Datasets:

johaness14
/

IMG2LATEX

Modalities:

Image

Text

Formats:

Size:

Libraries:

Dataset card Data Studio Files Files and versions

xet

Community

Dataset Viewer

Auto-converted to Parquet

Split (2)

train · 1.81M rows

image imagewidth (px) 15 6.82k	formula stringlengths 0 3.17k
	\begin{align}(K^{-1})^U_S = -\frac{z^{3}(1-A\|z\|^2)P'(y)} {e^{\tilde{K}/2}P''(y)},\mbox{ }(K^{-1})^U_T = \frac{(T+T^)z^{3}(1-A\|z\|^2)} {e^{\tilde{K}/2} (1-\frac{\bar{n}}{3}B(1-A\|z\|^2)\\|\Pi\\|)},\end{align*}
	\begin{align}[t_1v_1,t_2v_2]={}^{t_1}[v_1,t_2]\cdot {}^{t_1t_2}[v_1,v_2]\cdot{}^{t_2}[t_1,v_2]\in [^{t_1}v_1,t_2]\cdot[t_1,{}^{t_2}v_2]\,[N,N].\end{align}
	\begin{align}d(\phi\wedge \beta)=d\phi\wedge \beta-\phi\wedge d\beta=d\phi\wedge \beta=\alpha\wedge \beta.\end{align}
	\tau ^ { k } { } _ { B } ^ { A } \tau ^ { k } { } _ { D } ^ { C } = { \frac { 1 } { 4 } } ( \delta _ { D } ^ { A } \delta _ { B } ^ { C } - \epsilon ^ { A C } \epsilon _ { B D } ) ,
	L _ { a } = \frac { 1 } { 1 6 \pi } \mathrm { I m } \int d ^ { 2 } \theta W ^ { 2 } \tau ( z ^ { - 1 / 4 } ) ,
	\begin{align}D=-(v+A_2)(v^2-A_2^2+c)-8A_e^2,\end{align}
	{ \cal L } _ { p } = \sum _ { i = 0 } ^ { p } h ^ { a } \left( \phi _ { i } \right) \partial _ { a } ^ { ( i ) } .
	\Phi ^ { - } = - \frac { \partial ^ { I } } { \partial ^ { + } } \Phi ^ { I } + \frac { d - 3 } { \hat { \partial } ^ { + } } \Phi ^ { z } \, .
	F _ { t } ^ { 0 } ( s , u ) = Q _ { t } ( u + s ) \, , \; \; \; \; \; F _ { t } ^ { t } ( s , u ) = Q _ { t } ( u - s - t ) \, ,
	\begin{align}F^{}(t,x) = \int_{0}^{x}\int_{x-u}^{x_{}} u \beta(u,v)n(t, u) n(t, v) dv du, \end{align}
	1 \leq q , q ^ { \prime } \leq Q ^ { f }
	\begin{align}(\{\lambda_{j\|v}(s)\},\{\Gamma_{jk\|v}(s)\})=\sum_{u=1}^m h_{u\|v}(s)(\{\Lambda_{uj}'\},\{\theta_{ujk}\})-\delta_{HP}(\{g_{j\|v}(s)\})\end{align}
	\langle \Psi ^ { F _ { \pm } } \| \Psi ^ { F _ { \pm } } \rangle = \Bigl [ \mathrm { d e t } ( ( 1 - M ) ( 1 + M / 2 ) ^ { 2 } ) \Bigr ] ^ { 5 } .
	m \, \frac { d ^ { 2 } \mathbf { x } } { d t ^ { 2 } } = \frac { q \, g } { 4 \pi } \, \mathbf { v } \times \frac { \mathbf { r } } { r ^ { 3 } } .
	\begin{align} \epsilon \leq \langle e_i, g \cdot e_j \rangle= \frac{ \langle e_i, g e_j \rangle }{\\|g e_j\\|} = \frac{g^{i,j}}{ \sum_{i=1}^d \langle e_i, g e_j \rangle }= \frac{g^{i,j}}{ \sum_{i=1}^d g^{i,j} }.\end{align}
	\delta \dot { \mathbf { x } } ( t ) = \mathbf { J } ( { \bf x } , t ) \delta \mathbf { x } ( t ) \, ,
