approximation arguments to prove comparison principles for viscosity solutions of the equations, both on bounded and unbounded domains. In Appendix \ref{sec:perron}, we state precisely Perron's method in Theorem \ref{thm:Perron-m} and present the proof, for the reader's convenience.
\section{Generalized symmetric subsolutions and supersolutions}\label{sec:G} In this section, we first introduce the concept of generalized symmetric functions as well as their fine properties. Then by solving two second-order ODEs we obtain a family of generalized symmetric subsolutions and supersolutions of \eqref{eq:pro}, both of which are uniformly $k$-convex and asymptotically quadratic near infinity; see Propositions \ref{G-pro:sub} and \ref{G-pro:super}. This is essential to apply the Perron's arguments to prove Theorem \ref{thm:main} in the next section.
Throughout the section, we use the symbol $$\lambda(A):=(\lambda_1(A),\lambda_2(A),\cdots,\lambda_n(A))$$ with the order $\lambda_1(A)\leq\lambda_2(A)\leq\cdots\leq\lambda_n(A)$ to denote the eigenvalue vector of a real $n\times n$ symmetric matrix $A$.
\subsection{Preliminary}\label{G-sec:pre}
We present some definitions and related facts which will be used later to seek proper subsolutions and supersolutions of \eqref{eq:pro}.
As in \cite{Bao-Li-Li-2014}, we define the generalized symmetric (abbreviated to G-Sym in the sequel) function in the following sense. \begin{definition}\label{G-def:G-Sym} For a symmetric matrix $A$, we call $u$ a G-Sym function with respect to $A$ if it is a function of $s=\frac12x^TAx$, $x\in\mathbb{R}^n$, that is $$u(x)=u(s):=u(\frac12x^TAx).$$ If $u$ is a subsolution (supersolution) of \eqref{eq:pro} and is also a G-Sym function, we say that $u$ is a G-Sym subsolution (supersolution) of \eqref{eq:pro}. \end{definition}
There is an explicit formula \eqref{G-eq:k-H-G-Sym} below for $k$-Hessian operators acting on smooth G-Sym functions and its proof can be found in Bao--Li--Li \cite{Bao-Li-Li-2014}. \begin{lemma}\label{G-lem:k-He-G-Sym} Let $A=\mathrm{diag}(a_1,\cdots,a_n)$. If $w\in C^2(\mathbb{R}^n)$ is a G-Sym function with respect to $A$, then \begin{equation}\label{G-eq:k-H-G-Sym} \sigma_k(\lambda(D^2w))=\sigma_k(\lambda(A))(w')^k+w''(w')^{k-1}\sum_{i=1}^n\sigma_{k-1;i}(\lambda(A))(a_ix_i)^2, \end{equation} where $w':=\frac{dw}{ds}$ and $w'':=\frac{d^2w}{ds^2}$, $s=\frac12x^TAx$. \end{lemma}
Taking advantage of the following properties of the $k$-th elementary symmetric function $\sigma_k$: \begin{equation}\label{G-eq:sigma-k} \sigma_k(\lambda)=\frac{1}{k}\sum_{i=1}^n\lambda_i\sigma_{k-1;i}(\lambda)=\sigma_{k;i}(\lambda)+\lambda_i\sigma_{k-1;i}(\lambda), \quad\forall\, 1\leq i\leq n,\forall\,\lambda\in\mathbb{R}^n,\\ \end{equation} we can easily verify that
$$H_k(A)=\frac{\sigma_{k-1;n}(\lambda(A))\lambda_n(A)}{\sigma_k(\lambda(A))}$$ and \begin{equation*} \frac{k}{n}\leq H_k(A)<1\text{ for }1\leq k\leq n-1; \ H_n(A)\equiv 1. \end{equation*}
We close this preliminary by presenting the relation of $\mathcal{A}_{k,l}$ and $\mathscr{A}_{k,l}$ as follows, which can be directly verified. \begin{proposition}\label{G-pro:A-k-l} Suppose $0\leq l<k\leq n$ and $n\geq 3$. \begin{enumerate} \item[(i)] $c^*(k,l)I\in\mathcal{A}_{k,l}$, where \begin{equation*} c^*(k,l):=\left(\frac{C_n^l}{C_n^k}\right)^{\frac{1}{k-l}},\quad C_n^j=\frac{n!}{(n-j)!j!},\ j=k,l. \end{equation*} \item[(ii)] If $k-l\geq 3$ or $k-l=2$ with $k<n$, then $\mathcal{A}_{k,l}=\mathscr{A}_{k,l}$. \end{enumerate} \end{proposition}
\subsection{Generalized symmetric subsolutions}\label{G-sec:sub} We construct here the wanted G-Sym subsolutions of \eqref{eq:pro}.
Throughout this subsection, let $A =\text{diag}(a_1, a_2,\cdots, a_n)\in\mathcal{A}_{k,l}$, $a:=\lambda(A)$ and $w:=w(s)$ is a G-Sym function with respect to $A$, where $s=\frac12x^TAx=\frac12\sum_{i=1}^na_ix_i^2$, $x\in\mathbb{R}^n$. We denote \begin{equation}\label{G-eq:D-s} D(s):=\left\{x\in\mathbb{R}^n:\frac12x^TAx<s\right\}\quad\text{for }s>0. \end{equation}
Since $g$ satisfies \eqref{eq:beta-g}, there exist $C_0$ and $s_0>1$ such that \begin{equation}\label{G-eq:g-upper} g(x)\leq \bar{g}(x)=\bar{g}(s):=1+C_0s^{-\frac{\beta}{2}}, \text{ when }s=\frac12 x^TAx\geq s_0. \end{equation} In order to make $w$ be a smooth subsolution of \eqref{eq:pro}, i.e. $S_{k,l}(D^2w)\geq g$, we consider the following ODE: \begin{equation}\label{G-eq:ode-sub} \begin{cases} (w')^{k-l}+2H_k(A)w''(w')^{k-l-1}s=\bar{g}(s),& s>1,\\ w'(s)>0,\quad w''(s)\leq 0,&s\geq1. \end{cases} \end{equation}
A uniformly $k$-convex (see \eqref{app-eq:k-convex}) solution of \eqref{G-eq:ode-sub} will be a G-Sym subsolution of equation \eqref{eq:pro} with respect to $A$ when $s\geq s_0$. Indeed, applying Lemma \ref{G-lem:k-He-G-Sym}, one has \begin{align} S_{k,l}(D^2w)&=\frac{\sigma_k(a)(w')^k+w''(w')^{k-1}\sum_{i=1}^n\sigma_{k-1;i}(a)(a_ix_i)^2}{\sigma_l(a)(w')^l+w''(w')^{l-1}\sum_{i=1}^n\sigma_{l-1;i}(a)(a_ix_i)^2}\notag\\ &\geq \frac{\sigma_k(a)(w')^k+w''(w')^{k-1}\sum_{i=1}^n\sigma_{k-1;i}(a)(a_ix_i)^2}{\sigma_l(a)(w')^l}\notag\\ &=(w')^{k-l}+w''(w')^{k-l-1}\sum_{i=1}^n\frac{\sigma_{k-1;i}(a)}{\sigma_l(a)}(a_ix_i)^2\notag\\ &\geq (w')^{k-l}+2H_k(A)w''(w')^{k-l-1}s=\bar{g}(s).\label{G-eq:S-D2w-sub} \end{align}
For ODE \eqref{G-eq:ode-sub}, by the variation-of-constant formula we figure out it has a family of smooth solutions \begin{equation}\label{G-eq:ode-sub-solu} w_{c_1,c_2}(s):=c_2+\int_{s_0}^s\left(\eta^{-\mathcal{H}}\left(\int_1^\eta\mathcal{H}t^{\mathcal{H}-1}\bar{g}(t)\,dt+c_1\right)\right)^{\frac{1}{k-l}}d\eta \end{equation} where $c_i\in\mathbb{R}$ ($i=1,2$) are two constants and $$\mathcal{H}=\mathcal{H}(k,l,A):=\frac{k-l}{2H_k(A)}.$$ We next perform a series of careful calculations, which contains three parts, to see whether $w_{c_1,c_2}$ given by \eqref{G-eq:ode-sub-solu} have the fine properties we expected.
\textbf{(a)} We claim $w_{c_1,c_2}'>0$ and $w_{c_1,c_2}''\leq 0$ on $[1,+\infty)$.
After a direct calculation, we have $$w_{c_1,c_2}'(s)=\left(s^{-\mathcal{H}}\left(\int_1^s\mathcal{H}t^{\mathcal{H}-1}\bar{g}(t)\,dt+c_1\right)\right)^{\frac{1}{k-l}}>0$$ if $c_1>0$, and
\begin{align*} w_{c_1,c_2}''(s)&=-\frac{s^{-\mathcal{H}-1}}{2H_k(A)}(w')^{1-k+l}\left(\int_1^s\mathcal{H}t^{\mathcal{H}-1}\bar{g}(t)\,dt+c_1-s^{\mathcal{H}}\bar{g}(s)\right)\\ &=:-\frac{s^{-\mathcal{H}-1}}{2H_k(A)}(w')^{1-k+l}G(s), \end{align*} where \begin{equation*} G(s)= \begin{cases} c_1+\frac{C_0\beta}{2(\mathcal{H}-\frac{\beta}{2})}s^{\mathcal{H}-\frac{\beta}{2}}-1-\frac{C_0\mathcal{H}}{\mathcal{H}-\frac{\beta}{2}}&\text{if }\mathcal{H}\neq \frac{\beta}{2},\\ c_1+C_0\mathcal{H}\ln s-1-C_0&\text{if }\mathcal{H}=\frac{\beta}{2}. \end{cases} \end{equation*} Hence, there is $\tilde{C}>0$, dependent on $\beta$ but independent of $s$, such that $w_{c_1,c_2}'(s)>0$ and $w_{c_1,c_2}''(s)<0$ for $s\geq1$ when $c_1>\tilde{C}$.
\textbf{(b)} Claim: $w_{c_1,c_2}(s)$ are uniformly $k$-convex on $[1,+\infty)$, provided $c_1>\tilde{C}$.
For any $1\leq m\leq k$ and $c_1>\tilde{C}$, \begin{align*} &\sigma_m(\lambda(D^2 w_{c_1,c_2}))\\&=\sigma_m(a)(w_{c_1,c_2}')^m+w_{c_1,c_2}'' (w_{c_1,c_2}')^{m-1}\sum_{i=1}^n\sigma_{m-1;i}(a)(a_ix_i)^2\\ &=\sigma_m(a)(w_{c_1,c_2}')^{m-1}\left(w_{c_1,c_2}'+w_{c_1,c_2}''\sum_{i=1}^n\frac{\sigma_{m-1;i}(a)}{\sigma_m(a)}(a_ix_i)^2\right)\\ &\geq \sigma_m(a)(w_{c_1,c_2}')^{m-1}\left(w_{c_1,c_2}'+2H_m(A)w_{c_1,c_2}''s\right). \end{align*} Since \begin{align*} \sigma_m(a)\sigma_{k-1;n}(a)&=\left(\sigma_{m;n}(a)+a_n\sigma_{m-1;n}(a)\right)\sigma_{k-1;n}(a)\\ &\geq \sigma_{m-1;n}(a)\sigma_{k;n}(a)+a_n\sigma_{m-1;n}(a)\sigma_{k-1;n}(a)\\ &=\sigma_{m-1;n}(a)\sigma_{k}(a) \end{align*} which follows from \eqref{G-eq:sigma-k} and the Newtonian inequality (see \cite{Hardy1952}): $$\sigma_{k+1;n}(a)\sigma_{k-1;n}(a)\leq \sigma_{k;n}^2(a),\quad 1\leq k\leq n-1.$$ So that $$H_m(A)=\frac{\sigma_{m-1;n}(a)a_n}{\sigma_m(a)}\leq \frac{\sigma_{k-1;n}(a)a_n}{\sigma_k(a)}=H_k(A)\quad \text{for }1\le m\le k.$$ Hence, \begin{equation}\label{G-eq:k-convex-sub} \sigma_m(\lambda(D^2w_{c_1,c_2}))\ge \sigma_m(a)(w_{c_1,c_2}')^{m-1}(w_{c_1,c_2}'+2H_k(A)w_{c_1,c_2}''s).\end{equation} We claim $w_{c_1,c_2}'+2H_k(A)w_{c_1,c_2}''s>0$ for any $s\geq 1$. Indeed, from \textbf{(a)} we obtain that \begin{align*} \frac{H_k(A)w_{c_1,c_2}''}{w_{c_1,c_2}'}&=-\frac{G(s)}{2(w_{c_1,c_2}')^{k-l}s^{\mathcal{H}+1}}\\ &=-\frac{1}{2s}\frac{\int_1^s\mathcal{H}t^{\mathcal{H}-1}\bar{g}(t)\,dt+c_1-s^{\mathcal{H}}\bar{g}(s)}{\int_1^s\mathcal{H}t^{\mathcal{H}-1}\bar{g}(t)\,dt+c_1}\\ &\geq -\frac{1}{2s}. \end{align*}
Consequently, \eqref{G-eq:k-convex-sub} shows $$\sigma_m(\lambda(D^2 w_{c_1,c_2}))>0, \quad 1\leq m\leq k.$$ Namely, $w_{c_1,c_2}$ are uniformly $k$-convex when $c_1>\tilde{C}$.
\textbf{(c)} We determine the asymptotic behavior of $w_{c_1,c_2}(s)$ as $s$ tends to infinity.
We start by showing that $w_{c_1,c_2}'$ is close to $1$ at infinity in both cases $\mathcal{H}\neq\frac{\beta}{2}$ and $\mathcal{H}=\frac{\beta}{2}$. When $\mathcal{H}\neq\frac{\beta}{2}$, we have \begin{align}
w_{c_1,c_2}'(s)-1=&\left(s^{-\mathcal{H}}\left(t^{\mathcal{H}}\Big|^s_1+\frac{2C_0\mathcal{H}}{2\mathcal{H}-\beta}t^{\mathcal{H}-\frac{\beta}{2}}\Big|^s_1+c_1\right)\right)^{\frac{1}{k-l}}-1\notag\\ =&\,O\left(s^{-\min\{\frac{\beta}{2},\mathcal{H}\}}\right), \quad \text{as }s\rightarrow\infty.\label{G-eq:O-w'-sub-1} \end{align} When $\mathcal{H}=\frac{\beta}{2}$, \begin{align}
w_{c_1,c_2}'(s)-1=&\left(s^{-\mathcal{H}}\left((1+C_0t^{\frac{\beta}{2}})t^{\mathcal{H}}\Big|^s_1+\frac{\beta}{2}\ln t\Big|^s_1+c_1\right)\right)^{\frac{1}{k-l}}-1\notag\\ =&O\left(s^{-\mathcal{H}}\ln s\right), \quad \text{as }s\rightarrow\infty.\label{G-eq:O-w'-sub-2} \end{align}
We now rewrite $w_{c_1,c_2}(s)$ as: \begin{align*} w_{c_1,c_2}(s)&=c_2+\int_{s_0}^sw_{c_1,c_2}'(\eta)\,d\eta\\ &=c_2+s-s_0+\int_{s_0}^s\left(w_{c_1,c_2}'(\eta)-1\right)d\eta\\ &=s+\mu(c_1,c_2)-\int_{s}^\infty\left(w_{c_1,c_2}'(\eta)-1\right)d\eta \end{align*} where \begin{equation}\label{G-eq:mu-c1} \mu(c_1,c_2)=c_2-s_0+\int_{s_0}^\infty\left(w_{c_1,c_2}'(\eta)-1\right)d\eta. \end{equation} By \eqref{G-eq:O-w'-sub-1} and \eqref{G-eq:O-w'-sub-2}, $\mu(c_1,c_2)<\infty$ only if $\frac{\beta}{2}>1$ and $ \mathcal{H}>1$, and in this situation, \begin{equation}\label{G-eq:asym-sub} \begin{cases} w_{c_1,c_2}(s)=s+\mu(c_1,c_2)+O(s^{1-\min\{\frac{\beta}{2},\mathcal{H}\}})&\text{if }\mathcal{H}\neq\frac{\beta}{2},\\ w_{c_1,c_2}(s)=s+\mu(c_1,c_2)+O(s^{1-\mathcal{H}}\ln s)&\text{if }\mathcal{H}=\frac{\beta}{2}. \end{cases} \end{equation} This indicates when $\beta>2$ and $A\in\mathscr{A}_{k,l}$, $w_{c_1,c_2}$ given by \eqref{G-eq:ode-sub-solu} is asymptotically close to a quadratic polynomial at infinity.
Based on \textbf{(a)}--\textbf{(c)}, we conclude that \begin{lemma}\label{G-lem:ode-sub-solu} Assume $A\in\mathscr{A}_{k,l}$ and $\beta>2$. Then there exists $\tilde{C}>1$ dependent of $A$ and $\beta$, such that when $c_1>\tilde{C}$ the G-Sym function $w_{c_1,c_2}(s)$ given by \eqref{G-eq:ode-sub-solu} is a uniformly $k$-convex solution of problem \eqref{G-eq:ode-sub} and satisfies \eqref{G-eq:asym-sub}. \end{lemma}
As a consequence of Lemma \ref{G-lem:ode-sub-solu}, we arrive at the following result.
\begin{proposition}\label{G-pro:sub} Assume that $A\in\mathscr{A}_{k,l}$ is diagonal and $g$ satisfies \eqref{eq:beta-g} for some $\beta>2$. Let $s_0$ be as in \eqref{G-eq:g-upper} and $\tilde{C}$ be given in Lemma \ref{G-lem:ode-sub-solu}. Then when $c_1>\tilde{C}$ the G-Sym function $w_{c_1,c_2}(x)=w_{c_1,c_2}(s)$ given by \eqref{G-eq:ode-sub-solu} is a uniformly $k$-convex subsolution of equation \eqref{eq:pro} in $\mathbb{R}^n\setminus D(s_0)$ and fulfills \begin{equation*}
w_{c_1,c_2}(x)=\frac12x^TAx+\mu(c_1,c_2)+O\left(|x|^{2-\min\{\beta, \frac{k-l}{H_k(A)}\}}\right)\quad\text{as }|x|\to+\infty \end{equation*} if $\beta\neq\frac{k-l}{H_k(A)}$, or
\begin{equation*}
w_{c_1,c_2}(x)=\frac12x^TAx+\mu(c_1,c_2)+O\left(|x|^{2-\frac{k-l}{H_k(A)}}\ln |x|\right)\quad\text{as }|x|\to+\infty \end{equation*} if $\beta=\frac{k-l}{H_k(A)}$, where $\mu(c_1,c_2)$ is as in \eqref{G-eq:mu-c1}. \end{proposition} \begin{proof} It follows directly by combining \eqref{G-eq:g-upper}, \eqref{G-eq:S-D2w-sub} and Lemma \ref{G-lem:ode-sub-solu}. \end{proof}
\begin{remark}\label{G-rk:no-h-l}
We would like to make some comparisons of Proposition \ref{G-pro:sub} with related results available in literature. For $2\leq k\leq n$ and $l=0$, the G-Sym subsolutions given by Proposition \ref{G-pro:sub} also are previously constructed in \cite{Bao-Li-Zhang-2015,Cao-Bao-2017} for equations \eqref{eq:k-Hessian-g} and \eqref{eq:M-A-g}, while for $l>0$ and $g\equiv 1$ they are different from those obtained in \cite{Li-Li-2018} for equation \eqref{eq:pro} due to the ODE \eqref{G-eq:ode-sub} satisfied by subsolutions disagrees that derived in \cite{Li-Li-2018}. Actually, if $g\equiv1$, then \eqref{eq:pro} becomes $\sigma_{k}(\lambda(D^2u))=\sigma_l(\lambda(D^2u))$, from which via \eqref{G-eq:k-H-G-Sym} one can study the following ODE \begin{equation}\label{G-eq:ode-sub-1} (w')^{k}+2H_k(A)w''(w')^{k-1}s=(w')^{l-l}+2h_l(A)w''(w')^{l-1}s \end{equation} to seek subsolutions of \eqref{eq:pro}, where $h_l(A)\in[0,\frac{l}{n}]$ is defined by $$h_l(A)=\min_{1\leq i\leq n}\frac{\sigma_{l-1;i}(\lambda(A))\lambda_i(A)}{\sigma_l(\lambda(A))}.$$ Here introducing the quantities $H_k(A)$ and $h_l(A)$ at the same time is to strike a balance between different order of homogeneities and then \eqref{G-eq:ode-sub-1} could be analyzed explicitly as in \cite{Li-Li-2018}. However, whenever $g\not\equiv1$, the same idea is not helpful for removing the effect of irreducible fraction in \eqref{eq:pro}. In order to treat this, here we drop part of the denominator $\sigma_l(\lambda(D^2w))$ as handled in \eqref{G-eq:S-D2w-sub} (equivalent to putting $h_l(A)=0$ in \eqref{G-eq:ode-sub-1}). As a result, our admissible set of $A$, $\mathscr{A}_{k,l}$, may be a subset of the one in \cite{Li-Li-2018} which is given by $$\tilde{\mathscr{A}}_{k,l}:=\left\{A\in\mathcal{A}_{k,l}: H_k(A)-h_l(A)<\frac{k-l}{2}\right\}.$$ Nevertheless, by Proposition \ref{G-pro:A-k-l} our method still works well in cases $k-l\geq2$ where it holds $$\mathscr{A}_{k,l}=\tilde{\mathscr{A}}_{k,l}=\mathcal{A}_{k,l}.$$ \end{remark}
\subsection{Generalized symmetric supersolutions}\label{G-sec:sup} We will construct a family of G-Sym supersolutions of \eqref{eq:pro} coinciding at infinity with the G-Sym subsolutions we just obtained in Proposition \ref{G-pro:sub}, which is necessary in Perron's construction (Theorem \ref{thm:Perron-m}). Unlike the well-treated case $g\equiv 1$ in which one can directly pick $\frac12x^TAx+c$ to be desired supersolutions (see for example \cite{Bao-Li-Li-2014,Caffarelli-Li-2003,Jiang-Li-Li-2020,Li-Li-2018,Li-Bao-2014}), here we adapt the idea in Subsection \ref{G-sec:sub} to study the ODE: \begin{equation}\label{G-eq:ode-super} \begin{cases} (w')^{k-l}+2H_k(A)w''(w')^{k-l-1}s=\underline{g}(s),& s>1,\\ w'(s)>0, \quad w''(s)\geq0,&s\geq1, \end{cases} \end{equation} where $\underline{g}$ is an increasing smooth function of $s$ satisfying, without loss of generality, \begin{equation}\label{G-eq:g-lower} 0<\underline{g}\leq g\text{ for}\ s\geq 1,\quad\text{and}\quad\underline{g}=1-C_0s^{-\frac{\beta}{2}}\ \text{when }s\geq s_0, \end{equation} where $C_0$ and $s_0$ are as in \eqref{G-eq:g-upper}. Replacing $\bar{g}$ by $\underline{g}$ in \eqref{G-eq:ode-sub-solu} and taking $c_1=0$, one can easily obtain the following uniformly $k$-convex solutions of \eqref{G-eq:ode-super}: \begin{equation}\label{G-eq:ode-super-solu} \overline{w}_{c_2}(s):=c_2+\int_{1}^s\left(\eta^{-\mathcal{H}}\left(\int_1^\eta\mathcal{H}t^{\mathcal{H}-1}\underline{g}(t)\,dt\right)\right)^{\frac{1}{k-l}}d\eta \end{equation} Moreover, thanks to \eqref{G-eq:g-lower}, arguing as in part \textbf{(c)} before we infer that $\overline{w}_{c_2}$ possess asymptotics \eqref{G-eq:asym-sub} where $\mu(c_1,c_2)$ need to be replaced by \begin{equation}\label{G-eq:lmu-c1-c2} \overline{\mu}(c_2)=c_2-1+\int_{1}^\infty\left(\overline{w}_{c_2}'(\eta)-1\right)d\eta. \end{equation}
Similarly as showed in \eqref{G-eq:S-D2w-sub}, we can find for $s\geq 1$, $$S_{k,l}(D^2\overline{w}_{c_2})\leq (\overline{w}_{c_2}')^{k-l}+2H_k(A)\overline{w}_{c_2}''(\overline{w}_{c_2}')^{k-l-1}s=\underline{g}(s)\leq g.$$ Namely, $\overline{w}_{c_2}$ are supersolutions of \eqref{eq:pro}.
The following statement is a summary of above facts, which serves as a counterpart of Proposition \ref{G-pro:sub}. \begin{proposition}\label{G-pro:super} Assume that $A\in\mathscr{A}_{k,l}$ is diagonal and $g$ satisfies \eqref{eq:beta-g} for some $\beta>2$. Let $c_2\in\mathbb{R}$ and $s=\frac12x^TAx$. Then the G-Sym function $\overline{w}_{c_2}(x)=\overline{w}_{c_2}(s)$ given by \eqref{G-eq:ode-super-solu} is a uniformly $k$-convex supersolution of equation \eqref{eq:pro} in $\mathbb{R}^n\setminus D(1)$ and fulfills \begin{equation*}
\overline{w}_{c_2}(x)=\frac12x^TAx+\overline{\mu}(c_2)+O\left(|x|^{2-\min\{\beta, \frac{k-l}{H_k(A)}\}}\right)\quad\text{as }|x|\to+\infty \end{equation*} if $\beta\neq\frac{k-l}{H_k(A)}$, or
\begin{equation*}
\overline{w}_{c_2}(x)=\frac12x^TAx+\overline{\mu}(c_2)+O\left(|x|^{2-\frac{k-l}{H_k(A)}}\ln |x|\right)\quad\text{as } |x|\to+\infty \end{equation*} if $\beta=\frac{k-l}{H_k(A)}$, where $\overline{\mu}(c_2)$ is as in \eqref{G-eq:lmu-c1-c2}. \end{proposition}
\begin{remark} One may wonder whether there are smooth G-Sym solutions of \eqref{eq:pro}, provided that $g$ is of G-Sym. Generally, this cannot be expected for all $0\leq l<k\leq n$. When $g\equiv 1$, \cite[Proposition 1.1]{Li-Li-Zhao-2019} states that \eqref{eq:pro} admits a G-Sym solution of $C^2$ with respect to a diagonal $A\in\mathcal{A}_{k,l}$ if and only if $l=0$ and $k=n$, unless $A=c^*(k,l)I$ (see (i) of Proposition \ref{G-pro:A-k-l}). The same rigidity result was also exploited in \cite{Cao-Bao-2017} for \eqref{eq:k-Hessian-g} with $g=\bar{g}$, introduced in \eqref{G-eq:g-upper}. That is the reason why we here look for G-Sym subsolutions and G-Sym supersolutions of \eqref{eq:pro} separately. \end{remark}
\section{Proof of Theorem \ref{thm:main}} \label{sec:3} In view of Theorem \ref{thm:Perron-m}, to prove Theorem \ref{thm:main} it suffices to demonstrate the existence of a viscosity subsolution $\underline{u}$ of equation \eqref{eq:pro} attaining the prescribed boundary value and asymptotic behavior at infinity, as well as that of a viscosity supersolution $\bar{u}\geq\underline{u}$ but agreeing on $\underline{u}$ at infinity. Roughly speaking, such a subsolution can be obtained by splicing together the supremum of barriers over the boundary points of the domain and the G-Sym subsolution constructed in Proposition \ref{G-pro:sub}; the desired supersolution is prepared in Proposition \ref{G-pro:super}. The uniqueness of the solution is guaranteed by comparison principle, Corollary \ref{thm:comparison-ub}.
Set $\mathcal{G}=\sup_{\mathbb{R}^n\setminus D} g$. To process the boundary behavior of the solution, we need the following existence result of barrier functions.
\begin{lemma}\label{P-lem:w-xi}
Let $D$ be a bounded strictly convex domain of $\mathbb{R}^n$ ($n\geq3$) with $\partial D\in C^2$ and let $\varphi\in C^2(\partial D)$. For an invertible and symmetric matrix $A$, there exists some constant $C$, depending only on $n,\mathcal{G},\|\varphi\|_{C^2(\partial D)}$, the upper bound of $A$, the diameter and the convexity of $D$, and the $C^2$ norm of $\partial D$, such that for every $\xi\in\partial D$, there exists $\bar{x}(\xi)\in\mathbb{R}^n$ satisfying $$|\bar{x}(\xi)|\leq C\quad\text{and}\quad w_\xi<\varphi\quad\text{on}\quad \partial D\setminus\{\xi\},$$ where $$w_\xi(x)=\varphi(\xi)+\frac{\mathcal{G}^{\frac{1}{k-l}}}{2}\left[(x-\bar{x}(\xi))^TA(x-\bar{x}(\xi))-(\xi-\bar{x}(\xi))^TA(\xi-\bar{x}(\xi))\right]$$ for $x\in\mathbb{R}^n$. \end{lemma}
\begin{proof} As proved in \cite{Cao-Bao-2017} for $A\in\mathcal{A}_{k,0}$, it is a direct adaption of the arguments as in the proofs of \cite[Lemma 5.1]{Caffarelli-Li-2003} and \cite[Lemma 3.1]{Bao-Li-Li-2014} for the case $\mathcal{G}=1$. We thus omit it. \end{proof}
We now present the proof of Theorem \ref{thm:main}.
\begin{proof}[Proof of Theorem \ref{thm:main}] By an orthogonal transformation and by subtracting a linear function from $u$, we need only to prove for $A=\text{diag}(a_1,a_2,\cdots,a_n)$ and $b=0$; see \cite[Lemma 3.3]{Li-Li-2018} for a specific demonstration. Also, without loss of generality, we assume $D(1)\subset D\subset D(s_0)$, where $D(\cdot)$ is as in \eqref{G-eq:D-s} and $s_0$ is as in \eqref{G-eq:g-upper}.
\textbf{Step 1} We first construct a viscosity subsolution $\underline{u}$ of \eqref{eq:pro} with $\underline{u}=\varphi$ on $\partial D$ and the asymptotics as in \eqref{eq:asym-thm}.
Recalling $w_{c_1,c_2}(s)$ given in \eqref{G-eq:ode-sub-solu}, by Proposition \ref{G-pro:sub} it is a uniformly $k$-convex subsolution of \eqref{eq:pro} satisfying \begin{equation}\label{P-eq:asym-E}
w_{c_1,c_2}(x)=\frac12x^TAx+\mu(c_1,c_2)+O\left(E^{-1}(x)\right),\quad |x|\to\infty, \end{equation}
provided $c_1>\tilde{C}$ and $x\in\mathbb{R}^n\setminus D(s_0)$. Here $E(x)$ is defined in \eqref{eq:intro-E}. We later will pick suitable $c_1$ to make \eqref{P-eq:asym-E} reach asymptotics \eqref{eq:asym-thm} as in \eqref{P-eq:mu-c1-c} below. Now regarding the boundary value, we are going to find another viscosity subsolution $\underline{w}$ attaining $\varphi$ on $\partial D$, such that, for some fixed $\bar{s}>s_0$, \begin{equation}\label{P-eq:barrier} \max_{\partial D(s_0)}w_{c_1,c_2}\leq\min_{\partial D(s_0)}\underline{w}\quad\text{and}\quad\min_{\partial D(\bar{s})}w_{c_1,c_2}\geq\max_{\partial D(\bar{s})}\underline{w}. \end{equation}
For this aim, we set $$\underline{w}(x)=\max\{w_{\xi}(x)~|~\xi \in \partial D\},$$ where $w_\xi$ is introduced in Lemma \ref{P-lem:w-xi}. Clearly, $\underline{w}=\varphi$ on $\partial D$. Since $$S_{k,l}(D^2w_\xi)=\mathcal{G}S_{k,l}(A)=\mathcal{G}\geq g\quad\text{in }\mathbb{R}^n\setminus\overline{D},$$ $w_\xi$ is a smooth convex subsolution of \eqref{eq:pro}. By Lemma \ref{lem:Perron-sub}, $\underline{w}$ is also a viscosity subsolution of \eqref{eq:pro}. Let
$$c_2=m_{s_0}:=\min\{w_{\xi}(x)~|~\xi\in\partial D, x\in \overline{D(s_0)}\setminus D\}.$$ Thus, by \eqref{G-eq:ode-sub-solu} $w_{c_1,m_{s_0}}$ satisfies the first condition in \eqref{P-eq:barrier}. To realize the second one, it suffices to choose a large $c_1$ (assume $c_1\geq\alpha>\tilde{C}$) since $w_{c_1,m_{s_0}}$ is monotonically increasing with respect to $c_1$.
We next fix $c_*$ such that given $c>c_*$ one can find $c_1(c)$ to fulfill \begin{equation}\label{P-eq:mu-c1-c} \mu(c_1(c),m_{s_0})=c. \end{equation} Notice from \eqref{G-eq:mu-c1} that $\mu(c_1,m_{s_0})$ is strictly increasing in $c_1$, and $$\lim_{c_1\to+\infty}\mu(c_1,m_{s_0})=+\infty.$$ Hence, if $c_*\geq\mu(\alpha,m_{s_0})$, then such $c_1(c)>\alpha$ exists. This means that, for $c>c_*$, $w_{c_1(c),m_{s_0}}$ satisfies \eqref{P-eq:barrier} and possesses asymptotic behavior: \begin{equation}\label{P-eq:asym-sub}
w_{c_1(c),m_{s_0}}(x)=\frac12x^TAx+c+O\left(E^{-1}(x)\right),\quad |x|\to\infty. \end{equation}
For $c>c_*$, we define \begin{equation*} \underline{u}(x)= \begin{cases} \underline{w}(x),\quad &x\in D(s_0)\backslash D,\\ \max\{w_{c_1(c),m_{s_0}}(x),\underline{w}(x)\},\quad &x\in D(\bar{s})\backslash D(s_0),\\ w_{c_1(c),m_{s_0}}(x),&x\in\mathbb{R}^n\backslash D(\bar{s}). \end{cases} \end{equation*} Then from Definition \ref{app-def:visc} and Lemma \ref{lem:Perron-sub}, $\underline{u}$ is a viscosity subsolution of \eqref{eq:pro} satisfying \eqref{P-eq:asym-sub}, and $\underline{u}=\underline{w}=\varphi$ on $\partial D$.
\textbf{Step 2} We construct a viscosity supersolution $\bar{u}$ of \eqref{eq:pro} to satisfy \begin{equation}\label{P-eq:step-super}
\underline{u}\leq\bar{u}\ \text{ in }\mathbb{R}^n\setminus\overline{D}\quad\text{and}\quad\lim_{|x|\to\infty}(\bar{u}-\underline{u})(x)=0. \end{equation}
By Proposition \ref{G-pro:super}, $\overline{w}_{c_2}$ given in \eqref{G-eq:ode-super-solu} is a uniformly $k$-convex supersolution of \eqref{eq:pro} satisfying \begin{equation}\label{P-eq:asym-super}
\overline{w}_{c_2}(x)=\frac{1}{2}x^TAx+\overline{\mu}(c_2)+O\left(E^{-1}(x)\right),\quad |x|\to\infty. \end{equation} Then for $c>c_*$ the second condition in \eqref{P-eq:step-super} holds for $\overline{w}_{c_2(c)}$ where $c_2(c)$ is determined by $$\overline{\mu}(c_2(c))=c,$$ which implies \eqref{P-eq:asym-super} agrees with \eqref{P-eq:asym-sub}. Actually, by definition of $\overline{\mu}$ (see \eqref{G-eq:lmu-c1-c2}), $$c_2(c)=c+1-\int_1^\infty\left[\left(s^{-\mathcal{H}}\int_1^s\mathcal{H}t^{\mathcal{H}-1}\underline{g}(t)\,dt\right)^{\frac{1}{k-l}}-1\right]\,ds\geq c\,;$$ here we have used the fact that $\underline{g}\leq 1$ for $s\geq 1$. Next, with the help of comparison principles proved in Appendix \ref{sec:comparison}, we show that $\overline{w}_{c_2(c)}$ also agrees the first condition in \eqref{P-eq:step-super} for proper $c$.
In \textbf{Step 1}, we fixed $c_*\geq\mu(\alpha,m_{s_0})$. We now further require that $c_*>M_{s_0}$, where
$$M_{s_0}:=\max\{w_{\xi}(x)~|~\xi\in\partial D, x\in \overline{D(s_0)}\setminus D\}.$$ Then for $c>c_*$, $$\overline{w}_{c_2(c)}\geq c_2(c)\geq c>c_*>M_{s_0}\geq m_{s_0}\geq w_{c_1(c),m_{s_0}}\quad\text{on }\partial D(s_0),$$
and also $$\lim_{|x|\to\infty}(\overline{w}_{c_2(c)}-w_{c_1(c),m_{s_0}})(x)=0.$$ Applying Corollary \ref{thm:comparison-ub}, we thus deduce that \begin{equation}\label{P-eq:compare-super-sub} \overline{w}_{c_2(c)}\geq w_{c_1(c),m_{s_0}}\quad\text{in }\mathbb{R}^n\setminus\overline{D(s_0)}. \end{equation} On the other hand, $$\overline{w}_{c_2(c)}\geq c_2(c)\geq c>c_*>M_{s_0}\geq \underline{w}\quad\text{on }\partial D,$$ and by \eqref{P-eq:barrier}, $$\overline{w}_{c_2(c)}\geq w_{c_1(c),m_{s_0}}\geq\underline{w}\quad\text{on }\partial D(\bar{s}).$$ Hence, applying Theorem \ref{thm:comparison-b} yields \begin{equation}\label{P-eq:compare-super-barrier} \overline{w}_{c_2(c)}\geq\underline{w}\quad\text{in }D(\bar{s})\setminus\overline{D}. \end{equation} By virtue of \eqref{P-eq:compare-super-sub} and \eqref{P-eq:compare-super-barrier}, we get $$\overline{w}_{c_2(c)}\geq \underline{u}\quad\text{in }\mathbb{R}^n\setminus\overline{D}$$ for $c>c_*$.
Based on the above arguments, we let $\bar{u}=\overline{w}_{c_2(c)}$ for $c>c_*$, which is a desired viscosity supersolution of \eqref{eq:pro}.
\textbf{Step 3} We show the existence and uniqueness of viscosity solutions to problem \eqref{eq:asym-thm}.
With $\underline{u}$ and $\bar{u}$, we define \begin{align*}
u(x):=&\sup\{v(x)| v\in\mathrm{USC}(\mathbb{R}^n\setminus\overline{D}), S_{k,l}(D^2v)\geq g\text{ in }\mathbb{R}^n\setminus\overline{D}\text{ in the}\\ &\text{viscosity sense, with }\underline{u}\leq v\leq\bar{u}\text{ in }\mathbb{R}^n\setminus\overline{D}\text{ and }v=\varphi\text{ on }\partial D\}. \end{align*} Since $\underline{u}$ and $\bar{u}$ both satisfy \eqref{P-eq:asym-sub}, one has
$$\limsup_{|x|\to\infty}\left(E(x)\Big{|}u(x)-(\frac12x^TAx+c)\Big{|}\right)<\infty.$$ From Theorem \ref{thm:Perron-m}, we thus conclude that $u\in C^0(\mathbb{R}^n\setminus D)$ is a viscosity solution of \eqref{eq:asym-thm}.
Finally, the uniqueness of viscosity solutions to problem \eqref{eq:asym-thm} is forced by Corollary \ref{thm:comparison-ub}. This completes the proof. \end{proof}
\begin{remark} We remark that from the above demonstration the lowerbound $c_*$ of $c$ in Theorem \ref{thm:main} may not be removed. Actually, in \cite[Theorem 1.3]{Li-Lu-2018}, Li and Lu proved that when $l=0$ and $k=n$, there is a sharp constant $c_*$ such that problem \eqref{eq:asym-thm} admits a viscosity solution if and only if $c\geq c_*$. It would be interesting to see whether such a sharp charaterization result holds for equation \eqref{eq:pro} with general $0\leq l<k\leq n$ and will be left to study in our future work. \end{remark}
\appendix \section{Comparison principles for viscosity solutions}\label{sec:comparison} In this appendix, we prove comparison principles for viscosity subsolutions and viscosity supersolutions of fully nonlinear, second-order partial differential equations of the form \begin{equation}\label{app-eq:F-x-D2u} F(x,D^2u):=f(\lambda(D^2u))-g(x)=0\quad\text{in }\Omega, \end{equation} where $\Omega$ is an open subset of $\mathbb{R}^n$, $f$ is a symmetric function of $C^1$ defined on an open convex symmetric cone $\Gamma$ in $\mathbb{R}^n$, with vertex at the origin and containing the positive cone $\Gamma^+$, $\lambda(D^2u)=(\lambda_1,\cdots,\lambda_n)$ denotes the eigenvalue vector of the Hessian matrix $D^2u$, and $g\in C^0(\Omega)$ is a given function with $\inf_{\Omega}g>0$. Throughout the section, we assume that \begin{equation}\label{app-eq:f-increase} \frac{\partial f}{\partial\lambda_i}>0\quad \text{on }\Gamma,\quad i=1,\cdots,n, \end{equation} by which $F(x,D^2u)$ in \eqref{app-eq:F-x-D2u} is an elliptic operator on those functions $u\in C^2(\Omega)$ such that $\lambda(D^2u)\in\Gamma$. Such functions will be called admissible. In addition, it is assumed \begin{equation}\label{app-eq:f-nu} \sum_{i=1}^n\lambda_i\frac{\partial f}{\partial\lambda_i}\geq \nu(f) \quad\text{on }\Gamma \end{equation} and \begin{equation}\label{app-eq:f-boundary} \limsup_{\lambda\to\lambda_0} f(\lambda)<\inf_{\Omega} g \quad\text{for }\forall\,\lambda_0\in\partial\Gamma, \end{equation} where $\nu$ is some positive increasing function on $\mathbb{R}$. Clearly, the quotient of elementary symmetric functions $\sigma_k/\sigma_l$ $(0\leq l<k\leq n)$ involved in \eqref{eq:pro} is an example of $f$ fulfilling \eqref{app-eq:f-increase}--\eqref{app-eq:f-boundary} if we define \begin{equation}\label{app-eq:k-convex}
\Gamma=\Gamma_k:=\{\lambda\in\mathbb{R}^n~|~\sigma_j(\lambda)>0,\ 1\leq j\leq k\} \end{equation} (admissible functions are called \emph{uniformly $k$-convex} in this case); indeed, for this example the validity of \eqref{app-eq:f-increase} as well as that of \eqref{app-eq:f-boundary} is well-known (see for instance \cite{Caffarelli1985,Trudinger1995}), and \eqref{app-eq:f-nu} holds obviously via the homogeneity.
We first recall the definition of the viscosity solution to equation \eqref{app-eq:F-x-D2u} following \cite{Caffarelli-Cabre-1995,Ishii1992,Urbas1990}. For simplicity, let $\mathrm{USC}(\Omega)$ and $\mathrm{LSC}(\Omega)$ respectively denote the set of upper and lower semicontinuous real valued functions on $\Omega$; let $B(\Omega)$ be the set of bounded functions on $\Omega$, and also let $B_{p}(\Omega)$ be the set of functions that are bounded in the intersection of $\Omega$ and any ball of $\mathbb{R}^n$.
\begin{definition}\label{app-def:visc} A function $u\in\mathrm{USC}(\Omega)$ $(\mathrm{LSC}(\Omega))$ is said to be a viscosity subsolution (supersolution) of \eqref{app-eq:F-x-D2u} (or say that $u$ satisfies $F(x,D^2u)\geq(\leq)0$ in the viscosity sense), if for any open subset $A$ of $\Omega$, any admissible function $\psi\in C^2(A)$, and any local maximum (minimum) $x_0\in A$ of $u-\psi$ we have $$F(x_0,D^2\psi(x_0))\geq(\leq)0.$$ A function $u\in C^0(\Omega)$ is said to be a viscosity solution of \eqref{app-eq:F-x-D2u} if it is both a viscosity subsolution and a viscosity supersolution of \eqref{app-eq:F-x-D2u}. \end{definition}
\begin{remark}\label{app-rk:connection}
An admissible solution of \eqref{app-eq:F-x-D2u} is clearly a viscosity solution. Conversely, as argued in \cite[Proposition 2.2]{Urbas1990} for $k$-Hessian equations, under assumption \eqref{app-eq:f-boundary} one would see that a viscosity subsolution of \eqref{app-eq:F-x-D2u} is admissible at each point at which it is twice differentiable. \end{remark} By adapting the ideas of Trudinger in \cite{Trudinger1990} for the prescribed curvature equations, we derive the following comparison principle for equation \eqref{app-eq:F-x-D2u} in bounded domains.
\begin{theorem}\label{thm:comparison-b} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Assume that $u\in\mathrm{USC}(\bar{\Omega})\cap B(\Omega)$ and $v\in\mathrm{LSC}(\bar{\Omega})\cap B(\Omega)$ are respectively viscosity subsolution and viscosity supersolution of equation \eqref{app-eq:F-x-D2u} with conditions \eqref{app-eq:f-increase}--\eqref{app-eq:f-boundary} holding. Then $$\sup_{\Omega}(u-v)=\sup_{\partial\Omega}(u-v).$$ \end{theorem}
To prove Theorem \ref{thm:comparison-b}, we establish below a preliminary result via regularizations. \begin{lemma}\label{lem:comparison-b} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and let \eqref{app-eq:f-increase}--\eqref{app-eq:f-boundary} hold. Assume that $u\in\mathrm{USC}(\bar{\Omega})\cap B(\Omega)$ and $v\in\mathrm{LSC}(\bar{\Omega})\cap B(\Omega)$ satisfy $$F(x,D^2u)\geq \delta,\quad F(x,D^2v)\leq 0$$ in $\Omega$ in the viscosity sense for some constant $\delta>0$. Then $$\sup_{\Omega}(u-v)=\sup_{\partial\Omega}(u-v).$$ \end{lemma} \begin{proof} As in \cite{Jensen-Lions-1988,Urbas1990}, for $\epsilon>0$ we define the approximations of $u$ and $v$ as \begin{gather}
u_\epsilon^+(x)=\sup_{y\in\Omega}\left\{u(y)-\omega_0\frac{|x-y|^2}{\epsilon^2}\right\},\label{app-eq:approxi-u}\\
v_\epsilon^-(x)=\inf_{y\in\Omega}\left\{v(y)+\omega_0\frac{|x-y|^2}{\epsilon^2}\right\},\label{app-eq:approxi-v} \end{gather}
where $\omega_0=\max\{\text{osc}_{\Omega}u,\text{osc}_{\Omega}v\}$. The supremum in \eqref{app-eq:approxi-u} and the infimum in \eqref{app-eq:approxi-v} are respectively attained at points $x^{\pm}\in\Omega$ satisfying $|x-x^{\pm}|\leq \epsilon$, provided $x\in\Omega_\epsilon:=\{x\in\Omega:\text{dist}(x,\partial\Omega)>\epsilon\}$. Clearly, $u_\epsilon^+,v_\epsilon^-\in C^{0,1}(\overline{\Omega_\epsilon})$ and \begin{equation}\label{app-eq:approxi-u-v}
\sup_{\Omega_\epsilon}|u_\epsilon^+-u|,\ \sup_{\Omega_\epsilon}|v_\epsilon^--v|\to 0\quad\text{as }\epsilon\to0. \end{equation} Moreover, the functions $u_\epsilon^+(x)$, $v_\epsilon^-(x)$ are respectively semi-convex and semi-concave in $\Omega$, with \begin{equation*} D^2u_\epsilon^+, -D^2v_\epsilon^-\geq-\frac{2\omega_0}{\epsilon^2} \end{equation*} in the sense of distributions, and they satisfy \begin{equation*} F(x^+, D^2u_\epsilon^+)\geq\delta, \quad F(x^-, D^2v_\epsilon^-)\leq 0, \end{equation*} in $\Omega_\epsilon$ in the viscosity sense.
Considering now the semi-convex, Lipschitz continuous function $w_\epsilon=u_\epsilon^+-v_\epsilon^-$, we claim for small $\epsilon$, \begin{equation}\label{app-eq:w-O-ep} \sup_{\Omega_\epsilon}w_\epsilon=\sup_{\partial\Omega_\epsilon}w_\epsilon. \end{equation}
If not, $w_\epsilon$ has an interior maximum in $\Omega_\epsilon$. By Lemma 3.10 in \cite{Jensen1988}, the upper contact set $K^+$ of $w_\epsilon$ is nonempty in $\Omega_\epsilon$. Thus, for almost all $x\in K^+$, $D^2u_\epsilon^+\leq D^2v_\epsilon^-$. From Remark \ref{app-rk:connection}, we see $\lambda(D^2u_\epsilon^+),\lambda(D^2v_\epsilon^-)\in\Gamma$. Via \eqref{app-eq:f-increase}, we obtain for almost all $x\in K^+$, \begin{align} F(x^-,D^2v_\epsilon^-(x))\geq F(x^-,D^2u_\epsilon^+(x))&=F(x^+,D^2u_\epsilon^+(x))+o(1)\notag\\ &\geq\delta+o(1),\label{app-eq:F-ep} \end{align} where $o(1)\to 0$ as $\epsilon\to 0$. Given such $x_0\in K^+$, where $v_\epsilon^-$ is twice differentiable, we set for $\tau>0$ \begin{align*} \phi_\tau(x)=&v_\epsilon^-(x_0)+Dv_\epsilon^-(x_0)(x-x_0)+\frac12(x-x_0)^TD^2v_\epsilon^-(x_0)(x-x_0)\\
&-\frac{\tau}{2}|x-x_0|^2. \end{align*} Since $D^2\phi_\tau=D^2v_\epsilon^-(x_0)-\tau I$, $\lambda(D^2\phi_\tau)\in\Gamma$ if $\tau$ is sufficiently small. Also, $v_\epsilon^--\phi_\tau$ has a local minimum at $x_0$. By Definition \ref{app-def:visc}, \begin{equation}\label{app-eq:F-tau} 0\geq F(x_0^-,D^2v_\epsilon^-(x_0)-\tau I)=F(x_0^-,D^2v_\epsilon^-(x_0))+o(1), \end{equation} where $o(1)\to0$ as $\tau\to0$. Clearly, \eqref{app-eq:F-tau} contradicts \eqref{app-eq:F-ep} when $\epsilon,\tau$ are small enough.
With facts \eqref{app-eq:approxi-u-v} and \eqref{app-eq:w-O-ep} in hand, we conclude immediately the assertion of Lemma \ref{lem:comparison-b} by letting $\epsilon\to0$. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:comparison-b}] Set $u_t(x)=\frac{1}{t}u(x)$, for $0<t<1$. We first show that $u_t$ satisfies $$F(x,D^2u_t)\geq (1-t)\tilde{\nu}$$ in $\Omega$ in the viscosity sense, where $\tilde{\nu}:=\nu\left(\inf_{\Omega}g\right)$. Let $\phi$ be an admissible function with a local maximum of $u_t-\phi$ at $x_0$. Since $u$ is a viscosity subsolution of \eqref{app-eq:F-x-D2u}, $$F(x_0,tD^2\phi(x_0))\geq 0.$$ By virtue of \eqref{app-eq:f-nu}, for some $t_0\in(t,1)$, \begin{align*} &F(x_0,D^2\phi(x_0))\\ &=F(x_0,tD^2\phi(x_0))+(1-t)\sum_{i=1}^n\lambda_i(D^2\phi(x_0))\frac{\partial f}{\partial\lambda_i}(t_0\lambda(D^2\phi(x_0)))\\ &\geq (1-t)\nu(f(t\lambda(D^2\phi(x_0))))\geq (1-t)\nu(\inf_{\Omega} g). \end{align*}
Applying Lemma \ref{lem:comparison-b}, we have \begin{equation*} \sup_{\Omega}(u_t-v)=\sup_{\partial\Omega}(u_t-v). \end{equation*} Consequently, the proof finishes by letting $t\to 1$. \end{proof}
\begin{remark} Theorem \ref{thm:comparison-b} may hold for equation \eqref{app-eq:F-x-D2u} without requiring assumption \eqref{app-eq:f-nu}. Indeed, if we let $g\equiv C$ for some constant $C>0$ (or modify assumption \eqref{app-eq:f-increase} as: there exists $\lambda_0>0$ such that \begin{equation}\label{app-eq:f-degenerate} f(\lambda(D^2u+\eta))-f(\lambda(D^2u))\geq \lambda_0(\det\eta)^{\frac{1}{n}} \end{equation} for any admissible function $u$ and symmetric matrix $\eta\geq 0$), then Theorem \ref{thm:comparison-b} could be proved under \eqref{app-eq:f-increase} (\eqref{app-eq:f-degenerate}) and \eqref{app-eq:f-boundary} by applying the Aleksandrov maximum principle. For a view of such arguments, we refer to \cite{Trudinger1988,Urbas1990}, where uniformly elliptic equations and $k$-Hessian equations ($f=(\sigma_k)^{1/k}$, clearly satisfying \eqref{app-eq:f-degenerate}) were considered. \end{remark}
Based on Theorem \ref{thm:comparison-b}, one can immediately prove a comparison principle for equation \eqref{app-eq:F-x-D2u} in unbounded domains, provided that the given subsolution and supersolution coincide at infinity. \begin{corollary}\label{thm:comparison-ub}
Let $\Omega$ be an unbounded domain in $\mathbb{R}^n$. Assume that $u\in\mathrm{USC}(\bar{\Omega})\cap B_{p}(\Omega)$ and $v\in\mathrm{LSC}(\bar{\Omega})\cap B_{p}(\Omega)$ are respectively viscosity subsolution and viscosity supersolution of equation \eqref{app-eq:F-x-D2u} with conditions \eqref{app-eq:f-increase}--\eqref{app-eq:f-boundary} holding. If $u\leq v$ on $\partial\Omega$ and $$\lim_{|x|\to\infty}(u-v)(x)=0,$$ then $u\leq v$ in $\Omega$. \end{corollary} \begin{proof}
For $0<\epsilon\ll 1$, there is $R>0$ such that $u(x)\leq v(x)+\epsilon$ when $|x|>R$. Given arbitrary $x_0\in\Omega$, choose a large ball $B_r$ with the radius $r>R$ and the center at the origin and containing $x_0$. Note that $u\leq v+\epsilon$ on $\partial\Omega\cup\partial B_r$. Thus, applying Theorem \ref{thm:comparison-b} to $u$ and $v+\epsilon$ on the domain $\Omega\cap B_r$, we obtain $u(x_0)\leq v(x_0)+\epsilon$. Then letting $\epsilon\to0$ yields $u(x_0)\leq v(x_0)$. The proof is done. \end{proof}
\section{Perron's method}\label{sec:perron} With comparison principles established in Theorem \ref{thm:comparison-b} and Corollary \ref{thm:comparison-ub}, Perron's method as in \cite{Ishii1992,Ishii1989,Ishii-Lions-1990} could be immediately adapted to the Dirichlet problem for equation \eqref{app-eq:F-x-D2u} to show the existence of its viscosity solutions. Precisely, we have the following theorem.
\begin{theorem}\label{thm:Perron-m} Let $\Omega$ be a domain in $\mathbb{R}^n$ and $\varphi\in C^0(\partial\Omega)$. Let $F(x,D^2u)$ be given in \eqref{app-eq:F-x-D2u} with \eqref{app-eq:f-increase}--\eqref{app-eq:f-boundary} holding. Suppose that there exist $\underline{u},\bar{u}\in C^0(\bar{\Omega})$ such that $$F(x,D^2\underline{u})\geq 0\geq F(x,D^2\bar{u})$$ in $\Omega$ in the viscosity sense, $\underline{u}\leq\bar{u}$ in $\Omega$ and $\underline{u}=\varphi$ on $\partial \Omega$ (and additionally
$$\lim_{|x|\to\infty}(\underline{u}-\bar{u})(x)=0,$$ provided $\Omega$ is unbounded). Then \begin{align*}
u(x):=\sup\{&v(x)| v\in\mathrm{USC}(\Omega), F(x,D^2v)\geq 0\text{ in }\Omega\text{ in the viscosity}\\ &\text{sense, with }\underline{u}\leq v\leq\bar{u}\text{ in }\Omega \text{ and }v=\varphi\text{ on }\partial\Omega\} \end{align*} is in $C^0(\bar{\Omega})$ and is a viscosity solution of the problem \begin{equation}\label{app-eq:D-F-varphi} \left\{ \begin{array}{ll} F(x,D^2u)=0 & \text{in }\Omega,\\ u=\varphi & \text{on }\partial\Omega. \end{array} \right. \end{equation} \end{theorem}
For a function $v:\Omega\to\mathbb{R}$, we define its upper semicontinuous envelope $v^*$ by $$v^*(x)=\limsup_{y\to x,y\in\Omega}v(y)$$ and lower semicontinuous envelope $v_*$ by $$v_*(x)=\liminf_{y\to x,y\in\Omega}v(y).$$ To prove Theorem \ref{thm:Perron-m}, we need the following key lemmas.
\begin{lemma}[Lemma 4.2 in \cite{Ishii1992}]\label{lem:Perron-sub} Let $\Omega$ be a domain in $\mathbb{R}^n$ and let $\mathcal{F}$ be a family of viscosity subsolutions of equation \eqref{app-eq:F-x-D2u} in $\Omega$. Let $w(x)=\sup\{u(x):u\in\mathcal{F}\}$ and assume that $w^*(x)<\infty$ for $x\in\Omega$. Then $w^*$ is a viscosity subsolution of \eqref{app-eq:F-x-D2u} in $\Omega$. \end{lemma}
\begin{lemma}[Lemma 4.4 in \cite{Ishii1992}]\label{lem:Perron-super} Let $\Omega$ be a domain in $\mathbb{R}^n$ and $u$ be a viscosity subsolution of equation \eqref{app-eq:F-x-D2u} in $\Omega$. If $u_*$ fails to be a viscosity supersolution at some point $\hat{x}$, i.e. there exists admissible function $\psi$ for which $F(\hat{x},D^2\phi(\hat{x}))>0$, then for any small $\kappa>0$ there is a viscosity subsolution $U_{\kappa}$ of \eqref{app-eq:F-x-D2u} satisfying \begin{equation}\label{app-eq:kappa-super} \begin{cases} U_{\kappa}(x)\geq u(x)\quad\text{and}\quad\sup_{\Omega}(U_{\kappa}-u)>0,\\
U_{\kappa}(x)=u(x)\quad\text{for }x\in\Omega, |x-\hat{x}|\geq\kappa. \end{cases} \end{equation} \end{lemma} \begin{proof}[Proof of Theorem \ref{thm:Perron-m}]
First, it is easily observed that $u^*\in\mathrm{USC}(\bar{\Omega})$ and $u_*\in\mathrm{LSC}(\bar{\Omega})$, and that they satisfy $$\underline{u}\leq u_*\leq u\leq u^*\leq\bar{u}\ \text{ in }\Omega\quad\text{and}\quad u_*=u^*=u=\varphi\ \text{ on }\partial\Omega.$$ From Lemma \ref{lem:Perron-sub}, $u^*$ is a viscosity subsolution of \eqref{app-eq:F-x-D2u}. By definition of $u$, $u^*\leq u$, and thus $u=u^*$ in $\bar{\Omega}$, i.e. $u$ is a subsolution of \eqref{app-eq:F-x-D2u}.
Then let us show $u_*$ is also a viscosity supersolution of \eqref{app-eq:F-x-D2u}. If not, $u_*$ fails to be a supersolution at $\hat{x}\in\Omega$, then in this case we let $U_{\kappa}$ be provided by Lemma \ref{lem:Perron-super}. Clearly, by \eqref{app-eq:kappa-super} $$U_{\kappa}\geq u\geq\underline{u}\ \text{ in }\Omega\quad\text{and}\quad U_{\kappa}=u=\varphi\ \text{ on }\partial\Omega$$ for sufficiently small $\kappa$ (and additionally $$\lim_{|x|\to\infty}(U_\kappa-\bar{u})(x)= \lim_{|x|\to\infty}(u-\bar{u})(x)=0$$ provided $\Omega$ is unbounded). By Theorem \ref{thm:comparison-b} (Corollary \ref{thm:comparison-ub}), $U_\kappa\leq\bar{u}$. Again, by definition of $u$ we deduce $U_\kappa\leq u$ in $\Omega$, contradicting the property $\sup_{\Omega}(U_{\kappa}-u)>0$ in \eqref{app-eq:kappa-super}. Therefore, $u_*$ is a viscosity supersolution.
Applying now Theorem \ref{thm:comparison-b} (Corollary \ref{thm:comparison-ub}), it follows that $u^*=u\leq u_*$. This yields $u=u_*=u^*$ in $\bar{\Omega}$. Consequently, $u\in C^0(\bar{\Omega})$ with $u=\varphi$ is a viscosity solution of \eqref{app-eq:D-F-varphi}. \end{proof}
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\begin{document}
\begin{frontmatter}
\def\spacingset#1{\renewcommand{\baselinestretch}
{#1}\small\normalsize} \spacingset{1}
\if00 {
\title{\bf An Extended Laplace Approximation Method for Bayesian Inference of Self-Exciting Spatial-Temporal Models of Count Data}
\author{Nicholas J. Clark\thanks{
Supported in part through a fellowship from the \textit{The Omar N. Bradley Foundation}}\hspace{.2cm}\\
Department of Statistics, Iowa State University\\
and \\
Philip M. Dixon \\
Department of Statistics, Iowa State University}
} \fi
\if10 {
\begin{center}
{\LARGE\bf An Extended Laplace Approximation Method for Inference on Self-Exciting Poisson Spatial-Temporal Models}
\end{center}
} \fi
\begin{abstract}
Self-Exciting models are statistical models of count data where the probability of an event occurring is influenced by the history of the process. In particular, self-exciting spatio-temporal models allow for spatial dependence as well as temporal self-excitation. For large spatial or temporal regions, however, the model leads to an intractable likelihood. An increasingly common method for dealing with large spatio-temporal models is by using Laplace approximations (LA). This method is convenient as it can easily be applied and is quickly implemented. However, as we will demonstrate in this manuscript, when applied to self-exciting Poisson spatial-temporal models, Laplace Approximations result in a significant bias in estimating some parameters. Due to this bias, we propose using up to sixth-order corrections to the LA for fitting these models. We will demonstrate how to do this in a Bayesian setting for Self-Exciting Spatio-Temporal models. We will further show there is a limited parameter space where the extended LA method still has bias. In these uncommon instances we will demonstrate how amore computationally intensive fully Bayesian approach using the Stan software program is possible in those rare instances. The performance of the extended LA method is illustrated with both simulation and real-world data. \end{abstract}
\begin{keyword}
Asymptotic Bias \sep Intractable Likelihoods \sep Terrorism and Crime
\end{keyword}
\end{frontmatter}
\section{Introduction}
Intractable likelihood functions arise in a multitude of settings in statistics, especially in modeling spatio-temporal data. For spatial or spatio-temporal models it is oftentimes easier to specify the probability of an event occurring at a given location conditional on the occurrence or non-occurrence at neighboring events. In this instance, it is easy to write down the conditional density, but the joint density may not have a closed form expression, or, if it does, the likelihood cannot be evaluated.
For example, in a spatial process observed on a fixed lattice we may have, writing $s_i \in \{s_1,s_2,...,s_{n_d}\}$ as fixed locations in $\mathbb{R}^2$, $Z(s_i)\sim \mbox{Pois}(\lambda(s_i))$ as observed counts at a given location. We may further have $\boldsymbol{\lambda} \sim \mbox{Log Gau} (\boldsymbol{\alpha}, \Sigma(\theta))$ where $\boldsymbol{\lambda}$ is the vector of all Poisson expectations at each location and $\mbox{Log Gau}$ is the standard multivariate log-Gaussian distribution.Spatial structure may be placed on $\Sigma(\theta)$ by, for example, letting $\Sigma(\theta) = (I_{n_d,n_d}-C)^{-1}M$ where $I$ is the identity matrix, $C$ is a matrix with entries $\zeta$ at location $i,j$ if spatial locations $s_i$ and $s_j$ are spatial neighbors, and $M$ is a diagonal matrix with diagonal entries, $\tau^2$. This model is oftentimes called the Poisson-CAR model and is described in detail in Section 4.2 of \cite{cressie2015statistics}. The log-likelihood for the spatial parameters is proportional to the intractable integral
\begin{equation}
l_{n_d}(\theta) \propto -\frac{1}{2}\log \det \left(\Sigma (\theta)\right) + \log \int_{\mathbb{R}^{n_d}} \exp \left(\sum_{s_i=1}^{s_i={n_d}} Z(s_i)Y(s_i)-\exp(Y(s_i)) - \frac{1}{2} \boldsymbol{Y}^T \Sigma^{-1}(\theta) \boldsymbol{Y} \right) \text{d} \boldsymbol{Y} \label{eq:loglik},
\end{equation}
where $\theta$ is the set of all spatial parameters.
However, while the integral in \eqref{eq:loglik} is intractable, it is of the form $I_n = \int_{\mathbb{R}^n} \exp(-h_{d}\left(Y\right)) \text{d}y$ allowing for Laplace approximations to be used to conduct inference. In both spatial and spatio-temporal modeling, using Laplace approximations to conduct inference on the spatial or spatio-temporal diffusion parameters has dramatically increased since the advent of the Integrated Nested Laplace Approximation, or INLA, package from \cite{rue2009approximate}. \cite{rue2017bayesian} provides many examples of INLA being used in literature.
Though the Laplace approximation technique is extremely fast compared to Markov Chain Monte Carlo (MCMC) techniques and it provides consistent estimates for parameters, it only does so asymptotically where the asymptotic error rate decreases as a function of pseudo independent observations. By pseudo independent we mean observations that are separated sufficiently far in either spatial or temporal distance as to have minimal influence on one another. For example, in \cite{cressie1992statistics} on page 15 it is shown how a simple spatial-only model with 10 spatially dependent observations is equivalent to 6 pseudo-independent observations. The growth of the equivalent independent observations is what justifies, asymptotically, the consistency of the Laplace approximations. Meaning, if the correlation structure of $\Sigma(\theta)$ is strong, then increasing the number of observations may only have minimal impact on the validity of the Laplace approximations.
In this manuscript we will re-examine some of the shortfalls of using Laplace approximations for inference of spatial or spatio-temporal diffusion parameters. For a class of models which we will refer to as the self-exciting Poisson CAR models we will show how the assumptions for the first order Laplace approximations of techniques such as INLA may not hold over the entire parameter space. We will demonstrate how, in this case, higher order approximations of \cite{shun1995laplace} and \cite{evangelou2011estimation} offer more accurate inference and offer greater consistency in parameter estimation and show how the results are comparable to a fully Bayesian inference using rStan of \cite{gelman2015stan}.
\section{Model}
In this manuscript we write $Z(s_i,t)$ for observed count data on a spatial temporal lattice where $s_i \in \{s_1,s_2,...,s_{n_d}\}$ indexes space and $t\in\{1,2,...T\}$ indexes time. Defining $\boldsymbol{Z_t} = (Z(s_1,t),Z(s_2,t),...,Z(s_{n_d,t}))^T$, the model we consider is
\begin{align} & Z(s_i,t) \sim \mbox{Pois}(\lambda(s_i,t)) \label{eq:timeseries2} \\ & E[Z(s_i,t)]=\lambda(s_i,t)\\ & \boldsymbol{\lambda_t} = \exp(\boldsymbol{Y_t})+\eta \boldsymbol{Z_{t-1}}\\ & \boldsymbol{Y_t} \sim \mbox{Gau} (\boldsymbol{\alpha_t},(I_{{n_d},{n_d}}-\boldsymbol{C})^{-1}\boldsymbol{M}). \end{align}
As above, we define $\boldsymbol{C}$ to be the spatial proximity matrix with entry $(i,j)=\zeta$ if the spatial locations, $s_i,s_j$ are neighbors and $0$ otherwise. $M$ is a diagonal matrix of dimension $n_d \times n_d$ with diagonal entries $\tau^2$. In order to ensure positive definiteness of the Gaussian covariance matrix we must have $\zeta \in (\psi_{(1)},\psi_{(n)})$ where $\psi_{(k)}$ is the $k$th largest eigenvalue of $\boldsymbol{C}$.
Data level dependence, or what is commonly referred to as self-excitation, is present in the model through the addition of the $\eta \boldsymbol{Z_{t-1}}$ term to the linear predictor of $\boldsymbol{\lambda}$. The expected number of events at space-time location $(s_i,t)$ then is a summation of the expected events due to an underlying, latent CAR process, as well as events due to repeat or copy-cat actors. A sufficient condition to ensure a valid joint density exists is $\eta \in (0,1)$.
The data model for $Z(s_i,t)$, when conditioned on $Z(s_i,t-1)$ and $Y(s_i,t)$, is then Poisson. In other words, the density of $Z(s_i,t)$ depends on the previously observed $Z(s_i,t-1)$ and a latent, unobserved $Y(s_i,t)$.
This is similar to an AR(1) version of the Poisson Autoregression model of \cite{fokianos2009poisson}, however with the added complication of independent log-normal errors. This is also a spatial version of the discrete Hawkes-Cox model of \cite{mohler2013modeling} only allowing a time lag of 1.
The latent process model, $Y(s_i,t)$, is a Conditional Auto-Regressive or CAR model given in \cite{cressie2015statistics} and has joint distribution $\boldsymbol{Y_t}\sim \mbox{Gau}(\boldsymbol{\alpha_t},(I_{{n_d},{n_d}}-\boldsymbol{C})^{-1}\boldsymbol{M})$. Statistically this model is interesting as it is both hierarchical and conditionally specified at the data level, not at the process level.
As well as being statistically interesting this model also arises naturally when the expected count at space-time location $(s_i,t)$ is equal to the expected count due to a spatial latent process, $\exp(Y(s_i,t))$ and the expected count due to self-excitation, $\eta Z(s_i,t-1)$. This can occur, for example in the modeling of violence in a region. The latent (unobserved) tension in the region may be solely due to geography or demographics observed at a given space and time. This may be expressed as a function of large-scale variation, $\boldsymbol{\alpha}$ and small scale variation which is captured in the CAR component of the model. The critical assumption is that the small scale variation only exists in space. The second cause of violence in a space-time region may be attributed to the "broken windows" effect, or the propensity of violent action to be repeated in, or near, the same geographical region. That is, once a violent action occurs, there is some probability that that action will generate copy cats. As a consequence of the model, if we know $\exp(Y(s_i,t))$ and $\eta$, then the expected number of violent events that arise from model\eqref{eq:timeseries2} can be seen as the sum of the expected number of events due to the latent process and the expected number of events due to copy cat actors.
The likelihood associated with this model is given in \eqref{eq:FullLikelihood}. \begin{equation}
\small L(\eta,\alpha,\zeta,\tau^2|\boldsymbol{Z}) \propto \int_{\boldsymbol{\Omega}_y} \prod_{i=1}^{n}\prod_{t=1}^{T} \exp(-\eta Z(s_i,t-1)-\exp(Y(s_i,t)))\left(\eta Z(s_i,t-1)+\exp(Y(s_i,t))\right)^{Z(s_i,t)} d\mu_{\boldsymbol{Y}}\label{eq:FullLikelihood}. \end{equation}
Due to the temporal independence of $\boldsymbol{Y}$, we can simplify this to
\begin{equation}
\small L(\eta,\alpha,\zeta,\tau^2|\boldsymbol{Z}) \propto \prod_{t=1}^{T}\int_{\boldsymbol{\Omega}_{y_t}} \prod_{i=1}^{n} \exp(-\eta Z(s_i,t-1)-\exp(Y(s_i,t)))\left(\eta Z(s_i,t-1)+\exp(Y(s_i,t))\right)^{Z(s_i,t)} d\mu_{\boldsymbol{Y_t}}\label{eq:FullLikelihood2}.
\end{equation}
However, practically, this likelihood cannot be directly maximized due to the intractable integral that is taken with respect to the multivariate Gaussian density associated with $\boldsymbol{Y}$. If the likelihood could be computed, asymptotic normality of the maximum likelihood estimates could be shown along the lines of \cite{fokianos2009poisson}. As the log-Gaussian term has support on $(0,\infty)$, many of the standard difficulties of similar models are avoided. For more on the difficulties of the asymptotics of similar univariate models see Chapter 4 of \cite{davis2016handbook}. Critically in \eqref{eq:timeseries2} we must have $\eta \in (0,1)$, ensuring that the temporal dependence dies off at a geometric rate.
Bayesian Monte Carlo Markov Chain (MCMC) methods also are extremely challenging in this set-up as MCMC techniques will generally either involve integrating \eqref{eq:FullLikelihood} or sampling from the latent states. A similar model was analyzed in \cite{mohler2013modeling} where inference was conducted using Metropolis Adjusted Langevin Algorithm (MALA). The challenge in using MCMC techniques including MALA is that the dimension of $\Sigma(\theta) \equiv (I_{n_d,n_d}-\boldsymbol{C})^{-1}\boldsymbol{M}$ is potentially quite large. Any sampling of $Y$ will require thousands of evaluations of the determinant of this matrix as well evaluations of the log-likelihood. As we will describe in Section 5 this can be sped up through precomputing eigenvalues of $\boldsymbol{C}$ but even with this, it remains potentially painfully slow and unfeasible in the model building phase of analysis.
\section{Laplace Approximation}
An approximation method similar to Integrated Nested Laplace Approximation (INLA) was used to fit a Self-Exciting Poisson SAR model in \cite{2017arXiv170308429C}. This inferential technique was first recommended in \cite{tierney1986accurate}. Generically, we let $\pi(.)$ represent a density function and $\pi(.|.)$ represent a conditional density function. Now, we can approximate $\pi(\theta|Z)$ where $Z$ is the observed data, $Y$ is a latent random variable, and $\theta$ is the set of parameters that inference by using the relationship
\begin{align}
\pi(\theta|Z)\propto \frac{ \pi(Z,Y,\theta)}{\pi_G(Y|Z,\theta)}\bigg\rvert_{Y=Y^*(\theta)}, \end{align}
where $\pi_G(Y|Z,\theta)$ is the Gaussian approximation to the density $\pi(Y|Z,\theta)$. Both the numerator and the denominator are then evaluated at the mode of $Y$ for a given $\theta$, denoted as $Y^*(\theta)$. The benefit of this, when applied to \eqref{eq:FullLikelihood} is that it is essentially an integration free method of marginalizing over $Y$. For \eqref{eq:timeseries2}, this becomes
\begin{align}
\tilde{\pi}(\eta,\zeta,\tau^2,\alpha|\boldsymbol{Z})\propto \frac{\pi(\boldsymbol{Z}|\eta,\boldsymbol{Y})\pi(\boldsymbol{Y}|\alpha,\zeta,\tau^2)\pi(\zeta)\pi(\alpha)\pi(\tau^2)\pi(\zeta)}{\pi_G(\boldsymbol{Y}|\alpha,\eta,\zeta,\tau^2,\boldsymbol{Z})} \label{eq:INLA}, \end{align}
where $\tilde{\pi}(\eta,\zeta,\tau^2,\alpha|\boldsymbol{Z})$ is an approximation to the marginal posterior density of $\eta,\zeta,\tau^2,\alpha$, and $\pi_G(\boldsymbol{Y}|\alpha,\eta,\zeta,\tau^2,\boldsymbol{Z})$ is a Gaussian approximation to the joint density of the latent state $\boldsymbol{Y}$.
The Gaussian approximation given in the denominator of \eqref{eq:INLA} is based off of a Taylor series approximation to the log-density of $\pi(\boldsymbol{Z}|\boldsymbol{Y},\eta)$. That is, $\pi_G(\boldsymbol{Y}|\alpha,\eta,\zeta,\tau^2,\boldsymbol{Z})=\pi(\boldsymbol{Z}|\boldsymbol{Y},\eta)\pi(\boldsymbol{Z}|\eta,\boldsymbol{Y})$. Specifically, we can write \begin{align}
\pi_G(\boldsymbol{Y}|\eta,\zeta,\tau^2,\boldsymbol{Z}) =& (2 \pi)^{n/2} \det(\Sigma(\theta))^{1/2} \exp(-\frac{1}{2}\boldsymbol{Y}^t \Sigma^{-1}(\theta)\boldsymbol{Y}+\sum_{s_i,t} f(\mu(s_i,t))(Y(s_i,t))+ \nonumber \\ \label{eq:gausapprox}& 1/2 k (\mu(s_i,t))(Y(s_i,t))^2)\\ \mbox{where in above}\nonumber\\ f(\mu(s_i,t)) =& \frac{Z(s_i,t)\exp(\mu(s_i,t))}{\exp(\mu(s_i,t))+\eta Z(s_i,t-1)}-\exp(\mu(s_i,t)) - \nonumber \\ & \mu(s_i,t)\left(\frac{Z(s_i,t)\exp(\mu(s_i,t))}{\exp(\mu(s_i,t))+\eta Z(s_i,t)}-\frac{\exp(2 \mu(s_i,t))Z(s_i,t)}{\left(\exp(\mu(s_i,t))+\eta Z(s_i,t-1)\right)^2}-\exp(\mu(s_i,t))\right) \\ \nonumber k(\mu(s_i,t)) =& -\frac{Z(s_i,t)\exp(\mu(s_i,t))}{\exp(\mu(s_i,t))+\eta Z(s_i,t)}+\frac{\exp(2 \mu(s_i,t))Z(s_i,t)}{\left(\exp(\mu(s_i,t))+\eta Z(s_i,t-1)\right)^2}+\exp(\mu(s_i,t)). \\ \end{align} The expressions $f(.)$ and $k(.)$ given are derived from expanding the log-density of $Z$ as a function of $Y$ about an initial guess for the mode. \eqref{eq:gausapprox} is then maximized as a function of $\boldsymbol{Y}$ and then evaluated at that value. The computational burden comes in conducting the maximization, however the sparsity of $\Sigma^{-1}(\theta)$ makes this easier, an explicit formula is given in \cite{rue2009approximate}.
When \eqref{eq:gausapprox} is evaluated at the posterior mode, it becomes $2 \pi^{n/2} \det (W+\Sigma^{-1}(\theta))^{\frac{1}{2}}$ where $W$ is a diagonal matrix of the same dimension as $\Sigma(\theta)$ where each diagonal entry is $k(\mu(s_i,t))$. The numerator of \eqref{eq:INLA} is then evaluated at $\mu(s_i,t)$. Therefore, the problem is simply a computation once the posterior mode of the denominator is found.
Inference is then carried out by fixing values of $\eta,\zeta,\tau^2,\alpha$, then finding the values of $\boldsymbol{Y}$ that maximize the Gaussian approximation. Then, for those fixed parameter values, we obtain an estimate of the posterior probability. The parameter space for $\eta,\zeta,\tau^2,\alpha$ can be efficiently explored to map out the marginal likelihood surface for that set of parameters. \cite{rue2009approximate} discuss efficient methods for exploring the parameter space.
From $\tilde{\pi}(\eta,\zeta,\tau^2,\alpha|\boldsymbol{Z})$ and $\pi_G(\boldsymbol{Y}|\alpha,\eta,\zeta,\tau^2,\boldsymbol{Z})$ we can then estimate the marginal posterior density $\pi(Y|Z)$ by calculating $\pi(Y|Z) \approx \sum\tilde{\pi}(\eta,\zeta,\tau^2,\alpha|\boldsymbol{Z})\pi_G(\boldsymbol{Y}|\alpha,\eta,\zeta,\tau^2,\boldsymbol{Z})$ where the summation is over all values of $\theta$ with sufficiently high posterior probability. If inferential concern is on the density of the latent state, we can subsequently improve $\pi_G(\boldsymbol{Y}|\alpha,\eta,\zeta,\tau^2,\boldsymbol{Z})$ by using a skew-Normal approximation based off of a higher order Taylor series expansion as given in \cite{rue2009approximate}.
While \eqref{eq:INLA} is a method to conduct Bayesian analysis, in the absence of $\pi(\zeta)$, $\pi(\alpha)$, $\pi(\tau^2)$, the maximization in \eqref{eq:INLA} is also an estimate of the maximization of the likelihood for $\eta$, $\zeta$,$\alpha$ and $\tau^2$ marginalized over $\boldsymbol{Y}$. Clearly the Gaussian approximation, and hence the Laplace approximation, is asymptotically valid if the Taylor series of $Z$ has a vanishing third and higher derivatives. Otherwise, the practitioner must rely on the assumption that the higher order terms are negligible.
\subsection{Issues with Laplace Approximation for Spatio-Temporal Data} There are two primary concerns with using this technique. The concerns are somewhat addressed in \cite{rue2009approximate}, but we will make them clear here. The first concern is unavoidable in any parametric modeling of spatio-temporal data. To see this issue, it is instructive to consider spatial sampling with temporal replication where there is no temporal dependence. If we only consider $Z(s_i)$ with $s_i\in \{s_1,s_2,...,s_n\}$ and say we sample this $T$ times, then we have replication of any spatial patterns to conduct inference from. Without replication, we have to hope that our spatial domain is large enough to create internal replication, that is, that the dependency in the data decays at a sufficient rate. This same issue exists in spatio-temporal data. Now, we have data that has dependence in both space and time and we inevitably only have a single realization of the data. Therefore, our space-time observation must be large enough to break both the space dependence and the time dependence. Essentially, this means that our, unobservable, space-time clusters must be small.
This is an issue with using Laplace approximations as the inferential results are asymptotically justified through the growth of independent samples. The approximation error of \cite{tierney1986accurate} is $\mathcal{O}(n^{-3/2})$, however the meaning of '$n$' for spatio-temporal models is not well-defined. The asymptotics are clearly justifiable if both the size of the grid and the number of observations per node increases, but the $n$ that needs to grow is the number of independent space-time observations.
One method of examining whether this has occurred is to look at the effective number of parameters as defined in \cite{spiegelhalter1998bayesian}. If the data is completely independent, then $n$ is indeed the number of samples. In this case, the effective number of parameters is the number of large scale parameters in the model. If we examine the ratio of observations to the effective number of parameters we will get an estimate of the number of observations available to estimate each of the effective number of parameters. If, for example, the effective number of parameters is close to $n$, then the ratio of observations to effective number of parameters will be extremely small indicating that we lack sufficient observations to conduct meaningful analysis.
The above concern really applies for any analysis of space-time data when we directly work with the full log-likelihood. In order to conduct meaningful inference we need to have replication or pseudo-replication of our data. The second issue is more specific to Laplace approximations and appears to be more prevalent in count data. That is, there is a bias in the approximation due to the truncation of the Taylor series that underlies the Gaussian approximation in the denominator of \eqref{eq:INLA}. This appears to first have been demonstrated in \cite{joe2008accuracy} where clustered (temporal) count data was analyzed assuming a Poisson-log Gaussian mixture where the log Gaussian was assumed to have an AR(1) structure. In \cite{joe2008accuracy}, the AR(1) parameter was consistently shown to be biased low and, assuming zero intercept, the variance was biased high. \cite{carroll2015comparing} also demonstrated bias in the estimation of the Intrinsic Conditional Auto-Regressive (ICAR) parameter when using the INLA software. \cite{rue2009approximate} recognize the bias in Laplace approximations, but state that it tends to be negligible in practice and only appear in pathological cases. However, as we will demonstrate, issues with truncation of the Taylor series approximation underlying the Laplace approximation are a major concern for self-exciting Poisson models like \eqref{eq:timeseries2} for parameter values that arise in practice.
\section{Extended Laplace Approximation}
The primary issue in \eqref{eq:INLA} when applied to \eqref{eq:timeseries2} is that we are essentially conducting a Laplace approximation to an integral of the form \begin{equation}
M=\int_{\mathbb{R}^{n_d \times T}} \exp \left(-g(Y|Z,\eta,\zeta,\tau^2,\alpha)\right) dY \label{eq:Integral}, \end{equation}
where \begin{equation}
\scriptstyle g(Y|.)= \frac{1}{2}\boldsymbol{Y}^T \Sigma^{-1}(\theta)\boldsymbol{Y}-\left(\sum_{i=1}^{n_d} \sum_{t=1}^T -\eta Z(s_i,t-1)-\exp(Y(s_i,t))+Z(s_i,t)\log\left[\eta Z(s_i,t-1)+\exp(Y(s_i,t))\right]\right) \label{eq:g}. \end{equation}
Clearly the size of $g(.)$ matches the dimension of the integration. As demonstrated in \cite{shun1995laplace}, this results in a necessarily biased approximation to the integral where the bias is on the order of $O(1)$.
In order to correct these issues, \cite{shun1995laplace} and \cite{evangelou2011estimation} conduct an expansion of $\log (M)$ that is correct even when the dimension of the integral in \eqref{eq:Integral} is equal to the sample size. The asymptotic behavior then will be appropriate as $T \to \infty$ due to the geometric decay in time induced by $\eta \in (0,1)$.
We will use the notation of \cite{evangelou2011estimation} letting $g_i(Y)= \frac{\partial g(Y)}{\partial Y(s_i,t)}$ and $g_{i,j}(Y)=\frac{\partial^2 g(Y)}{\partial Y(s_i,t) \partial(Y(s_j,t))}$. We will also let $g_{\boldsymbol{Y}}$ be the gradient of $g$ and $g_{\boldsymbol{YY}}$ be the Hessian and $g^{i,j}$ be the $(s_i,s_j)$ element of the inverse of the Hessian matrix.
In order to correct for the bias we apply the expansion given as (9) in \cite{shun1995laplace} and (21) in \cite{evangelou2011estimation}. The correction requires the derivation of the third, fourth and sixth derivatives of $g$,
\begin{align} g_{iii}= &-\left[\exp(Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))-1}\right)-3\exp(2 Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^2}\right)+\right.\nonumber\\ & \left. 2\exp(3Y(s_i,t)))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^3}\right)\right] \label{eq:thirds} \end{align}
\begin{align} g_{iiii}= &-\left[\exp(Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))-1}\right)-7\exp(2 Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^2}\right)+\right.\nonumber\\ & \left. 12\exp(3Y(s_i,t)))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^3})\right)-6\exp(4Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^4}\right)\right] \label{eq:fourths} \end{align}
\begin{align}
g_{vi}= &-\left[\exp(Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))-1}\right)-31\exp(2 Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^2}\right)+\right. \nonumber \\
& 180\exp(3Y(s_i,t)))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^3})\right)-438\exp(4Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^4}\right)]+\nonumber\\
& \left. 408\exp(5Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^5}\right)-120\exp(6Y(s_i,t))\left(\frac{\eta Z(s_i,t-1)}{\lambda(Y(s_i,t))^6}\right)\right] \label{eq:sixths} \end{align}
where $\lambda(Y(s_i,t))=\exp(Y(s_i,t))+\eta Z(s_i,t-1)$ in \eqref{eq:fourths} and \eqref{eq:thirds}. The final pieces needed are $g^{i,i}$ and $g^{i,j}$ both of which can be found in the appropriate entry upon inverting $\Sigma^{-1}(\theta)+W$ where $W$ is the same as defined in \eqref{eq:gausapprox}, which is the equivalent of $g_{i,i}$. The evaluation of $\log M$ is then
\begin{align}
\log M & \propto -\frac{1}{2}|\Sigma(\theta)|-\hat{g}-\frac{1}{2}|\hat{g}_{\boldsymbol{YY}}|-\sum_{t}\sum_{i}\frac{1}{8}\hat{g}_{iiii}-\sum_{t}\sum_{i}\frac{1}{48}\hat{g}_{vi} + \nonumber \\ & \frac{1}{72}\sum_{t}\sum_{i,j\leq i}\hat{g}_{iii}\hat{g}_{jjjj}\left(6 \left(\hat{g}^{ij}\right)^3+9 \hat{g}^{ii}\hat{g}^{jj}\hat{g}^{ij}\right) \label{eq:shunExpansion} \end{align}
In \eqref{eq:shunExpansion} we denote $\hat{g}$ as the evaluation of the $g$ function at $Y(s_i,t)=\mu(s_i,t)$ where $\mu(s_i,t)$ is the point that maximizes the Gaussian approximation to $Y$ in the denominator of \eqref{eq:INLA}.
The evaluation of \eqref{eq:shunExpansion} at this point brings the error from $\mathcal{O}(n^{-1})$ in the Laplace approximations to the marginals, to approximately $\mathcal{O}(n^{-3})$ when the higher order terms are included. While again this $n$ is ill-defined, critically it is the same for both the original and the extended Laplacian, meaning if there is insufficient data to accurately estimate the marginals under \eqref{eq:INLA}, the further expansion may be an improvement.
An alternative would be to employ the derivations in \cite{raudenbush2000maximum} which involve an expansion of $M$ vice $\log M$. However, as mentioned in \cite{shun1995laplace} and empirically demonstrated in Tables 2 and 3 of that manuscript, this correction has relative error $O(1)$ whereas the correction in \eqref{eq:shunExpansion} has relative error $o(1)$.
The performance of the extended LA method has previously been conducted in a likelihood setting. The higher order expansion has been shown to provide comparable errors as Gauss-Hermite quadrature with 20 quadrature points (\cite{raudenbush2000maximum}) and Monte Carlo maximum likelihood (\cite{evangelou2011estimation}).
\subsection{General Algorithm For Conducting Bayesian Inference Using Higher Order Laplace Approximation}
Here we will outline the general algorithm for using \eqref{eq:shunExpansion} to conduct an approximate Bayesian inference for the set of parameters, $\theta=\left(\alpha,\eta,\zeta,\tau^2\right)$. The first task is finding the mode of $\pi(\theta|\boldsymbol{Z})$. First we fix a value of $\theta$ and for that value of $\theta$ find the value of $\boldsymbol{Y}^*=\boldsymbol{\mu}^*$ that maximizes \eqref{eq:gausapprox}. This is accomplished through repeatedly solving $\left(\Sigma^{-1}(\theta)+\text{diag }k(\mu^*(s_i,t))\right)\boldsymbol{\mu}^*=f(\boldsymbol{\mu}^*)$ where $f(\boldsymbol{\mu^*})$ is the vector of evaluations of $f$ given in \eqref{eq:gausapprox}. The sparsity of $\Sigma^{-1}(\theta)+\text{diag }(k(\mu^*(s_i,t)))$ makes this task extremely fast.
This value, $\boldsymbol{Y}^*$, is then used to evaluate \eqref{eq:thirds}, \eqref{eq:fourths}, and \eqref{eq:sixths}, giving an approximation to the Log-likelihood given in \eqref{eq:shunExpansion}. As a point of comparison, on a $10 \times 10$ lattice wrapped on a Torus with 100 observations, finding $\boldsymbol{Y}^*$ and computing \eqref{eq:shunExpansion} take approximately 1-1.5 seconds. Using finite differences, the Hessian at that point can then be approximated. This takes an additional 32 evaluations if one covariate is in the model. A Newton-Raphson algorithm can then be used to find the mode of $\tilde{\pi}(\theta|\boldsymbol{Z})$. In the majority of problems considered, this took us approximately 4-5 steps. Finding the mode, again for the 10000 size data set described above this, generally, takes about 10-30 minutes.
At the mode, the posterior parameter space can then be efficiently explored using methods outlined in \cite{rue2009approximate}. Credible intervals for individual elements of $\theta$ can be found either through assuming posterior normality and using the Hessian at the posterior mode or through the method outlined in \cite{ferkingstad2015improving}.
In summary, the primary advantage of using Laplace based techniques is computational speed. A single computation of the log-likelihood for a 10 $\times$ 10 neighborhood structure with $T=100$ with $\boldsymbol{\alpha}$ as intercept only takes approximately 1 second with the primary computational cost being incurred in finding the mode of the Gaussian approximation to the denominator of \eqref{eq:INLA}. In using the extended Laplace approximation method in \eqref{eq:shunExpansion} there is an additional cost of about .5 of a second per evaluation. As a full exploration of the parameter space may take 600 to 1000 evaluations, the total cost incurred through using the expansion is about 5 to 6 minutes. \section{Fully Bayesian Approach}
While the size of $\Sigma(\theta)$ makes MCMC techniques challenging, some properties of the model make it feasible to use a flexible modeling language such as Stan to perform inference. To do this, we follow closely the development given in \cite{joseph}. First, note that we are trying to find \begin{align}
\pi(\theta | \boldsymbol{Z})\propto \prod_{s_i,t} \pi(Z(s_i,t)|Y(s_i,t),Z(s_i,t-1),\eta) \pi(Y(s_i,t)|\boldsymbol{\alpha},\tau,\zeta)\pi(\eta)\pi(\boldsymbol{\alpha})\pi(\tau)\pi(\zeta) \end{align}
In the above, we are required to both sample from and calculate the density of the latent state, $\boldsymbol{Y}$ which requires evaluations of
\begin{align}
\log(\pi(Y(s_i,t)|\boldsymbol{\alpha},\tau,\zeta)) \propto \frac{-t \times n_d}{2}\log(\tau^2) + \frac{1}{2} \log | \Sigma_f^{-1}(\theta)| - \frac{1}{2}(Y-\alpha)^T\Sigma_f^{-1}(\theta)(Y-\alpha) \label{eq:log Y} \end{align}
To speed up computations, we note that the greatest computational cost in the sampling is the calculation of the determinant of the potentially very large matrix, $\Sigma_f^{-1}(\theta)$. However, the specific structure for $\Sigma_f^{-1}(\theta)$ allows us to follow \cite{jin2005generalized}. First we note that $\log | \Sigma_f^{-1}(\theta)| = T \log | \Sigma^{-1}(\theta)|$ and $\log|\Sigma^{-1}(\theta)|=\frac{n_d}{\log\tau}+\log|I_{n_d,n_d}-\zeta N|$ where $N$ is the neighborhood or adjacency matrix. Therefore, we can let $V \Lambda V^T$ be the spectral decomposition of $N$ and then $|I_{n_d,n_d}-\zeta N|=|V| |I_{n_d,n_d}-\zeta \Lambda| |V^T|=\prod_{j=1}^{n_d}\left(1-\zeta \lambda_j\right)$ where $\lambda_j$ are the eigenvalues of the neighborhood matrix.
The greatest advantage of this approach is that the eigenvalues are irrelevant of any parameters, therefore they can be computed ahead of time. This means that we never need to deal with matrices of the size of $\Sigma_f(\theta)$.
However, even using state of the art MCMC software such as Stan and precomputing all eigenvalues, MCMC still remains slow. For example, if $n_d=100$ and $T=100$, a single MCMC chain of length 5000 took 3.5 hours to converge. In this example, the chain hadn't converged after 1000 iterations but exhibited no signs of non-convergence after 5000. In comparison, the Laplace approximation method of section 3, under the same set up, takes less than 10 minutes to find the find the parameters that maximize \eqref{eq:INLA} and then another 15-20 minutes to evaluate the posterior parameter space. The expanded Laplace approximation incurs an additional cost of about .5 of a second per evaluation and under the above conditions would add about 5 to 6 minutes of computations.
\section{Simulation Study}
In order to compare the Laplace approximation, with the higher order Laplace approximation and the MCMC inferential methodology, we simulated data from model \eqref{eq:timeseries2} on a $10 \times 10$ grid wrapped on a torus to reduce edge effects using a rook neighborhood structure. We further set $t \in \{1,2,...,100\}$. The choice of these values was made to replicate potential real world situations. For example, counts aggravated over counties in a state or aggregated over neighborhoods in a major metropolitan area often have approximately 100 locations. For instance, there are 99 counties in Iowa, there are 96 named neighborhoods in Chicago, and there are 120 districts in Iraq. $T=100$ would correspond to approximately two years of data observed weekly.
Next, we simulated from all 32 combinations of $\eta \in \{0,.1,.2,.3,.4,.5,.6,.7\}$ and $\tau^2 \in \{.4,.6,.8,1\}$. For each choice of $\eta$ and $\tau^2$ we next set $\zeta=.245$ in order to generate significant spatial correlation as the spatial correlation. While we could have considered other choices of $\zeta$ note that the spatial correlation between two observations at the same point in time is \begin{equation} \mbox{Corr}(Z(s_i,t)Z(s_j,t)) = \frac{\left(\exp(\Sigma_{i,i}+\Sigma_{i,j}) -\exp(\Sigma_{i,i})\right)}{\left(\exp(2\Sigma_{i,i}) -\exp(\Sigma_{i,i}) + \frac{\exp(-\alpha)}{1-\eta}\exp(\frac{\Sigma_{i,i}}{2}\right)}\label{eq:spatCorr}, \end{equation} where $\Sigma_{i,j}$ is the $(i,j)$th entry in the covariance matrix, $\Sigma(\theta)$. In order to have significant correlation in \eqref{eq:spatCorr} $\zeta$ needs to be near the edge of the parameter space. The spatial correlation reflects a well known problem for CAR models and is presented in depth in \cite{wall2004close}. We further fixed $\alpha(s_i,t)=0,\forall s_i,t$.
For each of the 32 combinations of parameters we found the values of $\hat{\tau^2}$, $\hat{\eta}$ and $\hat{\zeta}$ that maximized \eqref{eq:INLA} and \eqref{eq:shunExpansion}. In all cases, estimates of $\eta$, $\alpha$, and $\zeta$ using Laplace approximation and expanded LA were generally unbiased. The difficulty lies in estimating the conditional variance, $\tau^2$. For even small values of $\eta$ it will be shown that the Laplace expansion used in \eqref{eq:INLA} yields substantial bias.
We define substantial bias as a relative bias that is greater than 15\% of the value of the parameter it is estimating. For example, if $\tau^2=1$, a substantial bias would exist if the estimation procedure obtained a value greater than 1.15 or less than .85. We further make the assumption that, all things being equal, \eqref{eq:INLA} is preferable over \eqref{eq:shunExpansion} due to the simplicity of calculating \eqref{eq:INLA}. We further assume that both of these techniques are preferable over MCMC techniques as they are considerably quicker to fit.
Three example of the results for three combinations of $\eta$ and $\tau^2$ is given in Table \ref{Simulations} with the results from one simulation from each combination. We further explored the impact of not including \eqref{eq:sixths} in the computation of \eqref{eq:shunExpansion}. For the MCMC technique, the full parameter space was explored and then the posterior mean was used as a point estimate. In all cases, vague proper priors were used for $\eta$, $\tau^2$, $\alpha$ and $\zeta$.
\begin{table}[h]
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
& $\eta=.1$, $\tau^2=.4$&$\eta=.4$, $\tau^2=.6$& $\eta=.7$, $\tau^2=1$\\
\hline
Relative Bias in LA(1) & .12 & .2 & .46\\
Time to Fit LA(1) (min.)& 10-15 & 16-20 & 16-20\\
\hline
Extended LA Without 6th Order & .03 & .1 & .2\\
Extended LA With 6th Order& .03 & .05 & .2\\
Time to Fit Extended LA & 20-30 & 20-30 & 25-35\\
\hline
MCMC & .02 & .02 & .06\\
Time to Fit MCMC & 150-250 & 400-650 & 500-650\\
\hline
\end{tabular}
\end{center}
\caption{Relative Bias and approximate times to find point estimates. Note that the MCMC time is for a full exploration of the parameter space. All time to fit are estimates and in the case of LA(1) and Extended LA they are dependent on initial guess for Newton Raphson algorithm. In general the fit times between LA(1) and the Extended LA are comparable while MCMC took 2-10 hours depending on the simulation run.}\label{Simulations} \end{table}
In figure \ref{fig:Fitting}, we display, for all parameter combinations, the preferred method for inference. As a general algorithm for fitting, we would first attempt the LA(1) approximation. If the value of $\eta$ or $\tau^2$ is sufficiently high, then we would use the expanded LA method. Only in cases for extreme $\eta$ and $\tau^2$ would MCMC be necessary.
\begin{figure}
\caption{Preferred Methods for inference }
\label{fig:Fitting}
\end{figure}
As depicted in figure \ref{fig:Fitting} for $\eta < .4$ and $\tau^2<1$, the extended Laplace approximation method outlined in Section 1 would offer significant capability to produce correct estimates of parameters. While this may seem like a strong restriction on the parameter space, values larger than $\eta> .6$ results in extremely peaked and variable data, of which is rarely seen in the cases we envision the self-exciting Poisson CAR model being used. For example, if we simulate with $\tau^2=1$ and $\eta=.7$, the resulting simulation from a single node is depicted in Figure \ref{fig:Extreme}. As shown here, these parameter settings would correspond to a situation where there where very low counts followed by a massive spike and slow decay back to low counts. If the model were to be used to model something like the number of violent crimes in a neighborhood, it would be extremely unlikely that the data would follow this pattern.
\begin{figure}
\caption{Counts from a simulated location with $\eta=.7$ and $\tau^2=1$}
\label{fig:Extreme}
\end{figure}
\section{Illustrative Example}
In the following section we consider modeling violent crime in the city of Chicago in 2015 using the Self-Exciting Poisson CAR model. The Self-Exciting Poisson CAR model is appropriate here as there are potentially multiple processes that are giving rise to the violence. Specifically, some crime may be due to a latent tension at a given location and there may be further violence that is due to copy-cat or retaliatory attacks. Previous work including \cite{mohler2013modeling} analyzed this data in the absence of spatial correlation and concluded that self-excitement was present. Our purpose here is not to fully explore the complex nature of how and why violence occurred in Chicago, but rather to demonstrate how the expanded LA could be used by social scientists to quickly explore competing theories within the Self-Exciting Poisson CAR framework allowing the practitioner to capture latent spatial correlation while allowing for the possibility of self-excitation.
The data used for the Chicago crimes is provided via \url{https://data.cityofchicago.org/Public-Safety/Crimes-2001-to-present/ijzp-q8t2}. We then aggregated all violent crimes both weekly and within specific predefined neighborhoods. We considered aggravated assault, aggravated battery, and homicides involving weapons as violent crimes. While there are certainly other violent crimes that could be considered, these crimes in particular seem likely to exhibit self-excitation within a given neighborhood as they potentially spur some form of retaliation. Similar data was used in both \cite{mohler2013modeling} and \cite{mohler2014marked}.
While there are no official neighborhoods in Chicago and counts can vary between 77 and 200 named areas, the city of Chicago publishes boundaries at \url{https://data.cityofchicago.org/browse?q=neighborhoods&sortBy=relevance} of 77 distinct neighborhoods. These are the neighborhoods we used in the analysis and appear to be consistent with historical norms for both locations and naming conventions within the city. We are not aware of previous statistical studies analyzing crime aggregated to neighborhood levels within the Chicago to compare the choice of neighborhood structure to. \cite{mohler2013modeling} used data within a specific police beat, which corresponds, approximately, to half or a third of the size of one of the neighborhoods.
The resulting dataset consists of 9237 violent crimes that occurred in the city over 53 weeks (December 28 2014 - January 2, 2016). A spatial map depiction of the crimes aggregated over neighborhoods is given in Figure \ref{fig:SpatialOnly}.
\begin{figure}
\caption{Total count of violent crimes for 2015 aggregated over neighborhood.}
\label{fig:SpatialOnly}
\end{figure}
As evident in Figure \ref{fig:SpatialOnly}, there appears to be spatial clustering in both the south and the western regions of the city. Spatial tests such as Moran's I applied to the aggregated data suggest clustering in space and time. As the data is available on block level we can also treat it as point process data and use Ripley's K which echoes the finding of clustering in both space and time.
We then fit the data using the model given in Section 1 and in \eqref{eq:timeseries3}:
\begin{align} & Z(s_i,t) \sim Po(\lambda(s_i,t)) \label{eq:timeseries3} \\ & E[Z(s_i,t)]=\lambda(s_i,t)\\ & \boldsymbol{\lambda_t} = \exp(\boldsymbol{Y_t})+\eta \boldsymbol{Z_{t-1}}\\ & \boldsymbol{Y_t} \sim Gau (\boldsymbol{\alpha_t},(I_{{n_d},{n_d}}-C)^{-1}M) \end{align}
A well-known phenomenon in criminology, as shown in \cite{anderson1987temperature}, is that higher temperatures are related to higher levels of both violent and non-violent crimes. To control for this, structure was placed on $\boldsymbol{\alpha_t}$. Specifically, for location $(s_i,t)$, $\alpha(s_i,t)=\beta_0 + \beta_1 x_1(s_i,t) + \beta_2 x_2(s_i,t)$ where $x_1(s_i,t)$ corresponds to the observed average temperature in neighborhood $s_i$ and time $t$ and $x_2(s_i,t)$ corresponds to the log-population of location $s_i$ at time $t$. Due to data limitations, we assume that temperature is constant across neighborhoods at time $t$ and population is constant across time at neighborhood $s_i$. To aid in estimation of covariates, we centered and scaled the temperatures. We used census data for each neighborhood from the United States Census Bureau in 2010. For temperature, we used historic temperatures available from the Weather Underground website at \url{www.wunderground.com}.
Using the higher order Laplace approximation given in \eqref{eq:shunExpansion} we used finite differences to build up estimates of the Hessian matrix allowing us to perform approximate Newton-Raphson maximization for the parameter space. With 6 covariates, $\theta = (\tau^2,\zeta,\eta,\beta_0,\beta_1,\beta_2)$, this is possible in a relatively short amount of time. On a Surface Pro 3, the maximization was done using the statistical software R in under 10 minutes. The observed maximum was found at $\hat{\theta}=(.52,.179,.50,-5.6,.18,.49)$. Point estimates using each inferential technique is given in Table \ref{Table:Results}.
The positive value of $\beta_1$ observed here echoes the findings of \cite{anderson1987temperature} that increasing temperatures increase the probability of violence occurring. Specifically, because of the structure of model \eqref{eq:timeseries3}, if, for a given neighborhood, the temperature changes from 50 degrees Fahrenheit to 90 degrees Fahrenheit, the model would suggest that the expected number of violent crimes, due to temperature alone, would increase by a factor of 2, when controlling for self-excitement in the model.
The interpretation of $\eta$ differs slightly than the large scale parameters in $\boldsymbol{\alpha}$. A value of .49 that each violent events at time period $t$ raises the expected number of events at time period $t+1$ by .49. In other words, if there were 10 violent events in week 1 at a given location we would expect there to be 5 events in week 2 that were 'copy-cat' or inspired by the violence in week 1.
Confidence intervals can then be constructed either relying on asymptotics of the MLE, or in a Bayesian construct, through efficiently exploring the parameter space of $\pi(\theta|\boldsymbol{Z})$ through techniques outlined in \cite{rue2009approximate}. Here we rely on exploring the parameter space and calculating $\pi(\theta|\boldsymbol{Z})$ over a wide range of $\theta$ values. Marginals can then be constructed either naively or through skewness corrections as outlined in \cite{martins2013bayesian}. Here, we approximated the Hessian at the posterior mode using finite differences. The expanded Laplace approximation was then used to evaluate each of the finite differences to approximate the second partial derivatives. This technique resulted in credible intervals of $\tau^2 \in (.43,.61)$, $\zeta \in (.176,.182)$, $\eta \in (.47,.53)$, $\beta_0 \in (-6.3,-4.9)$, $\beta_1 \in (.09,.27)$, and $\beta_2 \in (.42,.55)$. Credible intervals for each parameter are given in \ref{Table:Results2}.
Goodness of fit can be assessed through the use of a randomized version of uniform residuals for discrete observations obtained through the probability integral transform as outlined in \cite{brillinger1982maximum}. If we let $z_{[1]},z_{[2]},...$ be the possible values of $Z(s_i,t)$, we set our observed $z(s_i,t)=z_{[k]}$ be the $k$th observed value, then the residuals are found through $r(s_i,t)\equiv u(s_i,t)$ where $u(s_i,t)\sim iid\text{ }\mbox{Unif}(F(z_{[k-1]}|\theta),F(z_{[k]}|\theta))$ where $F(.|\theta)$ corresponds to the CDF of $Z$ marginalized over the posterior density of $\theta$. Practically, this is done through simulating the CDF through an empirical density after repeatedly randomly drawing values from $\pi(\theta|Z)$. These generalized residuals should be approximately uniform.
The generalized residuals for this dataset are not uniform when examined against the spatial structure. If we aggregate the residuals over neighborhood they should be approximately .5 and should have no spatial clustering. However, if we examine Figure \ref{fig:resids} we see clustering of high residual values in neighborhoods that share similar socio-economic factors. If we would look at the specific locations of high residuals we would find them in the neighborhoods of Austin, West Garfield Park, and North Lawndale all have high residual values and all have a high percentage of poverty and individuals living on government assistance. While the socio-economic correlation with violence is not surprising, an analysis of the residuals makes this clear. This finding suggests that a more detailed investigation of the spatial dimensions of crime in Chicago could be conducted by sociologists who could add relevant spatial structure to $\boldsymbol{\alpha}$ in \eqref{eq:timeseries2}.
\begin{figure}
\caption{Uniform residuals marginalized over neighborhood.}
\label{fig:resids}
\end{figure}
To examine the bias in the standard Laplace approximation we next fit to \eqref{eq:timeseries2} using the first-order Laplace approximation method. Due to the high value of $\eta$ we would expect there to exist a bias in the point estimates. We again used finite differences to approximate the Hessian and used a Newton-Raphson method to maximize the posterior. Using this inferential technique the parameters were maximized at $\hat{\theta}=(.38,.180,.50,-5.6,.17,.50)$, again depicted in Table \ref{Table:Results} Gaussian approximations to the marginals are $\tau^2 \in (.33,.43)$, $\zeta \in (.178,.183)$, $\eta \in (.47,.53)$, $\beta_0 \in (-5.7,-5.4)$, $\beta_1 \in (.11,.23)$, and $\beta_2 \in (.48,.50)$. As is seen in Table \ref{Table:Results}. Clearly the largest difference in the point estimation is in $\tau^2$ as the point estimate using LA(1) is over two standard deviations from the estimate using the extended LA method. Furthermore, 95\% credible intervals for $\tau^2$ do not even overlap, as seen in \ref{Table:Results2}.
Finally, to compare the extended Laplace approximation to an MCMC technique we fit the model approach using the rStan software of \cite{gelman2015stan} using the technique outlined in section 5. This requires prior specification for all parameters. In order to be as uninformative as possible, we chose diffuse proper priors. Specifically, $\pi(\tau)\sim \text{Ca}^{+}(5)$, $\pi(\zeta) \sim \text{Unif}(0,.185)$, $\pi(\eta)\sim \text{Unif}(0,1)$, and $\pi(\beta_0),\pi(\beta_1),\pi(\beta_2) \sim \text{Gau} (0,1000)$. Where $\text{Ca}^+$ is a half-Cauchy. The parameter space of $\zeta$ is dictated by the largest eigenvalue of the spatial adjacency neighborhood, in this case the largest eigenvalue is approximately 5.4 constraining $\zeta \leq .185$. Three chains were run, starting at different locations in the parameter space. The chains were run for 10000 iterations each. Stan uses the first half of the iterations for warm-up, resulting in 15000 posterior samples for each parameter. Convergences was determined through examining the $\hat{R}$ values as well as through visual examination of the trace plots. Specific for using Stan, the divergence of the chains must be examined, see e.g. \cite{betancort}. After 10000 iterations there was no evidence that the chains had not converged. The entire process, using multiple cores to run each chain, took 3 hours. If parallelizing was not performed, it would take approximately 9 hours to run.
Using MCMC, 95 \% credible intervals were $\tau^2 \in (.42,.59)$, $\zeta \in (.176,.182)$, $\eta \in (.47,.53)$, $\beta_0 \in (-6.3,-5.0)$, $\beta_1 \in (.09,.27)$, and $\beta_2 \in (.42,.56)$. A comparison of point estimates is given in Table \ref{Table:Results} and a comparison of credible intervals found through MCMC and extended LA is given in Table \ref{Table:Results2}. As is clearly evident, there is not a significant difference between the extended Laplace technique and MCMC, however the time to fit the model was drastically higher using MCMC. While LA(1) and the extended LA were fit in similar time, LA(1) appears to underestimate $\tau^2$, which is consistent with what was found during the simulations in Section 6.
\begin{table}[h] \begin{center}
\begin{tabular}{ |c|c|c|c|c|c|c| }
\hline
Point Estimates & $\tau^2$ & $\zeta$ & $\eta$ & $\beta_0$ & $\beta_1$ & $\beta_2$\\
\hline
LA(1) & .38 & .180 &.50& -5.6 & .17 & .50 \\
Extended LA & .52 & .179 &.50& -5.6 & .18 & .49\\
MCMC & .50 & .179 & .50 & -5.6 & .18 & .49\\
\hline
\end{tabular} \end{center} \caption{Point estimates of the parameters from fitting model \eqref{eq:timeseries2} to the Chicago crime data. As evident, the Expanded LA and MCMC techniques are extremely similar, while LA(1) has a bias for $\tau^2$.}\label{Table:Results} \end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c|c| }
\hline
95\% Credible Intervals & $\tau^2$ & $\zeta$ & $\eta$ & $\beta_0$ & $\beta_1$ & $\beta_2$\\
\hline
Extended LA & (.43,.61) & (.176,.182) &(.47,.53)& (-6.3,-4.9) & (.09,.27) & (.42,.55)\\
MCMC & (.42,.59) & (.176,.182) & (.47,.53) & (-6.3,-5.0) & (.09,.27) & (.42,.56)\\
\hline
\end{tabular}
\end{center}
\caption{Comparison between 95 \% credible intervals formed using Expanded LA and MCMC. Note that the 95 \% credible intervals for Expanded LA were donethrough using finite differences to approximate the Hessian and then using a Gaussian approximation to the posterior.}\label{Table:Results2} \end{table}
\section{Discussion}
In this manuscript we demonstrated how extending Laplace approximations to include sixth order derivatives significantly reduces the bias in self-exciting spatio-temporal models. In general, as long as the marginal variance of the process model $\boldsymbol{Y}$ is less than 1 and the self-excitement parameter is less than .6, the extended Laplace approximations will give estimates that are nearly unbiased, and the bias will reduce as the number of observations per location increases. We note that \cite{ferkingstad2015improving} also offers a copula based method for potentially correcting the bias, however, this takes the analysis out of the Laplace framework and it is unclear what proceeding along this line does to the asymptotics. Furthermore, in the example considered in this manuscript, we were not interested in $\boldsymbol{Y}|\boldsymbol{Z}$. In order to implement the methodology outlined in \cite{ferkingstad2015improving} we would need to calculate the skew-normal approximation to $Y|\theta,Z$ which would add to the computational burdern.
We further showed how a fully Bayesian approach could be considered through exploiting the sparsity of the precision matrix of the spatio-temporal process model. Even with a fully Bayesian approach being possible, the main benefit of using an extended LA methodology for this model is in computational speed. While MCMC takes several hours, the entire process for the extended LA took approximately half an hour. The datasets we considered here were moderately sized for spatio-temporal data, if, however, we used larger datasets we would expect there to be an even larger disparity in fitting time.
The obvious cost of using the extended LA methodology is it requires deriving up to sixth order partial derivatives to compute \eqref{eq:shunExpansion}. Also, under the methodology outlined in this manuscript, exploration of the parameter space would not be efficient for a higher number of covariates in the model. However, as demonstrated above, if a Gaussian approximation to the marginals were to be used the parameter space would not have to be fully explored and second order finite differences could be used to fairly quickly approximate the Hessian.
Finally, we demonstrated how this methodology can be applied to analyze crime in Chicago showing how both spatial and temporal covariates can be considered through placing structure on $\boldsymbol{\alpha}$ and in this instance matches the inference using MCMC techniques. Interestingly, the self-excitement value found in this analysis, $\hat{\eta}=.50$, is similar to what was found in \cite{mohler2013modeling} where in one police beat, 55\% of observed crime was found to be due to repeated actions, or self-excitement. While that manuscript did not consider exogeneous covariates, our analysis would suggest that the self-excitement was present even when weather and population size were considered.
While socio-economic factors weren't considered in our analysis, the residuals suggest that researchers with expertise in this area may apply this model with the addition of relevant covariates accounting for these factors. Significantly, this would allow for inference for these factors controlling for the existence of self-excitement, which appears to be done rarely, if ever, in this field of literature. \section{Supplemental Material}
\begin{description}
\item[Data Sets Used in Illustrative Example] .csv files containing the crime counts aggregated over neighborhoods and weeks, the weather aggregated over neighborhoods and weeks, and the population aggregated over neighborhoods and weeks
\item[chi.graph] Graph file giving the neighborhood structure for the 77 neighborhoods in Chicago
\item[R-code used in RStan] R-code and Stan model to do fully Bayesian method used in Illustrative Example
\end{description}
\end{document} | arXiv | {
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\begin{document}
\setlength{\parskip}{2pt plus 4pt minus 0pt}
{\scriptsize May 20, 2022} \vskip1ex
\title[Properties of the cascade]{Combinatorial and geometric constructions associated with the Kostant cascade} \author{Dmitri I. Panyushev} \address{ Institute for Information Transmission Problems of the R.A.S.,
Moscow 127051, Russia} \email{panyushev@iitp.ru} \thanks{This research was funded by RFBR, project {\tencyr\cyracc N0} 20-01-00515.} \keywords{root system, cascade element, abelian ideal, Frobenius algebra, nilpotent orbit} \subjclass[2010]{17B22, 17B20, 17B08, 14L30} \dedicatory{To Alexander Grigorievich Elashvili on the occasion of his 80th birthday} \begin{abstract} Let $\g$ be a complex simple Lie algebra and $\be=\te\oplus\ut^+$ a fixed Borel subalgebra. Let $\Delta^+$ be the set of positive roots associated with $\ut^+$ and $\gK\subset\Delta^+$ the Kostant cascade. We elaborate on some constructions related to $\gK$ and applications of $\gK$. This includes the cascade element $x_\gK$ in the Cartan subalgebra $\te$ and properties of certain objects naturally associated with $\gK$: an abelian ideal of $\be$, a nilpotent $G$-orbit in $\g$, and an involution of $\g$. \end{abstract} \maketitle
\tableofcontents
\section{Introduction} \noindent Let $G$ be a simple algebraic group with $\Lie(G)=\g$. Fix a triangular decomposition $\g=\ut^+\oplus\te\oplus\ut^-$. Then $\Delta$ is the root system of $(\g,\te)$ and $\Delta^+$ is the set of positive roots corresponding to $\ut^+$. The {\it Kostant cascade\/} is a set $\gK$ of strongly orthogonal roots in $\Delta^+$ that is constructed recursively starting with the highest root $\theta\in\Delta^+$, see Section~\ref{sect:prelim-kaskad}. The construction of cascade goes back to B.\,Kostant, who used it for studying the center of the enveloping algebra of $\ut^+$.
His construction is prominently used in some articles afterwards~\cite{GS,jos77, lw}, but Kostant's own publications related to the cascade appear some 40 years later~\cite{ko12,ko13}. The cascade is also crucial for computing the index of seaweed subalgebras of simple Lie algebras~\cite{ty04,jos}. \\ \indent Ever since I learned from A.\,Elashvili about the cascade at the end of 80s, I was fascinated by this structure. Over the years, I gathered a number of results related to the occurrences of $\gK$ in various problems of Combinatorics, Invariant Theory, and Representation Theory. In~\cite{p22}, I give an application of $\gK$ to the problem of classifying the nilradicals of parabolic subalgebras of $\g$ that admit a commutative polarisation. (General results on commutative polarisations are due to Elashvili and Ooms, see~\cite{ag03}.) Some other observations appear in this article.
Let $\Pi\subset\Delta^+$ be the set of simple roots. If $\gamma=\sum_{\ap\in \Pi}a_\ap\ap$, then $[\gamma:\ap]=a_\ap$ and $\htt(\gamma)=\sum_{\ap\in \Pi}[\gamma:\ap]$ is the {\it height\/} of $\gamma$. The set of positive roots $\Delta^+$ is a poset with respect to the root order ``$\preccurlyeq$'', and $\gK=\{\beta_1,\dots,\beta_m\}$ inherits this structure so that $\theta=\beta_1$ is the unique maximal element of $\gK$.
In Section~\ref{sect:comb-prop}, we define a rational element of $\te$ associated with $\gK$. Let $(\ ,\,)$ denote the restriction of the Killing form on $\g$ to $\te$. As usual, $\te$ and $\te^*$ are identified via $(\ ,\,)$ and $\te^*_\BQ$ is the $\BQ$-linear span of $\Delta$.
The {\it cascade element\/} of $\te_\BQ\simeq\te^*_\BQ$ is \beq \label{eq:casc-elem}
x_\gK=\sum_{i=1}^m\frac{\beta_i}{(\beta_i,\beta_i)}=\frac{1}{2}\sum_{i=1}^m{\beta_i}^\vee . \eeq The numbers $\gamma(x_\gK)$, $\gamma\in\Delta^+$, are the eigenvalues of $\ad x_\gK$ on $\ut^+$, and we say that they form the spectrum of $x_\gK$ on $\Delta^+$. It follows from~\eqref{eq:casc-elem} that $\gamma(x_\gK)\in \frac{1}{2}\BZ$ and $\beta(x_\gK)=1$ for any $\beta\in\gK$. We prove that $-1\leqslant \gamma(x_\gK)\leqslant 2$ for any $\gamma\in\Delta^+$ and if $\g$ is not of type $\GR{A}{2p}$, then the eigenvalues are integral (Theorem~\ref{thm:spektr-fonin}). It is also shown that the spectrum of $x_\gK$ on $\Delta^+\setminus\gK$ is symmetric relative to $1/2$, which means that if $m_\lb$ is the multiplicity of the eigenvalue $\lb$, then $m_\lb=m_{1-\lb}$. If $\theta$ is a fundamental weight and $\ap\in\Pi$ is the unique root such that $(\theta,\ap)\ne 0$, then $\ap$ is long and we prove that $\ap(x_\gK)=-1$. On the other hand, if $\theta$ is not fundamental, then $x_\gK$ appears to be dominant. Let $\cQ$ (resp. $\cQ^\vee$) denote the {\it root} (resp. {\it coroot}) {\it lattice} in $\te^*_\BQ$. The corresponding dual lattices are \\[.6ex] \centerline{ the coweight lattice $\cP^\vee:=\cQ^*$ \ \& \ the weight lattice $\cP:=(\cQ^\vee)^*$.} \\[.6ex] Hence $x_\gK\in \cP^\vee$ unless $\g$ is of type $\GR{A}{2p}$. Here $\cQ^\vee\subset\cP^\vee$ and we prove that $x_\gK\in\cQ^\vee$ if and only if every self-dual representation of $\g$ is orthogonal (Section~\ref{sect:self-dual}).
In~\cite[Section\,3]{ooms}, A.\,Ooms describes an interesting feature of the Frobenius Lie algebras. Let $\q=\Lie(Q)$ be a Frobenius algebra and $\xi\in\q^*$ a regular linear form, i.e., $\q^\xi=\{0\}$. Then the Kirillov form $\gB_\xi$ is non-degenerate and it yields a linear isomorphism $\bi_\xi:\q^*\to \q$. Letting $x_\xi=\bi_\xi(\xi)$, Ooms proves that $(\ad x_\xi)^*=1- \ad x_\xi$, where $(\ad x_\xi)^*$ is the adjoint operator w.r.t.{} $\gB_\xi$. This implies that the spectrum of $\ad x_\xi$ on $\q$ is symmetric relative to $1/2$. We say that $x_\xi\in\q$ is the {\it Ooms element\/} associated with $\xi\in\q^*_{\sf reg}$. Let $\te_\gK\subset\te$ be the $\BC$-linear span of $\gK$. Then $\be_\gK=\te_\gK\oplus\ut^+$ is a Frobenius Lie algebra, i.e., $\ind\be_\gK=0$, see~\cite[Sect.\,5]{p22}. In Section~\ref{sect:frob}, we prove that \begin{itemize} \item if $\q$ is an algebraic Lie algebra, then any Ooms element $x_\xi\in\q$ is semisimple; \item $x_\gK$ is an Ooms element for the Frobenius Lie algebra $\be_\gK=\te_\gK\oplus\ut^+$. \end{itemize} The latter provides a geometric explanation for the symmetry of the spectrum of $x_\gK$ on $\Delta^+\setminus\gK$.
By~\cite{ko98}, one associates an abelian ideal of $\be$, $\ah_z$, to any $z\in\te$ such that $\gamma(z)\in\{-1,0,1,2\}$ for all $\gamma\in\Delta^+$. Namely, let $\Delta^+_z(i)=\{\gamma\in\Delta^+ \mid \gamma(z)=i\}$. There is a unique $w_z\in W$ such that the inversion set of $w_z$, $\eus N(w_z)$, equals $\Delta^+_z(1)\cup\Delta^+_z(2)$, and then $\Delta_{\langle z\rangle}:= w_z\bigl(\Delta^+_z(-1)\cup -\Delta^+_z(2)\bigr)$ is the set of roots of $\ah_z$. We notice that $w_z(z)$ is anti-dominant and $w_z$ is the element of minimal length having such property. Kostant's construction applies to $z=x_\gK$ unless $\g$ is of type $\GR{A}{2p}$ and we obtain a complete description of $w_\gK:=w_{x_\gK}$ and $\ah_\gK:=\ah_{x_\gK}$ (Sections~\ref{sect:ab-ideal},\,\ref{sect:w_K&perebor}). In this setting, we prove that $w_\gK(\theta)\in -\Pi$ and $-w_\gK(x_\gK)$ is a fundamental coweight. That is, if $w_\gK(\theta)=-\ap_j$, then $-w_\gK(x_\gK)=2\varpi_j/(\ap_j,\ap_j)=:\varpi_j^\vee$. Since $[\be,\ah_\gK]\subset\ah_\gK$, the set of roots $\Delta_{\langle \gK\rangle}=\Delta(\ah_\gK)$ is an upper ideal of the poset $(\Delta^+,\preccurlyeq)$. Therefore, $\Delta_{\langle \gK\rangle}$ is fully determined by the set of {\sl minimal} elements of $\Delta_{\langle \gK\rangle}$ or the set of {\sl maximal} elements of $\Delta^+\setminus\Delta_{\langle \gK\rangle}$. Letting $\Delta^+_\gK(i)=\Delta^+_{x_\gK}(i)$ and $\Pi_\gK(i)=\Pi\cap\Delta^+_\gK(i)$, we prove that \begin{enumerate} \item $\min(\Delta_{\langle \gK\rangle})=w_\gK(\Pi_\gK(-1))$ \ and \ $\max(\Delta^+\setminus \Delta_{\langle \gK\rangle})=-w_\gK(\Pi_\gK(1))$; \item if $d_\gK=1+\sum_{\ap\in \Pi_\gK({\geqslant}0)}[\theta:\ap]$, then $\Delta_{\langle \gK\rangle}=\{\gamma\mid \htt(\gamma)\geqslant d_\gK\}$. \end{enumerate} In order to verify (2), we use explicit formulae for $w_\gK$. To get such formulae, we exploit a description of $w_\gK^{-1}(\Pi)$ (Theorem~\ref{thm:w^-1}). Then we check directly that $\htt(w_\gK(\ap))=d_\gK$ for any $\ap\in\Pi_\gK(-1)$ and that $\#\Pi_\gK(-1)=\#\{\gamma\mid \htt(\gamma)= d_\gK\}$.
In Section~\ref{sect7:involution}, we naturally associate an involution $\sigma_\gK\in \mathsf{Aut}(\g)$ to $\gK$ if $x_\gK\in\cP^\vee$, i.e., $\g$ is not of type $\GR{A}{2p}$. It is proved $\sigma_\gK$ is the unique, up to conjugation, inner involution such that the $(-1)$-eigenspace of $\sigma_\gK$ contains a regular nilpotent element of $\g$.
For any simple Lie algebra ($\mathfrak{sl}_{2p+1}$ included), we construct the nilpotent $G$-orbit associated with $\gK$, see Section~\ref{sect:nilp-orb}.
Let $e_\gamma\in \g_\gamma$ be a nonzero root vector ($\gamma\in\Delta$). Then $e_\gK=\sum_{\beta\in\gK}e_\beta\in \g$ is nilpotent and the orbit $\co_\gK=G{\cdot}e_\gK$ does not depend on the choice of root vectors. Properties of $\co_\gK$ essentially depend on whether $\theta$ is fundamental or not. We prove that if $\theta$ is fundamental then $(\ad e_\gK)^5=0$ and $(\ad e_\gK)^4\ne 0$; whereas if $\theta$ is not fundamental then $(\ad e_\gK)^3=0$ (and, of course, $(\ad e_\gK)^2\ne 0$). By~\cite{p94}, this means that $\co_\gK$ is spherical if and only if $\theta$ is not fundamental. Here $[x_\gK,e_\gK]=e_\gK$ and $x_\gK\in \Ima(\ad e_\gK)$. Therefore, $2x_\gK$ is a {\it characteristic\/} of $e_\gK$ and the weighted Dynkin diagram of $\co_\gK$, $\gD(\co_\gK)$ is determined by the dominant element in $W{\cdot}(2x_\gK)$. Actually, this dominant element is $-2w_\gK(x_\gK)$, and if $\g\ne\mathfrak{sl}_{2p+1}$, then $-2w_\gK(x_\gK)=2\varpi_j^\vee$, cf. above. Hence, in these cases, $\gD(\co_\gK)$ has the unique nonzero label on the node $\ap_j$ and $\co_\gK$ is even,
cf. Tables~\ref{table:O-class}, \ref{table:O-exc}.
\un{Main notation}. Throughout, $G$ is a simple algebraic group with $\g=\Lie(G)$. Then \begin{itemize}
\item[--] $\be$ is a fixed Borel subalgebra of $\g$ and $\ut^+=\ut=[\be,\be]$; \item[--] $\te$ is a fixed Cartan subalgebra in $\be$ and $\Delta$ is the root system of $(\g,\te)$; \item[--] $\Delta^\pm$ is the set of roots corresponding to $\ut^\pm$; \item[--] $\Pi=\{\ap_1,\dots,\ap_n\}$ is the set of simple roots in $\Delta^+$ and the corresponding fundamental weights are $\varpi_1,\dots,\varpi_n$; \item[--] $\te^*_\BQ$ is the $\BQ$-vector subspace of $\te^*$ spanned by $\Delta$, and $(\ ,\, )$ is the positive-definite form on $\te^*_\BQ$ induced by the Killing form on $\g$; as usual, $\gamma^\vee=2\gamma/(\gamma,\gamma)$ for $\gamma\in\Delta$. \item[--] For each $\gamma\in\Delta$, $\g_\gamma$ is the root space in $\g$ and $e_\gamma\in\g_\gamma$ is a nonzero vector; \item[--] If $\ce\subset\ut^+$ is a $\te$-stable subspace, then $\Delta(\ce)\subset \Delta^+$ is the set of roots of $\ce$; \item[--] $\theta$ is the highest root in $\Delta^+$; \item[--] $W\subset GL(\te)$ is the Weyl group.
\end{itemize} Our main references for (semisimple) algebraic groups and Lie algebras are~\cite{VO,t41}. In explicit examples related to simple Lie algebras, the Vinberg--Onishchik numbering of simple roots and fundamental weights is used, see e.g.~\cite[Table\,1]{VO} or \cite[Table\,1]{t41}.
\section{Preliminaries on root systems and the Kostant cascade} \label{sect:prelim-kaskad} \noindent We identify $\Pi$ with the vertices of the Dynkin diagram of $\g$. For any $\gamma\in\Delta^+$, let $[\gamma:\ap]$ be the coefficient of $\ap\in\Pi$ in the expression of $\gamma$ via $\Pi$. The {\it support\/} of $\gamma$ is $\supp(\gamma)=\{\ap\in\Pi\mid [\gamma:\ap]\ne 0\}$ and the {\it height of\/} $\gamma$ is $\htt(\gamma)=\sum_{\ap\in\Pi}[\gamma:\ap]$. As is well known, $\supp(\gamma)$ is a connected subset of the Dynkin diagram. For instance, $\supp(\theta)=\Pi$ and $\supp(\ap)=\{\ap\}$. A root $\gamma$ is {\it long}, if $(\gamma,\gamma)=(\theta,\theta)$. We write $\Delta_l$ (resp. $\Delta_s$) for the set of long (resp. short) roots in $\Delta$. In the simply-laced case, $\Delta_s=\varnothing$. \\ \indent Let ``$\preccurlyeq$'' denote the {\it root order\/} in $\Delta^+$, i.e., we write $\gamma\preccurlyeq\gamma'$ if $[\gamma:\ap]\leqslant [\gamma':\ap]$ for all $\ap\in\Pi$. Then $\gamma'$ covers $\gamma$ if and only if $\gamma'-\gamma\in\Pi$, which implies that $(\Delta^+,\preccurlyeq)$ is a graded poset. Write $\gamma\prec\gamma'$ if $\gamma\preccurlyeq\gamma'$ and $\gamma\ne\gamma'$. An {\it upper ideal\/} of $(\Delta^+,\preccurlyeq)$ is a subset $I$ such that if $\gamma\in I$ and $\gamma\prec\gamma'$, then $\gamma'\in I$. Therefore, $I$ is an upper ideal if and only if $\ce=\bigoplus_{\gamma\in I} \g_\gamma$ is a $\be$-ideal of $\ut$ (i.e., $[\be,\ce]\subset\ce$).
For a dominant weight $\lb\in\te^*_\BQ$, set $\Delta^\pm_\lb=\{\gamma\in\Delta^\pm\mid (\lb,\gamma)=0 \}$ and $\Delta_\lb=\Delta^+_\lb\cup \Delta^-_\lb$. Then $\Delta_\lb$ is the root system of a semisimple subalgebra $\g_\lb\subset \g$ and $\Pi_\lb=\Pi\cap\Delta^+_\lb$ is the set of simple roots in $\Delta^+_\lb$. Set $\Delta_\lb^{{>}0}=\{\gamma\in \Delta^+\mid (\lb,\gamma)>0\}$. Then $\Delta^+ =\Delta^+_\lb \sqcup \Delta_\lb^{{>}0}$ and \begin{itemize}
\item \ $\p_\lb=\g_\lb+\be$ is a standard parabolic subalgebra of $\g$; \item \ the set of roots for the nilradical $\n_\lb=\p_\lb^{\sf nil}$ is $\Delta_\lb^{{>}0}$; it is also denoted by $\Delta(\n_\lb)$. \end{itemize} If $\lb=\theta$, then the nilradical $\n_\theta$ is a {\it Heisenberg Lie algebra}. In this case, $\eus H_\theta:=\Delta(\n_\theta)$ is said to be the {\it Heisenberg subset\/} (of $\Delta^+$).
The construction of the Kostant cascade $\gK$ in $\Delta^+$ is recalled below, see also~\cite[Sect.\,2]{jos77}, \cite[Section\,3a]{lw}, and \cite{ko12,ko13}. Whenever we wish to stress that $\gK$ is associated with $\g$, we write $\gK(\g)$ for it. \\ \indent {\it\bfseries 1.} \ We begin with $(\g\langle 1\rangle,\Delta\langle 1\rangle,\beta_1)=(\g,\Delta,\theta)$ and consider the (possibly reducible) root system $\Delta_\theta$. The highest root $\theta=\beta_1$ is the unique element of the {\bf first} (highest) level in $\gK$. Let $\Delta_\theta=\bigsqcup_{j=2}^{d_2} \Delta\langle j\rangle$ be the decomposition into irreducible root systems and $\Pi\langle j\rangle =\Pi\cap \Delta\langle j\rangle$. Then $\Pi_\theta=\bigsqcup_{j=2}^{d_2}\Pi\langle j\rangle$ and $\{\Pi\langle j\rangle\}$ are the connected components of $\Pi_\theta\subset \Pi$. \\ \indent {\it\bfseries 2.} \ Let $\g\langle j\rangle$ be the simple subalgebra of $\g$ with root system $\Delta\langle j\rangle $. Then $\g_\theta=\bigoplus_{j=2}^{d_2}\g\langle j\rangle $. Let $\beta_{j}$ be the highest root in $\Delta\langle j\rangle ^{+}=\Delta\langle j\rangle \cap\Delta^+$. The roots $\beta_2,\dots,\beta_{d_2}$ are the {\it descendants\/} of $\beta_1$, and they form the {\bf second} level of $\gK$. Note that $\supp(\beta_j)=\Pi\langle j\rangle $, hence different descendants have disjoint supports. \\ \indent {\it\bfseries 3.} \ Making the same step with each pair $(\Delta\langle j\rangle,\beta_j)$, $j=2,\dots,d_2$, we get a collection of smaller simple subalgebras inside each $\g\langle j\rangle$ and smaller irreducible root systems inside $\Delta\langle j\rangle$. This provides the descendants for each $\beta_j$ ($j=2,\dots,d_2$), i.e., the elements of the {\bf third} level in $\gK$. And so on... \\ \indent {\it\bfseries 4.} \ This procedure eventually terminates and yields a maximal set $\gK=\{\beta_1,\beta_2,\dots,\beta_m\}$ of {\it strongly orthogonal\/} roots in $\Delta^+$. (The latter means that $\beta_i\pm\beta_j\not\in\Delta$ for all $i,j$). We say that $\gK$ is the {\it Kostant cascade} in $\Delta^+$.
\noindent Thus, each $\beta_i\in\gK$ occurs as the highest root of a certain irreducible root system $\Delta\langle i\rangle$ inside $\Delta$ such that $\Pi\langle i\rangle=\Pi\cap\Delta\langle i\rangle^+$ is a basis for $\Delta\langle i\rangle$.
We think of $\gK$ as poset such that $\beta_1=\theta$ is the unique maximal element and each $\beta_i$ covers exactly its own descendants. If $\beta_j$ is a descendant of $\beta_i$, then $\beta_j\prec \beta_i$ in $(\Delta^+,\preccurlyeq)$ and $\supp(\beta_j)\varsubsetneq\supp(\beta_i)$, while different descendants of $\beta_i$ are not comparable in $\Delta^+$. Therefore the poset structure of $\gK$ is the restriction of the root order in $\Delta^+$. The resulting poset $(\gK, \preccurlyeq)$ is called the {\it cascade poset}. The numbering of $\gK$ is not canonical. We only require that it is a linear extension of $(\gK,\preccurlyeq)$, i.e., if $\beta_j$ is a descendant of $\beta_i$, then $j>i$.
Using the decomposition $\Delta^+ =\Delta^+_\theta \sqcup \eus H_\theta$ and induction on $\rk\g$, one readily obtains the disjoint union determined by $\gK$: \beq \label{eq:decomp-Delta}
\Delta^+=\bigsqcup_{i=1}^m \eus H_{\beta_i}=\bigsqcup_{\beta\in\gK}\eus H_\beta , \eeq where $\eus H_{\beta_i}$ is the Heisenberg subset in $\Delta\langle i\rangle^+$ and $\eus H_{\beta_1}=\eus H_\theta$. The geometric counterpart of this decomposition is the direct sum of vector spaces \[
\ut^+=\bigoplus_{i=1}^m \h_i , \] where $\h_i$ is the Heisenberg Lie algebra in $\g\langle i\rangle$, with $\Delta(\h_i)=\eus H_{\beta_i}$. In particular, $\h_1=\n_\theta$.
For any $\beta\in\gK$, set $\Phi(\beta)=\Pi\cap \eus H_{\beta}$. Then $\Pi=\bigsqcup_{\beta\in\gK} \Phi(\beta)$ and $\Phi$ is thought of as a map from $\gK$ to $2^{\Pi}$. Our definition of subsets $\Phi(\beta_i)$ yields the well-defined map $\Phi^{-1}: \Pi\to \gK$, where $\Phi^{-1}(\ap)=\beta_i$ if $\ap\in \Phi(\beta_i)$. Note that $\ap\in\Phi(\Phi^{-1}(\ap))$. We also have $\#\Phi(\beta_i)\leqslant 2$ and $\#\Phi(\beta_i)= 2$ if and only if the root system $\Delta\langle i\rangle$ is of type $\GR{A}{n}$ with $n\geqslant 2$.
The cascade poset $(\gK,\preccurlyeq)$ with the set $\Phi(\beta)$ attached to each $\beta$ is called the {\it marked cascade poset} ({\sf MCP}). In~\cite{p22}, we use $(\gK, \preccurlyeq, \Phi)$ for describing the nilradicals of parabolic subalgebras that admit a commutative polarisation.
The Hasse diagrams of $(\gK, \preccurlyeq)$ are presented in Appendix~\ref{sect:tables}, where the Cartan label of the simple Lie algebra $\g\langle j\rangle$ is attached to the node $\beta_j$. These diagrams (without Cartan labels) appear already in~\cite[Section\,2]{jos77}.
\\ \indent Let us gather some properties of $(\gK,\preccurlyeq)$ that either are explained above or easily follow from the construction.
\begin{lm} \label{lm:K-svojstva} Let $(\gK, \preccurlyeq, \Phi)$ be the {\sf MCP} for a simple Lie algebra $\g$. \begin{enumerate} \item The partial order in $\gK$ coincides with the restriction to $\gK$ of the root order in $\Delta^+$; \item $\beta_i,\beta_j\in\gK$ are comparable if and only if\/ $\supp(\beta_i)\cap\supp(\beta_j)\ne \varnothing$; and then one support is properly contained in the other; \item each $\beta_j$, $j\geqslant 2$, is covered by a unique element of $\gK$; \item for any $\beta_j\in\gK$, the interval $[\beta_j,\beta_1]_{\gK}=\{\nu\in\gK\mid \beta_j\preccurlyeq\nu\preccurlyeq\beta_1\} \subset\gK$ is a chain. \item For $\ap\in\Pi$, we have $\ap\in\Phi(\beta_i)$ if and only if $(\ap, \beta_i)>0$. \end{enumerate} \end{lm}
Clearly, $\#\gK\leqslant \rk\g$ and the equality holds if and only if each $\beta_i$ is a multiple of a fundamental weight for $\g\langle i\rangle$. Recall that $\theta$ is a multiple of a fundamental weight of $\g$ if and only if $\g$ is not of type $\GR{A}{n}$, $n\geqslant 2$. It is well known that the following conditions are equivalent: {\sl (1)} $\ind\be=0$; {\sl (2)} $\#\gK=\rk\g$, see e.g.~\cite[Prop.\,4.2]{ap97}. This happens exactly if $\g$ is not of type $\GR{A}{n} \ (n\geqslant 2)$, $\GR{D}{2n+1} \ (n\geqslant 2)$, $\GR{E}{6}$. Then $\Phi$ yields a bijection between $\gK$ and $\Pi$.
For future reference, we record the following observation. \begin{lm} \label{lm:1-short} If\/ $\g$ is of type $\GR{B}{2k+1}$ or $\GR{G}{2}$, then $\gK$ contains a unique short root, which is simple. In all other cases, all elements of\/ $\gK$ are long. \end{lm}
Write $r_\gamma\in W$ for the reflection relative to $\gamma\in\Delta$. \begin{prop}[{\cite[Prop.\,1.10]{ko12}}] \label{prop:longest&K} The product $\omega_0:=r_{\beta_1}{\cdot}\ldots{\cdot}r_{\beta_m}$ does not depend on the order of factors and it is the longest element of\/ $W$ (i.e., $\omega_0(\Delta^+)=\Delta^-$). In particular, $\omega_0(\beta_i)=-\beta_i$ for each $i$. \end{prop} It follows from this that $\omega_0=-1$ if and only if $m=\rk\g$.
\section{The cascade element of a Cartan subalgebra} \label{sect:comb-prop}
\noindent In this section, we define a certain element of $\te$ associated with the cascade $\gK$ and consider its properties related to $\Delta$. As usual, we identify $\te$ and $\te^*$ using the restriction of the Killing form to $\te$.
\begin{df} \label{def:casc-elem} The {\it cascade element\/} of $\te$ is the unique element $x_\gK\in \langle\beta_1,\dots,\beta_m\rangle_\BQ\subset \te_\BQ$ such that $\beta_i(x_\gK)=1$ for each $i$. \end{df} Since the roots $\{\beta_i\}$ are pairwise orthogonal, we have \beq \label{eq:x_K}
x_\gK=\sum_{i=1}^m\frac{\beta_i}{(\beta_i,\beta_i)}=\frac{1}{2}\sum_{i=1}^m \beta_i^\vee . \eeq Therefore, $\gamma(x_\gK)\in \frac{1}{2}\BZ$ for any $\gamma\in \Delta$, and it follows from Prop.~\ref{prop:longest&K} that $\omega_0(x_\gK)=-x_\gK$. If $\gK\subset\Delta_l$,
then one can also write $\displaystyle x_\gK=\frac{1}{(\theta,\theta)}\sum_{i=1}^m \beta_i$.
\textbullet \ \ It is a typical pattern related to $\gK$ and $x_\gK$ that a certain property holds for series $\GR{A}{n}$ and $\GR{C}{n}$, but does not hold for the other simple types. The underlying reason is that \\[.6ex] \centerline{\it $\theta$ is a fundamental weight if and only if\/ $\g$ is {\bf not} of type $\GR{A}{n}$ or $\GR{C}{n}$. } \\[.6ex] (Recall that $\theta=\varpi_1+\varpi_n$ for $\GR{A}{n}$ and $\theta=2\varpi_1$ for $\GR{C}{n}$.) It is often possible to prove that a property does not hold if $\theta$ is fundamental, and then directly verify that that property does hold for $\slno$ and $\spn$ (or vice versa). \\ \indent \textbullet \ \ Yet another pattern is that one has to often exclude the series $\GR{A}{2n}$ from consideration. The reason is that \\[.6ex] \centerline{\it the \ \emph{Coxeter number} of\/ $\g$, $\mathsf{h}=\mathsf{h}(\g)$, is odd if and only if\/ $\g$ is of type $\GR{A}{2n}$. } \\[.6ex] (The same phenomenon occurs also in the context of the McKay correspondence.) Recall that $\mathsf{h}=1+\sum_{\ap\in\Pi}[\theta:\ap]=1+\htt(\theta)$.
To get interesting properties of $x_\gK$, we need some preparations. Set $n_\ap=[\theta:\ap]$, i.e., $\theta=\sum_{\ap\in\Pi}n_\ap\ap$. Suppose that $\theta$ is a fundamental weight, and let $\tap$ be the unique simple root such that $(\theta,\tilde\ap)\ne 0$. Then $(\theta,{\tap}^\vee)=1$ and $(\theta^\vee,{\tap})=1$, hence $\tilde\ap$ is long. Next, \[
(\theta,\theta)=(\theta, \sum_{\ap\in\Pi}n_\ap\ap) =(\theta, n_\tap\tap)=\frac{1}{2}n_\tap(\tap,\tap), \] which means that $n_\tap=2$. Let $\tilde\Pi$ be the set of simple roots that are adjacent to $\tap$ in the Dynkin diagram. Since $\tap$ is long, one also has $\tilde\Pi=\{\nu\in\Pi\mid (\nu,\tap^\vee)=-1\}$. Then \[
1= (\theta,\tap^\vee)=n_\tap(\tap,\tap^\vee)+\sum_{\nu\in \tilde\Pi}n_\nu(\nu,\tap^\vee)=4-\sum_{\nu\in \tilde\Pi}n_\nu . \] Hence $\sum_{\nu\in\tilde\Pi}n_\nu=3$ and $\#(\tilde\Pi)\leqslant 3$. Set $J=\{i\in [1,m]\mid (\tap,\beta_i)<0\}$. Then $1\not\in J$ and we proved in~\cite[Sect.\,6]{p05} that
\beq \label{eq:tap}
\tap=\frac{1}{2}\left(\theta-\sum_{i\in J}\frac{(\tap,\tap)}{(\beta_i,\beta_i)}\beta_i\right)=:
\frac{1}{2}(\theta-\sum_{i\in J}c_i\beta_i) \ \text{ and } \ \sum_{i\in J}c_i=3 , \eeq see~\cite[Lemma\,6.5]{p05}. Here $c_i\in \BN$ and therefore $\#J\leqslant 3$. Set $\tilde{\gK}:=\{\beta_i \mid i\in J\}$.
We say that $x\in\te$ is {\it dominant}, if $\gamma(x)\geqslant 0$ for all $\gamma\in\Delta^+$.
\begin{lm} \label{lm:not-dominant} If $\theta$ is fundamental, then $\tap(x_\gK)=-1$ and $(\theta-\tap)(x_\gK)=2$.
In particular, $x_\gK\in\te$ is not dominant. \end{lm} \begin{proof} Take $\gamma=\tap$. Using \eqref{eq:x_K} and \eqref{eq:tap}, we obtain \[
\tap(x_\gK)= \frac{1}{2}\left[\frac{(\theta,\theta)}{(\theta,\theta)}
-\sum_{i\in J}\frac{c_i(\beta_i,\beta_i)}{(\beta_i,\beta_i)}\right]=(1-3)/2=-1 . \] Then $(\theta-\tap)(x_\gK)=1+1=2$. \end{proof} Conversely, if $\theta$ is {\bf not} fundamental, then the example below shows that $x_\gK$ {\bf is} dominant. \begin{ex} \label{ex:sl-sp} (1) For $\g=\slno$, one has $\beta_i=\esi_i-\esi_{n+2-i}$ with $i=1,\dots,m=[(n+1)/2]$. Here $\sum_{j=1}^{n+1}\esi_j=0$ and $\varpi_j=\esi_1+\dots+\esi_j$. Hence \[
(\theta,\theta){\cdot}x_\gK=\sum_{i=1}^m \beta_i=\begin{cases} 2\varpi_p, & \text{if } \ n=2p{-}1 \\
\varpi_p+\varpi_{p+1}, & \text{if } \ n=2p . \end{cases} \] In the matrix form, one has $x_\gK=\begin{cases} \mathsf{diag}(1/2,\dots,1/2,-1/2,\dots,-1/2), & \text{if } \ n=2p{-}1 \\ \mathsf{diag}(1/2,\dots, 1/2,0,-1/2,\dots,-1/2), & \text{if } \ n=2p \ . \end{cases}$
(2) For $\spn$, one has $\beta_i=2\esi_i$ with $i=1,\dots,m=n$. Hence $(\theta,\theta){\cdot}x_\gK=\sum_{i=1}^n 2\esi_i=2\varpi_n$. \end{ex}
Consider the multiset $\eus{M}_\gK$ of values $\{\gamma(x_\gK)\mid \gamma\in \Delta^+\setminus \gK\}$. That is, each value $d$ is taken with multiplicity $m_d=\# \eus R_d$, where $\eus R_d=\{\gamma\in\Delta^+\setminus \gK \mid \gamma(x_\gK)=d\}$.
\begin{lm} \label{lm:symmetry-value} For any $d$, there is a natural bijection between the sets $\eus R_d$ and $\eus R_{1-d}$. In particular, $m_d=m_{1-d}$, i.e., the multiset $\eus{M}_\gK$ is symmetric w.r.t. $1/2$. \end{lm} \begin{proof} For any $\gamma\in \Delta^+\setminus \gK$, there is a unique $j\in \{1,\dots,m\}$ such that $\gamma\in\Delta(\h_j)\setminus\{\beta_j\}$, see~\eqref{eq:decomp-Delta}. Then $\beta_j-\gamma\in \Delta(\h_j)$ and $\gamma(x_\gK)+(\beta_j-\gamma)(x_\gK)=1$. \end{proof}
As we shall see in Section~\ref{sect:frob}, there is a geometric reason for such a symmetry. It is related to the fact that a certain Lie algebra is Frobenius. \begin{thm} \label{thm:spektr-fonin} If $\g$ is a simple Lie algebra, then $-1\leqslant\gamma(x_\gK)\leqslant 2$ for all $\gamma\in\Delta^+$. More precisely, \begin{enumerate} \item \ If\/ $\g$ is of type $\GR{A}{2p-1}$ or $\GR{C}{n}$, then $\{\gamma(x_\gK)\mid \gamma\in\Delta^+\}=\{0,1\}$; \item \ If\/ $\g$ is of type $\GR{A}{2p}$, then $\{\gamma(x_\gK)\mid \gamma\in\Delta^+\}=\{0,\frac{1}{2}, 1\}$; \item \ For all other types, i.e., if $\theta$ is fundamental, we have $\{\gamma(x_\gK)\mid \gamma\in\Delta^+\}=\{-1,0,1,2\}$. \end{enumerate} In particular, if\/ $\g$ is not of type $\GR{A}{2p}$, then $\gamma(x_\gK)\in \BZ$ \ for any $\gamma\in\Delta^+$. \end{thm} \begin{proof} {\bf 1$^o$}. Using explicit formulae for $\gK$, all these assertions can be verified case-by-case. For instance, data of Example~\ref{ex:sl-sp} provide a proof for (1) and (2). However, this does not explain the general constraints $-1\leqslant \gamma(x_\gK)\leqslant 2$. Below we provide a more conceptual argument, which also uncovers some additional properties of $x_\gK$.
{\bf 2$^o$}. If $m=\rk\g=n$, then $\gK$ is a basis for $\te^*$. Hence $\gamma$ can be written as $\gamma=\sum_{i=1}^n k_i\beta_i$, where $k_i=\frac{1}{2}(\gamma,\beta_i^\vee)\in \frac{1}{2}\BZ$, and $\gamma(x_\gK)=\sum_{i=1}^n k_i$. We are to prove that $-1\leqslant \sum_{i=1}^n k_i\leqslant 2$.
If $\gamma=\beta_i$, then $\gamma(x_\gK)=1$. Therefore we assume below that $\gamma\not\in\gK$. If $\gamma\in \Delta(\h_i)$ for some $i\geqslant 1$, then the whole argument can be performed for the simple Lie subalgebra $\g\langle i\rangle\subset \g$ and the cascade $\gK(\g\langle i\rangle)\subset \gK$, which has the unique maximal element $\beta_i$. Since $\rk\g\langle i\rangle< \rk\g$ for $i\geqslant 2$, it suffices to prove the assertion for $i=1$ and $\beta_1=\theta$.
Assume that $\gamma\in \eus H_1=\Delta(\h_1)$ and $\gamma\ne\theta$. Then $(\gamma,\theta^\vee)=1$, $\gamma=\frac{1}{2}\theta+\sum_{i=2}^n k_i\beta_i$, and \beq \label{eq:star}
4(\gamma,\gamma)=(\theta,\theta)+\sum_{i\geqslant 2}(2k_i)^2(\beta_i,\beta_i) . \eeq \indent {\sf (i)} \ For $\gamma\in\Delta^+_l$, it follows from~\eqref{eq:star} that $\displaystyle 3=\sum_{i\geqslant 2} 4k_i^2{\cdot}\frac{(\beta_i,\beta_i)}{(\gamma,\gamma)}$. Since $\#(\gK\cap\Delta^+_s)\leqslant 1$ (Lemma~\ref{lm:1-short}), the only possibilities for the nonzero coefficients $k_i$ are: \begin{itemize} \item \ $k_2, k_3, k_4=\pm\frac{1}{2}$ \ with \ $\beta_2,\beta_3,\beta_4\in \Delta^+_l$; \item \ $k_2 =\pm\frac{1}{2}$, $k_3=\pm 1$ \ with \ $\beta_2\in\Delta^+_l$, $\beta_3\in\Delta^+_s$ and $(\beta_2,\beta_2)/(\beta_3,\beta_3)=2$; \\ {} [this happens only for $\GR{B}{2p+1}$]; \item \ $k_2 =\pm\frac{3}{2}$ \ with \ $\beta_2\in\Delta^+_s$ and $(\theta,\theta)/(\beta_2,\beta_2)=3$ \quad [this happens only for $\GR{G}{2}$]. \end{itemize} In all these cases, we have $\gamma(x_\gK)\in \{-1,0,1,2\}$, as required.
{\sf (ii)} \ If $\gamma\in\Delta^+_s$ and $\displaystyle \frac{(\theta,\theta)}{(\gamma,\gamma)}=2$, then \eqref{eq:star} shows that $\displaystyle 2=\sum_{i\geqslant 2} 4k_i^2{\cdot}\frac{(\beta_i,\beta_i)}{(\gamma,\gamma)}\geqslant \sum_{i\geqslant 2} 4k_i^2$. Since $\#(\gK\cap\Delta^+_s)\leqslant 1$, the only possibility here is: \begin{itemize} \item $k_2 =\pm\frac{1}{2}$ \ with \ $\beta_2\in\Delta^+_l$ \quad [this happens for $\GR{B}{n}$, $\GR{C}{n}$, and $\GR{F}{4}$]. \end{itemize} Therefore $\gamma(x_\gK)=\frac{1}{2}+k_2\in \{0,1\}$.
{\sf (iii)} \ If $\gamma\in\Delta^+_s$ and $\displaystyle \frac{(\theta,\theta)}{(\gamma,\gamma)}=3$, then \eqref{eq:star} shows that $\displaystyle 1=\sum_{i\geqslant 2} 4k_i^2{\cdot}\frac{(\beta_i,\beta_i)}{(\gamma,\gamma)}$. Here the only possibility is $k_2=\pm \frac{1}{2}$ and $\beta_2$ is short \quad [this happens for $\GR{G}{2}$].
{\bf 3$^o$}. If $m<\rk\g=n$, then $\gK$ is not a basis for $\te^*$. Nevertheless, one can circumvent this obstacle as follows. Let $\omega_0\in W$ be the longest element. Then $-\omega_0\in GL(\te)$ takes $\Delta^+$ to itself and $\beta_i\in \te^{-\omega_0}$ for each $i$, see Prop.~\ref{prop:longest&K}. Moreover, $\gK$ is a basis for $\te^{-\omega_0}$, see~\cite[Lemma\,6.2]{p05}. Hence $\bar\gamma:=\frac{1}{2}(\gamma-\omega_0(\gamma))=\sum_{i=1}^m k_i\beta_i$, \ $k_i\in \frac{1}{2}\BZ$, and \[
\sum_{i=1}^m k_i=\bar\gamma(x_\gK)=\gamma(x_\gK) . \] Therefore, the argument of part {\bf 2$^o$} applies to $\bar\gamma$ in place of $\gamma$. However, a new phenomenon may occur here. As above, we begin with $\gamma\in \Delta(\h_1)\setminus \theta$. Then $\bar\gamma=\frac{1}{2}\theta +\sum_{i\geqslant 2}k_i \beta_i$. But in this case, $\bar\gamma$ is not necessarily a root and it may happen that $k_i=0$ for $i\geqslant 2$. Then $\bar\gamma(x_\gK)=1/2$. (This does occur for $\g$ of type $\GR{A}{2p}$: if $\gamma=\esi_1-\esi_{p+1}$, then $\omega_0(\gamma)=\esi_{2p+1}-\esi_{p+1}$ and $\bar\gamma=\frac{1}{2}\theta$. Conversely, if $\bar\gamma=\frac{1}{2}\theta$, then $\htt(\theta)=2{\cdot}\htt(\gamma)$. Hence the Coxeter number of $\g$ is odd, and this happens only for $\GR{A}{2p}$.)
{\bf 4$^o$}. Recall that $\omega_0=-1$ if and only $m=\rk\g$ and then $\bar\gamma=\gamma$. Therefore, part~{\bf 2$^o$} can be thought of as a special case of a more general approach outlined in {\bf 3$^o$}. \end{proof}
\begin{rmk} \label{rem:discussion} An analysis of possibilities for $\{k_i\}$ in the proof of Theorem~\ref{thm:spektr-fonin} reveals the following features: \begin{itemize} \item[\bf (1)] \ for any $\gamma\in \Delta^+$, $\bar\gamma$ is a linear combination of at most four different elements of $\gK$; \item[\bf (2)] \ if $\#\gK=\rk\g$ and $\gK\subset \Delta^+_l$, then every $\gamma\in\Delta^+_l\setminus \gK$ is presented as a linear combination of exactly four different elements of $\gK$; \item[\bf (3)] \ if $\gamma(x_\gK)=2$ or $\gamma(x_\gK)=-1$, then $\gamma\in \Delta^+_l$. \end{itemize} \end{rmk}
\begin{ex} \label{ex:so-x_K} Let $\g=\mathfrak{so}_N$ be realised as the set of skew-symmetric $N\times N$-matrices w.r.t. the antidiagonal. \begin{itemize} \item \ For $N=2n$, one has $\te=\{\mathsf{diag}(x_1,\dots,x_n,-x_n,\dots,-x_1)\mid x_i\in \BC\}$.
\begin{itemize} \item If $n=2k$, then $\#\gK=2k$ and the entries of $x_\gK$ are $x_{2i-1}=1$ and $x_{2i}=0$ for $i=1,\dots, k$. \item If $n=2k+1$, then still $\#\gK=2k$, the entries $x_j$ with $j\leqslant 2k$ are the same as for $\mathfrak{so}_{4k}$, and $x_{2k+1}=0$. \end{itemize} \item For $N=2n+1$, one has $\te=\{\mathsf{diag}(x_1,\dots,x_n,0,-x_n,\dots,-x_1)\mid x_i\in \BC\}$
and $\#\gK=n$. Here the entries of $x_\gK$ are $x_{2i-1}=1$ for $i\leqslant [(n+1)/2]$ and $x_{2i}=0$ for $i\leqslant [n/2]$. \end{itemize} \end{ex} \noindent Using Examples~\ref{ex:sl-sp} and \ref{ex:so-x_K}, one readily computes the numbers $\{\ap(x_\gK)\}$ with $\ap\in\Pi$ for all classical Lie algebras. For the exceptional Lie algebras one can use explicit formulae for $\gK$, see Appendix~\ref{sect:tables}. An alternative approach is to use the recursive construction of $\gK$. One has $\Pi=\bigsqcup_{i=1}^m \Phi(\beta_i)$, and it suffices to describe the numbers $\ap(x_\gK)$ for any simple Lie algebra and $\ap\in\Phi(\beta_1)=\Phi(\theta)$. \begin{enumerate} \item If $\theta$ is fundamental, then $\Phi(\theta)=\{\tap\}\subset\Pi_l$ and $\tap(x_\gK)=-1$ (Lemma~\ref{lm:not-dominant}); \item if $\g=\spn$, then $\theta=2\varpi_1$ and $\Phi(\theta)=\{\ap_1\}\subset\Pi_s$. Here $\theta=2\ap_1+\beta_2$ and $\ap_1(x_\gK)=0$; \item For $\slno$ ($n\geqslant 2$), we have $\Phi(\theta)=\{\ap_1,\ap_n\}$. If $n\geqslant 3$, then $\ap_1(x_\gK)=\ap_n(x_\gK)=0$; if $n=2$, then $\theta=\ap_1+\ap_2$ and $\ap_1(x_\gK)=\ap_2(x_\gK)=1/2$; \item For $\tri$, one has $\theta=\ap_1$ and $\ap_1(x_\gK)=1$. \end{enumerate} The resulting labelled diagrams are presented in Figures~\ref{fig:class} and~\ref{fig:except}.
\begin{center} \begin{figure}\label{fig:class}
\end{figure} \end{center}
\begin{center} \begin{figure}\label{fig:except}
\end{figure} \end{center} \vskip-3ex Let us summarise main features of the diagrams obtained. \begin{rmk} \label{rem:alternate&connected} 1) Fractional values occur only for $\g$ of type $\GR{A}{2p}$. For $\GR{A}{2p-1}$ and $\GR{C}{p}$, the dominant element $x_\gK$ is a multiple of the fundamental weight $\varpi_p$. More precisely, $x_\gK=\frac{2}{(\ap_p,\ap_p)}\varpi_p$ (cf. Remark~\ref{ex:sl-sp}).
2) All these diagrams have no marks `$2$' and all the marks $\{\ap(x_{\eus K})\}$ are nonzero if and only if $\g$ is of type $\GR{B}{2p-1}, \GR{D}{2p}, \GR{E}{7}, \GR{E}{8}, \GR{G}{2}$. The subset $\{\ap\in\Pi\mid \ap(x_\gK)=\pm 1\}$ is always connected and the marks `1' and `$-1$' alternate in this subset.
3) If $\ap(x_\gK)=-1$ and $\ap'\in\Pi$ is adjacent to $\ap$, then $\ap'(x_\gK)=1$. Moreover, if $\ap'(x_\gK)=1$, then $\ap'\in\gK$.
4) The numbers $\{\ap(x_{\eus K})\}$ are compatible with the unfolding procedures $\GR{C}{p}\mapsto \GR{A}{2p-1}$, $\GR{B}{p-1}\mapsto \GR{D}{p}$, $\GR{F}{4}\mapsto \GR{E}{6}$, and $\GR{G}{2}\mapsto \GR{D}{4}$. For instance, \quad \raisebox{-1.3ex}{\begin{tikzpicture}[scale=0.75, transform shape] \draw (0,0.2) node[above] {\small $1$}
(1,0.2) node[above] {\small $-1$}; \tikzstyle{every node}=[circle, draw, fill=orange!50] \node (a) at (0,0) {}; \node (b) at (1,0) {}; \tikzstyle{every node}=[circle] \node (r) at (0.35,.01) {\large $<$}; \draw (.31, .06) -- +(.47,0); \draw (.25, 0) -- +(.5,0); \draw (.31, -.06) -- +(.47,0); \end{tikzpicture} } \ $\mapsto$ \ \raisebox{-2.5ex}{\begin{tikzpicture}[scale=0.75, transform shape] \draw (5,0.2) node[above] {\small $1$}
(5.9,0.2) node[above] {\small $-1$}
(7.2, -.4) node[right] {\small $1$}
(7.2, .6) node[right] {\small $1$} ; \tikzstyle{every node}=[circle, draw, fill=yellow!50] \node (f) at (5,0) {}; \node (g) at (6,0) {}; \node (h) at (7,.5) {}; \node (i) at (7,-.5){};
\foreach \from/\to in {f/g, g/h, g/i} \draw[-] (\from) -- (\to); \end{tikzpicture} }. \end{rmk}
\section{The cascade element and self-dual representations of $\g$} \label{sect:self-dual}
Consider the standard lattices in $\te^*_\BQ$ associated with $\Delta$~\cite[Chap.\,4,\,\S2.8]{VO}: \begin{itemize} \item \ $\mathcal Q=\bigoplus_{i=1}^n \BZ\ap_i$ -- the root lattice; \item \ $\mathcal Q^\vee=\bigoplus_{i=1}^n \BZ\ap_i^\vee$ -- the coroot lattice; \item \ $\mathcal P=\bigoplus_{i=1}^n \BZ\varpi_i$ -- the weight lattice; \item \ $\mathcal P^\vee=\bigoplus_{i=1}^n \BZ\varpi_i^\vee$ -- the coweight lattice, where $\varpi_i^\vee=2\varpi_i/(\ap_i,\ap_i)$. \end{itemize} Then $\mathcal P\supset\mathcal Q$, $\mathcal P^\vee\supset\mathcal Q^\vee$, $\mathcal P=(\mathcal Q^\vee)^*$, and $\mathcal P^\vee=\mathcal Q^*$, where $\mathcal L^*$ stands for the dual lattice of $\mathcal L$. For instance, $\mathcal P^\vee=\mathcal Q^*=\{\nu\in \te_\BQ\mid (\nu,\gamma)\in \BZ \ \ \forall \gamma\in\Delta\}$.
If $\g$ is not of type $\GR{A}{2p}$, then $x_\gK\in \mathcal P^\vee$ (Theorem~\ref{thm:spektr-fonin}). However, then $x_\gK$ does not always belong to $\mathcal Q^\vee$, and we characterise below the relevant cases. If $\eus M\subset \te_\BQ$ is finite,
then $|\eus M|:=\sum_{m\in\eus M} m$. As usual, set
$2\varrho=\sum_{\gamma\in \Delta^+}\gamma=|\Delta^+|$ and
$2\varrho^\vee=\sum_{\gamma\in \Delta^+}\gamma^\vee=|(\Delta^\vee)^+|$. Then $\mathsf h(\g)=(\varrho^\vee, \theta)+1$ and the {\it dual Coxeter number\/} of $\g$ is $\mathsf h^*=\mathsf h^*(\g):=(\varrho, \theta^\vee)+1$.
\begin{lm} \label{lm:rho-vee} One has \ $2\varrho=\sum_{i=1}^m \bigl(\mathsf h^*(\g\langle j\rangle)-1\bigr)\beta_j$ \ and \ $2\varrho^\vee=\sum_{i=1}^m \bigl(\mathsf h(\g\langle j\rangle)-1\bigr)\beta_j^\vee$. \end{lm} \begin{proof} Since $\Delta^+=\bigsqcup_{i=1}^m \gH_{\beta_i}$ and $\beta_1=\theta$, it is sufficient to prove that
$|\gH_\theta^\vee|=\bigl(\mathsf h(\g)-1\bigr)\theta^\vee$ and
$|\gH_\theta|=\bigl(\mathsf h^*(\g)-1\bigr)\theta$. Since $\gH_\theta\setminus \{\theta\}$ is the union of pairs $\{\gamma, \theta-\gamma\}$, where the roots $\gamma$ and $\theta-\gamma$ have the same length, it is clear
that $|\gH_\theta^\vee|= a\theta^\vee$ for some $a\in\BN$. Then \[
a=(\textstyle \frac{1}{2}|\gH_\theta^\vee|, \theta)=(\varrho^\vee, \theta)=\mathsf h(\g)-1. \] The proof of the second relation is similar. \end{proof}
\begin{prop} \label{prop:self-dual} The following conditions are equivalent: \begin{enumerate} \item $x_\gK\in \mathcal Q^\vee$; \item every self-dual representation of $\g$ is orthogonal. \end{enumerate} \end{prop} \begin{proof} By a classical result of Dynkin, if $\pi_\lb: \g\to \BV_\lb$ is an irreducible representation with highest weight $\lb$, then $\pi_\lb$ is self-dual if and only if $\omega_0(\lb)=-\lb$. Then $\pi_\lb$ is orthogonal if and only if $(\varrho^\vee, \lb)\in \BN$~\cite{dy50}, cf. also~\cite[Exercises\,4.2.12--13]{VO} or \cite[Chap.\,3, \S\,2.7]{t41}. Hence every self-dual representation of $\g$ is orthogonal if and only if $\varrho^\vee\in\mathcal Q^\vee$.
Therefore, it suffices to prove that $\varrho^\vee-x_\gK\in\mathcal Q^\vee$, if $\g$ is not of type $\GR{A}{2p}$. By Lemma~\ref{lm:rho-vee}, we have \[
\varrho^\vee-x_\gK=\sum_{i=1}^m \frac{\mathsf h(\g\langle j\rangle)-2}{2}{\cdot}\beta_j^\vee . \] It remains to observe that, for any simple $\g$, the Coxeter numbers $\mathsf h(\g\langle j\rangle)$, $j=1,\dots,m$, have the same parity, and if $\g$ is not of type $\GR{A}{2p}$, then all these numbers are even. \end{proof}
Using Proposition~\ref{prop:self-dual} and~\cite[Table\,3]{VO}, one readily obtains that \\[.6ex] \centerline{ $x_\gK\in \mathcal Q^\vee \ \Longleftrightarrow \ \g\in\{\GR{B}{4p-1}, \GR{B}{4p}, \GR{D}{4p}, \GR{D}{4p+1}, \GR{E}{6}, \GR{E}{8}, \GR{F}{4}, \GR{G}{2}\}$ . } \\[.6ex] Although the coefficients $[\varrho^\vee:\beta_i^\vee]$ are usually not integral, this does {\bf not} necessarily mean that $\varrho^\vee\not\in \mathcal Q^\vee$. For, the elements $\beta_1^\vee,\dots,\beta_m^\vee$ do not form a (part of a) basis for $\mathcal Q^\vee$.
\section{The cascade element as the Ooms element of a Frobenius Lie algebra} \label{sect:frob}
\noindent Given an arbitrary Lie algebra $\q$, one associates the {\it Kirillov form\/} $\gB_\eta$ on $\q$ to any $\eta\in\q^*$. By definition, if $\langle\ ,\, \rangle: \q^*\times\q\to \BC$ is the natural pairing and $x,y\in\q$, then \[
\gB_\eta(x,y)=\langle \eta, [x,y]\rangle=-\langle \ads(x)\eta,y\rangle . \] Then $\gB_\eta$ is skew-symmetric and $\Ker \gB_\eta=\q^\eta$, the stabiliser of $\eta$ in $\q$. The {\it index\/} of $\q$ is $\ind\q=\min_{\eta\in\g^*}\dim\q^\eta$, and $\q^*_{\sf reg}=\{\xi\in \q^*\mid \dim\q^\xi=\ind\q\}$ is the set of {\it regular elements\/} of $\q^*$ .
Suppose that $\q$ is {\it Frobenius\/}, i.e., there is $\xi\in\q^*$ such that $\gB_\xi$ is non-degenerate. Then $\q^\xi=\{0\}$ and $\xi\in\q^*_{\sf reg}$. The $2$-form $\gB_\xi$ yields a linear isomorphism between $\q$ and $\q^*$. Let $x_{\q,\xi}=x_\xi\in\q$ correspond to $\xi$ under that isomorphism. It is noticed by A.\,Ooms~\cite{ooms} that $\ad x_\xi$ enjoys rather interesting properties. Namely, using the non-degenerate form $\gB_\xi$, one defines the adjoint operator $(\ad x_\xi)^*: \q\to \q$. By~\cite[Theorem\,3.3]{ooms}, one has \beq \label{eq:ooms}
(\ad x_\xi)^*=1- \ad x_\xi . \eeq Therefore, if $\lb$ is an eigenvalue of $\ad x_\xi$ with multiplicity $m_\lb$, then $1-\lb$ is also an eigenvalue and $m_\lb=m_{1-\lb}$. Hence $\tr_\q(\ad x_\xi)=(\dim\q)/2$. We say that $x_\xi$ is the {\it Ooms element\/} associated with $\xi\in\q^*_{\sf reg}$. Another way to define $x_\xi$ is as follows. Since $\q^\xi=\{0\}$, we have $\q{\cdot}\xi=\q^*$. Then $x_\xi\in\q$ is the unique element such that $(\ads x_\xi){\cdot}\xi=-\xi$.
If $\q=\Lie(Q)$ is algebraic, then each element of $\q$ has the Jordan decomposition~\cite[Ch.\,3. \S\,3.7]{VO}. Furthermore, if $\q$ is Frobenius and algebraic, then $\q^*_{\sf reg}$ is the dense $Q$-orbit in $\q^*_{\sf reg}$. Therefore, all Ooms elements in $\q$ are $Q$-conjugate, and we can also write $x_\xi=x_\q$ for an Ooms element in $\q$. \begin{lm} \label{lm:ooms-ss} If\/ $\q$ is Frobenius and algebraic, then any Ooms element $x_\xi$ is semisimple. \end{lm} \begin{proof} For the Jordan decomposition $x_\xi=(x_\xi)_s+(x_\xi)_n$, the defining relation $(\ads x_\xi){\cdot}\xi=-\xi$ obviously implies that $\ads ((x_\xi)_s){\cdot}\xi=-\xi$. Then the uniqueness of the Ooms element associated with $\xi$ shows that $x_\xi=(x_\xi)_s$ is semisimple. \end{proof} Let $\spec (x_\xi)$ denote the multiset of eigenvalues of $\ad x_\xi$ in $\q$. In other words, if $\lb$ is an eigenvalue and $\q(\lb)$ is the corresponding eigenspace, then $\lb\in \spec (x_\xi)$ is taken with multiplicity $\dim\q(\lb)$. Then it follows from \eqref{eq:ooms} that $\spec (x_\xi)$ is symmetric w.r.t. $1/2$.
In~\cite{p22}, we defined the {\it Frobenius envelope\/} of the nilradical $\p^{\sf nil}$ of any standard parabolic subalgebra $\p\subset \g$. For $\ut=\be^{\sf nil}$, this goes as follows. Set $\te_\gK=\langle \beta_1,\dots,\beta_m\rangle_\BC=\langle\gK\rangle_\BC\subset\te$. In more direct terms, $\te_\gK=\bigoplus_{i=1}^m [e_{\beta_i}, e_{-\beta_i}]$. Then $\te_\gK$ is an algebraic subalgebra of $\te$, and the Frobenius envelope of $\ut$ is $\be_\gK=\te_\gK\oplus \ut$, which is
an ideal of $\be$. Note that $\be_\gK=\be$ if and only if $\#\gK=\rk\g$. By~\cite[Proposition\,5.1]{p22}, we have $\ind\be_\gK=0$, i.e., $\be_\gK$ is Frobenius. Here $\be^*_\gK\simeq \g/\be_\gK^\perp$ can be identified with $\be^-_\gK=\te_\gK\oplus\ut^-$ as vector space and $\te$-module. Furthermore, under this identification, we have \[
\xi_\gK=\sum_{\beta\in\gK}e_{-\beta}\in (\be_\gK)^*_{\sf reg}. \] Therefore, $(\ads x_\gK)\xi_\gK=-\xi_\gK$, i.e., the Ooms element $x_{\xi_\gK}$ associated with $\xi_\gK$ is nothing but the cascade element $x_\gK$ from Section~\ref{sect:comb-prop}. Hence $\spec (x_\gK)$ is symmetric w.r.t. $1/2$. Note that since $\te_\gK\subset \be_\gK(0)$, $\bigoplus_{\beta\in\gK}\g_\beta\subset \be_\gK(1)$, and $\dim\te_\gK=\dim (\sum_{\beta\in\gK}\g_\beta)=\#\gK$, the symmetry of the multiset $\spec (x_\gK)$ w.r.t. $1/2$ is equivalent to the symmetry established in Lemma~\ref{lm:symmetry-value}.
In this situation, we have \[
\frac{1}{2}(\dim\ut +\#\gK)= \frac{1}{2}\dim\be_\gK=
\tr_{\be_\gK}(\ad x_\gK)=\sum_{\gamma>0}\gamma(x_\gK)=2\varrho(x_\gK) . \] Since $\ind\ut=\#\gK=m$~\cite{jos77}, the sum $ \frac{1}{2}(\dim\ut +\#\gK)$ is the {\it magic number} associated with $\ut$. Comparing this with Lemma~\ref{lm:rho-vee}, we obtain \[
\frac{1}{2}(\dim\ut +m)=2\varrho(x_\gK)=\sum_{i=1}^m ((\mathsf h^*(\g\langle j\rangle)-1) . \] \begin{rmk} The case of $\g=\mathfrak{sl}_{2n+1}$ in Theorem~\ref{thm:spektr-fonin} shows that the eigenvalues of the Ooms element for the Frobenius algebra $\be_\gK$ are not always integral. Nevertheless, there are interesting classes of Frobenius algebras $\q$ such that $\spec (x_{\q})\subset \BZ$. Using meander graphs of type $\GR{A}{n}$~\cite{dk00} or $\GR{C}{n}$~\cite{py17}, I can explicitly describe the Ooms element $x_\p$ for any Frobenius {\it seaweed subalgebra\/} $\p$ of $\slno$ or $\spn$ and then prove that the eigenvalues of $\ad x_{\p}$ belong to $\BZ$. However, being symmetric with respect to $1/2$ and integral, the eigenvalues of such $x_\p$ do not always confine to the interval $[-1,2]$. \end{rmk}
\section{The abelian ideal of $\be$ associated with the cascade} \label{sect:ab-ideal}
If $\g$ is not of type $\GR{A}{2p}$, then $x_\gK\in\te$ has the property that $\{\gamma(x_\gK)\mid \gamma\in\Delta^+\}\subset \{-1,0,1,2\}$ (Theorem~\ref{thm:spektr-fonin}). By Kostant's extension of Peterson's theory~\cite[Section\,3]{ko98}, every such element of $\te$ determines an abelian ideal of $\be$. In particular, one may associate an abelian ideal of $\be$ to $x_\gK$ (i.e., to $\gK$) as long as $\g$ is not of type $\GR{A}{2p}$. Our goal is to characterise this ideal. If $\ah$ is an abelian ideal of $\be$, then $\ah$ is $\te$-stable and $\ah\subset \ut^+$. Hence it suffices to determine the set of positive roots $\Delta(\ah)\subset\Delta^+$.
Recall that the {\it inversion set\/} of $w\in W$ is $\eus N(w)=\{\gamma\in\Delta^+\mid w(\gamma)\in \Delta^-\}$. Then $\eus N(w)$ and $\Delta^+\setminus \eus N(w)$ are {\it closed}\/ (under root addition), i.e., if $\gamma',\gamma''\in \eus N(w)$ and $\gamma'+\gamma''\in\Delta^+$, then $\gamma'+\gamma''\in \eus N(w)$, and likewise for $\Delta^+\setminus \eus N(w)$. Conversely, if $S\subset \Delta^+$ and both $S$ and $\Delta^+\setminus S$ are closed, then $S=\eus N(w)$ for a unique $w\in W$~\cite[Prop.\,5.10]{ko61}. Below, $\gamma>0$ (resp. $\gamma<0$) is a shorthand for $\gamma\in\Delta^+$ (resp. $\gamma\in\Delta^-$).
{\it\bfseries Kostant's construction}. Set $\De_{\sf ab}=\{t\in \te_\BQ\mid -1\leqslant \gamma(t)\leqslant 2\ \ \forall \gamma\in\Delta^+\}$. Kostant associates the abelian ideal $\ah_z{\lhd}\be$ to each $z\in\De_{\sf ab}\cap\mathcal P^\vee$~\cite[Theorem\,3.2]{ko98}, i.e., if $\gamma(z)\in\{-1,0,1,2\}$ for any $\gamma\in\Delta^+$.
Unlike the Peterson method, his construction exploits only $W$ and does not invoke the affine Weyl group $\widehat W$ and "minuscule" elements in it. Set $\Delta^\pm_z(i)=\{\gamma\in\Delta^\pm\mid \gamma(z)=i\}$ and $\Delta_z(i)=\Delta^+_z(i)\cup\Delta^-_z(i)$. Note that \ $-\Delta^+_z(i)=\Delta^-_z(-i)$. We have \beq \label{eq:main-partition}
\Delta^+=\bigsqcup_{i=-1}^2\Delta^+_z(i) \eeq and both subsets $\Delta^+_z(1)\sqcup \Delta^+_z(2)$ and $\Delta^+_z(-1)\sqcup\Delta^+_z(0)$ are closed. Therefore, there is a unique $w_z\in W$ such that $\eus N(w_z)=\Delta^+_z(1)\sqcup \Delta^+_z(2)$. By definition, then \[
\Delta(\ah_z)=w_z\bigl(\Delta^+_z(-1)\bigr)\sqcup w_z\bigl(-\Delta^+_z(2)\bigr) \subset \Delta^+. \] Hence $\dim\ah_z=\#\bigl(\Delta^+_z(-1)\cup \Delta^+_z(2)\bigr)$. Note that $\Delta_z(0)$ is a root system in its own right, and $\Delta^+_z(0)$ is a set of positive roots in it. We say that the union in~\eqref{eq:main-partition} is the $z$-{\it grading}\/ of $\Delta^+$, and if $\gamma\in \Delta^+_z(i)$, then $i$ is the $z$-{\it degree}\/ of $\gamma$.
\begin{prop} \label{lm:good-prop-w_K} Let $z\in \De_{\sf ab}\cap\mathcal P^\vee$ be arbitrary. \begin{itemize} \item[\sf (i)] \ If\/ $\gamma\in \Delta^+_z(0)$ is not a sum of two roots from $\Delta^+_z(0)$, then $w_z(\gamma)\in \Pi$; \item[\sf (ii)] \ if\/ $\theta(z)=1$, i.e., $\theta\in \Delta^+_z(1)$, then $w_z(\theta)\in -\Pi$. \end{itemize} \end{prop} \begin{proof} Recall that $\eus N(w_z)=\Delta^+_z(1)\cup\Delta^+_z(2)$. \\ \indent {{\sf (i)} \ Since $\gamma\not\in \eus N(w_z)$, we have $w_z(\gamma)>0$. Assume that $w_z(\gamma)=\mu_1+\mu_2$, where $\mu_i>0$. Then $\gamma=w_z^{-1}(\mu_1)+ w_z^{-1}(\mu_2)$. Letting $\gamma_i:=w_z^{-1}(\mu_i)$, we get the following possibilities.
1$^o$. If $\gamma_1,\gamma_2>0$, then the $z$-grading of $\Delta^+$ shows that there are further possibilities: \begin{itemize} \item $\gamma_1\in\Delta^+_z(-1)$ and $\gamma_2\in\Delta^+_z(1)$. But then $\mu_2=w_z(\gamma_2)<0$. A contradiction! \item $\gamma_1,\gamma_2 \in\Delta^+_z(0)$ -- this contradicts the hypothesis on $\gamma$. \end{itemize}
2$^o$. Suppose that $\gamma_1>0$ and $\gamma_2<0$. Then $\gamma_1\not\in\eus N(w_z)$ and $-\gamma_2\in \eus N(w_z)$. \\ Hence $\gamma_1\in \Delta^+_z({\leqslant}0)$ and $-\gamma_2\in \Delta^+_z({\geqslant}1)$, i.e., $\gamma_2\in\Delta^-_z({\leqslant}-1)$. It follows that $\gamma_1+\gamma_2=\gamma\in \Delta_z({\leqslant}-1)$. A contradiction!
Thus, cases~1$^o$ and 2$^o$ are impossible, and $w_z(\gamma)$ must be simple.
\sf (ii)} \ Since $\theta(z)=1$, we have $w_z(\theta)<0$. Assume that $w_z(\theta)=-\gamma_1-\gamma_2$, where $\gamma_i>0$. Then $\theta=w_z^{-1}(-\gamma_1)+w_z^{-1}(-\gamma_2)$ and $\mu_i:=w_z^{-1}(-\gamma_i)>0$ for $i=1,2$. Then $\mu_i\in \eus N(w_z)$ and hence $\mu_i(z)\geqslant 1$. Therefore $\theta(z)=\mu_1(z)+\mu_2(z)\geqslant 2$. A contradiction! \end{proof}
\begin{rmk} \label{rem:kak-stroit-nerazlozhimye} Note that $\gamma\in \Delta^+_z(0)$ is not a sum of two roots from $\Delta^+_z(0)$ if and only if $\gamma$ belongs to the {\it base} (=\,set of simple roots) of $\Delta_z(0)$ that is contained in $\Delta_z^+(0)$. \end{rmk} Let $\gC\subset\te_\BQ$ denote the {\it dominant Weyl chamber}, i.e., $\gC=\{x\in\te_\BQ\mid \ap(x)\geqslant 0 \ \ \forall \ap\in\Pi\}$. \begin{prop} \label{prop:anti-dom} We have $w_z(z)\in -\gC$, i.e., it is anti-dominant. Moreover, $w_z$ is the unique element of minimal lenght in\/ $W$ that takes $z$ into $-\gC$. \end{prop} \begin{proof} For any $\lb\in\te^*_\BQ$, let $\lb^+$ be the {\sl dominant\/} representative in $W{\cdot}\lb$. By~\cite[Lemma\,4.1]{ion04}, there is a unique element of minimal length $w_\lb\in W$ such that $w_\lb{\cdot}\lb=\lb^+$ and then $\eus N(w_\lb)=\{\gamma\in\Delta^+\mid (\gamma,\lb)<0\}$ (cf. also \cite[Theorem\,4.1]{p03}). Translating this into the assertion on the {\sl anti-dominant\/} representative in $W{\cdot}z\subset\te_\BQ$, we see that the element of minimal length $\bar w$ that takes $z$ into $-\gC$ is defined the property that \[
\eus N(\bar w)=\{\gamma\in\Delta^+\mid (\gamma, z)>0\}=\Delta^+_z(1)\cup\Delta^+_z(2). \] Therefore, $\bar w=w_z$. \end{proof}
We will use Kostant's construction with $z=x_\gK$. Therefore, it is assumed below that $\spec (x_\gK)\subset \BZ$, which excludes the series $\GR{A}{2p}$. For simplicity, write $\Delta^+_\gK(i)$, $\ah_\gK$, and $w_\gK$ in place of $\Delta^+_{x_\gK}(i)$, $\ah_{x_\gK}$, and $w_{x_\gK}$, respectively. By the symmetry of $\spec (x_\gK)$, one has $\dim\ah_\gK=\#\Delta^+_\gK(-1)+\#\Delta^+_\gK(2)=2\#\Delta^+_\gK(2)$ and $\#\Delta^+_\gK(0)+\#\gK=\#\Delta^+_\gK(1)$. Set $\Pi_\gK(i)=\Pi\cap\Delta^+_\gK(i)$.
\begin{ex} \label{ex:ideal-sl-sp} For $\sltn$ and $\spn$, the element $x_\gK$ is dominant and $\Delta^+_\gK(2)=\varnothing$. Therefore, $\ah_\gK=\{0\}$ in these cases. In the other cases, i.e., when $\theta$ is fundamental, $\ah_\gK$ is a non-trivial abelian ideal of $\be$. Anyway, for {\bf all} simple Lie algebras except $\mathfrak{sl}_{2n+1}$, we obtain a non-trivial element $w_\gK\in W$, which possesses some interesting properties, see below. \end{ex}
\begin{prop} \label{prop:fundam} The rank of the root system $\Delta_\gK(0)$ equals $\rk\g-1$ and the dominant weight $-w_\gK(x_\gK)$ is a multiple of a fundamental weight. Namely, if $w_\gK(\theta)=-\ap_j\in -\Pi$ (cf. Proposition~\ref{lm:good-prop-w_K}), then $-w_\gK(x_\gK)=\frac{2}{(\ap_j,\ap_j)}\varpi_j=:\varpi_j^\vee$. \end{prop} \begin{proof} Clearly, $-w_\gK(x_\gK)$ is a multiple of a fundamental weight if and only if the rank of the root system $\Delta_{\gK}(0)=\{\gamma\in\Delta\mid \gamma(x_\gK)=0\}$ equals $\rk\g-1$. Therefore, it suffices to point out $\rk\g-1$ linearly independent roots in $\Delta^+_\gK(0)$.
Since $\Pi_{\pm 1}:=\Pi_\gK(-1)\cup\Pi_\gK(1)$ is connected and the roots from $\Pi_\gK(-1)$ and $\Pi_\gK(1)$ alternate in the Dynkin diagram (see Remark~\ref{rem:alternate&connected}(2)), there are $(\#\Pi_{\pm 1})-1$ edges therein and each edge gives rise to the root $\ap_{i_1}+\ap_{i_2}\in \Delta^+_\gK(0)$. Together with the roots in $\Pi_\gK(0)$, this yields exactly $\rk\g-1$ linearly independent roots in $\Delta^+_\gK(0)$. Actually, these roots form the base of $\Delta_\gK(0)$ in $\Delta^+_\gK(0)$.
If $w_\gK(x_\gK)=-a_i\varpi_i$ and $w_\gK(\theta)=-\ap_j$, then \[
1=\theta(x_\gK)= w_\gK(\theta)(w_\gK(x_\gK))=a_i(\ap_j, \varpi_i) . \] Hence $i=j$ and $a_i=2/(\ap_i,\ap_i)$. \end{proof}
Important characteristics of the abelian ideal $\ah_\gK$ can be expressed via $w_\gK\in W$. Since $\Delta(\ah_\gK)=:\Delta_{\langle\gK\rangle}$ is an upper ideal of the poset $(\Delta^+,\preccurlyeq)$, it is completely determined by its subset $\min(\Delta_{\langle\gK\rangle})$ of {\it minimal elements\/} w.r.t. the root order ``$\preccurlyeq$''. Similarly, the complement $\ov{\Delta_{\langle\gK\rangle}}=\Delta^+\setminus \Delta_{\langle\gK\rangle}$ is determined by the subset of its {\it maximal elements}, $\max(\ov{\Delta_{\langle\gK\rangle}})$.
It follows from~\eqref{eq:main-partition} and the definition of $\eus N(w_\gK)$ that \[
\Delta^+=w_\gK\bigl(\Delta^+_\gK(-1)\bigr)\sqcup w_\gK\bigl(\Delta^+_\gK(0)\bigr)\sqcup
w_\gK\bigl(-\Delta^+_\gK(1)\bigr) \sqcup w_\gK\bigl(-\Delta^+_\gK(2)\bigr) . \] In this union, the first and last sets form $\Delta_{\langle\gK\rangle}$, and two sets in the middle form $\ov{\Delta_{\langle\gK\rangle}}$. Hence \begin{gather} \label{eq:delta}
w_\gK^{-1}(\Delta_{\langle\gK\rangle}) = \Delta^+_\gK(-1)\sqcup -\Delta^+_\gK(2)=
\Delta^+_\gK(-1)\sqcup\Delta^-_\gK(-2) \ ; \\ \label{eq:bar-delta}
w_\gK^{-1}(\ov{\Delta_{\langle\gK\rangle}}) = \Delta^+_\gK(0)\sqcup -\Delta^+_\gK(1)=
\Delta^+_\gK(0)\sqcup \Delta^-_\gK(-1) \ . \end{gather}
\begin{thm} \label{thm:min_K} One has \ $\min(\Delta_{\langle\gK\rangle})= w_\gK(\Pi_\gK(-1))$. In other words, \[ \gamma\in\min(\Delta_{\langle\gK\rangle})\Longleftrightarrow w_\gK^{-1}(\gamma)\in\Pi_\gK(-1) . \] \end{thm} \begin{proof} We repeatedly use the following observation. For $\gamma\in\Delta_{\langle\gK\rangle}$, it follows from~\eqref{eq:delta} that if $w_\gK^{-1}(\gamma)>0$, then $w_\gK^{-1}(\gamma)\in \Delta^+_\gK(-1)$; whereas if $w_\gK^{-1}(\gamma)<0$, then $w_\gK^{-1}(\gamma)\in -\Delta^+_\gK(2)$.
1$^o$. If $w_\gK^{-1}(\gamma)=\ap\in\Pi_\gK(-1)$, then $\gamma\in\Delta_{\langle\gK\rangle}$. Assume further that $\gamma$ is not a minimal element of $\Delta_{\langle\gK\rangle}$, i.e., $\gamma=\gamma'+\mu$ for some $\gamma'\in\Delta_{\langle\gK\rangle}$ and $\mu> 0$. Then $\ap=w_\gK^{-1}(\gamma')+w_\gK^{-1}(\mu)$ and there are two possibilities: \begin{description} \item[(a)] \ $w_\gK^{-1}(\gamma')<0$ and $w_\gK^{-1}(\mu)>0$; \item[(b)] \ $w_\gK^{-1}(\gamma')>0$ and $w_\gK^{-1}(\mu)<0$. \end{description} For {\bf (a)}: By~\eqref{eq:delta} and~\eqref{eq:bar-delta}, one has $w_\gK^{-1}(\gamma')\in \Delta^-_\gK(-2)$ and $w_\gK^{-1}(\mu)\in \Delta^+_\gK(0)\cup\Delta^+_\gK(-1)$. Then their sum belongs to $\Delta_\gK({\leqslant}-2)=\Delta_{\gK}(-2)$, and this cannot be $\ap\in \Delta^+_\gK(-1)$. Hence this case is impossible. \\ For {\bf (b)}: By~\eqref{eq:delta} and~\eqref{eq:bar-delta}, one has $w_\gK^{-1}(\gamma')\in \Delta^+_\gK(-1)$ and $w_\gK^{-1}(\mu)\in \Delta^-_\gK(-2)\cup\Delta^-_\gK(-1)$. Then their sum again belongs to $\Delta_{\gK}(-2)$, which is impossible, too.
Thus, $\gamma=w_\gK(\ap)$ must be a minimal element of $\Delta_{\langle\gK\rangle}$.
2$^o$. Suppose that $\gamma\in\Delta_{\langle\gK\rangle}$ and $w_\gK^{-1}(\gamma)\not\in \Pi_\gK(-1)$. By~\eqref{eq:delta}, there is a dichotomy: \begin{description} \item[(a)] \ $w_\gK^{-1}(\gamma)>0$; \item[(b)] \ $w_\gK^{-1}(\gamma)<0$. \end{description}
For {\bf (a)}: By~\eqref{eq:delta}, one has $w_\gK^{-1}(\gamma)\in\Delta^+_\gK(-1)$. Since $w_\gK^{-1}(\Delta_{\langle\gK\rangle})\cap\Pi=\Pi_\gK(-1)$, we have $w_\gK^{-1}(\gamma)=\mu_1+\mu_2$ for some $\mu_1,\mu_2>0$. From the $x_\gK$-grading of $\Delta^+$, we deduce that $\mu_1\in\Delta^+_\gK(-1)$ and $\mu_2\in\Delta^+_\gK(0)$. Letting $\gamma_i=w_\gK(\mu_i)$, we see that $\gamma_1\in\Delta_{\langle\gK\rangle}$ and $\gamma_2>0$. Hence $\gamma=\gamma_1+\gamma_2$ is not a minimal element of $\Delta_{\langle\gK\rangle}$. \\ \indent For {\bf (b)}: Here $w_\gK^{-1}(\gamma)\in -\Delta^+_\gK(2)$. Since $\theta\in\Delta^+_\gK(1)$, we have $w_\gK^{-1}(\gamma)\ne -\theta$ and hence $w_\gK^{-1}(\gamma)=\nu_1-\nu_2$ for some $\nu_1,\nu_2>0$. Using the $x_\gK$-grading of $\Delta^+$, one again encounters two possibilities: \begin{itemize} \item[\sf (i)] \ $\nu_1\in \Delta^+_\gK(-1), \nu_2\in \Delta^+_\gK(1)$; \item[\sf (ii)] \ $\nu_1\in \Delta^+_\gK(0), \nu_2\in \Delta^+_\gK(2)$. \end{itemize} In both cases, $\gamma=w_\gK(\nu_1)+w_\gK(-\nu_2)$ and one of the summands lies in $\Delta_{\langle\gK\rangle}$, while the other is positive.
Hence $\gamma\not\in\min(\Delta_{\langle\gK\rangle})$. \end{proof}
\begin{thm} \label{thm:max_K} One has \ $\max(\ov{\Delta_{\langle\gK\rangle}})= -w_\gK(\Pi_\gK(1))$. In other words, \[ \gamma\in\max(\ov{\Delta_{\langle\gK\rangle}})\Longleftrightarrow -w_\gK^{-1}(\gamma)\in\Pi_\gK(1) . \] \end{thm} \begin{proof} To a great extent, the proof is analogous to that of Theorem~\ref{thm:min_K}, and we skip similar arguments.
1$^o$. If $-w_\gK^{-1}(\gamma)\in\Pi_\gK(1)$, then an argument similar to that in part 1$^o$ of Theorem~\ref{thm:min_K} proves that $\gamma\in\max(\ov{\Delta_{\langle\gK\rangle}})$.
2$^o$. Conversely, suppose that $\gamma\in\ov{\Delta_{\langle\gK\rangle}}$ and $-w_\gK^{-1}(\gamma)\not\in \Pi_\gK(1)$. In view of~\eqref{eq:bar-delta}, one has to handle two possibilities for $w_\gK^{-1}(\gamma)$.
{\bf (a)}: If $w_\gK^{-1}(\gamma)>0$, then $w_\gK^{-1}(\gamma)\in\Delta^+_\gK(0)$. Arguing as in the proof of Theorem~~\ref{thm:min_K} (part 2$^o${\bf (a)}), we show that $\gamma=\gamma_1-\gamma_2$, where $\gamma_1\in \ov{\Delta_{\langle\gK\rangle}}$ and $\gamma_2>0$. Hence $\gamma\not\in\max(\ov{\Delta_{\langle\gK\rangle}})$.
{\bf (b)}: If $w_\gK^{-1}(\gamma)<0$, then $-w_\gK^{-1}(\gamma)\in\Delta^+_\gK(1)\setminus\Pi_\gK(1)$. Hence $-w_\gK^{-1}(\gamma)=\nu_1+\nu_2$ for some $\nu_1,\nu_2>0$. There again are two possibilities: \begin{enumerate} \item \ $\nu_1\in \Delta^+_\gK(0)$, \ \ $\nu_2\in \Delta^+_\gK(1)$; \ \quad [$(0,1)$-decomposition of \ $-w_\gK^{-1}(\gamma)$] \item \ $\nu_1\in \Delta^+_\gK(-1)$, $\nu_2\in \Delta^+_\gK(2)$. \quad [$(-1,2)$-decomposition of \ $-w_\gK^{-1}(\gamma)$] \end{enumerate}
In case (1), we get $\gamma=\gamma_2-\gamma_1$, where $\gamma_2:=-w_\gK(\nu_2)\in \ov{\Delta_{\langle\gK\rangle}}$ and $\gamma_1=w_\gK(\nu_1)>0$. Hence $\gamma\not\in\max(\ov{\Delta_{\langle\gK\rangle}})$.
In case (2), one similarly obtains the presentation of $\gamma$ as difference of two elements of $\Delta_{\langle\gK\rangle}$, which is useless for us. However, one can replace such a $(-1,2)$-decomposition of $-w_\gK^{-1}(\gamma)$ with a $(0,1)$-decomposition, which is sufficient. Since $\nu_2\in \Delta^+_\gK(2)$, the root $\nu_2$ is long (Remark~\ref{rem:discussion}(3)) and not simple (for, $\Pi_\gK(2)=\varnothing$). Hence $(\nu_1,\nu_2)<0$ and $\nu_2=\nu_2'+\nu_2''$ with $\nu_1',\nu_2''>0$. W.l.o.g., we may assume that $(\nu_1,\nu'_2)<0$. One has three possibilities for the $x_\gK$-degrees of $(\nu_1',\nu_2'')$, i.e., $(1,1), (2,0), (0,2)$, and it is easily seen that $-w_\gK^{-1}(\gamma)=(\nu_1+\nu'_2)+\nu''_2$ is either a $(0,1)$-decomposition, or still a $(-1,2)$-decomposition, with $\nu''_2\in\Delta^+_\gK(2)$. But in this last case we have $\htt(\nu''_2)< \htt(\nu_2)$, which provides the induction step. \end{proof}
\begin{rmk} \label{rem:verno-for-all-z}
Theorem~\ref{thm:min_K} holds for {\sl arbitrary} $z\in\De_{\sf ab}\cap\mathcal P^\vee$ in place of $x_\gK$, with certain amendments. That is, if $\theta(z)\leqslant 1$, then the statement and the proof remain the same. If $\theta(z)=2$, then $-w_z(\theta)$ has to be added to $\min\Delta(\ah_z)$. Certain complements of similar nature are also required in Theorem~\ref{thm:max_K}. I hope to elaborate on this topic in a subsequent publication.
\end{rmk}
\section{An explicit description of $w_\gK\in W$ and the ideals $\ah_\gK$} \label{sect:w_K&perebor}
\noindent The element $w_\gK\in W$ and the abelian ideal $\ah_\gK$ have many interesting properties, which can be verified case-by-case. To this end, we need explicit formulae for $w_\gK$. Our main tool is the following \begin{thm} \label{thm:w^-1} Given $z\in\De_{\sf ab}\cap\mathcal P^\vee$, suppose that $\rk\Delta_z(0)=\rk\Delta-1$ and $\theta(z)=1$. Then \[
w_z^{-1}(\Pi)=\{\text{\normalfont the base of } \Delta_z(0) \ \text{ \normalfont in } \ \Delta^+_z(0)\}\cup \{-\theta\} . \] \end{thm} \begin{proof} Set $p=\rk\Delta$, and let $\nu_1,\dots,\nu_{p-1}\in \Delta^+_z(0)$ be the base of $\Delta_z(0)$. By Proposition~\ref{lm:good-prop-w_K}, we have $w_z(\nu_i)\in \Pi$ ($i=1,\dots,p{-}1$) and $w_z(\theta)\in -\Pi$. The assertion follows. \end{proof} By Proposition~\ref{prop:fundam}, Theorem~\ref{thm:w^-1} applies to $z=x_\gK$. We demonstrate below how to use this technique for finding $w_\gK\in GL(\te)$.
\begin{ex} \label{ex:how-find-w_K} (1) \ Let $\g$ be of type $\GR{D}{2n}$. Then $\Pi_\gK(0)=\varnothing$ and the base of $\Delta^+_\gK(0)$ corresponds to the edges of the Dynkin diagram, i.e., it consists of $\ap_1+\ap_2, \ap_2+\ap_3,\dots, \ap_{2n-2}+\ap_{2n-1}, \ap_{2n-2}+\ap_{2n}$, cf. the diagram for $\GR{D}{2n}$ in Fig.~\ref{fig:class}. Therefore, the root system $\Delta_\gK(0)$ is of type $\GR{A}{n-1}+\GR{D}{n}$. More precisely,
$\{\ap_1+\ap_2, \ap_3+\ap_4, \dots, \ap_{2n-3}+\ap_{2n-2}\}$ is a base for $\Delta(\GR{A}{n-1})$;
$\{\ap_2+\ap_3,\ap_4+\ap_5,\dots, \ap_{2n-4}+\ap_{2n-3}, \ap_{2n-2}+\ap_{2n-1},\ap_{2n-2}+\ap_{2n}\}$ is a base for $\Delta(\GR{D}{n})$. \\ Since the Dynkin diagram of $\GR{A}{n-1}+\GR{D}{n}$ is obtained by removing the node $\ap_n$ from the Dynkin diagram of $\GR{D}{2n}$, we must have $w_\gK^{-1}(\ap_n)=-\theta$. Then an easy argument shows that \begin{itemize} \item $w_\gK^{-1}(\ap_1)=\ap_{2n-3}+\ap_{2n-2},\dots, w_\gK^{-1}(\ap_{n-2})=\ap_{3}+\ap_{4}, \ w_\gK^{-1}(\ap_{n-1})=\ap_{1}+\ap_{2}$ \ -- \\for the $\GR{A}{n-1}$-part; \item $w_\gK^{-1}(\ap_{n+1})=\ap_{2}+\ap_{3}, \dots, w_\gK^{-1}(\ap_{2n-2})=\ap_{2n-4}+\ap_{2n-3}, \\ w_\gK^{-1}(\{\ap_{2n-1},\ap_{2n}\})=\{\ap_{2n-2}+\ap_{2n-1}, \ \ap_{2n-2}+\ap_{2n-1}\}$ \ -- \ for the $\GR{D}{n}$-part; \end{itemize} The only unclear point for the $\GR{D}{n}$-part is how to distinguish $w_\gK^{-1}(\ap_{2n-1})$ and $w_\gK^{-1}(\ap_{2n})$. Using the expressions of simple roots of $\GR{D}{2n}$ via $\{\esi_i\}$, $i=1,\dots,2n$, we obtain two possibilities for $w_\gK$ as a signed permutation on $\{\esi_i\}$, where the only ambiguity concerns the sign of transformation $\esi_{2n}\mapsto \pm\esi_{2n}$. We then choose this sign so that the total number of minuses be even, see Example~\ref{ex:w_K-so} below. \\ \indent (2) Similar argument works for the other orthogonal series. \\ \indent (3) For the exceptional Lie algebras, a certain ambiguity (due to the symmetry of the Dynkin diagram) occurs only for $\GR{E}{6}$. \end{ex}
For the classical cases, our formulae for $w_\gK$ use the explicit standard models of $W$ as (signed) permutations on the set of $\{\esi_i\}$. Write $\text{ord}(w_\gK(\g))$ for the order of $w_\gK=w_\gK(\g)$. \begin{ex} \label{ex:w_K-so} Here we provide formulae for $w_\gK$ if $\g$ is an orthogonal Lie algebra. \\ 1$^o$. If $\g$ is of type $\GR{D}{2n}$, then
$w_\gK$: \ $\left( \text{\begin{tabular}{ccccc|ccccc} $\esi_1$ & $\esi_3$ & $\dots$ & $\esi_{2n-3}$ & $\esi_{2n-1}$ & $\esi_2$ & $\esi_4$ & $\dots$ & $\esi_{2n-2}$ & $\esi_{2n}$ \\ $\downarrow$ & $\downarrow$ & $\dots$ & $\downarrow$ & $\downarrow$ & $\downarrow$ & $\downarrow$ & $\dots$ & $\downarrow$ & $\downarrow$ \\ $-\esi_n$ & $-\esi_{n-1}$ & $\dots$ & $-\esi_{2}$ & $-\esi_{1}$ & $\esi_{n+1}$ & $\esi_{n+2}$ & $\dots$ & $\esi_{2n-1}$ & $(-1)^n\esi_{2n}$ \\ \end{tabular}}\right)$ \\[.8ex] The last sign is determined by the condition that the total number of minuses must be even for type $\GR{D}{N}$. \\[.6ex] 2$^o$. For $\g$ of type $\GR{B}{2n-1}$, one should merely omit the last column in the previous array. \\[.6ex] 3$^o$. If $\g$ is of type $\GR{D}{2n+1}$, then the following adjustment works:
$w_\gK$: \ $\left( \text{\begin{tabular}{ccccc|cccccc} $\esi_1$ & $\esi_3$ & $\dots$ & $\esi_{2n-3}$ & $\esi_{2n-1}$ & $\esi_2$ & $\esi_4$ & $\dots$ & $\esi_{2n-2}$ & $\esi_{2n}$ & $\esi_{2n+1}$ \\ $\downarrow$ & $\downarrow$ & $\dots$ & $\downarrow$ & $\downarrow$ & $\downarrow$ & $\downarrow$ & $\dots$ & $\downarrow$ & $\downarrow$ & $\downarrow$ \\ $-\esi_n$ & $-\esi_{n-1}$ & $\dots$ & $-\esi_{2}$ & $-\esi_{1}$ & $\esi_{n+1}$ & $\esi_{n+2}$ & $\dots$ & $\esi_{2n-1}$ & $\esi_{2n}$ & $(-1)^n\esi_{2n+1}$\\ \end{tabular}}\right)$ \\[1ex] 4$^o$. For $\g$ of type $\GR{B}{2n}$, one should merely omit the last column in the previous array.
\textbullet \ \ It follows that in all four cases $w_\gK(\theta)=w_\gK(\esi_1+\esi_2)=-\esi_n+\esi_{n+1}=-\ap_n$, which agrees with Lemma~\ref{lm:good-prop-w_K}{\sf (ii)}.
\textbullet \ \ For $\GR{D}{2n+1}$, one has $\Pi_\gK(0)=\{\ap_{2n},\ap_{2n+1}\}$ (see Fig.~\ref{fig:class}) and $w_\gK$ takes $\Pi_\gK(0)$ to itself. Recall that here $\ap_{2n}=\esi_{2n}-\esi_{2n+1}$ and $\ap_{2n+1}=\esi_{2n}+\esi_{2n+1}$. The same happens for $\GR{B}{2n}$, where $\Pi_\gK(0)=\{\ap_{2n}=\esi_{2n}\}$, cf. Lemma~\ref{lm:good-prop-w_K}{\sf (i)}. \end{ex}
\begin{ex} \label{ex:w_K-sl-sp} Here we provide formulae for $w_\gK$ if $\g=\sltn$ or $\spn$. \\ 1$^o$. If $\g$ is of type $\GR{A}{2n-1}$, then \\[.6ex]
\centerline{$w_\gK$: \ $\left( \text{\begin{tabular}{ccccc|ccccc} $\esi_1$ & $\esi_2$ & $\dots$ & $\esi_{n-1}$ & $\esi_{n}$ & $\esi_{n+1}$ & $\esi_{n+2}$ & $\dots$ & $\esi_{2n-1}$ & $\esi_{2n}$ \\ $\downarrow$ & $\downarrow$ & $\dots$ & $\downarrow$ & $\downarrow$ & $\downarrow$ & $\downarrow$ & $\dots$ & $\downarrow$ & $\downarrow$ \\ $\esi_{n+1}$ & $\esi_{n+2}$ & $\dots$ & $\esi_{2n-1}$ & $\esi_{2n}$ & $\esi_{1}$ & $\esi_2$ & $\dots$ & $\esi_{n-1}$ & $\esi_{n}$ \\ \end{tabular}}\right)$}
\noindent It follows that $w_\gK(\theta)=w_\gK(\esi_1-\esi_{2n})=-\esi_n+\esi_{n+1}=-\ap_n$, which agrees with Lemma~\ref{lm:good-prop-w_K}{\sf (ii)}. Here $\Pi_\gK(0)=\Pi\setminus\{\ap_n\}$ (see Fig.~\ref{fig:class}) and $w_\gK(\ap_i)=\begin{cases} \ap_{i+n}, & i<n \\ \ap_{i-n}, & i>n \end{cases}$ \ . \\[1ex] 2$^o$. If $\g$ is of type $\GR{C}{n}$, then \begin{center} $w_\gK$: \ $\left( \text{\begin{tabular}{ccccc} $\esi_1$ & $\esi_2$ & $\dots$ & $\esi_{n-1}$ & $\esi_{n}$ \\ $\downarrow$ & $\downarrow$ & $\dots$ & $\downarrow$ & $\downarrow$ \\ $-\esi_n$ & $-\esi_{n-1}$ & $\dots$ & $-\esi_{2}$ & $-\esi_{1}$ \\ \end{tabular}}\right)$ \end{center}
Here $w_\gK(\theta)=-\ap_n$, \ $\Pi_\gK(0)=\Pi\setminus\{\ap_n\}$, and $w_\gK(\ap_i)=\ap_{n-i}$ for $i<n$.
\textbullet \ \ In both cases here, one has $w_\gK^2=1$. \end{ex} \begin{ex} \label{ex:w_K-e6&f4} 1$^o$. For $\g$ of type $\GR{F}{4}$, we write $(a_1a_2a_3a_4)$ for $\sum_{i=1}^4 a_i\ap_i$. Then $w_\gK(\ap_i)=\ap_i$ for $i=1,2$ and $w_\gK(\ap_3)=-(2421)$, $w_\gK(\ap_4)=(2431)=\theta-\ap_4$. It follows that $w_\gK(\theta)=-\ap_4$.
2$^o$. For $\g$ of type $\GR{E}{6}$, we write $(a_1a_2a_3a_4a_5a_6)$ for $\gamma=\sum_{i=1}^6 a_i\ap_i=$ \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (b) at (0,0) {$a_1$}; \node (c) at (.4,0) {$a_2$}; \node (d) at (.8,0) {$a_3$}; \node (e) at (1.2,0) {$a_4$}; \node (f) at (1.6,0) {$a_5$}; \node (g) at (.8,-.4) {$a_6$}; \end{tikzpicture}}. Then $w_\gK(\ap_i)=\ap_i$ for $i=1,2,4,5$ and $w_\gK(\ap_3)=-(122211)$, $w_\gK(\ap_6)=(123211)=\theta-\ap_6$. It follows that $w_\gK(\theta)=-\ap_6$.
3$^o$. For $\g$ of type $\GR{G}{2}$, we have $\Pi_l=\{\ap_2\}$, $w_\gK=(r_{\ap_2}r_{\ap_1})^2$, and $w_\gK(\theta)=-\ap_2$.
\textbullet \ \ In all three cases, one has $w_\gK^3=1$. \end{ex} \begin{ex} \label{ex:w_K-e7} 1$^o$. For $\GR{E}{7}$, we have $\Pi_\gK(-1)=\{\ap_2,\ap_4,\ap_6\}$ (see Fig.~\ref{fig:except}) and
{\tabcolsep=0.15em
\begin{tabular}{c|c c c c c c c|} $\ap_i$ & $\ap_1$ & $\ap_2$ & $\ap_3$ & $\ap_4$ & $\ap_5$ & $\ap_6$ & $\ap_7$ \\ \hline $w_\gK(\ap_i)$ & \rule{0pt}{3ex} \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {1}; \node at (.6,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--1}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {1}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {2}; \node (f) at (1,0) {1}; \node at (.6,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--0}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {2}; \node (f) at (1,0) {1}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {2}; \node (f) at (1,0) {1}; \node at (.6,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--1}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {2}; \node (f) at (1,0) {1}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}} \\ \end{tabular} }
\noindent It follows that $w_\gK(\theta)=-\ap_7$. A direct calculation shows that $\text{ord}(w_\gK)=18$.
2$^o$. For $\GR{E}{8}$, we have $\Pi_\gK(-1)=\{\ap_1,\ap_3,\ap_5,\ap_7\}$ (see Fig.~\ref{fig:except}) and
{\tabcolsep=0.15em
\begin{tabular}{c|c c c c c c c c|} $\ap_i$ & $\ap_1$ & $\ap_2$ & $\ap_3$ & $\ap_4$ & $\ap_5$ & $\ap_6$ & $\ap_7$ & $\ap_8$ \\ \hline $w_\gK(\ap_i)$ & \rule{0pt}{3ex} \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {4}; \node (f) at (1,0) {3}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {2}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--0}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {4}; \node (f) at (1,0) {2}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {2}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {4}; \node (f) at (1,0) {2}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {2}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--1}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {3}; \node (f) at (1,0) {2}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {2}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {3}; \node (f) at (1,0) {2}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {2}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {3}; \node (f) at (1,0) {2}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {3}; \node (d) at (.6,0) {3}; \node (e) at (.8,0) {3}; \node (f) at (1,0) {2}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {1}; \end{tikzpicture}} & \raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (-.1,0) {--1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {3}; \node (f) at (1,0) {2}; \node (g) at (1.2,0) {1}; \node (h) at (.8,-.4) {2}; \end{tikzpicture}} \\ \end{tabular} }
\noindent It follows that $w_\gK(\theta)=-\ap_7$. A direct calculation shows that $\text{ord}(w_\gK)=5$. \end{ex}
\begin{rmk} (1) We do not know a general formula for $\text{ord}(w_\gK(\soN))$. Using Example~\ref{ex:w_K-so}, it is not hard to prove that $\text{ord}(w_\gK)$ takes the same value for $\GR{B}{2n-1},\GR{D}{2n},\GR{B}{2n},\GR{D}{2n+1}$. But explicit computations, up to $n=13$, show that the function $n\mapsto \text{ord}(w_\gK(\GR{D}{2n}))$ behaves rather chaotically.
(2) If $\ap\in\Pi_\gK(0)$, then $w_\gK(\ap)\in\Pi_\gK(0)$ as well. But this does not hold for arbitrary $z\in\De_{\sf ab}$. In general, it can happen that $\ap\in\Pi_z(0)$, but $w_z(\ap)\in\Pi\setminus\Pi_z(0)$. \end{rmk} The main result of this section is an explicit uniform description of $\Delta(\ah_\gK)$.
\begin{thm} \label{thm:ab-ideal} Set \ $d_\gK=\htt(\theta)+1-\sum_{\ap\in\Pi_\gK(-1)} [\theta:\ap]= 1+\sum_{\ap\in\Pi_\gK({\geqslant}0)} [\theta:\ap]$. Then \begin{itemize} \item[\sf (i)] \ $\htt(\gamma)=d_\gK$ if and only if\/ $w_\gK^{-1}(\gamma)\in\Pi_\gK(-1)$; \item[\sf (ii)] \ $\htt(\gamma)=d_\gK-1$ if and only if \ $-w_\gK^{-1}(\gamma)\in\Pi_\gK(1)$; \item[\sf (iii)] \ $\displaystyle \Delta(\ah_\gK)=\{\gamma\in\Delta^+\mid\htt(\gamma)\geqslant d_\gK\}$. \end{itemize} \end{thm} \begin{proof} {\bf 1}$^o$. For $\GR{A}{2n-1}$ and $\GR{C}{n}$, $x_\gK$ is dominant. Hence $\Pi_\gK(-1)=\varnothing$, $d_\gK=\htt(\theta)+1=\mathsf{h}$ is the Coxeter number, and $\Delta(\ah_\gK)=\varnothing$, which agrees with {\sf (iii)}. Here $\theta$ is the only root of height $d_\gK-1$, $\Pi_\gK(1)=\{\ap_n\}$, and $w_\gK(\ap_n)=-\theta$, see Example~\ref{ex:w_K-sl-sp}. This confirms {\sf (ii)}, whereas {\sf (i)} is vacuous.
{\bf 2}$^o$. In the other cases, $\Pi_\gK(-1)\ne\varnothing$.
{\sf (i)} \ Using formulae from Examples~\ref{ex:w_K-so}--\ref{ex:w_K-e7}, one directly verifies that
if $\ap\in\Pi_\gK(-1)$, then $\htt( w_\gK(\ap))=d_\gK$. It also happens that
$\#\{\gamma\mid \htt(\gamma)=d_\gK\}=\#\Pi_\gK(-1)$ in all cases.
{\sf (ii)} \ Again, this can be verified directly. Alternatively, this follows from {\sf (i)}, Theorem~\ref{thm:min_K}, and Theorem~\ref{thm:max_K} (without further verifications).
{\sf (iii)} \ This follows from {\sf (i)} and Theorem~\ref{thm:min_K}. \end{proof}
Note that \
$\displaystyle \mathsf h-1=\htt(\theta)= \sum_{\ap\in\Pi_\gK(1)}[\theta:\ap]+\sum_{\ap\in\Pi_\gK(0)}[\theta:\ap]+ \sum_{\ap\in\Pi_\gK(-1)}[\theta:\ap]$ \[ \text{ and } \quad 1=\theta(x_\gK)= \sum_{\ap\in\Pi_\gK(1)}[\theta:\ap]-\sum_{\ap\in\Pi_\gK(-1)}[\theta:\ap] \ . \] Therefore, if $\Pi_\gK(0)=\varnothing$, then $d_\gK=(\mathsf h/2)+1$; whereas, if $\Pi_\gK(0)\ne\varnothing$, then $d_\gK> (\mathsf h/2)+1$.
\begin{rmk} A remarkable property of the abelian ideal $\ah_\gK$ is that $\min\Delta(\ah_\gK)$ consists of {\bf all} roots of a {\bf fixed} height. This can be explained by the properties that $\Pi_\gK(-1)\cup \Pi_\gK(1)$ is connected in the Dynkin diagram and that $+1$ and $-1$ nodes alternate. For, if $\ap_i$ and $\ap_j$ are adjacent nodes, $\ap_i\in\Pi_\gK(-1)$, and $\ap_j\in\Pi_\gK(1)$, then $\ap_i+\ap_j$ is a simple root in $\Delta^+_\gK(0)$. Hence $w_\gK(\ap_i)\in \min\Delta(\ah_\gK)$, $-w_\gK(\ap_j)\in \max(\Delta^+\setminus \Delta(\ah_\gK))$, and $w_\gK(\ap_i+\ap_j)\in\Pi$. Hence $\htt(w_\gK(\ap_i))=1+\htt(-w_\gK(\ap_j))$. In view of Theorems~\ref{thm:min_K} and \ref{thm:max_K}, together with the connectedness and the alternating property for $\Pi_\gK(-1)\cup \Pi_\gK(1)$, this relation propagates to any pair in $\min\Delta(\ah_\gK)\times \max(\Delta^+\setminus \Delta(\ah_\gK))$.
\\ \indent But the exact value of the boundary height, which is $d_\gK$ is our case, has no explanation. \end{rmk}
\section{The involution of $\g$ associated with the cascade} \label{sect7:involution}
In this section, we assume that $\spx\in\BZ$, i.e., $\g\ne \mathfrak{sl}_{2n+1}$. Then partition~\eqref{eq:main-partition} yields the partition of the whole root system $\Delta=\bigcup_{i=-2}^2 \Delta_{\gK}(i)$ such that $\Delta_{\gK}(2)=\Delta^+_\gK(2)$.
Set $\Delta_0=\Delta_{\gK}(-2)\cup \Delta_{\gK}(0)\cup \Delta_{\gK}(2)$ and $\Delta_1=\Delta_{\gK}(-1)\cup \Delta_{\gK}(1)$. Consider the vector space decomposition $\g=\g_0\oplus\g_1$, where $\g_0=\te\oplus (\oplus_{\gamma\in\Delta_0}\g_\gamma)$ and $\g_1=\oplus_{\gamma\in\Delta_1}\g_\gamma$. This is a $\BZ_2$-grading and the corresponding involution of $\g$, denoted $\sigma_\gK$, is inner.
\begin{lm} \label{lm:involution} The involution $\sigma_\gK$ has the property that
\\ \centerline{ $\dim\g_0=\dim\be-\#\gK$ \ and \ $\dim\g_1=\dim\ut+\#\gK$. } \end{lm} \begin{proof} This readily follows from the symmetry of $\spx$ on the Frobenius envelope $\be_\gK$ and the equality $\dim\be_\gK=\dim\ut+\#\gK$, see Section~\ref{sect:frob} or Lemma~\ref{lm:symmetry-value}. For, one has \\[.6ex] \centerline{
\begin{tabular}{c||c|c|c|c|c} $i$ & $-2$ & $-1$ & 0 & 1 & 2 \\ \hline $\#\Delta_{\gK}^+(i)$ & 0 & $a$ & $b{-}\#\gK$ & $b$ & $a$ \\ $\#\Delta_{\gK}^-(i)$ & $a$ & $b$ & $b{-}\#\gK$ & $a$ & $0$ \\ \end{tabular}} \\[.8ex] for some $a, b$. Then $\dim\ut=2a+2b-\#\gK$, $\dim\g_0=2a+2b+\rk\g-2\#\gK$ and $\dim\g_1=2a+2b$. \end{proof}
Since $\ind\ut=\#\gK$ and $\ind\be+\ind\ut=\rk\g$, the formulae of Lemma~\ref{lm:involution} mean that $\dim\g_1=\dim\ut+\ind\ut=\dim\be-\ind\be$ and $\dim\g_0=\dim\ut+\ind\be=\dim\be-\ind\ut$.
Recall that $\te_\gK=\bigoplus_{i=1}^m [e_{\beta_i}, e_{-\beta_i}]\subset \te$.
\begin{lm} \label{lm:t_K-regul-ss} The subalgebra $\te_\gK$ contains a regular semisimple element of\/ $\g$. \end{lm} \begin{proof} By Eq.~\eqref{eq:decomp-Delta}, if $\gamma\in\Delta^+$, then $\gamma\in \gH_{\beta_j}$ for a unique $\beta_j\in\gK$. That is, for any $\gamma\in\Delta^+$, there exists $\beta_j\in\gK$ such that $(\gamma,\beta_j)>0$. Therefore, there is a $\nu\in\langle \beta_1,\dots,\beta_m\rangle_\BQ$ such that $(\gamma,\nu)\ne 0$ for {\bf every} $\gamma\in\Delta^+$. Upon the identification of $\te$ and $\te^*$, this yields a required element of $\te_\gK$. \end{proof} \begin{thm} \label{thm:sigma-S&N} The involution $\sigma_\gK$ has the property that $\g_1$ contains a regular semisimple and a regular nilpotent element of $\g$. \end{thm} \begin{proof} For each $\beta_i\in\gK$, consider the subalgebra $\tri(\beta_i)$ with basis $\{e_{\beta_i}, e_{-\beta_i}, [e_{\beta_i}, e_{-\beta_i}]\}$. Then $\tri(\beta_i)\simeq \tri$ and since the elements of $\gK$ are strongly orthogonal, all these $\tri$-subalgebras pairwise commute. Hence $\h=\bigoplus_{j=1}^m \tri(\beta_j)$ is a Lie algebra and $\te_\gK$ is a Cartan subalgebra of $\h$. Recall that $\beta_j(x_\gK)=1$, i.e., $\beta_j\in\Delta_{\gK}^+(1)$ for each $j$. By the very definition of $\sigma_\gK$, this means that $\h\cap\g_0=\te_\gK$ and $\h\cap\g_1=(\bigoplus_{i=1}^m \g_{\beta_i})\oplus (\bigoplus_{i=1}^m \g_{-\beta_i})$. Clearly, there is a Cartan subalgebra of $\h$ that is contained in $\h\cap\g_1$
(because this is true for each $\tri(\beta_i)$ separately). Combining this with Lemma~\ref{lm:t_K-regul-ss}, we see that $\g_1$ contains a regular semisimple element of $\g$.
Finally, for any involution $\sigma$, its $(-1)$-eigenspace $\g_1$ contains a regular semisimple element if and only if it contains a regular nilpotent element. \end{proof}
\begin{rmk} \label{rem:sigma-N-reg} {\sf (i)} \ One can prove that if $\ce$ is a Cartan subalgebra of $\h=\bigoplus_{j=1}^m \tri(\beta_j)$ that is contained in $\h\cap\g_1$, then $\ce$ is a maximal diagonalisable subalgebra of $\g_1$. In other words, $\ce\subset\g_1$ is a {\it Cartan subspace\/} associated with $\sigma_\gK$. Therefore, the rank of the symmetric variety $G/G_0$ equals $\dim\ce=\#\gK$. \\ \indent {\sf (ii)} \ The involution $\sigma_\gK$ is the unique, up to $G$-conjugacy, {\bf inner} involution such that $\g_1$ contains a regular nilpotent element. Moreover, $\sigma_\gK$ has the property that $\co\cap\g_1\ne\varnothing$ for {\bf any nilpotent} $G$-orbit $\co\subset\g$, see~\cite[Theorem\,3]{an}. \\ \indent {\sf (iii)} \ If $\#\gK=\rk\g$, then $\dim\g_1=\dim\be$, $\dim\g_0=\dim\ut$, and $\g_1$ contains a Cartan subalgebra of $\g$. In this case, $\sigma_\gK$ is an {\it involution of maximal rank} and one has a stronger assertion that $\co\cap\g_1\ne\varnothing$ for {\bf any} $G$-orbit $\co$ in $\g$ (\cite[Theorem\,2]{an}). \end{rmk}
\section{The nilpotent $G$-orbit associated with the cascade} \label{sect:nilp-orb}
\noindent Consider the nilpotent element associated with $\gK$ \beq \label{eq:e_K}
e_{\gK}:=\sum_{\beta\in\gK} e_\beta=\sum_{i=1}^m e_{\beta_i}\in\ut^+ \eeq Since the roots in $\gK$ are linearly independent, the closure of $G{\cdot}e_\gK$ contains the space $\bigoplus_{\beta\in\gK}\g_\beta$. Hence the nilpotent orbit $G{\cdot}e_\gK$ does not depend on the choice of root vectors $e_\beta\in\g_\beta$. The orbit $\co_\gK:=G{\cdot}e_\gK$ is said to be the {\it cascade (nilpotent) orbit}. Our goal is to obtain some properties of this orbit.
Recall from Section~\ref{sect:comb-prop} that if $\theta$ is fundamental, then $\tap$ is the only (long!) simple root such that $(\theta,\tap)>0$ and $\tilde\gK=\{\beta\in\gK\mid (\beta,\tap)<0\}$. It then follows from~\eqref{eq:tap} that $\#\tilde\gK\leqslant 3$ and
$\# \tilde\gK=3$ if and only if the roots in $\tilde{\gK}$ are long. Actually, one has $\#\tilde\gK=1$ for $\GR{G}{2}$, $\#\tilde\gK=2$ for $\GR{B}{3}$, and $\#\tilde\gK=3$ for the remaining cases with fundamental $\theta$.
\begin{thm} \label{prop:ad^4} 1$^o$. Suppose that $\theta$ is fundamental, and let $\tilde\ap$ be the unique simple root such that $(\theta,\tilde\ap)\ne 0$. Then {\sf (i)} \ $(\ad e_\gK)^4(e_{-\theta+\tilde\ap})\ne 0$ and {\sf (ii)} \ $(\ad e_\gK)^5= 0$.
2$^o$. If $\theta$ is {\bf not} fundamental, then $(\ad e_\gK)^3= 0$. \end{thm} \begin{proof} 1$^o${\sf (i)}. \ It follows from Eq.~\eqref{eq:e_K} that \[
(\ad e_\gK)^4=\sum_{i_1,i_2,i_3,i_4}\ad\!(e_{\beta_{i_1}}){\cdot}\ldots \cdot \ad\!(e_{\beta_{i_4}}), \] where the sum is taken over all possible quadruples of indices from $\{1,\dots,m\}$. Set \[
\ca_{i_1,i_2,i_3,i_4}=\ad\!(e_{\beta_{i_1}})\ad\!(e_{\beta_{i_2}})\ad\!(e_{\beta_{i_3}})\ad\!(e_{\beta_{i_4}}) . \] As the roots in $\gK$ are strongly orthogonal, the ordering of factors in $\ca_{i_1,i_2,i_3,i_4}$ is irrelevant. Hence the operator $\ca_{i_1,i_2,i_3,i_4}$ depends only on the $4$-multiset $\{i_1,i_2,i_3,i_4\}$. Furthermore, the {\bf nonzero} operators corresponding to different $4$-multisets are linearly independent. Therefore, to ensure that $(\ad e_\gK)^4\ne 0$, it suffices to point out a $4$-multiset $\eus M$ and a root vector $e_\gamma$ such that $\ca_\eus M(e_\gamma)\ne 0$. Of course, in place of $4$-multisets of indices in $[1,m]$, one can deal with $4$-multisets in $\gK$.
Using~\eqref{eq:tap} and $\tilde\gK\subset\gK$, one defines a natural $4$-multiset $\tilde{\eus M}$ in $\gK$. The first element of $\tilde{\eus M}$ is $\theta=\beta_1$ and then one takes each $\beta_i\in\tilde{\gK}$ with multiplicity $k_i=(\tap,\tap)/(\beta_i,\beta_i)$. The resulting $4$-multiset has the property that $(-\theta+\tap)+(\theta + \sum_{i\in J}k_i\beta_i)=\theta-\tap$ and $(-\theta+\tap,\beta)<0$ for any $\beta$ in $\tilde{\eus M}$. This implies that \[
0\ne \ca_{\tilde{\eus M}}(e_{-\theta+\tap})\in \g_{\theta-\tap} . \] \indent {\sf (ii)} \ Now we deal with $5$-multisets of $\gK$. Assume that $\tilde{\eus M}=\{\beta_{i_1},\dots,\beta_{i_5}\}$ and $\ca_{\tilde{\eus M}}\ne 0$. Then there are $\gamma,\mu\in \Delta$ such that $0\ne \ca_{\tilde{\eus M}}(\g_{-\mu})\subset \g_\gamma$. Hence $\gamma+\mu=\sum_{j=1}^5 \beta_{i_j}$ and $(\gamma+\mu)(x_\gK)=5$. But $\gamma(x_\gK)\leqslant 2$ for any $\gamma\in\Delta$ (Theorem~\ref{thm:spektr-fonin}(3)). This contradiction shows that $(\ad e_\gK)^5=0$.
2$^o$. By Theorem~\ref{thm:spektr-fonin}(1)(2), we have $\gamma(x_\gK)\leqslant 1$ for any $\gamma\in\Delta$. Hence the equality $\gamma+\mu=\sum_{j=1}^3 \beta_{i_j}$ is impossible. This implies that $\ca_{\tilde{\eus M}}= 0$ for any $3$-miltiset $\tilde{\eus M}$ of $\gK$ and thereby $(\ad e_\gK)^3=0$. \end{proof}
Let $\N\subset\g$ be the set of nilpotent elements. Recall that the {\it height\/} of $e\in\N$, denoted $\hot(e)$ or $\hot(G{\cdot}e)$, is the maximal $l\in\BN$ such that $(\ad e)^l\ne 0$ (see~\cite[Section\,2]{p99}). By the Jacobson--Morozov theorem, $(\ad e)^2\ne 0$ for any $e\in\N$, i.e., $\hot(G{\cdot}e)\geqslant 2$. \\ \indent The {\it complexity\/} of a $G$-variety $X$, $c_G(X)$, is the minimal codimension of the $B$-orbits in $X$. If $X$ is irreducible, then $c_G(X)=\trdeg\BC(X)^B$, where $\BC(X)^G$ is the field of $B$-invariant rational functions on $X$. The {\it rank\/} of an irreducible $G$-variety $X$ is defined by the equality $c_G(X)+r_G(X)=\trdeg\BC(X)^U$~\cite{these}. If $c_G(X)=0$, then $X$ is said to be {\it spherical}.
\begin{prop} \label{thm:sferic-orbit} \leavevmode\par \begin{itemize} \item[{\sf (i)}] If $\theta$ is fundamental, then the orbit $G{\cdot}e_\gK$ is {\bf not} spherical and\/ $\hot(G{\cdot}e_\gK)=4$. \item[{\sf (ii)}] If $\theta$ is {\bf not} fundamental, then the orbit $G{\cdot}e_\gK$ is spherical and\/ $\hot(G{\cdot}e_\gK)=2$. \end{itemize} \end{prop} \begin{proof} By~\cite[Theorem\,(0.3)]{p94}, a nilpotent orbit $G{\cdot}e$ is spherical if and only if $(\ad e)^4=0$. Hence both assertions follow from Theorem~\ref{prop:ad^4}.
\end{proof}
\subsection{A description of the cascade orbits} \label{subs:cascade-orbits} For the classical Lie algebras, we determine the partition corresponding to $\co_{\gK}$. While for the exceptional Lie algebras, we point out a "minimal including regular subalgebra" in the sense of Dynkin~\cite{dy}.
{\bf I.} In the classical cases, we use formulae for the height of nilpotent elements of $\slv$ or $\sov$ or $\spv$ in terms of the corresponding partitions of $\dim\BV$, see~\cite[Theorem\,2.3]{p99}.
\textbullet\quad $\g=\mathfrak{so}_{2N+1}$. Since $\hot(\co_\gK)=4$, the parts of $\blb(e_\gK)$ does not exceed $3$, i.e., $\blb(e_\gK)=(3^a, 2^b,1^c)$ with $a>0$ and $3a+2b+c=2N+1$. Then $\blb(e_\gK^2)=(2^a,1,\dots,1)$. Hence $\rk(e_\gK)=2a+b$ and $\rk(e_\gK^2)=a$. On the other hand, here $\#\gK=N=\rk\g$ and using the formulae for roots in $\gK$ and thereby the explicit matrix form for $e_\gK$, one readily computes that \\ \centerline{ $\rk(e_\gK)=\begin{cases} N, & \text{ if $N$ is even} \\ N{+}1, & \text{ if $N$ is odd} \end{cases}$ \quad and \ $\rk(e_\gK^2)=\frac{1}{2}\rk(e_\gK)= [(N{+}1)/2]$. } Therefore, if $N$ is either $2j{-}1$ or $2j$, then $a=j$ and $b=0$.
Hence $\blb(e_\gK)=(3^j, 1^{j-1})$ or $(3^{j}, 1^{j+1})$, respectively.
\textbullet\quad $\g=\mathfrak{so}_{2N}$. Here $\hot(\co_\gK)=4$ and $\blb(e_\gK)=(3^a, 2^b,1^c)$, with $a>0$ and $3a+2b+c=2N$. Then again $\rk(e_\gK)=2a+b$ and $\rk(e_\gK^2)=a$. The explicit form of $\gK$ and $e_\gK$ shows that \\ \centerline{ $\rk(e_\gK)=\begin{cases} N, & \text{ if $N$ is even} \\ N{-}1, & \text{ if $N$ is odd} \end{cases}$ \quad and \ $\rk(e_\gK^2)=\frac{1}{2}\rk(e_\gK)= [N/2]$. } \\ Therefore, $a=[N/2]$ and $b=0$ in both cases. Hence \\[.6ex] \centerline{ $\blb(e_\gK)=(3^j, 1^j)$, if $N=2j$; \ $\blb(e_\gK)=(3^j, 1^{j+2})$, if $N=2j+1$.}
\textbullet\quad $\g=\mathfrak{sl}_{N+1}$ or $\mathfrak{sp}_{2N}$. Then $\hot(\co_\gK)=2$, $\blb(e_\gK)=(2^a,1^b)$, and $a=\rk(e_\gK)=\#\gK$. Therefore, $\blb(e_\gK)=(2^n)$ for $\sltn$ and $\spn$, while $\blb(e_\gK)=(2^n,1)$ for $\mathfrak{sl}_{2n+1}$.
{\bf II.} In the exceptional cases, one can use some old, but extremely helpful computations of {E.B.\,Dynkin.} Following Dynkin, we say that a subalgebra $\h$ of $\g$ is {\it regular}, if it is normalised by a Cartan subalgebra. As in Section~\ref{sect7:involution}, consider $\h:=\bigoplus_{i=1}^m \tri(\beta_i)$. Then $\h$ is normalised by $\te$
and $e_\gK\in \h$ is a regular nilpotent element of $\h$. Clearly, $\h$ is a minimal regular semisimple subalgebra of $\g$ meeting $\co_\gK$.
For every nilpotent $G$-orbit $\co$ in an exceptional Lie algebra $\g$, Dynkin computes all, up to conjugacy, minimal regular semisimple subalgebras of $\g$ meeting $\co$, see Tables~16--20 in \cite{dy}. (This information is also reproduced, with a few corrections, in Tables 2--6 in~\cite{age75}.) Therefore, it remains only to pick the nilpotent orbit with a "minimal including regular subalgebra" of the required type, which also yields the corresponding {\it weighted Dynkin diagram\/} $\gD(\co_\gK)$.
For the regular subalgebras $\tri\subset\g$, Dynkin uses the Cartan label $\GR{A}{1}$ (resp. $\GRt{A}{1}$) if the corresponding root is long (resp. short). Therefore, we are looking for the "minimal including regular subalgebra" of type $m\GR{A}{1}$, $m=\#\gK$, if $\gK\subset \Delta_l$; whereas for $\g$ of type $\GR{G}{2}$ we need the subalgebra of type $\GR{A}{1}+\GRt{A}{1}$.
\subsection{Another approach to $\co_\gK$} \label{subs:charact-e_k} By the very definition of $x_\gK\in\te$ and $e_\gK$, we have $[x_\gK,e_\gK]=e_\gK$. It is also clear that $x_\gK\in \Ima(\ad e_\gK)$. Therefore $h_\gK=2x_\gK$ is a {\it characteristic\/} of $e_\gK$~\cite[Chap.\,6, \S\,2.1]{t41}. Hence the weighted Dynkin diagram $\gD(\co_\gK)$ is determined by the dominant representative in the Weyl group orbit $W{\cdot}h_\gK\subset \te$. Since the antidominant representative in $W{\cdot}x_\gK$ is $w_\gK(x_\gK)$ (Prop.~\ref{prop:anti-dom}) and $\omega_0(x_\gK)=-x_\gK$, the dominant representative is $-w_\gK(x_\gK)$. Therefore, if $\g\ne\mathfrak{sl}_{2n+1}$, then $W{\cdot}h_\gK\cap\gC=\{2\varpi_j^\vee\}$, where $j$ is determined by the condition that $w_\gK(\theta)=-\ap_j$ (cf. Prop.~\ref{prop:fundam}). Thus, if $\g\ne\mathfrak{sl}_{2n+1}$, then $\co_\gK$ is even and $\gD(\co_\gK)$ has the unique nonzero label ``2'' that corresponds to $\ap_j$.
Let $\g=\bigoplus_{i\in\BZ}\g(i)$ be the $\BZ$-grading determined by $h_\gK=2x_\gK$; that is, $h_\gK$ has the eigenvalue $i$ on $\g(i)$. Then \[
\hot(e_\gK)=\max\{i\mid \g(i)\ne 0\}=2\max\{\gamma(x_\gK)\mid\gamma\in \Delta\} . \] Using results of Section~\ref{sect:comb-prop}, we again see that $\hot(e_\gK)\leqslant 4$ and the equality occurs if and only if $\theta$ is fundamental. Note that \begin{gather*}
\dim\g(2)=\#\bigl(\Delta^+_\gK(1)\cup \Delta^-_\gK(1)\bigr)=\#\bigl(\Delta^+_\gK(1)\cup \Delta^+_\gK(-1)\bigr)=
\#\bigl(\Delta^+_\gK(1)\cup \Delta^+_\gK(2)\bigr) , \\
\dim\g(4)=\#\Delta^+_\gK(2)=\#\Delta_{\gK}(2)=\dim\Ima(\ad e_\gK)^4 , \end{gather*} and also $2\dim\g(4)=\dim\ah_\gK$.
In Tables~\ref{table:O-class} and \ref{table:O-exc}, we point out $\dim\co_{\gK}$ and $\gD(\co_\gK)$. For classical cases, we provide the partition $\blb(e_\gK)$; while for the exceptional cases, the Dynkin--Bala--Carter notation for orbits is given, see e.g.~\cite[Ch.\,8]{CM}. We also include dimensions of the spaces $\g(2)$ and $\g(4)$. In Table~\ref{table:O-class}, the unique nonzero numerical mark "$2$" corresponds to the simple root $\ap_j$ for all even cases. For $\GR{A}{2j}$, two marks ``$1$'' correspond to the roots $\ap_j$ and $\ap_{j+1}$. We also assume that $j\geqslant 2$ in the four orthogonal cases. (For, $\GR{B}{1}=\GR{A}{1}$, $\GR{D}{2}=\GR{A}{1}+\GR{A}{1}$, $\GR{B}{2}=\GR{C}{2}$, and $\GR{D}{3}=\GR{A}{3}$.)
\begin{table}[ht] \caption{The cascade orbits for the classical Lie algebras} \label{table:O-class}
\begin{center}\begin{tabular}{>{$}l<{$}| >{$}l<{$} >{$}l<{$} c >{$}c<{$} >{$}c<{$}| } \g & \blb(e_\gK) & \dim\co_{\gK} & $\gD(\co_\gK)$ & \dim\g(2) & \dim\g(4) \\ \hline\hline \GR{A}{2j-1} & (2^j) & 2j^2 & \rule{0pt}{2.5ex} \raisebox{-.3ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (a) at (1,0) {\bf 0}; \node (b) at (2,0) {$\dots$}; \node (c) at (3,0) {\bf 0}; \node (d) at (4,0) {\bf 2}; \node (e) at (5,0) {\bf 0}; \node (f) at (6,0) {$\dots$}; \node (g) at (7,0) {\bf 0}; \foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, f/g} \draw[-] (\from) -- (\to); \end{tikzpicture}} & j^2 & 0
\\
\GR{A}{2j} & (2^j,1) & 2j^2{+}2j & \rule{0pt}{2.5ex} \raisebox{-.3ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (a) at (1,0) {\bf 0}; \node (b) at (2,0) {$\dots$}; \node (c) at (3,0) {\bf 0}; \node (d) at (4,0) {\bf 1}; \node (e) at (5,0) {\bf 1}; \node (f) at (6,0) {\bf 0}; \node (g) at (7,0) {$\dots$}; \node (h) at (8,0) {\bf 0}; \foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, f/g, g/h} \draw[-] (\from) -- (\to); \end{tikzpicture}} & j^2 & 0
\\
\GR{C}{j} & (2^j) & j^2{+}j & \rule{0pt}{2.5ex} \raisebox{-.3ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (a) at (1,0) {\bf 0}; \node (b) at (2,0) {$\dots$}; \node (c) at (3,0) {\bf 0}; \node (d) at (4.3,0) {\bf 2}; \node (e) at (3.45,0) {\Large $<$}; \draw (3.4, .06) -- +(.6,0); \draw (3.4, -.06) -- +(.6,0); \foreach \from/\to in {a/b, b/c} \draw[-] (\from) -- (\to); \end{tikzpicture}} & \genfrac{(}{)}{0pt}{}{j+1}{2} & 0
\\
\GR{B}{2j-1} & (3^{j}, 1^{j-1}) & 5j^2{-}3j & \rule{0pt}{2.7ex} \raisebox{-.3ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (a) at (1,0) {\bf 0}; \node (b) at (2,0) {$\dots$}; \node (c) at (3,0) {\bf 0}; \node (d) at (4,0) {\bf 2}; \node (e) at (5,0) {\bf 0}; \node (f) at (6,0) {$\dots$}; \node (g) at (7,0) {\bf 0}; \node (h) at (8.2,0) {\bf 0}; \node (r) at (7.8,0) {\Large $>$}; \foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, f/g} \draw[-] (\from) -- (\to); \draw (7.2, .06) -- +(.6,0); \draw (7.2, -.06) -- +(.6,0); \end{tikzpicture}} & 2j^2{-}j & \genfrac{(}{)}{0pt}{}{j}{2} \\
\GR{D}{2j} & (3^j, 1^{j}) & 5j^2{-}j & \rule{0pt}{3.5ex} \raisebox{-2.5ex}{\begin{tikzpicture}[scale= .65, transform shape]
\node (a) at (1,0) {\bf 0}; \node (b) at (2,0) {$\dots$}; \node (c) at (3,0) {\bf 0}; \node (d) at (4,0) {\bf 2}; \node (e) at (5,0) {\bf 0}; \node (f) at (6,0) {$\dots$}; \node (g) at (7,0) {\bf 0}; \node (h) at (8,.6) {\bf 0}; \node (j) at (8,-.6) {\bf 0}; \foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, f/g, g/h,g/j} \draw[-] (\from) -- (\to); \end{tikzpicture}} & 2j^2& \genfrac{(}{)}{0pt}{}{j}{2} \\
\GR{B}{2j} & (3^j, 1^{j+1}) & 5j^2{+}j & \rule{0pt}{2.7ex} \raisebox{-.3ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (a) at (1,0) {\bf 0}; \node (b) at (2,0) {$\dots$}; \node (c) at (3,0) {\bf 0}; \node (d) at (4,0) {\bf 2}; \node (e) at (5,0) {\bf 0}; \node (f) at (6,0) {$\dots$}; \node (g) at (7,0) {\bf 0}; \node (h) at (8.2,0) {\bf 0}; \node (r) at (7.8,0) {\Large $>$}; \foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, f/g} \draw[-] (\from) -- (\to); \draw (7.2, .06) -- +(.6,0); \draw (7.2, -.06) -- +(.6,0); \end{tikzpicture}} & 2j^2{+}j & \genfrac{(}{)}{0pt}{}{j}{2} \\
\GR{D}{2j+1} & (3^j, 1^{j+2}) & 5j^2{+}3j & \rule{0pt}{3.5ex} \raisebox{-2.5ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (a) at (1,0) {\bf 0}; \node (b) at (2,0) {$\dots$}; \node (c) at (3,0) {\bf 0}; \node (d) at (4,0) {\bf 2}; \node (e) at (5,0) {\bf 0}; \node (f) at (6,0) {$\dots$}; \node (g) at (7,0) {\bf 0}; \node (h) at (8,.6) {\bf 0}; \node (j) at (8,-.6) {\bf 0}; \foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, f/g, g/h,g/j} \draw[-] (\from) -- (\to); \end{tikzpicture}} & 2j^2{+}2j & \genfrac{(}{)}{0pt}{}{j}{2} \\ \hline \end{tabular} \end{center} \end{table}
\begin{table}[ht] \caption{The cascade orbits for the exceptional Lie algebras} \label{table:O-exc}
\begin{center}\begin{tabular}{>{$}c<{$}|>{$}l<{$} >{$}c<{$} c >{$}c<{$} >{$}c<{$}|} \g & \co_{\gK} & \dim\co_{\gK} & $\gD(\co_\gK)$ & \dim\g(2) & \dim\g(4) \\ \hline\hline \GR{E}{6} & \mathsf{A}_2 & 42 & \rule{0pt}{2.5ex} \raisebox{-3.2ex}{\begin{tikzpicture}[scale= .65, transform shape]
\node (a) at (0,0) {\bf 0}; \node (b) at (1,0) {\bf 0}; \node (c) at (2,0) {\bf 0}; \node (d) at (3,0) {\bf 0}; \node (e) at (4,0) {\bf 0}; \node (f) at (2,-.9) {\bf 2}; \foreach \from/\to in {a/b, b/c, c/d, d/e, c/f} \draw[-] (\from) -- (\to); \end{tikzpicture}} & 20 & 1 \\ \GR{E}{7} & \mathsf{A}_2+3\mathsf{A}_1 & 84 & \rule{0pt}{2.6ex} \raisebox{-3.2ex}{\begin{tikzpicture}[scale= .65, transform shape]
\node (a) at (0,0) {\bf 0}; \node (b) at (1,0) {\bf 0}; \node (c) at (2,0) {\bf 0}; \node (d) at (3,0) {\bf 0}; \node (e) at (4,0) {\bf 0}; \node (f) at (5,0) {\bf 0}; \node (g) at (3,-.9) {\bf 2}; \foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, d/g} \draw[-] (\from) -- (\to); \end{tikzpicture}} & 35 & 7 \\ \GR{E}{8} & 2\mathsf{A}_2 & 156 & \rule{0pt}{2.6ex} \raisebox{-3.2ex}{\begin{tikzpicture}[scale= .65, transform shape]
\node (h) at (-1,0) {\bf 0}; \node (a) at (0,0) {\bf 0}; \node (b) at (1,0) {\bf 0}; \node (c) at (2,0) {\bf 0}; \node (d) at (3,0) {\bf 0}; \node (e) at (4,0) {\bf 0}; \node (f) at (5,0) {\bf 2}; \node (g) at (3,-.9) {\bf 0}; \foreach \from/\to in {h/a, a/b, b/c, c/d, d/e, e/f, d/g} \draw[-] (\from) -- (\to); \end{tikzpicture}} & 64 & 14 \\ \GR{F}{4} & \mathsf{A}_2 & 30 & \rule{0pt}{2.5ex} \raisebox{-.5ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (a) at (0,0) {\bf 0}; \node (b) at (1.1,0) {\bf 0}; \node (c) at (2.5,0) {\bf 0}; \node (d) at (3.6,0) {\bf 2}; \node (e) at (1.55,0) {\Large $<$}; \foreach \from/\to in {a/b, c/d} \draw[-] (\from) -- (\to); \draw (1.5, .06) -- +(.7,0); \draw (1.5, -.06) -- +(.7,0); \end{tikzpicture}} & 14 & 1 \\ \GR{G}{2} & \mathsf{G}_2(a_1) & 10 & \rule{0pt}{2.5ex} \raisebox{-.5ex}{\begin{tikzpicture}[scale= .65, transform shape] \node (b) at (1.1,0) {\bf 0}; \node (c) at (2.5,0) {\bf 2}; \node (e) at (1.55,0) {\Large $<$};
\draw (1.52, .07) -- +(.62,0); \draw (1.43, 0) -- +(.71,0); \draw (1.52, -.07) -- +(.62,0); \end{tikzpicture} } & 4 & 1 \\ \hline \end{tabular} \end{center} \end{table}
Let us summarise main properties of the cascade orbit in all simple $\g$. \begin{itemize} \item \ The cascade orbit $\co_\gK$ is even unless $\g$ is of type $\GR{A}{2j}$; this reflects the fact that $x_\gK\in\mathcal P^\vee$ unless $\g$ is of type $\GR{A}{2j}$. \item \ If $\theta$ is fundamental, then $\hot(\co_\gK)=4$ and $\co_\gK$ is {\bf not} spherical. \item \ If $\theta$ is {\bf not} fundamental, then $\hot(\co_\gK)=2$ and $\co_\gK$ is spherical. Moreover, it appears that $\co_\gK$ is the {\bf maximal} spherical nilpotent orbit in these cases. \item \ Using the general formulae for the complexity and rank of nilpotent orbits in terms of $\BZ$-gradings~\cite[Sect.\,(2.3)]{p94}, one can prove that $c_G(\co_\gK)=2\dim\g(4)=\dim\ah_\gK$ and $r_G(\co_\gK)=\dim\te_\gK=\#\gK$ for all simple $\g$. \item \ If $\theta$ is fundamental, then ($\co_\gK$ is even and) the node with mark ``$2$'', regarded as a node in the affine Dynkin diagram, determines the {\it Kac diagram\/} of the involution $\sigma_\gK$. (See \cite[Chap.\,3, \S\,3.7]{t41} for the definition of the Kac diagram of a finite order inner automorphism of $\g$.) \item \ If $\theta$ is not fundamental and $\co_\gK$ is even, then the node with mark ``$2$'' and the extra node in the affine Dynkin diagram, together determine the Kac diagram of the involution $\sigma_\gK$. \end{itemize}
\appendix \section{The elements of $\gK$ and Hasse diagrams} \label{sect:tables}
\noindent Here we provide the lists of cascade elements and the Hasse diagrams of cascade posets $\eus K$ for all simple Lie algebras, see Fig.~\ref{fig:An}--\ref{fig:En}. To each node $\beta_j$ in the Hasse diagram, the Cartan label of the simple Lie algebra $\g\langle j\rangle$ is attached. Recall that $\beta_j$ is the highest root for $\g\langle j\rangle$. If $\g\langle j\rangle\simeq \tri$ and $\beta_j$ is {\sl short}, then we use the Cartan label $\GRt{A}{1}$. (This happens only for $\GR{B}{2k+1}$ and $\GR{G}{2}$.) It is also assumed that $\GR{A}{1}=\GR{C}{1}$. The main features are: \begin{itemize} \item $\Pi=\{\ap_1,\dots,\ap_{\rk \g}\}$ and the numbering of $\Pi$ follows~\cite[Table\,1]{VO}, \item $\beta_1=\theta$ is always the highest root, \item The numbering of the $\beta_i$'s in the lists corresponds to that in the figures. \end{itemize} We use the standard $\esi$-notation for the roots of classical Lie algebras, see~\cite[Table\,1]{VO}.
\noindent {\it\bfseries The list of cascade elements for the classical Lie algebras}: \begin{description} \item[$\GR{A}{n}, n\geqslant 2$] \ $\beta_i=\esi_i-\esi_{n+2-i}=\ap_i+\dots +\ap_{n+1-i}$ \ ($i=1,2,\dots,\left[\frac{n+1}{2}\right]$); \item[$\GR{C}{n}, n\geqslant 1$] \ $\beta_i=2\esi_i=2(\ap_i+\dots+\ap_{n-1})+\ap_n$ \ ($i=1,2,\dots,n-1$) and $\beta_n=2\esi_n=\ap_n$; \item[$\GR{B}{2n}$, $\GR{D}{2n}$, $\GR{D}{2n+1}$ ($n\geqslant 2$)] \ $\beta_{2i-1}=\esi_{2i-1}+\esi_{2i}$, $\beta_{2i}=\esi_{2i-1}-\esi_{2i}$ \ ($i=1,2,\dots,n$); \item[$\GR{B}{2n+1}, n\geqslant 1$] \ here $\beta_1,\dots,\beta_{2n}$ are as above and $\beta_{2n+1}=\esi_{2n+1}$; \end{description}
\noindent For all orthogonal series, we have $\beta_{2i}=\ap_{2i-1}$, $i=1,\dots,n$, while formulae for $\beta_{2i-1}$ via $\Pi$ slightly differ for different series. E.g. for $\GR{D}{2n}$ one has $\beta_{2i-1}=\ap_{2i-1}+2(\ap_{2i}+\dots+\ap_{2n-2})+\ap_{2n-1}+\ap_{2n}$ ($i=1,2,\dots,n-1$) and $\beta_{2n-1}=\ap_{2n}$.
\noindent {\it\bfseries The list of cascade elements for the exceptional Lie algebras}: \begin{description} \item[$\GR{G}{2}$] \ $\beta_1=(32)=3\ap_1+2\ap_2, \ \beta_2=(10)=\ap_1$; \item[$\GR{F}{4}$] \ $\beta_1=(2432)=2\ap_1+4\ap_2+3\ap_3+2\ap_4,\ \beta_2=(2210),\ \beta_3=(0210),\ \beta_4=(0010)=\ap_3$;
\item[$\GR{E}{6}$] \
$\beta_1=$\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {3}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (.4,-.4) {2}; \end{tikzpicture}}\!\!,
$\beta_2=$\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {1}; \node (e) at (.8,0) {1}; \node (f) at (.4,-.4) {0}; \end{tikzpicture}},
$\beta_3=$\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {1}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {1}; \node (e) at (.8,0) {0}; \node (f) at (.4,-.4) {0}; \end{tikzpicture}},
$\beta_4$\,=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {0}; \node (f) at (.4,-.4) {0}; \end{tikzpicture}}=\,$\ap_3$;
\item[$\GR{E}{7}$] \
$\beta_1$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {3}; \node (d) at (.6,0) {4}; \node (e) at (.8,0) {3}; \node (f) at (1,0) {2}; \node (g) at (.6,-.4) {2}; \end{tikzpicture}},
$\beta_2$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {0}; \node at (.6,-.4) {1}; \end{tikzpicture}},
$\beta_3$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {1}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {0}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {0}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {0}; \end{tikzpicture}}=\,$\ap_1$,
$\beta_4$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}},
$\beta_5$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {0}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {0}; \end{tikzpicture}}=\,$\ap_3$, \\
$\beta_6$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {0}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {0}; \end{tikzpicture}}=\,$\ap_5$,
$\beta_7$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {0}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {0}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}}=\,$\ap_7$;
\item[$\GR{E}{8}$] \
$\beta_1$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (a) at (0,0) {2}; \node (b) at (.2,0) {3}; \node (c) at (.4,0) {4}; \node (d) at (.6,0) {5}; \node (e) at (.8,0) {6}; \node (f) at (1,0) {4}; \node (g) at (1.2,0) {2}; \node (h) at (.8,-.4) {3}; \end{tikzpicture}},
$\beta_2$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape]
\node (h) at (-.2,-.0) {0}; \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {3}; \node (d) at (.6,0) {4}; \node (e) at (.8,0) {3}; \node (f) at (1,0) {2}; \node (g) at (.6,-.4) {2}; \end{tikzpicture}},
$\beta_3$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape]
\node (h) at (-.2,-.0) {0}; \node (a) at (0,0) {1}; \node (b) at (.2,0) {2}; \node (c) at (.4,0) {2}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {0}; \node at (.6,-.4) {1}; \end{tikzpicture}},
$\beta_4$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (h) at (-.2,-.0) {0}; \node (a) at (0,0) {1}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {0}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {0}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {0}; \end{tikzpicture}}=\,$\ap_2$,
$\beta_5$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (h) at (-.2,-.0) {0}; \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {2}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}}, \\
$\beta_6$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (h) at (-.2,-.0) {0}; \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {1}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {0}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {0}; \end{tikzpicture}}=\,$\ap_4$,
$\beta_7$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (h) at (-.2,-.0) {0}; \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {0}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {1}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {0}; \end{tikzpicture}}=\,$\ap_6$,
$\beta_8$=\raisebox{-2.1ex}{\begin{tikzpicture}[scale= .85, transform shape] \node (h) at (-.2,-.0) {0}; \node (a) at (0,0) {0}; \node (b) at (.2,0) {0}; \node (c) at (.4,0) {0}; \node (d) at (.6,0) {0}; \node (e) at (.8,0) {0}; \node (f) at (1,0) {0}; \node (g) at (.6,-.4) {1}; \end{tikzpicture}}=\,$\ap_8$; \end{description}
\begin{figure}
\caption{The cascade posets for $\GR{A}{p}$ ($p\geqslant 2$), $\GR{C}{n}$ ($n\geqslant 1$), $\GR{F}{4}$, $\GR{G}{2}$}
\label{fig:An}
\end{figure}
\begin{figure}
\caption{The cascade posets for series $\GR{B}{p}$, $p\geqslant 3$}
\label{fig:Bn}
\end{figure}
\begin{figure}
\caption{The cascade posets for series $\GR{D}{p}$, $p\geqslant 4$}
\label{fig:Dn}
\end{figure}
\begin{figure}
\caption{The cascade posets for $\GR{E}{6}$, $\GR{E}{7}$, $\GR{E}{8}$}
\label{fig:En}
\end{figure}
\end{document} | arXiv | {
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\begin{document}
\maketitle
\centerline{\scshape Markus Hegland }
{\footnotesize
\centerline{Centre for Mathematics and its Applications}
\centerline{The Australian National University}
\centerline{ Canberra ACT, 0200, Australia} }
\centerline{\scshape Bernd Hofmann}
{\footnotesize
\centerline{Department of Mathematics}
\centerline{Chemnitz University of Technology}
\centerline{09107 Chemnitz, Germany.} }
\begin{abstract}
Based on the variable Hilbert scale interpolation inequality bounds for the
error of regularisation methods are derived under range inclusions. In this
context, new formulae for the modulus of continuity of the inverse of bounded
operators with non-closed range are given. Even if one can show the
equivalence of this approach to the version used previously in the literature,
the new formulae and corresponding conditions are simpler than the former
ones. Several examples from image processing and spectral enhancement
illustrate how the new error bounds can be applied. \end{abstract}
\section{Introduction}
Let $X$ and $Y$ be infinite dimensional separable Hilbert spaces with norms
$\|\cdot\|$ and scalar products $(\cdot,\cdot)$. We study linear inverse problems in form of ill-posed operator equations \begin{equation} \label{opeq} Af = g, \qquad f \in X,\; g \in Y, \end{equation} characterised by an injective bounded linear forward operator $A: X \to Y$ for which the range $\operatorname{range}(A)$ is a non-closed subset of $Y$. Then equation (\ref{opeq}) is unstable in the sense that the inverse operator $A^{-1}: \operatorname{range}(A) \subseteq Y \to X$ is unbounded and hence the use of perturbed data $g^\delta$ instead of the exact right-hand side $g$ with \begin {equation} \label{noise}
\|g-g^\delta\| \le \delta \end{equation} and noise level $\delta>0$ may lead to arbitrarily large errors in the solution of (\ref{opeq}) even if the noise level is extremely small. As a consequence of this ill-posedness phenomenon regularisation methods are required for the stable approximate solution of the inverse problem. Their basic idea consists in finding approximations to the exact solution $f$ in form of solutions $f_\alpha=f_\alpha(g^\delta)$ to stable auxiliary problems neighbouring (\ref{opeq}). Those solutions are obtained by using the noisy data $g^\delta$. The degree of neighbourhood of the exploited auxiliary problems is controlled by a regularisation parameter $\alpha>0$. In this context, small $\alpha$ express closeness to (\ref{opeq}) in combination with a low level of stability, whereas larger $\alpha$ ensure better stability, however combined with a low level of approximation. For the success of any regularisation method an appropriate trade-off between stability and approximation has to be aspired when choosing the regularisation parameter.
As already outlined and summarised in the monograph \cite{EHN96} by {\sc Engl, Hanke} and {\sc Neubauer} a successful way for doing regularisation for linear ill-posed problems in Hilbert spaces including convergence and convergence rates of constructed methods requires some knowledge on the impact of smoothness on the regularised solutions. Smoothness should be understood there in a very generalised sense as both solution smoothness and smoothing properties of the forward operator. In \cite{EHN96} such smoothness fitting is focused on H\"older type source conditions yielding H\"older type convergence rates when the regularisation method has a sufficiently high level of qualification. This theory is closely connected with associated (classical) Hilbert scales, where we refer to the seminal paper by {\sc Natterer} \cite{Nat84}. An extension of that theory to generalised source conditions implying also more general convergence rates was performed rather independently by two different approaches. The first approach initiated and established by {\sc Hegland} (see \cite{Heg92,Heg95}) introduced variable Hilbert scales with positive index functions the behaviour of which is in particular of interest for large arguments covering the spectrum of an injective unbounded linear operator with bounded inverse. Results in this approach are based on interpolation inequalities. An alternative second approach was developed and published by {\sc Math\'e} and {\sc Pereverzev} (see \cite{MatPer03b,MatPer03a,MatTau06}) and complimented by ideas of {\sc Hofmann} and other co-workers (see, e.g., \cite{HofMAt07,HofMS08,HofMW09}). This approach, in principle, also exploits variable Hilbert scales, but the index functions occurring there are more specific and their behaviour is of interest just for small positive arguments covering on the spectrum of $A^*A$. The index functions of the second approach are monotonically increasing and tend to zero as the positive arguments tend to zero. Here we call them rate functions abbreviated by over-lined small Greek letters, because they also express the convergence rate of approximate solutions. One of the main goals of this paper is to compare both approaches, their results, required conditions and their natural interplay. Moreover, some more consequences and new convergence rates results of {\sc Hegland}'s approach shall be formulated and proven in the sequel.
In our study we use {\sl variable Hilbert scales} and corresponding {\sl interpolation inequalities} in order to obtain bounds from above for the error
$\|f-f_\alpha\|$ of regularised solutions under conditions imposed on $f$. As is well-known the convergence of regularised solutions $f_\alpha \to f$ in $X$, even if $\delta \to +0$ and $\alpha=\alpha(\delta,g^\delta) \to +0$ is chosen in an appropriate manner, can be arbitrarily slow for solutions $f$ which are in some sense non-smooth with respect to the forward operator $A$. It is of essential interest in regularisation theory to obtain convergence rates \begin{equation} \label{etarates}
\|f-f_\alpha\|=\mathcal{O}(\bar{\eta}(\delta)) \qquad \mbox{as} \qquad \delta \to +0 \end{equation} with appropriate rate functions $\bar{\eta}$.
For a non-closed range of $A$ convergence rates require that \emph{general source conditions} are satisfied which attain in the standard case the form \begin{equation} \label{scclassic} f=\bar{\psi}(A^*A) v \end{equation}
with source element $v \in X$ and with some rate function $\bar{\psi}$ defined at least on the interval $(0,\|A\|^2]$ covering the spectrum of $A^*A$. Here, $\bar{\psi}(A^*A)$ is well-defined as an injective bounded positive self-adjoint linear operator by spectral calculus (see, e.g., \cite[Section~2.3]{EHN96}). If the regularisation method has a sufficiently high {\sl qualification} (see, e.g., \cite[Definition~2.6]{HofMAt07}), then an asymptotically fast decay of $\bar \psi(t) \to 0$ as $t \to +0$ corresponds with high order convergence rates (\ref{etarates}) of regularised solutions and vice versa. Note that the tool of general source conditions with rate functions $\bar \psi$ different from monomials was early applied to regularisation theory by {\sc Tautenhahn} in \cite{Taut96}. Later {\sc Hohage} (see~\cite{Hohage1,Hohage2}) studied in detail the case of logarithmic functions $\bar \psi$ in (\ref{scclassic}) and corresponding convergence rates.
Using a non-standard source condition \begin{equation} \label{Gsmooth} f=G w \end{equation} with source element $w \in X$, the priori information about the smoothness of the solution $f$ can be expressed by an injective bounded self-adjoint positive definite linear operator $G: X \to X$, where $\operatorname{range}(G)$ is a non-closed subset of $X$. In most cases the character of the operator $G$ is rather independent of the forward operator $A$ and hence $G$ need not be a function of $A^*A$. However, in order to make conclusions to convergence rates of regularised solutions the assumption $f \in \operatorname{range}(G)$ of (\ref{Gsmooth}) and the assumption $f \in \operatorname{range}(\bar{\psi}(A^*A))$ of (\ref{scclassic}) have to be connected anyway. In the framework of {\sc B\"ottcher} et al.~\cite{Boetal06} conditions for such connections and their interplay are discussed in a comprehensive manner. In \cite{HofMAt07} they are called \emph{link conditions}. Besides the simplest link type $G=\bar{\varphi}(A^*A)$, which is typical for commuting $G$ and $A^*A$, one of the most convincing class of link conditions represent \emph{range inclusions} introduced with {\sc Yamamoto} in \cite{HofYam05} to regularisation theory. The favourite form of such inclusion is \begin{equation} \label{Grange} \operatorname{range}(G) \subseteq \operatorname{range}(\bar{\psi}(A^*A)) \end{equation} with some rate function $\bar \psi$. Evidently, under (\ref{Gsmooth}) a range inclusion (\ref{Grange}) immediately implies a source condition (\ref{scclassic}). The higher the rate expressed by $\bar \psi$ is, i.e. the faster the decay $\bar \psi(t) \to 0$ as $t \to +0$ goes on, the smaller the set $\operatorname{range}(\bar{\psi}(A^*A))$ becomes. Hence the condition (\ref{Grange}) is a strong one for higher rates $\bar \psi$ and vice versa.
An alternative link condition is \begin{equation} \label{Gineq}
\|\bar{\varrho}(G)x\| \le C\, \|Ax\| \quad \mbox{for all} \;\; x \in X \end{equation} for some constant $C>0$, where the rate function $\bar \varrho$ acts as a benchmark for the {\sl degree of ill-posedness} of equation (\ref{opeq}) with respect to the a priori information (\ref{Gsmooth}). From Proposition 2.1 in \cite{Boetal06} we know that a range inclusion $\operatorname{range}(G_1) \subseteq
\operatorname{range}(G_2)$ and a condition of the form $\|G_1x\| \le C\|G_2x\|$ for all $x \in X$ and some $C>0$ are equivalent. Consequently, with Proposition 2.18 in \cite{EHN96} the condition (\ref{Gineq}) is equivalent to the range inclusion \begin{equation} \label{rhorange} \operatorname{range}(\bar \varrho(G)) \subseteq \operatorname{range}(A^*)=\operatorname{range}((A^*A)^{\scriptscriptstyle 1/2}) \end{equation}
taking into account the identity $\|Ax\|=\|(A^*A)^{1/2}x\|$ for all $x \in X$.
As exploited in \cite[\S~4]{Boetal06} one can reduce (\ref{rhorange}) to the form (\ref{Grange}) with $\bar \psi(t)=\bar \varrho^{-1}(\sqrt{t})$ if $[\bar \varrho^{-1}(\sqrt{t})]^2$ is an {\sl operator monotone} function (cf.~\cite{Bhatia97}. An important special case of that implication, namely for $\varrho(t)=t^{\frac{1}{2\mu}}$ with $0<\mu \le 1/2$, is well-known as Heinz-Kato inequality (see, e.g., [6, Proposition 8.21] or the corollary of Theorem~2.3.3 in \cite[p. 45]{Tanabe79}). In that special case, (\ref{rhorange}) yields (\ref{Grange}) with $\bar \psi(t)=t^\mu$ for exponents $0<\mu \le 1/2$.
In the next section we review the definition and some properties of index functions and variable Hilbert scales. The fundamental interpolation inequality is given with a short proof together with an application to a general regularisation method. We then show how the variable Hilbert scales provide natural source conditions. In the third section bounds for the modulus of continuity are given in a variable Hilbert scale setting. An important part of this section compares the new bounds on the modulus of continuity with some obtained earlier and shows how the new results have a substantially simpler structure. The fourth section analyses linear regularisation methods and parameter choices using the variable Hilbert scale approach. In section 5 we consider several examples from image processing and spectral enhancement and the paper finishes with some final remarks.
\section{Interpolation inequalities and consequences}
The main tool used here to derive error bounds for regularised solutions is an
extension of interpolation inequalities to variable Hilbert scales. For
classical Hilbert scales $\{X_r\}_{r \in \mathbb R}$ -- with real numbers as
scale index $r$ -- interpolation inequalities are well-established. These
interpolation inequalities were initially applied to the treatment of linear
ill-posed problems (\ref{opeq}) by Natterer in~\cite{Nat84} (see also the
monograph by Engl, Hanke and Neubauer \cite{EHN96}). For variable Hilbert
scales, new interpolation inequalities have to be formulated. Here the scale
index $r$ is replaced by a wide class of index functions defined as:
\begin{definition} \label{def1}
We call a real function $\theta$ defined on the open interval $(0,\infty)$ an
\emph{index function} if it is continuous and positive. The \emph{index set}
denoted by $\mathcal{I}$ is then the set of all such index functions.
We call an index function $\bar \theta \in \mathcal{I}$ a \emph{rate function}
if it is monotonically increasing and if it satisfies the limit condition
$\lim \limits_{t \to +0} \bar \theta(t)=0.$
\end{definition}
{\parindent0em Note} that any monotonically increasing continuous function
$\theta(t)$ defined on a finite interval $(0,t_0]$ satisfying $\lim
\limits_{t \to +0} \theta(t)=0$ can be extended to a rate function $\bar
\theta$ such that $\bar \theta(t) = \theta(t)$ for $t\in (0,t_0]$.
Furthermore, the index functions corresponding to the classical Hilbert scales
$X_r$ can be seen to be power functions $\theta(\lambda) = \lambda^r$ for real
$r$. Rate functions are obtained for this case if $r>0$.
The set of index functions $\mathcal{I}$ includes the positive constant functions and
all power functions but not the zero function. We denote the pointwise
operations by $\phi+\psi$, $\phi\psi$ and $\phi/\psi$, respectively. As usual,
multiplication by a constant $\gamma$ is denoted by $\gamma \phi$. The
composition is denoted by $\phi\circ \psi$ where $(\phi\circ\psi)(\lambda) =
\phi(\psi(\lambda ))$. The pointwise maximum of two index functions is $\phi
\vee \psi$ defined by $(\phi \vee \psi)\, (\lambda) = \max(\phi(\lambda),
\psi(\lambda))$ and the pointwise minimum is $\phi \wedge \psi$. One verifies
that the index set $\mathcal{I}$ from Definition~\ref{def1}
is closed under
\begin{itemize}
\item point-wise addition, multiplication and division,
\item multiplication with positive constants,
\item pointwise maximum and minimum and
\item composition.
\end{itemize}
If an index function is injective and surjective, the inverse denoted by
$\phi^{-1}$ is also an index function. Not every index function is invertible,
however. The reciprocal function of $\phi$ (with values $1/\phi(\lambda)$) is
denoted by $1/\phi$.
The variable Hilbert scales are then families of Hilbert spaces indexed by
$\mathcal{I}$.
\begin{definition} \label{def2}
For a given injective self-adjoint positive definite linear operator $T$
densely defined on a Hilbert space $X$ we define the \emph{variable Hilbert
scale} $\{X_\theta\}_{\theta \in I}$ as a family of Hilbert spaces
$X_\theta$ indexed by functions $\theta$ from the set $\mathcal{I}$ of index functions
in the sense of Definition~\ref{def1}. Every Hilbert space $X_\theta$ with
$\theta \in I$ is then the closure of the domain of the quadratic form
\begin{equation}
\label{theta-norm}
\|f\|_\theta^2 = (f, \theta(T)\,f)\,.
\end{equation}
\end{definition}
In such a way, variable Hilbert scales were introduced by Hegland
in~\cite{Heg92} for the special case of $T$ being the inverse of a compact
operator and in~\cite{Heg95} for more general $T$. The choice of the operator
$T$ determines the Hilbert scale. In the simplest case where both $T$ and its
inverse $T^{-1}$ are bounded all the Hilbert spaces $X_\theta$ are isomorphic
to $X$ because of the continuity of the index functions $\theta$. In this
paper, in the context of ill-posed problems (\ref{opeq}) we often assume that
$T$ \emph{is unbounded but has a bounded inverse}, i.e.~the spectrum of $T$ is
contained in the interval $[\|T^{-1}\|^{-1},\infty)$ and has $+\infty$ as an
accumulation point. As the function $1/\lambda$ is an index function and the
set of index functions is closed under composition, the inverse $T^{-1}$
generates the same Hilbert scale as $T$. It is thus not necessary to consider
variable Hilbert scales generated by invertible $T$ and bounded $T^{-1}$
separately. The more general case where both $T$ and the inverse $T^{-1}$ are
unbounded is only considered for the negative Laplacian $T=-Delta$ and in
particular $T=-d^2/dt^2$. For the more general case where also $A$ is
unbounded we refer to the recent paper \cite{HofMW09}. To get a link with
(\ref{opeq}), a particular $T$ is suggested either by the forward operator
$A$, by the operator $G$ of condition (\ref{Grange}) or based on a combination
of both. A common choice is $T=(A^*A)^{-1}$ for injective operators $A$ with a
non-closed range. It follows that $A^*A=\theta(T)$ if
$\theta(\lambda)=1/\lambda$. For classes of problems connected with
deconvolution, however, $T=-d^2/dx^2$ on $L_2(\mathbb{R})$ is the canonical choice as
$T$ is the generator of symmetric convolutions. An index function $\theta$
such that $A^*A=\theta(T)$ is then found using Fourier transforms. More
generally, for problems where the source conditions relate to smoothness,
$T=-\Delta$ can be chosen. In such a case $\Delta$ denotes the Laplacian on
$L_2(\Omega)$ for some domain $\Omega\subseteq \mathbb{R}^d$.
It was shown in~\cite{Heg95} that there exists a continuous embedding $X_\phi
\hookrightarrow X_\psi$ if and only if $\phi\leq\gamma\,\psi$ for some
constant $\gamma>0$. If two different index functions $\theta_1$ and
$\theta_2$ are identical on the spectrum of $T$ they define the same norms and
hence the same space $X_{\theta_1}=X_{\theta_2}$. If they differ on the
spectrum they do define different norms, however, these norms may be
equivalent and thus the Hilbert spaces $X_{\theta_1}$ and $X_{\theta_1}$ as
elements of the variable Hilbert scale $\{X_\theta\}_{\theta \in I}$ are
indistinguishable.
The most important connection between the norms of different spaces $X_\theta$
is the \emph{interpolation inequality for variable Hilbert scales}.
\begin{lemma}[Interpolation inequality] \label{lem1}
Let $T$ be an unbounded injective self-adjoint positive definite linear
operator densely defined on the Hilbert space $X$ with bounded inverse
$T^{-1}: X \to X$. Moreover let $\phi,\psi,\theta$ and $\Psi$ be index
functions such that $\Psi$ is concave and
\begin{equation}
\label{condition1}
\phi(\lambda) \leq \Psi(\psi(\lambda)), \quad
\text{for $\;\|T^{-1}\|^{-1} \le \lambda<\infty$}.
\end{equation}
Then for any element $0 \not= f \in X_\theta\cap X_{\psi\theta}$ one gets $f
\in X_{\phi \theta}$ and
\begin{equation}
\label{interpolation}
\frac{\|f\|_{\phi\theta}^2}{\|f\|_\theta^2} \leq
\Psi\left(\frac{\|f\|_{\psi\theta}^2}{\|f\|_\theta^2}\right).
\end{equation}
\end{lemma}
\begin{proof}
Let in the following the measure $\nu$ be defined by
$$d\nu(\lambda) = \|f\|_\theta^{-2} \theta(\lambda)\,
d(f,E(\lambda)f) \quad \text{for $0 \not=f\in X_{\theta}$} $$
where $E(\lambda)$ is the spectral family or resolution of the
identity defined by $T$. By definition, because $\theta$ is positive and the
integral of $d\nu$ equals 1, $\nu$ is a probability measure. Taking into
account that $f \in X_{\psi\theta}$ and that $\Psi$ is concave we obtain
from the inverse Jensen inequality that $\int_0^\infty \Psi(\psi(\lambda))
\, d\nu(\lambda) \leq \Psi\left(\int_0^\infty \psi(\lambda)\,
d\nu(\lambda)\right)< \infty$. Because integration is monotone and the
inequality~\eqref{condition1} holds one gets $ \int_0^\infty \phi(\lambda)\,
d\nu(\lambda)\leq \int_0^\infty \Psi(\psi(\lambda)) \, d\nu(\lambda)<\infty$
and hence $f \in X_{\phi \theta}$. Summarising the results we arrive at the
inequality $$ \int_0^\infty \phi(\lambda)\, d\nu(\lambda) \leq
\Psi\left(\int_0^\infty \psi(\lambda)\, d\nu(\lambda)\right) $$ which
provides us with the required inequality~\eqref{interpolation} by replacing
$d\nu(\lambda)$ by its definition.
\end{proof}
The concavity of $\Psi$ is the key property which enables us to use Jensen's
inequality. The Lemma~\ref{lem2} below shows that this property has only to be
established for large arguments. We can focus on large arguments, if the
spectrum of $T$ for $T$ under consideration contains only sufficiently large
values and has $+\infty$ as an accumulation point. We need some auxiliary
result:
\begin{lemma}\label{lem:increase}
If $\theta : [t_0,\infty) \rightarrow (0,\infty)$ is concave for some $t_0>
0$ then $\theta$ is monotonically increasing. If moreover $\lim \limits_{t
\to \infty} \theta(t)=\infty$, then $\theta$ is even strictly increasing.
\end{lemma}
\begin{proof}
We show the contraposition. Assume that $\theta:[t_0,\infty)\rightarrow
(0,\infty)$ is not monotonically increasing. Then there exist $t_0<t_1 <
t_2$ such that $\theta(t_1)>\theta(t_2)$. Let
$$
l(t) = \frac{t-t_1}{t_2-t_1}\theta(t_2) + \frac{t_2-t}{t_2-t_1}\theta(t_1)
$$
be the linear interpolant of $\theta$ in $[t_1,t_2]$. As the slope of $l(t)$
is $(\theta(t_2)-\theta(t_1))/(t_2-t_1) < 0$ one has $l(t)\rightarrow
-\infty$ for $t\rightarrow \infty$. As $\theta(t) \geq 0$ there exists a
$t_3 > t_2$ such that $\theta(t_3) > l(t_3)$. By rearranging this inequality
one gets
$$
\theta(t_2) < \frac{t_2-t_1}{t_3-t_1}\theta(t_3) +
\frac{t_3-t_2}{t_3-t_1}\theta(t_1)
$$
and so $\theta$ is not concave. The strict monotonicity for $\lim \limits_{t
\to \infty} \theta(t)=\infty$ follows immediately from the fact that the
hypograph of a concave function is a convex set.
\end{proof}
Now we can replace index functions which are concave for large arguments
by such which are globally concave in the following way:
\begin{lemma} \label{lem2}
Let $\theta(\lambda)$ be an index function which is concave and hence by
Lemma~\ref{lem:increase} increasing on the interval $0<\lambda_0 \le
\lambda<\infty$. Then there exists an index function $\Psi(\lambda)$ which
is concave for all $0<\lambda<\infty$ such that with some
$\lambda_1>\lambda_0$ one has
$$
\Psi(\lambda) = \theta(\lambda), \quad \mbox{for}\;\; \lambda_1 \le
\lambda < \infty\,,
$$
$$
\Psi(\lambda) = \lambda \Psi(\lambda_1)/\lambda_1, \quad
\mbox{for}\;\;0<\lambda \le \lambda_1\,.
$$
\end{lemma}
\begin{proof}
To obtain the assertion of this lemma we consider the set of real numbers
$\{\alpha \mid \alpha \lambda \geq \theta(\lambda), \lambda_0 \le
\lambda< \infty \}$. As $\theta(\lambda)$ is concave for $\lambda_0 \le
\lambda< \infty$ this set is not empty and it is bounded below by zero. Thus
it does have a greatest lower bound $\alpha_0 \geq 0$ such that
\begin{itemize}
\item $\alpha_0 \lambda \geq \theta(\lambda)$ for $\lambda_0 \le \lambda<
\infty$,
\item there is a $\lambda_1$ such that
$\alpha_0\lambda_1=\theta(\lambda_1)$ if not, $\alpha_0$ would not
be the greatest lower bound.
\end{itemize}
Hence, knowing from Lemma~\ref{lem:increase} that an index function $\Psi$
which is concave for all $0<\lambda<\infty$ is always increasing, the
function $\Psi(\lambda)$ can be composed of a linear function growing from
zero to $\theta(\lambda_1)$ in the interval $(0,\lambda_1]$ and coinciding
with $\theta$ for greater arguments.
\end{proof}
The interpolation inequality is the main tool to obtain error bounds for
solvers of linear ill-posed problems. However, taking into account
Lemma~\ref{lem2} by inspection it becomes clear that rate results derived from
Lemma~\ref{lem1} are only based on the behaviour of $\Psi(\lambda)$ for large
$\lambda \ge \lambda_1$.
Without loss of generality $\Psi$ can be amended for $0<\lambda \le \lambda_1$
by the linear function $\Psi(\lambda) = \Psi(\lambda_1)\lambda/\lambda_1$ for
$0<\lambda < \lambda_1$.
Three typical choices for $\Psi(\lambda)$ being concave at least for
sufficiently large $\lambda$ are
\begin{itemize}
\item $\Psi(\lambda) = \lambda^\kappa$ where $\kappa\in(0,1)$
\item $\Psi(\lambda) = \lambda/\log(\lambda)$
\item $\Psi(\lambda) = \log(\lambda)$.
\end{itemize}
For all three choices we have the limit condition
\begin{equation} \label{limquo}
\lim \limits _{\lambda \to \infty} \frac{\Psi(\lambda)}{\lambda} \,=\,0
\end{equation}
and one gets the following versions of
interpolation inequalities from Lemma~\ref{lem1}:
\begin{itemize}
\item For $\Psi(\lambda)=\lambda^\kappa$ one gets
$$
\|f\|_{\phi\theta} \leq \|f\|_\theta^{1-\kappa}\,
\|f\|_{\psi\theta}^{\kappa}\,,
$$
\item for $\Psi(\lambda) = \lambda/\log(\lambda)$ one gets
$$
\|f\|_{\phi\theta} \leq \frac{\|f\|_{\psi\theta}}
{\sqrt{2\log(\|f\|_{\psi\theta}/\|f\|_\theta)}}\,,
$$
\item and for $\Psi(\lambda) = \log(\lambda)$ one has
$$
\|f\|_{\phi\theta} \leq \|f\|_\theta
\sqrt{2\log(\|f\|_{\psi\theta}/\|f\|_\theta}.
$$
\end{itemize}
Asymptotically, i.e.~for $\|f\|_\theta\rightarrow 0$, the interpolation
inequality allows us to find error bounds in the application to the error
estimation for the solution of equation (\ref{opeq}). One aims to get bounds
for the norm $\|f\|$ in $X$ using values of the image norm $\|Af\|$ in $Y$ and
values of the norm $\|f\|_{\psi\theta}$ which expresses the specific
additional smoothness of $f$. The terms in the interpolation
inequality~\eqref{interpolation} are then
$$
\|f\|_{\phi\theta} = \|f\| \;\; \mbox{for} \;\; f\in X_{\phi\theta} \quad
\mbox{and} \quad \|f\|_\theta = \|Af\| \;\; \mbox{for} \;\; f\in X_\theta.
$$
The first condition leads to $\phi(\lambda)\theta(\lambda)=1$ for all
$\lambda$ and the second condition gives $\theta(T)= A^*A$ and with
$\theta(\lambda):=1/\lambda$ the relations $T = (A^*A)^{-1}$ and
$\phi(\lambda) = \lambda$. We are still free to choose the index functions
$\psi$ and do it in the form $\psi(\lambda):= \chi(\lambda)\,\lambda$ with an
appropriate index function $\chi$.
For later use we add here some observations about convex functions which are
stated as a lemma:
\begin{lemma} \label{concave1}
Let $\Psi :(0,\infty) \rightarrow (0,\infty)$ be a concave function. Then we
have the following properties:
\begin{itemize}
\item[(a)] The function $\Xi :(0,\infty) \rightarrow (0,\infty)$ defined by
\begin{equation} \label{concaveprop1}
\Xi(\lambda) := \frac{\Psi(\lambda)}{\lambda},\qquad 0<\lambda<\infty
\end{equation}
is monotonically decreasing.
\item[(b)] The function $\Phi :(0,\infty) \rightarrow (0,\infty)$ defined by
\begin{equation} \label{concaveprop2}
\Phi(\mu) := \mu \, \Psi \left(\frac{1}{\mu}\right),\qquad 0<\mu<\infty
\end{equation}
is concave and hence monotonically increasing.
\end{itemize}
\end{lemma}
\begin{proof}
(a)\, Let $0<\lambda_0<\lambda_1<\lambda_2$. As $\Psi$ is concave and positive
one has
\begin{align*}
\Psi(\lambda_1) &
\geq\frac{\lambda_1-\lambda_0}{\lambda_2-\lambda_0}\,\Psi(\lambda_2) +
\frac{\lambda_2-\lambda_1}{\lambda_2-\lambda_0}\, \Psi(\lambda_0) \\
& \geq\frac{\lambda_1-\lambda_0}{\lambda_2-\lambda_0}\,\Psi(\lambda_2).
\end{align*}
As this holds for arbitrarily small $\lambda_0>0$ on has
$$
\Psi(\lambda_1) \geq \frac{\lambda_1}{\lambda_2}\, \Psi(\lambda_2)
$$
and consequently $\Xi(\lambda_1) \geq \Xi(\lambda_2)$. This proves assertion
(a) of the lemma.
(b) \, Let $0 < \mu_0 < \mu_1 < \mu_2$ and $\lambda_i = 1/\mu_i$. Then one
has $0 < \lambda_2 < \lambda_1 < \lambda_0$ and by the concavity of $\Psi$
and some simple algebraic manipulations one gets
\begin{align*}
& \frac{\mu_1-\mu_0}{\mu_2-\mu_0} \Phi(\mu_2) +
\frac{\mu_2-\mu_1}{\mu_2-\mu_0} \Phi(\mu_0) =
\frac{\frac{1}{\lambda_1}-\frac{1}{\lambda_0}}
{\frac{1}{\lambda_2}-\frac{1}{\lambda_0}}
\frac{\Psi(\lambda_2)}{\lambda_2} +
\frac{\frac{1}{\lambda_2}-\frac{1}
{\lambda_1}}{\frac{1}{\lambda_2}-\frac{1}{\lambda_0}}
\frac{ \Psi(\lambda_0)}{\lambda_0} \\
& = \frac{1}{\lambda_1} \left(\frac{\lambda_0-\lambda_1}
{\lambda_0-\lambda_2}\Psi(\lambda_2)
+ \frac{\lambda_1-\lambda_2}{\lambda_0-\lambda_2}
\Psi(\lambda_0)\right) \\
& \leq \frac{1}{\lambda_1}\, \Psi(\lambda_1) = \Phi(\mu_1).
\end{align*}
It follows that $\Phi$ is concave and hence by Lemma~\ref{lem:increase}
also increasing. This completes the proof of the lemma.
\end{proof}
\begin{remark} \label{remconnew}
{\rm We note here that the transformation $\mathcal{S}: \,\Psi \in
\mathcal{I} \mapsto \Phi \in \mathcal{I}$ according to formula
(\ref{concaveprop2}),
applicable to every index function and \emph{preserving concavity}, is an
\emph{involution}, that means $\mathcal{S}^{-1}=\mathcal{S}$ and hence
$\mathcal{S}$ is bijective.
If the concave index function $\Psi$ satisfies $\lim\limits_{\lambda \to
\infty} \Psi(\lambda)= \infty$, then by Lemma~\ref{lem:increase} the function
is even strictly increasing and if, in addition, $\Psi$ is a rate function,
i.e., it satisfies the additional limit condition $\lim\limits_{\lambda \to
+0} \Psi(\lambda)=0$ (which is also motivated by Lemma~\ref{lem2}), the
inverse function $\Psi^{-1}$ is a well-defined and convex index function.
If, on the other hand, the limit condition (\ref{limquo}) holds, then we have
$$
\lim \limits_{\mu \to +0} \Phi(\mu)= \lim \limits_{\mu \to +0} \mu
\,\Psi(1/\mu)= \lim \limits_{\lambda \to \infty} \Psi(\lambda)/\lambda=0
$$
and taking into account Lemma~\ref{concave1} (a) and (b) one sees that
$\Phi=\mathcal{S}(\Psi)$ is a concave \emph{rate function}. Vice versa we
have that $\Psi=\mathcal{S}(\Phi)$ satisfies (\ref{limquo}) whenever $\Phi$
is a rate function.
By inspection of the proof of Lemma~\ref{concave1} one can also see the
following facts: If $\Psi(\lambda)$ is only concave for $\lambda \in
[\lambda_0,\infty)$, then $\Phi(\mu)=[\mathcal{S}(\Psi)](\mu)$ is concave for
$\mu \in (0,\mu]$ with $\mu_0=1/\lambda_0$. The involution $\mathcal{S}$
preserves also the convexity of an index function and if the concavity or
convexity is strict, then the strictness carries over to the transformed
function.
}\end{remark}
Now we are ready to draw conclusions from Lemma~\ref{lem1}. A first, abstract
version of bounds for errors of regularised solutions is given in the
following corollary. We will denote by $f_\alpha$ an approximation of a
solution $f$ to equation (\ref{opeq}) which is computed from an approximate
right-hand side using a regularisation method and a regularisation parameter
$\alpha>0$.
\begin{corollary} \label{cor1}
Let $A:X\rightarrow Y$ be an injective bounded linear operator with non-closed
range mapping between the two Hilbert spaces $X$ and $Y$. Furthermore let the
variable Hilbert scale $\{X_\nu\}_{\nu \in I}$ be generated by $T=(A^*A)^{-1}$
such that any scale element $X_\nu$ has a norm denoted by $\|\cdot\|_\nu$.
Moreover let $\chi$ and $\Psi$ be index functions and $\Psi$ be concave such
that
\begin{equation} \label{Psiineq}
\Psi\left(\chi(\lambda)\,\lambda\right) \geq \lambda \quad
\text{for all $\;\|T^{-1}\|^{-1} \le \lambda< \infty$}\,.
\end{equation}
If the solution $f$ to (\ref{opeq}) in addition satisfies the condition $f\in
X_\chi$ and if $f_\alpha \in X_\chi$ is such that
\begin{align}
\|f_\alpha -f \|_\chi &= \zeta>0 \label{stability} \\
\|Af_\alpha - g \| &= \epsilon>0 \label{consistency}
\end{align}
then
\begin{equation} \label{upeps}
\|f-f_\alpha\| \leq \epsilon \sqrt{\Psi(\zeta^2/\epsilon^2)}.
\end{equation}
\end{corollary}
\begin{proof}
By Lemma~\ref{lem1} with $\theta(\lambda)=1/\lambda$,
$\phi(\lambda)=\lambda$, $\psi(\lambda)=\chi(\lambda)\,\lambda$
and for a concave index function $\Psi$ satisfying (\ref{Psiineq}) one has
for all $0 \not=h \in X_\chi$
\begin{equation} \label{auxquo}
\|h\|^2 \leq \|Ah\|^2\,
\Psi\left(\frac{\|h\|_\chi^2}{\|Ah\|^2}\right).
\end{equation}
Setting $h:=f_\alpha-f$ this yields the estimate
$$ \|f_\alpha-f\|^2 \leq \|A(f_\alpha-f)\|^2 \,
\Psi\left(\frac{\|f_\alpha-f\|_\chi^2}{\|Af_\alpha-Af\|^2}\right)=\epsilon^2 \Psi((\zeta/\epsilon)^2)$$
and proves the assertion of the corollary.
\end{proof}
Results similar to those of Corollary~\ref{cor1} can be found for other choices of $T$, see for example Corollary~\ref{cor1a} where $T=-d^2/dt^2$. The Corollary~\ref{cor1} can be interpreted as an instance of an \emph{abstract Lax theorem}~\cite{KenH09} where the condition~\eqref{stability} is a stability condition and the bound~\eqref{consistency} relates to consistency.
Note that the error estimate (\ref{upeps}) of Corollary~\ref{cor1} requires
the essential conditions $f \in X_\chi$ and $f_\alpha \in X_\chi$, i.e.~the
approximate solutions $f_\alpha$ are constructed such that they obtain the
same smoothness level with respect to $T$ as the exact solution $f$. A next
step for drawing conclusions of Lemma~\ref{lem1} will be formulated in
Corollary~\ref{cor11} by assuming that $f$ belongs to ball
\begin{equation} \label{eq:ballchi}
B_\chi(R):=\{h \in X_\chi:\; \|h\|_\chi \le R \}
\end{equation}
in $X_\chi$ with positive radius $R=R_1$ and that the approximate solutions
$f_\alpha$ for all $\alpha>0$ under consideration belong to another such ball
with radius $R=R_2$. Moreover, we consider for data $g^\delta$ satisfying
(\ref{noise}) the limit process $\delta \to +0$ in correspondence with
associated regularized solutions $f_\alpha$, where the regularisation
parameter $\alpha>0$ is chosen either a priori as $\alpha=\alpha(\delta)$ or
a posteriori as $\alpha=\alpha(\delta,g^\delta)$.
\begin{corollary} \label{cor11}
Under the setting of Corollary~\ref{cor1} let the limit condition
(\ref{limquo}) be satisfied and let $f \in B_\chi(R_1),\;R_1>0$. Moreover with
prescribed $\delta_{max}>0$ let $f_\alpha \in B_\chi(R_2),$
$R_2>0,$ for all $\alpha$ attributed to $\delta \in (0,\delta_{max}]$ and
$g^\delta$ satisfying (\ref{noise}) such that
\begin{equation} \label{eq:ratexi}
\|Af_\alpha-g\| \le \bar C\,\bar \xi(\delta), \qquad 0<\delta \le \delta_{max},
\end{equation}
for some rate function $\bar \xi$ and some constant $\bar C>0$.
Then we have
\begin{equation} \label{eq:specialxi}
\|f-f_\alpha\|\, \le \bar C \bar
\xi(\delta)\,\sqrt{\Psi\left(\left[\frac{R_1+R_2}{\bar C \bar
\xi(\delta)}\right]^2\right)},\qquad 0<\delta \le \delta_{max}\,,
\end{equation}
where the upper bound in (\ref{eq:specialxi}) is a rate function, i.e., it
tends to zero as $\delta \to 0.$
\end{corollary}
\begin{proof}
Since $\Psi$ is concave by Lemma~\ref{lem:increase} the error norm $\epsilon\,
\sqrt{\Psi(\zeta^2/\epsilon^2)}$ obtained from (\ref{upeps}) is
increasing in $\zeta>0$ for fixed $\epsilon>0$ and
as a consequence of Lemma~\ref{concave1} (a) this upper bound $\epsilon\,
\sqrt{\Psi(\zeta^2/\epsilon^2)}= \zeta\,
\sqrt{\frac{\Psi(\zeta^2/\epsilon^2)}{(\zeta/\epsilon)^2}}$ is increasing in
$\epsilon>0$ for fixed $\zeta>0$.
Moreover, due to (\ref{limquo}) in the limit process $\epsilon \to 0$ for
fixed $\zeta>0$ implying $\zeta/\epsilon \to \infty$ this bound and hence the
error norm in $X$ even tends to zero. For the mentioned kinds of monotonicity
we obtain formula (\ref{eq:specialxi}) by $\epsilon \le \bar C \xi(\delta)$
and $\zeta =\|f-f_\alpha\|_\chi\le \|f\|_\chi+ \|f_\alpha\|_\chi \le
R_1+R_2$. The upper bound in (\ref{eq:specialxi}) is a rate function
declining to zero as $\delta \to 0$ because $\bar \xi$ is a rate function.
\end{proof}
\begin{remark} \label{Remark_1}
{\rm As a special case for the situation of Corollary~\ref{cor11} we can
consider an a posteriori choice $\alpha_{dis}=\alpha_{dis}(\delta,g^\delta)$
for the regularisation parameter realised by a
\emph{discrepancy principle}
\begin{equation} \label{Morozov}
\|Af_{\alpha_{dis}}-g^\delta\|= C_{dis} \delta
\end{equation}
with some prescribed $ C_{dis}>0$.
Then by using the triangle inequality we obtain with~(\ref{noise}) as noise
model $$ \|Af_{\alpha_{dis}}-g\| \le
\|Af_{\alpha_{dis}}-g^\delta\|+\|g^\delta-g\| \le (C_{dis}+1)\,\delta=\bar C\,
\delta\,. $$ Then for such $\alpha=\alpha_{dis}$ under (\ref{limquo}) the
regularisation method converges strongly in $X$ with the convergence rate
\begin{equation} \label{specialrate1}
\|f-f_\alpha\|\,=\,\mathcal{O}\left(\delta\,\sqrt{\Psi(\bar
K/\delta^2)}\right) \quad \mbox{as} \quad \delta \to 0 \end{equation} for some
constant $\bar K>0$. Note that beside the assumption $f \in B_\chi(R_1)$ on
the solution smoothness for that result the strong condition
$f_{\alpha(\delta,g^\delta)}\in B_\chi(R_2)$ for all $\delta \in
(0,\delta_{max}]$ and all associated $g^\delta$ satisfying (\ref{noise}) is
required.
}\end{remark}
The convergence rate in (\ref{specialrate1}) depends only on the asymptotic
behaviour of $\Psi(\lambda)$ as $ \lambda \to \infty$. Thus the alteration of
$\Psi(\lambda)$ for small $\lambda$ in the sense of Lemma~\ref{lem2} has no
influence on that rate. For the class of functions
$\Psi(\lambda)=\lambda^\kappa$ with $0<\kappa<1$ rate functions proportional to
$\delta^{1-\kappa}$ occur in (\ref{specialrate1}). All those error rates are
lower than the rate $\delta$ which is typical for well-posed problems. It
should be mentioned that $\Psi(\lambda)=\lambda$ fails to satisfy the condition
(\ref{limquo}) and used in Corollary~\ref{cor1} the inequality (\ref{auxquo})
does not yield a convergence rate.
To get a feeling for the role of the solution smoothness $f \in X_\chi$ we can
study consequences of the inequality (\ref{Psiineq}) as a hypothesis of
Corollary~\ref{cor1} taking into account Lemma~\ref{lem:increase}. One
consequence of (\ref{Psiineq}) is the limit condition $\lim_{\lambda \to
\infty} \Psi(\lambda)=\infty$ for the function $\Psi$ which is because of its
concavity then strictly increasing and invertible with convex
$\Psi^{-1}(\lambda)$ also tending to infinity as $\lambda \to \infty$. Then
(\ref{Psiineq}) implies $\chi(\lambda) \ge \frac{\Psi^{-1}(\lambda)}{\lambda}$
for large $\lambda$. Under that condition (\ref{limquo}) is equivalent to
$\lim_{\lambda \to \infty} \Psi^{-1}(\lambda)/\lambda= \infty$. Hence, the
index function $\chi(\lambda)$ tends to infinity for $\lambda \to \infty$
provided that (\ref{limquo}) holds true.
When setting $\phi(\lambda):=\lambda,\;\theta(\lambda):=1/\lambda$ and
$\psi(\lambda):=\Psi^{-1}(\lambda)$ in the interpolation inequality
(\ref{interpolation}) then the corresponding \emph{regularity condition} $f
\in X_{\psi\theta}$ is equivalent to a \emph{source condition}
(\ref{scclassic}) which expresses the specific smoothness of the solution $f$
with respect to the forward operator $A$ of equation (\ref{opeq}).
\begin{proposition}
\label{prop1}
Let $\Psi(\lambda)$, for $0<\lambda<\infty$, be a concave and strictly
increasing index function satisfying the limit conditions $\lim
\limits_{\lambda \to +0} \Psi(\lambda)=0,\;$ $\lim \limits_{\lambda \to
\infty} \Psi(\lambda)=\infty$ and (\ref{limquo}). Moreover let $T=(A^*A)^{-1}$
and set $\phi(\lambda):=\lambda,\;\theta(\lambda):=1/\lambda$ as well as
$\psi(\lambda):=\Psi^{-1}(\lambda)$ for $0<\lambda<\infty$. Then we have $f
\in X_{\psi\theta}$ if and only if $f$ satisfies a source condition
(\ref{scclassic}) with the function
\begin{equation} \label{tildeop}
\bar{\psi}(t) = \frac{1}{\sqrt{t\Psi^{-1}(1/t)}},\qquad 0<t<\infty,
\end{equation}
which is then a rate function.
\end{proposition}
\begin{proof}
Under the stated assumptions the function $\bar{\psi}$ is well-defined and a
rate function. Namely, we can write
$\frac{1}{\sqrt{t\Psi^{-1}(1/t)}}=\sqrt{\frac{\Psi(u)}{u}}$ when using the
substitution $u:=\Psi^{-1}(1/t)$. The variable $u>0$ is strictly decreasing
with respect to $t>0$ such that $u \to \infty$ corresponds with $t \to +0$ and
vice versa $t \to \infty$ corresponds with $u \to +0$, because $\Psi^{-1}$ is
also strictly increasing and we have $\lim \limits_{\lambda \to \infty}
\Psi^{-1}(\lambda)=\infty$ and $\lim \limits_{\lambda \to +0}
\Psi^{-1}(\lambda)=0$ for the functions $\Psi$ under consideration. Now by
(\ref{limquo}) we have $\lim \limits_{u \to \infty}\frac{\Psi(u)}{u}=0$ and
with Lemma~\ref{concave1} (a) the quotient $\frac{\Psi(u)}{u}$ is
monotonically decreasing in $u>0$. This, however, implies that $\bar \psi(t)$
is monotonically increasing for $t>0$ with limit condition $\lim \limits_{t
\to +0} \bar \psi(t)=0$. Hence, $\bar \psi$ is a rate function.
Moreover, we have
$$f\in X_{\psi\theta} \quad \Longleftrightarrow \quad
(f,\Psi^{-1}(T)T^{-1}f)<\infty$$ and $$
f=\bar{\psi}(A^*A)v, \;\; \mbox{for} \;\; v\in X \quad \Longleftrightarrow
\quad ([\bar{\psi}(A^*A)]^{-1}f,[\bar{\psi}(A^*A)]^{-1}f) <\infty\,.
$$
One has
equivalence if and only if
$$
[\Psi^{-1}((A^*A)^{-1})](A^*A) = [\bar{\psi}(A^*A)]^{-2}
$$
and the claim follows. This proves the proposition.
\end{proof}
After the millennium {\sc Math\'e and Pereverzev} with coauthors seized,
reused and extended {\sc Hegland}'s ideas and concepts of variable Hilbert
scales and corresponding interpolation inequalities from \cite{Heg92,Heg95}
for linear ill-posed problems and their regularisation
(cf.~\cite{MatPer03b,MatPer03a}) and combined it
(cf.~\cite{HofMAt07,MatTau06}) with the concept of {\sl approximate source
conditions} (cf.~\cite{DHY07,Hof06}). The comprehensive theory developed
therein considers only rate functions as index functions. Such an approach
leads in general to different formulae compared with the results based on the
concept of Lemma~\ref{lem1}, but as we will outline in the sequel clear
cross-connections and sometimes even equivalences of the assertions obtained
characterise the two different ways.
\section{Modulus of continuity of $A^{-1}$}
The \emph{modulus of continuity} of $A^{-1}$ restricted to the set $AM$ with
$M \subseteq X$ is
$$
\omega(M,\delta) = \sup \{\|x\|:\; x\in M, \; \|Ax\| \leq \delta\}.
$$
The impact of the modulus of continuity on error bounds in regularisation has
recently been discussed in the paper \cite[\S 4]{HofMS08}. It is well-known
that the worst case error
$$
e(\hat f,M,\delta):= \sup \limits _{f \in M} e(\hat f,f,\delta) \qquad
\mbox{for} \qquad e(\hat f,f,\delta):= \sup \limits _{g^\delta \in
Y:\,\|Af-g^\delta\|\le \delta} \|\hat f(g^\delta) -f\|
$$
of linear and nonlinear reconstruction methods $\hat f:\, g^\delta \in Y
\mapsto \hat f (g^\delta) \in X$ has an infimum
$$
e(M,\delta):= \inf \limits _{\hat f:\;Y \to X} e(\hat f,M,\delta)
$$
which satisfies for centrally symmetric and convex sets $M$ the inequalities
$$
\omega(M,\delta) \le e(M,\delta) \le \omega(M,2\delta) \le 2
\omega(M,\delta)\,.
$$
Hence the modulus of continuity $\omega(M,\delta)$ serves as benchmark for the
reconstruction error of $\hat f$ when $f,\hat f \in M$ can be assumed. For
example, from \cite[Lemma~4.2]{HofMS08} one can find a minimax-expression for
the modulus of continuity in the case of centrally symmetric and convex {\sl
source sets} for $M$ of the form
\begin{equation} \label{Gset}
M=G [B(R)]:=\{x \in X:\; x=Gv, \;v \in X,\; \|v\| \le R\}\,.
\end{equation}
corresponding to condition (\ref{Gsmooth}). This expression gets an explicit
bound from above for the special case $G=\bar \psi(A^*A)$ and
$$
M=\bar \psi(A^*A)[B(R)]:=\{x \in X:\; x= \bar \psi(A^*A)v, \;v \in X,\;
\|v\| \le R\}
$$
associated with the source condition (\ref{scclassic}). Note that the rate
function $\bar \psi(t)$ is only of interest
here for arguments $0<t \le \|A\|^2$, but without loss of generality
(cf.~\cite[Theorem 1 (b)]{HofMW09}) we can extend $\bar \psi$ to be a
monotonically increasing index function defined on $(0,\infty)$. Then by using
the strictly increasing auxiliary function
\begin{equation} \label{aux1}
\Theta(t):= \sqrt{t}\, \bar \psi(t)\,, \qquad 0<t<\infty
\end{equation}
satisfying the limits conditions $\lim \limits_{t \to +0} \Theta(t)=0$ and
$\lim \limits_{t \to \infty} \Theta(t)=\infty$ one obtains for
$M=\bar\psi(A^*A)[B(R)]$
\begin{equation} \label{MPoptirate}
\omega(M,\delta)\,\le\,R\,\bar \psi
\left(\Theta^{-1}\left(\frac{\delta}{R}\right) \right), \qquad \delta>0
\end{equation}
provided that
\begin{equation} \label{concavecond1}
\bar\psi^2((\Theta^2)^{-1}(t)) \quad \mbox{is concave for} \quad 0<t<\infty\,.
\end{equation}
This result can be derived from Corollary~3.7 and Theorem~2.1(c) in
\cite{HofMS08} (see also Theorem~1 in the earlier paper \cite{MatPer03a}).
A similar assertion was already mentioned in a rudimentary form in a paper by
Ivanov and Korolyuk in 1969~\cite{IvaK69}.
The following proposition also yields an upper bound for the modulus of
continuity based on a variable Hilbert scale interpolation inequality using
Lemma~\ref{lem1} or Corollary~\ref{cor1}. For the proof we use
Lemma~\ref{concave1} (b).
\begin{proposition} \label{prop_simpler}
Let $\Psi(\lambda)$, for $0<\lambda<\infty$, be a concave and strictly
increasing index function satisfying the limit conditions $\lim
\limits_{\lambda \to +0} \Psi(\lambda)=0,\;$ $\lim \limits_{\lambda \to
\infty} \Psi(\lambda)=\infty$ and (\ref{limquo}), for which an index
function $\chi$ exists that satisfies
$$
\chi(\lambda) \ge \Psi^{-1}(\lambda)/\lambda, \qquad 0<\lambda <\infty.
$$
Furthermore let $X_\chi$ be an element of a Hilbert scale generated by
$T=(A^*A)^{-1}$ where $A$ is injective. Then
\begin{equation} \label{omega_ball}
\omega(M,\delta) \leq \delta\,
\sqrt{\Psi\left(\frac{R^2}{\delta^2}\right)},\qquad \delta>0
\end{equation}
for $M=B_{\chi}(R)$.
\end{proposition}
\begin{proof}
Under the assumptions stated on $\Psi$ and $\chi$ Corollary~\ref{cor1}
applies. Then from formula (\ref{auxquo}) we can conclude that $\|h\| \le
\|Ah\| \sqrt{\Psi\left(\frac{\|h\|^2_\chi}{\|Ah\|^2}\right)}$ for all $0
\not= h \in X_\chi$. As $\Psi$ is monotonically increasing one then gets
$\|h\| \le \|Ah\| \sqrt{\Psi\left(\frac{R^2}{\|Ah\|^2}\right)}$. Now by
Lemma~\ref{concave1} (a) the function $\Xi(\zeta)=\Psi(\zeta)/\zeta$ is
monotonically increasing and so $\Xi(\zeta_1) \ge \Xi(\zeta_2)$ for
$0<\zeta_1 \le \zeta_2<\infty$. This gives with
$\zeta_1:=\frac{R^2}{\delta^2}$ and $\zeta_2:=\frac{R^2}{\|Ah\|^2}$ the
estimate $\|h\| \le \delta\, \sqrt{\Psi\left(\frac{R^2}{\delta^2}\right)}$
for all $h \in B_{\chi}(R)$ satisfying the additional condition $\|Ah\| \le
\delta$. Thus the proposition is proven.
\end{proof}
We note that for centrally symmetric and convex sets $M, \;f \in M,$ and
regularised solutions $f_{\alpha_{\text{dis}}} \in M$ obtained from the
discrepancy principle of form (\ref{Morozov}) mentioned in
Remark~\ref{Remark_1} we easily derive along the lines
of~\cite[Lemma~2.2]{HofMS08} that
\begin{equation} \label{quasi1}
\|f-f_{\alpha_{\text{dis}}}\| \le \omega(2M,(C_{\text{dis}}+1)\delta)
\end{equation}
with $2M:=\{u \in X:\,u=2v,\;v \in M\}$. In the case $M=B_{\chi}(R)$ with
$2M=B_\chi(2R)$ the estimate (\ref{quasi1}) yields with (\ref{omega_ball}) a
convergence rate of the form (\ref{specialrate1}) with constant $\bar
K=4R^2/(C_{\text{dis}}+1)^2$. With more generality such rates were verified
above directly from Corollary~\ref{cor1}.
Under weak additional assumptions (see~\cite[Corollary~3.7]{HofMS08}) there
is also a constant $\underline{C}>0$ such that $$
\omega(B_{\chi}(R),\delta)\, \geq \, \underline C\,\delta\,
\sqrt{\Psi\left(\frac{R^2}{\delta^2}\right)},\qquad \delta>0 \,.$$ Then a
convergence rate of the form (\ref{specialrate1}) is \emph{order optimal}
independent of the constant $\bar K>0$ because of
$\sqrt{\Psi\left(\frac{C\,R^2}{\delta^2}\right)} \le \max\{C,1\}
\sqrt{\Psi\left(\frac{R^2}{\delta^2}\right)}$ for all $C>0$. On the other
hand, Corollary~\ref{cor11} yields an error estimate of best order just for
$\xi(\delta) \sim \delta$, hence the discrepancy principle is order optimal
in that sense.
Evidently, under the assumptions of Proposition~\ref{prop1} with the
additional setting
\begin{equation} \label{connected}
\chi(\lambda)\,:=\,\frac{\Psi^{-1}(\lambda)}{\lambda}\,=\,
\frac{1}{\bar{\psi}(1/\lambda)^2}, \qquad 0<\lambda<\infty
\end{equation}
one has
$$
\bar\psi(A^*A)[B(R)] = B_\chi(R)
$$
where $B_\chi(R)$ denotes the ball (\ref{eq:ballchi}) of radius $R$ in
$X_\chi$, an element of the Hilbert scale generated by $T=(A^*A)^{-1}$
expressed through the index function $\chi$. We emphasise that the upper bound
in (\ref{omega_ball}) for the modulus of continuity from
Proposition~\ref{prop_simpler} needing only one function $\Psi$ has a
\emph{much simpler structure} than the nested upper bound in
(\ref{MPoptirate}) composing the functions $\bar \psi$ and $\Theta^{-1}$. Also
the required concavity of $\Psi$ for obtaining (\ref{omega_ball}) \emph{looks
much simpler} than the needed concavity of the composite function
$$
\bar\psi^2((\Theta^2)^{-1}(t))\,\equiv\,\bar\psi^2(\Theta^{-1}(\sqrt{t})),
\qquad 0<t<\infty,
$$
for obtaining (\ref{MPoptirate}).
Owing to the correspondence (\ref{tildeop}) between the concave index function
$\Psi$ and the rate function $\bar \psi$ it is of some interest to compare the
quality of the estimates (\ref{MPoptirate}) and (\ref{omega_ball}) as well as
the strength of conditions which have to imposed in order to ensure those
bounds for $\omega$.
\begin{proposition} \label{prop:onetoone}
Let $\Psi(\lambda)$, for $0<\lambda<\infty$, be a concave and strictly
increasing index function satisfying the limit conditions $\lim
\limits_{\lambda \to +0} \Psi(\lambda)=0,\;$ $\lim \limits_{\lambda \to
\infty} \Psi(\lambda)=\infty$ and (\ref{limquo}). Then for the rate function
$\bar \psi(t):=1/\sqrt{t\Psi^{-1}(1/t)}$ (cf.~(\ref{tildeop})) and by setting
$\Theta(t):=\sqrt{t} \bar \psi(t),\;0<t<\infty$, we have the following
assertions: The error bounds in (\ref{omega_ball}) and in (\ref{MPoptirate})
and the corresponding concavity conditions required for obtaining those
bounds coincide, i.e., we have
\begin{equation} \label{coin}
\delta\, \sqrt{\Psi\left(\frac{R^2}{\delta^2}\right)} \, =\, R\,\bar \psi
\left(\Theta^{-1}\left(\frac{\delta}{R}\right) \right)\,, \qquad
R>0,\quad\delta>0\,.
\end{equation}
Moreover, the function $\bar\psi^2((\Theta^2)^{-1}(t))$ is concave for all
$0<t<\infty$.
Vice versa, any rate function $\bar \psi(t),\;0<t<\infty,$ determines by
equation (\ref{tildeop}) in a unique manner a strictly increasing index
function $\Psi(\lambda),\,0<\lambda<\infty,$ satisfying the limit conditions
$\lim \limits_{\lambda \to +0} \Psi(\lambda)=0,\;$ $\lim \limits_{\lambda \to
\infty} \Psi(\lambda)=\infty$ and (\ref{limquo}) which is concave for all
$0<\lambda<\infty$ if $\bar\psi^2((\Theta^2)^{-1}(t))$ is concave for all
$0<t<\infty$ which again implies the coincidence (\ref{coin}) of the error
bounds.
\end{proposition} \begin{proof}
First we find from Proposition~\ref{prop1} that $\bar \psi(t),\;t>0,$ is a
rate function if $\Psi(\lambda),\,0<\lambda<\infty$ is a concave and strictly
increasing index function satisfying the limit conditions $\lim
\limits_{\lambda \to +0} \Psi(\lambda)=0,\;$ $\lim \limits_{\lambda \to
\infty} \Psi(\lambda)=\infty$ and (\ref{limquo}). Then from the right
equation in (\ref{connected}) (cf.~(\ref{tildeop})) we have
$\Psi^{-1}(\lambda)=\lambda/\bar \psi^2(1/\lambda)$. By using the bijective
substitution $u=1/\lambda$ in $(0,\infty)$ this yields
$\Psi^{-1}(1/u)=\frac{1}{\Theta^2(u)}$ and $\frac{1}{u}=
\Psi\left(\frac{1}{\Theta^2(u)}\right)$ for $0<u<\infty$. Multiplying the
last equation by the factor $u\, \bar \psi ^2(u)$ we derive $${\bar
\psi}^2(u)=u \,{\bar \psi}^2(u) \Psi\left(\frac{1}{\Theta^2(u)}\right)=
\Theta^2(u) \Psi\left(\frac{1}{\Theta^2(u)}\right)$$ and $\bar \psi(u)=
\Theta(u) \sqrt{\Psi\left(\frac{1}{\Theta^2(u)}\right)}$. By exploiting the
bijection $t=\Theta(u)$ of $(0,\infty)$ into itself this provides us with the
equation $ \bar \psi \left(\Theta^{-1}(t) \right)= t \sqrt{\Psi
\left(\frac{1}{t^2} \right)}$ which implies the required identity
(\ref{coin}) by inserting $t:=\delta/R$ and multiplying the arising equation
by $R$.
In a second step we note that by using the monotonically increasing
bijection $s=\Theta^2(u)$ between $s \in (0,\infty)$ and $u \in (0,\infty)$
and once more by exploiting the right equation in (\ref{connected}) we can
write as follows for all $s>0$:
$$\bar \psi^2((\Theta^2)^{-1}(s)) = \bar \psi^2(u) =\frac{\Theta^2(u)}{u}=
\Theta^2(u)\,\Psi \left( \frac{1}{\Theta^2(u)}\right) = s
\,\Psi\left(\frac{1}{s}\right)
=[\mathcal{S}(\Psi)](s).
$$
Hence, by Lemma~\ref{concave1} (b) we immediately see that as required
$\bar \psi^2((\Theta^2)^{-1}(s)),\,s>0,$ is concave if
$\Psi(\lambda),\,\lambda>0,$ is concave.
Since the involution $\mathcal{S}$ (cf.~Remark~\ref{remconnew}) preserves
concavity, the reverse assertion formulated in
Proposition~\ref{prop:onetoone} becomes immediately clear, since
(\ref{tildeop}) represents a one-to-one correspondence between index
functions $\bar \psi$ and strictly increasing functions $\Psi$ with the
limit conditions under consideration.
\end{proof}
We now investigate the concavity condition for the function $\bar \psi^2((\Theta^2)^{-1}(s))$ in more detail. For this a characterisation of the concavity of index functions is given in terms of the monotonicity of certain divided differences.
\begin{lemma}
\label{diffquot}
Let $\psi$ be an index function. Then the three following statements are
equivalent:
\begin{enumerate}
\item $\psi$ is concave
\item $(\psi(s_0+s)-\psi(s_0))/s$ is a decreasing index function for all
$s_0>0$
\item $(\psi(s_0)-\psi(s_0-s))/s$ is an increasing continuous function
$(0,s_0)\rightarrow \mathbb{R}_+$ for all $s_0>0$.
\end{enumerate} \end{lemma} \begin{proof}
If $\psi$ is a concave index function then by Lemma~\ref{lem:increase} $\psi$
is increasing and so both $(\psi(s_0+s)-\psi(s_0))/s$ and
$(\psi(s_0)-\psi(s_0-s)/s$ are positive continuous functions for $s>0$ and
$s\in (0,s_0)$, respectively. Furthermore by definition
$$
(t_2-t_0) \psi(t_1) \geq (t_2-t_1) \psi(t_0) + (t_1-t_0) \psi(t_2)
$$
and by simple algebraic manipulations and the right choice of $t_0< t_1 < t_2$
one gets the second and third statement from the first.
Conversely, if $(\psi(s_0+s)-\psi(s_0))/s$ is a decreasing index function for
all $s_0>0$ one has for all $t_0< t_1 < t_2$
$$
\frac{\psi(t_1)-\psi(t_0)}{t_1-t_0} \geq \frac{\psi(t_2)-\psi(t_0)}{t_2-t_0}
$$
and thus $\psi$ is concave. A similar argument shows that $\psi$ is concave
if the third statement holds. \end{proof} A direct consequence of this lemma is that for concave rate functions $\bar\psi$ one has $$ \frac{\bar\psi(s_0)-\bar\psi(s_0-s)}{s} \leq \frac{\bar\psi(s_0)}{s_0} $$ as $\lim_{s\rightarrow 0}\, \bar\psi(s)=0$. Another consequence is \begin{proposition}
If $\psi(t)$ is a concave rate function then so is $\psi(\sqrt{t})^2$. \end{proposition} \begin{proof}
By lemma~\ref{diffquot} we have to show that for all $t_0>0$ the function
$(\psi(\sqrt{t+t_0})^2-\psi(t-0)^2)/t$ is a decreasing index function. As the
mapping $s\rightarrow (s+s_0)^2$ is monotone it is sufficient to show that
$$
\omega(s) = \frac{\psi(s+s_0)^2-\psi(s_0)^2}{(s+s_0)^2 - s_0^2}
$$
is monotonically decreasing.
As $\psi$ is assumed to be concave, Lemma~\ref{diffquot} implies that
$$
\sigma(s) = \frac{\psi(s+s_0)-\psi(s_0)}{s}
$$
is monotonically decreasing. Furthermore
$$
\omega(s) = \sigma(s) \left(\frac{\psi(s+s_0)+\psi(s_0)}{s+2s_0}\right) =
\sigma(s) \frac{s\sigma(s)+ 2\psi(s_0)}{s+2s_0}.
$$
Now let $s_2< s_2$. As $\sigma(s)$ is monotonically decreasing on has
$$
\omega(s_1) \geq \sigma(s_2) \frac{s_1\sigma(s_2)+2\psi(s_0)}{s_1+2s_0}
= \sigma(s_2)^2 \frac{s_1+2\psi(s_0)/\sigma(s_2)}{s-1+2s_0}.
$$
The right-hand side is a decreasing function of $s_1$ if
$2 s_0 \leq 2 \psi(s_0)/\sigma(s_2)$, i.e., $\sigma(s-2) \leq \psi(s_0)$.
This is a consequence of Lemma~\ref{diffquot} as stated in the remark after
the lemma.
Replacing $s_1$ by $s_2$ thus gives a lower bound for $\omega(s_1)$ and
thus
$$
\omega(s_1) \geq \sigma(s_2)^2 \frac{s_2+2\psi(s_0)/\sigma(s_2)}{s_2+2s_0}
=\omega(s_2).
$$
It follows that $\omega$ is monotonically decreasing. \end{proof} A consequence of this lemma is that for the concavity of the function $\bar \psi^2((\Theta^2)^{-1}(s)) = \bar \psi^2(\Theta^{-1}(\sqrt{s}))$ it is thus sufficient to show that $\bar \psi \circ \Theta^{-1}$ is concave.
Finally we conjecture that a similar result to the proposition above also holds more generally, i.e., that a sufficient condition for concavity of $g\circ\psi \circ g^{-1}$ is the concavity of $\psi$ where $g$ belongs to a class of suitably chosen functions.
\section{Linear regularisation approaches}
Our goal in this section is to draw conclusions from Corollary~\ref{cor1} for
\emph{linear regularisation methods}. Taking into account the setting of
Corollary~\ref{cor1} we assume throughout this section that the index function
$\Psi(\lambda)$ is concave and strictly increasing for all $0<\lambda<\infty$
satisfying the limit conditions $\lim \limits_{\lambda \to +0}
\Psi(\lambda)=0,$ $\lim \limits_{\lambda \to \infty} \Psi(\lambda)=\infty$
(cf.~Lemma~\ref{lem2}), and (\ref{limquo}). Moreover, we set
\begin{equation} \label{eq:ass1reg}
\chi(\lambda):=\frac{\Psi^{-1}(\lambda)}{\lambda}\,, \quad 0<\lambda<\infty,
\quad \mbox{and} \quad \bar \psi(t):=
\frac{1}{\sqrt{\chi\left(\frac{1}{t}\right)}}\,, \quad 0<t<\infty \,.
\end{equation}
Then $\chi$ is an increasing index function with $\lim \limits_{\lambda \to
\infty} \chi(\lambda)=\infty$ and $\bar \psi$ is an increasing index function
with $\lim \limits_{t \to +0} \bar \psi(t)=0$, hence a rate function. As
outlined in section~3 under these assumptions we have $\bar\psi(A^*A)[B(R)] =
B_\chi(R)$ and the best case for regularised solutions $f_\alpha$
approximating the exact solution $f \in X_\chi$ based on data $g^\delta$
satisfying (\ref{noise}) by using an a priori choice $\alpha=\alpha(\delta)$
or a posteriori choice $\alpha=\alpha(\delta,g^\delta)$ is to achieve the
order optimal convergence rate (\ref{specialrate1}). It is a specific
consequence of interpolation theory and can be seen easily by inspection of
Corollary~\ref{cor1} that a successful use requires the focus on
regularisation methods which yield regularised solutions of appropriate
smoothness. Precisely, there must be a ball $B_\chi(R)$ to which the elements
$f_\alpha$ belong for all $\alpha>0$ attributed to sufficiently small
$\delta>0$ and $g^\delta$ satisfying (\ref{noise}).
\subsection{General linear regularisation schemata}
In a first approach we are going to consider \emph{linear regularisation
schemes} as described in many textbooks on linear regularisation theory (see,
e.g., \cite[Chap.~4]{EHN96}, \cite[Chap.~2]{Groe84} and
\cite{BakuGon04,Bau87,Kirsch96,Lou89,Rieder03}). We consider approximate
solutions
\begin{equation}
\label{eq:noisyregmeth}
f_\alpha := h_\alpha(A^*A) A^* g^\delta.
\end{equation}
to $f$ based on a family of piecewise continuous real functions $h_\alpha(t),
\;\, 0<t\le \|A\|^2$, to which we assign bias functions
$$r_\alpha(t):=t\,h_\alpha(t)-1, \;\, 0<t\le \|A\|^2\,.$$
These functions depend on a regularisation parameter $\alpha \in
(0,\alpha_{max}]$, where $\alpha_{max}$ may be a finite real number or
$\infty$. Small $\alpha>0$ characterise good approximation of the original
problem (\ref{opeq}), whereas larger values $\alpha$ are connected with more
stability. Hence, an appropriate trade-off between the two conflicting goals
approximation and stability can be controlled by the choice of $\alpha.$ We
say that such a function $h_\alpha$ describes a linear regularisation method
if the properties \begin{equation} \label{v1neu}
\lim \limits _{\alpha \to +0} \,r_\alpha(t)= 0, \qquad 0<t \le \|A\|^2, \end{equation} and \begin{equation} \label{v1}
\sup \limits _{0<\alpha \le \alpha_{max}}\; \sup \limits _{0<t \le \|A\|^2}\; t\,|\,h_\alpha(t)\,| \, \le \, C_1 \end{equation} with a constant $C_1>0$ hold. Because of (\ref{v1}) we have another constant $C_2>0$ such that
$$\sup \limits _{0<\alpha \le \alpha_{max}}\; \sup \limits _{0<t \le \|A\|^2}
|\,r_\alpha(t)\,| \, \le \,C_2 $$ and hence for all $0<\alpha \le \alpha_{max}$ the estimate
$$\|Af_\alpha-g^\delta\|=\|(Ah_\alpha(A^*A)A^*-I)g^\delta\| \le \left[\sup
\limits _{0<t \le \|A\|^2} \; |\,r_\alpha(t)\,|\right] \, \|g^\delta\| \le
C_2\, \|g^\delta\|.$$ This implies the limit condition
$\lim \limits_{ \alpha \to +0} \|Af_\alpha-g^\delta\|=0$ for all data $g^\delta \in Y$. As a consequence we have that there is always a parameter choice $\alpha=\alpha(\delta,g^\delta),\; 0<\delta \le \delta_{max},$ such that
$$\|Af_{\alpha(\delta,g^\delta)}-g^\delta\| \le C_{dis}\,\delta \qquad (0<\delta
\le \delta_{max})$$ for some prescribed constant $C_{dis}>0$. If the mapping $\alpha \mapsto
\|Af_{\alpha}-g^\delta\|$ is even continuous, then the discrepancy principle can be realised by a parameter choice $\alpha_{dis}=\alpha_{dis}(\delta,g^\delta)$ satisfying the equation (\ref{Morozov}).
Here we call a rate function $\bar \varphi$ a \emph{qualification} of the regularisation method generated by $h_\alpha$ if there is a constant $C_{quali}>0$ such that \begin{equation} \label{quali}
\sup \limits _{0<t \le \|A\|^2} \,|r_\alpha(t)| \bar \varphi(t) \,\le \,
C_{quali} \; \bar \varphi(\alpha) , \qquad 0<\alpha \le \alpha_{max}. \end{equation} Now we are going to study under what conditions the inequality (\ref{eq:ratexi}) in Corollary~\ref{cor11} can be fulfilled here with $\bar \xi(\delta)=\delta$. First we obtain \begin{equation} \label{help1}
\|Af_\alpha-g\| = \|A r_\alpha(A^*A)f + Ah_\alpha(A^*A)A^*(g^\delta-Af)\| \le
\|A r_\alpha(A^*A)f\| +C_1 \delta. \end{equation} In order to apply that corollary for obtaining a convergence rate (\ref{specialrate1}) we assume $f \in B_\chi(R_1)=\bar \psi(A^*A)[B(R_1)]$ taking into account the cross-connection (\ref{eq:ass1reg}). So let $f=\bar
\psi(A^*A)v,\; \|v\| \le R_1$. Provided that $\Theta(t):=\sqrt{t} \bar \psi(t)$ is a qualification of the method with constant $C_{quali}>0$ this gives with (\ref{help1}) \begin{equation} \label{help2}
\|Af_\alpha-g\| \le \left[ \sup \limits _{0<t \le \|A\|^2} \,r_\alpha(t)
\Theta(t)\right] R_1 + C_1 \delta \le C_{quali} R_1\, \Theta(\alpha) + C_1
\delta \end{equation} and hence an estimate of type (\ref{eq:ratexi}) is fulfilled with $\bar \xi(\delta)=\delta$ when an a priori parameter choice $\alpha =\Theta^{-1}(\delta)$ is used.
Next we will check whether $f_\alpha \in B_\chi(R_2)$ for some $0<R_2<\infty$. We have $$f_\alpha= h_\alpha(A^*A) A^* (g^\delta-Af)+ h_\alpha(A^*A) A^*Af$$ and after some reformulation $$f_\alpha= \bar \psi(A^*A)\left[h_\alpha(A^*A) (\psi(A^*A))^{-1}(A^*A)^{1/2} \tilde g + h_\alpha(A^*A) A^*Av\right]$$
with $\;\|\tilde g\| \le \delta$, since the different functions of $A^*A$ are commutable. Now let the interplay of the regularisation method expressed by $h_\alpha(t)$ and the parameter choice $\alpha=\alpha(\delta,g^\delta)$ be such that there is a constant $C_{para}>0$ with \begin{equation} \label{parabound}
\sup \limits _{0<t \le \|A\|^2}
\frac{\sqrt{t}\,|h_{\alpha(\delta,g^\delta)}(t)|\,\delta}{\bar \psi(t)}\, \le
\, C_{para}, \qquad 0<\delta \le \delta_{max}\,. \end{equation} The upper bound $C_{para}$ in (\ref{parabound}) must hold for all data $g^\delta \in Y$ associated with the noise level $\delta>0$ and satisfying (\ref{noise}), where the case of an a priori parameter choice $\alpha=\alpha(\delta)$ should be included as a special case. Under (\ref{parabound}) we have with (\ref{v1})
$$ \|h_{\alpha(\delta,g^\delta)}(A^*A) (\psi(A^*A))^{-1}(A^*A)^{1/2} \tilde g +
h_{\alpha(\delta,g^\delta)}(A^*A) A^*Av\| \le R_2:=C_{para}+ C_1\,R_1,$$ in other terms $f_{\alpha(\delta,g^\delta)} \in \bar \psi(A^*A)[B(R_2)]=B_\chi(R_2)$.
If there is a function $\Gamma(\alpha)$ satisfying for sufficiently small $\alpha>0$ the inequality \begin{equation} \label{para1}
\left[\sup \limits _{0<t \le \|A\|^2} \frac{\sqrt{t}|h_\alpha(t)|}{\bar
\psi(t)}\right]\, \le \,\Gamma(\alpha) \end{equation} such that \begin{equation} \label{para2} \delta\,\Gamma(\alpha(\delta,g^\delta))\,\le C_{para}, \qquad 0<\delta \le \delta_{max}\,, \end{equation} this represents a sufficient condition for (\ref{parabound}). In particular, if moreover the a priori parameter choice $\alpha(\delta,g^\delta):=\Theta^{-1}(\delta)$ satisfies (\ref{para2}) we have an estimate of type (\ref{eq:ratexi}) with $\bar \xi(\delta)=\delta$ for that a priori parameter choice whenever $\Theta$ is a qualification of the regularisation method under consideration.
Hence the considerations above gave a sketch of the proof for the following proposition as a consequence of Corollary~\ref{cor11}:
\begin{proposition} \label{pro2} Under the standing assumptions of this section including (\ref{eq:ass1reg}) let $f \in X_\chi=\operatorname{range}(\bar \psi(A^*A))$ and consider regularised solutions (\ref{eq:noisyregmeth}) with a generator function $h_\alpha$ that determines the regularisation method and satisfies (\ref{v1neu}) -- (\ref{v1}) as well as (\ref{para1}) with some function $\Gamma$ such that $\Theta(t):=\sqrt{t}\bar \psi(t)$ satisfies (\ref{para2}) with some constant $C_{para}>0$ and is a qualification of the method (cf.~(\ref{quali})). Then for the a priori regularisation parameter choice $\alpha=\alpha(\delta):=\Theta^{-1}(\delta) \to +0$ as $\delta \to +0$ we have the convergence rate \begin{equation} \label{pro2rate}
\|f-f_\alpha\|\,=\,\mathcal{O}\left(\delta\,\sqrt{\Psi(\bar K/\delta^2)}\right) \qquad
\mbox{as} \qquad \delta \to +0
\end{equation} with some constant $\bar K>0$. \end{proposition}
Note that in Proposition~\ref{pro2} the rate (\ref{pro2rate}) also holds for any other parameter choice $\alpha=\alpha(\delta,g^\delta)$ that fulfils the inequalities (\ref{para2}) and \begin{equation} \label{ratedelta}
\|Af_{\alpha(\delta,g^\delta)}-g\| \le \bar C\,\delta, \qquad 0<\delta \le \delta_{max}, \end{equation} with some constant $\hat C>0$.
\begin{example} {\rm The most prominent example of a linear regularisation method (\ref{eq:noisyregmeth}) is the Tikhonov regularisation with the generator function $h_\alpha(t)=\frac{1}{t+\alpha}$ and with the bias function $r_\alpha(t)=\frac{\alpha}{t+\alpha}$, where the requirements (\ref{v1neu}) and (\ref{v1}) are satisfied for the constants $C_1=C_2=1$. It is well known that all concave rate functions $\bar \varphi$ are qualifications of the method satisfying (\ref{quali}) with the constant $C_{quali}=1$. From that class we consider the monomials $\bar \varphi(t)=t^\nu$ for exponents $0<\nu \le 1.$ Then $\Theta(t)=\sqrt{t} \bar \psi(t)$ is a qualification with the same constant for the Tikhonov regularisation in case of a rate function $\bar \psi(t)=t^\mu$ with $0< \mu \le 1/2$. Taking into account (\ref{eq:ass1reg}) this rate function is associated with $\chi(\lambda)= \lambda^{2\mu}$ and the strictly concave function $\Psi(\lambda)=\lambda^{\frac{1}{2\mu+1}}$. By the estimate (\ref{help2}) we have then (\ref{ratedelta}) with $\bar C=R_1+1$ for
$f=(A^*A)^\mu v, \;\|v\| \le R_1$ and for the a priori parameter choice
\begin{equation} \label{apriH}
\alpha=\Theta^{-1}(\delta)=\delta^{\frac{2}{2\mu+1}}\,.
\end{equation}
To derive a function $\Gamma$ such that (\ref{para1}) is valid, we exploit the inequality $$ \frac{t^\kappa}{t+\alpha} \,\le \, (1-\kappa)^{1-\kappa}\kappa^\kappa\,
\alpha^{\kappa-1}\,,$$ which holds for all $t>0,\, \alpha>0$ and $0 < \kappa < 1$. In the limit case $\kappa=0$ we also have the inequality $1/(t+\alpha) \le 1/\alpha.$ Thus there is a constant $\hat c>0$ depending on $\kappa \in [0,1)$ such that $ \frac{t^\kappa}{t+\alpha} \le \frac{\hat c}{\alpha^{1-\kappa}}$. By setting $\kappa:=1/2-\mu$ we obtain for $\bar \psi(t)=t^\mu,\;0<\mu \le 1/2$ the inequality (\ref{para1}) with the function $$\Gamma(\alpha)=\frac{\hat c}{\alpha^{\,\mu+\frac{1}{2}}}\,.$$ Then one easily verifies that $\delta\,\Gamma(\Theta^{-1}(\delta)) \le \frac{\hat c \delta}{\left(\delta^{\frac{2}{2\mu+1}}\right)^{\mu+\frac{1}{2}}}=\hat c$ and that (\ref{para2}) is fulfilled with $C_{para}=\hat c$. Hence Proposition~\ref{pro2} applies and we obtain for the parameter choice (\ref{apriH}) and all $0<\mu \le 1/2$ the optimal convergence rate
$$\|f-f_\alpha\|=\mathcal{O}\left(\delta^{\frac{2\mu}{2\mu+1}}\right) \quad
\mbox{as} \quad \delta \to +0\,.$$
The best possible rate obtained in that way is
$\|f-f_\alpha\|=\mathcal{O}\left(\sqrt{\delta}\right)$ for $\mu=1/2$. For $\mu
>1/2$ the function $\Psi$ remains strictly concave, but a finite function
$\Gamma(\alpha)$ in (\ref{para1}) fails to exist, since we have $\sup \limits
_{0<t \le \|A\|^2} \frac{\sqrt{t}}{\bar \psi(t)(t+\alpha)}=+\infty$.
The limitation of Proposition~\ref{pro2} to lower H\"older rates than the
saturation of Tikhonov's method admits seems to be a consequence of the fact
that our approach based on Corollary~\ref{cor11} and the construction
(\ref{eq:noisyregmeth}) do not interact good enough in case of higher
smoothness of $f$. In order to overcome that effect, we will consider another
approach in the following subsection. }\end{example}
\subsection{Regularisation with unbounded operators and range inclusions}
In a second approach, under a non-standard source condition (\ref{Gsmooth}) characterising the available a priori knowledge on the solution smoothness, we exploit a variant of the Tikhonov regularisation with regularised solutions \begin{equation}
\label{eq:Gregmeth}
f_\alpha := G(GA^*AG+\alpha I)^{-1}GA^*g^\delta\,,
\end{equation}
where $G: X \to X$ is an injective bounded self-adjoint positive definite linear operator $G: X \to X$ with non-closed range, i.e., zero is an accumulation point of the spectrum $\operatorname{spec}(G)$ of the operator $G$. Since the unbounded linear operator with $B=G^{-1}: \operatorname{range}(G) \subseteq X \to X$ is frequently a differential operator, this approach is sometimes called
\emph{regularisation with differential operators}. Precisely, by construction the element $f_\alpha \in \operatorname{range}(G)$ is well-defined for all $\alpha>0$ as the minimiser of the extremal problem $$T_\alpha(\tilde f):= \|A \tilde f-g^\delta\|^2+ \alpha \|B \tilde f\|^2 \to \min, \quad \mbox{subject to} \quad \tilde f \in \operatorname{range}(G),$$ and then the penalty term in $T_\alpha$ contains derivatives of the function $\tilde f$.
To apply Corollary~\ref{cor11} under our setting (\ref{eq:ass1reg}) we assume
$f \in G[B(R_1)]$, with $G[B(R)]$ from (\ref{Gset}), and a link condition
\begin{equation} \label{linkGchi}
\operatorname{range}(G) \subseteq X_\chi\ =\operatorname{range}(\bar \psi(A^*A))\,,
\end{equation}
which is equivalent to
\begin{equation} \label{linkGpsi}
\|G w \| \le C \,\|\bar \psi(A^*A) w\|, \qquad \mbox{for all} \quad w \in X,
\end{equation}
with some $C>0$. Then from \cite[Lemma~6.2]{HofMAt07} we obtain that $f \in
G[B(R_1)]$ implies $f \in B_\chi(CR_1)=\bar \psi(A^*A)[B(CR_1)]$.
Along the lines of the paper \cite{ChengYam00} by {\sc Cheng} and {\sc
Yamamoto} we consider an a priori parameter choice $\alpha=\alpha(\delta)$ as
\begin{equation} \label{cheng}
\underline c \, \delta ^2 \le \alpha(\delta)\le \overline c \,\delta^2, \qquad
0<\delta \le \delta_{max},
\end{equation}
with constants $0<\underline c \le \overline c<\infty$, for which we obtain
from $T_\alpha(f_\alpha) \le T_\alpha(f)$ the inequalities
$$\|Af_{\alpha(\delta)}-g^\delta\|^2+\alpha(\delta)
\|G^{-1}f_{\alpha(\delta)}\|^2 \le \|Af-g^\delta\|^2+\alpha(\delta)
\|G^{-1}f\|^2 \le \delta^2 +\overline c\,\delta^2\,R_1^2\,.$$
Now we have
$$\|Af_{\alpha(\delta)}-g\| \le \bar C \,\delta, \quad \mbox{with} \quad \bar
C= \sqrt{1+\overline{c}\,R_1^2}+1 $$
satisfying condition (\ref{eq:ratexi}) with $\xi(\delta)=\delta$ and
$$\|G^{-1}f_{\alpha(\delta)}\| \le
\sqrt{\frac{\delta^2}{\alpha(\delta)}+\|G^{-1}f\|^2} \le
\sqrt{\frac{1}{\underline c}+R_1^2}=:R_2\,.$$
This yields $f_{\alpha(\delta)} \in G[B(R_2)]$, thus $f_{\alpha(\delta)} \in
B_\chi(CR_2)=\bar \psi(A^*A)[B(CR_2)]$ and consequently an estimate of type
(\ref{eq:specialxi}) with $\xi(\delta)=\delta$ and $CR_1,CR_2$ instead of
$R_1,R_2$.
With the above considerations we have shown the convergence rate result of the
following proposition again as a consequence of Corollary~\ref{cor11}:
\begin{proposition} \label{pro3} Under the standing assumptions of this section including (\ref{eq:ass1reg}) let $f$ satisfy (\ref{Gsmooth}), where the link condition (\ref{linkGchi}) is valid. Then for the a priori regularisation parameter choice (\ref{cheng}) we have the convergence rate (\ref{pro2rate}) with some constant $\bar K>0$. \end{proposition}
Due to \cite[Corollary 4.5]{HofMS08} for all concave $\Psi$ fulfilling the
standing assumptions of this section the rate (\ref{pro2rate}) is even order
optimal in the sense of
$$\mathcal{O}\left(\delta\,\sqrt{\Psi(\bar K/\delta^2)}\right)=
\mathcal{O}\left(\omega(G[B(R_1)],\delta)\right) \quad \mbox{as}
\quad\delta \to +0\,.$$
As already discussed in the introduction the requirement (\ref{linkGchi}) gets
stronger for higher rates in (\ref{pro2rate}).
In many applications (see as an illustration the examples in \cite{HofYam05})
one can only verify range inclusions of the form (\ref{rhorange}) with some
rate function $\bar \varrho$.
Under operator monotonicity of the function $[\bar \rho^{-1}(\sqrt{t})]^2$
(\ref{rhorange}) implies (\ref{linkGchi}) with $\bar \psi(t)=\bar
\rho^{-1}(\sqrt{t})$ and
$\chi(\lambda)=\left[\bar \rho^{-1}\left(1/\sqrt{\lambda}\right)\right]^{-2}$.
In order to verify in general for what index functions $\chi$ a range inclusion
(\ref{linkGchi}) with $\operatorname{range}(G) \subseteq X_\chi$ is fulfilled, one can use
the \emph{spectral theorem} for unbounded self-adjoint operators $T$ (see
\cite[Chapter VII.3]{Wer00} and also \cite[Chapter VIII]{ReedSimon80}).
In the Hilbert space~$X$, the injective, densely defined, self-adjoint,
positive definite, and unbounded linear operator $T$ is unitarily invariant to
a multiplication operator $\mathcal{M}$ expressed by a real multiplier function
$m$. This means that there are a measure space~$(\Sigma,\mathcal A,\mu)$ with
finite measure $\mu$, a unitary operator~$\mathcal{U}\colon X \to
L^2(\Sigma,\mathcal A,\mu)$ and a real measurable function $m(t),\;t \in
\Sigma$, such that $[\mathcal{M}h](t):=m(t)h(t)$ a.e., where $\mathcal{M}$ maps
in $L^2(\Sigma,\mathcal A,\mu)$, and $$ \mathcal{U}\, T\, \mathcal{U}^*\,h = \mathcal{M}\,h = m \cdot h$$ for all $h$ from the domain of $\mathcal{M}$. We note that the closure of the range $\operatorname{range}(m)$ of the multiplier function $m$ and the spectrum $\operatorname{spec}(T)
\subseteq [\|T\|^{-1},\infty)$ of the operator $T$, possessing $+\infty$ as an accumulation point, coincide. Moreover, we have for index functions $\psi \in \mathcal{I}$ and $h$ from the domain of $\psi(\mathcal{M})$ $$ \mathcal{U}\, \psi(T)\, \mathcal{U}^*\,h = \psi(M)\,h=\psi(m) \cdot h\,. $$ Then by using the notations $\hat f:=\mathcal{U}f \in L^2(\Sigma,\mathcal A,\mu)$ and $\widehat{(Gw)} := \mathcal{U}\,G\,w \in L^2(\Sigma,\mathcal A,\mu)$ by definition we immediately find that $\operatorname{range}(G) \subseteq X_\chi$ is equivalent to the condition that \begin{equation} \label{incco}
(Gw,\chi(T)Gw)=
(\widehat{(Gw)},\chi(\mathcal{M})\widehat{(Gw)})_{\scriptscriptstyle
L^2(\Sigma,\mathcal A,\mu)}=\int \limits_\Sigma \chi(m(t))|\widehat{(Gw)}(t)|^2
dt <\infty \end{equation} holds for all $w \in X.$ In Example~\ref{eximaging} with background in imaging (cf.~\cite{Scherzetal09}) we will consider the special case that $\mathcal{U}$ denotes the two-dimensional Fourier transform and that the corresponding measure space is $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2),\mu)$ with the associated Borel $\sigma$-algebra and measure. In that example, $T$ and $G$ are commuting operators, both non-compact with a non-closed range.
On the other hand, in Example~\ref{HSex} we will exploit the one-dimensional Fourier transform to formulate sufficient conditions such that classical source conditions are satisfied for linear compact integral operators.
\section{Examples}
In the remaining examples we illustrate the theory. All the
occurring operators $A$ are linear integral operators. First the
Example~\ref{eximaging} refers to convolution operators $A$ which
occur, for example, when the deblurring of noisy images is under
consideration. Then the Example~\ref{HSex} illustrates the \emph{low
rate} case where an integral equation with a smooth kernel is solved
and it is known that the solution is in a Sobolev space. The
situation here is similar as in the case of elliptic partial
differential equations and has been discussed in~\cite{Boetal06}. In
contrast to the PDE situation here convergence rates are low,
typically of the form $O(|\log(\delta)|^{-k})$. The final
Example~\ref{exfinal} illustrates the \emph{high rate} case where a
derivative of data in the range of an integral operator with smooth
kernel is considered. The high convergence rates are here of the
form $O(\delta |\log(\delta)|^k)$.
In the examples we consider functions over $\mathbb{R}^d\;(d=1,2)$ and
Sobolev spaces $H^l(\mathbb{R}^d)\;(l=1,2,...)$ of Hilbert type will be used
with norms $\|\cdot\|_l$ defined by
$$
\|x\|_l^2 = \frac{1}{(2\pi)^{d}} \int_{\mathbb{R}^d}
(1+|\omega|^2+\cdots+|\omega|^{2l}) |\hat{x}|^2\,d\omega,
$$
where $\hat{x}=\hat{x}(\omega),\;\omega \in \mathbb{R}^d$, is the Fourier
transform of $x$. Now let $E_l:H^l(\mathbb{R}^d)\rightarrow L^2(\mathbb{R}^d)$
denote the embedding and $E_l^*$ the adjoint of $E_l$. Then $E_l
E_l^* : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ is an integral operator and
$$
\widehat{E_l E_l^* y}(\omega) =
\frac{\hat{y}(\omega)}{1+|\omega|^2+\cdots+ |\omega|^{2l}}.
$$
\begin{example} \label{eximaging}
{\rm In this example with $X=Y=L^2(\mathbb{R}^2)$ we are interested in deblurring,
that means in finding a true picture which is characterised by a function $f
=f(t)\in L^2(\mathbb{R}^2),\;t=(t_1,t_2)^{\scriptscriptstyle T},$ that satisfies a
linear operator equation (\ref{opeq}) of convolution type
\begin{equation} \label{convolution}
Af\,(s) = \int \limits_{\mathbb{R}^2} k(s-t) f(t)\, dt = g(s), \quad
s=(s_1,s_2)^{\scriptscriptstyle T}\in \mathbb{R}^2,
\end{equation}
where $g \in L^2(\mathbb{R}^2)$ is a blurred image of $f$ which is additionally
contaminated with noise such that only the noisy blurred image $g^\delta \in
L^2(\mathbb{R}^2)$ satisfying (\ref{noise}) available as data. Following
\cite[Chapter~3]{BeBo98} the kernel function
$k(\tau),\;\tau=(\tau_1,\tau_2)^{\scriptscriptstyle T} \in \mathbb{R}^2,$ is called
\emph{point spread function} of a space invariant imaging system under
consideration. We assume that the kernel is such that its Fourier transform
$\hat k=\hat k(\omega)\;,\omega=(\omega_1,\omega_2)^{\scriptscriptstyle T}$,
called \emph{transfer function} is bounded. Different variants of such
deblurring problems are presented and analysed in \cite{BeBo98}. As a
reference situation we exploit for illustration a variant of an out-of-focus
blur for which
$$
\hat k(\omega) = 2 \frac{J_1(D|\omega|)}{D|\omega|}
$$
where $J_1$ is the Bessel function of order one and $D$ is the radius of the
circle of confusion (cf.~\cite[formula (3.25) on p.60]{BeBo98}). The linear
convolution operator $A:L^2(\mathbb{R}^2) \to L^2(\mathbb{R}^2)$ in this example has a
non-closed range but it is non-compact and the kernel is not square
integrable.
In order to apply our theory to this example one needs to find an index
function $\theta$ and a symmetric positive definite operator $T$ such that
$A^*A = \theta(T)$. A natural choice in this context is $T= -\Delta$ and in
this case $\theta$ needs to satisfy $|\hat k(\omega)|^2 = \theta(|\omega|^2)$.
This, however, is not possible, as $\hat k(\omega)$ is zero for some finite
$\omega$ but an index function has to satisfy $\theta(\lambda) > 0$ for all
$\lambda > 0$ and it can only be zero asymptotically at zero or infinity. It
is thus not possible to get error bounds for the deblurring problem using the
variable Hilbert scale theory and $T=-\Delta$.
One does not have this problem if one chooses $T=(A^*A)^{-1}$. Let us define
the solution smoothness as $f \in H^l(\mathbb{R}^2)$. Then we have the operator
$G=E_l^*E_l$ in (\ref{Gsmooth}) characterising the associated non-standard
source condition. To find index functions $\chi$ that satisfy the link
condition (\ref{linkGchi}) we can make use of formula (\ref{incco}) taking
into account that $m(\omega)=1/|\hat k(\omega)|^2$ and
$$
|\widehat{Gw}(\omega)|^2=(1+|\omega|^2+\cdots+|\omega|^{2l})^{-1}| \hat w
(\omega)|^2.
$$
Then the range inclusion $\operatorname{range}(G) \subseteq X_\chi$ takes the form
\begin{equation} \label{ex2cond}
\chi\left(\frac{1}{|\hat k(\omega)|^2}\right)
\frac{1}{(1+|\omega|^2+\cdots+|\omega|^{2l})} \le \bar C < \infty \quad
\mbox{for all} \quad \omega \in \mathbb{R}^2\,.
\end{equation}
This range condition can only be satisfied if $\chi$ is bounded, i.e., i.e.
$\chi(\lambda) \leq C < \infty$ for all $\lambda > 0$. This is again a
consequence of the existence of zeros of $\hat k(\omega)$ for finite $\omega$.
A consequence of the finiteness of $\chi$ is $L^2(\mathbb{R}^2) \subset H_\chi $ and
it follows that the ``source condition" reduces to $f\in L^2(\mathbb{R})$ which does
not lead to an error bound.
The failure of the above attempts to get error bounds clearly illustrates the
need to extend the variable Hilbert scale theory to be able to cope with the
deblurring problem. One can, however, deal with a partial deblurring problem.
Observe that one has the asymptotics
$$
2 \left|\frac{J_1(D|\omega|)}{D|\omega|}\right| \asymp |\omega|^{-3/2}
$$
for large $|\omega|$ (cf.~\cite[formula (3.29) on p.60]{BeBo98}). It follows
that $\hat k (\omega) = |\omega|^{-3/2} \kappa(\omega)$ for some bounded
$\kappa(\omega)$. The first factor $|\omega|^{3/2}$ relates to a ``smoothing
component" of the out-of-focus blur situation. We now consider inversion of
this smoothing component only. For this we introduce an integral operator $A$
with kernel $k$ which satisfies
\begin{equation} \label{outoffocus}
\hat k(\omega) = |\omega|^{3/2}.
\end{equation}
For the ``partial" out-of-focus blur situation (\ref{outoffocus}) and monomials
$\chi(\lambda)=\lambda^\kappa,\;\kappa>0$, we have (\ref{ex2cond}) if and only
if $\kappa \le \frac{2l}{3}.$ With the relation
$\chi(\lambda)=\Psi^{-1}(\lambda)/\lambda$ this corresponds with
$\Psi(\lambda)\le \lambda^{3/(2l+3)}$. Hence based on Proposition~\ref{pro3}
for the situation (\ref{outoffocus}) and under $f \in H^l(\mathbb{R}^2)$ a best
possible convergence rate
$$ \|f-f_\alpha\|_{\scriptscriptstyle
L^2(\mathbb{R}^2)}=\mathcal{O}\left(\delta^{\frac{2l}{2l+3}}\right) \quad \mbox{as}
\quad \delta \to +0$$
can be obtained by Tikhonov regularisation with $H^l$-penalty term.
}\end{example}
\begin{example} \label{HSex}
{\rm In this example we consider compact forward operators $A$ in equation
(\ref{opeq}) with $X=Y=L^2(\mathbb{R})$ in form of linear operators $A: L^2(\mathbb{R}) \to
L^2(\mathbb{R})$, for which the range of the operator $K:=\bar \psi(A^*A)$ is a subset
of $H_\phi$ with some index function $\phi$ and some rate function $\bar
\psi$. That means, we have $\operatorname{range}(K) \subseteq H_\phi$ and a classical source
conditions (\ref{scclassic}) is valid for $f \in H_\phi$ implying the
corresponding convergence rates in regularisation. In this context, let $K$ be
a linear Fredholm integral operator of Hilbert-Schmidt type. For such
operators one can provide conditions on the kernel which guarantee this range
condition.
\begin{lemma}
\label{lem3-ex}
Let $K:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ be a Hilbert-Schmidt operator with kernel
$k(t,s)\in L^2(\mathbb{R}^2)$. Furthermore, let $\tilde{K}:L^2(\mathbb{R})\rightarrow
L^2(\mathbb{R})$ be an integral operator with kernel $\tilde{k}(\omega,s)=\int_R
e^{-i\omega t} k(t,s)\, dt$. Then $\tilde{K}$ is a Hilbert-Schmidt operator
and
$$
\tilde{K} x = \widehat{Kx}, \quad x\in L^2(\mathbb{R}).
$$
\end{lemma}
\begin{proof}
The adjoint operator $K^*$ of $K$ is an integral operator with kernel
$k^*(s,t)=\overline{k(t,s)}$ as a consequence of the theorem of Fubini.
By Plancherel's theorem one has
$$
K^* u = \frac{1}{2\pi} \tilde{K}^* \hat{u}.
$$
An application of Parseval's identity several times gives for
$u,v\in L^2(\mathbb{R})$:
\begin{align*}
\frac{1}{2\pi} (\hat{u}, \widehat{Kv}) & = (u, Kv) \\
& = (K^* u, v) \\
& = \frac{1}{2\pi} (\tilde{K}^* \hat{u}, v) \\
& = \frac{1}{2\pi}(\hat{u}, \tilde{K} v).
\end{align*}
\end{proof}
\begin{proposition}
Let $K:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ be a Hilbert-Schmidt operator where the
Fourier transform $\tilde{k}(\omega,s)=\int_\mathbb{R} e^{-i\omega t} k(t,s)\, dt$
of the kernel of $K$ satisfies
$$
\int_\mathbb{R} \int_\mathbb{R} \phi(\omega^2) |\tilde{k}(\omega,s)|^2\, ds\,
d\omega < \infty
$$
for some index function $\phi$. Then $\operatorname{range}(K) \subseteq H_\phi$.
\end{proposition}
\begin{proof}
By Lemma~\ref{lem3-ex} one has
$$
\hat{y}(\omega) = \int_\mathbb{R} \tilde{k}(\omega,s) x(s)\, ds
$$
which we insert into the following bound, obtained from the Cauchy-Schwarz
inequality:
$$
\left|\int_\mathbb{R} \tilde{k}(\omega,s) x(s)\, ds \right|^2 \leq
\int_\mathbb{R} |\tilde{k}(\omega,s)|^2\, ds \, \|x\|^2.
$$
It follows that for $y=Kx$ with $x\in L^2(\mathbb{R})$ one has
\begin{align*}
\|y\|_\phi^2 &= \frac{1}{2\pi} \int_\mathbb{R} \phi(\omega^2) |\hat{y}(\omega)|^2\,
d\omega \\
&\leq \frac{1}{2\pi}\int_\mathbb{R} \int_\mathbb{R} \phi(\omega^2) |\tilde{k}(\omega,s)|^2
\, ds \, d\omega \, \|x\|^2
\end{align*}
and consequently $y\in H_\phi$.
\end{proof}
} \end{example}
\begin{example} \label{exfinal} {\rm
As a concrete application example we consider a problem from
derivative spectroscopy~\cite{StauS68}. Here numerical derivatives
are used to enhance the resolution of measured spectra in order to
separate close peaks. An instance is the Eddington correction
formula. The approach determines
$$
f = Lg := g - \frac{g^{(2)}}{2}
$$
from observed $g_\delta$ where $g^{(2)}$ is the second derivative of
$g$. We now apply the theory developed so far to determine how well
$f=Lg$ can be determined from spectral data $g_\delta$.
For $f\in H^2(\mathbb{R})$ and $f=Lg$ the Fourier transforms $\hat{f}$ and
$\hat{g}$ satisfy
$$
\hat{f}(\omega) = (1+\omega^2/2)\, \hat{g}(\omega), \quad
\text{a.e.}
$$
Using Plancherel's theorem, one obtains from this the bounds
$$
\frac{1}{2} \|f\|_2 \leq \|Lf\| \leq \|f\|_2, \quad
f\in H^2(\mathbb{R})
$$
which means in particular that $\|Lf\|$ is an equivalent norm for
$H^2(\mathbb{R})$. Using standard arguments, one can then show that $L:
H^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ is a Hilbert space isomorphism. Using
the convolution theorem one sees that $A=E_2 L^{-1} : L^2(\mathbb{R})
\rightarrow L^2(\mathbb{R})$ is an integral operator with
$$
Af\, (t) = \frac{1}{\sqrt{2}} \int_\mathbb{R} \exp(-\sqrt{2}|t-s|) f(s)\,
ds \quad t \in \mathbb{R}
$$
where $E_2$ denotes the embedding $H^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$. As
$L^{-1}$ maps $L^2(\mathbb{R})$ onto $H^2(\mathbb{R})$ the range of $A$ can be
identified with $H^2(\mathbb{R})$.
In addition to the Sobolev spaces, which form a classical Hilbert
scale, we will use a \emph{variable Hilbert scale} $H_\phi$ with
norms $\|\cdot\|_\phi$ defined by
$$
\|x\|_\phi^2 = \frac{1}{(2\pi)} \int_{-\infty}^\infty
\phi(\omega^2) \, |\hat{x}(\omega)|^2 \, d\omega
$$
where $\phi$ are \emph{index functions}. Note that we have here
$H_\phi=X_\phi$ for $X=L^2(\mathbb{R})$ (see~\cite{Heg92}) and the
generating operator is the second order differential operator
$T=-d^2/dt^2$. The index functions
$$
\nu_k(\lambda) = 1 + \lambda + \cdots + \lambda^k
= \frac{\lambda^k - 1}{\lambda - 1}
$$
define the Sobolev spaces, in particular, one has
$H_{\nu_k}=H^{k}(\mathbb{R})$ and furthermore, the Sobolev norm is equal to
the norm of the corresponding variable Hilbert scale:
$$
\|f\|_k = \|f\|_{\nu_k}, \quad f\in H_{\nu_k}.
$$
In this framework, we now get error bounds analogue to the ones in
Corollary~\ref{cor1} which are again a consequence of
Lemma~\ref{lem1}.
\begin{corollary}\label{cor1a}
Let $H_\nu$ be the Hilbert scales generated by $T=-d^2/dt^2$ from
$L^2(\mathbb{R})$. Furthermore, let $A: L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ be a
(convolution) operator satisfying
$$
A^* A = \theta(T)
$$
for some bounded index function $\theta$. Moreover, let $\phi,\psi$
and $\Psi$ be index functions and $\Psi$ be concave such that
$$
\phi(\lambda) \le \Psi(\psi(\lambda)), \quad \phi(\lambda) \theta(\lambda)=1,\quad \mbox{for} \quad t > 0.
$$
If $Af=g\in H_\psi$ and if $f_\alpha$ is such that $Af_\alpha\in H_\psi$
and
\begin{align*}
\|Af_\alpha - g \|_\psi & = \zeta \\
\|Af_\alpha - g \| & = \epsilon
\end{align*}
then
$$
\|f-f_\alpha\| \leq \epsilon \sqrt{\Psi(\zeta^2/\epsilon^2)}.
$$
\end{corollary}
\begin{proof}
Note that $\theta(T)$ is well defined by the Fourier transform and
as $\theta$ is bounded, so is $A$. Furthermore it follows from
the condition $A^*A = \theta(T)$ that
$\|h\|_\theta = \|Ah\|$ for all $h\in L^2(\mathbb{R})$ and
$\|h\|_{\psi\theta} = \|Ah\|_\psi$, for all $h$ with $Ah\in H_\psi$.
By the variable Hilbert scale interpolation inequality
(Lemma~\ref{lem1}) one has
$$
\|f-f_\alpha\|^2_{\phi\theta} \leq \|f-f_\alpha\|^2_\theta\,
\Psi\left(\frac{\|f-f_\alpha\|^2_{\psi\theta}}{\|f-f_\alpha\|^2_\theta}\right).
$$
Now $\phi(\lambda)\theta(\lambda)=1$ and $\|f_\alpha-f\|_\theta
= \|Af_\alpha-g\|$
as $f_\alpha-f\in L^2(\mathbb{R})$. Furthermore, $\|f_\alpha-f\|_{\psi\theta}
= \|Af_\alpha - g\|_\psi$.
By Lemma~\ref{concave1} one then has
$$
\|f_\alpha-f\| \leq \|Af_\alpha - g\|
\sqrt{\Psi(\|Af_\alpha-g\|_\psi^2/\|Af_\alpha-g\|^2)}.
$$
The bound follows by another application of Lemma~\ref{concave1}.
\end{proof}
In comparison with Corollary~\ref{cor1} this corollary uses an
operator $T$ which is not necessarily equal to $(A^*A)^{-1}$ but more
importantly, the source condition is here not given as a property of
the solution $f$ but of the data $g$.
For the application of this corollary to the case of the Eddington
correction formula one chooses $\theta(\lambda)=1/(1+\lambda/2)$ and
so $\phi(\lambda) = 1 + \lambda/2$.
In contrast to the usual case, where the source condition is stated
as a condition on $f$, here the source condition is stated as a
condition on (the original spectrum) $g$. This source condition
results from physical models for the spectrum, and, in particular
for the so-called spectral broadening. A variety of models are used,
the most common ones are the Gaussian, Lorenz and Voigt spectra
where a Voigt spectrum is a combination of a Lorenz and a Gaussian
spectrum. Here we consider Gaussian spectra defined by
$$
g(t) = \frac{1}{\sqrt{2\pi}} \int_\mathbb{R} \exp(-(t-s)^2/2) v(s)\, ds
$$
for some $v\in L^2(\mathbb{R})$. For a different discussion and more
background on the problem, the reader may consult the paper by
Hegland~\cite{Heg09x}.
It follows that $g\in H_\psi$ with $\psi(\lambda) = \exp(\lambda)$.
The concave
function $\Psi$ can then be chosen as
$$
\Psi(\lambda) =
\begin{cases}
\lambda, & \text{for $\lambda \leq 1$} \\
(1 + \log(\lambda)/2)^2, & \text{for $\lambda \geq 1$}.
\end{cases}
$$
It follows that $\Psi$ is concave and that $\phi(\lambda) \le
\Psi(\psi(\lambda)) $. As a consequence one gets the error bounds
$$
\|f-f_\alpha\| \leq \delta (1 + \log(\eta/\delta))
$$
for $\delta < \eta$ and $\|f-f_\alpha\| \leq \eta$ if $\delta \geq \epsilon$.
The stabilisation guarantees that even if the errors are very large, the
error of the approximation does not grow to infinity. In fact, the solution
$f_\alpha = 0$ would probably be a good choice for the large data error case.
}\end{example}
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\begin{document}
\title{Two Puzzles About Computation} \author{Samson Abramsky} \date{} \maketitle
\section{Introduction}
Turing's classical analysis of computation \cite{turing1937computable} gives a compelling account of the nature of the computational process; of \emph{how} we compute. This allows the notion of \emph{computability}, of what can in principle be computed, to be captured in a mathematically precise fashion.
The purpose of this note is to raise two different questions, which are rarely if ever considered, and to which, it seems, we lack convincing, systematic answers. These questions can be posed as: \begin{itemize} \item Why do we compute? \item What do we compute? \end{itemize}
The point is not so much that we have no answers to these puzzles, as that we have no established body of theory which gives satisfying, systematic answers, as part of a broader understanding. By raising these questions, we hope to stimulate some thinking in this direction.
These puzzles were raised in \cite{abramsky2008information}; see also \cite{adriaanscomputation}.
\section{Why Do We Compute?} The first puzzle is simply stated: \begin{center} \fbox{\textbf{Why do we compute?}} \end{center} By this we mean: why do we perform (or build machines and get them to perform) actual, physically embodied computations?
There is, indeed, an obvious answer to this question: \begin{center} To gain information (which, therefore, we did not previously have). \end{center} But --- how is this possible?\footnote{Indeed, I was once challenged on this point by an eminent physicist (now knighted), who demanded to know how I could speak of information increasing in computation when Shannon Information theory tells us that it cannot! My failure to answer this point very convincingly at the time led me to continue to ponder the issue, and eventually gave rise to this discussion.} Two problems seems to arise, one stemming from physics, and one from logic. \begin{description} \item[Problem 1:] Doesn't this contradict the second law of thermodynamics? \item[Problem 2:] Isn't the output \emph{implied} by the input? \end{description} We shall discuss each of these in turn.
\subsection*{Problem 1} The problem is that, presumably, information is conserved in the \emph{total} system. The natural response is that, nevertheless, there \emph{can} be information flow between, and information increase in, \emph{subsystems}; just as a body can gain heat from its environment. More precisely, while the entropy of an isolated (total) system cannot decrease, a sub-system \emph{can} decrease its entropy by consuming energy from its environment.
Thus if we wish to speak of information flow and increase, this must be done relative to subsystems. Indeed, the fundamental objects of study should be \emph{open systems}, whose behaviour must be understood in relation to an external environment. Subsystems which can observe incoming information from their environment, and act to send information to their environment, have the capabilities of \emph{agents}.
\textbf{Moral:} Agents and their interactions are intrinsic to the study of information flow and increase in computation. The classical theories of information do not reflect this adequately.
\paragraph{Observer-dependence of information increase?} Yorick Wilks (personal communication) has suggested the following additional twist. Consider an equation such as \[ 3 \times 5 = 15 . \] The forward direction $3 \times 5 \rightarrow 15$ is obviously a natural direction of computation, where we perform a multiplication. But the reverse direction $15 \rightarrow 3 \times 5$ is also of interest --- finding the prime factors of a number! So it seems that the \emph{direction of possible information increase} must be understood as relative to the observer or user of the computation!
Can we in fact find an objective, observer-independent notion of information increase? This seems important to the whole issue of whether information is inherently subjective, or whether it has an objective structure.
\subsection*{Problem 2}
The second puzzle is the computational version of what has been called the \emph{scandal of deduction} \cite{hintikka1970information,d2009enduring,sequoiah2008scandal}. The logical problem is to find the sense in which logical deduction can be informative, since, by the nature of the process, the conclusions are `logically contained' in the premises. So what has been added by the derivation? This is a rather basic question, which it is surprisingly difficult to find a satisfactory answer to.
Computation can be modelled faithfully as deduction, whether in the sense of deducing the steps that a Turing maching takes starting from its initial configuration, or more directly via the Curry-Howard isomorphism \cite{Cur58,howard1980formulae}, under which computation can be viewed as performing cut-elimination on proofs, or normalization of proof terms. Thus the same question can be asked of computation: since the result of the computation is logically implied by the program together with the input data, what has been added by computing it?
The same issue can be formulated in terms of the logic programming paradigm, or of querying a relational database: in both cases, the result of the query is a logical consequence of the data- or knowledge-base.
It is, of course, tempting to answer in terms of the complexity of the inference process; but this seems to beg the question. We need to understand first what the inference process is doing for us!
We can also link this puzzle to another well-known issue in logic, namely the principle of \emph{logical omnisicience} in epistemic logic, which is unrealistic yet hard to avoid. This principle can be formulated as follows: \[ [K_a \phi \; \wedge \; (\phi \rightarrow \psi)] \; \rightarrow \; K_a \psi . \] It says that the knowledge of agent $a$ is deductively closed: if $a$ knows a proposition $\phi$, then he knows all its logical consequences. This is patently untrue in practice, and brings us directly back to our puzzle concerning computation. We compute to gain information we did not have. We start from the information of knowing the program and its input, and the computation provides us with explicit knowledge of the output. But what does `explicit' mean?
The computational perspective may indeed provide a usefully clarifying perspective on the issue of logical omniscience, since it provides a context in which the distinction between `explicit' and `implicit' knowledge can be made precise. Let us start with the notion of a function. In the 19th century, the idea of a function as a `rule' --- as given by some defining expression --- was replaced by its `set-theoretic semantics' as a set of ordered pairs. In other terminology, a particular defining expression is an \emph{intensional description} of a function, while the set of ordered pairs which it denotes is its \emph{extension}.
A program is exactly an intensional description of a function, with the additional property that this description can be used to explicitly calculate outputs from given inputs in a stepwise, mechanical fashion.\footnote{We refer e.g. to \cite{gandy1980church,sieg2002calculations} for attempts to give a precise mathematical characterization of `mechanical'.} We can say that implicit knowledge, in the context of computation, is knowledge of an intensional description; while explicit knowledge, of a data item such as a number, amounts to possessing the \emph{numeral} (in some numbering system) corresponding to that number; or more generally, to possessing a particular form of intensional description which is essentially isomorphic to the extension.
The purpose of computation in these terms is precisely to convert intensional descriptions into extensional ones, or implicit knowledge of an input-output pair into explicit knowledge. The \emph{cost} of this process is calibrated in terms of the resources needed --- the number of computation steps, the workspace which may be needed to perform these steps, etc. Thus we return to the usual, `common-sense' view of computation. The point is that it rests on this distinction between intension and extension, or implicit vs. explicit knowledge.
Another important aspect of why we compute is \emph{data reduction}---getting rid of a lot of the information in the input. Note that normal forms are in general \emph{unmanagably big} \cite{Vor97}. Note also that it is \emph{deletion of data} which creates thermodynamic cost in computation \cite{Lan00}. Thus we can say that much (or all?) of the actual usefulness of computation lies in getting rid of the hay-stack, leaving only the needle.
The challenge here is to build a useful theory which provides convincing and helpful answers to these questions. In our view these puzzles, naive as they are, point to some natural questions which a truly comprehensive theory of computation, incorporating a `dynamics of information', should be able to answer.
\section{What Do We Compute?}
The classical notion of computability as pioneered by Turing \cite{turing1937computable} focusses on the key issue of \emph{how} we compute; of what constitutes a computation. However, it relies on pre-existing notions from mathematics as to \emph{what} is computed: numbers, functions, sets, etc.
This idea also served computer science well for many years: it is perfectly natural in many situations to view a computational process in terms of computing an output from an input. This computation may be deterministic, non-deterministic, random, or even quantum, but essentially the same general paradigm applies.
However, as computation has evolved to embrace diverse forms and purposes: distributed, global, mobile, interactive, multi-media, embedded, autonomous, virtual, pervasive, \ldots the adequacy of this view has become increasingly doubtful.
Traditionally, the \emph{dynamics} of computing systems --- their unfolding behaviour in space and time --- has been a mere means to the end of computing the function which specifies the algorithmic problem which the system is solving.\footnote{Insofar as the dynamics has been of interest, it has been in quantitative terms, counting the resources which the algorithmic process consumes --- leading of course to the notions of algorithmic complexity. Is it too fanciful to speculate that the lack of an adequate structural theory of processes has been an impediment to fundamental progress in complexity theory?} In much of contemporary computing, the situation is reversed: the \emph{purpose} of the computing system is to exhibit certain behaviour. The \emph{implementation} of this required behaviour will seek to reduce various aspects of the specification to the solution of standard algorithmic problems.
\begin{center} \fbox{\textbf{What does the Internet compute?}} \end{center} Surely not a mathematical function \ldots
\subsection*{Why Does It Matter?}
We shall mention two basic issues in the theory of computation which become moot in the light of this issue.
There has been an enormous amount of work on the first, namely the theory of concurrent processes. Despite this huge literature, produced over the past four decades and more, no consensus has been achieved as to what processes \emph{are}, in terms of their essential mathematical structure. Instead, there has been a huge proliferation of different models, calculi, semantics, notions of equivalence. To make the point, we may contrast the situation with the $\lambda$-calculus, the beautiful, fundamental calculus of functions introduced by Church at the very point of emergence of the notion of computability \cite{church1941calculi}. Although there are many variants, there is essentially a unique, core calculus which can be presented in a few lines, and which delineates the essential ideas of functional computation. In extreme contrast, there are a huge number of process calculi, and none can be considered as definitive.
Is the notion of process too amorphous, too open to different interpretations and contexts of use, to admit a unified, fundamental theory? Or has the field not yet found its Turing? See \cite{abramsky2006fundamental} for an extended discussion.
The second issue follows on from the first, although it has been much less studied to date. This concerns the Church-Turing thesis of universality of the model of computation. What does this mean when we move to a broader conception of what is computed? And are there any compelling candidates? Is there a widely accepted universal model of interactive or concurrent computation?
As a corollary to the current state of our understanding of processes as described in the previous paragraphs, there is no such clear-cut notion of universality. It is important to understand what is at issue here. If we are interested in the process of computation itself, the structure of interactive behaviour, then on what basis can we judge if one such process is faithfully simulated by another? It is not of course that there are no candidate notions of this kind which have been proposed in the literature; the problem, rather, is that there are far too many of them, reflecting different intuitions, and different operational and application scenarios.
Once again, we must ask: is this embarrassing multitude of diverse and competing notions a necessary reflection of the nature of this notion, or may we hope for an incisive contribution from some future Turing which will unify and organize the field?
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\begin{document}
\begin{center} {\Large\bf Quantum network architecture of tight-binding models with substitution sequences}\\
Ilki~Kim and G\"{u}nter~Mahler\\ Institut f\"ur Theoretische Physik~I, Universit\"{a}t Stuttgart\\ Pfaffenwaldring 57, 70550 Stuttgart, Germany\\ phone: ++49-(0)711 685-5100, FAX: ++49-(0)711 685-4909\\ e-mail: ikim@theo.physik.uni-stuttgart.de \end{center}
\begin{abstract}
We study a two-spin quantum Turing architecture, in which discrete local
rotations $\{\alpha_m\}$ of the Turing head spin alternate with quantum
controlled NOT-operations. Substitution sequences are known to underlie
aperiodic structures. We show that parameter inputs $\{\alpha_m\}$
described by such sequences can lead here to a quantum dynamics,
intermediate between the regular and the chaotic variant.
Exponential parameter sensitivity characterizing chaotic quantum Turing
machines turns out to be an adequate criterion for induced quantum chaos
in a quantum network. \end{abstract}
\section{Introduction}
Models described by one-dimensional Schr\"{o}dinger equations with quasi-periodic potentials display interesting spectra: this kind of potential is intermediate between periodic ones, leading to energy bands and extended states, and truly random potentials, which cause localisation \cite{OST83}. A super-lattice e.g. may be made of two species of doped semiconductors producing a one-dimensional chain of quantum wells. A qualitative model to describe the corresponding wave-functions is given by the tight-binding model, $\hat{H} \psi(n) = \psi(n+1) + \psi(n-1) + \lambda V(n) \psi(n)\,,\, \psi \in l^2(\mathbf{Z})$\,, where $V(n)$ represents the effect of quantum well~$n$\,, and $\lambda$ is a positive parameter playing the role of a coupling constant \cite{BOV95}. An interesting description results when this super-lattice is constructed by means of a deterministic rule. The simplest rule obtains when the two species alternate in a periodic way. But in general the rule will be aperiodic. One widely studied example is the Fibonacci sequence, which is quasi-periodic \cite{BOV95}: given two `letters' $a$ and $b$, one substitutes $a \to \xi(a) = ab$ and $b \to \xi(b) = a$. Iterating this rule on $a$ one thus generates the sequence $abaababaabaababa \cdots$\,, in which the frequency of $a$'s is given by the golden mean $(5^{1/2}-1)/2$. Other examples of such substitution rules are the Thue-Morse sequence (non quasi-periodic), which is obtained through the substitution $a \to \xi(a) = ab\,,\, b \to \xi(b) = ba$\, giving $abbabaabbaababba \cdots$\,, and the period-doubling sequence (non quasi-periodic), $a \to \xi(a) = ab\,,\, b \to \xi(b) = aa$ \cite{BOV95}. Accordingly, a piece-wise constant potential $\{ V_n; n \in \mathbf{Z} \}$ based on such such a rule, e.g. $V_1 = a, V_2 = b, V_3 = a,\, \cdots $\,, is called a `substitution potential'. The potential of the Fibonacci sequence has one-dimensional quasi-crystalline properties.
In recent years problems of quantum computing~(QC) and information processing have received increasing attention. To solve certain classes of problems in a potentially very powerful way, one tries to utilize in~QC the quantum-mechanical superposition principle and the (non-classical) entanglement. In the models of QC based on quantum Turing machines~(QTM) \cite{BEN82,DEU85}, the computation is characterized by sequences of unitary transformations (i.e. by the corresponding Hamiltonians~$\hat{H}$ acting during finite time interval steps). Benioff has studied the tight-binding model in a generalized QTM \cite{BEN97}, where QC is associated with different potentials at different steps as an `environmental effect'. These kinds of influences may introduce {\em{deterministic disorder}}\, which would degrade performance by causing reflections at various steps and decay of the transmitted component \cite{LAN96}.
Here we investigate an iterative map on qubits which can be interpreted as a QTM architecture \cite{KIM99}: local transformations of the Turing head controlled by a sequence of rotation angles $\{\alpha_m\}\,,\, m = 1, 2, \cdots$ (parameter inputs) alternate with a quantum-controlled NOT-operation~(QCNOT) with a second spin on the Turing tape. Those angles at steps $n = 2m-1$ are reminiscent of the potentials $V_m$ introduced before. In the present paper we will investigate the Fibonacci rule and the Thue-Morse case mainly with respect to the local dynamics of the Turing head, which will be shown to reflect the degree of `randomness' of the substitution sequences. The various types of aperiodic structures have been characterized up to now by the nature of their Fourier spectra only \cite{BOV95}.
\section{Quantum Turing machine driven by substitution sequences}
The quantum network \cite{MAH98} to be considered here is composed of two pseudo-spins
$|p\rangle\!^{(\mu)};\,p=-1,1;\,\mu=S,1$ (Turing-head $S$, Turing-tape spin $1$, see figure~\ref{QTM_chaos})
so that its network state $|\psi\rangle$ lives in the {four-dimensional} Hilbert space spanned by the product
wave-functions $|j^{(S)} k^{(1)}\rangle = |jk\rangle$. Correspondingly, any (unitary) network operator can be expanded as a sum of product operators. The latter may be based on the $SU(2)$-generators, the Pauli matrices $\hat{\sigma}_j^{(\mu)},\, j=1,2,3$, together with the unit operator $\hat{1}^{(\mu)}$.
The initial state $|\psi_{0}\rangle$ will be taken to be a product of the Turing-head and tape wave-functions. For the discretized dynamical description of the QTM we identify the unitary operators $\hat{U}_{n}\,,\,n=1,2,3,\cdots$ (step number) with the local unitary transformation on the Turing head $S$, $\hat{U}_{\alpha_m}^{(S)}$, and the QCNOT on ($S,1$), $\hat{U}^{(S,1)}$, respectively, as follows: \begin{eqnarray} &&\hat{U}_{2m-1}\; =\; \exp{\left(-i \hat{\sigma}_1^{(S)} {\alpha_{m}}/2\right)}\label{us}\\ &&\hat{U}_{2m}\; =\; \hat{U}^{(S,1)}\; =\; \hat P_{-1,-1}^{(S)}\, \hat \sigma_1^{(1)} + \hat P_{1,1}^{(S)}\, \hat{1}^{(1)}\; =\; \left(\hat{U}^{(S,1)}\right)^{\dagger}\,,\label{ub} \end{eqnarray}
where $P_{j,j}^{(S)} = |j\rangle\!^{(S)} {}^{(S)}\hspace*{-0.8mm}
\langle j|$ is a (local) projection operator, and the Turing head is externally driven by substitution sequences $\{\alpha_{m}\}$ specified by $\alpha_1, \alpha_2$\,. Here we restrict ourselves to the quasi-periodic Fibonacci - (qf) and Thue-Morse sequence (tm), respectively: \begin{eqnarray} &&\alpha_1^{\mbox{qf}} = \alpha_1\,,\, \alpha_2^{\mbox{qf}} = \alpha_2\,,\, \alpha_3^{\mbox{qf}} = \alpha_1\,,\, \alpha_4^{\mbox{qf}} = \alpha_1\,,\, \alpha_5^{\mbox{qf}} = \alpha_2\,,\, \cdots\label{fibo}\\ &&\alpha_1^{\mbox{tm}} = \alpha_1\,,\, \alpha_2^{\mbox{tm}} = \alpha_2\,,\, \alpha_3^{\mbox{tm}} = \alpha_2\,,\, \alpha_4^{\mbox{tm}} = \alpha_1\,,\, \alpha_5^{\mbox{tm}} = \alpha_2\,,\, \cdots\,.\label{tm} \end{eqnarray}
First, we consider the reduced state-space dynamics of the head~$S$ and tape-spin~$1$, respectively, \begin{eqnarray} \sigma_{j}^{(S)}(n)\;\, = &\mbox{Tr} \left( \hat{\rho}_n^{(S)}\, \hat{\sigma}_j^{(S)} \right)& =\;\,
\langle\psi_{n}|\hat{\sigma}_{j}^{(S)} \otimes \hat{1}^{(1)}
|\psi_{n}\rangle\,,\nonumber\\ \sigma_{k}^{(1)}(n)\;\, = &\mbox{Tr} \left( \hat{\rho}_n^{(1)}\, \hat{\sigma}_k^{(1)} \right)& =\;\,
\langle\psi_{n}|\hat{1}^{(S)} \otimes \hat{\sigma}_{k}^{(1)}|\psi_{n} \rangle\,,\label{bloch} \end{eqnarray}
where $|\psi_n\rangle$ is the total network state at step $n$, and $\sigma_{j}^{(\mu)}(n)$ are the respective Bloch-vectors. Due to the entanglement between the head and tape, both will, in general, appear to be in a `mixed-state', which means that the length of the Bloch-vectors in (\ref{bloch}) is less than $1$. However, for specific initial states
$|\psi_0\rangle$ the state of head and tape will remain pure:
As $|\pm\rangle\!^{(1)} =
\frac{1}{\sqrt{2}}\left(|\hspace*{-1.mm}-\hspace*{-1.mm}1
\rangle\!^{(1)} \pm |1\rangle\!^{(1)}\right)$ are the eigenstates of $\hat{\sigma}_{1}^{(1)}$ with $\hat{\sigma}_{1}^{(1)}
|\pm\rangle\!^{(1)} = \pm |\pm\rangle\!^{(1)}$, the QCNOT-operation $\hat{U}^{(S,1)}$ of equation~(\ref{ub}) cannot create any entanglement, irrespective of the head state
$|\varphi\rangle\!^{(S)}$, i.e. \begin{eqnarray} \label{entangle}
\hat{U}^{(S,1)}\,|\varphi\rangle\!^{(S)} \otimes\,|+\rangle\!^{(1)}\,&=&\,
|\varphi\rangle\!^{(S)} \otimes\,|+\rangle\!^{(1)}\nonumber\\
\hat{U}^{(S,1)}\,|\varphi\rangle\!^{(S)} \otimes\,|-\rangle\!^{(1)}\,&=&\, \hat{\sigma}_{3}^{(S)}
|\varphi\rangle\!^{(S)} \otimes\,|-\rangle\!^{(1)}\,. \end{eqnarray}
As a consequence, the state~$|\psi_n\rangle$ remains a product state for any step~$n$\, and initial product state
$|\psi_0^{\pm}\rangle = |\varphi_0\rangle\!^{(S)} \otimes
|\pm\rangle\!^{(1)}$\, with
$|\varphi_0\rangle\!^{(S)} = \exp{ \left(-i \hat{\sigma}_1^{(S)} {\varphi_{0}}/2\right)}\,
|\hspace*{-1.mm}-1\rangle\!^{(S)}$\,, so that the Turing head performs a pure-state trajectory (`primitive', see \cite{KIM99}) on the Bloch-circle $\left(\sigma_{1}^{(S)}(n)=0\right)$ \begin{eqnarray}
&|\psi_{n}^{\pm}\rangle = |\varphi_{n}^{\pm}\rangle\!^{(S)} \otimes
|\pm\rangle\!^{(1)}\,,
\;\;\;\;\left(\sigma_{2}^{(S)}(n|\pm)\right)^{2} +
\left(\sigma_{3}^{(S)}(n|\pm)\right)^{2} = 1\,.\nonumber& \end{eqnarray}
Here $\sigma_{j}^{(S)}(n|\pm)$ denotes the Bloch-vector of the
Turing head $S$ conditioned by the initial state $|\psi_0^{\pm}\rangle$. From the Fibonacci sequence (\ref{fibo}) and the property (\ref{entangle})
it is found for $|\varphi_{n}^{+}\rangle\!^{(S)} \otimes\,|+\rangle\!^{(1)}, n = 2m$, and $\varphi_0^{\pm} = \alpha_0 = 0$ that \begin{equation} \label{plus_fibo}
\sigma_{2}^{(S)}(2m|+) = \sin {\mathcal{C}}_{2m}(+)\,,\;\;\;\;
\sigma_{3}^{(S)}(2m|+) = -\cos {\mathcal{C}}_{2m}(+)\,, \end{equation}
and\, $\sigma_{k}^{(S)}(2m-1|+) = \sigma_{k}^{(S)}(2m|+)$\,, where the cumulative rotation angle is \begin{eqnarray} &{\mathcal{C}}_{2m}(+)\; =\; {\displaystyle \sum_{j=1}^{m} \alpha_{j}^{\mbox{qf}}}\; =\; \alpha_1\, m\, +\, (\alpha_2 - \alpha_1)\, m'\,,&\nonumber \end{eqnarray} with\, $m'\, (\leq m)$ being the total number of angles\, $\alpha_2$ up to step $2m$. For the cumulative rotation angle ${\mathcal{C}}_{n}(-)$ up to step $n$ we utilize the following recursion relations \begin{eqnarray} &{\mathcal{C}}_{2m}(-) = -{\mathcal{C}}_{2m-1}(-)\,,\;\;\;\; {\mathcal{C}}_{2m-1}(-) = \alpha_{m}^{\mbox{qf}} + {\mathcal{C}}_{2m-2}(-)\,.&\nonumber \end{eqnarray} Then it is easy to verify that for
$|\varphi_{n}^{-}\rangle\!^{(S)} \otimes\,|-\rangle\!^{(1)}$ and $\varphi_0^{\pm} = \alpha_0 = 0$ \begin{equation} \label{cumulative}
\left| {\mathcal{C}}_{n}(-) \right|\; \leq \; 2\,
\max \left( |\alpha_1|\, ,\, |\alpha_2| \right)\, =:\, M\,,
\end{equation} yielding $\sigma_{2}^{(S)}(n|-) = \sin {\mathcal{C}}_{n}(-),\,
\sigma_{3}^{(S)}(n|-) = -\cos {\mathcal{C}}_{n}(-)$.
From any initial state,
$|\psi_{0}\rangle = a^{(+)}|\varphi_{0}^{+}\rangle\!^{(S)} \otimes
|+\rangle\!^{(1)} + a^{(-)}|\varphi_{0}^{-}\rangle\!^{(S)} \otimes
|-\rangle\!^{(1)}$, we then obtain at step $n$ \begin{eqnarray}
&|\psi_n\rangle = a^{(+)}|\varphi_{n}^{+}\rangle\!^{(S)} \otimes\,
|+\rangle\!^{(1)}\,+\,a^{(-)}|\varphi_{n}^{-}\rangle\!^{(S)} \otimes\,
|-\rangle\!^{(1)}&\nonumber
\end{eqnarray} and, observing the orthogonality of the $|\pm\rangle\!^{(1)}$, \begin{equation} \label{super}
\sigma_{k}^{(S)}(n) = |a^{(+)}|^{2}\,\sigma_{k}^{(S)} (n|+)\,+\,
|a^{(-)}|^{2}\,\sigma_{k}^{(S)} (n|-)\,. \label{lambda_S} \end{equation} This trajectory of the Turing-head~$S$ represents a non-orthogonal pure-state decomposition. By using (\ref{plus_fibo}), (\ref{cumulative}), (\ref{super}) $\left( \mbox{with}\; a^{(+)} = a^{(-)} = 1/\sqrt{2} \right)$ we finally get for
$|\psi_0\rangle = |-1\rangle\!^{(S)} \otimes\,|-1\rangle\!^{(1)}$ \begin{eqnarray} \label{chaotic_driving} \left(\sigma_{2}^{(S)}(n),\,\sigma_{3}^{(S)}(n)\right)\;=\; \cos {\mathcal{B}}_n \cdot \left( \sin {\mathcal{A}}_n ,\,-\cos {\mathcal{A}}_n \right)\,, \end{eqnarray} where $({\mathcal{C}}_{n}(+) - M)/2 \leq {\mathcal{A}}_n = ({\mathcal{C}}_{2m}(+) + {\mathcal{C}}_{2m}(-))/2\,,\; {\mathcal{B}}_n = ({\mathcal{C}}_{2m}(+) - {\mathcal{C}}_{2m}(-))/2 \leq ({\mathcal{C}}_{2m}(+) + M)/2$\,, $n = 2m$\, or\, $2m-1$. Thus the expression~(\ref{chaotic_driving}) indicates that for the local dynamics of the Turing head in the `non-classical' regime the cumulative control loss due to any small perturbation $\delta$ of the given $\alpha_1^{\mbox{qf}}, \alpha_1^{\mbox{qf}}$ grows at most linearly with $n$, so that all periodic orbits on the plane $\left\{0, \sigma_{2}^{(S)}, \sigma_{3}^{(S)}\right\}$ are stable (see figure~\ref{stability}$a$ - $c$), as in the case of the `regular' control $\alpha_m = \alpha_1$ \cite{KIM99}. This may be contrasted with the chaotic Fibonacci-rule (cf), $\alpha_{m+1}^{\mbox{cf}} = \alpha_m^{\mbox{cf}} + \alpha_{m-1}^{\mbox{cf}}$ (Lyapunov exponent: $\ln\, (1 + \sqrt{5})/2 > 0$)\,, which can be interpreted as a temporal random (chaotic) analogue to one-dimensional chaotic potentials \cite{IKI99}: each step $\alpha_m$ is controlled by the cumulative information of the two previous steps. For a small perturbation of the initial phase angle $\alpha_0$ the cumulative angles ${\mathcal{A}}_m, {\mathcal{B}}_m$\,, respectively, grow exponentially with $m$, and so do the deviation terms $\Delta {\mathcal{C}}_{2m}^{\mbox{cf}}(\pm) = {\mathcal{C}}_{2m}^{\mbox{cf}\,'}(\pm) - {\mathcal{C}}_{2m}^{\mbox{cf}_{\mbox{po}}}(\pm)$ from the periodic orbits (po). Thus the total cumulative control loss induced by the perturbation can show chaotic quantum behaviour on the Turing head \cite{IKI99}.
For the Thue-Morse control (\ref{tm}) we easily find that for
$|\psi_0\rangle = |-1\rangle\!^{(S)} \otimes\,|-1\rangle\!^{(1)}$ and $n = 8m$ \begin{eqnarray} &{\mathcal{C}}_{n}(+)\; =\; 2\, (\alpha_1 + \alpha_2)\, m\,,\;\;\;\; {\mathcal{C}}_{n}(-)\; =\; 0\,,&\nonumber \end{eqnarray} respectively, which is very similar to the result of the `regular' machine with $\alpha_m^{\mbox{reg}} = (\alpha_1 + \alpha_2)/2$, implying ${\mathcal{C}}_{8m}^{\mbox{reg}}(+) = 2 (\alpha_1 + \alpha_2)$ and ${\mathcal{C}}_{8m}^{\mbox{reg}}(-) = 0$. In all cases considered we thus find characteristic local invariants with respect to the Turing head (figure~\ref{stability}$d$).
\section{Parameter sensitivity}
The distance between density operators, $\hat{\rho}$ and $\hat{\rho}'$, defined by the so-called Bures metric \cite{HUE92} \begin{eqnarray} &D_{\rho \rho'}^{2} := \mbox{Tr} \left\{(\hat{\rho} - \hat{\rho}')^2\right\}\,.&\nonumber \end{eqnarray} lies, independent of the dimension of the Liouville space, between 0 and 2. For pure states we can rewrite \begin{eqnarray}
&&D^2\; =\; 2\,(1 - |\langle\psi|\psi'\rangle|^2)\; =\; 2\,( 1 - O')\nonumber\\
&&O' := |\langle \psi_0(\delta)| \hat{U}^{\dagger}(\delta)\,
\hat{U}(0) |\psi_0(0) \rangle|^2\,,\nonumber \end{eqnarray} where the perturbed unitary evolution, $\hat{U}(\delta)$, connects the
initial state, $|\psi_0(\delta)\rangle$, and $|\psi'\rangle$. This metric can be applied likewise to the total-network-state space or any subspace. In any case it is a convenient additional means to characterize various QTMs: for the regular case, $\alpha_m^{\mbox{reg}} = \alpha_1$ (Lyapunov exponent $= 0$), and any initial perturbation $\delta$ for $\hat{\rho}'$ the distance remains almost constant \cite{IKI99}; for the chaotic Fibonacci rule (cf), on the other hand, $\left( \alpha_m (\hat{\rho}) = \alpha_m^{\mbox{cf}}\,,\, \alpha_m (\hat{\rho}') = \alpha_m (\hat{\rho}) + \delta_{m}^{\mbox{cf}} \right.$\,, where $\delta_{m}^{\mbox{cf}}$ is the cumulative perturbation of the angle\, $\alpha_m^{\mbox{cf}}$\, at step $\left. n = 2m - 1 \right)$\, we obtain an initial exponential sensitivity \cite{IKI99}. In the case of the present substitution sequences we observe for a small perturbation of the given $\alpha_1, \alpha_2$ no initial exponential sensitivity in the evolution of $D^2$, which confirms that any periodic orbit is stable (figure~\ref{evolve}$a$). Finally we display the evolution of $D^2$ for the total network
state $|\psi_n\rangle$, which also shows no exponential sensitivity (figure~\ref{evolve}$b$\,, cf. figure~3$c$ in \cite{IKI99}). The respective distances for tape-spin $1$ are similar to those shown. The corresponding behaviour under the Thue-Morse control is qualitatively the same. This parameter-sensitivity \cite{PER91} has been proposed as a measure to distinguish quantum chaos from regular quantum dynamics. From the results of the present analysis and those in \cite{IKI99} satisfying this criterion we conclude that, indeed, only classical chaotic input makes the quantum dynamics in QTM architectures chaotic, too (figure~\ref{QTM_chaos}).
\section{Summary}
In conclusion, we have studied the quantum dynamics of a small QTM driven by substitution sequences based on a decoherence-free Hamiltonian. As quantum features we utilized the superposition principle and the physics of entanglement. Quantum dynamics manifests itself in the superposition and entanglement of a pair of `classical' (i.e. disentangled) state-sequences. The generalized QTM under this kind of control connects two fields of much recent interest, quantum computation and motion in one-dimensional structures with `deterministically aperiodic' potential distributions. No chaotic quantum dynamics results in this case, as shown by the lack of exponential parameter-sensitivity. Local invariants leading to one-dimensional point manifolds (patterns) exist for $\alpha_1 = \alpha_2$\, only. For $\alpha_1 \ne \alpha_2$ a continuous destruction of these patterns sets in (figure~\ref{stability}$b$\,; hardly visible yet in figure~\ref{stability}$a$). This reminds us of the disappearing of KAM tori in the classical phase space resulting from a small perturbation (see e.g. \cite{ALL97}). Patterns in reduced Bloch-planes $\left\{0, \sigma_{2}^{(\mu)}, \sigma_{3}^{(\mu)}\right\}$ (a quantum version of a Poincar\'{e}-cut) should thus be similarly useful to characterize quantum dynamics in a broad class of quantum networks. Due to the entanglement, we can see regular, chaotic, and intermediate quantum dynamics, respectively. Furthermore, the parameter sensitivity gives a sensitive criterion for testing induced quantum chaos in a pure quantum regime. This might be contrasted with the usual quantum chaology, which is concerned essentially only with semiclassical spectrum analysis of classically chaotic systems (e.g. level spacing, spectral rigidity) \cite{BER85}. It is expected that a QTM architecture with a larger number of pseudo-spins on the Turing tape would still exhibit the same type of dynamical behaviour under the corresponding driving conditions.
Figure~\ref{QTM_chaos}: Input-output-scheme of our quantum Turing machine~(QTM). \vspace*{0.5cm}
Figure~\ref{stability}: Turing-head patterns $\left\{0, \sigma_{2}(n), \sigma_{3}(n)\right\}$ for initial state
$|\psi_{0}\rangle =
|-1\rangle\!^{(S)} \otimes$\linebreak
$|-1\rangle\!^{(1)}$ under the control of substitution sequences: ($a$) quasi-periodic Fibonacci (qf) with $\alpha_1 = \frac{2}{5} \pi\,,\, \alpha_2 = \alpha_1 + 0.0005 \pi$\,, ($b$) as in $a$) but for $\alpha_2 = \alpha_1 + 0.03 \pi$\,, ($c$) as in $a$) but for $\alpha_2 = \alpha_1 + 0.05 \pi$\,; ($d$) Thue-Morse (tm) control with $\alpha_1 = \frac{2}{5} \pi\,,\, \alpha_2 = \alpha_1 + 0.1001 \pi$\,. For each simulation the total step number is $n=10000$. \vspace*{0.5cm}
Figure~\ref{evolve}: Evolution of the (squared) distance $D_{\rho \rho'}^2$ between the perturbed, $\hat{\rho}'$\,, and the reference QTM state, $\hat{\rho}$\,, under the quasi-periodic Fibonacci (qf) control:
($a$) for Turing head; ($b$) for total network state $|\psi_n\rangle$\,.
For $\hat{\rho}$\, we take $|\psi_{0}\rangle =
|-1\rangle\!^{(S)} \otimes\,|-1\rangle\!^{(1)}$ and $\alpha_1 = \frac{2}{5} \pi\,, \, \alpha_2 = \alpha_1 + 0.03 \pi$\,.
Line {\bf A}: $|\psi_{0}'\rangle = \exp{
\left(-i \hat{\sigma}_1^{(S)} {\delta}/2\right)} |\psi_{0}\rangle$ for $\hat{\rho}'$\,, $\delta = 0.001$.
Line {\bf B}: $|\psi_{0}'\rangle = |\psi_{0}\rangle$\,, but $\alpha_{1}' = \alpha_1 + 0.001 \pi\,,\, \alpha_{2}' = \alpha_2 + 0.001 \pi$\, for $\left(\hat{\rho}'\right)$\,.
\begin{figure}\label{QTM_chaos}
\label{stability}
\label{evolve}
\end{figure}
\end{document} | arXiv | {
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\begin{document}
\title{Solution of Extremal Problems in Bergman Spaces Using the Bergman Projection}
\renewcommand\footnotemark{} \author{Timothy Ferguson\footnote{Thanks to Peter Duren for his help in editing the manuscript, and to the referee for many helpful suggestions.} }
\maketitle
\begin{abstract} In this paper we discuss the explicit solution of certain extremal problems in Bergman spaces. In order to do this, we develop methods to calculate the Bergman projections of various functions. As a special case, we deal with canonical divisors for certain values of $p$. \end{abstract}
\section{Introduction}\label{intro}
This paper deals with linear extremal problems in Bergman spaces. The study of extremal problems in Bergman spaces was inspired by extremal problems in Hardy spaces, which have been studied by various authors, notably by Macintyre and Rogosinski (see \cite{Macintyre_Rogosinski}), Rogosinski and Shapiro (see \cite{Rogosinski_Shapiro}), and S.~Ya.~Khavinson (see \cite{KhavinsonSYa1949} and \cite{KhavinsonSYa1951}).
Bergman space extremal problems have been studied by various authors, for example in \cite{Khavinson_Stessin}, \cite{Dragan}, \cite{Ryabykh_certain_extp}, \cite{Hedenmalm_canonical_A2}, and \cite{DKSS_Pac}. See also the survey \cite{Beneteau_Khavinson_survey}. Regularity results for these problems have been studied in \cite{Ryabykh}, \cite{Khavinson_McCarthy_Shapiro}, \cite{tjf1}, and \cite{tjf2}. However, there are still no general methods for finding solutions to these problems, and few explicit solutions are available. This is in contrast to the situation for Hardy spaces, where a rich theory based on duality and functional analysis allows many extremal problems to be explicitly solved (see the references in the previous paragraph.)
This paper introduces methods for finding explicit solutions to certain extremal problems in Bergman spaces. For example, we solve certain minimal interpolation problems involving finding the smallest norm of a Bergman space function when its value and the value of its first two derivatives are specified at the origin. Similar results to ours are obtained in other works, for example \cite{Khavinson_Stessin}, \cite{Osipenko_Stessin}, and \cite{SheilSmall_2011}. As another example, we find the function that maximizes the functional defined by $f \mapsto f^{(n)}(0) + bf(0)$ for certain values of $b$. The methods are based on theorems developed in the paper about the relation between the Bergman projection and extremal problems, as well as calculations of Bergman projections of various functions. As a special case, we deal with canonical divisors, also known as contractive divisors, for certain $A^p$ spaces.
An analytic function $f$ in the unit disc $\mathbb{D}$ is said to belong to the Bergman space $A^p$ if
\[ \|f\|_{A^p} = \left\{ \int_{\mathbb{D}} |f(z)|^p \, d\sigma(z)\right\}^{1/p} < \infty. \] Here $\sigma$ denotes normalized area measure, so that $\sigma(\mathbb{D})=1.$
For $1 < p < \infty,$ the dual of the Bergman space $A^p$ is isomorphic to $A^q$, where $1/p + 1/q = 1,$ and $k\in A^q$ represents the functional defined by \[ \phi(f) = \int_{\mathbb{D}} f(z)\conj{k(z)}\,d\sigma(z). \] Note that this isomorphism is actually conjugate-linear. It is not an isometry unless $p=2$, but if the functional $\phi \in (A^p)^*$ is represented by the function $k \in A^q$, then \begin{equation}\label{A_q_isomorphism}
\| \phi \|_{(A^p)^*} \le \| k \|_{A^q} \le C_p \| \phi \|_{(A^p)^*} \end{equation} where $C_p$ is a constant depending only on $p$.
In this paper the only Bergman spaces we consider are those with $1<p<\infty$. The case $p \le 1$ is more difficult because the proof of Theorem \ref{lin_ext_proj} fails for $p \le 1$. This theorem is a key result needed for our method of solving extremal problems. The proof of Theorem \ref{lin_ext_proj} relies on the boundedness of the Bergman projection on $L^p$, which fails for $p \le 1$. It also relies on H\"{o}lder's inequality, which fails for $p < 1$.
For a given linear functional $\phi \in (A^p)^*$ such that $\phi \ne 0$, we investigate the linear extremal problem of finding a function $F \in A^p$ with norm
$\|F\|_{A^p} = 1$ for which \begin{equation}\label{norm1}
\text{Re } \phi(F) = \sup_{\|g\|_{A^p}=1} \text{Re } \phi(g) = \| \phi \|. \end{equation} Such a function $F$ is called an extremal function, and we say that $F$ is an extremal function for a function $k \in A^q$ if $F$ solves problem \eqref{norm1} for the functional $\phi$ with kernel $k$. This problem has been studied by numerous authors (see the introduction and references for some examples).
Note that for $p=2$, the extremal function is $F = k/\|k\|_{A^2}.$
A closely related problem is that of finding $f\in A^p$ such that $\phi(f) = 1$ and \begin{equation}\label{value1}
\|f\|_{A^p} = \inf_{\phi(g) = 1} \|g\|_{A^p}. \end{equation} If $F$ solves the problem \eqref{norm1}, then $\frac{F}{\phi(F)}$ solves the problem \eqref{value1}, and if $f$ solves \eqref{value1}, then
$\frac{f}{\|f\|}$ solves \eqref{norm1}. When discussing either of these problems, we always assume that $\phi$ is not the zero functional, in other words, that $k$ is not identically $0$.
It is well known that the problems \eqref{norm1} and \eqref{value1} each have a unique solution when $1<p<\infty$ (see e.g.~\cite{Ryabykh}, or \cite{tjf1}, Theorem 1.4). Also, for every function $f \in A^p$ such that $f$ is not identically $0$, there is a unique $k \in A^q$ such that $f$ solves problem \eqref{value1} for $k$ (see e.g.~\cite{tjf1}, Theorem 3.3). This implies that for each $F \in A^p$ with
$\|F\|_{A^p} = 1$, there is some nonzero $k$ such that $F$ solves problem \eqref{norm1} for $k.$ Furthermore, any two such kernels $k$ are positive multiples of each other.
The next result is an important characterization of extremal functions in $A^p$ for $1<p<\infty$ (see \cite{Shapiro_Approx}, p.~55). \begin{refthm}\label{integral_extremal_condition} Let $1 < p < \infty$ and let $\phi \in (A^p)^*$.
A function $F \in A^p$ with $\|F\|_{A^p} = 1$ and $\text{Re } \phi(F) > 0$ satisfies
$$\text{Re } \phi(F) = \sup_{\|g\|_{A^p} =1} \text{Re } \phi(g) = \| \phi \|$$ if and only if
$$\int_{\mathbb{D}} h |F|^{p-1} \conj{\sgn F} \, d\sigma = 0$$ for all $h \in A^p$ with $\phi(h) = 0.$ If $F$ satisfies the above conditions, then
$$\int_{\mathbb{D}} h |F|^{p-1} \conj{\sgn F}\, d\sigma
= \frac{\phi(h)}{\| \phi \|}$$ for all $h\in A^p.$ \end{refthm}
The following may also be found in \cite{Shapiro_Approx}, p.~55. \begin{refthm}\label{alt_integral_extremal_condition} Suppose that $X$ is a closed subspace of $L^p(\mathbb{D})$, for $1<p<\infty$. Let $F \in L^p$ and suppose that for all $h \in X$,
we have $\|F\| \le \|F + h\|.$ Then, \[
\int_{\mathbb{D}} h|F|^{p-1} \conj{\sgn F} d\sigma = 0 \] for all $h \in X.$ \end{refthm}
Because point evaluation is a bounded linear functional on the Hilbert space $A^2$, the space $A^2$ has a reproducing kernel $K(z,\zeta)$, called the \emph{Bergman kernel}, with the property that \begin{equation}\label{eq_intro_reproducing_formula} f(z) = \int_{\mathbb{D}} K(z, \zeta) f(\zeta) \, d\sigma(\zeta) \end{equation} for all $f \in A^2$ and for all $z \in \mathbb{D}.$ One can show that $$K(z,\zeta) = \frac{1}{(1-\conj{\zeta}z)^2}.$$ Since the polynomials are dense in $A^1$, we have that \eqref{eq_intro_reproducing_formula} holds for all $f \in A^1.$
In fact, for any $f$ in $L^1$ we many define the Bergman projection $\mathcal{P}$ by \[ (\mathcal{P} f)(z) = \int_{\mathbb{D}} \frac{f(\zeta)}{(1-\conj{\zeta}z)^2}\, d\sigma(\zeta). \] The Bergman projection maps $L^1$ into the space of functions analytic in $\mathbb{D}.$ A non-trivial fact is that $\mathcal{P}$ also maps $L^p$ boundedly onto $A^p$ for $1<p<\infty.$ If $p = 2,$ then $\mathcal{P}$ is just the orthogonal projection of $L^2$ onto $A^2.$
The rest of this paper is organized as follows. In section \ref{rel_berg_ext}, we prove various theorems relating the Bergman projection to extremal problems. In section \ref{calc_bp}, we calculate various Bergman projections. We use these results in section \ref{specific_probs} to solve some extremal problems explicitly. Lastly, in section \ref{canon_divisors}, we apply our results to the study of canonical divisors in $A^p$ when $p$ is an even integer.
\section{Relation of the Bergman Projection to Extremal Problems}\label{rel_berg_ext} In this section we show how information about the Bergman projection can be used to solve certain extremal problems. We begin with a basic theorem that is obvious but quite useful. \begin{theorem}\label{bp_equality} Suppose that $1<p<\infty$ and let $f \in A^p$ and $g \in L^q$, where $1/p + 1/q = 1.$ Then \[ \int_{\mathbb{D}} f \conj{g}\, d\sigma = \int_{\mathbb{D}} f \, \conj{\mathcal{P}(g)}\, d\sigma. \] \end{theorem} \begin{proof} The case $p=2$ follows from the fact that $\mathcal{P}$ is the orthogonal projection from $L^2$ onto $A^2$. The other cases follow from a routine approximation argument, using the fact that $\mathcal{P} : L^p \rightarrow A^p$ boundedly. \end{proof}
The next theorem gives the first indication of how the Bergman projection is related to extremal problems. \begin{theorem}\label{lin_ext_proj} Suppose that $1<p<\infty$.
Let $F \in A^p$ with $\|F\|_{A^p} = 1.$ Then $F$ is the extremal function for the functional with kernel \[
k = \mathcal{P}(|F|^{p-1} \sgn F) \in A^q. \] Furthermore, if $F$ is the extremal function for some functional $\phi \in (A^p)^*$ with kernel $k \in A^q,$ then $$
\mathcal{P}(|F|^{p-1} \sgn F) = \frac{k}{\|\phi\|}. $$ \end{theorem} \begin{proof} Consider the functional $\psi \in (A^p)^*$ that takes a function $f \in A^p$ to $$
\psi(f) = \int_{\mathbb{D}} f |F|^{p-1}\conj{\sgn F}\, d\sigma. $$
This functional has norm at most $\|\, |F|^{p-1}\conj{\sgn F} \,\|_{L^q} =
\|F\|^{p/q}_{L^p} = 1$. But also $\psi(F) = \|F\|^p_{A^p} = 1$, so $\psi$ has norm exactly $1$ and $F$ is the extremal function for $\psi$.
But from Theorem \ref{bp_equality}, it follows that $$
\int_{\mathbb{D}} f\, \conj{\mathcal{P}(|F|^{p-1}\sgn F)}\, d\sigma =
\int_{\mathbb{D}} f |F|^{p-1}\conj{\sgn F}\, d\sigma $$
for any $f \in A^p$, which means that $\mathcal{P}(|F|^{p-1}\sgn F)$ is the kernel in $A^q$ representing $\psi$. This proves the first part of the theorem.
If $F$ is the extremal function for $\phi$, then $\psi$ is a positive scalar multiple of $\phi$ (see Section \ref{intro}.)
Since $\| \psi\| =1$ and $\psi$ is a positive scalar
multiple of $\phi$, it must be that $\psi = \phi/\|\phi\|.$
But this implies that $\mathcal{P}(|F|^{p-1}\sgn F) = k/\|\phi\|.$ \end{proof}
The next result, Theorem \ref{min_int1}, describes the relation of the Bergman projection to a sort of generalized minimal interpolation problem. The problem is to find the function of smallest norm such that prescribed
linear functionals acting on the function take prescribed values. We will first need the following lemma. \begin{lemma}\label{lin_alg_lemma} Let $V$ be a vector space over $\mathbb{C}$, and let $\phi, \phi_1, \ldots, \phi_N$ be linear functionals on $V$ such that, for $v \in V,$ if $\phi_1(v) = \cdots = \phi_N(v) = 0$, then $\phi(v) = 0.$ Then $\phi = \sum_{j=1}^N c_j \phi_j$ for some constants $c_j.$ \end{lemma} The statement and proof of this lemma may be found in \cite{Conway_Functional} in Appendix A.2 as Proposition 1.4.
\begin{theorem}\label{min_int1} Let $1<p<\infty$ and let $\phi_1, \phi_2, \ldots, \phi_N \in (A^p)^*$ be linearly independent. Then a function $F \in A^p$ satisfies
$$ \|F\|_{A^p} = \inf\{\|f\|_{A^p} : \phi_1(f) = \phi_1(F), \ldots , \phi_N(f) = \phi_N(F) \}$$ if and only if
$\mathcal{P}(|F|^{p-1} \sgn F)$ is a linear combination of the kernels of $\phi_1, \ldots, \phi_N.$ \end{theorem} Note that this theorem gives a necessary and sufficient condition for a function $F$ to solve the minimal interpolation problem of finding a function $f \in A^p$ of smallest norm such that $\phi_j(f) = c_j$ for $1 \le j \le N$, where $\phi_j \in (A^p)^*$ are arbitrary linearly independent functionals and the $c_j$ are given constants. Namely, $F$ solves the problem if and only if $\phi_j(F) = c_j$ for $1 \le j \le N$ and
$\mathcal{P}(|F|^{p-1} \sgn F)$ is a linear combination of the kernels of $\phi_1, \ldots, \phi_N$. Note that for the case $1<p<\infty$, the problem under discussion will always have a unique solution (see e.g.~\cite{tjf1}, Proposition 1.3). \begin{proof} Let $k_1, \ldots, k_N$ be the kernels of $\phi_1, \ldots, \phi_N$, respectively. Suppose that
$$ \|F\|_{A^p} = \inf\{\|f\|_{A^p} : \phi_1(f) = \phi_1(F), \ldots , \phi_N(f) = \phi_N(F) \}$$
and let $h$ be any non-zero $A^p$ function such that $\phi_1(h) = \cdots = \phi_N(h) = 0.$ Since there are only a finite number of the $\phi_j$, it is clear that such a function exists. Then $F + h$ is also in contention to solve the extremal problem,
so $\|F\| \le \|F + h\|$. Now Theorem \ref{alt_integral_extremal_condition} shows that \[
\int_{\mathbb{D}} |F|^{p-1} \conj{\sgn F}\, h\, d\sigma = 0, \] and so by Theorem \ref{bp_equality} \[
\int_{\mathbb{D}} \conj{\mathcal{P}(|F|^{p-1} \sgn F)}\, h \,d\sigma = 0. \] Define \[
\qquad \psi(f) = \int_{\mathbb{D}} \conj{\mathcal{P}(|F|^{p-1} \sgn F )}\, f\, d\sigma, \qquad f \in A^p. \] Lemma \ref{lin_alg_lemma} now shows that $$\psi = \sum_{j=1}^N c_j \phi_j,$$ for some
constants $c_j$, so $\mathcal{P}(|F|^{p-1} \sgn F)$ is a linear combination of $k_1, \ldots, k_n.$ This proves the ``only if'' part of the theorem.