	\begin{align}E = \prod_{i=1}^{s}\left[ \dfrac{a_{i}}{b^{d_{i}}},\dfrac{a_{i+1}}{b^{d_{i}}} \right]\end{align}
	\begin{gather}\left\\|\left(f(s) - \sum_{n=0}^{M} \frac{1}{(n+c)^s}\right)- \sum_{M<n \leq N_0} \frac{\beta_n}{(n+c)^s} \right\\|< \frac{\epsilon}{3}\end{gather}
	\displaystyle M _ { f } ^ { r } ( k ) ^ { \top } Q ( k { + } 1 ) M _ { f } ^ { r } ( k ) { - } M _ { f } ^ { r } ( k { - } 1 ) ^ { \top } Q ( k ) M _ { f } ^ { r } ( k { - } 1 )
	\displaystyle \frac { d N } { d t } =
	\begin{align}\varepsilon _{0}^{\beta }=\frac{\|e\|H}{(2\pi )^{2}} \sum_{n=0}^{\infty} \int_{-\infty }^{\infty} dp \;\epsilon _{n} \left\{ \frac{-2+{\rm e}^{\beta \epsilon _{n}}}{1-{\rm e}^{\beta \epsilon _{n}}}\right\} .\end{align}
	\begin{align}-\frac{b_1}{\varpi^{a_1}}\log_F(1+(y+\varpi^{\kappa} x)\varpi^{a_1-l}) = \sum_{j\geq 1} \frac{(-1)^{j} b_1}{j}(y+\varpi^{\kappa}x)^j\varpi^{(j-1)a_1-jl}. \end{align}
	\begin{align} r(p) \ =\ 2^{\gamma} 3^{\beta}\ \geq\ 2^{10^m-8}\cdot 3\ \geq\ 10^{m+1}\end{align}
	\sqrt { - G } T ^ { A B } ( X ) = \frac { 1 } { 2 \pi \alpha ^ { \prime } } \int d \sigma d \tau \, ( \dot { X } ^ { A } \dot { X } ^ { B } - X ^ { A } X ^ { B } ) \, \delta ^ { ( 3 ) } ( X - X ( \tau , \sigma ) ) .
	\begin{align}A_{i}^{j}=B_{i}^{j}\end{align}
	\begin{align} e^a_\mu (x)\, \frac{\partial x^\mu}{\partial q^\alpha}\, e^\alpha_b (q) \, V^\dagger (q) \, \sigma^b\, V (q) = \sigma^a.\end{align}
	\displaystyle d _ { h h ^ { \prime } ; b + }
	\begin{align} \left. \begin{array}{l l}\mathcal{E}_i= \{ x,y,z \ \| \ H_i > 0 \} \end{array} \right. \end{align}
	\begin{align} \mathcal F_{\delta} = \overline{\mathcal T}_{\delta}, \mathcal F = \bigcup_{0 < \delta < \delta_0} \mathcal F_{\delta},\end{align}
	\begin{align}\frac{\theta_\Gamma (c_{\mathbf{O}_{1m}})^2\theta_\Gamma(c_{\mathbf{O}_{2h}})^2}{\theta_\Gamma (c_{\mathbf{O}_{2m}})^2\theta_\Gamma(c_{\mathbf{O}_{1h}})^2}=\frac{\lambda_h-\lambda_m}{\lambda_h-\lambda_l}\frac{\lambda_k-\lambda_m}{\lambda_k-\lambda_l}\end{align}
	\begin{align}V(t,x_t,\dot x_t)=V_P(x(t))+V_U(t,\dot x_t)+V_X(t,x_t),\end{align}
	\begin{align}G(z,t)= e^{-t (c_{1}\{(\lambda_{1}+\mu(1-z))^{\alpha_{1}}-{\lambda_{1}}^{\alpha_{1}}\} +c_{2}\{(\lambda_{2}+\mu(1-z))^{\alpha_{2}}-{\lambda_{2}}^{\alpha_{2}}\})}, \; \|z\|\leq 1, \;\; \mu \leq \frac{\lambda_{i}}{2},\; i=1,2.\end{align}
	\begin{align} \langle f, g \rangle_{\mathrm{pet}}= \int_{\Gamma\backslash\mathbb{H}} f(z) \overline{g(z)}\, (\mathrm{Im}(z))^{k}\, \mu_{\mathrm{hyp}}(z)\,\ \,\ f, g\in S_{k}(\Gamma).\end{align}
	S _ { L 1 } = \sum _ { n = 1 } ^ { \infty } f _ { L q } ( n ) \, \left( { \frac { v ^ { 2 } } { R ^ { 2 } M _ { p } ^ { 9 } V _ { T } ~ b ^ { 7 - p } } } \right) ^ { 2 n } ~ \left( \frac { v ^ { 2 } } { R ^ { 2 } M _ { p } ^ { 6 } b ^ { 4 } } \right) ^ { - 2 n } = \sum _ { n = 1 } ^ { \infty } f _ { L q } ( n ) \, \left( { \frac { 1 } { M _ { P } ^ { 3 } V _ { T } ~ b ^ { 3 - p } } } \right) ^ { 2 n }
	\begin{align}\psi_1(h):[\textbf{q},\textbf{p}] \mapsto [\textbf{Q},\textbf{P}]:= [\textbf{q}+h \textbf{p}/m, \textbf{p}].\end{align}
	\begin{align}w(T)=\prod_{b \in T}w_T(b),\end{align}
	Y _ { L } ^ { \alpha } ( { \tilde { x } _ { L } } , { \tilde { \theta } } ) = \frac { 1 } { \sqrt { 2 } } ( { { \tilde { \theta } } ^ { \alpha } } + i \psi ^ { \alpha } ( { \tilde { x } _ { L } } , { \tilde { \theta } } ) ) .
	\begin{align} I(a,b) = \int_{\o} \psi(ax^2 + bx) \, dx\end{align}
	\rho _ { \mathrm { N } } ^ { \mathrm { t o t } } \equiv \rho _ { \mathrm { t o t } , + } + \mathrm { N } \cdot { \rho } _ { \mathrm { t o t } , - } ,
	{ \cal L } _ { \mathrm { B , c } } = - { \frac { 3 } { 2 } } \left[ S _ { 0 } \overline { S } _ { 0 } \, e ^ { - K / 3 } \right] _ { D } ,
	\begin{align}y \odot \lambda_a(x)=y\odot \lambda_b(x)\end{align}
	\exp ( x T r P ) = \sum _ { n _ { 1 } \geq n _ { 2 } \geq \cdots \geq n _ { N } \geq 0 } ^ { \infty } \det \left[ d _ { n _ { j } + N - j + 1 , i } \right] \chi _ { ( n _ { 1 } , n _ { 2 } , \ldots , n _ { N } ) } ( P ) ,
	\begin{align}\delta\int_{t_1}^{t_2}L_{\gamma_0}(\chi(t),\dot\chi(t))dt=0,\end{align}
	[ \{ - \partial _ { x } ^ { 2 } + ( q B x + k _ { y } ) ^ { 2 } \} + \{ - \partial _ { z } ^ { 2 } - ( \omega + q E z ) ^ { 2 } \} + m ^ { 2 } ] f _ { \omega k _ { y } } ( x , z ) = 0 .
	\begin{align}\omega(t,r,\theta) = A(r,\theta) a^{-3}(t).\end{align}
	\begin{align}\bar H := H \cup \{ (v_i, w_i) \}^m_{i=1}.\end{align}
	\sum _ { n , n ^ { \prime } } \delta _ { n , n ^ { \prime } } ( \rho _ { n n } ^ { L } + \rho _ { n n } ^ { R } ) ( \rho _ { n ^ { \prime } n ^ { \prime } } ^ { L } + \rho _ { n ^ { \prime } n ^ { \prime } } ^ { R } ) ,
	\: \sigma ^ { \pm } = z ^ { \pm } ( \tau ) . \:
	\begin{align} y_0 = x_0^2,~~ y_1 = x_0x_1,~~ y_2 = x_1^2,~~ y_3 = x_2 \end{align}