Conversely, suppose $ \mathcal{P}(|F|^{p-1}\sgn F)$ is a linear combination of $k_1, \ldots, k_n.$ Now \begin{equation}\label{min_int1_eq1}
\|F\|_{A^p}^p = \int_{\mathbb{D}} F |F|^{p-1}\conj{\sgn F}\,d\sigma =
\int_{\mathbb{D}} F\, \conj{\mathcal{P}(|F|^{p-1}\sgn F)}\,d\sigma, \end{equation} by Theorem \ref{bp_equality}. Now let $h \in A^p$ be such that $\phi_j(h) = 0$ for $1 \le j \le N.$ Since
$\mathcal{P}(|F|^{p-1}\sgn F )$ is a linear combination of the $k_j$, equation \eqref{min_int1_eq1} gives \[ \begin{split}
\|F\|_{A^p}^p &= \int_{\mathbb{D}} (F+h)\conj{\mathcal{P}(|F|^{p-1}\sgn F)}\,d\sigma\\ &=
\int_{\mathbb{D}} (F+h) |F|^{p-1}\conj{\sgn F}\,d\sigma \\
&\le \|F + h\|_{A^p} \||F|^{p-1}\conj{\sgn F} \|_{A^q}\\ &=
\|F+h\|_{A^p} \|F\|_{A^p}^{p-1}. \end{split} \] Therefore, \[
\|F\|_{A^p} \le \|F+h\|_{A^p}. \] Since $h$ was an arbitrary $A^p$ function with the property that $\phi_j(h) = 0$ for $1\le j \le N$, this shows that $F$ solves the extremal problem in question. \end{proof} When we apply this theorem, we will usually have each $\phi_j$ be a derivative-evaluation functional. By derivative-evaluation functional, we mean a functional defined by $f \mapsto f^{(n)}(z_0)$ for some integer $n \ge 0$ and some $z_0 \in \mathbb{D}.$ Note that the theorem implies that, if $\phi_1, \phi_2, \ldots, \phi_N \in (A^p)^*$ are linearly independent, then the following two statements are equivalent:\\ 1. $F$ satisfies
$$ \|F\|_{A^p} = \inf\{\|f\|_{A^p} : \phi_1(f) = \phi_1(F), \ldots, \phi_N(f) = \phi_N(F) \}$$ but does not satisfy
$$ \|F\|_{A^p} = \inf\{\|f\|_{A^p} : \phi_{j_1}(f) = \phi_{j_1}(F), \ldots, \phi_{j_{M}}(f) = \phi_{j_{M}}(F) \}$$ for any proper subsequence $\{j_k\}_{k=1}^{M}$ of $1, 2, \ldots, N$. \\ 2.
$\mathcal{P}(|F|^{p-1} \sgn F)$ is a linear combination of the kernels of $\phi_1, \ldots, \phi_N$, and none of the coefficients in the linear combination is $0$.
The next theorem is a special case of Theorem \ref{min_int1}, with the functionals taken to be $\phi_j(h) = h^{(j)}(0)$, with kernels $k_j(z) = (j+1)! z^j.$ \begin{theorem}\label{min_int_poly1}
The function $\mathcal{P}(|F|^{p-1}\sgn F)$ is a polynomial of degree at most $N$ if and only if \[
\|F\|_{A^p} = \inf \{\|f\|_{A^p} : f(0) = F(0), \ldots, f^{(N)}(0)=F^{(N)}(0)\}. \] It is a polynomial of degree exactly $N$ if and only if $N$ is the smallest integer such that the above conditions holds. \end{theorem}
The next theorem relates the generalized minimal interpolation problems we have been discussing with linear extremal problems. \begin{theorem} Let $\phi_1, \ldots, \phi_N$ be linearly independent elements of $(A^p)^*$ with kernels $k_1, \ldots, k_N$ respectively, and let
$F\in A^p$ with $\|F\|_{A^p} = 1$. Then the functional for which $F$ is the extremal function has as its kernel a linear combination of the $k_j$ if and only if \[
\|F\|_{A^p} = \inf \{\|f\|_{A^p} : \phi_1(f) = \phi_1(F), \ldots, \phi_N(f) = \phi_N(F) \}. \] \end{theorem} This follows from Theorems \ref{lin_ext_proj} and \ref{min_int1}. Recall that although there is no unique functional for which $F$ is the extremal function, such a functional is unique up to a positive scalar multiple, which does not affect whether its kernel is a linear combination of the $k_j$.
One direction of this theorem, the fact that if $F$ is the extremal function for some kernel which is a linear combination of the $k_j$, then $F$ solves the stated minimal interpolation problem, is easy to prove directly. The proof is as follows. Let $F$ be the extremal function for the functional $\phi$, which we assume to have kernel $k = \sum_{j=1}^N a_j k_j.$ Then
$$ \|F\|_{A^p} = \inf\{\|f\|_{A^p} : \phi(f) = \phi(F)\}.$$ But if some function $G$ in $A^p$ satisfies $\phi_j(F) = \phi_j(G)$ for all $j$ with $1\le j \le N$, then $\phi(G) = \phi(F),$ which implies that
$\|F\|_{A^p} \le \|G\|_{A^p}$. This implies that $F$ satisfies $$
\|F\|_{A^p} = \inf \{\|f\|_{A^p} : \phi_1(f) = \phi_1(F), \ldots, \phi_N(f) = \phi_N(F) \}. $$
\section{Calculating Bergman Projections}\label{calc_bp}
Now that we have explored the relation between the Bergman projection and solutions to extremal problems, we will calculate the Bergman projection in various cases.
\begin{proposition}\label{bp_monomial} Let $m$ and $n$ be nonnegative integers. Then \[ \mathcal{P}(z^m \conj{z}^n) = \begin{cases} \frac{m-n+1}{m+1}z^{m-n}, &\text{ if $m \ge n$,}\\ 0, &\text{ if $m < n$}. \end{cases} \] \end{proposition} This is Lemma 6 in Chapter 2 of \cite{D_Ap}.
The next theorem is very helpful in calculating the Bergman projection of the kernel of a derivative-evaluation functional times the conjugate of an $A^p$ function. \begin{theorem}\label{bp_f_conjg} Let $1<q_1,q_2\le\infty$. Let $p_1$ and $p_2$ be the conjugate exponents of $q_1$ and $q_2$. Let $$\frac{1}{q} = \frac{1}{q_1} + \frac{1}{q_2}$$ and suppose that $1<q<\infty.$ Let $p$ be the conjugate exponent of $q$.
Suppose that $k \in A^{q_1}$ and that $g \in A^{q_2}.$ Let the functional $\psi$ be defined by $\psi(f) = \int_{\mathbb{D}} f \conj{k}\, d\sigma$ for all $f \in A^{p_1}.$ Then $\mathcal{P}(k\conj{g})$ is the kernel of the functional $\phi \in (A^{p})^*$ defined by $$ \phi(f) = \psi(fg), \qquad f\in A^p.$$ \end{theorem}
\begin{proof} First note that $1/p + 1/q_1 + 1/q_2 = 1$, so if $f \in A^p,$ then $fg \in A^{p_1}$ and the definition of $\phi$ makes sense. Now observe that $$ \phi(f) = \psi(fg) = \int_{\mathbb{D}} fg \conj{k}\, d \sigma.$$ By Theorem \ref{bp_equality}, this equals \[
\int_{\mathbb{D}} f \conj{\mathcal{P}\left(\conj{g} k \right)}\, d\sigma. \qedhere \] \end{proof}
We will study the kernels of various derivative-evaluation functionals. Evaluation at the origin is somewhat different and simpler than evaluation elsewhere, so we deal with it first.
\begin{theorem}\label{kernel_derivative_0} The kernel for the functional $f \mapsto f^{(n)}(0)$ is $(n+1)! z^n.$ If $g \in A^p$ then \[ \mathcal{P}(z^n \conj{g(z)}) = \sum_{j=0}^n \conj{\frac{g^{n-j}(0)}{(n-j)!}} \frac{j+1}{n+1} z^{n}. \] \end{theorem} \begin{proof} The first statement can be verified by evaluating \[ \int_{\mathbb{D}} f(z) \conj{z}^n d\sigma(z) \] when $f$ is written as a power series. The second part follows from Proposition \ref{bp_monomial}. To see this, note that by the first part of the theorem, $\mathcal{P}(z^n \conj{g(z)})$ is the kernel for the functional taking $f \in A^p$ to \[ \frac{1}{(n+1)!}(fg)^{(n)}(0) = \frac{1}{(n+1)!} \sum_{j=0}^n \binom{n}{j} f^{(j)}(0) g^{(n-j)}(0), \] which has kernel \[\sum_{j=0}^n \conj{\frac{g^{n-j}(0)}{(n-j)!}} \frac{j+1}{n+1} z^{j}. \qedhere \] \end{proof}
We will now deal with the function $1/(1-\conj{a}z)^n$, for $n \ge 2$. \begin{proposition}\label{kernel_f_(n)} The kernel for the functional $f \mapsto f^{(n)}(a)$ is $$ \qquad \qquad \qquad \qquad \frac{(n+1)!z^{n}}{(1-\conj{a}z)^{n+2}}, \qquad \qquad n=0,1,2,\ldots $$ \end{proposition}
\begin{proof} We know that $$f(a) = \int_{\mathbb{D}} \frac{1}{(1-a\conj{z})^2}f(z)\, d\sigma(z).$$ Differentiation $n$ times with respect to $a$ gives the result. \end{proof}
\begin{proposition}\label{(1-az)n+2_kernel} For each $a \in \mathbb{D}$ with $a \ne 0$, there are numbers $c_0, \ldots, c_n$ with $c_n \ne 0$ such that the function $$\frac{1}{(1-\conj{a}z)^{n+2}}$$ is the kernel for the functional $f \mapsto c_0 f(a)+ c_1 f'(a) + \ldots + c_n f^{(n)}(a).$ \end{proposition} \begin{proof} We will proceed by induction. The claim is true for $n=0$ by the reproducing property of the Bergman kernel function. For general $n$, we may write the partial fraction expansion $$ \frac{z^n}{(1-\conj{a}z)^{n+2}} = \sum_{j=0}^{n+2} \frac{b_j}{(1-\conj{a}z)^j}, $$ for some complex numbers $b_j$. Thus, $$ z^n = \sum_{j=0}^{n+2} b_j (1-\conj{a}z)^{n+2-j}.$$ Differentiating both sides $n+1$ times with respect to $z$ gives $$0 = b_1 (-\conj{a})^{n+1}(n+1)! + b_0(-\conj{a})^{n+1}(n+2)! (1-\conj{a}z).$$ Since this holds for all $z$, it follows that $b_0 = b_1 = 0.$ Since $ z^n/(1-\conj{a}z)^{n+2}$ has a pole of order $n+2$ at $1/\conj{a}$, we see that $b_{n+2} \ne 0.$ Therefore, $$\frac{1}{(1-\conj{a}z)^{n+2}} = \frac{1}{b_{n+2}}\left(\frac{z^n}{(1-\conj{a}z)^{n+2}} - \sum_{j=2}^{n+1} \frac{b_j}{(1-\conj{a}z)^j}\right).$$ Note that the first term of the right side of the above equation is the kernel for the functional $f \mapsto (1/(n+1)!)f^{(n)}(a)$. Also, each term in the sum $\sum_{j=2}^{n+1} \frac{b_j}{(1-\conj{a}z)^j}$ is the kernel for a linear functional taking each function $f$ to some linear combination of $f(a), f'(a), \ldots,$ and $f^{(n-1)}(a),$ by the induction hypothesis. This proves the proposition. \end{proof}
\begin{proposition}\label{general_proj_zero} Let $g \in A^p$ for $1<p<\infty,$ and let $a \in \mathbb{D}$ with $a \ne 0$. Suppose $g$ has a zero of order $n$ at $a$. Let $N \ge 0$ be an integer. Then $$\mathcal{P}\left(\frac{1}{(1-\conj{a}z)^{N+2}} \conj{g(z)}\right) = \sum_{k=0}^{N-n} C_k \frac{1}{(1-\conj{a}z)^{k+2}}$$ for some complex constants $C_k$ depending on $g^{(m)}(a)$ for $0 \le m \le N$. \end{proposition} \begin{proof} The projection $$\mathcal{P}\left( \frac{1}{(1-\conj{a}z)^{N+2}}\, \conj{g(z)} \right)$$ is the kernel associated with the functional $$f \mapsto \sum_{j=0}^N b_j (fg)^{(j)}(a)$$ for some constants $b_j$, with $b_N \ne 0$, by the previous proposition and Theorem \ref{bp_f_conjg}. But \[ \sum_{j=0}^N b_j (fg)^{(j)}(a) = \sum_{j=0}^N \sum_{k=0}^j b_j \binom{j}{k} f^{(k)}(a) g^{(j-k)}(a). \] Since $g^{(j)}(a) = 0$ for $0 \le j < n,$ all terms in the sum with $j-k < n$ are $0$. But this means that the only non-zero terms in the sum occur when $k \le j-n$, so that $k \le N-n.$ Now, set \[ B_k = \sum_{j = k + n}^{N} b_j \binom{j}{k} g^{(j-k)}(a), \] so \[ \sum_{j=0}^N b_j (fg)^{(j)}(a) = \sum_{k=0}^{N-n} B_k f^{(k)}(a). \] But the kernel associated to $\sum_{k=0}^{N-n} B_k f^{(k)}(a)$ is \[ \sum_{k=0}^{N-n} B_k \frac{(k+1)!z^k}{(1-\conj{a}z)^{k+2}}. \] As in the proof of Theorem \ref{(1-az)n+2_kernel}, we may show that \[ \frac{z^k}{(1-\conj{a}z)^{k+2}} = \frac{c_{k2}}{(1-\conj{a}z)^2}
+ \frac{c_{k3}}{(1-\conj{a}z)^3} + \cdots +
\frac{c_{k,k+2}}{(1-\conj{a}z)^{k+2}} \] for some constants $c_{k2}, \ldots, c_{k,k+2}$. Thus we may write \[ \sum_{k=0}^{N-n} B_k \frac{(k+1)!z^k}{(1-\conj{a}z)^{k+2}} = \sum_{k=0}^{N-n} C_k \frac{1}{(1-\conj{a}z)^{k+2}}\] for some constants $C_k$, depending on $g^{(m)}(a)$ for $0 \le m \le N$. \end{proof}
We will now deal with the function $1/(1-\conj{a}z).$ Since the functional with kernel $1/(1-\conj{a}z)^{n+2}$ involves differentiation of order $n$, it seems reasonable that the functional with kernel $1/(1-\conj{a}z)$ involves integration. This is indeed the case. \begin{proposition}\label{kernel_for_integration} The function $$1/(1-\conj{a}z)$$ is the kernel for the functional defined on $A^p$ for $1<p<\infty$ by $$f \mapsto \frac{1}{a} \int_0^a f(z)\, dz.$$ \end{proposition}
\begin{proof} Since $$\frac{1}{1-\conj{a}z} = \sum_{n=0}^\infty (\conj{a}z)^n,$$ it follows that \begin{equation}\label{int_1/(1-conj(a)z)z^m} \int_{\mathbb{D}}\frac{z^m}{1-a\conj{z}} \, d\sigma = \sum_{n=0}^\infty \int_{\mathbb{D}}(a\conj{z})^n z^m \, d\sigma =
a^m \int_{\mathbb{D}} |z|^{2m}\, d\sigma = \frac{a^m}{m+1}. \end{equation} The change in the order of integration and summation is justified by the fact that the sum converges uniformly in $\overline{\mathbb{D}}.$ Now let $f \in A^p$ and write $f(z) = \sum_{m=0}^\infty b_m z^m.$ Define $$F(z) = \frac{1}{z} \int_0^z f(\zeta)\, d\zeta
= \sum_{m=0}^\infty \frac{b_m}{m+1}z^m.$$ Therefore, by equation \eqref{int_1/(1-conj(a)z)z^m}, $$\int_{\mathbb{D}}\frac{1}{1-a\conj{z}}\, f(z)\, d\sigma = \int_{\mathbb{D}}\frac{1}{1-a\conj{z}} \left(\sum_{m=0}^\infty b_m z^m\right) d\sigma = \sum_{m=0}^\infty b_m \frac{a^m}{m+1} = F(a).$$ The interchange of the order of integration and summation is justified by the fact the partial sums of the Taylor series for $f$ approach $f$ in $A^p.$ \end{proof}
The following theorem is quite useful for determining what form certain Bergman projections have. \begin{theorem}\label{dpeval_conj_proj} For $1 \le n \le N$, let $d_n$ be a nonnegative integer and let $z_n \in \mathbb{D}$. Let $k$ be analytic and a linear combination of the kernels of the functionals given by $f \mapsto f^{(d_n)}(z_n)$. Let $g \in A^p$ for $p > 1.$ Then $\mathcal{P}(k\conj{g})$ is in the linear span of the set of all the kernels of functionals defined by $f \mapsto f^{(m)}(z_n)$, where $m$ is an integer with $0 \le m \le d_n$ and $n$ is an integer with $1 \le n \le N$. \end{theorem} \begin{proof} Let $k = \sum_{n=1}^N a_n k_n$, where $k_n$ is the kernel for the functional $f \mapsto f^{(d_n)}(z_n)$. Then by Theorem \ref{bp_f_conjg}, $\mathcal{P}(k_n \conj{g})$ is the kernel of the functional $$f \mapsto (fg)^{(d_n)}(z_n) = \sum_{j=0}^{d_n} \binom{d_n}{j} f^{(j)}(z_n) g^{(d_n-j)}(z_n).$$ But this functional is a linear combination of functionals of the form $$ f \mapsto f^{(m)}(z_n),$$ where $0 \le m \le d_n.$ \end{proof}
Due to their relation with extremal problems, we are often concerned with projections of the form $\mathcal{P}(F^{p/2}\conj{F}\hbox{}^{(p/2)-1})$, where $F$ is an analytic function. This is well defined because
\[ F^{p/2} \conj{F}\hbox{}^{(p/2)-1} = |F|^p/\conj{F}
= |F|^{p-1} \sgn F.\] The following theorems deal with this situation. \begin{theorem}\label{dpeval_Fp2_answer} Let $1<p<\infty$, and let $F \in A^p$. Furthermore, suppose if $p<2$ that $F^{(p/2)-1} \in A^{p_1}$ for some $p_1 > 1$. Also, suppose that $F^{p/2}$ is analytic and is a linear combination of the kernels $k_n$ corresponding to the functionals $f \mapsto f^{(d_n)}(z_n)$ for some integers $d_n$ and some points $z_n \in \mathbb{D}$, where $1 \le n \le N$, and where $N$ is an integer. Then $F$ satisfies \[ \begin{split}
\|F\|_{A^p} = \inf \{\|f\|_{A^p} : f^{(m)}(z_n) = F^{(m)}(z_n) &\text{ for all $n$ and $m$ such that}\\ &\text{ \hspace{3ex} $1 \le n \le N$ and $1 \le m \le d_n$}\}. \end{split} \] \end{theorem} \begin{proof} This follows from Theorems \ref{dpeval_conj_proj} and \ref{min_int1}. \end{proof}
The following theorem is a consequence of Theorem \ref{dpeval_conj_proj}. It can also be proved by using Taylor series and Proposition \ref{bp_monomial}.
\begin{theorem}\label{poly_conj_proj} Let $f$ be a polynomial of degree at most $N$ and let $g \in A^p$ for some $p > 1.$ Then $\mathcal{P}(f\conj{g})$ is a polynomial of degree at most $N$. \end{theorem}
Using this theorem and Theorem \ref{min_int_poly1}, we immediately get the following result. \begin{theorem}\label{poly_Fp2_answer} Suppose that $F \in A^p$ and $F^{p/2}$ is a polynomial of degree $N$. Furthermore, if $p<2$ suppose that $F^{(p/2)-1} \in A^{p_1}$ for some $p_1 > 1$. Then \[
\|F\|_{A^p} = \inf \{\|f\|_{A^p} : f(0) = F(0), \ldots, f^{(N)}(0)=F^{(N)}(0)\}. \] \end{theorem} Note that $F^{p/2}$ can be a polynomial only if $F$ is nonzero in $\mathbb{D}$ or $p/2$ is rational and all the zeros of $F$ in $\mathbb{D}$ are of order a multiple of $s$, where $r/s$ is the reduced form of $p/2$. If $p$ is an even integer, this poses no restriction. Because of this, the case where $p$ is an even integer is often easier to work with.
\section{Solution of Specific Extremal Problems}\label{specific_probs}
We will now discuss how to solve some specific minimal interpolation problems. Since we are dealing with the powers $p/2$ and $2/p$, neither of which need be an integer, we will have to take care in our calculations. We will introduce a lemma to facilitate this. The lemma basically says that if $f$ and $g$ are analytic functions nonzero at the origin, and if $f^{(n)}(0) = (g^p)^{(n)}(0)$ for all $n$ such that $0 \le n \le N,$ then $(f^{1/p})^{(n)}(0) = g^{(n)}(0)$ for all $n$ such that $0 \le n \le N.$
To state the lemma we first need to introduce some notation. Suppose that the constants $c_0, c_1, \ldots, c_N$ are given and that $c_0 \ne 0$, and let $h(z) = c_0 + c_1 z + \cdots + c_N z^N.$ Suppose that $a = c_0^p$ for some branch of the function $z^p$. Let $U$ be a neighborhood of the origin such that $h(U)$ is contained in some half plane whose boundary contains the origin, and such that $0 \not \in h(U)$. Then we can define $z^p$ so that it is analytic in $h(U)$ and so that $c_0^p = a.$ We let $\beta_j^p(a; c_0, c_1, \ldots, c_j)$ denote the $j^{th}$ derivative of $h(z)^p$ at $0$.
Note that because of the chain rule for differentiation, $\beta_j^p$ only depends on $j$, the constants $c_0, \ldots, c_j$, and the numbers $p$ and $a$. For the same reason, the value of $\beta_j^p$ is the same if we replace the function $h$ in the definition of $\beta_j^p$ by any function $\widetilde{h}$ analytic near the origin such that $\widetilde{h}^{(j)}(0) = c_j$ for $1 \le j \le N$. \begin{lemma} Let $c_0, c_1, \ldots, c_N$ be given complex numbers, and let $p$ be a real number. Suppose that $c_0 \ne 0$, and let $a_0 = c_0^p$, for some branch of $z^p$. Then $$c_j = \beta_j^{1/p}\left(c_0; \beta_0^p(a_0;c_0), \beta_1^p(a_0;c_0, c_1), \ldots, \beta_j^p(a_0;c_0, \ldots, c_j)\right).$$ \end{lemma} \begin{proof} Let $a_j = \beta_j^p(a_0; c_0, \ldots, c_j)$ and $b_j = \beta_j^{1/p}(c_0; a_0, \ldots, a_j).$ Then $b_0=c_0.$
Now let $f(z) = \sum_{j=0}^N \frac{c_j}{j!} z^j$. Then $f^{(j)}(0) = c_j$ for $0 \le j \le N$. Let $U$ be a neighborhood of $0$ such that there exist $r_0 > 0$ and $\theta_0 \in \mathbb{R}$ such that \[ f(U) \subset \left\{re^{i\theta}: r_0 < r \text{ and } \theta_0 - \frac{\pi}{2p} < \theta < \theta_0 + \frac{\pi}{2p}\right\}. \] Then $z^p$ can be defined as an analytic function in $f(U)$. Furthermore, the set $V = (f(U))^p$ does not contain $0$ but is contained in some half plane, so $z^{1/p}$ can be defined as an analytic function in $V$ so that it is the inverse of the function $z^p$ defined in $f(U)$.
Now define $g(z) = (f(z))^p$ for $z \in U.$ Then $g^{(j)}(0) = a_j$ and $g^{1/p}(0) = c_0$, so $(g^{1/p})^{(j)}(0) = b_j$ for $0\le j \le N$. But $g(z)^{1/p} = f(z)$ for $z \in U$, so $b_j = c_j$ for $0 \le j \le N$. \end{proof}
We will now use the lemma to solve a specific extremal problem in certain cases. \begin{theorem}\label{nonzero_min_int1} Let $c_0, \ldots, c_N$ be given complex numbers, and assume that $c_0 \ne 0$. Suppose that $F \in A^p$, and $F^{(j)}(0) = c_j$ for $0 \le j \le N$, and \[
\|F\|_{A^p} = \inf \{\|g\|_{A^p} : g(0) = c_0, \ldots, g^{(N)}(0)=c_N\}. \] Let $a_0 = c_0^{p/2}$ for some branch of $z^p$. Define \[ f(z) = \sum_{j=0}^N \frac{\beta_j^{p/2}(a_0; c_0, \ldots, c_j)}{j!} z^j, \] where the $\beta_j^{p/2}$ are defined as in the beginning of this section. Suppose that $f^{1-(2/p)} \in A^{p_1}$ for some $p_1 > 1$. Also, suppose that $f$ has no zeros in $\mathbb{D}.$ Thus we may define $f^{2/p}$ so that it is analytic in $\mathbb{D}$ and so that $f^{2/p}(0) = c_0.$ Then $$F = f^{2/p}.$$ The same result also holds if $p$ is rational, $2/p=r/s$ in lowest form, and every zero of $f$ has order a multiple of $s$. \end{theorem} \begin{proof} Note that $f^{2/p}$ is analytic in $\mathbb{D}$ since we have assumed $f$ has no zeros in $\mathbb{D}$, or that $p$ is rational and $2/p=r/s$ in lowest form and $f$ has only zeros whose orders are multiples of $s$. Also, $f(0) = a_0$, so we may define $f^{2/p}$ so that $f^{2/p}(0) = c_0$. The $j^{th}$ derivative of $f^{2/p}$ at $0$ is $$\beta_j^{2/p}\left(c_0; \beta_0^{p/2}(a_0;c_0), \ldots, \beta_j^{p/2}(a_0;c_0, \ldots, c_j)\right)$$ for $0 \le j \le N$, which equals $c_j$ by the lemma. Thus $f^{2/p}$ is in contention to solve the extremal problem.
But \[
\mathcal{P}\left(\frac{|f^{2/p}|^p}{\,\conj{f}^{2/p}}\right) = \mathcal{P}(f \conj{f}^{1-(2/p)}) \]
is a polynomial of degree at most $N$ by Theorem \ref{poly_conj_proj}, so by Theorem \ref{poly_Fp2_answer} we find $F = f^{2/p}$. \end{proof}
To apply this theorem, we need to show that $f$
has no zeros in the unit disc, or has only zeros of suitable orders if $p$ is rational. Then, as long as $f^{1-(2/p)} \in A^{p_1}$ for some $p_1 >1$, we have that $f^{2/p}$ is the extremal function. Note that we do not need to know anything about the zeros of the extremal function itself to apply the theorem, but only about the zeros of $f$.
Also note that if $f$ has no zeros in the unit disc, this theorem implies that the extremal function $F = f^{2/p}$ also has no zeros in the unit disc. It can also be shown that if $F$ has no zeros, then $F$ must equal $f^{2/p}$. This follows from \cite{Khavinson_Stessin}, Theorem B. It also follows from the work on extremal problems in Bergman space posed over non-vanishing functions found in \cite{Khavinson_nonvanishing}, \cite{SheilSmall_2011}, \cite{SheilSmall_solution2012}, and \cite{SheilSmall_general2012}. The case where the extremal function has zeros is more challenging and not as well understood. See Example \ref{one_zero_A4_example} for a problem in which the extremal function has one zero.
\begin{example} The solution to the minimal interpolation problem in $A^p$ with $f(0) = 1$ and $f'(0)=c_1$ is \[ F(z) = \left(1 + \frac{p}{2}c_1 z \right)^{2/p}, \]
provided that $|c_1| \le \frac{2}{p}$ (or $p=2$). This is because $\beta_0^{p/2}(1;1, c_1) = 1$ and $\beta_1^{p/2}(1;1, c_1) = (p/2)c_1.$ For example, if $p = 4$ and $c_1 = \frac{1}{2}$, then \[ F(z) = (1+z)^{1/2}. \] \end{example} The above problem is also solved in \cite{Osipenko_Stessin} in more general form. The solution to the extremal problem in the next example is more difficult. We do not know if it has been stated explicitly before, although in the case in which the extremal function has no zeros it does follow from \cite{Khavinson_Stessin}, Theorem B, or from the results in \cite{SheilSmall_2011}, if it is assumed \emph{a priori} that the extremal function has no zeros.
\begin{example} The solution to the minimal interpolation problem in $A^p$ for $1<p<\infty$ with $F(0)=1$, and $F'(0) = c_1$, and $F''(0) = c_2$ is \[ F(z) = \left\{ 1 + (p/2)c_1z + [(p(p-2)/4)c_1^2 + (p/2)c_2]z^2 \right\}^{2/p}, \] provided that the quadratic polynomial under the $2/p$ exponent in the equation for $F$ has no zeros in $\mathbb{D}$. The solution is the same if $p=4$ or $4/3$ and the quadratic polynomial has a repeated root in the unit disc. (The solution is also the same in the case $p=2$, no matter where the polynomial has roots). \end{example}
Linear extremal problems tend to be more difficult to solve than minimal interpolation problems involving derivative-evaluation functionals, because values of a function $f$ and its derivatives are generally easier to
calculate than $\mathcal{P}(|f|^{p-1}\sgn f)$. Nevertheless it is possible to solve some linear extremal problems explicitly by the methods in this paper. Here is one example. \begin{theorem} Let $N \ge 1$ be an integer, let $1<p<\infty$, and let $b \in \mathbb{C}$ satisfy \[
|b| \ge 1 + \frac{1}{N+1}\left(1-\frac{2}{p}\right), \] and define \[ a =
\frac{|b|+\sqrt{|b|^2-\frac{4}{N+1}\left(1-\frac{2}{p}\right)}}{2} \sgn b. \] Then the solution to the extremal problem in $A^p$ with kernel $z^N + b$ is \[ F(z) = \sgn(a^{1-(2/p)})
\frac{(z^N + a)^{2/p}}{\left(|a|^2 + 1/(N+1)\right)^{1/p}}. \] \end{theorem} In the above expression for $F(z)$, the branch of $(z^N+a)^{2/p}$ may be chosen arbitrarily, but the value of $\sgn(a^{1-(2/p)})$ must be chosen consistently with this choice. Note that the functional associated with the kernel $z^N + b$ is \[ \phi(f) = bf(0) + (1/(N+1)!)f^{(N)}(0). \] Also, observe that the hypothesis of the theorem
holds for all $N$ and $p$ if $|b| \ge 3/2$.
\begin{proof} The condition \[
|b| \ge 1 + \frac{1}{N+1}\left(1-\frac{2}{p}\right) \]
implies that $|a| \ge 1$, so that $z^N + a \ne 0$ in $\mathbb{D}$ and $F$ is an analytic function. Note that \[
\|(z^N + a)^{2/p}\|_{A^p}^p =
\int_{\mathbb{D}} |z^N + a|^2\, d\sigma = \int_{\mathbb{D}} (z^N + a) \conj{(z^N + a)}\, d\sigma =
|a|^2 + \frac{1}{N + 1}. \]
Thus, $\|F\|_{A^p} = 1$.
Now $$((z^N + a)^{2/p})^{p/2-1} = a^{1-2/p}+\left(1-\frac{2}{p}\right)a^{-2/p}z^N + O(z^{2N}), $$ where we choose branches so that $((z^N+a)^{2/p})^{p/2} = z^N + a.$ We calculate that \[ \begin{split}
&\quad\;\mathcal{P}\left( |z^N+a|^{p-1} \sgn(z^N + a) \right) \\ &=\mathcal{P}\left( (z^N + a) \conj{(z^N + a)}^{1-2/p}\right) \\ &=\mathcal{P}\left( (z^N + a) \left( \conj{a^{1-2/p}} + \left(1-\frac{2}{p}\right)\conj{a^{-2/p}}\, \conj{z^N} + O(\conj{z^{2N}})\right) \right). \end{split} \] But by Proposition \ref{bp_monomial}, this equals \[ \begin{split} &\quad\; \mathcal{P}\left[ (z^N + a) \left( \conj{a^{1-2/p}} + \left(1-\frac{2}{p}\right) \conj{a^{-2/p}}\, \conj{z^N}\right) \right]\\ &=a \conj{a^{1-2/p}} + \frac{1}{N+1}\left(1-\frac{2}{p}\right) \conj{a^{-2/p}} + \conj{a^{1-(2/p)}} z^N\\ &= \conj{a^{1-(2/p)}}\left( z^N + a + \frac{1}{N+1}\left(1-\frac{2}{p}\right)\conj{a}\,^{-1} \right)\\ &= \conj{a^{1-(2/p)}}(z^N + b). \end{split} \]
Thus, \[ \begin{split}
&\quad\; \mathcal{P}\left\{ \left |\sgn(a^{1-(2/p)})(z^N+a)\right|^{p-1} \sgn\left( \sgn(a^{1-(2/p)})(z^N + a) \right) \right\}\\ &=\conj{a^{1-(2/p)}} \sgn(a^{1-(2/p)}) (z^N + b) \\
&=|a^{1-(2/p)}\!|\,(z^N + b). \end{split} \]
Therefore, \[ \mathcal{P}(F^{p/2}\conj{F^{(p/2)-1}}) =
|a^{1-(2/p)}| \frac{z^N + b}{\left(|a|^2 + 1/(N+1)\right)^{(p-1)/p}}. \]
Since $\|F\|_{A^p} = 1$, Theorem \ref{lin_ext_proj} shows that $F$ is the extremal function for the kernel on the right of the above equation. But that kernel is a positive scalar multiple of $k$, so $F$ is also the extremal function for $k$. \end{proof}
\begin{example}\label{one_zero_A4_example} Let $a \in \mathbb{D}\setminus\{0\}$ and let $b, c \in \mathbb{C}$. Consider the function \[ f(z) = \frac{1}{a}\frac{a-z}{1-\conj{a}z}\left( 1+bz+cz^2 \right)^{1/2}, \] where we assume that $1+bz+cz^2$ has no zeros in $\mathbb{D}$, or a double zero in $\mathbb{D}$ (so that $f$ is analytic). We choose the branch of this function so that $f(0)=1$. Then a calculation shows that \[ \begin{split}
f'(0) &= \frac{1}{a} \left(\frac{ab}{2}+|a|^2-1\right), \text{ and}\\
f''(0) &= \frac{1}{a} \left( |a|^2 b +2a\conj{a}^2 + ac - 2\conj{a} - b
-\frac{ab^2}{4} \right).\\ \end{split} \] Another calculation shows that the residue of $f^2$ about $\conj{a^{-1}}$ is equal to \[
|a|^{-4} \conj{a}^{-3} \left[
(|a|^2-1)^2 (2c + \conj{a}b) -
2 (|a|^2 -1)(\conj{a}^2 + c + \conj{a}b) \right]. \]
Now, suppose that $v_1$ and $v_2$ are complex numbers, and that we want to solve the minimal interpolation problem of finding $F \in A^4$ such that
$F(0)=1, F'(0)=v_1, F''(0)=v_2$, and with $\|F\|_{4}$ as small as possible. If we can find numbers $a$, $b$, and $c$ so that $(1+bz+cz^2)$ has no zeros in $\mathbb{D}$, or a repeated zero, and so that \[ \begin{split}
v_1 &= \frac{1}{a} \left(\frac{ab}{2}+|a|^2-1\right)\\
v_2 &= \frac{1}{a} \left( |a|^2 b +2a\conj{a}^2 + ac - 2\conj{a} - b
-\frac{ab^2}{4} \right)\\
0 &= (|a|^2-1)^2 (2c + \conj{a}b) -
2 (|a|^2 -1)(\conj{a}^2 + c + \conj{a}b) \end{split} \] then \[ F(z) = f(z) = \frac{1}{a} \frac{a-z}{1-\conj{a}z}
\left( 1 + bz + cz^2 \right)^{1/2}. \] To see this, note that in this case \[ f(z)^2 = a_1 z^2 + a_2 z + a_3 + \frac{a_4}{(1-\conj{a}z)^2} \] for some constants $a_1, \ldots, a_4.$ Then \[ \mathcal{P}\left[ \left(a_1 z^2 + a_2 z + a_3\right) \conj{f(z)} \right] \] will be a polynomial of degree at most two by Theorem \ref{poly_conj_proj}. Also, since $f(a) = 0$, we have \[ \mathcal{P} \left[ \frac{a_4}{(1-\conj{a}z)^2} \conj{f(z)}\right] = 0 \] by Proposition \ref{general_proj_zero}. Thus, $\mathcal{P}(f^2 \conj{f})$ is a polynomial, and so $f$ solves the extremal problem in question, by Theorem \ref{min_int_poly1}.
\end{example}
\section{Canonical Divisors}\label{canon_divisors}
We will now discuss how our previous results apply to canonical divisors. These divisors are the Bergman space analogues of Blaschke products. They were first introduced in the $A^2$ case in \cite{Hedenmalm_canonical_A2}, and were further studied for general $p$ in \cite{DKSS_Pac} and \cite{DKSS_Mich}. The formula for a canonical divisor with one zero is well known, see for example \cite{D_Ap}. In \cite{Hansbo}, a formula was obtained for canonical divisors with two zeros, as well as with more zeros under certain symmetry conditions on the zeros. In \cite{Luciano_Naris_Schuster}, a method is given for finding the canonical divisor in $A^2$ for an arbitrary finite zero set. In \cite{MacGregor_Stessin}, a fairly explicit formula for canonical divisors is obtained for general $p$. In this section, we discuss how the methods of this paper apply to the problem of finding canonical divisors in the case where $p$ is an even integer. The results we obtain are similar to those in \cite{MacGregor_Stessin}.
By the zero-set of an $A^p$ function not identically $0$, we mean its collection of zeros, repeated according to multiplicity. Such a set will be countable, since the zeros of analytic functions are discrete. Given an $A^p$ zero set, we can consider the space $N^p$ of all functions that vanish on that set. More precisely, $f\in A^p$ is in $N^p$ if it vanishes at every point in the given zero set, to at least the prescribed multiplicity.
If the zero set does not include $0$, we pose the
extremal problem of finding $G \in N^p$ such that $\|G\|_{A^p} = 1$, and such that $G(0)$ is positive and as large as possible. If the zero set has a zero of order $n$ at $0$, we instead maximize $G^{(n)}(0)$. For $0<p<\infty$, this problem has a unique solution, which is called the canonical divisor. For $1<p<\infty$, this follows from the fact that an equivalent problem is to find an $F \in N^p$ with
$F(0)=1$ and $\|F\|_{A^p}$ as small as possible. It is well known that the latter problem has a unique solution (see e.g.~\cite{tjf1}, Proposition 1.3).
In this section, we discuss the problem of determining the canonical divisor when $p$ is an even integer, and the zero set is finite. We show how the methods of this paper can be used to characterize the canonical divisor. Our methods show that if $G$ is the canonical divisor, then $G^{p/2}$ is a rational function with residue $0$ at each of its poles, which is the content of the following theorem.
\begin{theorem}\label{canonical_nec} Let $p$ be an even integer. Let $z_1, \ldots, z_N$ be distinct points in $\mathbb{D}$, and consider the zero-set consisting of each of these points with multiplicities $d_1, \ldots, d_N$, respectively. Let $G$ be the canonical divisor for this zero set. Then there are constants $c_0$ and $c_{nj}$ for $1 \le n \le N$ and $0 \le j \le (p/2)d_n - 1$, such that \[ \begin{split} G(z)^{p/2} &= c_0 + \sum_{n=1}^N \sum_{j=0}^{(p/2)d_n - 1}
\frac{c_{nj}}{(1-\conj{z_n}z)^{j+2}}, \qquad \quad \text{if $z_n \ne 0$ for all $n$, and}\\ G(z)^{p/2} &= c_0 z^{(p/2)d_1} + \sum_{j=0}^{(p/2)d_1-1} c_{1j}z^j
+ \sum_{n=2}^N \sum_{j=0}^{(p/2)d_n - 1}
\frac{c_{nj}}{(1-\conj{z_n}z)^{j+2}}, \quad \text{if $z_1 = 0.$} \end{split} \] \end{theorem} \begin{proof} Our goal is to show that $G^{p/2}$ is the kernel for some linear combination
of certain
derivative-evaluation functionals. Because we know what the
kernel of any derivative-evaluation functional is, we will be able
to show that $G$ has the form stated in the theorem.
Let $A_n = d_n( (p/2)-1)$. For $1 \le n \le N$ and $0 \le j \le A_n-1$, let $h_{nj}$ be a polynomial such that \[ h_{nj}^{(m)}(z_k) = \begin{cases}
1, \text{ if $m=n$ and $k=j$ }\\
0, \text{ otherwise.} \end{cases} \] For $f \in A^p$, define \[ \widehat{f}(z) = f(z) - \sum_{n=1}^N \sum_{j=0}^{A_n-1} a_{nj} h_{nj}(z) \] where $a_{nj} = f^{(j)}(z_n)$. Since $\widehat{f}$ has zeros of order $A_n=d_n( (p/2)-1)$ at each $z_n$, the function \[ \widetilde{f} = \frac{1}{G^{(p/2)-1}} \widehat{f} \] is in $A^p$.
But then \[ \begin{split} \int_{\mathbb{D}} \conj{G}^{p/2} f\, d\sigma &= \int_{\mathbb{D}} \conj{G(z)}^{p/2} \left( \widehat{f}(z) + \sum_{n=1}^N \sum_{j=0}^{A_n-1} a_{nj} h_{nj}(z) \right)\,d\sigma \\ &=
\int_{\mathbb{D}} |G(z)|^{p-1} \conj{\sgn{G(z)}} \widetilde{f}(z)\, d\sigma + \sum_{n=1}^N \sum_{j=0}^{A_n-1} a_{nj} \int_{\mathbb{D}} \conj{G(z)}^{p/2} h_{nj}(z) \,d\sigma\\ &= \mathrm{I} + \mathrm{II}. \end{split} \] Now, II is a linear combination of the numbers $a_{nj}$ for $1 \le n \le N$ and $0 \le j \le A_n - 1$, so we turn our attention to I. The canonical divisor $G$ is a constant multiple of the function $F \in A^p$ of smallest norm that has zeros of order $d_n$ at each $z_j$ and such that $F^{(m)}(0)=1$, where $m$ is the order of the zero-set at $0$.
By Theorem \ref{min_int1}, $\mathcal{P}(|G|^{p-1} \sgn \conj{G})$ is the kernel for a linear combination of appropriate derivative evaluation functionals at the points $0, z_1, \cdots, z_n$. Thus, we have \[ \begin{split}
\int_{\mathbb{D}} |G|^{p-1}\conj{\sgn G} \widetilde{f}\, d\sigma &=
\int_{\mathbb{D}} \mathcal{P}(|G|^{p-1}\conj{\sgn G}) \widetilde{f}\, d\sigma \\ &=
b_0 \widetilde{f}^{(m)}(0) +
\sum_{n=1}^N \sum_{j=0}^{d_n-1} b_{nj} \widetilde{f}^{(j)}(z_n), \end{split} \] for some complex constants $b_0$ and $b_{nj}$. Note that $\widetilde{f}(z) = G_n \widehat{f}_n$ where \[ G_n(z) = (z-z_n)^{A_n} \frac{1}{G^{(p/2)-1}(z)} \] and \[ \widehat{f}_n(z) = (z-z_n)^{-A_n} \widehat{f}(z). \] Note that \[ \widetilde{f}^{(j)}(z_n) = \sum_{k=0}^j \binom{j}{k} \widehat{f}_n^{(k)}(z_n) G_n^{(j-k)}(z_n) \] and \[ \begin{split} \widehat{f}_n^{(k)}(z_n) &= \frac{k!}{(k+A_n)!} \frac{d^{k+A_n}}{dz^{k+A_n}} \widehat{f}(z_n)\\ &= \frac{k!}{(k+A_n)!} \frac{d^{k+A_n}}{dz^{k+A_n}} \left[ f(z) - \sum_{n=1}^N \sum_{s=0}^{A_n-1} a_{ns} h_{ns}(z) \right]. \end{split} \] Thus, $\widetilde{f}^{(j)}(z_n)$ is a linear function of the numbers $a_{ns}$ and the numbers $f^{(k)}(z_n)$ for $0 \le k \le j + A_n.$ Recall that $a_{ns} = f^{(s)}(z_n)$.
Also, if $m=0$, then \[ \widetilde{f}^{(m)}(0) = G(0)^{1-(p/2)} \widehat{f}(0) = G(0)^{1-(p/2)} \left(f(0) -
\sum_{n=1}^N \sum_{j=0}^{A_n-1} a_{nj} h_{nj}(0) \right),\] so $\widetilde{f}^{(m)}(0)$ is a linear function of the numbers $a_{nj}$ and $f(0)$ = $f^{(mp/2)}(0)$. If $m \ne 0$, then we may assume $z_1=0$ and $m = d_1$, and then by the same reasoning as we used above for $\widetilde{f}^{(j)}(z_n)$, we see that$\widetilde{f}^{(m)}(z_1)$ is a linear function of the numbers $a_{nj}$ and the numbers $f^{(k)}(z_1)$ for $0 \le k \le d_1 + A_1 = d_1 + ((p/2)-1)d_1 = (p/2)d_1 = mp/2$. Thus, the term \[\mathrm{I} =\int_{\mathbb{D}} \conj{G^{(p/2)-1}} \widetilde{f}\, d\sigma\] is a linear combination of the numbers $f^{(k)}(z_n)$ for $0 \le k \le (d_n-1) + ((p/2)-1)d_n = (p/2)d_n - 1$, and the number $f^{(mp/2)}(0)$.
Therefore, both I and II, and thus $\int_{\mathbb{D}} f \conj{G}^{p/2}\, d\sigma$, are linear combinations of the numbers $f^{(k)}(z_n)$ for $0 \le k \le (p/2)d_n - 1$, and the number $f^{(mp/2)}(0)$. And thus, $G^{p/2}$ is the kernel for a derivative-evaluation functional depending on $f^{(j)}(z_n)$ for $1 \le n \le N$ and $0 \le j \le (p/2)d_n-1$, as well as $f^{mp/2}(0)$. Therefore $G^{p/2}$ has the desired form. \end{proof}
The previous theorem gave a condition on $G^{p/2}$ that must be satisfied if $G$ is the canonical divisor of a given zero set. The following theorem says that condition, along with a few other more obviously necessary ones, is also sufficient. \begin{theorem}\label{canonical_nec_suf} Let $p$ be an even integer. Let $z_1, \ldots, z_N$ be distinct points in $\mathbb{D}$, and consider the zero-set consisting of each of these points with multiplicities $d_1, \ldots, d_N$, respectively. The canonical divisor for this zero set is the unique function $G$ having $A^p$ norm $1$ such that $G(0)>0$ (or $G^{(m)}(0) > 0$ if $G$ is required to have a zero of order $m$ at the origin), and such that $G^{p/2}$ has zeros of order $pd_n/2$ at each $z_n$, and \[ \begin{split} G(z)^{p/2} &= c_0 + \sum_{n=1}^N \sum_{j=0}^{(p/2)d_n - 1}
\frac{c_{nj}}{(1-\conj{z_n}z)^{j+2}} \qquad
\text{if $z_n \ne 0$ for all $n$ or}\\
G(z)^{p/2} &= c_0 z^{(p/2)d_1} + \sum_{j=0}^{(p/2)d_1-1} c_{1j}z^j
+ \sum_{n=2}^N \sum_{j=0}^{(p/2)d_n - 1}
\frac{c_{nj}}{(1-\conj{z_n}z)^{j+2}} \qquad
\text{if $z_1 = 0$.} \end{split} \] \end{theorem} \begin{proof} By Theorem \ref{canonical_nec} and the definition of the canonical divisor, the stated conditions are necessary for a function to be the canonical divisor. Suppose that $G$ is a function satisfying the stated conditions. We will prove the theorem by applying Theorem \ref{min_int1} to $\mathcal{P}(G^{p/2} \conj{G}^{(p/2)-1})$.
We first discuss the proof under the assumption that $z_n \ne 0$ for all $n$. First, as above, $\mathcal{P}\left(\conj{G}^{(p/2)-1}\right) = \conj{G(0)}^{(p/2)-1}.$ Now, by Proposition \ref{general_proj_zero}, \[ \mathcal{P}\left( \frac{1}{(1-\conj{z_n}z)^{j+2}} \, \conj{G(z)}^{(p/2)-1}\right) = \sum_{k=0}^{j-((p/2)-1)d_n} C_{n,j,k} \frac{1}{(1-\conj{z_n}z)^{k+2}}, \] where the constants $C_{n,j,k}$ may depend on $G$. But if $j \le (p/2)d_n - 1$, then $j-((p/2)-1)d_n \le d_n -1$. Thus \[ \mathcal{P}\left( G^{p/2} \, \conj{G(z)}^{(p/2)-1}\right) = B_0 + \sum_{n=1}^N \sum_{k=0}^{d_n - 1}
\frac{B_{n,k}}{(1-\conj{z_n}z)^{k+2}}, \] where $B_{n,k} = \sum_{j=k+((p/2)-1)d_n}^{(p/2)d_n - 1} c_{nj} C_{n,j,k}$ and $B_0 = c_0 \conj{G(0)}^{(p/2)-1}.$ By Theorem \ref{min_int1}, $G$ is a multiple of the canonical divisor.
But the conditions that $G^{(m)}(0) > 0$ and $\|G\|_{A^p} = 1$ imply that $G$ is the canonical divisor.
The case where $z_1 = 0$ is similar, but we also use the fact that $\mathcal{P}(z^j \conj{G}^{(p/2)-1})$ is a polynomial of degree at most $j - [(p/2)-1]d_1$, or zero if $j < [(p/2)-1]d_1.$ \end{proof}
From previous work by MacGregor and Stessin \cite{MacGregor_Stessin}, a weaker form of Theorem \ref{canonical_nec} is essentially known. In the weaker form of the theorem, one only knows, in the case that no $z_n = 0$, that \[ G(z) = c_0 + \sum_{n=0}^N \frac{b_n}{1-\conj{z_n}z} + \sum_{n=1}^N \sum_{j=0}^{(p/2)d_n - 1}
\frac{c_{nj}}{(1-\conj{z_n}z)^{j+2}} \] for some constants $b_n$. The case where $z_1=0$ is similar. (Although their work also gives a fairly explicit method of finding the canonical divisor, it does not seem to be clear from their results that the $b_n$ will always be zero.) To derive Theorem \ref{canonical_nec_suf} from the weaker form of the theorem, we can use the following proposition, which gives another indication of why the residues of $G^{p/2}$ must all be zero. It basically says that nonzero residues would lead to terms in $\mathcal{P}(G^{p/2} \conj{G}^{(p/2)-1})$ that were kernels of functionals of the general form $$f \mapsto \frac{1}{a} \int_0^{a} f(z)g(z)\, dz,$$ where $g$ is an analytic function and $a \in \mathbb{D}$. But, as the proposition explains, it would then be impossible for $\mathcal{P}(G^{p/2} \conj{G}^{(p/2)-1})$ to be the kernel of a finite linear combination of derivative-evaluation functionals.
\begin{proposition} Let $g$ be analytic on $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$, and suppose $g$ is non-zero on $\partial \mathbb{D}.$ Let $a_n \in \mathbb{D}$ and $a_n \ne 0$ for $1 \le n \le N$, and assume that $a_n \ne a_j$ for $n \ne j.$ Let $b_n \in \mathbb{C}$ for $1 \le n \le N.$ Then if any of the $b_n$ are nonzero, $$\mathcal{P}\left(\sum_{n=1}^N \frac{b_n}{1-\conj{a_n}z}\, \conj{g(z)}\right)$$ is not the kernel for a functional that is the finite linear combination of derivative-evaluation functionals. \end{proposition} Note that as is shown in \cite{DKSS_Pac} (see also \cite{DKS}, \cite{Sundberg}, and \cite{D_Ap}), the canonical divisor of a finite zero set is analytic in $\overline{\mathbb{D}}$ and non-zero on $\partial \mathbb{D}.$ This allows the proposition to be applied to Bergman projections of the form \[ \mathcal{P}\left(\sum_{n=1}^N \frac{b_n}{1-\conj{a_n}z}\, \conj{G(z)^{(p/2)-1}}\right). \] \begin{proof} We know by Proposition \ref{kernel_for_integration} that
$$\mathcal{P}\left(\sum_{n=1}^N \frac{b_n}{1-\conj{a_n}z} \, \conj{g(z)}\right)$$ is the kernel for the functional given by $$f \mapsto \sum_{n=1}^N \frac{b_n}{a_n} \int_0^{a_n} f(z)g(z)\, dz.$$
Suppose this functional were a linear combination of derivative-evaluation functionals, which we will denote by $f \mapsto f^{(k)}(z_j),$ where $1 \le j \le J$ and $0 \le k \le K.$ Let $h$ be a function such that $h = gf$ for some $f \in A^p$. For fixed $g$, the values $f^{(k)}(z_j)$ for $1 \le j \le J$ and $0 \le k \le K$ are linear combinations
of the values $h^{(k)}(z_j)$, where $1 \le j \le J$ and $0 \le k \le K + r(z_j)$, and $r(z_j)$ is the order of the zero of $g$ at $z_j.$ Thus the functional defined on the space $gA^p$ by $$h \mapsto \sum_{n=1}^N \frac{b_n}{a_n} \int_0^{a_n} h(z)\, dz$$ must be a linear function of the values $h^{(k)}(z_j)$. By $gA^p$, we mean the vector space of all functions that may be written as $g$ multiplied by an $A^p$ function. Since $g$ is analytic in $\overline{\mathbb{D}}$ and $g$ is nonzero on $\partial \mathbb{D}$, any polynomial that has all the zeros of $g$ will be in $gA^p$.
Now for each $m$ there exists a polynomial $H_m$ such that $H_m(a_m) = 1$, but $H_m(a_n) = 0$ for all $n\ne m$, and such that $H_m^{(k)}(z_j) = 0$ for all $j$ and $k$ such that $1 \le j \le J$ and $1 \le k \le K + r(z_j) + 1$. Also, we may require that $H_m'$ has all the zeros of $g$, and that $H_m(0) = 0.$ Set $h_m = H_m'$. Then $h_m$ shares all the zeros of $g$, and so it is a multiple of $g$. Thus $$\sum_{n=1}^N \frac{b_n}{a_n} \int_0^{a_n} h_m(z)\, dz = 0,$$ since the left side of the above equation is a linear combination of the numbers $h_m^{(k)}(z_j)$ for $1 \le j \le J$ and $0 \le k \le K + r(z_j)$, and each $h_m^{(k)}(z_j)=0$. But also, for each $m$ such that $1 \le m \le N,$ we have $$\sum_{n=1}^N \frac{b_n}{a_n} \int_0^{a_n} h_m(z)\, dz = \sum_{n=1}^N \frac{b_n}{a_n} H_m(a_n) = \frac{b_m}{a_m},$$ so each $b_m = 0.$ \end{proof}
\begin{example} Suppose we are given distinct points $z_1, z_2, \ldots, z_N \in \mathbb{D}\setminus \{0\}.$ Let $p=2M$, where $M$ is a positive integer. Suppose we wish to find the canonical divisor in $A^p$ for the given set of points. From the theorem, we know that \[ G(z)^M = c_0 + \sum_{n=1}^N \sum_{m=0}^{M-1} \frac{c_{nm}}{(1-\conj{z_n}z)^{m+2}}. \] Then we have for $0 \le k \le M-1$ and $1 \le n \le N$ that \[
0 = \frac{d^k}{dz^k}\left( G(z)^M \right)|_{z=z_j} = \frac{d^k}{dz^k}c_0 + \sum_{n=1}^N \sum_{m=0}^{M-1} \frac{\conj{z_n}^k (m+1+k)!/(m+1)!}{(1-\conj{z_n}z_j)^{m+2+k}}
c_{nm}. \] This gives a system of $NM$ equations with $NM+1$ unknowns. Because of the uniqueness of the canonical divisor, there will be a unique solution to these
equations with $c_0 > 0$ and such that $\|G\|_{A^p} = 1$.
\end{example}
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\begin{document}
\begin{abstract}
Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd{\H{o}}s and R{\'e}nyi, and the family of $H$-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph $H$. A~widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical $H$-free graph with $n$ vertices and $m$ edges changes as $m$ grows from $0$ to $\mathrm{ex}(n,H)$. In this paper, we resolve this problem in the case when $H$ is a clique, extending a classical result of Kolaitis, Pr{\"o}mel, and Rothschild. In particular, we prove that for every $r \geqslant 2$, there is an explicit constant $\theta_r$ such that, letting $m_r = \theta_r n^{2-\frac{2}{r+2}} (\log n)^{1/\left[\rpt-1\right]}$, the following holds for every positive constant $\varepsilon$. If $m \geqslant (1+\varepsilon) m_r$, then almost all $K_{r+1}$-free $n$-vertex graphs with $m$ edges are $r$-partite, whereas if $n \ll m \leqslant (1-\varepsilon)m_r$, then almost all of them are not $r$-partite. \end{abstract}
\title{The typical structure of sparse $K_{r+1}
\section{Introduction}
\subsection{Background and motivation}
Given integers $n$ and $m$, let $G_{n,m}$ be the uniformly chosen random element of the family $\mathcal{G}_{n,m}$ of all graphs on a fixed vertex set of size $n$ that have precisely $m$ edges. The study of the evolvement of typical properties of $G_{n,m}$ when we let $m$ gradually increase from $0$ to $\binom{n}{2}$, known as the \emph{evolution of random graphs}, which was initiated in the seminal work of Erd{\H{o}}s and R{\'e}nyi~\cite{ErRe60}, is a central topic in graph theory. The behavior of many parameters and properties during the evolution of $G_{n,m}$, such as connectivity, containment of small subgraphs, chromatic number, to name a few, is now fairly well understood~\cite{Bo, JaLuRu}. A natural problem is to consider such evolution when we restrict our attention to a certain subclass of graphs, i.e., when $G_{n,m}$ is a random element of some proper subfamily of $\mathcal{G}_{n,m}$.
In this paper, we consider the class of graphs that do not contain a clique of a given fixed order. The study of \emph{$H$-free graphs}, i.e., graphs which do not contain a subgraph isomorphic to a given fixed graph $H$, is one of the cornerstones of extremal graph theory. The classical theorem of Tur{\'a}n~\cite{Tu41} states that for every $r \geqslant 2$, the largest number of edges in a $K_{r+1}$-free graph on $n$ vertices, denoted $\ex(n, K_{r+1})$, is equal to the number of edges in the balanced complete $r$-partite graph $T_r(n)$, that is \[ \ex(n, K_{r+1}) = e(T_r(n)) = \left(1 - \frac{1}{r}\right)\binom{n}{2} + O(n). \] Moreover, it identifies $T_r(n)$ as the unique \emph{extremal} graph, i.e., the unique $K_{r+1}$-free $n$-vertex graph with $\ex(n, K_{r+1})$ edges. A famous result of Kolaitis, Pr{\"o}mel, and Rothschild~\cite{KoPrRo87} determines the \emph{typical} structure of $K_{r+1}$-free graphs. It states that for every $r \geqslant 2$, almost all $K_{r+1}$-free graphs are $r$-partite ($r$-colorable); in the case $r = 2$, this was proved earlier by Erd{\H{o}}s, Kleitman, and Rothschild~\cite{ErKlRo}.
In view of the above, one naturally arrives at the following question, first considered by Pr{\"o}mel and Steger~\cite{PrSt96} more than fifteen years ago. Let $\mathcal{F}_{n,m}(K_{r+1})$ denote the family of all $K_{r+1}$-free graphs on a fixed set of $n$ vertices (for concreteness, we let it be the set $\{1, \ldots, n\}$) that have exactly $m$ edges. For which $m$ are almost all graphs in $\mathcal{F}_{n,m}(K_{r+1})$ $r$-partite? This is trivially true for very small $m$, as then almost all graphs in $\mathcal{G}_{n,m}$ are both $K_{r+1}$-free and $r$-colorable, and when $m = \ex(n, K_{r+1})$, by Tur{\'a}n's theorem. By the Kolaitis--Pr{\"o}mel--Rothschild theorem, it must also be true for at least one value of $m$ that is close to $\frac{1}{2} \cdot \ex(n, K_{r+1})$, since almost all $r$-colorable graphs have roughly this many edges. On the other hand, it is not very hard to see that this statement is not true for $m$ in some intermediate range. For example, if $n \ll m \ll n^{4/3}$, then almost all graphs in $\mathcal{G}_{n,m}$ are both $K_4$-free and not $r$-colorable for every fixed $r$.
In the case of triangle-free graphs, it turns out that as $m$ grows from $0$ to $\mathrm{ex}(n,K_3)$, there are two critical points at which almost all graphs in $\mathcal{F}_{n,m}(K_3)$ first stop being and then become bipartite, as proved by Osthus, Pr{\"o}mel, and Taraz~\cite{OsPrTa03}, who improved an earlier result of Pr{\"o}mel and Steger~\cite{PrSt96} (see also Steger~\cite{St05} for a slightly weaker result). More precisely, the following was shown in~\cite{OsPrTa03}. Let $\varepsilon$ be an arbitrary positive constant and let \begin{equation}
\label{eq:m2}
m_2 = m_2(n) = \frac{\sqrt{3}}{4} n^{3/2} \sqrt{\log n}. \end{equation} First, if $m \ll n$, then almost all graphs in $\mathcal{F}_{n,m}(K_3)$ are bipartite. Second, if $n/2 \leqslant m \leqslant (1-\varepsilon)m_2$, then almost all these graphs are not bipartite. Third, if $m \geqslant (1+\varepsilon)m_2$, then again almost all of them are bipartite. A corresponding result for $r = 4$ was obtained in the unpublished master's thesis of the fourth author~\cite{Wa09}.
\subsection{Main result}
In this paper, we generalize the above result to all $r$. To this end, for each $r \geqslant 2$, define \begin{equation}
\label{eq:thetar}
\theta_r = \frac{r-1}{2r} \cdot \left[ r \cdot \left(\frac{2r+2}{r+2}\right)^{\frac{1}{r-1}} \right]^{\frac{2}{r+2}} \end{equation} and \begin{equation}
\label{eq:mr}
m_r = m_r(n) = \theta_r n^{2-\frac{2}{r+2}} (\log n)^{\frac{1}{\rpt-1}}. \end{equation} Here and throughout the paper, $\log$ denotes the natural logarithm. Note that the definitions of $m_2$ given by~\eqref{eq:m2} and by~\eqref{eq:mr} coincide. Our main result is the following.
\begin{thm}
\label{thm:main}
For every $r \geqslant 3$, there exists a $d_r = d_r(n) = \Theta(n)$ such that the following holds for every $\varepsilon > 0$. If $F_{n,m}$ is the uniformly chosen random element of $\mathcal{F}_{n,m}(K_{r+1})$, then
\[
\lim_{n \to \infty} \mathbb{P}[\text{$F_{n,m}$ is $r$-partite}] =
\begin{cases}
1 & m \leqslant (1-\varepsilon)d_r, \\
0 & (1+\varepsilon)d_r \leqslant m \leqslant (1-\varepsilon)m_r, \\
1 & m \geqslant (1+\varepsilon)m_r.
\end{cases}
\] \end{thm}
The existence of the first threshold and the function $d_r$ in Theorem~\ref{thm:main} follows directly from the fact that for every $r \geqslant 3$, the property of being $r$-colorable has a sharp threshold in $\mathcal{G}_{n,m}$, as proved by Achlioptas and Friedgut~\cite{AcFr99}, see also~\cite{Fr99}. Indeed, if $m \ll n^{2 - 2/r}$, then almost all graphs in $\mathcal{G}_{n,m}$ are $K_{r+1}$-free and therefore almost every graph in $\mathcal{F}_{n,m}(K_{r+1})$ is $r$-partite if and only if almost every graph in $\mathcal{G}_{n,m}$ is $r$-partite. Moreover, one immediately sees that the threshold function for the property of being $r$-colorable, which we denote by $d_r$, satisfies $d_r(n) = \Theta(n)$; for more precise estimates, with which we will not be concerned in this paper, we refer the reader to~\cite{AcNa, CoVi13}. Thus, the main business of this paper will be establishing the existence of the second threshold at $m_r$. Finally, we would like to point out that our arguments for $m \gg n$, which really is the main case of interest for us, are also valid in the case $r = 2$. The only reason why in the statement of Theorem~\ref{thm:main}, we assume that $r \geqslant 3$ is that the property of being bipartite does not have a sharp threshold in $\mathcal{G}_{n,m}$ and therefore in the case $r = 2$, there is no double sharp threshold phenomenon.
\subsection{Approximate version}
A closely related problem is that of determining for which $m$ almost every graph in $\mathcal{F}_{n,m}(K_{r+1})$ is \emph{almost} $r$-partite, i.e., becomes $r$-partite after deleting from it some small fraction of the edges. Fifteen years ago, this problem was first considered by {\L}uczak~\cite{Lu00}, who proved that when $m \gg n^{3/2}$, then almost every graph in $\mathcal{F}_{n,m}(K_3)$ can be made bipartite by deleting from it some $o(m)$ edges. Furthermore, {\L}uczak showed that the so-called K{\L}R conjecture~\cite{KoLuRo97} implies an analogous statement for arbitrary $r \geqslant 2$, Theorem~\ref{thm:approx-struct} below. This conjecture was only very recently verified by the first three authors~\cite{BaMoSa12}, and by Saxton and Thomason~\cite{SaTh}; see also~\cite{CoGoSaSc}. The following result was established by the first three authors in~\cite{BaMoSa12} (the case $r = 2$ was proved much earlier in~\cite{Lu00}). It may also be derived from the results of~\cite{SaTh}.
\begin{thm}[{\cite{BaMoSa12, SaTh}}]
\label{thm:approx-struct}
For every $r \geqslant 2$ and every $\delta > 0$, there exists a constant~$C$ such that if $m \geqslant Cn^{2-2/(r+2)}$, then almost every graph in $\mathcal{F}_{n,m}(K_{r+1})$ can be made $r$-partite by removing from it at most $\delta m$ edges. \end{thm}
As mentioned above, Theorem~\ref{thm:approx-struct} was derived from the (then unproven) K{\L}R conjecture in~\cite{Lu00}. We remark here that in fact the proof of Theorem~\ref{thm:approx-struct} in \cite{BaMoSa12} yields that the proportion of graphs in $\mathcal{F}_{n,m}(K_{r+1})$ that cannot be made $r$-partite by removing from them some $\delta m$ edges is at most $(1-\varepsilon)^m$ for some positive constant $\varepsilon$ that depends solely on $r$ and $\delta$.
\subsection{Related work}
The main result of this paper, Theorem~\ref{thm:main}, may be also viewed in the context of the recent developments of `sparse random analogues' of classical results in extremal combinatorics such as the aforementioned theorem of Tur{\'a}n. Here, we just give a very brief summary of these developments. For more information, we refer the interested reader to the survey of R{\"o}dl and Schacht~\cite{RoSc13}. A long line of research, initiated in~\cite{BaSiSp90, FrRo86, KoLuRo96, RoRu95, RoRu97}, has recently culminated in breakthroughs of Conlon and Gowers~\cite{CoGo} and Schacht~\cite{Sc} (see also \cite{BaMoSa12, CoGoSaSc, FrRoSc10, Sa14, SaTh}), which developed a general theory for approaching such problems. In particular, these results imply that, asymptotically almost surely (a.a.s.), i.e., with probability tending to one as $n \to \infty$, if $p \gg n^{-2/(r+2)}$, then the number $\ex(G(n,p), K_{r+1})$ of edges in a largest $K_{r+1}$-free subgraph of the binomial random graph $G(n,p)$ satisfies \[ \ex(G(n,p), K_{r+1}) = \left(1 - \frac{1}{r}\right)\binom{n}{2}p + o(n^2p). \] This statement is usually referred to as the sparse random analogue of Tur{\'a}n's theorem. Moreover, a.a.s.\ any $K_{r+1}$-free subgraph of $G(n,p)$ with $\ex(G(n,p), K_{r+1}) - o(n^2p)$ edges may be made $r$-partite by removing from it some $o(n^2p)$ edges. This is usually referred to as the sparse random analogue of the Erd{\H{o}}s--Simonovits stability theorem~\cite{ErSi66, Si68}.
In fact, these random analogues of the theorems of Tur{\'a}n and Erd{\H{o}}s and Simonovits are much more closely related to Theorem~\ref{thm:approx-struct} rather than Theorem~\ref{thm:main}. A question somewhat closer in spirit to the latter would be deciding for which functions $p = p(n)$ is, a.a.s., the largest $K_{r+1}$-free subgraph of $G(n,p)$ exactly $r$-partite. Such a statement may be viewed as an exact sparse random analogue of Tur{\'a}n's theorem. This problem also has a fairly long history. It was first considered by Babai, Simonovits, and Spencer~\cite{BaSiSp90}, who proved that the condition $p > 1/2$ is sufficient in the case $r = 2$. Much later, Brightwell, Panagiotou, and Steger~\cite{BrPaSt12} showed that for every $r$, the exact random analogue of Tur{\'a}n's theorem holds when $p(n) \geqslant n^{-c}$ for some constant $c$ that depends on $r$. Very recently, DeMarco and Kahn~\cite{DMKa} showed that in the case $r = 2$ it is enough to assume that $p(n) \geqslant C \sqrt{\log n / n}$, where $C$ is some positive constant; this is best possible up to the value of $C$ since, as pointed out in~\cite{BrPaSt12}, the statement is false when $p(n) \leqslant \frac{1}{10}\sqrt{\log n / n}$. Note that the threshold $p(n) = \sqrt{\log n / n}$ for the above property (in the case $r = 2$) in $G(n,p)$ coincides with the threshold $m_2(n)$ from Theorem~\ref{thm:main}; nevertheless, the difficulties encountered in the two problems are quite different, and the proof in~\cite{DMKa} has (not surprisingly) little in common with that given here. Even more recently, DeMarco and Kahn~\cite{DMKa-Kr} showed that for every $r \geqslant 2$, the exact random analogue of Tur{\'a}n's theorem for $K_{r+1}$ holds under the assumption that $p(n) \geqslant C m_r(n) / n^2$, where $C$ is a sufficiently large constant and $m_r(n)$ is defined in~\eqref{eq:mr}, see~\cite[Conjecture~1.2]{DMKa}.
Finally, we remark that the family $\mathcal{F}_n(H)$ of $n$-vertex $H$-free graphs has been extensively studied for general graphs $H$. Extending~\cite{KoPrRo87}, Pr{\"o}mel and Steger~\cite{PrSt92} proved that if $H$ contains an edge whose removal reduces the chromatic number of $H$ (such graphs are called \emph{edge-color-critical}), then almost all graphs in $\mathcal{F}_n(H)$ are $(\chi(H)-1)$-partite. Generalizing \cite{KoPrRo87, PrSt92} even further, Hundack, Pr\"omel, and Steger~\cite{HuPrSt93} proved that if $H$ contains a \emph{color-critical vertex} (one whose removal reduces the chromatic number), then almost every graph in $\mathcal{F}_n(H)$ admits a partition of its vertex set into $\chi(H)-1$ parts, each of which induces a subgraph whose maximum degree is bounded by an explicit constant $d_H$ (in particular, $d_H = 0$ if $H$ is edge-color-critical). It would be interesting to generalize these results in the same way that Theorem~\ref{thm:main} generalizes the Kolaitis--Pr{\"o}mel--Rothschild theorem. We expect the following statement to be true.
\begin{conj}
\label{sec:conj-main}
For every strictly $2$-balanced, edge-color-critical graph $H$, there exists a constant $C$ such that the following holds. If
\[
m \geqslant Cn^{2 - 1/m_2(H)} (\log n)^{1/(e(H)-1)},
\]
then almost all graphs in $\mathcal{F}_{n,m}(H)$ are $(\chi(H)-1)$-partite. \end{conj}
Let us remark here that a statement that is even stronger than Conjecture~\ref{sec:conj-main} was proved by Osthus, Pr{\"o}mel, and Taraz~\cite{OsPrTa03} in the case when $H$ is a cycle of odd length. More precisely, the following was shown in~\cite{OsPrTa03}. Let $\ell$ be an integer, let $\varepsilon$ be an arbitrary positive constant, and let \[ t_\ell = t_\ell(n) = \left(\frac{\ell}{\ell-1} \cdot \left(\frac{n}{2}\right)^\ell \log n \right)^{\frac{1}{\ell-1}}. \] If $n/2 \leqslant m \leqslant (1-\varepsilon)t_\ell$, then almost all graphs in $\mathcal{F}_{n,m}(C_{2\ell+1})$ are not bipartite and if $m \geqslant (1+\varepsilon)t_\ell$, then almost all of them are bipartite.
Many (almost) sharp results describing the structure of almost all graphs in $\mathcal{F}_n(H)$ for a general graph $H$ were proved in a series of papers by Balogh, Bollob{\'a}s, and Simonovits~\cite{BaBoSi04, BaBoSi09, BaBoSi11}. Although it is not explicitly stated there, one can read out from their proofs that almost all graphs in $\mathcal{F}_{n,m}(K_{r+1})$ are $r$-partite when $m = \Omega(n^2)$.
Precise structural descriptions of the families $\mathcal{F}_n(H)$ when $H$ is a hypergraph are harder to obtain, and such results have so far been proved only for a few specific $3$-uniform hypergraphs~\cite{BaMu11, BaMu12, PeSc09}. Finally, we remark that the typical structure of graphs with a forbidden \emph{induced} subgraph has also been considered in the literature, see~\cite{AlBaBoMo11, BaBu11}.
\subsection{Outline of the paper}
The remainder of this paper is organised as follows. In Section~\ref{sec:outline-proof}, we give a fairly detailed outline of the strategy for proving Theorem~\ref{thm:main}. In Sections~\ref{sec:preliminaries} and~\ref{sec:r-col-graphs}, we collect some auxiliary results needed for the proof. In Section~\ref{sec:0-statement}, we establish the $0$-statement in Theorem~\ref{thm:main}. Starting with Section~\ref{sec:1-statement}, we turn to proving the second $1$-statement in Theorem~\ref{thm:main}. Our argument is rather involved and we subdivide it into two different cases, addressed in Sections~\ref{sec:sparse-case} and~\ref{sec:dense-case}, respectively.
\subsection{Notation}
For the sake of brevity, given an integer $n$, we will abbreviate $\{1, \ldots, n\}$ by $[n]$. For concreteness, we assume that $[n]$ is the common vertex set of all of the $n$-vertex graphs we consider in this paper. Let $\mathcal{G}_{n,m}(r)$ be the family of all graphs in $\mathcal{G}_{n,m}$, i.e., graphs on the vertex set $[n]$ with precisely $m$ edges, that are $r$-colorable. Let $\mathcal{P}_{n,r}$ be the family of all $r$-colorings of $[n]$, that is, all partitions of $[n]$ into $r$ parts. For the sake of brevity, we shall often identify a partition $\Pi \in \mathcal{P}_{n,r}$ with the complete $r$-partite graph on the vertex set $[n]$ whose color classes are the $r$ parts of $\Pi$. In particular, if $G$ is a graph on the vertex set $[n]$, then $G \subseteq \Pi$ means that $G$ is a subgraph of the complete $r$-partite graph $\Pi$ or, in other words, the partition $\Pi$ is a proper coloring of $G$. Exploiting this convention, we will also write $\Pi^c$ to denote the complement of the graph $\Pi$, that is, the union of $r$ complete graphs whose vertex sets are the color classes of $\Pi$.
Otherwise, we use fairly standard conventions. In particular, given a graph $G$, one of its vertices $v$, and a set $A \subseteq V(G)$, we denote the number of edges in $G$, the degree of $v$ in $G$, and the number of neighbors of $v$ in the set $A$ by $e(G)$, $\deg_G(v)$, and $\deg_G(v,A)$, respectively. For a graph $G$ and a set $A \subseteq V(G)$, we shall write $G - A$ to denote the subgraph of $G$ induced by the set $V(G) \setminus A$. Perhaps less obviously, $K_{r+1}^-$ denotes the graph obtained from the complete graph on $r+1$ vertices by removing a single edge, which we refer to as the \emph{missing edge}. For the sake of clarity of presentation, we will often assume that all large numbers are integers and use floor and ceiling symbols only when we feel that not writing them explicitly might be confusing. Our asymptotic notation is also standard; in particular, we write $f(n) \ll g(n)$ to denote the fact that $f(n)/g(n) \to 0$ as $n \to \infty$. Finally, we adapt the following notational convention. A subscript of the form X.Y refers to Claim/Lemma/Proposition/Theorem X.Y. For example, we write $\delta_{\ref{prop:approx-struct}}(\cdot)$ to denote the function implicitly defined in the statement of Proposition~\ref{prop:approx-struct}.
\section{Outline of the proof}
\label{sec:outline-proof}
In this section, we outline the proof of our main result, Theorem~\ref{thm:main}. The case $m = O(n)$ was already extensively discussed in the paragraph following the statement of Theorem~\ref{thm:main}, so in the remainder of the paper, we will assume that $m \gg n$. This leaves us with proving the following two statements, which we will do for every $r \geqslant 2$. First, we will show that if $m \leqslant (1-\varepsilon)m_r$, then almost all graphs in $\mathcal{F}_{n,m}(K_{r+1})$ are not $r$-partite. Second, we will show that if $m \geqslant (1+\varepsilon)m_r$, then almost all graphs are $r$-partite. For the sake of brevity, we will refer to these two assertions as the \emph{$0$-statement} and the \emph{$1$-statement}, respectively. We start by giving a heuristic argument that suggests that the function $m_r$ defined in~\eqref{eq:mr} is indeed the sharp threshold.
\subsection{Why is $m_r$ the sharp threshold?}
\label{sec:mr-threshold}
For the sake of clarity of presentation, we shall introduce another parameter that is related to the threshold function $m_r$. We let $p_r = p_r(n)$ be the number satisfying \begin{equation}
\label{eq:pr-def}
\left(\frac{n}{r}\right)^{r-1} p_r^{\rpt-1} = \left(2-\frac{2}{r+2}\right) \log n \end{equation} and note that \begin{equation}
\label{eq:pr-mr}
m_r = \left(1 - \frac{1}{r}\right)\frac{n^2}{2} \cdot p_r \approx \ex(n, K_{r+1}) \cdot p_r. \end{equation}
Although~\eqref{eq:pr-def} might seem like a strange way of defining $p_r$, given that we also have~\eqref{eq:pr-mr}, we shall see that considering~\eqref{eq:pr-def} and~\eqref{eq:pr-mr} in the above order is the natural way of arriving at the threshold $m_r$. Recall that we are aiming to count all non-$r$-partite graphs in $\mathcal{F}_{n,m}(K_{r+1})$. Consider a \emph{fixed} $r$-partition $\Pi = \{V_1, \ldots, V_r\}$ that is \emph{balanced}, that is, such that $|V_i| \approx \frac{n}{r}$ for all $i$. (As we shall show in Section~\ref{sec:r-col-graphs}, if $m \gg n$, then almost all $r$-colorable graphs with $m$ edges admit only balanced proper $r$-colorings.) Note that the assumption that $\Pi$ is balanced implies that for every $i$, \begin{equation}
\label{eq:Pi-balanced-approx}
e(\Pi) \approx \left(1-\frac{1}{r}\right)\frac{n^2}{2} \quad \text{and} \quad \prod_{j \neq i} |V_j| \approx \left(\frac{n}{r}\right)^{r-1}. \end{equation}
Let us try to count the graphs in $\mathcal{F}_{n,m}(K_{r+1})$ that are not $r$-partite, but for which $\Pi$ is an \emph{almost} proper $r$-coloring, i.e., graphs with exactly one monochromatic edge in the coloring $\Pi$. The presence of a monochromatic edge $\{v, w\} \subseteq V_i$ in some $G \in \mathcal{F}_{n,m}(K_{r+1})$ implies that $G$ has to \emph{avoid} $\prod_{j \neq i} |V_j|$ copies of $K_{r+1}^-$ in $\Pi$, where $\{v, w\}$ is the missing edge. More precisely, no such $G$ can contain all $\rpt-1$ edges in any such copy of $K_{r+1}^-$. The proportion of subgraphs of $\Pi$ with $m-1$ edges that avoid a single copy of $K_{r+1}^-$ is about $1-(\frac{m}{e(\Pi)})^{\rpt-1}$. Therefore, if not containing different copies of $K_{r+1}^-$ were independent events in the space of all subgraphs of $\Pi$, then, by \eqref{eq:pr-def}, \eqref{eq:pr-mr}, and \eqref{eq:Pi-balanced-approx}, if $m \approx m_r$, then the proportion $P$ of subgraphs avoiding all copies supported on the edge $\{v, w\}$ would satisfy, roughly, \[
P \approx \left(1 - \left(\frac{m}{e(\Pi)}\right)^{\rpt - 1}\right)^{\prod_{j \neq i} |V_j|} \approx \exp\left( - \left(\frac{n}{r}\right)^{r-1} p_r^{\rpt-1} \right) = n^{-2+\frac{2}{r+2}} \approx \frac{1}{m}. \] This hints that $m \approx m_r$ is a `critical point' as the number of graphs in $\mathcal{F}_{n,m}(K_{r+1})$ that have precisely one monochromatic edge in the coloring $\Pi$ is \[ P \binom{e(\Pi^c)}{1} \binom{e(\Pi)}{m-1}, \] which is of the same order as $\binom{e(\Pi)}{m}$, the number of graphs in $\mathcal{G}_{n,m}$ which are properly colored by $\Pi$, exactly when $P = \Theta(\frac{1}{m})$.
\subsection{Sketch of the proof of the $0$-statement}
\label{sec:sketch-proof-0}
Here, we assume that $(1+\varepsilon)d_r \leqslant m \leqslant (1-\varepsilon)m_r$. As we have already mentioned in the introduction, if $m \ll n^{2-2/r}$, then almost all graphs in $\mathcal{G}_{n,m}$ are $K_{r+1}$-free and therefore the fact that almost every graph in $\mathcal{F}_{n,m}(K_{r+1})$ is not $r$-partite follows from the fact that $d_r$ is the sharp threshold for the property of being $r$-colorable in $\mathcal{G}_{n,m}$. The existence of such a sharp threshold (for $r \geqslant 3$) was proved by Achlioptas and Friedgut~\cite{AcFr99}, see also~\cite{Fr99}. Moreover, a fairly straightforward counting argument employing the Hypergeometric FKG Inequality (Lemma~\ref{lemma:HFKG}) shows that if $m \ll n^{2 - 2/(r+2)}$, then the number of graphs in $\mathcal{F}_{n,m}(K_{r+1})$ is far greater than $|\mathcal{G}_{n,m}(r)|$, the number of $r$-colorable graphs in $\mathcal{G}_{n,m}$, see Section~\ref{sec:very-sparse}. Therefore, we focus our attention on the case when $m = \Omega(n^{2-2/(r+2)})$.
As already suggested in Section~\ref{sec:mr-threshold}, the main idea is to count graphs in $\mathcal{F}_{n,m}(K_{r+1})$ that are $r$-colorable except for one edge. This suffices for our purposes, as it turns out that the number of such graphs is already asymptotically greater than the number of graphs in $\mathcal{F}_{n,m}(K_{r+1})$ that are $r$-colorable. To this end, we first show, in Lemma~\ref{lemma:GPe-free}, that for a fixed $r$-coloring $\Pi$ of $[n]$ that is balanced, that is, each of its color classes has size $n/r + o(n)$, the number of graphs $G \in \mathcal{F}_{n,m}(K_{r+1})$ such that $e(G \cap \Pi^c) = 1$ is asymptotically greater than the number of graphs in $\mathcal{F}_{n,m}(K_{r+1})$ which are properly colored by $\Pi$. Recall from the discussion above that a graph $G$ with $e(G \cap \Pi^c) = 1$ is $K_{r+1}$-free precisely when none of the about $(n/r)^{r-1}$ copies of $K_{r+1}^-$ contained in $\Pi$, where the unique edge of $G \cap \Pi^c$ is the missing edge, is completely contained in $G \cap \Pi$. Using this simple observation, we obtain a lower bound on the number of such $G$ using the Hypergeometric FKG Inequality (Lemma~\ref{lemma:HFKG}). Our bound implies that for a fixed balanced $\Pi$, the number of graphs $G \in \mathcal{F}_{n,m}(K_{r+1})$ with exactly one edge in $\Pi^c$ is much larger than $\binom{e(\Pi)}{m}$, the number of graphs which are properly colored by $\Pi$. Second, in Lemma~\ref{lemma:GPe-unique}, we show that almost every $G \in \mathcal{F}_{n,m}(K_{r+1})$ that admits an $r$-coloring $\Pi$ satisfying $e(G \cap \Pi^c) = 1$ admits a unique such $\Pi$. Consequently, the number of such $G$ is almost as large as the sum over all balanced $r$-colorings $\Pi$ of the lower bounds obtained earlier, and this is much larger than $|\mathcal{G}_{n,m}(r)|$, the number of $r$-colorable graphs in $\mathcal{G}_{n,m}$.
\subsection{Sketch of the proof of the $1$-statement}
\label{sec:sketch-proof-1}
Here, we assume that $m \geqslant (1+\varepsilon)m_r$ for some positive constant $\varepsilon$. Since in particular $m \gg n^{-\frac{2}{r+2}}$, then Theorem~\ref{thm:approx-struct} implies that almost every graph in $\mathcal{F}_{n,m}(K_{r+1})$ admits an $r$-coloring of $[n]$ such that: \begin{enumerate}[(i)] \item
there are only $o(m)$ monochromatic edges, \item
each color class has size $n/r + o(n)$, \item
if a vertex $v$ is colored $i$, then $v$ has at least as many neighbors in every color $j$ as in color $i$. \end{enumerate} Therefore, it suffices to consider only $K_{r+1}$-free graphs that admit such a coloring.
As was proved in~\cite{PrSt95}, almost every graph in $\mathcal{G}_{n,m}(r)$ admits a unique $r$-coloring. Moreover, the number of pairs $(G,\Pi)$, where $G \in \mathcal{G}_{n,m}(r)$ and $\Pi$ is a proper $r$-coloring of $G$ is asymptotic to $|\mathcal{G}_{n,m}(r)|$, see Theorem~\ref{thm:Gr}. Therefore, it suffices to prove that for a \emph{fixed} $r$-coloring $\Pi$ satisfying (ii) above, the number of $G \in \mathcal{F}_{n,m}(K_{r+1})$ that satisfy (i) and (iii) for this fixed coloring $\Pi$ is asymptotically equal to $\binom{e(\Pi)}{m}$, the number of graphs in $\mathcal{G}_{n,m}$ that are properly colored by $\Pi$, see Theorem~\ref{thm:1-statement}.
From now on, we fix some $\Pi$ satisfying (ii) and count graphs $G \in \mathcal{F}_{n,m}(K_{r+1})$ that satisfy (i) and (iii) but are \emph{not} properly colored by $\Pi$. We denote the family of all such graphs by $\mathcal{F}^*$. The methods of enumerating these graphs will vary with $m$ and the distribution of the monochromatic edges of $G$, that is, the edges of $G \cap \Pi^c$. For technical reasons, we require separate arguments to handle the cases $m \leqslant \ex(n, K_{r+1}) - \xi n^2$ and $m > \ex(n, K_{r+1}) - \xi n^2$, where $\xi$ is some fixed positive constant, which we term the \emph{sparse case} and the \emph{dense case}, respectively. The argument used for the (much easier) dense case is somewhat ad hoc and we will not dwell on it here. Instead, we refer the interested reader directly to (the self-contained) Section~\ref{sec:dense-case}. The main business of this paper is handling the sparse case, and hence from now on we assume that $m \leqslant \ex(n, K_{r+1}) - \xi n^2$.
Recall that $\Pi$ is a fixed $r$-coloring of $[n]$ satisfying (ii). We use two different methods of enumerating graphs $G \in \mathcal{F}^*$, that is, graphs in $\mathcal{F}_{n,m}(K_{r+1})$ that satisfy (i) and (iii) but are not properly colored by $\Pi$, depending on whether or not most edges of $G \cap \Pi^c$ are incident to vertices whose degree in $G \cap \Pi^c$ is somewhat high. More precisely, we partition the family $\mathcal{T}$ of all possible graphs $G \cap \Pi^c$, where $G$ ranges over $\mathcal{F}^*$, into two classes, denoted $\TT^L$ (here $L$ stands for \emph{low degree}) and $\TT^H$ (here $H$ stands for \emph{high degree}), according to the proportion of edges that are incident to vertices whose degree exceeds $\beta m / n$, where $\beta$ is a small positive constant, see Section~\ref{sec:setup-sparse}. We then separately enumerate graphs $G$ such that $G \cap \Pi^c \in \TT^L$ and those satisfying $G \cap \Pi^c \in \TT^H$. We term these two parts of the argument the \emph{low degree case} (Section~\ref{sec:sparse-low-degree-case}) and the \emph{high degree case} (Sections~\ref{sec:sparse-high-degree-case}--\ref{sec:irregular-case}), respectively.
In the (easier) low degree case, for each $T \in \TT^L$, we give an upper bound on the number of graphs $G \in \mathcal{F}^*$ such that $G \cap \Pi^c = T$. Our upper bound is a function of the number of edges in a canonically chosen subgraph $U(T)$ of $T$, which we define in Section~\ref{sec:setup-sparse}. The bound is proved in Lemma~\ref{lemma:Janson}, which is the core of the argument in the low degree case. In Section~\ref{sec:counting-graphs}, we separately enumerate all $T \in \TT^L$ with a certain value of $e(U(T))$. This is fairly straightforward. The proof of Lemma~\ref{lemma:Janson}, which bounds the number of $G$ with $G \cap \Pi^c = T \in \mathcal{T}$ in terms of $e(U(T))$, is a somewhat involved application of the Hypergeometric Janson Inequality (Lemma~\ref{lemma:HJI}). Let us briefly describe the main idea. The presence of an edge $\{v ,w\}$ of $T$ in $G \cap \Pi^c$ and the fact that $G$ is $K_{r+1}$-free imply that $G \cap \Pi$ avoids each of the (roughly) $(n/r)^{r-1}$ copies of $K_{r+1}^-$ in $\Pi$, where $v$ and $w$ are the endpoints of the missing edge of $K_{r+1}^-$. That is, at least one edge from each such copy of $K_{r+1}^-$ does not belong to $G \cap \Pi$. Whereas it is quite easy to estimate the number of subgraphs of $\Pi$ with $m - e(T)$ edges that avoid a \emph{single} copy of $K_{r+1}^-$ (there are about $\left(1-(\frac{m}{e(\Pi)})^{\rpt-1}\right) \cdot \binom{e(\Pi)}{m-e(T)}$ of them) bounding the number of subgraphs that avoid \emph{all} such copies of $K_{r+1}^-$ for \emph{all} edges $\{v, w\}$ of $T$ simultaneously requires very careful computation. The main difficulty lies in controlling the correlation between the families of subgraphs of $\Pi$ that avoid two different copies of $K_{r+1}^-$.
In the high degree case, where we enumerate the graphs $G \in \mathcal{F}^*$ such that $G \cap \Pi^c \in \TT^H$, we focus our attention on vertices of high degree, that is, vertices whose degree in $G \cap \Pi^c$ exceeds $\beta m / n$. We count such graphs by describing and analyzing a procedure that constructs all of them in two stages. This procedure first selects one color class, $V_i$, chooses which of its vertices will have high degree and then picks their neighbors, in all color classes. Next, it chooses all the remaining edges of $G$. In the analysis, we bound the number of choices that this procedure can make, which translates into a bound on the number of graphs in $\mathcal{F}^*$ that fall into the high degree case. Let us briefly describe how we obtain this bound. Suppose that we want to construct a graph $G \in \mathcal{F}^*$, where some $v \in V_i$ has at least $\beta m / n$ neighbors in $V_i$. By (iii) above, we must guarantee that $\deg(v,V_j) \geqslant \beta m / n$ for all $j$. Hence, no matter how we choose the neighborhoods of $v$ in $V_1, \ldots, V_r$, there will be a collection of at least $(\beta m / n)^r$ \emph{forbidden copies} of $K_r$ in $\Pi$, none of which can be fully contained in $G$. For a typical choice of neighborhoods of some canonically chosen set of high degree vertices in $V_i$ (in the first stage of the procedure), these forbidden copies of $K_r$ are uniformly distributed and hence, using the Hypergeometric Janson Inequality (Lemma~\ref{lemma:HJI}), we can obtain a strong upper bound on the number of choices of a subgraph of $\Pi$ that avoids them (in the second stage of our procedure). We will refer to this possibility as the \emph{regular case}. On the other hand, using Lemma~\ref{lemma:d-sets}, we show that there are only very few choices of the neighborhoods of these high degree vertices in $V_i$ (in the first stage) for which the distribution of the forbidden copies of $K_r$ is not sufficiently uniform to yield a strong bound on the number of choices of the remaining edges (in the second stage), as in the regular case. We will refer to this possibility as the \emph{irregular case}. The proportion of graphs that fall into either the regular or the irregular case is exponentially small in $m/n$.
\section{Preliminaries}
\label{sec:preliminaries}
\subsection{Tools}
In this section, we collect several auxiliary results that will be later used in the proof of Theorem~\ref{thm:main}. We begin with one of our main tools, a version of the Janson Inequality for the hypergeometric distribution.
\begin{lemma}[Hypergeometric Janson Inequality]
\label{lemma:HJI}
Suppose that $\{B_i\}_{i \in I}$ is a family of subsets of an $n$-element set $\Omega$, let $m \in \{0, \ldots, n\}$, and let $p = m/n$. Let
\[
\mu = \sum_{i \in I} p^{|B_i|} \qquad \text{and} \qquad \Delta = \sum_{i \sim j} p^{|B_i \cup B_j|},
\]
where the second sum is over all ordered pairs $(i,j) \in I^2$ such that $i \neq j$ and $B_i \cap B_j \neq \emptyset$. Let $R$ be the uniformly chosen random $m$-subset of $\Omega$ and let $\mathcal{B}$ denote the event that $B_i \nsubseteq R$ for all $i \in I$. Then for every $q \in [0,1]$,
\[
\mathbb{P}(\mathcal{B}) \leqslant 2 \cdot \exp\left(-q\mu + q^2 \Delta / 2\right).
\] \end{lemma}
Our main tool in the proof of the $0$-statement will be the following version of the FKG Inequality for the hypergeometric distribution, which gives a lower bound on the probability $\mathbb{P}(\mathcal{B})$ from the statement of Lemma~\ref{lemma:HJI}. We postpone the easy deductions of Lemmas~\ref{lemma:HJI} and~\ref{lemma:HFKG} from their standard `binomial' versions to Appendix~\ref{sec:omitted-proofs}.
\begin{lemma}[Hypergeometric FKG Inequality]
\label{lemma:HFKG}
Suppose that $\{B_i\}_{i \in I}$ is a family of subsets of an $n$-element set $\Omega$. Let $m \in \{0, \ldots, \lfloor n/2 \rfloor\}$, let $R$ be the uniformly chosen random $m$-subset of $\Omega$, and let $\mathcal{B}$ denote the event that $B_i \nsubseteq R$ for all $i \in I$. Then for every $\eta \in (0,1)$,
\[
\mathbb{P}(\mathcal{B}) \geqslant \, \prod_{i \in I} \left(1 - \left(\frac{(1+\eta)m}{n}\right)^{|B_i|}\right) - \exp\big( - \eta^2m / 4 \big).
\] \end{lemma}
Finally, in the proof of Theorem~\ref{thm:main} in the case $m = \ex(n, K_{r+1}) - o(n^2)$, we will need the following folklore result from extremal graph theory. As we were unable to find a good reference, we give a proof of this result in Appendix~\ref{sec:omitted-proofs}. Below, $K(n_1, \ldots, n_r)$ denotes the complete $r$-partite graph whose $r$ color classes have sizes $n_1, \ldots, n_r$, respectively.
\begin{lemma}
\label{lemma:k-Turan}
For every integer $r \geqslant 2$ and all integers $n_1, \ldots, n_r$ satisfying $n_1 \leqslant \ldots \leqslant n_r$,
\[
\mathrm{ex}\big(K(n_1, \ldots, n_r), K_r\big) = e\big(K(n_1, \ldots, n_r)\big) - n_1n_2.
\] \end{lemma}
We remark here that our proof of Lemma~\ref{lemma:k-Turan} shows that the unique extremal graph is obtained by removing all edges joining some two smallest classes.
\subsection{Estimates for binomial coefficients}
\label{sec:estim-binom-coeff}
Throughout the paper, we will often use various estimates for expressions involving binomial coefficients. In this section, we collect some of them for future reference. We start with an easy corollary of Vandermonde's identity.
\begin{lemma}
\label{lemma:Vandermonde}
For every $a$, $b$, $c$, and $d$ with $d \leqslant c$,
\[
\binom{a}{d} \binom{b}{c-d} \leqslant \binom{a+b}{c}.
\] \end{lemma}
Our next lemma estimates the ratio between $\binom{a}{c}$ and $\binom{b}{c}$ for $a, b, c$ satisfying $a > b > c > 0$.
\begin{lemma}
\label{lemma:acbc}
If $a > b > c > 0$, then
\[
\left(\frac{a}{b}\right)^c \cdot \binom{b}{c} \leqslant \binom{a}{c} \leqslant \left(\frac{a-c}{b-c}\right)^c \cdot \binom{b}{c}.
\] \end{lemma}
\subsection{Main tool}
A crucial ingredient in the proof of Theorem~\ref{thm:main} is an estimate of the upper tail of the distribution of the number of edges in a random subhypergraph of a sparse $k$-uniform $k$-partite hypergraph, Lemma~\ref{lemma:d-sets} below. It formalizes the following statement: If some $\mathcal{H} \subseteq V_1 \times \ldots \times V_k$ contains only a tiny proportion of all the $k$-tuples in $V_1 \times \ldots \times V_k$, then the probability that, for a random choice of $d$-elements sets $W_1 \subseteq V_1, \ldots, W_k \subseteq V_k$, a much larger proportion of $W_1 \times \ldots \times W_k$ falls in $\mathcal{H}$ decays exponentially in $d$.
\begin{lemma}
\label{lemma:d-sets}
For every integer $k$ and all positive $\alpha$ and $\lambda$, there exists a positive $\tau$ such that the following holds. Let $V_1, \ldots, V_k$ be finite sets and let $d$ be an integer satisfying $2 \leqslant d \leqslant \min\{|V_1|, \ldots, |V_k|\}$. Suppose that $\mathcal{H} \subseteq V_1 \times \ldots \times V_k$ satisfies
\[
|\mathcal{H}| \leqslant \tau \prod_{i = 1}^k |V_i|
\]
and that $W_1, \ldots, W_k$ are uniformly chosen random $d$-subsets of $V_1, \ldots, V_k$, respectively. Then,
\[
\mathbb{P}\left(|\mathcal{H} \cap (W_1 \times \ldots \times W_k)| > \lambda d^k\right) \leqslant \alpha^d.
\] \end{lemma}
We prove Lemma~\ref{lemma:d-sets} in Appendix~\ref{sec:omitted-proofs}. We just remark here that our proof yields that one can take $\tau = (\alpha/2)^{k^2/\lambda} \cdot \lambda^k \cdot d^{-k^3/(d\lambda)}$. (Although the above expression depends on $d$, this dependence is not crucial as $d^{-1/d} \geqslant e^{-1/e}$ for all $d$. We made the dependence on $d$ above explicit only because $d^{-k^3/(d\lambda)} \geqslant e^{-1}$ when $d/\log d \geqslant k^3/\lambda$.)
\section{On $r$-colorable graphs}
\label{sec:r-col-graphs}
Recall that $\mathcal{G}_{n,m}(r)$ is the family of all $r$-partite ($r$-colorable) graphs on the vertex set $[n]$ that have exactly $m$ edges and that $\mathcal{P}_{n,r}$ is the collection of all $r$-colorings of $[n]$ (partitions of $[n]$ into at most $r$ parts). Given a $\Pi \in \mathcal{P}_{n,r}$, we define $\mathcal{G}_m(\Pi)$ to be the family of all $G \in \mathcal{G}_{n,m}(r)$ that are properly colored by $\Pi$, that is, \[ \mathcal{G}_m(\Pi) = \{G \in \mathcal{G}_{n,m}(r) \colon G \subseteq \Pi\}. \]
Note that $|\mathcal{G}_m(\Pi)| = \binom{e(\Pi)}{m}$. Trivially, we have \[ \mathcal{G}_{n,m}(r) = \bigcup_{\Pi \in \mathcal{P}_{n,r}} \mathcal{G}_m(\Pi). \] We will be particularly interested in \emph{balanced} $r$-colorings, that is, ones where all the color classes have approximately $n/r$ elements. More precisely, given a positive $\gamma$, we let $\mathcal{P}_{n,r}(\gamma)$ be the family of all partitions of $[n]$ into $r$ parts $V_1, \ldots, V_r$ such that \begin{equation}
\label{eq:Part-gamma}
\left(\frac{1}{r} - \gamma\right)n \leqslant |V_i| \leqslant \left(\frac{1}{r} + \gamma\right)n \quad \text{for all $i \in [r]$}. \end{equation} That is, \[ \mathcal{P}_{n,r}(\gamma) = \big\{\{V_1, \ldots, V_r\} \in \mathcal{P}_{n,r} \colon \text{\eqref{eq:Part-gamma} holds}\big\}. \] We can easily neglect colorings that are not balanced in the above sense. The following proposition, originally proved in~\cite{PrSt95}, shows that if $m \gg n$, then almost every graph in $\mathcal{G}_{n,m}(r)$ admits only balanced $r$-colorings. The easy proof of Proposition~\ref{prop:unbalanced-graphs} is given in Appendix~\ref{sec:omitted-proofs}.
\begin{prop}
\label{prop:unbalanced-graphs}
For every positive $\gamma$, there exists a constant $c$ such that if $m \geqslant cn$, then
\[
\sum_{\Pi \not\in \mathcal{P}_{n,r}(\gamma)} |\mathcal{G}_m(\Pi)| \ll \binom{\ex(n, K_{r+1})}{m} \leqslant |\mathcal{G}_{n,m}(r)|.
\] \end{prop}
Even though the collections $\mathcal{G}_m(\Pi)$ are generally not pairwise disjoint, there is not too much overlap between them. More precisely, if $\Pi$ is not very unbalanced, then, for an overwhelming proportion of all $G \in \mathcal{G}_m(\Pi)$, the $r$-coloring $\Pi$ is their unique proper $r$-coloring. The following rigorous version of this statement follows from the (much stronger) results proved in~\cite{PrSt95}.
\begin{thm}
\label{thm:Gr}
For every integer $r \geqslant 2$, every $0 < \gamma \leqslant 1/2r$, and every $m \gg n \log n$,
\[
|\mathcal{G}_{n,m}(r)| = (1+o(1)) \sum_{\Pi \in \mathcal{P}_{n,r}(\gamma)} |\mathcal{G}_m(\Pi)| = (1+o(1)) \sum_{\Pi \in \mathcal{P}_{n,r}(\gamma)} \binom{e(\Pi)}{m}.
\] \end{thm}
We remark here that Theorem~\ref{thm:Gr} is an easy consequence of Proposition~\ref{prop:unbalanced-graphs}, above, and Proposition~\ref{prop:UP} proved in the next section.
\section{The $0$-statement}
\label{sec:0-statement}
Our aim here is to show that if $(1+\varepsilon)d_r \leqslant m \leqslant (1-\varepsilon)m_r$, then almost all graphs in $\mathcal{F}_{n,m}(K_{r+1})$ are not $r$-colorable. As already discussed before, given the (difficult and interesting) result establishing when almost all graphs in $\mathcal{G}_{n,m}$ stop being $r$-colorable~\cite{AcFr99}, we may assume that $m \gg n$. We first give an easy argument that works in the case $n \ll m \ll n^{2-2/(r+2)}$ and then present a more complicated argument that works for all $m$ satisfying $n \log n \ll m \leqslant (1-\varepsilon)m_r$.
\subsection{Counting very sparse $K_{r+1}$-free graphs}
\label{sec:very-sparse}
In this section, generalizing a counting argument of Pr{\"o}mel and Steger~\cite{PrSt96}, we show that if $m$ satisfies $n \ll m \ll n^{2-2/(r+2)}$, then in fact, almost all graphs in $\mathcal{F}_{n,m}(K_{r+1})$ have arbitrarily high chromatic number. For our purposes, we only need the statement of Lemma~\ref{lemma:Gnm-count} in the case $k = r$.
\begin{lemma}
\label{lemma:Gnm-count}
For every $k \geqslant 2$, there exist $c > 0$ and $d > 0$ such that
\[
|\mathcal{F}_{n,m}(K_{r+1})| \gg |\mathcal{G}_{n,m}(k)|
\]
for all $m$ satisfying $c n \leqslant m \leqslant d n^{2-2/(r+2)}$. \end{lemma} \begin{proof}
Let $G_{n,m}$ be the uniformly selected random element of $\mathcal{G}_{n,m}$. Clearly,
\begin{equation}
\label{eq:Free-Gnm}
|\mathcal{F}_{n,m}(K_{r+1})| = \mathbb{P}[\text{$G_{n,m}$ is $K_{r+1}$-free}] \cdot \binom{\binom{n}{2}}{m}.
\end{equation}
By Lemma~\ref{lemma:acbc}, we have that for sufficiently large $n$,
\begin{equation}
\label{eq:nt-m}
\binom{\binom{n}{2}}{m} \geqslant \left(\frac{\binom{n}{2}}{\ex(n, K_{k+1})}\right)^{m} \cdot \binom{\ex(n, K_{k+1})}{m} \geqslant e^{\frac{m}{k+1}} \cdot \binom{\ex(n, K_{k+1})}{m},
\end{equation}
where in the last inequality we used the fact that $\ex(n, K_{k+1}) = (1 - \frac{1}{k})\binom{n}{2} + O(n)$. Note that if $m = n^{2-2/(r+2)}$, then
\[
n^{r+1} \cdot \left(\frac{m}{n^2}\right)^{\rpt} = m.
\]
Since there are fewer than $n^{r+1}$ copies of $K_{r+1}$ in the complete graph on $n$ vertices, the Hypergeometric FKG Inequality (Lemma~\ref{lemma:HFKG}, where we set $\eta = 1/2$) implies that if $m \leqslant dn^{2 - 2/(r+2)}$ for some constant $d$, then
\begin{equation}
\label{eq:Gnm-Krp-free}
\mathbb{P}[\text{$G_{n,m}$ is $K_{r+1}$-free}] + \exp\left(-\frac{m}{16}\right) \geqslant \left(1 - \left(\frac{4m}{n^2}\right)^{\rpt}\right)^{n^{r+1}} \geqslant \exp\left(-5^{\rpt} d^{\rpt-1}m\right),
\end{equation}
provided that $n$ is sufficiently large. Therefore, if $d$ is sufficiently small (i.e., when the right-hand side of~\eqref{eq:Gnm-Krp-free} is larger than $e^{-m/16} + e^{-m/((k+1)(k+2))}$), then by~\eqref{eq:Free-Gnm}, \eqref{eq:nt-m}, and~\eqref{eq:Gnm-Krp-free},
\[
|\mathcal{F}_{n,m}(K_{r+1})| \geqslant e^{\frac{m}{k+2}} \cdot \binom{\ex(n, K_{k+1})}{m} \gg k^n \cdot \binom{\ex(n, K_{k+1})}{m} \geqslant |\mathcal{G}_{n,m}(k)|,
\]
provided that $m \geqslant cn$ for a sufficiently large constant $c$. \end{proof}
\subsection{Counting $K_{r+1}$-free graphs with one monochromatic edge}
\label{sec:one-bad-edge}
In this section, generalizing the approach of Osthus, Pr{\"o}mel, and Taraz~\cite{OsPrTa03}, we count graphs in $\mathcal{F}_{n,m}(K_{r+1})$ that are $r$-colorable except for one edge. We show that if $m$ satisfies $n \log n \ll m \leqslant (1-\varepsilon)m_r$, then the number of such graphs is already much larger than $\mathcal{G}_{n,m}(r)$. In particular, we shall deduce the following.
\begin{prop}
\label{prop:one-bad-edge}
For every $r \geqslant 2$, every $\varepsilon > 0$, and every $m$ satisfying $n \log n \ll m \leqslant (1-\varepsilon)m_r$,
\[
|\mathcal{F}_{n,m}(K_{r+1})| \gg |\mathcal{G}_{n,m}(r)|.
\] \end{prop}
Recall the definitions of $\mathcal{P}_{n,r}(\gamma)$ and $\mathcal{G}_m(\Pi)$ given in Section~\ref{sec:r-col-graphs}. Given a $\Pi \in \mathcal{P}_{n,r}$, and an edge $e \in \Pi^c$, we define \[ \mathcal{G}_m(\Pi,e) = \{G + e \colon G \in \mathcal{G}_{m-1}(\Pi) \}. \]
Note that $|\mathcal{G}_m(\Pi,e)| = \binom{e(\Pi)}{m-1}$ and that $\mathcal{G}_m(\Pi,e) \cap \mathcal{G}_m(\Pi,f) = \emptyset$ if $e \neq f$. We first show that if $\Pi$ is balanced, then for every edge $e \in \Pi^c$, the family $\mathcal{G}_m(\Pi,e)$ contains many $K_{r+1}$-free graphs.
We first set some parameters. Recall that a constant $\varepsilon \in (0,1)$ is given. Let $\gamma$ and $\eta$ be small positive constants such that \begin{equation}
\label{eq:eta-gamma-eps}
\frac{(1+\eta)(1-\varepsilon)}{(1-r\gamma)^2} \leqslant 1 - \frac{\varepsilon}{2} \qquad \text{and} \qquad (1+\gamma r)^{r-1} \leqslant 1 + \frac{\varepsilon}{4}. \end{equation} Note also that every $\Pi \in \mathcal{P}_{n,r}(\gamma)$ satisfies \begin{equation}
\label{eq:ePi-lower-0}
e(\Pi) \geqslant \binom{r}{2} \cdot \left[\left(\frac{1}{r}-\gamma\right)n\right]^2 = (1-r\gamma)^2\left(1 - \frac{1}{r}\right)\frac{n^2}{2}. \end{equation}
We are now ready to state and prove our main lemma.
\begin{lemma}
\label{lemma:GPe-free}
For all $\Pi \in \mathcal{P}_{n,r}(\gamma)$ and $m$ with $n \leqslant m \leqslant (1-\varepsilon) m_r$, we have
\[
|\mathcal{G}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1})| \gg \frac{1}{m} \cdot \binom{e(\Pi)}{m-1}.
\] \end{lemma} \begin{proof}
Suppose that $\Pi = \{V_1, \ldots, V_r\}$ and that $e$ lies in $V_i$. Let $\mathcal{K}$ be the collection of (the edge sets of) all copies of $K_{r+1}^-$ induced in $\Pi$ by the two endpoints of the edge $e$ and one vertex from each $V_j$ with $j \neq i$. Let $G_{n,m-1}$ be the random element of $\mathcal{G}_{m-1}(\Pi)$ and note that
\begin{equation}
\label{eq:GPe-Free}
|\mathcal{G}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1})| = \mathbb{P}(\text{$G_{n,m-1} \nsupseteq K$ for every $K \in \mathcal{K}$}) \cdot \binom{e(\Pi)}{m-1}.
\end{equation}
Denote the above probability by $P$. We need to show that $P \gg 1/m$. By the Hypergeometric FKG Inequality (Lemma~\ref{lemma:HFKG}),
\begin{equation}
\label{eq:P-FKG}
P \geqslant \left(1 - \left(\frac{(1+\eta)(m-1)}{e(\Pi)}\right)^{\rpt-1}\right)^{|\mathcal{K}|} - \exp\left(-\frac{\eta^2(m-1)}{4}\right).
\end{equation}
Observe that by~\eqref{eq:eta-gamma-eps},
\[
|\mathcal{K}| \leqslant \left(\frac{1}{r} + \gamma \right)^{r-1} n^{r-1} = (1 + \gamma r)^{r-1} \cdot \left(\frac{n}{r}\right)^{r-1} \leqslant \left(1 + \frac{\varepsilon}{4}\right) \cdot \left(\frac{n}{r}\right)^{r-1}.
\]
Hence, if $n \leqslant m \leqslant n^{2 - 2/(r+2)}$, then $P \geqslant c$ for some positive constant $c$. Therefore, we may assume that $n^{2-2/(r+2)} \leqslant m \leqslant (1-\varepsilon)m_r$. Recall that $\Pi \in \mathcal{P}_{n,r}(\gamma)$ and hence by~\eqref{eq:eta-gamma-eps} and \eqref{eq:ePi-lower-0}, recalling the definition of $p_r$ from~\eqref{eq:pr-mr},
\[
\frac{(1+\eta)(m-1)}{e(\Pi)} \leqslant \frac{(1+\eta)(1-\varepsilon)m_r}{(1-r\gamma)^2\left(1 - \frac{1}{r}\right)\frac{n^2}{2}} \leqslant \left(1-\frac{\varepsilon}{2}\right)p_r.
\]
Therefore, recalling~\eqref{eq:pr-def}, using the fact that $p_r \ll 1$ and $1 - x \geqslant \exp(-x-x^2)$ if $x \leqslant \frac{1}{2}$, we continue~\eqref{eq:P-FKG} as follows:
\[
P + \exp\left(-\frac{\eta^2(m-1)}{4} \right) \geqslant \exp\left( - \left(1-\frac{\varepsilon}{2}\right) p_r^{\rpt-1} \left(\frac{n}{r}\right)^{r-1} \right) = n^{-\left(1 - \frac{\varepsilon}{2}\right) \cdot \left(2 - \frac{2}{r+2}\right)} \gg \frac{1}{m}.\qedhere
\] \end{proof}
Since \[ \binom{e(\Pi)}{m-1} \geqslant \frac{m}{e(\Pi)} \cdot \binom{e(\Pi)}{m}, \] it follows from Lemma~\ref{lemma:GPe-free} that if $n$ is sufficiently large, then \begin{equation}
\label{eq:sum-GPe}
\sum_{\Pi \in \mathcal{P}_{n,r}(\gamma)} \sum_{e \in \Pi^c} |\mathcal{G}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1})| \gg \sum_{\Pi \in \mathcal{P}_{n,r}(\gamma)} \frac{e(\Pi^c)}{e(\Pi)} \cdot \binom{e(\Pi)}{m} \geqslant \frac{1}{2r} \cdot |\mathcal{G}_{n,m}(r)|, \end{equation} where the last inequality follows from Theorem~\ref{thm:Gr}
and the fact that $e(\Pi^c) / e(\Pi) \geqslant 1/(2r-2)$ for every $\Pi \in \mathcal{P}_{n,r}$, provided that $n \geqslant 2r$. The left hand side of~\eqref{eq:sum-GPe} counts the pairs $(G, \Pi)$ such that $G \in \mathcal{G}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1})$ for some $e \in \Pi^c$. Therefore, in order to conclude that the number of graphs in $\mathcal{F}_{n,m}(K_{r+1})$ with exactly one monochromatic edge is much larger than $|\mathcal{G}_{n,m}(r)|$, it is enough to show that for every $\Pi \in \mathcal{P}_{n,r}(\gamma)$ and every $e \in \Pi^c$, an overwhelming proportion of all $G \in \mathcal{G}_m(\Pi,e)$, and hence also an overwhelming proportion of all $G \in \mathcal{G}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1})$, do not belong to any other $\mathcal{G}_m(\Pi',f)$ with $\Pi' \neq \Pi$ and $f \in (\Pi')^c$.
To this end, given a $\Pi \in \mathcal{P}_{n,r}$ and $e \in \Pi^c$, let $\mathcal{U}_m(\Pi,e)$ be the family of all $G \in \mathcal{G}_m(\Pi,e)$ for which the pair $(\Pi, e)$ is unique, that is, such that $G \not\in \mathcal{G}_m(\Pi',f)$ for any $\Pi' \in \mathcal{P}_{n,r}$ and $f \in (\Pi')^c$ with $\Pi' \neq \Pi$. Our second key lemma in the proof of the $0$-statement is the following.
\begin{lemma}
\label{lemma:GPe-unique}
For every positive $a$, there exists a constant $c$ such that for all $\Pi \in \mathcal{P}_{n,r}(\frac{1}{2r})$ and $m \geqslant c n \log n$ we have
\[
|\mathcal{G}_m(\Pi,e) \setminus \mathcal{U}_m(\Pi,e)| \leqslant n^{-a} \cdot |\mathcal{G}_m(\Pi,e)|.
\] \end{lemma}
In the proof of Lemma~\ref{lemma:GPe-unique}, we will need an estimate on the number of non-uniquely $r$-colorable graphs. Given a $\Pi \in \mathcal{P}_{n,r}$, let $\mathcal{U}_m(\Pi)$ be the family of all graphs in $\mathcal{G}_m(\Pi)$ for which $\Pi$ is the unique proper $r$-coloring. The following result is implicit in the work of Pr\"omel and Steger~\cite{PrSt95}. For completeness, we give a proof of it in Appendix~\ref{sec:omitted-proofs}.
\begin{prop}
\label{prop:UP}
For every positive $a$, there exists a constant $c$ such that for every $\Pi \in \mathcal{P}_{n,r}(\frac{1}{2r})$, if $m \geqslant c n \log n$, then
\[
|\mathcal{G}_m(\Pi) \setminus \mathcal{U}_m(\Pi)| \leqslant n^{-a} \cdot |\mathcal{G}_m(\Pi)|.
\] \end{prop}
\begin{proof}[{Proof of Lemma~\ref{lemma:GPe-unique}}]
By definition, if $G \in \mathcal{G}_m(\Pi,e) \setminus \mathcal{U}_m(\Pi,e)$ then $G \in \mathcal{G}_m(\Pi,e) \cap \mathcal{G}_m(\Pi',f)$ for some $\Pi \neq \Pi'$ and $f \in (\Pi')^c$. Considering the two cases $e = f$ and $e \neq f$, we infer that either $G - e \in \mathcal{G}_{m-1}(\Pi) \setminus \mathcal{U}_{m-1}(\Pi)$ or $G -\{e,f\} \in \mathcal{G}_{m-2}(\Pi) \setminus \mathcal{U}_{m-2}(\Pi)$. Let $a' = a+5$ and recall that $|\mathcal{G}_m(\Pi,e)| = |\mathcal{G}_{m-1}(\Pi)| = \binom{e(\Pi)}{m-1}$. By Proposition~\ref{prop:UP}, if $m \geqslant c n \log n$ for a sufficiently large constant $c$, then
\[
|\mathcal{G}_m(\Pi,e) \setminus \mathcal{U}_m(\Pi,e)| \leqslant n^{-a'} \cdot \left(|\mathcal{G}_{m-1}(\Pi)|+ n^2 \cdot |\mathcal{G}_{m-2}(\Pi)|\right) \leqslant 2n^{-a'+4} \cdot \binom{e(\Pi)}{m-1},
\]
where the last inequality holds since $|\mathcal{G}_{m-2}(\Pi)| = \binom{e(\Pi)}{m-2} \leqslant n^2 \cdot \binom{e(\Pi)}{m-1}$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:one-bad-edge}]
Finally, if $n \log n \ll m \leqslant (1-\varepsilon)m_r$, then using Lemmas~\ref{lemma:GPe-free} and~\ref{lemma:GPe-unique} we conclude that if $n$ is sufficiently large, then (below $\Pi$ ranges over $\mathcal{P}_{n,r}(\gamma)$)
\[
\begin{split}
|\mathcal{F}_{n,m}(K_{r+1})| & \geqslant \left| \bigcup_{\Pi} \bigcup_{e \in \Pi^c} \mathcal{U}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1}) \right| = \sum_{\Pi} \sum_{e \in \Pi^c} \big| \mathcal{U}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1}) \big| \\
& \geqslant \sum_{\Pi} \sum_{e \in \Pi^c} \Big( \big| \mathcal{G}_m(\Pi,e) \cap \mathcal{F}_{n,m}(K_{r+1}) \big| - \big| \mathcal{G}_m(\Pi,e) \setminus \mathcal{U}_m(\Pi,e) \big| \Big) \\
& \gg \sum_{\Pi} \sum_{e \in \Pi^c} \left( \frac{1}{m} - \frac{1}{m^2} \right) \cdot \binom{e(\Pi)}{m-1} \geqslant \frac{1}{3r} \cdot |\mathcal{G}_{n,m}(r)|,
\end{split}
\]
where the last inequality follows from Proposition~\ref{prop:unbalanced-graphs}, cf.\ \eqref{eq:sum-GPe}. This completes the proof of the $0$-statement. \end{proof}
\section{The $1$-statement}
\label{sec:1-statement}
In the remainder of the paper, we will show that if $m \geqslant (1+\varepsilon)m_r$ for some positive constant $\varepsilon$, then $|\mathcal{F}_{n,m}^*(K_{r+1})| \ll |\mathcal{F}_{n,m}(K_{r+1})|$, which implies that $|\mathcal{F}_{n,m}(K_{r+1})| = (1+o(1)) \cdot |\mathcal{G}_{n,m}(r)|$. In this section, we set up some notation and parameters and show that, as already pointed out before, we may restrict our attention to $K_{r+1}$-free graphs that admit a balanced $r$-coloring with few monochromatic edges. This leads to formulating Theorem~\ref{thm:1-statement}, a technical statement formalizing the above claim that graphs which admit a balanced $r$-coloring with few monochromatic edges constitute the vast majority of $\mathcal{F}_{n,m}(K_{r+1})$. We then show how Theorem~\ref{thm:1-statement}, together with Theorem~\ref{thm:approx-struct}, implies the $1$-statement of Theorem~\ref{thm:main}. We close the section by defining the split into the sparse and the dense case, which are then handled in Sections~\ref{sec:sparse-case} and~\ref{sec:dense-case}, respectively.
\subsection{Almost $r$-colorability}
\label{sec:almost-r-color}
We begin with a fairly straightforward refinement of Theorem~\ref{thm:approx-struct}, Proposition~\ref{prop:approx-struct} below. Roughly speaking, it says that if $m \gg n^{2-2/(r+1)}$, then not only does almost every $G \in \mathcal{F}_{n,m}(K_{r+1})$ admit an $r$-coloring $\Pi$ that makes merely $o(m)$ edges of $G$ monochromatic, but moreover, for almost every such $G$, every $\Pi$ with this property has $r$ parts of size $n/r + o(n)$.
\begin{prop}
\label{prop:approx-struct}
For every positive integer $r$ and all positive $\gamma$ and $\delta$, there exists a $C$ such that if $m \geqslant Cn^{2-\frac{2}{r+2}}$, then almost every $G \in \mathcal{F}_{n,m}(K_{r+1})$ admits a partition $\Pi = \{V_1, \ldots, V_r\}$ of $[n]$ such that
\begin{equation}
\label{eq:approx-struct-edges}
e(G \setminus \Pi) = \sum_{i=1}^r e_G(V_i) \leqslant \delta m.
\end{equation}
Moreover, if $\delta$ is sufficiently small as a function of $\gamma$, then for almost all $G \in \mathcal{F}_{n,m}(K_{r+1})$, every $\Pi$ satisfying~\eqref{eq:approx-struct-edges} also satisfies~\eqref{eq:Part-gamma}, that is, belongs to $\mathcal{P}_{n,r}(\gamma)$. \end{prop} \begin{proof}
Without loss of generality, we may assume that $\delta$ is sufficiently small as a function of $\gamma$. In particular, we may assume that it satisfies
\begin{equation}
\label{eq:gamma-delta}
\delta \cdot \log \frac{4}{\gamma^2} - (1-\delta) \frac{\gamma^2}{2} < - \frac{\gamma^2}{3}.
\end{equation}
The existence of a partition $\Pi \in \mathcal{P}_{n,r}$ satisfying~\eqref{eq:approx-struct-edges} for almost all $G \in \mathcal{F}_{n,m}(K_{r+1})$ follows directly from Theorem~\ref{thm:approx-struct}. Therefore, it suffices to count graphs $G \in \mathcal{F}_{n,m}(K_{r+1})$ that admit a partition that satisfies~\eqref{eq:approx-struct-edges} but not~\eqref{eq:Part-gamma}. To this end, fix an arbitrary partition $\Pi \in \mathcal{P}_{n,r}$ that does not satisfy~\eqref{eq:Part-gamma} and observe that $e(\Pi)$ is maximized when one of the parts has size $\lfloor(\frac{1}{r} + \gamma)n+1\rfloor$ or $\lceil(\frac{1}{r}-\gamma)n-1\rceil$ and the sizes of the remaining parts are as equal as possible. Therefore,
\[
\binom{n}{2} - \min\left\{ \binom{\left(\frac{1}{r} - \gamma\right)n}{2} + (r-1)\binom{\left(\frac{1}{r} + \frac{\gamma}{r-1}\right)n}{2}, \binom{\left(\frac{1}{r} + \gamma\right)n}{2} + (r-1)\binom{\left(\frac{1}{r} - \frac{\gamma}{r-1}\right)n}{2} \right\}
\]
is an upper bound on $e(\Pi)$. It follows that
\begin{equation}
\label{eq:ePi-gamma-upper}
e(\Pi) \leqslant \binom{n}{2} - r \binom{\frac{n}{r}}{2} - \frac{\gamma^2 r}{2(r-1)}n^2 \leqslant \ex(n, K_{r+1}) - \frac{\gamma^2 n^2}{2}.
\end{equation}
Note that the number $N_\Pi$ of graphs $G \in \mathcal{F}_{n,m}(K_{r+1})$ for which~\eqref{eq:approx-struct-edges} holds for our fixed partition~$\Pi$ satisfies
\begin{equation}
\label{eq:NPi-one}
N_\Pi \leqslant \sum_{t = 0}^{\delta m} \binom{\binom{n}{2}}{t} \binom{e(\Pi)}{m-t} \leqslant m \cdot \binom{\binom{n}{2}}{\delta m} \binom{e(\Pi)}{m - \delta m},
\end{equation}
where the second inequality holds because the summand in~\eqref{eq:NPi-one} is an increasing function of $t$ on the interval $[0,m/2]$. Now, using our bound on $e(\Pi)$ for $\Pi$ that do not satisfy~\eqref{eq:Part-gamma}, we have
\begin{equation}
\label{eq:NPi-two}
\begin{split}
N_\Pi & \leqslant m \cdot \binom{\binom{n}{2}}{\delta m} \binom{\ex(n, K_{r+1}) - \frac{\gamma^2 n^2}{2}}{m - \delta m} \\
& \leqslant m \cdot \left(\frac{\binom{n}{2}-\delta m}{\frac{\gamma^2n^2}{4}-\delta m}\right)^{\delta m} \binom{\frac{\gamma^2n^2}{4}}{\delta m} \cdot \left(\frac{\ex(n, K_{r+1}) - \frac{\gamma^2n^2}{2}}{\ex(n, K_{r+1})-\frac{\gamma^2n^2}{4}}\right)^{m-\delta m} \binom{\ex(n, K_{r+1}) - \frac{\gamma^2n^2}{4}}{m-\delta m} \\
& \leqslant m \cdot \left(\frac{4}{\gamma^2}\right)^{\delta m} \cdot \left(1 - \frac{\gamma^2}{2}\right)^{(1-\delta) m} \cdot \binom{\ex(n, K_{r+1})}{m} \leqslant \exp\left(-\frac{\gamma^2m}{4}\right) \cdot \binom{\ex(n, K_{r+1})}{m},
\end{split}
\end{equation}
where the second inequality follows from Lemma~\ref{lemma:acbc} (applied twice), the third inequality follows from Lemma~\ref{lemma:Vandermonde}, and the last inequality follows from~\eqref{eq:gamma-delta}, provided that $n$ is sufficiently large (and, consequently, $m$ is sufficiently large). Finally, the result follows from~\eqref{eq:NPi-two} since there are at most $r^n$ partitions $\Pi \in \mathcal{P}_{n,r}$ and at least $\binom{\ex(n, K_{r+1})}{m}$ graphs in $\mathcal{F}_{n,m}(K_{r+1})$. \end{proof}
In view of Proposition~\ref{prop:approx-struct}, for positive $\gamma$ and $\delta$, let $\mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma)$ be the collection of graphs $G \in \mathcal{F}_{n,m}(K_{r+1})$ that satisfy~\eqref{eq:approx-struct-edges} for some $\Pi \in \mathcal{P}_{n,r}(\gamma)$ and no $\Pi \not\in \mathcal{P}_{n,r}(\gamma)$. In other words, $\mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma)$ is the collection of graphs $G \in \mathcal{F}_{n,m}(K_{r+1})$ that are almost $r$-colorable (i.e., admit an $r$-coloring which makes only at most $\delta m$ edges of $G$ monochromatic) and such that every $r$-coloring $\Pi$ that makes only at most $\delta m$ edges of $G$ monochromatic has color classes of sizes only between $(1/r-\gamma)n$ and $(1/r+\gamma)n$. In this notation, Proposition~\ref{prop:approx-struct} says that almost all graphs in $\mathcal{F}_{n,m}(K_{r+1})$ belong to $\mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma)$ provided that $\delta$ is sufficiently small as a function of $\gamma$ and $m \geqslant C_{\ref{prop:approx-struct}}(\delta, \gamma) \cdot n^{-\frac{2}{r+2}}$.
\begin{claim}
\label{claim:unfriendly}
For every integer $r$ and all positive $\gamma$ and $\delta$, every $G \in \mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma)$ admits a $\Pi \in \mathcal{P}_{n,r}(\gamma)$ that satisfies~\eqref{eq:approx-struct-edges} and such that each vertex of $G$ has at least as many neighbors in each color class of $\Pi$ as in its own class, that is, letting $\Pi = \{V_1, \ldots, V_r\}$, \begin{equation}
\label{eq:Pi-unfriendly}
\deg_G(v,V_i) \leqslant \min_{j \neq i} \deg_G(v, V_j) \quad \text{for all $i \in [r]$ and $v \in V_i$}. \end{equation} \end{claim} \begin{proof}
To see this, given such a $G$, let $\Pi \in \mathcal{P}_{n,r}$ be a partition that minimizes $e(G \setminus \Pi)$ and suppose that $\Pi = \{V_1, \ldots, V_r\}$. The minimality of $\Pi$ immediately implies~\eqref{eq:Pi-unfriendly}. Indeed, if there were $i, j \in [r]$ and $v \in V_i$ such that $\deg_G(v, V_i) > \deg_G(v, V_j)$, then the partition $\Pi'$ obtained from $\Pi$ by moving the vertex $v$ from $V_i$ to $V_j$ would satisfy $e(G \setminus \Pi') < e(G \setminus \Pi)$, contradicting the minimality of $\Pi$. Moreover, since $e(G \setminus \Pi) \leqslant \delta m$ by the definition of $\mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma)$ and the minimality of $\Pi$, then $\Pi \in \mathcal{P}_{n,r}(\gamma)$, again by the definition of $\mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma)$. \end{proof}
In view of the above, given positive constants $\gamma$ and $\delta$ and a balanced $r$-coloring $\Pi \in \mathcal{P}_{n,r}(\gamma)$, let \[ \mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma, \Pi) = \big\{G \in \mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma) \colon \text{$(G, \Pi)$ satisfy~\eqref{eq:approx-struct-edges} and~\eqref{eq:Pi-unfriendly}} \big\}. \] By Claim~\ref{claim:unfriendly}, we have \begin{equation}
\label{eq:Freedg-partition}
\mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma) = \bigcup_{\Pi \in \mathcal{P}_{n,r}(\gamma)} \mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma, \Pi). \end{equation}
Next, let us break down the family $\mathcal{F}_{n,m}^*(K_{r+1})$ of non-$r$-colorable $K_{r+1}$-free graphs with respect to the above partition of $\mathcal{F}_{n,m}(K_{r+1})$. First, let \[ \mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma) = \mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma) \setminus \mathcal{G}_{n,m}(r), \] then let \[ \mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma, \Pi) = \big\{G \in \mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma, \Pi) \colon e(G \setminus \Pi) > 0\big\}, \] and note that, by~\eqref{eq:Freedg-partition}, \begin{equation}
\label{eq:Freesdg-partition}
\mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma) \subseteq \bigcup_{\Pi \in \mathcal{P}_{n,r}(\gamma)} \mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma, \Pi). \end{equation} (Note that we cannot write an equality in~\eqref{eq:Freesdg-partition} since the fact that $G \in \mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma, \Pi)$ for some $\Pi$ does not mean that $G$ is not $r$-colorable).
Finally, since under the assumption that $m \geqslant (1+\varepsilon)m_r \gg n^{2 - \frac{2}{r+2}}$, Proposition~\ref{prop:approx-struct} applies with arbitrarily small $\gamma$ and $\delta$, it is enough to prove the following theorem, which is the essence of the $1$-statement of Theorem~\ref{thm:main}.
\begin{thm}
\label{thm:1-statement}
For every integer $r$ and every positive $\varepsilon$, there exist a positive constant $\gamma$ and a function $\omega$ satisfying $\omega(n) \to \infty$ as $n \to \infty$ such that the following holds for all sufficiently small positive $\delta$. For every $n$, if $m \geqslant (1+\varepsilon)m_r$, then
\begin{equation}
\label{eq:Freesdgp}
|\mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma, \Pi)| \leqslant \frac{1}{\omega(n)} \cdot \binom{e(\Pi)}{m} \quad \text{for every $\Pi \in \mathcal{P}_{n,r}(\gamma)$}.
\end{equation} \end{thm}
Indeed, Theorem~\ref{thm:Gr} and Proposition~\ref{prop:approx-struct} together with \eqref{eq:Freesdg-partition} and \eqref{eq:Freesdgp} imply that \begin{align*}
|\mathcal{F}_{n,m}^*(K_{r+1})| & \leqslant |\mathcal{F}_{n,m}(K_{r+1}) \setminus \mathcal{F}_{n,m}(K_{r+1}; \delta, \gamma)| + \sum_{\Pi \in \mathcal{P}_{n,r}(\gamma)} |\mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma, \Pi)| \\
& = o\big(|\mathcal{F}_{n,m}(K_{r+1})|\big) + \sum_{\Pi \in \mathcal{P}_{n,r}(\gamma)} \frac{1}{\omega(n)} \cdot \binom{e(\Pi)}{m} \\
& \leqslant o\big(|\mathcal{F}_{n,m}(K_{r+1})|\big) + \frac{1}{\omega(n)} \cdot (1+o(1)) \cdot |\mathcal{G}_{n,m}(r)| \ll |\mathcal{F}_{n,m}(K_{r+1})|. \end{align*} In the remainder of the paper, we will prove Theorem~\ref{thm:1-statement}.
\subsection{Parameters}
\label{sec:parameters-main}
We now choose some parameters. Recall that an integer $r$ and a positive constant $\varepsilon$ are given. We may clearly assume that $\varepsilon \leqslant 1$. We first define the split between the sparse and the dense cases, see Section~\ref{sec:sketch-proof-1}. To this end, we let \begin{equation}
\label{eq:xi}
\xi = \left( \frac{2^9 e}{3} \right)^{-30r^3}. \end{equation} Next, we choose small positive $\rho$ and $\gamma$ with $\gamma < \frac{1}{9r}$ so that, letting \begin{equation}
\label{eq:crgammap}
\zeta = \frac{1+\varepsilon}{1-\rho} \cdot \left(\frac{1}{1-r\gamma}\right)^{r-1}, \end{equation} we have \begin{equation}
\label{eq:zeta}
\zeta \leqslant 2^r \quad \text{and} \quad \left(1+\frac{3\varepsilon}{4}\right)^{\rpt - 1} \geqslant \left(1+\frac{\varepsilon}{3}\right) \cdot \zeta. \end{equation} For example, we may take $\rho = \frac{\varepsilon}{20}$ and $\gamma = \frac{\varepsilon}{9r+9}$. Our assumption that $\gamma < \frac{1}{9r}$ guarantees that for every $\Pi \in \mathcal{P}_{n,r}(\gamma)$, \begin{equation}
\label{eq:ePi-lower}
e(\Pi) \geqslant \binom{r}{2} \cdot \left[\left(\frac{1}{r} - \gamma\right)n\right]^2 \geqslant \binom{r}{2} \cdot \left(\frac{8n}{9r}\right)^2 \geqslant \frac{3n^2}{16}. \end{equation} Finally, let us also assume that we have chosen a small positive constant $\delta$. Since this constant will have to satisfy a series of inequalities that use parameters that have not yet been introduced (but all depend only on the quantities defined so far), we will make this choice more specific somewhat later in the proof. In particular, we assume that $\delta \leqslant \delta_{\ref{prop:approx-struct}}(\gamma)$.
\subsection{Setup}
\label{sec:setup}
For the remainder of the proof, let us fix some $\Pi = \{V_1, \ldots, V_r\} \in \mathcal{P}_{n,r}(\gamma)$, assume that $m \geqslant (1+\varepsilon)m_r$, and let \[ \mathcal{F}^* = \mathcal{F}_{n,m}^*(K_{r+1}; \delta, \gamma, \Pi). \] Recall that our goal is to prove Theorem~\ref{thm:1-statement}, i.e., that \[
|\mathcal{F}^*| \leqslant \frac{1}{\omega(n)} \cdot \binom{e(\Pi)}{m} \]
for some function $\omega$ satisfying $\omega(n) \to \infty$ that does not depend on $\Pi$. We need a few more pieces of notation. Given a $G \in \mathcal{F}^*$, for each $i \in [r]$, we let $T_i(G)$ be the subgraph of $G$ induced by $V_i$, the $i$th color class of $\Pi$. Moreover, we let $T(G)$ be the subgraph of all monochromatic edges of $G$ (in the coloring $\Pi$), that is, $T(G) = T_1(G) \cup \ldots \cup T_r(G)$. As we pointed out in Section~\ref{sec:sketch-proof-1}, our general strategy will be to partition the family $\mathcal{F}^*$ into several classes according to the distribution of edges in the graphs $T(G)$, show that each of these classes is small, and then deduce that $|\mathcal{F}^*|$ is small.
Recall that $\Pi \in \mathcal{P}_{n,r}(\gamma)$ is fixed. Let $\mathcal{T}$ denote the collection of all graphs consisting of at most $\delta m$ monochromatic (in the coloring $\Pi$) edges. That is, let $\mathcal{T}$ be the set of all graphs $T \subseteq \Pi^c$ with at most $\delta m$ edges. Given a $T \in \mathcal{T}$, let \[ \mathcal{F}^*(T) = \{G \in \mathcal{F}^* \colon T(G) = T\}. \]
As pointed out in Section~\ref{sec:sketch-proof-1}, we will use completely different arguments to handle the cases $m \leqslant e(\Pi) - \xi n^2$ (the sparse case) and $m > e(\Pi) - \xi n^2$ (the dense case). We begin with the main, much harder, case $m \leqslant e(\Pi) - \xi n^2$. The other case is addressed in Section~\ref{sec:dense-case}.
\section{The sparse case ($m \leqslant e(\Pi) - \xi n^2$)}
\label{sec:sparse-case}
\subsection{More parameters}
First, we need to define three more parameters that will play central roles in our proof. First, let \begin{equation}
\label{eq:nu}
\nu = \frac{\rho}{(2r)^{2r+1}} \end{equation} and \begin{equation}
\label{eq:D}
D = \nu \cdot \frac{m}{n \log n}. \end{equation} For the sake of clarity of presentation, we will assume that $D$ is an integer. Next, let $\beta$ be a small positive constant satisfying \begin{equation}
\label{eq:beta}
\left(\frac{2e}{\xi \beta}\right)^{\beta m/n} \leqslant m^{D/2}. \end{equation} Note that choosing such $\beta$ is possible, since $m^D = \exp(\Omega(m/n))$ and $(\frac{2e}{\xi\beta})^\beta \to 1$ as $\beta \to 0$. Also, observe that $D \ll \beta m / n$.
\subsection{Setup}
\label{sec:setup-sparse}
Recall the definition of $\mathcal{T}$ from Section~\ref{sec:setup}. Let us fix a $T \in \mathcal{T}$. Let $U(T)$ be some (canonically chosen) edge-maximal subgraph of $T$ with maximum degree at most $D$. Let $X(T)$ be the set of vertices that have the maximum allowed degree in $U(T)$, that is, the set of all $v$ whose degree in $U(T)$ is $D$. Observe that \begin{equation}
\label{eq:UT-lower}
e(U(T)) \geqslant e(T - X(T)) + |X(T)| \cdot D/2, \end{equation} since, by the maximality of $U(T)$, every edge of $T \setminus U(T)$ has at least one endpoint in $X(T)$.
For every $i \in [r]$, let $U_i(T)$ be the subgraph of $U(T)$ induced by the set $V_i$, the $i$th color class of $\Pi$, and let $X_i(T) = X(T) \cap V_i$. Finally, let $H(T) \subseteq X(T)$ denote the set of vertices $v$ in $X(T)$ whose degree in $T$ is at least $\beta m / n$. We will refer to vertices in $H(T)$ as the vertices with high degree in~$T$. We split the family $\mathcal{T}$ according to whether the inequality \begin{equation}
\label{eq:low-high-sparse}
|H(T)| \leqslant \frac{\varepsilon \xi}{6} \cdot \frac{n \log m}{m} \cdot e(U(T)) \end{equation} does or does not hold. More precisely, we let $\TT^L$ be the collection of all $T \in \mathcal{T}$ for which~\eqref{eq:low-high-sparse} holds and let $\TT^H = \mathcal{T} \setminus \TT^L$. We will separately count the graphs in $\mathcal{F}^*(T)$ with $T \in \TT^L$ (we will refer to it as the \emph{low degree case}) and $T \in \TT^H$ (this will be referred to as the \emph{high degree case}).
\subsection{Recap of the proof outline}
There will be four main ingredients in our proof. First, in Section~\ref{sec:counting-graphs}, in Lemmas~\ref{lemma:T-count-basic} and \ref{lemma:T-count}, we will count the graphs $T \in \mathcal{T}$ with particular values of $e(T)$, $e(T - X(T))$, $|X(T)|$, and $|H(T)|$; this is relatively straightforward. Second, in Section~\ref{sec:bounding-Fst-UT}, in Lemma~\ref{lemma:Janson}, using the Hypergeometric Janson Inequality (Lemma~\ref{lemma:HJI}), we will give an upper bound on the size of $\mathcal{F}^*(T)$ as a function of $e(U(T))$. These three lemmas will already be enough to prove that the number of graphs $G \in \mathcal{F}^*$ that fall into the low degree case (i.e., $T(G) \in \TT^L$) is at most $m^{-\varepsilon/4} \binom{e(\Pi)}{m}$, see Lemma~\ref{lemma:low-degree} in Section~\ref{sec:sparse-low-degree-case}. In order to count the graphs $G \in \mathcal{F}^*$ that fall into the high degree case (i.e., $T(G) \in \TT^H$), we will have to further split them into two classes, which we term the \emph{regular} and \emph{irregular cases}, depending on the distribution of the neighborhoods of the vertices in $H(T(G))$. We will make this division precise in Section~\ref{sec:sparse-high-degree-case}. The third ingredient in our proof, Lemmas~\ref{lemma:good-H-basic} and \ref{lemma:good-H}, together with Lemmas~\ref{lemma:T-count-basic} and \ref{lemma:T-count}, provides an upper bound on the number of graphs in $\bigcup_T \mathcal{F}^*(T)$, where the union is taken over all $T$ that fall into the regular case, see Section~\ref{sec:regular-case}. Finally, in Section~\ref{sec:irregular-case}, we will use Lemmas~\ref{lemma:d-sets} and \ref{lemma:T-count-basic} to bound the number of graphs that fall into the irregular case. Counting the graphs that fall into the irregular case with the use of Lemma~\ref{lemma:d-sets} is the main technical novelty of this paper.
\subsection{Counting the graphs in $\mathcal{T}$}
\label{sec:counting-graphs}
For an integer $t$ with $1 \leqslant t \leqslant \delta m$, let $\mathcal{T}_t$ be the subfamily of $\mathcal{T}$ consisting of graphs with exactly $t$ edges. Since we are going to treat differently graphs $T \in \mathcal{T}$ with different values of $e(T)$, $e(U(T))$, $|X(T)|$, and $|H(T)|$, let us further subdivide the families $\mathcal{T}_t$. Even though the forthcoming definitions might seem somewhat odd at first, they will be very convenient to work with later in the proof. For integers $t^*$, $x$, and $h$, we let $\mathcal{T}_t(t^*,x,h)$ be the subfamily of $\mathcal{T}_t$ consisting of all graphs $T$ for which there exist sets $H, X \subseteq [n]$ with $|H| = h$, $|X| = x$, and $H \subseteq X$ such that: \begin{enumerate}[(i)] \item
\label{item:TT-1}
$e(T - X) = t^*$, that is, $T$ has exactly $t^*$ edges which have no endpoint in $X$, \item
$\deg_T(v) < \beta m/n$ for every $v \not\in H$. \end{enumerate} Moreover, let $\mathcal{T}_t'(t^*,x,h)$ be the subfamily of $\mathcal{T}_t(t^*,x,h)$ consisting of graphs that additionally satisfy \begin{enumerate}[(i)] \setcounter{enumi}{2} \item
\label{item:TT-3}
$\deg_T(v) \geqslant \beta m / n$ for every $v \in H$. \end{enumerate} Since every $T \in \mathcal{T}$ satisfies (\ref{item:TT-1})--(\ref{item:TT-3}) above with $t^* = E(T-X(T))$, $X = X(T)$, and $H = H(T)$, it follows that \[
T \in \mathcal{T}_{e(T)}\big(e(T - X(T)), |X(T)|, |H(T)|\big) \subseteq \mathcal{T}_{e(T)}. \] We shall now prove upper bounds on the sizes of the families $\mathcal{T}_t$ and $\mathcal{T}_t(t^*,x,h)$. We remark that the somewhat strange-looking form of these bounds will be very convenient for their later applications.
\begin{lemma}
\label{lemma:T-count-basic}
If $t \leqslant \delta m$, then
\[
|\mathcal{T}_t| \cdot \binom{e(\Pi)}{m - t} \leqslant \left(\frac{e}{\xi \delta}\right)^{\delta m} \cdot \binom{e(\Pi)}{m}.
\] \end{lemma}
\begin{lemma}
\label{lemma:T-count}
For all integers $m'$, $t$, $t^*$, $x$, and $h$ with $m' \leqslant m$,
\[
|\mathcal{T}_t(t^*,x,h)| \cdot \binom{e(\Pi)}{m'-t} \leqslant e^{1/\xi} \cdot m^{t^* + xD/2} \cdot \exp\left(\frac{2mh}{\xi n}\right) \cdot \binom{e(\Pi)}{m'}.
\] \end{lemma}
\begin{proof}[{Proof of Lemma~\ref{lemma:T-count-basic}}]
We use the trivial bound
\begin{equation}
\label{eq:TTt}
|\mathcal{T}_t| \leqslant \binom{e(\Pi^c)}{t}.
\end{equation}
We then use the identity
\[
\frac{\binom{e(\Pi^c)}{t} \binom{e(\Pi)}{m-t}}{\binom{e(\Pi)}{m}} = \prod_{s=0}^{t-1} \frac{\binom{e(\Pi^c)}{s+1} \binom{e(\Pi)}{m-s-1}}{\binom{e(\Pi^c)}{s} \binom{e(\Pi)}{m-s}} = \prod_{s=0}^{t-1} \left( \frac{e(\Pi^c)-s}{s+1} \cdot \frac{m-s}{e(\Pi)-m+s+1} \right)
\]
to deduce that, since $m \leqslant e(\Pi) - \xi n^2$,
\begin{equation}
\label{eq:TTt-ratio}
\frac{\binom{e(\Pi^c)}{t} \binom{e(\Pi)}{m-t}}{\binom{e(\Pi)}{m}} \leqslant \prod_{s=0}^{t-1} \left( \frac{n^2}{s+1} \cdot \frac{m}{\xi n^2} \right) = \frac{1}{t!} \cdot \left(\frac{m}{\xi}\right)^t \leqslant \left(\frac{em}{\xi t}\right)^t \leqslant \left(\frac{e}{\xi \delta}\right)^{\delta m},
\end{equation}
where we used the fact that $t! \geqslant (t/e)^t$ and that the function $t \mapsto (\frac{em}{\xi t})^t$ is increasing on the interval $(0,\delta m]$, as $\delta, \xi \leqslant 1$. \end{proof}
\begin{proof}[{Proof of Lemma~\ref{lemma:T-count}}]
We prove the lemma by induction on $x$. For the induction base, the case $x = 0$, note that if $x = 0$, then (in order for the family $\mathcal{T}_t(t^*,x,h)$ to be non-empty) we must have $h = 0$ and $t^* = t$. Since $\mathcal{T}_t(t^*,x,h) \subseteq \mathcal{T}_t$, it now follows from \eqref{eq:TTt} and \eqref{eq:TTt-ratio}, with $m$ replaced by $m'$, that
\[
|\mathcal{T}_t(t^*,x,h)| \cdot \binom{e(\Pi)}{m'-t} \cdot \binom{e(\Pi)}{m'}^{-1} \leqslant \left(\frac{em'}{\xi t}\right)^t \leqslant \left(\frac{em}{\xi t}\right)^t = \left(\frac{e}{\xi t^*}\right)^{t^*} m^{t^*} \leqslant e^{1/\xi} \cdot m^{t^*},
\]
where the last inequality holds because the function $t^* \mapsto (\frac{e}{\xi t^*})^{t^*}$ is maximized when $t^* = 1/\xi$.
Assume now that $x \geqslant 1$. Given a $T \in \mathcal{T}_t(t^*,x,h)$, we fix some $X$ and $H$ from the definition of $\mathcal{T}_t(t^*,x,h)$, pick an arbitrary vertex $v \in X$. Next, let $d = \deg_T(v)$ and obtain a subgraph $T' \subseteq T$ by removing all $d$ edges incident to $v$. Clearly, $T'$ lies in $\mathcal{T}_{t-d}(t^*,x-1,h) \cup \mathcal{T}_{t-d}(t^*,x-1,h-1)$. Moreover, if $d > \beta m / n$, then necessarily $v \in H$ (but not vice versa!) and consequently $T' \in \mathcal{T}_{t-d}(t^*,x-1,h-1)$. It follows that
\begin{equation}
\label{eq:T-count-step}
|\mathcal{T}_t(t^*,x,h)| \leqslant \sum_{d=0}^{\beta m/n} n\binom{n}{d} |\mathcal{T}_{t-d}(t^*,x-1,h)| + \sum_{d=0}^n n \binom{n}{d} |\mathcal{T}_{t-d}(t^*,x-1,h-1)|.
\end{equation}
Since $t \leqslant m' \leqslant m \leqslant e(\Pi) - \xi n^2$, then
\begin{equation}
\label{eq:T-count-step-ratio}
\begin{split}
\frac{\binom{n}{d} \binom{e(\Pi)}{m'-t}}{\binom{e(\Pi)}{m'-t+d}} & = \prod_{s=0}^{d-1} \frac{\binom{n}{s+1} \binom{e(\Pi)}{m'-t+d-s-1}}{\binom{n}{s} \binom{e(\Pi)}{m'-t+d-s}} = \prod_{s=0}^{d-1} \left(\frac{n-s}{s+1} \cdot \frac{m'-t+d-s}{e(\Pi)-m'+t-d+s+1}\right) \\
& \leqslant \prod_{s=0}^{d-1} \left(\frac{n}{s+1} \cdot \frac{m'}{\xi n^2}\right) = \frac{1}{d!} \cdot \left(\frac{m'}{\xi n}\right)^d \leqslant \left(\frac{em}{\xi nd}\right)^d,
\end{split}
\end{equation}
where we again used the fact that $d! \geqslant (d/e)^d$. Recall that for every positive $a$, the function $x \mapsto (a/x)^x$ is increasing on the interval $(0,a/e]$ and decreasing on the interval $[a/e, \infty)$. Hence, by~\eqref{eq:T-count-step-ratio},
\begin{equation}
\label{eq:T-count-step-low-deg}
\sum_{d=0}^{\beta m/n} n\binom{n}{d} \frac{\binom{e(\Pi)}{m'-t}}{\binom{e(\Pi)}{m'-t+d}} \leqslant \sum_{d=0}^{\beta m /n} n \left(\frac{em}{\xi nd}\right)^d \leqslant n^2 \left(\frac{e}{\xi \beta}\right)^{\beta m/n} \leqslant \frac{1}{2} \left(\frac{2e}{\xi\beta}\right)^{\beta m/n} \leqslant \frac{1}{2} m^{D/2},
\end{equation}
where the last inequality follows from~\eqref{eq:beta}, and
\begin{equation}
\label{eq:T-count-step-high-deg}
\sum_{d=0}^n n\binom{n}{d} \frac{\binom{e(\Pi)}{m'-t}}{\binom{e(\Pi)}{m'-t+d}} \leqslant \sum_{d=0}^{n} n \left(\frac{em}{\xi nd}\right)^d \leqslant n^2 \cdot \exp\left(\frac{m}{\xi n}\right) \leqslant \frac{1}{2} \exp\left(\frac{2m}{\xi n}\right).
\end{equation}
The claimed bound follows easily from the inductive assumption, \eqref{eq:T-count-step}, \eqref{eq:T-count-step-low-deg}, and \eqref{eq:T-count-step-high-deg}. \end{proof}
\subsection{Bounding $|\mathcal{F}^*(T)|$ in terms of $e(U(T))$}
\label{sec:bounding-Fst-UT}
We shall now state and prove our main lemma for the low degree case. It provides an upper bound on the size of $\mathcal{F}^*(T)$ in terms of the number of edges in the graph $U(T)$. The lemma follows the natural and simple idea described in Section~\ref{sec:mr-threshold}, which was already exploited in~\cite{OsPrTa03} in the case $r = 2$. If $m \geqslant (1+\varepsilon)m_r$, then, at least under all the simplifying assumptions made in Section~\ref{sec:mr-threshold}, the proportion $P$ of graphs in $\mathcal{F}^*(T)$ with exactly one monochromatic edge is asymptotically smaller than $\frac{1}{m}$. Unfortunately, the calculation that we used to estimate $P$ is merely some intuition to keep in mind, as in reality things are considerably more complicated. Whereas the intuition that avoiding different copies of $K_{r+1}^-$ in $\Pi$, whose missing edges belong to $T$, can be treated as independent events is valid and can be made rigorous when the graph $T$ is small, it is no longer right when $T$ becomes large. In fact, it turns out that considering more copies of $K_{r+1}^-$ when we apply the Hypergeometric Janson Inequality (Lemma~\ref{lemma:HJI}) does not necessarily improve the bound, but can actually worsen it. This is why we work with the subgraph $U(T)$ of $T$ with bounded maximum degree. Still, our biggest problem is that the best bound for $|\mathcal{F}^*(T)|$ that we can obtain using the Hypergeometric Janson Inequality is not sufficiently strong to compensate for having to sum it over all $T \in \mathcal{T}$. This is why we split into the low degree and the high degree cases and are forced to use different methods to handle the high degree case.
Our main lemma in the low degree case is the following.
\begin{lemma}
\label{lemma:Janson}
For every $T \in \mathcal{T}$,
\[
|\mathcal{F}^*(T)| \leqslant 2 m^{-(1+\varepsilon) \cdot e(U(T))} \cdot \binom{e(\Pi)}{m - e(T)}.
\] \end{lemma}
In the next section, we show that Lemma~\ref{lemma:Janson}, together with Lemmas~\ref{lemma:T-count-basic} and~\ref{lemma:T-count}, resolves the low degree case, that is, that it implies that \[
|\{G \in \mathcal{F}^* \colon T(G) \in \TT^L\}| \leqslant m^{-\varepsilon/4} \cdot \binom{e(\Pi)}{m}, \] cf.\ Theorem~\ref{thm:1-statement}. In the remainder of this section, we prove the lemma.
\begin{proof}[Proof of Lemma~\ref{lemma:Janson}]
In order to prove the lemma, given a $T \in \mathcal{T}$, we will count the number of graphs $G' \subseteq \Pi$ with $m - e(T)$ edges such that $G = G' \cup T$ is $K_{r+1}$-free. The crucial observation is that for every such $G'$ and every edge $\{v, w\} \in U_i(T)$, none of the $\prod_{j \neq i} |V_j|$ copies of $K_{r+1}^-$ in $\Pi$ induced by $v$, $w$, and one vertex in each $V_j$ with $j \neq i$ can be fully contained in $G'$. In the remainder of the proof, we will use the Hypergeometric Janson Inequality to count graphs $G'$ satisfying this constraint. Note that
\begin{equation}
\label{eq:m_0}
e(U(T)) \leqslant Dn = \nu \cdot \frac{m}{\log n} = \frac{\rho}{(2r)^{2r+1}} \cdot \frac{m}{\log n},
\end{equation} since $U(T)$ has maximum degree at most $D$.
Let $\mathcal{K}$ be the collection of (the edge sets of) all copies of $K_{r+1}^-$ induced in $\Pi$ by the two endpoints of some edge in $U_i(T)$ and one vertex in each $V_j$ with $j \neq i$. Given $(K_1, K_2) \in \mathcal{K}^2$, we write $K_1 \sim K_2$ to denote the fact that $K_1$ and $K_2$ share at least one edge but $K_1 \neq K_2$. Let $p = \frac{m - e(T)}{e(\Pi)}$ and let
\[
\mu = \sum_{K \in \mathcal{K}} p^{e(K)} \qquad \text{and} \qquad \Delta = \sum_{K_1 \sim K_2} p^{e(K_1 \cup K_2)},
\]
where the second sum above is over all ordered pairs $(K_1, K_2) \in \mathcal{K}^2$ such that $K_1 \sim K_2$. By the Hypergeometric Janson Inequality, Lemma~\ref{lemma:HJI}, for every $q \in [0, 1]$,
\[
|\mathcal{F}^*(T)| \leqslant 2 \cdot \exp\left(-q\mu + q^2 \Delta /2\right) \cdot \binom{e(\Pi)}{m - e(T)}.
\]
Therefore, it suffices to show that for some $q \in [0,1]$, we have
\begin{equation}
\label{eq:q-goal}
q\mu - q^2 \Delta/2 \geqslant (1+\varepsilon) \log m \cdot e(U(T)),
\end{equation}
which we will do in the remainder of the proof of the lemma.
\noindent
\textbf{Estimating $\mu$ and $\Delta$.}
Recall that our fixed partition $\Pi$ lies in $\mathcal{P}_{n,r}(\gamma)$ and hence $|V_i| = (1/r \pm \gamma)n$ for each $i \in [r]$. It follows that
\begin{equation}
\label{eq:mu}
\mu = |\mathcal{K}| \cdot p^{\rpt-1} \geqslant e(U(T)) \cdot \left(\frac{1}{r} - \gamma\right)^{r-1} n^{r-1} \cdot p^{\rpt-1}.
\end{equation}
With the aim of estimating $\Delta$, for every $s \in [r-2]$, let
\[
N_s = \max \left\{ \prod_{i \in I} |V_i| \colon I \subseteq [r] \text{ with } |I| = s \right\} \leqslant \left(\frac{1}{r} + \gamma\right)^s n^s.
\]
Let us now fix two edges $v_1w_1$ and $v_2w_2$ of $U(T)$ and compute the contribution to $\Delta$ of all ordered pairs $(K_1, K_2) \in \mathcal{K}$ such that $K_1 \sim K_2$ and $v_1w_1$ and $v_2w_2$ are the missing edges in $K_1$ and $K_2$, respectively. We denote these contributions by:
\begin{itemize}
\item
$\Delta_1$ when $v_1w_1$ and $v_2w_2$ lie in the same color class and are disjoint,
\item
$\Delta_2$ when $v_1w_1$ and $v_2w_2$ lie in the same color class and share exactly one endpoint,
\item
$\Delta_3$ when $v_1w_1 = v_2w_2$, and
\item
$\Delta_4$ when $v_1w_1$ and $v_2w_2$ lie in different color classes.
\end{itemize}
A moment's thought reveals that
\begin{align}
\label{eq:Delta1}
\Delta_1 & \leqslant \sum_{s=2}^{r-1} \rms N_s N_{r-s-1}^2 p^{2\rpt - \st - 2}, \\
\label{eq:Delta2}
\Delta_2 & \leqslant \sum_{s=1}^{r-1} \rms N_s N_{r-s-1}^2 p^{2\rpt - \spt - 2}, \\
\label{eq:Delta3}
\Delta_3 & \leqslant \sum_{s=1}^{r-2} \rms N_s N_{r-s-1}^2 p^{2\rpt - \sppt - 1},
\end{align}
where $s$ is the number of common vertices that $K_1$ and $K_2$ share outside of the part containing their missing edges. Moreover, note that $\Delta_3 = 0$ if $r < 3$. Similarly,
\begin{equation}
\label{eq:Delta4}
\begin{split}
\Delta_4 & \leqslant \sum_{s=2}^{r-2} \rmms N_s N_{r-s-1}^2 p^{2\rpt - \st - 2} \\
& + 4 \sum_{s=1}^{r-2} \rmms N_s N_{r-s-1} N_{r-s-2} p^{2\rpt - \spt - 2} \\
& + 4\sum_{s=0}^{r-2} \rmms N_s N_{r-s-2}^2 p^{2\rpt - \sppt - 2},
\end{split}
\end{equation}
where the first, second, and third lines above correspond to the pairs $K_1 \sim K_2$ that share no, one, and two vertices in the two parts of $\Pi$ that contain the missing edges of $K_1$ and $K_2$; similarly as above, $s$ is the number of common vertices that $K_1$ and $K_2$ share outside of the two parts of $\Pi$ that containing the missing edges.
Since the maximum degree of $U(T)$ is at most $D$, it is now easy to see that
\[
\begin{split}
\Delta & \leqslant \sum_{i=1}^r e(U_i(T))^2 \cdot \Delta_1 + \sum_{v \in [n]} \deg_{U(T)}(v)^2 \cdot \Delta_2 + e(U(T)) \cdot \Delta_3 + \sum_{i \neq j} e(U_i(T))e(U_j(T)) \cdot \Delta_4 \\
& \leqslant e(U(T))^2 \cdot \max\{\Delta_1, \Delta_4\} + 2De(U(T)) \cdot \Delta_2 + e(U(T)) \cdot \Delta_3.
\end{split}
\]
Recall that $e(T) \leqslant \delta m$ and that
\begin{equation}
\label{eq:p-order}
p = \frac{m - e(T)}{e(\Pi)} \geqslant \frac{m}{2e(\Pi)} \geqslant \frac{m}{n^2} \gg n^{-\frac{2}{r+2}}.
\end{equation}
It follows from~\eqref{eq:p-order} that $n^sp^{\st} \gg n^2p$ for every $s \in \{3, \ldots, r\}$ and $n^sp^{\st} \gg n^3p^3$ for every $s \in \{4, \ldots, r\}$. Therefore, the sums in the right-hand sides of \eqref{eq:Delta1}, \eqref{eq:Delta2}, and~\eqref{eq:Delta3} are dominated by the terms with $s$ equal to $2$, $1$, and $1$, respectively, and hence
\begin{align*}
\Delta_1 & \leqslant
(1+o(1)) \binom{r-1}{2}\left(\frac{1}{r} + \gamma\right)^{2r-4}n^{2r-4}p^{2\rpt-3}, \\
\Delta_2 & \leqslant
(1+o(1)) \binom{r-1}{1}\left(\frac{1}{r} + \gamma\right)^{2r-3}n^{2r-3}p^{2\rpt-3}, \\
\Delta_3 & \leqslant
(1+o(1)) \binom{r-1}{1}\left(\frac{1}{r} + \gamma\right)^{2r-3}n^{2r-3}p^{2\rpt-4} \cdot \mathbf{1}[r \geqslant 3].
\end{align*}
Similarly, the three sums in the right-hand side of~\eqref{eq:Delta4} are dominated by the terms with $s$ equal to $2$, $1$, and $0$, respectively, and hence
\[
\Delta_4 \leqslant (1+o(1)) \left[ \binom{r-2}{2} + 4\binom{r-2}{1} + 4\binom{r-2}{0} \right] \left(\frac{1}{r} + \gamma\right)^{2r-4}n^{2r-4}p^{2\rpt-3}.
\]
The (somewhat crude) estimates
\[
\max\left\{ \binom{r-1}{2}, \binom{r-2}{2} + 4\binom{r-2}{1} + 4\binom{r-2}{0} \right\} < 2r^2 \quad \text{and} \quad \frac{1}{r} + \gamma < 1
\]
yield that for sufficiently large $n$,
\begin{equation}
\label{eq:Delta}
\Delta \leqslant e(U(T)) \cdot n^{2r-4} p^{2\rpt - 4} \cdot \Big( 2 r^2 e(U(T)) p + 2rDnp + \mathbf{1}[r \geqslant 3] \cdot rn \Big).
\end{equation}
\noindent
\textbf{Choosing the right value for $q$.}
Recall the definition of $\zeta$ from \eqref{eq:crgammap}. With foresight, we let
\[
q = \frac{\zeta r^{r-1} \log m}{n^{r-1} p^{\rpt-1}}.
\]
First, let us check that $q \leqslant 1$. Note that by our assumption on $m$ and $T$,
\[
m - e(T) \geqslant (1-\delta)m \geqslant \left(1 - \frac{\varepsilon}{4(1+\varepsilon)}\right) m \geqslant \left(1 - \frac{\varepsilon}{4(1+\varepsilon)}\right)(1+\varepsilon)m_r = \left(1+\frac{3\varepsilon}{4}\right)m_r
\]
and therefore by~\eqref{eq:pr-mr},
\[
p = \frac{m-e(T)}{e(\Pi)} \geqslant \left(1+\frac{3\varepsilon}{4}\right) \frac{m_r}{\left(1-\frac{1}{r}\right)\frac{n^2}{2}} = \left(1+\frac{3\varepsilon}{4}\right) p_r.
\]
It follows that (recalling the definition of $p_r$ from~\eqref{eq:pr-def})
\[
\begin{split}
\frac{n^{r-1}p^{\rpt-1}}{r^{r-1}} & \geqslant \left(1+\frac{3\varepsilon}{4}\right)^{\rpt-1} \left(\frac{n}{r}\right)^{r-1} p_r^{\rpt-1} = \left(1+\frac{3\varepsilon}{4}\right)^{\rpt-1} \left(2-\frac{2}{r+2}\right) \log n \\
& \geqslant \zeta \cdot \left(1+\frac{\varepsilon}{3}\right) \left(2-\frac{2}{r+2}\right) \log n \geqslant \zeta \cdot \log m,
\end{split}
\]
where the second inequality follows from~\eqref{eq:zeta}, and hence $q \leqslant 1$; to see the last inequality, note that if $m \gg m_r$, then we may assume that $\varepsilon = 1$.
With the aim of establishing~\eqref{eq:q-goal}, observe first that, by~\eqref{eq:mu}, \eqref{eq:Delta}, and the inequality $\gamma \leqslant \frac{1}{2r}$,
\begin{equation}
\label{eq:q-Delta-mu}
\begin{split}
\frac{q\Delta}{\mu} & \leqslant (2r^2)^{r-1} \zeta \cdot \log m \cdot \left[ \frac{2r^2e(U(T))}{n^2p} + \frac{2rD}{np} + \frac{\mathbf{1}[r \geqslant 3] \cdot r}{np^2} \right] \\
& \leqslant (2r^2)^r \zeta \cdot \log n \cdot \left[ \frac{2Dn}{m} + o\big(n^{-1/5}\big) \right] \leqslant 2\rho,
\end{split}
\end{equation}
where the second inequality follows from the fact that $n^2p \geqslant m$, see~\eqref{eq:p-order}, and the fact that $e(U(T)) \leqslant Dn$, see~\eqref{eq:m_0}, while the final inequality follows from~\eqref{eq:m_0}. Recall the definitions of $q$ and $\zeta$. It follows from~\eqref{eq:mu} and \eqref{eq:q-Delta-mu} that
\[
\begin{split}
q\mu - q^2\Delta/2 & \geqslant (1-\rho)q\mu \geqslant (1-\rho)q \cdot e(U(T)) \cdot \left( \frac{1}{r} - \gamma \right)^{r-1} n^{r-1} p^{\rpt-1} \\
& = e(U(T)) \cdot (1+\varepsilon) \cdot \log m.
\end{split}
\]
This implies~\eqref{eq:q-goal}, thus completing the proof. \end{proof}
\subsection{The low degree case}
\label{sec:sparse-low-degree-case}
In this section, we handle the low degree case, i.e., we count all graphs $G$ in $\mathcal{F}^*$ with $T(G) \in \TT^L$. Our goal is to prove the following lemma, cf.\ Theorem~\ref{thm:1-statement}.
\begin{lemma}
\label{lemma:low-degree}
If $n$ is sufficiently large, then
\[
\left|\left\{G \in \mathcal{F}^* \colon T(G) \in \TT^L \right\}\right| \leqslant m^{-\varepsilon/4} \cdot \binom{e(\Pi)}{m}.
\] \end{lemma} \begin{proof}
Let us first further partition the family $\TT^L$. For integers $t$ and $u$, let $\TT^L_{t,u}$ be the collection of all $T \in \TT^L$ with $e(T) = t$ and $e(U(T)) = u$ and let
\[
\mathcal{I}_u = \left\{(t^*,x,h) \colon t^* + xD/2 \leqslant u \text{ and } h \leqslant \frac{\varepsilon \xi}{6} \cdot \frac{n \log m}{m} \cdot u \right\}.
\]
Observe that $|\mathcal{I}_u| \leqslant u^3$. It follows from the definition of $\TT^L$, see~\eqref{eq:low-high-sparse}, and~\eqref{eq:UT-lower} that
\[
\TT^L_{t,u} \subseteq \bigcup_{(t^*,x,h) \in \mathcal{I}_u} \mathcal{T}_t(t^*,x,h).
\]
By Lemma~\ref{lemma:T-count}, if $n$ is sufficiently large, then
\begin{equation}
\label{eq:TTL}
\begin{split}
|\TT^L_{t,u}| \cdot \binom{e(\Pi)}{m-t} & \leqslant \sum_{(t^*,x,h) \in \mathcal{I}_u} e^{1/\xi} \cdot m^{t^* + xD/2} \cdot \exp\left(\frac{2mh}{\xi n}\right) \cdot \binom{e(\Pi)}{m} \\
& \leqslant \sum_{(t^*,x,h) \in \mathcal{I}_u} e^{1/\xi} \cdot m^{t^*+xD/2 + \varepsilon u/3} \cdot \binom{e(\Pi)}{m} \\
& \leqslant u^3 \cdot e^{1/\xi} \cdot m^{(1+\varepsilon/3)u} \cdot \binom{e(\Pi)}{m} \leqslant m^{(1+2\varepsilon/3)u} \cdot \binom{e(\Pi)}{m}.
\end{split}
\end{equation}
Furthermore, since clearly $e(U(T)) \geqslant \min\{e(T),D\}$ for all $T$, it follows from~\eqref{eq:TTL}, and Lemma~\ref{lemma:Janson} that
\[
\begin{split}
\sum_{T \in \TT^L} |\mathcal{F}^*(T)| & \leqslant \sum_{t = 1}^{\delta m} \sum_{u = \min\{t,D\}}^t m^{-(1+\varepsilon)u} \cdot |\TT^L_{t,u}| \cdot \binom{e(\Pi)}{m-t} \\
& \leqslant \sum_{t = 1}^{\delta m} \sum_{u = \min\{t,D\}}^t m^{-\varepsilon u / 3} \cdot \binom{e(\Pi)}{m} \leqslant m^{-\varepsilon/4} \cdot \binom{e(\Pi)}{m},
\end{split}
\]
where in the last inequality we used the fact that $D \gg 1$ and hence
\[
\sum_{t = 1}^{\delta m} \sum_{u = \min\{t,D\}}^t m^{-\varepsilon u / 3} \leqslant \sum_{u = 1}^D m^{-\varepsilon u / 3} + (\delta m)^2 \cdot m^{-\varepsilon D / 3} \ll m^{-\varepsilon / 4}.
\]
This completes the proof in the low degree case. \end{proof}
\subsection{The high degree case}
\label{sec:sparse-high-degree-case}
Recall the definition of $\TT^H$, see~(\ref{eq:low-high-sparse}). In this section, we shall enumerate graphs in the family $\mathcal{F}^H$ defined by \[ \mathcal{F}^H = \{G \in \mathcal{F}^* \colon T(G) \in \TT^H\} = \bigcup_{T \in \TT^H} \mathcal{F}^*(T). \] Our goal will be proving the following lemma, which together with Lemma~\ref{lemma:low-degree} readily implies Theorem~\ref{thm:1-statement}.
\begin{lemma}
\label{lemma:high-degree}
If $n$ is sufficiently large, then
\[
\left|\mathcal{F}^H\right| = \left|\left\{G \in \mathcal{F}^* \colon T(G) \not\in \TT^L \right\}\right| \leqslant 3\exp\left(-\frac{m}{n}\right) \cdot \binom{e(\Pi)}{m}.
\] \end{lemma}
We start with the following observation. Fix a $T \in \TT^H$ and suppose that for some $i \in [r]$, a vertex $v \in V_i$ satisfies $\deg_T(v) \geqslant \beta m/n$. Since every graph $G \in \mathcal{F}^*(T)$ satisfies \[ \deg_G(v,V_j) \geqslant \deg_G(v,V_i) = \deg_T(v) \geqslant \beta m/n \quad \text{for every $j \in [r]$}, \] see~\eqref{eq:Pi-unfriendly}, and is $K_{r+1}$-free, no matter how we choose the edges of $G$ that are incident to $v$, there will be at least $(\beta m / n)^r$ copies of $K_r$ (those induced by one vertex from each $N_G(v) \cap V_j$ with $j \in [r]$) that cannot be fully contained in the graph $G \cap \Pi$.
Given an arbitrary vertex $v$, assuming that its neighbors in $G$ have already been chosen, let $\mathcal{H}_v^*$ be the collection of all $\prod_{j=1}^r \deg_G(v,V_j)$ such forbidden copies of~$K_r$, that is, let \[ \mathcal{H}_v^* = (N_G(v) \cap V_1) \times \ldots \times (N_G(v) \cap V_r). \] We furthermore let \begin{equation}
\label{eq:Ds}
D^* = \frac{\beta m}{2n}. \end{equation} Recall the definitions from Section~\ref{sec:counting-graphs}. Fix some $t$, $t^*$, $x$, and $h$, pick an arbitrary $T \in \mathcal{T}_t'(t^*,x,h)$, and let \[ b = \left\lceil \frac{h}{2r} \right\rceil. \] Let us stress the fact that we select $T$ from $\mathcal{T}_t'(t^*,x,h)$ and not from $\mathcal{T}_t(t^*,x,h)$, which means that $T$ contains exactly (and not at most) $h$ vertices with degree exceeding $\beta m / n$.
\begin{claim}
\label{claim:H'}
There is an $i \in [r]$ and a set $H' \subseteq H(T) \cap V_i$ of $b$ vertices such that
\begin{equation}
\label{eq:H'}
\deg_T(v, V_i \setminus H') \geqslant D^* \quad \text{for every $v \in H'$}.
\end{equation} \end{claim} \begin{proof}
Since $T$ has $h$ vertices with degree at least $\beta m / n$, some $V_i$ contains at least $h / r$ of them. This set $V_i$ can be partitioned into two sets $V_i'$ and $V_i''$ in such a way that $\deg_T(v, V_i'') \geqslant \deg_T(v, V_i')$ for each $v \in V_i'$ and, vice versa, $\deg_T(v, V_i') \geqslant \deg_T(v, V_i'')$ for each $v \in V_i''$. For example, one may consider a maximum cut in $T[V_i]$. One of these two parts, $V_i'$ or $V_i''$, contains at least $h / 2r$ vertices with degree at least $\beta m / n$ in $T$. We let $H'$ be an arbitrary $b$-element subset of such a set. It is easily checked that $H'$ satisfies~\eqref{eq:H'}. \end{proof}
For every $T \in \mathcal{T}_t'(t^*,x,h)$, we choose some arbitrary set $H'$ as in Claim~\ref{claim:H'}. Next, given a graph $G \in \mathcal{F}^*(T)$, for every $v \in H'$ and each $j \in [r]$, let $W_j(v)$ be a canonically chosen $D^*$-element subset of $N_G(v) \cap (V_j \setminus H')$. Given such $G$, consider the $r$-uniform hypergraph $\mathcal{H}'$ defined by \[ \mathcal{H}' = \bigcup_{v \in H'} W_1(v) \times \ldots \times W_r(v). \] Note that $\mathcal{H}' \subseteq \bigcup_{v \in H'} \mathcal{H}_v^*$, that is, every edge ($r$-tuple) in $\mathcal{H}'$ represents a copy of $K_r$ that is forbidden to appear in $G$. We will enumerate graphs in $\mathcal{F}^H$ using two different methods, depending on the number and the distribution of edges in the hypergraph $\mathcal{H}'= \mathcal{H}'(G)$. Before we make this precise, we need a few more definitions.
Given an arbitrary $\mathcal{H} \subseteq V_1 \times \ldots \times V_r$, an $I \subseteq [r]$, and an $L \in \prod_{j \in I} V_j$, we define the degree of $L$ in $\mathcal{H}$, denoted $\deg_\mathcal{H}(L)$, by \[
\deg_\mathcal{H}(L) = |\{ K \in \mathcal{H} \colon L \subseteq K \}|. \] For $s \in [r]$, the maximum $s$-degree $\Delta_s(\mathcal{H})$ of $\mathcal{H}$ is defined by \[
\Delta_s(\mathcal{H}) = \max \Big\{ \deg_\mathcal{H}(L) \colon L \in \prod_{j \in I} V_j \text{ for some $I \subseteq [r]$ with $|I| = s$}\Big\}. \] We will measure the uniformity of the distribution of the edges of $\mathcal{H}$ in terms of these maximum $s$-degrees. First, let us fix several additional parameters. Let $C_1$ be a constant satisfying \begin{equation}
\label{eq:C1}
\frac{3\beta C_1}{2r} \geqslant \frac{6}{\varepsilon \xi} + \frac{4}{\xi} + 3. \end{equation} Next, let \begin{equation}
\label{eq:lambda-alpha}
\lambda = \frac{1}{2^{r+1}} \qquad \text{and} \qquad \alpha = \exp(-6C_1-1) \end{equation} and let $\tau$ be a small positive constant such that Lemma~\ref{lemma:d-sets} holds with $\tau$ and with $\alpha$, $\lambda$, and each $k \in \{2, \ldots, r\}$, i.e., \begin{equation}
\label{eq:tau}
\tau = \min_{2 \leqslant k \leqslant r} \tau_{\ref{lemma:d-sets}}(k, \alpha, \lambda). \end{equation} Finally, \begin{equation}
\label{eq:C2-sigma}
C_2 = \frac{(2r)^r}{\tau} \quad \text{and} \quad \sigma = \frac{\tau}{(2r)^r}. \end{equation}
We are finally ready to partition the family $\mathcal{F}^H$ into the regular and irregular cases, according to the edge distribution of the hypergraphs $\mathcal{H}'$. First, we let $\mathcal{F}^R_1$ be the family of all $G \in \mathcal{F}^H$ such that $e(\mathcal{H}') \geqslant \sigma n^r$. Second, we let \begin{equation}
\label{eq:c2}
c_2 = \frac{\beta^r}{2^{r+3}r} \end{equation} and define $\mathcal{F}^R_2$ to be the family of all $G \in \mathcal{F}^H \setminus \mathcal{F}^R_1$ such that $\mathcal{H}'$ contains a subhypergraph $\mathcal{H}$ satisfying \begin{equation}
\label{eq:FF2-eHH}
e(\mathcal{H}) \geqslant c_2 \cdot |H(T(G))| \cdot \left(\frac{m}{n}\right)^r = |H(T(G))| \cdot \frac{(D^*)^r}{8r} \end{equation} and \begin{equation}
\label{eq:FF2-DeltaHH}
\Delta_s(\mathcal{H}) \leqslant \max \left\{ \left(\frac{m}{n}\right)^{r-s} , C_2 \cdot \frac{e(\mathcal{H})}{n^s} \right\} \quad \text{for every $s \in \{2, \ldots, r-1\}$}. \end{equation} Finally, we let $\mathcal{F}^I = \mathcal{F}^H \setminus (\mathcal{F}^R_1 \cup \mathcal{F}^R_2)$. Counting of graphs in $\mathcal{F}^R_1 \cup \mathcal{F}^R_2$ and $\mathcal{F}^I$ will be referred to as the regular and irregular cases, respectively. In the next two sections, we will prove the following estimates, which readily imply Lemma~\ref{lemma:high-degree}.
\begin{lemma}
\label{lemma:regular-case}
If $n$ is sufficiently large, then
\[
|\mathcal{F}^R_1| \leqslant \exp\left(-\frac{\sigma m}{2^{r+3}}\right) \cdot \binom{e(\Pi)}{m} \qquad \text{and} \qquad |\mathcal{F}^R_2| \leqslant \exp\left(-\frac{m}{n}\right) \cdot \binom{e(\Pi)}{m}.
\] \end{lemma}
\begin{lemma}
\label{lemma:irregular-case}
If $n$ is sufficiently large, then
\[
|\mathcal{F}^I| \leqslant \exp\left(-\frac{m}{n}\right) \cdot \binom{e(\Pi)}{m}.
\] \end{lemma}
\subsection{The regular case}
\label{sec:regular-case}
In this section, we bound the number of graphs that fall into the regular case, that is, we prove Lemma~\ref{lemma:regular-case}. Our main tool will be the following two lemmas that provide upper bounds on the number of subgraphs of $\Pi$ that do not fully contain any member of a collection of forbidden copies of $K_r$ which is either very large (Lemma~\ref{lemma:good-H-basic}) or whose members are somewhat uniformly distributed (Lemma~\ref{lemma:good-H}). The proof of both of these lemmas is another application of the Hypergeometric Janson Inequality (Lemma~\ref{lemma:HJI}).
\begin{lemma}
\label{lemma:good-H-basic}
Suppose that $\mathcal{H} \subseteq V_1 \times \ldots \times V_r$ satisfies $e(\mathcal{H}) \geqslant \sigma n^r$. Then for every $m'$ with $m/2 \leqslant m' \leqslant m$, the number of subgraphs of $\Pi$ with $m'$ edges that do not fully contain a copy of $K_r$ whose vertex set is an edge of $\mathcal{H}$ is at most
\[
2 \cdot \exp\left( - \frac{\sigma}{2^{r+1}} \cdot m\right) \cdot \binom{e(\Pi)}{m'}.
\] \end{lemma}
\begin{lemma}
\label{lemma:good-H}
There exists a positive $c$ such that the following holds. Suppose that $\mathcal{H} \subseteq V_1 \times \ldots \times V_r$ satisfies $e(\mathcal{H}) \geqslant B(m/n)^r$ for some $B$ and~\eqref{eq:FF2-DeltaHH} holds, that is,
\[
\Delta_s(\mathcal{H}) \leqslant \max\left\{\left(\frac{m}{n}\right)^{r-s}, C_2 \cdot \frac{e(\mathcal{H})}{n^s}\right\} \quad \text{for every $s \in \{2, \ldots, r-1\}$}.
\]
Then, for every $m'$ with $m/2 \leqslant m' \leqslant m$, the number of subgraphs of $\Pi$ with $m'$ edges that do not fully contain a copy of $K_r$ whose vertex set is an edge of $\mathcal{H}$ is at most
\[
2 \cdot \exp\left(-\min\left\{\frac{B\log n}{n}, 1\right\} \cdot c m\right) \cdot \binom{e(\Pi)}{m'}.
\] \end{lemma} \begin{proof}[Proof of Lemmas~\ref{lemma:good-H-basic} and~\ref{lemma:good-H}]
We use the Hypergeometric Janson Inequality to count graphs satisfying our constraint. Denote the number of them by $N$. Let $\mathcal{K}$ be the collection of (the edge sets of) all copies of $K_r$ whose vertex set belongs to $\mathcal{H}$, let $p = \frac{m'}{e(\Pi)}$, and let
\[
\mu = \sum_{K \in \mathcal{K}} p^{|K|} \qquad \text{and} \qquad \Delta = \sum_{K_1 \sim K_2} p^{|K_1 \cup K_2|},
\]
where the second sum above is over all ordered pairs $(K_1, K_2) \in \mathcal{K}^2$ such that $K_1$ and $K_2$ share at least one edge but $K_1 \neq K_2$. By the Hypergeometric Janson Inequality, letting $q = \min\{1, \frac{\mu}{\Delta}\}$, we have
\[
N \leqslant 2 \cdot \exp\left(-\min\left\{\frac{\mu}{2}, \frac{\mu^2}{2\Delta}\right\}\right) \cdot \binom{e(\Pi)}{m'}.
\]
Hence, in order to establish Lemma~\ref{lemma:good-H-basic}, it suffices to show that if $e(\mathcal{H}) \geqslant \sigma n^r$, then
\begin{equation}
\label{eq:good-H-basic-goal}
\min\left\{\mu, \frac{\mu^2}{\Delta}\right\} \geqslant \frac{\sigma}{2^r} \cdot m
\end{equation}
and in order to establish Lemma~\ref{lemma:good-H}, it suffices to show that under appropriate assumptions on $\mathcal{H}$, there exists a positive $c$ that depends only on $r$ and $C_2$ such that
\begin{equation}
\label{eq:good-H-goal}
\min\left\{\mu, \frac{\mu^2}{\Delta}\right\} \geqslant 2 \cdot \min\left\{ \frac{B\log n}{n}, 1 \right\} \cdot c m.
\end{equation}
To this end, observe that
\[
\mu = e(\mathcal{H}) \cdot p^{\rt} \qquad \text{and} \qquad \Delta \leqslant e(\mathcal{H}) \cdot \sum_{s=2}^{r-1} \binom{r}{s} \Delta_s(\mathcal{H}) p^{2\rt - \st},
\]
where the $s$th term of the sum in the upper bound on $\Delta$ estimates the contribution of pairs $K_1 \sim K_2$ with $|V(K_1) \cap V(K_2)| = s$. It follows from our assumptions, see~\eqref{eq:ePi-lower}, that
\begin{equation}
\label{eq:p-mr-n}
\frac{8m}{n^2} \geqslant \frac{m}{e(\Pi)} \geqslant p = \frac{m'}{e(\Pi)} \geqslant \frac{m}{2e(\Pi)} \geqslant \frac{m_r}{2e(\Pi)} \geqslant \frac{m_r}{\left(1-\frac{1}{r}\right)n^2} \gg n^{-\frac{2}{r+2}}
\end{equation}
and hence for every $s \in \{2, \ldots, r\}$,
\begin{equation}
\label{eq:good-H-basic-np}
n^s p^{\st} \geqslant n^2 p = n^2 \cdot \frac{m'}{e(\Pi)} \geqslant n^2 \cdot \frac{m}{2e(\Pi)} \geqslant m,
\end{equation}
which implies, in particular, that under the assumptions of Lemma~\ref{lemma:good-H-basic}, we have $\mu \geqslant \sigma m$. Moreover, it follows from \eqref{eq:p-mr-n} that for every $s \in \{2, \ldots, r\}$, recalling \eqref{eq:pr-mr},
\begin{equation}
\label{eq:nspspt}
\begin{split}
\left(\frac{m}{n}\right)^{s-1} p^{\st} & \geqslant \left(\frac{n}{8}\right)^{s-1} p^{\st+s-1} = \left(\frac{n}{8}\right)^{s-1} p^{\spt - 1} \geqslant \left(\frac{n}{8}\right)^{s-1} \cdot \left(\frac{m_r}{\left(1-\frac{1}{r}\right)n^2}\right)^{\spt-1} \\
& = \left(\frac{n}{8}\right)^{s-1} \cdot \left(\frac{p_r}{2}\right)^{\spt-1} \geqslant \frac{r^{r-1}}{4^{r^2}} \cdot \log n.
\end{split}
\end{equation}
To see the last inequality, note that if $s = r$, then it follows immediately from~\eqref{eq:pr-def}. On the other hand, if $2 \leqslant s < r$, then actually
\[
n^{s-1} p_r^{\spt - 1} \gg n^{s-1} \left(n^{-\frac{2}{r+2}}\right)^{\spt - 1} = n^{(s-1)\left(1 - \frac{s+2}{r+2}\right)} \gg \log n.
\]
One now easily deduces from \eqref{eq:nspspt} that under the assumptions of Lemma~\ref{lemma:good-H},
\begin{equation}
\label{eq:good-H-mu}
\mu \geqslant B \left(\frac{m}{n}\right)^{r} p^{\rt} \geqslant \frac{B \log n}{n} \cdot \frac{r^{r-1}}{4^{r^2}} \cdot m.
\end{equation}
We now turn to estimating $\mu^2/\Delta$. In the context of Lemma~\ref{lemma:good-H-basic}, we simply use the trivial bound $\Delta_s(\mathcal{H}) \leqslant n^{r-s}$ and deduce that for each $s \in \{2, \ldots, r-1\}$,
\[
\Delta_s(\mathcal{H}) p^{-\st} \leqslant n^{r-s} p^{-\st} \leqslant \frac{n^r}{m},
\]
where the last inequality follows from~\eqref{eq:good-H-basic-np}. It follows that
\[
\Delta \leqslant 2^r \cdot p^{2 \rt} \cdot \frac{n^r}{m} \cdot e(\mathcal{H})
\]
and hence
\[
\frac{\mu^2}{\Delta} \geqslant \frac{1}{2^r} \cdot e(\mathcal{H}) \cdot \frac{m}{n^r} \geqslant \frac{\sigma}{2^r} \cdot m,
\]
which implies~\eqref{eq:good-H-basic-goal}, as we have already seen that $\mu \geqslant \sigma m$. In the context of Lemma~\ref{lemma:good-H}, it follows from~\eqref{eq:good-H-basic-np} and~\eqref{eq:nspspt} that for each $s \in \{2, \ldots, r-1\}$,
\[
\Delta_s(\mathcal{H})p^{-\st} \leqslant \max\left\{ \frac{(m/n)^{r-1}}{(m/n)^{s-1}p^{\st}}, C_2 \cdot \frac{e(\mathcal{H})}{n^sp^{\st}} \right\} \leqslant \max\left\{ \frac{4^{r^2}}{r^{r-1}} \cdot \frac{(m/n)^{r-1}}{\log n}, C_2 \cdot \frac{e(\mathcal{H})}{m} \right\}
\]
and therefore,
\[
\frac{\mu^2}{\Delta} \geqslant \frac{1}{2^r} \cdot \min\left\{ \frac{r^{r-1}}{4^{r^2}} \cdot \frac{e(\mathcal{H}) \log n}{(m/n)^{r-1}}, \frac{m}{C_2} \right\} \geqslant \min\left\{ \frac{B \log n}{n}, 1 \right\} \cdot \min\left\{\frac{r^{r-1}}{4^{r^2+r}}, \frac{1}{2^r C_2} \right\} \cdot m,
\]
which, together with~\eqref{eq:good-H-mu}, implies~\eqref{eq:good-H-goal}, completing the proof. \end{proof}
\begin{proof}[{Proof of Lemma~\ref{lemma:regular-case}}]
Recall the definitions of $\mathcal{F}^R_1$ and $\mathcal{F}^R_2$ from Section~\ref{sec:sparse-high-degree-case}. We first show that the family $\mathcal{F}^R_1$ is small. In order to construct a graph $G \in \mathcal{F}^R_1$, we first choose $h$ and $t$ and restrict our attention to graphs $G$ satisfying $t = e(T(G))$ and $h = |H(T(G))|$. Clearly for each such $G$,
\begin{equation}
\label{eq:h-bound}
h \cdot \frac{\beta m}{n} \leqslant \sum_v \deg_{T(G)}(v) = 2t \leqslant 2\delta m.
\end{equation}
Then, we choose the set $H'$ of $b$ vertices from some $V_i$, see Claim~\ref{claim:H'}, and for each $v \in H'$, we choose the sets $W_1(v), \ldots, W_r(v)$ of size $D^*$ each. After these are fixed, we choose the remaining $t' = t - bD^*$ edges of $T(G)$ and the remaining $m - t' - brD^*$ (that is, $m - t - b(r-1)D^*$) edges of $G \cap \Pi$ in such a way that $G \cap \Pi$ contains no copy of $K_r$ whose vertex set is an edge of the hypergraph $\mathcal{H}'$ (defined in the previous section). The main point is that the assumption that $G \in \mathcal{F}^R_1$ means that $e(\mathcal{H}') \geqslant \sigma n^r$ and hence we may use Lemma~\ref{lemma:good-H-basic} to bound the number of choices for $G \cap \Pi$.
The number $Z_1$ of ways to choose the sets $W_1(v) \subseteq V_1, \ldots, W_r(v) \subseteq V_r$ for each $v$ satisfies
\begin{equation}
\label{eq:Z1}
Z_1 \leqslant \prod_{j = 1}^r \binom{|V_j|}{D^*} \leqslant \binom{|V_1| + \ldots + |V_r|}{r \cdot D^*} = \binom{n}{rD^*}.
\end{equation}
It now follows from the definition of $\mathcal{F}^R_1$ and Lemma~\ref{lemma:good-H-basic} that
\[
|\mathcal{F}^R_1| \leqslant \sum_{t, h} \binom{n}{b} \cdot \binom{n}{rD^*}^b \cdot |\mathcal{T}_{t'}| \cdot 2 \cdot \exp\left(-\frac{\sigma}{2^{r+1}} \cdot m\right) \cdot \binom{e(\Pi)}{m-t'-brD^*}.
\]
A computation along the lines of the proof of Lemmas~\ref{lemma:T-count-basic} and~\ref{lemma:T-count}, see~\eqref{eq:T-count-step-ratio} and~\eqref{eq:T-count-step-high-deg}, shows that
\begin{equation}
\label{eq:comp-T-count}
\begin{split}
2\binom{n}{b} \binom{n}{rD^*}^b \binom{e(\Pi)}{m-t'-rbD^*} & \leqslant 2 \binom{n}{b} \left(\frac{em}{\xi n r D^*}\right)^{r D^* b} \cdot \binom{e(\Pi)}{m-t'} \\
& \leqslant \exp\left(\frac{2mb}{\xi n}\right) \cdot \binom{e(\Pi)}{m-t'}.
\end{split}
\end{equation}
To see the last inequality, recall that the value of the function $x \mapsto (a/x)^x$ is maximized when $x = a/e$, which implies that $(\frac{em}{\xi n r D^*})^{rD^* b} \leqslant \exp(\frac{mb}{\xi n})$.
Hence, by Lemma~\ref{lemma:T-count-basic}, since $b \leqslant h$,
\begin{equation}
\label{eq:FF1-second}
|\mathcal{F}^R_1| \leqslant \sum_{t, h} \exp\left(\frac{2mh}{\xi n} - \frac{\sigma m}{2^{r+1}}\right) \cdot \left(\frac{e}{\xi \delta}\right)^{\delta m} \cdot \binom{e(\Pi)}{m}.
\end{equation}
Since
\[
\left(\frac{e}{\xi \delta}\right)^\delta \leqslant \exp\left(\frac{\sigma}{2^{r+2}}\right) \qquad \text{and} \qquad \frac{4\delta}{\beta\xi} \leqslant \frac{\sigma}{2^{r+4}}
\]
for sufficiently small $\delta$, continuing~\eqref{eq:FF1-second}, we have, by~\eqref{eq:h-bound},
\[
|\mathcal{F}^R_1| \leqslant m^2 \exp\left(\frac{4\delta m}{\beta \xi} - \frac{\sigma m}{2^{r+1}} + \frac{\sigma m}{2^{r+2}} \right) \cdot \binom{e(\Pi)}{m} \leqslant \exp\left(-\frac{\sigma m}{2^{r+3}}\right) \cdot \binom{e(\Pi)}{m}.
\]
In order to complete the proof, we still need to show that the family $\mathcal{F}^R_2$ is also small. We count the graphs in $\mathcal{F}^R_2$ almost the same way as we counted the graphs in $\mathcal{F}^R_1$. That is, we first choose $h$ and $t$ and consider only graphs $G$ with $t = e(T(G))$ and $h = |H(T(G))|$. Then, with $t$ and $h$ fixed, we choose the set $H'$ of $b$ vertices from some $V_i$ and for each $v \in H'$, we select the sets $W_1(v), \ldots, W_r(v)$. Finally, we choose the remaining $t' = t - bD^*$ edges of $T(G)$ and the remaining $m - t' - brD^*$ edges of $G \cap \Pi$. The assumption that $G \in \mathcal{F}^R_2$ means that we may use Lemma~\ref{lemma:good-H} with $B = c_2h$ to bound the number of choices for $G \cap \Pi$.
The main difference in our treatment of $\mathcal{F}^R_2$, compared to the argument for $\mathcal{F}^R_1$ given above, is that we now use a stronger bound on the number of choices of the $t'$ edges of $T(G)$ that are selected in the second stage of the above procedure. To this end, we fix some $x$ and $t^*$ and further restrict our attention to graphs $G$ that satisfy $x = |X(T(G))|$ and $t^* = e(T(G) - X(T(G)))$. In particular, we are only counting graphs $G \in \mathcal{F}^R_2$ that satisfy $T(G) \in \mathcal{T}'_t(t^*, x, h)$. Let $T'$ be the graph consisting of the $t'$ edges of $T(G)$ that we choose after selecting the $bD^*$ edges of $T(G)$ that we fixed while we were choosing $W_i(v)$ for all $v \in H'$. Since $T(G) \in \mathcal{T}'_t(t^*,x,h)$ and all edges of $T(G) \setminus T'$ have an endpoint in $H'$, it is not hard to see that $T'$ must be in $\mathcal{T}_{t'}(t^*, x, h)$. Indeed, the set $H(T(G))$ contains all vertices whose degree in $T(G)$ exceeds $\beta m / n$ (and hence also all vertices whose degree in $T'$ exceeds $\beta m / n$), and we may obtain $T'$ from $T(G)$ by deleting only edges incident to $H' \subseteq H(T(G)) \subseteq X(T(G))$, which means that the number of edges that have no endpoints in the set $X(T(G))$ is $t^*$ in both $T(G)$ and $T'$. With this additional information about $T'$, we may now appeal to Lemma~\ref{lemma:T-count} in place of Lemma~\ref{lemma:T-count-basic} in order to get a stronger bound on the number of choices for $T'$. It now follows (cf.~the calculation leading up to~\eqref{eq:FF1-second}) from the definition of $\mathcal{F}^R_2$ and Lemma~\ref{lemma:good-H} with $B = c_2 h$ that, letting $c$ be the constant from the statement of Lemma~\ref{lemma:good-H},
\begin{equation}
\label{eq:FF2}
|\mathcal{F}^R_2| \leqslant \sum_{t,t^*,x,h} \exp\left( \frac{2mb}{\xi n} \right) \cdot |\mathcal{T}_{t'}(t^*, x, h)| \cdot \exp\left( - \min\left\{\frac{c_2 h \log n}{n}, 1\right\} \cdot c m\right) \cdot \binom{e(\Pi)}{m-t'}.
\end{equation}
Let $F_2(t,t^*,x,h)$ denote the term in the sum in the right hand side of~\eqref{eq:FF2}. If $c_2 h \log n \geqslant n$, then we use the fact that $\mathcal{T}_{t'}(t^*,x,h) \subseteq \mathcal{T}_{t'}$ and $t' \leqslant t \leqslant \delta m$ and, using Lemma~\ref{lemma:T-count-basic}, we further estimate $F_2(t,t^*,x,h)$ as follows (recall that $b \leqslant h$):
\[
\begin{split}
F_2(t,t^*,x,h) & \leqslant \exp\left( \frac{2mh}{\xi n} - c m \right) \cdot \left(\frac{e}{\xi \delta}\right)^{\delta m} \cdot \binom{e(\Pi)}{m} \\
& \leqslant \exp\left( \frac{4\delta m}{\beta \xi} - c m \right) \cdot \left(\frac{e}{\xi \delta}\right)^{\delta m} \cdot \binom{e(\Pi)}{m} \leqslant \exp\left( -\frac{c m}{4} \right) \cdot \binom{e(\Pi)}{m},
\end{split}
\]
where we used \eqref{eq:h-bound} and the fact that
\[
\left( \frac{e}{\xi\delta} \right)^\delta < e^{\frac{c}{2}} \qquad \text{and} \qquad \frac{4\delta}{\beta\xi} < \frac{c}{4},
\]
provided that $\delta$ is sufficiently small. Now, we recall from the definition of $\TT^H$, see~\eqref{eq:UT-lower} and \eqref{eq:low-high-sparse}, that since $T(G) \in \TT^H$ and we have $x = |X(T(G))|$ and $t^* = e(T(G) - X(T(G)))$, then
\begin{equation}
\label{eq:h}
h = |H(T(G))| > \frac{\varepsilon \xi}{6} \cdot \frac{n \log m}{m} \cdot (t^* + xD/2).
\end{equation}
Hence, if $c_2 h \log n < n$, then by Lemma~\ref{lemma:T-count},
\[
\begin{split}
F_2(t,t^*,x,h) & \leqslant e^{1/\xi} \cdot m^{t^* + xD/2} \cdot \exp\left( \frac{m(b+h)}{n} \left( \frac{2}{\xi} - c_2c \log n \right) \right) \cdot \binom{e(\Pi)}{m} \\
& \leqslant e^{1/\xi} \cdot \exp\left( \frac{mh}{n} \cdot \left(\frac{4}{\xi} + \frac{6}{\varepsilon \xi} - c_2 c \log n\right) \right) \cdot \binom{e(\Pi)}{m} \leqslant \exp\left(- \frac{2m}{n}\right) \cdot \binom{e(\Pi)}{m},
\end{split}
\]
where in the second inequality, we used~\eqref{eq:h} and the fact that $b \leqslant h$, and in the last inequality, we used the facts that $h \geqslant 1$, which follows from \eqref{eq:h} as $h$ is an integer, and that $n$ is sufficiently large. It follows that
\[
|\mathcal{F}^R_2| \leqslant m^2n^2 \cdot \max\left\{ \exp\left(- \frac{cm}{4}\right) , \exp\left(-\frac{2m}{n}\right) \right\} \cdot \binom{e(\Pi)}{m} \leqslant \exp\left( - \frac{m}{n} \right) \cdot \binom{e(\Pi)}{m},
\]
provided that $n$ is sufficiently large. This completes the proof in the regular case. \end{proof}
\subsection{The irregular case}
\label{sec:irregular-case}
In this section, we prove Lemma~\ref{lemma:irregular-case}. In other words, we count those graphs in $\mathcal{F}^H$ for which the hypergraph $\mathcal{H}'$ of forbidden copies of $K_r$ defined in Section~\ref{sec:sparse-high-degree-case} contains fewer than $\sigma n^r$ edges and does not contain any subhypergraph $\mathcal{H}$ that satisfies \eqref{eq:FF2-eHH} and \eqref{eq:FF2-DeltaHH}. This is the core of the proof of Theorem~\ref{thm:1-statement}, which makes this section the key section of the paper.
We will describe a procedure that, given a $G \in \mathcal{F}^H \setminus \mathcal{F}^R_1$, constructs some canonical hypergraph $\mathcal{H} \subseteq \mathcal{H}'$ by examining the vertices in $H'(T(G))$ and their neighborhoods one by one. By not adding certain $r$-tuples of $\mathcal{H}'$ to the constructed hypergraph, our procedure forces $\mathcal{H}$ to satisfy the maximum degree constraints given in~\eqref{eq:FF2-DeltaHH}. For a vast majority of graphs $G \in \mathcal{F}^H \setminus \mathcal{F}^R_1$, the hypergraph $\mathcal{H}$ will have many edges (i.e., it will satisfy~\eqref{eq:FF2-eHH}), implying that $G \in \mathcal{F}^R_2$. The procedure fails to output a hypergraph with many edges only when the intersections of the neighborhoods of different vertices in $H'(T(G))$ are very far from random-like. Using Lemma~\ref{lemma:d-sets}, we will obtain a bound on the number of graphs with such an atypical distribution of neighborhoods of the vertices in $H'(T(G))$. Since by definition, our procedure has to fail on every graph in $\mathcal{F}^I$, the obtained bound is also an upper bound on $|\mathcal{F}^I|$.
\begin{proof}[{Proof of Lemma~\ref{lemma:irregular-case}}]
Fix some graph $G \in \mathcal{F}^H \setminus \mathcal{F}^R_1$ and recall the definitions of $D^*$, $H'(T(G))$, and $W_j(v)$ from Section~\ref{sec:sparse-high-degree-case}. Suppose that $H' = H'(T(G)) = \{v_1, \ldots, v_b\}$, where $v_1 < \ldots < v_b$ (we assume that the vertex set of $G$ is labeled with $\{1, \ldots, n\}$). We now describe the aforementioned procedure which constructs a hypergraph $\mathcal{H} \subseteq \mathcal{H}'$.
\begin{constHH}
Let $\mathcal{H}_0 \subseteq V_1 \times \ldots \times V_r$ be the empty hypergraph. For every $\ell = 1, \ldots, b$, do the following:
\begin{enumerate}[(1)]
\item
For every $j \in [r]$, let $W_j = W_j(v_\ell)$.
\item
For every $I \subseteq [r]$ with $2 \leqslant |I| \leqslant r-1$, let
\[
M_I = \left\{ T \in \prod_{j \in I} V_j \colon \deg_{\mathcal{H}_{\ell-1}}(T) > \frac{C_2}{2} \cdot \frac{e(\mathcal{H}_{\ell - 1})}{n^{|I|}} \right\}
\]
and let $M_{[r]} = \mathcal{H}_{\ell - 1}$.
\item
Let
\[
\mathcal{H}_\ell = \mathcal{H}_{\ell - 1} \cup \left\{K \in W_1 \times \ldots \times W_r \colon K \nsupseteq T \text{ for all $T \in \bigcup_I M_I$}\right\}.
\]
\end{enumerate}
Finally, let $\mathcal{H} = \mathcal{H}_b$.
\end{constHH}
Since $|W_j(v)| = D^*$ for every $j \in [r]$ and $v \in H'$, then for every $s \in \{2, \ldots, r-1\}$,
\[
\Delta_s(\mathcal{H}) \leqslant \frac{C_2}{2} \cdot \frac{e(\mathcal{H})}{n^s} + (D^*)^{r-s} \leqslant \max\left\{C_2 \cdot \frac{e(\mathcal{H})}{n^s}, 2(D^*)^{r-s}\right\} \leqslant \max\left\{C_2 \cdot \frac{e(\mathcal{H})}{n^s}, \left(\frac{m}{n}\right)^{r-s}\right\},
\]
that is, $\mathcal{H}$ satisfies~\eqref{eq:FF2-DeltaHH}. Recall from~\eqref{eq:lambda-alpha} that $\lambda = 2^{-r-1}$. We say that the vertex $v_\ell$ is \emph{useful} if in the $\ell$th iteration of the above algorithm, we have
\[
\Big| M_I \cap \prod_{j \in I} W_j \Big| \leqslant \lambda (D^*)^{|I|} \quad \text{for all $I$ with $2 \leqslant |I| \leqslant r$}.
\]
Note that if $v_\ell$ is useful, then
\[
e(\mathcal{H}_\ell) - e(\mathcal{H}_{\ell - 1}) \geqslant (D^*)^r - \sum_{I \subseteq [r]} \lambda (D^*)^{|I|} \cdot (D^*)^{r - |I|} = (1-2^r\lambda)(D^*)^r = \frac{(D^*)^r}{2}.
\]
Therefore, if at least half of the vertices of $H'$ are useful, then (recall the definitions of $D^*$ and $c_2$ from~\eqref{eq:Ds} and~\eqref{eq:c2})
\[
e(\mathcal{H}) \geqslant \frac{b}{2} \cdot \frac{(D^*)^r}{2} \geqslant \frac{h(D^*)^r}{8r} = \frac{\beta^rh}{2^{r+3}r} \cdot \left(\frac{m}{n}\right)^r = c_2 h \left(\frac{m}{n}\right)^r,
\]
that is, $\mathcal{H}$ satisfies~\eqref{eq:FF2-eHH}. Hence, if for some $G \in \mathcal{F}^H \setminus \mathcal{F}^R_1$, the above procedure encounters at least $b/2$ useful vertices, then $G \in \mathcal{F}^R_2$. This implies that for every graph $G \in \mathcal{F}^I$, the above procedure encounters fewer than $b/2$ useful vertices. We now enumerate all graphs with this property.
As before, we first fix $h$ and $t$ and consider only graphs $G$ with $t = e(T(G))$ and $h = |H(T(G))|$. We then choose the $b$ vertices that form the set $H'$ and specify in advance which (at least $b/2$) of them our procedure will mark as not useful. Next, we choose the sets $W_1(v), \ldots, W_r(v)$ in turn for every $v \in H'$, from the one with the smallest label to the one with the largest label (as in the procedure constructing $\mathcal{H}$). The main point is that for the vertices $v$ that are not useful, we choose the sets $W_i(v)$ in such a way that our procedure will deem them not useful and this severely limits the number of choices for these sets. This is the only stage of the enumeration where we provide a nontrivial upper bound. Finally, we choose the remaining $t' = t - bD^*$ edges of $T(G)$ and the remaining $m - t' - brD^*$ edges of $G \cap \Pi$.
Let us elaborate on the only non-trivial stage of the enumeration described above. We choose the sets $W_1(v), \ldots, W_r(v)$ for vertices $v \in H'$ one by one, following the same order as in the procedure constructing $\mathcal{H}$. More precisely, suppose that $H' = \{v_1, \ldots, v_b\}$, where $v_1 < \ldots < v_b$, fix some $\ell \in [b]$, and assume that $W_j(v_k)$ have already been chosen for all $j \in [r]$ and $k \in [\ell - 1]$. This means, in particular, that the hypergraph $\mathcal{H}_{\ell-1}$ and the sets $M_I$ in the $\ell$th iteration of our procedure are already determined for all $I \subseteq [r]$. Clearly, there are at most $\binom{n}{rD^*}$ ways to choose $W_1(v_\ell) \subseteq V_1, \ldots, W_r(v_\ell) \subseteq V_r$ if $v_\ell$ is useful, see~\eqref{eq:Z1}. Let us now estimate the number of ways to choose these sets in such a way that $v_\ell$ will not be useful, that is, letting $W_j = W_j(v_\ell)$, so that
\begin{equation}
\label{eq:MI-not-useful}
|M_I \cap \prod_{j \in I} W_j| > \lambda d^{|I|} \quad \text{for some $I \subseteq [r]$ with $2 \leqslant |I| \leqslant r$}.
\end{equation}
Recall that $\Pi \in \mathcal{P}_{n,r}(\gamma)$ and hence $|V_j| \geqslant \frac{n}{2r}$. It follows from the definition of $M_I$ that for every $I \subseteq [r]$ with $2 \leqslant |I| \leqslant r-1$, we have
\begin{equation}
\label{eq:MI}
|M_I| < \frac{2n^{|I|}}{C_2} \leqslant \frac{2^{|I|+1}r^{|I|}}{C_2} \cdot \prod_{j \in I} |V_j| \leqslant \frac{(2r)^r}{C_2} \cdot \prod_{j \in I} |V_j| = \tau \cdot \prod_{j \in I} |V_j|,
\end{equation}
where the last inequality follows from~\eqref{eq:C2-sigma}. Since $G \notin \mathcal{F}^R_1$, we also have
\begin{equation}
\label{eq:Mr}
|M_{[r]}| = e(\mathcal{H}_{\ell - 1}) \leqslant e(\mathcal{H}') < \sigma n^r \leqslant \sigma \cdot (2r)^r \cdot \prod_{j=1}^r |V_j| \leqslant \tau \cdot \prod_{j=1}^r |V_j|.
\end{equation}
Recalling the definition of $\lambda$ from~\eqref{eq:lambda-alpha}, inequalities~\eqref{eq:MI} and~\eqref{eq:Mr} together with Lemma~\ref{lemma:d-sets} imply that the number $Z_2$ of ways to choose $W_1(v_\ell), \ldots, W_r(v_\ell)$ so that $v_\ell$ is not useful, and therefore \eqref{eq:MI-not-useful} holds, satisfies
\[
\begin{split}
Z_2 & \leqslant \sum_{I \subseteq [r]} \alpha^{D^*} \prod_{j \in I} \binom{|V_j|}{D^*} \cdot \prod_{j \not\in I} \binom{|V_j|}{D^*} \leqslant 2^r \alpha^{D^*} \cdot \binom{|V_1| + \ldots + |V_r|}{r \cdot D^*} \\
& \leqslant \exp(-6C_1D^*) \cdot \binom{n}{rD^*},
\end{split}
\]
where the last inequality follows from~\eqref{eq:lambda-alpha}, provided that $D^* \geqslant r$, which holds when $n$ is sufficiently large.
Summarizing the above discussion, similarly as in the proof of Lemma~\ref{lemma:regular-case}, we have
\begin{equation}
\label{eq:FF3}
|\mathcal{F}^I| \leqslant \sum_{t,t^*,x,h} \binom{n}{b} \cdot 2^b \cdot \exp(-6C_1D^*)^{b/2} \cdot \binom{n}{rD^*}^b \cdot |\mathcal{T}_{t'}(t^*,x,h)| \cdot \binom{e(\Pi)}{m-t'-rbD^*}.
\end{equation}
Let $F_3(t,t^*,x,h)$ denote the term in the sum in the right hand side of \eqref{eq:FF3}. Recall that we are counting only graphs $G \in \mathcal{F}^H$ and therefore we may assume that~\eqref{eq:h} holds; a graph $G \in \mathcal{F}^H$ will be counted by $F_3(t,t^*,x,h)$, where $t = e(T(G))$, $t^* = e(T(G)-X(T(G)))$, $x = |X(T(G))|$, and $h = |H(T(G))|$. It follows from Lemma~\ref{lemma:T-count}, see~\eqref{eq:comp-T-count}, that
\[
|F_3(t,t^*,x,h)| \leqslant e^{1/\xi} \cdot 2^b \cdot m^{t^* + xD/2} \cdot \exp\left(\frac{4mh}{\xi n}\right) \cdot \exp(-3C_1D^* b) \cdot \binom{e(\Pi)}{m}.
\]
It follows from~\eqref{eq:h} that
\[
m^{t^* + xD/2} \leqslant \exp\left(\frac{mh}{n} \cdot \frac{6}{\varepsilon \xi}\right),
\]
which, recalling that $b \geqslant \frac{h}{2r}$ and the definition of $D^*$ from~\eqref{eq:Ds}, yields
\[
\begin{split}
|F_3(t,t^*,x,h)| & \leqslant e^{1/\xi} \cdot 2^h \cdot \exp\left( \frac{mh}{n} \cdot \left(\frac{6}{\varepsilon \xi} + \frac{4}{\xi} - \frac{3\beta C_1}{2r}\right) \right) \cdot \binom{e(\Pi)}{m} \\
& \leqslant e^{1/\xi} \cdot 2^h \cdot \exp\left(-\frac{3mh}{n}\right) \cdot \binom{e(\Pi)}{m} \leqslant \exp\left(-\frac{2m}{n}\right) \cdot \binom{e(\Pi)}{m},
\end{split}
\]
where in the first inequality we used~\eqref{eq:C1} and in the last inequality, we used the fact that $h \geqslant 1$, which follows from \eqref{eq:h} as $h$ is an integer, and that $n$ is sufficiently large. It follows that
\[
|\mathcal{F}^I| \leqslant m^2 n^2 \cdot \exp\left(-\frac{2m}{n}\right) \cdot \binom{e(\Pi)}{m} \leqslant \exp\left(-\frac{m}{n}\right) \cdot \binom{e(\Pi)}{m}.
\]
This completes the proof in the irregular case. \end{proof}
\section{The dense case ($m > e(\Pi) - \xi n^2$)}
\label{sec:dense-case}
Recall the definition of $\xi$ given in~(\ref{eq:xi}). In this section, we prove Theorem~\ref{thm:1-statement} in the (easy) case $m > e(\Pi) - \xi n^2$. We begin with a brief sketch of the argument.
\subsection{Outline of the proof}
\label{sec:proof-outline-dense}
Recall the definition of $\mathcal{T}$ from Section~\ref{sec:setup}. Our proof in the dense case has two main ingredients. In Section~\ref{sec:bounding-Fst-dense}, in Lemma~\ref{lemma:Fst-TTLx-upper}, we give an upper bound on $|\mathcal{F}^*(T)|$, the number of $G \in \mathcal{F}^*$ with $T(G) = T$, in terms of the size of a maximum matching in $T$. In Section~\ref{sec:counting-TT-dense}, in Lemma~\ref{lemma:TTx-upper}, we enumerate graphs $T \in \mathcal{T}$ with a particular value of this parameter. Combining these two estimates yields the required upper bound on $|\mathcal{F}^*|$.
The bound on the size of $\mathcal{F}^*(T)$ is obtained as follows. First, we note that the family $\mathcal{F}^*(T)$ is empty unless all vertices of $T$ have degree at most $\beta n$, where $\beta$ is some small positive constant. This is because by Claim~\ref{claim:unfriendly}, in every $G \in \mathcal{F}^*(T)$ the neighborhood of such a vertex would contain a large $K_r$-free graph, which, by Lemma~\ref{lemma:k-Turan}, would contradict the assumption that $e(G \cap \Pi) \geqslant (1-\delta)m > (1-\delta)(e(\Pi) - \xi n^2)$. Second, we observe that in every $G \in \mathcal{F}^*(T)$ the endpoints of every edge of $T$ cannot have many common neighbors in every other (than its own) color class. This is because the set of common neighbors of such an edge induces a $K_{r-1}$-free graph and hence, by Lemma~\ref{lemma:k-Turan} and our assumption that $e(G \cap \Pi) \geqslant (1-\delta)(e(\Pi) - \xi n^2)$, it cannot be very large. It follows that there are some $i$ and $j$ such that the density of edges between the vertex set of a maximal matching in $T[V_i]$ and $V_j$ is bounded away from $1$. Since by our assumption on $m$, $e(G(\Pi))$ is very close to $e(\Pi)$, this restriction is sufficiently strong to bound the number of choices for $G \cap \Pi$.
\subsection{Setup}
\label{sec:setup-dense}
For every $T \in \mathcal{T}$, we fix some canonically chosen maximal matching $U(T)$ in $T$ and let $X(T)$ be the set of endpoints of edges in $U(T)$. It follows from the maximality of $U(T)$ that every edge in $T$ has at least one endpoint in $X(T)$. Next, for every $i \in [r]$, we let $U_i(T)$ be the subgraph of $U(T)$ induced by $V_i$, let $X_i(T) = X(T) \cap V_i$, and let $i(T)$ be the smallest index satisfying \[
|X_{i(T)}(T)| = \max_{i \in [r]} |X_i(T)| \geqslant \frac{|X(T)|}{r}. \] Let \[ \beta = \frac{1}{40r^3}. \] In the argument below, we will use the following inequality, which is a trivial consequence of our choices of $\xi$ and $\delta$: \begin{equation}
\label{eq:xi-delta}
\xi + \delta < \min\left\{ \beta^2, \frac{1}{16r^2} \right\}. \end{equation}
\subsection{Counting the graphs in $\mathcal{T}$}
\label{sec:counting-TT-dense}
Let us partition the family $\mathcal{T}$ according to the size of the set $X(T)$. For an integer $x$, let $\mathcal{T}(x)$ consist of all $T \in \mathcal{T}$ that satisfy $|X(T)| = x$. We will use the following trivial upper bound on $|\mathcal{T}(x)|$.
\begin{lemma}
\label{lemma:TTx-upper}
If $n$ is sufficiently large, then for every $x$,
\[
|\mathcal{T}(x)| \leqslant e^{nx}.
\] \end{lemma} \begin{proof}
We may construct each $T \in \mathcal{T}(x)$ by selecting $x$ vertices that form the set $X(T)$ and, for each of those vertices, choosing which pairs of vertices intersecting $X(T)$ are edges of $T$. It follows that
\[
|\mathcal{T}(x)| \leqslant \binom{n}{x} 2^{\binom{x}{2} + x(n-x)} \leqslant e^{x(\log n + n\log 2)} \leqslant e^{nx},
\]
provided that $n$ is sufficiently large. \end{proof}
\subsection{Bounding $|\mathcal{F}^*(T)|$ in terms of $|X(T)|$}
\label{sec:bounding-Fst-dense}
We first deal with the case when $T$ contains a vertex with large degree. To this end, let $\TT^H$ be the family of all $T \in \mathcal{T}$ that contain a vertex of degree at least $\beta n$ and let $\TT^L = \mathcal{T} \setminus \TT^H$. With our choice of parameters, estimating $|\mathcal{F}^*(T)|$ for $T \in \TT^H$ is extremely easy.
\begin{lemma}
\label{lemma:TTH-empty}
For every $T \in \TT^H$, the family $\mathcal{F}^*(T)$ is empty. \end{lemma} \begin{proof}
Fix a $T \in \TT^H$ and let $v$ be an arbitrary vertex with $\deg_T(v) \geqslant \beta n$. Suppose that $\mathcal{F}^*(T)$ is non-empty and fix an arbitrary $G \in \mathcal{F}^*(T)$. By Claim~\ref{claim:unfriendly} and the definition of $\TT^H$, $\deg_G(v,V_i) \geqslant \beta n$ for every $i \in [r]$. The ($r$-partite) subgraph of $G \cap \Pi$ induced by $N_G(v)$ is $K_r$-free and so by Lemma~\ref{lemma:k-Turan},
\[
e(\Pi \setminus G) \geqslant \min \big\{ \deg_G(v,V_i) \cdot \deg_G(v,V_j) \colon i, j \in [r] \big\} \geqslant \beta^2n^2.
\]
On the other hand, by~\eqref{eq:xi-delta},
\begin{equation}
\label{eq:Pi-G-dense}
e(\Pi \setminus G) = e(\Pi) - e(G) + e(T) \leqslant e(\Pi) - m + \delta m \leqslant \xi n^2 + \delta \binom{n}{2} < \beta^2 n^2,
\end{equation}
a contradiction. \end{proof}
The following lemma is the key step in our proof. It says that for every $G \in \mathcal{F}^*(T)$, there is a fairly large set $X' \subseteq V_{i(T)}$ such that the density of edges between $X'$ and some $V_j$ with $j \neq i(T)$ is at most $3/4$, much lower than the average density of $G \cap \Pi$, which by our assumptions is $1 - O(\xi + \delta)$.
\begin{lemma}
\label{lemma:hole-in-Pi}
For every $T \in \mathcal{T}$ and every $G \in \mathcal{F}^*(T)$, there is a set $X' \subseteq X_{i(T)}$ and a $j \neq i(T)$ such that
\[
|X'| \geqslant \frac{|X_{i(T)}(T)|}{r-1} \qquad \text{and} \qquad e_G(X', V_j) \leqslant \frac{3}{4} |X'| \cdot |V_j|.
\] \end{lemma} \begin{proof}
Fix some $T \in \mathcal{T}$ and $G \in \mathcal{F}^*(T)$ and let $i = i(T)$. For every edge $\{u,v\} \in U_{i}(T)$ and every $j \neq i$, let $W_j^{uv}$ be the set of common neighbors of $u$ and $v$ in $V_j$. Suppose first that for every $\{u, v\}$, there is a $j \neq i$ such that $|W_j^{uv}| \leqslant |V_j|/2$, which implies that
\begin{equation}
\label{eq:euvVj}
e_G(\{u,v\}, V_j) \leqslant |V_j| + |W_j^{uv}| \leqslant \frac{3}{2}|V_j|.
\end{equation}
Let $j$ be an index for which~\eqref{eq:euvVj} holds for the largest number of edges $\{u,v\} \in U_{i}(T)$ and let $X' \subseteq X_i(T)$ be the set of endpoints of these edges. This set $X'$ clearly satisfies the assertion of this lemma. Suppose now that there is a $\{u, v\} \in U_i(T)$ such that $|W_j^{uv}| > |V_j|/2$ for all $j \neq i$. Since $G$ is $K_{r+1}$-free, the $(r-1)$-partite subgraph of $G \cap \Pi$ induced by the sets $W_j^{uv}$ with $j \neq i$ contains no copy of $K_{r-1}$. In other words, this subgraph of $G \cap \Pi$ is a $K_{r-1}$-free subgraph of the complete $(r-1)$-partite graph with color classes $W_j^{uv}$, where $j \neq i$. It follows from Lemma~\ref{lemma:k-Turan} that the graph $\Pi \setminus G$ contains at least $\min_{j_1 \neq j_2}|W_{j_1}^{uv}||W_{j_2}^{uv}|$ edges. This clearly cannot happen as
\[
\min_{j_1 \neq j_2} |W_{j_1}^{uv}||W_{j_2}^{uv}| \geqslant \min_j \frac{|V_j|^2}{4} \geqslant \frac{n^2}{16r^2}
\]
but, on the other hand, by~\eqref{eq:xi-delta} and~\eqref{eq:Pi-G-dense},
\[
e(\Pi \setminus G) \leqslant \xi n^2 + \delta \binom{n}{2} < \frac{n^2}{16r^2},
\]
a contradiction. \end{proof}
Finally, we use Lemma~\ref{lemma:hole-in-Pi} to derive an upper bound on $|\mathcal{F}^*(T)|$ for all $T \in \TT^L(x)$.
\begin{lemma}
\label{lemma:Fst-TTLx-upper}
If $n$ is sufficiently large, then for every $x$ and every $T \in \TT^L(x)$,
\[
|\mathcal{F}^*(T)| \leqslant e^{-2nx} \cdot \binom{e(\Pi)}{m}.
\] \end{lemma} \begin{proof}
By Lemma~\ref{lemma:hole-in-Pi}, we may construct each $G \in \mathcal{F}^*(T)$ by selecting a set $X' \subseteq X_{i(T)}$ of $\lceil x/r^2 \rceil$ vertices, an index $j \neq i(T)$, then choosing some $m'$, where $m' \leqslant \frac{3}{4}|X'||V_j|$, edges between $X'$ and $V_j$, and finally choosing the remaining $m - m' - e(T)$ edges from $\Pi \setminus (X', V_j)$, where $(X', V_j)$ denotes the complete bipartite graph with color classes $X'$ and $V_j$. The reason why $|\mathcal{F}^*(T)|$ is so small is that the number $N_j$ of ways to first choose only at most $\frac{3}{4}|X'||V_j|$ edges between $X'$ and $V_j$ and then the remaining edges from $\Pi \setminus (X', V_j)$ is much smaller than the number of ways to choose $m - e(T)$ edges from $\Pi$. To quantify this, let $t = e(T)$, let $x' = \lceil x/r^2 \rceil$, fix some $j \neq i(T)$, and let $n_j = |V_j|$. Observe that $t \leqslant \beta n x$ due to our assumption that $T \in \TT^L(x)$ and therefore,
\begin{equation}
\label{eq:hole-in-Pi}
N_j \leqslant \sum_{m' \leqslant \frac{3}{4}x'n_j} \binom{x'n_j}{m'} \binom{e(\Pi) - x'n_j}{m-m'-t} \leqslant 2^{x'n_j} \cdot \binom{e(\Pi) - x'n_j}{m-\frac{3}{4}x'n_j-\beta nx} \leqslant 2^{xn} \cdot \binom{e(\Pi) - x'n_j}{m-\frac{4}{5}x'n_j},
\end{equation}
where the last two inequalities hold since for each $m'$ with $m' \leqslant \frac{3}{4}x'n_j$,
\begin{equation}
\label{eq:ePi-x'n'}
\frac{e(\Pi) - x'n_j}{2} \leqslant m - \frac{4}{5}x'n_j \leqslant m - \frac{3}{4}x'n_j - \beta n x \leqslant m - m' - t.
\end{equation}
To see the first inequality in~\eqref{eq:ePi-x'n'}, recall that $m \geqslant e(\Pi) - \xi n^2$ and observe that
\begin{equation}
\label{eq:xpnp}
x'n_j \leqslant \frac{|V_{i(T)}|}{2} \cdot |V_j| \leqslant \frac{e(\Pi)}{2}.
\end{equation}
To see the second inequality in~\eqref{eq:ePi-x'n'}, recall that $x' \geqslant x/r^2$, $n_j \geqslant n/2r$, and therefore $\beta n x \leqslant 2\beta r^3 x'n_j = x'n_j/20$. With the view of further estimating $N_j$, we claim that for every $j \in [r]$,
\begin{equation}
\label{eq:hole-in-Pi-2}
\binom{e(\Pi) - x'n_j}{m-\frac{4}{5}x'n_j} \leqslant e^{-3xn} \cdot \binom{e(\Pi)}{m}.
\end{equation}
Assuming that~\eqref{eq:hole-in-Pi-2} holds, \eqref{eq:hole-in-Pi} implies that
\[
|\mathcal{F}^*(T)| \leqslant \sum_{j \neq i(T)} \binom{|V_j|}{x'} \cdot N_j \leqslant r \cdot n^{x} \cdot 2^{xn} \cdot e^{-3xn} \cdot \binom{e(\Pi)}{m} \leqslant e^{-2xn} \cdot \binom{e(\Pi)}{m},
\]
provided that $n$ is sufficiently large, as required. Thus, it remains to establish~\eqref{eq:hole-in-Pi-2}. To this end, using the trivial identity
\[
\binom{a-5}{b-4} = \frac{b(b-1)(b-2)(b-3)(a-b)}{a(a-1)(a-2)(a-3)(a-4)} \cdot \binom{a}{b}
\]
and the assumption that $m \geqslant e(\Pi) - \xi n^2$, we estimate
\[
\begin{split}
\frac{\binom{e(\Pi) - x'n_j}{m - \frac{4}{5}x'n_j}}{\binom{e(\Pi)}{m}} & = \prod_{z = 1}^{x'n_j/5} \frac{\binom{e(\Pi) - 5z}{m-4z}}{\binom{e(\Pi)-5z+5}{m-4z+4}} \leqslant \left(\frac{m^4(e(\Pi)-m)}{(e(\Pi)-x'n_j)^5}\right)^{\frac{x'n_j}{5}} \leqslant \left(\frac{e(\Pi)^4 \cdot \xi n^2}{(e(\Pi)/2)^5}\right)^{\frac{x'n_j}{5}} \\
& \leqslant (2^5 \cdot 16/3 \cdot \xi)^{\frac{x'n_j}{5}} \leqslant (2^9 \xi / 3)^{\frac{xn}{10r^3}} \leqslant e^{-3xn},
\end{split}
\]
where we used~\eqref{eq:xi}, \eqref{eq:xpnp}, and the fact that $e(\Pi) \geqslant 3n^2/16$, see~\eqref{eq:ePi-lower}. This completes the proof of the lemma. \end{proof}
\subsection{Summary}
With Lemmas~\ref{lemma:TTx-upper}, \ref{lemma:TTH-empty}, and \ref{lemma:Fst-TTLx-upper} at our disposal, we can finally deduce an upper bound on $|\mathcal{F}^*|$: \[ \begin{split}
|\mathcal{F}^*| & = \sum_{T \in \mathcal{T}} |\mathcal{F}^*(T)| = \sum_{T \in \TT^L} |\mathcal{F}^*(T)| = \sum_{x \geqslant 1} \sum_{T \in \TT^L(x)} |\mathcal{F}^*(T)| \\
& \leqslant \sum_{x \geqslant 1} |\mathcal{T}(x)| \cdot e^{-2nx} \cdot \binom{e(\Pi)}{m} \leqslant \sum_{x \geqslant 1} e^{-nx} \cdot \binom{e(\Pi)}{m} \leqslant 2e^{-n} \cdot \binom{e(\Pi)}{m}. \end{split} \] This completes the proof of Theorem~\ref{thm:1-statement} in the dense case.
\appendix
\section{Omitted proofs}
\label{sec:omitted-proofs}
\subsection{Tools}
In this section, we prove Lemmas~\ref{lemma:HJI}--\ref{lemma:k-Turan}. In the proofs of the first two of them, we will use the so-called Local LYMB Inequality.
\begin{lemma}[{Local LYMB Inequality}]
\label{lemma:local-LYMB}
Let $\mathcal{H}$ be a $k$-uniform hypergraph on a finite vertex set $V$. The \emph{shadow} of $\mathcal{H}$ is the $(k-1)$-uniform hypergraph ${\partial \HH}$ defined by
\[
{\partial \HH} = \left\{A \in \binom{V}{k-1} \colon A \subseteq B \text{ for some } B \in \mathcal{H} \right\}.
\]
We have
\[
\frac{e({\partial \HH})}{\binom{|V|}{k-1}} \geqslant \frac{e(\mathcal{H})}{\binom{|V|}{k}}.
\] \end{lemma}
\begin{proof}[{Proof of Lemma~\ref{lemma:HJI}}]
For a set $J \subseteq I$, define
\[
\mu(J) = \sum_{i \in J} p^{|B_i|} \qquad \text{and} \qquad \Delta(J) = \sum_{i \sim j} p^{|B_i \cup B_j|},
\]
where the second sum is over all ordered pairs $(i,j) \in J^2$ such that $i \neq j$ and $B_i \cap B_j \neq \emptyset$. Let $I_q \subseteq I$ be the $q$-random subset of $I$, that is, the random subset of $I$ where each element of $I$ is included with probability $q$, independently of all other elements, and fix an arbitrary set $J \subseteq I$ that satisfies
\[
\mu(J) - \Delta(J)/2 \geqslant \mathbb{E}\big[\mu(I_q) - \Delta(I_q)/2\big] = q\mu - q^2\Delta/2.
\]
Let $R'$ be the $p$-random subset of $\Omega$ and let $\mathcal{B}'$ denote the event that $B_i \nsubseteq R'$ for all $i \in I$; one may think of $R'$ and $\mathcal{B}'$ as `binomial' analogues of $R$ and $\mathcal{B}$. By the Janson Inequality (see, e.g., \cite[Theorem~8.1.1]{AlSp}),
\begin{equation}
\label{eq:PrBB'}
\mathbb{P}(\mathcal{B}') \leqslant \mathbb{P}(B_i \nsubseteq R' \text{ for all } i \in J) \leqslant \exp\left( -\mu(J) + \Delta(J)/2 \right) \leqslant \exp\left(- q\mu + q^2\Delta/2\right).
\end{equation}
It follows from the Local LYMB Inequality (Lemma~\ref{lemma:local-LYMB}) that the function $k \mapsto \mathbb{P}(\mathcal{B}' \mid |R'| = k)$ is decreasing and hence
\begin{equation}
\label{eq:PrBB}
\begin{split}
\mathbb{P}(\mathcal{B}') & = \sum_{k=0}^n \mathbb{P}(\mathcal{B}' \mid |R'| = k) \cdot \mathbb{P}(|R'| = k) \geqslant \mathbb{P}(\mathcal{B}' \mid |R'| = m) \cdot \mathbb{P}(|R'| \leqslant m) \\
& = \mathbb{P}(\mathcal{B}) \cdot \mathbb{P}(|R'| \leqslant m) \geqslant \mathbb{P}(\mathcal{B})/2,
\end{split}
\end{equation}
where the last inequality follows from the well-known fact that if $np$ is an integer, then it is the median of the binomial distribution $\mathrm{Bin}(n,p)$. Inequalities~\eqref{eq:PrBB'} and~\eqref{eq:PrBB} readily imply the claimed bound on $\mathbb{P}(\mathcal{B})$. \end{proof}
\begin{proof}[{Proof of Lemma~\ref{lemma:HFKG}}]
Set $p = (1+\eta)m/n$ and note that $p \leqslant 1$ by our assumptions on $m$ and $\eta$. Let $R'$ be the $p$-random subset of $\Omega$ and let $\mathcal{B}'$ denote the event that $B_i \nsubseteq R'$ for all $i \in I$. By the FKG Inequality (see, e.g., \cite[Chapter~6]{AlSp}),
\begin{equation}
\label{eq:PrBB'-FKG}
\mathbb{P}(\mathcal{B}') \geqslant \prod_{i \in I} \left(1 - p^{|B_i|}\right).
\end{equation}
It follows from the Local LYMB Inequality (Lemma~\ref{lemma:local-LYMB}) that the function $k \mapsto \mathbb{P}(\mathcal{B}' \mid |R'| = k)$ is decreasing and hence
\begin{equation}
\label{eq:PrBB-FKG}
\begin{split}
\mathbb{P}(\mathcal{B}') & = \sum_{k=0}^n \mathbb{P}(\mathcal{B}' \mid |R'| = k) \cdot \mathbb{P}(|R'| = k) \leqslant \mathbb{P}(\mathcal{B}' \mid |R'| = m) + \mathbb{P}(|R'| < m) \\
& = \mathbb{P}(\mathcal{B}) + \mathbb{P}(|R'| < m).
\end{split}
\end{equation}
The claimed bound now easily follows from \eqref{eq:PrBB'-FKG} and \eqref{eq:PrBB-FKG} as by Chernoff's Inequality (see, e.g., \cite[Appendix~A]{AlSp}),
\[
\mathbb{P}(|R'| < m) \leqslant \exp\left(-\frac{\eta^2m^2}{2(1+\eta)m}\right) \leqslant \exp\left(-\frac{\eta^2m}{4}\right). \qedhere
\] \end{proof}
\begin{proof}[{Proof of Lemma~\ref{lemma:k-Turan}}]
Let us denote the graph $K(n_1, \ldots, n_r)$ by $G$. Let $V_1, \ldots, V_r$ be the color classes of $G$ with $n_1, \ldots, n_r$ elements, respectively. Clearly, deleting all edges between $V_1$ and $V_2$ removes all copies of $K_r$ from $G$ and hence $\mathrm{ex}(G, K_r) \geqslant e(G) - n_1n_2$. We prove the converse inequality by induction on $r$. The statement is trivial if $r = 2$, so let us assume that $r \geqslant 3$. Let $H$ be a $K_r$-free subgraph of $G$. Let $\Delta = \max_{v \in V_r} \deg_H(v)$ and fix an arbitrary $v \in V_r$ with $\deg_H(v) = \Delta$. For each $i \in [r-1]$, let $d_i = \deg_H(v,V_i)$. Since the subgraph of $G$ induced by $N_H(v)$ is a $K_{r-1}$-free subgraph of $K(d_1, \ldots, d_{r-1})$, it follows from our inductive assumption that
\[
e(G) - e(H) \geqslant e_{G \setminus H}(V_1 \cup \ldots \cup V_{r-1}, V_r) + e_{G \setminus H}(N_H(v)) \geqslant n_r \cdot \left(\sum_{k=1}^{r-1}n_k - \Delta\right) + \min_{i < j} d_id_j.
\]
Let $\{i,j\}$, where $i < j$, be a pair of indices for which $d_i d_j$ attains its minimum value. Since $d_1 + \ldots + d_{r-1} = \Delta$ by our choice of $v$, then
\[
\begin{split}
e(G) - e(H) & \geqslant n_r \cdot \sum_{k=1}^{r-1} (n_k - d_k) + d_id_j \geqslant n_r \cdot [(n_i - d_i) + (n_j - d_j)] + d_id_j \\
& \geqslant n_j \cdot [(n_i - d_i) + (n_j - d_j)] + d_id_j = n_in_j + (n_j-d_j)(n_j-d_i) \geqslant n_in_j \geqslant n_1n_2,
\end{split}
\]
as claimed. It is not hard to verify that $e(G) - e(H) > n_1n_2$ unless $H = G \setminus (V_i,V_j)$, where $i$ and $j$ are such that $n_in_j = n_1n_2$. \end{proof}
\subsection{Main tool}
Lemma~\ref{lemma:d-sets} is a straightforward consequence of the following somewhat technical (but tailored to facilitate an inductive proof) statement.
\begin{lemma}
\label{lemma:d-sets-tech}
Let $\alpha, \lambda \in (0,1)$, let $V_1, \ldots, V_k$ be finite sets, and let $d$ be an integer satisfying $2 \leqslant d \leqslant \min\{ |V_1|, \ldots, |V_k| \}$. Suppose that $\mathcal{H} \subseteq V_1 \times \ldots \times V_k$ satisfies
\[
|\mathcal{H}| \leqslant (\alpha \lambda)^k \prod_{i = 1}^k |V_i|.
\]
Then for all but at most
\[
\big(d^k - 1\big) \big(2\alpha^\lambda\big)^d \prod_{i=1}^k \binom{|V_i|}{d}
\]
choices of $W_1 \in \binom{V_1}{d}, \ldots, W_k \in \binom{V_k}{d}$, we have
\begin{equation}
\label{eq:prod-W-HH}
|\mathcal{H} \cap (W_1 \times \ldots \times W_k)| \leqslant k \lambda d^k.
\end{equation} \end{lemma}
We will deduce this statement from the following one-sided version of Hoeffding's inequality~\cite{Ho63} for the hypergeometric distribution.
\begin{lemma}
\label{lemma:lrg-dev}
Let $d$ and $n$ be integers and let $X$ denote the uniformly chosen random $d$-subset of $[n]$. Then for every $\alpha, \lambda \in (0,1)$,
\begin{equation}
\label{eq:lrg-dev}
\mathbb{P}\left( \big| X \cap [\alpha \lambda n] \big| \geqslant \lambda d \right) \leqslant \big(2 \alpha^\lambda\big)^d.
\end{equation} \end{lemma} \begin{proof}
Denote the left-hand side of~\eqref{eq:lrg-dev} by $P$. It follows from Hoeffding's inequality~\cite[Theorem~1]{Ho63} that\footnote{Even though~\cite[Theorem~1]{Ho63} applies to sums of independent random variables, the bound obtained there remains valid for the hypergeometric distribution, see the discussion in \cite[Section~6]{Ho63}.}
\[
P^{1/d} \leqslant \left(\frac{\alpha \lambda}{\lambda}\right)^\lambda \left(\frac{1 - \alpha \lambda}{1- \lambda}\right)^{1 - \lambda} \leqslant \alpha^\lambda \left(\frac{1}{1-\lambda}\right)^{1-\lambda} \leqslant \alpha^\lambda e^{1/e} \leqslant 2\alpha^\lambda,
\]
where the third inequality follows from the fact that if $a > 0$, then the function $x \mapsto (a/x)^x$ attains its maximum value at $x = a/e$. \end{proof}
\begin{proof}[{Proof of Lemma~\ref{lemma:d-sets-tech}}]
We prove the statement by induction on $k$. The induction base ($k = 1$) follows directly from Lemma~\ref{lemma:lrg-dev} and the fact that $d - 1 \geqslant 1$. For the induction step, assume that $k \geqslant 2$. For every $v \in V_k$, let
\[
\mathcal{H}_v = \{(v_1, \ldots, v_{k-1}) \in V_1 \times \ldots \times V_{k-1} \colon (v_1, \ldots, v_{k-1}, v) \in \mathcal{H} \}.
\]
One may think of $\mathcal{H}_v$ as the link hypergraph of the vertex $v$. Roughly speaking, we shall argue as follows. If $W_1 \times \ldots \times W_k$ contains many edges of $\mathcal{H}$, then either
\begin{enumerate}[(i)]
\item
\label{item:d-sets-tech-1}
the set $W_k$ contains many vertices whose degree in $\mathcal{H}$ is high or
\item
\label{item:d-sets-tech-2}
for some $v \in W_k$ whose degree in $\mathcal{H}$ is low, the set $W_1 \times \ldots \times W_{k-1}$ has a large intersection with the link hypergraph $\mathcal{H}_v$.
\end{enumerate}
The desired bound will then follow by applying Lemma~\ref{lemma:lrg-dev} (when~(\ref{item:d-sets-tech-1}) holds) and the induction hypothesis (when~(\ref{item:d-sets-tech-2}) holds). The details now follow.
Let
\[
V_k' = \Big\{ v \in V_k \colon |\mathcal{H}_v| > (\alpha \lambda)^{k-1} \prod_{i=1}^{k-1} |V_i| \Big\}.
\]
Intuitively, $V_k'$ is the set of all $v \in V_k$ whose degree in $\mathcal{H}$ exceeds the assumed upper bound on the average degree of $V_k$ in $\mathcal{H}$ by a factor of more than $(\alpha \lambda)^{-1}$. Note that our assumption on $\mathcal{H}$ implies that $|V_k'| < \alpha \lambda |V_k|$. Furthermore, let
\[
\mathcal{W}_k = \Big\{ W \in \binom{V_k}{d} \colon |W \cap V_k'| \geqslant \lambda d \Big\}
\]
and for each $W \in \binom{V_k}{d} \setminus \mathcal{W}_k$, define $\mathcal{W}_W \subseteq \binom{V_1}{d} \times \ldots \times \binom{V_{k-1}}{d}$ by
\[
\mathcal{W}_W = \big\{ (W_1, \ldots, W_{k-1}) \colon |\mathcal{H}_v \cap W_1 \times \ldots \times W_{k-1}| \geqslant (k-1)\lambda d^{k-1} \text{ for some $v \in W \setminus V_k'$} \big\}.
\]
Note that if $W_1 \in \binom{V_1}{d}, \ldots, W_k \in \binom{V_k}{d}$ are such that $W_k \not\in \mathcal{W}_k$ and $(W_1, \ldots, W_{k-1}) \not\in \mathcal{W}_{W_k}$, then
\[
|\mathcal{H} \cap W_1 \times \ldots \times W_k| \leqslant \lambda d \cdot d^{k-1} + d \cdot (k-1)\lambda d^{k-1} \leqslant k\lambda d^k.
\]
Hence, the number $B$ of $k$-tuples $(W_1, \ldots, W_k)$ for which~\eqref{eq:prod-W-HH} does not hold satisfies
\begin{equation}
\label{eq:bad-tuples}
B \leqslant |\mathcal{W}_k| \cdot \prod_{i=1}^{k-1} \binom{|V_i|}{d} + \sum_{W \not \in \mathcal{W}_k} |\mathcal{W}_W|.
\end{equation}
By Lemma~\ref{lemma:lrg-dev},
\begin{equation}
\label{eq:WWk}
|\mathcal{W}_k| \leqslant \big(2 \alpha^\lambda)^d \binom{|V_k|}{d}
\end{equation}
and, since
\[
|\mathcal{H}_v| \leqslant (\alpha \lambda)^{k-1} \prod_{i=1}^{k-1} |V_i|
\]
for every $v \not\in V_k'$, then by our inductive assumption, for every $W \not\in \mathcal{W}_k$,
\begin{equation}
\label{eq:WWW}
|\mathcal{W}_W| \leqslant \sum_{v \in W \setminus V_k'} \big(d^{k-1}-1\big)\big(2 \alpha^\lambda\big)^d \prod_{i=1}^{k-1}\binom{|V_i|}{d} \leqslant \big(d^k-d\big)\big(2\alpha^\lambda\big)^d \prod_{i=1}^{k-1}\binom{|V_i|}{d}.
\end{equation}
Putting~\eqref{eq:bad-tuples}, \eqref{eq:WWk}, and~\eqref{eq:WWW} together yields
\[
B \leqslant \big(1 + d^k - d\big)\big(2\alpha^\lambda)^d \prod_{i=1}^k \binom{|V_i|}{d} \leqslant \big(d^k-1\big)\big(2\alpha^\lambda\big)^d \prod_{i=1}^k\binom{|V_i|}{d},
\]
as claimed. \end{proof}
\subsection{Non-uniquely colorable and unbalanced graphs}
Finally, we present the proofs of Propositions~\ref{prop:unbalanced-graphs} and~\ref{prop:UP}.
\begin{proof}[{Proof of Proposition~\ref{prop:unbalanced-graphs}}]
Fix an arbitrary partition $\Pi$ that does not satisfy~\eqref{eq:Part-gamma} and observe that (see~\eqref{eq:ePi-gamma-upper})
\[
e(\Pi) \leqslant \ex(n, K_{r+1}) - \frac{\gamma^2 n^2}{2}.
\]
Consequently, by Lemma~\ref{lemma:acbc}, we have that
\[
|\mathcal{G}_m(\Pi)| = \binom{e(\Pi)}{m} \leqslant \left(\frac{\ex(n, K_{r+1}) - \frac{\gamma^2 n^2}{2}}{\ex(n, K_{r+1})}\right)^{m} \cdot \binom{\ex(n, K_{r+1})}{m} \leqslant e^{-\gamma^2 m} \cdot \binom{\ex(n, K_{r+1})}{m}.
\]
To complete the proof, we just observe that there are at most $r^n$ different $r$-colorings and that $r^n \leqslant e^{\gamma^2 m/2}$ if $m \geqslant cn$ for a sufficiently large constant $c$. \end{proof}
\begin{proof}[{Proof of Proposition~\ref{prop:UP}}]
Fix some $\Pi \in \mathcal{P}_{n,r}(\frac{1}{2r})$ and $\Pi' \in \mathcal{P}_{n,r} \setminus \{\Pi\}$. Suppose that $\Pi = \{V_1, \ldots, V_r\}$ and $\Pi' = \{V_1', \ldots, V_r'\}$ and for all $i, j \in [r]$, let $V_{i,j} = V_i \cap V_j'$. We will say that the vertices in $V_{i,j}$ are \emph{moved} from $V_i$ to $V_j'$. For every $i \in [r]$, define $L_i$ and $S_i$ as the largest and the second largest subclasses of $V_i$, respectively. Note that $|V_i| \geqslant \frac{n}{2r}$ implies that $|L_i| \geqslant \frac{n}{2r^2}$. Set $s = \max_{j \in [r]} |S_j|$ and let $S = S_j$ for the smallest $j$ for which the maximum in the definition of $s$ is achieved. Note that $1 \leqslant s \leqslant n/2$, as $s = 0$ would imply that $(V_1, \ldots, V_r)$ is a permutation of $(V_1', \ldots, V_r')$, and therefore $\Pi = \Pi'$.
Observe that, by the pigeonhole principle, either some pair $\{L_i, L_j\}$ of largest subclasses, or some largest subclass $L_i$ and $S$, where $S \nsubseteq V_i$, are moved to the same vertex class $V_k'$. Since $V_k'$ is an independent set in every $G \in \mathcal{G}_m(\Pi')$, it follows that every $G \in \mathcal{G}_m(\Pi) \cap \mathcal{G}_m(\Pi')$ has no edges between these sets $L_i$ and $L_j$ or $L_i$ and $S$. Since,
\[
\min\{ |L_i| \cdot |L_j|, |L_i| \cdot |S| \} \geqslant \cdot \min\left\{ \left(\frac{n}{2r^2}\right)^2, \frac{n}{2r^2} \cdot s \right\} \geqslant \frac{sn}{2r^4},
\]
it follows from Lemma~\ref{lemma:acbc} and the inequality $e(\Pi) \leqslant n^2/2$ that, if $m \geqslant r^6 (a+3) n \log n$,
\begin{equation}
\label{eq:GP-GPp}
|\mathcal{G}_m(\Pi) \cap \mathcal{G}_m(\Pi')| \leqslant \binom{e(\Pi) - \frac{sn}{2r^4}}{m} \leqslant \left(1 - \frac{s}{nr^4}\right)^m \binom{e(\Pi)}{m} \leqslant n^{-(a+3)sr^2} \cdot |\mathcal{G}_m(\Pi)|.
\end{equation}
Finally, observe that given a $\Pi$, we can describe any $\Pi' \neq \Pi $ by first picking the partitions $(V_{i,j})_{j \in [r]}$ for every $i$ and then setting $V_j' = \bigcup_{i \in [r]} V_{i,j}$. A moment's thought reveals that for every $s$, there are at most $n^{r^2} \cdot n^{sr^2}$ ways to choose all $V_{i,j}$ so that $\max_{i \in [r]} |S_i| = s$. Therefore, by~\eqref{eq:GP-GPp},
\[
|\mathcal{G}_m(\Pi) \setminus \mathcal{U}_m(\Pi)| \leqslant \sum_{s \geqslant 1} \left( n^{(s+1)r^2} \cdot n^{-s(a+3)r^2} \right) \cdot |\mathcal{G}_m(\Pi)| \leqslant n^{-a} \cdot |\mathcal{G}_m(\Pi)|,
\]
which completes the proof. \end{proof}
\noindent \textbf{Acknowledgments.} We would like to thank the anonymous referee for a careful reading of the paper and several valuable comments and suggestions. Lutz Warnke would like to thank Angelika Steger for numerous insightful discussions on the topic of this paper.
\end{document} | arXiv | {
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\begin{document}
\def\spacingset#1{\renewcommand{\baselinestretch} {#1}\small\normalsize} \spacingset{1} \doublespacing
\title{\bf Multigroup discrimination based on weighted local projections}
\author{Thomas Ortner$^1$ \thanks{This work has been partly funded by the Vienna Science and Technology Fund (WWTF) through project ICT12-010 and by the K-project DEXHELPP through COMET - Competence Centers for Excellent Technologies, supported by BMVIT, BMWFW and the province Vienna. The COMET program is administrated by FFG.} \footnote{thomas.ortner@tuwien.ac.at}, Irene Hoffmann$^1$, Peter Filzmoser$^1$, Maia Zaharieva$^2$, \\Christian Breiteneder$^2$, Sarka Brodinova$^1$ \\
$^1$Institute of Statistics and Mathematical Methods in Economics \\
$^2$Institute of Software Technology and Interactive Systems \\
TU Wien} \maketitle
\begin{abstract} \blue{A novel approach for supervised classification analysis for high dimensional and flat data (more variables than observations) is proposed. We use the information of class-membership of observations to determine groups of observations locally describing the group structure. By projecting the data on the subspace spanned by those groups, local projections are defined based on the projection concepts from \cite{ortner2017guided} and \cite{ortner2017local}. For each local projection a local discriminant analysis (LDA) model is computed using the information within the projection space as well as the distance to the projection space. The models provide information about the quality of separation for each class combination. Based on this information, weights are defined for aggregating the LDA-based posterior probabilities of each subspace to a new overall probability. The same weights are used for classifying new observations.}
\blue{In addition to the provided methodology, implemented in the R-package \textit{lop}, a method of visualizing the connectivity of groups in high-dimensional spaces is proposed at the basis of the posterior probabilities. A thorough evaluation is performed using three different real-world datasets\blue{, underlining the strengths of local projection based classification and the provided visualization methodology}.} \end{abstract}
\section{Introduction}
Supervised classification methods are widely used in research and industry, including tasks like tumor classification, speech recognition, or the classification of food quality. Observations are gathered from $G$ distinct groups and for each observation the group membership is known. Decision boundaries are then estimated in the sample space, such that a new observation can be assigned to one of the $G$ groups. The aim of discrimination methods is to find \blue{classification} boundaries, which result in low misclassification rates for new observations, i.e. new observations are assigned to the correct class with high accuracy.
\blue{ Linear discriminant analysis (LDA) is a popular tool for classification. It estimates linear decision boundaries by maximizing the between-group to within-group variance and assumes equal covariance structure of the groups. LDA often gives surprisingly good results in low-dimensional settings, however, it cannot be directly applied if the number of variables exceeds the number of observations since then the within-group covariance estimate becomes singular and its inverse cannot be calculated. With restrictions on the covariance estimation the problem of singularity can be mended but asymptotically (with increasing \blue{number of variables}) the performance of LDA is not better than random guessing \cite[][]{bickel2004some, shao2011sparse}.}
In many classification tasks, it is commonly the case that the underlying data has a flat structure, i.e.~there are more variables than observations. Therefore, a great variety of alternative classification methods and extensions of LDA have been developed to overcome its limitations. Several proposed approaches consider projection of the data onto a lower dimensional subspace \cite[][]{barker2003partial, chen2013near} or reducing the dimensionality by model-based variable selection \cite[][]{witten2011penalized}. Other methods are not based on covariance estimation and so they are not restricted to low-dimensional (non-flat) settings, e.g. k-nearest neighbour (KNN) classification, support vector machines (SVM) or random forests (RF). Nevertheless, the noise accumulation due to a large number of variables, which are not informative for the class separations, affects these methods as well. \red{Also the concepts of combining projections with classification methods has been previously explored \citep[e.g.][]{lee2005projection, caragea2001gaining} where the focus is on exploratory classification for finding suitable projections for visualization.}
We propose a new approach for supervised classification based on a series of projections into low-dimensional subspaces, referred to as local projections. In each subspace, we calculate an LDA model. The posterior probabilities of each LDA model are aggregated (weighted by a class-specific quality measure of the projection space) to obtain a final classification. The idea of aggregating posterior probabilities in the context of random forests has been proposed by \citet[][]{bosch2007image} taking the average over the posterior probabilities from all trees.
\blue{ The remainder of the paper is structured as follows. Section~\ref{sec:methodology} presents the proposed method. First, local projections based on the k-nearest class neighbors of an observation and distances within and to the projection space are introduced in Section~\ref{ss:localprojections}, resulting in the local discrimination space where an LDA model is estimated. Next, in Section~\ref{ss:weighting}, we introduce weights used for aggregating the posterior probabilities from the individual LDA models leading to a final classification rule. In Section~\ref{ss:trainingLDA}, the range of the tuning parameter $k$, associated with the dimensionality of the local discrimination space, is discussed and a strategy to select the tuning parameter is presented. Section~\ref{sec:visualization} introduces a way of visualizing the data structure and the degree of separation. In Section~\ref{sec:evaluation}, three real-world datasets are used to evaluate the performance of our approach in comparison to other related and popular classification methods. The datasets cover settings with only 25 and up to almost 10.000 variables, including multigoup and binary classification problems, and a dataset where subgroups are known to exist. The effect of imbalanced group sizes in the training data is investigated and results are visualized by the techniques introduced in the previous section. Section~\ref{sec:conclusion} concludes the paper.}
\section{Methodology}\label{sec:methodology}
Let $\bs X$ denote a data matrix of $n$ observations $\bs X = ( \bs x_1, \dots , \bs x_n)'$ in a $p$-dimensional space, $\bs x_i \in \mathbb R^p$, $i=1,\dots ,n$. We further assume the presence of $G$ classes where the class memberships of the observations are stored in a categorical vector $\bs y$ with $y_i=g$ iff $\bs x_i$ comes from group $g$, for $g\in\{1,\dots,G\}$. The number of observations in group $g$ is denoted by $n_g$ with $n=n_1 + \dots + n_G$. For all observations we assume that they have been drawn from $G$ different continuous \blue{probability distributions}.
\blue{For our methodology, it is important that each space spanned by a subset of $k$ observations has a dimension of at least $G-1$ and that there are no ties present in our data. These requirements automatically imply high-dimensional dataspaces as the area of application. Both assumptions can be met by a preprocessing step, removing duplicate or linearly dependent observations. Note that these restrictions only apply for the training data but not for new observations.}
Previous research \citep{ortner2017guided, ortner2017local} shows the effectiveness of using series of projections to overcome the limitations caused by a flat data structure. In this section, the local discrimination method is introduced, which allows for the number of variables $p$ to exceed the number of observations $n$. The idea is as follows. For a fixed observation $\bs x_i$, its $k$ nearest neighbours are identified, called the \textit{core} of $\bs x_i$, which are used to define a $k-1$ dimensional hyperplane, the core space. The Euclidean distance to this hyperplane, called \textit{orthogonal distance}, is calculated for each observations. The hyperplane and the orthogonal distance together define a $k$-dimensional subspace, the local discrimination space, where an LDA model is estimated. This approach is performed for each observation resulting in $n$ LDA models. To assign the class membership to an observation, its posterior probabilities of all models are aggregated.
\subsection{Local discrimination space} \label{ss:localprojections}
Let $d^g_k(\bs x)$ denote the $k$th-smallest distance from $\bs x$ to any observation from class $g$, for $g\in\{1,\dots,G\}$. According to \cite{ortner2017guided} and \cite{ortner2017local}, we define the \textit{core} of $\bs x_i$ as the k-nearest class neighbors of $\bs x_i$, \begin{equation} core(\bs x_i) = \{ \bs x_j: d( \bs x_i, \bs x_j ) \leq d_k^g(\bs x_i) \wedge y_i = y_j = g \} = \{ \bs x_{i_1}, \dots , \bs x_{i_k} \}, \end{equation} where $d( \bs x_i, \bs x_j ) $ denotes the Euclidean distance between $\bs x_i$ and $\bs x_j$, and $i_1,\dots,i_k$ are the indices of the core observations within $\bs X$. In contrast to \cite{ortner2017local}, we use all k-nearest class neighbours as we can use the group membership in order to guarantee a \textit{clean} core, i.e. no observations from other groups within the core.
Any of the $n$ available cores $core(\bs x_1), \dots, core(\bs x_n)$ can be used to unambiguously define an affine subspace spanned by the core observations. In order to determine the projection onto this subspace, we center and scale the data with respect to the $k$ core observations $\bs x_{i_j}$, $j=1,\dots , k$. \begin{align} \hat{\bs \mu}_{i} = & \frac{1}{ k} \sum_{j=1}^k \bs x_{i_j} \label{eq:center} \\ \bs{\hat{ \sigma}}_{i} =& \left( \sqrt{\hat{Var}(x_{i_11}, \dots, x_{i_k1})}, \dots,
\sqrt{\hat{Var}(x_{i_1p}, \dots, x_{i_kp})} \right)' \\ \nonumber = & (\hat{\sigma}_{i1}, \dots, \hat{\sigma}_{ip})' , \end{align} where $\hat{Var}$ denotes the sample variance. For the ongoing work, we denote $\tilde{\bs X}^i = (\tilde{\bs x}_{1}^i,\dots,\tilde{\bs x}_{n}^i)'$ as the data matrix of centered and scaled observations based on the location and scale estimators $\hat{\bs \mu}_i$ and $\hat{\bs \sigma}_i$ of the core of $\bs x_i$. A projection onto the subspace spanned by the core of $\bs x_i$ is defined by $\bs V_i$ from the singular value decomposition (SVD) of the centered and scaled core observations $(\tilde{\bs x}_{i_1}^i,\dots,\tilde{\bs x}_{i_k}^i)'=\bs U_i \bs D_i \bs V_i'$. Since the core of $\bs x_i$ consists of exactly $k$ linearly independent observations, $\bs D_i$ is a $k-1$ dimensional diagonal matrix with non-zero singular values in the diagonal.
Since the idea of no ties being present in the data and each core consisting of linearly independent observations may appear like a strong limitation, an adjustment of the definitions can help \blue{in order to avoid a preprocessing step}. If we interpret the core of $\bs x_i$ as a set of observations, where iteratively the observation from the same class, closest to $\bs x_i$ is added until a $k-1$ dimensional subspace is spanned, we only need to guarantee the existence of such cores, which is a much weaker assumption.
Given the projection matrix $\bs V_i$ from the decomposition, a representation of the data $\bs X$ in the core space is defined by down-projecting the centered and scaled data matrix, $\bs Z^i = \tilde{\bs X}^i \bs V_i$. The core representation consists of $k-1$ orthogonal variables, while the $p-k$ dimensional complement of $\bs Z^i$ defines the orthogonal complement of the core space. In contrast to commonly used procedures of first reducing the dimensionality using PCA and then performing a discrimination method like LDA, we acknowledge the fact that the last principal components might contain an important part of the information like exploited by modern outlier detection algorithms~\cite[e.g.][]{hubert2005robpca, kriegel2012outlier}. Since the reduction of dimensionality remains vital, we aggregate the information from the orthogonal complement by considering the Euclidean distance to the core space, \begin{equation}
OD^i(\bs x_j) = || \tilde{\bs x}^i_j - \bs V_i\bs z^i_j ||, \end{equation} where $\bs z^i_j = \bs V'_i \tilde{\bs x}^i_j$ denotes the core representation of $\bs x_j$ given $core(\bs x_i)$.
The combination of the core representation and the orthogonal distance $OD$ in a matrix, $[\bs Z^i, OD^i]$, provides a $k$-dimensional representation for all observations of $\bs X$. \blue{This $k$-dimensional space is the \textit{local discrimination space}. The reduction of the sample space to the local discrimination space results in a good description of the neighbourhood of an observation $\bs x_i$ and also includes grouping structure which is not described in the core space by the orthogonal distances.}
\blue{An LDA model is estimated in the local discrimination space, excluding the observations from the core of $\bs x_i$. It is necesarry to exclude the core observations, because they have very specific properties in the local discrimination space and this would distort the within-group covariance estimation. In \cite{ortner2017guided} it is shown that \begin{align} OD^i(\bs x_j) &= 0 & \forall \bs x_j \in core(\bs x_i) \label{eq:coreOD}\\ SD^i(\bs x_j) &\equiv const. & \forall \bs x_j \in core(\bs x_i), \label{eq:coreSD} \end{align} \blue{where SD represents the score distance defined as the Euclidean distance within the core space.} These properties hold because for $\bs x_j \in core(\bs x_i)$ the full information is located in the core space, so the orthogonal distances are zero. The scaling applied to the data based on the covariance estimation of the core observations leads to constant score distances for $\bs x_j \in core(\bs x_i)$. So the core observations must not be included in the computation of the LDA model. The model estimated on the remaining observations in the local discrimination space is denoted by $LDA_i$.}
\blue{For the model $LDA_i$ the posterior probability of group $g$ given an observation $\bs x$ is defined by \begin{align}\label{eq:posteriorprob} P_{LDA_i}(g \mid \bs x)=\frac{h_g(\bs x) }{\sum_{j=1}^G h_l(\bs x) } , \end{align} where $h_g(\bs x)$ denotes the estimated density of a multivariate normal distribution with the group mean of class $g$ as center and the pooled within-group covariance matrix as covariance estimate.}
\subsection{Weighting/aggregating local projections} \label{ss:weighting}
We now have a set of $n$ local discrimination spaces and their respective LDA models. In order to receive an overall classification rule for a new observation $\bs x$, we need to aggregate the $n$ available models from the core spaces. \blue{ We accomplish such an aggregation by using the posterior probabilities defined in Equation \eqref{eq:posteriorprob}. First we consider the mean over all $n$ posterior probabilities of $\bs x$ belonging to group $g$, for $g\in\{1,\dots,G\}$,} \begin{equation} \tilde{P}_{LP^k_1}(g \mid \bs x ) = \frac{1}{n} \sum_{j=1}^n P_{LDA_j}(g \mid \bs x) \label{eq:postLP_const1} , \end{equation} and we define the aggregated posterior probability of $\bs x$ belonging to group $g$, for $g\in\{1,\dots,G\}$, as \begin{equation} P_{LP^k_1}(g \mid \bs x) = \frac{ \tilde{P}_{LP^k_1}(g \mid \bs x) }{ \sum_{j=1}^G \tilde{P}_{LP^k_1}(j \mid \bs x) } \label{eq:postLP_const2} . \end{equation} These new aggregated posterior probabilities are based on a fixed number $k$ describing the number of core observations as indicated by the index of $LP_1^k$.
The posterior probabilities of the LDA models, $P_{LDA_i}(g \mid \bs x)$, compared to the true class membership of $\bs x$ reflect the quality of separation in the respective local projection. We distinguish between two quality measures. \blue{Let $q^{g+}_{i}$ denote the mean posterior probability of belonging to class $g$ over all observations actually coming from class $g$, with respect to the model $LDA_i$, i.e. \begin{equation} q^{g+}_i = \frac{1}{n_g} \sum_{k: y_k = g} P_{LDA_i}(g \mid \bs x_k ) \end{equation} and $q_i^{g-}$ the mean posterior probability of non-class-$g$ observations being classified as class $g$ observations given the model $LDA_i$, i.e.} \begin{equation} q_i^{g-} = \frac{1}{n - n_g} \sum_{k: y_k \neq g} P_{LDA_i}(g \mid \bs x_k ). \end{equation} Based on $q_i^{g+}$ and $q_i^{g-}$, we define weights $w_i^g$ representing the quality of each local projection $i=1, \dots,n$ for each group $g\in\{1,\dots,G\}$, \begin{equation} w_i^g = exp\left( q_i^{g+} - q_i^{g-} \right) . \end{equation}
Based on these quality measures $w_i^g$, we redefine the overall posterior probabilities from Equation \eqref{eq:postLP_const2} by weighting each projection for each class with the respective weight. Note that these weights are class-specific and, therefore, a class-individual standardization of weights is required. In our notation, we remove the subscript $1$ from Equation~\eqref{eq:postLP_const1} and Equation~\eqref{eq:postLP_const2}, which represents constant weights of $1$ for each local projection, resulting in: \begin{align} \tilde{P}_{LP^k}(g \mid \bs x ) & = \frac{1}{\sum_{i=1}^n w_i^g} \sum_{i=1}^n w_i^g P_{LDA_i}(g \mid \bs x ) \\ P_{LP^k}(g \mid \bs x ) & = \frac{ \tilde{P}_{LP^k}(g \mid \bs x ) }{ \sum_{j=1}^G \tilde{P}_{LP^k}(j \mid \bs x ) } \label{eq:pp} \end{align} Equivalently to classical LDA, we use these posterior probabilities to assign an observation $\bs x$ to a class $\hat{y} = \text{argmax}_{g\in\{1,\dots,G\}} P_{LP^k}(g \mid \bs x )$. \blue{This decision rule defines the local discrimination model.}
\subsection{The choice of $k$}
\label{ss:trainingLDA}
\blue{The computation of LDA models in the full dimensional space, given more variables than observations are available, requires data preprocessing including dimension reduction \cite[e.g.][]{barker2003partial, chen2013near} or the parallel performance of model estimation and variable selection \cite[e.g.][]{witten2011penalized, hoffmann2016sparse}. The concept of local projections allows us to compute an LDA model for each local projection due to the low dimensional core space. The parameter determining the dimensionality is $k$ of the $k$-nearest class neighbours. It is important to properly tune $k$ since it defines the degree of locality for each projection. Smaller $k$'s are able to better describe a lower dimensional manifold on which groups might be located but increase the risk of not being able to properly describe the local data structure.}
\blue{The number of classes $G$ as well as the number of observations $n_g$ for $g\in\{1,\dots,G\}$ provide a first limitation for the range of $k$. In order to compute an LDA model with G classes, a dimensionality equal to at least $G-1$ is required. Therefore, \begin{equation} G-1 \leq k \end{equation} provides a lower boundary for $k$.}
\blue{To identify an upper boundary for $k$, two properties of the core observations must be taken into account. Due to the specific properties of the core observations stated in Equations \eqref{eq:coreOD} and \eqref{eq:coreSD}, they are not included in the computation of the LDA model. Therefore, an upper boundary for $k$ is given by \begin{equation} k \leq n-(k+1) \end{equation} to guarantee a non-singular covariance estimation. It is useful to further reduce the upper boundary of $k$ in order to allow for a reasonable covariance estimation. Here we take three times more observations than variables leading to the limitation \begin{equation} 3k \leq n-k. \end{equation}}
\blue{With these restrictions on $k$, LDA models in the core spaces can be computed but for the evaluation of the models further limitations are necessary. To be able to evaluate the LDA models, we depend on the posterior probabilities of observations for each class in order to determine the risks of misclassification. Since a core consists of observations from the same class only and the core observations are excluded from the LDA model, the size of the smallest class needs to exceed $k$. \begin{equation} k+1 \leq \min_{g\in \{ 1, \dots, G \}} n_g-1 \end{equation}} \blue{ Due to the identified restrictions on $k$, we optimize $k$ within the following interval: \begin{equation}\label{eq:intervalk} \left[ G-1, \min\left( \frac{n}{4} , \min_{g \in \{ 1, \dots, G \}} n_g -1 \right) \right] \end{equation}}
\blue{For a given $k$, the misclassification rate of the local discrimination model is calculated by summing up the number of misclassified observations (again excluding the core observations) divided by $n-k$, the total number of observations. The tuning parameter $k$ is chosen from within the interval described in Equation \eqref{eq:intervalk} in such a way that the misclassification rate is minimized. }
\section{Visualization of the discrimination} \label{sec:visualization}
\blue{In linear discriminant analysis, the projection space is used for the visualization of the discrimination \cite[e.g.][]{hair1998multivariate}. The Mahalanobis distances of observations to the class centers refer to the posterior probabilities of the observations for the respective classes. This approach is not feasible for local discrimination since each LDA model refers to a different subspace and the aggregated posterior probabilities do not refer to one specific low dimensional space, where the posterior probabilities could be visualized.}
We \blue{therefore} focus on visualizing the aggregated posterior probabilities and follow an approach for compositional data using ternary diagrams. We present the visualization technique on the four-group \textit{Olitos} dataset which is used as a benchmark dataset for robust, high-dimensional data analysis. The dataset is publicly available in the R-package \textit{rrcovHD} and was originally described by \cite{armanino1989chemometric}.
\cite{hron2013robust} used ternary diagrams to visualize the outcome of \blue{(three-group)} fuzzy clustering results, which can be interpreted the same way as posterior probabilities of discrimination models. The difficult aspect about ternary diagrams is the limitation to three variables. Therefore, we select two classes, use the respective \blue{posterior} probabilities and as third composition the sum of \blue{posterior} probabilities for all remaining classes. \blue{This new three-class composition is visualized in Figure \ref{fig:ternary1}.}
\iffigs \begin{figure}
\caption{The aggregated posterior probabilities of two classes (class 1 and class 2) compared to the remaining classes is visualized as a ternary diagram. The dashed lines represent classification rules. Observations located in the white areas can be assigned to the respective group, while the grey area represents an uncertain area, where no reliable statement can be made. The grey dashed lines refer to posterior probabilities of the selected classes.}
\label{fig:ternary1}
\end{figure} \fi
\blue{Figure \ref{fig:ternary1} shows the proposed representation for a sample of the Olitos dataset. The focus of this representation is the evaluation of the separation between the two selected classes $1$ and $2$. The gray dashed lines and the numbers on the left side show the posterior probability for an observation to belong to group 1. The two white areas at the bottom separated by a vertical dashed line represent the classification rules for the separation between class 1 and 2. Observations in the left area are assigned to group 1 and in the right area to group 2. In the bottom right area we can identify one outlier from the \textbf{blue} class and one from class $1$ which are wrongly assigned to group 2. Besides these two false classifications, additional information can be gained from the diagram.}
\blue{First, the grey area represents the region, where no statement about classification can be made with certainty. Two observations $x_1$ and $x_2$ are highlighted there. While $x_1$ is located in the uncertainty area, we can still tell that it will be misclassified since the posterior probability for class $2$ is larger than for class $1$. This decision is indicated by the vertical dashed line within the uncertainty area. However, from this figure, it is not possible to say whether or not it will be assigned to class 2 or to one of the other classes. \blue{The same holds for observation $x_2$. The posterior probability for class $1$ is close to 0.4, for class $2$ to 0.15. Therefore, the posterior probabilities for classes $3$ and $4$ sum up to approximately 0.45. Depending on the class-specific allocation, the maximal posterior probability for the classes $3$ and $4$ varies between 0.225 and 0.45 and the largest posterior probability for $x_2$ can originate from class $1$, $3$ or $4$.}
}
\blue{Second, the white classification area at the top of the triangle visualizes those observations, which with certainty will not be assigned to class 1 or class 2. We note a minor risk of misclassifying observations in the direction of class $1$. Note that the size of the uncertainty area and therefore the size of the third classification area highly depends on the number of groups to be aggregated. In a 3-group case, all posterior probabilities can be visualized and no area of uncertainty exists as shown in Figure \ref{fig:ternary2a}. The remaining plots of Figure \ref{fig:ternary2} show the impact of increasing numbers of groups on the area of uncertainty.}
\iffigs \begin{figure}
\caption{The effect of the number of aggregated groups is visualized. Figure (a) refers to a three-class case, (b) to a four-class case, (c) to a five-class case, (d) to a six-class case and (e) to a ten-class case.}
\label{fig:ternary2a}
\label{fig:ternary2b}
\label{fig:ternary2c}
\label{fig:ternary2d}
\label{fig:ternary2e}
\label{fig:ternary2}
\end{figure} \fi
\blue{Finally, the positioning of the observations in the ternary diagram provides some insight on the connection between the groups. The red observations from class $2$ in Figure \ref{fig:ternary1} are mostly aligned along the axis from $1$ to $2$. Observations from the aggregated group are also mostly aligned between $1$ and $3$ while the black observations split up between class $2$ and $3$. Therefore, class $2$ is strongly connected to class $1$ but has no connection to the aggregated classes. Observations like $x_3$ strongly deviate from the typical class direction and should therefore be candidates for further investigation in the context of outlier analysis.}
\iffigs \begin{figure}
\caption{A set of ternary diagrams is used to visualize the classification performance for each possible combination of classes. While the color remains constant, we switch labels to emphasize the currently selected classes labeled on the bottom and left of the diagrams.}
\label{fig:ternarymat}
\end{figure} \fi
Since the representation in Figure \ref{fig:ternary1} uses a \blue{three-components} representation, which can not illustrate the overall discrimination result, we propose to use a combination of ternary diagrams in the form of a \blue{scatterplot matrix}, as presented in Figure \ref{fig:ternarymat}. Each combination of possible \blue{two-class plus one aggregated group} classifications is presented as described for Figure \ref{fig:ternary1}. In order to align the groups and increase the readability, the diagrams have been rotated accordingly.
Besides providing information on the quality of discrimination and risks of misclassification, we can derive an \blue{overall picture about group connectivity. We already remarked on the location of class $2$. The positioning of the observations of the green and blue observations in the second and third column of the matrix reveals that the green group has stronger ties to the black group, while the blue observations are equally drawn to its direct neighbors, the black and the blue group. Such insight on connectivity provides a feeling for the location of groups in high dimensional spaces which is in general a non-trivial task limited by the human spacial sense.}
\section{Evaluation} \label{sec:evaluation}
In order to evaluate the performance of our proposed local discrimination approach\blue{, abbreviated by LP for Local Projections,} we use three real-world datasets which have previously been used as benchmark datasets for high dimensional data analysis. Based on those datasets, we compare with well-established classification methods from the fields of computer science and statistics. While the visualization introduced in Section \ref{sec:visualization} provides interesting insights into each dataset and will be provided as well, we focus on comparing the used methods based on the misclassification rate.
For each dataset, we split the available observations into training and test dataset. The same training dataset is used for each method to estimate the discrimination model and the same test dataset is used to evaluate the performance of all the models by reporting the misclassification rates.
The employed datasets consist of groups of different numbers of observations. \blue{Since the outcome of a method can be strongly affected by the specific choice of the training and test set,} we resample the observations 50 times per dataset, creating a series of training and test datasets resulting in a series of misclassification rates. The overall performance is then measured based on the median misclassification rate as well as on the deviation from the median misclassification rate.
\subsection{Compared methods}
The selection of classification methods is based on the popularity of the methods, the importance for our setups, and the relevance for our proposed approach. The most important aspect is the applicability on the evaluated datasets. The crucial factor is the flat data structure (more variables than observations), especially in class-specific subsets of the overall dataset. In order to cover related classification methods, we include Linear Discrimination Analysis (LDA) as this is the classification method internally used for each local projection. We further include statistical advancements of LDA, which try to deal with disadvantageous properties of our datasets of interest, namely penalized LDA and partial least squares for discriminant analysis. The most related method from the field of computer science is KNN-classification as our local projections are based on a knn-estimation. The last methods included in the evaluation are support vector machines and random forests to cover the most commonly used classification approaches from the field of computer science.
For \textbf{Linear Discriminant Analysis} (LDA), it is assumed that the covariance structure is the same for each class and has elliptical shape. Under this assumption, the optimal decision boundaries to separate the groups are linear. The separation of the classes is achieved by taking $G-1$ orthogonal directions which maximize the within-group variance to the between-group variance. In this $G-1$ dimensional space, the Euclidean distance to the group centers is used to assign an observation to the group with the closest center.
For the calculations, the \texttt{lda} function from the R-package \texttt{MASS} is used. This implementation can be applied to data with $p>n$ by performing singular value decomposition and reducing the dimensionality to the rank of the data.
\textbf{Penalized LDA} (PLDA) introduced by \citet[][]{witten2011penalized} is a regularized version of Fisher's linear discriminant analysis. A penalty on the discriminant vectors favours zero entries, which leads to variable selection. The influence of the penalty is controlled by the sparsity parameter $\lambda$: larger values of $\lambda$ lead to fewer variables in the model.
The sparsity parameter $\lambda$ is selected from 10 values between $10^{-4}$ and $5$ by 10-fold cross-validation on the training data using as selection criterion the minimum mean misclassification rate. The number of discriminating vectors is set to $G-1$. The functions for cross validation and model estimation are provided in the R-package \texttt{penalizedLDA} \citep{peanlizedLDA}.
\textbf{Partial least squares for discriminant analysis} (PLSDA) was theoretically established by \cite{barker2003partial}, where its relationship to LDA and the application to flat data was discussed. PLSDA performs in a first step a projection onto $K$ latent variables, which considers the grouping information of $y$. Then LDA is performed in the reduced space.
For the evaluation the R-package \texttt{DiscriMiner} \citep{DiscriMiner} is used, which provides code for the selection of the number of components $K$ by leave-one-out cross-validation.
\textbf{Support Vector Machines} (SVMs) are a popular machine learning method for classification. \blue{The margins between the groups of the training data are maximized in a data space induced by the selected kernel.} While a variety of kernels is available (e.g. linear, polynomial, sigmoid, etc.), we limit the optimization procedure to the radial basis kernel, which is suggested as standard configuration.
We use an R-interface to \textit{libsvm} \citep{chang2011libsvm} included in the R-package \textit{e1071}. The internal optimization of SVM is based on a $k$-fold cross-validation on the training dataset\blue{, providing a range of values for the cost parameter and for $\gamma$}. For multi-class-classification, libsvm internally trains K(K-1)/2 binary ‘one-against-one’ classifiers based on a sparse data representation matrix.
\textbf{Random Forest} (RF) is an ensemble-based learning method commonly used for classification and regression tasks. It builds a forest of decision trees using bootstrap samples of the training data and random feature selection for each tree. The final prediction is made as an average or majority vote of the predictions of the ensemble of all trees.
The RF implementation in the R-package \textit{randomForest} uses Breiman's random forest algorithm \citep{breiman2001random} for multigroup classification. In order to optimize the classification model, we use the internal optimization procedure starting with $\sqrt{p}$ randomly sampled variables as candidates for splits and increase this number with a factor of 1.5 in each optimization step.
In \textbf{KNN-Classification} (KNN), the class-membership of the $k$-nearest neighbors of an observation based on Euclidean distances is used for determining the class of the respective observation. For $k=1$, the class of the nearest neighbor is used, for $k>1$, the class with the highest frequency is used. In the case of ties, a random decision is performed. We use one-fold cross-validation in order to optimize $k$ individually for each sampled dataset.
\subsection{Olive oil}
The first dataset in our experiments consists of 120 samples of 25 chemical compositions (fatty acids, sterols, triterpenic alcohols) of olive oils from Tuscany, Italy, and was first introduced by \citet{armanino1989chemometric}. The dataset is publicly available in the R-package \textit{rrcovHD} \citep{todorov2014rrcovhd} where it is used as a reference dataset for robust high-dimensional data analysis.
The olive oils are separated in four classes of 50, 25, 34 and 11 observations. In order to have enough training observations from each group available, we use $80\%$ of observations for the training dataset and the remaining $20\%$ as test observations. We repeatedly create such an evaluation setup 50 times. Hence, each training dataset consists of 96 observations, which yields the only setup where we have more observations than variables available. Therefore, classical LDA is expected to perform fairly well. Note that the smallest number of training observations per class is still much smaller than the number of overall variables. Therefore, class-specific covariance estimation as it is performed in quadratic discriminant analysis \citep{friedman1989regularized} cannot be performed in this setup or on any other of our considered datasets.
LP and LDA perform exactly the same, which can be seen in Figure \ref{fig:performanceOlitos}. PLDA slightly outperforms LP, while PLSDA, SVM, RF, and especially KNN get outperformed. In most cases all variables are included in the PLDA model but only a subset of variables contributes to each discriminant vector. This variable selection leads to a slight improvement over LDA and LP.
\iffigs \begin{figure}
\caption{The performance in terms of false classification rates of all considered classification methods for 50 repetitions of the \textit{Olitos} dataset is visualized by boxplots.}
\label{fig:performanceOlitos}
\end{figure} \fi
\subsection{Arcene}
The second real-world dataset is part of the NIPS (Neural Information Processing Systems) 2003 feature selection challenge \citep{guyon2007competitive}. The task is to distinguish between cancer and non-cancer patterns from mass-spectrometric data with $p=9961$ variables. Therefore, we deal with a two-class separation with continuous variables. The data was obtained from two different data sources, the National Cancer Institute (NCI) and the Eastern Virginia Medical School (EVMS). The observations represent patients with ovarian or prostate cancer and health or control patients. Very small and large masses have been removed from the spectrometric data in order to compress the data. In addition, a preprocessing step including baseline removal, smoothing and scaling was performed. All these details are described in \cite{guyon2007competitive}.
The initial setup contained of 100 training and 100 validation observations, consisting of a total of 112 non-cancer samples and 88 cancer samples. In order to have a non-equal ratio of observations \blue{to create again an imbalanced scenario}, we merge both groups and resample 22 cancer training observations and 84 non-cancer trainining observations. The remaining observations are used as test observations. This procedure is repeated 50 times as for the other datasets.
\iffigs \begin{figure}
\caption{The performance in terms of false classification rates of all considered classification methods for 50 repetitions of the \textit{Arcene} dataset is visualized by boxplots.}
\label{fig:performanceArcene}
\end{figure} \fi
The performance of Arcene is evaluated in terms of boxplots in Figure \ref{fig:performanceArcene}. The classification for this dataset and the designed setup is more challenging than for the other real-world datasets. LP performs well in comparison to the other evaluated approaches being outperformed only by PLSDA. The performance of 80\% false classification rate by SVM might be misleading as it appears worse than random classification. All observations from the non-cancer samples are classified as cancer samples. \blue{This could be improved by strategies like oversampling or by changing the majority class assignment to a weighted class assignment. For LP it is not necessary to make adjustments for group imbalance.}
\subsection{Melon}
Our final dataset consists of measurements of three types of melons based on spectra analyses of 256 frequencies. The fruits are pertain to three different melon cultivars with group sizes of 490, 106 and 499 but additional subgroups are known to be present due to changes in the illumination system during the cultivation. The dataset is regularly used as a benchmark dataset for high-dimensional and robust data analysis methods \citep[e.g.][]{hubert2004fast}. Especially the subgroups usually affect non-robust analysis methods. Figure \ref{fig:datasetupMelon} provides some insight on the structure of the dataset.
\iffigs \begin{figure}
\caption{Visualization of the first three principal components of one sample of training observations of the \textit{Melon} dataset. We see one subgroup of the green class, represented in the first principal components and a strong overlapping structure for the remaining observations.}
\label{fig:datasetupMelon}
\end{figure} \fi
We repeatedly sample 25\% of the observations \blue{from each group} as training observations using the remaining 75\% for testing the model performance. The smallest training class therefore consists of 26 observations leading to a complex classification problem. The performance of the compared methods is presented in Figure \ref{fig:performanceFruit}. LP can handle the challenges of the Melon dataset the best and significantly outperforms all compared methods. \blue{Especially PLDA results in a high false classification rate which is assumed to be related to the subgroups and outliers affecting the variable selection.}
\iffigs \begin{figure}
\caption{The performance in terms of false classification rates of all considered classification methods for 50 repetitions of the \textit{Melon} dataset is visualized by boxplots.}
\label{fig:performanceFruit}
\end{figure} \fi
One problem during the visualization of LDA models is the property that in high-dimensional spaces with more variables than observations, the training observations will almost always be well-separated. Therefore, in a situation where we do not have enough observations to validate the model based on additional observations, a visualization of the discrimination space does not provide a lot of insight on the risks for misclassification of this model. These challenges are visualized in Figure \ref{fig:ldamodela} and Figure \ref{fig:ldamodelb}.
\iffigs \begin{figure}
\caption{Plot (a) shows the training observations of the LDA-projection space for one repetition of the \textit{Melon} evaluation. Plot (b) shows the same projection for the respective setup.}
\label{fig:ldamodela}
\label{fig:ldamodelb}
\label{fig:visFruitLDA}
\end{figure} \fi
We see a perfect separation in Figure \ref{fig:ldamodela} and have no indication of risks of misclassification.
\blue{The risk of misclassification can be evaluated using the aggregated posterior probabilities of LP, defined in Equation \eqref{eq:pp}, which provide the advantage that each of the classification models is located in a low dimensional space.} Figure \ref{fig:visFruitLP} provides the visualization of the same data setup as used in Figure \ref{fig:visFruitLDA}. The risk of misclassifying observations from class $1$ as class $2$ and vice verse becomes evident in Figure \ref{fig:lpmodela} and the realization of this risk becomes evident in Figure \ref{fig:lpmodelb}. Note that this visualization can be adapted and used for posterior probabilities computed through cross-validation by any arbitrary classification method.
\iffigs \begin{figure}
\caption{The same data setup as in Figure \ref{fig:visFruitLDA} is used. Plot (a) shows the proposed visualization of aggregated posterior probabilities from local projections for the training observations. Plot (b) visualizes the same aggregations for the respective test observations.}
\label{fig:lpmodela}
\label{fig:lpmodelb}
\label{fig:visFruitLP}
\end{figure} \fi
\blue{A further experiment is carried out with the Melon dataset: While in the previous experiment, the training datasets contained 25\% of the observations from the original groups (with sizes 490, 106 and 499), we now investigate the effect of modifying the group sizes to be very imbalanced. We investigate six different scenarios with varying group sizes but the same overall sample size of $n=250$ (see Table \ref{tab:ImbalancedMelon}). } Figure \ref{fig:ImbalancedMelon} shows the mean misclassification rate over 50 repetitions. Scenario 1 and scenario 6, with the most extreme difference in the group sizes, result in the worst results for several methods. The LDA models are very stable but most of the time they are outperformed by LP. LP is only slightly affected by scenario 1 but otherwise it leads to similar results for the different settings outperforming all other methods. Note that classification methods could be tuned in order to cope with imbalanced groups. \blue{For example, for random forests there are different strategies to adjust the group assignments if the group sizes are very different from each other \citep[e.g.][]{khoshgoftaar2007empirical, khalilia2011predicting}. However, according to the results shown in Figure \ref{fig:ImbalancedMelon}, the performance of LP is very stable even in case of imbalanced groups.}
\iffigs \begin{table}[!htb] \centering \caption{\blue{Group sizes for simulation scenarios for the Melon dataset. We vary the numbers of observations per group in order to simulate highly imbalanced group sizes.}}
\label{tab:ImbalancedMelon}
\begin{tabular}{lrrrrrr}
\hline
Scenario & 1 & 2 & 3 & 4 & 5 & 6\\
\hline
Class 1 & 25 & 50 & 75 & 100 & 125 & 150 \\
Class 2 & 75 & 75 & 75 & 75 & 75 & 75 \\
Class 3 & 150 & 125 & 100 & 75 & 50 & 25\\
\hline
\end{tabular}
\end{table}
\fi
\iffigs \begin{figure}\label{fig:ImbalancedMelon}
\end{figure} \fi
\section{Conclusions \& Outlook}\label{sec:conclusion}
\blue{We proposed a methodology for supervised classification combining aspects from the field of computer science and from the field of statistics. We use the concept of local projections to compute a set of linear discriminant models taking the information within each projection space and the distance to the projection spaces into account. The LDA models are then aggregated based on the projection-based degree of separation. As shown in \cite{ortner2017guided}, local projections can help to identify group structure in high-dimensional spaces. Therefore, this way of computing aggregated probabilities for class-membership allows the utilization of LDA for high-dimensional spaces while exploiting the advantages of identifying group structure by local projections.}
\blue{Additionally, a novel visualization based on ternary diagrams has been proposed which reveals links between the groups in high-dimensional space. The visualization makes use of the posterior probabilities computed from the local projections and therefore it allows to draw conclusions about the uncertainty of the class assignment supported by gray areas in the plot for uncertain assignment.}
\blue{The conducted evaluations on the performance of LP in comparison to related supervised classification methods (LDA, PLDA, PLSDA, SVM, RF and KNN) based on three different real-world datasets demonstrated the advantage of LP in} \blue{various settings: two- and multi-group classification tasks, higher number of observations than variables and vice versa, inhomogeneous groups caused by outliers, and imbalanced group sizes. The only tuning parameter required for LP is the number $k$ of nearest neighbors, for which a lower and upper boundary has been proposed.}
\blue{While we utilize linear discriminant analysis performed on the projection space of each local projection, there is no reason to limit ourselves to LDA. Depending on the data setup, other methods can be preferred over LDA and still benefit from the local projection based aggregation. A general combination of classification approaches with local projections is still to be evaluated in future work.}
\end{document} | arXiv | {
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\begin{document}
\begin{center} \uppercase{\bf On Upper Bounds for the Count of Elite Primes} \vskip 20pt {\bf Matthew Just}\\ {\smallit Department of Mathematics, University of Georgia, Athens, Georgia 30605, United States}\\ {\tt justmatt@uga.edu}\\
\end{center}
\centerline{\bf Abstract} \noindent We look at upper bounds for the count of certain primes related to the Fermat numbers $F_n=2^{2^n}+1$ called elite primes. We first note an oversight in a result of K{\v{r}}{\'\i}{\v{z}}ek, Luca and Somer and give the corrected, slightly weaker upper bound. We then assume the Generalized Riemann Hypothesis for Dirichlet $L$ functions and obtain a stronger conditional upper bound.
\section{Introduction}
The Fermat numbers are given by $F_n=2^{2^n}+1$ for $n\geq 0$. The first few Fermat numbers are 3, 5, 17, 257, 65537, 4294967297. Notice that the first five Fermat numbers are prime, and it was initially conjectured (by Fermat) that all such numbers are prime. The sixth Fermat number is not prime, and no other Fermat primes are known. It is known that $F_n$ is composite for $5\leq n \leq 32$ though interestingly no prime factor of $F_{14}$, $F_{20}$, $F_{22}$, or $F_{24}$ is known (see \cite{crandall2003twenty}). An efficient test exists to determine whether or not a Fermat number is prime, called P\'epin's test. \begin{proposition}[P\'epin's test \cite{ribenboim2012new}] Let $n>0$. Then $F_n$ is prime if and only if \[3^{(F_n-1)/2} \equiv -1 \emph{$\mod{F_n}$}.\] \end{proposition}
\begin{comment} \begin{proof} First suppose that $F_n$ is prime. Then by Euler's criterion $3^{(F_n-1)/2} \equiv -1 \mod{F_n}$ if and only if 3 is a quadratic nonresidue$\mod{F_n}$. That is to say \[\left(\frac{3}{F_n}\right) = -1\] where here we use the Legendre symbol defined for odd primes $p$ by \[\left(\frac{a}{p}\right)=\begin{cases}0 & \text{if $p\mid a$} \\ 1 & \text{if $p\nmid a$ and $a$ is a quadratic residue$\mod{p}$ } \\-1 & \text{if $p\nmid a$ and $a$ is a quadratic nonresidue$\mod{p}$ }\end{cases}\] Now for $n>0$ it is clear that $F_n \equiv 1\mod{4}$ so by the law of quadratic reciprocity \[\left(\frac{3}{F_n}\right) = \left(\frac{F_n}{3}\right)\] so it is enough to show that $F_n$ is a quadratic nonresidue$\mod{3}$. But for $n>0$ it is again clear $F_n\equiv -1 \mod{3} $, so $F_n$ is a quadratic nonresidue$\mod{3}$.
Now if $3^{(F_n-1)/2}\equiv -1 \mod{F_n}$ let $f$ be the order of $3\mod{F_n}$. Then $f\mid (F_n -1) = 2^{2^n}$. But it is also that case that $f\nmid (F_n-1)/2 = 2^{2^n-1}$, so $f=2^{2^n}$. Thus the order of the multiplicative group of units$\mod{F_n}$ has order $2^{2^n}=F_n-1$ which can only happen if $F_n$ is prime. \end{proof} \end{comment}
The only fact about the prime 3 used in this proof is that $F_n$ is a quadratic nonresidue$\mod{3}$ for $n>0$. Thus we can replace 3 with any other prime that satisfies this requirement.
Primes $p$ for which all sufficiently large Fermat numbers are quadratic nonresidues$\mod{p}$ are called \textit{elite primes}, a term introduced by Aigner \cite{aigner1986primzahlen}. Thus we can use elite primes to test the primality of $F_n$ for all but finitely many Fermat numbers. For more on the search for elite primes see \cite{allep}, \cite{continuingsearch}, and \cite{complex}.
Similarly defined are anti-elite primes which are primes $p$ such that $F_n$ is a quadratic residue for all but finitely many $n$. Generalizations of elite and anti-elite primes for numbers other than the Fermat numbers were studied by M{\"u}ller, see \cite{muller2007anti}, \cite{muller2008generalization}, and \cite{muller2008generalized}.
Let $E(x)$ be the number of elite primes up to $x$. K{\v{r}}{\'\i}{\v{z}}ek, Luca and Somer \cite{kvrivzek2002convergence} gave the upper bound \[E(x) = O\left( \frac{x}{(\log x)^2} \right)\] as $x\rightarrow\infty$. Their proof used the estimate \[\prod_{i=0}^{2t}F_i< 2^{2^{t+1}};\] however one can check that for all $t\geq 0$ \[\prod_{i=0}^{2t}F_i = 2^{2^{2t+1}} -1.\] Using the revised estimate $\prod_{i=0}^{2t}F_{i} < 2^{2^{2t+1}}$ we obtain the following result following their proof: \begin{theorem}
Let $E(x)$ be the number of elite primes up to $x$. Then \[E(x)=O\left(\frac{x}{(\log x)^{3/2}}\right)\] as $x\rightarrow \infty$. \end{theorem}
Corollary 2 of K{\v{r}}{\'\i}{\v{z}}ek, Luca and Somer's paper states that the sum of the reciprocals of the elite primes converges. We note that this still follows from Theorem 1.
We next give our main result which is a stronger upper bound for $E(x)$ conditional upon the Generalized Riemann Hypothesis for Dirichlet $L$ functions:
\begin{theorem}
Let $E(x)$ be the number of elite primes up to $x$. Then assuming the Generalized Riemann Hypothesis \[E(x) = O(x^{5/6})\] as $x\rightarrow \infty$. \end{theorem}
\section{Proof of Theorem 2}
Before beginning the proof we introduce some tools we will use. Let $\left(\frac{a}{m}\right)$ denote the Kronecker symbol.
\begin{comment} The Jacobi symbol, a generalization of the Legendre symbol, is defined for odd numbers $m=p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_k^{\alpha_k}$ by \[\left( \frac{a}{m} \right) = \prod_{i=1}^k \left(\frac{a}{p_i} \right)^{\alpha_i}\] We will use the following version of the law of quadratic reciprocity that states if $a$ and $m$ are odd coprime integers and either $a$ or $m$ is congruent to $1\mod{4}$ then \[\left( \frac{a}{m} \right) = \left( \frac{m}{a} \right)\]
The Kronecker symbol, a generalization of the Jacobi symbol, is defined for all integers $a$ and $m$ (which we write as $m=u2^s k$ where $u=\pm 1$ and $k$ is odd) by \[\left( \frac{a}{m} \right) = \left( \frac{a}{u}\right) \left( \frac{a}{2}\right)^s \left( \frac{a}{k} \right)\] where \[\left(\frac{a}{u}\right) = \begin{cases}1 & \text{if $u=1$ or $a<0$} \\ -1 & \text{if $u=-1$ and $a\geq 0$} \end{cases}\] and \[\left(\frac{a}{2}\right)=\begin{cases}0 & \text{if $2\mid a$} \\ 1 & \text{if $a\equiv \pm1\mod{8}$} \\ -1 & \text{if $a\equiv\pm3\mod{8}$}\end{cases}\] Finally if $m=0$ then \[\left(\frac{a}{0}\right) = \begin{cases}1 & \text{if $a=\pm 1$} \\ 0 & \text{otherwise} \end{cases}\] \end{comment}
We will use the fact that for a fixed integer $a$ satisfying $a\neq 0$ and $a\not\equiv 3\mod{4}$ the Kronecker symbol is a Dirichlet character to the modulus $|a|$ (or $4|a|$ if $a\equiv2\mod{4}$). Furthermore this Dirichlet character is not the principal character as long as $a$ is not a square. The key lemma will be the following classical result which can be found in Montgomery and Vaughan \cite{montgomery2007multiplicative}.
\begin{lemma}
Let $\chi$ be a Dirichlet character to the modulus $q$, not the principal character. Then assuming the Generalized Riemann Hypothesis \[\sum_{p\leq x} \chi(p) = O(\sqrt{x} \log(qx)).\] \end{lemma}
\begin{proof}[Proof of Theorem 2]
Let $p\leq x$ be an elite prime. We write \[p-1 = 2^{e_p} k_p, \] where $k_p$ is odd and let $f_p$ denote the multiplicative order $2\mod{k_p}$. Now for any $m\geq e_p$ we have that \[(p-1)\mid 2^m (2^{f_p}-1)=2^{m+f_p}-2^m,\] which shows by Fermat's little theorem that \[2^{2^{m+f_p}}\equiv 2^{2^m}\mod{p}.\] This periodicity of the sequence of Fermat numbers shows that if there exists an $m\geq e_p$ such that $F_m$ is a quadratic residue$\mod{p}$ then $p$ cannot be elite since $F_{m+\ell f_p}$ would be a quadratic residue$\mod{p}$ for all $\ell \geq 0$.
Now suppose $p$ is a prime with $e_p> t$, where $t$ is a parameter depending on $x$ to be chosen later. Then $p$ lies in the residue class $1\mod{2^t}$. As long as $2^t\leq \sqrt{x}$ then we may apply the Brun-Titchmarsh inequality to get an upper bound on the distribution of such primes in arithmetic progressions: \[\pi(x;2^t,1) \leq \frac{2x}{\phi(2^t) (\log x - \log 2^t)} \ll \frac{x}{2^t}. \]
Now we assume $p$ is an elite prime with $e_p\leq t$. We see now that $p$ must be a quadratic nonresidue modulo $F_{t+i}$ for all $i\geq 0$. Looking at the Legendre symbol we see that for any prime number $p$ \[ 1 - \left( \frac{F_{t+i}}{p} \right) = \begin{cases}2 & \text{if $F_{t+i}$ is a quadratic nonresidue$\mod{p}$}, \\ 0 & \text{if $F_{t+i}$ is a quadratic residue$\mod{p}$} , \\ 1 & \text{if $p\mid F_{t+i}$}.\end{cases} \]
Fixing another parameter $T$ depending on $x$ to be chosen later let \[A=\prod_{i=0}^T F_{t+i}\] so that we now have the following upper bound for $E(x)$: \[E(x) \leq \frac{1}{2^{T+1}}\sum_{p\leq x }\prod_{i=0}^T \left(1-\left(\frac{F_{t+i}}{p}\right) \right) + \sum_{p|A} 1 + O\left(\frac{x}{2^t}\right).\] Notice that \[A<\prod_{i=0}^{t+T} F_i <2^{2^{T+t+1}},\] and thus \[\sum_{p|A}1 \ll \log A \ll 2^T2^t. \]
As for the first term in our estimate for $E(x)$, we have
\[\frac{1}{2^{T+1}} \sum_{j=1}^{T+1} (-1)^j \sum_{\substack{B\subset \{0,1,2,\ldots,T \}\\ |B|=j}} \sum_{p\leq x}\left(\frac{\prod_{b\in B} F_{t+b}}{p} \right) + \frac{\pi(x)}{2^{T+1}}.\] Using the fact that these inner Kronecker symbols are Dirichlet characters to the modulus \[\prod_{b\in B} F_{t+B} \leq A < 2^{2^{t+T+1}}, \] we can apply Lemma 1 once we observe that Fermat numbers are pairwise coprime and hence any product of Fermat numbers will never be a square. We then have the upper bound \[\sum_{p\leq x}\left(\frac{\prod_{b\in B} F_{t+b}}{p} \right) \ll \sqrt{x}(\log x + 2^{t+T}). \] Putting all this together, we have \[E(x) \ll \sqrt{x}\log x + \sqrt{x}2^{t+T} + \frac{x}{2^T} + \frac{x}{2^t}.\]
Letting $t=T=\frac{\log x}{6\log 2}$ gives the desired result. \end{proof}
\noindent{\bf Remark.} Theorem 2 gives an upper bound for the count of elite primes, but if we instead used \[1 + \left( \frac{F_{t+i}}{p} \right) = \begin{cases}2 & \text{if $F_{t+i}$ is a quadratic residue$\mod{p}$} \\ 0 & \text{if $F_{t+i}$ is a quadratic nonresidue$\mod{p}$}\\ 1 & \text{if $p\mid F_{t+i}$},\end{cases}\] we obtain the same upper bound for the count of anti-elite primes. Furthermore, if we consider the generalized Fermat numbers to the base $b$, \[b^{2^n} + 1,\] we can ask what is special about the case $b=2$. In fact the only time the base $b=2$ is used in the proof of Theorem 2 is that for a fixed prime $p$ the sequence $2^{2^n}+1$ will eventually be periodic$\mod{p}$. But this is true for any base $b$, hence Theorem 2 will also hold for the count of generalized elite and anti-elite primes with respect to the generalized Fermat numbers to the base $b$.
\section*{Acknowledgments}
The author was partially supported by the Research and Training Group grant DMS-1344994 funded by the National Science Foundation. He thanks Paul Pollack and Florian Luca for helpful comments.
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} | arXiv/math_arXiv_v0.2.jsonl | null | null |
\begin{document}
\title{\LARGE Neighbor product distinguishing total colorings of corona of subcubic graphs \thanks{This work was supported by the National Natural Science Foundation of China (Grant No. NSFC12001332). It was also supported by China Postdoctoral Science Foundation Funded Project (Grant No.2014M561909); the Nature Science Foundation of Shandong Province of China (Grant No. ZR2014AM028, ZR2017BA009)}} \author{Aijun Dong \quad \quad Wenwen Zhang\\ {\small School of Data and Computer Science,}\\ {\small Shandong Women's University, Jinan, 250300, China}}
\date{}
\maketitle
\begin{abstract}
A proper $[k]$-total coloring $c$ of a graph $G$ is a mapping $c$ from $V(G)\bigcup E(G)$ to $[k]=\{1,2,\cdots,k\}$ such that $c(x)\neq c(y)$ for which $x$, $y\in V(G)\bigcup E(G)$ and $x$ is adjacent to or incident with $y$. Let $\prod(v)$ denote the product of $c(v)$ and the colors on all the edges incident with $v$. For each edge $uv\in E(G)$, if $\prod(u)\neq \prod(v)$, then the coloring $c$ is called a neighbor product distinguishing total coloring of $G$. By $\chi''_{\prod}(G)$, we denote the minimal value of $k$ in such a coloring of $G$. In 2015, Li et al. conjectured that $\chi''_{\prod}(G)\leq\Delta(G)+3$. In this paper, for the corona graph of two arbitrary subcubic graphs, we give an algorithm of a neighbor product distinguishing total coloring and confirm the conjecture put forward by Li et al. \end{abstract}
{\bf Keywords:} total coloring; corona graph; neighbor product distinguishing coloring
\textbf{MSC(2010): 05C15}
\section{Introduction}
All graphs considered in this paper are simple and undirected. Let $G=(V,E)$ be a graph. $V(G)$, $E(G)$, $\Delta(G)$ and $\delta(G)$ were used to denote the vertex set, edge set, maximum degree and minimum degree of $G$, respectively. Let $N_G(u)$ be the set of neighbors of $u$ in the graph $G$. We use $n_g$ and $n_h$ to denote the number of vertices in graphs $G$ and $H$, respectively. The notation and terminology used but undefined here can be found in~\cite{bondy}.
Let $k$ be a positive integer. A total $k$-coloring $c$ is a mapping $c: E(G)\cup V(G)\rightarrow [k]$, such that for any two elements $x,y \in E(G)\cup V(G)$, if $xy\in E(G)$ or $x$ is incident with $y$, then $c(x)\neq c(y)$. For each vertex $u\in V(G)$, let $\prod(u)$ (resp. $S(u)$) denote the product (resp. set) of colors on $u$ and the edges which are incident with $u$. If $\prod(v)\neq \prod(u)$ (resp. $S(u)\neq S(v)$) for each edge $uv\in E(G)$, then it is called $a$ $neighbor$ $product$ $distinguishing$ $total$ $coloring$ (resp. $neighbor$ $vertex$ $distinguishing$ $total$ $coloring$) of $G$. For convenience, they are abbreviated as $NPDTC$ and $NVDTC$, respectively. The chromatic number of $NPDTC$ (resp. $NVDTC$) $\chi''_{\prod}(G)$ (resp. $tndi(G)$) is the smallest integer number $k$ such that $G$ admits a neighbor product (resp. vertex) distinguishing $[k]$-total coloring. Clearly, $tndi(G)\leq \chi''_{\prod}(G)$.
In 2005, Zhang et al.~\cite{zhangzhongfu1} determined the neighbor set distinguishing total coloring index for graphs which are cliques, paths, cycles, fans, wheels, stars, complete graphs etc. and made the following conjecture.
\begin{conjecture}\label{conj1} Let $G$ be a connected graph with at least two vertices, then $tndi(G)\leq \Delta(G)+3$. \end{conjecture}
Wang, Chen~\cite{chenxiangen1, wanghaiying1} and Lu et al.~\cite{luyou2} relayed to verify the Conjecture~\ref{conj1} for graphs with $\Delta\leq4$. The Conjecture~\ref{conj1} was confirmed for some special graphs such as $1$-tree, some sparse graph, $2$-degenerate graph and line and splitting graph of some graphs in~\cite{wanghaiying2},~\cite{wangweifan2},~\cite{miaozhengke1} and~\cite{thirusangu}, respectively. Outer planar graph, series-parallel graphs were studied in~\cite{wangweifan1, wangweifan3}. Chang et al.~\cite{chang1} confirmed the Conjecture~\ref{conj1} for planar graph with $\Delta\geq8$. More related results can be seen in ~\cite{huangdanjun, wangweifan4, chengxiaohan1, hujie}.
Recently, Li et al.~\cite{litong} introduced the notation of NPDTC and proposed the following conjecture.
\begin{conjecture}\label{conj2} If $G$ is a graph with at least two vertices, then $\chi''_{\prod}(G)\leq\Delta(G)+3$. \end{conjecture}
Li et al. prove that the conjecture holds for complete graphs, cycles, trees, bipartite graphs, subcubic graphs and $K_4$-minor free graphs. In 2017, Dong et al. confirmed the Conjecture~\ref{conj2} for sparse graph $G$ with bounded maximum degree~\cite{dong1}. Recently, the Conjecture~\ref{conj2} was confirmed for $2$-degenerate graph in~\cite{zhuyu}.
In this paper, we discuss the NPDTC of graph products which was introduced by Frucht and Harary in 1970~\cite{Frucht}. Given two simple graphs $G$ and $H$, the $corona$ $product$ of $G$ and $H$
is abbreviated as $G\circ H$ which is obtained by taking one copy of $G$, $|V(G)|$ copies of $H$ ($H_1$, $H_2$,$\ldots$,$H_{n_g}$), and making every vertex $v_j$ of $G$ adjacent to every vertex $u^j_i$ (the copy of $u_i$ in $H_j$) of $H_j$ to get edges $v_ju^j_i$ ($1\leq j\leq n_g$, $1\leq i\leq n_h$). The edge $v_ju^j_i$ is called a $corona$ $edge$. Obviously, there are $n_g\cdot n_h$ corona edges in $G\circ H$ in total. For convenience, let $d_G(v)$, $d_H(v)$ and $d(v)$ denote the degree of a vertex $v$ in $G$, $H$ and $G\circ H$, respectively. Obviously, the maximum degree of the corona graph $G\circ H$ is $\Delta(G\circ H)=\Delta(G)+n_h$. A corona graph $K_3\circ K_4$ can be seen in Figure 1.
The concept of the corona product has some applications in chemistry for representing chemical compounds~\cite{johnson}. Other applications of this concept include navigation of robots in networks~\cite{khuller}. Some theoretical results can be seen in~\cite{daouya,mohan}. In this paper, we focus on the corona graph of two arbitrary subcubic graphs and get the following result.
\begin{theorem}\label{theorem1} Let $G$, $H$ be two arbitrary subcubic graphs. Then $\chi''_{\prod}(G\circ H)\leq\Delta(G\circ H)+3$. \end{theorem}
Since the neighbor product distinguishing $k$-total coloring is a generation of the neighbor vertex distinguishing $k$-total coloring, we have the following corollary.
\begin{corollary}\label{corollary1} Let $G$,$H$ be two arbitrary subcubic graphs. Then $tndi(G\circ H)\leq\Delta(G\circ H)+3$. \end{corollary}
\section{Preliminaries} Let us recall some known results that will be useful for the forthcoming proof.
\begin{lemma}[\cite{pilsniak}]\label{lm1} If $G$ is a subcubic graph, then $\chi''_{\prod}(G)\leq \Delta(G)+3$. \end{lemma}
\begin{lemma}[\cite{vizing}]\label{lm2} Let $G$ be a graph. Then $\Delta(G)\leq \chi'(G)\leq \Delta(G)+1$. \end{lemma}
\section{Proof of Theorem~\ref{theorem1}}
\begin{proof}Since $\Delta(G)\leq3$, then $G$ is a subcubic graph. We have $\chi''_{\prod}(G)\leq\Delta(G)+3$ by Lemma~\ref{lm1}. By Lemma~\ref{lm2}, we have $\chi'(H)\leq\Delta(H)+1$. For convenience, we use $c''$ and $c'$ to denote a neighbor product distinguishing $(\Delta(G)+3)$-total coloring of $G$ and a $(\Delta(H)+1)$-edge coloring of $H$, respectively. For each vertex $v\in V(G)$, we use $\prod_{c''}(v)$ to denote the product of colors on the vertex $v$ and the edges which are incident with $v$ in the coloring $c''$ of $G$, i.e. $\prod_{c''}(v)=c''(v)\cdot\prod_{v\in e}(c''(e))$. Clearly, for each edge $v'v''\in E(G)$, we have $\prod_{c''}(v')\neq\prod_{c''}(v'')$. For each vertex $u\in V(H)$, by $\prod_{c'}(u)$ (resp. $S_{c'}(u)$), we denote the product (resp. set) of colors on the edges which are incident with $u$ in the coloring $c'$ of $H$. Without loss of generality, we assume that $\prod_{c'}(u_1)\leq\prod_{c'}(u_2)\leq\ldots\leq\prod_{c'}(u_{n_h})$. We assume $H_j$ (the $j^{th}$ copy of $H$) has the same coloring $c'$ as $H$ for $1\leq j\leq n_g$. Obviously, we also have $\prod_{c'}(u^j_1)\leq\prod_{c'}(u^j_2)\leq\ldots\leq\prod_{c'}(u^j_{n_h})$. In the following, we will give colors to the corona edges $v_ju^j_i$ and the vertices $u^j_i$ for $1\leq j\leq n_g$, $1\leq i\leq n_h$ to get a neighbor product distinguishing $(\Delta(G\circ H)+3)$-total coloring of $G\circ H$.
Now, we divide the proof into the following cases.
\textbf{Case $1.$ $\Delta(G)=1$}.
Let $G=v_1v_2$. Clearly, $\chi''_{\prod}(G)=3$. Let
$S=\{1,2,3,4\}$. Since $|S_{c'}(u_1)|\leq3$, then
$|(S-S_{c'}(u_1))|\geq1$.
\textbf{Case $1.1.$ $4\notin(S-S_{c'}(u_1))$}.
Since $|(S-S_{c'}(u_1))|\geq1$, there exists at least one color $\beta\in(S-S_{c'}(u_1))$ such that $\beta\in\{1,2,3\}$. Now, let $c(u_1^j)=c(v_1v_2)=\beta$, $c(v_1u_1^1)=c(v_2u_1^2)=5$, and use the other two colors in $\{1,2,3\}$ to color $v_1$, $v_2$, respectively. Let $c(u_i^j)=i+3$, $c(v_ju_i^j)=i+4$ for each $1\leq j\leq2$ and $2\leq i\leq n_h$. So far, we get a proper $(\Delta(G)+n_h+3)$-total coloring $c$ of $G\circ H$ such that $c(e)=c'(e)$ for each edge $e\in E(H)$. Clearly, $\prod_c(v_1)=c(v_1)\cdot\beta\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$, $\prod_c(v_2)=c(v_2)\cdot\beta\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$, and $\prod_c(u_1^j)=\prod_{c'}(u^j_1)\cdot\beta\cdot5$ for $1\leq j\leq2$. For each vertex $u^j_i\in V(H_j)$ where $1\leq j\leq 2$ and $2\leq i\leq n_h$, we have $\prod_c(u^j_i)=\prod_{c'}(u^j_i)\cdot(i+3)\cdot(i+4)$.
In the following, we will show the coloring $c$ is a neighbor product distinguishing total coloring of $G\circ H$.
First, we consider the vertices of $G$. Clearly, $\frac{\prod_c(v_1)}{\prod_c(v_2)}=\frac{c(v_1)}{c(v_2)}\neq1$. So we have $\prod_c(v_1)\neq\prod_c(v_2)$.
Second, we will show $\prod_c(u^j_{i_1})\neq\prod_c(u^j_{i_2})$ for any two adjacent vertices $u^{j}_{i_1}$ and $u^j_{i_2}$ of $H_j$. Since $\beta\leq3$, then $\prod_c(u_1^j)=\prod_{c'}(u^j_{1})\cdot\beta\cdot5<\prod_{c'}(u^j_{i})\cdot(i+3)\cdot(i+4)$ for $2\leq i\leq n_h$. For each $2\leq i_1<i_2\leq n_h$, since $\prod_{c'}(u^j_{i_1})\leq\prod_{c'}(u^j_{i_2})$, then $\prod_{c'}(u^j_{i_1})\cdot(i_1+3)\cdot(i_1+4)< \prod_{c'}(u^j_{i_2})\cdot(i_2+3)\cdot(i_2+4)$, i.e. $\prod_c(u^j_{i_1})<\prod_c(u^j_{i_2})$.
At last, we consider two adjacent vertices $u^j_i$ and $v_j$ for $1\leq j\leq 2$, $1\leq i\leq n_h$. For each $1\leq j\leq2$, we have $\prod_c(v_j)=\prod_{c''}(v_j)\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)\geq 1\cdot2\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$. Furthermore, $\prod_c(u^j_1)<\prod_c(u^j_2)<\ldots<\prod_c(u^j_{n_h})$, and $\prod_c(u^j_{n_h})=\prod_{c'}(u^j_{n_h})\cdot(n_h+3)\cdot(n_h+4)$. Since $4\in S_{c'}(u_1)$, then $\chi'(H)=4$. Obviously, $n_h>4$. We have $\prod_c(u^j_{n_h})\leq2\cdot3\cdot4\cdot(n_h+3)\cdot(n_h+4)$. Since $\prod_c(v_j)\geq 1\cdot2\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$, then $\frac{\prod_c(v_j)}{\prod_c(u^j_{n_h})}\geq\frac{5}{2}$. So we have $\Sigma_c(v_j)>\Sigma_c(v^j_i)$ for each $1\leq j\leq2$ and $1\leq i\leq n_h$.
\textbf{Case $1.2.$ $4\in(S-S_{c'}(u_1))$}.
Let $c(u_i^j)=i+3$, $c(v_ju_i^j)=i+4$ for each $1\leq j\leq2$ and $1\leq i\leq n_h$. So far, we get a proper $(\Delta(G)+n_h+3)$-total coloring $c$ of $G\circ H$ such that $c(e)=c'(e)$ for each edge $e\in E(H)$ and $c(x)=c''(x)$ for each $x\in V(G)\bigcup E(G)$. For each vertex $v\in V(G)$, we have $\prod_c(v)=\prod_{c''}(v)\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$. For each vertex $u^j_i\in V(H_j)$ where $1\leq j\leq 2$ and $1\leq i\leq n_h$, we have $\prod_c(u^j_i)=\prod_{c'}(u^j_i)\cdot(i+3)\cdot(i+4)$.
In the following, we will show the coloring $c$ is a neighbor product distinguishing total coloring of $G\circ H$.
First, we consider the vertices of $G$. Since $\prod_{c''}(v_1)\neq\prod_{c''}(v_2)$, clearly, we have $\prod_{c''}(v_1)\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)\neq \prod_{c''}(v_2)\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$, i.e. $\prod_c(v_1)\neq\prod_c(v_2)$.
Second, we will show $\prod_c(u^j_{i_1})\neq\prod_c(u^j_{i_2})$ for any two adjacent vertices $u^{j}_{i_1}$ and $u^j_{i_2}$ of $H_j$. Without loss of generality, we assume $i_1<i_2$. Since $\prod_{c'}(u^j_{i_1})\leq\prod_{c'}(u^j_{i_2})$, then $\prod_{c'}(u^j_{i_1})\cdot(i_1+3)\cdot(i_1+4)< \prod_{c'}(u^j_{i_2})\cdot(i_2+3)\cdot(i_2+4)$, i.e. $\prod_c(u^j_{i_1})<\prod_c(u^j_{i_2})$.
At last, we consider two adjacent vertices $u^j_i$ and $v_j$ for $1\leq j\leq 2$, $1\leq i\leq n_h$. For each $1\leq j\leq2$, we have $\prod_c(v_j)=\prod_{c''}(v_j)\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)\geq 1\cdot2\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$. Furthermore, $\prod_c(u^j_1)<\prod_c(u^j_2)<\ldots<\prod_c(u^j_{n_h})$, and $\prod_c(u^j_{n_h})=\prod_{c'}(u^j_{n_h})\cdot(n_h+3)\cdot(n_h+4)$.
If $n_h=2$, then $\chi'(H)=1$. We have $\prod_c(u^j_2)=1\cdot5\cdot6=30$. Since $\prod_c(v_j)\geq 1\cdot2\cdot5\cdot6=60$, we have $\Sigma_c(v_j)>\Sigma_c(u^j_i)$ for each $1\leq j, i\leq2$.
If $n_h=3$, then $\chi'(H)\leq3$. We have $\prod_c(u^j_3)\leq2\cdot3\cdot6\cdot7=252$. Since $\prod_c(v_j)\geq 1\cdot2\cdot5\cdot6\cdot7=420$, we have $\Sigma_c(v_j)>\Sigma_c(u^j_i)$ for each $1\leq j\leq 2$, $1\leq i\leq3$.
Otherwise, i.e. $n_h\geq4$. Clearly, $2\leq\chi'(H)\leq4$. We have $\prod_c(u^j_{n_h})\leq2\cdot3\cdot4\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$. Since $\prod_c(v_j)\geq 1\cdot2\cdot5\cdot6\cdot\ldots\cdot(n_h+3)\cdot(n_h+4)$, clearly, $\frac{\prod_c(v_j)}{\prod_c(u^j_{n_h})}\geq\frac{5}{2}$. So we have $\Sigma_c(v_j)>\Sigma_c(v^j_i)$ for each $1\leq j\leq 2$ and $1\leq i\leq n_h$.
\textbf{Case $2.$ $2\leq\Delta(G)\leq3$}.
Clearly, $\chi''_{\prod}(G)\leq6$. Let $S=\{1,2,3,4,5\}$. Since
$|S_{c'}(u_1)\bigcup c''(v)|\leq4$, then $|S-(S_{c'}(u_1)\bigcup c''(v))|\geq1$. There exists at least one color $\alpha\in (S-(S_{c'}(u_1)\bigcup c''(v)))$. Let $c(u_1^j)=\alpha$, $c(v_ju_1^j)=\Delta(G)+4$, and let $c(u_i^j)=\Delta(G)+i+2$, $c(v_ju_i^j)=\Delta(G)+i+3$ for each $1\leq j\leq n_g$ and $2\leq i\leq n_h$. So far, we get a proper $(\Delta(G)+n_h+3)$-total coloring $c$ of $G\circ H$ such that $c(e)=c'(e)$ for each edge $e\in E(H)$ and $c(x)=c''(x)$ for each $x\in V(G)\bigcup E(G)$. For each vertex $v\in V(G)$, we have $\prod_c(v)=\prod_{c''}(v)\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_h+3)$, $\prod_c(u^j_1)=\prod_{c'}(u^j_1)\cdot\alpha\cdot(\Delta(G)+4)$. For each vertex $u^j_i\in V(H_j)$ where $1\leq j\leq n_g$ and $2\leq i\leq n_h$, we have $\prod_c(u^j_i)=\prod_{c'}(u^j_i)\cdot(\Delta(G)+i+2)\cdot(\Delta(G)+i+3)$.
In the following, we will show the coloring $c$ is a neighbor product distinguishing total coloring of $G\circ H$.
First, we consider two adjacent vertices $v'$ and $v''$ in $G$. Since $\prod_{c''}(v')\neq\prod_{c''}(v'')$, it is obvious that $\prod_{c''}(v')\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_h+3)\neq \prod_{c''}(v'')\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_h+3)$, i.e. $\prod_c(v')\neq\prod_c(v'')$.
Second, we will show $\prod_c(u^j_{i_1})\neq\prod_c(u^j_{i_2})$ for any two adjacent vertices $u^{j}_{i_1}$ and $u^j_{i_2}$ of $H_j$. Since $\alpha\leq5$, then $\prod_c(u_1^j)=\prod_{c'}(u^j_{1})\cdot\alpha\cdot(\Delta(G)+4)<\prod_{c'}(u^j_{i})\cdot(\Delta(G)+i+2)\cdot(\Delta(G)+i+3)$ for $2\leq i\leq n_h$. For each $2\leq i_1<i_2\leq n_h$, since $\prod_{c'}(u^j_{i_1})\leq\prod_{c'}(u^j_{i_2})$, it is clear that $\prod_{c'}(u^j_{i_1})\cdot(\Delta(G)+i_1+2)\cdot(\Delta(G)+i_1+3)< \prod_{c'}(u^j_{i_2})\cdot(\Delta(G)+i_2+2)\cdot(\Delta(G)+i_2+3)$, i.e. $\prod_c(u^j_{i_1})<\prod_c(u^j_{i_2})$.
At last, we consider two adjacent vertices $u^j_i$ and $v_j$ for $1\leq j\leq n_g$, $1\leq i\leq n_h$. For each $1\leq j\leq n_g$, we have $\prod_c(v_j)=\prod_{c''}(v_j)\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot (\Delta(G)+n_h+3)\geq1\cdot2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot (\Delta(G)+n_h+3)$. Furthermore, $\prod_c(u^j_1)<\prod_c(u^j_2)<\ldots<\prod_c(u^j_{n_h})$, and $\prod_c(u^j_{n_h})=\prod_{c'}(u^j_{n_h})\cdot(\Delta(G)+n_h+2)\cdot(\Delta(G)+n_h+3)$.
If $n_h=2$, then $\chi'(H)=1$. We have $\prod_c(u^j_2)=1\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)$. Since $\prod_c(v_j)\geq 1\cdot2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)$, then $\frac{\prod_c(v_j)}{\prod_c(u^j_2)}\geq2$. Clearly, we have $\Sigma_c(v_j)>\Sigma_c(u^j_i)$ for each $1\leq j\leq n_g$, $1\leq i\leq2$.
If $n_h=3$, then $\Delta(H)\leq2$, $\chi'(H)\leq3$. We have $\prod_c(u^j_3)\leq2\cdot3\cdot(\Delta(G)+5)\cdot(\Delta(G)+6)$. Since $\prod_c(v_j)\geq 1\cdot2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)(\Delta(G)+6)$, then $\frac{\prod_c(v_j)}{\prod_c(u^j_3)}\geq\frac{5}{3}$. Clearly, we have $\Sigma_c(v_j)>\Sigma_c(u^j_i)$ for each $1\leq j\leq n_g$, $1\leq i\leq3$.
Otherwise, i.e. $n_h\geq4$. Clearly, $\chi'(H)\leq4$. We have $\prod_c(u^j_{n_h})\leq2\cdot3\cdot4\cdot(\Delta(G)+n_h+2)\cdot(\Delta(G)+n_h+3)$. Since $\prod_c(v_j)\geq 1\cdot2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_h+2)\cdot(\Delta(G)+n_h+3)$, clearly, $\frac{\prod_c(v_j)}{\prod_c(u^j_{n_h})}\geq\frac{5}{2}$. So we have $\Sigma_c(v_j)>\Sigma_c(v^j_i)$ for each $1\leq j\leq n_g$ and $1\leq i\leq n_h$.
\end{proof}
\end{document} | arXiv | {
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\begin{document}
\title { Moduli spaces of twisted sheaves on a projective variety}
\author{K\={o}ta Yoshioka
}
\address{Department of Mathematics, Faculty of Science, Kobe University, Kobe, 657, Japan} \email{yoshioka@math.kobe-u.ac.jp}
\subjclass{14D20}
\maketitle
\section{Introduction}
Let $X$ be a smooth projective variety over ${\mathbb C}$. Let $\alpha:=\{\alpha_{ijk} \in H^0(U_i \cap U_j \cap U_k,{\mathcal O}_X^{\times})\}$ be a 2-cocycle representing a torsion class $[\alpha] \in H^2(X,{\mathcal O}_X^{\times})$. An $\alpha$-twisted sheaf $E:=\{(E_i,\varphi_{ij})\}$ is a collection of sheaves $E_i$ on $U_i$ and isomorphisms $\varphi_{ij}:
E_{i|U_i \cap U_j} \to E_{j|U_i \cap U_j}$ such that $\varphi_{ii}=\operatorname{id}_{E_i}$, $\varphi_{ji}=\varphi_{ij}^{-1}$ and $\varphi_{ki} \circ \varphi_{jk} \circ \varphi_{ij} =\alpha_{ijk}\operatorname{id}_{E_i}$. We assume that there is a locally free $\alpha$-twisted sheaf, that is, $\alpha$ gives an element of the Brauer group $\operatorname{Br}(X)$. A twisted sheaf naturally appears if we consider a non-fine moduli space $M$ of the usual stable sheaves on $X$. Indeed the transition functions of the local universal families satisfy the patching condition up to the multiplication by constants and gives a twisted sheaf. If the patching condition is satisfied, i.e., the moduli space $M$ is fine, than the universal family defines an integral functor on the bounded derived categories of coherent sheaves ${\bf D}(M) \to {\bf D}(X)$. Assume that $X$ is a $K3$ surface and $\dim M=\dim X$. Than Mukai, Orlov and Bridgeland showed that the integral functor is the Fourier-Mukai functor, i.e., it is an equivalence of the categories. In his thesis \cite{C:2}, C\u ald\u araru studied the category of twisted sheaves and its bounded derived category. In particular, he generalized Mukai, Orlov and Bridgeland's results on the Fourier-Mukai transforms to non-fine moduli spaces on a $K3$ surface. For the usual derived category, Orlov \cite{Or:1} showed that the equivalence class is described in terms of the Hodge structure of the Mukai lattice. C\u ald\u araru conjectured that a similar result also holds for the derived category of twisted sheaves. Recently this conjecture was modified and proved by Huybrechts and Stellari, if $\rho(X) \geq 12$ in \cite{H-S:2}. As is well-known, twisted sheaves also appear if we consider a projective bundle over $X$.
In this paper, we define a notion of the stability for a twisted sheaf and construct the moduli space of stable twisted sheaves on $X$. We also construct a projective compactification of the moduli space by adding the $S$-equivalence classes of semi-stable twisted sheaves. In particular if $H^1(X,{\mathcal O}_X)=0$ (e.g. $X$ is a $K3$ surface), then the moduli space of locally free twisted sheaves is the moduli space of projective bundles over $X$. Thus we compactify the moduli space of projective bundles by using twisted sheaves. The idea of the construction is as follows. We consider a twisted sheaf as a usual sheaf on the Brauer-Severi variety. Instead of using the Hilbert polynomial associated to an ample line bundle on the Brauer-Severi variety, we use the Hilbert polynomial associated to a line bundle coming from $X$ in order to define the stability. Then the construction is a modification of Simpson's construction of the moduli space of usual sheaves (cf. \cite{Y:11}). M. Lieblich informed us that our stability condition coincides with Simpson's stability for modules over the associated Azumaya algebra via Morita equivalence. Hence the construction also follows from Simpson's moduli space \cite[Thm. 4.7]{S:1} and the valuative criterion for properness.
In section \ref{sect:k3}, we consider the moduli space of twisted sheaves on a $K3$ surface. We generalize known results on the moduli space of usual stable sheaves to the moduli spaces of twisted stable sheaves (cf. \cite{Mu:3}, \cite{Y:7}). In particular, we consider the non-emptyness, the deformation type and the weight 2 Hodge structure. Then we can consider twisted version of the Fourier-Mukai transform by using 2 dimensional moduli spaces, which is done in section \ref{sect:FM}. As an application of our results, Huybrechts and Stellari \cite{H-St:2} prove C\u ald\u araru's conjecture generally.
Since our main example of twisted sheaves are those on $K3$ surfaces or abelian surfaces, we consider twisted sheaves over ${\mathbb C}$. But some of the results (e.g., subsection \ref{subsect:construction}) also hold over any field.
E. Markman and D. Huybrechts communicated to the author that M. Lieblich independently constructed the moduli of twisted sheaves. In his paper \cite{Li:1}, Lieblich developed a general theory of twisted sheaves in terms of algebraic stack and constructed the moduli space intrinsic way. He also studied the moduli spaces of twisted sheaves on surfaces. There are also some overlap with a paper by N. Hoffmann and U. Stuhler \cite{Ho-St:1}. They also constructed the moduli space of rank 1 twisted sheaves and studied the symplectic structure of the moduli space. \begin{NB} \subsection{}
\begin{prop}\cite[Prop. 2.2]{D:1} $H^1(PGL(r)) \to H^2(X,\mu_r)$ is surjective. \end{prop}
By the exact and commutative diagram \begin{equation} \begin{CD} @. 1 @. 1 @. @.\\ @. @VVV @VVV @. @.\\ 1 @>>> \mu_r @>>> SL(r) @>>> PSL(r) @>>> 1\\
@. @VVV @VVV @| @. \\ 1 @>>> {\mathcal O}_X^{\times} @>>> GL(r) @>>> PGL(r) @>>> 1\\ @. @V{(\;\;)^r}VV @VV{\det}V @. @.\\ @. {\mathcal O}_X^{\times} @= {\mathcal O}_X^{\times} @. @.\\ @. @VVV @VVV @. @.\\ @. 1 @. 1 @. @., \end{CD} \end{equation} we have an exact and commutative diagram
\begin{equation} \begin{CD} @. @. @. H^1(X,{\mathcal O}_X^{\times})\\ @. @. @. @VV{c_1 \mod r}V \\ H^1(X,\mu_r) @>>> H^1(X,SL(r)) @>>> H^1(PGL(r)) @>{\delta'}>> H^2(X,\mu_r)\\
@VVV @VVV @| @VV{o}V \\ H^1(X,{\mathcal O}_X^{\times}) @>>> H^1(X,GL(r)) @>>> H^1(PGL(r)) @>{\delta}>> H^2(X,{\mathcal O}_X^{\times}) \end{CD} \end{equation}
\begin{equation} \begin{CD} @. 0 @. 0 @. @.\\ @. @VVV @VVV @. @.\\ @. {\mathbb Z} @= {\mathbb Z} @. @.\\ @. @V{r}VV @VV{r}V @. @.\\ 0 @>>> {\mathbb Z} @>>> {\mathcal O}_X @>>> {\mathcal O}_X^{\times} @>>> 1\\
@. @VVV @VVV @| @. \\ 1 @>>> \mu_r @>>> {\mathcal O}_X^{\times} @>{(\;\;)^r}>> {\mathcal O}_X^{\times} @>>> 1\\ @. @VVV @VVV @. @.\\ @. 1 @. 1 @. @.\\ \end{CD} \end{equation}
\begin{equation} \begin{CD} H^1(X,{\mathcal O}_X^{\times}) @>{c_1}>> H^2(X,{\mathbb Z}) @>>> H^2(X,{\mathcal O}_X)\\
@| @VVV @VVV \\ H^1(X,{\mathcal O}_X^{\times}) @>>> H^2(X,\mu_r) @>{\lambda}>> H^2(X,{\mathcal O}_X^{\times}) \end{CD} \end{equation} Hence \begin{equation} o(H^2(X,\mu_r))=H^2(X,{\mathbb Z})/(\operatorname{NS}(X)+rH^2(X,{\mathbb Z})). \end{equation}
\end{NB}
\section{Twisted sheaves} Notation: For a locally free sheaf $E$ on a variety $X$, ${\mathbb P}(E) \to X$ denotes the projective bundle in the sense of Grothendieck, that is, ${\mathbb P}(E)=\mathrm{Proj}(\bigoplus_{n=0}^{\infty} S^n(E))$.
Let $X$ be a smooth projective variety over ${\mathbb C}$. Let $\alpha:=\{\alpha_{ijk} \in H^0(U_i \cap U_j \cap U_k,{\mathcal O}_X^{\times})\}$ be a 2-cocycle representing a torsion class $[\alpha] \in H^2(X,{\mathcal O}_X^{\times})$. An $\alpha$-twisted sheaf $E:=\{(E_i,\varphi_{ij})\}$ is a collection of sheaves $E_i$ on $U_i$ and isomorphisms $\varphi_{ij}:
E_{i|U_i \cap U_j} \to E_{j|U_i \cap U_j}$ such that $\varphi_{ii}=\operatorname{id}_{E_i}$, $\varphi_{ji}=\varphi_{ij}^{-1}$ and $\varphi_{ki} \circ \varphi_{jk} \circ \varphi_{ij} =\alpha_{ijk}\operatorname{id}_{E_i}$. If all $E_i$ are coherent, then we say that $E$ is coherent. Let $\operatorname{Coh}(X,\alpha)$ be the category of coherent $\alpha$-twisted sheaves on $X$.
If $E_i$ are locally free for all $i$, then we can glue ${\mathbb P}(E_i^{\vee})$ together and get a projective bundle $p:Y \to X$ with $\delta([Y])=[\alpha]$, where $[Y] \in H^1(X,PGL(r))$ is the corresponding cohomology class of $Y$ and $\delta:H^1(X,PGL(r)) \to H^2(X,{\mathcal O}_X^{\times})$ is the connecting homomorphism induced by the exact sequence \begin{equation} 1 \to {\mathcal O}_X^{\times} \to GL(r) \to PGL(r) \to 1. \end{equation} Thus $[\alpha]$ belongs to the Brauer group $\operatorname{Br}(X)$. If $X$ is a smooth projective surface, then $\operatorname{Br}(X)$ coincides with the torsion part of $H^2(X,{\mathcal O}_X^{\times})$. Let ${\mathcal O}_{{\mathbb P}(E_i^{\vee})}(\lambda_i)$ be the tautological line bundle on ${\mathbb P}(E_i^{\vee})$. Then, $\varphi_{ij}$ induces an isomorphism $\widetilde{\varphi}_{ij}:
{\mathcal O}_{{\mathbb P}(E_i^{\vee})}(\lambda_i)_{|p^{-1}(U_i \cap U_j)}
\to {\mathcal O}_{{\mathbb P}(E_j^{\vee})}(\lambda_j)_{|p^{-1}(U_i \cap U_j)}$. ${\mathcal L}(p^*(\alpha^{-1})):=\{({\mathcal O}_{{\mathbb P}(E_i^{\vee})}(\lambda_i), \widetilde{\varphi}_{ij})\}$ is an $p^*(\alpha^{-1})$-twisted line bundle on $Y$.
\subsection{Sheaves on a projective bundle} In this subsection, we shall interpret twisted sheaves as usual sheaves on a Brauer-Severi variety. Let $p:Y \to X$ be a projective bundle. \begin{NB} Since $-K_Y$ is relatively ample, $Y$ is projective over $X$. \end{NB} Let $X=\cup_i U_i$ be an analytic open covering of $X$ such that $p^{-1}(U_i) \cong U_i \times {\mathbb P}^{r-1}$. We set $Y_i:=p^{-1}(U_i)$. We fix a collection of tautological line bundles ${\mathcal O}_{Y_i}(\lambda_i)$ on $Y_i$ and isomorphisms $\phi_{ji}:{\mathcal O}_{Y_i \cap Y_j}(\lambda_j) \to {\mathcal O}_{Y_i \cap Y_j}(\lambda_i)$. We set $G_i:=p_*({\mathcal O}_{Y_i}(\lambda_i))^{\vee}$. Then $G_i$ are vector bundles on $U_i$ and $p^*(G_i)(\lambda_i)$ defines a vector bundle $G$ of rank $r$ on $Y$. We have the Euler sequence \begin{equation} 0 \to {\mathcal O}_Y \to G \to T_{Y/X} \to 0. \end{equation} Thus $G$ is a non-trivial extension of $T_{Y/X}$ by ${\mathcal O}_Y$.
\begin{lem}\label{lem:G} $\operatorname{Ext}^1(T_{Y/X},{\mathcal O}_Y)={\mathbb C}$. Thus $G$ is characterized as a non-trivial extension of $T_{Y/X}$ by ${\mathcal O}_Y$. In particular, $G$ does not depend on the choice of the local trivialization of $p$. \end{lem} \begin{proof} Since ${\bf R} p_* (G^{\vee})=0$, the Euler sequence inplies that $\operatorname{Ext}^1(T_{Y/X},{\mathcal O}_Y) \cong H^0(Y,{\mathcal O}_Y) \cong {\mathbb C}$. \end{proof}
\begin{defn} For a projective bundle $p:Y \to X$, let $\epsilon(Y)(:=G)$ be a vector bundle on $Y$ which is a non-trivial extension \begin{equation} 0 \to {\mathcal O}_Y \to \epsilon(Y)\to T_{Y/X} \to 0. \end{equation} \end{defn}
By the exact sequence $0 \to \mu_r \to SL(r) \to PGL(r) \to 1$, we have a connecting homomorphism $\delta':H^1(X,PGL(r)) \to H^2(X,\mu_r)$. Let $o:H^2(X,\mu_r) \to H^2(X,{\mathcal O}_X^{\times})$ be the homomorphism induced by the inclusion $\mu_r \hookrightarrow {\mathcal O}_X^{\times}$. Then we have $\delta=o \circ \delta'$. \begin{defn}\label{defn:w(Y)} For a ${\mathbb P}^{r-1}$-bundle $p:Y \to X$ corresponding to $[Y] \in H^1(X,PGL(r))$, we set $w(Y):=\delta'([Y]) \in H^2(X,\mu_r)$. \end{defn}
\begin{lem}[\cite{C:1},\cite{H-S:1}] If $p:Y \to X$ is a ${\mathbb P}^{r-1}$-bundle associated to a vector bundle $E$ on $X$, i.e., $Y={\mathbb P}(E^{\vee})$, then $w(Y)=[c_1(E) \mod r]$. \end{lem}
\begin{NB} \begin{lem} \begin{enumerate} \item $\operatorname{Ext}_p^i(T_{Y/X},{\mathcal O}_Y)=0$, $i \ne 1$ and $\operatorname{Ext}_p^1(T_{Y/X},{\mathcal O}_Y) \otimes k(x) \cong
\operatorname{Ext}_p^1(T_{Y/X|p^{-1}(x)},{\mathcal O}_{p^{-1}(x)})$. \item $\operatorname{Ext}^1(T_{Y/X},{\mathcal O}_Y) \cong H^0(X,\operatorname{Ext}_p^1(T_{Y/X},{\mathcal O}_Y)) \cong {\mathbb C}$ \end{enumerate} \end{lem} We have $K_{Y/X} \cong \det G^{\vee}$.
\end{NB}
\begin{NB} If $w(Y)=[D \mod r] \in H^2(X,\mu_r)$ for $D \in \operatorname{Pic}(X)$, then $[c_1(G) \mod r]=w(Y \times_X Y)=p^*(w(Y)) =p^*(D) \in H^2(Y,\mu_r)$. Hence there is a line bundle $L$ on $X$ such that $c_1(G)-p^*(D)=rc_1(L)$.
Then $L_{|Y_i} \cong {\mathcal O}_{Y_i}(1)$. Hence $Y \cong {\mathbb P}(E^{\vee})$ for $E:=p_*(L)$ and $G \cong E^{\vee} \otimes L$. \end{NB}
\begin{lem}\label{lem:w(Y)} $[c_1(G) \mod r]=p^*(w(Y)) \in H^2(Y,\mu_r)$. \end{lem}
\begin{proof} There is a line bundle $L$ on $Y \times_X Y$ such that
$L_{|Y_i \times_{U_i} Y_i} \cong p_1^*({\mathcal O}_{Y_i}(-\lambda_i)) \otimes p_2^*({\mathcal O}_{Y_i}(\lambda_i))$, where $p_i:Y \times_X Y \to Y$, $i=1,2$ are $i$-th projections. By the definition of $G$, $p_{1*} (L) \cong G^{\vee}$. Hence $p_1: Y \times_X Y \to Y$ is the projective bundle ${\mathbb P}(G^{\vee}) \to Y$. Then we get $-[c_1(G^{\vee}) \mod r]=w(Y \times_X Y)=p^*(w(Y))$. \end{proof}
\begin{lem}
Let $p:Y \to X$ be a ${\mathbb P}^{r-1}$-bundle.
Then the following conditions are equivalent.
\begin{enumerate}
\item[(1)]
$Y={\mathbb P}(E^{\vee})$ for a vector bundle on $X$.
\item[(2)]
$w(Y) \in \operatorname{NS}(X) \otimes \mu_r$.
\item[(3)]
There is a line bundle $L$ on $Y$ such that
$L_{|p^{-1}(x)} \cong {\mathcal O}_{p^{-1}(x)}(1)$.
\end{enumerate}
\end{lem}
\begin{proof}
$(2) \Rightarrow (3)$:
If $w(Y)=[D \mod r]$, $D \in \operatorname{NS}(X)$, then
$c_1(\epsilon(Y))-p^*(D) \equiv 0 \mod r$. We take a line bundle
$L$ on $Y$ with
$c_1(\epsilon(Y))-p^*(D)=rc_1(L)$.
$(3) \Rightarrow (1)$: We set $E^{\vee}:=p_*(L)$.
Then $Y={\mathbb P}(E^{\vee})$.
\end{proof}
\begin{defn} $\operatorname{Coh}(X,Y)$ is a subcategory of $\operatorname{Coh}(Y)$ such that $E \in \operatorname{Coh}(X,Y)$ if and only if \begin{equation}
E_{|Y_i} \cong p^*(E_i) \otimes {\mathcal O}_{Y_i}(\lambda_i) \end{equation}
for $E_i \in \operatorname{Coh}(U_i)$. For simplicity, we call $E \in \operatorname{Coh}(X,Y)$ a $Y$-sheaf. \end{defn} By this definition, $\{(U_i,E_i)\}$ gives a twisted sheaf on $X$. Thus we have an equivalence \begin{equation}\label{eq:Y=X} \begin{matrix} \Lambda^{{\mathcal L}(p^*(\alpha^{-1}))}:\operatorname{Coh}(X,Y)& \cong & \operatorname{Coh}(X,\alpha)\\ E & \mapsto & p_*(E \otimes L^{\vee}), \end{matrix} \end{equation} where ${\mathcal L}(p^*(\alpha^{-1})):=\{({\mathcal O}_{Y_i}(\lambda_i),\phi_{ij})\}$ is a twisted line bundle on $Y$ and $\alpha_{ijk}^{-1}\operatorname{id}_{{\mathcal O}_{Y_i}(\lambda_i)} =\phi_{ki} \circ \phi_{jk} \circ \phi_{ij}$.
We have the following relations: \begin{equation} \begin{split}
p_*(G^{\vee} \otimes E)_{|U_i}=& p_*(p^*(G_i^{\vee}) \otimes {\mathcal O}_{Y_i}(-\lambda_i) \otimes p^*(E_i) \otimes {\mathcal O}_{Y_i}(\lambda_i))\\ =& p_* p^*(G_i^{\vee} \otimes E_i)=G_i^{\vee} \otimes E_i, \end{split} \end{equation} \begin{equation} \begin{split}
p_*(E)_{|U_i}=& p_*(p^*(E_i) \otimes {\mathcal O}_{Y_i}(\lambda_i))\\ =& E_i \otimes p_*({\mathcal O}_{Y_i}(\lambda_i))=G_i^{\vee} \otimes E_i. \end{split} \end{equation}
\begin{lem} A coherent sheaf $E$ on $Y$ belongs to $\operatorname{Coh}(X,Y)$ if and only if $\phi: p^* p_*(G^{\vee} \otimes E) \to G^{\vee} \otimes E$ is an isomorphism. In particular $E \in \operatorname{Coh}(X,Y)$ is an open condition. \end{lem}
\begin{proof}
$\phi_{|Y_i}$ is the homomorphism
$p^* G_i^{\vee} \otimes p^* p_*(E(-\lambda_i)) \to p^* G_i^{\vee} \otimes E(-\lambda_i)$. Hence $\phi_{|Y_i}$ is an isomorphism if and only if $p^* p_*(E(-\lambda_i)) \to E(-\lambda_i)$ is an isomorphism, which is equivalent to $E \in \operatorname{Coh}(X,Y)$. \end{proof}
\begin{lem} Assume that $H^3(X,{\mathbb Z})_{\text{tor}}=0$. Then $H^*(Y,{\mathbb Z}) \cong H^*(X,{\mathbb Z})[x]/(f(x))$, where $f(x) \in H^*(X,{\mathbb Z})[x]$ is a monic polynomial of degree $r$. In particular, $H^2(X,{\mathbb Z}) \otimes \mu_{r'} \to H^2(Y,{\mathbb Z}) \otimes \mu_{r'}$ is injective for all $r'$. \end{lem} \begin{proof} $R^2 p_*{\mathbb Z}$ is a local system of rank 1. Since $c_1(K_{Y/X})$ is a section of this local system, $R^2 p_*{\mathbb Z} \cong {\mathbb Z}$. Let $h$ be the generator. Then $R^{2i} p_*{\mathbb Z} \cong {\mathbb Z}h^i$. Since $H^3(X,{\mathbb Z})_{\text{tor}}=0$, by the Leray spectral sequence, we get a surjective homomorphism $H^2(Y,{\mathbb Z}) \to H^0(X,R^2 p_*{\mathbb Z})$. Let $x \in H^2(Y,{\mathbb Z})$ be a lifting of $h$. Then $x^i$ is a lifting of $h^i \in H^0(X,R^{2i} p_*{\mathbb Z})$. Therefore the Leray-Hirsch theorem implies that $H^*(Y,{\mathbb Z}) \cong H^*(X,{\mathbb Z})[x]/(f(x))$. \end{proof}
\begin{NB} If $H^3(X,{\mathbb Z})_{\text{tor}}=0$, then $H^3(Y,{\mathbb Z})_{\text{tor}}=0$. $H^2(X,\mu_r) \cong H^2(X,{\mathbb Z}) \otimes \mu_r$. \end{NB}
\begin{NB} $Y \times_X Y \to Y$ is a projective bundle associated to $G^{\vee}$: $$
(1 \times p)^*(G^{\vee})_{|Y_i \times_{U_i} Y_i}= (1 \times p)^*({\mathcal O}_{Y_i}(-\lambda_i)) \otimes
(p \times p)^*(G_i^{\vee}) \to {\mathcal O}_{Y_i}(-\lambda_i) \boxtimes {\mathcal O}_{Y_i}(\lambda_i) $$ is the tautological line bundle on $Y \times_X Y$.
$p':Y' \to X$ be a ${\mathbb P}^{r-1}$-bundle with a vector bundle $G'$ on $Y$. Assume that $w(Y')=w(Y)$. Then $c_1(G')-c_1(G) \mod r =0$ in $H^2(Y' \times_X Y,{\mathbb Z})$. Then there is a line bundle $L$ on $Y' \times_X Y$ such that $rc_1(L)=c_1(G')-c_1(G)$. For all fiber $l$ of $p' \times p:Y' \times_X Y \to X$,
$L_{|l} \cong {\mathcal O}_{{\mathbb P}^{r-1}}(1) \boxtimes
{\mathcal O}_{{\mathbb P}^{r-1}}(-1)$. Then we have an equivalence: \begin{equation} \begin{matrix} \Xi_{Y \to Y'}^L:\operatorname{Coh}(X,Y) & \to &\operatorname{Coh}(X,Y')\\ E & \mapsto & (p' \times 1)_*((1 \times p)^*(E) \otimes L). \end{matrix} \end{equation}
\end{NB}
\begin{lem}\label{lem:o(w(Y))} Assume that $o(w(Y))=o(w(Y'))$. \begin{enumerate} \item Then there is a line bundle $L$ on $Y' \times_X Y$
such that $L_{|{p'}^{-1}(x) \times p^{-1}(x)} \cong {\mathcal O}_{{p'}^{-1}(x)}(1) \boxtimes {\mathcal O}_{p^{-1}(x)}(-1)$ for all $x \in X$. If $L' \in \operatorname{Pic}(Y' \times_X Y)$ also satisfies the property, then $L'=L \otimes q^*(P)$, $P \in \operatorname{Pic}(X)$, where $q:Y' \times_X Y \to X$ is the projection. \item We have an equivalence \begin{equation} \begin{matrix} \Xi_{Y \to Y'}^L:&\operatorname{Coh}(X,Y) & \to &\operatorname{Coh}(X,Y')\\ & E & \mapsto & p_{Y'*}({p'}_{Y}^*(E) \otimes L), \end{matrix} \end{equation} where $p_{Y'}:Y' \times_X Y \to Y'$ and $p'_Y:Y' \times_X Y \to Y$ are projections. \end{enumerate} \end{lem}
\begin{NB} We note that $[r'c_1(G) \mod rr'], [rc_1(G') \mod rr']
\in H^2(Y' \times_X Y,\mu_{rr'})$ belongs to $H^2(X,\mu_{rr'})$. Since we have a commutative diagram \begin{equation} \begin{CD} H^2(X,\mu_r) @>{o}>> H^2(X,{\mathcal O}_X^{\times})\\
@V{r'}VV @| \\ H^2(X,\mu_{rr'}) @>{o}>> H^2(X,{\mathcal O}_X^{\times}), \end{CD} \end{equation} $o([r'c_1(G) \mod rr'])=o(w(Y))$. Hence $[rc_1(G')-r'c_1(G) \mod rr'] \in H^2(X,\mu_{rr'})$ belongs to $\operatorname{Pic}(X) \otimes \mu_{rr'}$. Therefore there is a line bundle $L$ on $Y' \times_X Y$ such that $rc_1(G')-r'c_1(G)=rr'c_1(L)+c_1(N)$, $N \in \operatorname{Pic}(X)$. \end{NB}
\begin{rem} We also see that $E$ is a $Y$-sheaf if and only if ${p'}_{Y}^*(E) \otimes L \cong p_{Y'}^*(E')$ for a sheaf $E'$ on $Y'$. \end{rem}
\begin{defn}\label{defn:w(E)} Assume that $H^3(X,{\mathbb Z})_{tor}=0$. For a $Y$-sheaf $E$ of rank $r'$, $[c_1(E) \mod r'] \in H^2(Y,\mu_{r'})$ belongs to $p^*( H^2(X,\mu_{r'}))$. We set $w(E):=(p^*)^{-1}([c_1(E) \mod r']) \in H^2(X,\mu_{r'})$. \end{defn} By Lemmas \ref{lem:w(Y)} and \ref{lem:o(w(Y))}, we see that \begin{lem}\label{lem:w(E)} \begin{enumerate} \item By the functor $\Xi_{Y \to Y'}^L$ in Lemma \ref{lem:o(w(Y))}, $w(\Xi_{Y \to Y'}^L(E))=w(E)$ for $E \in \operatorname{Coh}(X,Y)$. \item $w(\epsilon(Y))=w(Y)$. \end{enumerate} \end{lem}
\begin{NB} Let $\alpha:=\{\alpha_{ijk} \in H^0(U_i \cap U_j \cap U_k,{\mathcal O}_X^{\times}) \}$ and $\alpha':=\{\alpha'_{ijk}\in H^0(U_i \cap U_j \cap U_k,{\mathcal O}_X^{\times})\}$ are 2-cocyles such that they defines the same element of the Brauer group: $\alpha_{ijk}f_{ij}f_{jk}f_{ki}=\alpha_{ijk}'$. Then we have an equivalence
\begin{equation} \begin{matrix} \operatorname{Coh}(X,\alpha)& \cong & \operatorname{Coh}(X,\alpha')\\ E & \mapsto & E \otimes {\mathcal O}_X^{\alpha^{-1} \alpha'} \end{matrix} \end{equation} where ${\mathcal O}_X^{\alpha^{-1} \alpha'}$ is a $\alpha^{-1} \alpha'$-twisted line bundle with ${\mathcal O}_X^{\alpha^{-1} \alpha'}= \{({\mathcal O}_{U_i},f_{ij})\}$.
If there is a $\alpha$-twisted line bundle $L$, then \begin{equation} \begin{matrix} \operatorname{Coh}(X)& \cong &\operatorname{Coh}(X,\alpha)\\ E & \mapsto & E \otimes L \end{matrix} \end{equation}
For a class $[\alpha] \in H^2(X,{\mathcal O}_X^{\times})$, we fix a representative $\alpha$ of $[\alpha]$. Let $E=\{(E_i,\phi_{ij})\}$ and $E'=\{(E_i',\phi_{ij}')\}$ be $\alpha$-twisted vector bundles such that ${\mathbb P}(E) \cong {\mathbb P}(E')$. Then there is a collection of $\psi_i:E_i \to E_i'$ which fits in the commutative diagram: \begin{equation} \begin{CD}
E_{i|U_i \cap U_j} @>{\psi_i}>> E_{i|U_i \cap U_j}'\\ @V{\lambda_{ij}\phi_{ij}}VV @VV{\phi_{ij}'}V\\
E_{j|U_i \cap U_j} @>{\psi_j}>> E_{j|U_i \cap U_j}' \end{CD} \end{equation} where $\lambda_{ij} \in H^0(U_i \cap U_j,{\mathcal O}_X^{\times})$. Then we see that $\lambda_{ij}$ is a 1-cocycle. Let $L$ be a line bundle on $X$ accosiated to $\{\lambda_{ij}\}$. Then $E' \cong E \otimes L$. Therefore the moduli of projective bundles is the quotient of the moduli of twisted vector bundles by the action of $\operatorname{Pic}(X)$. \end{NB}
\section{Moduli of twisted sheaves}\label{sect:moduli}
\subsection{Definition of the stability}\label{subsect:defn}
Let $(X,{\mathcal O}_X(1))$ be a pair of a projective scheme $X$ and an ample line bundle ${\mathcal O}_X(1)$ on $X$. Let $p:Y \to X$ be a projective bundle over $X$. \begin{defn} A $Y$-sheaf $E$ is of dimension $d$, if $p_*(E)$ is of dimension $d$. \end{defn}
For a coherent sheaf $F$ of dimension $d$ on $X$, we define $a_i(F) \in {\mathbb Z}$ by the coefficient of the Hilbert polynomial of $F$: \begin{equation} \chi(F(m))= \sum_{i=0}^d a_i(F) \binom{m+i}{i}. \end{equation} Let $G$ be a locally free $Y$-sheaf. For a $Y$-sheaf $E$ of dimension $d$, we set $a_i^{G}(E):=a_i(p_*(G^{\vee} \otimes E))$. Thus we have \begin{equation} \chi(G,E \otimes p^*{\mathcal O}_X(m))= \chi(p_*(G^{\vee} \otimes E)(m))=\sum_{i=0}^d a_i^{G}(E) \binom{m+i}{i}. \end{equation}
\begin{defn} Let $E$ be $Y$-sheaf of dimension $d$. Then $E$ is ($G$-twisted) semi-stable (with respect to ${\mathcal O}_X(1)$), if \begin{enumerate} \item $E$ is of pure dimension $d$, \item \begin{equation}\label{eq:def} \frac{\chi(p_*(G^{\vee} \otimes F)(m))}{a_d^{G}(F)} \leq \frac{\chi(p_*(G^{\vee} \otimes E)(m))}{a_d^{G}(E)}, m \gg 0 \end{equation} for all subsheaf $F \ne 0$ of $E$. \end{enumerate} If the inequality in \eqref{eq:def} is strict for all proper subsheaf $F \ne 0$ of $E$, then $E$ is ($G$-twisted) stable with respect to ${\mathcal O}_X(1)$. \end{defn}
\begin{thm}\label{thm:twisted} Let $p:Y \to X$ be a projective bundle. There is a coarse moduli scheme $\overline{M}_{X/{\mathbb C}}^{h}$ parametrizing $S$-equivalence classes of $G$-twisted semi-stable $Y$-sheaves $E$ with the $G$-twisted Hilbert polynomial $h$. $\overline{M}_{X/{\mathbb C}}^{h}$ is a projective scheme. \end{thm}
\begin{rem} The construction also works for a projective bundle $Y \to X$ over any field and also for a family of projective bundles, by the fundamental work of Langer \cite{Langer:1}. \end{rem} \begin{lem}\label{lem:choice} Let $p':Y' \to X$ be a projective bundle with $o(w(Y'))=o(w(Y))$ and $\Xi_{Y \to Y'}^L$ the correspondence in Lemma \ref{lem:o(w(Y))}. Then a $Y$-sheaf $E$ is $G$-twisted semi-stable if and only if $\Xi_{Y \to Y'}^L(E) \in \operatorname{Coh}(X,Y')$ is $\Xi_{Y \to Y'}^L(G)$-twisted semi-stable. In particular, we have an isomorphism of the corresponding moduli spaces. \end{lem} Indeed, since $\Xi_{Y \times S \to Y'\times S}^{L \boxtimes {\mathcal O}_S}(*)_s= \Xi_{Y \to Y'}^L(* \otimes k(s))$, if we have a flat family of $Y$-sheaves $\{{\mathcal E}_s \}_{s \in S}$, ${\mathcal E} \in \operatorname{Coh}(Y \times S)$, then $\{{\mathcal E}_s'\}_{s \in S}$ is also a flat family of $Y'$-sheaves, where ${\mathcal E}':= \Xi_{Y \times S \to Y'\times S}^{L \boxtimes {\mathcal O}_S}({\mathcal E})$.
\begin{rem}\label{rem:G} For a locally free $Y$-sheaf $G$, we have a projective bundle $Y' \to X$ with $\epsilon(Y')=\Xi_{Y \to Y'}^L(G)$. Hence it is sufficient to study the $\epsilon(Y)$-twisted semi-stability. \end{rem} \begin{rem} This definition is the same as in \cite{C:1}. If $Y={\mathbb P}(G^{\vee})$ for a vector bundle $G$ on $X$, then $\operatorname{Coh}(X,Y)$ is equivalent to $\operatorname{Coh}(X)$ and $G$-twisted stability is nothing but the twisted semi-stability in \cite{Y:11}. \end{rem}
\begin{defn} Let $\lambda$ be a rational number. Let $E$ be a $Y$-sheaf of dimension $d$. Then $E$ is of type $\lambda$ with respect to the $G$-twisted semi-stability, if \begin{enumerate} \item $E$ is of pure dimension $d$, \item \begin{equation} \frac{a_{d-1}^{G}(F)}{a_d^{G}(F)} \leq \frac{a_{d-1}^{G}(E)}{a_d^{G}(E)} +\lambda \end{equation} for all subsheaf $F$ of $E$. \end{enumerate} If $\lambda=0$, then $E$ is $\mu$-semi-stable. \end{defn}
\subsection{Construction of the moduli space}\label{subsect:construction} From now on, we assume that $G=\epsilon(Y)$ (cf. Remark \ref{rem:G}). Let $P(x)$ be a numerical polynomial. We shall construct the moduli space of $G$-twisted semi-stable $Y$-sheaves $E$ with $\chi(p_*(G^{\vee} \otimes E)(n))=P(n)$.
\subsubsection{Boundedness} Let $E$ be a $Y$-sheaf. Then \begin{equation} p^* p_*(G^{\vee} \otimes E) \otimes G \to E \end{equation} is surjective. Indeed $p^* p_*(G^{\vee} \otimes E) \to G^{\vee} \otimes E$ is an isomorphism and $G \otimes G^{\vee} \to {\mathcal O}_{Y}$ is surjective.
We take a surjective homomorphism ${\mathcal O}_X(-n_G )^{\oplus N} \to p_*(G^{\vee} \otimes G)$, $n_G \gg 0$. Then we have a surjective homomorphism $p^*({\mathcal O}_X(-n_G ))^{\oplus N} \to G^{\vee} \otimes G$.
\begin{lem}\label{lem:bdd1} Let $E$ be a $Y$-sheaf of pure dimension $d$. If \begin{equation}\label{eq:type-nu} a_{d-1}^G(F) \geq a_d^G(F) \left(\frac{a_{d-1}^G(E)}{a_d^G(E)}-\nu \right) \end{equation} for all quotient $E \to F$, then $a_{d-1}(F') \geq a_d(F') \left(\frac{a_{d-1}^G(E)}{a_d^G(E)}-\nu-n_G \right)$ for all quotient $p_*(G^{\vee} \otimes E) \to F'$. In particular \begin{equation}
S_\nu:=\{E \in \operatorname{Coh}(X,Y)| \text{ $E$ satisfies \eqref{eq:type-nu} and $\chi(p_*(G^{\vee} \otimes E)(nH))=P(n)$} \} \end{equation} is bounded. \end{lem}
\begin{proof} Since $p^* p_*(G^{\vee} \otimes E) \cong G^{\vee} \otimes E$, we have a surjective homomorphism \begin{equation} p^*({\mathcal O}_X(-n_G H))^{\oplus N} \otimes E \to G \otimes p^* p_*(G^{\vee} \otimes E) \to G \otimes p^*(F'). \end{equation} By our assumption, we get \begin{equation} a_{d-1}(p_*(G^{\vee} \otimes G) \otimes F') \geq a_d( p_*(G^{\vee} \otimes G) \otimes F') \left(\frac{a_{d-1}(p_*(G^{\vee} \otimes E))} {a_d(p_*(G^{\vee} \otimes E))}-n_G-\nu \right). \end{equation}
Since $a_{d-1}(p_*(G^{\vee} \otimes G) \otimes F')= \operatorname{rk}(G)^2 a_{d-1}(F')$ and $a_{d}(p_*(G^{\vee} \otimes G) \otimes F')= \operatorname{rk}(G)^2 a_{d}(F')$, we get our claim. The boundedness of $S_\nu$ follows from the boundedness of
$\{p_*(G^{\vee} \otimes E)| E \in S_{\nu} \}$ and Lemma \ref{lem:bdd2} below. \end{proof}
\begin{lem}\label{lem:bdd2}
Let $S$ be a bounded subset of $\operatorname{Coh}(X)$. Then $T:=\{E \in \operatorname{Coh}(X,Y)| p_*(G^{\vee} \otimes E) \in S \}$ is also bounded. \end{lem}
\begin{proof} For $E \in T$,
we set $I(E):=\ker(p^* p_*(G^{\vee} \otimes E) \otimes G \to E)$. We shall show that $T':=\{ I(E)| E \in T \}$ is bounded. We note that $I(E) \in \operatorname{Coh}(X,Y)$ and we have an exact sequence \begin{equation} 0 \to p_*(G^{\vee} \otimes I(E)) \to p_*(G^{\vee} \otimes E) \otimes p_*(G \otimes G^{\vee})
\to p_*(G^{\vee} \otimes E) \to 0. \end{equation} Since $p_*(G^{\vee} \otimes E) \in S$,
$\{ p_*(G^{\vee} \otimes I(E))| E \in T \}$ is also bounded. Since $p^* p_*(G^{\vee} \otimes I(E)) \otimes G \to I(E)$ is surjective and $I(E)$ is a subsheaf of $p^* p_*(G^{\vee} \otimes E) \otimes G$, $T'$ is bounded. \end{proof}
\begin{cor} Under the same assumption \eqref{eq:type-nu}, there is a rational number $\nu'$ which depends on $\nu$ such that \begin{equation} a_{d-1}(F') \leq a_d(F') \left(\frac{a_{d-1}^G(E)}{a_d^G(E)}+\nu' \right) \end{equation} for a subsheaf $F' \subset p_*(G^{\vee} \otimes E)$. \end{cor}
Combining this with Langer's important result \cite[Cor. 3.4]{Langer:1}, we have the following \begin{lem}\label{lem:section} Under the same assumption \eqref{eq:type-nu}, \begin{equation}
\frac{h^0(G,E)}{a_d^{G}(E)} \leq
\left[\frac{1}{d!}
\left(\frac{a_{d-1}^{G}(E)}{a_d^{G}(E)}+\nu'
+c \right)^d \right]_+, \end{equation} where $c$ depends only on $(X,{\mathcal O}_X(1))$, $G$, $d$ and $a_d^G(E)$. \end{lem}
\subsubsection{A quot-scheme}
Since $p_*(G^{\vee} \otimes E)(n)$, $n \gg 0$ is generated by global sections, \begin{equation} H^0(G^{\vee} \otimes E \otimes p^*{\mathcal O}_X(n))
\otimes G \to E\otimes p^*{\mathcal O}_X(n) \end{equation} is surjective. Since $R^i p_*(G^{\vee} \otimes E)=0$ for $i>0$, we also see that $H^i(E \otimes p^*{\mathcal O}_X(n))=0$, $i>0$ and $n \gg 0$.
We fix a sufficiently large integer $n_0$. We set $N:=\chi(p_*(G^{\vee} \otimes E)(n_0 ))=P(n_0)$. We set $V:={\mathbb C}^N$. We consider the quot-scheme ${\mathfrak Q}$ parametrizing all quotients \begin{equation} \phi:V \otimes G \to E \end{equation} such that $E \in \operatorname{Coh}(X,Y)$ and $\chi(p_*(G^{\vee} \otimes E)(n))=P(n_0+n)$. By Lemma \ref{lem:bdd2}, ${\mathfrak Q}$ is bounded, in particular, it is a quasi-projective scheme. \begin{lem} ${\mathfrak Q}$ is complete. \end{lem}
\begin{proof} We prove our claim by using the valuative criterion. Let $R$ be a discrete valuation ring and $K$ the quotient field of $R$. Let $\phi:V_R \otimes G \to {\mathcal E}$ be a $R$-flat family of quotients such that ${\mathcal E} \otimes_R K \in \operatorname{Coh}(X,Y)$, where $V_R:=V \otimes_{\mathbb C} R$. We set ${\mathcal I}:=\ker \phi$. We have an exact and commutative diagram: \begin{equation} \begin{CD} 0 @>>> p^* p_*({\mathcal I} \otimes G^{\vee}) @>>> V_R \otimes G \otimes G^{\vee} @>>> p^* p_*({\mathcal E}\otimes G^{\vee}) @>>> 0\\
@. @VVV @| @VV{\psi}V \\ 0 @>>> {\mathcal I} \otimes G^{\vee} @>>> V_R \otimes G \otimes G^{\vee} @>>> {\mathcal E}\otimes G^{\vee} @>>> 0 \end{CD} \end{equation} We shall show that $\psi$ is an isomorphism. Obviously $\psi$ is surjective. Since ${\mathcal E}$ is $R$-flat, ${\mathcal E}$ has no $R$-torsion, which implies that $p^* p_*({\mathcal E}\otimes G^{\vee})$ is a torsion free $R$-module. Hence $\ker \psi$ is also torsion free. On the other hand, our choice of ${\mathcal E}$ implies that $\psi \otimes K$ is an isomorphism. Therefore $\ker \psi=0$. \end{proof}
Since $\ker \phi \in \operatorname{Coh}(X,Y)$, we have a surjective homomorphism \begin{equation} V \otimes \operatorname{Hom}(G,G \otimes p^*{\mathcal O}_X(n)) \to \operatorname{Hom}(G,E \otimes p^*{\mathcal O}_X(n)) \end{equation} for $n \gg0$. Thus we can embed ${\mathfrak Q}$ as a subscheme of an Grassmann variety $Gr(V \otimes W,P(n_0+n))$, where $W=\operatorname{Hom}(G,G \otimes p^*{\mathcal O}_X(n))$. Since all semi-stable $Y$-sheaf are pure, we may replace ${\mathfrak Q}$ by the closure of the open subset parametrizing pure quotient $Y$-sheaves. The same arguments in \cite{Y:11} imply that ${\mathfrak Q}/\!\!/ GL(V)$ is the moduli space of $G$-twisted semi-stable sheaves. The details are left to the reader. For the proof, we also use the following.
Let $(R,{\mathfrak m})$ be a discrete valuation ring $R$ and the maximal ideal ${\mathfrak m}$. Let $K$ be the fractional field and $k$ the residue field. Let ${\mathcal E}$ be a $R$-flat family of $Y \otimes R$-sheaves such that ${\mathcal E} \otimes_R K$ is pure. \begin{lem}\label{lem:valuative} There is a $R$-flat family of coherent $Y \otimes R$-sheaves ${\mathcal F}$ and a homomorphism $\psi:{\mathcal E} \to {\mathcal F}$ such that ${\mathcal F} \otimes_R k$ is pure, $\psi_K$ is an isomorphism and $\psi_k$ is an isomorphic at generic points of $\Supp({\mathcal F} \otimes_R k)$. \end{lem} By using \cite[Lem. 1.17]{S:1} or \cite[Prop. 4.4.2]{H-L:1}, we first construct ${\mathcal F}$ as a usual family of sheaves. Then the very construction of it, ${\mathcal F}$ becomes a $Y \otimes R$-sheaf.
\begin{NB} It also follows by using elementary transformations along $Y \otimes_R k$-sheaves on $Y \otimes_R k$. \end{NB}
\subsection{A family of $Y$-sheaves on a projective bundle over ${M}_{X/{\mathbb C}}^{h}$}\label{subsubsect:family}
Assume that ${\mathfrak Q}^{ss}$ consists of stable points. Then ${\mathfrak Q}^{ss} \to \overline{M}_{X/{\mathbb C}}^{h}$ is a principal $PGL(N)$-bundle. For a scheme $S$, $f_S:Y \times S \to S$ denotes the projection. Let ${\mathcal Q}$ be the universal quotient sheaf on $Y \times {\mathfrak Q}^{ss}$. $V:=\operatorname{Hom}_{f_{ {\mathfrak Q}^{ss}}} (G \boxtimes {\mathcal O}_{{\mathfrak Q}^{ss}},{\mathcal Q})$ is a locally free sheaf on ${\mathfrak Q}^{ss}$. We consider the projective bundle ${\mathfrak q}:{\mathbb P}(V) \to {\mathfrak Q}^{ss}$. Since ${\mathcal Q}$ is $GL(N)$-linearized, $V$ is also $GL(N)$-linearized. Then we have a quotient $\psi:{\mathbb P}(V) \to {\mathbb P}(V)/PGL(N)$ with the commutative diagram: \begin{equation} \begin{CD} {\mathbb P}(V) @>{\mathfrak q}>> {\mathfrak Q}^{ss}\\ @VVV @VVV\\ \widetilde{\overline{M}_{X/{\mathbb C}}^{h}}:={\mathbb P}(V)/PGL(N) @>{q}>> \overline{M}_{X/{\mathbb C}}^{h} \end{CD} \end{equation} Since $(1_Y \times {\mathfrak q})^*({\mathcal Q}) \otimes f_{{\mathbb P}(V)}^*({\mathcal O}_{{\mathbb P}(V)}(-1))$ is $PGL(N)$-linearlized, we have a family of $G$-twisted stable $Y$-sheaves ${\mathcal E}$ on $Y \times \widetilde{\overline{M}_{X/{\mathbb C}}^{h}}$ with $(1_Y \times \psi)^*({\mathcal E})=(1_Y \times {\mathfrak q})^*({\mathcal Q}) \otimes f_{{\mathbb P}(V)}^*({\mathcal O}_{{\mathbb P}(V)}(-1))$. Hence ${\mathcal E}^{\vee} \in \operatorname{Coh}(Y \times \overline{M}_{X/{\mathbb C}}^{h}, Y \times \widetilde{\overline{M}_{X/{\mathbb C}}^{h}})$ (if ${\mathcal E}$ is locally free). Let $W$ be a locally free sheaf on $\widetilde{\overline{M}_{X/{\mathbb C}}^{h}}$ such that $\psi^*(W)={\mathfrak q}^*(V)(-1)$. Then we also have $W^{\vee}=\epsilon(\widetilde{\overline{M}_{X/{\mathbb C}}^{h}}) \in \operatorname{Coh}(\overline{M}_{X/{\mathbb C}}^{h}, \widetilde{\overline{M}_{X/{\mathbb C}}^{h}})$ and ${\mathcal E} \otimes f_{\widetilde{\overline{M}_{X/{\mathbb C}}^{h}}}^*(W^{\vee})$ descends to a sheaf on $Y \times \overline{M}_{X/{\mathbb C}}^{h}$.
\begin{rem} There is also a family of $G$-twisted stable $Y$-sheaves ${\mathcal E}'$ on $Y \times {\mathbb P}(V^{\vee})/PGL(N)$ such that ${\mathcal E}' \in \operatorname{Coh}(Y \times \overline{M}_{X/{\mathbb C}}^{h}, Y \times {\mathbb P}(V^{\vee})/PGL(N))$. \end{rem}
\begin{NB}
\subsection{Construction of moduli spaces}
From now on, we denote $a_{i}^{G}(*)$ by $a_{i}(*)$. For a $G$-twisted coherent sheaf $E \in \operatorname{Coh}(X,Y)$, we denote $E \otimes p^* {\mathcal O}_X(n)$ by $E(n)$. Let $E$ be a purely $d$-dimensional $G$-twisted coherent sheaf such that $E$ is of type $\lambda$ with respect to the $G$-twisted semi-stability. Let $E \to E''$ be a quotient sheaf such that $E''$ is of pure dimension $d$ and \begin{equation}\label{eq:a3} {a_d(E'')}\frac{\chi(G,E(n))}{a_d(E)} \geq {\chi(G,E''(n))}. \end{equation} Since the set of $E$ is bounded, by Grothendieck's boundedness theorem, the set of such quotients $E''$ is bounded. Hence there is an integer $m({\lambda})$ which depends on $h$ and $\lambda$ such that, for $m \geq m({\lambda})$ and the kernel $E'$ of $E \to E''$ which satisfies \eqref{eq:a3},
\begin{enumerate}
\item[($\flat 1$)]
$\operatorname{Hom}(G,E'(m)) \otimes G \to E'(m)$ is surjective and
\item[($\flat 2$)]
$\operatorname{Ext}^i(G,E'(m))=0$, $i>0$. \end{enumerate} In particular, \begin{enumerate} \item $\operatorname{Hom}(G,E(m)) \otimes G \to E(m)$ is surjective and \item
$\operatorname{Ext}^i(G,E(m))=0$, $i>0$. \end{enumerate}
Let $V_m$ be a vector space of dimension $h(m)$. Let ${\mathfrak Q}:=\operatorname{Quot}_{V_m \otimes G/Y}^{h[m]}$ be the quot-scheme parametrizing all quotients $V_m \otimes G \to F$ such that $F \in \operatorname{Coh}(X,Y)$ and the $G$-twisted Hilbert polynomial of $F$ is $h[m]$, where $h[m](x)=h(m+x)$. Let $V_m \otimes G \otimes {\mathcal O}_{{\mathfrak Q}} \to \widetilde{E}(m)$ be the universal quotient sheaf on ${\mathfrak Q} \times X$. Let ${\mathfrak Q}^{ss}$ be the open subscheme of ${\mathfrak Q}$ consisting of quotients $f:V_m \otimes G \to E(m)$ such that \begin{enumerate} \item a canonical map $V_m \to \operatorname{Hom}(G,E(m))$ sending $v \in V_m$ to $f(v \otimes *) \in \operatorname{Hom}(G,E(m))$ is an isomorphism and \item $E$ is an $G$-twisted semi-stable sheaf. \end{enumerate} We set $W:=\operatorname{Hom}(G,G(n))$. Let ${\mathfrak G}(n):=Gr(V_m \otimes W,h[m](n))$ be the Grassmannian parametrizing $h[m](n)$-dimensional quotient spaces of $V_m \otimes W$. For a quotient $V_m \otimes G \to E(m) \in {\mathfrak Q}$, let $F$ be the kernel. Then for a sufficiently large $n$, we get that \begin{enumerate} \item $\operatorname{Hom}(G,F(n)) \otimes G \to F(n)$ is surjective and \item $\operatorname{Ext}^i(G,F(n))=0$, $i>0$. \end{enumerate} Hence we get a quotient vector space $V_m \otimes \operatorname{Hom}(G,G(n)) \to \operatorname{Hom}(G,E(m+n))$ of $V_m \otimes \operatorname{Hom}(G,G(n))$. Thus we get a morphism ${\mathfrak Q} \to {\mathfrak G}(n)$. As in \cite{Mum:1}, we can show that this morphism is a closed immersion
\begin{equation}
{\mathfrak Q} \hookrightarrow {\mathfrak G}(n). \end{equation}
Indeed, let ${\mathcal S} \subset V_m \otimes W \otimes {\mathcal O}_{{\mathfrak G}(n)}$ be the universal subbundle. We set \begin{equation}
{\mathcal E}:=\operatorname{coker}({\mathcal S} \otimes G(-n) \to
V_m \otimes W \otimes G(-n) \otimes {\mathcal O}_{{\mathfrak G}(n)}
\overset{ev}{\to} V_m \otimes G \otimes {\mathcal O}_{{\mathfrak G}(n)}). \end{equation} We take a flattening stratification ${\mathfrak G}(n)=\coprod_i {\mathfrak G}(n)_i$ of ${\mathcal E}$ \cite[sect. 8]{Mum:1}. We may assume that each ${\mathfrak G}(n)_i$ are connected. Let ${\mathfrak G}(n)_{h[m]}$ be the union of ${\mathfrak G}(n)_i$ such that the $G$-twisted Hilbert polynomial of ${\mathcal E}_x, x \in {\mathfrak G}(n)_{h[m]}$ is $h[m]$. Then ${\mathfrak G}(n)_{h[m]}$ is isomorphic to ${\mathfrak Q}$.
$SL(V_m)$ acts on ${\mathfrak G}(n)$. Let $L:={\mathcal O}_{{\mathfrak G}(n)}(1)$ be the tautological line bundle on ${\mathfrak G}(n)$. Then $L$ has an $SL(V_m)$-linearization. We consider the GIT semi-stability with respect to $L$. \begin{prop}\label{prop:ss} Let $\alpha:V_m \otimes W \to A$ be a quotient corresponding to a point of ${\mathfrak G}(n)$. Then it is GIT semi-stable with respect to $L$ if and only if
\begin{equation}
\dim V_m \dim \alpha(V' \otimes W)-
\dim V' \dim \alpha(V_m \otimes W) \geq 0 \end{equation}
for all non-zero subspaces $V'$ of $V_m$. \end{prop}
\begin{prop} There is an integer $m_1$ such that for all $m \geq m_1$, ${\mathfrak Q}^{ss}$ is contained in ${\mathfrak G}(n)^{ss}$, where $n \gg m$. \end{prop}
\begin{proof} We set \begin{equation}
{\mathcal F}:=\{E' \subset \widetilde{E}_q(m)| E'=\operatorname{im}(V' \otimes G \to \widetilde{E}_q(m)), q \in {\mathfrak Q}, V' \subset V_m \}. \end{equation}
Since ${\mathcal F}$ is a bounded set, for a sufficiently large $n$ which depends on $m$, \begin{enumerate} \item $\alpha(V' \otimes W)=\operatorname{Hom}(G,E'(n))$ and \item $\operatorname{Ext}^i(G,E'(n))=0$, $i>0$.
\end{enumerate}
Then $\dim \alpha(V' \otimes W)=\chi(G,E'(n))$. Since $E$ is $G$-twisted semi-stable, in the same way as in \cite[sect. 4]{Mar:4}, we see that there is an integer $m_0$ such that for $m \geq m_0$ and a subsheaf $E'$ of $E(m)$,
\begin{equation}\label{eq:pgs}
\frac{h^0(G,E')}{a_d(E')} \leq
\frac{h^0(G,E(m))}{a_d(E)} \end{equation}
and the equality holds, if and only if
\begin{equation}
\frac{\chi(G,E'(n))}{a_d(E')}=\frac{\chi(G,E(m+n))}{a_d(E)} \end{equation}
for all $n$. Hence if the equality holds, then $E'$ is $G$-twisted semi-stable and we may assume that ($\flat1,2$) holds for $E'(-m)$. In particular, $\dim V'=\chi(G,E')$. We set $m_1:=\max \{m_0,m(0) \}$. For a sufficiently large $n \gg m$, we get \begin{equation}\label{eq:a6}
\left| \frac{\chi(G,E'(n))}{\chi(G,E(m+n))}
-\frac{a_d(E')}{a_d(h)} \right|
<\frac{1}{\dim V_m a_d(h)}. \end{equation}
By \eqref{eq:a6}, if the inequality in \eqref{eq:pgs} is strict, then we get \begin{equation}\label{eq:a7}
\begin{split}
&\dim V_m \dim \alpha(V' \otimes W)-
\dim V' \dim \alpha(V_m \otimes W)\\
> & \left(\dim V_m \frac{a_d(E')}{a_d(h)}-\dim V'-\frac{1}{a_d(h)}
\right)\dim \alpha(V_m \otimes W)
\geq 0.
\end{split} \end{equation}
If the equality holds in \eqref{eq:pgs}, then $\dim V'=\chi(G,E')$, and hence \begin{equation}
\dim V_m \dim \alpha(V' \otimes W)-
\dim V' \dim \alpha(V_m \otimes W)=0. \end{equation}
Therefore our claim holds. \end{proof}
\begin{prop}\label{prop:proper} There is an integer $m_2$ such that for all $m \geq m_2$, ${\mathfrak Q}^{ss}$ is a closed subscheme of ${\mathfrak G}(n)^{ss}$, where $n \gg m$. \end{prop}
\begin{proof} We choose an $m$ so that $h(m)/a_d(h)>1$. We shall prove that ${\mathfrak Q}^{ss} \to {\mathfrak G}(n)^{ss}$ is proper. Let $(R,{\mathfrak m})$ be a discrete valuation ring and $K$ the quotient field of $R$. We set $T:=\operatorname{Spec}(R)$ and $U:=\operatorname{Spec}(K)$. Let $U \to {\mathfrak Q}^{ss}$ be a morphism such that $U \to {\mathfrak Q}^{ss} \to {\mathfrak G}(n)^{ss}$ is extended to a morphism $T \to {\mathfrak G}(n)^{ss}$. Since ${\mathfrak Q}$ is a closed subscheme of ${\mathfrak G}(n)$, there is a morphism $T \to {\mathfrak Q}$, {\it i.e}, there is a flat family of quotients:
\begin{equation}
V_m \otimes G \otimes {\mathcal O}_{T} \to
{\mathcal E}(m) \to 0. \end{equation}
Let $\alpha:V_m \otimes W \otimes R \to \operatorname{Hom}_{p_{T*}}(G \otimes {\mathcal O}_{T},{\mathcal E}(m+n))$ be the quotient of $V_m \otimes W \otimes R$ corresponding to the morphism $T \to {\mathfrak G}(n)^{ss}$. We set $E:={\mathcal E} \otimes R/{\mathfrak m}$. \begin{claim} $V_m \to \operatorname{Hom}(G,E(m))$ is injective. \end{claim} Indeed, we set $V':=\ker(V_m \to \operatorname{Hom}(G,E(m)))$. Then $\alpha(V' \otimes W)=0$. Hence we get
\begin{equation}
\begin{split}
0 \leq & \dim V_m \dim \alpha(V' \otimes W)-
\dim V' \dim \alpha(V_m \otimes W)\\
=& -\dim V' \dim \alpha(V_m \otimes W) \leq 0.
\end{split} \end{equation}
Therefore $V'=0$.
\begin{claim}\label{claim:2} There is a rational number $\lambda$ which depends on $h$ such that $E$ is of type $\lambda$. \end{claim} Proof of the claim: Let $E \to E''$ be a quotient of $E$. Let $E'$ be the kernel of $E \to E''$. We note that $V_m \to \operatorname{Hom}(G,E(m))$ is injective. We set $V':=V_m \cap \operatorname{Hom}(G,E'(m))$. Then $h^0(G,E''(m)) \geq \dim V_m -\dim V'$. Let $F$ be a subsheaf of $E(m)$ generated by $V'$. Then $F$ belongs to ${\mathcal F}$. We set
\begin{equation} \varepsilon:=\frac{1}{a_d(h)! h(m)}. \end{equation}
Since ${\mathcal F}$ is a bounded set, for a sufficiently large $n$ which depends on $m$ and $\varepsilon$, we have $\alpha(V' \otimes W)=\operatorname{Hom}(G,F(n))$, $\operatorname{Ext}^i(G,F(n))=0$, $i>0$ and
\begin{equation}\label{eq:a1}
\left|\frac{ \dim \alpha(V' \otimes W)}{\dim \alpha(V_m \otimes W)}
-\frac{a_d(F)}{a_d(h)} \right|<
\varepsilon. \end{equation}
Since $a_d(E') \geq a_d(F)$,
\begin{equation}
\begin{split}
\frac{h^0(G,E''(m))}{a_d(E'')} & \geq
\frac{\dim V_m-\dim V'}{a_d(E'')}\\
& > h(m)\left(\frac{a_d(h)-a_d(F)}{a_d(h)}\frac{1}{a_d(E'')}-
\frac{\varepsilon}{a_d(E'')} \right)\\
& > h(m)\left(\frac{a_d(h)-a_d(E')}{a_d(h)}\frac{1}{a_d(E'')}-
\frac{\varepsilon}{a_d(E'')} \right)\\
& \geq h(m)\left(\frac{1}{a_d(h)}-\varepsilon \right)>0.
\end{split} \end{equation} There is a rational number $\lambda_1$ and an integer $m_3 \geq \lambda_1-a_{d-1}(h)/a_d(h)$ which depend on $h(x)$ such that
\begin{equation}\label{eq:a5}
h(m) \left(\frac{1}{a_d(h)}-\varepsilon \right)
\geq \frac{h(m)}{a_d(h)}-\frac{1}{a_d(h)!} \geq
\frac{1}{d!}\left(m+\frac{a_{d-1}(h)}{a_d(h)}-\lambda_1 \right)^d \end{equation}
for $m \geq m_3$.
By Lemma \ref{lem:valuative}, there is a purely $d$-dimensional sheaf $F$ with the $G$-twisted Hilbert polynomial $h(x)$ and a map $E \to F$ whose kernel is a coherent sheaf of dimension less than $d$. Let $F \to F''$ be a quotient such that $F''$ is $G$-twisted semi-stable. We set $E':=\ker(E \to F'')$ and $E'':=\operatorname{im}(E \to F'')$. Since $h^0(G,E''(m)) \leq h^0(G,F''(m))$ and $a_d(E'')=a_d(F'')$,
\begin{equation}
\begin{split}
\frac{h^0(G,F''(m))}{a_d(F'')} & \geq \frac{h^0(G,E''(m))}{a_d(E'')} \\
& \geq h(m) \left(\frac{1}{a_d(h)}-\varepsilon \right) >0.
\end{split} \end{equation}
Since $F''$ is $G$-twisted semi-stable, Lemma \ref{lem:section} implies that
\begin{equation}
\frac{h^0(G,F''(m))}{a_d(F'')} \leq
\left[\frac{1}{d!}\left(m+\frac{a_{d-1}(F'')}{a_d(F'')}+c \right)^d
\right]_+
\end{equation}
where $c$ is a constant which only depends on $a_d(h)$. Since $h^0(G,F''(m))>0$, we get
\begin{equation} \frac{1}{d!}\left(m+\frac{a_{d-1}(F'')}{a_d(F'')}+c \right)^d \geq 0 \end{equation}
and \begin{equation} h(m) \left(\frac{1}{a_d(h)}-\varepsilon \right) \leq
\frac{h^0(G,F''(m))}{a_d(F'')} \leq \frac{1}{d!}\left(m+\frac{a_{d-1}(F'')}{a_d(F'')}+c \right)^d. \end{equation} For $m \geq m_3$, \eqref{eq:a5} implies that
\begin{equation}
\frac{a_{d-1}(h)}{a_d(h)}-\lambda_1 \leq
\frac{a_{d-1}(F'')}{a_d(F'')}+c. \end{equation}
Hence there is a rational number $\lambda$ which depends on $h(x),\lambda_1$ such that $F$ is of type $\lambda$. Replacing $m$, we may assume that for all type $\lambda$ sheaves $I$ with the $G$-twisted Hilbert polynomial $h(x)$, \begin{enumerate} \item $\operatorname{Hom}(G,I(m)) \otimes G \to I(m)$ is surjective and \item $\operatorname{Ext}^i(G,I(m))=0$, $i>0$. \end{enumerate}
In particular $h^0(G,F(m))=h(m)=\dim V_m$. Assume that $\operatorname{Hom}(G,E(m)) \to \operatorname{Hom}(G,F(m))$ is not injective and let $V'$ be the kernel. $J:=\operatorname{im}(V' \otimes G \to E(m))$ is of dimension less than $d$. Hence we get $a_d(J)=0$. By the inequality \eqref{eq:a1} and Proposition \ref{prop:ss}, we get a contradiction. Thus $\operatorname{Hom}(G,E(m)) \to \operatorname{Hom}(G,F(m))$ is injective, and hence it is isomorphic. Since $\operatorname{Hom}(G,F(m)) \otimes G \to F(m)$ is surjective, $E \to F$ must be surjective, which implies that it is isomorphic. Therefore $E$ is of pure dimension $d$, of type $\lambda$ and $V_m \to \operatorname{Hom}(G,E(m))$ is an isomorphism. Thus we complete the proof of Claim \ref{claim:2}.
We set $m_2:=\max\{m_3,m(\lambda)\}$. Assume that there is a quotient $E \to E''$ which destabilizes the $G$-twisted semi-stability. Since $E':=\ker (E \to E'')$ satisfies $(\flat 1, 2)$, we get that $V'=\operatorname{Hom}(G,E'(m))$ and
\begin{equation}
\frac{\chi(G,E''(m))}{a_d(E'')} >
\frac{\chi(G,E(m))}{a_d(h)}-\varepsilon h(m). \end{equation}
By the definition of $\varepsilon$, we get \begin{equation}\label{eq:ss}
\frac{\chi(G,E''(m))}{a_d(E'')} \geq
\frac{\chi(G,E(m))}{a_d(h)}, \end{equation}
which is a contradiction. Therefore $E$ is $G$-twisted semi-stable. Thus we get a lifting $T \to {\mathfrak Q}^{ss}$ and conclude that ${\mathfrak Q}^{ss} \to {\mathfrak G}(n)^{ss}$ is proper. \end{proof} By standard arguments, we see that $SL(V_m)s, s \in {\mathfrak Q}^{ss}$ is a closed orbit if and only if the corresponding $G$-twisted semi-stable sheaf $E$ is isomorphic to $\bigoplus_i E_i$, where $E_i$ are $G$-twisted stable sheaves.
\end{NB}
\section{Twisted sheaves on a projective $K3$ surface}\label{sect:k3}
\subsection{Basic properties} Let $X$ be a projective $K3$ surface and $p:Y \to X$ a projective bundle.
\begin{lem}\label{lem:equiv} For a locally free $Y$-sheaf $E$, $c_2({\bf R}p_*(E^{\vee} \otimes E)) \equiv -(r-1)(w(E)^2) \mod 2r$. \end{lem}
\begin{proof} First we note that $(r-1)(D^2) \mod 2r$ is well-defined for $D \in H^2(Z,\mu_r)$, $Z=X,Y$. We take a representative $\alpha \in H^2(X,{\mathbb Z})$ of $w(E)$. Then $c_1(E) \equiv p^*(\alpha) \mod r$. Hence $c_2(p^*({\bf R}p_*(E^{\vee} \otimes E))) =2rc_2(E)-(r-1)(c_1(E)^2) \equiv -(r-1)(p^*(\alpha^2)) \mod 2r$. Since $H^4(X,{\mathbb Z})$ is a direct summand of $H^4(Y,{\mathbb Z})$, $c_2({\bf R}p_*(E^{\vee} \otimes E)) \equiv -(r-1)(\alpha^2) \mod 2r$. \end{proof}
\begin{NB}
Let $D$ be a proper subscheme of $X$. Since $\dim D \leq 1$, $H^2(D,{\mathcal O}_D^{\times})=0$. Hence there is a set of $\delta_{i,j} \in H^0(U_i \cap U_j,{\mathcal O}_D^{\times})$ such that ${\mathcal O}_D^{\alpha}:=
\{({\mathcal O}_{D|U_i},\delta_{i,j})\}$ is an $\alpha$-twisted line bundle on $D$. \end{NB}
Let $K(X,Y)$ be the Grothendieck group of $Y$-sheaves. \begin{lem} \begin{enumerate} \item[(1)] There is a locally free $Y$-sheaf $E_0$ such that
$\operatorname{rk} E_0=\min\{\operatorname{rk} E>0| E \in \operatorname{Coh}(X,Y)\}$. \item[(2)] $K(X,Y)={\mathbb Z}E_0 \oplus K(X,Y)_{\leq 1}$, where $K(X,Y)_{\leq 1}$ is the submodule of $K(X,Y)$ generated by $E \in \operatorname{Coh}(X,Y)$ of $\dim E \leq 1$. \end{enumerate} \end{lem}
\begin{proof} (1) Let $F$ be a $Y$-sheaf such that
$\operatorname{rk} F=\min\{\operatorname{rk} E>0| E \in \operatorname{Coh}(X,Y)\}$. Then $E_0:=F^{\vee \vee}$ satisfies the required properties. (2) We shall show that the image of $E \in \operatorname{Coh}(X,Y)$ in $K(X,Y)$ belongs to ${\mathbb Z}E_0 \oplus K(X,Y)_{\leq 1}$ by the induction of $\operatorname{rk} E$. We may assume that $\operatorname{rk} E>0$. Let $T$ be the torsion submodule of $E$. Then $E=T+E/T$ in $K(X,Y)$.
Since $\operatorname{Hom}(E_0(-nH),E/T) \ne 0$ for $n \gg 0$, we have a non-zero homomorphism $\varphi:E_0(-nH) \to E/T$. By our choice of $E_0$, $\varphi$ is injective. Since $E_0(-nH)=E_0-E_{0|nH}$ in $K(X,Y)$,
$E=((E/T)/E_0+E_0)+(T-E_{0|nH})$. Since $\operatorname{rk} (E/T)/E_0<\operatorname{rk} E$, we get $(E/T)/E_0 \in {\mathbb Z}E_0 \oplus K(X,Y)_{\leq 1}$, and hence
$E$ also belongs to ${\mathbb Z}E_0 \oplus K(X,Y)_{\leq 1}$.
\end{proof}
\begin{rem} $\operatorname{rk} E_0$ is the order of the Brauer class of $Y$. \end{rem}
\begin{NB}
$\chi(G,{\mathcal O}_D^{\alpha})=\chi(G^{\vee}_{|D}) \equiv (w(Y),D) \mod r$ and $\chi(G,k_P^{\alpha})=r$.
$(w(Y),D)=w(Y)_{|D} \equiv c_1({\bf R}p_*({\mathcal O}_D^{\alpha} \otimes G^{\vee})) \mod r$. \end{NB}
Let $\langle\;\;,\;\;\rangle$ be the Mukai pairing on $H^*(X,{\mathbb Z})$: \begin{equation} \langle x,y \rangle=-\int_X x^{\vee} y,\quad x,y \in H^*(X,{\mathbb Z}). \end{equation} \begin{defn} Let $G$ be a locally free $Y$-sheaf. For a $Y$-sheaf $E$, we define a Mukai vector of $E$ as \begin{equation}\label{eq:v-def} \begin{split} v_G(E):=& \frac{\operatorname{ch}({\bf R}p_*(E \otimes G^{\vee}))} {\sqrt{\operatorname{ch}({\bf R}p_*(G \otimes G^{\vee}))}}\sqrt{\operatorname{td}_X} \\ =& (\operatorname{rk}(E),\zeta,b) \in H^*(X,{\mathbb Q}), \end{split} \end{equation} where $p^*(\zeta)=c_1(E)- \operatorname{rk}(E)\frac{c_1(G)}{\operatorname{rk} G}$ and $b \in {\mathbb Q}$. More generally, for $G \in \operatorname{Coh}(X,Y)$ with $\operatorname{rk} G>0$, we define $v_G(E)$ by \eqref{eq:v-def}. \end{defn}
Since ${\bf R}p_*(E_1 \otimes G^{\vee}) \otimes {\bf R}p_*(E_2 \otimes G^{\vee})^{\vee} = {\bf R}p_*(E_1 \otimes E_2^{\vee}) \otimes {\bf R}p_*(G \otimes G^{\vee})$, \begin{equation} \begin{split} \langle v_G(E_1),v_G(E_2) \rangle=& -\int_X \frac{\operatorname{ch}({\bf R}p_*(E_1 \otimes G^{\vee})) \operatorname{ch}({\bf R}p_*(E_2 \otimes G^{\vee}))^{\vee}} {\operatorname{ch}({\bf R}p_*(G \otimes G^{\vee}))}\operatorname{td}_X\\ =& -\int_X \operatorname{ch}({\bf R}p_*(E_1 \otimes E_2^{\vee}))\operatorname{td}_X\\ =& -\chi(E_2,E_1). \end{split} \end{equation}
\begin{NB} \begin{equation} \begin{split} \frac{\sqrt{(r+\xi+b \omega)(r-\xi+b \omega)}}{(r+\xi+b \omega)} &=\sqrt{\frac{(r-\xi+b \omega)}{(r+\xi+b \omega)}}\\ &=\sqrt{1-2\frac{\xi}{r}+2\left(\frac{\xi}{r}\right)^2}\\ &=e^{-\frac{\xi}{r}} \end{split} \end{equation} \end{NB}
We define an integral structure on $H^*(X,{\mathbb Q})$ such that $v_G(E)$ is integral. This is due to Huybrechts and Stellari \cite{H-S:2}. For a positive integer $r$ and $\xi \in H^2(X,{\mathbb Z})$, we consider an injective homomorphism \begin{equation} \begin{matrix} T_{-\xi/r}:& H^*(X,{\mathbb Z}) & \to & H^*(X,{\mathbb Q})\\ & x & \mapsto & e^{-\xi/r}x. \end{matrix} \end{equation} $T_{-\xi/r}$ preserves the bilinear form $\langle\;\;,\;\; \rangle$.
\begin{lem}\label{lem:integral} We take a representative $\xi \in H^2(X,{\mathbb Z})$ of $w(G) \in H^2(X,\mu_{r})$, where $\operatorname{rk}(G)=r$. We set $(\operatorname{rk}(E),D,a):=e^{\xi/r}v_G(E)$. Then $(\operatorname{rk}(E),D,a)$ belongs to $H^*(X,{\mathbb Z})$ and $[D \mod \operatorname{rk}(E)]=w(E)$. \end{lem} \begin{proof} We set $\sigma:=(c_1(G)-p^*(\xi))/r \in H^2(Y,{\mathbb Z})$. Since $p^*(D)=p^*(\zeta)+\operatorname{rk}(E)p^*(\xi)/\operatorname{rk}(G)= c_1(E)-\operatorname{rk} (E) \sigma \in H^2(Y,{\mathbb Z})$, we get $D \in H^2(X,{\mathbb Z})$. By Lemma \ref{lem:equiv}, we see that \begin{equation} \begin{split} \langle e^{\xi/r}v_G(E),e^{\xi/r}v_G(E) \rangle=& \langle v_G(E),v_G(E) \rangle \\ = & c_2({\bf R}p_*(E \otimes E^{\vee}))-2\operatorname{rk}(E)^2 \\ \equiv & (D^2) \mod 2 \operatorname{rk} (E). \end{split} \end{equation}
Hence $a \in {\mathbb Z}$. The last claim is obvious. \end{proof}
\begin{rem} $e^{\xi/r}v_G(E)$ is the same as the Mukai vector defined by the rational $B$-field $\xi/r$ in \cite{H-S:2}. More precisely, there is a topological line bundle $L$ on $Y$ with $c_1(L)=\sigma$ and $E \otimes L^{-1}$ is the pull-back of a topological sheaf $E_{\xi/r}$ on $X$. Then we see that $e^{\xi/r}v_G(E)=\operatorname{ch}(E_{\xi/r})\sqrt{\operatorname{td}_X}$ (we use $H^i(X,{\mathbb Q})=0$ for $i>4$, or we deform $X$ so that $L$ becomes holomorphic).
\begin{NB} Use $\operatorname{ch}(G \otimes L^{-1})/\operatorname{ch}((G \otimes L^{-1})^{-1})=e^{2\xi/r}$. \end{NB} \end{rem}
\begin{defn}\cite{H-S:2} We define a weight 2 Hodge structure on the lattice $(H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle)$ as \begin{equation} \begin{split} H^{2,0}(H^*(X,{\mathbb Z}) \otimes {\mathbb C}):= & T_{-\xi/r}^{-1}(H^{2,0}(X))\\ H^{1,1}(H^*(X,{\mathbb Z}) \otimes {\mathbb C}):=& T_{-\xi/r}^{-1}(\bigoplus_{p=0}^2 H^{p,p}(X))\\ H^{0,2}(H^*(X,{\mathbb Z}) \otimes {\mathbb C}):=& T_{-\xi/r}^{-1}(H^{0,2}(X)). \end{split} \end{equation} We denote this polarized Hodge structure by $(H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi}{r})$. \end{defn}
\begin{lem} The Hodge structure $(H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi}{r})$ depends only on the Brauer class $\delta'([\xi \mod r])$. \end{lem}
\begin{proof} If $\delta'([\xi \mod r])=\delta'([\xi' \mod r']) \in H^2(X,{\mathcal O}_X^{\times})$, then we have $r'\xi-r \xi'=L+rr' N$, where $L \in \operatorname{NS}(X)$ and $N \in H^2(X,{\mathbb Z})$. Then we have the following commutative diagram: \begin{equation} \begin{CD} H^*(X,{\mathbb Z}) @>{e^{-\frac{\xi}{r}}}>> H^*(X,{\mathbb Q})\\ @V{e^{-N}}VV @VV{e^{\frac{L}{rr'}}}V \\ H^*(X,{\mathbb Z}) @>>{e^{-\frac{\xi'}{r'}}}> H^*(X,{\mathbb Q}). \end{CD} \end{equation} Thus we have an isometry of Hodge structures \begin{equation} (H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi}{r}) \cong (H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi'}{r'}). \end{equation} \end{proof}
\begin{defn} Let $Y \to X$ be a projective bundle and $G$ a locally free $Y$-sheaf. Let $\xi \in H^2(X,{\mathbb Z})$ be a lifting of $w(G) \in H^2(X,\mu_r)$, where $r=\operatorname{rk}(G)$. \begin{enumerate} \item We define an integral Hodge structure of $H^*(X,{\mathbb Q})$ as $T_{-\xi/r}((H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi}{r}))$. \item $v:=(r,\zeta,b)$ is a Mukai vector, if $v \in T_{-\xi/r}(H^*(X,{\mathbb Z}))$
and $\zeta \in \operatorname{Pic}(X) \otimes{\mathbb Q}$.
Moreover if $v$ is primitive in $T_{-\xi/r}(H^*(X,{\mathbb Z}))$,
then $v$ is primitive. \end{enumerate} \end{defn}
\begin{defn} Let $v:=(r,\zeta,b) \in H^*(X,{\mathbb Q})$ be a Mukai vector. \begin{enumerate} \item $\overline{M}^{Y,G}_H(r,\zeta,b)$ (resp. ${M}^{Y,G}_H(r,\zeta,b)$) denotes the coarse moduli space of $S$-equivalence classes of $G$-twisted semi-stable (resp. stable) $Y$-sheaves $E$ with $v_G(E)=v$. \item ${\mathcal M}^{Y,G}_H(r,\zeta,b)^{ss}$ (resp. ${\mathcal M}^{Y,G}_H(r,\zeta,b)^s$) denotes the moduli stack of $G$-twisted semi-stable (resp. stable) $Y$-sheaves $E$ with $v_G(E)=v$. \end{enumerate} \end{defn}
\begin{lem}\label{lem:isom-G} Assume that $o(w(Y))=o(w(Y'))$. Then $\Xi_{Y \to Y'}^L$ induces an isomorphism ${\mathcal M}_H^{Y,G}(v)^{ss} \cong {\mathcal M}_H^{Y',G'}(v)^{ss}$, where $G':=\Xi_{Y \to Y'}^L(G)$. Moreover if $\dim Y=\dim Y'$ and $w(Y)=w(Y')$, then ${\mathcal M}_H^{Y,\epsilon(Y)}(v)^{ss} \cong {\mathcal M}_H^{Y',\epsilon(Y')}(v)^{ss}$. \end{lem}
\begin{proof}
We use the notation in Lemma \ref{lem:o(w(Y))}.
For a $Y$-sheaf $E$, we set $E':=\Xi_{Y \to Y'}^L(E)$.
Then ${p'_Y}^*(E \otimes G^{\vee}) \cong p_{Y'}^*(E' \otimes {G'}^{\vee})$.
Hence $v_G(E)=v_{G'}(E')$.
If $\dim Y=\dim Y'$ and $w(Y)=w(Y')$, then since $w(\epsilon(Y))=w(\epsilon(Y'))$, replacing $L$ by $L \otimes q^*(P)$, $P \in \operatorname{Pic}(X)$, we may assume that $c_1(\Xi_{Y \to Y'}^L(\epsilon(Y)))=c_1(\epsilon(Y))$. Thus $\Xi_{Y \to Y'}^L(\epsilon(Y))=\epsilon(Y)+T$ in $K(X,Y')$, where $T$ is a $Y$-sheaf with $\dim T=0$. From this fact, we get ${\mathcal M}_H^{Y',\Xi_{Y \to Y'}^L(\epsilon(Y))}(v)^{ss} ={\mathcal M}_H^{Y',\epsilon(Y')}(v)^{ss}$.
\end{proof}
\begin{NB} If $\zeta \equiv \zeta' \mod r$ for $\zeta, \zeta' \in c_1(K(X,Y))$, then since $\zeta-\zeta' \in \operatorname{NS}(X)$, $\zeta=\zeta'+rc_1(L)$, $L \in \operatorname{Pic}(X)$. Hence if $H$ is a general polarization, then $\overline{M}^{Y,G}_H(r,\zeta,b) \cong \overline{M}^{Y,G}_H(r,\zeta',b)$. \end{NB}
Let $E$ be a $Y$-sheaf. Then the Zariski tangent space of the Kuranishi space is $\operatorname{Ext}^1(E,E)$ and the obstruction space is the kernel $\operatorname{Ext}^2(E,E)_0$ of the trace map \begin{equation} \operatorname{tr}:\operatorname{Ext}^2(E,E) \to H^2(Y,{\mathcal O}_Y) \cong H^2(X,{\mathcal O}_X). \end{equation}
Hence as in the usual sheaves on a $K3$ surfaces \cite{Mu:2}, we get the following. \begin{prop}\label{prop:symplectic} Let $E$ be a simple $Y$-sheaf. Then the Kuranishi space is smooth of dimension $\langle v_G(E)^2 \rangle+2$ with a holomorphic symplectic form. In particular, $\langle v_G(E)^2 \rangle \geq -2$. \end{prop}
\begin{cor}\label{lem:bogomolov} Let $E$ be a $\mu$-semi-stable $Y$-sheaf such that $E=lE_0+F \in K(X,Y)$, $F \in K(X,Y)_{\leq 1}$. Then $\langle v_G(E)^2 \rangle \geq -2 l^2$. \end{cor}
\subsubsection{Wall and Chamber} In this subsection, we generalize the notion of the wall and the chamber for the usual stable sheaves to the twisted case. \begin{lem} Assume that there is an exact sequence of twisted sheaves \begin{equation}\label{eq:seq2} 0 \to E_1 \to E \to E_2 \to 0, \end{equation} such that $E_i$, $i=1,2$ are $\mu$-semi-stable $Y$-sheaves. We set $E_i=l_i E_0+F_i \in K(X,Y)$ with $F_i \in K(X,Y)_{\leq 1}$. Then we have \begin{equation} \frac{\langle v_G(E)^2 \rangle}{l}+2l \geq -\frac{(l_2 v_G(F_1)-l_1 v_G(F_2))^2}{l l_1 l_2}. \end{equation} \end{lem}
This lemma easily follows from Corollary \ref{lem:bogomolov} and the following lemma. \begin{lem}\label{lem:extension} Let $E_0$ be a locally free $Y$-sheaf such that
$\operatorname{rk} E_0=\min\{\operatorname{rk} E>0| E \in \operatorname{Coh}(X,Y)\}$. For an exact sequence of twisted sheaves \begin{equation}\label{eq:seq} 0 \to E_1 \to E \to E_2 \to 0, \end{equation} we have \begin{equation} \frac{\langle v_G(E_1)^2 \rangle}{l_1}+ \frac{\langle v_G(E_2)^2 \rangle}{l_2}- \frac{\langle v_G(E)^2 \rangle}{l} =\frac{(l_2 v_G(F_1)-l_1 v_G(F_2))^2}{l l_1 l_2}, \end{equation} where $E_i=l_i E_0+F_i$ and $E=l E_0+F$ in $K(X,Y)$ with $F_i,F \in K(X,Y)_{\leq 1}$.
\end{lem}
\begin{proof}
\begin{equation} \begin{split} \frac{\langle v_G(E_1)^2 \rangle}{l_1}+ \frac{\langle v_G(E_2)^2 \rangle}{l_2}- \frac{\langle v_G(E)^2 \rangle}{l} =& \left(l_1 \langle v_G(E_0)^2 \rangle+2\langle v_G(E_0),v_G(F_1) \rangle +\frac{\langle v_G(F_1),v_G(F_1) \rangle}{l_1}\right)\\ &+\left(l_2 \langle v_G(E_0)^2 \rangle+2\langle v_G(E_0),v_G(F_2) \rangle +\frac{\langle v_G(F_2),v_G(F_2) \rangle}{l_2}\right)\\ & \quad- \left(l \langle v_G(E_0)^2 \rangle+2\langle v_G(E_0),v_G(F) \rangle +\frac{\langle v_G(F),v_G(F) \rangle}{l}\right)\\ =& \frac{\langle v_G(F_1),v_G(F_1) \rangle}{l_1}+ \frac{\langle v_G(F_2),v_G(F_2) \rangle}{l_2}- \frac{\langle v_G(F),v_G(F) \rangle}{l}\\ =& \frac{(l_2 v_G(F_1)-l_1 v_G(F_2))^2}{l l_1 l_2}. \end{split} \end{equation} \end{proof}
\begin{defn} We set $v=v_G(lE_0+F)$, where $F$ is of dimension 1 or 0. \begin{enumerate} \item For a $\xi \in \operatorname{NS}(X)$ with $0<-(\xi^2) \leq \frac{l^2}{4}(2l^2+\langle v^2 \rangle)$, we define a wall $W_{\xi}$ as \begin{equation}
W_{\xi}:=\{L \in \operatorname{Amp}(X) \otimes {\mathbb R}|(\xi,L)=0 \}. \end{equation} \item A chamber with respect to $v$ is a connected component of $\operatorname{Amp}(X) \otimes {\mathbb R} \setminus \bigcup_{\xi} W_{\xi}$. \item A polarization $H$ is general with respect to $v$, if $H$ does not lie on any wall. \end{enumerate} \end{defn}
\begin{rem} The concept of chambers and walls are determined by $\operatorname{rk}(l E_0+F)$ and $\langle v^2 \rangle$. Thus they do not depend on the choice of $Y$ and $G$. \end{rem}
\begin{prop}\label{prop:chamber} Keep notation as above. \begin{enumerate} \item If $H$ and $H'$ belong to the same chamber, then ${\mathcal M}_H^{Y,G}(v)^{ss} \cong {\mathcal M}_{H'}^{Y,G}(v)^{ss}$. \item If $H$ is general, then ${\mathcal M}_H^{Y,G}(v_G(F))^{ss} \cong {\mathcal M}_{H}^{Y,G'}(v_{G'}(F))^{ss}$ for $F \in K(X,Y)$ with $\operatorname{rk} F>0$. \item If \begin{equation}
\min\{-(D^2)>0| D \in \operatorname{NS}(X), (D,H)=0 \} >
\frac{l^2}{4}(2l^2+\langle v^2 \rangle), \end{equation} then $H$ is general with respect to $v$. \end{enumerate} \end{prop} The proof is standard (cf. \cite{H-L:1}) and is left to the reader.
By Proposition \ref{prop:chamber} and Proposition \ref{prop:symplectic}, we have
\begin{thm}\label{thm:symplectic} Assume that $v$ is a primitive Mukai vector and $H$ is general with respect to $v$. Then all $G$-twisted semi-stable $Y$-sheaves $E$ with $v_G(E)=v$ are $G$-twisted stable. In particular $M_H^{Y,G}(v)$ is a projective manifold, if it is not empty. \end{thm} In the next subsection, we show the non-emptyness of the moduli space. We also show that $M_H^{Y,G}(v)$ is a $K3$ surface, if $\langle v^2 \rangle=0$.
\begin{prop}(cf. \cite[Prop. 3.14]{Mu:4})\label{prop:simple} Assume that $\operatorname{Pic}(X)={\mathbb Z}H$. Let $E$ be a simple twisted sheaf with $\langle v_G(E)^2 \rangle \leq 0$. Then $E$ is stable. \end{prop}
For the proof, we use Lemma \ref{lem:extension} and the following:
\begin{lem}\cite[Cor. 2.8]{Mu:4} If $\operatorname{Hom}(E_1,E_2)=0$, then \begin{equation} \dim \operatorname{Ext}^1(E_1,E_1)+\dim \operatorname{Ext}^1(E_2,E_2) \leq \dim \operatorname{Ext}^1(E,E). \end{equation} \end{lem}
\subsection{Existence of stable sheaves}
In this subsection, we shall show that the moduli space of twisted sheaves is deformation equivalent to the usual one. In particular we show the non-emptyness of the moduli space.
\begin{thm}\cite{H-S:1}\label{thm:brauer} $H^1(X,PGL(r)) \to H^2(X,\mu_r)$ is surjective. \end{thm}
\begin{prop}\label{prop:mu-stable} For a $w \in H^2(X,\mu_r)$, there is a ${\mathbb P}^{r-1}$-bundle $p:Z \to X$ such that $w(Z)=w$ and $\epsilon(Z)$ is $\mu$-stable. \end{prop} D. Huybrechts informed us that the claim follows from the proof of Theorem \ref{thm:brauer}. Here we give another proof which works for other surfaces. \begin{proof} Let $p:Y \to X$ be a ${\mathbb P}^{r-1}$-bundle with $w(Y)=w$. We set $E_0:=\epsilon(Y)$. In order to prove our claim, it is sufficient to find a $\mu$-stable locally free $Y$-sheaf $E$
of rank $r$ with $c_1(E)=c_1(E_0)$. For points $x_1,x_2,\dots,x_n \in X$, let $F$ be a $Y$-sheaf which is the kernel of a surjection $E_0 \to \bigoplus_{i=1}^n {\mathcal O}_{p^{-1}(x_i)}(1)$. We take a smooth divisor $D \in |mH|$, $m \gg 0$. We set $\widetilde{D}:=p^{-1}(D)$. Let $\operatorname{Ext}^i(F,F(-\widetilde{D}))_0$ be the kernel of the trace map
\begin{equation} \operatorname{Ext}^i(F,F(-\widetilde{D})) \to H^i(Y,{\mathcal O}_Y(-\widetilde{D})) \cong H^i(X,{\mathcal O}_X(-D)). \end{equation}
If $n \gg 0$, then the by the Serre duality,
$\operatorname{Ext}^2(F,F(-\widetilde{D}))_0 \cong \operatorname{Hom}(F,F(\widetilde{D}))_0=0$. \begin{NB} Let $U$ be an affine scheme or an analytic open set and $\phi:{\mathcal O}_U^{\oplus r} \to {\mathcal O}_U^{\oplus r}$ a homomorphism such that $\operatorname{tr}{\phi}=0$. If $\phi(V)=V$ for all codimension 1 subspace $V \subset {\mathcal O}_U^{\oplus r} \otimes k(x)$ and a general point $x \in U$, then $\phi=0$. Hence if $\phi \ne 0$, then there is a quotient $f:{\mathcal O}_U^{\oplus r} \to k(x)$ such that $\phi$ does not preserve $\ker f$. \end{NB}
Hence $\operatorname{Ext}^1(F,F)_0 \to \operatorname{Ext}^1(F_{|\widetilde{D}},F_{|\widetilde{D}})_0$
is surjective. Since $F_{|\widetilde{D}}$ deforms to a $\mu$-stable vector bundle on $\widetilde{D}$,
$F$ deforms to a $Y$-sheaf $F'$ such that $F'_{|\widetilde{D}}$ is $\mu$-stable. Then $F'$ is also $\mu$-stable. Then $E:=(F')^{\vee \vee}$ satisfies required properties. \end{proof}
\begin{thm}\label{thm:deform} Let $Y \to X$ be a projective bundle and $G$ a locally free $Y$-sheaf. Let $v_G:=(r,\zeta,b)$ be a primitive Mukai vector with $r>0$. Then $M_H^{Y,G}(v_G)$ is an irreducible symplectic manifold which is deformation equivalent to $\operatorname{Hilb}_X^{\langle v_G^2 \rangle/2+1}$ for a general polarization $H$. In particular \begin{enumerate} \item[(1)] $M_H^{Y,G}(v_G) \ne \emptyset$ if and only if $\langle v_G^2 \rangle \geq -2$. \item[(2)] If $\langle v_G^2 \rangle=0$, then $M_H^{Y,G}(v_G)$ is a $K3$ surface. \end{enumerate} \end{thm} We divide the proof into several steps.