	\begin{align}([x,y],z,w)=0 \mbox{ if } \mathsf{char}(F)\neq 2, 3.\end{align}
	- \triangle \ = \ \nabla _ { i } ^ { * } \; g ^ { i j } \; \nabla _ { j } \; , \quad \nabla _ { j } \ = \ \partial _ { j } \ - \ \Gamma _ { j l } ^ { k } \; a ^ { * l } \; a _ { k } \ ,
	\begin{align}\frac{\partial e_j}{\partial [\tilde{v}_i]_{j,\alpha}}=\frac{[\tilde{w}_i]_{\alpha,j}}{m\|g\|\lambda_j}\bigg[c_{i-1}+\frac{i}{m}\|g\|-[\tilde{v}_i\tilde{w}_i]_{j,j}-\sum_{s=0}^{m-1}\frac{m-s}{m}[\tilde{v}_s\tilde{w}_s]_{j,j}\bigg].\end{align}
	\begin{align}u_{yy}+u_{xt}+u_x u_{xy}-u_y u_{xx}=0,\end{align}
	\begin{align} S_{a,1}:\,(r_1,y_1,z,\epsilon_1) = (r_1,\left(\frac{2(\vert \beta \vert+\mathcal O(z))}{\phi^{[k]}(\vert b\vert + \mathcal O(z))(-z) } \right)^{1/k} + \mathcal O(r_1),z,0).\end{align}
	\begin{align}<p_{1},...,p_{n}\|S\|q_{1},....,q_{m}>= lim_{T\rightarrow \infty}\frac{1}{T} \int_{0}^{T}d\tau E\Big[\prod_{j}\prod_{k}( \overline{f_{p_{j}}},\partial_{\tau}\phi_{\tau})( f_{q_{k}},\partial_{\tau}\phi_{\tau})\Big]\end{align}
	\begin{align} L(\gamma,Y)=-Y\|\eta\|+H^{1/q}&\bigg(\left(\frac{\gamma}{\gamma+1}\right)^{p-2}(1+Y)^{p-1}\left(Y-\frac{1}{p-1}\right)\\&+\frac{(2-p)\gamma^{p-1}+\gamma^{p-2}}{p-1}\frac{Z}{\|\zeta\|^p}\bigg)^{1/p},\end{align}
	q = e ^ { - \frac { \omega \hbar } { 4 m c ^ { 2 } } }
	\begin{align} G(F(z)) - G(F(0)) = \lambda G(z),\quad\|z\|<1.\end{align}
	x ( { \mathbf t } ) = x ( { \mathbf 0 } ) \cdot g ( { \mathbf t } ) .
	\begin{align} \{x^i,x^j\}=c^{ij}_kx^k.\end{align}
	\psi \to e ^ { i \omega _ { i j } \sigma _ { i j } / 4 } \psi , \qquad \sigma _ { i j } = \frac { i } { 2 } [ \gamma _ { i } , \gamma _ { j } ] .
	\begin{align} y\left[ i \right] = \sum\limits_{t = 0}^{{n'_{{\rm{span}}}}} {\left( {\sum\limits_{l' = 1}^{L'} {{\bf{g}}_{l'}^H\left[ t \right]{{\bf{\bar H}}_{l'}^ \bot {\bf{\bar X}}_{l'} }} } \right){{\bf{d}}\left[ {i - t } \right]}}+ z\left[ {i} \right]. \end{align}
	\begin{align} \varphi = -dr^1 \wedge dr^2 \wedge dr^3 + dr^1 \wedge \omega + dr^2 \wedge \Re (\Omega) + dr^3 \wedge \Im (\Omega).\end{align}
	\begin{align} AX(PX^T\!AXP)^\dagger X^T\!A &= AX(UU^TX^T\!AXUU^T)^\dagger X^T\!A \\ &= AXU(U^TX^T\!AXU)^\dagger U^TX^T\!A.\end{align}
	P ( \rho ) = \rho ^ { 2 } \alpha ^ { \, \prime } ( \rho ) .
	\mathcal { E } ^ { ( c l ) } ( C _ { r - 1 } ^ { \vee } ) = \mathcal { E } ^ { ( c l ) } ( B _ { r } ) = \frac { r m \cos ( \pi / 4 - \pi / 4 r ) } { \pi b ^ { 2 } \cos ( \pi / 4 r ) } ; \quad w _ { 0 } = w _ { r } = 0