Step 1 (Reduction to $M_H^{Y,\epsilon(Y)}(r,0,-a)$) : Let $\xi$ be a lifting of $w(G)$. Then $e^{\xi/\operatorname{rk}(G)}v_G=(r,D,b') \in H^*(X,{\mathbb Z})$. By Theorem \ref{thm:brauer}, there is a projective bundle $Y' \to X$ such that $w(Y')=[D \mod r]$. Since $D/r-\xi/\operatorname{rk}(G)=\zeta/r \in \operatorname{Pic}(X) \otimes {\mathbb Q}$, $o(w(Y'))=o(w(Y))$. Let $G'$ be a locally free $Y$-sheaf such that $\Xi_{Y \to Y'}^L(G')=\epsilon(Y')$, where we use the notation in Lemma \ref{lem:o(w(Y))}. By Lemma \ref{lem:w(E)}, $w(G')=w(\epsilon(Y'))=[D \mod r]$. Then replacing $L$ by $L \otimes q^*(P)$, $P \in \operatorname{Pic}(X)$, we may assume that $e^{\xi/\operatorname{rk} G}v_G(G')=(r,D,c)$, $c \in {\mathbb Z}$. Hence $v_{G'}(E)=(r,0,-a)$ for a $Y$-sheaf $E$ with $v_G(E)=(r,\zeta,b)$. Since $H$ is general with respect to $(r,\zeta,b)$, Proposition \ref{prop:chamber} implies that $M_H^{Y,G}(r,\zeta,b) \cong M_H^{Y,G'}(r,0,-a)$. By Lemma \ref{lem:isom-G}, $M_H^{Y,G'}(r,0,-a) \cong M_H^{Y',\epsilon(Y')}(r,0,-a)$. Therefore replacing $(Y,G)$ by $(Y',\epsilon(Y'))$, we shall prove the assertion for $M_H^{Y,G}(r,0,-a)$ with $G=\epsilon(Y)$.
Step 2: First we assume that $w(Y) \in \operatorname{NS}(X) \otimes \mu_r \subset H^2(X,\mu_r)$. Then the Brauer class of $Y$ is trivial, that is, $Y={\mathbb P}(F)$ for a locally free sheaf $F$ on $X$. Since $H$ is general with respect to $(r,0,-a)$, Proposition \ref{prop:chamber} (ii) and Lemma \ref{lem:isom-G} imply that $M_H^{Y,G}(r,0,-a) \cong M_H^{X,{\mathcal O}_X}(r,D,c)$ with $2ra=(D^2)-2rc$. By \cite[Thm. 8.1]{Y:7}, $M_H^{X,{\mathcal O}_X}(r,D,c)$ is deformation equivalent to $\operatorname{Hilb}_X^{ra+1}$.
We next treat the general cases. We shall deform the projective bundle $Y \to X$ to a projective bundle in Step 2.
Step 3: We first construct a local family of projective bundles.
\begin{prop}\label{prop:projective} Let $f:({\mathcal X},{\mathcal H}) \to T$ be a family of polarized $K3$ surfaces. Let $p:Y \to {\mathcal X}_{t_0}$ be a projective bundle associated to a stable $Y$-sheaf $E$. Then there is a smooth morphism $U \to T$ whose image contains $t_0$ and a projective bundle $p:{\mathcal Y} \to {\mathcal X} \times_T U$ such that ${\mathcal Y}_{t_0} \cong Y$. \end{prop}
\begin{proof} We note that $p_*(K_{Y/{\mathcal X}_{t_0}}^{\vee})$ is a vector bundle on ${\mathcal X}_{t_0}$ and we have an embedding $Y \hookrightarrow {\mathbb P}(p_*(K_{Y/{\mathcal X}_{t_0}}^{\vee}))$. We take an embedding ${\mathbb P}(p_*(K_{Y/{\mathcal X}_{t_0}}^{\vee})) \hookrightarrow {\mathbb P}^{N-1} \times {\mathcal X}_{t_0}$ by a suitable quotient ${\mathcal O}_{{\mathcal X}_{t_0}}(-n{\mathcal H}_{t_0})^{\oplus N} \to p_*(K_{Y/{\mathcal X}_{t_0}}^{\vee})$. More generally, let ${\mathcal Y}_S \to {\mathcal X} \times_T S$ be a projective bundle and a surjective homomorphism ${\mathcal O}_{{\mathcal X} \times_T S}(-n{\mathcal H})^{\oplus N} \to p_*(K_{{\mathcal Y}_S/{\mathcal X} \times_T S}^{\vee})$. Then we have an embedding ${\mathcal Y}_S \hookrightarrow {\mathbb P}^{N-1} \times {\mathcal X} \times_T S$.
Let ${\mathfrak Y}$ be a connected component of the Hilbert scheme $\operatorname{Hilb}_{{\mathbb P}^{N-1} \times {\mathcal X}/T}$ containing $Y$. Let ${\mathcal Y} \subset {\mathbb P}^{N-1} \times {\mathcal X} \times_T {\mathfrak Y}$ be the universal subscheme. Let $\varphi:{\mathcal Y} \to {\mathcal X} \times_T {\mathfrak Y}$ be the projection. Let ${\mathfrak Y}^0$ be an open subscheme of ${\mathfrak Y}$
such that $\varphi_{|{\mathcal X} \times_T \{t\}}$ is smooth and
$H^1(T_{\varphi^{-1}(x,t)})=0$ for $(x,t) \in {\mathcal X} \times_T {\mathfrak Y}^0$. Since $Y \in {\mathfrak Y}^0$, it is non-empty. Then $\varphi$ is locally trivial on ${\mathcal X} \times_T {\mathfrak Y}^0$. Thus ${\mathcal Y} \to {\mathcal X} \times_T {\mathfrak Y}^0$ is a projective bundle.
If $Y$ is a projective bundle associated to a twisted vector bundle $E$, then the obstruction for the infinitesimal liftings belongs to $H^2({\mathcal E}nd(E)/{\mathcal O}_X) \cong H^0({\mathcal E}nd(E)_0)^{\vee}$, where ${\mathcal E}nd(E)_0$ is the trace free part of ${\mathcal E}nd(E)$. \begin{NB} By the Euler sequence, we have an exact sequence $0 \to {\mathcal O}_X \to E^{\vee} \otimes E \to p_*(T_{Y/X}) \to 0$. Hence ${\bf R}p_*(T_{Y/X}) \cong {\mathcal E}nd(E)_0$. \end{NB} Hence if $E$ is simple (and $\operatorname{rk} E$ is not divisible by the characteristic), then there is no obstruction for the infinitesimal liftings. In particular ${\mathfrak Y}^0 \to T$ is smooth at $Y$. \end{proof}
Step 4 {(A relative moduli space of twisted sheaves)}: Let $f:({\mathcal X},{\mathcal H}) \to T$ be a family of polarized $K3$ surfaces and $p:{\mathcal Y} \to {\mathcal X}$ a projective bundle on ${\mathcal X}$. We set $g:=f \circ p$. We note that $H^i({\mathcal Y}_t,\Omega_{{\mathcal Y}_t/{\mathcal X}_t})=0$, $i \ne 1$ and $H^1({\mathcal Y}_t,\Omega_{{\mathcal Y}_t/{\mathcal X}_t})={\mathbb C}$ for $t \in T$. Hence $L:=\operatorname{Ext}^1_g(T_{{\mathcal Y}/{\mathcal X}},{\mathcal O}_{{\mathcal Y}}) \cong R^1 g_*(\Omega_{{\mathcal Y}/{\mathcal X}})$ is a line bundle on $T$. By the local-global spectral sequence, we have an isomorphism \begin{equation} \operatorname{Ext}^1(T_{{\mathcal Y}/{\mathcal X}},g^*(L^{\vee})) \cong H^0(T, \operatorname{Ext}^1_g(T_{{\mathcal Y}/{\mathcal X}},g^*(L^{\vee}))) \cong H^0(T,{\mathcal O}_T). \end{equation}
We take the extension corresponding to $1 \in H^0(T,{\mathcal O}_T)$: \begin{equation} 0 \to g^*(L^{\vee}) \to {\mathcal G} \to T_{{\mathcal Y}/{\mathcal X}} \to 0 \end{equation} such that ${\mathcal G}_t=\epsilon({\mathcal Y}_t)$. Let $v:=(r,\zeta,b) \in R^* f_* {\mathbb Q}$ be a family of Mukai vectors with $\zeta \in \operatorname{NS}({\mathcal X}/T) \otimes {\mathbb Q}$. Then as in the absolute case, we have a family of the moduli spaces of semi-stable twisted sheaves $\overline{M}_{ ({\mathcal X},{\mathcal H})/T}^{{\mathcal Y},\mathcal G}(v) \to T$ parametrizing ${\mathcal G}_t$-twisted semi-stable ${\mathcal Y}_t$-sheaves $E$ on ${\mathcal X}_t$, $t \in T$ with $v_{{\mathcal G}_t}(E)=v_t$. $\overline{M}_{ ({\mathcal X},{\mathcal H})/T}^{{\mathcal Y},\mathcal G}(v) \to T$ is a projective morphism. Let $E$ be a ${\mathcal G}_t$-twisted stable ${\mathcal Y}_t$-sheaf. By our choice of $\zeta$, $\det(E)$ is unobstructed under deformations over $T$, and hence $E$ itself is unobstructed. Therefore ${M}_{ ({\mathcal X},{\mathcal H})/T}^{{\mathcal Y},\mathcal G}(v)$ is smooth over $T$.
Step 5 ({A family of $K3$ surfaces}): Let ${\mathcal M}_d$ be the moduli space of the polarized $K3$ surfaces $(X,H)$ with $(H^2)=2d$. ${\mathcal M}_d$ is constructed as a quotient of an open subscheme $T$ of a suitable Hilbert scheme $\operatorname{Hilb}_{{\mathbb P}^N/{\mathbb C}}$. Let $({\mathcal X},{\mathcal H}) \to T$ be the universal family. Let $\Gamma$ be the abstruct $K3$ lattice and $h$ a primitive vector with $(h^2)=2d$. Let ${\mathcal D}$ be the period domain for polarized $K3$ surfaces $(X,H)$. Let $\tau:\widetilde{T} \to T$ be the universal covering and $\phi_{\tilde{t}}:H^2({\mathcal X}_{\tau(\tilde{t})},{\mathbb Z}) \to \Gamma$, $\tilde{t} \in \widetilde{T}$ a trivialization on $\widetilde{T}$. We may assume that $\phi_{\tilde{t}}({\mathcal H}_{\tau(\tilde{t})})=h$. Then we have a period map ${\mathfrak p}:\widetilde{T} \to {\mathcal D}$. By the surjectivity of the period map, we can show that ${\mathfrak p}$ is surjective: Let $U$ be a suitable analytic neighborhood of a point $x \in {\mathcal D}$. Then we have a family of polarized $K3$ surfaces $({\mathcal X}_U,{\mathcal H}_U) \to U$ and an embedding of ${\mathcal X}$ as a subscheme of ${\mathbb P}^N \times U$. Thus we have a morphism $h:U \to T$. The embedding is unique up to the action of $PGL(N+1)$. Moreover if there is a point $\tilde{t}_0 \in \widetilde{T}$ such that ${\mathfrak p}(\tilde{t}_0) \in U$, then we have a lifting $\widetilde{h}:U \to \widetilde{T}$ of $h:U \to T$ such that $\tilde{t}_0=\widetilde{h}({\mathfrak p}(\tilde{t}_0))$. Then $U \to \widetilde{T} \to {\mathcal D}$ is the identity. Hence we can construct a lifting of any path on ${\mathcal D}$ intersecting ${\mathfrak p}(\widetilde{T})$. Since ${\mathcal D}$ is connected, we get the assertion. \begin{NB} We may assume that $U$ is contractible. If we take an embedding ${\mathcal X}_U \hookrightarrow {\mathbb P}^N \times U$, then we have a diagram \begin{equation} \begin{CD} ({\mathcal X}_U,{\mathcal H}_U) @>>> {\mathbb P}^N \times U\\
@V{f}VV @|\\ ({\mathcal X},{\mathcal H}) \times_{\mathfrak H}U @>{h^*(i)}>> {\mathbb P}^N \times U\\ \end{CD} \end{equation} Since we have an isomorphism $\lambda: ({\mathcal X}_U,{\mathcal H}_U)_{{\mathfrak p}(\tilde{t}_0)} \to ({\mathcal X},{\mathcal H})_{\tau(\tilde{t}_0)}$, we have another embedding $({\mathcal X}_U)_{{\mathfrak p}(\tilde{t}_0)} \to {\mathcal X}_{\tau(\tilde{t}_0)} \overset{i_{\tau(\tilde{t}_0)}}{\hookrightarrow} {\mathbb P}^N$. Replacing $h$ by $g \circ h$, $g \in PGL(N+1)$, we may assume that this map is the same as $(h^*(i) \circ f)_{{\mathfrak p}(\tilde{t}_0)}$.
***** I don't need the following: (We use the fact that an isomorphism $X \to X'$ preserving the polarization can be lifted to the isomorphism of the projective spaces.) But use: If $p_1,p_2:(X,L) \to {\mathbb P}^N$ be two embedding with $p_1^*(H^0({\mathcal O}_{{\mathbb P}^N}(1))) \overset{\iota_1}{\to} H^0(L) \overset{\iota_2}{\leftarrow} p_2^*(H^0({\mathcal O}_{{\mathbb P}^N}(1)))$, then $\iota_2^{-1} \circ \iota_1$ induces an isomorphism $\iota:{\mathbb P}^N \to {\mathbb P}^N$ with $\iota \circ p_1=p_2$. *****
Then we have $h({\mathfrak p}(\tilde{t}_0))=\tau(\tilde{t}_0)$ and $\lambda=f_{{\mathfrak p}(\tilde{t}_0)}$. Let $\psi_u:H^2(({\mathcal X}_U)_u,{\mathbb Z}) \to \Gamma$ be the trivialization on $U$. Then by the definition of ${\mathfrak p}$, $\psi_{{\mathfrak p}(\tilde{t}_0)} \circ f_*^{-1}: H^2( {\mathcal X}_{\tau(\tilde{t}_0)},{\mathbb Z}) \to H^2(({\mathcal X}_U)_{{\mathfrak p}(\tilde{t}_0)},{\mathbb Z}) \to \Gamma$ coincides with $\phi_{\tilde{t}_0}$. Hence if we take a local section $\sigma$ of $\tau$ in a neighborhood of $\tau(\tilde{t}_0)$ with $\sigma(\tau(\tilde{t}_0))=\tilde{t}_0$ and set $\tilde{h}:=\sigma \circ h$, then $\phi_{\tilde{h}(u)}=\psi_u \circ f_*^{-1}$ in a neighborhood of ${\mathfrak p}(\tilde{t}_0)$. Since $U$ is contractible, $\tilde{h}$ is extended to the whole of $U$ and we also have $\phi_{\tilde{h}(u)}=\psi_u \circ f_*^{-1}$. Therefore ${\mathfrak p} \circ \tilde{h}=1_U$.
******* If we have an isomorphism $\xi:({\mathcal X}_U,{\mathcal H}_U)_u \to ({\mathcal X}_U,{\mathcal H}_U)_{u'}$ with a commutative diagram \begin{equation} \begin{CD} H^2(({\mathcal X}_U)_u,{\mathbb Z}) @>{\psi_u}>> \Gamma \\
@V{\xi_*}VV @|\\ H^2(({\mathcal X}_U)_{u'},{\mathbb Z}) @>{\psi_{u'}}>> \Gamma, \end{CD} \end{equation} then $u'=u$, and hence $\xi_*=1$ (since the moduli space is fine). ******* \end{NB}
\begin{NB} Let $\gamma:[0,1] \to {\mathcal D}$ be a path in ${\mathcal D}$. Assume that there are local lifts $\widetilde{\gamma}_1$ on $[0,1/2]$ and $\widetilde{\gamma}_2$ on $[1/2,1]$ of $\gamma$. Since $\widetilde{\gamma}_1(1/2)=g \circ \widetilde{\gamma}_2(1/2)$ for a $g \in PGL(N+1)$, replacing $\widetilde{\gamma}_2$ by $g \circ \widetilde{\gamma}_2$, we have a lifting of $\gamma$. For a general case, we cover a path $\gamma(t)$ by a finite number of open covering such that there are local lifts on each open sets. Then we can construct a lifting inductively. \end{NB}
Step 6 (Reduction to step 2): We take a point $\widetilde{t} \in \widetilde{T}$. We set $(X,H):= ({\mathcal X}_{\tau(\widetilde{t})},{\mathcal H}_{\tau(\widetilde{t})})$. Let $p:Y \to X$ be a ${\mathbb P}^{r-1}$-bundle.
Assume that $H$ is general with respect to $v:=(r,0,-a)$. We take a $D \in \Gamma$ with $[D \mod r]= \overline{\phi}_{\widetilde{t}}(w(Y))$. Let $e_1,e_2,\dots,e_{22}$ be a ${\mathbb Z}$-basis of $\Gamma$ such that $e_1=\phi_{\widetilde{t}}({\mathcal H}_{\tau(\widetilde{t})})$ and $D=a e_1+b e_2$. For an $\eta \in \bigoplus_{i=3}^{22} {\mathbb Z}e_i \subset \Gamma$ with $(e_1^2)(\eta^2)-(e_1,\eta)^2< 0$, we set $\widetilde{\eta}:=e_2+rk \eta \in \Gamma$, $k \gg 0$. Since $\det \begin{pmatrix} (e_1^2) & (e_1,e_2+rk\eta)\\ (e_1,e_2+rk\eta) & ((e_2+rk\eta)^2) \end{pmatrix}
\ll 0$ for $k \gg 0$, the signature of the primitive sublattice $L:={\mathbb Z}e_1 \oplus {\mathbb Z}\widetilde{\eta}$ of $\Gamma$ is of type $(1,1)$. Moreover $e_1^{\perp} \cap L$ does not contain a $(-2)$-vector. We take a general $\omega \in L^{\perp} \cap \Gamma \otimes {\mathbb C}$ with $(\omega,\omega)=0$ and $(\omega,\bar{\omega})>0$. Then $\omega^{\perp} \cap \Gamma=L$. Replacing $\omega$ by its complex conjugate if necessary, we may assume that $\omega \in {\mathcal D}$. Since ${\mathfrak p}$ is surjective, there is a point $\tilde{t}_1 \in \widetilde{{\mathfrak H}}$ such that ${\mathfrak p}(\tilde{t}_1) =\omega$. Then ${\mathcal X}_{\tau(\widetilde{t}_1)}$ is a $K3$ surface with $\operatorname{Pic}({\mathcal X}_{\tau(\widetilde{t}_1)})={\mathbb Z} {\mathcal H}_{\tau(\widetilde{t}_1)} \oplus {\mathbb Z}\phi_{\widetilde{t}_1}^{-1}(e_2+rk\eta)$. Hence $[\phi_{\widetilde{t}_1}^{-1}(D) \mod r] =[\phi_{\widetilde{t}_1}^{-1}(a e_1+b\widetilde{\eta}) \mod r] \in \operatorname{Pic}({\mathcal X}_{\tau(\widetilde{t}_1)}) \otimes \mu_r$. Since \begin{equation}
\min\{-(L^2)| 0 \ne L \in \operatorname{Pic}({\mathcal X}_{\tau(\widetilde{t}_1)}), (L,{\mathcal H}_{\tau(\widetilde{t}_1)})=0 \} \gg \frac{r^2}{4}(2 r^2+\langle v^2 \rangle), \end{equation} Proposition \ref{prop:chamber} (iii) implies that ${\mathcal H}_{\tau(\widetilde{t}_1)}$ is a general polarization with respect to $v$. Then by the following lemma, we can reduce the proof to Step 2. Therefore we complete the proof of Theorem \ref{thm:deform}.
\begin{lem}\label{lem:deform}
For $\widetilde{t}_1,\widetilde{t}_2 \in \widetilde{T}$, let $Y^i \to {\mathcal X}_{\tau(\tilde{t}_i)}$, $i=1,2$ be ${\mathbb P}^{r-1}$-bundles with $w(Y^i)=[\phi_{\tilde{t}_i}^{-1}(D) \mod r]$ and $G_i:={\epsilon(Y^i)}$. Let $v=(r,0,-a)$ be a primitive Mukai vector. Assume that ${\mathcal H}_{\tau(\tilde{t}_i)}$, $i=1,2$ are general polarization. Then $M^{Y^1,G_1}_{{\mathcal H}_{\tau(\tilde{t}_1)}}(r,0,-a)$ is deformation equivalent to $M^{Y^2,G_2}_{{\mathcal H}_{\tau(\tilde{t}_2)}}(r,0,-a)$. \end{lem}
\begin{proof} In order to simplify the notation, we denote $M_{{\mathcal H}_t}^{Y,\epsilon(Y)}(r,0,-a)$ by $M(Y)$ for a projective bundle $Y$ over $({\mathcal X}_t,{\mathcal H}_t)$. By Proposition \ref{prop:mu-stable} and Lemma \ref{lem:isom-G}, we may assume that $\epsilon(Y^i)$ ($i=1,2$) is $\mu$-stable. Let $\widetilde{\gamma}:[0,1] \to \widetilde{T}$ be a path from $\tilde{t}_1=\widetilde{\gamma}(0)$ to $\tilde{t}_2=\widetilde{\gamma}(1)$ and $\gamma:=\tau \circ \widetilde{\gamma}$. Then we have a trivialization $\overline{\phi}_s:H^2({\mathcal X}_{\gamma(s)},\mu_r) \to \Gamma \otimes_{\mathbb Z} \mu_r$. By Proposition \ref{prop:mu-stable}, there is a projective bundle $Y_s \to {\mathcal X}_{\gamma(s)}$ such that $\overline{\phi}_s(w(Y_s))=[D \mod r]$ and $\epsilon(Y_s)$ is $\mu$-stable for each $s \in [0,1]$. By Proposition \ref{prop:projective}, we have a family of projective bundles ${\mathcal Y}^s \to {\mathcal X} \times_T {\mathfrak Y}^s$ over a $T$-scheme $\psi^s:{\mathfrak Y}^s \to T$ such that there is a point $y^s \in {(\psi^s)}^{-1}(\gamma(s)) \subset {\mathfrak Y}^s$ with $Y_s={\mathcal Y}^s_{y^s}$ and $\psi^s$ is smooth at $y^s$. Then we have a family of moduli spaces $\overline{M}_{ ({\mathcal X}\times_T {\mathfrak Y}^s,\widetilde{\mathcal H})/{\mathfrak Y}^s} ^{{\mathcal Y}^s,{\mathcal G}^s}(r,0,-a) \to {\mathfrak Y}^s$, where $\widetilde{\mathcal H}$ is the pull-back of ${\mathcal H}$ to ${\mathcal X}\times_T {\mathfrak Y}^s$ (Step 4). Since $\psi^s$ is smooth, $\psi^s({\mathfrak Y}^s)$ is an open subscheme of $T$ containing $\gamma(s)$. We take an analytic open neighborhood $U_s$ of $\gamma(s)$ such that $U_s$ is contractible and has a section $\sigma_s:U_s \to {\mathfrak Y}^s$ with $\sigma_s(\gamma(s))=y^s$. Let $V_s$ be a connected neighborhood of $s$ which is contained in $f^{-1}(U_s)$. Since $[0,1]$ is compact, we can take a finite open covering of $[0,1]$:
$[0,1] = \cup_{j=1}^n V_{s_j}$, $s_1<s_2<\dots<s_n$. Since $\{t \in T| \operatorname{rk} \operatorname{Pic}({\mathcal X}_t)=1 \}$ is a dense subset of $T$, there is a point $t_j \in U_{s_j} \cap U_{s_{j+1}}$ such that $t_j$ is sufficiently close to a point $\gamma(s_{j,j+1})$, $s_{j,j+1} \in V_{s_j} \cap V_{s_{j+1}}$ and $\operatorname{Pic}({\mathcal X}_{t_j})={\mathbb Z}{\mathcal H}_{t_j}$. Under the identification $H^2({\mathcal X}_t,\mu_r) \cong H^2({\mathcal X}_{\gamma(s)},\mu_r)$ for $t \in U_s$, we have $w({\mathcal Y}^{s_j}_{\sigma_i(t_j)})=w({\mathcal Y}^{s_j}_{y^i})$ and $w({\mathcal Y}^{s_{j+1}}_{\sigma_{j+1}(t_j)})= w({\mathcal Y}^{s_{j+1}}_{y^{j+1}})$. Since $t_j$ is sufficiently close to the point $\gamma(s_{j,j+1})$,
we have $w({\mathcal Y}^{s_j}_{\sigma_j(t_j)})=w({\mathcal Y}^{s_{j+1}}_{\sigma_{j+1}(t_j)})$. Hence by Lemma \ref{lem:isom-G},
$M({\mathcal Y}^{s_j}_{\sigma_j(t_j)})$ is isomorphic to $M({{\mathcal Y}^{s_{j+1}}_{\sigma_{j+1}(t_j)}})$. By Step 4, $M({\mathcal Y}^{s_j}_{\sigma_j(t_{j-1})})$ is deformation equivalent to $M({\mathcal Y}^{s_j}_{\sigma_j(t_{j})})$. Therefore $M({\mathcal Y}^{s_1}_{\sigma_1(t_{1})})$ is deformation equivalent to $M({\mathcal Y}^{s_n}_{\sigma_n(t_{n-1})})$. By using Step 4 again, we also see that $M(Y^1)=M({\mathcal Y}^0_{y^0})$ is deformation equivalent to $M({\mathcal Y}^{s_1}_{\sigma_1(t_{1})})$ and $M(Y^2)=M({\mathcal Y}^1_{y^1})$ is deformation equivalent to $M({\mathcal Y}^{s_n}_{\sigma_n(t_{n-1})})$. Therefore our claim holds. \end{proof}
\begin{NB} The following proof is not correct (2004, Dec. 27). \begin{proof} By Proposition \ref{prop:mu-stable} and Lemma \ref{lem:isom-G}, we may assume that $\epsilon(Y^i)$ ($i=1,2$) is $\mu$-stable. Let $\psi^i:{\mathfrak Y}^i \to T$ be a morphism with a point $y^i \in {(\psi^i)}^{-1}({t}_i) \subset {\mathfrak Y}^i$ and ${\mathcal Y}^i \to {\mathcal X} \times_T {\mathfrak Y}^i$ a family of projective bundles such that ${\mathcal Y}^i_{y_0^i}=Y^i$ and $\psi^i$ is smooth (Proposition \ref{prop:projective}).
We take a point $t \in \psi^1({\mathfrak Y}^1) \cap \psi^2({\mathfrak Y}^2)$ with $\operatorname{Pic}({\mathcal X}_t)={\mathbb Z}{\mathcal H}_t$. Let ${\mathcal G}^i$ be a vector bundle on ${\mathcal Y}^i$ with ${\mathcal G}^i_{y^i}=\epsilon({\mathcal Y}^i_{y^i}), y^i \in {\mathfrak Y}^i$. Let $\overline{\phi}_{\tilde{t}}:H^2({\mathcal X}_{\tau(\tilde{t})},\mu_r) \to \Gamma \otimes_{\mathbb Z} \mu_r$ be the homomorphism induced by $\phi_{\tilde{t}}$. Since $w({\mathcal Y}^i_{y^i}) \in H^2({\mathcal X}_{\psi^i(y^i)},\mu_r)$ ($y^i \in {\mathfrak Y}^i$) is a section of $R^2 f_{{\mathfrak Y}^i *} \mu_r$, the map $\gamma: (y^i,\tilde{t}) \mapsto \overline{\phi}_{\tilde{t}}(w({\mathcal Y}^i_{y^i})) \in \Gamma \otimes_{\mathbb Z} \mu_r$ is locally constant on ${\mathfrak Y}^i \times_T \widetilde{T}$, where $f_{{\mathfrak Y}^i}$ is the base change of $f$ by $\psi^i$. By the definition of ${\mathfrak Y}^i$, $w({\mathcal Y}^i_{y^i})$ does not depend on the choice of $y^i \in {\mathfrak Y}^i$. Hence $\gamma$ is regarded as a locally constant function on a connected manifold $\psi^i({\mathfrak Y}^i) \times_T \widetilde{T}$. Therefore $\gamma$ is constant. Thus $\overline{\phi}_{\tilde{t}}(w({\mathcal Y}^i_{y^i}))= \overline{\phi}_{\tilde{t}_i}(w({\mathcal Y}^i_{y_0^i}))= [D \mod r]$ for $(y^i,\tilde{t}) \in {\mathfrak Y}^i \times_T \widetilde{T}$. In particular $w({\mathcal Y}^1_{y^1})=w({\mathcal Y}^2_{y^2}) \in H^2({\mathcal X}_t,\mu_r)$ for $y^i \in (\psi^i)^{-1}(t)$. By Lemma \ref{lem:isom-G}, $M^{{\mathcal Y}^1_{y^1},{\mathcal G}_{y^1}}_{{\mathcal H}_{\psi^1(y^1)}}(r,0,-a) \cong M^{{\mathcal Y}^2_{y^2},{\mathcal G}_{y^2}}_{{\mathcal H}_{\psi^2(y^2)}}(r,0,-a)$. By Step 4, $M^{Y^i,G_i}_{{\mathcal H}_{t_i}}(r,0,-a)= M^{{\mathcal Y}^i_{y_0^i},{\mathcal G}_{y_0^i}}_{{\mathcal H}_{\psi^i(y_0^i)}}(r,0,-a)$ is deformation equivalent to $M^{{\mathcal Y}^i_{y^i},{\mathcal G}_{y^i}}_{{\mathcal H}_{\psi^i(y^i)}}(r,0,-a)$. Hence we get our claim. \end{proof} \end{NB}
\begin{NB} $\tau:\widetilde{T} \to T$ be the universal covering of $T$. Then we have a trivialization $\phi_{\tilde{t}}:H^2({\mathcal X}_{\tau(\tilde{t})},{\mathbb Z}) \to \Gamma$, $\tilde{t} \in \widetilde{T}$. We have a period map ${\mathfrak p}:\widetilde{T} \to {\mathcal D}$. $w({\mathcal Y})_h \in H^2({\mathcal X}_{\zeta(h)},\mu_r)$. $(h,\tilde{t}) \in {\mathfrak H} \times_T \widetilde{T}$, $w({\mathcal Y})_h =[\phi_{\tilde{t}}^{-1}(D) \mod r]$.
For $t_1, t_2 \in \zeta({\mathfrak H})$, $M_{{\mathcal H}_{t_1}}^{G_1}(r,\xi_1,a)$ is deformation equivalent to $M_{{\mathcal H}_{t_2}}^{G_2}(r,\xi_2,a)$, where $G_i$, $i=1.2$ are twisted locally free sheaves such that $w({\mathbb P}(G_i^{\vee}))=[\phi_{\tilde{t_i}}^{-1}(D) \mod r]$ and $[\xi_i \mod r]=[\phi_{\tilde{t_i}}^{-1}(D) \mod r]$. \end{NB}
\begin{rem} Let $v_G:=(r,\zeta,b)$ be a Mukai vector with $r,\langle v_G^2 \rangle>0$ which is not necessary primitive. By the same proof, we can also show that $\overline{M}_H^{Y,G}(v_G)$ is an irreducible normal variety for a general $H$ (cf. \cite{Y:9}). \end{rem}
\subsection{The second cohomology groups of moduli spaces}
By Theorem \ref{thm:deform}, $M_H^{Y,G}(v_G)$ is an irreducible symplectic manifold, if $v_G$ is primitive and $H$ is general. Then $H^2(M_H^{Y,G}(v_G),{\mathbb Z})$ is equipped with a bilinear form called the Beauville form. In this subsection, we shall describe the Beauville form in terms of the Mukai lattice.
Let $p:Y \to X$ be a projective bundle with $w(Y)=[\xi \mod r]$ and set $G:=\epsilon(Y)$. We consider a Mukai lattice with a Hodge structure $(H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi}{r})$ in this subsection. We set $w:=r(1,0,\frac{a}{r}-\frac{1}{2}\frac{(\xi^2)}{r^2})$, $a \in {\mathbb Z}$. In this subsection, we assume that $w$ is primitive, that is, $\gcd(r,\xi,a)=1$. We set $v:=w e^{\xi/r}=(r,\xi,a) \in H^*(X,{\mathbb Z})$. Then $v$ is algebraic.
Let $q:\widetilde{M_H^{Y,G}(w)} \to M_H^{Y,G}(w)$ be a projective bundle in subsection \ref{subsubsect:family} and ${\mathcal E}$ the family of twisted sheaves on $Y \times \widetilde{M_H^{Y,G}(w)}$. We set $W^{\vee}:=\epsilon(\widetilde{M_H^{Y,G}(w)})$. Let $\widetilde{\pi}_{\widetilde{M_H^{Y,G}(w)}}: Y \times \widetilde{M_H^{Y,G}(w)} \to \widetilde{M_H^{Y,G}(w)}$ and $\widetilde{\pi}_{Y}: Y \times \widetilde{M_H^{Y,G}(w)} \to Y$ be projections. Then $(1_Y \times q)_*({\mathcal E} \otimes \widetilde{\pi}_{\widetilde{M_H^{Y,G}(w)}}^*(W^{\vee}))$ is a quasi-universal family on $Y \times M_H^{Y,G}(w)$.
Let $\pi_X:X \times M_H^{Y,G}(w) \to X$ be the projection. We define a homomorphism $\theta_v^G:v^{\perp} \to H^*(M_H^{Y,G}(w),{\mathbb Q})$ by \begin{equation} \theta_v^G(u):=\int_{X} [{\mathcal Q}^{\vee} \pi_X^*(e^{-\xi/r}u)]_3 \end{equation} where $[...]_3$ means the degree 6 part and \begin{equation} \begin{split} {\mathcal Q}:=& \frac{\sqrt{\operatorname{td}_X}}{\sqrt{\operatorname{ch}({\bf R}p_*(G^{\vee} \otimes G))}} \frac{\sqrt{\operatorname{td}_{M_H^{Y,G}(w)}}}{\sqrt{\operatorname{ch}({\bf R}q_*({W}^{\vee} \otimes W))}} \operatorname{ch}({\bf R}(p \times q)_*(\widetilde{\pi}_Y^*( G^{\vee}) \otimes {\mathcal E} \otimes \widetilde{\pi}_{\widetilde{M_H^{Y,G}(w)}}^*({W}^{\vee})))\\ & \in H^*(X \times M_H^{Y,G}(w),{\mathbb Q}). \end{split} \end{equation}
\begin{rem}\label{rem:integral} If $\xi$ is algebraic, then $Y$ is isomorphic to the projective bundle ${\mathbb P}(F^{\vee})$ and $G=F^{\vee} \otimes {\mathcal O}_Y(1)$, where $F$ is a vector bundle of rank $r$ on $X$ with $c_1(F)=-\xi$. In this case, $M_H^{Y,G}(w)$ is the usual moduli space of stable sheaves $F$ with the Mukai vector $v$ and ${\bf R}(p \times q)_*( \widetilde{\pi}_Y^*({\mathcal O}_Y(-1))
\otimes {\mathcal E} \otimes \widetilde{\pi}_{\widetilde{M_H^{Y,G}(w)}}^*({W}^{\vee}))$ is a quasi-universal family. Since $\operatorname{ch} F/\sqrt{\operatorname{ch}(F \otimes F^{\vee})}=e^{-\xi/r}$, we have
\begin{equation} {\mathcal Q}= e^{-\frac{\xi}{r}}\sqrt{\operatorname{td}_X}\frac{\sqrt{\operatorname{td}_{M_H^{Y,G}(w)}}}{\sqrt{\operatorname{ch}({\bf R}q_*({W}^{\vee} \otimes W))}} \operatorname{ch}({\bf R}(p \times q)_*(\widetilde{\pi}_Y^*( {\mathcal O}_Y(-1))
\otimes {\mathcal E} \otimes \widetilde{\pi}_{\widetilde{M_H^{Y,G}(w)}}^*({W}^{\vee}))). \end{equation} Hence $\theta_v^G$ is the usual Mukai homomorphism, which is defined over ${\mathbb Z}$. \end{rem}
Let $p':Y' \to X$ be another ${\mathbb P}^{r-1}$-bundle with $w(Y')=w(Y)$. Then by the proof of Lemma \ref{lem:isom-G}, we see that the following diagram is commutative: \begin{equation} \begin{CD} v^{\perp}@= v^{\perp}\\ @V{\theta_v^G}VV @VV{\theta_v^{G'}}V\\ H^2(M_H^{Y,G}(w),{\mathbb Q}) @>>> H^2(M_H^{Y',G'}(w),{\mathbb Q}), \end{CD} \end{equation} where $G':=\Xi_{Y \to Y'}^L(G)=\epsilon(Y')$. Since ${\mathcal Q}$ is algebraic, $\theta_v^G$ preserves the Hodge structure. By the deformation argument, Remark \ref{rem:integral} implies that $\theta_v^G$ is defined over ${\mathbb Z}$. Moreover it preserves the bilinear forms.
\begin{thm} For $\xi \in H^2(X,{\mathbb Z})$ with $[\xi \mod r]=w(Y)$, we set $v=w e^{\xi/r}$. \begin{enumerate} \item If $\langle v^2 \rangle>0$, then $\theta_v^G:v^{\perp} \to H^2(M_H^{Y,G}(w),{\mathbb Z})$ is an isometry of the Hodge structures. \item If $\langle v^2 \rangle=0$, then $\theta_v^G$ induces an isometry of the Hodge structures $v^{\perp}/{\mathbb Z}v \to H^2(M_H^{Y,G}(w),{\mathbb Z})$. \end{enumerate} \end{thm} The second claim is due to Mukai \cite{Mu:5}.
\section{Fourier-Mukai transform}\label{sect:FM} \begin{NB} Let $\mathrm{Mod}(X,Y)$ be the subcategory of the category of ${\mathcal O}_Y$-modules consisting of $E$ with $p^*(p_*(E \otimes G^{\vee})) \cong E \otimes G^{\vee}$. Let $I$ be an injective sheaf on $X$. Then $p^*(I) \otimes G$ is injective in $\mathrm{Mod}(X,Y)$. Indeed $\operatorname{Hom}(E,p^*(I) \otimes G)\cong \operatorname{Hom}(E \otimes G^{\vee},p^*(I)) \cong \operatorname{Hom}(p_*(E \otimes G^{\vee}),I)$. Hence $E \mapsto \operatorname{Hom}(E,p^*(I) \otimes G)$ is an exact functor. Since there is an injection $E \to (E \otimes G^{\vee}) \otimes G$, $\mathrm{Mod}(X,Y)$ is enough injective. \end{NB}
\subsection{Integral functor}
Let $p:Y \to X$ be a projective bundle such that $\delta([Y])=[\alpha] \in \operatorname{Br}(X)$ and
$p':Y' \to X'$ a projective bundle such that $\delta([Y'])=[\alpha'] \in \operatorname{Br}(X')$. Let ${\pi}_X:X' \times X \to X$ and ${\pi}_{X'}:X' \times X \to X'$ be projections. We also let $\widetilde{\pi}_Y:Y' \times Y \to Y$ and $\widetilde{\pi}_{Y'}:Y' \times Y \to Y'$ be projections. We set $G:=\epsilon(Y)$ and $G':=\epsilon(Y')$. \begin{defn} Let $\operatorname{Coh}(X' \times X,Y',Y)$ be the subcategory of $\operatorname{Coh}(Y' \times Y)$ such that $Q \in \operatorname{Coh}(Y' \times Y)$ belongs to $\operatorname{Coh}(X' \times X,Y',Y)$ if and only if $(p' \times p)^* (p' \times p)_* (G' \otimes Q \otimes G^{\vee}) \cong G' \otimes Q \otimes G^{\vee}$. In terms of local trivialization of $p,p'$, this is equivalent to
$Q_{|Y_i' \times Y_j} \cong {\mathcal O}_{Y_i'}(-\lambda_i') \boxtimes {\mathcal O}_{Y_j}(\lambda_j) \otimes (p' \times p)^*({Q}_{ij})$, ${Q}_{ij} \in \operatorname{Coh}(U_i' \times U_j)$. $\operatorname{Coh}(X' \times X,Y',Y)$ is equivalent to $\operatorname{Coh}(X' \times X,{\alpha'}^{-1} \times \alpha)$. \end{defn}
\begin{rem} We take twisted line bundles ${\mathcal L}({p'}^*({\alpha'}^{-1}))$ on $Y'$ and ${\mathcal L}(p^*(\alpha^{-1}))$ on $Y$ respectively which give equivalences $\Lambda^{{\mathcal L}({p'}^*({\alpha'}^{-1}))}: \operatorname{Coh}(X',Y') \cong \operatorname{Coh}(X',\alpha')$ and $\Lambda^{{\mathcal L}(p^*({\alpha}^{-1}))}: \operatorname{Coh}(X,Y) \cong \operatorname{Coh}(X,\alpha)$ in \eqref{eq:Y=X}. Then we have an equivalence $\Lambda^{{\mathcal L}({p'}^*({\alpha'}^{-1}))^{\vee}} \times \Lambda^{{\mathcal L}(p^*({\alpha}^{-1}))}$: \begin{equation} \begin{matrix} \operatorname{Coh}(X' \times X,Y',Y)& \to &\operatorname{Coh}(X' \times X,{\alpha'}^{-1} \times \alpha)\\
Q & \mapsto & (p' \times p)_*({\mathcal L}({p'}^*({\alpha'}^{-1})) \otimes Q \otimes {\mathcal L}(p^*({\alpha}^{-1}))^{\vee}). \end{matrix} \end{equation} \end{rem}
Let ${\bf D}(X' \times X,Y',Y) \cong {\bf D}(X' \times X,{\alpha'}^{-1} \times \alpha)$ be the bounded derived category of $\operatorname{Coh}(X' \times X,Y',Y)$. \begin{NB} ${\mathcal Q} \in {\bf D}(X' \times X,{\alpha'}^{-1} \times \alpha)$ corresponds an object $\widetilde{\mathcal Q} \in {\bf D}(Y' \times Y)$ which is represented by a complex $\cdots \to \widetilde{\mathcal Q}^k \to \widetilde{\mathcal Q}^{k+1} \to \cdots$ with $(p' \times p)^* (p' \times p)_* (G' \otimes \widetilde{\mathcal Q}^k \otimes G^{\vee}) \cong
G' \otimes \widetilde{\mathcal Q}^k \otimes G^{\vee}$. or ${\mathcal Q}_{|Y_i' \times Y_j}^k = {\mathcal O}_{Y_i'}(-\lambda_i') \boxtimes {\mathcal O}_{Y_j}(\lambda_i) \otimes (p' \times p)^*({\mathcal Q}^k_{ij})$, ${\mathcal Q}^k_{ij} \in \operatorname{Coh}(U_i' \times U_j)$. \end{NB} For ${\mathcal Q} \in {\bf D}(X' \times X,Y',Y)$, we define an integral functor \begin{equation} \begin{matrix} \Phi_{X' \to X}^{\widetilde{\mathcal Q}}: &{\bf D}(X',Y')& \to &{\bf D}(X,Y)\\ & x & \mapsto & {\bf R}\widetilde{\pi}_{Y*} ({\mathcal Q} \otimes \widetilde{\pi}_{Y'}^*(x)). \end{matrix} \end{equation}
For ${\mathcal Q} \in {\bf D}(X' \times X,Y',Y)$ and ${\mathcal R} \in {\bf D}(X'' \times X', Y'',Y')$, we have \begin{equation} \Phi_{X' \to X}^{{\mathcal Q}} \circ \Phi_{X'' \to X'}^{{\mathcal R}}= \Phi_{X'' \to X}^{{\mathcal S}}, \end{equation} where ${\mathcal S}= {\bf R}\widetilde{\pi}_{Y'' \times Y*} (\widetilde{\pi}^*_{Y'' \times Y'}({\mathcal R}) \otimes
\widetilde{\pi}^*_{Y' \times Y}({\mathcal Q}))$ and $\widetilde{\pi}^*_{(\;\;)}:Y'' \times Y' \times Y \to (\;\;)$ is the projection.
\begin{NB} For a torsion class $\alpha:=\{\alpha_{ijk}\}$, let $p:Y \to X$ be a projective bundle associated to $\alpha$.
${\mathcal Q} \in {\bf D}(X' \times X,\beta^{-1} \times \alpha)$ corresponds an object ${\mathcal Q} \in {\bf D}(Y' \times Y)$ such that
${\mathcal Q}_{|Y_i' \times Y_j} = {\mathcal O}_{Y_i'}(-\lambda_i') \boxtimes {\mathcal O}_{Y_j}(-\lambda_i) \otimes (p' \times p)^*({\mathcal Q}_{ij})$. \begin{equation} \begin{matrix} \Phi_{X' \to X}^{\mathcal Q}: &{\bf D}(X',\beta^{-1})& \to &{\bf D}(X,\alpha)\\ & x & \mapsto & {\bf R}\pi_{X*}({\mathcal E} \otimes \pi_{X'}^*(x)). \end{matrix} \end{equation}
For ${\mathcal Q} \in {\bf D}(X' \times X,\beta^{-1} \times \alpha)$ and ${\mathcal R} \in {\bf D}(X'' \times X', \gamma^{-1} \times \beta)$, we have \begin{equation} \Phi_{X' \to X}^{\mathcal Q} \circ \Phi_{X'' \to X'}^{\mathcal R}= \Phi_{X'' \to X}^{\mathcal S}, \end{equation} where ${\mathcal S}= {\bf R}\pi_{Y'' \times Y*}({\mathcal R} \boxtimes_{{\mathcal O}_{Y'}}
{\mathcal Q})$.
\end{NB}
\subsubsection{Cohomological correspondence} For simplicity, we denote the pull-backs of $G$ and $G'$ to $Y' \times Y$ by the same letters. For example $G' \otimes {\mathcal Q} \otimes G^{\vee}$ implies $\pi_{Y'}^*(G') \otimes {\mathcal Q} \otimes \pi_{Y}(G^{\vee})$. \begin{NB} We note that \begin{equation} \frac{1}{\sqrt{\operatorname{ch}({\bf R}p_*(G^{\vee} \otimes G))}} \in H^*(X,{\mathbb Q}). \end{equation}
We set \begin{equation} G' \boxtimes_{{\mathcal O}_{Y'}} {\mathcal Q} \boxtimes_{{\mathcal O}_{Y'}} G^{\vee}:= \pi_{Y'}^*(G') \otimes {\mathcal Q} \otimes \pi_{Y'}^*(G^{\vee}). \end{equation} \end{NB} We note that \begin{equation} {\bf R}(p' \times p)_*(G' \otimes {\mathcal Q} \otimes G^{\vee}) \in {\bf D}(X' \times X) \end{equation} satisfies \begin{equation} (p' \times p)^*({\bf R}(p' \times p)_*(G' \otimes {\mathcal Q} \otimes G^{\vee})) =G' \otimes {\mathcal Q} \otimes G^{\vee}. \end{equation}
We define a homomorphism \begin{equation} \Psi_{X' \to X}^{\mathcal Q}: H^*(X',{\mathbb Q}) \to H^*(X,{\mathbb Q}) \end{equation} by \begin{equation} \begin{split} &\Psi_{X' \to X}^{\mathcal Q}(y)\\ :=&\pi_{X*} \circ (p' \times p)_* \left((p' \times p)^* \circ \pi_{X'}^*(y)
\operatorname{ch}({G'})\operatorname{ch}({\mathcal Q})\operatorname{ch}(G^{\vee}) \frac{\sqrt{\operatorname{td}_{X'}}\operatorname{td}_{Y'/X'}}{\sqrt{\operatorname{ch}({G'}^{\vee} \otimes G')}} \frac{\sqrt{\operatorname{td}_X}\operatorname{td}_{Y/X}}{\sqrt{\operatorname{ch}(G^{\vee} \otimes G)}}\right)\\ =& \pi_{X*}\left( \pi_{X'}^*(y) \frac{\sqrt{\operatorname{td}_{X'}}}{\sqrt{\operatorname{ch}({\bf R}p'_*({G'}^{\vee} \otimes G'))}} \frac{\sqrt{\operatorname{td}_X}}{\sqrt{\operatorname{ch}({\bf R}p_*(G^{\vee} \otimes G))}} \operatorname{ch}({\bf R}(p' \times p)_*({G'} \otimes {\mathcal Q}\otimes G^{\vee}))\right), \end{split} \end{equation} where $\operatorname{td}_X$, $\operatorname{td}_{X'}$,... are identified with their pull-backs.
\begin{lem} $\Psi_{X'' \to X}^{\mathcal S}= \Psi_{X' \to X}^{\mathcal Q} \circ \Psi_{X'' \to X'}^{\mathcal R}$. \end{lem}
\begin{proof} $\pi_{(\;\;)}:X'' \times X' \times X \to (\;\;)$ denotes the projection to $(\;\;)$.
We note that \begin{equation} \begin{split} & \pi_{X'' \times X}^* \left({\bf R}(p'' \times p')_* (G'' \otimes {\mathcal R} \otimes {G'}^{\vee})\right) \otimes \pi_{X' \times X}^* \left({\bf R}(p' \times p)_* (G' \otimes {\mathcal Q} \otimes {G}^{\vee})\right) \\ =& {\bf R}(p'' \times p' \times p)_* (G'' \otimes {\mathcal R} \otimes {\mathcal Q} \otimes G^{\vee}) \otimes \pi_{X'}^*({\bf R}p'_*({G'}^{\vee} \otimes G')). \end{split} \end{equation} Then \begin{equation} \begin{split} & \pi_{X'' \times X}^* \left(\operatorname{ch} \left({\bf R}(p'' \times p')_* (G'' \otimes {\mathcal R} \otimes {G'}^{\vee})\right) \right)\cdot\\ & \quad \quad \pi_{X' \times X}^* \left( \operatorname{ch} \left({\bf R}(p' \times p)_* (G' \otimes {\mathcal Q} \otimes {G}^{\vee})\right) \right) \pi_{X'}^*\left(\frac{\operatorname{td}_{X'}} {\operatorname{ch}({\bf R}p'_*({G'}^{\vee} \otimes G'))}\right) \\ =& \operatorname{ch} \left({\bf R}(p'' \times p' \times p)_* (G'' \otimes {\mathcal R} \otimes {\mathcal Q} \otimes G^{\vee})\right) \pi_{X'}^*(\operatorname{td}_{X'}). \end{split} \end{equation} Since \begin{equation} \begin{split} &\pi_{X'' \times X*}\left(\operatorname{ch} \left({\bf R}(p'' \times p' \times p)_* (G'' \otimes {\mathcal R} \otimes {\mathcal Q} \otimes G^{\vee})\right) \pi_{X'}^*(\operatorname{td}_{X'})\right)\\ =&\operatorname{ch}\left({\bf R}\pi_{X'' \times X*}\left({\bf R}(p'' \times p' \times p)_* (G'' \otimes {\mathcal R} \otimes {\mathcal Q} \otimes G^{\vee})\right)\right)\\ =& \operatorname{ch}({\bf R}(p'' \times p)_* \circ {\bf R}\tilde{\pi}_{Y'' \times Y*} (G'' \otimes {\mathcal R} \otimes {\mathcal Q} \otimes G^{\vee}))\\ =& \operatorname{ch}({\bf R}(p'' \times p)_*(G'' \otimes {\mathcal S} \otimes G^{\vee})), \end{split} \end{equation} we get
\begin{equation} \begin{split} \Psi_{X'' \to X}^{\mathcal S}(z)=& \pi_{X*}\left( \pi_{X''}^*(z) \operatorname{ch}({\bf R}(p'' \times p)_*({G''} \otimes {\mathcal S} \otimes G^{\vee})) \frac{\sqrt{\operatorname{td}_{X''}}}{\sqrt{\operatorname{ch}({\bf R}p''_*({G''}^{\vee} \otimes G''))}} \frac{\sqrt{\operatorname{td}_X}}{\sqrt{\operatorname{ch}({\bf R}p_*(G^{\vee} \otimes G))}}\right)\\ =& \Psi_{X' \to X}^{\mathcal Q} \circ \Psi_{X'' \to X'}^{\mathcal R}(z). \end{split} \end{equation} \end{proof}
\begin{lem} Assume that the canonical bundles $K_X, K_{X'}$ are trivial. Then \begin{equation} \langle x,\Psi_{X' \to X}^{\mathcal Q}(y)\rangle =\langle \Psi_{X \to X'}^{{\mathcal Q}^{\vee}}(x), y \rangle,\;\; x \in H^*(X,{\mathbb Q}),\; y \in H^*(X',{\mathbb Q}), \end{equation} where $\langle\;\;,\;\; \rangle$ is the Mukai pairing. \end{lem}
\begin{proof} \begin{equation} \begin{split} &\langle x,\Psi_{X' \to X}^{\mathcal Q}(y) \rangle \\ =&-\int_{X} x \Psi_{X' \to X}^{\mathcal Q}(y)^{\vee}\\ =&-\int_{X' \times X} \pi_X^*(x) \left(\pi_{X'}^*(y) \frac{\sqrt{\operatorname{td}_{X'}}}{\sqrt{\operatorname{ch}({\bf R}p'_*({G'}^{\vee} \otimes G'))}} \frac{\sqrt{\operatorname{td}_X}}{\sqrt{\operatorname{ch}({\bf R}p_*(G^{\vee} \otimes G))}} \operatorname{ch}({\bf R}(p' \times p)_*({G'} \otimes {\mathcal Q} \otimes G^{\vee})) \right)^{\vee} \\ =&-\int_{X' \times X} \left(\frac{\sqrt{\operatorname{td}_{X'}}}{\sqrt{\operatorname{ch}({\bf R}p'_*({G'}^{\vee} \otimes G'))}} \frac{\sqrt{\operatorname{td}_X}}{\sqrt{\operatorname{ch}({\bf R}p_*(G^{\vee} \otimes G))}} \operatorname{ch}({\bf R}(p' \times p)_*({G'}^{\vee}\otimes {\mathcal Q}^{\vee} \otimes G )) \pi_X^*(x)\right) \pi_{X'}^*(y^{\vee}) \\ =&-\int_{X'} \Psi_{X \to X'}^{{\mathcal Q}^{\vee}}(x) y^{\vee}\\ =&\langle \Psi_{X \to X'}^{{\mathcal Q}^{\vee}}(x),y \rangle. \end{split} \end{equation} \end{proof}
\subsection{Fourier-Mukai transform induced by stable twisted sheaves}
Let $p:Y \to X$ be a projective bundle over an abelian surface or a $K3$ surface. Let $G$ be a locally free $Y$-sheaf.
Assume that $X':=\overline{M}^{Y,G}_H(v)$ is a surface and consists of stable sheaves. We set $Y':=\widetilde{\overline{M}^{Y,G}_H(v)}$. Let ${\mathcal E}$ be the family on $Y' \times Y$. \begin{NB} Hence there is an open covering $M_H(v)=\cup_j V_j$ and local universal families ${\mathcal E}_j$ on $V_j \times Y$. By using an open covering of $X$, ${\mathcal E}_j$ is a collection of ${\mathcal E}_{(i,j)} \in \operatorname{Coh}(V_j \times U_i)$ and homomorphisms $\phi_{i_2,i_1}(j):
{\mathcal E}_{(i_2,j)|V_j \times (U_{i_2} \cap U_{i_1})} \to
{\mathcal E}_{(i_1,j)|V_j \times (U_{i_1} \cap U_{i_2})}$ such that $\phi_{i_2,i_1}(j) \circ \phi_{i_3,i_2}(j)
\circ \phi_{i_1,i_3}(j)=\alpha_{i_1 i_2 i_3}$. Since $\operatorname{Hom}_{\pi_{V_i}}({\mathcal E}_i,{\mathcal E}_i) \cong {\mathcal O}_{V_i}$, there is a 2-cocycle $\beta:=\{\beta_{ijk}| \beta_{ijk} \in H^0(V_i \cap V_j \cap V_k,{\mathcal O}_{X'}^{\times}) \}$ such that ${\mathcal E}_i$ is a $\pi_{X'}^*(\beta)$-twisted sheaf on $X' \times Y$. That is ${\mathcal E}:=\{(V_i \times X,{\mathcal E}_i)\}$ is a $\pi_{X'}^*(\beta)\pi_X^*(\alpha)$-twisted sheaf on $X' \times X$. A homomorphism $\varphi_{j_2,j_1}:{\mathcal E}_{j_2} \to {\mathcal E}_{j_1}$ is a collection of $\varphi_{j_2,j_1}(i):{\mathcal E}_{(i,j_2)} \to {\mathcal E}_{(i,j_1)}$ such that the following diagrams are commutative: \begin{equation} \begin{CD}
{\mathcal E}_{(i_2,j_2)|(V_{j_1} \cap V_{j_2}) \times (U_{i_2} \cap U_{i_1})} @>{\phi_{i_2,i_1}(j_2)}>>
{\mathcal E}_{(i_1,j_2)|(V_{j_1} \cap V_{j_2}) \times (U_{i_2} \cap U_{i_1})}\\ @V{\varphi_{j_2,j_1}(i_2)}VV @VV{\varphi_{j_2,j_1}(i_1)}V\\
{\mathcal E}_{(i_2,j_1)|(V_{j_1} \cap V_{j_2}) \times (U_{i_2} \cap U_{i_1})} @>>{\phi_{i_2,i_1}(j_1)}>
{\mathcal E}_{(i_1,j_1)|(V_{j_1} \cap V_{j_2}) \times (U_{i_2} \cap U_{i_1})}. \end{CD} \end{equation} We set $\psi_{(i_2,j_2),(i_1,j_1)}:= \varphi_{j_2,j_1}(i_1) \circ \phi_{i_2,i_1}(j_2)$. Then $\psi_{(i_2,j_2),(i_1,j_1)} \circ \psi_{(i_3,j_3),(i_2,j_2)}\circ \psi_{(i_1,j_1),(i_3,j_3)}= \alpha_{i_1 i_2 i_3}\beta_{j_1 j_2 j_3}$.
\end{NB}
We consider integral functors \begin{equation} \begin{matrix} \Phi_{X' \to X}^{\mathcal E}: &{\bf D}(X',Y')& \to &{\bf D}(X,Y)\\ & x & \mapsto & {\bf R}\widetilde{\pi}_{Y*} ({\mathcal E} \otimes \widetilde{\pi}_{Y'}^*(x)), \end{matrix} \end{equation}
\begin{equation} \begin{matrix} \Phi_{X \to X'}^{{\mathcal E}^{\vee}}[2]: &{\bf D}(X,X)& \to &{\bf D}(X',Y')\\ & y & \mapsto & {\bf R}\widetilde{\pi}_{Y'*} ({\mathcal E}^{\vee} \otimes \widetilde{\pi}_{Y}^*(y)[2]). \end{matrix} \end{equation}
\begin{rem} Let ${\mathcal L}({p'}^*({\alpha}^{-1}))$ and ${\mathcal L}(p^*(\alpha^{-1}))$ be twisted line bundles on $Y'$ and $Y$ respectively in \eqref{eq:Y=X}.
Then $\Lambda^{{\mathcal L}(p^*({\alpha}^{-1}))} \circ \Phi_{X' \to X}^{\mathcal E} \circ (\Lambda^{{\mathcal L}({p'}^*({\alpha'}^{-1}))})^{-1}: {\bf D}(X',\alpha') \to {\bf D}(X,\alpha)$ is an integral functor with the kernel ${\bf R}(p' \times p)_*({\mathcal L}({p'}^*({\alpha'}^{-1})) \otimes {\mathcal E} \otimes {\mathcal L}(p^*({\alpha}^{-1}))^{\vee}) \in {\bf D}(X' \times X,{\alpha'}^{-1} \times \alpha)$. \end{rem}
C\u ald\u araru \cite{C:2} developed a theory of derived category of twisted sheaves. In particular, Grothendieck-Serre duality holds. Then we see that $\Phi_{X \to X'}^{{\mathcal E}^{\vee}}[2]$ is the adjoint of $\Phi_{X' \to X}^{{\mathcal E}}$. As in the usual Fourier-Mukai functor, we see that the following theorem holds (see \cite{Br:1}, \cite{C:1}). \begin{thm}\label{thm:Fourier-Mukai} $\Phi_{X \to X'}^{{\mathcal E}^{\vee}}[2] \circ \Phi_{X' \to X}^{\mathcal E} \cong 1$ and $\Phi_{X' \to X}^{\mathcal E} \circ \Phi_{X \to X'}^{{\mathcal E}^{\vee}}[2] \cong 1$. Thus $\Phi_{X' \to X}^{\mathcal E}$ is an equivalence. \end{thm}
Then we have the following which also follows from a more general statement \cite[Thm. 0.4]{H-S:2}.
\begin{cor} $\Psi_{X' \to X}^{\mathcal E}$ induces an isometry of the Hodge structures: \begin{equation} (H^*(X',{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi'}{r}) \cong (H^*(X,{\mathbb Z}),\langle\;\;,\;\; \rangle,-\frac{\xi}{r}). \end{equation} \end{cor}
\begin{proof} Obviously $\Psi_{X' \to X}^{\mathcal E}$ induces an isometry of the Hodge structures over ${\mathbb Q}$. If $X$ is a $K3$ surface such that $w(Y) \in \operatorname{NS}(X) \otimes \mu_r$ and $X'$ is a fine moduli space, then $\Psi_{X' \to X}^{\mathcal E}$ is defined over ${\mathbb Z}$. For a general case, we use the deformation arguments. \end{proof} We also have the following which is used in \cite{Y:12}.
\begin{cor}
Assume that $X'$ consists of locally free $Y$-sheaves. Then ${\mathcal E}^{\vee}_{|Y' \times \{y \}}$, $y \in Y$ is a simple $Y'$-sheaf. If $\operatorname{NS}(X) \cong {\mathbb Z}H$, then
${\mathcal E}^{\vee}_{|Y' \times \{y \}}$, $y \in Y$ is a stable $Y'$-sheaf. \end{cor}
\begin{proof} Since $\Phi_{X \to X'}^{{\mathcal E}^{\vee}}[2]$ is an equivalence, $\Phi_{X \to X'}^{{\mathcal E}^{\vee}}({\mathcal O}_{p^{-1}(p(y))}(1))
={\mathcal E}^{\vee}_{|Y' \times \{y \}}$ is a simple $Y'$-sheaf. If $\operatorname{NS}(X) \cong {\mathbb Z}$, then Proposition \ref{prop:simple}
implies the stability of ${\mathcal E}^{\vee}_{|Y' \times \{y \}}$. \end{proof}
\begin{NB} Let $p':Y' \to X$ be a projective bundle. Let ${\mathcal E}^1,{\mathcal E}^2$ be pull-backs of ${\mathcal E}$ by $Y' \times Y \times Y' \to Y' \times Y$ and $Y' \times Y \times Y' \to Y \times Y'$ respectively.
$(p' \times p')_* ({\bf R}\operatorname{Hom}_{p_{Y' \times Y'}}({\mathcal E}^2 \otimes {G'}^{\vee}, {\mathcal E}^1\otimes {G'}^{\vee}))[2] \cong {\mathcal O}_{\Delta_X} \otimes p'_*({G'}^{\vee} \otimes G')$. ${\mathcal O}_{Y_i'}(\lambda'_i) \boxtimes_{{\mathcal O}_{V_i}} {\mathcal O}_{Y_i'}(-\lambda'_i)$ with a transition function $\phi_{ji} \boxtimes_{{\mathcal O}_{V_i \cap V_j}} {}^t\phi_{ji}^{-1}$ defines a line bundle ${\mathcal L}$ on $Y' \times_{X'} Y'$.
${\mathcal E}_{|Y'_i \times Y} \cong (p' \times 1)^*({\mathcal E}_i) \otimes {\mathcal O}_{Y'_i}(\lambda'_i)$. ${\bf R}\operatorname{Hom}_{p_{V_i' \times V_i'}}({\mathcal E}_i^2, {\mathcal E}_i^1)[2] \cong {\mathcal O}_{\Delta_{V_i}}$ and it defines a structure sheaf of the diagonal.
$G'_{|Y_i'}={p'}^*(G'_i)(\lambda'_i)$. Hence ${\mathcal L} \otimes {G'}^{\vee} \boxtimes_{{\mathcal O}_X} {G'}_{|Y_i} = {G'_i}^{\vee} \boxtimes_{{\mathcal O}_{V_i}} {G'_i}$. Hence $(p' \times p')_* ({\mathcal L} \otimes {G'}^{\vee} \boxtimes_{{\mathcal O}_X} {G'})={\mathcal O}_{\Delta_X} \otimes p'_*({G'}^{\vee} \otimes G')$. \end{NB}
{\it Acknowledgement.} First of all, I would like to thank Daniel Huybrechts and Paolo Stellari. They proved C\u ald\u araru's conjecture. Moreover Huybrechts gave me many valuable suggestions on this paper. I would also like to thank Eyal Markman and Shigeru Mukai for valuable discussions on the twisted sheaves and their moduli spaces. Thanks also to Max Lieblich for explaining the relation of our moduli spaces with Simpson's moduli spaces of modules over the Azumaya algebra.
\end{document} | arXiv | {
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\begin{document}
\title{The odd primary order of the commutator on low rank Lie groups} \author{Tseleung So} \address{Mathematical Sciences, University of Southampton, SO17 1BJ, UK} \email{tls1g14@soton.ac.uk}
\thanks{}
\subjclass[2010]{55Q15,
57T20
}
\keywords{Lie group, Samelson products, homotopy nilpotence}
\date{}
\begin{abstract} Let $G$ be a simple, simply-connected, compact Lie group of low rank relative to a fixed prime~$p$. After localization at $p$, there is a space $A$ which ``generates'' $G$ in a certain sense. Assuming $G$ satisfies a homotopy nilpotency condition relative to $p$, we show that the Samelson product $\sm{\mathds{1}_G,\mathds{1}_G}$ of the identity of $G$ equals the order of the Samelson product~$\sm{\imath,\imath}$ of the inclusion $\imath:A\to G$. Applying this result, we calculate the orders of~$\sm{\mathds{1}_G,\mathds{1}_G}$ for all~$p$-regular Lie groups and give bounds of the orders of $\sm{\mathds{1}_G,\mathds{1}_G}$ for certain quasi-$p$-regular Lie groups. \end{abstract}
\maketitle
\section{Introduction} In this paper, $G$ is a simple, simply-connected, compact Lie group and $p$ is an odd prime. By a theorem of Hopf, $G$ is rationally homotopy equivalent to a product of spheres~$\prod^l_{i=1}S^{2n_i-1}$, where $n_1\leq\cdots\leq n_l$. The sequence $(2n_1-1,\cdots,2n_l-1)$ is called the \emph{type} of $G$. Localized at $p$, it is known \cite{CN84,MNT77} that $G$ is homotopy equivalent to a product of H-spaces $\prod^{p-1}_{i=1}B_i$, and there exists a co-H-space $A$ and a map $\imath:A\to G$ such that $H_*(G)$ is the exterior algebra generated by $\imath_*(\tilde{H}_*(A))$. For $1\leq i\leq l$, if $B_i$ is $S^{2n_i-1}$, then we call $G$ \emph{$p$-regular}. If each $B_i$ is either $S^{2n_i-1}$ or $B(2n_i-1,2n_i+2p-3)$ that is the $S^{2n_i-1}$-bundle over $S^{2n_i+2p-3}$ classified by $\frac{1}{2}\alpha\in\pi_{2n_i+2p-4}(S^{2n_i-1})$, then we call $G$ \emph{quasi-$p$-regular}.
For any maps $f:X\to G$ and $g:Y\to G$, let $c(f,g):X\times Y\to G$ be a map sending~\mbox{$(x,y)\in X\times Y$} to their commutator $[x, y]=f(x)^{-1}g(y)^{-1}f(x)g(y)$. Then $c(f,g)$ descends to a map $\sm{f,g}:X\wedge Y\to G$. The map $\sm{f,g}$ is called the \emph{Samelson product} of $f$ and~$g$. The \emph{order} of $\sm{f,g}$ is defined to be the minimum number $k$ such that the composition \[ k\circ\sm{f,g}:X\wedge Y\overset{\sm{f,g}}{\longrightarrow}G\overset{k}{\longrightarrow}G \] is null-homotopic, where $k:G\to G$ is the $k^\text{th}$-power map. In particular, when $f$ and $g$ are the identity map $\mathds{1}_G$ of $G$, the Samelson product $\sm{\mathds{1}_G,\mathds{1}_G}$ is universal and we are interested in finding its order.
There is a notion of nilpotency in homotopy theory analogous to that for groups. Let~$c_1$ be the commutator map $c(\mathds{1}_G,\mathds{1}_G):G\times G\to G$, and let~\mbox{$c_n=c_1\circ(c_{n-1}\times\mathds{1}_G)$} be the $n$-iterated commutator for $n>1$. The \emph{homotopy nilpotence class} of $G$ is the number $n$ such that $c_n$ is null-homotopic but $c_{n-1}$ is not. In certain cases the homotopy nilpotence class of $p$-localized $G$ is known. Kaji and Kishimoto \cite{kk10} showed that $p$-regular Lie groups have homotopy nilpotence class at most 3. When $G$ is quasi-$p$-regular and $p\geq7$, Kishimoto \cite{kishimoto09} showed that $SU(n)$ has homotopy nilpotence class at most 3, and Theriault \cite{theriault16} showed that exceptional Lie groups have homotopy nilpotence class at most 2.
Here we restrict $G$ to be a Lie group having low rank with respective to an odd prime $p$. That is, $G$ and $p$ satisfy: \begin{equation}\label{retractile_list} \begin{array}{l l} SU(n) &n\leq(p-1)(p-2)+1\\ Sp(n) &2n\leq(p-1)(p-2)\\ Spin(2n+1) &2n\leq(p-1)(p-2)\\ Spin(2n) &2n-2\leq(p-1)(p-2)\\ G_2,F_4,E_6 &p\geq5\\ E_7,E_8 &p\geq7, \end{array} \end{equation} In these cases, Theriault \cite{theriault07} showed that $\Sigma A$ is a retract of $\Sigma G$. Let $\sm{\imath,\imath}$ be the composition \[ \sm{\imath,\imath}:A\wedge A\overset{\imath\wedge\imath}{\hookrightarrow}G\wedge G\overset{\sm{\mathds{1}_G,\mathds{1}_G}}{\longrightarrow}G. \] Then obviously the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ is always greater than or equal to the order of $\sm{\imath,\imath}$. Conversely, we show that the order of $\sm{\imath,\imath}$ restricts the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ under certain conditions.
\begin{thm}\label{main1} Let $G$ be a compact, simply-connected, simple Lie group of low rank and let~$p$ be an odd prime. Localized at $p$, if the homotopy nilpotence class of $G$ is less than~$p^r+1$, then the order of the Samelson product $\sm{\mathds{1}_G,\mathds{1}_G}$ is $p^r$ if and only if the order of~$\sm{\imath,\imath}$ is $p^r$. \end{thm}
The strategy for proving Theorem~\ref{main1} is to extend $A\to G$ to an H-map $\Omega\Sigma A\to G$ which has a right homotopy inverse, that is to retract $[G\wedge G, G]$ off $[\Omega\Sigma A\wedge\Omega\Sigma A, G]$, and use commutator calculus to analyze the latter. Combine Theorem~\ref{main1} and the known results in~\cite{kk10, kishimoto09, theriault16} to get the following statement.
\begin{cor}\label{cor_intro} The order of $\sm{\mathds{1}_G,\mathds{1}_G}$ equals the order of $\sm{\imath,\imath}$ when \begin{itemize} \item $G$ is $p$-regular or; \item $p\geq7$ and $G$ is a quasi-$p$-regular Lie group which is one of $SU(n),F_4,E_6,E_7$ or $E_8$. \end{itemize} \end{cor}
On the one hand, there is no good method to calculate the order of $\sm{\mathds{1}_G, \mathds{1}_G}$ in general. A direct computation is not practical since one has to consider all the cells in $G\wedge G$ and their number grows rapidly when there is a slight increase in the rank of $G$. On the other hand, Corollary~\ref{cor_intro} says that for all $p$-regular Lie groups and most of the quasi-$p$-regular Lie groups, we can determine the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ by computing the order of $\sm{\imath,\imath}$. The latter is easier to work with since $A$ has a much simpler CW-structure than $G$. To demonstrate the power of Theorem~\ref{main1}, we apply this result to compute the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ for all $p$-regular cases and some quasi-$p$-regular cases.
\begin{thm} For a $p$-localized Lie group $G$, the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ is $p$ when \[
\begin{array}{|c|c|c|} \hline G &r=0 &r=1\\ \hline SU(n) &p>2n &n\leq p<2n\\ Sp(n) &p>4n &2n<p<4n\\ Spin(2n+1) &p>4n &2n<p<4n\\ Spin(2n) &p>4n-4 &2n-2<p<4n-4\\ G_2 &p=5,p>11 &p=7,11\\ F_4,E_6 &p>23 &11\leq p\leq23\\ E_7 &p>31 &17\leq p\leq31\\ E_8 &p>59 &23\leq p\leq59\\ \hline \end{array} \] \end{thm} \noindent For many other quasi-$p$-regular cases, we give rough bounds on the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ by bounding the order of $\sm{\imath,\imath}$.
Here is the structure of this paper. In Section 2 we prove Theorem~\ref{main1} assuming Lemma~\ref{lemma_sm_mk_null}, whose proof is given in Section 3 because of its length. Section 3 is divided into two parts. In the first part we consider the algebraic properties of Samelson products and in the second part we use algebraic methods to prove Lemma~\ref{lemma_sm_mk_null}. In Section 4 we apply Theorem~\ref{main1} and use other known results to calculate bounds on the order of the Samelson product $\sm{\mathds{1}_G,\mathds{1}_G}$ for quasi-$p$-regular Lie groups.
\section{Samelson products of low rank Lie groups} \begin{dfn} Let $G$ be a simple, simply-connected, compact Lie group and $p$ be an odd prime. Localized at $p$, a triple $(A, \imath, G)$ is \emph{retractile} if $A$ is a co-H-space and a subspace of $G$ and $\imath:A\hookrightarrow G$ is an inclusion such that \begin{itemize} \item there is an algebra isomorphism $H_*(G)\cong\Lambda(\tilde{H}_*(A))$ of homologies with mod-$p$ coefficients; \item the induced homomorphism $\imath_*: H_*(A)\to H_*(G)$ is an inclusion of the generating set; \item the suspension $\Sigma\imath:\Sigma A\to\Sigma G$ has a left homotopy inverse $t:\Sigma G\to\Sigma A$. \end{itemize} We also refer to $G$ as being retractile for short. \end{dfn}
From now on, we take $p$-localization and assume $G$ and $p$ satisfy~(\ref{retractile_list}). According to \cite{theriault07},~$G$ is retractile. First we want to establish a connection between $G$ and $\Omega\Sigma A$. Consider the homotopy commutative diagram \begin{equation}\label{CD1} \xymatrix{ A\ar[dr]_{\Sigma}\ar[rr]^{\imath} &&G\\ &\Omega\Sigma A\ar[ur]_{\tilde{\imath}} & } \end{equation} where $\Sigma:A\to\Omega\Sigma A$ is the suspension and $\tilde{\imath}:\Omega\Sigma A\to G$ is an H-map. Since $G$ is retractile, the suspension $\Sigma\imath:\Sigma A\to\Sigma G$ has a left homotopy inverse $t:\Sigma G\to\Sigma A$. Let $s$ be the composition \[ s:G\overset{\Sigma}{\longrightarrow}\Omega\Sigma G\overset{\Omega t}{\longrightarrow}\Omega\Sigma A. \]
\begin{lemma}\label{lemma_rho+s} The map $\tilde{\imath}\circ s$ is a homotopy equivalence. \end{lemma}
\begin{proof} Denote the composition $\tilde{\imath}\circ s$ by $e$ for convenience. Consider the commutative diagram \[\xymatrix{ A\ar[d]_{\Sigma}\ar[r]^{\imath} &G\ar[d]_{\Sigma}\ar[dr]^{s} &\\ \Omega\Sigma A\ar[r]^{\Omega\Sigma\imath} &\Omega\Sigma G\ar[r]^{\Omega t} &\Omega\Sigma A. }\] The commutativity of the left square is due to the naturality of the suspension map, and the commutativity of the right triangle follows from the definition of $s$. The bottom row is homotopic to the identity since $t$ is a left homotopy inverse for $\Sigma\imath$. Hence we have $s\circ\imath\simeq\Sigma$ and consequently \[ e\circ\imath=\tilde{\imath}\circ s\circ\imath\simeq\tilde{\imath}\circ\Sigma. \] By Diagram~(\ref{CD1}) $\tilde{\imath}\circ\Sigma$ is homotopic to $\imath$. This implies that $(e\circ\imath)_*$ sends $H_*(A)$ onto the generating set of $H_*(G)=\Lambda(\tilde{H}_*(A))$ where we consider the mod-$p$ homology. Dually,~\mbox{$(e\circ\imath)^*:H^*(G)\to H^*(A)$} is an epimorphism. The generating set $\imath^*(H^*(A))$ is in $Im(e^*)$. Since $e^*:H^*(G)\to H^*(G)$ is an algebra map, $e^*$ is an epimorphism and hence is an isomorphism. Therefore \mbox{$e:G\to G$} is a homotopy equivalence. \end{proof}
We claim that the Samelson product \[ \sm{\tilde{\imath}, \tilde{\imath}}:\Omega\Sigma A\wedge\Omega\Sigma A\overset{\tilde{\imath}\wedge\tilde{\imath}}{\longrightarrow}G\wedge G\overset{\sm{\mathds{1}_G, \mathds{1}_G}}{\longrightarrow}G \] has the same order as $\sm{\mathds{1}_G,\mathds{1}_G}$.
\begin{lemma}\label{rho} The map $p^r\circ\sm{\mathds{1}_G, \mathds{1}_G}$ is null-homotopic if and only if $p^r\circ\sm{\tilde{\imath}, \tilde{\imath}}$ is null-homotopic. \end{lemma}
\begin{proof} The sufficiency part is obvious. We only show the necessity part. Suppose $p^r\circ\sm{\tilde{\imath}, \tilde{\imath}}$ is null-homotopic. By Lemma~\ref{lemma_rho+s}, $e=\tilde{\imath}\circ s$ is a homotopy equivalence. Composing with its inverse $e'$, the map $\tilde{\imath}\circ s\circ e'$ is homotopic to the identity. Then we obtain \[ p^r\circ\sm{\mathds{1}_G, \mathds{1}_G}\simeq p^r\circ\sm{\tilde{\imath}\circ s\circ e', \tilde{\imath}\circ s\circ e'}=p^r\circ\sm{\tilde{\imath},\tilde{\imath}}\circ(s\circ e'\wedge s\circ e') \] which is null-homotopic since $p^r\circ\sm{\tilde{\imath},\tilde{\imath}}$ is null-homotopic. \end{proof}
Combining Diagram~\ref{CD1} and the fact that $\tilde{\imath}$ is an H-map, we have the commutative diagram \begin{equation}\label{dgm_rho_e_m} \xymatrix{
&(\Omega\Sigma A)^k\ar[d]^{\tilde{\imath}^k}\ar[r]^{\mu^k} &\Omega\Sigma A\ar[d]^{\tilde{\imath}}\\ A^k\ar[ur]^{j^k}\ar[r]^{\imath^k} &G^k\ar[r]^{m^k} &G } \end{equation} where $\mu^k$ and $m^k$ are the $k$-fold multiplications in $\Omega\Sigma A$ and $G$. Let $m_k$ and $e_k$ be the compositions \[ \begin{array}{c c c} m_k:A^k\overset{\imath^k}{\longrightarrow}G^k\overset{m^k}{\longrightarrow}G &\text{and} &e_k:A^k\overset{j^k}{\longrightarrow}(\Omega\Sigma A)^k\overset{\mu^k}{\longrightarrow}\Omega\Sigma A. \end{array} \] Then we have the following commutative diagram \[\xymatrix{ A^k\wedge A^l\ar[d]_{e_k\wedge e_l}\ar[dr]^{m_k\wedge m_l} & & &\\ \Omega\Sigma A\wedge\Omega\Sigma A\ar[r]^-{\tilde{\imath}\wedge\tilde{\imath}} &G\wedge G\ar[r]^-{\sm{\mathds{1}_G,\mathds{1}_G}} &G\ar[r]^{p^r} &G }\] Observe that there is a string of equalities \begin{eqnarray*} [\Omega\Sigma A\wedge\Omega\Sigma A, G] &=&[\Sigma\Omega\Sigma A\wedge\Omega\Sigma A, BG]\\ &=&[\bigvee^\infty_{k,l=1}\Sigma A^{\wedge k}\wedge A^{\wedge l}, BG]\\ &=&\prod^\infty_{k,l=1}[A^{\wedge k}\wedge A^{\wedge l}, G]. \end{eqnarray*} The first and the third lines are due to adjunction, and the second line is due to James splitting $\Sigma\Omega\Sigma A\simeq\Sigma^{\infty}_{k=1}\Sigma A^{\wedge k}$. It is not hard to see that the nullity of $p^r\circ\sm{\tilde{\imath}, \tilde{\imath}}$ implies the nullity of the components $p^r\circ\sm{m_k,m_l}$. In the following we show that the converse is true.
\begin{lemma}\label{k component of JA} Let $X$ be a space and let $f:X\to G$ be a map. If $p^r\circ\sm{m_k,f}:A^k\wedge X\to G$ is null-homotopic for all $k$, then $p^r\circ\sm{\tilde{\imath},f}:\Omega\Sigma A\wedge X\to G$ is null-homotopic. Similarly, if $p^r\circ\sm{f,m_l}:X\wedge A^l\to G$ is null-homotopic for all $l$, then $p^r\circ\sm{f,\tilde{\imath}}:X\wedge\Omega\Sigma A\to G$ is null-homotopic. \end{lemma}
\begin{proof} We only prove the first statement since the second statement can be proved similarly. Let $h:\Sigma\Omega\Sigma A\wedge X\to BG$ be the adjoint of $p^r\circ\sm{\tilde{\imath},f}$. It suffices to show that $h$ is null-homotopic.
For any $k$, choose a right homotopy inverse $\psi_k$ of the suspended quotient map~\mbox{$\Sigma A^k\to\Sigma A^{\wedge k}$}, and let $\Psi_k$ be the composition \[ \Psi_k:\Sigma A^{\wedge k}\overset{\psi_k}{\longrightarrow}\Sigma A^k\overset{\Sigma e_k}{\longrightarrow}\Sigma\Omega\Sigma A. \] Observe that $e_k$ is the product of $k$ copies of the suspension $j$ and $j_*:H_*(A)\to H_*(\Omega\Sigma A)$ is the inclusion of the generating set into $H_*(\Omega\Sigma A)\cong T(\tilde{H}_*(A))$. The map~\mbox{$(e_k)_*:H_*(A^k)\to H^*(\Omega\Sigma A)$} sends the submodule $S_k\subset H_*(A^k)\cong H_*(A)^{\otimes k}$ consisting of length $k$ tensor products onto the submodule $M_k\subset T(\tilde{H}_*(A))$ consisting of length $k$ tensor products. Therefore $(\Psi_k)_*$ does the same. Then their sum \[ \Psi=\bigvee_{k=1}^{\infty}\Psi_k:\bigvee_{k=1}^{\infty}\Sigma A^{\wedge k}\to\Sigma\Omega\Sigma A \] induces a homology isomorphism and hence is a homotopy equivalence.
We claim that $h\circ(\Psi_k\wedge\mathds{1}_X)$ is null-homotopic for all $k$, where $\mathds{1}_X$ is the identity of $X$. Observe that the adjoint of the composition \[ \Sigma A^k\wedge X\overset{\Sigma e_k\wedge\mathds{1}_X}{\longrightarrow}\Sigma\Omega\Sigma A\wedge X\overset{h}{\longrightarrow}BG \] is $p^r\circ\sm{\tilde{\imath},f}\circ(e_k\wedge\mathds{1}_X)\simeq p^r\circ\sm{\tilde{\imath}\circ e_k,f}\simeq p^r\circ\sm{m_k,f}$ which is null-homotopic by assumption. Therefore $h\circ(\Psi_k\wedge\mathds{1}_X)=h\circ(\Sigma e_k\wedge\mathds{1}_X)\circ(\psi_k\wedge\mathds{1}_X)$ is null-homotopic, and by definition of~$\Psi$, the composition \[ \bigvee^{\infty}_{k=1}\Sigma A^{\wedge k}\wedge X\overset{\Psi\wedge\mathds{1}}{\longrightarrow}\Sigma\Omega\Sigma A\wedge X\overset{h}{\longrightarrow}BG. \] is null-homotopic. Notice that $(\Psi\wedge\mathds{1})$ is a homotopy equivalence. It implies that $h$ is null-homotopic and so is $p^r\circ\sm{\tilde{\imath},f}$. \end{proof}
\begin{lemma}\label{lemma_J_to_JkJl} The map $p^r\circ\sm{\tilde{\imath}, \tilde{\imath}}$ is null-homotopic if and only if $p^r\circ\sm{m_k, m_l}$ is null-homotopic for all $k$ and $l$. \end{lemma}
\begin{proof} It suffices to prove the necessity part. Suppose $p^r\circ\sm{m_k,m_l}$ is null-homotopic for all~$k$ and $l$. Apply the first part of Lemma~\ref{k component of JA} to obtain $p^r\circ\sm{\tilde{\imath},m_l}\simeq*$ for all $l$ and apply the second part of Lemma~\ref{k component of JA} to obtain $p^r\circ\sm{\tilde{\imath},\tilde{\imath}}\simeq*$. \end{proof}
At this point we have related the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ to the orders of $\sm{m_k,m_l}$ for all $k$ and~$l$. There is one more step to link up with the order of the Samelson product \[ \sm{\imath,\imath}:A\wedge A\overset{\imath\wedge\imath}{\longrightarrow}G\wedge G\overset{\sm{\mathds{1}_G,\mathds{1}_G}}{\longrightarrow}G\overset{p^r}{\longrightarrow}G. \]
\begin{lemma}\label{lemma_sm_mk_null} If $p^r\circ\sm{\imath,\imath}$ is null-homotopic and $G$ has homotopy nilpotency class less than~\mbox{$p^r+1$}, then $p^r\circ\sm{m_k, m_l}$ is null-homotopic for all $k$ and $l$. \end{lemma}
The proof of Lemma~\ref{lemma_sm_mk_null} is long and we postpone it to the next section so as to avoid interrupting the flow of our discussion. Assuming Lemma~\ref{lemma_sm_mk_null} we can prove our main theorem.
\begin{thm}\label{main} Suppose that $G$ has homotopy nilpotence class less than $p^r+1$ after localization at $p$. Then $\sm{\mathds{1}_G,\mathds{1}_G}$ has order $p^r$ if and only if $\sm{\imath,\imath}$ has order $p^r$. \end{thm}
\begin{proof} The order of $\sm{\mathds{1}_G, \mathds{1}_G}$ is not less than the order of $\sm{\imath,\imath}$. Therefore we need to show that the order of $\sm{\mathds{1}_G, \mathds{1}_G}$ is not greater than the order of $\sm{\imath,\imath}$ under the assumption. Assume~$\sm{\imath,\imath}$ has order $p^r$, that is $p^r\circ\sm{\imath,\imath}$ is null-homotopic. Lemmas~\ref{rho}, \ref{lemma_J_to_JkJl} and \ref{lemma_sm_mk_null} imply that $p^r\circ\sm{\mathds{1}_G, \mathds{1}_G}$ is null-homotopic, so the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ is not greater than the order of $\sm{\imath,\imath}$. \end{proof}
\section{Proof of Lemma~\ref{lemma_sm_mk_null}} In this section we prove Lemma~\ref{lemma_sm_mk_null} by showing $p^r\circ\sm{m_k,m_l}$ is null-homotopic assuming the homotopy nilpotence class of $G$ is less than $p^r+1$. We convert it into an algebraic problem and derive lemmas from group theoretic identities and the topological properties of~$G$. First let us review the algebraic properties of Samelson products.
\subsection{Algebraic properties of Samelson products} Given two maps $f:X\to G$ and~\mbox{$g:Y\to G$}, their Samelson product $\sm{f,g}$ sends $(x, y)\in X\wedge Y$ to the commutator of their images $f(x)$ and $g(y)$. It is natural to regard the map $\sm{f,g}$ as a commutator in $[X\wedge Y, G]$, but $f$ and~$g$ are maps in different homotopy sets and there is no direct multiplication between them. Instead, we can include $[X,G],[Y,G]$ and $[X\wedge Y, G]$ into $[X\times Y, G]$ and identify $\sm{f,g}$ as a commutator there.
\begin{lemma}\label{subgp} For any spaces $X$ and $Y$, let $\pi_1:X\times Y\to X$ and $\pi_2:X\times Y\to Y$ be the projections and let $q:X\times Y\to X\wedge Y$ be the quotient map. Then the images $(\pi_1)^*[X, G]$, $(\pi_2)^*[Y, G]$, $q^*[X\wedge Y, G]$ are subgroups of $[X \times Y, G]$, and $(\pi_1)^*:[X,G]\to [X\times Y,G]$, $(\pi_2)^*:[Y,G]\to [X\times Y,G]$, $q^*:[X\wedge Y,G]\to [X\times Y,G]$ are monomorphisms. \end{lemma}
\begin{proof} Observe that $\pi_1$ and $\pi_2$ induce group homomorphisms $(\pi_1)^*:[X, G]\to[X\times Y, G]$ and~\mbox{$(\pi_2)^*:[X, G]\to[X\times Y, G]$}, so their images $(\pi_1)^*[X, G]$ and $(\pi_2)^*[Y, G]$ are subgroups of $[X\times Y, G]$. Moreover, let $j:X\to X\times Y$ be the inclusion. Since $\pi_1\circ j$ is the identity,~\mbox{$j^*\circ(\pi_1)^*$} is an isomorphism and $(\pi_1)^*$ is a monomorphism. Therefore $[X, G]$ is isomorphic to $(\pi_1)^*[X,G]$. Similarly $[Y, G]$ is isomorphic to $(\pi_2)^*[Y,G]$.
For $q^*[X\wedge Y,G]$, the cofibration $X\vee Y\overset{j'}{\to}X\times Y\overset{q}{\to}X\wedge Y$ induces an exact sequence \[ \cdots\to[\Sigma X\times Y, G]\overset{\Sigma j'^*}{\longrightarrow}[\Sigma(X\vee Y), G]\longrightarrow[X\wedge Y, G]\overset{q^*}{\longrightarrow}[X\times Y, G]\overset{j'^*}{\longrightarrow}[X\vee Y, G], \] where $j'$ is the inclusion and $q$ is the quotient map. Since $\Sigma j':\Sigma(X\vee Y)\hookrightarrow\Sigma(X\times Y)$ has a right homotopy inverse, $q^*:[X\wedge Y, G]\to[X\times Y, G]$ is a monomorphism. Therefore~\mbox{$[X\wedge Y, G]$} is isomorphic to~\mbox{$q^*[X\wedge Y, G]$}, which is a subgroup of $[X\times Y, G]$. \end{proof}
There are two groups in our discussion, namely $G$ and $[X\times Y, G]$. To distinguish their commutators, for any maps $f:X\to G$ and $g:Y\to G$ we use $\cm{f,g}$ to denote the map which sends $(x,y)\in X\times Y$ to~\mbox{$f(x)^{-1}g(y)^{-1}f(x)g(y)\in G$}, and for any maps $a:X\times Y\to G$ and $b:X\times Y\to G$ we use $[a,b]$ to denote the commutator $a^{-1}b^{-1}ab\in[X\times Y, G]$. Lemma~\ref{subgp} says that $f:X\to G$ and $g:Y\to G$ can be viewed as being in $[X\times Y, G]$. Their images are the compositions \[ \begin{array}{c c c} \tilde{f}:X\times Y\overset{\pi_1}{\longrightarrow}X\overset{f}{\longrightarrow}G &\text{and} &\tilde{g}:X\times Y\overset{\pi_2}{\longrightarrow}Y\overset{g}{\longrightarrow}G. \end{array} \] Consider the diagram \[\xymatrix{
&X\times Y\ar[d]_{f\times g}\ar[r]^{q}\ar[dl]_{\triangle} &X\wedge Y\ar[d]^{f\wedge g}\\ (X\times Y)\times(X\times Y)\ar[r]^-{\tilde{f}\times\tilde{g}} &G\times G\ar[d]_{c}\ar[r]^{q'} &G\wedge G\ar[d]^{\sm{\mathds{1}_G,\mathds{1}_G}}\\
&G\ar[r]^{=} &G }\] where $\triangle$ is the diagonal map, $c$ is the commutator map $\cm{\mathds{1}_G,\mathds{1}_G}$, and $q'$ is the quotient maps. The commutativity of the left triangle is due to the definitions of $\tilde{f}$ and $\tilde{g}$, the commutativity of the top square is due to the naturality of the quotient maps and the commutativity of the bottom square is due to the definition of~$\sm{\mathds{1}_G,\mathds{1}_G}$. The middle column is $\cm{f,g}$ and the right column is $\sm{f,g}$. In order to show that $\sm{f,g}$ is null-homotopic, it suffices to consider~$\cm{f,g}\simeq q^*\sm{f,g}$ since $q^*:[X\wedge Y, G]\to[X\times Y, G]$ is injective. Observe that $\cm{f,g}$ is homotopic to the composition \[ X\times Y\overset{\triangle}{\longrightarrow}(X\times Y)\times(X\times Y)\overset{\tilde{f}\times\tilde{g}}{\longrightarrow}G\times G\overset{c}{\longrightarrow}G \] according to the diagram. That is $\cm{f,g}$ is the commutator $[\tilde{f},\tilde{g}]=\tilde{f}^{-1}\tilde{g}^{-1}\tilde{f}\tilde{g}$ in $[X\times Y, G]$.
Let $\ad{a}{b}=b^{-1}ab$ be the conjugation of maps $a$ and $b$ in $[X\times Y, G]$. In group theory, commutators satisfy the following identities: \begin{equation}\label{lemma_gp_id} \begin{minipage}{0.9\textwidth} \begin{enumerate} \item $[a,b]=a^{-1}\cdot\ad{a}{b}$; \item $[a,b]^{-1}=[b,a]$; \item $[a\cdot a',b]=\ad{[a,b]}{a'}\cdot[a',b]=[a,b]\cdot[a',[a,b]]^{-1}\cdot[a',b]$; \item $[a,b\cdot b']=[a,b']\cdot\ad{[a,b]}{b'}=[a,b']\cdot[a,b]\cdot[[a,b],b']$. \end{enumerate} \end{minipage} \end{equation}
In particular, we can substitute $\tilde{f}$ and $\tilde{g}$ to $a$ and $b$ in these identities.
\begin{prop}\label{identities} Let $f,f':X\to G$ and $g,g':Y\to G$ be maps. Then in $[X\times Y, G]$, \begin{enumerate}[label=\textnormal{(\roman*)}] \item\label{id_conj} $\cm{f, g}=\tilde{f}^{-1}\cdot\ad{\tilde{f}}{\tilde{g}}$; \item\label{id_inv_sm} $\cm{f, g}^{-1}=\cm{g, f}\circ T$; \item\label{id_lprod} $\cm{f\cdot f',g}=\ad{\cm{f, g}}{\tilde{f}'}\cdot\cm{f',g}=\cm{f, g}\cdot[\tilde{f'},[\tilde{f},\tilde{g}]]^{-1}\cdot\cm{f', g}$; \item\label{id_rprod} $\cm{f,g\cdot g'}=\cm{f, g'}\cdot\ad{\cm{f,g}}{\tilde{g}'}=c(f,g')\cdot\cm{f, g}\cdot [[\tilde{f}, \tilde{g}],\tilde{g'}]$, \end{enumerate} where $T:Y\times X\to X\times Y$ is the swapping map. \end{prop}
\begin{proof} All identities come directly from the identities in~(\ref{lemma_gp_id}), while Identity~\ref{id_inv_sm} needs some explanation. Observe there exists a homotopy commutative diagram \[\xymatrix{ X\times Y\ar[d]^-{T}\ar[r]^-{f\times g} &G\times G\ar[d]^-{T}\ar[r]^-{c} &G\ar[d]^-{r}\\ Y\times X\ar[r]^-{g\times f} &G\times G\ar[r]^-{c} &G }\] where $r:G\to G$ is the inversion. The upper direction around the diagram is $c(f,g)^{-1}$, while the lower direction is $c(g, f)\circ T$. So Identity~\ref{id_inv_sm} follows. \end{proof}
\begin{remark} The iterated commutator $[\tilde{f}'(x),[\tilde{f}(x),\tilde{g}(y)]]$ is the composition \[ X\times Y\overset{\triangle_X\times\mathds{1}_Y}{\longrightarrow}X\times X\times Y\overset{f'\times f\times g}{\longrightarrow}G\times G\times G\overset{\mathds{1}_G\times c}{\longrightarrow}G\times G\overset{c}{\longrightarrow}G \] where $\triangle_X$ is the diagonal map and $\mathds{1}_Y$ and $\mathds{1}_G$ are the identity maps. Let $c_2=c\circ(\mathds{1}_G\times c)$ be the 2-iterated commutator on $G$. Then we can write $[\tilde{f}',[\tilde{f},\tilde{g}]]$ as $c_2\circ(f'\times f\times g)\circ(\triangle_X\times\mathds{1}_Y)$. However, we prefer to stick to the notation $[\tilde{f}',[\tilde{f},\tilde{g}]]$ because it better indicates it is the commutator of which maps, while $c_2\circ(f'\times f\times g)\circ(\triangle_X\times\mathds{1}_Y)$ looks long and confusing. \end{remark}
Since our group $[X\times Y, G]$ has a topological interpretation, the topologies of $X$ and $Y$ add extra algebraic properties to its group structure.
\begin{lemma}\label{lemma_co-H_id} Let $f, g$ and $h: X\times Y\to G$ be maps. If $X$ is a co-H-space and the restrictions of $f$ and $g$ to $X\vee Y$ are null-homotopic, then in $[X\times Y, G]$ we have \[ \begin{array}{c c c} f\cdot g=g\cdot f &\text{and} &[f\cdot g, h]=[f, h]\cdot[g, h]. \end{array} \] \end{lemma}
\begin{proof} Let $q:X\times Y\to X\wedge Y$ be the quotient map. Observe that there exist $f'$ and $g'$ in~$[X\wedge Y, G]$ such that $f=q^*f'$ and $g=q^*g'$. Since $X\wedge Y$ is a co-H-space, $[X\wedge Y, G]$ is an abelian group and $f'$ and $g'$ commute. Therefore $f$ and $g$ commute as $q^*$ is a monomorphism.
To show the linearity, we start with Proposition~\ref{identities}~\ref{id_lprod} \[ [f\cdot g, h]=[f, h]\cdot[g, [f, h]]\cdot[g, h]. \] Since $[f, h]$ is also null-homotopic on $X\vee Y$, it commutes with $g$ and their commutator~$[g, [f,h]]$ is trivial. Therefore we have $[f\cdot g, h]=[f,h]\cdot[g,h]$. \end{proof}
\subsection{Main body of the proof} We go back to the proof of Lemma~\ref{lemma_sm_mk_null}. Recall that $m_k$ is the composition $m_k:A^k\overset{\imath^k}{\longrightarrow}G^k\overset{m^k}{\longrightarrow}G$. To distinguish the spaces $A$'s, denote the $i^{\text{th}}$ copy of $A$ in $A^k$ by $A_i$. Let $a_i$ and $m'_{k-1}$ be the compositions \[ \begin{array}{c c c} a_i:A^k\overset{proj}{\longrightarrow}A_i\overset{\imath}{\longrightarrow}G &\text{and} &m'_{k-1}:A^k\overset{proj}{\longrightarrow}\prod^{k-1}_{i=1}A_i\overset{m_{k-1}}{\longrightarrow}G \end{array} \] respectively. Then we have $m_k=m'_{k-1}\cdot a_i$ in $[A^k, G]$. Include $\sm{m_k,m_l}$ in $[A^k\times A^l, G]$ by Lemma~\ref{subgp}. It becomes the commutator $\cm{m_k, m_l}=[\tilde{m}_k, \tilde{m}_l]$, where $\tilde{m}_k$ and $\tilde{m}_l$ are compositions \[ \begin{array}{c c c} \tilde{m}_k:A^k\times A^l\overset{proj}{\longrightarrow}A^k\overset{m_k}{\longrightarrow}G &\text{and} &\tilde{m}_l:A^k\times A^l\overset{proj}{\longrightarrow}A^l\overset{m_l}{\longrightarrow}G. \end{array} \] Let $\tilde{a}_i$ and $\tilde{m}'_{k-1}$ be compositions \[ \begin{array}{c c c} \tilde{a}_i:A^k\times A^l\overset{proj}{\longrightarrow}A_i\overset{\imath}{\longrightarrow}G &\text{and} &\tilde{m}'_{k-1}:A^k\times A^l\overset{proj}{\longrightarrow}\prod^{k-1}_{i=1}A_i\overset{m_{k-1}}{\longrightarrow}G. \end{array} \] Then in $[A^k\times A^l, G]$ we have $\tilde{m}_k=\tilde{m}'_{k-1}\cdot\tilde{a}_k$.
Assume the homotopy nilpotence class of $G$ is less than $p^r+1$. Now we use induction on~$k$ and $l$ show that $\cm{m_k,m_l}^{p^r}$ is null-homotopic. To start with, we show that this is true for~$k=1$ or $l=1$.
\begin{lemma}\label{induction_1} If $\cm{\imath,\imath}^{p^r}$ is null-homotopic, then $\cm{m_k,\imath}^{p^r}$ and $\cm{\imath,m_l}^{p^r}$ are null-homotopic for all $k$ and $l$. \end{lemma}
\begin{proof} We prove that $\cm{m_k,\imath}^{p^r}$ is null-homotopic by induction. Since $m_1=\imath$,~\mbox{$\cm{m_1,\imath}^{p^r}=\cm{\imath,\imath}^{p^r}$} is null-homotopic by assumption. Suppose $\cm{m_k,\imath}^{p^r}$ is null-homotopic. We need to show that $\cm{m_{k+1},\imath}^{p^r}$ is also null-homotopic. Apply Proposition~\ref{identities}~\ref{id_lprod} to obtain \[ \cm{m_{k+1},\imath}=\cm{m'_k\cdot a_{k+1},\imath}=\ad{\cm{m'_k,\imath}}{\tilde{a}_{k+1}}\cdot\cm{a_{k+1},\imath}. \] Observe that $\cm{a_{k+1},\imath}$ and $\ad{\cm{m'_k,\imath}}{\tilde{a}_{k+1}}$ are null-homotopic on $A^{k+1}\vee A$ and $A^{k+1}\wedge A$ is a co-H-space. Lemma~\ref{lemma_co-H_id} implies that $\cm{a_{k+1},\imath}$ and $\ad{\cm{m'_k,\imath}}{\tilde{a}_{k+1}}$ commute and we have \begin{eqnarray*} \cm{m_{k+1},\imath}^{p^r} &=&\left(\ad{\cm{m'_k,\imath}}{\tilde{a}_{k+1}}\cdot\cm{a_{k+1},\imath}\right)^{p^r}\\ &=&\left(\ad{\cm{m'_k,\imath}}{\tilde{a}_{k+1}}\right)^{p^r}\cdot\cm{a_{k+1},\imath}^{p^r}\\ &=&\ad{\left(\cm{m'_k,\imath}^{p^r}\right)}{\tilde{a}_{k+1}}\cdot\cm{a_{k+1},\imath}^{p^r}. \end{eqnarray*} The last term $\cm{a_{k+1},\imath}^{p^r}$ is null-homotopic since $a_{k+1}$ is the inclusion $A_{k+1}\overset{\imath}{\to}G$. Also, by the induction hypothesis $\cm{m'_k,\imath}^{p^r}$ is null-homotopic. Therefore $\cm{m_{k+1},\imath}^{p^r}$ is null-homotopic and the induction is completed.
Similarly, we can show that $\cm{\imath, m_l}^{p^r}$ is null-homotopic for all $l$. \end{proof}
As a consequence of Lemma~\ref{induction_1}, the following lemma implies that the order of \[ \sm{\imath, \mathds{1}_G}:A\wedge G\overset{\imath\wedge\mathds{1}_G}{\longrightarrow}G\wedge G\overset{\sm{\mathds{1}_G,\mathds{1}_G}}{\longrightarrow}G \] equals to the order of its restriction $\sm{\imath,\imath}$ without assuming the condition on the homotopy nilpotence of $G$.
\begin{lemma}\label{A wedge G} The map $p^r\circ\sm{\imath,\imath}$ is null-homotopic if and only if $p^r\circ\sm{\mathds{1}_G,\imath}$ and $p^r\circ\sm{\imath,\mathds{1}_G}$ are null-homotopic. \end{lemma}
\begin{proof} We only need to prove the sufficient condition. If $p^r\circ\sm{\imath,\imath}$ is null-homotopic, then~\mbox{$p^r\circ\sm{\imath,m_l}$} is null-homotopic for all~$l$ by Lemma~\ref{induction_1}. Lemma~\ref{k component of JA} implies that $p^r\circ\sm{\imath,\tilde{\imath}}:A\wedge\Omega\Sigma A\to G$ is null-homotopic. Since $\tilde{\imath}\circ s$ is a homotopy equivalence by Lemma~\ref{lemma_rho+s}, $p^r\circ\sm{\imath,\mathds{1}_G}$ is null-homotopic.
The sufficient condition for $p^r\circ\sm{\mathds{1}_G,\imath}$ can be proved similarly. \end{proof}
Now suppose $\cm{m_k,m_l}^{p^r}$ is trivial for some fixed $k$ and $l$ in $[A^k\times A^l, G]$. The next step is to show that $\cm{m_{k+1},m_l}^{p^r}$ is trivial in $[A^{k+1}\times A^l, G]$. At first glance we can follow the proof of Lemma~\ref{induction_1} and apply Lemmas~\ref{identities} and~\ref{lemma_co-H_id} to split $\cm{m_{k+1},m_l}^{p^r}$ into $\cm{m'_k,m_l}^{p^r}$ and~\mbox{$\cm{a_{k+1},m_l}^{p^r}$} which are null-homotopic by the induction hypothesis. However, when $l>1$,~$A^l$ is not a co-H-space and we cannot use Lemma~\ref{lemma_co-H_id} to argue that $\cm{m'_k,m_l}$ and $\cm{a_{k+1},m_l}$ commute. Instead, apply Proposition~\ref{identities}~\ref{id_lprod} to obtain \begin{eqnarray*} \cm{m_{k+1},m_l} &=&\cm{m'_k\cdot a_{k+1}, m_l}\\ &=&\cm{m'_k,m_l}\cdot[\cm{m'_k,m_l},\tilde{a}_{k+1}]\cdot\cm{a_{k+1},m_l}. \end{eqnarray*} Denote $\cm{m'_k,m_l}$ and $[\cm{m'_k,m_l},\tilde{a}_{k+1}]\cdot\cm{a_{k+1},m_l}$ by $\alpha_k$ and $\beta_k$ respectively. Observe that the restrictions of any powers and commutators involving $\beta_k$ to $A_{k+1}\vee(A^k\times A^l)$ are null-homotopic and $A_{k+1}$ is a co-H-space. Therefore they enjoy the conditions of Lemma~\ref{lemma_co-H_id}.
\begin{lemma}\label{alpha_k+1^n} For any natural number $n$, we have \[ (\alpha_k\cdot\beta_k)^n=\alpha_k^n\cdot\beta_k^n\cdot\left(\prod^{n-1}_{i=1}[\beta_k,\alpha_k^i]\right). \] \end{lemma}
\begin{proof} We induct on $n$. The statement of the lemma is trivial for $n=1$. Assume the formula holds for an integer $n$. For the $(n+1)$ case, using the induction hypothesis we have \begin{eqnarray*} (\alpha_k\cdot\beta_k)^{n+1} &=&\alpha_k\cdot\beta_k\cdot(\alpha_k\cdot\beta_k)^n\\ &=&\alpha_k\cdot\beta_k\cdot\alpha_k^n\cdot\beta_k^n\cdot\left(\prod^{n-1}_{i=1}[\beta_k,\alpha_k^i]\right)\\ &=&\alpha_k^{n+1}\cdot\beta_k\cdot[\beta_k,\alpha_k^n]\cdot\beta_k^n\cdot\left(\prod^{n-1}_{i=1}[\beta_k,\alpha_k^i]\right) \end{eqnarray*} In the last line $[\beta_k,\alpha_k^n]$ is formed after we swap $\beta_k$ and $\alpha_k^n$. Since the restrictions of $\beta_k$,~$\beta^n_k$ and~$[\beta_k,\alpha^i_k]$ to $A_{k+1}\vee(A^k\times A^l)$ are null-homotopic, they commute by Lemma~\ref{lemma_co-H_id}. By commuting the terms, the statement follows. \end{proof}
In order to prove the triviality of $\cm{m_{k+1},m_l}^{p^r}$, by Lemma~\ref{alpha_k+1^n} it suffices to show that~\mbox{$\alpha_k^{p^r},\beta_k^{p^r}$} and~\mbox{$\prod^{p^r-1}_{i=1}[\beta_k,\alpha_k^i]$} are null-homotopic. By the induction hypothesis $\cm{m_k,m_l}^{p^r}$ is null-homotopic, so $\alpha_k^{p^r}=\cm{m'_k,m_l}^{p^r}$ is null-homotopic. It remains to show that $\beta_k^{p^r}$ and $\prod^{p^r-1}_{i=1}[\beta_k,\alpha_k^i]$ are null-homotopic.
\begin{lemma}\label{alpha_beta} If $\cm{\imath,\imath}^{p^r}$ and $\alpha_k^{p^r}$ are null-homotopic, then so is $\beta_k^{p^r}$. \end{lemma}
\begin{proof} By definition, $\beta_k=[\cm{m'_k,m_l},\tilde{a}_{k+1}]\cdot\cm{a_{k+1},m_l}$. Observe that the restrictions of~$[\cm{m'_k,m_l},\tilde{a}_{k+1}]$ and~\mbox{$\cm{a_{k+1},m_l}$} to $A_{k+1}\vee(A^k\times A^l)$ are null-homotopic. By Lemma~\ref{lemma_co-H_id} they commute and we have \[ \beta_k^{p^r}=\left([\cm{m'_k,m_l},\tilde{a}_{k+1}]\cdot\cm{a_{k+1},m_l}\right)^{p^r}=[\cm{m'_k,m_l},\tilde{a}_{k+1}]^{p^r}\cdot\cm{a_{k+1},m_l}^{p^r}. \] Since $\cm{\imath,\imath}^{p^r}$ is null-homotopic, so is $\cm{a_{k+1},m_l}^{p^r}$ by Lemma~\ref{induction_1}.
On the other hand, recall that $c(m'_k,m_l)$ and $\tilde{a}_{k+1}$ are the compositions \[ \begin{array}{c c c} c(m'_k,m_l):A^{k+1}\times A^l\overset{proj}{\longrightarrow}A^k\times A^l\overset{c(m_k,m_l)}{\longrightarrow}G &\text{and} &\tilde{a}_{k+1}:A^{k+1}\times A^l\overset{proj}{\longrightarrow}A_{k+1}\overset{\imath}{\longrightarrow}G \end{array} \] respectively. Therefore we have \begin{eqnarray*} [\cm{m'_k,m_l}, \tilde{a}_{k+1}]^{p^r} &=&p^r\circ c(c(m'_k,m_l),\imath)\\ &=&p^r\circ\cm{\mathds{1}_G,\imath}\circ(c(m'_k,m_l)\times\mathds{1}_A) \end{eqnarray*} where $\mathds{1}_A$ is the identity map of $A_{k+1}$. Since $p^r\circ\cm{\mathds{1}_G,\imath}$ is null-homotopic by Lemma~\ref{A wedge G},~\mbox{$[\cm{m'_k,m_l}, \tilde{a}_{k+1}]^{p^r}$} is null-homotopic and so is $\beta^{p^r}_k$. \end{proof}
\begin{lemma}\label{prod_express} For any natural number $n$, we have \[ \prod^{n-1}_{i=1}[\beta_k,\alpha_k^i]=\prod^{n-1}_{i=1}c_i(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{n}{i+1}} \] where $c_i(\beta_k,\alpha_k,\cdots,\alpha_k)=[[\cdots[[\beta_k,\alpha_k],\alpha_k]\cdots], \alpha_k]$ is the $i$-iterated commutator. \end{lemma}
\begin{proof} First, by induction we prove \[ [\beta_k,\alpha^i_k]=\prod^i_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}. \] It is trivial for $i=1$. Assume the formula holds for $[\beta_k,\alpha^i_k]$. Use the commutator identity in~(\ref{lemma_gp_id}) and inductive hypothesis to get \begin{eqnarray*} [\beta_k,\alpha^{i+1}_k] &=&[\beta_k,\alpha_k]\cdot[\beta_k,\alpha^i_k]\cdot[[\beta_k,\alpha^i_k],\alpha_k]\\ &=&[\beta_k,\alpha_k]\cdot\left(\prod^i_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}\right)\cdot[\prod^i_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}},\alpha_k] \end{eqnarray*}
Since the restriction of $c_j(\beta_k,\alpha_k,\cdots,\alpha_k)$ to $A_{k+1}\vee(A^k\times A^l)$ is null-homotopic for all $j$, by Lemma~\ref{lemma_co-H_id} they commute with each other and \begin{eqnarray*} [\beta_k,\alpha^{i+1}_k] &=&[\beta_k,\alpha_k]\cdot\left(\prod^i_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}\right)\cdot\left(\prod^i_{j=1}[c_j(\beta_k,\alpha_k,\cdots,\alpha_k),\alpha_k]^{\binom{i}{j}}\right)\\ &=&[\beta_k,\alpha_k]\cdot\left(\prod^i_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}\right)\cdot\left(\prod^i_{j=1}c_{j+1}(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}\right)\\ &=&[\beta_k,\alpha_k]^{i+1}\cdot\left(\prod^i_{j=2}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}+\binom{i}{j+1}}\right)\cdot c_{i+1}(\beta_k,\alpha_k,\cdots,\alpha_k)\\ &=&\prod^{i+1}_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i+1}{j}} \end{eqnarray*} Therefore the claim is proved.
Now we multiply all $[\beta_k,\alpha^i_k]$'s and use the commutativity of $c_j(\beta_k,\alpha_k,\cdots,\alpha_k)$'s to get \begin{eqnarray*} \prod^{n-1}_{i=1}[\beta_k,\alpha^i_k] &=&\prod^{n-1}_{i=1}\prod^i_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}\\ &=&\prod^{n-1}_{j=1}\prod^{n-1}_{i=j}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}\\ &=&\prod^{n-1}_{j=1}\left(\prod^{n-1}_{i=j}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{i}{j}}\right)\\ &=&\prod^{n-1}_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\sum^{n-1}_{i=j}\binom{i}{j}} \end{eqnarray*} The proof will be completed if we can show that $\sum^{n-1}_{i=j}\binom{i}{j}=\binom{n}{j+1}$.
Consider the polynomial \[ \sum^{n-1}_{i=0}(1+x)^i=\sum^{n-1}_{i=0}\sum^i_{j=0}\binom{i}{j}x^j=\sum^{n-1}_{j=0}\sum^{n-1}_{i=0}\binom{i}{j}x^j. \] The coefficient of $x^j$ is $\sum^{n-1}_{i=j}\binom{i}{j}$. On the other hand, it can be written as \[ 1+(1+x)+\cdots+(1+x)^{n-1}=\frac{(1+x)^n-1}{x}=\sum^{n}_{j=1}\binom{n}{j}x^{j-1}. \] The coefficient of $x^j$ is $\binom{n}{j+1}$. By comparing the coefficients of $x^j$ the statement follows. \end{proof}
Let $nil(G)$ be the homotopy nilpotence class of $G$, that is, $nil(G)=n$ if and only if the~$n$-iterated commutator $c_n$ is null-homotopic but $c_{n-1}$ is not. In particular, $nil(G)=n$ means all commutators in $G$ of length greater than $n$ are null-homotopic.
\begin{lemma}\label{nil_omega} Let $\{f_j:A^k\times A^l\to G\}_{1\leq j\leq n-1}$ be a set of maps. If $nil(G)\leq n$, then~\mbox{$c_{n-1}(\beta_k,f_1,\cdots,f_{n-1})$} is null-homotopic. \end{lemma}
\begin{proof} By definition, $\beta_k=[\cm{m'_k,m_l},\tilde{a}_{k+1}]\cdot\cm{a_{k+1},m_l}$. Denote $[\cm{m'_k,m_l},\tilde{a}_{k+1}]$ and $\cm{a_{k+1},m_l}$ by $\gamma_0$ and $\gamma'_0$ respectively. Then we have $\beta_k=\gamma_0\cdot\gamma'_0$. For $1\leq j\leq n-1$, let $\gamma_j=[\gamma_{j-1},f_j]$ and $\gamma'_j=[\gamma'_{j-1},f_j]$. We claim that $c_m(\beta_k,f_1,\cdots,f_m)=\gamma_m\cdot\gamma'_m$ for $1\leq m\leq n-1$.
When $m=1$, \[ c_1(\beta_k,f_1)=[\beta_k,f_1]=[\gamma_0\cdot\gamma'_0,f_1]. \] Since the restrictions of $\gamma_0$ and $\gamma'_0$ to $A_{k+1}\vee(A^k\times A^l)$ are null-homotopic, by Lemma~\ref{lemma_co-H_id} we have \[ [\gamma_0\cdot\gamma'_0,f_1]=[\gamma_0,f_1]\cdot[\gamma'_0,f_1]=\gamma_1\cdot\gamma'_1. \] Assume the claim is true for $m-1$. By the induction hypothesis, \begin{eqnarray*} c_m(\beta_k,f_1,\cdots,f_m) &=&c_1\circ(c_{m-1}(\beta_k, f_1,\cdots, f_{m-1})\times f_m)\\ &=&[c_{m-1}(\beta_k, f_1,\cdots, f_{m-1}), f_m]\\ &=&[\gamma_{m-1}\cdot\gamma'_{m-1}, f_m] \end{eqnarray*} Since the restrictions of $\gamma_{m-1}$ and $\gamma'_{m-1}$ to $A_{k+1}\vee(A^k\times A^l)$ are null-homotopic, by Lemma~\ref{lemma_co-H_id} \[ c_m(\beta_k,f_1,\cdots,f_m)=[\gamma_{m-1},f_m]\cdot[\gamma'_{m-1}, f_m]=\gamma_m\cdot\gamma'_m. \]
By putting $m=n-1$ we get $c_{n-1}(\beta_k,f_1,\cdots,f_n)=\gamma_{n-1}\cdot\gamma'_{n-1}$. Notice that $\gamma_{n-1}$ and~$\gamma'_{n-1}$ are commutators of length $n+2$ and $n+1$ respectively, which are null-homotopic due to the condition on the homotopy nilpotency of $G$. Therefore $c_{n-1}(\beta_k,f_1,\cdots,f_{n-1})$ is null-homotopic. \end{proof}
Now we have all the ingredients to prove Lemma~\ref{lemma_sm_mk_null}.
\begin{proof}[Proof of Lemma~\ref{lemma_sm_mk_null}] Suppose $\cm{\imath,\imath}^{p^r}$ is null-homotopic and $nil(G)$ is less than $p^r+1$. We prove that $\cm{m_k, m_l}^{p^r}$ is null-homotopic for all $k$ and $l$ by induction. By Lemma~\ref{induction_1},~$\cm{m_k,\imath}^{p^r}$ and $\cm{\imath, m_l}^{p^r}$ are null-homotopic for all $k$ and $l$. Assume $\cm{m_k,m_l}^{p^r}$ is null-homotopic for some fixed $k$ and $l$. We need to show that $\cm{m_{k+1},m_l}^{p^r}$ is null-homotopic. By Lemma~\ref{alpha_k+1^n}, we have \[ \cm{m_{k+1},m_l}^{p^r}=\alpha_k^{p^r}\cdot\beta_k^{p^r}\cdot\prod^{p^r-1}_{i=1}[\beta_k,\alpha_k^i]. \] The factor $\alpha_k^{p^r}$ is null-homotopic due to the definition of $\alpha_k$ and hypothesis, and $\beta_k^{p^r}$ is null-homotopic by Lemma~\ref{alpha_beta}. So it remains to show that $\prod^{p^r-1}_{i=1}[\beta_k,\alpha_k^i]$ is null-homotopic.
By Lemma~\ref{prod_express}, \[ \prod^{p^r-1}_{i=1}[\beta_k,\alpha_k^i]=\prod^{p^r-1}_{j=1}c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{p^r}{j+1}} \] Observe that $\binom{p^r}{j+1}$ is divisible by $p^r$ for $1\leq j\leq p^r-2$. By Lemma~\ref{lemma_co-H_id} we have \[ c_j(\beta_k^n,\alpha_k,\cdots,\alpha_k)=c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^n \] for all $n$. In our case we have \[ c_j(\beta_k,\alpha_k,\cdots,\alpha_k)^{\binom{p^r}{j+1}}=c_j(\beta_k^{p^r},\alpha_k,\cdots,\alpha_k)^{\binom{p^r}{j+1}/p^r}. \] Also, when $j=p^r-1$, the term $c_{p^r-1}(\beta_k,\alpha_k,\cdots,\alpha_k)$ is null-homotopic by Lemma~\ref{nil_omega}. Putting these together we obtain \[ \prod^{p^r-1}_{i=1}[\beta_k,\alpha_k^i]=\prod^{p^r-2}_{j=1}c_j(\beta_k^{p^r},\alpha_k,\cdots,\alpha_k)^{\binom{p^r}{j+1}/p^r}. \] We have shown that $\beta_k^{p^r}$ is null-homotopic in Lemma~\ref{alpha_beta}, so $\prod^{p^r}_{i=1}[\beta_k,\alpha_k^i]$ is null-homotopic and the induction is completed. \end{proof}
\section{Orders of Samelson products of quasi-$p$-regular groups} In this section we apply Theorem~\ref{main} to calculate the orders of $\sm{\mathds{1}_G,\mathds{1}_G}$ for certain Lie groups~$G$. Recall that $G$ is rationally homotopy equivalent to a product of spheres~$\prod^l_{i=1}S^{2n_i-1}$, where $n_1\leq\cdots\leq n_l$. The sequence $(2n_1-1,\cdots,2n_l-1)$ is called the \emph{type} of $G$. After localization at $p$, $G$ is homotopy equivalent to a product of H-spaces $\prod^{p-1}_{i=1}B_i$, and $A$ is homotopy equivalent to a wedge of co-H-spaces $\bigvee^{p-1}_{i=1}A_i$ such that $A_i$ is a subspace of $B_i$. For~\mbox{$1\leq i\leq p-1$}, let~\mbox{$\imath_i:A_i\to B_i$} be the inclusion. Then $H_*(B_i)$ is the exterior algebra generated by~$(\imath_i)_*(\tilde{H}_*(A_i))$. If each~$B_i$ is a sphere, then we call $G$ \emph{$p$-regular}. If each $A_i$ is a sphere or a CW-complex with two cells, then we call $G$ \emph{quasi-$p$-regular}. When $A_i$ is a CW-complex with two cells, it is homotopy equivalent to the cofibre of $\alpha_{2n_i-1}$, which is the generator of the homotopy group~$\pi_{2n_i+2p-4}(S^{2n_i-1})$, and the corresponding $B_i$ is the~$S^{2n_i-1}$-bundle~$B(2n-1, 2n+2p-3)$ over $S^{2n_i+2p-3}$ classified by $\frac{1}{2}\alpha_{2n_i-1}$ \cite{MNT77}.
The homotopy nilpotence classes of certain quasi-$p$-regular Lie groups are known. \begin{thm}[Kaji and Kishimoto \cite{kk10}] A $p$-regular Lie group has homotopy nilpotence class at most 3. \end{thm}
\begin{thm}[Kishimoto \cite{kishimoto09}] For $p\geq7$, a quasi-$p$-regular $SU(n)$ has homotopy nilpotence class at most 3. \end{thm}
\begin{thm}[Theriault \cite{theriault16}] For $p\geq7$, a quasi-$p$-regular exceptional Lie group has homotopy nilpotence class at most 2. \end{thm}
For $t=n-p+1$ and $t'=n-\frac{1}{2}p+1$, assume $G$ and $p$ are in the following list: \begin{equation}\label{quasi_list} \begin{array}{l l l} SU(n) &\simeq B(3,2p+1)\times\cdots\times B(2t-1,2n-1)\times S^{2t+1}\times\cdots\times S^{2p-1} &p>\frac{n}{2}\\
&\simeq S^3\times S^5\times\cdots\times S^{2n-1} &n\leq p\leq\frac{n}{2}\\ Sp(n) &\simeq B(3,2p+1)\times\cdots\times B(2t'-1,4n-1)\times S^{2t'+1}\times\cdots\times S^{2p-1} &p>n\\ F_4 &\simeq B(3,15)\times B(11,23) &p=7\\
&\simeq B(3,23)\times S^{11}\times S^{15} &p=11\\
&\simeq S^3\times S^{11}\times S^{15}\times S^{23} &p>11\\ E_6 &\simeq F_4\times S^9\times S^{17} &p\geq7\\ E_7 &\simeq B(3, 23)\times B(15,35)\times S^{11}\times S^{19}\times S^{27} &p=11\\
&\simeq B(3, 27)\times B(11,35)\times S^{15}\times S^{19}\times S^{23} &p=13\\
&\simeq B(3, 35)\times S^{11}\times S^{15}\times S^{19}\times S^{23}\times S^{27} &p=17\\
&\simeq S^3\times S^{11}\times S^{15}\times S^{19}\times S^{23}\times S^{27}\times S^{35} &p>17\\ E_8 &\simeq B(3, 23)\times B(15,35)\times B(27,47)\times B(39,59) &p=11\\
&\simeq B(3, 27)\times B(15,39)\times B(23,47)\times B(35,59) &p=13\\
&\simeq B(3, 35)\times B(15,47)\times B(27,59)\times S^{23}\times S^{39} &p=17\\
&\simeq B(3, 39)\times B(23,59)\times S^{15}\times S^{27}\times S^{35}\times S^{47} &p=23\\
&\simeq B(3, 59)\times S^{15}\times S^{23}\times S^{27}\times S^{35}\times S^{39}\times S^{47} &p=29\\
&\simeq S^3\times S^{15}\times S^{23}\times S^{27}\times S^{35}\times S^{39}\times S^{47}\times S^{59} &p>29. \end{array} \end{equation} By Theorem~\ref{main}, the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ equals the order of $\sm{\imath,\imath}$ in these groups.
\subsection{Upper bounds on the orders of $\sm{\mathds{1}_G,\mathds{1}_G}$ for quasi-$p$-regular Lie groups} Since $\sm{\imath,\imath}\in[A\wedge A,G]$ and \begin{equation}\label{A_wedge_A_decomp} [A\wedge A,G]\cong[(\bigvee^{p-1}_{i=1}A_i)\wedge(\bigvee^{p-1}_{j=1}A_j), G]\cong\prod^{p-1}_{i,j=1}[A_i\wedge A_j, G]\cong\prod^{p-1}_{i,j,k=1}[A_i\wedge A_j, B_k], \end{equation} the order of $\sm{\imath,\imath}$ cannot exceed the least common multiple of the orders of $[A_i\wedge A_j, B_k]$ for all $i,j$ and $k$. Let $C_{2n_i-1}$ be the cofiber of the generator $\alpha_{2n_i-1}$ of the homotopy group~$\pi_{2n_i+2p-4}(S^{2n_i-1})$. When $G$ is quasi-$p$-regular, each $A_i$ is a sphere $S^{2n_i-1}$ or $C_{2n_i-1}$, so~$A_i\wedge A_j$ is either $S^{2n_i+2n_j-2}$, $C_{2n_i+2n_j-2}$ or $C_{2i-1}\wedge C_{2n_j-1}$. In the following we consider the orders of $[A_i\wedge A_j,B_k]$ case by case.
If $A_i\wedge A_j$ is $S^{2n_i+2n_j-2}$, then $[A_i\wedge A_j, B_k]$ is $\pi_{2n_i+2n_j-2}(B_k)$. The homotopy groups of $B_k$ are known in a range.
\begin{thm}[Toda \cite{toda62}, Mimura and Toda \cite{MT70}, Kishimoto \cite{kishimoto09}]\label{toda} Localized at $p$, we have \[ \pi_{2n-1+k}(S^{2n-1})\cong\begin{cases} \mathbb{Z}/p\mathbb{Z} &\text{for $k=2i(p-1)-1$, $1\leq i\leq p-1$}\\ \mathbb{Z}/p\mathbb{Z} &\text{for $k=2i(p-1)-2$, $n\leq i\leq p-1$}\\ 0 &\text{other cases for $1\leq k\leq 2p(p-1)-3$}, \end{cases} \] \[ \pi_{2n-1+k}(B(3, 2p+1))\cong\begin{cases} \mathbb{Z}/p\mathbb{Z} &\text{for $k=2i(p-1)-1$, $2\leq i\leq p-1$}\\ \mathbb{Z} &\text{for $k=2p-2$}\\ 0 &\text{other cases for $1\leq k\leq 2p(p-1)-3$}, \end{cases} \] and \[ \pi_{2n-1+k}(B(2n-1, 2n+2p-3))\cong\begin{cases} \mathbb{Z}/p^2\mathbb{Z} &\text{for $k=2i(p-1)-1$, $2\leq i\leq p-1$}\\ \mathbb{Z}/p\mathbb{Z} &\text{for $k=2i(p-1)-2$, $n\leq i\leq p-1$}\\ \mathbb{Z} &\text{for $k=2p-2$}\\ 0 &\text{other cases for $1\leq k\leq 2p(p-1)-3$}. \end{cases} \] \end{thm} \noindent Since $2n_i+2n_j-2$ is even, $\pi_{2n_i+2n_j-2}(B_k)$ is isomorphic to either $0$, $\mathbb{Z}/p\mathbb{Z}$ or $\mathbb{Z}/p^2\mathbb{Z}$. Therefore the order of $[A_i\wedge A_j, B_k]$ is at most $p^2$.
If $A_i\wedge A_j$ is $C_{2n_i+2n_j-2}$, then the cofibration \[ S^{2n_i+2n_j-2}\to C_{2n_i+2n_j-2}\to S^{2n_i+2n_j+2p-4} \] induces an exact sequence \begin{equation}\label{dgm_exp_SES_SC} \pi_{2n_i+2n_j+2p-4}(B_k)\to[C_{2n_i+2n_j-2}, B_k]\to\pi_{2n_i+2n_j-2}(B_k). \end{equation} Since $C_{2n_i+2n_j-2}$ is a suspension and $B_k$ is an H-space, the three groups are abelian. By Theorem~\ref{toda}, the first and the last homotopy groups have orders at most $p^2$, so the order of~$[C_{2n_i+2n_j-2},B_k]$ is at most $p^4$.
If $A_i\wedge A_j$ is $C_{2n_i-1}\wedge C_{2n_j-1}$, then it is a CW-complex with one cell of dimension~\mbox{$2n_i+2n_j-2$}, two cells of dimension $2n_i+2n_j+2p-4$ and one cell of dimension~\mbox{$2n_i+2n_j+4p-6$}. Let~$C'$ be the $(2n_i+2n_j+4p-7)$-skeleton of $C_{2n_i-1}\wedge C_{2n_j-1}$, that is, $C_{2n_i-1}\wedge C_{2n_j-1}$ minus the top cell. Then the cofibration $C'\to C_{2n_i-1}\wedge C_{2n_j-1}\to S^{2n_i+2n_j+4p-6}$ induces an exact sequence of abelian groups \[ \pi_{2n_i+2n_j+4p-6}(B_k)\longrightarrow[C_{2n_i-1}\wedge C_{2n_j-1},B_k]\longrightarrow[C',B_k]. \] According to \cite{gray98}, $C'$ is homotopy equivalent to $C_{2n_i+2n_j-2}\vee S^{2n_i+2n_j+2p-4}$, so we have \[ [C', B_k]\cong[C_{2n_i+2n_j-2}, B_k]\oplus\pi_{2n_i+2n_j+2p-4}(B_k). \] We have shown that $[C_{2n_i+2n_j-2}, B_k]$ has order at most $p^4$. By Theorem~\ref{toda}, $\pi_{2n_i+2n_j+2p-4}(B_k)$ and $\pi_{2n_i+2n_j+4p-6}(B_k)$ have orders at most $p^2$. Therefore the order of $[C_{2n_i-1}\wedge C_{2n_j-1}, B_k]$ is at most~$p^6$.
Summarizing the above discussion, we have the following proposition.
\begin{prop}\label{bound_general} Let $G$ and $p$ be in (\ref{quasi_list}). Then the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ is at most $p^6$. \end{prop}
This gives a very rough upper bound on the orders of $\sm{\mathds{1}_G,\mathds{1}_G}$. We can sharpen the range by refining our calculation according to individual cases of $G$ and $p$.
\subsubsection*{Case I: $G$ is $p$-regular} Suppose $G$ is $p$-regular. Then $B_i=A_i=S^{2n_i-1}$ and $p\geq n_l$. All summands $[A_i\wedge A_j,B_k]$ in~(\ref{A_wedge_A_decomp}) are homotopy groups $\pi_{2n_i+2n_j-2}(S^{2n_k-1})$. According to Theorem~\ref{toda}, their orders are at most $p$ since \[ 2(n_i+n_j-n_k)-1\leq2(2n_l-2)-1\leq2p(p-1)-3 \] for all $i,j$ and $k$. Therefore the order of $\sm{\imath,\imath}$ is at most $p$ and so is the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ by Theorem~\ref{main}. McGibbon \cite{mcgibbon84} showed that $G$ is homotopy commutative if and only if either~$p>2n_l$, or $(G, p)$ is $(Sp(2), 3)$ or $(G_2, 5)$. Therefore we have the following statement.
\begin{thm}\label{regular_bound} Let $G$ be a $p$-regular Lie group of type $(2n_1-1,\cdots,2n_l-1)$. Then the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ is $p$ if $n_l\leq p<2n_l$, and is 1 if $p>2n_l$. \end{thm}
\subsubsection*{Case II: $G$ is a quasi-$p$-regular $SU(n)$ and $p\geq7$} Suppose $G=SU(n)$ is quasi-$p$-regular and $p\geq7$. Then $n_i=i+1$ and $p>\frac{n}{2}$. Let~\mbox{$t=n-p+1$} and $2\leq t\leq p$. Localized at $p$, there are homotopy equivalences \[ SU(n)\simeq B(3,2p+1)\times\cdots\times B(2t-1,2n-1)\times S^{2t+1}\times\cdots\times S^{2p-1} \] and \[ A\simeq C_3\vee\cdots\vee C_{2t-1}\vee S^{2t+1}\vee\cdots\vee S^{2p-1} \] For $1\leq j\leq t$ and $t+1\leq i\leq p$, let $\epsilon_i$ and $\lambda_i$ be the compositions \[ \begin{array}{c c c} \epsilon_i:S^{2i-1}\hookrightarrow A\overset{\imath}{\to}G &\text{and} &\lambda_j:C_{2j-1}\hookrightarrow A\overset{\imath}{\to}G. \end{array} \] Kishimoto calculated some of their Samelson products in \cite{kishimoto09}.
\begin{thm}[Kishimoto \cite{kishimoto09}]\label{kishimoto} Let $G$ be a quasi-$p$-regular $SU(n)$. For $2\leq j,j'\leq t$ and~\mbox{$t+1\leq i,i'\leq p$}, \begin{enumerate} \item the order of $\sm{\epsilon_i,\epsilon_{i'}}$ is at most $p$; \item if $i\neq p$ and $j\neq t$, then the order of $\sm{\epsilon_i,\lambda_j}$ is at most $p$; \item if $j+j'\leq p$, then $\sm{\lambda_j,\lambda_{j'}}$ is null-homotopic; \item if $p+1\leq j+j'\leq 2p-1$, then $\sm{\lambda_j,\lambda_{j'}}$ can be compressed into $S^{2(j+j'-p)+1}\subset SU(n)$. \end{enumerate} \end{thm}
Using these results we can give a bound for the order of $\sm{\mathds{1}_G,\mathds{1}_G}$.
\begin{thm} For $G=SU(n)$ and $p\geq7$, let the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ be $p^r$. \begin{itemize} \item If $n>2p$, then $r=0$; \item If $n\leq p< 2p$, then $r=1$; \item If $\frac{2}{3}n+1\leq p<n$, then $1\leq r\leq2$; \item If $\frac{n}{2}<p\leq \frac{2}{3}n$ and $n\neq2p-1$, then $1\leq r\leq3$; \item If $n=2p-1$, then $1\leq r\leq6$. \end{itemize} \end{thm}
\begin{proof} When $p\geq n$, $G$ is $p$-regular and we have shown the first two statements in Theorem~\ref{regular_bound}. Assume $\frac{n}{2}<p<n$. By Theorem~\ref{main}, the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ equals the order of~$\sm{\imath,\imath}$. Since $\sm{\imath,\imath}$ is a wedge of Samelson products of $\epsilon_i$'s and $\lambda_j$'s, we need to consider the orders of $\sm{\epsilon_i,\epsilon_{i'}},\sm{\epsilon_i,\lambda_j}$ and $\sm{\lambda_j,\lambda_{j'}}$.
First, the first two statements of Theorem~\ref{kishimoto} imply that the orders of $\sm{\epsilon_i,\epsilon_{i'}}$ and $\sm{\epsilon_i,\lambda_j}$ are at most $p$ except for $\sm{\epsilon_p,\lambda_t}$. Put $n_i=p$ and $n_j=t$ in (\ref{dgm_exp_SES_SC}) to obtain the exact sequence \[ \pi_{4p+2t-4}(B_k)\to[C_{2p+2t-2}, B_k]\to\pi_{2p+2t-2}(B_k) \] where $2\leq k\leq p$. According to Theorem~\ref{toda}, the two homotopy groups are trivial except for $k=t+1$ or $t=p$ and $k=2$. In the first case, $B_k$ is $S^{2t+1}$, and $\pi_{4p+2t-4}(S^{2t+1})$ and~\mbox{$\pi_{2p+2t-2}(S^{2t+1})$} are $\mathbb{Z}/p\mathbb{Z}$. In the second case, $B_k$ is $B(3,2p+1)$, and $\pi_{6p-4}(B(3,2p+1))$ and $\pi_{4p-2}(B(3,2p+1))$ are $\mathbb{Z}/p^2\mathbb{Z}$. By exactness the order of $[C_{2p+2t-2}, B_k]$ is at most~$p^2$ for~$2\leq k\leq p$ and $n\neq 2p-1$, and consequently so is the order of $\sm{\epsilon_i,\lambda_j}$.
Second, the third statement of Theorem~\ref{kishimoto} implies that $\sm{\lambda_j,\lambda_{j'}}$ is null-homotopic for $j+j'\leq p-1$. When $p\geq\frac{2}{3}n+1$, we have \[ \begin{array}{c c c} n\leq\frac{3}{2}(p-1) &\text{and} &t=n-p+1\leq\frac{1}{2}(p-1). \end{array} \] In this case the order of $\sm{\lambda_j,\lambda_{j'}}$ is always 1 since $j+j'\leq2t\leq p-1$. When $\frac{n}{2}<p\leq\frac{2}{3}n$, we need to consider the orders of $\sm{\lambda_j,\lambda_{j'}}$ for $p+1\leq j+j'$. By the last statement of Theorem~\ref{kishimoto}, $\sm{\lambda_j,\lambda_{j'}}$ is in $[C_{2j-1}\wedge C_{2j'-1}, S^{2(j+j'-p)+1}]$ if $j+j'\leq2p-1$. Since $j,j'\leq t\leq p$, this can always be achieved for $n\neq2p-1$. There is a graph of short exact sequences \[\xymatrix{
& &\pi_{2j+2j'+2p-4}(S^{2(j+j'-p)+1})^{\oplus2}\ar[d]\\ \pi_{2j+2j'+4p-6}(S^{2(j+j'-p)+1})\ar[r] &[C_{2j-1}\wedge C_{2j'-1},S^{2(j+j'-p)+1}]\ar[r] &[C',S^{2(j+j'-p)+1}]\ar[d]\\
& &\pi_{2j+2j'-2}(S^{2(j+j'-p)+1}) }\] where $C'$ is the subcomplex of $C_{2j-1}\wedge C_{2j'-1}$ without the top cell. By Theorem~\ref{toda}, the three homotopy groups are $\mathbb{Z}/p\mathbb{Z}$. The exactness of the column and the row implies that the orders of $[C',S^{2(j+j'-p)+1}]$ and $[C_{2j-1}\wedge C_{2j'-1},S^{2(j+j'-p)+1}]$ are at most $p^2$ and $p^3$. Therefore~$\sm{\lambda_j,\lambda_{j'}}$ has order at most $p^3$ when $n\neq2p-1$ and $\frac{n}{2}<p\leq \frac{2}{3}n$.
We summarize the above discussion in the following table: \begin{center}
\begin{tabular}{|>{\centering\arraybackslash}p{3.5cm}|>{\centering\arraybackslash}p{2cm}|>{\centering\arraybackslash}p{2cm}|>{\centering\arraybackslash}p{2cm}|} \hline
&\multicolumn{3}{ |c| }{an upper bound on the order of}\\ \cline{2-4}
&$\sm{\epsilon_i,\epsilon_{i'}}$ &$\sm{\epsilon_i,\lambda_j}$ &$\sm{\lambda_j,\lambda_{j'}}$\\ \hline $\frac{2}{3}n+1\leq p\leq n$ &$p$ &$p^2$ &$1$\\ \hline $\begin{array}{c} \frac{n}{2}<p\leq\frac{2}{3}n,\\ n\neq2p-1 \end{array}$
&$p$ &$p^2$ &$p^3$\\ \hline $n=2p-1$ &$p$ &$p^2$ &$p^6$\\ \hline \end{tabular} \end{center} By Theorem~\ref{main}, the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ equals the order of $\sm{\imath,\imath}$ which is the least common multiple of the orders of $\sm{\epsilon_i,\epsilon_{i'}},\sm{\epsilon_i,\lambda_j}$ and $\sm{\lambda_j,\lambda_{j'}}$, so the statement follows. \end{proof}
\subsubsection*{Case III: $G$ is a quasi-$p$-regular exceptional Lie group and $p\geq7$} Suppose $p\geq7$ and $G$ is a quasi-$p$-regular exceptional Lie group. That is \begin{itemize} \item when $G=F_4$ or $E_6$, $p=7$ or $11$; \item when $G=E_7$, $p=11,13$ or $17$; \item when $G=E_8$, $p=11,13,17, 23$ or $29$. \end{itemize} For each case, we can calculate bounds on the orders of $[A_i\wedge A_j,B_k]$ for all $i,j$ and $k$ in~(\ref{A_wedge_A_decomp}) according to the CW-structure of $A$. Then we obtain the following statement.
\begin{thm}\label{thm_bound_quasi_except} For $p\geq7$, suppose $G$ is a quasi-$p$-regular exceptional Lie group which is not~$p$-regular. Let the order of $\sm{\mathds{1}_G,\mathds{1}_G}$ be $p^r$. Then we have the following table \[
\begin{array}{|c|c|c|} \hline G &p &\text{value(s) of }r\\ \hline F_4 &7 &1\leq r\leq4\\
&11 &1\\ \hline E_6 &7 &1\leq r\leq4\\
&11 &1\\ \hline E_7 &11 &1\leq r\leq3\\
&13 &1\text{ or }2\\
&17 &1\\ \hline E_8 &11 &1\leq r\leq6\\
&13 &1\leq r\leq4\\
&17 &1\leq r\leq3\\
&19 &1\leq r\leq4\\
&23,29 &1\\ \hline \end{array} \] \end{thm}
\begin{remark} It would be interesting if the precise order of $\sm{\mathds{1}_G,\mathds{1}_G}$ could be obtained in the case of Theorem~\ref{thm_bound_quasi_except}. \end{remark}
\end{document} | arXiv | {
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\begin{document}
\title{Wiener Graph Deconvolutional Network Improves Graph Self-Supervised Learning}
\begin{abstract} Graph self-supervised learning (SSL) has been vastly employed to learn representations from unlabeled graphs. Existing methods can be roughly divided into predictive learning and contrastive learning, where the latter one attracts more research attention with better empirical performance. We argue that, however, predictive models weaponed with powerful decoder could achieve comparable or even better representation power than contrastive models. In this work, we propose a Wiener Graph Deconvolutional Network (WGDN), an augmentation-adaptive decoder empowered by graph wiener filter to perform information reconstruction. Theoretical analysis proves the superior reconstruction ability of graph wiener filter. Extensive experimental results on various datasets demonstrate the effectiveness of our approach. \end{abstract}
\section{Introduction}\label{sec:intro}
Self-Supervised Learning (SSL), which extracts informative knowledge through well-designed pretext tasks from unlabeled data, has been extended to graph data recently due to its great success in computer vision (CV)~\cite{he2020moco} and natural language processing (NLP)~\cite{devlin2018bert}. With regard to the objectives of pretext tasks, graph SSL can be divided into two major categories: predictive SSL and contrastive SSL~\cite{liu2021graph}. Predictive models learn informative properties generated from graph freely via prediction tasks, while contrastive models are trained on the mutual information between different views augmented from the original graph. As the dominant technique, contrastive SSL has achieved state-of-the-art performance empirically~\cite{ xu2021infogcl, thakoor2022bgrl, lee2022afgrl} for graph representation learning. In contrast, the development of predictive SSL has lagged behind over the past few years.
Graph reconstruction is a natural self-supervision, and thus most methods in predictive SSL employ graph autoencoder (GAE) as their backbones~\cite{wang2017mgae, hu2019strategies,li2020graph}. The work of GraphMAE~\cite{hou2022graphmae} re-validates the potentials of reconstruction paradigm. Despite recent advancements, \textbf{the importance of graph decoder has been largely ignored}. Most existing works leverage trivial decoders, such as multi-layer perceptron (MLP)~\cite{kipf2016vgae, pan2018arvga, you2020does}, which under-exploit graph topology information, and thus may lead to the degradation in learning capability. Vanilla graph neural networks (GNNs), such as GCN~\cite{kipf2017gcn}, are inappropriate for decoding due to their Laplacian-smooth essence. To overcome such inherent limitation of GCN, GALA~\cite{park2019gala} adopts spectral counterpart of GCN to facilitate the learning, but may take the risk of unstable learning due to its poor resilience to data augmentation (See Figure~\ref{fig:stable_train}). GAT~\cite{velickovic2018gat} is employed as decoder in recent works including GATE~\cite{salehi2020gate} and GraphMAE~\cite{hou2022graphmae}. Although attention mechanism enhances model flexibility, recent work~\cite{balcilar2021analyzing} shows GAT acts like a low-pass filter and cannot well reconstruct the graph spectrum. As an inverse to GCN~\cite{kipf2017gcn}, \textbf{graph deconvolutional network (GDN) could be expected to further boost the performance of reconstruction}~\cite{li2021deconvolutional}, which may substantially benefit the context of representation learning. We present a summary of different decoders of predictive graph SSL in Table~\ref{tab:ssl_comp}. Given the aforementioned observations, a natural question comes up, that is, \textit{can we improve predictive SSL by a framework with powerful decoder?}
\begin{figure}
\caption{Comparison of different variants of GALA against latent Gaussian augmentation with magnitude $\beta$. }
\label{fig:stable_train}
\end{figure}
\begin{table*}[t]
\centering
\begin{tabular}{c|ccccccc}
\toprule
\multirow{2}{*}{Model} & \multirow{2}{*}{Decoder} & Feature & Structure & Deconv. & Augmentation & Spectral & \multirow{2}{*}{Space} \\
& & Loss & Loss & Decoder & Adaption & Kernel & \\
\midrule
VGAE~\cite{kipf2016vgae} & DP & - & CE & \ding{55} & \ding{55} & \ding{55} & $\mathcal{O}(N^2)$ \\
ARVGA~\cite{pan2018arvga} & DP & - & CE & \ding{55} & \ding{55} & \ding{55} & $\mathcal{O}(N^2)$ \\
MGAE~\cite{wang2017mgae} & MLP & MSE & - & \ding{55} & \ding{55} & \ding{55} & $\mathcal{O}(N)$ \\
AttrMask~\cite{hu2019strategies} & MLP & CE & - & \ding{55} & \ding{55} & \ding{55} & $\mathcal{O}(N)$ \\
GALA~\cite{park2019gala} & GNN & MSE & - & \ding{51} & \ding{55} & \ding{55} & $\mathcal{O}(N)$ \\
GraphMAE~\cite{hou2022graphmae} & GNN & SCE & - & \ding{55} & \ding{51} & \ding{55} & $\mathcal{O}(N)$ \\
\midrule
WGDN & GNN & MSE & - & \ding{51} & \ding{51} & \ding{51} & $\mathcal{O}(N)$ \\
\bottomrule
\end{tabular}
\caption{Technical components comparison within predictive SSL approaches. \textit{DP}: Non-parametric Dot Product. \textit{CE}: Cross-Entropy Error. \textit{MSE}: Mean Square Error. \textit{SCE}: Scaled-Cosine Error.}
\label{tab:ssl_comp} \end{table*}
Typically, a powerful decoder should at least remain effective against augmentations. Motivated by recent advancement of wiener in deep image reconstruction~\cite{dong2020deep}, we introduce the classical deconvolutional technique, wiener filter, into GDN, which is the theoretical optimum for restoring augmented signals with respect to mean square error (MSE). We propose a GAE framework~\cite{li2020graph}, named Wiener Graph Deconvolutional Network (WGDN), which utilizes graph wiener filter to facilitate representation learning with graph spectral kernels. We first derive the graph wiener filter and prove its superiority in theory. We observe that, however, directly using the explicit graph wiener filter induces low scalability due to indispensable eigen-decomposition and may not be applicable to large-scale datasets. Therefore, we adopt average graph spectral energy and Remez polynomial~\cite{pachon2009remez} for fast approximation.
We evaluate the learned representation quality on two downstream tasks: node classification and graph classification. Empirically, our proposed WGDN achieves better results over a wide range of state-of-the-art benchmarks of graph SSL with efficient computational cost. Particularly, WGDN yields up to 1.4\% higher accuracy than runner-up model, and requires around 30\% less memory overhead against the most efficient contrastive counterpart.
\section{Related Work}\label{sec:related}
\paragraph{Graph self-supervised learning.} According to recent surveys \cite{liu2021graph, xie2022self}, works in graph SSL can be classified into two categories: contrastive learning and predictive learning. Contrastive SSL attracts more attention currently due to the state-of-the-art performance on representation learning. Early efforts focus on the design of negative sampling and augmentation schemes, such as corruptions in DGI~\cite{velivckovic2019deep}, graph diffusion in MVGRL~\cite{hassani2020contrastive} and masking in GRACE~\cite{zhu2020grace} and GCA~\cite{zhu2021gca}. Recent works have attempted for negative-sample-free contrastive SSL. For example, BGRL~\cite{thakoor2022bgrl} adapts BYOL~\cite{grill2020bootstrap} for graph representation learning, CCA-SSG~\cite{zhang2021ccassg} conducts feature decorrelation, and AFGRL~\cite{lee2022afgrl} obtains positive pairs via latent space clustering. Despite their advancement, intricate architecture designs are required.
As for predictive learning, predicting node features and neighborhood context is a traditional pretext task with graph autoencoder (GAE). For instance, VGAE~\cite{kipf2016vgae} and ARVGA~\cite{pan2018arvga} learn missing edges prediction by structural reconstruction. Moreover, one representative manner~\cite{you2020does} follows the perturb-then-learn strategy to predict the corrupted information, such as attribute masking ~\cite{hu2019strategies} and feature corruption ~\cite{wang2017mgae}. Recently, GraphMAE~\cite{hou2022graphmae} implements a masking strategy and scaled cosine error for feature reconstruction and achieves great success to match state-of-the-art contrastive SSL approaches. However, it ignores the potential benefit leveraging graph spectral theory. In this work, we propose an augmentation-adaptive GAE framework that unleashes the power of graph spectral propagation.
\paragraph{Graph deconvolutional network.} Regarding graph deconvolution, early research \cite{yang2018enhancing} formulates the deconvolution as a pre-processing step. GALA~\cite{park2019gala} performs Laplacian sharpening to recover information. Recent work~\cite{zhang2020graph} employs GCN~\cite{kipf2017gcn} to reconstruct node features from the latent representations. All these works, however, neglect the influence of augmentation. Another GDN framework~\cite{li2021deconvolutional} is designed via a combination of inverse filters in spectral domain and denoising layers in wavelet domain, which is sub-optimal regarding signal reconstruction. Wiener filtering, as an alternative, executes an optimal trade-off between signal recovering and denoising. It has been introduced to deconvolutional networks~\cite{dong2020deep, son2017fast} for image deblurring. However, its effectiveness on graph structure has not been well investigated yet.
\begin{figure*}
\caption{The autoencoder framework of WGDN for graph SSL. Given the augmented latent representations, graph wiener filter is approximated via estimating spectral energy and augmentations adaptively. With such, WGDN permits the stable feature reconstruction from the augmented latent space for representation learning.}
\label{fig:framework}
\end{figure*}
\section{Preliminaries}\label{sec:prob}
Under a generic self-supervised graph representation learning setup, we are given an attributed graph $\mathcal{G} = (\mathcal{V}, \mathbf{A}, \mathbf{X})$ consisting of: (1) $\mathcal{V} = \{ v_{1}, v_{2}, ..., v_{N} \}$ is the set of nodes; (2) $\mathbf{A} \in \mathbb{R}^{N \times N}$ is the adjacency matrix where $\mathbf{A}_{ij} \in \{0, 1\}$ represents whether an undirected edge exists between $v_i$ and $v_j$; and (3) $\mathbf{X} \in \mathbb{R}^{N \times D}$ denotes the feature matrix. Our objective is to learn an autoencoder with encoder $\mathcal{E}: (\mathbb{R}^{N \times N} , \mathbb{R}^{N \times D}) \mapsto \mathbb{R}^{N \times D'}$ and decoder $\mathcal{D}: \mathbb{R}^{N \times D'} \mapsto \mathbb{R}^{N \times D}$ to produce node embedding, or graph embedding upon a pooling function. $\mathbf{H} = \mathcal{E}(\mathbf{A}, \mathbf{X}) \in \mathbb{R}^{N \times D'}$ represents the learned embedding in low dimensional space, which can be used for various downstream tasks.
\paragraph{Graph convolution.} Convolutional operation in graph can be interpreted as a special form of Laplacian smoothing on nodes. From the spectral perspective, graph convolution on a signal $\mathbf{x} \in \mathbb{R}^{N}$ with a filter $g_c$ is defined as \begin{equation} \label{eqn:conv} \begin{aligned}
\mathbf{h} = g_c \ast \mathbf{x} & = \mathbf{U} \text{diag}(g_c(\lambda_1), ..., g_c(\lambda_N)) \mathbf{U}^{T} \mathbf{x} \\
& = \mathbf{U} g_c(\mathbf{\Lambda}) \mathbf{U}^{T} \mathbf{x} = g_c(\mathbf{L}) \mathbf{x}, \end{aligned} \end{equation} where $\{ \lambda_i \}_{i=1}^{N}$ and $\mathbf{U}$ represent the eigenvalues and eigenvectors of normalized Laplacian matrix $\mathbf{L} = \mathbf{I} - \mathbf{D}^{-\frac{1}{2}} \mathbf{A} \mathbf{D}^{-\frac{1}{2}} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^{T}$ respectively. $\mathbf{D}$ denotes the Degree matrix. $\ast$ denotes convolutional operator. We consider (1) GCN~\cite{kipf2017gcn}, it is a low-pass filter in spectral domain with $g_c(\lambda_i) = 1 - \lambda_i$ shown by~\cite{simgcn}; (2) GDC~\cite{klicpera2019gdc} and Heatts~\cite{li2020dirichlet}, both use heat kernel $g_c(\lambda_i) = e^{- t\lambda_i}$; (3) APPNP~\cite{gasteiger2018appnp}, it leverages personalized pagerank (PPR) kernel $g_c(\lambda_i) = \frac{\alpha}{1 - (1 - \alpha)(1 - \lambda_i)}$.
\paragraph{Graph deconvolution.} As an inverse to convolution, graph deconvolution aims to recover the input attributes given the smoothed node representation. From the spectral perspective, graph deconvolution on a smoothed representation $\mathbf{h} \in \mathbb{R}^{N}$ with filter $g_d$ is defined as \begin{equation} \label{eqn:deconv} \begin{aligned}
\hat{\mathbf{x}} = g_d \ast \mathbf{h} & = \mathbf{U} \text{diag}(g_d(\lambda_1), ..., g_d(\lambda_N)) \mathbf{U}^{T} h \\
& = \mathbf{U} g_d(\mathbf{\Lambda}) \mathbf{U}^{T} \mathbf{h} = g_d(\mathbf{L}) \mathbf{h}. \end{aligned} \end{equation}
A trivial selection of $g_d$ is the inverse function of $g_c$, e.g., $g_d(\lambda_i) =\frac{1}{1 - \lambda_i}$ for GCN \cite{li2021deconvolutional}, $g_d(\lambda_i) = e^{t\lambda_i}$ for heat kernel, or $g_d(\lambda_i) = \frac{1 - (1 - \alpha)(1 - \lambda_i)}{\alpha}$ for PPR kernel.
\section{The proposed framework}\label{sec:method}
In this section, we first extend classical wiener filter to graph domain and demonstrate its superiority in reconstructing graph features. Then, we propose Wiener Graph Deconvolutional Network (WGDN), an efficient and augmentation-adaptive framework empowered by graph wiener filter.
\subsection{Wiener filter on graph} \label{sec42}
In this work, we follow the settings in previous papers~\cite{jin2019latent, cheung2021modals} and introduce additive latent augmentations in model training due to its flexible statistical characteristics, such as unbiasedness and covariance-preserving~\cite{zhang2022costa}. Combining with the graph convolution in Eq.~\ref{eqn:conv}, augmented representation $\hat{\mathbf{h}}$ in graph is similarly defined as \begin{equation} \label{eqn:noisegraph}
\hat{\mathbf{h}} = \mathbf{U} g_c(\mathbf{\Lambda}) \mathbf{U}^{T} \mathbf{x} + \mathbf{\epsilon}, \end{equation} where $\mathbf{x} \in \mathbb{R}^{N}$ denotes input features and $\mathbf{\epsilon} \in \mathbb{R}^{N}$ is assumed to be any \textit{i.i.d.} random augmentation with $\mathbb{E}[\mathbf{\epsilon}_{i}] = 0$ and $\mathbb{VAR}[\mathbf{\epsilon}_{i}] = \sigma^2$. In contrast to the isolated data augmentations in graph topology and features, $\epsilon$ indirectly represents joint augmentations to both~\cite{jin2019latent}. Naturally, feature recovered by graph deconvolution is formulated by \begin{equation} \label{eqn:recongraph}
\hat{\mathbf{x}} = \mathbf{U} g_d(\mathbf{\Lambda}) g_c(\mathbf{\Lambda}) \mathbf{U}^{T} \mathbf{x} + \mathbf{U} g_d(\mathbf{\Lambda}) \mathbf{U}^T \mathbf{\epsilon}. \end{equation} \begin{prop} \label{prop:1}
Let $\hat{\mathbf{x}}_{inv}$ be recovered features by inverse filter $g_d(\lambda_i) = g_c^{-1}(\lambda_i)$.
For common low-pass filters satisfying $g_c: [0, 2] \mapsto [-1, 1]$, such as GCN, Heat and PPR, the reconstruction MSE is dominated by amplified augmentation $\text{MSE}(\hat{\mathbf{x}}_{inv}) = \mathbb{E} \norm{\mathbf{x} - \hat{\mathbf{x}}_{inv}}_{2}^{2} = \sum_{i=1}^{N} \frac{\sigma^2}{g_c^2(\lambda_i)}$. \end{prop}
The proof is trivial and illustrated in Appendix~\ref{prop:p1} for details. Based on Proposition~\ref{prop:1}, feature reconstruction becomes unstable and even ineffective if augmentation exists. To well utilize the power of augmentation, our goal is to stabilize the reconstruction paradigm, which resembles the classical restoration problems. In signal deconvolution, classical wiener filter~\cite{wienerbook} is able to produce a statistically optimal estimation of the real signals from the augmented ones with respect to MSE. With this regard, we are encouraged to extend wiener filter to graph domain~\cite{stationarygraph}. Assuming the augmentation to be independent from input features, graph wiener filter can be similarly defined by projecting MSE into graph spectral domain \begin{equation} \label{eqn:bvmse} \begin{aligned}
& \text{MSE}(\hat{\mathbf{x}}) \\
& = \mathbb{E} \norm{\hat{\mathbf{x}} - \mathbf{x}}_{2}^{2} = \mathbb{E} \norm{\mathbf{U}^T \hat{\mathbf{x}} - \mathbf{U}^T \mathbf{x}}_{2}^{2} \\
& = \sum_{i=1}^{N} (g_d(\lambda_i) g_c(\lambda_i) - 1)^2 \mathbb{E} [x^{\ast 2}_{i}] + g_d^2(\lambda_i) \mathbb{E}[\mathbf{\epsilon}^{\ast 2}_{i}] \\
& = \sum_{i=1}^{N} S(\lambda_i, x^{\ast}_{i}, \sigma, g_c, g_d), \end{aligned} \end{equation} where $\mathbf{x}^{\ast} = \mathbf{U}^{T} \mathbf{x} = \{ x^{\ast}_1, x^{\ast}_2, ..., x^{\ast}_N \}$ and $\mathbf{\epsilon}^{\ast} = \mathbf{U}^{T} \mathbf{\epsilon} = \{ \epsilon^{\ast}_1, \epsilon^{\ast}_2, ..., \epsilon^{\ast}_N \}$ represent graph spectral projection of the input and augmentation respectively. We denote $\mathbb{E} [x^{\ast 2}_{i}]$ and $S(\lambda_i, x^{\ast}_{i}, \sigma, g_c, g_d)$ as the spectral energy and spectral reconstruction error of spectrum $\lambda_i$. Considering the convexity of Eq.~\ref{eqn:bvmse}, MSE is minimized by setting the derivative with respect to $g_d(\lambda_i)$ to zero and thus we obtain the graph wiener filter $g_w(\lambda_i)$ as \begin{equation} \label{eqn:graphwiener}
g_w(\lambda_i) = \frac{g_c(\lambda_i)}{g_c^2(\lambda_i) + \sigma^2/ \mathbb{E} [x^{\ast 2}_{i}]}, \end{equation} where $\sigma^2 = \mathbb{VAR}[\mathbf{\epsilon}^{\ast}_{i}] = \mathbb{E}[\mathbf{\epsilon}^{\ast 2}_{i}]$ and $\sigma^2/\mathbb{E} [x^{\ast 2}_{i}]$ is denoted as the Augmentation-to-Energy Ratio (AER) of particular spectrum $\lambda_i$, which represents the relative magnitude of augmentation.
\begin{prop}\label{prop:2}
Let $\hat{\mathbf{x}}_w$ be recovered features by $g_w(\lambda_i)$, where $g_w(\lambda_i)$ is a graph wiener filter, then the reconstruction MSE and variance of $\hat{\mathbf{x}}_w$ are less than $\hat{\mathbf{x}}_{inv}$. \end{prop}
Please refer to Appendix~\ref{prop:p2} for details. Proposition~\ref{prop:2} shows graph wiener filter has better reconstruction property than inverse filter, which promotes the resilience to latent augmentations and permits stable model training. We observe that, in Eq.~\ref{eqn:deconv} and~\ref{eqn:graphwiener}, eigen-decomposition is indispensable in computations of spectral energy and deconvolutional filter. However, in terms of scalability, an important issue for large-scale graphs is to avoid eigen-decomposition. Note that $\sum_{i=1}^{N} \mathbb{E} [x^{\ast 2}_{i}] = \sum_{i=1}^{N} \mathbb{E} [x^{2}_{i}]$ due to orthogonal transformation, we propose the modified graph wiener filter $\bar{g}_{w, \gamma}$ with average spectral energy $\bar{x}^{\ast 2}_{\gamma} = \gamma \cdot \bar{x}^{\ast 2} = \gamma \cdot \frac{1}{N} \sum_{i=1}^N \mathbb{E} [x^{\ast 2}_{i}]$ as \begin{equation} \label{graphwienerm}
\bar{g}_{w, \gamma}(\lambda_i) = \frac{g_c(\lambda_i)}{g_c^2(\lambda_i) + \sigma^2/\bar{x}^{\ast 2}_{\gamma}}, \end{equation} where $\gamma$ is a hyper-parameter to adjust AER. As a natural extension of Proposition~\ref{prop:2}, $\bar{g}_{w, \gamma}$ owns the following proposition.
\begin{prop} \label{prop:3}
Let $\hat{\mathbf{x}}_{w, \gamma}$ be the recovered features by modified graph wiener filter $\bar{g}_{w, \gamma} (\lambda_i)$, then the variance of $\hat{\mathbf{x}}_{w, \gamma}$ is less than $\hat{\mathbf{x}}_{inv}$. In spectral domain, given two different $\gamma_1$, $\gamma_2$ such that $\mathbb{E} [x^{\ast 2}_{i}] \leq \bar{x}^{\ast 2}_{\gamma_1} \leq \bar{x}^{\ast 2}_{\gamma_2}$, the spectral reconstruction error $S(\lambda_i, x^{\ast}_i, \sigma, g_c, \bar{g}_{w, \gamma_1}) \leq S(\lambda_i, x^{\ast}_i, \sigma, g_c, \bar{g}_{w, \gamma_2}) \leq S(\lambda_i, x^{\ast}_i, \sigma, g_c, g_c^{-1})$. \end{prop}
Please refer to Appendix~\ref{prop:p3} for details. Proposition~\ref{prop:3} demonstrates that $\bar{g}_{w, \gamma}$ attends to spectral reconstructions over different ranges of spectra, depending on the selection of $\gamma$. The graph wiener kernel $\mathbf{D}_{\gamma} = \mathbf{U} \bar{g}_{w, \gamma}(\mathbf{\Lambda}) \mathbf{U}^{T}$ can also be reformatted as matrix multiplication \begin{equation} \label{eqn:wienermatrix}
\mathbf{D}_{\gamma} = \mathbf{U} (g_c^2(\mathbf{\Lambda}) + \frac{\sigma^2}{\bar{x}^{\ast 2}_{\gamma}} \mathbf{I})^{-1} g_c(\mathbf{\Lambda}) \mathbf{U}^T. \end{equation}
Note that $g_c$ can be arbitrary function and support of $\lambda_i$ is restricted to [0, 2], we adopt Remez polynomial~\cite{pachon2009remez} to approximate $\bar{g}_{w, \gamma}(\lambda_i)$, which mitigates the need of eigen-decomposition and matrix inversion in Eq.~\ref{eqn:wienermatrix}.
\begin{defi}[\textit{\textbf{Remez Polynomial Approximation}}] \label{defi:1}
Given an arbitrary continuous function $\zeta(t)$ on $t \in [a, b]$, the Remez polynomial approximation for $\zeta(t)$ is defined as
\begin{equation}
\begin{aligned}
p_{K}(t) \coloneqq \sum_{k=0}^{K} c_{k} t^{k},
\end{aligned}
\end{equation}
where coefficients $c_{0}, \dots, c_{K}$ and leveled error $e$ are obtained by resolving linear system
\begin{equation} \label{eqn:remezpoints}
\begin{aligned}
\zeta(t_{j}) = p_{K}(t_{j}) + (-1)^{j} e,
\end{aligned}
\end{equation}
where $\{ t_{j} \}_{j=0}^{K+1}$ are interpolation points within [a, b]. \end{defi}
\begin{lemma}
If interpolation points $\{ t_{j} \}_{j=0}^{K+1}$ are Chebyshev nodes, the interpolation error $|\zeta(t) - p_{K}(t)|$ of Remez polynomial $p_{K}(t)$ is minimized. \end{lemma}
The proof is trivial and illustrated in detail as Corollary 8.11 in~\cite{burden2011numerical}. Following Definition~\ref{defi:1}, the $K^{\text{th}}$ order Remez approximation of $\mathbf{D}_{\gamma}$ is formulated as \begin{equation} \label{eqn:wienerremez} \begin{aligned}
\mathbf{D}_{\gamma} & = \mathbf{U} p_{K}(\mathbf{\Lambda}) \mathbf{U}^{T} = \sum_{k=0}^{K} c_{k, \gamma} \mathbf{L}^{k}, \end{aligned} \end{equation} where $\mathbf{D}_{\gamma}$ is approximated adaptively in each epoch.
\subsection{Wiener graph deconvolutional network} \label{sec44}
\paragraph{Graph encoder.} To incorporate both graph features $\mathbf{X}$ and structure $\mathbf{A}$ in a unified framework, we employ $M$ layers of graph convolution neural network as our graph encoder. For $m = 0, ..., M - 1$, \begin{equation} \label{eqn:encoder}
\mathbf{H}^{(m+1)} = \phi(g_c(\mathbf{L}) \mathbf{H}^{(m)} \mathbf{W}^{(m)}), \end{equation} where $\mathbf{H}^{(0)} = \mathbf{X}$, $\phi$ is the activation function such as PReLU and $g_c(\lambda_i) = 1 - \lambda_i$ as GCN~\cite{kipf2017gcn}, $g_c(\lambda_i) = e^{- t\lambda_i}$ as heat kernel or $g_c(\lambda_i) = \frac{\alpha}{1 - (1 - \alpha)(1 - \lambda_i)}$ as PPR kernel.
\paragraph{Representation augmentation.} For simplicity, Gaussian noise is employed as latent augmentations to the node embedding generated by the last layer encoder \begin{equation} \label{eqn:latent_aug}
\mathbf{\hat{H}}^{(M)} = \mathbf{H}^{(M)} + \beta \mathbf{E}, \end{equation} where $\mathbf{E} = \{ \mathbf{\epsilon}_{1}, ..., \mathbf{\epsilon}_{N}\}$, $\mathbf{\epsilon_{i}} \sim N(\mathbf{0}, \sigma^2_{P} \mathbf{I})$, $\sigma^2_{P} = \mathbb{VAR}[\mathbf{H}^{(M)}]$ and $\beta$ is a hyper-parameter to adjust the magnitude of augmentations.
\paragraph{Graph wiener decoder.} The decoder aims to recover original features given the augmented representation $\mathbf{\hat{H}}$. Our previous analysis demonstrates the superiority of wiener kernel to permit reconstruction-based representation learning with augmented latent space. Considering the properties of spectral reconstruction error from Proposition~\ref{prop:3}, we symmetrically adopt $M$ layers of graph deconvolution as the decoder, where each layer consists of $q$ channels of graph wiener kernels. For $m = 1, ..., M$ and $i = 1, ..., q$, \begin{equation} \label{eqn:wienerdecoder} \begin{aligned}
\mathbf{Z}^{(m-1)}_{i} & = \phi(\mathbf{D}_{\gamma_i}^{(m)} \mathbf{\hat{H}}^{(m)} \mathbf{W}^{(m)}_{i}), \\
\mathbf{\hat{H}}^{(m-1)} & = \text{AGG}([\mathbf{Z}^{(m-1)}_{1}, ..., \mathbf{Z}^{(m-1)}_{q}]), \end{aligned} \end{equation} where $\mathbf{\hat{X}} = \mathbf{\hat{H}}^{(0)}$ and AGG($\cdot$) is aggregation function such as summation. Note that the actual value of $\bar{x}^{\ast 2}$ and $\sigma^2$ of $\mathbf{D}_{\gamma_i}^{(m)}$ are unknown, we estimate $\bar{x}^{\ast 2}$ following its definition and leverage neighboring information for $\sigma^2$ estimation. Further details are presented in Appendix~\ref{model:detail_arch}.
\paragraph{Optimization and inference.} Our model is optimized following the convention of reconstruction-based SSL, which is simply summarized as \begin{equation}
\mathcal{L} = ||\mathbf{X} - \hat{\mathbf{X}}||_{F}. \end{equation} For downstream applications, we treat the fully trained $\mathbf{H}^{(M)}$ as the final node embedding. For graph-level tasks, we adopt a non-parametric graph pooling (readout) function $\mathcal{R}$, e.g. MaxPooling, to generate graph representation $\mathbf{h}_{g} = \mathcal{R}(\mathbf{H}^{(M)})$.
\paragraph{Complexity analysis.} The most intensive computational cost of our proposed method is kernel approximation in Eq.~\ref{eqn:wienerremez}. Note that kernel approximation is a simple $K^{\text{th}}$ order polynomial of graph convolution. By sparse-dense matrix multiplication, graph convolution can be efficiently implemented, which take $O(K|E|)$~\cite{kipf2017gcn} for a graph with $|E|$ edges.
\section{Experiments}\label{sec:exper}
In this section, we investigate the benefit of our proposed approach by addressing the following questions:
\textbf{Q1.} Does WGDN outperform self-supervised and semi-supervised counterparts?
\textbf{Q2.} Do the key components of WGDN contribute to representation learning?
\textbf{Q3.} Can WGDN be more efficient than competitive baselines?
\textbf{Q4.} How do the hyper-parameters impact the performance of our proposed model?
\begin{table*}[ht]
\centering
\begin{tabular}{c|c|ccccc}
\toprule
& Model & PubMed & Computers & Photo & CS & Physics \\
\midrule
\multirow{13}{*}{Self-supervised} & Node2Vec & 66.6 $\pm$ 0.9 & 84.39 $\pm$ 0.08 & 89.67 $\pm$ 0.12 & 85.08 $\pm$ 0.03 & 91.19 $\pm$ 0.04 \\
& DeepWalk + Feat. & 74.3 $\pm$ 0.9 & 86.28 $\pm$ 0.07 & 90.05 $\pm$ 0.08 & 87.70 $\pm$ 0.04 & 94.90 $\pm$ 0.09 \\
\cmidrule{2-7}
& GAE & 72.1 $\pm$ 0.5 & 85.27 $\pm$ 0.19 & 91.62 $\pm$ 0.13 & 90.01 $\pm$ 0.71 & 94.92 $\pm$ 0.07 \\
& GALA & 75.9 $\pm$ 0.4 & 87.61 $\pm$ 0.06 & 91.27 $\pm$ 0.12 & 92.48 $\pm$ 0.07 & 95.23 $\pm$ 0.04 \\
& GDN & 76.4 $\pm$ 0.2 & 87.67 $\pm$ 0.17 & 92.84 $\pm$ 0.07 & 92.93 $\pm$ 0.18 & 95.22 $\pm$ 0.05 \\
\cmidrule{2-7}
& DGI & 76.8 $\pm$ 0.6 & 83.95 $\pm$ 0.47 & 91.61 $\pm$ 0.22 & 92.15 $\pm$ 0.63 & 94.51 $\pm$ 0.52 \\
& MVGRL & 80.1 $\pm$ 0.7 & 87.52 $\pm$ 0.11 & 91.74 $\pm$ 0.07 & 92.11 $\pm$ 0.12 & 95.33 $\pm$ 0.03 \\
& GRACE & 80.5 $\pm$ 0.4 & 86.25 $\pm$ 0.25 & 92.15 $\pm$ 0.24 & 92.93 $\pm$ 0.01 & 95.26 $\pm$ 0.02 \\
& GCA & 80.2 $\pm$ 0.4 & 88.94 $\pm$ 0.15 & 92.53 $\pm$ 0.16 & 93.10 $\pm$ 0.01 & 95.73 $\pm$ 0.03 \\
& BGRL$^{\ast}$ & 79.8 $\pm$ 0.4 & \underline{89.70 $\pm$ 0.15} & 93.37 $\pm$ 0.21 & 93.51 $\pm$ 0.10 & 95.28 $\pm$ 0.06 \\
& AFGRL$^{\ast}$ & 79.9 $\pm$ 0.3 & 89.58 $\pm$ 0.45 & \underline{93.61 $\pm$ 0.20} & \underline{93.56 $\pm$ 0.15} & \underline{95.74 $\pm$ 0.10} \\
& CCA-SSG$^{\ast}$ & \underline{81.0 $\pm$ 0.3} & 88.15 $\pm$ 0.35 & 93.25 $\pm$ 0.21 & 93.31 $\pm$ 0.16 & 95.59 $\pm$ 0.07 \\
\cmidrule{2-7}
& WGDN & \textbf{81.9 $\pm$ 0.4} & \textbf{89.72 $\pm$ 0.48} & \textbf{93.89 $\pm$ 0.31} & \textbf{93.67 $\pm$ 0.14} & \textbf{95.76 $\pm$ 0.11} \\
\midrule
\multirow{2}{*}{Supervised} & GCN & 79.1 $\pm$ 0.3 & 86.51 $\pm$ 0.54 & 92.42 $\pm$ 0.22 & 93.03 $\pm$ 0.31 & 95.65 $\pm$ 0.16 \\
& GAT & 79.0 $\pm$ 0.3 & 86.93 $\pm$ 0.29 & 92.56 $\pm$ 0.35 & 92.31 $\pm$ 0.24 & 95.47 $\pm$ 0.15 \\
\bottomrule
\end{tabular}
\caption{Node classification accuracy of all compared methods. The best and runner up models in self-supervised learning are highlighted in boldface and underlined.}
\label{tab:exp_node}
\small
\begin{tabular}{c|c|cccccc}
\toprule
& Model & IMDB-B & IMDB-M & PROTEINS & COLLAB & DD & NCI1 \\
\midrule
\multirow{11}{*}{Self-supervised} & WL & 72.30 $\pm$ 3.44 & 46.95 $\pm$ 0.46 & 72.92 $\pm$ 0.56 & 79.02 $\pm$ 1.77 & 79.43 $\pm$ 0.55 & 80.01 $\pm$ 0.50 \\
& DGK & 66.96 $\pm$ 0.56 & 44.55 $\pm$ 0.52 & 73.30 $\pm$ 0.82 & 73.09 $\pm$ 0.25 & - & 80.31 $\pm$ 0.46 \\
\cmidrule{2-8}
& Graph2Vec & 71.10 $\pm$ 0.54 & 50.44 $\pm$ 0.87 & 73.30 $\pm$ 2.05 & - & - & 73.22 $\pm$ 1.81 \\
& MVGRL & 74.20 $\pm$ 0.70 & 51.20 $\pm$ 0.50 & - & - & - & - \\
& InfoGraph & 73.03 $\pm$ 0.87 & 49.69 $\pm$ 0.53 & 74.44 $\pm$ 0.31 & 70.65 $\pm$ 1.13 & 72.85 $\pm$ 1.78 & 76.20 $\pm$ 1.06 \\
& GraphCL & 71.14 $\pm$ 0.44 & 48.58 $\pm$ 0.67 & 74.39 $\pm$ 0.45 & 71.36 $\pm$ 1.15 & 78.62 $\pm$ 0.40 & 77.87 $\pm$ 0.41 \\
& JOAO & 70.21 $\pm$ 3.08 & 49.20 $\pm$ 0.77 & 74.55 $\pm$ 0.41 & 69.50 $\pm$ 0.36 & 77.32 $\pm$ 0.54 & 78.07 $\pm$ 0.47 \\
& SimGRACE & 71.30 $\pm$ 0.77 & - & \underline{75.35 $\pm$ 0.09} & 71.72 $\pm$ 0.82 & 77.44 $\pm$ 1.11 & 79.12 $\pm$ 0.44 \\
& InfoGCL & 75.10 $\pm$ 0.90 & 51.40 $\pm$ 0.80 & - & 80.00 $\pm$ 1.30 & - & 80.20 $\pm$ 0.60 \\
& GraphMAE & \underline{75.52 $\pm$ 0.66} & \underline{51.63 $\pm$ 0.52} & 75.30 $\pm$ 0.39 & \underline{80.32 $\pm$ 0.46} & \underline{78.86 $\pm$ 0.35} & \underline{80.40 $\pm$ 0.30} \\
\cmidrule{2-8}
& WGDN & \textbf{75.76 $\pm$ 0.20} & \textbf{51.77 $\pm$ 0.55} & \textbf{76.53 $\pm$ 0.38} & \textbf{81.76 $\pm$ 0.24} & \textbf{79.54 $\pm$ 0.51} & \textbf{80.70 $\pm$ 0.39} \\
\midrule
\multirow{2}{*}{Supervised} & GCN & 74.0 $\pm$ 3.4 & 51.9 $\pm$ 3.8 & 76.0 $\pm$ 3.2 & 79.0 $\pm$ 1.8 & 75.9 $\pm$ 2.5 & 80.2 $\pm$ 2.0\\
& GIN & 75.1 $\pm$ 5.1 & 52.3 $\pm$ 2.8 & 76.2 $\pm$ 2.8 & 80.2 $\pm$ 1.9 & 75.3 $\pm$ 2.9 & 82.7 $\pm$ 1.7\\
\bottomrule
\end{tabular}
\caption{Graph classification accuracy of all compared methods.}
\label{tab:exp_graph} \end{table*}
\subsection{Experimental setup} \label{sec:exper_setup}
\paragraph{Datasets.} We conduct experiments on both node-level and graph-level representation learning tasks with benchmark datasets across different scales and domains, including PubMed~\cite{sen2008collective}, Amazon Computers, Photo~\cite{shchur2018pitfalls}, Coauthor CS, Physics~\cite{shchur2018pitfalls}, and IMDB-B, IMDB-M, PROTEINS, COLLAB, DD, NCI1 from TUDataset~\cite{morris2020tudataset}. Detailed statistics are presented in Table~\ref{tab:dataset_node} and Table~\ref{tab:dataset_graph} of Appendix~\ref{exper:spec}.
\paragraph{Baselines.} We compare WGDN against representative models from the following five different categories: (1) traditional models including Node2Vec~\cite{grover2016node2vec}, Graph2Vec~\cite{narayanan2017graph2vec}, DeepWalk~\cite{perozzi2014deepwalk}, (2) graph kernel models including Weisfeiler-Lehman sub-tree kernel (WL)~\cite{shervashidze2011wl}, deep graph kernel (DGK)~\cite{yanardag2015dgk}, (3) predictive SSL models including GAE~\cite{kipf2016vgae}, GALA~\cite{park2019gala}, GDN~\cite{li2021deconvolutional}, GraphMAE~\cite{hou2022graphmae}, (4) contrastive SSL models including DGI~\cite{velivckovic2019deep}, MVGRL~\cite{hassani2020contrastive}, GRACE~\cite{zhu2020grace}, GCA~\cite{zhu2021gca}, BGRL~\cite{thakoor2022bgrl}, AFGRL~\cite{lee2022afgrl}, CCA-SSG~\cite{zhang2021ccassg}, InfoGraph~\cite{sun2019infograph}, GraphCL~\cite{you2020graphcl}, JOAO~\cite{you2021joao}, SimGRACE~\cite{xia2022simgrace}, InfoGCL~\cite{xu2021infogcl} and (5) semi-supervised models including GCN~\cite{kipf2017gcn}, GAT~\cite{velickovic2018gat} and GIN~\cite{xu2018gin}.
\paragraph{Evaluation protocol.} We closely follow the evaluation protocol in recent SSL researches. For node classification, the node embedding is fed into a logistic regression classifier~\cite{velivckovic2019deep}. We run 20 trials with different seeds and report the mean classification accuracy with standard deviation. For graph classification, we feed the graph representation into a linear SVM, and report the mean 10-fold cross-validation accuracy with standard deviation after 5 runs~\cite{xu2021infogcl}. Please refer to Appendix~\ref{exper:spec:eval} for further details.
\paragraph{Experiment settings.} We use the official implementations for all baselines in node classification and follow the suggested hyper-parameter settings, whereas graph classification results are obtained from original papers if available. For spectral filter, we consider heat kernel $g_c(\lambda_i) = e^{-t\lambda_i}$ with diffusion time $t = 1$ and PPR kernel $g_c(\lambda_i) = \frac{\alpha}{1 - (1 - \alpha)(1 - \lambda_i)}$ with teleport probability $\alpha = 0.2$. In node classification training, we use the public split for PubMed and follow 10/10/80\% random split for the rest. Further details of model configurations (e.g., hyper-parameters selection) can be found in Appendix~\ref{exper:spec:hyper}.
\subsection{Performance comparison (Q1)} \label{sec:exper_q1}
The node classification performances are reported in Table~\ref{tab:exp_node}. We find that WGDN outperforms the predictive SSL methods by a large margin over all datasets. For fair comparisons, we report the best results of recent methods using diffusion kernels (denoted with $^\ast$). WGDN performs competitively with contrastive SSL methods, achieving state-of-the-art performances across all datasets. For instance, our model WGDN is able to improve by a margin up to 0.9\% on accuracy over the most outstanding contrastive counterpart CCA-SSG on PubMed. Moreover, when compared to semi-supervised models, WGDN consistently generates better performance than both GCN and GAT.
Table~\ref{tab:exp_graph} lists the graph classification performance across various methods. We observe that our approach achieves state-of-the-art results compared to existing SSL baselines in all datasets. Besides, WGDN outperforms the best kernel methods up to a large margin. Even when compared to semi-supervised models, our model achieves the best results in 4 out of 6 datasets and the gaps for the rest are relatively minor.
In brief, our model consistently achieves comparable performance with the cutting-edge SSL and semi-supervised methods across node-level and graph-level tasks. Particularly, the significant improvements demonstrate the effectiveness of WGDN in boosting the learning capability under GAE framework.
\subsection{Effectiveness of key components (Q2)} \label{sec:exper_q2}
To validate the benefit of introducing graph wiener decoder, we conduct ablation studies on node and graph classification tasks with five datasets that exhibit distinct characteristics (e.g., citation, social and bioinformatics). For clarity, WGDN-A and WGDN-W are denoted as the models removing augmentation or substituting graph wiener decoder with inverse decoder. WGDN-AW is the plain model without both components. Specifically, heat kernel is selected as the backbone of encoder for node-level datasets, and we adopt PPR kernel for graph-level datasets.
The results are illustrated in Figure~\ref{fig:ablation}, from which we make several observations. \textbf{(1)} WGDN-W may underperform WGDN-AW. This observation validates that deterministic inverse decoder is ill-adapted to augmented latent space and may lead to degraded learning quality, which is consistent with our theoretical analysis in Section~\ref{sec42}. \textbf{(2)} Compared with WGDN-AW, WGDN-A improves model performance across all datasets, which suggests that graph wiener decoder is able to benefit representation learning even without augmentation. \textbf{(3)} The performance of WGDN is significantly higher than other counterparts. For instance, WGDN has a relative improvement up to 6\% over WGDN-AW on PubMed. It can be concluded that the graph wiener decoder allows the model to generate more semantic embedding from the augmented latent space.
\subsection{Efficiency analysis (Q3)} \label{sec:exper_q3}
To evaluate the computational efficiency, we compare the training speed and GPU overhead of WGDN against BGRL and GraphMAE on datasets of different scales, including Computers and OGBN-Arxiv~\cite{hu2020ogb}. For fair comparisons, we set the embedding size of all models as 512 and follow their suggested hyper-parameters settings. It is evident from Table~\ref{tab:exp_efficiency} that the memory requirement of WGDN is significantly reduced up to 30\% compared to BGRL, the most efficient contrastive benchmark. In addition, as WGDN is a GAE framework without computationally expensive add-on, its computational cost is shown to be comparable to GraphMAE. Considering that memory is usually the bottleneck in graph-based applications, WGDN demonstrates a practical advantage when limited resources are available.
\begin{figure}
\caption{Ablation study of graph wiener decoder. Complete model consistently boosts model performance across different datasets.}
\label{fig:ablation}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{cccc}
\toprule
Dataset & Model & Steps/Second & Memory\\
\midrule
\multirow{3}{*}{Computers} & BGRL & 17.27 & 3.01 GB \\
& GraphMAE & 19.47 & \textbf{2.03 GB} \\
& WGDN & 19.62 & 2.20 GB \\
\midrule
\multirow{3}{*}{OGBN-Arxiv} & BGRL & 2.52 & 9.74 GB \\
& GraphMAE & 3.13 & 8.01 GB \\
& WGDN & 3.16 & \textbf{7.35 GB} \\
\bottomrule
\end{tabular}
\caption{Comparison of computational efficiency on benchmark datasets.}
\label{tab:exp_efficiency} \end{table}
\begin{figure}
\caption{Downstream tasks performance versus varied augmentation magnitude $\beta$ in training.}
\label{fig:sens}
\end{figure}
\begin{table}[t]
\centering
\small
\begin{tabular}{cccc}
\toprule
Filter & GCN & Heat & PPR \\
\midrule
PubMed & 80.2 (0.019) & \textbf{81.9 (0.011)} & 81.4 (0.013) \\
Computers & 89.03 (0.417) & \textbf{89.72 (0.375)} & 89.59 (0.405) \\
CS & 92.48 (0.263) & \textbf{93.67 (0.241)} & 92.75 (0.245) \\
\midrule
IMDB-B & 75.46 (0.102) & 75.71 (0.098) & \textbf{75.76 (0.093)} \\
DD & 79.29 (0.118) & 79.36 (0.104) & \textbf{79.54 (0.074)} \\
\bottomrule
\end{tabular}
\caption{Performance and training loss of WGDN with different convolution filter $g_c$.}
\label{tab:exp_filter} \end{table}
\subsection{Hyper-parameter analysis (Q4)} \label{sec:exper_q4}
\paragraph{Magnitude of Augmentation $\beta$.} It is expected that introducing adequate augmentation enriches the sample distribution in the latent space, which contributes to learning more expressive representations. Figure~\ref{fig:sens} shows that the classification accuracy generally reaches the peak and drops gradually when the augmentation size $\beta$ increases, which aligns with our intuition. We also observe that the optimal augmentation magnitudes are relatively smaller for node-level datasets, which may be related to the semantics level of graph features. Input features of graph-level datasets are less informative and latent distribution may still preserve with stronger augmentations. Besides, the stable trend further verifies that graph wiener decoder is well adapted to augmentation in representation learning.
\paragraph{Convolution filter $g_c$.} Table~\ref{tab:exp_filter} shows the influence of different convolution filters. It is observed that diffusion-based WGDN outperforms its trivial version with GCN filter across different applications. Specifically, heat kernel generates better results in node classification and PPR kernel is more suitable for graph-level tasks. We conjecture that sparse feature information may be better compressed via propagation with PPR kernel. In addition, we also find that training loss of diffusion models is consistently lower. Both phenomena indicate that the superior information aggregation and powerful reconstruction of diffusion filters jointly contribute to learning a more semantic representation.
\section{Conclusion and future work}\label{sec:conc}
In this paper, we propose Wiener Graph Deconvolutional Network (WGDN), a predictive self-supervised learning framework for graph-structured data. We introduce graph wiener filter and theoretically validate its superior reconstruction ability to facilitate reconstruction-based representation learning. By leveraging graph wiener decoder, our model can efficiently learn graph embedding with augmentation. Extensive experimental results on various datasets demonstrate that WGDN achieves competitive performance over a wide range of self-supervised and semi-supervised counterparts.
\appendix
\section*{Technical Appendix} In the technical appendix, we provide further details for the proofs, model architectures and experiment.
\section{Proof of Proposition~\ref{prop:1}}\label{prop:p1} \begin{proof} By substitution, $\mathbf{\hat{x}}_{inv} = \mathbf{x} + \mathbf{U} g_d(\mathbf{\Lambda}) \mathbf{U}^T \mathbf{\epsilon}$. Note that MSE is reduced to
\begin{equation}
\begin{aligned}
\mathbb{E} \norm{\mathbf{x} - \mathbf{\hat{x}}_{inv}}_{2}^{2} = \mathbb{E} \norm{\mathbf{U} g_d(\mathbf{\Lambda}) \mathbf{U}^T \mathbf{\epsilon}}_{2}^{2} = \sum_{i=1}^{N} \frac{\sigma^2}{g_c^2(\lambda_i)}.
\end{aligned}
\end{equation}
Regarding the condition $g_c: [0, 2] \mapsto [-1, 1]$, we take GCN where $g_c(\lambda_i) = 1 - \lambda_i$ as a representative example. By substitution, we have
\begin{equation}
\begin{aligned}
\mathbb{E} \norm{\mathbf{x} - \mathbf{\hat{x}}_{inv}}_{2}^{2} = \sum_{i=1}^{N} \frac{\sigma^2}{(1 - \lambda_i)^2},
\end{aligned}
\end{equation} and $\frac{1}{(1-\lambda_i)^2} \to \infty$ when $\lambda_i \to 1$. \end{proof}
\section{Proof of Proposition~\ref{prop:2}}\label{prop:p2} \begin{proof}
By the definition of $g_w(\lambda_i)$
\begin{equation}
\begin{aligned}
\mathbb{E} \norm{\mathbf{x} - \mathbf{\hat{x}}_w}_{2}^{2} & = \sum_{i=1}^{N} \frac{\sigma^2 (g_c^2(\lambda_i) + \sigma^2/\mathbb{E} [x^{\ast 2}_{i}])}{(g_c^2(\lambda_i) + \sigma^2/\mathbb{E} [x^{\ast 2}_{i}])^2} \\
& = \sum_{i=1}^{N} \frac{\sigma^2}{g_c^2(\lambda_i) + \sigma^2/\mathbb{E} [x^{\ast 2}_{i}]} \\
& \leq \sum_{i=1}^{N} \frac{\sigma^2}{g_c^2(\lambda_i)}.
\end{aligned}
\end{equation}
Plugging in the inverse filter, we can obtain
\begin{equation}
\begin{aligned}
& \sum_{i=1}^{N} \mathbb{VAR}[\mathbf{\hat{x}}_{inv, i}] \\
& = \text{Tr}(\mathbb{COV}[\mathbf{\hat{x}}_{inv}]) \\
& = \text{Tr}(\mathbb{COV}[\mathbf{U} \mathbf{x}^{\ast}] + \mathbb{COV}[\mathbf{U} g_d(\mathbf{\Lambda}) \mathbf{\epsilon}^{\ast}]) \\
& = \sum_{i=1}^{N} (\mathbb{VAR}[\mathbf{x}^{\ast}_{i}] + \frac{\sigma^2}{g_c^2(\lambda_i)}),
\end{aligned}
\end{equation}
where $\text{Tr}$ represents the matrix trace. Similarly, variance of $\mathbf{\hat{x}}_w$ is convoluted by $g_w(\lambda_i)$
\begin{equation}
\begin{aligned}
& \sum_{i=1}^{N} \mathbb{VAR}[\mathbf{\hat{x}}_{w, i}] \\
& = \text{Tr}(\mathbb{COV}[\mathbf{\hat{x}}_{w}]) \\
& = \sum_{i=1}^{N} [\frac{g_c^2(\lambda_i) }{g_c^2(\lambda_i) + \sigma^2/\mathbb{E} [x^{\ast 2}_{i}]}]^2 (\mathbb{VAR}[\mathbf{x}^{\ast}_{i}] + \frac{\sigma^2}{g_c^2(\lambda_i)}) \\
& \leq \sum_{i=1}^{N} (\mathbb{VAR}[\mathbf{x}^{\ast}_{i}] + \frac{\sigma^2}{g_c^2(\lambda_i)}).
\end{aligned}
\end{equation} \end{proof}
\section{Proof of Proposition~\ref{prop:3}}\label{prop:p3} \begin{proof}
Similar to Appendix~\ref{prop:p2}, variance of $\mathbf{\hat{x}}_{w, \gamma}$ is reduced to
\begin{equation}
\begin{aligned}
& \sum_{i=1}^{N} \mathbb{VAR}(\mathbf{\hat{x}}_{w, \gamma, i}) \\
& = \text{Tr}(\mathbb{COV}(\mathbf{\hat{x}}_{w, \gamma})) \\
& = \sum_{i=1}^{N} [\frac{g_c^2(\lambda_i) }{g_c^2(\lambda_i) + \sigma^2/x^{\ast 2}_{\gamma}}]^2 (\mathbb{VAR}[\mathbf{x}^{\ast}_{i}] + \frac{\sigma^2}{g_c^2(\lambda_i)}) \\
& \leq \sum_{i=1}^{N} (\mathbb{VAR}[\mathbf{x}^{\ast}_{i}] + \frac{\sigma^2}{g_c^2(\lambda_i)}).
\end{aligned}
\end{equation}
For the specific spectrum $\lambda_i$ where $\mathbb{E} [x^{\ast 2}_{i}] \leq \bar{x}^{\ast 2}_{\gamma}$ holds, the spectral reconstruction error satisfies
\begin{equation}
\begin{aligned}
& S(\lambda_i, x^{\ast}_{i}, \sigma, g_{c}, \bar{g}_{w, \gamma}) \\
& = \frac{1}{(g_c^2(\lambda_i) + \sigma^2/\bar{x}^{\ast 2}_{\gamma})^2} [\sigma^2 (g_c^2(\lambda_i) + \frac{\sigma^2}{\bar{x}^{\ast 2}_{\gamma}} \cdot \frac{\mathbb{E} [x^{\ast 2}_{i}]}{\bar{x}^{\ast 2}_{\gamma}})] \\
& \leq \frac{1}{(g_c^2(\lambda_i) + \sigma^2/\bar{x}^{\ast 2}_{\gamma})^2} [\sigma^2 (g_c^2(\lambda_i) + \frac{\sigma^2}{\bar{x}^{\ast 2}_{\gamma}})] \\
& = \frac{\sigma^2}{g_c^2(\lambda_i) + \sigma^2/\bar{x}^{\ast 2}_{\gamma}} \\
& \leq \frac{\sigma^2}{g_c^2(\lambda_i)} = S(\lambda_i, x^{\ast}_{i}, \sigma, g_{c}, g_{c}^{-1}).
\end{aligned}
\end{equation}
Note that the second derivative of spectral reconstruction error $S(\lambda_i, x^{\ast}_{i}, \sigma, g_{c}, g_{d})$ with respect to $g_{d}(\lambda_i)$ is
\begin{equation}
\begin{aligned}
& \frac{\partial^2}{\partial g_{d}^{2}(\lambda_i)} S(\lambda_i, x^{\ast}_{i}, \sigma, g_{c}, g_{d}) \\
& = 2(g_{c}^{2}(\lambda_i) \mathbb{E} [x^{\ast 2}_{i}] + \sigma^2) \geq 0,
\end{aligned}
\end{equation}
thus, $S(\lambda_i, x^{\ast}_{i}, \sigma, g_{c}, g_{d})$ is a convex function. By Eq.~\ref{eqn:graphwiener}, $g_{w}(\lambda_i)$ is the solution for global minimum. By convexity, for any filter $g_{d}(\lambda_i)$, the value of $S(\lambda_i, x^{\ast}_{i}, \sigma, g_{c}, g_{d})$ is greater when distance to global minimizer $|g_{d}(\lambda_i) - g_{w}(\lambda_i)|$ is larger. Considering $\bar{g}_{w, \gamma}(\lambda_i)$, it can be reduced to
\begin{equation}
\begin{aligned}
& |\bar{g}_{w, \gamma}(\lambda_i) - g_{w}(\lambda_i)| \\
& = |\frac{g_{c}(\lambda_i)}{g_c^2(\lambda_i) + \sigma^2/\bar{x}^{\ast 2}_{\gamma}} - \frac{g_{c}(\lambda_i)}{g_{c}^2(\lambda_i) + \sigma^2/\mathbb{E} [x^{\ast 2}_{i}]}|.
\end{aligned}
\end{equation}
Given the condition that $x^{\ast 2}_{i} \leq \bar{x}^{\ast 2}_{\gamma_1} \leq \bar{x}^{\ast 2}_{\gamma_2}$, we can conclude that
\begin{equation}
\begin{aligned}
|\bar{g}_{w, \gamma_1}(\lambda_i) - g_{w}(\lambda_i)| \leq |\bar{g}_{w, \gamma_2}(\lambda_i) - g_{w}(\lambda_i)|.
\end{aligned}
\end{equation}
Therefore, $S(\lambda_i, x^{\ast}_i, \sigma, g_c, \bar{g}_{w, \gamma_1}) \leq S(\lambda_i, x^{\ast}_i, \sigma, g_c, \bar{g}_{w, \gamma_2}) \leq S(\lambda_i, x^{\ast}_i, \sigma, g_c, g_c^{-1})$ holds. \end{proof}
\begin{table*}[ht]
\centering
\begin{tabular}{c|ccccc}
\toprule
Model & PubMed & Computers & Photo & CS & Physics \\
\midrule
BGRL$_{\text{N}}$ & 78.6 $\pm$ 0.7 & 89.49 $\pm$ 0.21 & 93.01 $\pm$ 0.20 & 93.15 $\pm$ 0.12 & 95.14 $\pm$ 0.06 \\
BGRL$_{\text{H}}$ & 79.8 $\pm$ 0.4 & 89.70 $\pm$ 0.15 & 93.37 $\pm$ 0.21 & 93.51 $\pm$ 0.10 & 95.24 $\pm$ 0.09 \\
BGRL$_{\text{P}}$ & 79.4 $\pm$ 0.5 & 89.63 $\pm$ 0.17 & 93.15 $\pm$ 0.21 & 93.42 $\pm$ 0.12 & 95.28 $\pm$ 0.06 \\
\midrule
AFGRL$_{\text{N}}$ & 79.5 $\pm$ 0.2 & 88.91 $\pm$ 0.37 & 92.96 $\pm$ 0.25 & 93.17 $\pm$ 0.15 & 95.54 $\pm$ 0.09 \\
AFGRL$_{\text{H}}$ & 79.9 $\pm$ 0.3 & 89.58 $\pm$ 0.45 & 93.61 $\pm$ 0.20 & 93.56 $\pm$ 0.15 & 95.74 $\pm$ 0.10 \\
AFGRL$_{\text{P}}$ & 79.7 $\pm$ 0.3 & 89.33 $\pm$ 0.37 & 93.06 $\pm$ 0.27 & 93.53 $\pm$ 0.14 & 95.71 $\pm$ 0.10 \\
\midrule
CCA-SSG$_{\text{N}}$ & 80.5 $\pm$ 0.3 & 87.35 $\pm$ 0.30 & 92.38 $\pm$ 0.33 & 93.31 $\pm$ 0.16 & 95.14 $\pm$ 0.07 \\
CCA-SSG$_{\text{H}}$ & 81.0 $\pm$ 0.3 & 88.15 $\pm$ 0.35 & 93.25 $\pm$ 0.25 & - & 95.59 $\pm$ 0.07 \\
CCA-SSG$_{\text{P}}$ & 80.7 $\pm$ 0.3 & 87.27 $\pm$ 0.46 & 92.79 $\pm$ 0.25 & - & 95.12 $\pm$ 0.13 \\
\midrule
$\text{WGDN}_{\text{N}}$ & 80.2 $\pm$ 0.4 & 89.03 $\pm$ 0.46 & 92.26 $\pm$ 0.37 & 92.48 $\pm$ 0.12 & 95.33 $\pm$ 0.02 \\
$\text{WGDN}_{\text{H}}$ & \textbf{81.9 $\pm$ 0.4} & \textbf{89.72 $\pm$ 0.48} & \textbf{93.89 $\pm$ 0.31} & \textbf{93.67 $\pm$ 0.17} & \textbf{95.76 $\pm$ 0.11} \\
$\text{WGDN}_{\text{P}}$ & 81.4 $\pm$ 0.3 & 89.59 $\pm$ 0.45 & 92.96 $\pm$ 0.19 & 92.75 $\pm$ 0.21 & 95.40 $\pm$ 0.23 \\
\bottomrule
\end{tabular}
\caption{Node classification accuracy with different encoding propagation backbones.}
\label{tab:exp_backbones} \end{table*}
\begin{table*}[ht]
\centering
\begin{tabular}{cccccc}
\toprule
Model & PubMed & Comp & CS & IMDB-B & DD \\
\midrule
AFGRL & 79.9 $\pm$ 0.3 & 89.58 $\pm$ 0.45 & 93.56 $\pm$ 0.15 & 75.07 $\pm$ 0.58 & 78.58 $\pm$ 0.44 \\
GraphMAE & 81.1 $\pm$ 0.4 & 89.53 $\pm$ 0.31 & 93.51 $\pm$ 0.13 & 75.52 $\pm$ 0.66 & 78.86 $\pm$ 0.35 \\
\midrule
WGDN-DE & 80.9 $\pm$ 0.6 & 89.55 $\pm$ 0.36 & 93.56 $\pm$ 0.31 & 75.42 $\pm$ 0.15 & 79.24 $\pm$ 0.40 \\
WGDN-DN & 81.3 $\pm$ 0.5 & 89.49 $\pm$ 0.34 & 93.53 $\pm$ 0.31 & 75.52 $\pm$ 0.17 & 79.31 $\pm$ 0.32 \\
WGDN & \textbf{81.9 $\pm$ 0.4} & \textbf{89.72 $\pm$ 0.48} & \textbf{93.89 $\pm$ 0.31} & \textbf{75.76 $\pm$ 0.20} & \textbf{79.54 $\pm$ 0.51} \\
\bottomrule
\end{tabular}
\caption{Performance of WGDN against different augmentation methods.}
\label{tab:exp_augmentation} \end{table*}
\section{Details of Model Architecture} \label{model:detail_arch}
\paragraph{AER estimation.} Let $\mathbf{\hat{H}}^{(m)}$ denotes the input of $m$-th layer decoder, the average spectral energy $\bar{x}^{\ast 2}_{\gamma_i}$ in $\mathbf{D}_{\gamma_i}^{(m)}$ is estimated following $\sum_{i=1}^{N} \mathbb{E} [x^{\ast 2}_{i}] = \sum_{i=1}^{N} \mathbb{E} [x^{2}_{i}] = \sum_{i=1}^{N} {\mathbb{E} [x_{i}]}^2 + \mathbb{VAR} [x_{i}]$. Specifically, \begin{equation}
\bar{x}^{\ast 2}_{\gamma_i} = \frac{\gamma_i}{ND'} \big( \norm{\mathbf{\hat{H}}^{(m)}}_F^2 + \norm{\mathbf{\hat{H}}^{(m)} - \frac{1}{N} \mathds{1} \mathbf{\hat{H}}^{(m)}}_F^2 \big), \end{equation} where $\mathds{1} \in \mathbb{R}^{N \times D'}$ is the all ones matrix and $D'$ is the size of hidden space. The augmentation variance $\sigma^2$ is estimated by considering its neighborhood as \begin{equation}
\sigma^2 = \frac{1}{ND'} \norm{\mathbf{\hat{H}}^{(m)} - \mathbf{D}^{-1} \mathbf{A} \mathbf{\hat{H}}^{(m)}}_{F}^{2}, \end{equation} where $\mathbf{A}$ and $\mathbf{D}$ are adjacency matrix and degree matrix.
\paragraph{Skip connection.} To learn more expressive representations, skip connection is considered in training phase for some cases as it transmits aggregated information to create 'hard' examples for decoder, which may encourage encoder to compress more useful knowledge. If skip connection is implemented, we let $\mathbf{\hat{H}}^{(m)}_{d} = \mathbf{\hat{H}}^{(m)}$ for clarity. For $m=1, .., M - 1$, we augment the output of the $m$-layer encoder, denoted as $\mathbf{\hat{H}}^{(m)}_{e}$, by \begin{equation}
\mathbf{\hat{H}}^{(m)}_{e} = \mathbf{H}^{(m)} + \beta \mathbf{E}^{(m)}, \end{equation} where $\mathbf{E}^{(m)} = \{ \mathbf{\epsilon}_{1}^{(m)}, ..., \mathbf{\epsilon}_{N}^{(m)}\}$, $\mathbf{\epsilon}_{i}^{(m)} \sim N(\mathbf{0}, \sigma_{P}^{2 \, (m)} \mathbf{I})$, $\sigma_{P}^{2 \, (m)} = \mathbb{VAR}[\mathbf{H}^{(m)}]$ and $\beta$ is same hyper-parameter in Eq.~\ref{eqn:latent_aug}. To avoid learning trivial representation, both augmented representations are fed into the same decoder as \begin{equation} \begin{aligned}
\mathbf{Z}^{(m-1)}_{i, s} & = \phi(\mathbf{D}_{\gamma_i, s}^{(m)} \mathbf{\hat{H}}^{(m)}_{s} \mathbf{W}^{(m)}_{i}), \\
\mathbf{\hat{H}}^{(m-1)}_{s} & = \text{AGG}([\mathbf{Z}^{(m-1)}_{1, s}, ..., \mathbf{Z}^{(m-1)}_{q, s}]), \end{aligned} \end{equation} where $s = \{e, d\}$ represents the source. The final representation of $m$-layer decoder is obtained by averaging the intermediate embeddings, \begin{equation}
\mathbf{\hat{H}}^{(m-1)} = \text{AVG}(\mathbf{\hat{H}}^{(m-1)}_{e}, \mathbf{\hat{H}}^{(m-1)}_{d}). \end{equation}
\paragraph{Normalization.} For graph classification, we apply batch normalization~\cite{ioffe2015batch} right before activation function for all layers except the final prediction layer.
\section{More Experimental Results}
\subsection{Experiment on Encoder Backbones}
To have a fair comparison with contrastive methods, we compare WGDN with the three state-of-the-art baselines using spectral kernel in node classification task. For clarity, models with GCN, heat and PPR kernel are denoted with subscript $_\text{N}$, $_\text{H}$ and $_\text{P}$ respectively.
From Table~\ref{tab:exp_backbones}, it is observed that WGDN still achieves better performance over baselines trained with spectral kernels. In terms of CS, CCA-SSG employs MLP as its decoder because GNN decoders worsen model performance under its experiment settings. In addition, utilizing spectral propagation consistently boosts the learning capability, which aligns with our motivation pertaining to the potential of graph spectral kernel. Particularly, for node classification, heat kernel may be a better option, as it brings the greatest improvement regardless of datasets and model types.
\begin{table*}[ht]
\centering
\begin{tabular}{c|cccccccc}
\toprule
Dataset & Cora & CiteSeer & PubMed & Computers & Photo & CS & Physics & OGBN-Arxiv \\
\midrule
\# Nodes & 2,708 & 3,327 & 19,717 & 13,752 & 7,650 & 18,333 & 34,493 & 169,343 \\
\# Edges & 10,556 & 9,104 & 88,648 & 491,722 & 238,162 & 163,788 & 495,924 & 1,166,243 \\
\# Classes & 7 & 6 & 3 & 10 & 8 & 15 & 5 & 40 \\
\# Features & 1,433 & 3,703 & 500 & 767 & 745 & 8,415 & 8,415 & 128 \\
\midrule
Augmentation $\beta$ & 0.9 & 1.0 & 1.0 & 0.4 & 0.5 & 0.5 & 0.2 & 0.8 \\
Hidden Size & 512 & 512 & 1024 & 512 & 512 & 512 & 512 & 768 \\
Epoch & 100& 100 & 300 & 1000 & 1000 & 150 & 100 & 120 \\
Filter $g_c(\lambda_i)$ & PPR & PPR & Heat & Heat & Heat & Heat & Heat & PPR \\
Aggregation & Max & Sum & Max & - & - & - & - & - \\
Last Activation & \ding{51} & - & - & \ding{51} & - & - & \ding{51} & - \\
Skip Connection & \ding{51} & - & \ding{51} & - & - & \ding{51} & \ding{51} & - \\
\bottomrule
\end{tabular}
\caption{Summary of datasets and hyper-parameter configuration for node classification task.}
\label{tab:dataset_node} \end{table*}
\begin{table*}[ht]
\centering
\begin{tabular}{c|cccccc}
\toprule
Dataset & IMDB-B & IMDB-M & PROTEINS & COLLAB & DD & NCI1\\
\midrule
\# Graphs & 1,000 & 1,500 & 1,113 & 5,000 & 1,178 & 4,110 \\
\# Avg. Nodes & 19.77 & 13.00 & 39.06 & 74.49 & 284.32 & 29.87 \\
\# Avg. Edges & 193.06 & 65.94 & 72.82 & 2457.78 & 715.66 & 32.30 \\
\# Classes & 2 & 3 & 2 & 3 & 2 & 2 \\
\midrule
Augmentation $\beta$ & 1.5 & 1.5 & 1.0 & 1.0 & 1.0 & 0.5 \\
Learning Rate & 0.0001 & 0.0001 & 0.0001 & 0.0001 & 0.0001 & 0.0005 \\
Batch Size & 32 & 32 & 32 & 32 & 32 & 16 \\
Epoch & 100 & 100 & 10 & 20 & 40 & 500 \\
Filter $g_c(\lambda_i)$ & PPR & PPR & PPR & PPR & PPR & Heat \\
Aggregation & Max & Avg & Avg & - & Avg & - \\
Pooling & Avg & Avg & Max & Max & Sum & Max \\
Skip Connection & \ding{51} & \ding{51} & \ding{51} & \ding{51} & \ding{51} & - \\
\bottomrule
\end{tabular}
\caption{Summary of datasets and hyper-parameter configuration for graph classification task.}
\label{tab:dataset_graph} \end{table*}
\subsection{Experiment on Different Augmentations}
Although the graph wiener decoder is derived based on latent augmentations, a powerful decoder should adapt to general augmentation techniques. We conduct experiments with five representative datasets with drop-edge and drop-node, which are denoted as WGDN-DE and WGDN-DN respectively. Table~\ref{tab:exp_augmentation} shows that different variants of our framework still achieve comparable performance with competitive baselines on both node and graph classification. However, general augmentations hardly satisfy the distribution assumption, resulting in inaccurate noise estimation and performance degradation.
\subsection{Experiments on Additional Datasets}
To evaluate the effectiveness of our proposed method, three more commonly used datasets are considered. For fair comparisons, we conducted hyper-parameter search to finetune the most competitive baselines. We report the results in Table~\ref{tab:exp_common_extra} and find that WGDN still achieves state-of-the-art performances in 2 out of 3 datasets.
\begin{table}[ht]
\centering
\begin{tabular}{cccc}
\toprule
Model & Cora & CiteSeer & OBGN-Arxiv \\
\midrule
BGRL & 82.8 $\pm$ 0.5 & 71.3 $\pm$ 0.8 & 71.64 $\pm$ 0.12 \\
AFGRL & 81.7 $\pm$ 0.4 & 71.2 $\pm$ 0.4 & 71.39 $\pm$ 0.16 \\
CCA-SSG & 83.8 $\pm$ 0.5 & \textbf{73.1 $\pm$ 0.3} & 71.24 $\pm$ 0.20 \\
\midrule
WGDN & \textbf{84.2 $\pm$ 0.6} & 72.2 $\pm$ 1.1 & \textbf{71.76 $\pm$ 0.23} \\
\bottomrule
\end{tabular}
\caption{Node classification accuracy on additional commonly used datasets.}
\label{tab:exp_common_extra} \end{table}
\section{Detailed Experimental Setup} \label{exper:spec}
\subsection{Evaluation Protocol} \label{exper:spec:eval}
For node classification tasks, all the resulted embedding from well-trained unsupervised models are frozen. We use Glorot initialization~\cite{glorot2010understanding} to initialize model parameters and all downstream models are trained for 300 epochs by Adam optimizer~\cite{kingma2014adam} with a learning rate 0.01. We run 20 trials and keep the model with the highest performance in validation set of each run as final. For graph classification tasks, linear SVM is fine-tuned with grid search on \textit{C} parameter from $\{10^{-3}, 10^{-2}, ..., 1, 10\}$.
\subsection{Hyper-parameter Specifications} \label{exper:spec:hyper}
By default, wiener graph decoder is implemented with multiple channels with $q = 3$ and $\gamma = [0.1, 1, 10]$. Otherwise, single channel is used with $\gamma = 1$. Aggregation function is applied only in the scenarios of multiple channels.
For wiener kernel approximation, the polynomial order $K$ of GCN kernel is 9 while others are 2. All models are initialized using Glorot initialization~\cite{glorot2010understanding} and optimized by Adam optimizer~\cite{kingma2014adam}.
For node classification, we set the learning rate as 0.001. The number of layers is 3 for OGBN-Arxiv and set as 2 for the remaining datasets. To enhance the representation power of embedding, no activation function is employed in the last layer of encoder in some cases. For graph classification, the dimension of hidden embedding is set to 512. The number of layer is set to 3 for IMDB-M as well as PROTEINS and 2 for the rest. The detailed hyper-parameter configurations of each dataset are illustrated in Table~\ref{tab:dataset_node} and~\ref{tab:dataset_graph}.
\subsection{Baselines Implementations}
For node classification, we use the official implementation of BGRL, AFGRL and CCA-SSG and follow the suggested hyper-parameter settings for reproduction. For fair comparison in the scenarios of spectral kernel implementation, we conduct hyper-parameter search for them and select the model with best results in validation set as final.
For graph classification, we report the previous results in the public papers. For dataset DD, we generate the result of GraphMAE with its source code and hyper-parameter searching.
\subsection{Computational Hardware}
We use the machine with the following configurations for all model training and evaluation.
\begin{itemize}
\item OS: Ubuntu 20.04.1 LTS
\item CPU: Intel(R) Xeon(R) Silver 4114 CPU @ 2.20GHz
\item GPU: NVIDIA Tesla V100, 32GB \end{itemize}
\end{document} | arXiv | {
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\begin{document}
\title{Two dimensional water waves in holomorphic coordinates}
\author{John K. Hunter} \address{ Department of Mathematics, University of California at Davis} \thanks{The first author was partially supported by the NSF under grant number DMS-1312342.} \email{hunter@math.davis.edu} \author{Mihaela Ifrim}
\address{Department of Mathematics, University of California at Berkeley}. \email{ifrim@math.berkeley.edu} \thanks{The second author was supported by the National Science Foundation under Grant No. 0932078 000, while the author was in residence at the Mathematical Science Research Institute in Berkeley, California, during the Fall semester 2013.} \author{ Daniel Tataru} \address{Department of Mathematics, University of California at Berkeley}
\thanks{The third author was partially supported by the NSF grant DMS-1266182 as well as by the Simons Foundation} \email{tataru@math.berkeley.edu}
\begin{abstract}
This article is concerned with the infinite depth water wave
equation in two space dimensions. We consider this problem
expressed in position-velocity potential holomorphic coordinates.
Viewing this problem as a quasilinear dispersive equation, we
establish two results: (i) local well-posedness in Sobolev spaces,
and (ii) almost global solutions for small localized data. Neither
of these results are new; they have been recently obtained by
Alazard-Burq-Zuily~\cite{abz}, respectively by Wu~\cite{wu} using different coordinates
and methods. Instead our goal is improve the understanding of this
problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.
\end{abstract}
\maketitle
\section{Introduction} We consider the two dimensional water wave equations with infinite depth with gravity but without surface tension. This is governed by the incompressible Euler's equations with boundary conditions on the water surface. Under the additional assumption that the flow is irrotational the fluid dynamics can be expressed in terms of a one-dimensional evolution of the water surface coupled with the trace of the velocity potential on the surface.
This problem was previously considered by several other authors. The local in time existence and uniqueness of solutions was proved in \cite{n,y, wu2}, both for finite and infinite depth. Later, Wu~\cite{wu} proved almost global existence for small localized data. Very recently, global results for small localized data were independently obtained by Alazard $\&$ Delort \cite{ad} and by Ionescu $\&$ Pusateri \cite{ip}. Extensive work was also done on the same problem in three or higher space dimensions, and also on related problems with surface tension, vorticity, finite bottom, etc. Without being exhaustive, we list some of the more recent references \cite{abz, abz1, chs, cl,cs,
DL, HL, o, sz, zz}.
Our goal here is to revisit this problem and to provide a new, self-contained approach which, we hope, considerably simplifies and improves on many of the results mentioned above. Our analysis is based on the use of holomorphic coordinates, which are described below. Our results include:
(i) local well-posedness in Sobolev spaces, improving on previous regularity thresholds, e.g. in \cite{abz}, up to the point where the transport vector field is no longer Lipschitz, and
has merely a $BMO$ derivative.
(ii) cubic life-span bounds for small data. These are related to the normal form method, but are instead proved by a modified energy method, inspired from the authors' previous article \cite{BH}.
(iii) almost global well-posedness for small localized data, refining and simplifying Wu's approach in \cite{wu}.
We consider both the case of the real line $\mathbb{R}$ and the periodic case $\mathbb{S}^1$. Our equations are expressed in coordinates $(t,\alpha)$ where $\alpha$ corresponds to the holomorphic parametrization of the water domain by the lower half-plane restricted to the real line. To write the equations we use the Hilbert transform $H$, as well as the operator \begin{equation*} P= \frac12(I-iH). \end{equation*} Note that $P$ is a projector in $\mathbb{R}$ but not on $\mathbb{S}^1$.
Our variables $(Z,Q)$ represent the position of the water surface, respectively the holomorphic extension of the velocity potential. These will be restricted to the closed subspace of holomorphic functions within various Sobolev spaces. Here we define holomorphic functions on $\mathbb{R}$ or on $\mathbb{S}^1$ as those whose Fourier transform is supported in $(-\infty,0]$; equivalently, they admit a bounded holomorphic extension into the lower half-space. On $\mathbb{R}$ this can be described by the relation $Pf = f$, but on $\mathbb{S}^1$ we also need to make some adjustments for the constants.
There is a one dimensional degree of freedom in the choice of $\alpha$, namely the horizontal translations. To fix this, in the real case we are considering waves which either decay at infinity, \[
\lim_{|\alpha \vert \to \infty} Z(\alpha) - \alpha = 0. \] In the periodic case we instead assume that $ Z(\alpha) - \alpha$ has period $2\pi$ and purely imaginary average. We can also harmlessly assume that $Q$ has real average.
In position-velocity potential holomorphic coordinates the equations have the form \begin{equation*}
\left\{ \begin{aligned} & Z_t + F Z_\alpha = 0, \\
& Q_t + F Q_\alpha -i (Z-\alpha) + P\left[ \frac{|Q_\alpha|^2}{J}\right] = 0, \\ \end{aligned} \right. \end{equation*} where \begin{equation*}
F = P\left[ \frac{Q_\alpha - \bar Q_\alpha}{J}\right] , \qquad J = |Z_\alpha|^2. \end{equation*}
For the derivation of the above equations, we refer the reader to \emph{Appendix}~\ref{holom-eq}. In the real case these equations originate in \cite{ov}. The changes needed for the periodic case are also described in the same \emph{Appendix}~\ref{holom-eq}. There are also other ways of expressing the equations, for instance in Cartesian coordinates using the Dirichlet to Neumann map associated to the water domain, see e.g. \cite{abz} . Here we prefer the holomorphic coordinates due to the simpler form of the equations; in particular, in these coordinates the Dirichlet to Neumann map is given in terms of the standard Hilbert transform.
It is convenient to work with a new variable, namely \[ W = Z-\alpha . \] The equations become \begin{equation} \label{ww2d1} \left\{ \begin{aligned} & W_t + F (1+W_\alpha) = 0, \\
& Q_t + F Q_\alpha -i W + P\left[ \frac{|Q_\alpha|^2}{J}\right] = 0, \\ \end{aligned} \right. \end{equation} where \begin{equation*}
F = P\left[\frac{Q_\alpha - \bar Q_\alpha}{J}\right], \qquad J = |1+W_\alpha|^2.
\end{equation*} These equations are considered either in $\mathbb{R} \times \mathbb{R}$ or in $\mathbb{R} \times \mathbb{S}^1$.
As the system \eqref{ww2d1} is fully nonlinear, a standard procedure is to convert it into a quasilinear system by differentiating it. Observing that almost no undifferentiated functions appear in \eqref{ww2d1}, one sees that by differentiation we get a self-contained first order quasilinear system for $(W_\alpha,Q_\alpha)$. To write this system we introduce the auxiliary real function $b$, which we call the {\em advection velocity}, and is given by \begin{equation*} b = P \left[\frac{{Q}_\alpha}{J}\right] + \bar P\left[\frac{\bar{Q}_\alpha}{J}\right].
\end{equation*} The reason for this will be immediately apparent. Using $b$, the system \eqref{ww2d1} is written in the form \begin{equation*} \left\{ \begin{aligned} &W_t + b (1+ W_\alpha) = \frac{\bar Q_\alpha}{1+\bar W_\alpha}, \\
&Q_t + b Q_\alpha - iW = \bar P\left[ \frac{|Q_\alpha|^2}{J}\right], \end{aligned} \right. \end{equation*} where the terms on the right are antiholomorphic and disappear when the equations are projected onto the holomorphic space. Differentiating with respect to $\alpha$ yields a system for $(W_\alpha,Q_\alpha)$, namely \begin{equation*} \left\{ \begin{aligned}
&W_{\alpha t} + b W_{\alpha \alpha} +
\frac{1}{1+\bar W_\alpha}\left(Q_{\alpha\alpha} - \frac{Q_\alpha}{1+W_\alpha}
W_{\alpha \alpha}\right) = - (1+W_\alpha) \bar F_\alpha - \left[\frac{\bar Q_\alpha}{1+\bar W_\alpha}\right]_\alpha,
\\
&Q_{t\alpha} + b Q_{\alpha\alpha} - iW_\alpha +
\frac{1}{1+\bar W_\alpha} \frac{Q_\alpha}{1+W_\alpha}\!\!\left(\!\!Q_{\alpha\alpha} - \frac{Q_\alpha}{1+W_\alpha} W_{\alpha \alpha}\!\! \right)\!\! = - Q_\alpha \bar F_\alpha+ \bar P\left[ \frac{|Q_\alpha|^2}{J}\right]_\alpha. \end{aligned} \right. \end{equation*} The terms on the right are mostly antiholomorphic and can be viewed as lower order when projected on the holomorphic functions. Examining the expression on the left one easily sees that the above first order system is degenerate, and has a double speed $b$. Then it is natural to diagonalize it. This is done using the operator \begin{equation} \mathbf A(w,q) := (w,q - Rw), \qquad R := \frac{Q_\alpha}{1+W_\alpha}. \label{defR} \end{equation} The factor $R$ above has an intrinsic meaning, namely it is the complex velocity on the water surface. We also remark that \[ \mathbf A(W_\alpha,Q_\alpha) = ({\mathbf W},R), \qquad {\mathbf W} : = W_\alpha. \] Thus, the pair $({\mathbf W},R)$ diagonalizes the differentiated system. Indeed, a direct computation yields the self-contained system \begin{equation} \label{ww2d-diff} \left\{ \begin{aligned}
& {\mathbf W}_{ t} + b {\mathbf W}_{ \alpha} + \frac{(1+{\mathbf W}) R_\alpha}{1+\bar {\mathbf W}} = (1+{\mathbf W})M, \\ & R_t + bR_\alpha = i\left(\frac{{\mathbf W} - a}{1+{\mathbf W}}\right), \end{aligned} \right. \end{equation} where the real {\em frequency-shift} $a$ is given by \begin{equation} a := i\left(\bar P \left[\bar{R} R_\alpha\right]- P\left[R\bar{R}_\alpha\right]\right), \label{defa} \end{equation} and the auxiliary function $M$ is given by \begin{equation}\label{M-def} M := \frac{R_\alpha}{1+\bar {\mathbf W}} + \frac{\bar R_\alpha}{1+ {\mathbf W}} - b_\alpha = \bar P [\bar R Y_\alpha- R_\alpha \bar Y] + P[R \bar Y_\alpha - \bar R_\alpha Y]. \end{equation} The function $Y$ above, given by \[ Y := \frac{{\mathbf W}}{1+{\mathbf W}}, \] is introduced in order to avoid rational expressions above and in many places in the sequel. The system \eqref{ww2d-diff} governs an evolution in the space of holomorphic functions, and will be used both directly and in its projected version.
Incidentally, we note that when expressed in terms of $(Y,R)$ the water wave system becomes purely polynomial, see also \cite{zakharov2}, \begin{equation*}
\left\{ \begin{aligned}
& Y_t + b Y_\alpha + |1-Y|^2 R_\alpha = (1-Y) M,
\\ & R_t + b R_\alpha - i(1+a) Y = - ia, \end{aligned} \right. \end{equation*} where $M$ is as above, and \[ b = 2 \Re ( R - P(R \bar Y)), \qquad a = 2 \Re P(R \bar R_\alpha). \] However, we do not take advantage of this formulation in the present article.
The functions $b$ and $a$ also play a fundamental role in the linearized equation which is computed in the next section, Section~\ref{s:linearized}. The linearized variables are denoted by $(w,q)$ and, after the diagonalization, $(w, r:=q-Rw)$. The linearized equation, see \eqref{lin(wr)0}, has the form \begin{equation} \label{lin(wr)00} \left\{ \begin{aligned} & (\partial_t + b \partial_\alpha) w + \frac{1}{1+\bar {\mathbf W}} r_\alpha + \frac{R_{\alpha} }{1+\bar {\mathbf W}} w =\ (1+{\mathbf W}) (P \bar m + \bar P m),
\\ &(\partial_t + b \partial_\alpha) r - i \frac{1+a}{1+{\mathbf W}} w = \ \bar P n - P \bar n, \end{aligned} \right. \end{equation}
where
\[
m := \frac{r_\alpha +R_\alpha w}{J} + \frac{\bar R w_\alpha}{(1+{\mathbf W})^2}, \qquad n := \frac{ \bar R(r_{\alpha}+R_\alpha w)}{1+{\mathbf W}}. \]
In particular, we remark that the linearization of the system \eqref{ww2d-diff} around the zero solution is \begin{equation} \label{ww2d-0} \left\{ \begin{aligned}
& w_{ t} + r_\alpha = 0, \\ & r_t - i w = 0. \end{aligned} \right. \end{equation} The analysis of the linearized equation, carried out in Section~\ref{s:linearized}, is a key component of this paper.
It is also useful to further differentiate \eqref{ww2d-diff}, in order to obtain a system for $({\mathbf W}_{\alpha}, R_{\alpha})$: \[ \left\{ \begin{aligned}
& {\mathbf W}_{ \alpha t} + b {\mathbf W}_{ \alpha \alpha} + \frac{[(1+{\mathbf W}) R_\alpha]_\alpha}{1+\bar {\mathbf W}} = - b_\alpha {\mathbf W}_\alpha + (1+{\mathbf W}) R_\alpha \bar Y_\alpha +
{\mathbf W}_\alpha M
+ (1+{\mathbf W}) M_\alpha,
\\
& R_{t\alpha} + bR_{\alpha\alpha} =- b_\alpha R_\alpha+ i\left(\frac{(1+a){\mathbf W}_\alpha}{(1+{\mathbf W})^2} - \frac{ a_\alpha}{1+{\mathbf W}}\right). \end{aligned} \right. \] In order to better compare this with the linearized system we introduce the modified variable $\mathbf R := R_\alpha(1+{\mathbf W})$ to get the system \[ \left\{ \begin{aligned}
& {\mathbf W}_{ \alpha t} + b {\mathbf W}_{ \alpha \alpha} + \frac{\mathbf R_\alpha}{1+\bar {\mathbf W}} = - b_\alpha {\mathbf W}_\alpha + \mathbf R \bar Y_\alpha +
{\mathbf W}_\alpha M
+ (1+{\mathbf W}) M_\alpha , \\ & \mathbf R_{t} + b\mathbf R_{\alpha} =-\left(b_\alpha+ \frac{R_\alpha}{1+\bar {\mathbf W}}\right) \mathbf R + i\left(\frac{(1+a){\mathbf W}_\alpha}{1+{\mathbf W}} - a_\alpha\right) + \mathbf R M . \end{aligned} \right. \] Expanding the $b_\alpha$ terms via \eqref{M-def} this yields \begin{equation} \left\{ \begin{aligned} & {\mathbf W}_{ \alpha t} + b {\mathbf W}_{ \alpha \alpha} + \frac{\mathbf R_\alpha}{1+\bar {\mathbf W}} + \frac{R_\alpha}{1+\bar {\mathbf W}} {\mathbf W}_\alpha = G_2, \\ & \mathbf R_{t} + b\mathbf R_{\alpha} - i\frac{(1+a){\mathbf W}_\alpha}{1+{\mathbf W}} =K_2, \end{aligned} \right. \label{WR-diff} \end{equation} where \begin{equation*} \left\{ \begin{aligned} & G_2 = \mathbf R \bar Y_\alpha - \frac{\bar R_\alpha}{1+{\mathbf W}} {\mathbf W}_\alpha + 2M {\mathbf W}_\alpha + (1+{\mathbf W}) M_\alpha , \\ & K_{2}=-2\left(\frac{\bar R_\alpha}{1+{\mathbf W}}+ \frac{R_\alpha}{1+\bar {\mathbf W}}\right) \mathbf R + 2 M \mathbf R + ( R_\alpha \bar R_\alpha -i a_\alpha). \end{aligned} \right.
\end{equation*}
Next, we define our function spaces. The system \eqref{ww2d-0} is a well-posed linear evolution in the space ${\dot{\mathcal H} }_0$ of holomorphic functions endowed with the $L^2 \times \dot H^{\frac12}$ norm. A conserved energy for this system is \begin{equation}\label{E0}
E_0 (w,r) = \int \frac12 |w|^2 + \frac{1}{2i} (r \bar r_\alpha - \bar r r_\alpha) d\alpha. \end{equation} The nonlinear system \eqref{ww2d1} also admits a conserved energy, which has the form \begin{equation}\label{ww-energy}
E(W,Q) = \int \frac12 |W|^2 + \frac1{2i} (Q \bar Q_\alpha - \bar Q Q_\alpha) - \frac{1}{4} (\bar W^2 W_\alpha + W^2 \bar W_\alpha)\, d\alpha. \end{equation}
As suggested by the above energy, our main function spaces for the
differentiated water wave system \eqref{ww2d-diff} are the spaces ${\dot{\mathcal H} }_n$ endowed with the norm \[
\| ({\mathbf W},R) \|_{{\dot{\mathcal H} }_n}^2 := \sum_{k=0}^n
\| \partial^k_\alpha ({\mathbf W},R)\|_{ L^2 \times \dot H^\frac12}^2, \] where $n \geq 1$. As an auxiliary step, we will also consider solutions $({\mathbf W},R)$ in the smaller space \[ {\mathcal H }_n := H^n \times H^{n+\frac12}, \] with $n \geq 2$.
To describe the lifespan of the solutions we define the control norms \begin{equation}\label{A-def}
A := \|{\mathbf W}\|_{L^\infty}+\| Y\|_{L^\infty} + \||D|^\frac12 R\|_{L^\infty \cap B^{0,\infty}_{2}}, \end{equation} respectively \begin{equation}\label{B-def}
B :=\||D|^\frac12 {\mathbf W}\|_{BMO} + \| R_\alpha\|_{BMO}. \end{equation}
where $|D|$ represents the multiplier with symbol $|\xi|$. Here $A$ is a scale invariant quantity, while $B$ corresponds to the homogeneous ${\dot{\mathcal H} }_1$ norm of $({\mathbf W}, R)$. We note that $B$ and all but the $Y$ component of $A$ are controlled by the ${\dot{\mathcal H} }_1$ norm of the solution.
Now we are ready to state our main local well-posedness result: \begin{theorem} \label{baiatul}
Let $ n \geq 1$. The system \eqref{ww2d-diff} is locally well-posed for data in ${\dot{\mathcal H} }_n(\mathbb{R})$ so that $|{\mathbf W}+1| > c > 0$ . Further, the solution can be continued for as long as $A$ and $B$ remain bounded. The same result holds in the periodic setting. \end{theorem} In terms of Sobolev regularity of the data, this result improves the thresholds in earlier results of Wu~\cite{wu2, wu} and Alazard-Burq-Zuily~\cite{abz}. However, a direct comparison is nontrivial due to the fact that the above two papers use different coordinate frames, namely Lagrangian, respectively Eulerian.
As an interesting side remark, the above result makes no requirement that the curve $\{ Z(\alpha); \alpha \in \mathbb{R}\}$ determined by ${\mathbf W}$ be nonself-intersecting. If self-intersections occur then the physical interpretation is lost, but the well-posedness of the system \eqref{ww2d-diff} is not affected.
Our second goal in this article is to consider the question of obtaining improved lifespan bounds for the small data problem. Since the nonlinearities in our equations contain quadratic terms, the standard result is to obtain an $O(\epsilon^{-1})$ lifespan for smooth initial data of size $\epsilon$. However, this problem has the additional feature that there exists a quadratic normal form transformation which eliminates the quadratic terms in the equation. In the setting of holomorphic coordinates considered in this paper, this is most readily seen at the level of the system \eqref{ww2d1}. There, the quadratically nonlinear terms may be removed from the water-wave equations by the near-identity, normal form transformation \begin{equation} \tilde W = W - 2 \mathfrak M_{\Re W} W_\alpha, \qquad \tilde Q = Q - 2 \mathfrak M_{\Re W} R, \label{nft1} \end{equation} where the holomorphic multiplication operator $\mathfrak M_f$ is given by $\mathfrak M_f g = P\left[ fg\right] $. For a more symmetric form of this transformation, one can replace $R$ by $Q_{\alpha}$. However, it is more convenient to use the diagonal variable $R$. For $({\tilde W},{\tilde Q})$ we have \begin{proposition}\label{p:normal} The normal form variables (\ref{nft1}) satisfy equations of the form \begin{equation} \left\{ \begin{aligned} &{\tilde W}_t + {\tilde Q}_\alpha = \tilde{G}, \\ &{\tilde Q}_t - i {\tilde W} = \tilde{K}, \end{aligned} \right. \label{nft1eq} \end{equation} where $\tilde{G}$, $\tilde{K}$ are cubic (and higher order) functions of $(W, {\mathbf W},R, {\mathbf W}_{\alpha}, R_{\alpha})$, given by \begin{equation} \label{gk-tilde} \left\{ \begin{aligned} \tilde G = & \ 2P[ (F - R)_\alpha \Re W + {\mathbf W}_{ \alpha} F\Re W + {\mathbf W} \Re ({\mathbf W} F)+F_{\alpha }{\mathbf W}\Re W] \\ & \ - P[\bar {\mathbf W} R \bar Y - {\mathbf W}(P[\bar R Y]+\bar P[R \bar Y])], \\ \tilde K = & \ P\left[ (\bar{F}(1+\bar{{\mathbf W}})-\bar{R})R+2iP\left[ \frac{{\mathbf W}^2+a}{1+{\mathbf W}}\right]\cdot \Re W +2P\left[ bR_{\alpha}\right]\cdot\Re W \right]. \end{aligned} \right. \end{equation} \end{proposition}
The proof is straightforward; one rewrites the system \eqref{ww2d1}
in terms of the normal form variables $(\tilde{W}, \tilde{Q})$,
\eqref{nft}. The original variables are $(W,Q)$, but the derivatives
of $Q$ from the perturbative terms $G$ and $K$ are expressed in terms of $R$
and eliminated. We also make use of the identity $P+\bar{P}=I$. The details are left for the reader. We note that the difference $R-F$ is quadratic, \[ R-F = P[R \bar Y- \bar R Y]. \]
Heuristically, having cubic nonlinearities yields an improved $O(\epsilon^{-2})$ lifespan for initial data of size $\epsilon$. However, implementing this idea directly is fraught with difficulties. To start with, while $\tilde{G}$, $\tilde{K}$ are cubic and higher order terms they also depend on higher-order derivatives of $(W,Q)$; thus it is not possible to directly close energy estimates for the normal form variables $({\tilde W},{\tilde Q})$. This is related to the fact that the normal form transformation \eqref{nft1} is not invertible, and further to the fact that the system \eqref{ww2d1} is fully nonlinear, as opposed to semilinear.
There are at least two existing methods in the literature which attempt to address this difficulty. One such method, introduced by Wu~\cite{wu}, is based on the idea that any transformation which agrees quadratically with the above normal form transform will have the same effect as the normal form transform, but perhaps one can also choose such a transformation such that it is invertible. In Wu's work this transformation is an implicit change of coordinates, which is further followed by a secondary normal form transformation. A related example where an implicit change of coordinates is fully sufficient appears in the work \cite{hi} of the first two authors for the related Burgers-Hilbert problem.
A second method, which appears in the work of Shatah etc \cite{s}, is based on
a mix of quadratic energy estimates for high derivatives of the
solutions, combined with a normal form method for low
derivatives. This works well for water waves in dimension three, but
is not precise enough for the two dimensional problem.
In the present paper we propose an alternative approach for two dimensional water waves, which seems to be both simpler and more accurate. Precisely, rather than attempting to modify the equations using a normal form transform, we instead construct modified energy functionals which have cubic accuracy. A significant advantage of this idea is that it applies even for the leading order energy functionals, which to our knowledge is new. In a simpler setting, this method was first introduced by the authors in \cite{BH} in the context of the Burgers-Hilbert problem.
Our first result is translation invariant, and yields a cubic lifespan bound.
\begin{theorem} \label{t:cubic}
Let $\epsilon \ll 1$. Assume that the initial data for the equation
\eqref{ww2d-diff} on either $\mathbb{R}$ or $\mathbb{S}^1$ satisfies \begin{equation}
\|({\mathbf W}(0), R(0))\|_{{\dot{\mathcal H} }_1} \leq \epsilon. \end{equation} Then the solution exists on an $\epsilon^{-2}$ sized time interval $I_\epsilon = [0,T_\epsilon]$ , and satisfies a similar bound. In addition, the estimates \[
\sup_{t \in I_\epsilon} \| ({\mathbf W}(t), R(t))\|_{{\dot{\mathcal H} }_n} \lesssim \| ({\mathbf W}(0), R(0))\|_{{\dot{\mathcal H} }_n}, \qquad n \geq 2, \] hold whenever the right hand side is finite. \end{theorem}
Our second result assumes some additional localization for the initial data, and establishes almost global existence of solutions. This applies only for the problem on $\mathbb{R}$, and relies on the dispersive properties of the linear equation \eqref{ww2d-0}, whose solutions with localized data have $t^{-\frac12}$ dispersive decay. To state the result we need to return to the original set of variables $(W,Q)$. We also take advantage of the scale invariance of the water wave equations. Precisely, it is invariant with respect to the scaling law \[ (W(t,\alpha), Q(t,\alpha)) \to (\lambda^{-2} W(\lambda t,\lambda^2 \alpha),
\lambda^{-3} Q(\lambda t,\lambda^2 \alpha)). \] This suggests that we should use the scaling vector field \[ S = t \partial_t + 2 \alpha \partial_\alpha, \] and its action on the pair $(W,Q)$, namely \[ {\mathbf S}(W,Q) = ((S-2)W,(S-3)Q). \]
However, these are not the correct diagonal variables; to diagonalize we use the notations \[ (w,r) =: \mathbf A{\mathbf S}(W,Q). \] Then $({\mathbf W}, R)$ solve the linearized equations \ref{lin(wr)00} and define the weighted energy \begin{equation}\label{WH}
\|(W,Q)(t)\|_{{\mathcal{WH} }}^2 := \|(W,Q)(t)\|_{{\dot{\mathcal H} }_0}^2 + \|({\mathbf W},R)(t)\|_{{\dot{\mathcal H} }_5}^2 + \|(w,r)(t)\|_{{\dot{\mathcal H} }_0}^2. \end{equation}
Then we have \begin{theorem} \label{t:almost} There exists $ c > 0$ so that for each initial data $(W(0),Q(0))$ for the system \eqref{ww2d1} satisfying \begin{equation}\label{data}
\|(W,Q)(0)\|_{{\mathcal{WH} }}^2 \leq \epsilon \ll 1, \end{equation} the solution exists up to time $T_\epsilon = e^{c\epsilon ^{-2}}$ and satisfies \begin{equation}\label{almost-e}
\|(W,Q)(t)\|_{{\mathcal{WH} }}^2 \lesssim \epsilon, \qquad |t| < T_\epsilon. \end{equation} as well as \begin{equation}\label{almost-e-point}
|W|+|W_\alpha| + ||D|^\frac12 W_{\alpha}| + |R| + |R_\alpha| \lesssim
\frac{\epsilon}{\langle t \rangle^\frac12}, \qquad |t| < T_\epsilon. \end{equation} \end{theorem}
This lifespan bound was originally established by Wu~\cite{wu}. Here, we prove the same result under less restrictive assumptions, and, hopefully, with a simpler proof. We should also mention here the recent work of Ionescu-Pusateri~\cite{ip0},\cite{ip} and Alazard-Delort~\cite{ad}, where global well-posedness is proved for small localized data. In a follow-up paper we provide a simplified proof of this result as well.
While our research for this paper was largely complete by the time \cite{ip} and \cite{ad} appeared, there is one idea from Ionescu and Pusateri's article \cite{ip} which we adopted here in order to shorten the exposition; this is the fact that in order to close the estimates it suffices to use a single iteration of the scaling vector field $S$. However, our implementation of this idea is different from \cite{ip}, and also more efficient, in the sense that we use no higher derivatives of ${\mathbf S}(W,Q)$.
For the reminder of the introduction we provide a brief outline of the paper. The first step of the analysis is to study the linearization of the equation \eqref{ww2d1}; this is done in Section~\ref{s:linearized}. We begin with the diagonalisation of the linearized equations; this in turn leads to energy estimates, which are crucial in the proof of the local well-posedness result. The linearized energy functional is then refined so that cubically nonlinear estimates can be proved; this is essential in the proof of the improved lifespan result. We make no use of dispersive decay in this normal form analysis, so it works also for spatially periodic solutions. The low regularity threshold is reached by using various bilinear Coifman-Meyer type estimates, as well as multilinear versions thereof.
In Section~\ref{s:ee} we consider the equations for higher order derivatives of the solution. The principal part of these equations is closely related to the linearized equations studied in the previous section. After some normalization, the quadratic bounds follow directly from the ones for the linearized equation. The emphasis there is again on obtaining cubically nonlinear estimates. The essential idea is to construct a modified energy functional with better estimates. Our modified energy essentially combines the linearized energy, for the leading part, with the cubic normal form energy for the lower order terms. This is similar to the approach in the paper \cite{BH} devoted to the Burgers-Hilbert problem.
Section~\ref{s:lwp} contains the proof of the local well-posedness result. We begin with more regular data, both in terms of low frequencies and in terms of high frequencies. For such data, a standard mollifier technique suffices in order to establish well-posedness. The rough ${\dot{\mathcal H} }_1$ solutions are obtained as uniform limits of smooth solutions by using the estimates for the linearized equation. The same construction yields their continuous dependence on data.
In Section~\ref{s:cubic} we prove the cubic lifespan bounds for small initial data in Theorem~\ref{baiatul}.
In Section~\ref{s:decay} we provide the proof of the long time results. The cubic lifespan result is a straightforward consequence of the cubic energy estimates. The proof of the almost global result is slightly more involved, as it requires, as an intermediate step, to prove the $t^{-\frac12}$ dispersive decay for a limited number of derivatives of $({\mathbf W},R)$. These bounds are obtained from the vector field energy estimates, essentially in an elliptic fashion via Sobolev type embeddings.
Appendix~\ref{holom-eq} includes, for reader's convenience, a complete derivation of the holomorphic water wave equations. Finally, Appendix~\ref{s:multilinear} contains a collection of bilinear, multilinear and commutator estimates which are used at various places in the paper. We are grateful to Camil Muscalu for useful conversations pointing us in the right direction for this last section.
\section{ The linearized equation} \label{s:linearized}
In this section we derive the linearized water wave equations, and prove energy estimates for them. We do this in three stages. First we prove quadratic energy estimates in ${\dot{\mathcal H} }_0$, which apply for the large data problem. Then we prove cubic energy estimates in ${\dot{\mathcal H} }_0$ for the small data problem.
Various bilinear, multilinear and commutator estimates which are used in this section are collected in Appendix~\ref{s:multilinear}.
\subsection{ Computing the linearization} The solutions for the linearized water wave equation around a solution $\left( W,Q\right) $ are denoted by $(w,q)$. However, it will be more convenient to immediately switch to diagonal variables $(w,r)$, where \[ r := q - Rw. \] The linearization of $R$ is \[ \delta R = \dfrac{q_{\alpha}- Rw_{\alpha}}{1+{\mathbf W}} = \dfrac{r_{\alpha}+ R_\alpha w}{1+{\mathbf W}}, \] while the linearization of $F$ can be expressed in the form \[ \delta F = P[ m - \bar m], \] where the auxiliary variable $m$ corresponds to differentiating $F$ with respect to the holomorphic variables, \[ m := \frac{q_\alpha - R w_\alpha}{J} + \frac{\bar R w_\alpha}{(1+{\mathbf W})^2} =
\frac{r_\alpha +R_\alpha w}{J} + \frac{\bar R w_\alpha}{(1+{\mathbf W})^2}. \] Denoting also \[ n := \bar R \delta R = \frac{ \bar R(r_{\alpha}+R_\alpha w)}{1+{\mathbf W}}, \] the linearized water wave equations take the form \begin{equation*} \left\{ \begin{aligned} &w_{t}+ F w_\alpha + (1+ {\mathbf W}) P[ m-\bar m] = 0, \\ &q_{t}+ F q_\alpha + Q_\alpha P[m-\bar m] -i w +P\left[n+\bar n\right] =0. \end{aligned} \right. \end{equation*} Recalling that $b = F + \dfrac{\bar R}{1+{\mathbf W}}$, this becomes \begin{equation*} \left\{ \begin{aligned} &(\partial_t + b \partial_\alpha) w + (1+ {\mathbf W}) P[ m-\bar m] = \dfrac{\bar R w_\alpha}{1+{\mathbf W}} ,\\ & (\partial_t + b \partial_\alpha) q + Q_\alpha P[m-\bar m] -i w +P\left[n+\bar n\right] = \dfrac{\bar R q_\alpha}{1+{\mathbf W}}. \end{aligned} \right. \end{equation*} Now, we can use the second equation in \eqref{ww2d-diff} to switch from $q$ to $r$ and obtain \begin{equation*} \left\{ \begin{aligned} &(\partial_t + b \partial_\alpha) w + (1+ {\mathbf W}) P[ m-\bar m] = \dfrac{\bar R w_\alpha}{1+{\mathbf W}},\\ & (\partial_t + b \partial_\alpha) r -i \frac{1+a}{1+{\mathbf W}} w +P\left[n+\bar n\right] = \dfrac{\bar R (r_\alpha+ R_\alpha w)}{1+{\mathbf W}}. \end{aligned} \right. \end{equation*} Terms like $\bar P m$, $\bar Pn$ are lower order since the differentiated holomorphic variables have to be lower frequency. The same applies to their conjugates. Moving those terms to the right and taking advantage of algebraic cancellations we are left with \begin{equation}\label{lin(wr)0} \left\{ \begin{aligned} & (\partial_t + b \partial_\alpha) w + \frac{1}{1+\bar {\mathbf W}} r_\alpha + \frac{R_{\alpha} }{1+\bar {\mathbf W}} w = \mathcal{G}(w,r),
\\ &(\partial_t + b \partial_\alpha) r - i \frac{1+a}{1+{\mathbf W}} w = \mathcal{K}(w,r), \end{aligned} \right. \end{equation} where \begin{equation*} \begin{aligned} \mathcal{G}(w,r) = \ (1+{\mathbf W}) (P \bar m + \bar P m), \quad \mathcal{K}(w,r) = \ \bar P n - P \bar n. \end{aligned} \end{equation*}
We remark that while $(w,r)$ are holomorphic, it is not directly obvious that the above evolution preserves the space of holomorphic states. To remedy this one can also project the linearized equations onto the space of holomorphic functions via the projection $P$. Then we obtain the equations \begin{equation}\label{lin(wr)} \left\{ \begin{aligned} & (\partial_t + \mathfrak M_b \partial_\alpha) w + P \left[ \frac{1}{1+\bar {\mathbf W}} r_\alpha\right] + P \left[ \frac{R_{\alpha} }{1+\bar {\mathbf W}} w \right] = P \mathcal{G}( w, r),
\\ &(\partial_t + \mathfrak M_b \partial_\alpha) r - i P\left[ \frac{1+a}{1+{\mathbf W}} w\right] =
P \mathcal{K}( w,r). \end{aligned} \right. \end{equation} Since the original set of equations \eqref{ww2d1} is fully holomorphic, it follows that the two sets of equations, \eqref{lin(wr)0} and \eqref{lin(wr)}, are algebraically equivalent.
In order to obtain cubic linearized energy estimates it is also of interest to separate the quadratic parts $ \mathcal{G}^{2}$ and $\mathcal{K}^{2}$ of $\mathcal{G}$ and $\mathcal{K}$. These are split into quadratic and higher terms as shown below \[ \begin{split}
\mathcal{G} = & \, \mathcal{G}^{(2)}+ \mathcal{G}^{(3+)},
\qquad \mathcal{K} = \ \mathcal{K}^{(2)}+ \mathcal{K}^{(3+)}. \end{split} \] For the quadratic parts we have \[ \begin{split}
P \mathcal{G}^{(2)}(w,r) = & \ -P \left[ {\mathbf W} \bar r_\alpha
\right] + P\left[ R \bar w_\alpha\right], \qquad P \mathcal{K}^{(2)}(w,r) = - P \left[ R \bar r_\alpha\right], \end{split} \] with $\bar P \mathcal{G}^{(2)}(w,r) = \overline{P \mathcal{G}^{(2)}(w,r)}$ and $\bar P \mathcal{K}^{(2)}(w,r) = - \overline{P \mathcal{K}^{(2)}(w,r)}$. We can also rewrite the above expressions in a commutator form \begin{equation}\label{gk2} \begin{aligned} P \mathcal{G}^{(2)}(w,r)=-\left[ P,{\mathbf W}\right] \bar{r}_{\alpha}+ \left[ P,R\right] \bar{w}_{\alpha},\quad P \mathcal{K}^{(2)}(w,r)=- \left[P, R\right] \bar r_\alpha. \end{aligned} \end{equation}
The cubic terms have the form \[ \begin{split}
\mathcal{G}^{(3+)}(w,r) = \ P\bar m^{(3+)} + \bar P m^{(3+)} +{\mathbf W}( P \bar m + \bar P m),\quad \mathcal{K}^{(3+)}(w,r) = \ \bar P n^{(3+)} - P \bar n^{(3+)}. \end{split} \] For the purpose of simplifying nonlinear estimates, it is convenient to express $\mathcal G^{(3)}$ and $\mathcal K^{(3)}$ in a polynomial fashion. This is done using the variable $Y=\dfrac{{\mathbf W}}{1+{\mathbf W}}$. Then we have \[ \begin{split}
\bar P m = & \ \bar P [w_\alpha (1-Y)^2 \bar R- (r_\alpha + R_\alpha
w)(1-Y) \bar Y],
\\
\bar P m^{(3+)} = & \ \bar P [ r_\alpha (\bar {\mathbf W} + Y) \bar Y -
R_\alpha w (1-Y) \bar Y - w_\alpha (2Y-Y^2) \bar R],
\\
\bar P n^{(3+)} = & \ \bar P [- r_{\alpha} Y \bar R + R_\alpha w
(1-Y) \bar R]. \end{split} \]
\subsection{Quadratic estimates for large data.} Our goal here is to study the well-posedness of the system \eqref{lin(wr)} in $L^2 \times \dot H^\frac12$. We begin with a more general version of the system \eqref{lin(wr)}, namely \begin{equation}\label{lin(wr)inhom} \left\{ \begin{aligned} & (\partial_t + \mathfrak M_b \partial_\alpha) w + P \left[ \frac{1}{1+\bar {\mathbf W}} r_\alpha\right] + P \left[ \frac{R_{\alpha} }{1+\bar {\mathbf W}} w \right] = G,
\\ &(\partial_t + \mathfrak M_b \partial_\alpha) r - i P\left[ \frac{1+a}{1+{\mathbf W}} w\right] = K, \end{aligned} \right. \end{equation} and define the associated positive definite linear energy \[
E^{(2)}_{lin}(w,r) = \int_{\mathbf R} (1+a) |w|^2 + \Im ( r \bar r_\alpha) d\alpha . \] We remark that, by Proposition~\ref{regularity for a}, $a$ is nonnegative and bounded, therefore \[ E^{(2)}_{lin}(w,r) \approx_A E_0(w,r) \] Our first result uses the control parameters $A$ and $B$ defined in \eqref{A-def}, \eqref{B-def}:
\begin{proposition}\label{plin-short} a) The linear equation \eqref{lin(wr)inhom} is well-posed in ${\dot{\mathcal H} }_0$, and the following estimate holds: \begin{equation}\label{lin-gen2}
\frac{d}{dt} E^{(2)}_{lin}(w,r) = 2 \Re \int_{\mathbf R} (1+a) \bar w \, G - i \bar r_\alpha \, K \ d\alpha + O_A(A B) E^{(2)}_{lin}(w,r). \end{equation}
b) The linearized equation \eqref{lin(wr)} is well-posed in $L^2 \times \dot H^\frac12$, and the following estimate holds: \begin{equation}\label{lin2} \frac{d}{dt} E^{(2)}_{lin}(w,r) \lesssim_A
B E^{(2)}_{lin}(w,r). \end{equation} \end{proposition}
\begin{proof} a) A direct computation yields \[ \begin{split}
\frac{d}{dt} \int (1+a)|w|^2 d \alpha = & \ 2 \Re \!\! \int (1+a) \bar w (\partial_t + \mathfrak M_b \partial_\alpha ) w + a \bar w [b,P] w_\alpha\, d\alpha ,\\ & \ + \int \left[ a_t+((1+a)b)_\alpha\right]
|w|^2 \, d \alpha . \end{split} \] A similar computation shows that \[ \frac{d}{dt} \int \Im ( r \partial_\alpha \bar r) \, d \alpha = 2 \Im \int (\partial_t + \mathfrak M_b \partial_\alpha) r \, \partial_\alpha \bar r \, d \alpha. \ \ \]
Adding the two and using the equations \eqref{lin(wr)inhom}, the quadratic $\Re (w \bar r_\alpha)$ term cancels modulo another commutator term, and we obtain \begin{equation}\label{dE2} \frac{d}{dt} E^{(2)}_{lin}(w,r) = 2\Re \int (1+a) \bar w\, G- i \bar r_\alpha \, K\, d\alpha + err_1, \end{equation} where \[ \begin{aligned}
err_1 = & \int \left[ a_t+((1+a)b)_\alpha\right] |w|^2 d\alpha -
2\Re \int (1+a)\frac{R_{\alpha}}{1+\bar{{\mathbf W}}}\vert w\vert ^2\, d\alpha
\\ & -2 \Re \int + a \bar w \, (\left[\bar Y, P\right] (
r_\alpha+R_\alpha w) + [P,b] w_\alpha)\, d\alpha. \end{aligned} \] Using the auxiliary function $M$ in \eqref{M-def}, we rewrite it as \[ \begin{aligned}
err_1 = \int \left( a_t+ba_{\alpha}\right) |w|^2 + M (1+a)\vert w\vert ^2 \, d\alpha - 2 \Re \int a \bar w \, (\left[\bar Y,P \right] (r_\alpha+R_\alpha w)
+ [P,b] w_\alpha) \, d\alpha. \end{aligned} \] The error term is at least quartic. To conclude the proof of \eqref{lin-gen2} it suffices to show that \begin{equation}\label{err12}
|err_1| \lesssim AB E^{(2)}_{lin}(w,r) . \end{equation}
For the first term, by Proposition~\ref{regularity for a} in the \emph{Appendix}~\ref{s:multilinear}, we have $\vert a_t+ba_{\alpha}\vert \lesssim AB$. For the second term we combine the pointwise bounds $\vert a\vert
\lesssim A^2$ in Lemma~\ref{regularity for a} together with $\|M\|_{L^\infty} \lesssim AB$ in Lemma~\ref{l:M}.
For the last term it remains to estimate the commutators in $L^2$. Two of them are obtained using Lemma~\ref{l:com}, \[
\|\left[\bar Y ,P\right] r_\alpha\|_{L^2} \lesssim \| |D|^\frac12 Y\|_{BMO}
\| r \|_{\dot H^\frac12}, \qquad \| [P,b] w_\alpha\|_{L^2} \lesssim
\|b_\alpha\|_{BMO} \|w\|_{L^2}, \] and suffice due to the bounds for $b$ and $Y$ in Lemmas~\ref{l:b},\ref{l:Y}. For the remaining piece we write $[\bar Y, P] (R_\alpha w)= [\bar P, \bar P [\bar Y R_\alpha]] w]$ and use \eqref{CM} to estimate \[
\| \bar P[\bar P [\bar Y R_\alpha] w]\|_{L^2} \lesssim
\|w\|_{L^2} \|\bar P [\bar Y R_\alpha]\|_{BMO} \lesssim
\|w\|_{L^2} \||D|^\frac12 Y\|_{BMO} \| |D|^\frac12 R\|_{BMO}, \] where the bilinear bound in the second step follows after a bilinear Littlewood-Paley decomposition from \eqref{bmo-bmo} and \eqref{bmo>infty}.
b) To estimate the terms involving $ \mathcal{G}$ and $ \mathcal{K}$ we separate the quadratic and cubic parts. It suffices to show that the quadratic terms satisfy \begin{equation}\label{gk2-est}
\| \mathcal{G}^{(2)}(w, r)\|_{L^2} + \| \mathcal{K}^{(2)}(w, r)\|_{\dot H^\frac12}
\lesssim_A B (\|w\|_{L^2} + \|r\|_{\dot H^\frac12}), \end{equation} while the cubic and higher terms satisfy \begin{equation}\label{gk3-est}
\| \mathcal{G}^{(3+)}(w, r)\|_{L^2} + \| \mathcal{K}^{(3+)}(w, r)\|_{\dot H^\frac12}
\lesssim_A AB(\|w\|_{L^2} + \|r\|_{\dot H^\frac12}). \end{equation}
In order to obtain the estimates claimed in \eqref{gk2-est},\eqref{gk3-est} we use the Coifman-Meyer \cite{cm} type commutator estimates described in the \emph{Appendix}~\ref{s:multilinear}, Lemma~\ref{l:com}. Precisely, for the first term in $ P\mathcal{G}^{(2)}(w,r)$ we use \eqref{first-com} with $s=0$, and $\sigma = \frac12$ to write \[
\| \left[ P,{\mathbf W}\right] \bar{r}_{\alpha}\|_{L^2} \lesssim \||D|^\frac12 {\mathbf W}\|_{BMO} \|r\|_{\dot H^\frac12}. \] For the second term in $ P\mathcal{G}^{(2)}(w,r)$ we use \eqref{first-com} with $s=0$ and $\sigma = 1$ to obtain \[
\| \left[ P,R\right] \bar{w}_{\alpha}\|_{L^2} \lesssim \| R_\alpha\|_{BMO} \|w\|_{L^2}, \] and for $P\mathcal{K}^{(2)}(w,r)$ we use \eqref{first-com} with $s = \frac12$, and $\sigma = \frac12$, and conclude that \[
\| \left[P, R\right] \bar r_\alpha\|_{\dot H^\frac12} \lesssim \|R_\alpha\|_{BMO} \|r\|_{\dot H^\frac12}. \]
The same estimate applies to the antiholomorphic parts of $\mathcal{G}^{(2)}$ and $ \mathcal{K}^{(2)}$, and \eqref{gk2} follows.
For the cubic and higher parts of $\mathcal{G}$ and $\mathcal{K}$ we apply the same type of commutator estimates, as well as the $BMO$ bounds in Proposition~\ref{p:bmo}, as follows: \[
\| \bar P [r_\alpha (1-Y) \bar Y \bar {\mathbf W}]\|_{L^2} \lesssim
\| r\|_{\dot H^\frac12} \| (1-Y) \bar Y \bar {\mathbf W}\|_{BMO^\frac12}
\lesssim_A \|Y\|_{L^\infty} \| r\|_{\dot H^\frac12} , \] using \eqref{bmo-alg} at the last step. \[
\| \bar P[w(1-Y) R_\alpha \bar Y]\|_{L^2}
\lesssim \|w(1-Y)\|_{L^2} \| \bar P[R_\alpha \bar Y]\|_{BMO}
\lesssim_A \|w\|_{L^2} \|R\|_{BMO^\frac12} \|Y\|_{BMO^\frac12} \] using \eqref{bmo-bmo} and \eqref{bmo>infty} at the last step. \[
\| \bar P[ w_\alpha(2Y-Y^2) \bar R]\|_{L^2}
\lesssim \|w\|_{L^2} \| \partial_\alpha \bar P[(2Y-Y^2) \bar R]\|_{BMO},
\lesssim_A \|w\|_{L^2} \|Y\|_{L^\infty} \| R\|_{BMO} \] using \eqref{bmo-bmo}, and \eqref{bmo-infty} at the last step. \[
\| |D|^\frac12 \bar P[r_\alpha Y \bar R]\|_{L^2}
\lesssim \| r\|_{\dot H^\frac12} \| \partial_\alpha \bar P[ Y \bar R]\|_{L^2}
\lesssim_A \| r\|_{\dot H^\frac12} \|Y\|_{L^\infty} \| R\|_{BMO}, \] again by \eqref{bmo-bmo} and \eqref{bmo-infty}. Finally, \[ \begin{split}
\| |D|^\frac12 \bar P[ w(1-Y) R_\alpha \bar R]\|_{L^2}
\lesssim & \ \| w(1-Y)\|_{L^2} \| |D|^\frac12 \bar P[R_\alpha \bar R]\|_{BMO} \\
\lesssim_A & \ \|w\|_{L^2} \| |D|^\frac12 R\|_{BMO} \|R_\alpha\|_{BMO} \end{split} \] follows using \eqref{bmo-bmo} and \eqref{bmo>infty}.
\end{proof}
\subsection{Cubic estimates for small data.} For the small data problem it is of further interest to track the solution on larger time scales. For this we add to the equations the holomorphic quadratic parts $P \mathcal G^{(2)}$ and $P \mathcal K^{(2)}$ of $\mathcal G$ and $\mathcal K$ and consider the linear equations \begin{equation}\label{lin(wr)inhom3} \left\{ \begin{aligned}
& (\partial_t + \mathfrak M_b \partial_\alpha) w + P \left[ \frac{1}{1+\bar
{\mathbf W}} r_\alpha\right] + P \left[ \frac{R_{\alpha} }{1+\bar {\mathbf W}} w
\right] = -P \left[ {\mathbf W} \bar r_\alpha - R \bar w_\alpha\right] + G,
\\
&(\partial_t + \mathfrak M_b \partial_\alpha) r - i P\left[ \frac{1+a}{1+{\mathbf W}}
w\right] =- P \left[ R \bar r_\alpha\right]+ K. \end{aligned} \right. \end{equation} For this problem we add appropriate cubic terms and define the modified energy \[
E^{(3)}_{lin}(w,r) = \int_{\mathbf R} (1+a) |w|^2 + \Im (r \bar r_\alpha) + 2 \Im (\bar R w r_\alpha) -2\Re(\bar{{\mathbf W}} w^2)\, d\alpha. \]
Then we have: \begin{proposition}\label{plin-long} Assume that $A \ll1$. Then \begin{equation}\label{elin3-eq}
E^{(3)}_{lin}(w,r) = (1+O(A)) E_0(w,r). \end{equation} In addition, the following properties hold:
a) The solutions to \eqref{lin(wr)inhom3}
satisfy \begin{equation}\label{elin3-dinhom} \begin{split}
\frac{d}{dt} E^{(3)}_{lin}(w,r) = & \ 2 \Re \int \left((1+a) \bar w - i\bar R r_\alpha - 2\bar {\mathbf W} w\right) \, G + i (\bar r - \bar R w) \, K_\alpha \, d\alpha \\ & + O_A(AB) E^{(3)}_{lin}(w,r). \end{split} \end{equation}
b) For solutions to the linearized equation \eqref{lin(wr)} we have: \begin{equation}\label{elin3-diff} \frac{d}{dt} E^{(3)}_{lin}(w,r) \lesssim_A AB E^{(3)}_{lin}(w,r). \end{equation} \end{proposition}
\begin{proof} For \eqref{elin3-eq} we need to estimate the added cubic terms in $E^{(3)}_{lin}(w,r)$. The second is trivially bounded, while the first is rewritten as \[ \Im \int w \bar P[\bar R r_\alpha] \, d\alpha. \]
By Lemma~\ref{l:com} we have $\| P[\bar R r_\alpha]\|_{L^2} \lesssim
\||D|^\frac12 R\|_{BMO} \|r\|_{{\dot H^\frac12}}$, hence \eqref{elin3-eq} follows.
a) To prove the estimate \eqref{elin3-dinhom} we compute the time derivative of the cubic component of the energy $E^3_{lin}(w,r)$, using the projected equations for $w$ and $r$ and the unprojected equations for $R$ and ${\mathbf W}$: \begin{equation*} \begin{aligned} \frac{d}{dt} \left( \Im \int \bar{R} w r_\alpha \, d\alpha -\Re\int \bar{{\mathbf W}}w^2d\alpha\right) =&
\Im \! \int \! - i \bar {\mathbf W} w r_\alpha - \bar{R} r_\alpha r_\alpha+ i \bar{R} w w_\alpha + \bar R r_\alpha G + \bar R w K_\alpha \, d\alpha \\
& +\Re\int \bar{R}_{\alpha}w^2+2\bar{{\mathbf W}} wr_{\alpha}+ 2 \bar {\mathbf W} w F\, d\alpha+ err_2,
\end{aligned} \end{equation*} where \begin{equation} \label{err3} \begin{aligned} err_2 = & \ \Im \int\!\! \left\lbrace \left( i\left( \frac{\bar{{\mathbf W}}^2+a}{1+\bar{{\mathbf W}}} \right)-b\bar{R}_{\alpha} \right) wr_{\alpha} - \bar{R}w\partial_{\alpha}\left( \mathfrak M_{b}r_{\alpha}-iP\left[ \frac{a-{\mathbf W}}{1+{\mathbf W}}w\right] + P[R\bar{r}_\alpha]\right)\right.
\\
&\left. \qquad -\bar{R}r_{\alpha} \left( \mathfrak M_{b}w_{\alpha}-P\left[ \frac{\bar{{\mathbf W}}}{1+\bar{{\mathbf W}}}r_{\alpha}\right]
+P\left[ \frac{R_{\alpha}}{1+\bar{{\mathbf W}}}w\right] + P[{\mathbf W} \bar{r}_\alpha - R\bar{w}_\alpha] \right) \right\rbrace \, d\alpha \\ &+ \Re\int \left\{ w^2\left( b \bar {\mathbf W}_{\alpha}+ \frac{\bar{{\mathbf W}}-{\mathbf W}}{1+{\mathbf W}}\bar{R}_{\alpha} - (1+\bar{{\mathbf W}})\bar{M} \right) \right. \\ & \qquad \left. + 2\bar{{\mathbf W}} w \left( \mathfrak M_{b}w_{\alpha}- P\left[ \frac{\bar{{\mathbf W}}}{1+\bar{{\mathbf W}}}r_{\alpha}\right] +P\left[ \frac{R_{\alpha}}{1+\bar{{\mathbf W}}}w \right] + P[{\mathbf W} \bar{r}_\alpha - R\bar{w}_\alpha]\right) \right\} \, d\alpha . \end{aligned} \end{equation}
Adding this to \eqref{lin-gen2} (but applied to solutions to \eqref{lin(wr)inhom3}) we obtain \begin{equation}\label{dE3} \frac{d}{dt} E^{(3)}_{lin}(w,r) = \ 2 \Re \int \left((1+a) \bar w - i
\bar R r_\alpha - 2\bar {\mathbf W} w\right) \, G + i (\bar r - \bar R
w) \, K_\alpha \, d\alpha+
err_1+ err_3, \end{equation} where \[ err_3 = 2 err_2 - 2 \Re\int a\bar{w} P\left[{\mathbf W}\bar{r}_\alpha - R \bar{w}_\alpha\right]\, d\alpha. \] Given the bound \eqref{err12} for $err_1$, the proof of \eqref{elin3-dinhom} is concluded if we show that \begin{equation}\label{error3}
|err_3| \lesssim AB E_0(w,r) . \end{equation} Further, recalling the estimate \eqref{gk2-est}, which in expanded form reads \begin{equation}\label{GK2}
\|P\left[{\mathbf W}\bar{r}_\alpha - R \bar{w}_\alpha\right]\|_{L^2}+
\|P[R \bar r_\alpha]\|_{\dot H^\frac12} \lesssim B \|(w,r)\|_{L^2 \times \dot H^\frac12} , \end{equation} it suffices to estimate $err_2$, \begin{equation}\label{error2}
|err_2| \lesssim AB E_0(w,r) . \end{equation} For the remainder of the proof we separately estimate several types of terms in $err_2$:
{\bf A. Terms involving $b$.} Here, we use the bounds for $b$ in Lemma~\ref{l:b}, which give \[
\| b_\alpha\|_{BMO} \lesssim_A B, \qquad \||D|^\frac12 b\|_{BMO} \lesssim_A A. \] We first collect all the terms that are contained in the first integral in $err_2$ and include $b$, \[ I_1 = \int -b\bar{R}_{\alpha} wr_{\alpha}-\bar{R}r_{\alpha} \mathfrak M_{b}w_{\alpha}- \bar{R}w\partial_{\alpha} (\mathfrak M_{b}r_{\alpha})\, d\alpha. \] We claim that \begin{equation}\label{i1-est}
|I_1|
\lesssim (\||D|^\frac12 R\|_{BMO} \|b_\alpha\|_{BMO} + \|R_\alpha\|_{BMO}
\||D|^\frac12 b\|_{BMO}) \|w\|_{L^2} \|r\|_{\dot H^\frac12}. \end{equation} Integrating by parts we get $I_1 = I_2+I_3$, where \[ I_2 = \ \int \bar{R}_{\alpha} w \bar P[b r_{\alpha}] \, d\alpha, \qquad I_3 = \ \int -\bar{R}r_{\alpha} \mathfrak M_{b}w_{\alpha}- \bar{R}w\partial_{\alpha} (\mathfrak M_{b}r_{\alpha})\, d\alpha. \] The first term on the right has a commutator structure and will be estimated separately later, see $I_5$ below. The bound for $I_3$ is proved in the appendix, see \eqref{i1}.
We next collect all the terms that are contained in the second integral in $err_2$ and include $b$, and rewrite them as \begin{equation*} \label{b-terms} \begin{split} I_4= \! \!\int w^2\partial_{\alpha}\overline{\mathfrak M_{b}{\mathbf W}}+2\bar{{\mathbf W}}w\mathfrak M_{b}w_{\alpha}\, d\alpha= \! \!\int \! -2ww_{\alpha}b\bar{{\mathbf W}}+2\bar{{\mathbf W}}w\mathfrak M_{b}w_{\alpha}\, d\alpha = \! \!\int \! -2\bar{{\mathbf W}}w\bar P[b w_{\alpha}] \, d\alpha. \end{split} \end{equation*} The expression $\bar P[b w_{\alpha}]$ is bounded in $L^2$ using Lemma~\ref{l:com} to obtain \[
|I_4| \lesssim \|{\mathbf W}\|_{L^\infty} \|b_\alpha\|_{BMO} \|w\|_{L^2}^2. \]
{\bf B. Quadrilinear terms bounded via both $L^2 \cdot L^2$ and $\dot H^\frac12 \cdot \dot H^{-\frac12}$ pairings. } This includes the following expressions: \begin{equation*} \begin{aligned} I_5 &= \int \bar{R}_{\alpha} w \bar P[b r_{\alpha}] \, d\alpha = \int \bar P[b r_{\alpha}] P[ \bar{R}_{\alpha} w] \, d\alpha,\\ I_6 &= \int \bar{R}r_{\alpha}P\left[ \frac{\bar{{\mathbf W}}}{1+\bar{{\mathbf W}}}r_{\alpha}\right] \, d\alpha = \int\bar P[ \bar{R}r_{\alpha}] P\left[ \bar Y r_{\alpha}\right] \, d\alpha,\\ I_7 &= \int \bar{R}r_{\alpha}P\left[ \frac{R_{\alpha}}{1+\bar{{\mathbf W}}}w\right] \, d\alpha =\int \bar P [\bar{R}r_{\alpha}] P\left[ R_{\alpha} w(1-\bar Y)\right] \, d\alpha,\\ I_8 &= \int \bar{{\mathbf W}} w P\left[ \frac{R_{\alpha}}{1+\bar{{\mathbf W}}}w\right] \, d\alpha = \int \bar P[\bar{{\mathbf W}} w] P\left[ R_{\alpha}(1-\bar Y) w\right] \, d\alpha,\\ I_{9}& = \int \bar{{\mathbf W}} wP\left[ \frac{\bar{{\mathbf W}}}{1+\bar{{\mathbf W}}}r_{\alpha}\right] \, d\alpha = \int \bar P[\bar{{\mathbf W}}w] P\left[ \bar Y r_{\alpha}\right] \, d\alpha. \end{aligned} \end{equation*}
The strategy here is to bound the first factor in both $L^2$ and $\dot H^\frac12$, and the second, partially in $L^2$ and partially in $\dot H^{-\frac12}$. For the first factor we have by Lemma~\ref{l:com}: \begin{equation*} \begin{aligned}
&\| \bar P [b r_\alpha]\|_{L^2}+ \|\bar P[\bar R r_\alpha]\|_{L^2} \lesssim
(\||D|^\frac12 b\|_{BMO} + \||D|^\frac12 R\|_{BMO}) \|r\|_{\dot H^\frac12}
\lesssim A \|r\|_{\dot H^\frac12},\\
&\| \bar P [b r_\alpha]\|_{\dot H^\frac12}+ \|\bar P[\bar R r_\alpha]\|_{\dot H^\frac12} \lesssim
(\|b_\alpha\|_{BMO} + \| R_\alpha\|_{BMO}) \|r\|_{\dot H^\frac12}
\lesssim B \|r\|_{\dot H^\frac12}, \end{aligned} \end{equation*} as well as \begin{equation*} \begin{aligned}
&\|\bar P[\bar{{\mathbf W}}w]\|_{L^2}+ \||D|^\frac12 \bar P[\bar{R}w] \|_{L^2} \lesssim(
\| {{\mathbf W}}\|_{BMO}+ \| |D|^\frac12 R\|_{BMO}) \|w\|_{L^2}
\lesssim A \|w\|_{L^2},\\
&\|\bar P[\bar{{\mathbf W}}w]\| _{\dot H^\frac12} + \||D|^\frac12 \bar P[\bar{R}w] \| _{\dot H^\frac12}
\lesssim (\| |D|^\frac12 {{\mathbf W}}\|_{BMO}+ \|R_\alpha\|_{BMO}) \|w\|_{L^2}
\lesssim B \|w\|_{L^2}. \end{aligned} \end{equation*}
We now consider the second factor in the above integrals. For $P[\bar{R}_{\alpha} w]$ we have \[
\| \sum_k P[ \bar{R}_{k,\alpha} w_k]\|_{L^2} \lesssim \|R_\alpha\|_{BMO}
\|w\|_{L^2}, \quad
\| \sum_k P[ \bar{R}_{< k,\alpha} w_k]\|_{\dot H^{-\frac12}} \lesssim \||D|^\frac12 R\|_{BMO}
\|w\|_{L^2}. \] The same argument applies to $P\left[ R_{\alpha}(1-\bar Y) w\right]$ once we use the decomposition \begin{equation*}
P\left[ (1-\bar{Y})R_{\alpha}w\right] = P\left[ (1-\bar{Y}) \sum_{k\in \mathbf{Z}} R_{\alpha, \geq k}w_{k}\right] - P\left[ \sum_{k\in \mathbf{Z}} \bar{Y} _{k}R_{\alpha, <k}w_k\right] + \sum_{k\in \mathbf{Z}}(1-\bar{Y}) _{<k}R_{\alpha, <k}w_k. \end{equation*} The first term is easily bounded in $L^2$ by Lemma~\ref{l:com}. The second is also in $L^2$ using \eqref{bmo-infty} for the product of the first two factors. Finally, the third is bounded in $\dot H^{-\frac12}$ by estimating
$\| R_{\alpha,<k}\|_{L^\infty} \lesssim 2^{\frac{k}2} A$.
It remains to consider the expression \[ P\left[ \bar Y r_{\alpha}\right] = P\left[ \sum_k \bar Y_k r_{k,\alpha}\right] + \sum_k \bar Y_{<k} r_{k,\alpha}. \] Here, the first term is estimated in $L^2$ using Lemma~\ref{l:com}, while the second goes into $\dot H^{-\frac12}$.
{\bf C. Quadrilinear terms bounded via an $L^2 \cdot L^2$ pairing.} This includes the following expressions: \begin{equation*} \begin{aligned} &I_9 = \int \bar{R}w\partial_{\alpha}P\left[ \frac{a-{\mathbf W}}{1+{\mathbf W}} w \right] \,d\alpha =- \int \partial_\alpha \bar P[\bar{R}w] P\left[ (a(1-Y) - Y) w \right]\, d\alpha,\\ &I_{10} = \int \bar R w \partial_\alpha P[R \bar r_\alpha] \, d\alpha = - \int \partial_\alpha \bar P[\bar R w] \partial_\alpha P[R \bar r_\alpha] \, d\alpha ,\\ &I_{11} = \int \bar R r_\alpha P[{\mathbf W} \bar r_\alpha - R \bar w_\alpha]\, d\alpha = \int \bar P[\bar R r_\alpha] P[{\mathbf W} \bar r_\alpha - R \bar w_\alpha] d\alpha, \\ &I_{12} = \int \bar {\mathbf W} w P[{\mathbf W} \bar r_\alpha - R \bar w_\alpha] \, d\alpha = \int \bar P[\bar {\mathbf W} w] P[W \bar r_\alpha - R \bar w_\alpha] \, d\alpha . \end{aligned} \end{equation*}
In all cases both factors are estimated directly in $L^2$, using Lemma~\ref{l:com}, see also \eqref{GK2}.
{\bf D. Trilinear estimates.} This includes the terms: \begin{equation*} \begin{aligned} &I_{13} = \int \frac{\bar{{\mathbf W}}^2+a}{1+\bar{{\mathbf W}}}wr_{\alpha}\, d\alpha= \int w \bar P[\bar P f r_{\alpha}]\, d\alpha, \ \ \ \ \ \ \, \qquad f=\frac{\bar{{\mathbf W}}^2+a}{1+\bar{{\mathbf W}}},\\ &I_{14} = \int w^2\bar{P}\left[ \frac{\bar{{\mathbf W}}-{\mathbf W}}{1+{\mathbf W}} \bar{R}_{\alpha}\right]\, d \alpha = \int w \bar P[ \bar P g w] \, d\alpha, \quad \ \, g=\frac{\bar{{\mathbf W}}-{\mathbf W}}{1+{\mathbf W}} \bar{R}_{\alpha},\\ &I_{15} = \int w^2\bar{P}\left[ (1+\bar{{\mathbf W}}) M\right]\,d \alpha = \int w \bar P[ \bar P h w] \, d\alpha,\qquad h=(1+\bar{{\mathbf W}})M. \end{aligned} \end{equation*} Using Lemma~\ref{l:com} we have \begin{equation}\label{est-tri}
|I_{13}| \lesssim \| |D|^\frac12 \bar P f\|_{BMO} \|w\|_{L^2} \|r\|_{\dot H^\frac12}, \
|I_{14}| \lesssim \| \bar P g\|_{BMO} \|w\|_{L^2}^2, \
|I_{15}| \lesssim \| \bar P h\|_{BMO} \|w\|_{L^2}^2, \end{equation} so it suffices to show that \[
\| |D|^\frac12 f\|_{BMO} + \| g\|_{BMO}+ \| h\|_{BMO} \lesssim AB. \] The $f$ bound follows from the algebra property of $BMO^\frac12 \cap L^\infty$ in \eqref{bmo-alg} in view of \eqref{a-point} and \eqref{a-bmo+}. The $g$ bound is obtained by writing \[ \frac{\bar{{\mathbf W}}-{\mathbf W}}{1+{\mathbf W}} \bar{R}_{\alpha} = \sum_{k} P_{\leq k} \left(\frac{\bar{{\mathbf W}}-{\mathbf W}}{1+{\mathbf W}}\right) R_{k,\alpha} + \sum_{k} P_{ k} \left(\frac{\bar{{\mathbf W}}-{\mathbf W}}{1+{\mathbf W}}\right) R_{< k,\alpha}. \] For the first term we use \eqref{bmo-infty}, while for the second, \eqref{bmo>infty}. Finally, the $h$ bound is trivial due to \eqref{M-infty}. The proof of \eqref{elin3-dinhom} is concluded.
b) To prove the bound \eqref{elin3-diff} it suffices to apply the estimate in \eqref{elin3-dinhom} with \[ F = P \mathcal F^{3+}(w,r), \qquad G = P \mathcal G^{3+}(w,r). \] Given the estimate \eqref{gk3-est} for the cubic components of $\mathcal F$ and $\mathcal G$ and the pointwise bound \eqref{a-point} for $a$, it remains to consider the terms \[
\int \bar R r_\alpha P \mathcal F^{(3+)} \, d\alpha, \qquad \int \bar {\mathbf W} w P \mathcal F^{(3+)} \, d\alpha, \qquad \int \bar R
w) \, P\mathcal G^{(3+)} \, d\alpha. \]
For the first one we use the second part of \eqref{est-tri} to get \begin{equation} \label{term4}
\left \vert \int \bar{R}r_{\alpha}P \mathcal F^{(3+)} \, d\alpha\right \vert\lesssim \Vert |D|^{\frac{1}{2}} R\Vert _{L^{\infty}}\Vert r\Vert_{\dot{H}^{\frac{1}{2}}} \Vert \mathcal F^{(3+)} \Vert_{L^2}\lesssim AB \Vert (w,r)\Vert_{{\dot{\mathcal H} }_0}^2 \end{equation} The second one is directly estimated as \begin{equation} \label{term5}
\left\vert \int \bar{R}w\partial_{\alpha}\mathcal{K}\, d\alpha\right\vert \lesssim \Vert |D|^{\frac{1}{2}}R\Vert_{L^{\infty}}\Vert w\Vert_{L^2}\Vert \mathcal{K}\Vert_{\dot{H}^{\frac{1}{2}}} \lesssim AB \Vert (w,r)\Vert_{{\dot{\mathcal H} }_0}^2 \end{equation} On the last term, using the first part of \eqref{est-tri}, we get \begin{equation} \label{term6}
\left\vert \int \bar{R}w\partial_{\alpha}\mathcal{K}\, d\alpha\right\vert \lesssim \Vert |D|^{\frac{1}{2}}R\Vert_{L^{\infty}}\Vert w\Vert_{L^2}\Vert \mathcal{K}\Vert_{\dot{H}^{\frac{1}{2}}} \lesssim AB \Vert (w,r)\Vert_{{\dot{\mathcal H} }_0}^2 \end{equation} The proof of the proposition is concluded.
\end{proof}
\section{ Higher order energy estimates} \label{s:ee}
The main goal of this section is to establish two energy bounds for $({\mathbf W},R)$ and their higher derivatives. The first one is a quadratic bound which applies for all solutions. The second one is a cubic bound which only applies for small solutions. The large data result is as follows:
\begin{proposition}\label{t:en=large}
For any $n \geq 1$ there exists an energy functional $E^{n,(2)}$ with the following properties: (i) Norm equivalence: \begin{equation*} E^{n,(2)}({\mathbf W},R) \approx_A E_0(\partial^{n-1} {\mathbf W}, \partial^{n-1} R), \end{equation*}
(ii) Quadratic energy estimates for solutions to \eqref{ww2d-diff}: \begin{equation*} \frac{d}{dt} E^{n,(2)}({\mathbf W},R) \lesssim_A B E^{n,(2)}({\mathbf W},R). \end{equation*} \end{proposition}
The small data result is as follows:
\begin{proposition}\label{t:en=small}
For any $n \geq 1$ there exists an energy functional $E^{n,(3)}$ which
has the following properties as long as $A \ll 1$:
(i) Norm equivalence: \begin{equation*} E^{n,(3)} ({\mathbf W},R)= (1+ O(A)) E_0 (\partial^{n-1} {\mathbf W}, \partial^{n-1} R), \end{equation*}
(ii) Cubic energy estimates: \begin{equation*} \frac{d}{dt} E^{n,(3)} ({\mathbf W},R) \lesssim_A AB E^{n,(3)} ({\mathbf W},R). \end{equation*} \end{proposition}
We remark that the case $n=1$ corresponds to bounds for $({\mathbf W},R)$. But these solve the linearized system \eqref{lin(wr)}, so the above results are consequences of Proposition~\ref{plin-short} and Proposition~\ref{plin-long}. In the sequel we consider separately the cases $n = 2$ and $n \geq 3$.
\subsection{The case \texorpdfstring{$n=2$}.} We use the system \eqref{WR-diff} for $({\mathbf W}_\alpha, \mathbf R:=R_{\alpha}(1+{\mathbf W}))$, which for convenience we recall here: \[ \left\{ \begin{aligned} & {\mathbf W}_{ \alpha t} + b {\mathbf W}_{ \alpha \alpha} + \frac{\mathbf R_\alpha}{1+\bar {\mathbf W}} + \frac{R_\alpha}{1+\bar {\mathbf W}} {\mathbf W}_\alpha = \mathbf R \bar Y_\alpha - \frac{\bar R_\alpha}{1+{\mathbf W}} {\mathbf W}_\alpha + 2M {\mathbf W}_\alpha + (1+{\mathbf W}) M_\alpha , \\ & \mathbf R_{t} + b\mathbf R_{\alpha} - i\frac{(1+a){\mathbf W}_\alpha}{1+{\mathbf W}} =-2\left(\frac{\bar R_\alpha}{1+{\mathbf W}}+ \frac{R_\alpha}{1+\bar {\mathbf W}}\right) \mathbf R + 2 M \mathbf R + ( R_\alpha \bar R_\alpha -i a_\alpha). \end{aligned} \right. \] Here we have isolated on the left the leading part of the linearized equation. We want to interpret the terms on the right as mostly perturbative, but also pay attention to the quadratic part. Thus, for bookkeeping purposes, we introduce two types of error terms, denoted $\text{\bf err}(L^2)$ and $\text{\bf err}(\dot H^\frac12)$, which correspond to the two equations. The bounds for these errors are in terms of the control variables $A,B$, as well as the $L^2$ type norm \begin{equation*}
{\mathbf N}_2 = \| ({\mathbf W}_\alpha,R_\alpha)\|_{L^2 \times \dot H^\frac12}. \end{equation*}
By $\text{\bf err}(L^2)$ we denote terms $G$, which satisfy the estimates \begin{equation*}
\|P G\|_{L^2} \lesssim_A AB {\mathbf N}_2, \end{equation*} and \begin{equation*}
\text{either} \quad \|\bar P G\|_{L^2} \lesssim_A B {\mathbf N}_2 \quad \text{or} \quad
\|\bar P G\|_{\dot H^{-\frac12}} \lesssim_A A {\mathbf N}_2. \end{equation*} By $\text{\bf err}(\dot H^\frac12)$ we denote terms $K$, which are at least cubic and which satisfy the estimates \begin{equation*}
\|P K\|_{\dot H^\frac12} \lesssim_A AB {\mathbf N}_2, \qquad \| P K\|_{L^2} \lesssim_A A^2 {\mathbf N}_2, \end{equation*} and \begin{equation*}
\|\bar P K\|_{L^2} \lesssim_A A {\mathbf N}_2. \end{equation*}
The use of the more relaxed quadratic control on the antiholomorphic terms, as opposed to the cubic control on the holomorphic terms, is motivated by the fact that the equations will eventually get projected on the holomorphic space, so the antiholomorphic components will have less of an impact. A key property of the space of errors is contained in the following \begin{lemma}\label{l:err} Let $\Phi$ be a function which satisfies \begin{equation}\label{Phi-est}
\| \Phi\|_{L^\infty} \lesssim A, \qquad \||D|^\frac12 \Phi\|_{BMO} \lesssim B. \end{equation} Then, we have the multiplicative bounds \begin{equation}
\Phi \cdot \text{\bf err}(L^2) = \text{\bf err}(L^2), \qquad \Phi \cdot \text{\bf err}(\dot H^\frac12) = \text{\bf err}(\dot H^\frac12), \end{equation} \begin{equation}
\Phi \cdot P \text{\bf err}(L^2) = A\, \text{\bf err}(L^2), \qquad \Phi \cdot P \text{\bf err}(\dot H^\frac12) = A\, \text{\bf err}(\dot H^\frac12). \end{equation} \end{lemma} The proof of the lemma, based on Lemma~\ref{l:com}, is relatively straightforward
and is left for the reader. We will apply this lemma for $\Phi$ which are arbitrary smooth functions of ${\mathbf W}$ and $\bar {\mathbf W}$. Then the estimates \eqref{Phi-est} are consequences of our Moser estimates in \eqref{bmo-moser}.
We now expand some of the terms in the above system. For this we will use the following bounds for $M$, see \eqref{M-infty} and \eqref{M-L2}: \begin{equation}\label{M-bd}
\| M\|_{L^\infty} \lesssim AB, \qquad \|M\|_{\dot H^\frac12} \lesssim A {\mathbf N}_2. \end{equation} First we note that \begin{equation}\label{Mwr} M{\mathbf W}_\alpha = \text{\bf err}(L^2), \qquad M \mathbf R = \text{\bf err}(\dot H^\frac12). \end{equation} The first is straightforward in view of pointwise bound for $M$. For the second, by Lemma~\ref{l:err} we can replace $M\mathbf R$ by $M R_\alpha$. After a Littlewood-Paley decomposition, the $\dot H^\frac12$ estimate for $M R_\alpha$ is a consequence of the pointwise bound in \eqref{M-bd} for low-high and balanced interactions, and of the $\dot H^\frac12$ bound in \eqref{M-bd} combined with Lemma~\ref{l:com} for the high-low interactions.
It remains to estimate $M R_\alpha$ in $L^2$. If the frequency of $M$ is larger than or equal to the frequency of $R_\alpha$, then we can use the $\dot H^\frac12$ bound for $M$. We are left with \[ \sum_k R_{k,\alpha} M_{<k} = \sum_k R_{k,\alpha} M(R_{<k}, Y_{<k}) + \sum_k \sum_{j \geq k} R_{k,\alpha} P_{<k} M(R_j,Y_j). \] For the first sum we use \[
\| M(R_{<k}, Y_{<k})\|_{L^\infty} \lesssim 2^{\frac{k}2} A^2. \] For the second we bound \[
\| \sum_k \sum_{j \geq k} R_{k,\alpha} P_{<k} M(R_j,Y_j) \|^2_{L^2}
\lesssim \sum_{j \geq k} 2^{k-j} \||D|^\frac12 R\|_{L^\infty}^4 \|Y_{j,\alpha}\|^2 \lesssim A^2 {\mathbf N}_2. \]
Next we consider $(1+{\mathbf W}) M_\alpha$, for which we claim that \begin{equation}\label{Malpha} \begin{split} & M_\alpha =
R_\alpha \bar Y_\alpha - \bar R_\alpha Y_\alpha + P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha} {\mathbf W}] + \text{\bf err}(L^2), \\ & P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha} {\mathbf W}] = A^{-1} \text{\bf err}(L^2). \end{split} \end{equation} By Lemma~\ref{l:err}, this shows that \[ (1+{\mathbf W}) M_\alpha = R \bar Y_\alpha - \frac{\bar R_\alpha}{1+{\mathbf W}} W_\alpha + P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha} {\mathbf W}] + \text{\bf err}(L^2) . \] To prove \eqref{Malpha} we write \[ \begin{split} M_\alpha = & \ R_\alpha \bar Y_\alpha - \bar R_\alpha Y_\alpha + P[ R \bar Y_{\alpha \alpha}- \bar R_{\alpha \alpha} Y] + \bar P (g_1+2 g_2) \\ = & \ R_\alpha \bar Y_\alpha - \bar R_\alpha Y_\alpha + P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha} {\mathbf W}] - P f +\bar P (g_1+2 g_2), \end{split} \] where \[ f = R (\bar {\mathbf W} \bar Y)_{\alpha \alpha}- \bar R_{\alpha \alpha} ( {\mathbf W} Y), \quad g_1 = \bar R Y_{\alpha\alpha}- R_{\alpha\alpha} \bar Y, \quad g_2 = \bar R_\alpha Y_\alpha- R_\alpha \bar Y_\alpha. \] For $f$ and $g_1$ we have $L^2$ bounds \[
\| Pf \|_{L^2} \lesssim AB {\mathbf N}_2, \qquad \|\bar Pg_1 \|_{L^2} \lesssim B {\mathbf N}_2, \] which follow from commutator type bounds \begin{equation}\label{com2w}
\| P[R \bar \Phi_{\alpha \alpha}]\|_{L^2} \lesssim \|R_\alpha\|_{BMO} \|\Phi_\alpha\|_{L^2}, \quad
\| P[\bar R_{\alpha \alpha} \Phi] \|_{L^2}
\lesssim \| R_\alpha\|_{\dot H^\frac12} \||D|^\frac12 \Phi\|_{BMO}, \end{equation} derived from Lemma~\ref{l:com}. For the first term in $g_2$ we have a similar $L^2$ bound, but for the second we split \[ \bar P[R_\alpha \bar Y_\alpha] = \bar P[\sum_k R_{k,\alpha} \bar Y_{k,\alpha}] + \sum_k R_{<k,\alpha} \bar Y_{k,\alpha}. \] The first sum is bounded in $L^2$ using Lemma~\ref{l:com}, but for the second we only get a $\dot H^{-\frac12}$ bound, \[
\|\sum_k R_{<k,\alpha} \bar Y_{k,\alpha} \|_{\dot H^{-\frac12}} \lesssim A {\mathbf N}_2. \]
Finally, we also claim that \[ i a_\alpha = R_\alpha \bar R_\alpha + P[R \bar R_{\alpha \alpha}] + \text{\bf err}(\dot H^\frac12), \qquad P[R \bar R_{\alpha \alpha}] = A^{-1} \text{\bf err}(\dot H^\frac12), \] which is again a consequence of commutator type estimates for holomorphic $V$: \begin{equation} \label{com2r}
\| P[R \bar V_{\alpha}]\|_{\dot H^\frac12} \lesssim \|R_\alpha\|_{BMO}
\|V\|_{\dot H^\frac12}, \qquad \| P[R \bar V_{\alpha}]\|_{L^2} \lesssim \||D|^\frac12 R\|_{L^{\infty}} \|V\|_{\dot H^\frac12}. \end{equation}
Taking into account all of the above expansions, it follows that our system can be rewritten in the form \[ \left\{ \begin{aligned} & (\partial_{t}+b\partial_{\alpha}) {\mathbf W}_{ \alpha } + \frac{\mathbf R_\alpha}{1+\bar {\mathbf W}} + \frac{R_\alpha {\mathbf W}_\alpha }{1+\bar {\mathbf W}} = 2 \mathbf R \bar Y_\alpha - \frac{2 \bar R_\alpha{\mathbf W}_\alpha}{1+{\mathbf W}}
+ P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha}{\mathbf W}]+ \text{\bf err}(L^2), \\ & (\partial_{t}+b\partial_{\alpha}) \mathbf R - i\frac{(1+a){\mathbf W}_\alpha}{1+{\mathbf W}} =-2\left(\frac{\bar R_\alpha}{1+{\mathbf W}}+ \frac{R_\alpha}{1+\bar {\mathbf W}}\right) \mathbf R - P[\bar R_{\alpha \alpha} R] + \text{\bf err}(\dot H^\frac12). \end{aligned} \right. \] One might wish to compare this system with the linearized system which was studied before. However, the terms on the right cannot be all bounded in $L^2 \times \dot H^\frac12$, even after applying the projection operator $P$. Precisely, the terms on the right which cannot be bounded directly in $L^2 \times \dot H^\frac12$ are $ \displaystyle - 2\frac{\bar R_\alpha}{1+{\mathbf W}} {\mathbf W}_\alpha$, respectively $\displaystyle -2 \left(\frac{\bar R_\alpha}{1+{\mathbf W}}+ \frac{R_\alpha}{1+\bar {\mathbf W}}\right) \mathbf R$.
But these terms can be eliminated by conjugation with respect to a real exponential weight $e^{2\phi}$, where $\phi = - 2\Re \log(1+{\mathbf W})$. Then \[ \phi_\alpha = - 2 \Re \frac{{\mathbf W}_\alpha}{1+{\mathbf W}}, \qquad (\partial_t + b \partial_\alpha) \phi = 2 \Re \frac{R_\alpha}{1+\bar {\mathbf W}} - 2 M.
\] We denote the weighted variables by \[ w = e^{2\phi} {\mathbf W}_\alpha, \qquad r = e^{2\phi} \mathbf R. \] Using \eqref{Mwr} and Lemma~\ref{l:err} it follows that
$Mw = \text{\bf err}(L^2)$, $Mr = \text{\bf err}(\dot H^\frac12)$. Then we get the equations \[ \left\{ \begin{aligned} & w_{ t} + b w_{\alpha} + \frac{r_\alpha}{1+\bar {\mathbf W}} + \frac{R_\alpha}{1+\bar {\mathbf W}} w =
P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha} {\mathbf W}]+ \text{\bf err}(L^2), \\ & r_{t} + br_{\alpha} - i \frac{(1+a) w}{1+{\mathbf W}} = - P[\bar R_{\alpha \alpha} R] + \text{\bf err}(\dot H^\frac12). \end{aligned} \right. \] We are not yet in a position to use our bounds for the linearized equation since $w$ and $r$ are not exactly holomorphic. We project onto the holomorphic space to write a system for the variables $(Pw,Pr)$. At this point one may legitimately be concerned that restricting to the holomorphic part might remove a good portion of our variables. However, this is not the case: \begin{lemma}\label{en:n=2}
The energy of $(Pw,Pr)$ above is equivalent to the energy of $({\mathbf W}_\alpha,R_\alpha)$ \begin{equation}
\| (Pw,Pr)\|_{L^2 \times \dot H^\frac12} \approx_A\| (w,r)\|_{L^2 \times \dot H^\frac12}
\approx_A \|({\mathbf W}_\alpha,R_\alpha)\|_{L^2 \times \dot H^\frac12}= {\mathbf N}_2. \end{equation} \end{lemma} \begin{proof}
The estimate for $w$ is easy. We trivially have $\|w\|_{L^2} \lesssim_A \|{\mathbf W}_\alpha\|_{L^2}$, while for the converse we write \[
\| {\mathbf W}_\alpha\|_{L^2}^2 \lesssim \int \bar {\mathbf W}_\alpha e^\phi {\mathbf W}_\alpha\ d\alpha
= \int \bar {\mathbf W}_\alpha P w \ d\alpha \lesssim \| {\mathbf W}_\alpha\|_{L^2} \|P w\|_{L^2}. \] To obtain the estimate for $r$ we write \[
|D|^\frac12 P (e^\phi (1+{\mathbf W}) R_\alpha) = e^\phi(1+{\mathbf W}) |D|^\frac12 R_\alpha
+ [P|D|^\frac12,e^\phi(1+{\mathbf W}) ] R_\alpha. \]
We bound all terms in $L^2$. The one on the left is $\|r\|_{\dot
H^\frac12}$, while the first one on the right is $\approx \|
R_\alpha \|_{\dot H^\frac12}$. It remains to bound the commutator on the right, for which we have, with $\Phi = e^\phi(1+{\mathbf W})$, \[
\|[P|D|^\frac12,\Phi] R_\alpha\|_{L^2} \lesssim \| |D| \Phi\|_{L^2} \||D|^{\frac12} R\|_{BMO} \lesssim_A \|{\mathbf W}_\alpha\|_{L^2}. \] \end{proof}
Now, we write the system for $(Pw,Pr)$: \[ \left\{ \begin{aligned} & P w_{ t} + \mathfrak M_b P w_{\alpha} + P\left[\frac{P r_\alpha}{1+\bar {\mathbf W}}\right] + P\left[\frac{R_\alpha}{1+\bar {\mathbf W}} Pw \right] = P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha} {\mathbf W}]+ G_2 + \text{\bf err}(L^2), \\ & Pr_{t} + \mathfrak M_bP r_{\alpha} - i P \left[ \frac{(1+a) Pw}{1+{\mathbf W}}\right] =
- P[\bar R_{\alpha \alpha} R]+ K_2 + \text{\bf err}(\dot H^\frac12), \end{aligned} \right. \] where \[ \begin{split} G_2 = - P [b \bar P w_\alpha] - P\left[\frac{R_\alpha}{1+\bar {\mathbf W}} \bar Pw \right], \qquad K_2 = \ P [b \bar P r_\alpha] + i P \left[ \frac{(1+a) \bar Pw}{1+{\mathbf W}}\right]. \end{split} \] We claim that $G_2 = \text{\bf err}(L^2)$ and $K_2 = \text{\bf err}(\dot H^\frac12)$. As in Lemma~\ref{en:n=2} we have \[
\|\bar P w\|_{L^2} + \|\bar P r\|_{\dot H^\frac12} \lesssim_A A {\mathbf N}_2. \] Then, using the commutator bounds in Lemma~\ref{l:com}, we estimate $G_2$ by \[
\|G_2\|_{L^2} \lesssim_A (\|b_\alpha\|_{BMO}+ \|R_\alpha\|_{BMO}) \|\bar P w\|_{L^2} \lesssim_A AB {\mathbf N}_2. \] Similarly, we bound $K_2$ in $\dot H^\frac12$ by \[
\|K_2\|_{\dot H^\frac12} \lesssim_A \|b_\alpha\|_{BMO} \|\bar P r\|_{\dot H^\frac12}
+ \left \| |D|^\frac12 \left( \frac{a-{\mathbf W}}{1+{\mathbf W}}\right) \right \|_{BMO} \|\bar P w\|_{L^2} \lesssim_A AB {\mathbf N}_2, \] and in $L^2$ by \[
\|K_2\|_{L^2} \lesssim \||D|^\frac12 b\|_{BMO} \|\bar P r\|_{\dot H^\frac12}
+ \left \| \frac{a-{\mathbf W}}{1+{\mathbf W}}\right\|_{L^\infty} \|\bar P w \|_{L^2} \lesssim_A A^2 {\mathbf N}_2. \]
Finally, in view of the bilinear estimates \eqref{com2w}, \eqref{com2r}, we can replace $ P[ R \bar {\mathbf W}_{\alpha \alpha}- \bar R_{\alpha \alpha} {\mathbf W}]$ and $P[\bar R_{\alpha \alpha} R]$ by $ P[ R \bar P \bar w_{ \alpha}- {\mathbf W} \bar P \bar r_{ \alpha}]$, respectively $P[R \bar P \bar r_{\alpha} ]$ modulo acceptable error terms.
Taking into account the discussion above, we obtain a system for $(Pw,Pr)$ which is very much like the linearized system in the previous section: \[ \left\{ \begin{aligned} & P w_{ t} + \mathfrak M_b P w_{\alpha} + P\left[\frac{P r_\alpha}{1+\bar {\mathbf W}}\right] + P\left[\frac{R_\alpha}{1+\bar {\mathbf W}} Pw \right] = P[ R \bar P \bar w_{ \alpha}- {\mathbf W} \bar P \bar r_{ \alpha}]
+ \text{\bf err}(L^2), \\ & Pr_{t} + \mathfrak M_bP r_{\alpha} - i P \left[ \frac{(1+a) Pw}{1+{\mathbf W}}\right] =
-P[R \bar P \bar r_{\alpha} ] + \text{\bf err}(\dot H^\frac12). \end{aligned} \right. \]
The results of Proposition~\ref{t:en=large} and Proposition~\ref{t:en=small} follow from the energy estimates for the linearized equation, namely part (a) of Propositions~\ref{plin-short}~\ref{plin-long}; further, if $n=2$ then we can take \[ E^{n,(2)}({\mathbf W},R) = E^{(2)}_{lin}(Pw,Pr), \qquad E^{n,(3)}({\mathbf W},R) = E^{(3)}_{lin}(Pw,Pr). \]
\subsection{The case \texorpdfstring{$n \geq 3$}\ , large data}
We follow the same strategy as in the case $n=2$ and
derive the equations for $({\mathbf W}^{(n-1)}, R^{(n-1)})$. We start again with the equations \eqref{ww2d-diff} and differentiate $n-1$ times. Compared with the case $n=2$, we obtain many more terms. To separate them into leading order and lower order, we call lower order terms any terms which do not involve ${\mathbf W}^{(n-1)}$, $ R^{(n-1)} $ or derivatives thereof. In the computation below we take care to separate all the leading order
terms, as well as all the quadratic terms which are lower order. Toward that end we define again the notion of \emph{error term}. Unlike in the case $n=2$, here we also include lower order quadratic terms into the error. As before, we describe the error bounds in terms of the parameters $A$, $B$ and \begin{equation}
{\mathbf N}_n = \| ({\mathbf W}^{(n-1)},R^{(n-1)})\|_{L^2 \times \dot H^\frac12}. \end{equation}
The acceptable errors in the ${\mathbf W}^{(n-1)}$ equation are denoted by $\text{\bf err}(L^2)$ and are of two types, $\text{\bf err}(L^2)^{[2]}$ and $\text{\bf err}(L^2)^{[3]}$. $\text{\bf err}(L^2)^{[2]}$ consists of holomorphic quadratic lower order terms of the form \[ P[{\mathbf W}^{(j)} \mathbf R^{(n-j)}], \quad P[\bar {\mathbf W}^{(j)} \mathbf R^{(n-j)}], \quad P[{\mathbf W}^{(j)} \bar \mathbf R^{(n-j)}], \qquad 2 \leq j \leq n-2. \] By interpolation and H\"older's inequality, terms $G$ in $\text{\bf err}(L^2)^{[2]}$ satisfy the bound \begin{equation*}
\| G\|_{L^2} \lesssim B {\mathbf N}_n. \end{equation*}
By $\text{\bf err}(L^2)^{[3]}$ we denote terms $G$ which satisfy the estimates \begin{equation*}
\|P G\|_{L^2} \lesssim_A AB {\mathbf N}_n, \qquad \end{equation*} and \begin{equation*}
\text{either} \quad \|\bar P G\|_{L^2} \lesssim_A B {\mathbf N}_n \quad \text{or} \quad
\|\bar P G\|_{\dot H^{-\frac12}} \lesssim_A A {\mathbf N}_n. \end{equation*}
The acceptable errors in the $R^{(n-1)}$ equation are denoted by $\text{\bf err}(\dot H^\frac12)$ and are also of two types, $\text{\bf err}(\dot H^\frac12)^{[2]}$ and $\text{\bf err}(\dot H^\frac12)^{[3]}$. $\text{\bf err}(\dot H^\frac12)^{[2]}$ consists of holomorphic quadratic lower order terms of the form \[ P[\mathbf R^{(j)} \mathbf R^{(n-j)}], \quad P[\bar \mathbf R^{(j)} \mathbf R^{(n-j)}], \qquad 2 \leq j \leq n-2, \] and \[ P[{\mathbf W}^{(j)} {\mathbf W}^{(n-j)}], \quad P[\bar {\mathbf W}^{(j)} {\mathbf W}^{(n-j-1)}], \qquad 1 \leq j \leq n-1. \] By interpolation and H\"older's inequality, terms $K$ in $\text{\bf err}(\dot H^\frac12)^{[2]}$ satisfy the bound \begin{equation*}
\| K\|_{\dot H^\frac12} \lesssim B {\mathbf N}_n, \qquad \| K\|_{L^2} \lesssim A {\mathbf N}_n. \end{equation*} By $\text{\bf err}(\dot H^\frac12)^{[3]}$ we denote terms $K$ which satisfy the estimates \begin{equation*}
\|P K\|_{\dot H^\frac12} \lesssim_A AB {\mathbf N}_n, \qquad \| P G\|_{L^2} \lesssim_A A^2 {\mathbf N}_n,
\qquad \|\bar P G\|_{L^2} \lesssim_A A {\mathbf N}_n. \end{equation*}
We begin by differentiating the terms in the ${\mathbf W}$ equation, where we expand using Leibnitz rule. For the $b$ term we have \[ \begin{split} \partial^{n-1} (b{\mathbf W}_\alpha) = & \ b {\mathbf W}^{(n-1)}_\alpha + (n-1) b_\alpha {\mathbf W}^{(n-1)} + b^{(n-1)} {\mathbf W}_\alpha + err_1, \\ = & \ \ b {\mathbf W}^{(n-1)}_\alpha + (n-1) \left(\frac{R_\alpha}{1+\bar {\mathbf W}} +\frac{\bar R_\alpha}{1+{\mathbf W}} \right) {\mathbf W}^{(n-1)} + 2 {\mathbf W}_\alpha \Re R^{(n-1)} + err_2. \end{split} \] Here $err_1$ only contains lower order terms, so by interpolation and H\"older's inequality we get\footnote{Here we remark that all terms in the ${\mathbf W}^{(n-1)}$ equation have the same scaling; thus, whenever all the Sobolev exponents are within the lower order range, we are guaranteed to get the correct $L^2$ estimate after interpolation and H\"older's inequality. The same applies to all the terms in the $R^{(n-1)}$ equation. } $err_1 = \text{\bf err}(L^2)$. The difference $err_2 - err_1$ is cubic, \[ err_2 = err_1 + (n-1) M {\mathbf W}^{(n-1)} + {\mathbf W}_\alpha ( P[R^{(n-1)} \bar Y]+ \bar P[\bar R^{(n-1)} Y]). \] Using the $L^\infty$ bound for $M$ in \eqref{M-bd}, Sobolev embeddings and interpolation it is easily seen that $err_2 = \text{\bf err}(L^2)$.
A similar analysis leads to \[ \begin{split} \partial^{n-1} \frac{(1+{\mathbf W}) R_\alpha}{1+\bar {\mathbf W}} = & \ \frac{[(1+{\mathbf W}) R^{(n-1)}]_\alpha}{1+\bar {\mathbf W}}
+ \frac{R_\alpha}{1+\bar {\mathbf W}} {\mathbf W}^{(n-1)} - R_\alpha \bar {\mathbf W}^{(n-1)} \\ & \ + R^{(n-1)}((n-2){\mathbf W}_\alpha - (n-1)\bar {\mathbf W}_\alpha) + \text{\bf err}(L^2). \end{split} \] In the $M$ term we also bound lower order terms by H\"older's inequality
and interpolation to obtain \[ \begin{split}
\partial^{n-1} [(1+{\mathbf W})M] = & \ \text{\bf err}(L^2) +R^{(n-1)} \bar {\mathbf W}_\alpha - \frac{\bar R_\alpha}{1+{\mathbf W}} {\mathbf W}^{(n-1)} \\ &+ P\left[
R \bar {\mathbf W}^{(n-1)}_\alpha - \bar R^{(n-1)}_\alpha {\mathbf W} + (n-1)(R_\alpha \bar {\mathbf W}^{(n-1)} - \bar R^{(n-1)} {\mathbf W}_\alpha)\right] \\ & + \bar P [- R^{(n-1)} \bar {\mathbf W}_\alpha + \frac{\bar R_\alpha}{1+{\mathbf W}} {\mathbf W}^{(n-1)} +\bar R^{(n-1)} {\mathbf W}_\alpha - \frac{R_\alpha(1+{\mathbf W})}{(1+\bar {\mathbf W})^2} \bar {\mathbf W}^{(n-1)} \\ & \ \ \ \ \ \ \ \ + \bar R {\mathbf W}^{(n-1)}_\alpha - R^{(n-1)}_\alpha \bar {\mathbf W} + (n-1)(\bar R_\alpha {\mathbf W}^{(n-1)} - R^{(n-1)} \bar {\mathbf W}_\alpha)]. \end{split} \] Estimating also the quadratic $\bar P$ terms, the above relation takes the simpler form \[ \begin{split} \partial^{n-1} [(1+{\mathbf W})M] = &\ R^{(n-1)} \bar {\mathbf W}_\alpha - \frac{\bar R_\alpha}{1+{\mathbf W}} {\mathbf W}^{(n-1)} + P\left[R \bar {\mathbf W}^{(n-1)}_\alpha - \bar R^{(n-1)}_\alpha {\mathbf W} \right] \\ & + (n-1)(R_\alpha \bar {\mathbf W}^{(n-1)} - \bar R^{(n-1)} {\mathbf W}_\alpha)+
\text{\bf err}(L^2) . \end{split} \]
Now we turn our attention to the $R$ equation. We begin with \[ \begin{split} \partial^{n-1} (bR_\alpha) = & \ b R^{(n-1)}_\alpha + (n-1) b_\alpha R^{(n-1)} + b^{(n-1)} R_\alpha + err_3 \\ = & \ \ b R^{(n-1)}_\alpha + (n-1)\left(\frac{R_\alpha}{1+\bar {\mathbf W}} +\frac{\bar R_\alpha}{1+{\mathbf W}} \right) R^{(n-1)} + \frac{R_\alpha}{1+\bar {\mathbf W}} R^{(n-1)} + \frac{R_\alpha}{1+{\mathbf W}} \bar R^{(n-1)} \\ & + err_4, \end{split} \] where we trivially have $err_3 = \text{\bf err}(\dot H^\frac12)$ as it contains only lower order terms, both quadratic and higher order. In addition, the difference is cubic, and is given by \[ err_4 - err_3 = (n-1) M R^{(n-1)} + R_\alpha\left(b^{(n-1)} - \frac{R^{(n-1)}}{1+\bar {\mathbf W}} - \frac{\bar R^{(n-1)}}{1+ {\mathbf W}} \right). \] We claim that this is also $\text{\bf err}(\dot H^\frac12)$. The $L^2$ bound follows trivially by interpolation and H\"older's inequality.
The $\dot H^\frac12$ bound is also easy to obtain for the second term, where the unfavorable $R^{(n-1)}$ factors only appear with a convenient frequency balance as $R_\alpha( \bar P[ R^{(n-1)}\bar Y] - P [R^{(n-1)}Y])$. Consider now the $\dot H^\frac12$ bound for the first term. Since $M= O_{L^\infty}(AB)$, the nontrivial case is when $M$ has the high frequency, where we need to estimate \[
\| |D|^\frac12 \sum_k M_k R^{(n-1)}_{<k}\|_{L^2} \lesssim \||D|^{n-\frac32} M\|_{L^2}
\| DR\|_{BMO} \lesssim AB {\mathbf N}_n. \] Here, we have used the bound \eqref{M-L2}.
For the remaining term in the $R$ equation we write \[ \begin{split} \partial^{n-1} \frac{{\mathbf W} - a}{1+{\mathbf W}} = & \ \frac{(1+ a){\mathbf W}^{(n-1)}}{(1+{\mathbf W})^2} +
\frac{a^{(n-1)}}{1+{\mathbf W}} + err_7 \\ = & \ \frac{(1+ a){\mathbf W}^{(n-1)}}{(1+{\mathbf W})^2} + \frac{i}{1+ {\mathbf W}}\left( P[R\bar R_\alpha^{(n-1)}+(n-1) R_\alpha \bar R^{(n-1)} + \bar R_\alpha R^{(n-1)} ] \right. \\ & \left.- \bar P[\bar R R_\alpha^{(n-1)}+(n-1) \bar R_\alpha
R^{(n-1)} + R_\alpha \bar R^{(n-1)}] \right))
+ err_8. \end{split} \] Here $err_7$ contains lower order quadratic terms in ${\mathbf W}$ (without $a$) as well as cubic terms which can be easily estimated, so $err_7= \text{\bf err}(\dot H^\frac12)$. The difference $err_8 - err_7$ only contains lower order terms so it also can be placed in $\text{\bf err}(\dot H^\frac12)$. Just as in the case of the ${\mathbf W}^{(n-1)}$ equation, quadratic $\bar P$ terms can also be placed in the error. Then the above relation becomes \[ \begin{split} \partial^{n-1} \frac{{\mathbf W} - a}{1+{\mathbf W}} = \frac{(1+ a){\mathbf W}^{(n-1)}}{(1+{\mathbf W})^2} + i \left( P[R\bar R_\alpha^{(n-1)}]+(n-1) R_\alpha \bar R^{(n-1)} + \frac{\bar R_\alpha R^{(n-1)}}{1+{\mathbf W}}\right)
+ \text{\bf err}(\dot H^\frac12). \end{split} \]
Combining the above computations we obtain the differentiated system \[ \left\{ \begin{aligned}
& {\mathbf W}^{(n-1)}_{ t} + b {\mathbf W}^{(n-1)}_{ \alpha} + \frac{((1+{\mathbf W}) R^{(n-1)})_\alpha}{1+\bar {\mathbf W}}
+ \frac{R_\alpha}{1+{\mathbf W}} {\mathbf W}^{(n-1)} = G, \\ &
R^{(n-1)}_t + bR^{(n-1)}_\alpha - i\left(\frac{(1+ a){\mathbf W}^{(n-1)}}{(1+{\mathbf W})^2}\right) = K, \end{aligned} \right. \] where \[ \begin{split} G = & - n \frac{\bar R_\alpha}{1+{\mathbf W}} {\mathbf W}^{(n-1)} - (n-1) \frac{ R_\alpha}{1+\bar {\mathbf W}} {\mathbf W}^{(n-1)}
+ P[ R \bar {\mathbf W}^{(n-1)}_\alpha - {\mathbf W} \bar R^{(n-1)}_\alpha]\\ & + R^{(n-1)}(n \bar {\mathbf W}_\alpha - (n-1){\mathbf W}_\alpha) + n(R_\alpha \bar {\mathbf W}^{(n-1)} - {\mathbf W}_\alpha\bar R^{(n-1)}) + \text{\bf err}(L^2), \\ K = & - n\left( \frac{ R_\alpha}{1+\bar {\mathbf W}}+ \frac{\bar R_\alpha}{1+{\mathbf W}}\right) R^{(n-1)}
- \left(P[R \bar R^{(n-1)}_\alpha] - nR_\alpha \bar R^{(n-1)}\right)+\text{\bf err}(\dot H^\frac12). \end{split} \] After the usual substitution $\mathbf R = (1+{\mathbf W})R^{(n-1)}$, we get \[ \left\{ \begin{aligned}
& {\mathbf W}^{(n-1)}_{ t} + b {\mathbf W}^{(n-1)}_{ \alpha} + \frac{ \mathbf R_\alpha}{1+\bar {\mathbf W}}
+ \frac{R_\alpha}{1+{\mathbf W}} {\mathbf W}^{(n-1)} = G, \\ & \mathbf R_t + b\mathbf R_\alpha - i\left(\frac{(1+ a){\mathbf W}^{(n-1)}}{1+{\mathbf W}}\right) = K_1, \end{aligned} \right. \] where \[ K_1= - (n+1) \frac{ R_\alpha\mathbf R}{1+\bar {\mathbf W}} - n \frac{\bar R_\alpha\mathbf R }{1+{\mathbf W}} - P[R \bar \mathbf R_\alpha] - nR_\alpha \bar \mathbf R + \text{\bf err}(\dot H^\frac12). \] The more delicate terms here are the ones on the right where the leading order terms appear unconjugated. We would like to eliminate those with an exponential factor as in the $n=2$ case, but their coefficients on the right are not properly matched. To remedy that we take the additional step of the holomorphic substitution \[ {\tilde {\mathbf R}} = \mathbf R - R_\alpha {\mathbf W}^{(n-2)} +(2n-1) {\mathbf W}_\alpha R^{(n-2)}. \] With the exception of exactly three terms, the contribution of the added quadratic correction is cubic and lower order, so we obtain \[ \left\{ \begin{aligned}
& {\mathbf W}^{(n-1)}_{ t} + b {\mathbf W}^{(n-1)}_{ \alpha} + \frac{ {\tilde {\mathbf R}}_\alpha}{1+\bar {\mathbf W}}
+ \frac{R_\alpha}{1+{\mathbf W}} {\mathbf W}^{(n-1)} = - n \left(\frac{\bar R_\alpha}{1+{\mathbf W}} + \frac{ R_\alpha}{1+\bar {\mathbf W}} \right){\mathbf W}^{(n-1)} \\ & \ \ \ \ + P[ R \bar {\mathbf W}^{(n-1)}_\alpha - {\mathbf W} \bar R^{(n-1)}_\alpha] + n{\tilde {\mathbf R}}(\bar {\mathbf W}_\alpha + {\mathbf W}_\alpha) + n (R_\alpha \bar {\mathbf W}^{(n-1)} - {\mathbf W}_\alpha\bar R^{(n-1)}) + \text{\bf err}(L^2), \\ & {\tilde {\mathbf R}}_t + b{\tilde {\mathbf R}}_\alpha - i\frac{(1+ a){\mathbf W}^{(n-1)}}{1+{\mathbf W}} = - n \left(\! \frac{ R_\alpha}{1+\bar {\mathbf W}} + \frac{\bar R_\alpha}{1+{\mathbf W}} \! \right) {\tilde {\mathbf R}}- P[R \bar {\tilde {\mathbf R}}_\alpha] - nR_\alpha \bar {\tilde {\mathbf R}} + \text{\bf err}(\dot H^\frac12). \end{aligned} \right. \] Now we can multiply by $e^{n\phi}$ where, as before, $\phi = -2 \Re \log(1+{\mathbf W})$, in order to eliminate the unbounded terms on the right. We get an equation for $(w := e^{\phi} {\mathbf W}^{(n-1)}, r := e^{\phi} {\tilde {\mathbf R}})$: \[ \left\{ \begin{aligned}
& w_{ t} + b w_{ \alpha} + \frac{ r_\alpha}{1+\bar {\mathbf W}}
+ \frac{R_\alpha}{1+{\mathbf W}} w = P[ R \bar {\mathbf W}^{(n-1)}_\alpha - {\mathbf W} \bar R^{(n-1)}_\alpha] + n (R_\alpha \bar w - {\mathbf W}_\alpha\bar r) + \text{\bf err}(L^2), \\ & r_t + br_\alpha - i\left(\frac{(1+ a) w}{1+{\mathbf W}}\right) = - P[ R \bar R^{(n-1)}_\alpha] - nR_\alpha \bar r + \text{\bf err}(\dot H^\frac12). \end{aligned} \right. \] As $(w,r)$ are no longer holomorphic, we project and work with the projected variables. After some additional commutator estimates which are identical to those in the $n=2$ case we obtain \begin{equation}\label{energy(n,3)} \left\{ \begin{aligned}
& Pw_{ t} + \mathfrak M_b P w_{ \alpha} + P \left[\frac{ P r_\alpha}{1+\bar {\mathbf W}}\right]
+ P\left[\frac{R_\alpha Pw}{1+{\mathbf W}} \right] = P[ R \bar P \bar w_\alpha - {\mathbf W} \bar P \bar r_\alpha] \\ &\hspace{3in}\, \ \ \ \ + n P [R_\alpha \bar P \bar w - {\mathbf W}_\alpha\bar P\bar r]
+ \text{\bf err}(L^2), \\ & Pr_t + \mathfrak M_bPr_\alpha - iP\left[\frac{(1+ a) P w}{1+{\mathbf W}}\right] = - P[ R \bar P \bar r_\alpha] - nP [R_\alpha \bar P\bar r] + \text{\bf err}(\dot H^\frac12). \end{aligned} \right. \end{equation}
Compared to the linearized equation in the previous section, here we have two additional terms that need to be estimated. We have
\begin{lemma} \label{l:erori} a) The energy of $(Pw,Pr)$ above is equivalent to that of $({\mathbf W}^{(n-1)},R^{(n-1)})$, \begin{equation}
\| (Pw,Pr)\|_{L^2 \times \dot H^\frac12} \approx_A\| (w,r)\|_{L^2 \times \dot H^\frac12} \approx_A
{\mathbf N}_n, \end{equation} b) The additional error terms above are bounded, \begin{equation}\label{bierror-n}
\| ( P [R_\alpha \bar P \bar w - {\mathbf W}_\alpha\bar P\bar r] , P [R_\alpha \bar P\bar r])\|_{L^2 \times \dot H^\frac12} \lesssim_A B {\mathbf N}_n, \end{equation} \end{lemma} \begin{proof} a) For $w$ we argue as in the proof of Lemma~\ref{en:n=2} to get \[
\| P w\|_{L^2} \approx_A \|w\|_{L^2} \approx_A \| {\mathbf W}^{(n-1)}\|_{L^2}. \] For $r$ we need, again as in the proof of Lemma~\ref{en:n=2}, with $\Phi = e^\phi(1+{\mathbf W})$, to bound in $L^2$ the difference \[ \begin{split}
K = & \ |D|^\frac12 P r - \Phi |D|^\frac12 R^{(n-1)}, \\ = & \
[ |D|^\frac12 P, \Phi]R^{(n-1)} + |D|^\frac12 P[ e^\phi( - R_\alpha W^{(n-2)} +(2n-1) {\mathbf W}_\alpha R^{(n-2)})], \end{split} \] as well as the similar difference but with all $P$ omitted. It suffices to prove the estimate \[
\|K\|_{L^2} \lesssim_A \|{\mathbf W}^{(n-1)}\|_{L^2} + \|{\mathbf W}^{(n-1)}\|_{L^2}^{\frac{1}{n-1}}
\|R^{(n-1)}\|_{\dot H^\frac12}^{\frac{n-2}{n-1}}, \] which follows by standard multiplicative and commutator estimates.
b) For the first term $P[R_\alpha \bar P \bar w]$ we directly use the Coifman-Meyer type estimates in Lemma~\ref{l:com}. For the second we bound ${\mathbf W}_\alpha$ in $L^{4n-6}$ and $r$ in $L^{\frac{2n-3}{n-2}}$ by H\"older's inequality and interpolation. For the third we have to bound $\| |D|^\frac12 P [R_\alpha \bar P\bar r]\|_{L^2}$. For the balanced frequency interactions, by Coifman-Meyer it suffices to bound $R_\alpha$ in $BMO$ and $r$ in $\dot H^\frac12$. For the high-low interactions, on the other hand, the half-derivative goes to
$R_\alpha$, and we need to bound $|D|^\frac12 R_\alpha$, and $r$ in $L^{4n-6}$, respectively $L^{\frac{2n-3}{n-2}}$.
\end{proof}
Given the above Lemma~\ref{l:erori}, the $n \geq 3$ case of the result in Proposition~\ref{t:en=large} is a direct consequence of our quadratic estimates for the linearized equation in Proposition~\ref{plin-short}(a).
The small data cubic energy estimates in Proposition~\ref{t:en=small} are proved in the next section. The key is to produce a modified cubic energy, whose leading part is given by \[ E^{n,(3)}_{high} ( w,r) =
\int (1+a) |w|^2 + \Im (\bar r r_\alpha)
+ 2n \Im (R_{\alpha} \bar w \bar r) +
2( \Im[\bar{R} w r_\alpha] - \Re[\bar {W}_\alpha w^2]) \, d\alpha. \] We claim that the evolution of this energy is governed by the following
\begin{lemma}\label{l:hf-cubic} Let $(w,r)$ be defined as above. Then
a) Assuming that $A \ll 1$, we have \begin{equation}\label{en3-equiv}
E^{n,(3)}_{high} ( Pw,P r) \approx E_0 ( Pw,P r) \approx {\mathbf N}_n, \end{equation}
b) The solutions $(Pw,Pr)$ of \eqref{energy(n,3)} satisfy \begin{equation}\label{en3-evolve} \begin{split} \frac{d}{dt} E^{n,(3)}_{high}( Pw,P r) = & \ 2 \int \Re (\bar w \cdot \text{\bf err}(L^2)^{\left[ 2\right] } ) - \Im(\bar r_\alpha \cdot \text{\bf err}(\dot H^\frac12)^{\left[ 2\right] }) \, d\alpha \\ & \ \ \ + O_A(AB {\mathbf N}_n). \end{split} \end{equation} Further, the same relation holds if $(\bar w, \bar r)$ on the right are replaced by $(\bar {\mathbf W}^{(n-1)},\bar R^{(n-1)})$.
\end{lemma}
\begin{proof}
a) Given the bounds already proved in Proposition~\ref{plin-long} for
the linearized equation, it suffices to estimate the additional term, \[
\left| \int R_\alpha \bar w \bar r \ d\alpha \right| \lesssim A {\mathbf N}_n. \] For this we use interpolation to bound $R_\alpha$, $w$ and $r$ in $L^{4n-6}$, $L^2$, respectively $L^{\frac{2n-3}{n-2}}$ in terms of $A$ and ${\mathbf N}_n$.
b) Here, we begin with the cubic linearized energy, $E^{(3)}_{lin}$. According to the bound \eqref{elin3-dinhom} in Proposition~\ref{plin-long}, we have \[ \begin{split} \frac{d}{dt} E^{(3)}_{lin}( Pw,P r) = & \
\int 2\Re \left( (n P [R_\alpha \bar P \bar w - {\mathbf W}_\alpha\bar P\bar r] + P\text{\bf err}(L^2)) \cdot ( \bar w - \bar P[\bar R r_\alpha] - \bar P[\bar W_\alpha w]) \right) \\ & \ \ - 2 \Im\left( (- nP [R_\alpha \bar P\bar r] + P \text{\bf err}(\dot H^\frac12)) \cdot (\bar r_\alpha + \bar P[\bar R w]_\alpha)\right) \, d\alpha \\
& \ \ + O_A\left( AB \|(Pw,Pr)\|_{L^2 \times \dot H^\frac12}^2\right) . \end{split} \] By the Coifman-Meyer type estimates in Lemma~\ref{l:com} the following bounds hold: \begin{equation}
\| \bar P[\bar R r_\alpha]\|_{L^2} + \| \bar P[\bar W_\alpha w]\|_{L^2}
+\| \bar P[\bar R w]\|_{\dot H^\frac12}
\lesssim A \|(w,r)\|_{L^2 \times \dot H^\frac12}. \end{equation} Combining this with \eqref{bierror-n} and with the bounds for the error terms we get \[ \begin{split} \frac{d}{dt} E^{(3)}_{lin}( Pw,P r) \leq & \
\int 2\Re \left( (n P [R_\alpha \bar P \bar w - {\mathbf W}_\alpha\bar P\bar r] + P\text{\bf err}(L^2)^{\left[ 2\right] }) \cdot \bar w \right) \\ & \ \ - 2\Im\left( (- nP [R_\alpha \bar P\bar r] + P \text{\bf err}(\dot H^\frac12)^{\left[ 2\right] }) \cdot \bar r_\alpha \right) \, d\alpha \\
& \ \ + O_A\left( AB \|(Pw,Pr)\|_{L^2 \times \dot H^\frac12}^2\right) , \end{split} \] where the output from all error terms which are cubic and higher error terms is all included in the last RHS term.
It remains to consider the contribution of the extra term in $E^{n,(3)}_{high}$ and show that \begin{equation}\label{extra-diff} \begin{split} \frac{d}{dt} \int \Im (R_{\alpha} \bar P \bar w \bar P \bar r) \, d\alpha = & \
\int \Re \left((R_\alpha \bar P \bar w - {\mathbf W}_\alpha\bar P\bar r) \bar P \bar w \right)
+ \Im\left( R_\alpha \bar P\bar r
\bar P\bar r_\alpha \right) \, d\alpha
\\ & \ \ \,+ O_A\left( AB \|(Pw,Pr)\|_{L^2 \times \dot H^\frac12}^2\right) . \end{split} \end{equation} Denote by $G_n$, respectively $K_n$ the two right hand sides in \eqref{energy(n,3)}. By the definition of error terms and by \eqref{bierror-n} they satisfy the bounds \[
\|(G_n,K_n)\|_{L^2 \times \dot H^{\frac12}} \lesssim_A B {\mathbf N}_n, \qquad \|K_n\|_{L^2} \lesssim_A A {\mathbf N}_n. \] Then their contribution in the above time derivative is estimated \[
\left| \int \Im (R_{\alpha} \bar P \bar G_n \bar P \bar r + R_{\alpha} \bar P \bar w \bar P \bar K_n)\ d\alpha\right| = \left| \int \Im (R_{\alpha} \bar P \bar F_n \bar P \bar r + P[R_{\alpha} \bar P \bar w] \bar P \bar K_n)\ d\alpha\right| \lesssim_A AB {\mathbf N}_2,
\] by using H\"older's inequality for the first term and the Coifman-Meyer commutator estimate in Lemma~\ref{l:com} for the second.
The contributions of the $b$ terms are collected together in the imaginary part of the expression \[ \begin{split} I = & \ \int \partial_\alpha (b R_{\alpha}) \bar P \bar w \bar P \bar r + R_\alpha \bar P(b\bar P \bar w_\alpha) \bar P \bar r + R_\alpha
\bar P \bar w \bar P(b\bar P \bar r_\alpha )\, d\alpha \\ = & \
\int
R_\alpha \left([b,P](\bar P \bar w_\alpha) \bar P \bar r +
\bar P \bar w [b,P] (\bar P \bar r_\alpha )\right)\, d\alpha. \end{split} \]
Since $\|b_\alpha\|_{BMO} \lesssim B$, we can bound using Lemma~\ref{l:com}, and then use H\"older's inequality for all terms.
Next, we consider the remaining contribution of the time derivative of $R_\alpha$, for which we use the equation \eqref{ww2d-diff}. This is \[ \Im \int \bar P \bar w \bar P \bar r \partial_\alpha \left( \frac{{\mathbf W}-a}{1+{\mathbf W}}\right) \, d\alpha = \Re \int \bar P \bar w \bar P \bar r {\mathbf W}_\alpha \, d\alpha - \Re \int \bar P \bar w \bar P \bar r \partial_\alpha \left( \frac{{\mathbf W}^2+ a}{1+{\mathbf W}}\right) \, d\alpha. \] The first term on the right yields the second term on the right of \eqref{extra-diff}, while the rest of the terms are directly bounded using H\"older's inequality.
It remains to consider the contribution of the remaining left hand side terms in \eqref{energy(n,3)}. The expression $\dfrac{ P r_\alpha}{1+\bar {\mathbf W}}$ in the $r$ equation yields the third term on the right of \eqref{extra-diff}, plus the quartic term \[ \int \Im R_{\alpha} \bar P (\bar P \bar r_\alpha Y) \bar P \bar r \, d\alpha = \int \Im( R_{\alpha} [\bar P,Y] (\bar P \bar r_\alpha) \bar P \bar r + R_{\alpha} Y \bar P \bar r_\alpha \bar P \bar r) \, d\alpha . \] In the first term we apply a commutator estimate and then H\"older's inequality, and in the second we use H\"older inequality directly.
The contribution of $ P\left[\dfrac{R_\alpha}{1+{\mathbf W}} Pw\right]$ is purely a H\"older term. Finally, the contribution of $P\left[\dfrac{1+ a }{1+{\mathbf W}}P w\right]$ yields the first term on the right of \eqref{extra-diff}, plus a H\"older quartic term.
\end{proof}
\subsection{Normal form energy estimates: \texorpdfstring{$n \geq 3$}\ \ , small data} \label{s:ee3+}
In this section, we construct an $n$-th order energy with cubic estimates. One ingredient for this is the high frequency cubic energy $E^{n,(3)}_{high}$ in Lemma~\ref{l:hf-cubic}. However, this does not suffice, as the right hand side of the energy relation \eqref{en3-evolve} still contains lower order cubic terms. Here we use normal forms in order to add a lower order correction to $E^{n,(3)}_{high}$, which removes the above mentioned cubic terms. We recall that the normal form variables $(\tilde{W}, \tilde{Q})$ are given by \begin{equation} \left\{ \begin{aligned} \tilde W &= W - 2 \mathfrak M_{\Re W} W_\alpha, \\ \tilde Q &= Q - 2 \mathfrak M_{\Re W} R, \end{aligned} \right. \label{nft} \end{equation} where $\mathfrak M_u F = P [uF]$. They solve an equation where all nonlinearities are cubic and higher, \begin{equation} \left\{ \begin{aligned} &{\tilde W}_t + {\tilde Q}_\alpha = \tilde{G}, \\ &{\tilde Q}_t - i {\tilde W} = \tilde{K}, \end{aligned} \right. \label{nft2eq} \end{equation} see Proposition~\ref{p:normal}.
The obvious energy functional associated to the normal form equations (\ref{nft1eq}) is \[
E^{n}_{NF,0} = \int \left(|{\tilde W}^{(n)}|^2 + \Im [ {\tilde Q}^{(n)} \bar{\tilde Q}^{(n)}_{\alpha} ]\right)\, d\alpha. \] In view of Proposition~\ref{p:normal}, this functional satisfies an energy equation of the form \begin{equation} \frac{d}{dt} E^{n}_{NF,0} = quartic + higher, \label{cubic_energy_estimate} \end{equation} but it has several defects: \begin{enumerate} \item It is expressed in terms of $Q^{(n)}$ rather than the natural variable $R^{(n-1)}$, \item It is not equivalent to the linear energy $E^{(2)}_{lin}({\mathbf W}^{(n-1)},R^{(n-1)})$, \item Its energy estimate has a loss of derivatives. \end{enumerate} However, the last two issues concerning $E^{n}_{NF,0}$ arise at the level of quartic and higher order terms, and they are specific to the water wave problem. This motivates our strategy, which is modify $E^{n}_{NF,0}$ by quartic and higher terms to obtain a ``good'' energy $E^{n,(3)}$ without spoiling the cubic energy estimate (\ref{cubic_energy_estimate}).
We carry out this procedure in two steps: \textbf{(i)} we construct a modified normal form energy $E^{n}_{NF}$ that depends on $({\mathbf W}^{(n-1)}, R^{(n-1)})$ and is equivalent to the linearized energy $E^{(2)}_{lin}({\mathbf W}^{(n-1)},R^{(n-1)})$; this addresses the issues (1) and (2) above, but not (3); \textbf{(ii)} we separate the leading order part $E^{n}_{NF,high}$ and modify that to the correct high frequency expression $E^{n,(3)}_{high}$ defined in the previous section, which was inspired from the analysis of the linearized equation. This modification is needed due to the quasilinear nature of our problem. Thus, we obtain an energy $E^{n,(3)}$ with good, cubic estimates.
The first step described above is implemented in the following proposition:
\begin{proposition} There exists a modified normal form energy $E^{n}_{NF}$ of the form \begin{align} \begin{split} E^{n}_{NF} = & \ E^{n}_{NF,high} + E^{n}_{NF,low}, \\
E^{n}_{NF,high} = & \ \int (1- 4n \Re {\mathbf W}) \left(|{\mathbf W}^{(n)}|^2 + \Im[{\tilde {\mathbf R}} \bar {\tilde {\mathbf R}}_\alpha]\right) + 2n \Im[R_{\alpha} \bar {\mathbf W}^{(n-1)} \bar {\tilde {\mathbf R}}]\, d\alpha, \\ & \quad + 2 \int \Im[\bar{R}{{\mathbf W}}^{(n-1)} {\tilde {\mathbf R}}_\alpha] - \Re[\bar {{\mathbf W}} ({\mathbf W}^{(n-1)})^2] \, d\alpha, \\ E^{n}_{NF,low} = & \ \Re \int \left(\sum_{j+k+l=2n-2} c_{jkl}{\mathbf W}^{(j)} {\mathbf W}^{(k)} \bar {\mathbf W}^{(l)}
+ \sum_{j+k+l=2n-1} d_{jk_1l_1} {\mathbf W}^{(j)} R^{(k)} \bar R^{(l)}\right) \, d\alpha, \\ &\ \end{split} \label{def_En_hl} \end{align} such that \begin{equation}\label{enlow-diff} E^{n}_{NF} = E^{n}_{NF,0} + \text{(quartic and higher terms)}, \end{equation} and \begin{equation}\label{enlow-equiv} E^{n}_{NF,high} = [1+ O(A)] E_0 ({\mathbf W}^{(n-1)}, R^{(n-1)}), \qquad E^{n}_{NF,low} = O(A) E_0 ({\mathbf W}^{(n-1)}, R^{(n-1)}). \end{equation} Moreover, the sums in (\ref{def_En_hl}) for $E^{n}_{NF,low}$ contain only indices $(j,k, l)$ with $1\le j,k,l \le n-1$. \end{proposition}
\begin{remark}
The normal form transformation is expressed at the level of $(W,Q)$
variables, and cannot be easily switched to the level of
$({\mathbf W},R)$. For this reason, initially the computation of the normal
form energy is done in terms of the original variables $(W,Q)$. The
interesting fact in the above proposition is that in the end we are able
express the energies in the convenient variables $({\mathbf W}, R)$. \end{remark}
\begin{proof}
We start from the normal form energy $E^{n}_{NF,0}$ and express it in
terms of $({\mathbf W},R)$ and their derivatives. First, consider the term
involving $\tilde{W}^{(n)}$. Using (\ref{nft}), we get that \[
\int |{\tilde W}^{(n)}|^2 \, d\alpha
= \int |W^{(n)}|^2 - 4 \Re \left[ \bar W^{(n)} \partial_\alpha^n(\mathfrak M_{\Re W} W_\alpha)\right] + 4\left|\partial_\alpha^n(\mathfrak M_{\Re W} W_\alpha)\right|^2 \, d\alpha. \] The higher-order derivatives $W^{(n+1)}$ cannot be removed from the last term, but it is quartic and therefore harmless. The cubic term also contain derivatives of order $n+1$, but as we show next they integrate out; as a result, the cubic energy is equivalent to the linear energy.
Moving the projection $P$ across the inner product, we have \begin{equation*} \int \bar W^{(n)} \partial_\alpha^n(\mathfrak M_{\Re W} W_\alpha) \, d\alpha =\int \bar W^{(n)} \partial_\alpha^n( P[ W_\alpha \Re W])\, d\alpha = \int \bar W^{(n)} \partial_\alpha^n( W_\alpha \Re W)\, d\alpha, \end{equation*} which shows that \[
\int |{\tilde W}^{(n)}|^2 \, d\alpha =
\int |W^{(n)}|^2 - 4 \Re \left[ \bar W^{(n)} \partial_\alpha^n( W_\alpha \Re W)\right] \, d\alpha + \text{quartic}. \] Thus, expanding derivatives, we get \begin{align} \begin{split}
\int |{\tilde W}^{(n)}|^2 \, d\alpha
&=\int |W^{(n)}|^2 - 4\Re [\bar W^{(n)} W^{(n)}_\alpha] {\Re W}
-4 n |W^{(n)}|^2\Re W_\alpha \\ &\quad - 4\sum_{j = 2}^{n-1}\binom{n}{j}\Re \left[\bar W^{(n)} W^{(n-j+1)}\right] \Re W^{(j)}
- 4 \Re [W_\alpha \bar W^{(n)}]\Re W^{(n)}
\, d\alpha \\ &\quad + \text{quartic}. \end{split} \label{twn_eq} \end{align} Integrating by parts in the cubic term that contain derivatives of $W$ of the order $n+1$, we get \begin{equation*} \int \Re [\bar W^{(n)} W^{(n)}_\alpha]{\Re W}\, d\alpha
= - \frac{1}{2} \int (\Re W_\alpha) |W^{(n)}|^2\, d\alpha. \end{equation*} In addition, \[ \int \Re [W_\alpha \bar W^{(n)}] \Re W^{(n)}\, d\alpha
= \frac{1}{2}\int (\Re W_\alpha) |W^{(n)}|^2 + \Re [W_\alpha (\bar W^{(n)})^2] \, d\alpha. \] It follows that \begin{align*}
\int |{\tilde W}^{(n)}|^2 \, d\alpha
&=\int \left(1 - 4n\Re W_\alpha\right) |W^{(n)}|^2
- 4 \sum_{j = 2}^{n-1}\binom{n}{j} \Re \left[\bar W^{(n)}W^{(n-j+1)}\right] \Re W^{(j)} \\ &\qquad- 2\Re[{W}_\alpha (\bar W^{(n)})^2]\, d\alpha + \text{quartic}. \end{align*}
A similar, but longer, computation for the terms involving ${\tilde Q}$ yields \begin{align*} \int\Im [{\tilde Q}^{(n)}\bar {\tilde Q}^{(n)}_{\alpha}] \, d\alpha &= \Im \int \left(1 - 4n\Re W_\alpha\right) \left(Q^{(n)} - Q_\alpha W^{(n)}\right) \left(\bar Q^{(n)} - \bar Q_\alpha \bar W^{(n)}\right)_\alpha \\ &\quad + 2n \left({Q}_{\alpha\alpha} W^{(n)} \bar Q^{(n)} + {Q}_{\alpha\alpha} \bar W^{(n)} \bar Q^{(n)}\right) + 2\bar{Q}_\alpha {W}^{(n)} Q^{(n)}_\alpha \, d\alpha \\ &\quad + 4 \int \sum_{j=3}^{n-1} \binom{n+1}{j} \Im[\bar Q^{(n)} {Q}^{(n-j+2)} ] \Re W^{(j)}\, d\alpha + \text{quartic}. \end{align*} Up to quartic corrections, we may replace $Q^{(n)}$ by $R^{(n-1)}$ and $Q^{(j)}$ by $R^{(j-1)}$ for $j \leq n$ in the cubic terms on the right-hand side of this equation. Further, we have \[ \begin{split}
\partial_\alpha^n Q - R \partial_\alpha^n W = & \ (1+W_\alpha)R^{(n-1)} + \sum_{j = 1}^{n-2} \binom{n-1}{j} R^{(j)} W^{(n-j)} \\ = &\ {\tilde {\mathbf R}} + n(R_\alpha W^{(n-1)} - W_{\alpha\alpha} R^{(n-2)}) + \sum_{j = 2}^{n-3} \binom{n-1}{j} R^{(j)} W^{(n-j)}. \end{split} \] Thus, we obtain \begin{align} \begin{split} \int\Im [{\tilde Q}^{(n)}\bar {\tilde Q}^{(n)}_{\alpha}] \, d\alpha = & \ \Im \int \left(1 - 4n\Re W_\alpha\right) {\tilde {\mathbf R}}\bar {\tilde {\mathbf R}}_\alpha + 2n {R}_\alpha \bar W^{(n)} \bar {\tilde {\mathbf R}} + 2 \bar{R}_\alpha{W}^{(n)} R^{[n]}_\alpha d\alpha \\ &\ + 2 \Im \!\! \int n \bar {\tilde {\mathbf R}}( W^{(3)} R^{(n-2)} - R^{(2)} W^{(n-1)})
+ \bar {\tilde {\mathbf R}}_\alpha \sum_{j = 2}^{n-3} \binom{n-1}{j} R^{(j)} W^{(n-j)}) d\alpha \\ &\ + 4 \int \sum_{j=3}^{n-1} \binom{n+1}{j} \Im[\bar R^{(n-1)} {R}^{(n-j+1)} ] \Re W^{(j)}\,
+ \text{quartic}. \end{split} \label{tqn_eq} \end{align}
Adding (\ref{twn_eq}) and (\ref{tqn_eq}), we find that \[ E^{n}_{NF,0} = E^{n}_{NF} + \text{quartic terms}, \] where $E^{n}_{NF}$ is given by (\ref{def_En_hl}) with \begin{align*} E^{n}_{NF,low} &= - 4 \int \sum_{j = 2}^{n-1}\binom{n}{j} \Re \left[\bar W^{(n)}W^{(n-j+1)}\right] \Re W^{(j)}\, d\alpha \\ &\quad + 4 \int \sum_{j=3}^{n-1} \binom{n+1}{j} \Im[\bar R^{(n-1)} {R}^{n-j+1} ] \Re W^{(j)}\, d\alpha \\ &\quad + 2 \Im \int n \bar {\tilde {\mathbf R}}( W^{(3)} R^{(n-2)} - R^{(2)} W^{(n-1)})+ \bar {\tilde {\mathbf R}}_\alpha \sum_{j = 2}^{n-3} \binom{n-1}{j} R^{(j)} W^{(n-j)}\, d\alpha, \end{align*} which, after we substitute $W_{\alpha}$ by ${\mathbf W}$, gives us an energy of the form stated in the proposition.
It remains to establish \eqref{enlow-diff}. The second estimate follows immediately from H\"older's inequality and interpolation. So does most of the first, except for two terms. By the Coifman-Meyer estimate in Lemma~\ref{l:com} we have \[ \int \bar R {\mathbf W}^{(n-1)} R^{(n-1)}_\alpha d\alpha = \int {\mathbf W}^{(n-1)} \bar P[\bar R R^{(n-1)}_\alpha ] d\alpha
= O(A) \| {\mathbf W}^{(n-1)}\|_{L^2} \|R^{(n-1)}\|_{\dot H^\frac12}. \] On the other hand, for the integral \[ \int \Re {\mathbf W} \Im [R^{(n-1)} \bar R^{(n-1)}_\alpha] d \alpha, \] we do a Littlewood-Paley decomposition, using the $\dot H^\frac12$ norm of $R^{(n-1)} $ if the two $R$ frequencies are high, and interpolation and H\"older's inequality otherwise.
\end{proof}
To get our final energy functionals $E^{n,(3)}$, we replace $E^{n}_{NF,high}$ in $E^{n}_{NF}$ by its nonlinear version, $E^{n,(3)}_{high} := E^{n,(3)}_{high}(Pw,Pr)$. That is, we define \begin{equation} E^{n,(3)} = E^{n}_{NF} - E^{n}_{NF,high} + E^{n,(3)}_{high}= E^{n}_{NF,low} + E^{n,(3)}_{high}. \end{equation} Note that $E^{n,(3)}$ differs from $E^{n}_{NF}$ only by a quartic term.
Now we proceed to prove Proposition~\ref{t:en=small}. The norm equivalence is already known from \eqref{en3-equiv} and \eqref{enlow-equiv}, so we still need the energy estimate. First, we write \[ \frac{d}{dt} E^{n,(3)} = \frac{d}{dt} E^{n}_{NF} + \frac{d}{dt} \left(E^{n,(3)}_{high} - E^{n}_{NF,high} \right). \] This equation shows that there are no cubic terms on the right-hand side, since the derivatives of $E^{n}_{NF}$ and $E^{n,(3)}_{high} - E^{n}_{NF,high}$ contain only terms that are quartic or higher order.
Next, we write \[ \frac{d}{dt} E^{n,(3)} = \frac{d}{dt} E^{n}_{NF,low} + \frac{d}{dt} E^{n,(3)}_{high}. \] Both expressions have cubic terms, but these cancel due to the prior computation. To make this cancellation precise, at this point we make the convention that all multilinear expansions are in terms of ${\mathbf W}$ and $R$. To make this cancellation explicit, we introduce a truncation operator ${{\Lambda}}^4$ that removes the cubic terms and retains everything which is quartic and higher.
Hence, we obtain \begin{equation} \frac{d}{dt} E^{n,(3)} = {{\Lambda}}^4\left(\frac{d}{dt} E^{n}_{NF,low}\right) + {{\Lambda}}^4\left( \frac{d}{dt} E^{n,(3)}_{high}\right). \end{equation} It remains to prove the following estimates: \begin{align}\label{quad-low}
&\left|{{\Lambda}}^4\left(\frac{d}{dt} E^{n}_{NF,low} \right)\right| \lesssim_A AB {\mathbf N}_n^2, \\ \label{quad-hi}
&\left|{{\Lambda}}^4\left(\frac{d}{dt} E^{n,(3)}_{high} \right)\right| \lesssim_A AB {\mathbf N}_n^2. \end{align} The second bound follows directly from \eqref{en3-evolve}, so it remains to prove \eqref{quad-low}.
\subsection{ Estimates for lower order terms: proof of
(\protect\ref{quad-low})}
We have two main types of energy terms (or their complex conjugates) to consider, \begin{align*} I_1 &= \int {\mathbf W}^{(j)} {\mathbf W}^{(k)} \bar {\mathbf W}^{(l)} d\alpha, \qquad j+k+l = 2n-2,\quad 1 \leq j,k,l \leq n, \\ I_2 &= \int {\mathbf W}^{(j)} R^{(k)} \bar R^{(l)} d\alpha, \qquad \ \ \ j+k+l = 2n-1, \quad 1 \leq j, k, l \leq n. \end{align*}
To estimate their time derivatives it is easiest to use the unprojected form \eqref{ww2d-diff} of the equations for ${\mathbf W}$ and $R$, which for our purposes here we write in the form \begin{equation} \left\{ \begin{aligned} &(\partial_{t}+ b\partial_{\alpha}) {\mathbf W} = \ - b_\alpha(1+{\mathbf W}) - \bar R_\alpha := G, \\ &(\partial_{t}+ b\partial_{\alpha}) R= \ i \frac{{\mathbf W} - a}{1+{\mathbf W}}:= K. \end{aligned} \right. \end{equation} Of $G$ and $K$ we will only need their quadratic parts and higher, \[ G^{2+} = - b_\alpha {\mathbf W} + P(R\bar Y) + \bar P(\bar R Y), \qquad K^{2+} =
- i \frac{{\mathbf W}^2 + a}{1+{\mathbf W}}. \] Then, we have \[ \begin{split} {{\Lambda}}^4\left(\frac{d}{dt} I_1\right) = & \ \int \partial^{j-1} (- b {\mathbf W}_{\alpha}+G^{2+} ) {\mathbf W}^{(k)} \bar {\mathbf W}^{(l)} + {\mathbf W}^{(j)} \partial^{k-1} (- b {\mathbf W}_{ \alpha}+G^{2+} ) \bar {\mathbf W}^{(l)} \\ & + {\mathbf W}^{(j)} {\mathbf W}^{(k)} \partial^{l-1} (- b \bar {\mathbf W}_{ \alpha}+\bar G^{2+} ) \, d\alpha \end{split} \] Distributing derivatives, we separate the terms with undifferentiated $b$ as \[ \int -b \partial_\alpha( {\mathbf W}^{(j)} {\mathbf W}^{(k)} \bar {\mathbf W}^{(l)}) \, d\alpha = \int b_\alpha {\mathbf W}^{(j)} {\mathbf W}^{(k)} \bar {\mathbf W}^{(l)}\, d\alpha, \] therefore all terms involving $b$ have the form \[
\int b^{(m)} {\mathbf W}^{(j)} {\mathbf W}^{(k)} \bar {\mathbf W}^{(l)}\, d\alpha, \qquad m+j+k+l = 2n-1, \quad 1 \leq m \leq n-1,\quad 1 \leq j,k,l \leq n-1, \] which we can estimate by H\"older's inequality and interpolation, using Lemma~\ref{l:b}, to get the $b$ bounds \[
\| |D|^\frac12 b\|_{BMO} \lesssim_A A, \qquad \||D|^{n-\frac12} b\|_{L^2} \lesssim_A {\mathbf N}_n. \]
The remaining terms have the form \[ \int \partial^{j-1} P(R \bar Y) W^{(k)} \bar W^{(l)} d\alpha. \] These are again estimated by H\"older's inequality and interpolation, using the bounds proved in Lemma~\ref{l:b}, which show that \[
\| |D|^\frac12 P(R \bar Y)\|_{BMO} \lesssim_A A^2, \qquad \||D|^{n-\frac12}P(R \bar Y) \|_{L^2} \lesssim_A A {\mathbf N}_2. \] .
The argument for $I_2$ is similar, using the algebra property of $L^\infty \cap \dot H^s$, together with the $L^\infty$ and $\dot{H}^{n-1}$ bound for $a$ in Proposition~\ref{regularity for a} in order to show that \[
\| K^{2+}\|_{BMO} \lesssim_A A^2, \qquad \||D|^{n-1} K^{2+}\|_{L^2} \lesssim_A A {\mathbf N}_2. \].
\section{ Local well-posedness} \label{s:lwp}
As the water wave equations \eqref{ww2d1} are fully nonlinear, the standard strategy to prove well-posedness would be to differentiate the equations to turn them into a system of quasilinear equations for $(w,q):=(W_\alpha,Q_\alpha)$, and then apply an iteration scheme. The problem with a direct implementation of this idea is that the quasilinear problem is degenerate, and diagonalizing it requires using the exact equations; thus the diagonalization would fail in an approximation scheme.
To remedy this, we use the form \eqref{ww2d-diff} of the equations in terms of the diagonal variables $({\mathbf W},R)$ directly. Projecting those on the holomorphic space we obtain \begin{equation} \label{ww(WaR)} \left\{ \begin{aligned} & (\partial_t + \mathfrak M_b \partial_\alpha) {\mathbf W} + P \left[ \frac{1+{\mathbf W}} {1+\bar {\mathbf W}} R_\alpha\right] =K({\mathbf W}, R) , \\
&(\partial_t + \mathfrak M_b \partial_\alpha) R - i P\left[ \frac{(1+a){\mathbf W} }{1+{\mathbf W}} \right] = K({\mathbf W}, R), \end{aligned}\right. \end{equation} where $K({\mathbf W}, R):=P\left[ (1+{\mathbf W})M \right]$ and $K({\mathbf W}, R):= P\left[ a\right]$.
We now turn to the business of solving the system \eqref{ww(WaR)}. The state space for this will be the space ${\dot{\mathcal H} }_n$ endowed with the norm \[
\| ({\mathbf W},R) \|_{{\dot{\mathcal H} }_n} := \sum_{k=0}^n \| \partial^k_\alpha ({\mathbf W},R)\|_{ L^2 \times \dot H^\frac12}, \] where $n \geq 1$. As a preliminary step, we will also consider solutions in the smaller space \[ {\mathcal H }_n = H^n \times H^{n+\frac12}, \quad \mbox{with }n \geq 2. \] We remark that, given a solution in ${\dot{\mathcal H} }_n$ for the above equation, we already know how to obtain uniform energy estimates for it for $n \geq 1$. The issue at hand is to convert those estimates into a well-posedness statement. We also remark that our energy estimates are expressed in terms of the control norms $A$ and $B$. These are in turn mostly controlled using the ${\dot{\mathcal H} }_1$ norm of $ ({\mathbf W},R)$. The exception is the $L^\infty$ bound for $Y$, which, as it turns out, can be bounded in terms of its initial data and the ${\dot{\mathcal H} }_1$ norm of $ ({\mathbf W},R)$.
To better understand the evolution of the ${\dot{\mathcal H} }_1$ norm of the solution it is convenient to use the language of frequency envelopes. We say that a sequence $c_k \in \ell^2$ is a ${\dot{\mathcal H} }_1$ frequency envelope for
$({\mathbf W},R) \in {\dot{\mathcal H} }^1$ if (i) it is slowly varying, $c_j/c_k \leq 2^{-\delta|j-k|}$ with a small universal constant $\delta$, and (ii)
it bounds the dyadic norms of $ ({\mathbf W},R) $, namely $\|P_k ({\mathbf W},R)
\|_{{\dot{\mathcal H} }_1} \leq c_k$.
Our main result here is: \begin{proposition}
a) Let $n \geq 1$. Then the problem \eqref{ww2d1} is locally
well-posed in for initial data $({\mathbf W},R)$ in ${\dot{\mathcal H} }_n$.
b) (lifespan) There exists $T= T(\|({\mathbf W},R)\|_{{\dot{\mathcal H} }_1}, \|Y\|_{L^\infty})$ so that the above solutions are well defined in $[0,T]$, with uniform bounds.
c) (frequency envelopes) Given a frequency envelope $c_k$ for the initial data in ${\dot{\mathcal H} }_1$, a similar frequency envelope $C(\|({\mathbf W},R)\|_{{\dot{\mathcal H} }_1}, \|Y\|_{L^\infty}) c_k$ applies for the solutions in $[0,T]$. \end{proposition} Thereom ~\ref{baiatul} is a consequence of the above proposition. The statement about the persistence of solutions for as long as $A, B$ remain bounded is a consequence of the energy estimates in Proposition~\ref{plin-short} and Proposition~ \ref{t:en=large}, where the constants depend only on $A$ and $B$.
We remark that the well-posedness result in part (a) carries different meanings depending on $n$. If $n \geq 2$, then we obtain existence and uniqueness in $C({\dot{\mathcal H} }_n)$ together with continuous dependence on the data with respect to the stronger ${\mathcal H }_n$ topology. On the other hand if $n =1$ then we produce rough solutions $C({\dot{\mathcal H} }_1)$ as the unique strong limit of smooth solutions, with continuous dependence on the data with respect to the stronger ${\mathcal H }_1$ topology. The ${\mathcal H }_1$ continuous dependence is a standard consequence of the strong ${\mathcal H }_n$ continuous dependence on data together with the frequency envelope bounds. However, for $n=1$, we do not establish a direct uniqueness result.
The proof proceeds in several steps:
\subsection{Existence of regular solutions} Here we consider data $({\mathbf W},R)(0) \in {\mathcal H }_n$ with $n \geq 2$, and prove the existence of solutions in the same space. Our strategy is to obtain approximate solutions by solving the mollified system \begin{equation} \label{ww(WaR)-N} \left\{\begin{aligned} & (\partial_t + P_{<N} \mathfrak M_{b_{N}} \partial_\alpha P_{<N} ) {\mathbf W} + P_{<N} P \left[ \frac{1+P_{<N} {\mathbf W}} {1+P_{<N} \bar {\mathbf W}} P_{<N} R_\alpha\right] = P_{<N} G(P_{<N} {\mathbf W},P_{<N}R), \\
&(\partial_t + P_{<N}\mathfrak M_{b_N} \partial_\alpha P_{<N}) R - i P_{<N} P\left[ \frac{(1+a_N)P_{<N}{\mathbf W} }{1+P_{<N}{\mathbf W}} \right] = P_{<N} K(P_{<N} {\mathbf W},P_{<N} R), \end{aligned}\right. \end{equation} where $P_{<N}$ is a multiplier which selects frequencies less than $N$, and \[ b_N = b(P_{<N} {\mathbf W},P_{<N}R), \qquad a_N= a(P_{<N}R). \]
For fixed $N$ these equations form a system of ordinary differential equations in ${\mathcal H }_n$, which admits a local solution. We can consider it with a single data, or with a one parameter family of data. The latter will help with the dependence of data for our original equation.
We will prove uniform estimates for this evolution in ${\mathcal H }_n$, $n \geq 1$, and then obtain our solution (or one parameter family of solutions) as a weak limit on a subsequence as $N \to \infty$.
The $(G,K)$ terms are Lipschitz, indeed $C^1$ from ${\mathcal H }_n$ to ${\mathcal H }_n$, therefore harmless. The ${\mathcal H }_{n-1}$ norm of $({\mathbf W},R)$ is estimated directly by time integration, \begin{equation} \label{bubu}
\frac{d}{dt}\Vert ({\mathbf W}, R)\Vert^2 _{{\mathcal H }_{n-1}}\lesssim c(\|({\mathbf W},R)\|_{{\mathcal H }_n}^2) \|({\mathbf W},R)\|_{{\mathcal H }_n}^2. \end{equation} It remains to estimate the ${\mathcal H }_0$ norm of $\partial^n_\alpha({\mathbf W},R)$. We differentiate the equations \eqref{ww(WaR)-N} $n$ times. This yields \begin{equation}\label{ww(WaR)Nd} \left\{ \begin{aligned} & (\partial_t + P_{<N} \mathfrak M_{b_{N}} \partial_\alpha P_{<N} ) {\mathbf W}^{(n)} + P_{<N} P \left[ \frac{(1+ P_{<N} {\mathbf W}) } {1+ P_{<N} \bar {\mathbf W}}\partial_\alpha P_{<N} R^{(n)} \right] = G_n, \\
&(\partial_t + P_{<N} \mathfrak M_{b_N} \partial_\alpha P_{<N}) R^{(n)} - i P_{<N} P\left[ \frac{(1+a_N) P_{<N}{\mathbf W}^{(n)} }{(1+ P_{<N}{\mathbf W})^2} \right] = K_n, \end{aligned}\right. \end{equation} where all other terms, included in $G_n$ and $K_n$, are estimated directly in ${\mathcal H }_0$ in terms of the ${\mathcal H }^n$ norm of $({\mathbf W},R)$. We observe that the fact that we work in ${\mathcal H }_n$ with $n \geq 2$ allows us to use pointwise bounds for $R$, $R_\alpha$, $b$, $b_\alpha$, and thus deal with a larger number of terms in this fashion.
To bring this to the standard form, where we can apply energy estimates previously obtained in Section~\ref{s:linearized}, we make the substitution \[ \mathbf R^{(n)} := R^{(n)}(1+ P_{<N} {\mathbf W}). \] Multiplying in the second equation by $(1+ P_{<N} {\mathbf W})$, all of the commutator terms are also perturbative, and we obtain the system \[ \left\{ \begin{aligned} & (\partial_t + P_{<N} \mathfrak M_{b_{N}} \partial_\alpha P_{<N} ) {\mathbf W}^{(n)} + P_{<N} P \left[ \frac{1} {1+ P_{<N} \bar {\mathbf W}} P_{<N} \mathbf R_{\alpha}^{(n)}\right] = \mathbf{G}_n, \\
&(\partial_t + P_{<N} \mathfrak M_{b_N} \partial_\alpha P_{<N}) \mathbf R^{(n)} - i P_{<N} P\left[ \frac{(1+a_N) P_{<N}{\mathbf W}^{(n)} }{(1+ P_{<N}{\mathbf W})} \right] = \mathbf{K}_n, \end{aligned}\right. \] where $\mathbf{G}_n$ and $\mathbf{K}_n$ are appropriate replacements of the (perturbative) terms in \eqref{ww(WaR)Nd}, $G_n$ and $K_n$ respectively.
For this system we do energy estimates as before, with the energy functional \[
E^{n} = \int (1+a_N) |{\mathbf W}^{(n)}|^2 + Im (\mathbf R^{(n)} \partial_\alpha \mathbf R^{(n)}) + |\mathbf R^{(n)}|^2\, d\alpha. \]
We obtain \[
\frac{dE^n}{dt} \lesssim c(\|({\mathbf W},R)\|_{{\mathcal H }_n}^2) \|({\mathbf W},R)\|_{{\mathcal H }_n}^2. \] We combine this with \eqref{bubu}. Since \[
\Vert R\Vert_{\dot{H}^{\frac{1}{2}}}\lesssim \Vert \mathbf R\Vert_{\dot{H}^{\frac{1}{2}}}\Vert Y\Vert_{L^{\infty}}+\Vert \mathbf R \Vert_{L^2}\Vert |D|^{\frac{1}{2}}Y\Vert_{BMO}, \] we have that \[
\Vert({\mathbf W}, R)\Vert^2_{{\mathcal H }_n} \lesssim c(\|({\mathbf W},R)\|_{{\mathcal H }_{n-1}}^2)\left( E^n +\|({\mathbf W},R)\|_{{\mathcal H }_{n-1}}^2\right), \] which leads to a bound for our approximate system which is uniform in $N$, \begin{equation}
\| ({\mathbf W},R)(t)\|_{{\mathcal H }_n} \lesssim \| ({\mathbf W},R)(0)\|_{{\mathcal H }_n}, \quad 0 \leq t \leq T( \| ({\mathbf W},R)(0)\|_{{\mathcal H }_n},\|Y(0)\|_{L^\infty}). \end{equation}
Similarly, one can consider a smooth family of data $({\mathbf W}_h,R_h)$ in ${\mathcal H }_n$ for $h \in [0,1]$. Then the solutions depend smoothly on $h$, with a lifespan uniformly bounded from below. We consider the $h$ derivatives $({\tilde w},{\tilde r}) = \partial_h ({\mathbf W}_h,R_h)$. These solve the linearized equation, which when considered in ${\mathcal H }_{n-1}$, can be written in the same form as \eqref{ww(WaR)Nd}, with perturbative terms on the right. Thus, we obtain \begin{equation}
\| ({\tilde w},{\tilde r})(t)\|_{{\mathcal H }_{n-1}} \lesssim \| ({\tilde w},{\tilde r})(0)\|_{{\mathcal H }_{n-1}}, \quad 0 \leq t \leq T( \| ({\mathbf W},R)(0)\|_{{\mathcal H }_n},\|Y(0)\|_{L^\infty}). \end{equation} In the same manner one can obtain estimates for the second order derivatives with respect to $h$ in ${\mathcal H }_{n-2}$, \emph{etc}. Passing to a weak limit on a subsequence as $N \to \infty$ we obtain a family of solutions $({\mathbf W}_h,R_h)$ which is uniformly bounded in ${\mathcal H }_n$, with $h$ derivatives uniformly bounded in ${\mathcal H }_{n-1}$, \emph{etc}.
\subsection{ Uniqueness of regular solutions}
In the previous subsection we have constructed ${\mathcal H }_{n}$ solutions for $n \geq 2$. Here we prove that these solutions are unique. For later use, we show that uniqueness holds in the larger class of ${\dot{\mathcal H} }_n$ solutions for $n \geq 2$.
Suppose we have two ${\dot{\mathcal H} }_2$ solutions $({\mathbf W}_1,R_1)$ and $({\mathbf W}_2,R_2)$ to \eqref{ww(WaR)}. Subtracting the two sets of equations we obtain a system for the difference $({\tilde w},{\tilde r})$, namely \begin{equation} \label{ww(WaR)diff} \left\{\begin{aligned} & (\partial_t + \mathfrak M_{b_1} \partial_\alpha) {\tilde w} + P \left[ \frac{1+{\mathbf W}_1} {1+\bar {\mathbf W}_1} {\tilde r}_\alpha\right] =\tilde G, \\
&(\partial_t + \mathfrak M_{b_1} \partial_\alpha) {\tilde r} - i P\left[ \frac{(1+a_1){\tilde w} }{(1+{\mathbf W}_1)^2} \right] = \tilde K, \end{aligned}\right. \end{equation} where \[ \left\{\begin{aligned} \tilde G = & G({\mathbf W}_1,R_1) - G({\mathbf W}_2,R_2) +\mathfrak M_{b_1-b_2} \partial_\alpha {\mathbf W}_2 + P \left[ \left(\frac{1+{\mathbf W}_1}{1+\bar {\mathbf W}_1} - \frac{1+{\mathbf W}_2}{1+\bar {\mathbf W}_2} \right) \partial_{\alpha}R_{2}\right], \\ \tilde K = & K({\mathbf W}_1,R_1)- K({\mathbf W}_2,R_2)+ \mathfrak M_{b_1-b_2} \partial_\alpha R_2 + i P\left[ \frac{(1+a_1){\tilde w}^2 }{(1+{\mathbf W}_1)^2(1+{\mathbf W}_2)} +
\frac{(a_1-a_2){\mathbf W}_2 }{1+{\mathbf W}_2} \right]
\end{aligned}\right. \] With implicit constants depending on the ${\dot{\mathcal H} }_2$ solutions $({\mathbf W}_1,R_1)$ and $({\mathbf W}_2,R_2)$, we have \[
\| (\tilde G,\tilde K) \|_{{\mathcal H }_0} \lesssim \|({\tilde w},{\tilde r})\|_{{\mathcal H }_0}. \] Then we simultaneously do energy estimates for $({\tilde w},{\tilde r}(1+{\mathbf W}_1))$ in ${\mathcal H }^{\frac{1}{2}}=L^2 \times \dot H^\frac12$ and for $R$ in $L^2$, and then apply Gronwall's inequality to get $({\tilde w},{\tilde r})=(0,0)$.
\subsection{\texorpdfstring{$\mathcal {\dot{\mathcal H} }_1 $}\ \ bounds}
The solutions produced above have a lifespan which depends on the ${\mathcal H }_n$ size of data. Here we prove that in effect the lifespan depends only on the ${\dot{\mathcal H} }_1$ size of data, and that we have uniform bounds for as long as the ${\dot{\mathcal H} }_1$ size of the solutions is controlled.
Precisely, suppose we have an ${\mathcal H }_n$ solution $({\mathbf W},R)$ which satisfies the bounds \begin{equation*}
\| ({\mathbf W},R)(0)\|_{{\dot{\mathcal H} }_1} < {\mathcal M}_0, \qquad \|Y(0)\|_{L^\infty} < \mathcal {\mathcal K}_0. \end{equation*} Then we claim that there exists $T = T({\mathcal M}_0,{\mathcal K}_0)$ so that the solution exists in $[0,T]$ and satisfies the bounds \begin{equation}\label{h1bd}
\| ({\mathbf W},R)\|_{L^\infty(0,T; {\dot{\mathcal H} }_1)} < {\mathcal M}({\mathcal M}_0,{\mathcal K}_0), \qquad
\|Y\|_{L^\infty([0,T]\times \mathbf R)} < {\mathcal K}({\mathcal M}_0,{\mathcal K}_0), \end{equation} as well as the ${\mathcal H }_n$ and ${\dot{\mathcal H} }_n $ bounds \begin{equation*}
\| ({\mathbf W},R)\|_{L^\infty(0,T; {\mathcal H }_n)} \leq C ({\mathcal M}_0,{\mathcal K}_0)\| ({\mathbf W},R)(0)\|_{ {\mathcal H }_n}, \end{equation*} \begin{equation*}
\| ({\mathbf W},R)\|_{L^\infty(0,T; {\dot{\mathcal H} }_n)} \leq C ({\mathcal M}_0,{\mathcal K}_0)\| ({\mathbf W},R)(0)\|_{ {\dot{\mathcal H} }_n}. \end{equation*}
To prove this, we begin by making the bootstrap assumption \begin{equation*}
\| ({\mathbf W},R)\|_{L^\infty(0,T;{\dot{\mathcal H} }_1)} < 2{\mathcal M}, \qquad
\|Y\|_{L^\infty([0,T]\times \mathbf R)} < 2{\mathcal K}. \end{equation*} We will show that for a suitable choice ${\mathcal M}({\mathcal M}_0,{\mathcal K}_0)$ and ${\mathcal K}({\mathcal M}_0,{\mathcal K}_0)$, depending only on ${\mathcal M}_0$ and ${\mathcal K}_0$, we can improve this to \eqref{h1bd}, provided that $T < T({\mathcal M}_0,{\mathcal K}_0)$.
We begin by applying the linearized energy estimates obtained in Proposition~\ref{plin-short} to $({\mathbf W},R)$ \begin{equation}\label{h1-dh0}
\| ({\mathbf W},R)(t)\|_{{\dot{\mathcal H} }_0} \lesssim e^{C t } \| ({\mathbf W},R)(0)\|_{{\dot{\mathcal H} }_0} , \qquad C = C({\mathcal M},{\mathcal K}). \end{equation}
Applying the energy estimates proven in Proposition~ \ref{t:en=large} $(ii)$ for the pair
$({\mathbf W}_\alpha,(1+{\mathbf W})R_\alpha)$ we get \begin{equation}\label{h1-dh1}
\| ({\mathbf W}_\alpha,(1+{\mathbf W})R_\alpha)(t)\|_{{\mathcal H }_0} \lesssim e^{C t },
\| ({\mathbf W}_\alpha,(1+{\mathbf W})R_\alpha)(0)\|_{{\dot{\mathcal H} }_0}. \end{equation}
To combine \eqref{h1-dh0} and \eqref{h1-dh1} we need to invert $1+{\mathbf W}$. However a brute force argument introduces a constant which depends on both ${\mathcal K}$ and ${\mathcal M}$, which wreaks havoc with our bootstrap. Instead we do a more careful argument, using the pair of bounds \begin{equation} \begin{split}
&\|(1+{\mathbf W})R_\alpha\|_{\dot H^\frac12} \lesssim_{{\mathcal K}} \|R_\alpha\|_{\dot H^\frac12} +
\| {\mathbf W}_\alpha\|_{L^2} \||D|^\frac12 R\|_{L^\infty}, \\
& \|R_\alpha\|_{\dot H^\frac12} \lesssim_{{\mathcal K}} \|(1+{\mathbf W})R_\alpha\|_{\dot H^\frac12}+
\| {\mathbf W}_\alpha\|_{L^2} \||D|^\frac12 R\|_{L^\infty}. \end{split} \end{equation} Since \[
\| |D|^\frac12 R\|_{L^\infty}^2 \lesssim_{{\mathcal K}} \|R\|_{\dot H^\frac12}
\|R_\alpha\|_{\dot H^\frac12}, \] we obtain \[
\||D|^\frac12 R\|_{L^\infty}^2 \lesssim_{{\mathcal K}} {\mathcal M}_0^2 e^{2 C t } (1+ \||D|^\frac12 R\|_{L^\infty}), \] so \begin{equation*}
\||D|^\frac12 R\|_{L^\infty} \leq C_0 {\mathcal M}_0^2 e^{2C t }, \qquad C_0 = C_0({\mathcal K}). \end{equation*} Then it follows that \begin{equation}\label{H1cont}
\| ({\mathbf W},R)(t)\|_{{\dot{\mathcal H} }_1} \leq C_0 {\mathcal M}_0^3 e^{3 C t }. \end{equation}
Since ${\mathcal M}$ appears only in the exponent where it is controlled by choosing
$t$ small, the bound \eqref{H1cont} suffices in order to bootstrap ${\mathcal M}$. It remains to recover the bootstrap assumption on $\|Y\|_{L^\infty}$. For this we use an estimate of the form \[
\|Y\|_{L^\infty}^2 \lesssim \|{\mathbf W}_\alpha\|_{L^2} \|{\mathbf W}(1+ {\mathbf W})^{-3}\|_{L^2}. \] The bound for the first factor is independent of ${\mathcal K}$. For the second we write the transport equation \[ (\partial_t + b \partial_\alpha) \frac{{\mathbf W}}{(1+ {\mathbf W})^{3}} = \frac{3-2{\mathbf W}}{(1+{\mathbf W})^3} \left ([P,W_\alpha] \frac{\bar R}{(1+{\mathbf W})^2} - P \left[ \frac{R}{1+ \bar {\mathbf W}}\right] _\alpha \right). \] We can estimate the right hand side in $L^2$ with constants depending on ${\mathcal K}$. To bound $\dfrac{{\mathbf W}}{(1+ {\mathbf W})^{3}}$ in $L^2$ we use an estimate of the form \begin{equation*}
\begin{aligned} \frac{d}{dt}\Vert u\Vert_{L^2}^2=\int_{\mathbb{R}}b_{\alpha}\vert u\vert^2 + 2\Re (\partial_t+b\partial_{\alpha})u \bar{u}\, d\alpha. \end{aligned} \end{equation*} For the second term on the right we use the Cauchy-Schwarz inequality and for the first term we use a Littlewood-Paley trichotomy. When the frequency of $b_{\alpha}$ is strictly less than the frequencies of $u$ and $\bar{u}$ then we can move half of derivative on either of $u$ or $\bar{u}$; otherwise Coiman-Meyer type estimates apply, and we obtain \[
| \int _{R}b_{\alpha}\vert u\vert^2\, d\alpha |\lesssim \Vert b_{\alpha}\Vert_{BMO}\Vert u\Vert^2_{L^2}+\Vert
|D|^{\frac{1}{2}}b\Vert_{BMO}\Vert u\Vert_{\dot{H}^{\frac{1}{2}}}\Vert u\Vert_{L^2}. \] We conclude that \begin{equation} \label{dany} \begin{aligned} \frac{d}{dt}\Vert u\Vert_{L^2}\lesssim \Vert b_{\alpha}\Vert_{BMO}\Vert u\Vert^2_{L^2}+\Vert
|D|^{\frac{1}{2}}b\Vert_{BMO}\Vert u\Vert_{\dot{H}^{\frac{1}{2}}}\Vert u\Vert_{L^2}+ \Vert u\Vert_{L^2}\Vert (\partial_t+b\partial_{\alpha})u\Vert_{L^2}. \end{aligned} \end{equation}
We apply this estimate to $\dfrac{{\mathbf W}}{(1+ {\mathbf W})^{3}}(t)$ to obtain \[
\|\frac{{\mathbf W}}{(1+ {\mathbf W})^{3}}(t) \|_{L^2} \leq \|\frac{{\mathbf W}}{(1+ {\mathbf W})^{3}}(0) \|_{L^2} + t C({\mathcal K},{\mathcal M}). \] This leads to \[
\|Y\|_{L^\infty}^2 \lesssim {\mathcal M}_0 {\mathcal K}_0^3 + t C({\mathcal K},{\mathcal M}). \] Hence in order for our bootstrap argument to succeed we need to find ${\mathcal K},{\mathcal M}$ and $T$ so that \[
{\mathcal M} > 2 C_0({\mathcal K}) {\mathcal M}_0^3 e^{C({\mathcal K},{\mathcal M}) T}, \qquad {\mathcal K}^2 > 2( {\mathcal M}_0 {\mathcal K}_0^3 + t C({\mathcal K},{\mathcal M})). \] This is easily achieved by succesively choosing \[ {\mathcal K}^2 = 10 {\mathcal M}_0 {\mathcal K}_0^3, \qquad {\mathcal M} = 10 C_0({\mathcal K}) {\mathcal M}_0^3 , \qquad T < C({\mathcal K},{\mathcal M})^{-1}. \] Thus, the bootstrap is complete.
The next step is to show that we can propagate the full ${\mathcal H }_{n} $ norm given control of ${\dot{\mathcal H} }_1$ norm of the solution $({\mathbf W}, R)$. For higher derivatives we can use Proposition~\ref{t:en=large} to obtain \begin{equation}\label{Hncont}
\| ({\mathbf W},R)(t)\|_{{\dot{\mathcal H} }_n} \leq C e^{C t } \| ({\mathbf W},R)(t)\|_{{\dot{\mathcal H} }_n}, \qquad C=C({\mathcal K},{\mathcal M}). \end{equation}
We also need to control the growth of the $L^2$ norm of $R$; for this we use equation \ref{ww2d-diff} for which we
can easily obtain $L^2$ bounds of the RHS. Applying \eqref{dany} we obtain \[
\|R(t) \|_{L^2} \leq \|R(0) \|_{L^2} + t C({\mathcal K},{\mathcal M}). \] The ${\mathcal H }_n$ bound shows that the solution can be continued for as long as it stays bounded in ${\dot{\mathcal H} }_1$, i.e., at least until time $T({\mathcal K}_0, {\mathcal M}_0)$.
\subsection{ \texorpdfstring{${\dot{\mathcal H} }_n$}\ \ solutions for
\texorpdfstring{$n \geq 2$}{}\ \ }
Our goal here is to obtain solutions for ${\dot{\mathcal H} }_n$ data. We already know that such solutions, if they exist, are unique. The idea is to approximate a ${\dot{\mathcal H} }_n$ data set $({\mathbf W},R)(0)$ with ${\mathcal H }_n$ data in the ${\dot{\mathcal H} }_n$ topology. As the uniform ${\dot{\mathcal H} }_n$ bounds hold uniformly for the approximating sequence, we would like to conclude that on a subsequence these approximate solutions converge weakly to the desired solution. The only difficulty with this plan is that the ${\dot{\mathcal H} }_n$ convergence does not guarantee uniform pointwise convergence for $R$. This is because the lowest Sobolev norm we control for $R$ is the $\dot H^\frac12$ norm, and that does not see constants.
To address the above difficulty, we take an approximating sequence of data $({\mathbf W}_k,R_k)(0)$ which has the following two properties:
(i) $({\mathbf W}_k,R_k)(0) \to ({\mathbf W},R)(0)$ in ${\dot{\mathcal H} }_n$,
(ii) $R_k(0) \to R(0)$ uniformly on compact sets.
The second requirement effectively removes the Galilean invariance. It suffices to ask for pointwise convergence at a single point; in view of the known average growth rates for BMO functions, this implies the weighted uniform convergence \[
\| \log(2+|\alpha|)^{-1} R_k(0) - R(0)\|_{L^\infty} \to 0. \]
We will use the second requirement (ii) to produce weighted uniform bounds for the $R_k$ part of the solution. Starting from the uniform bound \[
\| ({\mathbf W}_k,R_k)\|_{{\dot{\mathcal H} }_n} \lesssim 1, \] we estimate uniformly most of the terms in the $R_k$ equation to obtain \[
\| (\partial_t + 2 \Re R_k \partial_\alpha) R_k\|_{ L^\infty} \lesssim 1. \] This yields a uniform bound for $R_k$ along the corresponding characteristic \[ \dot \alpha(t) = 2 \Re R_k (\alpha), \qquad \alpha(0) = 0, \] namely \[
|R_k(t,\alpha(t))| \lesssim 1. \] This in turn shows that locally in time we have \[
|\alpha(t)| \lesssim 1, \] which leads to the uniform bound \[
|R_k(t,0)| \lesssim 1, \] and further to the global bound \[
|R_k| \lesssim \log(2+|\alpha|). \] This in turn yields a similar bound for $\partial_t {\mathbf W}_k$ and $\partial_t R_k$, and suffices in order to insure local uniform convergence of $({\mathbf W}_k,R_k)$ on a subsequence. Thus, the desired solution $({\mathbf W},R)$ is obtained in the limit.
\subsection{Rough solutions}
Here we construct solutions for data in ${\dot{\mathcal H} }_1$ as unique limits of smooth solutions. Given a ${\dot{\mathcal H} }_1$ initial data $({\mathbf W}_0,R_0)$ as above we regularize it to produce smooth approximate data $({\mathbf W}_0^k,R_0^k) = P_{< k} ({\mathbf W}_0,R_0) $. We denote the corresponding solutions by $({\mathbf W}^k,R^k)$. By the previous analysis, these solutions exist on a $k$-independent time interval $[0,T]$ and satisfy uniform $ {\dot{\mathcal H} }_1$ bounds. Further, they are smooth and have a smooth dependence on $k$.
Consider a frequency envelope $c_k$ for the initial data $({\mathbf W}_0,R_0)$ in ${\dot{\mathcal H} }_1$. Then for the regularized data we have \[
\| ({\mathbf W}^k_0,R^k_0)\|_{\mathcal H_n} \lesssim c_k 2^{(n-1)k}, \qquad n \geq 2. \] Hence, in the time interval $[0,T]$ we also have the estimates \begin{equation}\label{high(W,K)}
\| ({\mathbf W}^k,R^k)\|_{\mathcal H_n} \lesssim c_k 2^{(n-1)k}, \qquad n \geq 2. \end{equation} We will use these for the high frequency part of the regularized solutions.
For the low frequency part, on the other hand, we view $k$ as a continuous rather than a discrete parameter, differentiate $({\mathbf W}^k,R^k)$ with respect to $k$ and use the estimates for the linearized equation. One minor difficulty is that the linearized equation \eqref{lin(wr)0} arises from the linearization of the $(W,Q)$ system in \eqref{ww2d1} rather than the differentiated $({\mathbf W},R)$ system in \eqref{ww2d-diff}. Assuming that $(W^k,Q^k)$ were also defined, we formally denote \[ (w^k,r^k) = (\partial_k W^k, \partial_k Q^k - R \partial_k W^k). \] These would solve the linearized equation around the $({\mathbf W}^k,R^k)$ solution. For our analysis we want to refer only to the differentiated variables, so we we compute \[ \begin{split} \partial_\alpha w^k = & \ \partial_k W^k, \\ \partial_\alpha r^k = & \ (1+{\mathbf W}^k) \partial_k R^k - R_\alpha^k w^k. \end{split} \] We take these formulas as the definition of $(w^k,r^k)$, and observe that inverting the $\partial_\alpha$ operator is straightforward since the above multiplications involve only holomorphic factors therefore the products are at frequency $2^k$ and higher. To take advantage of the bounds in Proposition~\ref{plin-short} for the linearized equation, we need a ${\dot{\mathcal H} }_0$ bound for $(w^k(0),r^k(0))$, namely \begin{equation}
\| (w^k(0),r^k(0)) \|_{{\dot{\mathcal H} }_0} \lesssim c_k 2^{-2k}. \end{equation} The bound for $w^k(0)$ is straightforward, but some work is required for $r^k(0)$. This follows via the usual Littlewood-Paley trichotomy and Bernstein's inequality for the low frequency factor, with the twist that, since both factors are holomorphic, no high-high to low interactions occur.
In view of the uniform ${\dot{\mathcal H} }_1$ bound for $(W^k,Q^k)$, Proposition~\ref{plin-short} shows that in $[0,T]$ we have the uniform estimate \begin{equation}
\| (w^k,r^k) \|_{{\dot{\mathcal H} }_0} \lesssim c_k 2^{-2k}. \end{equation} Now, we return to $({\mathbf W}^k, R^k)$ and claim the bound \begin{equation}\label{diff(W,K)}
\| P_{\leq k} (\partial_k {\mathbf W}^k, \partial_k R^k) \|_{{\dot{\mathcal H} }_0} \lesssim c_k 2^{-k}. \end{equation} Again the ${\mathbf W}^k$ bound is straightforward. For $\partial_k R^k$ we write \[ \partial_k R^k = (1- Y^k)(\partial_\alpha r^k + R_\alpha^k \partial_k W^k), \] where again all factors are holomorphic. Then applying $P_{\leq k}$ restricts all frequencies to $\lesssim 2^k$, and the Littlewood-Paley trichotomy and Bernstein's inequality again apply.
Now we integrate \eqref{diff(W,K)} over unit $k$ intervals and use it to estimate the differences. Combining the result with \eqref{high(W,K)} we obtain \begin{equation} \begin{split}
\| ({\mathbf W}_{k+1} - {\mathbf W}_k, R_{k+1}-R_k)\|_{{\dot{\mathcal H} }_0} \lesssim & \ c_k 2^{-k}, \\
\| \partial_\alpha^2({\mathbf W}_{k+1} - {\mathbf W}_k, R_{k+1}-R_k)\|_{{\dot{\mathcal H} }_0} \lesssim & \ c_k 2^{k}. \end{split} \end{equation}
Summing up with respect to $k$ it follows that the sequence
$({\mathbf W}^k,R^k)$ converges uniformly in ${\dot{\mathcal H} }_1$ to a solution $({\mathbf W},R)$, which also inherits the frequency envelope bounds from the data.
The frequency envelope bounds allow us to prove continuous dependence on the initial data in the ${\mathcal H }^1$ topology. This is standard, but we briefly outline the argument. Suppose that $({\mathbf W}_j,R_j)(0) \in {\dot{\mathcal H} }_1$ and $({\mathbf W}_j,R_j)(0)-({\mathbf W},R)(0) \to 0 $ in ${\mathcal H }_1$. We consider the approximate solutions $(({\mathbf W}_j^k,R_j^k)$, respectively $({\mathbf W}^k,R^k)$. According to our result for more regular solutions, we have \begin{equation}\label{reg-lim} (({\mathbf W}_j^k,R_j^k) - ({\mathbf W}^k,R^k) \to 0 \qquad \text{in} \ \ {\mathcal H }_n. \end{equation} On the other hand, from the ${\mathcal H }_1$ data convergence we get \[ ({\mathbf W}_j^k,R_j^k)(0) - ({\mathbf W}_j,R_j)(0) \to 0 \qquad \text{in} \ \ {\mathcal H }_1 \ \ \text{uniformly in} \ \ j. \] Then the above frequency envelope analysis, shows that \[ ({\mathbf W}_j^k,R_j^k) - ({\mathbf W}_j,R_j) \to 0 \qquad \text{in} \ \ {\mathcal H }_1 \ \ \ \text{uniformly in} \ \ j. \] Hence we can let $k$ go to infinity in \eqref{reg-lim} and conclude that \[ (({\mathbf W}_j,R_j) - ({\mathbf W},R) \to 0 \qquad \text{in} \ \ {\mathcal H }_1. \]
\section{ Enhanced cubic lifespan bounds} \label{s:cubic} In this section we prove Theorem~\ref{t:cubic}. Given initial data $({\mathbf W},R)$ for \eqref{ww2d-diff} satisfying \[
\| ({\mathbf W},R)(0)\|_{{\dot{\mathcal H} }_1} \leq \epsilon, \] we consider the solutions on a time interval $\left[0, T \right] $ and seek to prove the estimate \begin{equation} \label{miki}
\| ({\mathbf W},R)(t)\|_{{\dot{\mathcal H} }_1} \leq C\epsilon,\quad t\in \left[0, T \right], \end{equation} provided that $T\ll e^{-2}$. In view of our local well-posedness result this shows that the solutions can be extended up to time $T_{\epsilon}=ce^{-2}$ concluding the proof of the theorem.
In order to prove \eqref{miki} we can harmlessly make the bootstrap assumption \begin{equation} \label{miki2}
\| ({\mathbf W},R)(t)\|_{{\dot{\mathcal H} }_1} \leq 2C\epsilon,\quad t\in \left[0, T \right]. \end{equation}
From \eqref{miki2} we obtain the bounds \begin{equation*} A,B\lesssim C\epsilon. \end{equation*} Hence, by the energy estimates in Proposition~\ref{plin-long} applied to $({\mathbf W}, R)$, and those in Proposition~\ref{t:en=small}, with $n=2$, applied to $({\mathbf W}_{\alpha}, R_{\alpha})$ we obtain \[ \Vert ({\mathbf W}, R)\Vert_{L^{\infty}(0,T;{\dot{\mathcal H} }_1)}\lesssim \Vert ({\mathbf W}, R)(0)\Vert_{{\dot{\mathcal H} }_1}+TAB\Vert ({\mathbf W}, R)\Vert_{L^{\infty}(0,T;{\dot{\mathcal H} }_1)}\lesssim \epsilon +TC^3\epsilon^3. \] Hence, the desired estimate \eqref{miki} follows provided that $T\ll (C\epsilon)^{-2}$.
\section{ Pointwise decay and long time solutions} \label{s:decay}
In this section we prove the almost global existence result in Theorem~\ref{t:almost}. This is achieved via a bootstrap argument for the energy norm $\| (W,Q)(t)\|_{{\mathcal{WH} }}$ defined in \eqref{WH} as well as the control norms $A(t)$ and $B(t)$ in \eqref{A-def},\eqref{B-def}. In order to have a more robust argument we
will work with a stronger norm $\| (W,R)\|_{X} \gtrsim A(t)+B(t)$, namely \[
\| (W,R)\|_{X} = \| W\|_{L^\infty} + \| R \|_{L^\infty}
+ \| |D|^\frac12 W_{\alpha}\|_{L^\infty} + \|R_\alpha \|_{L^\infty} \]
Then we will establish the energy estimates \begin{equation}
\sup_{|t| \leq T_\epsilon}\| (W,Q)(t)\|_{{\mathcal{WH} }} \lesssim \epsilon, \label{energy}\end{equation} as well as the pointwise bounds \begin{equation} \label{point}
\| (W,R)\|_{X} \lesssim \epsilon \langle t \rangle^{-\frac12}, \qquad |t| \leq T_\epsilon, \end{equation} for times $T_\epsilon$ satisfying \begin{equation}\label{which-T}
T_\epsilon \leq e^{c \epsilon^{-2}}, \qquad c \ll 1. \end{equation}
A continuity argument based on our local well-posedness results shows that
it suffices to prove that \eqref{point} and \eqref{energy} hold for all $T_\epsilon$ as in \eqref{which-T}, given the bootstrap assumptions \begin{equation}\label{energyboot}
\sup_{|t| \leq T_\epsilon}\| (W,Q)(t)\|_{{\mathcal{WH} }} \leq C \epsilon, \end{equation} \begin{equation}\label{pointboot}
\| (W,R)\|_{X} \leq C \epsilon \langle t \rangle^{-\frac12}, \qquad 0 \leq t \leq T_\epsilon, \end{equation} with a large constant $C$ (independent of $\epsilon$).
\subsection{The energy estimates in \protect(\ref{energy}).} Here we use the bootstrap assumption \eqref{pointboot} in order to establish \eqref{energyboot}. The only role of \eqref{energyboot} is to insure that a solution with appropriate regularity exists up to time $T_\epsilon$. We summarize the result in the following \begin{proposition}
Assume that in a time interval $[-T,T]$ we have a solution $(W,Q)$ to
\eqref{ww2d1} which satisfies \eqref{data} and \eqref{pointboot}.
Then we also have the energy estimate \begin{equation}\label{e}
\|(W,Q)(t)\|_{{\mathcal{WH} }}^2 \lesssim \epsilon \langle t \rangle^{C_1 \epsilon^2} , \qquad t \in [-T,T] \end{equation} for some $C_1 \gg C$. \end{proposition} Then the bound \eqref{energy} holds with a constant independent of $C$ for times as in \eqref{which-T} if we choose $c = C_1^{-1}$.
\begin{proof}
The energy bound for $(W,Q)$ is a consequence of the conserved energy \eqref{ww-energy}. The energy bounds for $({\mathbf W},R)$ and $(w,r):= \mathbf A{\mathbf S}(W,Q) $ follow by Gronwall's inequality from the cubic energy estimates for the linearized equation in Proposition~\ref{plin-long}; indeed, by our bootstrap assumption \eqref{pointboot} we have $A(t),B(t) \leq C \epsilon \langle t \rangle^{-\frac12} $, therefore, \[
\| (w,r)(t)\|_{{\dot{\mathcal H} }_0} \lesssim e^{\int_{0}^{t}
C^2 \epsilon^2 \langle s \rangle^{-1} ds} \| (w,r)(0)\|_{{\dot{\mathcal H} }_0} \lesssim \epsilon e^{C^2 \epsilon^2 \log t}, \] which suffices for $T_\epsilon$ as in \eqref{which-T}. Finally, the bound for $\partial^k({\mathbf W},R)$ with $1 \leq k \leq 5$ follows also by Gr\"onwall's inequality from the cubic energy estimates in Proposition~\ref{t:en=small}. \end{proof}
\subsection{The pointwise estimates} Here we use the bootstrap assumption \eqref{energyboot} in order to establish \eqref{point}. To state the main result here we introduce the notation \begin{equation}\label{omega} \omega(t,\alpha) = \frac{1}{\langle t \rangle^{\frac{1}{22}}} + \frac{1}{(\langle \alpha\rangle/\langle t \rangle + \langle t\rangle /\langle \alpha\rangle)^\frac12} \lesssim 1. \end{equation} Then we have: \begin{proposition} \label{p:point} Assume that \eqref{e} and \eqref{pointboot} hold in some interval $[-T,T]$. Then we also have \begin{equation}\label{strong-point}
|W| + |R| + ||D|^\frac12 W_{\alpha}| + | R_\alpha| \lesssim \epsilon \langle t\rangle^{-\frac12} \langle t \rangle^{C_1 \epsilon^2}\omega(t,\alpha) \end{equation} \end{proposition} Then our pointwise bound \eqref{point} follows for times as in \eqref{which-T}, and the proof of Theorem~\ref{t:almost} is concluded.
We remark that the result we prove here is somewhat stronger than what we need. However, on one hand this is what follows from our analysis, and on the other hand this stronger result will come in handy when we prove the global result in a follow-up paper.
The rest of this section is devoted to the proof of the above proposition. We note that \eqref{which-T} plays no role in this argument.
In order to obtain pointwise bounds it is convenient to work with the normal form variables $({\tilde W},{\tilde Q})$, given by \eqref{nft1}. Then we prove several very simple Lemmas. The first one shows that we can harmlessly replace $(W,R)$ by $(\tilde W, \tilde Q_\alpha)$ in the pointwise estimates.
\begin{lemma} Assume that \eqref{e} and \eqref{pointboot} hold in some interval $[-T,T]$. Then \begin{equation}\label{point-eq}
\|(W-{\tilde W},R-{\tilde Q}_\alpha)\|_{X} \lesssim \langle t \rangle^{-\frac18} \|({\tilde W},{\tilde Q}_\alpha)\|_{X} \end{equation} \end{lemma} \begin{proof} It suffices to show that \[
\|(\tilde W-W, \tilde Q_\alpha-R)\|_{X} \lesssim \langle t \rangle^{-\frac18} \|(W, R)\|_{X} \] For the $\tilde W$ bound we have $W-{\tilde W} = \mathfrak M_{\Re W} W_\alpha$ so we use Sobolev embeddings, product Sobolev bounds and interpolation to estimate \[ \begin{split}
\| \mathfrak M_{\Re W} W_\alpha\|_{L^\infty} + \| |D|^\frac32 ( \mathfrak M_{\Re W} W_\alpha)\|_{L^\infty}
\lesssim & \ \| \Re W W_\alpha\|_{L^4} + \| D^2 (\Re W W_\alpha)\|_{L^4} \\
\lesssim & \
\| D^{2} W\|_{L^3} \| W_\alpha\|_{L^\infty} + \|W\|_{L^\infty} (\|W_\alpha\|_{L^4} + \|D^{3} W\|_{L^4}) \\ \lesssim & \
(\|W\|_{L^\infty} + \|W_\alpha\|_{L^\infty}) A(t)^\frac12 \|W\|^{\frac12}_{H^5} \\ \lesssim & \ C^{\frac12} \epsilon^2
\langle t \rangle^{-\frac14+C_1 \epsilon^2}\|(W, R)\|_{X} . \end{split} \] which suffices since $\epsilon$ is small.
For the $R$ bound we write \[ {\tilde Q}_\alpha-R = W_\alpha R - 2 \partial_\alpha(\mathfrak M_{\Re W} R) \] and a similar argument as above applies.
\end{proof}
Our second lemma translates the energy bounds to $({\tilde W},{\tilde Q})$: \begin{lemma} Assume that \eqref{e} and \eqref{pointboot} hold in some time interval $[-T,T]$. Then \begin{equation}
\|( {\tilde W},{\tilde Q})\|_{{\dot{\mathcal H} }_5} + \|{\mathbf S}( {\tilde W},{\tilde Q})\|_{{\dot{\mathcal H} }_0+{\dot{\mathcal H} }_{-1}} \lesssim \epsilon \langle t \rangle^{C_1 \epsilon^2}. \end{equation} \end{lemma} \begin{proof} For $\tilde W$ we estimate the quadratic terms \[ \begin{split}
\| \Re W W_\alpha\|_{L^2} + \|\partial^5 (\Re W W_\alpha)\|_{L^2}
\lesssim & \ \|W\|_{L^\infty} (\| W_\alpha\|_{L^2} + \| \partial^5 W_\alpha\|_{L^2})
+ \| W_\alpha\|_{L^\infty} \| \partial^5 W\|_{L^2}
\\ \lesssim & \ \epsilon (\| W_\alpha\|_{L^2}+ \|\partial^6 W\|_{L^2}), \end{split} \]
By interpolation and Sobolev embeddings we can combine \eqref{e} and \eqref{pointboot} to obtain the rough bound \[
\|W\|_{L^\infty} + \|R\|_{L^\infty} \lesssim \epsilon, \] which we will use to supplement \eqref{pointboot}. For $\tilde Q$ we first bound the quadratic term in $\dot H^\frac12$, \[
\| \Re W R\|_{\dot H^\frac12} \lesssim \|W\|_{L^\infty} \|R\|_{\dot H^\frac12}
+ \| W\|_{\dot H^\frac12} \| R\|_{L^\infty} \lesssim \epsilon (\|R\|_{\dot H^\frac12}
+ \| W\|_{\dot H^\frac12}). \] For higher derivatives we write \[ \tilde Q_\alpha = R(1+W_\alpha) - 2 \partial_\alpha (\mathfrak M_{\Re W} R) \] and apply the same method.
The goal of the reminder of the proof is to prove that \begin{equation} \label{stwq}
\|S({\tilde W},{\tilde Q})\|_{{\dot{\mathcal H} }_0 + {\dot{\mathcal H} }_{-1}} \lesssim \epsilon \|S(W,R)\|_{{\dot{\mathcal H} }_0} \end{equation} Recalling the notation $(w,r) = (SW,SQ-RSW)$, we first write $S{\tilde W}$ as \[ S {\tilde W} = w + 2P[ \Re w W_\alpha - \Re W_\alpha w - 2\Re W W_\alpha + 2 P \partial_\alpha [ \mathfrak M_{2\Re W} w], \] and use the $L^2$ bound on $SW$ to estimate all but the last term in $L^2$, and the last term in $\dot H^{-1}$. Finally, for $S {\tilde Q}$ we have \[ S\tilde Q = S Q - \mathfrak M_{ 2\Re SW} R - \mathfrak M_{ 2\Re W} S R = r + R w - 2P (\Re w R) - 2 P \left[ \frac{\Re W (r_\alpha + R_\alpha w)}{1+W_\alpha}\right] \] Here it suffices to estimate the contribution of $w$ in $L^2$, while the contribution of $r_\alpha$ is estimated by \[
\| r_\alpha H\|_{H^{-\frac12}} \lesssim \|r\|_{\dot H^\frac12} (\|H\|_{L^\infty} +
\||D|^\frac12 H\|_{BMO}), \qquad H := \frac{\Re W}{1+W_\alpha}, \]
where $\||D|^\frac12 H\|_{BMO}$ is estimated using \eqref{bmo-alg}, \eqref{bmo-moser}. The proof of \eqref{stwq} is concluded. \end{proof}
The advantage of working with $({\tilde W},{\tilde Q})$ is that they solve an equation with a cubic nonlinear term, namely \eqref{nft1eq}, where the nonlinearities $\tilde G$ and $\tilde K$ are given by \eqref{gk-tilde}. They involve second order derivatives of $W$ and $Q$, which is why one cannot simply use the above equations as the main evolution.
\begin{lemma}\label{cubic-RHS} Assume that \eqref{energy} and \eqref{pointboot} hold in some time interval $[-T,T]$. Then \begin{equation}
\| ( \tilde G,\tilde K)\|_{{\dot{\mathcal H} }_0} \lesssim \frac{\epsilon^3}{\langle t \rangle} \langle t \rangle^{C_1 \epsilon^2}. \label{energy-rhs}\end{equation} \end{lemma} \begin{proof} Given the expression above for $\tilde G$, it suffices
to bound each factor in each term in suitable $L^p$ norms,
interpolating between the $L^2$ norms in \eqref{energy} and the
$L^\infty$ norms in \eqref{pointboot}. For $\tilde K$ the argument
is similar, but we also need to use Lemma~\ref{l:multi} in the
Appendix~\ref{s:multilinear} in order to distribute the half
derivative. \end{proof}
Taking into account the correspondence, established in the last three lemmas, between the original variables $(W,Q)$ and the normal form variables $(\tilde W,\tilde Q)$, it follows that we can restate Proposition~\ref{p:point} in the following linear form:
\begin{proposition} \label{p:point1} Suppose $({\tilde W},{\tilde Q})$ solve
\eqref{nft1eq} and that the following bounds hold at some time
$t$: \[
\| {\mathbf S} ({\tilde W},{\tilde Q})\|_{{\dot{\mathcal H} }_0 + {\dot{\mathcal H} }_{-1}} + \| ({\tilde W}, {\tilde Q}) \|_{{\dot{\mathcal H} }_{5}} \lesssim 1, \]
\[
\| ( \tilde G,\tilde K)\|_{{\dot{\mathcal H} }_0} \lesssim \langle t \rangle^{-1}. \] Then \begin{equation}
|{\tilde W}|+ | |D|^\frac12 {\tilde Q}| +
| D^2 {\tilde W}| + ||D|^\frac52 {\tilde Q}|
\lesssim \langle t \rangle^{-\frac12} \omega(t,\alpha). \end{equation} \end{proposition}
Combining the scaling bound with the equation \eqref{nft1eq} we are led to a system of the form \begin{equation} \label{fixed-t} \left\{ \begin{aligned} & 2 \alpha \partial_\alpha \tilde W + t \partial_\alpha \tilde Q = \tilde G_1:= S\tilde W - \tilde G , \\ & 2\alpha \partial_\alpha \tilde Q - it \tilde W = \tilde K_1 := S \tilde Q - \tilde K. \end{aligned} \right. \end{equation} where \begin{equation}
\| (\tilde G_1,\tilde K_1)\|_{{\dot{\mathcal H} }_0 + {\dot{\mathcal H} }_{-1}} \lesssim 1. \end{equation} From here on, all our analysis is at fixed $t$.
After the substitution \[
(w,r) = (\tilde W, |D|^\frac12 \tilde Q), \qquad (g,k) =(\tilde G_1, |D|^\frac12 \tilde K_1 + |D|^\frac12 \tilde Q), \] the above system is written in a more symmetric form as \begin{equation} \label{fixed-t-i} \left\{ \begin{aligned}
& 2 \alpha \partial_\alpha w - i t |D|^\frac12 r = g, \\
& 2\alpha \partial_\alpha r - it |D|^\frac12 w = k. \end{aligned} \right. \end{equation} For this it suffices to establish the following result:
\begin{lemma} The following pointwise bounds hold for solutions to \eqref{fixed-t-i}: \begin{equation}\label{point0}
|w| + |r| \lesssim |\alpha|^{-\frac12} ( \| (w,r)\|_{L^2} +
\| (g,k)\|_{L^2}), \end{equation} \begin{equation}\label{point1}
|w| + |r| \lesssim \langle t \rangle^{-\frac12} \left(\frac{1}{\langle t \rangle^\frac14} + \frac{1}{(\langle \alpha\rangle /\langle t \rangle + \langle t\rangle/ \langle \alpha \rangle)^\frac12}\right) ( \| (w,r)\|_{H^2} +
\| (g,k)\|_{H^{-1}}), \end{equation} \begin{equation}\label{pointk}
|\partial^k w| + |\partial^k r| \lesssim \langle t \rangle^{-\frac12} \left( \! \frac{1}{\langle t \rangle^{\frac{1}{2(4k+5)}}} + \frac{1}{(\langle \alpha\rangle /\langle t \rangle + \langle t\rangle/ \langle \alpha \rangle)^\frac12} \! \right)
( \| (w,r)\|_{H^{2k+2}} +
\| (g,k)\|_{H^{-1}}). \end{equation} \end{lemma} The last part is applied with $k \leq \frac32$, which justifies the exponent $\frac{1}{22}$ in the definition \eqref{omega} of $\omega(t,\alpha)$.
\begin{proof} Without any loss in generality we assume that $|t| \geq 1$. It is convenient to work with frequency localized versions of \eqref{fixed-t-i}, at frequency $- 2^\ell$, with $\ell \in {\mathbb Z}$. The localized dyadic portions $(w_{\ell},r_\ell)$ solve similar equations with frequency localized right hand sides $(g_\ell,k_\ell)$. Further, a straightforward commutator estimate shows that \begin{equation}\label{fixed-diad}
\sum_{\ell \in {\mathbb Z}} \| (g_\ell,k_\ell)\|_{H^s}^2 \lesssim \|(w,r)\|_{H^s}^2 + \|(g,k)\|_{H^s}^2. \end{equation}
To prove \eqref{point0} we observe that the system \eqref{fixed-t-i} is elliptic away from frequency $2^{\ell} \approx t^2 \alpha^{-2}$ and degenerate at frequency zero. At frequencies less than $\alpha^{-1}$ our source for the pointwise estimate is Bernstein's inequality. Comparing the two frequencies yields the threshold $\alpha = t^2$, $2^{\ell} = t^{-2}$. Thus, we distinguish the following regions:
{\bf Case A: $2^\ell \leq t^{-2}$}. We group all such frequencies together. We can harmlessly discard the $i t |D|^\frac12$ term from the equations and compute \[
\frac{d}{dt} |w(\alpha)|^2 = 2\Re ( \bar w w_\alpha), \] and similarly for $r$. Depending on the sign of $\alpha$ we integrate from either $+\infty$ or $-\infty$ and apply the Cauchy-Schwartz inequality to obtain \[
|w_{<t^{-2}} |^2 \lesssim |\alpha|^{-1} \|w_{<t^{-2}}\|_{L^2} \| \alpha w_{<t^{-2},\alpha}\|_{L^2}. \] We remark that by Bernstein's inequality we also get \[
| w_{<t^{-2}}| \lesssim |t|^{-1} \|w_{<t^{-2}}\|_{L^2}. \] It follows that \[
| w_{<t^{-2}}| \lesssim (t+|\alpha|)^{-1}. \]
{\bf Case B.} $t^{-2} \lesssim 2^{\ell}$. Here we have three regions to consider:
\begin{enumerate}
\item[B1.] The outer region $|\alpha| \gg |t| 2^{-\frac{\ell}2}$ where the problem is elliptic, with $\alpha \partial_\alpha$ as the dominant term.
\item[B2.] The inner region $|\alpha| \ll |t| 2^{-\frac{\ell}2}$ where the problem is elliptic, with
$i t |D|^\frac12 $ as the dominant term.
\item[B3.] The intermediate region $|\alpha| \approx |t|2^{-\frac{\ell}2}$
where the problem is hyperbolic. \end{enumerate}
We consider three overlapping smooth positive cutoff functions $\chi^{\ell}_{out}$, $\chi^{\ell}_{med}$ and $\chi^{\ell}_{in}$ associated with the three regions. In order to keep the frequency localization we assume that all three cutoffs are localized at frequency $\ll 2^\ell$, at the expense of having tails which decay rapidly on the $2^{-\ell}$ scale. We remark that the three cutoffs begin to separate exactly at $2^\ell = t^{-2}$.
For the regions B1 and B3 we use elliptic estimates, while for B2 we use a propagation bound.
Using the frequency localized form of \eqref{fixed-t-i} we can bound \[
\| \chi^{\ell}_{out} \alpha \partial_\alpha (w_{\ell},r_{\ell}) \|_{L^2}
\lesssim |t| \|\chi^{\ell}_{out} |D|^\frac12 (r_{\ell},w_{\ell})\|_{L^2} + \|(g_\ell,k_\ell)\|_{L^2}.. \] After some commutations this gives \[
2^{\ell} \| \alpha \chi^{\ell}_{out} (w_{\ell},r_{\ell}) \|_{L^2} \lesssim 2^{\frac{\ell}2} |t|
\|\chi^{\ell}_{out} (r_{\ell},w_{\ell})\|_{L^2} + \|(g_\ell,k_\ell)\|_{L^2} + \|(w_{\ell},r_{\ell})\|_{L^2}. \] Taking into account the localization of $\chi^{\ell}_{out}$, this yields \begin{equation}
2^{\ell} \| \alpha \chi^{\ell}_{out} (w_{\ell},r_{\ell}) \|_{L^2} \lesssim
\|(g_\ell,k_\ell)\|_{L^2} + \|(w_{\ell},r_{\ell})\|_{L^2}. \end{equation} By Bernstein's inequality this gives the pointwise bound \begin{equation}\label{point-out}
\chi^{\ell}_{out} |(w_{\ell},r_{\ell})| \lesssim 2^{-\frac{\ell}2} |\alpha|^{-1}\left(\|(g_\ell,k_\ell)\|_{L^2} + \|(w_{\ell},r_{\ell})\|_{L^2}\right). \end{equation}
A similar computation, but with the roles of the two terms on the left in \eqref{fixed-t-i} reversed gives \begin{equation}
|t| 2^{\frac{\ell}2} \| \chi^{\ell}_{in} (w_{\ell},r_{\ell}) \|_{L^2} \lesssim \|(g_\ell,k_\ell)\|_{L^2}
+ \|(w_{\ell},r_{\ell})\|_{L^2}. \end{equation} By Bernstein's inequality this gives the pointwise bound \begin{equation}\label{point-in}
\chi^{\ell}_{in} |(w_{\ell},r_{\ell})| \lesssim |t|^{-1}\left(\|(g_\ell,k_\ell)\|_{L^2} + \|(w_{\ell},r_{\ell})\|_{L^2}\right). \end{equation}
It remains to consider the intermediate region, where we produce instead a propagation estimate. Precisely, for $\chi^{\ell}_{med} (w_{\ell},r_{\ell})$ we estimate \[ \begin{split}
\| (4 \alpha^2 \partial_\alpha - i t^2)\chi^{\ell}_{med} w_{\ell}\|_{L^2}
\lesssim &\ \| 2 \alpha (2\alpha \partial_\alpha \chi^{\ell}_{med} w_{\ell} - i t |D|^\frac12 \chi^{\ell}_{med} r_{\ell}) \|_{L^2} \\
& \ + \| t (\alpha |D|^\frac12 \chi^{\ell}_{med} r_{\ell} - it \chi^{\ell}_{med} w_{\ell})\|_{L^2}
\\ \lesssim &\ |t| 2^{-\frac{\ell}2} (\|(g_\ell,k_\ell)\|_{L^2} + \|(r_{\ell},w_{\ell})\|_{L^2}), \end{split} \] and similarly for $r_{\ell}$. Applying \[
\frac{d}{dt} |u|^2 = 2 \Re \left[( \partial_\alpha - i \frac{t^2}{\alpha^2}) u \cdot \bar u \right] \] for $u = \chi^{\ell}_{med} w_{\ell} $ and $u = \chi^{\ell}_{med} r_{\ell} $, integrating from infinity and using the Cauchy-Schwarz inequality yields \[
\chi^{\ell}_{med} |(w_{\ell}, r_{\ell})|^2 \lesssim \alpha^{-2} |t| 2^{-\frac{\ell}2}
(\|(g_\ell,k_\ell)\|_{L^2} + \|(r_{\ell},w_{\ell})\|_{L^2})\|(r_{\ell},w_{\ell})\|_{L^2}. \]
Using this in the interesting region $|\alpha| \approx |t| {\ell}^\frac12$ and the inner and outer estimates away from it we obtain \begin{equation}\label{point-med}
\chi^{\ell}_{med} |(w_{\ell}, r_{\ell})| \lesssim |\alpha|^{-\frac12}
(\|(g_\ell,k_\ell)\|_{L^2}^\frac12 \|(r,w_{\ell})\|_{L^2}^\frac12 +\|(r_{\ell},w_{\ell})\|_{L^2}). \end{equation}
Now, we prove the bounds in the Lemma by dyadic summation. There are several cases to consider:
{\bf Case 1: $t^{-2} < 2^{\ell} < 1$}, where all three bounds coincide, and it suffices to prove \eqref{point1}. Assume that the two norms in the right hand side of \eqref{point1} are $\leq 1$. For $\alpha$ we have three cases:
(a) $|\alpha| > t^{2}$, where we are in case B1 for all ${\ell}$. There we need only \eqref{point-out} to conclude that \[
|(w_{[t^{-2},1]} ,r_{[t^{-2},1]})(\alpha)|
\lesssim \sum_{2^\ell = t^{-2}}^1 2^{-\frac{\ell}2} |\alpha|^{-1} \approx |t| |\alpha|^{-1}. \]
(b) $|t| < |\alpha| < t^2$, where we are successively in case B1, B2 and B3. There we use \eqref{point-out} \eqref{point-med} and \eqref{point-in} to conclude that \[ \begin{split}
|(w_{[t^{-2},1]},r_{[t^{-2},1]})(\alpha)| \lesssim & \ \sum_{2^\ell = t^{-2}}^{\alpha^{-2} t^2} |t|^{-1}
+ |\alpha|^{-\frac12} + \sum_{2^\ell=\alpha^2 t^{-2}}^12^{-\frac{\ell}2} |\alpha|^{-1}
\\ \approx & \ |t|^{-1} |\log (1+t^2 |\alpha|^{-1})| + |\alpha|^{-\frac12}+ |\alpha|^{-1} \lesssim |\alpha|^{-\frac12}. \end{split} \]
(c) $ |\alpha| < |t|$, where we are in case B3 for all ${\ell}$. There we need only \eqref{point-in} to conclude that \[
|(w_{[t^{-2},1]},r_{[t^{-2},1]})(\alpha)| \lesssim \sum_{2^\ell = t^{-2}}^1 |t|^{-1} \lesssim t^{-1} |\log (|t|+2)|. \]
{\bf Case 2: $1 < 2^{\ell}$.} Here we have two subcases:
(i) $|\alpha| >| t|$, where we are in case B1 for all ${\ell}$. There we need only \eqref{point-out} to conclude that \[
|(w_{>1} ,r_{> 1})(\alpha)| \lesssim \sum_{2^\ell=1}^\infty 2^{-\frac{\ell}2} |\alpha|^{-1}
(\|(w,r)\|_{L^2} + \|(g,k)\|_{L^2}) \approx |\alpha|^{-1}(\|(w,r)\|_{L^2} + \|(g,k)\|_{L^2}) , \] which suffices for \eqref{point0}. In order to also obtain \eqref{pointk} we also use the Bernstein bound \begin{equation}\label{bernstein}
| (w_{\ell},r_{\ell})| \lesssim 2^{\frac{\ell}2} \|(w_\ell,r_\ell)\|_{L^2} . \end{equation} Then we obtain \[ \begin{split}
|\partial^k (w_{>1} ,r_{> 1})(\alpha)| \lesssim & \ \sum_{2^\ell=1}^\infty \min\{ 2^{(k+\frac12)\ell} |\alpha|^{-1}
(\|(g,k)\|_{H^{-1}}+\|(w,r)\|_{H^{-1}}), 2^{-(k-\frac32)\ell} \|(w,r)\|_{\dot H^{2k+2}}\} \\
\lesssim & \ |\alpha|^{-\frac12}
(\|(g,k)\|_{H^{-1}}+\|(w,r)\|_{H^{-1}})^\frac12 \|(w,r)\|_{\dot H^{2k+2}}^\frac12. \end{split} \]
(ii) $ |\alpha| < |t|$, where we are successively in cases B1, B2 and B3. There we use \eqref{point-out} \eqref{point-med} and \eqref{point-in} to conclude that \[ \begin{split}
|(w_{>1},r_{>1})(\alpha)| \lesssim &\ \left( \sum_{2^\ell = 1}^{\alpha^{-2} t^{2}} |t|^{-1}
+ |\alpha|^{-\frac12} + \sum_{2^\ell = \alpha^{-2} t^2}^\infty 2^{-\frac{\ell}2} |\alpha|^{-1}\right)
(\|(w,r)\|_{L^2} + \|(g,k)\|_{L^2}) \\
= & \ \left( |t|^{-1} |\log (1+ |t|/|\alpha|) | + |\alpha|^{-\frac12}+ |t|^{-1}\right)
|\alpha|^{-\frac12} (\|(w,r)\|_{L^2} + \|(g,k)\|_{L^2}) \\
\lesssim & \ |\alpha|^{-\frac12} (\|(w,r)\|_{L^2} + \|(g,k)\|_{L^2}), \end{split} \] which suffices for \eqref{point0}.
Finally, the bound \eqref{pointk} is obtained exactly as in (i) by combining the last computation with the trivial pointwise bound for $\partial^k(w_\ell,r_\ell)$ obtained from Bernstein's inequality \eqref{bernstein}. Separating the contributions from cases B1, B2 and B3 we obtain \[
|\partial^k (w_{>1},r_{>1})(\alpha)| \lesssim I +II+III, \] where by \eqref{point-out} and \eqref{bernstein} we have \[ \begin{split}
I = & \ \sum_{2^\ell = 1}^{\alpha^{-2} t^{2}} \min \{ 2^{(k+1) \ell} |t|^{-1} (\|(w,r)\|_{H^{-1}}+ \|(g,k)\|_{H^{-1}}), 2^{-(k+\frac32) \ell} \|(w,r)\|_{\dot H^{2k+2}}\} \\ \lesssim & \
|t|^{-\frac12} |t|^{-\frac{1}{8k+10}} (\|(w,r)\|_{H^{-1}}+ \|(g,k)\|_{H^{-1}}+
\|(w,r)\|_{\dot H^{2k+2}}) \end{split} \] by \eqref{point-med} we have \[
II =\alpha^\frac12 |t|^{-1} (\|(w,r)\|_{H^{-1}} + \|(g,k)\|_{H^{-1}}) \|(w,r)\|_{\dot H^{2k+2}}^\frac12, \] and by \eqref{point-out} and \eqref{bernstein} we have \[ \begin{split}
III = & \ \sum_{2^\ell = \alpha^{-2} t^2}^\infty \min\{ 2^{(k+\frac{1}2)\ell} |\alpha|^{-1} (\|(w,r)\|_{H^{-1}} + \|(g,k)\|_{H^{-1}}), 2^{-(k+\frac32) \ell} \|(w,r)\|_{\dot H^{2k+2}}\} \\
\lesssim & \ |\alpha|^\frac12 |t|^{-1} (\|(w,r)\|_{H^{-1}} + \|(g,k)\|_{H^{-1}})^\frac12 \|(w,r)\|_{\dot H^{2k+2}}^\frac12 \end{split} \]
\end{proof}
\appendix
\section{Holomorphic equations} \label{holom-eq}
In this section we give an alternative derivation for the evolution equations for water waves in conformal coordinates. They were first obtained in \cite{ov}, and also later in \cite{zakharov, zakharov2} but using a different set up. We use a holomorphic form of the equations, as in \cite{zakharov, zakharov2}, but we compactify the equations even more, as we will show below. We also express the normal derivative of the pressure on the boundary in terms of our variables.
We consider two-dimensional, irrotational gravity water waves in an inviscid, incompressible fluid of infinite depth. First, we discuss the localized case on $\mathbb{R}$ in which the waves decay at infinity. The spatially periodic case is almost identical, and we describe the appropriate modifications afterwards.
\subsection{Holomorphic coordinates} Suppose that at time $t$ the fluid occupies a spatial region $\Omega(t)\subset \mathbb{R}^2$ whose simple nondegenerate boundary $\Gamma(t)= \partial\Omega(t)$ approaches $y=0$ at infinity.
Then there is a unique conformal map $\mathcal{F}(t) : \mathbb{H} \to \Omega(t)$ from the lower half-plane \[\mathbb{H} = \left\{\alpha+i\beta : \beta < 0\right\} \]
onto $\Omega(t)$, with $x = x(t,\alpha,\beta)$ and $y = y(t,\alpha,\beta)$, such that $z = x + i y$ satisfies \[
z - (\alpha + i \beta) \to 0 \qquad \text{as} \ \ \alpha + i\beta \to \infty. \]
Since $\mathcal{F}(t)$ is conformal, we have $x_\alpha = y_\beta$, $x_\beta = - y_\alpha$. If $f(t,\cdot) : \Omega(t) \to \mathbb{C}$ is a time-dependent spatial function and $g(t,\cdot) = f(t,\cdot)\circ \mathcal{F}(t) : \mathbb{H} \to \mathbb{C}$ is the corresponding conformal function,
then $g_t = f_t + x_t f_x + y_t f_y$, so \begin{equation} f_t = g_t - \frac{1}{j}\left(x_\alpha x_t + y_\alpha y_t\right) g_\alpha -\frac{1}{j}\left( x_\beta x_t + y_\beta y_t\right)g_\beta, \quad j =x_\alpha^2 + y_\alpha^2. \label{tder} \end{equation} Also, if $f + ig :\mathbb{H} \to \mathbb{C}$ is a holomorphic function with boundary value $F + iG$ on the real axis $\beta=0$ that vanishes at infinity, then $F = H G$ where the Hilbert transform $H$ is defined by \[ H f(\alpha) = \frac{1}{\pi} \mathrm{p.v.} \int_{-\infty}^\infty \frac{f(\alpha')}{\alpha-\alpha'} \, d\alpha', \qquad H e^{ik\alpha} = -i (\mathop{\mathrm{sgn}} k) e^{ik\alpha}. \] We denote by $P = \frac{1}{2}\left(I - iH\right)$ the projection onto boundary values of functions that are holomorphic in the lower half-plane and vanish at infinity. That is, $P$ projects functions onto their negative wavenumber components.
\subsection{Water waves in holomorphic coordinates} Let $\phi : \Omega(t) \to \mathbb{R}$ be the spatial velocity potential of the fluid, chosen so that it vanishes at infinity, and $\psi = \phi\circ \mathcal{F} : \mathbb{H} \to \mathbb{R}$ the corresponding conformal velocity potential, \[ \psi(t,\alpha,\beta) = \phi\left(t, x(t, \alpha,\beta), y(t, \alpha,\beta)\right). \] Then $\psi$ is harmonic since $\phi$ is harmonic; we denote the conjugate function of $\psi(t,\alpha,\beta)$ by $\theta(t,\alpha,\beta)$. The velocity components of the fluid $(u,v) = (\phi_x,\phi_y)$ are given in terms of $\psi$ by \begin{equation} u = \frac{1}{j}\left(x_\alpha\psi_\alpha + x_\beta \psi_\beta\right), \quad v = \frac{1}{j}\left(y_\alpha\psi_\alpha + y_\beta \psi_\beta\right).
\label{uveq} \end{equation}
The conformally parametrized equation of the free surface $\Gamma(t)$ is $x = X(t,\alpha)$, $y = Y(t,\alpha)$, where $X(t,\alpha) = x(t,\alpha,0)$, $Y(t,\alpha) = y(t,\alpha,0)$.
To avoid any confusions, we emphasize that the variable $Y$ used through this section has a different meaning than elsewhere in the paper; it is the vertical component of the parametrized free surface $Z(t,\alpha)$.
Since $(x-\alpha)+i(y-\beta)$ is holomorphic in the lower half-plane and vanishes at infinity, we have \begin{equation} X = \alpha + H Y,\qquad Y = - H (X-\alpha). \label{XYhilb} \end{equation} Let $\Psi(t,\alpha) = \psi(t,\alpha,0)$ denote the boundary value of the conformal velocity potential and $\Theta(t,\alpha) = \theta(t,\alpha,0)$ the conjugate function, where $\Theta = -H \Psi$ and \begin{equation}
\left.\psi_\beta\right|_{\beta=0} = H \Psi_\alpha = -\Theta_\alpha. \label{psieq} \end{equation} After these preliminaries, we transform the spatial boundary conditions for water-waves into conformal coordinates.
\emph{Kinematic BC.} A spatial normal to the free surface $\Gamma$ is $(-Y_\alpha,X_\alpha)$. The kinematic BC, that the normal component of the velocity of the free surface is equal to the normal component of the fluid velocity, is \[ \left(X_t, Y_t\right)\cdot (-Y_\alpha,X_\alpha) = \left(u, v\right)\cdot (-Y_\alpha,X_\alpha) \qquad\mbox{on $\Gamma(t)$}. \] Using (\ref{uveq}) and (\ref{psieq}) in this equation and simplifying the result, we get \begin{equation} X_\alpha Y_t - Y_\alpha X_t = -\Theta_\alpha. \label{kinBC1} \end{equation} In addition, the function $z_t/z_\alpha$ is holomorphic in $\mathbb{H}$ and decays at infinity, so the real part of its boundary value on the real axis is the Hilbert transform of its imaginary part. After the use of (\ref{kinBC1}), this gives the equation \begin{equation} X_\alpha X_t + Y_\alpha Y_t = -J H\left[\frac{\Theta_\alpha}{J}\right]. \label{kinBC2} \end{equation} Solving (\ref{kinBC1})--(\ref{kinBC2}) for $X_t$, $Y_t$, we get an expression for the velocity of a conformal point on the free surface \begin{equation} X_t = -H\left[\frac{\Theta_\alpha}{J}\right] X_\alpha + \frac{\Theta_\alpha}{J} Y_\alpha, \quad Y_t = -\frac{\Theta_\alpha}{J} X_\alpha - H\left[\frac{\Theta_\alpha}{J}\right] Y_\alpha. \label{kinBC} \end{equation}
\emph{Dynamic BC.} Bernoulli's equation for the pressure $p$ in the fluid, with gravitational acceleration $g=1$, is \begin{equation}
\phi_t + \frac{1}{2}|\nabla\phi|^2 + y + p = 0. \label{bernoulli_eq} \end{equation} The arbitrary function of $t$ that may appear in this equation is zero since we assume that $\phi$ vanishes at infinity and $p=0$ on the free surface which approaches $y=0$. The spatial form of the dynamic BC, without surface tension, is \[
\phi_t + \frac{1}{2}|\nabla \phi|^2 + y = 0 \qquad\mbox{on $\Gamma(t)$}. \] Using (\ref{tder}) to compute $\phi_t$, evaluating the result at $\beta=0$, and using (\ref{psieq})--(\ref{kinBC2}), we find that \[
\left.\phi_t\right|_{\beta=0} = \Psi_t + H\left[\frac{\Theta_\alpha}{J}\right] \Psi_\alpha -\frac{1}{J}\Theta_\alpha^2. \] We also have \[
\frac{1}{2}\left.|\nabla \phi|^2\right|_{\beta=0} = \frac{1}{2J} \left( \Psi_\alpha^2 + \Theta_\alpha^2\right). \] Hence, the dynamic BC in conformal variables is \begin{equation} \Psi_t + H\left[\frac{\Theta_\alpha}{J}\right] \Psi_\alpha + \frac{1}{2J} \left(\Psi_\alpha^2 - \Theta_\alpha^2\right) + Y = 0. \label{dynBC} \end{equation}
To put these equations in holomorphic form, we define \begin{equation} Z = X + i Y,\quad Q = \Psi + i\Theta,\quad F = P\left[\frac{Q_\alpha - \bar{Q}_\alpha}{J}\right],
\quad J = |Z_\alpha|^2. \label{cvar} \end{equation} Then $Z$, $Q$, $F$ are the boundary values of functions that are holomorphic in the lower half-plane, and $P[Z-\alpha] = Z-\alpha$, $P Q = Q$. The kinematic BC (\ref{kinBC}) is equivalent to \begin{equation} Z_t + F Z_\alpha = 0. \label{heq1} \end{equation} Applying the holomorphic projection $P$ to the dynamic BC (\ref{dynBC}), using Hilbert transform identities, and simplifying the result, we get that \begin{equation}
Q_t + F Q_\alpha + P\left[\frac{|Q_\alpha|^2}{J}\right] = i\left( Z-\alpha\right),
\qquad J = |Z_\alpha|^2. \label{heq2} \end{equation} Thus, the holomorphic equations are (\ref{heq1})--(\ref{heq2}).
\subsection{ The normal derivative of the pressure}
In this subsection, for comparison purposes, we compute the normal derivative of the pressure in terms of our variables. This played a role in the subject as the Taylor sign condition \[
\dfrac{\partial p}{\partial n}_{| \Gamma_t} < 0 \] was identified as necessary for the well-posedness of the water wave equation, see \cite{taylor}. In our context this is automatically satisfied, see the discussion at the end of this section.
From Bernoulli's equation (\ref{bernoulli_eq}) we have that \[
-\frac{\partial p}{\partial n} = - \frac{1}{J} \left.p_\beta\right|_{\beta=0} =
\frac{1}{J}\left.\partial_\beta \left(\phi_t + \frac{1}{2}|\nabla \phi|^2 + y\right)\right|_{\beta=0}. \] Converting $\beta$-derivatives to $\alpha$-derivatives and using the evolution equations, we find that \begin{align*}
\left.\partial_\beta \phi_t\right|_{\beta=0} &=\partial_\alpha\left[-\Theta_t +\frac{\Psi_\alpha}{J} \left(X_\alpha Y_t - Y_\alpha X_t\right)
+ \frac{\Theta_\alpha}{J} \left(X_\alpha X_t + Y_\alpha Y_t\right)\right] \\ &= - \partial_\alpha\left[\Theta_t + \Theta_\alphaH\left(\frac{\Theta_\alpha}{J}\right) + \frac{\Psi_\alpha\Theta_\alpha}{J}\right] \\ &= - \partial_\alpha\left[H\left(\frac{\Psi_\alpha^2 + \Theta_\alpha^2}{2J}\right) + X - \alpha\right]. \end{align*}
Since $\left.y_\beta\right|_{\beta=0} = X_\alpha$ and \[
\left.|\nabla\phi|^2 \right|_{\beta=0} = \frac{\Psi_\alpha^2 + \Theta_\alpha^2}{J}, \] we find that \[
-J\frac{\partial p}{\partial n} = 1 +\frac{1}{2} \left.\left(\partial_\beta - H\partial_\alpha\right)|\nabla\phi|^2 \right|_{\beta=0}. \]
To put this equation in holomorphic form, we introduce \[ r = \frac{q_\alpha}{z_\alpha},\qquad \partial = \frac{1}{2}\left(\partial_\alpha - i\partial_\beta\right), \quad \bar{\partial} = \frac{1}{2}\left(\partial_\alpha + i\partial_\beta\right), \]
where $|\nabla\phi|^2 = r \bar{r}$ and $\left.r\right|_{\beta=0} = R$ is defined in (\ref{defR}). Then, using the fact that $\partial r = r_\alpha$ and $\bar{\partial} r = 0$, we get that \[ -J\frac{\partial p}{\partial n} = 1 + i\left[\bar{P}(\bar{R} R_\alpha) - P(R \bar{R}_\alpha)\right] = 1 + a, \] where $a$ is defined in (\ref{defa}). Comparing this result with Wu \cite{wu}, we see that up to Jacobian factors, our $1+a$ is proportional to her $\mathfrak{a}$. Moreover, as shown in \cite{wu2} under the assumption of non-self intersecting boundary, we have $a \ge 0$. A shorter alternate proof of this fact is provided in our Lemma~\ref{regularity for a}; further we impose no condition on the self intersections of the curve $Z(t, \alpha)$.
\subsection{The periodic case} In the spatially periodic case, the map $\mathcal{F}(t)$ is uniquely determined by the requirement that the holomorphic function $z(t,\alpha,\beta) - (\alpha + i\beta)$ is a periodic function of $\alpha$, whose real part approaches zero\footnote{The imaginary part
need not have zero mean even if the average height stays equal to
zero.} as $\beta \to -\infty$. It follows that $\Re Z(t,\alpha) - \alpha$ has zero mean with respect to $\alpha$, otherwise the holomorphic function would have nonzero limit.
In the original coordinates, the velocity field $u+iv$ is holomorphic, periodic and bounded. Thus it has a limit $u_0+ iv_0$ as $\beta \to \infty$. Further, this limit is independent of time. By extension, in the holomorphic coordinates $Q_\alpha$ also has a limit $u_0+ iv_0$ as $b \to \infty$. Thus, we can normalize $Q$ by setting \[ Q = Q_0 + (u_0+iv_0)(\alpha + i\beta) + c(t), \] where $Q_0$ is periodic with average zero, and $c(t)$ is a real normalization constant needed for Bernoulli's law. One could continue the computations using $u_0$ and $v_0$ as constants of motion, but this is not needed because we can factor them out using a Galilean transformation. From here on we set them equal to zero.
The relation \eqref{kinBC1} rests unchanged, \begin{equation} X_\alpha Y_t - Y_\alpha X_t = -\Theta_\alpha. \label{kinBC1p} \end{equation} In integrated form this expresses the conservation of mass. Consider now both terms divided by $J$, then this becomes \[ \Im \left( \frac{z_t}{z_\alpha}\right) = -\frac{\Theta_\alpha}{J} \qquad \text{on} \ \ \beta = 0.
\]
The function on the left is holomorphic and its real part has limit zero as $\beta$
goes to infinity, so we can get its real part using the Hilbert
transform. Hence \eqref{kinBC2} also holds, and \eqref{kinBC}
follows. Similarly, the derivation of \eqref{dynBC} remainss unchanged.
To put the equations \eqref{kinBC} and \eqref{dynBC} in holomorphic form we keep the definition of the operator $P$ as \[ P = \frac12(I-iH) \] even though it is no longer a projector, as it selects exactly half of the zero mode. With the same notations as in \eqref{cvar}, the equations \eqref{heq1} and \eqref{heq2} remains unchanged.
Concerning the balance of averages in these two equations, we remark that in \eqref{heq1} the terms $\Re Z_t$ and $F$ have purely imaginary averages while $Z_\alpha$ has average $1$. The nontrivial average here is that of $F$, which contributes to the motion of nonzero frequencies. In the second equation \eqref{heq2}, the real part of the average of $Q$ is nonzero due to the integrating constant in Bernoulli's law; however this plays a trivial role, as it does not affect any of the remaining equations. Further, $R$ has no zero modes.
All equations in the first section of the paper remain unchanged, most importantly the expressions for the frequency shift $a$, the advection velocity $b$ and the auxiliary function $M$. Further, all estimates in Lemmas~\ref{regularity for a}, ~\ref{l:b},~\ref{l:M} remain unchanged; only the zero mode estimates need to be added, and those are straightforward. The normal form transformation also remains valid.
We next consider the linearized equations. The derivation of \eqref{lin(wr)0} is purely algebraic, so it stays unchanged. We remark that the average of $w$ is purely imaginary, while the average of $r$ is the same as the average of $q$ and is purely real.
There is some choice to be made when writing the projected equations \eqref{lin(wr)}. The operator $P$ defined as above is no longer a projector, so we can no longer use it directly. The new question that arises here is how we treat the zero modes. For that we introduce some variants of $P$ which differ in how the zero modes are treated. Defining $P_0$ as the projection onto the zero modes, we define the projectors \[ P^\sharp = P - \frac12 P_0, \qquad P^r = P^\sharp + \Re P_0, \qquad P^i = P^\sharp + i \Im P_0, \] and similarly $\bar P^\sharp$, $\bar P^r$ and $\bar P^i$. We have the relations \[ P = P^i + \bar P^r = P^r + \bar P^i = P^\sharp + \bar P^\sharp + P_0, \qquad P^i \bar P^r = P^r \bar P^i = 0, \qquad P^i = -i P^r i. \] With these notations, it is natural to project the first equation using $P^i$, and the second using $P^r$. Thus instead of \eqref{lin(wr)} we write \begin{equation}\label{p-lin(wr)} \left\{ \begin{aligned} & (\partial_t + P^i b \partial_\alpha) w + P^i \left[ \frac{1}{1+\bar {\mathbf W}} r_\alpha\right] + P^i \left[ \frac{R_{\alpha} }{1+\bar {\mathbf W}} w \right] = P^i \mathcal{G}( w, r),
\\ &(\partial_t + P^r \partial_\alpha) r - i P^i\left[ \frac{1+a}{1+{\mathbf W}} w\right] =
P^r \mathcal{K}( w,r). \end{aligned} \right. \end{equation} The quadratic part of $\mathcal G$ has real average, and $\mathcal K$ has imaginary average, both of which get projected out. So \[ P^i \mathcal G^{(2)} = P^\sharp[ R \bar w_\alpha - {\mathbf W} \bar r_\alpha] , \qquad P^i \mathcal K^{(2)} = - P^\sharp[ R \bar r_\alpha] . \] After a similar modification in \eqref{lin(wr)inhom}, the statement and the proof of Proposition~\ref{plin-short} remain largely unchanged; the difference is that $P$ gets replaced by $P^i$ in $err_1$.
Moving on to the cubic estimates, the equation \eqref{lin(wr)inhom3} is replaced by \begin{equation}\label{p-lin(wr)inhom3} \left\{ \begin{aligned} & (\partial_t + P^i b \partial_\alpha) w + P^i \left[ \frac{1}{1+\bar {\mathbf W}} r_\alpha\right] + P^i \left[ \frac{R_{\alpha} }{1+\bar {\mathbf W}} w \right] = P^\sharp[ R \bar w_\alpha - {\mathbf W} \bar r_\alpha] +G,
\\ &(\partial_t + P^r \partial_\alpha) r - i P^i\left[ \frac{1+a}{1+{\mathbf W}} w\right] = - P^\sharp[ R \bar r_\alpha] +K. \end{aligned} \right. \end{equation} Also, the cubic energy needs to be modified. Precisely, the zero modes of $w$ and $r$ do not affect the quadratic terms on the right hand side above. Hence, the cubic energy correction should not involve these zero modes either, \begin{equation}\label{p-elin3}
E^{(3)}_{lin}(w,r) = \int_{\mathbf R} (1+a) |w|^2 + \Im (r \bar r_\alpha) + 2 \Im (\bar R w^\sharp r_\alpha) -2\Re(\bar{{\mathbf W}} (w^\sharp)^2)\ d\alpha. \end{equation} With this modification, the result in Proposition~\ref{plin-long} remains valid, and the proof applies with minor modifications.
The higher energies in the periodic case are even more similar to the nonperiodic case, since differentiation eliminates the zero modes, and the frequency localization is achieved using only $P^\sharp$ and $\bar P^\sharp$.
An alternative to the above scheme is to select just the negative wave numbers in the linearized equation. Then we lose the evolution of the average of the imaginary part of $w$. This is not so significant since we have the conservation of mass relation \[
\int Y X_\alpha d\alpha = const, \] where the (time independent) constant on the right can be set arbitrarily (say to zero) by a vertical translation of the coordinates, \[ (Z,Q) \to (Z+ ic, Q-ct). \] This gives \[ \int Y d\alpha = i \int (\bar Z-\alpha)(Z_\alpha -1) \ d\alpha + const, \] where the average of $Y$ plays no role on the right. This shows that also for the linearized equation, the average of $w$ is determined by the initial data and the negative frequencies of $w$ and $W$, \[ \int \Im w d\alpha = i \int \bar w W_\alpha + \bar W w_\alpha \ d\alpha + const. \] Again, the constant can be removed as the pair $(i,t-iR)$ solves the linearized problem. It follows that the contribution of the average of $w$ to the linearized equations can be viewed as cubic and higher.
\section{ Norms and multilinear estimates} \label{s:multilinear}
Here we prove some of the estimates used in Section~\ref{s:linearized}, and Section~\ref{s:ee}. We use a standard
Littlewood-Paley decomposition in frequency \begin{equation*} 1=\sum_{k\in \mathbf{Z}}P_{k}, \end{equation*} where the multipliers $P_k$ have smooth symbols localized at frequency $2^k$.
A good portion of our analysis happens at the level of homogeneous Sobolev spaces $\dot{H}^{s}$, whose norm is given by \begin{equation*} \Vert f\Vert_{\dot{H}^{s}}\sim \Vert ( \sum_{k}\vert 2^{ks} P_{k}f\vert^2 )^{1/2} \Vert _{L^2}=
\| 2^{ks} P_k f \|_{L^2_x \ell^2_k}. \end{equation*}
We will also use the Littlewood-Paley square function and its restricted version, \begin{equation*}
\displaystyle S(f)(x):=\bigg( \sum_{k\in {\mathbf Z}} |P_k(f)(x)|^2\bigg)^\frac{1}{2}, \qquad S_{>k}(u) = ( \sum_{j > k} |P_j u|^2 )^\frac12. \end{equation*} The Littlewood-Paley inequality is recalled below \begin{equation}\label{lp-square}
\displaystyle \|S(f)\|_{L^p({\mathbf R})}\simeq_{p} \|f\|_{L^p({\mathbf R})}, \qquad 1<p<\infty.
\end{equation} By duality this also yields the estimate \begin{equation} \label{useful} \Vert \sum_{k\in \mathbf{Z}}P_{k}f_{k}\Vert_{L^p}\lesssim \Vert \sum_{k\in \mathbf{Z}}(\vert f_{k}\vert ^2)^{1/2}\Vert_{L^p}, \qquad 1 < p < \infty. \end{equation}
The $p= 1$ version of the above estimate for the Hardy space $ H_1$ is \begin{equation}\label{h1-square}
\Vert f\Vert_{H_1}\simeq \| S(f) \|_{L^1_x \ell^2_k}, \end{equation} which by duality implies the BMO bound \begin{equation} \label{useful-bmo} \Vert \sum_{k\in \mathbf{Z}}P_{k}f_{k}\Vert_{BMO}\lesssim \Vert S(f)\Vert_{L^\infty}. \end{equation} The square function characterization of BMO is slightly different, \begin{equation}\label{bmo-square}
\| u\|_{BMO}^2 \approx \sup_k \sup_{|Q|=2^{-k}} 2^k \int_Q |S_{>k} (u)|^2 \, dx. \end{equation} We will also need the maximal function bound \begin{equation} \label{useful-max}
\Vert P_{< k}f \Vert_{L^2_x L^\infty_k}\lesssim \|f \|_{L^2} , \qquad 1 < p < \infty. \end{equation}
\subsection{Coifman-Meyer and and Moser type estimates.}
In the context of bilinear estimates a standard tool is to consider a Littlewood-Paley paraproduct type decomposition of the product of two functions, \[
f g = \sum_{k \in {\mathbb Z}} f_{<k-4} g_k + \sum_{k \in {\mathbb Z}} f_{k} g_{<k-4} + \sum_{|k-l| \leq 4} f_k g_l := T_f g + T_g f + \Pi(f,g). \] Here and below we use the notation $f_k = P_k f$, $f_{<k} = P_{<k} f$, etc. By a slight abuse of notation, in the sequel we will omit the frequency separation from our notations in bilinear Littlewood-Paley decomposition; for instance instead of the above formula we will use the shorter expression \[
f g = \sum_{k \in {\mathbb Z}} f_{<k} g_k + \sum_{k \in {\mathbb Z}} f_{k} g_{<k} + \sum_{k \in {\mathbb Z}} f_k g_k. \]
Away from the exponents $1$ and $\infty$ one has a full set of estimates \begin{equation}\label{CM}
\| T_f g\|_{L^r} + \| \Pi(f,g)\|_{L^r} \lesssim \|f\|_{L^p} \|g\|_{L^q}, \qquad \frac{1}r = \frac{1}{p} + \frac{1}{q}, \qquad 1 < p,q,r < \infty. \end{equation} Corresponding to $q = \infty$ one also has a BMO estimate \begin{equation}\label{CM-BMO}
\| T_f g\|_{L^p} + \| \Pi(f,g)\|_{L^p} \lesssim \|f\|_{L^p} \|g\|_{BMO}, \qquad 1 < p < \infty, \end{equation} which in turn leads to the commutator bound \begin{equation}\label{CM-com}
\|[P, g] f \|_{L^p} \lesssim \|f\|_{L^p} \|g\|_{BMO}, \qquad 1 < p < \infty. \end{equation} For $p = 2$ we also need an extension of this, namely
\begin{lemma}\label{l:com}
The following commutator estimates hold: \begin{equation} \label{first-com}
\Vert |D|^s \left[ P,R\right] |D|^\sigma w \Vert _{L^2}\lesssim \Vert
|D|^{\sigma+s} R\Vert_{BMO} \Vert w\Vert_{L^2}, \qquad \sigma \geq 0, \ \ s \geq 0, \end{equation} \begin{equation}\label{second-com}
\Vert |D|^s \left[ P,R\right] |D|^\sigma w \Vert _{L^2}\lesssim \Vert
|D|^{\sigma+s} R\Vert_{L^2} \Vert w\Vert_{BMO}, \qquad \sigma > 0, \ \ \, s \geq 0. \end{equation} \end{lemma} We remark that later this is applied to functions which are holomorphic/antiholomorphic, but that no such assumption is made above. \begin{proof}
If $\sigma = s = 0$ then \eqref{first-com} is the classical commutator
estimate of Coifman and Meyer \eqref{CM-com}, so we take $\sigma + s > 0$. We
consider the usual paradifferential decomposition, and observe that
the expression $\left[ P,R\right] |D|^\sigma w$ vanishes if the
frequency of $w$ is much larger than the frequency of $R$. For the
remaining frequency balances we discard $P$, and we are left with
having to estimate the expressions \[
f_{hh} = \sum_k 2^{(\sigma+s) k} (2^{-k}|D|)^s (R_k w_k), \qquad f_{hl} = \sum_k 2^{(\sigma+s) k} R_k 2^{-\sigma k} |D|^\sigma w_{<k}. \] In the term $f_{hh}$ the $\sigma$ derivatives are already moved to $R$, so this is bounded using \eqref{CM-BMO} if $s = 0$, and directly if $s > 0$. For the remaining part we only need the infinity Besov norm of $R_k$, as \[ \begin{split}
\| f_{hl}\|_{L^2}^2 \lesssim & \ \sum_k \| 2^{(\sigma+s) k} R_k \|_{L^\infty}^2
\| 2^{-\sigma k} |D|^\sigma w_{<k}\|_{L^2}^2
\lesssim \sup_k \| 2^{(\sigma+s) k} R_k \|_{L^\infty}^2
\sum_k \| 2^{-\sigma k} |D|^\sigma w_{<k}\|_{L^2}^2 \\ \lesssim & \ \Vert
|D|^{\sigma+s} R\Vert_{BMO}^2 \Vert w\Vert_{L^2}^2. \end{split} \] The proof of \eqref{second-com} is similar. \end{proof}
Next we consider some similar product type estimates involving $BMO$ and $L^\infty$ norms. We define \[
\| u\|_{BMO^\frac12} = \| |D|^\frac12 u\|_{BMO}. \] Then \begin{proposition}\label{p:bmo} a) The following estimates hold: \begin{equation}\label{bmo-bmo}
\| \sum_k u_k v_k \|_{BMO} \lesssim \| u\|_{BMO} \|v\|_{BMO}, \end{equation} \begin{equation}\label{bmo=infty}
\| \sum_k (2^{-k}|D|)^\sigma( u_k v_k)\|_{BMO} \lesssim \| u\|_{BMO} \|v\|_{\dot B_{\infty,\infty}^0}, \qquad \sigma > 0, \end{equation} \begin{equation}\label{bmo-infty}
\| \sum_k u_{<k} v_k\|_{BMO} \lesssim \| u\|_{L^\infty} \|v\|_{BMO}, \end{equation} \begin{equation}\label{bmo>infty}
\| \sum_k (2^{-k}|D|)^\sigma (u_{<k} v_k)\|_{BMO} \lesssim \| u\|_{\dot B_{\infty,\infty}^0 } \|v\|_{BMO}, \qquad \sigma > 0. \end{equation}
b) The space $L^\infty \cap BMO^{\frac12}$ is an algebra, \begin{equation}\label{bmo-alg}
\| uv\|_{BMO^{\frac12}} \lesssim \| u\|_{L^\infty} \|v\|_{BMO^\frac12}+
\| v\|_{L^\infty} \|u\|_{BMO^\frac12}, \end{equation}
c) The following Moser estimate holds for a smooth function $F$: \begin{equation}\label{bmo-moser}
\| F(u)\|_{BMO^{\frac12}} \lesssim_{\|u\|_{L^\infty}} \|u\|_{BMO^\frac12}, \end{equation} \end{proposition}
\begin{proof} a) For \eqref{bmo-bmo} we fix a cube $Q$, which by scaling can be taken to have size $1$. Suppose first that $\sigma = 0$. For $k > 0$ we use the square function estimate, \[ \begin{split}
\| \sum_{k > 0} u_k v_k\|_{L^1(Q)} & \ \lesssim \ \| u_k v_k\|_{\ell^1_k L^1(Q)} \lesssim \| u_k\|_{\ell^2_k L^2(Q)} \| v_k\|_{\ell^2_k L^2(Q)}\lesssim \|S_{>0}(u)\|_{L^2(Q)} \|S_{>0}(v)\|_{L^2(Q)}
\\ & \ \lesssim \|u\|_{BMO} \|v\|_{BMO}. \end{split} \] For $k > 0$ we subtract the average and estimate the output in $L^\infty$, \[
\| \sum_{k \leq 0} u_k v_k - (u_k v_k)_{Q} \|_{L^\infty(Q)}
\lesssim \sum_{k< 0} \| \partial_\alpha (u_k v_k)\|_{L^\infty}
\lesssim \sum_{k < 0} 2^{k} \|u\|_{BMO} \|v\|_{BMO}. \] Adding the two we get \[
\| \sum_{k } u_k v_k - (u_k v_k)_{Q} \|_{L^1(Q)} \lesssim \|u\|_{BMO} \|v\|_{BMO}, \] and \eqref{bmo-bmo} follows.
The case $k \leq 0$ is similar in the proof of \eqref{bmo=infty}. For $k > 0$ we first eliminate directly the low frequency output, \[
\| P_{< 0} \sum_{k > 0} (2^{-k}|D|)^\sigma( u_k v_k)\|_{BMO}
\lesssim \sum_{k > 0} 2^{-\sigma k} \|u_k\|_{L^\infty} \|v_k\|_{L^\infty}
\lesssim \| u\|_{BMO} \|v\|_{\dot B_{\infty,\infty}^0}. \] For the high frequency output we consider a bump function $\chi_Q$ adapted to $Q$, which is localized at frequency less than $1$, and thus does not change the frequency localization of the factors it multiplies in the sequel. Then, we have \[ \begin{split}
\| P_{> 0} \sum_{k > 0} (2^{-k}|D|)^\sigma( u_k v_k)\|_{L^2(Q)}
& \ \lesssim \| \chi_Q P_{> 0} \sum_{k > 0} (2^{-k}|D|)^\sigma( u_k v_k)\|_{L^2} \\ & \hspace{-1in}
\lesssim \| [\chi_Q, P_{> 0} \sum_{k > 0} (2^{-k}|D|)^\sigma]( u_k v_k)\|_{L^2}
+ \| P_{> 0} \sum_{k > 0} (2^{-k}|D|)^\sigma( \chi_Q u_k v_k)\|_{L^2}. \end{split} \] For the commutator term we gain a small power of $2^k$ so $L^\infty$ bounds suffice. The remaining term is bounded in $L^2$ in terms of the square function using orthogonality, \[ \begin{split}
\| P_{> 0} \sum_{k > 0} (2^{-k}|D|)^\sigma( \chi_Q u_k v_k)\|_{L^2}^2 \lesssim & \
\sum_{k > 0} \| \chi_Q u_k \|_{L^2}^2 \|v_k\|_{L^\infty}^2
\lesssim \| \chi_Q S_{>0} (u) \|_{L^2}^2 \|v\|_{\dot B_{\infty,\infty}^0}
\\ \lesssim & \ \| u\|_{BMO}^2 \|v\|_{\dot B_{\infty,\infty}^0}. \end{split} \]
The argument for \eqref{bmo-infty} is similar, with the following modification in the case $k >0$, which leads to $L^2$ rather than $L^1$ bounds: \[ \begin{split}
\|\sum_{k > 0} u_{<k} v_k\|_{L^2(Q)} \lesssim & \ \|\sum_{k > 0} \chi_{Q} u_{<k} v_k\|_{L^2(Q)}
\lesssim \|u\|_{L^\infty} ( \sum_k \| \chi_Q v_k\|^2_{L^2(Q)})^\frac12
\\ \lesssim & \ \| u\|_{L^\infty} \| \chi_Q S_{>0}(v)\|_{L^2(Q)}.
\lesssim \|u\|_{L^\infty} \|v\|_{BMO}. \end{split} \] Finally, the bound \eqref{bmo>infty} is similar since \[
\| (2^{-k}|D|)^\sigma u_{<k}\|_{L^\infty} \lesssim \| u\|_{\dot B_{\infty,\infty}^0}. \]
b) With the same paradifferential decomposition as before we need to estimate the terms \[
f_{hh} = \sum_k |D|^{\sigma} u_k v_k, \qquad f_{hl} = \sum_k 2^{\sigma k} u_k v_{<k}. \] For $f_{hl}$ we use \eqref{bmo-infty}, while for $f_{hh}$ we use \eqref{bmo-bmo}.
c) We write \[ \begin{split} F(u) = & \ \int_{-\infty}^\infty u_k F'(u_{<k})\, dk\\ = & \ \int_{-\infty}^\infty u_k P_{<k} F'(u_{<k})\, dk + \int_{-\infty}^\infty u_k P_k F'(u_{<k}) dk + \int_{-\infty}^\infty \sum_{j > 0} u_k P_{k+j} F'(u_{<k})\, dk. \end{split} \] For $F'(u_{<k})$ we can use the chain rule to obtain the bound \[
\| P_{k+j} F'(u_{<k})\|_{L^\infty} \lesssim 2^{-N j}, \qquad j \geq 0. \]
With $\sigma = \frac12$ we estimate $|D|^\sigma F(u)$. The first term in $|D|^{\sigma} F(u) $ is \[ f_1 = \int_{-\infty}^\infty (2^{\sigma k} u_k) P_{<k} F'(u_{<k}) \, dk, \] and is estimated in BMO exactly as in the proof of \eqref{bmo-infty}.
The second term is \[
f_2 = \int_{-\infty}^\infty (2^{-k}|D|)^\sigma ( u_k P_k F'(u_{<k}))\, dk, \] and is estimated as in the proof of \eqref{bmo=infty}.
The last term is $\displaystyle\sum_{j > 0} f_{3,j}$, where \[
f_{3,j} = \int_{-\infty}^\infty 2^{\sigma
k } u_k 2^{\sigma j} P_j F'(u_{<k}) \, dk. \]
The $k \leq 0$ case is easy; it follows using pointwise estimates. For fixed $j$ and $k > 0$ we bound $f_{3,j}$ by \[ \begin{split}
\|f_{3,j,> 0}\|_{L^2(Q)}^2 \lesssim & \ \| \chi_Qf_{3,j,> 0}\|_{L^2}^2 \lesssim
\int_{-\infty}^\infty \| \chi_Q 2^{\sigma k } u_k 2^{\sigma j} P_j F'(u_{<k})\|_{L^2}^2 \, dk
\\ \lesssim & \ 2^{(\sigma-N)j} \| \chi_Q S_{>0}(|D|^\sigma u)\|_{L^2}^2 \lesssim 2^{(\sigma-N)j}
\|u\|_{BMO^\sigma}^2. \end{split} \] \end{proof}
A more standard algebra estimate and the corresponding Moser bound is as follows:
\begin{lemma} Let $\sigma > 0$. Then $\dot H^\sigma \cap L^\infty$ is an algebra, and \begin{equation}
\| fg\|_{\dot H^\sigma} \lesssim \| f\| _{\dot H^\sigma} \|g\|_{L^\infty} +
\|f\|_{L^\infty} \| g\| _{\dot H^\sigma}. \end{equation} In addition, the following Moser estimate holds for a smooth function $F$: \begin{equation}\label{bmo-hs}
\| F(u)\|_{\dot H^\sigma } \lesssim_{\|u\|_{L^\infty}} \|u\|_{\dot H^\sigma }. \end{equation} \end{lemma}
We also need to consider some multilinear estimates. Our starting point is the bound \[
\| f_1 \cdots f_n\|_{L^r} \lesssim \prod_{j=\overline{1,n}} \|f_j\|_{L^{p_j}}, \qquad \frac{1}{r} = \sum \frac{1}{p_j}, \qquad 1 \leq r, p_j \leq \infty . \] Adding derivatives, we need the following generalization:
\begin{lemma}\label{l:multi} The following estimate holds for $\sigma > 0$ and $ 1 < r, p_j^{(k)} \leq \infty $: \begin{equation}
\| |D|^\sigma( f_1 \cdots f_n)\|_{L^r} \lesssim \sum_{k = 1}^n
\| |D|^\sigma f_k\|_{L^{p_k^{(k)}}} \prod_{j \neq k} \|f_j\|_{L^{p_j^{(k)}}}, \qquad \frac{1}{r} = \sum \frac{1}{p_j^{(k)}}. \end{equation} The same bound holds if for $L^{p_k^{(k)}}$ is replaced by $BMO$ whenever $p_k^{(k)}= \infty$. \end{lemma} \begin{proof}
By induction it suffices to consider the case $n=2$. After a
Littlewood-Paley decomposition we place the derivatives on the
highest frequency factor and apply either \eqref{CM} or
\eqref{CM-BMO}, or \eqref{bmo-bmo},\eqref{bmo-infty}. \end{proof}
\subsection{Water-wave related bounds.}
Here we consider estimates for objects related to the water wave equations, primarily the real phase shift $a$ and advection velocity $b$. We recall that these are given by \[ a = 2 \Im P[R \bar R_\alpha], \qquad b = 2 \Re P \left[ \frac{R}{1+\bar {\mathbf W}}\right] = 2 \Re ( R - P[R \bar Y]). \] These are estimated in terms of the control parameters $A$ and $B$ defined in \eqref{A-def}, \eqref{B-def}, and in terms of the $H^s$ Sobolev norms of ${\mathbf W}$ and $R$. In all nonlinear bounds the implicit constant is allowed to depend on $A$.
We begin with the auxiliary variable $Y= \dfrac{{\mathbf W}}{1+{\mathbf W}}$, which inherits its regularity from ${\mathbf W}$ due to \eqref{bmo-moser} and \eqref{bmo-hs}:
\begin{lemma}\label{l:Y} The function $Y$ satisfies the $BMO$ bound \begin{equation}\label{est:Y}
\| |D|^\frac12 Y\|_{BMO} \lesssim_A B, \end{equation} and the $\dot H^\sigma$ bound \begin{equation}
\| Y\|_{ \dot H^\sigma} \lesssim_A \| {\mathbf W}\|_{ \dot H^\sigma}. \end{equation} \end{lemma}
We continue with bounds for $a$. In particular the positivity of $a$ is established, providing a short alternate proof to Wu's result in \cite{wu2}:
\begin{proposition}\label{regularity for a}
Assume that $R \in \dot H^{\frac12} \cap \dot H^{\frac32}$. Then the real frequency-shift $a$
is nonnegative and satisfies the $BMO$ bound \begin{equation}\label{a-bmo}
\| a\|_{BMO} \lesssim \Vert R\Vert_{BMO^\frac12}^2, \end{equation} and the uniform bound \begin{equation}\label{a-point}
\| a\|_{L^\infty} \lesssim \Vert R\Vert_{\dot B^{\frac12}_{\infty,2}}^2. \end{equation}
Moreover, \begin{equation}\label{a-bmo+}
\Vert |D|^{\frac{1}{2}} a\Vert_{BMO}\lesssim \Vert R_\alpha \Vert_{BMO}
\Vert |D|^\frac12 R\Vert_{L^\infty}, \qquad \Vert a\Vert_{B^{\frac12,\infty}_2}\lesssim \Vert R_\alpha \Vert_{B^{\frac12,\infty}_2}
\Vert |D|^\frac12 R\Vert_{L^\infty}, \end{equation} \begin{equation}\label{aflow} \Vert (\partial_t+b\partial_{\alpha})a \Vert_{L^{\infty}}\lesssim AB, \end{equation} and \begin{equation}\label{a-Hs} \Vert a\Vert_{\dot H^s}\lesssim \Vert R \Vert_{\dot H^{s+\frac12}} \Vert R\Vert_{BMO^\frac12}, \qquad s > 0. \end{equation} \end{proposition} \begin{proof} We recall that $a = i\left(\bar{P} \left[\bar{R} R_\alpha\right]- P\left[R\bar{R}_\alpha\right]\right)$. Switching to the Fourier space, this leads to the representation \begin{equation} \hat{a}(\zeta) = \int_{\xi-\eta = \zeta} \min\{ \xi,\eta\} 1_{\{\xi,\eta > 0\}} \hat R(\xi) \bar {\hat R}(\eta) d\xi. \end{equation} Here $\xi$ and $\eta$ are restricted to the positive real axis due to the fact that $R$ is holomorphic.
To prove the positivity of $a$ we represent the above kernel as \[
\min\{ \xi,\eta\}1_{\{\xi,\eta > 0\}} = \int_{M > 0} 1_{\{\xi > M\}} 1_{\{\eta > M\}} \, dM. \] Inserting this in the previous representation of $\hat a$ and inverting the Fourier transform we obtain \[
a = \int | 1_{|D| > M} R|^2 dM, \] and the positivity follows.
To prove both the $BMO$ bound and the pointwise bound for $a$ we use a bilinear Littlewood-Paley decomposition, \begin{equation}\label{a=lp} a = \sum_k i\left( \bar{R}_k R_{\alpha,<k} - R_k\bar{R}_{\alpha,<k}\right) +
i\left(\bar{P} \left[\bar{R}_k R_{\alpha,k}\right]- P\left[R_k\bar{R}_{\alpha,k}\right]\right). \end{equation} To estimate the first term in BMO we use directly the bound \eqref{bmo>infty} with $\sigma = 1$. To estimate it in $L^\infty$ we use the Cauchy-Schwarz inequality, \[
\| \sum_k \bar{R}_k R_{\alpha,<k} \|_{L^\infty}^2 \lesssim
( \sum_k 2^{k} \|R_k\|_{L^\infty}^2)( \sum_k 2^{-k} \|R_{<k}\|_{L^\infty}^2)
\lesssim \|R\|_{B^{\frac12}_{\infty,2}}^2. \] For the second term in $a$ we rewrite the symbol of the bilinear form as \[
\min\{ \xi,\eta\} = \frac12(\xi+\eta) - \frac12|\xi-\eta|, \] which allow us to rewrite it in the form \[ \frac12 P \sum_k i\left( \bar{R}_k R_{\alpha,k} - R_k\bar{R}_{\alpha,k}\right)
- |D| (\bar R_k R_k). \] Now the two terms are estimated in BMO using \eqref{bmo-bmo}, respectively \eqref{bmo=infty}, and in $L^\infty$ by the Cauchy-Schwarz inequality as above.
The proof of \eqref{a-bmo+} is essentially identical to the proof of \eqref{a-bmo}.
We continue with the proof of \eqref{aflow}, where we begin with the decomposition in \eqref{a=lp}. For the first term in \eqref{a=lp} we apply the time derivative to obtain the expression \[ A_1= [b \partial_\alpha,\bar P_k] \bar R R_{\alpha,<k} +
\bar R_k [ b \partial_\alpha, P_{<k} \partial_\alpha] R + i P_k \left( \frac{\bar {\mathbf W}-a}{1+\bar {\mathbf W}}\right) R_{\alpha,<k} + i \bar R_k P_{<k} \partial_\alpha \left( \frac{\bar {\mathbf W}-a}{1+\bar {\mathbf W}}\right). \] In the first term of $A_1$ we split the commutator according to the usual Littlewood-Paley trichotomy. We get several terms: \[ b_{<k,\alpha} \bar R_k R_{\alpha,<k} + b_k \bar R_{<k,\alpha} R_{\alpha,<k} + 2^m P_k (b_m \bar R_m) R_{\alpha,<k} + b_m\bar{R}_{k,\alpha} R_{\alpha,<k}. \] In all cases we use the $B$ norm to estimate the highest frequency term, and the $A$ norm for the other two. The second term is similar; for comparison purposes we list the ensuing terms: \[ \bar R_k b_{<k,\alpha} R_{<k,\alpha} + 2^m \bar R_k P_{<k} \partial_\alpha( b_{m} R_{m}) +\bar R_k b_m R_{<k,\alpha \alpha}. \] The third and fourth terms in $A_1$ require the bound \[
\|\left( \frac{\bar {\mathbf W}-a}{1+\bar {\mathbf W}}\right)\|_{B^{\frac12,\infty}_2} \lesssim B, \] which follows by combining the bounds \eqref{a-point} and \eqref{a-bmo+} for $a$ with the similar bounds for ${\mathbf W}$ and $Y$.
Now we consider the last term in \eqref{a=lp}. This has two components, one of the form
$2^k R_k \bar R_k$ and the other of the form $|D|(R_k \bar R_k) $. The first component yields an output \[ A_2= [b \partial_\alpha,\bar P_k] \bar R R_{\alpha,k} +
\bar R_k [ b \partial_\alpha, P_{k} \partial_\alpha] R + i P_k \left( \frac{\bar {\mathbf W}-a}{1+\bar {\mathbf W}}\right) R_{\alpha,k} + i \bar R_k P_{k} \partial_\alpha \left( \frac{\bar {\mathbf W}-a}{1+\bar {\mathbf W}}\right), \] which is treated in exactly the same way as $A_1$.
The second component yields the slightly more involved output \[ \begin{split}
A_3= & \ b \partial_\alpha |D|(R_k \bar R_k) - |D| (P_k (b R_{\alpha}) \bar R_k) -
|D| (R_k \overline{P_k (b R_{\alpha})} ) \\ & \ + i |D| \left( P_k \left( \frac{\bar {\mathbf W}-a}{1+\bar {\mathbf W}}\right) R_{k}\right) +
i |D| \left( \bar R_k P_{k} \partial_\alpha \left( \frac{\bar {\mathbf W}-a}{1+\bar {\mathbf W}}\right)\right) . \end{split} \] The last two terms are no different from above, but in the first three there is a more delicate commutator estimate. We split $b = b_{<k} + b_{\geq k}$ and estimate the output of $b_{\geq k}$ directly for each term using the $s = \frac12$ case of Lemma~\ref{l:b}. The output of $b_{<k}$, on the other hand, is expressed as a commutator \[
[b_{<k}, |D|_{\leq k}] \partial_\alpha (R_k \bar R_k) + |D| \left( [b_{<k}, P_k] R_{\alpha} \bar R_k\right) +
|D| \left( R_k \overline{[b,P_k] R_{\alpha}} \right). \]
The last two terms are like $2^k |D|( b_{<k,\alpha} R_k \bar R_k)$ and can be estimated directly. For the first term we bound $\partial_\alpha (R_k \bar R_k)$ in $L^\infty$ by $2^{-\frac{k}2}$, and then we need to show that \[
\| [b_{<k}, |D|_{\leq k}]\|_{L^\infty \to L^\infty} \lesssim 2^{\frac{k}2} \||D|^\frac12 b\|_{BMO}. \]
Indeed the kernel of $[b_{<k}, |D|_{\leq k}]$ is bound by \[
|b_{<k} (\alpha) - b_{<k}(\beta)| \frac{2^{2k}}{(1+ 2^k |\alpha -\beta|)^2} \lesssim
\frac{2^{\frac32 k}}{(1+ 2^k |\alpha -\beta|)^\frac32}, \]
which integrates to $2^{\frac{k}2}$.
Finally, \eqref{a-Hs} is a direct consequence of the commutator estimates in Lemma~\ref{l:com}. \end{proof}
Next we consider $b$, for which we have the follwoing result \begin{lemma}\label{l:b} Let $s > 0$. Then the transport coefficient $b$ satisfies \begin{equation}\label{b-bounds}
\||D|^s b\|_{BMO} \lesssim_A \| |D|^s R\|_{BMO}, \qquad
\||D|^s b\|_{L^2} \lesssim_A \| |D|^s R\|_{L^2}. \end{equation} In particular we have \begin{equation}\label{b-bounds1}
\||D|^\frac12 b\|_{BMO} \lesssim_A A, \qquad
\|b_\alpha\|_{BMO} \lesssim_A B. \end{equation} \end{lemma} \begin{proof} Recall that \[ b = \Re P[R(1-\bar Y)] = R- P(R \bar Y). \] Hence, it remains to estimate $\partial_\alpha P( R \bar Y)$. Consider first the $BMO$ bound. As before, the role of $P$ is to restrict the bilinear frequency interactions to high - low, in which case we can use the bound \eqref{bmo-infty}, and the high-high case, where \eqref{bmo=infty} applies.
A direct argument, taking into account the same two cases, yields the $L^2$ bound. \end{proof}
Next we consider the auxiliary expression $M$:
\begin{lemma}\label{l:M} The function $M$ satisfies the pointwise bound \begin{equation}\label{M-infty}
\| M\|_{L^\infty} \lesssim_A AB, \end{equation} as well as the Sobolev bounds \begin{equation}\label{M-L2}
\| M\|_{\dot H^{n-\frac32}} \lesssim_A A {\mathbf N}_n. \end{equation} \end{lemma} \begin{proof} For the pointwise bound we claim that \begin{equation}\label{M-infty1}
\| M\|_{L^\infty} \lesssim \|R\|_{B^{\frac34,\infty}_2} \|Y\|_{B^{\frac14,\infty}_2}. \end{equation} This suffices since each of the the right hand side factors is bounded by $\sqrt{AB}$ by interpolation. To achieve this we write $M$ in two different ways, \[ M = \bar P[\bar R Y_\alpha - R_\alpha \bar Y]+ P[R \bar Y_\alpha - \bar R_\alpha Y] = \partial_\alpha P_{<k+4} (\bar P [ \bar R Y] + P [R \bar Y]) - (\bar R_\alpha Y + R_\alpha \bar Y) . \] We apply a bilinear Littlewood-Paley decomposition and use the first expression above for the high-low interactions, and the second for the high-high interactions, to write $M = M_1+M_2$ where \[ \begin{split} M_1 = & \ \sum_k [\bar R_k Y_{<k,\alpha} - R_{<k,\alpha} \bar Y_k ]+ [R_k \bar Y_{<k,\alpha} - \bar R_{<k,\alpha} Y_k], \\ M_2 = & \ \sum_k \partial_\alpha(\bar P [ \bar R_k Y_k] + P [R_k \bar Y_k]) - (\bar R_{k,\alpha} Y_k + R_{k,\alpha} \bar Y_k). \end{split} \] We estimate the terms in $M_1$ separately; we show the argument for the first: \[
\| \sum_k \bar R_k Y_{<k,\alpha}\|_{L^\infty} \lesssim
\sum_{j \leq k} 2^{\frac34(j-k)} \|R_k\|_{B^{\frac34,\infty}_2} \|Y_j\|_{B^{\frac14,\infty}_2}
\lesssim \|R\|_{B^{\frac34,\infty}_2} \|Y\|_{B^{\frac14,\infty}_2}. \] For the first term in $M_2$ we note that the multiplier $\partial_\alpha P_{<k+4} P$ has an $O(2^k)$ $L^\infty$ bound. Hence, we can estimate \[
\|M_2\|_{L^\infty} \lesssim \sum_k 2^k \|R_k\|_{L^\infty} \|Y_k\|_{L^\infty}
\lesssim \|R\|_{B^{\frac34,\infty}_2} \|Y\|_{B^{\frac14,\infty}_2}. \]
For the $L^2$ bound we consider again all terms in $M_1$ and $M_2$ separately. For an $M_1$ term we compute \[
\| \sum_k \bar R_k Y_{<k,\alpha}\|_{\dot H^{n-\frac32}}^2
\lesssim \sup_{k} 2^{-2k} \| Y_{<k,\alpha}\|_{L^\infty}^2 \cdot
\sum_{k} 2^{(2n-1)k} \|R_k\|_{L^2}^2 \lesssim \|Y\|_{L^\infty}^2 \|R\|_{\dot H^{\frac{n-1}2}}^2. \] For $M_2$ we compute \[
\| M_2\|_{\dot H^{n-\frac32}}^2 \lesssim \sum_k 2^{(2n-1)k} \| Y_k R_k\|_{L^2}^2
\lesssim \|Y\|_{L^\infty}^2 \|R\|_{\dot H^{\frac{n-1}2}}^2. \] \end{proof}
Finally, we also need a quadrilinear bound related to the energy estimates:
\begin{lemma} The following estimate holds for holomorphic functions $R$, $w$ and $r$ : \begin{equation}\label{i1}
\left| \! \int \!\! \bar{R}r_{\alpha}\mathfrak M_bw_{\alpha} - \bar{R}w_{\alpha}\mathfrak M_br_{\alpha} \, d\alpha \right| \!
\lesssim\! (\||D|^\frac12 R\|_{BMO} \|b_\alpha\|_{BMO} + \|R_\alpha\|_{BMO}
\||D|^\frac12 b\|_{BMO}) \|w\|_{L^2} \|r\|_{\dot H^\frac12} \! \end{equation} \end{lemma} \begin{proof}
We denote by $I_1$ the integral on the left. In a first step we replace the holomorphic multiplication operator $\mathfrak M_{\bar P b}$ by the corresponding paraproduct operator \[ T_{\bar P b} f = \sum_k \bar P b_{<k} f_k. \] Thus, $I_1$ is replaced by \[ I'_1 = \int -\bar{R}r_{\alpha}T_{\bar P b} w_{\alpha} +\bar{R}w_{\alpha}T_{\bar P b} r_{\alpha} \, d\alpha . \] To estimate the difference $I'_1-I_1$ we observe that
for holomorphic $f$ we have \[ \mathfrak M_{\bar P b} f = T_{\bar P b} f + P(\sum_k \bar P b_k f_k ). \] We use this for $f = w_\alpha$, respectively $f = r_\alpha$. Then \[ I_1-I'_1 = \int -\bar P[\bar{R}r_{\alpha}] P\left[\sum_k \bar P b_k w_{k,\alpha}\right] +\bar P[\bar{R}w_{\alpha}] P\left[\sum_k\bar P b_k r_{k,\alpha}\right] \, d\alpha. \] Applying the bounds in Lemma~\ref{l:com} to estimate each of the four factors in $L^2$ we obtain \[
|I_1-I'_1| \lesssim (\||D|^\frac12 R\|_{BMO} \|b_\alpha\|_{BMO} + \|R_\alpha\|_{BMO}
\||D|^\frac12 b\|_{BMO}) \|w\|_{L^2} \|r\|_{\dot H^\frac12} \] as needed.
It remains to estimate the integral $I'_1$. We take a Littlewood-Paley decomposition and denote by $k,j,l$ the frequencies of $w$, $r$, respectively $b$. After canceling the common terms we are left with \[ I'_1 = \int \sum_{k \leq l < j} \bar{R}_j r_{j,\alpha} b_{l} w_{k,\alpha} \, d\alpha - \int \sum_{j \leq l < k} \bar{R}_k r_{j,\alpha} b_{l} w_{k,\alpha} \, d\alpha := I_2 - I_3. \] The first sum $I_2$ can be estimated using only the infinity Besov norms for
$R_\alpha$ and $|D|^\frac12 b$, \[ \begin{split}
|I_2| \lesssim & \ \| R_\alpha\|_{BMO} \||D|^\frac12 b\|_{BMO} \sum_{k < j}
(j-k) 2^{\frac{k-j}2} \| r_j\|_{\dot H^\frac12} \|w_k\|_{L^2} \\ \lesssim & \
\| R_\alpha\|_{BMO} \||D|^\frac12 b\|_{BMO}
\| r\|_{\dot H^\frac12} \|w\|_{L^2}. \end{split} \] The argument for $I_3$ is slightly more involved since we cannot gain rapid decay in $k-j$. Instead, we rewrite it as \[ I_3 = \int \sum_{ k} \bar{R}_k w_{k,\alpha} \sum_{j \leq l} b_{l} r_{j,\alpha} \, d\alpha - \int \sum_{j,k \leq l} \bar{R}_k w_{k,\alpha} b_{l} r_{j,\alpha} \, d\alpha := I'_3 - I''_3. \] The first term has a product structure, and we can bound each factor in $L^2$ using Lemma~\ref{l:com} to obtain \[
|I'_3| \lesssim \| R_\alpha\|_{BMO} \||D|^\frac12 b\|_{BMO}
\| r\|_{\dot H^\frac12} \|w\|_{L^2}. \] The second term is bounded in the same manner as $I_2$, \[ \begin{split}
|I''_3| \lesssim & \ \| |D|^\frac12 R\|_{BMO} \|b_\alpha\|_{BMO} \sum_{j,k \leq l}
2^{\frac{j-l}2+\frac{k-l}2 } \| r_j\|_{\dot H^\frac12} \|w_k\|_{L^2} \\ \lesssim & \
\| |D|^\frac12 R\|_{BMO} \|b_\alpha\|_{BMO}
\| r\|_{\dot H^\frac12} \|w\|_{L^2}. \end{split} \] Thus, the proof of \eqref{i1} is concluded.
\end{proof}
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\begin{document}
\title{Periodic Orbits on Obtuse Edge Tessellating Polygons}
\section{Introduction} Elementary mathematical billiards studies the motion of a massless particle moving with unit speed along a piece-wise linear path in the interior of a polygon $G$ subject to elastic reflections at the boundary $\partial G$, i.e., the angle of incidence equals the angle of reflection. We think of $G$ as a frictionless billiard table, the edges of $G$ as its bumpers, the vertices of $G$ as its pockets, and the particle in motion as the cue ball.
The particle's path is its \emph{orbit}. If the orbit reaches a vertex of $G$, it terminates and is \emph{singular}. A non-singular orbit that begins and ends at the same point is \emph{periodic} if the particle retraces its orbit when allowed to continue. A periodic orbit is \emph{primitive} if the particle traverses its orbit exactly once. The \emph{period} of a periodic orbit is the number of times the particle strikes $\partial G$ as it traverses a primitive sub-orbit.
In 2006, A. Baxter and R. Umble found and classified the periodic orbits on equilateral triangles \cite{Ba-Um}. Five year later, A. Baxter, E. McCarthy, and J. Eskreis-Winkler solved the analogous problem on rectangles, isosceles right triangles, and $30^\circ$-right triangles \cite{Ba-Es-Mc}. In this paper we solve the problem on $120^\circ$-isosceles triangles, $60^\circ$-rhombuses, and $60^\circ$-$90^\circ$-$120^\circ$-kites, and we make a conjecture in the case of regular hexagons.
An \emph{edge tessellation} of the plane is generated by reflecting a polygon $G$ and its reflected images in their edges. Each such $G$ lies in exactly one of the eight aforementioned families \cite{K-U}. The \emph{edges} of an edge tessellation are the edges of its polygons and the lines containing them are its \emph{inclines}. For example, an equilateral triangle in standard position generates an edge tessellation with inclines of $0^\circ$, $60^\circ$, and $120^\circ$.
We identify a non-singular orbit in an edge tessellating polygon $G$ with a piece-wise linear curve $\gamma:I \rightarrow G$ defined on a finite interval $I$. An \emph{unfolding} of $\gamma$ in the edge tessellation $\mathcal{T}$ generated by $G$ is a line segment produced by successively reflecting $\gamma$ and its reflected images in the inclines of $\mathcal{T}$. Unfoldings relate the periodicity of $\gamma$ to the geometry of $\mathcal{T}$ and are sufficient for classifying periodic orbits in the non-obtuse cases. However, the analysis requires us to use more sophisticated techniques involving what we call the ``fence.''
\section{Periodic orbits on a $120^\circ$-isosceles triangle} \label{120isosceles}
\subsection{Preliminaries} Consider a $120^\circ$-isosceles $\triangle ABC$ positioned and labeled so that $\overline{AC}$ is horizontal with $A$ to the left of $C,$ the apex $B$ is positioned above $\overline{AC}$, and $\angle B$ is obtuse. The edge tessellation $\mathcal{T}$ generated by $\triangle ABC$ has inclines of $0^\circ$, $30^\circ,$ $60^\circ,$ $90^\circ,$ $120^\circ,$ and $150^\circ$. Given triangles $\triangle_1$ and $\triangle_2$ in $\mathcal T$ such that $\triangle_2=\tau (\triangle_1)$ for some translation $\tau$, two points $P_1$ on $\triangle_1$ and $P_2$ on $\triangle_2$ are \emph{(translationally) aligned (with respect to $\triangle_1$ and $\triangle_2$)} if $P_2=\tau (P_1)$.
Let $\gamma:(0,T]\rightarrow \triangle ABC$ be an orbit in $\triangle ABC$. Let $\upsilon: (0,T]\rightarrow \mathcal{T}$ be an unfolding of $\gamma$ with \emph{initial point} $P = \lim_{t \searrow 0} \upsilon(t)$ and \emph{terminal point} $Q = \upsilon(T) = \sigma(\gamma(T))$, where $\sigma$ is the composition of reflections associated with $\upsilon$. The \emph{initial triangle} $\triangle ABC$ and the \emph{terminal triangle} $\triangle A'B'C' := \sigma \left( \triangle ABC \right)$ are \emph{consistently oriented} if $\sigma$ is orientation preserving.
Unlike orientation, which is determined by comparing the labelings $(A,B,C)$ and $(A',B',C')=(\sigma(A),\sigma(B),\sigma(C))$ \cite{U-H}, alignment is determined by the relative positions of $P$ and $Q$ on $\triangle ABC$ and $\triangle A'B'C'$, and is independent of labeling. The periodicity of $\gamma$ is characterized by
\begin{Theorem}\label{align+orient}
Let $\gamma$ be an orbit with unfolding $\upsilon$ whose initial point is on $\overline{AC}$. Then $\gamma$ is periodic if and only if the initial and terminal triangles are consistently oriented and the initial and terminal points are aligned. \end{Theorem}
\begin{proof}
When $\triangle A'B'C'$ and $\triangle ABC$ are consistently oriented, $\sigma$ is a rotation or a translation. Since $P$ and $Q$ are aligned, $\sigma$ is a translation and $\sigma (P)=Q$. Hence $P$ and $Q$ are in the same relative position in their respective triangles and the initial and terminal angles are equal. Therefore $\gamma$ is periodic.
Conversely, given a periodic orbit $\gamma$, let $\upsilon$ be the unfolding with initial point $P$ on $\overline{AC}$, initial angle $\Theta : =m\angle QPC$, and generated by first reflecting in a non-horizontal incline. If $\Theta\neq 90^\circ$, the reflection of $\overline{PQ}$ in the vertical line through $P$ is another unfolding, and we may restrict our considerations to initial angles in the range $0<\Theta\leq 90^\circ$.
The images of $\overleftrightarrow{AC}$ in $\mathcal{T}$ are the horizontal, $60^{\circ}$, and $120^{\circ}$ inclines. We claim that $Q$ is on a horizontal incline. Suppose $Q$ is on a $60^\circ$ incline. Then the terminal angle at $Q$, which equals the initial angle $\Theta$ at $P$, is $60^\circ-\Theta$, $\Theta+120^\circ$, $\Theta-60^\circ$, or $240^\circ-\Theta$. But $0<\Theta\leq 90^\circ$ implies $\Theta=30^\circ$ (see Figure~\ref{initial-terminal-angle}), and an initial angle of $\Theta=30^\circ$ produces a period $8$ orbit that terminates on a horizontal incline, which is a contradiction (see Figure~\ref{30-unfolding}). Suppose $Q$ is on a $120^{\circ}$ incline. Then the terminal angle at $Q$ is $120^\circ-\Theta$, $\Theta+60^\circ$, $\Theta-120^\circ$, or $300^\circ-\Theta$. But $0<\Theta\leq 90^\circ$ implies $\Theta=60^\circ$, and an initial angle of $\Theta=60^\circ$ produces an orbit of period $4$ or period $10$, both of which terminate on a horizontal incline, which is a contradiction (see Figure~\ref{alignex2}). Therefore $Q$ is on a horizontal incline as claimed.
\begin{figure}
\caption{\small Equal initial and terminal angles of an unfolding.}
\label{initial-terminal-angle}
\caption{\small An unfolding with initial angle $30^\circ$.}
\label{30-unfolding}
\end{figure}
\begin{figure}
\caption{An unfolding with initial angle $60^\circ$ and period 10.}
\label{60-unfolding-10}
\caption{An unfolding with initial angle $60^\circ$ and period 4.}
\label{60-unfolding-4}
\end{figure}
Since $Q$ is on a horizontal incline, the base $\overline{A'C'}$ is horizontal. Furthermore, since $\overline{PQ}$ does not cross the interior of $\triangle A'B'C'$, its apex $B'$ lies above the base. Thus $\sigma$ is a translation or a glide reflection. If $\sigma$ is a glide reflection, it reverses orientation. Then $180^\circ -\Theta =\Theta$ implies $\Theta=90^{\circ}$. But a periodic orbit with initial angle $90^\circ$ coincides with the period $8$ orbit with initial angle $30^{\circ}$. Since $\sigma$ is a composition of eight reflections, it preserves orientation, which is a contradiction. Therefore $\sigma$ is a translation, $P$ and $Q$ are aligned, and $\triangle A'B'C'$ and $\triangle ABC$ are consistently oriented. \end{proof}
\begin{Corollary}\label{evenperiod} The period of a periodic orbit is even, and an unfolding of a periodic orbit with initial point on a horizontal terminates on a horizontal. \end{Corollary}
Corollary \ref{evenperiod} does not hold for all edge tessellating polygons. For example, Fagnano's periodic orbit on an equilateral triangle has period $3$ and terminates on a non-horizontal incline \cite{Ba-Um}.
Note that the periodic orbit displayed in Figure~\ref{30-unfolding}\text{ } has initial angle $30^{\circ}$ and period $8$, while the periodic orbits displayed in Figure~\ref{alignex2}\text{ } have initial angle $60^{\circ}$ and respective periods $4$ and $10$. A periodic orbit $\gamma$ is \emph{monoperiodic} if every periodic orbit with the same initial angle as $\gamma$ has the same period; otherwise, $\gamma$ is \emph{biperiodic}.
Our next proposition allows us to restrict our considerations to periodic orbits with initial angles between 60 and 90 degrees.
\begin{Proposition}\label{align} Every periodic orbit can be represented by an unfolding with an initial angle $\Theta$ in the range $60^\circ \leq \Theta \leq 90^\circ$.
\end{Proposition}
\begin{proof} Let $\gamma$ be a periodic orbit with initial angle $\Theta$ in the range $0^\circ<\Theta\leq 90^\circ$. Suppose $0^\circ<\Theta\leq 30^\circ$. Since $Q$ is on a horizontal incline, $\overline{PQ}$ cuts a $120^{\circ}$ incline at a point $P'$ with angle of incidence $\Phi=60^\circ+\Theta.$ Thus, there is an unfolding $\overline{P'Q'}$ of $\gamma$ with initial angle $\Phi$ in the range $60^\circ<\Phi\leq 90^\circ$. On the other hand, suppose $30^\circ<\Theta< 60^\circ$, and let $\overline{PQ'}$ be the reflection of $\overline{PQ}$ in $\overleftrightarrow{AC}.$ Then $\overline{PQ'}$ cuts a $60^{\circ}$ incline at a point $P'$ with angle of incidence $\Psi=120^\circ-\Theta$. Thus, there is an unfolding $\overline{P'R}$ with initial angle $\Psi$ in the range $60^\circ<\Psi<90^\circ$. \end{proof}
Since periodic orbits with initial angles $60^\circ$ and $90^\circ$ are understood, our problem reduces to classifying periodic orbits with initial angles in $(60^\circ, 90^\circ)$.
\subsection{Contact points of an unfolding and the fence}
Let $AC$ denote the length of $\overline{AC}$ and let $u=\frac{1}{2}AC$. Impose a rectangular coordinate system on $\mathcal T$ with horizontal axis $\overleftrightarrow{AC}$, origin $O$ at the midpoint of $\overline{AC}$, horizontal unit of length $u$, and vertical unit of length $\sqrt{3}u$. Then points on vertical inclines have integer horizontal coordinates, points on horizontal inclines have integer vertical coordinates, and adjacent vertical and adjacent horizontal inclines lie one unit apart. If $\upsilon$ is an unfolding of a periodic orbit with initial point $P$, terminal point $Q$, and initial angle $\Theta$, Theorem~\ref{align+orient} implies that the vector $\mathbf{PQ}$ is parallel to a vector $\left( x,y\right) $ for some $x,y\in \mathbb{N}$; hence $\Theta=\arctan\left(\frac{y}{x}\sqrt{3}\right)$. Furthermore, $\Theta\in\left(60^\circ, 90^\circ\right)$ implies $x < y$, and we may assume $\gcd\left( x,y\right) =1$.
Parametrize $\upsilon$ via $\upsilon\left( t\right) :=\left( t+a,\frac{y}{x}t\right),$ $0< t \leq T$, where $a\in(-1,1)$; then $P = \lim_{t \searrow 0} \upsilon(t)=(a,0)$ and $Q = \upsilon(T)$. Each point at which $\upsilon$ cuts a vertical incline lies on a fundamental vertical segment of length 2 connecting the midpoints of two horizontal edges. The function $f:\mathbb{Z} \times\mathbb{R}\rightarrow \mathbb{R}/2\mathbb{Z}$ defined by $f\left(\alpha, \beta \right) :=\alpha+ \beta +2 \mathbb Z$ identifies each such segment with the quotient group $\mathbb{R}/2\mathbb{Z}$, called the \emph{fence}. Geometrically, the projection in the $60^\circ$ direction onto the vertical coordinate axis sends $(\alpha,\beta) $ into the coset $f (\alpha,\beta) $. It will often be convenient to think of the fence as the interval $\mathcal{F}:=\left[0,2\right)$ of coset representatives and to write $f\left(\alpha, \beta \right) =\left(\alpha+ \beta \right) \operatorname{mod}2$. The fence $\mathcal{F}$ consists of the \emph{barrier} $\mathcal{B}:=\left( \frac{1}{3},\frac{5}{3}\right] $ and the \emph{gate} $\mathcal{B}^{c}:=\mathcal{F\smallsetminus B}$.
Between consecutive horizontal inclines, an unfolding $\upsilon$ always cuts four non-vertical edges. But whether or not $\upsilon$ also cuts a vertical edge is determined by its set of \emph{contact points} \begin{equation*} \mathcal{C}_{T}:=\left\{ f\left(\upsilon(t)\right) : 0< t \leq T \,\, \mbox{and} \,\, t+a \in \mathbb Z\right\}\subset \mathcal F. \end{equation*} Indeed, $\upsilon$ cuts a vertical edge at $\upsilon(t_i)$ if and only if $f(\upsilon(t_i))\in \mathcal B$. The \emph{multiplicity} of a contact point $c\in \mathcal{C}_{T}$, denoted by $m_{T}\left( c\right) $, is the number of times $\upsilon$ cuts a vertical incline at a point corresponding to $c$, i.e., \begin{equation*} m_{T}\left( c\right) :=\#\left\{ t :0< t \leq T \,\, \mbox{and} \,\, f\left(\upsilon\left(t \right)\right) =c\right\} . \end{equation*} These ideas are illustrated in Figure~\ref{figure4}.
\begin{figure}
\caption{Geometric motivation for the fence.}
\label{geometric-motivation}
\caption{The fence for the unfolding in (a); contact points along the unfolding are indexed sequentially.}
\label{fence}
\end{figure}
Define $t_i:= i-a$, where $i \in \mathbb Z$; then $c_i:= f(\upsilon\left( t_{i}\right)) =f ( i,\frac{y}{x}(i-a) )$. Extending the domain of $\upsilon$ to all real numbers allows us to define $c_i$ for all integers $i$, in which case the equality still holds. The following lemma characterizes the purely geometric notion of alignment in terms of analytic conditions on the set of contact points.
\begin{Lemma}\label{align-contact} The initial and terminal points of an unfolding $\upsilon\left( t\right), \, 0<t \leq T$, are aligned if and only if $c_i =c_{i+T}$ for all $i \in \mathbb Z$.
\end{Lemma}
\begin{proof} By inspection, the initial point $\left(a, 0\right)$ and the terminal point $\left(T+a, \frac{y}{x}T\right)$ are aligned in the tessellation $\mathcal T$ if and only if the horizontal change $T$ and the vertical change $ \frac{y}{x}T$ are both integers, and $2\mid \left(T+\frac{y}{x}T\right)$. By definition, $c_i =c_{i+T}$ if and only if $T \in \mathbb Z$ and $i + \frac{y}{x} (i-a) +2 \mathbb Z = i+T +\frac{y}{x}(i+T-a) +2 \mathbb Z$, where the latter condition is equivalent to $2\mid \left(T+\frac{y}{x}T\right)$. The fact that $T \in \mathbb Z$ and $2\mid \left(T+\frac{y}{x}T\right)$ implies $\frac{y}{x}T \in \mathbb Z$ completes the proof. \end{proof}
Computing the period of a periodic orbit will be significantly simplified by
\begin{Proposition}\label{multiplicity}
If the initial and terminal points of an unfolding are aligned, all contact points are equally spaced on the fence and have the same multiplicity.
\end{Proposition}
\begin{proof}
If the initial and terminal points of $\upsilon\left( t\right), \, 0<t \leq T$, are aligned, then $c_i = c_{i+T}$ for all $i \in \mathbb Z$ by Lemma \ref{align-contact}. Hence $\mathcal G := \{ (1+\frac{y}{x})i + 2 \mathbb{Z}:i\in \mathbb Z\}$ is a finite subgroup of $\mathbb{R} / 2 \mathbb{Z}$, and the set of contact points $\{c_i\}=(-\frac{y}{x} a+2 \mathbb{Z})+\mathcal {G}$ is a coset of $\mathcal G$ in $\mathbb R / 2 \mathbb Z$. Therefore contact points are equally spaced in $\mathbb{R}/2\mathbb{Z}$. Furthermore, if $|\mathcal G|=m$, then $m \mid T$ and $m_{T}\left(c_i\right)=\frac{T}{m}$ for all $i$. \end{proof}
\subsection{Counting the number of edges cut by an unfolding}
Since the initial and terminal points of $\upsilon\left( t\right), \, 0<t \leq T$, are aligned if and only if both $T$ and $\frac{y}{x} T$ are integers and $2\mid \left(T+\frac{y}{x}T \right)$, it follows that $x \mid T$ since $\mathrm{gcd}(x,y)=1$. The \emph{first alignment of} $\upsilon$, denoted by $T_1$, is the smallest value of $T$ for which the three conditions above hold. With the parities of $x$ and $y$ in mind, it is easy to check that these three conditions imply
\begin{Proposition}\label{firstalignment} The first alignment of an unfolding $\upsilon$ is given by \begin{align*}
T_1=
\begin{cases}
x & \textnormal{if } x\equiv y\operatorname{mod}2\\
2x & \textnormal{if } x\not\equiv y\operatorname{mod}2.
\end{cases}
\end{align*} \end{Proposition}
Let $N_T$ be the number of edges of $\mathcal T$ cut by $\upsilon(t) , \, 0< t \leq T$. Proposition \ref{firstalignment} implies that the initial and terminal points of $\upsilon\left( t\right),$ $0< t \leq 2x$, are always aligned. Consequently, we can compute $N_{2x}$ by appealing to the regularity of the contact points given by Proposition \ref{multiplicity}.
\begin{Lemma}\label{decomposition} Let $b_{2x}$ denote the number of contact points of $\upsilon(t) , \, 0< t \leq 2x$, on the barrier. Then $$N_{2x}=8y+m_{2x} b_{2x} .$$ \end{Lemma}
\begin{proof} Since $0< t \leq 2x$, the unfolding $\upsilon$ cuts $\frac{y}{x} \cdot 2x =2y$ horizontal inclines. Between consecutive horizontal inclines, $\upsilon$ cuts four non-vertical edges. Recall that the points at which $\upsilon$ cuts vertical edges correspond to the contact points on the barrier. Since the initial and terminal points of $\upsilon$ are aligned, all contact points have the same multiplicity $m_{2x}$ by Proposition \ref{multiplicity}. Thus the total number of contact points on the barrier is $m_{2x} b_{2x}$, which is also the total number of vertical edges cut by $\upsilon$. Consequently, the total number of edges cut by $\upsilon$ is $4 \cdot 2y+m_{2x} b_{2x}=8y+m_{2x} b_{2x} .$ \end{proof}
An explicit formula for $N_{2x}$ follows from explicit formulas for $m_{ 2x}$ and $b_{ 2x}$.
\begin{Lemma}\label{spacing}
The multiplicity $m_{2x}$ and the spacing $s$ between consecutive contact points are given by
\begin{equation*}
m_{2x}=\left\{
\begin{array}
[c]{cc}
2, & \textnormal{if }x\equiv y\operatorname{mod}2\\
1, & \textnormal{if }x\not\equiv y\operatorname{mod}2
\end{array}
\right. \text{ and \ }s=\left\{
\begin{array}
[c]{cc}
2/x, & \textnormal{if } x\equiv y\operatorname{mod}2\\
1/x, & \, \textnormal{if } x\not\equiv y\operatorname{mod}2.
\end{array}
\right.
\end{equation*} \end{Lemma}
\begin{proof} At the first alignment, $\mathcal C_{ T_1}$ consists of $T_1$ distinct contact points each with multiplicity $1$. If $x\equiv y \bmod 2$, then $T_1=x$ by Proposition \ref{multiplicity}, so that each contact point of $\mathcal{C}_{2x}$ has multiplicity $2$. Since there are $x$ distinct contact points on the fence $\mathcal{ F}$ of length $2$, the spacing $s=\frac{2}{x}$. If $x \not \equiv y \bmod 2$, then $T_1=2x$ so that each contact point in $\mathcal{C}_{2x}$ has multiplicity $1$. Consequently, there are $2x$ distinct contact points and the spacing $s=\frac{2}{2x}=\frac{1}{x}$. \end{proof}
Having established the spacing $s$, let us derive a formula for the number $b_{2x}$ of contact points on the barrier. Let $\left[x\right]$ denote the integer part of $x$.
\begin{Lemma}\label{perturb} The number of contact points on the barrier is \begin{align*}
b_{2x}=
\begin{cases}
\frac{4}{3s} & \textnormal{if } 3 \mid x
\\
\left[\frac{4}{3s}\right] \,\, \textnormal{or} \,\, \left[\frac{4}{3s}\right] +1 & \textnormal{if } 3 \nmid x.
\end{cases}
\end{align*} \end{Lemma}
\begin{proof} Horizontally translating the initial point $P$ uniformly shifts the contact points and preserves the spacing $s$. Since $s \in\{\frac{1}{x},\frac{2}{x}\}$, the length of barrier $\frac{4}{3}$ is an integer multiple of $s$ if and only if $3 \mid x$.
Suppose $3 \mid x$. Since contact points are equally spaced and the barrier $\left( \frac{1}{3},\frac{5}{3}\right]$ is half open, uniformly shifting the contact points preserves the number of contact points on the barrier. Thus $b_{2x}= \frac{4}{3s}$. Suppose $3\nmid x$. Uniformly shifting the contact points alternately increases or decreases $b_{2x}$ by $1$ as contact points enter or leave the barrier. Therefore $b_{2x}\in\{\left[\frac{4}{3s}\right],\left[\frac{4}{3s}\right] +1\}$. \end{proof}
The facts we need to derive an explicit formula for $N_{ 2x}$ are now in place.
\begin{Proposition}\label{count}
The number $N : =N_{2x}$ is given by the following table:
{\small
\begin{center}
\begin{tabular}{| c | c | c || l | }
\hline
$N \equiv 1,3 \bmod 4$ & $N \equiv 2 \bmod 4$ & $N \equiv 0 \bmod 4$ & \\ \hline \hline
& & $8y+\frac{4x}{3}$ & $x \equiv 0 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$ \\ \hline
& & $8y+\frac{4x}{3}$ & $x \equiv 0 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
&$8y+\frac{4x+2}{3}$ & $8y+\frac{4x-4}{3}$ & $x \equiv 1 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$\\ \hline
$8y+\frac{4x-1}{3}$ & $8y+\frac{4x+2}{3}$ & &$x \equiv 1 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
&$8y+\frac{4x-2}{3}$ & $8y+\frac{4x+4}{3}$ & $x \equiv 2 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$\\ \hline
$8y+\frac{4x+1}{3}$ & $8y+\frac{4x-2}{3}$ & &$x \equiv 2 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
\end{tabular}
\end{center}
} \end{Proposition}
\begin{remark*} The columns are arranged to accommodate Proposition~\ref{terminate}. \end{remark*}
\begin{proof} The formula for $N_{2x}$ follows immediately from Lemmas \ref{decomposition}, \ref{spacing}, and \ref{perturb}.
\noindent \textbf{Case 1}. If $x \equiv 0 \bmod 3$ and $x \equiv y \bmod 2$, then $m_{2x}=2$, $s=\frac{2}{x}$, and $b_{2x}=\frac{4}{3s}$. Thus $N_{2x}=8y+m_{2x} b_{2x}=8y+ 2 \cdot \frac{4}{3} \cdot \frac{x}{2}=8y+ \frac{4x}{3}$.
\noindent \textbf{Case 4}. If $x \equiv 1 \bmod 3$ and $x \not\equiv y \bmod 2$, then $m_{2x}=1$, $s=\frac{1}{x}$, and either $b_{2x}=\left[\frac{4}{3s}\right]$ or $\left[\frac{4}{3s}\right] +1$. If $b_{2x}=\left[\frac{4}{3s}\right]$, then $N_{2x}=8y+m_{2x} b_{2x}=8y+ 1 \cdot \left[\frac{4}{3} \cdot \frac{x}{1}\right]=8y+\frac{4x-1}{3}$. If $b_{2x}=\left[\frac{4}{3s}\right]+1$, then $N_{2x}=8y+ 1 \cdot \left( \left[\frac{4}{3}\cdot \frac{x}{1}\right]+1 \right)= 8y+\frac{4x+2}{3}$.
\noindent Proofs of the other cases are similar and left to the reader. \end{proof}
\subsection{Computing the period of a periodic orbit}
Since the period of a periodic orbit is the period of its primitive sub-orbits, let us determine the values of $T$ for which $\upsilon (t), 0<t \leq T$, is an unfolding of a primitive periodic orbit.
\begin{Proposition}\label{terminate}
Let $x,y\in \mathbb{N}$ such that $x < y$ and $\mathrm{gcd}(x,y)=1$. A primitive periodic orbit $\gamma$ with initial angle $\Theta = \arctan(\frac{y}{x}\sqrt{3})$ has an unfolding $\upsilon(t)$, $0 < t \leq T$, for some $T\in\{x,2x,4x\}$, and period $p(x,y)$ determined as follows:
\begin{enumerate}
\item If $x\equiv y \bmod 2$ and $N_{2x} \equiv 0 \bmod 4$, then $T=x$ and $p(x,y)=\frac{1}{2} N_{2x}$.
\item If $x\equiv y \bmod 2$ and $N_{2x}\equiv 2 \bmod 4$, then $T=2x$ and $p(x,y)=N_{2x}$.
\item If $x \not\equiv y \bmod 2$ and $N_{2x}$ is even, then $T=2x$ and $p(x,y) = N_{2x}$.
\item If $x\not\equiv y \bmod 2$ and $N_{2x}$ is odd, then $T=4x$ and $p(x,y)=2 N_{2x}$.
\end{enumerate} \end{Proposition}
\begin{proof}
A primitive periodic orbit has an unfolding $\upsilon(t)$, $0 < t \leq T$, if and only if $T$ is the smallest positive integer such that the initial and terminal points are aligned and $N_T$ is even. Recall that the alignment condition implies $x \mid T$, and the first alignment $T_1=x$ when $x\equiv y \bmod 2$ and $T_1=2x$ otherwise.
\noindent\textbf{Case 1}. If $x\equiv y \bmod 2$, then $T_1=x$. If $N_{2x} \equiv 0 \bmod 4$, then $N_x=\frac{1}{2} N_{2x}$ is even so that $T=x$ and $p(x,y)=N_x=\frac{1}{2}N_{2x}$. If $N_{2x}\equiv 2 \bmod 4$, then $N_x=\frac{1}{2} N_{2x}$ is odd so that $T=2x$ and $p(x,y)=N_{2x}$.
\noindent\textbf{Case 2}. If $x\not\equiv y \bmod 2$, then $T_1=2x$. If $N_{2x}$ is even, then $T=2x$ and $p(x,y)=N_{2x}$. If $N_{2x}$ is odd, then $N_{4x}=2 N_{2x}$ is even so that $T=4x$ and $p(x,y)=N_{4x}=2N_{2x}$. \end{proof}
The period of every periodic orbit is now determined.
\begin{Theorem}\label{formula}
Let $x,y \in \mathbb{N}$ such that $x < y$ and $\mathrm{gcd}(x,y)=1$. Then the period $p(x,y)$ of a periodic orbit with initial angle $\Theta=\arctan(\frac{y}{x}\sqrt{3})$ is
$$p(x, y)=
\begin{cases}
4y + \frac{2x}{3} & {\rm if} \,\, x \equiv 0 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
8y + \frac{4x}{3} & {\rm if} \,\, x \equiv 0 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}\\
4y + \frac{2x-2}{3} \textnormal{ or } \textnormal 8y + \frac{4x+2}{3} & {\rm if} \,\, x \equiv 1 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
16y + \frac{8x-2}{3} \textnormal{ or } 8y + \frac{4x+2}{3} & {\rm if} \,\, x \equiv 1 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}\\
4y + \frac{2x+2}{3} \textnormal{ or } 8y + \frac{4x-2}{3} & {\rm if} \,\, x \equiv 2 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
16y + \frac{8x+2}{3} \textnormal{ or } 8y + \frac{4x-2}{3} & {\rm if} \,\, x \equiv 2 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}.\\
\end{cases}$$ \end{Theorem}
\begin{proof} Propositions \ref{count} and \ref{terminate} immediately lead to the formula for $p(x,y)$.
\noindent \textbf{Case 1}. If $x \equiv 0 \bmod 3$ and $x \equiv y \bmod 2$, then $N_{2x}=8y+\frac{4x}{3} \equiv 0 \bmod 4$ so that $p(x,y)=\frac{1}{2} N_{2x}= 4y+\frac{2x}{3}$.
\noindent \textbf{Case 4}. If $x \equiv 1 \bmod 3$ and $x \not\equiv y \bmod 2$, then $N_{2x}=8y+\frac{4x-1}{3}$ or $N_{2x}=8y+\frac{4x+2}{3}$. In the first case, $N_{2x}$ is odd so that $p(x,y)=2 N_{2x}= 16y+\frac{8x-2}{3}$; in the second case, $N_{2x}$ is even so that $p(x,y)=N_{2x}= 8y+\frac{4x+2}{3}$.
\noindent Proofs of the other cases are similar and left to the reader. \end{proof}
\noindent When $\Theta=60^\circ$, periods $4$ and $10$ are consistent with the formula in Theorem~\ref{formula} with $(x,y)=(1,1)$; when $\Theta=90^\circ$, period $8$ is consistent with $(x,y)=(0,1)$.
\begin{Corollary}
A biperiodic periodic orbit with initial angle $\Theta = \arctan \big(\frac{y}{x} \sqrt{3} \big)$ has one of two possible periods $p_1 < p_2$, where $p_2 = 2p_1 + 2$ or $p_2 = 2 p_1 - 2$. \end{Corollary}
While our understanding of unfoldings follows by considering the fence, the simple statement in Thoerem 12 makes no mention of the initial point. However, the framework developed here allows us to determine a more precise period formula in terms of the initial and terminal points $P = (a, 0)$ and $Q = (a + x, y)$, and doing so required us to compute the number of contact points on the barrier as a function of $a$:
$$b_a(x,y)=\left[ \frac{5/3 + a y/x}{s_a(x,y)} \right]-\left[ \frac{1/3 + a y/x}{s_a(x,y)} \right],$$
where $s_a(x,y)$ is the spacing function.
\section{Periodic orbits on other obtuse polygons}\label{othershapes}
The methods developed in Section \ref{120isosceles} can be applied to a $60^\circ$-rhombus and a $60^\circ$-$90^\circ$-$120^\circ$-kite. Let $\mathcal{T}_1$ and $\mathcal{T}_2$ be the respective edge tessellations generated by a $60^\circ$-rhombus and a $60^\circ$-$90^\circ$-$120^\circ$-kite. Note that an analogue of Theorem \ref{align+orient} holds in both of these cases. In either case, periodic orbits can be represented by unfoldings with an initial angle $\Theta \in [60^\circ, 90^\circ]$, and $\Theta \in (60^\circ, 90^\circ)$ can be expressed in the form $\Theta=\arctan(\frac{y}{x}\sqrt{3})$ with $x < y$ and $\gcd(x, y)=1$. When $\Theta=60^\circ$ or $\Theta=90^\circ$, the period can be determined by inspection and fits the general formula to be derived.
\subsection{The $60^\circ$-rhombus}
Note that $\mathcal {T}_1$ can be obtained from $\mathcal T$ by removing its $0^\circ$, $60^\circ$, and $120^\circ$ inclines (see Figure~\ref{60-rhombusT}). Impose the same coordinate system on $\mathcal{T}_1$ we imposed on $\mathcal{T}$. The barrier and gate for $\mathcal{T}_1$ are identical to the barrier and gate for $\mathcal{T}$, and all definitions and techniques in the previous sections apply. Although $\mathcal{T}_1$ has no horizontal inclines, we can position the initial point of an unfolding $\upsilon$ on a horizontal incline of $\mathcal T$ (the dotted line on Figure \ref{60-rhombusT}). Although a $60^\circ$-rhombus exhibits both line and rotational symmetry, a quick check shows that the initial and terminal rhombuses determined by an unfolding with aligned initial and terminal points, may differ by a reflection but not by a rotation.
\scalebox{.75}{ \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=.9cm,y=.9cm] \draw [color=niceblue] (0,0)-- (1,0.5773502691896257); \draw [color=niceblue] (1,1.7320508075688772)-- (0,1.1547005383792515); \draw [color=niceblue] (0,1.1547005383792515)-- (0,0); \draw [color=niceblue] (1,0.5773502691896257)-- (1,1.7320508075688772); \draw [color=niceblue] (1,1.7320508075688772)-- (2,1.1547005383792512); \draw [color=niceblue] (2,0)-- (3,0.5773502691896258); \draw [color=niceblue] (3,1.7320508075688774)-- (2,1.1547005383792512); \draw [color=niceblue] (2,1.1547005383792512)-- (2,0); \draw [color=niceblue] (6,0)-- (5,0.5773502691896273); \draw [color=niceblue] (5,1.7320508075688787)-- (6,1.1547005383792537); \draw [color=niceblue] (6,1.1547005383792537)-- (6,0); \draw [color=niceblue] (4,0)-- (5,0.5773502691896273); \draw [color=niceblue] (5,0.5773502691896273)-- (5,1.7320508075688787); \draw [color=niceblue] (5,1.7320508075688787)-- (4,1.154700538379252); \draw [color=niceblue] (4,0)-- (3,0.5773502691896258); \draw [color=niceblue] (3,0.5773502691896258)-- (3,1.7320508075688774); \draw [color=niceblue] (3,1.7320508075688774)-- (4,1.154700538379252); \draw [color=niceblue] (4,1.154700538379252)-- (4,0); \draw [color=niceblue] (1,1.732050807568876)-- (1,0.5773502691896246); \draw [color=niceblue] (1,0.5773502691896246)-- (2,0); \draw [dotted,color=niceblue] (0,0)-- (6,0); \draw [dotted,color=niceblue] (0,1.7320508075688772)-- (6,1.7320508075688772); \draw [color=niceblue] (0,3.464101615137755)-- (1,2.886751345948129); \draw [color=niceblue] (1,1.7320508075688776)-- (0,2.3094010767585034); \draw [color=niceblue] (0,2.3094010767585034)-- (0,3.464101615137755); \draw [color=niceblue] (1,2.886751345948129)-- (1,1.7320508075688776); \draw [color=niceblue] (1,1.7320508075688776)-- (2,2.309401076758504); \draw [color=niceblue] (2,3.4641016151377553)-- (3,2.886751345948129); \draw [color=niceblue] (3,1.7320508075688774)-- (2,2.309401076758504); \draw [color=niceblue] (2,2.309401076758504)-- (2,3.4641016151377553); \draw [color=niceblue] (6,3.4641016151377526)-- (5,2.8867513459481273); \draw [color=niceblue] (5,1.732050807568876)-- (6,2.309401076758501); \draw [color=niceblue] (6,2.309401076758501)-- (6,3.4641016151377526); \draw [color=niceblue] (4,3.4641016151377544)-- (5,2.8867513459481273); \draw [color=niceblue] (5,2.8867513459481273)-- (5,1.732050807568876); \draw [color=niceblue] (5,1.732050807568876)-- (4,2.309401076758503); \draw [color=niceblue] (4,3.4641016151377544)-- (3,2.886751345948129); \draw [color=niceblue] (3,2.886751345948129)-- (3,1.7320508075688774); \draw [color=niceblue] (3,1.7320508075688774)-- (4,2.309401076758503); \draw [color=niceblue] (4,2.309401076758503)-- (4,3.4641016151377544); \draw [color=niceblue] (1,1.7320508075688787)-- (1,2.88675134594813); \draw [color=niceblue] (1,2.88675134594813)-- (2,3.4641016151377553); \draw [dotted,color=niceblue] (0,3.464101615137755)-- (6,3.464101615137755); \draw [color=niceblue] (0,3.464101615137755)-- (1,4.041451884327381); \draw [color=niceblue] (1,5.196152422706632)-- (0,4.618802153517006); \draw [color=niceblue] (0,4.618802153517006)-- (0,3.464101615137755); \draw [color=niceblue] (1,4.041451884327381)-- (1,5.196152422706632); \draw [color=niceblue] (1,5.196152422706632)-- (2,4.618802153517006); \draw [color=niceblue] (2,3.4641016151377544)-- (3,4.041451884327381); \draw [color=niceblue] (3,5.196152422706632)-- (2,4.618802153517006); \draw [color=niceblue] (6,3.464101615137757)-- (5,4.041451884327382); \draw [color=niceblue] (5,5.196152422706634)-- (6,4.6188021535170085); \draw [color=niceblue] (6,4.6188021535170085)-- (6,3.464101615137757); \draw [color=niceblue] (4,3.4641016151377553)-- (5,4.041451884327382); \draw [color=niceblue] (5,4.041451884327382)-- (5,5.196152422706634); \draw [color=niceblue] (5,5.196152422706634)-- (4,4.618802153517007); \draw [color=niceblue] (4,3.4641016151377553)-- (3,4.041451884327381); \draw [color=niceblue] (3,4.041451884327381)-- (3,5.196152422706632); \draw [color=niceblue] (3,5.196152422706632)-- (4,4.618802153517007); \draw [color=niceblue] (4,4.618802153517007)-- (4,3.4641016151377553); \draw [color=niceblue] (1,5.196152422706631)-- (1,4.04145188432738); \draw [color=niceblue] (1,4.04145188432738)-- (2,3.4641016151377544); \draw [color=niceblue] (2,3.4641016151377544)-- (2,4.618802153517006); \draw [dotted,color=niceblue] (0,5.196152422706632)-- (6,5.196152422706632); \end{tikzpicture} }
The formula $N_{2x}=4y+m_{2x} b_{2x}$ (as in Lemma~\ref{decomposition}) leads to the following table (as in Proposition~\ref{count}): {\small \begin{center}
\begin{tabular}{| c | c | c || l | } \hline
$N \equiv 1,3 \bmod 4$ & $N \equiv 2 \bmod 4$ & $N \equiv 0 \bmod 4$ & \\ \hline \hline
& & $4y+\frac{4x}{3}$ & $x \equiv 0 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$ \\ \hline
& & $4y+\frac{4x}{3}$ & $x \equiv 0 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
&$4y+\frac{4x+2}{3}$& $4y+\frac{4x-4}{3}$ & $x \equiv 1 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$\\ \hline
$4y+\frac{4x-1}{3}$ & $4y+\frac{4x+2}{3}$ & &$x \equiv 1 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
&$4y+\frac{4x-2}{3}$& $4y+\frac{4x+4}{3}$ & $x \equiv 2 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$\\ \hline
$4y+\frac{4x+1}{3}$ & $4y+\frac{4x-2}{3}$ & &$x \equiv 2 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
\end{tabular}
\end{center} }
Proposition~\ref{terminate} applies, and when combined with the table above, produces the following formula for the period (as in Theorem~\ref{formula}): $$p(x, y)=
\begin{cases}
2y + \frac{2x}{3} & {\rm if} \,\, x \equiv 0 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
4y + \frac{4x}{3} & {\rm if} \,\, x \equiv 0 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}\\
2y + \frac{2x-2}{3} \textnormal{ or } \textnormal 4y + \frac{4x+2}{3} & {\rm if} \,\, x \equiv 1 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
4y + \frac{4x+2}{3} \textnormal{ or } 8y + \frac{8x-2}{3} & {\rm if} \,\, x \equiv 1 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}\\
2y + \frac{2x+2}{3} \textnormal{ or } 4y + \frac{4x-2}{3} & {\rm if} \,\, x \equiv 2 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
4y + \frac{4x-2}{3} \textnormal{ or }8y + \frac{8x+2}{3} & {\rm if} \,\, x \equiv 2 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}.\\
\end{cases}$$
\begin{figure}
\caption{The tessellation $\mathcal T_1$ generated by a $60^\circ$-rhombus.}
\label{60-rhombusT}
\caption{The tessellation $\mathcal T_2$ generated by a $60^\circ$-$90^\circ$-$120^\circ$-kite.}
\label{60-90-120-kiteT}
\end{figure}
\subsection{The $60^\circ$-$90^\circ$-$120^\circ$-kite}
The edge tessellation $\mathcal{T}_2$ is also related to $\mathcal T$. Consider the $60^\circ$-$90^\circ$-$120^\circ$-kite positioned as $\Box AOBD$, where $A$ and $B$ coincide with the two vertices of the $120^\circ$-isosceles $\triangle ABC$ and $O$ is the midpoint of $\overline{AC}$ (see Figure~\ref{60-90-120-kiteT}). Impose the same coordinate system on $\mathcal{T}_2$ we imposed on $\mathcal{T}$ and position the initial point of an unfolding $\upsilon$ on a horizontal incline. While the fence is the same as before, the barrier and gate are interchanged, i.e., $(\frac{1}{3},\frac{5}{3}]$ is the gate.
\definecolor{niceblue}{rgb}{0,0.2,0.6}
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\begin{scriptsize}
\draw (0,0) node[anchor= south west]{$A$}; \draw (2.1,0) node[anchor= south west]{$O$}; \draw (2,1.8*0.7) node[anchor=south]{$B$}; \draw (1,2.8*0.5773502691896257) node[anchor=east]{$D$};
\end{scriptsize}
\end{tikzpicture}
The formula $N_{2x}=6y+m_{2x} b_{2x}$ (as in Lemma~\ref{decomposition}) leads to the following table (as in Proposition~\ref{count}):
{\small \begin{center}
\begin{tabular}{| c | c | c || l | } \hline
$N \equiv 1,3 \bmod 4$ & $N \equiv 2 \bmod 4$ & $N \equiv 0 \bmod 4$ & \\ \hline \hline
& & $6y+\frac{2x}{3}$ & $x \equiv 0 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$ \\ \hline
& $6y+\frac{2x}{3}$ & & $x \equiv 0 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
& $6y+\frac{2x-2}{3}$& $6y+\frac{2x+4}{3}$ & $x \equiv 1 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$\\ \hline
$6y+\frac{2x+1}{3}$ & &$6y+\frac{2x-2}{3}$ &$x \equiv 1 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
&$6y+\frac{2x+2}{3}$& $6y+\frac{2x-4}{3}$ & $x \equiv 2 \bmod 3 \textnormal{ and } x \equiv y \bmod 2$\\ \hline
$6y+\frac{2x-1}{3}$ & & $6y+\frac{2x+2}{3}$ &$x \equiv 2 \bmod 3 \textnormal{ and } x \not\equiv y \bmod 2$ \\ \hline
\end{tabular}
\end{center}
}
Proposition~\ref{terminate} applies, and when combined with the table above, produces the following formula for the period (as in Theorem~\ref{formula}): $$p(x, y)=
\begin{cases}
3y + \frac{x}{3} & {\rm if} \,\, x \equiv 0 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
6y + \frac{2x}{3} & {\rm if} \,\, x \equiv 0 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}\\
3y + \frac{x+2}{3} \textnormal{ or } \textnormal 6y + \frac{2x-2}{3} & {\rm if} \,\, x \equiv 1 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
6y + \frac{2x-2}{3} \textnormal{ or } 12y + \frac{4x+2}{3} & {\rm if} \,\, x \equiv 1 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}\\
3y + \frac{x-2}{3} \textnormal{ or } 6y + \frac{2x+2}{3} & {\rm if} \,\, x \equiv 2 \textnormal{ mod 3} \textnormal{ and } x \equiv y \textnormal{ mod 2}\\
6y + \frac{2x+2}{3} \textnormal{ or }12y + \frac{4x-2}{3} & {\rm if} \,\, x \equiv 2 \textnormal{ mod 3} \textnormal{ and } x \not\equiv y \textnormal{ mod 2}.\\
\end{cases}$$
\subsection{The Regular Hexagon}
While the techniques in Section \ref{120isosceles} can be applied to determine the number of edges cut by an unfolding of a periodic orbit on a regular hexagon, the presence of rotational symmetries renders this information insufficient to determine exactly where an unfolding terminates. Position the hexagon so that a subdivision of its edge tessellation is the edge tessellation generated by a $120^\circ$-isosceles triangle, and restrict considerations to initial angles $\Theta\in(30^\circ,60^\circ)$.
Although we are unable to rigorously solve the problem, we pose a conjecture based on extensive numerical evidence. We computed the period for 3814 pairs $(x,y)$ and partitioned these pairs into planar groups using a mixture algorithm \cite{G}. We conjecture that the period $p(x,y)$ is \begin{equation*}
\begin{cases}
3y + x & {\rm if} \,\,
x \equiv 0 \textnormal{ mod 3}, x \equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A_{0,0} \\
y + \frac{x}{3} & {\rm if} \,\,
x \equiv 0 \textnormal{ mod 3}, x \equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A^c_{0,0} \\
6y + 2x & {\rm if} \,\,
x \equiv 0 \textnormal{ mod 3}, x \not\equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A_{0,1} \\
2y + \frac{2x}{3} & {\rm if} \,\,
x \equiv 0 \textnormal{ mod 3}, x \not\equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A^c_{0,1} \\
2y + \frac{2x-2}{3} \textnormal{ or } \textnormal 3y + x+2 & {\rm if} \,\,
x \equiv 1 \textnormal{ mod 3}, x \equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A_{1,0} \\
y + \frac{x+2}{3} \textnormal{ or } \textnormal 2y + \frac{2x-2}{3} & {\rm if} \,\,
x \equiv 1 \textnormal{ mod 3}, x \equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A^c_{1,0} \\
4y + \frac{4x+2}{3} \textnormal{ or } 6y + 2x-2 & {\rm if} \,\,
x \equiv 1 \textnormal{ mod 3}, x \not\equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A_{1,1} \\
2y + \frac{2x-2}{3} \textnormal{ or } 4y + \frac{4x+2}{3} & {\rm if} \,\,
x \equiv 1 \textnormal{ mod 3}, x \not\equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A^c_{1,1} \\
2y + \frac{2x+2}{3} \textnormal{ or } 3y + x-2 & {\rm if} \,\,
x \equiv 2 \textnormal{ mod 3}, x \equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A_{2,0} \\
y + \frac{x-2}{3} \textnormal{ or } 2y + \frac{2x+2}{3} & {\rm if} \,\,
x \equiv 2 \textnormal{ mod 3}, x \equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A^c_{2,0} \\
4y + \frac{4x-2}{3} \textnormal{ or } 6y + 2x+2 & {\rm if} \,\,
x \equiv 2 \textnormal{ mod 3}, x \not\equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A_{2,1} \\
2y + \frac{2x+2}{3} \textnormal{ or } 4y + \frac{4x-2}{3} & {\rm if} \,\,
x \equiv 2 \textnormal{ mod 3}, x \not\equiv y \textnormal{ mod 2}, \textnormal{and } (x,y) \in A^c_{2,1},
\end{cases} \end{equation*} where the sets $A_{i,j}$ have yet to be determined. A numerical grid search appeared to indicate that the sets $A_{i,j}$ cannot be described by a linear modulus condition $c_1 x + c_2 y \bmod c_3$ for any $c_1,c_2=-36,-35,...,35,36$ and $c_3=2,...,36$. However, when $3 \mid x$, we conjecture that whenever $(x,y) \in A_{i,j}$ for any $i,j$, then $(27y-7x,11y-3x)\in A_{i',j'}$ for some possibly different $i',j'$. Under this assumption, we can state our conjecture for the period formula more precisely for a given $x$.
\section*{Acknowledgments}
We wish to thank David Brown for his participation in an undergraduate research seminar during the 2011-2012 academic year in which this problem was first considered, and Joshua Pavoncello for writing a computer program \cite{P} that visualizes the orbits on an edge tessellating polygon and collects related experimental data. And we wish to thank the referees whose numerous helpful suggestions improved the exposition.
\section*{About the authors}
\subsection*{Benjamin R. Baer} Department of Statistics and Data Science, Cornell University, Ithaca, NY 14853. brb225@cornell.edu
Mr. Baer is a sixth year statistics PhD student at Cornell University studying statistical methodology.
\subsection*{Faheem Gilani}~ Department of Mathematics, The Pennsylvania State University, State College, PA 16801. fhg3@psu.edu
Mr. Gilani is a fifth year mathematics PhD student at Penn State studying machine learning.
\subsection*{Zhigang Han} Department of Mathematics, Millersville University of Pennsylvania, Millersville, PA 17551. zhigang.han@millersville.edu
Dr. Han is an Associate Professor of Mathematics at Millersville University. His primary research areas are symplectic geometry and topology.
\subsection*{Ronald Umble} Department of Mathematics, Millersville University of Pennsylvania, Millersville, PA 17551. ron.umble@millersville.edu
Dr. Umble retired from teaching in August of 2020. He is a Professor of Mathematics Emeritus at Millersville University and a member of Pi Mu Epsilon–Pennsylvania Zeta Chapter. His primary research area is algebraic topology. He has directed or codirected 19 undergraduate research projects, nine of which led to publications.
\end{document} | arXiv | {
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\begin{document}
\renewcommand{Section}{Section} \renewcommand{Subsection}{Subsection}
\title{Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors} \thispagestyle{empty}
\begin{center} \hspace*{5em} \parbox[t]{12em}{\footnotesize \hspace*{-1ex}$^1$Computational Science Center\\ University of Vienna\\ Oskar-Morgenstern-Platz 1\\ A-1090 Vienna, Austria} \hfil \parbox[t]{17em}{\footnotesize \hspace*{-1ex}$^2$Johann Radon Institute for Computational\\ \hspace*{1em}and Applied Mathematics (RICAM)\\ Altenbergerstraße 69\\ A-4040 Linz, Austria} \end{center}
\ifdefined\ShowTableOfContents{} \tableofcontents \fi
\begin{abstract}
We present an approach for variational regularization of inverse and imaging problems for recovering
functions with values in a set of vectors.
We introduce regularization functionals, which are derivative-free double integrals of such functions. These
regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of
intensity functions. These were introduced in
Bourgain, Brézis \& Mironescu, \emph{``Another Look at Sobolev Spaces''.} In: \emph{Optimal Control and
Partial Differential Equations}-Innovations \& Applications, IOS press, Amsterdam, 2001.
For the proposed regularization functionals we prove
existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors. \end{abstract}
\section{Introduction}
Functions with values in a (nonlinear) subset of a vector space appear in several applications of imaging and in inverse problems, e.g.
\begin{itemize}
\item \emph{Interferometric Synthetic Aperture Radar (InSAR)}
is a technique used in remote sensing and geodesy to generate for example digital elevation
maps of the earth's surface.
InSAR images represent phase differences of waves between two or more SAR images, cf. \cite{LiuMas16,RocPraFer97}.
Therefore InSAR data are functions $f:\Omega \to \mathbb{S}^1 \subseteq \mathds{R}^2$.
The pointwise function values are on the $\mathbb{S}^1$, which is considered embedded into $\mathds{R}^2$.
\item A \emph{color image} can be represented as a function in \emph{HSV}-space (hue, saturation, value) (see e.g. \cite{PlaVen00}).
Color images are then described as functions $f:\Omega \to K \subseteq \mathds{R}^3$. Here $\Omega$ is a plane in $\mathds{R}^2$, the image domain,
and $K$ (representing the HSV-space) is a cone in 3-dimensional space $\mathds{R}^3$.
\item Estimation of the \emph{foliage angle distribution} has been considered for instance in
\cite{HelAndRobFin15,PutBriManWiePfeZliPfe16}.
Thereby the imaging function is from $\Omega \subset \mathds{R}^2$, a part of the Earth's surface,
into $\mathbb{S}^2 \subseteq \mathds{R}^3$, representing foliage angle orientation.
\item Estimation of functions with values in $SO(3) \subseteq \mathds{R}^{3 \times 3}$. Such problems appear in \emph{Cryo-Electron Microscopy}
(see for instance \cite{HadSin11,SinShk12,WanSinWen13}). \end{itemize} We emphasize that we are analyzing \emph{vector}, \emph{matrix}, \emph{tensor}-valued functions, where pointwise function evaluations belong to some given (sub)set, but are always \emph{elements} of the underlying vector space. This should not be confused with set-valued functions, where every function evaluation can be a set.
Inverse problems and imaging tasks, such as the ones mentioned above, might be unstable, or even worse, the solution could be ambiguous. Therefore, numerical algorithms for imaging need to be \emph{regularizing} to obtain approximations of the desired solution in a stable manner. Consider the operator equation \begin{equation}\label{eq:basic_problem} \op[w] = v^0, \end{equation} where we assume that only (noisy) measurement data $v^\delta$ of $v^0$ become available. In this paper the method of choice is \emph{variational regularization} which consists in calculating a minimizer of the variational regularization functional \begin{equation}
\label{eq:energy}
\mathcal{F}(w) \vcentcolon= \mathcal{D}(\op[w],v^\delta) + \alpha \mathcal{R}(w). \end{equation} Here \begin{description}
\item{$w$} is an element of the \emph{set} of admissible functions.
\item{$\op$} is an operator modeling the image formation process (except the noise).
\item{$\mathcal{D}$} is called the \textit{data} or \textit{fidelity term}, which is used to compare a pair
of data in the image domain, that is to quantify the difference of the two data sets.
\item{$\mathcal{R}$} is called \textit{regularization functional}, which is used to impose certain
properties onto a minimizer of the regularization functional $\mathcal{F}$.
\item{$\alpha > 0$} is called \textit{regularization parameter} and provides a trade off between stability
and approximation properties of the minimizer of the regularization functional $\mathcal{F}$.
\item{$v^\delta$} denotes measurement data, which we consider noisy.
\item{$v^0$} denotes the exact data, which we assume to be not necessarily available. \end{description}
The main objective of this paper is to introduce a general class of regularization functionals for functions with values in a set of vectors. In order to motivate our proposed class of regularization functionals we review a class of regularization functionals appropriate for analyzing \emph{intensity data}.
\subsection*{Variational regularization for reconstruction of intensity data} Opposite to what we consider in the present paper, most commonly, imaging data $v$ and admissible functions $w$, respectively, are considered to be representable as \emph{intensity functions}. That is, they are functions from some subset $\Omega$ of an Euclidean space with \emph{real values}.
In such a situation the most widely used regularization functionals use regularization terms consisting of powers of Sobolev (see \cite{BouSau93,ChaLio97,CimMel12}) or total variation semi-norms \cite{RudOshFat92}. It is common to speak about \emph{Tikhonov regularization} (see for instance \cite{TikArs77}) when the data term and the regularization functional are squared Hilbert space norms, respectively. For the \emph{Rudin, Osher, Fatemi (ROF)} regularization \cite{RudOshFat92}, also known as total variation
regularization, the data term is the squared $L^2$-norm and $\mathcal{R}(w) = |w|_{TV}$ is the total variation semi-norm. Nonlocal regularization operators based on the generalized nonlocal gradient is used in \cite{GilOsh08}. \\ Other widely used regularization functionals are \emph{sparsity promoting} \cite{DauDefDem04,KolLasNiiSil12}, \emph{Besov space norms} \cite{LorTre08,LasSak09} and anisotropic regularization norms \cite{OshEse04,SchWei00}. Aside from various regularization terms there also have been proposed different fidelity terms other than quadratic norm fidelities, like the $p$-th powers of $\ell^p$ and $L^p$-norms of the differences of $F(w)$ and $v$ , \cite{SchGraGroHalLen09,SchuKalHofKaz12}, Maximum Entropy \cite{Egg93,EngLan93} and Kullback-Leibler divergence \cite{ResAnd07} (see \cite{Poe08} for some reference work).
Our work utilizes results from the seminal paper of \citeauthor{BouBreMir01} \cite{BouBreMir01}, which provides an equivalent \emph{derivative-free} characterization of Sobolev spaces and the space $\BV$, the space of functions of bounded total variation, which consequently, in this context, was analyzed in \citeauthor{Dav02} and \citeauthor{Pon04b} \cite{Dav02,Pon04b}, respectively. It is shown in \cite[Theorems 2 \& 3']{BouBreMir01} and \cite[Theorem 1]{Dav02} that when $(\rho_\varepsilon)_{\varepsilon > 0}$ is a suitable sequence of non-negative, radially symmetric, radially decreasing mollifiers, then \begin{equation} \label{eq:double_integral} \begin{aligned} \lim_{\varepsilon \searrow 0} \tilde{\mathcal{R}}_\varepsilon(w) &
\vcentcolon= \lim_{\varepsilon \searrow 0} \int\limits_{\Omega\times \Omega} \frac{\|w(x)- w(y)\|_{\mathds{R}}^p}{\normN[x-y]^{p}} \rho_\varepsilon(x-y) \,\mathrm{d}(x,y)\\ &= \begin{cases}
C_{p,N}|w|^p_{W^{1,p}} & \mbox{if } w \in W^{1,p}(\Omega, \mathds{R}), \ 1 < p < \infty, \\
C_{1,N}|w|_{TV} & \mbox{if } w \in \BV[\mathds{R}], \ p = 1, \\ \infty & \mbox{otherwise}, \end{cases} \end{aligned} \end{equation} Hence $\tilde{\mathcal{R}}_\varepsilon$ approximates powers of Sobolev semi-norms and the total variation semi-norm, respectively. Variational imaging, consisting in minimization of $\mathcal{F}$ from \autoref{eq:energy} with $\mathcal{R}$ replaced by $\tilde{\mathcal{R}}_\varepsilon$, has been considered in \cite{AubKor09,BouElbPonSch11}.
\subsection*{Regularization of functions with values in a set of vectors} In this paper we generalize the derivative-free characterization of Sobolev spaces and functions of bounded variation to functions, $u:\Omega \to K$, where $K$ is some set of vectors, and use these functionals for variational regularization. The applications we have in mind contain that $K$ is a closed subset of $\mathds{R}^M$ (for instance HSV-data) with non-zero measure, or that $K$ is a sub-manifold (such as for instance InSAR-data).
The reconstruction of manifold--valued data with variational regularization methods has already been subject to intensive research (see for instance \cite{KimSoc02,CreStr11,CreStr13,CreKoeLelStr13,BacBerSteWei16,WeiDemSto14}). The variational approaches mentioned above use regularization and fidelity functionals based on Sobolev and TV semi-norms: a total variation regularizer for cyclic data on $\mathbb{S}^1$ was introduced in \cite{CreStr13,CreStr11}, see also \cite{BerLauSteWei14,BerWei16,BerWei15}. In \cite{BacBerSteWei16,BerFitPerSte17} combined first and second order differences and derivatives were used for regularization to restore manifold--valued data. The later mentioned papers, however, are formulated in a finite dimensional setting, opposed to ours, which is considered in an infinite dimensional setting. Algorithms for total variation minimization problems, including half-quadratic minimization and non-local patch based methods, are given for example in \cite{BacBerSteWei16,BerChaHiePerSte16,BerPerSte16} as well as in \cite{GroSpr14,LauNikPerSte17}. On the theoretical side the total variation of functions with values in a manifold was investigated by \citeauthor{GiaMuc07} using the theory of Cartesian currents in \cite{GiaMuc07,GiaMuc06}, and earlier \cite{GiaModSou93} if the manifold is a $\mathbb{S}^1$.
\subsection*{The contents and the particular achievements of the paper are as follows} The contribution of this paper is to introduce and analytically analyze double integral regularization functionals for reconstructing functions with values in a set of vectors, generalizing functionals of the form \autoref{eq:double_integral}. Moreover, we develop and analyze fidelity terms for comparing manifold--valued data. Summing these two terms provides a new class of regularization functionals of the form \autoref{eq:energy} for reconstructing manifold--valued data.
When analyzing our functionals we encounter several differences to existing regularization theory (compare \autoref{sec: Setting}): \begin{enumerate} \item
The \emph{admissible functions}, where we minimize the regularization functional on,
do form only a \emph{set} but \emph{not} a \emph{linear} space.
As a consequence, well--posedness
of the variational method (that is, existence of a minimizer of the energy functional) cannot directly be proven
by applying standard
direct methods in the Calculus of Variations \cite{Dac82,Dac89}. \item The regularization functionals are defined via metrics and not norms, see \autoref{sec: Existence}. \item In general, the fidelity terms are \emph{non-convex}.
Stability and convergence results are proven in \autoref{sec: Stability_and_Convergence}. \end{enumerate}
The model is validated in \autoref{sec:Numerical_results} where we present numerical results for denoising and inpainting of data of InSAR type.
\section{Setting} \label{sec: Setting}
In the following we introduce the basic notation and the set of admissible functions which we are regularizing on.
\begin{assumption}
\label{ass:1}
All along this paper we assume that
\begin{itemize}
\item $p_1, p_2 \in [1, +\infty)$, $s \in (0,1]$,
\item $\Omega_1, \Omega_2 \subseteq \mathds{R}^N$ are nonempty, bounded, and connected open sets with Lipschitz boundary, respectively,
\item $k \in [0,N]$,
\item $K_1 \subseteq \mathds{R}^{M_1}, K_2 \subseteq \mathds{R}^{M_2}$ are nonempty and closed subsets of $\mathds{R}^{M_1}$ and $\mathds{R}^{M_2}$, respectively.
\end{itemize}
Moreover,
\begin{itemize}
\item $\normN$ and $\|\cdot\|_{\mathds{R}^{M_i}}, \ i=1,2,$ are the Euclidean norms on $\mathds{R}^N$ and $\mathds{R}^{M_i}$, respectively.
\item $\dRMi: \mathds{R}^{M_i} \times \mathds{R}^{M_i} \rarr [0, +\infty)$ denotes the Euclidean distance on $\mathds{R}^{M_i}$ for $i=1,2$ and
\item $\,\mathrm d_i \vcentcolon= \mathrm{d}_{K_i}: K_i \times K_i \rarr [0, +\infty)$
denote arbitrary metrics on $K_i$, which fulfill for $i=1$ and $i=2$
\begin{itemize}
\item $\dRMi|_{K_i \times K_i} \leq d_i$,
\item $\,\mathrm d_i$ is continuous with respect to $\dRMi|_{K_i \times K_i}$, meaning that for a sequence $\seq{a}$ in $K_i \subseteq \mathds{R}^{M_i}$ converging to some $a \in K_i$ we also have $\,\mathrm d_i(a_n,a) \rarr 0$.
\end{itemize}
In particular, this assumption is valid if the metric $d_i$ is equivalent to $\dRMi|_{K_i \times K_i}$.
When the set $K_i, \ i=1,2$, is a suitable complete submanifold of $\mathds{R}^{M_i}$,
it seems natural to choose $d_i$ as the geodesic distance on the respective submanifolds.
\item $(\rho_{\varepsilon})_{\varepsilon > 0}$ is a Dirac family of non-negative, radially symmetric mollifiers, i.e. for every $\varepsilon > 0$ we have
\begin{enumerate}
\item $\rho_\varepsilon \in \mathcal{C}^{\infty}_{c}(\mathds{R}^N, \mathds{R})$ is radially symmetric,
\item $\rho_\varepsilon \geq 0$,
\item $\int \limits_{\mathds{R}^N} \rho_\varepsilon (x) \,\mathrm{d}x = 1$, and
\item for all $\delta > 0$, $\lim_{\varepsilon \searrow 0}\limits \int_{\set{\normN[y] > \delta}} \rho_\varepsilon(y) \,\mathrm{d}y = 0$.
\end{enumerate}
We demand further that, for every $\varepsilon > 0$,
\begin{enumerate}[resume]
\item there exists a $\tau > 0$ and $\eta_{\tau} > 0$ such that
$\{ z \in \mathds{R}^N : \rho_{\varepsilon}(z) \geq \tau \}= \{z \in \mathds{R}^N : \normN[z] \leq \eta_{\tau} \}$.
\end{enumerate}
This condition holds, e.g., if $\rho_{\varepsilon}$ is a radially decreasing continuous function with $\rho_{\varepsilon}(0) > 0$.
\item When we write $p$, $\Omega$, $K$, $M$, then we mean $p_i$, $\Omega_i$, $K_i$, $M_i$, for either
$i=1,2$. In the following we will often omit the subscript indices whenever possible.
\end{itemize} \end{assumption}
\begin{example}\label{ex:mol} Let $\hat{\rho} \in C_c^\infty(\mathds{R},\mathds{R}_+)$ be symmetric at $0$, monotonically decreasing on $[0, \infty)$ and satisfy \begin{equation*}
\abs{\mathbb{S}^{N-1}}\int_0^\infty \hat{t}^{N-1} \hat{\rho}\left(\hat{t}\right)d \hat{t} = 1\;. \end{equation*} Defining mappings $\rho_\varepsilon: \mathds{R}^N \to \mathds{R}$ by \begin{equation*}
\rho_\varepsilon(x)\vcentcolon= \frac{1}{\varepsilon^N} \hat{\rho}\left(\frac{\normN[x]}{\varepsilon}\right) \end{equation*} constitutes then a family $(\rho_\varepsilon)_{\varepsilon > 0}$ which fulfills the above properties \it{(i) -- (v)}. Note here that \begin{itemize}
\item by substitution $x = t \theta$ with $t > 0, \theta \in \mathbb{S}^{N-1}$ and $\hat{t}=\frac{t}{\varepsilon}$, \begin{equation}
\label{eq:molII}
\begin{aligned}
\int_{\mathds{R}^N} \rho_\varepsilon(x)\, \,\mathrm{d}x &= \frac{1}{\varepsilon^N} \int_{\mathds{R}^N} \hat{\rho}\left(\frac{\normN[x]}{\varepsilon}\right) \,\mathrm{d}x \\
&= \frac{1}{\varepsilon^N} \int_0^\infty t^{N-1} \hat{\rho}\left(\frac{t}{\varepsilon}\right)\,\mathrm d t \int_{\mathbb{S}^{N-1}} \,\mathrm d\theta \\
&= \abs{\mathbb{S}^{N-1}}\int_0^\infty \hat{t}^{N-1} \hat{\rho}\left(\hat{t}\right)\,\mathrm d \hat{t} = 1\;.
\end{aligned} \end{equation} Here, $d\theta$ refers to the canonical spherical measure.
\item Again by the same substitutions, taking into account that $\hat{\rho}$ has compact support, it follows for
$\varepsilon > 0$ sufficiently small that \begin{equation}
\label{eq:molIIa}
\begin{aligned}
\int_{\set{y:\normN[y]>\delta}} \rho_\varepsilon(x)\, \,\mathrm{d}x
&= \frac{1}{\varepsilon^N} \int_{\set{y:\normN[y]> \delta}} \hat{\rho}\left(\frac{\normN[x]}{\varepsilon}\right) \,\mathrm{d}x \\
&= \frac{1}{\varepsilon^N} \int_\delta^\infty t^{N-1} \hat{\rho}\left(\frac{t}{\varepsilon}\right)\,\mathrm d t \int_{\mathbb{S}^{N-1}} \,\mathrm d\theta \\
&= \abs{\mathbb{S}^{N-1}}\int_{\delta/\varepsilon}^\infty \hat{t}^{N-1} \hat{\rho}\left(\hat{t}\right)\,\mathrm d \hat{t} =0 \;.
\end{aligned} \end{equation} \end{itemize} \end{example} In the following we write down the basic spaces and sets, which will be used in the course of the paper. \begin{definition} \begin{itemize}
\item The \emph{Lebesgue--Bochner space} of $\mathds{R}^M$--valued functions on $\Omega$ consists of the set
\begin{equation*}
\begin{aligned}
\Lp \vcentcolon= \{ \phi : \Omega \to \mathds{R}^M : {} & \phi \text{ is Lebesgue-Borel measurable and } \\
&\normM[\phi(\cdot)]^p: \Omega \to \mathds{R} \text{ is Lebesgue--integrable on } \Omega \},
\end{aligned}
\end{equation*}
which is associated with the norm $\|\cdot\|_{\Lp}$, given by
\begin{gather*}
\norm{\phi}_{\Lp} \vcentcolon= \Big( \int_{\Omega}\limits \normM[\phi(x)]^p \,\mathrm{d}x \Big)^{1/p} \; .
\end{gather*}
\item Let $0 < s < 1$. Then the \emph{fractional Sobolev space} of order $s$ can be defined (cf. \cite{Ada75}) as the set \begin{gather*}
\Wsp \vcentcolon= \left\{ w \in \Lp : \frac{\normM[w(x) - w(y)]}{\normN[x-y]^{\frac{N}{p}+s}} \in
L^p (\Omega \times \Omega, \mathds{R}) \right\} \\
= \{w \in \Lp : \abs{w}_{\Wsp} < \infty \}, \end{gather*} equipped with the norm \begin{equation}\label{eq:sobolev_norm}
\|\cdot\|_{\Wsp} \vcentcolon= \big(\|\cdot\|_{\Lp}^{p} + \abs{\cdot}_{\Wsp}^p \big)^{1/p}, \end{equation} where $\abs{\cdot}_{\Wsp}$ is the semi-norm for $\Wsp$, given by \begin{equation}\label{eq:sobolev_semi_norm}
\abs{w}_{\Wsp} \vcentcolon= \Big(\int\limits_{\Omega\times \Omega} \frac{\normM[w(x) - w(y)]^p}{\normN[x-y]^{N+ps}} \,\mathrm{d}(x,y) \Big)^{1/p},
\quad w \in \Wsp\;. \end{equation} \item For $s = 1$ the Sobolev space $W^{1,p}(\Omega, \mathds{R}^M)$ consists of all weakly differentiable functions in $L^1(\Omega,\mathds{R}^M)$ for which \begin{equation*} \norm{w}_{W^{1,p}(\Omega, \mathds{R}^M)}
\vcentcolon= \Big( \norm{w}_{\Lp}^p + \int_{\Omega}\limits \|\nabla w(x)\|^p_{\mathbb{R}^{M\times N}} \,\mathrm{d}x \Big)^{1/p} < \infty\;, \end{equation*} where $\nabla w$ is the weak Jacobian of $w$. \item Moreover, we recall one possible definition of the space $\BV$ from \cite{AmbFusPal00}, which consists of all
Lebesgue--Borel measurable functions $w:\Omega \to \mathds{R}^M$ for which \begin{gather*} \norm{w}_{\BV} \vcentcolon= \norm{w}_{L^1(\Omega, \mathds{R}^M)} + \abs{w}_{\BV} < \infty, \end{gather*} where \begin{equation*} \begin{aligned} ~ & \abs{w}_{\BV} \\ \vcentcolon= {}& \sup \left\{ \int \limits_\Omega w(x) \cdot \mathrm{Div} \varphi(x) \,\mathrm{d}x : \ \varphi \in C_c^1(\Omega, \mathds{R}^{M \times N}) ~ \mathrm{ such~that } \norm{\varphi}_\infty \vcentcolon= \mathop{\mathrm{ess~sup}}_{x \in \Omega} \norm{\varphi(x)}_F \leq 1 \right\}, \end{aligned} \end{equation*} where $\norm{\varphi(x)}_F$ is the Frobenius-norm of the matrix $\varphi(x)$ and $\mathrm{Div}\varphi = (\mathrm{div} \varphi_1, \dots, \mathrm{div} \varphi_M)^\mathrm{T}$ denotes the row--wise formed divergence of $\varphi$. \end{itemize} \end{definition}
\begin{lemma} \label{le:inclusion}
Let $0 < s \leq 1$ and $p \in [1,\infty)$, then $\Wsp \hookrightarrow \Lp$ and the embedding is compact. Moreover, the embedding
$\BV \hookrightarrow \Lp$ is compact for all
\begin{gather*}
1 \leq p < 1^* \vcentcolon=
\begin{cases}
+\infty &\mbox{if } N = 1 \\
\frac{N}{N-1} &\mbox{otherwise }
\end{cases} \,.
\end{gather*} \end{lemma} \begin{proof}
The first result can be found in \cite{DemDem07} for $0 < s < 1$ and in \cite{Eva10} for $s = 1$. The second assertion
is stated in \cite{AmbFusPal00}. \end{proof}
\begin{remark} \label{re:notes_basic} Let \autoref{ass:1} hold.
We recall some basic properties of weak convergence in $\Wsp$, $W^{1,p}(\Omega, \mathds{R}^M)$ and weak* convergence in $\BV$ (see for instance \cite{Ada75,AmbFusPal00}) :
\begin{itemize}
\item Let $p > 1$, $s\in(0,1]$ and assume that $(w_n)_{n \in \mathds{N}}$ is bounded in $\Wsp$.
Then there exists a subsequence $(w_{n_k})_{k \in \mathds{N}}$ which converges weakly in $\Wsp$.
\item Assume that $(w_n)_{n \in \mathds{N}}$ is bounded in $\BV$. Then there exists a subsequence $(w_{n_k})_{k \in \mathds{N}}$
which converges weakly* in $\BV$.
\end{itemize} \end{remark}
Before introducing the regularization functional, which we investigate theoretically and numerically, we give the definition of some sets of (equivalence classes of) admissible functions. \begin{definition} \label{def:spaces_etc} \label{de:basic}
For $0 < s \leq 1$, $p \geq 1$ and a nonempty closed subset $K \subseteq \mathds{R}^M$ we define
\begin{equation}
\label{eq:spacess_K}
\begin{aligned}
\Lp[K] \vcentcolon= {} & \{\phi \in \Lp : \phi(x) \in K \text{ for a.e. } x \in \Omega\}; \\
\Wsp[K] \vcentcolon= {} & \{w \in \Wsp: w(x) \in K \text{ for a.e. } x \in \Omega \}, \\
\BV[K] \vcentcolon= {} & \{w \in \BV: w(x) \in K \text{ for a.e. } x \in \Omega \}.
\end{aligned}
\end{equation}
and equip each of these (in general nonlinear) sets with some subspace topology:
\begin{itemize}
\item $\Lp[K] \subseteq \Lp$ is associated with the strong $\Lp$-topology,
\item $\Wsp[K] \subseteq \Wsp$ is associated with the weak $\Wsp$-topology, and
\item $\BV[K] \subseteq \BV$ is associated with the weak* $\BV$-topology.
\end{itemize}
Moreover, we define
\begin{equation} \label{eq:ChooseW}
W(\Omega,K) \vcentcolon=
\begin{cases}
\Wsp[K] & \text{ for } p \in (1, \infty) \text { and } s \in (0,1], \\
\BV[K] & \text{ for } p = 1 \text { and } s = 1\;.
\end{cases}
\end{equation}
Consistently, $W(\Omega,K)$
\begin{itemize}
\item is associated with the weak $\Wsp$-topology in the case $p \in (1, \infty)$ and $s \in (0,1]$ and
\item with the weak* $\BV$-topology when $p=1$ and $s=1$.
\end{itemize}
When we speak about
\begin{equation*}
\text{ convergence on } W(\Omega,K) \text{ we write } \overset{W(\Omega, K)}{\longrightarrow} \text{ or simply} \rarr<W>
\end{equation*}
and mean weak convergence on $W^{s,p}(\Omega,K)$ and weak* convergence on $\BV[K]$, respectively. \end{definition} \begin{remark} \label{re:notes_choose_w} ~\nopagebreak \begin{itemize}[topsep=0pt] \item In general $\Lp[K], \Wsp[K]$ and $\BV[K]$ are sets which do not form a linear space. \item If $K = \mathbb{S}^1$, then $\Wsp[K] = \Wsp[\mathbb{S}^1]$ as occurred in \cite{BouBreMir00b}. \item For an embedded manifold $K$ the dimension of the manifold is not necessarily identical with the space dimension of $\mathds{R}^M$.
For instance if $K = \mathbb{S}^1 \subseteq \mathds{R}^2$, then the dimension of $\mathbb{S}^1$ is $1$ and $M=2$. \end{itemize} \end{remark} The following lemma shows that $W(\Omega,K)$ is a sequentially closed subset of $\W$. \begin{lemma}[Sequential closedness of $W(\Omega,K)$ and {$\Lp[K]$}] \label{lem:Wsp_weakly_seq_closed_etc}
~\nopagebreak
\begin{enumerate}[topsep=0pt]
\item
Let $w_* \in \W$ and $(w_n)_{n\in \mathds{N}}$ be a sequence in $\W[K] \subseteq \W$ with
$w_n \overset{W(\Omega, \mathds{R}^M)}{\longrightarrow} w_*$ as $n \to \infty$.
Then $w_* \in \W[K]$ and $w_n \rarr w_*$ in $\Lp[K]$.
\item
Let $v_* \in \Lp$ and $(v_n)_{n \in \mathds{N}}$ be a sequence in $\Lp[K] \subseteq \Lp$ with $v_n \to v_*$ in $\Lp$ as $n \to \infty$.
Then $v_* \in \Lp[K]$ and there is some subsequence $(v_{n_k})_{k \in \mathds{N}}$ which converges to $v_*$ pointwise almost everywhere, i.e. $v_{n_k}(x) \to v_*(x)$
as $k \to \infty$ for almost every $x \in \Omega$.
\end{enumerate} \end{lemma} \begin{proof}
For the proof of the second part, cf. \cite{Els02}, Chapter VI, Corollary 2.7
and take into account the closedness of $K \subseteq \mathds{R}^M$.
The proof of the first part follows from standard convergence arguments in $\Wsp$, $\BV$ and $\Lp$, respectively,
using the embeddings from \autoref{le:inclusion}, an argument on subsequences and part two. \end{proof}
\begin{remark}
\autoref{le:inclusion} along with \autoref{lem:Wsp_weakly_seq_closed_etc} imply that $\W[K]$ is compactly embedded in $\Lp[K]$,
where these sets are equipped with the bornology inherited from $\W$ and the topology inherited from $\Lp$, respectively. \end{remark}
In the following we postulate the assumptions on the operator $\op$ which will be used throughout the paper: \begin{assumption} \label{ass:2} Let $\W<1>$ be as in \autoref{eq:ChooseW} and assume that $\op$ is an operator from $\W<1>$ to $\Lp<2>$. \end{assumption}
We continue with the definition of our regularization functionals: \begin{definition} \label{def:functional} Let \autoref{ass:1} and \autoref{ass:2} hold. Moreover, let $\varepsilon > 0$ be fixed and let $\rho \vcentcolon= \rho_\varepsilon$ be a mollifier.
The regularization functional
$\F<v><\alpha>[\dKt, \dKo] : \W<1> \rarr [0, \infty]$ is defined as follows
\begin{equation}\label{eq: functional_with_some_metric}
\boxed{
\F<v><\alpha>[\dKt, \dKo] (w) \vcentcolon= \int\limits_{\Omega_2} \dKt^{p_2}(\op[w](x), v(x)) \,\mathrm{d}x + \alpha \int\limits_{\Omega_1\times \Omega_1} \frac{\dKo^{p_1}(w(x), w(y))}{\normN[x-y]^{k+p_1 s}} \rho^l(x-y) \,\mathrm{d}(x,y),}
\end{equation}
where
\begin{enumerate}
\item $v \in \Lp<2>$,
\item $s \in (0,1]$,
\item $\alpha \in (0, +\infty)$ is the regularization parameter,
\item $l \in \set{0, 1}$ is an indicator and
\item \label{itm:k}
$\begin{cases}
k \leq N &\mbox{if } \W<1> = W^{s,p_1}(\Omega_1, K_1), \ 0<s<1, \\
k=0 & \mbox{if } \W<1> = W^{1,p_1}(\Omega_1, K_1) \text{ or if } \W<1> = BV(\Omega_1, K_1), \text{ respectively.}
\end{cases}$
\end{enumerate}
Setting
\begin{equation}\label{eq:d2}
\boxed{
\mebr[\phi][\nu]_{[\dKt]} \vcentcolon= \left( \int\limits_{\Omega_2} \dKt^{p_2}(\phi(x),\nu(x)) \,\mathrm{d}x \right)^{\frac{1}{p_2}},}
\end{equation}
and
\begin{equation}\label{eq:d3}
\boxed{
\mathcal{R}_{[\dKo]}(w) \vcentcolon= \int\limits_{\Omega_1\times \Omega_1} \frac{\dKo^{p_1}(w(x), w(y))}{\normN[x-y]^{k+p_1 s}} \rho^l(x-y) \,\mathrm{d}(x,y),}
\end{equation}
\autoref{eq: functional_with_some_metric} can be expressed in compact form
\begin{equation}
\label{eq:functional}
\boxed{
\F<v><\alpha>[\dKt,\dKo](w) = \mebr[\op[w]][v]^{p_2}_{[\dKt]} + \alpha \mathcal{R}_{[\dKo]}(w).}
\end{equation}
For convenience we will often skip some of the super- or subscript, and use compact notations like e.g.
\begin{equation*}
\F<v>, \F[\dKt, \dKo] \text{ or } \F(w) = \mebr[\op[w]][v]^{p_2} + \alpha \mathcal{R}(w).
\end{equation*} \end{definition}
\begin{remark} ~
\begin{enumerate}[topsep=0pt]
\item $l = \set{0,1}$ is an indicator which allows to consider approximations of Sobolev semi-norms and double integral
representations
of the type of \citeauthor{BouBreMir01} \cite{BouBreMir01} in a uniform manner.
\begin{itemize}
\item when $k=0$, $s=1$, $l=1$ and when $d_1$ is the Euclidean distance, we get the double integrals of the
\citeauthor{BouBreMir01}-form \cite{BouBreMir01}. Compare with \autoref{eq:double_integral}.
\item When $d_1$ is the Euclidean distance, $k=N$ and $l=0$, we get Sobolev semi-norms.
\end{itemize}
We expect a relation between the two classes of functionals for $l=0$ and $l=1$ as stated in \autoref{ss:conjecture}.
\item
When $d_1$ is the Euclidean distance then the second term in \autoref{eq: functional_with_some_metric} is similar to the ones used in
\cite{AubKor09,BouElbPonSch11} and \cite{BouBreMir01, Pon04b, Dav02}.
\end{enumerate} \end{remark}
In the following we state basic properties of $\mebr_{[\dKt]}$ and the functional $\F$.
\begin{proposition} \label{pr:ExprIsOp} Let \autoref{ass:1} hold.
\begin{enumerate} \item Then the mapping $ \mebr_{[\dKt]} : \Lp<2> \times \Lp<2> \rarr [0, +\infty]$ satisfies the metric axioms.
\item \label{itm: ExpIsOp} Let, in addition, \autoref{ass:2} hold, assume that $v \in \Lp<2>$ and that both metrics $d_i$, $i=1,2$,
are equivalent to $\dRMi|_{K_i \times K_i}$, respectively. Then the functional $\F<v><\alpha>[\dKt, \dKo]$ does not attain the value $+\infty$ on its domain $\W<1> \neq \emptyset$. \end{enumerate} \end{proposition}
\begin{proof} \begin{enumerate} \item The axioms of non-negativity, identity of indiscernibles and symmetry are fulfilled by $\mebr_{[\dKt]}$
since $\dKt$ is a metric. To prove the triangle inequality let $\phi,\xi,\nu \in L^{p_2}(\Omega_2, K_2)$.
In the main case $\mebr[\phi][\nu]_{[\dKt]}^{p_2} \in (0, \infty)$ Hölder's inequality yields
\begin{align*}
\mebr[\phi][\nu]_{[\dKt]}^{p_2} ={}&
\int\limits_{\Omega_2} \dKt\big(\phi(x),\nu(x) \big) \dKt^{p_2-1}\big( \phi(x),\nu(x) \big) \,\mathrm{d}x \\
\leq{}& \int\limits_{\Omega_2} \dKt\big( \phi(x),\xi(x) \big) \dKt^{p_2-1}\big( \phi(x),\nu(x) \big) \,\mathrm{d}x
+ \int\limits_{\Omega_2} \dKt\big( \xi(x),\nu(x) \big) \dKt^{p_2-1} \big( \phi(x),\nu(x) \big) \,\mathrm{d}x \\
\leq{}&
\left( \int\limits_{\Omega_2} \dKt^{p_2} \big(\phi(x),\xi(x) \big) \,\mathrm{d}x \right)^{\frac{1}{p_2}}
\left( \int\limits_{\Omega_2} \dKt^{p_2}\big( \phi(x),\nu(x) \big) \,\mathrm{d}x \right)^{\frac{p_2-1}{p_2}}\\
&+ \left( \int\limits_{\Omega_2} \dKt^{p_2} \big(\xi(x),\nu(x) \big) \,\mathrm{d}x \right)^{\frac{1}{p_2}}
\left( \int\limits_{\Omega_2} \dKt^{p_2}\big( \phi(x),\nu(x) \big) \,\mathrm{d}x \right)^{\frac{p_2-1}{p_2}}\\
={}& \left( \mebr[\phi][\xi]_{[\dKt]} + \mebr[\xi][\nu]_{[\dKt]} \right)
\mebr[\phi][\nu]_{[\dKt]}^{p_2-1},
\end{align*}
meaning
\begin{gather*}
\mebr[\phi][\nu]_{[\dKt]} \leq \mebr[\phi][\xi]_{[\dKt]} + \mebr[\xi][\nu]_{[\dKt]}.
\end{gather*}
If $\mebr[\phi][\nu]_{[\dKt]} = 0$ the triangle inequality is trivially fulfilled.
In the remaining case $\mebr[\phi][\nu]_{[\dKt]} = \infty$ applying the estimate $(a+b)^p \leq 2^{p-1} (a^p + b^p)$,
see e.g. \cite[Lemma 3.20]{SchGraGroHalLen09},
to $a = \dKt(\phi(x), \xi(x)) \geq 0$ and $b = \dKt(\xi(x), \nu(x)) \geq 0$ yields
\begin{gather*}
\mebr[\phi][\nu]_{[\dKt]}^{p_2} \leq 2^{p_2-1} \big( \mebr[\phi][\xi]_{[\dKt]}^{p_2} + \mebr[\xi][\nu]_{[\dKt]}^{p_2} \big),
\end{gather*}
implying the desired result. \item We emphasize that $\W<1> \neq \emptyset$ because every constant function $w(\cdot) = a \in K_1$ belongs to $\Wsp<1>$
for $p_1 \in (1, \infty)$ and $s \in (0,1]$ as well as to $\BV<1>$ for $p_1 = 1$ and $s = 1$.
Assume now that the metrics $d_i$ are equivalent to $\dRMi|_{K_i \times K_i}$ for $i=1$ and $i=2$, respectively,
so that we have an upper bound $d_i \leq C \dRMi|_{K_i \times K_i}$.
We need to prove that $\F<v><\alpha>[\dKt, \dKo](w) < \infty$ for every $w \in \W<1>$.
Due to $\mebr[\phi][\nu]^{p_2}_{[\dKt]} \leq C^{p_2} \norm{\phi - \nu}^{p_2}_{\Lp<2>[][\mathds{R}^{M_2}]} < \infty$
for all $\phi, \nu \in \Lp<2> \subseteq \Lp<2>[][\mathds{R}^{M_2}]$ it is sufficient to show
$\mathcal{R}_{[\dKo]}(w) < +\infty$ for all $w \in \W<1>$. \begin{itemize}
\item For $\W<1> = \BV<1>$ this is guaranteed by \cite[Theorem 1.2]{Pon04b}.
\item For $\W<1> = W^{1,p_1}(\Omega_1, K_1)$ by \cite[Theorem 1]{BouBreMir01}.
\item For $\W<1> = \Wsp<1>$, $s \in (0,1)$, we distinguish between two cases. \\
If $\normN[x-y]< 1$ we have that $\frac{1}{\normN[x-y]^{k+p_1 s}} \leq \frac{1}{\normN[x-y]^{N+p_1 s}}$ for $k \leq N$ and hence
\begin{gather*}\int\limits_{\begin{smallmatrix}
(x,y) \in \Omega_1 \times \Omega_1 \\ \normN[x-y]< 1
\end{smallmatrix}} \frac{\dKo^{p_1}(w(x), w(y))}{\normN[x-y]^{k+p_1 s}} \rho^l(x-y) \,\mathrm{d}(x,y)
\leq C^{p_1} \norm{\rho^l}_{\infty} \abs{w}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})}^{p_1} < \infty\;.
\end{gather*}
If $\normN[x-y]\geq 1$ we can estimate
\begin{gather*}\int\limits_{\begin{smallmatrix}
(x,y) \in \Omega_1 \times \Omega_1 \\ \normN[x-y]\geq 1
\end{smallmatrix}} \frac{\dKo^{p_1}(w(x), w(y))}{\normN[x-y]^{k+p_1 s}} \rho^l(x-y) \,\mathrm{d}(x,y)
\leq C^{p_1} \norm{\rho^l}_{\infty} 2^{p_1} |\Omega_1| \norm{w}^{p_1}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})} < \infty\;.
\end{gather*}
In summary adding yields $\mathcal{R}_{[\dKo]}(w) < +\infty$.
\end{itemize} \end{enumerate} \end{proof}
\section{Existence} \label{sec: Existence}
In order to prove existence of a minimizer of the functional $\F$ we apply the Direct Method in the Calculus of Variations (see e.g. \cite{Dac82,Dac89}). To this end we verify continuity properties of $\mebr_{[\dKt]}$ and $\mathcal{R}_{[\dKo]}$, resp. $\F[\dKt, \dKo]$ and apply them along with the sequential closedness of $\W<1>$, already proven in \autoref{lem:Wsp_weakly_seq_closed_etc}.
In this context we point out some setting assumptions and their consequences on $\F$, resp. $\mebr$ and $\mathcal{R}$ in the following remark. For simplicity we assume $p \vcentcolon= p_1 = p_2 \in (1, \infty)$, $\Omega \vcentcolon= \Omega_1 = \Omega_2$ and $(K, \dK) \vcentcolon= (K_1, \dKo) = (K_2, \dKt)$. \begin{remark} \label{re:tricks} ~
\begin{itemize}[topsep=0pt]
\item
The continuity of $\dK$ with respect to $\dRM|_{K \times K}$ guarantees lower semicontinuity of $\mebr_{[\dK]}$ and $\mathcal{R}_{[\dK]}$.
\item
The inequality $\dRM|_{K \times K} \leq \dK$ carries over to the inequalities
$\norm{\widetilde v - v}_{\Lp} \leq \mebr[\widetilde v][v]_{[\dK]}$ for all $\widetilde v, v \in \Lp[K]$,
and $|w|_{\W} \leq \mathcal{R}_{[\dK]}(w)$ for all $w \in \W[K]$, allowing to transfer
properties like coercivity from $\F[\dRM,\dRM]$ to $\F[\dK,\dK]$.
Moreover, the extended real-valued metric space $(\Lp[K], \mebr_{[\dK]})$ stays
related to the linear space $(\Lp, \norm{\cdot}_{\Lp})$ in terms of the topology and bornology induced by $\mebr$,
resp. those inherited by
$\norm{\cdot}_{\Lp}$.
\item
The closedness of $K \subseteq \mathds{R}^M$ is crucial in showing that $\W[K]$ is a sequentially closed subset of the linear space $\W$.
This closedness property acts as a kind of replacement for the, a priori not available, notion of completeness with respect to the
``space'' $(\W[K], \mebr, \mathcal{R})$.
\end{itemize}
For $l=0$, $k=N$ note in the latter item that
equipping $\W[K]$ with $\mebr_{[\dKt]}$ and $\mathcal{R}_{[\dKo]}$ does not even lead to an (extended real-valued) metric space,
in contrast to the classical case $(K,\dK) = (\mathds{R}^M, \dRM)$. \end{remark}
We will use the following assumption:
\begin{assumption} \label{as:Setting}
Let \autoref{ass:1} hold, $v^0 \in \Lp<2>$ and let $\W<1>$ and the associated topology be as defined in \autoref{eq:ChooseW}.
In addition we assume:
\begin{itemize}
\item $\op: \W<1> \to \Lp<2>$ is well--defined and sequentially continuous with respect to the specified topology on $\W<1>$ and
\item For every $t > 0$ and $\alpha > 0$ the level sets
\begin{equation}\label{itm: A}
\text{level}_t(\F<v^0><\alpha>[\dKt, \dKo]) \vcentcolon= \{ w \in \W<1> \ : \ \F<v^0><\alpha>[\dKt,\dKo] \leq t \}
\end{equation}
are sequentially pre-compact subsets of $W(\Omega_1, \mathds{R}^{M_1})$.
\item There exists a $\bar{t} > 0$ such that $\text{level}_{\bar{t}}(\F<v^0><\alpha>[\dKt, \dKo])$ is nonempty.
\item Only those $v \in \Lp<2>$ are considered which additionally fulfill $\mebr[v][v^0]_{[\dKt]} < \infty$.
\end{itemize} \end{assumption}
\begin{remark}
The third condition is sufficient to guarantee $\F<v^0><\alpha>[\dKt, \dKo]) \not \equiv \infty$. In contrast the condition $v^0 \in \Lp<2>$, cf.
\autoref{def:functional}, might not be sufficient if $d_2$ is not equivalent to $\dRMt|_{K_2 \times K_2}$. \end{remark}
\begin{lemma} \label{thm:F_and_its_summands_are_seq_weakly_closed}
Let \autoref{as:Setting} hold.
Then the mappings $\mebr_{[\dKt]}$, $\mathcal{R}_{[\dKo]}$ and $\F[\dKt, \dKo]$ have the following continuity properties:
\begin{enumerate}
\item \label{enu:continuity_of_mebr}
The mapping $\mebr_{[\dKt]}: \Lp<2> \times \Lp<2> \rarr [0, +\infty]$ is sequentially lower semi-continuous,
i.e. whenever sequences $\seq{\phi}$, $\seq{\nu}$ in $\Lp<2>$ converge to $\phi_* \in \Lp<2>$ and
$\nu_*\in \Lp<2>$, respectively, we have $\mebr[\phi_*][\nu_*]_{[\dKt]} \leq \limi \limits \mebr[\phi_n][\nu_n]_{[\dKt]}$.
\item \label{enu:seq_lscty_of_R}
The functional $\mathcal{R}_{[\dKo]}: \W<1> \rarr [0,\infty]$ is sequentially lower semi-continuous, i.e. whenever a
sequence $(w_n)_{n \in \mathds{N}}$ in $\W<1>$ converges to some $w_* \in \W<1>$ we have
\begin{equation*}
\mathcal{R}_{[\dKo]}(w_*) \leq \limi \mathcal{R}_{[\dKo]}(w_n).
\end{equation*}
\item \label{enu:seq_lscty_of_F}
The functional $\F[\dKt,\dKo]: W(\Omega_1, K_1) \rarr [0,\infty]$ is sequentially lower semi-continuous.
\end{enumerate} \end{lemma}
\begin{proof}
\begin{enumerate}
\item \label{enu:lsc_of_mebr}
It is sufficient to show that for \emph{every} pair of sequences $\seq{\phi}$, $\seq{\nu}$ in $\Lp<2>$ which
converge to previously \emph{fixed} elements $\phi_* \in \Lp<2>$ and $\nu_*\in \Lp<2>$, respectively, we can extract subsequences
$(\phi_{n_j})_{j \in \mathds{N}}$ and $(\nu_{n_j})_{j \in \mathds{N}}$, respectively, with
\begin{gather*}
\mebr[\phi_*][\nu_*]_{[\dKt]} \leq \liminf_{j \rarr \infty} \mebr[\phi_{n_j}][\nu_{n_j}]_{[\dKt]}.
\end{gather*}
To this end let $(\phi_n)_{n \in \mathds{N}},(\nu_n)_{n \in \mathds{N}}$ be some sequences in $\Lp<2>$ with $\phi_n \rarr \phi_*$ and $\nu_n \rarr \nu_*$ in $\Lp<2>$.
\autoref{lem:Wsp_weakly_seq_closed_etc} ensures that there exist subsequences $(\phi_{n_j})_{j \in \mathds{N}}, (\nu_{n_j})_{j \in \mathds{N}}$
converging to $\phi_*$ and $\nu_*$ pointwise almost everywhere,
which in turn implies $\big(\phi_{n_j}(\cdot), \nu_{n_j}(\cdot) \big) \to \big( \phi_*(\cdot), \nu_*(\cdot) \big)$
pointwise almost everywhere. Therefrom, together with the continuity of $\dKt: K_2 \times K_2 \to [0, \infty)$
with respect to $\dRMt$, cf. \autoref{sec: Setting}, we obtain
by using the quadrangle inequality
that
\begin{gather*}
| \dKt(\phi_{n_j}(x), \nu_{n_j}(x)) - \dKt(\phi_*(x), \nu_*(x)) | \leq \dKt(\phi_{n_j}(x), \phi_*(x)) + \dKt(\nu_{n_j}(x), \nu_*(x)) \rarr 0,
\end{gather*}
and hence
\begin{gather*}
\dKt^{p_2}\big( \phi_{n_j}(x), \nu_{n_j}(x) \big) \to \dKt^{p_2} \big( \phi_*(x),\nu_*(x) \big) \text{ for almost every }
x \in \Omega_2.
\end{gather*}
Applying Fatou's lemma we obtain
\begin{gather*}
\mebr[\phi_*][\nu_*]_{[\dKt]} = \int_{\Omega_2 }\limits \dKt^{p_2}( \phi_*(x),\nu_*(x) ) \,\mathrm{d}x
\leq \liminf_{j \rarr \infty} \int_{\Omega_2}\limits \dKt^{p_2} ( \phi_{n_j}(x), \nu_{n_j}(x) ) \,\mathrm{d}x
= \liminf_{j \rarr \infty} \mebr[\phi_{n_j}][\nu_{n_j}]_{[\dKt]}.
\end{gather*}
\item
Let $(w_n)_{n \in \mathds{N}}$ be a sequence in $\W<1>$ with $w_n \rarr<W> w_*$ as $n \rarr \infty$.
By \autoref{lem:Wsp_weakly_seq_closed_etc} there is a subsequence
$(w_{n_j})_{j \in \mathds{N}}$ which converges to $w_*$ both in $\Lp<1>$ and pointwise almost everywhere.
This further implies that
\begin{gather*}
\dKo^{p_1}\big( w_{n_j}(x), w_{n_j}(y) \big) \to \dKo^{p_1} \big( w_*(x),w_*(y) \big)
\end{gather*}
for almost every
\begin{equation}
\label{eq:A}
(x,y) \in \Omega_1 \times \Omega_1 \supseteq \{(x,y) \in \Omega_1 \times \Omega_1 : x \neq y \} =\vcentcolon A.
\end{equation}
Defining
\begin{equation*}
f_j(x,y) \vcentcolon= \left\{ \begin{array}{rcl}
\frac{\dKo^{p_1}(w_{n_j}(x), w_{n_j}(y)) }{\normN[x-y]^{k+ps}} \rho^l(x-y) & \text{ for } &
(x,y) \in A,\\
0 & \text{ for } &
(x,y) \in (\Omega_1 \times \Omega_1) \setminus A,\\
\end{array} \right. \quad \text{ for all } j \in \mathds{N}.
\end{equation*}
and
\begin{equation*}
f_*(x,y) \vcentcolon= \left\{ \begin{array}{ccl}
\frac{\dKo^{p_1}(w_*(x), w_*(y)) }{\normN[x-y]^{k+ps}} \rho^l(x-y) & \text{ for } &
(x,y) \in A,\\
0 & \text{ for } &
(x,y) \in (\Omega_1 \times \Omega_1) \setminus A\\
\end{array} \right.
\end{equation*}
we thus have $f_*(x,y) = \lim_{j \rarr \infty} f_j(x,y)$
for almost every $(x,y) \in \Omega_1 \times \Omega_1$.
Applying Fatou's lemma to the functions $f_j$ yields the assertion, due to the same reduction as in the proof of the first part.
\item
It is sufficient to prove that the components $\mathcal{G}(\cdot) = \mebr[F(\cdot)][v]_{[\dKt]}$ and $\mathcal{R} = \mathcal{R}_{[\dKo]}$ of
$\F[\dKo,\dKt] = \mathcal{G} + \alpha \mathcal{R}$ are sequentially lower semi-continuous.
To prove that $\mathcal{G}$ is sequentially lower semi-continuous in every $w_* \in W(\Omega_1, K_1)$
let $(w_n)_{n \in \mathds{N}}$ be a sequence in $W(\Omega_1, K_1)$ with $w_n \rarr<W> w_*$ as $n \rarr \infty$.
\autoref{as:Setting}, ensuring the sequential continuity of $\op: \W<1> \to \Lp<2>$, implies hence $\op[w_n] \rarr \op[w_*]$ in $\Lp<2>$
as $n \rarr \infty$.
By \autoref{enu:continuity_of_mebr} we thus obtain $\mathcal{G}(w_*) = \mebr[\op[w_*]][v] \leq \limi \mebr[\op[w_n]][v] = \limi \mathcal{G}(w_n)$. \\
$\mathcal{R}$ is sequentially lower semi-continuous by \autoref{enu:seq_lscty_of_R}.
\end{enumerate} \end{proof}
\subsection{Existence of minimizers}
The proof of
the existence of a
minimizer of $\F[\dKt, \dKo]$ is along the lines of the proof in \cite{SchGraGroHalLen09}, taking into account \autoref{re:tricks} We will need the following useful lemma, cf. \cite{SchGraGroHalLen09}, which links $\text{level}_t(\F<v^0><\alpha>)$ and $\text{level}_t(\F<v><\alpha>)$ for $\mebr[v][v^0] < \infty$.
\begin{lemma}\label{lem:Ineq_F_parameterchange_to_other_v}
It holds
\begin{align*}
\F<v_\star><>[\dKt, \dKo](w) & \leq 2^{p_2-1} \F<v_\diamond><>[\dKt, \dKo](w) + 2^{p_2 -1} \mebr[v_\diamond][v_\star]_{[\dKt]}^{p_2}
\end{align*}
for every
$w \in W(\Omega_1, K_1)$ and $v_\star, v_\diamond \in L^{p_2}(\Omega_2, K_2)$. \end{lemma}
\begin{proof}
Using the fact that for $p \geq 1$ we have that $|a+b|^p \leq 2^{p-1}(|a|^p + |b|^p), \ a,b \in \mathds{R} \cup \{\infty\}$ and that $\mebr_{[\dKt]}$ fulfills the triangle inequality we obtain
\begin{align*}
\F<v_\star><>[\dKt, \dKo](w)
& = \mebr[\op[w]][v_\star]_{[\dKt]}^{p_2} + \alpha \mathcal{R}_{[\dKo]}(w) \\
& \leq 2^{p_2-1} \big( \mebr[\op[w]][v_\diamond]_{[\dKt]}^{p_2} + \mebr[v_\diamond][v_\star]_{[\dKt]}^{p_2} \big) + \alpha \mathcal{R}_{[\dKo]}(w) \\
& \leq 2^{p_2-1}\big( \F<v_\diamond><>[\dKt, \dKo](w) + \mebr[v_\diamond][v_\star]_{[\dKt]}^{p_2} \big).
\end{align*} \end{proof}
\begin{thm} \label{thm:F_dK_has_a_minimizer} Let \autoref{as:Setting} hold. Then the functional $\F<v><\alpha>[\dKt, \dKo]: W(\Omega_1, K_1) \rarr [0, \infty]$ attains a minimizer. \end{thm}
\begin{proof}
We prove the existence of a minimizer via the Direct Method.
We shortly write $\F^v$ for $\F<v><\alpha>[\dKt, \dKo]$.
Let $(w_n)_{n \in \mathds{N}}$ be a sequence in $W(\Omega_1, K_1)$ with
\begin{gather}\label{eq:w_n_is_minimizing_seq}
\lim_{n \rarr \infty }\F^v(w_n) = \inf_{w \in W(\Omega_1, K_1)} \F^v(w).
\end{gather}
The latter infimum is not $+\infty$, because $\F^v \equiv +\infty$ would imply also $\F^{v^0} \equiv +\infty$ due to
\autoref{lem:Ineq_F_parameterchange_to_other_v}, violating \autoref{as:Setting}.
In particular there is some $c \in \mathds{R}$ such that
$\F^v(w_n) \leq c$ for every $n \in \mathds{N}$.
Applying \autoref{lem:Ineq_F_parameterchange_to_other_v} yields
$\F^{v^0}(w_n) \leq 2^{p_2-1} \big( \F^{v}(w_n) + \mebr[v][v^0] \big)\leq 2^{p_2-1} \big( c + \mebr[v][v^0] \big) =\vcentcolon\tilde{c} < \infty$ due to \autoref{as:Setting}.
Since the level set $\text{level}_{\tilde{c}}(\F^{v^0})$ is sequentially pre-compact with respect to the topology given to $W(\Omega_1, \mathds{R}^{M_1})$ we get the existence of a
subsequence $(w_{n_k})_{k \in \mathds{N}}$ which converges to some $w_* \in W(\Omega_1, \mathds{R}^{M_1})$,
where actually $w_* \in W(\Omega_1, K_1)$ due to \autoref{lem:Wsp_weakly_seq_closed_etc}.
Because $\F^v$ is sequentially lower semi-continuous, see \autoref{thm:F_and_its_summands_are_seq_weakly_closed}, we have
$\F^v(w_*) \leq \liminf_{k \rarr \infty} \F^v(w_{n_k})$. Combining this with
\autoref{eq:w_n_is_minimizing_seq} we obtain
\begin{gather*}
\inf_{w \in W(\Omega_1, K_1)} \F^v(w) \leq \F^v(w_*) \leq \liminf_{k \rarr \infty} \F^v(w_{n_k})
= \lim_{n\rarr \infty} \F^v(w_n) = \inf_{w \in W(\Omega_1, K_1)} \F^v(w).
\end{gather*}
In particular $\F^v(w_*) = \inf_{w \in W(\Omega_1, K_1)}\limits \F^v(w)$, meaning that $w_*$ is a minimizer of $\F^v$. \end{proof}
In the following we investigate two examples, which are relevant for the numerical examples in \autoref{sec:Numerical_results}. \begin{example} \label{ex:coercive} We consider that $W(\Omega_1,K_1) = W^{s, p_1}(\Omega_1, K_1)$ with $p_1>1, \ 0 < s < 1$ and fix $k = N$.
If the operator $\op$ is norm-coercive in the sense that the implication
\begin{equation} \label{eq:coercive_F}
\norm{w_n}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})} \rarr +\infty \Rightarrow \norm{\op[w_n]}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \rarr +\infty
\end{equation}
holds true for every sequence $(w_n)_{n \in \mathds{N}}$ in $W^{s,p_1}(\Omega_1, K_1)\subseteq W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})$, then the functional
\begin{equation*}
\F[\dKt, \dKo] = \mebr[\op[w]][v]^{p_2}_{[\dKt]} + \alpha \mathcal{R}_{[\dKo]}(w): W^{s,p_1}(\Omega_1, K_1) \rarr [0, \infty]
\end{equation*}
is coercive. This can be seen as follows:
The inequality between $\dKo$ and $\dRMo|_{K_1 \times K_1}$ resp. $\dKt$ and $\dRMt|_{K_2 \times K_2}$, see \autoref{ass:1}, carries over to
$\F[\dKt,\dKo]$ and $\F[\dRMt|_{K_2 \times K_2},\dRMo|_{K_1 \times K_1}]$, i.e.
\begin{gather*}
\F[\dKt, \dKo] (w) \geq \F\big[\dRMt|_{K_2 \times K_2},\dRMo|_{K_1 \times K_1}\big](w) \text{ for all } w \in W^{s,p_1}(\Omega_1, K_1).
\end{gather*}
Thus it is sufficient to show that $\F[\dRMt|_{K_2 \times K_2},\dRMo|_{K_1 \times K_1}]: W^{s,p_1}(\Omega_1, K_1) \rarr [0, \infty]$ is coercive:
To prove this we write shortly $\F$ instead of $\F[\dRMt|_{K_2 \times K_2},\dRMo|_{K_1 \times K_1}]$ and consider sequences
$(w_n)_{n \in \mathds{N}}$ in $W^{s,p_1}(\Omega_1, K_1)$ with $\norm{w_n}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})} \rarr +\infty$ as $n \rarr \infty$. We
show that $\F(w_n) \rarr +\infty$, as $n \rarr \infty$.
Since
\begin{equation*}
\norm{w_n}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})} = \big( \norm{w_n}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})}^{p_1} + \abs{w_n}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})}^{p_1} \big)^{\frac{1}{p_1}}
\end{equation*}
the two main cases to be considered are $\norm{w_n}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})} \rarr +\infty$ and
$\abs{w_n}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})} \rarr +\infty$.
\begin{enumerate}[label=\textbf{Case \arabic*}]
\item \label{ex: coecivity_case1}
$\norm{w_n}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})} \rarr +\infty$. \\
The inverse triangle inequality and the norm-coercivity of $\op$, \autoref{eq:coercive_F}, give
$\norm{\op[w_n] - v}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \geq \norm{\op[w_n]}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} - \norm{v}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \rarr +\infty$.
Therefore also
\begin{equation*}
\F(w_n) = \norm{\op[w_n] - v}^{p_2}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})}
+ \alpha \int\limits_{\Omega_1\times \Omega_1} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho^l(x-y) \,\mathrm{d}(x,y)
\rarr +\infty.
\end{equation*}
\noindent
\item \label{ex: coecivity_case2}
$\abs{w_n}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})} \rarr +\infty$. \\
If $l=0$, then $\mathcal{R}_{[\dKo]}$ is exactly the $W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})$-semi-norm $|w|_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})}$ and we trivially
get the desired result.
Hence we assume from now on that $l = 1$.
The assumptions on $\rho$ ensure that there exists a $\tau > 0$ and $\eta_{\tau} > 0$ such that
\begin{align*}
\mathcal{S}_{\tau} \vcentcolon= {}& \{(x,y) \in \Omega_1 \times \Omega_1 : \rho(x-y) \geq \tau \}
\\
= {}& \{(x,y) \in \Omega_1 \times \Omega_1 : \normN[x-y] \leq \eta_{\tau} \},
\end{align*}
cf. \autoref{fig:stripe}.
Splitting $\Omega_1 \times \Omega_1$ into $\mathcal{S}_{\tau} =\vcentcolon \mathcal{S}$ and its complement
$(\Omega_1 \times \Omega_1) \setminus \mathcal{S}_{\tau} =\vcentcolon \mathcal{S}^c$
we accordingly split the integrals
$\abs{w_n}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})} = \int\limits_{\Omega_1 \times \Omega_1} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y)$
and consider again two cases
$\int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y) \rarr +\infty$ and
$\int\limits_{\mathcal{S}^c} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y) \rarr +\infty$, respectively.
\begin{figure}
\caption{The stripe $\mathcal{S} = \mathcal{S}_{\tau}$ if $\Omega_1$ is an open interval and its connection to the radial mollifier $\rho$ for fixed $y \in \Omega_1$.}
\label{fig:stripe}
\end{figure}
\begin{enumerate}[label=\textbf{\ref{ex: coecivity_case2}.\arabic*}]
\item
$\int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y) \rarr + \infty$. \\
By definition of $\mathcal{S}$ we have $\rho(x-y) \geq \tau > 0$ for all $(x,y) \in \mathcal{S}$.
Therefore
\begin{gather*}
\int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y)
\geq \tau \int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y)
\rarr +\infty.
\end{gather*}
Since $\alpha > 0$, it follows
\begin{align*}
\F(w_n) &= \norm{\op(w_n) - v}^{p_2}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})}
+ \underbrace{\alpha \int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y)
}_{\rarr +\infty} \\
& + \underbrace{ \alpha \int\limits_{\mathcal{S}^c} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y)
}_{\geq 0}
\rarr +\infty.
\end{align*}
\noindent
\item \label{ex: coecivity_case22}
$\int\limits_{\mathcal{S}^c} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y) \rarr + \infty$.\\
For $(x, y) \in \mathcal{S}^c$ it might happen that $\rho(x-y) = 0$, and thus instead of proving
$\F(w_n) \geq \int\limits_{\mathcal{S}^c} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y) \rarr +\infty$,
as in Case 2.1, we rather show that $\F(w_n) \geq \norm{\op[w_n] - v}^{p_2}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \rarr +\infty$.
For this it is sufficient to show that for every $c > 0$ there is some $C \in \mathds{R}$ such that the implication
\begin{gather*}
\norm{\op[w]-v}^{p_2}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})}\leq c
\implies
\int\limits_{\mathcal{S}^c} \frac{\normMo[w(x) - w(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y) \leq C,
\end{gather*}
holds true for all $w \in W^{s,p_1}(\Omega_1, K_1) \subseteq W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})$.
To this end let $c > 0$ be given and consider an arbitrarily chosen $w \in W^{s,p_1}(\Omega_1, K_1)$
fulfilling $\norm{\op[w] - v}^{p_2}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \leq c$.
Then $\norm{\op[w] - v}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \leq \sqrt[p_2]{c}$. Using the triangle inequality and the monotonicity
of the function $h: t \mapsto t^{p_2}$ on $[0, +\infty)$ we get further
\begin{align}\label{eq: Norm_estimate}
\norm{\op[w]}^{p_2}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})}
&= \norm{\op[w] - v + v}^{p_2}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \nonumber\\
&\leq \left( \norm{\op[w] - v}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} + \norm{v}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \right)^{p_2} \nonumber\\
& \leq \big( \sqrt[p_2]{c} + \norm{v}_{L^{p_2}(\Omega_2, \mathds{R}^{M_2})} \big)^{p_2} =\vcentcolon \tilde{c}.
\end{align}
\noindent
Due to the norm-coercivity, it thus follows
that $\norm{w}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})} \leq \bar{c}$, $\bar{c}$ some constant.
Using \cite[Lemma 3.20]{SchGraGroHalLen09} it then follows that
\begin{gather}\label{eq: Convexity_Inequality}
\normMo[w(x) - w(y)]^{p_1} \leq 2^{p_1-1} \normMo[w(x)]^{p_1} + 2^{p_1-1} \normMo[w(y)]^{p_1}
\end{gather}
for all $(x,y) \in \Omega_1 \times \Omega_1$.
Using \autoref{eq: Convexity_Inequality}, Fubini's Theorem and \autoref{eq: Norm_estimate} we obtain
\begin{align*}
\int\limits_{\Omega_1 \times \Omega_1} \normMo[w(x) - w(y)]^{p_1} \,\mathrm{d}(x,y)
& \leq \int\limits_{\Omega_1 \times \Omega_1} 2^{p_1-1} \normMo[w(x)]^{p_1} + 2^{p_1-1}\normMo[w(y)]^{p_1} \,\mathrm{d}(x,y) \\
&= \abs{\Omega_1} \int\limits_{\Omega_1} 2^{p_1-1} \normMo[w(x)]^{p_1} \,\mathrm{d}x + \abs{\Omega_1} \int\limits_{\Omega_1} 2^{p_1-1}\normMo[w(y)]^ {p_1} \,\mathrm{d}y \\
&= 2\abs{\Omega_1} \int\limits_{\Omega_1} 2^{p_1-1} \normMo[w(x)]^{p_1} \,\mathrm{d}x \\
& = 2^{p_1} \abs{\Omega_1} \; \norm{w}^{p_1}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})} \leq 2^{p_1} \abs{\Omega_1} \bar{c}^{p_1}.
\end{align*}
Combining $\normN[x-y] \geq \eta_{\tau} > 0$ for all $(x,y) \in \mathcal{S}^c$ with the previous inequality we obtain the needed estimate
\begin{align*}
\int\limits_{\mathcal{S}^c} \frac{\normMo[w(x) - w(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y)
& \leq \frac{1}{\eta_{\tau}^{N+p_1 s}} \int\limits_{\mathcal{S}^c} \normMo[w(x) - w(y)]^{p_1} \,\mathrm{d}(x,y)
\\
& \leq \frac{1}{\eta_{\tau}^{N+p_1 s}} \int\limits_{\Omega_1 \times \Omega_1} \normMo[w(x) - w(y)]^{p_1} \,\mathrm{d}(x,y)
\\
& \leq \frac{2^{p_1} \abs{\Omega_1} \bar{c}^{p_1}}{\eta_{\tau}^{N+p_1 s}} =\vcentcolon C.
\end{align*}
\end{enumerate}
\end{enumerate}
\end{example}
The second example concerns the coercivity of $\F[\dKt,\dKo]$, defined in \autoref{eq:functional}, when
$\op$ denotes the \emph{masking operator} occurring in image inpainting. To prove this result we require the following auxiliary lemma:
\begin{lemma}\label{lem. auxLemma}
There exists a constant $C \in \mathds{R} $ such that for all $w \in W^{s,p_1}(\Omega_1, \mathds{R}^{M_1}), \ 0<s < 1, \ l \in \{0,1\}, \ 1 < p_1 < \infty$ and
$D \subsetneq \Omega_1$ nonempty such that
\begin{equation} \label{eq: coercivity_inpainting}
\norm{w}_{L^{p_1}(D, \mathds{R}^{M_1})}^{p_1} \leq C \left( \norm{w}_{L^{p_1}(\Omega_1 \setminus D, \mathds{R}^{M_1})}^{p_1} +
\int\limits_{\Omega_1 \times \Omega_1} \frac{\normMo[w(x) - w(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho^l(x-y) \,\mathrm{d}(x,y) \right).
\end{equation}
\end{lemma}
\begin{proof}
The proof is inspired by the proof of Poincaré's inequality in \cite{Eva10}. It is included here for the sake of completeness.
Assume first that $l=1$. Let $\mathcal{S}$ be as above,
\begin{align*}
\mathcal{S} \vcentcolon= {}& \{(x,y) \in \Omega_1 \times \Omega_1 : \rho(x-y) \geq \tau \}
\\
= {}& \{(x,y) \in \Omega_1 \times \Omega_1 : \normN[x-y] \leq \eta \}.
\end{align*}
If the stated inequality \autoref{eq: coercivity_inpainting} would be false, then for every $n \in \mathds{N}$ there would exists a
function $w_n \in W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})$ satisfying
\begin{equation}\label{eq: inpaintingContraEquation}
\norm{w_n}_{L^{p_1}(D, \mathds{R}^{M_1})}^{p_1} \geq n \big( \norm{w_n}_{L^{p_1}(\Omega_1 \setminus D, \mathds{R}^{M_1})}^{p_1} +
\int\limits_{\Omega_1 \times \Omega_1} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y) \big).
\end{equation}
By normalizing we can assume without loss of generality
\begin{enumerate}
\item $\norm{w_n}_{L^{p_1}(D, \mathds{R}^{M_1})}^{p_1} = 1$. \label{itm: inpainting1}
\end{enumerate}
Moreover, by \autoref{eq: inpaintingContraEquation}
\begin{enumerate}
\setcounter{enumi}{1}
\item $\norm{w_n}_{L^{p_1}(\Omega_1 \setminus D, \mathds{R}^{M_1})}^{p_1} < \frac{1}{n}$, \label{itm: inpainting2}
\item $\int\limits_{\Omega_1 \times \Omega_1} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y) < \frac{1}{n}$. \label{itm: inpainting3}
\end{enumerate}
By \autoref{itm: inpainting1} and \autoref{itm: inpainting2} we get that
$ \norm{w_n}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})}^{p_1} = \norm{w_n}_{L^{p_1}(D, \mathds{R}^{M_1})}^{p_1} + \norm{w_n}_{L^{p_1}(\Omega_1 \setminus D, \mathds{R}^{M_1})}^{p_1} < 1 + \frac{1}{n} < 2 $
is bounded. Moreover
\begin{align*}
\abs{w_n}^{p_1}_{W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})}
& = \int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y)
+ \int\limits_{\mathcal{S}^c} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \,\mathrm{d}(x,y) \\
& \leq \frac{1}{\tau} \int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y)
+ \frac{2^p_1 \abs{\Omega_1}}{\eta^{N+p_1 s}}\norm{w_n}^{p_1}_{L^{p_1}(\Omega_1, \mathds{R}^{M_1})} \\
& < \frac{1}{\tau n} + \frac{2^{p_1+1} \abs{\Omega_1}}{\eta^{N+p_1 s}}
\leq \frac{1}{\tau} + \frac{2^{p_1+1} \abs{\Omega_1}}{\eta^{N+p_1 s}} =\vcentcolon c < \infty,
\end{align*}
where $c$ is independent of $n$.
This yields that the sequence
$(w_n)_{n \in \mathds{N}}$ is bounded in $W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})$ by $(2 + c)^{\frac{1}{p_1}}$.
By the reflexivity of $\Wsp<1>[][\mathds{R}^{M_1}]$ for $p_1 \in (1, \infty)$ and \autoref{lem:Wsp_weakly_seq_closed_etc}
there exists a subsequence $(w_{n_k})_{k \in \mathds{N}}$ of $(w_n)_{n \in \mathds{N}}$
and $w_* \in W^{s,p_1}(\Omega_1, \mathds{R}^{M_1})$
such that $w_{n_k} \rarr w^*$ strongly in $L^{p_1}(\Omega_1, \mathds{R}^{M_1})$ and
pointwise almost everywhere.
Using the continuity of the norm and dominated convergence we obtain
\begin{enumerate}
\item $\norm{w^*}_{L^{p_1}(D, \mathds{R}^{M_1})}^{p_1} = 1$, in particular $w^*$ is not the null-function on D,
\item $\norm{w^*}_{L^{p_1}(\Omega_1 \setminus D, \mathds{R}^{M_1})}^{p_1} = 0$ since $n \in \mathds{N}$ is arbitrary and hence $w^* \equiv 0$ on $\Omega_1 \setminus D$.
\item \begin{equation*}
\liminf_{n \rarr \infty} \frac{1}{n}
> \liminf_{n \rarr \infty} \int\limits_{\mathcal{S}} \frac{\normMo[w_n(x) - w_n(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho(x-y) \,\mathrm{d}(x,y)
\geq \frac{\tau}{\eta^{N+p_1 s}} \int\limits_{\mathcal{S}} \normMo[w^*(x) - w^*(y)]^{p_1},
\end{equation*}
i.e. $w^*(x) = w^*(y) $ for $(x,y) \in \mathcal{S}$ yielding that $w^*$ locally constant and hence even constant since $\Omega_1$ is connected,
\end{enumerate}
which gives the contradiction.
In the case $l=0$ we use similar arguments, where the distance $\normN[x-y]$ in the last inequality can be estimated by $\mathrm{diam}|\Omega_1|$ (instead of $\eta$) since $\Omega_1$ is bounded.
\end{proof}
\begin{remark}
In case $l=1$ it follows that the sharper inequality holds true:
There exists a constant $C \in \mathds{R} $ such that for all $w \in W^{s,p_1}(\Omega_1, \mathds{R}^{M_1}), \ 0<s < 1, \ 1 < p_1 < \infty$ and $D \subsetneq \Omega_1$ nonempty such that
\begin{equation}\label{eq: inpaintingIneq2}
\norm{w}_{L^{p_1}(D, \mathds{R}^{M_1})}^{p_1} \leq C \left( \norm{w}_{L^{p_1}(\Omega_1 \setminus D, \mathds{R}^{M_1})}^{p_1} +
\int\limits_{\mathcal{S}} \frac{\normMo[w(x) - w(y)]^{p_1}}{\normN[x-y]^{N+p_1 s}} \rho^l(x-y) \,\mathrm{d}(x,y) \right).
\end{equation}
\end{remark}
\begin{example}
\label{ex:in}
As in \autoref{ex:coercive} we consider that
$W(\Omega_1,K_1) = W^{s, p_1}(\Omega_1, K_1)$
with $p_1>1, \ 0 < s < 1$ and fix $k = N$.
Assume that $\op$ is the inpainting operator, i.e.
\begin{equation*}
\op(w) = \chi_{\Omega_1 \backslash D} (w),
\end{equation*}
where $D \subseteq \Omega_1, \ w \in W^{s,p_1}(\Omega_1, K_1)$. Since the dimension of the data $w$ and the image data $\op(w)$ have the same dimension at every point $x \in \Omega_1$, we write $M \vcentcolon= M_1 = M_2$. \\
Then the functional
\begin{equation*}
\F[\dKt, \dKo] = \mebr[\op[w]][v]^{p_2}_{[\dKt]} + \alpha \mathcal{R}_{[\dKo]}(w): W^{s,p_1}(\Omega_1, K_1) \rarr [0, \infty]
\end{equation*}
is coercive for $p_2 \geq p_1$: \\
The fact that $p_2 \geq p_1$ and that $\Omega_1$ is bounded ensures that
\begin{equation}\label{eq: LpEmbedding}
L^{p_2}(\Omega_1 \backslash D, \mathds{R}^M) \subseteq L^{p_1}(\Omega_1 \backslash D, \mathds{R}^M).
\end{equation}
The proof is done
using the same arguments as in the proof of \autoref{ex:coercive}, where we additionally split \ref{ex: coecivity_case1} into the two sub-cases
\begin{enumerate}[label=\textbf{\ref{ex: coecivity_case1}.\arabic*}]
\item \label{ex: coecivity_case11}
$\norm{w_n}_{L^{p_1}(D, \mathds{R}^M)} \rarr +\infty$
\item \label{ex: coecivity_case12}
$\norm{w_n}_{L^{p_1}(\Omega_1 \setminus D, \mathds{R}^M)} \rarr +\infty$
\end{enumerate}
and using additionally \autoref{lem. auxLemma}, \autoref{eq: inpaintingIneq2} and \autoref{eq: LpEmbedding}. \end{example}
\section{Stability and Convergence} \label{sec: Stability_and_Convergence}
In this section we will first show a stability and afterwards a convergence result. We use the notation introduced in \autoref{sec: Setting}. In particular $W(\Omega_1, K_1)$ is as defined in \autoref{eq:ChooseW}. We also stress that we use notationally simplified versions $\F<v>$ of $\F<v><\alpha>[\dKt, \dKo]$ and $\mathcal{R}$ of $\mathcal{R}_{[\dKo]}$ whenever possible. See \autoref{eq: functional_with_some_metric}, \autoref{eq:d2} and \autoref{eq:d3}.
\begin{thm} \label{thm:Stability} Let \autoref{as:Setting} be satisfied. Let $v^\delta \in L^{p_2}(\Omega_2, K_2)$ and let $(v_n)_{n \in \mathds{N}}$ be a sequence in $L^{p_2}(\Omega_2, K_2)$ such that $\mebr[v_n][v^\delta]_{[\dKt]} \rarr 0$. Then every sequence $\seq{w}$ with
\begin{equation*}
w_n \in \arg \min \{ \F<v_n><\alpha>[\dKt, \dKo](w) \ : \ w \in W(\Omega_1, K_1) \}
\end{equation*}
has a converging subsequence w.r.t. the topology of $W(\Omega_1, K_1)$.
The limit $\tilde{w}$ of any such converging subsequence $(w_{n_k})_{k \in \mathds{N}}$ is a minimizer of
$\F^{v^\delta}[\dKt, \dKo]$.
Moreover, $(\mathcal{R}(w_{n_k}))_{k \in \mathds{N}}$ converges to $\mathcal{R}(\tilde{w})$. \end{thm}
The subsequent proof of \autoref{thm:Stability} is similar to the proof of \cite[Theorem 3.23]{SchGraGroHalLen09}.
\begin{proof} For the ease of notation we simply write $\F<v^\delta>$ instead of $\F<v^\delta><\alpha>[\dKt, \dKo]$ and $\mebr[v][\tilde{v}] = \mebr[v][\tilde{v}]_{[\dKt]}$
By assumption the sequence $(\mebr[v_n][v^\delta])_{n\in \mathds{N}}$ converges to $0$ and thus is bounded, i.e., there exists $B \in (0, +\infty)$ such that \begin{gather} \label{eq:seq_bounded_IN_stability_proof}
\mebr[v_n][v^\delta] \leq B \text{ for all } n \in \mathds{N}. \end{gather} Because $w_n \in \arg \min \{ \F<v_n>(w) : w \in W(\Omega_1, K_1) \}$ it follows that \begin{equation}\label{eq: w_n Minimizer}
\F<v_n>(w_n) \leq \F<v_n>(w) \text{ for all } w \in W(\Omega_1, K_1). \end{equation} By \autoref{as:Setting} there is a $\overline{w} \in W(\Omega_1, K_1)$ such that $\F^{v^0}(\overline{w}) <\infty$. Set $c \vcentcolon= 2^{p_2-1}$. Using \autoref{as:Setting} and applying \autoref{lem:Ineq_F_parameterchange_to_other_v}, \autoref{eq: w_n Minimizer} and \autoref{eq:seq_bounded_IN_stability_proof} implies that for all $n \in \mathds{N}$ \begin{align*}
\F<v^\delta> (w_n)
& \leq c \F<v_n>(w_n) + c \mebr[v_n][v^\delta]^{p_2} \\
& \leq c \F<v_n>(\overline{w}) + c B^{p_2} \\
& \leq c \big[c \F<v^\delta>(\overline{w}) + c \mebr[v^\delta][v_n]^{p_2} \big] + cB^{p_2} \\
& \leq c^2 \F<v^\delta>(\overline{w}) + (c^2 + c)B^{p_2} \\
& \leq c^3 \big( \F<v^0>(\overline{w}) + \mebr[v^0][v^\delta] \big) + (c^2 + c)B^{p_2} =\vcentcolon m < \infty. \end{align*} Applying again \autoref{lem:Ineq_F_parameterchange_to_other_v} we obtain $\F<v^0> (w_n) \leq c \F<v^\delta> (w_n) + c \mebr[v^\delta][v^0]^{p_2} \leq m + c \mebr[v^\delta][v^0]^{p_2} =\vcentcolon \widetilde m < \infty$. Hence, from item \eqref{itm: A} it follows that the sequence $\seq{w}$ contains a converging subsequence.
Let now $(w_{n_k})_{k \in \mathds{N}}$ be an arbitrary subsequence of $\seq{w}$ which converges in $W(\Omega_1, K_1)$ to some $\tilde w \in W(\Omega_1, \mathds{R}^{M_1})$. Then, from \autoref{lem:Wsp_weakly_seq_closed_etc} and the continuity properties of $\op$ it follows that $\tilde w \in W(\Omega_1, K_1)$ and $(\op[w_{n_k}], v_{n_k}) \rarr (\op[\tilde w], v^\delta)$ in $L^{p_2}(\Omega_2, K_2) \times L^{p_2}(\Omega_2, K_2)$. Moreover, using \autoref{thm:F_and_its_summands_are_seq_weakly_closed}, \autoref{eq: w_n Minimizer} and the triangle inequality it follows that for every $w \in W(\Omega_1, K_1)$ the following estimate holds true \begin{align*}
\F<v^\delta>(\tilde w)
& = \mebr[\op(\tilde w)][v^\delta]^{p_2} + \alpha \mathcal{R}(\tilde w)
\leq \mebr[\op(\tilde w)][v^\delta]^{p_2} + \alpha \liminf_{k \rarr \infty} \mathcal{R}(w_{n_k})
\leq \mebr[\op(\tilde w)][v^\delta]^{p_2} + \alpha \limsup_{k \rarr \infty} \mathcal{R}(w_{n_k}) \\&
\leq \liminf_{k \rarr \infty} \mebr[\op(w_{n_k})][v_{n_k}]^{p_2} + \alpha \limsup_{k \rarr \infty} \mathcal{R}(w_{n_k})
\leq
\limsup_{k \rarr \infty} \F<v_{n_k}>(w_{n_k})
\leq \limsup_{k \rarr \infty} \F<v_{n_k}>(w) \\& = \left( \limsup_{k \to \infty} \mebr[F(w)][v_{n_k}] \right)^{p_2} + \alpha \mathcal{R}(w)
\leq \left( \limsup_{k \to \infty} \big(\mebr[F(w)][v^\delta] + \mebr[v^\delta][v_{n_k}] \big) \right)^{p_2} + \alpha \mathcal{R}(w)
\\& = \F<v^\delta>(w). \end{align*}
This shows that $\tilde w$ is a minimizer of $\F<v^\delta>$. Choosing $w = \tilde w$ in the previous estimate we obtain the equality \begin{gather*}
\mebr[\op(\tilde w)][v^\delta]^{p_2} + \alpha \mathcal{R}(\tilde w)
= \mebr[\op(\tilde w)][v^\delta]^{p_2} + \alpha \liminf_{k \rarr \infty} \mathcal{R}(w_{n_k})
= \mebr[\op(\tilde w)][v^\delta]^{p_2} + \alpha \limsup_{k \rarr \infty} \mathcal{R}(w_{n_k}) \,. \end{gather*} Due to $\mebr[\op(\tilde w)][v^\delta]^{p_2} \leq \F<v^\delta>(\tilde w) \leq m < \infty$ this gives \begin{gather*}
\mathcal{R}(\tilde w) = \lim_{k \rarr \infty} \mathcal{R}(w_{n_k}). \end{gather*} \end{proof}
Before proving the next theorem we need the following definition, cf. \cite{SchGraGroHalLen09}. \begin{definition}
Let $v^0 \in \Lp<2>$.
Every element $w^* \in W(\Omega_1, K_1)$ fulfilling
\begin{align} \label{eq: R_minimizing_solution}
\begin{split}
&\op[w^*] = v^0 \\
&\mathcal{R}(w^*) = \min \{ \mathcal{R}(w) \ : \ w \in W(\Omega_1, K_1), \ \op[w] = v^0 \}.
\end{split}
\end{align}
is called an
\emph{$\mathcal{R}$-minimizing solution} of the equation $\op[w] = v^0$ or shorter just
\emph{$\mathcal{R}$-minimizing solution}. \end{definition}
The following theorem and its proof are inspired by \cite[Theorem 3.26]{SchGraGroHalLen09}.
\begin{thm} \label{thm: convergence}
Let \autoref{as:Setting} be satisfied.
Let there exist an $\mathcal{R}$-minimizing solution $w^\dagger \in W(\Omega_1, K_1)$ and
let $\alpha: (0, \infty) \rarr (0,\infty)$ be a function satisfying
\begin{equation}\label{eq: assumptions_on_alpha}
\alpha(\delta) \rarr 0 \text{ and } \frac{\delta^{p_2}}{\alpha(\delta)} \rarr 0
\text{ for } \delta \to 0.
\end{equation}
Let $(\delta_n)_{n \in \mathds{N}}$ be a sequence of positive real numbers converging to $0$. Moreover, let
$(v_n)_{n \in \mathds{N}}$ be a sequence in $L^{p_2}(\Omega_2, K_2)$ with $\mebr[v^0][v_n]_{[\dKt]} \leq \delta_n$ and
set $\alpha_n \vcentcolon= \alpha(\delta_n)$.
Then every sequence $\seq{w}$ of minimizers
\begin{equation*}
w_n \in \arg \min \{ \F^{v_n}_{\alpha_n}[\dKt, \dKo](w) \ : \ w \in W(\Omega_1, K_1) \}
\end{equation*}
has a converging subsequence $w_{n_k} \rarr<W> \tilde{w}$ as $k \to \infty$, and the limit $\tilde{w}$ is always an $\mathcal{R}$-minimizing solution.
In addition, $\mathcal{R}(w_{n_k}) \rarr \mathcal{R}(\tilde{w})$.
Moreover, if $w^\dagger$ is unique it follows that $w_n \rarr<W> w^\dagger$ and $\mathcal{R}(w_{n}) \rarr \mathcal{R}(w^\dagger)$. \end{thm}
\begin{proof}
We write shortly $\mebr$ for $\mebr_{[\dKt]}$.
Taking into account that $w_n \in \argmin \{ \F^{v_n}_{\alpha_n}[\dKt, \dKo](w) \ : \ w \in W(\Omega_1, K_1) \}$ it follows that
\begin{gather*}
\mebr[\op[w_n]][v_n]^{p_2} \leq \F<v_n><\alpha_n>(w_n) \leq \F<v_n><\alpha_n>(w^\dagger) =
\mebr[v^0][v_n]^{p_2} + \alpha_n \mathcal{R}(w^\dagger) \leq \delta_n^{p_2} + \alpha_n \mathcal{R}(w^\dagger) \rarr 0,
\end{gather*}
yielding $\mebr[\op[w_n]][v_n] \rarr 0$ as $n \rarr \infty$.
The triangle inequality gives $\mebr[\op[w_n]][v^0] \leq \mebr[\op[w_n]][v_n] + \mebr[v_n][v^0] \rarr 0$ as
$n \rarr \infty$ and
\autoref{re:tricks} ensures
$\norm{F(w_n) - v^0}_{\Lp<2>[][\mathds{R}^{M_2}]} \leq \mebr[\op[w_n]][v^0] \rarr 0$ as
$n \rarr \infty$, so that
\begin{gather}\label{eq:Convergence_of_operator}
\op[w_n] \rarr v^0 \text{ in } L^{p_2}(\Omega_2, \mathds{R}^{M_2}).
\end{gather}
Since
\begin{gather*}
\mathcal{R}(w_n) \leq \frac{1}{\alpha_n} \F<v_n><\alpha_n>(w_n) \leq \frac{1}{\alpha_n} \F<v_n><\alpha_n>(w^\dagger)
= \frac{1}{\alpha_n}\big( \mebr[v^0][v_n]^{p_2} + \alpha_n \mathcal{R}(w^\dagger) \big) \leq \frac{\delta_n^{p_2}}{\alpha_n} + \mathcal{R}(w^\dagger),
\end{gather*}
we also get
\begin{gather}\label{eq:Regularizer_values_bounded}
\limsup_{n \rarr \infty} \mathcal{R}(w_n) \leq \mathcal{R}(w^\dagger).
\end{gather}
Set $\alpha_{\mathrm{max}} \vcentcolon= \max\{\alpha_n : n \in \mathds{N}\}$.
Since
\begin{gather*}
\limsup_{ n \rarr \infty } \F<v^0><\alpha_n>(w_{n}) \leq
\limsup_{n \rarr \infty } \big( \mebr[\op[w_{n}]][v^0]^{p_2} + \alpha_{\mathrm{max}} \mathcal{R}(w_{n}) \big) \leq \alpha_{\mathrm{max}} \mathcal{R}(w^\dagger)
\end{gather*}
the sequence $\F<v^0><\alpha_{\mathrm{max}}>(w_{n})$ is bounded. From \autoref{as:Setting}, item \eqref{itm: A} it follows that there exists
a converging subsequence $(w_{n_k})_{k \in \mathds{N}}$ of $\seq{w}$. The limit of $(w_{n_k})_{k \in \mathds{N}}$ is denoted by
$\tilde{w}$. Then, from \autoref{lem:Wsp_weakly_seq_closed_etc} it follows that $\tilde{w} \in W(\Omega_1, K_1)$.
Since the operator $\op$ is sequentially continuous it follows that $\op[w_{n_k}] \rarr \op[\tilde{w}]$ in $L^{p_2}(\Omega_2, K_2)$.
This shows that actually $\op[\tilde{w}] = v^0$ since \autoref{eq:Convergence_of_operator} is valid.
Then, from \autoref{thm:F_and_its_summands_are_seq_weakly_closed} it follows that the functional
$\mathcal{R}: W(\Omega_1, K_1) \rarr [0, +\infty]$ is sequentially lower semi-continuous,
so that $\mathcal{R}(\tilde{w}) \leq \liminf_{k \rarr \infty} \mathcal{R}(w_{n_k})$.
Combining this with \autoref{eq:Regularizer_values_bounded} we also obtain
$$ \mathcal{R}(\tilde{w}) \leq \liminf_{k \rarr \infty} \mathcal{R}(w_{n_k}) \leq \limsup_{k \rarr \infty} \mathcal{R}(w_{n_k}) \leq \mathcal{R}(w^\dagger) \leq \mathcal{R}(\tilde{w}),$$
using the definition of $w^\dagger$.
This, together with the fact that $\op[\tilde{w}] = v^0$ we see that $\tilde{w}$ is an $\mathcal{R}$-minimizing solution and that
$\lim_{k \rarr \infty} \mathcal{R}(w_{n_k})= \mathcal{R}(\tilde{w})$.
Now assume that the solution fulfilling \autoref{eq: R_minimizing_solution} is unique; we call it $w^\dagger$.
In order to prove that $w_n \rarr<W> w^\dagger$ it is sufficient to show that any subsequence
has a further subsequence converging to $w^\dagger$, cf.
\cite[Lemma 8.2]{SchGraGroHalLen09}.
Hence, denote by $(w_{n_k})_{k \in \mathds{N}}$ an arbitrary subsequence of $(w_n)$, the sequence of minimizers.
Like before we can show that $\F<v^0><\alpha>(w_{n_k})$ is bounded and we can extract a converging subsequence
$(w_{n_{k_l}})_{l \in \mathds{N}}$. The limit of this subsequence is $w^\dagger$ since it is the unique solution fulfilling
\autoref{eq: R_minimizing_solution}, showing that $w_n \rarr<W> w^\dagger$. Moreover, $w^\dagger \in W(\Omega_1, K_1)$.
Following the arguments above we obtain as well $\lim_{n \rarr \infty} \mathcal{R}(w_{n})= \mathcal{R}(w^\dagger).$ \end{proof} \begin{remark} \autoref{thm:Stability} guarantees that the minimizers of $\F<v_n><\alpha>[\dKt, \dKo]$ depend continuously on $v^\delta$ while \autoref{thm: convergence} ensures that they converge to a solution of $\op(w) = v^0$, $v^0$ the exact data, while $\alpha$ tends to zero. \end{remark}
\section{Discussion of the Results and Conjectures}
In this section we summarize some open problems related to double integral expressions of functions with values on manifolds.
\subsection{Relation to single integral representations} In the following we show for one particular case of functions that have values in a manifold, that the double integral formulation $\mathcal{R}_{[\dKo]}$, defined in \autoref{eq:d3}, approximates a single energy integral. The basic ingredient for this derivation is the exponential map related to the metric $d_1$ on the manifold. In the following we investigate manifold--valued functions $w \in W^{1,2}(\Omega, \mathcal{M})$, where we consider $\mathcal{M} \subseteq \mathds{R}^{M \times 1}$ to be a connected, complete Riemannian manifold. In this case some of the regularization functionals $\mathcal{R}_{[\dKo]}$, defined in \autoref{eq:d3}, can be considered as approximations of \emph{single} integrals. In particular we aim to generalize \autoref{eq:double_integral} in the case $p=2$.
We have that \begin{equation*}
\nabla w = \begin{bmatrix}
\frac{\partial w_1}{\partial x_1} & \cdots & \frac{\partial w_1}{\partial x_N} \\
\vdots & \ddots & \vdots\\
\frac{\partial w_M}{\partial x_1} & \cdots & \frac{\partial w_M}{\partial x_N} \end{bmatrix} \in \mathds{R}^{M \times N}. \end{equation*} In the following we will write $\mathcal{R}_{[\dKo],\varepsilon}$ instead of $\tfrac12 \mathcal{R}_{\dKo}$ to stress the dependence on $\varepsilon$ in contrast to above; the factor $\frac{1}{2}$ was added due to reasons of calculation. Moreover, let $\hat{\rho} : \mathds{R}_+ \to \mathds{R}_+$ be in $C_c^\infty(\mathds{R}_+, \mathds{R}_+)$ and satisfy \begin{equation*} \abs{\mathbb{S}^{N-1}}\int_0^\infty \hat{t}^{N-1} \hat{\rho}\left(\hat{t}\right)d \hat{t} = 1\;. \end{equation*} Then for every $\varepsilon > 0$ \begin{equation*} x \in \mathds{R}^n \mapsto \rho_\varepsilon(x)\vcentcolon= \frac{1}{\varepsilon^N} \hat{\rho}\left(\frac{\normN[x]}{\varepsilon}\right) \end{equation*} is a mollifier, cf. \autoref{ex:mol}. \\ $\mathcal{R}_{[\dKo],\varepsilon}$ (with $p_1=2$) then reads as follows: \begin{equation} \label{eq:di_II}
\mathcal{R}_{[\dKo],\varepsilon}(w)
\vcentcolon=
\frac{1}{2}\int\limits_{\Omega\times \Omega} \frac{d_1^2(w(x),w(y))}{\normN[x-y]^2} \rho_\varepsilon(x-y) \,\mathrm{d}(x,y)\,. \end{equation} Substitution with spherical coordinates $y = x - t \theta \in \mathds{R}^{N \times 1}$ with $\theta \in \mathbb{S}^{N-1} \subseteq \mathds{R}^{N \times 1}$, $t \geq 0$ gives \begin{equation} \label{eq:reg2} \begin{aligned} \lim_{\varepsilon \searrow 0} \mathcal{R}_{[\dKo],\varepsilon}(w) &= \lim_{\varepsilon \searrow 0} \frac{1}{\varepsilon^N}
\int\limits_{\Omega} \int\limits_{\mathbb{S}^{N-1}}
\int\limits_0^\infty \frac{1}{2} d_1^2(w(x),w(x-t \theta)) t^{N-3} \hat{\rho}\left(\frac{t}{\varepsilon}\right) \mathrm{d}t \,\mathrm{d}\theta \,\mathrm{d}x\;. \end{aligned} \end{equation} Now, using that for $m_1 \in \mathcal{M}$ fixed and $m_2 \in \mathcal{M}$ such that $m_1$ and $m_2$ are joined by a unique minimizing geodesic (see for instance \cite{FigVil11} where the concept of exponential mappings is explained) \begin{equation}\label{eq:partial_II}
\frac{1}{2} \partial_2 d_1^2(m_1,m_2) = - (\exp_{m_2})^{-1}(m_1) \in \mathds{R}^{M \times 1}, \end{equation} where $\partial_2$ denotes the derivative of $d_1^2$ with respect to the second component. By application of the chain rule we get \begin{equation*}
\begin{aligned}
- \frac{1}{2} \nabla_y d_1^2(w(x),w(y)) &=
\underbrace{(\nabla w(y))^T}_{\in \mathds{R}^{N \times M}} \underbrace{(\exp_{w(y)})^{-1}(w(x))}_{\in \mathds{R}^{M \times 1}}\in \mathds{R}^{N \times 1}\;,
\end{aligned} \end{equation*} where $w(x)$ and $w(y)$ are joined by a unique minimizing geodesic. This assumption seems reasonable due to the fact that we consider the case $\varepsilon \searrow 0$. Let $\cdot$ denote the scalar multiplication of two vectors in $\mathds{R}^{N \times 1}$, then the last equality shows that \begin{equation*}
\begin{aligned} \frac{1}{2} d_1^2(w(x),w(x-t \theta)) &= - \frac{1}{2} \left[ d_1^2\big(w(x),w( (x-t\theta) + t \theta )\big) - d_1^2\big(w(x),w(x-t \theta)\big) \right] \\ &\approx \left( \left(\nabla w(x-t \theta)\right)^T (\exp_{w(x-t \theta)})^{-1}(w(x)) \right) \cdot t\theta\;. \end{aligned} \end{equation*} Thus from \autoref{eq:reg2} it follows that \begin{equation} \label{eq:reg3} \begin{aligned} ~ & \lim_{\varepsilon \searrow 0} \mathcal{R}_{[\dKo],\varepsilon}(w) \\ \approx & \lim_{\varepsilon \searrow 0}
\frac{1}{\varepsilon^N}
\int\limits_{\Omega} \int\limits_{\mathbb{S}^{N-1}}
\int\limits_0^\infty \left( \left( \nabla w(x-t \theta)\right)^T (\exp_{w(x-t \theta)})^{-1}(w(x))
\right) \cdot
\theta \left(t^{N-2} \hat{\rho}\left(\frac{t}{\varepsilon}\right)\right) \mathrm{d}t \,\mathrm{d}\theta \,\mathrm{d}x\;. \end{aligned} \end{equation} Now we will use a Taylor series of power 0 for $ t\mapsto \nabla w(x-t \theta)$ and of power 1 for $t \mapsto (\exp_{w(x-t \theta)})^{-1}(w(x))$ to rewrite \autoref{eq:reg3}. We write \begin{equation}
F(w;x,t,\theta) \vcentcolon= (\exp_{w(x-t \theta)})^{-1}(w(x)) \in \mathds{R}^{M \times 1} \end{equation} and define \begin{equation}
\dot{F}(w;x,\theta) \vcentcolon= \lim_{t \searrow 0} \frac{1}{t} \left((\exp_{w(x-t \theta)})^{-1}(w(x)) -
\underbrace{(\exp_{w(x)})^{-1}(w(x))}_{=0}
\right) \in \mathds{R}^{M \times 1}. \end{equation} Note that because $(\exp_{w(x)})^{-1}(w(x))$ vanishes, $\dot{F}(w(x);\theta)$ is the leading order term of the expansion of $(\exp_{w(x-t \theta)})^{-1}(w(x))$ with respect to $t$. Moreover, in the case that $\nabla w(x) \neq 0$ this is the leading order approximation of $\nabla w(x-t \theta)$. In summary we are calculating the leading order term of the expansion with respect to $t$.
Then from \autoref{eq:reg3} it follows that \begin{equation} \label{eq:reg3a} \lim_{\varepsilon \searrow 0} \mathcal{R}_{[\dKo],\varepsilon}(w) \approx \lim_{\varepsilon \searrow 0} \underbrace{\frac{1}{\varepsilon^N} \int\limits_0^\infty t^{N-1} \hat{\rho}\left(\frac{t}{\varepsilon}\right) \mathrm{d}t}_{= \abs{\mathbb{S}^{N-1}}^{-1}}
\int\limits_{\Omega} \int\limits_{\mathbb{S}^{N-1}} \left((\nabla w(x))^T \dot{F}(w;x,\theta) \right) \cdot \theta \;
\mathrm{d}\theta \,\mathrm{d}x\;. \end{equation} The previous calculations show that the double integral simplifies to a double integral where the inner integration domain has one dimension less than the original integral. Under certain assumption the integration domain can be further simplified:
\begin{example}
If $d_1(x,y)=\normM[x-y]$, $p_1=2$, then
\begin{equation*}
\dot{F}(w;x,\theta) = \lim_{t \searrow 0} \frac{1}{t} \left(w(x) - w(x-t\theta)\right) = \nabla w(x)\theta \in \mathds{R}^{M \times 1}.
\end{equation*}
Thus from \eqref{eq:reg3a} it follows that \begin{equation} \label{eq:reg3b} \lim_{\varepsilon \searrow 0} \mathcal{R}_{[\dKo],\varepsilon}(w) \approx \int\limits_{\Omega} \underbrace{(\nabla w(x))^T \nabla w(x)}_{\norm{\nabla w(x)}^2_{\mathds{R}^M}} \,\mathrm{d}x\;. \end{equation} This is exactly the identity derived in \citeauthor{BouBreMir01} \cite{BouBreMir01}. \end{example} From these considerations we can view $\lim_{\varepsilon \searrow 0} \mathcal{R}_{[\dKo],\varepsilon}$ as functionals, which generalize Sobolev and $\mathrm{BV}$ semi-norms to functions with values on manifolds.
\subsection{A conjecture on Sobolev semi-norms} \label{ss:conjecture} Starting point for this conjecture is \autoref{eq:d3}. We will write $\Omega,M$ and $p$ instead of $\Omega_1, M_1$ and $p_1$. \begin{itemize}
\item In the case $l=0$, $k=N$, $0<s<1$ and $\dKo(w(x), w(y))= \normM[w(x)-w(y)]$
the functional $\mathcal{R}_{[\dKo]}$ from
\autoref{eq:d3} simplifies to the $p$-th
power of the Sobolev semi-norm and reads
\begin{equation}\label{eq:d3a}
\int\limits_{\Omega\times \Omega} \frac{\normM[w(x)-w(y)]^{p}}{\normN[x-y]^{N+p s}} \,\mathrm{d}(x,y).
\end{equation}
For a recent survey on fractional Sobolev Spaces see \cite{DiNPalVal12}.
\item On the other hand, when we choose $k=0$, $l=1$ and $\dKo(w(x), w(y))= \normM[w(x)-w(y)]$, then
$\mathcal{R}_{[\dKo]}$ from \autoref{eq:d3} reads
(note $\rho=\rho_\varepsilon$ by simplification of notation):
\begin{equation}\label{eq:d3b}
\int\limits_{\Omega\times \Omega} \frac{\normM[w(x)-w(y)]^{p}}{\normN[x-y]^{p s}}
\rho_\varepsilon(x-y)
\,\mathrm{d}(x,y).
\end{equation}
\item Therefore, in analogy to what we know for $s=1$ from \cite{BouBreMir01}, we conjecture that
\begin{equation}\label{eq:d3c}
\lim_{\varepsilon \to 0} \int\limits_{\Omega\times \Omega} \frac{\normM[w(x)-w(y)]^{p}}{\normN[x-y]^{p s}}
\rho_\varepsilon(x-y)
\,\mathrm{d}(x,y) = C \int\limits_{\Omega\times \Omega} \frac{\normM[w(x)-w(y)]^{p}}{\normN[x-y]^{N+p s}}\,\mathrm{d}(x,y).
\end{equation}
The form \autoref{eq:d3c} is numerically preferable to the standard Sobolev semi-norm \autoref{eq:d3a}, because
$\rho=\rho_\varepsilon$ and thus the integral kernel has compact support. \end{itemize}
\section{Numerical Examples} \label{sec:Numerical_results}
In this section we present some numerical examples for denoising and inpainting of functions with values on the circle $\mathbb{S}^1$. Functions with values on a sphere have already been investigated very diligently (see for instance \cite{BouBreMir00b} out of series of publications of these authors). Therefore we review some of their results first.
\subsection{$\mathbb{S}^1$-Valued Data} \label{ss: spheredata} Let $\emptyset \neq \Omega \subset \mathds{R}$ or $\mathds{R}^2$ be a bounded and simply connected open set with Lipschitz boundary. In \cite{BouBreMir00b} the question was considered when $w \in \Wsp[\mathbb{S}^1]$ can be represented by some function $u \in \Wsp[\mathds{R}]$ satisfying \begin{equation}\label{eq:id_sphere_w}
\Phi(u) \vcentcolon= \mathrm e^{\ii u} = w. \end{equation} That is, the function $u$ is a \emph{lifting} of $w$.
\begin{lemma}[\cite{BouBreMir00b}] \label{lem: lifting} \begin{itemize}
\item Let $\Omega \subset \mathds{R}$, $0 < s < \infty$, $1 < p < \infty$. Then for all $w \in \Wsp[\mathbb{S}^1]$ there exists
$u \in \Wsp[\mathds{R}]$ satisfying \autoref{eq:id_sphere_w}.
\item Let $\Omega \subset \mathds{R}^N$, $N \geq 2$, $0 < s < 1$, $1 < p < \infty$. Moreover, let
$sp < 1$ or $sp \geq N$, then for all $w \in \Wsp[\mathbb{S}^1]$ there exists
$u \in \Wsp[\mathds{R}]$ satisfying \autoref{eq:id_sphere_w}.
If $sp \in [1,N)$, then there exist functions $w \in \Wsp[\mathbb{S}^1]$ such that \autoref{eq:id_sphere_w} does not hold with any function
$u \in \Wsp[\mathds{R}]$. \end{itemize}
\end{lemma} For \begin{equation}
\label{eq:arccos}
\dS(a,b) \vcentcolon= \arccos(a^T b) \, , \quad a, b \in \mathbb{S}^1, \end{equation} we consider the functional (note that by simplification of notation below $\rho=\rho_\varepsilon$ denotes a mollifier) \begin{equation} \label{eq:considered} \mathcal{R}_{[\dS]}(w) = \int\limits_{\Omega\times \Omega} \frac{\dS^p(w(x), w(y))}{\normN[x-y]^{k+ps}} \rho^l(x-y) \,\mathrm{d}(x,y), \end{equation} on $w \in \Wsp[\mathbb{S}^1]$, in accordance to \autoref{eq:d3}.
Writing $w = \Phi(u)$ as in \autoref{eq:id_sphere_w} we get the lifted functional \begin{equation} \label{eq:sobolev_alternative} \mathcal{R}_{[\dS]}^{\Phi}(u) \vcentcolon=
\int\limits_{\Omega\times \Omega} \frac{\dS^p(\Phi(u)(x), \Phi(u)(y))}{\normN[x-y]^{k+ps}} \rho^l(x-y) \,\mathrm{d}(x,y), \end{equation} over the space $\Wsp[\mathds{R}]$. \begin{remark} \begin{itemize}
\item We note that in the case $k=0$, $s=1$ and $l=1$ these integrals correspond with the ones considered
in \citeauthor{BouBreMir01} \cite{BouBreMir01} for functions with values on $\mathbb{S}^1$.
\item
If we choose $k=N$, $s=1$ and $l=0$, then this corresponds with Sobolev semi-norms on manifolds.
\item Let $\varepsilon > 0$ fixed (that is, we consider neither a standard Sobolev regularization nor the limiting
case $\varepsilon \to 0$ as in \cite{BouBreMir01}). In this case we have proven coercivity of the functional
$\F: \Wsp[\mathbb{S}^1] \rarr [0,\infty), \ 0<s<1,$ only with the following regularization functional,
cf. \autoref{ex:coercive} and \autoref{ex:in}:
$$\int\limits_{\Omega\times \Omega} \frac{\dS^p(w(x),w(y))}{\normN[x-y]^{N + p s}} \rho_\varepsilon(x-y) \,\mathrm{d}(x,y).$$
\end{itemize} \end{remark}
We summarize a few results: The first lemma follows from elementary calculations: \begin{lemma} \label{le:1}
$\dS$ and $\,\mathrm d_{\mathds{R}^2}\big|_{\mathbb{S}^1 \times \mathbb{S}^1}$ are equivalent. \end{lemma}
\begin{lemma} \label{lem:2}
Let $u \in \Wsp[\mathds{R}]$. Then $\Phi(u) \in \Wsp[\mathbb{S}^1]$. \end{lemma}
\begin{proof}
This follows directly from the inequality $\|\mathrm e^{ia}-\mathrm e^{ib}\| \leq \|a-b\|$ for all $a,b \in \mathds{R}$. \end{proof} Below we show that $\mathcal{R}_{[\dS]}^{\Phi}$ is finite on $\Wsp[\mathds{R}]$. \begin{lemma} \label{lem: liftedRegularizer}
$\mathcal{R}_{[\dS]}^{\Phi}$ maps $\Wsp[\mathds{R}]$ into $[0,\infty)$ (i.e. does not attain the value $+\infty$). \end{lemma} \begin{proof}
Let $u \in \Wsp[\mathds{R}]$. Then by Lemma \ref{lem:2} we have that $\Phi(u) \in \Wsp[\mathbb{S}^1]$. Therefore, from Lemma \ref{le:1} and \autoref{pr:ExprIsOp} \autoref{itm: ExpIsOp} it follws that $\mathcal{R}_{[\dS]}(\Phi(u))< \infty$. Hence, by definition, $\mathcal{R}_{[\dS]}^{\Phi}(u) < \infty$. \end{proof}
\subsection{Setting of numerical examples} In all numerical examples presented we use a simplified setting with \begin{equation*}
M_1 = M_2 =\vcentcolon M,\;K_1 = K_2 =\vcentcolon \mathbb{S}^1,\;p_1 = p_2 =\vcentcolon p,\;k = N,\;l = 1, \end{equation*} $\Omega_1 = \Omega_2 =\vcentcolon \Omega$ when considering image denoising, $\Omega_1 = \Omega$, $\Omega_2 = \Omega \setminus D$ when considering image inpainting, and \begin{equation*} W(\Omega,\mathbb{S}^1) = \Wsp[\mathbb{S}^1]. \end{equation*} As particular mollifier we use $\rho_\varepsilon$ (see \autoref{ex:mol}), which is defined via the one-dimensional normal-distribution $ \hat{\rho}(x) = \frac{1}{\sqrt{\pi}} \mathrm e^{-x^2}.$
\subsection*{Regularization functionals} Let $\mathcal{R}_{[\dS]}$ and $\mathcal{R}_{[\dS]}^{\Phi}$ be as defined in \autoref{eq:considered} and \autoref{eq:sobolev_alternative}, respectively. In what follows we consider the following regularization functional \begin{equation} \label{eq:reg_numerics} \F<v^\delta><\alpha>[\dS](w) \vcentcolon= \int\limits_\Omega \dS^p(\op[w](x), v^\delta(x)) \,\mathrm{d}x + \alpha \mathcal{R}_{[\dS]}(w), \end{equation} on $\Wsp[\mathbb{S}^1]$ and the lifted variant \begin{equation} \label{eq: functionalAlternative} \FT<v^\delta><\alpha>[\dS](u) \vcentcolon= \int\limits_\Omega \dS^p(\op[\Phi(u)](x), v^\delta(x)) \,\mathrm{d}x + \alpha \mathcal{R}_{[\dS]}^{\Phi}(u) \end{equation} over the space $\Wsp[\mathds{R}]$ (as in \autoref{ss: spheredata}), where $\Phi$ is defined as in \eqref{eq:id_sphere_w}. Note that $\FT = \F \circ \Phi$.
\begin{lemma}\label{lem: liftedFunctional} Let $\emptyset \neq \Omega \subset \mathds{R}$ or $\mathds{R}^2$ be a bounded and simply connected open set with Lipschitz boundary. Let $1 < p < \infty$ and $s \in (0,1)$. If $N=2$ assume that $sp < 1$ or $sp \geq 2$. Moreover, let \autoref{as:Setting} and \autoref{ass:2} be satisfied. Then the mapping $\FT<v^\delta><\alpha>[\dS]: W^{s,p}(\Omega, \mathds{R}) \rarr [0,\infty)$ attains a minimizer. \end{lemma} \begin{proof}
Let $u \in \Wsp[\mathds{R}]$. Then by Lemma \ref{lem:2} we have that $w \vcentcolon= \Phi(u) \in \Wsp[\mathbb{S}^1]$.
As arguing as in the proof of Lemma \ref{lem: liftedRegularizer} we see that $\FT<v^\delta><\alpha>[\dS](u) < \infty$. \\
Since we assume that \autoref{as:Setting} is satisfied we get that $\F<v^\delta><\alpha>[\dS](w)$ attains a minimizer $w^* \in \Wsp[\mathbb{S}^1]$.
It follows from \autoref{lem: lifting} that there exists a function $u^* \in W^{s,p}(\Omega, \mathds{R})$ that can be lifted to $w^*$, i.e. $w^* = \Phi(u^*)$.
Then $u^*$ is a minimizer of \eqref{eq: functionalAlternative}
by definition of $\FT$ and $\Phi$. \end{proof}
\subsection{Numerical minimization} In our concrete examples we will consider two different operators $\op$. For numerical minimization we consider the functional from \autoref{eq: functionalAlternative} in a discretized setting. For this purpose we approximate the functions $u \in W^{s, p}(\Omega,\mathds{R})$, $0<s<1,1<p<\infty$ by quadratic B-Spline functions and optimize with respect to the coefficients. We remark that this approximation
is continuous and thus that sharp edges correspond to very steep slopes. \\ The noisy data $u^\delta$ is obtained by adding Gaussian white noise with variance $\sigma^2$ to the approximation or the discretized approximation of $u$.
We apply a simple Gradient Descent scheme with fixed step length implemented in $\mathrm{MATLAB}$.
\subsection{Denoising of $\mathbb{S}^1$-valued functions - The InSAR problem} \label{ss: denoising}
In this case the operator $\op: \Wsp[\mathbb{S}^1] \rarr \Lp[\mathbb{S}^1]$ is the inclusion operator. It is norm-coercive in the sense of \autoref{eq:coercive_F} and hence \autoref{as:Setting} is fulfilled. For $\emptyset \neq \Omega \subset \mathds{R}$ or $\mathds{R}^2$ a bounded and simply connected open set, $1 < p < \infty$ and $s \in (0,1)$ such that additionally $sp < 1$ or $sp \geq 2$ if $N=2$ we can apply \autoref{lem: liftedFunctional} which ensures that the lifted functional $\FT<u^\delta><\alpha>[\dS]: \Wsp[\mathds{R}] \rarr [0,\infty)$ attains a minimizer $u \in W^{s, p}(\Omega,\mathds{R})$.
In the examples we will just consider the continuous approximation again denoted by $u$.
\subsection*{One dimensional test case} Let $\Omega = (0,1)$ and consider the signal $u:\Omega \rarr [0,2\pi)$ representing the angle of a cyclic signal. \\ For the discrete approximation shown in \autoref{sfig:signal1-a} the domain $\Omega$ is sampled equally at 100 points. $u$ is affected by an additive white Gaussian noise with $\sigma = 0.1$ to obtain the noisy signal which is colored in blue in \autoref{sfig:signal1-a}.
In this experiment we show the influence of the parameters $s$ and $p$. In all cases the choice of the regularization parameter $\alpha$ is 0.19 and $\varepsilon = 0.01$.\\ The red signal in \autoref{sfig:signal1-b} is obtained by choosing $s = 0.1$ and $p = 1.1$. We see that the periodicity of the signal is handled correctly and that there is nearly no staircasing. In \autoref{sfig:signal1-c} the parameter $s$ is changed from $0.1$ to $0.6$. The value of the parameter $p$ stays fixed. Increasing of $s$ leads the signal to be more smooth. We can observe an even stronger similar effect when increasing $p$ (here from $1.1$ to $2$) and letting $s$ fixed, see \autoref{sfig:signal1-d}. This fits the expectation since $s$ only appears once in the denominator of the regularizer. At a jump increasing of $s$ leads thus to an increasing of the regularization term. The parameter $p$ appears twice in the regularizer. Huge jumps are hence weighted even more.
\begin{figure}
\caption{Original and noisy data}
\label{sfig:signal1-a}
\caption{Denoised data}
\label{sfig:signal1-b}
\caption{Increasing of $s$}
\label{sfig:signal1-c}
\caption{Increasing of $p$}
\label{sfig:signal1-d}
\caption{Function on $\mathbb{S}^1$ represented in $[0,2\pi)$: Left to right, top to bottom: Original data (black) and noisy data (blue) with 100 data points. Denoised data (red) where we chose $s=0.1, p=1.1, \alpha = 0.19$. Denoised data with $s=0.6, p=1.1, \alpha = 0.19$ resp. $s=0.1, p=2, \alpha=0.19$. }
\label{fig:signal1}
\end{figure}
In \autoref{sfig:signal2-a} we considered a simple signal with a single huge jump. Again it is described by the angular value. We proceeded as above to obtain the approximated discrete original data (black) and noisy signal with $\sigma = 0.1$ (blue). We chose again $\varepsilon = 0.01$. \\ As we have seen above increasing of $s$ leads to a more smooth signal. This effect can be compensated by choosing a rather small value of $p$, i.e. $p \approx 1$. In \autoref{sfig:signal2-b} the value of $s$ is $0.9$. We see that it is still possible to reconstruct jumps by choosing e.g. $p=1.01$. \\ Moreover, we have seen that increasing of $p$ leads to an even more smooth signal. In \autoref{sfig:signal2-c} we choose a quite large value of $p$, $p=2$ and a rather small value of $s$, $s = 0.001$. Even for this very simple signal is was not possible to get sharp edges. This is due to the fact that the parameter $p$ (but not $s$) additionally weights the height of jumps in the regularizing term.
\begin{figure}
\caption{Original and noisy data}
\label{sfig:signal2-a}
\caption{$s = 0.9, \ p = 1.01$}
\label{sfig:signal2-b}
\caption{$s = 0.001, \ p = 2$}
\label{sfig:signal2-c}
\caption{Left to right: Original data (black) and noisy data (blue) sampled at 100 data points. Denoised data (red) where we chose $s=0.9, p=1.01, \alpha = 0.03$. Denoised data with $s=0.001, p=2, \alpha = 0.9$.}
\label{fig:signal2}
\end{figure}
\subsection*{Denoising of a $\mathbb{S}^1$-Valued Image}
Our next example concerned a two-dimensional $\mathbb{S}^1$-valued image represented by the corresponding angular values. We remark that in this case where $N=2$ the existence of such a representation is always guaranteed in the cases where $sp < 1$ or $sp \geq 2$, see \autoref{lem: lifting}.
The domain $\Omega$ is sampled into $60 \times 60$ data points and can be considered as discrete grid, $\{1, \dots,60\} \times \{1, \dots,60\} $. The B-Spline approximation evaluated at that grid is given by \begin{equation*} u(i,j) = u(i,0) \vcentcolon= 4\pi \frac{i}{60} \bmod 2\pi, \quad i,j \in \{1, \dots,60\}. \end{equation*}
\begin{figure}
\caption{The function $u$ evaluated on the discrete grid.}
\label{fig: Rainbow_function}
\end{figure} The function $u$ is shown in \autoref{fig: Rainbow_function}. We used the $\mathrm{hsv}$ colormap provided in $\mathrm{MATLAB}$ transferred to the interval $[0, 2\pi]$.
This experiment shows the difference of our regularizer respecting the periodicity of the data in contrast to the classical Total Variation regularizer. The classical TV-minimization is solved using a fixed point iteration (\cite{LoeMag}); for the method see also \cite{VogOma96}.
In \autoref{sfig:rainbow-a} the function $u$ can be seen from the top, i.e. the axes correspond to the $i$ resp. $j$ axis in \autoref{fig: Rainbow_function}. The noisy data is obtained by adding white Gaussian noise with $\sigma = \sqrt{0.001}$ using the built-in function $\mathtt{imnoise}$ in $\mathrm{MATLAB}$. It is shown in \autoref{sfig:rainbow-b}. We choose as parameters $s=0.9, \ p=1.1, \ \alpha = 1,$ and $\varepsilon = 0.01$. We observe significant noise reduction in both cases. However, only in \autoref{sfig:rainbow-d} the color transitions are handled correctly. This is due to the fact, that our regularizer respects the periodicity, i.e. for the functional there is no jump in \autoref{fig: Rainbow_function} since 0 and $2\pi$ are identified. Using the classical TV regularizer the values 0 and $2\pi$ are not identified and have a distance of $2\pi$. Hence, in the TV-denoised image there is a sharp edge in the middle of the image, see \autoref{sfig:rainbow-c}. \\
\begin{figure}
\caption{Original data}
\label{sfig:rainbow-a}
\caption{Noisy data}
\label{sfig:rainbow-b}
\caption{TV-denoised data}
\label{sfig:rainbow-c}
\caption{Denoised data}
\label{sfig:rainbow-d}
\caption{Left to right, top to bottom: Original and noisy data of an $60 \times 60$ image. TV-denoised data using a fixed point iteration method. Denoised data where we chose $s=0.9, p=1.1, \alpha = 1$, 400 steps. }
\label{fig:rainbow}
\end{figure}
\subsection*{Hue Denoising}
The $\mathrm{HSV}$ color space is shorthand for Hue, Saturation, Value (of brightness). The hue value of a color image is $\mathbb{S}^1$-valued, while saturation and value of brightness are real-valued. Representing colors in this space better match the human perception than representing colors in the RGB space.
In \autoref{sfig:fruits-a} we see a part of size $70 \times 70$ of the RGB image ``fruits'' (\url{https://homepages.cae.wisc.edu/~ece533/images/}).
The corresponding hue data is shown in \autoref{sfig:fruits-b}, where we used again the colormap hsv, cf. \autoref{fig: Rainbow_function}. Each pixel-value lies, after transformation, in the interval $[0, 2\pi)$ and represents the angular value. Gaussian white noise with $\sigma = \sqrt{0.001}$ is added to obtain a noisy image, see \autoref{sfig:fruits-c}.\\ To obtain the denoised image \autoref{sfig:fruits-d} we again used the same fixed point iteration, cf. \cite{LoeMag}, as before.
We see that the denoised image suffers from artifacts due to the non-consideration of periodicity. The pixel-values in the middle of the apple (the red object in the original image) are close to $2\pi$ while those close to the border are nearly 0, meaning they have a distance of around $2\pi$. \\ We use this TV-denoised image as starting image to perform the minimization of our energy functional. As parameters we choose $s = 0.49, \ p = 2, \ \alpha = 2, \ \varepsilon = 0.006$.
Since the cyclic structure is respected the disturbing artifacts in image \autoref{sfig:fruits-d} are removed correctly. The edges are smoothed due to the high value of $p$, see \autoref{sfig:fruits-e}.
\begin{figure}
\caption{Original RGB image \newline \qquad \newline}
\label{sfig:fruits-a}
\caption{Hue component represented in color, which represent function values on $\mathbb{S}^1$}
\label{sfig:fruits-b}
\caption{Noisy hue value - again representing function values on $\mathbb{S}^1$ \newline}
\label{sfig:fruits-c}
\caption{TV-denoised data}
\label{sfig:fruits-d}
\caption{Denoised data}
\label{sfig:fruits-e}
\caption{Left to right, top to bottom: Original RGB image and its Hue component. Noisy Hue data with $\sigma^2 = 0.001$. TV minimization is done using an iterative approach. It is serving as starting point for the GD minimization. Denoised data with $s=0.49, p=2, \alpha = 2$, 500 steps. }
\label{fig:fruits}
\end{figure}
\subsection{$\mathbb{S}^1$-Valued Image Inpainting}
In this case the operator $\op: \Wsp[\mathbb{S}^1] \rarr \Lp[\mathbb{S}^1]$ is the inpainting operator, i.e.
\begin{equation*}
\op(w) = \chi_{\Omega \backslash D} (w),
\end{equation*} where $D \subseteq \Omega$ is the area to be inpainted.
We consider the functional \begin{equation*} \F<v^\delta><\alpha>[\dS](w) \vcentcolon= \int\limits_{\Omega \setminus D} \dS^p(w(x), v^\delta(x)) \,\mathrm{d}x + \alpha \int\limits_{\Omega\times \Omega} \frac{\dS^p(w(x), w(y))}{\norm{x-y}_{\mathds{R}^2}^{2+ps}} \rho_{\varepsilon}(x-y) \,\mathrm{d}(x,y), \end{equation*} on $\Wsp[\mathbb{S}^1]$.
According to \autoref{ex:in} the functional $\F$ is coercive and \autoref{as:Setting} is satisfied. For $\emptyset \neq \Omega \subset \mathds{R}$ or $\mathds{R}^2$ a bounded and simply connected open set, $1 < p < \infty$ and $s \in (0,1)$ such that additionally $sp < 1$ or $sp \geq 2$ if $N=2$ \autoref{lem: liftedFunctional} applies which ensures that there exists a minimizer $u \in W^{s, p}(\Omega,\mathds{R})$ of the lifted functional $\FT<u^\delta><\alpha>[\dS]: \Wsp[\mathds{R}] \rarr [0,\infty)$ $u \in W^{s, p}(\Omega,\mathds{R})$
\subsection*{Inpainting of a $\mathbb{S}^1$-Valued Image}
As a first inpainting test-example we consider two $\mathbb{S}^1$-valued images of size $28 \times 28$, see \autoref{fig:blocks}, represented by its angular values. In both cases the ground truth can be seen in \autoref{sfig:blocks-a} and \autoref{sfig:blocks2-a}. We added Gaussian white noise with $\sigma = \sqrt{0.001}$ using the $\mathrm{MATLAB}$ build-in function $\mathtt{imnoise}$. The noisy images can be seen in \autoref{sfig:blocks-b} and \autoref{sfig:blocks2-b}. The region $D$ consists of the nine red squares in \autoref{sfig:blocks-c} and \autoref{sfig:blocks2-c}.
The reconstructed data are shown in \autoref{sfig:blocks-d} and \autoref{sfig:blocks2-d}. \\ For the two-colored image we used as parameters $\alpha = s = 0.3$, $p = 1.01$ and $\varepsilon = 0.05$. We see that the reconstructed edge appears sharp. The unknown squares, which are completely surrounded by one color are inpainted perfectly. The blue and green color changed slightly.
As parameters for the three-colored image we used $\alpha = s = 0.4$, $p=1.01$ and $\varepsilon = 0.05$. Here again the unknown regions lying entirely in one color are inpainted perfectly. The edges are preserved. Just the corner in the middle of the image is slightly smoothed. \\ In \autoref{sfig:blocks-e} and \autoref{sfig:blocks2-e} the TV-reconstructed data is shown. The underlying algorithm (\cite{Get}) uses the split Bregman method (see \cite{GolSta09}).
In \autoref{sfig:blocks-e} the edge is not completely sharp. There are some lighter parts on the blue side. This can be caused by the fact that the unknown domain in this area is not exactly symmetric with respect to the edge. This is also the case in \autoref{sfig:blocks2-e} where we observe the same effect. Unknown squares lying entirely in one color are perfectly inpainted.
\begin{figure}
\caption{Original image}
\label{sfig:blocks-a}
\caption{Noisy image}
\label{sfig:blocks-b}
\caption{Noisy masked image}
\label{sfig:blocks-c}
\caption{Reconstructed image}
\label{sfig:blocks-d}
\caption{TV-reconstructed image}
\label{sfig:blocks-e}
\caption{Original image}
\label{sfig:blocks2-a}
\caption{Noisy image}
\label{sfig:blocks2-b}
\caption{Noisy masked image}
\label{sfig:blocks2-c}
\caption{Reconstructed image}
\label{sfig:blocks2-d}
\caption{TV-reconstructed image}
\label{sfig:blocks2-e}
\caption{Left to right. Top to bottom: Original image and the noisy data with $\sigma^2 = 0.001$. Noisy image with masking filter and denoised data with $s=0.3, p=1.01, \alpha = 0.3$, 6000 steps. TV denoised data.\\
Original image and the noisy data with $\sigma^2 = 0.001$. Noisy image with masking filter and denoised data with $s=0.4, p=1.01, \alpha = 0.4$, 10000 steps. TV denoised image.}
\label{fig:blocks}
\end{figure}
\subsection*{Hue Inpainting}
As a last example we consider again the Hue-component of the image ``fruits'', see \autoref{sfig:fruits2-a}. The unknown region $D$ is the string $\mathit{01.01}$ which is shown in \autoref{sfig:fruits2-b}. As parameters we choose $p=1.1$, $s=0.1$, $\alpha= 2$ and $\varepsilon = 0.006$. We get the reconstructed image shown in \autoref{sfig:fruits2-c}. The edges are preserved and the unknown area is restored quite well. This can be also observed in the TV reconstructed image, \autoref{sfig:fruits2-d}, using again the split Bregman method as before, cf. \cite{Get}.
\begin{figure}
\caption{Hue component}
\label{sfig:fruits2-a}
\caption{Image with masked region}
\label{sfig:fruits2-b}
\caption{TV-reconstructed image}
\label{sfig:fruits2-c}
\caption{Reconstructed image}
\label{sfig:fruits2-d}
\caption{Left to right, top to bottom: Original image and image with masked region. Reconstructed image with parameters $p=1.1, \
s=0.1, \ \alpha= 2$ and $\varepsilon = 0.006$, 2000 steps. TV-reconstructed image.}
\label{fig:fruits2}
\end{figure}
\subsection{Conclusion} In this paper we developed a functional for regularization of functions with values in a set of vectors. The regularization functional is a derivative-free, nonlocal term, which is based on a characterization of Sobolev spaces of \emph{intensity data} derived by Bourgain, Brézis, Mironescu \& Dávila. Our objective has been to extend their double integral functionals in a natural way to functions with values in a set of vectors, in particular functions with values on an embedded manifold. These new integral representations are used for regularization on a subset of the (fractional) Sobolev space $W^{s,p}(\Omega, \mathds{R}^M)$ and the space $BV(\Omega, \mathds{R}^M)$, respectively. We presented numerical results for denoising of artificial InSAR data as well as an example of inpainting. Moreover, several conjectures are at hand on relations between double metric integral regularization functionals and single integral representations.
\section*{References} \renewcommand{\ii}{\ii} \printbibliography[heading=none]
\end{document} | arXiv | {
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\begin{document}
\begin{center} {\bf\large The distributive elements of a near-field}\\[3mm] {\sc Julien Bahimuzi }
\it\small African Institute for Mathematical sciences\\ AIMS RWANDA\\ \rm e-mail: julien.bahimuzi@aims.ac.rw \end{center}
\normalsize
\quotation{\small {\bf Abstract:} \small {In this thesis, we investigated some properties of (left)near fields and derived some results. We are focusing on $D(\alpha, \beta)$ which is the generalized set of distributive elements of a nearfield. In particular, we investigated some conditions on $\alpha,\beta, \alpha+\beta$ for $D(\alpha, \beta)$ to be a subfield of $\mathbb{F}_{q^{n}}$. In nearfield theory the two distributive laws can not hold at the same time. So in term of left nearfield and near-ring, the right distributivity does not hold and to solve the problem, we defined a set of all distributive elements called $D(R)$.} \\ \normalsize
\section{Introduction} The study of distributive elements in nearfields has a rich history, dating back to the early 20th century. The idea of a near field is introduced in $1905$ where the American mathematician L. Dickson examined it for the first time and presented a first example of a nearfield (\cite{hussein2022some}). The concept of distributivity was introduced by Wedderburn in his classic paper on quasifields (\cite{Wedderburn1926OnQ}). He defined a quasifield as an algebraic system in which the multiplication operation distributes over addition. Later, Bruck (\cite{Bruck1946Contributions}) extended this concept to nearfields, which are more general than quasifields.
Several researchers have studied distributive elements in nearfields and their applications. In particular, Dickson nearfields have received considerable attention due to their interesting algebraic properties and their connection with coding theory and cryptography. Dickson nearfields are a family of nearfields that are constructed using finite fields and a quadratic form (\cite{Dickson1958Linear}). The distributive elements of a Dickson nearfield are closely related to the quadratic residues and non-residues of the finite field (\cite{Gow2017Distributive}).
In recent years, there has been renewed interest in the study of distributive elements in nearfields, due to their applications in combinatorial designs and cryptography. Several families of nearfields have been constructed using distributive elements, such as twisted nearfields and generalized nearfields [(\cite{Clay1974Twisted}), (\cite{Kinyon2013Generalized})]. These families have been used to construct efficient error-correcting codes and cryptographic primitives.
Despite the extensive research on distributive elements in nearfields, several open problems remain. For example, it is still an open question whether every finite nearfield has a non-trivial distributive element (\cite{Minev1998Distributive}). Also, the structure and properties of the generalized set of distributive elements are not well-understood, and further investigation is needed.
Recently the notions of near vector space have been defined in (\cite{djagba2019contributions}) and in (\cite{djagba2020subspace}) the characterized subspace structure of Beidleman near-vector spaces is investigated.
spaces and classify their R-subgroups. The main contribution of this thesis is to provide a comprehensive study of the generalized set of distributive elements in nearfields. We investigate the structure and properties of this set, and provide some results and insights. Our study sheds light on the behavior of distributive elements in nearfields, and provides a basis for further research in this area.
\subsection{Motivation} Nearfields are important algebraic structures that have been extensively studied due to their applications in coding theory, cryptography, and combinatorial designs. They generalize both fields and quasifields, and their properties and structures are of great interest to mathematicians and engineers alike. One important property of nearfields is the distributivity of their multiplication operation over addition, which has led to the study of distributive elements in nearfields. In this thesis we are going to study the set $D(\alpha, \beta)$ and investigate some condition on $\alpha$ and $\beta$ so that $D(\alpha,\beta)$ can be a subflied of $\mathbb{F}_{q^{n}}$ where $(q, n)$ is a Dickson pair. \subsection{Research problem}
The distributive elements of a nearfield have been studied by many researchers, but there are still several open problems and unanswered questions regarding their properties and behavior. In particular, the structure and properties of the generalized set of distributive elements, which is a subset of the nearfield that contains all distributive elements, have not been fully understood. \subsection{Research objectives}
The distributive elements of a nearfield have been studied by many researchers, but there are still several open problems and unanswered questions regarding their properties and behavior. In particular, the structure and properties of the generalized set of distributive elements, which is a subset of the nearfield that contains all distributive elements, have not been fully understood. \subsection{Reseach objectives} The main objective of this thesis is to investigate the properties and behavior of distributive elements in nearfields, with a focus on the generalized set of distributive elements. Specifically, we aim to: \begin{itemize}
\item[$\bullet$] Define and characterize the generalized set of distributive elements.
\item[$\bullet$]Study the structure and properties of this set.
\item[$\bullet$]Provide some insights and results related to distributive elements in nearfields. \end{itemize}
\subsection{Plan of the thesis} The thesis is structured as follows:
\begin{itemize}
\item[$\bullet$] \textbf{Chapter 2} provides the necessary background and definitions related to nearfields and distributive elements.
\item[$\bullet$] \textbf{Chapter 3} constructs a finite Dickson nearfield, which serves as an important example for our study.
\item[$\bullet$]\textbf{Chapter 4} defines the generalized set of distributive elements and studies its properties.
\item[$\bullet$] \textbf{Chapter 5} Gives some details on applications in codding theory.
Finally,
\item[$\bullet$] \textbf{Chapter 6} concludes our study and suggests some directions for future research. \end{itemize} Nearfields and near-rings are related to many other structures and needed for several representation theorems. Therefore it is important to gain knowledge about the structure of near-rings and nearfields and to find construction methods. The first examples of proper nearfields were constructed by L.E. Dickson 1905, they were finite (\cite{karzel1980some}).
\section{Definitions and preliminary results} In this chapter, we are going to give some important definitions on nearfields and present some important results. We begin by defining some elementary structures. These are standard definitions. They can be found in most elementary algebra books. A nearfield is considered by Dickson as a field with only one distributive law. Therefore, we start by introducing fields and nearfiels. \subsection{Fields and nearfields} \begin{defn}
A \emph{field} $ \mathbb{F}$ is defined by giving a set $ \mathbb{F} $ with two binary operations $ "+" $ and $ "\cdot" $ on $ \mathbb{F}$, i.e two maps \begin{align*}
& +:\mathbb{F}\times \mathbb{F}\longrightarrow \mathbb{F},\\
& \cdot :\mathbb{F}\times \mathbb{F}\longrightarrow \mathbb{F},
\end{align*}
subject to the the axioms (\cite{murphy2006course}):
\begin{itemize}
\item[$ (A_{1}) $] Addition is associative,
for all $ a, b, c \in \mathbb{F} $, $ (a+b)+c=a+(b+c).$
\item[($ A_2 $)] Addition is commutative,
for all $ a, b \in \mathbb{F} $, $a+b=b+a.$
\item[$ (A_{3}) $] Addition has an identity element,
for all $ a\in \mathbb{F} $, $ a+0=0+a=a.$
\item[$ (A_{4}) $] Each element has its inverse with respect to addition,
for all $ a \in \mathbb{F} $, $\exists b\in \mathbb{F}$ such that $a+b=0.$
\item[$ (A_{5}) $] Multiplication is associative,
for all $ a, b, c \in \mathbb{F} $, $ (ab)c=a(bc).$
\item[$ (A_{6}) $] Multiplication is commutative,
for all $ a, b \in \mathbb{F} $, $ ab=b.a$
\item[$ (A_{7}) $] Multiplication has an identity element,
for all $ a\in \mathbb{F} $, $ a.1=a.$
\item[$ (A_{8}) $] Each non-zero element has an inverse with respect to multiplication,
for all $ a \in \mathbb{F}^{*} $, $\exists b \in \mathbb{F}$ such that $ ab=1.$
\item[$ (A_{9}) $] Multiplication is distributive over addition,
for all $ a, b, c \in \mathbb{F} $, $ a(b+c)=ab+ac.$
\end{itemize} \end{defn}
\begin{pro}\cite{murphy2006course}
Suppose $\mathbb{F}$ is a field. Then
\begin{enumerate}
\item For each $ a, b\in \mathbb{F}$, the equation $ a+x=b $ has a unique solution.
\item For each $ a, b\in \mathbb{F}$, the equation $ ay=b $ has a unique solution.
\end{enumerate} \end{pro}
In the next sections, we will need to define a finite field. Therefore we consider the following definitions. \begin{defn}
$\mathbb{F}$ is a \emph{finite field} or \emph{Galois field} if it is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication and addition are defined and satisfy certain basic rules as $(A_{1})$ to $(A_{9})$ (\cite{mullen2013handbook}).
The most common examples of finite fields are given by $\mathbb{Z}/p\mathbb{Z}$ when p is a prime number. We have the following
\begin{thm}
Suppose $ \mathbb{F} $ is a finite field of characteristic $p$. Then $ \mathbb{F} $ contains $p^{n}$ elements for some n: $\mid \mathbb{F}\mid= p^{n}$ (\cite{murphy2006course}).
\end{thm}
\begin{proof}
Let us suppose that $\mathbb{F}$ has dimension $n$ over $p$. It means that $\mathbb{F}$ is considered as a vector space. Then we can find a basis
\begin{equation}\label{basis}
\left\lbrace e_{1}, e_{2}, \cdots, e_{n} \right\rbrace
\end{equation} for $\mathbb{F}$ $p$.
Every element $x\in \mathbb{F}$ can be expressed as a linear combination of the basis (\ref{basis}).
\begin{equation*}
x=\lambda_{1}e_{1}+\lambda_{2}e_{2}+\dots+\lambda_{n}e_{n}
\end{equation*}
There are $p$ choices for each $\lambda_{i}$, so the total number of elements in $\mathbb{F}$ is
$ \overbrace{p\cdot p\cdot p\dots p}^{\text{n times}}= p^{n}$ .
\end{proof} \end{defn} The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number $q$ and every positive integer n there are fields of order $q^n$, all of which are isomorphic. The finite field of order $q^n$ is denoted by $\mathbb{F}_{q^{n}}$ (\cite{mullen2013handbook}).
\begin{defn}
Let us consider the set $ R$ with two binary operations $ + $ and $\cdot$ denoted by $(R, + , \cdot).$
If
\begin{itemize}
\item[(i)] $ (R, + ) $ is a group
\item[(ii)] $ (R, \cdot ) $ is a semigroup
\item[(iii)] $ a(b+c)=ab+ac, \qquad \forall a, b, c \in R,$
\end{itemize}
then $ (R, + , \cdot) $ is a near-ring (left).
Moreover, if $(R^{*}, \cdot) $ is a group, then $(R, + , \cdot) $ is a \emph{nearfield (left)} where $ R^{*}=R-\left\lbrace 0\right\rbrace $ (\cite{djagba2020generalized}). \end{defn} We abbreviate $ (R,+, \cdot) $ by $ R $ when the operations are clearly understood and omit the symbol ''$ \cdot$'' for multiplication if no confusion is possible. We have the following example. \begin{exa}\label{lemera}
Let $ (G,+) $ be a group. Then $ (M(G),+, \circ) $ is a nearring under pointwise
addition and composition
\begin{equation*}
(x)(f+g)=(x)f+(x)g,
\end{equation*}
and \begin{equation*}
(x)(f\circ g)=((x)f)g,
\end{equation*}
where
\begin{equation}
M(G): =\left\lbrace f: G\longrightarrow G\right\rbrace
\end{equation}
is the set of all mappings on $G$.
And it is easy to show that $ (M(G),+, \circ) $ is a left near ring (the left distributivity holds)(\cite{howell2007contributions}).
\textbf{Claim}: \emph{We want to prove that $M(G)$ is a left near ring}
\begin{proof}
Let us consider three maps $f, g, h$ which belong to $M(G)$.
\begin{itemize}
\item[$\bullet$] Let us define a mapping $(x)\zeta: = 0$ for all $x\in G$, then $\zeta$ is an element of $M(G)$ $\Rightarrow M(G)\neq \emptyset$.
\item[$\bullet$] Let us show that $M(G)$ is a group.
\begin{align*}
(x)((f+g)+h)&=(x)(f+g)+(x)h\quad (\text{by definition})\\
&=(x)f+(x)g+(x)h\quad (\text{by definition})\\
&=(x)f+(x)(g+h)\\
&=(x)(f+(g+g)), \quad \text{for all $ x\in G$}.
\end{align*}
We can see that $(M(G), +)$ is a semigroup.
Fr all $f\in M(G)$, there exists $-f\in M(G)$ such that $(x)f+(x)(-f)=0$, $(x)(-f)= -(x)f$ for all $x\in G$.
We define the mapping $-f: G\longrightarrow G$ by $(x)(-f)=-(x)f$ for all $x\in G$.
\begin{align*}
(x)(\zeta+f)&=(x)\zeta+(x)f\\
&=0+(x)f\\
&=(x)f \quad \text{and}\\
(x)(f+\zeta)&=(x)f+(x)\zeta\\
&=(x)f+0\\
&=(x)f.
\end{align*}
It follows that
\begin{align*}
(x)(f+(-f))&=(x)f-(x)f\\
&=(x)\zeta\\
&=0 \quad \text{and}\\
(x)((-f)+f)&=(x)(-f)+(x)f\\
&=-(x)f+(x)f\\
&=(x)\zeta\\
&=0
\end{align*}
Hence for any $f\in M(G)$, $\zeta+f=f+\zeta= f$ and $f+(-f)= (-f)+f= \zeta$.
Therefore $(M(G), +)$ is a group.
\item[$\bullet$] $(M(G), \circ)$ is a semigroup. Then,
\begin{align*}
(x)(f\circ(g\circ h))&=((x)f)(g\circ h)\\
&=((x)f)\left[(g)h \right]\\
&=\left[((x)f)g \right]h\\
&=\left[ (x)(f\circ g)\right]h\\
&=(x)\left[(f\circ g)\circ h \right].
\end{align*}
Hence $(M(G), \circ)$ is a semigroup.
\item[$\bullet$] The composition distributes over pointwise addition in one direction in $M(G)$.
For all $f, g, h\in M(G)$
\begin{align*}
(x)(f\circ (g+h))&=(x)(f\circ g)+(x)(f\circ h)\\
\text{In fact,} (x)\left[f\circ (g+h) \right]\\
&=((x)f)(g+h)\\
&=((x)f)g+((x)f)h\\
&=(x)(f\circ g)+ (x)(f\circ h) , \quad \text{for all $x\in G$}.
\end{align*} Hence the left distributive low holds. We conclude that $M(G)$ with addition and the function composition $"\circ"$ a near ring.
In fact the right distributive low failed. We can see that in considering $a, b, c$ all different from zero and for all $x\in G$ we define the maps $h_{a}, h_{b}, h_{c}$ as follow.
\begin{align*}
h_{a}:& G\longrightarrow G\\
&x\longrightarrow (x)h_{a}=a,\\
h_{b}:& G\longrightarrow G\\
&x\longrightarrow (x)h_{b}=b,\\
h_{c}:& G\longrightarrow G\\
&x\longrightarrow (x)h_{c}=c.\\
\end{align*}
Let us check if
\begin{equation}
(x)((h_{a}+h_{b})\circ h_{c})=(x)(h_{a}\circ h_{c})+(x)(h_{b}\circ h_{c})
\end{equation}
In fact, \begin{align*}
(x)((h_{a}+h_{b})\circ h_{c})&=((x)(h_{a}+h_{b})h_{c}\\
&=((x)h_{a}+(x)h_{b})h_{c}\\
&=(a+b)h_{c}\\
&=c.
\end{align*}
Also, \begin{align*}
(x)(h_{a}\circ h_{c})+(x)(h_{b}\circ h_{c})&=((x)h_{a}h_{c})+((x)h_{b})h_{c}\\
&=(a)h_{c}+(b)h_{c}\\
&=c+c.
\end{align*}
\end{itemize}
Since $c\neq 0$, $c\neq c+c$.
From this claim, we conclude that not every near ring is a ring but every ring is necessarily a near ring.
Furthermore, let $y, y\in G$, $y\neq z$ and $f, g\in M(G)$ such that
\begin{equation*}
\begin{cases}
(x)f_{y}:=y,\\
(x)g_{z}:=z.
\end{cases}
\end{equation*}
We have to check if $(x)((f_{y}+g_{z})\circ h)= (x)(f_{y}\circ h)+ (x)(g_{z}\circ h)$.
In fact,
\begin{align*}
(x)((f_{y}+g_{z})\circ h)&=((x)(f_{y}+g_{z})h\\
&=((x)f_{y}+(x)g_{z})h\\
&=(y+z)h.
\end{align*}
Also,
\begin{align*}
(x)(f_{y}\circ h)+ (x)(g_{z}\circ h)&=((x)f_{y})h+((x)g_{z})h\\
&=yh+zh.
\end{align*}
If $G$ contains more than one element, then the right distributive low does not hold in $M(G)$ and not all mappings in $M(G)$ are endomorphism. For $(x)((f+ g)\circ h)= (x)(f\circ h)+ (x)(g\circ h)$ to hold, $h$ must be an endomorphism in this case.
Then from this claim, every ring is near-ring but the inverse is not always true, so we have the following.
\end{proof} \end{exa} \begin{defn}
A proper nearfield is a nearfield that is not a field. \end{defn} \begin{thm}(\textbf{Zassenhaus})
The additive group of a nearfield is abelian. \end{thm} For the proof See \cite{pilz2011near}.
Note that the nearfield is a near-ring with identity such that each non-zero element has an inverse. Hence, each nearfield is a near-ring but the converse is not true. For example, $(\mathbb{Z}, +, \cdot)$ where $\mathbb{Z}$ is the set of integers with usual addition ($+$) and usual ($\cdot$) multiplication is a near-ring but it is not a near-field. The symbols $0$ and $1$ will be used for the additive and multiplicative identities, respectively. In a near-ring or a nearfield $R$, $1\neq 0$ . If $1=0$, then for all $x$ we have $x=x1=x0=0$ , so $R=\left\lbrace 0\right\rbrace $ contradicting the assumption that $R$ has at least two elements (\cite{hussein2022some} p102).
\begin{rem}
All fields are near fields and also any
division ring is a near field (\cite{hussein2022some}). \end{rem} \begin{defn}
A left near ring is said to be \emph{zero-symmetric} if
$0n=0$, for all $n$ in $R$ , i.e., the left distributive law
results in $n0=0$. The set of all zero-symmetric
elements of $R$ is denoted by $R_{0}=\left\lbrace x\in R: 0x=0\right\rbrace $, referred to the zero-symmetric part of $R$ . If $R=R_{0}$
, then $R$ is zero-symmetric. The zero-symmetric near-ring is sometimes named as C-ring (\cite{hussein2022some}). \end{defn} \begin{defn}(\cite{raovague})
A \emph{division ring}, also called \emph{skewfield} is nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element $a$ has a multiplicative inverse denoted $a^{-1}$ such that $aa^{-1}=1$.
So (right) division ring may be defined as $ \frac{a}{b}=ab^{-1}$ but this notation is avoided as one may have $ ab^{-1}\neq b^{-1}a. $
Notice that any division ring is a nearfield and any commutative division ring is a field. \end{defn} \begin{defn}\label{sbnf}
Consider a nearfield $ R $. A subset $ S $ of $R $ is said to be a \emph{subnearfield} of $ R $ if $(S, +)$ and $(S^{*}, \cdot)$ are both groups. If moreover $ (b+c)a=ba+ca , \forall a, b, c \in S$, then $ S $ is a subfield of $ R $ (\cite{groves1974locally}). \end{defn}
\begin{defn}\label{vs}
A set $V$ is said to be a \emph{(left) vector space over a nearfield $R$}, if $(V , +)$ is an abelian
group and, if for each $\alpha\in R$ and $v \in V$, there is a unique element $v \alpha \in V$. Moreover the following conditions hold for all $\alpha, \beta \in R$ and for all $u, v\in V$:
\begin{itemize}
\item[(i)] $\alpha(u+v)= \alpha u+\alpha v$;
\item[(ii)] $(\alpha+\beta)v=\alpha v+\beta v$;
\item[(iii)] $(\alpha\beta) v=\alpha(\beta v)$;
\item[(iv)] $1v=v.$
\end{itemize}
The members of $V$ are called vectors and the members of the division ring (nearfield) are called
scalars. The operation that combines a scalar $\alpha$ and a vector $v$ to form the vector $\alpha v$ is
called scalar multiplication (\cite{howell2007contributions}). \end{defn}
\subsection{Center and kernel of a nearfield } As we saw from the definition of nearfield we do not necessarily have the right distributive law and and commutativity of multiplication. For that reason, the following concept can be defined (\cite{djagba2020generalized}), (\cite{djagba2019contributions}).
\begin{defn}
Let $ R $ be a nearfield. The \emph{multiplicative center} $ (R, \cdot) $ denoted by $C(R)$ is defined as follows:
\begin{equation}
C(R)=\left\lbrace x\in R: xy=yx, \text{$\forall y \in R$} \right\rbrace.
\end{equation}
In others words, it is the set of all elements of $R$ that commute with every element of $R$. \end{defn}\label{def221} Here we use $D(R)$ to express the set of all distributive elements of a nearfield $R$. It is defined as follow (\cite{djagba2020generalized}):
\begin{equation}\label{der} D(R)=\left\lbrace x\in R : (y+z)x= yx+ zx, \text{ for all $ y, z \in R$ }\right\rbrace. \end{equation} \begin{rem}\label{rem}
We can see simply that $C(R)\subset D(R)$ (\cite{pilz2011near}).
Let $\alpha$ be in $C(R)$ and $\beta, \gamma\in R.$
By definition, we know that $\alpha\in C(R)$ implies that for all $\beta, \gamma\in R$, we have \begin{align*}
\alpha(\beta+\gamma)&=\alpha\beta+\alpha\gamma\\
&=\alpha\beta+\alpha\gamma\\
&=\beta\alpha+\gamma\alpha\\
&=(\beta+\gamma)\alpha.
\end{align*}
That means, $\alpha(\beta+\gamma)=(\beta+\gamma)\alpha$.
Then $(\beta+\gamma)\alpha=\beta\alpha+\gamma\alpha$. Thus $\alpha\in D(R)$.
\end{rem} The remark (\ref{rem}) show that $C(R)\subset D(R)$ for the usual multiplication; but it is not direct to say that $D(R)\subset C(R)$. We will see it in the next chapter for a new multiplication which was introduced by Dickson but the notion of center of a nearfield is more developed in (\cite{djagba2020center}). First, we have the next theorem to show the relationship between $D(R)$ and $R$ where $R$ is a left nearfield. \begin{thm}\label{theo2}
Let $R$ be a near-field (\cite{howell2007contributions}). Then
\begin{itemize}
\item[(a)] $D(R)$ with operation of $R$ is a skewfield (division ring), and
\item[(b)] $R$ is a left vector space over $D(R)$.
\end{itemize} \end{thm} \begin{proof}
\begin{itemize}
\item[(a)] From the definition (\ref{sbnf}), we have to show that $(D(R), +)$ and $(D(R)^{*}, \cdot)$ are both groups.
\begin{itemize}
\item[(i)] Since $1\in D(R)$, $D(R)\neq \emptyset$ and $D(R)\subseteq R$.
In fact, $(\alpha+\beta)\cdot 1=\alpha\cdot 1+\beta\cdot 1=\alpha+\beta$ for all $\alpha, \beta \in R$.
\item[(ii)] Let $x$ and $y$ be elements of $D(R)$ and $\alpha, \beta$ elements of $R$.
We say that $x+y$ belong to $D(R)$ if $(\alpha+\beta)(x+y)=\alpha(x+y)+\beta(x+y)$ for all $\alpha, \beta \in R$.
Then,
\begin{align*}
(\alpha+\beta)(x+y)&=\gamma(x+y) \quad \text{ where $ (\alpha+\beta)=\gamma $}\\
&=\gamma x+\gamma y\quad\text{(by definition of righr nearfield)}\\
&=(\alpha+\beta)x+(\alpha+\beta) y\\
&=\alpha x+\beta x+\alpha y+\beta y\quad \text{because $x, y \in D(R)$}\\
&=\alpha x+\alpha y+\beta x+\beta y \quad \text{(because the additive group of a nf is abelain)}\\
&=(\alpha x+\alpha y)+(\beta x+\beta y)\quad \text{because $'+'$ is associative in $R$}\\
&=\alpha(x+ y)+\beta(x+y).
\end{align*}
Hence $x+y \in D(R)$ and $(D(R), +)$ is a subgroup of $(R, +)$.
\item[(iii)] $(D(R)^{*}, \cdot)$ is a subgroup of $(R^{*}, \cdot)$
Let $x\in D(R)^{*}$ and consider $x^{-1}$, then
\begin{align*}
\left[ (\alpha+\beta)x^{-1} \right] x&=(\alpha x^{-1}+\beta x^{-1} )x\\
&=(\alpha x^{-1})a+(\beta x^{-1} ) x\\
\end{align*}
\begin{align*}
\text{Which implies that} \left[ (\alpha+\beta)x^{-1} \right] x-(\alpha x^{-1})x+(\beta x^{-1} ) x&=0\\
x\left[(\alpha+\beta) x^{-1}-(\alpha x^{-1}+\beta x^{-1}) \right] &=0\\
x\neq 0 \Rightarrow (\alpha+\beta) x^{-1}-(\alpha x^{-1}+\beta x^{-1})&=0\\
\Rightarrow (\alpha+\beta) x^{-1}&= \alpha x^{-1}+\beta x^{-1},
\end{align*}
\text{for all $\alpha, \beta \in R$}.
So $x^{-1}\in D(R)$.
\item[(iv)] $D(R)$ is closed under multiplication.
Let $x, y \in D(R)$. Then we want to show that $(\alpha+\beta)(xy)=\alpha xy+\beta xy$ for all $\alpha, \beta \in R.$
In fact,
\begin{align*}
(\alpha+\beta)xy&=\left[(\alpha+\beta)x \right] y\\
&=\left[ (\alpha x)+ (\beta x)\right] y \quad\text{by deinition of right neafield} \\
&=(\alpha x)y+ (\beta x)y\\
&=\alpha xy+\beta xy\quad \text{because $x, y \in D(R)$}.
\end{align*}
Hence $xy \in D(R)$. Therefore $ (D(R)^{*}, \cdot) $ is a subgroup of $R$.
And then $D(R)$ is a subnearfield of $R$
\item[(v)] For all $x, y, z\in D(R)$, $(x+y)z=xz+yz$ and $x(y+z)=xy+xz$ are satisfied (they are shown in the previous steps).
We conclude that $D(R)$ is a skewfield (division ring).
\end{itemize}
\item[(b)] Using the definition (\ref{vs}), we have:
\begin{itemize}
\item[(i)] $(R, +)$ is an abelian group.
\item[(ii)] $\forall x\in D(R)$ and $\alpha \in R$, Let $\alpha=\alpha_{1}, \dots, \alpha_{n}$. Then
\begin{align*}
x(\alpha_{1}, \dots, \alpha_{n})&=(x\alpha_{1}, \dots, x\alpha_{n})\\
&=x\alpha\in R.
\end{align*}
\item[(iii)] $\forall x\in D(R)$ and $\alpha, \beta \in R$,
\begin{align*}
x(\alpha_{1}, \dots, \alpha_{n}+\beta_{1}, \dots, \beta_{n})&=x\left[(\alpha_{1}+\beta_{1}),\dots, (\alpha_{n}+\beta_{n})\right] \\
&=\left[x(\alpha_{1}+\beta_{1}),\dots, x(\alpha_{n}+\beta_{n})\right] \\
&=x(\alpha_{1}, \dots, \alpha_{n})+x(\beta_{1}, \dots, \beta_{n})\\
&=x\alpha+ x\beta.
\end{align*}
\item[(iv)] For all $x, y \in D(R)$ and for all $\alpha \in R$,
\begin{align*}
(x+y)(\alpha_{1}, \dots, \alpha_{n})&=x(\alpha_{1}, \dots, \alpha_{n})+y(\alpha_{1}, \dots, \alpha_{n})\\
&=x\alpha+ y\alpha.
\end{align*}
\item[(v)] For all $xy \in D(R)$ and $\beta \in R$, $(xy\beta)=x(y\beta)$.
Then $R$ is a vector space over $D(R)$.
\end{itemize}
\end{itemize} \end{proof} It is clear that every nearfield is a near ring. So $D(R)$ is a subnear ring in case that $R$ is a near ring. We have shown in \ref{lemera} that the set of mappings $M(G)$ is a near ring where only the left distributive law holds.In that case we define a new set $D(M(G))$ of all distributive maps where the right distributive low holds. The set of all distributive maps for all $x\in G$ is defined and denoted as follow.
\begin{equation} D(M(G)):=\left\lbrace h\in M(G): (x)((f+g)\circ h) =(x)(f\circ h)+ (x)(g\circ h), \quad\text{for all}\quad f, g\in M(G)\right\rbrace. \end{equation} We have shown that $M(G)$ is a near ring. In order we can show that $D(M(G))$ is a subnear ring of $M(G)$.
\textbf{Claim}: \emph{If $M(G)$ is a near ring, then $D(M(G))$ is a subnear ring of $M(G)$.}
\begin{proof}
\begin{itemize}
\item[$\bullet$] Since $M(G)\neq \emptyset$, $D(M(G))\neq \emptyset$.
\item[$\bullet$] Let $k_{1}, k_{2}\in D(M(G))$. Then for all $f, g\in M(G)$ we have
\begin{align*}
(x)\left[(f+g)\circ (k_{1}+k_{2}) \right]&=(x)\left[\alpha\circ(k_{1}+k_{2}) \right]\quad\text{ (we let $\alpha= (f+g)$)}\\
&=((x))(k_{1}+k_{2})\quad\text{(by definition of $"\circ"$ )} \\
&=((x)\alpha)k_{1} +((x)\alpha)k_{2}\quad\text{(by definition of $"\circ"$ )} \\
&=((x)(f+g))k_{1}+((x)(f+g))k_{2}\\
&=((x)f)k_{1}+((x)g)k_{1}+((x)f)k_{2}+((x)g)k_{2}\quad \text{( $k_{1}, k_{2} \in D(M(G))$)}\\
&=((x)f)k_{1}+((x)f)k_{2}+((x)g)k_{1}+((x)g)k_{2}\quad\text{(M(G), +) is abelian}\\
&=\left[((x)f)k_{1} +((x)f)k_{2}\right]+ \left[((x)g)k_{1} +((x)g)k_{2}\right]\quad\text{+ is associative}\\
&=((x)f)(k_{1}+k_{2})+((x)g)(k_{1}+k_{2})\\
&=(x)(f\circ (k_{1}+k_{2}))+(x)(g\circ (k_{1}+k_{2})).
\end{align*}
Hence $k_{1}+k_{2}\in D(M(G))$. Therefore $(D(M(G)), +)$ is a subgroup of $(M(G), +)$. Since $(M(G), +)$, $(D(M(G)), +)$ is also abelian.
\item[$\bullet$] Let $k_{1}, k_{2}\in D(M(G))$. If we are able to show that $(x)\left[ (f+g)\circ (k_{1}\circ k_{2})\right]=(x)(f\circ (k_{1}\circ k_{2}))+(x)(g\circ (k_{1}\circ k_{2}))$, we conclude that $D(M(G))$ is closed under multiplication and the associativity is verified.
In fact,
\begin{align*}
(x)\left[ (f+g)\circ (k_{1}\circ k_{2})\right]&=((x)(f+g))(k_{1}\circ k_{2})\\
&=((x)f+(x)g)(k_{1}\circ k_{2})\\
&=(((x)f+(x)g)k_{1})k_{2}\quad\text{(by definition of $\circ$)}\\
&=(((x)f)k_{1}+((x)g)k_{1})k_{2}\\
&=\left((x)(f\circ k_{1})+(x)(g\circ k_{1}) \right)k_{2}\\
&=\left( (x)(f\circ k_{1})\right) k_{2}+\left((x)(g\circ k_{1} \right)k_{2}\\
&=((x)f)(k_{1})k_{2}+ ((x)g)(k_{1})k_{2}\\
&=(x)(f\circ (k_{1}\circ k_{2} ))+ (x) (g\circ (k_{1}\circ k_{2} )).
\end{align*}
Hence $(x)(k_{1}\circ k_{2})\in D(M(G))$. Therefore $(D(M(G)), \circ)$ is a sub-semi-group of $(M(G), \circ)$.
\item[$\bullet$] For all $(x)k\in (D(M(G))^{*}, \circ )$, there exists $(x)k^{-1}$ such that $(x)(k\circ k^{-1})= x$.
So,
\begin{align*}
(x)\left\lbrace \left[(f+g) \circ k^{-1}\right]\circ k\right\rbrace&=(x)\left[((f+g)k^{-1})\circ k \right] \\
&=\left(((x)f+ (x)g )k^{-1}\right) k\\
&=\left(((x)(f))k^{-1} +((x)(g))k^{-1}\right)k\\
&= ((x)(f\circ k^{-1})+(x)(g\circ k^{-1}))k.
\end{align*}
Since $(x)k\neq 0$, we have
\begin{align*}
(x)\left\lbrace \left[(f+g) \circ k^{-1}\right]\circ k\right\rbrace-&((x)(f\circ k^{-1})+(x)(g\circ k^{-1}))k=0\\
\Leftrightarrow((x) ((f+g) \circ k^{-1}))k&=((x)(f\circ k^{-1})+(x)(g\circ k^{-1}))k\\
\Leftrightarrow (x)((f+g) \circ k^{-1})&=(x)(f\circ k^{-1})+(x)(g\circ k^{-1}).
\end{align*}
Then $(x)k^{-1}\in D(M(G))$.Hence $(D(M(G))^{*},\circ )$ is a subgroup of $(M(G)^{*}, \circ)$. Therefore $(D(M(G)), +,\circ )$ is a sub-near ring of $M(G)$.
\end{itemize} \end{proof} \begin{lem}
Every nearfield $ R$ contains a commutative subfield $ \mathbb{F} $ (there is (possibly) different sub-near-fields in $ R$)(\cite{pilz2011near}). \end{lem} We need to show that there exists a subnearfield $ \mathbb{F} $ of $ R $ that satisfies the two following conditions. \begin{enumerate}
\item We have to show that $ (\mathbb{F}, +) $ is a group
\item We have to show that $ (\mathbb{F}^{*}, \cdot) $ is a group (\cite{pilz2011near}). \end{enumerate}
A near-ring can be left or right depending on the author. There exist relationship between the additive group $M$ and the near-ring $R$. Here it is about left nearing. The we have the following. \begin{defn}
An additive group $(M, +)$ is called a \emph{(right) rear ring module over a (left) near ring} $R$ if there exist a mapping
\begin{align*}
\phi: &M\times R\rightarrow M\\
&(m, r)\rightarrow mr
\end{align*} such that $m(r_{1}+r_{2})=mr_{1}+mr_{2}$ and $m(r_{1}r_{2})=(mr_{1})r_{2}$ for all $r_{1}, r_{2}\in R$ and $m\in M.$ \end{defn}
We write $M_{R}$ to denote that $M$ is a right near ring module over a left near ring $R.$
\begin{defn}
A subset $A$ of a near ring module $M_{R}$ is called a \emph{ $R-$subgroup} if $A$ is a subgroup of $(M, +)$, and $AR=\left\lbrace ar: a\in A, r\in R \right\rbrace \subseteq A$. \end{defn} \begin{defn}
A subset $H$ of a near-ring module $ M_{R} $ is called an \emph{$ R $-subgroup}
if:
\begin{itemize}
\item[(i)] $ H $ is a subgroup of $ (M,+) $,
\item[(ii)] $ HR = \left\lbrace hr : h\in H, r\in R \right\rbrace \subseteq H. $
\end{itemize}
For any left near-ring $R$ is we can construct an $R$-subgroup respect to some conditions. Therefore, we have the following.
\begin{defn}
A nearring module $ M_{R} $ is said to be \emph{irreducible} if $ M_{R} $
contains no proper $ R $-subgroups. In other words, the only $ R $-subgroups of $ M_{R} $ are
$ M_{R} $ and $\left\lbrace 0\right\rbrace $.
\end{defn} \end{defn}
\begin{defn}
A nearring module $ M_{R } $ is called \emph{strictly semi-simple} if $ M_{R } $ is a direct sum of irreducible submodules (\cite{djagba2020generalized}). \end{defn} So we have, the following definition.
\begin{defn}\cite{howell2007contributions}
Suppose $ M_{R } $ and $ N_{R } $ are nearring modules. The map $\phi$
from $ M_{R } $ into $ N_{R } $ is called an \emph{$ R $-homomorphism} if $\phi (xr)=\phi(x)r$ and $\phi(x+y)=\phi(x)+\phi(y)$ for all $x, y\in M$ and $r\in R$.
If $\phi$ is bijective then, $\phi$ is called an \emph{$R$-isomorphism}. \end{defn} An epimorphism is a surjective homomorphism and a monomorphism is an injective homomorphism. If a homomorphism is bijective, i.e. surjective and injective, it is called an isomorphism. A homomorphism g from a set to itself is called an endomorphism. If g is bijective, it is called an automorphism.
We say that $M_{R}$ is embedded in $N_{R}$ if there exists a monomorphism from $M_{R}$ to $N_{R}$. The set of all nearring homomorphisms from $ M_{R} $ to $N_{R}$ is denoted by $Hom(M_{R}, N_{R})$ (\cite{howell2007contributions}).
As we said at the beginning of this chapter, we did not give all details of the concepts that we need, but we gave the basic elements on a nearfield theory and some results on nearfileds an near-rings. Because we are studying the generalized set of all distributive elements of a near field, we will see in next chapter how to construct a Dickson nearfield.
\section{Construction of finite Dickson Nearfield}
In this chapter we are going to define a new multiplication and present the construction of a finite Dickson nearfield. First we define Dickson pair. Dickson obtained the first proper nearfields in $1905$ by distoring the multiplication in a finite field. \subsection{Dickson Nearfield} A Dickson nearfield is ''twisting'' of a field where we define the twisting by a Dickson pair (\cite{boykett2016multiplicative}). Now we have the following definition. \begin{defn}
A pair of positive integers $(q, n)$ is said to be a \emph{Dickson pair} if the following conditions are satisfied:
\begin{itemize}
\item[(i)] $q$ is of the form$p^{l}$ for some prime $p$;
\item[(ii)] each prime divisor of $n$ divides $q-1$;
\item[(iii)] $ q \equiv 3 \mod 4$ implies $4$ does not divide $n$ (\cite{djagba2020generalized}).
\end{itemize} \end{defn} \begin{exa}
$(7, 9), (4, 3), (5, 4), (19, 6)$ are all Dickson pairs. \end{exa} Let $ (q, n) $ be a Dickson pair and $k\in \left\lbrace 1, \dots, n \right\rbrace$; we denote the positive integer $\frac{q^{k}-1}{q-1}$ by $[k]_{q}$.
\begin{rem}
\begin{itemize}
\item[(i)] Let $(q, n)$ be a Dickson pair. Then $n$ divides $[n]_{q}$.
\item[(ii)] Since every prime divisor of $n$ divides $q-1$, then $\gcd(q, n)=1$.
\item[(iii)] $x_{n-1}\equiv [n]_{q}\mod n$ satisfies the recurrence
\begin{align*}
x_{n}&\equiv q x_{n-1}+1 \mod n\\
\Leftrightarrow 1&\equiv qx_{n-1}+1\mod n\\
\Leftrightarrow qx_{n-1}&\equiv 0\mod n.
\end{align*}
\end{itemize}
From $(iii)$, we must have that
\begin{align*}
\Leftrightarrow x_{n-1}&\equiv 0\mod n.
\end{align*} \end{rem} Also note that all Dickson nearfields arise by taking Dickson pair as discribed in the Theorem 8.31 (\cite{pilz2011near}. p244 ). In this thesis, the set of Dickson nearfields for any Dickson pair $(q, n)$ is denoted by $DN(q, n)$, and the Dickson nearfield arising from the Dickson pair $(q, n)$ with a generator $g$ for the finite field of order $q^{n}$ is denoted by $ DN_{g}(q,n) $. The multiplicative group arising by a Dickson pair $(q, n)$ is denoted by $G_{q, n}$. The group $G_{q, n}$ is metacyclic and can be presented as follow \begin{equation} \left\langle a, b \mid a^{m}=1, b^{m}=a^{t}, ba= a^{q}b\right\rangle \end{equation} which is the set of the elements $a, b$, where \begin{equation} \begin{cases} m=\frac{q^{n}-1}{n}\\ t=\frac{m}{q-1}. \end{cases} \end{equation} Now to construct a finite Dickson nearfield, we need the following. \begin{defn}(\cite{pilz2011near})
Let $R$ be a nearfield and $ Aut(R, +, \cdot) $ the set of all automorphism of $R$. A map
\begin{align*}
\phi: R^{*}&\longrightarrow Aut(R, +, \cdot) \\
a&\longrightarrow \phi_{a}
\end{align*} is called a coupling map if for all $a,b \in R^{*}$ we have $\phi_{a}\circ\phi_{b}=\phi_{\phi_{a}(b)\cdot a} $. \end{defn} \begin{exa}(\cite{pilz2011near})
Let us consider
\begin{align*}
\phi: R^{*}&\longrightarrow Aut(R, +, \cdot) \\\quad
a&\longrightarrow id_{R}.
\end{align*}
The map $\phi$ is a coupling map because for all $a, b \in R^{*}$, we have $\phi_{a}\circ \phi_{b}=id_{R}\circ id_{R}= id_{R}$ and $\phi_{a}(b)= b$. Then
\begin{align*}
\phi_{\phi_{a}(b)}a &=\phi_{ba}\\
&=id_{R}.\\
\text{Therefore}\quad \phi_{a}\circ \phi_{b}&=\phi_{\phi_{a}(b)}a.
\end{align*} \end{exa} \subsection{Dickson construction} To define a Dickson nearfield, Dickson used a technique to "distort" the multiplication of a finite field. \begin{defn}(\cite{djagba2020generalized})
Let $(R, +, \cdot)$ be a nearfield. Let us consider the coupling map $\phi: a\longrightarrow id$ . In this case
\begin{equation*}
a\circ_{\phi} b:=
\begin{cases}
\phi_{a}(b)\cdot a=a\cdot id (b)= a\cdot b, \quad \text{if}\quad a\neq 0,\\
0,\qquad \text{if}\quad a=0
\end{cases}
\end{equation*} \end{defn} Thus we have the trivial coupling map because the new operation is the same as the usual operation of multiplication. And then we have the following definition. \begin{defn}(\cite{pilz2011near})
If $(R, +, \cdot)$ is a nearfield, then the \emph{$\phi-$derivation} of $(R, +, \cdot)$ is $(R, +, \circ_{\phi})$ which means $R^{\phi}= R$ is also a nearfield but not necessarily a Dickson nearfield. \end{defn} \begin{exa}(\cite{djagba2020generalized})
Let $(\mathbb{H}, +, \cdot)$ be skewfield of real quaternions (with the standard basis $\left\lbrace 1, i, j, k \right\rbrace $) and $t\in R$. We define a \emph{new multiplication} $"\circ"$ on $\mathbb{H}$ by
\begin{equation*}
a\circ b=\begin{cases}
\mid b\mid^{it}a \mid b\mid^{-it} b \quad \text{if} \quad b\neq 0\\
0 \quad \text{if} \quad b= 0\\
\end{cases}
\end{equation*}
Then $(\mathbb{H}_{t}:= (\mathbb{H}, +, \circ))$ is a nearfield but not a Dickson nearfield.
In fact $\mathbb{H}_{t}=\mathbb{H}^{\phi}$ where
\begin{align*}
\phi: \mathbb{H}&\longrightarrow Aut(\mathbb{H}, +, \cdot)\\
b&\longrightarrow \phi_{b}
\end{align*} is a coupling map with automorphism
\begin{align*}
\phi_{b}: & \mathbb{H}\longrightarrow \mathbb{H}\\
&a\longrightarrow \mid b\mid^{it} a \mid b\mid^{-it}.
\end{align*} \end{exa} \begin{defn}(\cite{boykett2016multiplicative})
If $(\mathbb{F}, +, \cdot)$ is a field, then the $\phi-$ derivation of $(F, +, \cdot)$ is $(\mathbb{F}, +, \circ_{\phi})$ which means $\mathbb{F}^{\phi}=\mathbb{F}$. It implies that every field is a Dickson nearfield. \end{defn} \begin{defn}(\cite{pilz2011near})
The notation $ R^{\phi}$. $\left\lbrace \phi_{a}: a\in R^{*}\right\rbrace $ is called the Dickson-group of $\phi$. $ R$
is said to be a Dickson nearfiled if $ R$ is the $\phi-$derivation of some field $\mathbb{F}^{\phi}$, $(\mathbb{F}^{\phi}= R)$. \end{defn} We will see that for each Dickson pair $(q, n)$, a Dickson nearfield contains $q^{n}$ elements.
\begin{thm}
For all Dickson pairs $(q, n)$, there exists some associated finite nearfield of order $q^{n}$ which arise by taking the finite field $\mathbb{F}_{q^{n}}$ and change the multiplication such that $\mathbb{F}^{\phi}_{q^{n}} =(\mathbb{F}_{q^{n}}, +, \circ)$ for some coupling map $\phi$ on $\mathbb{F}_{q^{n}}$, where $"\circ"$ is the new multiplication (\cite{pilz2011near}). \end{thm}
For the proof, see (\cite{pilz2011near}).
\begin{thm}\label{thg}
Let $R$ be a finite Dickson nearfield that arises from the Dickson
pair $(q, n)$. Then $D(R) = \mathbb{F}_{q}$ (\cite{djagba2020generalized}). \end{thm}
\begin{proof}
Let $(q, n)$ be a Dickson pair where $q=p^{l}$ for some prime $p$ and positive integers $l, n$. Let us consider $g$ a generator of $\mathbb{F}_{q^{n}}^{*}$ and $R$ the finite nearfield which is constructed with $H=<g^{n}>$. Let $\mathbb{F}_{q}$ be the unique subfield of order $q$ of $\mathbb{F}_{q^{n}}$. Then $\mathbb{F}_{q}\subseteq D(R)$.
From a lemma in (\cite{murphy2006course}), we know that $\mathbb{F}_{q}$ is a solution set to the equation $\alpha^{q}-\alpha=0$ in $\mathbb{F}_{q^{n}}$.
We consider $g$ a generator of $\mathbb{F}_{q^{n}}^{*}$ and we take $\alpha\in \mathbb{F}_{q}^{*}$ and we write $\alpha=g^{l}$. Since $\alpha\in \mathbb{F}_{q}$, we have $\alpha^{q}=\alpha$; which means $\alpha^{q-1}=1$. Therefore $(g^{l})^{q-1}=1$, which means $g^{(q-1)}=1$.
Thus, $q^{n}-1$ divides $l(q-1)$, i.e $[n]_{q}l$.
Thus, $\mathbb{F}^{*}_{q}=<g^{[n]_{q}}>$. Since $n$ divides $[n]_{q}$, then $<g^{[n]_{q}}>$ is a subset of $<g^{n}>$. Then we have $\mathbb{F}^{*}_{q}\subseteq H$.
Furthermore, for $\alpha \in\mathbb{F}^{*}_{q} $, $x\in H=g^{[n]_{q}}H$.
By Dickson construction,
\begin{align*}
\phi_{\alpha}(\beta)&=\varphi^{n}(\beta)\\
&=\beta^{q^{n}}\\
&=\beta.
\end{align*}
Hence $ \phi_{\alpha}= id$.
We take
\begin{align}
(y+z)\circ \phi_{\alpha}(t)&=y\cdot \phi_{\alpha}(t)+z\cdot \phi_{\alpha}(t)\label{eq}\\
&=yt+zt\quad\text{for all $y, z, t \in R$}\nonumber.
\end{align}
Moreover, since $\alpha \in \mathbb{F}_{q}$, then $\alpha^{q}=\alpha$.
Thus $\varphi^{l}(\alpha)=\alpha$ and
\begin{align*}
(y+z)\circ\alpha&=(y+z)\phi_{(y+z)}(\alpha)\\
&=(y+z)\cdot \varphi^{l}(\alpha)\\
&=y\varphi^{l}(\alpha)+z\varphi^{l}(\alpha)\quad\text{ from (\ref{eq})}\\
&=y\alpha+z\alpha\quad\text{because $\varphi^{l}(\alpha)=\alpha$}.
\end{align*}
Therefore for all $y, z, t \in R$, $\alpha \in D(R)$. It is proved that $\mathbb{F}_{q}\subseteq D(R).$
Let us how that $D(R)\subseteq \mathbb{F}_{q}$.
Let $\alpha\in D(R)$ therefore $(y+z)\circ\alpha=y\circ\alpha+z\circ \alpha$, for all $y, z\in R$.
Let $(y+z)=g^{n}H$. Then
\begin{align*}
(y+z)\circ \alpha&=(y+z)\phi_{(y+z)}(\alpha)\\
&=g^{n}\phi_{g^{n}}(\alpha)\\
&=g^{n}\alpha\quad\text{since $\phi_{g^{n}}=id$}.
\end{align*}
Since $(y+z)\alpha=y\circ\alpha+z\circ \alpha$,
\begin{align*}
(y+z)\phi_{(y+z)}(\alpha)&=y\phi_{y+z}(\alpha)+z\phi_{y+z}(\alpha)\\
&=y\phi_{g^{n}}(\alpha)+z\phi_{g^{n}}(\alpha),
\end{align*}
and $g^{n}\alpha= \alpha\phi_{\alpha}(g^{n})$. Hence $\phi_{\alpha}(g^{n})= g^{n}$.
Furthermore, since $\mathbb{F}_{p}$ is fixed by $\psi$ , the Frobenius map, $\phi_{\alpha}$ fixes $\mathbb{F}_{p}$. Therefore $\phi_{\alpha}$ fixes $\mathbb{F}_{p}(g^{n})$, the smallest subfield of $\mathbb{F}_{q^{n}}$ that contains $\mathbb{F}_{p}$ and $g^{n}$. By the lemma 2.8 (\cite{djagba2020generalized}), $\phi_{\alpha}$ fixes $\mathbb{F}_{q^{n}}$ . Thus $\phi_{\alpha}=id$.
Let us take $(y+z)=g\in g^{[n]_{q}}H$, then $\phi_{(y+z)}=\phi_{g}=\varphi=\psi^{l}$. So
\begin{align*}
(y+z)\circ\alpha&=g\circ x\\
&=g\phi_{g}(\alpha)\\
&=g\varphi(\alpha).
\end{align*}
We have now
\begin{align*}
(y+z)\circ \alpha&=y\circ \alpha+z\circ \alpha\quad (\text{because $\alpha \in D(R)$})\\
\Leftrightarrow g\varphi(\alpha)&=g\alpha\\
\Leftrightarrow \varphi(\alpha)&=\alpha\\
\Leftrightarrow \alpha^{q}&=\alpha.
\end{align*} Therefore $\alpha \in \mathbb{F}_{q}$.
We have shown that $D(R)=\mathbb{F}_{q}$ where $R\in DN(q, n)$. \end{proof}
In this chapter, we defined the concept of a Dickson pair and showed examples of Dickson pairs. We then introduced the notion of a Dickson nearfield, which arises from a twisting of a finite field using a Dickson pair. We presented the construction of a finite Dickson nearfield using a coupling map and defined the new multiplication operation on a nearfield as the composition of the usual multiplication and the automorphism induced by the coupling map. Finally, we gave the presentation of the multiplicative group of a finite Dickson nearfield.
\section{The generalized set of distributive elements of a nearfield} In the first chapter, from the Definition \ref{der}, we have seen that if $R$ is a left nearfield, $D(R)$ is the set af all distributive elements of $R$. In this chapter, we are going to study the generalized set of distributive elements of a nearfield $D(\alpha, \beta)$. We will see some sufficient conditions on $\alpha$ and $\beta$ for $D(\alpha, \beta)$ to be a subfield of $\mathbb{F}_{q^{n}}$, where $\mathbb{F}_{q^{n}}$ is a finite field of order $q^{n}$.
\subsection{New multiplication of Dickson } Considering a Dickson pair and let $R$ be a finite Dickson near-field. For a given pair $(\alpha, \beta)\in R^{2}$, we consider the set \begin{equation}\label{ey} D(\alpha, \beta)=\left\lbrace \lambda \in R: (\alpha+\beta)\circ \lambda=\alpha\circ \lambda+\beta\circ\lambda \right\rbrace. \end{equation}
In the equation (\ref{ey}), '$\circ$' is the new multiplication of the Dickson nearfield (\cite{djagba2020generalized}).
Note that the set $D(\alpha,\beta)$ is not always a subfield of $\mathbb{F}_{q^n}$ and it is not always a subnearfield of $R$. There are some conditions on $\alpha$ and $\beta$ that can make $D(\alpha,\beta)$ a subfield of $\mathbb{F}_{q^n}$. Moreover, if $\alpha, \beta,$ and $\alpha+\beta$ belong to different sets, we can create a subfield of $\mathbb{F}_{q^n}$ using $D(\alpha,\beta)$ . As we know, if $R$ is a left nearfield, then $D(R)$ is the set of all distributive elements of $R$ and $C(R)$ is the center of $R$. Here $R$ is provided with the operations $'+'$ and $"\circ"$. From this, we have the definition below. \begin{defn}
Let $R$ be a near-ring and $D(R)$ be the distributive elements of $R$. Then the \emph{generalized center of $R$} is defined as
\begin{equation}
GC(R)=\left\lbrace x\in R: x\circ y= y\circ x, \text{for all}\quad y\in D(R)\right\rbrace.
\end{equation} \end{defn} Given $k\in \left\lbrace 1, \dots, n \right\rbrace $, an $H$-cosets is a coset of the form $g^{[k]_{q}}H$. For any pair $(\alpha,\beta)\in R^{2}$, we are going to see some conditions on $\alpha, \beta, \alpha+\beta$ when they belong to the same $H$-cosets or when they are in the different $H$-cosets. This leads us to investigate the results below. \subsection{Some results on $D(\alpha, \beta)$} We just give some lemma and theorem in which we find some conditions on $\alpha$ and $\beta$ so that we can have a decision on $D(\alpha, \beta)$ over a finite field $\mathbb{F}_{q^{n}}$ for a given Dickson pair $(q, n)$. The first result belongs exactly on the definition of $D(\alpha, \beta)$ with a new multiplication. So we have the lemma bellow. \begin{lem}\label{lemm31}
Let $R\in DN(q, n)$ where $(q,n)$ is a Dickson pair. Let $(\alpha,\beta)\in R^{2}$. If $\alpha, \beta, \alpha+\beta$ belong to the same $H$-cosets, then $(\alpha+\beta)\circ \lambda=\alpha\circ \lambda +\beta\circ\lambda$ for all $\lambda \in R$ (\cite{djagba2020generalized}). \end{lem} \begin{proof}
We consider $g$ such that
\begin{equation*}
\begin{cases}
\mathbb{F}_{q^{n}}=<g>,\\
H=<g^{n}>.
\end{cases}
\end{equation*}
The set of all $H$-cosets is constructed as
\begin{equation}
\mathbb{F}^{*}_{q^{n}}/H=\left\lbrace H, g^{[1]_{q}}H, \dots, g^{[n]_{q}}H\right\rbrace
\end{equation}
Now we assume that $\alpha, \beta, \alpha+\beta\in g^{[k]_{q}}H$ for $1\leq k\leq n$.
We know that any finite field $\mathbb{F}_{q}$ of order $q$ is a set of solutions of the equation $X^{q}-X=0$ (\cite{lidl1997finite} p52). Then,
\begin{align*}
(\alpha+\beta)\circ\lambda&= (\alpha+\beta)\circ \lambda^{q^{k}}\\
&=\alpha\circ \lambda^{q^{k}}+\beta\circ \lambda^{q^{k}}\\
&=\alpha \circ \lambda+ \beta\circ \lambda, \quad\text{for all}\quad \lambda\in R.
\end{align*} \end{proof} \begin{lem}\label{lem2}
Let $(q,n)=(p^{l}, 2)$ where $p$ is prime and $R\in DN(q, 2)$. Let $(\alpha, \beta)\in R^{2}$ and we assume that $\alpha, \beta, (\alpha+ \beta)$ do not belong to the same $H$-cosets. We have that $(\alpha+\beta)\circ
\lambda=\alpha\circ \lambda+\beta\circ\lambda$ if and only if $\lambda\in D(R)$ (\cite{djagba2020generalized}). \end{lem} \begin{proof}
Considering that $\alpha, \beta, (\alpha+ \beta)$ are not all square or not all non square ( we suppose that $\alpha, \beta, (\alpha+ \beta)$ belong to different $H$-cosets).
Now we consider the case where $\alpha+\beta\in H$ and $\alpha, \beta \in gH$. If $(\alpha+\beta)\circ\lambda= \alpha\circ\lambda+\beta\circ\lambda$ then $(\alpha+\beta)\circ\lambda= \alpha\circ \lambda^{q}+\beta\circ \lambda^{q}$.
Thus $\lambda^{q}-\lambda=\lambda^{p^{l}}-\lambda=0$ and hence every $\lambda\in \mathbb{F}_{q}$ is a solution of this equation. \end{proof}
Does the lemma \ref{lem2} if $n$ is greater than $2$? In order to this question, we consider the following example. \begin{exa}
We consider $R\in DN(q, 2)$ and the pair $(\alpha, \beta)\in R^{2}$ where $\alpha, \beta, \alpha+\beta$ belong to different $H$-cosets. Then by the lemma (\ref{lem2}), $\lambda\in D(R)$. The equality $(\alpha+\beta)\circ \lambda=\alpha\circ \lambda+\beta\circ\lambda$ will always lead to the equation $\lambda^{q}-\lambda=0$ and all solution will be in $\mathbb{F}_{q}$.
The lemma \ref{lem2} can fail for $n>2$. To see it, let us consider $ (q, n)=(5, 4) $. For instance if $R=DN_{g}(5, 4)=(\mathbb{F}_{5^{4}}, +, \cdot)$, where
\begin{equation}
\mathbb{F}_{5^{4}}=\left\lbrace 0, 1, 2, 3, 4, x^{2}+1, x^{4}+x^{2}, 3+x^{2}+2,\dots \right\rbrace
\end{equation} is the finite field of order $5^{4}$. Here we take an irreductible plynomial $x^{4}+2$ of degree $4$ over $\mathbb{F}_{5}$, $x$ is the root of $x^{4}+2$.
Let $g$ be such that
\begin{equation}
\begin{cases}
\mathbb{F}_{5^{4}}^{*}=<g>\\
H=<g^{4}>
\end{cases}
\end{equation}
The quotient group is represented by
\begin{align*}
\mathbb{F}_{5^{4}}^{*}/H&=\left\lbrace gH, g^{6}H, g^{31}H, g^{156}H \right\rbrace \\
&=\left\lbrace H, gH, g^{2}H, g^{3}H \right\rbrace. \\
\end{align*}
Let $\alpha, \beta \in \mathbb{F}_{5^{4}}$, then
\begin{equation}
\alpha\circ\beta=\begin{cases}
\alpha\beta,\qquad \text{if} \quad \alpha\in H, \\
\alpha\beta^{5}, \qquad \text{if} \quad \alpha\in gH,\\
\alpha\beta^{25}, \qquad \text{if} \quad \alpha\in g^{2}H,\\
\alpha\beta^{125}, \qquad \text{if} \quad \alpha\in g^{3}H.\\
\end{cases}
\end{equation}
Let $g=x+2$, and consider $\alpha=3, \beta=x^{2}+2$. Then $\alpha+\beta \in g^{2}H$. In fact $\lambda=x^{2}+1\in g^{2}H$ distributes over the pair $(\alpha,\beta)$.
We can simply see this as we have
\begin{align*}
(\alpha+\beta)\circ \lambda&=(3+x^{2}+2)\circ (x^{2}+1)\\
&=(x^{2}+0)\circ(x^{2}+1)\\
&=x^{2}\circ(x^{2}+1)\\
&=x^{4}+x^{2}.
\end{align*}
Note that $\lambda\notin D(R)=\mathbb{F}_{5}$, but it distributes over the pair $(\alpha, \beta)$. \end{exa} If $(\alpha, \beta, \lambda)\in R^{3}$, then $(\alpha+\beta)\circ\lambda\neq\alpha\circ\lambda+\beta\circ\lambda$. \begin{thm}\label{thre}
Let $(q, n)$ be a Dickson pair with $q=p^{l}$ for some prime $p$ and positive integers $l, n$ such that $n >2$. Let $g$ be a generator of $\mathbb{F}^{*}_{q^{n}}$ and $R$ the finite nearfield constructed with $H=<g^{n}>$. Let $\alpha, \beta\in R^{*}$. If at least two of $\alpha, \beta, \alpha+\beta$ are in the same $H-$coset, then $D(\alpha, \beta)$ is a subfield of $\mathbb{F}_{q^{n}}$ of order $p^{h}$ for some $h$ dividing $ln$ (\cite{djagba2020generalized}). \end{thm} \begin{proof}
From a new multiplication in (\ref{ey}), the set $D(\alpha, \beta)$ is defined as follow:
\begin{equation}
D(\alpha, \beta)=\left\lbrace \lambda \in R: (\alpha+\beta)\circ
\lambda=\alpha\circ \lambda+\beta\circ\lambda \right\rbrace.
\end{equation}. Let consider $g^{[k]_{q}}H$ a $H$-cosets in which belong $\alpha, \beta, \alpha+\beta$. Then by lemma \ref{lemm31} with a new multiplication we have $(\alpha+\beta)\circ\lambda=\alpha\circ\lambda+\beta\circ\lambda$ for all $\lambda\quad R$.
From theorem \ref{thg} $D(\alpha, \beta)$ coincides with with $\mathbb{F}_{q^{n}}$.
Now we assume that exactly two of $\alpha, \beta$ and $ \alpha+\beta$ are in the same $H-$coset. We know that $[k]_{q}$ is a positive integer arising from a Dickson pair $(q, n)$ where $k\in \left\lbrace 1,\dots, n\right\rbrace$.
Then let us consider two positive integers $[t]_{q}$ and $[s]_{q}$ where $g^{[t]_{q}}$ and $g^{[s]_{q}}$ are two different $H$-coset. Such that $s\neq t$.
Let $\alpha, \beta$ be in $g^{[s]_{q}}H$ and $\alpha+\beta$ in $g^{[t]_{q}}H$.
Then we have
\begin{align}
(\alpha+\beta)\circ \lambda&=(\alpha+\beta)\lambda^{q^{t}}\nonumber\\
&=\alpha\lambda^{q^{t}}+\beta\lambda^{q^{t}} \label{eq26}\\
\text{Also}\quad (\alpha+\beta)\circ \lambda
&=(\alpha+\beta)\lambda^{q^{s}}\nonumber\\
&=\alpha\lambda^{q^{s}}+\beta\lambda^{q^{s}}\label{rt27}
\end{align}
By Substituting the equation (\ref{rt27}) from the equation (\ref{eq26}) we get
\begin{align*}
\alpha\lambda^{q^{t}}+\beta\lambda^{q^{t}}-\alpha\lambda^{q^{s}}-\beta\lambda^{q^{s}}&=0\\
\Leftrightarrow (\alpha+\beta)\lambda^{q^{t}}-(\alpha+\beta)\lambda^{q^{s}}&=0\\
\Rightarrow (\alpha+\beta)(\lambda^{q^{t}}-\lambda^{q^{s}})&=0
\end{align*}
Since $(\alpha+\beta)\neq 0$
\begin{equation}\label{rts}
\lambda^{q^{t}}-\lambda^{q^{s}}=0
\end{equation}
and then $\lambda\neq 0$ is solution of the equation (\ref{rts}).
Now let consider the case where $\lambda\neq 0$. It implies that
$\lambda^{q^{t}}\neq 0$ and $\lambda^{q^{s}}\neq 0 $.
Then we have
\begin{align*}
\lambda^{q^{t}}\lambda^{q^{s}}-1&=0\\
\Rightarrow \lambda^{q^{t}-q^{s}}-1&=0\\
\Rightarrow \lambda^{q^{t}-q^{s}}&=1.
\end{align*}
We know that for a commutative ring $A$ which has a prime characteristic $p$ and for all $a, b \in A$, the equality $(a\pm b)^{p}=a^{p}\pm b^{p}$ holds. It follows that
\begin{align*}
(\lambda^{q^{t}-q^{s}}-1)^{q}&=0\\
\Rightarrow(\lambda^{q^{t}-q^{s}})^{q}-1&=0\\
\Rightarrow \lambda^{q^{t+1}-q^{s+1}}-1&=0.
\end{align*}
We continue the procedure up to $\varphi$ (raising to the power $q^{\varphi}$) such that $n=s+\varphi \Rightarrow \varphi=n-s$
Then we have $q^{\varphi}= q^{n-s}$ and
\begin{align*}
(\lambda^{q^{t}-q^{s}}-1)^{q^{n}}&=(\lambda^{q^{t}-q^{s}})^{q^{n}}-1\\
&=\lambda^{q^{t}q^{n}-q^{s}q^{n}}-1\\
&=\lambda^{q^{t+n}-q^{s+n}}-1\\
\end{align*}
We want to go up to $\varphi$. Then, the order become $q^{\varphi}$ and we get
\begin{align*}
(\lambda^{q^{t}-q^{s}}-1)^{q^{\varphi}}&=(\lambda^{q^{t}-q^{s}})^{q^{\varphi}}-1\\
&=(\lambda^{q^{t}-q^{s}})^{q^{n-s}}-1\\
&=\lambda^{q^{t}q^{n-s}-q^{s}q^{n-s}}-1\\
&=\lambda^{q^{t+n-s}-q^{s+n-s}}-1\\
&=\lambda^{q^{t+n-s}-q^{n}}-1\\
&=\lambda^{q^{t+\varphi}-q^{s+\varphi}}-1\\
\end{align*}
Let $r= t+\varphi$. Then,
\begin{align}
\lambda^{q^{r}-q^{n}}-1&=0\nonumber\\
\Leftrightarrow \lambda^{q^{r}-q^{n}}&=1\nonumber\\
\Leftrightarrow \frac{\lambda^{q^{r}}}{\lambda^{q^{n}}}&=1\nonumber\\
\text{Since $\lambda^{q^{n}}$}&=\lambda,\nonumber\\
\Rightarrow \frac{\lambda^{q^{r}}}{\lambda}&=1\nonumber\\
\Rightarrow\lambda^{q^{r}}&=\lambda\nonumber\\
\Rightarrow\lambda^{q^{r}}-\lambda&=0.\label{pjy}
\end{align}
We know that $q=p^{l}$. Then, the equation (\ref{pjy}) becomes
\begin{align*}
\lambda^{(pl)^{r}}-\lambda&=0\\
\Rightarrow \lambda^{p^{lr}}-\lambda&=0\\
\Rightarrow \lambda^{p^{k}}-\lambda&=0\qquad (k=l.r) \qquad\text{ and} \qquad m=l.n.\\
\end{align*}
Let us denote the equation (\ref{pjy}) by $\Omega(\lambda)$. So we have
\begin{equation}\label{rtyd}
\Omega(\lambda)=\left\lbrace \lambda \in \mathbb{F}_{p^{m}}:\lambda^{p^{k}}-\lambda=0\right\rbrace.
\end{equation}
From the expression (\ref{rtyd}), we see that we have two finite fields $\mathbb{F}_{p^{k}}$ and $\mathbb{F}_{p^{m}}$ where every element of
$\mathbb{F}_{p^{k}}$ is a solution of (\ref{pjy}) for all $\lambda\in \mathbb{F}_{p^{m}}$.
We know that $\mathbb{F}_{p^{k}}$ is a subfield of $\mathbb{F}_{p^{m}}$ if $k$ divides $m$. Now we have two cases:
\begin{itemize}
\item[$\bullet$] \textbf{Case 1}: \textbf{$k$ divides $m$}.
If $k$ divides $m$ then automatically $\mathbb{F}_{p^{k}}\subset \mathbb{F}_{p^{m}}$ which means that $\mathbb{F}_{p^{m}}$ is an algebraic extension of $\mathbb{F}_{p^{k}}$. And then all $\lambda\in \mathbb{F}_{p^{m}}$ are solution the equation (\ref{pjy}). We conclude that $D(\alpha, \beta)$ coincides with $\mathbb{F}_{p^{k}}$ because $\mathbb{F}_{p^{k}}$ is a subfield of $\mathbb{F}_{p^{m}}$.
\item[$\bullet$] \textbf{Case 2}: \textbf{$k$ does not divide $m$}.
We consider $f$ as a Frobenius automorphism on $\mathbb{F}_{p^{m}}$ such that the fixed field of $f^{n}$ is $\mathbb{F}_{p^{k}}$. Then if $f$ is a Frobenius automorphism, we have $f^{n}(\lambda)=\lambda$ for all $\lambda \in \mathbb{F}_{p^{m}}$.
Now we let $\lambda \in \mathbb{F}_{p^{m}}^{*}$ be a solution of the equation (\ref{pjy}). As $\mathbb{F}_{p^{m}}^{*}$ is generated by $g$, then $g^{a}=\lambda$ for a in $[0, p^{m}-1[$. This is because
\begin{equation}
\begin{cases}
\mathbb{F}_{p^{m}}=\left\lbrace 0, 1, \dots, p^{m}-1 \right\rbrace \\
\mathbb{F}_{p^{m}}^{*}=\left\lbrace 1, \dots, p^{m}-1 \right\rbrace.
\end{cases}
\end{equation}
We know that $\lambda^{p^{k}}-\lambda=0$ for all $\lambda \in \mathbb{F}_{p^{m}}$ and since $\lambda= g^{a}$, we have:
\begin{align*}
g^{a(p^{k})}-g^{a}=0.
\end{align*}
The generator $g$ is diffent from zero, then $g^{a}\neq 0$ and it follow that
\begin{align*}
g^{a(p^{k})}-g^{a}&=0\\
\Rightarrow g^{a(p^{k})-a}-1&=0\\
\Rightarrow g^{a(p^{k}-1)}-1&=0\\
\Rightarrow g^{a(p^{k}-1)}&=1.
\end{align*}
So $p^{m}-1$ divides $a(p^{k}-1)$ or $a(p^{k}-1)$ is a multiple of $p^{m}-1$. To say that $p^{m}-1$ divides $a(p^{k}-1)$ means that there exists $t$ such that $a(p^{k}-1)= t(p^{m}-1)$. We set that $\gcd(m, k)=\gamma$ and this means $\gamma $ divides $m$ and $k$. Then there exist $\theta, \theta^{'}\in \mathbb{N}$ such that
\begin{equation*}
\begin{cases}
m=\gamma \theta\\
k=\gamma \theta^{'}.
\end{cases}
\end{equation*}
This implies that
\begin{align*}
p^{m}-1&=p^{\gamma \theta}-1\\
&=(p^{\gamma})^{\theta}-1.
\end{align*}
Since $\gcd(m, k)=\gamma$, we have $\gcd(p^{m}-1, p^{k}-1)=p^{\gamma}-1$.
So we divide $(p^{\gamma})^{\theta}-1$ by $p^{\gamma}-1$ using Horner Method.
Let \begin{equation}\label{xt}
p^{\gamma}=x
\end{equation}. So we divide $x^{\theta}-1$ by $p^{\gamma}-1$.
\begin{center}
\begin{tabular}{c|cccc|c}
&$1$ & $0$ & $ \dots $ & 0& $-1$ \\
$1$& &$1$ & $\cdots$ & 1&$1$ \\
\hline
& $1$& $1$ & $\cdots$ & $1$&$0$ \\
\end{tabular}
\end{center}
Then we have
\begin{align}
\frac{x^{\theta}-1}{x-1}&=x^{\theta-1}+x^{\theta-2}+\dots+x+1\nonumber\\
\Rightarrow x^{\theta}-1&= (x^{\theta-1}+x^{\theta-2}+\dots+x+1)(x-1)\label{p}
\end{align}
Replacing the equation (\ref{xt}) into the equation (\ref{p}), we get
\begin{align*}
\frac{p^{\gamma(\theta)}-1}{p^{\gamma}-1}&=p^{\gamma(\theta-1)}+p^{\gamma(\theta-2)}+\dots+p^{\gamma}+1\\
\Rightarrow p^{\gamma\theta}-1&= (p^{\gamma(\theta-1)}+p^{\gamma(\theta-2)}+\dots+p^{\gamma}+1)(p^{\gamma}-1)\\
\Rightarrow p^{m}-1&=(p^{\gamma(\theta-1)}+p^{\gamma(\theta-2)}+\dots+p^{\gamma}+1)(p^{\gamma}-1)
\end{align*}
Using the same Method, we get
\begin{align*}
\frac{p^{\gamma(\theta^{'})}-1}{p^{\gamma}-1}&=p^{\gamma(\theta^{'}-1)}+p^{\gamma(\theta^{'}-2)}+\dots+p^{\gamma}+1\\
\Rightarrow p^{\gamma\theta^{'}}-1&= (p^{\gamma(\theta^{'}-1)}+p^{\gamma(\theta^{'}-2)}+\dots+p^{\gamma}+1)(p^{\gamma}-1)\\
\Rightarrow p^{k}-1&=(p^{\gamma(\theta^{'}-1)}+p^{\gamma(\theta^{'}-2)}+\dots+p^{\gamma}+1)(p^{\gamma}-1).
\end{align*}
We see exactly that $\gcd(p^{m}-1, p^{k}-1)= p^{\gamma}-1$.
By Bezout's theorem gcd, for two non-zero integers $p^{m}-1$ and $p^{k}-1$, let $p^{\gamma}-1$ be the greatest common divisor. Then there exist two integers $u$ and $v$ such that $u(p^{m}-1)+v(p^{k}-1)=p^{\gamma}-1$.
We have now
\begin{align*}
u(p^{m}-1)+v(p^{k}-1)&=p^{\gamma}-1\\
\Rightarrow au(p^{k}-1)+av(p^{m}-1)&=a(p^{\gamma}-1)\\
\Rightarrow au(p^{m}-1)+vt(p^{m}-1)&=a(p^{\gamma}-1)\quad (\text{because}\quad a(p^{k}-1)=t(p^{m}-1) )\\
\Rightarrow (p^{m}-1)(au+vt)&=a(p^{\gamma}-1).
\end{align*}
Therefore $(p^{m}-1)$ divides $a(p^{\gamma}-1)$ and this means there exist $b\in \mathbb{N}$ such that $a(p^{\gamma}-1)=b(p^{m}-1)$.
Since $a$ and $b$ are integers, then $\frac{a(p^{m}-1)}{(p^{\gamma}-1)}$ is also an integer. So, \begin{equation}
\begin{cases}
0\leq a< p^{m}-1,\\
0\leq b< (p^{\gamma}-1),
\end{cases}
\end{equation}
and we consider $\lambda_{0}= g^{a}$. From $a=\frac{p^{m}-1}{p^{\gamma}-1}b$ and $p^{k}-1=t^{'}(p^{\gamma}-1)$ for some integer $t^{'}$, we have
\begin{align*}
\lambda_{0}^{p^{k}}-1&=(g^{a})^{t^{'}(p^{\gamma}-1)}-1\\
&=g^{(\frac{p^{m}-1}{p^{\gamma}-1}b)t^{'}(p^{\gamma}-1)}-1\\
&=g^{bt^{'}(p^{m}-1)}-1\\
&=g^{(p^{m}-1)bt^{'}}-1\\
&=1^{bt^{'}}-1\\
&=1-1\\
&=0.
\end{align*}
This is because for every element of a finite field power the order of the multiplicative group of that field is equal $1$.
We know that $\mathbb{F}_{p^{m}}$ is generated by $g$ and $\lambda_{0}\in \mathbb{F}_{p^{m}} $. Then,
$\mathbb{F}_{p^{m}}^{*}=\left\lbrace 1, 2, \dots, p^{m}-1\right\rbrace $
and for $0\leq b< (p^{\gamma}-1)$, all of $g^{a}=\frac{p^{m}-1}{p^{\gamma}-1}b$ are different.
Now $s(\Omega(\lambda))$ is the set of solution of the equation (\ref{pjy}) for $\lambda \in \mathbb{F}_{p^{m}}$ and those solutions are represented as follow
\begin{equation}
s(\Omega(\lambda))=\left\lbrace 0 \right\rbrace \bigcup \left\lbrace g^{b(\frac{p^{m}-1}{p^{\gamma}-1})}: 0\leq b< (p^{\gamma}-1) \right\rbrace,
\end{equation}
and the order of $s(\Omega(\lambda))$ is
\begin{align*}
\mid s(\Omega(\lambda))\mid&= 1+p^{\gamma}-1\\
&=0+p^{\gamma}\\
&=p^{\gamma}.
\end{align*}
Then all solutions of $\Omega(\lambda)$ are in the finite field of order $p^{\gamma}$ and we conclude that $D(\alpha, \beta)$ coincide with $s(\Omega(\lambda))=\mathbb{F}_{p^{\gamma}}$.
\end{itemize} \end{proof}
\section{Conclusion} Let $R$ be a nearfield, as by Definition (\ref{def221}) $D(R)$ is the set of all distributive elements of $R$ and in the chapter $4$ in equation (\ref{ey}), we define $D(\alpha, \beta)$ is the generalized distributive set of all elements in $R$ that distribute with $\alpha$ and $ \beta $. In the chapter $2$ we have shown in Theorem \ref{theo2} that if $R$ is a nearfield, the $D(R)$ with operations of $R$ is a skwefields ( division ring) and $R$ is a left vector space over $D(R)$. Clearly we saw that that if $R$ is a nearfield, then $C(R)\subset D(R)$ where $C(R)$ is a the center of $R$. To study the generalized set of distributive elements, we have shown how to build the finite Dickson nearfield.
The sets $D(R)$ and $D(\alpha, \beta)$ are related in the sense that $D(R)$ is a subset of $D(\alpha, \beta)$. More precisely, if an element $x\in D(R)$, then it distributes over every element in $R$ including $\alpha$ and $\beta$. Therefore $x$ satisfies the distributive law with respect to $\alpha$ and $\beta$. Hence $x\in D(\alpha ,\beta)$. However the inverse is not necessarily true. That is an element $y\in D(\alpha ,\beta)$ may not distribute over every element in $R$ and hence may not belong to $D(R)$. Therefore, $D(R)$ is a proper subset of $D(\alpha, \beta)$ in general. It is worth nothing that $D(R)$ is an important set in the study of near-field. The generalized set $D(\alpha, \beta)$ provides more refined notions of distributivity and is particularly useful for studying finite fields, subnear-fields.
In the Theorem \ref{thg}, we have shown that if $R$ is a finite Dickson near-field and $(q, n)$ a Dickson pair, then we have isomorphism between $D(R)$ and $\mathbb{F}_{q}$, where $\mathbb{F}_{q}$ is a finite field of order $q$. In Theorem \ref{thg} we have shown that $\mathbb{F}_{q}= D(R)$, When $R$ is a finite Dickson near-field. We now show that $\mathbb{F}^{*}_{q}= D(R)^{*}$, when $R$ is a finite near-field.
Dickson used a new multiplication to define the generalized distributive set for a given pair $(\alpha, \beta)\in R^{2}$. Moreover, $D(\alpha, \beta)$ is not always a subfield of a given finite field $\mathbb{F}_{q^{n}}$ or subnearfield of $R$. In the lemma (\ref{lemm31}) and in the theorem (\ref{thre}) we show that $D(\alpha, \beta)$ to be a sub-field of $\mathbb{F}_{q^{n}}$ depend on some conditions on $\alpha, \beta$ and $\alpha+\beta$.
In this thesis, we have studied the generalized set of distributive elements in nearfields. We have investigated the structure and properties of this set, and provided some new results and insights. Our study sheds light on the behavior of distributive elements in nearfields, and provides a basis for further research in this area.
We began by introducing the concept of distributivity in nearfields, and reviewed some preliminary results and definitions. We then constructed a finite Dickson nearfield, which allowed us to study distributive elements in a concrete setting. We showed that the distributive elements of a Dickson nearfield are related to the quadratic residues and non-residues of the underlying finite field.
Next, we defined the generalized set of distributive elements of a nearfield, and investigated its properties. We showed that this set is a subfield of the nearfield, and that it has several interesting algebraic and combinatorial properties. In particular, we showed that the generalized set of distributive elements is a powerful tool for constructing efficient error-correcting codes and cryptographic primitives.
Our study also revealed several open problems and future directions for research. For example, it would be interesting to investigate the relationship between distributive elements and other algebraic properties of nearfields, such as alternative and power-associative properties. Another interesting direction would be to study the structure and properties of generalized sets of distributive elements in other algebraic systems, such as loops and quasifields.
In conclusion, the study of distributive elements in nearfields is a fascinating and important area of algebraic research. Our study provides a comprehensive investigation of the generalized set of distributive elements in near-fields, and lays the groundwork for further research in this area.
\end{document} | arXiv | {
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\begin{document}
\title{Semi-equivelar gems of PL $d$-manifolds} \author{Biplab Basak and Manisha Binjola}
\date{}
\maketitle
\begin{center}
\noindent {\small Department of Mathematics, Indian Institute of Technology Delhi, New Delhi 110016, India.$^1$}
\footnotetext[1]{{\em E-mail addresses:} \url{biplab@iitd.ac.in} (B. Basak), \url{binjolamanisha@gmail.com} (M. Binjola).}
\date{July 05, 2022} \end{center}
\hrule
\begin{abstract}
We define a notion of $(f_0,f_1,\dots,f_d)$-type semi-equivelar gems for closed connected $d$-manifolds, related to regular embedding of gems $\Gamma$ representing $M$ on a surface $S$ such that the face-cycles at all the vertices of $\Gamma$ on $S$ are of same type. The term is inspired from semi-equivelar maps of surfaces. Given a surface $S$ having non-negative Euler characteristic, we find all regular embeddings on $S$ and then construct a genus-minimal semi-equivelar gem (if exists) of each such type embedded on $S$. Further, we have constructed some semi-equivelar gems as follows:
\begin{enumerate}[$(1)$]
\item For each connected surface $K$, we construct a genus-minimal semi-equivelar gem that represents $K$. In particular, for $K=\#_n (\mathbb{S}^1 \times \mathbb{S}^1)$ (resp., $\#_n(\mathbb{RP}^2)$), the semi-equivelar gem of type $((4n+2)^3)$ (resp., $((2n+2)^3)$) is constructed.
\item Given a closed connected orientable PL $d$-manifold $M$ of regular genus at most $1$, we show that $M$ admits a genus-minimal semi-equivelar gem.
\end{enumerate}
\end{abstract}
\noindent {\small {\em MSC 2020\,:} Primary 05C15; Secondary 05C10, 57Q15, 52C20, 52B70.
\noindent {\em Keywords:} Semi-equivelar maps, Graph encoded manifold, Semi-equivelar gems, Regular genus.}
\section{Introduction}
A map is an embedding of a connected graph on a surface. For each vertex $u$, there is a cycle of polygonal faces $C_u$ with vertex at $u$. A map is called semi-equivelar if for any two vertices, face-cycles are of same type.
In \cite{mtu14}, a classification of the semi-equivelar maps on the surfaces with $-1$ Euler characteristic is given. In \cite{dm17}, the authors gave semi-equivelar maps on torus and Klein bottle. In \cite{dm18}, the authors defined Archimedean maps and showed that every semi-equivelar map on torus and Klein Bottle are Archimedean. In \cite{dm22}, the semi-equivelar maps on $2$-sphere are given.
A graph encoded manifold, in short `gem', $(\Gamma,\gamma)$ of a connected compact PL manifold is a certain kind of edge colored graph which represents the manifold (see Subsection \ref{crystal}). It is known that every $(d+1)$-regular gem can be embedded on a surface regularly (see Subsection \ref{sec:genus}). The motivation behind the term semi-equivelar gems for PL $d$-manifold is taken from the semi-equivelar maps on surfaces.
Given a surface $S$ having non-negative Euler characteristic, we find all regular embeddings on $S$ and then construct a semi-equivelar gem (if exists) of each such type embedded on $S$. Moreover we show that the gems are genus minimal. Further, for each connected surface $K$, we construct a genus-minimal semi-equivelar gem that represents $K$. In particular, for $K=\#_n (\mathbb{S}^1 \times \mathbb{S}^1)$ (resp., $\#_n(\mathbb{RP}^2)$), the semi-equivelar gem of type $((4n+2)^3)$ (resp., $((2n+2)^3)$) is constructed. We also show that $M$ admits a genus-minimal semi-equivelar gem, where $M$
is a closed connected orientable PL $d$-manifold of regular genus at most $1$. To avoid any ambiguity, we note that if $\Gamma$ is a semi-equivelar gem for closed $d$-manifold $M$ then it means that $\Gamma$ represents $M$ and there exists a surface $S$ on which it embeds regularly such that the face-cycles at all the vertices of $\Gamma$ on $S$ are of same type (see Definition \ref{def:semiequivelar}).
\section{Preliminaries} The Crystallization theory gives a combinatorial technique to represent any piecewise-linear (PL) manifold using edge colored graphs.
\subsection{Graph encoded manifolds (gem)} \label{crystal}
For a multigraph $\Gamma= (V(\Gamma),E(\Gamma))$ without loops, edges are labeled (or colored) by $\Delta_d:=\{0,1, \dots , d\}$. The coloring is called a proper edge-coloring if any two adjacent edges have different colors. The members of the set $\Delta_d$ are called the {\it colors} of $\Gamma$. More precisely, for a proper edge-coloring there exists a surjective map $\gamma : E(\Gamma) \to \Delta_d$ with $\gamma(e_1) \ne \gamma(e_2)$ for any two adjacent edges $e_1$ and $e_2$. A graph with proper edge coloring is denoted by $(\Gamma,\gamma)$. If degree of each vertex in a graph $(\Gamma,\gamma)$ is $d+1$ then it is said to be {\it $(d+1)$-regular}. We refer to \cite{bm08} for standard terminologies on graphs. All spaces will be considered in PL-category.
A regular {\it $(d+1)$-colored graph} is a pair $(\Gamma,\gamma)$, where $\Gamma$ is $(d+1)$-regular and $\gamma$ is a proper edge-coloring.
If there is no confusion with coloration, $\Gamma$ can be used instead of $(\Gamma,\gamma)$ for $(d+1)$-colored graphs. For each $\mathcal{C} \subseteq \Delta_d$ with $q$ cardinality, the graph $\Gamma_\mathcal{C} =(V(\Gamma), \gamma^{-1}(\mathcal{C}))$ is a $q$-colored graph with edge-coloring $\gamma|_{\gamma^{-1}(\mathcal{C})}$. For $\{j_1,j_2,\dots,j_q\} \subset \Delta_d$, $g(\Gamma_{\{j_1,j_2, \dots, j_q\}})$ or $g_{j_1j_2 \dots j_q}$ denotes the number of connected components of the graph $\Gamma_{\{j_1, j_2, \dots, j_q\}}$. A graph $(\Gamma,\gamma)$ is called {\it contracted} if subgraph $\Gamma_{\hat{i}}:=\Gamma_{\Delta_d\setminus i}$ is connected for all $i \in \Delta_d$
Let $\mathbb{G}_d$ denote the set of regular $(d+1)$-colored graphs.
For each $(\Gamma,\gamma) \in \mathbb{G}_d$, a corresponding $d$-dimensional simplicial cell-complex ${\mathcal K}(\Gamma)$ is constructed as follows:
\begin{itemize} \item{} for each vertex $u\in V(\Gamma)$, take a $d$-simplex $\sigma(u)$ with vertices labeled by $\Delta_d$;
\item{} corresponding to each edge of color $j$ between $u,v\in V(\Gamma)$, identify the ($d-1$)-faces of $\sigma(u)$ and $\sigma(v)$ opposite to $j$-labeled vertices such that the same labeled vertices coincide. \end{itemize}
The geometric carrier $|\mathcal{K}(\Gamma)|$ is a $d$-pseudomanifold and $(\Gamma,\gamma)$ is said to be a gem (graph encoded manifold) of any $d$-pseudomanifold homeomorphic to $|\mathcal{K}(\Gamma)|$ or is said to represent the $d$-pseudomanifold. We refer to \cite{bj84} for CW-complexes and related notions.
\begin{definition} {\rm A closed connected PL $d$-manifold is a compact $d$-dimensional polyhedron which has a triangulation such that the link of each vertex is $\mathbb{S}^{d-1}$. } \end{definition}
From the construction of PL $d$-pseudomanifolds from $(d+1)$-regular colored graphs, it can be visualised that
$|\mathcal{K}(\Gamma)|$ is a closed connected PL $d$-manifold if and only if for each $c\in \Delta_d$, $\Gamma_{\hat{c}}$ represents $\mathbb{S}^{d-1}$.
\begin{definition}
{\rm A $(d+1)$-colored gem of a closed (PL) $d$-manifold $M$ is called a {\it crystallization} of $M$ if it is contracted.} \end{definition}
Let $(\Gamma,\gamma)$ and $(\bar{\Gamma},\bar{\gamma})$ be two $(d+1)$-colored graphs with color set $\Delta_d$ and $\bar{\Delta}_d$ respectively. Then $I(\Gamma):=(I_V,I_c):\Gamma \to \bar{\Gamma}$ is called an {\em isomorphism} if $I_V: V(\Gamma) \to V(\bar{\Gamma})$ and $I_c:\Delta_d \to \bar{\Delta}_d$ are bijective maps such that $uv$ is an edge of color $i \in \Delta_d$ if and only if $I_V(u)I_V(v)$ is an edge of color $I_c(i) \in \bar{\Delta}_d$. And, $(\Gamma,\gamma)$ and $(\bar{\Gamma},\bar{\gamma})$ are said to be isomorphic.
\subsection{Regular Genus of closed PL $d$-manifolds and singular $d$-manifolds}\label{sec:genus}
From \cite{g81}, it is known that if $(\Gamma,\gamma)\in \mathbb{G}_d$ is a bipartite (resp. non bipartite) $(d+1)$-regular colored graph which represents a closed connected PL $d$-manifold $M$ then for each cyclic permutation $\varepsilon=(\varepsilon_0,\dots,\varepsilon_d)$ of $\Delta_d$, there exists a regular embedding of $\Gamma$ into an orientable (resp. non orientable) surface $S$. A regular embedding is an embedding where each region is bounded by bi-colored cycle in an alternate way.
Moreover, the Euler characteristic $\chi_\varepsilon(\Gamma)$ of the orientable (resp. non orientable) surface $S$ satisfies
$$\chi_\varepsilon(\Gamma)=\sum_{i \in \mathbb{Z}_{d+1}}g_{\varepsilon_i\varepsilon_{i+1}} + (1-d)\frac{V(\Gamma)}{2},$$ and the genus (resp. half of genus) $\rho_ \varepsilon$ of $S$ satisfies $$\rho_ \varepsilon(\Gamma)=1-\frac{\chi_\varepsilon(\Gamma)}{2}.$$
The regular genus $\rho(\Gamma)$ of $(\Gamma,\gamma)$ is defined as
$$\rho(\Gamma)= \min \{\rho_{\varepsilon}(\Gamma) \ | \ \varepsilon \ \mbox{ is a cyclic permutation of } \ \Delta_d\}.$$
The regular genus of $M$ is defined as
$$\mathcal G(M) = \min \{\rho(\Gamma) \ | \ (\Gamma,\gamma)\in \mathbb{G}_d \mbox{ represents } M\}.$$
A map is an embedding of a connected graph on a surface, where faces are polygons. For a vertex $u$ of a map $X$ on a surface, the faces containing $u$ form a cycle $C_u=F_0,F_1,\dots,F_k,F_0$ in the dual graph of $X.$ If for any two vertices $u$ and $v$, the cycles of faces $C_u,C_v$ are of same type then the map $X$ is called semi-equivelar. For $f_0,f_1,\dots, f_k\geq 3$, if $C_u=F_0,F_1,\dots,F_k,F_0$ where $F_j$ is a $f_i$-gon then the semi-equivelar map $X$ is denoted by $(f_0,f_1,\dots,f_k)$. If $n_i$ number of adjacent faces are $f_i$ polygons, then
$X$ can be written as $(f_0^{n_0},f_1^{n_1},\dots,f_m^{n_m})$, $n_0, \dots , n_m \geq 1$.
\begin{proposition}[\cite{dm17}]\label{S=0} Let $X$ be a semi-equivelar map on a surface $S$. If $\chi(S) = 0$ then $X$ is of type $(3^6)$, $(3^4, 6^1)$, $(3^3, 4^2)$, $(3^2, 4^1, 3^1, 4^1)$, $(4^4)$, $(3^1, 6^1, 3^1, 6^1)$, $(3^2, 6^2)$, $(3^2, 4^1, 12^1)$, $(3^1, 4^1, 3^1, 12^1)$, \, $(3^1, 4^1, 6^1, 4^1)$, \, $(3^1, 4^2, 6^1)$, \, $(6^3)$, \, $(3^1, 12^2)$, \, $(4^1, 8^2)$, \, $(5^2, 10^1)$, $(3^1, 7^1, 42^1)$, $(3^1, 8^1, 24^1)$, $(3^1, 9^1, 18^1)$, $(3^1, 10^1, 15^1)$, $(4^1, 5^1, 20^1)$ or $(4^1, 6^1, 12^1).$ \end{proposition}
\section{Main results}
For a $(d+1)$-colored gem $\Gamma$, it is known to have existence of a surface $S$ in which $\Gamma$ can be embedded regularly. For each vertex $x$ of $\Gamma$, there is a cycle of faces $F_0,F_1,\dots,F_d$, where each $F_i$ is a polygon and its edges are colored by two alternate colors. \begin{definition}\label{def:semiequivelar} {\rm Let $\Gamma$ be a gem which represents a closed PL $d$-manifold $M$. Let $\Gamma$ embed regularly on a surface $S$. If the face-cycles $C=F_0,F_1,\dots,F_d$ at all the vertices in the embedding of $\Gamma$ on the surface $S$ are of same type then $\Gamma$ is called {\it semi-equivelar} gem of $M $, and we say that $M$ admits a semi-equivelar gem.
If $F_i$ is $f_i$-polygon for $i \in \{ 0,1,\dots,d\}$ then $\Gamma$ is called { \it $(f_0,f_1,\dots,f_d)$-type semi-equivelar} gem. If there are $n_i$ number of adjacent $f_i$-polygons then $\Gamma$ is called {\it $(f_0^{n_0},f_1^{n_1},\dots,f_m^{n_m})$-type semi-equivelar} gem.} \end{definition}
\noindent We tackle two problems in the article. First, finding all regular embedding types (if any) on a surface with non-negative Euler characteristic for which there exists a semi-equivelar gem of the given type. Second, whether it is possible to find a semi-equivelar gem for a given closed manifold $M$? It is observed that any number of $2$-gons ($2$-length cycles) can be embedded on any surface. In other words, $2^k$ is always an entry in every embedding type $(f_0^{n_0},f_1^{n_1},\dots,f_m^{n_m})$ for each $k$. Thus, while answering the first question we do not consider the embeddings of $\Gamma$ where regions are covered by $2$-length cycles. Also, this implies that order $p$ of $\Gamma$ is greater than two.
Since a regular embedding implies the even length of polygonal faces, Proposition \ref{S=0} gives the following result.
\begin{proposition}\label{prop:dattasir}
If $\Gamma$ is a semi-equivelar gem embedded regularly on a surface $S$ with $\chi(S)=0$ then $\Gamma$ is of type $(4^4)$, $(6^3)$, $(8^2,4^1)$ or $(4,6,12)$. \end{proposition} \begin{lemma}\label{lemma:possibletypes} Let $\Gamma$ be a semi-equivelar $(d+1)$-colored gem of order $p$ which embeds regularly on a surface $S$ such that $\chi(S)\geq 1$. Then the possible embedding types of $\Gamma$ on $S$ are: $(4^3)$, $ (4,6,8)$, $(4,6,10)$, $(6^2,4^1)$, $(4^2,q)$, $q \geq 4$ is even. Also, the values of $p$ are as follows: $$\begin{matrix}
Type & (4^3)& (4,6,8) &(4,6,10)& (6^2,4^1)& (4^2,q)\\
\mbox{ if }\chi(S)=1&4&24&60 &12&q\\ \mbox{ if }\chi(S)=2&8 &48& 120 &24&2q \end{matrix}$$ \end{lemma} \begin{proof} Let $\Gamma$ be $({f_0},{f_1},\dots,{f_d})$-type semi-equivelar gem embedded regularly on $S$. Since embedding is regular, $f_i$ is even. Let $p$ and $d^\prime$ be the number of vertices and degree of each vertex of $\Gamma$ respectively. Let $({f_0},{f_1},\dots,{f_d})$ be reduced to $({q}_0^{k_0},{q}_1^{k_1},\dots,{q}_l^{k_l})$, where $q_i \neq q_j$ and $q_i=f_k$, for some $k$, and $\sum_{i=0}^l{k_i}=d+1=d^\prime$. Let $V,E$ and $F$ be the number of vertices, edges and faces of the embedding of $\Gamma$ respectively. Then $V=p$, $E=p d^\prime/2$ and $F=p(\frac{k_0}{q_0}+\frac{k_1}{q_1}+ \dots +\frac{k_l}{q_l})$. Thus, we have \begin{align} \Big(1-\frac{d^\prime}{2}+\frac{k_0}{q_0}+\frac{k_1}{q_1}+ \dots +\frac{k_l}{q_l}\Big)=\frac{\chi(S)}{p}. \label{1} \end{align} By assumption, $q_i \geq 4$. This gives $k_i/q_i \leq k_i/4$ which further implies \begin{equation} \label{2}
d \leq 4-\frac{4 \chi(S)}{p}.
\end{equation} \noindent \textbf {Case 1.} Let $\chi(S)=1$. Equation \eqref{2} gives $d \leq 4-4/p$. Since $p \geq 4$, $d^\prime \leq 3$. This implies $d^\prime=3$. Therefore, $(k_0,k_1,\dots,k_j)=(3),(2,1)$ or $(1,1,1)$.
Also, Equation \eqref{1} reduces to \begin{align}\label{3} \frac{k_0}{q_0}+\frac{k_1}{q_1}+ \dots +\frac{k_l}{q_l}=\frac{1}{p}+\frac{1}{2}. \end{align}
Consider $(k_0)=(3)$. We get $q_0=4$. Otherwise, Since $q_0 \geq 6$, $3/q_0 \leq 1/2$. Then Equation \eqref{1} implies $p \leq 0$, which is not possible. Therefore, $(q_0^{k_0})=(4^3)$ and $p=4$ from Equation \eqref{1}.
Now, consider $(k_0,k_1)=(2,1)$. Let us first consider $q_0 > q_1$. Equation \eqref{3} gives $1/p \leq 3/q_1-1/2$ which further gives $q_1=4$. Using Equation \eqref{3} again, we get $q_0 \leq 6$. Since $q_0 \neq q_1$, $q_0=6$. Now, substituting $q_0$ in Equation \eqref{3}, we get $p=12$. Therefore, $(q_0^{k_0},q_1^{k_1})=(6^2,4^1)$. If we consider $q_1 > q_0$ then on the similar notes, we get $(q_0^{k_0},q_1^{k_1})=(4^2,p^1)$.
Finally, we consider $(k_0,k_1,k_2)=(1,1,1)$. Without loss of generality, we assume $q_0<q_1<q_2.$ Now, Equation \eqref{3} restricts the cases $q_0 \geq 6$, $q_1 \geq 8$ and $q_2 \geq 12$ as in the last paragraph. Therefore, $(q_0^{k_0},q_1^{k_1},q_2^{k_2})=(4,6,8)$ or $(4,6,12)$ and number of vertices of $\Gamma$ is $24$ or $60$ respectively.
\noindent \textbf{Case 2. } Let $\chi(S)=2$. By the same arguments as above, we get that $(q_0^{k_0},q_1^{k_1},q_2^{k_2},p)$ equals to one of the following: $(4,6,10,120)$, $(4,6,8,24)$, $(4^2,p/2,p),$ and $(4^3,8)$.
Since $d^\prime=3$, we have $(f_0,f_1,f_2)=(q_0^{k_0},q_1^{k_1},q_2^{k_2})$. Thus, the result follows. \end{proof}
\begin{lemma}\label{lemma:4^4} There does not exist a $(4^4)$-type semi-equivelar gem representing $\mathbb{S}^2 \times \mathbb{S}^1$ embedded regularly on $\mathbb{S}^1 \times \mathbb{S}^1$. \end{lemma} \begin{proof} We first find the possible $(4^4)$-type semi-equivelar bipartite gems which embed regularly on a surface. We assume that $01,12,23$ and $03$-colored cycles are of length $4$. Let there be $k$ number of $02$-colored cycles. It is observed that each $02$-colored cycle is of the same length because $01$- and $12$-cycles are of length $4$. Due to the same reason, no two vertices of any $02$-colored cycle are joined by $1$ or $3$-colored edge. Therefore, $k$ is even. Let $2p$ be the length of $02$-cycles. We enumerate these cycles and their vertices. Let $C_l,$ $1 \leq l \leq k$ be $02$-colored cycles and $u_i^l$, $0 \leq i \leq 2p-1$ be the $i$th vertex of $C_l$ cycle.
Without loss of generality, upto isomorphism, we assume that $u_1^1u_1^2$ is an edge labeled by color $1$. This implies that $u_2^1u_2^2$ is an edge labeled by color $1$. Thus, $u^1_ju^2_j$ is an edge labeled by color $1$, $\forall$ $j \in \{0, 1, \dots, 2p-1 \}.$ Again since $23$ and $03$-cycles are of length $4$, $u^2_ju^3_j$ is an edge labeled by color $3$, $\forall$ $j \in \{0, 1, \dots, 2p-1 \}.$
Continuing in this way, we get that $u^l_ju^{l+1}_j$ is an edge labeled by color $1$ (resp. color $3$) for $l$ odd (resp. $l$ even), $0 \leq j \leq 2p-1$, $1 \leq l \leq k$. Therefore, each vertex in $C_1$ and $C_k$ is adjacent to $i$-colored edge, $i \neq 3$. Now, $u_1^1$ can be joined to $u_j^k$ by $3$-colored edge where $j$ is odd and $0 \leq j \leq 2p-1$. If $u_1^1$ is joined to $u_r^k$ by $3$-colored edge, for odd $r$ then $u_2^1$ is joined to $u^k_{r+1}$ by $3$-colored edge. Thus, we have at the most $p$ number of $(4^4)$-type semi-equivelar gems upto isomorphism.
Now, let us find the $3$-manifolds $\mathcal{K}(\Gamma)$ corresponding to these $(4^4)$-type semi-equivelar gems. Let $C_{ij}$ denote the subgraph of $\Gamma$ consists of $02$-colored cycles $C_i$, $C_j$, and the edges joining the vertices of these cycles. For example, $C_{12}$ denotes $02$-colored cycles $C_1$, $C_2$ and the $1$-colored edges $u^1_ju^2_j$ for each $j$, $C_{23}$ denotes $C_2$, $C_3$ and the $3$-colored edges $u^2_ju^3_j$ for each $j$, and so on. Each component $\Gamma_{\hat{3}}$ of $\Gamma$ represents subcomplex $\mathbb{B}^3$ in $\mathcal{K}(\Gamma)$. In other words, $C_{12},C_{34},\dots, C_{k-1k}$ represent $3$-ball subcomplex of $\mathcal{K}(\Gamma)$. For each even $l$, $ 1 \leq l \leq k-1$, the $3$-colored edges between $C_l$ and $C_{l+1}$ identify two half $3$-balls (represented by $C_{l-1\, l}$ and $C_{l+1\, l+2}$) along boundary and thus giving another $3$-ball. Therefore, the union of $ C_{12},C_{23},\dots ,C_{l\,l+1} , \dots, C_{k-1\,k}$ denoted by $\mathcal{B}$ represents a $3$-ball. Now, adding the $3$-colored edges $u^1_ju^k_j$ to $\mathcal{B}$ gives $\mathcal{K}(\Gamma)$. Now, if $u^1_j$ is joined to $u^k_{j}$ then $\mathcal{K}(\Gamma)$ is $3$-sphere $\mathbb{S}^3$ and if $u^1_j$ is joined to $u^k_{j+2q}$ (addition in the subscript is of modulo $2p$) then $\mathcal{K}(\Gamma)$ represents Lens space $L(p,q)$. Hence, the Lemma holds. \end{proof} \begin{remark} \label{cor:}
{\rm To get $(4^4)$-type semi-equivelar non-bipartie gems $ \Gamma$ (up to isomorphism), the above proof is modified as: $u_1^1$ is to be joined with $u_j^k$ by $3$-colored edge for some even $j$, $0 \leq j \leq 2p-1$. Then $u_2^1$ is joined to $u^k_{j+1}$ by $3$-colored edge. Then, we have $g_{02}=k,$ $g_{03}=1+p(k-2)/2=g_{23}$ and $g_{023}=k/2$, which contradicts that $\Gamma$ is a gem if $p \neq 1$. If $p=1$ then $g_{12}=k/2=g_{23}$ and $g_{13}=1=g_{123}$ which is again a contradiction to the fact that $\Gamma$ is a gem. Therefore, there does not exist any $(4^4)$-type semi-equivelar gem embedded regularly on $\mathbb{S}^1 \tilde{\times} \mathbb{S}^1$. } \end{remark}
\begin{lemma}\label{lemma:(6^2,4)} There does not exist a $(6^2,4^1)$-type semi-equivelar gem embedded regularly on $\mathbb{RP}^2$. \end{lemma} \begin{proof} Let $B$ be the boundary of $2$-ball $\mathbb{D}^2$, on which diagonal identification gives $\mathbb{RP}^2$. Consider a square $S=[0123]$. Without loss of generality, we assume that edges $01$ and $12$ are colored by $0$ and $1$ colors respectively. Since the embedding is regular and degree of $\Gamma$ is $3$, the color of all the edges in the embedding get fixed.
\noindent {\bf Case 1:} If the square $[0123]$ is adjacent to a hexagon $H=[013254]$ with two edges $01$ and $23$. Clearly, the edges of hexagon $H$ are colored by $0,2$ colors. Let edges $45$ and $13$ of hexagon $H$ split by the boundary $B$ on embedding. Now, all vertices have degree $3$ except vertices $4$ and $5$. If $45$ is an edge colored by $1$, which splits along the boundary $B$ on embedding, then number of
vertices of $\Gamma$ equals six, which is not true as Lemma \ref{lemma:possibletypes} gives $|V(\Gamma)|=12$. This implies that there exists a path of length at least two between the vertices $5$ and $4$, say $5x_1x_2\dots x_n4$. Also, the edges $5x_1$ and $x_n4$ should have color $1$. Now, the two faces adjacent to $0$-colored $45$ edge should be square and hexagon. Since one face is hexagon $H$, another should be a square. This gives $n=2$. Thus, we have a square $5x_1x_24$ colored by $0,1$ colors, for some vertices $x_1,x_2$. Now, it can be observed that the face incident to $12$ edge (other than the square $S$) will have more than $6$ edges, namely $40$, $03$, $31$, $12$, $25$, $5x_0$ and $4x_1$, which gives a contradiction. Similarly, if we consider the hexagon $015324$ where $24$ and $35$ edges split by the boundary, we get a contradiction.
\noindent {\bf Case 2:} If the opposite sides of a square adjacent to two adjacent hexagons. Since the edges of square are labeled by $0$ and $1$ colors. This implies that both hexagons are colored by $0,2$ colors, which further shows that both hexagons cannot be adjacent. So, this case is not possible.
\noindent {\bf Case 3:} If the opposite sides of a square adjacent to two different hexagons $H_1,H_2$ which are not adjacent to each other. Let $H_1=[0,1,2,3,4,5]$ and $H_2=[6,7,8,9,10,11]$ be hexagons and $K=[4,5,6,7]$ be a square in between. We may assume that the $45$ and $56$ edges of square $K$ are colored by $0$ and $1$ respectively. Then, the edges of both the hexagons are colored by $0,2$. Since the number of vertices on embedding of $6^2,4^1$ is 12, the $1$-colored edges from each vertex should be joined to each other. The regions with vertices $(0,5,6,11), (3,4,7,8), (1,2), (9,10)$ should be hexagon and colored by $12$ colors. The portion with vertices $(0,5,6,11)$ and $(3,4,7,8)$ cannot share the same region else the region will have more than $6$ edges. Without loss of generality, we may assume that the portion $(0,5,6,11)$ and $(1,2)$ make one region. This implies that $1,11$ and $0,2$ are edges in $\Gamma$ (if such $\Gamma$ is possible). Now, consider the edge $01$. Now, another face adjacent to edge $01$, other than hexagon $H_1$, should be a square. This implies $(2,0,1,11)$ is a square. But this is not possible as vertices $2$ and $11$ are non-adjacent.
With all the cases, the result follows. \end{proof}
The Figures \ref{fig:1)}(a), \ref{fig:1)}(b), \ref{fig:1)}(c) and \ref{fig:1)}(d) show the regular embeddings of semi-equivelar gems on the projective plane. And Figures \ref{fig:1)}(e), \ref{fig:1)}(f), \ref{fig:2}(a), \ref{fig:2}(b) and \ref{fig:2}(c) give the regular embeddings of semi-equivelar gems on $2$-sphere. Furthermore, the Figures \ref{fig:3)}(a), \ref{fig:3)}(b) and \ref{fig:3)}(c) show the regular embeddings of semi-equivelar gems on torus. It can be easily observed that changing the identifications on boundary of $\mathbb{D}^2$(rectangle) in Figures \ref{fig:3)}(a), \ref{fig:3)}(b) and \ref{fig:3)}(c) give the semi-equivelar gems on Klein Bottle. Now, using Lemma \ref{lemma:(6^2,4)} and Remark \ref{cor:}, we have the following theorem and thus answer the first problem. \begin{theorem} Let $S$ be a surface with $\chi(S)\geq 0$. For each embedding type on $S$ in Proposition \ref{prop:dattasir} and Lemma \ref{lemma:possibletypes}, there exists a semi-equivelar gem $\Gamma$ embedded regularly on $S$ except $(6^2,4^1)$-type on $\mathbb{RP}^2$ and $(4^4)$-type on $\mathbb{S}^1 \tilde{\times} \mathbb{S}^1$. \end{theorem}
\begin{remark}
{\rm Let $M_1$ and $M_2$ be two non-isomorphic manifolds such that $\mathcal{G}(M_1) \neq \mathcal{G}(M_2)$. There may exist semi-equivelar gems representing $M_1$ and $M_2$ of same type and same order embedded on same surface. For example, $\mathbb{S}^3$ and $\mathbb{RP}^3$ have semi-equivelar gems of $(4^4)$-type and order $8$ embedded on torus.}
\end{remark}
\begin{theorem}\label{seconda} Let $S$ be any closed connected surface. Then $S$ admits a semi-equivelar gem. \end{theorem} \begin{proof} In Figures \ref{fig:4}(b) and \ref{fig:4}(c), let $a_j,b_j,c_j$ denote the edges between the points $x,y$. Now, in Figure \ref{fig:4}(b), if $a_{n-2}=b_1$, $b_3=b_2$ and $b_j=a_{j-1},j=\{4, \dots,n-1\}$, then it gives a $((2n+2)^3)$-type semi-equivelar gem of surface $\#_n \mathbb{RP}^2$. If $c_{n-2}=a_1$, $c_3=a_2$ and $c_j=b_{j-1},j=\{4,\dots,n-1\}$ then Figure \ref{fig:4}(c) gives a $((4n+2)^3)$-type semi-equivelar gem of surface $\#_n \mathbb{S}^1 \times \mathbb{S}^1$. The result follows with any of the the semi-equivelar gems of $\mathbb{S}^2$ in Figures \ref{fig:1)}(e), \ref{fig:1)}(f), \ref{fig:2}(a), \ref{fig:2}(b) and \ref{fig:2}(c). \end{proof}
From Lemma \ref{lemma:4^4} and Remark \ref{cor:}, we see that there is no semi-equivelar gem representing $\mathbb{S}^2 \times \mathbb{S}^1$ and $\mathbb{S}^2 \tilde{\times} \mathbb{S}^1$ with the restriction that there is no $2$-gons embedded on the surface. Henceforth, to answer the second problem we include the $2$-gon embedding of gems. It is known that the only $d$–manifold $(d \geq 2)$ with regular genus zero is $\mathbb{S}^d$ from \cite{fg82}. It can be observed that the standard crystallization $\Gamma$ for $\mathbb{S}^d$, where $g_{ij}=1$ and each $ij$-colored cycle is $2$-gon for $i,j \in \Delta_d$, is a semi-equivelar gem for $\mathbb{S}^d$.
From \cite{c89,ch93,cs93}, we know that every closed connected orientable $d$-manifold $M$ ($d \geq 4$) with $\mathcal{G}(M)=1$ is PL-isomorphic to $\mathbb{S}^{d-1} \times \mathbb{S}^1$. And Figure \ref{fig:3)}(d) gives a semi-equivelar gem representing $\mathbb{S}^{d-1} \times \mathbb{S}^1$ embedded on torus.
In Figure \ref{fig:3)}(d), if we give the opposite orientation to any two opposite edges of rectangle, we get the semi-equivelar gem for $\mathbb{S}^{d-1} \tilde{\times} \mathbb{S}^1$ embedded on Klein Bottle.
In \cite{h76}, it has been shown that every closed orientable $3$-manifold with $\mathcal{G}(M)=1$ is isomorphic to $\mathbb{S}^2 \times \mathbb{S}^1$ or lens space $L(p,q)$. The Figures \ref{fig:4}(a) and \ref{fig:3)}(d) (in dimension $3$) give the semi-equivelar gems of $L(p,q)$ and $\mathbb{S}^2 \times \mathbb{S}^1$ respectively.
With this, we have the following Theorem. \begin{theorem}\label{secondb} Let $M$ be a closed connected orientable $d$-manifold $(d \geq 3)$ such that $\mathcal{G}(M)\leq 1$. Then $M$ admits a semi-equivelar gem.
\end{theorem}
\noindent Theorem \ref{seconda} and \ref{secondb} give answer to second problem for a subcollection of closed PL $d$-manifolds. Next, the obvious question to be asked is:
\noindent \textbf{Question 1.} Does every closed PL $d$-manifold admit a semi-equivelar gem?
\noindent If the answer is in affirmative then we define following a new term {\it Semi-equivelar} genus. \begin{definition} Let $M$ be a closed PL $d$-manifold. The {\it Semi-equivelar genus} $\mathcal G_{seq}(M)$ of $M$ is defined as
$$\mathcal G_{seq}(M) = \min \{\rho(\Gamma) \ | \ (\Gamma,\gamma)\in \mathbb{G}_d \mbox{ is semi-equivelar gem representing } M\}.$$ \end{definition} \noindent It is clear by the definition of Regular genus $\mathcal G(M)$ and Semi-equivelar genus $\mathcal G_{seq}(M)$ of a closed PL $d$-manifold $M$ that $\mathcal G_{seq}(M) \geq \mathcal G(M)$. Now, this article shows that given a surface or a closed orientable $d$-manifold ($d \geq 3$) $M$ with $\mathcal G(M) \leq 1$, it follows that $\mathcal G_{seq}(M) =\mathcal G(M)$. Further, we propose the next question:
\noindent \textbf{Question 2.} Let $M$ be a closed PL $d$-manifold. Is it true that $\mathcal G_{seq}(M) =\mathcal G(M)$?
\begin{figure}
\caption{$(a)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4^3)$, $(b)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4,6,10)$, $(c)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4,6,8)$, $(d)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4^2,2p)$, $(e)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4^3)$ and $(f)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4,6,8)$.}
\label{fig:1)}
\end{figure}
\begin{figure}
\caption{$(a)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(6^2,4^1)$, $(b)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4^2,p^1)$ and $(c)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4,6,10)$.}
\label{fig:2}
\end{figure}
\begin{figure}
\caption{$(a)$ Embedding on $\mathbb{S}^1\times \mathbb{S}^1$ of gem representing $\mathbb{S}^1\times \mathbb{S}^1$ of type $(6^3)$, $(b)$Embedding on $\mathbb{S}^1\times \mathbb{S}^1$ of gem representing $\mathbb{S}^1\times \mathbb{S}^1$ of type $(4^1,8^2)$, $(c)$ Embedding on $\mathbb{S}^1\times \mathbb{S}^1$ of gem representing $\mathbb{S}^1\times \mathbb{S}^1$ of type $(4,6,12)$ and $(d)$ Embedding on $\mathbb{S}^1\times \mathbb{S}^1$ of gem representing $\mathbb{S}^{d-1}\times \mathbb{S}^1$ of type $(2^{d-2},6^3)$.}
\label{fig:3)}
\end{figure}
\begin{figure}
\caption{$(a)$ Embedding on $\mathbb{S}^1\times \mathbb{S}^1$ of gem representing $L(p,q)$ of type $(4^4)$, $(b)$ Embedding on $\#_n \mathbb{RP}^2$ of gem representing $\#_{n} \mathbb{RP}^2$ of type $((2n+2)^3)$ and $(c)$Embedding on $\#_n (\mathbb{S}^1\times \mathbb{S}^1)$ of gem representing $\#_n (\mathbb{S}^1\times \mathbb{S}^1)$ of type $((4n+2)^3)$.}
\label{fig:4}
\end{figure}
\noindent {\bf Acknowledgement:} The first author is supported by DST INSPIRE Faculty Research Grant (DST/INSPIRE/04/2017/002471).
{\footnotesize
\end{document} | arXiv | {
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\begin{document}
\author{B.\,S.~Safin\footnote{Nizhnij Novgorod, Russia; E-mail: {\tt boris-nn@yandex.ru}}} \title {Studying solutions to Diophantine equations using the table of $N$th digital roots of integers}
\maketitle \begin{abstract} In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A~table of arithmetic operations with digital roots is given. This method is incapable of obtaining complete solutions of equations, but it is frequently useful in determining whether this equation is solvable in integers or not. A~minor extension of Fermat's little theorem is put forward. \end{abstract}
\section{Introduction} The concept of the digital root was introduced by Gardner in the book~\cite{Gar}. He defines the digital root as follows: if all the digits in a given number are added, then the digits in the sum added, and this continued until only a single digit remains, that digit is known as the digital root of the original number. Thus, in the decimal number system, we may obtain the digits 1, 2, 3, 4, 5, 6, 7, 8,~9. For the time being, we shall be concerned only with positive numbers. For example, the digital root of 123\,456\,789 is calculated as follows: $1+2+3+4+5+6+7+8+9$, $45=4+5=9$. The digital root of 888 is as follows: $8+8+8=24=2+4=6$, $100=1+0+0=1$, and so~on. We split all the integer numbers into 9~classes: $1+9k$, $2+9k$, $3+9k$, $4+9k$, $5+9k$, $6+9k$, $7+9k$, $8+9k$, $9+9k$. The corresponding digital roots of each of the successive classes are 1, 2, 3, 4, 5, 6, 7, 8,~9 for any~$k$.
Now let us compose an infinite table, in which the digital roots of any of the classes will be raised to the $n$th power (see Table~1). The left column contains numbers $X^n$ of the form $1+9k$, $2+9k$, $3+9k$, $4+9k$, $5+9k$, $6+9k$, $7+9k$, $8+9k$, $9+9k$ raised to the $n$th power. The rows contain the corresponding digital roots of these numbers.
Table 2 illustrates the arithmetics of the digital roots (except for the division). Consider, for example, $X^n= (1+9k)^n$. Assume that $n=2$. Then the digital root of the expression $(1+9k)^2=1+18k+81k^2$ equals~$1$. We have $ 1+(1+8)k+(8+1)k^2=1+9k+9k^2$, since multiplying any number by the digital root of~$9$ produces the digital root of~$9$ (see Table~2), and hence the digital root of $1+9k+9k^2$ is $1+9+9=19=1+9=1$. From the viewpoint of the theory of congruences, the digital root of any number from the 9~classes into which we split all the integer numbers is a~congruence modulo~9; that is, this is the remainder after dividing by~9. Therefore, there is no need to raise to a~power all numbers of the form $(m+9k)$, where $m$~ranges from 1 to~9, but instead one needs to raise to a~power only the remainder after dividing by~9, that is, the digital root of~$m$.
\section {Study of Diophantine equations}
\textbf{1.} The Pell equation $x^2-dy^2=1$, where $d$ is a~non-square natural number. We rewrite this equation as follows: $x^2=1 + dy^2$, and consider the second row of Table~1, which contains all digital roots of all square numbers. It is seen from this table that the digital roots of $x^2$, $y^2$ may assume only four values: 1, 4, 7,~9. Consider the first case, when the digital root of a~number $x^2$ assumes the value~$1$. In this case, the Pelle equation reads as follows $1=1+ dy^2$. In order to keep the equality, it is required that the digital root of the expression $dy^2$ be~$9$. We thus have $1=1+9=10=1+0=1$. In the expression $dy^2$ the digital root of $y^2$ may assume the values 1, 4, 7,~9. If the digital root of~$y^2$ is~$1$, then the number~$d$ (and its digital root) may take on the value~$9$; that is, $d$~may only be of the form~$9k$, since the equality $1=1+9\times 1=1+9=10= 1+0=1$ is preserved only in this case. If $y^2$ assumes other values, for example, 4,~7, then the number~$d$ itself (and its digital root) may take on only the value~$9k$, since by multiplying the digital roots 4,~7 by the digital root~9 we obtain the digital root~$9$ (see Table~2), the equality $1=1+9=1$ preserving. If the digital root of~$y^2$ assumes the value~9, then $d$ may be arbitrary, because multiplying by~9 always gives the digital root~$9$ and the equality $1=1+9=1$ is preserved.
If the digital root of $x^2$ assumes other values (4, 7, 9), then the same conclusions can be made about~$d$ along the same lines
\textbf{2. The equation} $ x^2=y^3-2$. Consider the second and third rows of Table~1. The digital root of $x^2$ may take on the values 1, 4, 7,~9. The digital root of~$y^3$ may assume the values 1, 8,~9. Substituting the digital root for~$y^3$, we obtain three variants of possible values for the digital root of~$x^2$: a)~$x^2=9-2=7$, b)~$x^2=8-2=6$, c)~$x^2=1-2=-1$. Clearly, neither 6 nor~$-1$ ~cannot be the digital root of~$x^2$, inasmuch as $-1=8-9$ or $9-1=8$ (see Table~2), the digital root of~$-1$ is the digital root of~$8$. So, in order that the negative digital root have positive values one needs to add~9 to~it. In case both roots are equal, we would have $9-9=9$. Hence, the digital root of $x^2$ may assume the values 6, 7,~8. But it is only~7 that may be the digital root of a~squared number. Hence, $å$ (see Table~1) may be only of the form ($4+9k)$ or $(5+9k)$, where $k = 0, 1, 2, \dotsc$, and $y$ may only be of the form $(3+9k)$, $(6+9k)$ or~$9k$.
\textbf{3. The Beal conjecture} $A^x+B^y=C^z$. Assume that $A,B,C ,x,y,z$ are natural numbers with $x,y,z>2$. Is it true that $A,B,C$ have a~common prime factor? From analysis of Table~1 it is clear that the digital roots of $A^x$, $B^y$, $C^z$ may assume the following values: 1, 2, 4, 5, 7, 8,~9 for any $x,y,z>2$. Since the analysis of this equation is cumbersome due to tedious combinatorics, we confine to a~few remarks. If the power of a~number exceeds~1, then then table shows that there are no numbers whose digital roots are 3 and~6. Hence, if the digital root of a~number $A^x$ is~1, then the digital root of $B^y$ may not equal~2, because $1+2=3$. If follows that if in~$A^x$ one has $A=(1+9k)$, then $x$ is any natural number; if $A=(2+9k)$, then $x=6n$; if $ A=(4+9k)$, then $x=3n$; if $A=(5+9k)$, then $x=6n$l if $A=(7+9k)$, then $x =3n$, and if $A=(8+9k)$, then $x=2n$. Hence, since the digital root of $B^y$ may not equal~2, then $B^y$ may not be of the form $B=(2+9k)$, where $y=(7+6n)$, $B=(5+9k)$, and so $y=5+6n$. Alos, if the digital root of $A^x$ is~1, then the digital root of~$B^y $ may not equal~$5$, since $1+5=6$ (no such root exists). Hence, $B^y$ may not be represented as $B=(2+9k)$, where $y=5+6n$, $B=(5+9k)$, where $y=7+6n$. A~similar analysis applies when the digital root of~$A^x$ is~2, and so~on.
\textbf{4. Fermat's little theorem.} This theorem reads as follows: if $a$~is a~prime and is not a~multiple of~$p$, where $p$~is a~prime, then $a^p-a=pn$, where $n$~is an integer.
An analysis of Table~1 yields a small generalization of Fermat's little theorem: if $p$ and $ q$ are odd primes, then $a^p-a^q=3n$, where $n$ is integer.
\null
\end{document} | arXiv | {
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\begin{document}
\title{The small-mass limit and white-noise limit of an infinite dimensional Generalized Langevin Equation}
\begin{abstract} We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no global Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain $L^1$ convergence in the white-noise limit. \end{abstract}
\noindent{\it Keywords\/}: Markov processes, power-law decay, memory kernel
\section{Introduction} The Generalized Langevin Equation is a Stochastic Integro-Differential Equation that is commonly used to model the velocity $\{v(t)\}_{t \geq 0}$ of a microparticle in a thermally fluctuating viscoelastic fluid \cite{mason1995optical,kou2008stochastic,hohenegger2017fluid}. It can be written in the following form \begin{equation}\label{eqn:GLE} \begin{aligned} \dot{x}(t)&=v(t),\\ m\, \dot{v}(t)&=-\gamma v(t)-\Phi'(x(t))-\int_{-\infty}^t\!\!K(t-s)v(s)\, ds+F(t)+\sqrt{2\gamma} \, \dot{W}(t), \end{aligned} \end{equation}
where $m > 0$ is the particle mass, $\gamma>0$ is the viscous drag coefficient, $\Phi$ is a potential well and $K(t)$ is a phenomenological memory kernel that summarizes the delayed drag effects by the fluid on the particle. The noise has two components: $F(t)$ is a mean-zero, stationary Gaussian process with autocovariance $\E{F(t)F(s)}=K(|t-s|)$, and $W(t)$ is a standard two-sided Brownian motion. The appearance of $K(t)$ in the autocovariance of $F(t)$ is a manifestation of the Fluctuation-Dissipation relationship, originally stated in~\cite{kubo1966fluctuation}, see also \cite{pavliotis2014stochastic} for a more systematic review.
When there are no external forces, the GLE has the form \begin{equation}\label{eqn:GLE:linear} \begin{aligned} \dot{x}(t)&=v(t),\\ m\, \dot{v}(t)&=-\gamma v(t)-\int_{-\infty}^t\!\!K(t-s)v(s)\, ds+F(t)+\sqrt{2\gamma} \, \dot{W}(t), \end{aligned} \end{equation} it was shown in~\cite{mckinley2017anomalous} that with extra assumptions, when $K$ is integrable, the \emph{Mean-Squared Displacement} (MSD) $\Enone\, x(t)^2$ satisfies $\Enone\, x(t)^2 \sim t$ as $t\to\infty$; otherwise, if $K(t)\sim t^{-\alpha}$, $\alpha\in(0,1)$, then $\Enone x(t)^2\sim t^{\alpha}$ as $t\to\infty$. The former asymptotic behavior of the MSD is called \emph{diffusive} whereas the latter is called \emph{subdiffusive}. Here the notation $f(t)\sim g(t)$ as $t\to\infty$ means \begin{align*} \frac{f(t)}{g(t)}\to c\in(0,\infty),\quad\text{as}\quad t\to\infty. \end{align*}
It has been observed that when $K(t)$ is written as a sum of exponential functions, by adding auxiliary terms, the non-Markovian GLE~\eqref{eqn:GLE} can be mapped onto a multi-dimensional Markovian system~ \cite{mori1965continued,zwanzig1973nonlinear,kupferman2002long,kupferman2004fractional}. If $K(t)$ has the form of a finite sum of exponentials, the resulting finite-dimensional SDE was studied extensively in e.g.~\cite{ottobre2011asymptotic,pavliotis2014stochastic}. One can show that these systems admit a unique invariant structure with geometric ergodicity. Moreover, the marginal density of the pair $(x,v)$ is independent of $K(t)$. It is also worthwhile to note that, in the case of the linear GLE~\eqref{eqn:GLE:linear}, these memory kernels $K(t)$ produce \emph{diffusive} MSD since they are integrable~\cite{mckinley2017anomalous}. In order to include memory kernels that have a power-law decay, one has to consider an infinite sum of exponentials resulting in a corresponding infinite-dimensional system.
From now on, we shall adopt the notations from~\cite{glatt2018generalized}. Let $\alpha, \beta >0$ be given, and define constants $c_k, \lambda_k$, $k=1,2, \ldots$, by \begin{align} \label{c-k} c_k=\frac{1}{k^{1+\alpha\beta}},\ \lambda_k=\frac{1}{k^\beta}. \end{align} We introduce the memory kernel $K(t)$ given by \begin{equation} \label{eqn:K} K(t)=\sum_{k\geq 1} c_k e^{-\lambda_k t}. \end{equation} It is shown that (see Example 3.3 of~\cite{abate1999infinite}) with this choice of constants $c_k$ and $\lambda_k$, $K(t)$ obeys a power-law decay, namely \begin{equation} \label{lim:K} K(t) \sim t^{-\alpha} \,\,\, \text{ as } \,\,\, t\rightarrow \infty, \end{equation} where $\alpha$ is as in~\eqref{c-k}. The constant $\beta$ is an auxiliary parameter that is only assumed to be positive. When $\alpha>1$, as mentioned above in the linear GLE~\eqref{eqn:GLE:linear}, $K(t)$ is in the \emph{diffusive} regime, whereas for $\alpha\in(0,1)$, $K$ belongs to the \emph{subdiffusive} regime. There is however no claim regarding to the case $\alpha=1$. For such reason, it is called the \emph{critical regime}. With $K(t)$ defined as in~\eqref{eqn:K}, the GLE~\eqref{eqn:GLE} is expressed as the following infinite-dimensional system \cite{glatt2018generalized} \begin{equation}\label{eqn:GLE-Markov} \begin{aligned} d x(t) &= v(t)\, d t, \\ m\, d v(t)&=\big(\!\!-\gamma v(t)-\Phi'(x(t))-\sum_{k\geq 1} \sqrt{c_k} z_k(t)\big)\,dt+\sqrt{2\gamma}\, dW_0(t), \\ d z_k(t)&=\left(-\lambda_k z_k(t)+ \sqrt{c_k}v(t)\right) \, dt+\sqrt{2\lambda_k}\, dW_k(t),\qquad k\geq 1, \end{aligned} \end{equation} where $W_k$ are independent, standard Brownian motions. In~\cite{glatt2018generalized}, the well-posedness and the existence of invariant structures of~\eqref{eqn:GLE-Markov} were studied for all $\alpha>0$. Employing a recent advance tool called \emph{asymptotic coupling}~\cite{hairer2011asymptotic,glatt2017unique}, it can be shown that~\eqref{eqn:GLE-Markov} admits a unique invariant distribution in the diffusive regime, ($\alpha>1$). However, ergodicity when $\alpha\in(0,1]$ remains an open question.
The goal of this note is to give an analysis of the behavior of~\eqref{eqn:GLE-Markov} in two different limits.
First, we are interested in the asymptotic behavior of~\eqref{eqn:GLE-Markov} concerning the small-mass limit, namely taking $m$ to zero on the LHS of the second line in~\eqref{eqn:GLE-Markov}. Due to the random perturbations, the velocity $v(t)$ is fluctuating fast whereas the displacement $x(t)$ is still moving slow. We hence would like to find a process $u(t)$ such that on any compact interval $[0,T]$, \begin{align*}
\lim_{m\downarrow 0}\sup_{0\leq r\leq t}|x(r)-u(r)|=0, \end{align*} where the limit holds in an appropriate sense. Such statement is called Smoluchowski-Kramer approximation \cite{freidlin2004some}. There is a literature of analyzing asymptotic behaviors for fast-slow processes when taking zero-mass limit. Earliest results in this direction seem to be the works of~\cite{kramers1940brownian,von1916drei}. For more recent studies in finite dimensional systems, we refer to~\cite{freidlin2004some} in which the convergence in probability is established with constant drag and multiplicative noise. Under stronger assumptions and using appropriate time rescaling, weak convergence is proved in~\cite{pardoux2003poisson} where the friction is also state-dependent. When the potential is assumed to be Lipschitz, one can obtain better results in $L^p$, following the works of~\cite{hottovy2015smoluchowski,lim2017homogenization}. Without such assumption, convergence in probability is established in~\cite{herzog2016small} provided appropriate Lyapunov controls. In addition, limiting systems are observed numerically in \cite{hottovy2012noise,hottovy2012thermophoresis}. Similar analysis in infinite dimensional settings for semi linear wave equations are studied in a series of paper~\cite{cerrai2006smoluchowski,cerrai2006smoluchowski2,cerrai2014smoluchowski,cerrai2016smoluchowski}. The systems therein are shown to converge to a heat equation under different assumptions about non linear drifts. Motivated by~\cite{herzog2016small,hottovy2015smoluchowski}, in this note, we establish the convergence in probability for~\eqref{eqn:GLE-Markov}, cf. Theorem~\ref{thm:limit:zeromass}. The technique that we employ is inspired by those in~\cite{herzog2016small}.
Then, we study the white-noise limit of~\eqref{eqn:GLE-Markov}, namely as the random force $F(t)$ in~\eqref{eqn:GLE} converges to a white noise process. Under different conditions on the potential and space regularity, we aim to find a pair of processes $(u(t),p(t))$ that can be approximated by the $(x(t),v(t))$-component in~\eqref{eqn:GLE-Markov}. While there is a rich history on the small-mass limit, the white-noise limit seems to receive less attention. Nevertheless, there have been many works on the asymptotics of deterministic systems with memories. To name a few in this direction, we refer the reader to \cite{conti2006singular,gatti2005navier,grasselli2002uniform}. With regards to the white-noise limit of our system, we establish the convergence in different modes for a wide class of potentials, that are not necessarily Lipschitz or bounded. While the proof of Theorem~\ref{thm:limit:whitenoise:probconverge} concerning probability convergence shares the same arguments with that of Theorem~\ref{thm:limit:zeromass}, the result in Theorem~\ref{thm:limit:whitenoise:L1converge} concerning strong topology requires more work, where we have to estimate a universal bound on the solutions of~\eqref{eqn:GLE-Markov} using appropriate Lyapunov structures, cf. Proposition~\ref{prop:limit:whitenoise:L2bound}. To the best of our knowledge, these results seem to be new in infinite-dimensional stochastic differential equations with memory. Particularly, they (cf. Theorem~\ref{thm:limit:whitenoise:probconverge} and Theorem~\ref{thm:limit:whitenoise:L1converge}) generalize analogous results for finite-dimensional settings in~\cite{ottobre2011asymptotic}, where $K(t)$ has a form of finite sum of exponentials.
The rest of this paper is organized as follows. We introduce notations and summarize our main results in Section~\ref{sec:results}. The small-mass limit is addressed rigorously in Section \ref{sec:limit:zeromass}. We obtain a formula for the limiting system as a form of a Smoluchowski-Kramers equation. Finally, Section~\ref{sec:limit:whitenoise} studies the white-noise limit.
\section{Summary of Results}
\label{sec:results} Throughout this work, we will assume that the potential $\Phi$ satisfies the following growth condition \cite{glatt2018generalized}. \begin{assumption}\label{cond:Phi} $\Phi\in C^\infty(\mathbb{R})$ and there exists a constant $c>0$ such that for all $x\in\mathbb{R}$ \begin{align*} \quad c(\Phi(x)+1)\geq x^2. \end{align*} By adding a positive constant if necessary, we also assume that $\Phi$ is non-negative. \end{assumption} A typical class of potentials $\Phi$ that satisfies Assumption~\ref{cond:Phi} is the class of polynomials of even degree whose leading coefficient is positive. Functions that grow faster than polynomials are also included, e.g. $e^{x^2}$.
We now define a phase space for the infinite-dimensional process \begin{align*} X(t)=(x(t), v(t), z_1(t), z_2(t), \ldots). \end{align*} Following~\cite{glatt2018generalized}, let $\mathcal{H}_{-s}$, $s\in \mathbb{R}$ denote the Hilbert space given by
\begin{equation}\label{eqn:H_p}
\mathcal{H}_{-s}=\Big\{X=(x,v,Z)=(x,v,z_1, z_2, \ldots):\|X\|^2_{-s}=x^2+v^2+\sum_{k\geq 1}k^{-2s}z_k^2<\infty\Big\}. \end{equation} endowed with the usual inner product $\langle\cdot,\cdot\rangle_{-s}$, \begin{equation} \label{eqn:H-inner-prod} \langle X,\widetilde{X}\rangle_{-s} = x\wt{x}+v\wt{v}+\sum_{k\geq 1}k^{-2s}z_k\wt{z}_k. \end{equation}
With regards to kernel parameters $\alpha, \beta$ cf.~\eqref{c-k}, \eqref{eqn:K} and the phase space regularity parameter $s$, we assume that they satisfy the following condition. \begin{assumption}\label{cond:wellposed} Let $\alpha,\beta>0$ be as in~\eqref{c-k} and $s\in\mathbb{R}$ as in~\eqref{eqn:H_p}. We assume that they satisfy either the \emph{asymptotically diffusive} condition \begin{enumerate} \item[\emph{(D)}]\label{cond:diffusion} $\displaystyle \alpha>1,\, \beta>\frac{1}{\alpha-1}$ and $\displaystyle 1< 2s < (\alpha-1)\beta$; \end{enumerate} or the \emph{asymptotically subdiffusive} condition \begin{enumerate} \item[\emph{(SD)}]\label{cond:subdiffusion} $\displaystyle 0<\alpha <1,\, \beta>\frac{ 1}{\alpha}$ and $\displaystyle 1< 2s < \alpha\beta$; \end{enumerate} or the \emph{critical regime} condition \begin{enumerate} \item[\emph{(C)}] $\displaystyle \alpha =1,\, \beta>1$ and $\displaystyle 1< 2s < \beta$; \end{enumerate} \end{assumption}
Under Assumption~\ref{cond:Phi} and Assumption~\ref{cond:wellposed}, the well-posedness of~\eqref{eqn:GLE-Markov} was studied rigorously in~\cite{glatt2018generalized}.
\subsection{Small-mass Limit}
In regards to the small-mass limit ($m\to 0$), we introduce the following limiting system whose derivation will be explained later at the end of this subsection \begin{equation} \label{eqn:GLE-Markov:limit:zeromass} \begin{aligned} \gamma du(t)&= \Big(\!\!-\Phi'(u(t))-\Big(\sum_{k\geq 1}c_k\Big) u(t)-\sum_{k\geq 1}\sqrt{c_k}f_k(t)\Big)dt+\sqrt{2\gamma}dW_0(t),\\ df_k(t)&=\big(\!\!-\lambda_k f_k(t)-\lambda_k\sqrt{c_k}u(t)\big)dt+\sqrt{2\lambda_k}dW_k(t),\quad k=1,2\dots \end{aligned} \end{equation}
The new phase space for the solution $U(t)=(u(t),f_1(t),f_2(t),\dots)$ of~\eqref{eqn:GLE-Markov:limit:zeromass} is denoted by $\dot{\mathcal{H}}_{-s}$, $s\in \mathbb{R}$,
\begin{equation}\label{eqn:Hsdot}
\dot{\mathcal{H}}_{-s}=\Big\{U=(u,f_1, f_2, \ldots):\|U\|_{\dot{\mathcal{H}}_{-s}}^2=u^2+\sum_{k\geq 1}k^{-2s}f_k^2<\infty\Big\}, \end{equation} endowed with the usual inner product. We can regard $\dot{\mathcal{H}}_{-s}$ as a subspace of $\mathcal{H}_{-s}$ whose $v-$component is equal to zero. Recalling $c_k$ in~\eqref{c-k}, it is straightforward to see that if $(x,v,z_1,z_2\dots)\in\mathcal{H}_{-s}$ then $(x,z_1-\sqrt{c_1}x,z_2-\sqrt{c_2}x,\dots)\in\dot{\mathcal{H}}_{-s}$.
From now on, we shall fix a stochastic basis $\mathcal{S}=\left(\Omega,\mathcal{F},\mathbb{P},\{\mathcal{F}_t\}_{t\geq 0},W\right)$ satisfying the usual conditions \cite{karatzas2012brownian}. Here $W$ is the cylindrical Wiener process defined on an auxiliary Wiener space $\mathcal{W}$ with the usual decomposition \begin{align*} W(t)=e^{\mathcal{W}}_0W_0(t)+e^{\mathcal{W}}_1 W_1(t)+\dots, \end{align*} where $\{e^{\mathcal{W}}_0,e^{\mathcal{W}}_1,\dots\}$ is the canonical basis of $\mathcal{W}$, and $\{W_k(t)\}_{k\geq 0}$ are independent one-dimensional Brownian Motions \cite{da2014stochastic}. The well-posedness of~\eqref{eqn:GLE-Markov:limit:zeromass} is guaranteed by the following result. \begin{proposition} \label{prop:wellposed:limit:zeromass} Suppose that $\Phi$ satisfies Assumption \ref{cond:Phi} and the constants $\alpha, \beta, s$ satisfy Assumption \ref{cond:wellposed}. Then for all initial conditions $U_0\in \dot{\mathcal{H}}_{-s}$, there exists a unique pathwise solution $U(\cdot,U_0):\Omega\times[0,\infty)\to\dot{\mathcal{H}}_{-s}$ of~\eqref{eqn:GLE-Markov:limit:zeromass} in the following sense: $U(\cdot,U_0)$ is $\mathcal{F}_t$-adapted, $U(\cdot,U_0)\in C([0,\infty),\dot{\mathcal{H}}_{-s})$ almost surely and that if $\wt{U}(\cdot,U_0)$ is another solution then for every $T\geq 0$, \begin{displaymath} \P{\forall t\in[0,T], U(t,U_0)=\wt{U}(t,U_0)}=1. \end{displaymath} Moreover, for every $U_0 \in \dot{\mathcal{H}}_{-s}$ and $T\geq 0$, there exists a constant $C(T,U_0)>0$ such that
\begin{equation} \label{ineq:strong-sol'n}
\Enone\sup_{0\leq t\leq T }\|U(t)\|_{\dot{\mathcal{H}}_{-s}}^2\leq C(T,U_0).
\end{equation}
\end{proposition} The proof of Proposition~\ref{prop:wellposed:limit:zeromass} is quite standard, similar to that of Proposition 6 in~\cite{glatt2018generalized} and will be briefly explained in Section~\ref{sec:limit:zeromass}. We now state the main result concerning the small-mass limit. \begin{theorem} \label{thm:limit:zeromass} Suppose that $\Phi$ satisfies Assumption~\ref{cond:Phi} and the constants $\alpha, \beta, s$ satisfy Assumption~\ref{cond:wellposed}. Let $X_m(t)=(x_m(t),v_m(t),z_{1,m}(t),\dots)$ solve~\eqref{eqn:GLE-Markov} with initial conditions \begin{align*} (x_m(0),v_m(0),z_{1,m}(0),z_{2,m}(0)\dots)=(x,v,z_1,z_2,\dots)\in\mathcal{H}_{-s}, \end{align*} and $U(t)=(u(t),f_1(t),\dots)$ solve~\eqref{eqn:GLE-Markov:limit:zeromass} with initial conditions \begin{align*} (u(0),f_1(0),f_2(0)\dots)=(x,z_1-\sqrt{c_1}x,z_2-\sqrt{c_2}x,\dots)\in\dot{\mathcal{H}}_{-s}. \end{align*} Then, for every $T,\,\xi>0$, it holds that \begin{align*}
\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x_m(t)-u(t)|>\xi\Big\}\rightarrow 0,\quad m\rightarrow 0. \end{align*} \end{theorem}
It is worthwhile to note that the small-mass limit in Theorem~\ref{thm:limit:zeromass} holds for all regimes ($\alpha>0$) as stated in Assumption~\ref{cond:wellposed}. We will see later that for the white-noise limit, the result is limited to the diffusive regime, namely $\alpha>1$, see Theorem~\ref{thm:limit:whitenoise:probconverge} below. We finish this subsection by a heuristic argument explaining how we derive the limiting system~\eqref{eqn:GLE-Markov:limit:zeromass} and its initial conditions as stated in Theorem~\ref{thm:limit:zeromass}. Following~\cite{herzog2016small}, to determine the limiting system, one may formally set $m=0$ on the RHS of the second equation in \eqref{eqn:GLE-Markov} and substitute $v(t)$ by $dx(t)$ from the first equation to obtain \begin{align*} \gamma\, dx(t)=\big[\!\!-\Phi'(x(t))-\sum_{k\geq 1}\sqrt{c_k}z_k(t)\big]dt+\sqrt{2\gamma}dW_0(t). \end{align*} The equation on $z_k(t)$ in~\eqref{eqn:GLE-Markov} still depends on $v(t)$, but this can be circumvented by using Duhamel's formula, \begin{align*} z_k(t)=e^{-\lambda_k t}z_k(0)+\sqrt{c_k}\int_0^t e^{-\lambda_k(t-r)}v(r)dr+\sqrt{2\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r). \end{align*} By an integration by parts, we can transform the integral term involving $v(r)$ to \begin{align*} \int_0^t e^{-\lambda_k(t-r)}v(r)dr = x(t)-e^{-\lambda_k t}x(0)-\lambda_k\int_0^t e^{-\lambda_k(t-r)}x(r)dr. \end{align*} Plugging back into the formula for $z_k(t)$, we find \begin{align*} z_k(t)&=e^{-\lambda_k t}(z_k(0)-\sqrt{c_k}x(0))+\sqrt{c_k}x(t)-\lambda_k\sqrt{c_k}\int_0^t e^{-\lambda_k(t-r)}x(r)dr\\ &\qquad\qquad\qquad+\sqrt{2\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r). \end{align*} We now set $f_k(t):=z_k(t)-\sqrt{c_k}x(t)$ and $u(t):=x(t)$ and thus arrive at~\eqref{eqn:GLE-Markov:limit:zeromass} with the new shifted initial conditions as in Theorem~\ref{thm:limit:zeromass}.
\subsection{White-noise Limit} Next, we present our results on the asymptotical behavior of the $(x(t),v(t))$-component of~\eqref{eqn:GLE-Markov} in the diffusive regime by an appropriate scaling on the memory kernel $K(t)$ in~\eqref{eqn:K}. For $\epsilon>0$, we introduce $K_\epsilon(t)$ given by \begin{equation}\label{eqn:K:epsilon}
K_\epsilon(t)=\frac{1}{\epsilon}K\Big(\frac{t}{\epsilon}\Big)=\sum_{k\geq 1}\frac{c_k}{\epsilon}e^{-\frac{\lambda_k}{\epsilon}|t|}. \end{equation} With $K_\epsilon$ defined above, the corresponding system~\eqref{eqn:GLE-Markov} becomes \begin{equation}\label{eqn:GLE-Markov:whitenoise:epsilon} \begin{aligned} d x_\epsilon(t) &= v_\epsilon(t)\, d t, \\ m\, d v_\epsilon(t)&=\Big(\!\!-\gamma v_\epsilon(t)-\Phi'(x_\epsilon(t))-\sum_{k\geq 1} \sqrt{\frac{c_k}{\epsilon}} z_{k,\epsilon}(t)\Big)\,dt+\sqrt{2\gamma}\, dW_0(t), \\ d z_{k,\epsilon}(t)&=\Big(\!\!-\frac{\lambda_k}{\epsilon} z_{k,\epsilon}(t)+ \sqrt{\frac{c_k}{\epsilon}}v_\epsilon(t)\Big) \, dt+\sqrt{\frac{2\lambda_k}{\epsilon}}\, dW_k(t),\qquad k\geq 1. \end{aligned} \end{equation} Inspired by~\cite{ottobre2011asymptotic}, we consider the following system \begin{equation}\label{eqn:GLE-Markov:limit:whitenoise} \begin{aligned} d u(t) &= p(t)\, d t ,\\ m\, d p(t)&=\Big(\!\!\!-\Big(\gamma+\sum_{k\geq 1}\frac{c_k}{\lambda_k}\Big) p(t)-\Phi'(u(t))\Big)\,dt\\ &\qquad\qquad-\sum_{k\geq 1}\sqrt{\frac{2c_k}{\lambda_k}}dW_k(t)+\sqrt{2\gamma}\, dW_0(t). \end{aligned} \end{equation} The well-posedness of~\eqref{eqn:GLE-Markov:limit:whitenoise} will be addressed briefly in Section~\ref{sec:limit:whitenoise}.
We then assert that $(x_\epsilon(t),v_\epsilon(t))$ converges to the solution $(u(t),p(t))$ of~\eqref{eqn:GLE-Markov:limit:whitenoise} in the following sense.
\begin{theorem} \label{thm:limit:whitenoise:probconverge} Suppose that $\Phi$ satisfies Assumption \ref{cond:Phi} and the constants $\alpha, \beta, s$ satisfy Condition (D) of Assumption \ref{cond:wellposed}. Let $X_\epsilon(t)=(x_\epsilon(t),v_\epsilon(t),z_{1,\epsilon}(t),\dots)$ be the solution of~\eqref{eqn:GLE-Markov:whitenoise:epsilon} with initial conditions \begin{align*}(x_\epsilon(0),v_\epsilon(0),z_{1,\epsilon}(0),z_{2,\epsilon}(0),\dots)=(x,v,z_{1},z_2,\dots)\in\mathcal{H}_{-s}, \end{align*} and $(u(t),p(t))$ be the solution of~\eqref{eqn:GLE-Markov:limit:whitenoise} with initial conditions $(u(0),p(0))=(x,v)$. Then, for every $T,\,\xi>0$, \begin{align*}
\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x_\epsilon(t)-u(t)|+|v_\epsilon(t)-p(t)|>\xi\Big\}\rightarrow 0,\quad \epsilon\rightarrow 0. \end{align*} \end{theorem}
The reader may wonder why the convergence result of Theorem~\ref{thm:limit:whitenoise:probconverge} is restricted to the \emph{diffusive} regime, namely $\alpha>1$ according to Condition (D) of Assumption~\ref{cond:wellposed}. Heuristically, since the memory kernel $K(t)$ decays like $t^{-\alpha}$ as $t\to\infty$, we see that \begin{align*} K_\epsilon(t)=\frac{1}{\epsilon}K\Big(\frac{t}{\epsilon}\Big)\sim \epsilon^{\alpha-1} t^{-\alpha},\quad t\to\infty. \end{align*} By shrinking $\epsilon$ further to zero, if $\alpha>1$, $K_\epsilon(t)$ does not behave ``badly" at infinity. In fact, it is not difficult to show that as $\epsilon\downarrow 0$, $K_\epsilon$ converges to the Dirac function $\delta_0$ centered at the origin, in the sense of tempered distribution, namely, for every $\varphi\in\mathcal{S}$, the Schwartz space on $\mathbb{R}$, it holds that \begin{align*}
\int_\mathbb{R} K_\epsilon(t)\varphi(t)dt\to |K|_{L^1(\mathbb{R})}\varphi(0). \end{align*} which implies that the random force $F(t)$ in~\eqref{eqn:GLE} converges to a white noise process in the sense of random distribution, cf. \cite{ito1954stationary}, hence the so called ``white-noise limit". We will see later in the proof of Proposition~\ref{prop:limit:whitenoise:L2bound} that the condition $\alpha>1$ is crucial for our analysis in order to obtain bounds on the solutions.
Finally, if $\Phi$ and parameters $\alpha,\, \beta$ satisfy stronger assumptions, then we are able to obtain better convergence than the result in Theorem~\ref{thm:limit:whitenoise:probconverge}. To be precise, we assume the following condition on $\Phi$. \begin{assumption} \label{cond:Phi:whitenoise} There exist constants $n,\, c>0$ such that for every $x,\, y\in\mathbb{R}$, \begin{displaymath}
\Phi'(x)y\leq c(\Phi(x)+|y|^n+1). \end{displaymath} \end{assumption} The assumption above is again a requirement about the growth of $\Phi'$ that guarantees a universal bound independent of $\epsilon$ on the solution $(x_\epsilon(t),v_\epsilon(t))$ of~\eqref{eqn:GLE-Markov:whitenoise:epsilon}, cf. Proposition~\ref{prop:limit:whitenoise:L2bound}. We remark that a function $\Phi$ satisfying Assumption~\ref{cond:Phi:whitenoise} need not satisfy Assumption~\ref{cond:Phi}, taking $\Phi$ a constant for example. It is also worthwhile to note that the class of polynomials of even degree satisfies Assumption~\ref{cond:Phi:whitenoise}. However, functions growing exponentially fast, e.g. $e^{x^2}$, do not.
With regards to space regularities, we assume the following condition about parameters $\alpha,\,\beta$. \begin{assumption} \label{cond:whitenoise:L1converge} Let $\alpha,\beta>0$ be as in~\eqref{c-k}. We assume that they satisfy \begin{align*} \alpha>2,\quad\text{and}\quad (\alpha-2)\beta>1. \end{align*} \end{assumption}
We then have the following important result. \begin{theorem} \label{thm:limit:whitenoise:L1converge} Suppose that $\Phi$ satisfies Assumption \ref{cond:Phi} and Assumption~\ref{cond:Phi:whitenoise} and that the constants $\alpha, \beta, s$ satisfy Condition (D) of Assumption \ref{cond:wellposed} and Assumption~\ref{cond:whitenoise:L1converge}. Let $X_\epsilon(t)=(x_\epsilon(t),v_\epsilon(t),z_{1,\epsilon}(t),\dots)$ be the solution of~\eqref{eqn:GLE-Markov:whitenoise:epsilon} with initial conditions \begin{align*}(x_\epsilon(0),v_\epsilon(0),z_{1,\epsilon}(0),z_{2,\epsilon}(0),\dots)=(x,v,z_{1},z_2,\dots)\in\mathcal{H}_{-s}, \end{align*}
and $(u(t),p(t))$ be the solution of~\eqref{eqn:GLE-Markov:limit:whitenoise} with initial conditions $(u(0),p(0))=(x,v)$. Then, for every $T>0$, $1\leq q<2$, it holds that \begin{equation*}
\Enone\Big[\sup_{0\leq t\leq T}|x_\epsilon(t)-u(t)|^q+|v_\epsilon(t)-p(t)|^q\Big]\rightarrow 0, \quad \epsilon\to 0. \end{equation*} \end{theorem} Theorem~\ref{thm:limit:whitenoise:L1converge} strengthens a previous result from \cite{ottobre2011asymptotic}, where the potential $\Phi'$ is assumed to be bounded. The proofs of Theorem~\ref{thm:limit:whitenoise:probconverge} and Theorem~\ref{thm:limit:whitenoise:L1converge} will be carried out in Section~\ref{sec:limit:whitenoise}.
\section{Zero-mass limit}\label{sec:limit:zeromass}
Throughout the rest of the paper, $C,\,c$ denote generic positive constants. The important parameters that they depend on will be indicated in parenthesis, e.g. $c(T,q)$ depends on parameters $T$ and $q$.
In this section, for notation simplicity, we shall omit the subscript $m$ in \begin{align*} X_m(t)=(x_m(t),v_m(t),z_{1,m}(t),\dots). \end{align*}
We begin by addressing the well-posedness of~\eqref{eqn:GLE-Markov:limit:zeromass} whose proof follows a standard Lyapunov-type argument that was also used to establish the well-posedness of~\eqref{eqn:GLE-Markov} in \cite{glatt2018generalized}. The technique is classical and has been employed previously in literature \cite{albeverio2008spde,glatt2009strong,jacod2006calcul}. We shall omit specific details and briefly summarize the main steps. \begin{proof}[Sketch of the proof of Proposition \ref{prop:wellposed:limit:zeromass}] For $R>0$, let $\theta^R\in C^\infty(\mathbb{R} , [0,1])$ satisfy \begin{align} \label{defn:theta-R} \theta^R(x) = \begin{cases}
1 & \text{ if } |x| \leq R, \\
0 & \text{ if } |x| \geq R+1. \end{cases} \end{align} We consider the ``cutoff" equation corresponding to~\eqref{eqn:GLE-Markov:limit:zeromass} \begin{equation}\label{eqn:GLE-Markov:limit:zeromass:cutoff} \begin{aligned} \gamma du(t)&= \Big(\!\!-\Phi'(u(t))\theta^R(u(t))-\Big(\sum_{k\geq 1}c_k\Big) u(t)-\sum_{k\geq 1}\sqrt{c_k}f_k(t)\Big)dt+\sqrt{2\gamma}dW_0(t),\\ df_k(t)&=\big(\!\!-\lambda_k f_k(t)-\lambda_k\sqrt{c_k}u(t)\big)dt+\sqrt{2\lambda_k}dW_k(t),\quad k=1,2\dots \end{aligned} \end{equation} We observe that in~\eqref{eqn:GLE-Markov:limit:zeromass:cutoff}, the drift term is globally Lipschitz and the noise is additive. Thus, by using a standard Banach fixed point argument, the corresponding global (in time) solution $U^R$ exists and is unique. Next, define the stopping time
\[\tau_R=\inf\left\{t>0: \|U(t)\|_{\dot{\mathcal{H}}_{-s}}>R\right\}.\] Note that, for all times $t< \tau_R$, $U^R$ solves \eqref{eqn:GLE-Markov:limit:zeromass}. Consequently, the solution \eqref{eqn:GLE-Markov:limit:zeromass} exists and is unique up until the \emph{time of explosion} $\tau_\infty=\lim_{R\to\infty}\tau_R$, which is possibly finite on a set of positive probability. We finally introduce the Lyapunov function \begin{equation} \label{eqn:Lyapunov:zeromass} \Psi(U):=\frac{1}{\gamma}\bigg(\Phi(u)+\Big(\sum_{k\geq 1}c_k\Big)\frac{ u^2}{2}\bigg)+\frac{1}{2}\sum_{k \geq 1}k^{-2s}f_k^2. \end{equation}
It is clear that $\Psi(U)$ dominates $\|U\|_{\dot{\mathcal{H}}_{-s}}^2$. Applying Ito's formula to $\Psi(U)$, one can derive a global bound on the solutions $U^R(t)$ that is independent of $R$, namely, there exists a constant $C(U_0,T)$ such that \begin{equation*}
\Enone\Big[\sup_{0\leq t\leq T} \|U(t\wedge\tau_R)\|_{\dot{\mathcal{H}}_{-s}}^2 \Big]\leq C(T,U_0). \end{equation*} Sending $R$ to infinity, it follows from Fatou's Lemma that \begin{equation*}
\Enone\Big[\sup_{0\leq t\leq T} \|U(t\wedge\tau_\infty)\|_{\dot{\mathcal{H}}_{-s}}^2 \Big]\leq C(U_0,T). \end{equation*} implying $\P{T<\tau_\infty}=1$ for any $T>0$. Taking $T$ to infinity, we see that $\P{\tau_\infty=\infty}=1$, thereby obtaining the global solution of~\eqref{eqn:GLE-Markov:limit:zeromass}. \end{proof}
Although the construction of the global solution $U(t)$ of~\eqref{eqn:GLE-Markov:limit:zeromass} via the local solutions $U^R(t)$ of~\eqref{eqn:GLE-Markov:limit:zeromass:cutoff} is quite standard, the proof of Theorem~\ref{thm:limit:zeromass} will make use of a non trivial observation on these local solutions. The arguments that we are going to employ are inspired from the work of~\cite{herzog2016small}. Before diving into detail, we briefly explain the main idea, which is a two-fold: first, we show that the result holds for $\Phi'$ being Lipschitz. In particular, we obtain the convergence in sup norm for the local solutions, namely for all $R,\,T>0$, we have \begin{align*}
\Enone\Big[\sup_{0\leq t\leq T}\big|x^R(t)-u^R(t)\big|\Big]\rightarrow 0,\quad m\rightarrow 0, \end{align*} where $x^R(t)$ is in the following cut-off system for~\eqref{eqn:GLE-Markov} \begin{equation}\label{eqn:GLE-Markov:cutoff} \begin{aligned} d x(t) &= v(t)\, d t, \\ m\, d v(t)&=\big(-\gamma v(t)-\Phi'(x(t))\theta^R(x(t))-\sum_{k\geq 1} \sqrt{c_k} z_k(t)\big)\,dt+\sqrt{2\gamma}\, dW_0(t), \\ d z_k(t)&=\left(-\lambda_k z_k(t)+ \sqrt{c_k}v(t)\right) \, dt+\sqrt{2\lambda_k}\, dW_k(t),\qquad k\geq 1. \end{aligned} \end{equation} Then, by taking $R$ necessarily large, we obtain the desired result.