	\begin{align}\displaystyle X_V=\Xi-X_W+c(z_r,w)+\sum_{i=0}^r \lambda_i c(z_i,\eta z_i)+ \sum_{i=0}^{r-1} \mu_i c(z_i,z_{i+1})\end{align}
	\int _ { \mathbf { y } } ^ { y \prime } d z _ { i } \delta ^ { ( 3 ) } \left( \mathbf { x - z } \right) = (
	\begin{align}J(X)=\left\{ \delta_{ij}:1\leq i<j\leq n\right\} \cup\left\{ A_{1rs}:1<r<s\leq n\right\} . \end{align}
	K _ { \mu \nu } = - \frac { 1 } { 2 } g _ { \mu \nu , y } = \Sigma _ { \mu \nu } + { \frac { 1 } { 4 } } g _ { \mu \nu } K \ .
	\begin{align} u(\phi(\alpha(i_1,i_2)))=c(i_2+i_1+1)-(-1)=c(i_1+i_2)+c+1=u(\alpha(i_1,i_2))+c+1 \, . \end{align}
	\displaystyle \frac { \partial \sigma _ { i j } ^ { \mathrm { N } } } { \partial q ^ { 0 } }
	\begin{align} \operatorname{lev}\nolimits_{\varphi_{A,k},=}(t) = tk+ \operatorname{bd}A \forall\, t \in \mathbb{R}. \end{align}
	S U ( n ) \simeq S ^ { 2 n - 1 } \otimes S ^ { 2 n - 3 } \otimes \cdots \otimes S ^ { 3 } \; .
	R _ { 1 2 } R _ { 1 3 } R _ { 2 3 } = R _ { 2 3 } R _ { 1 3 } R _ { 1 2 } .
	\begin{align}\sum_{\lambda\in\{\lambda\|\lambda\vdash m, \ell(\lambda)\geq 1\}}\frac{(-1)^{\ell(\lambda)-1}}{\ell(\lambda)}\frac{\ell(\lambda)!}{\lambda_{1}!\cdot\cdot\lambda_{\ell(\lambda)}!\|Aut(\lambda)\|}=0\end{align}
	\begin{align} R_n^{(1)}(z) = zS_{n-1}(z) + S_{n-1}^{\ast}(z) \mbox{and} (z-1)R_n^{(2)}(z) = zS_{n}(z) - S_{n}^{\ast}(z), n\geq 1,\end{align}
	\begin{align}\widehat{A}&=a\cdot (y^{-1}x^2y)+ b \cdot (y^{-1})+c \cdot (x^2 y)+d\cdot (1)\\&=a \cdot (x^2)+ b \cdot (y) +c \cdot (x^2 y)+d \cdot (1),\\\widehat{B}&=e\cdot ((x^2y)^{-1} x^{-1}y)+ f \cdot ((x^2 y)^{-1} )+ g\cdot (x^{-1} y)+ h\cdot (1)\\&=e \cdot (x^3) + f\cdot (x^2 y) + g\cdot (x^3y)+ h\cdot (1).\end{align}
	\begin{align} \mathcal{K}^{\mu}(x,y):=\frac{G^{\mu}(x^{-1}y)}{G^{\mu}(y)}. \end{align}
	\begin{align}\bigcup_{n=1}^{\infty} S_{\mu}^n = \Gamma\,,\end{align}
	{ \cal I } _ { I J } \, \equiv \, { \hat { c } } _ { I } \, \cap \, { \hat { c } } _ { J } \ .
	\begin{align}K(N):&=\prod_{p\nmid N}\left( 1-\frac{(\frac{N-1}{p})^2p+1}{(p-1)^2(p+1)}\right)\prod_{{p\mid N}}\left(1-\frac{1}{p^{\nu_p(N)}(p-1)}\right) \end{align}
	\begin{align}b ( x ) ( \Delta_\lambda ( x ) - \Delta_\lambda ( q^{-1} x ) + c ) = 0 \end{align}
	\begin{align}\frac{a_y}{\sqrt B}\sin\phi+\sigma\cos\phi+\tilde\sigma=0.\end{align}
	\begin{align}\left\{\begin{array}{l}x_0=0\\begin{align}2mm]\displaystyle\sum_{i=1}^r d_i x_i+2x_{r+1}=0.\\\end{array}\right.\end{align*}