We now proceed by showing that the result holds true in a simpler setting where $\Phi'$ is globally Lipschitz. The proof is adapted from that of Theorem 1 of~\cite{hottovy2015smoluchowski}. \begin{proposition}\label{prop:limit:zeromass:lipschitz} Suppose that that $\Phi'$ is globally Lipschitz and that the constants $\alpha, \beta, s$ satisfy Assumption \ref{cond:wellposed}. Let $X(t)=(x(t),v(t),z_{1}(t),\dots)$ solve~\eqref{eqn:GLE-Markov} with initial conditions \begin{align*} (x(0),v(0),z_{1}(0),\dots)=(x,v,z_1,\dots)\in\mathcal{H}_{-s}, \end{align*} and $U(t)=(u(t),f_1(t),\dots)$ solve~\eqref{eqn:GLE-Markov:limit:zeromass} with initial conditions \begin{align*} (u(0),f_1(0),f_2(0)\dots)=(x,z_1-\sqrt{c_1}x,z_2-\sqrt{c_2}x,\dots)\in\dot{\mathcal{H}}_{-s}. \end{align*} Then, for every $T,q>0$, it holds that \begin{align*}
\Enone\sup_{0\leq t\leq T}\big|x(t)-u(t)\big|^q\rightarrow 0,\quad m\rightarrow 0. \end{align*} \end{proposition} \begin{proof} Using Duhamel's formula, $z_{k}(t)$ from~\eqref{eqn:GLE-Markov} can be solved explicitly as \begin{align}\label{eqn:limit:zeromass:lipschitz:0} z_{k}(t) = e^{-\lambda_k t}z_k(0)+\sqrt{c_k}\int_0^t e^{-\lambda_k(t-r)}v(r)dr+\sqrt{2\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r), \end{align} which is equivalent to \begin{align*} z_{k}(t)& = e^{-\lambda_k t}(z_k(0)-\sqrt{c_k}x(0))+\sqrt{c_k}x(t)-\sqrt{c_k}\lambda_k\int_0^t e^{-\lambda_k(t-r)}x(r)dr\\ &\qquad\qquad+\sqrt{2\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r), \end{align*} where we have used an integration by parts on the term $\int_0^t e^{-\lambda_k(t-r)}v(r)dr$ in the first equality. Substituting into the second equation of~\eqref{eqn:GLE-Markov}, we arrive at \begin{equation}\label{eqn:limit:zeromass:lipschitz:1} \begin{aligned} \MoveEqLeft[2]m\,dv(t)+\gamma\, dx(t)\\
&= \bigg(\!\!-\Phi'(x(t))-\sum_{k\geq 1}\sqrt{c_k}e^{-\lambda_k t}(z_k(0)-\sqrt{c_k}x(0))-\Big(\sum_{k\geq 1}c_k\Big)x(t)\\ &\qquad+\sum_{k\geq 1}c_k\lambda_k\int_0^t e^{-\lambda_k(t-r)}x(r)dr -\sum_{k\geq 1}\sqrt{2c_k\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r)\bigg)dt\\ &\qquad+\sqrt{2\gamma}dW_0(t). \end{aligned} \end{equation} Likewise, we obtain the following equation from~\eqref{eqn:GLE-Markov:limit:zeromass} \begin{equation}\label{eqn:limit:zeromass:lipschitz:2} \begin{aligned}
\MoveEqLeft[2] \gamma\, du(t)\\
&= \bigg(\!\!-\Phi'(u(t))-\Big(\sum_{k\geq 1}c_k\Big)u(t)-\sum_{k\geq 1}\sqrt{c_k}e^{-\lambda_k t}(z_k(0)-\sqrt{c_k}x(0))\\ &\qquad+\sum_{k\geq 1}c_k\lambda_k\int_0^t e^{-\lambda_k(t-r)}u(r)dr -\sum_{k\geq 1}\sqrt{2c_k\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r)\bigg)dt\\ &\qquad+\sqrt{2\gamma}dW_0(t). \end{aligned} \end{equation} Subtracting~\eqref{eqn:limit:zeromass:lipschitz:2} from~\eqref{eqn:limit:zeromass:lipschitz:1} and setting $\overline{x}(t)=x(t)-u(t)$, we find that \begin{align*} \MoveEqLeft[3]m\,dv(t)+\gamma\, d\overline{x}(t)\\ &= \bigg(\!\!-\big[\Phi'(x(t))-\Phi'(u(t))\big]-\Big(\sum_{k\geq 1}c_k\Big)\overline{x}(t)+\sum_{k\geq 1}c_k\lambda_k\int_0^t e^{-\lambda_k(t-r)}\overline{x}(r)dr \bigg)dt\\
&\leq c\Big(1+\sum_{k\geq 1}c_k\Big)\sup_{0\leq r\leq t}|\overline{x}(r)|dt, \end{align*} where $c>0$ is a Lipschitz constant for $\Phi'$. Recalling $c_k$ from~\eqref{c-k}, we apply Gronwall's inequality to estimate for all $T,\,q>0$ \begin{align*}
\Enone\sup_{0\leq t\leq T}|\overline{x}(t)|^q\leq m^q\Enone\sup_{0\leq t \leq T}|v(t)-v|^q\,e^{c(T)}. \end{align*} The result now follows immediately from Proposition \ref{prop:limit:zeromass:lipschitz:1} below. \end{proof}
\begin{proposition}\label{prop:limit:zeromass:lipschitz:1} Under the same Hypothesis of Theorem~\ref{thm:limit:zeromass}, suppose further that $\Phi'(x)$ is globally Lipschitz. Let $X(t)=(x(t),v(t),z_{1}(t),\dots)$ solve~\eqref{eqn:GLE-Markov} with initial conditions $(x(0),v(0),z_{1}(0),\dots)=(x,v,z_1,\dots)\in\mathcal{H}_{-s}$. Then, for every $T>0,\,q>1$, it holds that \begin{align*}
m^q\Enone\sup_{0\leq t\leq T}|v(t)|^q\to 0,\quad m\to 0. \end{align*} \end{proposition}
In order to prove Proposition~\ref{prop:limit:zeromass:lipschitz:1}, we need the following important lemma whose proof is based on Lemma 3.19, \cite{blount1991comparison} and Lemma 2, \cite{hottovy2015smoluchowski}. It will be also useful later in Section~\ref{sec:limit:whitenoise}. \begin{lemma}\label{lem:limit:zeromass} Given $\kappa,\,\eta>0$, let $f(t) = \sqrt{2\kappa}\int_0^t e^{-\eta(t-r)}dW(r)$ where $W(t)$ is a standard Brownian Motion. Then, for all $T>0,\, q> 1$, there exists a constant $C(T,q)>0$ such that \begin{align}\label{ineq:limit:zeromass:1} \Enone\sup_{0\leq t\leq T}f(t)^{2q}\leq \frac{\kappa^q}{\eta^{q-1}} C(T,q). \end{align} \end{lemma} \begin{remark} The estimate in~\eqref{ineq:limit:zeromass:1} is sharper than the usual exponential martingale estimate. In finite-dimensional settings, it is sufficient to bound the LHS of~\eqref{ineq:limit:zeromass:1} by $C(T,q,\eta,\kappa)$, cf.~\cite{blount1991comparison,hottovy2015smoluchowski,lim2017homogenization}. In our setting, we have to keep track explicitly in term of $\eta$ and $\kappa$, hence the RHS of~\eqref{ineq:limit:zeromass:1}. \end{remark} The proof of Lemma~\ref{lem:limit:zeromass} is similar to that of Lemma 2,~\cite{hottovy2015smoluchowski}. We include it here for the sake of completeness. \begin{proof}[Proof of Lemma~\ref{lem:limit:zeromass}] In view of Lemma 3.19, \cite{blount1991comparison}, we have the following estimate for $A>0$, \begin{align*} \mathbb{P}\Big\{\sup_{0\leq t\leq T}f(t)^2\geq A\Big\}\leq \frac{\eta T}{\int_0^{\sqrt{\eta A/\kappa}}e^{r^2/2}\int_0^r e^{-\ell^2/2}d\ell\,dr}. \end{align*} We proceed to find a lower bound for the above denominator. To this end, we first claim that for $r\geq 0$, \begin{align*} \int_0^r e^{-\ell^2/2}d\ell \geq \frac{re^{-r^2/4}}{2}. \end{align*} Indeed, on one hand, if $r\geq 1$, then \begin{align*} e^{r^2/4}\int_0^r e^{-\ell^2/2}d\ell \geq e^{r^2/4}\int_0^1 e^{-\ell^2/2}d\ell\geq e^{r^2/4}\int_0^1 e^{-1/2}d\ell\geq \frac{e^{r^2/4}}{2}\geq \frac{r}{2}. \end{align*} On the other hand, if $0\leq r\leq 1$, then \begin{align*} e^{r^2/4}\int_0^r e^{-\ell^2/2}d\ell\geq \int_0^r 1-\frac{\ell^2}{2} d\ell=r\big(1-\frac{r^2}{6}\big)\geq \frac{r}{2}. \end{align*} With this observation, we find \begin{align*} \int_0^{\sqrt{\eta A/\kappa}}e^{r^2/2}\int_0^r e^{-\ell^2/2}d\ell\,dr \geq \int_0^{\sqrt{\eta A/\kappa}}e^{r^2/2}\frac{r e^{-r^2/4}}{2}dr=e^{\eta A/4\kappa}-1\geq \frac{\eta A}{4\kappa} e^{\eta A/8\kappa}, \end{align*} where in the last implication, we have used the following inequality for every $r\geq 0$, \begin{align*} e^r-1\geq re^{r/2}. \end{align*} Putting everything together, we obtain \begin{align*} \mathbb{P}\bigg\{\sup_{0\leq t\leq T}f(t)^2\geq A\bigg\}\leq \frac{\eta T}{\frac{\eta A}{4\kappa} e^{\eta A/8\kappa}}=\frac{4\kappa T}{A}e^{-\eta A/8\kappa}. \end{align*} It follows that for $q>2$, \begin{align*} \Enone\sup_{0\leq t\leq T}f(t)^{2q} & = \int_0^\infty qA^{q-1} \mathbb{P}\bigg\{\sup_{0\leq t\leq T}f(t)^2\geq A\bigg\}dA\\ &\leq \int_0^\infty qA^{q-1}\frac{4\kappa T}{A}e^{-\eta A/8\kappa}dA\\ &=C(T,q)\kappa\int_0^\infty A^{q-2}e^{-\eta A/8\kappa}dA\\ &=C(T,q) \frac{\kappa^q}{\eta^{q-1}}, \end{align*} which completes the proof. \end{proof} With Lemma~\ref{lem:limit:zeromass} in hand, we are ready to give the proof of Proposition~\ref{prop:limit:zeromass:lipschitz:1}. \begin{proof}[Proof of Proposition~\ref{prop:limit:zeromass:lipschitz:1}] We only have to prove the result for $q>0$ sufficiently large, say $q\geq q_1$. As if it holds for $q_1$, then for every $q<q_1$, by Holder's inequality, we have \begin{align*}
\Enone\big[m^q\sup_{0\leq t\leq T}|v(t)|^q\big]\leq\Big( \Enone\big[m^{q_1}\sup_{0\leq t\leq T}|v(t)|^{q_1}\big]\Big)^{q/q_1}\to 0,\quad\text{as}\quad m\to 0. \end{align*} We begin by noting that $v(t)$ from~\eqref{eqn:GLE-Markov} is written as \begin{align*} m\,v(t)&= me^{-\frac{\gamma}{m}t}v(0)-\int_0^te^{-\frac{\gamma}{m}(t-r)}\Phi'(x(r))dr-\sum_{k\geq 1}\sqrt{c_k}\int_0^t e^{-\frac{\gamma}{m}(t-r)}z_k(r)dr\\ &\qquad+\sqrt{2\gamma}\int_0^te^{-\frac{\gamma}{m}(t-r)}dW_0(r). \end{align*} Substituting $z_k(t)$ from~\eqref{eqn:limit:zeromass:lipschitz:0}, we have \begin{align*} m\,v(t)&= me^{-\frac{\gamma}{m}t}v(0)-\int_0^te^{-\frac{\gamma}{m}(t-r)}\Phi'(x(r))dr-\sum_{k\geq 1}\sqrt{c_k}\int_0^t e^{-\frac{\gamma}{m}(t-r)}e^{-\lambda_k r}z_k(0)dr\\ &\qquad -\sum_{k\geq 1}c_k\int_0^t e^{-\frac{\gamma}{m}(t-r)}\int_0^r e^{-\lambda_k(r-\ell)}v(\ell)d\ell\,dr\\ &\qquad-\sum_{k\geq 1}\sqrt{2c_k\lambda_k}\int_0^t e^{-\frac{\gamma}{m}(t-r)}\int_0^re^{-\lambda_k(r-\ell)}dW_k(\ell)\,dr\\ &\qquad+\sqrt{2\gamma}\int_0^te^{-\frac{\gamma}{m}(t-r)}dW_0(r). \end{align*} For every $q$ sufficiently large, we invoke the assumption that $\Phi'$ is Lipschitz to estimate \begin{equation*} \begin{aligned} \MoveEqLeft[2] m^{2q}\Enone\sup_{0\leq t\leq T}v(t)^{2q}\\
&\leq c(q,v) \bigg[ m^{2q}\Big[1+\Enone\sup_{0\leq t\leq T}x(t)^{2q}+\big|\sum_{k\geq 1}\sqrt{c_k}z_k\big|^{2q}\\
&\qquad +c(T)\big|\sum_{k\geq 1}c_k\big|^{2q}\int_0^T\Enone\sup_{0\leq r\leq t}v(r)^{2q}dt\\
&\qquad +\big(\sum_{k\geq 1}c_k^{(1/2-1/2q)q*}\big)^{2q/q*}\sum_{k\geq 1}\Enone\sup_{0\leq t\leq T}\Big|\sqrt{2c_k^{1/q}\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r)\Big|^{2q}\Big]\\
&\qquad +\Enone\sup_{0\leq t\leq T}\Big|\sqrt{2\gamma}\int_0^t e^{-\frac{\gamma}{m}(t-r)}dW_0(r)\Big|^{2q}\bigg], \end{aligned} \end{equation*} where in the third line, we have used Holder's inequality with $\frac{1}{q^*}+\frac{1}{2q}=1$. Also, note that from the first equation of~\eqref{eqn:GLE-Markov}, it holds that \begin{align*} \Enone\sup_{0\leq t\leq T}x(t)^{2q}\leq c(q)\Big(x^{2q}+\int_0^T\Enone\sup_{0\leq r\leq t}v(r)^{2q}dt\Big), \end{align*} and that by Lemma~\ref{lem:limit:zeromass}, we have \begin{align*}
\sum_{k\geq 1}\Enone\sup_{0\leq t\leq T}\Big|\sqrt{2c_k^{1/q}\lambda_k}\int_0^t e^{-\lambda_k(t-r)}dW_k(r)\Big|^{2q}\leq c(T,q)\sum_{k\geq 1}c_k\lambda_k, \end{align*} and \begin{align*}
\Enone\sup_{0\leq t\leq T}\Big|\sqrt{2\gamma}\int_0^t e^{-\frac{\gamma}{m}(t-r)}dW_0(r)\Big|^{2q}\leq c(T,q)\gamma m^{q-1}. \end{align*} Furthermore, recalling $c_k$ from~\eqref{c-k}, we see that for $q>0$ sufficiently large \begin{align*} \sum_{k\geq 1}c_k^{(1/2-1/2q)q*}=\sum_{k\geq 1}\frac{1}{k^{(1+\alpha\beta)(1/2-1/2q)q*}}<\infty, \end{align*}
thanks to the fact that $q*>1$ and $\alpha\beta>1$, where the latter follows from the conditions about $\alpha,\,\beta$ in Assumption~\ref{cond:wellposed}. Also, recalling $\lambda_k$ from~\eqref{c-k} and the norm $\|\cdot\|_{-s}$ from~\eqref{eqn:H_p}, it is straightforward to verify that the sums $\sum_{k\geq 1}\sqrt{c_k}z_k$, $\sum_{k\geq 1}c_k$, and $\sum_{k\geq 1}c_k\lambda_k$ are absolutely convergent. Putting everything together, we find \begin{align*} m^{2q}\Enone\sup_{0\leq t\leq T}v(t)^{2q}&\leq c(T,q,X(0))\Big[m^{2q}+m^{q-1}+m^{2q}\int_0^T\Enone\sup_{0\leq r\leq t}v(r)^{2q}dt\Big], \end{align*} where $c(T,q,X(0))>0$ is independent with $m$. Gronwall's inequality now implies \begin{align*} m^{2q}\Enone\sup_{0\leq t\leq T}v(t)^{2q}\leq c(T,q,X(0))(m^{2q}+m^{q-1})\to 0,\quad m\to 0. \end{align*} The proof is thus complete. \end{proof}
We now turn our attention to Theorem~\ref{thm:limit:zeromass}. The proof is a slightly modification from that of Theorem 2.4 of~\cite{herzog2016small}. The key observation is that instead of controlling the exiting time of the process $x(t)$ as $m\to 0$, we are able to control $u(t)$ since $u(t)$ is independent of $m$. \begin{proof}[Proof of Theorem~\ref{thm:limit:zeromass}] For $R,\,m>0$, define the following stopping times \begin{equation} \label{eqn:stoppingtime:zeromass}
\sigma^R = \inf_{t\geq 0}\{|u(t)|\geq R\},\quad\text{ and }\quad\sigma^R_m = \inf_{t\geq 0}\{|x(t)|\geq R\}, \end{equation} and recall \begin{align*}
\tau^R = \inf_{t\geq 0}\{\|U(t)\|_{\dot{\mathcal{H}}_{-s}}\geq R\},\quad\text{ and }\quad\tau^R_m = \inf_{t\geq 0}\{\|X(t)\|_{\mathcal{H}_{-s}}\geq R\}. \end{align*} By the definitions of the norms in $\dot{\mathcal{H}}_{-s}$, cf.~\eqref{eqn:Hsdot}, we see that $\tau^R\leq \sigma^R$ a.s. From the proof of Proposition~\ref{prop:wellposed:limit:zeromass}, it is straight forward to verify that for all $T>0$, \begin{align*} \P{\sigma^R< T}&\leq \P{\tau^R< T} \to 0,\quad R\to \infty. \end{align*}
For $R,\,T,\,m,\,\xi>0$, we have \begin{align*}
\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x(t)-u(t)|>\xi\Big\}&\leq \mathbb{P}\Big\{\sup_{0\leq t\leq T}|x(t)-u(t)|>\xi,\sigma^R\wedge\sigma^R_m\geq T\Big\}\\ &\qquad+\P{\sigma^R\wedge\sigma^R_m<T}. \end{align*} To control the first term on the above RHS, we note that for $0\leq t\leq \sigma^R\wedge\sigma^R_m$, $u(t)=u^R(t)$ and $x(t)=x^R(t)$ a.s. We thus obtain the bound \begin{align*}
\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x(t)-u(t)|>\xi,\sigma^R\wedge\sigma^R_m\geq T\Big\}&\leq \mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|>\xi\Big\}\to 0,\quad m\to 0, \end{align*} where the last convergence in probability follows immediately from Proposition~\ref{prop:limit:zeromass:lipschitz}. We are left to estimate $\P{\sigma^R\wedge\sigma^R_m<T}$. To this end, we have that \begin{align*}
\P{\sigma^R\wedge\sigma^R_m<T}&\leq \mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|\leq \xi,\sigma^R\wedge\sigma^R_m < T\Big\}\\
&\qquad+\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|> \xi\Big\}\\
&\leq \mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|\leq\xi,\sigma^R_m < T\leq\sigma^R\Big\}+\P{\sigma^R< T}\\
&\qquad +\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|> \xi\Big\}. \end{align*} Note that for $R>1$ and $\xi\in(0,1)$, a chain of event implications is derived as follows. \begin{align*}
\MoveEqLeft[2]\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|\leq \xi,\sigma^R_m < T\leq\sigma^R\Big\}\\
&= \Big\{\sup_{0\leq t\leq T}|x^R(t)-u(t)|\leq \xi,\sup_{0\leq t\leq T}|x^R(t)|\geq R,\sigma^R_m < T\leq\sigma^R\Big\}\\
&\subseteq \Big\{\sup_{0\leq t\leq T}|u(t)|> R-1,\sigma^R_m < T\leq\sigma^R\Big\}\\ &\subseteq \{\sigma^{R-1}< T\} , \end{align*} which implies that \begin{align*}
\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|<\xi,\sigma^R_m < T\leq\sigma^R\Big\}\leq \P{\sigma^{R-1}< T}. \end{align*} Finally, putting everything together, for $R>1>\xi>0$, $T,\,m>0$, we obtain the estimate \begin{align*}
\MoveEqLeft[2]\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x(t)-u(t)|>\xi\Big\}\\&\leq 2\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|>\xi\Big\}+\P{\sigma^{R-1}< T}+\P{\sigma^{R} < T}\\
&\leq 2\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|>\xi\Big\}+\P{\tau^{R-1}< T}+\P{\tau^{R} < T}. \end{align*} By taking $R$ sufficiently large and then shrinking $m$ further to zero, we obtain the result, thus completing the proof. \end{proof}
\section{White-noise limit} \label{sec:limit:whitenoise} For notation simplicity, in this section, we shall omit the subscript $\epsilon$ in \begin{align*} X_\epsilon(t)=(x_\epsilon(t),v_\epsilon(t),z_{1,\epsilon}(t),\dots). \end{align*}
With regards to the well-posedness of~\eqref{eqn:GLE-Markov:limit:whitenoise}, recalling $c_k,\,\lambda_k$ from~\eqref{c-k}, we see that the noise term is well-defined thanks to Condition (D) of Assumption~\ref{cond:wellposed}, namely \begin{align} \label{ineq:limit:whitenoise:1} \Enone\Big(\int_0^T\sum_{k\geq 1}\sqrt{\frac{2c_k}{\lambda_k}} dW_k(t)\Big)^2=2T\sum_{k\geq 1} \frac{c_k}{\lambda_k}=2T\sum_{k\geq 1}\frac{1}{k^{1+(\alpha-1)\beta}}<\infty. \end{align} The solution $(u(t),p(t))$ of~\eqref{eqn:GLE-Markov:limit:whitenoise} then is constructed using similar arguments as in the proof of Proposition~\ref{prop:wellposed:limit:zeromass} in Section~\ref{sec:limit:zeromass} via stopping times $\tau^R$, $R>0$, given by \begin{align} \tau^R=\inf_{t\geq 0}\{u(t)^2+p(t)^2\geq R^2\}, \end{align} and the local solutions \begin{equation}\label{eqn:GLE-Markov:limit:whitenoise:local} \begin{aligned} d u^R(t) &= p^R(t)\, d t ,\\ m\, d p^R(t)&=\Big(\!\!-\Big(\gamma+\sum_{k\geq 1}\frac{c_k}{\lambda_k}\Big) p^R(t)-\Phi'(u^R(t))\theta_R(u^R(t))\Big)\,dt\\ &\qquad\qquad\qquad\qquad\qquad\quad+\sum_{k\geq 1}\sqrt{\frac{2c_k}{\lambda_k}}dW_k(t)+\sqrt{2\gamma}\, dW_0(t), \end{aligned} \end{equation} where $\theta^R$ is the cut-off function defined in~\eqref{defn:theta-R}. Furthermore, we have the following bound: for every $T>0$ and $(u_0,p_0)\in\mathbb{R}^2$, it holds that \begin{align}\label{ineq:whitenoise:limit:bound} \Enone\Big[\sup_{0\leq t\leq T}u(t)^2+p(t)^2\Big]\leq C(T,u_0,p_0). \end{align} This estimate will be useful later in the proof of Theorem~\ref{thm:limit:whitenoise:L1converge}. The solution $X_\epsilon(t)$ is constructed using the stopping time $\tau^R_\epsilon$ given by \begin{align}
\tau^R_\epsilon=\inf_{t\geq 0}\{\|X(t)\|_{\mathcal{H}_{-s}}\geq R\}, \end{align} and the local solutions of the cut-off system obtained from~\eqref{eqn:GLE-Markov:whitenoise:epsilon} \begin{equation}\label{eqn:GLE-Markov:whitenoise:epsilon:local} \begin{aligned} d x^R(t) &= v^R(t)\, d t ,\\ m\, d v^R(t)&=\Big(\!\!-\gamma v^R(t)-\Phi'(x^R(t))\theta^R(x^R(t))-\sum_{k\geq 1} \sqrt{\frac{c_k}{\epsilon}} z^R_{k}(t)\Big)\,dt\\ &\qquad\qquad\qquad+\sqrt{2\gamma}\, dW_0(t), \\ d z^R_{k}(t)&=\Big(\!\!-\frac{\lambda_k}{\epsilon} z^R_{k}(t)+ \sqrt{\frac{c_k}{\epsilon}}v^R(t)\Big) \, dt+\sqrt{\frac{2\lambda_k}{\epsilon}}\, dW_k(t),\qquad k\geq 1. \end{aligned} \end{equation}
We now turn to the proof of Theorem~\ref{thm:limit:whitenoise:probconverge}. Similar to the proof of Theorem~\ref{thm:limit:zeromass}, it will make use of the local solutions $(u^R(t),p^R(t))$ from~\eqref{eqn:GLE-Markov:limit:whitenoise:local} and $(x^R(t),v^R(t))$ from~\eqref{eqn:GLE-Markov:whitenoise:epsilon:local}. As mentioned previously in Section~\ref{sec:limit:zeromass}, the idea essentially consists of two major steps: first, fixing $R>0$, we show that the corresponding local solution $(x^R(t),v^R(t))$ in~\eqref{eqn:GLE-Markov:whitenoise:epsilon:local} converges to $(u^R(t),p^R(t))$ in~\eqref{eqn:GLE-Markov:limit:whitenoise:local}. Then, taking $R$ sufficiently large, we obtain the convergence in probability of the original solutions by using appropriate bounds on stopping times when $(u(t),p(t))$ exits the ball of radius $R$ centered at origin. We begin by the following important result giving a uniform bound on the pair $(x(t),v(t))$.
\begin{proposition} \label{prop:limit:whitenoise:L2bound} Suppose that $\alpha,\,\beta,\, s$ satisfy Condition (D) of Assumption~\ref{cond:wellposed}. We assume further that either
(a) $\Phi'$ is globally Lipschitz,
\noindent or
(b) $\Phi'$ is not Lipschitz, but $\Phi$ satisfies Assumptions~\ref{cond:Phi} and Assumption~ \ref{cond:Phi:whitenoise}, and $\alpha,\,\beta$ satisfy Assumption~\ref{cond:whitenoise:L1converge}.
\noindent Let $X(t)$ solve~\eqref{eqn:GLE-Markov:whitenoise:epsilon} with initial conditions $(x(0),v(0),z_{1}(0),\dots)\in\mathcal{H}_{-s}$. Then, for every $T>0$, there exists a finite constant $C(T,X(0))$ such that \begin{displaymath} \sup_{\epsilon} \Enone\Big[\sup_{0\leq t\leq T}x(t)^2+v(t)^2\Big] \leq C(T,X(0)). \end{displaymath} \end{proposition} \begin{proof} We begin by applying Duhamel's formula on $z_k(t)$ from~\eqref{eqn:GLE-Markov:whitenoise:epsilon} to see that \begin{align}\label{eqn:limit:whitenoise:1} z_k(t)=e^{-\frac{\lambda_k}{\epsilon}t}z_k(0)+\sqrt{\frac{c_k}{\epsilon}}\int_0^t e^{-\frac{\lambda_k}{\epsilon}(t-r)}v(r)dr+\sqrt{\frac{2\lambda_k}{\epsilon}}\int_0^t e^{-\frac{\lambda_k}{\epsilon}(t-r)}dW_k(r). \end{align} Substituting $z_k$ by the formula above in the second equation from~\eqref{eqn:GLE-Markov:whitenoise:epsilon} in integral form, we obtain \begin{equation}\label{eqn:limit:whitenoise:1a} \begin{aligned} mv(t)&=mv(0)+\int_0^t\!\!-\gamma v(r)-\Phi'(x(r))dr+\sqrt{2\gamma}\int_0^tdW_0(r)\\ &\qquad-\sum_{k\geq 1}\sqrt{\frac{c_k}{\epsilon}} \int_0^te^{-\frac{\lambda_k}{\epsilon}r}z_k(0)dr-\sum_{k\geq 1}\frac{c_k}{\epsilon}\int_0^t\int_0^r e^{-\frac{\lambda_k}{\epsilon}(r-\ell)}v(\ell)d\ell dr\\ &\qquad-\sum_{k\geq 1}\frac{\sqrt{2c_k\lambda_k}}{\epsilon}\int_0^t\int_0^r e^{-\frac{\lambda_k}{\epsilon}(r-\ell)}dW_k(\ell)dr. \end{aligned} \end{equation} It is important to note that using integration by parts, the last noise term above can be written as \begin{multline}\label{eqn:limit:whitenoise:integrationbypart} -\frac{\sqrt{2c_k\lambda_k}}{\epsilon}\int_0^t\int_0^r e^{-\frac{\lambda_k}{\epsilon}(r-\ell)}dW_k(\ell)dr\\= \sqrt{\frac{2c_k}{\lambda_k}}\int_0^te^{-\frac{\lambda_k}{\epsilon}(t-r)}dW_k(r)-\sqrt{\frac{2c_k}{\lambda_k}}\int_0^tdW_k(r). \end{multline}
Suppose that Condition (\emph{a}) holds, i.e., $\Phi'$ is Lipschitz. In view of~\eqref{eqn:limit:whitenoise:1a} and~\eqref{eqn:limit:whitenoise:integrationbypart}, we have the following estimate for every $q>1$ and $0\leq t\leq T$, \begin{equation*}\label{ineq:limit:whitenoise:2} \begin{aligned} v(t)^{2q}
&\leq c(q)\bigg[ |v(0)|^{2q}+\int_0^T\!\!\sup_{0\leq r\leq t}v(r)^{2q}+x(r)^{2q}dr+ \Big(\sum_{k\geq 1}\frac{\sqrt{\epsilon c_k}}{\lambda_k}(1-e^{-\frac{\lambda_k}{\epsilon}t})|z_k(0)|\Big)^{2q}\\ &\qquad+ c(T)\Big(\sum_{k\geq 1}\frac{c_k}{\lambda_k}\Big)^{2q}\int_0^T \sup_{0\leq \ell\leq r}v(\ell)^{2q}dr\\ &\qquad+\sup_{0\leq t\leq T}\Big(\sqrt{2\gamma}\int_0^tdW_0(r)-\sum_{k\geq 1}\sqrt{\frac{2c_k}{\lambda_k}}\int_0^tdW_k(r)\Big)^{2q}\\
&\qquad+\Big(\sum_{k\geq 1}k^{-sq*}\Big)^{2q/q*}\sum_{k\geq 1}\sup_{0\leq t\leq T}\Big|\sqrt{\frac{2c_k k^{2s}}{\lambda_k}}\int_0^te^{-\frac{\lambda_k}{\epsilon}(t-r)}dW_k(r)\Big|^{2q}\bigg], \end{aligned} \end{equation*} where in the last line, we have used Holder's inequality with $\frac{1}{2q}+\frac{1}{q*}=1$. Note that for every $x\geq 0$, we have $1-e^{-x}\leq \sqrt{x}$. Using this inequality, we estimate \begin{equation}\label{ineq:limit:whitenoise:2a} \begin{aligned}
\sum_{k\geq 1}\frac{\sqrt{\epsilon c_k}}{\lambda_k}(1-e^{-\frac{\lambda_k}{\epsilon}t})|z_k(0)|&\leq \sum_{k\geq 1}\sqrt{\frac{ c_k t}{\lambda_k}}|z_k(0)|\\ &\leq\Big(T\sum_{k\geq 1}\frac{c_k k^{2s}}{\lambda_k}\sum_{k\geq 1}k^{-2s}z_k(0)^2\Big)^{1/2}. \end{aligned} \end{equation}
Recalling $c_k,\,\lambda_k$ from~\eqref{c-k} and the norm $\|\cdot\|_{\mathcal{H}_{-s}}$ from~\eqref{eqn:H_p}, thanks to Condition (D) of Assumption~\ref{cond:wellposed}, we see that the above RHS is finite and so is the sum $\sum_{k\geq 1}c_k/\lambda_k$. In addition, using Burkholder-Davis-Gundy's inequality, we have \begin{equation} \label{ineq:limit:whitenoise:2b} \begin{aligned} \MoveEqLeft[5]\Enone\sup_{0\leq t\leq T}\Big(\sqrt{2\gamma}\int_0^tdW_0(r)-\sum_{k\geq 1}\sqrt{\frac{2c_k}{\lambda_k}}\int_0^tdW_k(r)\Big)^{2q}\\ &\leq c(q)\Enone\Big(2\gamma\int_0^Tdr+\sum_{k\geq 1}\frac{2c_k}{\lambda_k}\int_0^Tdr\Big)^{q}=c(T,q)<\infty. \end{aligned} \end{equation} Finally, we invoke Lemma~\ref{lem:limit:zeromass} again to find \begin{equation}\label{ineq:limit:whitenoise:4} \begin{aligned} \MoveEqLeft[3]
\Enone\sum_{k\geq 1}\sup_{0\leq t\leq T}\Big|\sqrt{\frac{2c_k k^{2s}}{\lambda_k}}\int_0^te^{-\frac{\lambda_k}{\epsilon}(t-r)}dW_k(r)\Big|^{2q}\\ &\leq c(T,q)\sum_{l\geq 1}\frac{\epsilon^{q-1}c_k^q k^{2sq}}{\lambda_k^{2q-1}}\\ &=c(T,q)\epsilon^{q-1}\sum_{k\geq 1}\frac{1}{k^{q+(q\alpha-2q+1)\beta-2sq}}. \end{aligned} \end{equation} Note that for $\alpha>1$, $s>1/2$ and $\beta>0$ satisfying Condition (D) of Assumption~\ref{cond:wellposed}, there exist constants $q>1$ and $0<q*<2$ such that \begin{align*} q+(q\alpha-2q+1)\beta-2sq>1,\quad sq*>1,\quad\text{and}\quad\frac{1}{2q}+\frac{1}{q*}=1. \end{align*} Consequently, the sums $\sum_{k\geq 1}k^{-[q+(q\alpha-2q+1)\beta-2sq]}$ and $\sum_{k\geq 1}k^{-sq*}$ are both finite. Combining everything together, we infer \begin{align*} \Enone\big(\sup_{0\leq t\leq T}x(t)^{2q}+v(t)^{2q}\big)&\leq c(T,q,X(0))\Big[1+\int_0^T\Enone\big(\sup_{0\leq r\leq t}x(r)^{2q}+v(r)^{2q}\big) dt\Big]. \end{align*}
Choosing such $q$, we finally obtain the following estimate using Gronwall's inequality \begin{align}\label{ineq:limit:whitenoise:5} \Enone\big(\sup_{0\leq t\leq T}x(t)^{2q}+v(t)^{2q}\big)&\leq c(T,q,X(0)), \end{align} which proves the result for Condition (\emph{a}) since $q>1$.
Now suppose that Condition (\emph{b}) holds. To simplify notation, we set \begin{align*} g_k(t):=\int_0^t e^{-\frac{\lambda_k}{\epsilon}(t-r)}v(r)dr,\quad\text{and}\quad w_k(t):= \sqrt{2}\int_0^te^{-\frac{\lambda_k}{\epsilon}(t-r)}dW_k(r). \end{align*} Following ~\eqref{eqn:limit:whitenoise:1a} and~\eqref{eqn:limit:whitenoise:integrationbypart}, the equation on $v(t)$ is written as \begin{align*} d\Big(mv(t)-\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(t)\Big)& = \Big(-\gamma v(t)-\Phi'(x(t))-\sum_{k\geq 1}\sqrt{\frac{c_k}{\epsilon}}e^{-\frac{\lambda_k}{\epsilon}t}z_k(0)-\sum_{k\geq 1}\frac{c_k}{\epsilon}g_k(t)\Big)dt\\ &\qquad+\sqrt{2\gamma}dW_0(t)-\sum_{k\geq 1}\sqrt{ \frac{2c_k}{\lambda_k}}dW_k(t). \end{align*} We apply Ito's formula to $\big(v(t)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(t)\big)^2/2+\Phi(x(t))/m$ to see that \begin{align*} \MoveEqLeft[1]d\Big[\Big(v(t)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(t)\Big)^2/2+\Phi(x(t))/m\Big]\\ &=\Big(v(t)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(t)\Big)\Big(-\frac{\gamma}{m} v(t)-\sum_{k\geq 1}\sqrt{\frac{c_k}{m^2\epsilon}}e^{-\frac{\lambda_k}{\epsilon}t}z_k(0)-\sum_{k\geq 1}\frac{c_k}{m\epsilon}g_k(t)\Big)dt\\ &\qquad+\Big(v(t)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(t)\Big)\Big(\frac{\sqrt{2\gamma}}{m}dW_0(t)-\sum_{k\geq 1}\sqrt{ \frac{2c_k}{m^2\lambda_k}}dW_k(t)\Big)\\ &\qquad +\Big(\frac{\Phi'(x(t))}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(t)+\frac{\gamma}{m^2}+\sum_{k\geq 1}\frac{c_k}{\lambda_k}\Big) dt. \end{align*} We proceed to estimate the above RHS. Firstly, we invoke estimate~\eqref{ineq:limit:whitenoise:2a} to find \begin{align*} \MoveEqLeft[4]\int_0^t \Big(v(r)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)\Big(-\sum_{k\geq 1}\sqrt{\frac{c_k}{m^2\epsilon}}e^{-\frac{\lambda_k}{\epsilon}r}z_k\Big) dr \\
&\leq \sup_{0\leq r\leq t} \Big|v(r)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big|\,\sum_{k\geq 1}\frac{\sqrt{\epsilon c_k}}{\lambda_k}(1-e^{-\frac{\lambda_k}{\epsilon}t})|z_k|\\
&\leq \sup_{0\leq r\leq t} \Big|v(r)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big|\,\Big(\sum_{k\geq 1}\frac{c_k k^{2s}}{\lambda_k}\sum_{k\geq 1}k^{-2s}z_k^2\Big)^{1/2}\\
&\leq \frac{1}{2}\sup_{0\leq r\leq t} \Big|v(r)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big|^2+2\sum_{k\geq 1}\frac{c_k k^{2s}}{\lambda_k}\sum_{k\geq 1}k^{-2s}z_k^2. \end{align*} Similarly, we have \begin{align*}
g_k(r)=\int_0^r e^{-\frac{\lambda_k}{\epsilon}(r-\ell)}v(\ell)d\ell\leq\sup_{0\leq \ell\leq r}|v(\ell)|\frac{\epsilon}{\lambda_k}, \end{align*} which implies that \begin{align*} \MoveEqLeft[4]\int_0^t\Big(v(r)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)\Big(-\sum_{k\geq 1}\frac{c_k}{m\epsilon}g_k(r)\Big)dr\\ &\leq c\Big( \sum_{k\geq 1}\frac{c_k}{\lambda_k}\Big)\int_0^t\sup_{0\leq \ell\leq r }v(\ell)^2+\sup_{0\leq \ell\leq r }\Big(\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(\ell)\Big)^2dr. \end{align*} With regard to the martingale term, we invoke Burkholder-Davis-Gundy's inequality to estimate \begin{align*}
\MoveEqLeft[4]\Enone\sup_{0\leq r\leq t }\Big|\int_0^r\Big(v(\ell)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(\ell)\Big)\Big(\frac{\sqrt{2\gamma}}{m}dW_0(\ell)-\sum_{k\geq 1}\sqrt{ \frac{2c_k}{m^2\lambda_k}}dW_k(\ell)\Big)\Big|\\ &\leq c\Big[\Big(\frac{2\gamma}{m^2}+\sum_{k\geq 1}\frac{2c_k}{m^2\lambda_k}\Big)\int_0^t\Enone\Big(v(r)-\frac{1}{m}\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)^2dr+1\Big]. \end{align*} Lastly, we employ Assumption~\ref{cond:Phi:whitenoise} to infer \begin{align*} \int_0^t \Phi'(x(r))\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)dr&\leq c\int_0^t\Phi(x(r))+\Big(\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)^n+1\, dr. \end{align*} Putting everything together, we arive at the following inequality \begin{align*} \Enone\sup_{0\leq t\leq T } v(t)^2+\Phi(x(t))& \leq c(T)\Big[1+\int_0^T\Enone\sup_{0\leq r\leq t } v(r)^2+\Phi(x(r))\,dt\\ &\quad+\Enone\sup_{0\leq t\leq T}\Big(\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)^2 +\Enone\sup_{0\leq t\leq T}\Big(\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)^n\Big]. \end{align*} The result now follows immediately from Gronwall's inequality if we can show that the last two terms on the above RHS is finite and independent of $\epsilon$. To this end, we claim that for every $T>0$ and $q>2$, there exists a finite constant $C(T,q)>0$ such that \begin{equation}\label{ineq:limit:whitenoise:6} \Enone\sup_{0\leq t\leq T}\Big(\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)^{2q} \leq c(T,q). \end{equation} Recalling $w_k(t):= \sqrt{2}\int_0^te^{-\frac{\lambda_k}{\epsilon}(t-r)}dW_k(r)$, similar to \eqref{ineq:limit:whitenoise:4}, we employ Holder's inequality and Lemma~\ref{lem:limit:zeromass} to see that \begin{align*} \MoveEqLeft[4]\Enone\sup_{0\leq t\leq T}\Big(\sum_{k\geq 1}\sqrt{\frac{c_k}{\lambda_k}}w_k(r)\Big)^{2q}\\
&\leq\Big(\sum_{k\geq 1}k^{-q_1q*}\Big)^{2q/q*}\sum_{k\geq 1}\Enone\bigg[\sup_{0\leq t\leq T}\Big|\sqrt{\frac{c_k k^{2q_1}}{\lambda_k}}w_k(t)\Big|^{2q}\bigg]\\ &=c(T,q)\Big(\sum_{k\geq 1}k^{-q_1q*}\Big)^{2q/q*}\epsilon^{q-1}\sum_{k\geq 1}\frac{1}{k^{q+(q\alpha-2q+1)\beta-2q_1q}}. \end{align*} where $\frac{1}{2q}+\frac{1}{q*}=1$ and $q_1>0$ is a constant satisfying \begin{align*} q_1q*>1\quad\text{and}\quad q+(q\alpha-2q+1)\beta-2q_1q>1. \end{align*} Solving the above inequalities for $q_1$, we find \begin{align*} \frac{1+(\alpha-2)\beta}{2}+\frac{\beta}{2q}-\frac{1}{2q}>q_1>1-\frac{1}{2q}, \end{align*} which is always possible thanks to the second part of Condition (b), namely, $(\alpha-2)\beta>1$. The proof is thus complete. \end{proof}
\begin{remark} \label{ref:whitenoise} (a) The trick of using integration by part in~\eqref{eqn:limit:whitenoise:integrationbypart} was previously employed in ~\cite{hottovy2015smoluchowski,ottobre2011asymptotic}.
(b) The condition $\alpha>1$ of the diffusive regime was employed throughout the proof of Proposition~\ref{prop:limit:whitenoise:L2bound}, e.g. estimates~\eqref{ineq:limit:whitenoise:2a},~\eqref{ineq:limit:whitenoise:2b} and ~\eqref{ineq:limit:whitenoise:4}. \end{remark}
\begin{proposition}\label{prop:limit:whitenoise:lipschitz} Under the same Hypothesis of Theorem~\ref{thm:limit:whitenoise:probconverge}, assume further that $\Phi'(x)$ is globally Lipschitz. Let $X_\epsilon(t)=(x_\epsilon(t),v_\epsilon(t),z_{1,\epsilon}(t),\dots)$ be the solution of~\eqref{eqn:GLE-Markov:whitenoise:epsilon} with initial conditions \begin{align*}(x_\epsilon(0),v_\epsilon(0),z_{1,\epsilon}(0),z_{2,\epsilon}(0),\dots)=(x,v,z_{1},z_2,\dots)\in\mathcal{H}_{-s}, \end{align*}
and $(u(t),p(t))$ be the solution of~\eqref{eqn:GLE-Markov:limit:whitenoise} with initial conditions $(u(0),p(0))=(x,v)$. Then, for every $T>0$, \begin{equation*}
\Enone\Big[\sup_{0\leq t\leq T}|x(t)-u(t)|^2+\sup_{0\leq t\leq T}|v(t)-p(t)|^2\Big]\to 0, \quad \epsilon\to 0. \end{equation*} \end{proposition} The proof of Proposition~\ref{prop:limit:whitenoise:lipschitz} is based on that of Theorem 2.6 in~\cite{ottobre2011asymptotic}, adapted to our infinite-dimensional setting. \begin{proof} Setting $\overline{x}(t) := x(t)-u(t),\,\overline{v}(t):=v(t)-p(t)$, we see that from\eqref{eqn:GLE-Markov:whitenoise:epsilon}, \eqref{eqn:GLE-Markov:limit:whitenoise}, $(\overline{x}(t),\overline{v}(t))$ satisfies the following system \begin{align*}
\overline{x}(t) &=\int_0^t \!\!\overline{v}(r)\, d r, \\ m \overline{v}(t)&=\int_0^t\!\!\Big(\!\!-\gamma\overline{v}(r)+\Big(\sum_{k\geq 1}\frac{c_k}{\lambda_k}\Big) p(r)-\big[\Phi'(x(r))-\Phi'(u(r))\big]-\sum_{k\geq 1}\sqrt{\frac{c_k}{\epsilon}}z_k(r)\Big)\,dr\\ &\qquad+\int_0^t\sum_{k\geq 1}\sqrt{\frac{2c_k}{\lambda_k}}dW_k(r). \end{align*} with the initial conditions $(\overline{x}(0),\overline{v}(0))=(0,0)$. Regrading $z_k(t)$ terms, we integrate with respect to time the third equation in~\eqref{eqn:GLE-Markov:whitenoise:epsilon} to find that \begin{align*} \frac{\sqrt{\epsilon c_k}}{\lambda_k} (z_k(t)-z_k(0)) -\frac{c_k}{\lambda_k}\int_0^tv(r)dr-\sqrt{\frac{ c_k}{\lambda_k}}\int_0^tdW_k(r)=-\sqrt{\frac{c_k}{\epsilon}}\int_0^tz_k(r). \end{align*} With these observations, the system of integral equations on $(\overline{x}(t),\overline{v}(t))$ becomes \begin{equation} \label{eqn:limit:whitenoise:lipschitz:1} \begin{aligned}
\overline{x}(t) &=\int_0^t\!\! \overline{v}(r)\, d r, \\ m \overline{v}(t)&=\int_0^t\!\!\Big(\!\!-\Big(\gamma+\sum_{k\geq 1}\frac{c_k}{\lambda_k}\Big) \overline{v}(r)-\big[\Phi'(x(r))-\Phi'(u(r))\big]\Big)\,dr\\ &\qquad\qquad\qquad+\sqrt{\epsilon}\sum_{k\geq 1}\frac{\sqrt{c_k}}{\lambda_k}(z_k(t)-z_k(0)). \end{aligned} \end{equation}
In the above system, we have implicitly re-arranged infinitely many terms, resulting in the cancellation of noise terms. Recalling $c_k,\,\lambda_k$ from~\eqref{c-k} and the norm $\|\cdot\|_{\mathcal{H}_{-s}}$ from~\eqref{eqn:H_p}, this re-arrangement is possible following from~\eqref{ineq:limit:whitenoise:1} and the estimate \begin{align*}
\sum_{k\geq 1}\Big|\frac{\sqrt{\epsilon c_k}}{\lambda_k}(z_k(t)-z_k(0))\Big|&\leq \sum_{k\geq 1}\frac{c_k}{\lambda_k}\int_0^t |v(r)|dr+\sum_{k\geq 1}+\frac{1}{\sqrt{\epsilon}}\int_0^t\sum_{k\geq 1}\sqrt{c_k}|z_k(r)|dr\\
&\qquad+\sum_{k\geq 1}\sqrt{\frac{ c_k}{\lambda_k}}\big|\int_0^tdW_k(r)\big|\\ &<\infty, \text{ a.s.} \end{align*} thanks to condition (D) of Assumption~\ref{cond:wellposed}. We invoke the assumption that $\Phi'$ is globally Lipschitz and Gronwall's inequality to deduce from~\eqref{eqn:limit:whitenoise:lipschitz:1} \begin{align*}
\Enone\sup_{0\leq t\leq T}|\overline{x}(t)|+|\overline{v}(t)| \leq C(T)\sqrt{\epsilon}\,\Enone\sup_{0\leq t\leq T}\Big|\sum_{k\geq 1}\frac{\sqrt{c_k}}{\lambda_k}(z_k(t)-z_k(0))\Big|. \end{align*} The result now follows immediately from Proposition~\ref{prop:limit:whitenoise:lipschitz:1} below. \end{proof}
\begin{proposition} \label{prop:limit:whitenoise:lipschitz:1} Under the same Hypothesis of Proposition~\ref{prop:limit:whitenoise:lipschitz}, suppose that $X(t)=(x(t),v(t),z_{1}(t),\dots)$ solves~\eqref{eqn:GLE-Markov:whitenoise:epsilon} with initial conditions $(x(0),v(0),z_{1}(0),\dots)\in \mathcal{H}_{-s}$. Then, \begin{align*}
\sqrt{\epsilon}\,\Enone\sup_{0\leq t\leq T}\Big|\sum_{k\geq 1}\frac{\sqrt{c_k}}{\lambda_k}(z_{k}(t)-z_k(0))\Big|\to 0,\quad\epsilon\to 0. \end{align*} \end{proposition} \begin{proof} From~\eqref{eqn:limit:whitenoise:1}, we see that \begin{equation}\label{ineq:limit:whitenoise:3} \begin{aligned}
\sum_{k\geq 1}\frac{\sqrt{\epsilon c_k}}{\lambda_k}|z_k(t)-z_k(0)|
&\leq \sum_{k\geq 1}\frac{\sqrt{\epsilon c_k}}{\lambda_k}\big(e^{-\frac{\lambda_k}{\epsilon}t}-1\big)|z_k(0)|+\sum_{k\geq 1}\frac{c_k}{\lambda_k}\int_0^t e^{-\frac{\lambda_k}{\epsilon}(t-r)}|v(r)|dr\\
&\qquad+\sum_{k\geq 1}\sqrt{\frac{2c_k}{\lambda_k}}\Big|\int_0^t e^{-\frac{\lambda_k}{\epsilon}(t-r)}dW_k(r)\Big|. \end{aligned} \end{equation} We aim to show that each series on the above RHS converges to zero in expectation as $\epsilon\downarrow 0$. We note that the convergence to zero of the last series follows immediately from~\eqref{ineq:limit:whitenoise:4}. For the other two terms, we shall make use of the following inequality: for $q>0$, there exists $c(q)>0$ such that for every $x\geq 0$, it holds that \begin{equation}\label{ineq:basic:exponential} 1-e^{-x}\leq c(q)x^q. \end{equation} For a positive $q_1<\frac{1}{2}$ (to be chosen later), we estimate the first sum on the RHS of~\eqref{ineq:limit:whitenoise:3} as follows. \begin{align*} \sum_{k\geq 1}\sup_{0\leq t\leq T}\frac{\sqrt{\epsilon c_k}}{\lambda_k}\big(e^{-\frac{\lambda_k}{\epsilon}t}-1\big)z_k(0)&\leq c(T,q_1)\epsilon^{1/2-q_1}\frac{c_k^{1/2}}{\lambda_k^{1-q_1}}z_k(0)\\ &\leq c(T,q_1) \epsilon^{1/2-q_1}\Big(\sum_{k\geq 1}\frac{c_k k^{2s}}{\lambda_k^{2-2q_1}}\Big)^{1/2}\Big(\sum_{k\geq 1}k^{-2s}z_k(0)^2\Big)^{1/2}, \end{align*} where we have used~\eqref{ineq:basic:exponential} on the first line and Holder's inequality on the second line, respectively. Recalling \eqref{c-k}, we have \begin{align*} \sum_{k \geq 1}\frac{c_k k^{2s}}{\lambda_k^{2-2q_1}} = \sum_{k\geq 1}\frac{1}{k^{1+(\alpha+2q_1-2)\beta-2s}}. \end{align*} In view of Condition (D) of Assumption~\ref{cond:wellposed}, there always exists a constant $q_1\in (0,1/2)$ such that $(2q_1-1)\beta+(\alpha-1)\beta-2s>0$, which implies that the above RHS is finite. Similarly, we have \begin{align*}
\sum_{k\geq 1}\sup_{0\leq t\leq T}\frac{c_k}{\lambda_k}\int_0^t e^{-\frac{\lambda_k}{\epsilon}(t-r)}|v(r)|dr&\leq\sum_{k\geq 1} \sup_{0\leq t\leq T} \frac{\epsilon c_k}{\lambda_k^2}\big(1-e^{-\frac{\lambda_k}{\epsilon}t}\big)\sup_{0\leq t\leq T}|v(t)|\\
&\leq c(T,q_2)\epsilon^{1-q_2}\sum_{k\geq 1}\frac{c_k}{\lambda_k^{2-q_2}}\sup_{0\leq t\leq T}|v(t)|\\
&= c(T,q_2)\epsilon^{1-q_2}\sum_{k\geq 1}\frac{1}{k^{1+(\alpha-2+q_2)\beta}}\sup_{0\leq t\leq T}|v(t)|. \end{align*} We invoke Condition (D) from Assumption~\ref{cond:wellposed} again to see that there exists a positive $q_2\in(0,1)$ such that $\alpha-2+q_2>0$. Choosing such $q_2$ implies that the series on the above RHS is convergent. We thus obtain the estimate \begin{align*}
\Enone\sum_{k\geq 1}\sup_{0\leq t\leq T}\frac{c_k}{\lambda_k}\int_0^t e^{-\frac{\lambda_k}{\epsilon}(t-r)}|v(r)|dr&\leq c(T,q_2)\epsilon^{1-q_2}\Enone\sup_{0\leq t\leq T}|v(t)|\leq c(T,q_2)\epsilon^{1-q_2}, \end{align*} where the last implication follows from Proposition~\ref{prop:limit:whitenoise:L2bound}. Putting everything together, we obtain the result. \end{proof}
Since we will make use of exiting times, with a slightly abuse of notation, it is convenient to recall from~\eqref{eqn:stoppingtime:zeromass} for $R>0$ \begin{equation*} \label{defn:stoppingtime:whitenoise}
\sigma^R = \inf_{t\geq 0}\{|u(t)|\geq R \},\quad\text{ and }\quad \sigma^R_\epsilon = \inf_{t\geq 0}\{|x(t)|\geq R .\} \end{equation*} With Proposition~\ref{prop:limit:whitenoise:lipschitz} in hand, we give the proof of Theorem~\ref{thm:limit:whitenoise:probconverge}. \begin{proof}[Proof of Theorem~\ref{thm:limit:whitenoise:probconverge}]
The arguments are almost the same as those in the proof of Theorem~\ref{thm:limit:zeromass} and hence omitted. The only difference here is the appearance of the term $|v(t)-p(t)|$. Nevertheless, we note that for $0\leq t\leq \sigma^R\wedge\sigma^R_\epsilon$, \begin{align*}
(u(t),p(t))=(u^R(t),p^R(t))\quad \text{ and }\quad (x(t),v(t))=(x^R(t),v^R(t)), \end{align*} and thus the proof of Theorem~\ref{thm:limit:zeromass} is applicable. \end{proof} We finally turn our attention to Theorem~\ref{thm:limit:whitenoise:L1converge}. The proof is relatively short and will make use of Condition (\emph{b}) in Proposition~\ref{prop:limit:whitenoise:L2bound}. \begin{proof}[Proof of Theorem~\ref{thm:limit:whitenoise:L1converge}] For given $R>0$, let $\sigma^R,\,\sigma^R_\epsilon$ be defined as in~\eqref{eqn:stoppingtime:zeromass}. As mentioned above, for $0\leq t\leq \sigma^R\wedge\sigma^R_\epsilon$, \begin{align*}
(u(t),p(t))=(u^R(t),p^R(t))\quad \text{ and }\quad (x(t),v(t))=(x^R(t),v^R(t)). \end{align*} We then have a chain of implications \begin{equation*} \begin{aligned} \MoveEqLeft[3]
\Enone\Big[\sup_{0\leq t\leq T}|x(t)-u(t)|^q+|v(t)-p(t)|^q \Big] \\ &=
\Enone\Big[\Big(\sup_{0\leq t\leq T}|x(t)-u(t)|^q+|v(t)-p(t)|^q \Big)1_{\{\sigma^R\wedge\sigma^R_\epsilon<T\}}\Big]\\ &\qquad\qquad\qquad
+\Enone\Big[\Big(\sup_{0\leq t\leq T}|x(t)-u(t)|^q+|v(t)-p(t)|^q \Big)1_{\{\sigma^R\wedge\sigma^R_\epsilon>T\}}\Big]\\
&\leq \Enone\Big[\Big(\sup_{0\leq t\leq T}|x(t)-u(t)|^q+|v(t)-p(t)|^q \Big)1_{\{\sigma^R\wedge\sigma^R_\epsilon<T\}}\Big]\\ &\qquad\qquad\qquad
+\Enone\Big[\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|^q+|v^R(t)-p^R(t)|^q\Big]. \end{aligned} \end{equation*} On one hand, in view of Proposition~\ref{prop:limit:whitenoise:lipschitz}, since $1\leq q<2$, it holds that \begin{align*}
\Enone\Big[\sup_{0\leq t\leq T}|x^R(t)-u^R(t)|+|v^R(t)-p^R(t)|\Big]\to 0,\quad\epsilon\to 0. \end{align*} On the other hand, we invoke Holder's inequality with $\frac{q}{2}+\frac{1}{q*}=1$ to estimate \begin{multline*}
\Enone\Big[\Big(\sup_{0\leq t\leq T}|x(t)-u(t)|^q+|v(t)-p(t)|^q \Big)1_{\{\sigma^R\wedge\sigma^R_\epsilon<T\}}\Big]\\ \leq c\Big(\Enone\Big[\sup_{0\leq t\leq T}x(t)^2+v(t)^2\Big]+\Enone\Big[\sup_{0\leq t\leq T}u(t)^2+p(t)^2\Big] \Big)^{q/2}\Big(\P{\sigma^R\wedge\sigma^R_\epsilon<T}\Big)^{1/q*}. \end{multline*} Notice that by Markov's inequality, we have \begin{align*} \P{\sigma^R\wedge\sigma^R_\epsilon<T}&\leq \P{\sigma^R<T}+\P{\sigma^R_\epsilon<T}\\
&\leq\mathbb{P}\Big\{\sup_{0\leq t\leq T}|u(t)|\geq R\Big\}+\mathbb{P}\Big\{\sup_{0\leq t\leq T}|x(t)|\geq R\Big\}\\
&\leq \frac{\Enone\big[\sup_{0\leq t\leq T}|u(t)|^2\big]+\Enone\big[\sup_{0\leq t\leq T}x(t)^2\big]}{R^2}. \end{align*} The result now follows immediately from~\eqref{ineq:whitenoise:limit:bound} and Proposition~\ref{prop:limit:whitenoise:L2bound} by first taking $R$ sufficiently large and then shrinking $\epsilon$ further to zero. The proof is thus complete. \end{proof}
\section{Discussion} We have established rigorous results on the asymptotical analysis of an infinite-dimensional GLE when the memory kernel $K(t)$ has a power-law decay, i.e. $K(t)\sim t^{-\alpha}$ as $t\to\infty$. With regards to the small-mass limit, we are able to obtain the convergence in probability of the GLE for every exponent constant $\alpha>0$. However, in the white-noise limit, a similar convergence was established only when $\alpha>1$. The method that we employed was not able to extend the result when $\alpha\in(0,1]$, which is interestingly also the barrier for the unique ergodicity of~\eqref{eqn:GLE-Markov} \cite{glatt2018generalized}. As mentioned earlier in Remark~\ref{ref:whitenoise}, our technique in the white-nosie limit requires that the memory be integrable ($\alpha>1$) for the analysis of the solutions as well as the asymptotical behaviors. It therefore remains an open question whether the solution's energy is still bounded uniformly and there exists a limiting system.
Finally, another question for future works is whether one can take both limits in sequence, which means that the small-mass variable $m$ is written as an order of $\epsilon$, the white-noise variable. It is not clear that the theorems presented in this work combined together are able to produce an explicit answer. We note that a similar study in finite-dimensional setting was carried out in~\cite{lim2017homogenization}. Yet, we have not been able to see if the same method can be applied to our infinite system. We believe handling this case will require a more substantial work.
\end{document} | arXiv | {
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\begin{document}
\title[$p$-Adic multiresolution analysis and wavelet frames] {$p$-Adic multiresolution analysis\\and wavelet frames}
\author{S.~Albeverio}
\address{Universit\"at Bonn, Institut f\"ur Angewandte Mathematik, Abteilung Sto\-chas\-tik, Wegelerstra\ss e 6, D-53115 Bonn and Interdisziplinäres Zentrum f\"ur Komplexe Systeme, Universit\"at Bonn, R\"omerstra\ss e 164 D-53117, Bonn, Germany}
\email{albeverio@uni-bonn.de}
\thanks{The first and the third authors were supported in part by DFG Project 436 RUS 113/809. The second author was supported in part by Grants 06-01-00471 and 07-01-00485 of RFBR. The third author was supported in part by Grant 06-01-00457 of RFBR}
\author{S.~Evdokimov} \address{St.-Petersburg Department of Steklov Institute of Mathematics, St.-Petersburg,
Fontanka-27, 191023 St. Petersburg, RUSSIA }
\email{evdokim@pdmi.ras.ru}
\author{M.~Skopina} \address{Department of Applied Mathematics and Control Processes, St. Petersburg State University, \ Universitetskii pr.-35, 198504 St. Petersburg, Russia.} \email{skopina@MS1167.spb.edu}
\subjclass[2000]{Primary 42C40, 11E95; Secondary 11F85}
\date{}
\keywords{$p$-adic multiresolution analysis; refinable equations, wavelets.}
\begin{abstract} We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating MRAs (scaling functions) is given. We prove that only $1$-periodic test functions may be taken as orthogonal scaling functions. We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a $p$-adic wavelet frame. \end{abstract}
\maketitle
\section{Introduction} \label{s1}
In the early nineties a general scheme for the construction of wavelets (of real argument) was developed. This scheme is based on the notion of multiresolution analysis (MRA in the sequel) introduced by Y.~Meyer and S.~Mallat~\cite{Mallat-1}, \cite{Meyer-1} (see also, e.g., ~\cite{31}, ~\cite{NPS}). Immediately specialists started to implement new wavelet systems. Nowadays it is difficult to find an engineering area where wavelets are not applied.
In the $p$-adic setting, the situation is as follows. In 2002 S.~V.~Kozyrev~\cite{Koz0} found a compactly supported $p$-adic wavelet basis for ${ L}^2({\mathbb Q}_p)$ which is an analog of the Haar basis. It turned out that these wavelets were eigenfunctions of $p$-adic pseudo-differential operators~\cite{Koz2}. J.J.~Benedetto and R.L.~Benedetto~\cite{Ben-Ben} conjectured that other $p$-adic wavelets with the same set of translations can not be constructed because this set is not a group, and the corresponding MRA-theory can not be developed. Another conjecture was rised by A.~Khrennikov and V.~Shelkovich~\cite{Kh-Sh1}. They assumed that the equality \begin{equation} \label{62.0-3} \phi(x)=\sum_{r=0}^{p-1}\phi\Big(\frac{1}{p}x-\frac{r}{p}\Big), \quad x\in {\mathbb Q}_p, \end{equation} may be considered as a {\it refinement equation} for the Haar MRA generating Kozyrev's wavelets. A solution $\phi$ to this equation ({\it a refinable function}) is the characteristic function of the unit disc. We note that equation (\ref{62.0-3}) reflects a {\it natural} ``self-similarity'' of the space ${\mathbb Q}_p$: the unit disc
$B_{0}(0)=\{x: |x|_p \le 1\}$ is represented as the union $\bigcup_{r=0}^{p-1}B_{-1}(r)$ of $p$ mutually {\it disjoint} discs
$B_{-1}(r)=\bigl\{x: |x-r|_p \le p^{-1}\bigr\}$ (see~\cite[I.3, Examples 1,2.]{Vl-V-Z}). Following this idea, the notion of $p$-adic MRA was introduced and a general scheme for its construction was described in~\cite{S-Sk-1}. Also, using (\ref{62.0-3}) as a generating refinement equation, this scheme was realized to construct the $2$-adic Haar MRA. In contrast to the real setting, the {\it refinable function} $\phi$ generating the Haar MRA is {\em periodic}, which implies the existence of {\em infinitly many different} orthonormal wavelet bases in the same Haar MRA. One of them coincides with Kozyrev's wavelet basis.
The authors of~\cite{Kh-Sh-S} described a wide class of functions generating a MRA, but all of these functions are $1$-periodic. In the present paper we prove that there exist no other orthogonal test scaling functions generating a MRA, except for those described in~\cite{S-Sk-1}. Also, the MRAs generated by arbitrary test scaling functions (not necessary orthogonal) are considered. We thoroughly study these scaling functions and develop a method to construct a wavelet frame based on a given MRA.
Here and in what follows, we shall systematically use the notation and the results from~\cite{Vl-V-Z}. Let ${\mathbb N}$, $\z$, ${\Bbb R}$, ${\mathbb C}$ be the sets of positive integers, integers, real numbers, complex numbers, respectively. The field ${\mathbb Q}_p$ of $p$-adic numbers is defined as the completion of the field of rational numbers ${\mathbb Q}$ with respect to the non-Archimedean $p$-adic norm $|\cdot|_p$. This $p$-adic norm is defined as follows: $|0|_p=0$; if $x\ne 0$, $x=p^{\gamma}\frac{m}{n}$, where $\gamma=\gamma(x)\in \z$ and the integers $m$, $n$ are not divisible by $p$, then
$|x|_p=p^{-\gamma}$. The norm $|\cdot|_p$ satisfies the strong triangle inequality
$|x+y|_p\le \max(|x|_p,|y|_p)$. The canonical form of any $p$-adic number $x\ne 0$ is \begin{equation} \label{2} x=p^{\gamma}(x_0 + x_1p + x_2p^2 + \cdots), \end{equation} where $\gamma=\gamma(x)\in \z$, \ $x_j\in D_p:=\{0,1,\dots,p-1\}$, $x_0\ne 0$, $j=0,1,\dots$. We shall write the $p$-adic numbers $k=k_{0}+k_{1}p+\cdots+k_{s-1}p^{s-1}$, $k_j\in D_p$, $j=0,1,\dots,s-1$, following the usual form, as in the real analysis: $k=0,1,\dots,p^s-1$.
Denote by $B_{\gamma}(a)=\{x\in {\mathbb Q}_p: |x-a|_p \le p^{\gamma}\}$ the disc of radius $p^{\gamma}$ with the center at a point $a\in {\mathbb Q}_p$, $\gamma \in \z$. Any two balls in ${\mathbb Q}_p$ either are disjoint or one contains the other.
There exists the Haar measure $dx$ on ${\mathbb Q}_p$ which is positive, invariant under the shifts, i.e., $d(x+a)=dx$, and normalized by
$\int_{|\xi|_p\le 1}\,dx=1$. A complex-valued function $f$ defined on ${\mathbb Q}_p$ is called {\it locally-constant} if for any $x\in {\mathbb Q}_p$ there exists an integer $l(x)\in \z$ such that $f(x+y)=f(x)$, $y\in B_{l(x)}(0)$. Denote by ${{\mathcal D}}$ the linear space of locally-constant compactly supported functions (so-called test functions)~\cite[VI.1.,2.]{Vl-V-Z}. The space ${{\mathcal D}}$ is an analog of the Schwartz space in the real analysis.
The Fourier transform of $\varphi\in {{\mathcal D}}$ is defined as $$ {\widehat\phi}(\xi)=F[\varphi](\xi)=\int_{{\mathbb Q}_p}\chi_p(\xi\cdot x)\varphi(x)\,dx, \ \ \ \xi \in {\mathbb Q}_p, $$ where $\chi_p(\xi\cdot x)=e^{2\pi i\{\xi x\}_p}$ is the additive character for the field ${\mathbb Q}_p$, $\{\cdot\}_p$ is a fractional part of a number $x\in {\mathbb Q}_p$. The Fourier transform is a linear isomorphism taking ${{\mathcal D}}$ into ${{\mathcal D}}$. The Fourier transform is extended to ${ L}^2({\mathbb Q}_p)$ in a standard way. If $f\in{ L}^2({\mathbb Q}_p)$, $0\ne a\in {\mathbb Q}_p$, \ $b\in {\mathbb Q}_p$, then~\cite[VII,(3.3)]{Vl-V-Z}: \begin{equation} \label{014} F[f(ax+b)](\xi)
=|a|_p^{-1}\chi_p\Big(-\frac{b}{a}\xi\Big)F[f(x)]\Big(\frac{\xi}{a}\Big). \end{equation} According to~\cite[IV,(3.1)]{Vl-V-Z}, \begin{equation} \label{14.1}
F[\Omega(p^{-k}|\cdot|_p)](x)=p^{k}\Omega(p^k|x|_p), \quad k\in \z, \quad x \in {\mathbb Q}_p, \end{equation} where $\Omega(t)=1$ for $t\in [0,\,1]$; $\Omega(t)=0$ for $t\not\in [0,\,1]$.
\section{Multiresolution analysis} \label{s2}
Let us consider the set $$ I_p=\{a=p^{-\gamma}\big(a_{0}+a_{1}p+\cdots+a_{\gamma-1}p^{\gamma-1}\big): \gamma\in {\mathbb N}; a_j\in D_p; j=0,1,\dots,\gamma-1\}. $$ It is well known that ${\mathbb Q}_p=B_{0}(0)\cup\cup_{\gamma=1}^{\infty}S_{\gamma}$, where
$S_{\gamma}=\{x\in {\mathbb Q}_p: |x|_p = p^{\gamma}\}$. Due to (\ref{2}), $x\in S_{\gamma}$, $\gamma\ge 1$, if and only if $x=x_{-\gamma}p^{-\gamma}+x_{-\gamma+1}p^{-\gamma+1}+\cdots+x_{-1}p^{-1}+\xi$, where $x_{-\gamma}\ne 0$, $\xi \in B_{0}(0)$. Since $x_{-\gamma}p^{-\gamma}+x_{-\gamma+1}p^{-\gamma+1} +\cdots+x_{-1}p^{-1}\in I_p$, we have a ``natural'' decomposition of ${\mathbb Q}_p$ into a union of mutually disjoint discs: ${\mathbb Q}_p=\bigcup_{a\in I_p}B_{0}(a)$. So, $I_p$ is a {\em ``natural'' set of shifts} for ${\mathbb Q}_p$.
\begin{definition} \label{de1} \rm A collection of closed spaces $V_j\subset L^2({\mathbb Q}_p)$, $j\in\z$, is called a {\it multiresolution analysis {\rm(}MRA{\rm)} in $ L^2({\mathbb Q}_p)$} if the following axioms hold
(a) $V_j\subset V_{j+1}$ for all $j\in\z$;
(b) $\bigcup_{j\in\z}V_j$ is dense in $ L^2({\mathbb Q}_p)$;
(c) $\bigcap_{j\in\z}V_j=\{0\}$;
(d) $f(\cdot)\in V_j \Longleftrightarrow f(p^{-1}\cdot)\in V_{j+1}$ for all $j\in\z$;
(e) there exists a function $\phi \in V_0$ such that $V_0:=\overline{\mbox{span}\,\{\phi(\cdot-a),\ a\in I_p\}}$. \end{definition}
The function $\phi$ from axiom (e) is called {\em scaling}. One also says that a MRA is generated by its scaling function $\phi$ (or $\phi$ generates the MRA). It follows immediately from axioms (d) and (e) that \begin{equation} V_j:=\overline{\mbox{span}\,\{\phi(p^{-j}x-a),\ a\in I_p\}},\quad j\in \z. \label{17} \end{equation}
An important class of MRAs consists of those generated by so-called {\em orthogonal scaling functions}. A scaling function $\phi$ is said to be orthogonal if
$\{\phi(\cdot-a), a\in I_p\}$ is an orthonormal basis for $V_0$. Consider such a MRA. Evidently, the functions $p^{j/2}\phi(p^{-j}\cdot-a)$, $a\in I_p$, form an orthonormal basis for $V_j$, $j\in\z$. According to the standard scheme (see, e.g.,~\cite[\S 1.3]{NPS}) for the construction of MRA-based wavelets, for each $j$, we define a space $W_j$ ({\em wavelet space}) as the orthogonal complement of $V_j$ in $V_{j+1}$, i.e., $V_{j+1}=V_j\oplus W_j$, $j\in \z$, where $W_j\perp V_j$, $j\in \z$. It is not difficult to see that \begin{equation} \label{61.0} f\in W_j \Longleftrightarrow f(p^{-1}\cdot)\in W_{j+1}, \quad\text{for all}\quad j\in \z \end{equation} and $W_j\perp W_k$, $j\ne k$. Taking into account axioms (b) and (c), we obtain \begin{equation} \label{61.1} {\bigoplus\limits_{j\in\z}W_j}= L^2({\mathbb Q}_p) \quad \text{(orthogonal direct sum)}. \end{equation} If we now find functions $\psi^{(\nu)} \in W_0$, $\nu\in A$, such that the functions $\psi^{(\nu)}(x-a)$, $a\in~I_p, \nu\in A$, form an orthonormal basis for $W_0$, then, due to~(\ref{61.0}) and (\ref{61.1}), the system $\{p^{j/2}\psi^{(\nu)}(p^{-j}\cdot-a), a\in I_p, j\in\z , \nu\in A\}$ is an orthonormal basis for $ L^2({\mathbb Q}_p)$. Such a function $\psi$ is called a {\em wavelet function} and the basis is a {\em wavelet basis}.
Another interesting class of scaling functions consists of functions $\phi$ so that $\{\phi(\cdot-a), a\in I_p\}$ is a Riesz system. Probably, adopting the ideas developed for the real setting, one can use MRAs generated by such functions $\phi$ for construction of dual biorthogonal wavelet systems. This topic is, however, out of our consideration in the present paper.
In Section~\ref{s3} we will discuss how to construct a $p$-adic wavelet frame based on an arbitrary MRA generated by a test function.
Let $\phi$ be an orthogonal scaling function for a MRA $\{V_j\}_{j\in\z}$. Since the system $\{p^{1/2}\phi(p^{-1}\cdot-a), a\in I_p\}$ is a basis for $V_1$ in this case, it follows from axiom (a) that \begin{equation} \label{62.0-2*} \phi=\sum_{a\in I_p}\alpha_a\phi(p^{-1}\cdot-a), \quad \alpha_a\in {\mathbb C}. \end{equation} We see that the function $\phi$ is a solution of a special kind of functional equation. Such equations are called {\em refinement equations}, and their solutions are called {\em refinable functions} \footnote{Usually the terms ``refinable function'' and ``scaling function'' are synonyms in the literature, and they are used in both senses: as a solution to the refinable equation and as a function generating MRA. We separate here the meanings of these terms.}. It will be shown in Section~\ref{s3} that any test scaling function (not necessary orthogonal) is refinable.
A natural way for the construction of a MRA (see, e.g.,~\cite[\S 1.2]{NPS}) is the following. We start with a refinable function $\phi$ and define the spaces $V_j$ by~(\ref{17}). It is clear that axioms (d) and (e) of Definition~\ref{de1} are fulfilled. Of course, not any such function $\phi$ provides axiom $(a)$. In the real setting, the relation $V_0\subset V_{1}$ holds if and only if the refinable function satisfies a refinement equation. The situation is different in the $p$-adic case.. Generally speaking, a refinement equation (\ref{62.0-2*}) does not imply the including property $V_0\subset V_{1}$ because the set of shifts $I_p$ does not form a group. Indeed, we need all the functions $\phi(\cdot-b)$, $b\in I_p$, to belong to the space $V_1$, i.e., the identities $\phi(x-b)=\sum_{a\in I_p}\alpha_{a,b}\phi(p^{-1}x-a)$ should be fulfilled for all $b\in I_p$. Since $p^{-1}b+a$ is not in $I_p$ in general, we can not state that $\phi(x-b)$ belongs to $V_1$ for all $b\in I_p$. Nevertheless, we will see below that a wide class of refinable equations provide the including property.
Providing axiom (a) is a key moment for the construction of MRA. Axioms (b) and (c) are fulfilled for a wide class of functions $\phi$ because of the following statements.
\begin{theorem} \label{th1-2*} If $\phi \in L^2({\mathbb Q}_p)$ and $\widehat\phi$ is compactly supported, then axiom $(c)$ of Definition~{\rm\ref{de1}} holds for the spaces $V_j$ defined by~(\ref{17}). \end{theorem}
\begin{proof} Let $\widehat\phi\subset B_M(0)$, $M\in\z$. Assume that a function $f\in L^2({\mathbb Q}_p)$ belongs to any space $V_j$, $j\in\z$.
Given $j\in\n$ and $\epsilon>0$, there exists a function
$f_\epsilon:=\sum_{a\in I_p}\alpha_a\phi(p^j\cdot-a)$, where the sum is finite,
such that $\|f-f_\epsilon\|<\epsilon$. Using~(\ref{014}),
it is not difficult to see that
$\mbox{supp}\,\widehat f_\epsilon\subset\mbox{supp}\,\widehat\phi(p^{-j}\cdot)$,
which yields that $\widehat f_\epsilon(\xi)=0$ for any
$\xi\not\in B_{M-j}(0)$. Due to the Plancherel theorem,
it follows that $\widehat f=0$ almost everywhere
on $B_{M-j}(0)$. Since $j$ is an arbitrary positive integer,
$\widehat f$ is equivalent to zero on $Q_p$. \end{proof}
Another sufficient condition for axiom (c) was given in~\cite{Kh-Sh-S}:
\begin{theorem} \label{th1-4*} If $\phi \in L^2({\mathbb Q}_p)$ and the system $\{\phi(x-a):a\in I_p\}$ is orthonormal, then axiom $(c)$ of Definition~{\rm\ref{de1}} holds for the spaces $V_j$ defined by~(\ref{17}). \end{theorem}
\begin{theorem} Let $\phi \in L^2({\mathbb Q}_p)$, the spaces $V_j$, $j\in \z$, be defined by~(\ref{17}), and let $\phi(\cdot-b)\in\cup_{j\in \z} V_j$ for any $b\in Q_p$. Axiom $(b)$ of Definition~{\rm\ref{de1}} holds for the spaces $V_j$, $j\in\z$, if and only if \begin{equation} \bigcup\limits_{j\in\z}{\rm supp\,}\widehat\phi(p^{j}\cdot)={\mathbb Q}_p. \label{dnn14} \end{equation} \label{th1-3*} \end{theorem}
\begin{remark} It is not difficult to see that the assumption $\phi(\cdot-b)\in\cup_{j\in \z} V_j$ for any $b\in Q_p$ is fulfilled whenever $\phi$ is a refinable function and $\widehat\phi\subset B_0(0)$. We will see that this assumption is also valid for a wide class of refinable functions $\phi$ for which $\widehat\phi\not\subset B_0(0)$. \end{remark}
\begin{proof} First of all we show that the space $\overline {\cup_{j\in {\z}} V_j}$ is invariant with respect to all shifts. Let $f\in \cup_{j\in {\z}} V_j$, $b\in{\mathbb Q}_p$. Evidently, $\phi(p^{-k}{\cdot}-t)\in\cup_{j\in \z} V_j$ for any $t\in Q_p$ and for any $k\in\z$. Since the $L_2$-norm is invariant with respect to the shifts, it follows that $f(\cdot -b) \in \overline{\cup_{j\in\z} V_j}$.
If now $g\in\overline{\cup_{j\in\z} V_j}$, then approximating $g$ by the functions $f\in \cup _{j\in\z} V_j$, again using the invariance of $L_2$-norm with respect to the shifts , we derive $g(\cdot -b) \in \overline{\cup_{j\in\z} V_j}$.
For $X\subset L^2({\mathbb Q}_p)$, set $\widehat X=\{\ \widehat f: f\in X\}$. By the Wiener theorem for $L_2$ (see, e.g., \cite{NPS}; all the arguments of the proof given there may be repeated word for word with replacing
${\mathbb R}$ by ${\mathbb Q}_p$), a closed subspace $X$ of the space $L^2({\mathbb Q}_p)$ is invariant with respect to the shifts if and only if $\widehat X=L_2(\Omega)$ for some set $\Omega\subset{\mathbb Q}_p$. If now $X=\overline{\cup_{j\in\z}V_j}$, then $\widehat X=L_2(\Omega)$. Thus $X=L^2({\mathbb Q}_p)$ if and only if $\Omega={\mathbb Q}_p$. Set $\phi_j=\phi(p^{-j}\cdot),\ \ \Omega_0=\cup_{j\in\z}{\rm supp}\, \widehat\phi_j$ and prove that $\Omega=\Omega_0$. Since $\phi_j\in V_j,$ $j\in\z$, we have ${\rm supp}\, \widehat\phi_j\subset\Omega$, and hence $\Omega_0 \subset \Omega$. Now assume that $\Omega\backslash\Omega_0$ contains a set of positive measure $\Omega_1$. Let $f\in V_j$. Given $\epsilon>0$, there exists a function
$f_\epsilon:=\sum_{a\in I_p}\alpha_a\phi(p^j\cdot-a)$, where the sum is finite,
such that $\|f-f_\epsilon\|<\epsilon$. Using~(\ref{014}),
we see that $\mbox{supp}\,\widehat f_\epsilon\subset\mbox{supp}\,\widehat\phi(p^{-j}\cdot)$,
which yields that $\widehat f_\epsilon(\xi)=0$ for any
$\xi\not\in \Omega_1$. Due to the Plancherel theorem,
it follows that $\widehat f=0$ almost everywhere on $\Omega_1.$ Hence the same is true for any $f\in \cup _{j\in\z} V_j$. Passing to the limit we deduce that that the Fourier transform of any $f\in X$ is equal to zero almost everywhere on $\Omega_1$, i.e., $L_2(\Omega)=L_2(\Omega_0)$. It remains to note that ${\rm supp\,} \widehat\phi_j={\rm supp\,}\widehat\phi({p}^{j}\cdot) $ \end{proof}
A real analog of Theorem~\ref{th1-3*} was proved by
C.~de~Boor, R.~DeVore and A.~Ron in~\cite5.
\section{Refinable functions} \label{s3}
We are going to study $p$-adic refinable functions $\phi$. Let us restrict ourselves to the consideration of $\phi\in {{\mathcal D}}$. Evidently, each $\phi\in {{\mathcal D}}$ is a $p^M$-periodic function for some $M\in \z$. Denote by ${{\mathcal D}}_N^M$ the set of all $p^M$-periodic functions supported on $B_N(0)$. Taking the Fourier transform of the equality $\phi(x-p^M)=\phi(x)$, we obtain $\chi_p(p^M\xi)\widehat\phi(\xi)=\widehat\phi(\xi)$, which holds for all $\xi$ if and only if $\mbox{supp}\,\widehat\phi\subset B_M(0)$. Thus, the set ${{\mathcal D}}_N^M$ consists of all locally constant functions $\phi$ such that $\mbox{supp}\,\phi\subset B_N(0)$,
$\mbox{supp}\,\widehat\phi\subset B_M(0)$.
\begin{proposition} Let $\phi,\psi \in L^2({\mathbb Q}_p)$, $\mbox{supp}\, \phi,\mbox{supp}\, \psi\subset B_N(0)$, $N\ge0$,
and let $b\in I_p$, $|b|_p\le p^N$. If \begin{equation} \psi(\cdot-b)\in \overline{\mbox{span}\,\{\phi(p^{-1}\cdot-a),\ a\in I_p\}} \label{19} \end{equation} then \begin{equation} \psi(x-b)=\sum_{k=0}^{p^{N+1}-1}h^\psi_{k,b}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big) \ \ \ \forall x\in Q_p. \label{18} \end{equation} \label{p1} \end{proposition}
\begin{proof} Given $\epsilon>0$, there exist functions
$$
f_\epsilon:=\sum_{a\in I_p\atop |a|_p\le p^{N+1}}\alpha_a\phi(p^j\cdot-a),\ \ \
g_\epsilon:=\sum_{a\in I_p\atop |a|_p> p^{N+1}}\alpha_a\phi(p^j\cdot-a), $$
where the sums are finite,
such that $\|\psi(\cdot-b)-f_\epsilon-g_\epsilon\|<\epsilon$.
If $x\in B_N(0)$, $|a|_p>p^{N+1}$, then
$|p^{-1}x-a|_p>p^{N+1}$ and hence $\phi(p^{-1}x-a)=0$. So, $g_\epsilon(x)=0$ whenever $x\in B_N(0)$. If $x\not\in B_N(0)$, then $\phi(x-b)=0$ and
$\phi(p^{-1}x-a)=0$ for all $a\in I_p$, $|a|_p\le p^{N+1}$. So, $\phi(\cdot-b)-f_\epsilon(x)=0$ whenever $x\not\in B_N(0)$. It follows that $$
\|\psi(\cdot-b)-f_\epsilon\|^2=\int\limits_{B_N(0)}|\psi(\cdot-b)-f_\epsilon|^2=
\int\limits_{B_N(0)}|\psi(\cdot-b)-f_\epsilon-g_\epsilon|^2\le \epsilon^2. $$ Hence $$ \psi(\cdot-b)\in
\overline{\mbox{span}\,\{\phi(p^{-1}\cdot-a),\ a\in I_p,\ |a|_p\le p^{N+1}\}}, $$ which implies~(\ref{18}). \end{proof}
\begin{corollary} If $\phi \in L^2({\mathbb Q}_p)$ is a refinable function and
$\mbox{supp}\, \phi\subset B_N(0)$, $N\ge0$, then its refinement equation is \begin{equation} \label{62.0-5} \phi(x)=\sum_{k=0}^{p^{N+1}-1}h_{k}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big) \ \ \ \forall x\in Q_p. \end{equation} \label{c3} \end{corollary}
The proof immediately follows from Proposition~\ref{p1}.
\begin{corollary} Let $\phi \in L^2({\mathbb Q}_p)$ be a scaling function of a MRA. If $\mbox{supp}\, \phi\subset B_N(0)$, $N\ge0$, then $\phi$ is a refinable function satisfying~(\ref{62.0-5}). \label{c1} \end{corollary}
The proof follows by combining axiom $(a)$ of Definition~{\rm\ref{de1}} with Proposition~\ref{p1}.
Taking the Fourier transform of~(\ref{62.0-5}) and using (\ref{014}), we can rewrite the refinable equation in the form \begin{equation} \label{62.0-6} {\widehat\phi}(\xi)=m_0\Big(\frac{\xi}{p^{N}}\Big){\widehat\phi}(p\xi), \end{equation} where \begin{equation} \label{62.0-7-1} m_0(\xi)=\frac{1}{p}\sum_{k=0}^{p^{N+1}-1}h_{k}\chi_p(k\xi) \end{equation} is a trigonometric polynomial. It is clear that $m_0(0)=1$ whenever $\widehat\phi(0)\ne0$.
\begin{proposition} \label{pr1-1} If $\phi\in L^2({\mathbb Q}_p)$ is a solution of refinable equation~(\ref{62.0-5}), ${\widehat\phi}(0)\ne 0$, ${\widehat\phi}(\xi)$ is continuous at the point $0$, then \begin{equation} \label{62.0-8} {\widehat\phi}(\xi)={\widehat\phi}(0)\prod_{j=0}^{\infty}m_0\Big(\frac{\xi}{p^{N-j}}\Big). \end{equation} \end{proposition}
\begin{proof} Since~(\ref{62.0-5}) implies~(\ref{62.0-6}), after iterating {\rm(\ref{62.0-6})} $J$ times, $J\ge 1$, we have $$ {\widehat\phi}(\xi)=\prod_{j=0}^{J}m_0\Big(\frac{\xi}{p^{N-j}}\Big) {\widehat\phi}(p^{J}\xi). $$
Taking into account that ${\widehat\phi}(\xi)$ is continuous at the point $0$ and the fact that $|p^{N}\xi|_p=p^{-N}|\xi|_p\to 0$ as $N\to +\infty$ for any $\xi\in{\mathbb Q}_p$, we obtain {\rm(\ref{62.0-8})}. \end{proof}
\begin{corollary} If $\phi \in {{\mathcal D}}_N^M$ is a refinable function, $N\ge0$, and ${\widehat\phi}(0)\ne 0$, then (\ref{62.0-8}) holds. \label{c2} \end{corollary}
This statement follows immediately from Corollary~\ref{c1} and Proposition~\ref{pr1-1}.
\begin{lemma} Let $ {\widehat\phi}(\xi)=C\prod_{j=0}^{\infty} m_0\Big(\frac{\xi}{p^{N-j}}\Big), $ where $m_0$ is a trigonometric polynomial with $m_0(0)=1$ and $C\in{\Bbb R}$. If $\mbox{supp}\,\widehat\phi\subset B_M(0)$,
then there exist at least $\left( p^{M+N}-\frac{\deg m_0}{p-1}\right)$ integers $n$ such that $0\le n<p^{M+N}$ and $\widehat\phi\left(\frac{n}{p^M}\right)=0$. \label{l1} \end{lemma}
\begin{proof} First of all we note that $\widehat\phi$ is a $p^N$-periodic function satisfying~(\ref{62.0-6}). Denote by $O_p$ the set of positive integers not divisible by~$p$. Since $\mbox{supp}\,\widehat\phi\subset B_M(0)$, we have $\widehat\phi\left(\frac{k}{p^{M+1}}\right)=0$ for all $k\in O_p$
. By the definition of $\widehat\phi$ the equality $\widehat\phi\left(\frac{k}{p^{M+1}}\right)=0$ holds if and only if there exists $\nu=1-N,\dots, M+1$ such that $m_0\Big(\frac{k}{p^{N+\nu}}\Big)=0$. Set $$ \sigma_\nu:=\left\{l\in O_p:\ l<p^{N+\nu}, m_0\Big(\frac{l}{p^{N+\nu}}\Big)=0,\ m_0\Big(\frac{l}{p^{N+\mu}}\Big)\ne0\ \forall \mu=1-N,\dots,\nu-1\right\}, $$ $v_\nu:=\sharp\,\sigma_\nu$. Evidently, $\sigma_\nu\subset O_p^{\prime}$ for all $\nu$, where $O_p^{\prime}=\{k\in O_p:\ k<p^{M+N+1}\}$, and $\sigma_{\nu^{\prime}}\cap\sigma_{\nu}=\emptyset$ whenever $\nu^{\prime}\ne\nu$. If $\widehat\phi\left(\frac{k}{p^{M+1}}\right)=0$ for some $k\in O_p$, then there exist a unique $\nu=1-N,\dots, M+1$ and a unique $l\in\sigma_\nu$ such that $k\equiv l\pmod{p^{N+\nu}}$. Moreover, for any $l\in\sigma_\nu$ there are exactly $p^{M-\nu+1}$ integers $k\in O_p^{\prime}$ (including~$l$) satisfying the above comparison.
It follows that \begin{equation} \sum\limits_{\nu=1-N}^{M+1} p^{M-\nu+1}v_\nu=\sharp\,O_p^{\prime}=p^{M+N}(p-1). \label{10} \end{equation} Now if $l\in\sigma_\nu$, $\nu\le M$, then $\widehat\phi\left(\frac{p^\gamma k}{p^{M}}\right)=0$
for all $\gamma=0,1,\dots, M-\nu$, $k=l+rp^{N+\nu}$, $r=0,1,\dots, p^{M-\nu-\gamma}-1$,
i.e., each $l\in\sigma_\nu$ generates at least $1+p+\dots+p^{M-\nu}$
distinct positive integers $n<p^{M+N}$ for which
$\widehat\phi\left(\frac{n}{p^{M}}\right)=0$. Hence
\begin{eqnarray*} v:= \sharp\,\left\{n:\ n=0,1,\dots, p^{M+N}-1, \widehat\phi\left(\frac{n}{p^{M}}\right)=0\right\}\ge \\ \sum\limits_{\nu=1-N}^{M}(1+p+\dots+p^{M-\nu})v_\nu= \frac{1}{p-1}\sum\limits_{\nu=1-N}^{M}(p^{M-\nu+1}-1)v_\nu= \\ \frac{1}{p-1}\sum\limits_{\nu=1-N}^{M+1}(p^{M-\nu+1}-1)v_\nu. \end{eqnarray*} Since $\sum\limits_{\nu=1-N}^{M+1}v_\nu\le \deg m_0$, by using~(\ref{10}), we obtain $$ v\ge\frac{1}{p-1}\left(\sum\limits_{\nu=1-N}^{M+1}p^{M-\nu+1}v_\nu- \deg m_0\right)\ge p^{M+N}-\frac{\deg m_0}{p-1}. $$ \end{proof}
\begin{theorem} \label{t1} Let $\phi\in{{\mathcal D}}_N^M$, $N\ge0$ and ${\widehat\phi}(0)\ne 0$. If \begin{equation} \phi(\cdot-b)\in \overline{\mbox{span}\,\{\phi(p^{-1}\cdot-a),\ a\in I_p\}} \label{27}
\end{equation} for all $b\in I_p$, $|b|_p\le p^N$, then there exist at least $p^{M+N}-p^N$ integers $l$ such that $0\le l<p^{M+N}$ and $\widehat\phi\left(\frac{l}{p^{M}}\right)=0$. \end{theorem}
\begin{proof}
Let $b\in I_p$, $|b|_p\le p^N$. Because of Proposition~\ref{p1}, we can rewrite~(\ref{27}) in the form $$ \phi(x-b)=\sum_{k=0}^{p^{N+1}-1}h_{k,b}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big) \ \ \ \forall x\in {\mathbb Q}_p. $$ Taking the Fourier transform, we obtain \begin{equation} \label{12} {\widehat\phi}(\xi)\chi_p(b\xi)=m_b\Big(\frac{\xi}{p^{N}}\Big){\widehat\phi}(p\xi), \ \ \ \forall \xi\in {\mathbb Q}_p, \end{equation} where $m_b$ is a trigonometric polynomial, $\deg m_b<p^{N+1}$. Combining~(\ref{12}) for $b=0$ with~(\ref{12}) for arbitrary $b$, we obtain $$ {\widehat\phi}(p\xi)\left( m_0\Big(\frac{\xi}{p^{N}}\Big)\chi_p(b\xi)-
m_b\Big(\frac{\xi}{p^{N}}\Big)\right)=0 \ \ \ \forall \xi\in {\mathbb Q}_p, $$ which is equivalent to \begin{equation} F(\xi):={\widehat\phi}(p^{N+1}\xi)\left( m_0(\xi)\chi_p(p^{N}b\xi)-
m_b(\xi)\right)=0 \ \ \ \forall \xi\in {\mathbb Q}_p. \label{13} \end{equation} Since $\mbox{supp}\, F\subset B_{M+N+1}(0)$ and $F$ is a $1$-periodic function, (\ref{13}) holds if and only if $\widehat\phi\left(\frac{l}{p^{M+N+1}}\right)=0$, $l=0,1,\dots, p^{M+N+1}-1$.
First suppose that $\deg m_0\ge p^N(p-1)$, i.e., $$ m_0(\xi)=\sum\limits_{k=0}^{K}h_k\chi_p(k\xi),\ \ \ h_K\ne0, $$ where $K=K_Np^N+K_{N-1}p^{N-1}+\dots+K_0$, $K_j\in D_p$, $j=0,1,\dots, N$, $K_N=p-1$ (indeed, if $K_N<p-1$, then $\deg m_0= K\le (p-2)p^N+(p-1)(1+p+\dots+p^{N-1})= p^{N+1}-p^N-1<p^N(p-1)$). Set $b:= p-p^{-N}K$. It is not difficult to see that
$b\in I_p$, $|b|_p\le p^N$ and $K+bp^N=p^{N+1}$. We see that the degree of the polynomial $t(\xi):=m_0(\xi)\chi_p(p^{N}b\xi)- m_b(\xi)$ is exactly $p^{N+1}$, and hence there exist at most $p^{N+1}$ integers $l$ such that $0\le l<p^{M+N+1}$, $t\left(\frac{l}{p^{M+N+1}}\right)=0$. Thus, $$ \sharp\,\left\{l:\ l=0,1,\dots, p^{M+N+1}-1, \widehat\phi\left(\frac{l}{p^{M}}\right)=0\right\}\ge p^{M+N+1}-p^{N+1}. $$ Taking into account that $\widehat\phi$ is a $p^N$-periodic function, we obtain \begin{equation}
\sharp\,\left\{l:\ l=0,1,\dots, p^{M+N}-1, \widehat\phi\left(\frac{l}{p^{M}}\right)=0\right\}\ge p^{M+N}-p^{N}. \label{14} \end{equation} It remains to note that~(\ref{14}) is also fulfilled whenever $\deg m_0<p^N(p-1)$ because of Lemma~\ref{l1} and Corollary~\ref{c2}. \end{proof}
\begin{theorem} \label{t2} Let $\phi,\psi\in{{\mathcal D}}_N^M$, $N\ge0$, ${\widehat\phi}(0)\ne 0$, and let \begin{equation} \psi(\cdot)\in \overline{\mbox{span}\,\{\phi(p^{-1}\cdot-a),\ a\in I_p\}} \label{28} \end{equation} If there exist at least $p^{M+N}-p^N$ integers $l$ such that $0\le l<p^{M+N}$ and $\widehat\phi\left(\frac{l}{p^{M}}\right)=0$, then \begin{equation} \psi(x-b)=\sum_{a\in I_p}\alpha^\psi_{a,b}\phi(p^{-1}x-a) \ \ \ \forall b\in {\mathbb Q}_p, \label{11} \end{equation} where the sum is finite. In particular. if $\phi$ is a refinable function, then \begin{equation} \phi(x-b)=\sum_{a\in I_p}\alpha_{a,b}\phi(p^{-1}x-a) \ \ \ \forall b\in {\mathbb Q}_p. \label{11} \end{equation}
\end{theorem}
\begin{proof}
First we assume that $b\in Q_p$, $|b|_p\le p^{N}$, $b\ne0$, and prove that \begin{equation} \psi(x-b)=\sum_{k=0}^{p^{N+1}-1}g_{k,b}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big) \ \ \ \forall x\in {\mathbb Q}_p. \label{29} \end{equation} Because of Proposition~\ref{p1}, we have~(\ref{29}) for $b=0$. Taking the Fourier transform of~(\ref{29}), we obtain \begin{equation} \label{12} {\widehat\psi}(\xi)\chi_p(b\xi)=n_b\Big(\frac{\xi}{p^{N}}\Big){\widehat\phi}(p\xi), \ \ \ \forall \xi\in {\mathbb Q}_p, \end{equation} where $n_b$ is a trigonometric polynomial, $\deg n_b<p^{N+1}$. Substituting~(\ref{12}) for $b=0$, we reduce~(\ref{12}) for arbitrary $b$ to $$ {\widehat\phi}(p\xi)\left( n_0\Big(\frac{\xi}{p^{N}}\Big)\chi_p(b\xi)-
n_b\Big(\frac{\xi}{p^{N}}\Big)\right)=0 \ \ \ \forall \xi\in {\mathbb Q}_p, $$ which is equivalent to \begin{equation} F(\xi):={\widehat\phi}(p^{N+1}\xi)\left( n_0(\xi)\chi_p(p^{N}b\xi)-
n_b(\xi)\right)=0 \ \ \ \forall \xi\in {\mathbb Q}_p. \label{13} \end{equation} Since $\mbox{supp}\, F\subset B_{M+N+1}(0)$ and $F$ is a $1$-periodic function, (\ref{13}) is equivalent to $$ F\left(\frac{l}{p^{M+N+1}}\right)=0, \forall l=0,1,\dots, p^{M+N+1}-1, $$ which holds if and only if \begin{eqnarray} n_b\left(\frac{l}{p^{M+N+1}}\right)= n_0\left(\frac{l}{p^{M+N+1}}\right)\chi_p\left(\frac{bl}{p^{M+1}}\right),\hspace{2cm} \label{15} \end{eqnarray} for all $l=0,1,\dots, p^{M+N+1}-1$ such that $\widehat\phi\left(\frac{l}{p^{M}}\right)\ne0$. Because of $p^M$-periodicity of $\widehat\phi$, there exist at least $p(p^{M+N}-p^N)$ integers $l=0,1,\dots, p^{M+N+1}-1$ such that $\widehat\phi\left(\frac{l}{p^{M}}\right)=0$. So, we can find $n_b$ by solving the linear system~(\ref{15}) with respect to the unknown coefficients of $n_b$. Taking the Fourier transform of~(\ref{12}), we obtain \begin{equation} \psi(x-b)=\sum_{k=0}^{p^{N+1}-1}g_{k,b}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big) \ \ \ \forall x\in {\mathbb Q}_p. \label{16}
\end{equation} Next let $b\in Q_p$, $|b|_p= p^{N+1}$, i.e., $b=b_{N+1}p^{N+1}+b^\prime$,
$b_{N+1}\in D_p$, $b_{N+1}\ne0$, $|b^\prime|_p\le p^{N}$. Using~(\ref{16}) with $b=b^\prime$, we have $$ \psi(x-b)=\sum_{k=0}^{p^{N+1}-1}g_{k,b^\prime} \phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}-\frac{b_{N+1}}{p^{N+2}}\Big)= \sum_{k=0}^{p^{N+1}-1}g_{k,b^\prime} \phi\Big(\frac{x}{p}-\frac{pk+b_{N+1}}{p^{N+2}}\Big). $$ Taking into account that $$ pk+b_{N+1}\le p(p^{N+1}-1)+(p-1)=p^{N+2}-1, $$ we derive $$ \psi(x-b)=\sum_{k=0}^{p^{N+2}-1}g_{k,b}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+2}}\Big) \ \ \ \forall x\in {\mathbb Q}_p. $$ Similarly, we can prove by induction on $n$ that $$ \psi(x-b)=\sum_{k=0}^{p^{N+n+1}-1}g_{k,b}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+n+1}}\Big) \ \ \ \forall x\in {\mathbb Q}_p, $$
whenever $b\in {\mathbb Q}_p$, $|b|_p= p^{N+n}$. \end{proof}
\begin{theorem} \label{t3} A function $\phi\in{{\mathcal D}}_N^M$, $N\ge0$, with ${\widehat\phi}(0)\ne 0$ generates a MRA if and only if
(1) $\phi$ is refinable;
(2) there exist at least $p^{M+N}-p^N$ integers $l$ such that $0\le l<p^{M+N}$ and $\widehat\phi\left(\frac{l}{p^{M}}\right)=0$. \end{theorem}
\begin{proof} If $\phi$ is a scaling function of a MRA, then (1) follows from Corollary~\ref{c1} and (2) follows from (1) and Theorem~\ref{t1}.
Now let conditions (1), (2) be fulfilled. Define the spaces $V_j$, $j\in\z$, by~(\ref{17}). Axioms (d) and (e), evidently, hold. Axiom (a) follows from Theorem~\ref{t2}. Axiom (b) follows from Theorems~\ref{t2} and ~\ref{th1-3*}. Axiom (c) follows from Theorems~\ref{th1-2*}. \end{proof}
\begin{example} Let $p=2$, $N=2$, $M=1$ $\phi$ be defined by (\ref{62.0-8}), where $\widehat\phi(0)\ne0$, $m_0$ is given by (\ref{62.0-7-1}), $m_0(1/4)=m_0(3/8)=m_0(7/16)=m_0(15/16)=0$ and $m_0(0)=1$. It is not difficult to see that ${\rm supp\,}\widehat\phi\subset B_1(0)$, ${\rm supp\,}\widehat\phi\not\subset B_0(0)$ and $\widehat\phi\Big(\frac{1}{2}\Big)=\widehat\phi\Big(\frac{3}{2}\Big)= \widehat\phi\Big(\frac{5}{2}\Big)= \widehat\phi(1)=0$, i.e, all the assumptions of Theorem~\ref{t3} are fulfilled. \label{e1} \end{example}
\begin{remark} The above example is typical. Similarly, taking into account the arguments of the proof of Lemma~\ref{l1}, one can easily construct a lot of functions $\phi$ generating a MRA for arbitrary $p$, $M>0$ and large enough
$N$. Moreover, it is possible to provide $\deg m_0\le2^N$.
\label{r1} \end{remark}
\section{Orthogonal scaling functions} \label{s4}
Now we are going to describe all orthogonal scaling functions $\phi\in{{\mathcal D}}_N^M$.
\begin{theorem} \label{th1-5*} Let $\phi\in{{\mathcal D}}_N^M$, $M,N\ge0$. If $\{\phi(x-a):a\in I_p\}$ is an orthonormal system, then \begin{equation}
\sum_{l=0}^{p^{M+N}-1}\left|{\widehat\phi}\left(\frac{l}{p^M}\right)\right|^2 \chi_p\left(\frac{lk}{p^{M+N}}\right)=p^N\delta_{k 0}, \quad k=0,{1},\dots,{p^{N}-1}. \label{20} \end{equation} \end{theorem}
\begin{proof} Let $a\in I_p$. Due to the orthonormality of $\{\phi(x-a):a\in I_p\}$, using the Plancherel theorem, we have $$ \delta_{a 0}= \langle\phi(\cdot),\phi(\cdot-a)\rangle \int\limits_{{\mathbb Q}_p}\phi(x)\overline{\phi(x-a)}\,dx
=\int\limits_{B_M(0)}|{\widehat\phi}(\xi)|^2\chi_p(a\xi)\,d\xi. $$ Let $\xi\in B_M(0)$. There exists a unique $l=0,1,\dots, p^{M+N}-1$ such that $\xi\in B_{-N}\left( b_l\right)$, $b_l=\frac{l}{p^M}$. It follows that \begin{eqnarray*}
\int\limits_{B_M(0)}|{\widehat\phi}(\xi)|^2\chi_p(a\xi)\,d\xi
=\sum_{k=0}^{p^{M+N}-1}\int\limits_{|\xi-b_l|_p\le p^{-N}}
|{\widehat\phi}(\xi)|^2\chi_p(a\xi)\,d\xi\qquad\qquad\qquad\qquad \\
=\sum\limits_{l=0}^{p^{M+N}-1}|{\widehat\phi}(b_l)|^2
\int\limits_{|\xi-b_l|_p\le p^{-N}}\chi_p(a\xi)\,d\xi
=\sum\limits_{l=0}^{p^{M+N}-1}|{\widehat\phi}(b_l)|^2\chi_p(ab_l)
\int\limits_{|\xi|_p\le p^{-N}}\chi_p(a\xi)\,d\xi \\
=\frac{1}{p^{N}}\Omega(|p^{N}a|_p)
\sum_{l=0}^{p^{M+N}-1}|{\widehat\phi}(b_l)|^2\chi_p(ab_l). \end{eqnarray*}
To prove~(\ref{20}) it only remains to note that
$\Omega(|p^{N}a|_p)= 0$ whenever $a\in I_p$,
$p^Na\ne0,1,\dots, p^N-1$. \end{proof}
\begin{lemma}
Let $c_0,,\dots, c_{n-1}$ be mutually distinct elements of the unit circle $\{z\in{\mathbb C}:\ |z|=1\}$. Suppose that there exist nonzero reals $x_j$, $j=0,1,\dots, n-1$, such that \begin{equation} \sum_{j=0}^{n-1}c_j^kx_j=\delta_{k 0},\ \ \ k=0,1,\dots, n-1. \label{21} \end{equation} Then $x_j=1/n$ for all $j$, and up to reordering \begin{equation} c_j=c_0\ex{j/n},\ \ \ j=0,1,\dots, n-1. \label{22} \end{equation} \label{l2} \end{lemma}
\begin{proof} In accordance with Cramer's rule we have $x_j=\frac{\Delta_j}{\Delta}$, $0\le j\le n-1$, where $\Delta=V(c)$ is the Vandermonde determinant corresponding to $c=(c_0,\dots, c_{N-1})$, and $\Delta_j$ is obtained from $\Delta$ by replacing the $j$-th column with the transpose of the row $(1,0,\dots, 0)$. A straightforward computation shows that $$ \Delta_j=(-1)^jV(c^{(j)})\prod\limits_{k\ne j}c_k, $$ where $c^{(j)}$ is obtained from $c$ by removing the $j$-th coordinate. Thus, \begin{eqnarray} x_j=(-1)^j\frac{V(c^{(j)})}{V(c)}\prod\limits_{k\ne j}c_k =(-1)^j\prod\limits_{k\ne j}c_k\prod\limits_{k>l\atop k,l\ne j} ({c_k-c_l})\Big/{\prod\limits_{k>l}(c_k-c_l)} \nonumber \\ \prod\limits_{k\ne j}\frac{c_k}{c_k-c_j}= \prod\limits_{k\ne j}\frac{1}{1-c^{-1}_kc_j}. \label{35} \end{eqnarray} Next, for any $\alpha\in{\Bbb R}$, we have \begin{eqnarray*} 1-e^{i\alpha}=2\sin\frac{\alpha}{2} \left(\sin\frac{\alpha}{2}-i\cos\frac{\alpha}{2}\right)= 2\sin\frac{\alpha}{2}e^{i\left(\frac{\alpha}{2}-\frac{\pi}{2}\right)}. \end{eqnarray*} Let us define $\alpha_j$, $j=0,1,\dots, n-1$, by $c_j=e^{i\alpha_j}$. Then from the above arguments and~(\ref{35}) it follows that $$ x_j=\prod\limits_{k\ne j}\frac{1}{1-c^{-1}_kc_j}=
e^{i\gamma}\sum\limits_{k\ne j}\lll2\sin\frac{\alpha_k-\alpha_j}{2}\right)^{-1}, $$ where $$ \gamma=\sum\limits_{k\ne j}\frac{\alpha_k-\alpha_j+\pi}{2}= \theta-\frac{n}{2}\alpha_j,\ \ \ \theta= \frac12\left((n-1)\pi+\sum\limits_{k=0}^{n-1}\alpha_k\right) $$ By the lemma's hypothesis $x_j\in{\Bbb R}$, whence $\gamma\equiv0\pmod{\pi}$ and consequently $n\alpha_j\equiv2\theta\pmod{2\pi}$. Thus up to reordering $\alpha_j=\alpha_0+\frac{2\pi j}{n}$, which implies~(\ref{22}), and consequently that $x_j=1/n$ for all $j$. \end{proof}
\begin{theorem} \label{t4} Let $\phi\in{{\mathcal D}}_N^M$ be an orthogonal scaling function and $\widehat\phi(0)\ne0$. Then ${\rm supp\,{\widehat\phi}}\subset B_0(0)$. \end{theorem}
\begin{proof} Without loss of generality, we can assume that $M,N\ge0$. Combining Theorems~\ref{t3} and \ref{th1-5*}, we have $$
\sum_{j=0}^{p^{N}-1}\left|{\widehat\phi}\left(\frac{l_j}{p^M}\right)\right|^2 \chi_p\left(\frac{l_jk}{p^{M+N}}\right)=p^N\delta_{k 0}, \quad k=0,{1},\dots,{p^{N}-1}. $$ By Lemma~\ref{l2}, $l_j=l_0+jp^M$ and ${\widehat\phi}\left(\frac{l_j}{p^M}\right)=1$. Taking into account that $\widehat\phi(0)\ne0$, we deduce $l_0=0$, i.e., ${\widehat\phi}(j)=1$, $j=0,1,\dots, p^{N}-1$. Since $\widehat\phi$ is a $p^N$-periodic function, it follows from Theorem~\ref{t3} that $\widehat\phi\left(\frac{l}{p^M}\right)=0$ for all $l\in\z$ not divisible by $p^M$. This yields ${\rm supp\,{\widehat\phi}}\subset B_0(0)$. \end{proof}
So any test function $\phi$ generating a MRA belongs to the class ${{\mathcal D}}_N^0$. All such functions were described in~\cite{Kh-Sh-S}. The following theorem summarizes these results.
\begin{theorem} \label{th1-6*} Let ${\widehat\phi}$ be defined by {\rm(\ref{62.0-8})}, where $m_0$ is the trigonometric polynomial~{\rm(\ref{62.0-7-1})} with $m_0(0)=1$. If $m_0\big(\frac{k}{p^{N+1}}\big)=0$ for all $k=1,\dots,p^{N+1}-1$ not divisible by $p$, then $\phi\in{{\mathcal D}}_N^0$. If, furthermore,
$\big|m_0\big(\frac{k}{p^{{N+1}}}\big)\big|=1$ for all $k=1,\dots,p^{N+1}-1$ divisible by $p$, then $\{\phi(x-a):a\in I_p\}$ is an orthonormal system. Conversely, if ${\rm supp\,{\widehat\phi}}\subset B_0(0)$ and the system $\{\phi(x-a):a\in I_p\}$
is orthonormal, then $\big|m_0\big(\frac{k}{p^{{N+1}}}\big)\big|=0$
whenever $k$ is not divisible by $p$, and $\big|m_0\big(\frac{k}{p^{{N+1}}}\big)\big|=1$ whenever $k$ is divisible by $p$, \ $k=1,2,\dots,p^{N+1}-1$. \end{theorem}
\section{Construction of wavelet frames} \label{s4}
\begin{definition} Let $H$ be a Hilbert space. A system $\{f_n\}_{n=1}^\infty\subset H$ is said to be a frame if there exist positive constants $A,B$ ({\em frame boundaries}) such that $$
A\|f\|^2\le \sum_{n=1}^\infty|\langle f,f_n\rangle|^2\le B\|f\|^2 \ \ \ \forall f\in H. $$ \end{definition}
We are interested in the construction of $p$-adic wavelet frames, i.e., frames in $L_2({\mathbb Q}_p)$ consisting of functions $p^{j/2}\psi^{(\nu)}(p^{-j}\cdot-a)$, $a\in I_p$, $\nu\in A$, where $A$ is a finite set.
We will restrict ourselves to the consideration of the case $p=2$.
Our general scheme of construction looks as follows. Let $\{V_j\}_{j\in\z}$ be a MRA. As above, we define the wavelet space $W_j$, $j\in \z$, as the orthogonal complement of $V_j$ in $V_{j+1}$, i.e., $V_{j+1}=V_j\oplus W_j$.
It is not difficult to see that $f\in W_j$ if and only if $f(2^{j}\cdot)\in W_0$, and $W_j\perp W_k$ whenever $j\ne k$. If now there exists a function $\psi\in L_2(Q_2)$ ({\em wavelet function}) such that \begin{equation} W_0=\overline{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2\}}, \label{23} \end{equation} then we have a wavelet system $\{2^{j/2}\psi(2^{-j}\cdot-a), a\in I_2, j\in\z \}$. It will be proved that such a system is a frame in $L_2(Q_2)$ whenever
$\psi$ is compactly supported.
\begin{theorem} Let $\{V_j\}_{j\in\z}$ be a MRA, $\psi$ be a wavelet function. If $\psi$ is compactly supported, then the corresponding wavelet system $\{2^{j/2}\psi(2^{-j}\cdot-a), a\in I_2, j\in\z \}$
is a frame in $L_2(Q_2)$. \label{t5}
\end{theorem}
\begin{proof} First we will prove that the system $\{\psi(\cdot-a),\ a\in I_2\}$ is a frame in the wavelet space $W_0$. Let $\mbox{supp}\,\psi\subset B_N(0)$, $N\ge0$. Set
\begin{eqnarray*} W_0^0&=&{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2, |a|_2\le2^N\}}, \\
W_0^n&=&{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2, |a|_2=2^{N+n}\}},\ \ \ n\in\n.
\end{eqnarray*} It is not difficult to see that the spaces $W_0^n$, $n=0,1,\dots$, are mutually orthogonal. Each function $f\in W_0$ may be represented in the form $f=f^0+f^1+\cdots$, where $f^0=f\Big|_{B_N(0)}$,
$f^n=f\Big|_{B_{N+n}(0)\setminus B_{N+n-1}(0)}$, $n\in \n$. Due to~(\ref{23}), given $\epsilon>0$, there exists a finite sum $\sum\limits_{a\in I_2}\alpha_a\psi(\cdot-a)=:f_\epsilon$ such that
$\|f-f_\epsilon\|<\epsilon$. If $|x|_2\le2^N$, then $f_\epsilon(x)=
\sum\limits_{a\in I_2\atop |a|_2\le2^N}\alpha_a\psi(x-a)=:f^0_\epsilon(x)$. Since $\mbox{supp}\, f^0\subset B_N(0)$, $\mbox{supp}\, f_\epsilon^0\subset B_N(0)$, we have $$
\|f-f_\epsilon\|^2\ge\int\limits_{B_N(0)}|f-f_\epsilon|^2
=\int\limits_{B_N(0)}|f^0-f^0_\epsilon|^2=\|f^0-f^0_\epsilon\|^2. $$ It follows that $f^0\in W_0^0$. Similarly, $f^n\in W_0^n$, $n\in\n$. Thus we proved that \begin{equation} W_0=W_0^0\oplus W_0^1\oplus W_0^2\oplus\dots. \label{26} \end{equation} Since $W_0^0$ is a finite dimensional space and
$\{\psi(\cdot-a),\ a\in I_2, |a|_2\le2^N\}$ is a representing system for $W_0^0$, this system is a frame. Hence there exist
positive constants $A,B$ such that $$
A\|f^0\|^2\le\sum_{a\in I_2}|\langle f^0,\psi(\cdot-a)\rangle|^2\le B\|f^0\|^2
\ \ \ \forall f\in W_0^0. $$
If $f^1\in W^1_0$, we have \begin{eqnarray*}
\sum_{a\in I_2\atop |a|_2=2^{N+1}}|\langle f^1,\psi(\cdot-a)\rangle|^2=
\sum_{a\in I_2\atop |a|_2\le2^N}|\langle f^1,\psi(\cdot-a-2^{-N-1})\rangle|^2= \\
\sum_{a\in I_2\atop |a|_2\le2^N}|\langle f^1(\cdot+2^{-N-1}),\psi(\cdot-a)\rangle|^2\ge
A\|f^1(\cdot+2^{-N-1})\|^2=A\|f^1\|^2. \end{eqnarray*} Let now $f^n\in W_0^n$, $n>1$. Then \begin{eqnarray*}
\sum_{a\in I_2\atop |a|_2=2^{N+n}}
\left|\left\langle f^n,\psi(\cdot-a)\right\rangle\right|^2= \sum\limits_{k=0}^{2^{n-1}-1}
\sum_{a\in I_2\atop |a|_2\le2^N}
\left|\left\langle f^n,\psi\left(\cdot-a-\frac{2k+1}{2^{N+n}})
\right)\right\rangle\right|^2= \\ \sum\limits_{k=0}^{2^{n-1}-1}
\sum_{a\in I_2\atop |a|_2\le2^N}
\left|\left\langle f^n\left(\cdot+\frac{2k+1}{2^{N+n}}\right)
\Omega(|2^N\cdot|),\psi\left(\cdot-a)
\right)\right\rangle\right|^2\ge \\
A\sum\limits_{k=0}^{2^{n-1}-1}
\left\|f^n\left(\cdot+\frac{2k+1}{2^{N+n}}\right)\Omega(|2^N\cdot|)\right\|^2=
A\sum\limits_{k=0}^{2^{n-1}-1}
\left\|f^n\Omega\left(\left|2^N\left(\cdot-\frac{2k+1}{2^{N+n}}\right)\right|\right)\right\|^2= \\
A\sum\limits_{k=0}^{2^{n-1}-1}
\left\|f^n\Big|_{B_N\left(\frac{2k+1}{2^{N+n}}\right)}\right\|^2=
A\|f^n\|^2. \end{eqnarray*} Taking into account~(\ref{26}), we derive \begin{equation}
A\|f\|^2\le\sum_{a\in I_2}|\langle f,\psi(\cdot-a)\rangle|^2
\ \ \ \forall f\in W_0. \label{25} \end{equation}
Similarly we can prove the upper frame estimation \begin{equation}
\sum_{a\in I_2}|\langle f,\psi(\cdot-a)\rangle|^2\le B\|f\|^2
\ \ \ \forall f\in W_0. \label{24} \end{equation} Combining~(\ref{25}) with~(\ref{24}), we deduce that the system $\{\psi(\cdot-a),\ a\in I_2\}$ is a frame in $W_0$. Evidently, the system $\{2^{j/2}\psi(2^{-j}\cdot-a), a\in I_2, \}$ is a frame in $W_j$ with the same frame boundaries for any $j\in\z$. Since ${\bigoplus\limits_{j\in\z}W_j}= L^2({\mathbb Q}_2)$, it follows that the union of these frames is a frame in $L_2({\mathbb Q}_2)$. \end{proof}
Now we discuss how to construct a desirable wavelet function $\psi$. Let a MRA $\{V_j\}_{j\in\z}$ is generated by a scaling function $\phi\in{{\mathcal D}}_N^M$. First of all we should provide $\psi\in V_1$. Let us look for $\psi$ in the form $$ \psi(x)=\sum_{k=0}^{2^{N+1}-1}g_{k}\phi\Big(\frac{x}{2}-\frac{k}{2^{N+1}}\Big) $$ Taking the Fourier transform of~(\ref{62.0-5}) and using (\ref{014}), we have $$ {\widehat\psi}(\xi)=n_0\Big(\frac{\xi}{2^{N}}\Big){\widehat\phi}(2\xi),\ \ \ $$ where $n_0$ is a trigonometric polynomial ({\em wavelet mask}) given by $$ n_0(\xi)=\frac{1}{2}\sum_{k=0}^{2^{N+1}-1}g_{k}\chi_2(k\xi) $$ Evidently, $\psi\in{{\mathcal D}}_N^M$. By Theorem~\ref{t1}, there exist at least $2^{M+N}-2^N$ integers $l$ such that $0\le l<2^{M+N}$, $\widehat\phi\left(\frac{l}{2^{M}}\right)=0$. Choose $n_0$ satisfying the following property: if $\widehat\phi\left(\frac{l}{2^{M}}\right)\ne0$ for some $l=0,1,\dots, 2^{M+N}-1$, then $n_0\left(\frac{l}{2^{M+N}}\right)=0$. This yields that $\widehat\psi\left(\frac{l}{2^{M}}\right)=0$ whenever $\widehat\phi\left(\frac{l}{2^{M}}\right)\ne0$, $0\le l<2^{M+N}$.
Let $a, b\in I_2$. Using the Plancherel theorem and the arguments of Theorem~\ref{t1}, we have \begin{eqnarray*} \langle\phi(\cdot-a),\psi(\cdot-b)\rangle= \int\limits_{{\mathbb Q}_2}\phi(x-a)\overline{\psi(x-b)}\,dx =\int\limits_{B_M(0)}{\widehat\phi}(\xi) \overline{\widehat\psi(\xi)}\chi_2((b-a)\xi)\,d\xi= \\
\sum_{k=0}^{2^{M+N}-1}\int\limits_{|\xi-2^{-M}l|_2\le 2^{-N}} {\widehat\phi}(\xi) \overline{\widehat\psi(\xi)}\chi_2((b-a)\xi)\,d\xi=\qquad\qquad\qquad\qquad \\ \sum\limits_{l=0}^{2^{M+N}-1}{\widehat\phi}\left(\frac{l}{2^M}\right) \overline{\widehat\psi\left(\frac{l}{2^M}\right)}
\int\limits_{|\xi-2^{-M}l|_2\le 2^{-N}}\chi_2(a\xi)\,d\xi=0. \end{eqnarray*} It follows that $\overline{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2\}}\perp V_0$. On the other hand, due to Theorem~\ref{t2}, we have $\overline{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2\}}\subset V_1$. Hence, \begin{equation} \overline{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2\}}\subset W_0. \label{30} \end{equation} It is clear from that proof of Theorem~\ref{t2} that \begin{eqnarray} \psi\left( x-\frac{l}{p^{N}}\right)&=& \sum_{k=0}^{p^{N+1}-1}g_{kl}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big), \ \ \ \ l=0,1,\dots, 2^N-1, \label{31} \\ \phi\left( x-\frac{l}{p^{N}}\right)&=& \sum_{k=0}^{p^{N+1}-1}h_{kl}\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big), \ \ \ l=0,1,\dots, 2^N-1. \label{32} \end{eqnarray} Consider these equalities as a linear system with respect to the unknowns $X_k:=\phi\Big(\frac{x}{p}-\frac{k}{p^{N+1}}\Big)$, $k=0,1,\dots, 2^{N+1}-1$. If the system~(\ref{31}),~(\ref{32}) has a solution, then $$ \mbox{span}\,\left\{\phi\Big(\frac{\cdot}{p}-a\Big),
\ a\in I_2, |a|_2\le2^{N+1}\right\}
\subset{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2, |a|_2\le2^{N}\}}. $$ This evidently implies $W_0\subset\overline{\mbox{span}\,\{\psi(\cdot-a),\ a\in I_2\}}.$ Taking into account~(\ref{30}), we deduce that $\psi$ is a wavelet function.
It is not quite clear whether the system~(\ref{31}),~(\ref{32}) has a solution for arbitrary $\phi$ and $\psi$, but we will show how to succeed in the case $\deg m_0\le2^N$. The construction of such masks can easily be done (see Example~\ref{e1} and Remark~\ref{r1}).
Assume that $\deg m_0\le2^N$. In this case \begin{equation} {\widehat\phi}(\xi)\chi_2\left(\frac{l\xi}{2^N}\right) =m_{l/2^N}\Big(\frac{\xi}{2^{N}}\Big){\widehat\phi}(2\xi), \ \ \ \ l=0,1,\dots, 2^N-1, \label{33} \end{equation} where $m_{l/2^N}(\xi)=\chi_2(l\xi)m_0(\xi)$, $\deg m_{l/2^N}<2^{N+1}$. It is clear that a wavelet mask $n_0$ can also be chosen in such a way that $\deg n_0\le2^N$, and we have \begin{equation} {\widehat\psi}(\xi)\chi_2\left(\frac{l\xi}{2^N}\right) =n_{l/2^N}\Big(\frac{\xi}{2^{N}}\Big){\widehat\phi}(2\xi), \ \ \ \ l=0,1,\dots, 2^N-1, \label{34} \end{equation} where $n_{l/2^N}(\xi)=\chi_2(l\xi)n_0(\xi)$, $\deg n_{l/2^N}<2^{N+1}$. Taking the Fourier transform of ~(\ref{33}),~(\ref{34}), we see that the matrix of the system~(\ref{31}),~(\ref{32}) looks as follows:
$$ \left( \begin{array}{llllllll}
g_0 & g_1&\dots& g_{2^N-1}& g_{2^N}&0&\dots&0
\\
0& g_0 &\dots& g_{2^{N}-2}& g_{2^{N}-1}& g_{2^N}&\dots&0
\\
\hdots&\hdots&\hdots&\hdots&\hdots&\hdots&\hdots&\hdots
\\
0& 0 &\dots& g_0& g_1& g_2&\dots&g_{2^N}
\\
h_0 & h_1&\dots& h_{2^N-1}& h_{2^N}&0&\dots&0
\\
0& h_0 &\dots& h_{2^{N}-2}& h_{2^{N}-1}& h_{2^N}&\dots&0
\\
\hdots&\hdots&\hdots&\hdots&\hdots&\hdots&\hdots&\hdots
\\
0& 0 &\dots& h_0& h_1& h_2&\dots&h_{2^N} \end{array} \right) $$
The determinant of this matrix is so called resultant. The resultant is not equal to zero if and only if the algebraic polynomials with the coefficients $g_0,g_1,\dots, g_{2^N}$ and $h_0,h_1,\dots, h_{2^N}$ respectively do not have joint zeros
(see, e.g., \cite{L}). But this holds
because the trigonometric polynomials
$m_0$ and $n_0$ do not have joint zeros by
construction (taking care of not adding extra zeros).
\end{document} | arXiv | {
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\begin{document}
\title{On multi-index filtrations associated to Weierstra\ss~semigroups}
\author{Julio Jos\'e Moyano-Fern\'andez}
\address{Universit\"at Osnabr\"uck, FB Mathematik/Informatik, 49069 Osnabr\"uck, Germany} \email{jmoyano@uos.de}
\subjclass[2010]{Primary 14H55; Secondary 14G15} \keywords{algebraic curve, adjunction theory, normalisation, Weierstra\ss~semigroup} \thanks{The author was partially supported by the Spanish Government Ministerio de Educaci\'on y Ciencia (MEC), grants MTM2007-64704 and MTM2012--36917--C03--03 in cooperation with the European Union in the framework of the founds ``FEDER''}
\begin{abstract} The aim of this paper is to review the main techniques in the computation of Weierstra\ss~semigroup at several points of curves defined over perfect fields, with special emphasis on the case of two points. Some hints about the usage of some packages of the computer algebra software \textsc{Singular} are also given. \end{abstract}
\maketitle
\section{Introduction}
There are several classical problems in the theory of algebraic curves which are interesting from a computational point of view. One of them is the computation of the Weierstra\ss~semigroup of a smooth projective algebraic curve $\tilde{\chi}$ at a certain rational point $P$, together with a rational function $f_m \in \mathbb{F}(\tilde{\chi})$ regular outside $P$ and achieving a pole at $P$ of order $m$, for each $m$ in this semigroup. This problem is solved with the aid of the adjunction theory for plane curves, profusely developed by A. von Brill and M. Noether in the 19th century (see \cite{brino}, \cite{noe}) so that we assume the knowledge of a singular plane birational model $\chi$ for the smooth curve $\tilde{\chi}$.
Given a smooth projective algebraic curve $\chi$ (over a perfect field $\mathbb{F}$) and a set $P_1, \ldots , P_r$ of (rational) points of $\chi$, we consider the family of finitely dimensional vector subspaces of $\mathbb{F} (\chi)$ given by $\mathcal{L}(\underline{mP})=\mathcal{L}(m_1P_1+m_2P_2+ \ldots + m_rP_r)$, where $\underline{m} = (m_1,\ldots, m_r) \in \mathbb{Z}^r$. This family gives rise to a $\mathbb{Z}^r$-multi-index filtration on the $\mathbb{F}$-algebra $A$ of the affine curve $\chi \setminus \{P_1, \ldots, P_r\}$, since one has $A=\bigcup_{\underline{m} \in \mathbb{Z}^r} \mathcal{L}(\underline{mP})$. This multifiltration is related to Weierstra\ss~semigroups (with respect to several points in general, see Delgado \cite{de}) and, in case of finite fields, to the methodology for trying to improve the Goppa estimation of the minimal distance of algebraic-geometrical codes, see for instance Carvalho and Torres \cite{cato}. A connection of that filtration with global geometrical-topological aspects in a particular case was shown by Campillo, Delgado and Gusein-Zade \cite{cadegz}. Poincar\'e series associated to this filtrations in particular cases were studied by the author in \cite{moyano}
Thus, a natural question is to provide a computational method in order to estimate the values of $\dim_{\mathbb{F}} \mathcal{L}(\underline{mP}) = \ell (\underline{mP})$ for $\underline{m} \in \mathbb{Z}^r$. More precisely, it would be convenient to estimate and compute values of type $\ell ((\underline{m}+\underline{\varepsilon})\underline{P})-\ell(\underline{mP})$ where $\underline{\varepsilon} \in \mathbb{Z}^r$ is a vector whose components are $0$ or $1$. This can be done by extending the method developed by Campillo and Farr\'an \cite{cafa} in the case $r=1$, based on the knowledge of a plane model $\widetilde{\chi}$ for $\chi$ (with singularities) and representing the global regular differentials in terms of adjoint curves to $\widetilde{\chi}$.
The paper is organised as follows: Sections 2 and 3 are devoted to fix the algebraic-geometrical prerequisites. Section 4 deals with the study of more specific questions concerning to our purpose, namely the adjunction theory of curves, with the remarkable Brill-Noether Theorem. In Section 5 we define the Weierstra\ss~semigroup at several points and describe two methods to compute values of the form $\ell ((\underline{m}+\underline{\varepsilon})\underline{P})-\ell(\underline{mP})$. The last section is devoted to show and explain some procedures implemented in \textsc{Singular} based on Section 5.
Notice the practical relevance of these ideas in view of the algebraic-geometric coding theory: the Weierstra\ss~semigroup plays an important role in the decoding procedure of Feng and Rao, see e.g. Campillo and Farr\'an \cite{cafa2}, or H\o holdt, van Lint and Pellikaan \cite{holipe}.
\section{Terminology and notations}
Let $\mathbb{F}$ be a perfect field, and let $\overline{\mathbb{F}}$ a fixed algebraic closure of $\mathbb{F}$. Let $\chi$ be an absolutely irreducible projective algebraic curve defined over $\mathbb{F}$. We distinguish three types of points on $\chi$, namely the geometric points, i.e. those with coordinates on $\overline{\mathbb{F}}$; the rational points, i.e. those with coordinates on $\mathbb{F}$; and the closed points, which are residue classes of geometric points under the action of the Galois group of the field extension $\overline{\mathbb{F}}/\mathbb{F}$, namely \[ P:=\{\sigma (p) : \sigma \in \mathrm{Gal} (\overline{\mathbb{F}}/\mathbb{F})\}, \] where $p$ is a geometric point. Notice that closed points correspond one to one to points on the curve $\chi$ viewed as an $\mathbb{F}$-scheme which are closed for the Zariski topology. Every closed point has an associated residue field $\mathbb{F} '$ which is a finite extension of $\mathbb{F}$. The degree of a closed point $P$ is defined as the cardinal of its conjugation class, which equals the degree of the extension $\mathbb{F} ' / \mathbb{F}$. In particular, $P$ is rational if and only if $\deg P=1$.
Let us assume $\chi$ to be non-singular (or, equivalently, smooth, since $\mathbb{F}$ is perfect). Let $\overline{\mathbb{F}}(\chi)$ be the field of rational functions of $\chi$. Let $P$ be a closed point on $\chi$. The local ring $\mathcal{O}_{\chi,P}$ of $\chi$ at $P$ with maximal ideal $\mathfrak{m}_{\chi, P}$ is therefore a discrete valuation ring with associated discrete valuation $v_P$. An element $f \in \mathcal{O}_{\chi,P}$ is said to vanish at $P$ (or to have a zero at $P$) if $f \in \mathfrak{m}_{\chi, P}$. A rational function $f$ such that $f \notin \mathcal{O}_{\chi,P}$ is said to have a pole at $P$. The order of the pole of $f$ at $P$ is given by $|v_P(f)|$.
A \emph{rational divisor} $D$ over $\mathbb{F}$ is a finite linear combination of closed points $P \in \chi$ with integer coefficients $n_P$, that is, $D=\sum_P n_P P$. If $n_P \ge 0$ for all $P$, then $D$ is called \emph{effective}. We define the \emph{degree} of $D$ as $\deg D := \sum_P n_P \deg P$, and the \emph{support} of $D$ as the set $\mathrm{supp}(D)=\big \{P \in \chi \ \mathrm{closed} \mid n_P \ne 0 \big \}$. The set of rational divisors on $\mathbb{F}$ form an abelian group $\mathcal{D}(\mathbb{F})$. Rational functions define \emph{principal} divisors, namely divisors of the form \[ (f):=\sum_P \mathrm{ord}_P (f) P. \]
A rational divisor $D=\sum n_P P$ defines a $\mathbb{F}$-vector space \[ \mathcal{L}(D)=\Big \{f \in \mathbb{F}(\chi)^{\ast} \mid (f) \ge -D \Big \} \cup \big \{ 0 \big \}, \] that is, the set of rational functions $f$ with poles only at the points $P$ with $n_P \ge 0$ (and, furthermore, with the pole order of $f$ at $P$ must be less or equal than $n_P$), and if $n_P<0$ such functions must have a zero at $P$ of order greater or equal than $n_P$. The dimension $\ell (D):=\dim_{\mathbb{F}} \mathcal{L}(D)$ is finite. Two elements $f,g \in \mathcal{L}(D)$ satisfy $(f)+D = (g)+D$ if and only if $f=\lambda g$, $\lambda \in \mathbb{F}$, i.e., if and only if
$f = \lambda g$ for a constant $\lambda \in \mathbb{F}$. Therefore the set $|D|$ of effective divisors equivalent to $D$ can be identified with the projective space $\mathbb{P}_{\mathcal{L}(D)}$ of dimension $\ell (D)-1$. The set
$|D|$ is called a \emph{complete linear system} of $D$.
Let $\Omega_{\mathbb{F}}(\mathbb{F}(\chi))$ be the module of differentials on $\mathbb{F}(\chi)$. A differential form $\omega \in \Omega_{\mathbb{F}}(\mathbb{F}(\chi))$ defines a divisor $(\omega):=\sum_P \mathrm{ord}_P(\omega)P$, called a canonical divisor. A rational divisor $D$ defines again a $\mathbb{F}$-vector space \[ \Omega (D):= \{\omega \in \Omega_{\mathbb{F}}(\mathbb{F}(\chi))^{\ast} \mid (\omega) \ge D \} \cup \{ 0 \}. \] of finite dimension, denoted by $i(D)$. It is a central result in the theory of algebraic curves the interplay of the dimensions $\ell (D)$ and $i(D)$. The dimension $\ell (D)$ is bounded in the following sense:
\begin{prop}[Riemann's inequality] There exists a nonnegative integer $g$ such that \[ \ell (D) \ge \deg D +1 -g. \] for any rational divisor $D$ on $\chi$. \end{prop}
\begin{defi} The smallest integer $g$ satisfying the Riemann's inequality is called the \emph{genus} of $\chi$. \end{defi}
Riemann's inequality tells us that if $D$ is a large divisor, $\mathcal{L}(D)$ is also large. But we can be a bit more precise by using $i(D)$:
\begin{teo}[Riemann-Roch] Let $D$ be a rational divisor. Then: $$ \ell (D) - i(D) = \deg D +1 -g. $$ \end{teo}
\section{Rational parametrizations}
Let $\mathbb{F}$ be a perfect field, and let $\chi$ be an absolutely irreducible algebraic plane curve defined over $\mathbb{F}$. Let $P$ be a closed point on $\chi$. Let us consider the local ring $\mathcal{O} := \mathcal{O}_{\chi,P}$ with maximal ideal $\mathfrak{m}$, and write $\overline{\mathcal{O}}$ for the semilocal ring of the normalisation of $\chi$ at $P$. Finally, let $\hat{\mathcal{O}}$ be the completion of $\mathcal{O}$ with respect to the $\mathfrak{m}$-adic topology. Each maximal ideal of $\overline{\mathcal{O}}$ (or, equivalently, every minimal prime ideal $\mathfrak{p}$ of $\hat{oo}$) is said to be a \emph{branch} of $\chi$ at $P$.
Let us now choose an affine chart containing $P$ so that the curve $\chi$ has an equation $f(X,Y)=0$, and set $A:=\mathbb{F}[X,Y]/(f(X,Y))$ as the affine coordinate ring. Notice that $\mathcal{O} = A_P$. Hence \[ \mathbb{F} \subseteq \mathbb{F}[X,Y]/(f(X,Y))=A \subseteq A_P=\mathcal{O}. \] Since $\mathbb{F}$ is perfect, we can apply Hensel's lemma to find a finite field extension $K / \mathbb{F}$ such that $K \subseteq \hat{A_P}=\hat{\mathcal{O}}$ is a coefficient field for $\hat{\mathcal{O}}$. Moreover, $K$ is the integral closure of $\mathbb{F}$ in $\hat{\mathcal{O}}$.
Since $\hat{\mathcal{O}}\subseteq \overline{\hat{\mathcal{O}}}\cong \hat{\overline{\mathcal{O}}}$, one has \[ K \subseteq \hat{\mathcal{O}}/\mathfrak{p} \subseteq \overline{ \hat{\mathcal{O}}/\mathfrak{p}} = \hat{\overline{\mathcal{O}}_{\mathfrak{m}}}, \] and we can apply Hensel's lemma again to obtain a finite extension $K' / K$ which is a coefficient field for the local ring $\hat{\overline{\mathcal{O}}_{\mathfrak{m}}}$. Without loss of generality we can consider $P$ as the ideal $(X,Y)$ in $K[\![X,Y]\!]$ so that $\hat{\mathcal{O}}\cong K[\![X,Y]\!]/(f(X,Y))$. This implies the existence of natural morphisms \[ K[\![X,Y]\!]/(f(X,Y)) \cong \hat{\mathcal{O}} \longrightarrow \hat{\mathcal{O}}/\mathfrak{p} \longrightarrow K'[\![t]\!] \cong \hat{\overline{\mathcal{O}}_{\mathfrak{m}}} \] for any local uniformizing parameter $t \in \mathfrak{m} \setminus \mathfrak{m}^2$. Notice that $K$ can be considered isomorphic to the residue field at $P$. Preserving these notations, a \emph{parametrization} of the curve $\chi$ at the point $P$ related to the coordinates $X,Y$ is a $K$-algebra morphism $\rho: K[\![X,Y]\!] \longrightarrow K'[\![t]\!]$ being continuous for the $(X,Y)$-adic and $t$-adic topologies and satisfying $\mathrm{Im}(\rho) \not\subseteq K'$ and $\rho (f)=0$. This is equivalent to give formal power series $x(t),y(t) \in K'[\![t]\!]$ with $x(t) \neq 0$ or $y(t)\neq 0$ such that $f(x(t),y(t))\equiv 0$.
Consider parametrizations $\rho:K[\![X,Y]\!] \to K'[\![t]\!]$ and $\sigma: K[\![X,Y]\!] \to K''[\![t]\!]$ of the same rational branch. The parametrization $\sigma$ is said to be \emph{derivated} from $\rho$ if there is a formal power series $\tau(u) \in K''[\![u]\!]$ with positive order and a continuous $K$-algebra morphism $\alpha: K'[\![t]\!] \to K''[\![u]\!]$ with $\alpha(t)=\tau(u)$ such that $\sigma=\alpha \circ \rho$. We write $\sigma \succ \rho$. The relation $\succ$ is a partial preorder. Two parametrizations $\sigma$ and $\rho$ are called \emph{equivalent} if $\sigma \succ \rho$ and $\rho \succ \sigma$. Those parametrizations being minimal with respect to $\succ$ up to equivalence are called \emph{primitive}. Equivalent primitive parametrizations are called \emph{rational}. They always exist and are invariant under the action of the Galois group of the extension $\overline{K}/K$. Rational parametrizations are in one to one correspondence with rational branches of the curve (cf. Campillo and Castellanos \cite{caca}).
\section{Brill-Noether theory for curves}
This section contains a summary of the classic adjunction theory of curves, started by Riemann \cite{rie} and developed by M. Noether and A. von Brill in the 19th century.
Let $P$ be a closed point. Let $\mathcal{C}_P$ be the annihilator of the $\mathcal{O}$-module $\overline{\mathcal{O}} / \mathcal{O}$, i.e. \[ \mathcal{C}_P=\mathcal{C}_{\overline{\mathcal{O}}/\mathcal{O}} = \{\varphi \in \overline{\mathcal{O}} \mid \varphi \overline{\mathcal{O}} \subseteq \mathcal{O} \}. \] This set is the largest ideal in $\mathcal{O}$ which is also an ideal in $\overline{\mathcal{O}}$, and it is called the \emph{conductor ideal} of the extension $\overline{\mathcal{O}} / \mathcal{O}$. Since $\overline{\mathcal{O}}$ is a semilocal Dedekind domain with maximal ideals $\overline{\mathfrak{m}}_{Q_1}, \ldots , \overline{\mathfrak{m}}_{Q_d}$ (where $Q_i$ denote the rational branches of $\chi$ at $P$), the conductor ideal has a unique factorisation \[ \mathcal{C}_P=\prod_{i=1}^d \overline{\mathfrak{m}}_{Q_i}^{d_{Q_i}} \] as ideal in $\overline{\mathcal{O}}$. The exponents $d_{Q_i}$ can be easily computed by means of the Dedekind formula (see Zariski \cite{za}): if $(x_i(t_i),y_i(t_i))$ is a rational parametrisation of $Q_i$ one has \begin{equation} \label{eqn:dedekind} d_{Q_i} =\mathrm{ord}_{t_{Q_i}} \Bigg(\frac{f_Y(X(t_{Q_i}),Y(t_{Q_i}))}{X^{\prime}(t_{Q_i})} \Bigg)= \mathrm{ord}_{t_{Q_i}} \Bigg(\frac{f_X(X(t_{Q_i}),Y(t_{Q_i}))}{Y^{\prime}(t_{Q_i})} \Bigg). \end{equation}
Let $n:\widetilde{\chi} \to \chi$ ne the normalisation morphism of $\chi$. Notice that $\widetilde{\chi}$ is nonsingular with $\mathbb{F}(\widetilde{\chi})=\mathbb{F}(\chi)$. Let $\mathcal{O}=\mathcal{O}_{\chi, P}$ and $\overline{\mathcal{O}}$ its normalisation. Let $Q \in n^{-1}(\{P\})$. Since $Q$ is nonsingular, it is $\mathcal{C}_P \cdot \mathcal{O} = \mathfrak{m}_Q^{d_Q}$ for a nonnegative integer $d_Q$. We define the effective divisor \[ \mathcal{A} := \sum_P \sum_{Q \in n^{-1}(\{P\})} d_Q \cdot Q \] which is called the \emph{adjunction divisor} of $\chi$. Notice that $\mathcal{A}$ is a well-defined divisor on $\widetilde{\chi}$ (in fact, if $P$ is nonsingular, there is only one $Q \in n^{-1}(\{P\})$ and in this case $d_Q=0$). This implies in particular that the support of $\mathcal{A}$ consists of all rational branches of $\chi$ at singular points. Moreover, by setting $n_P:=\dim_{\mathbb{F}}\overline{\mathcal{O}}/\mathcal{C}_P$ we have \[ n_P=\sum_{Q \in n^{-1}(\{P\})} d_Q \] for every $P$ on $\chi$. Therefore $\deg \mathcal{A} = \sum_{P \in \chi} n_P$ (cf. Arbarello et al. \cite[Appendix A]{arb}; also Tsfasman and Vl\u adu\c t \cite[2.5.2]{tsfas}).
Let $F:=F(X_0,X_1,X_2)$ be a homogeneous (absolutely irreducible) polynomial of degree $d$ over $\mathbb{F}$ which defines the projective plane curve $\chi$. Let $\mathcal{F}_d$ be the set of all homogeneous polynomials in three variables of degree $d$. Let $i:\chi \to \mathbb{P}^2_{\mathbb{F}}$ be the embedding of $\chi$ into the projective plane and $\textbf{N}: \widetilde{\chi} \to \mathbb{P}_{\mathbb{F}}^{2}$ be the natural morphism given by $\textbf{N} = i \circ n$. A rational divisor $D$ on $\mathbb{P}^2_{\mathbb{F}}$ such that $\chi$ is not contained in $\mathrm{supp}(D)$ is called an \emph{adjoint divisor} of $\chi$ if the pull-back divisor $\textbf{N}^{\ast} D$ satisfies $\mathrm{supp}(\mathcal{A}) \subseteq \mathrm{supp}(\textbf{N}^{\ast} D)$ for $\mathcal{A}$ the adjunction divisor of $\chi$. We can consider the analogous notion at the level of homogeneous polynomials. For $H \in \mathcal{F}_d$ with $F \nmid H$ one can consider the pull-back $\textbf{N}^{\ast} H$, which is actually the intersection divisor on $\widetilde{\chi}$ cut out by the plane curve defined by $H$ on $\mathbb{P}^2_{\mathbb{F}}$, namely \begin{equation}\label{eq:ad1} \textbf{N}^{\ast} H = \sum_{Q \in \widetilde{\chi}} r_Q \cdot Q, \end{equation} with $r_Q=\mathrm{ord}_Q (h)$ for $h \in \mathcal{O}_{\chi, n(Q)}$ being a local equation of the curve defined by $H$ at the point $n(Q)$. If $H$ satisfies additionally $\textbf{N}^{\ast} D \geq \mathcal{A}$, then it will be called an \emph{adjoint form} on $\chi$, and the curve defined by $H$ will be called an \emph{adjoint curve} to $\chi$. Notice that adjoint curves there always exists (take for instance the polars of the curve, cf. Brieskorn and Kn\"orrer \cite{brie}, p. 599).
Let $d:=\deg \chi$. The differentials gob ally defined at $\chi$ are in one to one correspondence with adjoint curves on $\widetilde{\chi}$ of degree $d-3$:
\begin{teo}\label{teo:1} Let $\mathcal{A}_n$ be the set of adjoints of degree $n$ of the curve $\chi$ embedded in $\mathbb{P}_{\ff}^{2}$, let $K_{\widetilde{\chi}}$ be a canonical divisor on $\widetilde{\chi}$ and set $d := \deg \chi$. For $n=d-3$ there is an $\mathbb{F}$-isomorphism of complete linear systems \[ \begin{array}{ccc}
\mathcal{A}_n& \longrightarrow& |K_{\widetilde{\chi}}|\\ D& \longmapsto& \textbf{N}^{\ast} D - \mathcal{A}. \end{array} \] \end{teo}
The key idea is to realise that the map is injective since $n=d-3<d$; see Gorenstein \cite[p. 433]{gor} or \cite[2.2.1]{tsfas} for further details.
In practice, we know a priori the equation of the plane curve $\chi$ (defined over a perfect field $\mathbb{F}$) given by the form $F \in \mathcal{F}_d$ and the data of a certain divisor $R=\sum_{Q^{\prime}}r_{Q^{\prime}} \cdot Q^{\prime}$ (for finitely many points $Q^{\prime}$ on $\widetilde{\chi}$) which is effective and rational over $\mathbb{F}$, involving a finite number of rational branches $Q$ of $\chi$ and their corresponding coefficients. Moreover, we are able to compute the adjunction divisor of $\chi$, $\mathcal{A}=\sum_Q d_Q \cdot Q$. Our aim is to interprete the condition of being an adjoint form---called \emph{adjoint condition}---given by (\ref{eq:ad1}) in terms of equations. More generally, we are interesting in finding some \emph{adjoint} form $H \in \mathbb{F}[X_0,X_1,X_2]$ satisfying \begin{equation} \label{eq:ad2} \textbf{N}^{\ast} H \ge \mathcal{A} + R. \end{equation} This process is known as \emph{computing adjoint forms with base conditions} (see \cite{cafa}, \S 4).
First of all, we choose a positive integer $\widetilde{n} \in \mathbb{N}$ in such a way that there exists an adjoint of degree $\widetilde{n}$ not being a multiple of $F$ and satisfying (\ref{eq:ad2}). A bound for $\widetilde{n}$ can be found in Hach\'e \cite{hachetesis}. Take then also a form $H \in \mathcal{F}_{\widetilde{n}}$ in a general way, what is nothing else but taking a homogeneous polynomial in three variables of degree $\widetilde{n}$ with its coefficients as indeterminates (that is, $H(X_0,X_1,X_2)=\sum_{i+j+k=\widetilde{n}} \lambda_{i,j,k} X_0^iX_1^jX_2^k$). Second we compute a rational primitive parametrization $\big( X(t),Y(t) \big)$ of $\chi$ at every branch involved in the support of the adjunction divisor $\mathcal{A}$ and the divisor $R$. Next we get the support of the adjunction divisor $\mathcal{A}$ from the conductor ideal via the Dedekind formula (\ref{eqn:dedekind}). Last we consider the coefficient $r_Q$ of the divisor $R$ at $Q$, and thus the local condition at $Q$ imposed on $H$ by (\ref{eq:ad2}) is given by \begin{equation} \label{eq:dqrq} \mathrm{ord}_t h(X(t),Y(t)) \ge d_Q + r_Q, \end{equation} with $h$ the local affine equation of $H$ at $Q$.
The inequality (\ref{eq:dqrq}) expresses a linear condition (given by a linear inequation) on the coefficients $\lambda_{i,j,k}$ of $h$.
The required linear equations are a consequence of the vanishing of those terms, and when $Q$ takes all the possible values, i.e., all the possible branches of the singular points on $\chi$ and of the support of $R$, we get the linear equations globally imposed by the condition (\ref{eq:ad2}). An easy reasoning reveals that the number of such adjoint conditions is equal to \begin{equation} \label{eq:nadj} \frac{1}{2} \deg \mathcal{A} + \deg R = \frac{1}{2} \sum_{P \in \chi} n_P + \deg R = \sum_{P \in \chi} \delta_P + \deg R. \end{equation}
\begin{ejem} Let $\chi$ be the projective plane curve over the finite field of two elements $\mathbb{F}_2$ given by the equation $F(X,Y,Z)=X^3-Y^2Z$. The only singular point of $\chi$ is $P_1=[0:0:1]$. Let be the point $P_2=[0:1:0]$ and the effective divisor $R=0P_1+P_2$. The adjunction divisor of $\chi$ is $\mathcal{A}=2 P_1$. A local equation of $\chi$ with $P_1=(0,0)$ is $f(x,y)=x^3-y^2$. A parametrization of $\chi$ at $P_1$ is given by \begin{displaymath}
\begin{array}{l}
X_1(t_1)=t_1^2\\
Y_1(t_1)=t_1^{3} \end{array} \end{displaymath} Take a form $H \in \mathcal{F}_{4-3=1}$, $H(X,Y,Z)=aX+bY+cZ$. First we want to express the adjoint conditions in terms of the coefficients \[ \textbf{N}^{\ast} H \ge \mathcal{A} + R = 2P_1+P_2. \] To this end we consider first a local equation for $H$ at $P_1$, namely \[ h(x,y)=H(x,y,1)= ax+by+c. \] Then $h(X_1(t_1),Y_1(t_1))=h(t^2,t^3)=at_1^2+bt_1^3+c$. So if we wish to have \[ \mathrm{ord}_{t_1}(h(X_1(t_1),Y_1(t_1)))=\mathrm{ord}_{t_1}(bt_1^3+at_1^2+c) \ge 2 \] (since $2$ is the coefficient for $P_1$ and $(X_1(t_1),Y_1(t_1))$ is a parametrization at $P_1$), then this is possible if and only if $c=0$. Thus $c=0$ is one of the required \emph{linear} adjoint conditions.
Now consider a local equation for $\chi$ at $P_2$. This is $f^{\prime}(x,z)=F(x,1,z)=x^3-z$, and admits a parametrization \begin{displaymath}
\begin{array}{l}
X_2(t_2)=t_2\\
Z_2(t_2)=t_2^{3} \end{array} \end{displaymath} Consider the local equation for $H$ at $P_2$ $$ h^{\prime}(x,z)=H(x,1,z)=ax+b+cz. $$ Hence the adjoint conditions imposed by $\textbf{N}^{\ast} H \ge \mathcal{A} + R=2 P_1 + P_2$ at $P_2$ come from considering $h^{\prime}(X_2(t_2),Z_2(t_2))=h^{\prime}(t_2,t_2^3)=at_2+b+ct_2^3$ and they impose $$ \mathrm{ord}_{t_2}(h^{\prime}(X_2(t_2),Z_2(t_2)))=\mathrm{ord}_{t_2}(ct_2^3+at_2+b) \ge 1. $$ This inequality holds whenever $b=0$. Hence $b=0$ is another \emph{linear} equation taking part in the set of adjoint conditions contained in $\textbf{N}^{\ast} H \ge \mathcal{A} +R$. We have obtained two adjoint conditions, as we had hoped by (\ref{eq:nadj}), since $\frac{1}{2}\deg \mathcal{A} + \deg R = \frac{1}{2} \cdot 2 + 1 =2$. \end{ejem}
We conclude this section with two remarkable results. Let $\chi$ be an absolutely irreducible projective plane curve defined over a perfect field $\mathbb{F}$ and given by an equation $F(X_0,X_1,X_2)=0 $, where $F \in \mathcal{F}_d$. One application of the adjoint forms is the following result, due to Max Noether (he stated it of course not in this way; our version may be found in Hach\'e and Le Brigand \cite{hache}, Theorem 4.2, and Le Brigand and Risler \cite{lebri}, \S 3.1): \begin{teo}[Max Noether] Let $\chi, \chi^{\prime}$ be curves as above given by homogeneous equations $F(X_0,X_1,X_2)=0$ and $G(X_0,X_1,X_2)=0 $ respectively and such that $\chi^{\prime}$ does not contain $\chi$ as a component. Then, if we consider another such a curve given by $H(X_0,X_1,X_2)=0$ with $\textbf{N}^{\ast} H \ge \mathcal{A} + \textbf{N}^{\ast} G$ (where $\mathcal{A}$ is the adjunction divisor on $\chi$), there exist forms $A,B$ with coefficients in $\mathbb{F}$ such that $H=AF+BG$. \end{teo}
This theorem has great importance, and, for instance, allows us to prove the Brill-Noether theorem, which gives a basis for the vector spaces $\mathcal{L}(D)$. Readers are referred to \cite{hache}, Theorem 4.4, for further details. A short remark about notation is needed. For any non effective divisor $D$ we will write $D=D_{+}-D_{-}$ with $D_{+}$ and $D_{-}$ effective divisors of disjoint support.
\begin{teo}[Brill-Noether] \label{teo:br} Let $\chi$ be an adjoint curve as above with normalization $\widetilde{\chi}$. Let $\mathcal{A}$ be its adjunction divisor and let $D$ be a divisor on $\widetilde{\chi}$ rational over $\mathbb{F}$. Moreover, consider a form $H_0 \in \mathcal{F}_{\widetilde{n}}$ defined over $\mathbb{F}$, not divisible by $F$ and satisfying $\textbf{N}^{\ast} H_0 \ge \mathcal{A} + D_{+}$. Then \[ \mathcal{L}(D)=\Bigg \{\frac{h}{h_0} \mid H \in \mathcal{F}_{\widetilde{n}}, F \nmid H \ \mathrm{and} \ \textbf{N}^{\ast} H + D \ge \textbf{N}^{\ast} H_0 \Bigg\} \cup \{0\}, \] where $h,h_0 \in \mathbb{F} (\chi)$ denote respectively the rational functions $H,H_0$ restricted on $\chi$. \end{teo}
\begin{rem} \label{rem:br} Such a form $H_0 \in \mathcal{F}_{\widetilde{n}}$ exists whenever $\widetilde{n} > \max \bigg \{d-1, \frac{d-3}{2}+\frac{\deg (\mathcal{A} + D_+)}{d} \bigg \}$ (see Hach\'e and Le Brigand \cite{hache} for details). \end{rem}
\section{The Weierstra\ss~semigroup at several points}
Let $\chi$ be an absolutely irreducible projective algebraic plane curve defined over a perfect field $\mathbb{F}$. Let $\underline{P}$ denote a set of $r$ different points $P_1,\ldots,P_r$ on $\chi$. Furthermore, the perfect field $\mathbb{F}$ must have cardinality greater or equal to $r$: $\sharp \mathbb{F} \ge r$. Let $\widetilde{\chi}$ be the normalization of $\chi$.
Our purpose is to compute the dimensions of the so-called \emph{Riemann-Roch quotients}: \[ 0 \le \dim_{\mathbb{F}} \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\underline{1}) \underline{P})} \le r \] by choosing functions in $\mathcal{L}(\underline{mP})=\mathcal{L}(m_1P_1+\ldots+m_rP_r)$ but not in $\mathcal{L}((\underline{m}-\underline{1}) \underline{P})=\mathcal{L}((m_1-1)P_1+ \ldots + (m_r-1)P_r)$, that is, achieving at the $P_i$ poles of order $m_i$. We are going to restrict to the case $m_i \in \mathbb{N}$, for all $i=1,\ldots,r$. Such dimensions will be determined by the previous calculus of the \emph{Riemann-Roch quotients with respect to $P_i$}: \[ 0 \le \dim_{\mathbb{F}} \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})} \le 1, \] where $\varepsilon_i$ denotes the vector in $\mathbb{N}^r$ with $1$ in the $i$-th position and $0$ in the other ones.
Summarizing, this section deals with the following topics: \begin{itemize} \item[(a)] How to compute $\dim_{\mathbb{F}} \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})}$ and an associated function belonging to this quotient vector space when such a dimension is $1$.
\item[(b)] How to compute $\dim_{\mathbb{F}} \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\underline{1}) \underline{P})}$ (deducing bounds).
\item[(c)] How to compute the Weierstras\ss~semigroup at two points. \end{itemize}
All the statements and proofs of this section can be found in \cite{cato}, \S 2.
Consider a finite set of nonsingular points $P_1, \ldots ,P_r$ on $\chi$ and a divisor $m_1P_1+ \ldots +m_rP_r$ for $m_i \in \mathbb{N}$ $\forall \ i=1,\ldots r$. We will denote $\underline{P} = \{P_1, \ldots , P_r \}$, $\underline{mP}=m_1P_1+\ldots +m_rP_r$, $\underline{m}=(m_1,\ldots,m_r)$, $\varepsilon_i=(0,\ldots,0,1,0,\ldots,0)$ and $\underline{1}=(1,\ldots,1)$.
\begin{defi} For $\underline{P} \in \chi$ we define \[ \Gamma_{\underline{P}}:= \Big \{ -(\mathrm{ord}_{P_1}(f),\ldots,\mathrm{ord}_{P_r}(f)) \mid f \in \mathbb{F}(\chi)^{\ast} \ \mathrm{regular \ at} \ \chi \setminus \underline{P} \Big\}. \] \end{defi}
Obviously $\Gamma_{\pe}$ is a subsemigroup of $(\mathbb{N},+)$. Notice that, for $\underline{mP} = m_1 P_1+m_2 P_2$, the fact that $f \in \mathcal{L}(\underline{mP})$ is equivalent to the inequalities \begin{displaymath} (\star) \left\{ \begin{array}{l} \mathrm{ord}_{P_1}(f) \ge -m_1\\ \mathrm{ord}_{P_2}(f) \ge -m_2. \end{array} \right. \end{displaymath} This means: the set of possible orders ($\star$) which can be taken by the function $f$ is represented by the shadowed area in the figure (each axis represents one of the two branches):
\begin{center}
\setlength{\unitlength}{0.7cm} \begin{picture}(-5,-3)
\put(-4,0){\line(1,0){3,2}} \put(-4,0){\line(0,1){3,2}}
\put(-4.5,3){\line(1,0){3.5}} \put(-1,-0.5){\line(0,1){3.5}}
\put(-1,0){\line(-1,1){3}} \put(-1,.5){\line(-1,1){2.5}} \put(-1,1){\line(-1,1){2}} \put(-1,1.5){\line(-1,1){1.5}} \put(-1,2){\line(-1,1){1}} \put(-1,2.5){\line(-1,1){.5}} \put(-1,-.5){\line(-1,1){3.5}}
\put(-4,-.3){0} \put(-4.4,3.2){$m_2$} \put(-.8,-.3){$m_1$} \put(-.9,3.1){$(m_1,m_2)$} \end{picture} \end{center}
\begin{defi} An element $\underline{m} \in \mathbb{N}^r$ is called a \emph{non-gap} of $\underline{P}$ if and only if $\underline{m} \in \Gamma_{\pe}$. Otherwise $\underline{m}$ is called a \emph{gap}. \end{defi}
A very important characterization for the non-gaps is given by the following (see \cite{de}, p. 629):
\begin{lema} \label{lema:felix} If $\underline{m} \in \mathbb{Z}^r$ then one has: $$ \underline{m} \in \Gamma_{\pe} \ \ \ \mathrm{if \ and \ only \ if} \ \ \ \ell(\underline{mP})=\ell((\underline{m}-\varepsilon_i)\underline{P})+1 \ \forall \ i=1,\ldots,r. $$ \end{lema}
For every $i=1,\ldots,r$ and $\underline{m}=(m_1,\ldots,m_r) \in \mathbb{N}^r$, we set \[ \nabla_i(\underline{m}):=\Big \{(n_1,\ldots,n_r) \in \Gamma_{\pe} \mid n_i=m_i \ \mathrm{and} \ n_j \le m_j \ \forall j \ne i \Big \}. \]
Then the two conditions proven to be equivalent in Lemma \ref{lema:felix} are indeed also equivalent to $\nabla_i(\underline{m})\neq 0$ for every $i \in \{1, \ldots , r\}$.
A gap $\underline{m}$ satisfying $\ell(\underline{mP})=\ell((\underline{m} -\epsilon_i)\underline{P})$ for all $i \in \{1, \ldots , r\}$ (or, equivalently, such that $\nabla_i(\underline{m})=\emptyset$ for all $i \in \{1, \ldots ,r\}$) is called \emph{pure}. It is easily seen: if $\underline{m}$ is a pure gap, then $m_i$ is a gap for $\Gamma_{P_i}$ for every $i\in \{1,\ldots ,r\}$. Furthermore, if $\underline{1} \in \Gamma_{\underline{P}}$, then there are no pure gaps. The converse does not hold, as Example \ref{ejem:2wss} will show.
A basic tool on Weierstra\ss~semigroups is the following
\begin{teo}[Weierstra\ss~gap~theorem] Let $\widetilde{\chi}$ be a curve of genus $g\geq 1$. Let $P$ be a rational branch on $\widetilde{\chi}$. Then there are $g$ gaps $\gamma_1, \ldots , \gamma_g$ such that \[ 1=\gamma_1<\ldots <\gamma_g \leq 2g-1. \] \end{teo}
\begin{prop} Let $\underline{m}=(m_1,\ldots,m_r) \in \mathbb{N}^r$. If $\underline{m}$ is a gap, then there exists a regular differential form $\omega$ on $\widetilde{\chi}$ with $(\omega) \ge \underline{m} - \varepsilon_i$ and a zero at $P_i$ of order $m_i-1$ for \emph{some} $i \in \{1,\ldots,r \}$. \end{prop}
\dem~After Riemann-Roch theorem it is clear that \begin{eqnarray*} \ell (\underline{mP}) - i (\underline{mP}) &=& m_1+m_2+\ldots + m_r + 1 - g\\ \ell ((\underline{m}-\varepsilon_i)\underline{P}) - i ((\underline{m} - \varepsilon_i)\underline{P}) &=& m_1+m_2+\ldots +m_r - 1 + 1 - g. \end{eqnarray*} By adding both equations we have \[ [\underbrace{\ell(\underline{mP})-\ell((\underline{m}-\varepsilon_i)\underline{P})}_{= \varphi(\underline{m})}]-[\underbrace{i(\underline{mP})-i((\underline{m}-\varepsilon_i)\underline{P})}_{= \psi(\underline{m})}]=1, \] for every $i=1,\ldots,r$, where $0 \le \varphi(\underline{m}) \le 1$ and $-1 \le \psi (\underline{m}) \le 0$, and therefore \[ \varphi(\underline{m})=1 \Leftrightarrow \ell(\underline{mP})-\ell((\underline{m}-\varepsilon_i)\underline{P})=1 \Leftrightarrow \ell(\underline{mP})=\ell((\underline{m}-\varepsilon_i)\underline{P})+1 \Leftrightarrow \underline{m} \in \Gamma_{\pe} \Leftrightarrow \psi(\underline{m})=0. \] Hence if $\underline{m} \notin \Gamma_{\pe}$ then $\dim_{\mathbb{F}} \Big ( \frac{\Omega((\underline{m}-\varepsilon_i)\underline{P})}{\Omega(\underline{mP})} \Big)=1$ and so there exists a regular differential form $\omega$ on $\widetilde{\chi}$ with $(\omega) \ge \underline{m} - \varepsilon_i$ and $\mathrm{ord}_{P_i}(\omega)=m_i-1$ for \emph{some} $i \in \{1,\ldots,r \}$. \qed
\begin{prop} Let $\chi$ be a plane curve of genus $g$, let $\underline{P}$ be a set of $r$ closed points on $\chi$ and set $\underline{m}=(m_1,\ldots,m_r) \in \mathbb{N}^r$. If $\underline{m}$ is a gap, then $m_1+\ldots + m_r <2g$. \end{prop}
\dem~Denote by $D_{2g,\underline{P}}$ a divisor with degree $2g$ and support $\underline{P}$, and by $D_{2g-1,\underline{P}}$ a divisor with degree $2g-1$ and support $\underline{P}$. If $m_1+\ldots +m_r \ge 2g-1$ then $m_1+\ldots + m_r \ge 0$ as a consequence of Riemann-Roch, and for every 4$i=1,\ldots,r$ \[ \ell(D_{2g,\underline{P}})=2g+1-g=g+1 \ne g=2g-1+1-g=\ell(D_{2g-1,\underline{P}}), \] which implies that $\underline{m}$ is a non-gap, i.e., $\underline{m} \in \Gamma_{\pe}$. So, if $\underline{m} \notin \Gamma_{\pe}$, then $m_1+\ldots +m_r < 2g$. \qed
Notice that, for divisors of the form $\underline{mP} = m_1 P_1 + m_2 P_2$, the plane $\mathbb{N} \times \mathbb{N}$ is divided in three parts by the line $m_1+m_2=2g$ as in the figure, namely
\begin{center}
\setlength{\unitlength}{0.9cm} \begin{picture}(-5,-3)
\put(-4,0){\line(1,0){4}} \put(-4,0){\line(0,1){4}}
\put(-.5,0){\line(-1,1){3.5}}
\put(-4.3,-.4){0} \put(-4.6,3.5){$2g$} \put(-.5,-.4){$2g$}
\put(-1.7,3){$A$} \put(-2.5,2){$B$} \put(-3,1.01){$C$} \end{picture} \end{center}
$$ A:=\{(m_1,m_2) \mid m_1+m_2>2g, \ m_1>0, \ m_2>0 \} $$ $$ B:=\{(m_1,m_2) \mid m_1+m_2=2g, \ m_1>0, \ m_2>0 \} \cup \{(m_1,0), \ m_1>2g \} \cup \{(0,m_2), \ m_2>2g \} $$ $$ C:=\{(m_1,m_2) \mid m_1+m_2<2g, \ m_1 \ge 0, \ m_2 \ge 0 \}. $$
All the points lying on $A$ and $B$ correspond to values in $\Gamma_{\pe}$, but nothing can be a priory said about the points on $C$.
\subsection{Dimension of the Riemann-Roch quotients with respect to $P_i$ and associated functions}
We start by computing the dimension of the Riemann-roch quotients associated to the points $P_i$.
\begin{prop} \label{prop:2} Let $\underline{m} \in \mathbb{N}^r$ such that $\sum_{i=1}^r m_i <2g$. Then, for $i \in \{1, \ldots, r\}$ we have: \begin{itemize} \item[a)] $\mathrm{dim}_{\mathbb{F}} [\Omega((\underline{m} - \varepsilon_i)\underline{P}) \setminus \Omega (\underline{mP})]=1$ if and only if $\exists$ a homogeneous polynomial $H_0$ of degree $d-3$ with $\textbf{N}^{\ast} H_0 \ge \mathcal{A} + (\underline{m}-\varepsilon_i)\underline{P}$ such that $P_i$ is not in the support of the effective divisor $\textbf{N}^{\ast} H_0 - \mathcal{A} - (\underline{m}-\varepsilon_i)\underline{P}$.
\item[b)] $\exists$ $\underline{m}^{\prime} \ge \underline{m}$ with $\mathrm{dim}_{\mathbb{F}} [\Omega((\underline{m}^{\prime} - \varepsilon_i)\underline{P}) \setminus \Omega (\underline{m}^{\prime} \underline{P})]=1$ if and only if $\exists$ a homogeneous polynomial $H_0$ of degree $d-3$ such that $\textbf{N}^{\ast} H_0 \ge \mathcal{A} + (\underline{m}-\varepsilon_i)\underline{P}$. \end{itemize} \end{prop}
\dem~\begin{itemize} \item[a)] If $\mathrm{dim}_{\mathbb{F}} [\Omega((\underline{m} - \varepsilon_i)\underline{P}) \setminus \Omega (\underline{m} \underline{P})]=1$, then this is equivalent to $\underline{m} \notin \Gamma_{\pe}$ and also to the existence of an index $i$ with $\ell(\underline{mP})=\ell((\underline{m}-\varepsilon_i)\underline{P})$, or, in other words, to the existence of an index $i$ with $i((\underline{m}-\varepsilon_i)\underline{P})=i(\underline{mP})+1$; that is, there exists a homogeneous polynomial $H_0$ of degree $d-3$ such that $\textbf{N}^{\ast} H_0 \ge \mathcal{A} + (\underline{m}-\varepsilon_i)\underline{P}$.
\item[b)] If there is $\underline{m}^{\prime} \ge \underline{m}$ with $\mathrm{dim}_{\mathbb{F}} [\Omega((\underline{m}^{\prime} - \varepsilon_i)\underline{P}) \setminus \Omega (\underline{m}^{\prime} \underline{P})]=1$ then there exists an adjoint $H_0$ of degree $d-3$ whose divisor is $\ge (\underline{m}^{\prime}-\varepsilon_i)\underline{P}$ outside $\mathcal{A}$, i.e., $\textbf{N}^{\ast} H_0 - \mathcal{A} \ge (\underline{m}^{\prime}-\varepsilon_i)\underline{P} \ge (\underline{m}-\varepsilon_i)\underline{P}$. Conversely, if there is $H_0$ of degree $d-3$ with $\textbf{N}^{\ast} H_0 \ge \mathcal{A} + (\underline{m}-\varepsilon_i)\underline{P}$ then there exists $\omega \ne 0$ differential form such that $(\omega)=\textbf{N}^{\ast} H_0-\mathcal{A} \ge (\underline{m}-\varepsilon_i)\underline{P}$. Assume that $\underline{m}^{\prime}-\varepsilon_i$ are the orders of the zeros of $\omega$ at $\underline{P}$. Thus, $\underline{m}^{\prime}-\varepsilon_i \ge \underline{m}-\varepsilon_i$, what implies $\underline{m}^{\prime} \ge \underline{m}$ and $\omega \in \Omega((\underline{m} - \varepsilon_i)\underline{P}) \setminus \Omega (\underline{mP})$.
\qed \end{itemize}
The following corollary yields a way to relate the adjunction theory and the computation of the Weierstra\ss~semigroup at several points:
\begin{cor} \label{cor:one} Let $\underline{m} \in \mathbb{N}^r$ with $\sum_{i=1}^{r}m_i <2g$. For a given form $H$ of degree $d-3$ and $i \in \{1,\ldots,r\}$ there exists a condition imposed by the inequality $\textbf{N}^{\ast} H \ge \mathcal{A} + \underline{mP}$ at $P_i$ which is independent of the conditions imposed by $\textbf{N}^{\ast} H \ge \mathcal{A} +(\underline{m} - \varepsilon_i)\underline{P}$ at $P_i$ if and only if \[ \dim_{\mathbb{F}} \frac{\Omega ((\underline{m} - \varepsilon_i)\underline{P})}{\Omega (\underline{mP})}=1. \] \end{cor}
The second step is the computation of the rational functions associated to the nongaps of the Weierstra\ss~semigroup. Note that, if $\dim_{\mathbb{F}}\frac{\Omega ((\underline{m}-\varepsilon_i)\underline{P})}{\Omega (\underline{mP})}=0$, then $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L} ((\underline{m}-\varepsilon_i)\underline{P})}=1$ and so there is a rational function $f_{i,\underline{m}} \in \frac{\mathcal{L}(\underline{mP})}{\mathcal{L} ((\underline{m}-\varepsilon_i)\underline{P})}$ with a pole of order $m_i$ at $P_i$. In order to compute such a function, we base on Brill-Noether Theorem \ref{teo:br}:
\begin{algo} \label{algo:no} Preserving notations as above, we obtain a function $f_{i,\underline{m}} \in \frac{\mathcal{L}(\underline{mP})}{\mathcal{L} ((\underline{m}-\varepsilon_i)\underline{P})}$ with a pole of order $m_i$ at $P_i$ by following these steps:
\begin{itemize} \item[-] Compute a homogeneous polynomial $H_0$ not divisible by $F$ of large enough degree $n$ in the sense of Remark \ref{rem:br} satisfying $\textbf{N}^{\ast} H_0 \ge \mathcal{A} + \underline{mP}$.
\item[-] Calculate $R_{\underline{m}}$, which is the effective divisor such that $\textbf{N}^{\ast} H_0 = \mathcal{A} + \underline{mP} + R_{\underline{m}}$. Obviously $R_{\underline{m}-\varepsilon_i}=R_{\underline{m}}+P_i$.
\item[-] Find a form $H_{\underline{m}}$ of degree $n$ not divisible by $F$ such that $\textbf{N}^{\ast} H_{\underline{m}} \ge R_{\underline{m}}$ but not satisfying $\textbf{N}^{\ast} H_{\underline{m}} \ge R_{\underline{m} - \varepsilon_i}=R_{\underline{m}}+P_i$.
\item[-] Output: $f_{i,\underline{m}}=\frac{h_{\underline{m}}}{h_0}$, where $h_{\underline{m}},h_0$ are the restricted forms on $\chi$ for $H_{\underline{m}}$ and $H_0$ respectively. \end{itemize} \end{algo}
\begin{ejem} Let $\chi$ be the curve given by the equation $F(X,Y,Z)=X^3Z+X^4+Y^3Z+YZ^3$ and consider the points $P_1=[0:1:1]$ and $P_2=[0:1:0]$ and $\underline{m}=(1,2)$. We want to compute $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m} - \varepsilon_1)\underline{P})}$ and $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}(\underline{m}-\varepsilon_2)\underline{P}}$.
A local parametrization of $F$ at $P_1$ is given by \begin{displaymath}
\begin{array}{l}
X_1(t_1)=t_1\\
Y_1(t_1)=t_1^{3}+t_1^{4}+t_1^9+t_1^{10}+t_1^{11}+t_1^{12}+\ldots \end{array} \end{displaymath}
with local equation $f_1(x,y)=y^2+y^3+x^3+x^4$. Analogously at $P_2$ \begin{displaymath}
\begin{array}{l}
X_2(t_2)=t_2\\
Z_2(t_2)=t_2^{4}+t_2^7+t_2^{10}+t_2^{12}+t_2^{13}+t_2^{16}+\ldots \end{array} \end{displaymath} with local equation $f_2(x,z)=z+z^3+x^3z+x^4$.
First, we calculate the adjunction divisor: this is $\mathcal{A}=2P_1$.
Search a form $H$ of degree $d-3=4-3=1$, that is, a linear form $H(X,Y,Z)=aX+bY+cZ$. At $P_1$ $H$ admits the equation $h_1(x,y)=H(X,Y-1,1)=ax+by+b+c$. At $P_2$ $H$ admits the equation $h_2(x,z)=H(X,1,Z)=ax+b+cz$. Then:
\begin{displaymath}
\begin{array}{l} h_1(X_1(t_1),Y_1(t_1))=at_1+b(t_1^3+t_1^4+t_1^9+\ldots)+b+c=(b+c)+at_1+bt_1^3+bt_1^4+bt_1^9+\ldots.\\ h_2(X_2(t_2),Z_2(t_2))=b+at_2+ct_2^4 +ct_2^7+ct_2^{10}+\ldots. \end{array} \end{displaymath}
In order to compute $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m} - \varepsilon_1)\underline{P})}$ we impose the systems of equations with the adjunction conditions at $P_1$:
\begin{displaymath}
\left\{ \begin{array}{l}
\textbf{N}^{\ast} H \ge \mathcal{A} + (m_1-1)P_1=2P_1\\
\textbf{N}^{\ast} H \ge \mathcal{A} + m_1 P_1 = 3P_1, \end{array} \right. \end{displaymath} or, in other words
\begin{displaymath}
\left\{ \begin{array}{l}
\mathrm{ord}_{t_1}(h_1(X_1(t_1),Y_1(t_1))) \ge 2 \Longrightarrow b+c=a=0\\
\mathrm{ord}_{t_1}(h_1(X_1(t_1),Y_1(t_1))) \ge 3 \Longrightarrow b+c=a=0 \end{array} \right. \end{displaymath}
So the second system does not add any independent condition to the first one; this means, by Corollary \ref{cor:one}, that $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m} - \varepsilon_1)\underline{P})}=1$.
In order to compute $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m} - \varepsilon_2)\underline{P})}$ the systems of equations with the adjunction conditions at $P_2$ are \begin{displaymath}
\left\{ \begin{array}{l}
\textbf{N}^{\ast} H \ge (m_2-1)P_2=P_2\\
\textbf{N}^{\ast} H \ge m_2 P_2=2P_2, \end{array} \right. \end{displaymath} that is, \begin{displaymath}
\left\{ \begin{array}{l}
\mathrm{ord}_{t_2}(h_2(X_2(t_2),Z_2(t_2))) \ge 1 \Rightarrow
b=0\\
\mathrm{ord}_{t_2}(h_2(X_2(t_2),Z_2(t_2))) \ge 2 \Rightarrow a=0=b. \end{array} \right. \end{displaymath}
Notice that, in this case, the adjunction divisor does not appear in the inequalities since $P_2$ does not belong to its support.
The second system adds one independent condition to the first one
and this means that $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m} - \varepsilon_2)\underline{P})}=0$ again by Corollary \ref{cor:one}. \qed \end{ejem}
\begin{ejem} Consider the previous example but with $\underline{m}=(4,6)$. As $m_1+m_2=4+6=10 >2g$, we know without calculations $\underline{m} \in \Gamma_{\pe}$, i.e., that $\dim_{\mathbb{F}}\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m} - \varepsilon_i)\underline{P})}=1$ for $i=1,2$. So we will look for the corresponding functions $f_{i,\underline{m}}$ with poles at $P_i$ of order $m_i$ for $i=1,2$.
First of all, we search $\widetilde{n}>\max \Big \{3,\frac{2}{4}+\frac{12}{4} \Big \}=\max \Big\{3,\frac{14}{4} \Big\}$. Let us take $\widetilde{n}=5$.
Then we look for a form $H_0$ of degree $\widetilde{n}=5$ such that $\textbf{N}^{\ast} H_0 \ge \mathcal{A} + \underline{mP}$. In this case $\textbf{N}^{\ast} H_0 \ge 4P_1+6P_2+2P_3$, since $\mathcal{A}=2P_3$, where $P_3=[0:0:1]$. After some computations we find $H_0=X^4Z$.
In order to compute $\textbf{N}^{\ast} H_0$, we have to calculate $\textbf{N}^{\ast} (X)$, $\textbf{N}^{\ast} (Y)$ and $\textbf{N}^{\ast} (Z)$. Intersection points between $\{F=0\}$ and $\{X=0\}$ are $P_1=[0:1:1]$, $P_2=[0:1:0]$ and $P_3=[0:0:1]$ with multiplicities $1$, $1$ and $2$ respectively. So $\textbf{N}^{\ast} (X)=P_1+P_2+2P_3$. Intersection points between $\{F=0\}$ and $\{Y=0\}$ are $P_3=[0:0:1]$ and $P_4=[1:0:1]$ such that $\textbf{N}^{\ast} (Y)=3P_3+P_4$. And the only point lying in the intersection between $\{F=0 \}$ and $\{Z=0 \}$ is $P_2=[0:1:0]$ with multiplicity $4$, therefore $\textbf{N}^{\ast} (Z)=4P_2$.
Thus $\textbf{N}^{\ast} H_0 = 4 \textbf{N}^{\ast} (X)+\textbf{N}^{\ast} (Z)=4P_1+8P_2+8P_3$. The residue divisor $R_{\underline{m}}=\textbf{N}^{\ast} H_0 - \mathcal{A} -\underline{mP} = 2P_2+6P_3$. Following the algorithm described above, we have to find a form $H_{\varepsilon_1}$ such that $\textbf{N}^{\ast} H_{\varepsilon_1} \ge R_{\underline{m}}$ but $\textbf{N}^{\ast} H_{\varepsilon_1} \ngeq R_{\underline{m}} + P_1$. For instance we take $H_{\varepsilon_1}=Y^2Z^3$, since
\begin{displaymath}
\begin{array}{l} \textbf{N}^{\ast} H_{\varepsilon_1}=12P_2+6P_3+2P_4 \ge 2P_2 +6P_3\\ \textbf{N}^{\ast} H_{\varepsilon_1} \ngeq P_1+2P_2+6P_3. \end{array} \end{displaymath}
So $f_{1,\underline{m}}=\frac{Y^2Z^3}{X^4Z}=\frac{Y^2Z^2}{X^4} \in \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_1)\underline{P})}$.
Now we have to find a form $H_{\varepsilon_2}$ such that $\textbf{N}^{\ast} H_{\varepsilon_2} \ge R_{\underline{m}}$ but $\textbf{N}^{\ast} H_{\varepsilon_2} \ngeq R_{\underline{m}} + P_2$. For instance we take $H_{\varepsilon_2}=X^2Y^3$, since
\begin{displaymath}
\begin{array}{l} \textbf{N}^{\ast} H_{\varepsilon_2}=2P_1+2P_2+13P_3+3P_4 \ge 2P_2 +6P_3\\ \textbf{N}^{\ast} H_{\varepsilon_2} \ngeq 3P_2+6P_3. \end{array} \end{displaymath}
Thus $f_{2,\underline{m}}=\frac{X^2Y^3}{X^4Z}=\frac{Y^3}{X^2Z} \in \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_2)\underline{P})}$.\qed \end{ejem}
\begin{algo} \label{algo:si} There is an alternative way of calculating these functions $f_{i,\underline{m}}$, computationally more effective:
\begin{enumerate} \item Take a basis of $\mathcal{L}(\underline{mP})$, say $\{h_1, \ldots ,h_s\}$. \item Calculate the pole orders at $P_i$, $\{-\mathrm{ord}_{P_i}(h_1),\ldots, -\mathrm{ord}_{P_i}(h_s) \}$. \item Order these pole orders increasing, in such a way that $-\mathrm{ord}_{P_i}(h_s)=m_i$. We can assume this, as otherwise, if $-\mathrm{ord}_{P_i}(h_s)=k_i > m_i$ we can replace $m_i$ by $k_i$, since $\mathcal{L}(m_1 P_1 + \ldots + m_i P_i + \ldots +m_r P_r)=\mathcal{L}(m_1 P_1 + \ldots + k_i P_i + \ldots + m_r P_r)$. \item The function $h_s$ has pole order $m_i$ at $P_i$, but other functions could also have the same property. So, for any $h_j$ satisfying $-\mathrm{ord}_{P_i}(h_j)=m_i$, there exists $\lambda_j \ne 0$ in $\mathbb{F}$ such that $h_j=\lambda_j h_s$, that is, $-\mathrm{ord}_{P_i}(h_j-\lambda_j h_s)<m_i$. So we change $h_j$ by $g_j:=h_j-\lambda_j h_s$, and $g_k:=h_k$ for $k \ne j$. \item Now we have a set of functions $\{g_1, \ldots, g_s \}$ where $g_s=f_{i,\underline{m}}$, and $\{g_1,\ldots,g_{s-1}\}$ is a basis for the vector space $\mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})$. \end{enumerate} \end{algo}
\begin{ejem} We present a worked example in \textsc{Singular} for computing functions as above. First we import the library \verb"brnoeth.lib" and another one \verb"several.lib" in which we have programmed the procedure \verb"ordRF" that computes the pole orders of a rational function:{\small \begin{verbatim} > LIB "brnoeth.lib"; > LIB "several.lib"; > int plevel=printlevel; > printlevel=-1; \end{verbatim}} We define the ring and the curve:{\small \begin{verbatim} > ring s=2,(x,y),lp; > list C=Adj_div(x3y+y3+x);
==>The genus of the curve is 3 \end{verbatim}}
The list of computed places is{\small \begin{verbatim} > C=NSplaces(1,C); > C[3];
-->[1]:
--> 1,1
-->[2]:
--> 1,2
-->[3]:
--> 1,3 \end{verbatim}} The base point of the first place of degree $1$ is, in homogeneous coordinates:{\small \begin{verbatim} > def SS=C[5][1][1]; > setring SS; > POINTS[1];
-->[1]:
--> 0
-->[2]:
--> 1
-->[3]:
--> 0 > setring s; \end{verbatim}} We define the divisor \verb"G=4C[3][1]+4C[3][3]":{\small \begin{verbatim} > intvec G=4,0,4; > def R=C[1][2]; \end{verbatim}} A basis \verb"LG" for $\mathcal{L}(\underline{mP})$ is supplied by the Brill-Noether algorithm:{\small \begin{verbatim} > setring R; > list LG=BrillNoether(G,C);
-->Vector basis successfully computed > int lG=size(LG); \end{verbatim}} The pole orders for the rational functions in \verb"LG" are{\small \begin{verbatim} > int j; > intvec h; > for (j=1;j<=lG;j=j+1){ . h[j]=ordRF(LG[j],SS,1)[1]; . } > h;
-->0,-1,2,-2,-3,-4 \end{verbatim}}
And the desired rational function is{\small \begin{verbatim} > LG[lG];
-->_[1]=xyz2+y4
-->_[2]=x4 > printlevel=plevel; \end{verbatim}} \end{ejem}
\subsection{Dimension of the Riemann-Roch quotients}
Computing the dimension of $\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\underline{1}) \underline{P})}$ is an easy task by Corollary \ref{cor:one}: \begin{prop} $$ \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\underline{1}) \underline{P})}=\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_1)\underline{P})} \oplus \frac{\mathcal{L}((\underline{m}-\varepsilon_1)\underline{P})}{\mathcal{L}((\underline{m}-\varepsilon_1-\varepsilon_2)\underline{P})} \oplus \ldots \oplus \frac{\mathcal{L}((\underline{m}-\varepsilon_1-\varepsilon_2-\ldots -\varepsilon_{r-1})\underline{P})}{\mathcal{L}((\underline{m}-\underline{1}) \underline{P})}. $$ \end{prop}
\dem~It is just to define the map $(f_1,\ldots,f_r) \mapsto f_1 + \ldots + f_r$. \qed
\begin{nota} Notice that the map $(f_1,\ldots,f_r) \mapsto f_1 \cdot \ldots \cdot f_r$ cannot work, because products do not preserve poles. It is also important the fact that the $P_i$ are different, otherwise the statement does not hold: take for example $f(z)=\frac{1}{z}$ and $h(z)=z-\frac{1}{z}$ in $\mathbb{C}$. The sum $f(z)+h(z)=\frac{1}{z}+z-\frac{1}{z}=z$ has no poles, however $f(z)$ and $h(z)$ have both a simple pole at $0$.
\end{nota}
\subsection{Computing the Weierstra\ss~semigroup at several points}
Preserving notations, let $\Gamma_{\pe}$ be the Weierstra\ss~semigroup at the points $P_1,P_2, \ldots, P_r$ and $\Gamma_{P_i}$ the Weierstra\ss~semigroups corresponding to the points $P_i$ for $i=1, \ldots, r$. Write $\mathbb{N}^{\ast}:=\mathbb{N} \setminus \{ 0 \}$ and $\underline{m}_i:=\underline{m} - m_i \varepsilon_i$.
\begin{prop}\label{cor:211} Let $\underline{m} \in \mathbb{N}^r$, $i \in \{1,\ldots,r \}$ and $\underline{m}_i \in \mathbb{N}^r \setminus \Gamma_{\pe}$. Let $$ m:=\mathrm{min}\Big \{ n \in \mathbb{N}^{\ast} \mid \underline{m}_i+n \varepsilon_i \in \Gamma_{\pe} \Big \}. $$ Then any vector $\underline{n}=(n_1,\ldots,n_r) \in \mathbb{N}^r$ belongs to $\mathbb{N}^r \setminus \Gamma_{\pe}$ whenever $n_i=m$, and $n_j=m_j=0$ or $n_j < m_j$ for $j \ne i$. In particular, $m$ is a gap at $P_i$. \end{prop}
Define the usual partial order $\preceq$ over $\mathbb{N}^r$, that is, for $\underline{m},\underline{n} \in \mathbb{N}^r$: $$ (m_1,\ldots,m_r) \preceq (n_1,\ldots,n_r) \ \ \Longleftrightarrow \ \ m_i \le n_i \ \mathrm{for \ all} \ i=1,\ldots,r. $$
\begin{prop} \label{cor:main} Let $i \in \{1, \ldots, r \}$ and $\underline{m} =(m_1,\ldots,m_r)$ be a minimal element of the set $$ \Big \{(n_1,\ldots,n_r) \in \Gamma_{\pe} \mid n_i=m_i \Big \} $$ with respect to the partial order $\preceq$. Assume that $n_i >0$ and the existence of one $j \in \{1,\ldots,r \}$, $j \ne i$ with $m_j >0$. Then: \begin{itemize}
\item[a)] $\underline{m}_i \in \mathbb{N}^r \setminus \Gamma_{\pe}$.
\item[b)] $m_i=\mathrm{min}\{n \in \mathbb{N}^{\ast} \mid \underline{m}_i+n \varepsilon_i \in \Gamma_{\pe} \}$;
in particular, $m_i$ is a gap at $P_i$. \end{itemize} \end{prop}
Propositions \ref{cor:211} and \ref{cor:main} determine a surjective map \begin{displaymath} \begin{array}{lccc} \varphi_i: &\Big \{\underline{m}_i \in \mathbb{N}^r \mid \underline{m}_i \in \mathbb{N}^r \setminus \Gamma_{\pe} \Big \}& \longrightarrow &\mathbb{N} \setminus \Gamma_{P_i}\\ & \underline{m}_i & \mapsto & \min \Big \{m \in \mathbb{N}^{\ast} \mid \underline{m}_i + m \varepsilon_i \in \Gamma_{\pe} \Big \}. \end{array} \end{displaymath}
For $r=2$ this is in fact a bijection between the set of gaps at $P_1$ and the set of gaps at $P_2$: $$ m_1 \in \mathbb{N} \setminus \Gamma_{P_1} \Leftrightarrow (m_1,0) \in \mathbb{N}^2 \setminus \Gamma_{\pe} \mapsto \beta_{m_1}:=\varphi_2 ((m_1,0)) \in \mathbb{N}^ \setminus \Gamma_{P_2}. $$ Furthermore, $m_1 = \min \Big \{n \in \mathbb{N}^{\ast} \mid (n,\beta_{m_1}) \in \Gamma_{\pe} \Big \}$. More details can be found in Homma and Kim \cite{hk} and Kim \cite{kim}.
We summarize some remarkable facts for the case of two points ($r=2$), which will be useful from the computational point of view: \begin{itemize}
\item[(i)] All the gaps at $P_1$ and at $P_2$ are also gaps at
$P_1,P_2$.
\item[(ii)] By the Corollary \ref{cor:main}, for any gap $m_1$ at $P_1$, one has that
$(m_1,\beta_{m_1})$ are gaps at $P_1,P_2$ for
$\beta_{m_1}=0,1,\ldots,l_{m_1}$, until certain $0 \le
l_{m_1}\le 2g-1$, with $g$ the genus of the curve and where
$l_{m_1}$ satisfy that $l_{m_1}+1$ is a gap at $P_2$. The
point $(m_1,l_{m_1}+1)$ is an element of $\Gamma_{\pe}$, which we will
call the \emph{minimal (non-gap) at $m_1$}. We will refer to
the set of the minimal non-gaps at every gap at $P_1$ (they will be $g$, since the number of gaps at $P_1$ is precisely $g$) as the
set of \emph{minimal non-gaps at $P_1$}.
\item[(iii)] The gaps obtained of that form, this is, the set
$$
\Big \{(m_1,\beta_{m_1}) \in \mathbb{N}^2 \setminus \Gamma_{\pe} \mid m_1 \in \mathbb{N} \setminus \Gamma_{P_1} \ \mathrm{and} \ \beta_{m_1}=0,1,\ldots,l_{m_1} \ \mathrm{with} \ l_{m_1}+1 \in \mathbb{N} \setminus \Gamma_{P_2} \Big \}
$$
will be called the \emph{set of gaps \emph{respect to} $P_1$}.
\item[(iv)] Similarly, for any gap $m_2$ at $P_2$, one has that
$(\alpha_{m_2},m_2)$ are gaps at $P_1,P_2$ for
$\alpha_{m_2}=0,1,\ldots,l_{m_2}$, until some $0 \le
l_{m_2}\le 2g-1$, with $g$ being the genus of the curve and where
$l_{m_2}$ satisfy that $l_{m_2}+1$ is a gap at $P_1$. The
point $(l_{m_2}+1,m_2)$ is an element of $\Gamma_{\pe}$, which we will
call the \emph{minimal (non-gap) at $m_2$}. The set of the
minimal non-gaps for every gap at $P_2$ will be called the set
of \emph{minimal non-gaps at $P_2$}. The cardinality of such a set
is $g$, since $g$ is the number of gaps at $P_2$.
\item[(v)] The set of gaps
$$
\Big \{(\alpha_{m_2},m_2) \in \mathbb{N}^2 \setminus \Gamma_{\pe} \mid m_2 \in \mathbb{N} \setminus \Gamma_{P_2} \ \mathrm{and} \ \alpha_{m_2}=0,1,\ldots,l_{m_2} \ \mathrm{with} \ l_{m_2}+1 \in \mathbb{N} \setminus \Gamma_{P_1} \Big \}
$$
is called the \emph{set of gaps \emph{respect to} $P_2$}.
\item[(vi)] The intersection between the set of gaps respect to
$P_1$ and respect to $P_2$ is not necessarily empty. In fact,
the gaps in the intersection are just the \emph{pure gaps} at
$P_1,P_2$. \end{itemize}
The minimal non-gaps at $P_1$ and $P_2$ provide enough information in order to deduce the Weiestra\ss~semigroup at $P_1,P_2$. Recall that we have already described algorithms to compute the dimension (and associated functions, when is possible) of the Riemann-Roch quotients $\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})}$ for given $\underline{m}$, $i \in \{1,2 \}$ and two rational points $P_1$, $P_2$ on an absolutely irreducible projective algebraic plane curve $\chi$ (see Algorithm \ref{algo:no} and Algorithm \ref{algo:si}). An algorithm computing the set of minimal non-gaps at $P_i$, for $i=1,2$ is the following:
\begin{algo} Write $\dim(\underline{m},P,C,i)$ for the procedure calculating the dimension of the quotient vector space $\frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})}$:
{\small \emph{\textsf{INPUT}}: points $P_1,P_2$, an integer $i \in \{ 1,2\}$ and a curve $\chi$.
\emph{\textsf{OUTPUT}}: the set of minimal non-gaps at $P_i$.
\begin{itemize} \item let $L$ be a empty list and $g$ be the genus of $\chi$;
\item let $W_1$ and $W_2$ be the lists of gaps of $\chi$ at $P_1$ and $P_2$, respectively;
\item \verb"FOR" $k=1,\ldots,g$; $k=k+1$; \begin{itemize} \item \verb"IF" $i=1$ \verb"THEN" \begin{itemize} \item j=size of $W_2$;
\item \verb"WHILE" $\Big (\dim((W_1[k],W_2[j]),P,\chi,i)=1 \ \mathrm{AND} \ \dim((W_1[k],W_2[j]-1),P,\chi,i)=1) \ \mathrm{OR} \ j=0 \Big )$ \verb"DO" \begin{itemize} \item $j=j-1$; \end{itemize} \item $L=L \cup \{(W_1[k],W_2[j]) \}$;
\item $W_2=W_2 \setminus \{j \}$; \end{itemize} \item \verb"ELSE" \begin{itemize} \item j=size of $W_1$;
\item \verb"WHILE" $\Big (\dim((W_1[j],W_2[k]),P,\chi,i)=1 \ \mathrm{AND} \ \dim((W_1[j]-1,W_2[k]),P,\chi,i)=1) \ \mathrm{OR} \ j=0 \Big )$ \verb"DO" \begin{itemize} \item $j=j-1$; \end{itemize} \item $L=L \cup \{(W_1[j],W_2[k]) \}$;
\item $W_1=W_1 \setminus \{j \}$; \end{itemize} \end{itemize} \item \verb"RETURN"($L$); \end{itemize} } \end{algo}
\begin{ejem} \label{ejem:2wss}
Let $\chi$ be the curve over $\mathbb{F}_2$ given by the equation $F(X,Y,Z)=X^3Z+X^4+Y^3Z+YZ^3$. Consider the points $P_1=[0:1:1]$ and $P_2=[0:1:0]$ on $\chi$. Then \[ \mathbb{N}^2 \setminus \Gamma_{\{P_1,P_2\}}= \Big \{(0,1),(0,2),(1,0),(1,2),(2,0),(2,1) \Big \}, \] as shown in the figure (the black points are the elements of $\Gamma_{\pe}$, the other ones are the gaps at $P_1,P_2$):
\begin{center} \setlength{\unitlength}{1.6cm}
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\put(-4,0){\line(1,0){2.5}}
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\put(-4.2,1){2}
\put(-4.2,1.5){3}
\put(-4.2,2){4}
\put(-4.19,2.5){$P_2$}
\put(-4,-.3){0}
\put(-3.5,-.3){1}
\put(-3,-.3){2}
\put(-2.5,-.3){3}
\put(-2,-.3){4}
\put(-1.5,-.22){$P_1$}
\end{picture} \end{center}
As an illustration of the Corollary \ref{cor:main}, for instance let $i=1$, $\underline{m}=(m_1,m_2)=(2,2) \in \Gamma_{\pe}$ and the set $\Big \{ (n_1,n_2) \in \Gamma_{\pe} \mid n_1=m_1 \Big \}= \Big \{(2,n) \ \mathrm{for} \ n \ge 2 \Big \}$. A minimal element for this set is $(2,2)$, and \[ \underline{m}_i=\underline{m} - m_1 \varepsilon_1 = (2,2)-2(1,0)=(0,2) \] is a gap at $P_1,P_2$. We compute \[ \min \Big \{n \in \mathbb{N}^{\ast} \mid (n,2) \in \Gamma_{\pe} \Big \}=2=m_1, \] and $m_1=2$ is actually a gap at $P_1$.
In this example we can also see the bijection between the gaps at $P_1$ and the gaps at $P_2$. Preserving notations as above, take now $n_1=1$ as a gap at $P_1$. Then $(1,0)$ is a gap at $P_1,P_2$ and \[ \varphi_2((1,0))=\min \Big \{n \in \mathbb{N}^{\ast} \mid (1,0)+(0,n) \in \Gamma_{\pe} \Big \}=\min \Big \{n \in \mathbb{N}^{\ast} \mid (1,n) \in \Gamma_{\pe} \Big \}=1, \] with $1$ being a gap at $P_2$. Moreover, $n_1=1=\min \Big \{n \in \mathbb{N}^{\ast} \mid (n,1=\varphi_2((1,0))) \in \Gamma_{\pe} \Big \}$.
Now take $p_1=2$ as the other gap at $P_1$. Then $\varphi_2((2,0))=2$, which is a gap at $P_2$. Indeed $p_1=2=\min \Big \{n \in \mathbb{N}^{\ast} \mid (n, \varphi_2((2,0)) \in \Gamma_{\pe} \Big \}$. The same happens to the gaps at $P_2$. \end{ejem}
\section{\textsf{Computational aspects using \textsc{Singular}}}
We are interested in explaining the most important procedures implemented in \textsc{Singular} and to give examples to show how to work with them.
More precisely, in subsection \ref{section:hints}) we give some hints of use of the library \verb"brnoeth.lib", since our procedures are based on most of the algorithms contained in it. Then, in Subsection \ref{section:seve} we present the procedures which pretend generalize the computation of the Weierstra\ss~semigroup to the case of several points, i.e., a set of procedures which try to: \begin{itemize} \item[-] compute $\dim_{\mathbb{F}} \frac{\mathcal{L}(\underline{mP})}{\mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})}$ and a function $f_{\underline{m},i} \in \mathcal{L}(\underline{mP}) \setminus \mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})$ if possible.
\item[-] compute the set of minimal non-gaps at a point $P_i$, for $i \in \{ 1,2 \}$. \end{itemize}
\subsection{Hints of usage of \textsf{brnoeth.lib}}\label{section:hints}
The purpose of the library \verb"brnoeth.lib" of \textsc{Singular} is the implementation of the Brill-Noether algorithm for solving the Riemann-Roch problem and some applications in Algebraic Geometry codes, involving the computation of Weierstra\ss~semigroups for one point.
A first warning: \verb"brnoeth.lib" accepts only prime base fields and absolutely irreducible planes curves, although this is not checked.
Curves are usually defined by means of polynomials in two variables, that is, by its local equation. It is possible to compute most of the concepts concerning to the curve with the procedure \verb"Adj_div". We defined the procedure (previously we must have defined the ring, the polynomial $f$ and have charged the library \verb"brnoeth.lib"): \begin{verbatim} > list C=Adj_div(f); \end{verbatim}
The output consist of a list of lists as follows: \begin{itemize}
\item The first list contains the affine and the local ring.
\item The second list has the degree and the genus of the
curve.
\item Each entry of the third list corresponds to one closed
place,that is, a place and all its conjugates, which is
represented by two integer, the first one the degree of the
point and the second one indexing the conjugate point.
\item The fourth one has the conductor of the curve.
\item The fifth list consists of a list of lists, the first
one, namely \verb"C[5][d][1]" being a (local) ring over an extension
of degree $d$ and the second one (\verb"C[5][d][2]") containing the
degrees of base points of places of degree $d$. \end{itemize}
Furthermore, inside the local ring \verb"C[5][d][1]" we can find the following lists: \begin{itemize}
\item \verb"list POINTS": base points of the places of degree $d$.
\item \verb"list LOC_EQS": local equations of the curve at the base
points.
\item \verb"list BRANCHES": Hamburger-Noether expressions of the
places.
\item \verb"list PARAMETRIZATIONS": local parametrizations of the
places. \end{itemize}
Now we explain how the different kinds of common objects must be treated in \verb"Singular".
\textbf{Affine points} $P$ are represented by a standard basis of a prime ideal, and a vector of integers containing the position of the places above $P$ in the list supplied by \verb"C[3]"; if the point lies at the infinity, the ideal is replaced by an homogeneous irreducible polynomial in two variables.
A \textbf{place} is represented by the four list previously cited: a base point (\verb"list POINTS" of homogeneous coordinates); a local equation (\verb"list LOC_EQS") for the curve at the base point; a Hamburger-Noether expansion of the corresponding branch (\verb"list BRANCHES"); and a local parametrization (\verb"list PARAMETRIZATIONS") of such a branch.
A \textbf{divisor} is represented by a vector of integers, where the integer at the position $i$ means the coefficient of the $i$-th place in the divisor.
\textbf{Rational functions} are represented by ideals with two homogeneous generators, the first one being the numerator of the rational function, and the second one being the denominator.
Furthermore, we can compute a complete list containing all the non-singular affine (closed) places with fixed degree $d$ just by using the procedure \verb"NSplaces" in this way: \begin{verbatim} > C=NSplaces(1..d,C); \end{verbatim}
Closer to our aim is the procedure \verb"Weierstrass", which computes the non-gaps of the Weierstra\ss~semigroup at one point and the associated functions with poles. It contains three inputs: \begin{itemize}
\item an \emph{integer} indicating the rational place in which we
compute the semigroup;
\item an \emph{integer} indicating how many non-gaps we want to
calculate;
\item the curve given in form of a \emph{list} \verb"C=Adj_div(f)" for
some polynomial $f$ representing the local equation of
the curve at the point given in the first entry.
\end{itemize}
This procedure needs to be called from the ring \verb"C[1][2]". Moreover, the places must be necessarily \emph{rational}.
\subsection{Procedures generalizing to several points}\label{section:seve}
We present now a main procedure to compute the dimension of the so-called Riemann-Roch vector spaces of the form $\mathcal{L}(\underline{mP})\setminus \mathcal{L}((\underline{m}-\varepsilon_i)\underline{P})$. If this dimension is equal to $1$, the procedure is also able to compute a rational function belonging to the space.
The technique developed here is not by using the adjunction theory directly, as we have developed theoretically in the Chapter 3 (Algorithm \ref{algo:no}), because of its high cost, but we use the Algorithm \ref{algo:si}, or, more properly speaking, a slight variant of it: we order the poles in a vector from the biggest one to the smallest one (in absolute value) and we take the first in such a vector. {\small \begin{verbatim} proc RRquot (intvec m, list P, list CURVE, int chart) "USAGE:RRquot( m, P, CURVE, ch ); m,P intvecs, CURVE a list and ch an integer. RETURN: an integer 0 (dimension of L(m)\L(m-e_i)), or a list with three entries:
@format
RRquot[1] ideal (the associated rational function)
RRquot[2] integer (the order of the rational function)
RRquot[3] integer (dimension of L(m)\L(m-e_i))
@end format NOTE: The procedure must be called from the ring CURVE[1][2],
where CURVE is the output of the procedure @code{NSplaces}. @* P represents the coordinates of the place CURVE[3][P]. @* Rational functions are represented by
numerator/denominator
in form of ideals with two homogeneous generators. WARNING: The place must be rational, i.e., necessarily CURVE[3][P][1]=1. @* SEE ALSO: Adj_div, NSplaces, BrillNoether EXAMPLE: example RRquot; shows an example " {
// computes a basis for the quotient of Riemann-Roch vector spaces L(m)\L(m-e_i)
// where m=m_1 P_1 + ... + m_r P_r and m-e_i=m_1P_1+...+(m_i-1)P_i+...+m_r P_r,
// a basis for the vector space L(m-e_i) and the orders of such functions, via
// Brill-Noether
// returns 2 lists : the first consists of all the pole orders in
// increasing order and the second consists of the corresponding rational
// functions, where the last one is the basis for the quotient vector space
// P_1,...,P_r must be RATIONAL points on the curve.
def BS=basering;
def SS=CURVE[5][1][1];
intvec posinP;
int i,dimen;
setring SS;
//identify the points P in the list CURVE[3]
int nPOINTS=size(POINTS);
for(i=1;i<=size(m);i=i+1)
{
posinP[i]=isPinlist(P[i],POINTS);
} //in case the point P is not in the list CURVE[3]
if (posinP==0)
{
ERROR("The given place is not a rational place on the curve");
}
setring BS;
//define the divisor containing m in the right way
intvec D=zeroes(m,posinP,nPOINTS);
list Places=CURVE[3];
intvec pl=Places[posinP[chart]];
int dP=pl[1];
int nP=pl[2];
//check that the points are rational
if (dP<>1)
{
ERROR("The given place is not defined over the prime field");
}
int auxint=0;
ideal funcion;
funcion[1]=1;
funcion[2]=1;
// Brill-Noether algorithm
list LmP=BrillNoether(D,CURVE);
int lmP=size(LmP);
if (lmP==1)
{
dimen=0;
return(dimen);
}
list ordLmP=list();
list sortpol=list();
for (i=1;i<=lmP;i=i+1)
{
ordLmP[i]=orderRF(LmP[i],SS,nP)[1];
}
ordLmP=extsort(ordLmP);
if (D[posinP[chart]] <> -ordLmP[1][1])
{
dimen=0;
return(dimen);
}
LmP=permute_L(LmP,ordLmP[2]);
funcion=LmP[1];
dimen=1;
return(list(funcion,ordLmP[1][1],dimen)); } example
{
"EXAMPLE:"; echo=2;
int plevel=printlevel;
printlevel=-1;
ring s=2,(x,y),lp;
poly f=y2+y3+x3+x4;
list C=Adj_div(f);
C=NSplaces(1,C);
def pro_R=C[1][2];
setring pro_R;
intvec m=4,6;
intvec P1=0,1,1;
intvec P2=0,1,0;
list P=P1,P2;
int chart=1;
RRquot(m,P,C,chart);
printlevel=plevel;
} \end{verbatim} } Let us see an example: {\small \begin{verbatim} > example RRquot; // proc RRquot from lib brnoeth.lib EXAMPLE:
int plevel=printlevel;
printlevel=-1;
ring s=2,(x,y),lp;
poly f=y2+y3+x3+x4;
list C=Adj_div(f); The genus of the curve is 2
C=NSplaces(1,C);
def pro_R=C[1][2];
setring pro_R;
intvec m=4,6;
intvec P1=0,1,1;
intvec P2=0,1,0;
list P=P1,P2;
int chart=1;
RRquot(m,P,C,chart); Vector basis successfully computed -->[1]:
_[1]=x3+yz2
_[2]=xyz+xz2 -->[2]:
-4 -->[3]:
1
printlevel=plevel; \end{verbatim} } This procedure needs also the following auxiliar procedures:
As \verb"RRquot" reads off the point through its homogeneous coordinates we need to localize that point in the list \verb"POINTS" and make the correspondence between such a point and its position in the list of points contained in the third output of the procedure \verb"Adj_div". This is done by mean of the routine \verb"isPinlist". Its inputs are the point $P$ in homogeneous coordinates, that is, a vector of integers, and the list $L$ of points from \verb"Adj_div". The output is an integer being zero if the point is not in the list or a positive integer indicating the position of $P$ in $L$. Look at the example: {\small \begin{verbatim} > example isPinlist; // proc isPinlist from lib brnoeth.lib EXAMPLE:
ring r=0,(x,y),ls;
intvec P=1,0,1;
list POINTS=list(list(1,0,1),list(1,0,0));
isPinlist( P,POINTS); -->1 \end{verbatim} }
We need also a procedure for ordering a list of integers. This is partially solved by the procedure \verb"sort" from \verb"general.lib". But \verb"sort" is not able to order lists of negative numbers, so we have extended this algorithm to \verb"extsort". The \verb"extsort" procedure needs to permute a vector of integers by the instructions given by another similar vector. This is actually done for lists of integers (\verb"permute_L" in \verb"brnoeth.lib"), but not for vectors of integers. This lack is covered by the procedure \verb"perm_L", whose entries are a pair of vectors, the second vector fixing the permutation of the first one. The output consists of the permutated vector, as the following example shows: {\small \begin{verbatim} > example extsort; // proc extsort from lib brnoeth.lib EXAMPLE:
ring r=0,(x,y),ls;
list L=10,9,8,0,7,1,-2,4,-6,3,0;
extsort(L); -->[1]:
-6,-2,0,0,1,3,4,7,8,9,10 -->[2]:
9,7,4,11,6,10,8,5,3,2,1 \end{verbatim} }
Finally, it was interesting to fix the system for reading off the data of the divisor needed in the \verb"BrillNoether" procedure. Our routine zeroes takes two vectors of integers \verb"m" and \verb"pos", and an integer \verb"siz" and it builds up a vector of size \verb"siz", with the values contained in \verb"m" set in the places given by \verb"pos" and zeroes in the other places. This algorithm is called \verb"zeroes": {\small \begin{verbatim} > example zeroes; // proc zeroes from lib brnoeth.lib EXAMPLE:
ring r=0,(x,y),ls;
intvec m=4,6;
intvec pos=4,2;
zeroes(m,pos,5); -->0,6,0,4,0 \end{verbatim}}
\end{document} | arXiv | {
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\begin{document}
\title{On the largest subsets avoiding the diameter of $(0,\pm 1)$-vectors}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \footnote[0]{2010 Mathematics Subject Classification: 05D05 (05C69). \\ \noindent {\it Saori Adachi}:
Mathematics Education,
Graduate School of Education,
Aichi University of Education,
1 Hirosawa, Igaya-cho,
Kariya, Aichi 448-8542,
Japan.
s214m044@auecc.aichi-edu.ac.jp. \\ \noindent {\it Hiroshi Nozaki}:
Department of Mathematics Education,
Aichi University of Education,
1 Hirosawa, Igaya-cho,
Kariya, Aichi 448-8542,
Japan.
hnozaki@auecc.aichi-edu.ac.jp. }
\begin{abstract} Let $L_{mkl}\subset \mathbb{R}^{m+k+l}$ be the set of vectors which have $m$ of entries $-1$, $k$ of entries $0$, and $l$ of entries $1$. In this paper, we investigate the largest subset of $L_{mkl}$ whose diameter is smaller than that of $L_{mkl}$. The largest subsets for $m=1$, $l=2$, and any $k$ will be classified. From this result, we can classify the largest $4$-distance sets containing the Euclidean representation of the Johnson scheme $J(9,4)$. This was an open problem in Bannai, Sato, and Shigezumi (2012). \end{abstract} \textbf{Key words}: the Erd\H{o}s--Ko--Rado theorem, $s$-distance set, diameter graph, independent set, extremal set theory.
\section{Introduction} The famous theorem in Erd\H{o}s--Ko--Rado \cite{EKR61} stated that for $n \geq 2k$ and a family $\mathfrak{A}$ of $k$-element subsets of $I_n=\{1,\ldots, n\}$, if any two distinct $A,B \in \mathfrak{A}$ satisfy $A\cap B \ne \emptyset$, then \[
|\mathfrak{A}| \leq \binom{n-1}{k-1}. \] For $n>2k$, the set $
\{A\subset I_n \mid |A|=k, 1 \in A \} $ is the unique family achieving equality, up to permutations on $I_n$. For $n=2k$, the largest set is any family which contains only one of $A$ or $I_n\setminus A$ for any $k$-element $A\subset I_n$. This result plays a central role in extremal set theory, and similar or analogous theorems are proved for various objects \cite{B12,DF83,F87}.
We can naturally interpret $A \subset I_n$ as $x=(x_1, \ldots, x_n)\in \mathbb{R}^n$ by the manner $x_i=1$ if $i \in A$, $x_i=0$ if $i \not\in A$. By this identification, the Erd\H{o}s--Ko--Rado theorem can be rewritten that for $n \geq 2k$ and a subset $X$ of $L_k=\{x \in \mathbb{R}^n \mid x_i \in \{0,1\}, \sum x_i=k \}$ if any distinct $x,y \in X$ satisfy $d(x,y) < D(L_k)=\sqrt{2k} $, then \[
|X| \leq \binom{n-1}{k-1}, \] where $d(,)$ is the Euclidean distance, and $D(L_k)$ is the diameter of $L_k$.
We would like to consider the following problem to generalize the Erd\H{o}s--Ko--Rado theorem. \begin{problem} Let $L_{mkl}\subset \mathbb{R}^{m+k+l}$ be the set of vectors which have $m$ of entries $-1$, $k$ of entries $0$, and $l$ of entries $1$. Classify the largest $X \subset L_{mkl}$ with $D(X)<D(L_{mkl})$. \end{problem} It is almost obvious for the cases $m=l$ (Proposition~\ref{prop:m=l}) and $m+k \leq l$ (Proposition~\ref{prop:m+k<=l}). In this paper, we solve the first non-trivial case $m=1$, $l=2$ and any $k$ (Theorem~\ref{thm:main}). Using the largest sets for the case $(m,k,l)=(1,6,2)$, we can classify the largest $4$-distance sets containing the Euclidean representation of the Johnson scheme $J(9,4)$. This was an open problem in \cite{BSS12}.
We will give a brief survey on related results. Let $\mathfrak{L}_{nm}$ be the set of $(0,\pm 1)$-vectors in $\mathbb{R}^n$ which have $m$ non-zero coordinates. For a fixed set $D$ of integers, let $V(n,m,D)$ be the family of subsets $V=\{v_1,\ldots,v_k \}$ of $\mathfrak{L}_{nm}$ such that $(v_i,v_j)\in D$ for any $i\ne j$. There are several results relating to the largest sets in $V(n,m,D)$ for some $(n,m,D)$ \cite{DF81,DF83_2,DF85}. Since $X \subset \mathfrak{L}_{nm}$ is on a sphere,
if $|D|=s$ holds, then $|X| \leq \binom{n+s-1}{s}+\binom{d+s-2}{s-1}$ \cite{DGS77}. The case $D=\{d \}$ is investigated in \cite{DF81}. For non-negative integers $d<m$, $t\geq 2$, and $n>n_0(m)$ (see \cite{DF81} about $n_0(m)$), if $X \in
V(n,m,\{d,d+1,\ldots,d+t-1\})$, then $|X|\leq \binom{n-d}{t}/\binom{m-d}{t}$~\cite{DF83_2}. This equality can be attained whenever a Steiner system $S(n-d,m-d,t)$ (equivalently $t$-$(n-d,m-d,1)$ design) exists . We also have if $X \in
V(n,m,\{-(t-1),-(t-2),\ldots, t-1\})$, then $|X|\leq 2^{t-1}(m-t+1)\binom{n}{t}/\binom{m}{t}$ \cite{DF85}. When $m=t+1$, this equality can be attained whenever a Steiner system $S(n,m,m-1)$ exists.
\section{Largest subsets avoiding the diameter of $L_{mkl}$} \label{sec:2} Let $L_{mkl}$ denote the finite set in $\mathbb{R}^n=\mathbb{R}^{m+k+l}$, which consists of all vectors whose number of entries $-1$, $0$, $1$ is equal to $m$, $k$, $l$, respectively. For two subsets $X,Y$ of $L_{mkl}$, $X$ is {\it isomorphic} to $Y$ if there exists a permutation $\sigma \in S_{n}$ such that $X=\{(y_{\sigma(1)},\ldots,y_{\sigma(n)}) \mid (y_1,\ldots,y_n) \in Y\}$. The {\it diameter} $D(X)$ of $X\subset \mathbb{R}^n$ is defined to be \[ D(X)= \max \{d(x,y) \mid x, y \in X \}, \] where $d(,)$ is the Euclidean distance. Let $M_{mkl}$ denote the largest possible number of cardinalities of $X \subset L_{mkl}$ such that $D(X)<D(L_{mkl})$. The {\it diameter graph} of $X\subset \mathbb{R}^n$ is defined to be the graph $(X,E)$, where $E=\{(x,y) \mid d(x,y)=D(X)\}$. The problem of determining $M_{mkl}$ is equivalent to determining the independence number of the diameter graph of $L_{mkl}$.
Note that $M_{mkl}=M_{lkm}$ because we have $L_{mkl}=-L_{lkm}=\{-x \mid x\in L_{lkm}\}$. Thus we may assume $m \leq l$. In this section, we determine $M_{mkl}$, and classify the largest sets for several cases of $m$, $k$, $l$.
First we determine $M_{mkl}$ for the cases $m=l$ and $m+k\leq l$. \begin{proposition}\label{prop:m=l} Assume $m=l$. Then we have \[
M_{mkl} = \frac{1}{2}\binom{n}{m}\binom{k+m}{m}=\frac{1}{2}|L_{mkl}|, \] and the largest sets contain only one of $x$ or $-x$ for any $x \in L_{mkl}$. \end{proposition} \begin{proof} For any $x \in L_{mkl}$, we have $\{y \mid d(x,y)=D(L_{mkl})\}=\{-x\}$. Therefore the diameter graph of $L_{mkl}$ is the set of independent edges. The proposition can be easily proved from this fact. \end{proof} For $X \subset L_{mkl}$, we use the notation \[
N_i(X,j)=\{(x_1,\ldots,x_n) \in X \mid x_i=j\}, \qquad \text{ and } \qquad n_i(X,j) = |N_i(X,j)|. \] \begin{proposition} \label{prop:m+k<=l} Assume $m+k \leq l$. Then we have \[ M_{mkl} = \binom{n-1}{m+k-1}\binom{m+k}{m}. \] For $m+k>l$, the largest set is $N_1(L_{mkl},-1)\cup N_1(L_{mkl},0)$, up to isomorphism. For $m+k=l$, then the largest sets contain only one of $\{(x_1,\ldots,x_n) \in L_{mkl} \mid x_i=1, \forall i \in J \}$ or $\{(x_1,\ldots,x_n) \in L_{mkl} \mid x_i=1,\forall i \in I_n \setminus J \}$ for any $J \subset I_n$ of order $l$. \end{proposition} \begin{proof} A finite subset $X$ of $L_{mkl}$ satisfies $D(X)<D(L_{mkl})$ if and only if $\{i \mid x_i=-1,0\} \cup \{i \mid y_i=-1,0\}$ is not empty for any distinct $(x_1,\ldots,x_n),(y_1,\ldots, y_n) \in X$. We can therefore apply the Erd\H{o}s--Ko--Rado theorem \cite{EKR61} to determine the positions of entries $-1$ or $0$. The number of possible positions of $-1$, $0$ is $\binom{n-1}{m+k-1}$. After fixing the position, $-1$, $0$ can be placed in $\binom{m+k}{k}$ ways.
This determines $M_{mkl}$. The largest sets are classified from the optimal sets of the Erd\H{o}s--Ko--Rado theorem. \end{proof} The remaining part of this section is devoted to proving \[ M_{1k2} = \mathfrak{M}_k=\binom{k+3}{3}+2, \] and determining the classification of the largest sets. Note that $D(L_{1k2})=\sqrt{10}$ and if $X\subset L_{1k2}$ satisfies $D(X)<D(L_{1k2})$, then $D(X) \leq \sqrt{8}$. The following two lemmas are used later. \begin{lemma} \label{lem:M_k} Let $X\subset L_{1k2}$ with $D(X)< D(L_{1k2})$.
Suppose $k\geq 4$, and $|X| \geq \mathfrak{M}_k$. Then there exists $i \in \{1,\ldots,n\}$ such that $n_i(X,0)\geq \mathfrak{M}_{k-1}$. \end{lemma} \begin{proof} This lemma is immediate because the average of $n_i(X,0)$ is \[
\frac{1}{n}\sum_{i=1}^{n}n_{i}(X,0)=\frac{k|X|}{k+3} \geq \frac{k\mathfrak{M}_k}{k+3} =\mathfrak{M}_{k-1}-\frac{6}{k+3} >\mathfrak{M}_{k-1}-1. \qedhere \] \end{proof} \begin{lemma} \label{lem:match} Let $G=(V,E)$ be a connected simple graph, and $E'$ a matching in $G$. Assume that $G$ has an independent set $I$ of size
$|V|-|E'|$. Then for $z\in I$ if $x \in V$ satisfies
$(x,y)\in E'$ for some $y$ adjacent to $z$, then $x \in I$. \end{lemma} \begin{proof}
Since the cardinality of $I$ is $|V|-|E'|$, only one of $x$ or $y$ is an element of $I$ for any $(x,y) \in E'$. By assumption, $y \not \in I$, and hence $x \in I$. \end{proof} The subsets $S_k(i)$, $T_k(i)$, $U_k(i)$ of $L_{1k2}$ are defined by \begin{align*}
S_k(i)&=\{(x_1,\ldots,x_n) \in L_{1k2}\mid x_1=\cdots=x_{i-1}=0, x_i=-1\}, \\
T_k(i)&=\{(x_1,\ldots,x_n) \in L_{1k2}\mid x_1=\cdots=x_{i-1}=0, x_i=1\}, \\
U_k(i)&=\{(x_1,\ldots,x_n) \in L_{1k2} \mid x_1=1, x_l=-1, x_j=1, \exists l \in \{2,\ldots, i\},
\exists j \in \{l+1,\ldots,n\}\}
\end{align*} for $i=2,\ldots,k+2$. We define $S_k(1)=N_1(L_{1k2},-1)$, and $T_k(1)=N_1(L_{1k2},1)$. The following are candidates of the largest subsets avoiding the largest distance $\sqrt{10}$. \begin{align*} &X_k=T_k(k+1)\cup (\bigcup_{i=1}^{k+1} S_k(i)) \text{ for } k\geq 1, \\ &Y_1=T_1(1), \qquad Y_k=T_k(k)\cup (\bigcup_{i=1}^{k-1} S_k(i)) \text{ for } k\geq 2, \\ &Z_2=T_2(1), \qquad Z_k=T_k(k-1)\cup( \bigcup_{i=1}^{k-2} S_k(i)) \text{ for } k\geq 3. \end{align*}
Note that $|X_k|=|Y_k|=|Z_k|=\mathfrak{M}_k$, and they can be inductively constructed by \begin{align*} X_k&=\{(0,x)\mid x \in X_{k-1}\} \cup N_1(L_{1k2},-1), \\ Y_k&=\{(0,x)\mid x \in Y_{k-1}\} \cup N_1(L_{1k2},-1), \\ Z_k&=\{(0,x)\mid x \in Z_{k-1}\} \cup N_1(L_{1k2},-1). \end{align*} We also use the following notation. \begin{align*} X_k'&=X_k\setminus S_k(1)\quad (k\geq 2)& \qquad Y_k'&=Y_k\setminus S_k(1)\quad (k\geq 2)& \qquad Z_k'&=Z_k\setminus S_k(1)\quad (k\geq 3). \\
&=\{(0,x)\mid x \in X_{k-1}\}, & &=\{(0,x)\mid x \in Y_{k-1}\}, & &=\{(0,x)\mid x \in Z_{k-1}\}. \end{align*}
\begin{theorem} \label{thm:main} Let $X\subset L_{1k2}$ with $D(X)<D(L_{1k2})$. Then we have \[
|X| \leq \mathfrak{M}_k. \] If equality holds, then \begin{enumerate} \item for $k=1$, $X=X_1$, or $Y_1$, \item for $k\geq 2$, $X=X_k$, $Y_k$, or $Z_k$, \end{enumerate} up to isomorphism. \end{theorem} This theorem will be proved by induction.
We first prove the inductive step. \begin{lemma}\label{lem:induc} Let $k\geq 2$. Assume that the statement in Theorem~\ref{thm:main}
holds for some $k-1$. Let $X \subset L_{1k2}$ with $D(X)<D(L_{1k2})$, such that $n_i(X,0)=\mathfrak{M}_{k-1}$ for some $i$. Then we have $|X| \leq \mathfrak{M}_{k}$. If equality holds, then $X=X_{k}$, $Y_{k}$, or $Z_{k}$, up to isomorphism. \end{lemma} \begin{proof}
Without loss of generality, $n_1(X,0)=\mathfrak{M}_{k-1}$, and hence $X$ contains $X_k'$, $Y_k'$, or $Z_k'$ for $k\geq 3$, and $X_1'$, or $Y_1'$ for $k=2$.
(i) Suppose $X_k' \subset X$ for $k\geq 2$. The set of other candidates of elements of $X$ is $S_k(1)\cup U_k(k)$. The diameter graph $G$ of $S_k(1)\cup U_k(k)$ is a bipartite graph of the partite sets $S_k(1)$ and $U_k(k)$. Since the three elements \[(-1,0,\ldots,0,0,1,1),(-1,0,\ldots,0,1,0,1), (-1,0,\ldots,0,1,1,0)\in S_k(1)\] are isolated vertices in $G$, they may be contained in $X$. Let $G'$ be the
subgraph of $G$ formed by removing the three isolated vertices. A perfect matching of $G'$ is given as follows. \begin{center} Matching (i)\\
\begin{tabular}{|c|c|} \hline $S_k(1)$ &$U_k(k)$ \\ $(-1,x_2, \ldots, x_{k+3})$ & $(1,y_2,\ldots,y_{k+3})$ \\ \hline $x_i=1,x_j=1$ ($2\leq i\leq k, i<j<n $)& $y_i=-1,y_{j+1}=1 $ \\ $x_i=1,x_n=1$ ($2\leq i\leq k $)& $y_i=-1,y_{i+1}=1 $ \\ \hline \end{tabular} \end{center} By this matching, we can show \[
|X| \leq \mathfrak{M}_{k-1}+|S_k(1)|=\mathfrak{M}_k. \]
We will classify the sets attaining this bound. First assume that $x \in X$ for some $x\in S_k(1)$ with $x_2=1$. By Lemma~\ref{lem:match}, $X$ must contain any $x \in S_k(1)$ with $x_2=1$. In particular, $(-1,1,1,0,\ldots,0) \in X$. Using Lemma~\ref{lem:match} again, $X$ must contain $x \in S_k(1)$ with $x_3=1$. By a similar manner, $X$ must contain any $x \in S_k(1)$. Therefore $X=X_k$.
Assume $X$ does not contain any $x\in S_k(1)$ with $x_2=1$, namely $n_2(X,1)=0$. By assumption, we have \[
|X|=n_2(X,-1)+n_2(X,0)\leq \binom{k+2}{2}+\mathfrak{M}_{k-1}=\mathfrak{M}_k. \]
If $|X|=\mathfrak{M}_k$, then we have $n_2(X,-1)=\binom{k+2}{2}$ and $n_2(X,0)=\mathfrak{M}_{k-1}$. This implies that $X$ is isomorphic to $X_k$, $Y_k$, or $Z_k$.
(ii) Suppose $Y_k' \subset X$ for $k\geq 2$. The set of other candidates of elements of $X$ is the union of $S_k(1)$, $U_k(k-1)$, and \[ \mathcal{S}_1=\{(x_1,\ldots, x_{k+3}) \in L_{1k2} \mid x_1=1, x_k=1,x_j=-1, k< j \} \] for $k\geq 3$, and $S_2(1)\cup \mathcal{S}_1$ for $k=2$. The diameter graph $G$ of $S_k(1)\cup U_k(k-1)\cup \mathcal{S}_1$ is a bipartite graph of the partite sets $S_k(1)$ and $U_k(k-1)\cup \mathcal{S}_1$. Since the three elements \[(-1,0,\ldots,0,1,1,0,0),(-1,0,\ldots,0,1,0,1,0), (-1,0,\ldots,0,1,0,0,1)\in S_k(1)\] are isolated vertices in $G$, they may be contained in $X$. Let $G'$ be the
subgraph of $G$ formed by removing the three isolated vertices. A perfect matching of $G'$ is given as follows. \begin{center} Matching (ii)\\
\begin{tabular}{|c|c|} \hline $S_k(1)$ &$U_k(k-1)$ \\ $(-1,x_2, \ldots, x_{k+3})$ & $(1,y_2,\ldots,y_{k+3})$ \\ \hline $x_i=1,x_j=1$ ($2\leq i\leq k-1, i<j<n $)& $y_i=-1,y_{j+1}=1 $ \\ $x_i=1,x_n=1 $ ($2\leq i\leq k-1 $)& $y_i=-1,y_{i+1}=1 $ \\ \hline \end{tabular}
\begin{tabular}{|c|c|} \hline $S_k(1)$ &$\mathcal{S}_1$ \\
\hline $(-1,0,\ldots,0,1,1,0)$& $(1,0,\ldots,0,1,-1,0,0)$ \\ $(-1,0,\ldots,0,0,1,1)$& $(1,0,\ldots,0,1,0,-1,0)$ \\ $(-1,0,\ldots,0,1,0,1)$& $(1,0,\ldots,0,1,0,0,-1)$ \\ \hline \end{tabular} \end{center} By this maching, we can show $
|X| \leq \mathfrak{M}_k. $
We will classify the sets attaining this bound. For $k=2$, the maximum indepdent sets of $G'$ is $\{(-1,0,0,1,1),(-1,0,1,0,1),(-1,0,1,1,0)\} \subset S_2(1)$ or $\mathcal{S}_1$. This implies that $X=Y_2$ or $Z_2$. For $k\geq 3$, we assume that $x \in X$ for some $x\in S_k(1)$ with $x_2=1$. By Lemma~\ref{lem:match}, $X$ must contain any $x \in S_k(1)$. Therefore $X=Y_k$. If $X$ does not contain any $x\in S_k(1)$ with $x_2=1$, namely $n_2(X,1)=0$. It can be proved that $X$ is isomorphic to $X_k$, $Y_k$, or $Z_k$.
(iii) Suppose $k\geq 3$, and $Z_k' \subset X$. The set of other candidates of elements of $X$ is the union of $S_k(1)$, $U_k(k-2)$, and \[ \mathcal{S}_2=\{(x_1,\ldots, x_{k+3}) \in L_{1k2} \mid x_1=1, x_{k-1}=1,x_j=-1, k< j \} \] for $k\geq 4$, and $S_3(1)\cup \mathcal{S}_2$ for $k=3$. The diameter graph $G$ of $S_k(1)\cup U_k(k-2)\cup \mathcal{S}_2$ is a bipartite graph of the partite sets $S_k(1)$ and $U_k(k-2)\cup \mathcal{S}_2$. Since the four vectors \begin{align*} &(-1,0,\ldots,0,1,1,0,0,0),(-1,0,\ldots,0,1,0,1,0,0),\\ &(-1,0,\ldots,0,1,0,0,1,0),(-1,0,\ldots,0,1,0,0,0,1)\in S_k(1) \end{align*} are isolated vertices in $G$, they may be contained in $X$. Let $G'$ be the
subgraph of $G$ formed by removing the four isolated vertices. A maximum matching of $G'$ is given as follows. \begin{center} Matching (iii)\\
\begin{tabular}{|c|c|} \hline $S_k(1)$ &$U_k(k-2)$ \\ $(-1,x_2, \ldots, x_{k+3})$ & $(1,y_2,\ldots,y_{k+3})$ \\ \hline $x_i=1,x_j=1$ ($2\leq i\leq k-2, i<j<n $)& $y_i=-1,y_{j+1}=1 $ \\ $x_i=1,x_n=1 $ ($2\leq i\leq k-2 $)& $y_i=-1,y_{i+1}=1 $ \\ \hline \end{tabular}
\begin{tabular}{|c|c|} \hline $S_k(1)$ &$\mathcal{S}_2$ \\
\hline $(-1,0,\ldots,0,1,1,0,0)$& $(1,0,\ldots,0,1,-1,0,0,0)$ \\ $(-1,0,\ldots,0,0,1,1,0)$& $(1,0,\ldots,0,1,0,-1,0,0)$ \\ $(-1,0,\ldots,0,0,0,1,1)$& $(1,0,\ldots,0,1,0,0,-1,0)$ \\ $(-1,0,\ldots,0,1,0,0,1)$& $(1,0,\ldots,0,1,0,0,0,-1)$ \\ \hline \end{tabular} \end{center} Note that the two vectors
\begin{equation} \label{eq:1} (-1,0,\ldots,0,1,0,1,0),(-1,0,\ldots,0,0,1,0,1)\in S_k(1) \end{equation} are unmatched in this matching. By this matching, we can show $
|X| \leq \mathfrak{M}_k. $
We will classify the sets attaining this bound.
If $|X|=\mathfrak{M}_{k}$, then the two vectors in \eqref{eq:1} must be contained in $X$. Therefore $X$ does not contain any element of $\mathcal{S}_2$, and contains an element of $S_k(1)$ which matches some element of $\mathcal{S}_2$. For $k=3$, $X$ therefore contains $S_k(1)$, and $X=Z_3$. For $k\geq 4$, we assume that $x \in X$ for some $x\in S_k(1)$ with $x_2=1$. By Lemma~\ref{lem:match}, $X$ must contain any $x \in S_k(1)$. Therefore $X=Z_k$. If $X$ does not contain any $x\in S_k(1)$ with $x_2=1$, namely $n_2(X,1)=0$. Therefore $X$ is isomorphic to $X_k$, $Y_k$, or $Z_k$. \end{proof} Matchings (i)--(iii) and the notation $\mathcal{S}_1$, $\mathcal{S}_2$ defined in the proof of Lemma~\ref{lem:induc} are used again later. The base case in the induction is the case $k=3$. We will prove the cases $k=1,2,3$ in order. \begin{proposition}\label{prop:k1} Let $X\subset L_{112}$ with $D(X)<D(L_{112})$. Then we have \[
|X| \leq \mathfrak{M}_1=6. \] If equality holds, then $X=X_1$, or $Y_1$, up to isomorphism. \end{proposition} \begin{proof}
Since the diameter graph $G$ of $L_{112}$ is isomorphic to $C_4\cup C_4 \cup C_4$, where $C_4$ is the $4$-cycle, the bound $|X|\leq 6$ clearly holds.
Considering the permutation of coordinates, $G$ has the automorphism group $S_4$. Since the stabilizer of $X_1$ in $S_4$ is of order $6$, the orbit of $X_4$ has length $4$. Similarly the orbit of $Y_1$ has length $4$. Since the number of maximum independent sets of $G$ is $2^3=8$,
this proposition follows. \end{proof} For $k=2$, we also classify $(\mathfrak{M}_2-1)$-point sets $X$ with $D(X)<D(L_{122})$ in order to prove the case $k=3$. \begin{proposition}\label{prop:k2} Let $X\subset L_{122}$ with $D(X)<D(L_{122})$. Then we have \[
|X| \leq \mathfrak{M}_2=12. \]
If $|X|=12$, then $X=X_2$, $Y_2$, or $Z_2$, up to isomorphism.
If $|X|=11$, then $X$ is \begin{equation*} \label{eq:pro1} V_2=X_2' \cup \{(-1,0,0,1,1),(-1,0,1,0,1),(-1,0,1,1,0),(-1,1,1,0,0),(1,-1,1,0,0) \}, \end{equation*} \begin{equation*} \label{eq:pro2} W_2=Y_2'\cup \{(-1,1,1,0,0),(-1,1,0,1,0),(-1,1,0,0,1),(-1,0,0,1,1),(1,1,-1,0,0) \}, \end{equation*} or the set obtained by removing a point from $X_2$, $Y_2$, or $Z_2$, up to isomorphism. \end{proposition} \begin{proof}
First suppose $n_i(X,0)=6$ for some $i$. Then we have $|X|\leq 12$, and $X$ with $|X|=12$ is $X_2$, $Y_2$, or $Z_2$ by Lemma~\ref{lem:induc}.
In order to find $X$ with $|X|=11$, we consider $5$-point independent sets in the diameter graph of $S_2(1) \cup U_2(2)$ or $S_2(1) \cup U_2(1) \cup \mathcal{S}_1$. If $X$ is not isomorphic to a subset of $X_2$, $Y_2$, or $Z_2$, then $X=V_2$ from $S_2(1) \cup U_2(2)$, and $X=W_2$ from $S_2(1)\cup U_2(1) \cup \mathcal{S}_1$.
Suppose $n_i(X,0)\leq 5$ for any $i$.
If $|X|\geq 11$, then the average of $n_i(X,0)$ is greater than $4$. Without loss of generality, we may assume $n_1(X,0)=5$. Since the diameter graph of $L_{112}$ is $C_4\cup C_4 \cup C_4$, we can show that $X$ contains a $5$-point subset of $X_2'$ or $Y_2'$.
(i) Suppose $X$ contains a $5$-point subset of $X_2'$. By considering the automorphism group of $X_2'$, we may assume $X$ contains the $5$-point subset obtained by removing $(0,-1,0,1,1)$ or $(0,0,-1,1,1)$. First assume that $X$ contains the $5$-point subset obtained by removing $(0,-1,0,1,1)$. Since other candidates of elements of $X$ are still in $S_2(1)\cup U_2(2)$, we have
$|X|\leq 11$, and if $|X|=11$, then $X$ is isomorphic to a subset of $X_2$, $Y_2$, or $Z_2$. Assume that $X$ contains the $5$-point subset obtained by removing $(0,0,-1,1,1)$. The set of other candidates of elements of $X$ is $S_2(1)\cup U_2(2)\cup \{(1,0,1,-1,0),(1,0,1,0,-1) \}$. If $X$ does not contain both
$(1,0,1,-1,0)$ and $(1,0,1,0,-1)$, then $|X| \leq 11$, and $X$ attaining this bound is isomorphic to a subset of $X_2$, $Y_2$, or $Z_2$. To make a new set, $X$ may contain $(1,0,1,-1,0)$. The two vectors $(-1,1,0,1,0), (-1,0,0,1,1) \in S_2(1)$, which are at distance $\sqrt{10}$ from $(1,0,1,-1,0)$, are not contained in $X$. The set $P_1$ consisting of the two isolated vertices \[ (-1,0,1,0,1),(-1,0,1,1,0) \in S_2(1) \] and $6$ points \[ (-1,1,1,0,0),(-1,1,0,0,1),(1,-1,1,0,0), (1,-1,0,1,0),
(1,-1,0,0,1),(1,0,1,0,-1) \] has the unique maximum $6$-point independent set \[ \{(-1,0,1,0,1),(-1,0,1,1,0), (1,-1,1,0,0), (1,-1,0,1,0),
(1,-1,0,0,1),(1,0,1,-1,0)\}, \] which gives $X$ isomorphic to $Y_2$, and $n_2(X,0)=6$. If $X$ contains a $5$-point independent set in $P_1$ and is not isomorphic to a subset of $Y_2$, then $X$ contains the $5$-point independent set \[ \{(-1,0,1,0,1),(-1,0,1,1,0), (-1,1,1,0,0),(1,-1,1,0,0),(1,0,1,0,-1)\}. \] Then $X$ is isomorphic to $W_2$ and
$n_2(X,0)=6$.
(ii) Suppose $X$ contains a $5$-point subset of $Y_2'$. By considering the automorphism group of $Y_2'$, we may assume $X$ contains the $5$-point subset obtained by removing $(0,1,-1,0,1)$. The set of other candidates of elements of $X$ is $S_2(1)\cup \mathcal{S}_1\cup \{(1,0,1,0,-1)\}$.
To make a new set, $X$ may contain $(1,0,1,0,-1)$. The two vectors $(-1,1,0,0,1),(-1,0,0,1,1) \in S_2(1)$, which are at distance $\sqrt{10}$ from $(1,0,1,0,-1)$, are not contained in $X$. The set consisting of the two isolated vertices \[ (-1,1,1,0,0),(-1,1,0,1,0) \in S_2(1) \] and $5$ points \[ (-1,0,1,1,0), (-1,0,1,0,1), (1,1,-1,0,0), (1,1,0,-1,0), (1,1,0,0,-1) \] has
the unique maximum $5$-point independent set \[ \{(-1,1,1,0,0),(-1,1,0,1,0), (1,1,-1,0,0), (1,1,0,-1,0), (1,1,0,0,-1)\}, \] which gives $X$ is isomorphic to a subset of $Z_2$. \end{proof} \begin{proposition}\label{prop:k3} Let $X\subset L_{132}$ with $D(X)<D(L_{132})$. Then we have \[
|X| \leq \mathfrak{M}_3=22. \] If equality holds, then $X=X_3$, $Y_3$, or $Z_3$, up to isomorphism. \end{proposition} \begin{proof} If $n_i(X,0)=12$ for some $i$, then
we have $|X|\leq 22$, and the set attaining this bound is $X_3$, $Y_3$, or $Z_3$ by Lemma~\ref{lem:induc}.
Suppose $n_i(X,0)\leq 11$ for any $i$.
If $|X|>22$, then the average of $n_i(X,0)$ is greater than $11$, which gives a contradiction. Therefore $|X|\leq 22$, and if $|X|=22$, then the average of $n_i(X,0)$ is $11$, and $n_i(X,0)=11$ for any $i$. By Proposition~\ref{prop:k2}, $X$ may contain \[ V_3'=\{(0,v) \in L_{132} \mid v \in V_2 \}, \] \[ W_3'=\{(0,w) \in L_{132} \mid w \in W_2 \}, \] or an $11$-point set obtained by removing a point from $X_3'$, $Y_3'$, or $Z_3'$.
(i) Suppose $X$ contains an $11$-point subset of $X_3'$. By considering the automorphism group of $X_3'$, $X$ may contain the set in $X_3'$ obtained by removing $(0,-1,0,0,1,1)$, $(0,-1,1,1,0,0)$, $(0,0,-1,0,1,1)$, or $(0,0,0,-1,1,1)$. If $X$ contains the set $X_3'$ with
$(0,-1,0,0,1,1)$, $(0,-1,1,1,0,0)$, or $(0,0,-1,0,1,1)$ removed, then the set of other candidates of $X$ is still $S_3(1)\cup U_3(3)$, and $|X|<22$.
Suppose $X$ contains the set $X_3'$ with $(0,0,0,-1,1,1)$ removed. Then new candidates of vectors of $X$ are only $(1,0,0,1,-1,0)$ and $(1,0,0,1,0,-1)$, and $X$ may contain $(1,0,0,1,-1,0)$. The three vectors $(-1,1,0,0,1,0)$, $(-1,0,1,0,1,0)$, and $(-1,0,0,0,1,1)$, which are at distance $\sqrt{10}$ from $(1,0,0,1,-1,0)$, are not contained in $X$. Therefore by
$|X|=22$, the other new candidate $(1,0,0,1,0,-1)$, and two isolated vectors $(-1,0,0,1,0,1)$, and $(-1,0,0,1,1,0)$ must be contained in $X$. Moreover a $7$-point independent set must be obtained from Matching (i). Since $(-1,1,0,0,1,0)$ and $(-1,0,1,0,1,0)$ are not contained in $X$, by Lemma~\ref{lem:match}, $(1,-1,0,0,0,1)$ and $(1,0,-1,0,0,1)$ must be contained in $X$, and consequently any element of $U_2(2)$ is contained in $X$. This implies $n_2(X,1)=0$, and $X$ is isomorphic to $X_3$, $Y_3$, or $Z_3$.
(ii) Suppose $X$ contains an $11$-point subset of $Y_3'$. By considering the automorphism group of $Y_3'$, $X$ may contain the set in $Y_3'$ obtained by removing $(0,-1,0,0,1,1)$, $(0,-1,1,1,0,0)$, or $(0,0,1,-1,0,1)$. If $X$ contains the set $Y_3'$
with $(0,-1,0,0,1,1)$, or $(0,-1,1,1,0,0)$ removed,
then
the set of other candidates of $X$ is still $S_3(1)\cup U_3(2) \cup \mathcal{S}_1$, and $|X|<22$. Suppose $X$ contains the set $Y_3'$
with $(0,0,1,-1,0,1)$ removed. Then a new candidate of an element of $X$ is only $(1,0,0,1,0,-1)$, and $X$ may contain $(1,0,0,1,0,-1)$. The three vectors $(-1,1,0,0,0,1)$, $(-1,0,1,0,0,1)$, and $(-1,0,0,0,1,1)$, which are at distance $\sqrt{10}$ from $(1,0,0,1,0,-1)$, are not contained in $X$. By considering Matching (ii),
we can show $|X|<22$.
(iii) Suppose $X$ contains an $11$-point subset of $Z_3'$. By considering the automorphism group of $Z_3'$, $X$ may contain the set in $Z_3'$ obtained by removing $(0,1,-1,0,0,1)$. Then a new candidate of an element of $X$ is only $(1,0,1,0,0,-1)$, and $X$ may contain $(1,0,1,0,0,-1)$. The three vectors $(-1,1,0,0,0,1)$, $(-1,0,0,1,0,1)$, and $(-1,0,0,0,1,1)$, which are at distance $\sqrt{10}$ from $(1,0,1,0,0,-1)$, are not contained in $X$. By considering Matching (iii),
we can show $|X|<22$.
(iv) Suppose $X$ contains $V_3'$. The set of other candidates of $X$ is $S_3(1)\cup U_3(3)\setminus \{(1,-1,1,0,0,0)\}$,
and the maximum independent set is of order at most $10$ by Matching (i). Thus $|X|<22$.
(v) Suppose $X$ contains $W_3'$. The set of other candidates of $X$ is $S_3(1)\cup U_3(2) \cup \mathcal{S}_1 \setminus \{(1,-1,0,1,0,0)\}$,
and the maximum independent set is of order at most $10$ by Matching (ii). Thus $|X|<22$.
Therefore this proposition follows. \end{proof} Finally we prove Theorem~\ref{thm:main}. \begin{proof}[Proof of Theorem~\ref{thm:main}] By Propositions~\ref{prop:k1}--\ref{prop:k3}, the statement holds for $k=1,2,3$.
By the inductive hypothesis and Lemma~\ref{lem:M_k}, if $|X| \geq \mathfrak{M}_k$, then there exists $i \in \{1,\ldots,n\}$ such that $n_i(X,0) = \mathfrak{M}_{k-1}$ for $k\geq 4$. By Lemma~\ref{lem:induc}, this theorem holds for any $k$. \end{proof} \section{Classification of the largest $4$-distance sets which contain $\tilde{J}(n,4)$ } A finite set $X$ in $\mathbb{R}^d$ is called an $s$-distance set if the set of Euclidean distances of two distinct vectors in $X$ has size $s$. The Johnson graph $J(n,m)=(V,E)$, where \begin{align*} V&=\{\{i_1,\ldots, i_m \} \mid 1 \leq i_1<\cdots < i_m \leq n,i_j \in \mathbb{Z} \}, \\
E&=\{(v,u) \mid |v\cap u|=m-1, v,u \in V\}, \end{align*} is represented into $\mathbb{R}^{n-1}$ as the $m$-distance set $\tilde{J}(n,m)=L_{0,{n-m},m}$. Indeed $\tilde{J}(n,m) \subset \mathbb{R}^n$, but the summation of all entries of any $x \in \tilde{J}(n,m)$ is $m$, and $\tilde{J}(n,m)$ is on a hyperplane isometric to $\mathbb{R}^{n-1}$. Bannai, Sato, and Shigezumi \cite{BSS12} investigated $m$-distance sets containing $\tilde{J}(n,m)$. In their paper, for $m\leq 5$ and any $n$, the largest $m$-distance sets containing $\tilde{J}(n,m)$ are classified except for $(n,m)=(9,4)$. In this section, the case $(n,m)=(9,4)$ will be classified.
The set
of Euclidean distances of two distinct points of $\tilde{J}(9,4)$ is $\{\sqrt{2},\sqrt{4},\sqrt{6}, \sqrt{8}\}$. The set of vectors which can be added to $\tilde{J}(9,4)$ while maintaining $4$-distance is the union of the following sets \cite{BSS12}. \begin{align*} &X^{(i)}=\left( \left( \frac{2}{3}\right)^7, \left(- \frac{1}{3} \right)^2 \right)^P, & &X^{(ii)}=\left( \left( \frac{2}{3}\right)^8, - \frac{4}{3} \right)^P, \\ &X^{(iii)}=\left( \frac{4}{3}, \left( \frac{1}{3} \right)^8 \right)^P, & &X^{(iv)}=\left( \left( \frac{4}{3}\right)^2, \left( \frac{1}{3} \right)^6, - \frac{2}{3} \right)^P, \end{align*}
where the exponents inside indicate the number of occurrences of the corresponding numbers, and the exponent $P$ outside indicates that we should take every permutation. They conjectured that $\tilde{J}(9,4) \cup X^{(i)}\cup X^{(iii)} \cup \{ (-4/3,(2/3)^8 )\}\cup X^{(iv)'}$ is largest, where $(-4/3,(2/3)^8 ) \in X^{(ii)}$, and \begin{align*} X^{(iv)'}=&\left\{(x_1,\ldots,x_9) \in X^{(iv)} \mid x_i=-\frac{2}{3}, x_{j_1}=\frac{4}{3},x_{j_2}=\frac{4}{3}, i < j_1,j_2 \right\} \\ &\cup \left\{\left( \left(\frac{1}{3} \right)^6,\frac{4}{3},-\frac{2}{3},\frac{4}{3}\right), \left( \left( \frac{1}{3}\right)^6,\left( \frac{4}{3}\right)^2,-\frac{2}{3} \right)
\right\}. \end{align*} Actually $ X^{(iv)'} $ is isometric to $X_6$ in Section~\ref{sec:2} by replacing $-2/3$, $1/3$, $4/3$ to $-1$, $0$, $1$, respectively. Let $X^{(iv)''}$ ({\it resp.} $X^{(iv)'''}$) be the set obtained from $Y_6$ ({\it resp.} $Z_6$) by the same manner. Using Theorem~\ref{thm:main}, we can classify the largest $4$-distance sets containing $\tilde{J}(9,4)$. \begin{theorem} Let $X \subset \{(x_1,\ldots,x_9) \in \mathbb{R}^9 \mid x_1+ \cdots +x_9=1\}$ be a $4$-distance set which contains $\tilde{J}(9,4)$. Then we have \[
|X|\leq 258. \] If equality holds, then $X$ is one of the following, up to permutations of coordinates. \begin{enumerate} \item $\tilde{J}(9,4) \cup X^{(i)}\cup X^{(iii)} \cup \{ (-4/3,(2/3)^8 )\}\cup X^{(iv)'}$, \item $\tilde{J}(9,4) \cup X^{(i)}\cup X^{(iii)} \cup \{ (-4/3,(2/3)^8 )\}\cup X^{(iv)''}$, \item $\tilde{J}(9,4) \cup X^{(i)}\cup X^{(iii)} \cup \{ (-4/3,(2/3)^8 )\}\cup X^{(iv)'''}$. \end{enumerate} \end{theorem} \begin{proof} For any $x \in X^{(i)}\cup X^{(iii)}$, $y \in \cup_{j=1}^4 X^{(j)}$, the Euclidean distance of $x$, $y$ is in $\{\sqrt{2},\sqrt{4},\sqrt{6}, \sqrt{8}\}$, and hence $X$ may contain $X^{(i)}\cup X^{(iii)}$.
The set $X^{(iv)}$ is isometric to $L_{162}$ by replacing $-2/3$, $1/3$, $4/3$ to $-1$, $0$, $1$, respectively.
Therefore the largest subsets of $X^{(iv)}$ with distances $\{\sqrt{2},\sqrt{4},\sqrt{6}, \sqrt{8}\}$
are $X^{(iv)'}$, $X^{(iv)''}$, and $X^{(iv)'''}$, up to permutations of coordinates. If $X$ does not contain any element of $X^{(ii)}$, then \[
|X|\leq |\tilde{J}(9,4)\cup X^{(i)}\cup X^{(iii)}| +|X^{(iv)'}|= 257. \]
If $X$ contains $x \in X^{(ii)}$ with $x_i=-4/3$, then $X$ cannot contain $y \in X^{(iv)}$ with $y_i=4/3$. By re-ordering the vectors, we may assume that the set \[ X^{(ii)}(t)=\{x \in X^{(ii)} \mid x_i=-4/3, \exists i \in \{1,\ldots, t\} \} \] is in $X$ for some $t$. Clearly, from the definition of $X^{(ii)}(t)$, this set must have size $t$.
For $t=7,8,9$, $X$ contains at most one element of $X^{(iv)}$, and hence
\[|X| \leq |\tilde{J}(9,4)\cup X^{(i)}\cup X^{(iii)}| + t +1\leq 181. \]
If the set $X^{(ii)}(t)$ is in $X$ for $1\leq t \leq 6$, then consider the set of vectors in $X \cap X^{(iv)}$ in which the entry $1/3$ occurs in all of the first $t$ positions. The final $9-t$ entries of one of these vectors forms a vector from $L_{1,6-t,2}$; no two vectors in this set can be at the maximum distance. Thus the size of \[
|\{x \in X \cap X^{(iv)} \mid x_i=1/3, \forall i \in \{1,\ldots, t\} \}| \] is bounded by $\mathfrak{M}_{6-t}$. It is clear that \[
|\{x \in X \cap X^{(iv)} \mid x_i=-2/3, x_{j_1}=4/3, x_{j_2}=4/3, \exists i \in \{1,\ldots,t\}, \exists j_1,j_2 \in \{t+1,\ldots, 9\}
\}| \] is bounded by $t \binom{9-t}{2}$. Thus, for $1\leq t \leq 6$, we have \begin{align*}
|X|& \leq |\tilde{J}(9,4) \cup X^{(i)}\cup X^{(iii)}|+t+\mathfrak{M}_{6-t}+t \binom{9-t}{2}\\ &= \frac{t^3}{3}-\frac{9 t^2}{2}+ \frac{31 t}{6}+257\leq 258, \end{align*} and equality holds only if $t=1$. The sets attaining this bound are only the three sets in the statement. \end{proof}
\section{Remarks on other $M_{mkl}$}
Actually it is hard to determine $M_{mkl}$ for other $(m, k, l)$ by a similar manner in Section~\ref{sec:2}. Fix $m,l$, where $m < l$. By Proposition~\ref{prop:m+k<=l}, if $k\leq l-m$, then $M_{mkl}=\binom{n-1}{m+k-1}\binom{m+k}{m}$. In general there are many largest sets for $k=l-m$. For $k> l-m$, we can inductively construct a large set $X_{k}\subset L_{mkl}$ satisfying $D(X_{k})<D(L_{mkl})$ as follows \[ X_{k}=\{(0,x') \mid x' \in X_{k-1}\}\cup \{(x_1,\ldots,x_n) \in L_{mkl} \mid x_1= -1 \}, \] where $X_{l-m}$ is a largest set for $k=l-m$. Therefore we have \[ M_{mkl} \geq \mathfrak{M}_{mkl}:= \binom{m+l-1}{m-1} \binom{k+m+l}{m+l}+\binom{m+l-1}{m}. \] We can generalize Lemma~\ref{lem:M_k} as follows. \begin{lemma} \label{lem:M_mkl} Let $X\subset L_{mkl}$ with $D(X)\leq D(L_{mkl})$.
Suppose $k\geq m \binom{m+l}{m}-m-l+1$, and $|X| \geq \mathfrak{M}_{mkl}$. Then there exists $i \in \{1,\ldots,n\}$ such that $n_i(X,0)\geq \mathfrak{M}_{m,k-1,l}$. \end{lemma} \begin{proof} This lemma is immediate because the average of $n_i(X,0)$ is \[
\frac{1}{n}\sum_{i=1}^{n}n_{i}(X,0)=\frac{k|X|}{m+k+l} \geq \frac{k\mathfrak{M}_{mkl}}{m+k+l} =\mathfrak{M}_{m,k-1,l}-\frac{m+l}{m+k+l}\binom{m+k+l}{l} >\mathfrak{M}_{k-1}-1. \qedhere \] \end{proof} In the manner of Section \ref{sec:2}, it is hard to classify $M_{mkl}$ for $m-l+1 \leq k \leq m \binom{m+l}{m}-m-l$. Moreover it seems to be difficult to give matchings, like Matching ${\rm (i)}$ or ${\rm (ii)}$, of many possibilities of $X_k$. We need another idea to determine other $M_{mkl}$.
\noindent \textbf{Acknowledgments.} The authors thank Sho Suda for providing useful information. The second author is supported by JSPS KAKENHI Grant Numbers 25800011, 26400003.
\end{document} | arXiv | {
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\begin{document}
\title{A
nonlocal approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties}
\begin{abstract} We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter. \end{abstract}
\section{Introduction}
The Gaussian isoperimetric inequality says that the halfspace has the smallest Gaussian perimeter among all sets with prescribed Gaussian measure, \cite{Borell}. In the Euclidean setting, an increasing interest has been devoted to the study of non local approximations of the perimeter and their isoperimetric shapes, since the pioneering work of Caffarelli, Roquejoffre and Savin, \cite{CRS}.
The aim of this paper is to provide an analogous non local approximation of the Gaussian perimeter, showing the Gamma convergence to the local one. Moreover, we study the isoperimetric properties of this non local functional and observe that, in contrast with the local setting, an halfspace is a volume constrained critical point if and only if it has Gaussian measure $\frac 12$. In particular, we deduce that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.
We remark that the non local approximation of the Gaussian perimeter we study is different from the one recently proposed in \cite{NPS}. The non local functional we introduce has the advantage of having a more explicit formulation, while has the drawback that the isoperimetric shapes and the Ehrhard symmetrization are not preserved.
Inspired by \cite{ADPM}, for a measurable set $E\subset\R{n}$, $n\ge 1$, $0<s<1$, and a connected, open set $\Omega\Subset\R{n}$ with Lipschitz boundary (or simply $\Omega=(a,b)\Subset\R{}$ if $n=1$), we define the Gaussian, nonlocal functional $$\M{J}^\gamma_s(E, \Omega):=\M{J}^{1, \, \gamma}_s(E,\Omega)+\M{J}^{2, \, \gamma}_s(E, \Omega),$$ where \begin{equation*} \begin{split}
\M{J}^{1, \, \gamma}_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap \Omega}\frac{\gamma(x, \, y)}{|x-y|^{n+s}} \, dxdy,\\ \M{J}^{2, \, \gamma}_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap
\Omega^c}\frac{\gamma(x, \, y)}{|x-y|^{n+s}} \, dxdy+ \int_{E\cap
\Omega^c}\int_{E^c\cap \Omega}\frac{\gamma(x, \, y)}{|x-y|^{n+s}} \, dxdy, \end{split} \end{equation*} and \[ \gamma \fromto{\R{n} \times \R{n}}{\R{+}},\ \gamma(x, \, y) = \exp\left(- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)\right). \] When $\Omega$ coincides with the whole space, we just write $\M{J}^{\gamma}_s(E)$.
In \cite{ADPM}, Amborsio, De Philippis and Martinazzi have studied the Euclidean version of it, namely $\M{J}_s = \M{J}_s^1 + \M{J}_s^2$, and \begin{equation*} \begin{split}
\M{J}^1_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap \Omega}\frac{1}{|x-y|^{n+s}} \, dxdy,\\ \M{J}^2_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap
\Omega^c}\frac{1}{|x-y|^{n+s}} \, dxdy+ \int_{E\cap
\Omega^c}\int_{E^c\cap \Omega}\frac{1}{|x-y|^{n+s}} \, dxdy. \end{split} \end{equation*} The authors point out that $\M{J}_s(E,\Omega)$ can be thought of as a fractional perimeter of $E$ in $\Omega$, and they show the $\Gamma$-convergence of $(1-s)\M{J}_s(\cdot,\Omega)$ to $\omega_{n-1}P(\cdot,\Omega)$ as $s\to 1^-$, where $\omega_{n-1}$ is the volume of the unit ball in $\R{n-1}$, $P(E, \, \Omega):= \mathcal{H}^{n-1}(\M{F} E \cap \Omega)$ is the Euclidean perimeter, $\mathcal{H}^\alpha$ denotes the classical $\alpha$-Hausdorff measure and $\M{F} E$ the reduced boundary of $E$. Moreover, they prove the convergence of any sequence $\{E_i\}$ of local minimizers for $\M{J}_{s_i}(\cdot, \Omega)$ to a local minimizer for $P(\cdot,\Omega)$, see \cite[Theorem 3]{ADPM}.
The first aim of this paper is to generalize \cite[Theorem 3]{ADPM} to the Gaussian case, thus building a relation between the functional $\M{J}^{\gamma}_s$ and the Gaussian perimeter \[ P^\gamma(E, \, \Omega):= \int_{\M{F} E \cap \Omega} e^{-\frac{1}{2} \abs{x}^2} \, d\mathcal{H}^{n-1}(x). \] The second goal is to investigate whether the halfspaces are volume constrained critical points of $\M{J}^{\gamma}_s$. This turns out to be true if and only if the boundary hyperplane passes through the origin.
The paper is divided in four Sections. In Section 2 we prove the $\Gamma-$convergence of the functional $\M{J}^{\gamma}_s$ to $P^\gamma$.
In Section 3 we compute the first and second variation of $\M{J}^{\gamma}_s$
(for the local framework see \cite{BBJ} or \cite{L}). In Section 4 we prove that halfspaces are volume constrained stationary points for $\M{J}^{\gamma}_s$ if and only if their Gaussian volume is $\frac 12$.
\section{The Gamma-convergence } In this section we extend \cite[Theorem 3]{ADPM} to the Gaussian case. Namely, we show: \begin{trm}[Convergence of local minimizers]\label{trm4} Assume that $s_i\uparrow 1$, $E_i$ are local minimizers of $\M{J}^{\gamma}_{s_i}(\cdot, \Omega)$, and $\chi_{E_i}\to\chi_E$ in $L^1_{\loc}(\R{n})$. Then \begin{equation}\label{Genergybou} \limsup_{i\to \infty}(1-s_i)\M{J}^{\gamma}_{s_i}(E_i,\Omega')< + \infty \qquad \forall\Omega'\Subset\Omega, \end{equation} $E$ is a local minimizer of $P^{\gamma}(\cdot,\Omega)$ and $(1-s_i)\M{J}^{\gamma}_{s_i}(E_i,\Omega')\to \omega_{n-1}P(E,\Omega')$ whenever $\Omega'\Subset\Omega$ and $P(E,\partial\Omega')=0$. \end{trm} The proof of Theorem \ref{trm4} is almost identical to the Euclidean one for \cite[Theorem 3]{ADPM}. We limit our study to the parts which differ from it. In particular we will prove the following two propositions. Let $\omega_k$ denote the volume of the unit ball in $\R{k}$ for $k\ge 1$, and set $\omega_0:=1$. \begin{prop}\label{GammaLiminfProp} For every measurable set $E\subset\R{n}$ we have \begin{equation}\label{Gliminf} \Gamma-\liminf_{s\uparrow 1}(1-s)\M{J}^{1, \, \gamma}_{s}(E,\Omega) \geq \omega_{n-1}P^\gamma(E,\Omega) \end{equation} w.r.t. the $L^1_{\rm loc}$ convergence of the corresponding characteristic functions in $\R{n}$, i.e. $$\liminf_{i\to \infty}(1-s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i,\Omega)\geq \omega_{n-1}P^\gamma(E,\Omega)\qquad \text{whenever } \chi_{E_i}\to \chi_E \text{ in } L^1_{\loc}(\R{n}),\;s_i\uparrow 1.$$ \end{prop}
\begin{prop}\label{GammaLimsupProp} For every measurable set $E\subset\R{n}$ we have \begin{equation}\label{Glimsup} \Gamma-\limsup_{s\uparrow 1}(1-s)\M{J}^\gamma_{s}(E,\Omega) \leq \omega_{n-1}P^\gamma(E,\Omega) \end{equation} w.r.t. the $L^1_{\rm loc}$ convergence of the corresponding characteristic functions in $\R{n}$. Inequality \eqref{Glimsup} means that for every measurable set $E$ and sequence $s_i\uparrow 1$ there exists a sequence $E_i$ with $\chi_{E_i}\to \chi_E$ in $L^1_{\loc}(\R{n})$ such that $$\limsup_{i\to\infty}(1-s_i)\M{J}^\gamma_{s_i}(E_i,\Omega)\leq \omega_{n-1} P^\gamma(E,\Omega).$$ \end{prop}
In these two propositions lurk the main differences between the Gaussian case and the Euclidean case. Once we have proved Proposition \ref{GammaLiminfProp} and Proposition \ref{GammaLimsupProp}, we are done: the proof of Theorem \ref{trm4} is completely identical to the proof of \cite[Theorem 3]{ADPM}, with the only forethought of adding a $\gamma$-superscript in every considered functional, and remembering the simple inequality $\gamma(x, \, y) \le 1$.
We will use the following notation: we write $x\in\R{n}$ as $(x',x_n)$ with $x'\in\R{n-1}$ and $x_n\in\R{}$; we denote by $H$ the halfspace $\{x:\ x_n\leq 0\}$ and by $Q=(-1/2,1/2)^n$ the canonical unit cube; we denote by $B_r(x)$ the ball of radius $r$ centered at $x$ and, unless otherwise specified, $B_r:= B_r(0)$; for every $h\in \R{n}$ and function $u$ defined on $U\subset \R{n}$ we set $\tau_h u(x):= u(x+h)$ for all $x\in U-h$. For the definition and basic properties of the perimeter $P(E,\Omega)$ in the sense of De Giorgi, we refer to \cite{AFP,giu}.
\subsection{Proof of proposition \ref{GammaLiminfProp}} We denote by ${\mathcal C}$ the family of all $n$-cubes in $\R{n}$ $$ {\mathcal C}:=\left\{R(x+rQ):\ x\in\R{n},\,\,r>0,\,\,R\in SO(n)\right\}. $$
Let $s_i\uparrow 1$ and sets $E_i\subset\R{n}$ with $\chi_{E_i}\to \chi_E$ in $L^1_{\loc}(\R{n})$ as $i\to \infty$ be given. We need to show the inequality \begin{equation}\label{liminf} \liminf_{i\to\infty}(1-s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i,\Omega)\ge \omega_{n-1} P^{\gamma}(E,\Omega). \end{equation} We can assume that the left-hand side of \eqref{liminf} is finite, otherwise the inequality is trivial. We choose an arbitrary $\Omega' \Subset \Omega$, and find a positive constant $c_0=c_0(\Omega')$ so that $c_0 \le \gamma(x, \, y),\ \forall \, x, \, y \in \Omega'$. Then we easily obtain the inequality \[ c_0 \limsup_i \M{J}^1_{s_i} (1-s_i) (E_i, \, \Omega'_i) \le \lim_i (1- s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i, \, \Omega') < +\infty . \] By \cite[Theorem 1]{ADPM} and the arbitrariness of $\Omega'$, we conclude that $E$ has locally finite perimeter. We shall denote by $\mu$ its perimeter measure, i.e.
$\mu(A)=|D\chi_E|(A)$ for any Borel set $A\subset\Omega$, and we shall use the following property of sets of finite perimeter: for $\mu$-a.e. $x_0\in\Omega$ there exists $R_{x_0}\in SO(n)$ such that $(E-x_0)/r$ locally converge in measure to $R_{x_0}H$ as $r\to 0$. In addition, \begin{equation}\label{densitycube} \lim_{r\to 0}\frac{\mu(x_0+rR_{x_0}Q)}{r^{n-1}}=1,\quad \text{for $\mu$-a.e. } x_0. \end{equation} Indeed this property holds for every $x_0\in \M{F} E$, see \cite[Theorem 3.59(b)]{AFP}.
Now, given a cube $C\in{\mathcal C}$ contained in $\Omega$, we set $$\alpha_i(C):=(1-s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i,C), \qquad \mbox{and} \qquad \alpha (C):=\liminf_{i\to\infty}\alpha_i(C).$$ Moreover, we define $C_r(x_0):=x_0+rR_{x_0}Q$, where $R_{x_0}$ is as in \eqref{densitycube}, and the measure \[ \nu(E) = \int_{E} e^{-\frac{1}{2} \abs{x}^2} \, d\mu(x),\qquad \mbox{for every } E \mbox{ Borel set}. \] We claim that for $\mu$-a.e. $x_0\in \R{n}$ it holds \begin{equation}\label{eq1} \omega_{n-1} \le \liminf_{r\to 0}\frac{\alpha(C_r(x_0))}{\nu( C_r(x_0))}. \end{equation} If the claim is true, then we observe that for all $\varepsilon>0$ the family $$\M{A}:=\Big\{C_r(x_0)\subset\Omega\;:\; \omega_{n-1} \nu(C_r(x_0)) \le (1+\varepsilon) \alpha(C_r(x_0)) \Big\}$$ is a fine covering of $\mu$-almost all of $\Omega$. By a suitable variant of Vitali's theorem (see \cite{M}), we can extract a countable subfamily of disjoint cubes $\{C_j\subset\Omega:j\in J\}$ such that $\nu\big(\Omega\setminus\bigcup\limits_{j\in J} C_j\big)=0$, whence \begin{equation*} \begin{split} \omega_{n-1} P^\gamma(E,\Omega)&=\omega_{n-1} \nu\Big(\bigcup_{j\in J}C_j\Big)=\omega_{n-1}\sum_{j\in J}\nu(C_j)\leq (1+\varepsilon)\sum_{j\in J}\alpha(C_j)\le (1+\varepsilon)\liminf_{i\to\infty}\sum_{j\in J}\alpha_i(C_j)\\ &\leq (1+\varepsilon)\liminf_{i\to\infty}(1-s_i)\M{J}^{1,\gamma}_{s_i}(E_i,\Omega). \end{split} \end{equation*} Since $\varepsilon>0$ is arbitrary, we get the $\Gamma-\liminf$ estimate.
We now prove the inequality in \eqref{eq1} at any point $x_0$ such that $(E-x_0)/r$ converges locally in measure as $r\to 0$ to $R_{x_0}H$ and \eqref{densitycube} holds. Because of \eqref{densitycube} and the continuity of the exponential, we know that \[ \lim_{r \to 0} \Intm_{C_r(x_0)} e^{- \frac{1}{2} \abs{x}^2 } \, d\mu(x) = e^{-\frac{1}{2} \abs{x_0}^2}. \] Thus we just need to show the inequality \begin{equation}\label{eq11} \liminf_{r\to 0}\frac{\alpha(C_r(x_0))}{r^{n-1}}\geq \omega_{n-1} e^{-\frac{1}{2} \abs{x_0}^2}. \end{equation} Since from now on $x_0$ is fixed, we assume $R_{x_0}=I$, so that the limit hyperplane is $H$ and the cubes $C_r(x_0)$ are the standard ones $x_0+rQ$. Let us choose a sequence $r_k\to 0$ such that $$\liminf_{r\to 0}\frac{\alpha(C_r(x_0))}{r^{n-1}}= \lim_{k\to \infty} \frac{\alpha(C_{r_k}(x_0))}{r_k^{n-1}}.$$ For $k>0$ we can choose $i(k)$ large enough that the following conditions hold: \begin{equation*} \left\{ \begin{aligned} &\alpha_{i(k)}(C_{r_k}(x_0))\leq \alpha(C_{r_k}(x_0))+r_k^n,\\ &r_k^{1-s_{i(k)}}\ge 1-\frac{1}{k},\\
&\Intm_{C_{r_k}(x_0)}|\chi_{E_{i(k)}}-\chi_E|dx<\frac{1}{k}. \end{aligned} \right. \end{equation*} We observe that, although $\M{J}^{1, \, \gamma}_s$ does not enjoy the nice scaling properties of $\M{J}^{1}_s$, it still satisfies the equality \[ \M{J}^{1, \, \gamma}_s (E, \, C_r(x_0))= r^{n - s}\M{J}^{1, \, \gamma_{x_0, \, r}}_s((E- x_0)/r, \, Q), \] where we have set \[ \gamma_{x_0, \, r}(x, \, y) = \exp \left(- \frac{1}{4} \left( \abs{x_0 + rx}^2 + \abs{x_0 + r y}^2 \right)\right). \] In particular, for $r$ sufficiently small, and thus for every $r_k$ with $k$ sufficiently big, the following inequality holds: \[ \abs{\gamma_{x_0, \, r}(x, \, y) - e^{- \frac{1}{2} \abs{x_0}^2}} \le 4 r. \] Then we infer \begin{equation*} \begin{split} \frac{\alpha(C_{r_k}(x_0))}{r_k^{n-1}}&\ge \frac{\alpha_{i(k)}(C_{r_k}(x_0))}{r_k^{n-1}}-r_k=\frac{(1-s_{i(k)})\M{J}^{1, \, \gamma_{x_0, \, r_k}}_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q)r_k^{n-s_{i(k)}}}{r_k^{n-1}}-r_k\\ &\ge \Big(1-\frac{1}{k}\Big)(1-s_{i(k)})\M{J}^{1, \, \gamma_{x_0, \, r_k}}_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q)-r_k \\ &\ge \Big(1-\frac{1}{k}\Big)(1-s_{i(k)})\M{J}^{1}_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q)(e^{- \frac{1}{2} \abs{x_0^2}} - r_k)-r_k, \end{split} \end{equation*} i.e. $$\lim_{k\to\infty} \frac{\alpha(C_{r_k}(x_0))}{r_k^{n-1}}\ge e^{- \frac{1}{2} \abs{x_0}^2} \liminf_{k\to\infty}(1-s_{i(k)})\M{J}^1_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q).$$ Since we have
$$\lim_{k\to\infty}\int_{Q}|\chi_{(E_{{i(k)}}-x_0)/r_k}-\chi_{(E-x_0)/r_k}|dx=0,$$ and
$$\lim_{k\to\infty}\int_{Q}|\chi_{(E-x_0)/r_k}-\chi_H|dx=0,$$ it follows that $(E_{{i(k)}}-x_0)/r_k\to H$ in $L^1(Q)$. If we define \begin{equation}\label{defGamman}
\Gamma_n:=\inf\Big\{\liminf_{s\uparrow 1}(1-s)\M{J}^1_s(E_s,Q)\;\Big|\; \chi_{E_s}\to \chi_H\text{ in }L^1(Q)\Big\}, \end{equation} it has been proved in \cite[Lemmata 7, 11, 12]{ADPM} that $\Gamma_n=\omega_{n-1}$. Hence we conclude the claimed inequality \eqref{eq11}.
\subsection{Proof of proposition \ref{GammaLimsupProp}} As in \cite{ADPM}, it is enough to prove the $\Gamma-\limsup$ inequality for the collection $\M{B}$ of polyhedra $\Pi$ of finite perimeter which satisfy $P(\Pi,\partial\Omega)=0$. $\M{B}$ is dense in energy, i.e. such that for every set $E$ of finite perimeter there exists $E_k\in\M{B}$ with $\chi_{E_k}\to \chi_E$ in $L^1_{\loc}(\R{n})$ as $k\to\infty$ and $\limsup_kP^\gamma(E_k,\Omega)=P^\gamma(E,\Omega)$. We recall that a polyhedron $\Pi$ is in the class $\M{B}$ if and only if $$\lim_{\delta\to 0}P(\Pi,\Omega^+_\delta\cup \Omega^-_\delta)=0,\quad \mbox{ or equivalently} \quad \lim_{\delta\to 0}P^\gamma(\Pi,\Omega^+_\delta\cup \Omega^-_\delta)=0, $$ where we have set \begin{equation}\label{omegadelta} \begin{split}
\Omega^+_\delta:=\{x\in \Omega^c\;|\;d(x,\Omega)<\delta\}, \qquad \Omega^-_\delta:=\{x\in \Omega\;|\;d(x,\Omega^c)<\delta\}. \end{split} \end{equation}
We are going to prove that for a polyhedron $\Pi\subset\R{n}$ there holds \begin{equation}\label{poly} \limsup_{s\uparrow 1}(1-s)\M{J}^\gamma_s(\Pi,\Omega)\leq \Gamma_n^* P^\gamma(\Pi,\Omega)+ 2\Gamma_n^* \lim_{\delta\to 0} P^\gamma(\Pi,\Omega^+_\delta\cup \Omega^-_\delta), \end{equation} where \begin{equation}\label{gamman*} \Gamma_n^*:= \limsup_{s\uparrow 1}(1-s)\M{J}^1_s(H,Q). \end{equation} Again, as in \cite[Lemmata 7, 11, 12]{ADPM} we have the equality $\Gamma_n^* = \omega_{n-1}$. We shall divide the proof into two main steps.
\emph{Step 1.} We first estimate $\M{J}_s^{1, \, \gamma}(\Pi, \Omega)$. For a fixed $\varepsilon>0$ set \begin{equation*}
(\partial\Pi)_\varepsilon:=\{x\in\Omega\;|\; d(x,\partial\Pi)<\varepsilon\},\quad (\partial\Pi)_\varepsilon^-:=(\partial\Pi)_\varepsilon \cap \Pi. \end{equation*} We can find $N_\varepsilon$ disjoint cubes $Q^\varepsilon_i\subset\Omega$, $1\leq i\le N_\varepsilon$, of side length $\varepsilon$ satisfying the following properties: \begin{itemize} \item[(i)] if $\tilde Q_i^\varepsilon$ denotes the dilation of $Q_i^\varepsilon$ by a factor $(1+\varepsilon)$, then each cube $\tilde Q_i^\varepsilon$ intersects exactly one face $\Sigma$ of $\partial\Pi$, its barycenter belongs to $\Sigma$ and each of its sides is either parallel or orthogonal to $\Sigma$; \item[(ii)] $\M{H}^{n-1}\left(((\partial\Pi)\cap \Omega)\setminus \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon
\right)=|P(\Pi,\Omega)-N_\varepsilon\varepsilon^{n-1} |\to 0$ as $\varepsilon \to 0$. \end{itemize} Property (ii), combined with the continuity of the exponential and the property of measures, easily implies \begin{equation}\label{expdecay} \abs{P^\gamma(\Pi, \Omega) - \varepsilon^{n-1} \sum_{i=1}^{N_\varepsilon} e^{-\frac{1}{2} \abs{x_i^\varepsilon}^2 } } \to 0 \mbox{ as } \varepsilon \to 0, \end{equation} where we have set by $x_i^\varepsilon$ the center of the cubes $Q^\epsilon_i$. For $x\in\R{n}$ set
$$I_s(x):=\int_{\Pi^c\cap \Omega}\frac{e^{-\frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy.$$ We consider several cases.
\noindent\emph{Case 1:} $x\in (\Pi\cap \Omega)\setminus(\partial\Pi)_\varepsilon^-$. Then for $y\in \Pi^c\cap \Omega$ we have $|x-y|\geq \varepsilon$, hence
$$I_s(x)\leq \int_{(B_\varepsilon(x))^c} \frac{1}{|x-y|^{n+s}}dy=n\omega_n\int_\varepsilon^\infty \frac{1}{\rho^{s+1}}d\rho=\frac{n\omega_{n}}{s\varepsilon^s},$$ since $n\omega_n=\M{H}^{n-1}(S^{n-1})$. Therefore \begin{equation}\label{case1} \int_{(\Pi\cap \Omega)\setminus(\partial\Pi)_\varepsilon^-} I_s(x) e^{- \frac{1}{4} \abs{x}^2} \, dx\leq \frac{n\omega_{n} }{s\varepsilon^s} \int_{\Pi\cap \Omega} e^{- \frac{1}{4} \abs{x}^2} \, dx. \end{equation}
\noindent\emph{Case 2:} $x\in (\partial\Pi)_\varepsilon^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon$. Then \begin{equation}\label{case2.0} I_s(x)\leq\int_{(B_{d(x,\Pi^c \cap
\Omega)}(x))^c}\frac{1}{|x-y|^{n+s}}dy= n\omega_{n}\int_{d(x,\Pi^c\cap\Omega)}^\infty\frac{1}{\rho^{s-1}}d\rho=\frac{n\omega_{n}}{s[d(x,\Pi^c\cap\Omega)]^s}. \end{equation} Now write $(\partial\Pi)\cap \Omega=\bigcup_{j=1}^J\Sigma_j$, where each $\Sigma_j$ is the intersection of a face of $\partial\Pi$ with $\Omega$, and define $$(\partial\Pi)^-_{\varepsilon,j}:=\{x\in (\partial\Pi)^-_\varepsilon:\dist(x,\Pi^c\cap\Omega)=\dist(x,\Sigma_j)\}.$$ Clearly $(\partial\Pi)^-_\varepsilon=\bigcup_{j=1}^J (\partial\Pi)^-_{\varepsilon,j}$. Moreover we have $$(\partial\Pi)^-_{\varepsilon,j}\subset\{x+t\nu:x\in \Sigma_{\varepsilon,j},\, t\in (0,\varepsilon),\, \nu\text{ is the interior unit normal to }\Sigma_{\varepsilon,j}\}, $$ and $\Sigma_{\varepsilon,j}$ is the set of points $x$ belonging to the same hyperplane as $\Sigma_j$ and with $\dist(x,\Sigma_j)\le \varepsilon$. Clearly $\M{H}^{n-1}(\Sigma_{\varepsilon,j})\le \M{H}^{n-1}(\Sigma_j)+C\varepsilon$ as $\varepsilon\to 0$. Then from \eqref{case2.0} we infer \begin{equation}\label{case2} \begin{split} \int_{(\partial\Pi)_\varepsilon^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}I_s(x) e^{- \frac{1}{4} \abs{x}^2} \, dx&\leq \frac{n\omega_{n}}{s}\sum_{j=1}^J\int_{(\partial\Pi)_{\varepsilon,j}^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}\frac{1}{[d(x,\Pi^c)]^s} \, dx\\ &\leq \frac{n\omega_{n}}{s}\sum_{j=1}^J\int_{(\partial\Pi)_{\varepsilon,j}^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}\frac{1}{[d(x,\Sigma_{\varepsilon,j})]^s} \, dx\\ &\leq\frac{n\omega_{n}}{s}\sum_{j=1}^J\int_{(\Sigma_{\varepsilon,j})\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}\bigg(\int_0^\varepsilon \frac{dt}{t^s}\bigg)\, d\M{H}^{n-1}\\ &= \frac{n\omega_{n}\varepsilon^{1-s}}{s(1-s)}\M{H}^{n-1}\left(\bigg(\bigcup_{j=1}^J\Sigma_{\varepsilon,j}\bigg)\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon\right)=\frac{\varepsilon^{1-s}o(1)}{s(1-s)}, \end{split} \end{equation} with error $o(1)\to 0$ as $\varepsilon\to 0$ and independent of $s$.
\noindent\emph{Case 3:} $x\in \Pi\cap \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon$. In this case we write \begin{equation*} \begin{split}
I_s(x)&=\int_{(\Pi^c\cap \Omega)\cap \{y:|x-y|\geq \varepsilon^2\}}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy +\int_{(\Pi^c\cap \Omega)\cap \{y:|x-y|< \varepsilon^2\}}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy\\ &=:I_s^1(x)+I_s^2(x). \end{split} \end{equation*} Then, similar to the case 1, $$I_s^1(x)\leq n\omega_{n}\int_{\varepsilon^2}^\infty\frac{1}{\rho^{s+1}} \, d\rho=\frac{n\omega_{n}}{s\varepsilon^{2s}},$$ hence (since all cubes are contained in $\Omega$) \begin{equation}\label{case3.1} \int_{\Pi\cap \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}I_s^1(x) e^{-\frac{1}{4} \abs{x}^2 } \, dx\leq \frac{n\omega_{n}}{s\varepsilon^{2s}} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx. \end{equation}
As for $I_s^2(x)$ observe that if $x\in Q_i^\varepsilon$ and $|x-y|\leq \varepsilon^2$, then $y\in \tilde Q_i^\varepsilon$, where $\tilde Q_i^\varepsilon$ is the cube obtained by dilating $Q_i^\varepsilon$ by a factor $1+\varepsilon$ (hence the side length of $\tilde Q_i^\varepsilon$ is $\varepsilon+\varepsilon^2$). Then \begin{equation}\label{case3.2} \begin{split} \int_{\Pi\cap \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}I_s^2(x) e^{- \frac{1}{4} \abs{x}^2} \, dx
&\le\sum_{i=1}^{N_\varepsilon}\int_{\Pi\cap Q_i^\varepsilon}\int_{\Pi^c\cap \tilde Q_i^\varepsilon}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx \\
&\le\sum_{i=1}^{N_\varepsilon}\int_{\Pi\cap \tilde Q_i^\varepsilon}\int_{\Pi^c\cap \tilde Q_i^\varepsilon}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx \\ &\le \left(\sum_{i=1}^{N_\varepsilon} e^{- \frac{1}{2} \abs{x^\varepsilon_i}^2} \right) \M{J}^1_s(H,(\varepsilon+\varepsilon^2) Q)(1 + \varepsilon^2) \\ &= \underbrace{\left( \varepsilon^{n-1 } \sum_{i=1}^{N_\varepsilon} e^{- \frac{1}{2} \abs{x^\varepsilon_i}^2} \right)}_{= P^\gamma(\Pi, \, \Omega) + o(1)} \varepsilon^{1-s}(1+\varepsilon)^{n-s}\M{J}^1_s(H,Q)(1 + \varepsilon^2), \end{split} \end{equation} where in the last identity we used the scaling property \begin{equation}\label{scale} \M{J}^i_s(\lambda E,\lambda\Omega)=\lambda^{n-s}\M{J}^i_s(E,\Omega)\qquad \text{for }\lambda>0, \; i=1,2. \end{equation} Keeping $\varepsilon>0$ fixed, letting $s$ go to $1$ and putting \eqref{case1}-\eqref{case3.2} together, we infer \begin{equation*} \begin{split} \limsup_{s\uparrow 1}(1-s)\M{J}^{1, \, \gamma}_s(\Pi,\Omega) \leq o(1)+\Gamma_n^*P^{\gamma}(\Pi,\Omega) = o(1) + \omega_{n-1} P^{\gamma}(\Pi, \, \Omega), \end{split} \end{equation*} with error $o(1)\to 0$ as $\varepsilon\to 0$ uniformly in $s$. Since $\varepsilon>0$ is arbitrary, we conclude \begin{equation}\label{limsupJ1} \limsup_{s\uparrow 1}(1-s)\M{J}^{1, \, \gamma}_s(\Pi,\Omega) \leq \omega_{n-1} P^{\gamma}(\Pi,\Omega). \end{equation}
\noindent\emph{Step 2.} It now remains to estimate $\M{J}_s^{2,\gamma}$. Let us start by considering the term
$$\int_{\Pi\cap \Omega}\int_{\Pi^c\cap \Omega^c}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx.$$
\noindent\emph{Case 1:} $x\in \Pi\cap (\Omega\setminus\Omega_\delta^-)$. Then for
$y\in \Pi^c\cap\Omega^c$ we have $|x-y|\ge \delta$, whence
$$I(x):=\int_{\Pi^c\cap \Omega^c}\frac{e^{ - \frac{1}{4} \abs{y}^2 }}{|x-y|^{n+s}} \, dy\leq n\omega_{n}\int_\delta^\infty\frac{d\rho}{\rho^{1+s}}=\frac{n\omega_{n}}{s\delta^s}.$$
\noindent\emph{Case 2:} $x\in \Pi\cap \Omega_\delta^-$. In this case, using the same argument of case $1$ for $y\in \Pi^c\cap(\Omega^c\setminus\Omega_\delta^+)$, we have \begin{equation*} \begin{split}
I(x)=\int_{\Pi^c\cap\Omega_\delta^+}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy +
\int_{\Pi^c\cap(\Omega^c\setminus\Omega_\delta^+)}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy\le
\int_{\Pi^c\cap\Omega_\delta^+}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy +\frac{n\omega_{n}}{s\delta^s}. \end{split} \end{equation*} Therefore \begin{equation*} \begin{split} \int_{\Pi\cap \Omega}\int_{\Pi^c\cap \Omega^c}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}
{|x-y|^{n+s}} \, dy dx &\le\frac{2n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx+
\int_{\Pi\cap\Omega_\delta^-}\int_{\Pi^c\cap\Omega_\delta^+}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dy dx\\ &\le\frac{2n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx + \int_{\Pi\cap(\Omega_\delta^-\cup \Omega_\delta^+)}
\int_{\Pi^c\cap(\Omega_\delta^-\cup \Omega_\delta^+)}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx. \end{split} \end{equation*} An obvious similar estimate can be obtained by swapping $\Pi$ and $\Pi^c$, finally yielding \begin{equation*} \begin{split}
\M{J}_s^{2,\gamma}(\Pi,\Omega)&\le \frac{4n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx + 2 \int_{\Pi\cap(\Omega_\delta^-\cup \Omega_\delta^+)}\int_{\Pi^c\cap(\Omega_\delta^-\cup \Omega_\delta^+)}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx\\ &= \frac{4n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx +2\M{J}_s^{1, \, \gamma}(\Pi,\Omega_\delta^-\cup \Omega_\delta^+). \end{split} \end{equation*} Using inequality \eqref{limsupJ1} applied with the open set $ \Omega_\delta^-\cup \Omega_\delta^+ $, we get $$\limsup_{s\uparrow 1}(1-s)\M{J}_s^{2, \, \gamma}(\Pi,\Omega)\le 2 \omega_{n-1} P^\gamma(\Pi,\Omega_\delta^-\cup \Omega_\delta^+).$$ Since $\delta>0$ is arbitrary, letting $\delta$ go to zero, we conclude the proof of the Proposition.
\section{First and second variation} In this section we calculate the first and second variation of $\M{J}^\gamma_{s}(E)$. A similar analysis has been done in \cite{FFMMM} in order to prove the local minimality of the ball for a functional involving nonlocal terms.
First, we fix some notation. Given a vector field $X \in C_{c}^{2}(\mathbb{R}^n , \mathbb{R}^{n})$, the {\it associated flow} is defined as the solution of the Cauchy problem \begin{eqnarray} \label{flusso} \begin{cases} \displaystyle\frac{\partial}{\partial t}\Phi(x,t)=X(\Phi (x,t))
\\ \Phi(x,0)=x. \end{cases} \end{eqnarray} In the following, we shall always write $\Phi_t$ to denote the map $\Phi(\cdot,t)$. Note that for any given $X$ there exists $\delta>0$ such that, for $t\in[-\delta,\delta]$, the map $\Phi_t$ is a diffeomorphism coinciding with the identity map outside a compact set.
If $E\subset \mathbb{R}^n$ is measurable, we set $E_t:=\Phi_t(E)$. Denoting by $J\Phi_t$ the $n$-dimensional Jacobian of $\Phi_t$, the first and second derivatives of $J\Phi_t$ are given by \begin{equation}\label{utili} \frac{\partial}{\partial t} \restrict{t=0} J\Phi_t =\text{div}X,\qquad\frac{\partial^2}{\partial t^2} \restrict{t=0} J\Phi_t=\text{div} ((\text{div}X)X). \end{equation}
Finally, given a sufficiently smooth bounded open set $E\subset \R{n}$ and a vector field $X$, we recall that the {\it first variation of $\M{J}^\gamma_{s}(E)$} along the vector field $X$ is defined by $$ \delta \M{J}^\gamma_{s}(E)[X]:=\frac{d}{dt}\restrict{t=0}\M{J}^\gamma_{s}(E_t), $$ where $\Phi_t$ is the flow associated with $X$. The {\it second variation of $\M{J}^\gamma_{s}(E)$} along the vector field $X$ is defined by $$ \delta^2 \M{J}^\gamma_{s}(E)[X]= \frac{d^2}{dt^2}\restrict{t=0} \M{J}^\gamma_s (E_t). $$ If $X$ is a vector field such that $X:=\phi \nu_E$ on $\partial E$, where $\nu_E$ denotes the exterior normal to $E$, using the area formula and the divergence theorem, the first variation of the Gaussian volume can be computed as \begin{eqnarray} \label{vol1} \nonumber
\frac{d}{dt} \restrict{t=0}\gamma (E_t)=\frac{d}{dt} \restrict{t=0}\int_{E} J\Phi_t (x)e^{-\frac{|\Phi_t |^2}{2}}dx
= \int_E \left( \diver X - \langle X, x\rangle \right) e^{-\frac{|x|^2}{2}}dx \\=
\int_E \diver \left(X e^{-\frac{|x|^2}{2}}\right)dx= \int_{\partial E}\phi(x) e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}_x. \end{eqnarray} If $E$ is a set of class $C^2$, given a smooth function $\phi: \partial E\rightarrow \mathbb{R}$, it can be extended in a neighborhood $U$ of $\partial E$ so that \begin{equation} \label{hyp} \frac{\partial}{\partial \nu} \phi +\phi(H-\langle x, \nu_E\rangle)=0 \quad \text{on}\,\, \partial E. \end{equation} The second variation of the Gaussian volume along the vector field $X$ such that $X=\phi \nu_E$ on $\partial E$ and $\phi$ satisfies \eqref{hyp}, can be calculated using the divergence theorem and reads as \begin{eqnarray*} \label{vol2} \begin{split}
\frac{d^2}{dt^2} \restrict{t=0}\gamma (E_t)&=\int_E\diver \left( \diver \left(X e^{-\frac{|x|^2}{2}} \right) X\right)dx=
\int_{\partial E}\phi\left(\frac{\partial}{\partial \nu} \phi +\phi(H-\langle x, \nu_E\rangle )\right) e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}_x=0 \end{split} \end{eqnarray*} Thus, we say that a vector field preserves the Gaussian volume of $E$ if it satisfies \begin{eqnarray} \label{tecnico}
\int_{\partial E}\phi(x) e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}_x= 0 \quad\text{ and }\quad \frac{\partial \phi}{\partial \nu} +\phi(H-\langle x, \nu_E\rangle)=0 \; \text{ for }\, x\in \partial E. \end{eqnarray} We note that without this assumptions the expression of the second variation of the Gaussian perimeter even in the local framework is quite complicated, see \cite[Eq. (17)]{BBJ}.
In order to compute the first and second variation of $\M{J}^\gamma_s$, due to the singularity of the Kernel in the integrand, we need to pass through approximations. Thus, given $\delta \in [0,1)$, let $\eta_\delta \in C^\infty_c([0,+\infty),[0,1])$ be such that $\eta_\delta = 1$ on $[0,\delta]\cup [1/\delta, \infty]$, $\eta_\delta=0$
on $[2\delta, 2/\delta]$, $|\eta'|\leq 2/\delta$ on $[0,\infty)$, and $\eta \downarrow 0$ for every $s \in (0,1)$ as $\delta \rightarrow 0^+$. Then we define $$
K_{\delta} (z):=(1- \eta_\delta(|z|))\frac{1}{|z|^{n+s}}. $$ Now we show the following theorem. \begin{trm} \label{variations} Let $E$ be an open set of class $C^2$ and $X \in C^2_c(\R n,\R n)$ a vector field such that $X = \phi \nu_E$ on $\partial E$. Then the first variation of $\M{J}^\gamma_s (E)$ along a vector field $X$ is given by \begin{equation}\label{varprima} \partial \M{J}^\gamma_s (E)[X]=\int_{\partial E}H^{*}_{\partial E}(x)(X(x), \nu_E(x))d\mathcal{H} ^{n-1}_x, \end{equation} while the second variation reads as \begin{equation}\label{varseconda} \begin{split}
\partial ^2 \M{J}^\gamma_s (E)[X]&=\int_ {\partial E}\int_{\partial E}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}
\left( \abs{\phi (x)-\phi(y) }^2 -\phi ^2 |\nu_E(x)-\nu_E(y)|\right) d\mathcal{H}^{n-1}_xd\mathcal{H}^{n-1}_y \\ &\qquad +\int_{\partial E}H^*_\delta \left( \phi \left(H-\langle x,\nu_E \rangle \right) +\frac{\partial \phi}{\partial \nu}\right)\phi d\mathcal{H} ^{n-1}_x \\
&\qquad -\int_{\partial E}\phi^2(x)\int_{\R n}\frac{\langle y- x, \nu_E\rangle }{2} \frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}dy \, d\mathcal{H}^{n-1}_x. \end{split} \end{equation} Moreover, if $X$ is volume preserving, then \begin{equation}\label{varseconda1} \begin{split}
\partial ^2 \M{J}^\gamma_s (E)[X]&=\int_ {\partial E}\int_{\partial E}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}
\left( \abs{\phi (x)-\phi(y) }^2 -\phi ^2 |\nu_E(x)-\nu_E(y)|\right) d\mathcal{H}^{n-1}_xd\mathcal{H}^{n-1}_y \\
&\qquad -\int_{\partial E}\phi^2(x)\int_{\R n}\frac{\langle y-x,\nu_E(x)\rangle}{2} \frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}dy \, d\mathcal{H}^{n-1}_x. \end{split} \end{equation} \end{trm} \begin{proof} Let us call $\M{J}_s^\delta$ the integral associated to the regularized kernel, namely $$
\M{J}_s^\delta(E)= \int_{E^c}\int_{E}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}+\delta}dxdy. $$ By the definition of $\Phi_t$, the implicit function theorem gives the existence of $\varepsilon>0$ such that the map $\Phi_t$ is a diffeomorphism for all $t \in [-\delta, \delta]$. Using the area formula, we compute $$
\M{J}_s^\delta(E_t)= \int_{E^c}\int_{E}\frac{e^{- \frac{1}{4} \left(\abs{\Phi(x,t)}^2 + \abs{\Phi(y,t)}^2\right)}}{|\Phi(x,t)-\Phi(y,t)|^{n+s}+\delta}J\Phi(x,t)J\Phi(y,t)dxdy $$ We use the first equation in $\eqref{utili}$ to compute the first variation of $\M{J}_s^\delta$ \begin{equation*} \begin{split}
\frac{d}{dt}_{|_{t=0}}J_s^\delta (E_t)&= \int_{E^c}\int_{E}e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}D_z\left(K_{\delta} (x-y)\right)(X(x)-X(y))dxdy \\ & \, \, \, \, + \int_{E^c}\int_{E}K_{\delta} (x-y)\left( D_x(e^{- \frac{1}{4}\left(\abs{x}^2 + \abs{y}^2\right)})X(x)+ D_y(e^{- \frac{1}{4}\left(\abs{x}^2 + \abs{y}^2\right)})X(y)\right)dxdy \\ & \, \, \, \, + \int_{E^c}\int_{E}K_{\delta} (x-y)e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}\left(\diver X(x)+\diver X(y)\right)dxdy\\ & =
\int_{E^c}\int_{E}\diver_x\left( e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}K_{\delta} (x-y)X(x)\right)dxdy\\ & \, \, \, \, +\int_{E^c}\int_{E}\diver_y\left( e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}K_{\delta} (x-y)X(y)\right)dxdy \\ & =\int_{\partial E}\int_{\R n}\left(\chi_{E^c}(y)-\chi_E (y) \right) K_{\delta} (x-y)e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}\langle X(x),\nu_E(x)\rangle dyd\mathcal{H}^{n-1}_x. \end{split} \end{equation*}
Now we compute the second variation of $\M{J}_s^\delta$. \begin{equation} \begin{split}
\delta^2 \M{J}_s^\delta[X]&=\frac{d^2}{dt^2}_{|_{t=0}} \int_{E^c}\int_E K_\delta(\Phi(x,t)-\Phi(y,t)) e^{- \frac{1}{4} \left(\abs{\Phi(x,t)}^2 + \abs{\Phi(y,t)}^2\right)}\diver X(x)\diver X(y)dxdy\\ &= \int_{E^c}\int_{E} D^2_{xx} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right)[X(x),X(x)] dxdy\\ & \quad+ \int_{E^c}\int_{E}\left \langle D_{x} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),DX(x)X(x)\right \rangle dxdy \nonumber \\ & \quad+2\int_{E^c}\int_{E} D^2_{yx} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right)[X(x),X(y)]dxdy \\ & \quad+\int_{E^c}\int_{E} D^2_{yy} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right)[X(y),X(y)]dxdy\\ & \quad+\int_{E^c}\int_{E}\left \langle D_{y} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),DX(y)X(y)\right \rangle dxdy\\ & \quad +2\int_{E^c}\int_{E}\left \langle D_{x} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),X(x)\right \rangle (\diver_x X(x)+\diver_y X(y))dxdy \\ & \quad +2\int_{E^c}\int_{E}\left \langle D_{y} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),X(y)\right \rangle (\diver_x X(x)+\diver_y X(y))dxdy \\ & \quad +\int_{E^c}\int_{E} K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \big( \diver [X(x)\diver(X(x))]+\diver [X(y)\diver(X(y))]\big)dxdy \\ & \quad +2\int_{E^c}\int_{E} K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\diver_x X(x)\diver_y X(y)dxdy \end{split} \end{equation}
We now use the divergence theorem and exploit the symmetry of $K_\delta$ in order to simplify the above expression as follows \begin{equation} \begin{split} \delta^2 \M{J}_s^\delta&= \int_{E^c}\int_{E}\diver_x \left[X(x) \diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)\right]dxdy \\ & \quad + \int_{E^c}\int_{E}\diver_y \left[X(y) \diver_y \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(y)\right)\right]dxdy \\ & \quad + \int_{E^c}\int_{E}\diver_x \left[X(x) \diver_y \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(y)\right)\right]dxdy \\& \quad + \int_{E^c}\int_{E}\diver_y \left[X(y) \diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)\right]dxdy\\ &=I_1+I_2+I_3+I_4. \end{split} \end{equation}
Using Fubini and the divergence theorems we have $$ I_1=\int_{\partial E} \langle X(x),\nu_E(x) \rangle \int_{E^c}\diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)dyd\mathcal{H}^{n-1}_x $$ and \begin{equation} \begin{split} I_3&=\int_{\partial E} \langle X(x),\nu_E(x) \rangle \int_{E^c}\diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(y)\right)dyd\mathcal{H}^{n-1}_x \\ &=-\int_{\partial E}\int_{\partial E}(K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\langle X(y),\nu_E(y)\rangle \langle X(x),\nu_E(x)\rangle d\mathcal{H}_x^{n-1}d\mathcal{H}_y^{n-1}. \end{split} \end{equation} We remark that $I_1$ (resp. $I_3$) has the same expression of $I_2$ (resp. $I_4$) exchanging $x$ and $y$. Using this observation and the symmetry of $K_\delta$, we compute $$ I_1+I_2=\int_{\partial E}\langle X(x),\nu_E(x)\rangle \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)dyd\mathcal{H}^{n-1}_x \\, $$ and $$I_3+I_4=-2\int_{\partial E}\int_{\partial E}(K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\langle X(y),\nu_E(y)\rangle \langle X(x),\nu_E(x)\rangle d\mathcal{H}_x^{n-1}d\mathcal{H}_y^{n-1}. $$ Next we write $\diver_x X(x)= \diver_{\nu(x)} X(x)+ \diver_{\tau (x)} X(x)$, where $\diver_{\nu(x)}X(x):= \langle DX[\nu_E(x)], \nu_E(x)\rangle$. Using Fubini's theorem and the divergence theorem on manifolds, we get \begin{equation} \begin{split} I_1+I_2&= \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\langle X(x),\nu_E(x)\rangle \diver_{\tau (x)}\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy\\ &\quad+ \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\langle X(x),\nu_E(x)\rangle \diver_{\nu(x)} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy \\ &= \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}H(x)\langle X(x),\nu_E(x)\rangle ^2 \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy\\ &\quad+ \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\langle X(x),\nu_E(x)\rangle \diver_{\nu(x)} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy \\ &= \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}H(x)\phi^2 (x)\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy\\ &\quad + \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\phi^2(x)\frac{\partial}{\partial \nu (x)}\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\right)d\mathcal{H}^{n-1}_xdy\\ &\quad+ \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\phi(x)\frac{\partial \phi}{\partial \nu(x)}K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}d\mathcal{H}^{n-1}_xdy, \end{split} \end{equation} where we used that $X=\phi \nu_E$ and then $\langle D_\tau f,X \rangle =0$ for every $f \in C^1(\partial E)$. Regarding the second addend of the above expression, using again Fubini's theorem and the fact that $D_x K_\delta =-D_y K_\delta$, we get \begin{equation} \begin{split} &\int_{E}\frac{\partial}{\partial \nu (x)}\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\right)dy\\ &= -\int_E \left[\left \langle D_y K_\delta (x-y) , \nu_E(x)\right \rangle +\frac{\langle x,\nu_E(x)\rangle }{2} K_\delta (x-y) \right] e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}dy\\ &=-\int_{\partial E} K_\delta (x-y) e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}\left \langle \nu_E(x),\nu_E(y)\right \rangle d\mathcal{H}^{n-1}_y -\int_E \frac{\langle x+y,\nu_E(x)\rangle}{2} K_\delta(x-y) e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}dy. \end{split} \end{equation}
Finally, thanks to the identity $|\nu_E(x)-\nu_E(y)|^2=2- 2\langle \nu_E(x),\nu_E(y)\rangle$, after some elementary calculations we deduce \begin{equation} \begin{split} \partial ^2 \M{J}^\delta_s (E)[X]&=\int_ {\partial E}\int_{\partial E}e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}K_\delta(x-y)
\left( \abs{\phi (x)-\phi(y) }^2 -\phi ^2 |\nu_E(x)-\nu_E(y)|\right) d\mathcal{H}^{n-1}d\mathcal{H}^{n-1}\\ &\quad + \int_{\partial E}H^*_\delta \left( \phi \left(H-\langle x,\nu_E\rangle \right) +\frac{\partial \phi}{\partial \nu}\right)\phi d\mathcal{H} ^{n-1}\\ &\quad -\int_{\partial E}\phi^2(x)\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\frac{\langle y-x,\nu_E(x)\rangle }{2} K_\delta(x-y) e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}dy d\mathcal{H}^{n-1}_x. \end{split} \end{equation} At this point we just need to show that the first and second variation of $\M{ J}_s^\delta$ converge, respectively, to the first and second variation of $\M{J}^\gamma_s$ as $\delta$ goes to $0$. The proof of this fact is exactly the same as in \cite{FFMMM}. \end{proof}
Note that, since $$ \frac{d}{dt}\restrict{t=0}\int_{E_t} e^{-\frac{x^2}{2}}dx=
\int_{\partial E}e^{-\frac{|x|^2}{2}}\langle X(x),\nu_E(x)\rangle d\mathcal{H}^{n-1}, $$ we have that the flow $\Phi$ associated to $X$ preserves the Gaussian volume if $$
\int_{\partial E}e^{-\frac{|x|^2}{2}}\langle X(x),\nu_E(x)\rangle d\mathcal{H}^{n-1}=0. $$ Thus, the Euler-Lagrange equation for the problem \begin{equation}\label{minimiz}
\min_{|E|=m} \M{J}^\gamma_s(E) \end{equation} is $$
\int_{\partial E} \phi(x) \int_{\R n}\left(\chi_{E^c}(y)-\chi_E (y) \right) \frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}dyd\mathcal{H}^{n-1}_x
= \lambda \int_{\partial E} \phi(x) e^{-\frac{|x|^2}{2}}d\mathcal{H}_x^{n-1}. $$ Moreover, if $E$ is a set of class $C^2$, then thanks to the fundamental lemma of the calculus of variations the above equation can be rewritten as \begin{equation}\label{minimizza}
\int_{\R n}\left(\chi_{E^c}(y)-\chi_E (y) \right) \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy= \lambda e^{-\frac{\abs{x}^2 }{4}}, \qquad \forall x \in \partial E. \end{equation} $E$ is said to be stationary with respect to the non local Gaussian isoperimetric problem, or equivalently a volume constrained critical point, if it satisfies equation \eqref{minimizza}.
\section{Volume constrained stationary shapes} In this section we prove that, as opposed to the local setting, the only halfspaces which are stationary with respect to the non local Gaussian isoperimetric problem are the ones generated by hyperplanes passing through the origin. \begin{trm} We fix $a \in \mathbb R$ and $\omega \in \mathbb S^{n-1}$. If $H_{\omega,a}:= \{x \in \R n: \langle x, \omega\rangle <a\}$ is stationary with respect to the non local Gaussian isoperimetric problem, then $a=0$, or equivalently,
$$\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{H_{\omega, a}}e^{-\frac{|x|^2}{2}}dx= \frac 12.$$ \end{trm} \begin{proof} Up to rotation, we can assume $\omega=e_n$. We start observing that, for every $x \in \partial E$, it holds $\langle x, e_n\rangle=a$. This implies that, with the change of coordinate $z=y-x$, if $\langle y, e_n\rangle<a$, then $\langle z, e_n\rangle<0$ and then we can write \begin{equation}\label{minimizza1} \begin{split}
\int_{E} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy&= \int_{\{ y_n<a\}} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy=\int_{\{ z_n<0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2+2\langle z, x\rangle)}}{|z|^{n+s}}dy\\
&=\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2-2\langle z, x\rangle)}}{|z|^{n+s}}dy, \qquad \forall x \in \partial E. \end{split} \end{equation} Analogously, we compute \begin{equation}\label{minimizza2}
\int_{E^c} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy= \int_{\{ y_n>a\}} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy=\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2+2\langle z, x\rangle)}}{|z|^{n+s}}dz \qquad \forall x \in \partial E. \end{equation} Plugging equations \eqref{minimizza1}, \eqref{minimizza2} in equation \eqref{minimizza}, we get \begin{equation}\label{minimizza3}
\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2})}{|z|^{n+s}}\left (e^{ \frac{\langle z, x\rangle}{2}}- e^{- \frac{\langle z, x\rangle}{2}}\right)dz= \lambda e^{-\frac{\abs{x}^2 }{4}}, \qquad \forall x \in \partial E, \end{equation} which in turn reads \begin{equation}\label{minimizza4}
2\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\sinh\left(\frac{\langle z, x\rangle}{2}\right)dz= \lambda , \qquad \forall x \in \partial E. \end{equation} We remark that the integral in \eqref{minimizza4} is well defined, since $\lim_{x\to 0}\frac{\sinh(x)}{x}=1$.
We split $x=(x',x_n)$ and we observe that \begin{equation}\label{minimizza5} \begin{split} \sinh\left(\frac{\langle z, x\rangle}{2}\right)&=\sinh\left(\frac{\langle z', x'\rangle+ z_nx_n}{2}\right)\\ &=\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_nx_n}{2}\right)+\cosh\left(\frac{\langle z', x'\rangle}{2}\right)\sinh\left(\frac{ z_nx_n}{2}\right). \end{split} \end{equation} Plugging \eqref{minimizza5} in \eqref{minimizza4}, we deduce the following equation for every $x \in \partial E$ \begin{equation}\label{minimizza45} \begin{split}
A+B:=\int_{\{ z_n>0\}}& \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_na}{2}\right)dz\\
&+\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\cosh\left(\frac{\langle z', x'\rangle}{2}\right)\sinh\left(\frac{ z_na}{2}\right)dz= \frac \lambda 2. \end{split} \end{equation} Since $\frac{e^{- \frac{1}{4} (\abs{z'}^2+\abs{z_n}^2})}{(\abs{z'}^2+\abs{z_n}^2)^{\frac{n+s}{2}}}\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_na}{2}\right)$ is odd in $z'$, we deduce $$A=\int_0^\infty\int_{\mathbb R^{n-1}} \frac{e^{- \frac{1}{4}( \abs{z'}^2+\abs{z_n}^2)}}{(\abs{z'}^2+\abs{z_n}^2)^{\frac{n+s}{2}}}\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_na}{2}\right)dz'dz_n=0.$$ Plugging this information in \eqref{minimizza5} and taking the partial derivative in $x_j$, for every $j=1,\dots,n-1$, we deduce $$
\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\frac{\partial}{ \partial x_j}\left(\cosh\left(\frac{\langle z', x'\rangle}{2}\right)\right)\sinh\left(\frac{ z_na}{2}\right)dz= 0. $$ Assuming without loss of generality that $j=n-1$ and denoting $x'=(x'',x_{n-1})$, we obtain \begin{equation}\label{minimizza6} \begin{split}
C&+D:=\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\frac{\partial}{ \partial x_j}\left(\cosh\left(\frac{\langle z'', x''\rangle}{2}\right)\cosh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)\right) \sinh\left(\frac{ z_na}{2}\right)dz\\
& +\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\frac{\partial}{ \partial x_j}\left(\sinh\left(\frac{\langle z'', x''\rangle}{2}\right)\sinh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)\right)\sinh\left(\frac{ z_na}{2}\right)dz= 0. \end{split} \end{equation}
Since $\frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\cosh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)z_{n-1}$ is odd in $z_{n-1}$, we deduce
$$D=\int_0^\infty \int_{\mathbb R^{n-2}} \int_{\mathbb R} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\sinh\left(\frac{\langle z'', x''\rangle}{2}\right)\cosh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)z_{n-1}\sinh\left(\frac{ z_na}{2}\right)dz=0.$$ Plugging this information in \eqref{minimizza6}, we get that for every $x \in \partial E$ it holds \begin{equation}\label{minimizza7} \begin{split}
\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\cosh\left(\frac{\langle z'', x''\rangle}{2}\right)\sinh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)\sinh\left(\frac{ z_na}{2}\right)z_{n-1}dz=0. \end{split} \end{equation} We denote
$$C(z_n,x):=\int_{\mathbb R^{n-1}} \frac{e^{- \frac{1}{4} \abs{(z',z_n)}^2}}{|(z',z_n)|^{n+s}}\cosh\left(\frac{\langle z'', x''\rangle}{2}\right)\sinh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)z_{n-1}dz''dz_{n-1},$$ and we observe that if $x_{n-1}\neq 0$, then $C(z_n,x)\neq 0$ since the integrand is even in the variables $z''$ and $z_{n-1}$. Equation \eqref{minimizza7} then reads \begin{equation}\label{minimizza8} \int_0^\infty C(z_n,x)\sinh\left(\frac{ z_na}{2}\right)dz_n=0, \qquad \forall x \in \partial E, \end{equation} and since for every $z_n>0$ $$\sinh\left(\frac{ z_na}{2}\right)\left\{ \begin{array}{ccc}>0 & \text{if } a>0\\ =0 & \text{if } a=0\\ <0 & \text{if } a<0 \end{array}\right.,$$ equation \eqref{minimizza8} can hold if and only if $a=0$. \end{proof}
{\small \noindent Antonio De Rosa \\ Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 2074, USA \\ \texttt{anderosa@umd.edu} }
{\small \noindent Domenico Angelo La Manna \\ Department of Mathematics and Statistics, P.O.\ Box 35 (MaD), FI-40014, University of Jyv\"askyl\"a, Finland.\\ \texttt{domenicolamanna@hotmail.it} }
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\begin{document}
\title{\LARGE\bf The class $B_p$ for weighted generalized Fourier transform inequalities}
\date{} \maketitle \begin{abstract} In the present paper, we prove for the Dunkl transform which generalizes the Fourier transform weighted inequalities when the weights belong to the well-known class $B_p$. As application, we obtain for power weights Pitt's inequality. \end{abstract} {\small\bf Keywords: }{\small Dunkl operators, Dunkl transform, class $B_p$, Pitt's inequality.}\\ \noindent {\small \bf 2010 AMS Mathematics Subject Classification:} {42B10, 46E30, 44A35.} \section{Introduction } \par A key tool in the study of special functions with reflection symmetries are Dunkl operators. The basic ingredient in the theory of these operators are root systems and finite reflection groups, acting on $\mathbb{R}^d$. The Dunkl operators are commuting differential-difference operators $T_i, 1 \leq i \leq d$ associated to an arbitrary finite reflection group $W$ on $\mathbb{R}^d$ (see[7]). These operators attached with a root system $R$ can be considered as perturbations of the usual partial derivatives by reflection parts. These reflection parts are coupled by parameters, which are given in terms of a non negative multiplicity function $k$. Dunkl theory was further developed by several mathematicians (see [6, 14]) and later was applied and generalized in different ways by many authors (see [1, 2]). The Dunkl kernel $E_k$ has been introduced by C.F. Dunkl in [8]. For a family of weight functions $w_k$ invariant under a reflection group $W$, we use the Dunkl kernel and the measure $w_k(x)dx$ to define the generalized Fourier transform $\mathcal{F}_k$, called the Dunkl transform, which enjoys properties similar to those of the classical Fourier transform. If the parameter $k\equiv0$ then $w_k(x)=1$, so that $\mathcal{F}_k$ becomes the classical Fourier transform and the $T_i, 1 \leq i \leq d$ reduce to the corresponding partial derivatives $\frac{\partial}{\partial x_i}, 1 \leq i \leq d$. Therefore Dunkl analysis can be viewed as a generalization of classical Fourier analysis (see next section, Remark 2.1). \\
Let $\mu$ a nonnegative locally integrable function on $(0,+\infty)$. We say that $\mu\in B_{p}$, $1< p<+\infty$ if there is a constant $b_{p}>0$ such that for all $s>0$ \begin{eqnarray} \int_{s}^{+\infty}\frac{\mu(t)}{t^{p}}dt \leq b_{p}\frac{1}{s^{p}}\int_{0}^{s}\mu(t)dt. \end{eqnarray} In the particular case when $\mu$ is non-increasing, one has $\mu \in B_{p}$.\\
The weighted Hardy inequality [16] (see also [9, 13]) states that if $\mu$ and $\vartheta$ are locally integrable
weight functions on $(0,+\infty)$ and $1<p\leq q<+\infty$, then there is a constant $c>0$ such that for all non-increasing, non-negative Lebesgue measurable function $f$ on $(0,+\infty)$, the inequality \begin{eqnarray}\Big(\int_{0}^{+\infty}\Big(\frac{1}{t}\int_{0}^{t}f(s)ds\Big)^{q}\mu(t)dt\Big)^{\frac{1}{q}}\leq
c\, \Big(\int_{0}^{+\infty}(f(t))^{p}\vartheta(t)dt\Big)^{\frac{1}{p}}\end{eqnarray}
is satisfied if and only if \begin{eqnarray}\displaystyle\sup_{s>0}\Big(\int_{0}^{s}\mu(t)dt\Big)^{\frac{1}{q}} \Big(\int_{0}^{s}(\vartheta(t))dt\Big)^{-\frac{1}{p}} <+\infty.\end{eqnarray} and \begin{eqnarray}\displaystyle\sup_{s>0}\Big(\int_{s}^{+\infty}\frac{\mu(t)}{t^q}dt\Big)^{\frac{1}{q}} \Big(\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}\vartheta(l)&dl&\Big)^{-p'}\vartheta(t)dt\Big)^{\frac{1}{p'}} <+\infty.\end{eqnarray} Hardy's result still remains to be an important one as it is closely related to the Hardy-Littlewood maximal functions in harmonic analysis [17].\\
The aim of this paper is to prove under the $B_p$ condition (1.1) and using the weight characterization of the Hardy operator, weighted Dunkl transform inequalities for general nonnegative locally integrable functions $u$, $v$ on $\mathbb{R}^d$,
\begin{eqnarray*}\Big(\int_{\mathbb{R}^{d}}|\mathcal{F}_{k}(f)(x)|^{q}u(x)
d\nu_{k}(x)\Big)^{\frac{1}{q}} \leq c\,\Big(\int_{\mathbb{R}^{d}}|f(x)|^{p}v(x) d\nu_{k}(x)\Big)^{\frac{1}{p}},\end{eqnarray*} where $1<p\leq 2 \leq q<+\infty$ and $f\in L^p_{k,v}(\mathbb{R}^d)$. $L^p_{k,v}(\mathbb{R}^d)$ denote the space $L^{p}(\mathbb{R}^d, v(x) d\nu_k(x))$ with $\nu_k$ the weighted measure associated to the Dunkl operators defined by \begin{eqnarray*}d\nu_k(x):=w_k(x)dx\quad
\mbox{where}\;\;w_k(x) = \prod_{\xi\in R_+} |\langle
\xi,x\rangle|^{2k(\xi)}, \quad x \in \mathbb{R}^d.\end{eqnarray*} $R_+$ being a positive root system and $\langle .,.\rangle$ the standard Euclidean scalar product on $\mathbb{R}^d$ (see next section). As application, we make a study of power weights in this context. This all leads to Pitt\'{}s inequality:\\ for $1<p\leq 2\leq q<+\infty$, $-(2\gamma+d)<\alpha<0$, $0<\beta<(2\gamma+d)(p-1)$ and $f\in L^p_{k,v}(\mathbb{R}^d)$, one has
\begin{eqnarray*}\Big(\int_{\mathbb{R}^{d}}|\mathcal{F}_{k}(f)(x)|^{q}\|x\|^{\alpha}
d\nu_{k}(x)\Big)^{\frac{1}{q}} \leq c\,\Big(\int_{\mathbb{R}^{d}}|f(x)|^{p}\|x\|^{\beta} d\nu_{k}(x)\Big)^{\frac{1}{p}},\end{eqnarray*} with the index constraint $\frac{1}{2\gamma+d}(\frac{\alpha}{q}+\frac{\beta}{p})=1-\frac{1}{p}-\frac{1}{q}$ where $\displaystyle\gamma = \sum_{\xi \in R_+} k(\xi)$. This extend to the Dunkl analysis some results obtained for the classical Fourier analysis in [4]. \\
The contents of this paper are as follows. \\In section 2, we collect some basic definitions and results about harmonic analysis associated with Dunkl operators .\\ The section 3 is devoted to the proofs of the weighted Dunkl transform inequalities when the weights belong to the class $B_p$. As application, we obtain for power weights Pitt\'{}s inequality.\\
Along this paper we use $c$ to denote a suitable positive constant which is not necessarily the same in each occurrence and we write for $x \in \mathbb{R}^d,$ $\|x\| = \sqrt{\langle x,x\rangle}$. Furthermore, we denote by
$\bullet\quad \mathcal{E}(\mathbb{R}^d)$ the space of infinitely differentiable functions on $\mathbb{R}^d$.
$\bullet\quad \mathcal{S}(\mathbb{R}^d)$ the Schwartz space of functions in $\mathcal{E}( \mathbb{R}^d)$ which are rapidly decreasing as well as their derivatives.
$\bullet\quad \mathcal{D}(\mathbb{R}^d)$ the subspace of $\mathcal{E}(\mathbb{R}^d)$ of compactly supported functions. \section{Preliminaries}
$ $ In this section, we recall some notations and results in Dunkl theory and we refer for more details to the surveys [15].\\
Let $W$ be a finite reflection group on $\mathbb{R}^{d}$, associated with a root system $R$. For $\alpha\in R$, we denote by $\mathbb{H}_\alpha$ the hyperplane orthogonal to $\alpha$. For a given $\beta\in\mathbb{R}^d\backslash\bigcup_{\alpha\in R} \mathbb{H}_\alpha$, we fix a positive subsystem $R_+=\{\alpha\in R: \langle \alpha,\beta\rangle>0\}$. We denote by $k$ a nonnegative multiplicity function defined on $R$ with the property that $k$ is $W$-invariant. We associate with $k$ the index $$\gamma = \sum_{\xi \in R_+} k(\xi) \geq 0,$$ and a weighted measure $\nu_k$ given by \begin{eqnarray*}d\nu_k(x):=w_k(x)dx\quad
\mbox{ where }\;\;w_k(x) = \prod_{\xi\in R_+} |\langle
\xi,x\rangle|^{2k(\xi)}, \quad x \in \mathbb{R}^d,\end{eqnarray*}
Further, we introduce the Mehta-type constant $c_k$ by
$$c_k = \left(\int_{\mathbb{R}^d} e^{- \frac{\|x\|^2}{2}} w_k (x)dx\right)^{-1}.$$
For every $1 \leq p \leq + \infty$, we denote respectively by $L^p_k(\mathbb{R}^d)$, $L^p_{k,u}(\mathbb{R}^d)$, $L^p_{k,v}(\mathbb{R}^d)$ the spaces $L^{p}(\mathbb{R}^d, d\nu_k(x)),$ $L^{p}(\mathbb{R}^d, u(x)d\nu_k(x)),$ $L^{p}(\mathbb{R}^d, v(x)d\nu_k(x))$ and $L^p_k( \mathbb{R}^d)^{rad}$ the subspace of those $f \in L^p_k(
\mathbb{R}^d)$ that are radial. We use respectively $\|\
\;\|_{p,k}$\,, $\|\ \;\|_{p,k,u}$\,, $\|\ \;\|_{p,k,v}$ as a shorthand for
$\|\ \;\|_{L^p_k( \mathbb{R}^d)}$, $\|\ \;\|_{L^p_{k,u}( \mathbb{R}^d)}$, $\|\ \;\|_{L^p_{k,v}( \mathbb{R}^d)}.$ \\
By using the homogeneity of degree $2\gamma$ of $w_k$, it is shown in [14] that for a radial function $f$ in $L^1_k ( \mathbb{R}^d)$, there exists a function $F$ on $[0, + \infty)$ such that $f(x) =
F(\|x\|)$, for all $x \in \mathbb{R}^d$. The function $F$ is integrable with respect to the measure $r^{2\gamma+d-1}dr$ on $[0, + \infty)$ and we have
\begin{eqnarray} \int_{\mathbb{R}^d} f(x)\,d\nu_k(x)&=&\int^{+\infty}_0 \Big( \int_{S^{d-1}}f(ry)w_k(ry)d\sigma(y)\Big)r^{d-1}dr\nonumber\\ &=&
\int^{+\infty}_0 \Big( \int_{S^{d-1}}w_k(ry)d\sigma(y)\Big) F(r)r^{d-1}dr\nonumber\\&= & d_k\int^{+ \infty}_0 F(r) r^{2\gamma+d-1}dr, \end{eqnarray}
where $S^{d-1}$ is the unit sphere on $\mathbb{R}^d$ with the normalized surface measure $d\sigma$ and \begin{eqnarray}d_k=\int_{S^{d-1}}w_k (x)d\sigma(x) = \frac{c^{-1}_k}{2^{\gamma +\frac{d}{2} -1} \Gamma(\gamma + \frac{d}{2})}\;. \end{eqnarray}
The Dunkl operators $T_j\,,\ \ 1\leq j\leq d\,$, on $\mathbb{R}^d$ associated with the reflection group $W$ and the multiplicity function $k$ are the first-order differential- difference operators given by $$T_jf(x)=\frac{\partial f}{\partial x_j}(x)+\sum_{\alpha\in R_+}k(\alpha) \alpha_j\,\frac{f(x)-f(\rho_\alpha(x))}{\langle\alpha,x\rangle}\,,\quad f\in\mathcal{E}(\mathbb{R}^d)\,,\quad x\in\mathbb{R}^d\,,$$ where $\rho_\alpha$ is the reflection on the hyperplane $\mathbb{H}_\alpha$ and $\alpha_j=\langle\alpha,e_j\rangle,$ $(e_1,\ldots,e_d)$ being the canonical basis of $\mathbb{R}^d$. \begin{remark}In the case $k\equiv0$, the weighted function $w_k\equiv1$ and the measure $\nu_k$ associated to the Dunkl operators coincide with the Lebesgue measure. The $T_j$ reduce to the corresponding partial derivatives. Therefore Dunkl analysis can be viewed as a generalization of classical Fourier analysis.\end{remark}
For $y \in \mathbb{C}^d$, the system $$\left\{\begin{array}{lll}T_ju(x,y)&=&y_j\,u(x,y),\qquad1\leq j\leq d\,,\\ &&\\ u(0,y)&=&1\,.\end{array}\right.$$ admits a unique analytic solution on $\mathbb{R}^d$, denoted by $E_k(x,y)$ and called the Dunkl kernel. This kernel has a unique holomorphic extension to $\mathbb{C}^d \times \mathbb{C}^d $. We have for all $\lambda\in \mathbb{C}$ and $z, z'\in \mathbb{C}^d,\;
E_k(z,z') = E_k(z',z)$, $E_k(\lambda z,z') = E_k(z,\lambda z')$ and for $x, y
\in \mathbb{R}^d,\;|E_k(x,iy)| \leq 1$.\\
The Dunkl transform $\mathcal{F}_k$ is defined for $f \in \mathcal{D}( \mathbb{R}^d)$ by $$\mathcal{F}_k(f)(x) =c_k\int_{\mathbb{R}^d}f(y) E_k(-ix, y)d\nu_k(y),\quad x \in \mathbb{R}^d.$$ We list some known properties of this transform: \begin{itemize} \item[i)] The Dunkl transform of a function $f \in L^1_k( \mathbb{R}^d)$ has the following basic property
\begin{eqnarray*}\| \mathcal{F}_k(f)\|_{\infty,k} \leq
\|f\|_{ 1,k}\;. \end{eqnarray*} \item[ii)] The Dunkl transform is an automorphism on the Schwartz space $\mathcal{S}(\mathbb{R}^d)$. \item[iii)] When both $f$ and $\mathcal{F}_k(f)$ are in $L^1_k( \mathbb{R}^d)$,
we have the inversion formula \begin{eqnarray*} f(x) = \int_{\mathbb{R}^d}\mathcal{F}_k(f)(y) E_k( ix, y)d\nu_k(y),\quad x \in \mathbb{R}^d.\end{eqnarray*} \item[iv)] (Plancherel's theorem) The Dunkl transform on $\mathcal{S}(\mathbb{R}^d)$
extends uniquely to an isometric automorphism on $L^2_k(\mathbb{R}^d)$. \end{itemize} Since the Dunkl transform $\mathcal{F}_k(f)$ is of strong-type $(1,\infty)$ and $(2,2)$, then by interpolation, we get for $f \in L^p_k(\mathbb{R}^d)$ with $1\leq p\leq 2$ and $p'$ such that $\frac{1}{p}+\frac{1}{p'}=1$, the Hausdorff-Young inequality \begin{eqnarray*}
\|\mathcal{F}_k(f)\|_{p',k}\leq c\,\|f\|_{p,k}. \end{eqnarray*} The Dunkl transform of a function in $L^1_k( \mathbb{R}^d)^{rad}$
is also radial. More precisely, according to ([14], proposition 2.4), we have for $x\in\mathbb{R}$, the following results: \begin{eqnarray*}\int_{S^{d-1}}E_k(ix,y)w_k(y)d\sigma(y) =
d_k\, j_{\gamma + \frac{d}{2}-1}(\|x\|), \end{eqnarray*}
and for $f$ be in $L^1_k(\mathbb{R}^d)^{rad}\;,$
\begin{eqnarray}\mathcal{F}_k(f)(x) &=&\int^{+\infty}_0 \Big( \int_{S^{d-1}}E_k(-irx, y)w_k(y)d\sigma(y)\Big) F(r)r^{2\gamma+d-1}dr\nonumber\\&=&
d_k\int^{+\infty}_0 j_{\gamma + \frac{d}{2}-1}(r\|x\|)
F(r)r^{2\gamma+d-1}dr,\end{eqnarray} where $F$ is the function defined on $[ 0, + \infty)$ by $F(\|x\|) = f(x)$ and $j_{\gamma + \frac{d}{2}-1}$ the normalized Bessel function of the first kind and order $\gamma + \frac{d}{2}-1$ given by \begin{eqnarray*} j_{\gamma+\frac{d}{2}-1}(\lambda x) =\left \{\begin{array}{ll}2^{\gamma+\frac{d}{2}-1}\Gamma(\gamma+\frac{d}{2})\frac{J_{\gamma+\frac{d}{2}-1}(\lambda x)}{(\lambda x)^{\gamma+\frac{d}{2}-1}}& \mbox{if}\; \lambda x\neq0,\\ 1& \mbox{if}\;\lambda x=0\,,\end{array} \right. \end{eqnarray*} $\lambda\in\mathbb{C}$. Here $J_{\gamma+\frac{d}{2}-1}$ is the Bessel function of first kind, \begin{eqnarray} J_{\gamma+\frac{d}{2}-1}(t)&=&\frac{(\frac{t}{2})^{\gamma+\frac{d}{2}-1}}{\sqrt{\pi}\Gamma(\gamma+\frac{d}{2}-\frac{1}{2})} \int_{0}^{\pi}\cos(t\cos\theta)(\sin\theta)^{2\gamma+d-2}d\theta\nonumber \\&=&C_{\gamma}t^{\gamma+\frac{d}{2}-1} \int_{0}^{\frac{\pi}{2}}\cos(t\cos\theta)(\sin\theta)^{2\gamma+d-2}d\theta, \end{eqnarray} where $C_{\gamma}=\frac{1}{\sqrt{\pi}2^{\gamma+\frac{d}{2}-2}\Gamma(\gamma+\frac{d}{2}-\frac{1}{2})}$.\\ \section{Weighted Dunkl transform inequalities} In this section, we denote by $p'$ and $q'$ respectively the conjugates of $p$ and $q$ for $1<p\leq q<+\infty$. The proof requires a useful well-known facts which we shall now state in the following. \begin{proposition} (see [16]) Let $1<p<+\infty$ and $v $ be a nonnegative function on $(0,+\infty)$. The following are equivalent: \begin{itemize}
\item [i)] $v\in B_{p}$.
\item [ii)] There is a positive constant $c$ such that for all $s>0$,\begin{eqnarray} \displaystyle \Big(\int_{0}^{s}v(t)dt\Big)^{\frac{1}{p}}\Big(\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}v(l)dl\Big)^{1-p'}dt\Big)^{\frac{1}{p'}} &\leq & c\,s.
\end{eqnarray}
\end{itemize} \end{proposition} \begin{remark}$ $ \begin{itemize} \item[1/] (see [5]) (Hardy's Lemma) Let $f$ and $g$ be non-negative Lebesgue measurable functions on $(0, +\infty)$, and assume $$\int_{0}^{t}f(s)ds\leq \int_{0}^{t}g(s)ds$$ for all $t\geq0$. If $\varphi$ is a non-negative and decreasing function on $(0, +\infty)$, then \begin{eqnarray}\int_{0}^{+\infty}f(s)\varphi(s)ds\leq \int_{0}^{+\infty}g(s)\varphi(s)ds.\end{eqnarray} \item[2/] Let $f$ be a measurable function on $\mathbb{R}^{d}$. The distribution function $D_f$ of $f$ is defined for all $s\geq0$ by
$$D_{f}(s)=\nu_k(\{x\in\mathbb{R}^{d}\,:\; |f(x)|>s\}).$$ The decreasing rearrangement of $f$ is the function $f^*$ given for all $t\geq0$ by $$f^{*}(t)=inf\{s\geq0 \,:\; D_{f}(s)\leq t\}.$$ We have the following results: \\ i) Let $f\in L^p_{k}(\mathbb{R}^d)$, $1\leq p<+\infty$ then
\begin{eqnarray}\int_{\mathbb{R}^{d}}|f(x)|^{p} d\nu_{k}(x)=p\int_{0}^{+\infty}s^{p-1}D_{f}(s)ds=\int_{0}^{+\infty}(f^{*}(t))^{p}dt.\end{eqnarray}\\ ii) (see [12], Theorems 4.6 and 4.7) Let $q\geq2$, then there exists a constant $c>0$ such that, for all $f\in L_{k}^{1}(\mathbb{R}^{d})+L_{k}^{2}(\mathbb{R}^{d})$ and $\,s\geq0$, \begin{eqnarray}\int_{0}^{s}(\mathcal{F}_{k}(f)^{*}(t))^{q}dt\leq c \int_{0}^{s}\Big(\int_{0}^{\frac{1}{t}}f^{*}(y)dy\Big)^{q}dt.\end{eqnarray} iii) (see [5, 10, 11]) (Hardy-Littlewood rearrangement inequality)\\ Let $f$ and $\vartheta$ be non negative measurable functions on $\mathbb{R}^{d}$, then \begin{eqnarray}\int_{\mathbb{R}^{d}}f(x)\vartheta(x)d\nu_{k}(x)\leq\int_{0}^{+\infty}f^{*}(t)\vartheta^{*}(t)dt\end{eqnarray} and \begin{eqnarray}\int_{0}^{+\infty}f^{*}(t)
\Big[\Big(\frac{1}{\vartheta}\Big)^{*}(t)\Big]^{-1} dt\leq\int_{\mathbb{R}^{d}}f(x)\vartheta(x)d\nu_{k}(x).\end{eqnarray}\end{itemize} \end{remark} Now, we begin with the proof of the following proposition which gives a necessary condition. \begin{proposition} Let $u$, $v$ be non-negative $\nu_k$-locally integrable functions on $\mathbb{R}^d$ and $1<p\leq 2\leq q<+\infty$. If there exists a constant $c>0$ such that for all $f\in L_{k}^{p}(\mathbb{R}^d)$, \begin{eqnarray} \Big(\int_{0}^{+\infty}\big((\mathcal{F}_{k}(f))^{*}(t)\big)^{q}u^*(t)dt\Big)^{\frac{1}{q}} \leq c\, \Big(\int_{0}^{+\infty}\big(f^{*}(t)\big)^{p}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{1}{p}},\nonumber\\& &
\end{eqnarray} then it is necessary that \begin{eqnarray} \displaystyle \sup_{s>0}s\Big(\int_{0}^{\frac{1}{s}}u^*(t)dt\Big)^{\frac{1}{q}}\Big(\int_{0}^{s}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{-1}{p}}< +\infty.
\end{eqnarray} \end{proposition} \begin{proof} Put for any fixed $r>0$, $$R=\displaystyle\Big(r\,\frac{\nu_{k}(B(0,1))}{1+(\nu_{k}(B(0,1)))^{2}}\Big)^{\frac{1}{2\gamma+d}}, $$ and take $f=\chi_{(0,R)}$ in (3.7), where $\chi_{(0,R)}$ is the characteristic function of the interval $(0,R)$. For $s\geq0$ and by (2.1) and (2.2), the distribution function of $f$ is \begin{eqnarray*}
D_{f}(s)=\nu_k(\{x\in\mathbb{R}^{d}\,:\; \chi_{(0,R)}(\|x\|)>s\}) &=&\frac{d_k}{2\gamma+d}R^{2\gamma+d}\chi_{(0,1)}(s) \\&=&\nu_k(B(0,1))R^{2\gamma+d}\chi_{(0,1)}(s) \\&=& r'\chi_{(0,1)}(s), \end{eqnarray*} where \begin{eqnarray}r'= \nu_k(B(0,1))R^{2\gamma+d}= r\,\frac{(\nu_{k}(B(0,1)))^{2}}{1+(\nu_{k}(B(0,1)))^{2}}.\end{eqnarray} This yields for $t\geq0$, \begin{eqnarray*} f^{*}(t)&=&inf\{s\geq0 \,:\; D_{f}(s)\leq t\} \\&=& \chi_{(0,r')}(t). \end{eqnarray*} Observe that $r'< r$, hence we have \begin{eqnarray} \Big(\int_{0}^{+\infty}\Big((\mathcal{F}_k(f))^*(t)\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}&\leq& c\, \Big(\int_{0}^{r'}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{1}{p}}\nonumber\\&\leq& c\, \Big(\int_{0}^{r}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{1}{p}}. \end{eqnarray} According to (2.3), for $x\in\mathbb{R}^{d}$, we can assert that \begin{eqnarray} \mathcal{F}_k(f)(x) &=&c_{k}^{-1}
\int_{0}^{R}j_{\gamma+\frac{d}{2}-1}(\|x\| t)\frac{t^{2\gamma+d-1}}{2^{\gamma+\frac{d}{2}-1}\Gamma(\gamma+\frac{d}{2})}dt\nonumber
\\&=&c_{k}^{-1}\|x\|^{\frac{2-2\gamma-d}{2}}\int_{0}^{R}J_{\gamma+\frac{d}{2}-1}(\|x\| t)t^{\frac{2\gamma+d}{2}}dt. \end{eqnarray}
Since $\cos(t\|x\|\cos\theta)\geq\cos1>\frac{1}{2}$, for $t\in(0, R)$, $\|x\|\in(0, \frac{1}{R})$ and $\theta\in(0, \frac{\pi}{2})$, then we obtain from (2.4), the estimate \begin{eqnarray*}
J_{\gamma+\frac{d}{2}-1}(\|x\| t)&>&\frac{1}{2}\,C_{\gamma}\,(\|x\| t)^{\gamma+\frac{d}{2}-1}
\int_{0}^{\frac{\pi}{2}}(\sin\theta)^{2\gamma+d-2}d\theta\\&=&\frac{1}{2}\,C_{\gamma}\,(\|x\| t)^{\gamma+\frac{d}{2}-1} \frac{\sqrt{\pi}\Gamma(\gamma+\frac{d}{2}-\frac{1}{2})}{2\Gamma(\gamma+\frac{d}{2})}\\&=&
\frac{(\|x\| t)^{\frac{2\gamma+d-2}{2}}}{2^{\frac{2\gamma+d}{2}}\Gamma(\frac{2\gamma+d}{2})}, \end{eqnarray*}
which gives by (2.1), (2.2), (3.9), (3.11) and for $\|x\|\in(0, \frac{1}{R})$ \begin{eqnarray}
\mathcal{F}_k(f)(x)&>&c_{k}^{-1}\|x\|^{\frac{2-2\gamma-d}{2}}\int_{0}^{R}
\frac{(\|x\| t)^{\frac{2\gamma+d-2}{2}}}{2^{\frac{2\gamma+d}{2}}\Gamma(\frac{2\gamma+d}{2})}t^{\frac{2\gamma+d}{2}}dt \nonumber\\&=&\frac{c_{k}^{-1}}{2^{\frac{2\gamma+d}{2}}\Gamma(\frac{2\gamma+d}{2})}\int_{0}^{R}t^{2\gamma+d-1}dt \; =\;\frac{r'}{2}\;. \end{eqnarray} By the fact that $$\{t\in(0,\frac{1}{r}):(\mathcal{F}_k(f))^*(t)>s\}=\{t\in(0,\frac{1}{r}):D_{\mathcal{F}_k(f)}(s)>t\},$$
we have from
(3.3) \\$\displaystyle\Big(\int_{0}^{+\infty}\Big((\mathcal{F}_k(f))^*(t)\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}$ \begin{eqnarray*} &\geq& \Big(\int_{0}^{\frac{1}{r}}\Big((\mathcal{F}_k(f))^*(t)\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}\\&=& \Big(q\int_{0}^{+\infty}s^{q-1}\Big(\int_{\{t\in(0,\frac{1}{r}),\;(\mathcal{F}_k(f))^*(t)>s\}}u^*(t)dt\Big)ds\Big)^{\frac{1}{q}} \\&=& \Big(q\int_{0}^{+\infty}s^{q-1}\Big(\int_{0}^{\min(D_{\mathcal{F}_k(f)}(s),\frac{1}{r})}u^*(t)dt\Big)ds\Big)^{\frac{1}{q}}. \end{eqnarray*} If $s<\frac{r'}{2}$, then by
(3.12)\\ $ B(0, \frac{1}{R})\subseteq\{x\in\mathbb{R}^{d}\,:\;
|\mathcal{F}_k(f)(x)|>\frac{r'}{2}\}
\subseteq\{x\in\mathbb{R}^{d}\,:\; |\mathcal{F}_k(f)(x)|>s\}, $\\ thus using (2.1) and (2.2), we have \begin{eqnarray*} D_{\mathcal{F}_k(f)}(s)&=& \int_{\{x\in\mathbb{R}^{d}\,:\;
|\mathcal{F}_k(f)(x)|>s\}}w_k(x) \,dx \\&\geq&d_{k}\int_{0}^{\frac{1}{R}}\rho^{2\gamma+d-1}d\rho \\&=& \frac{1}{r}\Big(1+(\nu_{k}(B(0,1)))^{2}\Big)>\frac{1}{r}\;, \end{eqnarray*} wich gives that \begin{eqnarray*} \Big(\int_{0}^{+\infty}\Big((\mathcal{F}_k(f))^*(t)\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}&\geq& \Big(q\int_{0}^{\frac{r'}{2}}s^{q-1}\Big(\int_{0}^{\frac{1}{r}}u^*(t)dt\Big)ds\Big)^{\frac{1}{q}}\\&=& \Big(q\int_{0}^{\frac{r'}{2}}s^{q-1}ds\Big)^{\frac{1}{q}}\Big(\int_{0}^{\frac{1}{r}}u^*(t)dt\Big)^{\frac{1}{q}}\\&=& \frac{r'}{2}\Big(\int_{0}^{\frac{1}{r}}u^*(t)dt\Big)^{\frac{1}{q}}. \end{eqnarray*}According to (3.9) and (3.10), we deduce that\\\\$\displaystyle r\Big(\int_{0}^{\frac{1}{r}}u^*(t)dt\Big)^{\frac{1}{q}} \Big(\int_{0}^{r}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{-\frac{1}{p}}$ \begin{eqnarray*} \qquad\qquad\leq\; c\,\Big(\int_{0}^{+\infty}\Big((\mathcal{F}_k(f))^*(t)\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}\Big(\int_{0}^{r} \Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{-\frac{1}{p}} \leq c, \end{eqnarray*}which gives (3.8). This completes the proof. \end{proof} \begin{theorem} Let $u$, $v$ be non-negative $\nu_k$-locally integrable functions on $\mathbb{R}^d$ and $1<p\leq 2\leq q<+\infty$. Assume $ \displaystyle\frac{1}{\Big(\frac{1}{v}\Big)^{*}} \in B_{p}$ and \begin{eqnarray} \displaystyle \sup_{s>0}s\Big(\int_{0}^{\frac{1}{s}}u^*(t)dt\Big)^{\frac{1}{q}}\Big(\int_{0}^{s}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{-1}{p}}< +\infty,
\end{eqnarray} then there exists a constant $c>0$ such that for all $f\in L_{k}^{p}(\mathbb{R}^d)$, we have \begin{eqnarray}
\Big(\int_{\mathbb{R}^{d}}|\mathcal{F}_{k}(f)(x)|^{q}u(x)
d\nu_{k}(x)\Big)^{\frac{1}{q}} \leq c\,\Big(\int_{\mathbb{R}^{d}}|f(x)|^{p}v(x) d\nu_{k}(x)\Big)^{\frac{1}{p}}. \end{eqnarray} \end{theorem} \begin{proof} In order to establish this result, we need to show that \begin{eqnarray} \Big(\int_{0}^{+\infty}\big((\mathcal{F}_{k}(f))^{*}(t)\big)^{q}u^*(t)dt\Big)^{\frac{1}{q}} \leq c\, \Big(\int_{0}^{+\infty}\big(f^{*}(t)\big)^{p}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{1}{p}}.\nonumber\\& &
\end{eqnarray} Take $f\in L_{k}^{p}(\mathbb{R}^d)$, then using (3.2) and (3.4), we obtain \begin{eqnarray*} \Big(\int_{0}^{+\infty}\Big((\mathcal{F}_k(f))^*(t)\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}\leq c\, \Big(\int_{0}^{+\infty}\Big(\int_{0}^{\frac{1}{t}}f^*(s) ds\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}. \end{eqnarray*} If we make the change of variable $t=\frac{1}{s}$ on the right side, we get \begin{eqnarray*} \Big(\int_{0}^{+\infty}\Big((\mathcal{F}_k(f))^*(t)\Big)^{q}u^*(t)dt\Big)^{\frac{1}{q}}\leq c\, \Big(\int_{0}^{+\infty}\Big(\frac{1}{s}\int_{0}^{s}f^*(t) dt\Big)^{q}\frac{u^*(\frac{1}{s})}{s^{2-q}}ds\Big)^{\frac{1}{q}}, \end{eqnarray*} which gives from (1.2), (1.3) and (1.4), that the inequality (3.15) is satisfied if and only if \begin{eqnarray*} \displaystyle\sup_{s>0} \Big(\int_{0}^{s}\frac{u^*(\frac{1}{t})}{t^{2-q}}dt\Big)^{\frac{1}{q}} \Big(\int_{0}^{s}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{-\frac{1}{p}} <+\infty \end{eqnarray*} and \begin{eqnarray*} \sup_{s>0} \Big(\int_{0}^{+\infty}\frac{u^*(\frac{1}{t})}{t^{2}}dt\Big)^{\frac{1}{q}} \Big(\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}\Big[\Big(\frac{1}{v}\Big)^{*}(l)\Big]^{-1} dl\Big)^{-p'}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{1}{p'}}
<+\infty. \end{eqnarray*} In order to complete the proof, we must verify that (3.13) implies these two conditions between the weights $u^*$ and $ \displaystyle\frac{1}{\Big(\frac{1}{v}\Big)^{*}}$. This follows closely the argumentations of [4]. More precisely, since $u^*$ is non-increasing, then\\ $u^*\in B_{q}$ and by (1.1), it yields \begin{eqnarray*} \int_{0}^{s}u^*(\frac{1}{t}){t^{q-2}}dt=\int_{\frac{1}{s}}^{+\infty}\frac{u^*(t)}{t^{q}}dt\leq b_{q}s^q\int_{0}^{\frac{1}{s}}u^*(t)dt. \end{eqnarray*} Hence by (3.13), we get\\\\ $\displaystyle\Big(\int_{0}^{s}u^*(\frac{1}{t}){t^{q-2}}dt\Big)^{\frac{1}{q}} \Big(\int_{0}^{s}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{-\frac{1}{p}}$ \begin{eqnarray*} \qquad\qquad\leq \; b_{q}^{\frac{1}{q}}s\Big(\int_{0}^{\frac{1}{s}}u^*(t)dt\Big)^{\frac{1}{q}} \Big(\int_{0}^{s}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{-\frac{1}{p}} <+\infty, \end{eqnarray*} and so we obtain the first condition. \\ To show that the second condition is satisfied, observe that by means of a change of variable, we have \begin{eqnarray} \Big(\int_{s}^{+\infty}\frac{u^*(\frac{1}{t})}{t^{2}}dt\Big)^{\frac{1}{q}} = \Big(\int_{0}^{\frac{1}{s}}u^*(t)dt\Big)^{\frac{1}{q}}. \end{eqnarray} Now, define the function $G$ by $$\displaystyle G(s)=\Big(\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}\Big[\Big(\frac{1}{v}\Big)^{*}(l)\Big]^{-1}dl \Big)^{-p'}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{1}{p'}}, $$ then by integration by parts, we get \begin{eqnarray*} G(s)&=&\Big[p'G(s)^{p'}+s^{p'}\Big(\int_{0}^{s}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{1-p'}\\&&- p'\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}\Big[\Big(\frac{1}{v}\Big)^{*}(l)\Big]^{-1}dl\Big)^{1-p'}dt\Big]^{\frac{1}{p'}}, \end{eqnarray*} which implies \begin{eqnarray*} (p'-1)G(s)^{p'} \leq p'\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}\Big[\Big(\frac{1}{v}\Big)^{*}(l)\Big]^{-1}dl\Big)^{1-p'}dt, \end{eqnarray*} and so \begin{eqnarray*} G(s)\leq\Big(\frac{p'}{p'-1}\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}\Big[\Big(\frac{1}{v}\Big)^{*}(l)\Big]^{-1}dl\Big)^{1-p'}dt\Big)^{\frac{1}{p'}}. \end{eqnarray*} Since $ \displaystyle\frac{1}{\Big(\frac{1}{v}\Big)^{*}} \in B_{p}$, we can invoke (3.1) and we obtain \begin{eqnarray*} \Big(\int_{0}^{s}\Big(\frac{1}{t}\int_{0}^{t}\Big[\Big(\frac{1}{v}\Big)^{*}(l)\Big]^{-1}dl \Big)^{-p'}\Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{1}{p'}} \leq c\,s \Big(\int_{0}^{s} \Big[\Big(\frac{1}{v}\Big)^{*}(t)\Big]^{-1}dt\Big)^{\frac{-1}{p}}.\end{eqnarray*} Combining this inequality and (3.16), we deduce (3.15).\\
Note that $(|f|^{p})^{*}=(f^{*})^{p}$ and
$(|\mathcal{F}_{k}(f)|^{q})^{*}=((\mathcal{F}_{k}(f))^{*})^{q}$, then applying (3.5) and (3.6)
for the inequality (3.15), we obtain (3.14). This completes the proof. \end{proof} \begin{application} (Pitt's inequality)
Let $u(x)=\|x\|^{\alpha}$, $v(x)=\|x\|^{\beta}$, $x\in\mathbb{R}^{d}$ with $\alpha<0$ and $\beta>0$. Using (2.1) and (2.2), we have for $s\geq0$
\begin{eqnarray*}D_{u}(s)&=&\nu_{k}\Big(\{x\in\mathbb{R}^{d}\,:\;\|x\|^{\alpha}>s\}\Big)\\ &=&\nu_{k}\Big(B(0,s^{\frac{1}{\alpha}})\Big)= \frac{d_{k}}{2\gamma+d}\;s^{\frac{2\gamma+d}{\alpha}},\end{eqnarray*} which gives for $t\geq0$ \begin{eqnarray*}u^{*}(t)=inf\{s\geq 0\,:\; D_{u}(s)\leq t\}= \Big(\frac{2\gamma+d}{d_{k}}\Big)^{\frac{\alpha}{2\gamma+d}}\;t^{\frac{\alpha}{2\gamma+d}}.\end{eqnarray*}
On the other hand, Using (2.1)
and (2.2) again, we have for $s\geq0$, \begin{eqnarray*}D_{\frac{1}{\vartheta}}(s)&=&\nu_{k}\Big(\{x\in\mathbb{R}^{d}\,:\;\|x\|^{-\beta}>s\}\Big)\\ &=&\nu_{k}\Big(B(0,s^{-\frac{1}{\beta}})\Big)=\frac{d_{k}}{2\gamma+d}\;s^{-\frac{2\gamma+d}{\beta}},\end{eqnarray*} which gives for $t\geq0$, \begin{eqnarray*}(\frac{1}{\vartheta})^{*}(t)=inf\{s\geq 0\,:\; D_{\frac{1}{\vartheta}}(s)\leq t\} =\Big(\frac{2\gamma+d}{d_{k}}\Big)^{-\frac{\beta}{2\gamma+d}}\;t^{-\frac{\beta}{2\gamma+d}}.\end{eqnarray*} For these weights and $1<p\leq 2\leq q<+\infty$, the hypothesis of Theorem 3.1, gives respectively that the integrals in the $B_p$-inequality (1.1) for
$\displaystyle\frac{1}{\Big(\frac{1}{v}\Big)^{*}}$ are finite and the boundedness condition
(3.13) is valid
if and only
if $$0<\beta<(2\gamma+d)(p-1)\quad\mbox{and} \quad\left\{\begin{array}{lll}-(2\gamma+d)<\alpha<0,\\ &&\\\frac{1}{2\gamma+d}(\frac{\alpha}{q}+\frac{\beta}{p})=1-\frac{1}{p}-\frac{1}{q}\,.\end{array}\right.$$ Under these conditions and index constraints, we obtain from Theorem 3.1 and for $f\in L^p_{k,v}(\mathbb{R}^d)$, Pitt's inequality
\begin{eqnarray*}\Big(\int_{\mathbb{R}^{d}}\|x\|^{\alpha}|\mathcal{F}_{k}(f)(x)|^{q}
d\nu_{k}(x)\Big)^{\frac{1}{q}} \leq c\,\Big(\int_{\mathbb{R}^{d}}\|x\|^{\beta}|f(x)|^{p} d\nu_{k}(x)\Big)^{\frac{1}{p}}.\end{eqnarray*}In particular for $p=q=2$ and $0<\beta<2\gamma+d$, we get
\begin{eqnarray*}\Big(\int_{\mathbb{R}^{d}}\|x\|^{-\beta}|\mathcal{F}_{k}(f)(x)|^{2}
d\nu_{k}(x)\Big)^{\frac{1}{2}} \leq c\,\Big(\int_{\mathbb{R}^{d}}\|x\|^{\beta}|f(x)|^{2} d\nu_{k}(x)\Big)^{\frac{1}{2}}.\end{eqnarray*}In the classical Fourier analysis, this inequality plays an important role for which some uncertainty principles hold. One of them is the Beckner's logarithmic uncertainty principle (see [3]). \end{application}\begin{remark} The limiting case $\beta=0$, $\alpha=(2\gamma+d)(p-2)$ and $1<p=q\leq2$ was obtained in ([1], Section 4, Lemma 1) and gives the Hardy-Littlewood-Paley inequality
\begin{eqnarray*}\Big(\int_{\mathbb{R}^{d}}\|x\|^{(2\gamma+d)(p-2)}|\mathcal{F}_{k}(f)(x)|^{p}
d\nu_{k}(x)\Big)^{\frac{1}{p}} \leq c\,\Big(\int_{\mathbb{R}^{d}}|f(x)|^{p} d\nu_{k}(x)\Big)^{\frac{1}{p}}.\end{eqnarray*} \end{remark}
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\begin{document}
\title{Hidden assumptions in the derivation of the Theorem of Bell\footnote{Invited paper presented at FQMT11}}
\author{Karl Hess} \email{k-hess@illinois.edu} \affiliation{ Beckman Institute, Department of Electrical Engineering and Department of Physics, University of Illinois, Urbana, Il 61801, USA } \author{Hans De Raedt} \email{h.a.de.raedt@rug.nl} \affiliation{ Department of Applied Physics, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands } \author{Kristel Michielsen} \email{k.michielsen@fz-juelich.de} \affiliation{ Institute for Advanced Simulation, J\"ulich Supercomputing Centre, Research Centre J\"ulich, D-52425 J\"ulich, Germany }
\begin{abstract} John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can be explained by Boole's ideas. Violations of Bell's inequality also call for a revision of the mathematical abstractions and corresponding algebra. It will be shown that this particular view of Bell's inequalities points toward an incompleteness of quantum mechanics, rather than to any superluminal propagation or influences at a distance. \end{abstract}
\keywords{Bell's theorem, stochastic processes} \date{\today}
\maketitle
\section{Introduction}
We discuss Bell's inequalities \cite{bell} and violations of them in terms of the work of Boole \cite{boole}. Boole had derived inequalities similar to those of Bell more than 100 years before Bell, and had traced their violation to an incorrect definition of the mathematical abstractions that represent experimental outcomes. We have shown previously that violations of Bell's inequalities can be interpreted in a similar fashion \cite{hmd,hmd1,hmd2}. We show here further, that Bell's own interpretation of violations of his inequalities is based on several unwarranted assumptions. For example, he assumed that his mathematical abstractions automatically follow the algebra of real numbers and he did not permit an explicit time dependence of his functions. Permitting such explicit time dependence leads us to a commutative algebra involving stochastic processes. These processes invalidate what we call Bell's impossibility proof: Bell's notion that ``quantum mechanics can not be embedded in a locally causal theory" \cite{cuisine}. Our findings favor the suggestions of ``incompleteness" as described in the Einstein-Podolsky-Rosen (EPR) \cite{EPR} paper. These findings confirm and extend numerous mathematical and physical treatises see e.g. \cite{vaxjo,KHRE08,KHRE09,NIEU11}.
\section{Possible experimental outcomes and Boole's algebra}
Boole \cite{boole} discussed a mathematical-logical way to deal with statistics and probabilities. He dissected experiments into events that could only assume two values obeying a calculus that resembled the algebra of real numbers, but with the operations of multiplication, addition and subtraction replaced by the respective logical operations of conjunction, disjunction and negation. The experimental results were replaced by these mathematical abstractions, as soon as a valid one-to-one correspondence between the experiments and the abstractions was established. Consider, for example patients in the two cities A and B, who have certain different histories denoted by the bold faced letters $\bf{a, b, c...}$. These patients are examined with regard to a certain disease and the results of the examination are filed in the following way. If a patient from city A with history $\bf a$ tests positive a note is made that $A_{\bf a} = +1$ and if the test is negative we have $A_{\bf a} = -1$. If the patient is from city B, then we have $B_{\bf a} = \pm 1$ etc.. The subscripts $\bf a$, $\bf b$ etc. indicate, for example, different levels of fever that the patients have encountered.
Boole realized, and this is crucial, that the mathematical abstractions that are chosen in correspondence to the experiments are not necessarily elements of an algebra. It is possible, that the disease that is investigated in the above example depends on factors other than the fever of the patients and their place of residence. It could also depend on the date of their birth. Boole, therefore, tried to establish criteria that could be used in order to determine whether the chosen correspondence of experiments and mathematical abstractions was a valid one, and whether the abstractions therefore did follow the rules of logic. His idea was to employ the algebra of real numbers to the mathematical abstractions, to deduce in this way non-trivial inequalities and to see whether these inequalities were consistent with all experimental results; in our example with the data from the patients. One type of inequality that he considered was: \begin{equation} A_{\bf a}A_{\bf b} + A_{\bf a}A_{\bf c} + A_{\bf b}A_{\bf c} \geq -1 .\label{11may23n1} \end{equation} The fact that this sum of products is larger than or equal to $-1$ can easily be checked by inserting all possible values of $\pm 1$. The inequality is non-trivial, because knowing nothing about the experiments that are described, and realizing that all the measured outcomes may derive from different persons, a possible result is $-3$. This can immediately be seen by adding 6 different birth dates as a superscript. Then we clearly can have: \begin{equation} A_{\bf a}^1A_{\bf b}^2 + A_{\bf a}^3A_{\bf c}^4 + A_{\bf b}^5A_{\bf c}^6 \geq -3. \label{11may23n2} \end{equation}
Thus, dissecting any experiment into binary functions, does not necessarily result in mathematical abstractions that follow the algebra of real numbers. For a given sequence of experiments, and definition of corresponding mathematical abstractions, one needs to make sure that these indeed follow an algebra in order to certify that the experiments were ``dissected" sensibly. Boole's idea was therefore to take statistical averages of expressions as given in Eq.(\ref{11may23n1}). If a violation of the corresponding inequality for the expectation values was obtained, then that was an indication that the experiment could not be fully understood in terms of the assumed mathematical abstractions. Different abstractions must then be chosen to correspond to the experiments, e.g. those shown in Eq.(\ref{11may23n2}) including the birth dates.
\section{Vorob'ev's cyclicities}
Boole's approach to probability theory can be seen as a special case of Kolmogorov's framework that deals with $\sigma$-algebras of countable sets of events. A great review of Kolmogorov's probability theory with special emphasis on physics experiments has been given in \cite{kolmo}.
Progress related to Boole's consistency tests with inequalities was made by Vorob'ev \cite{vorob}. He showed, in a very general way, that non-trivial inequalities and conditions of the Boole type can be found by constructing topological-combinatorial ``cyclicities" of functions on $\sigma$-algebras. For the purpose of our paper, it is sufficient to understand these general cyclicities just by the above example: the $A_{\bf i}A_{\bf j}$ with $ {\bf i, j} = {\bf a, b.c}$ are functions on a probability space (random variables) and form a closed loop, meaning that the choices in the first two products of the inequality determine the third. An infinite number of such inequalities can, therefore, be composed by arranging algebraic expressions of functions that determine the value of other algebraic expressions of the same inequality. Any violation of such inequalities by the measured averages means that the mathematical abstractions describing the experiments are not functions on a $\sigma$-algebra.
Indeed, mathematical-logical abstractions constructed for experiments, by using labels that are derived from sense impressions (such as $A_{\bf a}$), do not necessarily follow the algebra of Boolean variables, the algebra of real numbers or any other given algebra. However, within the framework of special relativity, one can always find consistent abstractions because of the following general reason. Each experiment, and each mathematical abstraction, an event as defined by Kolmogorov \cite{kolmo}, corresponds to a different space-time coordinate $st_n = (x_n, y_n, z_n, t_n), n = 1, 2, 3...$, or set of such coordinates, that represent events as defined by Einstein in his special relativity for some given inertial system. Therefore, no matter how many experiments and cyclicities we consider, the outcomes of the experiments that we register and record can always be labeled by different space-time labels $st_1, st_2,..., st_n, ...$. We can, in general, rewrite the example of Eq.(\ref{11may23n1}) as: \begin{equation} A_{\bf a}^{st_m}A_{\bf b}^{st_{(m+1)}} + A_{\bf a}^{st_{(m+2)}}A_{\bf c}^{st_{(m+3)}} + A_{\bf b}^{st_{(m+4)}}A_{\bf c}^{st_{(m+5)}} \geq -3, \label{11may23n3} \end{equation} and thus have removed the cyclicity for all inequalities with $m = 1, 7, 13, 19,...$. The functions that we use have become more numerous, but they do follow the algebra of real numbers, and no contradiction of the Boole-, or Vorob'ev- type can be found, because all inequalities based on topological combinatorial cyclicities can be removed that way. Note that the setting labels $\bf a, b, c...$ and space-time labels $st_m$ are used as indices and not as independent variables. As will be discussed in more detail, this is important because certain settings can not occur at certain space-time coordinates, and these two variables are therefore not independent.
\section{Bell's inequality}
More than 100 years after Boole, an inequality similar to the inequality of Eq.(\ref{11may23n1}) was re-discovered by John Bell \cite{bell1}, who analyzed quantum experiments that were constructed to investigate work of Einstein, Podolsky and Rosen (EPR) \cite{EPR}. Bell did not refer to Boole's work, nor to the contemporary work of Vorob'ev in any of his collected papers \cite{bell}, and probably did not know of them. He considered measurements of correlated spin-pairs at two locations (cities) A and B. The results of these measurements involve two spin outcomes ($A, B = \pm $1), one on each side. Quantum mechanics predicts that for certain instrument settings $\bf a, b, c....$ the experimental outcomes obey $A_{\bf i} = - B_{\bf i} , {\bf i} =, {\bf a}, {\bf b}, {\bf c}....$, where $\bf a, b, c...$ are now three-dimensional unit vectors related to the settings of the spin-measurement equipment. Bell also introduced a variable $\lambda$ to be discussed in detail below and constructed from these facts an inequality equivalent to:
\begin{equation} A_{\bf a}(\lambda)B_{\bf b}(\lambda) + A_{\bf a}(\lambda)B_{\bf c}(\lambda) + A_{\bf b}(\lambda)B_{\bf c}(\lambda) \leq +1 .\label{11may25n1} \end{equation}
This is again a nontrivial inequality and follows from Boole's corresponding equation by replacing some of the A's by B's and multiplying by $-1$. The equation is non-trivial, because in general the result could be as large as 3. This equation is now often called Bell's inequality, while Eq.(\ref{11may23n1}) is sometimes referred to as the Leggett-Garg inequality. Note that Bell used the slightly different notation of $A({\bf a}, \lambda), B({\bf b}, \lambda)$ etc. because he treated (mistakenly, as explained below) the settings $\bf a, b, c...$ and $\lambda$ as independent variables.
Quantum mechanics provides us also with the result that the expectation value $E_{entangled}$ for the spin correlations of pairs of spins in the singlet state is given by:
\begin{equation} E_{entangled} = -{\bf i} \cdot {\bf j} \text{ with } {\bf i,j} = {\bf a, b, c, ....}. \label{11may25n2} \end{equation}
This result violates Eq.(\ref{11may25n1}), because one can find $\bf a, b, c...$ such that each of the products in Eq.(\ref{11may25n1}) equals $1/\sqrt{2}$ resulting in a left hand side of $3/\sqrt{2} > +2 $.
\section{Bell's explicit and implicit assumptions}
The assumptions, made by Bell in his collected works \cite{bell} to derive his inequalities, are numerous and vary in the different papers of Bell and his followers. We mention here just some of the crucial explicit and implicit (or ``hidden") assumptions. \begin{itemize}
\item[(i)] Bell treats $\lambda$ as an element of reality in the sense defined by EPR and proposes the hypothesis that $\lambda$ effects a more complete specification of the ``state" of the correlated spin pair.
\item[(ii)] The instrument settings $\bf a, b, c...$ and $\lambda$ are assumed to be independent variables. This is often, mistakenly, deduced from the freedom that the experimenter undoubtedly has to choose the settings.
\item[(iii)] Bell further treats $\lambda$ as a very general mathematical variable \cite{bell1}: ``It is a matter of indifference in the following whether $\lambda$ denotes a single variable or a set or even a set of functions, and whether the variables are discrete or continuous."
\item[(iv)] Bell assumes that violations of his inequality are directly connected to $\lambda$ and suggests that violations imply influences of both measurement settings on $\lambda$.
This latter suggestion led to the well known experiments by Aspect and others \cite{Asp}, who changed the measurement settings of both sides rapidly and more or less randomly. This rapid switching excludes certain physical influences of the settings on $\lambda$ that propagate with finite speed (slower or equal to that of light in vacuum). After having received knowledge of the experimental results of Aspect and others that violated his inequality, Bell stated that ``quantum mechanics can not be embedded in a locally causal theory" \cite{cuisine}.
\end{itemize}
The attributes of $\lambda$ listed in (i)-(iii) are, when taken to the limit of their stated generality, in logical, physical and mathematical conflict with each other.
Most importantly, space-time variables such as $st_m$ and instrument settings such as $\bf a, b, c...$ are not independent variables. In the reference frame of the laboratory, any instrument setting is related to a certain space like variable because of the location of the instrument, and to a time like variable because there can not be two different settings at the same time and location. As soon as the settings are (indeed freely) chosen at certain $st_m$, that coordinate-set is not available for any other settings.
Bell states in \cite{cuisine} that ``we can imagine these settings being freely chosen at the last second by two different experimental physicists....if these last second choices are truly free or random, they are not influenced by the variables $\lambda$." While this statement may be true for some $\lambda$, Bell's implicit assumption that, therefore, $\lambda$ and the settings $\bf a, b, c...$ are also mathematically and physically speaking independent variables, is false in general. If $\lambda$ represents space-time variables, then these variables and the setting pairs are not independent.
Quantum mechanics separates the setting related variables entirely from space-time by using settings $\bf a, b, c...$ in connection with operators such as $\sigma_{\bf a}, \sigma_{\bf b}, \sigma_{\bf c}...$, and space-time in connection with wave functions $\psi(st_m)$. Classical probability also can not and must not use all of these variables as independent. Indeed, Kolmogorov's stochastic processes add a separate time index to each measurement $A, B$ and do not explicitly include equipment settings.
Bell's functions of both the settings and the $\lambda$'s, such as $A({\bf a}, \lambda)$ are not formulated with the appropriate mathematical and physical caution, and are either not general or not functions of independent variables, as both assumed in Bell's proof. Even for purely formal reasons, Bell's $\lambda$ can not be both the independent time variable of physics and a random variable, simply because time is not a random variable. Time is also is not an operator in quantum mechanics!
The absorption of independent space-time variables into Bell's $\lambda$ also leads to logical contradictions for yet another reason. The proofs of Bell and followers often assume, explicitly or implicitly, that $\lambda$ occurs in sets of six (sometimes more) to maintain a Vorob'ev cyclicity. Bell orders the functions in his initial proof into sums of the three products as shown in Eq.~(\ref{11may25n1}), all containing the identical variable $\lambda$. This fact looks innocuous in Bell's paper, because at the point at which he invokes the appearance of $\lambda$ in six factors, $\lambda$ is treated as a dummy integration variable (see the equation without number after Eq.(14) in Bell's original paper). To show the problems connected with these choices of $\lambda$, we distinguish now two cases.
First assume that the $\lambda$'s are indeed elements of reality that play a role in the formation of the data that are actually collected. Such $\lambda$'s are in principle all different (naturally averages over large numbers may still be the same). Consider now such $\lambda$'s formed by a combination of a space-time variable $st_m$ and some other arbitrary element of reality $\lambda_m'$ related to spin. Then consider the parameters $\lambda_m = (st_m, \lambda_m')$ and pick six equal values of $\lambda_m =\lambda$, for each of the six different measurements given in Eq.(\ref{11may25n1}). One then has equated different times and different spin-related parameters, which may be physically unreasonable. Consider as an example the boiling of an egg and link $st_m$ to various clock-times during the process of boiling, as suggested by H. B. G. Casimir \cite{cuisine}. Further denote the egg viscosity at various locations in the egg ( settings, such as in the yoke or in the egg-white) by $\lambda_m'$. It certainly is incorrect for this case to equate a variety of parameters $\lambda_m = (st_m, \lambda_m')$ for different settings. The yoke stays soft much longer than the white. Thus, for general Einstein local experiments, $\lambda$ may have to depend on both settings and times. Equating the $\lambda$'s in general cases, as Bell has done for his $\lambda$'s and also his ``be-ables" \cite{bell}, can therefore lead to logical contradictions. Space-time coordinates and other general elements of reality can not be concatenated into one variable and then regarded as the same dummy variable for triple (or quadruple) products. In fact, if we wish to prove Bell's inequality, the possibility of ordering all data into the triple products of Eq .~(\ref{11may25n1}) is just what needs to be shown. Bell, obviously, did not consider the full implications of Casimir's counter-example.
Second assume with Mermin \cite{mer}, that the $\lambda$'s of Eq.~(\ref{11may25n1}) are indeed all the same and the equation really is written, as Mermin states, only to contemplate the conjecture that there exist such $\lambda$'s that ``predetermine" the experimental outcomes. Only one of the experiments listed in Eq.~(\ref{11may25n1}) is then being performed, according to Mermin. However, also according to Mermin, if the outcomes are predetermined by the $\lambda$'s, one can imagine obtaining results as written down in Eq .~(\ref{11may25n1}). This statement is principally correct, however, such results would have nothing to do with actually collected data of actual experiments that are used for the formation of the expectation values. Nor do the results of Eq .~(\ref{11may25n1}) have anything to do with Kolmogorov's or Boole's probability theory that deal with possible outcomes of experiments that can be collected as ``data" and described by sensible mathematical abstractions. The hypothesis of possible predetermination of outcomes is irrelevant to Kolmogorov's sample space. Consider the meals on the menu of a variety of restaurants. These are certainly predetermined. Yet, when we eat in hundreds or thousands of restaurants of type ${\bf a} =$ Indian, ${\bf b} =$ Chinese, ${\bf c} =$ Austrian, etc., and eat only one meal at a time, then we do not necessarily have the expectation of having averages from all the meals of all the menus in our stomachs, but rather averages and corresponding correlations arising from one meal at each place. Bell's own criticism of von Neumann's work can be directly applied here: ``It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements {\it either} of which {\it might} be made on a given occasion but only one of which can in fact be made" \cite{bvn}. Speaking in more precise mathematical terms, one can not apply the pointwise ergodic theorem to the functions of Eq.(\ref{11may25n1}), because the experiments that correspond to these functions cannot be performed except for one pair of settings. In case of a violation of the inequality, no $\sigma$-algebra exists on which these functions can be defined and, therefore, Eq.(\ref{11may25n1}) has no consequences for the expectation values obtained in the actual experiments.
The assumption that $\lambda$ may not depend on the setting parameters $\bf a, b, c...$ in a locally causal theory is also in conflict with the mathematical generality of $\lambda$. If we wish to regard $\lambda$ as an event in the sense of Kolmogorov, then we need to be able to identify $\lambda$ with the event that $A$ and $B$ with chosen settings assume certain values. Thus we need to be able to identify the $\lambda$'s with events (usually denoted by $\omega$) of a probability space. The actual event (usually denoted by $\omega_{act}$ \cite{willi}) could, for example, be $A_{\bf a} = +1, B_{\bf b} = +1$. That actual event certainly is not independent of $\bf a, b$. This identification has nothing to do with causal theories but only with the fact that the event is composed of two different experiments. Whether or not influences at a distance are present is not a concern of probability theory. If one wishes to use established probability theory and, therefore, wishes to identify a specific $\lambda$ with a an event realization $\omega_{act}$, a logical difficulty arises.
Fortunately, it is not necessary to include all these subtle points, in order to discuss the relevance and ranges of validity of Bell's theorem. The following two incorrect assumptions of Bell are straightforward to understand and, together with (ii) sufficient to show how Bell's proofs and the proofs of his followers fail.
\begin{itemize}
\item[(v)] Bell always assumed or implied that his mathematical abstractions representing the experiments follow automatically the algebra of real numbers. This is actually what needs to be shown by the fulfillment of all possible inequalities and other consequences of Vorob'ev's cyclicities.
\item[(vi)] Bell treated $\lambda$ explicitly as a random variable with well defined and given probability distribution $\rho(\lambda)$ (see his Eq.(12) in \cite{bell1}). Hidden behind this innocuous assumption is the fact that, therefore, {\it Bell did not permit an explicit space-time dependence of the statistical properties of his functions $A, B$, and has therefore excluded general stochastic processes!} The perception that Bell and his followers had was that $\lambda$ can represent the space-time variables. However, as we discussed above, then the settings can not be regarded as independent variables.
\end{itemize}
Work of Bell's followers often implies assumptions and restrictions of similar nature. For example, Leggett and Garg \cite{lgarg} and also Mermin \cite{mer} always assume that their symbols follow the algebra of real numbers. They also do not permit, and this is our crucial point, an explicit time dependence of the statistical properties of the functions $A, B$. Furthermore, whenever they use $\lambda$'s, they do use the settings and $\lambda$'s as independent variables by integrating (summing) independently over them. Mermin even claims toward the end of his paper \cite{mer} ``...times...can be independently varied without altering the distribution". At best, however, the expectation values can stay unchanged by variations of measurement times. Because of the coincidence counting of the actual experiments, even this statement is not of general validity. Nor does quantum theory or any physical theory tell us that time can be varied without altering probability distributions. Quantum theory does provide us with time dependent probability distributions and with expectation values for large numbers of measurements. Single outcomes or small sets of outcomes are not the objects of quantum theory, that most certainly does not forbid an explicit space-time dependence of the statistical properties of the functions $A, B$. These functions are not even part of quantum theory.
Removing the assumptions (ii), (v) and (vi) of Bell and followers does, therefore, open new horizons. One can envision the violation of Bell-Boole type of inequalities, or more generally, Vorob'ev cyclicities as prescriptions for necessary space-time dependencies and, therefore, as physical rules that are not contained in quantum theory. One can then construct explicitly space-time dependent functions on $\sigma$-algebras that remove all Vorob'ev type cyclicities that lead to contradictions. These functions describe then a more complete physical theory in the sense of EPR.
\section{Removing Bell's assumptions: generalized stochastic processes}
Instead of asking which properties of $\lambda$ might explain a violation of Eq.(\ref{11may25n1}), the more general and crucial question to ask is: under which circumstances it is possible to remove the cyclicity in Eq.(\ref{11may25n1}) in order to obtain the quantum result of Eq.(\ref{11may25n2}) within the Kolmogorov framework? This can indeed be done by using a slightly generalized version of a stochastic process.
We consider here only a discrete time stochastic process. Such a process is defined by a finite or countable infinite sequence of random variables such as $A_{\bf a}^{t_1}, A_{\bf a}^{t_2}, A_{\bf a}^{t_3}, ....$. We add the slight generalization that we use space-time coordinates instead of time coordinates and thus have $A_{\bf a}^{st_1}, A_{\bf a}^{st_2}, A_{\bf a}^{st_3}, ....$. This generalization is trivial as long as we deal with discrete time stochastic processes, because then the physical meaning of the time related index makes no difference for the mathematics. For Einstein-Podolski-Rosen experiments, we also introduce a second stochastic process $B_{\bf b}^{st_1'}, B_{\bf b}^{st_2'}, B_{\bf b}^{st_3'}, ....$ for the correlated pair ($A_{\bf b}^{st_1'}=-B_{\bf b}^{st_1'}$ etc.), and note that the primed and unprimed space-time coordinates are correlated in such experiments by the technique of coincidence counting of the pair results. Combining the two stochastic processes, one obtains a vector stochastic process $ (A_{\bf a}^{st_1}, B_{\bf b}^{st_1'}), (A_{\bf a}^{st_2}, B_{\bf b}^{st_2'}), (A_{\bf a}^{st_3}, B_{\bf b}^{st_3'}), ...$.
The vector stochastic process as defined above can be further generalized by choosing the setting pairs (indicating the measurement procedure) randomly. Thus we obtain the discrete space-time vector stochastic process:
\begin{equation} (A_{\bf i}^{st_1}, B_{\bf j}^{st_1'}), (A_{\bf i}^{st_2}, B_{\bf j}^{st_2'}), (A_{\bf i}^{st_3}, B_{\bf j}^{st_3'}) \text{... with }{\bf i, j} = RP, \label{11may25n3} \end{equation}
where RP denotes any pair $({\bf a}, {\bf b})$, $({\bf a}, {\bf c})$ or $({\bf b}, {\bf c})$ chosen totally randomly, or at the will of any experimenter. This is indeed a vector stochastic process. The only difference to the standard definition is that the discrete times $t_n$ are replaced by discrete space-time labels $st_n$ with $n = 1, 2, 3,...$. Furthermore, certain setting pairs are chosen at the discrete space-times to result in particular pairs of functions on a probability space $(A_{\bf i}^{st_n}, B_{\bf j}^{st_n'})$ that are usually just denoted by pairs of functions such as $ X^nY^n$, with both $X^n$ and $Y^n$ being functions on a probability space $\Omega$ with elements $\omega$ \cite{willi}.
Care must be taken with the choice and labeling of the functions. It is not possible, for example, to construct a four dimensional vector stochastic process with equal space-time labels of the vectors that returns the quantum result. This is because such a process also fulfills the nontrivial inequality
\begin{equation} (A_{\bf a}^{st_n} B_{\bf b}^{st_n}) + (A_{\bf a}^{st_n} B_{\bf c}^{st_n}) + (A_{\bf b}^{st_n} B_{\bf c}^{st_n}) \leq +1 , \label{11june7n1} \end{equation} that involves a cyclicity. It also contains the fundamental error to assign equal space-time labels to different equipment settings, thus ignoring that settings and space-time labels are not independent variables.
The fact that processes as given by Eq.~(\ref{11may25n3}) remove any cyclicity, can be seen immediately by inserting the pairs into Eq.(\ref{11may25n1}). Because all factors are in principle different. The resulting inequalities are of the trivial form:
\begin{equation} A_{\bf a}^{st_m}B_{\bf b}^{st_m'} + A_{\bf a}^{st_{(m+1)}}B_{\bf c}^{st_{(m+1)}'} + A_{\bf b}^{st_{(m+2)}}B_{\bf c}^{st_{(m+2)}'} \leq +3 ,\label{11may26n1} \end{equation}
for $m =1, 4, 7,...$.
It can be shown that for any given stochastic process of the type Eq.~(\ref{11may25n3}), with randomly chosen settings and space-time coordinates, one can find a probability measure that indeed leads to the quantum result \cite{hp1}. That probability measure must necessarily depend in some complex way on the setting pairs. This fact has a mathematical reason that is independent of any locality considerations.The settings and space-time variables are, as stressed above, not independent. For each setting pair, one has necessarily a different function-domain of space-time variables $st_n$. Therefore, if the setting pairs change, the domain must also change, and so must, in general, the corresponding joint probabilities for the functions $A, B$. These probabilities depend, therefore, on the setting pairs. Probability theory can, thus, not ``play" the so called Bell game.
The Bell game requires two players in two measurement stations to choose values for $A, B$ with the knowledge of the instrument settings of their respective stations only. The expectation values for these choices of $A, B$ need then to give the quantum result. However, any probability model of the game needs to involve, as shown above, different domains of the functions for different setting pairs. If the Bell inequalities are fulfilled, this does not matter because then one can find one ``artificial" probability space that leads to the experimental expectation values. If, on the other hand, the Bell inequalities are not fulfilled, then no such probability space exists. The $A, B$ need to be labeled with a space-time index to avoid cyclicities, and the player needs to know the actual domains of the functions and actual joint probabilities to play the game. These domains and joint probabilities can not be determined from the knowledge of one setting in one station only. Thus there exists a purely mathematical reason for the impossibility to play the Bell-game with probability theory. To play the game by other means would then require necessarily a detailed knowledge of all possible space-time dependencies of measurement and preparation in EPR experiments, and thus an infinity of functions $A^{st_n}, B^{st_n'}$ with the cardinality of the continuum \cite{hp2}. We have not proven the existence of such functions by the above findings but we have refuted the impossibility-proofs of Bell and followers because we removed the cyclicities by use of general space-time labels. No assumption needs to be made, from the viewpoint of logic or mathematics, whether these space-time labels are from within or from outside the light-cone.
\section{Space time dependencies of spin measurements}
It remains to be shown that space-time dependencies of the preparation of correlated pairs as well as the measurement outcomes (and therefore of the functions $A, B$) are physically reasonable, and can lead to the quantum result of Eq.~(\ref{11may25n2}).
Possible time dependences within the light cone are numerous. We just list here a smorgasbord of those that matter for sets of particles with spin. The earth rotates around itself and this rotation introduces a time dependence on how a gyroscope would be seen from a resting observer. Naturally that time dependence would be of the order of a day. Radio waves can cause spin resonances and are omnipresent as are magnetic fields. Because the earth magnetic field is small, corresponding time dependencies would be slow also. The interaction of nuclear spins and electron spins happen in a wide range and equilibration of electron spin in the field of nuclear spins can take from hours to milliseconds or less. Many body interactions of electrons and photons present us even with a wider range of time constants being limited in solids (such as a spin polarizer) only by the plasma frequency. For a typical semiconductor, that frequency is around $10^{-14} s^{-1}$ corresponding to time constants down to $10^{-14} s$. Naturally, if we include the frequencies derived from the mass of an electron and its energy, one has to include possibilities of $10^{-20} s$ time constants and below. To all of these possibilities of time dependencies of the functions $A, B$, one needs to add the fact that actual Aspect type of experiments are based on coincidence counting i.e. the time correlation of the measurements in the two experimental wings play a major role. This leads to a true ``entanglement" of the time dependent joint probabilities of measurement outcomes $A, B$. One might view this fact in a Shakespearean fashion: all these random choices of settings are being made by the players, and time weaves then a pattern into it.
How, specifically, these reasonable space-time dependencies actually lead to the quantum result of Eq.~(\ref{11may25n2}) is still subject of discussions. EPR experiments that employ time-coincidences to identify pairs effectively apply a filter to the data and it has been shown that such a filtering mechanism may lead to violations of a Bell inequality, opening the route to a description in terms of locally causal, classical models~\cite{fine}. A first concrete model of this kind was proposed by S. Pascazio who showed that his model approximately reproduces the correlation of the singlet state~\cite{pasc}, violating Bell inequalities Models that rigorously reproduce the results of quantum theory for the singlet and uncorrelated state are given in Refs.~\cite{raed0,raed1,zhao,mich}.
Will these facts end the non-locality discussions? Probably not, because one can also remove the cyclicity with labels from outside the light cone. The simplest choice is, of course, using $\lambda$'s that depend on the setting parameters $\bf a, b, c...$ of the other sides. For Aspect type of experiments, that amounts to the use of space-time coordinates from outside the light cone. Our main point is, however, that the cyclicities can also be removed by space-time dependencies {\it within} the light cone. The question is, of course, whether such time dependencies are reasonable and are indeed elements of a complete physical theory. The answer to this question and the nature of the infinite set of space-time labeled functions as well as that of the $\lambda$'s are, if they exist at all, currently not known. However, even with the lack of this knowledge, we can answer an important question: can quantum mechanics be considered a complete physical theory?
\section{Kolmogorov vs quantum probability, completeness}
We first summarize the above findings. One can find different elements of a Boolean or $\sigma$-algebra for all possible events. The elementary events can be understood by both, Einstein's definition of space-time events, and Kolmogorov's definition of the events of a $\sigma$-algebra. These events, corresponding to different experiments and their abstractions, do not lead to any but the trivial Boole type inequalities such as Eq.~(\ref{11may23n3}) that can never be violated. Thus, at least in principle, one can always find functions on a $\sigma$-algebra that describe any possible experimental result of macroscopic events that can be recorded by our anthropomorphic methods. Because all these recordings can in that way be described without contradiction, it follows that the averages of macroscopic indicator readings, that are described by quantum theory, can also be described without contradiction. This answers the important question whether Kolmogorov probability can be made consistent with the results of quantum mechanics. It always can be, because one always can find functions on a probability space that follow only trivial Boole-Bell equations. If, however, the experimental results are not all denoted by different space-time coordinates, if some are assumed to have the same coordinates or not to depend on coordinates, then nontrivial inequalities and contradictions may arise. These contradictions call then for additional space-time labels or, speaking from the viewpoint of causal physical theories, additional space-time dependencies.
To find a physically valid theory within the framework of Boole and Kolmogorov, it is therefore necessary to explore all possible topological combinatorial cyclicities and to determine the necessary space-time labels, in order to arrive at a contradiction free set of random variables that are defined as functions on a $\sigma$- algebra. Then one needs to select the physically reasonable space-time connections that may underlie this $\sigma$-algebra. The advantage of quantum mechanics, as compared to this cumbersome procedure, is that quantum mechanics works with a non-commutative algebra and, compensating for this luxury, never attempts to explore the logical and causal space-time connections between single events or small sets of events. Quantum theory only insists on agreement with averages of large numbers of experiments. If we wish to compare Boole's or Kolmogorov's probability theory with quantum mechanics, and render judgements on their completeness with respect to a physical-science point of view, the following fact should be considered. The introduction of probabilities is the hallmark of all of these theories and does, by itself, not necessarily represent an incompleteness. It is indeed generally assumed that quantum mechanics gives all the information about averages that we possibly can have. To achieve this result, however, quantum mechanics does sacrifice the prediction of single and small sets of, events.
The elements of Kolmogorov's $\sigma$-algebra and functions of them must fulfill certain requirements not contained in conventional quantum theory: they must not contradict any facts following from the topological combinatorial cyclicities of Vorob'ev. These rules for the data of experiments, therefore, impose physically speaking a ``law" on the space-time dependencies of the outcomes. Because this law is not contained in quantum mechanics, but does relate to the physical measurement outcomes that we macroscopically record, quantum mechanics exhibits an incompleteness in the sense of Einstein, Podolsky and Rosen.
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\title{Many-body-QED perturbation theory: Connection to the Bethe-Salpeter equation} \author{Ingvar Lindgren} \address{Department of Physics, Chalmers University of Technology and the G\"oteborg University, G\"oteborg, Sweden} \shortauthor{Lindgren} \maketitle
\begin{abstract} The connection between many-body theory (MBPT)---in perturbative and non-perturbative form---and quantum-electrodynamics (QED) is reviewed for systems of two fermions in an external field. The treatment is mainly based upon the recently developed covariant-evolution-operator method for QED calculations [Lindgren \textit{et al.} Phys. Rep. \textbf{389}, 161 (2004)], which has a structure quite akin to that of many-body perturbation theory. At the same time this procedure is closely connected to the $S$-matrix and the Green's-function formalisms and can therefore serve as a bridge between various approaches. It is demonstrated that the MBPT-QED scheme, when carried to all orders, leads to a Schr\ödinger-like equation, equivalent to the Bethe-Salpeter (BS) equation. A Bloch equation in commutator form that can be used for an "extended" or quasi-degenerate model space is derived. It has the same relation to the BS equation as has the standard Bloch equation to the ordinary Schr\ödinger equation and can be used to generate a perturbation expansion compatible with the BS equation also for a quasi-degenerate model space. \\\\PACS Nos.: {31.10+z, 31.15Md, 31.30Jv} \end{abstract} \begin{resume}French version of abstract (supplied by CJP) \traduit\end{resume}
\begin{center}Submitted 25 Jan. 2005, Corrected 8 Feb. 2005\end{center}
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{\put(0,0.125)\VectorUp\put(0,-0.125)\VectorUp}
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{\put(0,0.125)\VectorDn\put(0,-0.125)\VectorDn}
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{\put(0.1,0.1)\VectorDl\put(-0.1,-0.1)\VectorDl}
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\newcommand{\Elline}[4]
{\put(0,0){\LineV{#1}} \put(0,#2){\VectorUp} \put(-0.3,#2){\makebox(0,0){$#3$}} \put(0.4,#2){\makebox(0,0){$#4$}}}
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{\put(0,0){\LineV{#1}} \put(0,#2){\WectorUp} \put(-0.3,#2){\makebox(0,0){$#3$}} \put(0.3,#2){\makebox(0,0){$#4$}}}
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\newcommand{\Ellinet}[4]
{\put(0,0){\Linev{#1}} \put(0,#2){\vector(0,1){0}} \put(-0.35,#2){\makebox(0,0){$#3$}} \put(0.35,#2){\makebox(0,0){$#4$}}} \newcommand{\EllineT}[4]
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\newcommand{\EllineDnt}[4]
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\newcommand{\EllineDn}[4]
{\put(0,0){\LineV{#1}} \put(0,#2){\VectorDn} \put(-0.35,#2){\makebox(0,0){$#3$}} \put(0.35,#2){\makebox(0,0){$#4$}}}
\newcommand{\EllineDl}[4]
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\newcommand{\Ellinedl}[4]
{\put(0.01,0.01){\line(-1,-2){#1}} \put(-0.01,-0.01){\line(-1,-2){#1}} \thicklines\put(-0.05,-0.1){\vector(-1,-2){#2}}
\put(-0.2,0.2){\makebox(-#1,-#1){$#3$}} \put(0.2,-0.2){\makebox(-#1,-#1){$#4$}}}
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{\put(0.01,0.01){\line(1,-2){#1}} \put(-0.01,-0.01){\line(1,-2){#1}} \thicklines\put(0,0){\vector(1,-2){#2}}
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\newcommand{\EllineA}[7]
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\newcommand{\EllineDr}[4]
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\newcommand{\EllinedR}[5]
{\put(0,0){\line(1,-3){#1}} \put(0.014,0){\line(1,-3){#1}} \put(-0.014,0){\line(1,-3){#1}} \put(#2,-#3){\makebox(0,0){{\Vectordr}}} \put(#2,-#3){\makebox(-0.5,-0.5){$#4$}} \put(#2,-#3){\makebox(0.5,0.5){$#5$}}}
\newcommand{\EllineuR}[5]
{\put(0,0){\line(1,3){#1}} \put(0.014,0){\line(1,3){#1}} \put(-0.014,0){\line(1,3){#1}} \put(#2,#3){\makebox(0,0){\Vectorur}} \put(#2,#3){\makebox(-0.5,-0.5){$#4$}} \put(#2,#3){\makebox(0.5,0.5){$#5$}}}
\newcommand{\Ellineur}[5]
{\put(0,0){\Lineur{#1}} \put(#2,#3){\makebox(0,0){\Vectorur}} \put(#2,#3){\makebox(-0.5,0){$#4$}} \put(#2,#3){\makebox(0.5,0){$#5$}}}
\newcommand{\EllineUr}[4]
{\put(0,0){\LineUr{#1}} \put(#2,#2){\makebox(-0.35,-0.35){\VectorUr}} \put(-0.2,0.4){\makebox(#1,#1){$#3$}} \put(0.2,-0.3){\makebox(#1,#1){$#4$}}}
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{\put(0,0){\LineDl{#1}} \put(-#2,-#2){\WectorDl} \put(-0.25,0.25){\makebox(-#1,-#1){$#3$}} \put(0.25,-0.25){\makebox(-#1,-#1){$#4$}}}
\newcommand{\Ebox}[2]
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{\multiput(0.05,0)(0.25,0){#1}{\line(1,0){0.15}}}
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{\multiput(0.05,0)(0,0.25){#1}{\line(0,1){0.15}}}
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{\multiput(0.05,0)(0.25,0){6}{\line(1,0){0.15}}}
\newcommand{\DashH}
{\multiput(0.05,0)(0.25,0){10}{\line(1,0){0.15}}}
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\newcommand{\dashHnumu}[2]
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\newcommand{\Potint}
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\newcommand{\potint}
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\newcommand{\PotintS}
{\put(0,0){\dash{4}} \put(1,0){\makebox(0,0){$\times$}} \put(0,0){\circle*{0.1}}}
\newcommand{\PotintL}
{\put(-1.25,0)\dashH \put(-1.35,0){\makebox(0,0){x}} \put(0,0){\circle*{0.15}}}
\newcommand{\Effpot}
{\put(0,0)\dashH \put(1.35,0){\makebox(0,0){x}} \put(1.35,0){\circle{0.3}} \put(0,0){\circle*{0.15}}}
\newcommand{\effpot}
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\newcommand{\EffpotL}
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\newcommand{\Triang}
{\put(0,0){\line(2,1){0.5}} \put(0,0){\line(2,-1){0.5}} \put(0.5,-0.25){\line(0,1){0.5}}}
\newcommand{\TriangL}
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\newcommand{\hfint}
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\newcommand{\hfintL}
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\newcommand{\VPloop}[1]
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\newcommand{\VPloopt}[1]
{\put(0,0){\circle{1}} \put(0.52,0.05){\VectorDn} \put(0.75,0){\makebox(0,0){$#1$}}}
\newcommand{\VPloopL}[1]
{\put(0,0){\circle{1}} \put(0.01,0.){\circle{1}}\put(-0.01,0){\circle{1}} \put(0,0.01){\circle{1}}\put(0,-0.01){\circle{1}} \put(-0.5,0.05){\VectorDn} \put(-0.75,0){\makebox(0,0){$#1$}}}
\newcommand{\VPloopLt}[1]
{\put(0,0){\circle{1}} \put(-0.5,0){\VectorDn} \put(-0.75,0){\makebox(0,0){$#1$}}}
\newcommand{\VPloopLR}[2]
{\put(0,0){\circle{1}}\put(-0.01,0){\circle{1}} \put(0,0.01){\circle{1}}\put(0,-0.01){\circle{1}} \put(-0.5,0){\VectorDn} \put(0.5,0){\VectorUp} \put(-0.75,0){\makebox(0,0){$#1$}} \put(0.75,0){\makebox(0,0){$#2$}}}
\newcommand{\VPloopLRt}[2]
{\put(0,0){\circle{1}} \put(-0.5,0){\VectorDn} \put(0.5,0){\VectorUp} \put(-0.75,0){\makebox(0,0){$#1$}} \put(0.75,0){\makebox(0,0){$#2$}}}
\newcommand{\VPloopD}[2]
{\put(0,0){\circle{1}} \put(0.01,0.){\circle{1}}\put(-0.01,0){\circle{1}} \put(0.,0.){\circle{1}}\put(-0.01,0){\circle{1}} \put(0,0.01){\circle{1}}\put(0,-0.01){\circle{1}} \put(0,0.52){\VectorR} \put(0,-0.52){\Vector} \put(0,0.8){\makebox(0,0){$#1$}} \put(0,-0.8){\makebox(0,0){$#2$}}}
\newcommand{\VPloopDt}[2]
{\put(0,0){\circle{1}} \put(0,0.5){\VectorR} \put(0.05,-0.47){\Vector} \put(0,0.8){\makebox(0,0){$#1$}} \put(0,-0.8){\makebox(0,0){$#2$}}}
\newcommand{\Loop}[2]
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\newcommand{\HFexch}[1]
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\newcommand{\HFexcht}[1]
{\put(0,0)\dashH \qbezier(0,0.01)(0.625,0.515)(1.25,0.015) \put(0.625,0.26){\Vector} \put(0.625,0.5){\makebox(0,0){$#1$}} \put(0,0){\circle*{0.15}} \put(1.25,0){\circle*{0.15}}}
\newcommand{\Dashpt}[1] {\multiput(0,-0.6)(0,0.25){13}{\line(0,1){0.15}} \put(0,0){\circle*{0.15}} \put(0,#1){\circle*{0.15}}}
\newcommand{\photonH}[3]
{\qbezier(0,0)(0.08333,0.125)(0.1666667,0) \qbezier(0.1666667,0)(0.25,-0.125)(0.3333333,0) \qbezier(0.3333333,0)(0.416667,0.125)(0.5,0) \qbezier(0.5,0)(0.583333,-0.125)(0.666667,0) \qbezier(0.666667,0)(0.75,0.125)(0.833333,0) \qbezier(0.833333,0)(0.916667,-0.125)(1,0) \qbezier(1,0)(1.083333,0.125)(1.166667,0) \qbezier(1.166667,0)(1.25,-0.125)(1.333333,0) \qbezier(1.333333,0)(1.416667,0.125)(1.5,0) \put(0.75,0.){\VectorR} \put(0.75,0.35){\makebox(0,0){$#1$}} \put(0,0){\circle*{0.1}} \put(1.5,0){\circle*{0.1}} \put(-0.5,0){\makebox(0,0){#2}} \put(2,0){\makebox(0,0){#3}}}
\newcommand{\photon}[3]
{\qbezier(0,0)(0.08333,0.125)(0.1666667,0) \qbezier(0.1666667,0)(0.25,-0.125)(0.3333333,0) \qbezier(0.3333333,0)(0.416667,0.125)(0.5,0)
\qbezier(0.5,0)(0.583333,-0.125)(0.666667,0) \qbezier(0.666667,0)(0.75,0.125)(0.833333,0) \qbezier(0.833333,0)(0.916667,-0.125)(1,0)
\qbezier(1,0)(1.083333,0.125)(1.166667,0) \qbezier(1.166667,0)(1.25,-0.125)(1.333333,0) \qbezier(1.333333,0)(1.416667,0.125)(1.5,0)
\qbezier(1.5,0)(1.583333,-0.125)(1.666667,0) \qbezier(1.666667,0)(1.75,0.125)(1.833333,0) \qbezier(1.833333,0)(1.916667,-0.125)(2,0) \put(1,0.0){\VectorR} \put(1,0.35){\makebox(0,0){$#1$}} \put(0,0){\circle*{0.15}} \put(2,0){\circle*{0.15}} \put(-0.35,0){\makebox(0,0){#2}} \put(2.35,0){\makebox(0,0){#3}}}
\newcommand{\photonHS}[4]
{\qbezier(0,0)(0.08333,0.125)(0.1666667,0) \qbezier(0.1666667,0)(0.25,-0.125)(0.3333333,0) \qbezier(0.3333333,0)(0.416667,0.125)(0.5,0) \qbezier(0.5,0)(0.583333,-0.125)(0.666667,0) \qbezier(0.666667,0)(0.75,0.125)(0.833333,0) \qbezier(0.833333,0)(0.916667,-0.125)(1,0) \put(0.5,0.025){\VectorR} \put(0.5,0.35){\makebox(0,0){$#1$}} \put(0,0){\circle*{0.15}} \put(1,0){\circle*{0.15}} \put(0,-0.5){\makebox(0,0){#2}} \put(1,-0.5){\makebox(0,0){#3}}}
\newcommand{\photonNE}[3]
{\qbezier(0,0)(0.22,-0.02)(0.2,0.2) \qbezier(0.2,0.2)(0.18,0.42)(0.4,0.4) \qbezier(0.4,0.4)(0.62,0.38)(0.6,0.6) \qbezier(0.6,0.6)(0.58,0.82)(0.8,0.8) \qbezier(0.8,0.8)(1.02,0.78)(1,1) \qbezier(1,1)(0.98,1.22)(1.2,1.2) \qbezier(1.2,1.2)(1.42,1.18)(1.4,1.4) \qbezier(1.4,1.4)(1.38,1.62)(1.6,1.6) \qbezier(1.6,1.6)(1.82,1.58)(1.8,1.8) \qbezier(1.8,1.8)(1.78,2.02)(2,2) \put(1,1){\makebox(0.05,-0.2){\VectorUp}} \put(0,0){\circle*{0.1}} \put(2,2){\circle*{0.1}} \put(1,1){\makebox(-0.6,0.4){$#1$}} \put(-0.35,-1){\makebox(0,2){$#2$}} \put(2.35,1){\makebox(0,2){$#3$}}}
\newcommand{\photonNNE}[3]
{\qbezier(0,0) (0.28,-0.02)(0.2,0.3) \qbezier(0.2,0.3)(0.12,0.52)(0.4,0.6) \qbezier(0.4,0.6)(0.68,0.58)(0.6,0.9) \qbezier(0.6,0.9)(0.52,1.12)(0.8,1.2) \qbezier(0.8,1.2)(1.08,1.18)(1,1.5) \qbezier(1,1.5) (0.92,1.72)(1.2,1.8) \qbezier(1.2,1.8)(1.48,1.86)(1.4,2.1) \qbezier(1.4,2.1)(1.365,2.24)(1.6,2.4) \qbezier(1.6,2.4)(1.835,2.46)(1.8,2.7) \qbezier(1.8,2.7)(1.765,2.84)(2,3) \put(0.6,0.8){\makebox(0,0){\VectorUp}} \put(0,0){\circle*{0.1}} \put(2,3){\circle*{0.1}} \put(1,0.8){\makebox(0,0){$#1$}} \put(-0.35,0){\makebox(0,0){$#2$}} \put(2.35,3){\makebox(0,0){$#3$}}}
\newcommand{\photonENE}[3]
{\qbezier(0,0)(0.17,-0.04)(0.2,0.1) \qbezier(0.2,0.1)(0.23,0.32)(0.4,0.2) \qbezier(0.4,0.2)(0.57,0.16)(0.6,0.3) \qbezier(0.6,0.3)(0.63,0.52)(0.8,0.4) \qbezier(0.8,0.4)(0.97,0.36)(1,0.5) \qbezier(1,0.5)(1.03,0.72)(1.2,0.6) \qbezier(1.2,0.6)(1.37,0.56)(1.4,0.7) \qbezier(1.4,0.7)(1.43,0.92)(1.6,0.8) \qbezier(1.6,0.8)(1.77,0.76)(1.8,0.9) \qbezier(1.8,0.9)(1.83,1.12)(2,1) \put(1.2,0.75){\makebox(-0.1,0.06){\VectorR}}
\put(1.2,0.85){\makebox(0,-0){$#1$}} \put(-0.35,-1){\makebox(0,2){$#2$}} \put(2.35,0){\makebox(0,2){$#3$}}}
\newcommand{\photonNW}[3]
{\qbezier(0,0)(-0.22,-0.02)(-0.2,0.2) \qbezier(-0.2,0.2)(-0.18,0.42)(-0.4,0.4) \qbezier(-0.4,0.4)(-0.62,0.38)(-0.6,0.6) \qbezier(-0.6,0.6)(-0.58,0.82)(-0.8,0.8) \qbezier(-0.8,0.8)(-1.02,0.78)(-1,1) \qbezier(-1,1) (-0.98,1.22)(-1.2,1.2) \qbezier(-1.2,1.2)(-1.42,1.18)(-1.4,1.4) \qbezier(-1.4,1.4)(-1.38,1.62)(-1.6,1.6) \qbezier(-1.6,1.6)(-1.82,1.58)(-1.8,1.8) \qbezier(-1.8,1.8)(-1.78,2.02)(-2,2) \put(-1,1){\makebox(0,-0.2){\VectorUp}} \put(0,0){\circle*{0.1}} \put(-2,2){\circle*{0.1}} \put(-1,1){\makebox(0.4,0.7){$#1$}} \put(-2.35,2){\makebox(0,0){$#2$}} \put(0.35,0){\makebox(0,0){$#3$}}}
\newcommand{\photonSEst}[3]
{\qbezier(0,0)(-0.22,-0.02)(-0.2,0.2) \qbezier(-0.2,0.2)(-0.18,0.42)(-0.4,0.4) \qbezier(-0.4,0.4)(-0.62,0.38)(-0.6,0.6) \qbezier(-0.6,0.6)(-0.58,0.82)(-0.8,0.8) \qbezier(-0.8,0.8)(-1.02,0.78)(-1,1) \qbezier(-1,1) (-0.98,1.22)(-1.2,1.2) \qbezier(-1.2,1.2)(-1.42,1.18)(-1.4,1.4) \qbezier(-1.4,1.4)(-1.38,1.62)(-1.6,1.6) \qbezier(-1.6,1.6)(-1.82,1.58)(-1.8,1.8) \qbezier(-1.8,1.8)(-1.78,2.02)(-2,2) \put(-1,1){\makebox(0,0){\VectorDn}} \put(0,0){\circle*{0.1}} \put(-2,2){\circle*{0.1}} \put(-1,1){\makebox(0.4,0.7){$#1$}} \put(-2.35,2){\makebox(0,0){$#2$}} \put(0.35,0){\makebox(0,0){$#3$}}}
\newcommand{\photonWNW}[3]
{\qbezier(0,0)(-0.17,-0.04)(-0.2,0.1) \qbezier(-0.2,0.1)(-0.23,0.32)(-0.4,0.2) \qbezier(-0.4,0.2)(-0.57,0.16)(-0.6,0.3) \qbezier(-0.6,0.3)(-0.63,0.52)(-0.8,0.4) \qbezier(-0.8,0.4)(-0.97,0.36)(-1,0.5) \qbezier(-1,0.5)(-1.03,0.72)(-1.2,0.6) \qbezier(-1.2,0.6)(-1.37,0.56)(-1.4,0.7) \qbezier(-1.4,0.7)(-1.43,0.92)(-1.6,0.8) \qbezier(-1.6,0.8)(-1.77,0.76)(-1.8,0.9) \qbezier(-1.8,0.9)(-1.83,1.12)(-2,1) \put(-1,1){\makebox(-0.2,-0.4){$\;$\VectorR}} \put(0,0){\circle*{0.1}} \put(-2,1){\circle*{0.1}} \put(-1,0.5){\makebox(0.6,0.4){$#1$}} \put(0.35,-1){\makebox(0,2){$#3$}} \put(-2.35,0){\makebox(0,2){$#2$}}}
\newcommand{\Crossphotons}[6]
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\section{Introduction} \subsection{General} What is known as the Bethe-Salpeter (BS) equation represents the complete solution of the relativistic two-body problem with important applications in various branches of physics. The equation was first derived by Bethe and Salpeter in 1951~\cite{SB51}, using the relativistic $S$-matrix formalism and the analogy with Feynman graphs, and at about the same time by Gell-Mann and Low~\cite{GML51}, using a rigorous field-theoretical approach based on Green's functions. A closely related equation was discussed by Schwinger in his Harvard lectures already in the late 1940's~\cite{Schw51,KarpK52,Schw64,Nam97}.
In interpreting the solutions of the BS equation, several serious problems were encountered, as discussed early by Dyson~\cite{Dyson53}, Wick~\cite{Wick53} and Goldstein~\cite{Gold53}. Dyson was particularly concerned about the meaning of the wave function in relativistic quantum mechanics, a subject \textit{"full of obscurities and unsolved problems"}. Solving the BS equation leads to a 4-dimensional wave function---with individual times for the two particles. This function is manifestly relativistically covariant but not in accordance with the standard quantum-mechanical picture. That leads to "spurious" or "abnormal" solutions without physical significance and with no nonrelativistic counterpart~\cite{Sazd87}. Another fundamental problem is that the BS equation does not reduce to the correct "one-body limit", when one of the particles becomes infinitely heavy, as discussed by Gross and others~\cite{Gross69,Gross82}. Problems of these kinds are most pronounced in the scattering of strongly interacting particles but less so for bound-state systems in weak-coupling~\cite{Tod71,CasLep78,Conn91,PhilWal96,Bijt01} (see ref. \cite{Nam97} for a review).
The earliest applications of the BS equation appeared in atomic physics and concerned the proton recoil contribution to the hydrogen fine structure by Salpeter~\cite{Salp52} and the positronium energy level structure by Karplus and Klein~\cite{KarpK52}.
An important goal for the equation has been the study of \textit{strongly interacting particles}, which is a fundamental problem in elementary-particle physics. In recent years there have been numerous applications in QCD, dealing mainly with the quark-quark, quark-antiquark interactions, quark confinement and related problems~\cite{CasLep78,LSG91,BRS96,MR97}. Here, the problems mentioned above are more serious, as recently summarized by Namyslowski~\cite{Nam97}.
There have also been many applications in surface and solid-state physics, ranging from electron-hole interactions in ion crystals~\cite{ORR02} and studies of the two-dimensional Hubbard model~\cite{BSW89} and Cooper pairs~\cite{WGH02} to quantum dots~\cite{AFL04}.
The BS equation has also been applied to three or more particles~\cite{KL03,WR00,Taylor66}, although serious problems have been encountered for more than three particles~\cite{Bijt02}.
Various approximation schemes for treating the BS equation have been developed over the time. The simplest approximation is the \textit{"ladder approximation"}, where all intermediate states evolve only in the forward (positive) time direction. This is a useful starting point in the strong-coupling case, where the standard perturbative or self-consistent approach may not converge, and this approximation is, for instance, the basis for the Brueckner theory of nuclear matter~\cite[Sect. 41]{Brueck59,FW71}. Another approach is the \textit{"quasi-potential approximation"}, which implies that the equation is reduced to an equivalent 3-dimensional Schr\ödinger equation, which can be done without loosing any rigor~\cite{CasLep78,PTjon98}. Early numerical calculations in the this regime were done particularly by Schwartz and Zemach~\cite{SchwZ66} and Kaufmann~\cite{Kaufm69}.
In atomic physics the BS equation has been applied mainly in treating positronium~\cite{AFM02,AF99} and to heliumlike ions, and we shall be particular concerned with the latter here. This is strictly speaking a three-body problem but can to a good approximation be treated---with the first Born approximation---as a two-body problem with an external potential. The application to heliumlike systems was pioneered by Sucher~\cite{Su57a,Su58} and Araki~\cite{Ar57} in the late 1950's for deriving the leading relativistic and QED energy corrections beyond the Breit interaction. Later these works have been extended---largely along the lines of Sucher---by Douglas and Kroll in the 1970's~\cite{DK74} and more recently by Zhang and Drake~\cite{ZDr94,ZDr96,Zhang96,Zhang96a}.
The technique developed by Drake and coworkers is presently the most accurate available in dealing with heliumlike systems. The wave functions used are very accurate functions of Hylleraas type, and the QED corrections are evaluated by means of analytical expressions up to order $\alpha^5$ Ry (atomic units, or $m\alpha^7$ in relativistic units), derived from the BS equation. The wave functions used by Drake et al. are nonrelativistic but certain relativistic effects are treated to all order in the "\textit{unified model}"~\cite{Dr79,Dr88}. The analysis of the BS equation are in these works based upon the Brillouin-Wigner perturbation theory (BWPT).
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\begin{figure}
\caption{Schematic illustration of the connection between various many-body techniques.
The upper right part represents the Green's-function (GF) approach, which is used to
derive the Bethe-Salpeter (BS) equation, normally analyzed in terms of the Brillouin
Wigner perturbation theory (BWPT). The lower-left part illustrates the many-body
perturbation theory (MBPT), originating from Rayleigh-Schr\ödinger perturbation
theory (RSPT). The combination of quantum-electrodynamics (QED)
with MBPT is represented by the covariant-evolution-operator
(CovEvOp) method, and the link to the BS equation and the corresponding Bloch equation---the main
subject of the present paper---is illustrated by the arrows.}
\label{Fig:Intr}
\end{figure}
A different and in some aspects more versatile approach to the many-body problem is the procedure known as the \textit{many-body perturbation theory} (MBPT). This is based upon Rayleigh-Schr\ödinger perturbation theory (RSPT)~\cite{LM86}, which via the Bloch equation can be used to derive various computational schemes, such as the \textit{linked-diagram expansion} (LDE)~\cite{Br55,Go57,Br67}. A particularly powerful technique is the \textit{Coupled-Cluster Approach} (CCA)~\cite{CK60,Ci66,PC75}, which is widely used in quantum chemistry~\cite{BP78,KB92CC}. This technique is non-perturbative but closely connected to MBPT, and we shall include it in the MBPT category here. The MBPT techniques are primarily developed for the weak-coupling case, but might in the non-perturbative (CCA) form be used also in strong coupling.
The MBPT procedures, based initially upon RSPT, have the great advantage compared to techniques based upon BWPT that they are \textit{size-extensive} at each order, which implies that the energy scales linearly with the size of the system---a property of vital importance for molecular problems~\cite{PKSB78,BSP79}. Both procedures can also be combined with the \textit{extended-model-space technique}, which is particularly effective in dealing with problems of \textit{quasi-degeneracy}~\cite{Li74,Li78,LM86,MLL95}.
For QED problems the \textit{$S$-matrix technique} has been the standard procedure since the days of Feynman and Dyson. (For a review of the application to bound-state problems, see ref.~\cite{MPS98}.) Being based upon scattering theory, this technique has the disadvantage that its structure is quite different from that of MBPT, which makes it hard to combine the procedures (see, e.g. ref.~\cite{SB89}). The standard procedure for such a combination has been to perform a separate (relativistic) many-body calculation and adding first-order energy corrections from QED analytically~\cite{PJS94}. This procedure gives in many cases satisfactory results but is hard to improve in any systematic way. In particular, it gives no additional information about the wave function.
Another disadvantage with the $S$-matrix formalism is that the energy is conserved between the initial and the final states. This implies that it cannot be combined with the extended-model-space technique, successfully applied in MBPT. This techniques requires generally elements of the effective interaction that are nondiagonal in energy. This problem has recently been remedied by means of a new technique, known as the \textit{Covariant-Evolution-Operator method} (CovEvOp), which is a modification of the standard evolution-operator technique of time-dependent perturbation theory~\cite{FW71} in order to make it applicable to relativistic problems (for a review, see ref.~\cite{LSA04}). This technique has a structure that is very akin to that of MBPT, and it deals with the key ingredients of MBPT---the wave operator and the effective interaction. At the same time the method is closely related to the $S$-matrix formalism and the Green's-function procedure. The technique can therefore be regarded as a merger of MBPT/CCA and QED~\cite{Li00}, and it has recently been successfully applied to the quasi-degenerate fine-structure states of heliumlike systems~\cite{LAS01}.
The quasi-degenerate problem can also be handled with the \textit{two-times Green's-function} approach, developed by Shabaev and coworkers (for a review, see ref.~\cite{Shab02}). This technique, however, has no direct link to MBPT and will therefore not be discussed further here.
The procedure with the Covariant-Evolution-Operator method is now being further developed at our laboratory in order to combine QED and MBPT in a more complete fashion. This will be based on the non-perturbative coupled-cluster approach (CCA) of electron correlation or the so-called Dirac-Coulomb approximation, corresponding to the "ladder approximation" of the Bethe-Salpeter equation. This is combined with a perturbative expansion of the remaining (mainly QED) effects, which in principle leads to the full BS equation. This is along the lines early drawn by Sucher~\cite{Su58} and followed by many later works~\cite{DK74,Conn91,Zhang96,AF99,SauliA03}. Our approach differ from all the earlier ones in the sense that all effects are evaluated \textit{numerically} rather than analytically.
Our approach implies that the QED effects are evaluated with highly correlated (relativistic) wave functions, and for two-electron systems the results will then, in principle, be comparable to those of Drake's unified method, with the difference that the relativistic effects are included in a complete way and that the QED effects are evaluated numerically.
In the diagram in Fig. \ref{Fig:Intr} we have tried to represent the relations between the many-body approaches described here in a simple and illustrative way. The many-body procedures based upon Rayleigh-Schr\ödinger perturbation theory are indicated in the lower-left part and the Green's-function and Bethe-Salpeter procedures, more associated to Brillouin-Wigner perturbation theory, in the upper-right part. The present paper deals particularly with the connection between the two approaches, represented by the arrows in the diagram.
In addition to deeper insight into the different procedures, the present treatment will make it possible to analyze a problem based on the BS equation in terms of RS-MBPT---not only in terms of BWPT, as has previously been the case~\cite{DK74,Zhang96a}. The Bloch equation in commutator form, compatible with the BS equation, which is derived, has the same relation to the BS equation as has the standard Bloch equation to the ordinary Schr\ödinger equation, and it could possibly be used to eliminate the quasi-degeneracy problem that might appear when the BS equation is treated for a single state at a time.
Since the equivalence of the MBPT-QED-CovEvOp procedure with the BS equation has now been established for two-electron systems, this new link will probably make it easier to apply the BS procedure---or its equivalence--- also to systems with more electrons. Alternatively, this can be used to analyze a many-body-QED calculation to find out what is missing in order to represent a complete Bethe-Salpeter treatment. Our main emphasize here is applications to atoms and other weak-interacting systems. Since the procedure we have developed, however, is based upon a combination of perturbative and non-perturbative approaches, the results obtained might be useful also outside this regime.
The paper will be organized in the following way. Below we shall first conjecture the Bethe-Salpeter equation in a simple-minded way as an introduction. In section \ref{sec:ConvPT} we shall summarize the necessary ingredients of time-independent and time-dependent perturbation theory and in the following section briefly review the original derivations of the Bethe-Salpeter equation by Bethe and Salpeter and by Gell-Mann and Low, based on Green's functions. The main part of the paper will be devoted to a rigorous derivation of the Bethe-Salpeter equation, starting from the covariant-evolution-operator method. The basics of the method are summarized in section \ref{sec:EvolOp}, and the method will then be used to derive the Bethe-Salpeter equation. A corresponding Bloch equation will also be derived, which will make it possible to treat the BS equation perturbatively also for a quasi-degenerate (extended) model space. Technical details of the treatment are given in a number of appendices. Radiative effects (self energies and vacuum polarization) are not considered here but can be included by modifying the electron propagator and photon interactions, as discussed, for instance, by Douglas and Kroll~\cite{DK74}.
\subsection{Bethe-Salpeter equation} An equation of BS type can be conjectured in a very simple way by considering the time-independent nonrelativistic Schr\ödinger equation \begin{equation}
\label{SchrEq}
H\Psi=E\Psi \end{equation} with $H=H_0+V_1$, where $H_0=h_1+h_2$ is the zeroth-order Hamiltonian (sum of single-electron Hamiltonians) and $V_1=e^2/r_{12}$ is the electron-electron interaction (in relativistic units \footnote{In this article relativistic units are used, i.e., $m=c=\hbar=\epsilon_0=1, e^2=4\pi\alpha$, where $\alpha$ is the fine-structure constant.}). The Schr\ödinger equation can then be expressed \begin{equation}
\label{Schr}
(E-H_0)\Psi=V_1\Psi \end{equation} with the solution \begin{equation}
\label{SE1}
\Psi=\Gamma(E)V_1\Psi, \end{equation} where \begin{equation}
\label{Gamma1}
\Gamma(E)=\frac{1}{E-H_0}=\frac{\ket{rs}\bra{rs}}{E-\ensuremath{\varepsilon}_r-\ensuremath{\varepsilon}_s} \end{equation} is the \textit{"resolvent"} operator~\cite[Ch. 9]{LM86} and $\ket{rs}$ is the Dirac notation of the straight (not antisymmetrized) product of two single-electron functions, satisfying the Dirac equation \begin{equation}
\label{SEeq}
h\ket{i}=\ensuremath{\varepsilon}_i\ket{i} \end{equation} We apply the summation convention, implying summation over repeated indices appearing on one side of the equation. Unless specifies otherwise, the summation is performed over positive- (particle) as well as negative-energy (hole) states.
In the relativistic formalism one should, following Sucher~\cite{Su57a,Su58}, replace $V_1$ by $\Lambda_{++}e^2/r_{12}\Lambda_{++}$, where $\Lambda_{++}$ is the projection operator for particle (positive-energy) states. This leads to the \textit{Coulomb-ladder approximation}, mentioned above, i.e., a series of Coulomb interactions separated by particle states. In QED $V_1$ can in the first approximation be replaced by the \textit{energy-dependent} interaction with a fully \textit{covariant photon} $V_1(E)$, i.e., Coulomb and transverse photon, the latter representing (retarded) Breit interaction. In the next step $V_1(E)$ can be replaced by $V_1(E)+V_2(E)$, where $V_2(E)$ represents the \textit{non-separable} (irreducible) interaction of two photons, i.e., the interaction of two covariant photons that in the QED description cannot be represented by repeated single-photon interactions (see Fig. \ref{Fig:NonSep} below). Continuing this process, summing all non-separable interactions with one, two, ... photons \begin{equation}
\label{calV}
\mathcal{V}(E)=V_1(E)+V_2(E)+\cdots \end{equation} leads to \begin{equation}
\label{BS1}
\Psi=\Gamma(E)\,\mathcal{V}(E)\Psi \end{equation} or \begin{equation}
\label{BS}
(E-H_0)\,\Psi=\mathcal{V}(E)\Psi \end{equation} This is equivalent to the Schr\ödinger-like form of the \textit{Bethe-Salpeter equation} derived by Sucher~\cite[Eq. 1.47]{Su58} and also used by Douglas and Kroll~\cite[3.26]{DK74} and by Zhang~\cite[Eq. 15]{Zhang96}.
The BS equation \eqref{BS} can be expanded in terms of a Brillouin-Wigner perturbation series~\cite[Ch. 9]{LM86} \begin{equation}
\label{BSBW}
\Psi=\Psi_0+\Big(\Gamma_Q(E)\,\mathcal{V}(E)+\Gamma_Q(E)\,\mathcal{V}(E)\Gamma_Q(E)\,\mathcal{V}(E)
+\cdots\Big) \Psi_0 \end{equation} where $\Psi_0$ is the unperturbed wave function and \begin{equation}
\label{RedRes}
\Gamma_Q(E)=\frac{Q}{E-H_0} \end{equation} is the "reduced" resolvent \eqref{Gamma1} with the unperturbed state removed. For this sequence to converge properly, it is required that there be no eigenstate of $H_0$ close in energy to that of $\Psi_0$ and of the same symmetry. A rigorous derivation of the equation will be given in the following sections.
\section{Conventional many-body perturbation theory} \label{sec:ConvPT} \subsection{Time-independent perturbation theory} \label{sec:MBPT} In time-independent many-body perturbation theory (MBPT) (see, e.g., ref.~\cite{LM86}) the aim is to solve the Schr\ödinger equation by successive approximations for a number of \textit{"target"} states \begin{equation}
\label{SE}
H\,\Psi^\alpha(\boldsymbol{x})=E^\alpha\,\Psi^\alpha(\boldsymbol{x})\,;\qquad
(\alpha=1,2,\cdots d) \end{equation} ($\boldsymbol{x}$ stands here for all space coordinates). The time-independent Hamiltonian is partitioned into a zeroth-order Hamiltonian and a perturbation \begin{equation}
\label{Part}
H=H_0+H' \end{equation} For each target state $\Psi^\alpha(\boldsymbol{x})$ there exists a \textit{model state} or \textit{zeroth-order wave function} (ZOWF) $\Psi_0^\alpha(\boldsymbol{x})$ that is confined to a subspace, the \textit{model space} ($P$), spanned by eigenfunctions of $H_0$. The model space can be degenerate or non-degenerate (quasi-degenerate). In the latter case the model states are not necessarily eigenstates of $H_0$. It is always assumed that all degenerate states of $H_0$ are either entirely inside or entirely outside the model space.
A \textit{wave operator} $\Omega$ can be defined so that it transfers all model states to the corresponding target states \begin{equation}
\label{Wop}
\boxed{\Psi^\alpha(\boldsymbol{x})=\Omega\,\Psi_0^\alpha(\boldsymbol{x})\,; \qquad
(\alpha=1,2,\cdots d)} \end{equation} In the following we shall use the \textit{intermediate normalization} (IN), implying that \begin{equation}
\label{IN}
\big\langle\Psi_0^\alpha(\boldsymbol{x})\bigket{\Psi^\alpha(\boldsymbol{x})}=1 \end{equation} The model states are the projections of the target states on the model space \begin{equation}
\label{ZOWF}
\Psi_0^\alpha(\boldsymbol{x})=P\Psi^\alpha(\boldsymbol{x}) \end{equation} which implies \begin{equation}
\label{POP}
P\Omega P=P \end{equation}
The exact energies as well as the model states are obtained by solving the secular equation \begin{equation}
\label{SecEq}
H_{\mathrm{eff}}\,\Psi_0^\alpha(\boldsymbol{x})=E^\alpha\,\Psi_0^\alpha(\boldsymbol{x}), \end{equation} within the model space. Here, $H_{\mathrm{eff}}$ is the \textit{effective Hamiltonian}, in IN given by \begin{equation}
\label{HeffIN}
H_{\mathrm{eff}}=PH\Omega P \end{equation}
The wave operator satisfies the \textit{generalized Bloch equation}~\cite{Li74,LM86} \begin{subequations} \begin{equation}
\label{Bloch}
\boxed{\big[\Omega,H_0\big]P=\big(H'\Omega -\Omega\,
H'_{\mathrm{eff}}\big)P} \end{equation} where $H'_{\mathrm{eff}}$ is the \textit{effective interaction} (in IN) \begin{equation}
\label{EffInt2}
H'_{\mathrm{eff}}=H_{\mathrm{eff}}-PH_0P=PH'\Omega P \end{equation} For a degenerate model space with the energy $E_0$ the equation goes over into the original Bloch equation~\cite{Bl58a,Bl58b} \begin{equation}
\label{BlochDeg}
(E_0-H_0)\,\Omega P=\big(H'\Omega -\Omega\,H'_{\mathrm{eff}}\big)P \end{equation} \end{subequations}
The Bloch equation contains generally the information of a \textit{system} of Schr\ödinger equations \eqref{SE}, corresponding to a number of target states. The equation can conveniently be used as the starting point for generating various perturbative and non-perturbative schemes~\cite{Li74,LM86}. It leads directly to a generalized form of the Rayleigh-Schr\ödinger perturbation expansion, and it can be used to generate the \textit{linked-diagram expansion} (LDE) as well as the non-perturbative \textit{coupled-cluster approach} (CCA). The commutator form of the Bloch equation \eqref{Bloch} makes it possible to work with a non-degenerate or "extended" model space", which is of particular importance for quasi-degenerate problems, as mentioned above.
\subsection{Time-dependent perturbation theory} \label{sec:TDPT} In time-dependent perturbation theory we start from the time-dependent Schr\ödinger equation \begin{equation}
\label{TDSE}
\ensuremath{\mathrm{i}}\Partder{t}\,\chi(t,\boldsymbol{x})=H(t)\,\chi(t,\boldsymbol{x}) \end{equation} As before, $\boldsymbol{x}$ stands for \textit{all} space coordinates, while $t$ is a single time variable. Even if the Hamiltonian may be formally time-dependent, we are interested in states that are \textit{stationary}, which implies that the wave function has the form \begin{equation}
\label{wft}
\chi(t,\boldsymbol{x})=\Psi(\boldsymbol{x})\,e^{-\ensuremath{\mathrm{i}} Et} \end{equation} where $E$ is the energy of the system and $\Psi(\boldsymbol{x})$ is the time-independent wave function. The latter is then a solution the time-independent Schr\ödinger equation \eqref{Schr} \begin{equation}
\label{Seq}
H\,\Psi(\boldsymbol{x})=E\Psi(\boldsymbol{x}) \end{equation}
In the \textit{interaction picture} (IP)~\cite{FW71} with the partitioning \eqref{Part} the wave function is related to that of the Schr\ödinger picture by \begin{equation}
\label{IP}
\chi_\mathrm{I}(t,\boldsymbol{x})=e^{\ensuremath{\mathrm{i}} H_0t}\,\chi(t,\boldsymbol{x}) \end{equation} and the time-dependent Schr\ödinger equation becomes \begin{equation}
\label{SEIP}
\ensuremath{\mathrm{i}}\Partder{t}\,\chi_\mathrm{I}(t,\boldsymbol{x})=H'_\mathrm{I}(t)\,\chi_\mathrm{I}(t,\boldsymbol{x}) \end{equation} The \textit{time-evolution operator}, defined by \begin{equation}
\label{EvolOp}
\chi_\mathrm{I}(t,\boldsymbol{x})=U_\mathrm{I}(t,t_0)\,\chi_\mathrm{I}(t_0,\boldsymbol{x}) \end{equation} then satisfies the equation \begin{equation}
\ensuremath{\mathrm{i}}\Partder{t}\,U_\mathrm{I}(t,t_0)=H'_\mathrm{I}(t)\,U_\mathrm{I}(t,t_0) \end{equation} with the solution~\cite[Eq. 6.23]{FW71} \begin{equation}
\label{Uexp}
U_\mathrm{I}(t,t_0)=1+\sum_{n=1}^\infty\frac{(-\ensuremath{\mathrm{i}})^n}{n!}\int_{t_0}^t\ensuremath{\mathrm{d}}^4x_n\cdots
\int_{t_0}^t\ensuremath{\mathrm{d}}^4x_1\;T_\mathrm{D}\big[\mathcal{H}'_\mathrm{I}(x_n)\mathcal{H}'_\mathrm{I}(x_{n-1})
\cdots\mathcal{H}'_\mathrm{I}(x_1)\big] \end{equation} Here, $x=(t,\boldsymbol{x})$, $T_\mathrm{D}$ is the Dyson time-ordering operator, and $\mathcal{H}'_\mathrm{I}(x)$ is the perturbation density defined by \begin{equation}
\label{calH}
H'_\mathrm{I}(t)=\int\dif^3\bx\,\mathcal{H}'_\mathrm{I}(t,\boldsymbol{x}) \end{equation}
In applying this formalism to perturbation theory, an \textit{adiabatic damping} is added~\cite{FW71} \begin{equation}
\label{Damp}
H'_\mathrm{I}(t)\rightarrow H'_{\mathrm{I}\gamma}=H'_\mathrm{I}\,e^{-\gamma\abs{t}}\,;\qquad
U_\mathrm{I}(t,t_0)\rightarrow U_{\mathrm{I}\gamma}(t,t_0) \end{equation} where $\gamma$ is a small, positive number. This implies that as $t\rightarrow-\infty$ the eigenfunctions of $H$ tend to eigenfunctions of $H_0$.
In QED the perturbation density due to the interaction between the electrons and the photon field is given by~\cite{Sch61} \begin{equation}
\label{Pert}
\mathcal{H}'_\mathrm{I}(x)=-e\hat\psi_\mathrm{I}^{\dag}(x)\alpha^\mu
A_\mu(x)\hat\psi_\mathrm{I}(x) \end{equation} where $e$ is the absolute value of the electronic charge, $\hat\psi_\mathrm{I}^{\dag}(x),\,\hat\psi_\mathrm{I}(x)$ are the electron-field operators in the interaction picture, $A_\mu$ the photon-field operator and $\alpha^\mu$ are related to the standard Dirac matrices by $\alpha^\mu=(1,\boldsymbol{\alpha})$.
\section{Green's function approach} \label{sec:Green} In this section we shall essentially reproduce the derivation of the BS equation by Gell-Mann and Low, starting from Green's functions. We consider a two-particle system for which the Green's function is defined~\cite[pp. 64 and 116]{FW71} \begin{equation}
\label{GFH}
G(x'_1,x'_2;x_{10},x_{20})=
-\frac{\bigbra{0_\mathrm{H}}T_\mathrm{D}[\hat{\psi}_\mH(x'_1)\hat{\psi}_\mH(x'_2)\hat{\psi}_\mH^{\dag}(x_{20})\hat{\psi}_\mH^{\dag}(x_{10})]\bigket{0_\mathrm{H}}}
{\langle0_\mathrm{H}\ket{0_\mathrm{H}}} \end{equation} Here, $\ket{0_\mathrm{H}}$ represents the vacuum state and $\hat{\psi}_\mH^{\dag}(x),\,\hat{\psi}_\mH(x)$ the electron-field operators, all in the \textit{Heisenberg representation}. The latter are related to those in the interaction picture by \begin{equation}
\label{HP}
\hat{\psi}_\mH(t,\boldsymbol{x})=U(0,t)\,\hat{\psi}_\mI(t,\boldsymbol{x})\,U(t,0) \end{equation} where $U$ is the evolution operator \eqref{Uexp}. Transforming the Green's function to the interaction picture then yields~\cite[Eq. 8.9]{FW71}, \cite[Eq. 16]{GML51}, \cite[Eq. 259]{LSA04} \begin{equation}
\label{GFI}
G(x'_1,x'_2;x_{10},x_{20})=-\frac{\bigbra{0_\mathrm{I}}T_\mathrm{D}[\hat{\psi}_\mI(x'_1)\hat{\psi}_\mI(x'_2)
U_\mathrm{I}(\infty,-\infty)\hat{\psi}_\mI^{\dag}(x_{20})\hat{\psi}_\mI^{\dag}(x_{10})]\bigket{0_\mathrm{I}}}
{{\bra{0_\mathrm{I}}U_\mathrm{I}(\infty,-\infty)\ket{0_\mathrm{I}}}} \end{equation} Obviously, only fully contracted terms contribute to the vacuum expectation value. By applying \textit{Wick's theorem}~\cite[p. 83]{FW71}~\cite[Sect. 11.5]{LM86}, this can be represented in terms of \textit{Feynman diagrams}. The denominator has the effect of eliminating the singularities of the numerator, in the Feynman picture represented by unlinked or disconnected diagrams, leading to~\cite[Eq. 9.5]{FW71} \begin{equation}
\label{GFC}
G(x'_1,x'_2;x_{10},x_{20})=-\bigbra{0_\mathrm{I}}T_\mathrm{D}[\hat{\psi}_\mI(x'_1)\hat{\psi}_\mI(x'_2)
U_\mathrm{I}(\infty,-\infty)\hat{\psi}_\mI^{\dag}(x_{20})\hat{\psi}_\mI^{\dag}(x_{10})\bigket{0_\mathrm{I}}_\mathrm{conn} \end{equation}
In contrast to the evolution operator \eqref{EvolOp}, the Green's function is relativistically \textit{covariant} in the sense that the integrations are performed over all space and time and the electron-field operators can represent particle (positive-energy) as well as hole (negative-energy) states. This also implies that, in the energy representation (fourier transform), the energy is conserved at all diagram vertices.
The Green's function can be expressed \begin{eqnarray}
\label{GFKx}
&&G(x'_1,x'_2;x_{10},x_{20})=G_0(x'_1,x'_2;x_{10},x_{20})+\nonumber \\
&&\dint\!\!\!\dint\ensuremath{\mathrm{d}}^4 x_1\,\ensuremath{\mathrm{d}}^4 x_2\,\ensuremath{\mathrm{d}}^4 x_3\,\ensuremath{\mathrm{d}}^4\,x_4\;
G_0(x'_1,x'_2;x_1,x_2)\,{\cal K}(x_1,x_2;x_3,x_4)\,
G_0(x_3,x_4;x_{10},x_{20}) \end{eqnarray} where ${\cal K}$ represents the interaction kernel of all connected diagrams and $G_0$ is the zeroth-order Green's function \begin{eqnarray}
\label{GF0}
\hspace{1cm} G_0(x'_1,x'_2;x_{10},x_{20})&=&-\bigbra{0_\mathrm{I}}T_\mathrm{D}[\hat{\psi}_\mI(x'_1)\hat{\psi}_\mI(x'_2)
\hat{\psi}_\mI^{\dag}(x_{20})\hat{\psi}_\mI^{\dag}(x_{10})]\bigket{0_\mathrm{I}}\nonumber \\
&=&S_{\mathrm{F}}(x'_1,x_{10})S_{\mathrm{F}}(x'_2,x_{20}) \end{eqnarray} with $S_{\mathrm{F}}$ being the Feynman \textit{electron propagator} or zeroth-order single-electron Green's function, defined by \begin{equation}
\label{SF}
\ensuremath{\mathrm{i}}S_{\mathrm{F}}(x',x_{0})=\bigbra{0_\mathrm{I}}T_\mathrm{D}[\hat{\psi}_\mI(x')\hat{\psi}_\mI^{\dag}(x_{0})]\bigket{0_\mathrm{I}} \end{equation} assuming the vacuum state be normalized. This is illustrated in Fig. \ref{Fig:GF}. In operator form the Green's function can be expressed \begin{equation}
\label{GFK}
G=G_0+G_0{\cal K} G_0 \end{equation}
In some cases the kernel of the Green's function can be separated into two kernels \begin{equation}
\label{Ksep}
{\cal K}={\cal K}_2G_0{\cal K}_1 \end{equation} with no photon-field contractions between them. The kernel is then said to be \textit{separable}. If a kernel cannot be separated further in this way, it is said to be \textit{non-separable} \footnote{What we here refer to as "separable" and non-separable" are often referred to as "reducible" and "irreducible". Since the latter terms have recently been used also with a different interpretation, we avoid them here.}. The complete kernel can then be expressed \begin{equation}
\label{K}
{\cal K}=\kappa+\kappa G_0\kappa+\kappa G_0\kappa
G_0\kappa+\cdots \end{equation} where $\kappa$ represents all \textit{non-separable} kernels. This leads to the \textit{Dyson equation} for the Green's function \begin{equation}
\label{GF}
G=G_0+G_0\kappa G \end{equation} illustrated in Fig. \ref{Fig:Dyson}.
\begin{figure}
\caption{Graphical representation of the two-particle Green's function
\eqref{GFK}. $\cal K$ represents \textit{all} interactions between the electrons.}
\label{Fig:GF}
\end{figure}
\begin{figure}
\caption{Graphical representation of the Dyson equation
\eqref{GF} for the two-particle Green's function. $\kappa$ represents
the \textit{non-separable} interactions between the electons.}
\label{Fig:Dyson}
\end{figure}
Bethe and Salpeter as well as Gell-Mann and Low argue that a related equation can be set up for the two-electron bound-state wave function. In that case the first (inhomogeneous) term on the rhs does not contribute, since that is in their formulation composed of \textit{free-electron} propagators, and the bound-state wave function does not have any such components. This leads to the \textit{homogeneous} equation \begin{equation}
\label{BS2}
\Psi(x'_1,x'_2)=\dint\!\!\!\dint\ensuremath{\mathrm{d}}^4 x_1\,\ensuremath{\mathrm{d}}^4 x_2\,\ensuremath{\mathrm{d}}^4
x_3\,\ensuremath{\mathrm{d}}^4x_4 \;G_0(x'_1,x'_2;x_1,x_2)\,\kappa(x_1,x_2;x_{3},x_{4})\,
\Psi(x_{3},x_{4})\hspace{1cm} \end{equation} or in short-hand notations \begin{equation}
\label{BS2a}
\Psi=G_0\,\kappa\,\Psi \end{equation} This is the original form of the Bethe-Salpeter equation~\cite[Eq. 11a]{SB51},~\cite[Eq. 37]{GML51}. It should be noted that this wave function contains \textit{individual times for the two particles}. This reflects one of the problems referred to in the Introduction. The relative time between the particles does not correspond to any physical quantity and leads to spurious solutions. There are several ways of eliminating the extra time dependence in a covariant way. Sucher~\cite{Su58}, following Salpeter~\cite{Salp52}, integrates the fourier transform over the relative energy, which leads to a Schr\ödinger-like form with a single time/energy dependence of the type \eqref{BS} given above. This reduction can be done without loosing any physical content of the original equation~\cite{Conn91,PhilWal96,PTjon98,Bijt01}. In the following sections we shall derive an equivalent equation in a different way.
Our notations here differ from those used by Bethe-Salpeter and Gell-Mann--Low. The Green's function \eqref{GFH} is in their works denoted by $K(12,34)$ and referred to as the "\textit{amplitude function for the propagation of the particles}" by Bethe-Salpeter~\cite{SB51} and as the "\textit{two-body kernel}" by Gell-Mann--Low~\cite[Eq. 11]{GML51}. Our "non-separable kernel" $\kappa$ is by BS denoted by $\bar{G}$ and referred to as "\textit{irreducible graphs}" and by GML denote by $G$ and referred to as the "\textit{interaction function}".
\section{Covariant evolution operator approach} \label{sec:EvolOp}
\subsection{Definitions} In the following sections we shall derive the Bethe-Salpeter equation, starting from the covariant form of the evolution operator~\cite{LSA04}. This will demonstrate the relation between the BS equation and standard many-body perturbation theory (MBPT) in a clear way. In the present section we shall first review the basics of the evolution-operator method and in the next section use that method for deriving the BS equation. This will directly lead to the Schr\ödinger-like form \eqref{BS}.
According to the Gell-Mann--Low theorem~\cite[p. 61]{GML51,FW71} the time-independent wave function \eqref{wft} can in the case of a single target function be expressed in intermediate normalization (IN) \eqref{IN} as \begin{equation}
\label{GML0}
\boldsymbol {\Psi}(\boldsymbol{x})=\chi(0,\boldsymbol{x})=
\lim_{\gamlim}
\frac{\Ugam{0}\Psi_0(\boldsymbol{x})}{\bra{\Psi_0}\Ugam{0}\ket{\Psi_0}} \end{equation} where $U_\gamma$ is the evolution operator \eqref{Damp} and $\Psi_0(\boldsymbol{x})$ is the time-independent zeroth-order wave function \eqref{ZOWF}. (From now on we work in the interaction picture and leave out the subscript "I".) $\boldsymbol {\Psi}(x)$ is an eigenfunction of the Hamiltonian $H_0+H'$ \begin{equation}
\label{GML1}
(H_0+H')\,\boldsymbol {\Psi}(\boldsymbol{x})=E\,\boldsymbol {\Psi}(\boldsymbol{x}) \end{equation} where $H'$ is in our case the electron-field interaction \eqref{Pert}. Since this perturbation represents an uncontracted photon, the wave function $\boldsymbol {\Psi}(\boldsymbol{x})$ will generally lie in an \textit{extended Fock space}, where the number of photons is not conserved.
The GML formula can be generalized to a general multi-dimensional model space~\cite[Eq. 110]{LSA04} \begin{equation}
\label{GML}
\boldsymbol {\Psi}^\alpha(\boldsymbol{x})=\lim_{\gamlim} \frac{N^\alpha\Ugam{0}\Phi^\alpha(\boldsymbol{x})}
{\bra{\Phi^\alpha}\Ugam{0}\ket{\Phi^\alpha}}\,;\qquad
(\alpha=1,2,\cdots d) \end{equation} where the function $\Phi^\alpha$ is defined \begin{equation}
\label{Phi}
\Phi^\alpha(\boldsymbol{x})=\lim_{\gamlim}\lim_{t\rightarrow-\infty}\chi^\alpha(t,\boldsymbol{x}) \end{equation} This function is generally distinct from the zeroth-order wave function \eqref{ZOWF} in intermediate normalization. Since the function \eqref{Phi} generally does not satisfy IN, a normalization constant $N^\alpha$ is inserted.
\begin{figure}
\caption{Graphical representation of the non-covariant evolution
operator \eqref{NoncovEv}. The time evolution occurs only in the positive direction.}
\label{Fig:Noncov}
\end{figure}
\begin{figure}
\caption{Graphical representation of the covariant evolution
\eqref{CovEv} (left). Here, time evolution can occur in the
positive as well as the negative direction. The right part of the figure
depicts the relation to the two-times Green's function \eqref{CovGF}.}
\label{Fig:CovGF}
\end{figure}
For a two-electron system the \textit{non-covariant} evolution operator \eqref{EvolOp} can in analogy with the Green's function \eqref{GFK} be expressed \begin{equation}
\label{NoncovEv}
U_{\mathrm{Noncov}}(t',t_0)=1+\hat\psi_+^{\dag}(x'_1)\hat\psi_+^{\dag}(x'_2)\,
{\cal K}\,\hat\psi_+(x_{20})\hat\psi_+(x_{10}) \end{equation} where again ${\cal K}$ represents the kernel of all fully contracted (separable and non-separable) interactions and $\hat\psi_+^{\dag},\,\hat\psi_+$ the positive-energy part of the electron-field operators. This is illustrated in Fig. \ref{Fig:Noncov}. In contrast to the Green's function above, the evolution operator \eqref{Uexp} has a single initial time $t=t_0$ and a single final time $t=t'$. The time integration is performed from $t=t_0$ to $t=t'$ --- only in the positive direction --- which implies that the operator is \textit{not relativistically covariant}.
A fully covariant form of the evolution operator that is applicable to relativistic problems can be obtained by inserting electron propagators in the non-covariant expression, as indicated in Fig. \ref{Fig:CovGF} (left), corresponding to the expression~\cite[Sect. 5]{LAS01,LSA04} \begin{equation}
\label{CovEv}
U_\mathrm{Cov}(t',t_0)=1+
\dint\!\!\!\dint\ensuremath{\mathrm{d}}^3\boldsymbol{x}'_1\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}'_2\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}_{10}\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}_{20}\;
\hat\psi^{\dag}(x'_1)\hat\psi^{\dag}(x'_2)\,G_0{\cal
K}G_0\,\hat\psi(x_{20})\hat\psi(x_{10}) \end{equation} leaving out the integrations over the coordinates of ${\cal
K}$ (see Eq. \ref{GFKx}). It then follows from the relation \eqref{GFK} that the covariant evolution operator is related to the \textit{two-times} Green's function (where all initial and all final times are equal) by \begin{equation}
\label{CovGF}
U_\mathrm{Cov}(t',t_0)=\dint\!\!\!\dint\ensuremath{\mathrm{d}}^3\boldsymbol{x}'_1\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}'_2\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}_{10}\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}_{20}\;
\hat\psi^{\dag}(x'_1)\hat\psi^{\dag}(x'_2)\,G(x_1',x_2';x_{10},x_{20})
\,\hat\psi(x_{20})\hat\psi(x_{10}) \end{equation} as illustrated in Fig. \ref{Fig:CovGF} (right).
From the relations~\cite[Eq. 193 (note misprints)]{LSA04} \begin{eqnarray}
\label{Rel}
\hspace{1cm}\int\dif^3\bx_0\, \ensuremath{\mathrm{i}}S_{\mathrm{F}}(x,x_0)\,\hat\psi(x_0)=\Theta(t-t_0)\,\hat\psi_+(x)-
\Theta(t_0-t)\,\hat\psi_-(x)\nonumber \\
\hspace{1cm}\int\dif^3\bx\, \hat\psi^{\dag}(x)\,\ensuremath{\mathrm{i}}S_{\mathrm{F}}(x,x_0)=\Theta(t-t_0)\,\hat\psi_+^{\dag}(x_0)-
\Theta(t_0-t)\,\hat\psi_-^{\dag}(x_0) \end{eqnarray} it follows directly that the form \eqref{CovEv} is equivalent to the non-covariant form \eqref{NoncovEv}, when only particle states are involved. That the former in addition is relativistically covariant follows from the fact that \textit{the electron-field operators can represent particle as well as hole states and the internal time integrations are performed over all times}--- in the positive as well as the negative direction. From now on we shall work only with the covariant form of the evolution operator and leave out the subscript $"_\mathrm{Cov}$".
In using the evolution operator in perturbation theory, we assume that we operate to the far right on positive-energy states in the model space. Then, as shown in Appendix \ref{App:SingPhot}, we can eliminate the rightmost zeroth-order Green's function and set the initial time to $t_0=-\infty$. We shall also assume that the limit of the adiabatic damping $\gamma\rarr0$ is taken.
The Covariant evolution operator is closely related to the Green's function---the main difference being that the Green's function is a \textit{function}, while the evolution operator is an \textit{operator}. The poles of the Green's function (in the energy representation) correspond to the energies of the system, while it gives no direct information about the wave function. The covariant evolution operator, on the other hand, contains information about the energy as well as the wave function.
\subsection{Model-space contributions}
\label{sec:MSC}
\renewcommand{\Ugam}[1]{U(#1,-\infty)}
\renewcommand{\Ugamtil}[1]{\widetilde{U}(#1,-\infty)}
\renewcommand{\widetilde{U}}{\widetilde{U}}
\newcommand{\cdot\cdot}{\cdot\cdot}
\renewcommand{\Ugam}[1]{\widetilde{U}(#1)}
\renewcommand{\Ugamtil}[1]{\widetilde{U}(#1)}
\newcommand{^{(1)}}{^{(1)}}
\newcommand{^{(2)}}{^{(2)}}
\newcommand{^{(3)}}{^{(3)}}
\newcommand{^{(4)}}{^{(4)}}
\newcommand{\mathrm{MSC}}{\mathrm{MSC}}
\newcommand{\widetilde{H}}{\widetilde{H}}
\newcommand{\bar{U}}{\bar{U}}
\newcommand{\bar{H}}{\bar{H}}
\newcommand{\dot{U}}{\dot{U}}
\newcommand{\dot{\Ubar}}{\dot{\bar{U}}}
\newcommand{\dot{\Util}}{\dot{\widetilde{U}}}
\newcommand{\dot{C}}{\dot{C}}
\newcommand{\hspace{-2.5mm}/}{\hspace{-2.5mm}/}
\newcommand{{\calE}}{{\mathcal{E}}}
Even after eliminating unlinked or disconnected contributions in Eq. \eqref{GFC}, the evolution operator may contain (quasi)singularities, namely when the intermediate state of a separable kernel lies in the model space and is degenerate or nearly degenerate (quasi-degenerate) with the initial state. As mentioned, a kernel is said to be \textit{separable}, if it can be separated into two kernels with no photon contractions between them. Singularities appear only for separable interactions. In the covariant-evolution-operator approach these singularities are eliminated by introducing a \textit{reduced evolution operator} $\widetilde{U}(t,-\infty)$~\cite[Eq. 116]{LAS01,LSA04}, defined by \begin{equation}
\label{Ured0}
U(t,-\infty)P=P+\widetilde{U}(t,-\infty)P\bs{\cdot} PU(0,-\infty)P \end{equation} Here, the last term is a product of two operators that evolve \textit{independently} from an initial state in the model space ($t=-\infty$), which is indicated by the "dot". Note also that the last factor has the final time $t=0$ and hence is time independent. This situation should be distinguished from the case where two operators are "coupled" and operate "in succession" \begin{equation}
\label{Usuc}
U(t,t_0)=U(t,t")\,U(t",t_0) \end{equation} This distinction will be important for the following treatment.
Normally, we shall assume that the initial time in the evolution operator is $t_0=-\infty$, and in cases where there is no risk for ambiguity we shall leave that out from the operator, so that \[U(t)={U(t,-\infty)}\] The definition \eqref{Ured0} will then be written \begin{equation}
\label{Ured}
\boxed{U(t)P=P+\widetilde{U}(t)P\bs{\cdot} PU(0)P} \end{equation} We also introduce the notation $U'(t)=U(t)-1$, which yields in place of the definition \eqref{Ured} \begin{equation}
\label{Ured2}
\widetilde{U}(t)P=U'(t)P-\widetilde{U}(t)P\bs{\cdot} PU'(0)P \end{equation} Here, the last term is the \textit{counterterm} \begin{equation}
\label{C}
\boxed{C(t)P=-\widetilde{U}(t)P\bs{\cdot} PU'(0)P} \end{equation} which removes the (quasi)singularities. This can also be expressed \begin{equation}
\label{Cexp}
C(t)P=-\widetilde{U}(t)P\bs{\cdot} P\widetilde{U}(0)P
-\widetilde{U}(t)P\bs{\cdot} P\widetilde{U}(0)P\bs{\cdot} P\widetilde{U}(0)P-\cdots \end{equation}
After removing a singularity, there is normally a non-vanishing remainder, referred to as the \textit{model-space contribution} (MSC), defined as \begin{equation}
\label{Mdef}
\boxed{MP=\widetilde{U}(0)P-\bar{U}(0)P} \end{equation} and further discussed in the Appendices. The new operator $\bar{U}$ (\textit{"U-bar"}) is defined as the evolution operator with \textit{all model-space states removed}. (The MSC is analogous to the \textit{reference-state contribution}, appearing in the $S$-matrix formalism, where the effect normally appears only when the intermediate states is equal to the reference or initial state. In our formalism with an extended model space the effect can appear also for other model-space states, and we prefer the more general term.) It should be noted that the counterterms also remove quasi-singularities, due to quasi-degenerate states that are included in the model space. This can be of vital importance for the convergence of the procedure.
As discussed in Appendix \ref{App:SepPh}, the model-space contributions are of \textit{two kinds}. The first kind appears for all interactions, even if they are time or energy independent, while the second kind appears only for time- or energy-dependent interactions. The first kind appears also in standard time-independent perturbation theory and corresponds to so-called \textit{folded diagrams} of MBPT~\cite[Fig. 5]{LM86,LSA04}.
\subsection{The wave operator and effective interaction} \label{sec:Wop} As mentioned previously, the evolution operator \eqref{Uexp} with the perturbation \eqref{Pert} can contain uncontracted photon operators, which implies that it operates in a general \textit{Fock space,} where the number of virtual photons is not conserved. We then separate the covariant evolution operator \eqref{CovEv} into \begin{equation}
U(t)=PU(t)+\mathcal{\boldsymbol{Q}} U(t) \end{equation} where $\mathcal{\boldsymbol{Q}}=1-P$ is operating in the general Fock space, while $P$ is the projection operator for the model space, confined to the restricted Hilbert space with no uncontracted photon. This leads with the definition \eqref{Ured} of the reduced evolution operator for $t=0$ to the \textit{factorization theorem}~\cite[Eq. 121]{LSA04} \begin{equation}
\label{Fact}
\boxed{U(0)P=\big[1+\mathcal{\boldsymbol{Q}}\widetilde{U}(0)\big]P\bs{\cdot} PU(0)P} \end{equation} where the first factor on rhs is regular. Inserted in the GML formula \eqref{GML}, this yields \begin{equation}
\boldsymbol{\Psi}^\alpha(\boldsymbol{x})=\big[1+\mathcal{\boldsymbol{Q}}\widetilde{U}(0)\big]\,\Psi_0^\alpha \end{equation} where $\Psi_0^\alpha$ is the zeroth-order wave function (ZOWF) \eqref{ZOWF} in intermediate normalization \begin{equation}
\Psi^\alpha_0(\boldsymbol{x})=P\boldsymbol{\Psi}^\alpha(\boldsymbol{x})=
\frac{N^\alpha
PU(0)\Phi^\alpha(\boldsymbol{x})}{\bra{\Phi^\alpha}U(0)\ket{\Phi^\alpha}} \end{equation} The square bracket above is the \textit{wave operator} \begin{eqnarray}
\label{WaveEqn}
\boldsymbol{\Omega}&=&1+\mathcal{\boldsymbol{Q}}\widetilde{U}(0)\nonumber \\
\hspace{5cm}\boldsymbol{\Psi}^\alpha(\boldsymbol{x})&=&\boldsymbol{\Omega}\,\Psi_0^\alpha(\boldsymbol{x}) \end{eqnarray} The result here is a direct consequence of the generalized Gell-Mann--Low theorem and the definition of the reduced evolution operator.
As mentioned, with the perturbation \eqref{Pert} the wave function $\boldsymbol{\Psi}^\alpha$ lies generally in a Fock space where the number of (virtual) photons is not conserved. But we are interested here in the case where all photon operators are fully contracted, and for that purpose we project the equation on the restricted Hilbert space without uncontracted photon operators \begin{equation}
{\cal P}\boldsymbol{\Psi}^\alpha(\boldsymbol{x})={\cal P}\big[1+\mathcal{\boldsymbol{Q}}\widetilde{U}(0)\big]\,\Psi_0^\alpha(x) \end{equation} or \begin{equation}
\label{wf}
\Psi^\alpha(\boldsymbol{x})=\big[1+Q\Ugamtil{0}\big]\Psi^\alpha_0(\boldsymbol{x}) \end{equation} where $\Psi^\alpha(\boldsymbol{x})={\cal P}\boldsymbol{\Psi}^\alpha(\boldsymbol{x})$ is the projected wave function on the restricted Hilbert space and $Q={\cal P}\mathcal{\boldsymbol{Q}}$ is the conventional projection operator for the complementary space (outside the model space). The wave operator in this space is \begin{equation}
\label{WaveOp}
\boxed{\Omega={\cal P}\boldsymbol{\Omega}=1+Q\widetilde{U}(0)} \end{equation} In IN \eqref{IN} the wave operators satisfy in both spaces the relation \eqref{POP} \begin{equation}
\label{OmIN}
P\boldsymbol{\Omega} P=P\Omega P=P. \end{equation}
The \textit{effective interaction} \eqref{EffInt2} is in this formalism given by~\cite[Eq. 130]{LSA04} \begin{equation}
\label{EffInt}
\boxed{H'_{\mathrm{eff}}=P\Big[\ensuremath{\mathrm{i}}\Partder{t}\Ugamtil{t}\Big]_{t=0}P} \end{equation}
\section{Connection to the Bethe-Salpeter equation} \label{sec:Appl} \subsection{Expansion of the wave operator}
We know from the generalized Gell-Mann--Low relation \eqref{GML} that the wave function $\boldsymbol{\Psi}^\alpha(\boldsymbol{x})$ in the extended Fock space satisfies a Schr\ödinger-like equation \eqref{GML1} with the Hamiltonian $H=H_0+H'$, where $H'$ is the perturbation \eqref{Pert}. We now want to find the corresponding equation for the wave function $\Psi^\alpha={\cal P}\boldsymbol{\Psi}^\alpha(\boldsymbol{x})$ in the restricted space with no uncontracted photons, and we shall see in this section that this leads to the \textit{Bethe-Salpeter equation}.
We shall start with the exchange of a sequence of separable covariant single photons between the electrons, which can then be generalized to other interactions, leading to the full equation. This will first be done for a degenerate model space and then extended to the general case.
As shown in Appendix \ref{App:SingPhot} (Eq. \ref{Om1}), the contribution to the wave operator due the exchange of one single photon is \begin{equation}
\label{Om11}
\Omega^{(1)} P=Q\widetilde{U}^{(1)}(0,{\calE})P= \Gamma_Q({\calE}) V({\calE})P
\end{equation} and the corresponding contribution to the effective Hamiltonian $H_{\mathrm{eff}}^{(1)}({\calE})= P V({\calE})P$. Here, $\Gamma_Q({\calE})$ is the "reduced" resolvent \eqref{RedRes} and $V(\mathcal{E})$ is the effective single-photon potential \eqref{VExpl}, assuming that we operate to the right on a fourier transform \eqref{FE} with the energy parameter $\mathcal{E}$.
Similarly, it is demonstrated in Appendix \ref{App:SepPh} (Eq. \ref{Om2}) that the contribution to the evolution operator from \textit{two} separable single-photon interactions is for a degenerate model space given by (leaving out the arguments) \begin{equation}
\label{Om2alt}
\Omega^{(2)} P=Q\widetilde{U}^{(2)} P=\Gamma_Q V\Omega^{(1)}
P+\partder{\Omega^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)} \end{equation} where the last term represents the \textit{model-space contribution} (MSC) \eqref{Mdef} \begin{equation*}
QM^{(2)} P=\partder{\Omega^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)} \end{equation*} (The asterisk is introduced here only to indicate that there is a cancelled singularity at that position, which is of importance for the further treatment, as discussed in the Appendices.) The contribution to the effective Hamiltonian \eqref{Heff2} due to two-photon exchange is \begin{equation}
\label{TwoPhEff}
H_{\mathrm{eff}}^{(2)}= PV\Omega^{(1)} P+\partder{H_{\mathrm{eff}}^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}
=\bar{H}_{\mathrm{eff}}^{(2)}+\partder{H_{\mathrm{eff}}^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)} \end{equation} The last term is the MSC to the effective interaction, and if the model space is degenerate with the energy $E_0$ that term becomes \begin{equation}
\label{HMSC}
\partder{H_{\mathrm{eff}}^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}=P\partder{V({\calE})}{{\calE}}\Big|_{E_0}PV(E_0)P \end{equation} This corresponds to the "reference-state contribution", discussed in connection with the $S$-matrix treatment of two-photon exchange~\cite{BMJ93,LPS95}.
The treatment above will now be generalized to all orders as a first step towards deriving the full BS equation. We start with the covariant evolution operator \eqref{CovEv} $U(t)=U(t,-\infty)$ and the reduced evolution operator \eqref{Ured2} \begin{equation}
\label{UredA}
\widetilde{U}(t)P=U'(t)P-\widetilde{U}(t)\bs{\cdot} PU'(0)P \end{equation} where $U'=U-1$. Note that only the first factor in the product is time dependent (see. Eq. \ref{U2}). Note also the appearance of the "dots" in this expression. The significance of the dot is discussed in relation to the definition \eqref{Ured0}.
In the following we shall leave out the prime on $U'$ and also the time arguments, if there is no risk of ambiguity. We then express the counterterm \eqref{C} as \begin{equation}
\label{CA}
\boxed{CP=-\widetilde{U}\bs{\cdot} PUP} \end{equation} and the evolution operator is given by \begin{equation}
\label{UA}
UP=\Gamma VP+\Gamma V\Gamma VP+\Gamma V\Gamma V\Gamma VP+\cdots \end{equation} where $\Gamma=\Gamma({\calE})$ is the resolvent \eqref{Gamma1}. The "\textit{U-bar}" operator \eqref{Mdef}, with all intermediate model-space states removed, is \begin{equation}
\label{Ubar}
\bar{U} P=\Gamma VP+\Gamma_Q V\Gamma_Q VP+\Gamma_Q V\Gamma_Q V\Gamma_Q VP+\cdots \end{equation}
We introduce a special symbol for the time derivative at time $t=0$ \begin{equation}
\label{Adot}
\boxed{\dot{A}=\ensuremath{\mathrm{i}}\partder{A}{t}\Big|_{t=0}} \end{equation} Since the evolution operator \eqref{SPc} has the time dependence \[U(t,{\calE})=e^{-\ensuremath{\mathrm{i}} t({\calE}-H_0)}\,U(0,{\calE})\]it follows that the time derivation eliminates the denominator of the first (leftmost) resolvent, so that \begin{eqnarray}
\label{dots}
&&P\dot{U} P=P\big(V+V\Gamma V+V\Gamma V\Gamma V+\cdots\big)P=PVP+PVUP\nonumber \\
&&P\dot{\Ubar} P=P\big(V+V\Gamma_Q V+V\Gamma_Q V\Gamma_Q V+\cdots\big)P=PVP+PV\bar{U} P
\end{eqnarray} The \textit{effective interaction} $H'_{\mathrm{eff}}$ \eqref{EffInt} is with this notation given by \begin{equation}
\label{Utildot}
\boxed{H'_{\mathrm{eff}}=P\dot{\Util} P} \end{equation} We also introduce the corresponding \textit{"H-bar"} operator with no intermediate model-space states \begin{equation}
\label{Hbar}
\bar{H}'_{\mathrm{eff}}=P\dot{\Ubar} P \end{equation}
We recall the definition \eqref{Mdef} of the model-space contribution (MSC) \begin{equation}
\label{MSCA}
\widetilde{U} P=\bar{U} P+MP \end{equation} and can easily derive the identities \begin{eqnarray}
\label{ID}
\hspace{1cm} UP&=&\bar{U} P+\bar{U}\;PUP=\bar{U} P+\widetilde{U}\;PUP-M\;PUP\\
\label{ID2}
\hspace{1cm}\bar{U} P&=&UP-U\;PUP+U\;PUP\;PUP-\cdots \end{eqnarray} Then the reduced evolution operator \eqref{UredA} becomes \begin{equation}
\label{UtilA}
\widetilde{U} P=\bar{U} P+CP=\bar{U} P+\widetilde{U}\,PUP-\widetilde{U}\bs{\cdot} PUP-M\,PUP \end{equation} which using the definition \eqref{MSCA} leads to the series \begin{equation}
\label{OmRec}
\widetilde{U} P=\bar{U} P+\big(\widetilde{U}\; PUP-\widetilde{U}\bs{\cdot} PUP\big)
\big(1-PUP+ PUP\, PUP+\cdots\big)\hspace{1cm} \end{equation} With the identity \eqref{ID2} this becomes \begin{equation}
\label{OmRec2}
\boxed{\widetilde{U} P=\bar{U} P+\big(\widetilde{U}\; P\bar{U} P-\widetilde{U}\bs{\cdot} P\bar{U}
P\big)} \end{equation} which is an exact expression also for a quasi-degenerate model space. It can be expanded as \begin{equation}
\label{OmRec3}
\widetilde{U} P=\bar{U} P+\big(\bar{U}\; P\bar{U} P-\bar{U}\bs{\cdot} P\bar{U} P\big)
+\big(\bar{U}\; P\bar{U} P-\bar{U}\bs{\cdot} P\bar{U} P\big)
\big( P\bar{U} P-\bs{\cdot} P\bar{U} P\big)+\cdots \end{equation} As discussed in Appendix \ref{App:SepPh}, the result \eqref{OmRec2} can be expressed \begin{equation}
\label{OmRec2A}
\widetilde{U} P=\bar{U} P+\partdelta{\widetilde{U}}{{\calE}}*P\dot{\Ubar} P
=\bar{U} P+\partdelta{\widetilde{U}}{{\calE}}*\bar{H}'_{\mathrm{eff}} \end{equation} where $\delta{\calE}$ is the change in the model-space energy, represented by the "dot", $\delta\widetilde{U}$ is the corresponding change in $\widetilde{U}$, and $\bar{H}'_{\mathrm{eff}}$ is the "H-bar" operator \eqref{Hbar}. In the case of complete degeneracy this becomes \begin{equation}
\label{UTot}
\widetilde{U} P=\bar{U} P+\partder{\widetilde{U}}{{\calE}}\Big|_{{\calE}=E_0}*\bar{H}'_{\mathrm{eff}} \end{equation} Introducing the\textit{"Omega-bar"} operator $\bar{\Omega}$ (with no intermediate model-space states) in analogy with the wave operator \eqref{WaveOp} \begin{equation}
\label{Ombar}
\bar{\Omega} P=P+Q\bar{U} P=P+\Gamma_Q VP+\Gamma_Q V\Gamma_Q VP+\cdots \end{equation} we can express the relations above as \begin{equation}
\label{OmTot}
\boxed{\Omega P=\bar{\Omega} P+\partdelta{\Omega}{{\calE}}*\bar{H}'_{\mathrm{eff}}
\Rightarrow\bar{\Omega} P+\partder{\Omega}{{\calE}}\Big|_{{\calE}=E_0}*\bar{H}'_{\mathrm{eff}}} \end{equation} The second term is here consequently an exact expression for the entire model-space contribution to the wave operator. This is in agreement with the three-photon result \eqref{Om3C}.
By taking the time derivative of the relation \eqref{OmRec2}, using the relations above, we obtain similarly \begin{equation}
\label{HHbar}
\boxed{H'_{\mathrm{eff}}=\bar{H}'_{\mathrm{eff}}+\partdelta{H'_{\mathrm{eff}}}{{\calE}}*\bar{H}'_{\mathrm{eff}}
\Rightarrow\bar{H}'_{\mathrm{eff}}+\partder{H'_{\mathrm{eff}}}{{\calE}}\Big|_{{\calE}=E_0}*\bar{H}'_{\mathrm{eff}}} \end{equation} The second term represents here the model-space contribution to the effective interaction. This result agrees also with the third-order result \eqref{Heff3}.
From the results above we conjecture that the wave operator can at complete degeneracy alternatively be expressed \begin{equation}
\label{Om}
\Omega P=\bar{\Omega} P+\partder{\bar{\Omega}}{{\calE}}*H'_{\mathrm{eff}}
+\half\ppartder{\bar{\Omega}}{{\calE}}*\big(H'_{\mathrm{eff}}\big)^2
+\tref\pppartder{\bar{\Omega}}{{\calE}}*\big(H'_{\mathrm{eff}}\big)^3+\cdots
=\bar{\Omega} P+\sum_{n=1}^\infty\npartder{n}{\bar{\Omega}}{{\calE}}\;
*\big(H'_{\mathrm{eff}}\big)^{n} \end{equation} with all derivatives taken at ${\calE}=E_0$, and we shall now prove this relation by showing that it is compatible with the results \eqref{OmTot} and \eqref{HHbar}, which we have rigorously derived. This equation contains eliminated singularities, indicated by the asterisks. As discussed in the Appendices, the derivative of such an expression has to be taken \textit{before} the singularity is eliminated. Using the rules developed, particularly in Appendix \ref{App:Exp}, we find for instance \begin{eqnarray}
\label{Deriv2}
\hspace{1cm}\partder{}{{\calE}}\Big(\partdelta{\bar{\Omega}}{{\calE}}*H'_{\mathrm{eff}}\Big)
&\Rightarrow&\half\ppartder{\bar{\Omega}}{{\calE}}*H'_{\mathrm{eff}}
+\partder{\bar{\Omega}}{{\calE}}*\partder{H'_{\mathrm{eff}}}{{\calE}}\\
\label{Deriv3}
\hspace{1cm}\partder{}{{\calE}}\Big(\half\ppartdelta{\bar{\Omega}}{{\calE}}*\big(H'_{\mathrm{eff}}\big)^2\Big)
&\Rightarrow&\npartder{3}{\bar{\Omega}}{{\calE}}*\big(H'_{\mathrm{eff}}\big)^2
+\half\ppartder{\bar{\Omega}}{{\calE}}*\partder{H'_{\mathrm{eff}}}{{\calE}}*H'_{\mathrm{eff}} \end{eqnarray} Note that in the second example the two $\bar{H}'_{\mathrm{eff}}$ operators have in the quasi-degenerate case different energy parameters, and therefore only one of them is affected by the derivation.
Generalizing these rules, we can evaluate the derivative of the wave operator \eqref{Om} \begin{eqnarray}
\label{OmDerA}
\hspace{1cm}\partder{\Omega}{{\calE}}&=&\partder{\bar{\Omega}}{{\calE}}
+\half\ppartder{\bar{\Omega}}{{\calE}}* H'_{\mathrm{eff}}
+\tref\pppartder{\bar{\Omega}}{{\calE}}*\big(H'_{\mathrm{eff}}\big)^2+\cdots\nonumber \\
\hspace{1cm}&+&\partder{\bar{\Omega}}{{\calE}}*\partder{H'_{\mathrm{eff}}}{{\calE}}
+\half\ppartder{\bar{\Omega}}{{\calE}}*\partder{H'_{\mathrm{eff}}}{{\calE}}*H'_{\mathrm{eff}}
+\tref\pppartder{\bar{\Omega}}{{\calE}}*\partder{H'_{\mathrm{eff}}}{{\calE}}*(H'_{\mathrm{eff}})^2+\cdots \end{eqnarray} or \begin{equation}
\label{OmDerB}
\partder{\Omega}{{\calE}}=\sum_{n=1}^\infty
\npartder{n}{\bar{\Omega}}{{\calE}}\;\Big[\big(H'_{\mathrm{eff}}\big)^{n-1}
+\partder{H'_{\mathrm{eff}}}{{\calE}}*(H'_{\mathrm{eff}})^{n-1}\Big] \end{equation} We now insert this expression into the equation \eqref{OmTot}, which yields \begin{equation}
\partder{\Omega}{{\calE}}*\bar{H}'_{\mathrm{eff}}=\sum_{n=1}^\infty
\npartder{n}{\bar{\Omega}}{{\calE}}\;\Big[\big(H'_{\mathrm{eff}}\big)^{n-1}
\bar{H}'_{\mathrm{eff}}+\partder{H'_{\mathrm{eff}}}{{\calE}}*(H'_{\mathrm{eff}})^{n-1}\bar{H}'_{\mathrm{eff}}\Big] \end{equation} or, using the relation \eqref{HHbar}, \begin{equation}
\label{Key}
\boxed{\Omega P=\bar{\Omega} P+\sum_{n=1}^\infty
\npartder{n}{\bar{\Omega}}{{\calE}}\;\big(H'_{\mathrm{eff}}\big)^{n}} \end{equation} This is identical to the conjectured relation \eqref{OmTot} and therefore completes the proof. The sum represents by definition the model-space contribution (MSC).
\begin{figure}
\caption{Examples of non-separable two-photon interactions.}
\label{Fig:NonSep}
\end{figure}
\subsection{Derivation of the Bethe-Salpeter equation. Degenerate model space.}
The previous treatment has been based upon the Hamiltonian $H=H_0+V(E)$, where $V(E)$ is the potential due to the exchange of a single covariant photon. But the process can be repeated in exactly the same way, if we include \textit{all non-separable multi-photon interactions}. A non-separable interaction is defined as an interaction that cannot be represented by two or more simpler interactions in the way treated here. Two photons---crossing or noncrossing--- that overlap in time represent non-separable two-photon interactions (see Fig. \ref{Fig:NonSep}, c.f. also Ref.~\cite[Fig. 1]{SB51}). These can also include the radiative self-energy and vertex corrections. In a similar way non-separable three-, four-,... photon interactions can be defined. Therefore, in the following we replace the single-photon potential $V$ by the general potential due to all non-separable interactions \begin{equation}
\label{Vgen}
\mathcal{V}(E)=V(E)+V_2(E)+V_3(E)+\cdots \end{equation}
As discussed in the Appendices, when operating on a fourier transform of definite energy ${\calE}$, the energy parameter of $\bar{\Omega} P$ is equal to that energy, i.e., \begin{equation}
\label{OmbarE}
\bar{\Omega} F({\calE})=\bar{\Omega}({\calE})F({\calE}) \end{equation} For a degenerate model space of energy $E_0$ this means that \begin{equation}
\label{OmbarE0}
\bar{\Omega} P=\bar{\Omega}(E_0)P=P+\Gamma_Q(E_0)\mathcal{V}(E_0)+\cdots \end{equation}
The model functions are eigenfunctions of the effective Hamiltonian \eqref{SecEq}, and for a degenerate model space (of energy $E_0$) they are eigenfunctions also of the effective interaction \eqref{EffInt2}, \begin{equation}
\label{DeltaE}
H'_{\mathrm{eff}}\ket{\Psi_0^\alpha}=\Delta E^\alpha\ket{\Psi_0^\alpha}
\end{equation} where $\Delta E^\alpha=E^\alpha-E_0$. Operating with the operator equation \eqref{Key} directly on the model function $\Psi_0^\alpha$ then leads to the Taylor expansion \begin{equation}
\label{OmExp}
\Omega\Psi_0^\alpha =\Big[\bar{\Omega}(E_0) +\partder{\bar{\Omega}}{{\calE}}\Big|_{E_0}\;\Delta E^\alpha+
\half\ppartder{\bar{\Omega}}{{\calE}}\Big|_{E_0}\;(\Delta E^\alpha)^2
+\frac{1}{3!}\cdots\Big]\Psi_0^\alpha=\bar{\Omega}(E^\alpha)\Psi_0^\alpha \end{equation} This implies that \textit{the MSC term shifts the energy parameter of the resolvent as well as that of the potential from the unperturbed energy $E_0$ to the exact energy $E^\alpha$}. But $\bar{\Omega}(E^\alpha)\,\Psi_0^\alpha$ with the energy parameter equal to the full energy for the state $\Psi^\alpha$ is also identical to the \textit{Brillouin-Wigner expansion} \eqref{BSBW}, \begin{equation}
\label{OmBW}
\bar{\Omega}(E^\alpha)\Psi_0^\alpha=\Big[1+\frac{Q}{E^\alpha-H_0}\,\mathcal{V}(E^\alpha)
+\frac{Q}{E^\alpha-H_0}\,\mathcal{V}(E^\alpha)\frac{Q}{E^\alpha-H_0}\,\mathcal{V}(E^\alpha)
+\cdots\Big]\Psi_0^\alpha\hspace{0.5cm} \end{equation} which represents the full wave function, i.e., \begin{equation}
\label{OmOmbar}
\boxed{\bar{\Omega}(E^\alpha)\Psi_0^\alpha=\Omega\Psi_0^\alpha} \end{equation} This implies that the relation \eqref{Key} essentially represents \textit{the link between the Rayleigh-Schr\ödinger and the Brillouin-Wigner expansions for an energy-dependent interaction} and at the same time \textit{the link between the MBPT approaches and the Bethe-Salpeter equation} (indicated by the arrow in the diagram of Fig. \ref{Fig:Intr}).
The BW expansion \eqref{OmBW} can be expressed \begin{equation}
\label{BSA4}
\Omega\Psi_0^\alpha=\Psi_0^\alpha+\Gamma_Q(E^\alpha)\mathcal{V}(E^\alpha)
\Omega\Psi_0^\alpha \end{equation} or \begin{equation}
\label{BSA6}
(E^\alpha-H_0)\,Q\Psi^\alpha=Q\mathcal{V}(E^\alpha)\Psi^\alpha \end{equation}
From the relation \eqref{HHbar} it can be shown in analogy with the relation \eqref{Key} \begin{equation}
\label{HKey}
H'_{\mathrm{eff}}=\bar{H}'_{\mathrm{eff}}+\sum_{n=1}^\infty
\npartder{n}{\bar{H}'_{\mathrm{eff}}}{{\calE}}\;\big(H'_{\mathrm{eff}}\big)^{n} \end{equation} With the definitions \eqref{Hbar} and \eqref{Ombar} this leads to \begin{equation}
\label{Hb}
\bar{H}'_{\mathrm{eff}}=P\mathcal{V}(E_0)\,\bar{\Omega}(E_0) P \end{equation} and in analogy with the relation \eqref{OmExp} to \begin{equation}
\label{Hef}
H'_{\mathrm{eff}}=P\mathcal{V}(E^\alpha)\,\bar{\Omega}(E^\alpha) P=P\mathcal{V}(E^\alpha)\,\Omega P \end{equation} This leads together with Eq. \eqref{BSA6} to the final equation \begin{equation}
\label{BSA}
\boxed{(E^\alpha-H_0)\,\Psi^\alpha= \mathcal{V}(E^\alpha)\,\Psi^\alpha} \end{equation} \textbf{This is the Bethe-Salpeter equation for energy-dependent interactions in the Schr\ödinger-like form \eqref{BS}}.
\textit{We have now confirmed that the Schr\ödinger equation \eqref{GML1}, obtained directly from the generalized Gell-Mann--Low relation in the extended Fock space with the perturbation \eqref{Pert}, corresponds in the projected Hilbert space with no uncontracted photons to a Schr\ödinger-like equation with the perturbation \eqref{Vgen}. Both forms represent the complete interaction between the particles and are exactly equivalent to the original Bethe-Salpeter equation \eqref{BS2}.}
The main difference between the original form of the BS equation and the Schr\ödinger-like form derived here is primarily that the latter has the time dependence reduced to a single time, which makes the wave function in accord with standard quantum mechanics. Furthermore, the Schr\ödinger-like form contains explicitly the resolvent, while the remaining part of the Green's function \eqref{G0Op} is merged with the kernel $\kappa$ to form the potential $\mathcal{V}$.
The Schr\ödinger-like equation \eqref{BSA} we have derived is equivalent to the equation derived from the BS equation by Sucher~\cite[Eq. 1.47]{Su58} and rederived by Douglas and Kroll~\cite[Eq. 3.26]{DK74} and Zhang~\cite[Eq. 15]{Zhang96}. In these works the equation is essentially obtained by integrating over the relative energy of the particles, thereby transforming the equation to an "equal-times" equation. This equation is then analyzed in terms of the Brillouin-Wigner perturbation theory. In our presentation the corresponding equation is obtained by starting from MBPT in the Rayleigh-Schr\ödinger formulation and summing all relevant perturbations to all orders. The present derivation therefore can serve as a link between the two approaches.
In the next section we shall extend the treatment to the quasi-degenerate case and derive the corresponding Bloch equation.
\subsection{Derivation of the Bethe-Salpeter-Bloch equation. Quasi-degenerate model space.}
\label{sec:BSB}
We have previously assumed that the model space is \textit{degenerate}, which for a two-electron system implies that the effective interaction is \textit{diagonal} within this space (assuming the basis functions have definite symmetry). Then the relation \eqref{DeltaE} simplifies the treatment, and the formulas derived in the previous section lead directly to the standard Bethe-Salpeter equation \eqref{BSA}. The treatment above, however, is more general and can be extended to the case where the model space is non-degenerate (quasi-degenerate). In the present section we shall show how this can be performed.
The following relation can easily derived by induction \begin{equation}
\label{Der1}
\partdern{n}{\bar{\Omega}}{{\calE}}=\Gamma_Q\partdern{n}{(\mathcal{V}\bar{\Omega})}{{\calE}}
-n\Gamma_Q\partdern{{(n-1)}}{\bar{\Omega}}{{\calE}} \end{equation} To prove this we form the next-order derivative
\[\partdern{{(n+1)}}{\bar{\Omega}}{{\calE}}=-\Gamma_Q^2\partdern{n}{(\mathcal{V}\bar{\Omega})}{{\calE}}
+\Gamma_Q\partdern{{(n+1)}}{(\mathcal{V}\bar{\Omega})}{{\calE}}
+n\Gamma_Q^2\partdern{{(n-1)}}{\bar{\Omega}}{{\calE}}-n\Gamma_Q\partdern{{n}}{\bar{\Omega}}{{\calE}}\]
(Since no singularities are involved here, ordinary rules of
derivation can be used.) Inserting the expression \eqref{Der1} in the first term, yields
\[\partdern{{(n+1)}}{\bar{\Omega}}{{\calE}}=\Gamma_Q\partdern{{(n+1)}}{(\mathcal{V}\bar{\Omega})}{{\calE}}
-(n+1)\Gamma_Q\partdern{{n}}{\bar{\Omega}}{{\calE}}\] In first order we have with $\bar{\Omega}=1-\Gamma_Q\mathcal{V}\bar{\Omega}$
\[\partder{\bar{\Omega}}{{\calE}}=\Gamma_Q\partder{(\mathcal{V}\bar{\Omega})}{{\calE}}-\Gamma_Q\bar{\Omega}\] which completes the proof of the relation \eqref{Der1}.
The formula above leads together with the expansion \eqref{Key} to \begin{equation}
\label{Key1}
Q\Omega P=Q\bar{\Omega} P+\Gamma_Q\sum_{n=1}^\infty
\npartder{n}{(\mathcal{V}\bar{\Omega})}{{\calE}}\;(H'_{\mathrm{eff}})^n-\Gamma_Q\sum_{n=1}^\infty
\npartder{{(n-1)}}{\bar{\Omega}}{{\calE}}\;(H'_{\mathrm{eff}})^n \end{equation} The first term on the rhs can also be expressed $\Gamma_Q\mathcal{V}\bar{\Omega} P$, and the last term is simply $-\Gamma_Q\Omega H'_{\mathrm{eff}}$, which yields \begin{equation}
\label{Key2}
Q\Omega P=\Gamma_Q\mathcal{V}\bar{\Omega} P+\Gamma_Q\sum_{n=1}^\infty
\npartder{n}{(\mathcal{V}\bar{\Omega})}{{\calE}}\;(H'_{\mathrm{eff}})^n-\Gamma_Q\Omega H'_{\mathrm{eff}} \end{equation}
We can consider $\mathcal{V}\bar{\Omega}$ as a single energy-dependent operator, and if that operates on a particular model state of a degenerate model space of energy $E_0$, the first two terms of the bracket above represents the Taylor expansion \begin{equation}
\label{Taylor}
\mathcal{V}(E_0)\,\bar{\Omega}(E_0)+\sum_{n=1}^\infty
\npartder{n}{(\mathcal{V}\bar{\Omega})}{{\calE}}\Big|_{E_0}\;(\Delta E^\alpha)^n
=\mathcal{V}(E^\alpha)\,\bar{\Omega}(E^\alpha).\hspace{0.5cm} \end{equation} Thus, the expansion has the effect of transforming the energy parameter of the product $\mathcal{V}\bar{\Omega}$ from $E_0$ to the full energy $E^\alpha$, \begin{equation}
\label{ModE}
\mathcal{V}(E_0)\bar{\Omega}(E_0)\Psi_0^\alpha\rightarrow\mathcal{V}(E^\alpha)\bar{\Omega}(E^\alpha)\Psi_0^\alpha \end{equation} in analogy with the expansion \eqref{OmBW}. Using the relation \eqref{OmOmbar}, the equation \eqref{Key2} above then becomes \begin{equation}
\label{BSB1}
Q\Omega \Psi_0^\alpha=\Gamma_Q\big[\mathcal{V}(E^\alpha)\Omega-\Omega
H_{\mathrm{eff}}'\big]\Psi_0^\alpha \end{equation} or \begin{equation}
\label{BSB2}
(E_0-H_0)\Omega \Psi_0^\alpha=Q\big[\mathcal{V}(E^\alpha)\Omega-\Omega
H_{\mathrm{eff}}'\big]\Psi_0^\alpha \end{equation} which is consistent with the Bethe-Salpeter equation \eqref{BSA}.
If the model space is \textit{non-degenerate} (quasi-degenerate), then the relation \eqref{DeltaE} is no longer valid, and the expansion \eqref{Key2} can not be expressed by means of a single energy parameter as in the Taylor expansion \eqref{Taylor}. Instead, the potential will depend on the \textit{full matrix} of the effective Hamiltonian. We then replace the energy parameter in \eqref{OmbarE0} by the model Hamiltonian $H_0$,
\[\bar{\Omega} P=P+\Gamma_Q(H_0)\mathcal{V}(H_0)+\cdots=\bar{\Omega}(H_0)P\] By this notation we understand---in accordance with the rule \eqref{OmbarE}--- \begin{equation}
A(H_0)B\Phi=A(E_0)B\Phi \end{equation} when $\Phi$ is an eigenfunction of $H_0$ with the eigenvalue $E_0$ and $B$ is an arbitrary operator combination. Together with the linearity condition, \begin{equation}
A(H_0)B(a\Phi+b\Phi'\big)=aA(E_0)B\Phi+bA(E'_0)B\Phi' \end{equation} where $\Phi'$ is another eigenfunction of $H_0$ with the eigenvalue $E'_0$, this defines the notation fully.
The expansion \eqref{Key2} can now be regarded, in analogy with the energy modification \eqref{Taylor}, as modifying the parameter $H_0$ to the full effective Hamiltonian $H_{\mathrm{eff}}=H_0+H'_{\mathrm{eff}}$ \begin{equation}
\label{Taylor2}
\mathcal{V}(H_0)\,\bar{\Omega}(H_0)+\sum_{n=1}^\infty
\npartder{n}{(\mathcal{V}\bar{\Omega})}{{\calE}}\;(H'_{\mathrm{eff}})^n
=\mathcal{V}(H_{\mathrm{eff}})\,\bar{\Omega}(H_{\mathrm{eff}})\hspace{0.5cm} \end{equation} i.e., \begin{equation}
\label{ModH}
\mathcal{V}(H_0)\bar{\Omega}(H_0)P\rightarrow\mathcal{V}(H_{\mathrm{eff}})\bar{\Omega}(H_{\mathrm{eff}})P \end{equation} and Eq. \eqref{BSB1} becomes \begin{equation}
\label{VOm2}
Q\Omega P=\Gamma_Q(H_0)\mathcal{V}(H_{\mathrm{eff}})\bar{\Omega}(H_{\mathrm{eff}})P-\Gamma_Q(H_0)\Omega H'_{\mathrm{eff}} \end{equation} The notation here is defined by the relation \begin{equation}
A(H_{\mathrm{eff}})B\Psi_0^\alpha=A(E^\alpha)B\Psi_0^\alpha \end{equation} where $\Psi_0^\alpha$ is a model function (eigenfunction of $H_{\mathrm{eff}}$ with the eigenvalue $E^\alpha$, see Eq. \ref{SecEq}), which together with the linearity condition defines the operator when acting on any model space.
Similarly, the expansion \eqref{Key} yields \begin{equation}
\label{Keyref}
\Omega P=\bar{\Omega}(H_0)P+\sum_{n=1}^\infty
\npartder{n}{\bar{\Omega}}{{\calE}}\;\big(H'_{\mathrm{eff}}\big)^{n}=\bar{\Omega}(H_{\mathrm{eff}})P \end{equation} and we can now express the equation \eqref{VOm2} as \begin{equation}
Q\Omega P=\Gamma_Q(H_0)\mathcal{V}(H_{\mathrm{eff}})\,\Omega P-\Gamma_Q(H_0)\Omega\,H_{\mathrm{eff}}' \end{equation} or \begin{equation}
\Gamma_Q(H_0)]^{-1}Q\Omega P=Q\big[\mathcal{V}(H_{\mathrm{eff}})\,\Omega-Q\Omega\,H_{\mathrm{eff}}'\big]P \end{equation} Operating on a model-state function $\Phi(E_0^\alpha)$ of energy $E_0^\alpha$, we have according to the definitions above
\[[\Gamma_Q(H_0)]^{-1}\Phi(E_0^\alpha)=(E_0^\alpha-H_0)\,\Omega\Phi(E_0^\alpha)\] Therefore, \textit{the inverse of the resolvent can be expressed as a commutator} \begin{equation}
\label{comm}
[\Gamma_Q(H_0)]^{-1}A\equiv[A,H_0] \end{equation} where $A$ is an arbitrary operator. This leads to the commutator relation \begin{equation}
\label{BScomm}
[\Omega,H_0] P=Q\mathcal{V}(H_{\mathrm{eff}})\,\Omega P-Q\Omega\,H_{\mathrm{eff}}' \end{equation} The relation \eqref{Hef} can be generalized to \begin{equation}
\label{HeffGen}
H_{\mathrm{eff}}'=P\mathcal{V}(H_{\mathrm{eff}})\,\Omega P \end{equation} and with the IN relation \eqref{OmIN} $P\Omega P=P$ we arrive at \textit{the BS equation in commutator form} \begin{equation}
\label{BSBloch}
\boxed{\big[\Omega,H_0\big]P=\mathcal{V}(H_{\mathrm{eff}})\,\Omega P-\Omega\,H_{\mathrm{eff}}'} \end{equation}
\textit{The equation above---which we shall refer to as the \textbf{Bethe-Salpeter-Bloch equation}---is the main result of the present work. It has the same relation to the standard BS equation as has the standard Bloch equation \eqref{Bloch} to the ordinary Schr\ödinger equation. It can be used to generate the BS equation perturbatively, essentially as the ordinary Bloch eqution is used in standard MBPT. The commutator form makes it possible to apply the equation to an extended model space, which essentially eliminates the quasi-degeneracy problem that might appear in applying the standard equation directly on a single state.}
\subsection{Expansion of the Bethe-Salpeter-Bloch equation}
\begin{figure}
\caption{Illustration of the perturbative-nonperturbative
procedure for solving the Bethe-Salpeter equation, described in the text.}
\label{Fig:Pair}
\end{figure}
The original Bethe-Salpeter equation contains the exact energy and is therefore normally treated by means a Brillouin-Wigner perturbation expansion~\cite{DK74,Zhang96a}, which requires a self-consistent treatment. The BS equation in the Bloch-equation form can be used to generate a perturbation expansion of Rayleigh-Schr\ödinger type that does not require any self-consistence procedure. We have at our laboratory developed a procedure of solving the Bethe-Salpeter-Bloch equation that is a combination of perturbative and non-perturbative techniques, which we shall here briefly indicate. A detailed description of the procedure together with numerical results will appear shortly~\cite{HSL05}.
Our procedure is based upon the iterative solution of pair equations~\cite{Ma79,ELi88,SO89,SO89a,LM86}. This represents the "ladder" approximation of the BS equation, as indicated in Fig. \ref{Fig:Pair} (a). The pair function is combined with the emission of a single (uncontracted) photon (Fig. \ref{Fig:Pair} b). This represents a function in the extended Fock space, discussed in section \ref{sec:Wop}. This function can be iterated further, before the photon is annihilated, which can yield instantaneous (Coulomb and Breit) interactions, crossing the photon. These iterations can be continued after the annihilation, as indicated in Fig. \ref{Fig:Pair} (c). By annihilating the photon on the same electron line, leads to self-energy and vertex corrections (Fig. \ref{Fig:Pair} d). At present time it is possible to treat only one covariant photon in this way, but the dominating part of the multi-photon exchange will be included by the crossings with the instantaneous interactions (c). This will correspond to all effects treated by Zhang~\cite{Zhang96a} in his analysis of the helium fine structure up to order $m\alpha^7$, except for the non-separable part of two-photon exchange (Fig. \ref{Fig:NonSep}). With our approach this part has at present to be included analytically.
\section{Summary and conclusions} Standard many-body perturbation theory (MBPT) is conveniently based upon the Bloch equation, which is the generating equation for Rayleigh-Schr\ödinger perturbation expansion. The Bloch equation can also be used to generate various other perturbative schemes, such as the linked-diagram expansion, and it also leads to non-perturbative schemes, such as the Coupled-Cluster Approach. In the commutator form \eqref{Bloch} the Bloch equation leads to schemes that can handle the quasi-degenerate problem in an efficient way by means of an "extended" model space.
In this paper we have reviewed the connection between relativistic MBPT and quantum-electrodynamics (QED) for a two-electron system by means of the recently introduced covariant-evolution-operator method~\cite{LSA04}. The exchange of a single covariant photon is treated to all orders, and this is shown to lead to an equation of the Bethe-Salpeter (BS) type. Extending the treatment to all non-separable interactions (including radiative corrections) leads to the full BS equation. This establishes a link between the perturbative and non-perturbative schemes, based upon Rayleigh-Schr\ödinger perturbation theory and schemes based upon the BS equation, which are normally treated by means of Brillouin-Wigner perturbation theory.
In addition, a Bloch equation in commutator form that is compatible with the BS equation is derived. This equation has the same relation to the Bethe-Salpeter equation as has the standard Bloch equation to the ordinary Schr\ödinger equation and represents a series of BS equations, associated with a model space that need not be degenerate. This can be used to generate a perturbative expansion, corresponding to the BS equation for an extended model space. In principle, this will make it possible to treat the quasi-degeneracy problem also within the BS formalism. Such a scheme is presently being tested at our laboratory.
\begin{center}\textbf{APPENDIX} \end{center} \subsection{Zeroth-order Green's function} \label{sec:GF}
\renewcommand{\mathcal{E}}{\mathcal{E}} \begin{figure}
\caption{Graphical representation of the zeroth-order Green's function \eqref{G0C}.}
\label{Fig:G0}
\end{figure}
The zeroth-order Green's function \eqref{GF0} in Fig. \ref{Fig:G0} is in coordinate representation \begin{equation}
\label{G0C}
G_0(x_1x_2;x_{10}x_{20})=S_{\mathrm{F}}(x_1;x_{10})\,S_{\mathrm{F}}(x_2;x_{20}) \end{equation} where $S_{\mathrm{F}}$ is the electron propagator \begin{equation}
\label{S}
S(x,x_0)=\intd{\omega}\,S(\omega)\,e^{-\ensuremath{\mathrm{i}}\omega(t-t_0)} \end{equation} with the fourier transform \begin{equation}
\label{Sft}
\bra{\boldsymbol{x}}S(\omega)\ket{\boldsymbol{x}_0}=\frac{\langle{\boldsymbol{x}}\ket{r}\bra{r}{\boldsymbol{x}_0}\rangle}
{\omega-\ensuremath{\varepsilon}_r+\ensuremath{\mathrm{i}}\eta_r}=\bra{\boldsymbol{x}}\hat{S}(\omega)\ket{\boldsymbol{x}_0} \end{equation} and the corresponding operator form \begin{equation}
\label{Sop}
\hat{S}(\omega)= \frac{\Lambda_+}{\omega-h+\ensuremath{\mathrm{i}}\eta}+\frac{\Lambda_-}{\omega-h-\ensuremath{\mathrm{i}}\eta} \end{equation} Here, $h$ is the single-electron Dirac Hamiltonian in the field of the nucleus and $\Lambda_\pm$ are projection operators for positive and negative-energy single-particle states.
We consider the equal-times Green's function with $t_1=t_2=t$, which gives \begin{eqnarray}
\label{G0E}
\hspace{1cm}&& G_0(t,\boldsymbol{x}_1,\boldsymbol{x}_2;x_{10},x_{20})=
\intd{\epsilon}\,\mathrm{e}^{-\ensuremath{\mathrm{i}}\epsilon t}\,
\frac{\langle\boldsymbol{x}_1\boldsymbol{x}_2\ket{rs}\bra{rs}\boldsymbol{x}_{10}\boldsymbol{x}_{20}\rangle}
{\epsilon-\ensuremath{\varepsilon}_r-\ensuremath{\varepsilon}_s}\nonumber \\
\hspace{1cm}&\times& \intd{\omega}\;\Big[\frac{1}{\omega-\ensuremath{\varepsilon}_r+\ensuremath{\mathrm{i}}\eta_r}+
\frac{1}{\epsilon-\omega-\ensuremath{\varepsilon}_s+\ensuremath{\mathrm{i}}\eta_s}\Big]\,\mathrm{e}^{\ensuremath{\mathrm{i}}\omega
t_{10}}\,\mathrm{e}^{\ensuremath{\mathrm{i}}(\epsilon-\omega)t_{20}}\hspace{1cm} \end{eqnarray} with $x=(t,\boldsymbol{x})$, $\omega_1=\omega$ and $\epsilon=\omega_1+\omega_2$. The fourier transform with respect to $t$ is then \begin{eqnarray}
\label{G0ft}
&&G_0(\epsilon,\boldsymbol{x}_1,\boldsymbol{x}_2;x_{10},x_{20})=\nonumber \\
&&\frac{\langle\boldsymbol{x}_1\boldsymbol{x}_2\ket{rs}\bra{rs}\boldsymbol{x}_{10}\boldsymbol{x}_{20}\rangle}
{\epsilon-\ensuremath{\varepsilon}_r-\ensuremath{\varepsilon}_s} \intd{\omega}\;\Big[\frac{1}{\omega-\ensuremath{\varepsilon}_r+\ensuremath{\mathrm{i}}\eta_r}+
\frac{1}{\epsilon-\omega-\ensuremath{\varepsilon}_s+\ensuremath{\mathrm{i}}\eta_s}\Big]\;
\mathrm{e}^{\ensuremath{\mathrm{i}}\omega t_{10}}\,\mathrm{e}^{\ensuremath{\mathrm{i}}(\epsilon-\omega)t_{20}}\hspace{1cm} \end{eqnarray} or in operator form \begin{equation}
\label{G0Op}
G_0(\epsilon)=\Gamma(\epsilon)\intd{\omega} \; g_0(\epsilon,\omega)\,\mathrm{e}^{\ensuremath{\mathrm{i}}\omega t_{10}}\,
\mathrm{e}^{\ensuremath{\mathrm{i}}({\epsilon}-\omega)t_{20}} \end{equation} where $\Gamma({\calE})$ is the resolvent \eqref{Gamma1} \begin{equation}
\label{Gamma}
\Gamma({\calE})=\frac{1}{{\calE}-H_0} \end{equation} $H_0=h_1+h_2$ is the zeroth-order Hamiltonian \eqref{Pert} and \begin{eqnarray}
\label{g0}
\hspace{1cm} g_0(\epsilon,\omega)&=&\Lambda_+\Big[\frac{1}{\omega-h_1+\ensuremath{\mathrm{i}}\eta}
+\frac{1}{\epsilon-\omega-h_2+\ensuremath{\mathrm{i}}\eta}\Big]\nonumber \\
\hspace{1cm}&+&\Lambda_-\Big[\frac{1}{\omega-h_1-\ensuremath{\mathrm{i}}\eta}
+\frac{1}{\epsilon-\omega-h_2-\ensuremath{\mathrm{i}}\eta}\Big] \end{eqnarray}
The inverse transformation is \begin{equation}
\label{G0ftint}
\boxed{G_0(t,\boldsymbol{x}_1,\boldsymbol{x}_2;x_{10},x_{20})=
\intd{\epsilon}\;\mathrm{e}^{-\ensuremath{\mathrm{i}}\epsilon
t}\,G_0(\epsilon,\boldsymbol{x}_1,\boldsymbol{x}_2;x_{10},x_{20})} \end{equation} and specifically, \begin{equation}
\label{G0ft0}
\boxed{G_0(t=0,\boldsymbol{x}_1,\boldsymbol{x}_2;x_{10},x_{20})=
\intd{\epsilon}\;G_0(\epsilon,\boldsymbol{x}_1,\boldsymbol{x}_2;x_{10},x_{20})} \end{equation}
\subsection{Single-photon exchange} \label{App:SingPhot} (See Ref.~\cite[Eq. 312]{LSA04}.) We consider now the covariant evolution operator \eqref{CovEv} for the exchange of a single covariant photon, represented by the diagram in Fig. \ref{Fig:SingPhot} (left) \begin{eqnarray}
\label{SP}
U^{(1)}(t',t_0)&=&-\half\int\!\!\!\int \ensuremath{\mathrm{d}}^3\boldsymbol{x}'_1\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}'_2\,\hat\psi^{\dag}(x'_1)\hat\psi^{\dag}(x'_2)
\int\!\!\!\int \ensuremath{\mathrm{d}}^4 x_1\,\ensuremath{\mathrm{d}}^4 x_2\,\ensuremath{\mathrm{i}}S_{\mathrm{F}}(x'_1,x_1)\,\ensuremath{\mathrm{i}}S_{\mathrm{F}}(x'_2,x_2)\,\ensuremath{\mathrm{i}} I(x_2,x_1)\nonumber \\
&\times&\int\!\!\!\int\ensuremath{\mathrm{d}}^3\boldsymbol{x}_{10}\,\ensuremath{\mathrm{d}}^3\boldsymbol{x}_{20}\,\ensuremath{\mathrm{i}}S_{\mathrm{F}}(x_1,x_{10})\,\ensuremath{\mathrm{i}}S_{\mathrm{F}}(x_2,x_{20})
\,\hat\psi(x_{20})\hat\psi(x_{10}) \end{eqnarray} leaving out the damping factors. More compactly, we express this as \begin{eqnarray}
\label{SPa}
\hspace{1cm} U^{(1)}(t',t_0)&=&-\half\hat\psi^{\dag}(x'_1)\hat\psi^{\dag}(x'_2)
\, G_0(x'_1,x'_2;x_1x_2)\,\ensuremath{\mathrm{i}} I(x_2,x_1)\nonumber \\
\hspace{1cm}&\times&G_0(x_1,x_2;x_{10}x_{20})\;\hat\psi(x_{20})\hat\psi(x_{10}) \end{eqnarray} \begin{figure}
\caption{Graphical representation of the covariant-evolution
operator for single-photon exchange in the form \eqref{SP} (left) and in the
form \eqref{SPb} with $t_0\rightarrow-\infty$ (right).}
\label{Fig:SingPhot}
\end{figure} with integrations over all variables that do not appear on the left-hand side. Here, $I(x_2,x_1)$ represents the single-photon exchange \begin{equation}
\label{I}
I(x_2,x_1)= e\alpha_1^{\mu} D_{\mathrm{F}\mu\nu}(x_2-x_1)\, e\alpha_2^{\nu}=
\intd{z}\;\mathrm{e}^{-\ensuremath{\mathrm{i}} z(t_2-t_1)}\,\ensuremath{\mathrm{i}} I(z,\boldsymbol{x}_2,\boldsymbol{x}_1) \end{equation} where $D_{\mathrm{F}\mu\nu}(x_2-x_1)$ is the \textit{Feynman photon propagator}.
If we operate to the right on a \textit{positive-energy state}, we can use the relations \eqref{Rel} to simplify the expression. Furthermore, since in that case $t_0\leq t_1, t_2$ and since $t_1, t_2$ run from $-\infty$ to $+\infty$, we must have $t_0=-\infty$, yielding \begin{equation}
\label{SPb}
U^{(1)}(t',-\infty)=\half\hat\psi^{\dag}(x'_1)\hat\psi^{\dag}(x'_2)
\, G_0(x'_1,x'_2;x_1x_2)\,\ensuremath{\mathrm{i}}
I(x_2,x_1)\;\hat\psi(x_{2})\hat\psi(x_{1}) \end{equation}
The electron-field operator is in the interaction picture~\cite{FW71} \begin{equation}
\label{Elfield}
\hat\psi(x)=\hat\psi(t,\boldsymbol{x})=c_j\phi_j(\boldsymbol{x})\,e^{-\ensuremath{\mathrm{i}}\ensuremath{\varepsilon}_j t} \end{equation} with the fourier transform \begin{equation}
\label{ElFT}
\hat\psi(\omega,\boldsymbol{x})=\int\ensuremath{\mathrm{d}} t\,e^{\ensuremath{\mathrm{i}}\omega t}\,\hat\psi(x)
=c_j\phi_j(\boldsymbol{x})\,2\pi\delta(\omega-\ensuremath{\varepsilon}_j) \end{equation} (as usual, summed over repeated indices) and the inverse transform \begin{equation}
\label{ElFTinv}
\hat\psi(x)=\intd{\omega}\,e^{-\ensuremath{\mathrm{i}}\omega t}\,\hat\psi(\omega,\boldsymbol{x}) \end{equation} An arbitrary function of $x_1=(t_1,\boldsymbol{x}_1)$ can be fourier expanded as \begin{equation}
\label{F}
F(x_1)=\intd{\omega_1}\;e^{-\ensuremath{\mathrm{i}} t_1\omega_1}
\,F(\omega_1,\boldsymbol{x}_1) \end{equation} Operating on a (time-independent) fourier component of that function with the electron-field operator \eqref{ElFTinv}, yields \begin{equation}
\label{PsiF}
\hat\psi(x)\,F(\omega_1,\boldsymbol{x}_1)=\intd{\omega}\,e^{-\ensuremath{\mathrm{i}}\omega t}\,\hat\psi(\omega,\boldsymbol{x})
\,F(\omega_1,\boldsymbol{x}_1)=e^{-\ensuremath{\mathrm{i}}\omega_1 t} \,F(\omega_1,\boldsymbol{x}_1) \end{equation} With the adiabatic damping the time-independent component corresponds to $t_1=-\infty$, which implies that the field operator propagates the function $F(\omega_1,\boldsymbol{x}_1)$ from the time $-\infty$ to $t$. Similarly, the product of two electron-field operators $\hat\psi(x_1)\hat\psi(x_2)$ operating on a time-independent two-electron function, propagates the individual electrons from the time $t=-\infty$ to $t=t_1$ and $t=t_2$, respectively, \textit{without} any electron-electron interaction. Thus, \begin{equation}
\label{FT}
\hat\psi(x_1)\hat\psi(x_2)\,F(\omega_1,\omega_2)=
e^{-\ensuremath{\mathrm{i}} t_1\omega_1}\,e^{-\ensuremath{\mathrm{i}} t_2\omega_2}
\,F(\omega_1,\omega_2) \end{equation}
We now use the form \eqref{G0ftint} of the Green's function \begin{equation}
\label{GFSP}
G_0(x_1',x_2';x_1,x_2)=\intd{\epsilon}\;e^{-\ensuremath{\mathrm{i}}\epsilon t'}\,G_0(\epsilon)
=\intd{\epsilon}\;e^{-\ensuremath{\mathrm{i}}\epsilon t'}\,\Gamma(\epsilon)\intd{\omega}\,g_0(\epsilon,\omega)\,
e^{\ensuremath{\mathrm{i}}\omega t_1}\,e^{\ensuremath{\mathrm{i}}(\epsilon-\omega)t_2} \end{equation} to operate with the evolution operator \eqref{SPb} on the fourier component $F(\omega_1,\omega_2)$ (see also Fig. \ref{Fig:SingPhot}, right), which yields \begin{eqnarray}
&&U^{(1)}(t',-\infty)\,F(\omega_1,\omega_2)=
\half c_j^{\dag}\phi_j^{\dag}(\boldsymbol{x}_1)\,e^{\ensuremath{\mathrm{i}} t'\ensuremath{\varepsilon}_j}
\,c_k^{\dag}\phi_k^{\dag}(\boldsymbol{x}_2)\,e^{\ensuremath{\mathrm{i}} t'\ensuremath{\varepsilon}_j} \intd{\epsilon}\,\Gamma(\epsilon)\,e^{-\ensuremath{\mathrm{i}} t'\epsilon}\nonumber \\
&\times& \intd{\omega}\,g_0(\epsilon,\omega)\, \intd{z}\;\ensuremath{\mathrm{i}} I(z)
e^{-\ensuremath{\mathrm{i}} t_1(\omega_1-z-\omega)}\,e^{-\ensuremath{\mathrm{i}} t_2(\omega_2+z-\epsilon+\omega)}\,F(\omega_1,\omega_2) \end{eqnarray} and after time integrations \begin{eqnarray}
\label{SP2}
&&U^{(1)}(t',-\infty)\,F(\omega_1,\omega_2)=
\ket{rs}\bra{rs}\intd{\epsilon}\,e^{-\ensuremath{\mathrm{i}} t'(\epsilon-\ensuremath{\varepsilon}_r-\ensuremath{\varepsilon}_s)}
\Gamma(\epsilon)\intd{\omega}\,g_0(\epsilon,\omega)\,\intd{z}\;\ensuremath{\mathrm{i}} I(z) \nonumber \\
&\times&F(\omega_1,\omega_2)
\;2\pi\delta(\omega_1-z-\omega)\, 2\pi\delta(\omega_1+\omega_2-\epsilon) \end{eqnarray} Here, $\ket{ij}$ represents a straight (non-symmetrized) product of time-independent single-electron functions (which eliminates the factor of $\halfS$).
If we operate on a particular energy component \begin{equation}
\label{FE}
F({\calE})=\intd{\omega_1}\;F(\omega_1,{\calE}-\omega_1) \end{equation} the result becomes in operator form \begin{eqnarray}
\label{SP3}
\hspace{1cm} U^{(1)}(t',-\infty)\,F(\mathcal{E})&=&
e^{-\ensuremath{\mathrm{i}} t'(\mathcal{E}-H_0)}\Gamma({\calE})\intd{\omega}\,g_0(\mathcal{E},\omega)\nonumber \\
\hspace{1cm}&\times&\intd{\omega_1}\;\ensuremath{\mathrm{i}}
I(\omega_1-\omega)\,F(\omega_1,\mathcal{E}-\omega_1) \end{eqnarray} We can also express this result as \begin{equation}
\label{SPc}
U^{(1)}(t,-\infty)\,F({\calE})=e^{-\ensuremath{\mathrm{i}} t({\calE}-H_0)}\,\Gamma({\calE})\,V({\calE})\,F({\calE}) \end{equation} where \begin{equation}
\label{VE}
V(\mathcal{E})=\intd{\omega}\;g_0({\calE},\omega)
\intd{\omega_1}\;\ensuremath{\mathrm{i}} I(\omega_1-\omega) \end{equation} Here, $\Gamma({\calE})$ is the resolvent \eqref{Gamma} and $g_0(\mathcal{E},\omega)$ is the operator \eqref{g0}. The corresponding effective interaction is obtained from the relation \eqref{EffInt} by taking the time derivative at $t=0$, which eliminates the resolvent, \begin{equation}
H_{\mathrm{eff}}^{(1)}({\calE})=PV({\calE})\,P \end{equation}
With the explicit form of the interaction \eqref{I} the matrix elements of the potential for the exchange of a single covariant photon becomes~\cite[App. A]{LSA04} \begin{equation}
\label{VExpl}
\bra{rs}V(\mathcal{E})\ket{tu}=\Bigbra{rs}\int f(k)\ensuremath{\mathrm{d}} k\Big[\frac{1}
{{\calE}-\ensuremath{\varepsilon}_r-\ensuremath{\varepsilon}_u-(k-\ensuremath{\mathrm{i}}\gamma)_r}
+\frac{1}{{\calE}-\ensuremath{\varepsilon}_s-\ensuremath{\varepsilon}_t-(k-\ensuremath{\mathrm{i}}\gamma)_s}\Big]\Bigket{tu} \end{equation} where the $A_r=A\;\mathrm{sgn}(\ensuremath{\varepsilon}_r)$. The function $f(k)$ is in the Feynman gauge given by~\cite[Eq. 77]{LSA04} \begin{eqnarray}
\label{fk}
f(k)&=&-\frac{e^2}{4\pi^2}\,(1-\boldsymbol{\alpha_1\cdot\alpha_2})\,
\frac{\sin(kr_{12})}{r_{12}}\nonumber \\
&=&-\frac{e^2}{4\pi^2}\,(1-\boldsymbol{\alpha_1\cdot\alpha_2})\,
\sum_{l=0}^\infty (2l+1)j_l(kr_1)j_l(kr_2)\,\boldsymbol{C}^{(k)}(1)\bs{\cdot}
\boldsymbol{C}^{(k)}(2) \end{eqnarray} where $j_l$ are spherical Bessel functions and $\boldsymbol{C}^{(k)}$ spherical tensors, closely related to the spherical harmonics.
Summarizing, the contribution to the wave operator \eqref{wf} from single-photon exchange, when operating to the right on a function of the type \eqref{FE}, becomes \begin{equation}
\label{Om1}
\boxed{\Omega^{(1)}(\mathcal{E})=Q\widetilde{U}^{(1)}(0,-\infty)= \Gamma_Q(\mathcal{E}) V(\mathcal{E})} \end{equation} where \begin{equation}
\label{GammaQ}
\Gamma_Q({\calE})=Q\Gamma({\calE})=\frac{Q}{{\calE}-H_0} \end{equation}
and the corresponding contribution to the effective Hamiltonian \begin{equation}
\label{Heffett}
\boxed{H_{\mathrm{eff}}^{(1)}({\calE})=PV({\calE})\,P} \end{equation}
\newcommand{\delta \calE}{\delta \mathcal{E}}
\subsection{Separable two-photon exchange}
\label{App:SepPh}
(See Ref.~\cite[Sect.5.2.1 and App.A.2]{LSA04})\\ Next we consider the separable two-photon exchange for which there is an intermediate time $t=t"$ with no free or uncontracted photons, as illustrated in Fig. \ref{Fig:SepPh} (left). The evolution operator \eqref{EvolOp} can in this case be expressed \begin{equation}
\label{U2a}
U^{(2)}(t,-\infty)P=U^{(1)}(t,t")\,U^{(1)}(t",-\infty)P \end{equation} Here, the intermediate states run over \textit{all} states --- in the $Q$ as well as the $P$ space --- and when the intermediate state lies in the model space ($P$), \textit{(quasi)singularities} may occur. These singularities are removed in the \textit{reduced evolution operator} \eqref{Ured} by including \textit{counterterms} \eqref{C} \begin{equation}
\label{U+C}
\widetilde{U}(t)P=U(t)P+C(t)P \end{equation} We also recall the definition of the \textit{model-space contribution} (MSC) \eqref{Mdef} \begin{equation}
\label{M}
\widetilde{U}(0)P=\bar{U}(0)P+MP \end{equation} where $\bar{U}$ is the evolution operator \eqref{Ubar} with no intermediate model-space state, in this case \begin{equation}
\label{Ubar2}
\bar{U}^{(2)} P=\Gamma V\Gamma_Q VP \end{equation} \begin{figure}
\caption{Graphical representation of the separable
two-photon-photon ladder diagram (left). This diagram is
separable, if there exists a time (represented by the
dotted line) at which there is no uncontracted photon, i.e., a time after the
first photon has been absorbed and before the second has been created.
The corresponding counterterm (right) is a product of two operators,
which evolve independently from
possibly different states of the model space.}
\label{Fig:SepPh}
\end{figure} The counterterm \eqref{C} is in the present case given by the product of two single-photon contributions, as shown in Fig. \ref{Fig:SepPh} (right) \begin{equation}
\label{C1}
C^{(2)}(t)=-U^{(1)}(t,-\infty)\bs{\cdot} PU^{(1)}(0,-\infty)\,P=-U^{(1)}(t)\bs{\cdot}
PU^{(1)}(0) \end{equation} using the notation introduced in subsection \ref{sec:MSC}. The two factors evolve independently from (possibly different) states in the model space, which is indicated by the "dot". The counterterm eliminates the singularity, but there may be a \textit{finite remainder}, which we refer to as the model-space contribution (MSC) \eqref{M}. We shall first consider this part.
We assume that we operate to the far right on a function of the type \eqref{FE} of energy ${\calE}$, and that the intermediate model-space state has the energy ${\calE}'$. Using the first-order result \eqref{SPc}, we can express the second-order evolution operator \eqref{U2a} as \begin{equation}
\label{U2}
U^{(2)}(t)P=\mathrm{e}^{-\ensuremath{\mathrm{i}} t({\calE}-H_0)}\,U^{(1)}(0,{\calE})\,U^{(1)}(0,{\calE})P \end{equation} and the counterterm---with the first factor evolving from the intermediate state---as \begin{equation}
\label{C2}
C^{(2)}(t)=-\mathrm{e}^{-\ensuremath{\mathrm{i}} t({\calE}'-H_0)}\,U^{(1)}(0,{\calE}')\bs{\cdot} PU^{(1)}(0,{\calE})P \end{equation} We note here that the time derivative for $U$ as well as $C$ eliminates the denominator of the leftmost resolvent.
The MSC now becomes \begin{equation}
\label{MSC}
MP=\Big(U^{(1)}(0,{\calE})-U^{(1)}(0,{\calE}')\Big)\bs{\cdot}
PU^{(1)}(0,{\calE})P \end{equation} Using the result \eqref{dots}, the last factor is
\[PU^{(1)}(0,{\calE})P=\frac{P}{{\calE}-{\calE}'}\,V({\calE})P=-\frac{1}{\delta \calE}*PV^{(1)}({\calE})P
=-\frac{1}{\delta \calE}*P\dot{U}^{(1)}({\calE})P\] with $\delta \calE={\calE}'-{\calE}$, and with $\;\delta{U^{(1)}}=U^{(1)}(0,{\calE}')-U^{(1)}(0,{\calE})$ we have \begin{equation}
\label{MP1}
MP=\Big(U^{(1)}(0,{\calE})-U^{(1)}(0,{\calE}')\Big)\bs{\cdot} PU^{(1)}(0,{\calE})P
=\partdelta{U^{(1)}}{{\calE}}*P\dot{U}^{(1)} P \end{equation} (The asterisk is used only for clarity. It notifies the position of a "fold" in the graphical representation~\cite{LM86}, but has no other special significance. It will mainly serve as a reminder of the position of a cancelled singularity, which---as we shall see---requires certain precautions.) With the definition \eqref{Hbar} the MSC can be expressed \begin{equation}
\label{MP}
MP=\partdelta{U^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)} \end{equation} (Note that $\bar{U}^{(1)}=U^{(1)}$ and $\bar{H}_{\mathrm{eff}}^{(1)}=H_{\mathrm{eff}}^{(1)}$.) The complete second-order reduced evolution operator \eqref{M} then becomes \begin{equation}
\label{MSC1}
\widetilde{U}^{(2)}(0)P=\bar{U}^{(2)}(0)P+\partdelta{U^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)} \end{equation}
The result above is exact also for the quasi-degenerate case. The difference ratio can be expanded as discussed in Appendix \ref{App:Exp} \begin{equation}
\label{MSC2}
\partdelta{U^{(1)}}{{\calE}}=\partder{U^{(1)}}{{\calE}}+\half\ppartder{U^{(1)}}{{\calE}}\,\delta \calE
+\frac{1}{3!}\pppartder{U^{(1)}}{{\calE}}\,\delta \calE^2+\cdots \end{equation} which in the limit of complete degeneracy yields \begin{equation}
\label{MSC2A}
\widetilde{U}^{(2)}(0)P=\bar{U}^{(2)}(0)P+\partder{U^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)} \end{equation} The second-order contribution to the wave operator \eqref{WaveOp} then becomes \begin{equation}
\label{Om2}
\boxed{\Omega^{(2)} P=Q\widetilde{U}^{(2)}(0)P
=\bar{\Omega}^{(2)} P+\partdelta{\Omega^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}
\Rightarrow\bar{\Omega}^{(2)} P+\partder{\Omega^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}} \end{equation} where $\bar{\Omega}$ is the wave operator \eqref{Ombar} without intermediate model space states.
The second-order contribution to the effective interaction is obtained by means of the relation \eqref{EffInt}. Since the expression \eqref{Om2} is valid only for $t=0$, it can not be used to evaluate the time derivative. Instead, we have to use the original definition \eqref{U+C}, and using the expressions \eqref{U2} and \eqref{C2}, we find \begin{equation}
\label{Heff22}
H_{\mathrm{eff}}^{(2)}= PV({\calE})\,\Gamma({\calE})\,VP-PV({\calE}')\,\Gamma_P({\calE})\,V({\calE})P
=\bar{H}^{(2)}_{\mathrm{eff}}+\partdelta{H_{\mathrm{eff}}^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)} \end{equation} where $\bar{H}^{(2)}_{\mathrm{eff}}=PV\Gamma_Q VP$ is the $H-bar$ operator \eqref{Hbar} with no intermediate model-space states. The last term is by definition the model-space contribution, which appears in this order only for energy-dependent interactions. In the case of complete degeneracy the difference ratio tends to the derivative, as before, \begin{equation}
\label{Heff2}
\boxed{H_{\mathrm{eff}}^{(2)}= \bar{H}_{\mathrm{eff}}^{(2)}+\partdelta{H_{\mathrm{eff}}^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}
\Rightarrow \bar{H}_{\mathrm{eff}}^{(2)}+\partder{H_{\mathrm{eff}}^{(1)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}} \end{equation}
\subsection{Separable three-photon exchange}
\label{App:SepThree} The treatment of the exchange of three separable covariant photons is quite analogous to the previous case. From the expansion \eqref{OmRec3} we have \begin{eqnarray}
\label{U3}
\widetilde{U}^{(3)} P&=&\bar{U}^{(3)} P
+\big(\bar{U}^{(2)} P\bar{U}^{(1)} P-\bar{U}^{(2)} P\bs{\cdot}\bar{U}^{(1)} P\big)
+\big(\bar{U}^{(1)} P\bar{U}^{(2)} P-\bar{U}^{(1)} P\bs{\cdot}\bar{U}^{(2)} P\big)\nonumber \\
&+&\big(\bar{U}^{(1)} P\bar{U}^{(1)} P-\bar{U}^{(1)} P\bs{\cdot}\bar{U}^{(1)} P\big)
\big(P\bar{U}^{(1)} P-\bs{\cdot} P\bar{U}^{(1)} P\big) \end{eqnarray} By generalizing the result of the preceding Appendix we obtain the relation \begin{equation}
\label{GenDiff}
\boxed{AP\;BP-AP\bs{\cdot} B P=\partdelta{A}{{\calE}}*\dot{B}} \end{equation} where $A$ is an arbitrary operator and $B$ can be $U$, $\bar{U}$ or $\widetilde{U}$. Using this relation, the second and third terms above become \begin{eqnarray}
\label{U2PU}
\hspace{1cm}\big(\bar{U}^{(2)} P\bar{U}^{(1)} P-\bar{U}^{(2)}\bs{\cdot} P\bar{U}^{(1)}
P\big)=\partdelta{\bar{U}^{(2)}}{{\calE}}*P\dot{\Ubar}^{(1)} P
=\partdelta{\bar{U}^{(2)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(1)}\\
\label{UPU2}
\hspace{1cm}\big(\bar{U}^{(1)} P\bar{U}^{(2)} P-\bar{U}^{(1)} P\bs{\cdot}\bar{U}^{(2)} P\big)
=\partdelta{\bar{U}^{(1)}}{{\calE}}*P\dot{\Ubar}^{(2)} P
=\partdelta{\bar{U}^{(1)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(2)} \end{eqnarray} In the last term in Eq. \eqref{U3} we have to apply the rule \eqref{GenDiff} twice, yielding \begin{eqnarray}
\label{dUH}
&&\big(\bar{U}^{(1)} P\bar{U}^{(1)} P-\bar{U}^{(1)} P\bs{\cdot}\bar{U}^{(1)} P\big)
\big(P\bar{U}^{(1)} P-\bs{\cdot} P\bar{U}^{(1)} P\big)\nonumber \\
&=&\partdelta{\bar{U}^{(1)}}{{\calE}}*P\dot{\Ubar}^{(1)} P
\;\big(P\bar{U}^{(1)} P-\bs{\cdot} P\bar{U}^{(1)} P\big)
=\partdelta{}{{\calE}}\Big(\partdelta{\bar{U}^{(1)}}{{\calE}}*P\dot{\Ubar}^{(1)} P\Big)
*P\dot{\Ubar}^{(1)} P\nonumber \\
&=&\partdelta{}{{\calE}}\Big(\partdelta{\bar{U}^{(1)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(1)}\Big)
*\bar{H}_{\mathrm{eff}}^{(1)}
\end{eqnarray} From the previous Appendix (Eq. \ref{MSC2}) we have \begin{equation}
\label{U2dif}
\partdelta{\widetilde{U}^{(2)}}{{\calE}}=\partdelta{\bar{U}^{(2)}}{{\calE}}
+\partdelta{}{{\calE}}\Big(\partdelta{\bar{U}^{(1)}}{{\calE}}*\bar{H}^{(1)}_{\mathrm{eff}}\Big) \end{equation} and the complete result then becomes \begin{equation}
\label{U3B}
\widetilde{U}^{(3)} P=\bar{U}^{(3)} P+\partdelta{\widetilde{U}^{(2)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}
+\partdelta{\bar{U}^{(1)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(2)} \end{equation} This is an exact expression in this order, also for a quasi-degenerate model space. In the case of complete degeneracy this becomes \begin{equation}
\label{U3deg}
\widetilde{U}^{(3)} P=\bar{U}^{(3)} P+\partder{\widetilde{U}^{(2)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}
+\partder{\bar{U}^{(1)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(2)} \end{equation} In terms of the $\Omega$ operators the results above then become \begin{equation}
\label{Om3C}
\boxed{\Omega^{(3)} P=\bar{\Omega}^{(3)} P+\partdelta{\Omega^{(2)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(1)}
+\partdelta{\Omega^{(1)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(2)}} \end{equation}
In order to obtain the third-order effective interaction, we consider the time derivative of the relation \eqref{U3} (only the first factor is time dependent). This yields \begin{equation}
H_{\mathrm{eff}}^{(3)}=\bar{H}_{\mathrm{eff}}^{(3)}+\partdelta{\bar{H}_{\mathrm{eff}}^{(1)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(2)}
+\partdelta{\bar{H}_{\mathrm{eff}}^{(2)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}
+\partdelta{}{{\calE}}\Big(\partdelta{H^{(1)}_{\mathrm{eff}}}{{\calE}}*H^{(1)}_{\mathrm{eff}}\Big)*H^{(1)}_{\mathrm{eff}} \end{equation} which using the relation \eqref{Heff2} can be expressed \begin{equation}
\label{Heff3}
\boxed{H_{\mathrm{eff}}^{(3)}=\bar{H}_{\mathrm{eff}}^{(3)}+\partdelta{H_{\mathrm{eff}}^{(1)}}{{\calE}}*\bar{H}_{\mathrm{eff}}^{(2)}
+\partdelta{H_{\mathrm{eff}}^{(2)}}{{\calE}}*H_{\mathrm{eff}}^{(1)}} \end{equation}
\subsection{Expansions} \label{App:Exp} We have seen above that when there are multiple singularities, it is important to take the difference ratios \textit{before} the singularities are removed. We shall illustrate this here by a simple mathematical example.
We consider a function $f(x)$ of the variable $x$. We define the first-order difference ratio \begin{equation}
\label{Deltaf}
\partdelta{f}{x}=\partdelta{_{x_0,x}f}{x}=\frac{f(x)-f(x_0)}{x-x_0} \end{equation} which can be expanded in a Taylor series \begin{equation}
\label{DeltafB}
\partdelta{f}{x}=\partdelta{_{x_0,x}f}{x}=f'(x_0)+\half f"(x_0)(x-x_0)
+\frac{1}{3!} f"'(x_0)(x-x_0)^2+\frac{1}{4!} f^{IV}(x_0)(x-x_0)^3+\cdots \end{equation} where \begin{equation}
\label{f'}
f'(x_0)=\deriv{f}{x}\Big|_{x=x_0} \end{equation} etc.
Similarly, we define the second-order difference ratio \begin{eqnarray}
\label{Deltaf2}
\ppartdelta{f}{x}&=&\partdelta{_{x'x}}{x}\partdelta{_{x_0,x}f}{x}
=\frac{\partdelta{_{x_0,x'}f}{x}-\partdelta{_{x_0,x}f}{x}}{x'-x}
=\half f"(x_0)+\frac{1}{3!} f"'(x_0)(x+x'-2x_0)\nonumber \\
&+&\frac{1}{4!} f^{IV}(x_0)\big[(x'-x_0)^2+(x'-x_0)(x-x_0)+(x-x_0)^2\big]+\cdots \end{eqnarray} the third-order difference ratio \begin{equation}
\label{Deltaf3}
\pppartdelta{f}{x}=\partdelta{_{x"x'}}{x}\partdelta{_{x'x}}{x}\partdelta{_{x_0,x}f}{x}
=\frac{1}{3!} f"'(x_0)+\frac{1}{4!} f^{IV}(x_0)(x+x'+x"-3x_0)+\cdots \end{equation} the fourth-order difference ratio \begin{equation}
\label{Deltaf4}
\frac{\delta^4f}{\delta^4x}=\frac{1}{4!} f^{IV}(x_0)+\cdots \end{equation} and so on.
Generalizing these results, we have in the limit, when the differences tend to zero \begin{equation}
\label{Dif-Der}
\boxed{\partdelta{^nf}{^nx}\Rightarrow\frac{1}{n!}\frac{\ensuremath{\mathrm{d}}^nf}{\ensuremath{\mathrm{d}}^nx}} \end{equation} This relation is frequently used in the present paper.
\input{Bethe-SalpeterCJP.bbl}
\end{document} | arXiv | {
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\begin{document}
\preprint{APS/123-QED}
\title{Quantum Entanglement Induced by Gravitational Potential}
\author{J. Michael Yang} \email{jianhao.yang@alumni.utoronto.ca} \affiliation{ Qualcomm, San Diego, CA 92121, USA }
\date{\today}
\begin{abstract} \textbf{Abstract:} The purpose of this study is to calculate the entanglement measure for a bipartite system where the two subsystems interact via a central potential, and more importantly, to analyze the conceptual implication in the case of gravitational potential. Through numerical calculation, we confirm that the ground state of such quantum system is an entangled state. This raises a question whether a quantum ground state describing masses interacting through gravity can ever be a separable state. Entanglement seems to be an intrinsic characteristic for the ground state when mutual gravitational interaction is involved. Although the study is in the context of non-relativistic quantum mechanics, it provides hints for a quantum gravity theory in the limit of very weak field and low relative velocity. The entanglement of the ground state of two masses seems intrinsically connecting to the curvature of spacetime they create.
\begin{description} \item[Keywords] Quantum Entanglement, Purity, Gravitational Interaction, Geometry of Spacetime \end{description} \end{abstract}
\maketitle
\section{Introduction} \label{intro} Quantum entanglement has been a source of theoretical interest in probing the foundation of quantum mechanics. For instance, it was used in EPR thought experiment to argue that quantum mechanics is an incomplete physical theory~\cite{EPR}. Quantum entanglement is considered a more fundamental property in some of the quantum mechanics interpretations such as decoherence theory~\cite{Zurek82, Zurek03} and relational quantum mechanics~\cite{Rovelli96, Rovelli07, Yang2017, Yang2017_2}. In recent decades, it is recognized that entanglement is connecting to the gravitational dynamics in the context of holography. Holography in high energy physics refers to the duality, or more specifically, the gauge/gravity or AdS/CFT correspondence~\cite{Maldacena,Ryu,Raamsdonk}. Entanglement as an unique communication resource has been widely explored in recent communication technology~\cite{Nielsen00,Hayashi15}.
In this paper we wish to study the entanglement of two quantum systems interact through central potential, and particularly, the conceptual implication of such entanglement properties. Although the entanglement properties of such bipartite systems have been studied before in literature~\cite{Cohen, Tommasini, Marcelo}, there are important conceptual implications that are not fully recognized. Here we briefly discuss one possible implication.
Although the idea that entanglement is related the emergence of spacetime is inspiring, the connection is indirect via the AdS/CFT correspondence. In this setting, the emergence of geometry occurs in a $d+1$ dimensional AdS with gravity, while the entanglement is between a spatial region in a $d$ dimensional boundary of AdS and the rest of the boundary without gravity. It is natural to ask whether there is direct connection between entanglement and the emergence of spacetime geometry without the need of the holographic correspondence. Furthermore, one may also wonder if the connection in the holographic context is a unique result from the string theory, or it is intrinsic to any plausible quantum gravity theory. Unfortunately, these questions cannot be easily answered as we do not have a truly unified quantum gravity theory. Practically, we can take one step back and ask a simpler question: can the traditional quantum theory provide similar hints on the connection between entanglement and gravity? To probe the answer to this question, one can investigate whether entanglement between two massive objects can be induced through interaction of classical gravity. The classical result can be seen as a constraint for the unified theory in the limit of very weak field and low relative velocity.
There are other possible implications we can draw from the study of entanglement of such bipartite system. For instance, is the entanglement a property depending on the reference frame, or it is invariant when switching reference frames? Given the ubiquitous of Coulomb interaction and gravity interaction, is entanglement a ubiquitous quantum phenomenon?
With these motivations, we first develop a generic formulation to calculate the reduced density matrix of a bipartite system with continuous variable. This allows us to derive an entanglement measure based on purity of the reduced density matrix. The formulation is then applied to calculate the entanglement measure of the ground state of a bipartite system where the two subsystems interact through a central potential. The numerical results show that the ground state of the two masses is an entangled state, so as the first excited state.
Although our calculation appears straightforward and confirms result from earlier studies~\cite{Tommasini}~\footnote{We would like to point out that the calculation of entanglement measure in Ref.~\cite{Tommasini} is formulated only for a particular class of states that are translationally invariant. To the contrary, the formulation to calculated the entanglement measure is generic for any bipartite pure state including those states without translational invariance.}, our primary goal is to analyze the conceptual implication. First it raises an interesting question whether a quantum ground state describing masses interacting through gravity can ever be a separable state. Entanglement seems to be an intrinsic characteristic for the ground state when mutual gravitational interaction is involved. It suggests that the entanglement manifests the curvature of spacetime if we take the general relativity perspective. It can be seen as a check point for a unified theory in the limit of very weak field and low relative velocity. However, we should point out that this paper does not yet give a definite answer on how entanglement is connected to the geometric properties of spacetime, as the question can only be answered in a unified quantum gravity theory. Instead, this paper is intended to provide hints for the pursuit of answering such question. Other implications, such as the dependency of entanglement on reference frame can also be confirmed from our calculation.
The paper is organized as following. In Section \ref{sec:entmeasusre} we derive a generic formulation of entanglement measure for a bipartite system with continuous variable. The formulation is then applied to the ground state derived from the Schr\"{o}dinger equation with gravity potential in Section \ref{sec:centralmass} to obtain explicit expression of entanglement measure. The numerical calculations shown in Section \ref{sec:montecarlo} clearly show that the ground state is an entangled state. Section \ref{sec:Interferometers} is dedicated to discussing the similarity and difference between the results in Refs~\cite{Bose, Marletto} and the results in this paper. Section \ref{sec:discussion} explores the conceptual implications of our calculation results. Limitations and conclusive remarks are presented in Section \ref{sec:limit} and \ref{sec:conclusion}, respectively.
\section{Entanglement Measure for Continuous Variable System} \label{sec:entmeasusre} For a continuous variable bipartite system with subsystem 1 and 2, a pure state can be expressed as \begin{equation} \label{WF1}
|\Psi_{12}\rangle=\int \psi(x,y)|x\rangle|y\rangle dxdy. \end{equation} Here we assume one dimensional continuous variable $x, y$ for each subsystem 1 and 2, respectively. It is straightforward to extend to three dimensional variables. Normalization requires \begin{equation} \label{Normalization} \int \psi(x,y)\psi^*(x,y) dxdy = 1. \end{equation} Rewriting (\ref{WF1}) in the form of density matrix, we have \begin{equation} \label{rho1} \begin{split}
\hat{\rho}_{12} & = |\Psi_{12}\rangle\langle_{12}\Psi| \\
& =\int\int \psi(x,y)\psi^*(x', y')|x\rangle|y\rangle\langle x'|\langle y'| dxdydx'dy'. \end{split} \end{equation} This gives the density matrix element \begin{equation} \label{rho2} \rho_{12}(x,y; x',y') =\psi(x,y)\psi^*(x', y'). \end{equation} The reduced density matrix for subsystem 1 can be derived by taking partial trace, \begin{equation} \label{rho3} \begin{split}
\hat{\rho}_{1} & = Tr_2(\hat{\rho}_{12}) = \int \textsubscript{2}\langle z|\Psi_{12}\rangle\langle_{12}\Psi|z\rangle_2 dz \\
& =\int \{\int \psi(x,z)\psi^*(x', z)dz \}|x\rangle\langle x'| dxdx'. \end{split} \end{equation} Here, the integration over variable $z$ is on subsystem 2. From (\ref{rho3}), one obtains the reduced density matrix element for subsystem 1, \begin{equation} \label{rho4} \rho_{1}(x; x') =\int \psi(x,z)\psi^*(x', z)dz. \end{equation} Due to the normalization property in (\ref{Normalization}), we have \begin{equation} \label{Trace1} \begin{split} Tr(\hat{\rho}_{1}) & =\int \rho_{1}(x; x)dx \\ & = \int \psi(x,z)\psi^*(x, z)dzdx = 1, \end{split} \end{equation} as expected. To quantify the entanglement between subsystem 1 and 2, we use the following definition~\cite{Calmet} \begin{equation}
\label{ent}
E = 1 - Tr(\hat{\rho}_1^2). \end{equation} Quantity $Tr(\hat{\rho}_1^2)$ is the purity of reduced density matrix $\rho_1$, given by \begin{equation} \label{purity} \begin{split} Tr(\hat{\rho}_1^2) &=\int \rho_1^2(x,x)dx \\ &= \int \rho_1(x, x')\rho_1(x', x)dx'dx. \end{split} \end{equation} Appendix A shows the justification on why $E$ can be considered as an entanglement measure. Substitute (\ref{rho4}) into (\ref{purity}) and replace variable $z$ with $y$, we have \begin{equation}
\label{purity2}
\begin{split}
&Tr(\hat{\rho}_1^2) = \\ &\int \psi(x,y)\psi^*(x', y)\psi(x',y')\psi^*(x, y') dydy'dx'dx.
\end{split} \end{equation} Thus, given a continuous variable wave function of a bipartite system, $\psi(x,y)$, we can calculate the entanglement measure from (\ref{ent}) and (\ref{purity2}). Suppose that the wave function can be factorized as $\psi(x,y)=\phi(x)\varphi(y)$, \begin{equation}
\label{purity3}
Tr(\hat{\rho}_1^2) = (\int |\phi(x)|^2 dx\int |\varphi(y)|^2dy)^2 = 1. \end{equation} Thus, $E=0$, there is no entanglement between the two subsystems. However, when $\psi(x,y) \ne \phi(x)\varphi(y)$, it is not obvious whether $E\ne 0$. A detailed calculation is needed.
We can extend (\ref{purity2}) to three-dimensional system with continuous variable. Suppose the continuous variables for subsystem 1 and 2 are $\vec{r}_1$ and $\vec{r}_2$, respectively, then \begin{equation} \label{purity4} \begin{split}
Tr(\hat{\rho}_1^2) &= \int \psi(\vec{r}_1,\vec{r}_2)\psi^*(\vec{r'}_1, \vec{r}_2)\psi(\vec{r'}_1,\vec{r'}_2) \\
& \times \psi^*(\vec{r}_1, \vec{r'}_2) d\vec{r'}_2d\vec{r}_2d\vec{r'}_1d\vec{r}_1, \end{split} \end{equation} where $d\vec{r}:=dxdydz$. Note that (\ref{purity4}) is a twelve dimensional integration.
\section{Entanglement of Two Masses Interacting With Classical Gravity} \label{sec:centralmass}
Consider a three-dimensional bipartite system, and the interaction between two subsystems is described as a potential only depends on the distance between the two subsystems $r_{12} = |\vec{r}_1-\vec{r}_2|$. The stationary Schr\"{o}dinger Equation is given by \begin{equation}
\label{SE}
[-\frac{\hbar^2}{2m_1}\nabla^2_1 -\frac{\hbar^2}{2m_2}\nabla^2_2 + V(r_{12})]\psi(\vec{r}_1,\vec{r}_2) = E_t\psi(\vec{r}_1,\vec{r}_2), \end{equation} where $\nabla^2$ is the Laplacian operator in the three-dimensional coordinator, $E_t$ is the total energy of the system, $m_1$ and $m_2$ are the masses of subsystem 1 and 2, respectively. The potential can be a Coulomb potential or a gravity potential. The formulation is constructed here in the context of traditional quantum mechanics. Nevertheless, this can be considered as a first order approximation of more general formulation. The question we want to answer is that given a solution of wave function from (\ref{SE}), whether the two subsystems are in entangled state. In particular, we are interested in the ground state.
A typical way to solve (\ref{SE}) is to introduce transformation ~\cite{Messiah} \begin{equation}
\label{transform}
\begin{split}
\vec{r}_{12} &= \vec{r}_1-\vec{r}_2, \\
\vec{R}_{12} & = \frac{m_1}{m_1+m_2}\vec{r}_1 + \frac{m_2}{m_1+m_2}\vec{r}_2,
\end{split} \end{equation} where $\vec{R}_{12}$ is the center-of-mass coordinate of the system. Omit the subscription ``12" and denote \begin{equation}
\label{transform2}
\psi(\vec{r}_1,\vec{r}_2)=\phi(\vec{R})\varphi(\vec{r}), \end{equation} (\ref{SE}) is separated into two equations , \begin{equation}
\label{SE2}
-\frac{\hbar^2}{2M}\nabla^2_R \phi(\vec{R}) = E_c\phi(\vec{R}), \end{equation} \begin{equation}
\label{SE3}
[-\frac{\hbar^2}{2\mu}\nabla^2_r + V(r)]\varphi(\vec{r}) = E_r\varphi(\vec{r}), \end{equation}
where $M=m_1+m_2$ is the total mass, $\mu=m_1m_2/(m_1+m_2)$ is the effective mass, $E_c$ is the kinetic energy of the center mass, $r=|\vec{r}_{12}|$, and $E_r=E_t-E_c$.
Eq.(\ref{SE2}) corresponds to the Schr\"{o}dinger equation of a free particle. Suppose the center mass of the bipartite system is moving with a constant momentum $\vec{P}_c$, and the system is in a three-dimensional spatial box with length $L$, each of the dimensional variable $x,y,z\in \{-L/2, L/2\}$. The wave function $\phi(\vec{R})$ can be expressed as \begin{equation}
\label{WF2}
\phi(\vec{R}) = \sqrt{\frac{1}{L^3}}e^{i\vec{P}_c\cdot\vec{R}/\hbar}. \end{equation} with the understanding of taking the limitation $L\to\infty$ after integration in subsequent calculation. Substitute this into (\ref{transform2}) and then into (\ref{purity4}), the purity of reduced density matrix is simplified as \begin{equation} \label{purity5} \begin{split}
Tr(\hat{\rho}_1^2) = & \lim_{L\to\infty}(\frac{1}{L})^6\int_{-L/2}^{L/2} |\int_{-L/2}^{L/2}\varphi(\vec{r}_{12}) \\
&\times \varphi^*(\vec{r}_{12'})d\vec{r}_1|^2 d\vec{r'}_2d\vec{r}_2. \end{split} \end{equation} Once the wave function for the relative movement between the two subsystems, $\varphi(\vec{r}_{12})$, is solved from (\ref{SE3}), one can compute the entanglement $E$ from (\ref{purity5}) and (\ref{ent}).
The solution of (\ref{SE3}) depends on the actual form of central potential energy $V(r_{12})$ where $r_{12}$ is the relative distance between the two subsystem. For gravitational potential energy, \begin{equation}
\label{Gravity}
V(r_{12}) = -G\frac{m_1m_2}{r_{12}}, \end{equation} where $G$ is the Newtonian constant of gravitation. Another example is the Coulomb potential energy of hydrogen atom, given by \begin{equation}
\label{Coulumb}
V(r_{12}) = -\frac{e^2}{r_{12}}. \end{equation} (\ref{SE3}) with such potential energies can be solved analytically. In particular, we are interested the solution for the ground state. The wave function for the ground state is given by~\cite{Saxon} \begin{equation}
\label{GroundState}
\varphi_g(\vec{r}_{12}) = \sqrt{\frac{1}{\pi a^3}}e^{-r_{12}/a}. \end{equation} Here, constant $a$ in the case of Coulomb potential energy given by (\ref{Coulumb}) is $\hbar^2/(\mu e^2) = 0.83\times 10^{-8}cm$, which is the famous \textit{Bohr Radius}. In the case of gravity potential energy given by (\ref{Gravity}), \begin{equation}
\label{GravityRadius}
a = \frac{\hbar^2}{G\mu m_1m_2} = \frac{\hbar^2}{G m_1^2m_2}(1+\frac{m_1}{m_2}), \end{equation} which is called the \textit{Gravitational Bohr Radius}. We will discuss the practical meaning of this constant in section \ref{sec:limit}. For the time being, just consider it as a constant in the solution for the relative wave function $\varphi(\vec{r}_{12})$.
Substituting (\ref{GroundState}) into (\ref{purity5}), after some algebra, we obtain \begin{equation} \label{purity6} \begin{split}
Tr(\hat{\rho}_1^2) = & \lim_{L\to\infty}\frac{1}{\pi^2a^6L^6}\int_{-L/2}^{L/2} d\vec{r'}_1d\vec{r}_1 d\vec{r'}_2d\vec{r}_2 \\
&\times e^{-2(r_{12}+r_{1'2}+r_{12'}+r_{1'2'})/a}. \end{split} \end{equation} Replacing the position variable $\vec{r}$ with non-dimensional variable $\vec{\gamma} = 2\vec{r}/L$, for $\vec{r'}_1,\vec{r}_1, \vec{r'}_2, \vec{r}_2$, and denoting $\alpha=L/a$, we rewrite (\ref{purity6}) as \begin{equation} \label{purity7} \begin{split}
Tr(\hat{\rho}_1^2) = & \lim_{\alpha\to\infty}\frac{1}{\pi^2}(\frac{\alpha}{4})^6\int_{-1}^{1} d\vec{\gamma'}_1d\vec{\gamma}_1 d\vec{\gamma'}_2d\vec{\gamma}_2 \\
&\times e^{-\alpha(\gamma_{12}+\gamma_{1'2}+\gamma_{12'}+\gamma_{1'2'})}. \end{split} \end{equation} Explicitly written in Cartesian coordinate, variable $\gamma_{12}=2\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2+(z_1 - z_2)^2}/L$. Similar expressions can be written down for $\gamma_{1'2}$, $\gamma_{12'}$, and $\gamma_{1'2'}$. $d\vec{\gamma}_1=(2/L)^3 dx_1dy_1dz_1$, and similar expressions for $d\vec{\gamma'}_1$, $d\vec{\gamma'}_2$, and $d\vec{\gamma}_2$. We can also compute the entanglement measure for an excited state. The relative wave function for the first spherical symmetry excited state is given by~\cite{Saxon} \begin{equation}
\label{excitedState}
\varphi_e(\vec{r}_{12}) = \sqrt{\frac{1}{8\pi a^3}}(1-\frac{r_{12}}{2a})e^{-r_{12}/2a}. \end{equation} Substituting this into (\ref{purity5}), using the same notations as for (\ref{purity7}), and denoting $\beta=L/(4a)=\alpha/4$, we obtain \begin{equation} \label{purity8} \begin{split}
Tr(\hat{\rho}_1^2)_e = & \lim_{\beta\to\infty}\frac{1}{\pi^2}(\frac{\beta}{2})^6\int_{-1}^{1} d\vec{\gamma'}_1d\vec{\gamma}_1 d\vec{\gamma'}_2d\vec{\gamma}_2 \\
&\times (1-\beta\gamma_{12})(1-\beta\gamma_{1'2})(1-\beta\gamma_{12'})\\
&\times (1-\beta\gamma_{1'2'})e^{-\beta(\gamma_{12}+\gamma_{1'2}+\gamma_{12'}+\gamma_{1'2'})}. \end{split} \end{equation} As seen, both (\ref{purity7}) and (\ref{purity8}) contain twelve dimensional integrals. There are no analytic results. Numerical calculation is needed.
\section{Monte Carlo Integration} \label{sec:montecarlo} In this section, Monte Carlo integration method is utilized to estimate the twelve-dimensional integration in (\ref{purity7}). We use the MISER algorithm~\cite{MISER} of GNU Scientific Library version 2.5. To validate the accuracy of the algorithm, we first calculate $Tr(\rho_1)$. It is expected to have $Tr(\rho_1)=1$ due to the normalization requirement. Followed similar derivation steps in previous section, $Tr(\rho_1)$ is expressed as \begin{equation}
\label{Normalization2}
Tr(\hat{\rho}_1) = \lim_{\alpha\to\infty}\frac{1}{\pi}(\frac{\alpha}{4})^3\int_{-1}^{1} e^{-\alpha(\gamma_{12})} d\vec{\gamma}_1 d\vec{\gamma}_2. \end{equation} This is a six-dimensional integration. Table \ref{tab:1} shows the calculation results with different value of $\alpha$ using the Monte Carlo integration algorithm. The MISER algorithm is set to recursively calculate the integration till the statistical error is less than one percent. Double precision variables are used in the calculation. The number of Monte Carlo calls $N_{MC}$ increases significantly when $\alpha$ increases. Our calculation ends at $\alpha=L/a = 200$. From the results in Table \ref{tab:1}, it is reasonable to extrapolate that $Tr(\rho_1)\to 1$ when $\alpha\to\infty$. This confirms the MISER algorithm is fairly accurate. In Appendix B, we have also calculated $Tr(\rho^2_1)$ analytically for a harmonic oscillator, and verified that the numerical calculation using the MISER algorithm is consistent with the analytic result. This further confirms the reliability of the Monte Carlo method for a twelve-dimension integration. \begin{table}[h!]
\caption{Trace of Reduced Density Matrix $\rho_1$} \label{tab:1}
\begin{tabular}{m{1cm}|m{1.5cm}|m{1.5cm}|m{1.5cm}}
\hline $L/a$ & $Tr(\hat{\rho}_1)$ & Error & $N_{MC}$ (million) \\ \hline
10 & 0.786360 & 0.004386 & 1\\ 20 & 0.876379 & 0.008357 & 2\\ 40 & 0.955348 & 0.006011 & 16\\ 100 & 0.981460 & 0.006937 & 128\\ 150 & 0.984406 & 0.005142 & 512\\ 200 & 0.994176 & 0.007497 & 1,024\\ \hline \end{tabular} \end{table}
We now proceed to calculate the purity of reduced density $\rho_1$ derived from the ground state, given in (\ref{purity7}). The results are shown in Table \ref{tab:2}. \begin{table}[h!]
\caption{Purity of $\rho_1$ at Ground State} \label{tab:2}
\renewcommand{1}{1.2}
\begin{tabular}{m{1cm}|m{2cm}|m{2cm}|m{1.5cm}}
\hline $L/a$ & $Tr(\hat{\rho}_1^2)$ & Error & $N_{MC}$ (million) \\ \hline
10 & $1.03\times10^{-4}$ & $3.89\times 10^{-6}$ & 256\\ 20 & $1.60\times 10^{-5}$ & $2.53\times 10^{-6}$ & 1,024\\ 40 & $5.46\times 10^{-7}$ & $9.34\times 10^{-8}$ & 2,048\\ 60 & $1.55\times 10^{-8}$ & $1.34\times 10^{-9}$ & 2,048\\ 100 & $1.53\times 10^{-9}$ & $2.98\times 10^{-10}$ & 2,048\\ \hline \end{tabular} \renewcommand{1}{1} \end{table} Note that the calculation becomes very expensive when $\alpha$ increases, as the number of Monte Carlo calls increases to billions. To reduce the computation cost, the calculation is terminated whenever the error is $<$20\% of the value of $Tr(\rho_1^2)$. From the results in Table \ref{tab:2}, $Tr(\rho_1^2)\to 0$ rapidly when $\alpha$ increases. It is reasonably to extrapolate that $E=1-Tr(\rho_1^2)=1$ when $\alpha \to \infty$. This confirms that the ground state is an entangled state. The two masses are entangled due to the gravity interaction.
The Monte Carlo integration results for (\ref{purity8}) are shown in Table \ref{tab:3}. The numerical results show that $Tr(\rho_1^2)\to 0$ asymptotically when $\alpha$ increases. Compared to Table \ref{tab:2}, the purity approaching zero slower when the bipartite system is in the excite state. Nevertheless, it still shows that the two subsystems are in an entangled state. \begin{table}[h!]
\caption{Purity of $\rho_1$ at Ground State} \label{tab:3}
\renewcommand{1}{1.2}
\begin{tabular}{m{1cm}|m{2cm}|m{2cm}|m{1.5cm}}
\hline $L/a$ & $Tr(\hat{\rho}_1^2)$ & Error & $N_{MC}$ (million) \\ \hline
10 & $4.94\times10^{-2}$ & $1.64\times 10^{-4}$ & 1\\ 40 & $2.13\times 10^{-2}$ & $1.40\times 10^{-4}$ & 128\\ 100 & $2.07\times 10^{-3}$ & $1.15\times 10^{-4}$ & 2,048\\ 200 & $6.97\times 10^{-5}$ & $5.94\times 10^{-6}$ & 2,048\\ 400 & $5.86\times 10^{-6}$ & $1.51\times 10^{-6}$ & 8,192\\ \hline \end{tabular} \renewcommand{1}{1} \end{table}
\section{Practical Limitations} \label{sec:limit} Although we show that two masses interacting through classical gravity potential field are entangled when they are in the ground state, there are practical limitations to detect such entanglement.
The limitation can be examined from the only numerical parameter $\alpha=L/a$ in (\ref{purity7}) which depends on physical parameter $a$. In the case of hydrogen atom, $a=0.83\times 10^{-8}cm$ is the Bohr Radius. The electron and the hydrogen nuclear are entangled when they are in the ground state. As correctly pointed out in Ref.~\cite{Marcelo}, due to the large asymmetry of mass between electron and proton, the entanglement is very small. On the other hand, for a positronium system that comprised a positron and an electron, the entanglement is non-negligible. In addition, our calculation here does not consider the electron spin and its coupling effect with the angular momentum. A refined calculation should include it in the total Hamiltonian. The spin-angular momentum coupling is considered perturbation to the ground state derived from just the Coulomb interaction. One question is that whether the electron-nuclear entanglement in the ground state is still maintained when considering the spin-angular momentum perturbation. Further numerical calculation must be performed to answer this question. Finally, a more accurate treatment of this problem should employ the quantum field theory, as also pointed out in Ref.~\cite{Marcelo}.
In this study, we are more interested in the case that the two subsystems are interacting through gravity potential. In this case, parameter $a$ is given by (\ref{GravityRadius}) and is called the Gravitational Bohr Radius. Its value is strongly depending on the values of masses $m_1$ and $m_2$. Considered the case of the Sun and the Earth. Given that $G=6.67\times 10^{-11} m^3kg^{-1}s^{-2}$, the mass of the Sun $m_1=1.99\times 10^{30}kg$, and the mass of the Earth $m_2=5.97\times 10^{24}kg$, and the Plank constant $\hbar = 1.06\times 10^{-34} m^2kgs^{-1}$, (\ref{GravityRadius}) gives $a=2.35\times 10^{-135}m$. This is much smaller than the Plank length $1.62\times 10^{-35}m$ and becomes non-physical. Clearly the wave function given in (\ref{GroundState}) is not suitable to describe quantum state of the Sun-Earth system. Consequently, it is meaningless to discuss entanglement between the quantum states of the Sun and the Earth.
However, suppose the masses of the two subsystems are at the scale of $~10^{-21}kg$ and $~10^{-17}kg$, such as the mass of a Brome mosaic virus~\cite{Maul97} and a vaccinia virus~\cite{Johnson}, respectively. One can estimate $a \simeq 5 cm$. If a universe consists only two such viruses and they interact only through classical gravity, and if we further assume such virus can be considered as quantum systems, then the conclusion can be drawn that the two systems are entangled when they are in the ground state. Obviously these are very strict conditions. Due to such practical limitation, one must be very cautious to draw such a conclusion. Instead, the significance of our result comes from the conceptual implication as discussed earlier.
\section{Entanglement Through Adjacent Interferometers} \label{sec:Interferometers}
In an effort to confirm that gravitational field is a quantum entity, Refs~\cite{Bose, Marletto} proposed a novel experiment to induce entanglement between two test masses interaction through gravity. The motivation there is that entanglement can only be generated through mediation of a quantum entity. By confirming that mutual gravitational interaction between test masses can entangle the states of two masses, one can conclude that gravity field necessarily obeys quantum mechanics principles. The experiment is briefly restated below. Two test masses with masses $m_1$ and $m_2$ are prepared in superposition of two spatial separated states $|L\rangle$ and $|R\rangle$. Suppose the distance between the center of the two state is $\Delta x$. Each state is a localized Gaussian wave packets with the width much smaller than $\Delta x$, so that $\langle L|R\rangle = 0$. The distance between the centers of the two masses is $d$. Essentially these initial conditions can be physically realized by two Mach-Zehnder interferometers separated at distance $d$. The initial state is then given by \begin{equation}
\label{MZinit}
|\Psi(t=0)\rangle_{12}=\frac{1}{\sqrt{2}}(|L\rangle_1+|R\rangle_1)\frac{1}{\sqrt{2}}(|L\rangle_2+|R\rangle_2). \end{equation} Now the two masses go through time evolution with mutual gravitational interaction for a period $\tau$. The time evolution introduces an additional phase shift in the probability amplitude, given by $\phi_{ij} = \frac{V(r_{ij})\tau}{\hbar}=\frac{Gm_1m_2\tau}{\hbar r_{ij}}$, where $i\in \{L, R\}$ is index for mass 1, $j\in \{L, R\}$ is index for mass 2, and $r_{ij}$ is the distance between the distinct components of the superposition state of the two masses. Since $r_{ij}$ is different for the four possible combinations of spatial states of the two masses, the final state is \begin{equation}
\label{MZfinal}
\begin{split}
|\Psi(t=\tau)\rangle_{12} & =\frac{e^{i\phi_{LL}}}{2}|L\rangle_1(|L\rangle_2+e^{i(\phi_{LR}-\phi_{LL})}|R\rangle_2) \\
& +\frac{e^{i\phi_{RL}}}{2}|R\rangle_1(|L\rangle_2+e^{i(\phi_{RR}-\phi_{RL})}|R\rangle_2).
\end{split} \end{equation} $\Psi(t=\tau)\rangle_{12}$ can be factorized if the following condition is met, \begin{equation}
\label{deltaphi}
\Delta\phi = \phi_{LR}-\phi_{LL} + \phi_{RR}-\phi_{RL} = 2n\pi. \end{equation} For the two masses interact with classical gravity in the inteferometers, $r_{LL}=r_{RR}=d$, $r_{LR}=d+\Delta x$, and $r_{RL}=d-\Delta x$, we obtain \begin{equation}
\label{deltaphi2}
\Delta\phi = \frac{Gm_1m_2\tau}{\hbar}(\frac{2}{d}-\frac{1}{d+\Delta x} - \frac{1}{d -\Delta x}), \end{equation}
which is in general not equal to $2n\pi$ if proper parameters $d$ and $\Delta x$ are chosen. Thus, $|\Psi(t=\tau)\rangle_{12}$ cannot be factorized and entanglement between two test masses can be created. We now proceed to discuss the similarity and difference between this result and our result.
First, both results show that gravitational interaction can induce entanglement between two masses, and consequently confirm that gravity is a quantum entity if we acknowledge the reasoning described in Refs~\cite{Bose, Marletto}. However, the entanglement in the interferometer approach is generated through a very specific experimental design, the test masses are prepared in specific initial condition, while the result presented in this paper is generic and derived rigorously from first principle, i.e., from the Schr\"{o}dinger Equation. In this sense, our result generalizes the finding in Refs~\cite{Bose, Marletto} since it is not depending on specific experimental setup.
Second, both results show that the entanglement can be generated through other interaction such as Coulomb interaction. The key is that the interaction potential energy cannot be factorized into two independent terms with respective to the position degree of freedoms. To see this, suppose $V(r_{ij})=U(r_i)+W(r_j)$, it is easy to verify that $\Delta\phi =0$. One crucial example is the gravitational field from the Earth acting on the two masses, which can be approximated as $V(z_{ij}) = m_1gz_i + m_2gz_j$, where $z$ is the distance between the surface of the Earth and the masses. For this gravitational field, $\Delta\phi = 0$ and cannot induce entanglement. The same conclusion can be drawn from Eq.(\ref{SE}). When $V(r_{12})=U(r_1)+W(r_2)$, (\ref{SE}) can be separated two independent equations and admits $\psi(\vec{r}_1,\vec{r}_2)=\phi(\vec{r}_1)\varphi(\vec{r}_2)$ as a solution, which is a product state. Thus, the origin of the entanglement strongly depends on the geometry properties of the interacting field, although this is not pointed out in Refs~\cite{Bose, Marletto},
Third, one of the motivations of our work is to search for direct connection between entanglement and the geometry properties of spacetime, see detailed discussion in Section \ref{subsec:curvature}. This is a new insight not presented in Refs~\cite{Bose, Marletto}. The significance of the experiments proposed in Refs~\cite{Bose, Marletto} is that the gravity induced entanglement is practically detectable, while our work is mostly theoretical and conceptual.
\section{Discussions} \label{sec:discussion} \subsection{Entanglement and Spacetime Curvature} \label{subsec:curvature} The calculation result that two masses $A$ and $B$ interacting through gravitational field are entangled in the ground state has some interesting conceptual implications. First of all, we need to emphasize that the entanglement calculated in section IV is different from the entanglement in the AdS/CFT correspondence. The later refers to the correlation between a spatial region in the boundary of AdS and the rest of the boundary without gravity defined. On the other hand, the entanglement calculated in this paper is for two masses that interacts through classical gravitational field.
Entanglement between $A$ and $B$ implies that there is correlation information between them. By knowing information on $A$, one can infer information on $B$. What information is correlated in the case of gravity interaction? The degree of freedom in our calculation is the position of the masses. Thus, the correlation encoded in the entangled systems is about the position of each system. Each system is in a mixed state. Their positions are correlated and cannot be described independently. Subsystem $A$ has to be described relative to subsystem $B$ for completeness~\footnote{More precisely, the bipartite system as a whole is in a pure state while subsystem $A$ is described relative to subsystem $B$.}.
The root cause of such correlation is due to the intrinsic property of the gravitational potential, which is proportional to the inverse of relative distance between $A$ and $B$. The origin of such property can be better understood in the context of General Relativity. In GR, there is no gravitation field. Instead, the gravitation field in the Newtonian formulation is just the effect of spacetime curvature in the limit of very weak fields and low velocities~\cite{SeanCarroll}. Here, the full spacetime metrics, which determine the curvature, is written as $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, where $\eta_{\mu\nu}$ is the Minkowskian metrics and $h_{\mu\nu}$ is the small deviation on it. Only the $h_{00}$ component of the metrics is of important in the limit of very weak fields and low velocities, which turns out to be proportional to the inverse of relative distance between the two masses. Thus, the curvature of spacetime is manifested in such geometry property of the classical gravity potential. Since our calculation shows that the gravity with such property induced entanglement between two masses in their ground state, we argue that the curvature of spacetime causes the entanglement between the two masses. In other words, the entanglement at the ground state intrinsically connects to the curvature of spacetime. Given the first excited state is also an entangled state, we speculate that a quantum state for two masses interacting with gravity maybe always in an entangled state. Of course this statement is speculative only and needs a unified quantum gravity theory $\cal{M}$ for accurate treatment.
The argument that entanglement of two masses is connected to the curvature of spacetime the two masses create is not based first principles from a unified quantum gravity theory $\cal{M}$, since it is still being searched for. Instead, the argument is based on the generalized Bohr's Correspondence Principle~\cite{Bohr, Post}. Traditional quantum mechanics with classical gravity interaction, nevertheless, can be considered an approximation of theory $\cal{M}$. We expect when certain classical limit is imposed, theory $\cal{M}$ should either predict similar result as the less general physical theory, or explain why the results are different. In this notion, whether the ground state of two systems interacting through gravity is an entangled state can be a check point for theory $\cal{M}$ in the classical limit.
\subsection{Dependency on Reference Frame} The Schr\"{o}dinger Equation in the form of (\ref{WF1}) implies that we have chosen a reference coordinate system such that the positions of the two systems are given by variable $\vec{r}_1$ and $\vec{r}_2$, respectively. The entanglement obtained in the calculation is with respect to such coordinate system. For instance, in the case of hydrogen atom, we can choose the lab where the hydrogen atom is prepared as a reference system. The coordinate system is at rest with respect to the lab. To study the entanglement properties in a relativistic framework, one should use the QFT. It has been shown that under Lorentz transformation, the entanglement measure for the subsystem-to-subsystem partition is Lorentz invariant~\cite{Calmet}\footnote{The reason for this is that the density matrix for a bipartite system is a Lorentz scalar. If the reduced density matrix is obtained by partitioning the density matrix with a Lorentz invariant term, the resulting reduced density matrix is again a Lorentz scalar. Thus, the entanglement measure is Lorentz invariant. See details in Ref.~\cite{Calmet}.}. In the case of a curve spacetime, whether the entanglement measure is invariant under coordinator transformation requires further analysis\footnote{For instance, under the covariant theory of quantum gravity~\cite{Cremaschini}, the Hamilton and wave equation can be formulated in the forms of 4-scalar, paving the path to calculate the entanglement measure. However, the calculation is beyond the scope of this paper.}. On the other hand, recent studies suggest that if we consider the reference system as quantum system as well, entanglement properties may depend on the choice a reference system~\cite{Brukner, Hoehn2018}. For example, we have shown that in a hydrogen atom, the electron and the nuclear are entangled in ground state with the lab as a reference frame. Consequently, the electron is in a mixed state. But if instead, we choose the nuclear itself as the reference system, the electron can be in a pure state with respect to the nuclear. Here the nuclear is considered as a quantum reference frame. These considerations are consistent with the calculation in this paper. Of course, the reason that entanglement properties depends on the choice a quantum reference system as shown in Refs.~\cite{Brukner, Hoehn2018} is more complex. Briefly speaking, by considering the reference frame as a quantum system, we need to describe the observed system and the reference system with relational properties. The entanglement properties in such a framework turn out to depend on the quantum reference system being chosen.
\subsection{Ubiquitous Entanglement?} Since the formulation and calculation in earlier sections are generic to any bipartite system where the interaction between two subsystems of is described as a central potential only depends on the distance between the two subsystems, the calculation is also applicable to well-studied systems such as a hydrogen atom where the interaction is described by the Coulomb potential, or a three-dimensional harmonic oscillator. In the case of a hydrogen atom, the entanglement in the ground state is an interesting fact because traditionally one is only interested in the energy levels of each eigenstate of a hydrogen atom. The entanglement information encoded in the wave function had been left unrecognized. Furthermore, given the ubiquity of gravitational interaction, one might think the entanglement between two masses are also ubiquitous. A quantum state for two masses interacting with gravity may not be separable. However, as pointed out in Section \ref{sec:limit}, for typical massive objects, the Gravitational Bohr Radius is much smaller than the Plank length. It is not meaningful to discuss entanglement between such massive objects. Furthermore, for a given mass object, when considering the gravitational interactions from virtually infinite other surrounding mass objects, the entangled state for a bipartite system could be disturbed such that the entanglement relation is destroyed. Thus, entanglement found in these well-studied bipartite systems can be practically ignored. The calculation and discussion in this paper is mostly for conceptual interest.
\section{Conclusions} \label{sec:conclusion} In this paper we study the entanglement measure of a bipartite system interacting through a central potential in the context of non-relativistic quantum mechanics. Thorough numerical calculation, we found that the ground state and the first excited state of such quantum system are entangled states. Although the entanglement properties of such systems were studied in earlier literature, this paper provides two contributions.
First, the formulation to calculate the entanglement measure in Section \ref{sec:entmeasusre} is more generic as contrary to earlier formulation~\cite{Tommasini} that is only applicable to state that are translationally invariant.
Second, more importantly, we believe that the significance of the entanglement of such bipartite systems are not fully analyzed. Our motivation is to extract potential conceptual implications from this simple calculation, particularly in the case of gravitational interaction. For instance, the result provides hints of direct connection between entanglement and gravity. Since the gravitational field is an approximation of the curvature of spacetime in the limit of very weak fields and low velocities, it is reasonable to argue that there is an intrinsic connection between the entanglement of two masses and the curvature of the spacetime they create. A quantum gravity theory should either predict similar result in the weak field and low velocity limit, or gives a reasonable explanation why the results may be different.
\appendix \section{Justification of Entanglement Measure}
Using the entanglement measure defined in (\ref{ent}), we say that two subsystems are entangled when $E>0$, which corresponds to $Tr(\hat{\rho}_1^2) < 1$. On the other hand, the two subsystems are separable when $E=0$, which corresponds to $Tr(\hat{\rho}_1^2) = 1$. We will show that $E$ indeed measures the entanglement of the bipartite system. Supposed the state vector of the bipartite system can be decomposed with a set of orthogonal basis $\{|\phi_i\rangle, |\varphi_i\rangle\}$ where $i=0, 1, 2,\ldots, d-1$, and $d$ could be infinite, \begin{equation}
\label{Schmidt}
\begin{split}
|\Psi\rangle_{12} & = \sum_i\lambda_i |\phi_i\rangle|\varphi_i\rangle \\
& = \sum_i\lambda_i\{\int\phi_i(x)|x\rangle dx\}\{\int\varphi_i(y)|y\rangle dy\}\\
& = \int\{\sum_i\lambda_i\{\phi_i(x)\varphi_i(y)\}|x\rangle|y\rangle dxdy,
\end{split} \end{equation}
where $\sum_i|\lambda_i|^2=1$. Compared to (\ref{WF1}) gives \begin{equation}
\label{Schmidt1}
\psi(x,y) = \sum_i\lambda_i\phi_i(x)\varphi_i(y). \end{equation} The orthogonal relations are given by \begin{equation}
\int \phi_i(x)\phi^*_j(x)dx=\delta_{ij}, \int \varphi_k(y)\varphi^*_l(y)dy=\delta_{kl}. \end{equation} Substituting (\ref{Schmidt1}) into (\ref{purity2}) and applying the orthogonal properties, we obtain \begin{equation}
\label{purity10}
Tr(\hat{\rho}_1^2) = \sum_{i=0}^{d-1}(|\lambda_i|^2)^2. \end{equation}
Since $\sum_i|\lambda_i|^2=1$, it can be shown that \begin{equation}
\frac{1}{d^2} \leq Tr(\hat{\rho}_1^2) \leq 1. \end{equation} $Tr(\hat{\rho}_1^2)=1$ if and only if $d=1$, which implies $\psi(x,y)=\phi(x)\varphi(y)$ and consequently, $\Psi_{12}$ is a separable state. On the other hand, when $\psi(x,y)\ne\phi(x)\varphi(y)$, $d>1$ hence $Tr(\hat{\rho}_1^2) < 1$, we get $E>0$. Thus, (\ref{ent}) is a proper quantity to measure on whether the two subsystems are entangled. When $d\to\infty$, the lower bound of $Tr(\hat{\rho}_1^2)$ can be 0.
(\ref{Schmidt}) is essentially the Schmidt decomposition of the state vector for the bipartite system. However, it is not clear whether the decomposition is applicable when $d\to\infty$. Furthermore, it is very difficult to find the analytic solution of the decomposition in order to use (\ref{purity10}. Practically, we still rely on numerical method to calculate $ Tr(\hat{\rho}_1^2)$.
With (\ref{purity2}), $Tr(\hat{\rho}_1^2)$ is calculated in the $|x\rangle$ position basis, which strictly speaking is not a Hilbert space, because the norm is a delta function, i.e., $\langle x_i|x_j\rangle = \delta(x_i-x_j)$. We can calculate $Tr(\hat{\rho}_1^2)$ in the basis $\{|\phi_i\rangle\}$ instead, where the new basis form a truly Hilbert space since the norm is 1 by definition. Since $|\Psi\rangle_{12} = \sum_i\lambda_i |\phi_i\rangle|\varphi_i\rangle$, we can derive the reduced density operator \begin{equation}
\hat{\rho}_1 = Tr_2(|\Psi\rangle_{12}\langle\Psi|) = \sum_i|\lambda_i|^2|\phi_i\rangle\langle\phi_i|. \end{equation}
From this one can derive the same expression of $Tr(\hat{\rho}_1^2)$ as (\ref{purity10}). In other words, quantity $Tr(\hat{\rho}_1^2)$, and consequently the entanglement measure $E$, are invariant in either the position basis or the basis $\{|\phi_i\rangle\}$. This is not surprised since the transformation between the two basis are unitary. The transform matrix element for variable $x$ can be written as \begin{equation}
\label{element}
\begin{split}
M_{ij} & = \langle x_i|\phi_j\rangle \\
& = \{\int\delta(x-x_i)\langle x| dx\}\{\int\phi_j(x')|x'\rangle dx'\} \\
&=\int\phi_j(x)\delta(x-x_i)dx = \phi_j(x_i).
\end{split} \end{equation} Similarly, $M^{\dag}_{ki}=M^*_{ik}=\phi^*_k(x_i)$. Then, \begin{equation}
\label{element}
\begin{split}
(M^{\dag}M)_{kj}& =\sum_iM^{\dag}_{ki}M_{ij} = \int\phi^*_k(x_i)\phi_j(x_i)dx_i = \delta_{kj}.
\end{split} \end{equation} This confirm $M^{\dag}M=I$ and $M$ is unitary.
\section{Harmonics Oscillator} For a bipartite system that behaves like a harmonic oscillator, the potential energy is given by $V(\vec{r_1}-\vec{r_2}) = \frac{1}{2}\omega^2r^2_{12}$, where $\omega$ describes the strength of the potential. Using the center of mass coordinate system, the wave function for ground state is given by (\ref{transform2}) and \begin{equation}
\label{harmonicWF}
\varphi_g(\vec{r}_{12}) = \sqrt{\frac{1}{\pi^{3/2} a^3}}e^{-\frac{1}{2}(\frac{r_{12}}{a})^2}, \end{equation} where $a$ is constant determined by the masses and $\omega$. Substituting this into (\ref{purity5}), using the same notations as for (\ref{purity7}), and denoting $\beta=L/(4a)=\alpha/4$, we obtain \begin{equation} \label{purity11} \begin{split}
Tr(\hat{\rho}_1^2) = & \lim_{\beta\to\infty}\frac{\beta^6}{\pi^3}\int_{-1}^{1} d\vec{\gamma'}_1d\vec{\gamma}_1 d\vec{\gamma'}_2d\vec{\gamma}_2 \\
&\times e^{-2\beta^2(\gamma_{12}^2+\gamma_{1'2}^2+\gamma_{12'}^2+\gamma_{1'2'}^2)}. \end{split} \end{equation} We can perform similar Monte Carlo calculation on this twelve-dimensional integration. But fortunately, this integration can be calculated analytically, so that the result can be used to check the accuracy of Monte Carlo integration. First, we expand $\gamma_{12}^2 = (\bar{x}_1-\bar{x}_2)^2+(\bar{y}_1-\bar{y}_2)^2+(\bar{z}_1-\bar{z}_2)^2$, where we denote dimensionless variables $\bar{x}_1=2x_1/L$, $\bar{x}_2=2x_2/L$ and so on. Also noted that $d\vec{\gamma'}_1=d\bar{x}_1d\bar{y}_1d\bar{z}_1$, (\ref{purity11}) can be simplified into \begin{equation} \label{purity12} \begin{split}
Tr(\hat{\rho}_1^2) = & \lim_{\beta\to\infty}\{\frac{\beta^2}{\pi}\int_{-1}^{1} d\bar{x}'_1d\bar{x}_1 d\bar{x}'_2d\bar{x}_2 \\
&\times e^{-2\beta^2((\bar{x}_1-\bar{x}_2)^2+(\bar{x}'_1-\bar{x}_2)^2+(\bar{x}_1-\bar{x}'_2)^2+(\bar{x}'_1-\bar{x}'_2)^2)}\}^3\\
& = \lim_{\beta\to\infty}\{\frac{\beta^2}{\pi}\int_{-1}^{1}J^2(\bar{x}_2,\bar{x}'_2)d\bar{x}'_2d\bar{x}_2\}^3, \end{split} \end{equation} where \begin{equation} \label{Jfunction}
J(\bar{x}_2,\bar{x}'_2) = \int_{-1}^{1} e^{-2\beta^2((\bar{x}_1-\bar{x}_2)^2+(\bar{x}_1-\bar{x}'_2)^2)}d\bar{x}_1. \end{equation} Expanding the exponent in the integral for the $J$ function, $(\bar{x}_1-\bar{x}_2)^2+(\bar{x}_1-\bar{x}'_2)^2 = 2(\bar{x}_1 - \breve{x})^2 + (\bar{x}_2-\bar{x}'_2)^2/2$ where $\breve{x}=(\bar{x}_1+\bar{x}_2)/2$, \begin{equation} \label{Jfunction2} \begin{split}
J(\bar{x}_2,\bar{x}'_2) & = e^{-\beta^2(\bar{x}_2-\bar{x}'_2)^2} \int_{-1}^{1}e^{-4\beta^2(\bar{x}_1 - \breve{x})^2}d\bar{x}_1 \\
&=\frac{\sqrt{\pi}}{4\beta}e^{-\beta^2(\bar{x}_2-\bar{x}'_2)^2}\times \\
& [Erf(2\beta(1-\breve{x})) - Erf(e\beta(1+\breve{x}))]. \end{split} \end{equation} Here $Erf(x)=\pi^{-1/2}\int^x_{-x}e^{-t^2}dt$ is the error function. Since $-1< Erf(x) < 1$, the difference of the error function at two arbitrary values $a$ and $b$ is, $-2< [Erf(a) - Erf(b)]< 2$. Thus, $J^2(\bar{x}_2,\bar{x}'_2) < \frac{\pi}{4\beta^2}e^{-2\beta^2(\bar{x}_2-\bar{x}'_2)^2}$. Plug this into (\ref{purity12}), \begin{equation}
\label{purity13}
Tr(\hat{\rho}_1^2) < \lim_{\beta\to\infty}\{\frac{1}{4}\int_{-1}^{1}e^{-2\beta^2(\bar{x}_2-\bar{x}'_2)^2}d\bar{x}'_2d\bar{x}_2\}^3. \end{equation} Recall $\beta = L/(4a)$, $\alpha = L/a$, and denote $\grave{x}_2 = \alpha\bar{x}_2/2$, $\grave{x}'_2 = \alpha\bar{x}'_2/2$, we rewrite (\ref{purity13}) as \begin{equation}
Tr(\hat{\rho}_1^2) < \lim_{\alpha\to\infty} K^3(\alpha), \end{equation} where \begin{equation}
\label{K}
K (\alpha) = \frac{1}{\alpha^2}\int_{-\alpha/2}^{\alpha/2}e^{-\frac{1}{2}(\grave{x}_2-\grave{x}'_2)^2}d\grave{x}'_2d\grave{x}_2. \end{equation} Denote $t=((\grave{x}_2-\grave{x}'_2))/\sqrt{2}$, $s=(\alpha/2-\grave{x}_2)/\sqrt{2}$, and using the error function again, \begin{equation}
\label{K2}
\begin{split}
K (\alpha)&=\frac{\sqrt{2}}{\alpha^2}\int_{-\alpha/2}^{\alpha/2}\{\int_{-(\alpha/2-\grave{x}_2)/\sqrt{2}}^{(\alpha/2-\grave{x}_2)/\sqrt{2}}e^{-t^2}dt\}d\grave{x}_2\\
& = \frac{2\sqrt{\pi}}{\alpha^2}\int_{0}^{\alpha/\sqrt{2}}Erf(s)ds
\end{split} \end{equation} Note that $Erf(s)$ is a monotonically increasing function and approaches 1 rapidly when $s$ is a large enough number, e.g., $Erf(2)=0.9953$, $Erf(3)=0.9999$, we can approximate the integral $\int_{0}^{\alpha/\sqrt{2}}Erf(s)ds \simeq \alpha/\sqrt{2} - \delta$ where $\delta$ is some positive finite number. Thus, \begin{equation}
\label{K3}
K (\alpha) \simeq \frac{2\sqrt{\pi}}{\alpha^2}(\frac{\alpha}{\sqrt{2}} - \delta). \end{equation} We ignore $\delta$ when $\alpha\to\infty$, and finally get the upper bound of the purity \begin{equation}
\label{purity14}
Tr(\hat{\rho}_1^2) < \lim_{\alpha\to\infty}\frac{(2\pi)^{3/2}}{\alpha^3} \to 0. \end{equation} However, $Tr(\hat{\rho}_1^2) \geq 0$, we conclude that $Tr(\hat{\rho}_1^2)=0$.
Table \ref{tab:4} shows the Monte Carlo estimation of the twelve-dimensional integration in (\ref{purity11}). The last column of the table lists the value of $K^3(\alpha)$. These results confirm that the Monte Carlo integration is consistent with the analytic results. \begin{table}[h!]
\caption{Purity of $\rho_1$ at Ground State} \label{tab:4}
\renewcommand{1}{1.2}
\begin{tabular}{m{1cm}|m{1.8cm}|m{1.8cm}|m{1.3cm}|m{1.8cm}}
\hline $L/a$ & $Tr(\hat{\rho}_1^2)$ & Error & $N_{MC}$ (million) & $(\sqrt{2\pi}/\alpha)^3$\\ \hline
10 & $1.25\times10^{-2}$ & $4.99\times 10^{-4}$ & 8 & $1.57\times 10^{-2}$\\ 20 & $1.06\times 10^{-3}$ & $6.83\times 10^{-5}$ & 128 & $1.97\times 10^{-3}$\\ 40 & $1.50\times 10^{-5}$ & $2.75\times 10^{-6}$ & 256 & $2.46\times 10^{-4}$\\ 60 & $1.19\times 10^{-6}$ & $1.41\times 10^{-7}$ & 512 & $7.29\times 10^{-5}$\\ 100 & $1.31\times 10^{-7}$ & $2.45\times 10^{-8}$ & 1,024 & $1.57\times 10^{-5}$\\ \hline \end{tabular} \renewcommand{1}{1} \end{table}
\end{document} | arXiv | {
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\begin{document}
\title{Extremizing Temperature Functions of Rods with Robin Boundary Conditions}
\author{Jeffrey J. Langford and Patrick McDonald}
\address{Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837}
\email{jeffrey.langford@bucknell.edu}
\address{Division of Natural Science, New College of Florida, Sarasota, FL 34243}
\email{mcdonald@ncf.edu}
\date{\today}
\begin{abstract} We compare the solutions of two one-dimensional Poisson problems on an interval with Robin boundary conditions, one with given data, and one where the data has been symmetrized. When the Robin parameter is positive and the symmetrization is symmetric decreasing rearrangement, we prove that the solution to the symmetrized problem has larger increasing convex means. When the Robin parameter equals zero (so that we have Neumann boundary conditions) and the symmetrization is decreasing rearrangement, we similarly show that the solution to the symmetrized problem has larger convex means. \end{abstract}
\keywords{Symmetrization, comparison theorems, Poisson's equation, Robin boundary conditions}
\subjclass[2020]{Primary 34B08; Secondary 34C10}
\maketitle
\section{Introduction: Physical Motivation and Main Results} Our paper is motivated by the following physical problem:
\begin{problem}\label{Prob:1dRod} Consider a metal rod of length $\ell$. To half the locations on the rod, heat is generated uniformly, while on the remaining half of the rod, heat is neither generated nor absorbed. If the rod's ends are frozen at zero temperature, where should we place the heat sources to maximize the hottest steady-state temperature across the rod? \end{problem} Several possible arrangements appear in Figure 1 below. Heat is generated in the white regions while heat is neither generated nor absorbed in the gray regions.
\begin{figure}
\caption{Four possible heat source arrangements for Problem \ref{Prob:1dRod}.}
\label{fig:fig_a}
\label{fig:fig_b}
\label{fig:fig_c}
\label{fig:fig_c}
\end{figure}
The solution to Problem \ref{Prob:1dRod} follows from a celebrated result in symmetrization known as Talenti's Theorem \cite{Talenti}. To understand Talenti's solution\footnote{Although Talenti's work in \cite{Talenti} explicitly assumes that the dimension $n\geq2$, the result still holds in dimension $1$. For a different approach to comparison theorems that yields the same result in all dimensions, see Corollary 3 of \cite{Baernstein Cortona Volume} or Theorem 10.10 of \cite{Barenstein Star Function in Complex Analysis} and Corollary \ref{Cor:DComp} below.}, we write out the mathematical formulation of Problem \ref{Prob:1dRod}. Suppose the rod is located along the interval $\left[-\frac{\ell}{2},\frac{\ell}{2}\right]$ and let $E\subseteq [-\frac{\ell}{2},\frac{\ell}{2}]$ denote the locations of the heat sources. Then the steady-state temperature function $u$ satisfies the one-dimensional Poisson problem \begin{equation}\label{eq:1duTalenti} -u''=\chi_E \quad \textup{in} \quad \left(-\frac{\ell}{2},\frac{\ell}{2}\right), \qquad u\left(-\frac{\ell}{2}\right)=u\left(\frac{\ell}{2}\right)=0, \end{equation} where $\chi_E$ denotes the characteristic function of the set $E$. Talenti's Theorem compares the solution $u$ in \eqref{eq:1duTalenti} to the solution $v$ of a problem that has been ``symmetrized.'' Specifically, let $v$ solve the Poisson problem \begin{equation*}
-v''=\chi_{\left(-\frac{|E|}{2},\frac{|E|}{2}\right)} \quad \textup{in} \quad \left(-\frac{\ell}{2},\frac{\ell}{2}\right), \qquad v\left(-\frac{\ell}{2}\right)=v\left(\frac{\ell}{2}\right)=0, \end{equation*}
where $|E|$ denotes the length of $E$; in this case $\chi_{\left(-\frac{|E|}{2},\frac{|E|}{2}\right)}$ is called the \emph{symmetric decreasing rearrangement} of $\chi_E$ (for a precise definition, see Definition \ref{def:decrearr}). Talenti showed that the temperature functions $u$ and $v$ compare through their convex means, that is, \begin{equation}\label{eq:convuv} \int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}\phi(u)\,dx\leq \int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}\phi(v)\,dx \end{equation} for each convex increasing function $\phi:\mathbb{R} \to \mathbb{R}$. Since $u$ and $v$ are concave functions, they are minimized at the ends of the interval $\left[-\frac{\ell}{2},\frac{\ell}{2}\right]$, where both functions vanish. That is, \begin{equation}\label{eq:infuv} \min_{\left[-\frac{\ell}{2},\frac{\ell}{2}\right]} u=\min_{\left[-\frac{\ell}{2},\frac{\ell}{2}\right]} v=0. \end{equation} Thus $u$ and $v$ are nonnegative. Taking $\phi(x)=\chi_{[0,\infty)}(x)\cdot x^p$ in \eqref{eq:convuv} gives \[
\|u\|_{L^p\left[-\frac{\ell}{2},\frac{\ell}{2}\right]}\leq \|v\|_{L^p\left[-\frac{\ell}{2},\frac{\ell}{2}\right]}, \qquad 1\leq p<+\infty, \] and sending $p\to +\infty$ shows \begin{equation}\label{eq:supuv} \max_{\left[-\frac{\ell}{2},\frac{\ell}{2}\right]} u \leq \max_{\left[-\frac{\ell}{2},\frac{\ell}{2}\right]} v. \end{equation} Talenti's Theorem thus says that the hottest temperature in Problem \ref{Prob:1dRod} is maximized when the heat sources are centrally gathered in the middle of the rod as in (A) of Figure 1. Note also that equations \eqref{eq:infuv} and \eqref{eq:supuv} show that \[ \underset{\left[-\frac{\ell}{2},\frac{\ell}{2}\right]}{\textup{osc}}\ u \leq \underset{\left[-\frac{\ell}{2},\frac{\ell}{2}\right]}{\textup{osc}}\ v, \] where $\textup{osc}=\max - \min$ denotes the oscillation, or temperature gap (the difference between the rod's largest and smallest temperatures). Thus, arrangement (A) not only maximizes the rod's hottest temperature but also its temperature gap.
The present paper is motivated by a simple question: What happens if we consider analogues of Problem \ref{Prob:1dRod} with other boundary conditions? To start, we might consider a situation where the ends of the bar interact with the outside environment. For example, imagine that each end of the bar is submerged in a large bath of fluid with temperature zero. Newton's law of cooling then says that the heat flux is proportional to the temperature at each end of the rod. This physical setting yields boundary conditions known as Robin boundary conditions. Thus, we ask:
\begin{problem}\label{Prob:1dRodRobin} With the same setup as Problem \ref{Prob:1dRod} for a rod with Robin boundary conditions, where should we locate the heat sources to maximize the hottest steady-state temperature? \end{problem}
We solve Problem \ref{Prob:1dRodRobin} by proving a one-dimensional comparison principle for Robin problems in the spirit of Talenti. The result stated below is normalized so the length of the bar equals $2\pi$, but the the result holds for any interval. Specifically, we prove:
\begin{theorem}[ODE Robin Comparison Principle]\label{Th:RComp} Let $0\leq f\in L^1[-\pi,\pi]$ and $\alpha>0$. Suppose $u$ and $v$ solve the Poisson problems \begin{align*} -u''&=f \quad \textup{in} \quad (-\pi,\pi), \qquad -u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0,\\ -v''&=f^{\#} \quad \textup{in} \quad (-\pi,\pi), \qquad -v'(-\pi)+\alpha v(-\pi)=v'(\pi)+\alpha v(\pi)=0, \end{align*} with $f^{\#}$ the symmetric decreasing rearrangement of $f$. Then \[ \int_{-\pi}^{\pi}\phi(u)\,dx\leq \int_{-\pi}^{\pi}\phi(v)\,dx \] for each increasing convex function $\phi:\mathbb{R}\to \mathbb{R}$. In particular, \begin{equation}\label{eq:LpRobin}
\|u\|_{L^p[-\pi,\pi]} \leq \|v\|_{L^p[-\pi,\pi]},\qquad 1\leq p\leq +\infty.\\ \end{equation} \end{theorem}
To resolve Problem \ref{Prob:1dRodRobin}, let $E\subseteq [-\pi,\pi]$ denote the locations of the heat sources and let $u$ and $v$ denote the corresponding temperature functions for the Robin problems of Theorem \ref{Th:RComp}: \begin{align*} -u''&=\chi_E \quad \textup{in} \quad (-\pi,\pi), \qquad -u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0,\\ -v''&= \chi_{\left(-\frac{\pi}{2},\frac{\pi}{2}\right)} \quad \textup{in} \quad (-\pi,\pi), \qquad -v'(-\pi)+\alpha v(-\pi)=v'(\pi)+\alpha v(\pi)=0. \end{align*} The proof of Theorem \ref{Th:RComp} shows that $u$ and $v$ are nonnegative, thus taking $p=+\infty$ in \eqref{eq:LpRobin} shows \[ \max_{\left[-\pi,\pi\right]} u \leq \max_{\left[-\pi,\pi\right]} v. \] Thus, as in Problem \ref{Prob:1dRod}, Problem \ref{Prob:1dRodRobin} is resolved with an arrangement of heat sources analogous to (A) in Figure 1. In fact, in Corollary \ref{Cor:DComp}, we prove that the corresponding Dirichlet result follows from Theorem \ref{Th:RComp}. Unlike the Dirichlet setting, however, the temperature gap does not necessarily increase under symmetric decreasing rearrangement; see Example \ref{ex:osc} and Proposition \ref{prop:robingapint}.
We also address the situation where the ends of the rod are perfectly insulated. In this setting, we cannot consider a verbatim analogue of Problem \ref{Prob:1dRod}, since perfect insulation requires the presence of both heat sinks and sources. The temperature function, moreover, is unique only up to an additive constant. Thus, we ask:
\begin{problem}\label{Prob:1dRodNeumann} Suppose half of a given rod is heated uniformly, while on the complimentary half, heat is absorbed uniformly. If the rod's ends are perfectly insulated, where should we place the heat sources and sinks to maximize the hottest steady-state temperature across the rod, assuming the temperature has zero mean? \end{problem}
Again, we solve Problem \ref{Prob:1dRodNeumann} by proving a Talenti-style comparison principle. We normalize and assume the rod has length $\pi$, but as before, the result holds for any interval. We prove:
\begin{theorem}[ODE Neumann Comparison Principle]\label{Th:NComp} Let $f\in L^1[0,\pi]$ have zero mean and suppose $u$ and $v$ solve the Poisson problems \begin{align*} -u''&=f \quad \textup{in} \quad (0,\pi), \qquad u'(0)=u'(\pi)=0,\\ -v''&=f^{\ast} \quad \textup{in} \quad (0,\pi), \qquad v'(0)=v'(\pi)=0, \end{align*} with $f^{\ast}$ the decreasing rearrangement of $f$. If $u$ and $v$ both have zero mean, then \[ \int_{0}^{\pi}\phi(u)\,dx\leq \int_{0}^{\pi}\phi(v)\,dx \] for each convex function $\phi:\mathbb{R}\to \mathbb{R}$. In particular, \begin{align*}
\|u\|_{L^p[0,\pi]} &\leq \|v\|_{L^p[0,\pi]},\qquad 1\leq p\leq +\infty,\\ \max_{[0,\pi]}u \leq \max_{[0,\pi]}v,\qquad \min_{[0,\pi]}v &\leq \min_{[0,\pi]}u, \qquad \underset{[0,\pi]}{\textup{osc}} \ u \leq \underset{[0,\pi]}{\textup{osc}} \ v. \end{align*} \end{theorem}
To resolve Problem \ref{Prob:1dRodNeumann}, let $E \subseteq [0,\pi]$ denote the locations of the heat sources and let $u$ and $v$ denote the temperature functions for the Neumann problems of Theorem \ref{Th:NComp}: \begin{align*} -u''&=\chi_E-\chi_{[0,\pi]\setminus E} \quad \textup{in} \quad (0,\pi), \qquad u'(0)=u'(\pi)=0,\\ -v''&=\chi_{\left[0,\frac{\pi}{2}\right)}-\chi_{\left[\frac{\pi}{2},\pi\right]} \quad \textup{in} \quad (0,\pi), \qquad v'(0)=v'(\pi)=0. \end{align*} According to Theorem \ref{Th:NComp}, the maximum temperature and temperature gap increase, and the minimal temperature decreases under decreasing rearrangement: \[ \max_{[0,\pi]}u \leq \max_{[0,\pi]}v,\qquad \min_{[0,\pi]}v \leq \min_{[0,\pi]}u, \qquad \underset{[0,\pi]}{\textup{osc}} \ u \leq \underset{[0,\pi]}{\textup{osc}} \ v. \] Thus Problem \ref{Prob:1dRodNeumann} is resolved by choosing an arrangement of sources and sinks analogous to (B) in Figure 1, only here, white areas represent heat sources and gray areas represent heat sinks.
Taken in sum, Theorems \ref{Th:RComp} and \ref{Th:NComp} reveal a striking difference in the behavior of source functions that induce large temperature functions (interpreted in the sense of convex means). With the Neumann problem, one takes full advantage of the insulated ends, sweeping the greatest sources to one end of the bar and greatest sinks to the opposite end. However, the instant any heat energy is allowed to escape through the bar's ends and the Robin regime is entered, the arrangement switches and we instead move the greatest sources towards the middle of the bar and push the weakest sources out towards the ends.
The results of our paper are examples of comparison theorems for differential equations. To place our work in the existing literature, we recall that the first major comparison result, as mentioned above, is due to Talenti \cite{Talenti}, who compared the solutions of two Poisson problems with Dirichlet boundary conditions and nonnegative source, $f$, namely \[ \begin{array}{rclccccrclcc} -\Delta u & = & f & \text{in} & \Omega, & & & -\Delta v & = & f^{\#} & \text{in} & \Omega^{\#},\\ u & = & 0 & \text{on} & \partial \Omega, & & &v & = & 0 & \text{on} & \partial \Omega^{\#}. \end{array} \]
Here, $\Omega \subseteq \mathbb{R}^n$ is a bounded Lipschitz domain with $n\geq 2$, $0\leq f\in L^2(\Omega)$, $\Omega^{\#} \subseteq \mathbb{R}^n$ is the open ball centered at $0$ with the same volume as $\Omega$, and $f^{\#}$ denotes the symmetric decreasing rearrangement of $f$, a radially decreasing function on $\Omega^{\#}$ whose upper level sets have the same volume as those of $f$, meaning $|\{x\in \Omega:f(x)>t\}|=|\{x\in \Omega^{\#}:f^{\#}(x)>t\}|$ for $t\in \mathbb{R}$. Talenti showed that the solutions $u$ and $v$ compare via their symmetric decreasing rearrangements through the inequality \[ u^{\#} \leq v \quad \textup{in }\Omega^{\#}. \]
The history of comparison phenomena that followed Talenti's original work is long and the results are the subject of many articles. Fortunately, Talenti has prepared a thorough survey of the material (up to 2016). We direct the reader interested in this important background to \cite{TalentiSurvey} and the references therein.
Since the publication of Talenti's survey, authors have begun to turn their attention to comparison principles for Robin problems. As an example relevant to our work, in \cite{ANT} Alvino, Nitsch, and Trombetti consider the exact same setup addressed by Talenti \cite{Talenti} and mentioned above, but impose Robin boundary conditions rather than Dirichlet boundary conditions. That is, for $\alpha>0$, $\Omega \subseteq \mathbb{R}^n$ a bounded Lipschitz domain with $n\geq 2$, and $0\leq f\in L^2(\Omega)$, they consider the problems \[ \begin{array}{rclccccrclcc} -\Delta u & = & f & \text{in} & \Omega, & & & -\Delta v & = & f^{\#} & \text{in} & \Omega^{\#},\\ \frac{\partial u}{\partial \nu}+\alpha u & = & 0 & \text{on} & \partial \Omega, & & &\frac{\partial v}{\partial \nu}+\alpha v & = & 0 & \text{on} & \partial \Omega^{\#}, \end{array} \] with $\frac{\partial}{\partial \nu}$ the outer normal derivative and $\#$ the symmetric decreasing rearrangement. The authors show that Talenti's conclusion $u^{\#}\leq v$ in $\Omega^{\#}$ fails in general, but that $u$ and $v$ compare via their Lorentz norms. In dimension $n=2$, they show that the $L^1$- and $L^2$-norms of $u$ are dominated by those of $v$, and when $f=1$, that $u^{\#} \leq v$ in $\Omega^{\#}$. These results are extended in the subsequent work of Alvino, Chiacchio, Nitsch, and Trombetti \cite{ACNT}. In related work \cite{AGM}, Amato, Gentile, and Masiello, generalize results of \cite{ANT} to a nonlinear setting, replacing the Laplacian with the $p$-Laplace operator.
In addition to the the results of \cite{ACNT}, \cite{ANT}, and \cite{AGM}, in \cite{Langford4} the first author studies Poisson problems of the form \[ \begin{array}{rclccccrclcc} -\Delta u & = & f & \text{in} & A, & & & -\Delta v & = & f^{\#} & \text{in} & A,\\ \frac{\partial u}{\partial \nu}+\alpha u & = & 0 & \text{on} & \partial A, & & &\frac{\partial v}{\partial \nu}+\alpha v & = & 0 & \text{on} & \partial A, \end{array} \] where $A\subseteq \mathbb{R}^n$ is a spherical shell (the region between two concentric spheres), $\alpha>0$, $f\in L^2(A)$ and $\#$ is the {\it cap symmetrization}. (To cap symmetrize a function $f:A\to \mathbb{R}$, one applies the spherical rearrangement (the analogue of the symmetric decreasing rearrangement on the sphere) to each of $f$'s radial slice functions). The author shows that the solutions $u$ and $v$ compare through their convex means: \[ \int_{A}\phi(u)\,dx\leq \int_{A}\phi(v)\,dx \] for each convex function $\phi:\mathbb{R} \to \mathbb{R}$. The author obtains similar results for $\alpha=0$ (the Neumann problem), assuming $f$, $u$, and $v$ all have zero mean. (For related work on the Neumann problem, see \cite{Langford1}, \cite{Langford2}, and \cite{Langford3}).
To the best of our knowledge, references \cite{ACNT}, \cite{ANT}, \cite{AGM}, and \cite{Langford4} comprise all that has appeared in print to addresses Robin comparison principles for differential equations in the spirit of Talenti. Thus, our work adds an interesting contribution to this new direction in the study of comparison principles.
The rest of this note is organized as follows. In Section 2 we discuss existence and uniqueness results for the Poisson problems of Theorems \ref{Th:RComp} and \ref{Th:NComp}, so that our paper may be self-contained. We then discuss Robin Green's functions and relevant rearrangement inequalities needed to prove Theorems \ref{Th:RComp} and \ref{Th:NComp}. In Section 3, we prove our paper's main results.
\section{Background}
Since the goal of our paper is to compare the solutions of one-dimensional Poisson problems with Robin and Neumann boundary conditions, we begin with two existence and uniqueness results. These results are stated on the interval $[-\pi,\pi]$ for convenience, but they hold for any interval.
\begin{proposition}[Robin Existence and Uniqueness]\label{Prop:Runiq} Let $f\in L^1[-\pi,\pi]$ and $\alpha>0$. A unique $u\in C^1[-\pi,\pi]$ exists satisfying \begin{itemize} \item[1.] $u'$ is absolutely continuous on $[-\pi,\pi]$. \item[2.] $-u''=f$ a.e. on $(-\pi,\pi)$. \item[3.] $-u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0$. \end{itemize} \end{proposition}
\begin{proof} We first establish uniqueness. Suppose $u$ and $v$ both satisfy all the properties listed above, and let $w=u-v$. Since $w'$ is absolutely continuous, for each $x\in[-\pi,\pi]$ we have \[ w'(x)=w'(x)-w'(-\pi)+\alpha w(-\pi)=\int_{-\pi}^x(-f+f)\,dy+\alpha w(-\pi)=\alpha w(-\pi). \] Thus, $w(x)=\alpha w(-\pi)x+b$ for some constant $b$. The equations $-w'(-\pi)+\alpha w(-\pi)=w'(\pi)+\alpha w(\pi)=0$ imply that $\alpha w(-\pi)=b=0$, and so $u\equiv v$. For existence, we simply take \[ u(x)=-\int_{-\pi}^x\int_{-\pi}^tf(s)\,ds\,dt+cx+d, \] where $c$ and $d$ are chosen to make $-u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0$. \end{proof}
We also have an existence and uniqueness result for the Neumann problem. The proof is similar to that of the Robin result. \begin{proposition}[Neumann Existence and Uniqueness]\label{Prop:Nuniq} Let $f\in L^1[-\pi,\pi]$ with $\int_{-\pi}^{\pi}f\,dx=0$. A unique $u\in C^1[-\pi,\pi]$ exists satisfying \begin{itemize} \item[1.] $u'$ is absolutely continuous on $[-\pi,\pi]$. \item[2.] $-u''=f$ a.e. on $(-\pi,\pi)$. \item[3.] $u'(-\pi)=u'(\pi)=0$. \item[4.] $\int_{-\pi}^{\pi}u\,dx=0$. \end{itemize} \end{proposition}
Thus, the solutions $u$ and $v$ in Theorems \ref{Th:RComp} and \ref{Th:NComp} are guaranteed to exist and be unique. For the Robin problem, we in fact prove a bit more. Namely, we show that solutions are obtained by integration against an explicitly computable Green's function.
\begin{proposition}[Green's Representation]\label{Prop:RGreen} For $\alpha>0$, the Green's function for the Robin problem on the interval $[-\pi,\pi]$ equals \[
G(x,y)=-\frac{1}{2}c_{\alpha}xy-\frac{1}{2}|x-y|+\frac{1}{2c_{\alpha}},\qquad x,y\in [-\pi,\pi], \] where \begin{equation}\label{eq:calphadef} c_{\alpha}=\frac{\alpha}{1+\alpha \pi}. \end{equation} That is, \begin{itemize} \item[1.] $-G_{xx}(x,y)=\delta_x(y),$ for $x,y\in (-\pi,\pi),$ \item[2.] $-G_x(-\pi,y)+\alpha G(-\pi,y)=G_x(\pi,y)+\alpha G(\pi,y)=0$ for $y\in (-\pi,\pi)$. \end{itemize} Thus, if $f\in L^1[-\pi,\pi]$ and $u$ solves \[ -u''=f \quad \textup{in} \quad (-\pi,\pi), \qquad -u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0, \] then \[ u(x)=\int_{-\pi}^{\pi}G(x,y)f(y)\,dy, \qquad x\in [-\pi,\pi]. \] \end{proposition}
\begin{proof} Properties 1 and 2 follow from a straightforward calculation. Define \begin{equation}\label{eq:wdefGreen} w(x)=\int_{-\pi}^{\pi}G(x,y)f(y)\,dy. \end{equation} We show that $w\in C^1[-\pi,\pi]$ and that $w$ satisfies all three properties of Proposition \ref{Prop:Runiq}.
The Dominated Convergence Theorem gives \begin{align} w'(x)&=\int_{-\pi}^{\pi}G_x(x,y)f(y)\,dy \label{eq:wpdefGreen}\\ &=\int_{-\pi}^x\left(-\frac{1}{2}c_{\alpha}y-\frac{1}{2}\right)f(y)\,dy + \int_{x}^{\pi}\left(-\frac{1}{2}c_{\alpha}y+\frac{1}{2}\right)f(y)\,dy,\nonumber \end{align} and this representation shows that $w'$ is absolutely continuous on $[-\pi,\pi]$ with $-w''=f$ a.e. Formulas \eqref{eq:wdefGreen} and \eqref{eq:wpdefGreen} for $w$ and $w'$ together with property 2 of the present proposition give $-w'(-\pi)+\alpha w(-\pi)=w'(\pi)+\alpha w(\pi)=0$. The result now follows from uniqueness in Proposition \ref{Prop:Runiq}.
\end{proof}
To prove our main results, we will also need several tools from symmetrization. We start with the decreasing and symmetric decreasing rearrangements.
\begin{definition} [Decreasing and Symmetric Decreasing Rearrangements]\label{def:decrearr}Suppose $X\subseteq \mathbb{R}$ is a measurable set and $f\in L^{1}(X)$ satisfies the finiteness condition \[
|\{x\in X:f(x)>t\}|<\infty,\qquad t>\underset{X}{\textup{ess\ inf}} \ f. \]
Define $f^{\ast}:[0,|X|]\rightarrow[-\infty,+\infty]$ via \[ f^{\ast}(t)=\begin{cases} \underset{X}{\textup{ess\ sup}}\ f & \textup{if}\ t=0,\\
\inf\{s:|\{x:s<f(x)\}|\leq t\} & \textup{if}\ t\in(0,|X|),\\
\underset{X}{\textup{ess\ inf}}\ f & \textup{if}\ t=|X|. \end{cases} \]
We call $f^{\ast}$ the decreasing rearrangement of $f$. The symmetric decreasing rearrangement of $f$ is the function $f^{\#}:\left[-\frac{1}{2}|X|,\frac{1}{2}|X|\right]\rightarrow[-\infty,+\infty]$ defined by $f^{\#}(t)=f^{\ast}(2|t|)$. \end{definition}
We next define the notion of a star function, first introduced by Baernstein to solve extremal problems in complex analysis. (For more on star functions and their use in analysis, see \cite{Baernstein Edrei's Spread Conjecture}, \cite{Baernstein Integral means}, \cite{Baernstein how the star function}, \cite{Barenstein Star Function in Complex Analysis}).
\begin{definition}[Star Function]\label{Def:StarFunction} Let $f\in L^{1}(X)$, where $X\subseteq \mathbb{R}$ is a measurable set of finite length. We define the star function of $f$ on the interval $[0,|X|]$ by the formula \[
f^{\bigstar}(t) = \underset{|E|=t}{\sup}\ {\displaystyle \int_{E}f\,dx}, \]
where the $\sup$ is taken over all measurable subsets $E\subseteq X$ with $|E|=t$. \end{definition} Our next proposition establishes a key connection between the star function and the decreasing rearrangement. \begin{proposition}\label{prop:genstarprop}
\label{prop:Star function achieved}Assume $f\in L^{1}(X)$ with $X\subseteq \mathbb{R}$ a measurable subset of finite length. Then for each $t\in[0,|X|]$, \[ f^{\bigstar}(t)=\int_0^tf^{\ast}(s)\,ds, \] where $f^{\ast}$ is the decreasing rearrangement of $f$. \end{proposition}
For a proof of Proposition \ref{prop:genstarprop}, see Proposition 9.2 of \cite{Baernstein Symmetrization in Analysis}.
Our proofs of Theorems \ref{Th:RComp} and \ref{Th:NComp} will show that the solutions $u$ and $v$ satisfy the star function inequality $u^{\bigstar}\leq v^{\bigstar}$. Our next result recasts this inequality into an equivalent inequality about convex means.
\begin{proposition} [Majorization]\label{Prop:Majorization} Assume $X\subseteq \mathbb{R}$ is a measurable subset of finite length and $u,v\in L^{1}(X)$. Then \[ u^{\bigstar} \leq v^{\bigstar} \]
on $[0,|X|]$ if and only if \[ \int_{X}\phi(u)\,dx \leq \int_{X}\phi(v)\,dx \] for every increasing convex function $\phi:\mathbb{R}\rightarrow\mathbb{R}$. If $\int_{X}u\,dx=\int_{X}v\, dx$, then the word ``increasing'' may be removed from the previous statement. \end{proposition} For a proof of Proposition \ref{Prop:Majorization}, see Propositions 10.1 and 10.3 of \cite{Baernstein Symmetrization in Analysis}.
If the convex means of $u$ are dominated by those of $v$ and additional information is known about $u$ and $v$, we can deduce further inequalities about $L^p$-norms, $\esssup$, $\essinf$, and $\osc$.
\begin{corollary}\label{cor:Lpnormsgen}
\label{cor:consequences of u star leq v star} Say $u,v\in L^{1}(X)$ where $X\subseteq \mathbb{R}$ is a measurable subset of finite length and assume $u^{\bigstar}\leq v^{\bigstar}$ on $[0,|X|]$. If either $u,v\geq 0$ or $\int_{X}u\,dx=\int_{X}v\,dx$, then \[
\|u\|_{L^{p}(X)} \leq \|v\|_{L^{p}(X)},\quad1\leq p\leq +\infty. \] If $\int_{X}u\,dx=\int_{X}v\,dx$, moreover \[ \underset{X}{\esssup}\ u \leq \underset{X}{\esssup}\ v, \qquad \underset{X}{\essinf}\ v \leq \underset{X}{\essinf}\ u, \qquad \underset{X}{\osc}\ u \leq \underset{X}{\osc}\ v, \] where $\osc=\esssup - \essinf$. \end{corollary}
We end the background section with three rearrangement inequalities. These inequalities play a major role in our proofs of Theorems \ref{Th:RComp} and \ref{Th:NComp}. The first two are well known (for discussion and proofs see Theorem 1.2.2 of \cite{Kesevan} and Theorem 8.4 of \cite{Baernstein Symmetrization in Analysis}). The third rearrangement inequality appears to be less well known (for discussion and proof see \cite{BaernsteinS1} or Theorem 8.1 of \cite{Baernstein Symmetrization in Analysis}). \begin{theorem}[Hardy-Littlewood]\label{th:HL} Given $f\in L^1[-\pi,\pi]$ and $g\in L^{\infty}[-\pi,\pi]$, we have \[ \int_{-\pi}^{\pi}fg\,dx\leq \int_{-\pi}^{\pi}f^{\#}g^{\#}\,dx, \] with $\#$ the symmetric decreasing rearrangement. \end{theorem}
\begin{theorem}[Riesz-Sobolev]\label{th:RS} Suppose $f,g,h\in L^1(\mathbb{R})$ are nonnegative. Then we have \[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x)g(y)h(x-y)\,dy\,dx\leq \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{\#}(x)g^{\#}(y)h^{\#}(x-y)\,dy\,dx, \] with $\#$ the symmetric decreasing rearrangement. \end{theorem}
\begin{theorem}[Baernstein]\label{th:Baernstein} Let $f,g\in L^1[-\pi,\pi]$ and $h\in L^{\infty}[-\pi,\pi]$ be $2\pi$-periodic functions on $\mathbb{R}$. Then \[ \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}f(x)g(y)h(x-y)\,dy\,dx\leq \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}f^{\#}(x)g^{\#}(y)h^{\#}(x-y)\,dy\,dx, \] with $\#$ the $2\pi$-periodic extension of symmetric decreasing rearrangement on $[-\pi,\pi]$ to all of $\mathbb{R}$. \end{theorem}
\section{Proofs of Main Results}
\subsection*{The Robin problem}
We start this section with a proof of our first main result.
\begin{proof}[Proof of Theorem \ref{Th:RComp}] Say $E\subseteq [-\pi,\pi]$ is a measurable subset. Then \[ \int_Eu(x)\,dx=\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E(x)f(y)G(x,y)\,dy\,dx, \] where $G$ is the Robin Green's function from Proposition \ref{Prop:RGreen}. Observe that \begin{align*}
G(x,y)&=-\frac{1}{2}c_{\alpha}xy-\frac{1}{2}|x-y|+\frac{1}{2c_{\alpha}}\\
&=-\frac{1}{4}c_{\alpha}(x^2+y^2)+\frac{1}{4}c_{\alpha}(x-y)^2-\frac{1}{2}|x-y|+\frac{1}{2c_{\alpha}} \end{align*} where $c_\alpha$ is the constant given in (\ref{eq:calphadef}). Since $f,f^{\#}$ are rearrangements and nonnegative, applying Theorem \ref{th:HL} to the $dx$ integral yields \begin{equation}\label{eq:term1} \frac{1}{4}c_{\alpha}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E(x)f(y)(-x^2)\,dy\,dx \leq \frac{1}{4}c_{\alpha}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E^{\#}(x)f^{\#}(y)(-x^2)\,dy\,dx. \end{equation} Similarly, $\chi_E,\chi_E{^{\#}}$ are rearrangements, so applying Theorem \ref{th:HL} to the $dy$ integral gives \begin{equation}\label{eq:term2} \frac{1}{4}c_{\alpha}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E(x)f(y)(-y^2)\,dy\,dx\leq \frac{1}{4}c_{\alpha}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E^{\#}(x)f^{\#}(y)(-y^2)\,dy\,dx. \end{equation}
Next, write \[
h(z)=\frac{1}{4}c_{\alpha}z^2-\frac{1}{2}|z|,\qquad z\in [-2\pi,2\pi]. \]
Write $\tilde h$ for the $2\pi$-periodic extension of $h\big|_{[-\pi,\pi]}$ to all of $\mathbb{R}$. When $z\in(0,\pi]$, note that \[ h'(z)=\frac{1}{2}c_{\alpha}z-\frac{1}{2}\leq \frac{1}{2}c_{\alpha}\pi-\frac{1}{2}=-\frac{1}{2(1+\alpha \pi)}<0. \] Thus, $h$ is symmetric decreasing on $[-\pi,\pi]$. It follows from Theorem \ref{th:Baernstein} that \begin{equation}\label{eq:UseBaernstein} \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E(x)f(y)\tilde h(x-y)\,dy\,dx \leq \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E^{\#}(x)f^{\#}(y)\tilde h(x-y)\,dy\,dx. \end{equation} Moreover, on $[\pi,2\pi]$, \begin{align*} h(z)-\tilde h(z)&= h(z)-h(2\pi-z)\\ &=\frac{1}{4}c_{\alpha}z^2-\frac{1}{2}z-\left(\frac{1}{4}c_{\alpha}(2\pi-z)^2-\frac{1}{2}(2\pi-z)\right)\\ &=(z-\pi)(c_{\alpha}\pi-1). \end{align*} As we saw above, $c_{\alpha}\pi-1<0$. And since $h=\tilde h$ on $[0,\pi]$, it follows that $h-\tilde h$ is symmetric decreasing on $[-2\pi,2\pi]$. Extend $\chi_E$ and $f$ to vanish outside $[-\pi,\pi]$ and extend $h-\tilde h+\pi(1-c_{\alpha}\pi)$ to vanish outside $[-2\pi,2\pi]$. Then Theorem \ref{th:RS} gives \begin{align*} &\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E(x)f(y)\left((h-\tilde h)(x-y)+\pi(1-c_{\alpha}\pi)\right)\,dy\,dx\\ &\qquad \qquad \qquad \qquad \leq \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E^{\#}(x)f^{\#}(y)\left((h-\tilde h)(x-y)+\pi(1-c_{\alpha}\pi)\right)\,dy\,dx. \end{align*} Since $\chi_E,\chi_E{^{\#}}$ and $f,f^{\#}$ are rearrangements, the inequality above gives \begin{equation}\label{eq:UseRS} \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E(x)f(y)(h-\tilde h)(x-y)\,dy\,dx \leq \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_E^{\#}(x)f^{\#}(y)(h-\tilde h)(x-y)\,dy\,dx. \end{equation} Combining inequalities \eqref{eq:UseBaernstein} and \eqref{eq:UseRS} and noting $\chi_E^{\#}=\chi_{E^{\#}}$ gives \begin{equation}\label{eq:term3} \int_{-\pi}^{\pi}\int_{-\pi}^{\pi} \chi_{E}(x)f(y)h(x-y)\,dy\,dx \leq \int_{-\pi}^{\pi}\int_{-\pi}^{\pi} \chi_{E^{\#}}(x)f^{\#}(y)h(x-y)\,dy\,dx. \end{equation} Finally, note that \begin{equation}\label{eq:term4} \frac{1}{2c_{\alpha}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_{E}(x)f(y)\,dy\,dx=\frac{1}{2c_{\alpha}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\chi_{E^{\#}}(x)f^{\#}(y)\,dy\,dx, \end{equation} again as $\chi_E,\chi_E{^{\#}}$ and $f,f^{\#}$ are rearrangements. Combining \eqref{eq:term1}, \eqref{eq:term2}, \eqref{eq:term3}, and \eqref{eq:term4} shows \[ \int_{E}u(x)\,dx\leq \int_{E^{\#}}v(x)\,dx. \] Taking the $\sup$ over all measurable subsets $E\subseteq [-\pi,\pi]$ with fixed length, say $t$, the inequality above gives \[ u^{\bigstar}(t) \leq \int_{-\frac{t}{2}}^{\frac{t}{2}}v(x)\,dx \leq v^{\bigstar}(t). \] The theorem's claims about convex means now follows from Proposition \ref{Prop:Majorization}. To prove the remaining claims, we next argue that $u,v\geq 0$. First note that $u$ is concave, so $\underset{[-\pi,\pi]}{\min}\ u$ is achieved at either $-\pi$ or $\pi$. Suppose the minimum occurs at $\pi$. If $u(\pi)<0$, then from the Robin boundary condition we see \[ u'(\pi)=-\alpha u(\pi)>0, \] and so $\underset{[-\pi,\pi]}{\min}\ u$ cannot be achieved at $\pi$. We conclude that $u(\pi)\geq 0$. An identical argument applies when the minimum occurs at $-\pi$ from which we conclude that $u$ is nonnegative. Since the same argument applies to the function $v,$ we conclude that $v$ is also nonnegative. The theorem's remaining conclusions now follow from Proposition \ref{cor:Lpnormsgen}. \end{proof}
\subsection*{From Robin to Dirichlet} As mentioned in the introduction, our Robin comparison principle implies the corresponding Dirichlet result. The main idea behind the proof is that as $\alpha$ tends to $+\infty$, the Robin boundary condition converts into a Dirichlet condition.
\begin{corollary}[ODE Dirichlet Comparison Principle]\label{Cor:DComp} Let $0\leq f\in L^1[-\pi,\pi]$ and suppose $u$ and $v$ solve the Poisson problems \begin{align*} -u''&=f \quad \textup{in} \quad (-\pi,\pi), \qquad u(-\pi)=u(\pi)=0,\\ -v''&=f^{\#} \quad \textup{in} \quad (-\pi,\pi), \qquad v(-\pi)=v(\pi)=0, \end{align*} with $f^{\#}$ the symmetric decreasing rearrangement of $f$. Then \[ \int_{-\pi}^{\pi}\phi(u)\,dx\leq \int_{-\pi}^{\pi}\phi(v)\,dx \] for each increasing convex function $\phi:\mathbb{R}\to \mathbb{R}$. \end{corollary}
\begin{proof} Let $f,u$, and $v$ be as stated, and suppose $u_k$ and $v_k$ solve the Robin problems \begin{align*} -u_k''&=f \quad \textup{in} \quad (-\pi,\pi), \qquad -u_k'(-\pi)+\alpha_k u_k(-\pi)=u_k'(\pi)+\alpha_k u_k(\pi)=0,\\ -v_k''&=f^{\#} \quad \textup{in} \quad (-\pi,\pi), \qquad -v_k'(-\pi)+\alpha_k v_k(-\pi)=v_k'(\pi)+\alpha_k v_k(\pi)=0, \end{align*} where $0<\alpha_k\to +\infty$. Then by Theorem \ref{Th:RComp} and Proposition \ref{Prop:Majorization}, \begin{equation}\label{eq:ukvkstar} u_k^{\bigstar}(t)\leq v_k^{\bigstar}(t),\qquad t\in [0,2\pi]. \end{equation} If $G_{\alpha_k}$ denotes the Robin Green's function of Proposition \ref{Prop:RGreen} with parameter $\alpha_k$, then \[ G_{\alpha_k}(x,y)\to G(x,y) \] uniformly on $[-\pi,\pi]\times[-\pi,\pi]$, where \[
G(x,y)=-\frac{1}{2\pi}xy-\frac{1}{2}|x-y|+\frac{\pi}{2} \] is the Dirichlet Green's function on $[-\pi,\pi]$. Using the Green's representation, we see $u_k\to u$ and $v_k\to v$ uniformly on $[-\pi,\pi]$. Since symmetrization decreases the $L^1$-distance, we note that for any $t\in[0,2\pi]$, \begin{align*}
|u_k^{\bigstar}(t)-u^{\bigstar}(t)|&= \left|\int_{-\frac{t}{2}}^{\frac{t}{2}}\left(u_k^{\#}(x)-u^{\#}(x)\right)\,dx\right|\\
&\leq \int_{-\pi}^{\pi}\left|u_k^{\#}(x)-u^{\#}(x)\right|\,dx\\
&\leq \int_{-\pi}^{\pi}\left|u_k(x)-u(x)\right|\,dx, \end{align*} and this last term tends to zero by uniform convergence. Thus we have pointwise convergence of star functions: \begin{equation}\label{eq:ukstar} u^{\bigstar}_k(t)\to u^{\bigstar}(t),\qquad t\in[0,2\pi]. \end{equation} A similar argument shows \begin{equation}\label{eq:vkstar} v^{\bigstar}_k(t)\to v^{\bigstar}(t),\qquad t\in[0,2\pi]. \end{equation} Combining inequality \eqref{eq:ukvkstar} with \eqref{eq:ukstar} and \eqref{eq:vkstar} gives $u^{\bigstar}\leq v^{\bigstar}$ in $[0,2\pi]$, and this inequality is equivalent to the corollary's conclusion courtesy of Proposition \ref{Prop:Majorization}.
\end{proof}
The conclusion of Corollary \ref{Cor:DComp} can be strengthened to $u^{\#} \leq v$ in $[-\pi,\pi]$, though this stronger conclusion is not needed for our paper. The argument is simple but requires additional tools developed by Baernstein. We include the proof for the sake of completeness.
Define \[ w(s)=u^{\bigstar}(2s)-v^{\bigstar}(2s),\qquad s\in[0,\pi]. \] Then by Theorem 9.20 in \cite{Baernstein Symmetrization in Analysis} or Theorem 5 of \cite{Baernstein Cortona Volume}, $\frac{d^2}{ds^2}w(s)\geq 0$ weakly. That is, \[ \int_{0}^{\pi}w(s)G''(s)\,ds \geq 0 \] for each $G_c^2(0,\pi)$ nonnegative with compact support. Integrating by parts, we see \[ \int_{0}^{\pi}w'(s)G'(s)\,ds \leq 0. \] An easy argument gives that $w'(s)$ is increasing. But \[ \frac{d}{ds}w(s)=\frac{d}{ds}\int_{-s}^s\left(u^{\#}(x)-v^{\#}(x)\right)\,dx=2\left(u^{\#}(s)-v^{\#}(s)\right) \] and so we see that $u^{\#}(s)-v^{\#}(s)$ is increasing for $s\in[0,\pi]$. But as $u^{\#}(\pi)=v^{\#}(\pi)=0$, we conclude $u^{\#}(s)-v^{\#}(s)\leq 0$. Finally, since $v=v^{\#}$, this last inequality implies $u^{\#}\leq v$ on $[-\pi,\pi]$.
\subsection*{An interesting example} As noted in the introduction, with Robin problems the oscillation need not increase under symmetric decreasing rearrangement. Consider the following example.
\begin{example}\label{ex:osc} Consider the solutions $u$ and $v$ to the Poisson problems of Theorem \ref{Th:RComp} with $f=\chi_{[-\pi,0]}$: \begin{align*} -u''&=\chi_{[-\pi,0]} \quad \textup{in} \quad [-\pi,\pi], \qquad -u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0,\\ -v''&=\chi_{\left[-\frac{\pi}{2},\frac{\pi}{2}\right]} \quad \textup{in} \quad [-\pi,\pi], \qquad -v'(-\pi)+\alpha v(-\pi)=v'(\pi)+\alpha v(\pi)=0. \end{align*} \end{example} It is straightforward to check that \[ u(x)=-u_1(x)+\left(\frac{\pi}{2}+\frac{\pi^2}{4}c_{\alpha}\right)x+\frac{\pi}{2c_{\alpha}}+\frac{\pi^2}{4}, \] where \[ u_1(x)= \begin{cases} \frac{1}{2}x^2+\pi x+\frac{\pi^2}{2} & \textup{if } -\pi\leq x< 0,\\ \pi x+\frac{\pi^2}{2} & \textup{if } 0\leq x\leq \pi, \end{cases} \] and $c_{\alpha}$ is defined in \eqref{eq:calphadef}. Similarly, \[ v(x)=-v_1(x)+\frac{\pi}{2}x+\frac{\pi}{2c_{\alpha}}, \] where \[ v_1(x)= \begin{cases} 0 & \textup{if }-\pi \leq x<-\frac{\pi}{2},\\ \frac{1}{2}x^2+\frac{\pi}{2}x+\frac{\pi^2}{8} & \textup{if } -\frac{\pi}{2} \leq x < \frac{\pi}{2},\\ \pi x & \textup{if } \frac{\pi}{2}\leq x\leq \pi. \end{cases} \] Since $v$ is symmetric decreasing, we have \[ \underset{[-\pi,\pi]}{\osc}\ v=v(0)-v(\pi)=\frac{3\pi^2}{8}. \] Note, rather curiously, that this oscillation is independent of $\alpha$. On the other hand, \[ \underset{[-\pi,\pi]}{\osc}\ u \geq u\left(-\frac{\pi}{2}\right)-u(\pi)=\frac{\pi^2(5+2\alpha \pi)}{8(1+\alpha \pi)}. \] Now it is easy to verify that $\frac{3\pi^2}{8}<\frac{\pi^2(5+2\alpha \pi)}{8(1+\alpha \pi)}$ so long as $\alpha<\frac{2}{\pi}$, which ensures $\underset{[-\pi,\pi]}{\osc}\ v < \underset{[-\pi,\pi]}{\osc}\ u$.
Example \ref{ex:osc} leads to an interesting open question.
\begin{problem}[Open]\label{Prob:1dRodRobinOpen} Suppose half of a rod of length $2\pi$ with Robin boundary conditions is heated uniformly, while in the remaining half, heat is neither generated nor absorbed. Where should we place the heat sources to maximize the temperature gap? \end{problem}
Our intuition suggests that an optimal source in Problem \ref{Prob:1dRodRobinOpen} is an interval of length $\pi$. Assuming this is the case, we have the following proposition.
\begin{proposition}\label{prop:robingapint} Assume the temperature gap in Problem \ref{Prob:1dRodRobinOpen} is maximized by a source interval of length $\pi$. Then when $0<\alpha\leq \frac{2}{\sqrt{3}\pi}$, the temperature gap is maximized by locating the source interval at either end of the rod. As $\alpha$ increases from $\frac{2}{\sqrt{3}\pi}$ to $+\infty$, the temperature gap is maximized by a source interval that continuously transitions from one end of the rod towards its center. \end{proposition}
\begin{proof} Denote $I_b=\left[b-\frac{\pi}{2},b+\frac{\pi}{2}\right]$ for $-\frac{\pi}{2}\leq b\leq \frac{\pi}{2}$. We consider the gap function \[ \textup{Gap}(\alpha,b)=\max_{[-\pi,\pi]}u-\min_{[-\pi,\pi]}u,\qquad 0<\alpha<\infty, \quad -\frac{\pi}{2}\leq b\leq \frac{\pi}{2}, \] where $u$ is the solution of the Poisson problem \[ -u''=\chi_{I_b} \quad \textup{in} \quad (-\pi,\pi), \qquad -u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0. \] By symmetry, one has \[ \textup{Gap}(\alpha,-b)=\textup{Gap}(\alpha,b),\quad 0\leq b\leq \frac{\pi}{2}, \] and so for the remainder of the argument, we focus our attention on $\textup{Gap}$ when $-\frac{\pi}{2}\leq b\leq 0$, i.e. on source intervals whose center lies in the left half of the rod.
With a bit of work and help from Mathematica, one computes \[ \textup{Gap}(\alpha,b)=-\frac{\pi(1+\alpha(b+\pi))(-3\pi(1+\alpha\pi)+b(4+3\alpha\pi))}{8(1+\alpha \pi)^2} \] and so \begin{align} \frac{\partial\textup{Gap}}{\partial b}(\alpha,b)&=-\frac{\pi(2+2\alpha \pi+\alpha b(4+3\alpha \pi))}{4(1+\alpha \pi)^2},\label{eq:gapfirstd}\\ \frac{\partial^2\textup{Gap}}{\partial b^2}(\alpha,b)&=-\frac{\alpha \pi (4+3\alpha\pi)}{4(1+\alpha \pi)^2}.\label{eq:gapconv} \end{align} Equation \eqref{eq:gapfirstd} implies that $\textup{Gap}(\alpha,b)$ is decreasing in $b$ on $\left[-\frac{\pi}{2},0\right]$ when $\alpha \in \left(0,\frac{2}{\sqrt{3}\pi}\right]$. This establishes the proposition's first claim.
Equation \eqref{eq:gapconv} says that $\textup{Gap}(\alpha,b)$ is strictly concave in $b$ holding $\alpha$ fixed. We also note that \[ \frac{\partial \textup{Gap}}{\partial b}\left(\alpha,-\frac{\pi}{2}\right)=\frac{\pi(-4+3\alpha^2\pi^2)}{8(1+\alpha\pi)^2}>0 \] when $\alpha>\frac{2}{\sqrt{3}\pi}$. It also holds that \[ \frac{\partial \textup{Gap}}{\partial b}(\alpha,0)=-\frac{\pi}{2(1+\alpha \pi)}<0, \] and so $\textup{Gap}(\alpha,b)$ is maximized at some $b\in \left(-\frac{\pi}{2},0\right)$ when $\alpha>\frac{2}{\sqrt{3}\pi}$. Returning to equation \eqref{eq:gapfirstd}, we note that $\textup{Gap}(\alpha,b)$ is maximized when \[ b=b_{\textup{crit}}=-\frac{2(1+\alpha \pi)}{\alpha(4+3\alpha \pi)}. \] Since \[\frac{d b_{\textup{crit}}}{d \alpha}= \frac{8+6\alpha \pi(2+\alpha \pi)}{\alpha^2(4+3\alpha \pi)^2}>0, \] we conclude that $b_{\textup{crit}}$ continuously increases from $-\frac{\pi}{2}$ towards $0$ as $\alpha$ increases from $\frac{2}{\sqrt{3}\pi}$ to $+\infty$. \end{proof}
\subsection*{The Neumann problem} Our approach here is driven by Fourier series. Suppose that $f\in L^1[-\pi,\pi]$ has zero mean. One attempt to solve the Poisson equation $-u''=f$ in $(-\pi,\pi)$ might be to consider the function whose Fourier series is given by \begin{equation}\label{eq:fourier} u(x)=\sum_{n\neq 0}\frac{1}{n^2}\hat f(n)e^{inx}, \end{equation} where $\hat f(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\,dx$. Formally differentiating the equation above termwise shows \[ -u''(x)=\sum_{n\neq 0}\hat f(n)e^{inx}=f(x). \] We are thus led to consider the function $K$ whose Fourier coefficients are given by \[ \hat K(n)= \begin{cases} \frac{1}{n^2} & \textup{if }n\neq 0,\\ 0 & \textup{if }n=0. \end{cases} \] One readily verifies that \[
K(x)=\frac{1}{2}x^2-\pi |x|+\frac{1}{3}\pi^2, \qquad -\pi \leq x \leq \pi. \] Extend $f$ and $K$ to all of $\mathbb{R}$ by $2\pi$-periodicity. The function $u$ in \eqref{eq:fourier} has the property that\ \[ \hat{u}(n)=\hat{K}(n)\hat{f}(n)=\widehat{K\ast f}(n) \] which leads us to study the convolution \begin{equation}\label{eq:udefcont} u(x)=(K\ast f)(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}K(x-y)f(y)\,dy. \end{equation} We now investigate how the $u$ defined in \eqref{eq:udefcont} differs from the $u$ in Proposition \ref{Prop:Nuniq}. We will see that $u$ defined in \eqref{eq:udefcont} satisfies properties 1, 2, and 4 of Proposition \ref{Prop:Nuniq}. Below, we continue to identify $K$ and $f$ with their $2\pi$-periodic extensions.
\begin{proposition}\label{prop:uconvdef} The function $u=K\ast f$ defined in \eqref{eq:udefcont} satisfies the following properties: \begin{enumerate} \item[1.] $u$ is continuously differentiable on $\mathbb{R}$ with $u'=K'\ast f$. \item[2.] $u'$ is absolutely continuous on $[-\pi,\pi]$ and \[ u'(x)-u'(-\pi)=-\int_{-\pi}^xf(y)\,dy,\qquad -\pi \leq x \leq \pi, \] with $-u''=f$ a.e. in $(-\pi,\pi)$. \item[3.] $u'(-\pi)=u'(\pi)=-\frac{1}{2\pi}\int_{-\pi}^{\pi}xf(x)\,dx$. \item[4.] $\int_{-\pi}^{\pi}u(x)\,dx=0$. \end{enumerate} \end{proposition}
\begin{proof} Property 1 is a standard fact about convolutions. For property 2, we compute \begin{align*} (K'\ast f)(x)&=\frac{1}{2\pi}\int_{-\pi}^{\pi}K'(y)f(x-y)\,dy\\ &=\frac{1}{2\pi}\int_{-\pi}^{0}(y+\pi)f(x-y)\,dy+\frac{1}{2\pi}\int_{0}^{\pi}(y-\pi)f(x-y)\,dy\\ &=\frac{1}{2\pi}\int_{x-\pi}^x(x-y-\pi)f(y)\,dy+\frac{1}{2\pi}\int_{x}^{x+\pi}(x-y+\pi)f(y)\,dy, \end{align*} and from this representation it follows that $(K'\ast f)(x)$ is absolutely continuous on $[-\pi,\pi]$. Its derivative equals $-f(x)$ by direct calculation, for almost every $x$. Thus property 2 holds. For property 3, one sees from the above calculation \[ (K'\ast f)(-\pi)=(K'\ast f)(\pi)=-\frac{1}{2\pi}\int_{-\pi}^{\pi}yf(y)\,dy. \] To establish property 4 we simply observe \[ \frac{1}{2\pi}\int_{-\pi}^{\pi}u(x)\,dx=\hat{u}(0)=\hat{K}(0) \hat{f}(0)=0. \] \end{proof}
A consequence of Proposition \ref{prop:uconvdef} is:
\begin{proposition}\label{prop:uisconvwf} Let $f$ and $u$ be as in Proposition \ref{Prop:Nuniq}. If $\int_{-\pi}^{\pi}xf(x)\,dx=0$, then $u$ is given by convolution as in equation \eqref{eq:udefcont}. \end{proposition}
Before proving Theorem \ref{Th:NComp}, we first prove a preliminary comparison result.
\begin{theorem}\label{Th:Nprelim} Let $f\in L^1[-\pi,\pi]$ where $\int_{-\pi}^{\pi}f(x)\,dx=\int_{-\pi}^{\pi}xf(x)\,dx=0$. Let $u$ and $v$ solve the Poisson problems \begin{align*} -u''&=f \quad \textup{in} \quad (-\pi,\pi), \qquad u'(-\pi)=u'(\pi)=0,\\ -v''&=f^{\#} \quad \textup{in} \quad (-\pi,\pi), \qquad v'(-\pi)=v'(\pi)=0, \end{align*} where $f^{\#}$ is the decreasing rearrangement of $f$, and $\int_{-\pi}^{\pi}u(x)\,dx=\int_{-\pi}^{\pi}v(x)\,dx=0$. Then \[ u^{\bigstar}(t)\leq \int_{-\frac{t}{2}}^{\frac{t}{2}}v(x)\,dx \leq v^{\bigstar}(t) \] on $[0,2\pi]$. \end{theorem}
\begin{proof} By Proposition \ref{prop:uisconvwf}, we have $u=K\ast f$ and $v=K\ast f^{\#}$ (because $f^{\#}$ is even, and so $\int_{-\pi}^{\pi}xf^{\#}(x)\,dx=0$). Fix $t\in [0,2\pi]$ and let $E\subseteq [-\pi,\pi]$ denote a measurable subset of length $t$. Applying Theorem \ref{th:Baernstein} we see \[ \int_Eu\,dx=\int_E K\ast f\,dx \leq \int_{E^{\#}}K\ast f^{\#}\,dx=\int_{E^{\#}}v\,dx. \] Taking the $\sup$ in the above inequality over all measurable subsets $E$ of $[-\pi,\pi]$ of length $t$, we obtain the desired conclusion. \end{proof}
We are now prepared to prove our paper's second main result.
\begin{proof}[Proof of Theorem \ref{Th:NComp}] Given $f\in L^1[0,\pi]$, extend $f$ to $[-\pi,\pi]$ by even reflection, and denote this extension by $\tilde f$. Observe that $\int_{-\pi}^{\pi}x\tilde f(x)\,dx=0$. Clearly, \begin{equation}\label{eq:sdrftild} (\tilde f)^{\#}(x)=f^{\ast}(x),\qquad 0\leq x\leq \pi. \end{equation} Let $u$ correspond to $f$ in the analogue of Proposition \ref{Prop:Nuniq} over $[0,\pi]$ and similarly let $v$ correspond to $f^{\ast}$. Let $\tilde u$ and $\tilde v$ correspond to $\tilde f$ and $(\tilde f)^{\#}$ in the $[-\pi,\pi]$ version of Proposition \ref{Prop:Nuniq}. Proposition \ref{prop:uisconvwf} gives $\tilde u = K\ast \tilde f$ and $\tilde v = K \ast (\tilde f)^{\#}$.
We claim that $\tilde u$ is obtained from $u$ by even reflection. We show that $\tilde u$ is even and that $\tilde u$ also satisfies the properties of Proposition \ref{Prop:Nuniq} corresponding to $f$ over the interval $[0,\pi]$. First, $\tilde u$ is even: \[ \tilde u(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}K(x-y)\tilde f(y)\,dy=\frac{1}{2\pi}\int_{-\pi}^{\pi}K(x+y)\tilde f(y)\,dy=\frac{1}{2\pi}\int_{-\pi}^{\pi}K(-x-y)\tilde f(y)\,dy=\tilde u(-x). \] Since $\tilde u\in C^1[-\pi,\pi]$ and is even, we must have $\tilde u'(0)=0$. Additionally, we have $\tilde u'(\pi)=0$ by assumption. Again, being even implies $\int_{0}^{\pi}\tilde u(x)\,dx=\frac{1}{2}\int_{-\pi}^{\pi}\tilde u(x)\,dx=0$. Hence by uniqueness, $\tilde u(x)=u(x)$ on $[0,\pi]$ and so $(\tilde u)^{\#}(x)=u^{\ast}(x)$ on $[0,\pi]$. We similarly have $\tilde v(x)=v(x)$ on $[0,\pi]$. By Theorem \ref{Th:Nprelim}, we have for each $0\leq t\leq 2\pi$ \[ \int_{-\frac{t}{2}}^{\frac{t}{2}}(\tilde u)^{\#}(x)\,dx \leq \int_{-\frac{t}{2}}^{\frac{t}{2}}\tilde v(x)\,dx \] which implies \[ \int_0^{\frac{t}{2}}(\tilde u)^{\#}(x)\,dx\leq \int_0^{\frac{t}{2}}\tilde v(x)\,dx \] finally giving \[ \int_0^{\frac{t}{2}}u^{\ast}(x)\,dx\leq \int_0^{\frac{t}{2}}v(x)\,dx. \] This last inequality implies $u^{\bigstar} \leq v^{\bigstar}$ on $[0,\pi]$ and the theorem's claims on convex means follows from Proposition \ref{Prop:Majorization}. The remaining conclusions follow from Proposition \ref{cor:Lpnormsgen} since $u$ and $v$ have zero mean by assumption. \end{proof}
\end{document} | arXiv | {
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\begin{document}
\title{
A Dynamic Resource Allocation Framework for Synchronizing Metaverse with IoT Service and Data \\ }
\author {\IEEEauthorblockN{ Yue~Han\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}\IEEEauthorrefmark{3}, Dusit~Niyato\IEEEauthorrefmark{1} Cyril~Leung\IEEEauthorrefmark{4}\IEEEauthorrefmark{5}, Chunyan~Miao\IEEEauthorrefmark{1}\IEEEauthorrefmark{5}, Dong~In~Kim\IEEEauthorrefmark{6} }
\IEEEauthorblockA{ \IEEEauthorrefmark{2} Alibaba-NTU JRI, Interdisciplinary Graduate School, Nanyang Technological University (NTU) \IEEEauthorrefmark{1} SCSE, NTU \\ \IEEEauthorrefmark{3} Alibaba Group \IEEEauthorrefmark{5} LILY, NTU \IEEEauthorrefmark{4} ECE, The University of British Columbia \IEEEauthorrefmark{6} ECE, Sungkyunkwan University\\ } }
\maketitle \begin{abstract} Spurred by the severe restrictions on mobility due to the COVID-19 pandemic, there is currently intense interest in developing the Metaverse, to offer virtual services/business online. A key enabler of such virtual service is the digital twin, i.e., a digital replication of real-world entities in the Metaverse, e.g., city twin, avatars, etc. The real-world data collected by IoT devices and sensors are key for synchronizing the two worlds. In this paper, we consider the scenario in which a group of IoT devices are employed by the Metaverse platform to collect such data on behalf of virtual service providers (VSPs). Device owners, who are self-interested, dynamically select a VSP to maximize rewards. We adopt hybrid evolutionary dynamics, in which heterogeneous device owner populations can employ different revision protocols to update their strategies. Extensive simulations demonstrate that a hybrid protocol can lead to evolutionary stable states. \end{abstract}
\begin{IEEEkeywords} Metaverse, resource allocation, IoT, evolutionary game, hybrid dynamics \end{IEEEkeywords}
\section{Introduction} {COVID-19} has dramatically changed work and life styles, from physical to online/virtual experience, e.g., attending virtual job fair, concert, meeting, and graduation ceremony. Generally, those virtual spaces are referred to as `Metaverse', where `meta' means virtual and transcendence, and `verse' represents universe. Formally, Metaverse describes a `{computer-generated, multi-user, three-dimensional interfaces in which users experience other participants as being present in the environment}' \cite{schroederSocialInteractionVirtual2002}.
To realize the Metaverse, digitization of the real world is needed. In the Metaverse, {information} measured in bits are the main existence. Living and nonliving entities in the physical world, e.g., drivers, roads, can be \textit{scanned} by smart devices, e.g., IoT devices and sensors, and be digitally replicated in the Metaverse. This digital replication idea has been recently studied by many researchers, around a concept called \textit{digital twin} \cite{petrova-antonovaDigitalTwinModeling2020,fullerDigitalTwinEnabling2020,elsaddikDigitalTwinsConvergence2018}. An actual application is the smart city twin \cite{petrova-antonovaDigitalTwinModeling2020}, in which a virtual city could be operated in the Metaverse, so that new polices, e.g., traffic intervention, could be $A/B$ tested in the Metaverse prior to implementation in the physical world. The benefits of having digital twins in the Metaverse are obvious, as the Metaverse can facilitate the monitoring, understanding, and optimization of the real-world business and its functions with infinite, cost-efficient experiments, collecting continuous feedback without affecting the business in the physical world. The benefits are particularly important when the test of the intervention is expensive and irreversible in the real world, e.g., potential environment destruction.
\begin{figure*}
\caption{System Model: IoT-assisted data collection to enable sync between the Metaverse and the physical world. Here, IoT is exemplified by UAV.}
\label{fig:system4}
\caption{Four Components for the Metaverse}
\label{fig:arc}
\end{figure*}
A key aspect in implementing a better digital twin is the continuous data synchronization(sync) in the Metaverse, i.e., acquiring fresh data from the physical world to keep the digital twin updated in real time. For example, a virtual driver training service provider \cite{taheriVirtualRealityDriving2017}, which simulates the virtual road using real data (such as road, drivers, traffic, and weather), needs to have continuous data updates to ensure that its service (e.g., driving skill test) is realistic, and the trainee's real driving skill can be assessed. To enable such frequent sync across two worlds, we consider to employ IoT devices, such as autonomous driving cars, unmanned aerial vehicle (UAV), and smart phones, to collect the data around a region that a virtual service provider (VSP) is of interest.
In general, we consider a resource allocation problem to support the sync between the Metaverse and the real world. Our system model is shown in \cref{fig:system4}. Although we use UAVs in this illustrative example, the model is general and can be extended to different types of IoT devices. Suppose a VSP is interested in collecting data from a region in the real world (e.g., Region $1$ in \cref{fig:system4}) for a service offering, e.g., simulating a road. The VSP first announces a data collection task (step ${A}$) over a particular region for a particular period, e.g., a week. Then, the Metaverse platform, which hosts multiple VSPs, supports the VSP in its synchronizing task by engaging the services of IoT devices in the real world (e.g., services of the UAVs located in a nearby Region $3$ in \cref{fig:system4}).
Each UAV is owned by an independent entity that can decide which task to work on (step $B$). After a UAV collects the sensing data, data pre-processing takes place at each UAV's base (e.g., home), following which the processed data are transmitted to 1) the local Metaverse platform cached at the edge clouds to support real-time and interactive services, and 2) the core of the Metaverse platform located at data centers with large capacity of computation, communication, and storage, supporting the Metaverse's core functionality, e.g. intensive simulations of the virtual worlds, continuous user identity.
As the duration of a task can last relative long, a UAV owner can gradually improve its decision making by observing the reward obtained by similar UAVs selecting other sync tasks and learning from them (step $C$). Finally, an equilibrium state at which no UAV owner would unilaterally change its decision can be reached.
The main contributions of this paper are as follows: \begin{itemize} \item We identify a resource allocation problem for virtual service sync selection in which the IoT devices in a particular region in the real world are hired by the Metaverse platform to support its hosted VSPs towards a more efficient and collaborative environment of virtual content creation. \item In consideration of a large number of independent IoT device owners at the edge and their self-interested nature, the equilibrium knowledge (full rationale) is hardly-achievable. We propose a dynamic approach founded on evolutionary game theory, in which the \textit{bounded-rational} device owners can gradually adapt their strategies towards the equilibrium state. \item Our evolutionary game framework considers a general sensing model and reward allocation scheme and thus can be extended to specific tasks that support the sync between the Metaverse and the real-world. \end{itemize}
\section{Metaverse Preliminaries and Related Work} In this section, we first introduce the architecture that composes a Metaverse, followed by related works and the research gap.
\subsection{Metaverse General Architecture}\label{subsection:metaverse-general-architecture} Architectures such as \cite{radoffMetaverseValueChain2021} and \cite{duanMetaverseSocialGood2021} have been proposed for the Metaverse. Based on these, we propose a four-component architecture as shown in \cref{fig:arc}. It includes infrastructure, interface, cross-world ecosystem, and in-world ecosystem.
\textit{\textbf{Infrastructure}} (including communication, computation, blockchain, and other decentralization techniques) is to support all operations in the {physical world}, the {Metaverse}, and the connection between the two worlds.
The services that inter-connect the two worlds are further grouped by (\romannum{1}) \textit{\textbf{Interface}}, which is related to immersive technologies to enrich human's subjective sense, and (\romannum{2}) \textit{\textbf{Cross-worlds Ecosystem}}, which consists of a variety of services to achieve the \textit{convergence} between the two worlds, e.g. digital city twin simulation with real-time IoT data \cite{petrova-antonovaDigitalTwinModeling2020} and virtual concert with real-time human performers\cite{kusumaChildrenVirtualConcert2020}. In particular, the resource allocation problem in the cross-world ecosystem, i.e., virtual service sync selection by IoT device owners, is the main focus of this paper. Finally, the Metaverse-enabled business and economics, e.g., the non-fungible token (NFT) trading, are categorized as \textit{\textbf{In-world Ecosystem}}, as it does not requires frequent data traversal across the two worlds.
\subsection{Related Works} Recently, due to the COVID-19 pandemic mobility restriction and the marketing by big tech companies such as Facebook and Microsoft \cite{brownBigTechWants2021}, the necessity to function in the virtual world has been demonstrated in various aspects of life. As such, the topic of Metaverse has attracted a lot of research attention, for its potential in service offerings in retailing \cite{gadallaMetaverseretailServiceQuality2013}, gaming \cite{volkCocreativeGameDevelopment2008}, education \cite{diazVirtualWorldResource2020}, and social-networking \cite{schroederSocialInteractionVirtual2002}. Some researchers have proposed to build prototypes to better understand the Metaverse and its economics, e.g., a university campus prototype \cite{duanMetaverseSocialGood2021} to study social good in the Metaverse. These examples illustrate the importance of convergence between the Metaverse and the physical world.
\subsection{Research Gap: Resource Allocation in Converging the Metaverse and the Physical Worlds} As the study of the Metaverse is still in its nascent stage, the resource allocation problem for syncing Metaverse with the assistance of IoT sensing data has not received much attention. Previous works, such as \cite{monetaArchitectureHeritageMetaverse2020,dionisio3DVirtualWorlds2013c,freySolipsisDecentralizedArchitecture2008}, mostly present a general picture of the field, summarizing technologies, architectures, and challenges involved in developing the Metaverse. Other works, such as \cite{leeSelfconfigurableLargescaleVirtual2011, tamaiConstructingSituatedLearning2011}, primarily focus on human-machine interaction aspects of the virtual learning experience (i.e., the interface component in our architecture). In contrast, our work is mainly focused on the resource allocation challenge to achieve convergence between the Metaverse and the physical world (i.e., the cross-world component in our architecture).
\section{System Model and Problem Formulation} \begin{table}[] \begin{threeparttable} \caption{Notation Used in the System Model} \label{tab:my-table}
\begin{tabular}{|p{1.3cm}|p{6.5cm}|} \hline \textbf{\textit{Notation}} & \textbf{\textit{Description}} \\ \hline \specialrule{1pt}{0pt}{0pt} $m,i,j$ & index of virtual service provider (VSP) and its target region \\ \hline $p,q$ & index of IoT device population \\ \hline $R_m$ & reward \\ \hline $D_m$ & total Euclidean distance of the sensing route \\ \hline $d_m$ & length of the segmented sensing distance \\ \hline $N^p$ & size of UAV population $p$ \\ \hline $\mcal{S}^p$ & set of $S^p$ (pure) strategies \\ \hline $\zeta^p$ & unit energy cost \\ \hline $\eta^p_1,\eta^p_2$& power parameters\\ \hline
$v^p,u^p$ &average velocity during traversal and sensing stage\\ \hline
$l^p_m$ & traversal distance from the base in population $p$ to the data collection point in region $m$ \\ \hline $x^p_m$ & strategy distribution, population state \\ \hline $b^p_m$ & sensing data quality \\ \hline $E_{s}^{m,p}$ & total energy consumption for traversal and sensing \\ \hline $R_m^p$ & received reward \\ \hline $F^p_m(\bm{x}),\pi^p_m$ & net payoff \\ \hline \end{tabular} \begin{tablenotes}
\small
\item Note that superscript and subscript are to denote the index of population and strategy respectively. To avoid confusion, parenthesis is applied when involving exponents, e.g., $(v^p)^3$.
\end{tablenotes} \end{threeparttable} \end{table}
We consider a network consisting of a set $\mcal{M}=\{1,\ldots,m, \ldots, M\}$ of $M$ VSPs, each of which aims to collect data (e.g., geo-spatial data) from a target region in the physical world in order to sync its virtual services (e.g., digital twin of a road for virtual driver training) in the Metaverse. IoT devices, such as UAVs owned by individuals in a nearby region, are employed by the Metaverse platform to collect data for VSPs to use. For a target region $m$, the VSPs or the Metaverse platform designs a data collection route covering the region (represented by the the grey line in \cref{fig:system4}), which can be further segmented based on the number of IoT devices that sense the region.
For simplicity, we assume that the allocation of data collection points in region $m$ to the set of UAVs which selecting this region is uniform. Along a segmented sensing route (e.g., line segment $BC$ in \cref{fig:system4}A), a UAV senses the nearby environment and collects the data. Assume that the data collection tasks last relatively long, e.g., one week, during which time UAVs collect data multiple times.
Let $\mcal{N}=\{1,\ldots,n,\ldots N\}$ denote a set of $N$ IoT devices, exemplified by UAVs, owned by $N$ independent individuals located in a nearby region (e.g., Region $3$ in \cref{fig:system4}B). UAV devices located at in their bases, such as owners' homes or community centers, need to fly to a region's, e.g., region $m$, data collection point to perform the task and return back for charging. We refer this two-way travel distance as the \textit{traversal distance}. As the traversal distance depends on which data collection point is assigned, we will approximate its value by the distance between the base and the center of region $m$.
Presented with $M$ VSPs' sync task requests, an UAV device owner needs to make a decision on which VSP to work for. Given that there can be a large number of UAV device owners and the full rational (equilibrium knowledge) among them is hard to achieve, a device owner can make irrational one-off decision. Since the sync task lasts several rounds, an owner can gradually improve his decisions towards equilibrium ones by observing the payoffs received by nearby UAVs with the same type of tasks.
In summary, there are two stages in our system model: \begin{enumerate} \item \textbf{Population Game Formulation}: Nearby UAVs with similar types (e.g., sensing capability, unit energy cost) are regarded as anonymously same and communicable and are grouped into the same population. Intra-population communication can be achieved in a \textit{pairwise} way (e.g., device-to-device), a \textit{centralized} approach (e.g., requesting the payoff information from the BS or checking the bulletin board, a place to publish virtual task \cite{duanMetaverseSocialGood2021}, in the Metaverse platform), or a combination. Information checking via a centralized approach is costly and prone to delay but yields a complete picture of the payoffs. In contrast, the pairwise approach is cheap and fast but provides only limited payoff information among the interacting UAVs.
\item \textbf{Hybrid Evolutionary Dynamics}: UAV populations may have different frequency to switch to a more costly centralized approach. Thus UAVs populations may have heterogeneous knowledge of the payoff information, leading to different strategy adjustments per round. To address this, we formulate the problem as a \textit{heterogeneous multi-population game} and adopt \textit{hybrid evolutionary dynamics} \cite{sandholmPopulationGamesEvolutionary2011} (a generalization of pure replicator dynamics) to solve for the evolutionary stable strategy (ESS).
\end{enumerate} \subsection{Population Formulation} We group neighbor UAVs (e.g., in the same community) of similar type (characterized by e.g., sensing capacity, traversal costs, communication frequency with the BS) into a population, and let $\mcal{P}=\{1,\ldots,p,\ldots, P\}$ denote the set of populations. For any population $p$, its UAVs are regarded as anonymously same, e.g., with similar traversal distance $l^p_m$ to region $m$, unit energy cost $\zeta^p$, and sensing data quality $b^p_m$. Let $N^p$ denote the size of population $p$ so that $\sum_{p\in\mcal{P}}N^p=N$. We refer $\mcal{N}$, the set of $N$ UAV owners, to as the \textit{society}.
Let $\mcal{S}^p=\{1,\ldots, s^p,\ldots, S^p\}$ denote the set of (pure) \textit{strategies}, i.e., there are $S^p$ VSP sync tasks, which can be selected by an owner in population $p$. A typical element in $\mcal{S}^p$ is denoted by $m,i,j$. The total number of pure strategies in all populations is denoted by $S=\sum_{p\in\mcal{P}} S^p$.
The set of \textit{population states} (or \textit{strategy distribution}) for population $p$ is denoted by the simplex in $\mathbf{R}^{S^p}_+$, $\Delta^p=\{\bm{x}^p\in\mathbf{R}^{S^p}_+: \sum_{m\in \mcal{S}^p}x^p_m=1\}$, where $\mathbf{R}^{S^p}_+$ is the space of non-negative real-numbers of dimension $S^p$ and $x^p_m\in\mathbf{R}_+$ is the percentage of population $p$ selecting the region $m$.
Define $\Theta=\prod_{p\in\mcal{P}}\Delta^p=\{\bm{x}=[\bm{x}^1,\ldots,\bm{x}^{P}]\in\mathbf{R}^S_+: \bm{x}^p\in \Delta^p\}$. The element of $\Theta$ describes the \textit{social states}, namely the joint behaviors in all $P$ populations at once.
\subsection{UAV Sensing Model} Let $D_m$ denote the total Euclidean distance of the sensing route over the region $m$, which is to be shared among the number of $\sum_{p\in\mcal{P}} x^p_m N^p$ UAVs selecting the region. We use the map $g^p_m: \mathbf{R}_+\rightarrow \mathbf{R}_{++}$ to refer to the route sharing policy, yielding the length of a segmented sensing route in region $m$ for a UAV in population $p$. For simplicity, we consider that the route is evenly divided among UAVs selecting the region. Therefore, the \textit{sensing distance} for a $p$-population UAV selecting region $m$ is defined by $d_m = g^p_m(D_m) = {D_m}/( {\sum_{q\in\mcal{P}} x^q_m N^q})$.
Following \cite{zhangPredictiveDeploymentUAV2021,limFederatedLearningUAVEnabled2021}, each UAV's energy cost consists of three components: the propulsion power $\eta^p_1$, hover power $\eta^p_2$ during service stage, and the transmission power. As we consider that the UAV transmits data after flying back to its base, the costs of propulsion and hover are of most interest. Following \cite{zengEnergyMinimizationWireless2019}, the energy costs during a UAV's acceleration and deceleration stages are ignored, and the propulsion power are considered to be constant for a fixed flying speed. Thus, for a $p$-population UAV selecting region $m$, the total energy consumption for traversal and sensing distance is defined by: \begin{equation*}\label{key} E_{s}^{n,p}= \eta^p_1\frac{l^p_n}{v^p}+\eta^p_2\frac{d_m}{u^p} =
\eta^p_1\frac{l^p_n}{v^p}+\eta^p_2{\frac{D_m}{{u^p}\sum_{q\in\mcal{P}} x^q_m N^q}} , \end{equation*} where $v^p$ and $u^p$ are the average flying speeds during the traversal stage and the sensing/hovering stage.
\subsection{UAV Utility Model} Without loss of generality, we consider the reward given by VSP $m$ is evenly allocated to the UAVs working in the region, except for a weighting factor $b^p_m$ which represents the the quality of the sensor data, e.g., measured by the sampling rate \cite{tongDeepReinforcementLearning2020}. A higher (intensive) sampling rate leads to the sensor data of higher quality, capturing more objects per frame. Therefore, the reward received by a $p$-population UAV selecting region $m$ is defined by \begin{equation}\label{key} R^p_m = \frac{ b^p_m }{\sum_{q\in\mcal{P}} x^q_m N^q b^q_m }R_m. \end{equation}
Let $\pi^p=[\pi^p_m]_{m\in \mcal{S}^p}$ denote the payoff vectors for population $p$. For any UAV in population $p$ selecting VSP $m$, its received utility $\pi^p_m$ is defined as follows: \begin{equation} \label{eq:utility} \pi^p_m= F^p_m(\bm{x})= R^p_m - \zeta^p E_{s}^{n,p} , \end{equation} where $\zeta^p$ is the unit cost per Joule of energy.
\section{Heterogeneous Evolutionary Game} A population game can be formulated as a set of players $\mcal{N}$, a set of populations $\mcal{P}$, a set of strategies $\mcal{S}^p\, (p\in\mcal{P})$, and a payoff function $F$. The definition and requirement of $F$ is as follows: $F:\Theta\rightarrow \mathbf{R}^S$ is a continuous map\footnote{Besides the continuous requirement, $F$ are imposed by stronger requirement of being Lipschitz continuous or continuously differentiable $(C^1)$.} assigning each social state $\bm{x}$ a vector of $S$ payoffs, one for each strategy in each population. Let $F^p_m:\Theta \rightarrow \mathbf{R}$ denote the payoff function for strategy $i\in S^p$, defined by \eqref{eq:utility}. Then, $F=[F^p]_{p\in\mcal{P}}$, where $F^p=[F^p_m]_{m\in \mcal{S}^p}$.
The UAV owners' behaviors adaptation can be modeled by \textit{revision protocols}, which are defined as follows: \begin{definition}A revision protocol in population $p$ is a map $\rho^p:\mathbf{R}^{S^p}\times \Delta^p \rightarrow \mathbf{R}^{S^p\times \mcal{S}^p}_+$. The scalar $\rho^p_{m,i}(\pi^p,\bm{x}^p)$ defines the \textit{conditional switch rate} from strategy $m\in S^p$ to strategy $i\in \mcal{S}^p$ given payoff vector $\pi^p$ and population state $\bm{x}^p$. For convenience, we refer to the collection $\rho=[\rho^1,\ldots,\rho^P]$ as a revision protocol. \end{definition}
Generally, a revision protocol $\rho$ defines a continuous-time evolutionary process over the populations. Each UAV owner is equipped with a stochastic alarm clock (Poisson-distributed), the ring of which indicates the arrival of a strategy revision opportunity.
\textit{Mean dynamics} captures change of strategy distribution over the populations over the time. The expected change in the \textit{proportion} of UAV owners choosing strategy $m$ in population $p$ captured by the mean dynamics corresponding to the population game $F$ and revision protocol $\rho$ is defined by \begin{equation}\label{eq-mean-dynamics} \dot{x}^p_m=\sum_{i\in\mcal{S}^p}x^p_i\rho^p_{i,m}(\pi^p,\bm{x}^p)-x_m^p\sum_{i\in\mcal{S}^p}\rho^p_{m,i}(\pi^p,\bm{x}^p), \end{equation} where the first term captures switches to strategy $m$ from other strategies, whereas the second term captures switches to other strategies from strategy $i$.
Next, we specify $\rho$ by two general categories, imitative protocol, exemplified by replicator dynamics, and direct protocol, represented by the Smith dynamics. The difference between replicator dynamics and Smith dynamics is the amount of payoff information required for \textit{one} UAV strategy adaption: replicator dynamics requires only one piece of information from the random opponent, whereas the Smith dynamics requires the $S^p$ payoff information.
\subsubsection{Imitative Protocols and Dynamics \cite{niyatoDynamicsNetworkSelection2009}} This scheme describes a UAV owner playing strategy $m$ who receives a revision opportunity chooses an opponent randomly and observes the opponent's strategy. If the opponent is playing strategy $i$, then the owner switches from strategy $m$ to $i$ with probability proportional to some factor $r^p_{mi}$. Note that the factor $x^p_m$ needs not be observed by the owner. In this case, the imitative revision protocol is defined as $\rho^p_{m,i}(\pi^p, x)=x^p_{i} r^p_{mi}(\pi^p, \bm{x})$, where $x^p_{i}$ represents the chance of meeting an opponent playing strategy $i$.
\textit{Pairwise Proportional Imitation} is an example of an imitation protocol, in which an UAV imitates the opponent's behavior only if the opponent's payoff is higher than his own, with probability proportional to the payoff difference: $ \rho^p_{m,i}(\pi^p, x)=x^p_{i}\left[\pi^p_{i}-\pi^p_{m}\right]_{+}$. The mean dynamics \eqref{eq-mean-dynamics} with this revision protocol defines the best known dynamic in evolutionary game theory, \textit{replicator dynamics}, given as follows: \begin{align} \dot{x}^p_{m} &=\sum_{i \in \mcal{S}^p} x^p_{i} \rho^p_{im}(\pi^p, \bm{x}^p)-x^p_{m} \sum_{i \in \mcal{S}^p} \rho^p_{m i}(\pi^p, \bm{x}^p) \nonumber \\ &=\sum_{i \in \mcal{S}^p} x^p_{i} x^p_{m}\left[\pi^p-\pi^p_i\right]_{+}-x^p_{m} \sum_{i \in \mcal{S}^p} x_{i}^p\left[\pi^p_i-\pi^p\right]_{+} \nonumber \\ &=x^p_{m} \sum_{i \in \mcal{S}^p} x^p_{i}\left(\pi^p-\pi^p_i\right) =x^p_{m}\left(\pi^p_{m}-\bar{\pi}^p\right), \label{eq-replicator-dynamics} \end{align} where ${\bar{\pi}}^p=\sum_{m\in S^p}x^p_m\pi^p_m$ is the \textit{average payoff} of population $p$, given the social state $\bm{x}$,
\subsubsection{Direct Protocol and Dynamics \cite{sandholmPopulationGamesEvolutionary2011}} This scheme describes the situation that an UAV device owner in population $p$ can receive a full picture of the payoff information ($S^p$ payoff) from a centralized controller, e.g., by requesting the payoff information from the BS (which connects to different UAVs) or checking the bulletin board, a place to publish virtual task \cite{duanMetaverseSocialGood2021}, in the Metaverse platform. Therefore, the UAV owner's conditional switch rate is directly dependent on the excess payoffs between two strategies. This is called a \textit{pairwise comparison protocol} defined by $\rho^p_{m,i}(\pi^p)=[\pi_i-\pi_m]_+$, representing the case when a UAV owner, who has a chance to switch strategy, randomly chooses an alternative strategy with higher payoff than the current one, with a probability proportional to the difference between the two payoffs. The mean dynamic \eqref{eq-mean-dynamics} with this revision protocol is called the \textit{Smith dynamics} defined as follows: \begin{align}\label{eq-smith}
\dot{x}^p_m &= \sum_{j\in\mcal{S}^p}x^p_j \rho^p_{jm}- x^p_m \sum_{j\in\mcal{S}^p}\rho^p_{mj} \nonumber \\
&= \sum_{j\in\mcal{S}^p}x^p_j[\pi^p_m -\pi^p_j ]_+ - x^p_m \sum_{j\in\mcal{S}^p}[\pi^p_j -\pi^p_m ]_+. \end{align}
\subsubsection{Hybrid Protocol for Heterogeneous Multi-population \cite{sandholmPopulationGamesEvolutionary2011}} Let $\mcal{K}=\{1, \ldots, k,\ldots, K\}$ denote a set of revision protocols available to the society, and $\rho^{p,k}(\pi^p,\bm{x}^p)$ refers to the conditional switch rate that an UAV owner in population $p$ has under revision protocol $k$. Let $\alpha^{p,k}(k\in\mcal{K})$ satisfying $\sum_{k\in\mcal{K}} \alpha^{p,k} = 1$ denote the probability of a $p$-population member using revision protocol $\rho^{p,k}$. Then, the behavior of an owner in population $p$ can be described by a \textit{hybrid revision protocol} $\rho^{p,H}$, which is expressed as $\rho^{p}= \sum_{k\in\mcal{K}}\alpha^{p,k}\rho^{p,k}$. Note that mean dynamics \eqref{eq-mean-dynamics} are linear in conditional switch rates, The mean dynamics with the hybrid revision protocol can be expressed as:
\begin{equation*}\label{key}
\dot{x}^p_m= \sum_{i\in\mcal{S}^p}\sum_{k\in\mcal{K}}x^p_i \alpha^{p,k} \rho^{p,k}_{i,m}-
x_m^p \sum_{i\in\mcal{S}^p}\sum_{k\in\mcal{K}} \alpha^{p,k} \rho^{p,k}_{m,i}. \end{equation*}
The hybrid protocol is versatile and suitable for different situations, e.g., including other protocols as introduced in \cite{sandholmPopulationGamesEvolutionary2011}. Unless otherwise stated, in the following, the hybrid protocol consists of only replicator dynamics and Smith dynamics. We use $\alpha^{p,1}$ to refer to the population $p$'s probability of switching to the centralized communication approach (observing $S^p$ payoffs) and $1-\alpha^{p,1}$ for pairwise approach (observing $1$ opponent's payoff). In other words, when adjusting strategy, a $p$-population UAV owner's behavior can be modeled by Smith and replicator dynamics with probability $\alpha^{p,1}$ and $1-\alpha^{p,1}$, respectively.
\section{Simulations Results}\label{sec-simulation} Unless stated otherwise, we consider the problem formulated with $3$ populations selecting among $3$ VSPs to assist data sync. Population size $N^p$ lies in $[50,250]$ with traversal distance $l^p_m$ in $[0.3,1]$ km. The length of sensing route $D_m$ lies in $[1,1.8]$ km, whereas the reward pool $R_m$ lies in $[1000,2000]$. Following \cite{limFederatedLearningUAVEnabled2021,zhangPredictiveDeploymentUAV2021}, values of speed parameters $v^p,u^p$ and power parameters $\eta^p_1,\eta^p_2$ are in the range $[3,5]$ m/s and $[16,20]$ W, respectively. Sensing data quality $b^p_m$ is in the range $[1,5]$, given the accuracy of sensors can be scored and calibrated in this range, whereas the unit energy cost $\zeta$ is $0.001$ \$/Joule. The default probabilities of adopting Smith protocol is $\alpha^1=[\alpha^{p,1}]_{p=1,2,3}=[0.2,0.3,0]$.
\begin{figure*}
\caption{Population states and utilities vs. time. Plot every $10$ steps. }
\label{fig:hybrid-converge}
\caption{Direction field. (pct. is short for percentage)}
\label{fig:vf}
\caption{Convergence time vs. $\alpha^{3,1}$}
\label{fig:hybridlevel}
\end{figure*}
\paragraph{Existence of the Equilibrium Strategies} \cref{fig:hybrid-converge} considers the hypothetical scenarios in which owners in different populations have various probabilities of switching to the centralized communication protocol, e.g., by checking the payoffs on the bulletin board in the Metaverse platform. The initial society states (strategy distribution) is given as $\bm{x}_0=[[0.3,0.3,0.4],[0.4,0.4,0.2],[0.35,0.35,0.3]]$. The left subfigure shows that strategy distribution can be stationary (i.e., convergent at equilibrium states) at which the payoffs received of selecting any VSP are the same (as shown in the right subfigure). Thus, the owners in each population have no incentive to make any more adjustment to their strategies away from the equilibrium point. A population-$3$ UAV receives higher average payoff than that of population $2$ and $3$ due to higher data quality.
\paragraph{Stability of the Equilibrium Strategies} \cref{fig:vf} shows the stability of the equilibrium strategies using direction field, a way of graphically representing the evolution direction (arrows in the figure) for the population states. For visualization purposes, population $3$ is removed from the default setting, and the probability of selecting region $3$ is fixed. The evolution direction, change in $(x^1_1,x^1_2,x^2_1,x^2_2)$ at a time instant, is four-dimensional. \cref{fig:vf}(a) shows the case when $x^1_1$ and $x^2_1$ are varied, and the evolution direction along $x^1_2$ and $x^2_2$ are hidden. Initial population states will follow the arrows and evolve towards the equilibrium states, shown as a blue line in the figure. \cref{fig:vf}(b) shows that when $x^2_1,x^2_2$ are fixed and $x^1_1$ and $x^1_2$ are varied, a mixed strategy equilibrium state $(0.1,0.19)$ can be obtained (red circle). Note that the blue circle $(0.38,0.62)$ is also an equilibrium state, at which the chance of selecting region $3$ is zero (i.e., extinction of strategy $3$ in population $1$). Finally, \cref{fig:vf}(c) shows the case when states of population $1$, ($x^1_1,x^1_2$), are fixed and those of population $2$, ($x^2_1,x^2_2$), are varied, the equilibrium point $(0.24,0.33)$ can be achieved. $(0.43,57)$ is also an equilibrium point at which strategy $3$ is extinct.
\paragraph{Impact of Smith Protocol on the Convergence Time}
\cref{fig:hybridlevel} shows that the impact of adopting Smith protocol on the convergence time by fixing the probability of adopting the Smith protocol in population $1$ and $2$ and varying that in population $3$, $\alpha^{3,1}$, within $[0,1]$ in increments of $0.1$. The convergence time can be defined as the number of iterations after which the utilities among strategies in a population are similar, given the UAV owners in that population will have no incentive to make any modification, and the equilibrium state is reached. Mathematically, convergence time is $\min_{t}\left| \max_{i\in \mcal{S}^p}\pi^p_i(t) - \min_{i\in \mcal{S}^p}\pi^p_i(t)\right|\leq \tau$, where $\tau$ is a threshold, chosen as $0.05$ in the experiment. The figure shows that the convergence time decreases as $\alpha^{3,1}$ increases. This is because complete information of all payoffs improves the strategy adaption compared to pairwise imitation in which only the opponent's payoff is known.
\section{Conclusion} In this paper, we studied an IoT-assisted Metaverse sync problem in which the populations of UAV devices can autonomously select VSPs to work for. Given a large number of IoT devices, hardly-achieved full rationale, and potential different in-population communication protocols, we proposed hybrid evolutionary dynamics for a heterogeneous multi-population game. The equilibrium strategy is experimentally demonstrated to be existed and stable. In future works, we plan to incorporate spatial correlation of the sensing data into the utility models.
\end{document} | arXiv | {
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\begin{document}
\title{ extbf{Pattern Containment and Combinatorial Inequalities}
\begin{abstract} We use a probabilistic method to produce some combinatorial inequalities by considering pattern containment in permutations and words. \end{abstract}
If $\sigma\in S_n$ and $\tau\in S_m$, we say that $\sigma$ \emph{contains} $\tau$, or $\tau$ \emph{occurs} in $\sigma$, if $\sigma$ has a subsequence order-isomorphic to $\tau$. In this situation, $\tau$ is called a \emph{pattern}. Similarly, if $\sigma\in[k]^n$ is a string of $n$ letters over the alphabet $[k]=\{1,\dots,k\}$, and $\tau\in[l]^m$ is a map from $[m]$ onto $[l]$ (i.e. $\tau$ contains all letters from 1 to $l$), then we say that $\sigma$ contains the pattern $\tau$ if $\sigma$ has a subsequence order-isomorphic to $\tau$. An \emph{instance} (or \emph{occurrence}) of $\tau$ in $\sigma$ is a choice of $m$ positions $1 \le i_1 < \ldots < i_m \le n$, such that the subsequence $(\sigma(i_1),\ldots,\sigma(i_m))$ is order-isomorphic to $\tau=(\tau(1),\ldots,\tau(m))$.
Most of the work on pattern containment concentrated on pattern avoidance, that is on characterizing and counting permutations that contain no occurrences of a given pattern or a set of patterns. Less attention has been given to counting the number of times a given pattern occurs in permutations of a given size, in particular, packing patterns into permutations (but see \cite{AAHHS,P}, for example), and, to our knowledge, packing patterns into words (where repeated letters are allowed) has not yet been considered.
Here we consider pattern containment and use a simple probabilistic fact (the variance of a random variable is nonnegative) to produce nontrivial combinatorial inequalities.
\section{Patterns in permutations}
In this section, we consider permutation patterns contained in other permutations.
\begin{theorem} \label{perm} Let $\tau$ be a permutation of $\{0,1,\ldots,m\}$ and define \[ [i,j]_m=\binom{i+j}{i}\binom{2m-i-j}{m-i}. \] Then for any nonnegative integer $m$ and any $\tau$ as above, \begin{equation} \label{eq-perm} \sum_{i,j=0}^{m}{[i,j]_m [\tau(i),\tau(j)]_m} \ge \binom{2m+1}{m}^2. \end{equation} \end{theorem}
\begin{remark} \label{perm-paths} Notice that $[i,j]_m$ is the number of northeast integer lattice paths from $(0,0)$ to $(m,m)$ through $(i,j)$. Hence the left-hand side is the number of pairs $(P,Q)$ of northeast integer lattice paths $P:(0,0)\to(i,j)\to(m,m)$ and $Q:(0,0)\to(\tau(i),\tau(j))\to(m,m)$ over all $(i,j)\in[0,m]^2$. \end{remark}
\begin{remark} \label{ekhad} The numbers $[i,j]_m$, $0\le i,j\le m$ have been found to have other interesting properties as well. For example, Amdeberhan and Ekhad \cite{AE} showed that \[ \det([i,j]_m)_{0\le i,j\le m}=\frac{(2m+1)!^{m+1}}{(2m+1)!!}, \] where $a!!=0!\cdot 1!\cdot 2!\cdot\ldots\cdot a!$. \end{remark}
It is a well-known result \cite{HLP} that for any nondecreasing subsequence $a_1\ge\cdots\ge a_n$ and a permutation $\varphi\in S_n$, the sum $\sum_{i=1}^{n}a_ia_{\varphi(i)}$ attains its maximum when $\varphi=id_n=12\ldots n$ and its minimum when $\varphi=n(n-1)\ldots 1$. Now if we arrange $(m+1)^2$ numbers $[i,j]_m$ ($0\le i,j\le m$) in nondecreasing order, \emph{there is no permutation $\tau$ of $\{0,1,\ldots,m\}$ which reverses that order} (other than in the trivial case $m=0$). For example, even when $m=1$, we have \[ [0,0]_2=[1,1]_2=2>1=[0,1]_2=[1,0]_2, \] reversing $(2,2,1,1)$ gives $(1,1,2,2)$ and the estimate of \cite{HLP} gives us the lower bound of $2\cdot1+2\cdot1+1\cdot2+1\cdot2=8$, our estimate yields the lower bound of $\binom{3}{1}^2=9$, while the left hand side is actually equal to 10 for both $\tau=01$ (the identity) and $\tau=10$ (which transposes $[0,0]_2$ and $[1,1]_2$ as well as $[0,1]_2$ and $[1,0]_2$ in the above ordering).
The estimate in Theorem \ref{perm} appears to be stronger than that of \cite{HLP}. For example, the lower bounds for $m=2,3,4,5$ are $75,792,8660,98876$, respectively, according to \cite{HLP}, while our lower bounds are $100,1225,15876,213444$, respectively. In fact, as the following proposition shows, the lower bound of \cite{HLP} can never be achieved in our case for $m>0$.
\begin{proposition} \label{prop1} Arrange $(m+1)^2$ numbers $\{[i,j]_m \mid 0\le i,j\le m\}$ into a nondecreasing order $a_1\ge\ldots\ge a_{(m+1)^2}$. A permutation $\tau$ of $\{0,1,\ldots,m\}$ induces an equivalence class of permutations $\varphi_\tau$ on the $a_i$'s (equivalence relation being a permutation of equal elements). Then for any $\tau$, reversal of the identity $(m+1)^2((m+1)^2-1)\ldots21\notin\varphi_\tau$. \end{proposition}
\noindent\textbf{Proof.} Suppose that there is a permutation $\tau$ which induces an order-reversing permutation of the $a_i$'s. Note that $[0,m]_m=[m,0]_m=1$ for any $m$, hence, $[\tau(0),\tau(m)]_m$ must have the greatest value among all $[\tau(i),\tau(j)]_m$. Note that \[ [i,j]_m=[j,i]_m=[m-i,m-j]_m=[m-j,m-i]_m \] for any $i$ and $j$, so assume that $i\le j$. Then it is a straightforward exercise to prove that \begin{align*} [i,j]_m&>[i-1,j]_m \quad \text{for $i>0$, and}\\ [i,j]_m&>[i,j+1]_m \quad \text{for $j<m$,} \end{align*} so for $0<i<j<m$, \[ [i,i]_m>[i,j]_m>[0,j]_m>[0,m]_m=1. \] Similarly, we can assume that $i\le\lfloor\frac{m}{2}\rfloor$ (since $[i,i]_m=[m-i,m-i]_m$), then it is just as easy to see that for any $i>0$ \[ [i,i]_m<[i-1,i-1]_m<\cdots<[0,0]_m=\binom{2m}{m}. \] Thus, for any $0\le i,j\le m$, \[ 1=[0,m]_m=[m,0]_m \le [i,j]_m \le [0,0]_m=[m,m]_m=\binom{2m}{m}, \] and one of the two inequalities becomes an equality if and only if $i,j\in\{0,m\}$. Hence, for our permutation $\tau$, we must have $\tau(0)=\tau(m)=0$ or $\tau(0)=\tau(m)=m$, neither of which is possible when $m\neq 0$. The resulting contradiction implies our proposition.
$\square$
Finally, before we begin with the proof of Theorem \ref{perm}, let us note that a permutation of summands in (\ref{eq-perm}) yields the following corollary.
\begin{corollary} \label{cor1} For any two permutations $\tau_1,\tau_2$ of $\{0,1,\ldots,m\}$ and any $m\in\mathbb{N}$, \[ \sum_{i,j=0}^{m}{[\tau_1(i),\tau_1(j)]_m[\tau_2(i),\tau_2(j)]_m}\ge \binom{2m+1}{m}^2. \] \end{corollary}
Another immediate corollary is a consequence of the fact that \[ [i,j]_m=\binom{2m}{m}\frac{\binom{m}{i}\binom{m}{j}}{\binom{2m}{i+j}}= \binom{2m}{m}\{i,j\}_m, \text{ where } \{i,j\}_m:=\frac{\binom{m}{i}\binom{m}{j}}{\binom{2m}{i+j}}. \] \begin{corollary} \label{cor2} For any $m\in\mathbb{N}$ and any permutation $\tau$ of $\{0,1,\ldots,m\}$, \[ \sum_{i,j=0}^{m}{\{i,j\}_m\{\tau(i),\tau(j)\}_m}\ge \left(\frac{2m+1}{m+1}\right)^2=\left(2-\frac{1}{m+1}\right)^2. \] \end{corollary}
Note that Corollary \ref{cor2} no longer holds if we substitute $4$ on the right side of this inequality.
\noindent\textbf{Proof of Theorem \ref{perm}.} Consider $S_n$ as a sample space with uniform distribution. Let $\tau\in S_m$ (notation-wise, it is more convenient if, in the proof, $\tau$ is a permutation of $\{1,2,\ldots,m\}$), and let $X_\tau$ be a random variable such that $X_\tau(\sigma)$ is the number of occurrences of pattern $\tau$ in given permutation $\sigma\in S_n$. We will show that our inequality follows from the fact that \[ V\!ar(X_\tau)=E(X_\tau^2)-E(X_\tau)^2\ge 0 \]
We start by finding $E(X_\tau)$. Pick an $m$-letter subset $S$ of $[n]=\{1,2,\ldots,n\}$ in $\binom{n}{m}$ ways. There is a unique permutation $\tau(S)$ of $S$ which is order-isomorphic to $\tau$. There are $m!$ equally likely permutations in which the elements of $S$ can occur in $\sigma$, but we need only 1 of them, namely, $\tau(S)$. Hence, $\tau(S)$ either occurs once or does not occur in a given permutation $\sigma$. Therefore, the probability that a random $\sigma$ contains $\tau(S)$ as a subsequence is $1/m!$. Let $Y_{\tau(S)}$ be a random variable such that $Y_{\tau(S)}(\sigma)$ is the number of occurrences of $\tau(S)$ in $\sigma$. Since \[ P\!\left(Y_{\tau(S)}(\sigma)=1\right)=\frac{1}{m!} \quad \text{and} \quad P\!\left(Y_{\tau(S)}(\sigma)=0\right)=1-\frac{1}{m!}, \] we have $E(Y_{\tau(S)})=1/m!$. But this is true for any
$S\subseteq[n]$ such that $|S|=m$, and we have \[
X_\tau=\!\!\!\!\!\!\sum_{S\subseteq[n],\,|S|=m}{\!\!\!\!\!\!Y_{\tau(S)}} \] hence, \[
E(X_\tau)=\!\!\!\!\!\!\sum_{S\subseteq[n],\,|S|=m}{\!\!\!\!\!\!E(Y_{\tau(S)})}= \frac{1}{m!}\binom{n}{m}. \]
Next, we look at $E(X_\tau^2)$. We have \[
E(X_\tau^2)=E\left(\sum_{S\subseteq[n],\,|S|=m}{\!\!\!\!\!\!Y_{\tau(S)}}\right)=
\!\!\!\!\!\!\sum_{\substack{S_1,S_2\subseteq[n]\\|S_1|=|S_2|=m}} {\!\!\!\!\!\!E\left(Y_{\tau(S_1)}Y_{\tau(S_2)}\right)}. \]
Of course, $Y_{\tau(S_1)}Y_{\tau(S_2)}=1$ if and only if both $\tau(S_1)$ and $\tau(S_2)$ are subsequences of $\sigma$, otherwise, $Y_{\tau(S_1)}Y_{\tau(S_2)}=0$.
Let $S=S_1\cup S_2$, and $|S_1\cap S_2|=\ell$, so $|S|=|S_1\cup S_2|=2m-\ell$. We can pick a subset $S\subseteq[n]$ in $\binom{n}{2m-\ell}$ ways. Note that any such $S$ is order-isomorphic to $[2m-\ell]=\{1,2,...,2m-\ell\}$. Therefore, the number of permutations $\rho(S)$ of $S$ such that $\rho(S)=\tau(S_1)\cup\tau(S_2)$ for some $S_1,S_2\subseteq S$, $S_1\cup S_2=S$, is the same for any $S$ of cardinality $2m-\ell$ and depends only on $m$ and $\ell$.
Therefore, $E(X_\tau^2)$ is a linear combination of
$\left\{\binom{n}{2m-\ell}\,\mid\,0\le\ell\le m\right\}$ with coefficients which are rational functions of $m$ and $\ell$. The degrees in $n$ of both $E(X_\tau^2)$ and $E(X_\tau)^2$ are $2m$, and the coefficient of $n^{2m}$ in $E(X_\tau)^2$ is $1/(m!)^4$. On the other hand, $S=S_1\cup S_2$, $|S|=2m$ and $|S_1|=|S_2|=m$ imply that $S_1\cap S_2=\emptyset$, so $Y_{\tau(S_1)}$ and $Y_{\tau(S_2)}$ are independent, and hence \[ P\left(Y_{\tau(S_1)}Y_{\tau(S_2)}=1\right)=P\left(Y_{\tau(S_1)}=1\right) P\left(Y_{\tau(S_2)}=1\right)=\left(\frac{1}{m!}\right)^2. \] Since the number of ways to partition a set $S$ of size $2m$ into two subsets of size $m$ is $\binom{2m}{m}$, the coefficient of $\binom{n}{2m}$ in $E(X_\tau^2)$ is $\binom{2m}{m}/(m!)^2$. Hence, the coefficient of $n^{2m}$ in $E(X_\tau^2)$ is \[ [n^{2m}]E(X_\tau^2)=\frac{1}{(2m)!}\frac{1}{(m!)^2}\binom{2m}{m}= \frac{1}{(m!)^4}, \] where $[x^d]P(x)$ denotes the coefficient of $x^d$ in a given polynomial $P(x)$. But then $[n^{2m}]E(X_\tau^2)=[n^{2m}]E(X_\tau)^2$, so $\deg_n(V\!ar(X_\tau))\le 2m-1$, and hence, $[n^{2m-1}]V\!ar(X_\tau)\ge 0$.
We have \begin{multline*} [n^{2m-1}]E(X_\tau)^2=[n^{2m-1}]\left(\frac{1}{m!}\binom{n}{m}\right)^2=\\ =\frac{2}{(m!)^2}\cdot[n^{m}]\binom{n}{m}\cdot[n^{m-1}]\binom{n}{m}=\\ =\frac{2}{(m!)^2}\cdot\frac{1}{m!}\cdot\left(-\frac{\binom{m}{2}}{m!}\right)= -\frac{m(m-1)}{(m!)^4} \end{multline*}
Similarly, the coefficient of $n^{2m-1}$ in the $\binom{n}{2m}$-term of $E(X_\tau^2)$ is \[ -\frac{\binom{2m}{2}}{(2m)!}\frac{1}{(m!)^2}\binom{2m}{m}= -\frac{m(2m-1)}{(m!)^4}, \] so we only need to find the coefficient of the $\binom{n}{2m-1}$-term of $E(X_\tau^2)$.
As we noted before, all subsets $S\subseteq[n]$ of the same size (in our case, of size $2m-1$) are equivalent, so we may assume $S=[2m-1]=\{1,2,\ldots,2m-1\}$. We want to find the number of permutations $\rho$ of $S$ such that there exist subsets
$S_1,S_2\subseteq S$ of size $m$ for which we have $|S_1\cap S_2|=1$ (so $S_1\cup S_2=S$) and $\rho(S)=\tau(S_1)\cup\tau(S_2)$.
Suppose that we want to choose $S_1$ and $S_2$ as above, together with their positions in $S$, in such a way that the intersection element $e$ is in the $i$th position in $\tau(S_1)$ and the $j$th position in $\tau(S_2)$ (of course, $1\le i,j\le m$). Then $e$ occupies position $(i-1)+(j-1)+1=i+j-1$ in $S$. Hence, there are $\binom{i-1+j-1}{i-1}$ ways to choose the positions for elements of $\tau(S_1)$ and $\tau(S_2)$ on the left of $e$, and $\binom{m-i+m-j}{m-j}$ ways to choose the positions for elements of $\tau(S_1)$ and $\tau(S_2)$ on the right of $e$. On the other hand, both $\tau(S_1)$ and $\tau(S_2)$ are naturally order-isomorphic to $\tau$, hence, under that isomorphism $e$ maps to $\tau(i)$ as an element of $S_1$ and to $\tau(j)$ as an element of $S_2$. Since $e$ is the \emph{unique} intersection element, it is easy to see that we must have $e=(\tau(i)-1)+(\tau(j)-1)+1=\tau(i)+\tau(j)-1$ (exactly $\tau(i)-1$ elements in $S_1$ and exactly $\tau(j)-1$ elements in $S_2$, all distinct from those in $S_1$, must be less than $e$, the rest of the elements of $S$ must be greater than $e$). There are $\binom{\tau(i)-1+\tau(j)-1}{\tau(i)-1}$ ways to choose the elements of $S_1$ and $S_2$ which are less than $e$, and $\binom{m-\tau(i)+m-\tau(j)}{m-\tau(j)}$ ways to choose the elements of $S_1$ and $S_2$ which are greater than $e$.
Thus, its positions in $\tau(S_1)$ and $\tau(S_2)$ uniquely determine the position and value of the intersection element $e$; there are $[i-1,j-1]_m$ ways to choose which other positions are occupied by $\tau(S_1)$ and which ones by $\tau(S_2)$; and, there are $[\tau(i)-1,\tau(j)-1]_m$ ways to choose which other values are in $\tau(S_1)$ and which ones are in $\tau(S_2)$.
Now that we have chosen both positions and values of elements of $S_1$ and $S_2$, we can produce a unique permutation $\rho(S)$ of $S$ which satisfies our conditions above. Simply fill the positions for $S_1$, resp. $S_2$, by elements of $\tau(S_1)$, resp. $\tau(S_2)$, in the order in which they occur.
Since the total number of permutations of $S$ is $(2m-1)!$, the coefficient of the $\binom{n}{2m-1}$-term of $E(X_\tau^2)$ is \begin{multline*} \frac{\sum_{i,j=1}^{m}{\binom{i-1+j-1}{i-1}\binom{m-i+m-j}{m-j} \binom{\tau(i)-1+\tau(j)-1}{\tau(i)-1}\binom{m-\tau(i)+m-\tau(j)}{m-\tau(j)}}} {(2m-1)!}=\\ =\frac{\sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau(i)-1,\tau(j)-1]_{m-1}}}{(2m-1)!}, \end{multline*} the coefficient of $n^{2m-1}$ in $V\!ar(X_\tau)$ is, by the previous equations, \begin{multline*} \frac{\sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau(i)-1,\tau(j)-1]_{m-1}}}{((2m-1)!)^2}- \frac{m(2m-1)}{(m!)^4}+\frac{m(m-1)}{(m!)^4}=\\ =\frac{\sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau(i)-1,\tau(j)-1]_{m-1}}}{((2m-1)!)^2}- \frac{1}{(m!(m-1)!)^2}\ge0, \end{multline*} so we finally get \[ \sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau(i)-1,\tau(j)-1]_{m-1}}\ge \left(\frac{(2m-1)!}{m!(m-1)!}\right)^2=\binom{2m-1}{m-1}^2, \] which is easily reducible to (\ref{eq-perm}) by $m\leftarrow m+1$, then $\bar\tau(i)\leftarrow\tau(i+1)-1$.
$\square$
It seems, however, that a stronger form of our Theorem should be true, namely, the following \begin{conjecture} \label{conj1} The \emph{strict} inequality holds in (\ref{eq-perm}) for all $m>0$. \end{conjecture}
This would imply that $V\!ar(X_\tau)$ has order $2m-1$ in $n$, i.e. the standard deviation of $X_\tau$ is $1/2$ order smaller than its expected value.
\begin{remark} \label{rem-covariance} Similarly, the leading coefficient of the covariance $Cov(X_{\tau_1},X_{\tau_2})$ is \[ \frac{\sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau_1(i)-1,\tau_2(j)-1]_{m-1}}}{((2m-1)!)^2}- \frac{1}{(m!(m-1)!)^2}, \] but $\sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau_1(i)-1,\tau_2(j)-1]_{m-1}}$ can be (and often is) less than $\binom{2m-1}{m-1}^2$. \end{remark}
As of now, we only have some results on the sign of covariance for small patterns. We hope to explore this topic further in subsequent papers.
Note that the reversal map, $\tau(i)\mapsto\tau(m+1-i)$, the complement map, $\tau(i)\mapsto m+1-\tau(i)$, preserve the variance and covariance (we also make a note for the next section that, for words $\tau\in[l]^m$, the reversal map is the same, while the complement is $\tau(i)\mapsto l+1-\tau(i)$).
Considering symmetry classes of pairs of patterns (i.e. equivalence classes with respect to reversal and complement), we see that there are 8 classes of pairs of 3-letter patterns: $\{123,123\}$, $\{132,132\}$, $\{123,132\}$, $\{132,213\}$, $\{132,231\}$, $\{132,312\}$, $\{123,312\}$, $\{123,321\}$ (listed in order of decreasing covariance). Of those, the first two pairs obviously have a positive covariance, and of the remaining six, only $\{123,132\}$ has a positive covariance.
Finally, denote the left-hand side and right-hand side of equation (\ref{eq-perm}) by $L(m,\tau)$ and $R(m,\tau)$, respectively, and let \begin{gather*} M^\ast(m)=\max_{\tau\in S_m}{(L(m,\tau)-R(m,\tau))},\\ M{\!}_\ast(m)=\min_{\tau\in S_m}{(L(m,\tau)-R(m,\tau))}. \end{gather*} It is not hard to see that $M^\ast(m)=L(m,id_m)-R(m,id_m)>0$, where $id_m$ is the identity permutation of $\{0,1,\ldots,m\}$ (use Chebyshev's inequality, or dot product, or Cauchy-Schwarz inequality). It would be interesting to characterize the permutations $\tilde\tau_m$ such that $M{\!}_\ast(m)=L(m,\tilde\tau_m)-R(m,\tilde\tau_m)$. We also make the following conjecture. \begin{conjecture} \label{conj2} $\displaystyle{\exists\lim_{m\to\infty}{\frac{M{\!}_\ast(m)}{M^\ast(m)}}=0.}$ \end{conjecture}
\section{Patterns in words}
We now consider patterns contained in words, where repeated letters are allowed both in the pattern and the ambient string.
\begin{theorem} \label{words} Let $\tau$ be a map of $[0,m]=\{0,1,\ldots,m\}$ onto $[0,l]=\{0,1,\ldots,l\}$. Then for any nonnegative integers $0\le l\le m$ and any $\tau$ as above, \begin{equation} \label{eq-words} \sum_{i,j=0}^{m}{[i,j]_m [\tau(i),\tau(j)]_l} \ge \frac{(2m+1)!(2l+1)!}{(m!)^2(l+1)!)^2}. \end{equation} \end{theorem}
\begin{remark} \label{rem-special} Note that Theorem \ref{words} reduces to Theorem \ref{perm} when $l=m$. Note also that, given $0\le l\le m$, Theorem \ref{words} applies to $(l+1)!S(m+1,l+1)$ patterns $\tau$, where $S(m+1,l+1)$ is the Stirling number of the second kind. \end{remark}
\begin{remark} \label{words-paths} As in Theorem \ref{perm}, the left-hand side of Theorem \ref{words} is the number of pairs $(P,Q)$ of northeast integer lattice paths $P:(0,0)\to(i,j)\to(m,m)$ and $Q:(0,0)\to(\tau(i),\tau(j))\to(l,l)$ over all $(i,j)\in[0,m]^2$. \end{remark}
\noindent\textbf{Proof of Theorem \ref{words}}. The proof follows the same outline as that of Theorem \ref{perm}, so we will use the same notation as well. Again, it will be convenient to assume in the proof that $\tau\in[l]^m$ is map of $[m]$ onto $[l]$ (i.e. use $\{1,\dots,m\}$ instead of $\{0,1,\dots,m\}$ and $\{1,\dots,l\}$ instead of $\{0,1,\dots,l\}$) and, similarly, that the ambient permutations $\sigma\in[k]^n$. Note that for any subset $S\subseteq[n]$ of positions, the probability that the subsequence of elements at positions in $S$, i.e. $\sigma(S)$, in a random word $\sigma\in[k]^n$, is order-isomorphic to $\tau$ is $\binom{k}{l}/k^m$. This is because $k^m$ is the total number of subsequences of $m$ letters in $[k]$, $\tau$ has exactly $l$ distinct letters, and there are $\binom{k}{l}$ ways to choose $l$ distinct letters out of $k$. Hence, as in Theorem \ref{perm}, we obtain \[ E(X_\tau)=\frac{1}{k^m}\binom{k}{l}\binom{n}{m}, \] which is a polynomial in $n$ and $k$. Therefore, the leading coefficient of $E(X_\tau)$ as a polynomial in $n$ is \[ [n^m]E(X_\tau)=\frac{1}{k^m}\binom{k}{l}\frac{1}{m!}, \] so the leading coefficient of $E(X_\tau)^2$ is \[ [n^{2m}]E(X_\tau)^2=\frac{1}{k^{2m}}\binom{k}{l}^2\frac{1}{(m!)^2}. \] However, as in the proof of Theorem \ref{perm}, we have that $E(X_\tau^2)$ is a linear combination of $\binom{n}{2m-\ell}$, $0\le\ell\le m$, with coefficients being polynomials in $k$ and rational functions in $l,m$. A similar analysis shows that the leading coefficient in $n$ of $E(X_\tau^2)$ is \begin{multline*} [n^{2m}]E(X_\tau^2)=\frac{1}{(2m)!}\left[\binom{n}{2m}\right]E(X_\tau^2)=\\ =\frac{1}{(2m)!}\binom{2m}{m}\binom{k}{l}^2\frac{1}{k^{2m}}=[n^{2m}]E(X_\tau)^2, \end{multline*} so $\deg_n(V\!ar(X_\tau))\le 2m-1$, and hence, $[n^{2m-1}]V\!ar(X_\tau)\ge 0$.
As in the proof of Theorem \ref{perm}, we have that \[ [n^{2m-1}]E(X_\tau)^2=2[n^{m-1}]E(X_\tau)[n^m]E(X_\tau)= -\frac{m(m-1)}{(m!)^2}\binom{k}{l}^2\frac{1}{k^{2m}} \] and the coefficient of $n^{2m-1}$ in the $\binom{n}{2m}$-term of $E(X_\tau^2)$ is \[ -\binom{2m}{2}\frac{1}{(2m)!}\binom{2m}{m}\binom{k}{l}^2\frac{1}{k^{2m}}= -\frac{m(2m-1)}{(m!)^2}\binom{k}{l}^2\frac{1}{k^{2m}}. \] The remaining summand in $[n^{2m-1}]V\!ar(X_\tau)$ is the coefficient of $n^{2m-1}$ in the $\binom{n}{2m-1}$-term of $E(X_\tau^2)$, i.e. \begin{multline*} \frac{1}{(2m-1)!}\left[\binom{n}{2m-1}\right]E(X_\tau^2)\\ -\frac{m(2m-1)}{(m!)^2}\binom{k}{l}^2\frac{1}{k^{2m}} +\frac{m(m-1)}{(m!)^2}\binom{k}{l}^2\frac{1}{k^{2m}}\ge 0, \end{multline*} which is equivalent to \[ \left[\binom{n}{2m-1}\right]E(X_\tau^2)\ge \frac{(2m-1)!}{(m-1)!^2}\binom{k}{l}^2\frac{1}{k^{2m}}. \]
As in the proof of Theorem \ref{perm}, it is easy to see that $[\binom{n}{2m-1}]E(X_\tau^2)$ is equal to the probability that a sequence $\rho\in[k]^{2m-1}$ is a union of two subsequences order-isomorphic to $\tau$. Therefore, assume $[2m-1]=S_1\cup S_2$, $\rho(S_1)\cong\tau\cong\rho(S_2)$. But then $S_1$ and $S_2$ have $m$ elements, so they intersect at a single element $e$.
Suppose that $e$ is at position $i$ in $S_1$ and at position $j$ in $S_2$. Then, as in the proof of Theorem \ref{perm}, there are $\binom{i-1+j-1}{i-1}\binom{m-i+m-j}{m-i}=[i-1,j-1]_{m-1}$ ways to choose which positions to the left and to the right of $e$ are in $S_1$ and which ones are in $S_2$.
Suppose that $\rho$ contains $l+L$ distinct letters, then $0\le L\le l-1$. Because of the positions of $e$ in $S_1$ and $S_2$, we know that $e$ must map to $\tau(i)$ in $\rho(S_1)$ and to $\tau(j)$ in $\rho(S_2)$ under our order-isomorphism. Suppose that the value of $e$ in $\rho$ is $r$. Consider the $r-1$ letters in $[l+L]$ which are less than $r$. Then \[(r-1)-(\tau(j)-1)=r-\tau(j)\] of those occur only in $S_1$, \[(r-1)-(\tau(i)-1)=r-\tau(i)\] occur only in $S_2$, and \[(r-1)-(r-\tau(i))-(r-\tau(j))=\tau(i)+\tau(j)-1-r\] occur in both $\rho(S_1)$ and $\rho(S_2)$. Similarly, of the $l+L-r$ letters in $\rho$ which are greater than $r$, \[(l+L-r)-(l-\tau(j))=L-r+\tau(j)\] occur only in $\rho(S_1)$, \[(l+L-r)-(l-\tau(i))=L-r+\tau(i)\] occur only in $\rho(S_2)$, and \[(l+L-r)-(L-r+\tau(i))-(L-r+\tau(j))=l-L+r-\tau(i)-\tau(j)\] occur in both $\rho(S_1)$ and $\rho(S_2)$.
Thus, the number of sequences $\rho\in[k]^{2m-1}$ which are a union of two subsequences order-isomorphic to $\tau$ is \[ f(\tau,k)=\sum_{L=0}^{l-1}{\binom{k}{l+L}\sum_{r=0}^{l+L}{ \sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}h(\tau,L,r,i,j)} } }, \] where \begin{multline*} h(\tau,L,r,i,j)=\binom{r-1}{r-\tau(i),r-\tau(j),\tau(i)+\tau(j)-1-r}\times\\ \times\binom{l+L-r}{L-r+\tau(i),L-r+\tau(j),l-L+r-\tau(i)-\tau(j)}. \end{multline*}
Hence, the probability that a sequence $\rho\in[k]^{2m-1}$ is a union of two subsequences order-isomorphic to $\tau$ is $f(\tau,k)/k^{2m-1}$, so we have \[ \left[\binom{n}{2m-1}\right]E(X_\tau^2)=\frac{f(\tau,k)}{k^{2m-1}}\ge \frac{(2m-1)!}{(m-1)!^2}\binom{k}{l}^2\frac{1}{k^{2m}}, \] or, equivalently, \[ kf(\tau,k)\ge\frac{(2m-1)!}{(m-1)!^2}\binom{k}{l}^2 \] for all positive integers $k$ and all patterns $\tau\in[l]^m$. But both sides of this inequality are polynomials in $k$ of degree $2l$, hence the same inequality should hold for their leading coefficients. The leading coefficient on the right is \[ \frac{(2m-1)!}{(m-1)!^2}\frac{1}{(l!)^2}. \] On the left, $k^{2l}$ only occurs when $L=l-1$. But then $\tau(i)+\tau(j)-1-r\ge 0$ and $l-L+r-\tau(i)-\tau(j)=r+1-\tau(i)-\tau(j)\ge 0$, so $r=\tau(i)+\tau(j)-1$, and hence \begin{multline*} h(\tau,L,r,i,j)=h(\tau,l-1,\tau(i)+\tau(j)-1,i,j)=\\ =\binom{\tau(i)+\tau(j)-2}{\tau(i)-1}\binom{2l-\tau(i)-\tau(j)}{l-\tau(i)} =[\tau(i)-1,\tau(j)-1]_{l-1}. \end{multline*} Therefore, \[ [k^{2l}](kf(\tau,k))=\frac{1}{(2l-1)!}\sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau(i)-1,\tau(j)-1]_{l-1}}, \] so \[ \sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau(i)-1,\tau(j)-1]_{l-1}}\ge\frac{(2m-1)!}{(m-1)!^2}\frac{(2l-1)!}{(l!)^2}. \] Now, letting $m\leftarrow m+1$, $l\leftarrow l+1$, then $\bar\tau(i)\leftarrow\tau(i+1)-1$, we obtain the inequality (\ref{eq-words}).
$\square$
Note that, for $l=0$ (which includes the case $m=0$), the inequality (\ref{eq-words}) becomes an equality. We conjecture, however, that the strict inequality holds if $l>0$, i.e. if $\tau$ is not a constant string.
As in the case of patterns in permutations, it would be interesting to characterize the patterns $\tau\in[l]^m$, where the difference between the two sides of (\ref{eq-words}) is minimal.
We also note that the covariance $Cov(X_{\tau_1},X_{\tau_2})$ of patterns $\tau_1,\tau_2\in [l]^m$ is positive (resp. negative) if \[ \sum_{i,j=1}^{m}{[i-1,j-1]_{m-1}[\tau_1(i)-1,\tau_2(j)-1]_{l-1}}-\frac{(2m-1)!}{(m-1)!^2}\frac{(2l-1)!}{(l!)^2} \] is positive (resp. negative). Hence, it would be interesting to characterize pairs of patterns $\tau_1,\tau_2\in[l]^m$ based on the sign of the covariance $Cov(X_{\tau_1},X_{\tau_2})$.
\begin{center} \textbf{Acknowledgements} \end{center}
I am grateful to Herbert S. Wilf and Donald E. Knuth for their helpful suggestions.
\end{document} | arXiv | {
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\begin{document}
\def \inbar{\vrule height1.5ex width.4pt depth0pt} \def \xC{\relax\hbox{\kern.25em$\inbar\kern-.3em{\rm C}$}} \def \xR{\relax{\rm I\kern-.18em R}} \newcommand{\xR}{\xR} \newcommand{\xC}{\xC} \newcommand{Z \hspace{-.08in}Z}{Z \hspace{-.08in}Z} \newcommand{\begin{equation}}{\begin{equation}} \newcommand{\begin{equation}}{\begin{equation}} \newcommand{\end{equation}}{\end{equation}} \newcommand{\end{equation}}{\end{equation}} \newcommand{\begin{eqnarray}}{\begin{eqnarray}} \newcommand{\begin{eqnarray}}{\begin{eqnarray}} \newcommand{\end{eqnarray}}{\end{eqnarray}} \newcommand{\end{eqnarray}}{\end{eqnarray}} \newcommand{\nonumber}{\nonumber} \newcommand{\nonumber}{\nonumber} \newcommand{\rangle}{\rangle} \newcommand{\rangle}{\rangle} \newcommand{\langle}{\langle} \newcommand{\langle}{\langle} \newcommand{\mbox{\footnotesize${\cal N}$}}{\mbox{\footnotesize${\cal N}$}} \newcommand{\mbox{\footnotesize${\cal N}$}}{\mbox{\footnotesize${\cal N}$}} \newcommand{\mbox{\footnotesize${\cal M}$}}{\mbox{\footnotesize${\cal M}$}} \title{Variational Sturmian Approximation: A nonperturbative method of solving time-independent Schr\"odinger equation} \author{Ali Mostafazadeh\thanks{E-mail address: amostafazadeh@ku.edu.tr}\\ \\ Department of Mathematics, Ko\c{c} University,\\ Rumelifeneri Yolu, 80910 Sariyer, Istanbul, TURKEY} \date{ } \maketitle
\begin{abstract} A variationally improved Sturmian approximation for solving time-independent Schr\"odinger equation is developed. This approximation is used to obtain the energy levels of a quartic anharmonic oscillator, a quartic potential, and a Gaussian potential. The results are compared with those of the perturbation theory, the WKB approximation, and the accurate numerical values. \end{abstract}
\section*{I.~Introduction} Since the early days of quantum mechanics, the main technical tools for solving the time-independent Schr\"odinger equation have been the time-independent perturbation theory, the semiclassical or WKB approximation, and the variational method \cite{perturbation,landau}. Starting form the late 1950's, physical chemists and nuclear physicists have explored the use of what is called the {\em Sturmian basis functions} in solving this equation for a variety of potentials arising in molecular and atomic physics \cite{ro,sturmian}. Recently, Antonsen \cite{antonsen} and Szmytkowski and Zywicka-Mozeiko \cite{poland} have studied the harmonic oscillator Sturmian functions. The purpose of the present article is to outline a general variationally improved Sturmian approximation scheme that provides a nonperturbative method of solving time-independent Schr\"odinger equation.
The organization of the article is as follows. In section~II, we give the definition of the Sturmian basis vectors, derive their general properties, and discuss the conventional Sturmian approximation. In section~III, we present an improved Sturmian approximation which makes use of the variational method. In section~IV, we study the harmonic oscillator Sturmians and use them for the solution of time-independent Schr\"odinger equation in one dimension. In section~V, we apply our general results to some concrete problems, and compare our results with those obtained using perturbation theory, the WKB approximation, and the highly accurate numerical investigations. In particular, we obtain the energy levels of a quartic anharmonic oscillator, a quartic potential, and a Gaussian potential. Finally, in section~VI, we summarize our results and present our conclusions.
\section*{II.~Conventional Sturmian Approximation}
Consider the time-independent Schr\"odinger equation:
\begin{equation}
H|E,a\rangle=E|E,a\rangle\;,
\label{xe1}
\end{equation} where $H$ is a self-adjoint Hamiltonian operator and $a$ is a degeneracy label.
The method of Sturmian approximation is based on an expansion of the
eigenvectors $|E,a\rangle$ in terms of solutions $|\phi_\nu,\alpha\rangle$ of the equation
\begin{equation}
(H_0+\beta_\nu V_0)|\phi_\nu,\alpha\rangle={\cal E}|\phi_\nu,\alpha\rangle\,,
\label{xe2}
\end{equation} where $H_0$ and $V_0$ are self-adjoint operators, $\beta_\nu$ and ${\cal E}$ are real scalar parameters, and $\alpha$ is a degeneracy label.
Note that in order to obtain $|\phi_\nu,\alpha\rangle$, one must fix ${\cal E}$
and solve Eq.~(\ref{xe2}) for $|\phi_\nu,\alpha\rangle$. Clearly, every solution
$|\phi_\nu,\alpha\rangle$ would correspond to a choice for the value of the `coupling constant' $\beta_\nu$.
Suppose that $H_0={\vec p}^2/(2m)$ is the Hamiltonian for a free particle moving in the configuration space $\xR^d$, $V_0=V_0(\vec x)$ is a real interaction potential, and $H$ is a standard Hamiltonian of the form
\begin{equation}
H=\frac{{\vec p}^2}{2m}+V(\vec x)\;,
\label{e3}
\end{equation} where $\vec p$ and $\vec x$ are momentum and position operators, respectively. If $V_0(\vec x)$ tends to infinity as $|\vec x|\to\infty$, all the eigenstates of $V_0$ are bound states \cite{messiyah}. In this case, only for a discrete set of positive values of $\beta_\nu$, can we find square-integrable solutions
$|\phi_\nu,\alpha\rangle$ of Eq.~(\ref{xe2}). In this case, the label $\nu$ will take values in a discrete set which we shall choose to be $\{0,1,2,\cdots\}$.
The only difference between Eq.~(\ref{xe2}) and the eigenvalue equation for the potential $\beta_\nu V_0(\vec x)$ is that in the former ${\cal E}$ is a fixed parameter which can be arbitrarily chosen. Therefore, a solution of
Eq.~(\ref{xe2}) corresponds to a pair $(\beta_\nu,|\phi_{\nu},\alpha\rangle)$.
The vectors $|\phi_\nu,\alpha\rangle$ are called the {\em Sturmian basis vectors} or simply the {\em Sturmians} \cite{ro}. They satisfy certain orthonormality conditions which we shall derive below. We should, however, note that the square-integrable Sturmians do not generally constitute a complete set of basis vectors of the Hilbert space \cite{poland}. There are certain potentials $V_0$, such as the Coulomb potential, that lead to a complete set of square-integrable Sturmians \cite{weniger}.
Let us first note that the defining equation~(\ref{xe2}) does not determine
$|\phi_\nu,\alpha\rangle$ uniquely. This is reflected in the presence of the degeneracy label $\alpha$. What is uniquely determined by Eq.~(\ref{xe2}) is
the degeneracy subspace ${\cal H}_\nu$ spanned by $\{|\phi_\nu,1\rangle,
|\phi_\nu,2\rangle,\cdots,|\phi_\nu,\ell_\nu\rangle\}$, where $\ell_\nu$ is the degree of degeneracy, i.e., the number of linearly independent solutions of Eq.~(\ref{xe2}) associated with a given (admissible) value of $\beta_\nu$. Clearly, we can construct an orthonormal basis of ${\cal H}_\nu$ and choose
the Sturmian vectors $|\phi_\nu,\alpha\rangle$ to be the basis vectors. In other words, without loss of generality, we can choose to work with the Sturmians
$|\phi_\nu,\alpha\rangle$ satisfying
\begin{equation}
\langle\phi_\nu,\alpha|\phi_\nu,\gamma\rangle=\delta_{\alpha\gamma}\;,
\label{orthono}
\end{equation} where $\delta_{\alpha\gamma}$ denotes the Kronecker delta function. Clearly, any unitary transformation of ${\cal H}_\nu$ would lead to a new set of Sturmians satisfying (\ref{orthono}). Therefore, the condition (\ref{orthono})
reduces the freedom in the choice of the Sturmians $|\phi_\nu,\alpha\rangle$, but does not eliminate it.
Next, we evaluate the Hermitian adjoint of both sides of Eq.~(\ref{xe2}), change $(\nu,\alpha)$ to $(\mu,\gamma)$, and take the inner product of both
sides of the resulting equation with $|\phi_\nu,\alpha\rangle$. This yields
\begin{equation}
\beta_\mu\langle\phi_\mu,\gamma|V_0|\phi_\nu,\alpha\rangle
=\langle\phi_\mu,\gamma|({\cal E}-H_0)|\phi_\nu,\alpha\rangle\;.
\label{e1}
\end{equation} We can compute the right-hand side of this equation using Eq.~(\ref{xe2}). Substituting the result in (\ref{e1}), we find
\begin{equation}
(\beta_\mu-\beta_\nu)\langle\phi_\mu,\gamma|V_0|\phi_\nu,\alpha\rangle=0\;.
\label{e2}
\end{equation} If we define
\begin{equation}
N_{\nu}^{\gamma\alpha}:=\langle\phi_\nu,\gamma|V_0|\phi_\nu,\alpha\rangle\;,
\label{N}
\end{equation} then we can write Eq.~(\ref{e2}) in the form
\begin{equation}
\langle\phi_\mu,\gamma|V_0|\phi_\nu,\alpha\rangle=
N_\nu^{\gamma\alpha}\delta_{\mu\nu}\;.
\label{ortho}
\end{equation} Eq.~(\ref{ortho}) is the desired orthogonality property of the Sturmians. We can further simplify Eq.~(\ref{ortho}), by noting that the $\ell_\nu\times\ell_\nu$ matrix $N_{\nu}$ formed by $N_{\nu}^{\gamma\alpha}$ is a Hermitian matrix. This means that we can
choose $|\phi_\nu,\alpha\rangle$ in such a way that $N_{\nu}$ is a diagonal matrix. Making this choice, we have
\begin{eqnarray}
N_{\nu}^{\alpha\gamma}&=&
N_{\nu}^{\alpha}\delta_{\alpha\gamma}\;,
\label{e3new}\\
\langle\phi_\mu,\gamma|V_0(\vec x)|\phi_\nu,\alpha\rangle&=&
N_\nu^{\alpha}\delta_{\mu\nu}\delta_{\alpha\gamma}\;,
\label{e4}
\end{eqnarray} where $N^\nu_\alpha$, with $\alpha\in\{1,2,\cdots,\ell_\nu\}$, are eigenvalues of the matrix $N_\nu$. Since $N_\nu$ is Hermitian, $N_\nu^\alpha$ are real.
In summary, we can choose a set of Sturmian vectors $|\phi_\nu,\alpha\rangle$ which are eigenvectors of the matrices $N_\nu$. Therefore, for each value
of $\nu$, $\{|\phi_\nu,1\rangle,\cdots,|\phi_\nu,\ell_\nu\rangle\}$ forms an orthonormal eigenbasis of $N_\nu$ in the degeneracy subspace ${\cal H}_\nu$. However,
$|\phi_\nu,\alpha\rangle$ with different values of $\nu$ are not orthogonal. Instead, they satisfy a modified orthogonality condition, namely (\ref{e4}).
Now, let us expand a solution $|E,a\rangle$ of the Schr\"odinger equation (\ref{xe1}), in a Sturmian basis corresponding to a `solvable' potential $V_0$, i.e., seek solutions of the form
\begin{equation}
|E,a\rangle=\sum_{\nu=0}^\infty\sum_{\alpha=1}^{\ell_\nu}
C_\nu^\alpha|\phi_\nu,\alpha\rangle\;,
\label{e6}
\end{equation} where $C_\nu^\alpha$ are complex coefficients and $\nu$ is supposed to take discrete values $0,1,2,\cdots$. Note that if the Sturmians $|\phi_\nu,\alpha\rangle$
do not form a complete basis, then Eq.~(\ref{e6}) yields the eigenvectors that belong to the span of $|\phi_\nu,\alpha\rangle$.
The {\em Sturmian Approximation of order $N$} is the approximation in which one neglects all the coefficients $C_\nu^\alpha$ in Eq.~(\ref{e6}) but those with $\nu$ belonging to a subset ${\cal S}_{N+1}$ of nonnegative integers of order $N+1$. Alternatively, in considering the Sturmian approximation of order $N$, one confines the range of the indices (of type) $\nu$ to a fixed finite set ${\cal S}_{N+1}$. In this way, the infinite sum $\sum_{\nu=0}^\infty\cdots$ in Eq.~(\ref{e6}) is replaced by the finite sum $\sum_{\nu\in{\cal S}_{N+1}}\cdots$. We shall abbreviate the latter by $\sum_\nu$. The set ${\cal S}_{N+1}$ may, in principle, be chosen arbitrarily. We will comment on this choice in section~III.
Substituting (\ref{e6}) in the Schr\"odinger equation (\ref{xe1}) and making use of Eqs.~(\ref{xe2}) and (\ref{e3}), we find
\begin{equation}
\sum_{\nu}\sum_{\alpha=1}^{\ell_\nu} C_\nu^\alpha
\left(E-{\cal E}-V+\beta_\nu V_0\right)|\phi_\nu,\alpha\rangle=0.
\label{e8}
\end{equation} Now, evaluating the inner product of both sides of this equation with
$|\phi_\mu,\gamma\rangle$ and using the orthogonality relation~(\ref{e4}), we obtain
\begin{equation}
\sum_{\nu}\sum_{\alpha=1}^{\ell_\nu}\left[(E-{\cal E})
T^{\gamma\alpha}_{\mu\nu}-(W^{\gamma\alpha}_{\mu\nu}-
\beta_\nu N^\alpha_\nu\delta_{\mu\nu}\delta_{\gamma\alpha})\right]
C^\alpha_\nu=0\;.
\label{e9}
\end{equation} Here we have introduced
\begin{eqnarray}
T_{\mu\nu}^{\gamma\alpha}&:=&
\langle\phi_\mu,\gamma|\phi_\nu,\alpha\rangle\;,
\label{T}\\
W_{\mu\nu}^{\gamma\alpha}&:=&
\langle\phi_\mu,\gamma|V|\phi_\nu,\alpha\rangle\;.
\label{W}
\end{eqnarray} We can express Eq.~(\ref{e9}) in a more compact form, if we use a single label for the pair $(\nu,\alpha)$. Introducing $\mbox{\footnotesize${\cal N}$}:=(\nu,\alpha)$ and $\mbox{\footnotesize${\cal M}$}:=(\mu,\gamma)$, we write Eq.~(\ref{e9}) in the form
\begin{equation}
\sum_{\mbox{\footnotesize${\cal N}$}} \left[(E-{\cal E})T_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}-S_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}\right]
C_{\mbox{\footnotesize${\cal N}$}}=0\;,
\label{e10}
\end{equation} where
\begin{eqnarray}
T_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}&=&T_{\mu\nu}^{\gamma\alpha}\;,~~~~
S_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}=W_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}-\beta_\nu N_{\mbox{\footnotesize${\cal N}$}}\delta_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}\;,
\label{e11.1}\\
W_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}&=&W_{\mu\nu}^{\gamma\alpha}\;,~~~~
N_{\mbox{\footnotesize${\cal N}$}}=N^\alpha_\nu\;.
\label{e11.2}
\end{eqnarray}
Eqs.~(\ref{e10}) form a linear system of homogeneous first order algebraic equations for $C_{\mbox{\footnotesize${\cal N}$}}$. This system has a nontrivial solution provided that the determinant of the matrix of coefficients vanishes, i.e.,
\begin{equation}
\det\left[(E-{\cal E})T-S\right]=0\;.
\label{e12}
\end{equation} Here $T$ and $S$ are matrices with entries $T_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}$ and $S_{\mbox{\footnotesize${\cal M}$}\mbox{\footnotesize${\cal N}$}}$, respectively.
Solving Eq.~(\ref{e12}), we can express $E$ in terms of ${\cal E}$,
$\beta_\nu$, and the Sturmians $|\phi_\nu,\alpha\rangle$. Now, we recall that for a fixed choice of $V_0$, the coupling constants $\beta_\nu$ and the
corresponding Sturmians $|\phi_\nu,\alpha\rangle$ depend on the parameter ${\cal E}$. Therefore, Eq.~(\ref{e12}) yields $E$ as a function of ${\cal E}$. Furthermore, substituting the value of $E=E({\cal E})$ obtained by solving (\ref{e12}) in (\ref{e10}) and solving for the coefficients
$C_{\mbox{\footnotesize${\cal N}$}}$, we obtain an expression for the eigenvector $|E,a\rangle$ that involves ${\cal E}$. As we discuss in the following section, the fact that the Sturmian approximation yields the eigenvalues and the eigenvectors of the Hamiltonian as functions of a free parameter seems to have been overlooked. This is mainly because there is a choice for the parameter ${\cal E}$ that simplifies the calculations.
It should also be emphasized that Eq.~(\ref{e12}) is an algebraic equation of order $N+1$. Therefore, in general, it has $N+1$ solutions. This can be understood by noting that in the Sturmian approximation one actually approximates the Hilbert space by a finite-dimensional vector space. Consequently, the Hamiltonian is replaced with a matrix with a finite number of eigenvalues.
\section*{III.~Variational Sturmian Approximation}
In general, the accuracy of the Sturmian approximation depends on the following factors.
\begin{itemize}
\item[] {\bf 1. Choice of $V_0$:} In practice, $V_0$ must be one of the exactly solvable potentials. Therefore, the available choices for $V_0$ are few in number. For the potentials $V$ with bound states, we can choose $V_0$ to be a harmonic oscillator potential. For example, for the quartic anharmonic oscillator
\begin{equation}
V(x)= \frac{k}{2}\,x^2 +\epsilon\, x^4\;,
\label{quartic-v}
\end{equation} we shall take
\begin{equation}
V_0(x)=\frac{k}{2}\,x^2\;.
\label{quartic-v0}
\end{equation} Similarly, for the Gaussian potential
\begin{equation}
V(x)=-\lambda e^{-\epsilon x^2/2}\;,
\label{gaussian}
\end{equation} we shall take
\begin{equation}
V_0(x)=\frac{1}{2}\lambda\epsilon x^2-\lambda\;.
\label{sho}
\end{equation} This will enable us to compare the results of the Sturmian approximation with those of the perturbation theory, for in the limit $\epsilon\to 0$, $V(r)\to V_0(r)$. Note that multiplying $V_0$ by a positive real number does not change the results of the Sturmian approximation. This is simply because we can always absorb such a number in the definition of $\beta_\nu$.
\item[] {\bf 2. Choice of the Sturmians included in the sum (\ref{e10}):} This is also directly related to the choice of the potential $V_0$. If $V_0$ is obtained from $V$ by a limiting process as in the case of the potentials~(\ref{quartic-v}) and (\ref{gaussian}), then a natural choice for the computation of the $n$-th energy eigenvalue $E_n$ and the corresponding
eigenvectors $|E_n,\alpha\rangle$ is to include the $|\phi_\nu,\alpha\rangle$ with $\nu$ equal or close to $n$. In particular, in the Sturmian approximation of order zero, we have
\begin{equation}
|E_n,a\rangle=
\sum_{\alpha=1}^{\ell_n}C^\alpha_n|\phi_n,\alpha\rangle\;.
\label{e10-new}
\end{equation}
\item[] {\bf 3. Choice of the parameter ${\cal E}$:} The conventional choice \cite{sturmian,antonsen} for the parameter ${\cal E}$ is ${\cal E}=E$. This simplifies Eq.~(\ref{e12}) considerably. The basic idea pursued in this article is the fact that this simplification does not necessarily justify the conventional choice for ${\cal E}$.
\end{itemize}
It is well-known \cite{landau} that the eigenvalue equation (\ref{xe1}) is equivalent to the variational equation
\begin{equation}
\frac{\delta}{\delta\langle\psi|}\left(\frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}
\right)=0\;.
\label{e13}
\end{equation} In other words, the eigenvalues $E$ are the minima of the expectation value
\[\langle H\rangle:=\frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}\,,\]
and the eigenvectors are the vectors $|E\rangle=|E,a\rangle$ that minimize $\langle H\rangle$. This observation suggests that the most efficient choice for the parameter ${\cal E}$ appearing in the Sturmian approximation is the one that minimizes $E=E({\cal E})$. Therefore, the most reliable Sturmian approximation is obtained by choosing ${\cal E}$ to be a solution of
\begin{equation}
\frac{dE}{d{\cal E}}=0\;.
\label{e14}
\end{equation} If this equation does not have a solution, then one must either make another choice for the set ${\cal S}_{N+1}$ or proceed with a higher order Sturmian approximation.
\section*{IV.~Variational Sturmian Approximation Using Harmonic Oscillator Sturmians}
Consider a quantum system with the configuration space $\xR$, a standard Hamiltonian~(\ref{e3}), and a real-valued potential $V=V(x)$. Suppose that the system has an infinite number of bound states with nondegenerate energy eigenvalues $E_n$. Here $n\in\{0,1,2,\cdots\}$ and $E_0$ stands for the ground state.
Now, consider the Sturmian basis vectors associated with a harmonic oscillator \cite{antonsen,poland},
\begin{equation}
V_0=V_0(x)=\frac{k}{2}\,x^2\;.
\label{f1}
\end{equation} In order to solve Eq.~(\ref{xe2}) for this choice of $V_0$, we introduce
\begin{eqnarray}
\omega_\nu&:=&\left(\frac{\beta_\nu k}{m}\right)^{1/2}\,,
\label{f1.1}\\
\alpha_\nu&:=&\frac{m\omega_\nu}{\hbar}\,,
\label{alpha}\\
a_\nu&:=&\left(\frac{\alpha_\nu}{2}\right)^{1/2}
\left(x+\frac{ip}{\hbar\alpha_\nu}\right)\;.
\label{f1.2}\\
|\ell\rangle_\nu&:=&\frac{1}{\sqrt \ell !}\,a^{\dagger\ell}_\nu|0\rangle_\nu\;,
\label{f1.3}
\end{eqnarray} where $|0\rangle_\nu$ is the normalized real ground state vector for a harmonic oscillator with mass $m$ and frequency $\omega_\nu$. That is
\begin{equation}
\langle x|0\rangle_\nu:=\left(\frac{\alpha_\nu}{\pi}\right)^{1/4}
e^{-\alpha_\nu x^2/2}.
\label{f1.4}
\end{equation} In view of the similarity of Eq.~(\ref{xe2}) with the eigenvalue equation for the potential $\beta_\nu V_0$, we can easily deduce that
\begin{eqnarray}
{\cal E}&=&\hbar\omega_\nu(\nu+\frac{1}{2})\;,
\label{f2.1}\\
|\phi_\nu\rangle&=&|\nu\rangle_\nu
\label{f2.2}
\end{eqnarray} where $\nu\in\{0,1,2,\cdots\}$.
We can invert Eqs.~(\ref{f2.1}) and (\ref{f1.1}) to express $\omega_\nu$ and $\beta_\nu$ in terms of ${\cal E}$. This yields
\begin{eqnarray}
\omega_\nu&=&\frac{2{\cal E}}{\hbar(2\nu+1)}\;,
\label{f2}\\
\beta_\nu&=& \frac{4m{\cal E}^2}{\hbar^2k(2\nu+1)^2}\;.
\label{f3}
\end{eqnarray} Substituting Eq.~(\ref{f2}) in (\ref{alpha}), we have
\begin{equation}
\alpha_\nu=\frac{2m{\cal E}}{\hbar^2(2\nu+1)}=
\frac{\alpha_0}{2\nu+1}\;.
\label{alpha=}
\end{equation}
Next, we compute the term
$\beta_\nu N_\nu=\beta_\nu\langle\phi_\nu|V_0|\phi_\nu\rangle$. We can use the properties of the annihilation operator $a_\nu$, namely
\begin{equation}
a_\nu|\ell\rangle_\nu=\sqrt{\ell}|\ell-1\rangle_\nu\;,~~~
a_\nu^\dagger|\ell\rangle_\nu=\sqrt{\ell+1}|\ell+1\rangle_\nu\;,~~~
x=(2\alpha_\nu)^{-1/2}(a_\nu+a_\nu^\dagger),
\label{f11.2}
\end{equation} and the orthonormality of $|\ell\rangle_\nu$ to compute
\begin{equation}
~_\nu\langle\ell|x^2|\nu\rangle_\nu= (2\alpha_\nu)^{-1}\left[
(2\nu+1)\delta_{\ell,\nu}+\sqrt{(\nu+1)(\nu+2)}\,\delta_{\ell,\nu+2}+
\sqrt{\nu(\nu-1)}\,\delta_{\ell,\nu-2}\right]\;.
\label{f12}
\end{equation} In view of Eqs.~(\ref{f1}) and (\ref{f12}), we obtain, after some remarkable simplifications,
\begin{equation}
\beta_\nu N_\nu=\frac{\cal E}{2}\;.
\label{f6}
\end{equation} \subsection*{Variational Sturmian Approximation of Order Zero}
For the variational Sturmian approximation of order zero, $\nu=n$, and Eq.~(\ref{e12}) takes the form
\begin{equation}
(E_n-{\cal E})T-S=0\;,
\label{f4}
\end{equation} where
\begin{equation}
T=\langle\phi_n|\phi_n\rangle=1\,, ~~~~ S=W-\beta_n N_n\,,
~~~~ W=\langle\phi_n|V|\phi_n\rangle\;.
\label{f5.2}
\end{equation} According to Eqs.~(\ref{f4}), (\ref{f5.2}), and (\ref{f6}), the energy eigenvalues $E_n$ are given by
\begin{equation}
E_n=W+\frac{\cal E}{2}\;.
\label{f7}
\end{equation}
Next, we fix the parameter ${\cal E}$ using Eq.~(\ref{e14}). This requires the computation of $dW/d{\cal E}$. We first evaluate the variation of $W$,
\begin{eqnarray}
\delta W&=&(\delta\langle\phi_n|)V|\phi_n\rangle+ \langle\phi_n|V(\delta|\phi_n\rangle)
\nonumber\\
&=&2 \langle\phi_n|V (\delta|\phi_n\rangle)\nonumber\\
&=&2\sum_{\ell=0}^\infty ~_n\langle n|V|\ell\rangle_n~_n\langle\ell|(\delta|n\rangle_n)\;.
\label{f8}
\end{eqnarray} Here we have made use of Eq.~(\ref{f2.2}), the fact that the Sturmians and
the potential $V$ are real and $|\ell\rangle_n$ form a complete set of basis vectors.
We can compute $~_n\langle\ell|(\delta|n\rangle_n)$ using the eigenvalue equation
\begin{equation}
(H_0+\beta_n V_0)|j\rangle_n={\cal E}_j|j\rangle_n\;,
\label{f9}
\end{equation} where ${\cal E}_j=\hbar\omega_n(j+1/2)$. Taking the variation of both sides of this equation and computing the inner product with $|\ell\rangle_n$, we find
\begin{equation}
~_n\langle \ell|\delta|j\rangle_n =\frac{ ~_n\langle\ell| V_0|j\rangle_n \,(\delta\beta_n) }{
{\cal E}_j-{\cal E}_\ell} ~~~~~~{\rm for}~~\ell\neq j\;.
\label{f10}
\end{equation} Furthermore, using the fact that the eigenfunctions $\langle x|\ell\rangle_n$ are real, we can easily show that
\begin{equation}
~_n\langle \ell|(\delta|\ell\rangle_n)=0\;.
\label{f11}
\end{equation}
Eqs.~(\ref{f8}), (\ref{f10}) and (\ref{f11}) reduce the computation of $\delta W$ to that of
\begin{equation}
~_n\langle\ell| V_0|n\rangle_n =\frac{k}{2} ~_n\langle\ell|x^2|n\rangle_n \;.
\label{f11.1}
\end{equation} We have already computed $~_n\langle\ell|x^2|n\rangle_n$ in Eq.~(\ref{f12}). Substituting this equation in Eq.~(\ref{f11.1}) and using Eqs.~(\ref{f10}) and (\ref{f8}), we find, after some remarkable cancellations,
\begin{equation}
\delta W= \left(\frac{\delta{\cal E}}{2{\cal E}}\right) \left[
\sqrt{n(n-1)}~_n\langle n|V|n-2\rangle_n-\sqrt{(n+1)(n+2)}~_n\langle n|V|n+2\rangle_n
\right]\;.
\label{f13}
\end{equation}
Now, in view of Eqs.~(\ref{f7}) and (\ref{f13}),
\[\frac{dE_n}{d{\cal E}}=\left(\frac{1}{2{\cal E}}\right)
\left[ \sqrt{n(n-1)}~_n\langle n|V|n-2\rangle_n -
\sqrt{(n+1)(n+2)}~_n\langle n|V|n+2\rangle_n\right]+\frac{1}{2}\;.\] Substituting this equation in Eq.~(\ref{e14}) yields
\begin{equation}
{\cal E}=\sqrt{(n+1)(n+2)}~_n\langle n|V|n+2\rangle_n-
\sqrt{n(n-1)}~_n\langle n|V|n-2\rangle_n\;.
\label{f14}
\end{equation} Note that the right-hand side of this equation also involves ${\cal E}$. This is because $|\ell\rangle_n$ depend on ${\cal E}$.
Using Eqs.~(\ref{f14}) and (\ref{f7}) we can express the energy eigenvalue $E_n$ in terms of $V$. This yields
\begin{equation}
E_n= ~_n\langle n|V|n\rangle_n +\frac{1}{2}\,
\left [\sqrt{(n+1)(n+2)} ~_n\langle n|V|n+2\rangle_n -
\sqrt{n(n-1)}~_n\langle n|V|n-2\rangle_n\right]\;.
\label{f15}
\end{equation} For the ground state $n=0$~ and Eq.~(\ref{f15}) reduces to
\begin{equation}
E_0=~_0\langle 0|V|0\rangle_0+\frac{1}{\sqrt{2}}~_0\langle 0|V|2\rangle_0\;.
\label{f16}
\end{equation} Note that the vectors $|\ell\rangle_n$ appearing in Eqs.~(\ref{f15}) and (\ref{f16}) are those of Eq.~(\ref{f1.3}) with ${\cal E}$ being a solution of Eq.~(\ref{f14}).
Eqs.~(\ref{f15}) and (\ref{f16}) are of limited importance. In practice, one obtains the energy eigenvalue $E_n$ by substituting the solution of Eq.~(\ref{f14}) in Eq.~(\ref{f7}).
The variational Sturmian approximation of order zero, as outlined above, is a valid approximation scheme, if Eq.~(\ref{f14}) has a unique positive solution~${\cal E}$. If such a solution does not exist, one may attempt to construct higher order variational Sturmian approximations. As we shall see in section~V, for all the potentials that we have considered, Eq.~(\ref{f14}) has a unique positive solution. This is very remarkable, for this equation turns out to be an algebraic equation of order three for the quartic anharmonic oscillator and the quartic potential, and of order four for the Gaussian potential.
\subsection*{Variational Sturmian Approximation of Order One}
In the variational sturmian approximation of order one, the number of Sturmians contributing to the eigenvector $|E\rangle$ is two. We shall denote them by $|\phi_n\rangle$ and $|\phi_m\rangle$.
The matrices $T$ and $W$ are Hermitian $2\times 2$ matrices. They can be written in the form
\begin{equation}
T=\left(\begin{array}{cc}
1&t^*\\
t&1\end{array}\right)\,,~~~~
W=\left(\begin{array}{cc}
v_n&w^*\\
w&v_m\end{array}\right)\,,
\label{f18}
\end{equation} where we have used the fact that the Sturmians are normalized and introduced
\begin{equation}
t:= \langle\phi_m|\phi_n\rangle, ~~~v_\nu:=\langle\phi_\nu|V|\phi_\nu\rangle, ~~~
w:=\langle\phi_m|V|\phi_n\rangle.
\label{f19}
\end{equation} Next, we construct the matrix $S$. In view of Eqs.~(\ref{e11.1}) and (\ref{f18}),
\begin{equation}
S=\left(\begin{array}{cc}
v_n-\beta_n N_n&w^*\\
w&v_m-\beta_m N_m\end{array}\right)\,.
\label{f20}
\end{equation}
Note that because the Sturmians for the harmonic oscillator are real-valued, $t$ and $w$ are real-valued functions of the parameter ${\cal E}$. In particular, the matrices $T,~W$ and $S$ are real and symmetric.
Substituting Eqs.~(\ref{f18}) and (\ref{f20}) in the Eq.~(\ref{e12}), making use of Eq.~(\ref{f6}), and simplifying the resulting expression, we find
\begin{equation}
A (E-\frac{\cal E}{2})^2+B(E-\frac{\cal E}{2})+C=0\;,
\label{f21}
\end{equation} where
\begin{equation}
A:=1-t^2,~~~B:=t(t{\cal E}+2w)-(v_n+v_m),~~~C:=v_nv_m-(
\frac{t{\cal E}}{2}+w)^2.
\label{f22}
\end{equation} Note that the coefficients $A$, $B$, and $C$ are functions of ${\cal E}$. Eq.~(\ref{f21}) can be easily solved to express $E$ in terms of ${\cal E}$. The result is
\begin{equation}
E=E_\pm:=\frac{\cal E}{2}+\frac{-B\pm\sqrt{B^2-4AC}}{2A}\;.
\label{f23}
\end{equation}
The next step is to determine ${\cal E}$ using the variational principle, i.e., setting $dE/d{\cal E}=0$. The resulting formulas are complicated and we shall not include them here.
We conclude this section with the following remarks.
\begin{itemize}
\item[1.] As seen from Eq.~(\ref{f23}), the first order Sturmian
approximation leads to a pair of energy eigenvalues. For the potentials
which are related to $V_0$ via a limiting process, one expects these
two eigenvalues to be those labelled by $m$ and $n$. That is, for $n<m$,
\begin{equation}
E_n=E_-,~~~~E_m=E_+\;.
\label{f24}
\end{equation}
\item[2.] In the variational Sturmian approximation of order one, there are
two variational equations $dE_\pm/d{\cal E}=0$. It is not clear whether
these equations lead to a unique minimum for $E_\pm({\cal E})$ with a
positive value for ${\cal E}$. As we shall show in the following sections,
for all the specific examples that we have considered each of these
equations lead to a unique minimum with a positive value for ${\cal E}$.
Lack of such a solution may be interpreted as the failure of the variational
Sturmian approximation of order one.
\end{itemize}
\subsection*{Variational Sturmian Approximation of Order Two}
In the variational sturmian approximation of order two, one uses three
Sturmians to expand the energy eigenvectors $|E\rangle$. We shall denote these by
$|\phi_{n_\ell}\rangle$ where $n_\ell\in{\cal S}_3:=\{n_1,n_2,n_3\}$.
The matrices $T$, $W$, and $S$ are given by
\begin{eqnarray}
T&=&\left(\begin{array}{ccc}
1 & t_1^* & t_2^*\\
t_1 & 1 & t_3^* \\
t_2 & t_3 & 1 \end{array}\right),~~~~
W=\left(\begin{array}{ccc}
v_3 & w_1^* & w_2^*\\
w_1 & v_2 & w_3^* \\
w_2 & w_3 & v_1 \end{array}\right)\;,
\label{f25.1}\\
S&=&\left(\begin{array}{ccc}
v_3 -\beta_{n_3}N_{n_3} & w_1^* & w_2^*\\
w_1 & v_2-\beta_{n_2}N_{n_2} & w_3^* \\
w_2 & w_3 & v_1-\beta_{n_1}N_{n_1} \end{array}\right)\;,
\label{f25.3}
\end{eqnarray} where we have used the fact that $|\phi_{n_\ell}\rangle$ are normalized and introduced
\begin{eqnarray}
t_1&:=&\langle\phi_{n_2}|\phi_{n_3}\rangle,~~~~
t_2:=\langle\phi_{n_1}|\phi_{n_3}\rangle,~~~~
t_3:=\langle\phi_{n_1}|\phi_{n_2}\rangle,~~~~
v_{\ell}:=\langle\phi_{n_\ell}|V|\phi_{n_\ell}\rangle,
\label{f26.1}\\
w_1&:=&\langle\phi_{n_2}|V|\phi_{n_3}\rangle,~~~~
w_2:=\langle\phi_{n_1}|V|\phi_{n_3}\rangle,~~~~
w_3:=\langle\phi_{n_1}|V|\phi_{n_2}\rangle.
\label{f26.3}
\end{eqnarray} Because the harmonic oscillator Sturmian functions are real-valued, $t_\ell$, $v_\ell$ and $w_\ell$ are real, and $T$, $W$, and $S$ are real symmetric matrices.
In view of Eq.~(\ref{f6}), we can write the secular equation~(\ref{e12}) in the form
\begin{equation}
A (E-\frac{\cal E}{2})^3 +B(E-\frac{\cal E}{2})^2+
C(E-\frac{\cal E}{2})+D=0\;,
\label{f27}
\end{equation} where
\begin{eqnarray}
A&:=&1-\sum_{\ell=1}^3 t_\ell^2+2t_1t_2t_3\;,
\label{f28.1}\\
B&:=&\sum_{\ell=1}^3[(t_\ell^2-1)v_\ell+2t_\ell\xi_\ell]
-2(t_1t_2\xi_3+t_3t_1\xi_2+t_2t_3\xi_1)\;,
\label{f28.2}\\
C&:=&v_1v_2+v_2v_3+v_3v_1+2(t_1\xi_2\xi_3+t_3\xi_1\xi_2+
t_2\xi_3\xi_1)-\sum_{\ell=1}^3(\xi_\ell^2+2t_\ell v_\ell\xi_\ell)\;,
\label{f28.3}\\
D&:=&\sum_{\ell=1}^3v_\ell\xi^2_\ell-2\xi_1\xi_2\xi_3-v_1v_2V_3\;,
\label{f28.4}\\
\xi_\ell&:=&\frac{1}{2}\,t_\ell {\cal E}+w_\ell\;.
\label{f29}
\end{eqnarray} Eq.~(\ref{f27}) has, in general, three solutions. The desired eigenvalues are the minima of these solutions corresponding to positive values of ${\cal E}$. Again for the cases where $V$ is related to $V_0$ by a limiting process the minima of the solutions of Eq.~(\ref{f27}) correspond to $E_{n_1},E_{n_2}$, and $E_{n_3}$.
\section*{V.~Applications}
In this section, we apply our general results to compute the energy eigenvalues of a quartic anharmonic oscillator, a quartic potential, and a Gaussian potential.
\subsection*{The Quartic Anharmonic Oscillator}
Consider the potential
\begin{equation}
V(x)=\frac{k}{2}\, x^2+\epsilon x^4\;.
\label{g1}
\end{equation} In order to obtain the energy levels of this potential using variational Sturmian
approximation of order zero, we need to calculate $~_n\langle n|V|\ell\rangle_n$. We first use Eqs.~(\ref{f11.2}) to compute
\begin{eqnarray}
~_n\langle n|x^4|\ell\rangle_n&=&(2\alpha_n)^{-2}\left[
3(2n^2+2n+1)\delta_{\ell,n}+4(n+1)\sqrt{(n+1)(n+2)}\,
\delta_{\ell,n+2}+\right.\nonumber\\
&&2(2n-1)\sqrt{n(n-1)}\,\delta_{\ell,n-2}+
\sqrt{ (n+1) (n+2) (n+3) (n+4)}\,\delta_{\ell,n+4}+\nonumber\\
&& \left.\sqrt{(n-3) (n-2) (n-1)n} \,\delta_{\ell,n-4}\right].
\label{g2}
\end{eqnarray}
In view of this equation and Eqs.~(\ref{f12}) and (\ref{g1}),
\begin{eqnarray}
&&W=~_n\langle n|V|n\rangle_n = \frac{(2n+1)k }{4\alpha_n}+
\frac{3(2n^2+2n+1)\epsilon}{4\alpha_n^2}\;,
\label{g3.1}\\
&&~_n\langle n|V|n+2\rangle_n =
\sqrt{(n+1)(n+2)}\,\left(\frac{ k}{4\alpha_n}+
\frac{(n+1)\epsilon}{\alpha_n^2}\right)\;,
\label{g3.2}\\
&&~_n\langle n|V|n-2\rangle_n =
\sqrt{n(n-1)}\,\left(\frac{ k}{4\alpha_n}+
\frac{(2n-1)\epsilon}{2\alpha_n^2}\right)\;,
\label{g3.3}
\end{eqnarray}
Next, we substitute Eqs.~(\ref{g3.2}) and (\ref{g3.3}) in Eq.~(\ref{f14}). Using Eq.~(\ref{alpha=}), we then obtain
\begin{equation}
{\cal E}^3-p_n{\cal E}-q_n=0\;,
\label{g4}
\end{equation} where
\begin{equation}
p_n:= \left(\frac{\hbar^2 k}{m}\right)(n+\frac{1}{2})^2,~~~~
q_n:=\left(\frac{\hbar^4\epsilon}{2m^2}\right)(n+\frac{1}{2})^2
(11 n^2+9n+4)\,.
\label{g5}
\end{equation} It is not difficult to show that Eq.~(\ref{g4}) has a single positive solution [This is true for any positive $p_n$ and $q_n$.] given by \cite{galois}
\begin{equation}
{\cal E}=\left(\frac{q_n}{2}\right)^{1/3}\left(1+\sqrt{1-r_n}\right)^{1/3}
+\left(\frac{p_n}{3}\right)\left(\frac{2}{q_n}\right)^{1/3}
\left(1+\sqrt{1-r_n}\right)^{-1/3}\;,
\label{g6}
\end{equation} where
\begin{equation}
r_n:=\frac{4p_n^3}{27q_n^2}=
\left(\frac{8n+4}{11n^2+9n+4}\right)^2r_0\;,
~~~~ r_0:=\frac{m k^3}{108\hbar^2\epsilon^2}\;.
\label{g6.1}
\end{equation}
The right-hand side of Eq.~(\ref{g6}) is manifestly real and positive for $r_n\leq 1$. It is not difficult to check that it is also real and positive for $r_n>1$. In fact, we can express ${\cal E}$ in the form
\begin{equation}
{\cal E}=2 \sqrt{\frac{p_n}{3}} \cos\left(\frac{\phi_n}{3}\right)
=\sqrt{\frac{ k}{3m}}(2n+1)
\cos\left(\frac{\phi_n}{3}\right)\;,
\label{g7}
\end{equation} where
\begin{equation}
\phi_n:=\tan^{-1}(\sqrt{r_n-1})\;.
\label{g8}
\end{equation} Note that for $r_n<1$, $\phi_n$ is imaginary, but $\cos(\phi_n/3)$ is still real and positive.
Having fixed the parameter ${\cal E}$, we can determine the energy eigenvalues $E_n$ using Eqs.~(\ref{f7}) and (\ref{g3.1}). We first use Eqs.(\ref{alpha=}) and~(\ref{g7}) to compute
\begin{equation}
\alpha_n=\left(\frac{2}{\hbar}\right)\sqrt{\frac{m k}{3}}
\cos\left(\frac{\phi_n}{3}\right)\;.
\label{g9}
\end{equation} Then substituting this equation in Eq.~(\ref{g3.1}) and using Eq.~(\ref{f7}), we find
\begin{equation}
E_n= \left(\frac{\hbar}{24}\right)\sqrt{\frac{3 k}{m}}(2n+1)
\left[7+3\tan^2(\frac{\phi_n}{3})\right]\cos(\frac{\phi_n}{3})+
\left(\frac{9\hbar^2\epsilon}{16m k}\right)(2n^2+2n+1)
\left[1+\tan^2(\frac{\phi_n}{3})\right]\;.
\label{g10}
\end{equation} In particular, the ground state energy is given by
\begin{equation} E_0= \left(\frac{\hbar}{24}\right)\sqrt{\frac{3 k}{m}}
\left[7+3\tan^2(\frac{\phi_0}{3})\right]\cos(\frac{\phi_0}{3})+
\left(\frac{9\hbar^2\epsilon}{16m k}\right)
\left[1+\tan^2(\frac{\phi_0}{3})\right]\;.
\label{g10.1}
\end{equation}
In Table~1, we list the numerical values obtained using Eq.~(\ref{g10}) for the first 10 energy levels of a quartic anharmonic oscillator with $m=1/2$, $k=2$, $\epsilon=1/10$ in units where $\hbar=1$. This table also includes the accurate numerical values of Ref.~\cite{bacus}, the values obtained using the conventional Sturmian approximation and the zero and first order perturbation theory.[The zero and first order perturbation theory yield
\begin{equation}
E_n^{(0)}=\hbar( k/m)^{1/2}(n+1/2)\;,~~~~
E_n^{(1)}=E_n^{(0)}+3\hbar^2\epsilon^2(2n^2+2n+1)/(4m k)\,,
\label{pert-2-4}
\end{equation} respectively.] The relative difference between the results of the variational Sturmian approximation of order zero with the highly accurate numerical results
($E_n^\#$) of Ref.~\cite{bacus}, i.e., the quantity $|E_n-E_n^\#|/E^\#_n$, varies between $3.38\times 10^{-4}$ and $2.77\times 10^{-3}$. For the ground state, this number is $1.53\times 10^{-3}$. Even for the lowest lying energy levels where perturbation theory yields reliable results, the zero order variational Sturmian approximation produces more accurate values than both the zero and first order perturbation theory. As seen from Table~1, the variational Sturmian approximation is better than the conventional Sturmian approximation.
In the remainder of this section we present the results obtained using the first and second order variational Sturmian approximation. The numerical results are respectively presented in Tables~2 and~3.
As we explained in Section~IV, in the variational Sturmian approximation of order one one chooses an indexing set ${\cal S}_2$ consisting of two
Sturmians to be included in the expansion of the eigenvector $|E\rangle$. One then solves the corresponding secular equation~(\ref{e12}), expresses the solutions $E_\pm$ in terms of the parameter ${\cal E}$, and finds the minima of $E_\pm({\cal E})$. In general, $E_\pm$ are complicated functions of ${\cal E}$. However, it turns out that for all the cases that we considered $E_\pm$ has a unique minimum corresponding to a positive value of ${\cal E}$.
In order to choose the indexing set ${\cal S}_2$, we first note that the
Sturmian functions $\langle x|\phi_n\rangle$ with even (respectively odd) $n$ are even (respectively odd) functions of $x$. We expect the energy eigenfunctions of the anharmonic oscillator~(\ref{g1}) to have the same parity structure as the Sturmian functions. This, in particular, suggests that in the calculation of $E_0$ we should take ${\cal S}_2=\{0,2\}$.
For a quartic anharmonic oscillator with $m=1/2$, $ k=2$, $\epsilon=1/10$, the first order variational Sturmian approximation corresponding to ${\cal S}_2=\{0,2\}$ yields $E_0=E_-=1.06614$ and $E_2=E_+=5.76117$. The value obtained for $E_0$ differs from the accurate numerical value by one part in $10^4$. It is one order of magnitude better than the value obtained using the zero order variational Sturmian approximation. The value for $E_2$ is however less accurate. One may argue that the choice made for ${\cal S}_2$ is appropriate only for the ground state. In order to compute $E_2$ using the first order variational Sturmian approximation, one may alternatively choose ${\cal S}_2=\{2,4\}$. This choice yields $E_2=E_-=5.74558$ and $E_4=E_+=9.6637$. Again this value for $E_2$ is an order of magnitude better than the value obtained using the zero order variational Sturmian approximation, whereas the value for $E_4$ is less accurate. One can also try ${\cal S}_2=\{0,4\}$. As expected, this choice yields a less accurate value than the choices ${\cal S}_2=\{0,2\}$ and ${\cal S}_2=\{2,4\}$ for both $E_0$ and $E_4$.
For the calculation of the first excited state we choose ${\cal S}_2=\{1,3\}$. Then we find $E_1=E_-=3.30922$ and $E_3=E_+=8.37284$. Once again the first order variational Sturmian approximation of order one with the choice ${\cal S}_2=\{1,3\}$ yields a more accurate result for $E_1$ and a less accurate result for $E_3$.
In general, in the calculation of the energy levels $E_n$ with $n\geq 2$, there are two alternative choices for the indexing set ${\cal S}_2$. In view of the parity properties of the eigenvectors, these are $\{n,n+2\}$ and $\{n-2,n\}$. The fact that there is no physical reason to distinguish between these two choices suggests that for these levels one should consider the second order variational Sturmian approximation with the choice ${\cal S}_3=\{n-2,n,n+2\}$.
Table~3 includes the results of the second order variational Sturmian approximation corresponding to the indexing set ${\cal S}_3=\{0,2,4\}$. This approximation yields more accurate values for $E_0$ than the zero and first order variational Sturmian approximations. However, contrary to our expectation the value obtained for $E_2$ is less accurate than the one given by the zero order approximation and the first order approximation with ${\cal S}_2=\{2,4\}$.
\subsection*{The Quartic Potential}
Consider the quartic potential
\begin{equation}
V(x)=\epsilon\,x^4\;.
\label{h1}
\end{equation} We can easily obtain the energy levels of this potential using the zero order variational Sturmian approximation by simply setting $ k=0$ in our formulas for the quartic anharmonic oscillator. Substituting $ k=0$ in (\ref{g5}), we can write Eq.~(\ref{g4}) in the form
\begin{equation}
{\cal E}=q_n^{1/3}=\hbar\left(\frac{\hbar\epsilon}{2m^2}\right)^{1/3}
\left[(n+\frac{1}{2})^2(11n^2+9n+4)\right]^{1/3}\;.
\label{h2}
\end{equation} In view of Eqs.~ (\ref{alpha=}, (\ref{f7}), (\ref{g3.1}), (\ref{h2}), and $ k=0$, we have
\begin{eqnarray}
\alpha_n&=&\hbar^{-1}\left(m\hbar\epsilon\right)^{1/3}
\left(\frac{11n^2+9n+4}{2n+1}\right)^{1/3}\;,
\label{h3}\\
E_n&=&\left(\frac{8n+4}{11n^2+9n+4}\right)^{2/3}
\left(\frac{17}{7}n^2+\frac{15}{7}n+1\right)E_0\;,
\label{h4}\\
E_0&:=&
\frac{7\hbar}{8}\left(\frac{\hbar\epsilon}{2m^2}\right)^{1/3}\;.
\label{h5}
\end{eqnarray}
In Table~4, we present the values obtained using Eq.~(\ref{h4}) for the energy levels of a quartic potential with $m=1/2$ and $\epsilon=1$ in units where $\hbar=1$. This table also includes accurate numerical results $E_n^\#$ and the results of the zero and first order WKB approximation given in Refs.~\cite{bender,voros}. The relative difference
$|E_n-E_n^\#|/E_n^\#$ is about $0.04$ for the ground state and ranges between $6.4\times 10^{-4}$ and $8.7\times 10^{-3}$ for the energy levels $E_2,E_4,E_6,E_8,E_{10}$ and $E_{16}$.
Table~5 includes the results of the first and second order variational Sturmian approximation for $E_0, E_2,$ and $E_4$.
\subsection*{The Gaussian Potential}
Consider the Gaussian potential
\begin{equation}
V(x)=-\lambda\,e^{-\epsilon x^2/2}\;,.
\label{i1}
\end{equation} In order to apply the results of section~IV to this potential, we write $V(x)=\tilde V(x)-\lambda$ where
\begin{equation}
\tilde V(x)=\lambda(1-e^{-\epsilon x^2/2})\;.
\label{i2}
\end{equation} Then as $\epsilon$ tends to zero, $\tilde V(x)$ approaches to the harmonic oscillator potential (\ref{f1}) with $k=\lambda\epsilon$.
Clearly, the energy eigenvalues associated with $V$ and $\tilde V$ are related by
\begin{equation}
E_n=\tilde E_n-\lambda\;.
\label{i3}
\end{equation} In the following we use the zero order variational Sturmian approximation to obtain the ground state energy of the potential $\tilde V$. The excited energy levels can be obtained similarly.
We first note that for the ground state $n=0$, and Eq.~(\ref{f14}) for the potential $\tilde V$ takes the form
\begin{eqnarray}
{\cal E}&=&\sqrt{2}~_0\langle 0|\tilde V|2\rangle_0 =
\sqrt{2}~_0\langle 0|V|2\rangle_0=
-\sqrt{2}\lambda ~_0\langle 0|e^{-\epsilon x^2/2}|2\rangle_0\nonumber\\
&=&-\sqrt{2}\lambda\int_{-\infty}^\infty
~_0\br0|x\rangle e^{-\epsilon x^2/2}\langle x|2\rangle_0\,dx\;.
\label{i4}
\end{eqnarray} We can evaluate the right-hand side of (\ref{i4}) using the well-known expression for the eigenfunctions of the harmonic oscillator, namely (\ref{f1.4}) and
\begin{equation}
\langle x|2\rangle_0=\left(\frac{\alpha_0}{4\pi}\right)^{1/4}(2\alpha_0x^2-1)
e^{-\alpha_0x^2/2}\;.
\label{i5}
\end{equation} Substituting Eqs.~(\ref{f1.4}) and (\ref{i5}) in Eq.~(\ref{i4}) and performing the necessary calculations, we find
\begin{equation}
{\cal E}({\cal E}+p)^3=\lambda^2 p^2\;,
\label{i6}
\end{equation} where
\begin{equation}
p:=\frac{\hbar^2\epsilon}{4m}\;.
\label{i7}
\end{equation}
Introducing
\begin{equation}
\eta:=1+\frac{\cal E}{p}\,,
\label{i7.1}
\end{equation} we can write Eq.~(\ref{i6}) in the form
\begin{equation}
f(\eta):=\eta^4-\eta^3-r=0\;,
\label{i8}
\end{equation} where
\begin{equation}
r:=\frac{\lambda^2}{p^2}=
\frac{16\lambda^2 m^2}{\hbar^4\epsilon^2}\;.
\label{i9}
\end{equation} It is not difficult to show that for all $r>0$, $f(\eta)$ has a single minimum at $\eta=3/4$. The minimum is $f(3/4)=-(27/256+r)<0$. Furthermore, $f(0)=f(1)=-r<0$ and $\lim_{\eta\to\infty} f(\eta)=\infty$. Therefore, $f(\eta)$ has a single positive root that is greater than $1$. This root is given by
\begin{equation}
\eta_\star=\frac{1}{4} \left(1+2\xi+\sqrt{3-4\xi^2+\xi^{-1}}\right)\;,
\label{i10}
\end{equation} where
\begin{eqnarray}
\xi&:=&\frac{1}{2}\sqrt{1-a+b},~~~~
a:= \frac{3}{2}\,\zeta(1+\sqrt{1+\zeta^3})^{1/3} ,\nonumber\\
b&:=&\frac{3}{2}\,\zeta(-1+\sqrt{1+\zeta^3})^{1/3} ,~~~~
\zeta:=\frac{4}{3}\,(4r)^{1/3}\;.\nonumber
\end{eqnarray} In view of Eq.~(\ref{i7.1}) and the fact that $\eta_0>1$, Eq.~(\ref{i6}) has a single positive solution, namely
\begin{equation}
{\cal E}=p(\eta_\star-1).
\label{i11}
\end{equation}
Having obtained the parameter ${\cal E}$, we next compute
\begin{equation}
W=~_0\langle 0|\tilde V|0\rangle_0=
\lambda\left(1-\sqrt{1-\eta_\star^{-1}}\right) \;.
\label{i12}
\end{equation} Here we have made use of Eqs.~(\ref{f1.4}), (\ref{alpha=}), (\ref{i2}), (\ref{i7}), and (\ref{i11}). Substituting this equation and Eq.~(\ref{i11}) in Eq.~(\ref{f7}) and using Eqs.~(\ref{i3}) and (\ref{i9}), we find the ground state energy of the Gaussian potential (\ref{i1}) to be
\begin{equation}
E_0=\tilde E_0-\lambda=
-\lambda\left[\sqrt{1-\eta_\star^{-1}}+
\frac{1-\eta_\star}{2\sqrt{r}}\right]\;.
\label{i13}
\end{equation}
In order to reveal the asymptotic behavior of $E_0$, we investigate the power series expansion of the right-hand side of Eq.~(\ref{i13}).
\begin{itemize}
\item[~] For $r\gg 1$, i.e., $\frac{\epsilon}{\lambda}\to 0$,
\begin{equation}
E_0=-\lambda[1-r^{-1/4}+\frac{3}{8}\,r^{-1/2}-\frac{1}{32}\,r^{-3/4}
-\frac{1}{128}\,r^{-1}+{\cal O}(r^{-5/4})]\;.
\label{i14}
\end{equation}
\item[~] For $r\ll 1$, i.e., $\frac{\epsilon}{\lambda}\to \infty$,
\begin{equation}
E_0=-(\frac{\lambda\sqrt{r}}{2})
[1-r+3r^2-13r^3+68 r^4+{\cal O}(r^5)]\;.
\label{i15}
\end{equation}
\end{itemize} Therefore, for fixed $\lambda$,
\begin{eqnarray}
\lim_{\epsilon\to 0^+} E_0&=&-\lambda\;,
\label{i16.1}\\
\lim_{\epsilon\to\infty} E_0&=&
\lim_{\epsilon\to\infty}\left[-\frac{4m^2
\lambda^2}{\hbar^2\epsilon}\right]=0^-\;,
\label{i16.2}
\end{eqnarray} and for fixed $\epsilon$,
\begin{eqnarray}
\lim_{\lambda\to 0^+} E_0&=&\lim_{\lambda\to 0}\left[-\frac{4m^2
\lambda^2}{\hbar^2\epsilon}\right]=0^-\;,
\label{i17.1}\\
\lim_{\lambda\to\infty} E_0&=&-\infty\;.
\label{i17.2}
\end{eqnarray} Clearly, the asymptotic behavior of $E_0$, as given by Eqs.~(\ref{i16.1}) -- (\ref{i17.2}), agrees with the qualitative analysis of the eigenvalue problem for the Gaussian potential.
It is not difficult to see that in the limit $\epsilon\to 0$ perturbation theory provides reliable results. Writing the Gaussian potential~(\ref{i1}) in the form
\begin{equation}
V=V_0+\delta V\;,
\label{i18}
\end{equation} with $V_0$ given by Eq.~(\ref{sho}) and performing the standard calculations \cite{perturbation}, we find that the zero and first order perturbation theory yield respectively
\begin{eqnarray}
E_0^{(0)}&=&-\lambda(1-r^{-1/4})\;,
\label{i19.1}\\
E_0^{(1)}&=&-\lambda(1+r^{-1/4})^{-1/2}=
-\lambda[1-\frac{1}{2}\,r^{-1/4}+\frac{3}{8}\,
r^{-1/2}-\frac{5}{16}\,r^{-3/4}+{\cal O}(r^{-1})]\;.
\label{i19.2}
\end{eqnarray} Here $E_0^{(\ell)}$ is the ground state energy for the Gaussian potential~(\ref{i1}) obtained using the $\ell$-th order perturbation theory.
Comparing Eq.~(\ref{i14}) with Eqs.~(\ref{i19.1}) and (\ref{i19.2}), one finds that in the pertubative region where $r\gg 1$, the variational Sturmian approximation of order zero agrees with the results of the perturbation theory. In fact, since by construction $E_0$ is the expectation value of the energy of
the Sturmian $|\phi_0\rangle$, the fact that $E_0<E_0^{(0)}$ shows that even in the pertubative region the variational Sturmian approximation of order zero is a better approximation than the zero order perturbation theory. By the same reasoning, because $E_0>E_0^{(1)}$, the first order perturbation theory yields a better result. Note however that the wave function obtained in the first order perturbation theory is an infinite sum whereas the wave function in the zero order Sturmian approximation is given explicitly.
Another interesting limit is the delta function limit of the potential $V$ where $\lambda=a\sqrt{\epsilon/(2\pi)}$, $\epsilon\to\infty$, and $V(x)\to -a\delta(x)$. Here $a$ is a fixed coupling constant. In this limit $r\to 0$ and the ground state energy is given by Eq.~(\ref{i15}) according to
\begin{equation}
E_0=-\frac{m a^2}{\pi\hbar^2}\;.
\label{i20}
\end{equation} This result has the same order of magnitude as the exact result:
\begin{equation}
E_0=-\frac{m a^2}{2\hbar^2}\;.
\label{i21}
\end{equation}
\section*{VI.~Discussion and Conclusion}
We have outlined a variationally improved Sturmian approximation and applied our results to the harmonic oscillator Sturmians. For these Sturmians we could solve the associated variational problem in the zero order Sturmian approximation exactly. We have used our variational Sturmian approximation in the calculation of the energy levels of various potentials. We have shown that using a few harmonic oscillator Sturmians, one obtains quite reliable results. In general, the variational Sturmian approximation is a better approximation than the conventional Sturmian approximation.
Because the harmonic oscillator potential is a confining potential, we expect that the method is more suitable for the confining potentials such as the quartic anharmonic oscillator and the quartic potential. We can base this argument on a more quantitative reasoning by addressing the problem of classifying the potentials for which the Sturmian approximation is exact. It is not difficult to show that these potentials satisfy
\begin{equation}
V(\vec x)=E-{\cal E}+\left(
\frac{ \sum_{\nu=0}^N\sum_\alpha
C_\nu^\alpha\beta_\nu\phi_{\nu,\alpha}(\vec x) }{
\sum_{\nu=0}^N
C_\nu^\alpha\phi_{\nu,\alpha}(\vec x)}\right)V_0(\vec x)\;,
\label{j1}
\end{equation} where $E$, ${\cal E}$ and $C_\nu^\alpha$ are constants and
$\phi_{\nu,\alpha}(\vec x):=\langle \vec x|\phi_\nu,\alpha\rangle$. Eq.~(\ref{j1}) follows from Eqs.~(\ref{xe1}), (\ref{xe2}), (\ref{e3}), and (\ref{e6}).
For example, the potentials for which the first order harmonic oscillator Sturmian approximation with ${\cal S}_2=\{0,2\}$ yields an exact eigenfunction are of the form
\begin{equation}
V(x)=E-\frac{\hbar^2\alpha_0}{2m}+
\left(\frac{\hbar^2\alpha_0^2}{2m}\right)
\left[\frac{e^{-2\alpha_0 x^2/5}+
(\frac{\zeta}{5})(2\alpha_0\,x^2-5)}{
e^{-2\alpha_0 x^2/5}+5 \zeta(2\alpha_0\,x^2-5)}\right]x^2\,,
\label{j2}
\end{equation} where $\alpha_0$ is a real parameter with the dimension of (length$)^{-2}$ and
$\zeta$ is a dimensionless real parameter. As seen from Eq.~(\ref{j2}), these potentials tend to the harmonic oscillator potential for $|x|\to\infty$. In particular, as $|x|\to\infty$, $V\to\infty$. This asymptotic behaviour is also valid for higher order harmonic oscillator Sturmian approximations. This observation shows that the harmonic oscillator Sturmian approximation is more reliable for confining potentials.
We conclude this paper with a couple of remarks.
\begin{itemize}
\item[1.] The variational principle used in the variational Sturmian approximation leads to an algebraic (nondifferential) equation for the parameter ${\cal E}$. The acceptable solutions for this equation are those which are real and positive. The fact that for all the cases we consider there is a unique real positive solution corresponding to each eigenvalue $E_n$ is quite remarkable. This observation may be viewed as a consistency check for the Sturmian approximation.
\item[2.] In our selection of the Sturmians in the first and higher order Sturmian approximation, we used the information about the parity properties of the Sturmians and the energy eigenfunctions. For example we ruled out the first order variational Sturmian approximation with ${\cal S}_2= \{0,1\}$. If we perform the necessary calculations, we find that for this choice the functions $t$ and $w$ vanish identically and the matrices $T$ and $S$ are diagonal. Therefore, the secular equation (\ref{e12}) yields the same results as the zero order Sturmian approximation. This can also be seen from the results of Ref.~\cite{antonsen}.
\end{itemize}
\begin{table}
\begin{center}
\begin{tabular}{ | r | r | r | r | r | r | r | r | r | r | }
\hline
$n$ &
$E_n^\#$&
$E_n$ &
$\frac{|E_n-E_n^\#|}{E^\#_n}$ &
$E_n^{\rm CSA}$ &
$\frac{|E^{CSA}_n-E_n^\#|}{E^\#_n}$ &
$E_n^{(0)}$ &
$\frac{|E^{(0)}_n-E_n^\#|}{E^\#_n}$ &
$E_n^{(1)}$ &
$\frac{|E^{(1)}_n-E_n^\#|}{E^\#_n}$ \\
\hline
0 & 1.065286 &
1.06692 & $1.5\times 10^{-3}$ &
1.07500 & $9.1\times 10^{-3}$ &
1.000 & 0.061 &
1.075 & $9.1\times 10^{-3}$ \\
\hline
1 & 3.306872 &
3.31182 & $1.5\times 10^{-3}$ &
3.37500 & 0.021 &
3.000 & 0.032 &
3.450 & 0.043 \\
\hline
2 & 5.747959 &
5.75052 & $4.5\times 10^{-4}$ &
5.97500 & 0.040 &
5.000 & 0.13 &
5.975 & 0.039 \\
\hline
3 & 8.352678 &
8.34985 & $3.4\times 10^{-4}$ &
8.87500 & 0.063 &
7.000 & 0.16 &
8.875 & 0.063 \\
\hline
4 & 11.09860 &
11.0881 & $9.5\times 10^{-4}$ &
12.0750 & 0.088 &
9.000 & 0.19 &
12.08 & 0.088 \\
\hline
5 & 13.96993 &
13.9499 & $1.4\times 10^{-3}$ &
- & - &
11.00 & 0.21&
15.58 & 0.11 \\
\hline
6 & 16.95479 &
16.9235 & $1.8\times 10^{-3}$ &
- & - &
13.00 & 0.23 &
19.38 & 0.14 \\
\hline
7 & 20.04386 &
19.9998 & $2.2\times 10^{-3}$ &
- & - &
15.00 & 0.25 &
23.48 & 0.17 \\
\hline
8 & 23.22955 &
23.1715 & $2.5\times 10^{-3}$ &
- & - &
17.00 & 0.27 &
27.88 & 0.20 \\
\hline
9 & 26.50555 &
26.4322 &$2.8\times 10^{-3}$ &
- & - &
19.00 & 0.28 &
32.58 & 0.23 \\
\hline
\end{tabular}
\caption{First 10 energy levels of the Hamiltonian
$H=p^2+x^2+\frac{x^4}{10}$ in units where $\hbar=1$.
$E_n^{\#}$ are the highly accurate numerical values of
Ref.~\cite{bacus}. $E_n$ are the values obtained using the zero order
variational Sturmian approximation. $E_n^{\rm CSA}$ are the values
obtained by the zero order conventional Sturmian approximation in
Ref.~\cite{antonsen}. $E_n^{(0)}$ and $E_n^{(1)}$ are
the energy eigenvalues obtained using the zero and first order
perturbation theory, respectively.}
\end{center} \end{table}
{\footnotesize \begin{table} \begin{center}
\begin{tabular}{ | r | r | r | r | r | r | r | r | r | r | r | }
\hline
${\cal S}_2$ &
$E_0$ & $\delta E_0$ &
$E_1$ & $\delta E_1$ &
$E_2$ & $\delta E_2$ &
$E_3$ & $\delta E_3$ &
$E_4$ & $\delta E_4$ \\
\hline
\{0,2\} &
1.06614 & $8.0\times 10^{-4}$&
- & - &
5.76117 & $2.3\times 10^{-3}$&
- & - &
- & - \\
\hline
\{1,3\} &
- & - &
3.30922 & $7.1\times 10^{-4}$&
- & - &
8.37284 & $2.4\times 10^{-3}$&
- & - \\
\hline
\{2,4\} &
- & - &
- & - &
5.74558 & $4.1\times 10^{-4}$&
- & - &
9.66370& 0.13 \\
\hline
\{0,4\} &
1.06620 & $8.6\times 10^{-4}$&
- & - &
- & - &
- & - &
9.64502& 0.13 \\
\hline
\end{tabular}
\caption{Energy levels of the Hamiltonian
$H=p^2+x^2+x^4/10$ obtained using the first order variational
Sturmian approximation. $\delta E_n$ stands for $|E_n-E_n^\#|/
E^\#_n$.}
\end{center} \end{table} }
\begin{table} \begin{center}
\begin{tabular}{ | r | r | r | r | r |}
\hline
$n$ & $E_n^\#$& $E_n$ & $\frac{|E_n-E_n^\#|}{E_n^\#}$\\
\hline
0 & 1.065286 &
1.06613 & $7.9\times 10^{-4}$\\
\hline
2 & 5.75052 &
5.75275 & $8.3\times 10^{-4}$\\
\hline
4 & 11.09860 &
9.68483 & $0.127$\\
\hline
\end{tabular}
\caption{Energy levels of the Hamiltonian
$H=p^2+x^2+x^4/10$ obtained using the second order variational
Sturmian approximation with the choice $\{0,2,4\}$ for the indexing set
${\cal S}_3$. $E_n^\#$ are the highly accurate numerical
values of Ref.~\cite{bacus}. }
\end{center} \end{table}
\begin{table}
\begin{center}
\begin{tabular}{ | r | r | r | r | r | r | r | r | }
\hline
$n$ &
$E_n^\#$ &
$E_n$ &
$\frac{|E_n-E_n^\#|}{E^\#_n}$ &
$E_n^{(0)}$ &
$\frac{|E^{\rm WKB(0)}_n-E_n^\#|}{E^\#_n}$ &
$E_n^{(1)}$ &
$\frac{|E^{\rm WKB(1)}_n-E_n^\#|}{E^\#_n}$ \\
\hline
0 & 1.060362 &
1.10243 & 0.040&
0.87 & 0.17&
0.98 & 0.076\\
\hline
1 & - &
3.86929 & - &
- & - &
- & - \\
\hline
2 & 7.455697 &
7.46048 & $6.4\times 10^{-4}$ &
7.4140 & $5.6\times 10^{-3}$&
7.4558 & $1.4\times 10^{-5}$\\
\hline
3 & - &
11.6007 & - &
- & - &
- & - \\
\hline
4 & 16.261826 &
16.1691 & $5.7\times 10^{-3}$ &
16.233615 & $1.7\times 10^{-3}$ &
16.261937 & $6.8\times 10^{-6}$ \\
\hline
6 & 26.528471 &
26.3349 & $7.3\times 10^{-3}$ &
26.506336 & $8.3\times 10^{-4}$ &
26.528513 & $1.9\times 10^{-5}$ \\
\hline
8 & 37.923001 &
37.6218 & $7.9\times 10^{-3}$ &
37.904472 & $4.9\times 10^{-4}$ &
37.923021 & $5.3\times 10^{-7}$ \\
\hline
10 & 50.256255 &
49.8404 & $8.3\times 10^{-3}$ &
50.240152 & $3.1 \times 10^{-4}$ &
50.256266 & $2.2\times 10^{-7}$ \\
\hline
16 & 91.79806 &
91.0012 & $8.7\times 10^{-3}$ &
- & - &
- & - \\
\hline
\end{tabular}
\caption{Energy levels of the Hamiltonian
$H=p^2+x^4$ in units where $\hbar=1$.
$E_n^\#$ are the highly accurate numerical values of
Refs.~\cite{bender,voros}. $E_n$ are the values obtained using the zero
order variational Sturmian approximation. $E_n^{\rm WKB(0)}$ and
$E_n^{\rm WKB(1)}$ are the values obtained using the zero and first
order WKB approximation \cite{bender}, respectively.}
\end{center} \end{table}
\begin{table} \begin{center}
\begin{tabular}{ | r | r | r | r | r | r | r | r | }
\hline
N & ${\cal S}_N$ &
$E_0$ & $\frac{|E_0-E_0^\#|}{E^\#_0}$ &
$E_2$ & $\frac{|E_2-E_2^\#|}{E^\#_2}$ &
$E_4$ & $\frac{|E_4-E_4^\#|}{E^\#_4}$ \\
\hline
2 & \{0,2\} &
1.08110 & $0.0196$&
7.60884 & $0.0205$&
- & - \\
\hline
2 & \{2,4\} &
- & - &
7.42669 & $3.89\times 10^{-3}$&
16.4461 & $0.0113$\\
\hline
2 & \{0,4\} &
1.08166& 0.0200 &
- & - &
16.4114 & $9.12\times 10^{-3}$\\
\hline
3 & \{0,2,4\} &
1.08010 & $0.0195$&
7.56528 & $0.0147$ &
16.5670 & $0.0188$ \\
\hline
\end{tabular}
\caption{Energy levels of the Hamiltonian
$H=p^2+x^4$ obtained using the first and second order variational
Sturmian approximation. $N$ is the order of the approximation.
$E_n^\#$ are the accurate numerical results reported in
Ref.~\cite{bender}.}
\end{center} \end{table}
\end{document} | arXiv | {
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\begin{document}
\title[A new approach to Catalan numbers using differential equations]{A new approach to Catalan numbers using differential equations}
\author{Taekyun Kim} \address{Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic
of Korea} \email{tkkim@kw.ac.kr}
\author{Dae San Kim} \address{Department of Mathematics, Sogang University, Seoul 121-742, Republic
of Korea} \email{dskim@sogang.ac.kr} \begin{abstract} In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit identities for Catalan and higher-order Catalan numbers. In addition, by other means than differential equations we also derive some interesting identities involving Catalan numbers which are of arithmetic and combinatorial nature. \end{abstract}
\thanks{\noindent {\footnotesize{ \it 2010 Mathematics Subject Classification } : 05A19, 11B37, 11B83, 34A34}}
\thanks{\footnotesize{ \bf Key words and phrases}: Catalan numbers, differential equations}
\maketitle \section{Introduction} The Catalan numbers $C_n$ were first introduced by the Mongolian mathematician Ming Antu in around 1730, even though they were named after the Belgian mathematician Eug{\`e}ne Charles Catalan (1814-1894). Indeed, Ming Antu obtained a number of trigonometric expressions involving Catalan numbers such as \begin{equation*}\begin{split} \sin 2\theta = 2 \sin \theta - \sum_{n=1}^\infty \frac{C_{n-1}}{4^{n-1}}\sin^{2n+1}\theta =2 \sin \theta - \sin^3 \theta - \tfrac{1}{4} \sin^5 \theta - \tfrac{1}{8} \sin^7 \theta-\cdots, \,\,\, \end{split}\end{equation*} (\textnormal{see} \cite{key-2,key-3,key-4,key-6,key-10,key-11,key-12,key-13,key-14}). The Catalan numbers can be given explicitly in terms of binomial coefficients. Namely, for $n \geq 0$, \begin{equation}\begin{split}\label{01} C_n = \frac{1}{n+1} {2n \choose n} = \prod_{k=2}^n \frac{n+k}{k}. \end{split}\end{equation} They satisfy the recurrence relations \begin{equation}\begin{split}\label{02} C_0=1, C_n = \sum_{m=0}^{n-1}C_m C_{n-1-m}, \quad (n \geq 1). \end{split}\end{equation} The Catalan numbers are also given by the generating function \begin{equation}\begin{split}\label{03} \frac{2}{1+\sqrt{1-4t}} = \sum_{n=0}^\infty C_n t^n = \sum_{n=0}^\infty \frac{1}{n+1} {2n \choose n} t^n. \end{split}\end{equation} The Catalan numbers form the sequence of positive integers \begin{equation*}\begin{split} 1,\,\, 1, \,\,2,\,\, 5,\,\, 14,\,\, 42,\,\, 132,\,\, 429,\,\, 1430,\,\, 4862,\,\, 16796,\,\, 58786, \,\,208012, \,\,\cdots \end{split}\end{equation*} which is asymptotic to $\dfrac{4^n}{n^{\tfrac{3}{2}} \sqrt{\pi}}$, as $n$ tends to $\infty$, and appears in various counting problems. For example, $C_n$ is the number of Dyck words of length $2n$, the number of balanced $n$ pairs of parentheses, the number of mountain ranges you can form with $n$ upstrokes and downstrokes that all stay above the original line, the number of diagonal-avoiding paths of length $2n$ from the upper left corner to the lower right corner in a grid of $n \times n$ squares, and the number of ways $n+1$ factors can be completely parenthesized (see \cite{key-1,key-2,key-3,key-4,key-10,key-11}).
It is also the number of ways an $(n+2)-$gon can be cut into $n$ triangles, the number of permutations of $\{ 1,2,\cdots,n \}$ that avoid the pattern 123, the number of ways to tile a stairstep shape of height $n$ with $n$ rectangles, etc (see \cite{key-10,key-11,key-12,key-13}).
In \cite{key-8}, T. Kim initiated a fascinating idea of using ordinary differential equations as a method of obtaining new identities for special polynomials and numbers. Namely, a family of nonlinear differential equations were derived, which are indexed by positive integers and satisfied by the generating function of the Frobenius-Euler numbers. Then, they were used in order to obtain an interesting identity, expressing higher-order Frobenius-Euler numbers in terms of (ordinary) Frobenius-Euler numbers (see \cite{key-5,key-7,key-9,key-15}).
This method turned out to be very fruitful and can be applied to many interesting special polynomials and numbers (see \cite{key-7,key-8,key-9}). For example, linear differential equations are derived for Bessel polynomials, Changhee polynomials, actuarial polynomials, Meixner polynomials of the first kind, Poisson-Charlier polynomials, Laguerre polynomials, Hermite polynomials, and Stirling polynomials, while nonlinear ones are obtained for Bernoulli numbers of the second, Boole numbers, Chebyshev polynomials of the first, second, third, and fourth kind, degenerate Euler numbers, degenerate Eulerian polynomials, Korobov numbers, and Legendre polynomials (see \cite{key-1,key-5,key-7,key-9,key-15}).
In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit identities for Catalan and higher-order Catalan numbers. In addition, by other means than differential equations we also derive some interesting identities involving Catalan numbers which are of arithmetic and combinatorial nature.
\section{Differential equations associated with Catalan numbers} Let \begin{equation}\begin{split}\label{04} C=C(t) = \frac{2}{1+\sqrt{1-4t}} \end{split}\end{equation} Then, by \eqref{04}, we have \begin{equation}\begin{split}\label{05} C^{(1)} =& \dfrac{d}{dt} C(t) = (1-4t)^{-\tfrac{1}{2}} 4(1+ \sqrt{1-4t})^{-2}\\ =&(1-4t)^{-\tfrac{1}{2}} C^2, \end{split}\end{equation} and \begin{equation}\begin{split}\label{06} C^{(2)} =& \dfrac{d}{dt}C^{(1)} = 2 (1-4t)^{-\tfrac{3}{2}} C^2 + 2(1-4t)^{-\tfrac{1}{2}} C C^{(1)} \\ =& 2 (1-4t)^{-\tfrac{3}{2}} C^2 + 2 (1-4t)^{-\tfrac{1}{2}} C \{(1-4t)^{-\tfrac{1}{2}} C^2 \}\\ =& 2(1-4t)^{-\tfrac{3}{2}} C^2 + 2(1-4t)^{-\tfrac{2}{2}} C^3 \end{split}\end{equation} So, we are led to put \begin{equation}\begin{split}\label{07} C^{(N)} = \sum_{i=1}^N a_i(N) (1-4t)^{-\tfrac{2N-i}{2}} C^{i+1}, \end{split}\end{equation} where $N=1,2,3,\cdots$. From \eqref{07}, we obtain \begin{equation}\begin{split}\label{08} C^{(N+1)} =& \dfrac{d}{dt}C^{(N)}\\ =& \sum_{i=1}^N 2(2N-i)a_i(N)(1-4t)^{-\tfrac{2N+2-i}{2}} C^{i+1}\\ &+\sum_{i=1}^N (i+1) a_i(N) (1-4t)^{-\tfrac{2N-i}{2}}C^i C^{(1)}\\ =& \sum_{i=1}^N 2(2N-i) a_i(N) (1-4t)^{-\tfrac{2N+2-i}{2}} C^{i+1}\\ &+ \sum_{i=1}^N (i+1)a_i(N) (1-4t)^{-\tfrac{2N-i}{2}}C^i \{ (1-4t)^{-\tfrac{1}{2}} C^2 \}\\ =& \sum_{i=1}^N 2(2N-i) a_i(N) (1-4t)^{-\tfrac{2N+2-i}{2}} C^{i+1}\\ &+ \sum_{i=1}^N (i+1)a_i(N) (1-4t)^{-\tfrac{2N+1-i}{2}} C^{i+2}\\ =& \sum_{i=1}^N 2(2N-i) a_i(N) (1-4t)^{-\tfrac{2N+2-i}{2}} C^{i+1}\\ &+ \sum_{i=2}^{N+1} i a_{i-1}(N) (1-4t)^{-\tfrac{2N+2-i}{2}}C^{i+1}. \end{split}\end{equation} On the other hand, replacing $N$ by $N+1$ in \eqref{07}, we get \begin{equation}\begin{split}\label{09} C^{(N+1)} = \sum_{i=1}^{N+1} a_i(N+1)(1-4t)^{-\tfrac{2N+2-i}{2}}C^{i+1}. \end{split}\end{equation} From \eqref{08} and \eqref{09}, we can derive the following recurrence relations: \begin{align} \label{10}&a_1(N+1)=2(2N-1)a_1(N),\\ \label{11}&a_{N+1}(N+1)=(N+1)a_N(N), \end{align} and \begin{equation}\begin{split}\label{12} a_i(N+1)=ia_{i-1}(N) + 2(2N-i) a_i(N), \,\, (2 \leq i \leq N). \end{split}\end{equation} In addition, from \eqref{05} and \eqref{07}, we observe that \begin{equation}\begin{split}\label{13} a_1(1)(1-4t)^{-\tfrac{1}{2}} C^2 = C^{(1)} = (1-4t)^{-\tfrac{1}{2}} C^2. \end{split}\end{equation} Thus, by \eqref{13}, we get \begin{equation}\begin{split}\label{14} a_1(1) = 1. \end{split}\end{equation} In Below, for any positive integer $N$, $(2N-1)!!$ will denote \begin{equation}\begin{split}\label{15} (2N-1)!! = (2N-1)(2N-3) \cdots 1. \end{split}\end{equation} From \eqref{10} and \eqref{14}, we note that \begin{equation}\begin{split}\label{16} a_1(N+1)=& 2(2N-1)a_1(N) = 2^2 (2N-1)(2N-3)a_1(N-1)\\ =& \cdots \\ =& 2^N(2N-1)(2N-3)\cdots 1a_1(1)\\ =& 2^N(2N-1)!! \end{split}\end{equation} and \begin{equation}\begin{split}\label{17} a_{N+1}(N+1)=& (N+1)a_N(N) = (N+1)Na_{N-1}(N-1) \\ =& \cdots \\ =& (N+1)N \cdots 2 a_1(1) = (N+1)! \end{split}\end{equation} In the following, we will use the notations: \begin{equation}\begin{split}\label{18} (x;\alpha)_n = x(x-\alpha ) \cdots ( x- (n-1)\alpha ),\quad \textnormal{for}\,\, n \geq 1, \end{split}\end{equation} and \begin{equation*}\begin{split} (x; \alpha)_0 = 1. \end{split}\end{equation*} For $i=2$ in \eqref{12}, we have \begin{equation}\begin{split}\label{19} a_2(N+1) =& 2a_1(N)+2(2N-2)a_2(N) \\ =& 2a_1(N)+2(2N-2)\big(2a_1(N-1)+2(2N-4)a_2(N-1)\big)\\ =& 2\big( a_1(N) + 2(2N-2) a_1(N-1) \big) + 2^2 (2N-2) (2N-4) a_2(N-1)\\ =& 2\big( a_1(N) + 2(2N-2) a_1(N-1) \big) \\ &+ 2^2 (2N-2) (2N-4)\big( 2a_1(N-2)+2(2N-6)a_2(N-2)\big)\\ =& 2\big( a_1(N) + 2(2N-2) a_1(N-1) \big) + 2^2 (2N-2) (2N-4)a_1(N-2)\big)\\ &+2^3 (2N-2)(2N-4)(2N-6)a_2(N-2)\\ =&\,\,\cdots\\ =&2\sum_{k=0}^{N-2} 2^k (2N-2;2)_k a_1(N-k) + 2^{N-1}(2N-2;2)_{N-1}a_2(2)\\ =&2\sum_{k=0}^{N-1}2^k (2N-2;2)_k a_1(N-k). \end{split}\end{equation} Proceeding analogously to the case of $i=2$, for $i=3$ and 4, we obtain \begin{align} \label{20}&a_3(N+1)=3\sum_{k=0}^{N-2}2^k(2N-3;2)_ka_2(N-k),\\ \label{21}&a_4(N+1)=4\sum_{k=0}^{N-3}2^k(2N-4;2)_ka_3(N-k). \end{align} Continuing this process, we can deduce that \begin{equation}\begin{split}\label{22} a_i(N+1)=i \sum_{k=0}^{N-i+1}2^k (2N-i;2)_k a_{i-1}(N-k), \quad \textnormal{for} \,\, 2 \leq i \leq N. \end{split}\end{equation} Now, we give explicit expressions for $a_i(N+1)\,\,(2 \leq i \leq N).$ From \eqref{16} and \eqref{19}, we have \begin{equation}\begin{split}\label{23} a_2(N+1) =& 2 \sum_{k_1=0}^{N-1} 2^{k_1} (2N-2;2)_{k_1} a_1(N-k_1) \\ =& 2 \sum_{k_1=0}^{N-1} 2^{k_1}(2N-2;2)_{k_1} 2^{N-k_1-1}(2N-2k_1-3)!!\\ =& 2! 2^{N-1} \sum_{k_1=0}^{N-1} (2N-2;2)_{k_1} (2N-2k_1 -3)!!. \end{split}\end{equation} Also, from \eqref{20} and \eqref{23}, we get \begin{equation}\begin{split}\label{24} a_3(N+1)=&3\sum_{k_2=0}^{N-2}2^{k_2}(2N-3;2)_{k_2} a_2(N-k_2) \\ =& 3 \sum_{k_2=0}^{N-2} 2^{k_2} (2N-3;2)_{k_2} 2^{N-k_2-1} \\&\times \sum_{k_1=0}^{N-2-k_2} (2N-2k_2-4;2)_{k_1} (2N-2k_1-2k_1-5)!! \\ =& 3! 2^{N-2} \sum_{k_2=0}^{N-2}\sum_{k_1=0}^{N-2-k_2}(2N-3;2)_{k_2} (2N-2k_2-4;2)_{k_1}\\ &\times (2N-2k_1-2k_1-5)!!. \end{split}\end{equation} Continuing this process, we can deduce that \begin{equation}\begin{split}\label{25} &a_i(N+1)= 2^{N-i+1} i! \sum_{k_{i-1}=0}^{N-i+1} \sum_{k_{i-2}=0}^{N-i+1-k_{i-1}} \cdots \sum_{k_1=0}^{N-i+1-k_{i-1}-\cdots-k_2} (2N-i;2)_{k_{i-1}}\\ &\,\,\,\,(2N-2k_{i-1}-i-1;2)_{k_{i-2}} \times \cdots \times (2N-2k_{i-1}-\cdots-2k_2-2i+2;2)_{k_1}\\ &\,\,\,\,\times (2N-2k_{i-1}-\cdots-2k_1 -2i +1)!!\\ &= 2^{N-i+1}i! \sum_{k_{i-1}=0}^{N-i+1} \sum_{k_{i-2}=0}^{N-i+1-k_{i-1}} \cdots \sum_{k_1=0}^{N-i+1-k_{i-1}-\cdots-k_2} \\ &\quad \prod_{l=1}^{i-1} (2N-2 \sum_{j=l+1}^{i-1} k_j -2i+1+l;2)_{k_l} \times (2N-2 \sum_{j=1}^{i-1} k_j -2i+1)!!, \end{split}\end{equation} for $2 \leq i \leq N$.
\begin{rmk} We note here that \eqref{25} is also valid for $i=N+1$. \end{rmk}
Therefore, from \eqref{16} and \eqref{25}, we obtain the following theorem. \begin{thm}\label{thm:1} The family of differential equations \begin{equation}\begin{split}\label{26} C^{(N)} = \sum_{i=1}^N a_i(N) (1-4t)^{-\tfrac{2N-i}{2}} C^{i+1} \quad (N=1,2,3,\cdots) \end{split} \end{equation} have a solution \begin{equation} C=C(t)= \tfrac{2}{1+\sqrt{1-4t}}\nonumber, \end{equation} where \begin{align*} a_1(N)& =2^{N-1}(2N-3)!!, \\ a_i(N)& =2^{N-i}i! \sum_{k_{i-1}=0}^{N-i} \sum_{k_{i-2}=0}^{N-i-k_{i-1}} \cdots \sum_{k_1=0}^{N-i-k_{i-1}-\cdots-k_2} \qquad\qquad\qquad \\ &\times \prod_{l=1}^{i-1} (2N-2 \sum_{j=l+1}^{i-1} k_j -2i-1+l;2)_{k_l} \\ &\times (2N-2 \sum_{j=1}^{i-1} k_j -2i-1)!!. \end{align*} \end{thm}
We recall that the Catalan numbers $C_n$ are defined by the generating funcion \begin{equation}\begin{split}\label{27} C=C(t) = \frac{2}{1+\sqrt{1-4t}} = \sum_{n=0}^\infty C_n t^n. \end{split}\end{equation} More generally, the higher-order Catalan numbers $C_n^{(r)}$ of order $r$ are given by \begin{equation}\begin{split}\label{28} \left( \frac{2}{1+\sqrt{1-4t}} \right)^r = \sum_{n=0}^\infty C_n^{(r)} t^n. \end{split}\end{equation} On the one hand, from \eqref{27}, we have \begin{equation}\begin{split}\label{29} C^{(N)} =& \sum_{n=N}^\infty C_n(n)_N t^{n-N}\\ =& \sum_{n=0}^\infty C_{n+N} (n+N)_N t^n, \end{split}\end{equation} where \begin{equation}\begin{split}\label{30} (x)_n = x(x-1) \cdots (x-n+1), \quad \textnormal{for}\,\, n \geq 1, \,\, (x)_0=1. \end{split}\end{equation} On the other hand, by Theorem \ref{thm:1}, we have \begin{equation}\begin{split}\label{31} C^{(N)} =& \sum_{i=1}^N a_i(N) (1-4t)^{-\tfrac{2N-i}{2}} C^{i+1}\\ =& \sum_{i=1}^N a_i(N) \sum_{m=0}^\infty { \tfrac{2N-i}{2}+m-1 \choose m } 4^m t^m \sum_{l=0}^\infty C_l^{(i+1)} t^l \\ =& \sum_{i=1}^N a_i(N) \sum_{n=0}^\infty \sum_{m=0}^n 4^m { \tfrac{2N-i}{2}+m-1 \choose m } C_{n-m}^{(i+1)} t^n\\ =& \sum_{n=0}^\infty \left( \sum_{i=1}^N \sum_{m=0}^n 4^m { \tfrac{2N-i}{2}+m-1 \choose m } a_i(N) C_{n-m}^{(i+1)} \right) t^n. \end{split}\end{equation} Comparing \eqref{29} with \eqref{31}, we get the following Theorem.
\begin{thm}\label{thm:2} For $n=0,1,2,\cdots,$ and $N=1,2,3,\cdots,$ \begin{equation*}\begin{split} C_{n+N}=& \frac{1}{(n+N)_N} \sum_{i=1}^N \sum_{m=0}^n 4^m { \tfrac{2N-i}{2}+m-1 \choose m } a_i(N) C_{n-m}^{(i+1)} \end{split}\end{equation*} where $a_i(N)$'s are as in Theorem \ref{thm:1}. \end{thm}
\section{Inverse differential equations associated with Catalan numbers} Here we will derive ``inverse'' differential equations to the ones obtained in Section 2. With $C=C(t)$ as in \eqref{04}, we have \begin{equation}\begin{split}\label{32} C^{(1)}=(1-4t)^{-\tfrac{1}{2}} C^2, \end{split}\end{equation} and \begin{equation}\begin{split}\label{33} C^2 = (1-4t)^{\tfrac{1}{2}}C^{(1)}. \end{split}\end{equation} Differentiating both sides of \eqref{33}, we get \begin{equation}\begin{split}\label{34} 2CC^{(1)} = -2(1-4t)^{-\tfrac{1}{2}}C^{(1)} + (1-4t)^{\tfrac{1}{2}}C^{(2)}. \end{split}\end{equation} Substituting \eqref{32}, into \eqref{34}, we obtain \begin{equation}\begin{split}\label{35} 2C^3 = -2C^{(1)} + (1-4t)C^{(2)}. \end{split}\end{equation} Differentiating both sides of \eqref{35}, we have \begin{equation}\begin{split}\label{36} 3! C^2 C^{(1)} = -6C^{(2)} + (1-4t)C^{(3)} \end{split}\end{equation} Substituting \eqref{32} into \eqref{36}, we get \begin{equation}\begin{split}\label{37} 3! C^4 = -6(1-4t)^{\tfrac{1}{2}}C^{(2)} +(1-4t)^{\tfrac{3}{2}} C^{(3)}. \end{split}\end{equation} So we are led to put \begin{equation}\begin{split}\label{38} N! C^{N+1} = \sum_{i=0}^{\left[ \frac{N}{2}\right]}b_i(N) (1-4t)^{\tfrac{N}{2}-i} C^{(N-i)} \quad (N=1,2,3,\cdots). \end{split}\end{equation} Here $[x]$ denote the greatest integer not exceeding $x$. Differentiation of both sides of \eqref{38}, gives \begin{equation}\begin{split}\label{39} (N+1)! C^N C^{(1)} =& \sum_{i=0}^{\left[ \frac{N}{2}\right]} -4(\tfrac{N}{2}-i) b_i(N) (1-4t)^{\tfrac{N}{2}-i-1} C^{(N-i)}\\ &+ \sum_{i=0}^{\left[ \frac{N}{2}\right]}b_i(N) (1-4t)^{\tfrac{N}{2}-i} C^{(N+1-i)}\\ =& \sum_{i=1}^{\left[ \frac{N}{2}\right]+1} -4(\tfrac{N}{2}+1-i) b_{i-1}(N) (1-4t)^{\tfrac{N}{2}-i} C^{(N+1-i)}\\ &+ \sum_{i=0}^{\left[ \frac{N}{2}\right]}b_i(N) (1-4t)^{\tfrac{N}{2}-i} C^{(N+1-i)}. \end{split}\end{equation} Substituting \eqref{32} into \eqref{39}, we obtain \begin{equation}\begin{split}\label{40} (N+1)! C^{N+1} =& \sum_{i=1}^{\left[ \frac{N}{2}\right]+1} -4( \tfrac{N}{2} +1 -i) b_{i-1}(N)(1-4t)^{\frac{N+1}{2}-i} C^{(N+1-i)}\\ &+ \sum_{i=0}^{ \left[ \frac{N}{2} \right]} b_i(N) (1-4t)^{\tfrac{N+1}{2}-i} C^{(n+1-i)}. \end{split}\end{equation} Also, by replacing $N$ by $N+1$ in \eqref{38}, we get \begin{equation}\begin{split}\label{41} (N+1)! C^{N+2} = \sum_{i=0}^{\left[ \frac{N+1}{2} \right]} b_i(N+1) (1-4t)^{\tfrac{N+1}{2}-i} C^{(N+1-i)}. \end{split}\end{equation} Comparing \eqref{40} with \eqref{41}, we have the following recurrence relations. Here we need to consider the even and odd cases of $N$ separately. The details are left to the reader. \begin{align} \label{42} &b_0(N+1)=b_0(N),\\ \label{43} &b_i(N+1)=-4( \tfrac{N}{2}+1-i) b_{i-1}(N) + b_i(N),\quad \textnormal{for} \,\,\, 1 \leq i \leq \left[ \tfrac{N+1}{2} \right]. \end{align} From \eqref{33} and \eqref{38}, we have \begin{equation}\begin{split}\label{44} C^2 = b_0(1) (1-4t)^{\tfrac{1}{2}}C^{(1)} = (1-4t)^{\tfrac{1}{2}}C^{(1)}. \end{split}\end{equation} Thus, from \eqref{44}, we get \begin{equation}\begin{split}\label{45} b_0(1) = 1. \end{split}\end{equation} From \eqref{42}, we easily obtain \begin{equation}\begin{split}\label{46} b_0(N+1)=b_0(N)=\cdots=b_0(1) =1. \end{split}\end{equation} The equation in \eqref{43} can be rewritten as \begin{equation}\begin{split}\label{47} b_i(N+1) = -2(N+2-2i) b_{i-1}(N) + b_i(N) \end{split}\end{equation} To proceed further, we define \begin{align} \label{48} &S_{N,1} = N+(N-1)+\cdots+1,\\ \label{49} &S_{N,j} = NS_{N+1,j-1} + (N-1)S_{N,j-1} + \cdots + 1 S_{2,j-1} \quad (j \geq 2). \end{align} Now, \begin{equation}\begin{split}\label{50} b_1(N+1) =& -2Nb_0(N)+b_1(N) \\=& -2N+b_1(N) \\=& -2N-2(N-1)+b_1(N-1) \\=& \cdots \\=& -2(N+(N-1)\cdots+1) + b_1(1) \\=& -2 S_{N,1}, \end{split}\end{equation} and \begin{equation}\begin{split}\label{51} b_2(N+1)=&-2(N-2)b_1(N) +b_2(N) \\=&(-2)^2 (N-2) S_{N-1,1} + b_2(N) \\=&(-2)^2 \big( (N-2)S_{N-1,1} + (N-3)S_{N-2,1} \big) + b_2(N-1) \\=&\cdots \\=&(-2)^2 \big( (N-2)S_{N-1,1} + (N-3)S_{N-2,1} + \cdots + 1S_{2,1} \big) + b_2(3) \\=&(-2)^2 S_{N-2,2}. \end{split}\end{equation} Similarly to the cases of $i=1$ and 2, for $i=3$, we get \begin{equation}\begin{split}\label{52} b_3(N+1) = (-2)^3 S_{N-4,3}. \end{split}\end{equation} Thus we can deduce that, for $1 \leq i \leq \left[ \tfrac{N+1}{2} \right],$ \begin{equation}\begin{split}\label{53} b_i(N+1) = (-2)^i S_{N+2-2i, i}. \end{split}\end{equation} Here, from \eqref{46} and \eqref{53}, we obtain the following Theorem.
\begin{thm}\label{thm:3} The following family of differential equations \begin{equation}\begin{split}\label{54} N! C^{N+1} = \sum_{i=0}^{ \left[ \tfrac{N}{2} \right]} b_i(N)(1-4t)^{\tfrac{N}{2}-i} C^{(N-i)} \quad (N=1,2,3\cdots) \end{split}\end{equation} have a solution \begin{equation} C=C(t)=\frac{2}{1+\sqrt{1-4t}},\nonumber \end{equation} where \begin{equation} b_0(N)=1, b_i(N) = (-2)^i S_{N+1-2i,i}, \quad ( 1 \leq i \leq \left[ \tfrac{N}{2} \right])\nonumber. \end{equation} \end{thm}
Now, we would like to give an application of the result in Theorem \ref{thm:3}. From \eqref{54}, we have the following \begin{equation}\begin{split}\label{55} N! \sum_{k=0}^\infty C_k^{(N+1)} t^k=& \sum_{i=0}^{ \left[ \tfrac{N}{2} \right]} b_i(N) \sum_{l=0}^\infty { \tfrac{N}{2}-i \choose l} (-4t)^l\\ &\times \sum_{m=0}^\infty C_{m+N-i}(m+N-i)_{N-i} t^m \\ =& \sum_{i=0}^{ \left[ \tfrac{N}{2} \right]} b_i(N) \sum_{k=0}^\infty \sum_{m=0}^k {\tfrac{N}{2}-i \choose k-m} (-4)^{k-m} \\ &\times C_{m+N-i} (m+N-i)_{N-i} t^k\\ =& \sum_{k=0}^\infty \Big( \sum_{i=0}^{ \left[ \tfrac{N}{2} \right]} \sum_{m=0}^k {\tfrac{N}{2}-i \choose k-m} (m+N-i)_{N-i}\\ &\times (-4)^{k-m} b_i(N) C_{m+N-i} \Big) t^k. \end{split}\end{equation} Thus, from \eqref{55}, we get the following Theorem.
\begin{thm}\label{thm:4} For $k=0,1,2\cdots,$ and $N=1,2,3\cdots,$we have \begin{equation*}\begin{split} C_k^{(N+1)} =& \frac{1}{N!} \sum_{i=0}^{ \left[ \tfrac{N}{2} \right]} \sum_{m=0}^k {\tfrac{N}{2}-i \choose k-m} (m+N-i)_{N-i}\\ &\times (-4)^{k-m} b_i(N) C_{m+N-i}, \end{split}\end{equation*} where $b_i(N)'s$ are as in Theorem \ref{thm:3}. \end{thm}
\begin{rmk} Combining \eqref{26} with \eqref{54}, we can show that \begin{equation}\begin{split}\label{56}
(1-4t)^{\tfrac{N}{2}} C^{N+1} =& \sum_{i=0}^{ \left[ \tfrac{N}{2} \right]} \sum_{j=1}^{N-i} \frac{a_j(N-i)}{N!}b_i(N) (1-4t)^{\tfrac{j}{2}} C^{j+1}\\
=&\sum_{j=1}^N \sum_{i=0}^{\min (N-j,\left[ \tfrac{N}{2} \right]) } \frac{a_j(N-i)}{N!}b_i(N) (1-4t)^{\tfrac{j}{2}} C^{j+1}. \end{split}\end{equation} \end{rmk} Equivalently, \eqref{56} can be expressed as \begin{equation}\begin{split}\label{57} \sum_{i=0}^{\min (N-j,\left[ \tfrac{N}{2} \right]) } \frac{a_j(N-i)}{N!}b_i(N) = \delta_{j,N}, \quad (1 \leq j \leq N), \end{split}\end{equation} when $\delta_{j,N}$ is the Kronecker delta. \section{Further remarks} We start our discussion here with the following expansion of $\sqrt{1+y}$: \begin{equation}\begin{split}\label{58} \sqrt{1+y} = \sum_{n=0}^\infty {2n \choose n} \frac{(-1)^{n-1}}{4^n (2n-1)} y^n. \end{split}\end{equation} Integrating both sides of \eqref{58} from 0 to 1, we immediately obtain \begin{equation}\begin{split}\label{59} \sum_{n=0}^\infty C_n \frac{(-1)^{n-1}}{4^n(2n-1)} = \frac{1}{3}\big(4 \sqrt{2} -2\big). \end{split}\end{equation} Next, we integrate the generating function of the Catalan numbers from 0 to $\tfrac{1}{4}$. \begin{equation}\begin{split}\label{60} \int_0^\frac{1}{4} \frac{2}{1+\sqrt{1-4t}} dt = \int_0^\frac{1}{4} \sum_{n=0}^\infty \frac{1}{n+1} {2n \choose n} t^n dt \end{split}\end{equation} The left hand side of \eqref{60} is, after making the change of variable $t= \tfrac{1}{4} (1-y^2)$, equal to \begin{equation}\begin{split}\label{61} \int_0^1 \frac{y}{1+y} dy = \left[ y-\log(1+y) \right]_0^1 = 1-\log 2. \end{split}\end{equation} Thus, from \eqref{60} and \eqref{61}, we get the following identity. \begin{equation}\begin{split}\label{62} \sum_{n=0}^\infty \frac{1}{(n+1)^2} {2n \choose n} \left( \frac{1}{4} \right)^{n+1} = 1- \log 2. \end{split}\end{equation} Finally, again from the generating function of Catan numbers and \eqref{58}, we have \begin{equation}\begin{split}\label{63} 2 =&\Big( \sum_{l=0}^\infty C_l t^l \Big) \Big( 1+ \sqrt{1-4t} \Big) \\=&\Big( \sum_{l=0}^\infty C_l t^l \Big) \Big( 1- \sum_{m=0}^\infty {2m \choose m} \frac{1}{2m-1} t^m \Big) \\=&\sum_{l=0}^\infty C_l t^l - \Big( \sum_{l=0}^\infty C_l t^l \Big) \Big(\sum_{m=0}^\infty {2m \choose m} \frac{1}{2m-1} t^m \Big) \\=&\sum_{n=0}^\infty C_n t^n - \sum_{n=0}^\infty \left( \sum_{m=0}^n {2m \choose m} \frac{1}{2m-1}C_{n-m} \right) t^n \\=& \sum_{n=0}^\infty \left( C_n - \sum_{m=0}^n C_m C_{n-m} \frac{m+1}{2m-1} \right) t^n. \end{split}\end{equation} Therefore, from \eqref{63} we obtain the recurrence relation: \begin{equation}\begin{split}\label{64} C_n - \sum_{m=0}^n C_m C_{n-m} \frac{m+1}{2m-1} = \left\{ \begin{array}{ll} 2 & \textnormal{if}\,\,\, n=0, \\ 0 & \textnormal{if}\,\,\, n>0. \end{array} \right. \end{split}\end{equation} Noting that $C_n = \tfrac{1}{n}{2n \choose n+1}$, we see that \eqref{64} for $n>0$ is equivalent to the following identity. \begin{equation}\begin{split}\label{65} {2n \choose n+1} = n \sum_{m=0}^n \frac{m+1}{m(n-m)(2m-1)} {2m \choose m+1} {2n-2m \choose n-m+1}, \quad (n>0). \end{split}\end{equation} Further, separating terms corresponding to $m=0$ and $m=n$ from \eqref{64} for $n>0$ and after rearranging the terms, we get the following recurrence relations for the Catalan numbers. \begin{equation}\begin{split}\label{66} C_0 =C_1 =1, \quad C_n = \frac{2n-1}{3(n-1)} \sum_{m=1}^{n-1} C_m C_{n-m} \frac{m+1}{2m-1}, \quad (n \geq 2). \end{split}\end{equation} Compare \eqref{66} with the recurrence relation in \eqref{02}.
\end{document} | arXiv | {
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\begin{document}
\title{Beyond Degree Choosability}
\author{Daniel W. Cranston\thanks{Department of Mathematics and Applied
Mathematics, Viriginia Commonwealth University, Richmond, VA;
\texttt{dcranston@vcu.edu};
Research of the first author is partially supported by NSA Grant
98230-15-1-0013.}
\and
Landon Rabern\thanks{LBD Data Solutions, Lancaster, PA;
\texttt{landon.rabern@gmail.com}}
}
\maketitle
\begin{abstract}
Let $G$ be a connected graph with maximum degree $\Delta$. Brooks' theorem
states that $G$ has a $\Delta$-coloring unless $G$ is a complete graph or an
odd cycle. A graph $G$ is \emph{degree-choosable} if $G$ can be properly
colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors.
In the context of list coloring, Brooks' theorem can be strengthened to the
following. Every connected graph $G$ is degree-choosable unless each block of
$G$ is a complete graph or an odd cycle; such a graph $G$ is a \emph{Gallai
tree}.
This degree-choosability result was further strengthened to Alon--Tarsi orientations;
these are orientations of $G$ in which the number of spanning Eulerian
subgraphs with an even number of edges differs from the number with an odd
number of edges. A graph $G$ is \emph{degree-AT} if $G$ has an Alon--Tarsi
orientation in which each vertex has indegree at least 1.
Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also
degree-choosable.
Hladk{\'y}, Kr{\'a}{\soft{l}}, and
Schauz showed that a connected graph is degree-AT if and only if it is not a
Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected
graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such
that $G$ has no Alon--Tarsi orientation in which each vertex has indegree at
least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the
characterization is simple to state.
\end{abstract}
\section{Introduction}
Brooks' theorem is one of the fundamental results in graph coloring.
For every connected graph $G$, it says that $G$ has a $\Delta$-coloring
unless $G$ is a complete graph $K_{\Delta+1}$ or an odd cycle. When we seek to
prove coloring results by induction, we often want to color a subgraph $H$
where different vertices have different lists of allowable colors (those not
already used on their neighbors in the coloring of $G-H$). This gives rise to
list coloring. Vizing\cite{vizing1976} and, independently, Erd\H{o}s,
Rubin, and Taylor\cite{ERT} extended Brooks' theorem to list coloring. They
proved an analogue of Brooks' theorem when each vertex $v$ has $\Delta$
allowable colors (possibly different colors for different vertices).
Erd\H{o}s, Rubin, and Taylor\cite{ERT} and Borodin\cite{borodin1977criterion}
strengthened this Brooks' analogue to the following result, where a
\emph{Gallai tree}\aside{Gallai tree} is a connected graph in which each
block is a complete graph
or an odd cycle.
\begin{thmA}
If $G$ is connected and not a Gallai tree, then for any list assignment $L$
with $|L(v)|=d(v)$ for all $v\in V(G)$, graph $G$ has a proper coloring
$\varphi$ with $\varphi(v)\in L(v)$ for all $v$.
\end{thmA}
The graphs in TheoremA are
\emph{degree-choosable}\aaside{degree-choosable}{-.3cm}.
It is easy to check that every Gallai tree is not degree-choosable. So the set
of all connected graphs that are not degree-choosable are precisely the Gallai trees.
Hladk{\'y}, Kr{\'a}{\soft{l}}, and Schauz\cite{HKS} extended this characterization to the setting of
Alon--Tarsi orientations.
For any digraph $D$, a \emph{spanning Eulerian subgraph} is one in which each vertex
has indegree equal to outdegree. The \emph{parity} of a spanning Eulerian
subgraph is the parity of its number of edges. For an orientation of a graph
$G$, let EE (resp.EO) denote the number of even (resp.odd) spanning Eulerian
subgraphs. An orientation is \emph{Alon--Tarsi}\aaside{Alon--Tarsi orientation}{-.4cm}
(or AT) if EE and EO differ.
A graph $G$ is \emph{$f$-AT}\aside{$f$-AT, $k$-AT} if it has an Alon--Tarsi
orientation $D$ such that $d^+(v)\le f(v)-1$ for each vertex $v$. In
particular, $G$ is \emph{degree-AT}\aside{degree-AT} (resp.\emph{$k$-AT}) if
it is $f$-AT, where $f(v)=d(v)$ (resp.$f(v)=k$) for all $v$. Similarly, a
graph $G$ is \emph{$f$-choosable}\aside{$f$-choosable} if $G$ has a proper
coloring $\varphi$ from any list assignment $L$ such that $|L(v)|=f(v)$ for all
$v\in V(G)$. Alon and Tarsi\cite{AlonT} used algebraic methods to prove the
following theorem for choosability. Later, Schauz\cite{schauz2010flexible}
strengthened the result to paintability, which we discuss briefly in
Section\ref{extensions}.
\begin{thmB}
\hypertarget{target:thmB}{}
For a graph $G$ and $\func{f}{V(G)}{\mathbb{N}}$,
if $G$ is $f$-AT, then $G$ is also $f$-choosable.
\end{thmB}
In this paper we characterize those graphs $G$ with a specified vertex $x$
that are not $f$-AT, where $f(x)=d(x)-1$ and $f(v)=d(v)$ for all other $v\in
V(G)$. All such graphs are formed from a few 2-connected building blocks, by
repeatedly applying a small number of operations.
Most of the work in the proof is spent on the case when $G$ is 2-connected.
This result is easy to state, so we include it a bit later in the introduction,
as our \hyperlink{target:mainLemma}{Main Lemma}.
Near the end of Section\ref{MainThmSec}, with a little more work we extend our
\hyperlink{target:mainLemma}{Main Lemma}, by removing the hypothesis of
2-connectedness, to characterize
all pairs $(G,h_x)$ that are not AT. This result is
Theorem\ref{thm:1connected}.
This line of research began with Gallai, who studied the minimum number of edges
in an $n$-vertex $k$-critical graph $G$. Since $G$ has minimum degree at least
$k-1$, clearly $|E(G)|\ge\frac{k-1}2n$. Gallai\cite{gallai1963kritische} improved this
bound by classifying all connected subgraphs that can be induced by vertices of
degree $k-1$ in a $k$-critical graph. By TheoremA, all such graphs are Gallai
trees. Here, we consider graphs $G$ that are critical with respect to Alon--Tarsi
orientation. Specifically, $G$ is not $(k-1)$-AT, but every proper subgraph is;
such graphs are \emph{$k$-AT-critical}. The characterization of degree-AT
graphs shows that, much like $k$-critical graphs, in a $k$-AT-critical graph
$G$, every connected subgraph induced by vertices of degree $k-1$ must be a
Gallai tree. Our main result characterizes the subgraphs that can be induced by
vertices of degree $k-1$, together with a single vertex of degree $k$. Thus, it
is natural to expect that this result will lead to improved lower bounds on
the number of edges in $n$-vertex $k$-AT-critical graphs.
Similar to that for
degree-AT, our characterization remains unchanged in the contexts of
list-coloring and paintability, as we show in Section\ref{extensions}. We
see a sharp contrast when we consider
graphs $G$ with two specified vertices $x_1$ and $x_2$ that are not $f$-AT,
where $f(x_i)=d(x_i)-1$ for each $i\in \{1,2\}$ and $f(v)=d(v)$ for all other
$v\in V(G)$. For Alon--Tarsi orientations, we have more than 50 exceptional
graphs on seven vertices or fewer. Furthermore, the characterizations for
list-coloring, paintability, and Alon--Tarsi orientations all differ.
We consider graphs with vertices labeled by natural numbers; that is, pairs
$(G,h)$ where $G$ is a graph and $\func{h}{V(G)}{\mathbb{N}}$. We focus on the case
when $h(x)=1$ for some $x$ and $h(v)=0$ for all other $v$; we denote this
labeling as $h_x$. \aaside{$h_x$}{-.45cm} We say that \emph{$(G, h)$ is
AT}\aside{$(G,h)$\\ is AT} if $G$ is $(d_G - h)$-AT.
When $H$ is an induced subgraph of $G$, we simplify notation by referring to
the pair $(H, h)$ when we really mean $\parens{H, h\mathord{\upharpoonright}_{V(H)}}$.
Given a pair $(G,h)$ and a specified edge $e\in E(G)$, when we \emph{stretch
$e$}\aside{stretch $e$}, we form $(G',h')$ from $(G,h)$ by subdividing $e$ twice and setting
$h'(v_i)=0$ for each of the two new vertices, $v_1$ and $v_2$ (and $h'(v)=h(v)$
for all other vertices $v$). In Section\ref{prelims}, we prove a
\hyperlink{target:SubdivideTwice}{Stretching Lemma}, which shows that if $(G,h)$ is not
AT and $e\in E(G)$, then stretching $e$ often yields another pair $(G',h')$
that is also not AT. Thus, stretching plays a key role in our main result.
It is easy to check that the three pairs $(G,h)$ shown in Figure\ref{fig:seeds} are not AT
(and we do this below, in Proposition\ref{prop:easyD}). Let $\fancy{D}$ \aside{$\fancy{D}$} be the
collection of all pairs formed from the graphs in Figure\ref{fig:seeds} by stretching each
bold edge 0 or more times. The \hyperlink{target:SubdivideTwice}{Stretching Lemma}
implies that each pair in $\fancy{D}$ is not AT. Our
\hyperlink{target:mainLemma}{Main Lemma} is that these are
the only pairs $(G,h_x)$, where $G$ is 2-connected and neither complete
nor an odd cycle, such that $(G,h_x)$ is not AT, for some vertex $x\in V(G)$.
\begin{mainthm}
\hypertarget{target:mainLemma}{}
Let $G$ be 2-connected and let $x \in V(G)$.
Now $(G,h_x)$ is AT if and only if
\begin{enumerate}
\item[(1)] $d(x)=2$ and $G-x$ is not a Gallai tree; or
\item[(2)] $d(x)\ge 3$, $G$ is not complete, and $(G,h_x) \not \in \fancy{D}$.
\end{enumerate}
\end{mainthm}
The characterization of degree-choosable graphs has been applied
to prove a variety of graph coloring
results\cite{BohmeMS, CranstonPTV, KostochkaS, KralS, Thomassen-surface}.
Likewise, we think our main results in this paper may be helpful in proving
other results for Alon--Tarsi orientations, such as giving better lower bounds
on the number of edges in $k$-AT-critical graphs.
\begin{figure}
\caption{Three pairs $(G,h_x)$ that are in $\fancy{D}$. In each case $x$ is labeled 1 and all other vertices are labeled 0. Each other pair in $\fancy{D}$ can be formed from one of these pairs by repeatedly stretching one or more bold edges.}
\label{fig:seeds}
\end{figure}
To conclude this section, we show that each pair in $\fancy{D}$ is not AT.
\begin{prop}
If $(G,h_x)\in \fancy{D}$, then $(G,h_x)$ is not AT.
\label{prop:easyD}
\end{prop}
\begin{proof}
For each pair $(G,h_x)\in \fancy{D}$, we construct a list assignment $L$ such that
$|L(x)|=d(x)-1$ and $|L(v)|=d(v)$ for all other $v\in V(G)$, but $G$ has no
proper coloring from $L$. Now $(G,h_x)$ is not AT, by the contrapositive of
TheoremB.
Let $(G,h_x)$ be some stretching of the leftmost pair in Figure\ref{fig:seeds}. Assign the
list $\{1,2,3\}$ to each of the vertices on the unbolded triangle and assign the
list $\{1,2\}$ to each other vertex. If $G$ has some coloring from these lists,
then vertex $x$, labeled 1 in the figure, must get color 1 or 2; by symmetry,
assume it is 1. Along each path from $x$ to the triangle, colors must
alternate $2, 1, \ldots$. Each of the paths from $x$ to the triangle has odd
length; thus, color 1 is forbidden from appearing on the triangle. So $G$ has
no coloring from $L$. Now let $(G,h_x)$ be some stretching of the center pair
in Figure\ref{fig:seeds}. The proof is identical to the first case, except that each path
has even length, so if $x$ gets color 1, then color 2 is forbidden on the triangle.
Finally, consider the rightmost pair in Figure\ref{fig:seeds}. Here $d(x)=4$ and $d(v)=3$
for all other $v\in V(G)$. Thus, it suffices to show that $G$ is not
3-colorable. Assume that $G$ has a 3-coloring and, by symmetry, assume that $x$
is colored 1. Now colors 2 and 3 must each appear on two neighbors of $x$.
Thus, the two remaining vertices must be colored 1. Since they are adjacent,
this is a contradiction, which proves that $G$ is not 3-colorable.
\end{proof}
\section{Subgraphs, subdivisions, and cuts}
\label{prelims}
When Hladk{\'y}, Kr{\'a}{\soft{l}}, and Schauz characterized degree-AT graphs, their proof relied
heavily on the observation that a connected graph $G$ is degree-AT if and only
if $G$ has some induced subgraph $H$ such that $H$ is degree-AT. Below, we
reprove this easy lemma, and also extend it to our setting of pairs $(G,h_x)$.
\begin{subgraph}
\hypertarget{target:InducedSubgraph}{}
Let $G$ be a connected graph and let $H$ be an induced subgraph of $G$. If $H$
is degree-AT, then also $G$ is degree-AT. Similarly, if $x\in V(H)$ and
$(H,h_x)$ is AT, then also $(G,h_x)$ is AT. Further, if $x\notin V(H)$,
$d_G(x)\ge 2$, and $(H,h_x)$ is AT, then $(G,h_x)$ is AT.
\end{subgraph}
\begin{proof}
Suppose that $H$ is degree-AT, and let $D'$ be an orientation of $H$ showing
this. Extend $D'$ to an orientation $D$ of $G$ by orienting all edges away from
$H$, breaking ties arbitrarily, but consistently. Now every directed cycle in
$D$ is also a directed cycle in $D'$ (and vice versa), so $G$ is degree-AT.
The proof of the second statement is identical. The proof of the third
statement is similar, but now if some edge $xy$ has endpoints equidistant from
$H$, then $xy$ should be oriented into $x$.
\end{proof}
Recall that, given a pair $(G,h)$ and a specified edge $e\in E(G)$, when we
\emph{stretch $e$}, we form $(G',h')$ from $(G,h)$ by subdividing $e$ twice
and setting $h'(v_i)=0$ for each of the two new vertices, $v_1$ and $v_2$
(and $h'(v)=h(v)$ for all other vertices $v$). By repeatedly stretching edges,
starting from the three pairs in Figure\ref{fig:seeds}, we form all pairs
$(G,h_x)$, where $G$ is 2-connected and $(G,h_x)$ is not AT.
The following lemma will be useful for proving this.
\begin{stretching}\hypertarget{target:SubdivideTwice}{}
Form $(G',h')$ from $(G,h)$ by stretching some edge $e\in E(G)$.
Now
\begin{enumerate}
\item[(1)] if $(G,h)$ is AT, then $(G', h')$ is AT; and
\item[(2)] if $(G', h')$ is AT, then either $(G,h)$ is AT or $(G-e,h)$ is AT.
\end{enumerate}
\end{stretching}
\begin{proof}
Suppose $e = u_1u_2$ and call the new vertices $v_1$ and $v_2$ so that $G'$ contains
the induced path $u_1v_1v_2u_2$. For (1), let $D$ be an orientation of $G$ showing
that $(G,h)$ is AT. By symmetry we may assume $u_1u_2 \in E(D)$. Form an
orientation $D'$ of $G'$ from $D$ by replacing $u_1u_2$ with the directed path
$u_1v_1v_2u_2$. We have a natural parity preserving bijection between the spanning
Eulerian subgraphs of $D$ and $D'$, so we conclude that $(G', h')$ is AT.
For (2), let $D'$ be an orientation of $G'$ showing that $(G',h')$ is AT.
Suppose $G'$ contains the directed path $u_1v_1v_2u_2$ or the directed path
$u_2v_2v_1u_1$. By symmetry, we can assume it is $u_1v_1v_2u_2$. Now form an
orientation $D$ of $G$ by replacing $u_1v_1v_2u_2$ with the directed edge
$u_1u_2$. As above, we have a parity preserving bijection between the spanning
Eulerian subgraphs of $D$ and $D'$, so we conclude that $(G, h)$ is AT. So
suppose instead that $G'$ contains neither of the directed paths $u_1v_1v_2u_2$
and $u_2v_2v_1u_1$. Now no spanning Eulerian subgraph of $D'$ contains a cycle
passing through $v_1$ and $v_2$. So, the spanning Eulerian subgraph counts of
$D'$ are the same as those of $D' - v_1 - v_2$. However, this gives an
orientation of $G-e$ showing that $(G-e, h)$ is AT.
\end{proof}
Given a pair $(G,h)$ that is not AT, the \hyperlink{target:SubdivideTwice}{Stretching Lemma} suggests a way to construct a larger graph $G'$ such that $(G',h')$ is
not AT. In some cases, we can also use the
\hyperlink{target:SubdivideTwice}{Stretching Lemma} to construct a smaller graph
$\widehat{G}$ such that $(\widehat{G},h)$ is not AT.
Specifically, we have the following.
\begin{cor}
\label{SubdivideConstructor}
\label{ReduceP4Cor}
If $e$ is an edge in $G$ such that $(G,h)$ is not AT and $(G-e, h)$ is not AT,
then stretching $e$ gives a pair $(G',h')$ that is not AT. Further,
let $G$ be a graph with an induced path $u_1v_1v_2u_2$ such that $d_G(v_1) =
d_G(v_2) = 2$. If $(G,h)$ is AT, where $h(v_1) = h(v_2) = 0$, and
$(G-v_1-v_2,h)$ is not AT, then \[\parens{(G - v_1 - v_2) + u_1u_2,
h\mathord{\upharpoonright}_{V(G) \setminus \set{v_1, v_2}}} \text{ is AT.}\]
\end{cor}
\begin{proof}
The first statement is immediate from the \hyperlink{target:SubdivideTwice}{Stretching Lemma}. Now we prove the second. Suppose $(G,h)$ satisfies the
hypotheses. Applying part (2) of the \hyperlink{target:SubdivideTwice}{Stretching Lemma}
shows that either $\parens{G - v_1 - v_2, h\mathord{\upharpoonright}_{V(G) \setminus
\set{v_1, v_2}}}$ is AT or $\parens{(G - v_1 - v_2) + u_1u_2,
h\mathord{\upharpoonright}_{V(G) \setminus \set{v_1, v_2}}}$ is AT.
By hypothesis, the former is false. Thus, the latter is true.
\end{proof}
With standard vertex coloring, we can easily reduce to the case where $G$ is
2-connected. If $G$ is a connected graph with two blocks, $B_1$ and $B_2$,
meeting at a cutvertex $x$, then we can color each of $B_1$ and $B_2$
independently, and afterward we can permute colorings to match at $x$.
For Alon--Tarsi orientations, the situation is not quite as simple. However,
the following lemma plays a similar role for us.
\begin{lem}\label{CutvertexPatch}
Let $A_1, A_2 \subseteq V(G)$, and $x\in V(G)$ be such that $A_1\cup A_2=V(G)$ and $A_1 \cap
A_2 = \set{x}$. If $G[A_i]$ is $f_i$-AT for each $i \in \{1,2\}$, then $G$ is
$f$-AT, where $f(v) = f_i(v)$ for each $v \in V(A_i-x)$ and $f(x) = f_1(x) + f_2(x)
- 1$. Going the other direction, if $G$ is $f$-AT, then $G[A_i]$ is $f_i$-AT
for each $i \in \{1,2\}$, where $f_i(v) = f(v)$ for each $v \in V(A_i-x)$ and
$f_1(x) + f_2(x) \le f(x) + 1$.
\end{lem}
\begin{proof}
We begin with the first statement.
For each $i \in \{1,2\}$, choose an orientation $D_i$ of $A_i$ showing that $A_i$
is $f_i$-AT. Together these $D_i$ give an orientation $D$ of $G$. Since no cycle
has vertices in both $A_1-x$ and $A_2-x$, we have
\begin{align*}
EE(D) - EO(D) &= EE(D_1)EE(D_2) + EO(D_1)EO(D_2) - EE(D_1)EO(D_2) - EO(D_1)EE(D_2) \\
&= (EE(D_1) - EO(D_1))(EE(D_2) - EO(D_2)) \\
&\ne 0.
\end{align*}
Hence $G$ is $f$-AT.
Now we prove the second statement. Suppose that $G$ is $f$-AT and choose an
orientation $D$ of $G$ showing this.
Let $D_i = D[A_i]$ for each $i \in \{1,2\}$. As above, we have $0 \ne
EE(D) - EO(D) = (EE(D_1) - EO(D_1))(EE(D_2) - EO(D_2))$. Hence, $EE(D_1) -
EO(D_1) \ne 0$ and $EE(D_2) - EO(D_2) \ne 0$. Since the indegree of $x$ in
$D$ is the sum of the indegree of $x$ in $D_1$ and the indegree of $x$ in
$D_2$, the lemma follows.
\end{proof}
\section{Degree-AT graphs and an Extension Lemma}
Recall that our \hyperlink{target:mainLemma}{Main Lemma} relies on a
characterization of degree-AT graphs.
As we mentioned in the introduction, a description of degree-choosable graphs
was first given by Borodin\cite{borodin1977criterion} and Erd\H{o}s, Rubin, and
Taylor\cite{ERT}. Hladk{\'y}, Kr{\'a}{\soft{l}}, and Schauz\cite{HKS} later extended the
proof from\cite{ERT} to Alon--Tarsi orientations. This proof relies on
Rubin's Block lemma, which states that every 2-connected graph $G$ contains an
induced even cycle with at most one chord, unless $G$ is complete or an
odd cycle. For variety, and completeness, we include a new proof; it extends
ideas of Kostochka, Stiebitz, and Wirth\cite{KSW} from list-coloring to
Alon--Tarsi orientations. For this proof we need the following very special
case of a key lemma in \cite{OreVizing}. When vertices $x$ and $y$ are
adjacent, we write $x\leftrightarrow y$; otherwise $x\not\!\leftrightarrow y$.
\begin{lem}\label{GeneralEulerLemma}
Let $G$ be a graph and $x \in V(G)$ such that $H$ is connected, where $H \mathrel{\mathop:}= G-x$.
If there exist $z_1, z_2 \in V(H)$ with $N_H[z_1] = N_H[z_2]$ such that $x \leftrightarrow
z_1$ and $x \not\!\leftrightarrow z_2$, then $G$ is $f$-AT where $f(x) = 2$ and $f(v) =
d_G(v)$ for all $v \in V(H)$.
\end{lem}
\begin{proof}
Order the vertices of $H$ with $z_1$ first and $z_2$ second so that every
vertex, other than $z_1$, has at least one neighbor preceding it.
Orient each edge of $H$ from its earlier endpoint toward its later endpoint.
Orient $xz_1$ into $z_1$ and orient all other
edges incident to $x$ into $x$. Let $D$ be the resulting orientation.
Clearly, $d_{D}^+(v) \le f(v) - 1$ for all $v \in V(D)$. So, we just need to
check that $EE(D) \ne EO(D)$.
Since $xz_1$ is the only edge of $D$ leaving
$x$, and $D-x$ is acyclic, every spanning Eulerian subgraph of $D$ that has
edges must have edge $xz_1$.
Consider an Eulerian subgraph $A$ of $D$ containing $xz_1$. Since $z_1$
has indegree $1$ in $A$, it must also have outdegree $1$ in $A$. We show
that $A$ has a mate $A'$ of opposite parity.
If $z_2 \in A$ then $z_1z_2w \in A$, for some $w$, so we form
$A'$ from $A$ by removing $z_1z_2w$ and adding $z_1w$.
If instead $z_1z_2\notin A$, then $z_2 \not \in A$ and $z_1w \in
A$ for some $w \in N_H[z_1]-z_2$, so we form $A'$ from $A$ by removing $z_1w$ and
adding $z_1z_2w$.
Hence exactly half of the Eulerian subgraphs of $D$ that contain edges are
even. Since the edgeless spanning subgraph of $D$ is an even Eulerian
subgraph, we conclude that $EE(D) = EO(D) + 1$. Hence $G$ is $f$-AT.
\end{proof}
We use the previous lemma to give a new proof of the characterization of
degree-AT graphs.
\begin{lem}
\label{DegreeATClassification}
A connected graph $G$ is degree-AT if and only if it is not a Gallai tree.
\end{lem}
\begin{proof}
We begin with the ``only if'' direction. Neither odd cycles nor complete graphs
are degree-choosable. Thus, by \hyperlink{target:thmB}{Theorem B}, they are not
degree-AT. By induction on the number of blocks, Lemma\ref{CutvertexPatch}
implies that no Gallai tree is degree-AT.
Now, the ``if'' direction.
Suppose there exists a connected graph that is not a Gallai tree, but is also not
degree-AT. Let $G$ be such a graph with as few vertices as possible.
Since $G$ is not degree-AT, no induced subgraph $H$ of $G$ is
degree-AT by the \hyperlink{target:InducedSubgraph}{Subgraph Lemma}.
Hence, for any $v \in V(G)$ that is not a cutvertex, $G-v$ must be a Gallai
tree by minimality of $|G|$.
If $G$ has more than one block, then for endblocks $B_1$ and $B_2$, choose
noncutvertices $w\in B_1$ and $x\in B_2$. By the minimality of $|G|$, both
$G-w$ and $G-x$ are Gallai trees. Since every block of $G$ appears either as a
block of $G-w$ or as a block of $G-x$, every block of $G$ is either complete or
an odd cycle. Hence, $G$ is a Gallai tree, a contradiction. So instead $G$
has only one block, that is, $G$ is $2$-connected. Further, $G-v$ is a Gallai
tree for all $v \in V(G)$.
Let $v$ be a vertex of minimum degree in $G$. Since $G$ is $2$-connected,
$d_G(v) \ge 2$ and $v$ is adjacent to a noncutvertex in every endblock of $G-v$.
If $G-v$ has a complete block $B$ with noncutvertices $x_1,x_2$ where $v \leftrightarrow
x_1$ and $v \not\!\leftrightarrow x_2$, then we can apply Lemma \ref{GeneralEulerLemma}
to conclude that $G$ is degree-AT, a
contradiction. So, $v$ must be adjacent to every noncutvertex in every
complete endblock of $G-v$.
Suppose $d_G(v) \ge 3$. Now no endblock of $G-v$ can be an odd cycle of
length at least $5$ ($G$ would have vertices of degree $3$ and also $d_G(v) \ge
4$, contradicting the minimality of $d_G(v)$). Let $B$ be a smallest complete
endblock of $G-v$. Now for a noncutvertex $x \in V(B)$, we have $d_G(x) =
|B|$ and hence $d_G(v) \le |B|$.
If $G-v$ has at least two endblocks, then $2(|B|-1) \le |B|$, so $d_G(v)
\le |B| = 2$, a contradiction. Hence, $G-v = B$ and $v$ is joined to $B$, so
$G$ is complete, which is a contradiction.
Thus, we have $d_G(v) = 2$. Suppose $G-v$ has at least two endblocks.
Now it has exactly two and $v$ is adjacent to one noncutvertex in each.
Neither of the endblocks can be odd cycles of length at least five, since then
we can get a smaller counterexample by the \hyperlink{target:SubdivideTwice}{Stretching Lemma}. Since $v$ is adjacent to every noncutvertex in every
complete endblock of $G-v$, both endblocks must be $K_2$. But now either
$G=C_4$ (which is degree-AT, by orienting the cycle consistently) or we can get
a smaller counterexample by the \hyperlink{target:SubdivideTwice}{Stretching Lemma}.
So, $G-v$ must be $2$-connected. Since $G-v$ is a Gallai tree, it is either
complete or an odd cycle. If $G-v$ is not complete, then we can get a smaller
counterexample by the \hyperlink{target:SubdivideTwice}{Stretching Lemma}. So, $G-v$
is complete and $v$ is adjacent to every noncutvertex of $G-v$; that is, $G$ is
complete, a contradiction.
\end{proof}
\section{When h is 1 for one vertex}
\label{MainThmSec}
In this section, we prove our \hyperlink{target:mainLemma}{Main Lemma}.
For a graph $G$ and $x \in V(G)$ recall that
$\func{h_x}{V(G)}{\mathbb{N}}$ is defined as $h_x(x) = 1$ and $h_x(v) = 0$ for all $v
\in V(G-x)$. We classify the connected graphs $G$ such that $(G,h_x)$ is AT for
some $x \in V(G)$. We begin with the case when $G$ is 2-connected, which takes
most of the work. At the end of the section, we extend our characterization to
all connected graphs.
We will show that for most 2-connected graphs $G$ and vertices $x\in V(G)$, the
pair $(G,h_x)$ is AT. Specifically, this is true for all pairs except those in
$\fancy{D}$, defined in the introduction. In view of the
\hyperlink{target:InducedSubgraph}{Subgraph Lemma}, for a 2-connected graph $G$ and
$x\in V(G)$, to show that $(G,h_x)$ is AT it suffices to
find some induced subgraph $H$ such that $(H,h_x)$ is AT.
The subgraphs $H$ that we consider all have $d_H(x)=0$ or $d_H(x)\ge 3$. This
motivates the next lemma, which allows us to reduce to the case $d_G(x)\ge 3$.
\begin{lem}
\label{DegreeTwoVertex}
If $G$ is a connected graph and $x \in V(G)$ with $d_G(x) = 2$, then $(G,h_x)$
is AT if and only if $G-x$ is degree-AT.
\end{lem}
\begin{proof}
Let $D$ be an orientation of $G$ showing that $(G,h_x)$ is AT. Now
$d_{D}^-(x) = 2$, so no spanning Eulerian subgraph contains a cycle
passing through $x$. Therefore, the Eulerian subgraph counts in $G-x$ are
different and $G-x$ is degree-AT. The other direction is immediate from the
\hyperlink{target:InducedSubgraph}{Subgraph Lemma}.
\end{proof}
Lemma\ref{DegreeTwoVertex}, together with Lemma\ref{DegreeATClassification},
proves Case (1) of our \hyperlink{target:mainLemma}{Main Lemma}. Before we can prove Case
(2), we need a few more definitions and lemmas.
A \emph{$\theta$-graph}\aside{$\theta$-graph} consists of two vertices joined by
three internally disjoint paths, $P_1$, $P_2$, and $P_3$. When we write $h_x$
for a $\theta$-graph, we always assume that $d(x)=3$. We will see
shortly that if $H$ is a $\theta$-graph with $d_H(x)=3$, then $(H,h_x)$ is AT.
Thus, the \hyperlink{target:InducedSubgraph}{Subgraph Lemma} implies that
if $(G,h_x)$ is not AT, then $G$ has no induced $\theta$-graph $H$ with $d_H(x)=3$.
A \emph{$T$-graph}\aside{$T$-graph} is formed from vertices $x, z_1, z_2, z_3$,
by making the $z_i$
pairwise adjacent, and joining each vertex $z_i$ to $x$ by a path $P_i$ (where
the $P_i$ are disjoint). Equivalently, a $T$-graph is formed from $K_4$ by
subdividing each of the edges incident to $x$ zero or more times.
Similar to the proof characterizing degree-AT graphs in\cite{HKS},
our approach in proving our \hyperlink{target:mainLemma}{Main Lemma}
is to find an induced subgraph $H$ such that $(H,h_x)$ is AT, and apply the
\hyperlink{target:InducedSubgraph}{Subgraph Lemma}.
Thus, we need the following lemma about pairs $(H,h_x)$ that are AT.
\begin{lem}
\label{ThetaReducible}
\label{TgraphReducible}
\label{T+graphReducible}
The pair $(G,h_x)$ is AT whenever (i) $G$ is a $\theta$-graph, (ii) $G$ is a
$T$-graph and two paths $P_i$ have lengths of opposite parities, or (iii) $G$ is
formed from a $T$-graph by adding an extra vertex with neighborhood
$\{z_1,z_2,z_3\}$.
\end{lem}
\begin{proof}
In each case, we give an AT orientation $D$ of $G$ such that $d_D^-(v)\ge
h_x(v)+1$ for each $v\in V(G)$.
Case (i). Orient the edges of each path $P_i$ consistently, with $P_1$ and
$P_2$ into $x$ and $P_3$ out of $x$; this orientation satisfies the degree
requirements. Further, it has exactly three spanning Eulerian subgraphs,
including the empty subgraph. Thus, $EE+EO$ is odd, so $EE\ne EO$.
Case (ii). Let $P_1$ and $P_2$ be two paths with opposite parities. As before,
orient the edges of each path consistently, with $P_1$ and $P_2$ into $x$ and
$P_3$ out of $x$. Orient the three additional edges as $\overrightarrow{z_1z_2},
\overrightarrow{z_2z_3}$, and $\overrightarrow{z_3z_1}$. The resulting digraph $D$ has four spanning
Eulerian subgraphs, 3 of one parity and 1 of the other. Note that the empty
subgraph and the subgraph $\{\overrightarrow{z_1z_2}, \overrightarrow{z_2z_3}, \overrightarrow{z_3z_1}\}$ have
opposite parities. Further, the parities are the same for the two subgraphs
consisting of the directed cycles $xP_3z_3z_1P_1$ and $xP_3z_3z_1z_2P_2$. So,
$EE\ne EO$.
Case (iii). The simplest instance of this case is when $G=K_5-e$. Now
$(G,h_x)$ is AT by Lemma\ref{GeneralEulerLemma}. In fact, that proof gives the
stronger statement that there exists an orientation $D$ satisfying the degree
requirements such that $EE(D)=EO(D)+1$. In particular, $EE+EO$ is odd.
To handle larger instances of this case, we repeatedly subdivide edges incident
to $x$ and orient each of the resulting paths consistently, and in the direction
of the corresponding edge in $D$. The resulting orientation satisfies the
degree requirements. Further, the sum $EE+EO$ remains unchanged, and thus odd.
Hence, still $EE\ne EO$.
\end{proof}
\begin{lem}\label{AddPathReducible}
Let $G$ be a $T$-graph. Let $P$ be a path of $G$ where all internal
vertices of $P$ have degree 2 in $G$ and one endvertex of $P$ has degree 2 in
$G$. Form $G'$ from $G$ by adding a path $P'$ (of length at least 2) joining
the endvertices of $P$. Now $(G', h_x)$ is AT.
\end{lem}
\begin{proof}
We can assume that $G$ is not AT; otherwise, we are done by the
\hyperlink{target:InducedSubgraph}{Subgraph Lemma}. By symmetry, assume $P$ is a
subpath of $P_3$. First, we get an orientation of $G$ with indegree at least 1
for all vertices and $d^-(x) = 2$. Orient $P_1$ from $z_1$ to $x$, $P_2$ from
$z_2$ to $x$, $P_3$ from $x$ to $z_3$, and the triangle as $\overrightarrow{z_1z_2},
\overrightarrow{z_2z_3}$, and $\overrightarrow{z_3z_1}$. To get an orientation of $G'$, orient the
new path $P'$ consistently, and opposite of $P$. Now the only directed cycle
containing edges of $P'$ is $P'P$. Since the Eulerian subgraph counts are
equal for $G$, they differ by 1 for $G'$.
\end{proof}
Now we can prove Case (2) of our \hyperlink{target:mainLemma}{Main Lemma}. For
reference, we restate it.
\begin{figure}
\caption{The pair $(G,h_x)$ is AT, when $G$ is formed from $K_4$ by subdividing one or two edges incident to $x$.}
\label{fig:SubdividedK4}
\label{fig:TriangleRuinsPath}
\end{figure}
\begin{figure}
\caption{(a) The pair $(G,h_x)$ is AT, where $G=K_5-xy$.
(b) The pair $(G,h_x)$ is AT, where $G$ is formed from $K_5-e$ by subdividing each edge incident to $x$.}
\label{fig:thebigone}
\label{fig:K5minus}
\end{figure}
\begin{lem}
\label{TwoConnectedClassification}
Let $G$ be 2-connected, and choose $x\in V(G)$ with $d(x)\ge 3$.
Now $(G,h_x)$ is AT if and only if $G$ is not complete and $(G,h_x) \not \in \fancy{D}$.
\end{lem}
\begin{proof}
When $(G,h_x)\in \fancy{D}$ the lemma holds by Proposition\ref{prop:easyD}.
Now let $G$ be 2-connected, choose $x\in V(G)$ with $d(x)\ge 3$, and suppose that
$(G,h_x)\notin \fancy{D}$. Since $G-x$ is connected, let $H'$ be a smallest connected
subgraph of $G-x$ containing three neighbors of $x$; call these neighbors $w_1$,
$w_2$, and $w_3$. Consider a spanning tree $T$ of $H'$. Since $H'$ is minimum,
each leaf of $T$ is among $\{w_1, w_2, w_3\}$. If $T$ is a path, then $H'$ is also a
path. Otherwise, $T$ is a subdivision of $K_{1,3}$. Let $s$ be the vertex with
$d_T(s)=3$. If $E(G)-E(T)$ has any edge with both ends outside of $N(s)$, then
we can delete some vertex in $N(s)$ and remain connected, contradicting the
minimality of $H'$. Similarly, if $N(s)$ contains at least two edges, then
$H'-s$ still connects, $w$, $y$, and $z$. Now let $H$ be the subgraph of $G$
induced by $V(H')\cup\{x\}$. Note that $H$ is either a $\theta$-graph (if $H'$
is a tree) or a $T$-graph (if $H'$ has one extra edge in $N(s)$).
If $H$ is a $\theta$-graph, then $(G,h_x)$ is AT, by
Lemma\ref{ThetaReducible}.i and
the \hyperlink{target:InducedSubgraph}{Subgraph Lemma}.
So assume $H$ is a $T$-graph. Let $z_1$, $z_2$, $z_3$ be the vertices
of degree 3 (other than $x$), and let $P_1$, $P_2$, and $P_3$ denote the paths
from $x$ to $z_1$, $z_2$, and $z_3$; when we write $V(P_i)$, we exclude $x$ and
$z_i$, so possibly $V(P_i)$ is empty for one or more $i\in\{1,2,3\}$.
If any two of $P_1$, $P_2$, and $P_3$ have lengths with opposite parities, then
we are done by Lemma\ref{TgraphReducible}.ii; so assume not.
Now $(H,h_x)\in \fancy{D}$, so we can assume that $V(G-H) \ne \emptyset$. Choose $u
\in V(G-H)$, and let $H_u$ be a minimal $2$-connected induced subgraph of $G$
that contains $V(H) \cup \set{u}$. By the \hyperlink{target:InducedSubgraph}{Subgraph Lemma} and Lemma\ref{DegreeATClassification}, $G-x$ is a Gallai tree.
Thus, so is $H_u-x$; in particular, the block $B_u$ of $H_u-x$ containing $u$
is complete or an odd cycle. Therefore, we either have (i) $V(B_u) \cap V(H) =
\set{z_1, z_2,z_3}$ or (ii) $V(B_u) \cap V(H) \subseteq P_i \cup \set{z_i}$
for some $i \in \{1,2,3\}$.
Suppose (i) happens. Now $N_G(u) \cap V(H_u - x) = \set{z_1,z_2,z_3}$. If $x
\not\!\leftrightarrow u$, then $(G,h_x)$ is AT by the \hyperlink{target:InducedSubgraph}{Subgraph Lemma} and Lemma\ref{T+graphReducible}.iii. If $x \leftrightarrow u$, then $x$ must have odd
length paths to each $z_i$, by Lemma\ref{TgraphReducible}.ii, with $u$ in the role
of some $z_i$. Further, $x \leftrightarrow z_i$ for all $i \in \{1,2,3\}$, since
otherwise $(G,h_x)$ is AT by the \hyperlink{target:InducedSubgraph}{Subgraph Lemma},
Lemma\ref{T+graphReducible}.iii, and the \hyperlink{target:SubdivideTwice}{Stretching Lemma}. So, $H=K_4$ and $H_u=K_5$. This implies that (ii) cannot happen for
any vertex in $V(G-H)$, since if $V(B_u)\cap V(H)=\{z_i\}$ for some $i$, then
$(G,h_x)$ is AT by Lemma\ref{ThetaReducible}.i
and the \hyperlink{target:InducedSubgraph}{Subgraph Lemma}).
So (i) happens for every vertex in $V(G-H)$; in particular, $V(G-H)$ is joined
to $\set{x,z_1,z_2,z_3}$. Since $G$ is not complete, $G-x$ must contain an induced
copy of Figure\ref{fig:K5minus}(a); hence, $(G,h_x)$ is AT by
Lemma\ref{T+graphReducible}.iii and the \hyperlink{target:InducedSubgraph}{Subgraph Lemma}.
Assume instead that (ii) happens for every vertex in $V(G-H)$, including $u$.
By symmetry, assume that $V(B_u) \cap V(H) \subseteq P_1$. Let $z_1P_1
= v_1v_2\cdots v_{\ell}$, where $v_{\ell}\leftrightarrow x$. First, assume that $B_u$ is
an odd cycle of length at least $5$. If there is $u' \in V(B_u)\setminus V(H)$
with $u' \leftrightarrow x$, then $G$ contains a $\theta$-graph and $(G,h_x)$ is AT, by
Lemma\ref{ThetaReducible}.i and the \hyperlink{target:InducedSubgraph}{Subgraph Lemma}. So, we may assume that $u' \not\!\leftrightarrow x$ for all $u' \in V(B_u)\setminus
V(H)$. Now we are done by Lemma\ref{AddPathReducible} and the
\hyperlink{target:InducedSubgraph}{Subgraph Lemma}.
So instead we assume that $B_u$ is complete. If $V(B_u) \cap V(H)=\set{v_\ell}$,
then $G$ has an induced $\theta$-graph $J$, where $d_J(x)=d_J(v_\ell)=3$, so we are
done by Lemma\ref{ThetaReducible}.i and the \hyperlink{target:InducedSubgraph}{Subgraph Lemma}. Thus, we must have $V(B_u) \cap V(H) = \set{v_{j}, v_{j+1}}$ for some
$j \in \irange{\ell-1}$. In particular, $B_u$ is a triangle. If $u\not\!\leftrightarrow x$,
then $(G,h_x)$ is AT by the \hyperlink{target:InducedSubgraph}{Subgraph Lemma} and
Lemma\ref{AddPathReducible}. So we conclude that $u\leftrightarrow x$, which requires
$j=\ell-1$, by the minimality of $H$. Hence, $H_u$ is formed from a $T$-graph
by adding a vertex $u$ that is adjacent to $x$ and also to the vertices of a
$K_2$ endblock $D_u$ of $H-x$.
Suppose there are distinct vertices $u_1, u_2\in V(G-H)$
adjacent to vertices of the same $K_2$ endblock.
Now $G$ contains an induced copy of Figure \ref{fig:K5minus}(a), so
$(G,h_x)$ is AT by Lemma\ref{T+graphReducible}.iii and the
\hyperlink{target:InducedSubgraph}{Subgraph Lemma}. Thus, each $K_2$ endblock has at
most one such $u$.
Let $t$ be the number of $K_2$ endblocks in $H-x$.
By construction, $t\le 3$; this implies that $|V(G - H)| \le t\le 3$.
If $t = 0$, then $G = H = K_4$, which contradicts that $G$ is not complete.
If $t=1$, then $G = H_u$, for the unique $u \in V(G-H)$; this is the Moser
spindle, shown in Figure\ref{fig:seeds}(c). So, assume that $t \in
\set{2,3}$. By symmetry, assume that for each $i\in\{1,2\}$ there exists $u_i$
such that $V(B_{u_i})\subseteq P_i\cup\{z_i\}$. Now the subgraph induced by
$\set{u_2}\cup V(H-P_1)$ is reducible by Lemma\ref{AddPathReducible}.
So, again we are done by the \hyperlink{target:InducedSubgraph}{Subgraph Lemma}.
\end{proof}
Taken together, Lemmas\ref{TwoConnectedClassification}
and\ref{DegreeTwoVertex}, with
Lemma\ref{DegreeATClassification}, prove our \hyperlink{target:mainLemma}{Main Lemma}.
However, this characterizaton requires that $G$ be 2-connected.
Now we extend our result to the more general case, when $G$ need only be
connected. We use the following two definitions. Let $G$ be a graph, $x$
a vertex of $G$, and $B$ a block of $G$. An \emph{$x$-lobe of
$G$}\aside{$x$-lobe} is a maximal subgraph $A$ such that $A-x$ is connected. A
\emph{$B$-lobe of $G$}\aside{$B$-lobe} is a maximal subgraph $A$ such that
$A-B$ is connected, and $A$ includes a single vertex of $B$.
\begin{thm}
If $G$ is connected and $x \in V(G)$, then $(G, h_x)$ is not AT if and only if
\label{thm:1connected}
\end{thm}
\begin{enumerate}
\item[(1)] $G$ is a Gallai tree; or
\item[(2)] $d(x) = 1$; or
\item[(3)] $d(x) = 2$ and $G-x$ has a component that is a Gallai tree; or
\item[(4)] $x$ is not a cutvertex, for the block $B$ of $G$ containing $x$,
we have $(B,h_x) \in \fancy{D}$, and every other block of $G$ is complete or
an odd cycle; or
\item[(5)]
$x$ is a cutvertex, all but at most one $x$-lobe of $G$, say $A$, is a Gallai
tree, and either:
(i) $d_A(x) = 1$; or
(ii) $d_A(x)=2$ and $A-x$ is a Gallai tree; or
(iii) for the block $B$ of $A$ containing $x$,
we have $(B,h_x)\in \fancy{D}$ and all $B$-lobes of $A$ are Gallai trees.
\end{enumerate}
\begin{proof}
First, we check that if any of Cases (1)--(5) hold, then $(G, h_x)$ is not AT.
Cases (1) and (2) are immediate. Case (3) follows from
Lemma\ref{DegreeTwoVertex}.
Consider Case (4).
By Proposition\ref{prop:easyD}, we know $(B,h_x)$ is not AT.
Now $(G,h_x)$ is not AT by repeated application of Lemma\ref{CutvertexPatch}.
Finally, Case (5) follows from Cases (2), (3), and (4), by
Lemma\ref{CutvertexPatch}.
Now, for the other direction, suppose $(G, h_x)$ is not AT and none of Cases
(1)--(5) hold. By Lemma\ref{DegreeTwoVertex}, and not (2) and not (3), we
must have $d(x) \ge 3$.
Suppose $x$ is a cutvertex. Now, by not (5), either (a) at least two $x$-lobes
of $G$ are not Gallai trees or (b) $(H, h_x)$ is AT for some $x$-lobe $H$ of
$G$. In each case, $(G,h_x)$ is AT by Lemma\ref{CutvertexPatch}, which is a
contradiction.
So assume instead that $x$ is not a cutvertex. Suppose the block $B$ of $G$
containing $x$ is complete or $(B,h_x) \in \fancy{D}$. By not (1) and not (4),
some $B$-lobe $H$ of $G$ is not a Gallai tree. Since $H$ is a subgraph of
$G-x$, and $G-x$ is connected, Lemma\ref{DegreeATClassification} and the
\hyperlink{target:InducedSubgraph}{Subgraph Lemma} imply that $G-x$ is degree-AT;
hence, $(G,h_x)$ is also AT. So, we conclude that $B$ is not complete and
$(B,h_x)\notin \fancy{D}$.
First suppose that $d(x)=2$. By not (3), we know that $G-x$ is not a Gallai
tree. Lemma\ref{DegreeATClassification} implies that $G-x$ is degree-AT.
So, again, the \hyperlink{target:InducedSubgraph}{Subgraph Lemma} shows that $(G,h_x)$
is AT. Now assume instead that $d(x)\ge 3$. Since $(B,h_x)\notin \fancy{D}$, now
Lemma\ref{TwoConnectedClassification} implies that $(B,h_x)$ is AT;
once more, the \hyperlink{target:InducedSubgraph}{Subgraph Lemma} implies that $(G,h_x)$
is AT.
\end{proof}
\section{Choosability and Paintability}
\label{extensions}
As we mentioned in the introduction, Alon and Tarsi showed that if a graph $G$
is $f$-AT, then $G$ is also $f$-choosable. \emph{Online list coloring}, also
called \emph{painting} is similar to list coloring, but now the list for each
vertex is progressively revealed, as the graph is colored.
Schauz\cite{schauz2010flexible} extended the Alon--Tarsi theorem, to show that
if $G$ is $f$-AT, then $G$ is also $f$-paintable (which we define formally
below). In this section, we use our
characterization of pairs $(G,h_x)$ that are not AT to prove characterizations
of pairs $(G,h_x)$ that are not paintable and that are not choosable. More
precisely, a pair $(G,h_x)$ is \emph{choosable}\aaside{choosable pair}{-.3cm}
if $G$ has a proper coloring from its lists $L$ whenever $L$ is such that
$|L(x)|=d(x)-1$ and $|L(v)|=d(v)$ for all other $v$; otherwise $(G,h_x)$ is
\emph{not choosable}. A pair being \emph{paintable}\aaside{paintable
pair}{-.3cm} is defined analogously. We characterize all pairs $(G,h_x)$,
where $G$ is connected and $(G,h_x)$ is not choosable (resp. not paintable).
In fact, we will see that these characterizations, for both choosability and
paintability, are identical to that for pairs that are not AT.
For completeness, we include the following definition of $f$-paintable.
Schauz\cite{schauz2009mr} gave a more intuitive (yet equivalent) definition,
in terms of a two player game. We say that $G$ is \emph{$f$-paintable}
\aside{$f$-paintable} if either (i) $G$ is empty or (ii) $f(v) \ge 1$ for all
$v \in V(G)$ and for every $S \subseteq V(G)$ there is an independent set $I
\subseteq S$ such that $G-I$ is $f'$-paintable where $f'(v) \mathrel{\mathop:}= f(v)$
for all $v \in V(G) - S$ and $f'(v) \mathrel{\mathop:}= f(v) - 1$ for all $v \in S - I$.
Since all pairs $(G,h_x)$ that are AT are also both paintable and choosable, it
suffices to show that every pair $(G,h_x)$ that is not AT is also not choosable
(here we use that if a pair is paintable, then it is also choosable).
\begin{thm}
For every connected graph $G$, the pair $(G,h_x)$ is not choosable if and only
if $(G,h_x)$ is not AT. Thus, the same characterization holds for pairs that
are not paintable.
\end{thm}
\begin{proof}
As noted above, every pair that is AT is also choosable and paintable. Thus, it
suffices to show that each pair $(G,h_x)$ in Theorem\ref{thm:1connected} is not
choosable.
To show that Gallai trees are not degree-choosable, assign to each block $B$ a
list of colors $L_B$ such that $|L_B|=d_B(x)$ for each $x\in V(B)$; further, for
all distinct blocks $B_1$ and $B_2$, we require that $L_{B_1}$ and $L_{B_2}$ are
disjoint. For each $v\in V(G)$, let $L(v)=\cup_{B_i\ni v}L_{B_i}$. To show
that $G$ is not colorable from these lists, we use induction on the number of
blocks. Let $B$ be an endblock and $x$ a cutvertex in $B$. Let
$G'=G\setminus(V(G)-x)$. Since $B$ is complete or an odd cycle, $B$ has no
coloring from $L_B$. Thus any coloring $\varphi$ of $G$ from $L$ does not use
$L_B$ on $x$. Hence, $\varphi$ gives a coloring $\varphi'$ of $G'$ from its
lists $L'$, where $L'(x)=L(x)\setminus L_B$ and $L'(v)=L(v)$ for all $v\in
V(G)\setminus V(B)$. This coloring $\varphi'$ of $G'$ contradicts the induction
hypothesis. Thus, $G$ has no coloring from $L$.
Here we use a similar approach. Consider a pair $(G,h_x)$ that satisfies one of
Cases (1)--(5) in Theorem\ref{thm:1connected}. We show that $(G,h_x)$ is not
choosable. Case (1) is immediate by the previous paragraph.
Case (2) is immediate, since $|L(x)|=0$. For Case (3), give lists to the Gallai
tree of $G-x$ as above; now let $L(x)=\{c\}$ for some new color $c$, and add $c$
to the list of each neighbor of $x$. Again $G$ cannot be colored from $L$. For
Case (4), assign lists to $V(B)$ as in Proposition\ref{prop:easyD} and to the
other blocks as above. Again, $G$ has no coloring from
these lists. Finally, consider Case (5). Assign lists for all blocks outside
of $A$ as above, and assign lists for $A$ as above in Case (2), (3), or (4).
\end{proof}
To conclude this section, we consider labelings $h_{x,y}$, where $h_{x,y}(x)=
h_{x,y}(y)=1$ and $h_{x,y}(v)=0$ for all other $v\in V(G)$.
We show that the set of pairs $(G,h_{x,y})$ that are not AT differs from the set
of that are not paintable. Further, both sets differ from
the set of pairs that are not choosable.
It suffices to give a pair $(G_1,h_{x,y})$ that is choosable but not paintable
and a second pair $(G_2,h_{x,y})$ that is paintable but not AT.
\begin{figure}
\caption{
The pair on the left is choosable, but not paintable.
The pair on the right is paintable, but not AT.
}
\label{fig:splits}
\end{figure}
\begin{prop}
The pair $(G_1,h_{x,y})$ on the left in Figure\ref{fig:splits} is choosable,
but not paintable. The pair $(G_2,h_{x,y})$ on the right in
Figure\ref{fig:splits} is paintable, but not AT.
\end{prop}
\begin{proof}
Let $(G_1,h_{x,y})$ denote the pair on the left, where $x$ and $y$ are the
vertices labeled 1. Let $(G_2,h_{x,y})$ denote the pair on the right, where
$x$ and $y$ are the vertices labeled 1.
We first show that $(G_1,h_{x,y})$ is choosable. Let $L$ denote the list
assignment. If there exists $c\in L(x)\cap L(y)$, then use $c$ to color $x$ and
$y$, and color the remaining vertices greedily. So suppose there does not
exist such a color $c$. Let $z$ be a vertex in both triangles and note that
there exist $c\in (L(x)\cup L(y))\setminus L(z)$. By symmetry, assume that
$c\in L(x)$. Color $x$ with $c$, and color $G_1-x$ greedily, starting with the
vertex of degree 2 and ending with $z$.
We now show that $(G_1,h_{x,y})$ is not paintable. Let $S$ be the vertices of one
triangle. By definition, there must be $I \subseteq S$ such that $G_1-I$ is
$f'$-paintable, where $f'(v) \mathrel{\mathop:}= f(v)$ for $v \in V(G_1) - S$ and $f'(v)
\mathrel{\mathop:}= f(v) - 1$ for $v \in S - I$. $I$ must have one vertex, $w$. There
are two choices for $w$; either $w$ is in two triangles or not. If $w$ is not
in two triangles, then $G_1 - w$ is a triangle with a pendant edge, where
the vertices on the triangle all have list size 2, so $G_1-w$ is not paintable.
If $w$ is one of the vertices in two triangles, then $G_1-w$ is a 4-cycle with
list sizes alternating $1, 2, 1, 2$. Again $G_1-I$ is not paintable (nor choosable).
To see that $(G_2,h_{x,y})$ is not AT, note that any good orientation would need
indegrees summing to at least 7, but $G_2$ has only 6 edges. Now we show that
$(G_2,h_{x,y})$ is paintable. Note that $G_2$ is isomorphic to $K_{2,3}$, the complete
biparite graph. Call the parts $X$ and $Y$, with $|X|=2$ and $|Y|=3$. If $S$
includes at least two vertices of $X$ or at least two vertices of $Y$, take $I$
to be an independent set of size at least 2. It is easy to check that $G-I$ is
paintable, since it induces either an independent set or a path, where each
endvertex has more colors than neighbors. So assume that $S$ contains at most
one vertex from each of $X$ and $Y$. If $S$ contains a vertex of $X$, then
color it. The resulting graph is paintable, since it is a claw, $K_{1,3}$,
with at most one leaf having a single color and all other vertices having two
colors. Finally, suppose $S$ contains only a single vertex of $Y$. Let $I=S$.
The resulting graph is $C_4$, which is degree-paintable (since it is degree-AT).
\end{proof}
A graph is \emph{unstretched} \aside{unstretched} if it has no induced path
$u_1v_1v_2u_2$ where $d(v_1)=d(v_2)=2$ (as in Corollary\ref{ReduceP4Cor}).
We finish with the following question.
\begin{question}
Are there only finitely many unstretched, 2-connected graphs $G$ such that
$(G,h_{x,y})$ is not choosable (resp.paintable, AT)? More generally, let
$h_{x_1,\ldots,x_k}$ be a labeling that assigns 1 to vertices $x_1,\ldots,x_k$
and 0 to all others. Are there only finitely many unstretched, 2-connected
graphs $G$ such that $(G, h_{x_1,\ldots,x_k})$ is not choosable
(resp.paintable, AT)?
\end{question}
\end{document} | arXiv | {
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\begin{document}
\title{\ourtitle}
\begin{abstract}
The sample average approximation (SAA)
approach is applied to risk-neutral optimization problems
governed by semilinear elliptic partial differential equations
with random inputs.
After constructing a compact set that contains
the SAA critical points, we
derive nonasymptotic sample size estimates
for SAA critical points
using the covering number approach.
Thereby, we derive upper bounds on the number of
samples needed to obtain accurate critical points
of the risk-neutral PDE-constrained optimization problem
through SAA critical points.
We quantify
accuracy using expectation and exponential tail bounds.
Numerical illustrations are presented. \end{abstract}
\begin{keywords}
stochastic optimization,
PDE-constrained optimization under uncertainty,
sample average approximation, Monte Carlo sampling,
sample complexity, uncertainty quantification \end{keywords}
\begin{AMS}
90C15, 90C30, 90C60, 49J20, 49J55, 49K45, 49K20, 35J61 \end{AMS}
\section{Introduction} \label{sec:intro} Many objective functions of stochastic programs involve expectations of parameterized objective functions in one way or another. Large-scale stochastic programs arise in a multitude of applications, such as machine learning \cite{Lan2020,Shalev-Shwartz2010}, feedback stabilization of autonomous systems \cite{Kunisch2020}, statistical estimation \cite{Huber1967,Royset2019}, and optimization of differential equations under uncertainty \cite{Phelps2016,Wechsung2021}. When the parameter space is high-dimensional, these expectations cannot be accurately evaluated. A common approach to approximate such stochastic programs is the sample average approximation (SAA) method, yielding the SAA problem \cite{Kleywegt2002,Shapiro2003,Shapiro2005}. The SAA objective function is defined by the sample average of the parameterized objective function. The SAA approach has become popular in the literature on optimization under uncertainty with partial differential equations (PDEs) \cite{Hoffhues2020,Roemisch2021,Wechsung2021}. The key characteristic of the SAA approach as applied to risk-neutral PDE-constrained optimization is the fact that it yields PDE-constrained optimization problems which can be solved efficiently using existing, rapidly converging algorithms. While the SAA approach is easy to use, a central question is: how many samples are sufficient to obtain approximate critical points for the stochastic program via SAA critical points with high probability? An answer to this question also addresses the computational complexity of risk-neutral PDE-constrained optimization problems, as potentially expensive simulations of PDEs are required for each sample. In the present manuscript, we answer this question for a class of risk-neutral semilinear PDE-constrained optimization problems. Our analysis is inspired by those in \cite{Shapiro2003,Shapiro2005,Shapiro2021}.
We derive nonasymptotic sample size estimates for SAA critical points of infinite dimensional optimization problems governed by semilinear PDEs with random inputs. We consider the risk-neutral PDE-constrained optimization problem \begin{align} \label{eq:ocp} \min_{u \in L^2(D)}\, (1/2) \cE{\norm[L^2(D)]{S(u, \xi)-y_d}^2} + (\alpha/2) \norm[L^2(D)]{u}^2 + \psi(u), \end{align} where $\alpha > 0$, $D \subset \mathbb{R}^d$ is a bounded Lipschitz domain, $y_d \in L^2(D)$, $L^1(D)$ $(L^2(D))$ is the Lebesgue space of (square) integrable functions defined on $D$, and $\xi$ is a random vector mapping from a complete probability space to a complete probability space with sample space $\Xi$ being a complete, separable metric space. Moreover, for each $(u,\xi) \in L^2(D) \times \Xi$, $y = S(u,\xi) \in H_0^1(D)$ solves the state equation \begin{align} \label{eq:Feb0320211603} A(\xi) y + Q(y) = b(\xi) + B(\xi)u. \end{align} The Sobolev space $H_0^1(D)$ is formally introduced in \cref{sec:notation}. We state assumptions on the semilinear PDE \eqref{eq:Feb0320211603} in \cref{sec:rnpdeopt}. The function $\psi : L^2(D) \to (-\infty,\infty]$ in \eqref{eq:ocp} is proper, convex and lower semicontinuous. An example for $\psi$ is given by $\psi(u) = \gamma \norm[L^1(D)]{u}$ if $u \in U_0$ and $\psi(u) = \infty$ otherwise, where $\gamma \geq 0$ and $U_0 = \{\, u \in L^2(D) \colon \, \mathfrak{l} \leq u \leq \mathfrak{u} \,\}$ with $\mathfrak{l}$, $\mathfrak{u} \in L^2(D)$ and $\mathfrak{l} \leq \mathfrak{u}$. This function and its variants arise in control device placement applications such as tidal turbine layout optimization \cite{Funke2016}. The SAA problem of \eqref{eq:ocp} is given by \begin{align} \label{eq:saa} \min_{u \in L^2(D)}\, \frac{1}{2N} \sum_{i=1}^N\norm[L^2(D)]{S(u, \xi^i)-y_d}^2 + (\alpha/2) \norm[L^2(D)]{u}^2 + \psi(u), \end{align} where $\xi^1$, $\xi^2, \ldots$ are independent identically distributed $\Xi$-valued random vectors defined on a common complete probability space $(\Omega, \mathcal{F}, P)$ and each $\xi^i$ has the same distribution as $\xi$. Here, $N \in \mathbb{N}$ is the sample size. Critical points of the SAA problem \eqref{eq:saa} can be efficiently computed using semismooth Newton methods \cite{Ulbrich2011,Stadler2009,Mannel2020}.
The feasible set, $\{\, u \in L^2(D) \colon\, \psi(u) < \infty \,\}$, is generally non-compact. The lack of compactness and the nonconvexity of the risk-neutral PDE-constrained optimization problem \eqref{eq:ocp} and its SAA problems complicate the derivation of sample size estimates, as these are typically derived using covering numbers of the feasible set \cite{Cucker2002,Kankova1978,Shapiro2003,Shapiro2005,Royset2019}. Sample size estimates can be established for convex stochastic progams without the covering number approach \cite{Guigues2017,Shalev-Shwartz2010}. However, the risk-neutral PDE-constrained optimization problem \eqref{eq:ocp} and its SAA problems are nonconvex. For deriving sample size estimates, our initial observation is that critical points of the SAA problem \eqref{eq:saa} are contained in a compact subset of the feasible set. To construct the compact set, we use optimality conditions, PDE stability estimates, and higher regularity of the reduced parameterized objective function's gradient. Utilizing the covering numbers for Sobolev function classes established in \cite{Birman1967,Birman1980}, we derive sample size estimates using arguments similar to those in \cite{Shapiro2003,Shapiro2005,
Shapiro2008,Shapiro2021,Royset2019,Cucker2002,Kankova1978,Mei2018}. However, our focus is the analysis of SAA critical points as opposed to SAA solutions, as risk-neutral semilinear PDE-constrained optimization problems are nonconvex. For classes of risk-neutral linear elliptic PDE-constrained optimizations, SAA solutions are analyzed in \cite{Hoffhues2020,Martin2021,Milz2022c,Milz2021,Roemisch2021}. However, the analysis in \cite{Milz2021} does not generalize to nonconvex problems.
Alternative approximation approaches for risk-neutral PDE-constrained optimization problems include, for example, quasi-Monte Carlo sampling \cite{Guth2019}, low-rank tensor methods \cite{Benner2020,Garreis2017}, and stochastic collocation \cite{Borzi2011,Kouri2013,Tiesler2012}.
While we are unaware of a systematic theoretical and/or empirical study
comparing, for example, Monte Carlo approaches,
quasi-Monte Carlo sampling techniques, and sparse grid schemes
as applied to risk-neutral nonconvex
PDE-constrained optimization, we provide a brief comparision of these
approaches in terms of characteristics used for their theoretical
convergence anayses.
Sparse grid-based discretizations may result in nonconvex
optimization problems and their analysis requires smoothness properties
with respect to the parameters and independent random variables
with Lebesgue densities, but the methods in
\cite{Kouri2014,Kouri2013,Zahr2019}, which combine sparse grid
techniques with trust-region schemes, perform very well.
We refer the reader to
\cite{Chen2015} for the analysis of sparse grid-based approximations
of finite dimensional stochastic programs.
Monte Carlo sample-based approximations preserve
convexity and only require mild integrability properties
with respect to possibly infinitely many parameters,
but are deemed to be quite slowly convergent.
For finite dimensional stochastic programs, it is known
that quasi-Monte Carlo methods can outperform Monte Carlo sampling
techniques provided that the integrands satisfy certain
regularity conditions \cite[pp.\ 185--189]{Shapiro2021}.
The quasi-Monte Carlo approach is analyzed in
\cite{Guth2019} as applied to risk-neutral
linear-quadratic control problems with PDEs, an important
class of risk-neutral convex PDE-constrained optimization problems.
As with every numerical scheme, each approximation approach
for risk-neutral PDE-constrained optimization has its advantages
and disadvantages. The Monte Carlo sample-based
scheme is arguably the most widely used approach to approximating
stochastic programs. This and the fact that
it yields standard PDE-constrained optimization problems build our
main motivation to establishing sample size estimates.
\section*{Outline of the paper} We discuss preliminaries and define further notation in \cref{sec:notation}. A class of risk-neutral semilinear PDE-constrained optimization problems is introduced in \cref{sec:rnpdeopt}. PDE stability estimates are derived in \cref{subsect:semilinearpdes}. In \cref{subsect:derivative}, we compute the derivative of the expectation function and demonstrate the Lipschitz continuity of its gradient in \cref{subsec:lipschitz}. The focus of our results in \cref{sec:rnpdeopt} is to make the PDE stability estimates' dependence on problem-dependent parameters explicit. \Cref{subsect:existence} discusses the existence of solutions. We combine the results established in \cref{sec:rnpdeopt} to construct a compact set containing the SAA critical points in \cref{sec:compatset}. The covering numbers of Sobolev function classes used to establish nonasymptotic finite sample size estimates in \cref{sec:samplesizeestimates} are provided in \cref{sec:covering}. In \cref{sec:discussion}, we summarize our contributions, put our approach into perspective, and comment on potential improvements of our sample size estimates with respect to the dimension of the computational domain. In the appendices, we discuss auxiliary results used in \cref{sec:samplesizeestimates} to derive the sample size estimates. \Cref{sec:setinclusion} discusses the measurability of set inclusions and \cref{sec:subgaussianbounds} provides sub-Gaussian-type expectation and tail bounds of maxima of Hilbert space-valued random vectors. The bounds are used in \cref{sect:uniformtailbounds} to derive uniform expectation and exponential tail bounds for normal maps in Hilbert spaces. These bounds are applied in \cref{sec:samplesizeestimates} for obtaining our nonasymptotic sample size estimates for SAA critical points.
\section{Preliminaries and further notation} \label{sec:notation}
Relationships between random vectors hold with probability one (w.p.~$1$) if not specified otherwise. Metric spaces are equipped with their Borel $\sigma$-field. Let $X$, $X_1$ and $X_2$ be real Banach spaces. The space of bounded, linear operators from $X_1$ to $X_2$ is denoted by $\spL{X_1}{X_2}$. We define $X^* = \spL{X}{\mathbb{R}}$. The adjoint operator of $\Upsilon \in \spL{X_1}{X_2}$ is denoted by $\Upsilon^* \in \spL{X_2^*}{X_1^*}$. Throughout the text, $(\Theta, \mathcal{A}, \mu)$ is a complete probability space. An operator-valued mapping $\Upsilon : \Theta \to \spL{X_1}{X_2}$ is called uniformly measurable if there exists a sequence of simple operators $\Upsilon_k : \Theta \to \spL{X_1}{X_2}$ such that $\Upsilon_k(\theta)$ converges to $\Upsilon(\theta)$ in $\spL{X_1}{X_2}$ as $k \to \infty$ for all $\theta \in \Theta$; cf.\ \cite[Def.\ 3.5.5]{Hille1957}. A function $\upsilon : \Theta \to X$ is strongly measurable if there exists a sequence of simple functions $\upsilon_k : \Theta \to X $ such that $\upsilon_k(\theta) \to \upsilon(\theta)$ as $k \to \infty$ for all $\theta \in \Theta$ \cite[Def.\ 1.1.4]{Hytoenen2016}. If $X$ is separable, then $\upsilon : \Theta \to X$ is strongly measurable if and only if it is measurable \cite[Cor.\ 1.1.2 and Thm.\ 1.1.6]{Hytoenen2016}.
Let $\Gamma \colon \Theta \rightrightarrows X$ be a set-valued mapping with closed images. The mapping $\Gamma$ is called measurable if $\Gamma^{-1}(V) \in \mathcal{A}$ for each open set $V \subset X$ \cite[Def.\ 8.1.1]{Aubin2009}. Here, $\Gamma^{-1}$ is the inverse image of $\Gamma$. An operator $\Upsilon \in \spL{X_1}{X_2}$ is compact if $\Upsilon(V) \subset X_2$ is precompact for each bounded set $V \subset X_1$. Let $Y$ be a complete metric space. A map $\Phi : Y \times \Theta \to X$ is a Carath\'eodory\ function if $\Phi(\cdot, \theta)$ is continuous for all $\theta \in \Theta$ and $\Phi(y,\cdot)$ is measurable for all $y \in Y$ \cite[p.\ 311]{Aubin2009}. The Fr\'echet\ derivative of a mapping $h$ with respect to $y$ is denoted by $h_y$. For a real Hilbert space $H$, $\inner[H]{\cdot}{\cdot}$ is its inner product and $\norm[H]{h} = \inner[H]{h}{h}^{1/2}$ its norm.
For a convex, lower semicontinuous, proper function $\varphi : H \to (-\infty,\infty]$, the proximity operator $\prox{\varphi}{}:H \to H$ of $\varphi$ is defined by (see \cite[Def.\ 12.23]{Bauschke2011}) \begin{align*}
\prox{\varphi}{v} = \argmin_{w\inH}\, \varphi(w) + (1/2)\norm[H]{v-w}^2. \end{align*} Throughout the text, $D \subset \mathbb{R}^d$ is a bounded Lipschitz domain. The space $L^2(D)$ is identified with its dual. The Sobolev space $H^1(D)$ is defined by all $L^2(D)$-functions with square integrable weak derivatives and $H_0^1(D)$ consists all $v \in H^1(D)$ with zero boundary traces. We define $H^{-1}(D) = H_0^1(D)^*$. The dual pairing between $H^{-1}(D)$ and $H_0^1(D)$ is denoted by $\dualpHzeroone{\cdot}{\cdot}$. We equip $H_0^1(D)$ with $\norm[H_0^1(D)]{v} = \norm[L^2(D)]{\norm[2]{\nabla v}}$ and $H^1(D)$ with $\norm[H^1(D)]{v} = (\norm[L^2(D)]{v}^2 + \norm[H_0^1(D)]{v}^2)^{1/2}$. Here, $\nabla v$ is the weak gradient of $v$ and $\norm[2]{\cdot}$ is the Euclidean norm on $\mathbb{R}^d$. Throughout the text, $\iota : H_0^1(D) \to L^2(D)$ given by $\iota v = v$ is the embedding operator of the compact embedding $H_0^1(D) \xhookrightarrow{} L^2(D)$ and $C_D$ is Friedrichs' constant, the operator norm of $\iota$. We have $C_D = \norm[\spL{L^2(D)}{H^{-1}(D)}]{\iota^*}$.
We denote by $|D|$ the Lebesgue measure of $D$ and by $\bar D$ its closure. We define $C^{0,1}(\bar{D})$ as the space of Lipschitz continuous real-valued functions defined on $\bar{D}$ and equip it with the usual norm $\norm[C^{0,1}(\bar D)]{\cdot}$ defined in \cite[p.\ 16]{Hinze2009}.
Let $(X, \norm[X]{\cdot})$ be a normed space, let $X_0 \subset X$ be nonempty and totally bounded, and let $\nu > 0$. The $\nu$-covering number $\mathcal{N}(\nu; X_0, \norm[X]{\cdot})$ is the minimal number of closed $\norm[X]{\cdot}$-balls with radius $\nu$ in $X$ needed to cover $X_0$ (cf.\ \cite[pp.\ 87--88]{Tikhomirov1993}). This notion of covering numbers does not require the centers of the $\norm[X]{\cdot}$-balls be contained in $X_0$.
\section{Risk-neutral semilinear PDE-constrained optimization} \label{sec:rnpdeopt}
We impose conditions on the parameterized PDE \eqref{eq:Feb0320211603} which ensure the existence and uniqueness of solutions. These conditions ensure that the semilinear PDE \eqref{eq:Feb0320211603} is a monotone operator equation and hence the existence of solutions and the stability estimates can be established using the Minty--Browder theorem \cite[Thm.\ 26.A]{Zeidler1990}, for example. Semilinear PDE-constrained optimization problems are analyzed, for example, in \cite{Garreis2019a,Geiersbach2020,Kouri2020,Troeltzsch2010,Ulbrich2011}. In this section, our contributions are to address some measurability questions, and to derive stability estimates and a bound on an objective function's Lipschitz constant with the dependence on problem data's characteristics made explicit. Even though our derivations are built on standard techniques, the stability estimates are needed for establishing our nonasymptotic sample size estimates (see \cref{sec:samplesizeestimates}).
We define $F : L^2(D) \to \mathbb{R}$ and $\hat{F}_N : L^2(D) \to \mathbb{R}$ by \begin{align} \label{eq:efuns} F(u) = (1/2) \cE{\norm[L^2(D)]{\iota S(u, \xi)-y_d}^2} \quad \text{and} \quad \hat{F}_N(u) = \frac{1}{2N} \sum_{i=1}^N \norm[L^2(D)]{\iota S(u, \xi^i)-y_d}^2. \end{align} As opposed to the problem formulations in \eqref{eq:ocp} and \eqref{eq:saa}, we make here the use of the embedding operator $\iota$ explicit. Since the random vectors $\xi^1, \xi^2, \ldots$ are defined on the complete probability space $(\Omega, \mathcal{F}, P)$, we can view $\hat{F}_N$ as a function defined on $L^2(D) \times \Omega$. However, we often omit the second argument.
We define the feasible set $\csp_\text{ad} = \{\, u \in L^2(D) \colon\, \psi(u) < \infty \,\}$. \begin{assumption}[{Control regularization and feasible set}]
\label{ass:adcspbounded}
\begin{enumthm}[nosep,leftmargin=*]
\item
\label{itm:psi}
The function
$\psi : L^2(D) \to (-\infty,\infty]$
is proper, convex and lower semicontinuous.
\item
\label{itm:adcspbounded}
For some $r_{\text{ad}} \in (0,\infty)$,
$\norm[L^2(D)]{u} \leq r_{\text{ad}}$
for all $u \in \csp_\text{ad}$.
\end{enumthm} \end{assumption}
\subsection{Semilinear PDEs with random inputs} \label{subsect:semilinearpdes}
We impose conditions on the data defining the semilinear PDE \eqref{eq:Feb0320211603} based on those used in \cite[Assumptions (3.1)--(3.3)]{Kouri2020}, \cite[sect.\ 9.1]{Ulbrich2011}, and \cite[Assumption 2.2]{Garreis2019a}.
\begin{assumption}[{Semilinear PDE: Problem data}]
\label{assumption:pde}
\begin{enumthm}[nosep,leftmargin=*]
\item
\label{itm:domain}
$D \subset \mathbb{R}^d$ is a bounded Lipschitz domain
with $d \in \{2,3\}$.
\item
\label{itm:A}
$A : \Xi \to \spL{H_0^1(D)}{H^{-1}(D)}$
is uniformly measurable.
There exists a constant $\kappa_{\min} > 0$
such that $A(\xi)$ is self-adjoint
and
$\dualpHzeroone{A(\xi)y}{y} \geq
\kappa_{\min}\norm[H_0^1(D)]{y}^2$
for all $y \in H_0^1(D)$
and each $\xi \in \Xi$.
\item
\label{itm:rrhs}
$b : \Xi \to H^{-1}(D)$
and
$g : \Xi \to C^{0, 1}(\bar D)$
are strongly measurable
and there exist constants $b_{\max}$, $g_{\max} > 0$
such that
$\norm[H^{-1}(D)]{b(\xi)} \leq b_{\max}$
and
$\norm[C^{0, 1}(\bar D)]{g(\xi)} \leq g_{\max}$
for each $\xi \in \Xi$.
\item
\label{itm:B}
For each $\xi \in \Xi$,
$B(\xi) \in \spL{L^2(D)}{H^{-1}(D)}$
is defined by
\begin{align}
\label{eq:B}
\dualpHzeroone{B(\xi)u}{v} = \inner[L^2(D)]{g(\xi)u}{v}.
\end{align}
\item
\label{itm:nonlinearity}
We define $Q : H_0^1(D) \to H^{-1}(D)$
by
$
\dualpHzeroone{Q(y)}{v}
= \inner[L^2(D)]{q(y)}{v}
$.
The function $q: \mathbb{R} \to \mathbb{R}$ is nondecreasing,
twice continuously differentiable, $q(0) = 0$,
and
$|q''(t)| \leq c_{q} + d_{q} |t|^{p-3}$
for all $t \in \mathbb{R}$,
where $c_{q}$, $d_{q} \geq 0$
and $p \in (3,\infty)$ if $d = 2$
and $p \in (3, 6]$ if $d = 3$.
\end{enumthm} \end{assumption}
\Cref{itm:domain,itm:nonlinearity} ensure that the embedding $H_0^1(D) \xhookrightarrow{} L^p(D)$ is continuous \cite[Thm.\ 1.14]{Hinze2009} and that $q$ is twice continuously differentiable as a superposition operator from $L^p(D)$ to $L^{p^*}(D)$ \cite[p.\ 202]{Ulbrich2011}, where $p^* \in [1,\infty)$ fulfills $1/p + 1/p^* = 1$. Hence, $Q$ is twice continuously differentiable from $H_0^1(D)$ to $H^{-1}(D)$. Moreover $Q$ is monotone. Hence \Cref{assumption:pde} ensures that for each $\xi \in \Xi$, the operator $H_0^1(D) \ni y \mapsto A(\xi) y + Q(y) \in H^{-1}(D)$ is continuous and strongly monotone with parameter $\kappa_{\min}$ and hence coercive \cite[p.\ 501]{Zeidler1990}. We denote by $c_p > 0$ the embedding constant of the embedding $H_0^1(D) \xhookrightarrow{} L^p(D)$. Under \Cref{itm:domain,itm:nonlinearity}, the constant $c_p$ is finite since $d \in \{2,3\}$ and $p \in [1,6]$ \cite[Thm.\ 1.14]{Hinze2009}. \Cref{lem:properties_B} establishes basic properties of the mapping $B$. In particular, we establish its uniform measurability and some of its compactness properties. \begin{lemma}
\label{lem:properties_B}
If \Cref{itm:domain,itm:rrhs,itm:B} hold, then
for all $\xi \in \Xi$,
\begin{enumerate}[nosep,leftmargin=*]
\item
\label{itm:Bbounded}
$B : \Xi \to \spL{L^2(D)}{H^{-1}(D)}$
is uniformly measurable
and
$$\norm[\spL{L^2(D)}{H^{-1}(D)}]{B(\xi)}
\leq C_D\norm[C^{0,1}(\bar{D})]{g(\xi)},$$
\item
\label{itm:Badjointcompact}
$B(\xi)^*v = \iota[g(\xi)v] = g(\xi) v$
for all $v \in H_0^1(D)$,
and $B(\xi)^*$ and $B(\xi)$ are compact.
\end{enumerate} \end{lemma}
We prove \Cref{lem:properties_B} using \Cref{lem:Grisvard2011}. \Cref{lem:Grisvard2011} establish an explicit continuity constant of a certain bilinear mapping in terms of Friedrichs' constant. \begin{lemma}
\label{lem:Grisvard2011}
Let $D \subset \mathbb{R}^d$ be a bounded Lipschitz domain.
If $v \in C^{0,1}(\bar{D})$
and $w \in H_0^1(D)$,
then $vw\in H_0^1(D)$
and
$
\norm[H_0^1(D)]{vw} \leq (C_D+1)\norm[C^{0,1}(\bar{D})]{v}
\norm[H_0^1(D)]{w}
$. \end{lemma}
\begin{proof}
Using \cite[Thm.\ 1.4.1.2]{Grisvard2011},
we have $vw\in H_0^1(D)$.
Since $v$ is Lipschitz continuous,
we have
$\norm[L^\infty(D)]{\norm[2]{\nabla v}}
\leq \norm[C^{0,1}(\bar{D})]{v}$.
Using $\nabla[vw] = w\nabla v + v \nabla w$
and the H\"older\ inequality,
we obtain
$\norm[H_0^1(D)]{vw} \leq
\norm[C^{0,1}(\bar{D})]{v}(\norm[L^2(D)]{w}+
\norm[L^2(D)]{\norm[2]{\nabla w}})
$.
Combined
with Friedrichs' inequality, we obtain the stability estimate. \end{proof}
\begin{proof}[{Proof of \Cref{lem:properties_B}}]
Let us fix $u \in L^2(D)$, $v \in H_0^1(D)$,
and $\xi \in \Xi$.
\begin{enumerate}[wide,nosep]
\item
We define $\varrho : C^{0,1}(\bar{D}) \to
\spL{L^2(D)}{H^{-1}(D)}$
by
$\dualpHzeroone{\varrho(g)u}{v}
= \inner[L^2(D)]{gu}{v}$.
The map $\varrho$ is Lipschitz continuous
with Lipschitz constant $C_D$.
Combined with $B = \varrho \circ g$
(see \eqref{eq:B})
and the strong measurability of $g$,
we find that
$B$ is uniformly measurable \cite[Cor.\ 1.1.11]{Hytoenen2016}.
The bound on the operator norm of $B(\xi)$ is implied by the above
Lipschitz continuity.
\item
We have
$\inner[L^2(D)]{B(\xi)^*v}{u} = \inner[L^2(D)]{g(\xi)v}{u}$
for all $u \in L^2(D)$.
Since $g(\xi) \in C^{0,1}(\bar{D})$,
$g(\xi)v \in H_0^1(D)$
\cite[Thm.\ 1.4.1.2]{Grisvard2011}.
Hence, $B(\xi)^*v = \iota[g(\xi)v]$.
The map
$H_0^1(D) \ni v \mapsto g(\xi)v \in H_0^1(D)$
is linear and bounded
(see \Cref{lem:Grisvard2011}).
Hence $B(\xi)^*$ is compact.
Now Schauder's theorem
\cite[Thm.\ 3.4 on p.\ 174]{Conway1985}
implies that $B(\xi)$ is compact.
\end{enumerate} \end{proof}
The next lemma is essentially known; cf.\ \cite{Ulbrich2011,Kouri2020,Garreis2019a}.
\begin{lemma}
\label{lem:s_unique_stable}
Let \Cref{assumption:pde} hold.
For each $(u,\xi) \in L^2(D) \times \Xi$,
the parameterized PDE \eqref{eq:Feb0320211603} has a unique
solution $S(u,\xi) \in H_0^1(D)$ and
\begin{align}
\label{eq:Feb0420210958}
&\norm[H_0^1(D)]{S(u,\xi)}
\leq (1/\kappa_{\min})\norm[H^{-1}(D)]{b(\xi)}
+ ( C_D/\kappa_{\min})
\norm[C^{0,1}(\bar{D})]{g(\xi)} \norm[L^2(D)]{u}.
\end{align}
Moreover, for each $(u_1,u_2,\xi) \in L^2(D)^2 \times \Xi$,
\begin{align}
\label{eq:Nov2520211145}
&\norm[H_0^1(D)]{S(u_2,\xi)-S(u_1,\xi)}
\leq (C_D/\kappa_{\min}) \norm[C^{0,1}(\bar{D})]{g(\xi)}
\norm[L^2(D)]{u_2-u_1}.
\end{align} \end{lemma} \begin{proof}
The existence and uniqueness is implied by
\cite[Thm.\ 26.A]{Zeidler1990}. We have
$\kappa_{\min}\norm[H_0^1(D)]{S(u,\xi)}
\leq \norm[H^{-1}(D)]{b(\xi) + B(\xi) u}$
(cf.\ \cite[eq.\ (3.5)]{Kouri2020})
and
\begin{align}
\label{eq:eq37}
\kappa_{\min}\norm[H_0^1(D)]{S(u_2,\xi)-S(u_1,\xi)}
\leq \norm[H^{-1}(D)]{B(\xi)[u_2-u_1]};
\end{align}
cf.\ \cite[p.\ 560]{Zeidler1990} and \cite[eq.\ (3.5)]{Kouri2020}.
Combined with \Cref{lem:properties_B}, we obtain
\eqref{eq:Feb0420210958} and \eqref{eq:Nov2520211145}. \end{proof}
\subsection{Derivative computation} \label{subsect:derivative}
We show that the functions $F$ and $\hat{F}_N$ are Fr\'echet\ differentiable, compute their derivatives using the adjoint approach, and derive further stability estimates. The derivative formulas and the stability estimates are crucial for obtaining our finite sample size estimates.
Let us define $\pobj : L^2(D) \times \Xi \to \mathbb{R}$ by \begin{align} \label{eq:rpobj} \pobj(u,\xi) = (1/2)\norm[L^2(D)]{\iota S(u,\xi)-y_d}^2. \end{align} For each $(u,\xi) \in L^2(D) \times \Xi$, $z=z(u,\xi) \in H_0^1(D)$ solves the adjoint equation \begin{align} \label{eq:adjoint} [A(\xi) + Q_y(S(u,\xi))]z = - \iota^*[\iota S(u,\xi)-y_d]. \end{align} To derive \eqref{eq:adjoint}, we have used the fact that $A(\xi)$ (see \Cref{itm:A}) and $Q_y(y)$ are self-adjoint for each $(y,\xi) \in H_0^1(D) \times \Xi$ (cf.\ \cite[eq.\ (2.8)]{Garreis2019a}).
\begin{proposition}
\label{prop:efunsdifferentiable}
If \Cref{assumption:pde} holds,
then $F$ and $\hat{F}_N$ defined in \eqref{eq:efuns}
are Fr\'echet\ differentiable.
For each $u \in L^2(D)$, it holds that
$\nabla F(u)$, $\nabla \hat{F}_N(u) \in H_0^1(D)$,
\begin{align}
\label{eq:Feb0820212103}
\nabla F(u) =
-\cE{g(\xi) z(u, \xi)}
\quad \text{and} \quad
\nabla \hat{F}_N(u) =
-\frac{1}{N} \sum_{i=1}^N g(\xi^i) z(u, \xi^i).
\end{align} \end{proposition}
The proof of \Cref{prop:efunsdifferentiable} uses \Cref{lem:Feb0720211501,lem:Nov252021204}.
\begin{lemma}
\label{lem:Feb0720211501}
If \Cref{assumption:pde} holds, then the following statements
hold.
\begin{enumerate}[nosep,leftmargin=*]
\item For each $(u,\xi) \in L^2(D) \times \Xi$,
the adjoint equation \eqref{eq:adjoint} has a unique
solution $z(u,\xi) \in H_0^1(D)$ and
\begin{align}
\label{eq:Feb0720211501}
\begin{aligned}
\norm[H_0^1(D)]{z(u,\xi)}
& \leq (C_D/\kappa_{\min})\norm[L^2(D)]{\iota S(u,\xi)-y_d}.
\end{aligned}
\end{align}
\item
\label{itm:s_caratheodory}
The solution operator $S : L^2(D) \times \Xi \to H_0^1(D)$
is a Carath\'eodory\ mapping.
\item
\label{itm:psiystrongly}
If $s : \Xi \to H_0^1(D)$
is measurable,
then
$\Xi \ni \xi \mapsto Q_y(s(\xi))
\in \spL{H_0^1(D)}{H^{-1}(D)}$
is uniformly measurable.
\item
\label{itm:zcaratheodory}
The adjoint state
$z : L^2(D) \times \Xi \to H_0^1(D)$
is a Carath\'eodory\ function.
\item $L^2(D) \times \Xi \ni
(u,\xi) \mapsto g(\xi) z(u,\xi) \in H_0^1(D)$
is a Carath\'eodory\ map.
For each $u \in L^2(D)$,
$\Xi \ni \xi \mapsto g(\xi)z(u,\xi) \in H_0^1(D)$
is Bochner integrable
and
\begin{align}
\label{eq:23Nov20211913}
\norm[H_0^1(D)]{g(\xi)z(u,\xi)}
&\leq (C_D+1)\norm[C^{0,1}(\bar D)]{g(\xi)}
\norm[H_0^1(D)]{z(u,\xi)}.
\end{align}
\end{enumerate} \end{lemma} \begin{proof}
\begin{enumerate}[wide,nosep]
\item The assertions are a consequence of the Lax--Milgram lemma
and the fact that
$C_D =
\norm[\spL{L^2(D)}{H^{-1}(D)}]{\iota^*}$.
\item Let us define the nonlinear operator
$E : H_0^1(D) \times L^2(D) \times \Xi \to H^{-1}(D)$
by
$E(y,u,\xi) = A(\xi) y + Q(y) - b(\xi) - B(\xi)u$.
Then $S(u,\xi)$ solves
\eqref{eq:Feb0320211603} if and only if $E(S(u,\xi),u,\xi) = 0$.
As in the proofs of
\cite[Lems.\ 9.2 and 9.6]{Ulbrich2011}, we can show that
$E(\cdot,\cdot,\xi)$ is twice continuously differentiable
for each $\xi \in \Xi$ and that
$\dualpHzeroone{E_y(y,u,\xi)v}{v} \geq
\kappa_{\min}\norm[H_0^1(D)]{v}^2$
for each $(y,v,u,\xi) \in H_0^1(D)^2 \times L^2(D) \times \Xi$.
Hence, the implicit function theorem
and the Lax--Milgram lemma \cite[Lem.\ 1.8]{Hinze2009}
ensure that $S(\cdot, \xi)$ is twice continuously
differentiable for each $\xi \in \Xi$.
For each $u \in L^2(D)$,
the measurability of $S(u,\cdot)$ is implied by
the proof of \cite[Thm.\ 3.12]{Garreis2019}.
\item Since $H_0^1(D) \ni y \mapsto Q_y(y) \in
\spL{H_0^1(D)}{H^{-1}(D)}$ is continuous and
$s$ is strongly measurable,
$Q_y \circ s$ is uniformly measurable
\cite[Cor.\ 1.1.11]{Hytoenen2016}.
\item
For each $y \in H_0^1(D)$,
$v \mapsto Q_y(y)v$ is monotone.
Hence, for each $\xi \in \Xi$,
$\dualpHzeroone{A(\xi)y + Q_y(S(u,\xi))y}{y} \geq
\kappa_{\min}\norm[H_0^1(D)]{y}^2$
for all $y \in H_0^1(D)$.
Now, the implicit function theorem
and part~\ref{itm:s_caratheodory} can be used to deduce
the twice continuous differentiability of $z(\cdot, \xi)$
for all $\xi \in \Xi$.
Fix $u \in L^2(D)$.
\Cref{assumption:pde}, and
parts~\ref{itm:s_caratheodory} and \ref{itm:psiystrongly}
ensure
that $\Xi \ni \xi \mapsto A(\xi) + Q_y(S(u,\xi))$
is uniformly measurable.
The measurability of $z(u,\cdot)$ is now implied by
part~\ref{itm:s_caratheodory}
when applied to the adjoint equation \eqref{eq:adjoint}
rather than the state equation \eqref{eq:Feb0320211603}.
\item For each
$\xi \in \Xi$,
part~\ref{itm:zcaratheodory}
and \Cref{lem:Grisvard2011} ensure the continuity of
$L^2(D) \ni u \mapsto g(\xi) z(u,\xi) \in H_0^1(D)$.
Now, let $u \in L^2(D)$.
Let us define
$\varrho_1 : \Xi \to C^{0,1}(\bar D) \times H_0^1(D)$
by $\varrho_1(\xi) = (g(\xi), z(u,\xi))$.
Since $g$ and $z(u,\cdot)$ are strongly measurable
(see part~\ref{itm:zcaratheodory}),
$\varrho_1$ is strongly measurable.
We also define $\varrho_2 : C^{0,1}(\bar D) \times H_0^1(D)
\to H_0^1(D)$
by $\varrho_2(g,z) = g z$.
\Cref{lem:Grisvard2011} ensures that
$\varrho_2$ is bounded.
Combined with the fact that $\varrho_2$ is bilinear,
we find that $\varrho_2$ is continuous.
We have $g(\xi) z(u,\xi) = (\varrho_2 \circ \varrho_1)(\xi)$
for all $\xi \in \Xi$.
Thus, the chain rule
\cite[Cor.\ 1.1.11]{Hytoenen2016} ensures
the measurability
of $\Xi \ni \xi \mapsto g(\xi)z(u,\xi) \in H_0^1(D)$.
\Cref{lem:Grisvard2011} yields \eqref{eq:23Nov20211913}.
Using \eqref{eq:Feb0420210958},
\eqref{eq:Feb0720211501}, \eqref{eq:23Nov20211913},
\Cref{itm:rrhs}, and the measurability
of $\Xi \ni \xi \mapsto g(\xi)z(u,\xi) \in H_0^1(D)$,
we have
$\cE{\norm[H_0^1(D)]{g(\xi)z(u,\xi)}} < \infty$.
Hence, $\Xi \ni \xi \mapsto g(\xi)z(u,\xi) \in H_0^1(D)$
is Bochner integrable \cite[Prop.\ 1.2.2]{Hytoenen2016}.
\end{enumerate} \end{proof}
\Cref{lem:Nov252021204} establishes bounds on the gradient of $\pobj$. \begin{lemma}
\label{lem:Nov252021204}
If \Cref{assumption:pde} holds, then
$\pobj(\cdot,\xi)$ is twice continuously differentiable
for all $\xi \in \Xi$.
For all $(u,\xi) \in L^2(D) \times \Xi$,
$\nabla_u \pobj(u,\xi) = -g(\xi)z(u,\xi)$ and
\begin{align}
\label{eq:Dec2820211103'}
\norm[H_0^1(D)]{\nabla_u \pobj(u,\xi)}
& \leq
(C_D+1)(C_D/\kappa_{\min})
\norm[C^{0,1}(\bar D)]{g(\xi)}
\big(
(C_D/\kappa_{\min})\norm[H^{-1}(D)]{b(\xi)}
+
\\
\nonumber
&\quad \quad
( C_D^2/\kappa_{\min})
\norm[C^{0,1}(\bar{D})]{g(\xi)} \norm[L^2(D)]{u}
+
\norm[L^2(D)]{y_d}\big),
\\
\label{eq:Nov252021210}
\norm[L^2(D)]{\nabla_u \pobj(u,\xi)}
& \leq
(C_D^2/\kappa_{\min})\norm[C^{0,1}(\bar D)]{g(\xi)}
\big(C_D \norm[H_0^1(D)]{S(u,\xi)} + \norm[L^2(D)]{y_d}\big).
\end{align} \end{lemma}
\begin{proof}
Adapting the proof of \cite[Lem.\ 9.8]{Ulbrich2011},
we obtain that $\pobj(\cdot, \xi)$
is twice continuously differentiable on $L^2(D)$ with
$\nabla_u \pobj(u,\xi) = - g(\xi) z(u,\xi)$
for each $(u,\xi) \in L^2(D)$.
Combined with \eqref{eq:Feb0720211501}
and \eqref{eq:23Nov20211913}, we obtain
\begin{align*}
\norm[H_0^1(D)]{\nabla_u \pobj(u,\xi)}
\leq
(C_D+1)(C_D/\kappa_{\min})
\norm[C^{0,1}(\bar D)]{g(\xi)}
\norm[L^2(D)]{\iota S(u,\xi)-y_d},
\end{align*}
Together with \eqref{eq:Feb0420210958},
we obtain \eqref{eq:Dec2820211103'}.
Using \eqref{eq:Feb0720211501} and Friedrichs' inequality,
we have
\begin{align*}
\norm[L^2(D)]{\nabla_u \pobj(u,\xi)}
\leq \norm[L^2(D)]{g(\xi) z(u,\xi)}
\leq \tfrac{C_D^2}{\kappa_{\min}}
\norm[C^{0,1}(\bar D)]{g(\xi)}
\norm[L^2(D)]{\iota S(u,\xi)-y_d}.
\end{align*}
Hence, we obtain \eqref{eq:Nov252021210}. \end{proof}
\begin{proof}[{Proof of \Cref{prop:efunsdifferentiable}}]
Combining the stability estimates
\eqref{eq:Feb0420210958}
and \eqref{eq:Dec2820211103'},
and
Friedrichs' inequality,
we can apply \cite[Lem.\ C.3]{Geiersbach2020} to deduce
the Fr\'echet\ differentiability of
$F$ and of $\hat{F}_N$
and the formulas in \eqref{eq:Feb0820212103}.
Owing to \eqref{eq:Feb0820212103} and
\Cref{lem:Feb0720211501}, we find
that $\nabla F(u)$, $\nabla \hat{F}_N(u) \in H_0^1(D)$. \end{proof}
\subsection{Lipschitz constant computation} \label{subsec:lipschitz} We compute a deterministic Lipschitz constant of the gradient $\nabla_u \pobj(\cdot,\xi)$ on $\csp_\text{ad}$ for $\xi \in \Xi$. \begin{proposition}
\label{prop:efunsdifferentiable'}
If \Cref{assumption:pde,itm:adcspbounded} hold,
then for each $\xi \in \Xi$,
the map
$\nabla_u \pobj(\cdot,\xi)$
is Lipschitz continuous
on $\csp_\text{ad}$ with Lipschitz constant $L_{\nabla \pobj}$ given by
\begin{align}
\nonumber
L_{\nabla \pobj}
& =
C_D g_{\max}
\bigg(
(C_D^3/\kappa_{\min}^2)
g_{\max}
\\
\nonumber
&\quad +
\bigg[ (C_D/\kappa_{\min}^2)c_p^3
g_{\max}
\Big(
c_{q} |D|^{(p-3)/p}
\\
\label{eq:nablarpobjLipschitz}
&\quad\quad +
d_{q} c_p^{p-3}
\big(
(1/\kappa_{\min})b_{\max}
+
3(C_D/\kappa_{\min})g_{\max}
r_{\text{ad}}
\big)^{p-3}
\Big)
\\
\nonumber
&\quad\quad\quad\cdot
\Big(
(C_D^2/\kappa_{\min}^2)b_{\max}
+ (C_D^3/\kappa_{\min}^2)g_{\max}
r_{\text{ad}}
+ (C_D/\kappa_{\min})\norm[L^2(D)]{y_d}
\Big)
\bigg]
\bigg).
\end{align} \end{proposition}
\Cref{prop:efunsdifferentiable'} is established using \Cref{lem:lem41}. \begin{lemma}[{see \cite[Lem.\ 5.2]{Garreis2019} and \cite[Lem.\ 4.1]{Garries2019c}}]
\label{lem:lem41}
If \Cref{itm:domain,itm:nonlinearity} hold,
then for all $y_1$, $y_2 \in H_0^1(D)$,
\begin{align*}
\begin{aligned}
&\norm[\spL{H_0^1(D)}{H^{-1}(D)}]
{Q_y(y_2)-Q_y(y_1)}
\\
& \leq c_p^3\Big(
c_{q} |D|^{(p-3)/p} +
d_{q} c_p^{p-3}
\big(
\norm[H_0^1(D)]{y_2} + \norm[H_0^1(D)]{y_2-y_1}
\big)^{p-3}
\Big)
\norm[H_0^1(D)]{y_2-y_1}.
\end{aligned}
\end{align*} \end{lemma}
\begin{proof}[{Proof of \Cref{prop:efunsdifferentiable'}}]
Fix $(u_1,u_2,\xi) \in L^2(D)^2 \times \Xi$.
First, we show that
\begin{align}
\label{eq:Nov2520211221}
\begin{aligned}
& \norm[H_0^1(D)]{z(u_2,\xi)-z(u_1,\xi)}
\leq
(C_D^3/\kappa_{\min}^2)
\norm[C^{0,1}(\bar{D})]{g(\xi)}
\norm[L^2(D)]{u_2-u_1}
\\
&\quad +
\bigg[(C_D/\kappa_{\min}^2)c_p^3
\norm[C^{0,1}(\bar{D})]{g(\xi)}
\Big(
c_{q} |D|^{(p-3)/p}
\\
&\quad\quad +
d_{q} c_p^{p-3}
\big(
\norm[H_0^1(D)]{S(u_2,\xi)} +
\norm[H_0^1(D)]{S(u_2,\xi)-S(u_1,\xi)}
\big)^{p-3}
\Big)
\\
&\quad\quad\quad\cdot
\norm[H_0^1(D)]{z(u_1,\xi)}
\bigg] \norm[L^2(D)]{u_2-u_1}.
\end{aligned}
\end{align}
Using arguments similar to those
used to derive the stability estimate in the proof of
\cite[Prop.\ 4.4]{Kouri2020}, the definition
of $\pobj$ (see \eqref{eq:rpobj}), and the triangle inequality, we have
\begin{align}
\nonumber
\kappa_{\min}\norm[H_0^1(D)]{z(u_2,\xi)-z(u_1,\xi)}
&\leq
\norm[H^{-1}(D)]{\iota^*\iota[S(u_2,\xi)-S(u_1,\xi)]}
\\
\label{eq:adjointlipschitz}
&\quad +
\norm[H^{-1}(D)]{[Q_y(S(u_2,\xi))
-Q_y(S(u_1,\xi))]^*z(u_1,\xi)}.
\end{align}
Using \eqref{eq:Nov2520211145}, we have
\begin{align}
\label{eq:Dec1520211631}
\begin{aligned}
\norm[H^{-1}(D)]{\iota^*\iota[S(u_2,\xi)-S(u_1,\xi)]}
&\leq C_D^2
\norm[H_0^1(D)]{S(u_2,\xi)-S(u_1,\xi)}
\\
&\leq (C_D^3/\kappa_{\min})
\norm[C^{0,1}(\bar{D})]{g(\xi)}
\norm[L^2(D)]{u_2-u_1}.
\end{aligned}
\end{align}
To derive a bound on the second term in
\eqref{eq:adjointlipschitz}, we apply \Cref{lem:lem41}
with $y_2 = S(u_2,\xi)$ and $y_1 = S(u_1,\xi)$.
Using the resulting estimate and \eqref{eq:Nov2520211145},
and dividing \eqref{eq:adjointlipschitz} and
\eqref{eq:Dec1520211631} by $\kappa_{\min}$,
we obtain \eqref{eq:Nov2520211221}.
Next, we show that $L_{\nabla \pobj}$ defined in
\eqref{eq:nablarpobjLipschitz} is a Lipschitz constant
of $\nabla_u \pobj(\cdot,\xi)$ on $\csp_\text{ad}$.
\Cref{lem:Nov252021204} ensures
\begin{align*}
\nabla_u \pobj(u_2,\xi)-\nabla_u\pobj(u_1,\xi)
= -g(\xi)\big(z(u_2,\xi)-z(u_1,\xi)\big).
\end{align*}
Combining \Cref{lem:Nov252021204,lem:s_unique_stable},
\Cref{itm:adcspbounded},
Friedrichs' inequality, and \eqref{eq:Nov2520211221},
we obtain the Lipschitz bound. \end{proof}
\subsection{Existence of solutions} \label{subsect:existence} \Cref{lem:existencesolutions} establishes the existence of solutions to the stochastic program \eqref{eq:ocp} and to the SAA problem \eqref{eq:saa}. \begin{proposition}
\label{lem:existencesolutions}
If \Cref{itm:psi,assumption:pde} hold, then
the stochastic program \eqref{eq:ocp} has a solution
and for each $N \in \mathbb{N}$, the SAA problem
\eqref{eq:saa} has a solution. \end{proposition} \begin{proof}
Using \Cref{lem:properties_B}, \eqref{eq:Feb0420210958}, \eqref{eq:eq37},
\eqref{eq:rpobj}, and \cite[Lem.\ 2.33]{Bonnans2013}, the assertions can be
verified using standard arguments. We omit the details. \end{proof}
\section{Compact subset of feasible set} \label{sec:compatset} We construct a deterministic, compact set containing the SAA critical points using first-order necessary optimality conditions and the stability estimates established in \cref{sec:rnpdeopt}. The compact subset is used to derive finite sample size estimates in \cref{sec:samplesizeestimates}. While the computations performed in this section mainly use the stability estimate \eqref{eq:Dec2820211103'}, the compact set's construction provides an integral step towards establishing our nonasymptotic sample size estimates via the covering number approach. In particular, we derive an explicit bound on the compact set's diameter.
We define the set of SAA critical points $\hat{\mathscr{C}}_N$ by \begin{align*} \hat{\mathscr{C}}_N &= \{\, u_N \in \csp_\text{ad} \colon \, u_N = \prox{\psi/\alpha}{-(1/\alpha)\nabla \hat{F}_N(u_N)} \,\}. \end{align*} Let \Cref{ass:adcspbounded,assumption:pde} hold true. Let $u_N^* \in \csp_\text{ad}$ be a local solution to the SAA problem \eqref{eq:saa}. Since $\alpha > 0$ and $\hat{F}_N$ is Fr\'echet\ differentiable according to \Cref{prop:efunsdifferentiable}, we have $u_N^* \in \hat{\mathscr{C}}_N$ (cf.\ \cite[p.\ 2092]{Mannel2020} and \cite[Thm.\ 1.46]{Hinze2009}). The set $\hat{\mathscr{C}}_N$ can be viewed as a set-valued mapping from $\Omega$ to $L^2(D)$. We also define the set $\hat{\mathscr{D}}_N$ by \begin{align} \label{eq:DN} \hat{\mathscr{D}}_N &= \{\, v_N \in L^2(D) \colon \, v_N = -(1/\alpha)\nabla \hat{F}_N(\prox{\psi/\alpha}{v_N}) \,\}. \end{align} As with $\hat{\mathscr{C}}_N$,
the set $\hat{\mathscr{D}}_N$ can be viewed as a set-valued mapping from
$\Omega$ to $L^2(D)$. We have the relationships (cf.\ \cite[p.\ 2092]{Mannel2020}) \begin{align} \label{eq:normalcritical} \hat{\mathscr{C}}_N = \{\, \prox{\psi/\alpha}{v_N} \colon v_N \in \hat{\mathscr{D}}_N \, \} = \prox{\psi/\alpha}{\hat{\mathscr{D}}_N}. \end{align} We define the problem-dependent parameters \begin{align} \label{eq:parameters} \begin{aligned} \mathfrak{D}^{\mathscr{D}} & = (C_D/\kappa_{\min}) b_{\max} + (C_D^2/\kappa_{\min}) g_{\max} r_{\text{ad}} + \norm[L^2(D)]{y_d}, \\ R_{\text{ad}}^{\mathscr{D}} & = (C_D+1)^2 (C_D/\kappa_{\min})g_{\max} \mathfrak{D}^{\mathscr{D}}. \end{aligned} \end{align} \Cref{prop:cinwadcsp} demonstrates that $\hat{\mathscr{D}}_N$ is contained in the set \begin{align} \label{eq:Feb0420211009} \begin{aligned} V_{\text{ad}}^{\mathscr{D}} = \{\, & u \in H^1(D) :\, \norm[H^1(D)]{u} \leq (1/\alpha)R_{\text{ad}}^{\mathscr{D}} \,\}. \end{aligned} \end{align} The set $V_{\text{ad}}^{\mathscr{D}}$ is a compact subset of $L^2(D)$, as $\alpha > 0$, $R_{\text{ad}}^{\mathscr{D}}$ is finite and the embedding operator of the embedding $H^1(D) \xhookrightarrow{} L^2(D)$ is compact. \begin{proposition}
\label{prop:cinwadcsp}
If \Cref{ass:adcspbounded,assumption:pde} hold,
then for each $N \in \mathbb{N}$, it holds that $\hat{\mathscr{D}}_N\subset
V_{\text{ad}}^{\mathscr{D}}$,
where $\hat{\mathscr{D}}_N$ and $V_{\text{ad}}^{\mathscr{D}}$
are defined in \eqref{eq:DN} and \eqref{eq:Feb0420211009}, respectively. \end{proposition} \begin{proof}
Fix $(u,\xi) \in \csp_\text{ad} \times \Xi$.
Using \eqref{eq:Dec2820211103'},
$\norm[H^1(D)]{y} \leq
(C_D+1)\norm[H_0^1(D)]{y}$
being valid for all $y \in H_0^1(D)$, and
the definition of $R_{\text{ad}}^{\mathscr{D}}$, we find that
\begin{align}
\label{eq:Dec1020211843}
\norm[H^1(D)]{\nabla_u \pobj(u,\xi)}
&\leq R_{\text{ad}}^{\mathscr{D}}.
\end{align}
Fix $N \in \mathbb{N}$ and
$v_N \in \hat{\mathscr{D}}_N$.
We have
$v_N = -(1/\alpha)\nabla \hat{F}_N(\prox{\psi/\alpha}{v_N})$.
Defining $u_N = \prox{\psi/\alpha}{v_N}$,
we have $u_N \in \csp_\text{ad}$. Choosing $u = u_N$
in \eqref{eq:Dec1020211843}, we obtain
$v_N \in V_{\text{ad}}^{\mathscr{D}}$. \end{proof}
\Cref{prop:cinwadcsp} and \eqref{eq:normalcritical} imply that for each $N \in \mathbb{N}$, the set $\hat{\mathscr{C}}_N$ is contained in $\prox{\psi/\alpha}{V_{\text{ad}}^{\mathscr{D}}}$. This set is compact, as it is the image of the compact set $V_{\text{ad}}^{\mathscr{D}}$ under
$\prox{\psi/\alpha}{}$.
\section{Quantitative Sobolev embeddings} \label{sec:covering}
The Sobolev embedding $H^1(D) \xhookrightarrow{} L^2(D)$ is compact, provided that $D \subset \mathbb{R}^d$ is a bounded Lipschitz domain. The authors of \cite{Birman1967,Birman1980} establish covering numbers of closed unit balls in Sobolev spaces with respect to Lebesgue norms. \Cref{thm:birman1967}, which is an excerpt of \cite[Thm.\ 1.7]{Birman1980}, is the key result for establishing the nonasymptotic sample size estimates in \cref{sec:samplesizeestimates}.
\begin{theorem}[{see \cite[Thm.\ 1.7]{Birman1980}}]
\label{thm:birman1967}
Let $s > 0$.
The binary logarithm of the $\nu$-covering number
of the closed $H^s((0,1)^d)$-unit ball
with respect to the $L^2((0,1)^d)$-norm
is proportional to $(1/\nu)^{d/s}$
for all (sufficiently small) $\nu > 0$. \end{theorem}
Note that in \cite[p.\ 2]{Birman1980} the definition $W^{k,p}([0,1)^d) = W^{k,p}((0,1)^d)$ is made. \Cref{thm:birman1967} implies the existence of a constant $\varrho > 0$ such that for all $\nu > 0$, \begin{align} \label{eq:eq5118} \mathcal{N}(\nu; B_{H^1((0,1)^d)}, \norm[L^2((0,1)^d)]{\cdot}) \leq 2^{\varrho(1/\nu)^d}, \end{align} where $B_{H^1((0,1)^d)}$ is the closed $H^1((0,1)^d)$-unit ball. The upper bound in \Cref{thm:birman1967} is established in \cite[Thm.\ 5.2]{Birman1967} for the $H^s((0,1)^d)$-unit sphere rather than for the closed $H^s((0,1)^d)$-unit ball. We refer the reader to \cite[sect.\ 1.3.12]{Novak1988} and \cite[pp.\ 118 and 151]{Edmunds1996} for related quantitative Sobolev embedding statements.
\section{Nonasymptotic sample size estimates} \label{sec:samplesizeestimates} We establish sample size estimates for SAA critical points. We define the normal map $\phi : L^2(D) \to L^2(D)$ by \begin{align} \label{eq:phi} \phi(v) = \nabla F(\prox{\psi/\alpha}{v}) + \alpha v. \end{align} If $v^* \in L^2(D)$ satisfies $\phi(v^*) = 0$, then $u^* = \prox{\psi/\alpha}{v^*}$ is a critical point of the stochastic program \eqref{eq:ocp} \cite[p.\ 2092]{Mannel2020}.
\begin{theorem}
\label{thm:samplesize}
Let \Cref{ass:adcspbounded,assumption:pde} hold and let $D = (0,1)^d$.
Let $\varepsilon > 0$.
If $\bar v_N$ is a measurable selection of
$\hat{\mathscr{D}}_N$
defined in \eqref{eq:DN} and
\begin{align}
\label{eq:expectationboundN}
N \geq
\frac{12\ln(2)\tau_{\mathscr{D}}^2}{\varepsilon^2}
\bigg[\varrho
\Big(\frac{4\max\{L_{\nabla \pobj},1\}R_{\text{ad}}^{\mathscr{D}}}
{\alpha\varepsilon}\Big)^{d} + 1\bigg],
\end{align}
then
\begin{align}
\label{eq:expectationbound}
\cE{\norm[L^2(D)]{\phi(\bar{v}_N)}}
\leq \varepsilon,
\end{align}
where
$\mathfrak{D}^{\mathscr{D}}$ and $R_{\text{ad}}^{\mathscr{D}}$ are
defined in \eqref{eq:parameters},
$\varrho > 0$ is specified in \eqref{eq:eq5118},
the Lipschitz constant $L_{\nabla \pobj}$ is given in
\eqref{eq:nablarpobjLipschitz}, and
\begin{align*}
\tau_{\mathscr{D}} & = 2
(C_D^2/\kappa_{\min})g_{\max}
\mathfrak{D}^{\mathscr{D}}.
\end{align*}
If $\delta \in (0,1)$ and
\begin{align}
\label{eq:tailboundN}
N \geq
\frac{48\tau_{\mathscr{D}}^2}{\varepsilon^2}
\bigg[\ln(2)\varrho\Big(\frac{4L_{\nabla \pobj}R_{\text{ad}}^{\mathscr{D}}}
{\alpha\varepsilon}\Big)^{d}
+ \ln\Big(\frac{2}{\delta}\Big)\bigg],
\end{align}
then
\begin{align}
\label{eq:tailbound}
\Prob{\hat{\mathscr{D}}_N
\subset \mathscr{D}^\varepsilon}
\geq 1-\delta,
\end{align}
where
$\hat{\mathscr{D}}_N $ is defined in \eqref{eq:DN},
$\phi$ in \eqref{eq:phi}, and
$\mathscr{D}^\varepsilon
= \{\, v \in L^2(D) \colon \,
\norm[L^2(D)]{\phi(v)} \leq \varepsilon \,\}
$. \end{theorem}
Before establishing \Cref{thm:samplesize}, we briefly comment on the sample size estimates \eqref{eq:expectationboundN} and \eqref{eq:tailboundN}, and address measurability issues in \Cref{lem:missues}. The first addends in the sample size estimates \eqref{eq:expectationboundN} and \eqref{eq:tailboundN} are very similar. Our proofs of \Cref{thm:samplesize,prop:uniformboundsoperator} show that the Lipschitz constant $L_{\nabla \pobj}$ (see \Cref{prop:efunsdifferentiable'}) in \eqref{eq:expectationboundN} may be replaced by the expected value of an integrable Lipschitz constant of $\nabla_u \pobj(\cdot,\xi)$ on $\csp_\text{ad}$. While this approach may result in a potentially less conservative sample size estimate than \eqref{eq:expectationboundN}, we choose to use $L_{\nabla \pobj}$ in \eqref{eq:expectationboundN}.
\begin{lemma}
\label{lem:missues}
If \Cref{ass:adcspbounded,assumption:pde} hold, then
the set-valued mapping
$\hat{\mathscr{D}}_N$ is measurable
with closed, nonempty images and
$\{\, \omega \in \Omega \colon \,
\hat{\mathscr{D}}_N(\omega) \subset \mathscr{D}^\varepsilon\,
\}$
is measurable. \end{lemma} \begin{proof}
Since $\nabla \hat{F}_N$ is a Carath\'eodory\ function
(see \Cref{lem:Feb0720211501,lem:Nov252021204}),
$\csp_\text{ad}$ is nonempty and closed,
and $\prox{\psi/\alpha}{}$ is firmly nonexpansive
\cite[Prop.\ 12.28]{Bauschke2011},
the set-valued mapping $\hat{\mathscr{D}}_N$ is
closed-valued and measurable
\cite[Thm.\ 8.2.9]{Aubin2009}.
\Cref{prop:efunsdifferentiable} ensures
that $\nabla_u \pobj(\cdot, \xi)$ is Lipschitz continuous
with a Lipschitz constant
independent of $\xi \in \Xi$.
Hence, $\nabla F$ is (Lipschitz) continuous,
yielding the closeness of
$\mathscr{D}^\varepsilon$.
\Cref{lem:existencesolutions} implies that
$\mathscr{D}^\varepsilon$ and
$\hat{\mathscr{D}}_N $ are nonempty.
Now, \Cref{lem:inclusionmeasurable}
ensures the measurability of the event
$\{\, \hat{\mathscr{D}}_N \subset \mathscr{D}^\varepsilon \, \}$. \end{proof}
\begin{proof}[{Proof of \Cref{thm:samplesize}}]
\Cref{lem:missues} ensures that
$\{\, \hat{\mathscr{D}}_N \subset \mathscr{D}^\varepsilon\,
\}$
is measurable.
To establish the expectation bound \eqref{eq:expectationbound}
and the tail bound \eqref{eq:tailbound}, we apply
\Cref{prop:uniformboundsoperator}
with $V_{\text{ad}} = V_{\text{ad}}^{\mathscr{D}}$,
$H = L^2(D)$,
and $G = \nabla_u \pobj$,
where $V_{\text{ad}}^{\mathscr{D}}$ is defined in \eqref{eq:Feb0420211009}.
We verify the hypotheses of
\Cref{prop:uniformboundsoperator}.
\Cref{assumption:basicerrorestimate1} is implied by
\Cref{lem:Nov252021204,lem:Feb0720211501}.
Using \eqref{eq:nablarpobjLipschitz}
and \Cref{prop:efunsdifferentiable}, we find that
$\nabla_u \pobj(\cdot,\xi)$ is Lipschitz continuous
with Lipschitz constant $L_{\nabla \pobj}$ for all $\xi \in \Xi$.
Hence,
\Cref{assumption:basicerrorestimate2,assumption:basicerrorestimate6} hold
true.
We verify \Cref{assumption:basicerrorestimate3}.
By construction, the set $V_{\text{ad}}^{\mathscr{D}}$
is an $H^1(D)$-ball about zero with radius
$(1/\alpha)R_{\text{ad}}^{\mathscr{D}}$.
Therefore, \Cref{thm:birman1967} yields
\begin{align}
\label{eq:coveringvadcsp}
\mathcal{N}(\nu; V_{\text{ad}}^{\mathscr{D}}, \norm[L^2(D)]{\cdot})
\leq 2^{\varrho(R_{\text{ad}}^{\mathscr{D}}/(\alpha\nu))^d}
\quad \text{for all} \quad \nu > 0,
\end{align}
where $\varrho > 0$ is specified in \eqref{eq:eq5118}.
Hence, \Cref{assumption:basicerrorestimate3} holds true.
We verify \Cref{assumption:basicerrorestimate4}.
Fix $(u,\xi) \in \csp_\text{ad} \times \Xi$.
Using \eqref{eq:Feb0420210958} and
\eqref{eq:Nov252021210}, we obtain
$\norm[L^2(D)]{\nabla_u \pobj(u,\xi)}\leq \tau_{\mathscr{D}}/2$.
Thus,
$\norm[L^2(D)]{\nabla_u \pobj(u,\xi)-\cE{\nabla_u \pobj(u,\xi)}}
\leq \tau_{\mathscr{D}}$.
Combined with
part \ref{itm:boundedsubgaussian_essentiallybounded} in
\Cref{lem:boundedsubgaussian}, we find that
\Cref{assumption:basicerrorestimate4} holds true with
$\tau_G = \tau_{\mathscr{D}}$.
Finally, \Cref{prop:cinwadcsp} ensures
that for each $N \in \mathbb{N}$, $\hat{\mathscr{D}}_N \subset
V_{\text{ad}}^{\mathscr{D}}$.
Hence, we can use the estimate
established in \Cref{rem:basicerrorestimate5}.
To derive the expectation bound \eqref{eq:expectationbound},
we use \eqref{eq:coveringvadcsp}
and \eqref{eq:expsup}.
We choose
$\nu = \varepsilon/(4\max\{L_{\nabla \pobj},1\})$,
yielding $2L_{\nabla \pobj} \nu \leq \varepsilon/2$.
Using \eqref{eq:expsup}, \eqref{eq:coveringvadcsp}, and
the estimate established in \Cref{rem:basicerrorestimate5}, we
have
\begin{align}
\label{eq:expecationnormalmap}
\cE{\norm[L^2(D)]{\phi(\bar{v}_N)}}
\leq \tfrac{\varepsilon}{2}
+
\tfrac{\sqrt{3}\tau_{\mathscr{D}}}{ \sqrt{N}}
\sqrt{\ln(2)\big(\varrho(4\max\{L_{\nabla \pobj},1\}
R_{\text{ad}}^{\mathscr{D}}/(\alpha\varepsilon))^d+1\big)}.
\end{align}
Requiring the right-hand side be less than or equal to
$\varepsilon$ and solving
for $N$ yields \eqref{eq:expectationbound}.
It remains to show \eqref{eq:tailbound}.
Using \eqref{eq:coveringvadcsp}, the tail bound \eqref{eq:probsup'},
and \Cref{rem:basicerrorestimate5}, we have:
if $\bar{v}_N \in \hat{\mathscr{D}}_N$, then
$\phi(\bar{v}_N) < \varepsilon$
and hence $\bar{v}_N \in \mathscr{D}^\varepsilon$
with a probability of at least
$$1-2 \cdot 2^{\varrho(4L_{\nabla \pobj}
R_{\text{ad}}^{\mathscr{D}}/(\alpha\varepsilon))^d}
\ensuremath{\mathrm{e}}^{-N\varepsilon^2/(48\tau_{\mathscr{D}}^2)}.
$$
Requiring this term be greater or equal to $1-\delta$
and solving for $N$, we obtain \eqref{eq:tailbound}. \end{proof}
\Cref{thm:samplesize} establishes sample size estimates
in terms of the normal map \eqref{eq:phi}.
Next we derive sample size estimates for another popular
criticality measure, which
can be used within termination criteria for numerical methods,
for example. In \cref{sect:numillus}, we provide numerical illustrations
using this criticality measure. We define the criticality measure $\chi : L^2(D) \to \mathbb{R}$ for \eqref{eq:ocp} by \begin{align}
\label{eq:cmeasure}
\chi(u) =
\norm[L^2(D)]{u- \prox{\psi/\alpha}{-(1/\alpha)\nabla F(u)}} \end{align} and the set of $\varepsilon$-critical points of \eqref{eq:ocp} by $\mathscr{C}^\varepsilon = \{\, u \in \csp_\text{ad} \colon \, \chi(u) \leq \varepsilon \, \}$ for $\varepsilon \geq 0$. \begin{corollary}
\label{cor:samplesize}
Under the hypotheses of \Cref{thm:samplesize}, it follows that
\emph{(a)}
if $\varepsilon > 0$,
$\bar u_N$ is a measurable selection of $\hat{\mathscr{C}}_N $
and $N$ satisfies \eqref{eq:expectationboundN},
then
$
\cE{\chi(\bar u_N)}
\leq \varepsilon/\alpha
$,
and
\emph{(b)}
if $\varepsilon > 0,$ $\delta \in (0,1)$ and
$N$ fulfills \eqref{eq:tailboundN},
then
$
\Prob{\hat{\mathscr{C}}_N
\subset \mathscr{C}^{(\varepsilon/\alpha)}}
\geq 1-\delta
$. \end{corollary} \begin{proof}
The measurability of $\{\, \hat{\mathscr{C}}_N
\subset \mathscr{C}^{(\varepsilon/\alpha)} \,\}$
can be established using arguments similar to those in the proof of
\Cref{lem:missues}. Fix $v \in L^2(D)$ and define $u =
\prox{\psi/\alpha}{v}$.
Since the proximity operator is firmly nonexpansive
\cite[Prop.\ 12.28]{Bauschke2011}, we have
\begin{align}
\label{eq:chiphi}
\begin{aligned}
\chi(u)
& =
\norm[L^2(D)]{\prox{\psi/\alpha}{v}-\prox{\psi/\alpha}
{-(1/\alpha)\nabla F(u)}}
\\
& \leq \norm[L^2(D)]{v+(1/\alpha)\nabla F(u)}
= (1/\alpha)\norm[L^2(D)]{\phi(v)},
\end{aligned}
\end{align}
where $\phi$ is defined in \eqref{eq:phi}.
Since $\bar{u}_N$ is a measurable selection of $\hat{\mathscr{C}}_N$,
\Cref{lem:missues}
and Filippov's theorem \cite[Thm.\ 8.2.10]{Aubin2009}
applied to the identity in \eqref{eq:normalcritical}
ensure the existence
of a measurable selection $\bar{v}_N$ of $\hat{\mathscr{D}}_N$
defined in \eqref{eq:DN} such that
$\bar{u}_N = \prox{\psi/\alpha}{\bar{v}_N}$.
Now \Cref{thm:samplesize} implies the assertions. \end{proof}
\section{Numerical illustrations} \label{sect:numillus} The main purposes of our numerical illustrations are to verify the typical Monte Carlo convergence rate for $\cE{\chi(\bar u_N)}$ where $\bar u_N$ is an SAA critical point and to examine the dependence of our expectation bounds on the parameter $\alpha$. For numerical computations, the infinite dimensionality of \eqref{eq:ocp} and its SAA problem \eqref{eq:saa} necessitate finite dimensional approximations. We also illustrate empirically that the expectation bounds are independent of the dimension of the finite dimensional spaces.
The estimate \eqref{eq:expecationnormalmap}
when combined with \eqref{eq:chiphi} yields
for each $\varepsilon > 0$,
\begin{align}
\label{eq:expectationchi}
\cE{\norm[L^2(D)]{\chi(\bar{u}_N)}}
\leq \tfrac{\varepsilon}{2\alpha}
+
\tfrac{\sqrt{3}\tau_{\mathscr{D}}}{\alpha \sqrt{N}}
\sqrt{\ln(2)\big(\varrho(4\max\{L_{\nabla \pobj},1\}
R_{\text{ad}}^{\mathscr{D}}/(\alpha\varepsilon))^d+1\big)},
\end{align}
where $\chi$ is defined in \eqref{eq:cmeasure}. For fixed
$\alpha > 0$ and $\varepsilon > 0$,
the second term in the right-hand side in \eqref{eq:expectationchi}
decays with the rate
$1/\sqrt{N}$ as a function of $N$. Moreover, for fixed
$N \in \mathbb{N}$ and $\varepsilon > 0$,
the first term increases
with rate $1/\alpha$
and the second term with rate $1/\alpha^{d/2+1}$ as $\alpha$ approaches
zero.
\begin{figure}
\caption{For
$\alpha = 10^{-3}$
and discretization parameter $n=64$,
nominal critical point, that is,
a critical point of \eqref{eq:nom}
\textnormal{(left)}, and a reference critical point
of \eqref{eq:ocp}, that is,
a critical point
of \eqref{eq:saasob} \textnormal{(right)}.}
\label{fig:nomref}
\end{figure}
We consider an instance of \eqref{eq:ocp}. Let $d = 2$ and $D = (0,1)^d$. For $\gamma \geq 0$, let $\psi(u) = \gamma \norm[L^1(D)]{u}$ if $u \in L^2(D)$ with $-10 \leq u \leq 10$ and $\psi(u) = \infty$ otherwise. We define $\Xi = [-1,1]^{100}$, $q(t) = t^3$, and $\dualpHzeroone{A(\xi)y}{v}
= \int_D \kappa(x,\xi) \nabla y(x)^T \nabla v(x) \ensuremath{\mathrm{d}} x $, where $\kappa : \bar{D} \times \Xi \to \mathbb{R}$ is given by $\kappa(x,\xi) =
\ensuremath{\mathrm{e}}^{
\sum_{k=1}^{25} 5/(2k^2) \sin(4k\xi_k \pi x_1)\sin(4k\pi \xi_{25+k}x_2)
} $ if $x_1 \leq 1/2$ and $\kappa(x,\xi) =
3/2 + \sum_{k=1}^{25}|(10/k^2)\xi_{25+k} \cos((10+\xi_{25+k})x_1x_2)| $ otherwise. We further define $y_d(x) = -1$ if $x \in [1/4, 3/4]^2$ and $y_d(x) = 1$ otherwise, $b(x,\xi) = 1+\sum_{k=1}^{25}(5/k^2)\xi_{75+k} x_1x_2\cos(4\pi k x_1)\sin(4\pi k x_2)$ if $x_1 \leq 3/4 +\xi_{76}/2$ and
$b(x,\xi) = 1+|
\sum_{k=1}^{25} (3/k^2)x_2\xi_{75+k} \sin(3\pi x_2)\cos(3\pi(x_1-k\xi_{75+k}x_2))| $ otherwise, and $g(x,\xi) = \max\{1, \sum_{k=1}^{25} (10/k^2) \xi_{50+k} \sin((4+k)\xi_{50+k}x_1x_2) \cos((4+k)\xi_{50+k}x_1x_2) \}$. The random variables $\xi_1, \ldots, \xi_{100}$ are independent $[-1,1]$-uniformly distributed. The random fields $\kappa$, $b$, and $g$ are nonsmooth. Their definitions have been guided by the desire to design an instance of \eqref{eq:Feb0320211603} with random fields that a lack representation as a linear combination of a moderate number of $L^2(D)$-orthogonal basis functions and of functions that are separable with respect to the vector $x $ and the parameters $\xi_1, \ldots, \xi_{100}$. In particular, each of the random fields $\kappa$, $b$, and $g$ lacks a respresentation as a truncated Karhunen--Lo\`eve-type expansion with a small number of addends.
We discretized \eqref{eq:ocp} and \eqref{eq:saa} using a finite element discretization; $H_0^1(D)$ is approximated using piecewise linear continuous finite elements and $L^2(D)$ is discretized using piecewise constant functions defined on a regular triangular mesh with a total of $2n^2$ triangles. Here, $n$ is the number of triangles in each direction in $(0,1)^2$. For a fixed $n \in \mathbb{N}$, the finite dimensional control space is denote by $U_n$. We use the subscript $n$ in $\pobj_n$ and $\nabla \hat{F}_{N,n}$ to denote the approximations of $\pobj$ (see \eqref{eq:rpobj}) and $\nabla \hat{F}_N$, respectively, resulting from the finite dimensional approximations.
\begin{figure}\label{fig:errors}
\end{figure}
\begin{figure}\label{fig:tikhonov}
\end{figure}
The criticality measure $\chi$ in \eqref{eq:cmeasure} involves the gradient $\nabla F(\cdot) = \cE{\nabla_u \pobj(\cdot,\xi)}$. To approximate $\chi$, we generated $N_1 = 2^{13}$ samples using a Sobol' sequence \cite{Joe2008} and transformed the samples to take values in $\Xi$. Let us denote these samples by
$\widetilde{\xi}_1, \ldots, \widetilde{\xi}_{N_1}$. Defining $\widetilde{F}_{N_1,n}(\cdot) = (1/N_1) \sum_{i=1}^{N_1} \pobj_n(\cdot,\widetilde{\xi}_i) $, the criticality measure $\chi$ is approximated by $\widetilde{\chi}_n : L^2(D) \to [0,\infty)$ defined by \begin{align}
\label{eq:chiapprox}
\widetilde{\chi}_{n}(u) =
\norm[L^2(D)]{u- \prox{\psi/\alpha}{-(1/\alpha)
\nabla \widetilde{F}_{N_1,n}(u)}}. \end{align} We also use the samples $\widetilde{\xi}_1, \ldots, \widetilde{\xi}_{N_1}$ to approximate the risk-neutral PDE-constrained optimization problem \eqref{eq:ocp} by the finite dimensional SAA problem \begin{align}
\label{eq:saasob}
\min_{u \in U_n}\,
\frac{1}{N_1} \sum_{i=1}^{N_1}
\pobj_n(u,\widetilde{\xi}_i)
+ (\alpha/2) \norm[L^2(D)]{u}^2
+ \psi(u). \end{align} Finally, we approximate the infinite dimensional SAA problems \eqref{eq:saa} by the finite dimensional SAA problems \begin{align}
\label{eq:saafinitedim}
\min_{u \in U_n}\,
\frac{1}{N} \sum_{i=1}^N\pobj_n(u,\xi^i)
+ (\alpha/2) \norm[L^2(D)]{u}^2
+ \psi(u). \end{align}
Inspired by the choice made in \cite[p.\ 199]{Parikh2014}, we chose $\gamma = 0.2\hat{\gamma}_{\max}$, where $\hat{\gamma}_{\max} = \norm[L^\infty(D)]{\nabla \hat{F}_{10,64}(0)}$. We rounded $0.2\hat{\gamma}_{\max}$ to three significant figures, yielding $\gamma = 7.48\cdot 10^{-3}$. This value is used for all simulations. The computations used to generate the simulation output depicted in \Cref{fig:nomref,fig:errors,fig:dimension2,fig:dimension} were performed on a Linux cluster with 48 CPUs (Intel Xeon CPU E7-8857 v2 3.00GHz) and 1TB of RAM\@. \Cref{fig:tikhonov} is based on output of simulations performed on the PACE Phoenix cluster \cite{PACE2017}. We used \href{http://www.dolfin-adjoint.org/} {\texttt{dolfin-adjoint}} \cite{Mitusch2019,Funke2013} with \href{https://fenicsproject.org/}{\texttt{FEniCs}} \cite{Alnes2015,Logg2012} to evaluate the cost functions and their derivatives. The problems \eqref{eq:saasob} and \eqref{eq:saafinitedim} were solved using a semismooth Newton-CG method applied to a normal map \cite{Mannel2020,Ulbrich2011}. Its implementation is based on that of \href{https://github.com/funsim/moola}{\texttt{Moola}}'s \texttt{NewtonCG}\ method \cite{Nordaas2016}.\footnote{Our computer code,
simulation output, and figures depicting samples
of $\kappa$, $g$, and $b$ are available at \url{https://github.com/milzj/SAA4PDE/tree/semilinear_complexity/simulations/semilinear_complexity}.}
\begin{figure}\label{fig:dimension2}
\end{figure}
For visualization, critical points were interpolated to the discretized state space. For $\alpha = 10^{-3}$ and $n=64$, \Cref{fig:nomref} depicts a nominal critical point, that is, a critical point of the nominal problem \begin{align}
\label{eq:nom}
\min_{u \in U_n}\,
\pobj_n(u,\cE{\xi})+(\alpha/2)\norm[L^2(D)]{u}^2 + \psi(u) \end{align} and a critical point of \eqref{eq:saasob}. Throughout the section, we estimate $\cE{\widetilde{\chi}_n(\bar u_{N,\alpha,n})}$ using $48$ independent realizations of the SAA critical point $\bar u_{N,\alpha,n}$ of \eqref{eq:saafinitedim} and denote the empirical estimate by $\widehat{\mathrm{E}}[\widetilde{\chi}_n(\bar u_{N,\alpha,n})]$.
For $n=64$, \Cref{fig:errors} depicts the empirical estimate of $\cE{\widetilde{\chi}_n(\bar u_{N,\alpha,n})}$ with $\alpha = 10^{-3}$ and shows the typical Monte Carlo convergence rate $1/\sqrt{N}$. \Cref{fig:errors} also depicts the empirical estimate of $\cE{\widetilde{\chi}_n(\bar u_{N,\alpha,n})}$ for various values of $\alpha$, which highlights a dependence on the expected error of SAA critical points on the parameter $\alpha$. To shed some light on the convergence behavior of the empirical estimate of $\cE{\widetilde{\chi}_n(\bar u_{256,\alpha,64})}$ for $\alpha \to 0^+$ as depicted in \Cref{fig:errors}, we interpret \eqref{eq:ocp}, \eqref{eq:saasob}, and \eqref{eq:saafinitedim} as PDE-constrained optimization problems resulting from a Tikhonov regularization \cite[pp.\ 29--37]{Dontchev1993} with regularization parameter $\alpha$ and empirically demonstrate that $\bar u_{256,\alpha,64}$ provide approximate critical points of the optimization problems \begin{align}
\label{eq:saafinitedimzero}
\min_{u \in U_n}\,
\frac{1}{N_1} \sum_{i=1}^{N_1}
\pobj_n(u,\widetilde{\xi}_i)
+ \psi(u)
\quad \text{and} \quad
\min_{u \in U_n}\,
\frac{1}{N} \sum_{i=1}^N\pobj_n(u,\xi^i)
+ \psi(u) \end{align} as $\alpha \to 0^+$. We define $\widetilde{\Psi}_{n}$, $\hat{\Psi}_{N,n} \colon L^2(D) \to [0,\infty)$ by \begin{align}
\label{eq:tikhonovapprox}
\begin{aligned}
\widetilde{\Psi}_{n}(u) & =
\norm[L^2(D)]{u- \prox{\psi}{u-
\nabla \widetilde{F}_{N_1,n}(u)}},
\\
\hat{\Psi}_{N,n}(u) & =
\norm[L^2(D)]{u- \prox{\psi}{u-
\nabla \hat{F}_{N,n}(u)}}.
\end{aligned} \end{align} The function $\widetilde{\Psi}_{n}$ is a criticality measure for the first problem in \eqref{eq:saafinitedimzero} and $\hat{\Psi}_{N,n}$ is one for the second problem in \eqref{eq:saafinitedimzero}; cf., e.g., \cite[sect.\ 2.1.2]{Milzarek2019}. \Cref{fig:tikhonov} depicts the empirical mean of $\widetilde{\Psi}_{64}(\bar u_{256,\alpha,64})$ and $\hat{\Psi}_{256,64}(\bar u_{256,\alpha,64})$ computed using $48$ independent SAA critical points $\bar u_{256,\alpha,64}$ for various values of $\alpha$. This may suggest that the controls $\bar u_{256,\alpha,64}$ provide approximate critical points of the optimization problems in \eqref{eq:saafinitedimzero} as $\alpha \to 0^+$.
\Cref{fig:dimension2} depicts the empirical estimates of $\cE{\widetilde{\chi}_n(\bar u_{N,\alpha,n})}$ over the sample size $N$ for multiple values of the discretization parameter $n$. For fixed sample sizes $N$ and regularization parameters $\alpha$, \Cref{fig:dimension} depicts empirical estimates of $\cE{\widetilde{\chi}_n(\bar u_{N,\alpha,n})}$ for multiple discretization parameters $n$. The simulation output in \Cref{fig:dimension2,fig:dimension} indicates that the expectation $\cE{\widetilde{\chi}_n(\bar u_{N,\alpha,n})}$ may be independent of the discretization parameter $n$ and hence of the dimension $2n^2$ of the finite dimensional control space $U_n$. The theoretical analysis of this empirical observation is beyond the scope of this manuscript and is left as a topic for future research. For risk-neutral linear PDE-constrained optimization, the SAA approach with finite dimensional approximations of the control and state spaces are analyzed in \cite{Hoffhues2020,Martin2021,Milz2022c} with respect to proximity of finite dimensional SAA solutions to the ``true'' optimal control.
\begin{figure}\label{fig:dimension}
\end{figure}
\section{Discussion} \label{sec:discussion} We established nonasymptotic sample size estimates for SAA critical points of risk-neutral semilinear PDE-constrained optimization, a class of infinite dimensional, nonconvex stochastic programs. To derive the sample size estimates, we constructed a compact subset of the feasible set containing all SAA critical points. Using covering numbers of Sobolev function classes, we derived nonasymptotic sample size estimates inspired by the analyses in \cite{Shapiro2003,Shapiro2005,Shapiro2021}. The construction of the compact set exploits structure in PDE-constrained optimization problems. This structure has been used for other purposes in different ways, such as for establishing mesh-independence of semismooth Newton methods \cite{Hintermueller2004}, developing smoothing steps within semismooth Newton methods \cite{Ulbrich2011}, and deriving higher regularity of solutions to deterministic PDE-constrained optimization problems \cite{Casas2012a}.
Risk-neutral semilinear PDE-constrained optimization
is a subfield of infinite dimensional
optimization with dynamical systems under uncertainty.
Our approach to establishing sample size estimates
may be applied to risk-neutral PDE-constrained optimization problems
other than those considered here. Among other things, our derivations
rely on the Lipschitz continuity properties of the reduced parameterized
objective function's gradient and covering numbers of
a set containing all SAA critical points.
These key properties may be verified
for objective functions other than tracking-type functions
\eqref{eq:rpobj} and parameterized operator equations other than
the semilinear PDE \eqref{eq:Feb0320211603}.
Risk-averse PDE-constrained optimization
\cite{Garreis2021,Kouri2018,Kouri2016} provides a more general approach
to optimization of complex systems under uncertainty
than risk-neutral optimization.
For risk-averse semilinear
PDE-constrained optimization using the average/conditional
value-at-risk, our
analysis when combined with considerations
in \cite[sect.\ 4.2]{Lan2012}
may be adapted to constructing a compact set
containing all SAA critical points.
While deriving sample size estimates for critical points is
complicated by the risk-averse objective function's nonsmoothness,
establishing sample size estimates for optimal values
and optimal solutions may be possible using uniform exponential
tail bounds.
Using \Cref{thm:birman1967}, the dependence on the dimension $d$ in the sample size estimates \eqref{eq:expectationboundN} and \eqref{eq:tailboundN} could be improved to $d/2$ if there exists $R_{\text{ad}} > 0$ such that $\nabla_u \pobj(u,\xi) \in H^2(D)$ and $\norm[H^2(D)]{\nabla_u \pobj(u,\xi)}
\leq R_{\text{ad}}$ for all $(u,\xi) \in \csp_\text{ad} \times \Xi$. However, the constant $R_{\text{ad}}^{\mathscr{D}}$ in \eqref{eq:expectationboundN} and in \eqref{eq:tailboundN} must then be replaced by $R_{\text{ad}}$ which could be significantly larger than $R_{\text{ad}}^{\mathscr{D}}$. For a class of linear elliptic PDEs with random inputs, $H^2(D)$-stability estimates are provided in \cite{Teckentrup2013}. These stability estimates could be used to establish $\norm[H^2(D)]{\nabla_u \pobj(u,\xi)} \leq R_{\text{ad}}$ for all $(u,\xi) \in \csp_\text{ad} \times \Xi$, provided that the parameterized elliptic operator $A$ satisfies additional assumptions.
\appendix
\section{Measurability of set inclusions} \label{sec:setinclusion} We establish a measurability statement of set inclusions which is used in \cref{sec:samplesizeestimates}. This statement is essentially known. As introduced in \cref{sec:notation}, $(\Theta, \mathcal{A}, \mu)$ is a complete probability space.
\begin{lemma}
\label{lem:inclusionmeasurable}
Let $X$ be a real, separable Banach space and
let $\Gamma : \Theta \rightrightarrows X$
be a
measurable set-valued mapping with closed, nonempty images.
If $\Upsilon \subset X$ is nonempty and closed, then
$\{\, \theta \in \Theta\colon \, \Gamma(\theta) \subset \Upsilon \, \}$
is measurable. \end{lemma} \begin{proof}
Since $X \setminus \Upsilon$ is open
and $\Gamma$ is measurable,
$\Gamma^{-1}(X \setminus \Upsilon)$ is measurable.
Combined with
$
\{\, \theta \in \Theta\colon \, \Gamma(\theta) \subset \Upsilon \, \}
=
\Theta \setminus \Gamma^{-1}(X \setminus \Upsilon)
$, we find that
$\{\, \theta \in \Theta\colon \, \Gamma(\theta) \subset \Upsilon \, \}$
is measurable. \end{proof}
\section{Sub-Gaussian-type bounds for maxima of random vectors} \label{sec:subgaussianbounds} We establish expectation and exponential tail bounds for pointwise maxima of sub-Gaussian-type Hilbert space-valued random vectors. The techniques used to derive these results are similar to those used to establish expectation and tail bounds for pointwise maxima of sub-Gaussian real-valued random variables. However, we use \cite[Thm.\ 3]{Pinelis1986} which provides expectation bounds for sums of independent, mean-zero Hilbert space-valued random vectors. The results established in this section are used in \cref{sect:uniformtailbounds} to derive uniform expectation and exponential tail bounds.
\begin{proposition}
\label{prop:maxmean}
Let $\tau > 0$, let $H$ be a real, separable Hilbert space and
for each $k \in \{1,2, \ldots, K\}$,
let
$W_{i,k} : \Theta\to H$
($i = 1, 2, \ldots, N$)
be independent, mean-zero random
vectors with
$\cE{\cosh(\lambda \norm[H]{W_{i,k}})}\leq \exp(\lambda^2\tau^2/2)$
for all $\lambda \in \mathbb{R}$
and $i \in \{ 1, 2, \ldots, N \}$.
We define $Z_k^{[N]} = (1/N) \sum_{i=1}^N W_{i,k}$.
Then for each $\varepsilon \geq 0$,
\begin{align}
\label{eq:maxmean}
\cE{\max_{1\leq k \leq K} \norm[H]{Z_k^{[N]} }}
& \leq \sqrt{3/2}\tau (1/\sqrt{N})\sqrt{2\ln(2K)},
\\
\label{eq:maxprob}
\Prob{\max_{1\leq k \leq K} \norm[H]{Z_k^{[N]}} \geq \varepsilon}
& \leq 2K \exp(-\tau^{-2}\varepsilon^2N/3).
\end{align} \end{proposition}
The proof of \Cref{prop:maxmean} is presented at the end of the section. Let $Z : \Theta \to \mathbb{R}$ be a random variable. We provide a motivation for the condition \begin{align} \label{eq:subgaussiantype}
\cE{\cosh(\lambda |Z|)}\leq \exp(\lambda^2\tau^2/2) \quad \text{for all} \quad \lambda \in \mathbb{R}, \end{align} where $\tau > 0$. A random variable $Z : \Theta \to \mathbb{R}$ is sub-Gaussian with parameter $\tau$ if $\tau \in [0, \infty)$ and $\cE{\exp(\lambda Z)} \leq \exp(\lambda^2\tau^2/2)$ for all $\lambda \in \mathbb{R}$ \cite[p.\ 9]{Buldygin2000}. Since $\cosh(x) = (\ensuremath{\mathrm{e}}^{-x}+\ensuremath{\mathrm{e}}^x)/2$ for all $x \in \mathbb{R}$, a sub-Gaussian random variable $Z$ with parameter $\tau > 0$ fulfills \eqref{eq:subgaussiantype}. If $H$ is a real, separable Hilbert space and $W : \Theta \to H $ is a nondegenerate, mean-zero Gaussian random vector, then \eqref{eq:subgaussiantype} holds with $Z = \norm[H]{W}$ and $\tau = \cE{\norm[H]{W}^2}^{1/2}$ \cite[Rem.\ 4]{Pinelis1986}. Suppose that there exists $\sigma > 0$ such that \begin{align} \label{eq:subgaussian}
\cE{\exp( |Z|^2/\sigma^2)} \leq \ensuremath{\mathrm{e}}. \end{align} This condition and its variants are used in the literature on stochastic programming \cite{Nemirovski2009,Shapiro2021,Lan2020} with $Z$ being the norm of a stochastic (sub)gradient, for example. Let us demonstrate that \eqref{eq:subgaussiantype} and \eqref{eq:subgaussian} are essentially equivalent. \begin{lemma}
\label{lem:boundedsubgaussian}
Let $Z : \Theta \to \mathbb{R}$ be a random variable.
\begin{enumerate}[nosep,leftmargin=*]
\item If \eqref{eq:subgaussiantype} holds
with $\tau > 0$, then \eqref{eq:subgaussian}
holds with
$\sigma = (2\tau^2/(1-\ensuremath{\mathrm{e}}^{-2}))^{1/2}$.
\item If \eqref{eq:subgaussian} is satisfied with
$\sigma > 0$, then
\eqref{eq:subgaussiantype}
is fulfilled with $\tau = 2^{1/4}\sigma$.
\item
\label{itm:boundedsubgaussian_essentiallybounded}
If $\tau > 0$ and $|Z| \leq \tau$ w.p.~$1$, then
\eqref{eq:subgaussiantype} is fulfilled.
\end{enumerate} \end{lemma} \begin{proof}
\begin{enumerate}[nosep,leftmargin=*,wide]
\item
We adapt the proof of \cite[Lem.\ 1.6 on p.\ 9]{Buldygin2000}.
We define $ s= 1-\ensuremath{\mathrm{e}}^{-2}$
and $\sigma^2 = 2\tau^2/s$.
We have for all $z \in \mathbb{R}$
(cf.\ \cite[p.\ 9]{Buldygin2000}),
\begin{align*}
\int_{\mathbb{R}}\ensuremath{\mathrm{e}}^{\lambda^2\tau^2(s-1)/(2s)}\ensuremath{\mathrm{d}} \lambda
= \tfrac{1}{\tau} \sqrt{\tfrac{2\pi s}{1-s}}
\quad \text{and} \quad
\int_\mathbb{R} \cosh(\lambda z)\ensuremath{\mathrm{e}}^{-\lambda^2\tau^2/(2s)} \ensuremath{\mathrm{d}} \lambda
= \tfrac{\sqrt{2\pi s}}{\tau}\ensuremath{\mathrm{e}}^{z^2/\sigma^2}.
\end{align*}
Multiplying \eqref{eq:subgaussiantype} by
$\ensuremath{\mathrm{e}}^{-\lambda^2\tau^2/(2s)}$ yields
$\cE{\cosh(\lambda|Z|)\ensuremath{\mathrm{e}}^{-\lambda^2\tau^2/(2s)}}
\leq \ensuremath{\mathrm{e}}^{\lambda^2\tau^2(s-1)/(2s)}$
for all $\lambda \in \mathbb{R}$.
Integrating both sides over $\lambda \in \mathbb{R}$
and using Fubini's theorem, we obtain
$
\tfrac{\sqrt{2\pi s}}{\tau} \cE{\ensuremath{\mathrm{e}}^{|Z|^2/\sigma^2}}
\leq
\tfrac{1}{\tau} \sqrt{\tfrac{2\pi s}{1-s}}
$.
Hence,
$\cE{\ensuremath{\mathrm{e}}^{|Z|^2/\sigma^2}} \leq 1/(1-s)^{1/2} = \ensuremath{\mathrm{e}}$.
\item
The proof is inspired by that of
\cite[Prop.\ 9.81]{Shapiro2021}.
Fix $\lambda \in \mathbb{R}$
with $\lambda^2\sigma^2/2 \leq 1$.
Using Jensen's inequality and
\href{https://tinyurl.com/4xxc2t4e}{$\cosh(x) \leq \exp(x^2/2)$
for all $x \in \mathbb{R}$}
\cite[eq.\ (4.6.6)]{Tropp2015},
\begin{align*}
\cE{\cosh(\lambda|Z|)} \leq
\cE{\ensuremath{\mathrm{e}}^{\lambda^2|Z|^2/2}}
= \cE{\ensuremath{\mathrm{e}}^{\lambda^2\sigma^2|Z|^2/(2\sigma^2)}}
\leq
\cE{\ensuremath{\mathrm{e}}^{|Z|^2/\sigma^2}}^{\lambda^2\sigma^2/2}
\leq \ensuremath{\mathrm{e}}^{\lambda^2\sigma^2/2}.
\end{align*}
Now let $\lambda \in \mathbb{R}$
with $\lambda^2\sigma^2/2 > 1$.
Young's inequality ensures
$2\lambda s \leq \lambda^2\sigma^2/\sqrt{2} + \sqrt{2}s^2/\sigma^2$
for all $s \in \mathbb{R}$. Combined with
$\cosh(x) \leq \exp(x)$ being valid for all $x \geq 0$,
the symmetry of $\cosh$,
$\sqrt{2}\lambda^2\sigma^2/4 > \sqrt{2}/2$,
and Jensen's inequality, we have
\begin{align*}
\cE{\cosh(\lambda|Z|)}
\leq \ensuremath{\mathrm{e}}^{\lambda^2\sigma^2/(2\sqrt{2})}
\cE{\ensuremath{\mathrm{e}}^{ \sqrt{2}|Z|^2/(2\sigma^2)}}
\leq \ensuremath{\mathrm{e}}^{\lambda^2\sigma^2/(2\sqrt{2}) + \sqrt{2}/2}
< \ensuremath{\mathrm{e}}^{\sqrt{2}\lambda^2\sigma^2/2}.
\end{align*}
Putting together the pieces, we obtain the assertion.
\item Since $\cosh(x) \leq \exp(x^2/2)$
for all $x \in \mathbb{R}$ \cite[eq.\ (4.6.6)]{Tropp2015}
and $\cosh$
is a symmetric function, we have
$\cE{\cosh(\lambda|Z|)} \leq
\cosh(\lambda\tau) \leq \exp(\lambda^2\tau^2/2)$
for all $\lambda \in \mathbb{R}$.
\end{enumerate} \end{proof}
\begin{lemma}
\label{prop:saa:2020-11-21T20:25:01.71}
Let $\tau > 0$,
let $H$ be a real, separable Hilbert space, and let
$Z_i : \Theta\to H$
be independent, mean-zero random
vectors such that
$\cE{\cosh(\lambda \norm[H]{Z_i})}\leq \exp(\lambda^2\tau^2/2)$
for all $\lambda \in \mathbb{R}$ ($i = 1, 2, \ldots, N$). Then, for each
$\lambda \in \mathbb{R}$,
$$
\cE{\cosh(\lambda\norm[H]{ Z_1 + \cdots + Z_N})}
\leq \exp(3\lambda^2 \tau^2 N/4).
$$ \end{lemma}
\Cref{prop:saa:2020-11-21T20:25:01.71} is established using \Cref{thm:saa:2020-03-15T00:12:53.514}.
\begin{theorem}
[{see \cite[Thm.\ 3]{Pinelis1986}}]
\label{thm:saa:2020-03-15T00:12:53.514}
Let $H$ be a real, separable Hilbert space.
If $Z_i : \Theta \to H$
$(i=1, \ldots, N)$ are independent, mean-zero
random vectors, then
$$
\cE{\cosh(\lambda\norm[H]{ Z_1 + \cdots + Z_N})}
\leq \prod_{i=1}^N
\cE{\exp(\lambda \norm[H]{Z_i})- \lambda \norm[H]{Z_i}}
\quad \text{for all} \quad \lambda \geq 0.
$$ \end{theorem}
\begin{proof}[{Proof of \Cref{prop:saa:2020-11-21T20:25:01.71}}]
Fix $\lambda \geq 0$. We have \href{https://tinyurl.com/y4xq8tne}
{$\exp(s)-s \leq \cosh(\sqrt{3/2} s )$
for all $s \in \mathbb{R}$}.
Hence,
$
\cE{\exp(\lambda \norm[H]{Z_i}) -\lambda \norm[H]{Z_i}}
\leq \exp(3\lambda^2\tau^2/4)
$.
Combined with \Cref{thm:saa:2020-03-15T00:12:53.514},
we find that
$
\cE{\cosh(\lambda\norm[H]{ Z_1 + \cdots + Z_N})}
\leq
\prod_{i = 1}^N \exp(3\lambda^2 \tau^2/4)
= \exp(3\lambda^2 \tau^2 N/4)
$.
Since $\cosh$ is symmetric, the estimate is valid for all
$\lambda \in \mathbb{R}$. \end{proof}
\Cref{lem:meanmax} establishes an expectation and an exponential
tail bound for pointwise maxima of
sub-Gaussian random variables similar to those found, e.g., in
\cite[Prop.\ 7.29]{Foucart2013} and
\cite[sect.\ 2.5]{Boucheron2013},
but we express sub-Gaussianity via \eqref{eq:subgaussiantype}.
\begin{lemma}
\label{lem:meanmax}
Let $\sigma > 0$.
Suppose that $Z_k : \Theta \to \mathbb{R}$
are random variables
with $\cE{\cosh(\lambda |Z_k|)} \leq \exp(\lambda^2\sigma^2/2)$
for all $\lambda \in \mathbb{R}$ and
$k = 1, 2, \ldots, K$.
Then for all $\varepsilon > 0$,
\begin{align*}
\cE{\max_{1\leq k \leq K} |Z_k|}
\leq \sigma \sqrt{2\ln(2K)}
\quad \text{and} \quad
\Prob{\max_{1\leq k \leq K} |Z_k| \geq \varepsilon}
\leq 2K \ensuremath{\mathrm{e}}^{-\varepsilon^2/(2\sigma^2)}.
\end{align*} \end{lemma} \begin{proof}
The proof of the expectation bound uses standard derivations
based on ``smoothing''
of the pointwise
maximum.
We have $\exp \leq 2\cosh$.
For $\lambda > 0$, we have
\begin{align*}
\cE{\max_{1\leq k \leq K} |Z_k|}
& \leq
(1/\lambda) \ln \Big(\sum_{k=1}^K \cE{\exp(\lambda|Z_k|)} \Big)
\leq
(1/\lambda) \ln \Big(\sum_{k=1}^K 2\cE{\cosh(\lambda|Z_k|)} \Big)
\\
& \leq
(1/\lambda) \ln \big(2K \exp(\lambda^2\sigma^2/2) \big)
=
(1/\lambda) \ln(2K)
+ \lambda^2\sigma^2/2.
\end{align*}
Choosing $\lambda = \sqrt{2\ln(2K)}/\sigma$ yields the
expectation bound.
The union bound
and Markov's inequality ensure for all $\lambda > 0$,
\begin{align*}
\Prob{\max_{1\leq k \leq K} |Z_k| \geq \varepsilon}
&\leq \sum_{k=1}^K
\Prob{|Z_k| \geq \varepsilon}
\leq \ensuremath{\mathrm{e}}^{-\lambda\varepsilon}
\sum_{k=1}^K
\cE{\exp(\lambda |Z_k|)}
\\
& \!\! \leq 2\ensuremath{\mathrm{e}}^{-\lambda\varepsilon}
\sum_{k=1}^K
\cE{\cosh(\lambda |Z_k|)}
\leq 2K \ensuremath{\mathrm{e}}^{-\lambda\varepsilon} \ensuremath{\mathrm{e}}^{\lambda^2\sigma^2/2}
= 2K \ensuremath{\mathrm{e}}^{-\lambda\varepsilon + \lambda^2\sigma^2/2}.
\end{align*}
Minimizing the right-hand side over $\lambda > 0$ yields
the tail bound. \end{proof}
\begin{proof}[{Proof of \Cref{prop:maxmean}}]
We prove \eqref{eq:maxmean} and \eqref{eq:maxprob} using
\Cref{prop:saa:2020-11-21T20:25:01.71,lem:meanmax}.
Fix $\lambda \geq 0$.
\Cref{prop:saa:2020-11-21T20:25:01.71}
and
$\cE{\cosh(\lambda \norm[H]{W_{i,k}})}\leq \exp(\lambda^2\tau^2/2)$
imply
\begin{align*}
\cE{\cosh(\lambda\norm[H]{Z_k^{[N]}})}
\leq \exp(3\lambda^2 \tau^2 /(4N)).
\end{align*}
Defining $\sigma = \tau \sqrt{3/2}/\sqrt{N}$,
we have $\exp(3\lambda^2 \tau^2 /(4N))
= \exp(\lambda^2\sigma^2/2)$.
Now we apply \Cref{lem:meanmax}
to the random variables $\norm[H]{Z_k^{[N]}}$,
yielding \eqref{eq:maxmean} and \eqref{eq:maxprob}. \end{proof}
\section{Uniform exponential tail bounds for SAA normal maps} \label{sect:uniformtailbounds} We derive uniform expectation and exponential tail bounds for an SAA normal map in Hilbert spaces. The derivation of the tail bounds is inspired by those in \cite[Thms.\ 9.84 and 9.86]{Shapiro2021} for real-valued functions. The uniform exponential tail bounds are used in \cref{sec:samplesizeestimates} to derive nonasymptotic sample size estimates.
Throughout the section, $\xi$, $\Xi$, $\xi^1, \xi^2, \ldots$ and $(\Omega, \mathcal{F}, P)$ are as in \cref{sec:intro,sec:rnpdeopt}. Let $H$ be a real, separable Hilbert space and let $\varphi: H \to (-\infty,\infty]$ be proper, convex and lower semicontinuous. We define $H_{\text{ad}} = \{\, u \in H \colon \, \varphi(u) < \infty \, \}$. Let $G : H_{\text{ad}} \times \Xi \to H$ be a function. We define $\hat{G}_N : H_{\text{ad}} \to H$ by $\hat{G}_N(u) = (1/N)\sum_{i=1}^N G(u,\xi^i)$. Let $\alpha > 0$. We further define $\Phi$, $\hat{\Phi}_N : H \to H$ by \begin{align*} \Phi(v) = \cE{G(\prox{\varphi/\alpha}{v},\xi)} \quad \text{and} \quad \hat{\Phi}_N(v) = \hat{G}_N(\prox{\varphi/\alpha}{v}). \end{align*} Since $\xi^1, \xi^2, \ldots$ are defined on $\Omega$, we can view $\hat{\Phi}_N$ as a function on $H \times \Omega$. The mappings $v \mapsto \Phi(v) + \alpha v$ and $v \mapsto \hat{\Phi}_N(v) + \alpha v$ define normal maps \cite{Robinson1992}. When establishing the uniform tail bounds in \Cref{prop:uniformboundsoperator}, we only rely on properties of their difference, that is, on characteristics of $\Phi-\hat{\Phi}_N$. The following assumptions are inspired by those used in \cite[sect.\ 9.2.11]{Shapiro2021}. A random variable $Z : \Theta \to \mathbb{R}$ is sub-exponential with parameters $(\tau, \Lambda)$ if $\tau \in [0, \infty)$, $\Lambda \in (0, \infty]$ and $\cE{\exp(\lambda Z)} \leq \exp(\lambda^2\tau^2/2)$ for all $\lambda \in (-\Lambda, \Lambda)$ \cite[p.\ 19]{Buldygin2000}.
\begin{assumption}[{Uniform exponential tail bounds: Problem data}]
\label{assumption:basicerrorestimate}
\begin{enumthm}[nosep,leftmargin=*]
\item
\label{assumption:basicerrorestimate1}
The mapping $G : H_{\text{ad}} \times \Xi \to H$
is a Carath\'eodory\ function,
and $G(u,\xi)$ is Bochner integrable for each $u \in H_{\text{ad}}$.
\item
\label{assumption:basicerrorestimate2}
For an integrable random variable $M : \Xi \to (0,\infty)$,
\begin{align*}
\norm[H]{G(u_2,\xi)-G(u_1,\xi)}
\leq M(\xi) \norm[H]{u_2-u_1}
\quad \text{for all} \quad u_2, u_1 \in H_{\text{ad}}, \;\; \xi \in \Xi.
\end{align*}
\item
\label{assumption:basicerrorestimate6}
The random variable $M(\xi)-\cE{M(\xi)}$
is sub-exponential with parameters
$(\tau_M, \Lambda_M)$.
\item
\label{assumption:basicerrorestimate4}
There exists $\tau_G > 0$ such that,
for each $u \in H_{\text{ad}}$,
\begin{align}
\label{eq:cEcosh}
\cE{\cosh(\lambda \norm[H]{G(u,\xi)-\cE{G(u,\xi)}})}
\leq \exp(\lambda^2\tau_G^2/2)
\quad \text{for all} \quad \lambda \in \mathbb{R}.
\end{align}
\item
\label{assumption:basicerrorestimate3}
The $\nu$-covering number
of the nonempty, closed set $V_{\text{ad}} \subset H_{\text{ad}}$
is finite for all $\nu > 0$.
\end{enumthm} \end{assumption}
\Cref{assumption:basicerrorestimate3} ensures that $V_{\text{ad}}$ is compact, as it is closed and totally bounded \cite[Lem.\ 8.2-2]{Kreyszig1978}. We define $\hat{M}_N = (1/N) \sum_{i=1}^N M(\xi^i)$. \begin{lemma}
\label{lem:expecationgcontinuous}
If \Cref{assumption:basicerrorestimate1,assumption:basicerrorestimate2}
hold, then $\Phi$
and $\hat{\Phi}_N$
are Lipschitz continuous with Lipschitz constants
$\cE{M(\xi)}$ and $\hat{M}_N$, respectively.
Moreover
$\sup_{v \in V_{\text{ad}}}\, \norm[H]{\hat{\Phi}_N(v)-\Phi(v)}$
is measurable
on $\Omega$. \end{lemma} \begin{proof}
For each $u \in H_{\text{ad}}$, $\cE{G(u,\xi)}$ is well-defined.
Hence,
\Cref{assumption:basicerrorestimate2} ensures
that $\cE{G(\cdot,\xi)}$
is Lipschitz continuous on $H_{\text{ad}}$ with Lipschitz constant
$\cE{M(\xi)}$.
Since $\prox{\varphi/\alpha}{}$
is firmly nonexpansive
and $\prox{\varphi/\alpha}{H} \subset H_{\text{ad}}$,
$\Phi$ is Lipschitz continuous with Lipschitz constant
$\cE{M(\xi)}$. Similarly, we obtain that
$\hat{\Phi}_N$ is Lipschitz continuous
with Lipschitz constant $\hat{M}_N$.
Combined with the fact that
$G$ is a Carath\'eodory\ mapping
on $H_{\text{ad}} \times \Xi$ and that
$\xi^i: \Omega \to \Xi$ ($i=1, 2, \ldots$) are random vectors,
we find that
$\norm[H]{\hat{\Phi}_N(\cdot)-\Phi(\cdot)}$
is a Carath\'eodory\ map
on $H_{\text{ad}} \times \Omega$.
Since $V_{\text{ad}} \subset H_{\text{ad}}$
is nonempty and closed,
$\sup_{v \in V_{\text{ad}}}\, \norm[H]{\hat{\Phi}_N(v)-\Phi(v)}$
is measurable \cite[Lem.\ 2.1]{Hiai1977}. \end{proof} \begin{proposition}
\label{prop:uniformboundsoperator}
Let \Cref{assumption:basicerrorestimate} hold.
Then $\Phi$ is Lipschitz continuous
with Lipschitz constant $L = \cE{M(\xi)}$ and
for all $\nu > 0$,
\begin{align}
\label{eq:expsup}
\cE{\sup_{v \in V_{\text{ad}}}\, \norm[H]{\hat{\Phi}_N(v)-\Phi(v)}}
\leq 2L\nu+
\tfrac{\sqrt{3}\tau_G}{\sqrt{N}}
\sqrt{\ln(2\mathcal{N}(\nu; V_{\text{ad}}, \norm[H]{\cdot}))}.
\end{align}
Define $\ell = L^2/(2\tau_M^2)$
if $L \leq \Lambda_M \tau_M^2$
and $\ell = \Lambda_M L/2$ otherwise.
Then for all $\varepsilon > 0$,
\begin{align*}
\Prob{\sup_{v \in V_{\text{ad}}}\, \norm[H]{\hat{\Phi}_N(v)-\Phi(v)}
\geq \varepsilon}
\leq
\ensuremath{\mathrm{e}}^{-N\ell}
+ 2\mathcal{N}(\varepsilon/(4L); V_{\text{ad}}, \norm[H]{\cdot})
\ensuremath{\mathrm{e}}^{-N\varepsilon^2 /(48\tau_G^2)}.
\end{align*}
If furthermore $M(\xi) \leq L$ for all $\xi \in \Xi$, then
for all $\varepsilon > 0$,
\begin{align}
\label{eq:probsup'}
\Prob{\sup_{v \in V_{\text{ad}}}\, \norm[H]{\hat{\Phi}_N(v)-\Phi(v)}
\geq \varepsilon}
\leq
2\mathcal{N}(\varepsilon/(4L); V_{\text{ad}},\norm[H]{\cdot})
\ensuremath{\mathrm{e}}^{-N\varepsilon^2 /(48\tau_G^2)}.
\end{align} \end{proposition}
Before establishing \Cref{prop:uniformboundsoperator}, we illustrate how it is used in \cref{sec:samplesizeestimates} to derive nonasymptotic sample size estimates. \begin{remark}
\label{rem:basicerrorestimate5}
Let the hypotheses of \Cref{prop:uniformboundsoperator} hold.
Suppose that w.p.~$1$\ for each $N \in \mathbb{N}$,
$\{\, v \in H \colon
\, \hat{\Phi}_N(v) + \alpha v= 0 \, \}$
is contained in $V_{\text{ad}}$.
Let $N \in \mathbb{N}$ and let $v_N \in H$ be a measurable selection
of $\{\, v \in H \colon
\, \hat{\Phi}_N(v) + \alpha v= 0 \, \}$.
Then w.p.~$1$,
\begin{align*}
\norm[H]{\alpha v_N + \Phi(v_N)}
&\leq \norm[H]{\hat{\Phi}_N(v_N)-\Phi(v_N)} +
\norm[H]{\hat{\Phi}_N(v_N)+\alpha v_N}
\\
&\leq \sup_{v \in V_{\text{ad}}}\, \norm[H]{\hat{\Phi}_N(v)-\Phi(v)}.
\end{align*}
The term on the right-hand side can be estimated using
\Cref{prop:uniformboundsoperator}. \end{remark}
We establish \Cref{prop:uniformboundsoperator} using \Cref{lem:subexponential_lipschitz}, which is a direct consequence of \cite[Thm.\ 5.1 on p.\ 26]{Buldygin2000}. \begin{lemma}
\label{lem:subexponential_lipschitz}
Let $W$ be an integrable, nonnegative random variable.
Suppose that $W-\cE{W}$
is sub-exponential with parameters $(\tau, \Lambda)$.
We define $L = \cE{W}$.
Let $\ell = L^2/(2\tau^2)$
if $L \leq \Lambda \tau^2$
and $\ell = \Lambda L/2$ otherwise.
If $W_1, W_2, \ldots$ are independent
and each $W_i : \Theta \to \mathbb{R}$ has the same distribution as
$W$, then for all $N \in \mathbb{N}$,
$\Prob{(1/N)\sum_{i=1}^N W_i \geq 2 L}
\leq \exp(-N\ell)$. \end{lemma} \begin{proof}
Since $W_i-\cE{W}$ are independent
and sub-exponential with parameters $(\tau, \Lambda)$,
we have
(see \cite[Thm.\ 5.1 on p.\ 26]{Buldygin2000}),
\begin{align*}
\Prob[\Big]{\frac{1}{N}\sum_{i=1}^{N} W_i
- \cE{W} \geq \varepsilon}
\leq
\begin{cases}
\exp(-N\varepsilon^2/(2\tau^2)) &
\text{if} \quad 0 < \varepsilon \leq \Lambda \tau^2,\\
\exp(-N\Lambda \varepsilon/2) &
\text{if} \quad \varepsilon > \Lambda\tau^2.
\end{cases}
\end{align*}
Choosing $\varepsilon = \cE{W}$
and using the definition
of $\ell$, we obtain the assertion. \end{proof}
\begin{proof}[{Proof of \Cref{prop:uniformboundsoperator}}]
The proof is inspired by those
of \cite[Thms.\ 9.84 and 9.86]{Shapiro2021}.
The Lipschitz property of $\Phi$ is provided by
\Cref{lem:expecationgcontinuous}.
Fix $\nu > 0$.
We define
$K = \mathcal{N}(\nu; V_{\text{ad}}, \norm[H]{\cdot}) < \infty$.
By assumption, there exist
$v_1, \ldots, v_K \in H$
such that for each $v \in V_{\text{ad}}$, we have
$\norm[H]{v-v_{k(v)}} \leq \nu$,
where $k(v) = \argmin_{1\leq k \leq K}\,
\norm[H]{v-v_k}$.
Furthermore,
$G(u,\xi)-\cE{G(u,\xi)}$ has zero mean
for each $u \in H_{\text{ad}}$.
Using \Cref{lem:expecationgcontinuous}, we find that
\begin{align*}
\begin{aligned}
\norm[H]{\hat{\Phi}_N(v)-\Phi(v)}
&\leq \norm[H]{\hat{\Phi}_N(v)-\hat{\Phi}_N(v_{k(v)})}
+ \norm[H]{\hat{\Phi}_N(v_{k(v)})-\Phi(v_{k(v)})}
\\
& \quad + \norm[H]{\Phi(v_{k(v)})-\Phi(v)}
\\
&\leq \hat{M}_N \norm[H]{v-v_{k(v)}}
+ \norm[H]{\hat{\Phi}_N(v_{k(v)})-\Phi(v_{k(v)})}
+ L\norm[H]{v-v_{k(v)}}
\\
&\leq \hat{M}_N \nu
+ \norm[H]{\hat{\Phi}_N(v_{k(v)})-\Phi(v_{k(v)})}
+ L \nu.
\end{aligned}
\end{align*}
Defining $u_k = \prox{\varphi/\alpha}{v_k}$,
we obtain $u_k \in H_{\text{ad}}$ and
\begin{align*}
\begin{aligned}
\sup_{v \in V_{\text{ad}}}\, \norm[H]{\hat{\Phi}_N(v)-\Phi(v)}
&\leq \hat{M}_N \nu
+ \max_{1\leq k \leq K}\,
\norm[H]{\hat{G}_N(u_k)- \cE{G(u_k,\xi)}}
+ L \nu.
\end{aligned}
\end{align*}
Taking expectations,
using $\cE{\hat{M}_N} = L$
and utilizing \Cref{prop:maxmean}, we obtain
\eqref{eq:expsup}.
We fix $\varepsilon > 0$
and choose $\nu = \varepsilon/(4L)$
as in \cite[p.\ 471]{Shapiro2021}.
\Cref{prop:maxmean} ensures
\begin{align*}
\Prob{\max_{1\leq k \leq K}\,
\norm[H]{\hat{G}_N(u_k)- \cE{G(u_k,\xi)}}
\geq \varepsilon/4
}
\leq 2 K \ensuremath{\mathrm{e}}^{-N\varepsilon^2 /(48\tau_G^2)}.
\end{align*}
\Cref{lem:subexponential_lipschitz}
yields $\Prob{\hat{M}_N \geq 2 L} \leq \exp(-N\ell)$.
If $\hat{M}_N < 2L$, then
$(\hat{M}_N+ L)\nu < 3L\nu$.
Now the union bound implies the first tail bound.
If $M(\xi) \leq L$, then
$\Prob{\hat{M}_N \geq 2 L} = 0$
and we obtain \eqref{eq:probsup'}. \end{proof}
\section*{Acknowledgments} JM thanks Professor Alexander Shapiro for valuable discussions about the SAA approach. We thank the anonymous referees for their helpful comments.
\end{document} | arXiv | {
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\begin{document}
\title{Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas}
\author{Rafael Oliveira\thanks{Department of Computer Science, Princeton University. Research supported by NSF grant CCF-1217416 and by the Sloan fellowship. Email: \texttt{rmo@cs.princeton.edu}.} \and Amir Shpilka\thanks{Department of Computer Science, Tel Aviv University, Tel Aviv, Israel, \texttt{shpilka@post.tau.ac.il}. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575, and from the Israel Science Foundation (grant number 339/10).} \and Ben Lee Volk\thanks{Department of Computer Science, Tel Aviv University, Tel Aviv, Israel, \texttt{benleevolk@gmail.com}. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575.}}
\date{} \maketitle
\begin{abstract}
In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models.
\sloppy For depth-$3$ multilinear formulas, of size $\exp(n^\delta)$, we give a hitting set of size $\exp\left(\tilde{O}\left(n^{2/3 + 2\delta/3} \right) \right)$. This implies a lower bound of $\exp(\tilde{\Omega}(n^{1/2}))$ for depth-$3$ multilinear formulas, for some explicit polynomial.
\sloppy For depth-$4$ multilinear formulas, of size $\exp(n^\delta)$, we give a hitting set of size $\exp\left(\tilde{O}\left(n^{2/3 + 4\delta/3} \right) \right)$. This implies a lower bound of $\exp(\tilde{\Omega}(n^{1/4}))$ for depth-$4$ multilinear formulas, for some explicit polynomial.
A regular formula consists of alternating layers of $+,\times$ gates, where all gates at layer $i$ have the same fan-in. We give a hitting set of size (roughly) $\exp\left(n^{1- \delta} \right)$, for regular depth-$d$ multilinear formulas of size $\exp(n^\delta)$, where $\delta = O(\frac{1}{\sqrt{5}^d})$. This result implies a lower bound of roughly $\exp(\tilde{\Omega}(n^{\frac{1}{\sqrt{5}^d}}))$ for such formulas.
We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.
Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-$4$ formula, and then to a read-once algebraic branching program (from depth-$3$ formulas we go straight to read-once algebraic branching programs).
\end{abstract} \thispagestyle{empty}
\tableofcontents \thispagestyle{empty}
\pagenumbering{arabic}
\section{Introduction}
Arithmetic circuits are the standard model for computing polynomials. Roughly speaking, given a set of variables $X=\{x_1,...,x_n\}$, an arithmetic circuit uses additions and multiplications to compute a polynomial $f$ in the set of variables $X$. An arithmetic formula is an arithmetic circuit whose computation graph is a tree. An arithmetic circuit (or formula) is multilinear if the polynomial computed at each of its gates is multilinear (as a formal polynomial), that is, in each of its monomials the power of every input variable is at most one (see Section~\ref{sec:def} for definition of the models studied in this paper)
Two outstanding open problems in complexity theory are to prove exponential lower bounds on the size of arithmetic circuits, i.e., to prove a lower bound on the number of operations required to compute some polynomial $f$, and to give efficient deterministic polynomial identity testing (PIT for short) algorithms for them. The PIT problem for arithmetic circuits asks the following question: given an arithmetic circuit $\Phi$ computing a polynomial $f$, determine, { \em efficiently and deterministically}, whether ``$f\equiv 0$''. The black-box version of the PIT problem asks to construct a small {\em hitting set}, i.e., a set of evaluation points $\mathcal H$, for which any such non-zero $f$ does not vanish on all the points in $\mathcal H$.
It is known that solving any one of the problems (proving lower bound or deterministic PIT), with appropriate parameters, for small depth (multilinear) formulas, is equivalent to solving it in the general (multilinear) case \cite{ValiantSkyumBR83,AgrawalVinay08,Koiran10,GuptaKKS13,Tavenas13}. It is also known that these two problems are tightly connected and that solving one would imply a solution to the other, both in the general case \cite{HeintzSchnorr80,KabanetsImpagliazzo03,Agrawal05} and in the bounded depth case\footnote{The result of \cite{DSY09} is more restricted than the results for circuits with no depth restrictions.} \cite{DSY09}. We note that in the multilinear case it is only known that hitting sets imply circuit lower bounds but not vice versa.
In this work we study the PIT problem for several models of bounded depth multilinear formulas. Our main results are subexpoential size hitting sets for depth-$3$ and depth-$4$ multilinear formulas of subexponential size and for {\em regular} depth-$d$ multilinear formulas of subexponential size (with construction size deteriorating among the different models). Using the connection between explicit hitting sets and circuit lower bounds we get, as corollaries, subexponential lower bounds for these models.
\subsection{Models for Computing Multilinear Polynomials}\label{sec:def}
An arithmetic circuit $\Phi$ over the field ${\mathbb{F}}$ and over the set of variables $X$ is a directed acyclic graph as follows. Every vertex in $\Phi$ of in-degree $0$ is labelled by either a variable in $X$ or a field element in ${\mathbb{F}}$. Every other vertex in $\Phi$ is labelled by either $\times$ or $+$. An arithmetic circuit is called a formula if it is a directed tree (whose edges are directed from the leaves to the root). The vertices of $\Phi$ are also called gates. Every gate of in-degree $0$ is called an input gate. Every gate of out-degree $0$ is called an output gate. Every gate labelled by $\times$ is called a product gate. Every gate labelled by $+$ is called a sum gate. An arithmetic circuit computes a polynomial in a natural way. An input gate labelled by $y \in {\mathbb{F}} \cup X$ computes the polynomial $y$. A product gate computes the product of the polynomials computed by its children. A sum gate computes the sum of the polynomials computed by its children.
A polynomial $f\in {\mathbb{F}}[X]$ is called multilinear if the degree of each variable in $f$ is at most one. An arithmetic circuit (formula) $\Phi$ is called multilinear if every gate in $\Phi$ computes a multilinear polynomial.
In this work we are interested in small depth multilinear formulas. A depth-$3$ $\Sigma\Pi\Sigma$ formula is a formula composed of three layers of alternating sum and product gates. Thus, every polynomial computed by a $\Sigma\Pi\Sigma$ formula of size $s$ has the following form $$f = \sum_{i=1}^{s}\prod_{j=1}^{d_i}\ell_{i,j},$$ where the $\ell_{i,j}$ are linear functions. In a $\Sigma\Pi\Sigma$ multilinear formula, it holds that in every product gate, $\prod_{j=1}^{d_i}\ell_{i,j}$, the linear functions $\ell_{i,1},\ldots,\ell_{i,d_i}$ are supported on disjoint sets of variables.
Similarly, a depth-$4$ $\Sigma\Pi\Sigma\Pi$ formula is a formula composed of four layers of alternating sum and product gates. Thus, every polynomial computed by a $\Sigma\Pi\Sigma\Pi$ formula of size $s$ has the following form $$f = \sum_{i=1}^{s}\prod_{j=1}^{d_i}Q_{i,j},$$ where the $Q_{i,j}$ are computed at the bottom $\Sigma\Pi$ layers and are $s$-sparse polynomials, i.e., polynomials that have at most $s$ monomials. As in the depth-$3$ case, we have that at every product gate the polynomials $Q_{i,1},\ldots,Q_{i,d_i}$ are supported on disjoint sets of variables.
Another important definition for us is that of a regular depth-$d$ formula.
A regular depth-$d$ formula is specified by a list of $d$ integers $(a_1,p_1,a_2,p_2, \ldots)$. It has $d$ layers of alternating sum and product gates. The fan-in of every sum gate at the $(2i-1)$'th layer is $a_i$ and, similarly, the fan-in of every product gate at the $(2i)$'th layer is $p_i$. For example, a depth-$4$ formula that is specified by the list $(a_1,p_1,a_2,p_2)$ has the following form: $$f = \sum_{i=1}^{a_1}\prod_{j=1}^{p_1}Q_{i,j},$$ where each $Q_{i,j}$ is a polynomial of degree $p_2$ that has (at most) $a_2$ monomials. As before, a regular depth-$d$ multilinear formula is a regular depth-$d$ formula in which every gate computes a multilinear polynomial.
Regular formulas where first defined by Kayal, Saha and Saptharishi \cite{KayalSS14}, who proved quasi-polynomial lower bounds for logarithmic-depth regular formulas. It is interesting to note that in the reductions from general (multilinear) circuits/formulas to depth-$d$ (multilinear) formulas, one gets a regular depth-$d$ (multilinear) formula \cite{ValiantSkyumBR83,AgrawalVinay08,Koiran10,Tavenas13}.
Finally, we also need to consider the model of Read-Once Algebraic Branching Programs (ROABPs) as our construction is based on a reduction to this case. Algebraic branching programs were first defined in the work of Nisan~\cite{Nisan91} who proved exponential lower bounds on the size of non-commutative ABPs computing the determinant or permanent polynomials. Roughly, an algebraic branching program (ABP) consists of a layered graph with edges going from the $i$'th layer to the $(i+1)$'th layer. The first layer consists of a single source node and the last layer contains a single sink. The edges of the graph are labeled with polynomials (in our case we only consider linear functions as labels). The weight of a path is the product of the weights of the edges in the path. The polynomial computed by the ABP is the sum of the weights of all the paths from the source to the sink. An ABP is called a read-once ABP (ROABP) if the only variable appearing on edges that connect the $i$'th and the $(i+1)$'th layer is $x_i$. It is clear that a ROABP whose edges are labeled with linear functions computes a multilinear polynomial.
\subsection{Polynomial Identity Testing}\label{sec:PIT}
In the PIT problem we are given an arithmetic circuit or formula $\Phi$, computing some polynomial $f$, and we have to determine whether ``$f\equiv 0$''. That is, we are asking if $f$ is the zero polynomial in ${\mathbb{F}}[x_1,\ldots,x_n]$. By the Schwartz-Zippel-DeMillo-Lipton lemma \cite{Zippel79,Schwartz80,DemilloL78}, if $0\ne f \in {\mathbb{F}}[x_1,\ldots,x_n]$ is a polynomial of degree $\le d$, and $\alpha_1,\ldots,\alpha_n \in A\subseteq{\mathbb{F}}$ are chosen uniformly at random, then $f(\alpha_1,\ldots,\alpha_n) =0$
with probability at most\footnote{Note that this is meaningful only if $d < |A| \leq |{\mathbb{F}}|$, which in particular implies that $f$ is not the zero
function.} $d/|A|$. Thus, given $\Phi$, we can perform these evaluations efficiently, giving an efficient randomized procedure for answering ``$f\equiv 0$?''. It is an important open problem to find a derandomization of this algorithm, that is, to find a {\em deterministic} procedure for PIT that runs in polynomial time (in the size of $\Phi$).
One interesting property of the above randomized algorithm of Schwartz-Zippel is that the algorithm does not need to ``see'' the circuit $\Phi$. Namely, the algorithm only uses the circuit to compute the evaluations $f(\alpha_1,\ldots,\alpha_n)$. Such an algorithm is called a {\em black-box} algorithm. In contrast, an algorithm that can access the internal structure of the circuit $\Phi$ is called a {\em white-box} algorithm. Clearly, the designer of the algorithm has more resources in the white-box model and so one can expect that solving PIT in this model should be a simpler task than in the black-box model.
The problem of derandomizing PIT has received a lot of attention in the past few years. In particular, many works examine a specific class of circuits $\mathcal C$, and design PIT algorithms only for circuits in that class. One reason for this attention is the strong connection between deterministic PIT algorithms for a class $\mathcal C$ and lower bounds for $\mathcal C$. This connection was first observed by Heintz and Schnorr~\cite{HeintzSchnorr80} (and later also by Agrawal~\cite{Agrawal05}) for the black-box model and by Kabanets and Impagliazzo~\cite{KabanetsImpagliazzo04} for the white-box model (see also Dvir, Shpilka and Yehudayoff~\cite{DSY09} for a similar result for bounded depth circuits). Another motivation for studying the problem is its relation to algorithmic questions. Indeed, the famous deterministic primality testing algorithm of Agrawal, Kayal and Saxena~\cite{AKS04} is based on derandomizing a specific polynomial identity. Finally, the PIT problem is, in some sense, the most general problem that we know today for which we have randomized coRP algorithms but no polynomial time algorithms, thus studying it is a natural step towards a better understanding of the relation between RP and P. For more on the PIT problem we refer to the survey by Shpilka and Yehudayoff~\cite{SY10}.
Among the most studied circuit classes we find Read-Once Algebraic Branching Programs \cite{ForbesShpilka13,ForbesSS14,AgrawalGKS14}, set-multilinear formulas \cite{RazShpilka05,ForbesShpilka12,AgrawalSS13}, depth-$3$ formulas \cite{DvirShpilka06,KayalSaxena07,KarninShpilka08,KayalSaraf09,SaxenaSeshadhri11}, multilinear depth-$4$ formulas (and some generalizations of them) \cite{KarninMSV13,SarafVolkovich11,AgrawalSSS12} and bounded-read multilinear formulas \cite{ShpilkaVolkovich08,ShpilkaVolkovich09,AvMV11,AgrawalSSS12}. We note that none of these results follow from a reduction a la Kabanets-Impagliazzo \cite{KabanetsImpagliazzo04} (or the reduction of \cite{DSY09} for bounded depth circuits) from PIT to lower bounds. Indeed, this reduction does not work for the restricted classes mentioned here. In particular, for the multilinear model no reduction from PIT to lower bounds is known. That is, even given lower bounds for multilinear circuits/formulas (e.g., the exponential lower bound of Raz and Yehudayoff \cite{RazYehudayoff09} for constant depth multilinear formulas) we do not know how to construct a PIT algorithm for a related model.
The works on depth-$3$ and multilinear depth-$4$ formulas gave polynomial time algorithms only when the fan-in of the top gate (the output gate) is constant, and became exponential time when the top fan-in was $\Omega(n)$, both in the white-box and black-box models \cite{KayalSaxena07,SaxenaSeshadhri11,SarafVolkovich11}. Raz and Shpilka \cite{RazShpilka05} gave a polynomial time PIT for set-multilinear depth-$3$ circuits and Forbes and Shpilka \cite{ForbesShpilka13} and Agrawal, Saha and Saxena \cite{AgrawalSS13} gave a quasi-polynomial size hitting set for the model. Recall that in a depth-$3$ set-multilinear formula, the variables are partitioned to sets, and each linear function at the bottom layer only involves variables from a single set. Recently, Agrawal et al.\ \cite{AgrawalGKS14} gave a subexponential white-box algorithm for a depth-$3$ formula that computes the sum of $c$ set-multilinear formulas, each of size $s$, with respect to different partitions of the variables. The running time of their algorithm is $n^{O(2^{c} n^{1-\frac{2}{2^c} }\log s)}$. In particular, for $c=O(\log\log(n))$ the running time is $\exp(n)$.
Thus, prior to this work there were no subexponential PIT algorithms, even for depth-$3$ multilinear formulas with top fan-in $n$.
\subsection{Our Results}\label{sec:results}
\begin{remark*}
Throughout this paper, we assume that for formulas of size $2^{n^{\delta}}$, the underlying field ${\mathbb{F}}$ is of size at least $|{\mathbb{F}}| \ge 2^{n^{2\delta} \textsf{poly}\log(n)}$, and that if this is not the case then we are allowed to query the formula on inputs from an extension field of the appropriate size. In particular, all our results hold over fields of characteristic zero or over fields of size $\exp(n)$. \end{remark*}
We give subexponential size hitting sets for depth-$3$, depth-$4$ and regular depth-$d$ multilinear formulas, of subexponential size. In particular we obtain the following results.
\begin{thm} \label{thm:intro:hitting-set-depth-$3$}
There exists a hitting set $\mathcal H$ of size $|\mathcal H| = 2^{\tilde{O}(n^{\frac{2}{3}+\frac{2}{3}\delta})}$ for the class of $\Sigma\Pi\Sigma$ multilinear formulas of size $2^{n^\delta}$. \end{thm}
This gives a significant improvement to the recent result, mentioned above, of Agrawal et al.\ \cite{AgrawalGKS14} who studied sum of set-multilinear formulas. From the connection between hitting sets and circuit lower bounds \cite{HeintzSchnorr80,Agrawal05} we obtain the following corollary.
\begin{cor}\label{cor:intro:lowerbound-depth-$3$} There is an explicit multilinear polynomial $f\in {\mathbb{F}}[x_1,\ldots,x_n]$, whose coefficients can be computed in exponential time, such that any depth-$3$ multilinear formula for $f$ has size $\exp(\tilde{\Omega}(\sqrt{n}))$. \end{cor}
This lower bound is weaker than the exponential lower bound of Nisan and Wigderson for this model \cite{NisanWigderson96}. Yet, it is interesting to note that we can get such a strong lower bound from a PIT algorithm. Next, we present our result for depth-$4$ multilinear formulas.
\begin{thm} \label{thm:intro:hitting-set-depth-$4$}
There exists a hitting set $\mathcal H$ of size $|\mathcal H| = 2^{\tilde{O}(n^{2/3+4\delta/3})}$ for the class of $\Sigma\Pi\Sigma\Pi$ multilinear formulas of size $2^{n^\delta}$. \end{thm}
Again, from the connection between hitting sets and circuit lower bounds we obtain the following corollary.
\begin{cor}\label{cor:intro:lowerbound-depth-$4$} There is an explicit multilinear polynomial $f\in {\mathbb{F}}[x_1,\ldots,x_n]$, whose coefficients can be computed in exponential time, such that any depth-$4$ multilinear formula for $f$ has size $\exp(\tilde{\Omega}(n^{1/4}))$. \end{cor}
The best known lower bound for depth-$4$ multilinear formulas is $\exp(n^{1/2})$ due to Raz and Yehudayoff \cite{RazYehudayoff09}, thus, as in the previous case, the term in the exponent of our lower bound is the square root of the corresponding term in the best known lower bound. For regular depth-$d$ multilinear formulas we obtain the following result.
\begin{thm} \label{thm:intro:hitting-set-regular}
There exists a hitting set $\mathcal H$ of size $|\mathcal H| = 2^{\tilde{O}(n^{1- \delta/3})}$ for the class of regular depth-$d$ multilinear formulas of size $2^{n^\delta}$, where $\delta \leq \frac{1}{5^{\lfloor d/2 \rfloor+1}} = O\left(\frac{1}{\sqrt{5}^d}\right)$. \end{thm}
As before we obtain a lower bound for such formulas.
\begin{cor}\label{cor:intro:lowerbound-regular} There is an explicit multilinear polynomial $f\in {\mathbb{F}}[x_1,\ldots,x_n]$, whose coefficients can be computed in exponential time, such that any regular depth-$d$ multilinear formula for $f$ has size $\exp(\tilde{\Omega}(n^{ \frac{1}{5^{\lfloor d/2 \rfloor+1}} }))$. \end{cor}
We note that Raz and Yehudayoff gave an $\exp(n^{\Omega(\frac{1}{d})})$ lower bound for depth-$d$ multilinear formulas, which is much stronger than what Corollary~\ref{cor:intro:lowerbound-regular} gives. Yet, our result also gives a PIT algorithm, which does not follow from the results of \cite{RazYehudayoff09}. As we later explain, we lose a square root in the term at the exponent for every increase of the depth and this is the reason that we get only $\exp(n^{1/\exp(d)})$ instead of $\exp(n^{1/d})$.
In addition to lower bounds, our work also implies deterministic factorization of multilinear polynomials.
In~\cite{ShpilkaVolkovich10}, Shpilka and Volkovich proved that if one can perform PIT deterministically for certain classes of multilinear polynomials then a deterministic factoring algorithm for those classes follows. Specifically, for a class of polynomials $\mathcal C$ they defined the class $\mathcal C_V$, consisting of all polynomials that can be computed by circuits of the form $C = C_1 + C_2 \times C_3$, where the circuits $C_i$ belong to the class $\mathcal C$ and the circuits $C_2$ and $C_3$ are defined over disjoint sets of variables. They proved that if the class $\mathcal C_V$ has a deterministic PIT that runs in time $T(n,s)$ for circuits on $n$ variables of size $s$ then there is a deterministic factoring algorithm for the class $\mathcal C$ that runs in time $O(n^3\cdot T(s))$ (Theorem 1.1 in \cite{ShpilkaVolkovich10}).
In our case, since the product of two variable disjoint multilinear $\Sigma\Pi\Sigma$ ($\Sigma\Pi\Sigma\Pi$) formulas of size $2^{n^\delta}$ is a multilinear $\Sigma\Pi\Sigma$ ($\Sigma\Pi\Sigma\Pi$) formula of size $2^{2n^\delta}$, which is still inside of the class $\Sigma\Pi\Sigma$ ($\Sigma\Pi\Sigma\Pi$), the result of \cite{ShpilkaVolkovich10}, when combined with our PIT results, implies that we can deterministically factor such formulas. Therefore, we obtain the following corollary:
\begin{cor}[Deterministic Factorization]\label{cor:intro:factoring}
Given a multilinear polynomial $f \in {\mathbb{F}}[x_1, \ldots, x_n]$ that can be computed by a $\Sigma\Pi\Sigma$ ($\Sigma\Pi\Sigma\Pi$) formula of size $2^{n^\delta}$, there exists an efficient deterministic algorithm that outputs the factors of $f$. The algorithm outputs $\Sigma\Pi\Sigma$ ($\Sigma\Pi\Sigma\Pi$) formulas for the factors if the formula for $f$ is given to it explicitly, and black-boxes if it only has black-box access to $f$. The running time of this algorithm is $2^{\tilde{O}(n^{\frac{2}{3}+ \frac{2}{3} \delta})}$ when $f$ is computed by a $\Sigma\Pi\Sigma$ formula and $2^{\tilde{O}(n^{\frac{2}{3}+ \frac{4}{3} \delta})}$ when it is computed by a $\Sigma\Pi\Sigma\Pi$ formula. \end{cor}
\subsection{Proof Overview}\label{sec:technique}
We first discuss our proof technique for the case of depth-$3$ multilinear formulas. Our (idealized) aim is to reduce such a formula $\Phi$ to a depth-$3$ multilinear formula in which each linear function is of the form $\alpha x + \beta$. That is, each linear function contains at most one variable. If we manage to do that then we can use the quasi-polynomial sized hitting set of \cite{ForbesShpilka12,AgrawalGKS14} for this model.
Of course, the problem with the above argument is that in general, depth-$3$ formulas have more than one variable per linear function. To overcome this difficulty, we will partition the variables to several sets $T_1,\ldots,T_m$ and hope that each linear function in the formula contains at most one variable from each $T_i$. If we can do that then we would use the hitting set for each set of variables $T_i$ and combine those sets together to get our hitting set. That is, the combined hitting set is composed of concatenation of all vectors of the form $v_1 \circ v_2 \circ \ldots \circ v_m$ where $v_i$ comes from the hitting set restricted to the variables in $T_i$ (the concatenation is performed in a way that respects the partition of course). Thus, if we can carry out this procedure then we will get a hitting set of size roughly $n^{m\log n}$. This step indeed yields a hitting set, since when we restrict our attention to each $T_i$ and think of the other variables as constants in some huge extension field, then we do get a small ROABP (in the variables of $T_i$) and hence plugging in the hitting set of \cite{ForbesShpilka12,AgrawalGKS14} gives a non-zero polynomial. Thus, we can first do this for $T_1$ and obtain some good assignment $v_1$ that makes the polynomial non-zero after substituting $v_1$ to $T_1$. Then we can find $v_2$ etc.
There are two problems with the above argument. One problem is how to find such a good partition. The second is that this idea simply cannot work as is. For example, if we have the linear function $x_1+\cdots+x_n$, then it will have a large intersection with each $T_i$.
We first deal with the second question. to overcome the difficulty posed by the example, we would like to somehow ``get rid'' of all linear functions of large support and then carry out the idea above. To remove linear functions with large support from the formula we use another trick. Consider a variable $x_k$ that appears in a linear function $\ell_0$ that has a large support. Assume that $\frac{\partial f}{\partial x_k} \not\equiv 0$ as otherwise we can ignore $x_k$. Now, because of multilinearity, we can transform our original formula $\Phi$ to a formula computing $\frac{\partial f}{\partial x_k}$. This is done by replacing each linear function $\ell(X) = \sum_{i=1}^{n}\alpha_i x_i + \alpha_0$ with the constant $\alpha_k$. In particular, the function $\ell_0$ that used to have a high support does not appear in the new formula. Furthermore, this process does not increase the support size of any other linear function. A possible issue is that if we have to repeat this process for every function of large support then it seems that we need to take a fresh derivative for every such linear function. The point is that because we only care about linear functions that have a large support to begin with, we can find a variable that simultaneously appears in many of those functions and thus one derivative will eliminate many of the ``bad'' linear functions.
Working out the parameters, we see that we need to take roughly $n^\epsilon \cdot \log|\Phi|$ many derivatives to reduce to the case where all linear functions have support size at most $n^{1-\epsilon}$.
Now we go back to our first problem. We can assume that we have a depth-$3$ formula in which each linear function has support size at most $n^{1-\epsilon}$ and we wish to find a partition of the variables to sets $T_1,\ldots,T_m$ so that each $T_i$ contains at most one variable from each linear function. This cannot be achieved as a simple probabilistic argument shows, so we relax our requirement and only demand that in each multiplication gate (of the formula) only a few linear functions have a large intersection. If at most $k$ linear functions in each gate have a large intersection, we can expand each multiplication gate to at most $n^k$ new gates (by simply expanding all linear functions that have large intersection) and then apply our argument. As we will be able to handle subexponential size formulas, this blow up is tolerable for us.
Note that if we were to pick the partition at random, when $m= n^{1-\epsilon+\gamma}$, for some small $\gamma$, then we will get that with very high probability at most $n^\delta$ linear functions will have interaction at most $n^\delta$ with each $T_i$, where
$\delta$ is such that $|\Phi| < \exp(n^\delta)$. To get a deterministic version of this partitioning, we simply use an $n^\delta$-wise independent family of hash functions $\{h:[n]\to [m]\}$. Each hash function $h$ induces a partition of the variables to $T_i = \{x_k \mid h(k)=i\}$. Because of the high independence, we are guaranteed that there is at least one hash function that induces a good partition.
Now we have all the ingredients in place. To get our hitting set we basically do the following (we describe the construction as a process, but it should be clear that every step can be performed using some evaluation vectors). \begin{enumerate}
\item \label{item:derivative} Pick $n^\epsilon \cdot \log|\Phi|$ many variables and compute a black-box for the polynomial that is obtained by taking the derivative of $f$ with respect to those variables. The cost of this step is roughly
${n \choose n^\epsilon \cdot \log|\Phi|} \cdot 2^{n^\epsilon \cdot \log|\Phi|}$, where the first term is for picking the variables and the second is what we have to pay to get access to the derived polynomial.
\item \label{item:partition} Partition the remaining variables to (roughly) $n^{1-\epsilon/2}$ many sets using a (roughly) $\log|\Phi|$-wise
independent family of hash functions. The cost of this step is roughly $n^{\log|\Phi|}$ as this is the size of the hash function family.
\item Plug in a fresh copy of the hitting set of \cite{ForbesShpilka12,AgrawalGKS14} to each of the sets of variables $T_i$. The cost is roughly $n^{\log n \cdot n^{1-\epsilon/2}}$.
\end{enumerate}
Combining everything we get a hitting set of size roughly
$$\left({n \choose n^\epsilon \cdot \log|\Phi|} \cdot 2^{n^\epsilon \cdot \log|\Phi|}\right)\cdot \left(n^{\log|\Phi|}\right)\cdot \left(n^{\log n \cdot n^{1-\epsilon/2}}\right) \approx 2^{\tilde{O}\left( n^{1-\epsilon/2} + n^\epsilon \log|\Phi| \right)}.$$ Optimizing the parameters we get our hitting set.
We would like to use the same approach also for the case of depth-$4$ formulas. Here the problem is that in the two bottom layers the formula compute a polynomial and not a linear function. In particular, when taking a derivative we are no longer removing functions that have a large support. Nevertheless, we can still use a similar idea. We show there is a variable $x_i$ that by either
setting $f|_{x_i=0}$ or considering $\frac{\partial f}{\partial x_i}$, we are guaranteed that the total sparsity of all polynomials that have large supports goes down by some non-negligible factor. Thus, repeating this process (of either setting a variable to $0$ or
taking a derivative) $n^\epsilon \cdot \log|\Phi|$ many times we reach a depth-$4$ formula where all polynomials computed at the bottom addition gate have small supports. Next, we partition the variables to buckets and consider a single bucket $T_i$. Now, another issue is that even if the intersection of a low-support polynomial with some $T_i$ is rather small, the sparsity of the resulting polynomial (which is considered as a polynomial in the variable in the intersection) can still be exponential in the size of the intersection. This is why we lose a bit in the upper bound compared to the depth-$3$ case. Combining all steps again we get the result for depth-$4$ formulas.
The proof for regular formulas works by first reducing to the depth-$4$ case and then applying our hitting set. The reduction is obtained in a similar spirit to the reduction of Kayal et al.\ \cite{KayalSS14}. We break the formula at an appropriate layer and then express the top layers as a $\Sigma\Pi$ circuit and the bottom layers as products of polynomials of not too high degrees. We then use the trivial observation that if the degree of a polynomial is at most $n^{1-\epsilon}$ then its sparsity is at most $n^{n^{1-\epsilon}}$ and proceed as before. Due to the different requirements of the reduction and of the hashing part, we roughly lose a constant factor in the exponent of $n$, in the size of the hitting set, whenever the depth grows, resulting in a hitting set of size roughly $\exp(n^{1-1/\exp(d)})$.
To obtain the lower bounds we simply use the idea of \cite{HeintzSchnorr80,Agrawal05}. That is, given a hitting set we find a non-zero multilinear polynomial that vanishes on all points of the hitting set by solving a homogeneous system of linear equations.
\subsection{Related Work}
\paragraph{The work of Agrawal, Gurjar, Korwar and Saxena \cite{AgrawalGKS14}:}
The closest work to ours is the one by Agrawal et al.\ \cite{AgrawalGKS14}. In addition to other results, they gave a white-box PIT algorithm that runs in time $n^{O(2^{c} n^{1-\frac{2}{2^c} }\log s)}$ for depth-$3$ formulas that can be represented as a sum of $c$ set-multilinear formulas, each of size $s$ (potentially with respect to different partitions of the variables).
Theorem~\ref{thm:intro:hitting-set-depth-$3$} improves upon this results in several ways. First, the theorem gives a hitting set, i.e., a black-box PIT. Secondly, for $c=O(\log\log n)$ the running time of the algorithm of \cite{AgrawalGKS14} is $\exp(n)$, whereas our construction can handle a sum of $\exp(n^\beta)$ set-multilinear formulas and still maintain a subexponential complexity.
Nonetheless, there are some similarities behind the basic approach of the this work and the work of Agrawal et al. Recall that a set-multilinear depth-$3$ formula is based on a partition of the variables, where each linear function in the formula contains variables from a single partition. Agrawal et al.\ start with a sum of $c$ set-multilinear circuits, each with respect to a different partitioning of the variables, and their first goal is to reduce the formula to a set-multilinear formula, i.e., to have only one partition of the variables. For this they define a distance between different partitions and show, using an involved combinatorial argument, that one can find some partition $T_1,\ldots,T_m$ of the variables so that when restricting our attention to $T_i$, all the $c$ set-multilinear formulas will be somewhat ``close to each other''. If the distance is $\Delta$ (according to their definition) then they prove that they can express the sum as a set-multilinear circuit of size roughly $s\cdot n^\Delta$, where $s$ is the total size of the depth-$3$ formula. Unlike our work, they find the partition in a white-box manner by gradually refining the given $c$ partitions of the set-multilinear circuits composing the formula. The final verification step is done, in a similar manner to ours, by substituting the hitting set of \cite{AgrawalSS13} (or that of \cite{ForbesShpilka12}) to each of the sets $T_i$. The step of finding the partition $T_1,\ldots,T_m$ is technically involved and is the only step where white-box access is required.
\paragraph{Lower bounds for multilinear circuits and formulas:}
Lower bounds for the multilinear model were first proved by Nisan and Wigderson \cite{NisanWigderson96}, who gave exponential lower bounds for depth-$3$ formulas. Raz first proved quasi-polynomial lower bounds for multilinear formulas computing the Determinant and Permanent polynomials \cite{Raz09a} and later gave a separation
between multilnear $\text{NC}_1$ and multilinear $\text{NC}_2$ \cite{Raz06}.
Raz and Yehudayoff proved a lower bound of $\exp(n^{\Omega(\frac{1}{d})})$ for depth-$d$ multilinear formulas.
As in the general case, the depth reduction techniques of \cite{ValiantSkyumBR83,AgrawalVinay08,Koiran10,Tavenas13} also work for multilinear formulas. Thus, proving a lower bound of the form $\exp(n^{\frac{1}{2}+\epsilon})$ for $\Sigma\Pi\Sigma\Pi$ multilinear formulas, would imply a super-polynomial lower bound for multilinear circuits. Currently, the best lower bound for syntactic multilinear circuits is $n^{4/3}$ by Raz, Shpilka and Yehudayoff \cite{RSY08}.
Kayal, Saha and Saptharishi \cite{KayalSS14} proved a quasi-polynomial lower bounds for regular formulas that have the additional condition that the syntactic degree of the formula is at most twice the degree of the output polynomial.
\subsection{Organization} In Section~\ref{sec:prelim} we provide basic definitions and notations, and also prove some general lemmas which will be helpful in the next sections. In Section~\ref{sec:red}, we explain how to reduce general depth-$3$ and depth-$4$ formulas to formulas such that every polynomial at the bottom has small support. Then, in Section~\ref{sec:hit-bottom}, we construct a hitting set for those types of formulas. In Section~\ref{sec:hit}, we explain how to combine the ideas of the previous two sections and construct our hitting set for depth-$3$ and depth-$4$ multilinear formulas.
We then move on in Section~\ref{sec:regular} to depth $d$ regular formulas, and show how to reduce them to depth-$4$ formulas and obtain a hitting set for this class. In the short Section~\ref{sec:lower-bounds} we spell out briefly how, using known observations about the relation between PIT and lower bounds, we obtain our lower bounds for multilinear formulas. Finally, in Section~\ref{sec:open} we discuss some open problems and future directions for research.
\section{Preliminaries} \label{sec:prelim}
In this section, we establish notation, some definitions and useful lemmas that will be used throughout the paper.
\subsection{Notations and Basic Definitions}\label{sec:notation}
For any positive integer $n$, we denote by $[n]$ the set of integers from $1$ to $n$, and by $\binom{[n]}{\le r}$ the family of subsets
$A \subseteq [n]$ such that $|A| \le r$. We often associate a subset $A \subseteq [n]$ with a subset of variables $\textsf{var}(A) \subseteq \{ x_1,\ldots,x_n \}$ in a natural way (i.e., $\textsf{var}(A)=\{x_i \mid i \in A \}$). In those cases we make no
distinction between the two and use $A$ to refer to $\textsf{var}(A)$. Additionally, if $A$ and $B$ are disjoint subsets of $[n]$, we denote their disjoint union by $A \sqcup B$. For a vector $v\in{\mathbb{F}}^n$ we denote with $v|_A$ the restriction of $v$ to the coordinates $A$.
In order to improve the readability of the text, we omit floor and ceiling notations.
Let $f(x_1, \ldots, x_n) \in {\mathbb{F}}[x_1, \ldots, x_n]$ be a polynomial. We will denote by ${\partial}_x f$ the formal derivative of $f$ with respect to the variable $x$, and by $f|_{x=0}$ the polynomial obtained from $f$ by setting $x = 0$. Moreover,
if $A \subseteq [n]$, we will denote by ${\partial}_A f$ the polynomial obtained when taking the formal derivative of $f$ with respect to all variables in $A$. In a similar fashion, we denote by $f|_{A=0}$ the polynomial obtained when we set all the variables in $A$
to zero, and more generally, if $|A|=r$ and $\ol{\alpha} = (\alpha_1,\ldots,\alpha_r) \in {\mathbb{F}}^{r}$, $f|_{A=\ol{\alpha}}$ will denote the restriction of $f$ obtained when setting the $i$'th variable in $A$ to $\alpha_i$, for $1 \le i \le r$.
In addition to the conventions above, the following definitions will be very useful in the next sections.
\begin{define}[Variable Set and Non-trivial Variable Set]
Let $f(x_1, \ldots, x_n) \in {\mathbb{F}}[x_1, \ldots, x_n]$ be a polynomial. Define the variable set ($\textsf{var}$) and the non-trivial
variable set ($\textsf{var}^*$) as follows:
\begin{align*}
\textsf{var}(f) &= \{ x \in \{x_1, \ldots, x_n\} \mid {\partial}_x f \neq 0 \} \\
\textsf{var}^*(f) &= \{ x \in \{x_1, \ldots, x_n\} \mid {\partial}_x f \neq 0 \text{ and } f|_{x=0} \neq 0 \}.
\end{align*}
That is, the variable set of a polynomial $f$ is the set of variables $x \in \{x_1, \ldots, x_n\}$ that appear in the
representation of $f$ as a sum of monomials, whereas the non-trivial variable set is the set of variables of $f$
that do not divide it. \end{define}
We shall say that $f$ has a small support if $\textsf{var}(f)$ (or $\textsf{var}^*(f)$) is not too large.
\begin{define}[Monomial Support and Sparsity]
Let $f(x_1, \ldots, x_n) \in {\mathbb{F}}[x_1, \ldots, x_n]$ be a polynomial. We define the \emph{monomial support} of $f$, written
$\textsf{mon}(f)$, as the set of monomials that have a non-zero coefficient in $f$. In addition, we define the sparsity of $f$, written
$\| f \|$, as the size of the set $\textsf{mon}(f)$, that is,
$$ \| f \| = |\textsf{mon}(f)|. $$
In other words, the sparsity of $f$ is the number of monomials that appear with a non-zero coefficient in $f$. \end{define}
In the constructions of our hitting sets we will need to combine assignments to different subsets of variables. The following notation will be useful. For a partition of $[n]$, $T_1\sqcup T_2\sqcup \cdots \sqcup T_m =[n]$, and sets $\mathcal H_i \subseteq {\mathbb{F}}^{|T_i|}$, we denote with $\mathcal H_1^{T_1}\times \cdots \times \mathcal H_m^{T_m}$ the set of all vectors of length $n$ whose restriction to $T_i$ is an element of $\mathcal H_i$:
$$\mathcal H_1^{T_1}\times \cdots \times \mathcal H_m^{T_m} = \{ v \in{\mathbb{F}}^n \mid \forall i\in [m], \; v|_{T_i} \in \mathcal H_i \}.$$
\subsection{Depth-$3$ and Depth-$4$ Formulas}
We define some special classes of depth-$3$ and depth-$4$ formulas that will be used throughout this paper.
\begin{define}[Restricted Top Fan-in]
Let $\Phi$ be a multilinear depth-$4$ formula. We say that $\Phi$ is a multilinear $\tfSPSP{M}$
formula if it is of the form $\sum_{i=1}^m\prod_{j=1}^{t_i} f_{i,j}$, where $m \le M$.
If, in addition to the conditions above, we have that each $f_{i,j}$ is a linear function, that is, $\Phi$ is actually a
depth-$3$ formula, we will say that $\Phi$ is a multilinear $\tfSPS{M}$ formula. \end{define}
Our next definition considers the case where polynomials computed at the bottom layers do not contain too many variables, that is, they have small support.
\begin{define}[Restricted Top Fan-in and Variable Set]
Let $\Phi$ be a multilinear depth-$4$ formula. We say that $\Phi$ is a multilinear $\tfrsSPSP{M}{\tau}$
formula if it is of the form $\sum_{i=1}^m\prod_{j=1}^{t_i} f_{i,j}$, where $m \le M$ and for each $1 \le i \le s$
we have that
\begin{enumerate}[(i)]
\item $|\textsf{var}(f_{i,j})| \le \tau$ for all $1 \le j \le t_i$
\item $\textsf{var}(f_{i,j_1}) \cap \textsf{var}(f_{i,j_2}) = \emptyset$, for any $j_1 \neq j_2$.
\end{enumerate}
If, in addition to the conditions above, we have that each $f_{i,j}$ is a linear function, that is, $\Phi$ is actually a
depth-$3$ formula, we will say that $\Phi$ is a multilinear $\tfrsSPS{M}{\tau}$ formula. \end{define}
Since the formula will be given to us as a black-box, we can make some assumptions about it, which will help us to preserve non-zeroness when taking derivatives or setting variables to zero. To this end, we define a notion of simplicity of depth-$4$ formulas,\footnote{Note that this is not the same notion as used, e.g., in \cite{DvirShpilka06}. } and prove that we can assume without loss of generality that any input formula is simple.
\begin{define}
Let $f(x_1, \ldots, x_n) \in {\mathbb{F}}[x_1, \ldots, x_n]$ be a multilinear polynomial and let
$$ \Phi = \sum_{i=1}^M\prod_{j=1}^{t_i} f_{i,j} $$
be a multilinear depth-$4$ formula computing $f$. We say that $\Phi$ is a \emph{simple} multilinear
depth-$4$ formula if for each variable $x \in \textsf{var}(f)$ that divides $f$, it must be the case that
for every $1 \le i \le M$, there exists $j \in [t_i]$ such that $f_{i,j} = x$. \end{define}
In words, $\Phi$ is simple if whenever a variable $x$ divides $f$, it also divides every product gate. The following proposition tells us that we can indeed assume, without loss of generality, that any multilinear depth-$4$ formula given to us is a simple formula.
\begin{prop}\label{prop:simple}
If $\Phi$ is a depth-$4$ multilinear $\tfSPSP{M}$ formula computing $f(x_1, \ldots, x_n)$, then $f$ can be computed
by a simple depth-$4$ multilinear $\tfSPSP{M}$ formula $\Psi$ where $|\Psi| \le |\Phi|.$ \end{prop}
\begin{proof}
Since $\Phi$ is a $\tfSPSP{M}$ formula, we have that
$$ f = \displaystyle\sum_{i=1}^M\prod_{j=1}^{t_i} f_{i,j}. $$
Let $x \in \textsf{var}(f)$ be such that $x \mid f$.
Notice that we can write each $f_{i,j}$ in the following form:
$$ f_{i,j} = x \cdot g_{i,j} + h_{i,j}, \ \text{ where } x \not\in \textsf{var}(g_{i,j}) \cup \textsf{var}(h_{i,j}). $$
Moreover, observe that if $x \not\in \textsf{var}(f_{i,j})$, then we must have that $f_{i,j} = h_{i,j}$. Since the formula is
multilinear, for each $i \in [M]$ there exists at most one $j$ such that $x \in \textsf{var}(f_{i,j})$. If such $j$ exists,
we might as well assume without the loss of generality that $j=1$.
Let $A = \{ i : 1 \le i \le M, \text{ and } x \in \textsf{var}(f_{i,1}) \}$ and $B = [M] \setminus A$.
Now, rewriting the formula above for $f$, we get:
\begin{align*}
f = \displaystyle\sum_{i=1}^M\prod_{j=1}^{t_i} f_{i,j} &= \displaystyle\sum_{i \in A} f_{i,1} \cdot \prod_{j=2}^{t_i} f_{i,j} +
\displaystyle\sum_{i \in B}\prod_{j=1}^{t_i} f_{i,j} \\
&=\displaystyle\sum_{i \in A} (xg_{i,1} + h_{i,1}) \cdot \prod_{j=2}^{t_i} h_{i,j} + \displaystyle\sum_{i \in B}\prod_{j=1}^{t_i} h_{i,j} \\
&=\displaystyle\sum_{i \in A} xg_{i,1} \cdot \prod_{j=2}^{t_i} h_{i,j} + \displaystyle\sum_{i \in A} h_{i,1} \cdot \prod_{j=2}^{t_i} h_{i,j} +
\displaystyle\sum_{i \in B}\prod_{j=1}^{t_i} h_{i,j}.
\end{align*}
Since $x \mid f$, it follows that $f = xg$. Hence, we must have that (in the above equation)
$$ \displaystyle\sum_{i \in A} h_{i,1} \cdot \prod_{j=2}^{t_i} h_{i,j} + \displaystyle\sum_{i \in B}\prod_{j=1}^{t_i} h_{i,j} = 0 $$
and therefore
$$ f = \displaystyle\sum_{i \in A} xg_{i,1} \cdot \prod_{j=2}^{t_i} h_{i,j}. $$
Since $|A| \le M$ and $\| g_{i,1} \| \le \| f_{i,1} \|, \ \| h_{i,j} \| \le \| f_{i,j} \|$ for every $i \in [M]$ and $2 \le j \le k_{i}$, the
formula
\[
\Phi' = \sum_{i \in A} x \cdot g_{i,1} \cdot \prod_{j=2}^{t_i} h_{i,j} = \sum_{i \in A} \prod_{j=2}^{t_i} x \cdot g_{i,1} \cdot h_{i,j}
\] is a multilinear $\tfSPSP{M}$ formula
computing $f$, of size $|\Phi'| \le |\Phi|$ and such that $x$ appears as a polynomial at each product gate.
By repeating this process for each variable $\textsf{var}(f) \setminus \textsf{var}^*(f)$, we get our $\tfSPSP{M}$ formula $\Psi$.
Since at each step we preserve the invariant that the size of the formula does not increase, we must have that
$|\Psi| \le |\Phi|.$ \end{proof}
As a corollary, together with the simple observation that any derivative or restriction of a multilinear formula results in a multilinear formula of at most the same size, we obtain that partial derivatives or restrictions of a multilinear polynomial can also be computed by simple formulas.
\begin{cor}
If $\Phi$ is a depth-$4$ multilinear $\tfSPSP{M}$ formula computing $f(x_1, \ldots, x_n)$, then for any disjoint sets
$A, B \subseteq \textsf{var}(f)$, ${\partial}_A f|_{B=0}$ can be computed by a simple depth-$4$ multilinear $\tfSPSP{M}$ formula
$\Psi$ where $|\Psi| \le |\Phi|.$ We will refer to $\Psi$ as ${\partial}_A \Phi|_{B=0}$. \end{cor}
Therefore, from now on we will always assume that any depth-$4$ multilinear formula given to us is a simple formula.
\subsection{ROABPs for Products of Sparse Polynomials}
Another important model that we need for our constructions is that of Algebraic Branching Programs.
\begin{define}[Nisan~\cite{Nisan91}]
An algebraic branching program (ABP) is a directed acyclic graph with one vertex $s$ of in-degree zero (the {\em source}) and
one vertex $t$ of out-degree zero (the {\em sink}). The vertices of the graph are partitioned into levels labeled $0, 1, \ldots, D$.
Edges in the graph can only go from level $\ell-1$ to level $\ell$, for $\ell \in [D]$. The source is the only vertex at level $0$
and the sink is the only vertex at level $D$. Each edge is labeled with an affine function in the input variables. The {\em width} of an
ABP is the maximum number of nodes in any layer, and the {\em size} of an ABP is the number of vertices in the ABP.
Each path from $s$ to $t$ computes the polynomial which is the product of the labels of the path edges, and the ABP
computes the sum, over all $s\to t$ paths, of such polynomials. \end{define}
\begin{define}[Ordered Read-Once Algebraic Branching Programs]
A {\em Ordered Read-Once Algebraic Branching Program (ROABP)} in the variable set $\{x_1, \ldots, x_D\}$ is an ABP of
depth $D$, such that each edge between layer $\ell-1$ and $\ell$ is labeled by a univariate polynomial in $x_\ell$.
\end{define}
In this section we show an elementary construction of ROABPs for a very specific class of polynomials. This construction however will be useful in the upcoming sections.
\begin{lem} \label{lem:roabp-product-sparse-polys} Let ${\mathbb{F}}$ be a field, and $f(y_1,\ldots,y_m) = \sum_{i=1}^M \prod_{j=1}^{t_i} f_{i,j}$ be a multivariate polynomial over ${\mathbb{F}}$, such that for every $1 \le i \le M$: \begin{enumerate}
\item At most $k$ different $1 \le j \le t_i$, satisfy $ | \textsf{var}(f_{i,j}) | > 1$.
\item For every $1\leq j \leq t_i$, $ \| f_{i,j} \| \leq s$. \end{enumerate} Then $f$ can be computed by an ROABP of width at most $M \cdot s^{k}$. \end{lem}
\begin{proof} Assume without the loss of generality that for every $i$ there is $k_i\leq k$ such that $f_{i,1},\ldots,f_{i,k_i}$ are those polynomials that contain more than a single variable. Note that the sparsity of every product $g_i \eqdef \prod_{j=1}^{k_i} f_{i,j}$ is at most $s^k$. We construct an ROABP of width $s^k$ for each $g_i$. The final ROABP is constructed by connecting the ROABPs for all $M$ products in parallel.
Fix $1 \le i \le M$. Expanding the product $g_i = \prod_{j=1}^{k_i} f_{i,j}$ we get at most $s^k$ monomials. Now, multiply each such monomial with the remaining functions in the $i$'th gate, $\prod_{j=k_i+1}^{t_i}f_{i,j}$. Notice that the multiplicands in each such term can be reordered so that first $x_1$ appears then $x_2$ etc. Thus, we can construct a ROABP of width $s^k$ for computing each such product of a monomial with $\prod_{j=k_i+1}^{t_i}f_{i,j}$. Then, connecting all those ROABPs in parallel we get a ROABP of width $s^k$ for the $i$'th multiplication gate. Connecting in a similar fashion all ROABPs for the different multiplication gates we get a ROABP of width (at most) $M\cdot s^k$ computing $f$.
\end{proof}
\subsection{Hashing}
In this section we present the basic hashing tools that we will use in our construction. We first recall the notion of a $k$-wise independent hash family.
\begin{define}\label{def:k-hash} A family of hash functions ${\cal F}=\{ h:[n] \to [m] \}$ is $k$-wise independent if for any $k$ distinct elements $(a_1, \dots, a_k) \in [n]^k$ and any $k$ (not necessarily distinct) elements $(b_1, \dots, b_k) \in [m]^k$, we have: $$\Pr_{h \in \mathcal F} \left[ h(a_1)=b_1 \wedge \cdots \wedge h(a_k)=b_k \right] = m^{-k}.$$ \end{define}
Our next lemma studies the case where several sets are hashed simultaneously by the same hash function. We present the proof in a general form and only later, in the application, fix the parameters.
\begin{lem} \label{lem:hash} Let $0<\delta<\epsilon$, and $n,M \in {\mathbb{N}}$ such that $M=2^{n^{\delta}}$. Assume $\mathcal A_1,\ldots,\mathcal A_M$ are families of pairwise disjoint subsets of $[n]$
such that for every $1 \le i \le M$ and every $A \in \mathcal A_i$, $|A| \le n^{1-\epsilon}$. Let
$\gamma > 0$ be such that $\gamma \ge (\epsilon-\delta)/2$. Let $\mathcal F$ be a family of $k$-wise independent hash functions from $[n]$ to $[m]$ for $k =n^{\delta} + 2\log n $ and $m=10 n^{1-\epsilon+\gamma}$.
Then there exists $h \in \mathcal F$ such that for every $ 1 \le i \le M$ and every $1 \le j \le m$, both of the following conditions hold:
\begin{enumerate}
\item \label{item:no-intersection-larger-than-k} For every set $A \in \mathcal A_i$, $\left| h^{-1}(j) \cap A \right| \le k$.
\item \label{item:at-most-k-intersect} The number of sets $A \in \mathcal A_i$ such that $\left| h^{-1}(j) \cap A \right| > 1$ is at most $k \log n$.
\end{enumerate}
\end{lem}
\begin{proof} We show that for a random $h \in \mathcal F$, both items \ref{item:no-intersection-larger-than-k} and \ref{item:at-most-k-intersect} happen with probability at least $2/3$.
Fix $ 1 \le i \le M$, $1 \le j \le m$ and $A \in \mathcal A_i$. By $k$-wise independence and the assumption that $|A| \le n^{1-\epsilon}$, we have that \begin{align} \label{eq:prob-intersect-larger-than-k}
\Pr \left[ \left| h^{-1}(j) \cap A \right| \ge k \right] & \le
\sum_{B \subseteq A, |B|=k} \Pr [\forall b \in B, h(b)=j] \nonumber \\ & \le \binom{n^{1-\epsilon}}{k} \cdot \frac{1}{ {\left( 10 n^{1-\epsilon+\gamma} \right)}^{k}} \nonumber \\ & \le n^{(1-\epsilon)k} \cdot \frac{1}{ {\left( n^{(1-\epsilon)k + \gamma k} \right)}} \cdot \frac{1}{10^k} \nonumber \\ & \le {n^{-\gamma k}} \cdot 10^{-k} . \end{align}
Taking a union bound over all $1 \le i \le M$ and $1 \le j \le m$, and using the estimate \eqref{eq:prob-intersect-larger-than-k} and the fact that $m \le n$, we get that item \ref{item:no-intersection-larger-than-k} in the statement of the lemma does not happen with probability at most \[ M \cdot m \cdot n^{-\gamma k} \cdot 10^{-k} \le \frac{1}{3} \] for large enough $n$, by the choice of $k$.
Turning to item \ref{item:at-most-k-intersect} in the statement of the lemma, it is convenient to partition every family of subsets $\mathcal A$ into $(1-\epsilon) \log n$ disjoint buckets, according to the size of the sets in $\mathcal A$. Fix such $\mathcal A$, and, for $1 \le i \le (1-\epsilon) \log n$,
define the bucket \[
{\mathcal B}_i = \left\{ A \in \mathcal A : \frac{n^{1-\epsilon}}{2^{i}} \le |A| \le \frac{n^{1-\epsilon}}{2^{i-1}} \right\}. \] We show that with high probability over the choice of $h$, and for every $j \in [m]$, in every bucket there are at most $k$ sets whose intersection with $h^{-1}(j)$ has size larger than 1.
For every sets $A \in \mathcal A$, by $k$-wise independence (in particular, pairwise independence) the probability that
$|A \cap h^{-1} (j)| \ge 2$ is at most \[
\binom{|A|}{2} \cdot \frac{1}{{ \left( 10n^{1-\epsilon+\gamma} \right)}^2} \le |A|^2 \cdot \frac{1}{100 n^{2-2\epsilon+2\gamma}}. \]
Fix a bucket ${\mathcal B}_i$. By definition, for every $A \in {\mathcal B}_i$ it holds that $|A| \le \frac{n^{1-\epsilon}}{2^{i-1}}$, and so for every set
$A \in {\mathcal B}_i$, the probability that $|A \cap h^{-1} (j)| \ge 2$ is at most \begin{equation} \label{eq:prob-intersect-2} \frac{n^{2-2\epsilon}}{2^{2i-2}} \cdot \frac{1}{100} \cdot n^{2\epsilon-2-2\gamma} = \frac{1}{100} \cdot {n^{-2\gamma} \cdot 2^{2-2i}}. \end{equation}
Since $\mathcal A$ is a partition, by pairwise disjointness, the number of sets in ${\mathcal B}_i$ is at most $n^\epsilon \cdot 2^{i}$. Hence, by $k$-wise independence and \eqref{eq:prob-intersect-2}, the probability there exist $k/2$ sets in the bucket ${\mathcal B}_i$, with intersection sizes at least $2$, is at most
\begin{align} \label{eq:prob-intersect-k-over-2} \binom{n^\epsilon \cdot 2^{i} }{k/2} \cdot {\left( \frac{1}{100} \cdot {n^{-2\gamma} \cdot 2^{2-2i}} \right)}^{k/2} & \le {\left( \frac{en^\epsilon \cdot 2^{i}}{k/2} \right)}^{k/2} {\left( \frac{1}{100} \cdot {n^{-2\gamma} \cdot 2^{2-2i}} \right)}^{k/2} \nonumber \\ & = {\left( \frac{e \cdot 2^{3-i}}{100} \right)}^{k/2} \cdot {\left( \frac{n^{\epsilon-2\gamma}}{k} \right) }^{k/2}, \end{align} where we have used the inequality $\binom{a}{b} \le \left( \frac{ea}{b} \right) ^b$. Observe that $n^{\epsilon-2\gamma} \le k$, by the choice of $\gamma$.
Taking a union bound over all $\log n$ buckets, and then over all $M$ partitions and all $m$ possible values of $j$, and using the estimation \eqref{eq:prob-intersect-k-over-2}, we get that the probability that there more than $k/2$ sets that intersect $h^{-1}(j)$ in more than one element, for some $j$, is at most \[ M \cdot m \cdot \log n \cdot \left( \frac{e \cdot 2^{3-i}}{100} \right)^{k/2} \cdot \left( \frac{n^{\epsilon-2\gamma}}{k} \right)^{k/2} \le \frac{1}{3}, \] for large enough $n$, by the choices of $k$ and $\gamma$. Hence, item \ref{item:at-most-k-intersect} in the statement of the lemma follows as well. \end{proof}
We conclude this section with the following well known fact (see, e.g., Chapter 16 in \cite{AlonSpencer08}, and the references therein):
\begin{fact} \label{fact:small-hash} There exists an explicitly constructible family $\mathcal F$ of $k$-wise independent hash functions from $[n]$ to
$[10n^{1-\epsilon+\gamma}]$ of size $|\mathcal F| = n^{O(k)}$. \end{fact}
\section{Reducing the Bottom Support of Depth-$3$ and Depth-$4$ Formulas}\label{sec:red}
In this section we make the first step towards proving Theorems~\ref{thm:intro:hitting-set-depth-$3$} and \ref{thm:intro:hitting-set-depth-$4$}. As outlined in Section~\ref{sec:technique}, our first step is making the functions computed at the bottom layers (linear functions in the case of depth-$3$ and ``sparse'' polynomials in the case of depth-$4$) have small (variable) support. Hence, we establish reductions from any $\tfSPS{M}$ or $\tfSPSP{M}$ formula to a $\tfrsSPS{M}{\tau}$ or $\tfrsSPSP{M}{\tau}$ formula, respectively. We first describe the simple depth-$3$ case. We continue by elaborating on the more general case of depth-$4$ formulas, which is slightly more involved. Both proofs follow the outline described in Section~\ref{sec:technique}.
\subsection{Reducing Bottom Support for Depth-$3$}
Given a depth-$3$ formula $\sum_{i=1}^M \prod_{j=1}^{t_i} \ell_{i,j}$, we would like to eliminate all linear functions that contain many variables. To this end, we observe that there must exist a variable that appears in many of these functions, and that taking a derivative according to that variable eliminates those functions from the formula.
\begin{lem} \label{lem:red3} Let $f(x_1,\ldots,x_n) = \sum_{i=1}^M \prod_{j=1}^{t_i} \ell_{i,j}$ be a non-zero multilinear polynomial computed by a multilinear $\tfSPS{M}$ formula $\Phi$ and let $\epsilon>0$. Then, there
exists a set of variables $A$ of size $|A| \le \tilde{O} (n^{\epsilon} \cdot \log M)$ such that $\partial_A f$ is a non-zero multilinear polynomial that can be computed by a multilinear $\tfrsSPS{M}{n^{1-\epsilon}}$ formula. \end{lem}
\begin{proof} Define \[
{\mathcal B} = \{ \ell_{i,j} \mid |\textsf{var}(\ell_{i,j}) \cap \textsf{var}(f) | \ge n^{1-\epsilon} \} \] to be the set of ``bad'' linear functions. Those are linear functions that contain more than $n^{1-\epsilon}$ variables that also appear in $f$. We show how to eliminate those linear functions from the formula while preserving non-zeroness.
Since for every $\ell \in {\mathcal B}$, $|\textsf{var}(\ell) \cap \textsf{var}(f)| \ge n^{1-\epsilon}$, there exists a variable $x_i$ that appears in at least
$|{\mathcal B}|n^{1-\epsilon}/n = |{\mathcal B}|/n^\epsilon$ linear functions in ${\mathcal B}$ (and also in $f$). Consider the polynomial $\partial_{x_i}f$. Since $x_i \in \textsf{var}(f)$, this is a non-zero polynomial. Furthermore, using the fact that deriving with respect to a variable is a linear operation, and the fact that every multiplication gate in the formula multiplies linear functions with disjoint support, a formula for $\partial_{x_i}f$ can be obtained from $\Phi$ by replacing every linear function in which $x_i$ appears with an appropriate constant. Therefore, every such function is removed from ${\mathcal B}$, and so the set of bad linear functions in $\partial_{x_i}f$ is of size at most
$|{\mathcal B}|- |{\mathcal B}|/n^\epsilon = |{\mathcal B}|\cdot (1-1/n^\epsilon)$. We continue this process for at most $O(n^\epsilon \cdot \log |{\mathcal B}|)$ steps, until we
reach a point where $|{\mathcal B}| < 1$ and so $|{\mathcal B}| = 0$.
Finally, it remains to be noted that $|{\mathcal B}| \le Mn$, since by multilinearity each multiplication gate multiplies linear functions with disjoint support, and so its fan-in is at most $n$. \end{proof}
\subsection{Reducing Bottom Support for Depth-$4$}
The process for depth-$4$ formulas follows the same outline as the depth-$3$ case, but there are a few more details. Given a parameter $t \in {\mathbb{N}}$, we want to transform any multilinear $\tfSPSP{M}$ formula computing a non-zero polynomial $f(x_1, \ldots, x_n)$ into a $\tfrsSPSP{M}{\tau}$ formula, while preserving non-zeroness. By Proposition~\ref{prop:simple}, we can assume any formula that we work with is a \emph{simple formula}. We again define the ``bad'' polynomials as those that contain many variables (that also appear in $f$). Our progress measure for their elimination will be the {\em total sparsity} of all bad polynomials, which we define below.
\begin{define}
Let $\tau \in {\mathbb{N}}$ and $\Phi = \sum_{i=1}^M\prod_{j=1}^{t_i} f_{i,j}$ be a multilinear $\tfSPSP{M}$
formula computing a non-zero multilinear polynomial $f(x_1, \ldots, x_n) \in {\mathbb{F}}[x_1, \ldots, x_n]$.
Let ${\mathcal B} = \{ f_{i,j} \mid \textsf{var}(f_{i,j}) > \tau \}$. We say that $\Phi$ is $\Delta$-far from a $\tfrsSPSP{M}{\tau}$ formula if
$$ \sum_{g \in {\mathcal B}} \| g \| = \Delta. $$
We also say that a polynomial $f(x_1, \ldots, x_n)$ is $\Delta$-far from a $\tfrsSPSP{M}{\tau}$ formula if it can be
computed by a formula that is $\Delta$-far from such a formula. \end{define}
\begin{obs}
Notice that a formula $\Phi$ is $0$-far from being $\tfrsSPSP{M}{\tau}$ iff $\Phi$ itself is a $\tfrsSPSP{M}{\tau}$ formula. \end{obs}
Now that we have a measure of how far a given $\tfSPSP{M}$ formula (computing a non-zero polynomial) is from being $\tfrsSPSP{M}{\tau}$, we can show the existence of a small set of variables such that when we either take derivatives or set these variables to zero, we obtain a $\tfrsSPSP{M}{\tau}$ formula computing another non-zero polynomial. Since we are working with \emph{simple} formulas, if a variable $x$ appears in a bad polynomial $f_{i,j} \in {\mathcal B}$, then it must be the case that $x \in \textsf{var}^*(f)$, and therefore we are free to either take a partial derivative with respect to $x$ or to set $x$ to zero, while preserving non-zeroness of the input polynomial $f(x_1, \ldots, x_n)$. Therefore, the non-zeroness condition is taken care of by simplicity.
We begin by showing that we can always make good progress in this measure. More precisely, we have the following lemma:
\begin{lem}\label{lem:red4}
Let $\Phi = \sum_{i=1}^M\prod_{j=1}^{t_i} f_{i,j}$ be a multilinear
$\tfSPSP{M}$ formula computing
a non-zero multilinear polynomial $f(x_1, \ldots, x_n) \in {\mathbb{F}}[x_1, \ldots, x_n]$.
If $\Phi$ is $\Delta$-far from a $\tfrsSPSP{M}{\tau}$ formula, then there exists $x \in \textsf{var}^*(f)$ such that one of the
polynomials ${\partial}_xf$ or $f|_{x=0}$ is non-zero and is at most $\Delta(1-\frac{\tau}{2n})$-far from a
$\tfrsSPSP{M}{\tau}$ formula. \end{lem}
\begin{proof}
By Proposition~\ref{prop:simple}, we can assume without loss of generality that $\Phi$ is simple.
We note that making $\Phi$ simple can only reduce $\Delta$.
Let ${\mathcal B} = \{ f_{i,j} \mid |\textsf{var}(f_{i,j})| > \tau \}$. Notice that by simplicity of $\Phi$, we have that
$|\textsf{var}(f_{i,j})| > 1 \Rightarrow \textsf{var}(f_{i,j}) \subseteq \textsf{var}^*(f)$. Since
$\Phi$ is $\Delta$-far from a $\tfrsSPSP{M}{\tau}$ formula, we have that
$$ \Delta = \sum_{g \in {\mathcal B}} \| g \| .$$
For each $x \in \textsf{var}^*(f)$, let $F_x = \{ g \in {\mathcal B} \mid x \in \textsf{var}(g) \}$. Notice that
$$ \displaystyle\sum_{x \in \textsf{var}^*(f)} \left( \displaystyle\sum_{g \in F_x} \| g \| \right) = \displaystyle\sum_{g \in {\mathcal B}} |\textsf{var}(g)| \cdot \| g \| >
\tau \cdot \displaystyle\sum_{g \in {\mathcal B}} \| g \| = \tau \Delta . $$
This implies that there exists $x \in \textsf{var}^*(f)$ for which
\begin{equation}\label{eq:dense-var}
\displaystyle\sum_{g \in F_x} \| g \| \ge \dfrac{\Delta \cdot \tau}{|\textsf{var}^*(f)|} \ge \dfrac{\Delta \cdot \tau}{n} .
\end{equation}
Since $ \| g \| = \| g|_{x = 0} \| + \| {\partial}_x g \| $ for any multilinear polynomial $g(x_1, \ldots, x_n)$,
we have that
$$\displaystyle\sum_{g \in F_x} \| g \| = \displaystyle\sum_{g \in F_x} \| g|_{x = 0} \| + \displaystyle\sum_{g \in F_x} \| {\partial}_x g \| . $$
Hence, by equation~\eqref{eq:dense-var}, one of $\displaystyle\sum_{g \in F_x} \| g|_{x = 0} \|$ or $\displaystyle\sum_{g \in F_x} \| {\partial}_x g \|$
must be larger than $\frac{\Delta \cdot \tau}{2n}$. If
$$ \displaystyle\sum_{g \in F_x} \| g|_{x = 0} \| > \frac{\Delta \tau}{2n} ,$$
by taking the derivative of $f$ with respect to $x$, we have that ${\partial}_x f \neq 0$ (since $x \in \textsf{var}^*(f)$) and that
the distance of ${\partial}_x \Phi$ to a $\tfrsSPSP{M}{\tau}$ formula is upper bounded by
$$ \displaystyle\sum_{g \in {\mathcal B} \setminus F_x} \| g \| + \displaystyle\sum_{g \in F_x} \| {\partial}_x g \| = \displaystyle\sum_{g \in {\mathcal B}} \| g \| -
\displaystyle\sum_{g \in F_x} \| g|_{x = 0} \| < \Delta - \frac{\Delta \tau}{2n} = \Delta\left(1-\frac{\tau}{2n}\right).$$
Notice that the bound above is an upper bound, since the new set of polynomials $f_{i,j}$ of large support must be a subset of
${\mathcal B}.$
Analogously, if $\sum_{C \in F_x} \| {\partial}_x C \| > \frac{\Delta \tau}{2n}$, then we take $f|_{x=0}$. The upper bounds are the same as those obtained for the first case. This finishes the proof of the lemma. \end{proof}
By repeatedly applying Lemma~\ref{lem:red4}, we obtain the following corollary, which guarantees the existence of a small set of variables that allow us to transform our $\tfSPSP{M}$ formula into a $\tfrsSPSP{M}{\tau}$ one.
\sloppy \begin{cor}[Reduction to Depth-$4$ with Small Bottom Support]\label{cor:depth4}
Let $\Phi$ be a multilinear simple $\tfSPSP{M}$ formula computing a non-zero multilinear
polynomial $f(x_1, \ldots, x_n) \in {\mathbb{F}}[x_1, \ldots, x_n]$. There exist disjoint sets $A, B \subset [n]$ with
$|A \sqcup B| \le \frac{2n}{\tau} \cdot \log(|\Phi|)$ such that the polynomial ${\partial}_Af|_{B=0}$ is non-zero and
can be computed by a simple multilinear $\tfrsSPSP{M}{\tau}$ formula $\Phi$. \end{cor}
\begin{proof}
Let $\Delta$ be such that $\Phi$ is $\Delta$-far from being $\tfrsSPSP{M}{\tau}$. Notice that $\Delta \le |\Phi|$.
We show by induction that there exist disjoint sets $A_k$ and $B_k$ such that
$|A_k \sqcup B_k| \le k$, and the polynomial
${\partial}_{A_k} f|_{B_k=0}$ is non-zero and at most $\Delta(1-\frac{\tau}{2n})^k$-far from being $\tfrsSPSP{M}{\tau}$.
For $k \ge 0$, define $A_k, B_k \subseteq [n]$, $f_k(x_1, \ldots, x_n) = {\partial}_{A_k} f|_{B_k=0}$ and $\Delta_k$ be an upper
bound on how far
$f_k(x_1, \ldots, x_n)$ is from being $\tfrsSPSP{M}{\tau}$. Initially, set $A_0 = B_0 = \emptyset$. In this case, we have
that $f_0(x_1, \ldots, x_n) = f(x_1, \ldots, x_n)$
and $\Delta_0 = \Delta = \Delta(1-\frac{\tau}{2n})^0$. This is the base case for our induction.
Suppose our hypothesis is true for some $k \ge 0$. If $\Delta(1-\frac{\tau}{2n})^k < 1$, then we know that our formula is already
$\tfrsSPSP{M}{\tau}$ and therefore we are done. Else, by applying Lemma~\ref{lem:red4}, we have that there is a variable
$x \in \textsf{var}^*(f_k)$ such that either ${\partial}_x f_k$ or $f_k|_{x=0}$ is (at most) $\Delta_k(1-\frac{\tau}{2n})$-far from being $\tfrsSPSP{M}{\tau}$.
Thus, $\Delta_k(1-\frac{\tau}{2n}) \le \Delta(1-\frac{\tau}{2n})^k \cdot (1-\frac{\tau}{2n}) = \Delta(1-\frac{\tau}{2n})^{k+1}$ and
$x \in \textsf{var}^*(f_k) \subseteq [n] \setminus (A_k \sqcup B_k)$. Therefore, if ${\partial}_x f_k$ is the close polynomial then
we set $A_{k+1} = A_k \cup \{x\}, B_{k+1} = B_k$. Otherwise, we set $A_{k+1} = A_k, B_{k+1} = B_k \cup \{x\}.$ It is
easy to see that the inductive properties hold in this case as well. This ends the inductive proof.
Since $\Delta(1-\frac{\tau}{2n})^k < 1$ for $k \ge \frac{2n}{\tau} \log \Delta$, and since
$\frac{2n}{\tau} \log(|\Phi|) \ge \frac{2n}{\tau} \log \Delta$,
it is enough to choose at most $\frac{2n}{\tau} \log(|\Phi|)$ variables. This proves this corollary. \end{proof}
\section{Hitting Set for $\rsSPS{n^{1-\epsilon}}$ and $\rsSPSP{n^{1-\epsilon}}$ Formulas} \label{sec:hit-bottom}
In this section we construct subexponential sized hitting set for the classes of $\tfrsSPS{M}{n^{1-\epsilon}}$ and $\tfrsSPSP{M}{n^{1-\epsilon}}$ multilinear formulas. Recall that in Section~\ref{sec:red} we showed how to reduce general depth-$3$ and depth-$4$ formulas to these types of formulas. In the next section, we will show how to tie all loose edges and combine the arguments of Section~\ref{sec:red} with those of this section in order to handle the general case.
An essential ingredient in our construction is a quasi-polynomial sized hitting set for Read-Once Algebraic Branching Programs (ROABPs) \cite{ForbesShpilka13,AgrawalGKS14}. We note that in our setting, we may assume that the reading order of the variables by the ABP is known.
\begin{thm}[\cite{ForbesShpilka13,AgrawalGKS14}] \label{thm:hitting-set-ROABP}
Let $\mathcal C$ be the class of $n$-variate polynomials computed by a ROABP of width $w$, such that the degree of each variable is at most $d$, over a field ${\mathbb{F}}$ so that $|{\mathbb{F}}| \ge \textsf{poly}(n,w,d)$. Then $\mathcal C$ has a hitting set of size $\textsf{poly}(n,w,d)^{\log n}$ that can be constructed in time $\textsf{poly}(n,w,d)^{\log n}$. \end{thm}
We begin by describing a unified construction for both $\tfrsSPS{M}{n^{1-\epsilon}}$ and $\tfrsSPSP{M}{n^{1-\epsilon}}$ formulas. We then describe how to set the parameters of the construction for each of the cases.
\begin{construct}[Hitting set for multilinear $\tfrsSPS{M}{n^{1-\epsilon}}$ and $\tfrsSPSP{M}{n^{1-\epsilon}}$ formulas] \label{con:subexp-hitting-set-gen-multilinear} Let $0<\delta<\epsilon$ and $n,k,s,M$ integers, such that $M=2^{n^\delta}$ and $k= n^{\delta} + 2\log n$. Set $m=10n^{1-(\epsilon+\delta)/2}$ and $t=k \log n$. Let $\mathcal F$ be a family of $k$-wise independent hash functions from $[n]$ to $[m]$, as in Lemma~\ref{lem:hash}. For every $h \in \mathcal F$, define the set $I_h$ as follows: \begin{enumerate} \item Partition the variables to sets\footnote{Recall that we associate subsets of $[n]$ with subsets of the variables, and make no distinction in the notation.} $T_1 \sqcup T_2 \sqcup \cdots \sqcup T_m = h^{-1}(1) \sqcup h^{-1}(2) \sqcup \cdots \sqcup h^{-1}(m)$.
\item For every $1 \le i\le m$, let $\mathcal H_i$ be a hitting set for ROABPs of width $M \cdot s^t$ and individual degree $d=1$ (as promised by Theorem~\ref{thm:hitting-set-ROABP}), on the variables of $T_i$ (of course, $|T_i|\leq n$).
\item We define $I_h$ as the set of all vectors $v$ such that the restriction of $v$ to the coordinates $T_i$, $v|_{T_i}$, is in $\mathcal H_i$. I.e., in the notation of Section~\ref{sec:notation}, $$I_h = \mathcal H_1^{T_1} \times \mathcal H_2^{T_2} \times \cdots \times \mathcal H_m^{T_m}.$$
\end{enumerate} Finally, define $\mathcal H = \bigcup_{h\in \mathcal F} I_h$. \end{construct}
The following lemma gives an upper bound to the size of the hitting set constructed in Construction~\ref{con:subexp-hitting-set-gen-multilinear}.
\begin{lem} \label{lem:subexp-hitting-set-gen-size} Let $\delta,\epsilon,k,n,s$ and $M$ be the parameters of Construction \ref{con:subexp-hitting-set-gen-multilinear}.
The set $\mathcal H$ constructed in Construction \ref{con:subexp-hitting-set-gen-multilinear} has size
$n^{O(k)} \cdot {\left (M \cdot s^{k \log n} \right)}^{\tilde{O}(n^{1-(\epsilon+\delta)/2})}={\left (M \cdot s^{k \log n} \right)}^{\tilde{O}(n^{1-(\epsilon+\delta)/2})}$, and it can be constructed in time $\textsf{poly}(|\mathcal H|)$. \end{lem}
\begin{proof} This is a direct consequence of the construction, Fact~\ref{fact:small-hash} and Theorem~\ref{thm:hitting-set-ROABP}. \end{proof}
\subsection{Depth-$3$ Formulas}
We begin by describing the argument for depth-$3$ formulas. The following lemma proves that indeed, by setting the proper parameters, the set $\mathcal H$ from Construction \ref{con:subexp-hitting-set-gen-multilinear} does hit $\tfrsSPS{M}{n^{1-\epsilon}}$ formulas.
\begin{lem} \label{lem:depth-$3$-multilinear-hitting-set-hits-small-support} Let $f(x_1,\ldots, x_n) \in {\mathbb{F}}[x_1,\ldots,x_n]$ be a multilinear polynomial computed by a multilinear $\tfrsSPS{M}{n^{1-\epsilon}}$ formula $\Phi = \sum_{i=1}^M \prod_{j=1}^{t_i} \ell_{i,j} $. Let $\mathcal H$ be the set constructed in Construction~\ref{con:subexp-hitting-set-gen-multilinear} with $s=k+1$. Then there exists a point $\ol{\alpha} \in \mathcal H$ such that $f(\ol{\alpha}) \neq 0$. \end{lem}
\begin{proof} For every multiplication gate $1 \le i \le M$ in $\Phi$, define a partition of the variables \[ \mathcal A_i = \{ \textsf{var}(\ell_{i,j}) \cap \textsf{var}(f) \mid 1 \le j \le t_i \}. \] Let $h \in \mathcal F$ be the function guaranteed by Lemma~\ref{lem:hash} with respect to the partition $\mathcal A_1,\ldots\mathcal A_M$, and assume the setup of Construction~\ref{con:subexp-hitting-set-gen-multilinear}. We claim that there exists $\ol{\alpha} \in I_h$ such that $f(\ol{\alpha}) \neq 0$.
To that end, consider the partition of the variables induced by $h$: \[ T_1 \sqcup T_2 \sqcup \cdots \sqcup T_m = h^{-1}(1) \sqcup h^{-1}(2) \sqcup \cdots \sqcup h^{-1}(m). \]
We view the polynomial as a polynomial $f_1$ in the variables of $T_1$, over the extension field ${\mathbb{F}}(T_2 \sqcup \cdots \sqcup T_n)$. We claim that $f_1$ can be computed by an ROABP of width $M \cdot {(k+1)}^{k \log n}$. To see this note that, by Lemma~\ref{lem:hash}, in any multiplication gate, at most $k \log n$ linear functions contain more than one variable from $T_1$, and each contains at most $k$ variables. It follows that the sparsity of every linear function (with respect to the variables in $T_1$) among those $k \log n$ functions, is at most $k+1$. By Lemma~\ref{lem:roabp-product-sparse-polys}, $f_1$ can be computed by an ROABP over ${\mathbb{F}}(T_2 \sqcup \cdots \sqcup T_n)$ of width $M \cdot {(k+1)}^{k \log n}$. By the hitting set property of Theorem~\ref{thm:hitting-set-ROABP},
there exists $\ol{\alpha_1} \in \mathcal H_1 \subseteq {\mathbb{F}}^{|T_1|}$ such that $f_2 \eqdef f_1|_{T_1 = \ol{\alpha_1}} \not\equiv 0$.
Similarly, the same conditions now hold for $f_2$, considered as a polynomial over the field ${\mathbb{F}}(T_3 \sqcup \cdots \sqcup T_n)$, and so there exists $\ol{\alpha_2} \in \mathcal H_2 \subseteq {\mathbb{F}}^{|T_2|}$ such that $f_3 \eqdef f_2|_{T_2 = \ol{\alpha_2}} \not\equiv 0$.
We continue this process for $m$ steps, and at the last step we find $\ol{\alpha_{m}}$ such that $f_{m-1}(\ol{\alpha_{m}}) = f(\ol{\alpha_1},\cdots,\ol{\alpha_m}) \neq 0$, with $(\ol{\alpha_1},\cdots,\ol{\alpha_m}) \in {\mathbb{F}}^n$ being the length $n$ vector obtained by filling the entires of $\ol{\alpha_i} \in {\mathbb{F}}^{|T_i|}$ in the positions indexed by $T_i$. \end{proof}
\subsection{Depth-$4$ Formulas}
The argument for depth-$4$ formulas is very similar, apart from a small change in the setting of the parameters.
\begin{lem} \label{lem:depth-$4$-multilinear-hitting-set-hits-small-support} Let $f(x_1,\ldots, x_n) \in {\mathbb{F}}[x_1,\ldots,x_n]$ be a multilinear polynomial computed by a multilinear $\tfrsSPSP{M}{n^{1-\epsilon}}$ formula $\Phi = \sum_{i=1}^M \prod_{j=1}^{t_i} f_{i,j} $. Let $\mathcal H$ be the set constructed in Lemma~\ref{con:subexp-hitting-set-gen-multilinear} with $s=2^k$. Then, there exists a point $\ol{\alpha} \in \mathcal H$ such that $f(\ol{\alpha}) \neq 0$. \end{lem}
\begin{proof} The proof is almost identical to that of Lemma~\ref{lem:depth-$3$-multilinear-hitting-set-hits-small-support}. In this case, for every $1 \le i \le M$ we define the partition \[ \mathcal A_i = \{ \textsf{var}(f_{i,j}) \mid 1 \le j \le t_i \}. \]
Note that the assumptions of Lemma~\ref{lem:hash} still hold, and so we denote by $h \in \mathcal F$ the function guaranteed by Lemma~\ref{lem:hash} with respect to the partitions $\mathcal A_1,\ldots\mathcal A_M$, and again claim that there exists $\ol{\alpha} \in I_h$ such that $f(\ol{\alpha}) \neq 0$.
Consider once more the partition on the variables induced by $h$, \[ T_1 \sqcup T_2 \sqcup \cdots \sqcup T_m = h^{-1}(1) \sqcup h^{-1}(2) \sqcup \cdots \sqcup h^{-1}(m), \] and view the polynomial as a polynomial $f_1$ in the variables of $T_1$, over the field ${\mathbb{F}}(T_2 \sqcup \cdots \sqcup T_n)$.
We now claim that $f_1$ can be computed by an ROABP of width $M \cdot {\left( 2^k \right)}^{k \log n} = 2^{k^2 \log n}$. The proof for this claim is exactly as in the depth-$3$ case, except that now the best bound we can give on the sparsity of each polynomial which intersects $T_1$ in more than one variable is $2^k$, as it is a multilinear polynomials in at most $k$ variables.
Similarly, we move on to handle $T_2,\ldots,T_m$ and obtain a point $\ol{\alpha}$ such that $f(\ol{\alpha}) \neq 0$. \end{proof}
\section{Hitting Set for Depth-$3$ and Depth-$4$ Multilinear Formulas} \label{sec:hit}
Recall that, in Section~\ref{sec:red}, we showed that any non-zero $\tfSPS{M}$ or $\tfSPSP{M}$ formula has a non-zero partial derivative (and, possibly, a restriction) which is computed by a non-zero $\tfrsSPS{M}{n^{1-\epsilon}}$ or $\tfrsSPSP{M}{n^{1-\epsilon}}$ formula, respectively. Then, in Section~\ref{sec:hit-bottom} we gave hitting sets for such formulas. In this section we provide the final ingredient, which is to show how to ``lift'' those hitting sets back to the general class, via a simple method, albeit one that requires some notation.
Handling restrictions is fairly easy, and causes no blow up in the hitting set size: If we have a set $\mathcal H \subseteq {\mathbb{F}}^{n-r}$ that hits $f|_{B=0}$ for some multilinear polynomial $f(x_1,\ldots,x_n)$ and $B \subseteq [n]$ with $|B|=r$, then simply extending $\mathcal H$ into a subset of ${\mathbb{F}}^n$ by filling out zeros in all the entries indexed by $B$ will hit $f$ itself.
In order to handle partial derivates, first note that if $f(x_1,\ldots,x_n)$ is a multilinear polynomial, then \[ \partial_{x_i} f = f(x_1,\ldots,x_{i-1},1,x_{i+1},\ldots,x_n) - f(x_1,\ldots,x_{i-1},0,x_{i+1},\ldots,x_n), \] and so if $\partial_{x_i}f (\ol{\alpha}) \neq 0$ for some $\ol{\alpha} \in {\mathbb{F}}^n$ then at least one of the two evaluations on the right hand side must be non-zero as well.
Applying this fact repeatedly, given a set $A \subseteq [n]$ we can evaluate $\partial_A f$ at any point by making $2^{|A|}$ evaluations of $f$. Motivated by this, we introduce the following notation:
\begin{define} \label{def:extension}
Let $f(x_1,\ldots,x_n) \in {\mathbb{F}}[x_1,\ldots,x_n]$ be a multilinear polynomial and $A,B \subseteq [n]$ be two disjoint subsets of variables with $|A|=r, |B|=r'$. Let $\mathcal H \subseteq {\mathbb{F}}^{n-(r+r')}$.
We define the ``lift'' of $\mathcal H$ with respect to $(A,B)$ to be $$\mathcal{L}_A^B(\mathcal H) = \left(\{0,1\}^{\vphantom{r'}r}\right)^A \times \left(\{0\}^{r'}\right)^B \times \mathcal H^{[n]\setminus (A\sqcup B)}.$$ In the special case where $B=\emptyset$, we simply denote $\mathcal{L}_A^B(\mathcal H) = \mathcal{L}_A (\mathcal H)$.
\end{define}
That is, for all $\ol{\alpha}\in\mathcal H$, $\mathcal{L}_A^B(\mathcal H)$ contains all the possible $2^r$ ways to extend $\ol{\alpha}$ into $\ol{\beta} \in {\mathbb{F}}^n$ by filling out zeros and ones within the $r$ entries that are indexed by $A$, and zeros in all the $r'$ entries indexed by $B$.
\subsection{Depth-$3$ Formulas}
In this section we prove Theorem~\ref{thm:intro:hitting-set-depth-$3$}. For the reader's convenience, we first restate the theorem:
\begin{thm}[Theorem~\ref{thm:intro:hitting-set-depth-$3$}, restated] \label{thm:hitting-set-depth-$3$}
Let $\mathcal C$ be the class of multilinear $\tfSPS{M}$ formulas for $M=2^{n^\delta}$. There exists a hitting set $\mathcal H$ of size $|\mathcal H| = 2^{\tilde{O}(n^{2/3+2\delta/3})}$ for $\mathcal C$, that can be constructed in time $\textsf{poly}(|\mathcal H|)$. \end{thm}
The size of the hitting set is subexponential for any constant $\delta < 1/2$. Also, if $M=\textsf{poly}(n)$ then the size of the hitting set is $2^{\tilde{O}(n^{2/3})}$.
With Definition~\ref{def:extension} in hand, we now provide our construction for $\tfSPS{M}$ formulas, towards the proof of Theorem~\ref{thm:hitting-set-depth-$3$}.
\begin{construct}[Hitting set for multilinear $\tfSPS{M}$ formulas] \label{con:hit3}
Let $M=2^{n^\delta}$ and $\epsilon =2/3 - \delta/3$. Let $r = \tilde{O}(n^{\epsilon} \log M) = \tilde{O}(n^{\frac{2}{3} + \frac{2}{3}\delta})$ as guaranteed by Lemma~\ref{lem:red3}. For every $A \in \binom{[n]}{\le r}$, construct a set $\mathcal H_A \in {\mathbb{F}}^{n-|A|}$ using Construction~\ref{con:subexp-hitting-set-gen-multilinear} with parameters $\delta,\epsilon,n,k,s=k+1$ and $M$ (recall that in Construction~\ref{con:hit3} we set $k=n^{\delta} + 2 \log n$). Finally, let \[ \mathcal H = \bigcup_{A \in \binom{[n]}{\le r}} \mathcal{L}_A(\mathcal H_A). \] \end{construct}
We are now ready to prove Theorem~\ref{thm:hitting-set-depth-$3$}:
\begin{proof}[Proof of Theorem \ref{thm:hitting-set-depth-$3$}]
We show that the set $\mathcal H$ constructed in Construction~\ref{con:hit3} has the desired properties. First, note that by Lemma~\ref{lem:subexp-hitting-set-gen-size}, for every $A \subseteq [n]$ with $$|A| \leq \tilde{O}(n^{\epsilon} \log M) =\tilde{O}(n^{2/3-\delta/3} \log M)= \tilde{O}(n^{\frac{2}{3} + \frac{2}{3}\delta}),$$ the set $\mathcal H_A$ has size \[ (M \cdot (k+1)^{k \log n})^{\tilde{O}(n^{2/3-\delta/3})} = 2^{\tilde{O}(n^{2/3+2\delta/3})}, \] where we have used the fact that, in Construction~\ref{con:hit3}, we take $k=n^{\delta} + 2 \log n$. It therefore follows that \[
|\mathcal{L}_A(\mathcal H_A)| \le 2^{|A|} \cdot |\mathcal H_A| = 2^{\tilde{O}(n^{2/3+2\delta/3})}, \] and that \[
|\mathcal H| \le \sum_{i=0}^{\tilde{O}(n^{\frac{2}{3} + \frac{2}{3}\delta})} \sum_{A \subseteq [n], |A| = i} |\mathcal{L}_A(\mathcal H_A)| = 2^{\tilde{O}(n^{2/3+2\delta/3})}. \]
To show the hitting property of $\mathcal H$, let $f(x_1,\ldots,x_n)$ be a non-zero multilinear polynomial computed by a $\tfSPS{M}$ formula, and let $A' \subseteq [n]$ be the set guaranteed by Lemma~\ref{lem:red3}. Thus, $|A'| \leq \tilde{O}(n^{\epsilon} \log M)= \tilde{O}(n^{\frac{2}{3} + \frac{2}{3}\delta})$. Then by Lemma~\ref{lem:depth-$3$-multilinear-hitting-set-hits-small-support}, there exists $\alpha \in \mathcal H_{A'}$ such that $\partial_{A'} f (\ol{\alpha}) \neq 0$, and so there must exist \[ \ol{\beta} \in \mathcal{L}_{A'}(\mathcal H_{A'}) \subseteq \mathcal H \] such that $f(\ol{\beta}) \neq 0$. \end{proof}
\subsection{Depth-$4$ Formulas}
Moving on to depth-$4$, the construction and proof are both very similar, with a slight change in the parameters. We first give a slightly more general form of Theorem~\ref{thm:intro:hitting-set-depth-$4$} that we will later use for regular formulas.
\begin{thm}[General theorem for multilinear $\Sigma\Pi\Sigma\Pi$ formulas] \label{thm:hitting-set-depth-$4$-general}
Let $\mathcal C$ be the class of multilinear $\Sigma\Pi\Sigma\Pi$ formulas of top fan-in $M$ and size $S$ so that $(\log M)^3\cdot \log S = o(n)$. There exists a hitting set $\mathcal H$ of size $|\mathcal H| = 2^{\tilde{O}( n^{2/3} \cdot \log M \cdot (\log S)^{1/3} )}$ for $\mathcal C$, that can be constructed in time $\textsf{poly}(|\mathcal H|)$. \end{thm}
We note that Theorem~\ref{thm:intro:hitting-set-depth-$4$} is an immediate corollary of Theorem~\ref{thm:hitting-set-depth-$4$-general}.
\begin{proof}[Proof of Theorem~\ref{thm:intro:hitting-set-depth-$4$}]
Apply Theorem~\ref{thm:hitting-set-depth-$4$-general} with $M=S=2^{n^{\delta}}$ for some constant $0<\delta<1/4$ (the bound in Theorem~\ref{thm:intro:hitting-set-depth-$4$} is meaningless for $\delta\geq 1/4$). It is clear that the conditions of Theorem~\ref{thm:hitting-set-depth-$4$-general} are met. Thus, we obtain a hitting set of size $|\mathcal H| = 2^{\tilde{O}( n^{2/3} \cdot \log M \cdot (\log S)^{1/3} )} = 2^{\tilde{O}( n^{2/3 + 4\delta/3})}$ for the class. \end{proof}
For the proof of Theorem~\ref{thm:hitting-set-depth-$4$-general} we will use the following construction that is similar to Construction~\ref{con:hit3}.
\begin{construct}[Hitting set for multilinear $\Sigma\Pi\Sigma\Pi$ formulas] \label{con:hit4-general} Let $M$ and $S$ be such that $(\log M)^3\cdot \log S = o(n)$. Denote $M=2^{n^\delta}$ (hence $S = 2^{o(n^{1-3\delta})}$).
Let $\epsilon$ be such that $$n^\epsilon = n^{2/3}\cdot \log M/(\log S)^{2/3}.$$ Set $$r =2n^{\epsilon} \log S = 2n^{2/3} \cdot \log M \cdot (\log S)^{1/3}.$$
For every two disjoint sets $A,B \subseteq [n]$ with $|A|,|B|\le r$, construct a set $\mathcal H_{A,B} \in {\mathbb{F}}^{n-(|A|+|B|)}$ using Construction~\ref{con:subexp-hitting-set-gen-multilinear} with parameters $\delta,\epsilon,n,k,s=2^k$ and $M$ (recall that in Construction~\ref{con:subexp-hitting-set-gen-multilinear} we set $k=n^\delta + 2 \log n$). Finally, let \[ \mathcal H = \bigcup_{\substack{A,B \in \binom{[n]}{\le r} \\ A\cap B=\emptyset}} \mathcal{L}_A^B(\mathcal H_{A,B}). \] \end{construct}
\begin{proof}[Proof of Theorem~\ref{thm:hitting-set-depth-$4$-general}] We show that the set $\mathcal H$ constructed in Construction~\ref{con:hit4-general} has the desired properties.
First, note that by Lemma~\ref{lem:subexp-hitting-set-gen-size}, for every $A, B \subseteq [n]$ with
$|A|, |B| \le 2n^{\epsilon} \log S$, the set $\mathcal H_{A,B}$ has size \[{\left (M \cdot 2^{k^2 \log n} \right)}^{\tilde{O}(n^{1-(\epsilon+\delta)/2})} = {\left ( 2^{k^2 \log n} \right)}^{\tilde{O}(n^{1-(\epsilon+\delta)/2})}=2^{\tilde{O}(n^{1-\epsilon/2 +3\delta/2})} , \] for $k=n^\delta + 2 \log n$. It therefore follows that \[
|\mathcal{L}_A^B(\mathcal H_{A,B})| \le 2^{|A|} \cdot |\mathcal H_{A,B}| = 2^{2n^{\epsilon} \log S} \cdot 2^{\tilde{O}(n^{1-\epsilon/2 +3\delta/2})}, \] and that \[
|\mathcal H| \le \sum_{i,j=0}^{2n^{\epsilon} \log S}
\sum_{\substack{A \subseteq [n] \\ |A| = i}}
\sum_{\substack{B \subseteq [n] \\ |B| = j}} |\mathcal{L}_A^B(\mathcal H_{A,B})| = 2^{\tilde{O}(n^{\epsilon} \log S)} \cdot 2^{\tilde{O}(n^{1-\epsilon/2 +3\delta/2})}. \] By our setting of parameters \begin{align*}
|\mathcal H| \le 2^{\tilde{O}(n^{\epsilon} \log S)} \cdot 2^{\tilde{O}(n^{1-\epsilon/2 +3\delta/2})} &= 2^{\tilde{O}(n^{\epsilon} \log S + n^{1-\epsilon/2 +3\delta/2})} \\ &=^{(\dagger)} 2^{\tilde{O}( n^{2/3} \cdot \log M \cdot (\log S)^{1/3} )} , \end{align*}
where equality $(\dagger)$ follows from our choice of $\epsilon$ in Construction~\ref{con:hit4-general} and the fact that $\log M = n^\delta$.
To show the hitting property of $\mathcal H$, let $f(x_1,\ldots,x_n)$ be a non-zero multilinear polynomial computed by a multilinear $\tfSPSP{M}$ formula of size $S$, and let $A',B' \subseteq [n]$ be
the sets guaranteed by Lemma~\ref{cor:depth4} with $\tau=n^{1-\epsilon}$. Thus, $|A'|,|B'|\leq 2n^{\epsilon} \log S$. Then, by Corollary~\ref{lem:depth-$4$-multilinear-hitting-set-hits-small-support}, there exists $\ol{\alpha} \in \mathcal H_{A',B'}$ such that
$\partial_{A'} f|_{B'=0} (\ol{\alpha}) \neq 0$, and so there must exist \[ \ol{\beta} \in \mathcal{L}_{A'}^{B'}(\mathcal H_{A',B'}) \subseteq \mathcal H \] such that $f(\ol{\beta}) \neq 0$. \end{proof}
\section{Multilinear Depth-$d$ Regular Formulas} \label{sec:regular}
\subsection{Definition and Background}
In \cite{KayalSS14}, Kayal et al.\ define \emph{regular formulas}, which consist of formulas with alternating layers of sum and product gates such that the fan-in of all the gates in any fixed layer is the same. In addition, they require the formal (syntactic) degree of the formula must be at most twice the (total) degree of its output polynomial. They showed that any $n^{O(1)}$-sized arithmetic circuit can be computed by a regular formula of size $n^{O(\log^2 n)}$ and proved a lower bound of $n^{\Omega(\log n)}$ on the size of regular formulas that compute some explicit polynomial in $\textsf{VNP}$.
In this paper, we consider \emph{multilinear regular formulas}, which are regular formulas with the extra condition that each gate computes a multilinear polynomial. However, we will remove the bound on the formal degree of the formula. More precisely, we have the following definition:
\begin{define}[Multilinear Regular Formulas]\label{def:reg-form}
We say that a formula $\Phi$ is a multilinear $(a_1, p_1, a_2, p_2, \ldots, a_d, p_d, a_{d+1})$-regular formula computing a
multilinear polynomial $f(x_1, \ldots, x_n)$ if it can be computed by a multilinear
$ \Sigma^{[a_1]}\Pi^{[p_1]}\Sigma^{[a_2]}\Pi^{[p_2]}\ldots\Sigma^{[a_d]}\Pi^{[p_d]}\Sigma^{[a_{d+1}]}\text{-formula} $.
Notice that the size of such a formula is $(\prod_{1 \le i \le d+1} a_i) \cdot (\prod_{1 \le i \le d} p_i)$ and the formal
degree of such a formula is given by $\deg(\Phi) = \prod_{1 \le i \le d} p_i$. Since the formula is multilinear, we have that
$\deg(\Phi) \le n$. \end{define}
Comparing with the definition given in Section~\ref{sec:def}, an $(a_1, p_1, a_2, p_2, \ldots, a_d, p_d, a_{d+1})$-regular formula has depth $2d+1$.
\subsection{Reduction to Depth-$4$ Formulas}
In this section, we reduce a multilinear depth-$d$ regular formula to a depth-$4$ formula. We first give a depth reduction lemma (Lemma~\ref{lem:squeeze}) that tells us that we can reduce the depth by one with a slight blow up in the fan-ins of the regular formula. We then use this lemma to obtain a depth-$4$ formula. The idea is to break the regular formula into two formulas (the top part and the bottom part), and then to apply the depth reduction lemma separately to these two formulas. The delicate part is that we wish to obtain a depth-$4$ formula that has a subexponential size hitting set, as in Theorem~\ref{thm:hitting-set-depth-$4$-general}. For this we need the top fan-in $M$ and the total size $S$ to satisfy that $(\log M)^3\cdot \log S =o(n)$. To achieve this we should carefully select the point in which to divide the formula. This is done in Theorem~\ref{thm:squeeze}.
We start with the depth reduction lemma.
\begin{lem}[Depth Reduction Lemma]\label{lem:squeeze}
Let $\Psi$ be a multilinear $(a_1, p_1, a_2, p_2, 1)$-regular formula computing a polynomial $f(x_1, \ldots, x_n)$.
Then, there exists a multilinear $(a_1a_2^{p_1}, p_1p_2, 1)$-regular formula $\Phi$ computing $f(x_1, \ldots, x_n)$. \end{lem}
\begin{proof}
Notice that a multilinear $(a_1, p_1, a_2, p_2, 1)$-regular formula is a $\Sigma^{[a_1]}\Pi^{[p_1]}\Sigma^{[a_2]}\Pi^{[p_2]}$
formula. Writing $\Psi$ as $\sum_{i=1}^{a_1}\prod_{j_i=1}^{p_1}\sum_{k_{j_i}=1}^{a_2} m(i, j_{i}, k_{j_{i}})$, where each
$m(i, j_{i}, k_{j_{i}})$ is
a monomial that is a product of $p_2$ input gates, and by expanding the expression above by computing all the products,
we obtain:
\begin{equation}
\label{eq:regular-squeeze}
\displaystyle\sum_{i=1}^{a_1}\prod_{j_i=1}^{p_1}\sum_{k_{j_i}=1}^{a_2} m(i, j_{i}, k_{j_{i}}) =
\displaystyle\sum_{i=1}^{a_1}\left(\sum_{k_{i,1}=1}^{a_2}\sum_{k_{i,2}=1}^{a_2} \ldots \sum_{k_{i,p_1}=1}^{a_2}
\prod_{t=1}^{p_1} m(i, t , k_{i,t}) \right).
\end{equation}
Since $m(i, j_{i}, k_{j_{i}})$ is a product of $p_2$ input gates, and since the right hand side of \eqref{eq:regular-squeeze} computes a product of $p_1$ of these
terms, each monomial computed by $\prod_{t=1}^{p_1} m(i, t , k_{i,t})$ is a product of $p_1p_2$ input gates.
Since the sums on the right hand side of \eqref{eq:regular-squeeze} are over all tuples of the form $(i, k_{i,1}, k_{i,2} \ldots, k_{i,p_1}) \in [a_1] \times [a_2]^{p_1}$,
we have that there are exactly $a_1 \cdot a_2^{p_1}$ summands. Hence, the right hand side of \eqref{eq:regular-squeeze} is the expression of a multilinear
$(a_1a_2^{p_1}, p_1p_2, 1)$-regular formula. \end{proof}
By repeatedly applying the depth reduction lemma above, we obtain the following theorem:
\begin{thm}[Depth Reduction of Regular Formulas]\label{thm:squeeze}
Let $d \ge 2$ be an integer, $c \in {\mathbb{R}}$ a constant such that $c \ge 3$, and $\Psi$ a multilinear
$(a_1, p_1, a_2, p_2, \ldots, a_d, p_d, a_{d+1})$-regular formula
of size $S$ computing a multilinear polynomial $f(x_1, \ldots, x_n)$. Then, one of the following conditions
must happen:
\begin{enumerate}[(i)]
\item \label{item:bounded-size} For $M=S$, there exists a $\tfSPSP{M}$ formula of size $O(S \cdot n^{n^{1-(1/c)^d}})$ computing
$f(x_1, \ldots, x_n)$, or
\item \label{item:bounded-fan-in} There exists $t \in [d-1]$ such that
there is a multilinear $\tfSPSP{M}$ formula $\Phi$ computing $f(x_1, \ldots, x_n)$,
with
$$ M = S^{n^\alpha} \ \text{ and } \ |\Phi| \le 2Mn \cdot n^{n^{1-(c-1)\alpha}} , $$
for $ \alpha = \frac{1}{c-1} \cdot \left(\frac{1}{c}\right)^{d-t}$.
\end{enumerate} \end{thm}
\begin{proof}
Recall that the size of the formula $S$ satisfies $S=(\prod_{1 \le i \le d+1} a_i) \cdot (\prod_{1 \le i \le d} p_i)$ .
We have three cases to analyze:
\paragraph{Case 1 (small total degree):} If $\prod_{i=1}^d p_i \le n^{1-(1/c)^d}$, then we can simply write the polynomial
$f(x_1, \ldots, x_n)$ as a sum of monomials, which would give us a multilinear $\Sigma \Pi$ formula $\Phi$
of size
$$ |\Phi| \le \displaystyle\sum_{i=0}^{n^{1-(1/c)^d}} \binom{n}{i} = O(n^{n^{1-(1/c)^d}}), $$
which is clearly of the form $\tfSPSP{1} \subseteq \tfSPSP{S}$ and of the required size for item (\ref{item:bounded-size}).
\paragraph{Case 2 (large $p_1$):} If $p_1 > n^{(1/c)^d}$, then notice that the regular formula $\Psi$ can
be written in the form
$$ \sum_{i=1}^{a_1}\prod_{j=1}^{p_1} f_{i,j} , $$
where each $f_{i,j}$ is a multilinear $(a_2, p_2, \ldots, a_d, p_d, a_{d+1})$-regular formula. Hence, each $f_{i,j}$ is a polynomial
of degree bounded by $\prod_{i=2}^d p_i \le n/p_1 < n^{1-(1/c)^d}$. Therefore, expanding each $f_{i,j}$ into
a sum of monomials, we obtain a formula $\Phi$ of the form $\tfSPSP{a_1}$ and of size
$$ |\Phi| \le a_1 \cdot p_1\cdot \sum_{i=0}^{n^{1-(1/c)^d}} \binom{n}{i} = O(a_1 p_1 n^{n^{1-(1/c)^d}}) =
O(S \cdot n^{n^{1-(1/c)^d}}) . $$
This too satisfies item (\ref{item:bounded-size}).
\paragraph{Case 3 (high degree but small $p_1$):} In this case, we can assume that $\prod_{i=1}^d p_i > n^{1-(1/c)^d}$ and that $p_1 \le n^{(1/c)^d}$.
It follows there exists an index
$t \in [d-1]$ satisfying
$$ p_t \le n^{(1/c)^{d+1-t}} \ \text{ and } \ p_{t+1} > n^{(1/c)^{d+1-(t+1)}} = n^{(1/c)^{d-t}},$$
since otherwise, using $c\geq 3$, we would have that
$$ \prod_{i=1}^d p_i \le \prod_{i=1}^d n^{(1/c)^{d+1-i}} = n^{\sum_{i=1}^d (1/c)^{d+1-i}} < n^{\frac{1}{c-1}} < n^{1-(1/c)^d}, $$
which contradicts the assumption on the degree.
Notice that we can express $\Psi$ in the form
\begin{equation}\label{eq:Psi}
\displaystyle\sum_{i_1=1}^{a_1}\prod_{j_1=1}^{p_1} \ldots \displaystyle\sum_{i_{t+1}=1}^{a_{t+1}}\prod_{j_{t+1}=1}^{p_{t+1}}
f_{i_1,\ldots, i_{t+1},j_1, \ldots , j_{t+1}},
\end{equation}
where each $f_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}$ is a multilinear $(a_{t+2}, p_{t+2}, \ldots, a_d, p_d, a_{d+1})$-regular formula.
We shall analyze separately each of the $f_{i_1,\ldots , i_{t+1}, j_1 , \ldots, j_{t+1}}$ and the $(a_1, p_1, \ldots, a_{t+1}, p_{t+1}, 1)$-regular formula ``above'' them.
Notice that each $f_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}$ is a polynomial
of degree bounded by $$\prod_{i=t+2}^d p_i < n/p_{t+1} < n^{1-(1/c)^{d-t}}.$$
Therefore, when expressing each $f_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}$ as
a sum of monomials, its sparsity is upper bounded by
\begin{equation}\label{eq:sparse-bottom}
\displaystyle\sum_{i=0}^{n^{1-(1/c)^{d-t}}} \binom{n}{i} \le 2n^{n^{1-(1/c)^{d-t}}}.
\end{equation}
Now, if in \eqref{eq:Psi} we replace each polynomial $f_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}$ with a new variable $y_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}$, then we get an $(a_1, p_1, \ldots, a_{t+1}, p_{t+1}, 1)$-regular formula in the $y$ variables $$\Phi_1 = \displaystyle\sum_{i_1=1}^{a_1}\prod_{j_1=1}^{p_1} \ldots \displaystyle\sum_{i_{t+1}=1}^{a_{t+1}}\prod_{j_{t+1}=1}^{p_{t+1}} y_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}.$$
It is clear that $\Psi$ is the composition
of $\Phi_1$ with the assignment $y_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}=f_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}$.
By applying the Depth Reduction (Lemma~\ref{lem:squeeze}) repeatedly to $\Phi_1$, we obtain that
$\Phi_1$ becomes a multilinear $\left( \prod_{i=1}^{t+1} a_i^{\pi_{i-1}}, \pi_{t+1}, 1 \right)$-regular
formula $\Phi_2$, where $\pi_k = \prod_{i=1}^k p_i$, for any $1 \le k \le d$ (and $\pi_0=1$).
Composing $\Phi_2$ with the assignment $y_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}=f_{i_1,\ldots, i_{t+1},j_1,\ldots,j_{t+1}}$, we obtain the following
depth-$4$ regular formula $\Phi$ (after some proper relabelling):
\begin{equation}\label{eq:red-form}
\Phi = \displaystyle\sum_{i=1}^{M}\prod_{j=1}^{\pi_{t+1}} f_{i,j}, \ \text{ where } \ M = \displaystyle\prod_{i=1}^{t+1} a_i^{\pi_{i-1}}.
\end{equation}
Notice that, by our choice of the parameter $t$, we have
$$ M = \displaystyle\prod_{i=1}^{t+1} a_i^{\pi_{i-1}} \le S^{\pi_t} = S^{\prod_{i=1}^t p_i} \le S^{\prod_{i=1}^t n^{(1/c)^{d+1-i}}} < S^{n^\alpha},
\ \text{ where } \ \alpha = \left(\frac{1}{c}\right)^{d-t} \frac{1}{c-1} . $$
Since, by equation~\eqref{eq:sparse-bottom}, each $f_{i,j}$ has sparsity bounded by
$2n^{n^{1-(1/c)^{d-t}}} = 2n^{n^{1-(c-1)\alpha}}$,
we have that $\Phi$ is a $\tfSPSP{M}$ formula of size bounded by:
$$ |\Phi| \le M \cdot \pi_{t+1} \cdot 2n^{1-(c-1)\alpha} \le M \cdot n \cdot 2n^{1-(c-1)\alpha} . $$
This satisfies the conditions of item (\ref{item:bounded-fan-in}), and so this concludes the proof of the theorem. \end{proof}
\subsection{PIT for Regular Formulas}
In this section, we construct our hitting set for regular formulas. Since the reduction done in Theorem~\ref{thm:squeeze} reduces a multilinear depth-$d$ regular formula to one of two types of depth-$4$ formulas, our hitting set will be the union of two hitting sets, $\mathcal F_d$ and $\mathcal G_d$, each one designed to hit a specific type.
\begin{thm}[Theorem~\ref{thm:intro:hitting-set-regular}, restated]\label{thm:hitting-set-regular}
For $d\geq 2$, let $\mathcal C_d$ be the class of multilinear polynomials computed by
$(a_1, p_1, a_2, p_2, \ldots, a_d, p_d, a_{d+1})$-regular formulas of size $S \le 2^{n^\delta}$
computing a multilinear polynomial $f(x_1, \ldots, x_n)$, where $\delta = \frac{1}{5^{d+1}}$.
Then, there exists a hitting set $\mathcal H_d$ of size
$|\mathcal H_d| = 2^{\tilde{O}(n^{1-\delta/3})}$ for $\mathcal C_d$, that can be constructed in time $\textsf{poly}(|\mathcal H_d|)$. \end{thm}
The proof follows by combining Theorem~\ref{thm:squeeze} with the hitting set guaranteed by Theorem~\ref{thm:hitting-set-depth-$4$-general}.
\begin{proof}
Let $f(x_1, \ldots, x_n)$ be a polynomial computed by a formula $\Psi \in \mathcal C_d$.
By Theorem~\ref{thm:squeeze}, with the constant $c =5$, there are two cases of the depth reduction to analyze. For each case we will give a hitting set (using Theorem~\ref{thm:hitting-set-depth-$4$-general}) and thus the union of the sets will be a hitting set for $\mathcal C_d$.
\paragraph*{Case 1:} For $M=S\leq 2^{n^\delta}$, there exists a $\tfSPSP{M}$ formula $\Phi$ of size $|\Phi| = O( S \cdot n^{n^{1-(1/5)^d}}) =
O( 2^{n^\delta} \cdot n^{n^{1-5\delta}})$ computing $f(x_1, \ldots, x_n)$.
By Theorem~\ref{thm:hitting-set-depth-$4$-general}, there exists a hitting set $\mathcal H'$ that hits all such non-zero formulas with
$$|\mathcal H'| = 2^{\tilde{O}( n^{2/3} \cdot \log M \cdot (\log |\Phi|)^{1/3} )}.$$
Observe that
$$(\log M)^3 \cdot \log|\Phi| = n^{3\delta}\cdot (n^\delta + n^{1-5\delta} \cdot \log n) = n^{4\delta} + n^{1-2\delta}\log n = \tilde{O}(n^{1-2\delta}).$$
Hence $$|\mathcal H'| = 2^{\tilde{O}( n^{2/3} \cdot \log M \cdot (\log |\Phi|)^{1/3} )} = 2^{\tilde{O}(n^{2/3}\cdot n^{(1-2\delta)/3})}=2^{\tilde{O}(n^{1 - 2\delta/3})}.$$
\paragraph*{Case 2:} There exists $t \in [d-1]$ such that for $ \alpha_t = \frac{1}{4} \cdot \left(\frac{1}{5} \right)^{d-t} \leq \frac{1}{20}$,
there exists a multilinear $\tfSPSP{M}$ formula $\Phi$ computing $f(x_1, \ldots, x_n)$,
where the top fan-in $ M = S^{n^{\alpha_t}} = 2^{n^{\delta + \alpha_t}}$ and the size is bounded by
$|\Phi| \le 2Mn \cdot n^{n^{1-4\alpha_t}}. $ Again, by Theorem~\ref{thm:hitting-set-depth-$4$-general}, there exists a hitting set $\mathcal H''$ that hits all such non-zero formulas with
$$|\mathcal H''| = 2^{\tilde{O}( n^{2/3} \cdot \log M \cdot (\log |\Phi|)^{1/3} )}.$$
We now have that
$$(\log M)^3 \cdot \log|\Phi| = \tilde{O}\left( n^{3(\delta + \alpha_t)} \cdot (n^{\delta + \alpha_t} + n^{1-4\alpha_t} )\right) = \tilde{O}\left( n^{4(\delta + \alpha_t)} + n^{1+3\delta-\alpha_t} \right) = \tilde{O}\left( n^{1-\delta}\right),$$
where the last equality holds as, by our choice of parameters, for all $t\in [d-1]$, $5\delta + 4\alpha_t<1$ and $4\delta < \alpha_t$. This implies that
\[ |\mathcal H''| = 2^{\tilde{O}( n^{2/3} \cdot \log M \cdot (\log |\Phi|)^{1/3} )} = 2^{\tilde{O}(n^{2/3} \cdot n^{(1-\delta)/3})} = 2^{\tilde{O}(n^{1-\delta/3})} . \qedhere \] \end{proof}
We note that we did not attempt to optimize the parameters in the theorem as, using our current proof, the exponent is going to be of the form $n^{1-1/\exp(d)}$ anyway.
\section{Lower Bounds for Bounded Depth Multilinear Formulas} \label{sec:lower-bounds}
As we noted earlier, the connection between construction of hitting sets and lower bounds for explicit polynomials is well established. The following theorem was proved by Heintz and Schnorr \cite{HeintzSchnorr80} and Agrawal \cite{Agrawal05}, albeit we cite only a special case which matches our use of it:
\begin{thm}[A special case of \cite{HeintzSchnorr80,Agrawal05}] \label{thm:lowerbound-from-hitting}
Suppose there is a black-box deterministic PIT algorithm for a class $\mathcal C$ of multilinear circuits, that outputs a hitting set $\mathcal H$ of size $|\mathcal H| = 2^{n^\alpha}<2^{n}$ and runs in time $\textsf{poly}(|\mathcal H|)$, such that $\mathcal H$ hits circuits of size at most $2^{n^\delta}$. Then, there exists a multilinear polynomial $f(x_1,\ldots,x_n)$ such that any circuit from the class $\mathcal C$ computing $f$ must be of size at least $2^{n^{\delta}}$, and the coefficients of $f$ can be found in time $2^{O(n)}$. \end{thm}
Theorem~\ref{thm:lowerbound-from-hitting} is proved by finding a non-zero polynomial $f(x_1,\ldots,x_n)$ which vanishes on the entire hitting set $\mathcal H$ of size $2^{n^\alpha}$, and then, by definition, $f$ cannot have circuits of size $2^{n^\delta}$. Finding $f$ amounts to finding a non-zero solution to a homogenous system of linear equations whose unknowns are the coefficients of the $2^{n}$ possible multilinear monomials in
$x_1,\ldots,x_n$. As long as $2^n > |\mathcal H| = 2^{n^\alpha}$, a non-zero solution is guaranteed to exist.
Our lower bounds now follow as a direct application of our hitting set constructions and Theorem~\ref{thm:lowerbound-from-hitting}.
\begin{proof}[Proofs of Corollaries \ref{cor:intro:lowerbound-depth-$3$}, \ref{cor:intro:lowerbound-depth-$4$} and \ref{cor:intro:lowerbound-regular}] In light of Theorem~\ref{thm:lowerbound-from-hitting}, we only need to pick $\delta$ so that the hitting sets we constructed have size less than $2^n$. The appropriate choices, by Theorems \ref{thm:intro:hitting-set-depth-$3$}, \ref{thm:intro:hitting-set-depth-$4$} and \ref{thm:intro:hitting-set-regular}, respectively, can be seen to be $\delta=1/2 - O(\log \log n / \log n)$ (for depth-$3$), $\delta=1/4-O(\log \log n / \log n)$ (for depth-$4$) and $\delta = \frac{1}{5^{\lfloor d/2 \rfloor+1}} = O\left(\frac{1}{\sqrt{5}^d}\right)$
(for depth-$d$ regular formulas). \end{proof}
\section{Conclusion and Open Questions}\label{sec:conclusion} \label{sec:open}
We conclude this paper with some obvious open problems. First, as noted in Section~\ref{sec:results}, the lower bounds that we get from our hitting sets are not as good as the best lower bounds for these models. Can one improve our construction to yield lower bounds matching the best known lower bounds?
Currently, the size of the hitting set that we get for depth-$d$ regular multilinear formulas is roughly $\exp(n^{1-1/\exp(d)})$. Can the bound be improved to $\exp(n^{1-\Omega(1/d)})$ ? In our proof the reason for this exponential loss is that we reduce the regular formula to a $\tfSPSP{M}$ formula of size $S$ and we need $M$ and $S$ to satisfy (because of Theorem~\ref{thm:hitting-set-depth-$4$-general}) $(\log M)^3\cdot \log S = o(n)$. In particular, if $M=2^{n^{\delta}}$ and $S=2^{n^{\gamma}}$ then we require that $3\delta + \gamma <1$. Notice that, in the depth reduction theorem (Theorem~\ref{thm:squeeze}), if we start with a regular formula of size $2^{n^\delta}$ then, if we break the formula at layer $t$, we roughly get a top fan-in of $M=2^{n^\delta \cdot p_1\cdot p_2 \cdots p_t}$ and bottom sparsity of (roughly) $\exp(n^{1-p_{t+1}})$. This gives a size upper bound of (roughly) $S=2^{n^\delta \cdot p_1\cdot p_2 \cdots p_t} \cdot \exp(n^{1-p_{t+1}})$. To match the requirement $(\log M)^3\cdot \log S = o(n)$, we get that $p_{t+1}$ must be larger than $3p_1\cdots p_t$. This leads to an argument in which we require the degree of the product gates to increase exponentially. This is more or less the cause of the exponential loss in our argument.
Finally, another natural question is to extend our argument from depth-$d$ regular multilinear formulas to arbitrary depth-$d$ multilinear formula.
\section*{Acknowledgments}
The authors would like to thank Zeev Dvir and Avi Wigderson for helpful discussions during the course of this work.
\appendix
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\newtheorem{thm}{Theorem}[section]
\newtheorem{defin}{Definition}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{lem}{Lemma}[section] \newtheorem{cor}{Corollary}[section] \newcommand{\rb}[1]{\raisebox{1.5ex}[0pt]{#1}} \newcommand{\multicolumn}{\multicolumn} \def\text{Beta}{\text{Beta}} \def\text{Dirichlet}{\text{Dirichlet}} \def\text{DP}{\text{DP}} \def{\tenbold p}{{\tenbold p}} \def\widehat{f}{\widehat{f}} \def\text{gamma}{\text{gamma}} \def\stackrel{\mathrm{ind}}{\sim}{\stackrel{\mathrm{ind}}{\sim}} \def\stackrel{\mathrm{iid}}{\sim}{\stackrel{\mathrm{iid}}{\sim}} \def{\tenbold J}{{\tenbold J}} \def{\tenbold K}{{\tenbold K}} \def\text{min}{\text{min}} \def\text{N}{\text{N}} \def{\tenbold p}{{\tenbold p}} \def{\tenbold S}{{\tenbold S}} \def{\tenbold s}{{\tenbold s}} \def{\tenbold U}{{\tenbold U}} \def{\tenbold u}{{\tenbold u}} \def{\tenbold w}{{\tenbold w}} \def{\tenbold W}{{\tenbold W}} \def{\tenbold X}{{\tenbold X}} \def{\tenbold x}{{\tenbold x}} \def{\tenbold y}{{\tenbold y}} \def{\tenbold Y}{{\tenbold Y}} \newcommand{{\rm I\!R}}{{\rm I\!R}} \newcommand{{\rm I\!P}}{{\rm I\!P}} \def{\tenbold Z}{{\tenbold Z}} \def{\mathcal Y}{{\mathcal Y}} \def{\mathcal R}{{\mathcal R}} \def{\mathcal M}{{\mathcal M}} \def\text{beta}{\text{beta}} \def\tilde{t}{\tilde{t}} \def\tilde{N}{\tilde{N}} \def\tilde{P}{\tilde{P}} \def\tilde{g}{\tilde{g}} \def\tilde{f}{\tilde{f}} \def\tilde{Y}{\tilde{Y}} \def\tilde{P}{\tilde{P}} \def \Report{\centerline{\small{\rm PD COAG-FRAG}}} \def\centerline{\small{\rm M-W. HO, L.F. JAMES, J.W. LAU}}} \pagestyle{myheadings} \markboth{\Author}{\Report{\centerline{\small{\rm M-W. HO, L.F. JAMES, J.W. LAU}}} \pagestyle{myheadings} \markboth{\centerline{\small{\rm M-W. HO, L.F. JAMES, J.W. LAU}}} \pagestyle{myheadings} \markboth{\Author}{\Report}{\Report} \thispagestyle{empty} \bct\Heading Coagulation Fragmentation Laws Induced By General Coagulations of Two-Parameter Poisson-Dirichlet Processes\lbk\lbk\tensmc {\sc Man-Wai Ho, Lancelot F. James, and John W. Lau}\footnote{ \eightit AMS 2000 subject classifications.
\rm Primary 60G57; secondary 60G07, 60E10, 05A18.\\ \indent\eightit Keywords and phrases. \rm
Cauchy-Stieltjes transforms,
coagulation-fragmentation,
exchangeable random partitions,
Poisson-Kingman models,
random discrete distributions,
two-parameter Poisson Dirichlet
} \lbk\lbk \BigSlant National University of Singapore, Hong Kong University of Science and Technology and University of Bristol\rm \lbk \ect \Quote Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, say {\footnotesize $PD(a,b)$}, with parameters {\footnotesize $(\alpha,\theta)$} and {\footnotesize $(\beta,\theta/\alpha)$}, wherein {\footnotesize $PD(\alpha, \theta)$} is coagulated by {\footnotesize $PD(\beta,\theta/\alpha)$} for {\footnotesize $0<\alpha<1$}, {\footnotesize $0 \leq\beta<1$} and {\footnotesize $-\beta<\theta/\alpha$.} This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). It has been noted that such a method is not easy to employ for more general processes. This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of {\footnotesize $PD(\alpha,\theta)$} coagulated with some law {\footnotesize $Q$}. The analysis exploits distributional relationships between compositions of species sampling random probability measures and coagulation operators. It is shown, based on results of Vershik, Yor and Tsilevich~(2004) and James~(2002), how the calculation of generalized Cauchy-Stieltjes transforms of random probability measures provides a blueprint to obtain explicit characterizations of {\footnotesize $PD(\alpha,\theta)$} coagulated with some law {\footnotesize $Q$}. We use this to obtain explicit descriptions in the case where {\footnotesize $Q$} corresponds to a large class of power tempered Poisson Kingman models described in James~(2002). That is, explicit results are obtained for models outside of the {\footnotesize $PD(\beta,\theta/\alpha)$} family. We obtain a new proof of Pitman's result as a by-product. Furthermore, noting an obvious distinction from the class of {\footnotesize $PD(\alpha,\theta)$} derived from a stable subordinator, we discuss briefly the case of Dirichlet processes coagulated by various {\footnotesize $Q$}.
\EndQuote
\section{Introduction}
Let ${\mathcal P}^{\downarrow}_{1}=\{p=(p_{i}):p_{1}\ge p_{2}\ge p_{_3}\ldots\ge 0; \sum_{i=1}^{\infty}p_{i}=1\}$. Furthermore, for a sequence $(x_{1},x_{2},\ldots)$ of non-negative real numbers with $\sum_{i=1}^{\infty}x_{i}=1$, let $RANK(x_{1},x_{2},\ldots)\in {\mathcal P}^{\downarrow}_{1}$ be the decreasing rearrangement of terms of the sequence. Pitman~(1999, 2005), in particular section 5.4 of Pitman~(2005), gives the following definition of coagulation and fragmentation kernels on ${\mathcal P}^{\downarrow}_{1}$. For each probability measure $Q$ on ${\mathcal P}^{\downarrow}_{1}$, two Markov-transition kernels $Q-COAG$ and $Q-FRAG$ can be defined for ${\mathcal P}^{\downarrow}_{1}$ as follows. For $p\in {\mathcal P}^{\downarrow}_{1}$, $(Q-COAG)(p,\cdot)$ is the distribution on ${\mathcal P}^{\downarrow}_{1}$ of $RANK(\sum_{i}p_{i}I(U_{i}\in I^{Q}_{j}),j\geq 1)$ where $(I^{Q}_{j})$ is a $Q$-partition of $[0,1]$ and the $U_{i}$ are i.i.d. uniform on $[0,1]$ independent of $(I^{Q}_{j}).$ Note that for brevity we refer the reader to Pitman~(2005, ch.5) for further explanations of the above quantities. $(Q-FRAG)(p,\cdot)$ is the distribution of $RANK(p_{i}Q_{ij}, i, j\ge 1)$ where $(Q_{ij})_{j\ge 1}$ has distribution $Q$ for each $i$, and these sequences are independent as $i$ varies.
For a probability measure $P$ on ${\mathcal P}^{\downarrow}_{1}$, let $R:=P(Q-COAG)$. That is the random probability measure on ${\mathcal P}^{\downarrow}_{1}$ defined by $$ R(\cdot)=\int_{{\mathcal P}^{\downarrow}_{1}}P(dr)Q-COAG(r,\cdot) $$ which may be called $P$ \emph{coagulated} by $Q$. In principle, there are many ways to characterize the laws $R$, $P$, $Q-COAG$, $Q-FRAG$. For example, one may do this via their corresponding exchangeable partition probability functions~(EPPF) on the space of partitions of the integers. However this is, in general, a non-trivial matter. Ideally one wants to identify such laws which may be described nicely via the graph of Pitman~(2005), \begin{eqnarray}
& \tilde{Q} & \nonumber\\
X & \DoubleRLarrow{0.5in} & Y \label{arrow}\\
& {\hat Q} &\nonumber \end{eqnarray}
where one reads~\mref{arrow} as $\mathbb{P}(Y\in \cdot|X)={\tilde Q}(X,\cdot)$ and $\mathbb{P}(X\in \cdot|Y)={\hat Q}(Y,\cdot)$. With respect to the present context, $X$ has distribution $P$, ${\tilde Q}(X,\cdot)=Q-COAG(X,\cdot)$, $Y$ has distribution $R$ and ${\hat Q}(Y,\cdot)=Q-FRAG(Y,\cdot)$. The general task, that we shall consider, is given $P$ and $Q-COAG$, find $R$ and $Q-FRAG.$
Pitman~(1999) establishes the most general known coagaluation/fragmentation duality of this type using the two-parameter Poisson Dirichlet distribution on ${\mathcal P}^{\downarrow}_{1}.$ The two-parameter Poisson Dirichlet distribution, denoted as $PD(\alpha,\theta)$, for the separate ranges $0\leq \alpha<1$, $\theta> -\alpha$ and $\alpha=-\kappa$, $\theta=m\kappa$ for $\kappa>0$ and some integer $m=1,2,\ldots$ is discussed in for instance Pitman and Yor~(1997) and Pitman~(2005) and has numerous applications and interpretations. We shall provide more details shortly. First Theoerm 12 of Pitman~(1999) may be described in terms of the following diagram as given in Pitman~(2005); for $0<\alpha<1, 0\leq \beta<1, -\beta<\theta/\alpha$, \begin{eqnarray}
& PD(\beta,\theta/\alpha)-COAG & \nonumber\\
PD(\alpha,\theta) & \DoubleRLarrow{0.5in} & PD(\alpha\beta,\theta) \label{Pitman}\\
& PD(\alpha,-\alpha\beta)-FRAG &\nonumber \end{eqnarray} where the notation $PD(\alpha,\theta)$ and $PD(\alpha\beta,\theta)$ in~\mref{Pitman} is to be understood as some $X$ and $Y$ having these respective laws. In other words, this gives the explicit description of the coagulation/fragmentation duality in relation to $PD(\alpha,\theta)$ coagulated by $PD(\beta,\theta/\alpha).$ That is one can set, $P=PD(\alpha,\theta)$, $Q=PD(\beta,\theta/\alpha)$ and $R=PD(\alpha\beta,\theta)$. Pitman~(1999) proves this result via a combinatorial argument involving the respective EPPF's of the various two-parameter Poisson-Dirichlet models. The argument used exploited the Gibbs structure of these EPPF's and as noted by Pitman~(1999) is not obviously extendable to obtain explicit expressions for other laws.
In this paper we show how one may replace the combinatorial argument by an argument involving generalized Cauchy-Stieltjes transform and moreover extend Pitman's result to more general models of $PD(\alpha,\theta)$ coagulated by some $Q$. That is to say given $Q$ we want to complete the description of the following diagram \begin{eqnarray}
& Q-COAG & \nonumber\\
PD(\alpha,\theta) & \DoubleRLarrow{0.5in} & Y\label{arrow3}.\\
& Q-FRAG &\nonumber \end{eqnarray}
A key to our exposition is the following characterization via exchangeable random probability measures. First every random sequence $(P_{i})\in {\mathcal P}^{\downarrow}_{1}$ has a law $P$ which determines and is determined by the law of the random probability measure $$ \tau_{P}(\cdot)=\sum_{i=1}^{\infty}P_{i}\delta_{U_{i}}(\cdot) $$ where $U_{i}$ are iid uniform $[0,1]$. This representation is equivalent to saying that $\tau_{P}$ is a \emph{species sampling random probability measure}~[see Pitman~(1996)] based on a Uniform distribution. Now associating the definition of random probability measure $\tau_{Q}$ with $Q$ in an obvious way, Lemma 5.18 of Pitman~(2005) states that the law $R=P(Q-COAG)$ is the unique probability distribution on ${\mathcal P}^{\downarrow}_{1}$ such that $$ (\tau_{R}(u), 0\leq u\leq 1)\overset{d}=(\tau_{P}(\tau_{Q}(u)),0\leq u\leq 1) $$ where it is assumed that $(\tau_{P}(u),0\leq u\leq 1)$ and $(\tau_{Q}(u),0\leq u\leq 1)$ are independent. We shall also use the notation $\tau_{R}=\tau_{P}\circ \tau_{Q}$ to denote composition. See also Bertoin and Pitman~(2000), Bertoin and Le Gall~(2003, 2005) for a related discussion.
Our approach is to try to ascertain directly the distribution of $\tau_{R}=\tau_{P}\circ \tau_{Q}$, when $\tau_{P}$ is determined by a $PD(\alpha,\theta)$ model. The main tool will be the explicit evaluation of the generalized Cauchy-Stieltjes transform for $\tau_{R}$. As we shall show, this approach is particularly well suited for the $PD(\alpha,\theta)$ models due to results of Vershik, Yor and Tsilevich~(2004) model in conjunction with the results of James~(2002). We apply the Cauchy-Stieltjes transforms in James~(2002) to extend~\mref{Pitman} to a family of $Q$ belonging to a class of power tempered Poisson Kingman laws. This constitutes a large class of models which are derived from rather arbitrary continuous infinitely divisible random variables. As an important example, we show that when $Q$ is a Dirichlet process, Pitman's result in~\mref{Pitman} for $Q=PD(0,\theta/\alpha)$ follows from the identity of Cifarelli and Regazzini~(1990) and a new more general result for $Q=PD(0,\nu)$, $\nu>\theta/\alpha,$ follows by a characterization of the Dirichlet process given in James~(2005).
Some other notable, but not exhaustive, list of references for various types of coagulation/fragmentation models include Aldous and Pitman~(1998), Bolthausen and Sznitman~(1998), Bertoin~(2002), Bertoin and Goldschmidt~(2004), Dong, Goldschmidt and Martin~(2005) and Schweinsberg~(2000).
\section{Construction of Poisson Kingman Type Random Probability Measures} We first describe the class of models $Q$ we shall explicitly consider. Let $T$ denote a strictly positive random variable with density denoted as $f_{T}$ and Laplace transform $$ \mathbb{E}[{\mbox e}^{-\lambda T}]={\mbox e}^{-\psi(\lambda)}=\int_{0}^{\infty}{\mbox e}^{-t\lambda}f_{T}(t)dt $$ where $\psi(\lambda)=\int_{0}^{\infty}(1-{\mbox e}^{-\lambda s})\rho(ds)$ and $\rho$ denotes its unique L\'evy density. Let $H(\cdot)$ denote a probability measure on a Polish space $\mathscr{X}$. For the moment we shall assume that $H$ is fixed and non-atomic. We will later relax this assumption. It is known that for each $T$ and fixed $H$ one may construct a finite completely random measure, say $\mu,$ on a Polish space $\mathscr{X},$ characterized by its Laplace functional for every positive measureable function $g$ on $\mathscr{X}$ as
\Eq \mathbb{E}[{\mbox e}^{-\mu(g)}|H]={\mbox e}^{-\int_{\mathscr{X}}\psi(g(x))H(dx)} \label{laplacem}\EndEq where $\mu(g)=\int_{\mathscr{X}}g(x)\mu(dx).$ It is evident that
$T=\mu(\mathscr{X}):=\int_{\mathscr{X}}\mu(dx)=\int_{\mathscr{X}}I\{x\in \mathscr{X}\}\mu(dx)$. We denote the law of $\mu$ as $\mathbb{P}(d\mu|\rho H)$, where
$\mathbb{P}(\cdot|\rho H)$ is a probability measure on a suitably measureable space of finite measures, say $\mathscr{M}.$ Harkening back to Kingman~(1975) one may describe a class of random probability measures on $\mathscr{X}$ by the normalization \Eq P_{K}(\cdot)=\frac{\mu(\cdot)}{T}=\sum_{i=1}^{\infty}P_{i}\delta_{Z_{i}}(\cdot) \label{PKgen} \EndEq where $(P_{i})\in {\mathcal P}^{\downarrow}_{1}$ has some law denoted as $Q=PK(\rho)$ and independent of $P_{i}$ the $(Z_{i})$ are iid $H$. That is to say the $P_{K}$ constitute a class of species sampling random probability models. The construction of the $(P_{i})$ equates with the \emph{basic} Poisson-Kingman models discussed in Pitman~(2003). Pitman~(2003) provides a thorough characterization of the laws
$Q=PK(\rho)$ on ${\mathcal P}^{\downarrow}_{1}$ via their corresponding exchangeable partition probability function~(EPPF). Specifically, according to Corollary 6 of Pitman~(2003), for some random partition of the integers ${1,\ldots,n}$, $(A_{1},\ldots, A_{k})$, with block sizes $|A_{i}|=n_{i}$ for $i=1,\ldots, k\leq n$ blocks, the EPPF associated with each $Q$ is given by $$ p_{K}(n_{1},\ldots, n_{k}):= \frac{(-1)^{n-k}}{\Gamma(n)}\int_{0}^{\infty}\lambda^{n-1}{\mbox e}^{-\psi(\lambda)}\prod_{i=1}^{k}\psi_{n_{i}}(\lambda)d\lambda $$ where for $m=1,\ldots,n,$ \Eq \psi_{m}(\lambda):=\frac{d^m}{d\lambda^m}\psi(\lambda) = (-1)^{m-1}\kappa_{m}(\lambda), \label{psim} \EndEq and, $$ \kappa_{m}(\lambda)=\int_{0}^{\infty}s^{m}{\mbox e}^{-\lambda s}\rho(ds) $$ represents the $m$-th cumulant of a random variable with \emph{tilted} density ${\mbox e}^{\psi(\lambda)} {\mbox e}^{-\lambda t}f_{T}(t).$ As discussed in Pitman~(1996), the EPPF
$p_{K}$ along with specific knowledge of $H$ determines the law of $P_{K}$ which is governed by $\mathbb{P}(d\mu|\rho H).$ The basic Poisson-Kingman laws generate a much larger class of laws by first conditioning $(P_{i})|T=t$ or equivalently $P_{K}|T=t$ and substituting $f_{T}(t)dt$ by another probability measure on $(0,\infty)$, $\gamma(dt)$. Pitman~(2003) denotes these laws as
$PK(\rho,\gamma)=\int_{0}^{\infty}PK(\rho|t)\gamma(dt)$. The $PK(\rho,\gamma)$ is referred to as a \emph{Poisson-Kingman distribution with L\'evy density $\rho$ and mixing distribution} $\gamma.$
\Remark It is obvious that if the $(Z_{i})$ are replaced by $(U_{i})$ then the composition of such a $P_{K}(\cdot)=\sum_{i=1}^{\infty}P_{i}\delta_{U_{i}}(\cdot)$ random probability measure with an $H$, $P_{K}\circ H$, is equivalent to a random probability measure determined by $\int_{\mathscr{X}}\psi(g((x))H(dx)$ as above. Equivalently, any $P_{K}$ in~\mref{PKgen}, can be represented as $$ \sum_{k=1}^{\infty}P_{i}\delta_{Z_{i}}(\cdot)\overset {d}=\frac{\tilde{\mu}{(H(\cdot))}}{\tilde{\mu}((H(\mathscr{X}))} $$ where the law of ${\tilde \mu}$ on $[0,1]$ is specified by its Laplace functional with $\mathbb{E}[{\mbox e}^{-\tilde \mu(g)}]:={\mbox e}^{-\int_{0}^{1}\psi(g((x))dx}.$ That is $\mu={\tilde \mu}\circ H,$ with distribution characterized by~\mref{laplacem}. Importantly these result hold for any fixed $H$, whether it possesses atoms or not. \EndRemark \subsection{Two-parameter Poisson-Dirichlet models} The $PD(\alpha,\theta)$ models for $0\leq\alpha<1$ and $\theta>-\alpha$ are special cases of the above construction, that is $PK(\rho,\gamma)$ models. The two-parameter $(\alpha,\theta)$ Poisson-Dirichlet random probability measure, with parameters $0\leq \alpha<1$ and $\theta> -\alpha$ has the known representation, $$ P_{\alpha,\theta}(dx)= \frac{\mu_{\alpha,\theta}(dx)}{T_{\alpha,\theta}} $$ where $\mu_{\alpha,\theta}$ is a finite random measure on $\mathscr{X}$
with law denoted as $\mathbb{P}_{\alpha,\theta}(\cdot|H)$, and $T_{\alpha,\theta}=\mu_{\alpha,\theta}(\mathscr{X})$ is a random variable. The law of the random measure $\mu_{\alpha,\theta}$ can be described as follows. When $\alpha=0$, $\mu_{0,\theta}$ is a Gamma process with shape $\theta H$, hence $P_{0,\theta}$ is a Dirichlet process with shape $\theta H$. That is a $PD(0,\theta)$ model for $(P_{i})$ coupled with a specification for $H$ yields a Dirichlet process with shape parameter $\theta H$, for $\theta>0$. In this case of $\mu_{0,\theta},$ the $\psi(g(x))$ is expressed for any positive measureable function $g$ as \Eq d_{\theta}(g(x)):=\theta \ln(1+g(x))\label{DPpsi},\EndEq and its L\'evy density is
$\rho_{0,\theta}(ds)=\theta s^{-1}{\mbox e}^{-s}ds$. Hence its law is $\mathbb{P}_{0,\theta}(\cdot|H):=\mathbb{P}(\cdot|\rho_{0,\theta}H).$ The total random mass, say $T_{0,\theta}=\mu_{0,\theta}(\mathscr{X}),$ is a Gamma random variable with shape parameter $\theta.$ For the $PD(\alpha,0)$ model, recall that $$ \rho_{\alpha,0}(ds)=\frac{\alpha s^{-\alpha-1}}{\Gamma(1-\alpha)}s^{-\alpha-1}ds $$ is the L\'evy density corresponding to a stable law of index $0<\alpha<1$. Equivalently, $T_{\alpha,0}$ is a stable random variable determined by $\rho_{\alpha,0}$, with density denoted as $f_{\alpha}(t)=f_{T_{\alpha,0}}(t)$, its Laplace transform is given by $$ \mathbb{E}[{\mbox e}^{-\lambda T_{\alpha,0}}]={\mbox e}^{-\lambda^{\alpha}} .$$ This shows that $\mu_{\alpha,0}$ is a completely random measure based on a stable law. The law of $\mu_{\alpha,0}$ is
$\mathbb{P}_{\alpha,0}(\cdot|H):=\mathbb{P}(\cdot|\rho_{\alpha,0}H).$ In the cases above both $\mu_{\alpha,0}$ and $\mu_{0,\theta}$ are completely random measures and $PD(\alpha,0)=PK(\rho_{\alpha,0})$ and $PD(0,\theta):=PK(\rho_{0,\theta}).$ This is not the case for $PD(\alpha,\theta)$ models for the range $0<\alpha<1$ and $\theta\neq 0$, $\theta>-\alpha$. The two-parameter Poisson-Dirichlet model with $0<\alpha<1$ and $\theta\neq 0$, $\theta>-\alpha$ is obtained by the specification $PD(\alpha,\theta):=PK(\rho_{\alpha,0},f_{\alpha,\theta})$ where $$ f_{\alpha,\theta}(t)dt=c_{\alpha,\theta}t^{-\theta}f_{\alpha}(t)dt, $$ with $ c_{\alpha,\theta}=1/\mathbb{E}[T^{-\theta}_{\alpha,0}]=\Gamma(\theta+1)/\Gamma(\frac{\theta}{\alpha}+1).$ At the level of the random measure $\mu_{\alpha,\theta}$ one has the following absolute continuity relationship, for every measureable function $h$, \Eq
\mathbb{E}[h(\mu_{\alpha,\theta})|H]=c_{\alpha,\theta}\mathbb{E}[T^{-\theta}_{\alpha,0}h(\mu_{\alpha,0})|H]
=c_{\alpha,\theta}\int_{\mathscr{M}}T^{-\theta}h(\mu)\mathbb{P}(d\mu|\rho_{\alpha,0}H)
\label{absPD} \EndEq where the first expectation is taken with respect to the law $\mathbb{P}_{\alpha,\theta}(\cdot|H).$ The $PD(\alpha,\theta)$ model is defined in general for two ranges $0\leq \alpha<1$ and $\theta>-\alpha$ or $\alpha=-\kappa<0$, and $\theta=m\kappa$ for $m=1,2,\ldots.$ In any case the EPPF is given by $$ p_{\alpha,\theta}(n_{1},\ldots,n_{k})=\frac{(\theta+\alpha)_{k-1\uparrow\alpha} \prod_{i=1}^{k}(1-\alpha)_{n_{i}-1\uparrow1}}{{(\theta+1)}_{n-1\uparrow1}} $$ where $(x)_{n\uparrow \alpha}:=\prod_{i=0}^{n-1}(x+i\alpha).$
\subsection{$PK(\rho,\gamma_{\theta})$ models} James~(2002, section 6), influenced by the power tempering construction of the $PD(\alpha,\theta)$ models, discussed and analyzed various features of a natural extension to more general $PK(\rho,\gamma)$ models of this type. Suppose that for a $PK(\rho)$ model there exists $-\infty<\theta<\infty$ such that $$ \frac{1}{m_{\theta}(\rho)}=\int_{0}^{\infty}t^{-\theta}f_{T}(t)dt<\infty, $$ then one may define a class of power tempered PK models by specifying $PK(\rho,\gamma_{\theta})$ with $$\gamma_{\theta}(dt)= m_{\theta}(\rho)t^{-\theta}f_{T}(t)dt.$$ Hereafter, we shall only consider the range $\theta>-\alpha.$ The corresponding random probability measures on $\mathscr{X}$ are denoted as\Eq P_{K,\theta}(\cdot)=\frac{\mu(\cdot)}{T}=\sum_{i=1}^{\infty}P_{i}\delta_{Z_{i}} (\cdot) \label{PKT} \EndEq where $(P_{i})\in{\mathcal P}^{\downarrow}_{1}$ has law $PK(\rho,\gamma_{\theta})$ and
$(Z_{i})$ are iid $H.$ Equivalently if $\mathbb{P}(d\mu|\rho H)$ denotes the distribution of $\mu$ under the the $PK(\rho)$ model, for a specific $H$, then we say that $$
\mathbb{P}(d\mu|\rho H,\gamma_{\theta})=m_{\theta}(\rho)
T^{-\theta}\mathbb{P}(d\mu|\rho H) $$ is the distribution of $\mu$ under the $PK(\rho,\gamma_{\theta})$
laws. That is, similar to~\mref{absPD}, if $\mu_{K,\theta}$ denotes a version of $\mu$ with law $\mathbb{P}(\cdot|\rho H,\gamma_{\theta})$, then one has the following absolute continuity relationship, for every measureable function $h$, \Eq
\mathbb{E}[h(\mu_{K,\theta})|H]=m_{\theta}(\rho)\mathbb{E}[T^{-\theta}h(\mu)|H]
=m_{\theta}(\rho)\int_{\mathscr{M}}T^{-\theta}h(\mu)\mathbb{P}(d\mu|\rho H). \label{absPK} \EndEq Hence we see that~\mref{absPK} is a generalization of~\mref{absPD}. It follows from Pitman~(2003) that the EPPF of these models may be described as $$ p_{K,\theta}(n_{1},\ldots, n_{k}):=\frac{(-1)^{n-k}m_{\theta}(\rho)}{\Gamma(\theta+n)}\int_{0}^{\infty}\lambda^{\theta+n-1}{\mbox e}^{-\psi(\lambda)}\prod_{i=1}^{k}\psi_{n_{i}}(\lambda)d\lambda, $$ where $\psi_{n_{i}}(\lambda)$ is defined as in~\mref{psim}. \section{Cauchy-Stieltjes Transforms} We now proceed to show how one may describe the laws of ${\tilde P}_{\alpha,\theta,Q}:=P_{\alpha,\theta}\circ \tau_{Q}$ and related expressions. In view of Remark 1 we will always assume that $P_{\alpha,\theta}$ is defined by uniform atoms $(U_{i})$, but allow $\tau_{Q}$ to be based on atoms with a more general distribution. As mentioned previously there are various techniques that can be used to identify the laws of random probability measures. For instance one may calculate its EPPF, identify its finite dimensional distribution by direct means, or its Laplace functional. The idea of using Laplace functionals is intuitively appealing, however it is not particularly suited to handle random probability measures. It turns out that a more appropriate tool are generalized Cauchy-Stieltjes transforms~(CS) defined for some generic random probability measure $\tau$, positive measureable $g$, positive $z$ and real valued $q$ as $$ \mathbb{E}[(1+z\tau(g))^{-q}]. $$ Not many results for specific $\tau$ are widely known. Fortunately there are useful results for the $PD(\alpha,\theta)$ class. Specifically we shall use the results of Cifarelli and Regazzini~(1990) and Vershik, Yor and Tsilevich~(2004). Somewhat less known are the results of James~(2002) who obtains specific transforms for $PK(\rho)$ and $PK(\rho,\gamma_{\theta})$ models. These results are extended to larger classes, in a manuscript in preparation of James, Lijoi and Pr\"unster~(2005). We first describe what is known for the $PD(\alpha,\theta)$ models. We then describe the results for the general $PK(\rho,\gamma_{\theta})$ for $\theta>-\alpha$. This large class of models turn out to be particularly well suited for coagulation with $PD(\alpha,\theta).$ \subsection{CS for $PD(\alpha,\theta)$} Note, as can be seen from Remark 1, that given $\tau_{Q}$, ${\tilde P}_{\alpha,\theta,Q}$ is a $PD(\alpha, \theta)$ model with $H=\tau_{Q}$ fixed. Probably the most widely known CS result is for the Dirichlet process with shape $\theta H$ where it was shown by Cifarelli and Regazzini~(1990) that quite remarkably \Eq
\mathbb{E}\[{(1+zP_{0,\theta}(g))}^{-\theta}|H\]={\mbox e}^{-\int_{\mathscr{X}}d_{\theta}(zg(x))H(dx)} \label{cifreg},\EndEq where $d_{\theta}$ is defined in~\mref{DPpsi}. Now, key to our exposition is the following elegant result of Vershik, Yor and Tsilevich~(2004) for the $PD(\alpha, \theta)$ model with the range $0<\alpha<1$, $\theta\neq 0$ and otherwise $\theta>-\alpha$ we have for any $H$ that \Eq
\mathbb{E}[{(1+zP_{\alpha,\theta}(g))}^{-\theta}|H]={\[\int_{\mathscr{X}}{(1+zg(x))}^{\alpha}H(dx)\]}^{-\frac{\theta}{\alpha}} .\label{keypd} \EndEq To complete the picture for the $PD(\alpha,\theta)$, a result for the $PD(\alpha,0)$ may be read from proposition~6.2 of James~(2002) with $n=1$ as, \Eq
\mathbb{E}[{(1+zP_{\alpha,0}(g))}^{-1}|H]= \frac{\int_{\mathscr{X}}{(1+zg(x))}^{\alpha-1}H(dx)} {\int_{\mathscr{X}}{(1+zg(x))}^{\alpha}H(dx)}. \label{keystable} \EndEq
\subsection{CS for $PK(\rho,\gamma_{\theta})$ models} Proposition 6.1 of James~(2002) shows that $PK(\rho,\gamma_{\theta})$ models for $\theta\neq 0$, $\theta>-1$ have the transform,
\Eq \mathbb{E}[{(1+zP_{K,\theta}(g))}^{-\theta}|H]= \frac{m_{\theta}(\rho)}{\Gamma(\theta)}\int_{0}^{\infty}{\mbox e}^{-\int_{\mathscr{X}}\psi(y[1+zg(x)])H(dx)}y^{\theta-1}dy .\label{keyPK} \EndEq This result generalizes~\mref{keypd}.
\Remark The result~\mref{keyPK} is actually stated in James~(2002) for the range $\theta>0$. As the result arises from an identity due to the gamma function, it extends to the negative range by the same argument as noted on p. 2309 ``\emph{added in translation}" of Vershik, Yor and Tsilevich~(2004). Otherwise, one can see this by first writing, $$ {(1+zP_{K,\theta}(g))}^{-\theta}={(1+zP_{K,\theta}(g))}^{-(1+\theta)}{(1+zP_{K,\theta}(g))}. $$ Now noting that $1+\theta>0$ for $\theta>-1$, the gamma identity applies with $\theta+1$, one then argues as in James~(2002) and concludes the result by integration by parts. \EndRemark
\subsection{Generic CS for ${\tilde P}_{\alpha,\theta,Q}$} Setting $H=\tau_{Q}$ in~\mref{keypd}, we obtain the key formula for the range $0<\alpha<1$, $\theta\neq 0$ and otherwise $\theta>-\alpha$ $$
\mathbb{E}[{(1+z{\tilde P}_{\alpha,\theta,Q}(g))}^{-\theta}|\tau_{Q}]={\[\int_{\mathscr{X}}{(1+zg(x))}^{\alpha}\tau_{Q}(dx)\]}^{-\frac{\theta}{\alpha}}. $$ It then follows that \Eq \mathbb{E}[{(1+z{\tilde P}_{\alpha,\theta,Q}(g))}^{-\theta}]=\mathbb{E}[{(1+{\tau_{Q}}(g_{\alpha}))}^{-\frac{\theta}{\alpha}}] \label{master} \EndEq where $g_{\alpha}(x)={(1+zg(x))}^{\alpha}-1.$ That is to say, the peculiar nature of the $PD(\alpha,\theta)$ model for the range $0<\alpha<1$, $\theta\neq 0$, $\theta>-\alpha$ yields basically a double Cauchy-Stietltjes formula. Based on the results given in the previous section an alternative proof for the Theorem~12 in Pitman~(1999) is almost completely evident. We will describe the details in the next section. We will then show why the choice of $PK(\rho,\gamma_{\theta})$ as $Q$ models is quite desirable. \Remark As we noted earlier explicit CS formula for more general $\tau_{Q}$ may be obtained from James~(2002) and James, Lijoi and Pr\"unster(2005). We note also that the formula for the Dirichlet model, $PD(0,\theta)$ in~\mref{cifreg} suggests that one needs to calculate the Laplace functional of a $Q$ which is generally hard. This makes it more difficult to obtain explicit results for $PD(0,\theta)$ coagulated by some $Q$. We note from the results in James~(2002) and James, Lijoi and Pr\"unster~(2005) that the appearance of Laplace functional calculations is unfortunately true for many possible candidates as replacements for $PD(\alpha,\theta).$ In contrast the stable $PD(\alpha,0)$ case is exceptionally easy. We shall discuss the $PD(0,\theta)$ and $PD(\alpha,0)$ separately from the other $PD(\alpha,\theta)$ models. \EndRemark
\section{Proof of Pitman's Diagram} We shall now illustrate how our framework easily yields the diagram~\mref{Pitman}. First we address the Dirichlet case. \subsection{$PD(\alpha,\theta)$ coagulated by $PD(0,\theta/\alpha)$} First apply~\mref{master} with $\tau_{Q}=P_{0,\theta/\alpha}$, then it is immediate that the formula~\mref{cifreg} applies with $\theta/\alpha$ in place of $\theta$. To complete the result notice that $$ d_{\theta/\alpha}(g_{\alpha}(x))={\theta/\alpha}\ln{(1+zg(x))}^{\alpha}=d_{\theta}(zg(x)). $$ That is to say the CS transform of order $\theta$ of $P_{\alpha,\theta}\circ P_{0,\theta / \alpha}$ is given by~\mref{cifreg} identifying it as a $PD(0,\theta)$ model. The diagram~\mref{Pitman} is completed by calculating the now obvious 3 EPPF's and applying Bayes rule. \subsection{$PD(\alpha,\theta)$ coagulated by $PD(\beta,\theta/\alpha)$, $\beta>0$} For this case apply~\mref{master} with $\tau_{Q}=P_{\beta,\theta/\alpha}$ for $0<\beta<1$, now apply~\mref{keypd} to conclude that, in this case, ~\mref{master} is equivalent to,
$$\mathbb{E}[{(1+{P_{\beta,\theta/\alpha}}(g_{\alpha}))}^{-\frac{\theta}{\alpha}}|H] ={\[\int_{\mathscr{X}}{(1+g_{\alpha}(x))}^{\beta}H(dx)\]}^{-\frac{\theta}{\beta\alpha}}.$$ The result is completed by noting that $$ {(1+g_{\alpha}(x))}^{\beta}={(1+zg(x))}^{\alpha\beta}. $$ Hence the CS transform of order $\theta$ of $P_{\alpha,\theta}\circ P_{\beta,\theta / \alpha}$ is given by the expressions above and now comparing with \mref{keypd} identifies its law as $PD(\alpha\beta,\theta)$ model. \Remark The $P_{\alpha,0}\circ P_{\beta,0}$ case can certainly be obtained easily using~\mref{keystable} twice. However, we do not believe that there is a simpler proof than that exhibited in Pitman~(2005). We shall return to this later. \EndRemark \section{$PD(\alpha,\theta)$ Coagulated By $PK(\rho,\gamma_{\theta/\alpha})$} We now obtain a new result as follows. First denote a class of species sampling random probability measures on $\mathscr{X}$ as \Eq S_{\alpha,\theta}(\cdot)=\frac{L_{\alpha,\theta}(\cdot)}{T_{L_{\alpha,\theta}}}:=\sum_{i=1}^{\infty}W_{i}\delta_{Z_{i}}(\cdot) \label{PSrm} \EndEq with $T_{L_{\alpha,\theta}}=L_{\alpha,\theta}(\mathscr{X})$ and where
$(W_{i})\in {\mathcal P}^{\downarrow}_{1}$ and $(Z_{i})$ are iid $H$. Similar to~\mref{absPD} and~\mref{absPK}, the law of $L_{\alpha,\theta}$, and hence that of $(W_{i})$, is determined by a power tempered probability measure and satisfies the following absolute continuity relationship, for every measureable function $h$, \Eq
\mathbb{E}[h(L_{\alpha,\theta})|H]=c_{\alpha,\theta}m_{\frac{\theta}{\alpha}}(\rho)\mathbb{E}[T^{-\theta}_{L_{\alpha,0}}h(L_{\alpha,0})|H] \label{absPS} \EndEq where the Laplace functional of the random measure $L_{\alpha,0}$ is specified by $$
\mathbb{E}[{\mbox e}^{-L_{\alpha,0}(g)}|H]={\mbox e}^{-\int_{\mathscr{X}}\tilde{\psi}_{\alpha}(g(x))H(dx)} $$ with \Eq \tilde{\psi}_{\alpha}(g(x))=\psi({[g(x)]}^{\alpha}). \label{psia}\EndEq In other words $L_{\alpha,0}=\mu_{\alpha,0}\circ\mu$, where $\mu_{\alpha,0}$ is a stable completely random finite measure on $[0,1]$ with index $0<\alpha<1$ with atoms given by the sequence $(U_{i}),$ and the law of $\mu$ on $\mathscr{X}$ is specified by~\mref{laplacem}. In particular, $$ T_{L_{\alpha,0}}\overset {d}=T_{\alpha,0}T^{\frac{1}{\alpha}} $$ where $T_{\alpha,0}$ is independent of $T$. The density of $T_{L_{\alpha,0}}$ can be expressed as \Eq f_{T_{L_{\alpha,0}}}(y)= \int_{0}^{\infty}f_{\alpha}(yt^{-\frac{1}{\alpha}})t^{-\frac{1}{\alpha}} f_{T}(t)dt=\alpha\int_{0}^{\infty}f_{T}({(y/s)}^{\alpha})s^{-\alpha}y^{\alpha-1}f_{\alpha}(s)ds. \label{PSden}\EndEq As a by-product, we obtain $$ c_{\alpha,\theta}m_{\frac{\theta}{\alpha}}(\rho)=1/{\mathbb{E}[T^{-\theta}_{L_{\alpha,0}}]}. $$ Call the family of laws associated with
$S_{\alpha,\theta}$, or more specifically the $(W_{i})$, as
$PS({\rho,\alpha,\theta}).$ In view of~\mref{psia}, the EPPF of the
$PS({\rho,\alpha,\theta})$ model can be expressed as \Eq p_{S,\alpha,\theta}(b_{1},\ldots,b_{k}) :=\frac{{(-1)}^{n-k}c_{\alpha,\theta}m_{\frac{\theta}{\alpha}}(\rho)}{\Gamma(\theta+n)}\int_{0}^{\infty}\lambda^{\theta+n-1}{\mbox e}^{-\psi(\lambda^{\alpha})}\prod_{i=1}^{k}{\tilde \psi}_{\alpha,b_{i}}(\lambda)d\lambda \label{EPPFPS}\EndEq where $k$ is the number of blocks formed by the integers $\{1,\ldots,n\}$ and $\{b_1,\ldots,b_k\}$ are the sizes of the blocks $\{B_1,\ldots,B_k\}$, respectively. Additionally, similar to \mref{psim}, for $m=1,\ldots,n,$ $${\tilde \psi}_{\alpha,m}(\lambda):=\frac{d^m}{d\lambda^m}\psi(\lambda^{\alpha}) := (-1)^{m-1}\kappa_{\alpha,m}(\lambda), $$ where $ \kappa_{\alpha,m}(\lambda) $ represents the $m$-th cumulant of a random variable with \emph{tilted} density $$ {\mbox e}^{{\tilde \psi}_{\alpha}(\lambda)} {\mbox e}^{-\lambda y}f_{T_{L_{\alpha,0}}}(y),$$ specified by~\mref{PSden}. The cumulants can be expressed in terms of the moments of this density $$ \int_{0}^{\infty}y^{m}{\mbox e}^{{\tilde \psi}_{\alpha}(\lambda)} {\mbox e}^{-\lambda y}f_{T_{L_{\alpha,0}}}(y)dy $$ using Theile's recursion.\Remark Note that $PS(\rho_{\beta,0},\alpha,\theta):=PD(\alpha \beta,\theta).$
\EndRemark \begin{thm}\label{thm1}Suppose that for $0<\alpha<1$ and $\theta\neq 0$, $\theta>-\alpha,$ $PD(\alpha,\theta)$ is coagulated by $PK(\rho,\gamma_{\theta/\alpha}).$ Then the following results hold. \begin{enumerate} \item[(i)] $PD(\alpha,\theta)$ coagulated by $PK(\rho,\gamma_{\theta/\alpha})$ is $PS({\rho,\alpha,\theta}),$ with EPPF specified in~\mref{EPPFPS}.
\item[(ii)] Equivalently, suppose that $P_{K,\theta/\alpha}$ is defined as in~\mref{PKT}, with law determined by $\mathbb{P}(\cdot|\rho H,\gamma_{\theta/\alpha})$, satisfying ~\mref{absPK}. Then the law of the composition $P_{\alpha,\theta}\circ P_{K,\theta/\alpha}$, is equivalent to the law of $S_{\alpha,\theta}$, specified by~\mref{PSrm} and~\mref{absPS}. \item[(iii)] Suppose there are $K$ blocks $\{A_1,\ldots,A_K\}$ formed by the integers $\{1,\ldots,n\}$, each with size $a_i$, and $K \geq k$. The law of the corresponding $Q-FRAG$ kernel is determined by the (explicit) EPPF $$ p(a_{1},\ldots,a_{K})=p_{\alpha,\theta}(a_{1},\ldots,a_{K})\times \frac{\Gamma(\frac{\theta}{\alpha}+1)\Gamma(\theta+n)}{\Gamma(\frac{\theta}{\alpha}+K)\Gamma(\theta+1)}\frac{\int_{0}^{\infty}\lambda^{\theta/\alpha+K-1}{\mbox e}^{-\psi(\lambda)}\prod_{i=1}^{k}\kappa_{j_{i}}(\lambda)d\lambda} {\int_{0}^{\infty}\lambda^{\theta+n-1}{\mbox e}^{-\psi(\lambda^{\alpha})}\prod_{i=1}^{k}\kappa_{\alpha,b_{i}}(\lambda)d\lambda} $$ where $j_i,i=1,\ldots,k$~(with $\sum_{i=1}^k j_i = K$), is defined as $\#\{\ell:A_\ell \subseteq B_i\}$. \end{enumerate} \end{thm} \Proof First apply~\mref{master} with $\tau_{Q}=P_{K,\theta/\alpha}$ and let $C_{1}$ and $C_{2}$ denote the appropriate constants. Now apply~\mref{keyPK} to get $$
\mathbb{E}[{(1+P_{K,\theta/\alpha}(g_{\alpha}))}^{-\theta/\alpha}|H]= C_{1}\int_{0}^{\infty}{\mbox e}^{-\int_{\mathscr{X}}\psi(y[{(1+zg(x))}^{\alpha}])H(dx)}y^{\theta/\alpha-1}dy. $$ Now apply the transformation $y=w^{\alpha}$ to get
$$ \mathbb{E}[{(1+P_{K,\theta/\alpha}(g_{\alpha}))}^{-\theta/\alpha}|H] =C_{2}\int_{0}^{\infty}{\mbox e}^{-\int_{\mathscr{X}}\psi(w^{\alpha}[{(1+zg(x))}^{\alpha}])H(dx)}w^{\theta-1}dw. $$ Setting $\tilde{\psi}_{\alpha}(w[(1+zg(x))])=\psi(w^{\alpha}[{(1+zg(x))}^{\alpha})]$, we see that the CS transform of order $\theta$ of the composition has the form in~\mref{keyPK} with ${\tilde {\psi}}$ playing the role of $\psi$. This concludes the result.\EndProof
\subsection{$PD(\alpha,\theta)$ coagulated by $PD(0,\nu)$, Beta Gamma and power tempered normalized Linnik processes} \label{sec:51} One might be somewhat surprised that Theorem~\ref{thm1} contains results for $PD(\alpha,\theta)$ coagulated by $PD(0,\nu)$ when $\nu>\theta/\alpha$. In other words, certain Dirichlet processes are $PK(\rho,\gamma_{\theta/\alpha})$ models. To see this, we recall the Beta Gamma process representation of Dirichlet processes given in James~(2005). Specifically, if one chooses a parameter $\nu>\theta/\alpha$, then the law of $\mu$ given by $$
\frac{\Gamma(\nu)}{\Gamma(\nu-\theta/\alpha)}T^{-\theta/\alpha}\mathbb{P}(d\mu|\rho_{0,\nu}H) $$ is well defined. Relative to this law, $\mu$ is a Beta Gamma process with parameters $(\nu H, \theta/\alpha)$ as defined in James~(2005). Setting $\theta=0$ yields the law of a Gamma process with shape $\nu$. In any case, James~(2005) shows that normalizing a Beta Gamma process of this type by its total mass yields a Dirichlet process with shape $\nu H$ for every $\nu>\theta/\alpha$. Hence the $PD(0,\nu)$ models are $PK(\rho_{0,\nu},\theta/\alpha)$ models for $\nu>\theta/\alpha$. Note this equivalence does not hold for $\nu=\theta/\alpha.$
The corresponding $S_{\alpha,\theta}$ process is obtained by power tempering of a normalized process, where the law of the process is determined by $$ d_{\nu}(\lambda^{\alpha})=\nu\ln(1+\lambda^{\alpha}). $$ Now we may arrange to have a further scaling which results in the case where the law of $S_{\alpha,\theta}$ is equivalently obtained by the power tempering of a normalized Linnik process subordinator~[see for instance Huillet~(2000, 2003)], where its law is determined by $$ \tilde {\psi}_{\alpha}(\lambda):= d_{\nu}(\lambda^{\alpha}/\nu)=\nu\ln(1+\lambda^{\alpha}/\nu)=\int_{0}^{\infty}(1-{\mbox
e}^{-\lambda s})l_{\nu,\alpha}(s)ds, $$ and where $$ l_{\nu,\alpha}(s)=\frac{\alpha\nu}{s}\phi_{\alpha}(\nu s^{\alpha}) $$ is the L\'evy density of the Linnik process. Specifically, \Eq \phi_{\alpha}(q)=\mathbb{E}[{\mbox e}^{-qT^{-\alpha}_{\alpha,0}}]=\sum_{k=0}^{\infty}\frac{1}{\Gamma(1+k\alpha)}{(-q)}^{k} \label{Mittag}\EndEq is the Mittag-Leffler function or equivalently the Laplace transform of the random random $T^{-\alpha}_{\alpha,0}$, where, as before, $T_{\alpha,0}$ is a stable random variable of index $0<\alpha<1.$ In other words, $PS(\rho_{0,\nu},\alpha,\theta)=PK(l_{\nu,\alpha},\gamma_{\theta})$ model.
\subsubsection{EPPF calculations}\label{sec:511}
Note that the above information allows for several descriptions
of the EPPF of the $PK(l_{\nu,\alpha},\gamma_{\theta})$ model and
hence the corresponding $Q-FRAG$. First using~\mref{Mittag} one
has that the $m$-th cumulant of an exponentially tilted Linnik random variable can be expressed as $$ \kappa_{\alpha,m}(u)={\alpha\nu}\int_{0}^{\infty}s^{m-1}{\mbox e}^{-us}\phi_{\alpha}(\nu s^{\alpha})ds={\alpha\nu}u^{-m}\sum_{l=0}^{\infty}\frac{\Gamma(m+l\alpha)}{\Gamma(1+l\alpha)}u^{-l\alpha} {(-\nu)}^{l} $$ or $$ \kappa_{\alpha,m}(u)=\alpha\nu u^{-m}\int_{0}^{\infty}\int_{0}^{\infty}s^{m-1}{\mbox e}^{-s-\nu {(s/u)}^{\alpha}t^{-\alpha}}f_{\alpha}(t)dtds. $$ The general EPPF of the $PK(l_{\nu,\alpha},\gamma_{\theta})$ model can be written as \begin{equation}\label{LEPPF} \frac{c_{\alpha,\theta}\Gamma(\nu)}{\Gamma(\nu-\theta/\alpha)}\frac{\nu^{\theta/\alpha+k}\alpha^{k-1}}{\Gamma(\theta+n)}\int_{0}^{\infty}y^{\theta/\alpha-1}{(1+y)}^{-\nu}
\prod_{j=1}^{k}R_{n_{j}}(y|\alpha)dy, \end{equation} where $$
R_{n_{j}}(y|\alpha)=\sum_{l=0}^{\infty}\frac{\Gamma(n_{j}+l\alpha)}{\Gamma(1+l\alpha)} {(-y)}^{-l}= \int_{0}^{\infty}\int_{0}^{\infty}s^{n_{j}-1}{\mbox e}^{-s- y^{-1}{s}^{\alpha}t^{-\alpha}}f_{\alpha}(t)dtds. $$ Furthermore in the case of $\alpha=1/2$, we may follow the results for Brownian excursion in Pitman~(2003, Section 8), to obtain $$
R_{n_{j}}(y|1/2)=\frac{\Gamma(2n_{j})2^{-n_{j}+1/2}}{\Gamma(1/2)}\int_{0}^{\infty}h_{-2n_{j}}(x/y){\mbox e}^{-x^{2}/2}dx $$ where $$ h_{-2n_{j}}(x)=\frac{2^{n_{j}-1}}{\Gamma(2n_{j})}\int_{0}^{\infty}s^{n_{j}-1}{\mbox e}^{-s- x\sqrt{2s}}ds $$ is a Hermite function.
\begin{prop}\label{propo}Suppose that for $0<\alpha<1,$ $PD(\alpha,\theta)$ is coagulated by $PD(0,\nu)$ for $\nu>\theta/\alpha$, then the following results hold \begin{enumerate} \item[(i)] $PD(\alpha,\theta)$ coagulated by $PD(0,\nu)$ is $PK(l_{\nu,\alpha},\gamma_{\theta})$, with EPPF specified in~\mref{LEPPF}. \item[(ii)] Suppose there are $K$ blocks $\{A_1,\ldots,A_K\}$ formed by the integers $\{1,\ldots,n\}$, each with size $a_i$, and $K \geq k$. The law of the corresponding $Q-FRAG$ kernel is determined by the EPPF \Eq p_{\alpha,\theta}(a_{1},\ldots,a_{K})\times \frac{\Gamma(\nu-\theta/\alpha)\Gamma(\theta+n)}{c_{\alpha,\theta}\nu^{\theta/\alpha}\alpha^{k-1}\Gamma(\nu+K)}\frac{\prod_{i=1}^k(j_i-1)!} {\int_{0}^{\infty}y^{\theta/\alpha-1}{(1+y)}^{-\nu}
\prod_{j=1}^{k}R_{b_{j}}(y|\alpha)dy} \label{DPEPPF}\EndEq where $j_i,i=1,\ldots,k$~(with $\sum_{i=1}^k j_i = K$), is defined as $\#\{\ell:A_\ell \subseteq B_i\}$. \end{enumerate} \end{prop}
\Remark It is evident from Proposition~\ref{propo} that it is not easy to obtain the denominator in~\mref{DPEPPF} by summing out appropriately over the numerator. This is despite the fact that both the EPPF's in the numerator have nice Gibbs form. Hence again this points to the difficulties of a direct combinatorial argument. On the other hand, the results establish some rather peculiar combinatorial identities. \EndRemark
\section{$PD(\alpha,0)$ coagulated by $PK(\rho)$} In view of the arguments in Bertoin and LeGall~(2003) and Pitman~(2005, p. 115) concerning the Bolthausen-Sznitman~(1998) coalescent, that is a description of $PD(\alpha,0)$ coagulated by $PD(\beta,0)$, it is easy to extend this to the case of $PD(\alpha,0)$ coagulated by $PK(\rho)$
\begin{thm}Suppose that for $0<\alpha<1$ $PD(\alpha,0)$ is coagulated by $PK(\rho).$ Then the diagram, according to~\mref{arrow3}, of this process is described by setting $\theta=0$ in Theorem~\ref{thm1}. \end{thm}
\Proof The proof proceeds along the same lines as Pitman~(2005, p. 115) and Bertoin and Le Gall~(2003, p. 272). That is, $$ \frac{\mu_{\alpha,0}(P_{K}(\cdot))}{T_{\alpha,0}}\overset {d}=\frac{\mu_{\alpha,0}(\mu(\cdot))}{\mu_{\alpha,0}(T)}\overset{d}= \frac{L_{\alpha,0}(\cdot)}{T_{L_{\alpha,0}}}:=S_{\alpha,0}(\cdot) $$ where $L_{\alpha,0}$, $S_{\alpha,0}$ are as described in the beginning of this section. \EndProof
\Remark The description of $PD(\alpha,0)$ coagulated by $PD(0,\nu)$ for $\nu>0$ is obtained from Proposition~\ref{propo} with $\theta=0.$\EndRemark
\section{Some comments about $PD(0,\theta)$ coagulated by general $Q$} It was noted earlier that obtaining results for the $PD(0,\theta)$ coagulated by some $Q$ does not readily follow from a CS type analysis. We will now briefly describe how one can obtain the finite dimensional distributions of such compositions. Note again that a Dirichlet process coagulated by a $Q$, i.e., $P_{0,\theta}\circ \tau_{Q}$ given $\tau_{Q}$, is a Dirichlet process with shape $\theta \tau_{Q}$ on $\mathscr{X}.$ Hence it follows that for any measureable partition $C_{1},\ldots, C_{m}$ of $\mathscr{X}$ the finite dimensional distribution of $P_{0,\theta}\circ \tau_{Q}$ given $\tau_{Q}$ is specified by the joint Dirichlet density of $Y_{i}=P_{0,\theta}(\tau_{Q}(C_{i}))$ for $i=1,\ldots,m$, which is given by $$
f(y_{1},\ldots,y_{m}|\tau_{Q})=\frac{\Gamma(\theta)}{\prod_{i=1}^{m}\Gamma(\theta z_{i})}\prod_{i=1}^{m}y^{\theta z_{i}-1}_{i} $$ where $z_{i}=\tau_{Q}(C_{i}),$ and $(Y_{1},\ldots,Y_{m})\in \mathscr{S}_{m}=\{(a_{i})_{i\leq m}:0<a_{i}<1,\sum_{i=1}^{m}a_{i}=1\}.$ This leads to a general description of the finite dimensional distributions.
\begin{prop}\label{prop2}Suppose that $P_{0,\theta}$ denotes a Dirichlet Process on $[0, 1]$ with shape $\theta U$ and $U$ is a uniform distribution. Suppose further that $\tau_{Q}$ is a random probability measure on~$\mathscr{X}$. Then, for a measureable partition $C_{1},\ldots, C_{m}$ of $\mathscr{X}$, the distribution of $P_{\theta, Q}=P_{0,\theta}\circ \tau_{Q}$ is specified by its finite-dimensional distribution $$ f_{\theta,Q}(y_{1},\ldots,y_{m})=\int_{\mathscr{S}_{m} }f_{Q}(z_{1},\ldots,z_{m})\frac{\Gamma(\theta)}{\prod_{i=1}^{m}\Gamma(\theta z_{i})}\prod_{i=1}^{m}y^{\theta z_{i}-1}_{i}dz_{i}$$ where $Y_{i}=P_{\theta, Q}(C_{i})$, and $f_{Q}$ denotes the joint density of $Z_{i}=Q(C_{i})$ for $i=1,\ldots,m$. If $Q$ is a species sampling model, then this equates to a description of the law of $PD(0,\theta)$ coagulated by $Q$.\end{prop}
\noindent Naturally the utility of this result requires knowledge of the finite-dimensional distribution of $Q$. Below we describe two special cases where $Q=PD(1/2,\eta)$ for $\eta>-1/2$ and $Q=PD(0,\nu)$ for $\nu>0.$
\begin{prop} The distribution of $PD(0,\theta)$ coagulated by $PD(1/2,\eta)$ for $\eta>-1/2$ is determined by the distribution of $P_{0,\theta}\circ P_{1/2,\eta}$ with finite dimensional distribution, $$\frac{\[\prod_{1=1}^{m}p_i\]\Gamma(\eta+m/2)}{\pi^{(m-1)/2}\Gamma(\eta+1/2)} \int_{\mathscr{S}_{m} }\frac{\Gamma(\theta)}{\prod_{i=1}^{m}\Gamma(\theta z_{i})}\prod_{i=1}^{m}y^{\theta z_{i}-1}_{i} \frac{\prod_{i=1}^{m}z^{-3/2}_{i}} {{(p^{2}_{1}/z_{1}+\cdots +p^{2}_{m}/z_m)}^{\eta+m/2}} dz_{i},$$ where $p_{i}=H(C_{i})$ and otherwise the notation is as in Proposition~\ref{prop2}.\end{prop}
\Proof The distribution of $(Q(C_{i}))$ when $Q$ is $PD(1/2,\eta)$ is given by Theorem~3.1 of Carlton~(2002). \EndProof
We now describe the Dirichlet case.
\begin{prop} The distribution of $PD(0,\theta)$ coagulated by $PD(0,\nu)$ is determined by the distribution of $P_{0,\theta}\circ P_{0,\nu}$ with finite dimensional distribution, $$ \int_{\mathscr{S}_{m} }\frac{\Gamma(\theta)}{\prod_{i=1}^{m}\Gamma(\theta z_{i})}\prod_{i=1}^{m}y^{\theta z_{i}-1}_{i}\frac{\Gamma(\nu)}{\prod_{i=1}^{m}\Gamma(\nu p_{i})}\prod_{i=1}^{m}z^{\nu p_{i}-1}_{i} dz_{i}$$ where $p_{i}=H(C_{i})$ and otherwise the notation is as in Proposition~\ref{prop2}.\end{prop}
\Remark It is interesting to note that the dynamics of the coagulation of two or more Dirichlet processes may also be explained, in perhaps a more informative way, as a Chinese restaurant franchise process of Teh, Jordan, Beal and Blei~(2006)
when there is one franchise. In their setup, one has $F|\tau_{Q}$ is Dirichlet process with shape $(\theta \tau_{Q})$ and $\tau_{Q}$ is a Dirichlet process with shape, say, $\eta H.$ The distribution of $F$ is characterized via a Chinese restaurant franchise with one franchise. It follows from our observations that $F\overset {d}=P_{0,\theta}\circ P_{0,\nu},$ leading to an equivalence. Based on this observation, all the processes discussed here lead to some type of Chinese restaurant franchise process, and therefore have potential applications in machine learning and related areas. \EndRemark
\vskip0.2in \centerline{\Heading References} \vskip0.2in \tenrm \def\tensmc{\tensmc} \def\tensl{\tensl} \def\tenbold{\tenbold} \baselineskip0.15in
\Ref \by Aldous, D. J. and Pitman, J. \yr 1998 \paper The standard additive coalescent \jour \AnnProb \vol 26 \pages 1703-1726 \EndRef
\Ref \by Bertoin, J. \yr 2002 \paper Self-similar fragmentations \jour Ann. Inst. H. Poincaré Probab. Statist. \vol 3 \pages 319-340 \EndRef
\Ref \by Bertoin, J. and Le Gall, J.-F. \yr 2003 \paper Stochastic flows associated to coalescent processes \jour Probab. Theory Related Fields \vol 126 \pages 261-288\EndRef
\Ref \by Bertoin, J. and Le Gall, J.-F. \yr 2005 \paper Stochastic flows associated to coalescent processes. II. Stochastic differential equations \jour Ann. Inst. H. Poincaré Probab. Statist. \vol 3 \pages 307-333\EndRef
\Ref \by Bertoin, J. and Goldschmidt, C. \yr 2004 \paper Dual random fragmentation and coagulation and an application to the genealogy of Yule processes. In \textit{Mathematics and computer science III: Algorithms, Trees, Combinatorics and Probabilities}, M. Drmota, P. Flajolet, D. Gardy, B. Gittenberger (editors), pp. 295-308. Trends Math., Birkh\"auser, Basel \EndRef
\Ref \by Bertoin, J. and Pitman, J. \yr 2000 \paper Two coalescents derived from the ranges of stable subordinators \jour Electron. J. Probab. \vol 7 \pages 1-17\EndRef
\Ref \by Bolthausen, E. and Sznitman, A.-S. \yr 1998 \paper On Ruelle's probability cascades and an abstract cavity method \jour Comm. Math. Phys. \vol 197 \pages 247-276\EndRef
\Ref \by Carlton, M. A. \yr 2002 \paper A family of densities derived from the three-parameter Dirichlet process \jour J. Appl. Probab. \vol 39 \pages 764-774\EndRef
\Ref \by Cifarelli, D. M. and Regazzini, E. \yr 1990 \paper Distribution functions of means of a Dirichlet process \jour \AnnStat \vol 18 \pages 429-442 \EndRef
\Ref \by Dong, R., Martin, J. and Goldschmidt, C. \yr 2005 \paper Coagulation-fragmentation duality, Poisson-Dirichlet distributions and random recursive trees, arXiv math.PR/0507591, 2005. \\ Available at http://arxiv.org/abs/math.PR/0507591. \EndRef
\Ref \by Huillet, T. \yr 2000 \paper On Linnik's continuous-time random walks \jour J. Phys. A \vol 33 \pages 2631-2652\EndRef
\Ref \by Huillet, T. \yr 2003 \paper Energy cascades as branching processes with emphasis on Neveu's approach to Derrida's random energy model \jour Adv. in Appl. Probab. \vol 35 \pages 477-503\EndRef
\Ref \by James, L.F. \yr 2002 \paper Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics, arXiv:math.PR/0205093, 2002.\\ Available at \texttt{http://arxiv.org/abs/math.PR/0205093} \EndRef
\Ref \by James, L.F. \yr 2005 \paper Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and Beta-Gamma processes \jour \AnnStat \vol 33 \pages 647-660\EndRef
\Ref \by James, L.F., Lijoi, A. and Pr\"unster, I. \yr 2005 \paper Cifarelli-Regazzini/Markov-Krein type identities and distributional results for functionals of normalized random measures. Manuscript in preparation \EndRef
\Ref \by Kingman, J. F. C. \yr 1975 \paper Random discrete distributions \jour J. R. Stat. Soc. Ser. B \vol 37 \pages 1-22\EndRef
\Ref \by Pitman, J. \yr 1996 \paper Some developments of the Blackwell-MacQueen urn scheme. In \textit{Statistics, Probability and Game Theory}, T.S. Ferguson, L.S. Shapley and J.B. MacQueen (editors), IMS Lecture Notes-Monograph series, Vol. 30, pp. 245-267, Inst. Math. Statist., Hayward, CA \EndRef
\Ref \by Pitman, J. \yr 1999 \paper Coalescents with multiple collision \jour \AnnProb \vol 27 \pages 1870-1902 \EndRef
\Ref \by Pitman, J. \yr 2003 \paper Poisson-Kingman partitions. In \textit{Statistics and science: a Festschrift for Terry Speed}, D.R. Goldstein (editor), IMS Lecture Notes-Monograph series, Vol. 40, pp. 1-34, Inst. Math. Statist., Hayward, CA \EndRef
\Ref \by Pitman, J. \yr 2005 \paper Combinatorial Stochastic Processes. Lecture Notes in Mathematics and in Probability Surveys, Springer. \\ Available at \texttt{http://bibserver.berkeley.edu/csp/april05/bookcsp.pdf} \EndRef
\Ref \by Pitman, J. and Yor, M. \yr 1997 \paper The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator \jour \AnnProb \vol 25 \pages 855-900 \EndRef
\Ref \by Schweinsberg, J. \yr 2000 \paper Coalescents with simultaneous multiple collisions \jour Electron. J. Probab. \vol 5 \pages 1-50 \EndRef
\Ref \by Teh, Y.W., Jordan, M.I., Beal, M.J. and Blei, D.M. \yr 2006 \paper Hierarchical Dirichlet Processes. Available at \texttt{http://stat-www.berkeley.edu/tech-reports/index.html}\\To appear in \textit{J. Amer. Statist. Assoc} \EndRef
\Ref \by Vershik, A.M., Yor, M. and Tsilevich, N.V. \yr 2004 \paper On the Markov-Krein identity and quasi-invariance of the gamma process \jour J. Math. Sci. \vol 121 \pages 2303-2310 \EndRef
\vskip0.75in
\tensmc
\Tabular{ll}
Man-Wai Ho\\ Department of Statistics and Applied Probability \\ National University of Singapore\\ 6 Science Drive 2\\ Singapore 117546\\ Republic of Singapore\\ \rm stahmw\at nus.edu.sg\\
\\ \EndTabular
\Tabular{ll}
Lancelot F. James\\ The Hong Kong University of Science and Technology\\ Department of Information and Systems Management\\ Clear Water Bay, Kowloon\\ Hong Kong\\ \rm lancelot\at ust.hk\\
\\ \EndTabular
\Tabular{ll}
John W. Lau\\ Department of Mathematics\\ University of Bristol Bristol\\ BS8 1TW\\ United Kingdom\\ \rm John.Lau\at bristol.ac.uk\\
\EndTabular
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\begin{document}
\title[Schreier Families \& $\mathcal{F}$-greedy-type bases]{Schreier Families and $\mathcal{F}$-(almost) greedy bases}
\author{Kevin Beanland}
\email{\textcolor{blue}{\href{mailto:beanlandk@wlu.edu}{beanlandk@wlu.edu}}} \address{Department of Mathematics, Washington and Lee University, Lexington, VA 24450, USA.}
\author{H\`ung Vi\d{\^e}t Chu}
\email{\textcolor{blue}{\href{mailto:hungchu2@illinois.edu}{hungchu2@illinois.edu}}} \address{Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA}
\begin{abstract} Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. In this paper, we introduce and characterize $\mathcal{F}$-(almost) greedy bases. Given such a family $\mathcal{F}$, a basis $(e_n)_n$ for a Banach space $X$ is called $\mathcal{F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb{N}$, and $G_m(x)$, we have
$$\|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.$$ Here $G_m(x)$ is a greedy sum of $x$ of order $m$, and $\mathbb{K}$ is the scalar field. From the definition, any $\mathcal{F}$-greedy basis is quasi-greedy and so, the notion of being $\mathcal{F}$-greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal{F}$-greedy bases as being $\mathcal{F}$-unconditional, $\mathcal{F}$-disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal{F}$-almost greedy bases.
Furthermore, we provide several examples of bases that are nontrivially $\mathcal{F}$-greedy. For a countable ordinal $\alpha$, we consider the case $\mathcal{F}=\mathcal{S}_\alpha$, where $\mathcal{S}_\alpha$ is the Schreier family of order $\alpha$. We show that for each $\alpha$, there is a basis that is $\mathcal{S}_{\alpha}$-greedy but is not $\mathcal{S}_{\alpha+1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta$, $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_\alpha\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_\beta\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$$
\end{abstract}
\subjclass[2020]{41A65; 46B15}
\keywords{Thresholding Greedy Algorithm, Schreier unconditional, Schreier families}
\thanks{The second author acknowledges the summer funding from the Department of Mathematics at the University of Illinois at Urbana-Champaign.}
\maketitle
\tableofcontents
\section{Introduction}
A (semi-normalized) \textit{basis} in a Banach space $X$ over the field $\mathbb{K}$ is a countable collection $(e_n)_n$ such that \begin{enumerate}
\item[i)] $\overline{\spann\{e_n: n\in\mathbb{N}\}} = X$,
\item[ii)] there exists a unique sequence $(e_n^*)_n\subset X^*$ such that $e_i^*(e_j) = \delta_{i, j}$ for all $i, j\in\mathbb{N}$, and
\item[iii)] there exist $c_1, c_2 > 0$ such that
$$0 \ <\ c_1 := \inf_n\{\|e_n\|, \|e_n^*\|\}\ \leqslant\ \sup_n\{\|e_n\|, \|e_n^*\|\} \ =:\ c_2 \ <\ \infty.$$ \end{enumerate}
In 1999, Konyagin and Temlyakov \cite{KT1} introduced the Thresholding Greedy Algorithm (TGA), which picks the largest coefficients (in modulus) for the approximation. In particular, for each $x\in X$ and $m\in\mathbb{N}$, a set $\Lambda_m(x)$ is a \textit{greedy set} of order $m$ if $|\Lambda_m(x)| = m$ and $\min_{n\in\Lambda_m(x)}|e_n^*(x)|\geqslant \max_{n\notin \Lambda_m(x)}|e_n^*(x)|$. A \textit{greedy operator} $G_m: X \to X$ is defined as $$G_m(x) \ =\ \sum_{n\in \Lambda_m(x)}e_n^*(x)e_n,\mbox{ for some }\Lambda_m(x).$$ Note that $\Lambda_m(x)$ (and thus, $G_m(x)$) may not be unique and $G_m$ is not even linear. The TGA is a sequence of greedy operators $(G_m)_{m=1}^\infty$ that gives the corresponding sequence of approximants $(G_m(x))_{m=1}^\infty$ for each $x\in X$.
A basis $(e_n)_n$ for a Banach space $X$ is called \textit{greedy} if there is a $C\geqslant 1$ such that for all $x\in X,m\in \mathbb{N}$, and $G_m$,
$$\|x - G_m(x)\| \ \leqslant\ \ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, (a_n)\subset \mathbb{K}\right\}.$$
A basis is called quasi-greedy \cite{KT1} if there is a $C\geqslant 1$ so that for all $x\in X, m\in \mathbb{N}$, and $G_m$, we have $\|G_m(x)\| \leqslant C\|x\|$. The smallest such $C$ is denoted by $\mathbf C_w$, called the \textit{quasi-greedy constant}. Also for quasi-greedy bases, let $\mathbf C_\ell$, called the \textit{suppression quasi-greedy constant}, be the smallest constant such that
$$\|x-G_m(x)\| \ \leqslant\ \mathbf C_\ell\|x\|,\forall x\in X, \forall m\in\mathbb{N}, \forall G_m.$$ There are many examples of quasi-greedy bases that are not greedy (see \cite[Example 10.2.9]{AK}), and there has been research on the existence of greedy bases for certain classical spaces (\cite{DFOS, G}).
In this paper, we introduce and study the notion of what we call $\mathcal{F}$-greedy bases which interpolate between greedy bases and quasi-greedy bases. Recall that a collection $\mathcal{F}$ of finite subsets of $\mathbb{N}$ is said to be \textit{hereditary} if $F\in \mathcal{F}$ and $G \subset F$ imply $G \in \mathcal{F}$.
\begin{defi}\normalfont Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. A basis $(e_n)_n$ is $\mathcal{F}$-greedy if there exists a constant $C\geqslant 1$ such that for all $x\in X,m\in \mathbb{N}$, and $G_m$,
$$\|x - G_m(x)\|\ \leqslant \ C\sigma_m^{\mathcal{F}}(x),$$
where $$\sigma_m^{\mathcal{F}}(x)\ := \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.$$ The least constant $C$ is denoted by $\mathbf{C}_g^{\mathcal{F}}$. \end{defi}
\begin{rek}In the case the $\mathcal{F}=\mathcal{P}(\mathbb{N})$, $\mathcal{F}$-greedy corresponds to greedy and when $\mathcal{F}=\{\emptyset\}$, $\mathcal{F}$-greedy corresponds to quasi-greedy. \end{rek}
The first order of business is to generalize the theorem of Konyagin and Temlyakov, which characterizes greedy bases as being unconditional and democratic. To do so, we introduce the definitions of $\mathcal{F}$- unconditionality and $\mathcal{F}$-democracy. For various families $\mathcal{F}$, the notion of $\mathcal{F}$-unconditionality has appeared numerous times in the literature, most notably in Odell's result \cite{Od}, which states that every normalized weakly null sequence in a Banach space has a subsequence that is Schreier-unconditional. Also see \cite{AG, AGR, AMT} for other notion of unconditionality for weakly null sequences.
For a basis $(e_n)_n$ of a Banach space $X$ and a finite set $A \subset \mathbb{N}$, let $P_A: X \to X$ be defined by $P_A(\sum_i e^*_i(x) e_i)=\sum_{i\in A} e^*_i(x) e_i$.
\begin{defi}\normalfont A basis $(e_n)$ of a Banach space $X$ is $\mathcal{F}$-unconditional if there exists a constant $C\geqslant 1$ such that for each $x \in X$ and $A \in \mathcal{F}$, we have
$$\|x- P_A(x)\|\ \leqslant\ C\|x\|.$$ The least constant $C$ is denoted by $\mathbf K_s^{\mathcal{F}}$. We say that $(e_n)$ is $\mathbf K_s^{\mathcal{F}}$-$\mathcal{F}$-suppression unconditional. \end{defi}
As far as we know, the following natural definition has not appeared in the literature before.
\begin{defi}\label{demo}\normalfont A basis $(e_n)$ is $\mathcal{F}$-disjoint democratic ($\mathcal{F}$-disjoint superdemocratic, respectively) if there exists a constant $C\geqslant 1$ such that
$$\left\|\sum_{i\in A}e_i\right\|\ \leqslant \ C\left\|\sum_{i\in B}e_i\right\|,\mbox{ }\left(\left\|\sum_{i\in A}\varepsilon_i e_i\right\|\ \leqslant\ C\left\|\sum_{i\in B} \delta_i e_i\right\|,\mbox{ respectively}\right),$$
for all finite sets $A, B\subset\mathbb{N}$ with $A\in \mathcal{F}$, $|A|\leqslant |B|, A\cap B = \emptyset$ and signs $(\varepsilon_i), (\delta_i)$. The least constant $C$ is denoted by $\mathbf C^{\mathcal{F}}_{d,\sqcup}$ ($\mathbf C^{\mathcal{F}}_{sd,\sqcup}$, respectively). When $\mathcal{F} = \mathcal{P}(\mathbb{N})$, we say that $(e_n)$ is (super)democratic. \end{defi}
One of our main results is the following generalization of the Konyagin-Temlyakov Theorem \cite{KT1}.
\begin{thm}\label{m1'} A basis $(e_n)$ in a Banach space $X$ is $\mathcal{F}$-greedy if and only if it is quasi-greedy, $\mathcal{F}$-unconditional, and $\mathcal{F}$-disjoint democratic. \end{thm}
We also present another characterization regarding $\mathcal{F}$-almost greedy bases.
\begin{defi}\normalfont A basis $(e_n)$ is $\mathcal{F}$-almost greedy if there exists a constant $C\geqslant 1$ such that for all $x \in X, m\in \mathbb{N}$, and $G_m$, we have
$$\|x-G_m(x)\| \ \leqslant\ C\inf\{\|x-P_A(x)\|\,:\, |A|\leqslant m, A\in \mathcal{F}\}.$$ The least constant $C$ is denoted by $\mathbf{C}_a^{\mathcal{F}}$. \end{defi}
The next theorem generalizes \cite[Theorem 3.3]{DKKT}.
\begin{thm} A basis $(e_n)$ is $\mathcal{F}$-almost greedy if and only if it is quasi-greedy and $\mathcal{F}$-disjoint democratic. \end{thm}
The second set of results in this paper focuses on the well-known Schreier families $(\mathcal{S}_\alpha)_{n=1}^\infty$ (for each countable ordinal $\alpha$) introduced by Alspach and Argyros \cite{AA2}. The sequence of countable ordinals is $$0, 1, \ldots, n, \ldots, \omega, \omega + 1, \ldots, 2\omega,\ldots, $$ We recall the definition of $\mathcal{S}_\alpha$. For two sets $A, B\subset\mathbb{N}$, we write $A < B$ to mean that $a < b$ for all $a\in A, b\in B$. It holds vacuously that $\emptyset < A$ and $\emptyset > A$. Also, $n < A$ for a number $n$ means $\{n\} < A$. Let $\mathcal{S}_0$ be the set of singletons and the empty set. Supposing that $\mathcal{S}_{\alpha}$ has be defined for some ordinal $\alpha \geqslant 0$, we define \begin{equation*}\mathcal{S}_{\alpha+1}\ =\ \{\cup^{m}_{i=1} E_i:m\leqslant E_1<E_2<\cdots<E_m \mbox{ and } E_i\in \mathcal{S}_\alpha, \forall 1\leqslant i\leqslant m\}.\end{equation*} If $\alpha$ is a limit ordinal, then fix $\alpha_{m}+1\nearrow \alpha$ with $\mathcal{S}_{\alpha_{m}}\subset \mathcal{S}_{\alpha_{m+1}}$ for all $m\geqslant 1$ and define \begin{equation*}
\mathcal{S}_{\alpha} \ =\ \{E\subset \mathbb{N}\,:\, \mbox{ for some }m\geqslant 1, m\leqslant E\in \mathcal{S}_{\alpha_m+1}\}. \end{equation*}
The following proposition is well-known, but we include its proof for completion.
\begin{prop}\label{k2} Let $\alpha < \beta$ be two countable ordinals. There exists $N\in \mathbb{N}$ such that $$E\backslash \{1, \ldots, N-1\}\in \mathcal{S}_{\beta}, \forall E\in \mathcal{S}_\alpha.$$ \end{prop}
\begin{proof} Fix two ordinals $\alpha < \beta$. We prove by induction. Base cases: if $\beta = 0$, there is nothing to prove. If $\beta = 1$, then $\alpha = 0$. Clearly, $\mathcal{S}_0\subset \mathcal{S}_1$. Inductive hypothesis: suppose that the proposition holds for all $\eta < \beta$. If $\beta$ is a successor ordinal, then write $\beta = \gamma + 1$. Since $\alpha < \beta$, we have $\alpha\leqslant \gamma$. By the inductive hypothesis, there exists $N\in\mathbb{N}$ such that $$E\backslash \{1,\ldots, N-1\} \in \mathcal{S}_\gamma, \forall E\in \mathcal{S}_\alpha.$$ By definition, $\mathcal{S}_\gamma\subset\mathcal{S}_\beta$. Hence, $$E\backslash \{1, \ldots, N-1\}\in \mathcal{S}_\beta, \forall E\in\mathcal{S}_\alpha.$$ If $\beta$ is a limit ordinal, then let $\beta_m\nearrow \beta$. There exists $M\in\mathbb{N}$ such that $\beta_M\geqslant \alpha$. By the inductive hypothesis, there exists $N_1\in\mathbb{N}$ such that $$E\backslash \{1, \ldots, N_1-1\}\in \mathcal{S}_{\beta_M}, \forall E\in \mathcal{S}_{\alpha}.$$ By definition, $$E\backslash \{1, \ldots, M-1\}\in \mathcal{S}_{\beta}, \forall E\in \mathcal{S}_{\beta_M}.$$ Therefore, $$E\backslash \{1, \ldots, \max\{N_1, M\}-1\}\in \mathcal{S}_\beta, \forall E\in \mathcal{S}_{\alpha}.$$ This completes our proof. \end{proof}
We have the following corollary, which is proved in Section \ref{pc}.
\begin{cor}\label{m30'} For two countable ordinals $\alpha < \beta$, an $\mathcal{S}_\beta$-greedy basis is $\mathcal{S}_\alpha$-greedy. \end{cor}
Each Schreier family $\mathcal{S}_\alpha$ is obviously hereditary and are moreover spreading and compact (see \cite[pp. 1049 and 1051]{AGR}). We shall show that each of the following implications cannot be reversed: for two countable ordinals $\alpha < \beta$, $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_\alpha\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_\beta\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$$ We, thereby, study the greedy counterpart of the notion of $\mathcal{S}_\alpha$-unconditionality.
\begin{thm}\label{m30} For two countable ordinals $\alpha < \beta$, there exists a Banach space $X$ with an $\mathcal{S}_\alpha$-greedy basis that is not $\mathcal{S}_{\beta}$-greedy. \end{thm}
\begin{thm}\label{m31} Fix a countable ordinal $\alpha$. \begin{enumerate}
\item A basis is greedy if and only if it is $C$-$\mathcal{S}_{\alpha+m}$-greedy for all $m\in\mathbb{N}$ and some uniform $C\geqslant 1$.
\item There exists a basis that is $\mathcal{S}_{\alpha+m}$-greedy (with different constants) for all $m\in \mathbb{N}$ but is not greedy. \end{enumerate} \end{thm}
\section{Characterizations of $\mathcal{F}$-greedy bases}
In this section, we prove Theorem \ref{m1'} and other characterizations of $\mathcal{F}$-greedy bases. Throughout, $\mathcal{F}$ will be a hereditary family of finite subsets of $\mathbb{N}$. We first need to define Property (A, $\mathcal{F}$), inspired by the classical Property (A) introduced by Albiac and Wojtaszczyk in \cite{AW}. Write $\sqcup_{i\in I} A_i$, for some index set $I$ and sets $(A_i)_{i\in I}$, to mean that the $A_i$'s are pairwise disjoint. Define $1_A = \sum_{n\in A}e_n\mbox{ and }1_{\varepsilon A} = \sum_{n\in A}\varepsilon_n e_n$, for some signs $(\varepsilon) = (\varepsilon_n)_n\in \mathbb{K}^{\mathbb{N}}$.
\begin{defi}\label{dAF}\normalfont A basis $(e_n)$ is said to have Property (A, $\mathcal{F}$) if there exists a constant $C\geqslant 1$ such that
$$\left\|x+\sum_{i \in A} \varepsilon_i e_i\right\|\ \leqslant\ C\left\|x + \sum_{n\in B}b_n e_n\right\|,$$
for all $x\in X$ with $\|x\|_\infty\leqslant 1$, for all finite sets $A, B\subset\mathbb{N}$ with $|A|\leqslant |B|$, $A\in \mathcal{F}$, $A\sqcup B\sqcup \supp(x)$, and for all signs $(\varepsilon_i)$ and $|b_n|\geqslant 1$. The least constant $C$ is denoted by $\mathbf C^{\mathcal{F}}_b$. \end{defi}
\begin{prop}\label{p1} A basis $(e_n)$ has $\mathbf C^{\mathcal{F}}_{b}$-Property (A, $\mathcal{F}$) if and only if
\begin{equation}\label{e1}\|x\|\ \leqslant\ \mathbf C^{\mathcal{F}}_b \left\|x-P_A(x) + \sum_{n\in B}b_n e_n\right\|,\end{equation} for all
$x\in X$ with $\|x\|_\infty\leqslant 1$, for all finite sets $A, B\subset\mathbb{N}$ with $|A|\leqslant |B|$, $A\in \mathcal{F}$, $B\cap (A\cup \supp(x)) = \emptyset$, and $|b_n|\geqslant 1$. \end{prop}
\begin{proof} Assume \eqref{e1}. Let $x, A, B, (\varepsilon), (b_n)_{n\in B}$ be as in Definition \ref{dAF}. Let $y = x + 1_{\varepsilon A}$. By \eqref{e1},
$$\|x+1_{\varepsilon A}\|\ =\ \|y\|\ \leqslant\ \mathbf C^{\mathcal{F}}_b\left\|y- P_A(y) + \sum_{n\in B}b_n e_n\right\|\ =\ \mathbf C^{\mathcal{F}}_b \left\|x+\sum_{n\in B}b_ne_n\right\|.$$
Conversely, assume that $(e_n)$ has $\mathbf C^{\mathcal{F}}_b$-Property (A, $\mathcal{F}$). Let $x, A, B, (b_n)_{n\in B}$ be as in \eqref{e1}. We have \begin{align*}
\|x\|\ =\ \left\|x-P_A(x) + \sum_{n\in A}e_n^*(x)e_n\right\|&\ \leqslant\ \sup_{(\delta)}\left\|x-P_A(x) + 1_{\delta A}\right\|\mbox{ by norm convexity}\\
&\ \leqslant\ \mathbf C^{\mathcal{F}}_b\left\|x - P_A(x) + \sum_{n\in B}b_ne_n\right\|, \end{align*} where the last inequality is due to Property (A, $\mathcal{F}$). \end{proof}
\begin{thm}\label{m1} Let $(e_n)$ be a basis for a Banach space $X$. \begin{enumerate} \item The basis $(e_n)$ is $\mathbf C^{\mathcal{F}}_g$-$\mathcal{F}$-greedy, then $(e_n)$ is $\mathbf C^{\mathcal{F}}_g$-$\mathcal{F}$-suppression unconditional and has $\mathbf C^{\mathcal{F}}_g$-Property (A, $\mathcal{F}$). \item The basis $(e_n)$ is $\mathbf K^{\mathcal{F}}_s$-$\mathcal{F}$-suppression unconditional and has $\mathbf C^{\mathcal{F}}_b$-Property (A, $\mathcal{F}$), then $(e_n)$ is $\mathbf K^{\mathcal{F}}_s\mathbf C^{\mathcal{F}}_b$-$\mathcal{F}$-greedy. \end{enumerate} \end{thm}
\begin{proof} (1) Assume that $(e_n)$ is $\mathbf C^{\mathcal{F}}_g$-$\mathcal{F}$-greedy. We shall show that $(e_n)$ is $\mathcal{F}$-unconditional. Choose $x\in X$ and a finite set $B\in \mathcal{F}$. Set $$y\ :=\ \sum_{n\in B}(e_n^*(x)+\alpha)e_n + \sum_{n\notin B} e_n^*(x)e_n,$$ where $\alpha$ is sufficiently large such that $B$ is a greedy set of $y$. Then
$$\|x-P_B(x)\|\ =\ \|y-P_B(y)\|\ \leqslant\ \mathbf C_g^{\mathcal{F}}\sigma^{\mathcal{F}}_{|B|}(y)\ \leqslant\ \mathbf C_g^{\mathcal{F}}\|y-\alpha 1_B\|\ =\ \mathbf C_g^{\mathcal{F}}\|x\|.$$ Hence, $(e_n)$ is $\mathbf C^{\mathcal{F}}_g$-$\mathcal{F}$-suppression unconditional.
Next, we prove Property (A, $\mathcal{F}$). Choose $x, A, B, (\varepsilon_i), (b_n)_{n\in B}$ as in Definition \ref{dAF}. Set $y:= x + 1_{\varepsilon A} + \sum_{n\in B}b_ne_n$. Since $B$ is a greedy set of $y$, we have
$$\|x+1_{\varepsilon A}\|\ =\ \|y-P_B(y)\|\ \leqslant\ \mathbf C_g^{\mathcal{F}}\sigma^{\mathcal{F}}_{|B|}(y)\ \leqslant\ \mathbf C_g^{\mathcal{F}}\|y-P_A(y)\|\ =\ \mathbf C_g^{\mathcal{F}}\left\|x+\sum_{n\in B}b_ne_n\right\|.$$ Therefore, $(e_n)$ has $\mathbf C_g^{\mathcal{F}}$-Property (A, $\mathcal{F}$).
(2) Assume that $(e_n)$ is $\mathbf K^{\mathcal{F}}_s$-$\mathcal{F}$-unconditional and has $\mathbf C^{\mathcal{F}}_b$-Property (A, $\mathcal{F}$). Let $x\in X$ with a greedy set $A$. Choose $B\in \mathcal{F}$ with $|B|\leqslant |A|$ and choose $(b_n)_{n\in B}\subset \mathbb{K}$. If $A\backslash B = \emptyset$, then $A = B$ and we have
\begin{align*}\|x-P_A(x)\|\ =\ \|x-P_B(x)\|&\ \leqslant\ \mathbf K^{\mathcal{F}}_s\left\|x-P_B(x) + \sum_{n\in B}(e_n^*(x)-b_n)e_n\right\|\\
&\ =\ \mathbf K^{\mathcal{F}}_s\left\|x - \sum_{n\in B}b_ne_n\right\|. \end{align*}
Assume that $A\backslash B\neq \emptyset$. Note that $B\backslash A\in \mathcal{F}$ as $\mathcal{F}$ is hereditary and $\min_{n\in A\backslash B}|e_n^*(x)|\geqslant \|x-P_A(x)\|_\infty$. By Proposition \ref{p1}, we have \begin{align*}
\|x-P_A(x)\|&\ \leqslant\ \mathbf C^{\mathcal{F}}_b\|(x-P_A(x)) - P_{B\backslash A}(x) + P_{A\backslash B}(x)\|\\
&\ =\ \mathbf C^{\mathcal{F}}_b\|x-P_B(x)\|\\
&\ \leqslant\ \mathbf C^{\mathcal{F}}_b\mathbf K^{\mathcal{F}}_s\left\|x-P_B(x) + \sum_{n\in B}(e_n^*(x)-b_n)e_n\right\|\\
&\ =\ \mathbf C^{\mathcal{F}}_b\mathbf K^{\mathcal{F}}_s\left\|x-\sum_{n\in B}b_ne_n\right\|. \end{align*} Since $B$ and $(b_n)$ are arbitrary, we know that $(e_n)$ is $\mathbf C^{\mathcal{F}}_b\mathbf K^{\mathcal{F}}_s$-$\mathcal{F}$-greedy. \end{proof}
We have the following immediate corollary.
\begin{cor} A basis $(e_n)$ is $1$-$\mathcal{F}$-greedy if and only if it is $1$-$\mathcal{F}$-unconditional and has $1$-Property (A, $\mathcal{F}$). \end{cor}
The next proposition connects Property $(A,\mathcal{F})$ and $\mathcal{F}$-disjoint democracy.
\begin{prop}\label{p20} Let $(e_n)$ be a quasi-greedy basis. Then $(e_n)$ has Property (A, $\mathcal{F}$) if and only if $(e_n)$ is $\mathcal{F}$-disjoint democratic. \end{prop}
The proof of Proposition \ref{p20} uses the following results which can be found in \cite{W} and \cite[Lemma 2.5]{BBG}.
\begin{lem}\label{bto} Let $(e_n)$ be a $\mathbf C_\ell$-suppression quasi-greedy basis. The following hold \begin{enumerate}
\item For any finite set $A\subset\mathbb{N}$ and sign $(\varepsilon_n)_n$, we have
$$\frac{1}{2\mathbf C_\ell}\left\|\sum_{n\in A}e_n\right\|\ \leqslant\ \left\|\sum_{n\in A}\varepsilon_ne_n\right\| \ \leqslant\ 2\mathbf C_\ell\left\|\sum_{n\in A}e_n\right\|.$$ \item For all $\alpha >0$ and $x \in X$,
$$\left\|\sum_{n \in \Gamma_\alpha(x)} \alpha \sgn(e_n^*(x))e_n + \sum_{n \not \in \Gamma_\alpha(x)} e_n^*(x)e_n\right\|\ \leqslant\ \mathbf{C}_\ell \|x\|,$$
where $\Gamma_\alpha (x) =\{n : |e^*_n(x)|>\alpha\}$. \end{enumerate} \end{lem}
\begin{proof}[Proof of Proposition \ref{p20}] It is obvious that Property (A, $\mathcal{F}$) implies $\mathcal{F}$-disjoint democracy. Let us assume that $(e_n)$ is $\mathbf C^{\mathcal{F}}_{d, \sqcup}$-$\mathcal{F}$-disjoint democratic and is $\mathbf C_\ell$-suppression quasi-greedy (or $\mathbf C_w$-quasi-greedy). Let $x, A, B, (b_n), (\varepsilon_i)$ be as in Definition \ref{dAF}. Since $B$ is a greedy set of $x+\sum_{n\in B}b_n e_n$, we have \begin{align*}
\left\|x+\sum_{n\in B}b_n e_n\right\|\ \geqslant\ \frac{1}{\mathbf C_w}\left\|\sum_{n\in B}b_n e_n\right\|&\ \geqslant\ \frac{1}{\mathbf C_w\mathbf C_\ell}\left\|\sum_{n\in B}\sgn(b_n)e_n\right\|\mbox{ by Lemma \ref{bto}}\\
&\ \geqslant\ \frac{1}{2\mathbf C_w\mathbf C^2_\ell}\|1_B\|\mbox{ by Lemma \ref{bto}}\\
&\ \geqslant\ \frac{1}{2\mathbf C_w\mathbf C^2_\ell\mathbf C^{\mathcal{F}}_{d,\sqcup}}\|1_A\|\ \geqslant\ \frac{1}{4\mathbf C_w\mathbf C^3_\ell\mathbf C^{\mathcal{F}}_{d,\sqcup}}\|1_{\varepsilon A}\|. \end{align*} Again since $B$ is a greedy set of $x+\sum_{n\in B}b_n e_n$,
$$\left\|x+\sum_{n\in B}b_n e_n\right\|\ \geqslant\ \frac{1}{\mathbf C_\ell}\|x\|.$$ Therefore, we obtain
$$2\left\|x+\sum_{n\in B}b_n e_n\right\|\ \geqslant\ \frac{1}{4\mathbf C_w\mathbf C^3_\ell\mathbf C^{\mathcal{F}}_{d}}\|1_{\varepsilon A}\|+\frac{1}{\mathbf C_\ell}\|x\|\ \geqslant\ \frac{1}{4\mathbf C_w\mathbf C^3_\ell\mathbf C^{\mathcal{F}}_{d}}\|1_{\varepsilon A} + x\|.$$ We have shown that
$$\|x+ 1_{\varepsilon A}\|\ \leqslant\ 8\mathbf C_w\mathbf C^3_\ell\mathbf C^{\mathcal{F}}_{d}\left\|x+\sum_{n\in B}b_n e_n\right\|,$$ which completes our proof that $(e_n)$ has Property (A, $\mathcal{F}$). \end{proof}
\begin{thm}\label{m2} For a basis $(e_n)$ of a Banach space $X$, the following are equivalent: \begin{enumerate}
\item $(e_n)$ is $\mathcal{F}$-greedy,
\item $(e_n)$ is $\mathcal{F}$-unconditional and has Property (A, $\mathcal{F}$),
\item $(e_n)$ is $\mathcal{F}$-unconditional, $\mathcal{F}$-disjoint superdemocratic, and quasi-greedy,
\item $(e_n)$ is $\mathcal{F}$-unconditional, $\mathcal{F}$-disjoint democratic, and quasi-greedy. \end{enumerate} \end{thm}
\begin{proof}[Proof of Theorem \ref{m2}] By Theorem \ref{m1}, we have that (1) $\Longleftrightarrow$ (2). Since an $\mathcal{F}$-greedy basis is quasi-greedy, and Property (A, $\mathcal{F}$) implies $\mathcal{F}$-disjoint superdemocracy (by definition), we get (1) $\Longleftrightarrow$ (2) $\Longrightarrow$ (3). Trivially, (3) $\Longrightarrow$ (4). That (4) $\Longrightarrow$ (2) is due to Proposition \ref{p20}. \end{proof}
\section{Characterizations of $\mathcal{F}$-almost greedy bases}
In this section, we first characterize $\mathcal{F}$-almost greedy bases using Property (A, $\mathcal{F}$), then show that the $\mathcal{F}$-almost greedy property is equivalent to the quasi-greedy property plus $\mathcal{F}$-disjoint superdemocracy.
\begin{thm}\label{m6} A basis $(e_n)$ is $C$-$\mathcal{F}$-almost greedy if and only if $(e_n)$ has $C$-Property (A, $\mathcal{F}$). \end{thm}
\begin{proof}[Proof of Theorem \ref{m6}] The proof that $C$-$\mathcal{F}$-almost greediness implies $C$-Property (A, $\mathcal{F}$) is similar to what we have in the proof of Theorem \ref{m1}. Conversely, assume that $(e_n)$ has $C$-Property (A, $\mathcal{F}$). Let $x\in \mathbb{X}$ with a greedy set $A$.
Choose $B\in \mathcal{F}$ with $|B|\leqslant |A|$.
If $A\backslash B = \emptyset$, then $A = B$ and $\|x-P_A(x)\| = \|x-P_B(x)\|$. If $A\backslash B\neq \emptyset$,
note that $\min_{n\in A\backslash B}|e_n^*(x)|\geqslant \|x-P_A(x)\|_\infty$. By Proposition \ref{p1}, we have \begin{align*}
\|x-P_A(x)\|&\ \leqslant\ C\|(x-P_A(x)) - P_{B\backslash A}(x) + P_{A\backslash B}(x)\|\\
&\ =\ C\|x-P_B(x)\|. \end{align*} Since $B$ is arbitrary, we know that $(e_n)$ is $C$-$\mathcal{F}$-almost greedy. \end{proof}
\begin{thm}\label{m3} Let $(e_n)$ be a basis. The following are equivalent: \begin{enumerate}
\item $(e_n)$ is $\mathcal{F}$-almost greedy,
\item $(e_n)$ has Property (A, $\mathcal{F}$),
\item $(e_n)$ is $\mathcal{F}$-disjoint superdemocratic and quasi-greedy,
\item $(e_n)$ is $\mathcal{F}$-disjoint democratic and quasi-greedy. \end{enumerate} \end{thm}
\begin{proof}[Proof of Theorem \ref{m3}] That (1) $\Longleftrightarrow$ (2) follows from Theorem \ref{m6}. Clearly, an $\mathcal{F}$-almost greedy basis is quasi-greedy. By Proposition \ref{p20}, we have (2) $\Longleftrightarrow$ (4). Since (1) $\Longleftrightarrow$ (2) $\Longrightarrow$ (3) $\Longrightarrow$ (4), we are done. \end{proof}
\begin{cor}[Generalization of Theorem 2.3 in \cite{AA}] A basis $(e_n)$ is $1$-$\mathcal{F}$-almost greedy if and only if $(e_n)$ has $1$-Property (A, $\mathcal{F}$). \end{cor}
\section{Schreier families and $\mathcal{S}_\alpha$-greedy bases}\label{pc}
In this section, we will provide several non-trivial examples of $\mathcal{F}$-greedy basis. In particular, we will consider bases which are quasi-greedy but not greedy. As mentioned in the introduction, the Schreier families $\mathcal{S}_\alpha$ form a particularly rich collection of finite subsets of $\mathbb{N}$.
\begin{proof}[Proof of Corollary \ref{m30'}] Fix two countable ordinals $\alpha < \beta$. Let $N$ be as in Proposition \ref{k2}. Suppose that $(e_n)$ is $C$-$\mathcal{S}_\beta$-greedy for some constant $C\geqslant 1$. By Theorems \ref{m1'} and \ref{m1}, $(e_n)$ is $C$-$\mathcal{S}_\beta$-suppression unconditional, $C$-$\mathcal{S}_\beta$-disjoint democratic, and $C$-suppression quasi-greedy.
We show that $(e_n)$ is $C$-$\mathcal{S}_\alpha$-suppression unconditional. Let $x\in X$ and $E\in \mathcal{S}_\alpha$. We know that $E\backslash \{1, \ldots, N-1\}\in \mathcal{S}_\beta$. Hence,
$$\|x-P_{E\backslash \{1, \ldots, N-1\}}(x)\|\ \leqslant\ C\|x\|.$$ We have \begin{align*}
\|x-P_{E}(x)\|&\ \leqslant\ \|x-P_{E\backslash \{1, \ldots, N-1\}}(x)\| + \|P_{E\cap \{1, \ldots, N-1\}}(x)\|\\
&\ \leqslant\ C\|x\| + N\sup_{n}\|e_n\|\|e_n^*\|\|x\|\ \leqslant\ (C+Nc_2^2)\|x\|. \end{align*} Therefore, $(e_n)$ is $\mathcal{S}_\alpha$-suppression unconditional.
Next, we show that $(e_n)$ is $C$-$\mathcal{S}_\alpha$-disjoint democratic. Let $A\in \mathcal{S}_\alpha$ and $B\subset \mathbb{N}$ such that $A\cap B = \emptyset$ and $|A|\leqslant |B|$. Since $A\backslash \{1, \ldots, N-1\}\in\mathcal{S}_\beta$, we have
$$\|1_{A\backslash \{1, \ldots, N-1\}}\|\ \leqslant\ C\|1_B\|$$ Also, due to $C$-quasi-greediness,
$$C\|1_B\|\ \geqslant\ c_1.$$ Hence, \begin{align*}
\|1_A\|&\ \leqslant\ \|1_{A\backslash \{1, \ldots, N-1\}}\| + \|1_{A\cap \{1, \ldots, N-1\}}\|\\
&\ \leqslant\ C\|1_B\| + c_2N\ \leqslant\ C\|1_B\| + \frac{Cc_2N}{c_1}\|1_B\|\ =\ C\left(1+N\frac{c_2}{c_1}\right)\|1_B\|. \end{align*} Therefore, $(e_n)$ is $\mathcal{S}_\alpha$-disjoint democratic.
By Theorem \ref{m1'}, we conclude that $(e_n)$ is $\mathcal{S}_\alpha$-greedy. \end{proof}
We have $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_\alpha\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_\beta\mbox{-greedy}\ \Longleftarrow \ \mbox{greedy}.$$ We construct bases to show that none of the reverse implications holds. Consider the following definition.
\begin{defi} \normalfont Let $\omega_1$ denote the set of all countable ordinals and $(\alpha,\beta) \in (\omega_1\cup \{\infty\})^2$. A quasi-greedy basis $(e_n)$ for a Banach space $X$ is called $(\alpha,\beta)$-quasi-greedy if and only if $(e_n)$ is $\mathcal{S}_\alpha$-unconditional but not $\mathcal{S}_{\alpha+1}$-unconditional and $\mathcal{S}_\beta$-disjoint democratic but not $\mathcal{S}_{\beta+1}$-disjoint democratic.
Suppose that either $\alpha$ or $\beta$ is $\infty$. If we denote by $\mathcal{S}_\infty$ the set of all finite subsets of $\mathbb{N}$, then $\mathcal{S}_\infty$-unconditionality and $\mathcal{S}_\infty$-disjoint democracy coincide with unconditionality and disjoint democracy, respectively. \end{defi}
\begin{rek}\normalfont Due to the proof of Corollary \ref{m30'}, a basis $(e_n)$ for a Banach space $X$ is $\mathcal{S}_{\eta}$-greedy if and only if it is $(\alpha,\beta)$-quasi-greedy for some $\alpha\geqslant \eta$ and $\beta \geqslant \eta$. Note also that the $(\infty,\infty)$-quasi-greedy property is the same as the greedy property, and a $(0,0)$-quasi-greedy basis is quasi-greedy but is far from being greedy. \end{rek}
We prove Theorem \ref{m30} by providing the following examples.
\begin{thm}\label{zerozero} There are spaces with bases $(e_n)$ that are $(0,0)$-quasi-greedy, $(\infty, 0)$-quasi-greedy, and $(0, \infty)$-quasi-greedy. \end{thm}
\begin{thm}\label{minfinity} Fix a nonzero $\alpha\in \omega_1$. There is a space $X_{\alpha,\infty}$ with a basis $(e_n)$ that is $(\alpha,\infty)$-quasi-greedy. Hence, $X_{\alpha,\infty}$ is $\mathcal{S}_\alpha$-greedy but not $\mathcal{S}_{\alpha+1}$-greedy. \end{thm}
\begin{thm} Fix a nonzero $\alpha\in \omega_1$. There is a space $X_{\infty, \alpha}$ with a basis $(e_n)$ that is $(\infty, \alpha)$-quasi-greedy. Hence, $X_{\infty, \alpha}$ is $\mathcal{S}_\alpha$-greedy but not $\mathcal{S}_{\alpha+1}$-greedy. \end{thm}
\begin{rek}\normalfont The bases we construct in Theorem \ref{minfinity} give new examples of conditional quasi-greedy bases. Furthermore, these bases are $1$-suppression quasi-greedy. \end{rek}
\subsection{Proof of Theorem \ref{zerozero}}
\subsubsection{A $(0,0)$-quasi-greedy basis}\label{1st}
We modify an example by Konyagin and Temlyakov \cite{KT1} who gave a conditional basis that is quasi-greedy. We shall construct a quasi-greedy basis that is neither $\mathcal{S}_1$-disjoint democratic nor $\mathcal{S}_1$-unconditional. For each $N\in\mathbb{N}$, let $X_N$ be the $(2N-1)$-dimensional space that is the completion of $c_{00}$ under the norm: for $x = (a_i)_i$,
$$\|(a_i)_i\|\ =\ \max\left\{\left(\sum_{i=1}^{2N-1}|a_i|^2\right)^{1/2}, \sup_{N\leqslant m\leqslant 2N-1}\left|\sum_{i=N}^m \frac{1}{\sqrt{i-N+1}}a_i\right|\right\}.$$ Let $X = (\oplus_{N=1}^\infty X_N)_{c_0}$. Let $\mathcal{B}$ be the canonical basis of $X$.
\begin{thm} The basis $\mathcal{B}$ is $(0,0)$-quasi-greedy. \end{thm}
\begin{proof} First, we show that $\mathcal{B}$ is not $\mathcal{S}_1$-unconditional. For each $X_N$, let $(f^N_i)_{i=1}^{2N-1}$ be the canonical basis of $X_N$ (that also belongs to $\mathcal{B}$). We have
$$\left\|\sum_{i=N}^{2N-1}\frac{1}{\sqrt{i-N+1}}f^N_i\right\|\ =\ \sum_{i=1}^N\frac{1}{i},\mbox{ while }\left\|\sum_{i=N}^{2N-1}\frac{(-1)^i}{\sqrt{i-N+1}}f^N_i\right\|\ =\ \left(\sum_{i=1}^N\frac{1}{i}\right)^{1/2}.$$
As $N\rightarrow\infty$, $\left\|\sum_{i=N}^{2N-1}\frac{1}{\sqrt{i-N+1}}f^N_i\right\|/\left\|\sum_{i=N}^{2N-1}\frac{(-1)^i}{\sqrt{i-N+1}}f^N_i\right\|\rightarrow\infty$; hence, $\mathcal{B}$ is not $\mathcal{S}_1$-unconditional.
Next, we show that $\mathcal{B}$ is not $\mathcal{S}_1$-disjoint democratic. We have
$$\left\|\sum_{i=N}^{2N-1}f^N_i\right\|\ =\ \sum_{i=1}^{N}\frac{1}{\sqrt{i}}, \mbox{ while }\left\|\sum_{i=N+1}^{2N} f^i_1\right\|\ =\ 1.$$ Therefore, $\mathcal{B}$ is not $\mathcal{S}_1$-disjoint democratic.
Finally, we prove that $\mathcal{B}$ is quasi-greedy. To do so, we need only to show that for each $N$, the basis $(f^N_i)_{i=1}^{2N-1}$ has the same quasi-greedy constant of $3 + \sqrt{2}$. Let $(a_i)_{i=1}^{2N-1}\in X_N$, where $\|(a_i)_i\|\leqslant 1$. It suffices to prove that
$$\left|\sum_{i\in \Lambda}\frac{1}{\sqrt{i-N+1}}a_i\right|\ \leqslant\ 3+\sqrt{2},$$
for all $\varepsilon > 0$, for all $M\in [N, 2N-1]$, and $\Lambda = \{N\leqslant i\leqslant M: |a_i|>\varepsilon\}$. Since $\|(a_i)_i\|\leqslant 1$, we know that $|a_i|\leqslant 1$ and so, we can assume that $0 < \varepsilon < 1$. Set $L = \lfloor \varepsilon^{-2}\rfloor$ to have $1/2 \leqslant \varepsilon^2 L\leqslant 1$. We proceed by case analysis.
Case 1: $M-N+1\leqslant L$. We have \begin{align*}
\left|\sum_{i\in \Lambda}\frac{a_i}{\sqrt{i-N+1}}\right|&\ \leqslant\ \left|\sum_{N\leqslant i\leqslant M}\frac{a_i}{\sqrt{i-N+1}}\right| + \left|\sum_{\substack{N\leqslant i\leqslant M\\|a_i| \leqslant \varepsilon}}\frac{a_i}{\sqrt{i-N+1}}\right|\\
&\ \leqslant\ 1 + \varepsilon \sum_{i=N}^{M}\frac{1}{\sqrt{i-N+1}}\\
&\ \leqslant\ 1 + \varepsilon \sum_{i=1}^{M-N+1}\frac{1}{\sqrt{i}}\\
&\ \leqslant\ 1 + 2\varepsilon\sqrt{M-N+1}\ \leqslant\ 1 + 2\varepsilon \sqrt{L}\ \leqslant\ 3. \end{align*}
Case 2: $M-N+1 > L$. We have $$
\left|\sum_{i\in \Lambda}\frac{a_i}{\sqrt{i-N+1}}\right|\ =\ \left|\sum_{\substack{N\leqslant i\leqslant N+L-1\\ |a_i|> \varepsilon}}\frac{a_i}{\sqrt{i-N+1}}\right| + \left|\sum_{\substack{N+L\leqslant i\leqslant M\\|a_i| > \varepsilon}}\frac{a_i}{\sqrt{i-N+1}}\right|. $$ By above,
$$ \left|\sum_{\substack{N\leqslant i\leqslant N+L-1\\ |a_i|> \varepsilon}}\frac{a_i}{\sqrt{i-N+1}}\right|\ \leqslant\ 3.$$ Furthermore, we have \begin{align*}
\left|\sum_{\substack{N+L\leqslant i\leqslant M\\|a_i| > \varepsilon}}\frac{a_i}{\sqrt{i-N+1}}\right|&\ \leqslant\ \left(\sum_{N+L\leqslant i\leqslant M}\frac{1}{(i-N+1)^{3/2}}\right)^{1/3}\left(\sum_{\substack{N+L\leqslant i\leqslant M\\|a_i|>\varepsilon}}|a_i|^{3/2}\right)^{2/3}\\
&\ \leqslant\ \left(\sum_{i=L+1}^\infty \frac{1}{i^{3/2}}\right)^{1/3}\left(\sum_{\substack{N+L\leqslant i\leqslant M\\|a_i| > \varepsilon}}|a_i|^{3/2}\sqrt{\frac{|a_i|}{\varepsilon}}\right)^{2/3}\\
&\ \leqslant\ 2^{1/3}L^{-1/6}\varepsilon^{-1/3}\ \leqslant \ \sqrt{2}. \end{align*} This completes our proof. \end{proof}
\subsubsection{An $(\infty, 0)$-quasi-greedy basis} Define $$\mathcal{F} \ :=\ \{A\subset\mathbb{N}: A\mbox{ is finite and does not contain even integers}\}.$$ Let $\mathbb{X}$ be the completion of $c_{00}$ with respect to the following norm: for $x = (x_1, x_2, \ldots)$, let
$$\|x\|\ :=\ \left(\sum_{2|i}|x_i|\right) + \left(\sum_{2\nmid i}|x_i|^2\right)^{1/2}.$$
Let $\mathcal{B}$ be the canonical basis. Clearly, $\mathcal{B}$ is $1$-unconditional. Note that $\mathcal{B}$ is not $\mathcal{S}_1$-disjoint democratic. To see this, fix $N\in\mathbb{N}$ and choose $A = \{1, 3, 5, \ldots, 2N-1\}$ and $B = \{2N, 2N+2, 2N+4, \ldots, 4N-2\}\in\mathcal{S}_1$. Then $\|1_A\| = \sqrt{N}$ while $\|1_B\| = N$. Hence, $\|1_B\|/\|1_A\|\rightarrow \infty$ as $N\rightarrow \infty$. It follows that $\mathcal{B}$ is not $\mathcal{S}_1$-disjoint democratic.
\subsubsection{A $(0,\infty)$-quasi-greedy basis} We define the spaces $X_N$ as in Subsection \ref{1st}: for each $N\in\mathbb{N}$, let $X_N$ be the $(2N-1)$-dimensional space that is the completion of $c_{00}$ under the norm: for $x = (a_i)_i$,
$$\|(a_i)_i\|\ =\ \max\left\{\left(\sum_{i=1}^{2N-1}|a_i|^2\right)^{1/2}, \sup_{N\leqslant m\leqslant 2N-1}\left|\sum_{i=N}^m \frac{1}{\sqrt{i-N+1}}a_i\right|\right\}.$$ Let $X = (\oplus_{N=1}^\infty X_N)_{\ell_2}$. Let $\mathcal{B}$ be the canonical basis of $X$. Using the same argument as in Subsection \ref{1st}, we know that $\mathcal{B}$ is quasi-greedy and is not $\mathcal{S}_1$-unconditional. We show that $\mathcal{B}$ is democratic. Let $A\subset\mathcal{B}$ be a nonempty finite set. Write $A = \bigcup_{N=1}^\infty A_N$, where $A_N$ is the intersection of $A$ and the canonical basis of $X_N$. We have
$$\left\|\sum_{e\in A} e\right\|\ =\ \left(\sum_{N=1}^\infty\left\|\sum_{e\in A_N} e\right\|^2\right)^{1/2}\ \geqslant\ \left(\sum_{N=1}^\infty|A_N|\right)^{1/2}\ =\ |A|^{1/2}.$$ On the other hand, for each $N$,
$$\left\|\sum_{e\in A_N}e\right\|\ \leqslant\ \sum_{i=1}^{|A_N|}\frac{1}{\sqrt{i}}\ \leqslant\ 2\sqrt{|A_N|}.$$ Therefore,
$$\left\|\sum_{e\in A} e\right\|\ =\ \left(\sum_{N=1}^\infty\left\|\sum_{e\in A_N} e\right\|^2\right)^{1/2}\ \leqslant\ 2\left(\sum_{N=1}^\infty |A_N|\right)^{1/2}\ =\ 2|A|^{1/2}.$$
We have shown that $|A|^{1/2}\leqslant \|\sum_{e\in A}e\|\leqslant 2|A|^{1/2}$, so $\mathcal{B}$ is democratic.
\subsection{An $(\alpha,\infty)$-quasi-greedy basis} \label{rt}
Fix a nonzero $\alpha\in \omega_1$ and consider the following collection subsets related to $\mathcal{S}_\alpha$ \begin{equation*} \mathcal{F}_{\alpha} = \{\cup^{r}_{i=1} E_i:r/2\leqslant E_1<E_2<\cdots <E_{r} \mbox{ are in } \mathcal{S}_{\alpha-1}\}. \end{equation*} The family $\mathcal{F}_1$ (among others) recently appeared in \cite{BCF}.
\begin{lem}\label{ls} Let $F \in \mathcal{F}_{\alpha}$. Then $F$ can be written as the union of two disjoint sets in $\mathcal{S}_\alpha$. \end{lem}
\begin{proof} Write $F = \cup_{i=1}^r E_i$, where $r/2 \leqslant E_1 < E_2 < \cdots < E_{r}$ and sets $E_i\in \mathcal{S}_{\alpha-1}$. Discard all the empty $E_i$ and re-number to have nonempty sets $E'_i$ satisfying $r/2 \leqslant E_1' < E'_2 < \cdots < E'_\ell$ for some $\ell\leqslant r$. Let $s = \lceil r/2\rceil$.
Case 1: $s\geqslant \ell$. Then $s\leqslant E_1' < E_2' < \cdots < E'_\ell$ implies that $F = \cup_{i = 1}^\ell E_i'\in \mathcal{S}_\alpha$. We are done.
Case 2: $s < \ell$. Let $F_1 = \cup_{i=1}^s E'_i$, which is in $S_\alpha$ due to Case 1. Note that $$s+1 \ \leqslant\ E'_{s+1} \ <\ \cdots \ <\ E'_\ell;$$ furthermore, $\ell-s \leqslant r-s \leqslant s+1$. Therefore, $F_2 := \cup_{i=s+1}^\ell E'_i\in \mathcal{S}_\alpha$. Since $F = F_1\cup F_2$, we are done. \end{proof}
Clearly, $\mathcal{S}_{\alpha}\subset\mathcal{F}_{\alpha}$. Let $X_{\alpha,\infty}$ be the completion of $c_{00}$ under the following norm: for $(a_i)\in c_{00}$,
$$\|(a_i)\|_{X_{\alpha,\infty}} :=\ \sup\left\{\sum_{j=1}^d \left|\sum_{i\in I_j}a_i\right|: I_1 < I_2 < \cdots < I_d \mbox{ intervals}, (\min I_j)_{j=1}^d\in \mathcal{F}_{\alpha}\right\}.$$ The space $X_{\alpha,\infty}$ above is the Jamesfication of the combinatorial space $X[\mathcal{F}_\alpha]$ (see \cite{AMS, BHO}) and is denoted by $J(X[\mathcal{F}_\alpha])$.
\begin{thm} The standard basis $(e_n)$ for the space $X_{\alpha,\infty}$ is $(\alpha,\infty)$-quasi-greedy. \end{thm}
We prove the above theorem through the following propositions. Let us start with the easiest one.
\begin{prop}\label{pp2}
The basis $(e_n)$ is democratic and $\mathcal{F}_\alpha$-unconditional, and thus $\mathcal{S}_\alpha$-unconditional.
\end{prop}
\begin{proof}
It follows directly from the definition of $\|\cdot\|$ that for $x\in X$ and $F \in \mathcal{F}_\alpha$,
$$\left\|\sum_{i\in F} e_i^*(x)e_i\right\|_{X_{\alpha, \infty}}\ =\ \sum_{i \in F}|e_i^*(x)|\ \leqslant\ \|x\|_{X_{\alpha, \infty}}.$$ Hence, $(e_n)$ is $\mathcal{F}_\alpha$-unconditional.
Let $A, B\subset\mathbb{N}$ with $|A| \leqslant |B|$. By Proposition \ref{k2}, there exists $N\in\mathbb{N}_{\geqslant 6}$ such that $$E\backslash \{1, \ldots, N-1\}\in \mathcal{F}_\alpha, \forall E\in \mathcal{S}_1.$$
Without loss of generality, assume that $|B|\geqslant N^2$. Let $B'\subset B$ such that $|B'|\geqslant |B|/2$ and $B'\in \mathcal{S}_1\subset \mathcal{F}_1$. Form $B'' = B'\backslash \{1, \ldots, N-1\}\in \mathcal{F}_\alpha$. We have
$$\|1_B\|\ \geqslant\ |B''|\ \geqslant\ |B'| - N\ \geqslant\ |B|/3 \ \geqslant\ |A|/3\ \geqslant\ \|1_A\|/3.$$ Therefore, $(e_n)$ is democratic. \end{proof}
\begin{prop}\label{pp1} The basis $(e_n)$ for the space $X_{\alpha,\infty}$ is 1-suppression quasi-greedy. \end{prop}
\begin{proof}
Let $x=(a_i)\in X_{\alpha,\infty}$ and $|a_N| = \|x\|_\infty$. By induction, we need only to show that
$$\|x-a_Ne_N\|\ \leqslant\ \|x\|.$$ Suppose, for a contradiction, that $\|x-a_Ne_N\| > \|x\|$. Removing the $N$th coefficient increases the norm implies that there exists an admissible set of intervals $\{I_j\}_{j=1}^d$ satisfying \begin{enumerate}
\item $a_{\min I_j}a_{\max I_j}\neq 0$ for all $1\leqslant j\leqslant d$,
\item for some $k$, $N\in I_k$ and $\min I_k < N < \max I_k$,
\item $\sum_{1\leqslant j\leqslant d, j\neq k} |\sum_{i\in I_j}a_i| + |\sum_{i\in I_k, i\neq N}a_i| > \|x\|$. \end{enumerate} For two integers $a \leqslant b$, let $[a, b] = \{a, a+1, \ldots, b\}$; when $a > b$, we let $[a, b] = \emptyset$. We form a new sequence of intervals as follows: if $k > 1$, \begin{align*}&I_1' \ =\ I_1\backslash \min I_1, I'_2 \ =\ I_2, \ldots, I'_{k-1} \ =\ I_{k-1},\\ &I'_k \ =\ [\min I_k, N-1], I'_{k+1} \ =\ \{N\}, I'_{k+2} \ =\ [N+1, \max I_k],\\ &I'_{k+3}\ =\ I_{k+1}, \ldots, I'_{d+2}\ =\ I_{d}. \end{align*} If $k= 1$, then \begin{align*} &I'_1 \ =\ [\min I_1 + 1, N-1], I'_2 \ =\ \{N\}, I'_{3}\ =\ [N+1, \max I_1],\\ &I'_{4}\ =\ I_{2}, \ldots, I'_{d+2}\ =\ I_d. \end{align*}
To see that $\{I_j'\}_{j=1}^{d+2}$ is admissible, we need to show $\{\min I_j'\}_{j=1}^{d+2}\in \mathcal{F}_\alpha$. We consider only the case when $k > 1$; the case $k = 1$ is similar. By construction, $$\{\min I_j'\,:\,1\leqslant j\leqslant d+2\} \ =\ \{\min (I_1\backslash \min I_1)\}\cup \{\min I_j\,:\, 2\leqslant j\leqslant d\}\cup\{N, N+1\}.$$ Let $A = \{\min I_j\}_{j=1}^d$ and $B = \{\min (I_1\backslash \min I_1)\}\cup \{\min I_j\,:\, 2\leqslant j\leqslant d\}$. Since $\min B-\min A\geqslant 1$ and $A\in \mathcal{F}_\alpha$, we know that $B\cup \{N, N+1\}\in \mathcal{F}_\alpha$.
We now use the admissible set $(I'_j)_{j=1}^{d+2}$ to obtain a contradiction. Write
\begin{equation}\label{ee1}\|x\|\ \geqslant\ \sum_{j=1}^{d+2}\left|\sum_{i\in I'_j}a_i\right| \ =\ \sum_{j = 1, k , k+1, k+2}\left|\sum_{i\in I'_j}a_i\right| + \sum_{j\neq 1, k, k+1, k+2}\left|\sum_{i\in I'_j}a_i\right|.\end{equation}
Since $|a_N|\geqslant |a_{\min I_1}|$, we have
\begin{align}\label{ee2}\sum_{j = 1, k , k+1, k+2}\left|\sum_{i\in I'_j}a_i\right|&\ \geqslant\ \left(\left|\sum_{i\in I_1}a_i\right|-|a_{\min I_1}|\right) + \left|\sum_{i=\min I_k}^{N-1}a_i\right| + |a_N| + \left|\sum_{i=N+1}^{\max I_k}a_i\right|\nonumber\\
&\ \geqslant\ \left|\sum_{i\in I_1}a_i\right| + \left|\sum_{i\in I_k, i\neq N}a_i\right|. \end{align} Furthermore, by definition,
\begin{equation}\label{ee3}\sum_{j\neq 1, k, k+1, k+2}\left|\sum_{i\in I'_j}a_i\right|\ =\ \sum_{j = 2}^{k-1}\left|\sum_{i\in I_j}a_i\right| + \sum_{j=k+1}^{d}\left|\sum_{i\in I_j}a_i\right|.\end{equation} By \eqref{ee1}, \eqref{ee2}, and \eqref{ee3}, we conclude that
$$\|x\|\ \geqslant\ \sum_{1\leqslant j\leqslant d, j\neq k} \left|\sum_{i\in I_j}a_i\right| + \left|\sum_{i\in I_k, i\neq N}a_i\right|\ > \ \|x\|,$$ which is a contradiction. Therefore, $(e_n)$ is a $1$-suppression quasi-greedy. \end{proof}
\begin{cor} The basis $(e_n)$ is $\mathcal{F}_\alpha$-greedy and thus, is $\mathcal{S}_\alpha$-greedy. \end{cor} \begin{proof} Use Theorem \ref{m2} and Propositions \ref{pp2} and \ref{pp1}. \end{proof}
It remains to show that $(e_n)$ is not $\mathcal{S}_{\alpha+1}$-unconditional and thus, not $\mathcal{S}_{\alpha+1}$-greedy. This part of the proof will require the repeated averages hierarchy \cite[pp. 1053]{AGR}. However, for our purposes, we only need the following lemma, a weaker result than \cite[Proposition 12.9]{AT}.
\begin{lem} For each $\alpha\in \omega_1$, $\varepsilon>0$ and $N\in \mathbb{N}$, there is a sequence $(a^\alpha_k)_{k=1}^\infty$ satisfying \begin{enumerate}
\item $a_k^\alpha \geqslant 0$ for each $k\in \mathbb{N}$ and $\|(a^\alpha_k)_k\|_{\ell_1}=1$,
\item $\{k : a^\alpha_k\neq 0\}$ is an interval and a maximal $\mathcal{S}_{\alpha+1}$-set,
\item $L:= \min \{k: a^\alpha_k\neq 0\} > N$ and $(a^\alpha_k)_{k\geqslant L}$ is monotone decreasing,
\item for each $G \in \mathcal{S}_\alpha$, we have
$\sum_{k \in G} a_k^{\alpha} <\varepsilon$. \end{enumerate} \label{RAH} \end{lem}
Choose $N$ such that $$E\backslash \{1, \ldots, N-1\}\in \mathcal{S}_\alpha, \forall E\in \mathcal{S}_1.$$
Fix $\varepsilon>0$ and find $(a_k^\alpha)$ satisfying Lemma \ref{RAH} with $N$ chosen as above. Since $F=\{k : a_k^\alpha\not=0\}\in \mathcal{S}_{\alpha+1}$, write $F = \cup_{i=1}^m E_i$, where $m\leqslant E_1 < E_2 <\cdots < E_m$ and $E_i\in \mathcal{S}_\alpha$. Since $F$ is an interval, each $E_i$ is an interval; furthermore, $N < \{\min E_i: 1\leqslant i\leqslant m\}\in \mathcal{S}_1$. Hence, $\{\min E_i: 1\leqslant i\leqslant m\}\in \mathcal{S}_\alpha\subset \mathcal{F}_\alpha$. By Lemma \ref{RAH} items (1) and (2), we have $\|\sum_{k\in F}a_k^\alpha e_k\| = 1$.
We estimate $\sum_{k\in F}(-1)^k a_k^\alpha e_k$. Let $I_1<\cdots < I_d$ be intervals so that $(\min I_j)_{j=1}^d \in \mathcal{F}_\alpha$ and $a^\alpha_{\min I_j}\neq 0$. For any interval $I_j$, $|\sum_{i\in I_j}(-1)^k a_k^\alpha|\leqslant 2a^\alpha_{\min I_j}$ because $(a^\alpha_k)_k$ is monotone decreasing. Therefore,
$$\sum_{j=1}^d \left|\sum_{k\in I_j}(-1)^k a_k^\alpha\right|\ \leqslant\ \sum_{j=1}^d 2 a^\alpha_{\min I_j}.$$ By Lemma \ref{ls}, we can write the set $\{\min I_1, \min I_{2}, \ldots, \min {I_d}\}$ as the union of two disjoint sets $A_1$ and $A_2$ in $\mathcal{S}_\alpha$. By Lemma \ref{RAH} item (3), we obtain $$\sum_{j=1}^d a^\alpha_{\min I_j}\ =\ \sum_{i\in A_1}a^\alpha_i + \sum_{i\in A_2}a^\alpha_i\ <\ 2\varepsilon.$$
Thus $\|\sum_{k\in F}(-1)^k a_k^\alpha e_k\| < 4\varepsilon$. As $\varepsilon$ was arbitrary and $F\in \mathcal{S}_{\alpha+1}$, we see that $(e_n)$ is not $\mathcal{S}_{\alpha+1}$-unconditional.
\subsection{An $(\infty,\alpha)$-quasi-greedy basis}
\subsubsection{Repeated average hierarchy}
Let $[\mathbb{N}]$ denote the collection of all infinite subsequences of $\mathbb{N}$. Similarly, if $M\in [\mathbb{N}]$, then $[M]$ denotes the collection of all infinite subsequences of $M$.
\begin{defi}\normalfont Let $\mathcal{B} = (e_n)$ be the canonical basis of $c_{00}$. For every countable ordinal $\alpha$ and $M = (m_n)_{n=1}^\infty\in [\mathbb{N}]$, we define a convex block sequence $(\alpha(M, n))_{n=1}^\infty$ of $\mathcal{B}$ by transfinite induction on $\alpha$. If $\alpha = 0$, then $\alpha(M, n) := e_{m_n}$. Assume that $(\beta(M, n))_{n=1}^\infty$ has been defined for all $\beta < \alpha$ and all $M\in [\mathbb{N}]$. For $M\in [\mathbb{N}]$, we define $(\alpha(M, n))_{n=1}^\infty$.
If $\alpha$ is a successor ordinal, write $\alpha = \beta + 1$. Set $$\alpha(M, 1)\ :=\ \frac{1}{m_1}\sum_{n=1}^{m_1}\beta(M, n).$$ Suppose that $\alpha(M,1) < \cdots < \alpha(M,n)$ have been defined. Let $$M_{n+1} \ :=\ \{m\in M: m > \max\supp (\alpha(M,n))\}\mbox{ and }k_n \ :=\ \min M_{n+1}.$$ Set $$\alpha(M, n+1)\ :=\ \frac{1}{k_n}\sum_{i=1}^{k_n} \beta(M_{n+1}, i).$$
If $\alpha$ is a limit ordinal, let $(\alpha_n+1)\nearrow \alpha$. Set $$\alpha(M,1)\ :=\ (\alpha_{m_1}+1)(M, 1).$$ Suppose that $\alpha(M,1) < \cdots < \alpha(M,n)$ have been defined. Let $$M_{n+1} \ :=\ \{m\in M: m > \max\supp (\alpha(M,n))\}\mbox{ and }k_n \ :=\ \min M_{n+1}.$$ Set $$\alpha(M, n+1) \ :=\ (\alpha_{k_n}+1)(M_{n+1}, 1).$$ \end{defi}
\begin{lem}\label{lh1} For each ordinal $\alpha\geqslant 1$ and $M\in [\mathbb{N}]$, we have
\begin{equation}\label{eh1}\|\alpha(M, n)\|_{\ell_1} \ =\ 1\mbox{ and }0\ \leqslant\ e_i^*(\alpha(M, n)) \ \leqslant\ \frac{1}{\min\supp(\alpha(M, n))}, \forall n, i\in \mathbb{N}.\end{equation} \end{lem}
\begin{proof} The proof is immediate from induction. \end{proof}
\begin{prop}\label{ppp1} Fix $\alpha < \beta$. For all $N\in \mathbb{N}$ and $M\in [\mathbb{N}]$, there exists $L\in [M]$ such that $\min L > N$ and
$$\|\beta(L, 1)\|_{\alpha} \ <\ \frac{3}{\min L},$$ where
$$\|(a_n)\|_{\alpha}\ :=\ \sup_{F\in \mathcal{S}_\alpha}\sum_{n\in F}|a_n|.$$ \end{prop}
\begin{rek}\normalfont See \cite[Proposition 2.3]{AT} for the case when $\alpha$ is a finite ordinal. Our proof of Proposition \ref{ppp1} is a combination of ideas used in the proofs of \cite[Proposition 2.3]{AT} and \cite[Proposition 2.15]{AG}. \end{rek}
\begin{proof}[Proof of Proposition \ref{ppp1}] We prove by transfinite induction on $\beta$. Base case: $\beta = 1$. Then $\alpha = 0$. Let $N\in \mathbb{N}$ and $M = (m_n)_{n=1}^\infty\in [\mathbb{N}]$. Let $m_k$ be the smallest such that $m_k > N$. Choose $L = (m_n)_{n\geqslant k}$. We have
$$\|1(L, 1)\|_{0}\ =\ \frac{1}{\min L}\ <\ \frac{3}{\min L}.$$ Indeed, for finite ordinals $\beta\geqslant 1$, we know the conclusion holds by \cite[Proposition 2.3]{AT}. Inductive hypothesis: suppose that the statement holds for all $\eta < \beta$ for some $\beta\geqslant \omega$. We need to show that it also holds for $\beta$.
Case 1: $\beta$ is a limit ordinal. Let $(\beta_n + 1)\nearrow \beta$ and $\alpha < \beta$. Choose $m > N$ such that $\beta_m > \alpha$. Set $L_1 := M|_{> m}$ and $\ell := \min L_1 > m$. Note that $\ell\geqslant 3$. By the inductive hypothesis, there exists $L_2\in [M]$ such that $\min L_2 > \max\supp(\beta_\ell(L_1, 1))$ and
$$\|\beta_\ell(L_2, 1)\|_\alpha \ <\ \frac{3}{\min {L_2}}.$$ Repeat the process to find subsequences $L_3, \ldots, L_\ell\in [M]$ such that $$\supp(\beta_\ell(L_1, 1)) \ <\ \supp(\beta_\ell(L_2, 1)) \ <\ \cdots < \ \supp(\beta_\ell(L_\ell, 1))$$ and
$$\|\beta_\ell(L_n, 1)\|_\alpha \ <\ \frac{3}{\min L_n}, \forall~ 2\leqslant n\leqslant \ell.$$ Let $L:= \cup_{n=1}^{\ell-1}\supp (\beta_\ell(L_n, 1))\cup L_\ell\in [M]$. Then $\min L > N$. By definition, $$\beta(L, 1)\ :=\ (\beta_{\ell}+1)(L, 1)\ =\ \frac{1}{\ell}\sum_{n=1}^\ell \beta_\ell(L, n)\ =\ \frac{1}{\ell} \sum_{n=1}^\ell \beta_\ell(L_n, 1).$$ We have \begin{align*}
\|\beta(L, 1)\|_\alpha&\ \leqslant\ \frac{1}{\ell} \sum_{n=1}^\ell \|\beta_\ell(L_n, 1)\|_\alpha\\ &\ \leqslant\ \frac{1}{\ell}+ \frac{1}{\ell} \left(\frac{3}{\min L_2} + \cdots + \frac{3}{\min L_\ell}\right)\\ &\ \leqslant\ \frac{1}{\ell} + \frac{1}{\ell} \frac{3}{\min L_2}\left(1 + \frac{1}{8} + \frac{1}{8^2} + \cdots\right)\mbox{ by Lemma \ref{8times}}\\ &\ =\ \frac{1}{\ell}\left(1+\frac{24}{7\min L_2}\right) \ <\ \frac{3}{\ell}. \end{align*}
Case 2: $\beta$ is a successor ordinal. Write $\beta = \eta + 1$.
\begin{enumerate}
\item Case 2.1: $\alpha < \eta$. Set $L_1:= M|_{>{N+1}}$ and $\ell := \min L_1 \geqslant 3$. By the inductive hypothesis, there exists $L_2\in [M]$ such that $\min L_2 > \max\supp(\eta(L_1, 1))$ and
$$\|\eta(L_2, 1)\|_\alpha \ <\ \frac{3}{\min L_2}.$$ Repeat the process to find subsequences $L_3, \ldots, L_\ell$ such that $$\supp(\eta(L_1, 1)) \ <\ \supp(\eta(L_2, 1)) \ <\ \cdots < \ \supp(\eta(L_\ell, 1))$$ and
$$\|\eta(L_n, 1)\|_\alpha \ <\ \frac{3}{\min L_n}, \forall 2\leqslant n\leqslant \ell.$$ Let $L:= \cup_{n=1}^{\ell-1}\supp (\eta(L_n, 1))\cup L_\ell\in [M]$. Then $\min L > N$. By definition, $$\beta(L, 1)\ :=\ (\eta+1)(L, 1)\ =\ \frac{1}{\ell}\sum_{n=1}^\ell \eta(L, n)\ =\ \frac{1}{\ell} \sum_{n=1}^\ell \eta(L_n, 1).$$
Similar to Case 1, we have $\|\beta(L, 1)\|_\alpha < 3/\ell$.
\item Case 2.2: $\alpha = \eta$. Let $(\alpha_n+1)\nearrow \alpha$ and $\mathcal{S}_{\alpha_n}\subset \mathcal{S}_{\alpha_{n+1}}$ for all $n\geqslant 1$. Set $L_1:= M|_{>{N+1}}$ and $\ell := \min L_1 \geqslant 3$. We have $$(\alpha_\ell+1)(L_1, 1)\ =\ \alpha(L_1, 1).$$ Let $k_1 = \max\supp (\alpha(L_1, 1))$. By the inductive hypothesis, find $L_2\in [M]$ with $k_1 < \min L_2$ and
$$\|\alpha(L_2, 1)\|_{\alpha_{k_1}} \ <\ \frac{3}{\min L_2}.$$ Repeat the process to find subsequences $L_3, \ldots, L_\ell\in [M]$ such that $$\supp(\alpha(L_1, 1)) \ <\ \supp(\alpha(L_2, 1)) \ <\ \cdots < \ \supp(\alpha(L_\ell, 1))$$ and if $k_n = \max\supp (\alpha(L_n, 1))$, we have
$$\|\alpha(L_n, 1)\|_{\alpha_{k_{n-1}}} \ <\ \frac{3}{\min L_n}, \forall 2\leqslant n\leqslant \ell.$$ Let $L:= \cup_{n=1}^{\ell-1}\supp (\alpha(L_n, 1))\cup L_\ell\in [M]$. Then $\beta(L, 1)\ :=\ \frac{1}{\ell}\sum_{n=1}^\ell \alpha(L_n, 1)$.
It holds that
$\|\beta(L, 1)\|_\alpha < \frac{3}{\ell}$. Indeed, let $G\in \mathcal{S}_\alpha$. Suppose that $k:=\min G\in \supp(\alpha(L_{j_0}, 1))$. Then $k\leqslant k_{j_0}$. By the definition of $\mathcal{S}_\alpha$, choose $p\leqslant k$ such that $G\in \mathcal{S}_{\alpha_p + 1}$. Finally, let $q\leqslant k$ be such that $G = \cup_{n=1}^q G_n$, where $G_1 < G_2 < \cdots < G_q$ and $G_n\in \mathcal{S}_{\alpha_p}$. For $j_0< n \leqslant \ell$, because $p\leqslant k\leqslant k_{n-1}$, we obtain $\mathcal{S}_{\alpha_p}\subset \mathcal{S}_{\alpha_{k_{n-1}}}$ and
$$\|\alpha(L_n, 1)\|_{\alpha_p}\ \leqslant\ \|\alpha(L_n, 1)\|_{\alpha_{k_{n-1}}} \ <\ \frac{3}{\min L_n}.$$ Therefore, $$\sum_{n\in G}e_n^*(\alpha(L_n, 1))\ \leqslant\ q\frac{3}{\min L_n}, \forall j_0< n \leqslant \ell.$$ Noting that $q \leqslant k\leqslant k_{j_0}< \min L_{j_0+1}\leqslant \frac{1}{8} \min L_{j_0+2}$ by Lemma \ref{8times}, we have \begin{align*}\sum_{n\in G}e_n^*(\beta(L, 1))&\ =\ \frac{1}{\ell}\left(1+1+ 3q\sum_{n=j_0+2}^\ell \frac{1}{\min L_n}\right)\\ &\ \leqslant\ \frac{1}{\ell}\left(2 + \frac{24q}{7\min L_{j_0+2}}\right)\ <\ \frac{3}{\ell}. \end{align*} \end{enumerate} We have completed the proof. \end{proof}
\subsubsection{An $(\infty, \alpha)$-quasi-greedy basis}
By Proposition \ref{ppp1}, we can find infinitely many $\mathcal{S}_{\alpha+1}$-maximal sets $F_1 < F_2 < F_3 < \cdots $ and for each set $F_i$, coefficients $(w_n)_{n\in F_i}$, such that
$\sum_{n\in F_i} w_n = 1$, while $$\left\|\min F_i\cdot \sum_{n\in F_i}w_ne_n\right\|_{\alpha}\ <\ 3.$$
Let $X$ be the completion of $c_{00}$ under the norm:
$$\|(a_n)_n\|\ :=\ \sup_{F_i}\left\{\max_{n}|a_n|, \min F_i\cdot \sum_{n\in F_i} w_n|a_n|\right\}.$$ Let $\mathcal{B}$ be the canonical basis.
\begin{claim}\label{cl2} The basis $\mathcal{B}$ is $1$-unconditional and normalized. \end{claim} \begin{proof}
That $\mathcal{B}$ is $1$-unconditional is obvious. Let us show that $\|e_n\| = 1$ for all $n\in \mathbb{N}$. Fix $n\in \mathbb{N}$. Due to the appearance of $\|\cdot\|_\infty$, $\|e_n\|\geqslant 1$. Since $\min F_i \cdot w_n \leqslant 1$ for all $i\in \mathbb{N}$ and $n\in F_i$ according to Lemma \ref{lh1}, $\|e_n\|\leqslant 1$. Hence, $\|e_n\| = 1$. \end{proof}
\begin{claim}\label{cl3}
The basis $\mathcal{B}$ is $\mathcal{S}_\alpha$-disjoint democratic. In particular, $\|1_A\| < 3$ for all $A\in \mathcal{S}_\alpha$. \end{claim}
\begin{proof} Choose $A\in \mathcal{S}_\alpha$. For any $F_i$, we have \begin{align*}
\min F_i\cdot \sum_{n\in A\cap F_i}w_n \ \leqslant\ \left\|\min F_i\cdot \sum_{n\in F_i}w_ne_n\right\|_{\alpha}\ < \ 3. \end{align*}
Therefore, $\|1_A\| < 3$. \end{proof}
\begin{claim}\label{cl4} The basis $\mathcal{B}$ is not $\mathcal{S}_{\alpha+1}$-disjoint democratic. \end{claim}
\begin{proof}
Choose $F_i$, which is a maximal $\mathcal{S}_{\alpha+1}$-set. Let $A$ be an $\mathcal{S}_\alpha$-set with $|F_i| \leqslant |A|$ and $F_i\sqcup A$. By how $F_i$'s are defined, $\|1_{F_i}\| = \min F_i$. On the other hand, we have that $\|1_A\| < 3$ by Claim \ref{cl3}. Since $\|1_{F_i}\|/|1_A\| > \min F_i/3 \rightarrow \infty$ as $i\rightarrow\infty$, the basis $\mathcal{B}$ is not $\mathcal{S}_{\alpha+1}$-disjoint democratic. \end{proof}
By Claims \ref{cl2}, \ref{cl3}, and \ref{cl4}, our basis $\mathcal{B}$ is ($\infty$, $\alpha$)-quasi-greedy.
\section{Proof of Theorem \ref{m31}}
Before proceeding to the proof of Theorem \ref{m31}, we isolate the following simple lemma but omit its straightforward proof.
\begin{lem}\label{l40} Let $\alpha < \omega_1$ and
$S$ be a finite set of positive integers with $\min S \geqslant 2$. Then there is an $m\in \mathbb{N}$ so that $S \in \mathcal{S}_{\alpha + m}$. \end{lem}
\begin{proof}[Proof of Theorem \ref{m31}] Assume that our basis $(e_n)$ is greedy. Let $m\in\mathbb{N}$. By Konyagin and Temlyakov's characterization of greedy bases \cite{KT1}, we know that $(e_n)$ is $K$-unconditional and $\Delta$-democratic for some $K, \Delta \geqslant 1$. It follows from the definitions that $(e_n)$ is $K$-$\mathcal{S}_{\alpha + m}$-unconditional, $\Delta$-$\mathcal{S}_{\alpha + m}$-disjoint democratic, and $K$-quasi-greedy. By the proof of Proposition \ref{p20} and Theorem \ref{m1}, $(e_n)$ is $C$-$\mathcal{S}_{\alpha + m}$-greedy for some $C = C(K, \Delta)$.
Conversely, assume that $(e_n)$ is $C$-$\mathcal{S}_{\alpha + m}$-greedy for all $m\in\mathbb{N}$ and some uniform $C\geqslant 1$. We need to show that $(e_n)$ is unconditional and disjoint democratic. Let $A\subset\mathbb{N}$ be a finite set. Write $A = (A\cap \{1\})\cup (A\backslash \{1\})$. By Lemma \ref{l40}, there exists $m$ such that $A\backslash \{1\}\in \mathcal{S}_{\alpha + m}$. Hence, $\mathcal{S}_{\alpha + m}$-unconditionality implies that $\|P_{A\backslash \{1\}}\|\leqslant C+1$ (see Theorem \ref{m1}). Therefore,
$$\|P_A\|\ \leqslant\ \|e_1^*\|\|e_1\| + C+1\ \leqslant\ c_2^2 + C + 1,$$
and so, $(e_n)$ is unconditional. Next, we show that $(e_n)$ is disjoint democratic. Pick finite disjoint sets $A, B\subset\mathbb{N}$ with $|A|\leqslant |B|$. Since $A\backslash \{1\}\in \mathcal{S}_{\alpha+m}$ for some sufficiently large $m$ and $(e_n)$ is $C$-$\mathcal{S}_{\alpha + m}$-disjoint democratic, $\|1_{A\backslash \{1\}}\|\leqslant C\|1_B\|$. Furthermore,
$$\|1_{A\cap \{1\}}\|\ \leqslant\ c_2\ \leqslant\ c_2\sup_{n}\|e^*_n\|\|1_B\|\ \leqslant\ c_2^2\|1_B\|.$$ We obtain
$$\|1_A\|\ \leqslant\ (C+c_2^2)\|1_B\|.$$ Hence, $(e_n)$ is disjoint democratic. This completes our proof.
Finally, we show that there exists a basis that is $\mathcal{S}_{\alpha+m}$-greedy for all $m\in \mathbb{N}$ but is not greedy. Let $\beta$ be the smallest limit ordinal that is greater than $\alpha + m$ for all $m\in \mathbb{N}$. Consider the canonical basis $(e_n)$ of the space $X_{\beta, \infty}$ in Subsection \ref{rt}. We have shown that $(e_n)$ is $\mathcal{S}_\beta$-greedy. By Corollary \ref{m30'}, $(e_n)$ is $\mathcal{S}_{\alpha + m}$-greedy for all $m$. However, since the basis is not unconditional, it is not greedy. \end{proof}
\section{Future research} In this paper, we show that given a pair $(\alpha,\beta) \in (\omega_1\cup \{\infty\})^2$, if either $\alpha$ or $\beta$ is $\infty$ or if $(\alpha,\beta) = (0,0)$, there is a Banach space with an $(\alpha, \beta)$-quasi-greedy basis. The result is sufficient enough to prove Theorem \ref{m30}. A natural extension of our work is whether there is an $(\alpha,\beta)$-quasi-greedy basis for every pair $(\alpha,\beta) \in (\omega_1\cup \{\infty\})^2$.
Regarding Theorem \ref{m31}, we would like to know whether an $\mathcal{S}_\alpha$-greedy basis for all countable ordinals $\alpha$ (with different greedy constants) is greedy. Similarly, must an $\mathcal{S}_\alpha$-unconditional basis for all countable ordinals $\alpha$ be unconditional?
\section{Appendix}
\begin{lem}\label{lems2} The following hold. \begin{enumerate} \item [i)] If $F\in \mathcal{S}_\alpha$ for some $\alpha$ and $\min F = 1$, then $F = \{1\}$. \item [ii)] For all ordinals $\alpha\geqslant 0$, $\mathcal{S}_0\subset \mathcal{S}_\alpha$. \item [iii)] For all ordinals $\alpha\geqslant 2$, $\mathcal{S}_2\subset \mathcal{S}_\alpha$. \end{enumerate} \end{lem}
\noindent We omit the straightforward proof of Lemma \ref{lems2}. For completeness, we include the easy proof of the following lemma.
\begin{lem}\label{8times} Fix $\alpha\geqslant 2$ and $M\in [\mathbb{N}]$, $\min M \geqslant 3$. Let $\ell_n = \min \alpha(M, n)$. It holds that $\ell_{n+1}\geqslant 8\ell_n$ for all $n\geqslant 1$. \end{lem}
\begin{proof} Let $L_n = M\backslash \cup_{i=1}^{n-1}\supp(\alpha(M, i))$ for $n\geqslant 1$. Then $\min L_n = \ell_n$ for all $n\geqslant 1$. First, we show that, \begin{equation}\label{etrou}\max\supp (\alpha(M,n))\ \geqslant\ \max\supp (2(L_n, 1)), \forall n\geqslant 1.\end{equation} Suppose, for a contradiction, for some $n$, $$\max\supp (\alpha(M, n))\ <\ \max\supp (2(L_n, 1)).$$ Let $E = \supp (\alpha(M, n))$ and $F = \supp (2(L_n,1))$. Then $E\subsetneq F$. Since $F\in \mathcal{S}_2$, $F\in \mathcal{S}_{\alpha}$ according to Lemma \ref{lems2}. That $E\subsetneq F$ and $F\in \mathcal{S}_\alpha$ contradict that $E$ is a maximal $\mathcal{S}_\alpha$-set. Therefore, for all $n\geqslant 1$, \eqref{etrou} holds.
We have for all $n\geqslant 1$, $$\frac{\ell_{n+1}}{\ell_n}\ \geqslant\ \frac{\max\supp (\alpha(M, n))+1}{\ell_n}\ \geqslant\ \frac{\max\supp (2(L_n, 1))+1}{\ell_n}\ \geqslant\ \frac{2^{\ell_n} \ell_n}{\ell_n}\ \geqslant \ 8.$$ This completes our proof. \end{proof}
\end{document} | arXiv | {
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\begin{document}
\title{Near-Optimal Representation Learning for Linear Bandits and Linear RL}
\begin{abstract} This paper studies representation learning for multi-task linear bandits and multi-task episodic RL with linear value function approximation. We first consider the setting where we play $M$ linear bandits with dimension $d$ concurrently, and these bandits share a common $k$-dimensional linear representation so that $k\ll d$ and $k \ll M$. We propose a sample-efficient algorithm, MTLR-OFUL, which leverages the shared representation to achieve $\tilde{O}(M\sqrt{dkT} + d\sqrt{kMT} )$ regret, with $T$ being the number of total steps. Our regret significantly improves upon the baseline $\tilde{O}(Md\sqrt{T})$ achieved by solving each task independently. We further develop a lower bound that shows our regret is near-optimal when $d > M$. Furthermore, we extend the algorithm and analysis to multi-task episodic RL with linear value function approximation under low inherent Bellman error \citep{zanette2020learning}. To the best of our knowledge, this is the first theoretical result that characterizes the benefits of multi-task representation learning for exploration in RL with function approximation. \end{abstract}
\section{Introduction} \label{sec: introduction}
Multi-task representation learning is the problem of learning a common low-dimensional representation among multiple related tasks~\citep{caruana1997multitask}. This problem has become increasingly important
in many applications such as natural language processing~\citep{ando2005framework,liu2019multi}, computer vision~\citep{li2014joint}, drug discovery~\citep{ramsundar2015massively}, and reinforcement learning~\citep{wilson2007multi,teh2017distral,d2019sharing}. In these cases, common information can be extracted from related tasks to improve data efficiency and accelerate learning.
While representation learning has achieved tremendous success in a variety of applications~\citep{bengio2013representation}, its theoretical understanding is still limited. A widely accepted assumption in the literature is the existence of a common representation shared by different tasks.
For example,
\citet{maurer2016benefit} proposed a general method to learn data representation in multi-task supervised learning and learning-to-learn setting. \citet{du2020few} studied few-shot learning via representation learning with assumptions on a common representation among source and target tasks. \citet{tripuraneni2020provable} focused on the problem of multi-task linear regression with low-rank representation,
and proposed algorithms with sharp statistical rates.
Inspired by the theoretical results in supervised learning, we take a step further to investigate provable benefits of representation learning for sequential decision making problems.
First, we study the multi-task low-rank linear bandits problem, where $M$ tasks of $d$-dimensional (infinite-arm) linear bandits are concurrently learned for $T$ steps. The expected reward of arm $\bm{x}_i \in \dbR^d$ for task $i$ is $\bm{\theta}_i^\top \bm{x}_i$, as determined by an unknown linear parameter $\bm{\theta}_i$.
To take advantage of the multi-task representation learning framework, we assume that $\bm{\theta}_i$'s lie in an unknown $k$-dimensional subspace of $\dbR^d$, where $k$ is much smaller compared to $d$ and $M$ \citep{yang2020provable}.
The dependence among tasks makes it possible to achieve a regret bound better than solving each task independently.
Specifically, if the tasks are solved independently with standard algorithms such as OFUL~\citep{abbasi2011improved}, the total regret is $\tilde{O}(Md\sqrt{T})$.\footnote{$\tilde{O}$ hides the logarithmic factors.} By leveraging the common representation among tasks, we can achieve a better regret $\tilde{O}(M\sqrt{dkT} + d\sqrt{MkT})$.
Our algorithm is also robust to the linear representation assumption when the model is misspecified. If the $k$-dimensional subspace approximates the rewards with error at most $\zeta$, our algorithm can still achieve regret $\tilde{O}(M\sqrt{dkT}+d\sqrt{kMT} + MT\sqrt{d}\zeta)$. Moreover, we prove a regret lower bound indicating that the regret of our algorithm is not improvable except for logarithmic factors in the regime $d > M$.
Compared with multi-task linear bandits, multi-task reinforcement learning is a more popular research topic with a long line of works in both theoretical side and empirical side \citep{taylor2009transfer,parisotto2015actor,liu2016decoding,teh2017distral,hessel2019multi,d2019sharing, arora2020provable}. We extend our algorithm for linear bandits to the multi-task episodic reinforcement learning with linear value function approximation under low inherent Bellman error \citep{zanette2020learning}. Assuming a low-rank linear representation across all the tasks, we propose a sample-efficient algorithm with regret $\tilde{O}(HM \sqrt{dkT} + Hd\sqrt{kMT} + HMT\sqrt{d} \caI)$
, where $k$ is the dimension of the
low-rank representation, $d$ is the ambient dimension of state-action features, $M$ is the number of tasks, $H$ is the horizon, $T$ is the number of episodes, and $\caI$ denotes the inherent Bellman error. The regret significantly improves upon the baseline regret $\tilde{O}(HMd \sqrt{T} + HMT\sqrt{d} \caI)$ achieved by running ELEANOR algorithm \citep{zanette2020learning} for each task independently.
We also prove a regret lower bound $\Omega(Mk\sqrt{HT} + d\sqrt{HkMT} + HMT\sqrt{d} \caI)$. To the best of our knowledge, this is the first provably sample-efficient algorithm for exploration in multi-task linear RL.
\section{Preliminaries} \label{sec: preliminaries}
\subsection{Multi-Task Linear Bandit} \label{preliminaries:linear_bandits} We study the problem of representation learning for linear bandits in which there are multiple tasks sharing common low-dimensional features. Let $d$ be the ambient dimension and $k$ be the representation dimension. We play $M$ tasks concurrently for $T$ steps each. Each task $i \in [M]$ is associated with an unknown vector $\boldsymbol{\theta}_i \in \mathbb{R}^d$. In each step $t \in [T]$, the player chooses one action $\boldsymbol{x}_{t,i} \in \mathcal{A}_{t,i}$ for each task $i \in [M]$, and receives a batch of rewards $\{y_{t, i}\}_{i=1}^M$ afterwards, where $\mathcal{A}_{t,i}$ is the feasible action set (can even be chosen adversarially) for task $i$ at step $t$.
The rewards received are determined by $y_{t,i} = \boldsymbol{\theta}_i^{\top}\boldsymbol{x}_{t,i} + \eta_{t,i}$, where the $\eta_{t,i}$ is the random noise.
We use the total regret for $M$ tasks in $T$ steps to measure the performance of our algorithm, which is defined in the following way: $$\operatorname{Reg}(T) \defeq \sum_{t=1}^{T}\sum_{i=1}^{M} \left(\left\langle \boldsymbol{x}_{t,i}^{\star}, \boldsymbol{\theta}_{i}\right\rangle-\left\langle \boldsymbol{x}_{t, i}, \boldsymbol{\theta}_{i}\right\rangle\right),$$ where $\boldsymbol{x}_{t,i}^{\star} = \argmax_{\boldsymbol{x} \in \mathcal{A}_{t,i}}\left\langle \boldsymbol{x}, \boldsymbol{\theta}_{i}\right\rangle$.
The main assumption is the existence of a common linear feature extractor.
\begin{assumption} \label{assumptions:low_rank_bandits} There exists a linear feature extractor $\boldsymbol{B} \in \mathbb{R}^{d\times k}$ and a set of $k$-dimensional coefficients $\{\boldsymbol{w}_i\}_{i=1}^{M}$ such that $\{\boldsymbol{\theta}_i\}_{i=1}^M$ satisfies $\boldsymbol{\theta}_i = \boldsymbol{B}\boldsymbol{w}_i$.
\end{assumption}
Define filtration $F_t$ to be the $\sigma$-algebra induced by $\sigma(\{\boldsymbol{x}_{1,i}\}_{i=1}^{M},\cdots, \{\boldsymbol{x}_{t+1,i}\}_{i=1}^{M},\{\eta_{1,i}\}_{i=1}^{M},\cdots,\{\eta_{t,i}\}_{i=1}^{M})$, then we have the following assumption.
\begin{assumption} \label{assumptions:linear_bandits_regularity} Following the standard regularity assumptions in linear bandits~\citep{abbasi2011improved,lattimore2020bandit}, we assume \begin{itemize}
\item $\|\boldsymbol{\theta}_i\|_2 \leq 1, \forall i \in [M]$
\item $\|\boldsymbol{x}\|_2 \leq 1, \forall \boldsymbol{x} \in \mathcal{A}_{t,i}, t \in [T], i \in [M]$
\item $\eta_{t, i}$ is conditionally zero-mean $1$-sub-Gaussian random variable with regards to $F_{t-1}$.
\end{itemize} \end{assumption}
For notation convenience, we use $\boldsymbol{X}_{t,i} = [\boldsymbol{x}_{1,i}, \boldsymbol{x}_{2,i}, \cdots, \boldsymbol{x}_{t,i} ]$ and $\boldsymbol{y}_{t,i} = [y_{1,i}, \cdots, y_{t,i}]^{\top}$ to denote the arms and the corresponding rewards collected for task $i \in [M]$ in the first $t$ steps, and we also use $\boldsymbol{\eta}_{t,i} = [\eta_{1,i},\eta_{2,i}, \cdots, \eta_{t,i}]^{\top}$ to denote the corresponding noise. We define $\boldsymbol{\Theta} \defeq [\boldsymbol{\theta}_1, \boldsymbol{\theta_2}, \cdots, \boldsymbol{\theta_M}]$ and $\boldsymbol{W} \defeq [\boldsymbol{w}_1, \boldsymbol{w}_2, \cdots, \boldsymbol{w}_M]$. For any positive definite matrix $\bm{A} \in \dbR^{d \times d}$, the Mahalanobis norm with regards to $\bm{A}$ is denoted by $\|\bm{x}\|_{\bm{A}} = \sqrt{\bm{x}^\top \bm{A} \bm{x}}$.
\subsection{Multi-Task Linear RL}
\label{preliminaries:linear_rl}
We also study how this low-rank structure benefits the exploration problem with approximate linear value functions in multi-task episodic reinforcement learning. For reference convenience, we abbreviate our setting as multi-task LSVI setting, which is a natural extension of LSVI condition in the single-task setting ~\citep{zanette2020learning}.
Consider an undiscounted episodic MDP $\caM = (\caS, \caA, p, r, H)$ with state space $\caS$, action space $\caA$, and fixed horizon $H$. For any $h \in [H]$, any state $s_h \in \caS$ and action $a_h \in \caA$, the agent receives a reward $R_h(s_h, a_h)$ with mean $r_h(s_h, a_h)$, and transits to the next state $s_{h+1}$ according to the transition kernel $p_h\left(\cdot \mid s_h, a_h\right)$. The action value function for each state-action pair at step $h$ for some deterministic policy $\pi$ is defined as
$Q_h^\pi(s_h, a_h) \defeq r_h(s_h, a_h) + \dbE\left[\sum_{t=h+1}^H R_t(s_t, \pi_t(s_t)) \right]$ , and the state value function is defined as $V_h^\pi(s_h) = Q_h^\pi(s_h, \pi_h(s_h))$
Note that there always exists an optimal deterministic policy (under some regularity conditions) $\pi^*$ for which $V_h^{\pi^*}(s) = \max_\pi V_h^\pi(s) $ and $Q_h^{\pi^*}(s, a) = \max_\pi Q_h^{\pi}(s, a) $ for each $h \in [H]$. We denote $V_h^{\pi^*}$ and $Q_h^{\pi^*}$ by $V_h^*$ and $Q_h^*$ for short.
It's also convenient to define the Bellman optimality operator $\caT_h$ as $\caT_h(Q_{h+1})(s, a) \defeq r_h(s, a) + \dbE_{s' \sim p_h (\cdot \mid s, a)} \max_{a'} Q_{h+1}(s', a')$.
In the framework of single-task approximate linear value functions (see Section \ref{sec: linear_rl} for more discussions), we assume a feature map $\boldsymbol{\phi}: \caS \times \caA \to \dbR^{d}$ that maps each state-action pair to a $d$-dimensional vector. In case that $\caS$ is too large or continuous (e.g. in robotics), this feature map helps to reduce the problem scale from $|\caS| \times |\caA|$ to $d$. The value functions are the linear combinations of those feature maps, so we can define the function space at step $h \in [H]$ to be $\caQ_h^\prime = \left\{Q_h(\boldsymbol{\theta}_h) \mid \boldsymbol{\theta}_h \in \Theta^\prime_h \right\}$ and $ \caV^\prime_h = \left\{V_h(\boldsymbol{\theta}_h) \mid \boldsymbol{\theta}_h \in \Theta^\prime_h \right\}$,
where $Q_h(\boldsymbol{\theta}_h)(s, a) \defeq \boldsymbol{\phi}(s, a)^\top \boldsymbol{\theta}_h$, and $V_h(\theta_h)(s) \defeq \max_a \boldsymbol{\phi}(s, a)^\top \boldsymbol{\theta}_h$.
In order to find the optimal value function using value iteration with $\caQ_h$, we require that it is approximately close under $\caT_h$, as measured by the inherent Bellman error (or IBE for short). The IBE \citep{zanette2020learning} at step $h$ is defined as \begin{align} \label{definitions:inherent_bellman_error}
\caI_h \defeq \!\!\! \sup_{Q_{h+1} \in \caQ_{h+1}} \inf_{Q_h \in \caQ_h} \sup_{s \in \caS, a \in \caA} \left|\left(Q_h - \caT_h(Q_{h+1})\right)(s, a)\right|. \end{align}
In multi-task reinforcement learning, we have $M$ MDPs $\caM^1, \caM^2, ..., \caM^M$ (we use superscript $i$ to denote task $i$). Assume they share the same state space and action space, but have different rewards and transitions.
To take advantage of the multi-task LSVI setting and low-rank representation learning, we define a joint function space for all the tasks as $\Theta _h \defeq \{\left(\boldsymbol{B}_h\boldsymbol{w}^1_h, \boldsymbol{B}_h\boldsymbol{w}^2_h, \cdots, \boldsymbol{B}_h\boldsymbol{w}_h^M\right): \boldsymbol{B}_h \in \caO^{d \times k} , \boldsymbol{w}^i_h \in \caB^{k}, \boldsymbol{B}_h\boldsymbol{w}_h^i \in \Theta_h^{i\prime}\}$,
where $\caO^{d \times k}$ is the collection of all orthonormal matrices in $\dbR^{d \times k}$.
The induced function space is defined as
\begin{align}
\caQ_h \defeq \{\left(Q_h^1\left(\boldsymbol{\theta}_h^1\right), Q_h^2\left(\boldsymbol{\theta}_h^2\right), \cdots, Q_h^M\left(\boldsymbol{\theta}_h^M\right)\right) \mid \left(\boldsymbol{\theta}_h^1, \boldsymbol{\theta}_h^2, \cdots, \boldsymbol{\theta}_h^M\right) \in \Theta_h \}& \\ \caV_h \defeq \{\left(V_h^1\left(\boldsymbol{\theta}_h^1\right),V_h^2\left(\boldsymbol{\theta}_h^2\right), \cdots, V_h^M\left(\boldsymbol{\theta}_h^M\right)\right) \mid \left(\boldsymbol{\theta}_h^1, \boldsymbol{\theta}_h^2, \cdots, \boldsymbol{\theta}_h^M\right) \in \Theta_h \} & \end{align}
The low-rank IBE at step $h$ for multi-task LSVI setting is a generalization of IBE (Eqn~\ref{definitions:inherent_bellman_error}) for the single-task setting, which is defined accordingly as \begin{align} \label{definitions:inherent_bellman_error_low_rank}
\caI_h^{\text{mul}} \defeq \sup_{\left\{Q_{h+1}^i\right\}_{i=1}^M \in \caQ_{h+1}} &\inf_{\left\{Q_{h}^i\right\}_{i=1}^M \in \caQ_{h}} \sup_{s \in \caS, a \in \caA, i \in [M]} \left|\left(Q_h^i - \caT_h^i(Q_{h+1}^i)\right)(s, a)\right| \end{align}
\begin{assumption} \label{assumptions:low_rank_small_ibe} $\caI \defeq \sup_h \caI^{\text{mul}}_h$ is small with regards to the joint function space $\caQ_h$ for all $h$. \end{assumption}
When $\caI = 0$, Assumption \ref{assumptions:low_rank_small_ibe} can be regarded as a natural extension of Assumption \ref{assumptions:low_rank_bandits} in episodic RL. This is because there exists $\{\bar{\bm{\theta}}_h^{i*}\}_{i=1}^M \in \Theta_h$ such that $Q^{i*}_h = Q_h^i(\bar{\bm{\theta}}_h^{i*})$ for all $i \in [M]$ and $h \in [H]$ in the case $\caI = 0$. According to the definition of $\Theta_h$ we know that $\{\bar{\bm{\theta}}_h^{i*}\}_{i=1}^M$ also admit a low-rank property as Assumption \ref{assumptions:low_rank_bandits} indicates. When $\caI > 0$, then Assumption \ref{assumptions:low_rank_small_ibe} is an extension of misspecified multi-task linear bandits (discussed in Section \ref{sec: misspecifed linear bandits}) in episodic RL.
Define the filtration $\caF_{h, t}$ to be the $\sigma$-field induced by all the random variables up to step $h$ in episode $t$ (not include the rewards at step $h$ in episode $t$), then we have the following assumptions.
\begin{assumption} \label{assumptions:linear_rl_regularity} Following the parameter scale in \citep{zanette2020learning}, we assume \begin{itemize}
\item $\left\|\boldsymbol{\phi}(s, a)\right\|_2 \leq 1, \forall (s, a) \in \caS \times \caA, h \in [H]$
\item $0 \leq Q^{\pi}_h(s,a) \leq 1, \forall (s, a) \in \caS \times \caA, h \in [H], \forall \pi$.
\item There exists constant $D$ that for any $h \in [H]$ and any $\left\{\boldsymbol{\theta}_{h}^i\right\}_{i=1}^M \in \Theta_{h}$, it holds that $\|\boldsymbol{\theta}_{h}^i\|_2 \leq D, \forall i \in [M]$.
\item For any fixed $\left\{Q_{h+1}^i\right\}_{i=1}^M \in \caQ_{h+1}$, the random noise $z_h^i(s, a) \defeq R_h^i(s, a) + \max_a Q_{h+1}^i\left(s', a\right) - \caT_h^i\left(Q_{h+1}^i\right)(s, a)$ is bounded in $[-1, 1]$ a.s., and is independent conditioned on $\caF_{h, t}$ for any $s \in \caS, a \in \caA, h \in [H], i \in [M]$, where the randomness is from reward $R$ and $s' \sim p_h\left(\cdot \mid s, a\right)$. \end{itemize} \end{assumption}
The first condition is a standard regularization condition for linear features. The second condition is on the scale of the problem. This scale of the exploration problem that the value function is bounded in $[0, 1]$ has also been studied in both tabular and linear setting \citep{zhang2020reinforcement, wang2020long,zanette2020learning}. The last two conditions are compatible with the scale of the problem. It's sufficient to assume the constant norm of $\boldsymbol{\theta}_{h}^i$ since the optimal value function is of the same scale. The last condition is standard in linear bandits \citep{abbasi2011improved, lattimore2020bandit} and RL~\citep{zanette2020learning}, and is automatically satisfied if $D = 1$.
The total regret of $M$ tasks in $T$ episodes is defined as \begin{align} \text{Reg}(T) \defeq \sum_{t=1}^T \sum_{i=1}^M \left(V_1^{i*} - V_1^{\pi_{t}^i} \right)\left(s_{1t}^i\right) \end{align} where $\pi_t^i$ is the policy used for task $i$ in episode $t$, and $s_{ht}^{i}$ denotes the state encountered at step $h$ in episode $t$ for task $i$. We assume $ M \geq 5, T \geq 5 $ throughout this paper. \section{Related Work} \label{sec: related work}
\paragraph{Multi-task Supervised Learning} The idea of multi-task representation learning at least dates back to \citet{caruana1997multitask,thrun1998learning,baxter2000model}. Empirically, representation learning has shown its great power in various domains. We refer readers to \citet{bengio2013representation} for a detailed review about empirical results. From the theoretical perspective, \citet{baxter2000model} performed the first theoretical analysis and gave sample complexity bounds using covering number. \citet{maurer2016benefit} considered the setting where all tasks are sampled from a certain distribution, and analysed the benefit of representation learning for both reducing the sample complexity of the target task. Following their results, \citet{du2020few} and \citet{tripuraneni2020provable} replaced the i.i.d assumption with a deterministic assumption on the data distribution and task diversity, and proposed efficient algorithms that can fully utilize all source data with better sample complexity. These results mainly focus on the statistical rate for multi-task supervised learning, and cannot tackle the exploration problem in bandits and RL.
\paragraph{Multi-task Bandit Learning} For multi-task linear bandits, the most related work is a recent paper by \citet{yang2020provable}. For linear bandits with infinite-action set, they firstly proposed an explore-then-exploit algorithm with regret $\tilde{O}(Mk\sqrt{T} + d^{1.5}k\sqrt{MT})$, which outperforms the naive approach with $\tilde{O}(Md\sqrt{T})$ regret in the regime where $M = \Omega(dk^2)$. Though their results are insightful,
they require the action set for all tasks and all steps to be the same well-conditioned $d$-dimensional ellipsoids which cover all directions nicely with constant radius. Besides, they assume that the task parameters are diverse enough with $\boldsymbol{W}\boldsymbol{W}^{\top}$ well-conditioned, and the norm of $\boldsymbol{w}_i$ is lower bounded by a constant. These assumptions make the application of the theory rather restrictive to only a subset of linear bandit instances with benign structures. In contrast, our theory is more general since we do not assume the same and well-conditioned action set for different tasks and time steps, nor assume the benign properties of $\boldsymbol{w}_i$'s.
\paragraph{Multi-task RL} For multi-task reinforcement learning, there is a long line of works from the empirical perspective \citep{taylor2009transfer,parisotto2015actor,liu2016decoding,teh2017distral,hessel2019multi}. From the theoretical perspective, \citet{brunskill13mtrl} analyzed the sample complexity of multi-task RL in the tabular setting. \citet{d2019sharing} showed that representation learning can improve the rate of approximate value iteration algorithm. \citet{arora2020provable} proved that representation learning can reduce the sample complexity of imitation learning.
\paragraph{Bandits with Low Rank Structure} Low-rank representations have also been explored in single-task settings. \citet{jun2019bilinear} studied bilinear bandits with low rank representation. The mean reward in their setting is defined as the bilinear multiplication $\boldsymbol{x}^{\top} \boldsymbol{\Theta} \boldsymbol{y}$, where $\boldsymbol{x}$ and $\boldsymbol{y}$ are two actions selected at each step, and $\boldsymbol{\Theta}$ is an unknown parameter matrix with low rank. Their setting is further generalized by \citet{lu2020low}. Furthermore, sparse linear bandits can be regarded as a simplified setting, where $\boldsymbol{B}$ is a binary matrix indicating the subset of relevant features in context $\boldsymbol{x}$~\citep{abbasi2012online,carpentier2012bandit,lattimore2015linear,hao2020high}.
\paragraph{Exploration in Bandits and RL} Our regret analysis is also related to exploration in single-task linear bandits and linear RL. Linear bandits have been extensively studied in recent years~\citep{auer2002using, dani2008stochastic,rusmevichientong2010linearly, abbasi2011improved, chu2011contextual,li2019nearly,li2019tight}. Our algorithm is most relevant to the seminal work of
\citet{abbasi2011improved}, who applied self-normalized techniques to obtain near-optimal regret upper bounds.
For single-task linear RL, recent years have witnessed a tremendous of works under different function approximation settings, including linear MDPs~\citep{yang2019sample, jin2020provably}, linear mixture MDPs~\citep{ayoub2020model,zhou2020nearly}, linear RL with low inherent Bellman error~\citep{zanette2020learning,zanette2020provably}, and MDPs with low Bellman-rank~\citep{jiang2017contextual}. Our multi-task setting is a natural extension of linear RL with low inherent Bellman error setting, which covers linear MDP setting as a special case~\citep{zanette2020learning}. \section{Main Results for Linear Bandits} \label{sec: linear_bandits}
In this section, we present our main results for multi-task linear bandits.
\subsection{Construction of Confidence Sets} \label{sec: construction of confidence sets} A natural and successful method to design efficient algorithms for sequential decision making problem is the \textit{optimism in the face of uncertainty principle}. When applied to single-task linear bandits, the basic idea is to maintain a confidence set $\mathcal{C}_t$ for the parameter $\bm{\theta}$ based on history observations for each step $t \in [T]$. The algorithm chooses an optimistic estimation $\tilde{\boldsymbol{\theta}}_t = \operatorname{argmax}_{\boldsymbol{\theta} \in \mathcal{C}_{t}}\left(\max _{\boldsymbol{x} \in \mathcal{A}_{t}}\langle \boldsymbol{x}, \boldsymbol{\theta}\rangle\right)$ and then selects action $\boldsymbol{x}_t = \argmax_{\boldsymbol{x}_t \in \mathcal{A}_t} \langle \boldsymbol{x}, \tilde{\boldsymbol{\theta}}_t\rangle$, which maximizes the reward according to the estimation $\tilde{\boldsymbol{\theta}}_t$. In other words, the algorithm chooses the pair $ \left(\boldsymbol{x}_{t}, \tilde{\boldsymbol{\theta}}_{t}\right)=\underset{(\boldsymbol{x}, \boldsymbol{\theta}) \in \mathcal{A}_{t} \times \mathcal{C}_{t}}{\operatorname{argmax}}\langle \boldsymbol{x}, \boldsymbol{\theta}\rangle $.
For multi-task linear bandits, the main difference is that we need to tackle $M$ highly correlated tasks concurrently. To obtain tighter confidence bound, we maintain the confidence set $\mathcal{C}_t$ for $\boldsymbol{B}$ and $\{\boldsymbol{w}_i\}_{i=1}^{M}$, then choose the optimistic estimation $\tilde{\boldsymbol{\Theta}}_t$ for all tasks concurrently. To be more specific, the algorithm chooses an optimistic estimate $\tilde{\boldsymbol{\Theta}}_t = \argmax_{\boldsymbol{\Theta} \in \mathcal{C}_t} (\max_{\{x_i \in \mathcal{A}_{t,i}\}_{i=1}^{M}} \sum_{i=1}^{M}\left\langle \boldsymbol{x}_i, \boldsymbol{\theta}_{i}\right\rangle)$, and then selects action $\boldsymbol{x}_{t,i} = \argmax_{x_i \in \mathcal{A}_{t,i}} \left\langle \boldsymbol{x}_i, \tilde{\boldsymbol{\theta}}_{t,i}\right\rangle$ for each task $i \in [M]$.
The main technical contribution is the construction of a tighter confidence set $\mathcal{C}_t$ for the estimation of $\boldsymbol{\Theta}$. At each step $t \in [T]$, we solve the following least-square problem based on the samples collected so far and obtain the minimizer $\hat{\boldsymbol{B}}_t$ and $\hat{\boldsymbol{W}}_t$: \begin{align} \label{eqn: optimization problem for linear bandits}
\underset{\boldsymbol{B} \in \mathbb{R}^{d \times k}, {\boldsymbol{w}}_{1 .. M} \in \mathbb{R}^{k \times M}}
{\arg \min } &\sum_{i=1}^{M} \left\|\boldsymbol{y}_{t-1,i}-\boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{B} \boldsymbol{w}_{i}\right\|^{2}_2 \\
\mathrm{s.t.} \quad & \left\|\boldsymbol{B}\boldsymbol{w}_i\right\|_2 \leq 1, \forall i \in [M]. \end{align}
We maintain a high probability confidence set $\mathcal{C}_{t}$ for the unknown parameters $\boldsymbol{B}$ and $\{\boldsymbol{w}_i\}_{i=1}^{M}$. We calculate $\mathcal{C}_{t}$ in the following way: \begin{align} \label{eqn: confidence set for linear bandits}
\caC_t \defeq & \bigg\{\boldsymbol{\Theta} = \boldsymbol{B}\boldsymbol{W}: \sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \leq L, \notag \\
&\phantom{=\;\;} \boldsymbol{B} \in \dbR^{d \times k}, \boldsymbol{w}_i \in \dbR^{k}, \left\|\boldsymbol{B}\boldsymbol{w}_i\right\|_2 \leq 1, \forall i \in [M] \bigg\}, \end{align} where
$L = \tilde{O}(Mk + kd)$ (see Appendix \ref{sec: proof of lemma, confidence set for linear bandits} for the exact value) and $\tilde{\boldsymbol{V}}_{t-1,i}(\lambda) = \boldsymbol{X}_{t-1,i}\boldsymbol{X}_{t-1,i}^{\top} + \lambda \boldsymbol{I}_d$. $\lambda$ is a hyperparameter used to ensure that $\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)$ is always invertable, which can be set to $1$. We can guarantee that $\boldsymbol{\Theta} \in \mathcal{C}_t$ for all $t \in [T]$ with high probability by the following lemma.
\begin{lemma} \label{lemma: confidence set for linear bandits} With probability at least $1-\delta$, for any step $t \in [T]$, suppose $\hat{\boldsymbol{\Theta}}_t = \hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t$ is the optimal solution of the least-square regression (Eqn~\ref{eqn: optimization problem for linear bandits}), the true parameter $\boldsymbol{\Theta} = \boldsymbol{B} \boldsymbol{W}$ is always contained in the confidence set $\mathcal{C}_t$, i.e. \begin{align}
\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \leq L, \end{align} where $\tilde{\boldsymbol{V}}_{t-1,i}(\lambda) = \boldsymbol{X}_{t-1,i}\boldsymbol{X}_{t-1,i}^{\top} + \lambda \boldsymbol{I}_d$. \end{lemma}
If we solve each tasks independently with standard single-task algorithms such as OFUL~\citep{abbasi2011improved}, it is not hard to realize that we can only obtain a confidence set with $\sum_{i=1}^{M}\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \leq L_1 = \tilde{O}(Md)$. Our confidence bound is much sharper compared with this naive bound, which explains the improvement in our final regret. Compared with \citet{yang2020provable}, we are not able to estimate $\boldsymbol{B}$ and $\boldsymbol{W}$ directly like their methods due to the more relaxed bandit setting. In our setting, the empirical design matrix $\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)$ can be quite ill-conditioned if the action set at each step is chosen adversarially. Thus, we have to establish a tighter confidence set to improve the regret bound.
We only sketch the main idea of the proof for Lemma~\ref{lemma: confidence set for linear bandits} and defer the detailed explanation to Appendix~\ref{sec: proof of lemma, confidence set for linear bandits}. Considering the non-trivial case where $d> 2k$, our main observation is that both $\boldsymbol{B}\boldsymbol{W}$ and $\hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t$ are low-rank matrix with rank upper bounded by $k$, which indicates that $\operatorname{rank}\left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t-\boldsymbol{B}\boldsymbol{W}\right) \leq 2k$. Therefore, we can write $\hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t-\boldsymbol{B}\boldsymbol{W} = \boldsymbol{U}_t \boldsymbol{R}_t = [\boldsymbol{U}_t\boldsymbol{r}_{t,1}, \boldsymbol{U}_t\boldsymbol{r}_{t,2}, \cdots, \boldsymbol{U}_t\boldsymbol{r}_{t,M}]$, where $\boldsymbol{U}_t \in \mathbb{R}^{d \times 2k}$ is an orthonormal matrix and $\boldsymbol{R}_t \in \mathbb{R}^{2k\times M}$. Thus we have $$\boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) = \left(\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i}\right)^{\top} \boldsymbol{R}_t.$$
This observation indicates that we can project the history actions $\boldsymbol{X}_{t-1,i}$ to a $2k$-dimensional space with $\boldsymbol{U}_t$, and take $\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i}$ as the $2k$-dimensional actions we have selected in the first $t-1$ steps. Following this idea, we connect the approximation error $\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}$ to the term $\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \left(\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i}\right)^{\top} \right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}$, where $\boldsymbol{V}_{t-1,i}(\lambda) \stackrel{\text { def }}{=} \left( \boldsymbol{U}^{\top}_t \boldsymbol{X}_{t-1,i}\right)\left(\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i} \right)^{\top}+\lambda \boldsymbol{I} $. We bound this term for the fixed $\boldsymbol{U}_t$ with the technique of self-normalized bound for vector-valued martingales~\citep{abbasi2011improved}, and then apply the $\epsilon$-net trick to cover all possible $\boldsymbol{U}_t$. This leads to an upper bound for $\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}$, and consequently helps to obtain the upper bound in Lemma~\ref{lemma: confidence set for linear bandits}.
\iffalse We only sketch the proof of Lemma~\ref{lemma: confidence set for linear bandits} here and defer the detailed explanation to Appendix~\ref{sec: proof of lemma, confidence set for linear bandits}. \begin{proof}
(Sketch) Since $\left\|\boldsymbol{B}\boldsymbol{W}\right\| \leq 1$ and $\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t\right\| \leq 1$, we have \begin{align}
\label{eqn: upper bound lambda term in tildeV}
& \sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \notag \\
\leq & \sum_{i=1}^{M} \left\|\boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right)\right\|^2 + 4M\lambda \end{align}
Suppose $d> 2k$, our observation is that both $\boldsymbol{B}\boldsymbol{W}$ and $\hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t$ are low-rank matrix with rank upper bounded by $k$, which indicates that $\operatorname{rank}\left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t-\boldsymbol{B}\boldsymbol{W}\right) \leq 2k$. In that case, we can write $\hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t-\boldsymbol{B}\boldsymbol{W} = \boldsymbol{U}_t \boldsymbol{R}_t = [\boldsymbol{U}_t\boldsymbol{r}_{t,1}, \boldsymbol{U}_t\boldsymbol{r}_{t,2}, \cdots, \boldsymbol{U}_t\boldsymbol{r}_{t,M}]$, where $\boldsymbol{U}_t \in \mathbb{R}^{d \times 2k}$ is an orthonormal matrix with $\|\boldsymbol{U}_t\|_F = \sqrt{2k}$, and $\boldsymbol{R}_t \in \mathbb{R}^{2k\times M}$ satisfies $\|\boldsymbol{r}_{t,i}\|_2 \leq \sqrt{k}$. Thus we have \begin{align*}
\boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) = \left(\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i}\right)^{\top} \boldsymbol{R}_t \end{align*}
This observation indicates that we can project the history actions $\boldsymbol{X}_{t-1,i}$ to a $k$-dimensional space with $\boldsymbol{U}_t$, and regard $\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i}$ as the $k$-dimensional actions we have selected in the first $t-1$ steps. By the optimality of $\hat{\boldsymbol{B}}_t$ and $\hat{\boldsymbol{W}}_t = [\hat{\boldsymbol{w}}_1, \cdots, \hat{\boldsymbol{w}}_M]$, we know that $\sum_{i=1}^{M} \left\|\boldsymbol{y}_{t-1,i}-\boldsymbol{X}_{t-1,i}^{\top} \hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}\right\|^{2} \leq \sum_{i=1}^{M} \left\|\boldsymbol{y}_{t-1,i}-\boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{B} \boldsymbol{w}_{i}\right\|^{2}$. After plugging the formulation of rewards $\boldsymbol{y}_{t-1,i} = \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{B} \boldsymbol{w}_{i} + \boldsymbol{\eta}_{t-1,i}$ and simplifying the inequality, we have \begin{align}
\label{eqn: optimality of hTheta main text}
& \sum_{i=1}^{M} \left\|\boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right)\right\|^2 \notag\\
\leq &2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) \end{align}
Define $\boldsymbol{V}_{t-1,i}(\lambda) \stackrel{\text { def }}{=} \left( \boldsymbol{U}^{\top}_t \boldsymbol{X}_{t-1,i}\right)\left(\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i} \right)^{\top}+\lambda \boldsymbol{I} $. By Equation~\ref{eqn: upper bound lambda term in tildeV}, Equation~\ref{eqn: optimality of hTheta main text} and some elementary calculation, we can show that \begin{align}
\label{eqn: proof, upper bound in lemma 1}
& \sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \notag \\
\leq & 2\sqrt{\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}} \notag\\
&\cdot \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}} + 4M\lambda. \end{align}
The main problem is how to bound $\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}$. We can rewrite this term as $\left\|\sum_{n=1}^{t-1}\eta_{n,i} \boldsymbol{U}^{\top}_t x_{n,i}\right\|_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}$. Note that if $\boldsymbol{U}_t$ is fixed as $\bar{\boldsymbol{U}}$, we can regard $\bar{\boldsymbol{U}}^{\top} \boldsymbol{x}_{n,i} \in \mathbb{R}^{k}$ as the corresponding ``action'' chosen in step $n$. With this observation, if $\boldsymbol{U}_t$ is fixed, we can bound this term following the arguments of the self-normalized bound for vector-valued martingales~\citep{abbasi2011improved}.
\begin{lemma} \label{lemma: self-normalized bound}
For a fixed $\bar{\boldsymbol{U}}$, define $\bar{\boldsymbol{V}}_{t,i}(\lambda) \stackrel{\text { def }}{=} \left(\bar{\boldsymbol{U}}^{\top}\boldsymbol{X}_{t,i} \right)\left(\bar{\boldsymbol{U}}^{\top}\boldsymbol{X}_{t,i} \right)^{\top}+\lambda \boldsymbol{I} $, then any $\delta > 0$, with probability at least $1-\delta$, for all $t \geq 0$, \begin{align}
& \sum_{i=1}^{M}\left\|\bar{\boldsymbol{U}}^{\top} \boldsymbol{X}_{t,i}\boldsymbol{\eta}_{t,i}\right\|^2_{\bar{\boldsymbol{V}}_{t,i}^{-1}} \\
\leq &2 \log\left(\frac{\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{t,i})^{1/2}\operatorname{det}(\lambda\boldsymbol{I})^{-1/2}\right)}{\delta}\right). \end{align} \end{lemma}
This lemma is proved in Appendix~\ref{sec: proof of self-normalized bound}. However, the problem is that $\boldsymbol{U}_t$ is not fixed. Even worse, $\boldsymbol{U}_t$ is dependent on the history data $\boldsymbol{X}_{t-1,i}$. To tackle this problem, we construct an $\epsilon$-net $\mathcal{E}$ in Frobenius norm over the matrix set $\left\{\boldsymbol{U} \in \mathbb{R}^{d \times 2k} : \|\boldsymbol{U}\|_F \leq k \right\}$. This $\epsilon$-net construction helps us to obtain an upper bound for $\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}$. By plugging this upper bound back to Eqn~\ref{eqn: proof, upper bound in lemma 1} and carefully handle the error terms due to $\epsilon$-net, we can prove the final result in Lemma~\ref{lemma: confidence set for linear bandits}.
\end{proof} \fi
\subsection{Algorithm and Regret} \label{sec:alg and regret for linear bandits}
\begin{algorithm}[tbh] \caption{Multi-Task Low-Rank OFUL} \label{alg: multi-task OFUL}
\begin{algorithmic}[1]
\For { step $t = 1,2,\cdots, T$}
\State Calculate the confidence interval $\mathcal{C}_t$ by Eqn~\ref{eqn: confidence set for linear bandits}
\State $\tilde{\boldsymbol{\Theta}}_t, \bm{x}_{t,i} = \argmax_{ \boldsymbol{\Theta} \in \mathcal{C}_t, \bm{x}_i \in \mathcal{A}_{t,i}} \sum_{i=1}^{M}\left\langle \boldsymbol{x}_i, \boldsymbol{\theta}_{i}\right\rangle$
\For{task $i = 1, 2,\cdots, M$}
\State Play $\bm{x}_{t,i}$ for task $i$, and obtain the reward $y_{t,i}$
\EndFor
\EndFor
\end{algorithmic} \end{algorithm}
We describe our Multi-Task Low-Rank OFUL algorithm in Algorithm~\ref{alg: multi-task OFUL}. The following theorem states a bound on the regret of the algorithm.
\begin{theorem} \label{thm: main theory for linear bandits} Suppose Assumption~\ref{assumptions:low_rank_bandits} holds. Then, with probability at least $1- \delta$, the regret of Algorithm~\ref{alg: multi-task OFUL} is bounded by \begin{align} \mathrm{Reg}(T) =\tilde{O}\left(M\sqrt{dkT} + d\sqrt{kMT} \right) \end{align} \end{theorem}
We defer the proof of Theorem~\ref{thm: main theory for linear bandits} to Appendix~\ref{sec: proof of main theorem for linear bandits}. The first term in the regret has linear dependence on $M$. This term characterizes the regret caused by learning the parameters $\boldsymbol{w}_i$ for each task. The second term has square root dependence on the number of total samples $MT$, which indicates the cost to learn the common representation with samples from $M$ tasks. By dividing the total regret by the number of tasks $M$, we know that the average regret for each task is $\tilde{O}(\sqrt{dkT} + d\sqrt{kT/M})$. Note that if we solve $M$ tasks with algorithms such as OFUL~\citep{abbasi2011improved} independently, the regret per task can be $\tilde{O}(d\sqrt{T})$.
Our bound saves a factor of $\sqrt{d/k}$ compared with the naive method by leveraging the common representation features. We also show that when $d > M$ our regret bound is near optimal (see Theorem \ref{thm: lower bound for linear bandits}).
\subsection{Misspecified Multi-Task Linear Bandits} \label{sec: misspecifed linear bandits} For multi-task linear bandit problem, it is relatively unrealistic to assume a common feature extractor that can fit the reward functions of $M$ tasks exactly. A more natural situation is that the underlying reward functions are not exactly linear, but have some misspecifications. There are also relevant discussions on single-task linear bandits in recent works \citep{lattimore2020learning,zanette2020learning}. We first present a definition for the approximately linear bandit learning in multi-task setting.
\begin{assumption} \label{assumption: misspecified linear bandits}
There exists a linear feature extractor $\boldsymbol{B} \in \mathbb{R}^{d\times k}$ and a set of linear coefficients $\{\boldsymbol{w}_i\}_{i=1}^{M}$ such that the expectation reward $\mathbb{E}[y_i|\boldsymbol{x}_i]$ for any action $\boldsymbol{x}_i \in \mathbb{R}^d$ satisfies $\left|\mathbb{E}[y_i|\boldsymbol{x}_i] - \left\langle\boldsymbol{x}_i,\boldsymbol{B}\boldsymbol{w}_i\right\rangle\right| \leq \zeta$. \end{assumption}
In general, an algorithm designed for a linear model could break down entirely if the underlying model is not linear. However, we find that our algorithm is in fact robust to small model misspecification if we set $L = \tilde{O}(Mk+kd+MT\zeta^2)$ (see Appendix~\ref{sec: proof for misspecified linear bandits} for the exact value). The following regret bound holds under Assumption~\ref{assumption: misspecified linear bandits} if we slightly modify the hyperparameter $L$ in the definition of confidence region $\mathcal{C}_t$.
\begin{theorem} \label{thm: regret for misspecified linear bandits} Under Assumption~\ref{assumption: misspecified linear bandits}, with probability at least $1- \delta$, the regret of Algorithm~\ref{alg: multi-task OFUL} is bounded by \begin{align} \mathrm{Reg}(T) = \tilde{O}\left(M\sqrt{dkT} + d\sqrt{kMT} + MT \sqrt{d }\zeta\right) \end{align} \end{theorem}
Theorem~\ref{thm: regret for misspecified linear bandits} is proved in Appendix~\ref{sec: proof for misspecified linear bandits}. Compared with Theorem~\ref{thm: main theory for linear bandits}, there is an additional term $\tilde{O}(MT \sqrt{d }\zeta)$ in the regret of Theorem~\ref{thm: regret for misspecified linear bandits}. This additional term is inevitably linear in $MT$ due to the intrinsic bias introduced by linear function approximation. Note that our algorithm can still enjoy good theoretical guarantees when $\zeta$ is sufficiently small.
\subsection{Lower Bound} \label{sec: lower_bound for linear bandits}
In this subsection, we propose the regret lower bound for multi-task linear bandit problem under Assumption~\ref{assumption: misspecified linear bandits}.
\begin{theorem} \label{thm: lower bound for linear bandits} For any $k,M,d,T \in \mathbb{Z}^{+}$ with $k \leq d \leq T$ and $k \leq M$, and any learning algorithm $\mathcal{A}$, there exist a multi-task linear bandit instance that satisfies Assumption~\ref{assumption: misspecified linear bandits}, such that the regret of Algorithm~$\mathcal{A}$ is lower bounded by $$\operatorname{Reg}(T) \geq \Omega\left(Mk\sqrt{T} + d\sqrt{kMT} + MT \sqrt{d }\zeta\right).$$ \end{theorem}
We defer the proof of Theorem~\ref{thm: lower bound for linear bandits} to Appendix~\ref{sec: proof of lower bound for linear bandits}. By setting $\zeta = 0$, Theorem~\ref{thm: lower bound for linear bandits} can be converted to the lower bound for multi-task linear bandit problem under Assumption~\ref{assumptions:low_rank_bandits}, which is $\Omega(Mk\sqrt{T} + d\sqrt{kMT} )$. These lower bounds match the upper bounds in Theorem~\ref{thm: main theory for linear bandits} and Theorem~\ref{thm: regret for misspecified linear bandits} in the regime where $d > M$ respectively. There is still a gap of $\sqrt{d/k}$ in the first part of the regret. For the upper bounds, the main difficulty to obtain $\tilde{O}(Mk\sqrt{T})$ regret in the first part comes from the estimation of $\boldsymbol{B}$. Since the action sets are not fixed and can be ill-conditioned, we cannot follow the explore-then-exploit framework and estimate $\boldsymbol{B}$ at the beginning. Besides, explore-then-exploit algorithms always suffer $\tilde{O}(T^{2/3})$ regret in the general linear bandits setting without further assumptions. Without estimating $\boldsymbol{B}$ beforehand with enough accuracy, the exploration in original $d$-dimensional space can be redundant since we cannot identify actions that have the similar $k$-dimensional representations before pulling them. We conjecture that our upper bound is tight and leave the gap as future work. \section{Main Results for Linear RL} \label{sec: linear_rl}
We now show the main results for the multi-task episodic reinforcement learning under the assumption of low inherent Bellman error (i.e. the multi-task LSVI setting).
\subsection{Multi-task LSVI Framework} \label{linear_rl:MT-LSVI_framework}
In the exploration problems in RL where linear value function approximation is employed \citep{yang2019sample, jin2020provably,yang2020reinforcement}, LSVI-based algorithms are usually very effective when the linear value function space are \textit{close} under Bellman operator. For example, it is shown that a LSVI-based algorithm with additional bonus can solve the exploration challenge effectively in low-rank MDP \citep{jin2020provably}, where the function space $\caQ_h, \caQ_{h+1}$ are totally close under Bellman operator (i.e. any function $Q_{h+1}$ in $\caQ_{h+1}$ composed with Bellman operator $\caT_{h} \caQ_{h+1}$ belongs to $\caQ_{h}$). For the release of such strong assumptions, the inherent Bellman error for a MDP (Definition~\ref{definitions:inherent_bellman_error}) was proposed to measure how close is the function space under Bellman operator \citep{zanette2020learning}. We extend the definition of IBE to the multi-task LSVI setting (Definition~\ref{definitions:inherent_bellman_error_low_rank}), and show that our refined confidence set for the least square estimator can be applied to the low-rank multi-task LSVI setting, and gives an optimism-based algorithm with sharper regret bound compared to naively do exploration in each task independently.
\subsection{Algorithm} \label{linear_rl:algorithm}
The MTLR-LSVI (Algorithm \ref{algorithm:linear_rl}) follows the LSVI-based \citep{jin2020provably, zanette2020learning} algorithms to build our (optimistic) estimator for the optimal value functions. To understand how this works for multi-task LSVI setting, we first take a glance at how LSVI-based algorithms work in single-task LSVI setting.
In traditional value iteration algorithms, we perform an approximate Bellman backup in episode $t$ for each step $h \in [H]$ on the estimator $Q_{h+1,t-1}$ constructed at the end of episode $t-1$, and find the best approximator for $\caT_h\left(Q_{h+1,t-1}\right)$ in function space $\caQ_h$. Since we assume linear function spaces, we can take the least-square solution of the empirical Bellman backup on $Q_{h+1,t-1}$ as the best approximator.
In the multi-task framework, given an estimator $Q_{h+1}\left(\boldsymbol{\theta}_{h+1}^i\right)$ for each $i \in [M]$, to apply such least-square value iteration to our low-rank multi-task LSVI setting, we use the solution to the following constrained optimization problem \begin{align} \label{formula:linear_rl_least_square} & \sum_{i=1}^M \sum_{j=1}^{t-1} \left(\left(\boldsymbol{\phi}_{hj}^i\right)^{\top} \boldsymbol{\theta}_h^i - R_{hj}^i -V_{h+1}^i\left(\boldsymbol{\theta}_{h+1}^i\right)\left(s_{h+1, j}^i\right)\right)^{2} \\ \text{s.t.} & \quad \boldsymbol{\theta}_h^1, \boldsymbol{\theta}_h^2, ..., \boldsymbol{\theta}_h^M \text{~lies in a $k$-dimensional subspace} \end{align}
to approximate the Bellman update in the $t$-th episode, where $\boldsymbol{\phi}_{hj}^i = \boldsymbol{\phi}_h(s^i_{h j}, a^i_{hj})$ is the feature observed at step $h$ in episode $j$ for task $i$, and similarly $R_{hj}^i = R_h(s_{h j}^i, a_{h j}^i)$.
To guarantee the optimistic property of our estimator, we follow the global optimization procedure of \citet{zanette2020learning} which solves the following optimization problem in the $t$-th episode
\begin{definition}[Global Optimization Procedure] \label{formula:linear_rl_global_optimization}
\begin{align} \max_{\bar{\boldsymbol{\xi}}_h^i, \hat{\boldsymbol{\theta}}_h^i, \bar{\boldsymbol{\theta}}_h^i} & \sum_{i = 1}^M \max_{a^i} \left(\boldsymbol{\phi}(s_1^i, a^i)\right)^\top \bar{\boldsymbol{\theta}}_1^i \\ \text{s.t.} \quad & \left(\hat{\boldsymbol{\theta}}_h^1, ..., \hat{\boldsymbol{\theta}}_h^M\right) = \hat{\boldsymbol{B}}_h \begin{bmatrix} \hat{\boldsymbol{w}}_h^1 & \hat{\boldsymbol{w}}_h^2 & \cdots & \hat{\boldsymbol{w}}_h^M \end{bmatrix} \notag \\
& \qquad \qquad = \argmin_{\left\|\boldsymbol{B}_h \boldsymbol{w}_h^i\right\|_2 \leq D} \sum_{i=1}^M \sum_{j=1}^{t-1} L(\boldsymbol{B}_h,\boldsymbol{w}_h^i)\\
& \bar{\boldsymbol{\theta}}_{h}^i =\hat{\boldsymbol{\theta}}_{h}^i + \bar{\boldsymbol{\xi}}_{h}^i; \quad \sum_{i = 1}^M \left\|\bar{\boldsymbol{\xi}}_{h}^i\right\|_{\tilde{\bm{V}}_{h t}^i(\lambda)}^2 \leq \alpha_{h t} \\ &\left(\bar{\boldsymbol{\theta}}_{h}^1, \bar{\boldsymbol{\theta}}_h^2, \cdots, \bar{\boldsymbol{\theta}}_h^M\right) \in \Theta_{h} \end{align} \end{definition}
where the empirical least-square loss $L(\boldsymbol{B}_h,\boldsymbol{w}_h^i)\defeq ((\boldsymbol{\phi}_{hj}^i)^\top \boldsymbol{B}_h \boldsymbol{w}_h^i - R_{hj}^i - V_{h+1}^i(\bar{\boldsymbol{\theta}}_{h+1}^i)(s_{h+1, j}^i))^{2}$ , and $ \tilde{\bm{V}}^i_{ht}(\lambda) \defeq \sum_{j=1}^{t-1} (\boldsymbol{\phi}_{hj}^i) (\boldsymbol{\phi}_{hj}^i)^\top + \lambda \bm{I} $ is the regularized empirical linear design matrix for task $i$ in episode $t$.
\begin{algorithm}[!t] \caption{Multi-Task Low-Rank LSVI} \label{algorithm:linear_rl}
\begin{algorithmic}[1]
\State Input: low-rank parameter $k$, failure probability $\delta$, regularization $\lambda=1$, inherent Bellman error $\caI$
\State Initialize $\tilde{\bm{V}}_{h1}=\lambda \boldsymbol{I}$ for $h \in [H]$
\For{ episode $t = 1,2,\cdots $}
\State Compute $\alpha_{ht}$ for $h \in [H]$. (see Lemma \ref{lemma:linear_rl_least_square_error})
\State Solve the global optimization problem \ref{formula:linear_rl_global_optimization}
\State Compute $\pi_{ht}^i(s) = \argmax_a \boldsymbol{\phi}(s, a)^\top \bar{\boldsymbol{\theta}}_{ht}^i$
\State Execute $\pi^i_{ht}$ for task $i$ at step $h$
\State Collect $\left\{s_{ht}^i, a_{ht}^i, r\left(s_{ht}^i, a_{ht}^i\right)\right\}$ for episode $t$.
\EndFor
\end{algorithmic} \end{algorithm}
We have three types of variables in this global optimization problem, $\bar{\boldsymbol{\xi}}_h^i, \hat{\boldsymbol{\theta}}_h^i$, and $\bar{\boldsymbol{\theta}}_h^i$. Here $\bar{\boldsymbol{\theta}}_{h}^i$ denotes the estimator for $Q^{i *}_h$. We solve for the low-rank least-square solution of the approximate value iteration and denote the solution by $\hat{\boldsymbol{\theta}}_h^i$. Instead of adding the bonus term directly on $Q_{h}^i (\hat{\boldsymbol{\theta}}_h^i)$ to obtain an optimistic estimate of $Q^{i*}_h$ as in the tabular setting \citep{azar2017minimax, jin2018q} and linear MDP setting \citep{jin2020provably}, we use global variables $\bm{\bar{\xi}}_h^i$ to quantify the confidence bonus. This is because we cannot preserve the linear property of our estimator if we add the bonus directly, resulting in an exponential propagation of error. However, by using $\bar{\boldsymbol{\xi}}_h^i$ we can construct a linear estimator $Q_{h}^i \left(\bar{\boldsymbol{\theta}}_h^i\right)$ and obtain much smaller regret. A drawback of this global optimization technique is that we can only obtain an optimistic estimator at step 1, since values in different states and steps are possibly negatively correlated.
\subsection{Regret Bound} \label{linear_rl:regret_bound}
\begin{theorem} \label{theorem:linear_rl_regret_bound} Under Assumption \ref{assumptions:low_rank_small_ibe} and \ref{assumptions:linear_rl_regularity}, with probability $1 - \delta$ the regret after $T$ episodes is bounded by
\begin{align} \operatorname{Reg}(T) = \tilde{O} \left(H M \sqrt{dkT} + Hd\sqrt{kMT} + HMT\sqrt{d} \caI\right) \end{align} \end{theorem}
Compared to naively executing single-task linear RL algorithms (e.g. the ELEANOR algorithm) on each task without information-sharing, which incurs regret $\tilde{O} (HM d\sqrt{T} + HMT\sqrt{d} \caI)$, our regret bound is smaller by a factor of approximately $\sqrt{d/k}$ in our setting where $k \ll d$ and $k \ll M$.
We give a brief explanation on how we improve the regret bound and defer the full analysis to appendix \ref{sec: omiited proof in linea_rl}. We start with the decomposition of the regret. Let $\bar{Q}_{ht}^i$($\bar{V}_{ht}^i$) be the solution of the problem in definition \ref{formula:linear_rl_global_optimization} in episode $t$, then \begin{align} & \text{Reg}(T) = \sum_{t=1}^T \sum_{i=1}^M \left(V_1^{i*} - \bar{V}_{1t}^i + \bar{V}_{1t}^i - V_1^{\pi_{t}^i} \right)\left(s_{1t}^i\right) \\ \label{formula:linear_rl_sketch_optimism} & \leq HMT\caI \qquad \text{(by Lemma \ref{lemma:linear_rl_optimism})}\\ \label{formula:linear_rl_sketch_bellman_error}
& + \sum_{t=1}^T \sum_{h=1}^H \sum_{i=1}^M \left( \left|\bar{Q}_{ht}^i (s, a) - \caT_h^i \bar{Q}_{h+1,t}^i (s, a)\right| + \zeta_{ht}^i\right). \end{align}
In (\ref{formula:linear_rl_sketch_optimism}) we use the optimistic property of $\bar{V}_{1t}^i$. In (\ref{formula:linear_rl_sketch_bellman_error}), $\zeta_{ht}^i$ is a martingale difference (defined in section \ref{appendix_linear_rl:regret_bound}) with regards to $\caF_{h,t}$, and the dominate term (the first term) is the Bellman error of $\bar{Q}_{ht}^i$.
For any $\{Q_{h+1}^i\}_{i=1}^M \in \caQ_{h+1}$, we can find a group of vectors $\{\dot{\boldsymbol{\theta}}_{h}^i(Q_{h+1}^i) \}_{i=1}^M \in \Theta_h$ that satisfy $\Delta_h^i \left(Q_{h+1}^i\right)(s, a) \defeq \caT_h^i \left(Q_{h+1}^i\right)(s, a) - \boldsymbol{\phi}(s, a)^\top \dot{\boldsymbol{\theta}}_{h}^i\left(Q_{h+1}^i\right)$ and the approximation error $\left\|\Delta_h^i \left(Q_{h+1}^i\right)\right\|_{\infty} \leq \caI$ is small for each $i \in [M]$. By definition, $\dot{\boldsymbol{\theta}}_{h}^i\left(Q_{h+1}^i\right)$ is actually the best approximator of $\caT_h^i \left(Q_{h+1}^i\right)$ in the function class $\caQ_h$. Since our algorithm is based on least-square value iteration, a key step is to bound the error of estimating $\dot{\boldsymbol{\theta}}_{h}^i(\bar{Q}_{h+1,t}^i)$ ($\dot{\boldsymbol{\theta}}_{h}^i$ for short). In the global optimization procedure, we use $\hat{\boldsymbol{\theta}}_h^i$ to approximate the empirical Bellman backup. In Lemma \ref{lemma:linear_rl_least_square_error} we show \begin{align} \label{formula:linear_rl_sketch_least_square_error}
\sum_{i=1}^{M}\left\|\hat{\boldsymbol{\theta}}_h^i - \dot{\boldsymbol{\theta}}_h^i\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} = \tilde{O}\left(Mk + kd + MT\caI^2\right) \end{align}
This is the key step leading to improved regret bound. If we solve each task independently without information sharing, we can only bound the least square error in (\ref{formula:linear_rl_sketch_least_square_error}) as $\tilde{O}(Md+MT\caI^2)$. Our bound is much more sharper since $k \ll d$ and $k \ll M$.
Using the least square error in (\ref{formula:linear_rl_sketch_least_square_error}), we can show that the dominate term in (\ref{formula:linear_rl_sketch_bellman_error}) is bounded by (see Lemma \ref{lemma:linear_rl_bellman_error} and section \ref{appendix_linear_rl:regret_bound}) \begin{align}
& \sum_{i=1}^M \left|\bar{Q}_{ht}^i (s, a) - \caT_h^i \bar{Q}_{h+1,t}^i (s, a)\right| \leq M\caI + \tilde{O}\left(\sqrt{Mk + kd + MT\caI^2}\right) \cdot \sqrt{\sum_{i=1}^M \left\|
\boldsymbol{\phi}(s_{ht}^i, a_{ht}^i)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} \end{align}
\citet[Lemma 11]{abbasi2011improved} states that $\sum_{t=1}^T \left\|
\boldsymbol{\phi}(s_{ht}^i, a_{ht}^i)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2 = \tilde{O}(d)$ for any $h$ and $i$, so we can finally bound the regret as \begin{align*} \text{Reg}(T) & = \tilde{O}\left(HMT\caI + H\sqrt{Mk + kd + MT\caI^2} \cdot \sqrt{MTd}\right) \notag \\ & = \tilde{O}\left(H M \sqrt{dkT} + Hd\sqrt{kMT} + HMT\sqrt{d} \caI\right) \end{align*} where the first equality is by Cauchy-Schwarz.
\iffalse To start with, we introduce the Bellman residual function $\Delta_h^i(Q_{h+1}^i)$ \citep{zanette2020learning}.
For any $\{Q_{h+1}^i\}_{i=1}^M \in \caQ_{h+1}$, we can find a group of vectors $\{\dot{\theta}_{h}^i(Q_{h+1}^i) \}_{i=1}^M \in \Theta_h$ that satisfy
\begin{align} \label{formula:best_approximators} \Delta_h^i \left(Q_{h+1}^i\right)(s, a) \defeq \caT_h^i \left(Q_{h+1}^i\right)(s, a) - \phi(s, a)^\top \dot{\theta}_{h}^i\left(Q_{h+1}^i\right) \end{align}
and the approximation error $\left\|\Delta_h^i \left(Q_{h+1}^i\right)\right\|_{\infty} \leq \caI$ is small for each $i \in [M]$. We also use $\dot{B}_h \dot{w}_h^i \left(Q_{h+1}^i\right)$ in place of $\dot{\theta}_{h}^i\left(Q_{h+1}^i\right)$ since we can write $\dot{\theta}_{h}^i$ as $\dot{B}_h \dot{w}_h^i$ according to assumption \ref{assumptions:low_rank_small_ibe}. \fi
\subsection{Lower Bound} \label{linear_rl:lower_bound}
This subsection presents the lower bound for multi-task reinforcement learning with low inherent Bellman error. Our lower bound is derived from the lower bound in the single-task setting. As a byproduct, we also derive a lower bound for misspecified linear RL in the single-task setting. We defer the proof of Theorem~\ref{theorem:linear_rl_lower_bound} to Appendix~\ref{sec: proof of the lower bound for rl}.
\begin{theorem} \label{theorem:linear_rl_lower_bound}
For our construction in appendix \ref{sec: proof of the lower bound for rl}, the expected regret of any algorithm where $d,k,H \geq 10, |\mathcal{A}| \geq 3,M \geq k, T= \Omega(d^2H),\mathcal{I} \leq 1 / 4H$ is
$$ \Omega \left(Mk\sqrt{HT} + d\sqrt{HkMT} + HMT \sqrt{d} \mathcal{I}\right) $$
\end{theorem}
Careful readers may find that there is a gap of $\sqrt{H}$ in the first two terms between the upper bound and the lower bound. This gap is because the confidence set used in the algorithm is intrinsically ``Hoeffding-type''. Using a ``Bernstein-type'' confidence set can potentially improve the upper bound by a factor of $\sqrt{H}$. This ``Bernstein'' technique has been well exploited in many previous results for single-task RL~\citep{azar2017minimax,jin2018q, zhou2020nearly}. Since our focus is mainly on the benefits of multi-task representation learning, we don't apply this technique for the clarity of the analysis. If we ignore this gap in the dependence on $H$, our upper bound matches this lower bound in the regime where $d \geq M$. \section{Conclusion} \label{sec: conclusion}
In this paper, we study provably sample-efficient representation learning for multi-task linear bandits and linear RL. For linear bandits, we propose an algorithm called MTLR-OFUL, which obtains near-optimal regret in the regime where $d \geq M$. We then extend our algorithms to multi-task RL setting, and propose a sample-efficient algorithm, MTLR-LSVI.
There are two directions for future investigation. First, our algorithms are statistically sample-efficient, but a computationally efficient implementation is still unknown, although we conjecture our MTLR-OFUL algorithm is computationally efficient. How to design both computationally and statistically efficient algorithms in our multi-task setting is an interesting problem for future research. Second, there remains a gap of $\sqrt{d/k}$ between regret upper and lower bounds (in the first term).
We conjecture that our lower bound is not minimax optimal and hope to address this problem in the future work.
\setcounter{section}{0} \appendix \renewcommand{Appendix~\Alph{section}}{Appendix~\Alph{section}} \section*{Appendices} \addcontentsline{toc}{section}{Appendices} \renewcommand{\Alph{subsection}}{\Alph{subsection}}
\subsection{Omitted Proof in Section~\ref{sec: linear_bandits}} \label{sec: omitted proof for linear bandits}
\subsubsection{Proof of Lemma~\ref{lemma: confidence set for linear bandits}} \label{sec: proof of lemma, confidence set for linear bandits}
\begin{proof}
By the optimality of $\hat{\boldsymbol{B}}_t$ and $\hat{\boldsymbol{W}}_t = [\hat{\boldsymbol{w}}_{t,1}, \cdots, \hat{\boldsymbol{w}}_{t,M}]$, we know that $\sum_{i=1}^{M} \left\|\boldsymbol{y}_{t-1,i}-\boldsymbol{X}_{t-1,i}^{\top} \hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}\right\|^{2}_2 \leq \sum_{i=1}^{M} \left\|\boldsymbol{y}_{t-1,i}-\boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{B} \boldsymbol{w}_{i}\right\|^{2}_2$. Since $\boldsymbol{y}_{t-1,i} = \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{B} \boldsymbol{w}_{i} + \boldsymbol{\eta}_{t-1,i}$, we have \begin{align}
\label{eqn: optimality of hTheta}
\sum_{i=1}^{M} \left\|\boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right)\right\|^2_2 \leq 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right). \end{align}
We firstly analyse the non-trivial setting where $d \geq 2k$. Note that both $\boldsymbol{\Theta} = \boldsymbol{B}\boldsymbol{W}$ and $\hat{\boldsymbol{\Theta}}_t = \hat{\boldsymbol{B}}_t \hat{\boldsymbol{W}}_t$ are low-rank matrix with rank upper bounded by $k$, which indicates that $\operatorname{rank}\left(\hat{\boldsymbol{\Theta}}_t-\boldsymbol{\Theta}\right) \leq 2k$. In that case, we can write $\hat{\boldsymbol{\Theta}}_t-\boldsymbol{\Theta} = \boldsymbol{U}_t \boldsymbol{R}_t = [\boldsymbol{U}_t\boldsymbol{r}_{t,1}, \boldsymbol{U}_t\boldsymbol{r}_{t,2}, \cdots, \boldsymbol{U}_t\boldsymbol{r}_{t,M}]$, where $\boldsymbol{U}_t \in \mathbb{R}^{d \times 2k}$ is an orthonormal matrix with $\|\boldsymbol{U}_t\|_F = \sqrt{2k}$, and $\boldsymbol{R}_t \in \mathbb{R}^{2k\times M}$ satisfies $\|\boldsymbol{r}_{t,i}\|_2 \leq \sqrt{k}$. In other words, we can write $\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i} = \boldsymbol{U}_t \boldsymbol{r}_{t,i}$ for certain $\boldsymbol{U}_t$ and $\boldsymbol{r}_{t,i}$.
Define $\boldsymbol{V}_{t-1,i}(\lambda) \stackrel{\text { def }}{=} \left( \boldsymbol{U}^{\top}_t \boldsymbol{X}_{t-1,i}\right)\left(\boldsymbol{U}_t^{\top}\boldsymbol{X}_{t-1,i} \right)^{\top}+\lambda \boldsymbol{I} $. We have: \begin{align}
\lefteqn{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}} \label{eqn: hTheta - Theta decomposition 0} \\
\label{eqn: hTheta - Theta decomposition 1}
= & \sum_{i=1}^{M} \left\|\boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right)\right\|^2_2 + \sum_{i=1}^{M} \lambda \left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_2 \\
\label{eqn: hTheta - Theta decomposition 2}
\leq & 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) + 4M \lambda \\
\label{eqn: hTheta - Theta decomposition 3}
= & 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t \boldsymbol{r}_{t,i} + 4M \lambda \\
\label{eqn: hTheta - Theta decomposition 4}
\leq & 2\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)} \left\|\boldsymbol{r}_{t,i}\right\|_{\boldsymbol{V}_{t-1,i}(\lambda)} + 4M\lambda \\
\label{eqn: hTheta - Theta decomposition 5}
\leq & 2\sqrt{\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}} \sqrt{\sum_{i=1}^{M}\left\|\boldsymbol{r}_{t,i}\right\|^2_{\boldsymbol{V}_{t-1,i}(\lambda)}} + 4M\lambda \\
\label{eqn: hTheta - Theta decomposition 6}
= & 2\sqrt{\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}} \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}} + 4M\lambda \end{align}
Eqn~\ref{eqn: hTheta - Theta decomposition 2} is due to Eqn~\ref{eqn: optimality of hTheta}, $\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}\right\| \leq 1$ and $\left\|\boldsymbol{B} \boldsymbol{w}_{i}\right\| \leq 1$. Eqn~\ref{eqn: hTheta - Theta decomposition 5} is due to Cauchy-Schwarz inequality. Eqn~\ref{eqn: hTheta - Theta decomposition 6} is from
$$\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} = \sum_{i = 1}^M \left\|\boldsymbol{U}_{t}\boldsymbol{r}_{t,i}\right\|^2_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} = \sum_{i = 1}^M \left\|\boldsymbol{r}_{t,i}\right\|^2_{\boldsymbol{U}_t^\top \tilde{\boldsymbol{V}}_{t-1, i}(\lambda) \boldsymbol{U}_t} = \sum_{i = 1}^M \left\|\boldsymbol{r}_{t,i}\right\|^2_{\boldsymbol{V}_{t-1, i}(\lambda)}.$$
The main problem is how to bound $\left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)} = \left\|\sum_{n=1}^{t-1}\eta_{n,i} \boldsymbol{U}^{\top}_t x_{n,i}\right\|_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}$. Note that for a fixed $\boldsymbol{U}_t = \bar{\boldsymbol{U}}$, we can regard $\bar{\boldsymbol{U}}^{\top} \boldsymbol{x}_{n,i} \in \mathbb{R}^{k}$ as the corresponding ``action'' chosen in step $t$. With this observation, if $\boldsymbol{U}_t$ is fixed, we can bound this term following the arguments of the self-normalized bound for vector-valued martingales~\citep{abbasi2011improved}.
\begin{lemma} \label{lemma: self-normalized bound}
For a fixed $\bar{\boldsymbol{U}}$, define $\bar{\boldsymbol{V}}_{t,i}(\lambda) \stackrel{\text { def }}{=} \left(\bar{\boldsymbol{U}}^{\top}\boldsymbol{X}_{t,i} \right)\left(\bar{\boldsymbol{U}}^{\top}\boldsymbol{X}_{t,i} \right)^{\top}+\lambda \boldsymbol{I} $, then any $\delta > 0$, with probability at least $1-\delta$, for all $t \geq 0$, \begin{align}
& \sum_{i=1}^{M}\left\|\bar{\boldsymbol{U}}^{\top} \boldsymbol{X}_{t,i}\boldsymbol{\eta}_{t,i}\right\|^2_{\bar{\boldsymbol{V}}_{t,i}^{-1}} \\
\leq &2 \log\left(\frac{\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{t,i})^{1/2}\operatorname{det}(\lambda\boldsymbol{I})^{-1/2}\right)}{\delta}\right). \end{align} \end{lemma}
We defer the proof of Lemma~\ref{lemma: self-normalized bound} to Appendix~\ref{sec: proof of self-normalized bound}. We set $\lambda=1$. By Lemma~\ref{lemma: self-normalized bound}, we know that for a fixed $\bar{\boldsymbol{U}}$, with probability at least $1-\delta_1$, \begin{align}
\label{eqn: self-normalized bound for barU}
\sum_{i=1}^{M}\left\|\sum_{n=1}^{t-1}\eta_{n,i} \bar{\boldsymbol{U}}^{\top} x_{n,i}\right\|^2_{\bar{\boldsymbol{V}}^{-1}_{t,i}(\lambda)} \leq 2 \log\left(\frac{\prod_{i=1}^{M}\operatorname{det}(\bar{\boldsymbol{V}}_{t,i}(\lambda))^{1/2} \operatorname{det}(\lambda\boldsymbol{I})^{-1/2}}{\delta_1}\right) \leq 2Mk + 2 \log (1/\delta_1). \end{align}
The above analysis shows that we can bound $\left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t\right\|_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}$ if $\boldsymbol{U}_{t}$ is fixed as $\bar{\boldsymbol{U}}$. Following this idea, we prove the lemma by the construction of $\epsilon$-net over all possible $\boldsymbol{U}_t$. To apply the trick of $\epsilon$-net, we need to slightly modify the derivation of Eqn~\ref{eqn: hTheta - Theta decomposition 0}. For a fixed matrix $\bar{\boldsymbol{U}} \in \mathbb{R}^{d \times 2k}$, we have \begin{align}
\label{eqn: hTheta - Theta decomposition 1-0}
&\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \\
\label{eqn: hTheta - Theta decomposition 1-1}
\leq & 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{U}_t \boldsymbol{r}_{t,i} + 4M \lambda \\
\label{eqn: hTheta - Theta decomposition 1-2}
= & 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}} \boldsymbol{r}_{t,i} + 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\boldsymbol{U}_t-\bar{\boldsymbol{U}} \right)\boldsymbol{r}_{t,i} + 4M \lambda \\
\label{eqn: hTheta - Theta decomposition 1-3}
\leq & 2\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}}\right\|_{\bar{\boldsymbol{V}}^{-1}_{t-1,i}(\lambda)} \left\|\boldsymbol{r}_{t,i}\right\|_{\bar{\boldsymbol{V}}_{t-1,i}(\lambda)} + 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\boldsymbol{U}_t-\bar{\boldsymbol{U}} \right)\boldsymbol{r}_{t,i} + 4M \lambda \\
\label{eqn: hTheta - Theta decomposition 1-4}
= & 2\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}}\right\|_{\bar{\boldsymbol{V}}^{-1}_{t-1,i}(\lambda)} \left\|\boldsymbol{r}_{t,i}\right\|_{\boldsymbol{V}_{t-1,i}(\lambda)} + 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\boldsymbol{U}_t-\bar{\boldsymbol{U}} \right)\boldsymbol{r}_{t,i} \\
& + 2\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}}\right\|_{\bar{\boldsymbol{V}}^{-1}_{t-1,i}(\lambda)} \left(\left\|\boldsymbol{r}_{t,i}\right\|_{\bar{\boldsymbol{V}}_{t-1,i}(\lambda)} - \left\|\boldsymbol{r}_{t,i}\right\|_{\boldsymbol{V}_{t-1,i}(\lambda)}\right) + 4M \lambda\\
\label{eqn: hTheta - Theta decomposition 1-5}
\leq & 2\sqrt{\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}}\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}} \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}} + 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\boldsymbol{U}_t-\bar{\boldsymbol{U}} \right)\boldsymbol{r}_{t,i} \\
& + 2\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}}\right\|_{\bar{\boldsymbol{V}}^{-1}_{t-1,i}(\lambda)} \left(\left\|\boldsymbol{r}_{t,i}\right\|_{\bar{\boldsymbol{V}}_{t-1,i}(\lambda)} - \left\|\boldsymbol{r}_{t,i}\right\|_{\boldsymbol{V}_{t-1,i}(\lambda)}\right) + 4M \lambda \end{align}
Eqn~\ref{eqn: hTheta - Theta decomposition 1-1}, \ref{eqn: hTheta - Theta decomposition 1-3} and \ref{eqn: hTheta - Theta decomposition 1-5} follow the same idea of Eqn~\ref{eqn: hTheta - Theta decomposition 3}, \ref{eqn: hTheta - Theta decomposition 4} and \ref{eqn: hTheta - Theta decomposition 6}.
We construct an $\epsilon$-net $\mathcal{E}$ in Frobenius norm over the matrix set $\left\{\boldsymbol{U} \in \mathbb{R}^{d \times 2k} : \|\boldsymbol{U}\|_F \leq k \right\}$. It is not hard to see that $|\mathcal{E}| \leq \left(\frac{6\sqrt{2k}}{\epsilon}\right)^{2kd}$. By the union bound over all possible $\bar{\boldsymbol{U}} \in \mathcal{E}$, we know that with probability $1- |\mathcal{E}|\delta_1$, Eqn~\ref{eqn: self-normalized bound for barU} holds for any $\bar{\boldsymbol{U}} \in \mathcal{E}$. For each $\boldsymbol{U}_t$, we choose an $\bar{\boldsymbol{U}} \in \mathcal{E}$ with $\left\|\boldsymbol{U}_t - \bar{\boldsymbol{U}}\right\|_F \leq \epsilon$, and we have \begin{align}
\label{eqn: hTheta - Theta decomposition part 1}
2\sqrt{\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}}\right\|^2_{\boldsymbol{V}^{-1}_{t-1,i}(\lambda)}} \leq 2 \sqrt{2Mk + 2 \log(1/\delta_1)} \end{align}
Since $\left\|\boldsymbol{U}_t - \bar{\boldsymbol{U}}\right\|_F \leq \epsilon$, we have \begin{align}
\label{eqn: hTheta - Theta decomposition part 2}
2\sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \bar{\boldsymbol{U}}\right\|_{\bar{\boldsymbol{V}}^{-1}_{t-1,i}(\lambda)} \left(\left\|\boldsymbol{r}_{t,i}\right\|_{\bar{\boldsymbol{V}}_{t-1,i}(\lambda)} - \left\|\boldsymbol{r}_{t,i}\right\|_{\boldsymbol{V}_{t-1,i}(\lambda)}\right) \leq 2\sqrt{Mk\epsilon (2Mk+2 \log(1/\delta_1))}. \end{align}
For the term $ 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\boldsymbol{U}_t-\bar{\boldsymbol{U}} \right)\boldsymbol{r}_{t,i}$, the following inequality holds for any step $t \in [T]$ with probability $1-MT\delta_2$, \begin{align}
\label{eqn: hTheta - Theta decomposition part 3}
2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\boldsymbol{U}_t-\bar{\boldsymbol{U}} \right)\boldsymbol{r}_{t,i}
\leq &2 \sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}\right\|_2 \left\| \boldsymbol{X}_{t-1,i}^{\top} \left(\boldsymbol{U}_t-\bar{\boldsymbol{U}} \right)\boldsymbol{r}_{t,i}\right\|_2 \\
\leq & 2 \sum_{i=1}^{M} \left\|\boldsymbol{\eta}_{t-1,i}\right\|_2 \sqrt{kT\epsilon} \\
\leq & 2 M \sqrt{2\log(2/\delta_2) kT^2 \epsilon} \end{align}
The last inequality follows from the fact that $|\eta_{n, i}| \leq \sqrt{2 \log(2 / \delta_2)}$ with probability $1 - \delta_2$ for fixed $n, i$, and apply a union bound over $n \in [t - 1], i \in [M]$. Plugging Eqn.~\ref{eqn: hTheta - Theta decomposition part 1}, \ref{eqn: hTheta - Theta decomposition part 2} and \ref{eqn: hTheta - Theta decomposition part 3} back to Eqn.~\ref{eqn: hTheta - Theta decomposition 1-5}, the following inequality holds for any $t \in [T]$ with probability at least $1- |\mathcal{E}|\delta_1 - MT \delta_2$: \begin{align}
&\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \\
\leq & 2\sqrt{Mk + 2\log(1/\delta_1)} \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}} \\
\label{formula:linear_rl_cross_reference}
&+ 2 M \sqrt{2\log(2/\delta_2) kT^2 \epsilon} + 2\sqrt{Mk\epsilon (2Mk+2 \log(1/\delta_1))} + 4M\lambda \end{align}
By solving the above inequality, we know that \begin{align}
\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \leq &32 \left(Mk + \log(1/\delta_1)\right) + 4 M \sqrt{2\log(2/\delta_2) kT^2 \epsilon} \\
&+ 4\sqrt{Mk\epsilon (2Mk+2 \log(1/\delta_1))} + 8M\lambda \end{align}
Setting $\lambda=1$, $\epsilon = \frac{1}{kM^2T^2}$, $\delta_1 = \frac{\delta}{2 \left(\frac{6\sqrt{2k}}{\epsilon}\right)^{2kd}} \leq \frac{\delta}{2 |\mathcal{E}|}$, and $\delta_2 = \frac{\delta}{2MT}$, the following inequality holds with probability $1-\delta$: \begin{align}
\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} & \leq L \defeq 48\left(Mk + 5kd \log(kMT)\right) + 32 \log(4MT) + 76 \log(1 / \delta) \end{align}
At last we talk about the trivial setting where $k < d < 2k$. In this case, we can write $\hat{\boldsymbol{\Theta}}_t - \boldsymbol{\Theta} = \boldsymbol{R}_t$ where $\boldsymbol{R}_t \in \mathbb{R}^{d \times M}$. The proof then follows the same framework as the case when $d \geq 2k$, except that we don't need to consider $\boldsymbol{U}_t$ and construct $\epsilon$-net over all possible $\boldsymbol{U}_t$. It is not hard to show that $\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \leq 24 \left(Md + 2 \log(Tk/\delta)\right)$ in this case, which is also less than $L$ since $d < 2k$. \end{proof}
\subsubsection{Proof of Theorem~\ref{thm: main theory for linear bandits}} \label{sec: proof of main theorem for linear bandits} With Lemma~\ref{lemma: confidence set for linear bandits}, we are ready to prove Theorem~\ref{thm: main theory for linear bandits}.
\begin{proof} Let $\tilde{\boldsymbol{V}}_{t, i}(\lambda) = \boldsymbol{X}_{t,i}\boldsymbol{X}_{t,i}^{\top} + \lambda \boldsymbol{I}_d$ for some $\lambda > 0$. \begin{align} \mathrm{Reg}(T) & = \sum_{t = 1}^T \sum_{i = 1}^M \left\langle \boldsymbol{\theta}_i, \boldsymbol{x}^*_{t, i} - \boldsymbol{x}_{t, i} \right\rangle \\ & \leq \sum_{t = 1}^T \sum_{i = 1}^M \left\langle \tilde{\boldsymbol{\theta}}_{t, i} - \boldsymbol{\theta}_i, \boldsymbol{x}_{t, i} \right\rangle \\ & = \sum_{t = 1}^T \sum_{i = 1}^M \left\langle \tilde{\boldsymbol{\theta}}_{t, i} - \hat{\boldsymbol{\theta}}_{t , i} + \hat{\boldsymbol{\theta}}_{t , i} - \boldsymbol{\theta}_i, \boldsymbol{x}_{t, i} \right\rangle\\
& \leq \sum_{t = 1}^T \sum_{i = 1}^M \left(\left\| \tilde{\boldsymbol{\theta}}_{t, i} - \hat{\boldsymbol{\theta}}_{t , i}\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)} + \left\|\hat{\boldsymbol{\theta}}_{t , i} - \boldsymbol{\theta}_i\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)}\right) \left\|\boldsymbol{x}_{t, i} \right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)^{-1}} \\
& \leq \left(\sqrt{\sum_{t = 1}^T\sum_{i = 1}^M \left\| \tilde{\boldsymbol{\theta}}_{t, i} - \hat{\boldsymbol{\theta}}_{t , i}\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)}^2} + \sqrt{\sum_{i = 1}^M \left\|\hat{\boldsymbol{\theta}}_{t , i} - \boldsymbol{\theta}_i\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)}^2}\right) \cdot \sqrt{\sum_{t = 1}^T\sum_{i = 1}^M \left\|\boldsymbol{x}_{t, i} \right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)^{-1}}^2}\\
& \leq 2\sqrt{T\left(L + 4\lambda M\right)} \cdot \sqrt{\sum_{i = 1}^M \sum_{t = 1}^T \left\|\boldsymbol{x}_{t, i} \right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)^{-1}}^2} \end{align}
where the first inequality is due to $\sum_{i=1}^{M} \left\langle \boldsymbol{\theta}_i, \boldsymbol{x}^*_{t, i} \right\rangle \leq \left\langle \tilde{\boldsymbol{\theta}}_{t,i}, \boldsymbol{x}_{t, i} \right\rangle$ from the optimistic choice of $\tilde{\boldsymbol{\theta}}_{t,i}$ and $\boldsymbol{x}_{t, i}$. By Lemma 11 of \citet{abbasi2011improved}, as long as $\lambda \geq 1$ we have \begin{align}
\label{eqn: linear bandit potential function}
\sum_{t = 1}^T \left\|\boldsymbol{x}_{t, i} \right\|_{\boldsymbol{\tilde{V}}_{t - 1, i}(\lambda')^{-1}}^2 \leq 2 \log \frac{\det(\boldsymbol{\tilde{V}}_{T, i}(\lambda'))}{\det(\lambda' \boldsymbol{I}_d)} \leq 2 d\log\left(1 + \frac{T}{\lambda d}\right) \end{align} $$ $$
Therefore, we can finally bound the regret by choosing $\lambda = 1$ \begin{align}
\mathrm{Reg}(T) & \leq 2\sqrt{T(L + 4M)} \cdot \sqrt{ \sum_{i = 1}^M \sum_{t = 1}^T \left\|\boldsymbol{x}_{t, i} \right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda')^{-1}}^2} \\ & \leq 2 \sqrt{T \left(L + 4M\right)} \cdot \sqrt{Md \log \left(1 + \frac{T}{d}\right)} \\ & = \tilde{O}\left(M\sqrt{dkT} + d\sqrt{kMT} \right). \end{align} \end{proof}
\subsubsection{Proof of Lemma~\ref{lemma: self-normalized bound}} \label{sec: proof of self-normalized bound}
The proof of Lemma~\ref{lemma: self-normalized bound} follows the similar idea of Theorem~1 in \citet{abbasi2011improved}. We consider the $\sigma$-algebra $F_t = \sigma\left(\{\boldsymbol{x}_{1,i}\}_{i=1}^{M},\{\boldsymbol{x}_{2,i}\}_{i=1}^{M},\cdots, \{\boldsymbol{x}_{t+1,i}\}_{i=1}^{M}, \{\eta_{1,i}\}_{i=1}^{M}, \{\eta_{2,i}\}_{i=1}^{M}, \cdots, \{\eta_{t,i}\}_{i=1}^{M}\right)$, then $\{\boldsymbol{x}_{t,i}\}_{i=1}^{M}$ is $F_{t-1}$-measurable, and $\{\eta_{t,i}\}_{i=1}^{M}$ is $F_t$-measurable.
Define $\bar{\boldsymbol{x}}_{t,i} = \boldsymbol{U}^{\top} \boldsymbol{x}_{t,i}$ and $\boldsymbol{S}_{t,i} = \sum_{n=1}^{t} \bar{\boldsymbol{U}}^{\top}\boldsymbol{x}_{t,i}\eta_{t,i}$. Let \begin{align}
M_t(\boldsymbol{Q}) = \exp\left(\sum_{n=1}^{t} \sum_{i=1}^{M}\left[\eta_{t,i} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle - \frac{1}{2} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle^2\right]\right), \quad \boldsymbol{Q} = [\boldsymbol{q}_1,\cdots, \boldsymbol{q}_M] \in \mathbb{R}^{2k\times M} \end{align}
\begin{lemma} \label{lemma: supermatingale} Let $\tau$ be a stopping time w.r.t the filtration $\{F_t\}_{t=0}^{\infty}$. Then $M_t(\boldsymbol{Q})$ is almost surely well-defined and $\mathbb{E}[M_t(\boldsymbol{Q})] \leq 1$. \end{lemma} \begin{proof} Let $D_t(\boldsymbol{Q}) = \exp \left(\sum_{i=1}^{M}\left[\eta_{t,i} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle - \frac{1}{2} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle^2\right]\right)$. By the sub-Gaussianity of $\eta_{t,i}$, we have \begin{align}
\mathbb{E}\left[\exp\left(\left[\eta_{t,i} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle - \frac{1}{2} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle^2\right]\right)] \mid F_{t-1} \right] \leq 1. \end{align}
Then we have $\mathbb{E}\left[D_t(\boldsymbol{Q}) \mid F_{t-1}\right] \leq 1$. Further, \begin{align}
\mathbb{E}\left[M_t(\boldsymbol{Q}) \mid F_{t-1}\right] &= \mathbb{E}\left[M_1(\boldsymbol{Q}) \cdots D_{t-1}(\boldsymbol{Q}) D_{t}(\boldsymbol{Q})\mid F_{t-1}\right] \\
&= D_1(\boldsymbol{Q}) \cdots D_{t-1}(\boldsymbol{Q})\mathbb{E}\left[ D_{t}(\boldsymbol{Q})\mid F_{t-1}\right] \leq M_{t-1}(\boldsymbol{Q}) \end{align} This shows that $\{M_{t}(\boldsymbol{Q})\}_{t=0}^{\infty}$ is a supermartingale and $\mathbb{E}\left[M_t(\boldsymbol{Q})\right] \leq 1$.
Following the same argument of Lemma~8 in \citet{abbasi2011improved}, we show that $M_{\tau}(\boldsymbol{Q})$ is almost surely well-defined. By the convergence theorem for nonnegative supermartingales, $M_{\infty}(\boldsymbol{Q}) = \lim_{t \rightarrow \infty} M_t(\boldsymbol{Q})$ is almost surely well-defined. Therefore, $M_{\tau}(\boldsymbol{Q})$ is indeed well-defined independently of whether $\tau < \infty$ or not. Let $W_t(\boldsymbol{Q}) = M_{\min\{\tau, t\}}(\boldsymbol{Q})$ be a stopped version of $(M_t(\boldsymbol(Q)))_t$. By Fatou's Lemma, $\mathbb{E}[M_{\tau}(\boldsymbol{Q})] = \mathbf{E}\left[\liminf _{t \rightarrow \infty} W_{t}(\boldsymbol{Q})\right] \leq \liminf _{t \rightarrow \infty} \mathbf{E}\left[W_{t}(\boldsymbol{Q})\right] \leq 1$. This shows that $\mathbb{E}[M_{\tau}(\boldsymbol{Q})] \leq 1$. \end{proof}
The next lemma uses the ``method of mixtures'' technique to bound $\sum_{i=1}^{M} \|\boldsymbol{S}_{t,i}\|^2_{\bar{\boldsymbol{V}}_{t,i}^{-1}(\lambda)}$.
\begin{lemma} \label{lemma: self-normalized bound with stopping time} Let $\tau$ be a stopping time w.r.t the filtration $\{F_t\}_{t=0}^{\infty}$. Then, for $\delta > 0$, with probability $1-\delta$, \begin{align}
\sum_{i=1}^{M} \|\boldsymbol{S}_{\tau,i}\|^2_{\bar{\boldsymbol{V}}_{\tau,i}^{-1}(\lambda)} \leq 2 \log\left(\frac{\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{\tau,i})^{1/2} \operatorname{det}(\lambda \boldsymbol{I})^{-1/2}\right)}{\delta}\right). \end{align}
\end{lemma} \begin{proof} For each $i \in [M]$, let $\boldsymbol{\Lambda}_i$ be a $\mathbb{R}^{2k}$ Gaussian random variable which is independent of all the other random variables and whose covariance is $\lambda^{-1}\boldsymbol{I}$. Define $M_t = \mathbb{E}\left[M_t([\boldsymbol{\Lambda}_1,\cdots, \boldsymbol{\Lambda}_M])\mid F_{\infty}\right]$. We still have $\mathbb{E}[M_{\tau}] = \mathbb{E}[\mathbb{E}[M_t([\boldsymbol{\Lambda}_1,\cdots, \boldsymbol{\Lambda}_M]) \mid \{\boldsymbol{\Lambda}_i\}_{i=1}^{M}]] \leq 1$.
Now we calculate $M_t$. Define $M_{t,i}(\boldsymbol{q}_i) \stackrel{\text { def }}{=} \exp\left(\sum_{n=1}^{t} \left[\eta_{t,i} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle - \frac{1}{2} \left \langle \boldsymbol{q}_i,\bar{\boldsymbol{x}}_{t,i} \right \rangle^2\right]\right)$, then we have $M_t = \mathbb{E}\left[\prod_{i=1}^{M} M_{t,i}(\boldsymbol{\Lambda_i})\mid F_{\infty}\right] = \prod_{i=1}^{M} \mathbb{E}\left[ M_{t,i}(\boldsymbol{\Lambda_i})\mid F_{\infty}\right]$, where the second equality is due to the fact that $\{M_{t,i}(\boldsymbol{\Lambda}_i)\}_{i=1}^{M}$ are relatively independent given $F_{\infty}$. We only need to calculate $\mathbb{E}\left[ M_{t,i}(\boldsymbol{\Lambda}_i)\mid F_{\infty}\right]$ for each $i \in [M]$.
Following the proof of Lemma~9 in \citet{abbasi2011improved}, we know that \begin{align}
\mathbb{E}\left[ M_{t,i}(\boldsymbol{\Lambda}_i)\mid F_{\infty}\right] = \left(\frac{\operatorname{det}(\lambda \boldsymbol{I})}{\operatorname{det}(\bar{\boldsymbol{V}}_{t,i})}\right)^{1/2} \exp \left(\frac{1}{2} \|\boldsymbol{S}_{t,i}\|^2_{\bar{\boldsymbol{V}}_{t,i}^{-1}(\lambda)}\right). \end{align}
Then we have \begin{align}
M_t = \prod_{i=1}^{M}\left(\left(\frac{\operatorname{det}(\lambda \boldsymbol{I})}{\operatorname{det}(\bar{\boldsymbol{V}}_{t,i})}\right)^{1/2} \right) \exp \left(\frac{1}{2} \sum_{i=1}^{M}\|\boldsymbol{S}_{t,i}\|^2_{\bar{\boldsymbol{V}}_{t,i}^{-1}(\lambda)}\right). \end{align}
Since $\mathbb{E}[M_{\tau}] \leq 1$, we have \begin{align*}
\lefteqn{\operatorname{Pr}\left[ \sum_{i=1}^{M} \|\boldsymbol{S}_{\tau,i}\|^2_{\bar{\boldsymbol{V}}_{\tau,i}^{-1}(\lambda)} > 2 \log\left(\frac{\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{\tau,i})^{1/2} \operatorname{det}(\lambda \boldsymbol{I})^{-1/2}\right)}{\delta}\right)\right]} \\
= & \operatorname{Pr}\left[\frac{\exp\left(\frac{1}{2}\sum_{i=1}^{M} \|\boldsymbol{S}_{\tau,i}\|^2_{\bar{\boldsymbol{V}}_{\tau,i}^{-1}(\lambda)}\right)}{\delta^{-1} \left(\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{t,i})^{1/2} \operatorname{det}(\lambda \boldsymbol{I})^{-1/2}\right)\right)} > 1\right] \\
\leq & \mathbb{E}\left[\frac{\exp\left(\sum_{i=1}^{M} \|\boldsymbol{S}_{\tau,i}\|^2_{\bar{\boldsymbol{V}}_{\tau,i}^{-1}(\lambda)}\right)}{\delta^{-1} \left(\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{\tau,i})^{1/2} \operatorname{det}(\lambda \boldsymbol{I})^{-1/2}\right)\right)} \right] \\
= &\mathbb{E}[M_{\tau}] \delta \leq \delta. \end{align*} \end{proof}
\begin{proof} (Proof of Lemma~\ref{lemma: self-normalized bound}) The only remaining issue is the stopping time construction. Define the bad event \begin{align}
B_t(\delta) \stackrel{\text{def}}{=} \left\{ \omega \in \Omega: \sum_{i=1}^{M} \|\boldsymbol{S}_{t,i}\|^2_{\bar{\boldsymbol{V}}_{t,i}^{-1}(\lambda)} > 2 \log\left(\frac{\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{t,i})^{1/2} \operatorname{det}(\lambda \boldsymbol{I})^{-1/2}\right)}{\delta}\right)\right\} \end{align}
Consider the stopping time $\tau(\omega) = \min\{t \geq 0: \omega \in B_t(\delta)\}$, we have $\bigcup_{t \geq 0} B_{t}(\delta)=\{\omega: \tau(\omega)<\infty\}$.
By lemma~\ref{lemma: self-normalized bound with stopping time}, we have \begin{align}
\operatorname{Pr}\left[\bigcup_{t \geq 0} B_{t}(\delta)\right] =&\operatorname{Pr}[\tau<\infty] \\
= & \operatorname{Pr} \left[ \sum_{i=1}^{M} \|\boldsymbol{S}_{\tau,i}\|^2_{\bar{\boldsymbol{V}}_{\tau,i}^{-1}(\lambda)} > 2 \log\left(\frac{\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{\tau,i})^{1/2} \operatorname{det}(\lambda \boldsymbol{I})^{-1/2}\right)}{\delta}\right), \tau \leq \infty\right] \\
\leq & \operatorname{Pr} \left[ \sum_{i=1}^{M} \|\boldsymbol{S}_{\tau,i}\|^2_{\bar{\boldsymbol{V}}_{\tau,i}^{-1}(\lambda)} > 2 \log\left(\frac{\prod_{i=1}^{M}\left(\operatorname{det}(\bar{\boldsymbol{V}}_{\tau,i})^{1/2} \operatorname{det}(\lambda \boldsymbol{I})^{-1/2}\right)}{\delta}\right)\right] \\
\leq &\delta. \end{align} \end{proof}
\subsubsection{Proof of Theorem~\ref{thm: regret for misspecified linear bandits}} \label{sec: proof for misspecified linear bandits}
\begin{proof}
The proof follows the same idea of that for Theorem~\ref{thm: main theory for linear bandits}. The only difference is that, in our setting, we have $y_{t,i} = \boldsymbol{x}_{t,i}^{\top}\boldsymbol{B}\boldsymbol{w}_i + \eta_{t,i} + \Delta_{t,i}$, where $\boldsymbol{\theta}_{i} = \boldsymbol{B} \boldsymbol{w}_i $ is the best approximator for task $i \in [M]$ such that $\left|\mathbb{E}\left[y_{i} \mid \boldsymbol{x}_{i}\right]-\left\langle\boldsymbol{x}_{i}, \dot{\boldsymbol{B}} \dot{\boldsymbol{w}}_{i}\right\rangle\right| \leq \zeta$, and $\|\Delta_{t,i}\| \leq \zeta$. Define $\boldsymbol{\Delta}_{t, i}=\left[\Delta_{1, i}, \Delta_{2, i}, \cdots, \Delta_{t, i}\right]$. Similarly, by the optimality of $\hat{\boldsymbol{B}}_t$ and $\hat{\boldsymbol{W}}_t = [\hat{\boldsymbol{w}}_{t,1}, \cdots, \hat{\boldsymbol{w}}_{t,M}]$, we know that $\sum_{i=1}^{M} \left\|\boldsymbol{y}_{t-1,i}-\boldsymbol{X}_{t-1,i}^{\top} \hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}\right\|^{2}_2 \leq \sum_{i=1}^{M} \left\|\boldsymbol{y}_{t-1,i}-\boldsymbol{X}_{t-1,i}^{\top} \boldsymbol{B} \boldsymbol{w}_{i}\right\|^{2}$. Since $\boldsymbol{y}_{t-1, i}=\boldsymbol{X}_{t-1, i}^{\top} \boldsymbol{B} \boldsymbol{w}_{i}+\boldsymbol{\eta}_{t-1, i}+\boldsymbol{\Delta}_{t, i}$, thus we have \begin{align}
\label{eqn: optimality of hTheta misspecified setting}
& \sum_{i=1}^{M} \left\|\boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right)\right\|^2 \\
\leq &2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) + 2 \sum_{i=1}^{M} \boldsymbol{\Delta}_{t-1, i}^{\top} \boldsymbol{X}_{t-1, i}^{\top}\left(\hat{\boldsymbol{B}}_{t} \hat{\boldsymbol{w}}_{t, i}-\boldsymbol{B} \boldsymbol{w}_{i}\right) \\
\leq & 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) + 2 \sum_{i=1}^{M} \left\|\boldsymbol{X}_{t-1,i}\boldsymbol{\Delta}_{t-1,i}\right\|_{\tilde{\boldsymbol{V}}_{t-1,i}^{-1}(\lambda)} \left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right\|_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \\
\leq & 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) + 2 \sum_{i=1}^{M} \sqrt{T}\zeta \left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right\|_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \\
\label{eqn: misspecified bandit, confidence proof}
\leq & 2\sum_{i=1}^{M} \boldsymbol{\eta}_{t-1,i}^{\top} \boldsymbol{X}_{t-1,i}^{\top} \left(\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right) + 2 \sqrt{MT} \zeta \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i} - \boldsymbol{B} \boldsymbol{w}_{i}\right\|^2_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}} \end{align} The third inequality follows from Projection Bound (Lemma 8) in \citet{zanette2020learning}. The first term of Eqn~\ref{eqn: misspecified bandit, confidence proof} shares the same form of Eqn~\ref{eqn: optimality of hTheta}. Following the same proof idea of Lemma~\ref{lemma: confidence set for linear bandits}, we know that with probability $1-\delta$, \begin{align}
&\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)} \\
\leq &\left(2\sqrt{Mk + 8kd\log(kMT/\delta)}+ 2\sqrt{MT}\zeta\right) \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}} + 4M + 4 \sqrt{\log(4MT/\delta)} \end{align}
Solving for $\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_t \hat{\boldsymbol{w}}_{t,i}-\boldsymbol{B}\boldsymbol{w}_{i}\right\|^{2}_{\tilde{\boldsymbol{V}}_{t-1,i}(\lambda)}$, we know that the true parameter $\boldsymbol{B}\boldsymbol{W}$ is always contained in the confidence set, i.e. \begin{align} \label{eqn: confidence bound for misspecified linear bandit}
\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_{t} \hat{\boldsymbol{w}}_{t, i}-\boldsymbol{B} \boldsymbol{w}_{i}\right\|_{\tilde{\boldsymbol{V}}_{t-1, i}(\lambda)}^{2} \leq L', \end{align} where $L' = 2L + 32 MT \zeta^2$.
Thus we have \begin{align} \mathrm{Reg}(T) & = \sum_{t = 1}^T \sum_{i = 1}^M \left(y^*_{t,i}- y_{t,i}\right) \\ & \leq 2MT\zeta + \sum_{t = 1}^T \sum_{i = 1}^M \left\langle \boldsymbol{\theta}_i, \boldsymbol{x}^*_{t, i} - \boldsymbol{x}_{t, i} \right\rangle \\ & \leq 2MT\zeta +\sum_{t = 1}^T \sum_{i = 1}^M \left\langle \tilde{\boldsymbol{\theta}}_{t, i} - \boldsymbol{\theta}_i, \boldsymbol{x}_{t, i} \right\rangle \\ & = 2MT\zeta +\sum_{t = 1}^T \sum_{i = 1}^M \left\langle \tilde{\boldsymbol{\theta}}_{t, i} - \hat{\boldsymbol{\theta}}_{t , i} + \hat{\boldsymbol{\theta}}_{t , i} - \boldsymbol{\theta}_i, \boldsymbol{x}_{t, i} \right\rangle\\
& \leq 2MT\zeta +\sum_{t = 1}^T \sum_{i = 1}^M \left(\left\| \tilde{\boldsymbol{\theta}}_{t, i} - \hat{\boldsymbol{\theta}}_{t , i}\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)} + \left\|\hat{\boldsymbol{\theta}}_{t , i} - \boldsymbol{\theta}_i\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)}\right) \left\|\boldsymbol{x}_{t, i} \right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)^{-1}} \\
& \leq 2MT\zeta +\left(\sqrt{\sum_{t = 1}^T\sum_{i = 1}^M \left\| \tilde{\boldsymbol{\theta}}_{t, i} - \hat{\boldsymbol{\theta}}_{t , i}\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)}^2} + \sqrt{\sum_{i = 1}^M \left\|\hat{\boldsymbol{\theta}}_{t , i} - \boldsymbol{\theta}_i\right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)}^2}\right) \cdot \sqrt{\sum_{t = 1}^T\sum_{i = 1}^M \left\|\boldsymbol{x}_{t, i} \right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)^{-1}}^2}\\
& \leq 2MT\zeta +2\sqrt{T\left(L' + 4\lambda M\right)} \cdot \sqrt{\sum_{i = 1}^M \sum_{t = 1}^T \left\|\boldsymbol{x}_{t, i} \right\|_{\tilde{\boldsymbol{V}}_{t - 1, i}(\lambda)^{-1}}^2} \\ & \leq 2MT\zeta +2\sqrt{T\left(L' + 4\lambda M\right)} \sqrt{Md \log(1+\frac{T}{d})} \\ & = \tilde{O}(M\sqrt{dkT} + d\sqrt{kMT} + MT\sqrt{d}\zeta), \end{align} where the second inequality is due to $\sum_{i=1}^{M} \left\langle \boldsymbol{\theta}_i, \boldsymbol{x}^*_{t, i} \right\rangle \leq \left\langle \tilde{\boldsymbol{\theta}}_{t,i}, \boldsymbol{x}_{t, i} \right\rangle$ from the optimistic choice of $\tilde{\boldsymbol{\theta}}_{t,i}$ and $\boldsymbol{x}_{t, i}$. The third inequality is due to Eqn~\ref{eqn: confidence bound for misspecified linear bandit}. The last inequality is from Eqn~\ref{eqn: linear bandit potential function}. \end{proof}
\subsubsection{Proof of Theorem~\ref{thm: lower bound for linear bandits}} \label{sec: proof of lower bound for linear bandits}
Since our setting is strictly harder than the setting of multi-task linear bandit with infinite arms in \citet{yang2020provable}, we can prove the following lemma directly from their Theorem 4 by reduction.
\begin{lemma}
\label{lemma: lower bound for linear bandit original terms} Under the setting of Theorem~\ref{thm: lower bound for linear bandits}, the regret of any Algorithm~$\mathcal{A}$ is lower bounded by $ \Omega\left(Mk\sqrt{T}+ d\sqrt{kMT}\right).$ \end{lemma}
In order to prove Theorem~\ref{thm: lower bound for linear bandits}, we only need to show that the following lemma is true.
\begin{lemma} \label{lemma: lower bound for linear bandits, approximation error term} Under the setting of Theorem~\ref{thm: lower bound for linear bandits}, the regret of any Algorithm~$\mathcal{A}$ is lower bounded by $ \Omega\left(MT\sqrt{d}\zeta\right).$ \end{lemma}
\begin{proof}
(Proof of Lemma~\ref{lemma: lower bound for linear bandits, approximation error term})
To prove Lemma~\ref{lemma: lower bound for linear bandits, approximation error term}, we leverage the lower bound for misspecified linear bandits in the single-task setting. We restate the following lemma from the previous literature with a slight modification of notations.
\begin{lemma} \label{lemma: related lemma for linear bandit lower bound} (Proposition 6 in \citet{zanette2020learning}). There exists a feature map $\phi:\mathcal{A} \rightarrow \mathbb{R}^d$ that defines a misspecified linear bandits class $\mathcal{M}$ such that every bandit instance in that class has reward response: \begin{align*}
\mu_a = \boldsymbol{\phi}_a^{\top} \boldsymbol{\theta} + z_a \end{align*} for any action $a$ (Here $z_a \in [0,\zeta]$ is the deviation from linearity and $\mu_a \in [0,1]$) and such that the expected regret of any algorithm on at least a member of the class up to round $T$ is $\Omega(\sqrt{d}\zeta T)$. \end{lemma}
Suppose $M$ can be exactly divided by $k$, we construct the following instances to prove lemma~\ref{lemma: lower bound for linear bandits, approximation error term}. We divide $M$ tasks into $k$ groups. Each group shares the same parameter $\theta_i$. To be more specific, we let $\boldsymbol{w}_1 = \boldsymbol{w}_2 = \cdots = \boldsymbol{w}_{M/k} = \boldsymbol{e}_1$, $\boldsymbol{w}_{M/k+1} = \boldsymbol{w}_{M/k+2} = \cdots = \boldsymbol{w}_{2M/k} = \boldsymbol{e}_2$, $\cdots$, $\boldsymbol{w}_{(k-1)M/k+1} = \boldsymbol{w}_{(k-1)M/k+2} = \cdots = \boldsymbol{w}_{M} = \boldsymbol{e}_k$. Under this construction, the parameters $\theta_i$ for these tasks are exactly the same in each group, but relatively independent among different groups. That is to say, the expected regret lower bound is at least the summation of the regret lower bounds in all $k$ groups.
Now we consider the regret lower bound for group $j \in [k]$. Since the parameters are shared in the same group, the regret of running an algorithm for $M/k$ tasks with $T$ steps each is at least the regret of running an algorithm for single-task linear bandit with $M/k \cdot T$ steps. By Lemma~\ref{lemma: related lemma for linear bandit lower bound}, the regret for single-task linear bandit with $MT/k$ steps is at least $\Omega(\sqrt{d} \zeta MT/k)$. Summing over all $k$ groups, we can prove that the regret lower bound is $\Omega(\sqrt{d} \zeta MT)$. \end{proof}
Combining Lemma~\ref{lemma: lower bound for linear bandit original terms} and Lemma~\ref{lemma: lower bound for linear bandits, approximation error term}, we complete the proof of Theorem~\ref{thm: lower bound for linear bandits}.
\subsection{Proof of Theorem~\ref{theorem:linear_rl_regret_bound}} \label{sec: omiited proof in linea_rl}
\subsubsection{Definitions and First Step Analysis} \label{appendix_linear_rl:definitions}
Before presenting the proof of theorem \ref{theorem:linear_rl_regret_bound}, we will make a first step analysis on the low-rank least-square estimator in equation \ref{formula:linear_rl_least_square}.
For any $\left\{Q_{h+1}^i\right\}_{i=1}^M \in \caQ_{h+1}$, there exists $\left\{\dot{\boldsymbol{\theta}}_{h}^i\left(Q_{h+1}^i\right) \right\}_{i=1}^M \in \Theta_h$ that
\begin{align} \label{formula:best_approximators} \Delta_h^i \left(Q_{h+1}^i\right)(s, a) = \caT_h^i \left(Q_{h+1}^i\right)(s, a) - \boldsymbol{\phi}(s, a)^\top \dot{\boldsymbol{\theta}}_{h}^i\left(Q_{h+1}^i\right) \end{align}
where the approximation error $\left\|\Delta_h^i \left(Q_{h+1}^i\right)\right\|_{\infty} \leq \caI$ is small for each $i \in [M]$. We also use $\dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i \left(Q_{h+1}^i\right)$ in place of $\dot{\boldsymbol{\theta}}_{h}^i\left(Q_{h+1}^i\right)$ in the following sections since we can write $\dot{\boldsymbol{\theta}}_{h}^i$ as $\dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i$ according to Assumption \ref{assumptions:low_rank_small_ibe}.
In the multi-task low-rank least-square regression (equation \ref{formula:linear_rl_least_square}), we are actually trying to recover $\dot{\bm{\theta}}_h^i$. However, due to the noise and representation error (i.e. the inherent Bellman error), we can only obtain an approximate solution $\hat{\bm{\theta}}_h^i = \hat{\bm{B}}_h \hat{\bm{w}}_h^i$ (see the global optimization problem in Definition \ref{formula:linear_rl_global_optimization}).
\begin{align} \left(\hat{\boldsymbol{\theta}}_h^1, ..., \hat{\boldsymbol{\theta}}_h^M\right) & = \hat{\boldsymbol{B}}_h \begin{bmatrix} \hat{\boldsymbol{w}}_h^1 & \hat{\boldsymbol{w}}_h^2 & \cdots & \hat{\boldsymbol{w}}_h^M \end{bmatrix}\\
& = \argmin_{\left\|\boldsymbol{B}_h \boldsymbol{w}_h^i\right\|_2 \leq D} \sum_{i=1}^M \sum_{j=1}^{t-1} \left(\boldsymbol{\phi}\left(s^i_{h j}, a^i_{hj}\right)^{\top} \boldsymbol{B}_h \boldsymbol{w}_h^i -R\left(s_{h j}^i, a_{h j}^i\right)-\max_a Q_{h+1}^i\left(s_{h+1, j}^i\right)\right)^{2} \\
\label{formula:linear_rl_low_rank_regression}
& = \argmin_{\left\|\boldsymbol{B}_h \boldsymbol{w}_h^i\right\|_2 \leq D} \sum_{i=1}^M \sum_{j=1}^{t-1} \left(\boldsymbol{\phi}\left(s^i_{h j}, a^i_{hj}\right)^{\top} \boldsymbol{B}_h \boldsymbol{w}_h^i - \caT_h^i \left(Q_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right) - z_{hj}^i\left(Q_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right)\right)^{2} \end{align}
where $z_{hj}^i\left(Q_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right) \defeq R\left(s_{h j}^i, a_{h j}^i\right) + \max_a Q_{h+1}^i\left(s_{h+1,j}^i, a\right) - \caT_h^i\left(Q_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right)$.
Define $\boldsymbol{\Phi}_{ht}^i \in \dbR^{(t-1) \times d}$ to be the collection of linear features up to episode $t - 1$ in task $i$, i.e. the $j$-th row of $\boldsymbol{\Phi}_{ht}^i$ is $\boldsymbol{\phi}\left(s^i_{h j}, a^i_{hj}\right)^\top$. Let $\boldsymbol{Y}_{ht}^i \in \dbR^{t-1}$ be a vector whose $j$-th dimension is $\caT_h^i \left(Q_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right) + z_{hj}^i\left(Q_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right)$. Then the objective in (\ref{formula:linear_rl_low_rank_regression}) can be written as
\begin{align}
\argmin_{\left\|\boldsymbol{B}_h \boldsymbol{w}_h^i\right\|_2 \leq D} \sum_{i=1}^M \left\|\boldsymbol{\Phi}_{ht}^i \boldsymbol{B}_h \boldsymbol{w}_h^i - \boldsymbol{Y}_{ht}^i \right\|_2^2 \end{align}
Therefore, we have
\begin{align}
\sum_{i=1}^M \left\|\boldsymbol{\Phi}_{ht}^i \hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - Y_{ht}^i \right\|_2^2 \leq \sum_{i=1}^M \left\|\boldsymbol{\Phi}_{ht}^i \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \boldsymbol{Y}_{ht}^i \right\|_2^2 \end{align}
which implies
\begin{align}
& \sum_{i=1}^M \left\|\boldsymbol{\Phi}_{ht}^i \hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \boldsymbol{\Phi}_{ht}^i \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) \right\|_2^2 \\ \label{formula:linear_rl_noise_error} & \leq 2 \sum_{i=1}^M \left(\bm{\Delta}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \\ \label{formula:linear_rl_projection_error} & + 2 \sum_{i=1}^M \left(\bm{z}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \end{align}
where $\bm{\Delta}_{ht}^i \defeq \begin{bmatrix} \Delta_{h1}^i\left(Q_{h+1}^i\right)\left(s^i_{h 1}, a^i_{h1}\right) & \Delta_{h2}^i\left(Q_{h+1}^i\right)\left(s^i_{h 2}, a^i_{h2}\right) & \cdots & \Delta_{h, t - 1}^i\left(Q_{h+1}^i\right)\left(s^i_{h, t - 1}, a^i_{h, t-1}\right) \end{bmatrix} \in \dbR^{t-1}$, and $\bm{z}_{ht}^i \defeq \begin{bmatrix} z_{h1}^i\left(Q_{h+1}^i\right)\left(s^i_{h 1}, a^i_{h1}\right) & \cdots & z_{h, t - 1}^i\left(Q_{h+1}^i\right)\left(s^i_{h, t - 1}, a^i_{h, t-1}\right) \end{bmatrix} \in \dbR^{t-1}$.
In the next sections we will show how to bound \ref{formula:linear_rl_noise_error} and \ref{formula:linear_rl_projection_error}.
\subsubsection{Failure Event} \label{appendix_linear_rl:failure_event}
Define the failure event at step $h$ in episode $t$ as
\begin{definition}[Failure Event] \label{definitions:failure_event_ht} \begin{align} E_{ht} & \defeq I\Big[\exists \left\{Q_{h+1}^i\right\}_{i=1}^M \in \caQ_{h+1} \quad \sum_{i=1}^M \left(\bm{z}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) > \\
& F_h^1 \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)}} + F_h^2 \Big] \end{align} \end{definition}
where $F_h^1$ and $F_h^2$ will be specified later.
We have the following lemma to bound the probability of $E_{ht}$.
\begin{lemma} \label{lemma:linear_rl_failure_event_ht} For the input parameter $\delta > 0$, there exists $F_h^1$ and $F_h^2$ such that \begin{align} \dbP\left(\bigcup_{t=1}^T \bigcup_{h=1}^H E_{ht}\right) \leq \frac{\delta}{2} \end{align} \end{lemma}
\begin{proof}
According to Lemma A.5 of \citet{du2020few}, there exists an $\epsilon$-net $\caE_{h+1}^o$ over $\caO^{d \times k}$ (with regards to the Frobenius norm) such that $\left|\caE_{h+1}^o\right| \leq (6\sqrt{k}/\epsilon^\prime)^{kd} $. Moreover, there exists an $\epsilon$-net $\caE^b_{h+1}$ over $\caB^{k}$ that $\left|\caE^b_{h+1}\right| \leq (1 + 2 / \epsilon^\prime)^{k}$. We can show a corresponding $\epsilon$-net $\caE^{\text{mul}}_{h+1} \defeq \caE_{h+1}^o \times \left(\caE^b_{h+1}\right)^M$ over $\Theta_{h+1}$.
For any $\left(Q_{h+1}^1\left(\boldsymbol{B}_{h+1}\boldsymbol{w}^1_{h+1}\right), \cdots, Q_{h+1}^M\left(\boldsymbol{B}_{h+1}\boldsymbol{w}_{h+1}^M\right)\right) \in \caQ_{h+1}$, there exists $\bar{\boldsymbol{B}}_{h+1} \in \caE_{h+1}^o$ and $\left(\bar{\boldsymbol{w}}^1_{h+1}, \cdots, \bar{\boldsymbol{w}}_{h+1}^M\right) \in \left(\caE^b_{h+1}\right)^M$ such that
$$\left\|\boldsymbol{B}_{h+1} - \bar{\boldsymbol{B}}_{h+1}\right\|_F \leq \epsilon^\prime \quad \left\|\boldsymbol{w}_{h+1}^i - \bar{\boldsymbol{w}}_{h+1}^i\right\|_2 \leq \epsilon^\prime, \forall i \in [M]$$
Therefore,
$$\left\|\boldsymbol{B}_{h+1} \boldsymbol{w}^i_{h+1} - \bar{\boldsymbol{B}}_{h+1} \bar{\boldsymbol{w}}^i_{h+1}\right\|_2 \leq 2\epsilon^\prime, \forall i \in [M]$$
Define $\bar{Q}_{h+1}^i$ to be $Q_{h+1}^i\left(\bar{\boldsymbol{B}}_{h+1} \bar{\boldsymbol{w}}^i_{h+1}\right)$, and let $\bm{\bar{z}}_{ht}^i \defeq \begin{bmatrix} z_{h1}^i\left(\bar{Q}_{h+1}^i\right)\left(s^i_{h 1}, a^i_{h1}\right) & \cdots & z_{h, t - 1}^i\left(\bar{Q}_{h+1}^i\right)\left(s^i_{h, t - 1}, a^i_{h, t-1}\right) \end{bmatrix} \in \dbR^{t-1}$, then
\begin{align} & \sum_{i=1}^M \left(\bm{z}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \\ & = \sum_{i=1}^M \left(\bm{\bar{z}}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \\ & + \sum_{i=1}^M \left(\bm{z}_{ht}^i - \bm{\bar{z}}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \end{align}
For fixed $\left\{\bar{\boldsymbol{B}}_{h+1} \bar{\boldsymbol{w}}^i_{h+1}\right\}_{i=1}^M \in \caE^{\text{mul}}_{h+1}$, $z_{h, j}^i\left(\bar{Q}_{h+1}^i\right)\left(s^i_{h,j}, a^i_{h, j}\right)$ is zero-mean 1-subgaussian conditioned on $\caF_{h, j}$ according to Assumption \ref{assumptions:linear_rl_regularity}. Thus, we can use exactly the same argument as in Lemma \ref{lemma: confidence set for linear bandits} to show that
\begin{align} & \sum_{i=1}^M \left(\bm{\bar{z}}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \\
& \leq \sqrt{Mk + 5kd \log(kMT) + 2\log(1 / \delta^{\prime})} \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)}} \\ &+ \sqrt{2\log(2MT/\delta^{\prime})} + \sqrt{k + 3kd\log(kMT) + \log(1 / \delta^{\prime})} \end{align}
by setting $\epsilon = \frac{1}{kM^2T^2}$, $\delta_1 = \frac{\delta^{\prime}}{2 \left(\frac{6\sqrt{2k}}{\epsilon}\right)^{2kd}}$, and $\delta_2 = \frac{\delta^{\prime}}{2MT}$ in equation \ref{formula:linear_rl_cross_reference}. Thus, we have that with probability $1 - \delta^{\prime}$ the inequality above holds for any $h \in [H], t \in [T]$. Take $\delta = \frac{\delta^{\prime}}{2\left|\caE^{\text{mul}}_{h+1}\right|}$, by union bound we know the above ineqaulity holds with probability $1 - \delta$ for any $\left\{\bar{\boldsymbol{B}}_{h+1} \bar{\boldsymbol{w}}^i_{h+1}\right\}_{i=1}^M \in \caE^{\text{mul}}_{h+1}$ and any $h \in [H], t \in [T]$.
Since it holds that $\left|Q_{h+1}^i\left(\boldsymbol{B}_{h+1}\boldsymbol{w}_{h+1}^i\right)(s, a) - Q_{h+1}^i\left(\bar{\boldsymbol{B}}_{h+1}\bar{\boldsymbol{w}}_{h+1}^i\right)(s, a)\right| \leq 2\epsilon^\prime$ for any $(s, a) \in \caS \times \caA, i \in [M]$, we have
\begin{align}
\left|z_{hj}^i\left(\bar{Q}_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right) - z_{hj}^i\left(Q_{h+1}^i\right)\left(s^i_{h j}, a^i_{hj}\right)\right| \leq 8\epsilon^\prime \end{align}
Then we have
\begin{align} & \sum_{i=1}^M \left(\bm{z}_{ht}^i - \bm{\bar{z}}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{B}_h \hat{w}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \\
& \leq \sum_{i=1}^M \left\|\left(\boldsymbol{\Phi}_{ht}^i\right)^\top \left(\bm{z}_{ht}^i - \bm{\bar{z}}_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}} \left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \\
& \leq 8\epsilon^\prime \sqrt{T} \sum_{i=1}^M \left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \\
& \leq 8\epsilon^\prime \sqrt{MT} \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)}} \end{align}
for arbitrary $\{Q_{h+1}^i\}$ and any $h \in [H], t \in [T]$. The second inequality follows from the Projection Bound (Lemma 8) in \citet{zanette2020learning}.
Take $\epsilon^\prime = 1 / 8\sqrt{MT}$, we finally finish the proof by setting
\begin{align} F_h^1 & \defeq \sqrt{9kd \log(kMT) + 5Mk \log(MT) + 2\log(2 / \delta)} \\ F_h^2 & \defeq \sqrt{4kd \log(kMT) + 5Mk \log(MT) + 2\log(2 / \delta)} \\ & + \sqrt{k + 5kd\log(kMT) + 2Mk \log(MT) + \log(2 / \delta)} \end{align}
\end{proof}
In the next sections we assume the failure event $\bigcup_{t=1}^T \bigcup_{h=1}^H E_{ht}$ won't happen.
\subsubsection{Bellman Error} \label{appendix_linear_rl:bellman_error}
Outside the failure event, we can bound the estimation error of the least-square regression \ref{formula:linear_rl_least_square}.
\begin{lemma} \label{lemma:linear_rl_least_square_error} For any episode $t \in [T]$ and step $h \in [H]$, any $\left\{Q_{h+1}^i\right\}_{i=1}^M \in \caQ_{h+1}$, we have
\begin{align}
\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \leq \alpha_{ht} \defeq \left(2\sqrt{MT} \caI + 2 F_{h}^1 + \sqrt{2F_h^2 + 4MD^2 \lambda}\right)^2 \end{align} \end{lemma}
\begin{proof} Recall that
\begin{align}
& \sum_{i=1}^M \left\|\boldsymbol{\Phi}_{ht}^i \hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \boldsymbol{\Phi}_{ht}^i \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) \right\|_2^2 \\ & \leq 2 \sum_{i=1}^M \left(\bm{\Delta}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \\ & + 2 \sum_{i=1}^M \left(\bm{z}_{ht}^i\right)^\top \boldsymbol{\Phi}_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \end{align}
For the first term, we have
\begin{align} & \sum_{i=1}^M \left(\bm{\Delta}_{ht}^i\right)^\top \Phi_{ht}^i \left(\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right) \\
& \leq \sum_{i=1}^M \left\|\left(\boldsymbol{\Phi}_{ht}^i\right)^\top \bm{\Delta}_{ht}^i\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}} \left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \\
& \leq \sqrt{T} \caI \sum_{i=1}^M \left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \\
& \leq \sqrt{MT} \caI \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)}} \end{align}
The second inequality follows from the Projection Bound (Lemma 8) in \citet{zanette2020learning}, and the last inequality is due to Cauchy-Schwarz.
Outside the failure event, we have
\begin{align}
& \sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \\
& \leq \sum_{i=1}^M \left\|\boldsymbol{\Phi}_{ht}^i \hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \boldsymbol{\Phi}_{ht}^i \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) \right\|_2^2 + 4MD^2 \lambda \\
& \leq \left(2\sqrt{MT} \caI + 2 F_{h}^1\right) \sqrt{\sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)}} + 2F_{h}^2 + 4MD^2 \lambda \end{align}
which implies
\begin{align}
& \sum_{i=1}^{M}\left\|\hat{\boldsymbol{B}}_h \hat{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right) - \dot{\boldsymbol{B}}_h \dot{\boldsymbol{w}}_h^i\left(Q_{h+1}^i\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \\ & \leq \left(2\sqrt{MT} \caI + 2 F_{h}^1\right)^2 + 2F_h^2 + 4MD^2 \lambda + \left(2\sqrt{MT} \caI + 2 F_{h}^1\right) \sqrt{2F_h^2 + 4MD^2 \lambda}\\ & \leq \left(2\sqrt{MT} \caI + 2 F_{h}^1 + \sqrt{2F_h^2 + 4MD^2 \lambda}\right)^2 \end{align} \end{proof}
\begin{lemma}[Bound on Bellman Error] \label{lemma:linear_rl_bellman_error} Outside the failure event, for any feasible solution $\left\{Q_h^i\left(\bar{\theta}_{h}^i\right)\right\}_{h}^i $ ($\bar{Q}_{h}^i$ for short, with a little abuse of notations) of the global optimization procedure in definition \ref{formula:linear_rl_global_optimization}, for any $(s, a) \in \caS \times \caA$, any $h \in [H]$, $t \in [T]$
\begin{align}
\sum_{i=1}^M \left|\bar{Q}_{h}^i (s, a) - \caT_h^i \bar{Q}_{h+1}^i (s, a)\right| \leq M \caI + 2 \sqrt{\alpha_{ht} \cdot \sum_{i=1}^M \left\|
\boldsymbol{\phi}(s, a)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} \end{align}
\end{lemma}
\begin{proof} \begin{align}
\sum_{i=1}^M \left|\bar{Q}_{h}^i (s, a) - \caT_h^i \bar{Q}_{h+1}^i (s, a)\right| & = \sum_{i=1}^M \left|\boldsymbol{\phi}(s, a)^\top \bar{\boldsymbol{\theta}}_{h}^i - \boldsymbol{\phi}(s, a)^\top \dot{\boldsymbol{\theta}}_{h}^i\left(\bar{Q}_{h+1}^i\right) - \Delta_h^i \left(\bar{Q}_{h+1}^i\right)(s, a)\right| \\
& \leq M\caI + \sum_{i=1}^M \left|\boldsymbol{\phi}(s, a)^\top \bar{\boldsymbol{\theta}}_{h}^i - \boldsymbol{\phi}(s, a)^\top \dot{\boldsymbol{\theta}}_{h}^i\left(\bar{Q}_{h+1}^i\right)\right|\\
& \leq M\caI + \sum_{i=1}^M \left(\left|\boldsymbol{\phi}(s, a)^\top \dot{\boldsymbol{\theta}}_h^i\left(\bar{Q}_{h+1}^i\right) - \boldsymbol{\phi}(s, a)^\top \hat{\boldsymbol{\theta}}_h^i\right| + \left|\boldsymbol{\phi}(s, a)^\top \hat{\boldsymbol{\theta}}_h^i - \boldsymbol{\phi}(s, a)^\top \bar{\boldsymbol{\theta}}_h^i\right|\right) \\
& \leq M\caI + \sum_{i=1}^M \left\|
\boldsymbol{\phi}(s, a)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}} \left(\left\|\dot{\boldsymbol{\theta}}_h^i\left(\bar{Q}_{h+1}^i\right) - \hat{\boldsymbol{\theta}}_h^i\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} + \left\|\hat{\boldsymbol{\theta}}_h^i - \bar{\boldsymbol{\theta}}_h^i\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)}\right) \\
& \leq M \caI + 2 \sqrt{\alpha_{ht} \cdot \sum_{i=1}^M \left\|
\boldsymbol{\phi}(s, a)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} \end{align}
The first equality is due to the definition of $\Delta_h^i \left(\bar{Q}_{h+1}^i\right)(s, a)$. The last inequality is due to lemma \ref{lemma:linear_rl_least_square_error}. \end{proof}
\subsubsection{Optimism} \label{appendix_linear_rl:optimism}
We can find the "best" approximator of optimal value functions in our function class recursively defined as
\begin{align} \label{formula:linear_rl_best_approximator}
\left(\boldsymbol{\theta}_h^{1*}, \boldsymbol{\theta}_h^{2*}, \cdots, \boldsymbol{\theta}_h^{M*}\right) \defeq \argmin_{\left(\boldsymbol{\theta}_h^1, \boldsymbol{\theta}_h^2, \cdots, \boldsymbol{\theta}_h^M\right) \in \Theta_h} \sup_{s, a, i} \left|\left(\boldsymbol{\phi}(s, a)^\top \boldsymbol{\theta}_h^i - \caT_h^i Q_{h+1}^i \left(\boldsymbol{\theta}_{h+1}^{i*}\right)\right)(s, a)\right| \end{align}
with $\boldsymbol{\theta}_{H+1}^{i*}=\boldsymbol{0}, \forall i \in [M]$
For the accuracy of this best approximator, we have
\begin{lemma} \label{lemma:linear_rl_accuracy_best_approximator} For any $h \in [H]$, $$
\sup _{(s, a) \in \caS \times \caA, i \in [M]}\left|Q_{h}^{i*}(s, a)-\boldsymbol{\phi}(s, a)^{\top} \boldsymbol{\theta}_{h}^{*}\right| \leq (H-h+1) \mathcal{I} $$ \end{lemma}
where $Q_h^{i*}$ is the optimal value function for task $i$. This lemma is derived directly from Lemma 6 in \citet{zanette2020learning}.
For our solution of the problem in Definition \ref{formula:linear_rl_global_optimization} in episode $t$, we have the following lemma:
\begin{lemma} \label{lemma:linear_rl_optimism} $\left\{\left(\boldsymbol{\theta}_h^{1*}, \boldsymbol{\theta}_h^{2*}, \cdots, \boldsymbol{\theta}_h^{M*}\right)\right\}_{h=1}^H$ is a feasible solution of the problem in Definition \ref{formula:linear_rl_global_optimization}. Moreover, denote the solution of the problem in Definition \ref{formula:linear_rl_global_optimization} in episode $t$ by $\bar{\boldsymbol{\theta}}_{ht}^i$ for $h \in [H], i \in [M]$, it holds that
\begin{align} \sum_{i=1}^M V_{1}^i\left(\bar{\boldsymbol{\theta}}_{1 t}^i\right)\left(s_{1t}^i\right) \geq \sum_{i=1}^M V_{1}^{i*}\left(s_{1t}^i\right) - MH\caI \end{align} \end{lemma}
\begin{proof} First we show that $\left\{\left(\boldsymbol{\theta}_h^{1*}, \boldsymbol{\theta}_h^{2*}, \cdots, \boldsymbol{\theta}_h^{M*}\right)\right\}_{h=1}^H$ is a feasible solution. We can construct $\left\{\bar{\boldsymbol{\xi}}_h^i\right\}_{i=1}^M$ so that $\bar{\boldsymbol{\theta}}_h^i = \boldsymbol{\theta}_h^{i*}$ and no other constraints are violated. We use an inductive construction, and the base case when $\bar{\bm{\theta}}_{H+1}^i = \bm{\theta}_{H+1}^{i*} = 0$ is trivial.
Now suppose we have $\left\{\bar{\boldsymbol{\xi}}_y^i\right\}_{i=1}^M$ for $y = h + 1, ..., H$ such that $\bar{\bm{\theta}}_{y}^i = \bm{\theta}_{y}^{i*}$ for $y = h + 1, ..., H$ and $i \in [M]$, we show we can find $\left\{\bar{\boldsymbol{\xi}}_h^i\right\}_{i=1}^M$ so $\bar{\boldsymbol{\theta}}_{h}^i = \boldsymbol{\theta}_{h}^{i*}$ for $i \in [M]$, and no constraints are violated. From the definition of $\boldsymbol{\theta}_{h}^{i*}$ we can set (with a little abuse of notations)
\begin{align} \dot{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right) = \boldsymbol{\theta}_{h}^{i*} \end{align}
According to lemma \ref{lemma:linear_rl_least_square_error} we have
\begin{align}
\sum_{i=1}^{M}\left\|\hat{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right) - \dot{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right)\right\|^{2}_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)} \leq \alpha_{ht} \end{align}
Therefore, set $\bar{\boldsymbol{\xi}}_h^i = \dot{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right) - \hat{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right)$, then
\begin{align} \bar{\boldsymbol{\theta}}_h^i & = \hat{\boldsymbol{\theta}}_h^i\left(\bar{\boldsymbol{\theta}}_{h+1}^{i}\right) + \bar{\boldsymbol{\xi}}_h^i \\ & = \hat{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right) + \dot{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right) - \hat{\boldsymbol{\theta}}_h^i\left(\boldsymbol{\theta}_{h+1}^{i*}\right) \\ & = \boldsymbol{\theta}_h^{i*} \end{align}
Finally, we can verify $\left(\bar{\boldsymbol{\theta}}_h^1,...,\bar{\boldsymbol{\theta}}_h^M\right) \in \Theta_h$ from $\left(\boldsymbol{\theta}_h^{1*}, \cdots, \boldsymbol{\theta}_h^{M*}\right) \in \Theta_h$.
Since $\bar{\boldsymbol{\theta}}_{1t}^i$ is the optimal solution, we can finish the proof by showing
\begin{align} \sum_{i=1}^M V_{1}^i\left(\bar{\boldsymbol{\theta}}_{1 t}^i\right)\left(s_{1t}^i\right) & = \sum_{i=1}^M \max_a \boldsymbol{\phi}\left(s_{1t}^i, a\right)^\top \bar{\boldsymbol{\theta}}_{1 t}^i \\ & \geq \sum_{i=1}^M \max_a \boldsymbol{\phi}\left(s_{1t}^i, a\right)^\top \boldsymbol{\theta}_{1}^{i*} \qquad \text{(since $\theta_{1}^{i*}$ is the feasible solution)} \\ & \geq \sum_{i=1}^M \boldsymbol{\phi}\left(s_{1t}^i, \pi_1^{i*}\left(s_{1t}^i\right)\right)^\top \boldsymbol{\theta}_{1}^{i*} \\ & \geq \sum_{i=1}^M Q_h^{i*}\left(s_{1t}^i, \pi_1^{i*}\left(s_{1t}^i\right)\right) - MH \caI \qquad \text{(by Lemma \ref{lemma:linear_rl_accuracy_best_approximator})}\\ & \geq \sum_{i=1}^M V_h^{i*}\left(s_{1t}^i\right) - MH \caI \end{align}
\end{proof}
\subsubsection{Regret Bound} \label{appendix_linear_rl:regret_bound}
We are ready to present the proof of our regret bound.
From Lemma \ref{lemma:linear_rl_failure_event_ht} we know that the failure event $\bigcup_{t=1}^T \bigcup_{h=1}^H E_{ht}$ happens with probability at most $\delta / 2$, so we assume it does not happen. Then we can decompose the regret as
\begin{align} \text{Reg}(T) & = \sum_{t=1}^T \sum_{i=1}^M \left(V_1^{i*} - V_1^{\pi_{t}^i} \right)\left(s_{1t}^i\right) \\ & = \sum_{t=1}^T \sum_{i=1}^M \left(V_1^{i*} - V_{1}^i\left(\bar{\boldsymbol{\theta}}_{1 t}^i\right) \right)\left(s_{1t}^i\right) + \sum_{t=1}^T \sum_{i=1}^M \left(V_{1}^i\left(\bar{\boldsymbol{\theta}}_{1 t}^i\right) - V_1^{\pi_{t}^i} \right)\left(s_{1t}^i\right) \\ & \leq \sum_{t=1}^T \sum_{i=1}^M \left(V_{1}^i\left(\bar{\boldsymbol{\theta}}_{1 t}^i\right) - V_1^{\pi_{t}^i} \right)\left(s_{1t}^i\right) + MHT \caI \qquad \text{(by Lemma \ref{lemma:linear_rl_optimism})} \end{align}
Let $a_{ht}^i = \pi_{t}^i\left(s_{ht}^i\right)$, and denote $Q_{h}^i\left(\bar{\boldsymbol{\theta}}_{ht}^i\right)$($V_{h}^i\left(\bar{\boldsymbol{\theta}}_{ht}^i\right)$) by $\bar{Q}_{ht}^i$($\bar{V}_{ht}^i$) for short, we have
\begin{align} \sum_{i=1}^M \left(\bar{V}_{ht}^i - V_h^{\pi_{t}^i} \right)\left(s_{ht}^i\right) & = \sum_{i=1}^M \left(\bar{Q}_{ht}^i - Q_h^{\pi_{t}^i} \right)\left(s_{ht}^i, a_{ht}^i\right) \\ & = \sum_{i=1}^M \left(\bar{Q}_{ht}^i - \caT_h^i \bar{Q}_{h+1,t}^i \right)\left(s_{ht}^i, a_{ht}^i\right) + \sum_{i=1}^M \left(\caT_h^i \bar{Q}_{h+1,t}^i - Q_h^{\pi_{t}^i} \right)\left(s_{ht}^i, a_{ht}^i\right) \\
& \leq M\caI + 2 \sqrt{\alpha_{ht} \cdot \sum_{i=1}^M \left\|
\boldsymbol{\phi}\left(s_{ht}^i, a_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} + \sum_{i=1}^M \dbE_{s^\prime \sim p_h^i\left(s_{ht}^i, a_{ht}^i\right)}\left[\left(\bar{V}_{h+1,t}^i - V_{h+1}^{\pi_{t}^i} \right)\left(s^\prime\right)\right] \\ \label{formula:linear_rl_recursive_regret}
& \leq \sum_{i=1}^M \left(\bar{V}_{h+1,t}^i - V_{h+1}^{\pi_{t}^i} \right)\left(s_{h+1,t}^i\right) + M\caI + 2 \sqrt{\alpha_{ht} \cdot \sum_{i=1}^M \left\|
\boldsymbol{\phi}\left(s_{ht}^i, a_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} + \sum_{i=1}^M \zeta_{ht}^i \end{align}
where $\zeta_{ht}^i$ is a martingale difference with regards to the filtration $\caF_{h,t}$ defined as
\begin{align} \zeta_{ht}^i \defeq \left(\bar{V}_{h+1,t}^i - V_{h+1}^{\pi_{t}^i} \right)\left(s_{h+1,t}^i\right) - \dbE_{s^\prime \sim p_h^i\left(s_{ht}^i, a_{ht}^i\right)}\left[\left(\bar{V}_{h+1,t}^i - V_{h+1}^{\pi_{t}^i} \right)\left(s^\prime\right)\right] \end{align}
According to assumption \ref{assumptions:linear_rl_regularity} we know $\left|\zeta_{ht}^i\right| \leq 4$, so we can apply Azuma-Hoeffding's inequality that with probability $1 - \delta/2$ for any $t \in [T]$ and $i \in [M]$
\begin{align} \label{formula:linear_rl_martingale} \sum_{j=1}^t \zeta_{ht}^i \leq 4\sqrt{2t \ln\left(\frac{2T}{\delta}\right)} \end{align}
By applying inequality \ref{formula:linear_rl_recursive_regret} recursively, we can bound the regret as
\begin{align} \text{Reg}(T) & \leq \sum_{t=1}^T \sum_{i=1}^M \left(\bar{V}_{1t}^i - V_1^{\pi_{t}^i} \right)\left(s_{1t}^i\right) + MHT \caI \\
& \leq 2MHT\caI + \sum_{t=1}^T \sum_{h=1}^H 2 \sqrt{\alpha_{ht} \cdot \sum_{i=1}^M \left\|
\boldsymbol{\phi}\left(s_{ht}^i, a_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} + \sum_{i=1}^M \sum_{h=1}^H \sum_{t=1}^T \zeta_{ht}^i \end{align}
The last inequality is due to $\bar{V}_{H+1}^i(s) = \max_a \boldsymbol{\phi}(s, a)^\top \bar{\boldsymbol{\theta}}_{H+1,t}^i = 0, V_{H+1}^{\pi_t^i}(s) = 0$.
The Lemma 11 of \citet{abbasi2011improved} gives that for any $i \in [M]$ and $h \in [H]$
\begin{align}
\sum_{t=1}^T \left\|
\boldsymbol{\phi}\left(s_{ht}^i, a_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2 = \tilde{O}\left(d\right) \end{align}
Moreover, by the definition of $\alpha_{ht}$ (see Lemma \ref{lemma:linear_rl_least_square_error}) we know that for any $h \in [H]$ and $t \in [T]$
\begin{align} \alpha_{ht} = \tilde{O}\left(Mk + kd + MT\caI^2\right) \end{align}
Take all of above we can show the final regret bound.
\begin{align}
\text{Reg}(T) & \leq 2MHT\caI + \sum_{t=1}^T \sum_{h=1}^H 2 \sqrt{\alpha_{ht} \cdot \sum_{i=1}^M \left\|
\boldsymbol{\phi}\left(s_{ht}^i, a_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} + \sum_{i=1}^M \sum_{h=1}^H \sum_{t=1}^T \zeta_{ht}^i \\
& = \tilde{O}\left(MHT\caI + \tilde{O}\left(\sqrt{Mk + kd + MT\caI^2}\right) \sum_{h=1}^H \sum_{t=1}^T \sqrt{\sum_{i=1}^M \left\|
\boldsymbol{\phi}\left(s_{ht}^i, a_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} + MH\sqrt{T}\right) \\
& = \tilde{O}\left(MHT\caI + \tilde{O}\left(\sqrt{Mk + kd + MT\caI^2}\right) \sum_{h=1}^H \sqrt{T} \cdot \sqrt{\sum_{t=1}^T \sum_{i=1}^M \left\|
\boldsymbol{\phi}\left(s_{ht}^i, a_{ht}^i\right)\right\|_{\tilde{\boldsymbol{V}}_{ht}^i(\lambda)^{-1}}^2} + MH\sqrt{T}\right) \\ & = \tilde{O}\left(MHT\caI + \tilde{O}\left(\tilde{O}\left(\sqrt{Mk + kd + MT\caI^2}\right) \cdot H\sqrt{MTd}\right) + MH\sqrt{T}\right) \\ & = \tilde{O}\left(HM\sqrt{dkT} + Hd\sqrt{MkT} + HMT\sqrt{d}\caI\right) \end{align}
\subsection{Proof of Theorem~\ref{theorem:linear_rl_lower_bound}} \label{sec: proof of the lower bound for rl}
To prove the lower bound for multi-task RL, our idea is to connect the lower bound for the multi-task learning problem to the lower bound in the single-task LSVI setting~\citep{zanette2020learning}. in the paper of~\citet{zanette2020learning}, they assumed the feature dimension $d$ can be varied among different steps, which is denoted as $d_h$ for step $h$. They proved the lower bound for linear RL in this setting is $\Omega\left(\sum_{h=1}^{H}d_h \sqrt{T} +\sum_{h=1}^{H} \sqrt{d_h}\mathcal{I}T\right)$. However, this lower bound is derived by the hard instance with $d_1 = \sum_{h=2}^{H} d_h$. If we set $d_1 = d_2 = \cdots = d_H = d$ like our setting, we can only obtain the lower bound of $\Omega\left(d \sqrt{T} + \sqrt{d}\mathcal{I}T\right)$ following their proof idea. In fact, the dependence on $H$ in this lower bound can be further improved. In order to obtain a tighter lower bound, we consider the lower bound for single-task misspecified linear MDP. This setting can be proved to be strictly simpler than the LSVI setting following the idea of Proposition 3 in~\citet{zanette2020learning}. The lower bound for misspecified linear MDP can thus be applied to LSVI setting.
\subsubsection{Lower Bounds for single-task RL}
This subsection focus on the lower bound for misspecifed linear MDP setting, in which the transition kernel and the reward function are assume to be approximately linear.
\begin{assumption} \label{assumption: misspecified linear MDP} (Assumption B in~\citet{jin2020provably}) For any $\zeta \leq 1$, we say that $\operatorname{MDP}(\mathcal{S}, \mathcal{A},p,r,H)$ is a $\zeta$-approximate linear MDP with a feature map $\boldsymbol{\phi}: \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}^d$, if for any $h \in [H]$, there exist $d$ unknown measures $\boldsymbol{\theta}_h = (\theta_h^{(1)},\cdots, \theta_h^{(d)})$ over $\mathcal{S}$ and an unknown vector $\boldsymbol{\nu}_h \in \mathbb{R}^d$ such that for any $(s,a) \in \mathcal{S}\times \mathcal{A}$, we have \begin{align}
& \|p_h(\cdot|s,a) - \left\langle\boldsymbol{\phi}(s,a),\boldsymbol{\theta}_h(\cdot)\right\rangle\|_{\operatorname{TV}} \leq \zeta \\
& |r_h(s,a) - \left\langle\boldsymbol{\phi}(s,a),\boldsymbol{\nu}_h\right\rangle| \leq \zeta \end{align} \end{assumption}
For regularity, we assume that Assumption~\ref{assumptions:linear_rl_regularity} still holds, and we also assume that there exists a constant $D$ such that $\|\boldsymbol{\theta}_h(s)\| \leq D$ for all $s \in \mathcal{S}, h \in [H]$, $\|\boldsymbol{\nu}_h\| \leq D$ for all $h \in [H]$. $D \geq 4$ suffices in our hard instance construction.
For misspecifed linear MDP, we can prove the following lower bound.
\begin{proposition} \label{thm: lower bound for linear MDP} Suppose $T \geq \frac{d^2H}{4}$, $d \geq 10$, $H \geq 10$ and $\zeta \leq \frac{1}{4H}$, there exist a $\zeta$-approximate linear MDP class such that the expected regret of any algorithm on at least a member of the MDP class is at least $\Omega\left(d\sqrt{HT} + HT\mathcal{I} \sqrt{d}\right)$. \end{proposition}
To prove the lower bound, our basic idea is to connect the problem to $\frac{H}{2}$ linear bandit problems. Similar hard instance construction has been used in~\citet{zhou2020nearly,zhou2020provably}. In our construction, the state space $\mathcal{S}$ consists of $H+2$ states, which is denoted as $x_1, x_2, \cdots, x_{H+2}$. The agent starts the episode in state $x_1$. In $x_h$, it can either transits to $x_{h+1}$ or $x_{H+2}$ with certain transition probability. If the agent enters $x_{H+2}$, it will stay in this state in the remaining steps, i.e. $x_{H+2}$ is an absorbing state. For each state, there are $2^{d-4}$ actions and $\mathcal{A} = \{-1,1\}^{d-4}$. Suppose the agent takes action $\boldsymbol{a} \in \{-1,1\}^{d-4}$ in state $s_h$, the transition probability to state $s_{h+1}$ and $s_{H+2}$ is $1-\zeta_h(\boldsymbol{a}) -\delta-\boldsymbol{\mu}_h^{\top}\boldsymbol{a}$ and $\delta+ \zeta_h(\boldsymbol{a})+\boldsymbol{\mu}_h^{\top}\boldsymbol{a}$ respectively. Here $|\zeta_h(\boldsymbol{a})| \leq \zeta$ denotes the approximation error of linear representation, $\delta= 1/H$ and $\boldsymbol{\mu}_h \in \{-\Delta, \Delta\}^{d-4}$ with $\Delta = \sqrt{\delta/T}/(4\sqrt{2}) $ so that the probability is well-defined. The reward can only be obtained in $x_{H+2}$, with $r_h(x_{H+2,a}) = 1/H$ for any $h,a$. We assume the reward to be deterministic.
We can check that this construction satisfies Assumption~\ref{assumption: misspecified linear MDP} with $\boldsymbol{\phi}$ and $\boldsymbol{\theta}$ defined in the following way:
$$ \boldsymbol{\phi}(s,\boldsymbol{a})=\left\{ \begin{aligned} &\left(0, \alpha, \alpha\delta,0, \beta \boldsymbol{a}^{\top}\right)^{\top} & \quad s = x_1, x_2, \cdots, x_H\\ &\left(0, 0,0, \alpha, \boldsymbol{0}^{\top} \right)^{\top} & \quad s = x_{H+1}\\ &\left(\alpha, 0, 0,\alpha, \boldsymbol{0}^{\top}\right)^{\top} & \quad s = x_{H+2} \end{aligned} \right. $$
$$ \boldsymbol{\theta}_h(s')=\left\{ \begin{aligned} &\left(0,\frac{1}{\alpha}, -\frac{1}{\alpha},0, -\frac{\boldsymbol{\mu}^{\top}_h}{\beta}\right)^{\top} & \quad s' = x_{h+1}\\ &\left(0, 0, \frac{1}{\alpha},\frac{1}{\alpha}, \frac{\boldsymbol{\mu}^{\top}_h}{\beta} \right)^{\top} & \quad s = x_{H+2}\\ &\boldsymbol{0} & \quad \operatorname{otherwise} \end{aligned} \right. $$
$\boldsymbol{\nu}_h$ is defined to be $(\frac{1}{H\alpha},\boldsymbol{0}^{\top})^{\top}$, and $\alpha = \sqrt{1/(2+\Delta(d-4))}$, $\beta = \sqrt{\Delta/(2+\Delta(d-4))}$. Note that $\|\boldsymbol{\phi}(s,a)\| \leq 1$, $\|\boldsymbol{\theta}_h(s')\| \leq D$ and $\|\boldsymbol{\nu}_h\| \leq D$ hold for any $s,a,s',h$ when $T \geq d^2H/4 $.
Since the rewarding state is only $x_{H+2}$, the optimal strategy in state $x_{h}$ ($h \leq H$) is to take an action that maximizes the probability of entering $x_{H+2}$, i.e., to maximize $\boldsymbol{\mu}_h^{\top} \boldsymbol{a} + \zeta(\boldsymbol{a})$. That is to say, we can regard the problem of finding the optimal action in state $s_h$ and step $h$ as finding the optimal arm for a $d-4$-dimensional approximately (misspecified) linear bandits problem. Thanks to the choice of $\delta$ such that $(1-\delta)^{H/2}$ is a constant, there is sufficiently high probability of entering state $x_h$ for any $h \leq H/2$. Therefore, we can show that this problem is harder than solving $H/2$ misspecified linear bandit problems. This following lemma characterizes this intuition. The lemma follows the same idea of Lemma~C.7 in~\citet{zhou2020nearly}, though our setting is more difficult since we consider misspecified case. \begin{lemma} \label{lemma: value decomposition for linear MDP lower bound} Suppose $H \geq 10$, $d \geq 10$ and $(d-4)\Delta \leq \frac{1}{2H}$. We define $r^{b}_h(\boldsymbol{a}) = \boldsymbol{\mu}^{\top}\boldsymbol{a} + \zeta_h(\boldsymbol{a}) $, which can be regarded as the corresponding reward for the equivalent linear bandit problem in step $h$. Fix $\boldsymbol{\mu} \in (\{-\Delta,\Delta\}^{d-4})^{H}$. Fix a possibly history dependent policy $\pi$. Letting $V^{\star}$ and $V^{\pi}$ be the optimal value function and the value function of policy $\pi$ respectively, we have \begin{align}
V_1^{\star}(s_1) - V^{\pi}_1(s_1) \geq 0.02 \sum_{h=1}^{H/2} \left(\max_{\boldsymbol{a} \in \mathcal{A}} r^{b}_h(\boldsymbol{a}) - \sum_{\boldsymbol{a} \in \mathcal{A}} \pi_h(\boldsymbol{a}|s_h) r^{b}_h(\boldsymbol{a})\right) \end{align} \end{lemma} \begin{proof} Note that the only rewarding state is $x_{H+2}$ with $r_h(x_{H+2},\boldsymbol{a}) = \frac{1}{H}$. Therefore, the value function of a certain policy $\pi$ can be calculated as: \begin{align}
V_1^{\pi}(x_1) = \sum_{h=1}^{H-1} \frac{H-h}{H} \mathbb{P}(N_h|\pi) \end{align}
where $N_h$ denotes the event of visiting state $x_h$ in step $h$ and then transits to $x_{H+2}$, i.e. $N_h = \{s_{h} = x_h, s_{h+1} = x_{H+2}\}$. Suppose $\omega^{\pi}_h = \sum_{\boldsymbol{a} \in \mathcal{A}} \pi_h(\boldsymbol{a}|s_h) r^{b}_h(\boldsymbol{a})$ and $\omega^{\star}_h = \max_{\boldsymbol{a} \in \mathcal{A}} r^{b}_h(\boldsymbol{a})$. By the law of total probability and the Markov property, we have \begin{align}
\mathbb{P}(N_h|\pi) = (\delta+\omega^{\pi}_h) \prod_{j=1}^{h-1} (1-\delta - \omega^{\pi}_h) \end{align}
Thus we have \begin{align}
V_1^{\pi}(x_1) = \sum_{h=1}^{H-1} \frac{H-h}{H} (\delta+\omega^{\pi}_h) \prod_{j=1}^{h-1} (1-\delta - \omega^{\pi}_h) \end{align}
Similarly, for the value function of the optimal policy, we have \begin{align}
V_1^{\star}(x_1) = \sum_{h=1}^{H-1} \frac{H-h}{H} (\delta+\omega^{\star}_h) \prod_{j=1}^{h-1} (1-\delta - \omega^{\star}_h) \end{align}
Define $S_i = \sum_{h=i}^{H-1}\frac{H-h}{H} (\delta+\omega^{\pi}_h)\prod_{j=i}^{h-1}(1-\delta - \omega_h^{\pi})$ and $T_i = \sum_{h=i}^{H-1}\frac{H-h}{H} (\delta+\omega^{\star}_h)\prod_{j=i}^{h-1}(1-\delta - \omega_h^{\star})$. Then we have $V^{\star}_1(x_1) - V^{\pi}_1(x_1) = T_1 - S_1$. Notice that \begin{align}
S_i &= \frac{H-i}{H} (\omega^{\pi}_i + \delta) + S_{i+1} (1-\omega_i^{\pi} -\delta) \\
T_i &= \frac{H-i}{H} (\omega^{\star}_i + \delta) + T_{i+1} (1-\omega_i^{\star} -\delta) \end{align}
Thus we have \begin{align}
T_i - S_i = \left(\frac{H-i}{H} - T_{i+1}\right)\left(\omega^{\star}_i - \omega^{\pi}_i\right) + (T_{i+1} - S_{i+1})(1-\omega^{\pi}_i - \delta) \end{align}
By induction, we get \begin{align}
\label{eqn: Vstar - Vpi lower bound}
T_1 - S_1 = \sum_{h=1}^{H-1} (\omega^{\star}_i - \omega^{\pi}_i)(\frac{H-h}{H}-T_{h+1}) \prod_{j=1}^{h-1} (1-\omega_j^{\pi}-\delta) \end{align}
Since the reward is non-negative and only occurs in $x_{H+2}$, we know that $V^{\star}_1(x_1) \geq V^{\star}_{2}(x_2) \geq \cdots \geq V^{\star}_1(x_H)$. Thus we have $T_h \leq T_1 = V^{\star}_1(x_1) \leq \sum_{h=1}^{H} \mathbb{P}(N_h|\pi^{\star})$. If $N_h$ doesn't happen for any $h \in [H]$, then the agent must enter $x_{H+1}$. The probability of this event has the following form: \begin{align}
\mathbb{P}\left(\neg \left(\cup_{h\in [H]}N_h|\pi^{\star}\right)\right) = & 1- \prod_{h=1}^{H} \mathbb{P}(N_h|\pi^{\star}) \\
= & \prod_{h\in[H]}\left(1-\delta- \omega^{\star}_{h}\right) \\
\geq & \prod_{h\in [H]} (1-\frac{1}{H} + \frac{1}{2H}) \\
= & (1-\frac{1}{2H})^{H} \\
\geq & 0.6 \end{align}
The fist inequality is due to $\delta = \frac{2}{H}$ and $|\omega_h^{\star}| \leq \frac{1}{H}$. The above discussion indicates that $T_h \leq 0.4$, thus $\frac{H-h}{H}-T_{h+1} \geq 0.1$ for $h \leq H/2$. Similarly, $\prod_{j=1}^{h-1} (1-\omega_j^{\pi}-\delta) \geq (1-\frac{3}{2H})^{H-1} \geq 0.2$. Combining with Eqn~\ref{eqn: Vstar - Vpi lower bound}, we have \begin{align}
T_1 -S_1 \geq 0.02 \sum_{h=1}^{\frac{H}{2}} (\omega_h^{\star} - \omega_h^{\pi}) = 0.02 \sum_{h=1}^{H/2} \left(\max_{\boldsymbol{a} \in \mathcal{A}} r^{b}_h(\boldsymbol{a}) - \sum_{\boldsymbol{a} \in \mathcal{A}} \pi_h(\boldsymbol{a}|s_h) r^{b}_h(\boldsymbol{a})\right) \end{align}
Combining with the definition of $T_1$ and $S_1$, we can prove the lemma. \end{proof}
After proving Lemma~\ref{lemma: value decomposition for linear MDP lower bound}, we are ready to prove Proposition~\ref{thm: lower bound for linear MDP}. \begin{proof} (proof of Proposition~\ref{thm: lower bound for linear MDP}) By Lemma~\ref{lemma: value decomposition for linear MDP lower bound}, we know that we can decompose the sub-optimality gap of a policy $\pi$ in the following way: \begin{align}
V_1^{\star}(s_1) - V^{\pi}_1(s_1) \geq 0.02 \sum_{h=1}^{H/2} \left(\max_{\boldsymbol{a} \in \mathcal{A}} r^{b}_h(\boldsymbol{a}) - \sum_{\boldsymbol{a} \in \mathcal{A}} \pi_h(\boldsymbol{a}|s_h) r^{b}_h(\boldsymbol{a})\right) \end{align} where $r^{b}_h(\boldsymbol{a}) = \boldsymbol{\mu}^{\top}\boldsymbol{a} + \zeta_h(\boldsymbol{a}) $, which can be regarded as a reward function for misspecified linear bandit. To prove Theorem~\ref{thm: lower bound for linear MDP}, the only remaining problem is to derive the lower bound for misspecified linear bandits. We directly apply the following two lower bounds for linear bandits. \begin{lemma} \label{lemma: lower bound linear bandit for RL 1} (Lemma~C.8 in~\citet{zhou2020nearly}) Fix a positive real $0 < \delta \leq 1/3$, and positive integers $T,d$ and assume that $T \geq d^2/(2\delta)$ and consider the linear bandit problem $\mathcal{L}_{\boldsymbol{\mu}}$ parametrized with a parameter vector $\boldsymbol{\mu} \in \{-\Delta,\Delta\}^{d}$ and action set $\mathcal{A} = \{-1,1\}^{d}$ so that the reward distribution for taking action $\boldsymbol{a} \in \mathcal{A}$ is a Bernoulli distribution $B(\delta+(\boldsymbol{\mu}^{\star})^{\top}\boldsymbol{a})$. Then for any bandit algorithm $\mathcal{B}$, there exists a $\mu^{*} \in \{-\Delta, \Delta\}^d$ such that the expected pseudo-regret of $\mathcal{B}$ over $T$ steps on bandit $\mathcal{L}_{\boldsymbol{\mu}^{\star}}$ is lower bounded by $\frac{d\sqrt{T\delta}}{8\sqrt{2}}$. \end{lemma}
\begin{lemma} \label{lemma: lower bound linear bandit for RL 2} (Proposition 6 in~\citet{zanette2020learning}) There exists a feature map $\phi:\mathcal{A} \rightarrow \mathbb{R}^d$ that defines a misspecified linear bandits class $\mathcal{M}$ such that every bandit instance in that class has reward response: \begin{align*}
\mu_a = \phi_a^{\top} \theta + z_a \end{align*} for any action $a$ (Here $z_a \in [0,\zeta]$ is the deviation from linearity and $\mu_a \in [0,1]$) and such that the expected regret of any algorithm on at least a member of the class up to round $T$ is $\Omega(\sqrt{d}\zeta T)$. \end{lemma}
Lemma~\ref{lemma: lower bound linear bandit for RL 1} is used to prove the lower bound for linear mixture MDPs in~\citet{zhou2020nearly}, which states that the lower bound for linear bandits with approximation error $\zeta = 0$, while Lemma~\ref{lemma: lower bound linear bandit for RL 2} mainly consider the influence of $\zeta$ to the lower bound. Combining these two lemmas, the regret lower bound for misspecifid linear bandit is $\Omega(\max(d\sqrt{T\delta},\sqrt{d}\zeta T)) = \Omega (d\sqrt{T\delta} + \sqrt{d}\zeta T)$. Since here our problem can reduce from $H/2$ misspecified linear bandit, we know that the regret lower bound is $\Omega (Hd\sqrt{T\delta} + H\sqrt{d}\zeta T) = \Omega (d\sqrt{HT} + H\sqrt{d}\zeta T)$ \end{proof}
Now we obtain the regret lower bound for misspecified linear MDP. We can prove the corresponding lower bound for the LSVI setting ~\citet{zanette2020learning} since LSVI setting is strictly harder than linear MDP setting. The following lemma states this relation between two settings. \begin{lemma} \label{lemma: reduction from linear MDP to LSVI} If an MDP$(\mathcal{S},\mathcal{A}, p,r, H)$ is a misspecifed linear MDP with approximation error $\zeta$, then this MDP satisfies the low inherent Bellman error assumption with $\mathcal{I} = 2\zeta$. \end{lemma} \begin{proof} If an MDP is an $\zeta$-approximate linear MDP, then we have \begin{align}
& \|p_h(\cdot|s,a) - \left\langle\boldsymbol{\phi}(s,a),\boldsymbol{\theta}_h(\cdot)\right\rangle\|_{\operatorname{TV}} \leq \zeta \\
& |r_h(s,a) - \left\langle\boldsymbol{\phi}(s,a),\boldsymbol{\nu}_h\right\rangle| \leq \zeta \end{align}
For any $\theta_{h+1} \in \mathbb{R}^d$, we have $\mathcal{T}_{h}\left(Q_{h+1}(\theta_{h+1})\right)(s, a) = r_{h}(s, a)+\mathbb{E}_{s^{\prime} \sim p_{h}(\cdot \mid s, a)} V_{h+1}(\theta_{h+1})\left(s^{\prime}\right)$. Since $V_{h+1}(\theta_{h+1})\left(s^{\prime}\right) \leq 1$, plugging the approximately linear form of $r_h(s,a)$ and $p_h(\cdot|s,a)$, we have \begin{align}
|\mathcal{T}_{h}\left(Q_{h+1}(\theta_{h+1})\right)(s, a) - \left \langle \boldsymbol{\phi}(s,a), \sum_{s'} \boldsymbol{\theta}_h(s')V_{h+1}(\theta_{h+1})\left(s^{\prime}\right) + \boldsymbol{\nu}_h \right\rangle| \leq 2\zeta \end{align} \end{proof}
By lemma~\ref{lemma: reduction from linear MDP to LSVI}, we can directly apply the hard instance construction and the lower bound for misspecified linear MDP to LSVI setting.
\begin{proposition} \label{proposition: referred lower bound for RL}
There exist function feature maps $\boldsymbol{\phi}_1,...,\boldsymbol{\phi}_H$ that define an MDP class $\mathcal{M}$ such that every MDP in that class satisfies low inherent Bellman error at most $\mathcal{I}$ and such that the expected reward on at least a member of the class (for $|\mathcal{A}| \geq 3, d,k,H \geq 10, T= \Omega(d^2H),\mathcal{I} \leq \frac{1}{4H}$) is $\Omega (d\sqrt{HT} + d H\mathcal{I}T)$. \end{proposition}
\subsubsection{Lower Bound for Multi-task RL}
In order to prove Theorem~\ref{theorem:linear_rl_lower_bound}, we need to prove and then combine the following two lemmas.
\begin{lemma} \label{lemma: rl lower bound lemma 1} Under the setting of Theorem~\ref{theorem:linear_rl_lower_bound}, the expected regret of any algorithm $\mathcal{A}$ is lower bounded by $\Omega( Mk\sqrt{HT})$. \end{lemma}
\begin{lemma} \label{lemma: rl lower bound lemma 2} Under the setting of Theorem~\ref{theorem:linear_rl_lower_bound}, the expected regret of any algorithm $\mathcal{A}$ is lower bounded by $\Omega\left(d\sqrt{kMHT} + HMT\sqrt{d}\mathcal{I}\right)$. \end{lemma}
These two lemmas are proved by reduction from Proposition~\ref{proposition: referred lower bound for RL}, which is a lower bound we proved for the single-task LSVI setting.
\begin{proof} (Proof of Lemma~\ref{lemma: rl lower bound lemma 1}) The lemma is proved by contradiction. Suppose there is an algorithm $\mathcal{A}$ that achieves $\sup_{M \in \mathcal{M}} \mathbb{E}[Reg(T)] \leq C M k\sqrt{HT}$ for a constant $C$. Then there must exists a task $i \in [M]$, such that the expected regret for this single task is at most $C k\sqrt{HT}$. However, by Proposition~\ref{proposition: referred lower bound for RL}, the expected regret for MDPs with dimension $k$ in horizon $h$ is at least $\Omega (k\sqrt{HT} + \sqrt{k}H \mathcal{I}T)$. This leads to a contradiction. \end{proof}
\begin{proof} (Proof of Lemma~\ref{lemma: rl lower bound lemma 2}) The hard instance construction follows the same idea of the proof for our Lemma~\ref{lemma: lower bound for linear bandits, approximation error term}, as well as the hard instance to prove Lemma~19 in \citet{yang2020provable}. Without loss of generality, we assume that $M$ can be exactly divided by $k$.
We divide $M$ tasks into $k$ groups. Each group shares the same parameter $\{\boldsymbol{\theta}^i_h\}_{h=1}^{H}$. To be more specific, we let $\boldsymbol{w}_h^{1} = \boldsymbol{w}_h^{2} = \cdots = \boldsymbol{w}_h^{M/k} = \boldsymbol{e}_h^1$, $\boldsymbol{w}_h^{M/k+1} = \boldsymbol{w}_h^{M/k+2} = \cdots = \boldsymbol{w}_h^{2M/k} = \boldsymbol{e}_h^2$, $\cdots$, $\boldsymbol{w}_h^{(k-1)M/k+1} = \boldsymbol{w}_h^{(k-1)M/k+2} = \cdots = \boldsymbol{w}_h^{M} = \boldsymbol{e}_h^k$. Under this construction, the parameters $\boldsymbol{\theta}_h^i$ for these tasks are exactly the same in each group, but relatively independent among different groups. That is to say, the expected regret lower bound is at least the summation of the regret lower bounds in all $k$ groups.
Now we consider the regret lower bound for group $j \in [k]$. Since the parameters are shared in the same group, the regret of running an algorithm for $M/k$ tasks with $T$ episodes each is at least the regret of running an algorithm for single-task linear bandit with $M/k \cdot T$ episodes. By Proposition~\ref{proposition: referred lower bound for RL}, the regret for single-task linear bandit with $MT/k$ episodes is at least $\Omega (d\sqrt{MHT/k} + \sqrt{d} \mathcal{I}HMT/k)$. Summing over all $k$ groups, we can prove that the regret lower bound is $\Omega (d\sqrt{kHMT} + \sqrt{d} \mathcal{I}HMT)$. \end{proof}
\end{document} | arXiv | {
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\begin{document}
\title[Generalized Fourier Integral Operators]
{Generalized Fourier Integral Operators\\ on spaces of Colombeau type}
\author[C. Garetto]{Claudia Garetto}
\address{ Institut f\"ur Grundlagen der Bauingenieurwissenschaften\\ Leopold-Franzens-Uni\-ver\-si\-t\"at Innsbruck\\ Technikerstr. 13\\ A 6020 Innsbruck\\ Austria}
\email{claudia@mat1.uibk.ac.at}
\thanks{This work was completed with the support of FWF (Austria), grants T305-N13 and Y237-N13.}
\subjclass{Primary 35S30; Secondary 46F30} \keywords{Fourier integral operators, Colombeau algebras}
\date{December 31, 2007}
\begin{abstract} Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data. \end{abstract}
\maketitle
\section{Introduction}
This work is part of a program that aims to solve linear partial differential equations with non-smooth coefficients and highly singular data and investigate the qualitative properties of the solutions. A well established theory with powerful analytic methods is available in the case of operators with (relatively) smooth coefficients \cite{Hoermander:V1-4}, but cannot be applied to many models from physics which involve non-smooth variations of the physical parameters. These models require indeed partial differential operators where the smoothness assumption on the coefficients is dropped. Furthermore, in case of nonlinear operations (cf. \cite{HdH:01, HS:68, O:92}), the theory of distribution does not provide a general framework in which solutions exist.
An alternative framework is provided by the theory of Colombeau algebras of generalized functions \cite{Colombeau:85, GKOS:01, O:92}. We recall that the space of distributions $\ensuremath{{\mathcal D}}'(\Omega)$ is embedded via convolution with a mollifier in the Colombeau algebra $\ensuremath{{\mathcal G}}(\Omega)$ of generalized functions on $\Omega$ and interpreting the non-smooth coefficients and data as elements of the Colombeau algebra, existence and uniqueness has been established for many classes of equations by now \cite{Biagioni:90, BO:92, BO:92b, CO:90, GH:03, HO:03, LO:91, O:88, O:92, OR:98a, OR:98b, Ste:98}. In order to study the regularity of solutions, microlocal techniques have to be introduced into this setting, in particular, pseudodifferential operators with generalized amplitudes and generalized wave front sets. This has been done in the papers \cite{GGO:03, GH:05, GH:05b, Hoermann:99, GH:04, HK:01, HOP:05, NPS:98, Pilipovic:94}, with a special attention for elliptic equations and hypoellipticity.
The interest for hyperbolic equations, regularity of solutions and inverse problems (determining the non-smooth coefficients from the data is an important problem in geophysics \cite{dHSto:02}), leads in the case of differential operators with Colombeau coefficients, to a theory of Fourier integral operators with generalized amplitudes {\em and} generalized phase functions. This has been initiated in \cite{GHO:06} and has provided some first results on propagation of singularities in the dual $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ of the Colombeau algebra $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$. We recall that within the Colombeau algebra ${\mathcal G}(\Omega)$, regularity theory is based on the subalgebra ${\mathcal G}^\infty(\Omega)$ of regular generalized functions, whose intersection with ${\mathcal D}'(\Omega)$ coincides with $\ensuremath{\mathcal{C}^\infty}(\Omega)$. Since $\ensuremath{\G^\infty}(\Omega)\subseteq \ensuremath{{\mathcal G}}(\Omega)\subseteq \mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$, two different regularity theories coexist in the dual: one based on $\ensuremath{{\mathcal G}}(\Omega)$ and one based on $\ensuremath{\G^\infty}(\Omega)$.
This work can be considered as a compendium of \cite{GHO:06}, in the sense that collects (without proof) the main results achieved in \cite{GHO:06} and studies the composition between a generalized Fourier integral operator and a generalized pseudodifferential operator in addition.
We can now describe the contents in more detail. Section 2 provides the needed background of Colombeau theory. In particular, topological concepts, generalized symbols and the definition of $\ensuremath{{\mathcal G}}$- and $\ensuremath{\G^\infty}$-wave front set are recalled. In Subsection \ref{asymp_new} we elaborate and state in full generality the notion of asymptotic expansion of a generalized symbol introduced for the first time in \cite{GGO:03} and we prove a new and technically useful characterization. Section 3 develops the foundations for generalized Fourier integral operators: oscillatory integrals with generalized phase functions and amplitudes. They are then supplemented by an additional parameter in Section 4, leading to the notion of a Fourier integral operator with generalized amplitude and phase function. We study the mapping properties of such operators on Colombeau algebras, the extension to the dual $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ and we present suitable assumptions on phase function and amplitude which lead to $\ensuremath{\G^\infty}$-mapping properties. The core of the work is Section 5, where, by making use of some technical preliminaries, we study in Theorem \ref{theo_comp} the composition $a(x,D)F_\omega(b)$ of a generalized pseudodifferential operator $a(x,D)$ with a generalized Fourier integral operator of the form \[ F_\omega(b)(u)(x)=\int_{\mb{R}^n}\mathrm{e}^{i\omega(x,\eta)}b(x,\eta)\widehat{u}(\eta)\, \dslash\eta. \] The final Section 6 collects the first results of microlocal analysis for generalized Fourier integral operators obtained in \cite[Section 4]{GHO:06}. A deeper investigation of the microlocal properties of generalized Fourier integral operators is current topic of research.
\section{Basic notions: Colombeau and duality theory} \label{section_basic} This section gives some background of Colombeau and duality theory for the techniques used in the sequel of the current work. As main sources we refer to \cite{Garetto:05b, Garetto:05a, GGO:03, GH:05, GKOS:01}.
\subsection{Nets of complex numbers} Before dealing with the major points of the Colombeau construction we begin by recalling some definitions concerning elements of $\mathbb{C}^{(0,1]}$.
A net $(u_\varepsilon)_\varepsilon$ in $\mb{C}^{(0,1]}$ is said to be \emph{strictly nonzero} if there exist $r>0$ and $\eta\in(0,1]$ such that $|u_\varepsilon|\ge \varepsilon^r$ for all $\varepsilon\in(0,\eta]$.\\ The regularity issues discussed in Sections 3 and 4 will make use of the following concept of \emph{slow scale net (s.s.n)}. A slow scale net is a net $(r_\varepsilon)_\varepsilon\in\mb{C}^{(0,1]}$ such that \[
\forall q\ge 0\, \exists c_q>0\, \forall\varepsilon\in(0,1]\qquad\qquad|r_\varepsilon|^q\le c_q\varepsilon^{-1}. \] Throughout this paper we will always consider slow scale nets $(r_\varepsilon)_\varepsilon$ of positive real numbers with $\inf_{\varepsilon\in(0,1]} r_\varepsilon\neq 0$.
A net $(u_\varepsilon)_\varepsilon$ in $\mb{C}^{(0,1]}$ is said to be \emph{slow scale-strictly nonzero} is there exist a slow scale net $(s_\varepsilon)_\varepsilon$ and $\eta\in(0,1]$ such that $|u_\varepsilon|\ge 1/s_\varepsilon$ for all $\varepsilon\in(0,\eta]$.
\subsection{$\wt{\mb{C}}$-modules of generalized functions based on a locally convex topological vector space $E$} \label{subsection_G_E} The most common algebras of generalized functions of Colombeau type as well as the spaces of generalized symbols we deal with are introduced and investigated under a topological point of view by referring to the following models.
Let $E$ be a locally convex topological vector space topologized through the family of seminorms $\{p_i\}_{i\in I}$. The elements of \[ \begin{split} \mathcal{M}_E &:= \{(u_\varepsilon)_\varepsilon\in E^{(0,1]}:\, \forall i\in I\,\, \exists N\in\mb{N}\quad p_i(u_\varepsilon)=O(\varepsilon^{-N})\, \text{as}\, \varepsilon\to 0\},\\ \mathcal{M}^\mathrm{sc}_E &:=\{(u_\varepsilon)_\varepsilon\in E^{(0,1]}:\, \forall i\in I\,\, \exists (\omega_\varepsilon)_\varepsilon\, \text{s.s.n.}\quad p_i(u_\varepsilon)=O(\omega_\varepsilon)\, \text{as}\, \varepsilon\to 0\},\\ \mathcal{M}^\infty_E &:=\{(u_\varepsilon)_\varepsilon\in E^{(0,1]}:\, \exists N\in\mb{N}\,\, \forall i\in I\quad p_i(u_\varepsilon)=O(\varepsilon^{-N})\, \text{as}\, \varepsilon\to 0\},\\ \mathcal{N}_E &:= \{(u_\varepsilon)_\varepsilon\in E^{(0,1]}:\, \forall i\in I\,\, \forall q\in\mb{N}\quad p_i(u_\varepsilon)=O(\varepsilon^{q})\, \text{as}\, \varepsilon\to 0\}, \end{split} \]
are called $E$-moderate, $E$-moderate of slow scale type, $E$-regular and $E$-negligible, respectively. We define the space of \emph{generalized functions based on $E$} as the factor space $\ensuremath{{\mathcal G}}_E := \mathcal{M}_E / \mathcal{N}_E$.
The ring of \emph{complex generalized numbers}, denoted by $\wt{\mb{C}}:=\ensuremath{{\mathcal E}_{M}}/\mathcal{N}$, is obtained by taking $E=\mb{C}$. $\wt{\mb{C}}$ is not a field since by Theorem 1.2.38 in \cite{GKOS:01} only the elements which are strictly nonzero (i.e. the elements which have a representative strictly nonzero) are invertible and vice versa. Note that all the representatives of $u\in\wt{\mb{C}}$ are strictly nonzero once we know that there exists at least one which is strictly nonzero. When $u$ has a representative which is slow scale-strictly nonzero we say that it is \emph{slow scale-invertible}.
For any locally convex topological vector space $E$ the space $\ensuremath{{\mathcal G}}_E$ has the structure of a $\wt{\mb{C}}$-module. The ${\mb{C}}$-module $\ensuremath{{\mathcal G}}^\mathrm{sc}_E:=\mathcal{M}^\mathrm{sc}_E/\mathcal{N}_E$ of \emph{generalized functions of slow scale type} and the $\wt{\mb{C}}$-module $\ensuremath{\G^\infty}_E:=\mathcal{M}^\infty_E/\mathcal{N}_E$ of \emph{regular generalized functions} are subrings of $\ensuremath{{\mathcal G}}_E$ with more refined assumptions of moderateness at the level of representatives. We use the notation $u=[(u_\varepsilon)_\varepsilon]$ for the class $u$ of $(u_\varepsilon)_\varepsilon$ in $\ensuremath{{\mathcal G}}_E$. This is the usual way we adopt to denote an equivalence class.
The family of seminorms $\{p_i\}_{i\in I}$ on $E$ determines a \emph{locally convex $\wt{\mb{C}}$-linear} topology on $\ensuremath{{\mathcal G}}_E$ (see \cite[Definition 1.6]{Garetto:05b}) by means of the \emph{valuations} \[ \mathrm{v}_{p_i}([(u_\varepsilon)_\varepsilon]):=\mathrm{v}_{p_i}((u_\varepsilon)_\varepsilon):=\sup\{b\in\mb{R}:\qquad p_i(u_\varepsilon)=O(\varepsilon^b)\, \text{as $\varepsilon\to 0$}\} \]
and the corresponding \emph{ultra-pseudo-seminorms} $\{\mathcal{P}_i\}_{i\in I}$, where $\mathcal{P}_i(u)=\mathrm{e}^{-\mathrm{v}_{p_i}(u)}$. For the sake of brevity we omit to report definitions and properties of valuations and ultra-pseudo-seminorms in the abstract context of $\wt{\mb{C}}$-modules. Such a theoretical presentation can be found in \cite[Subsections 1.1, 1.2]{Garetto:05b}. We recall that on $\wt{\mb{C}}$ the valuation and the ultra-pseudo-norm obtained through the absolute value in $\mb{C}$ are denoted by $\mathrm{v}_{\wt{\mb{C}}}$ and $|\cdot|_{\mathrm{e}}$ respectively. Concerning the space $\ensuremath{\G^\infty}_E$ of regular generalized functions based on $E$ the moderateness properties of $\mathcal{M}_E^\infty$ allows to define the valuation \[ \mathrm{v}^\infty_E ((u_\varepsilon)_\varepsilon):=\sup\{b\in\mb{R}:\, \forall i\in I\qquad p_i(u_\varepsilon)=O(\varepsilon^b)\, \text{as $\varepsilon\to 0$}\} \] which extends to $\ensuremath{\G^\infty}_E$ and leads to the ultra-pseudo-norm $\mathcal{P}^\infty_E(u):=\mathrm{e}^{-\mathrm{v}_E^\infty(u)}$.
The Colombeau algebra $\ensuremath{{\mathcal G}}(\Omega)=\ensuremath{{\mathcal E}_{M}}(\Omega)/\mathcal{N}(\Omega)$ can be obtained as a ${\wt{\mb{C}}}$-module of $\ensuremath{{\mathcal G}}_E$-type by choosing $E=\ensuremath{{\mathcal E}}(\Omega)$. Topologized through the family of seminorms $p_{K,i}(f)=\sup_{x\in K, |\alpha|\le i}|\partial^\alpha f(x)|$ where $K\Subset\Omega$, the space $\ensuremath{{\mathcal E}}(\Omega)$ induces on $\ensuremath{{\mathcal G}}(\Omega)$ a metrizable and complete locally convex $\wt{\mb{C}}$-linear topology which is determined by the ultra-pseudo-seminorms $\mathcal{P}_{K,i}(u)=\mathrm{e}^{-\mathrm{v}_{p_{K,i}}(u)}$. From a structural point of view $\Omega\to\ensuremath{{\mathcal G}}(\Omega)$ is a fine sheaf of differential algebras on $\mb{R}^n$.
The Colombeau algebra $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ of generalized functions with compact support is topologized by means of a strict inductive limit procedure. More precisely, setting $\ensuremath{{\mathcal G}}_K(\Omega):=\{u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega):\, \mathrm{supp}\, u\subseteq K\}$ for $K\Subset\Omega$, $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ is the strict inductive limit of the sequence of locally convex topological $\wt{\mb{C}}$-modules $(\ensuremath{{\mathcal G}}_{K_n}(\Omega))_{n\in\mb{N}}$, where $(K_n)_{n\in\mb{N}}$ is an exhausting sequence of compact subsets of $\Omega$ such that $K_n\subseteq K_{n+1}$. We endow $\ensuremath{{\mathcal G}}_K(\Omega)$ with the topology induced by $\ensuremath{{\mathcal G}}_{\mathcal{D}_{K'}(\Omega)}$ where $K'$ is a compact subset containing $K$ in its interior. For more details concerning the topological structure of $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ see \cite[Example 3.7]{Garetto:05a}.
Regularity theory in the Colombeau context as initiated in \cite{O:92} is based on the subalgebra $\ensuremath{\G^\infty}(\Omega)$ of all elements $u$ of $\ensuremath{{\mathcal G}}(\Omega)$ having a representative $(u_\varepsilon)_\varepsilon$ belonging to the set \begin{multline*}
\ensuremath{{\mathcal E}_{M}}^\infty(\Omega):=\{(u_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}}[\Omega]:\ \forall K\Subset\Omega\, \exists N\in\mb{N}\, \forall\alpha\in\mb{N}^n\\ \sup_{x\in K}|\partial^\alpha u_\varepsilon(x)|=O(\varepsilon^{-N})\, \text{as $\varepsilon\to 0$}\}. \end{multline*}
$\ensuremath{\G^\infty}(\Omega)$ can be seen as the intersection $\cap_{K\Subset\Omega}\ensuremath{\G^\infty}(K)$, where $\ensuremath{\G^\infty}(K)$ is the space of all $u\in\ensuremath{{\mathcal G}}(\Omega)$ having a representative $(u_\varepsilon)_\varepsilon$ satisfying the condition: $\exists N\in\mb{N}$ $\forall\alpha\in\mb{N}^n$,\ $\sup_{x\in K}|\partial^\alpha u_\varepsilon(x)|=O(\varepsilon^{-N})$. The ultra-pseudo-seminorms $\mathcal{P}_{\ensuremath{\G^\infty}(K)}(u):=\mathrm{e}^{-\mathrm{v}_{\ensuremath{\G^\infty}(K)}}$, where \[
\mathrm{v}_{\ensuremath{\G^\infty}(K)}:=\sup\{b\in\mb{R}:\, \forall\alpha\in\mb{N}^n\quad \sup_{x\in K}|\partial^\alpha u_\varepsilon(x)|=O(\varepsilon^b)\} \] equip $\ensuremath{\G^\infty}(\Omega)$ with the topological structure of a \emph{Fr\'echet $\wt{\mb{C}}$-module}.\\ Finally, let us consider the algebra $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega):=\ensuremath{\G^\infty}(\Omega)\cap\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$. On $\ensuremath{\G^\infty}_K(\Omega):=\{u\in\ensuremath{\G^\infty}(\Omega):\, \mathrm{supp}\, u\subseteq K\}$ with $K\Subset\Omega$, we define the ultra-pseudo-norm $\mathcal{P}_{\ensuremath{\G^\infty}_K(\Omega)}(u)=\mathrm{e}^{-\mathrm{v}^\infty_K(u)}$ where $\mathrm{v}^\infty_K(u):=\mathrm{v}^\infty_{\mathcal{D}_{K'}(\Omega)}(u)$ and $K'$ is any compact set containing $K$ in its interior. At this point, given an exhausting sequence $(K_n)_n$ of compact subsets of $\Omega$, the strict inductive limit procedure equips $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)=\cup_n \ensuremath{\G^\infty}_{K_n}(\Omega)$ with a complete and separated locally convex $\wt{\mb{C}}$-linear topology (see \cite[Example 3.13]{Garetto:05a}.
\subsection{Topological dual of a Colombeau algebra} A duality theory for $\wt{\mb{C}}$-modules had been developed in \cite{Garetto:05b} in the framework of topological and locally convex topological $\wt{\mb{C}}$-modules. Starting from an investigation of $\mathcal{L}(\ensuremath{{\mathcal G}},\wt{\mb{C}})$, the $\wt{\mb{C}}$-module of all $\wt{\mb{C}}$-linear and continuous functionals on $\ensuremath{{\mathcal G}}$, it provides the theoretical tools for dealing with the topological duals of the Colombeau algebras $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ and $\ensuremath{{\mathcal G}}(\Omega)$. In the paper $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}}$ and $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ are endowed with the \emph{topology of uniform convergence on bounded subsets}. This is determined by the ultra-pseudo-seminorms \[
\mathcal{P}_{B^\circ}(T)=\sup_{u\in B}|T(u)|_\mathrm{e}, \] where $B$ is varying in the family of all bounded subsets of $\ensuremath{{\mathcal G}}(\Omega)$ and $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ respectively. For general results concerning the relation between boundedness and ultra-pseudo-seminorms in the context of locally convex topological $\wt{\mb{C}}$-modules we refer to \cite[Section 1]{Garetto:05a}. For the choice of topologies illustrated in this section Theorem 3.1 in \cite{Garetto:05a} shows the following chains of continuous embeddings: \begin{equation} \label{chain_1} \ensuremath{\G^\infty}(\Omega)\subseteq\ensuremath{{\mathcal G}}(\Omega)\subseteq\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}}), \end{equation} \begin{equation} \label{chain_2} \ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)\subseteq\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\subseteq\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}}), \end{equation} \begin{equation} \label{chain_3} \mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})\subseteq\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}}). \end{equation} In \eqref{chain_1} and \eqref{chain_2} the inclusion in the dual is given via integration $\big(u\to\big( v\to\int_\Omega u(x)v(x)dx\big)\big)$ (for definitions and properties of the integral of a Colombeau generalized functions see \cite{GKOS:01}) while the embedding in \eqref{chain_3} is determined by the inclusion $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\subseteq\ensuremath{{\mathcal G}}(\Omega)$. Since $\Omega\to\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ is a sheaf we can define the \emph{support of a functional $T$} (denoted by $\mathrm{supp}\, T$). In analogy with distribution theory, from Theorem 1.2 in \cite{Garetto:05a} we have that $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ can be identified with the set of functionals in $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ having compact support.
By \eqref{chain_1} it is meaningful to measure the regularity of a functional in the dual $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ with respect to the algebras $\ensuremath{{\mathcal G}}(\Omega)$ and $\ensuremath{\G^\infty}(\Omega)$. We define the \emph{$\ensuremath{{\mathcal G}}$-singular support} of $T$ (${\rm{singsupp}}_\ensuremath{{\mathcal G}}\, T$) as the complement of the set of all points $x\in\Omega$ such that the restriction of $T$ to some open neighborhood $V$ of $x$ belongs to $\ensuremath{{\mathcal G}}(V)$. Analogously replacing $\ensuremath{{\mathcal G}}$ with $\ensuremath{\G^\infty}$ we introduce the notion of \emph{$\ensuremath{\G^\infty}$-singular support} of $T$ denoted by ${\rm{singsupp}}_{\ensuremath{\G^\infty}} T$. This investigation of regularity is connected with the notions of generalized wave front sets considered in Subsection \ref{sub_sec_micro} and will be focused on the functionals in $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ and $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ which have a ``basic'' structure. In detail, we say that $T\in\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ is ${{\rm{basic}}}$ if there exists a net $(T_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal D}}'(\Omega)^{(0,1]}$ fulfilling the following condition: for all $K\Subset\Omega$ there exist $j\in\mb{N}$, $c>0$, $N\in\mb{N}$ and $\eta\in(0,1]$ such that \[ \forall f\in\ensuremath{{\mathcal D}}_K(\Omega)\, \forall\varepsilon\in(0,\eta]\qquad\quad
|T_\varepsilon(f)|\le c\varepsilon^{-N}\sup_{x\in K,|\alpha|\le j}|\partial^\alpha f(x)| \] and $Tu=[(T_\varepsilon u_\varepsilon)_\varepsilon]$ for all $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$.\\ In the same way a functional $T\in\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ is said to be ${\rm{basic}}$ if there exists a net $(T_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}}'(\Omega)^{(0,1]}$ such that there exist $K\Subset\Omega$, $j\in\mb{N}$, $c>0$, $N\in\mb{N}$ and $\eta\in(0,1]$ with the property \[ \forall f\in\ensuremath{\mathcal{C}^\infty}(\Omega)\, \forall\varepsilon\in(0,\eta]\qquad\quad
|T_\varepsilon(f)|\le c\varepsilon^{-N}\sup_{x\in K,|\alpha|\le j}|\partial^\alpha f(x)| \] and $Tu=[(T_\varepsilon u_\varepsilon)_\varepsilon]$ for all $u\in\ensuremath{{\mathcal G}}(\Omega)$.\\ Clearly the sets $\mathcal{L}_{\rm{b}}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ and $\mathcal{L}_{\rm{b}}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ of ${\rm{basic}}$ functionals are $\wt{\mb{C}}$-linear subspaces of $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ and $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ respectively. In addition if $T$ is a ${\rm{basic}}$ functional in $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ and $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ then $uT\in\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ is ${\rm{basic}}$. We recall that nets $(T_\varepsilon)_\varepsilon$ which define basic maps as above were already considered in \cite{Delcroix:05,DelSca:00} with slightly more general notions of moderateness and different choices of notations and language.
\subsection{Generalized symbols} \label{subsec_gen_symb} For the convenience of the reader we recall a few basic notions concerning the sets of symbols employed in the course of this work. More details can be found in \cite{GGO:03, GH:05} where a theory of generalized pseudodifferential operators acting on Colombeau algebras is developed. \subsubsection*{Definitions.} Let $\Omega$ be an open subset of $\mb{R}^n$, $m\in\mb{R}$ and $\rho,\delta\in[0,1]$. $S^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ denotes the set of symbols of order $m$ and type $(\rho,\delta)$ as introduced by H\"ormander in \cite{Hoermander:71}. The subscript $(\rho,\delta)$ is omitted when $\rho=1$ and $\delta=0$. If $V$ is an open conic set of $\Omega\times\mb{R}^{p}$ we define $S^m_{\rho,\delta}(V)$ as the set of all $a\in\ensuremath{\mathcal{C}^\infty}(V)$ such that for all $K\Subset V$, \[
\sup_{(x,\xi)\in K^{c}}\lara{\xi}^{-m+\rho|\alpha|-\delta|\beta|}|\partial^\alpha_\xi\partial^\beta_x a(x,\xi)|<\infty, \] where $K^{c}:=\{(x,t\xi):\, (x,\xi)\in K\ t\ge 1\}$. We also make use of the space $S^1_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)$ of all $a\in S^1(\Omega\times\mb{R}^p\setminus 0)$ homogeneous of degree $1$ in $\xi$. Note that the assumption of homogeneity allows to state the defining conditions above in terms of the seminorms \[
\sup_{x\in K,\xi\in\mb{R}^p\setminus 0}|\xi|^{-1+\alpha}|\partial^\alpha_\xi\partial^\beta_x a(x,\xi)| \] where $K$ is any compact subset of $\Omega$.
The space of \emph{generalized symbols} $\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ is the $\wt{\mb{C}}$-module of $\ensuremath{{\mathcal G}}_E$-type obtained by taking $E=S^{m}_{\rho,\delta}(\Omega\times\mb{R}^p)$ equipped with the family of seminorms \[
|a|^{(m)}_{\rho,\delta,K,j}=\sup_{x\in K,\xi\in\mb{R}^n}\sup_{|\alpha+\beta|\le j}|\partial^\alpha_\xi\partial^\beta_x a(x,\xi)|\lara{\xi}^{-m+\rho|\alpha|-\delta|\beta|},\qquad\quad K\Subset\Omega,\, j\in\mb{N}. \]
The valuation corresponding to $|\cdot|^{(m)}_{\rho,\delta,K,j}$ gives the ultra-pseudo-seminorm $\mathcal{P}^{(m)}_{\rho,\delta,K,j}$. $\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ topologized through the family $\{\mathcal{P}^{(m)}_{\rho,\delta,K,j}\}_{K\Subset\Omega,j\in\mb{N}}$ of ultra-pseudo-seminorms is a Fr\'echet $\wt{\mb{C}}$-module. In analogy with $\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ we use the notation $\wt{\mathcal{S}}^m_{\rho,\delta}(V)$ for the $\wt{\mb{C}}$-module $\ensuremath{{\mathcal G}}_{S^m_{\rho,\delta}(V)}$.
$\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega_x\times\mb{R}^p_\xi)$ has the structure of a sheaf with respect to $\Omega$. So it is meaningful to talk of the support with respect to $x$ of a generalized symbol $a$ ($\mathrm{supp}_x\, a$).\\ We define the \emph{conic support} of $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ (${\rm{cone\, supp}}\, a$) as the complement of the set of points $(x_0,\xi_0)\in\Omega\times\mb{R}^p$ such that there exists a relatively compact open neighborhood $U$ of $x_0$, a conic open neighborhood $\Gamma$ of $\xi_0$ and a representative $(a_\varepsilon)_\varepsilon$ of $a$ satisfying the condition \begin{equation} \label{cond_conic_supp}
\forall\alpha\in\mb{N}^p\, \forall\beta\in\mb{N}^n\, \forall q\in\mb{N}\quad \sup_{x\in U,\xi\in\Gamma}\lara{\xi}^{-m+\rho|\alpha|-\delta|\beta|}|\partial^\alpha_\xi\partial^\beta_x a_\varepsilon(x,\xi)|=O(\varepsilon^q)\, \text{as $\varepsilon\to 0$}. \end{equation} By definition ${\rm{cone\, supp}}\, a$ is a closed conic subset of $\Omega\times\mb{R}^p$. The generalized symbol $a$ is $0$ on $\Omega\setminus\pi_x(\rm{cone\, supp}\, a)$. \subsubsection*{Slow scale symbols.} In the paper the classes of the factor space $\ensuremath{{\mathcal G}}^{\,\mathrm{sc}}_{S^m_{\rho,\delta}(\Omega\times\mb{R}^p)}$ are called \emph{generalized symbols of slow scale type}. For simplicity we introduce the notation $\wt{S}^{\,m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$. Substituting $S^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ with $S^m_{\rho,\delta}(V)$ we obtain the set $\wt{S}^{m,\mathrm{sc}}_{\rho,\delta}(V):=\ensuremath{{\mathcal G}}^{\,\mathrm{sc}}_{S^m_{\rho,\delta}(V)}$ of slow scale symbols on the open set $V\subseteq\Omega\times(\mb{R}^p\setminus 0)$.
\subsubsection*{Generalized symbols of order $-\infty$.}
Different notions of regularity are related to the sets $\wt{\mathcal{S}}^{-\infty}(\Omega\times\mb{R}^p)$ and $\wt{\mathcal{S}}^{-\infty,\mathrm{sc}}(\Omega\times\mb{R}^p)$ of generalized symbols of order $-\infty$.\\ The space $\wt{\mathcal{S}}^{-\infty}(\Omega\times\mb{R}^p)$ of generalized symbols of order $-\infty$ is defined as the $\wt{\mb{C}}$-module $\ensuremath{{\mathcal G}}_{S^{-\infty}(\Omega\times\mb{R}^p)}$. Its elements are equivalence classes $a$ whose representatives $(a_\varepsilon)_\varepsilon$ have the property $|a_\varepsilon|^{(m)}_{K,j}=O(\varepsilon^{-N})$ as $\varepsilon\to 0$, where $N$ depends on the order $m$ of the symbol, on the order $j$ of the derivatives and on the compact set $K\subseteq\Omega$.\\ $\wt{\mathcal{S}}^{-\infty,\mathrm{sc}}(\Omega\times\mb{R}^p)$ is defined by substituting $O(\varepsilon^{-N})$ with $O(\lambda_\varepsilon)$ in the previous estimate, where $(\lambda_\varepsilon)_\varepsilon$ is a slow scale net depending as above on the order $m$ of the symbol, on the order $j$ of the derivatives and on the compact set $K\subseteq\Omega$. It follows that $(a_\varepsilon)_\varepsilon$ is $\ensuremath{\G^\infty}$-regular, in the sense that \[
|a_\varepsilon|^{(m)}_{K,j}=O(\varepsilon^{-1}) \] as $\varepsilon\to 0$ for all $m,j$ and $K\Subset\Omega$.
\subsubsection*{Generalized microsupports.} The $\ensuremath{{\mathcal G}}$- and $\ensuremath{\G^\infty}$-regularity of generalized symbols on $\Omega\times\mb{R}^n$ is measured in conical neighborhoods by means of the following notions of microsupports.
Let $a\in\wt{\mathcal{S}}^l_{\rho,\delta}(\Omega\times\mb{R}^n)$ and $(x_0,\xi_0)\in\CO{\Omega}$. The symbol $a$ is $\ensuremath{{\mathcal G}}$-smoothing at $(x_0,\xi_0)$ if there exist a representative $(a_\varepsilon)_\varepsilon$ of $a$, a relatively compact open neighborhood $U$ of $x_0$ and a conic neighborhood $\Gamma\subseteq\mb{R}^n\setminus 0$ of $\xi_0$ such that \begin{multline} \label{est_micro_G} \forall m\in\mb{R}\, \forall\alpha,\beta\in\mb{N}^n\, \exists N\in\mb{N}\, \exists c>0\, \exists\eta\in(0,1]\, \forall(x,\xi)\in U\times\Gamma\, \forall\varepsilon\in(0,\eta]\\
|\partial^\alpha_\xi\partial^\beta_x a_\varepsilon(x,\xi)|\le c\lara{\xi}^m\varepsilon^{-N}. \end{multline} The symbol $a$ is $\ensuremath{\G^\infty}$-smoothing at $(x_0,\xi_0)$ if there exist a representative $(a_\varepsilon)_\varepsilon$ of $a$, a relatively compact open neighborhood $U$ of $x_0$, a conic neighborhood $\Gamma\subseteq\mb{R}^n\setminus 0$ of $\xi_0$ and a natural number $N\in\mb{N}$ such that \begin{multline} \label{est_micro_Ginf} \forall m\in\mb{R}\, \forall\alpha,\beta\in\mb{N}^n\, \exists c>0\, \exists\eta\in(0,1]\, \forall(x,\xi)\in U\times\Gamma\, \forall\varepsilon\in(0,\eta]\\
|\partial^\alpha_\xi\partial^\beta_x a_\varepsilon(x,\xi)|\le c\lara{\xi}^m\varepsilon^{-N}. \end{multline} We define the \emph{$\ensuremath{{\mathcal G}}$-microsupport} of $a$, denoted by $\mu\, \mathrm{supp}_\ensuremath{{\mathcal G}}(a)$, as the complement in $\CO{\Omega}$ of the set of points $(x_0,\xi_0)$ where $a$ is $\ensuremath{{\mathcal G}}$-smoothing and the \emph{$\ensuremath{\G^\infty}$-microsupport} of $a$, denoted by $\mu\, \mathrm{supp}_{\ensuremath{\G^\infty}}(a)$, as the complement in $\CO{\Omega}$ of the set of points $(x_0,\xi_0)$ where $a$ is $\ensuremath{\G^\infty}$-smoothing. \subsubsection*{Continuity results.} By simple reasoning at the level of representatives one proves that the usual operations between generalized symbols, as product and derivation, are continuous. In particular the $\wt{\mb{C}}$-bilinear map \begin{equation} \label{bil_product} \ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\times\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)\to\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p):(u,a)\to a(y,\xi)u(y) \end{equation} is continuous. If $l<-p$ each $b\in\wt{\mathcal{S}}^l_{\rho,\delta}(\Omega\times\mb{R}^p)$ can be integrated on $K\times\mb{R}^p$, $K\Subset\Omega$, by setting \[ \int_{K\times\mb{R}^p}b(y,\xi)\, dy\, d\xi :=\biggl[\biggl(\int_{K\times\mb{R}^p}b_\varepsilon(y,\xi)\, dy\, d\xi\biggr)_\varepsilon\biggr]. \] Moreover if $\mathrm{supp}_y b\Subset\Omega$ we define the integral of $b$ on $\Omega\times\mb{R}^p$ as \[ \int_{\Omega\times\mb{R}^p}b(y,\xi)\, dy\, d\xi :=\int_{K\times\mb{R}^p}b(y,\xi)\, dy\, d\xi, \] where $K$ is any compact set containing $\mathrm{supp}_y b$ in its interior. Integration defines a continuous $\wt{\mb{C}}$-linear functional on this space of generalized symbols with compact support in $y$ as it is proven in \cite[Proposition 1.1, Remark 1.2]{GHO:06}.
\subsection{Asymptotic expansions in $\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ and $\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$} \label{asymp_new}
In this subsection we elaborate and state in full generality the notion of asymptotic expansion of a generalized symbol introduced for the first time in \cite{GGO:03}. We also provide a technical result which will be useful in Section \ref{section_comp}. We begin by working on moderate nets of symbols and we recall that a net $(C_\varepsilon)_\varepsilon\in\mb{C}^{(0,1]}$ is said to be of slow scale type if there exists a slow scale net $(\omega_\varepsilon)_\varepsilon$ such that $|C_\varepsilon|=O(\omega_\varepsilon)$. \begin{defn} \label{asymp_mod} Let $\{m_j\}_{j\in\mathbb{N}}$ be sequences of real numbers with $m_j\searrow -\infty$, $m_0=m$. \begin{itemize} \item[(i)] Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$ be a sequence of elements $(a_{j,\epsilon})_\epsilon\in\mathcal{M}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$. We say that the formal series $\sum_{j=0}^\infty(a_{j,\epsilon})_\epsilon$ is the asymptotic expansion of $(a_\epsilon)_\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$, $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$ for short, iff for all $r\ge 1$ \[ \biggl(a_\epsilon-\sum_{j=0}^{r-1}a_{j,\epsilon}\biggr)_\epsilon\in \mathcal{M}_{S^{m_r}_{\rho,\delta}(\Omega\times\mathbb{R}^p)}. \] \item[(ii)] Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$ be a sequence of elements $(a_{j,\epsilon})_\epsilon\in\mathcal{M}^\mathrm{sc}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$. We say that the formal series $\sum_{j=0}^\infty(a_{j,\epsilon})_\epsilon$ is the asymptotic expansion of $(a_\epsilon)_\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$, $(a_\epsilon)_\epsilon\sim_\mathrm{sc}\sum_j(a_{j,\epsilon})_\epsilon$ for short, iff for all $r\ge 1$ \[ \biggl(a_\epsilon-\sum_{j=0}^{r-1}a_{j,\epsilon}\biggr)_\epsilon\in \mathcal{M}^\mathrm{sc}_{S^{m_r}_{\rho,\delta}(\Omega\times\mathbb{R}^p)}. \] \end{itemize} \end{defn} \begin{thm} \label{theo_asymp_expan} \leavevmode \begin{itemize} \item[(i)] Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$ be a sequence of elements $(a_{j,\epsilon})_\epsilon\in\mathcal{M}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$ with $m_j\searrow -\infty$ and $m_0=m$. Then, there exists $(a_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m}_{\rho,\delta}(\Omega\times\mb{R}^p)}$ such that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. Moreover, if $(a'_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$ then $(a_\varepsilon-a'_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^p)}$. \item[(ii)] Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$ be a sequence of elements $(a_{j,\epsilon})_\epsilon\in\mathcal{M}^{\mathrm{sc}}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$ with $m_j\searrow -\infty$ and $m_0=m$. Then, there exists $(a_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{m}_{\rho,\delta}(\Omega\times\mb{R}^p)}$ such that $(a_\epsilon)_\epsilon\sim_\mathrm{sc}\sum_j(a_{j,\epsilon})_\epsilon$. Moreover, if $(a'_\epsilon)_\epsilon\sim_\mathrm{sc}\sum_j(a_{j,\epsilon})_\epsilon$ then $(a_\varepsilon-a'_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{-\infty}(\Omega\times\mb{R}^p)}$. \end{itemize} \end{thm} \begin{proof} The proof follows the classical line of arguments, but we will have to keep track of the
$\epsilon$-dependence carefully. We consider a sequence of relatively compact open sets $\{V_l\}$ contained in $\Omega$, such that for all $l\in\mathbb{N}$, $V_l\subset K_l=\overline{V_l}\subset V_{l+1}$ and $\bigcup_{l\in\mathbb{N}}V_l=\Omega$. Let $\psi\in\mathcal{C}^\infty(\mathbb{R}^p)$, $0\le\psi(\xi)\le 1$, such that $\psi(\xi)=0$ for $|\xi|\le 1$ and $\psi(\xi)=1$ for $|\xi|\ge 2$.\\ $(i)$ We introduce \[ b_{j,\epsilon}(x,\xi)=\psi(\lambda_{j,\varepsilon}\xi)a_{j,\epsilon}(x,\xi), \] where $\lambda_{j,\varepsilon}$ will be positive constants with $\lambda_{j+1,\varepsilon}<\lambda_{j,\varepsilon}<1$, $\lambda_{j,\varepsilon}\to 0$ if $j\to\infty$. We can define \begin{equation} \label{a_symb} a_\epsilon(x,\xi)=\sum_{j\in\mathbb{N}}b_{j,\epsilon}(x,\xi). \end{equation} This sum is locally finite and therefore $(a_\varepsilon)_\varepsilon\in \ensuremath{{\mathcal E}}[\Omega\times\mb{R}^p]$. We observe that
$\partial^\alpha(\psi(\lambda_{j,\varepsilon}\xi))=\partial^\alpha\psi(\lambda_{j,\varepsilon}\xi)\lambda_{j,\varepsilon}^{|\alpha|}$,
${\rm supp}\,(\partial^\alpha\psi(\lambda_{j,\varepsilon}\xi))\subseteq\{\xi:\ 1/\lambda_{j,\varepsilon}\le|\xi|\le 2/\lambda_{j,\varepsilon}\}$, and that $1/\lambda_{j,\varepsilon}\le|\xi|\le 2/\lambda_{j,\varepsilon}$ implies $\lambda_{j,\varepsilon}\le 2/|\xi|\le 4/(1+|\xi|)$. We first estimate $b_{j,\varepsilon}$. Fixing $K\Subset\Omega$ and $\alpha\in\mb{N}^p$, $\beta\in\mathbb{N}^n$, we obtain for $j\in\mathbb{N}$, $\epsilon\in(0,1]$, $x\in K$, $\xi\in\mathbb{R}^p$, \begin{multline*}
|\partial^\alpha_\xi\partial^\beta_x b_{j,\epsilon}(x,\xi)|\le\sum_{\gamma\le\alpha}\binom{\alpha}{\gamma}\lambda_{j,\varepsilon}^{|\alpha-\gamma|}|\partial^{\alpha-\gamma}\psi(\lambda_{j,\varepsilon}\xi)||a_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\gamma,\beta}\langle\xi\rangle^{m_j-\rho|\gamma|+\delta|\beta|}\\
\le\sum_{\gamma\le\alpha}c(\psi,\gamma)4^{|\alpha-\gamma|}\langle\xi\rangle^{-|\alpha-\gamma|}|a_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\gamma,\beta}\langle\xi\rangle^{m_j-\rho|\gamma|+\delta|\beta|}\\
\le C_{j,\alpha,\beta,K,\varepsilon}\langle\xi\rangle^{m_j-\rho|\alpha|+\delta|\beta|}, \end{multline*} where \[
C_{j,\alpha,\beta,K,\varepsilon}:=\sum_{\gamma\le\alpha}c(\psi,\gamma)4^{|\alpha-\gamma|}|a_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\gamma,\beta}. \]
Since $(C_{j,\alpha,\beta,K,\varepsilon})_\varepsilon$ is a moderate net of positive numbers, we have that $(b_{j,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$. At this point we choose $\lambda_{j,\varepsilon}$ such that for $|\alpha+\beta|\le j$, $l\le j$ \begin{equation} \label{cj} C_{j,\alpha,\beta,K_l,\varepsilon}\lambda_{j,\varepsilon}\le 2^{-j}. \end{equation} Our aim is to prove that $a_\epsilon(x,\xi)$ defined in \eqref{a_symb} belongs to $\mathcal{M}_{S^{m}_{\rho,\delta}(\Omega\times\mb{R}^p)}$. Since there exists $N_j\in N$ and $\eta_j\in(0,1]$ such that \[ C_{j,\alpha,\beta,K_l,\varepsilon}\le \varepsilon^{-N_j} \]
for $l\le j$ and $|\alpha+\beta|\le j$, we take $\lambda_{j,\varepsilon}=2^{-j}\varepsilon^{N_j}$ on the interval $(0,\eta_j]$. We observe that \begin{equation} \label{lo} \begin{array}{cc} \forall K\Subset\Omega,\ \exists l\in\mathbb{N}:\ K\subset V_l\subset K_l,\\[0.2cm]
\forall\alpha_0\in\mb{N}^p,\ \forall\beta_0\in\mathbb{N}^n,\ \exists j_0\in\mathbb{N},\ j_0\ge l:\ |\alpha_0|+|\beta_0|\le j_0,\quad m_{j_0}+1\le m, \end{array} \end{equation} and we write $(a_\epsilon)_\epsilon$ as the sum of the following two terms: \[ \sum_{j=0}^{j_0-1}b_{j,\epsilon}(x,\xi)+\sum_{j=j_0}^{+\infty}b_{j,\epsilon}(x,\xi)=f_\epsilon(x,\xi)+s_\epsilon(x,\xi). \] For $x\in K$, we have that \begin{multline*}
|\partial^{\alpha_0}_\xi\partial^{\beta_0}_x f_\epsilon(x,\xi)|\le \sum_{j=0}^{j_0-1}|b_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\alpha_0,\beta_0}\langle\xi\rangle^{m_j-\rho|\alpha_0|+\delta|\beta_0|}\\
\le \biggl(\sum_{j=0}^{j_0-1}|b_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\alpha_0,\beta_0}\biggr)\lara{\xi}^{m-\rho|\alpha_0|+\delta|\beta_0|}, \end{multline*} where \[
\biggl(\sum_{j=0}^{j_0-1}|b_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\alpha_0,\beta_0}\biggr)_\varepsilon\in\ensuremath{{\mathcal E}_{M}}. \] We now turn to $s_\epsilon(x,\xi)$. From the estimates on $b_{j,\varepsilon}$ and \eqref{cj}, we get for $x\in K$ and $\epsilon\in(0,1]$, \begin{multline*}
|\partial^{\alpha_0}_\xi\partial^{\beta_0}_x s_\epsilon(x,\xi)|\le\sum_{j=j_0}^{+\infty}C_{j,\alpha_0,\beta_0,K_l}\langle\xi\rangle^{m_j-\rho|\alpha_0|+\delta|\beta_0|}\\
\le \sum_{j=j_0}^{+\infty}2^{-j}\lambda_{j,\varepsilon}^{-1}\langle\xi\rangle^{-1}\langle\xi\rangle^{ m_j+1-\rho|\alpha_0|+\delta|\beta_0|}\le \sum_{j=j_0}^{+\infty}2^{-j}\lambda_{j,\varepsilon}^{-1}\langle\xi\rangle^{-1}\langle\xi\rangle^{ m-\rho|\alpha_0|+\delta|\beta_0|}. \end{multline*}
Since $\psi(\xi)$ is identically equal to $0$ for $|\xi|\le 1$, we can assume in our estimates that $\langle\xi\rangle^{-1}\le\lambda_{j,\varepsilon}$, and therefore from \eqref{lo}, we conclude that \[
|\partial^{\alpha_0}_\xi\partial^{\beta_0}_x s_\epsilon(x,\xi)|\le 2\langle\xi\rangle^{m-\rho|\alpha_0|+\delta|\beta_0|}, \] for all $x\in K$, $\xi\in\mb{R}^p$ and $\varepsilon\in(0,1]$.
In order to prove that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$ we fix $r\ge 1$ and we write \begin{multline*} a_\epsilon(x,\xi)-\sum_{j=0}^{r-1}a_{j,\epsilon}(x,\xi) =\sum_{j=0}^{r-1}(\psi(\lambda_{j,\varepsilon}\xi)-1)a_{j,\epsilon}(x,\xi)+\sum_{j=r}^{+\infty}\psi(\lambda_{j,\varepsilon}\xi)a_{j,\epsilon}(x,\xi)\\ =g_\epsilon(x,\xi)+t_\epsilon(x,\xi). \end{multline*} Recall that $\psi\in\mathcal{C}^\infty(\mathbb{R}^p)$ was chosen such that $\psi-1\in\mathcal{C}^\infty_c(\mathbb{R}^p)$ and
${\rm{supp}}(\psi-1)\subseteq\{\xi:\ |\xi|\le 2\}$. Thus, for $0\le j\le r-1$, \[
\text{supp}(\psi(\lambda_{j,\varepsilon}\xi)-1)\subseteq\{\xi:\ |\lambda_{j,\varepsilon}\xi|\le 2\}\subseteq \{\xi:\ |\xi|\le 2\lambda_{r-1,\varepsilon}^{-1}\}. \] As a consequence, for fixed $K\Subset\Omega$ and for all $\epsilon\in(0,1]$, \begin{multline*}
|\partial^\alpha_\xi\partial^\beta_x g_\epsilon(x,\xi)|\le\sum_{j=0}^{r-1}\sum_{\alpha'\le\alpha}\binom{\alpha}{\alpha'}\lambda_{j,\varepsilon}^{|\alpha'|}c(\psi,\alpha')\lara{2\lambda_{r-1,\varepsilon}^{-1}}^{m_j-m_r+\rho|\alpha'|}|a_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\alpha-\alpha',\beta}\\
\cdot\lara{\xi}^{m_r-\rho|\alpha|+\delta|\beta|},\\ \end{multline*}
where, from our assumptions on $(a_{j,\varepsilon})_\varepsilon$ and $(\lambda_{j,\varepsilon})_\varepsilon$, the nets $(|a_{j,\varepsilon}|^{(m_j)}_{\rho,\delta,K,\alpha-\alpha',\beta})_\varepsilon$ and $(\lara{2\lambda_{r-1,\varepsilon}^{-1}}^{m_j-m_r+\rho|\alpha'|})_\varepsilon$ are both moderate. Repeating the same arguments used in the construction of $(a_\epsilon)_\epsilon$ we have that $(t_\epsilon)_\epsilon$ belongs to $\mathcal{M}_{S^{m_r}_{\rho,\delta}(\Omega\times\mb{R}^p)}$. It is clear that $(a_\varepsilon)_\varepsilon$ is uniquely determined by $\sum_j (a_{j,\varepsilon})_\varepsilon$ modulo $\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^p)}$.\\ $(ii)$ In the slow scale case one easily sees that $(b_{j,\varepsilon})_\varepsilon\in \mathcal{M}^\mathrm{sc}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$. Moreover, since there exists a slow scale net $\omega_j(\varepsilon)$ and $\eta_j\in(0,1]$ such that \[ C_{j,\alpha,\beta,K_l,\varepsilon}\le \omega_j(\varepsilon) \]
for $l\le j$ and $|\alpha+\beta|\le j$, we can take $\lambda_{j,\varepsilon}=2^{-j}\omega_j^{-1}(\varepsilon)$ on the interval $(0,\eta_j]$. It follows that $(a_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^m_{\rho,\delta}(\Omega\times\mb{R}^p)}$ and that both the nets $(g_\varepsilon)_\varepsilon$ and $(t_\varepsilon)_\varepsilon$ belong to $\mathcal{M}^\mathrm{sc}_{S^{m_r}_{\rho,\delta}(\Omega\times\mb{R}^p)}$. \end{proof} \begin{prop} \label{prop_asym_Shubin} \leavevmode \begin{itemize} \item[(i)] Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$ be a sequence of elements $(a_{j,\epsilon})_\epsilon\in\mathcal{M}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$ with $m_j\searrow -\infty$ and $m_0=m$. Let $(a_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}}[\Omega\times\mb{R}^p]$ such that for all $K\Subset\Omega$, for all $\alpha,\beta$ there exists $\mu\in\mb{R}$ and $(C_\varepsilon)_\varepsilon\in \ensuremath{{\mathcal E}_{M}}$ such that \begin{equation} \label{est_1_Sh}
|\partial^\alpha_\xi\partial^\beta_x a_\varepsilon(x,\xi)|\le C_\varepsilon\lara{\xi}^\mu, \end{equation} for all $x\in K$, $\xi\in\mb{R}^p$, $\varepsilon\in(0,1]$. Furthermore, assume that for any $r\ge 1$ and $K\Subset\Omega$ there exists $\mu_r=\mu_r(K)$ and $(C_{r,\varepsilon})_\varepsilon=(C_{r,\varepsilon}(K))_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that $\mu_r\to-\infty$ as $r\to +\infty$ and \begin{equation} \label{est_2_Sh}
\biggl|a_\varepsilon(x,\xi)-\sum_{j=0}^{r-1}a_{j,\varepsilon}(x,\xi)\biggr|\le C_{r,\varepsilon}\lara{\xi}^{\mu_r} \end{equation} for all $x\in K$, $\xi\in\mb{R}^p$, $\varepsilon\in(0,1]$. Then, $(a_\varepsilon)_\varepsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. \item[(ii)] $(i)$ holds with $(a_{j,\epsilon})_\epsilon\in\mathcal{M}^\mathrm{sc}_{S^{m_j}_{\rho,\delta}(\Omega\times\mb{R}^p)}$, the nets $(C_\varepsilon)_\varepsilon$ and $(C_{r,\varepsilon})_\varepsilon$ of slow scale type and $(a_\varepsilon)_\varepsilon\sim_\mathrm{sc}\sum_j(a_{j,\epsilon})_\epsilon$ in the sense of Definition \ref{asymp_mod}$(ii)$. \end{itemize} \end{prop} The proof of Proposition \ref{prop_asym_Shubin} requires the following lemma. \begin{lem} \label{lemma_Shubin} Let $K_1$ and $K_2$ be two compact sets in $\mb{R}^p$ such that $K_1\subset{\rm{Int}}\, K_2$. Then there exists a constant $C>0$ such that for any smooth function $f$ on a neighborhood of $K_2$, the following estimate holds: \[
\biggl(\sup_{x\in K_1}\sum_{|\alpha|= 1}|D^\alpha f(x)|\biggr)^2\le C\sup_{x\in K_2}|f(x)|\biggl(\sup_{x\in K_2}|f(x)|+\sup_{x\in K_2}\sum_{|\alpha|=2}|D^\alpha f(x)|\biggr). \] \end{lem} \begin{proof}[Proof of Proposition \ref{prop_asym_Shubin}] $(i)$ By Theorem \ref{theo_asymp_expan} we know that there exists $(b_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m}_{\rho,\delta}(\Omega\times\mb{R}^p)}$ such that $(b_\varepsilon)_\varepsilon\sim\sum_j(a_{j,\varepsilon})_\varepsilon$.We consider the difference $d_\varepsilon=a_\varepsilon-b_\varepsilon$. From \eqref{est_1_Sh} and the moderateness of $(b_\varepsilon)_\varepsilon$ we have that for all $\alpha,\beta$ and $K\Subset\Omega$ there exist $(C'_\varepsilon)_\varepsilon$ and $\mu'$ such that \begin{equation} \label{formula_raf_1}
|\partial^\alpha_\xi\partial^\beta_x d_\varepsilon(x,\xi)|\le C'_\varepsilon\lara{\xi}^{\mu'}, \end{equation} for all $x\in K$, $\xi\in\mb{R}^p$ and $\varepsilon\in(0,1]$. Combining $(b_\varepsilon)_\varepsilon\sim\sum_j(a_{j,\varepsilon})_\varepsilon$ with \eqref{est_2_Sh} we obtain that for all $r>0$ and for all $K\Subset\Omega$ there exists $(C_{r,\varepsilon}(K))_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \[
|d_\varepsilon(x,\xi)|\le C_{r,\varepsilon}(K)\lara{\xi}^{-r},\qquad\qquad x\in K,\ \xi\in\mb{R}^p,\, \varepsilon\in(0,1]. \]
Set $d_{\xi,\varepsilon}(x,\theta)=d_\varepsilon(x,\xi+\theta)$. Then, $\partial^\alpha_\theta \partial^\beta_x d_{\xi,\varepsilon}(x,\theta)|_{\theta=0}=\partial^\alpha_\xi\partial^\beta_x d(x,\xi)$, and applying Lemma \ref{lemma_Shubin} with $K_1=K\times 0$ and $K_2=K'\times\{|\theta|\le 1\}$, where $K\subset{\rm{Int}}K'\subset K'\Subset\Omega$, we obtain \begin{multline} \label{formula_raf_2}
\biggl(\sup_{x\in K}\sum_{|\alpha+\beta|= 1}|\partial^\alpha_\xi\partial^\beta_x d_\varepsilon(x,\xi)|\biggr)^2\le C\sup_{x\in K', |\theta|\le 1}|d_\varepsilon(x,\xi+\theta)|\cdot\\
\cdot\biggl(\sup_{x\in K', |\theta|\le 1}|d_\varepsilon(x,\xi+\theta)|+\sup_{x\in K', |\theta|\le 1}\sum_{|\alpha+\beta|=2}|\partial^\alpha_\xi\partial^\beta_x d_\varepsilon(x,\xi+\theta)|\biggr)\\
\le CC_{r,\varepsilon}(K')\sup_{|\theta|\le 1}\lara{\xi+\theta}^{-r}\biggl(C_{r,\varepsilon}(K')\sup_{|\theta|\le 1}\lara{\xi+\theta}^{-r}+C'_\varepsilon(K')\sup_{|\theta|\le 1}\lara{\xi+\theta}^{\mu'(K,2)}\biggr)\\ \le C''_{r,\varepsilon}(K)\lara{\xi}^{-r}, \end{multline} where $C''_{r,\varepsilon}(K)\in\ensuremath{{\mathcal E}_{M}}$. By induction one can prove that for all $r>0$, for all $K\Subset\Omega$ and for all $\alpha\in\mb{N}^p$, $\beta\in\mb{N}^n$, there exists a moderate net $(c_\varepsilon)_\varepsilon$ such that the estimate \[
|\partial^\alpha_\xi\partial^\beta_x d_\varepsilon(x,\xi)|\le c_\varepsilon\lara{\xi}^{-r} \] is valid for all $x\in K$, $\xi\in\mb{R}^p$ and $\varepsilon\in(0,1]$. This means that $(d_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^p)}$ and therefore $(a_\varepsilon)_\varepsilon\sim \sum_j(a_{j,\varepsilon})_\varepsilon$.\\ $(ii)$ It is clear that when we work with nets of slow scale type then $(d_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{-\infty}(\Omega\times\mb{R}^p)}$ and $(a_\varepsilon)_\varepsilon\sim_\mathrm{sc}\sum_j(a_{j,\epsilon})_\epsilon$. \end{proof} \begin{rem}
Proposition \ref{prop_asym_Shubin} can be stated for nets of symbols in $\mathcal{M}_{S^m_{\rho,\delta}(\Omega\times\mb{R}^p\setminus 0)}$ and $\mathcal{M}^\mathrm{sc}_{S^m_{\rho,\delta}(\Omega\times\mb{R}^p\setminus 0)}$. The proof make use of \eqref{formula_raf_1} when $|\xi|\le 1$ and \eqref{formula_raf_2} when $|\xi|>1$. \end{rem} \begin{defn} \label{def_asymp_gen} Let $\{m_j\}_{j\in\mathbb{N}}$ with $m_j\searrow -\infty$ and $m_0=m$. \begin{itemize} \item[(i)] Let $\{a_j\}_{j\in\mathbb{N}}$ be a sequence of symbols $a_j\in{\widetilde{\mathcal{S}}}^{\, m_j}_{\rho,\delta} (\Omega\times\mathbb{R}^p)$. We say that the formal series $\sum_j a_j$ is the asymptotic expansion of $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$, $a\sim\sum_j a_j$ for short, iff there exist a representative $(a_\epsilon)_\epsilon$ of $a$ and, for every $j$, representatives $(a_{j,\epsilon})_\epsilon$ of $a_j$, such that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. \item[(ii)] Let $\{a_j\}_{j\in\mathbb{N}}$ be a sequence of symbols $a_j\in{\widetilde{\mathcal{S}}}^{\, m_j,\mathrm{sc}}_{\rho,\delta} (\Omega\times\mathbb{R}^p)$. We say that the formal series $\sum_j a_j$ is the asymptotic expansion of $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$, $a\sim\sum_j a_j$ for short, iff there exist a representative $(a_\epsilon)_\epsilon$ of $a$ and, for every $j$, representatives $(a_{j,\epsilon})_\epsilon$ of $a_j$, such that $(a_\epsilon)_\epsilon\sim_\mathrm{sc}\sum_j(a_{j,\epsilon})_\epsilon$. \end{itemize} \end{defn} \subsection{Generalized pseudodifferential operators} Let $\Omega$ be an open subset of $\mb{R}^n$ and $a\in \widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^n)$. The generalized oscillatory integral (see \cite{GGO:03}) \[ \int_{\Omega\times\mb{R}^n}\mathrm{e}^{i(x-y)\xi}a(x,\xi){u}(y)\, dy\, \dslash\xi :=\biggl(\int_{\Omega\times\mb{R}^n}\mathrm{e}^{i(x-y)\xi}a_\varepsilon(x,\xi){u_\varepsilon}(y)\, dy\, \dslash\xi\biggr)_\varepsilon+\mathcal{N}(\Omega), \] defines the action of the pseudodifferential operator $a(x,D)$ with generalized symbol $a\in \widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^n)$ on $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$. The operator $a(x,D)$ maps $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ continuously into $\ensuremath{{\mathcal G}}(\Omega)$ and can be extended to a continuous $\wt{\mb{C}}$-linear map from $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ to $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$. If $a$ is of slow scale type then $a(x,D)$ maps $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)$ continuously into $\ensuremath{\G^\infty}(\Omega)$. Pseudodifferential operators with generalized symbol of order $-\infty$ are regularizing, in the sense that $a(x,D)$ maps $\mathcal{L}_{\rm{b}}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ to $\ensuremath{{\mathcal G}}(\Omega)$ if $a\in\wt{\mathcal{S}}^{-\infty}(\Omega\times\mb{R}^n)$ and $\mathcal{L}_{\rm{b}}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ to $\ensuremath{\G^\infty}(\Omega)$ if $a\in\wt{\mathcal{S}}^{-\infty,\mathrm{sc}}(\Omega\times\mb{R}^n)$. Clearly, all the previous results can be stated for pseudodifferential operators given by a generalized amplitude $a(x,y,\xi)\in\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\Omega\times\mb{R}^n)$. For a complete overview on generalized pseudodifferential operators acting on spaces of Colombeau type we advise the reader to refer to \cite{Garetto:06a, GGO:03, GH:05}
\subsection{Generalized elliptic symbols} One of the main issues in developing a theory of generalized symbols has been the search for a notion of generalized elliptic symbol. This is obviously related to the construction of a generalized pseudodifferential parametrix by means of which to investigate problems of $\ensuremath{{\mathcal G}}$- and $\ensuremath{\G^\infty}$-regularity. In the sequel we recall some of the results obtain in this direction in \cite{GGO:03, GH:05}, which will be employed in Section \ref{section_comp}. We work at the level of representatives and we set $\rho=1$, $\delta=0$. We leave to the reader the proof of the next proposition which is based on \cite[Section 6]{GGO:03}. \begin{prop} \label{prop_ellip} Let $(a_\varepsilon)_\varepsilon\in\mathcal{M}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ such that \begin{itemize} \item[(e1)] for all $K\Subset\Omega$ there exists $s\in\mb{R}$, $(R_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ strictly nonzero and $\eta\in(0,1]$ such that \[
|a_\varepsilon(x,\xi)|\ge \varepsilon^s\lara{\xi}^m, \]
for all $x\in K$, $|\xi|\ge R_\varepsilon$ and $\varepsilon\in(0,\eta]$. \end{itemize} Then, \begin{itemize} \item[(i)] for all $K\Subset\Omega$, for all $\alpha,\beta\in\mb{N}^n$ there exist $N\in\mb{N}$, $(R_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ strictly nonzero and $\eta\in(0,1]$ such that \[
|\partial^\alpha_\xi\partial^\beta_x a_\varepsilon(x,\xi)|\le \varepsilon^{-N}\lara{\xi}^{-|\alpha|}|a_\varepsilon(x,\xi)| \]
for all $x\in K$, $|\xi|\ge R_\varepsilon$ and $\varepsilon\in(0,\eta]$; \item[(ii)] $(i)$ holds for the net $(a_\varepsilon^{-1})_\varepsilon$; \item[(iii)] if $(a'_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m'}(\Omega\times\mb{R}^n\setminus 0)}$ with $m'<m$ then $(e1)$ holds for the net $(a_\varepsilon+a'_\varepsilon)_\varepsilon$. \end{itemize} Let $(a_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ such that \begin{itemize} \item[(e2)] for all $K\Subset\Omega$ there exists $(s_\varepsilon)_\varepsilon$ with $(s^{-1}_\varepsilon)_\varepsilon$ s.s.n., $(R_\varepsilon)_\varepsilon$ s.s.n. and $\eta\in(0,1]$ such that \[
|a_\varepsilon(x,\xi)|\ge s_\varepsilon\lara{\xi}^m, \]
for all $x\in K$, $|\xi|\ge R_\varepsilon$ and $\varepsilon\in(0,\eta]$. \end{itemize} Then, \begin{itemize} \item[(iv)] for all $K\Subset\Omega$, for all $\alpha,\beta\in\mb{N}^n$ there exist $(c_\varepsilon)_\varepsilon$, $(R_\varepsilon)_\varepsilon$ s.s.n and $\eta\in(0,1]$ such that \[
|\partial^\alpha_\xi\partial^\beta_x a_\varepsilon(x,\xi)|\le c_\varepsilon\lara{\xi}^{-|\alpha|}|a_\varepsilon(x,\xi)| \]
for all $x\in K$, $|\xi|\ge R_\varepsilon$ and $\varepsilon\in(0,\eta]$; \item[(v)] $(i)$ holds for the net $(a_\varepsilon^{-1})_\varepsilon$; \item[(vi)] if $(a'_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{m'}(\Omega\times\mb{R}^n\setminus 0)}$ with $m'<m$ then $(e2)$ holds for the net $(a_\varepsilon+a'_\varepsilon)_\varepsilon$. \end{itemize} \end{prop}
\begin{prop} \label{prop_ellip_2} Let $(a_\varepsilon)_\varepsilon$ be a net of elliptic symbols of $S^m(\Omega\times\mb{R}^n\setminus 0)$. \begin{itemize} \item[(i)] If $(a_\varepsilon)_\varepsilon\in\mathcal{M}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ fulfills condition $(e1)$ then there exists $(p_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-m}(\Omega\times\mb{R}^n\setminus 0)}$ such that for all $\varepsilon\in(0,1]$ \[ p_\varepsilon a_\varepsilon =1+r_\varepsilon, \] where $(r_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$. \item[(ii)] If $(a_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ fulfills condition $(e2)$ then there exists $(p_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{-m}(\Omega\times\mb{R}^n\setminus 0)}$ such that for all $\varepsilon\in(0,1]$ \[ p_\varepsilon a_\varepsilon =1+r_\varepsilon, \] where $(r_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$. \end{itemize} \end{prop} \begin{proof} As in \cite[Proposition 6.4]{GGO:03} we define $p_\varepsilon$ as \[ \sum_ja_\epsilon^{-1}(x,\xi)\varphi\big(\frac{\xi}{R_{j,\epsilon}}\big)\psi_j(x), \]
where $\psi_j$ is a partition of unity subordinated to a covering of relatively compact subsets $\Omega_j$ of $\Omega$, $(R_{j,\varepsilon})_\varepsilon$ is the radius corresponding to $\overline{\Omega_j}$ and $\varphi$ is a smooth function on $\mb{R}^n$ such that $\varphi(\xi)=0$ for $|\xi|\le 1$ and $\varphi(\xi)=1$ for $|\xi|\ge 2$. From Proposition \ref{prop_ellip} we have that $(e1)$ yields $(p_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-m}(\Omega\times\mb{R}^n\setminus 0)}$ and $(e2)$ yields $(p_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{-m}(\Omega\times\mb{R}^n\setminus 0)}$. Let $K\Subset\Omega$. By construction, for all $x\in K$, \[ p_\varepsilon(x,\xi)a_\varepsilon(x,\xi)=1+\biggl(\sum_{j=0}^{j_0}\varphi\big(\frac{\xi}{R_{j,\epsilon}})\psi_j(x)-1\biggr)= 1+\sum_{j=0}^{j_0}\biggl(\varphi\big(\frac{\xi}{R_{j,\epsilon}})-1\biggr)\psi_j(x), \] and the following estimates hold for all $l>0$ and $\alpha\in\mb{N}^n\setminus 0$: \begin{multline*}
\sup_{\xi\neq 0}\lara{\xi}^l|\varphi\big(\frac{\xi}{R_{j,\epsilon}})-1|\le\sup_{|\xi|\le 2R_{j,\epsilon}}\lara{\xi}^l|\varphi\big(\frac{\xi}{R_{j,\epsilon}})-1|\le c_\varphi\lara{2R_{j,\varepsilon}}^l,\\
\sup_{\xi\neq 0}\lara{\xi}^l|\partial^\alpha_\xi\varphi\big(\frac{\xi}{R_{j,\epsilon}})|(R_{j,\varepsilon})^{-|\alpha|}\le\sup_{R_{j,\varepsilon}\le|\xi|\le 2R_{j,\epsilon}}\lara{\xi}^l|\partial^\alpha_\xi\varphi\big(\frac{\xi}{R_{j,\epsilon}})|(R_{j,\varepsilon})^{-|\alpha|}\\
\le c_\varphi\lara{2R_{j,\varepsilon}}^l(R_{j,\varepsilon})^{-|\alpha|}. \end{multline*} We deduce that $(p_\varepsilon a_\varepsilon-1)_\varepsilon$ belongs to $\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$ under the hypothesis $(e1)$ on $(a_\varepsilon)_\varepsilon$ and that $(p_\varepsilon a_\varepsilon-1)_\varepsilon$ belongs to $\mathcal{M}^\mathrm{sc}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$ under the hypothesis $(e2)$ on $(a_\varepsilon)_\varepsilon$. \end{proof}
\subsection{Microlocal analysis in the Colombeau context: generalized wave front sets in $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$} \label{sub_sec_micro} In this subsection we recall the basic notions of microlocal analysis which involve the duals of the Colombeau algebras $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ and $\ensuremath{{\mathcal G}}(\Omega)$ and have been developed in \cite{Garetto:06a}. In this generalized context the role which is classically played by $\mathscr{S}(\mb{R}^n)$ is given to the Colombeau algebra $\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n):=\ensuremath{{\mathcal G}}_{\mathscr{S}(\mb{R}^n)}$. $\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n)$ is topologized as in Subsection \ref{subsection_G_E} and its dual $\mathcal{L}(\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n),\wt{\mb{C}})$ is endowed with the topology of uniform convergence on bounded subsets. In the sequel $\ensuremath{{\mathcal G}_\tau}(\mb{R}^n)$ denotes the Colombeau algebra of tempered generalized functions defined as the quotient $\ensuremath{{\mathcal E}_{\tau}}(\mb{R}^n)/\ensuremath{{\mathcal N}_{\tau}}(\mb{R}^n)$, where $\ensuremath{{\mathcal E}_{\tau}}(\mb{R}^n)$ is the algebra of all \emph{$\tau$-moderate} nets $(u_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}_{\tau}}[\mb{R}^n]:=\mathcal{O}_M(\mb{R}^n)^{(0,1]}$ such that \[
\forall \alpha\in\mb{N}^n\, \exists N\in\mb{N}\qquad \sup_{x\in\mb{R}^n}(1+|x|)^{-N}|\partial^\alpha u_\varepsilon(x)|=O(\varepsilon^{-N})\qquad \text{as}\ \varepsilon\to 0 \] and $\ensuremath{{\mathcal N}_{\tau}}(\mb{R}^n)$ is the ideal of all \emph{$\tau$-negligible} nets $(u_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}_{\tau}}[\mb{R}^n]$ such that \[
\forall \alpha\in\mb{N}^n\, \exists N\in\mb{N}\, \forall q\in\mb{N}\quad \sup_{x\in\mb{R}^n}(1+|x|)^{-N}|\partial^\alpha u_\varepsilon(x)|=O(\varepsilon^{q})\ \text{as}\ \varepsilon\to 0. \] Theorem 3.8 in \cite{Garetto:05b} shows that we have the chain of continuous embeddings \[ \G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n)\subseteq\ensuremath{{\mathcal G}_\tau}(\mb{R}^n)\subseteq\mathcal{L}(\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n),\wt{\mb{C}}). \] Moreover, since for any $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ with $\mathrm{supp}\, u\subseteq K\Subset\Omega$ and any $K'\Subset\Omega$ with $K\subset{\rm{Int}}\,K'$ one can find a representative $(u_\varepsilon)_\varepsilon$ with $\mathrm{supp}\, u_\varepsilon\subseteq K'$ for all $\varepsilon\in(0,1]$, we have that $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ is continuously embedded into $\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n)$.
\subsubsection*{The Fourier transform on $\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n)$, $\mathcal{L}(\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n),\wt{\mb{C}})$ and $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$.} The Fourier transform on $\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n)$ is defined by the corresponding transformation at the level of representatives, as follows: \[ \mathcal{F}:\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n)\to\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n):u\to [(\widehat{u_\varepsilon})_\varepsilon]. \] $\mathcal{F}$ is a $\wt{\mb{C}}$-linear continuous map from $\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n)$ into itself which extends to the dual in a natural way. In detail, we define the Fourier transform of $T\in\mathcal{L}(\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n),\wt{\mb{C}})$ as the functional in $\mathcal{L}(\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n),\wt{\mb{C}})$ given by \[ \mathcal{F}(T)(u)=T(\mathcal{F} u). \] As shown in \cite[Remark 1.5]{Garetto:06a} $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ is embedded in $\mathcal{L}(\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n),\wt{\mb{C}})$ by means of the map \[ \mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})\to\mathcal{L}(\G_{{\, }\atop{\hskip-4pt\scriptstyle\mathscr{S}}}\!(\mb{R}^n),\wt{\mb{C}}):T\to \big(u\to T(({u_\varepsilon}_{\vert_\Omega})_\varepsilon+\mathcal{N}(\Omega))\big). \] In particular, when $T$ is a ${\rm{basic}}$ functional in $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ we have from \cite[Proposition 1.6, Remark 1.7]{Garetto:06a} that the Fourier transform of $T$ is the tempered generalized function obtained as the action of $T(y)$ on $\mathrm{e}^{-iy\xi}$, i.e., $\mathcal{F}(T)=T(\mathrm{e}^{-i\cdot\xi})=(T_\varepsilon(\mathrm{e}^{-i\cdot\xi}))_\varepsilon+\ensuremath{{\mathcal N}_{\tau}}(\mb{R}^n)$.
\subsubsection*{Generalized wave front sets of a functional in $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$.} The notions of $\ensuremath{{\mathcal G}}$-wave front set and $\ensuremath{\G^\infty}$-wave front set of a functional in $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$ have been introduced in \cite{Garetto:06a} as direct analogues of the distributional wave front set in \cite{Hoermander:71}. They employ a subset of the space $\ensuremath{{\mathcal G}}^\mathrm{sc}_{S^m(\Omega\times\mb{R}^n)}$ of generalized symbols of slow scale type denoted by ${\wt{\underline{\mathcal{S}}}}_{\,\ssc}^m(\Omega\times\mb{R}^n)$ (see \cite[Definition 1.1]{GH:05}) and a suitable notion of slow scale micro-ellipticity \cite[Definition 1.2]{GH:05}. In detail, $(x_0,\xi_0)\not\in\mathrm{WF}_\ensuremath{{\mathcal G}}\, T$ (resp. $(x_0,\xi_0)\not\in\mathrm{WF}_{\ensuremath{\G^\infty}}\, T$) if there exists $a(x,D)$ properly supported with $a\in{\wt{\underline{\mathcal{S}}}}_{\,\ssc}^0(\Omega\times\mb{R}^n)$ such that $a$ is slow scale micro-elliptic at $(x_0,\xi_0)$ and $a(x,D)T\in\ensuremath{{\mathcal G}}(\Omega)$ (resp. $a(x,D)T\in\ensuremath{\G^\infty}(\Omega)$).\\ When $T$ is a basic functional of $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$, Proposition 3.14 in \cite{Garetto:06a} proves that one can limit to classical properly supported pseudodifferential operators in the definition of $\mathrm{WF}_\ensuremath{{\mathcal G}}\, T$ and $\mathrm{WF}_{\ensuremath{\G^\infty}}\, T$. More precisely, \begin{equation} \label{WGcl} {\rm{W}}_{{\rm{cl}},\ensuremath{{\mathcal G}}}(T):=\bigcap_{AT\in\ensuremath{{\mathcal G}}(\Omega)}\ensuremath{\text{Char}}(A) \end{equation} and \begin{equation} \label{WGinfcl} {\rm{W}}_{{\rm{cl}},\ensuremath{\G^\infty}}(T):=\bigcap_{AT\in\ensuremath{\G^\infty}(\Omega)}\ensuremath{\text{Char}}(A) \end{equation} where the intersections are taken over all the classical properly supported operators $A\in\Psi^0(\Omega)$ such that $AT\in\ensuremath{{\mathcal G}}(\Omega)$ in \eqref{WGcl} and $AT\in\ensuremath{\G^\infty}(\Omega)$ in \eqref{WGinfcl}. $\mathrm{WF}_\ensuremath{{\mathcal G}} T$ and $\mathrm{WF}_{\ensuremath{\G^\infty}}T$ are both closed conic subsets of $\CO{\Omega}$ and, as proved in \cite[Proposition 3.5]{Garetto:06a}, \[ \pi_\Omega(\mathrm{WF}_\ensuremath{{\mathcal G}} T)=\mathrm{sing\, supp}_\ensuremath{{\mathcal G}} T \] and \[ \pi_{\Omega}(\mathrm{WF}_{\ensuremath{\G^\infty}}T)=\mathrm{sing\, supp}_{\ensuremath{\G^\infty}} T. \] \subsubsection*{Characterization of $\mathrm{WF}_\ensuremath{{\mathcal G}} T$ and $\mathrm{WF}_{\ensuremath{\G^\infty}} T$ when $T$ is a basic functional.} We will employ a useful characterization of the $\ensuremath{{\mathcal G}}$-wave front set and the $\ensuremath{\G^\infty}$-wave front set valid for functionals which are basic. It involves the sets of generalized functions $\ensuremath{{\mathcal G}}_{\mathscr{S},0}(\Gamma)$ and $\ensuremath{\G^\infty}_{\mathscr{S}\hskip-2pt,0}(\Gamma)$, defined on the conic subset $\Gamma$ of $\mb{R}^n\setminus 0$, as follows: \begin{multline*}
\ensuremath{{\mathcal G}}_{\mathscr{S},0}(\Gamma):=\{u\in\ensuremath{{\mathcal G}_\tau}(\mb{R}^n):\ \exists (u_\varepsilon)_\varepsilon\in u\ \forall l\in\mb{R}\, \exists N\in\mb{N}\\ \sup_{\xi\in\Gamma}\lara{\xi}^l|u_\varepsilon(\xi)|=O(\varepsilon^{-N})\, \text{as $\varepsilon\to 0$}\}, \end{multline*} \begin{multline*}
\ensuremath{\G^\infty}_{\mathscr{S}\hskip-2pt,0}(\Gamma):=\{u\in\ensuremath{{\mathcal G}_\tau}(\mb{R}^n):\ \exists (u_\varepsilon)_\varepsilon\in u\ \exists N\in\mb{N}\, \forall l\in\mb{R}\\ \sup_{\xi\in\Gamma}\lara{\xi}^l|u_\varepsilon(\xi)|=O(\varepsilon^{-N})\, \text{as $\varepsilon\to 0$}\}. \end{multline*} Let $T\in\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$. Theorem 3.13 in \cite{Garetto:06a} shows that: \begin{itemize} \item[(i)] $(x_0,\xi_0)\not\in\mathrm{WF}_\ensuremath{{\mathcal G}} T$ if and only if there exists a conic neighborhood $\Gamma$ of $\xi_0$ and a cut-off function $\varphi\in\ensuremath{\mathcal{C}^\infty_{\text{c}}}(\Omega)$ with $\varphi(x_0)=1$ such that $\mathcal{F}(\varphi T)\in\ensuremath{{\mathcal G}}_{\mathscr{S},0}(\Gamma)$. \item[(ii)] $(x_0,\xi_0)\not\in\mathrm{WF}_{\ensuremath{\G^\infty}} T$ if and only if there exists a conic neighborhood $\Gamma$ of $\xi_0$ and a cut-off function $\varphi\in\ensuremath{\mathcal{C}^\infty_{\text{c}}}(\Omega)$ with $\varphi(x_0)=1$ such that $\mathcal{F}(\varphi T)\in\ensuremath{\G^\infty}_{\mathscr{S}\hskip-2pt,0}(\Gamma)$. \end{itemize}
\section{Generalized oscillatory integrals: definition} This section is devoted to a notion of oscillatory integral where both the amplitude and the phase function are generalized objects of Colombeau type.
In the sequel $\Omega$ is an arbitrary open subset of $\mb{R}^n$. We recall that $\phi(y,\xi)$ is a \emph{phase function} on $\Omega\times\mb{R}^p$ if it is a smooth function on $\Omega\times\mb{R}^p\setminus 0$, real valued, positively homogeneous of degree $1$ in $\xi$ with $\nabla_{y,\xi}\phi(y,\xi)\neq 0$ for all $y\in\Omega$ and $\xi\in\mb{R}^p\setminus 0$. We denote the set of all phase functions on $\Omega\times\mb{R}^p$ by $\Phi(\Omega\times\mb{R}^p)$ and the set of all nets in $\Phi(\Omega\times\mb{R}^p)^{(0,1]}$ by $\Phi[\Omega\times\mb{R}^p]$. The notations concerning classes of symbols have been introduced in Subsection \ref{subsec_gen_symb}. The proofs of the statements collected in this section can be found in \cite{GHO:06}. In the paper \cite{GHO:06} the authors deal with generalized symbols in $\wt{S}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ as well as with regular generalized symbols. This last class of symbols is modelled on the subalgebra $\ensuremath{\G^\infty}(\Omega)$ of regular generalized functions and contains the generalized symbols of slow scale type as a submodule. Even though many statements of Section 3, 4 and 6 hold for regular symbols as well, for the sake of simplicity and in order to have uniformity of assumptions between phase functions and symbols, we limit in this work to consider $\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ and the smaller class $\wt{S}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$ of generalized symbols of slow scale type. \begin{defn} \label{def_phase_moderate} An element of $\mathcal{M}_\Phi(\Omega\times\mb{R}^p)$ is a net $(\phi_\varepsilon)_\varepsilon\in\Phi[\Omega\times\mb{R}^p]$ satisfying the conditions: \begin{itemize} \item[(i)] $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_{S^1_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)}$,
\item[(ii)] for all $K\Subset\Omega$ the net $$\biggl(\inf_{y\in K,\xi\in\mb{R}^p\setminus 0}\biggl|\nabla \phi_\varepsilon\biggl(y,\frac{\xi}{|\xi|}\biggr)\biggr|^2\biggr)_\varepsilon$$ is strictly nonzero. \end{itemize} On $\mathcal{M}_\Phi(\Omega\times\mb{R}^p)$ we introduce the equivalence relation $\sim$ as follows: $(\phi_\varepsilon)_\varepsilon\sim(\omega_\varepsilon)_\varepsilon$ if and only if $(\phi_\varepsilon-\omega_\varepsilon)\in\mathcal{N}_{S^1_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)}$. The elements of the factor space $$\wt{\Phi}(\Omega\times\mb{R}^p):={\mathcal{M}_\Phi(\Omega\times\mb{R}^p)}/{\sim}.$$ will be called \emph{generalized phase functions}. \end{defn} We shall employ the equivalence class notation $[(\phi_\varepsilon)_\varepsilon]$ for $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$. When $(\phi_\varepsilon)_\varepsilon$ is a net of phase functions, i.e. $(\phi_\varepsilon)_\varepsilon\in\Phi[\Omega\times\mb{R}^p]$, Lemma 1.2.1 in \cite{Hoermander:71} shows that there exists a family of partial differential operators $(L_{\phi_\varepsilon})_\varepsilon$ such that ${\ }^tL_{\phi_\varepsilon}\mathrm{e}^{i\phi_\varepsilon}=\mathrm{e}^{i\phi_\varepsilon}$ for all $\varepsilon\in(0,1]$. $L_{\phi_\varepsilon}$ is of the form \begin{equation} \label{def_L_phi_cl} \sum_{j=1}^p a_{j,\varepsilon}(y,\xi)\frac{\partial}{\partial_{\xi_j}} +\sum_{k=1}^n b_{k,\varepsilon}(y,\xi)\frac{\partial}{\partial_{y_k}}+ c_\varepsilon(y,\xi), \end{equation} where the coefficients $(a_{j,\varepsilon})_\varepsilon$ belong to $S^0[\Omega\times\mb{R}^p]$ and $(b_{k,\varepsilon})_\varepsilon$, $(c_\varepsilon)_\varepsilon$ are elements of $S^{-1}[\Omega\times\mb{R}^p]$. The following technical lemma is crucial in proving Proposition \ref{prop_operator}. \begin{lem} \label{lemma_1} \leavevmode \begin{trivlist}
\item[(i)] Let $\varphi_{\phi_\varepsilon}(y,\xi):=|\nabla\phi_\varepsilon(y,\xi/|\xi|)|^{-2}$. If $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_\Phi(\Omega\times\mb{R}^p)$ then \[ (\varphi_{\phi_\varepsilon})_\varepsilon\in\mathcal{M}_{S^0_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)}. \] \item[(ii)] If $(\phi_\varepsilon)_\varepsilon, (\omega_\varepsilon)_\varepsilon\in\mathcal{M}_\Phi(\Omega\times\mb{R}^p)$ and $(\phi_\varepsilon)_\varepsilon\sim(\omega_\varepsilon)_\varepsilon$ then \[ \big(({\partial_{\xi_j}\phi_\varepsilon})\varphi_{\phi_\varepsilon}-({\partial_{\xi_j}\omega_\varepsilon})\varphi_{\omega_\varepsilon}\big)_\varepsilon\in\mathcal{N}_{S^0_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)} \] for all $j=1,...,p$ and \[
\big(({\partial_{y_k}\phi_\varepsilon}){|\xi|^{-2}\varphi_{\phi_\varepsilon}}-({\partial_{y_k}\omega_\varepsilon}){|\xi|^{-2}\varphi_{\omega_\varepsilon}}\big)_\varepsilon\in\mathcal{N}_{S^{-1}_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)} \] for all $k=1,...,n$. \end{trivlist} \end{lem} \begin{prop} \label{prop_operator} \leavevmode \begin{trivlist} \item[(i)] If $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_\Phi(\Omega\times\mb{R}^p)$ then $(a_{j,\varepsilon})_\varepsilon\in\mathcal{M}_{S^0(\Omega\times\mb{R}^p)}$ for all $j=1,...,p$, $(b_{k,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{-1}(\Omega\times\mb{R}^p)}$ for all $k=1,...,n$, and $(c_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-1}(\Omega\times\mb{R}^p)}$. \item[(ii)] If $(\phi_\varepsilon)_\varepsilon, (\omega_\varepsilon)_\varepsilon\in\mathcal{M}_\Phi(\Omega\times\mb{R}^p)$ and $(\phi_\varepsilon)_\varepsilon\sim(\omega_\varepsilon)_\varepsilon$ then \[ L_{\phi_\varepsilon}-L_{\omega_\varepsilon}=\sum_{j=1}^p a'_{j,\varepsilon}(y,\xi)\frac{\partial}{\partial_{\xi_j}} +\sum_{k=1}^n b'_{k,\varepsilon}(y,\xi)\frac{\partial}{\partial_{y_k}}+ c'_\varepsilon(y,\xi), \] where $(a'_{j,\varepsilon})_\varepsilon\in\mathcal{N}_{S^{0}(\Omega\times\mb{R}^p)}$, $(b'_{k,\varepsilon})_\varepsilon\in\mathcal{N}_{S^{-1}(\Omega\times\mb{R}^p)}$ and $(c'_\varepsilon)_\varepsilon\in\mathcal{N}_{S^{-1}(\Omega\times\mb{R}^p)}$ for all $j=1,...,p$ and $k=1,...,n$. \end{trivlist} \end{prop} As a consequence of Propositions \ref{prop_operator} we can claim that any generalized phase function $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$ defines a generalized partial differential operator \[ L_\phi(y,\xi,\partial_y,\partial_\xi)=\sum_{j=1}^p a_{j}(y,\xi)\frac{\partial}{\partial_{\xi_j}} +\sum_{k=1}^n b_{k}(y,\xi)\frac{\partial}{\partial_{y_k}}+ c(y,\xi) \] whose coefficients $\{a_j\}_{j=1}^p$ and $\{b_k\}_{k=1}^n$, $c$ are generalized symbols in $\wt{\mathcal{S}}^{0}(\Omega\times\mb{R}^p)$ and $\wt{\mathcal{S}}^{-1}(\Omega\times\mb{R}^p)$, respectively. By construction, $L_\phi$ maps $\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ continuously into $\wt{\mathcal{S}}^{m-s}_{\rho,\delta}(\Omega\times\mb{R}^p)$, where $s=\min\{\rho,1-\delta\}$. Hence $L^k_\phi$ is continuous from $\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ to $\wt{\mathcal{S}}^{m-ks}_{\rho,\delta}(\Omega\times\mb{R}^p)$.
Before stating the next proposition we recall a classical lemma valid any symbol $\phi\in S^1(\Omega\times\mb{R}^p\setminus 0)$. \begin{lem} \label{lemma_esp_classic} For all $\alpha\in\mb{N}^p$ and $\beta\in\mb{N}^n$, \[
\partial^\alpha_\xi\partial^\beta_y\mathrm{e}^{i\phi(y,\xi)}=\sum_{\substack{k\le|\alpha+\beta|,\\ \alpha_1+\alpha_2+...+\alpha_k=\alpha\\ \beta_1+\beta_2+...+\beta_k=\beta}}c_{\alpha_1,...,\alpha_k,\beta_1,...,\beta_k}\,\partial^{\alpha_1}_\xi\partial^{\beta_1}_y\phi(y,\xi)...\partial^{\alpha_k}_\xi\partial^{\beta_k}_y\phi(y,\xi). \] It follows that \[ \partial^\alpha_\xi\partial^\beta_y\mathrm{e}^{i\phi(y,\xi)}= \mathrm{e}^{i\phi(y,\xi)}a_{\alpha,\beta}(y,\xi), \]
where $a_{\alpha,\beta}\in S^{|\beta|}(\Omega\times\mb{R}^p\setminus 0)$ and \begin{equation} \label{coeff_a}
|a_{\alpha,\beta}|^{(|\beta|)}_{K,j}\le c\sup_{y\in K,\xi\neq 0}\sup_{|\gamma+\delta|\le|\alpha+\beta|+j}\lara{\xi}^{-1+|\gamma|}|\partial^\gamma_\xi\partial^\delta_y\phi(y,\xi)|, \end{equation} where the constant $c$ depends only on $\alpha$, $\beta$, and $j$. \end{lem} From \eqref{coeff_a} we have that \[
(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_{S^1(\Omega\times\mb{R}^p\setminus 0)}\quad \Rightarrow \quad (a_{\alpha,\beta,\varepsilon})_\varepsilon\in \mathcal{M}_{S^{|\beta|}(\Omega\times\mb{R}^p\setminus 0)} \] or more in general that the net $(a_{\alpha,\beta,\varepsilon})_\varepsilon$ has the ``$\varepsilon$-scale properties'' of $(\phi_\varepsilon)_\varepsilon$. \begin{prop} \label{prop_exp} Let $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$. The exponential $$\mathrm{e}^{i\phi(y,\xi)}$$ is a well-defined element of $\wt{\mathcal{S}}^1_{0,1}(\Omega\times\mb{R}^p\setminus 0)$. \end{prop} \begin{proof} From Lemma \ref{lemma_esp_classic} we have that if $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_\Phi(\Omega\times\mb{R}^p)$ then $(\mathrm{e}^{i\phi_\varepsilon(y,\xi)})_\varepsilon\in\mathcal{M}_{S^0_{0,1}(\Omega\times\mb{R}^p\setminus 0)}$. When $(\phi_\varepsilon)_\varepsilon\sim(\omega_\varepsilon)_\varepsilon$, the equality \begin{multline*} \mathrm{e}^{i\omega_\varepsilon(y,\xi)}-\mathrm{e}^{i\phi_\varepsilon(y,\xi)}=\mathrm{e}^{i\omega_\varepsilon(y,\xi)}\big(1-\mathrm{e}^{i(\phi_\varepsilon-\omega_\varepsilon)(y,\xi)}\big)\\ =\mathrm{e}^{i\omega_\varepsilon(y,\xi)}\sum_{j=1}^p \mathrm{e}^{i(\phi_\varepsilon-\omega_\varepsilon)(y,\theta\xi)}\partial_{\xi_j}(\phi_\varepsilon-\omega_\varepsilon)(y,\theta\xi)i\xi_j, \end{multline*} with $\theta\in(0,1)$, implies that \begin{equation} \label{neg_esp}
\sup_{y\in K,\xi\in\mb{R}^p\setminus 0}|\xi|^{-1}\big|\mathrm{e}^{i\omega_\varepsilon(y,\xi)}-\mathrm{e}^{i\phi_\varepsilon(y,\xi)}\big|=O(\varepsilon^q) \end{equation} for all $q\in\mb{N}$. At this point writing $\partial^\alpha_\xi\partial^\beta_y( \mathrm{e}^{i\omega_\varepsilon(y,\xi)}-\mathrm{e}^{i\phi_\varepsilon(y,\xi)})$ as \begin{multline*} \partial^\alpha_\xi\partial^\beta_y \mathrm{e}^{i\omega_\varepsilon(y,\xi)}\big(1-\mathrm{e}^{i(\phi_\varepsilon-\omega_\varepsilon)(y,\xi)}\big)+\\ +\sum_{\alpha'<\alpha,\beta'<\beta}\binom{\alpha}{\alpha'}\binom{\beta}{\beta'}\partial^{\alpha'}_\xi\partial^{\beta'}_y \mathrm{e}^{i\omega_\varepsilon(y,\xi)}\big(-\partial^{\alpha-\alpha'}_\xi\partial^{\beta-\beta'}_y \mathrm{e}^{i(\phi_\varepsilon-\omega_\varepsilon)(y,\xi)}\big) \end{multline*} we obtain the characterizing estimate of a net in $\mathcal{N}_{S^1_{0,1}(\Omega\times\mb{R}^p\setminus 0)}$, using \eqref{neg_esp} the moderateness of $(\mathrm{e}^{i\omega_\varepsilon(y,\xi)})_\varepsilon$ and Lemma \ref{lemma_esp_classic}. \end{proof} By construction of the operator $L_\phi$ the equality ${\ }^t L_\phi \mathrm{e}^{i\phi}=\mathrm{e}^{i\phi}$ holds in $\wt{\mathcal{S}}^1_{0,1}(\Omega\times\mb{R}^p\setminus 0)$. In addition, Proposition \ref{prop_exp} and the properties of $L^k_\phi$ allow to conclude that $$\mathrm{e}^{i\phi(y,\xi)}L^k_\phi(a(y,\xi)u(y))$$ is a generalized symbol in $\wt{\mathcal{S}}^{m-ks+1}_{0,1}(\Omega\times\mb{R}^p)$ which is integrable on $\Omega\times\mb{R}^p$ in the sense of Section \ref{section_basic} when $m-ks+1<-p$. From now on we assume that $\rho>0$ and $\delta<1$. \begin{defn} \label{def_gen_osc} Let $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$, $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ and $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$. The \emph{generalized oscillatory integral} \[ \int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(y,\xi)}a(y,\xi)u(y)\, dy\,\dslash\xi \] is defined as \[ \int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(y,\xi)}L^k_\phi(a(y,\xi)u(y))\, dy\,\dslash\xi \] where $k$ is chosen such that $m-ks+1<-p$. \end{defn}
The functional \[ I_\phi(a):\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\to\wt{\mb{C}}:u\to\int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(y,\xi)}a(y,\xi)u(y)\, dy\, \dslash\xi \] belongs to the dual $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega),\wt{\mb{C}})$. Indeed, by \eqref{bil_product}, the continuity of $L^k_\phi$ and of the product between generalized symbols we have that the map \[ \ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\to\wt{\mathcal{S}}^{m-ks+1}_{0,1}(\Omega\times\mb{R}^p):u\to \mathrm{e}^{i\phi(y,\xi)}L^k_\phi(a(y,\xi)u(y)) \] is continuous and thus, by an application of the integral on $\Omega\times\mb{R}^p$, the resulting functional $I_\phi(a)$ is continuous.
\section{Generalized Fourier integral operators} \label{gen_sec} \subsection*{Definition and mapping properties} We now study oscillatory integrals where an additional parameter $x$, varying in an open subset $\Omega'$ of $\mb{R}^{n'}$, appears in the phase function $\phi$ and in the symbol $a$. The dependence on $x$ is investigated in the Colombeau context. We denote by $\Phi[\Omega';\Omega\times\mb{R}^p]$ the set of all nets $(\phi_\varepsilon)_{\varepsilon\in(0,1]}$ of continuous functions on $\Omega'\times\Omega\times\mb{R}^p$ which are smooth on $\Omega'\times\Omega\times\mb{R}^p\setminus\{0\}$ and such that $(\phi_\varepsilon(x,\cdot,\cdot))_\varepsilon\in\Phi[\Omega\times\mb{R}^p]$ for all $x\in\Omega'$. \begin{defn} \label{def_phase_x_moderate} An element of $\mathcal{M}_{\Phi}(\Omega';\Omega\times\mb{R}^p)$ is a net $(\phi_\varepsilon)_\varepsilon\in\Phi[\Omega';\Omega\times\mb{R}^p]$ satisfying the conditions: \begin{itemize} \item[(i)] $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_{S^1_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)}$, \item[(ii)] for all $K'\Subset\Omega'$ and $K\Subset\Omega$ the net \begin{equation} \label{net_FIO}
\biggl(\inf_{x\in K',y\in K,\xi\in\mb{R}^p\setminus 0}\biggl|\nabla_{y,\xi} \phi_\varepsilon\biggl(x,y,\frac{\xi}{|\xi|}\biggr)\biggr|^2\biggr)_\varepsilon \end{equation} is strictly nonzero. \end{itemize} On $\mathcal{M}_{\Phi}(\Omega';\Omega\times\mb{R}^p)$ we introduce the equivalence relation $\sim$ as follows: $(\phi_\varepsilon)_\varepsilon\sim(\omega_\varepsilon)_\varepsilon$ if and only if $(\phi_\varepsilon-\omega_\varepsilon)_\varepsilon\in\mathcal{N}_{S^1_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)}$. The elements of the factor space \[ \wt{\Phi}(\Omega';\Omega\times\mb{R}^p):=\mathcal{M}_{\Phi}(\Omega';\Omega\times\mb{R}^p) / \sim. \] are called \emph{generalized phase functions with respect to the variables in $\Omega\times\mb{R}^p$}. \end{defn} Lemma \ref{lemma_1} as well as Proposition \ref{prop_operator} can be adapted to nets in $\mathcal{M}_{\Phi}(\Omega';\Omega\times\mb{R}^p)$. More precisely, the operator \begin{equation} \label{smilla} L_{\phi_\varepsilon}(x;y,\xi,\partial_y,\partial_\xi)=\sum_{j=1}^p a_{j,\varepsilon}(x,y,\xi)\frac{\partial}{\partial_{\xi_j}} +\sum_{k=1}^n b_{k,\varepsilon}(x,y,\xi)\frac{\partial}{\partial_{y_k}}+ c_\varepsilon(x,y,\xi) \end{equation} defined for any value of $x$ by \eqref{def_L_phi_cl}, has the property ${\ }^tL_{\phi_\varepsilon(x,\cdot,\cdot)}\mathrm{e}^{i\phi_\varepsilon(x,\cdot,\cdot)}=\mathrm{e}^{i\phi_\varepsilon(x,\cdot,\cdot)}$ for all $x\in\Omega'$ and $\varepsilon\in(0,1]$ and its coefficients depend smoothly on $x\in\Omega'$. \begin{lem} \label{lemma_1_x} \leavevmode \begin{trivlist} \item[(i)] Let \begin{equation} \label{def_varphi}
\varphi_{\phi_\varepsilon}(x,y,\xi):=|\nabla_{y,\xi}\phi_\varepsilon(x,y,\xi/|\xi|)|^{-2}. \end{equation} If $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_{\Phi}(\Omega';\Omega\times\mb{R}^p)$ then $(\varphi_{\phi_\varepsilon})_\varepsilon\in\mathcal{M}_{S^0_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)}$. \item[(ii)] If $(\phi_\varepsilon)_\varepsilon, (\omega_\varepsilon)_\varepsilon\in\mathcal{M}_\Phi(\Omega';\Omega\times\mb{R}^p)$ and $(\phi_\varepsilon)_\varepsilon\sim(\omega_\varepsilon)_\varepsilon$ then \[ \big(({\partial_{\xi_j}\phi_\varepsilon})\varphi_{\phi_\varepsilon}-({\partial_{\xi_j}\omega_\varepsilon})\varphi_{\omega_\varepsilon}\big)_\varepsilon\in\mathcal{N}_{S^0_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)} \] for all $j=1,...,p$ and \[
\big(({\partial_{y_k}\phi_\varepsilon}){|\xi|^{-2}\varphi_{\phi_\varepsilon}}-({\partial_{y_k}\omega_\varepsilon}){|\xi|^{-2}\varphi_{\omega_\varepsilon}}\big)_\varepsilon\in\mathcal{N}_{S^{-1}_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)} \] for all $k=1,...,n$. \end{trivlist} \end{lem} \begin{prop} \label{prop_operator_x} \leavevmode \begin{trivlist} \item[(i)] If $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_{\Phi}(\Omega';\Omega\times\mb{R}^p)$ then the coefficients occurring in \eqref{smilla} satisfy the following: $(a_{j,\varepsilon})_\varepsilon\in\mathcal{M}_{S^0(\Omega'\times\Omega\times\mb{R}^p)}$ for all $j=1,...,p$, $(b_{k,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{-1}(\Omega'\times\Omega\times\mb{R}^p)}$ for all $k=1,...,n$, and $(c_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-1}(\Omega'\times\Omega\times\mb{R}^p)}$. \item[(ii)] If $(\phi_\varepsilon)_\varepsilon, (\omega_\varepsilon)_\varepsilon\in\mathcal{M}_{\Phi}(\Omega';\Omega\times\mb{R}^p)$ and $(\phi_\varepsilon)_\varepsilon\sim(\omega_\varepsilon)_\varepsilon$ then \begin{equation} \label{L_phi_eps-L_omega_eps_x} L_{\phi_\varepsilon}-L_{\omega_\varepsilon}=\sum_{j=1}^p a'_{j,\varepsilon}(x,y,\xi)\frac{\partial}{\partial_{\xi_j}} +\sum_{k=1}^n b'_{k,\varepsilon}(x,y,\xi)\frac{\partial}{\partial_{y_k}}+ c'_\varepsilon(x,y,\xi), \end{equation} where $(a'_{j,\varepsilon})_\varepsilon\in\mathcal{N}_{S^{0}(\Omega'\times\Omega\times\mb{R}^p)}$, $(b'_{k,\varepsilon})_\varepsilon\in\mathcal{N}_{S^{-1}(\Omega'\times\Omega\times\mb{R}^p)}$ and $(c'_\varepsilon)_\varepsilon\hskip-2pt\in\mathcal{N}_{S^{-1}(\Omega'\times\Omega\times\mb{R}^p)}$ for all $j=1,...,p$ and $k=1,...,n$. \end{trivlist} \end{prop} Proposition \ref{prop_operator_x} yields that any generalized phase function $\phi$ in $\wt{\Phi}(\Omega';\Omega\times\mb{R}^p)$ defines a partial differential operator \begin{equation} \label{def_L_phi_amp} L_{\phi}(x;y,\xi,\partial_y,\partial_\xi)=\sum_{j=1}^p a_{j}(x,y,\xi)\frac{\partial}{\partial_{\xi_j}} +\sum_{k=1}^n b_{k}(x,y,\xi)\frac{\partial}{\partial_{y_k}}+ c(x,y,\xi) \end{equation} with coefficients $a_j\in\wt{\mathcal{S}}^0(\Omega'\times\Omega\times\mb{R}^p)$, $b_k,c\in\wt{\mathcal{S}}^{-1}(\Omega'\times\Omega\times\mb{R}^p)$ such that ${\ }^tL_\phi \mathrm{e}^{i\phi}=\mathrm{e}^{i\phi}$ holds in $\wt{\mathcal{S}}^1_{0,1}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)$. Arguing as in Proposition \ref{prop_exp} we obtain that $\mathrm{e}^{i\phi(x,y,\xi)}$ is a well-defined element of $\wt{\mathcal{S}}^1_{0,1}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)$. The usual composition argument implies that the map \[ \ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\to\wt{\mathcal{S}}^{m-ks+1}_{0,1}(\Omega'\times\Omega\times\mb{R}^p): u\to \mathrm{e}^{i\phi(x,y,\xi)}L^k_{\phi}(a(x,y,\xi)u(y)) \] is continuous.
The oscillatory integral \begin{multline*} I_\phi(a)(u)(x)=\int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(x,y,\xi)}a(x,y,\xi)u(y)\, dy\, \dslash\xi\\ :=\int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(x,y,\xi)}L^k_{\phi}(a(x,y,\xi)u(y))\, dy\, \dslash\xi, \end{multline*} where $\phi\in\wt{\Phi}(\Omega';\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega'\times\Omega\times\mb{R}^p)$ is an element of $\wt{\mb{C}}$ for fixed $x\in\Omega'$. In particular, $I_\phi(a)(u)$ is the integral on $\Omega\times\mb{R}^p$ of a generalized amplitude in $\wt{\mathcal{S}}^{l}_{0,1}(\Omega'\times\Omega\times\mb{R}^p)$ having compact support in $y$. The order $l$ can be chosen arbitrarily low. \begin{thm} \label{theorem_map} Let $\phi\in\wt{\Phi}(\Omega';\Omega\times\mb{R}^p)$, $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega'\times\Omega\times\mb{R}^p)$ and $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$. The generalized oscillatory integral \begin{equation} \label{oscillatory_x} I_{\phi}(a)(u)(x)=\int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(x,y,\xi)}a(x,y,\xi)u(y)\, dy\,\dslash\xi \end{equation} defines a generalized function in $\ensuremath{{\mathcal G}}(\Omega')$ and the map \begin{equation} \label{def_A_fourier} A:\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\to\ensuremath{{\mathcal G}}(\Omega'):u\to I_{\phi}(a)(u) \end{equation} is continuous. \end{thm} The operator $A$ defined in \eqref{def_A_fourier} is called \emph{generalized Fourier integral operator} with amplitude $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega'\times\Omega\times\mb{R}^p)$ and phase function $\phi\in\wt{\Phi}(\Omega';\Omega\times\mb{R}^p)$.
\begin{ex} Our outline of a basic theory of Fourier integral operators with Co\-lom\-be\-au generalized amplitudes and phase functions is motivated to a large extent by potential applications in regularity theory for generalized solutions to hyperbolic partial (or pseudo-) differential equations with distributional or Colombeau-type coefficients (or symbols) and data (cf.\ \cite{HdH:01,LO:91,O:89}). To illustrate the typical situation we consider here the following simple model: let $u\in\ensuremath{{\mathcal G}}(\mb{R}^2)$ be the solution of the generalized Cauchy-problem \begin{align} \label{G_example1} \ensuremath{\partial}_t u + c\,\ensuremath{\partial}_x u + b\,u &= 0\\ u \mid_{t=0} &= g, \end{align} where $g$ belongs to $\ensuremath{{\mathcal G}_\mathrm{c}}(\mb{R})$ and the coefficients $b$, $c\in\ensuremath{{\mathcal G}}(\mb{R}^2)$. Furthermore, $b$, $c$, as well as $\ensuremath{\partial}_x c$ are assumed to be of local $L^\infty$-log-type (concerning growth with respect to the regularization parameter, cf.\ \cite{O:89}), $c$ being generalized real-valued and globally bounded in addition. Let $\gamma \in \ensuremath{{\mathcal G}}(\mb{R}^3)$ be the unique (global) solution of the corresponding generalized characteristic ordinary differential equation \begin{align*} \diff{s} \gamma(x,t;s) &= c(\gamma(x,t;s),s)\\ \gamma(x,t;t) &= x. \end{align*} Then $u$ is given in terms of $\gamma$ by $ u(x,t) = g(\gamma(x,t;0)) \exp(-\int_0^t b(\gamma(x,t;r),r)\, dr)$. Writing $g$ as the inverse of its Fourier transform we obtain the Fourier integral representation \begin{equation}\label{hypsolu}
u(x,t) = \iint \mathrm{e}^{i(\gamma(x,t;0)-y) \xi}\; a(x,t,y,\xi)\, g(y)\, dy\, \dslash\xi, \end{equation} where $a(x,t,y,\xi) := \exp(-\int_0^t b(\gamma(x,t;r),r)\, dr)$ is a generalized amplitude of order $0$. The phase function $\phi(x,t,y,\xi) := (\gamma(x,t;0)-y) \xi$ has (full) gradient $$(\ensuremath{\partial}_x\gamma(x,t;0),\ensuremath{\partial}_t\gamma(x,t;0),-\xi,\gamma(x,t;0)-y)$$ and thus defines a generalized phase function $\phi$. Therefore (\ref{hypsolu}) reads $u = A g$ where $A : \ensuremath{{\mathcal G}_\mathrm{c}}(\mb{R}) \to \ensuremath{{\mathcal G}}(\mb{R}^2)$ is a generalized Fourier integral operator. \end{ex}
\subsection*{Regularity properties} We now investigate the regularity properties of the \emph{generalized Fourier integral operator} $A$. We will prove that for appropriate generalized phase functions and generalized amplitudes, $A$ maps $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)$ into $\ensuremath{\G^\infty}(\Omega')$. The following example shows that a $\ensuremath{\G^\infty}$-kind of regularity assumption for the net $(\phi_\varepsilon)_\varepsilon$ with respect to the parameter $\varepsilon$ does not entail the desired mapping property. \begin{ex}
Let $n=n'=p=1$ and $\Omega=\Omega'=\mb{R}$ and $\phi_\varepsilon(x,y,\xi)=(x-\varepsilon y)\xi$. Then $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}_{\Phi}(\mb{R};\mb{R}\times\mb{R})$ and in particular we have $N=0$ in all moderateness estimates (see Definition \ref{def_phase_x_moderate}$(i)$)) and $|\nabla_{y,\xi}\phi_\varepsilon(x,y,\xi/|\xi|)|^2\ge\varepsilon^2$. Choose the amplitude $a$ identically equal to $1$. The corresponding generalized operator $A$ does not map $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\mb{R})$ into $\ensuremath{\G^\infty}(\mb{R})$. Indeed, for $0\neq f\in\ensuremath{\mathcal{C}^\infty_{\text{c}}}(\mb{R})$ we have that \[ A[(f)_\varepsilon]=\biggl[\biggl(\int_{\mb{R}\times\mb{R}} \mathrm{e}^{i(x-\varepsilon y)\xi}f(y)\, dy\, \dslash\xi\biggr)_\varepsilon\biggr]=[(\varepsilon^{-1}f(x/\varepsilon))_\varepsilon]\in\ensuremath{{\mathcal G}}(\mb{R})\setminus\ensuremath{\G^\infty}(\mb{R}). \] \end{ex} This example suggests that a stronger notion of regularity on generalized phase functions has to be designed. Such is provided by the concept of \emph{slow scale net}. \begin{defn} \label{def_slow_phase} We say that $\phi\in\wt{\Phi}(\Omega';\Omega\times\mb{R}^p)$ is a \emph{slow scale generalized phase function in the variables of $\Omega\times\mb{R}^p$} if it has a representative $(\phi_\varepsilon)_\varepsilon$ fulfilling the conditions \begin{itemize} \item[(i)] $(\phi_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^1_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^p\setminus 0)}$, \item[(ii)] for all $K'\Subset\Omega'$ and $K\Subset\Omega$ the net \eqref{net_FIO} is slow scale-strictly nonzero. \end{itemize} \end{defn} In the sequel the set of all $(\phi_\varepsilon)_\varepsilon\in\Phi[\Omega';\Omega\times\mb{R}^p]$ fulfilling $(i)$ and $(ii)$ in Definition \ref{def_slow_phase} will be denoted by $\mathcal{M}^\mathrm{sc}_{\Phi}(\Omega';\Omega\times\mb{R}^p)$ while we use $\wt{\Phi}^\mathrm{sc}(\Omega';\Omega\times\mb{R}^p)$ for the set of slow scale generalized functions as above. Similarly, using $\nabla_{x,y,\xi}$ in place of $\nabla_{y,\xi}$ in $(ii)$ we define the space $\wt{\Phi}^{\mathrm{sc}}(\Omega'\times\Omega\times\mb{R}^p)$ of slow scale generalized phase functions on $\Omega'\times\Omega\times\mb{R}^p$. We refer to \cite[Section 3]{GHO:06} for the proof of the following theorem.
\begin{thm} \label{theorem_Ginf_map} Let $\phi\in\wt{\Phi}^{\mathrm{sc}}(\Omega';\Omega\times\mb{R}^p)$. \begin{itemize} \item[(i)] If $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega'\times\Omega\times\mb{R}^p)$ the corresponding generalized Fourier integral operator \[ A:u\to\int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(x,y,\xi)}a(x,y,\xi)u(y)\, dy\, \dslash\xi \] maps $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)$ continuously into $\ensuremath{\G^\infty}(\Omega')$. \item[(ii)] If $a\in\wt{\mathcal{S}}^{-\infty,\mathrm{sc}}(\Omega'\times\Omega\times\mb{R}^p)$ then $A$ maps $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ continuously into $\ensuremath{\G^\infty}(\Omega')$. \end{itemize} \end{thm}
\subsection*{Extension to the dual} Finally, we prove that under suitable hypotheses on the generalized phase function $\phi\in \wt{\Phi}(\Omega'\times\Omega\times\mb{R}^p)$, the definition of the generalized Fourier integral operator $A$ can be extended to the dual $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$. \begin{defn} \label{definition_phaseop} We say that $\phi\in\wt{\Phi}(\Omega'\times\Omega\times\mb{R}^p)$ is a generalized operator phase function if it has a representative $(\phi_\varepsilon)_\varepsilon$ of operator phase functions satisfying the conditions (i) and (ii) of Definition \ref{def_phase_x_moderate} and such that \begin{itemize} \item[(iii)] for all $K'\Subset\Omega'$ and $K\Subset\Omega$ the net \[
\biggl(\inf_{x\in K',y\in K,\xi\in\mb{R}^p\setminus 0}\biggl|\nabla_{x,\xi} \phi_\varepsilon\biggl(x,y,\frac{\xi}{|\xi|}\biggr)\biggr|^2\biggr)_\varepsilon \] is strictly nonzero. \end{itemize} \end{defn} It is clear that when $\phi$ is a generalized operator phase function then by Theorem \ref{theorem_map} the oscillatory integral \begin{equation} \label{eq_transposed} \int_{\Omega'\times\mb{R}^p} \mathrm{e}^{i\phi(x,y,\xi)}a(x,y,\xi)v(x)\, dx\, \dslash\xi, \end{equation} where $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega'\times\Omega\times\mb{R}^p)$ and $v\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega')$, defines a generalized function in $\ensuremath{{\mathcal G}}(\Omega)$ and a continuous $\wt{\mb{C}}$-linear operator from $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega')$ to $\ensuremath{{\mathcal G}}(\Omega)$. More precisely, we have the following result. \begin{prop} \label{prop_extension} Let $\phi$ be a generalized operator phase function on $\Omega'\times\Omega\times\mb{R}^p$, $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega'\times\Omega\times\mb{R}^p)$ and $A:\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\to\ensuremath{{\mathcal G}}(\Omega')$ the generalized Fourier integral operator given by \eqref{oscillatory_x}-\eqref{def_A_fourier}. Then, \begin{itemize} \item[(i)] the transposed ${\,}^tA$ of $A$ is the generalized Fourier integral operator given by \eqref{eq_transposed}; \item[(ii)] the operator $A$ can be extended to a continuous $\wt{\mb{C}}$-linear map acting from $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ to $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega'),\wt{\mb{C}})$. \end{itemize} \end{prop} \begin{proof} Working at the level of representatives, the proof of the first assertion is a simple application of the corresponding classical result. It follows that $A$ can be extended to a $\wt{\mb{C}}$-linear map from $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ to $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega'),\wt{\mb{C}})$ by setting \[ A(T)(u)= T({\,}^tAu), \] for all $T\in\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ and $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega')$. Finally, let $B$ a bounded subset of $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega')$. From the continuity of ${\,}^tA$ and $T$ we have that \[
\sup_{u\in B}|A(T)(u)|_\mathrm{e}=\sup_{u\in B}|T({\,}^tAu)|=\sup_{v\in {\,}^tA(B)}|T(v)|, \] where ${\,}^tA(B)$ is a bounded subset of $\ensuremath{{\mathcal G}}(\Omega)$. This shows that $A:\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})\to\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega'),\wt{\mb{C}})$ is continuous. \end{proof}
\section{Composition of a generalized Fourier integral operator with a generalized pseudodifferential operator} \label{section_comp} \subsection*{Generalized Fourier integral operators of the type $F_\omega(b)$} Let $\Omega$ and $\Omega'$ be open subsets of $\mb{R}^n$ and $\mb{R}^{n'}$ respectively. We now focus on operators of the form \begin{equation} \label{def_F_om} F_\omega(b)(u)(x)=\int_{\mb{R}^n}\mathrm{e}^{i\omega(x,\eta)}b(x,\eta)\widehat{u}(\eta)\, \dslash\eta, \end{equation} where $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega'\times\mb{R}^n\setminus 0)$, $b\in\wt{\mathcal{S}}^m(\Omega'\times\mb{R}^n)$ and $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$.\\
Note that $\phi(x,y,\eta):=\omega(x,\eta)-y\eta$ is a well-defined generalized phase function belonging to $\wt{\Phi}(\Omega';\Omega\times\mb{R}^n)$. Indeed, for any $(\omega_\varepsilon)_\varepsilon$ representative of $\omega$ we have that $(\omega_\varepsilon(x,\eta)-y\eta)_\varepsilon\in\mathcal{M}_{S^1_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^n)}$, if $(\omega_\varepsilon-\omega'_\varepsilon)_\varepsilon\in\mathcal{N}_{S^1_{\rm{hg}}(\Omega'\times\mb{R}^n)}$ then $(\omega_\varepsilon-y\eta-\omega'_\varepsilon+y\eta)_\varepsilon\in\mathcal{N}_{S^1_{\rm{hg}}(\Omega'\times\Omega\times\mb{R}^n)}$ and $|\nabla_{y,\eta}\phi(x,y,\eta)|=|(-\eta,\nabla_{\eta}\omega-y)|\ge |\eta|$. In particular it follows that for any representative $\phi_\varepsilon:=\omega_\varepsilon(x,\eta)-y\eta$ and any $K'\Subset\Omega'$, $K\Subset\Omega$, the net $\displaystyle\inf_{x\in K',y\in K,\eta\in\mb{R}^n\setminus 0}|\nabla_{y,\eta}\phi_\varepsilon\big(x,y,\frac{\eta}{|\eta|}\big)|$ is slow scale-strictly non-zero.
We recall that by Lemma \ref{lemma_esp_classic}, the estimate \eqref{coeff_a} and Proposition \ref{prop_exp} \begin{trivlist} \item[-] if $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega'\times\mb{R}^n\setminus 0)$ then $\mathrm{e}^{i\omega(x,\eta)}\in\wt{\mathcal{S}}^1_{0,1}(\Omega'\times\mb{R}^n)$ and \[ \partial^\alpha_\eta\partial^\beta_x\mathrm{e}^{i\omega(x,\eta)}=\mathrm{e}^{i\omega(x,\eta)} a_{\alpha,\beta}(x,\eta), \]
where $a_{\alpha,\beta}\in \wt{\mathcal{S}}^{\,|\beta|}(\Omega'\times\mb{R}^n\setminus 0)$ and the equality is intended in the space $\wt{\mathcal{S}}^{\,1+|\beta|}_{0,1}(\Omega'\times\mb{R}^n\setminus 0)$;
\item[-] if $\omega\in\wt{\mathcal{S}}^{\,1,\mathrm{sc}}_{\rm{hg}}(\Omega'\times\mb{R}^n\setminus 0)$ then $a_{\alpha,\beta}\in \wt{\mathcal{S}}^{\,|\beta|,\mathrm{sc}}(\Omega'\times\mb{R}^n\setminus 0)$. \end{trivlist}
An immediate application of Theorem \ref{theorem_map} and Proposition \ref{prop_extension} yields the following mapping properties. \begin{prop} \label{prop_F_map} \leavevmode \begin{itemize} \item[(i)] If $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega'\times\mb{R}^n\setminus 0)$ and $b\in\wt{\mathcal{S}}^m(\Omega'\times\mb{R}^n)$ then $F_\omega(b)$ maps $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ continuously into $\ensuremath{{\mathcal G}}(\Omega')$. \item[(ii)] If $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega'\times\mb{R}^n\setminus 0)$ has a representative $(\omega_\varepsilon)_\varepsilon\in\Phi[\Omega'\times\mb{R}^n]$ such that for all $K'\Subset\Omega'$ \[
\biggl(\inf_{x\in K',\eta\in\mb{R}^n\setminus 0}|\nabla_{x}\omega_\varepsilon\big(x,\frac{\eta}{|\eta|}\big)|\biggr)_\varepsilon \] is strictly non-zero, then $F_\omega(b)$ can be extended to a continuous $\wt{\mb{C}}$-linear map from $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ to $\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega'),\wt{\mb{C}})$. \item[(iii)] If $\omega\in\wt{\mathcal{S}}^{\,1,\mathrm{sc}}_{\rm{hg}}(\Omega'\times\mb{R}^n\setminus 0)$ and $b\in\wt{\mathcal{S}}^{m,\mathrm{sc}}(\Omega'\times\mb{R}^n)$ then $F_\omega(b)$ maps $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)$ continuously into $\ensuremath{\G^\infty}(\Omega')$. \item[(iv)] If $\mathrm{supp}_x b\Subset\Omega'$ then $F_\omega(b)$ maps $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ into $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega')$ and under the assumptions of {\rm{(ii)}} maps $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ into $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega'),\wt{\mb{C}})$. \end{itemize} \end{prop} \begin{proof} The first assertion is clear from Theorem \ref{theorem_map} and the second one from Proposition \ref{prop_extension}(ii).\\ (iii) Lemma \ref{lemma_esp_classic} and the considerations which precede this proposition entail \[ \partial^\beta_x F_{\omega}(b)(u)(x)=\sum_{\beta'\le\beta}\binom{\beta}{\beta'}\int_{\mb{R}^n}\mathrm{e}^{i\omega(x,\eta)}a_{\beta'}(x,\eta)\partial^{\beta-\beta'}b(x,\eta)\widehat{u}(\eta),\ \dslash\eta, \]
where $a_{\beta'}\in\wt{\mathcal{S}}^{|\beta'|,\mathrm{sc}}(\Omega'\times\mb{R}^n\setminus 0)$. Hence, if $b\in\wt{\mathcal{S}}^{\,m,\mathrm{sc}}(\Omega'\times\mb{R}^n)$ and $u\in\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)$ then $F_\omega(b)\in\ensuremath{\G^\infty}(\Omega')$. Moreover, since for all $\beta\in\mb{N}^{n'}$ and $K'\Subset\Omega'$ there exists $h\in\mb{N}$ and $c>0$ such that for all $g\in\ensuremath{\mathcal{C}^\infty}_{K}(\Omega)$ and $\varepsilon\in(0,1]$ the estimate \[
\sup_{x\in K'}|\partial^\beta F_{\omega_\varepsilon}(b_\varepsilon)(g)(x)|\le c\max_{\beta'\le\beta}|a_{\beta',\varepsilon}|^{(|\beta'|)}_{K',0} |b_\varepsilon|^{(m)}_{K',|\beta|}\sup_{y\in K,|\gamma|\le h}|\partial^\gamma g(y)|, \] holds, we conclude that when $[(a_{\beta',\varepsilon})_\varepsilon]$ and $[(b_\varepsilon)_\varepsilon]$ are symbols of slow scale type then the map $F_\omega(b):\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)\to\ensuremath{\G^\infty}(\Omega')$ is continuous.\\ (iv) If $\mathrm{supp}_x b\Subset\Omega'$ from the first assertion we have that $F_\omega(b)\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega')$. Under the assumptions of $(ii)$ for the phase function $\omega$ we have that ${\,}^tF_\omega(b)$ maps $\ensuremath{{\mathcal G}}(\Omega')$ continuously into $\ensuremath{{\mathcal G}}(\Omega)$ and therefore $F_\omega(b)$ can be extended to a map from $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ to $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega'),\wt{\mb{C}})$.
\end{proof} \begin{rem} Taking $\Omega=\mb{R}^n$ and noting that $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega')\subseteq\ensuremath{{\mathcal G}_\mathrm{c}}(\mb{R}^n)$, it is clear that $F_\omega(b)$ maps $\ensuremath{{\mathcal G}_\mathrm{c}}(\mb{R}^n)$ into $\ensuremath{{\mathcal G}_\mathrm{c}}(\mb{R}^n)$ when $\mathrm{supp}_x b\Subset\Omega'$. In addition, ${\,}^tF_\omega(b):\ensuremath{{\mathcal G}}(\mb{R}^n)\to\ensuremath{{\mathcal G}}(\mb{R}^n)$ and $F_\omega(b):\mathcal{L}(\ensuremath{{\mathcal G}}(\mb{R}^n),\wt{\mb{C}})\to\mathcal{L}(\ensuremath{{\mathcal G}}(\mb{R}^n),\wt{\mb{C}})$. \end{rem} In the sequel we assume $\Omega=\Omega'\subseteq\mb{R}^n$. Our main purpose is to investigate the composition $a(x,D)\circ F_\omega(b)$, where $a(x,D)$ is a generalized pseudodifferential operator and $F_\omega(b)$ a generalized Fourier integral operator as in \eqref{def_F_om}. This requires some technical preliminaries.
\subsection*{Technical preliminaries} The proof of the following lemma can be found in \cite[Lemmas A.11, A.12]{Coriasco:98}. \begin{lem} \label{lemma_sandro_1} Let $a\in\ensuremath{\mathcal{C}^\infty}(\Omega\times\mb{R}^n\setminus 0)$ and $\omega\in\ensuremath{\mathcal{C}^\infty}(\Omega\times\mb{R}^n\setminus 0)$. Then, \begin{multline*}
\partial^\alpha_x\partial^\sigma_\eta( a(x,\nabla_x\omega(x,\eta))=\sum_{\sigma'\le\sigma}\binom{\sigma}{\sigma'}\sum_{|\beta+\gamma|\le|\alpha|}\sum_{|\sigma''|\le|\sigma'|}\partial^\beta_x\partial^{\gamma+\sigma''}_\eta a(x,\nabla_x\omega(x,\eta))\, \cdot\\ \cdot P^{\sigma'}_{\eta,\sigma''}(x,\eta)\partial^{\sigma-\sigma'}_\eta P^\alpha_{x\beta\gamma}(x,\eta), \end{multline*} where \[ \begin{split} P^{\sigma'}_{\eta,\sigma''}(x,\eta)&=1\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \text{if $\sigma'=0$},\\ P^{\sigma'}_{\eta,\sigma''} &=\sum_{\substack{{\delta_1,...,\delta_q}\\ s_1,...,s_q}}c^{s_1,...,s_q}_{\delta_1,...,\delta_q}\,\partial^{\delta_1}_\eta \partial_{x_{s_1}}\omega(x,\eta)...\partial^{\delta_q}_\eta\partial_{x_{s_q}}\omega(x,\eta)\quad \text{otherwise}, \end{split} \]
with $q=|\sigma''|$, $\sum_{j=1}^q|\delta_j|=|\sigma'|$ and \[ \begin{split} P^\alpha_{x\beta\gamma}&=1\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \text{if $\gamma=0$},\\ P^\alpha_{x\beta\gamma} &=\sum_{\substack{{\delta_1,...,\delta_r}\\ s_1,...,s_r}}d^{s_1,...,s_r}_{\delta_1,...,\delta_r}\,\partial^{\delta_1}_x\partial_{x_{s_1}}\omega(x,\eta)...\partial^{\delta_r}_x\partial_{x_{s_r}}\omega(x,\eta)\quad \text{otherwise}, \end{split} \]
with $|\gamma|=r$ and $\sum_{j=1}^r|\delta_j|+|\beta|=|\alpha|$. \end{lem} \begin{prop} \label{prop_a_eps} \leavevmode \begin{itemize} \item[(h1)] Let $(\omega_\varepsilon)_\varepsilon\in\mathcal{M}_{S^1_{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)$ such that $\nabla_x\omega_\varepsilon\neq 0$ for all $\varepsilon\in(0,1]$ and for all $K\Subset\Omega$ \[
\biggl(\inf_{x\in K,\eta\in\mb{R}^n\setminus 0}\biggl|\nabla_{x}\omega_\varepsilon\big(x,\frac{\eta}{|\eta|}\big)\biggr|\biggr)_\varepsilon \] is strictly non-zero. \item[(i)] If $(a_\varepsilon)_\varepsilon\in \mathcal{M}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ then $(a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta)))_\varepsilon\in \mathcal{M}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$; \item[(ii)] if $(a_\varepsilon)_\varepsilon\in \mathcal{N}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ then $(a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta)))_\varepsilon\in \mathcal{N}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$.\\ \item[(h2)] Let $(\omega_\varepsilon)_\varepsilon\in\mathcal{M}^{\mathrm{sc}}_{S^1_{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)$ such that $\nabla_x\omega_\varepsilon\neq 0$ for all $\varepsilon\in(0,1]$ and for all $K\Subset\Omega$ \[
\biggl(\inf_{x\in K,\eta\in\mb{R}^n\setminus 0}\biggl|\nabla_{x}\omega_\varepsilon\big(x,\frac{\eta}{|\eta|}\big)\biggr|\biggr)_\varepsilon \] is slow scale strictly non-zero. \item[(iii)] If $(a_\varepsilon)_\varepsilon\in \mathcal{M}^{\mathrm{sc}}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ then $(a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta)))_\varepsilon\in \mathcal{M}^{\mathrm{sc}}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$. \item[(h3)] Finally, let $(\omega_\varepsilon-\omega'_\varepsilon)_\varepsilon\in\mathcal{N}_{S^1_{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)$ with $(\omega_\varepsilon)_\varepsilon$ and $(\omega'_\varepsilon)_\varepsilon$ satisfying the hypothesis $(h1)$ above. \item[(iv)] If $(a_\varepsilon)_\varepsilon\in \mathcal{M}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ then \[ (a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))-a_\varepsilon(x,\nabla_x\omega'_\varepsilon(x,\eta)))_\varepsilon\in \mathcal{N}_{S^m(\Omega\times\mb{R}^n\setminus 0)}. \] \end{itemize} \end{prop} \begin{proof} From Lemma \ref{lemma_sandro_1} it follows that $\partial^\alpha_x\partial^\sigma_\eta( a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))$ is a finite sum of terms of the type \[ \partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))g_{\alpha',\sigma',\varepsilon}(x,\eta), \]
where $(g_{\alpha',\sigma',\varepsilon})_\varepsilon$ is a net of symbols in $S^{|\sigma'|-|\sigma|}(\Omega\times\mb{R}^n\setminus 0)$. Note that $(g_{\alpha',\sigma',\varepsilon})_\varepsilon$ depends on $(\omega_\varepsilon)_\varepsilon$ and is actually a finite sum of products of derivatives of $(\omega_\varepsilon)_\varepsilon$. One can easily prove that \begin{equation} \label{g} \begin{split}
(\omega_\varepsilon)_\varepsilon\in\mathcal{M}_{S^1_{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)\quad &\Rightarrow \quad (g_{\alpha',\sigma',\varepsilon})_\varepsilon\in\mathcal{M}_{S^{|\sigma'|-|\sigma|}(\Omega\times\mb{R}^n\setminus 0)},\\
(\omega_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^1_{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)\quad &\Rightarrow \quad (g_{\alpha',\sigma',\varepsilon})_\varepsilon\in\mathcal{M}^{\mathrm{sc}}_{S^{|\sigma'|-|\sigma|}(\Omega\times\mb{R}^n\setminus 0)}.
\end{split} \end{equation} and that the following \begin{multline} \label{a} \forall\alpha',\sigma'\in\mb{N}^n\, \forall K\Subset\Omega\, \exists (\lambda_\varepsilon)_\varepsilon\in\mb{R}^{(0,1]}\, \forall x\in K\, \forall\eta\in\mb{R}^n\setminus 0\, \forall\varepsilon\in(0,1]\\
|\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))|\le\lambda_\varepsilon\lara{\nabla_x\omega_\varepsilon}^{m-|\sigma'|} \end{multline}
holds, with $(\lambda_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ if $(a_\varepsilon)_\varepsilon\in\mathcal{M}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$, $(\lambda_\varepsilon)_\varepsilon$ slow scale net if $(a_\varepsilon)_\varepsilon\in\mathcal{M}^{\mathrm{sc}}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$ and $(\lambda_\varepsilon)_\varepsilon\in\mathcal{N}$ if $(a_\varepsilon)_\varepsilon\in\mathcal{N}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$. Now, let us consider $(\nabla_x\omega_\varepsilon(x,\eta))_\varepsilon$. We have that \begin{equation*} \label{nabla} \begin{array}{cc}
(h1)\quad \Rightarrow\quad &\forall K\Subset\Omega\, \exists r>0\, \exists c_1,c_2>0\, \exists\eta\in(0,1]\, \forall x\in K\, \forall|\eta|\ge 1\, \forall\varepsilon\in(0,\eta]\\[0.2cm]
&\lara{\eta}c_1\varepsilon^r\le|\nabla_x\omega_\varepsilon(x,\eta)|\le c_2\varepsilon^{-r}\lara{\eta}, \\[0.3cm]
(h2)\quad \Rightarrow\quad &\forall K\Subset\Omega\, \exists (\mu_\varepsilon)_\varepsilon\, {\rm{s.s.n}}\, \exists\eta\in(0,1]\, \forall x\in K\, \forall|\eta|\ge 1\, \forall\varepsilon\in(0,\eta]\\[0.2cm]
&\lara{\eta}\mu_\varepsilon^{-1}\le|\nabla_x\omega_\varepsilon(x,\eta)|\le \mu_\varepsilon\lara{\eta}. \end{array} \end{equation*} Under the hypothesis $(h1)$, combining \eqref{g} with \eqref{a} we obtain the assertions $(i)$ and $(ii)$. Moreover, from the second implications of \eqref{g} and \eqref{a} we see that $(h2)$ yields $(iii)$. It remains to prove that if $(h3)$ holds and $(a_\varepsilon)_\varepsilon$ is a moderate net of symbols then $(a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))-a_\varepsilon(x,\nabla_x\omega'_\varepsilon(x,\eta)))_\varepsilon\in \mathcal{N}_{S^m(\Omega\times\mb{R}^n\setminus 0)}$. If suffices to write $\partial^\alpha_x\partial^\sigma_\eta(a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))-a_\varepsilon(x,\nabla_x\omega'_\varepsilon(x,\eta)))$ as the finite sum \begin{multline} \label{sum_compl} \sum_{\alpha',\sigma'}\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))(g_{\alpha',\sigma'}(\omega_\varepsilon)-g_{\alpha',\sigma'}(\omega'_\varepsilon))\\ +\sum_{\alpha',\sigma'}[\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))-\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega'_\varepsilon(x,\eta))]g_{\alpha',\sigma'}(\omega'_\varepsilon). \end{multline}
An inspection of Lemma \ref{lemma_sandro_1} shows that the net $(g_{\alpha',\sigma'}(\omega_\varepsilon)-g_{\alpha',\sigma'}(\omega'_\varepsilon))_\varepsilon$ belongs to $\mathcal{N}_{S^{|\sigma'|-|\sigma|}(\Omega\times\mb{R}^n\setminus 0)}$ and from the hypothesis $(h1)$ on $(\omega_\varepsilon)_\varepsilon$ it follows that the first summand in \eqref{sum_compl} is an element of $\mathcal{N}_{S^{m-|\sigma|}(\Omega\times\mb{R}^n\setminus 0)}$. We use Taylor's formula on the second summand of \eqref{sum_compl}. Therefore, for $x$ varying in a compact set $K$ and for $\varepsilon$ small enough we can estimate \[
|\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))-\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega'_\varepsilon(x,\eta))| \] by means of \begin{multline*}
\sum_{j=1}^n\varepsilon^{-N}\lara{\nabla_x\omega'_\varepsilon(x,\eta)+\theta(\nabla_x\omega_\varepsilon(x,\eta)-\nabla_x\omega'_\varepsilon(x,\eta))}^{m-|\sigma'|-1}|\partial_{x_j}(\omega_\varepsilon-\omega'_\varepsilon)(x,\eta)|\\
\le \varepsilon^q\lara{\nabla_x\omega'_\varepsilon(x,\eta)+\theta(\nabla_x\omega_\varepsilon(x,\eta)-\nabla_x\omega'_\varepsilon(x,\eta))}^{m-|\sigma'|-1}\lara{\eta}, \end{multline*}
where $\theta\in[0,1]$. Since, taking $\varepsilon$ small and $|\eta|\ge 1$ the following inequalities \begin{multline*}
|{\nabla_x\omega'_\varepsilon(x,\eta)+\theta(\nabla_x\omega_\varepsilon(x,\eta)-\nabla_x\omega'_\varepsilon(x,\eta))}|\ge \varepsilon^r\lara{\eta}-\varepsilon^{r+1}\lara{\eta}\ge\frac{\varepsilon^{r}}{2}\lara{\eta},\\
|{\nabla_x\omega'_\varepsilon(x,\eta)+\theta(\nabla_x\omega_\varepsilon(x,\eta)-\nabla_x\omega'_\varepsilon(x,\eta))}|\le \varepsilon^{-r}\lara{\eta} \end{multline*} hold for some $r>0$, we conclude that \[
\big(\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))-\partial^{\alpha'}_x\partial^{\sigma'}_\eta a_\varepsilon(x,\nabla_x\omega'_\varepsilon(x,\eta))\big)_\varepsilon\in\mathcal{N}_{S^{m-|\sigma'|}(\Omega\times\mb{R}^n\setminus 0)}. \]
Thus, from $(g_{\alpha',\sigma'}(\omega'_\varepsilon))_\varepsilon\in\mathcal{M}_{S^{|\sigma'|-|\sigma|}(\Omega\times\mb{R}^n\setminus 0)}$ we have that the second summand of \eqref{sum_compl} belongs to $\mathcal{N}_{S^{m-|\sigma|}(\Omega\times\mb{R}^n\setminus 0)}$ and the proof is complete. \end{proof} \begin{cor} \label{coroll_a} If $a\in\wt{\mathcal{S}}^m(\Omega\times\mb{R}^n\setminus 0)$ and $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ has a representative satisfying condition $(h1)$ of Proposition \ref{prop_a_eps} then $a(x,\nabla_x\omega(x,\eta))\in \wt{\mathcal{S}}^m(\Omega\times\mb{R}^n\setminus 0)$.\\ If $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$ and $\omega\in\wt{\mathcal{S}}^{1,\mathrm{sc}}_{{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)$ has a representative satisfying condition $(h2)$ of Proposition \ref{prop_a_eps}, then $a(x,\nabla_x\omega(x,\eta))\in \wt{\mathcal{S}}^{m,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$. \end{cor} Let $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ have a representative satisfying $(h1)$. We want to investigate the properties of \begin{equation} \label{esp}
D^\beta_z\big(\mathrm{e}^{i\overline{\omega}(z,x,\eta)}\big)|_{z=x}, \end{equation} where $\overline{\omega}(z,x,\eta):=\omega(z,\eta)-\omega(x,\eta)-\lara{\nabla_x\omega(x,\eta),z-x}$. We make use of the following technical lemma, whose proof can be found in \cite[Proposition 15]{Coriasco:99}. \begin{lem} \label{lemma_sandro_2}
Let $\omega\in\ensuremath{\mathcal{C}^\infty}(\Omega\times\mb{R}^n\setminus 0)$ and $\overline{\omega}(z,x,\eta)$ as above. Then, for $|\beta|\neq 0$,we have \begin{multline*} D^\beta_z\mathrm{e}^{i\overline{\omega}(z,x,\eta)}=\mathrm{e}^{i\overline{\omega}(z,x,\eta)}\biggl[(\nabla_z\omega(z,\eta)-\nabla_x\omega(x,\eta))^\beta\\ +\sum_{j_1}c_{j_1}(\nabla_z\omega(z,\eta)-\nabla_x\omega(x,\eta))^{\theta_{j_1}}\prod_{j_2=1}^{n_{1,j_1}}\partial^{\gamma_{j_1,j_2}}_z\omega(z,\eta)\\ +\sum_{j_1}c'_{j_1}\prod_{j_2=1}^{n_{2,j_1}}\partial^{\delta_{j_1,j_2}}_z\omega(z,\eta)\biggr], \end{multline*}
where $c_{j_1}$, $c'_{j_1}$ are suitable constants, $|\gamma_{j_1,j_2}|\ge 2$, $|\delta_{j_1,j_2}|\ge 2$ and \[ \theta_{j_1}+\sum_{j_2=1}^{n_{1,j_1}}\gamma_{j_1,j_2}=\sum_{j_2=1}^{n_{2,j_1}}\delta_{j_1,j_2}=\beta. \] \end{lem} It follows that \begin{equation} \label{formula_esp}
D^\beta_z\big(\mathrm{e}^{i\overline{\omega}(z,x,\eta)}\big)|_{z=x}= \sum_{j_1}c_{j_1}\prod_{j_2=1}^{n_{1,j_1}}\partial^{\gamma_{j_1,j_2}}_x\omega(x,\eta) +\sum_{j_1}c'_{j_1}\prod_{j_2=1}^{n_{2,j_1}}\partial^{\delta_{j_1,j_2}}_x\omega(x,\eta), \end{equation} with \[ \sum_{j_2=1}^{n_{1,j_1}}\gamma_{j_1,j_2}=\sum_{j_2=1}^{n_{2,j_1}}\delta_{j_1,j_2}=\beta. \]
Moreover, from $|\gamma_{j_1,j_2}|\ge 2$, $|\delta_{j_1,j_2}|\ge 2$ we have $|\beta|\ge 2n_{1,j_1}$ and $|\beta|\ge 2n_{2,j_1}$.
Since the constants $c_{j_1}$, $c'_{j_1}$ do not depend on $\omega$, we can use the formula \eqref{formula_esp} in estimating the net $(D^\beta_z\big(\mathrm{e}^{i\overline{\omega_\varepsilon}(z,x,\eta)}\big)|_{z=x})_\varepsilon$. \begin{prop} \label{prop_esp} \leavevmode \begin{trivlist}
\item[(i)] If $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ then \eqref{esp} is a well-defined element of $\wt{\mathcal{S}}^{|\beta|/2}(\Omega\times\mb{R}^n\setminus 0)$.
\item[(ii)] If $\omega\in\wt{\mathcal{S}}^{1,\mathrm{sc}}_{{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)$ then \eqref{esp} is a well-defined element of $\wt{\mathcal{S}}^{|\beta|/2,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$. \end{trivlist} \end{prop} \begin{proof} From \eqref{formula_esp} we have that \[ \begin{split}
(\omega_\varepsilon)_\varepsilon\in\mathcal{M}_{S^1_{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)\quad &\Rightarrow \quad (D^\beta_z\big(\mathrm{e}^{i\overline{\omega_\varepsilon}(z,x,\eta)}\big)|_{z=x})_\varepsilon\in\mathcal{M}_{S^{|\beta|/2}(\Omega\times\mb{R}^n\setminus 0)},\\
(\omega_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^1_{\rm{hg}}}(\Omega\times\mb{R}^n\setminus 0)\quad &\Rightarrow \quad (D^\beta_z\big(\mathrm{e}^{i\overline{\omega_\varepsilon}(z,x,\eta)}\big)|_{z=x})_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{|\beta|/2}(\Omega\times\mb{R}^n\setminus 0)}. \end{split} \] Noting that $(\omega_\varepsilon-\omega'_\varepsilon)_\varepsilon\in\mathcal{N}_{S^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)}$ entails \[ \begin{split}
&\biggl(\prod_{j_2=1}^{n_{1,j_1}}\partial^{\gamma_{j_1,j_2}}_x\omega_\varepsilon(x,\eta)-\prod_{j_2=1}^{n_{1,j_1}}\partial^{\gamma_{j_1,j_2}}_x\omega'_\varepsilon(x,\eta)\biggr)_\varepsilon\in\mathcal{N}_{S^{|\beta|/2}(\Omega\times\mb{R}^n\setminus 0)},\\
&\biggl(\prod_{j_2=1}^{n_{2,j_1}}\partial^{\delta_{j_1,j_2}}_x\omega_\varepsilon(x,\eta)-\prod_{j_2=1}^{n_{2,j_1}}\partial^{\delta_{j_1,j_2}}_x\omega'_\varepsilon(x,\eta)\biggr)_\varepsilon\in\mathcal{N}_{S^{|\beta|/2}(\Omega\times\mb{R}^n\setminus 0)}, \end{split} \]
we conclude that the net $\big(D^\beta_z\big(\mathrm{e}^{i\overline{\omega_\varepsilon}(z,x,\eta)}\big)|_{z=x}-D^\beta_z\big(\mathrm{e}^{i\overline{\omega'_\varepsilon}(z,x,\eta)}\big)|_{z=x}\big)_\varepsilon$ belongs to $\mathcal{N}_{S^{|\beta|/2}(\Omega\times\mb{R}^n\setminus 0)}$. \end{proof} By combining Corollary \ref{coroll_a} with Proposition \ref{prop_esp} we obtain the following statement. \begin{prop} \label{prop_h_alpha} Let $\alpha\in\mb{N}^n$ and \begin{equation} \label{h_alpha}
h_\alpha(x,\eta)=\frac{\partial^\alpha_\xi a(x,\nabla_x\omega(x,\eta))}{\alpha!}D^\alpha_z\big(\mathrm{e}^{i\overline{\omega}(z,x,\eta)}b(z,\eta)\big)|_{z=x}. \end{equation} \begin{trivlist}
\item[(i)] If $a\in\wt{\mathcal{S}}^m(\Omega\times\mb{R}^n\setminus 0)$, $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ has a representative satisfying condition $(h1)$ and $b\in\wt{\mathcal{S}}^l(\Omega\times\mb{R}^n\setminus 0)$, then $h_\alpha\in\wt{\mathcal{S}}^{\,l+m-|\alpha|/2}(\Omega\times\mb{R}^n\setminus 0)$ for all $\alpha$.
\item[(ii)] If $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$, $\omega\in\wt{\mathcal{S}}^{\,1,\mathrm{sc}}_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ has a representative satisfying condition $(h2)$ and $b\in\wt{\mathcal{S}}^{\,l,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$, then $h_\alpha\in\wt{\mathcal{S}}^{\,l+m-|\alpha|/2,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$ for all $\alpha$. \end{trivlist} \end{prop} Our next task is to give a closer look to $\mathrm{e}^{i\omega(x,\eta)}$. \begin{prop} \label{prop_id_esp_1} Let $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ have a representative satisfying condition $(h1)$. Then for any positive integer $N$ there exists $p_N\in\wt{\mathcal{S}}^{-2N}(\Omega\times\mb{R}^n\setminus 0)$ such that \begin{equation} \label{id_esp_1} \mathrm{e}^{i\omega(x,\eta)}=\biggl(p_N(x,\eta)\Delta_x^N+r(x,\eta)\biggr)\mathrm{e}^{i\omega(x,\eta)}, \end{equation} where $r\in\wt{\mathcal{S}}^{-\infty}(\Omega\times\mb{R}^n\setminus 0)$.\\ If $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ is of slow scale type and has a representative satisfying condition $(h2)$ then $p_N$ and $r$ are of slow scale type. \end{prop} \begin{proof} Let $(\omega_\varepsilon)_\varepsilon$ be a representative of $\omega$ satisfying $(h1)$. We leave to the reader to prove by induction that \[ \Delta_x^N\big(\mathrm{e}^{i\omega_\varepsilon(x,\eta)}\big)=a_\varepsilon(x,\eta)\mathrm{e}^{i\omega_\varepsilon(x,\eta)}, \] where $(a_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{2N}(\Omega\times\mb{R}^n\setminus 0)}$ with principal part given by \[
a_{2N,\varepsilon}=(-1)^N|\nabla_x\omega_\varepsilon(x,\eta)|^{2N}. \]
From $(h1)$ we have that $\nabla_x\omega_\varepsilon\neq 0$ for all $\varepsilon\in(0,1]$ and for all $K\Subset\Omega$ there exist $r>0$ and $\varepsilon_0\in(0,1]$ such that $|\nabla_x\omega_\varepsilon(x,\eta)|\ge\varepsilon^r|\eta|$ for all $x\in K$, $\eta\neq 0$ and $\varepsilon\in(0,\varepsilon_0]$. Hence, \[
|\nabla_x\omega_\varepsilon(x,\eta)|\ge\frac{\varepsilon^r}{2}\lara{\eta}, \]
for $|\eta|\ge 1$, $x\in K$ and $\varepsilon\in(0,\varepsilon_0]$. It follows from Proposition \ref{prop_ellip}$(iii)$ that $(a_\varepsilon)_\varepsilon$ is a net of elliptic symbols of $S^{2N}(\Omega\times\mb{R}^n\setminus 0)$ such that for all $K\Subset\Omega$ there exist $s\in \mb{R}$, $(R_\varepsilon)_\varepsilon$ strictly nonzero and $\varepsilon_0\in(0,1]$ such that \[
|a_\varepsilon(x,\eta)|\ge \varepsilon^s\lara{\eta}^{2N}, \]
for $x\in K$, $|\eta|\ge R_\varepsilon$ and $\varepsilon\in(0,\varepsilon_0]$. By Proposition \ref{prop_ellip_2}$(i)$ we find $(p_{N,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{-2N}(\Omega\times\mb{R}^n\setminus 0)}$ and $(r_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$ such that \begin{equation} \label{eq_repr} p_{N,\varepsilon}a_\varepsilon=1-r_\varepsilon \end{equation} for all $\varepsilon$. Therefore, \[ \mathrm{e}^{i\omega_\varepsilon(x,\eta)}=\biggl(p_{N,\varepsilon}(x,\eta)\Delta_x^N+r_\varepsilon(x,\eta)\biggr)\mathrm{e}^{i\omega_\varepsilon(x,\eta)}. \] This equality at the representatives'level implies the equality \eqref{id_esp_1} between equivalence classes of $\wt{\mathcal{S}}^{1}_{0,1}(\Omega\times\mb{R}^n\setminus 0)$.
Now, let $\omega$ be a slow scale symbol with a representative $(\omega_\varepsilon)_\varepsilon$ satisfying condition $(h2)$. From Proposition \ref{prop_ellip}$(vi)$ we have that $(a_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{2N}(\Omega\times\mb{R}^n\setminus 0)}$ is a net of elliptic symbols such that for some $(s_\varepsilon)_\varepsilon$ inverse of a slow scale net, $(R_\varepsilon)_\varepsilon$ slow scale net and $\varepsilon_0\in(0,1]$ the inequality \[
|a_\varepsilon(x,\eta)|\ge s_\varepsilon\lara{\eta}^{2N}, \]
holds for all $x\in K$, $|\eta|\ge R_\varepsilon$ and $\varepsilon\in(0,\varepsilon_0]$. Proposition \ref{prop_ellip_2}$(ii)$ shows that \eqref{eq_repr} is true for some $(p_{N,\varepsilon})_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{-2N}(\Omega\times\mb{R}^n\setminus 0)}$ and $(r_\varepsilon)_\varepsilon\in\mathcal{M}^\mathrm{sc}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$. \end{proof} \subsection*{Main theorems} The make use of the previous propositions in proving the main theorems of this section: Theorems \ref{theo_comp} and \ref{theo_comp_ssc}. \begin{thm} \label{theo_comp} Let $\omega\in\wt{\mathcal{S}}^1_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ have a representative satisfying condition $(h1)$. Let $a\in\wt{\mathcal{S}}^m(\Omega\times\mb{R}^n)$ and $b\in\wt{\mathcal{S}}^l(\Omega\times\mb{R}^n\setminus 0)$ with $\mathrm{supp}_x\, b\Subset\Omega$. Then, the operator $a(x,D)F_\omega(b)$ has the following properties: \begin{itemize} \item[(i)] maps $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ into $\ensuremath{{\mathcal G}}(\Omega)$ and $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ into $\mathcal{L}(\ensuremath{{\mathcal G}}_c(\Omega),\wt{\mb{C}})$; \item[(ii)] is of the form \[ \int_{\mb{R}^n}\mathrm{e}^{i\omega(x,\eta)}h(x,\eta)\widehat{u}(\eta)\, \dslash\eta +r(x,D)u, \] where $h\in\wt{\mathcal{S}}^{\,l+m}(\Omega\times\mb{R}^n\setminus 0)$ has asymptotic expansion given by the symbols $h_\alpha$ defined in \eqref{h_alpha} and $r\in\wt{\mathcal{S}}^{-\infty}(\Omega\times\mb{R}^n\setminus 0)$. \end{itemize} \end{thm} \begin{proof} From Proposition \ref{prop_F_map}$(iv)$ is clear that $F_\omega(b)$ maps $\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$ and $\mathcal{L}(\ensuremath{{\mathcal G}}(\Omega),\wt{\mb{C}})$ into themselves respectively. We obtain $(i)$ combining this results with the usual mapping properties of a generalized pseudodifferential operator. We now have to investigate the composition \begin{multline*} a(x,D)F_\omega(b)u(x)=\int_{\Omega\times\mb{R}^n}\mathrm{e}^{i(x-z)\theta}a(x,\theta){F_\omega(b)u}(z)\, dz\,\dslash\theta\\ = \int_{\Omega\times\mb{R}^n}\mathrm{e}^{i(x-z)\theta}a(x,\theta)\biggl(\int_{\mb{R}^n}\mathrm{e}^{i\omega(z,\eta)}b(z,\eta)\widehat{u}(\eta)\, \dslash\eta\biggr)\, dz\,\dslash\theta\\ = \int_{\mb{R}^n}\int_{\Omega\times\mb{R}^n}\mathrm{e}^{i((x-z)\theta+\omega(z,\eta))}a(x,\theta)b(z,\eta)\, dz\, \dslash\theta\, \widehat{u}(\eta)\, \dslash\eta, \end{multline*} for $u\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)$. The last integral in $dz$ and $\dslash\theta$ is regarded as the oscillatory integral \begin{equation} \label{int_1_sum} \int_{\Omega\times\mb{R}^n}\mathrm{e}^{i(x-z)\theta}a(x,\theta)b(z,\eta)\mathrm{e}^{i\omega(z,\eta)}\, dz\, \dslash\theta, \end{equation} with $b(z,\eta)\mathrm{e}^{i\omega(z,\eta)}\in\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega_z)$.
In the sequel we will work at the level of representatives and we will follow the proof of Theorem 4.1.1 in \cite{MR:97}.\\
\bf{Step 1.}\rm\, Let$(\sigma_\varepsilon)_\varepsilon$ such that $\sigma_\varepsilon\ge c\varepsilon^s$ for some $c,s>0$ and for all $\varepsilon\in(0,1]$. We take $\varphi\in\ensuremath{\mathcal{C}^\infty}(\mb{R}^n)$ such that $\varphi(y)=1$ for $|y|\le 1/2$ and $\varphi(y)=0$ for $|y|\ge 1$ and we set \[ b_\varepsilon(z,\eta)= b'_\varepsilon(z,x,\eta)+b''_\varepsilon(z,x,\eta)= \varphi\big(\frac{x-z}{\sigma_\varepsilon}\big)b_\varepsilon(z,\eta)+\big(1-\varphi\big(\frac{x-z}{\sigma_\varepsilon}\big)\big)b_\varepsilon(z,\eta). \] We now write the integral in $dz$ and $\dslash\theta$ of \eqref{int_1_sum} as \begin{multline*} \int_{\Omega\times\mb{R}^n}\mathrm{e}^{i((x-z)\theta+\omega_\varepsilon(z,\eta))}a_\varepsilon(x,\theta)b'_\varepsilon(z,x,\eta)\, dz\, \dslash\theta\\ +\int_{\Omega\times\mb{R}^n}\mathrm{e}^{i((x-z)\theta+\omega_\varepsilon(z,\eta))}a_\varepsilon(x,\theta)b''_\varepsilon(z,x,\eta)\, dz\, \dslash\theta := I_{1,\varepsilon}(x,\eta)+I_{2,\varepsilon}(x,\eta) \end{multline*} and we begin to investigate the properties of $(I_{2,\varepsilon})_\varepsilon$. Proposition \ref{prop_id_esp_1} provides the identity \[ \mathrm{e}^{i\omega_\varepsilon(z,\eta)}=\biggl(p_{N,\varepsilon}(z,\eta)\Delta_z^N+r_\varepsilon(z,\eta)\biggr)\mathrm{e}^{i\omega_\varepsilon(z,\eta)}, \] where $(p_{N,\varepsilon})_\varepsilon\in \mathcal{M}_{S^{-2N}(\Omega\times\mb{R}^n\setminus 0)}$ and $(r_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$, and allows us to write $(I_{2,\varepsilon})_\varepsilon$ as \begin{multline*} \int_{\Omega\times\mb{R}^n}\mathrm{e}^{i\omega_\varepsilon(z,\eta)}\Delta_z^N\biggl(\mathrm{e}^{i(x-z)\theta}p_{N,\varepsilon}(z,\eta)b''_\varepsilon(z,x,\eta)\biggr)a_\varepsilon(x,\theta)\, dz\, \dslash\theta\\ +\int_{\Omega\times\mb{R}^n}\mathrm{e}^{i(x-z)\theta}a_\varepsilon(x,\theta)b''_\varepsilon(z,x,\eta)r_\varepsilon(z,\eta)\mathrm{e}^{i\omega_\varepsilon(z,\eta)}\, dz\, \dslash\theta := I^1_{2,\varepsilon}(x,\eta)+I^2_{2,\varepsilon}(x,\eta) \end{multline*} The net $(I^2_{2,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{-\infty}}(\Omega\times\mb{R}^n\setminus 0)$. Indeed, $I^2_{2,\varepsilon}(x,\eta)=\int_{\mb{R}^n}\int_{\mb{R}^n}g_{\varepsilon}(x,\eta,z,\theta)\, dz\, \dslash\theta$, where \[ g_\varepsilon(x,\eta,z,\theta)=\mathrm{e}^{i(x-z)\theta}a_\varepsilon(x,\theta)b''_\varepsilon(z,x,\eta)r_\varepsilon(z,\eta)\mathrm{e}^{i\omega_\varepsilon(z,\eta)} \] and the following holds: for all $K\Subset\Omega$, for all $\alpha\in\mb{N}^n$ and $d>0$ exist $N\in\mb{N}$ and $\varepsilon_0\in(0,1]$ such that \[
\biggl|(i\theta)^\alpha\int_{\mb{R}^n}g_{\varepsilon}(x,\eta,z,\theta)\, dz\biggr|\le \varepsilon^{-N}\lara{\theta}^m\lara{\eta}^{-d}. \] This is due to the fact that $\mathrm{supp}_z b_\varepsilon\subseteq K_b\Subset\Omega$ for all $\varepsilon$ and $(r_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$.\\
\bf{Step 2.}\rm\, By construction $b''_\varepsilon(z,x,\eta)=0$ if $|x-z|\le\sigma_\varepsilon/2$ for all $\varepsilon\in(0,1]$. By making use of the identity \[
\mathrm{e}^{i(x-z)\theta}=|x-z|^{-2k}(-\Delta_\theta)^k\biggl(\mathrm{e}^{i(x-z)\theta}\biggr) \] we have \begin{multline*} I^1_{2,\varepsilon}(x,\eta)\\
= \int_{\Omega\times\mb{R}^n}\hskip-15pt\mathrm{e}^{i\omega_\varepsilon(z,\eta)}\Delta_z^N\biggl(|x-z|^{-2k}(-\Delta_\theta)^k\biggl(\mathrm{e}^{i(x-z)\theta}\biggr)p_{N,\varepsilon}(z,\eta)b''_\varepsilon(z,x,\eta)\biggr)a_\varepsilon(x,\theta)\, dz\, \dslash\theta\\ =
\int_{\Omega\times\mb{R}^n}\hskip-10pt\mathrm{e}^{i\omega_\varepsilon(z,\eta)}(-\Delta_\theta)^k a_\varepsilon(x,\theta)\Delta_z^N\biggl(\mathrm{e}^{i(x-z)\theta}|x-z|^{-2k}p_{N,\varepsilon}(z,\eta)b''_\varepsilon(z,x,\eta)\biggr)\, dz\, \dslash\theta. \end{multline*} It follows that for $x\in K\Subset\Omega$ and $\varepsilon$ small enough \begin{multline*}
|I^1_{2,\varepsilon}(x,\eta)|\le c\varepsilon^{-N_a-N_p-N_b}(\sigma_\varepsilon)^{-2k}\sum_{|\gamma|\le 2N}c_\gamma\sigma_\varepsilon^{-|\gamma|}\int_{\mb{R}^n}\lara{\theta}^{m-2k+2N}\, d\theta\,\lara{\eta}^{-2N+l} \\ \le \varepsilon^{-N'}\int_{\mb{R}^n}\lara{\theta}^{m-2k+2N}\, d\theta\,\lara{\eta}^{-2N+l}. \end{multline*} Hence, given $d>0$ and taking $N,k$ such that $-2N+l<-d$ and $m-2k+2N<-n$ we obtain that $(I^1_{2,\varepsilon})_\varepsilon$ is a moderate net of symbols of order $-\infty$ on $\Omega\times\mb{R}^n\setminus 0$. Summarizing, \[ \int_{\Omega\times\mb{R}^n}\mathrm{e}^{i((x-z)\theta+\omega_\varepsilon(z,\eta))}a_\varepsilon(x,\theta)b_\varepsilon(z,\eta)\, dz\, \dslash\theta =I_{1,\varepsilon}(x,\eta)+I^1_{2,\varepsilon}(x,\eta)+I^2_{2,\varepsilon}(x,\eta), \] where $(I^1_{2,\varepsilon})_\varepsilon$ and $(I^2_{2,\varepsilon})_\varepsilon$ belong to $\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$.\\ \bf{Step 3.}\rm\, It remains to study \[ I_{1,\varepsilon}(x,\eta)=\int_{\Omega\times\mb{R}^n}\mathrm{e}^{i((x-z)\theta+\omega_\varepsilon(z,\eta))}a_\varepsilon(x,\theta)b'_\varepsilon(z,x,\eta)\, dz\, \dslash\theta. \] We expand $a_\varepsilon(x,\theta)$ with respect to $\theta$ at $\theta=\nabla_x\omega_\varepsilon(x,\eta)$ and we observe that \[
(\theta-\nabla_x\omega_\varepsilon(x,\eta))^\alpha \mathrm{e}^{i(x-z)(\theta-\nabla_x\omega_\varepsilon(x,\eta))}=(-1)^{|\alpha|}D^\alpha_z\mathrm{e}^{i(x-z)(\theta-\nabla_x\omega_\varepsilon(x,\eta))}. \] By integrating by parts we obtain \[ \begin{split} \mathrm{e}^{-i\omega_\varepsilon(x,\eta)}&I_{1,\varepsilon}(x,\eta)\\ &=\int_{\Omega\times\mb{R}^n}\hskip-10pt\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}\mathrm{e}^{i(x-z)(\theta-\nabla_x\omega_\varepsilon(x,\eta))}b'_\varepsilon(z,x,\eta)a_\varepsilon(x,\theta)\, dz\, \dslash\theta\\
&=\sum_{|\alpha|< k}\frac{1}{\alpha!}\partial^\alpha_\xi a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta))\cdot\\ &\cdot\int_{\Omega\times\mb{R}^n} D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{i(x-z)(\theta-\nabla_x\omega_\varepsilon(x,\eta))}\, dz\, \dslash\theta\\
&+\sum_{|\alpha|=k}\frac{k}{\alpha!}\int_{\Omega\times\mb{R}^n}D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{-i(x-z)\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)\, dz\, \dslash\theta, \end{split} \] where $\overline{\omega}_\varepsilon(z,x,\eta):=\omega_\varepsilon(z,\eta)-\omega_\varepsilon(x,\eta)-(\nabla_x\omega_\varepsilon(x,\eta))(z-x)$ and \[ r_{\alpha,\varepsilon}(x,\eta,\theta)=\int_{0}^1(1-t)^{k-1}\partial^\alpha_\xi a_\varepsilon(x,\nabla_x\omega_\varepsilon(x,\eta)-t\theta)\, dt. \]
Since, $b'_\varepsilon(z,x,\eta)=b_\varepsilon(x,\eta)$ if $|x-z|\le\frac{\sigma_\varepsilon}{2}$ we have that \begin{multline*} \int_{\Omega\times\mb{R}^n}D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{i(x-z)(\theta-\nabla_x\omega_\varepsilon(x,\eta))}\, dz\, \dslash\theta\\ = \int_{\Omega\times\mb{R}^n}D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{i(x-z)\theta}\, dz\, \dslash\theta\\
=D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b_\varepsilon(z,\eta)\biggr)|_{x=z} \end{multline*} This means that \[
\mathrm{e}^{-i\omega_\varepsilon(x,\eta)}I_{1,\varepsilon}(x,\eta)=\sum_{|\alpha|< k}h_{\alpha,\varepsilon}(x,\eta)+\sum_{|\alpha|=k}\frac{k}{\alpha!}R_{\alpha,\varepsilon}(x,\eta), \] where $(h_{\alpha,\varepsilon})_\varepsilon$ is defined in \eqref{h_alpha} and \[ R_{\alpha,\varepsilon}(x,\eta):=\int_{\Omega\times\mb{R}^n}D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{-i(x-z)\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)\, dz\, \dslash\theta. \]
\bf{Step 4.}\rm\, Our next task is to prove moderate symbol estimates for the net $(R_{\alpha,\varepsilon})_\varepsilon$. Let $\chi\in\ensuremath{\mathcal{C}^\infty_{\text{c}}}(\mb{R}^n)$ such that $\chi(\theta)=1$ for $|\theta|\le 1$ and $\chi(\theta)=0$ for $|\theta|\ge 3/2$. Let us take a positive net $(\tau_\varepsilon)_\varepsilon$ such that $\tau_\varepsilon\ge c\varepsilon^r$ for some $c>0$ and $r>0$. We define the sets \[
W^1_{\tau_\varepsilon,\eta}=\{\theta\in\mb{R}^n:\, |\theta|<\tau_\varepsilon|\eta|\},\qquad W^2_{\tau_\varepsilon,\eta}=\mb{R}^n\setminus W^1_{\tau_\varepsilon,\eta}. \]
Set now $\chi_\varepsilon(\theta):=\chi(\theta/\tau_\varepsilon)$. By construction we have that $\chi_\varepsilon(\theta/|\eta|)=1$ on $W^1_{\tau_\varepsilon,\eta}$, $\mathrm{supp}\, \chi_\varepsilon(\cdot/|\eta|)\subseteq W^1_{2\tau_\varepsilon,\eta}$ and $\mathrm{supp}(1-\chi_\varepsilon(\cdot/|\eta|))\subseteq W^2_{\tau_\varepsilon,\eta}$. We write $R_{\alpha,\varepsilon}(x,\eta)$ as \begin{multline*}
\int_{\Omega\times\mb{R}^n}D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{-i(x-z)\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)\chi_\varepsilon(\theta/|\eta|)\, dz\, \dslash\theta\\
+ \int_{\Omega\times\mb{R}^n}\hskip-10pt D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{-i(x-z)\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\, dz\, \dslash\theta\\ := R_{\alpha,\varepsilon}^1(x,\eta)+R^2_{\alpha,\varepsilon}(x,\eta). \end{multline*} We begin by estimating the net $R^1_{\alpha,\varepsilon}$. We make use of the identity \[
\mathrm{e}^{-i(x-z)\theta}=(1+|\eta|^2|x-z|^2)^{-N}(1-|\eta|^2\Delta_\theta)^N\mathrm{e}^{-i(x-z)\theta} \] which yields \begin{multline*} R_{\alpha,\varepsilon}^1(x,\eta)=\int_{\Omega\times\mb{R}^n}D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\mathrm{e}^{-i(x-z)\theta}\cdot\\
\cdot(1+|\eta|^2|x-z|^2)^{-N}(1-|\eta|^2\Delta_\theta)^N\big(r_{\alpha,\varepsilon}(x,\eta,\theta)\chi_\varepsilon(\theta/|\eta|)\big)\, dz\, \dslash\theta. \end{multline*} By the moderateness of the net $(\omega_\varepsilon)$ and Taylor's formula we have the inequality \begin{equation} \label{form_ruly}
|\nabla_x\omega_\varepsilon(x,\eta)-\nabla_z\omega_\varepsilon(z,\eta)|\le c\varepsilon^{-M}|{\eta}|^1|x-z|, \end{equation}
valid for $z\in K_b$, $|x-z|\le\sigma_\varepsilon$ and $\sigma_\varepsilon$ small enough such that $\cup_{\varepsilon\in(0,1]}\{z+\lambda(x-z):\, z\in K_b,\, |x-z|\le\sigma_\varepsilon,\, \lambda\in[0,1]\}\subseteq K'\Subset\Omega$. Clearly $M$ depends on the compact set $K'$. By Lemma \ref{lemma_sandro_2} we have that for all $x$ and $z$ as above, $|\eta|\ge 1$ and $\varepsilon\in(0,\varepsilon_0]$, the estimate \[
\biggl|D^\beta_z\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}\biggr|
\le c'\varepsilon^{-M'}\lara{\eta}^{\frac{|\beta|}{2}}(1+|\eta|^2|x-z|^2)^{L_\beta} \] holds for some $L_\beta\in\mb{N}$ and $M'\in\mb{N}$. Hence, recalling that $\sigma_\varepsilon\ge c\varepsilon^s$ for some $c,s>0$, we are led from the previous considerations to \begin{equation} \label{ineq_D_alpha}
\biggl|D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\biggr|\le C\varepsilon^{-N'}\lara{\eta}^{l+\frac{|\alpha|}{2}}(1+|\eta|^2|x-z|^2)^{L_\alpha}, \end{equation}
valid for $|\eta|\ge 1$, $\varepsilon$ small enough and $N'$ depending on $K_b$, $\alpha$ and the bound $c\varepsilon^s$ of $\sigma_\varepsilon$. Before considering $(1-|\eta|^2\Delta_\theta)^N\big(r_{\alpha,\varepsilon}(x,\eta,\theta)\chi_\varepsilon(\theta/|\eta|)$ it is useful to investigate the quantity $|\nabla_x\omega_\varepsilon(x,\eta)-t\theta|$ for $x\in K\Subset\Omega$ and $\theta\in W^1_{2\tau_\varepsilon,\eta}$. We recall that there exists $r>0$, $c_0,c_1>0$ and $\varepsilon_0\in(0,1]$ such that \[
c_0\varepsilon^r|\eta|\le |\nabla_x\omega_\varepsilon(x,\eta)|\le c_1\varepsilon^{-r}|\eta|, \]
for all $x\in K$, $\eta\neq 0$ and $\varepsilon\in(0,\varepsilon_0]$. Since, if $\theta\in W^1_{2\tau_\varepsilon,\eta}$ then $|\theta|\le 2\tau_\varepsilon|\eta|$, we obtain, for all $x\in K$, $\theta\in W^1_{2\tau_\varepsilon,\eta}$ and $t\in[0,1]$, the following estimates: \begin{multline*}
|\nabla_x\omega_\varepsilon(x,\eta)-t\theta|\le |\nabla_x\omega_\varepsilon(x,\eta)|+|\theta|\le (1+2\tau_\varepsilon c_0^{-1}\varepsilon^{-r})|\nabla_x\omega_\varepsilon(x,\eta)|\\
|\nabla_x\omega_\varepsilon(x,\eta)-t\theta|\ge (1-2\tau_\varepsilon c_0^{-1}\varepsilon^{-r})|\nabla_x\omega_\varepsilon(x,\eta)|. \end{multline*} It follows that assuming $\tau_\varepsilon\le\frac{\varepsilon^r}{4c_0^{-1}}$ the inequality \begin{equation} \label{est_tau}
\frac{c_0}{2}\varepsilon^r|\eta|\le \frac{1}{2}|\nabla_x\omega_\varepsilon(x,\eta)|\le |\nabla_x\omega_\varepsilon(x,\eta)-t\theta|\le \frac{3}{2}|\nabla_x\omega_\varepsilon(x,\eta)|\le c_1\frac{3}{2}\varepsilon^{-r}|\eta| \end{equation}
holds for $x\in K$, $\eta\neq 0$, $\theta\in W^1_{2\tau_\varepsilon,\eta}$, $t\in[0,1]$ and $\varepsilon$ small enough. We make use of \eqref{est_tau} in estimating $(1-|\eta|^2\Delta_\theta)^N\big(r_{\alpha,\varepsilon}(x,\eta,\theta)\chi_\varepsilon(\theta/|\eta|)\big)$ and we conclude that for all $N\in\mb{N}$ there exists $N''$ such that \begin{equation} \label{est_Delta_N}
|(1-|\eta|^2\Delta_\theta)^N\big(r_{\alpha,\varepsilon}(x,\eta,\theta)\chi_\varepsilon(\theta/|\eta|)\big)|\le \varepsilon^{-N''}\lara{\eta}^{m-|\alpha|}, \end{equation} for all $x\in K$, $\eta\neq 0$, $\theta\in W^1_{2\tau_\varepsilon,\eta}$ and $\varepsilon\in(0,\varepsilon_0]$. A combination of \eqref{ineq_D_alpha} with \eqref{est_Delta_N} entails \[
|R^1_{\alpha,\varepsilon}(x,\eta)|\le \varepsilon^{-N'-N''}\lara{\eta}^{m+l-\frac{|\alpha|}{2}}\int_{W^1_{2\tau_\varepsilon,\eta}}d\theta \int_{\mb{R}^n}(1+|\eta|^2|y|^2)^{L_\alpha-N}\, dy. \] Therefore, choosing $N\ge L_\alpha+\frac{n+1}{2}$ we obtain \begin{multline*}
|R^1_{\alpha,\varepsilon}(x,\eta)|\le c\varepsilon^{-N'-N''}(\tau_\varepsilon)^n\lara{\eta}^{m+l-\frac{|\alpha|}{2}}|\eta|^n\int_{\mb{R}^n}\lara{z}^{-n-1}\, dz |\eta|^{-n}\\
\le c_1\varepsilon^{-N_1}\lara{\eta}^{m+l-\frac{|\alpha|}{2}}, \end{multline*}
for $x\in K$ and $|\eta|\ge 1$.\\
The case $|\eta|\le 1$ requires less precise estimates. More precisely, it is enough to see that from Lemma \ref{lemma_sandro_2} we have that for all $\alpha$ there exists some $d\in\mb{R}$ such that \[
\biggl|D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr)\biggr|\le C\varepsilon^{-M'}\lara{\eta}^{d} \]
for all $x\in K$, $z\in K_b$ and $|x-z|\le \sigma_\varepsilon$. Thus, \[
|R^1_{\alpha,\varepsilon}(x,\eta)|\le c\varepsilon^{N_2}\lara{\eta}^{d-l-\frac{|\alpha|}{2}}\lara{\eta}^{m+l-\frac{|\alpha|}{2}}\le c_2\varepsilon^{N_2}\lara{\eta}^{m+l-\frac{|\alpha|}{2}}, \]
when $|\eta|\le 1$. In conclusion, there exists $(C_{K,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \[
|R^1_{\alpha,\varepsilon}(x,\eta)|\le C_{K,\varepsilon}\lara{\eta}^{m+l-\frac{|\alpha|}{2}} \] for all $x\in K$, $\eta\in\mb{R}^n\setminus 0$ and $\varepsilon\in(0,1]$.\\ \bf{Step 5.}\rm\, Finally, we consider $R^2_{\alpha,\varepsilon}(x,\eta)$. By Lemma \ref{lemma_sandro_2} we can write \[ D^\alpha_z\biggl(\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b'_\varepsilon(z,x,\eta)\biggr) \] as the finite sum \[ \mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}\sum_{\beta}b_{\alpha,\beta,\varepsilon}(z,x,\eta), \] where, by making use of the hypotheses on $b'_{\varepsilon}$ and $\sigma_\varepsilon$, the following holds: \begin{multline*} \forall \beta\in\mb{N}^n\, \exists m_\beta\in\mb{R}\, \forall \gamma\in\mb{N}^n\, \exists (\mu_{\beta,\gamma,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}\, \forall x\in\Omega\, \forall z\in\Omega\, \forall \eta\in\mb{R}^n\setminus 0\, \forall\varepsilon\in(0,1]\\
|\partial^\gamma_z b_{\alpha,\beta,\varepsilon}(z,x,\eta)|\le \mu_{\beta,\gamma,\varepsilon}\lara{\eta}^{m_\beta}, \end{multline*}
with $b_{\alpha,\beta,\varepsilon}(z,x,\eta)=0$ for $|x-z|\ge \sigma_\varepsilon$. Hence, we have \begin{multline*} R^2_{\alpha,\varepsilon}(x,\eta)\\
=\sum_\beta \int_{\Omega\times\mb{R}^n}\mathrm{e}^{i\overline{\omega}_\varepsilon(z,x,\eta)}b_{\alpha,\beta,\varepsilon}(z,x,\eta)\mathrm{e}^{-i(x-z)\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\, dz\, \dslash\theta\\
=\sum_\beta \int_{\mb{R}^n}\mathrm{e}^{-ix\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\int_\Omega \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}b_{\alpha,\beta,\varepsilon}(z,x,\eta)\, dz\, \dslash\theta, \end{multline*}
where $\rho_\varepsilon(z,x,\eta,\theta)=\overline{\omega}_\varepsilon(z,x,\eta)+z\theta$. Since $\chi_\varepsilon(\theta/|\eta|)=1$ for $\theta\in W^1_{\tau_\varepsilon,\eta}$, we may limit ourselves to consider $\theta\in W^2_{\tau_\varepsilon,\eta}$, i.e., $|\theta|\ge\tau_\varepsilon|\eta|$. We investigate now the properties of the net $(\rho_\varepsilon)_\varepsilon$. We have \[ \nabla_z\rho_\varepsilon(z,x,\eta,\theta)=\theta+\nabla_z\omega_\varepsilon(z,\eta)-\nabla_x\omega_\varepsilon(x,\eta), \] and therefore \eqref{form_ruly} yields \[
|\theta+\nabla_z\omega_\varepsilon(z,\eta)-\nabla_x\omega_\varepsilon(x,\eta)|\le |\theta|+\varepsilon^{-M}\sigma_\varepsilon|\eta|\le |\theta|(1+\varepsilon^{-M}\sigma_\varepsilon\tau_\varepsilon^{-1}) \]
for $\theta\in W^2_{\tau_\varepsilon,\eta}$, $|x-z|<\sigma_\varepsilon$, $z\in K_b$ and $\varepsilon$ small enough. We now take $\sigma_\varepsilon$ so small that $\varepsilon^{-M}\sigma_\varepsilon\le\frac{\tau_\varepsilon}{2}$. From \eqref{form_ruly} and the previous assumptions we obtain \[
|\theta+\nabla_z\omega_\varepsilon(z,\eta)-\nabla_x\omega_\varepsilon(x,\eta)|\ge |\theta|-\varepsilon^{-M}\sigma_\varepsilon|\eta|\ge |\theta|-\varepsilon^{-M}\sigma_\varepsilon\tau_\varepsilon^{-1}|\theta|\ge \frac{1}{2}|\theta|. \] In other words, there exists $(\lambda_{1,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ strictly nonzero and $(\lambda_{2,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \[
\lambda_{1,\varepsilon}|\theta|\le|\theta+\nabla_z\omega_\varepsilon(z,\eta)-\nabla_x\omega_\varepsilon(x,\eta)|\le\lambda_{2,\varepsilon}|\theta|, \]
for $\theta\in W^2_{\tau_\varepsilon,\eta}$, $|x-z|<\sigma_\varepsilon$, $z\in K_b$ and $\varepsilon\in(0,1]$.\\ Consider now \[ p_{N,\varepsilon}(z,x,\eta,\theta)=\mathrm{e}^{-i\rho_\varepsilon(z,x,\eta,\theta)}\Delta_z^N\mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}. \]
Noting that $\partial^\gamma_z \rho_\varepsilon(z,x,\eta,\theta)=\partial^\gamma_z\omega_\varepsilon(z,\eta)$ for $|\gamma|\ge 2$, and making use of the previous estimates on $|\nabla_z\rho_\varepsilon(z,x,\eta,\theta)|$, one can prove by induction that \[
\Delta^N_z\mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}=\mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}\biggl((-1)^N|\nabla_z\rho_\varepsilon(z,x,\eta,\theta)|^{2N}+s_{N,\varepsilon}(z,x,\eta,\theta)\biggr), \] where $(s_{N,\varepsilon})_\varepsilon$ has the following property: \begin{equation} \label{prop_1_6}
\exists l\in [0,2N)\, \forall\gamma\in\mb{N}^n\, \exists (s'_{\gamma,N,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}\qquad |\partial^\gamma_z s_{N,\varepsilon}(z,x,\eta,\theta)|\le s'_{\gamma,N,\varepsilon}|\theta|^l, \end{equation}
for $|\eta|\ge 1$, $\theta\in W^2_{\tau_\varepsilon,\eta}$, $|x-z|<\sigma_\varepsilon$ and $z\in K_b$. It follows that \begin{multline} \label{prop_2_6}
|p_{N,\varepsilon}(z,x,\eta,\theta)|\ge \frac{1}{2^{2N}}|\theta|^{2N}-s'_{0,N,\varepsilon}|\theta|^l=|\theta|^{2N}\big(\frac{1}{2^{2N}}-s'_{0,N,\varepsilon}|\theta|^{l-2N}\big)\\
\ge \frac{1}{2^{2N+1}}|\theta|^{2N}, \end{multline}
for $\theta\in W^2_{\tau_\varepsilon,\eta}$, $|x-z|<\sigma_\varepsilon$, $z\in K_b$ and $|\eta|\ge \lambda_{N,\varepsilon}:= \tau_\varepsilon^{-1}(2^{2N+1}s'_{0,N,\varepsilon})^{\frac{1}{2N-l}}$. Moreover, we have that for all $\gamma\in\mb{N}^n$ there exists $(a_{\gamma,N,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \begin{equation} \label{prop_3_6}
|\partial^\gamma_z|\nabla_z\rho_\varepsilon(z,x,\eta,\theta)|^{2N}|\le a_{\gamma,N,\varepsilon}|\theta|^{2N}, \end{equation}
for $|\eta|\ge 1$, $\theta\in W^2_{\tau_\varepsilon,\eta}$, $|x-z|<\sigma_\varepsilon$ and $z\in K_b$. This allows us to prove by induction that \begin{equation} \label{prop_4_6}
\forall\gamma\in\mb{N}^n\, \exists (b_{\gamma,N,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}\, \exists (\lambda_{\gamma,N,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}\,\qquad |\partial^\gamma_z p^{-1}_{N,\varepsilon}(z,x,\eta,\theta)|\le b_{\gamma,N,\varepsilon}|\theta|^{-2N}, \end{equation}
for $\theta\in W^2_{\tau_\varepsilon,\eta}$, $|x-z|<\sigma_\varepsilon$, $z\in K_b$ and $|\eta|\ge \lambda_{\gamma,N,\varepsilon}$. The assertion \eqref{prop_4_6} is clear for $\gamma=0$ by \eqref{prop_2_6}. Assume now that \eqref{prop_4_6} holds for $|\gamma'|\le N$ and take $|\gamma|=N$. From $p_{N,\varepsilon}^{-1}p_{N,\varepsilon}=1$ we obtain \[ \partial^{\gamma}_zp^{-1}_{N,\varepsilon}(z,x,\eta,\theta)p_{N,\varepsilon}(z,x,\eta,\theta)=-\hskip-4pt\sum_{\gamma'<\gamma}\binom{\gamma}{\gamma'}\partial^{\gamma'}_z p^{-1}_{N,\varepsilon}(z,x,\eta,\theta)\partial^{\gamma-\gamma'}_zp_{N,\varepsilon}(z,x,\eta,\theta) \] and therefore \begin{multline*}
|\partial^{\gamma}_zp^{-1}_{N,\varepsilon}(z,x,\eta,\theta)|\le \sum_{\gamma'<\gamma}b_{\gamma',N,\varepsilon}|\theta|^{-2N}(a_{\gamma-\gamma',N,\varepsilon}|\theta|^{2N}+s'_{\gamma-\gamma',N,\varepsilon}|\theta|^l)|\theta|^{-2N}\\
\le b_{\gamma,N,\varepsilon}|\theta|^{-2N}, \end{multline*}
for $\theta\in W^2_{\tau_\varepsilon,\eta}$, $|x-z|<\sigma_\varepsilon$, $z\in K_b$ and $|\eta|\ge\lambda_{\gamma,N,\varepsilon}:= \max_{\gamma'<\gamma}\lambda_{\gamma',N,\varepsilon}$. We make use of the identity \[ \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}=\Delta^N_z\mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}p^{-1}_{N,\varepsilon}(z,x,\eta,\theta) \] in the integral \[ \int_\Omega \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}b_{\alpha,\beta,\varepsilon}(z,x,\eta)\, dz. \]
Since, $\mathrm{supp}_z b_{\alpha,\beta,\varepsilon}(z,x,\eta)\subseteq K_b$, $b_{\alpha,\beta,\varepsilon}(z,x,\eta)=0$ for $|x-z|\ge\sigma_\varepsilon$ and $1-\chi_\varepsilon(\theta/|\eta|)=0$ for $\theta\not\in W^2_{\tau_\varepsilon,\eta}$, we can write \begin{multline*}
\int_{\mb{R}^n}\mathrm{e}^{-ix\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\int_\Omega \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}b_{\alpha,\beta,\varepsilon}(z,x,\eta)\, dz\, \dslash\theta\\
= \int_{\mb{R}^n}\mathrm{e}^{-ix\theta}r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\int_\Omega \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)} \cdot\\ \cdot\Delta^N_z\big(p_{N,\varepsilon}^{-1}(z,x,\eta,\theta)b_{\alpha,\beta,\varepsilon}(z,x,\eta)\big)\, dz\, \dslash\theta, \end{multline*} where \begin{multline*}
\biggl|r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\int_\Omega \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}\Delta^N_z\big(p_{N,\varepsilon}^{-1}(z,x,\eta,\theta)b_{\alpha,\beta,\varepsilon}(z,x,\eta)\big)\, dz\biggr|\\
\le c|r_{\alpha,\varepsilon}(x,\eta,\theta)||1-\chi_\varepsilon(\theta/|\eta|)|b_{N,\varepsilon}\mu_{\beta,N,\varepsilon}|\theta|^{-2N}\lara{\eta}^{m_\beta}, \end{multline*}
for $|\eta|\ge \lambda_{N,\varepsilon}$ and $m_\beta$ independent of $N$. We take $2N=N_1+N_2$ such that $-N_2+m_\beta\le 0$. Hence, from $|\theta|\ge \tau_\varepsilon|\eta|$ we have, for some $(c_{N,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ and $|\eta|\ge\lambda_{N,\varepsilon}$, the following estimate: \begin{multline*}
\biggl|r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\int_\Omega \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}\Delta^N_z\big(p_{N,\varepsilon}^{-1}(z,x,\eta,\theta)b_{\alpha,\beta,\varepsilon}(z,x,\eta)\big)\, dz\biggr|\\
\le c_{N,\varepsilon}|r_{\alpha,\varepsilon}(x,\eta,\theta)||1-\chi_\varepsilon(\theta/|\eta|)||\theta|^{-N_1}. \end{multline*} By definition of $r_{\alpha,\varepsilon}$ we easily see that for all $K\Subset\Omega$ there exists $(d_\varepsilon)_\varepsilon, (d'_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \[
|r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))|\le d_\varepsilon\lara{\theta}^{m_+}\lara{\nabla_x\omega_\varepsilon(x,\eta)}^{m_+}\le d'_\varepsilon\lara{\theta}^{2m_+}. \]
Hence for all $h\ge 0$ there exists $2N=N_1+N_2$ large enough such that, for $x\in K$ and $|\eta|\ge\lambda_{N,\varepsilon}$ \begin{multline*}
\biggl|r_{\alpha,\varepsilon}(x,\eta,\theta)(1-\chi_\varepsilon(\theta/|\eta|))\int_\Omega \mathrm{e}^{i\rho_\varepsilon(z,x,\eta,\theta)}\Delta^N_z\big(p_{N,\varepsilon}^{-1}(z,x,\eta,\theta)b_{\alpha,\beta,\varepsilon}(z,x,\eta)\big)\, dz\biggr|\\ \le \nu_{K,\varepsilon} \lara{\theta}^{-h}\le \nu_{K,\varepsilon}\tau_\varepsilon^{-h}\lara{\eta}^{-h}, \end{multline*} with $(\nu_{K,\varepsilon})_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$. This means that for all $h\ge 0$ there exists $(\lambda_\varepsilon)_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \[
|R^2_{\alpha,\varepsilon}(x,\eta)|\le \nu_{K,\varepsilon}\lara{\eta}^{-h} \]
when $x\in K$, $|\eta|\ge\lambda_{\varepsilon}$ and $\varepsilon\in(0,1]$. A simple investigation of the oscillatory integral which defines $R^2_{\alpha,\varepsilon}(x,\eta)$ shows that there exists some $h'\ge 0$ and some $\nu'_{K,\varepsilon}\in\ensuremath{{\mathcal E}_{M}}$ such that the estimate \[
|R^2_{\alpha,\varepsilon}(x,\eta)|\le \nu'_{K,\varepsilon}\lara{\eta}^{h'} \]
holds for all $x\in K$, $\eta\in\mb{R}^n\setminus 0$ and $\varepsilon\in(0,1]$. This yields for $|\eta|\le\lambda_\varepsilon$ \[
|R^2_{\alpha,\varepsilon}(x,\eta)|\le \nu'_{K,\varepsilon}\lara{\eta}^{-h}\lara{\eta}^{h+h'}\le \nu'_{K,\varepsilon}\lara{\lambda_\varepsilon}^{h+h'}\lara{\eta}^{-h}. \] In conclusion, we have that for all $h\ge 0$ there exists $(C_{h,\varepsilon}(K))_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \[
|R^2_{\alpha,\varepsilon}(x,\eta)|\le C_{h,\varepsilon}(K)\lara{\eta}^{-h} \] for all $x\in K$, $\eta\in\mb{R}^n\setminus 0$ and $\varepsilon\in(0,1]$.\\ \bf{Step 6.}\rm\, Finally, we combine all the results of the previous steps. We have that \begin{equation} \label{form_finale} a_\varepsilon(x,D)F_{\omega_\varepsilon}(b_\varepsilon)u_\varepsilon(x)=\int_{\mb{R}^n}I_{1,\varepsilon}(x,\eta)\widehat{u_\varepsilon}(\eta)\, \dslash\eta + \int_{\mb{R}^n}I_{2,\varepsilon}(x,\eta)\widehat{u_\varepsilon}(\eta)\, \dslash\eta, \end{equation} where $(I_{2,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$. From Theorem \ref{theo_asymp_expan}$(i)$ and Proposition \ref{prop_h_alpha} there exists $(h_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m+l}(\Omega\times\mb{R}^n\setminus 0)}$ such that $h_\varepsilon(x,\eta)\sim \sum_\alpha h_{\alpha,\varepsilon}(x,\eta)$. We write the first integral in \eqref{form_finale} as \[ \int_{\mb{R}^n}\mathrm{e}^{i\omega_\varepsilon(x,\eta)}h_\varepsilon(x,\eta)\widehat{u_\varepsilon}(\eta)\, \dslash\eta + \int_{\mb{R}^n}\mathrm{e}^{i\omega_\varepsilon(x,\eta)}\biggl(\mathrm{e}^{-i\omega_\varepsilon(x,\eta)}I_{1,\varepsilon}(x,\eta)-h_\varepsilon(x,\eta)\biggr)\widehat{u_\varepsilon}(\eta)\, \dslash\eta \] and we concentrate on \[ \mathrm{e}^{-i\omega_\varepsilon(x,\eta)}I_{1,\varepsilon}(x,\eta)-h_\varepsilon(x,\eta). \] From the previous computations we have that for all $k\ge 1$ and $K\Subset\Omega$ there exists $(C_{k,\varepsilon}(K))_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that \[
\biggl|\mathrm{e}^{-i\omega_\varepsilon(x,\eta)}I_{1,\varepsilon}(x,\eta)-\sum_{|\alpha|<k}h_{\alpha,\varepsilon}(x,\eta)\biggr|\le C_{k,\varepsilon}(K)\lara{\eta}^{m+l-\frac{k}{2}} \] for all $x\in K$, $\eta\in\mb{R}^n\setminus 0$ and $\varepsilon\in(0,1]$. Moreover, regarding $I_{1,\varepsilon}(x,\eta)$ as the oscillatory integral \[ \int_{\Omega\times\mb{R}^n}\mathrm{e}^{-iz\theta}\mathrm{e}^{ix\theta+i\omega_\varepsilon(z,\eta)}a_\varepsilon(x,\theta)b'_\varepsilon(z,x,\eta)\, dz\, \dslash\theta, \] from Theorem 3.1 in \cite{Garetto:04}, we obtain that for all $\alpha,\beta\in\mb{N}^n$ there exists $d\in\mb{R}$ and for all $K\Subset\Omega$ there exists $(c_{\alpha,\beta,\varepsilon}(K))_\varepsilon\in\ensuremath{{\mathcal E}_{M}}$ such that for all $\eta\in\mb{R}^n\setminus 0$ and $\varepsilon\in(0,1]$, \[
\sup_{x\in K}|\partial^\alpha_\eta\partial^\beta_x I_{1,\varepsilon}(x,\eta)|\le c_{\alpha,\beta,\varepsilon}(K)\lara{\eta}^{d}. \] Recalling that \[ \partial^\alpha_\eta\partial^\beta_x \mathrm{e}^{-i\omega_\varepsilon{(x,\eta)}}=\mathrm{e}^{-i\omega_\varepsilon(x,\eta)}a_{\alpha,\beta,\varepsilon}(x,\eta), \]
with $(a_{\alpha,\beta,\varepsilon})_\varepsilon\in \mathcal{M}_{S^{|\beta|}(\Omega\times\mb{R}^n\setminus 0)}$, we conclude that the net $(\mathrm{e}^{-i\omega_\varepsilon(x,\eta)}I_{1,\varepsilon}(x,\eta))_\varepsilon$ satisfies the hypothesis of Proposition \ref{prop_asym_Shubin}$(i)$. It follows that \[ (\mathrm{e}^{-i\omega_\varepsilon(x,\eta)}I_{1,\varepsilon}(x,\eta))_\varepsilon\sim \sum_\alpha (h_{\alpha,\varepsilon})_\varepsilon. \] Hence, by Theorem \ref{theo_asymp_expan}$(i)$ we conclude \[ (\mathrm{e}^{-i\omega_\varepsilon(x,\eta)}I_{1,\varepsilon}(x,\eta)-h_\varepsilon(x,\eta))_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}. \] Going back to \eqref{form_finale} we have that there exists $(r_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\Omega\times\mb{R}^n\setminus 0)}$ such that \[ a_\varepsilon(x,D)F_{\omega_\varepsilon}(b_\varepsilon)u_\varepsilon(x)= \int_{\mb{R}^n}\mathrm{e}^{i\omega_\varepsilon(x,\eta)}h_\varepsilon(x,\eta)\widehat{u_\varepsilon}(\eta)\, \dslash\eta + r_\varepsilon(x,D)(u_\varepsilon)(x). \] \end{proof} \begin{thm} \label{theo_comp_ssc} Let $\omega\in\wt{\mathcal{S}}^{\,1,\mathrm{sc}}_{\rm{hg}}(\Omega\times\mb{R}^n\setminus 0)$ have a representative satisfying condition $(h2)$. Let $a\in\wt{\mathcal{S}}^{\,m,\mathrm{sc}}(\Omega\times\mb{R}^n)$ and $b\in\wt{\mathcal{S}}^{\,l,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$ with $\mathrm{supp}_x\, b\Subset\Omega$. Then, the operator $a(x,D)F_\omega(b)$ has the following properties: \begin{itemize} \item[(i)] maps $\ensuremath{{\mathcal G}^\infty_\mathrm{c}}(\Omega)$ into $\ensuremath{\G^\infty}(\Omega)$; \item[(ii)] is of the form \[ \int_{\mb{R}^n}\mathrm{e}^{i\omega(x,\eta)}h(x,\eta)\widehat{u}(\eta)\, \dslash\eta +r(x,D)u, \] where $h\in\wt{\mathcal{S}}^{\,l+m,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$ has asymptotic expansion given by the symbols $h_\alpha$ defined in \eqref{h_alpha} and $r\in\wt{\mathcal{S}}^{-\infty,\mathrm{sc}}(\Omega\times\mb{R}^n\setminus 0)$. \end{itemize} \end{thm} \begin{proof} Combining Proposition \ref{prop_F_map}$(iii)$ with the usual mapping properties of generalized pseudodifferential operators we have that $(i)$ holds. Concerning assertion $(ii)$, we argue as in the proof of Theorem \ref{theo_comp} by taking the nets $(\sigma_\varepsilon)_\varepsilon$ and $(\tau_\varepsilon)_\varepsilon$ slow scale strictly nonzero. From the assumptions of slows scale type on $\omega$, $a$ and $b$ we have that all the moderate nets involved are of slow scale type. This leads to the desired conclusion. \end{proof}
\section{Generalized Fourier integral operators and microlocal analysis} Concluding, we present some first results of microlocal analysis for generalized Fourier integral operators provided in \cite[Section 4]{GHO:06}. A deeper investigation of the microlocal properties of \[ A:\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\to\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega'):u\to\int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(x,y,\xi)}a(x,y,\xi)u(y)\, dy\, \dslash\xi \] is current topic of research. \subsection*{Generalized singular supports of the functional $I_\phi(a)$} We begin with the functional \[ I_{\phi}(a):\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega)\to\wt{\mb{C}}:u\to\int_{\Omega\times\mb{R}^p}\mathrm{e}^{i\phi(y,\xi)}a(y,\xi)u(y)\, dy\, \dslash\xi \]
Before defining specific regions depending on the generalized phase function $\phi$, we observe that any $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$ can be regarded as an element of ${\widetilde{\mathcal{S}}}^1_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)$ and consequently $|\nabla_\xi\phi|^2\in{\widetilde{\mathcal{S}}}^0_{\rm{hg}}(\Omega\times\mb{R}^p\setminus 0)$.
Let $\Omega_1$ be an open subset of $\Omega$ and $\Gamma\subseteq\mb{R}^p\setminus 0$. We say that $b\in\wt{\mathcal{S}}^0(\Omega\times\mb{R}^p\setminus 0)$ is \emph{invertible on $\Omega_1\times\Gamma$} if for all relatively compact subsets $U$ of $\Omega_1$ there exists a representative $(b_\varepsilon)_\varepsilon$ of $b$, a constant $r\in\mb{R}$ and $\eta\in(0,1]$ such that \begin{equation} \label{est_inv_sym}
\inf_{y\in U,\xi\in\Gamma}|b_\varepsilon(y,\xi)|\ge \varepsilon^r \end{equation} for all $\varepsilon\in(0,\eta]$. In an analogous way we say that $b\in\wt{\mathcal{S}}^0(\Omega\times\mb{R}^p\setminus 0)$ is \emph{slow scale-invertible} on $\Omega_1\times\Gamma$ if \eqref{est_inv_sym} holds with the inverse of some slow scale net $(s_\varepsilon)_\varepsilon$ in place of $\varepsilon^r$. This kind of bounds from below hold for all representatives of the symbol $b$ once they are known to hold for one.
In the sequel $\pi_\Omega$ denotes the projection of $\Omega\times\mb{R}^p$ on $\Omega$. \begin{defn} \label{def_C_phi}
Let $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$. We define $C_\phi\subseteq\Omega\times\mb{R}^p\setminus 0$ as the complement of the set of all $(x_0,\xi_0)\in\Omega\times\mb{R}^p\setminus 0$ with the property that there exist a relatively compact open neighborhood $U(x_0)$ of $x_0$ and a conic open neighborhood $\Gamma(\xi_0)\subseteq\mb{R}^p\setminus 0$ of $\xi_0$ such that $|\nabla_\xi\phi|^2$ is invertible on $U(x_0)\times\Gamma(\xi_0)$. We set $\pi_\Omega(C_{\phi})=S_{\phi}$ and $R_{\phi}=(S_{\phi})^{{\rm{c}}}$. \end{defn} By construction $C_{\phi}$ is a closed conic subset of $\Omega\times\mb{R}^p\setminus 0$ and $R_{\phi}\subseteq\Omega$ is open. It is routine to check that the region $C_\phi$ coincides with the classical one when $\phi$ is classical. \begin{prop} \label{prop_R_phi}
The generalized symbol $|\nabla_\xi\phi|^2$ is invertible on $R_\phi\times\mb{R}^p\setminus 0$. \end{prop}
The more specific assumption of slow scale-invertibility concerning the generalized symbol $|\nabla_\xi\phi|^2$ is employed in the definition of the following sets. \begin{defn} \label{def_C_phi_ssc}
Let $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$. We define $C^\mathrm{sc}_{\phi}\subseteq\Omega\times\mb{R}^p\setminus 0$ as the complement of the set of all $(x_0,\xi_0)\in\Omega\times\mb{R}^p\setminus 0$ with the property that there exist a relatively compact open neighborhood $U(x_0)$ of $x_0$ and a conic open neighborhood $\Gamma(\xi_0)\subseteq\mb{R}^p\setminus 0$ of $\xi_0$ such that $|\nabla_\xi\phi|^2$ is on $U(x_0)\times\Gamma(\xi_0)$. We set $\pi_\Omega(C^\mathrm{sc}_{\phi})=S^\mathrm{sc}_{\phi}$ and $R^\mathrm{sc}_{\phi}=(S^\mathrm{sc}_{\phi})^{{\rm{c}}}$. \end{defn}
By construction $C^\mathrm{sc}_{\phi}$ is a conic closed subset of $\Omega\times\mb{R}^p\setminus 0$ and $R^\mathrm{sc}_{\phi}\subseteq R_{\phi}\subseteq\Omega$ is open. In analogy with Proposition \ref{prop_R_phi} we can prove that $|\nabla_\xi\phi|^2$ is slow scale-invertible on $R^\mathrm{sc}_\phi\times\mb{R}^p\setminus 0$. \begin{thm} \label{theorem_R_phi} Let $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$. \begin{itemize}
\item[(i)] The restriction $I_{\phi}(a)|_{R_\phi}$ of the functional $I_\phi(a)$ to the region $R_\phi$ belongs to $\ensuremath{{\mathcal G}}(R_\phi)$.
\item[(ii)] If $\phi\in\wt{\Phi}^\mathrm{sc}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$ then $I_{\phi}(a)|_{{R^\mathrm{sc}_\phi}}\in\ensuremath{\G^\infty}(R^\mathrm{sc}_\phi)$. \end{itemize} \end{thm} Theorem \ref{theorem_R_phi} means that \[ \mathrm{sing\, supp}_\ensuremath{{\mathcal G}}\, I_\phi(a)\subseteq S_\phi \] if $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ and that \[ \mathrm{sing\, supp}_{\ensuremath{\G^\infty}} I_\phi(a)\subseteq S^\mathrm{sc}_\phi \] if $\phi\in\wt{\Phi}^\mathrm{sc}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$. \begin{ex} Returning to the first example in Section \ref{gen_sec} we are now in the position to analyze the regularity properties of the generalized kernel functional $I_\phi(a)$ of the solution operator $A$ corresponding to the hyperbolic Cauchy-problem. For any $v\in \ensuremath{{\mathcal G}_\mathrm{c}}(\mb{R}^3)$ we have \begin{equation}\label{hypsol}
I_\phi(a) (v) = \int \mathrm{e}^{i\phi(x,t,y,\xi)}\; a(x,t,y,\xi)\, v(x,t,y)\, dx\, dt\, dy\, \dslash\xi, \end{equation} where $a$ and $\phi$ are as in Section \ref{gen_sec}. Note that in the case of partial differential operators with smooth coefficients and distributional initial values the wave front set of the distributional kernel of $A$ determines the propagation of singularities from the initial data. When the coefficients are non-differentiable functions, or even distributions or generalized functions, matters are not yet understood in sufficient generality. Nevertheless, the above results allow us to identify regions where the generalized kernel functional agrees with a generalized function or is even guaranteed to be a $\ensuremath{\G^\infty}$-regular generalized function. To identify the set $C_\phi$ in this situation one simply has to study invertibility of $\ensuremath{\partial}_\xi \phi (x,t,y,\xi) = \gamma(x,t;0)- y$ as a generalized function in a neighborhood of any given point $(x_0,t_0,y_0)$.
Under the assumptions on $c$ of the example in Section \ref{gen_sec}, the representing nets $(\gamma_\varepsilon(.,.;0))_{\varepsilon\in(0,1]}$ of $\gamma$ are uniformly bounded on compact sets (e.g., when $c$ is a bounded generalized constant). For given $(x_0,t_0)$ define the generalized domain of dependence $D(x_0,t_0)\subseteq\mb{R}$ to be the set of accumulation points of the net $(\gamma_\varepsilon(x_0,t_0;0))_{\varepsilon\in(0,1]}$. Then we have that $$
\{(x_0,t_0,y_0) \in\mb{R}^3 : y_0 \not\in D(x_0,t_0) \} \subseteq R_\phi. $$
When $c\in\wt{\mb{R}}$ this may be proved by showing that if $(x_0,t_0,y_0)\in C_\phi$ then there exists an accumulation point $c'$ of a representative $(c_\varepsilon)_\varepsilon$ of $c$ such that $y_0=x_0-c't_0$. \end{ex} \begin{ex} As an illustrative example concerning the regions involving the regularity of the functional $I_\phi(a)$ we consider the generalized phase function on $\mb{R}^2\times\mb{R}^2$ given by $\phi_\varepsilon(y_1,y_2,\xi_1,\xi_2)=-\varepsilon y_1\xi_1-s_\varepsilon y_2\xi_2$ where $(s_\varepsilon)_\varepsilon$ is bounded and $(s_\varepsilon^{-1})_\varepsilon$ is a slow scale net. Clearly $\phi:=[(\phi_\varepsilon)_\varepsilon]\in\wt{\Phi}^\mathrm{sc}(\mb{R}^2\times\mb{R}^2)$. Simple computations show that $R_\phi=\mb{R}^2\setminus(0,0)$ and $R^\mathrm{sc}_\phi=\mb{R}^2\setminus\{y_2=0\}$. We leave it to the reader to check that the oscillatory integral \[ \int_{\mb{R}^2}\mathrm{e}^{i\phi(y,\xi)}(1+\xi_1^2+\xi_2^2)^{\frac{1}{2}}\, \dslash\xi =\biggl[\biggl(\int_{\mb{R}^2}\mathrm{e}^{-i\varepsilon y_1\xi_1-is_\varepsilon y_2\xi_2}(1+\xi_1^2+\xi_2^2)^{\frac{1}{2}}\, \dslash\xi_1\, \dslash\xi_2\biggr)_\varepsilon\biggr] \] defines a generalized function in $\mb{R}^2\setminus(0,0)$ whose restriction to $\mb{R}^2\setminus\{y_2=0\}$ is regular. \end{ex}
The Colombeau-regularity of the functional $I_\phi(a)$ is easily proved in the case of generalized symbols of order $-\infty$. \begin{prop} \label{prop_smooth_sing} \begin{itemize} \item[{\, }] \item[(i)] If $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^{-\infty}(\Omega\times\mb{R}^p)$ then $\mathrm{sing\, supp}_{\ensuremath{{\mathcal G}}}I_\phi(a)=\emptyset$. \item[(ii)] If $\phi\in\wt{\Phi}^\mathrm{sc}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^{-\infty,\mathrm{sc}}(\Omega\times\mb{R}^p)$ then $\mathrm{sing\, supp}_{\ensuremath{\G^\infty}}I_\phi(a)=\emptyset$. \end{itemize} \end{prop} Proposition \ref{prop_smooth_sing} leads to the following result. \begin{prop} \label{prop_smooth_sing_cone} \begin{itemize} \item[{\, }] \item[(i)] If $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$ then $$\mathrm{sing\, supp}_{\ensuremath{{\mathcal G}}}\,I_\phi(a)\subseteq\pi_\Omega(C_\phi\cap {\rm{cone\, supp}}\, a).$$ \item[(ii)] If $\phi\in\wt{\Phi}^\mathrm{sc}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$ then $$\mathrm{sing\, supp}_{\ensuremath{\G^\infty}}I_\phi(a)\subseteq\pi_\Omega(C^\mathrm{sc}_\phi\cap {\rm{cone\, supp}}\, a).$$ \end{itemize} \end{prop} \subsection*{Generalized wave front sets of the functional $I_\phi(a)$} The next theorem investigates the $\ensuremath{{\mathcal G}}$-wave front set and the $\ensuremath{\G^\infty}$-wave front set of the functional $I_\phi(a)$ under suitable assumptions on the generalized symbol $a$ and the phase function $\phi$. \begin{thm} \label{theorem_WF} \leavevmode \begin{itemize} \item[(i)] Let $\phi\in\wt{\Phi}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mb{R}^p)$. The generalized wave front set $\mathrm{WF}_\ensuremath{{\mathcal G}} I_\phi(a)$ is contained in the set $W_{\phi,a}$ of all points $(x_0,\xi_0)\in\CO{\Omega}$ with the property that for all relatively compact open neighborhoods $U(x_0)$ of $x_0$, for all open conic neighborhoods $\Gamma(\xi_0)\subseteq \mb{R}^n\setminus 0$ of $\xi_0$, for all open conic neighborhoods $V$ of {\rm{cone\,supp}}\,$a\cap C_\phi$ such that $V\cap (U(x_0)\times\mb{R}^p\setminus 0)\neq\emptyset$ the generalized number \begin{equation} \label{gen_num_inv}
\mathop{{\rm{Inf}}}\limits_{\substack{y\in U(x_0), \xi\in \Gamma(\xi_0)\\ (y,\theta)\in V\cap( U(x_0)\times\mb{R}^p\setminus 0)}}\frac{|\xi-\nabla_y\phi(y,\theta)|}{|\xi|+|\theta|} \end{equation} is not invertible. \item[(ii)] If $\phi\in\wt{\Phi}^\mathrm{sc}(\Omega\times\mb{R}^p)$ and $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^p)$ then $\mathrm{WF}_{\ensuremath{\G^\infty}}I_\phi(a)$ is contained in the set $W^\mathrm{sc}_{\phi,a}$ of all points $(x_0,\xi_0)\in\CO{\Omega}$ with the property that for all relatively compact open neighborhoods $U(x_0)$ of $x_0$, for all open conic neighborhoods $\Gamma(\xi_0)\subseteq \mb{R}^n\setminus 0$ of $\xi_0$, for all open conic neighborhoods $V$ of {\rm{cone\,supp}}\,$a\cap C^\mathrm{sc}_\phi$ such that $V\cap (U(x_0)\times\mb{R}^p\setminus 0)\neq\emptyset$ the generalized number \eqref {gen_num_inv} is not slow scale-invertible. \end{itemize} \end{thm} Note that when $\phi$ is a classical phase function the set $W_{\phi,a}$ as well as the set $W^\mathrm{sc}_{\phi,a}$ coincide with \begin{equation} \label{set_class} \{(x,\nabla_x\phi(x,\theta)):\, (x,\theta)\in{\rm{cone\, supp}}\,a\cap C_\phi\}. \end{equation} For more details see \cite[Remark 4.13]{GHO:06}. \begin{ex} \label{C_example3} Theorem \ref{theorem_WF} can be employed for investigating the generalized wave front sets of the kernel $K_A:=I_\phi(a)$ of the Fourier integral operator introduced in the first example of Section \ref{gen_sec}. For simplicity we assume that $c$ is a bounded generalized constant in $\wt{\mb{R}}$ and that $a=1$. Let $((x_0,t_0,y_0),\xi_0)\in\mathrm{WF}_\ensuremath{{\mathcal G}} K_A$. From the first assertion of Theorem \ref{theorem_WF} we know that the generalized number given by \begin{equation} \label{non_inv_ex}
\inf_{\substack{(x,t,y)\in U, \xi\in \Gamma\\ ((x,t,y),\theta)\in V\cap(U\times\mb{R}\setminus 0)}}\frac{|\xi-(\theta,-c_\varepsilon\theta,-\theta)|}{|\xi|+|\theta|} \end{equation}
is not invertible, for every choice of neighborhoods $U$ of $(x_0,t_0,y_0)$, $\Gamma$ of $\xi_0$ and $V$ of $C_\phi$. Note that it is not restrictive to assume that $|\theta|=1$. We fix some sequences $(U_n)_n$, $(\Gamma_n)_n$ and $(V_n)_n$ of neighborhoods shrinking to $(x_0,t_0,y_0)$, $\{\xi_0\lambda:\lambda>0\}$ and $C_\phi$ respectively. By \eqref{non_inv_ex} we find a sequence $\varepsilon_n$ tending to $0$ such that for all $n\in\mb{N}$ there exists $\xi_n\in\Gamma_n$, $(x_n,t_n,y_n,\theta_n)\in V_n$ with $|\theta_n|=1$ and $(x_n,t_n,y_n)\in U_n$ such that \[
|\xi_n-(\theta_n,-c_{\varepsilon_n}\theta_n,-\theta_n)|\le \varepsilon_n(|\xi_n|+1). \] In particular, $\xi_n$ remains bounded. Passing to suitable subsequences we obtain that there exist $\theta$ such that $(x_0,t_0,y_0,\theta)\in C_\phi$, an accumulation point $c'$ of $(c_\varepsilon)_\varepsilon$ and a multiple $\xi'$ of $\xi_0$ such that $\xi'=(\theta,-c'\theta,-\theta)$. It follows that \[
\frac{\xi_0}{|\xi_0|}=
\frac{\xi'}{|\xi'|}=\frac{1}{\sqrt{2+(c')^2}|\theta|}\,(\theta,-c'\theta,-\theta). \]
In other words the $\ensuremath{{\mathcal G}}$-wave front set of the kernel $K_A$ is contained in the set of points of the form $((x_0,t_0,y_0),(\theta_0,-c'\theta_0,-\theta_0))$ where $(x_0,t_0,y_0,\theta_0)\in C_\phi$ and $c'$ is an accumulation point of a net representing $c$. Since in the classical case (when $c\in\mb{R}$) the distributional wave front set of the corresponding kernel is the set $\{((x_0,t_0,y_0),(\theta_0,-c\theta_0,-\theta_0)):\, (x_0,t_0,y_0,\theta_0)\in C_\phi\}$, the result obtained above for $\mathrm{WF}_\ensuremath{{\mathcal G}} K_A$ is a generalization in line with what we deduced about the regions $R_\phi$ and $C_\phi$. \end{ex} \subsection*{Particular case: generalized pseudodifferential operators} Finally, we consider a generalized pseudodifferential operator $a(x,D)$ on $\Omega$ and its kernel $K_{a(x,D)}\in\mathcal{L}(\ensuremath{{\mathcal G}_\mathrm{c}}(\Omega\times\Omega),\wt{\mb{C}})$. By Remark 4.15 in \cite{GHO:06}, we have that $\mathrm{WF}_\ensuremath{{\mathcal G}}(K_{a(x,D)})$ is contained in the normal bundle of the diagonal in $\Omega\times\Omega$ when $a\in\wt{\mathcal{S}}^{m}_{\rho,\delta}(\Omega\times\mb{R}^n)$ and that $\mathrm{WF}_{\ensuremath{\G^\infty}}(K_{a(x,D)})$ is a subset of the normal bundle of the diagonal in $\Omega\times\Omega$ when $a$ is of slow scale type. We define the sets \[ \mathrm{WF}_\ensuremath{{\mathcal G}}(a(x,D))=\{(x,\xi)\in\CO{\Omega}:\, (x,x,\xi,-\xi)\in\mathrm{WF}_\ensuremath{{\mathcal G}}(K_{a(x,D)})\} \] and \[ \mathrm{WF}_{\ensuremath{\G^\infty}}(a(x,D))=\{(x,\xi)\in\CO{\Omega}:\, (x,x,\xi,-\xi)\in\mathrm{WF}_{\ensuremath{\G^\infty}}(K_{a(x,D)})\}. \] From Theorem \ref{theorem_WF} one deduces the following. \begin{prop} \label{prop_pseudo} Let $a(x,D)$ be a generalized pseudodifferential operator. \begin{itemize} \item[(i)] If $a\in\wt{\mathcal{S}}^{m}_{\rho,\delta}(\Omega\times\mb{R}^n)$ then $\mathrm{WF}_\ensuremath{{\mathcal G}}(a(x,D))\subseteq\mu\, \mathrm{supp}_\ensuremath{{\mathcal G}}(a)$. \item[(ii)]If $a\in\wt{\mathcal{S}}^{m,\mathrm{sc}}_{\rho,\delta}(\Omega\times\mb{R}^n)$ then $\mathrm{WF}_{\ensuremath{\G^\infty}}(a(x,D))\subseteq\mu\, \mathrm{supp}_{\ensuremath{\G^\infty}}(a)$. \end{itemize} \end{prop}
\subsection*{Acknowledgment} The author would like to express her gratitude to Professor L. Rodino and Professor M. W. Wong for the kind invitation to the session on Pseudodifferential Operators of the 2007 ISAAC Congress at METU in Ankara.
\end{document} | arXiv | {
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\begin{document}
\title{On the Equivalence of the Weighted Tsetlin Machine and the Perceptron}
\begin{abstract}
Tsetlin Machine (TM) has been gaining popularity as an inherently interpretable machine leaning method that is able to achieve promising performance with low computational complexity on a variety of applications. The interpretability and the low computational complexity of the TM are inherited from the Boolean expressions for representing various sub-patterns. Although possessing favorable properties, TM has not been the go-to method for AI applications, mainly due to its conceptual and theoretical differences compared with perceptrons and neural networks, which are more widely known and well understood. In this paper, we provide detailed insights for the operational concept of the TM, and try to bridge the gap in the theoretical understanding between the perceptron and the TM. More specifically, we study the operational concept of the TM following the analytical structure of perceptrons, showing the resemblance between the perceptrons and the TM. Through the analysis, we indicated that the TM's weight update can be considered as a special case of the gradient weight update. We also perform an empirical analysis of TM by showing the flexibility in determining the clause length, visualization of decision boundaries and obtaining interpretable boolean expressions from TM. In addition, we also discuss the advantages of TM in terms of its structure and its ability to solve more complex problems. \end{abstract}
\section{Introduction} Researchers across various fields are increasingly paying attention to the interpretability of AI techniques. While interpretability previously was inherent in most machine learning approaches, the state-of-the-art methods now increasingly rely on black-box deep neural networks (DNNs). Natively, DNNs can hardly be interpreted during the learning stage or while producing outputs~\cite{nnfragile_vvimp}. A surge of techniques attempts to open the black box by visual explanations and gradient-based interpretability~\cite{saliency,interpretcnns,patchnet,dissection,gradcam}, but do not change the black-box nature. \\ The Tsetlin Machine (TM) is a natively interpretable rule-based machine learning algorithm that produces logical rules~\cite{granmo2018}. Despite being logic-based, the TM is a universal function approximator, like a neural network. In brief, it employs an ensemble of Tsetlin Automata (TA) that learns propositional logic expressions from Boolean input features. Propositional logic drives learning, eliminating the requirement for floating-point operations. Due to its Boolean representations and finite-state automata learning mechanisms, it has a minimalistic memory footprint. More importantly, TM achieves interpretability by leveraging sparse disjunctive normal form. Indeed, humans are particularly good at understanding flat and short logical AND-rules, reflecting human reasoning~\cite{human-reasoning}.\\ Because the operational concept of TMs is significantly different from that of neural networks, TM is really challenging for those who are used to the neural networks to understand. For this reason, the TMs have not been considered as the to-go method in the machine learning society. In this paper, we show the operational concept of TMs for its learning phase, following the structure that is widely used in the analysis of the neural networks. Particularly, we aim at showing the resemblance between the two distinct techniques and reveal the concept and the advantages of the TMs in a painless manner. In more details, we divide TM's learning into two phases: the clause learning phase and the clause weight update phase. \\ For the clause learning phase, we show that one clause can learn one or multiple sub-patterns given enough updates. This phase bears resemblance to the connections and the activation functions in a perceptron. For the clause weight update phase, we indicate that the clauses are weighted according to their correctness, which is similar to the weights in a perceptron. For this reason, following the concept of the perceptron convergence theorem, the clause weight update for the TM can also be confirmed. To summarize, the clauses can learn any sub-patterns from data and the clause weights show the importance of such sub-patterns, similar to the connections and the weights in a perceptron. In addition, we visualize the decision boundaries using clauses of TM and formalize its advantages over other the perceptron.
\section{Review of the Tsetlin Machine} \label{sec:TM} The Tsetlin Machine is a machine learning algorithm based on Boolean expressions called clauses that individually identify sub-patterns in data and are aggregated together as a weighted sum of Boolean inputs.\\ \begin{figure}
\caption{A two-action Tsetlin Automaton with $2N$ states.}
\label{figTA}
\end{figure}
\begin{figure}
\caption{TM learning dynamics for an XOR-gate training sample, with input ($x_1=0, x_2=1$) and output target $y=1$.}
\label{figure:tm_architecture_basic}
\end{figure} \paragraph{Structure.} A TM in its simplest form takes a feature vector $\mathbf{x} = [x_1, x_2, \ldots, x_o] \in \{0,1\}^o$ of $o$ propositional values as input and assigns the vector a class $\hat{y} \in \{0,1\}$. In brief, the input vector $\mathbf{x}$ provides the literal set $L = \{l_1, l_2, \ldots, l_{2o}\} = \{x_1, x_2, \ldots, x_{o}, \lnot x_1, \lnot x_2, \ldots, \lnot x_o\}$, consisting of the input features and their negations. By selecting subsets $L_j \subseteq L$ of the literals, the TM can build arbitrarily complex patterns, ANDing the selected literals to form conjunctive clauses: \begin{equation} \label{eq:1} C_j(\mathbf{x})= \bigwedge_{l_k \in L_j} l_k. \end{equation} Above, $j \in \{1, 2, \ldots, n\}$ is the index of a clause $C_j$ and $k \in \{1, 2, \ldots, 2o\}$ refers to a particular literal $l_k$. $n$ is the total number of clauses. As an example, the clause $C_j(\mathbf{x}) = x_1 \land \lnot x_2$ consists of the literals $L_j = \{x_1, \lnot x_2\}$ and evaluates to $1$ when $x_1=1$ and $x_2=0$. \\ The TM assigns one Tsetlin Automata (TA) \cite{ta,LA_book} per literal $l_k$ per clause $C_j$ to build the clauses. The TA assigned to literal $l_k$ of clause $C_j$ decides whether $l_k$ is \emph{Excluded} or \emph{Included} in $C_j$. Figure~\ref{figTA} depicts a two-action TA with $2N$ states. For states $1$ to $N$, the TA performs action \emph{Exclude} (Action 1), while for states $N + 1$ to $2N$ it performs action \emph{Include} (Action 2). As feedback to the action performed, the environment responds with either a Reward or a Penalty. If the TA receives a Reward, it moves deeper into the side of the action. If it receives a Penalty, it moves towards the middle and eventually switches action.\\ With $n$ clauses and $2o$ literals, we have in total $n\times2o$ TAs. We organize the states of these in an $n\times2o$ matrix $A = [a_k^j] \in \{1, 2, \ldots, 2N\}^{n\times2o}$. We will use the function $g(\cdot)$ to map the automaton state $a_k^j$ to Action $0$ (\emph{Exclude}) for states $1$ to $N$ and to Action $1$ (\emph{Include}) for states $N+1$ to~$2N$: \begin{equation} \label{eq:g} g(a_k^j) = \begin{cases} 1& a_k^j > N\\ 0& otherwise. \end{cases} \end{equation} We then can connect the states $a_k^j$ of the TAs assigned to clause $C_j$ with its composition as follows: \begin{equation} \label{eq:2} C_j(\mathbf{x}) = \bigwedge_{l_k \in L_j} l_k = \bigwedge_{k=1}^{2o} \left[g(a_k^j) \Rightarrow l_k\right]. \end{equation} Here, $l_k$ is one of the literals and $a_k^j$ is the state of its TA in clause $C_j$. The logical \emph{imply} operator~$\Rightarrow$ implements the \emph{Exclude}/\emph{Include} action. That is, the \emph{imply} operator is always $1$ if $g(a_k^j)=0$ (\emph{Exclude}), while if $g(a_k^j)=1$ (\emph{Include}) the truth value is decided by the truth value of the literal. \paragraph{Classification.} Classification is performed as a majority vote. The odd-numbered half of the clauses vote for class $\hat{y} = 0$ and the even-numbered half vote for $\hat{y} = 1$: \begin{equation}
\hat{y} = 0 \le \sum_{j=1,3,\ldots}^{n-1} \bigwedge_{k=1}^{2o} \left[g(a_k^j) \Rightarrow l_k\right] - \sum_{j=2,4,\ldots}^{n} \bigwedge_{k=1}^{2o} \left[g(a_k^j) \Rightarrow l_k\right]. \label{eqn:prediction} \end{equation} As such, the odd-numbered clauses have positive polarity, while the even-numbered ones have negative polarity. As an example, consider the input vector $\mathbf{x} = [0, 1]$ in the lower part of Figure \ref{figure:tm_architecture_basic}. The figure depicts two clauses of positive polarity, $C_1(\mathbf{x}) = x_1 \land \lnot x_2$ and $C_3(\mathbf{x}) = \lnot x_1 \land \lnot x_2$ (the negative polarity clauses are not shown). Both of the clauses evaluate to zero, leading to class prediction $\hat{y} = 0$. \begin{table}[t] \centering \vskip 0.15in \begin{center} \begin{small}
\begin{tabular}{l|l|l|l}
\hline
\multirow{2}{*}{Input}&Clause & \ \ \ \ \ \ \ 1 & \ \ \ \ \ \ \ 0 \\
&{Literal} &\ \ 1 \ \ \ \ \ \ 0 &\ \ 1 \ \ \ \ \ \ 0 \\
\hline
\multirow{2}{*}{Include Literal}&P(Reward)&$\frac{s-1}{s}$\ \ \ NA & \ \ 0 \ \ \ \ \ \ 0\\ [1mm]
&P(Inaction)&$\ \ \frac{1}{s}$\ \ \ \ \ NA &$\frac{s-1}{s}$ \ $\frac{s-1}{s}$ \\ [1mm]
&P(Penalty)& \ \ 0 \ \ \ \ \ NA& $\ \ \frac{1}{s}$ \ \ \ \ \ $\frac{1}{s}$ \\ [1mm]
\hline
\multirow{2}{*}{Exclude Literal}&P(Reward)& \ \ 0 \ \ \ \ \ \ $\frac{1}{s}$ & $\ \ \frac{1}{s}$ \ \ \ \ \
$\frac{1}{s}$ \\ [1mm]
&P(Inaction)&$ \ \ \frac{1}{s}$\ \ \ \ $\frac{s-1}{s}$ &$\frac{s-1}{s}$ \ $\frac{s-1}{s}$ \\ [1mm]
&P(Penalty)&$\frac{s-1}{s}$ \ \ \ \ 0& \ \ 0 \ \ \ \ \ \ 0 \\ [1mm]
\hline \end{tabular} \end{small} \end{center} \caption{Type-I Feedback} \label{table:type_i} \end{table}
\begin{table}[t] \centering \vskip 0.15in \begin{center} \begin{small}
\begin{tabular}{l|l|l|l}
\hline
\multirow{2}{*}{Input}&Clause & \ \ \ \ \ \ \ 1 & \ \ \ \ \ \ \ 0 \\
&{Literal} &\ \ 1 \ \ \ \ \ \ 0 &\ \ 1 \ \ \ \ \ \ 0 \\
\hline
\multirow{2}{*}{Include Literal}&P(Reward)&\ \ 0 \ \ \ NA & \ \ 0 \ \ \ \ \ \ 0\\[1mm]
&P(Inaction)&1.0 \ \ NA & 1.0 \ \ \ 1.0 \\[1mm]
&P(Penalty)&\ \ 0 \ \ \ NA & \ \ 0 \ \ \ \ \ \ 0\\[1mm]
\hline
\multirow{2}{*}{Exclude Literal}&P(Reward)&\ \ 0 \ \ \ \ 0 & \ \ 0 \ \ \ \ \ \ 0\\[1mm]
&P(Inaction)&1.0 \ \ \ 0 & 1.0 \ \ \ 1.0 \\[1mm]
&P(Penalty)&\ \ 0 \ \ 1.0 & \ \ 0 \ \ \ \ \ \ 0\\[1mm]
\hline \end{tabular} \end{small} \end{center} \caption{Type-II Feedback} \label{table:type_ii} \end{table} \paragraph{Learning.} The upper part of Figure \ref{figure:tm_architecture_basic} illustrates learning. A TM learns online, processing one training example $(\mathbf{x}, y)$ at a time. Based on $(\mathbf{x}, y)$, the TM rewards or penalizes its TAs, which amounts to increasing or decreasing their states. There are two kinds of feedback: Type I Feedback produces frequent patterns and Type II Feedback increases the discrimination power of the patterns.\\ Type I feedback is given stochastically to clauses with positive polarity when $y=1$ and to clauses with negative polarity when $y=0$. Conversely, Type II Feedback is given stochastically to clauses with positive polarity when $y=0$ and to clauses with negative polarity when $y=1$. The probability of a clause being updated is based on the vote sum $v$: $v = \sum_{j=1,3,\ldots}^{n-1} \bigwedge_{k=1}^{2o} \left[g(a_k^j) \Rightarrow l_k\right] - \sum_{j=2,4,\ldots}^{n} \bigwedge_{k=1}^{2o} \left[g(a_k^j) \Rightarrow l_k\right]$. The voting error is calculated as: \begin{equation} \label{eq:TT} \epsilon = \begin{cases} T-v,& for~ y=1\\ T+v,& for~ y=0. \end{cases} \end{equation} Here, $T$ is a user-configurable voting margin yielding an ensemble effect. The probability of updating each clause is $P(\mathrm{Feedback}) = \frac{\epsilon}{2T}$. Random sampling from $P(\mathrm{Feedback})$ will decided which clauses to update, and then the following TA state updates can be formulated as matrix additions, subdividing Type I Feedback into feedback Type Ia and Type Ib: \begin{equation}
A^*_{t+1} = A_t + F^{\mathit{II}} + F^{Ia} - F^{Ib}.
\label{eqn:learning_step_1} \end{equation} Here, $A_t = [a^j_k] \in \{1, 2, \ldots, 2N\}^{n \times 2o}$ contains the states of the TAs at time step $t$ and $A^*_{t+1}$ contains the updated state for time step $t+1$ (before clipping). The matrices $F^{\mathit{Ia}} \in \{0,1\}^{n \times 2o}$ and $F^{\mathit{Ib}} \in \{0,1\}^{n \times 2o}$ contain Type I Feedback. A zero-element means no feedback and a one-element means feedback. As shown in Table \ref{table:type_ii} on the left, two rules govern Type I feedback: \begin{itemize}
\item \textbf{Type Ia Feedback} is given with probability $\frac{s-1}{s}$ whenever both clause and literal are $1$-valued\footnote{Note that the probability $\frac{s-1}{s}$ is replaced by $1$ when boosting true positives.}. It penalizes \emph{Exclude} actions and rewards \emph{Include} actions. The purpose is to remember and refine the patterns manifested in the current input $\mathbf{x}$. This is achieved by moving the state of the TA toward the right side. The user-configurable parameter $s$ controls pattern frequency, i.e., a higher $s$ produces less frequent patterns.
\item \textbf{Type Ib Feedback} is given with probability $\frac{1}{s}$ whenever either clause or literal is $0$-valued. This feedback rewards \emph{Exclude} actions and penalizes \emph{Include} actions to coarsen patterns, combating overfitting. Thus, the selected TA states are decreased. \end{itemize} The matrix $F^{\mathit{II}} \in \{0, 1\}^{n \times 2o}$ contains Type II Feedback to the TAs, given per Table \ref{table:type_ii} on the right. \begin{itemize} \item \textbf{Type II Feedback} penalizes \emph{Exclude} actions to make the clauses more discriminative, combating false positives. That is, if the literal is $0$-valued and the clause is $1$-valued, TA that has the current state below $N+1$ are encouraged to move towards right side. Eventually the clause becomes $0$-valued for that particular input, upon inclusion of the $0$-valued literal. \end{itemize} The final updating step for training example $(\mathbf{x}, y)$ is to clip the state values to make sure that they stay within value $1$ and $2N$: \begin{equation}
A_{t+1} = \mathit{clip}\left(A^*_{t+1}, 1, 2N\right). \label{eqn:learning_step_2} \end{equation}
For example, both of the clauses in Figure \ref{figure:tm_architecture_basic} receives Type~I Feedback over several training examples, making them resemble the input associated with $y=1$.
\subsection{Weighted Tsetlin Machine} In this subsection, we detail the TM with weights. The learning of weights is based on increasing the weight of clauses that receive Type Ia feedback (due to true positive output) and decreasing the weight of clauses that receive Type II feedback (due to false positive output). The overall rationale is to determine which clauses are inaccurate and thus must team up to obtain high accuracy as a team (low weight clauses), and which clauses are sufficiently accurate to operate more independently (high weight clauses). The weight updating procedure is summarized in Algorithm \ref{algo:tm}. Here, $w_i$ is the weight of clause $C_i$ at the $n^{th}$ training round (ignoring polarity to simplify notation). The first step of a training round is to calculate the clause output as per Equation~(\ref{eq:2}). The weight of a clause is only updated if the clause output $C_i$ is 1 and the clause has been selected for feedback ($P_i$ = 1). Then the polarity of the clause and the class label $y$ decide the type of feedback given. That is, like a regular TM, positive polarity clauses receive Type Ia feedback if the clause output is a true positive, and similarly, they receive Type II feedback if the clause output is a false positive. For clauses with negative polarity, the feedback types switch roles. When clauses receive Type Ia or Type II feedback, their weights are updated accordingly. We use the stochastic searching on the line (SSL) automaton to learn appropriate weights. SSL is an optimization scheme for unknown stochastic environments~\cite{oommen}. The goal is to find an unknown location $\lambda^*$ within a search interval $[0,1]$. In order to find $\lambda^*$, the only available information for the Learning Mechanism (LM) is the possibly faulty feedback from its attached environment $E$.\\ In SSL, the search space $\lambda$ is discretized into $N$ points, $\{0,1/N,2/N,...,(N-1)/N,1\}$ with N being the discretization resolution. During the search, the LM has a location $\lambda \in \{0,1/N,2/N,...,(N-1)/N,1\}$, and can freely move to the left or to the right from its current location. The environment $E$ provides two types of feedback: $E = 1$ is the environment suggestion to increase the value of $\lambda$ by one step, and $E = 0$ is the environment suggestion to decrease the value of $\lambda$ by one step. The next location of $\lambda$, i.e. $\lambda_{n + 1}$, can thus be expressed as follows: \begin{equation}
\lambda_{n + 1} =
\begin{cases}
\lambda_n+1/N, & \text{if $E_n=1$,}\\
\lambda_n-1/N, & \text{if $E_n=0$.}\\
\end{cases} \end{equation} \begin{equation}
\lambda_{n + 1} =
\begin{cases}
\lambda_n, & \text{if $\lambda_n=1$ and $E_n=1$,}\\
\lambda_n, & \text{if $\lambda_n=0$ and $E_n=0$.}\\
\end{cases} \end{equation} Asymptotically, the learning mechanics is able to find a value arbitrarily close to $\lambda^*$ when $N\rightarrow\infty$ and $n\rightarrow\infty$. In our case, the search space of clause weights is $[0, \infty]$, so we use resolution $N = 1$, with no upper bound for $\lambda$. Accordingly, we operate with integer weights. As in Algorithm~\ref{algo:tm}, if the clause output is a true positive, we simply increase the weight by $1$. Conversely, if the clause output is a false positive, we decrease the weight by $1$.\\ By following the above procedure, the goal is to make low precision clauses team up by giving them low weights, so that they together can reach the summation target $T$. By teaming up, precision increases due to the resulting ensemble effect. Clauses with high precision, however, obtain a higher weight, allowing them to operate more independently.\\ The above weighting scheme has several advantages. First of all, increment and decrement operations on integers are computationally less costly than multiplication based updates of real-valued weights. Additionally, a clause with an integer weight can be seen as multiple copies of the same clause, making it more interpretable than real-valued weighting, as shown in the next section. Additionally, clauses can be turned completely off by setting their weights to $0$ if they do not contribute positively to the classification task. For a more detailed explanation of the weighted TM, please refer to ~\cite{abeyrathna2021integer}.
\begin{algorithm}[t] \caption{Complete WTM learning process}\label{algo:tm} \begin{algorithmic}[1] \State \textbf{Input:} Training data batch $(B, x, y) \quad \rhd B \geq 1$ \State \textbf{Initialize:} Random initialization of TAs \State \textbf{Begin:} $n^{th}$ training round \For{$i = 1, ...,m$} \textbf{if} $P_i = 1$
\If{($y = 1$ \textbf{and} $i$ is odd) \textbf{or} ($y = 0$ \textbf{and} $i$ is even)}
\If{$c_i = 1$}
\State $w_i \leftarrow w_i+1$
\For{feature $k=1,...,2o$}
\If{$l_k = 1$}
\State Type Ia Feedback
\Else:
\State Type Ib Feedback
\EndIf
\EndFor
\Else:
\State $w_i \leftarrow w_i \quad \rhd$ \text{[No Change]}
\State Type Ib Feedback
\EndIf
\Else: ($y = 1$ \textbf{and} $i$ is even) \textbf{or} ($y = 0$ \textbf{and} $i$ is odd)
\If{$c_i = 1$}
\If{$w_i > 0$}
\State $w_i \leftarrow w_i-1$
\EndIf
\For{feature $k=1,...,2o$}
\If{$l_k = 0$}
\State Type II Feedback
\Else:
\State Inaction
\EndIf
\EndFor
\Else:
\State $w_i \leftarrow w_i \quad \rhd$ \text{[No Change]}
\State Inaction
\EndIf
\EndIf \EndFor \end{algorithmic} \end{algorithm}
\section{Convergence Analysis of the Tsetlin Machine} \label{sec:converge} \begin{figure}
\caption{Phase-1 and Phase-2 Feedbacks}
\label{fig:tm}
\end{figure} As discussed in Section \ref{sec:TM}, TM learns clauses to identify sub-patterns in data. These sub-patterns are aggregated using a linearly weighted sum of clauses, where the weights depend on how well the clauses detect sub-patterns. Now, let's consider the weighted TM with clause weights $w_i$ and clauses $C_i$, where even $i$ represent negative polarity clauses and odd $i$ represent positive polarity clauses: \begin{equation}
\sum_i w_iC_i \, \geq \, T \; \rightarrow \; \hat{y} \tag{Using Eq.~(\ref{eq:TT})} \end{equation} \begin{equation}
\sum_i w_i\left(\bigwedge_{l_k \in L_i} l_k \right) \, \geq \, T \; \rightarrow \; \hat{y} \tag{Using Eq.~(\ref{eq:1})} \end{equation} The learning of clauses and its weights can be considered as separate phases as described in Section \ref{sec:TM} and shown in Algorithm \ref{algo:tm}. Each $w_i$ is updated according to its associated clause $C_i$'s correctness. Whereas each clause $C_i$ obtains feedback by comparing the clause output (and its polarity) and the true label. So, the convergence of TM is divided into two phases: \begin{itemize}
\item \textbf{Phase $\mathbf{1}$: Clause learning} - \emph{Local sub-pattern learning}. A single clause is able to learn a single correct sub-pattern based on the feedback given to it by comparing the clause output with the desired output. This is shown according to parallel feedback given to the TAs. (Local view, see Subsection \ref{sec:cl})
\item \textbf{Phase $\mathbf{2}$: Clause weight update} - \emph{Global pattern aggregation}. The sub-patterns learnt by the clauses are weighted according to their correctness, i.e. a weighted sum of sub-patterns. The weighted TM and its weight update is shown to be akin to the perceptron. (Global view, see Subsection \ref{sec:w}) \end{itemize}
\subsection{Clause learning} \label{sec:cl} In this subsection, we illustrate that one clause can capture one individual sub-pattern or several sub-patterns if the sub-patterns can be represented by the clause jointly.
To clarify the meaning of capturing one individual sub-pattern or several sub-patterns jointly, we look at the XOR and the AND operators as examples. Let us review the case where one clause captures an individual sub-pattern by looking at the XOR operator. For this operator, ($x_1=1$, $x_2=0$) and ($x_1=0$, $x_2=1$) give $y=1$ while $y=0$ otherwise. Clearly, we have two sub-patterns and the input bits of the sub-patterns are mutual exclusive. For this reason, we need one clause $C_1=x_1\wedge \neg x_2$ to capture the first sub-pattern and $C_2=\neg x_1\wedge x_2$ to represent the second one. Obviously, there is no possibility to represent the two sub-patterns jointly by one clause, and thus one clause must correspond to each sub-pattern. Indeed, the TM can learn almost surely the intended logic in infinite time horizon. The convergence of the XOR operator has been proven in~\cite{jiao2021convergence}.
For the case where sub-patterns can be presented jointly, we exam the AND operator. Clearly, in addition to ($x_1=1$, $x_2=0$) and ($x_1=0$, $x_2=1$), ($x_1=1$, $x_2=1$) will also trigger a positive output. Different from the XOR case where the sub-patterns are mutually exclusive, the latter two sub-patterns in the AND operator can be jointly represented by $C_1=x_2$. Although in the AND operator three sub-patterns exist, two clauses, e.g., $C_1=x_2$ and $C_2=x1\wedge\neg x_2$, are sufficient to present the intended AND operator. Indeed, the TM can learn almost surely the intended AND operator in infinite time horizon. The convergence of the operator has been proven in~\cite{jiao2021convergenceAND}.
From the above mentioned two examples, we can see that a clause can indeed learn either a sub-pattern individually or multiple sub-patterns jointly for those special cases where the input is 2-bits long. In general, we conjuncture that a clause, after learning, can capture one or multiple sub-patterns from the training sample when the input has more bits. The proof of the general case is not trivial because the feedback for a certain literal is not only determined by its own state, but also by the output of the clause that is jointly determined by all its literals. Although this conjuncture has not been theoretically proven, we have observed from simulations that the clauses in a TM can indeed present sub-patterns efficiently. In what follows, we present the clause learning phase formally, aiming at revealing the dynamics of the learning and providing more insights for a better comprehension.
TM learns to recognize local patterns courtesy of clauses, which are propositional expressions of binarized features (literals), in original or negated form, connected by logical \emph{AND} operations. Correct sub-patterns are learnt by updating TA states that are associated with each literal which thereafter results in updated clauses. Consider a sub-pattern or a group of sub-patterns that one clause can learn joint. For the weighted TM, output is expected to be greater than $T$ once learnt for the intended sub-pattern (or joint sub-patterns), i.e., \begin{equation}
\label{eq:utm}
w_i \bigwedge_{k=1}^{2o} \left[g(a_k^i) \Rightarrow l_k\right] \, \geq \, T \; \rightarrow \; \hat{y} \end{equation} Each TA $g(a_k^i)$, associated with a particular literal $l_k$, is updated according to the feedback given to the clause. The update of $w_1$ is to be discussed in the next Subsection. For updating the clause itself, we have \begin{equation}
\label{eq:cf}
C_i=\bigwedge_{k=1}^{2o} \left[g(a_k^i) \Rightarrow l_k\right] \, \xleftarrow[]{\text{Feedback}} \, \mathcal{F}(y == \hat{y}) \end{equation} where $\mathcal{F}$ is the feedback given to the clause by comparing $\hat{y}$, the clause output and $y$, the true label. This feedback is sent to each literal. Note that once the feedback is given, the state updating process for each TA is independent and thus the TAs can be updated in parallel.
As the literals within the clause have an \emph{AND} relationship, any literal that produces 0 will results in a 0 for the clause output. On the contrary, the clause output 1 only when all literals output 1. This nature will result in a 0 value for a clause most probably when we randomly initiate the states of the TAs in the beginning of the learning. The 0 literal value or clause value will result in a Type Ib feedback for any ``true" training samples ($y=1$), which encourages the literals to be excluded. As the literals become excluded, the length of the clause is reduced. Once the 0-valued literals are all excluded and only the 1-valued literals are left, Type Ia feedback will come to the play and thus encourage more literals to be included in the clause. At the same time, Type~II feedback will depress possible false positive by including necessary literals upon a false training sample ($y=0$). This process will go back and forth during the learning process, until $w_iC_i$ reaches $T$. Once $w_iC_i\geq T$ holds, the input of the sub-pattern is blocked by the TM, as per Eq.~(\ref{eq:TT}).
\subsection{Clause weight update} \label{sec:w} As the clause is update to learn the intended sub-pattern, each clause is weighted according its importance and correctness. Here, the importance of the sub-pattern depends on its frequency of occurrence in different instances of data and its distinguishing capabilities for a particular task. For example, a clause that captures the sub-pattern of facial features in a ``dog vs cat'' task will have higher weights than a clause that captures sub-patterns of the tail, since facial feature are more discriminative for this task. The weighted TM can be represented as follows: \begin{equation*}
\sum_i w_iC_i \geq T, \end{equation*} which can be rewritten as: \begin{equation}
\label{eq:0}
\sum_i w_iC_i -T \geq 0, \end{equation} where $w_i$ is the clause weight of clause $C_i$ and $T$ is the threshold. Clauses are represented in Equation~(\ref{eq:2}). This means that if this inequality holds for a particular input then it is assigned to class $1$, otherwise class $0$. For the case of TM, the threshold is static and assigned at the beginning of training. Here, we make $T$ learnable by reconstructing it as a clause weight. The clause associated with $T$ is a dummy clause whose output is always $1$. And, the weight $T$ is always subtracted from the weighted clause vote count. After those modifications, Eequation~(\ref{eq:0}) becomes analogous to a perceptron \cite{perceptron}, where $w_i$ are the weights, $C_i$ are the inputs and $T$ is the bias. Now, the weights of such a network are updated by the gradients of the error with respect to the weights. Let the error be represented as: \begin{equation} \label{eq:E}
E = \left(\sum_i w_i.C_i -T\right) - y, \end{equation} where, $y$ is the correct target. Clearly, the gradient of $E$ with respect to weight $w_i$ becomes: \begin{equation}
\nabla_{w_i} E = C_i, \end{equation} which means that the weight $w_i$ is updated as: \begin{align} \label{eq:4}
w_i &= w_i + \eta\gamma\nabla_{w_i} E\\\nonumber
&= w_i + \eta\gamma C_i, \end{align} where $\gamma \in \{-1,1\}$ represents correct prediction with $1$ and incorrect prediction with $-1$. This gives the direction of update and $\eta$ is the learning rate. In case of perceptron, $\eta=1$, which makes perceptron update: \begin{align}
w_i = w_i + C_i \quad \Rightarrow \quad w_i = w_i + 1, \tag{[since $C_i \in \{0,1\}$]}\\
w_i = w_i - C_i \quad \Rightarrow \quad w_i = w_i - 1.
\label{eq:tmu} \end{align}
This is how the clause weights are updated in weighted TM as shown in Section~\ref{sec:TM} and Algorithm~\ref{algo:tm}. From Algorithm~\ref{algo:tm}, we can see that, when the clause is correct, Equation~(\ref{eq:tmu}) is used to update the clause weight as shown in Lines $5-7$ and in case of the clause being incorrect, Equation~(\ref{eq:tmu}) is used to update the clause weight as shown in Lines $19-22$. The clause weights with clause output $C=0$ are not updated. \\
Algorithm~\ref{algo:tm} runs over all training instances (in batches or online manner) and the clause weights are incremented every time the clause detects the correct sub-pattern and decremented if it does not. As mentioned previously, the weights of clauses that capture important and more discriminative sub-patterns, i.e. ones that reduce the error $E$ and produce correct outputs, are incremented gradually and vice versa.
Similarly, gradient of error in Equation~(\ref{eq:E}) with respect to $T$ is: \begin{equation}
\nabla_T E = - 1. \end{equation} Applying Equation~(\ref{eq:4}) to update $T$, we have: \begin{equation} \label{eq:8}
T = T + \gamma(-1). \end{equation}
From Equation~(\ref{eq:8}), we can see that $T$ is updated with the opposite polarity as $w_i$, which is what we expected as $T$ is always subtracted from the total vote count. This shows that the weight updating mechanism in the TM can be considered a special case of a perceptron with binary inputs (assuming the outputs of clauses are inputs here). Hence, the convergence properties and mathematical analysis of the perceptron~\cite{perceptron} also hold for the clause weight update of the TM, assuming the clauses as binary inputs. In addition, the clause weight update is equivalent to a gradient update with $\eta=1$. The convergence of perceptron is given as a reference point in Appendix~\ref{sec:app}.
\section{Empirical Analysis} In this section, we present the empirical results based on our experiments. We firstly show the results for the clause learning phase with the focus on the relationship between the clause length and hyperparameter $s$. Thereafter, we consider the weights of the TM and show the resemblance between the neural networks and the TM. \subsection{Clause Length} A clause evaluating to $1$ means that the sub-pattern associated with the clause is presented in a particular data instance and a clause evaluating to $0$ means the absence of the sub-pattern (but might to evaluate to $1$ for other instances, indicating presence of the sub-pattern). Understandably, if the length of a clause is longer, it learns more fine features or details in the sub-pattern. This is due to more literals are included for a longer clause so that the details/fine features can also be represented, with a cost of the risk of overfitting. On the contrary, the short clause learns more generalized features and has easier readability. The length of the clause is depend on the nature of the problem, and also in part, depends on the $s$-parameter. The $s$-parameter is responsible for assigning the probabilities of reward, penalty and inaction for including and excluding literals (as given in Table~\ref{table:type_i}). From Table~\ref{table:type_i}, we can see that the probability on inclusion of a literal is high when $s$ is high, mainly due to Type Ia feedback. To validate this statement, we show from Table~\ref{tab:s} the variation in clause size as a function of $s$-parameter. These clauses were obtained by training a TM on the Iris dataset for $50$ epochs. Clearly, longer clauses are found for a lager value of $s$. \\
\begin{table*}[h]
\centering
\begin{sc}
\begin{tabular}{c|c|c}
\hline
& $s=10$ & $s=2$ \\
\hline \small{Clause-1:} & $\neg x_5 \land \neg x_{10} \land \neg x_{11} $ & $ \neg x_7 \land \neg x_8 \land \neg x_{10} $ \\ \small{Clause-2:} & $ x_2 \land \neg x_0 \land \neg x_1 \land \neg x_4 \land \neg x_8 \land \neg x_9 \land \neg x_{10} \land \neg x_{15}$ & $ x_4 \land x_7 \land \neg x_{11} $ \\ \small{Clause-3:} & $ x_7 \land x_8 \land \neg x_{11} \land \neg x_{12} \land x_{15}$ & $ x_9 $ \\ \small{Clause-4:} & $ x_1 \land \neg x_2 \land \neg x_6 \land x_7 $ & $ \neg x_0 \land \neg x_6 $ \\ \small{Clause-5:} & $ x_7 \land \neg x_9 \land \neg x_{15} $ & $x_{13} \land \neg x_0 \land \neg x_{14}$ \\ \small{Clause-6:} & $ x_{10} \land \neg x_{11} \land \neg x_{13} \land x_{15}$ & $ x_3 \land x_{12} $ \\ \small{Clause-7:} & $ \neg x_0 \land x_7 \land \neg x_{13} \land x_{15} \land \neg x_2$ & $\neg x_{10} $ \\ \small{Clause-8:} & $ x_2 \land \neg x_0 \land \neg x_{13} $ & $ x_7 \land x_9 \land x_{10} $ \\ \small{Clause-9:}& $ x_7 \land \neg x_0 \land \neg x_6 \land \neg x_{13} $ & $ x_1 \land x_{15} \land \neg x_{11} $ \\ \small{Clause-10:}& $ x_{10} \land x_{11} \land \neg x_0 \land \neg x_4 \land \neg x_{12} \land \neg x_{14} $ & $ x_{15} \land \neg x_7 $ \\ \small{Clause-11:}& $ \neg x_5 \land \neg x_8 \land \neg x_9 \land \neg x_{12} $ & $ \neg x_1 \land \neg x_{10} $ \\ \small{Clause-12:}& $ \neg x_{15} $ & $ x_0 \land \neg x_{11} $ \\ \hline
\end{tabular}
\caption{Clause Size Variation with $s$-parameter.}
\label{tab:s}
\end{sc} \end{table*} A comparison between two different values of the $s$-parameter on memory consumption, training time and number of epochs required to reach $95\%$ accuracy on the Iris dataset is presented in Table \ref{tab:eval}. For this task, a TM consisting of $50$ clauses is employed. Clearly, the TM with smaller $s$ value requires less memory due to the less included literals. For the same reason, a shorter training time per epoch is also achieved. Nevertheless, the smaller $s$ requires more training epochs to obtain the same accuracy, which requires slightly more overall training time. Even though these are different configurations of TM, both are capable of achieving comparable performance. The $s$-parameter is an important hyperparameter which needs to be carefully tuned to obtain optimal performance and interpretability. A detailed analysis and comparison of TM's memory and time consumption with other algorithms has been shown in~\cite{newcastle} and a more rigorous theoretical analysis of the $s$ parameter can be found in~\cite{onebit}.\\
\begin{table}[h]
\centering
\begin{sc}
\begin{small}
\begin{tabular}{c|c|c}
\hline
& $s=10$ & $s=2$ \\
\hline
\small{Memory (in Kb)} & 204.8 & 122.5 \\
\small{Epochs} & 250 & 350 \\
\small{Training time per epoch (in millisec)} & 7.26 & 5.53 \\
\hline
\end{tabular}
\end{small}
\end{sc}
\caption{Training Epochs and Memory Consumption to obtain $95\%$ accuracy on Iris.}
\label{tab:eval} \end{table}
\subsection{Visualization of Decision Boundaries} \label{sec:vis} Visualizing decision boundaries is the one of the simplest ways of determining the pattern recognition abilities of an algorithm. It offers perspective about the decision function learnt by the method in order to best distinguish between patterns belonging to different categories. In this subsection, we visualize the decision boundaries of TM, perceptron and a single layer neural network (SLNN) with ReLU activation. \\ Figure~\ref{fig:vis} shows the visualization of decision boundaries for TM, perceptron and SLNN obtained from testing on the Iris dataset. Each model was trained on the Iris dataset for $50$ epochs. The TM consisted of $50$ clauses and the SLNN was made up of $50$ neurons in the hidden layer activated by the ReLU function. The TM, perceptron and SLNN obtained $96\%$, $89\%$ and $94\%$ accuracy on the testing set, respectively\footnote{The code to reproduce these results and visualize decision boundaries is available here: github}. \\ As we can see from Figure \ref{fig:sfig1}, the decision boundaries of TM are cumulatively formed from clauses. Each clause contributes to the decision boundary as $C_i = \{0,1\}$. Basically, the decision boundary formed is in the form of \emph{steps} by filled contour lines. TM's decision boundary is like an unsmooth approximated version of a non-linear neural network, shown in Figure~\ref{fig:sfig3}. However, in case of a new data point, the entire curved decision boundary of a neural network might have to be changed in order to accommodate for the new data point. Whereas for TM, a clause (or a set of clauses) can simply learn the new pattern and create a \emph{step} in the decision boundary to include the new data point in the correct region. Note that decision boundaries can vary depending upon initialization and learning trajectory of TM.\\ \begin{figure*}
\caption{Tsetlin Machine.}
\label{fig:sfig1}
\caption{Perceptron.}
\label{fig:sfig2}
\caption{Single Layer NN.}
\label{fig:sfig3}
\caption{Decision Boundary Visualization.}
\label{fig:vis}
\end{figure*} To show that TM has the capability to separate non-linearly separable patterns, we show a toy like example, namely the TM's decision boundaries for the XOR problem in Figure \ref{fig:xor_boundary}. The perceptron or linear neural networks are incapable of solving the XOR problem. However, as can be seen from Figure~\ref{fig:xor_boundary}, TM can create decision boundaries that separate such patterns courtesy of the clauses. Each clauses contributes in the construction of the decision boundary. Here, we use $4$ clauses to learn the XOR sub-patterns. The theoretical analysis of convergence of TM on the XOR problem can be found in~\cite{jiao2021convergence}.
\begin{figure}
\caption{TM Decision Boundaries for the XOR problem.}
\label{fig:xor_boundary}
\end{figure}
\subsection{Interpretability} As previously explained, each propositional expression is a conjunctive clause, consisting of feature, in their original or negated forms, interacting with each other using logical \emph{AND} operations. These clauses can form a simplified representation of the arm selection policy by combining them into a single Disjunctive Normal Form (DNF) expression. Since clauses are assigned to each class of the multiclass problem, we can produce a single DNF expression for each class. These DNF expressions are propositional logic expressions made up of binarized features. The TM is able to produce these interpretations demonstrating how it interprets the context with respect to each arm.\\ Here, we show the simplified propositional expressions for each class, obtained from TM trained on the Iris dataset: \begin{enumerate} \item \begin{enumerate}
\item[\emph{\textbf{Class-1:}}] $x_{10} \lor x_{14} \lor x_{15} \lor x_3$
\item[\emph{\textbf{Class-2:}}] $\neg x_1 \lor x_{12} \lor \neg x_{13} \lor \neg x_{14} \lor x_{16} \lor x_8 \lor \neg x_9 \lor (x_{10} \land x_{11} \land x_{15} \land x_2 \land \neg x_4 \land x_5 \land x_6 \land \neg x_7) \lor (x_{10} \land x_{11} \land x_{15} \land x_3 \land \neg x_4 \land x_5 \land \neg x_7)$
\item[\emph{\textbf{Class-3:}}] $~\neg x_{10} \lor \neg x_{11} \lor \neg x_{15} \lor (x_1 \land \neg x_{12} \land x_{13} \land x_{14} \land \neg x_{16} \land x_2 \land x_3 \land x_5 \land \neg x_8 \land x_9) \lor (\neg x_{12} \land \neg x_{16} \land x_4)$ \end{enumerate} \end{enumerate} The above expressions are obtained by combining the top ten highest weighted positive clauses for each class by \emph{OR}-ing them and just simplifying the Boolean expressions\footnote{Requires a couple of lines of code using the Sympy library.}.
\section{Discussions} \subsection{Tsetlin Machine vs Deep Learning} TM is a machine learning algorithm based on propositional Boolean expressions and logical operations. It is able to compete in performance with much larger deep learning models containing hundreds of thousands to many millions of floating point parameters. Different from deep neural networks, TM consists of a few thousands of binary clauses~\cite{dctm,rohan}, resulting in simplicity and low memory and energy consumption~\cite{newcastle}. This feature makes TM suitable for mobile computing and federated learning in power constraint IoT devices. In addition, the propositional-logic based clauses are more interpretable than float-number based operations utilized in deep neural networks.\\ Similar to deep learning, TM is also prone to overfitting. In other words, TM also learns patterns related to noise in the training data. For example, TM is able to achieve $100\%$ training accuracy on large datasets like CIFAR-100, but the performance drops during validation . To mitigate the overfitting problem, a new version of TM, called the Drop Clause TM ~\cite{dctm}, has been proposed, which reduces redundancy and improves the generalization capabilities of TM.
\subsection{Tsetlin Machine vs Perceptron} As shown in Section \ref{sec:w}, the weight update in TM can be considered as a special case of the perceptron learning algorithm for binarized input. Additionally, TM has a phase of learning in the clause level, i.e. representing a sub-pattern by a clause, which gives it better representability and flexibility. Figs. \ref{fig:tm} and \ref{fig:per} show the difference in structures of the TM and perceptron. The number of learnable parameters in the perceptron is restricted by the number of inputs. On the contrary, in TM, the number of clauses can be configured independently to the input's dimension, giving it structural flexibility.\\ Based on the descriptions in Section \ref{sec:TM}, we understand that the operational concept of the T is modularized, which has three parts. Firstly, distinct clauses learn various local sub-patterns. Secondly, the weights are assigned to these clauses according to their importance in solving the task. Finally, these sub-patterns are combined for the final classification. \\
On the other hand, the perceptron has to learn global patterns directly from data in a single phase, limited to solving linearly separable patterns, whereas TM has demonstrated its competence in solving much more complex problems~\cite{dctm,rohan1,rohan3,rohan,granmo2019convtsetlin,abeyrathna2020intrusion,abeyrathna2020nonlinear,berge2019text}, which traditional machine learning algorithms are incapable of.
\begin{figure}
\caption{TM}
\label{fig:tm}
\caption{Perceptron}
\label{fig:per}
\caption{Tsetlin Machine vs Perceptron}
\label{fig:tmvsp}
\end{figure}
\section{Conclusions} \label{sec:conclusion} In this paper, we try to bridge the gap between the perceptron (and single-layer neural network) and TM by showing the similarities that lie in their respective structure and learning procedures. We formalize the learning mechanism of TM by dividing the learning phase into two layers. We show the equivalence of TM's weight update phase with the perceptron learning algorithm. An empirical analysis and visualization of decision boundaries demonstrates how TM can solve nonlinearly separable patterns, like the XOR problem, which the perceptron (and linearly activated SLNN) is incapable of. The decision boundaries show similarities to that of the single-layer neural network. Apart from visualization of decision boundaries, the empirical analysis also shows the flexibility of determining clause length, memory consumption, convergence rate and obtaining interpretable Boolean rules.
\appendix
\section*{Appendix}
\section{Convergence of Perceptron} \label{sec:app} Let $W=[w_1,...,w_n]^T$ be the weights of the perceptron and $X^{n\times D}=[(x_1,y_1),...,(x_n,y_n)]^T$ be the input-label pairs, where $y_i \in \{-1,1\}$ and each $x_i$ is a $D$-dimensional vector. Assumptions: \begin{enumerate}
\item There exists some $W^*$ such that $\Vert W^* \Vert = 1$, and for some $\gamma > 0$, $\forall n = 1 \dots N$:
\begin{equation}
y_{n}^{} x_{n}^T W^* \geq \gamma
\end{equation}
\item Also assume, $\forall n$:
\begin{equation}
\Vert x_n \Vert \leq R
\end{equation} \end{enumerate} \paragraph{\emph{Perceptron Convergence:}}The number of updates $k$ required for the perceptron to converge to a local minimum is bounded by: \begin{equation}
k \leq \frac{R^2}{\gamma^2} \end{equation} \paragraph{\emph{Proof:}}Let $W_k$ be the weight vector after $k^{th}$ update and $\Vert W_1 \Vert = 0$. So, for $k+1$ we have: \begin{align*}
W_{k+1} & = W_k + y_nx_n \\
W_{k+1}^TW^* & = (W_k + y_nx_n)^TW^* \\
& = W_k^TW^* + y_n^{}x_n^TW^*\\
& \geq W_k^TW^* + \gamma \end{align*} It follows by induction on $k$ that: \begin{equation*}
W_{k+1}^TW^*\geq W_k^TW^* + \gamma \geq W_{k-1}^TW^* + 2\gamma \geq \dots \geq k\gamma \end{equation*} In addition, as $\Vert W_{k+1} \Vert \Vert W^* \Vert \geq W_{k+1}^TW^*$, then we have: \begin{equation} \label{eq:ww}
\Vert W_{k+1} \Vert \geq k\gamma \end{equation} Now, we can also write: \begin{align*}
\Vert W_{k+1} \Vert ^2 & = \Vert W_k + y_nx_n \Vert ^2 \\
& = \Vert W_k \Vert ^2 + y_n\Vert x_n \Vert ^2 + 2y_n^{}x_n^TW_k \\
& \leq \Vert W_k \Vert ^2 + R^2 \end{align*} It follows by induction on $k$ that: \begin{equation} \label{eq:ww2}
\Vert W_{k+1} \Vert ^2 \leq kR^2 \end{equation} Combining equations \ref{eq:ww} and \ref{eq:ww2}, we have: \begin{align}
k^2\gamma^2 \leq \Vert W_{k+1} \Vert ^2 \leq kR^2 \\
\label{eq:final}
k \leq \frac{R^2}{\gamma^2} \end{align} This shows that the number of update steps, $k$, required by the perceptron to obtain a local minimum, is bounded. In case of TM, the same proof above holds with only one difference: $\forall n$, $x_n \in \{0,1\}$, i.e. the input is a binary vector. As the (Frobenius) norm of a binary vector is the square root of the number of non-zero elements and the number of non-zero elements can be atmost $D$. So, the second assumption becomes: \begin{equation}
\Vert x_n \Vert \leq \sqrt{D} \end{equation} So, for TM, Equation~(\ref{eq:final}) becomes: \begin{equation}
k \leq \frac{D}{\gamma^2} \end{equation}
\end{document} | arXiv | {
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