	\begin{align}\omega_\alpha( \textit{\textbf{p}}_\lambda, \textit{\textbf{p}}_\mu)=\max\Big\{\frac{s_\lambda}{s_\mu},\frac{s_\mu}{s_\lambda}\Big\}(1+d_\alpha(\textit{\textbf{p}}_\lambda,\textit{\textbf{p}}_\mu))\,,\end{align}
	\begin{align}\{ \ ( \sum_{(\lambda,\mu)} \phi \mid_{k,m} [\lambda,\mu] )\ \mid U_l , \psi \} &= \{ \phi, \sum_{(\lambda,\mu)} \psi \mid A_l \mid_{k,m} [-\lambda,-\mu] \}\end{align}
	\forall a , b \in { \cal G } \equiv s l ( N , { \mathbb C } ) \; , \; \mathrm { T r } _ { { \cal G } \otimes { \cal G } } \; { \cal C } ^ { s l ( N ) } \: a \otimes b = \mathrm { T r } _ { \cal G } \; a b
	\begin{align}\left[\mathcal{J}_n^T[\mathbf{u}^{\mathcal{L}}(x_j^{(n)})]^T\right]^T \mathbf{u}^{\mathcal{R}}(x_k^{(n)})=[ \mathbf{u}^{\mathcal{R}}(x_k^{(n)})]^T \mathcal{J}_n^T [\mathbf{u}^{\mathcal{L}}(x_j^{(n)})]^T = \omega_{n,j,k}^{-1}\delta_{j,k}.\end{align}
	\begin{align}[L_{-2}, \, X_{2, \, j+2, \, 13}] = 0.\end{align}
	\displaystyle q ^ { 2 } \left[ \Pi _ { A } ( q ^ { 2 } ) - \Pi _ { V } ( q ^ { 2 } ) \right]
	\begin{align} \lim_{\tau \downarrow 0}\sup_{\|x\| > \rho \zeta_{\tau}} h(x,\tau) = 0.\end{align}
	\begin{align}L_3=\{(\gamma_1, \gamma_2, \gamma_3)\in T^(PM)^3 \mid \gamma_1* \gamma_2 \sim \gamma_3 \}.\end{align*}
	\begin{align} \langle \omega_{n-1}\otimes\omega_{n-2}\otimes\dots\otimes\omega_{0},\tilde\omega_{n-1}\otimes\tilde\omega_{n-2}\otimes\dots\otimes\tilde\omega_{0}\rangle_{F^{\otimes n}}=\prod \langle \omega_{a},\tilde \omega_{a}\rangle_{F}.\end{align}
	\overrightarrow { J } = - i \overrightarrow { P } \times \overrightarrow { \bigtriangledown } + ( \overrightarrow { S } + \overrightarrow \Sigma )
	\delta ( \partial _ { \mu } \phi + A _ { \mu } \phi + A _ { \mu } \zeta ) = \epsilon ( \partial _ { \mu } \phi + A _ { \mu } \phi + A _ { \mu } \zeta ) .
	\phi _ { \pm } ^ { \mu , k } ( \sigma , \tau ) = \frac { 1 } { \sqrt 2 } \sum _ { r \in Z + \frac { 1 } { 2 } } b _ { r } ^ { \prime \, \mu , \; k } e ^ { - i r ( \tau \pm \sigma ) }
	\begin{align}\\|u\\|_{X^{s,b}_{\|\tau\|=\|\xi\|}}=\\|\left<\xi\right>^s\left<\|\tau\|-\|\xi\|\right>^{b}\hat{u}(\tau,\xi)\\|_{L_{\tau,\xi}^2},\end{align}
	c h _ { G } ^ { g } ( E ) = \sum _ { i = 1 } ^ { s } \lambda _ { i } c h ( E ^ { i } )
	\rho = \frac { r } { r _ { 0 } } \ll 1 \qquad \frac { 1 } { 2 } N ^ { 2 } \rho ^ { D } \gg 1 ,

End of preview. Expand in Data Studio

Dataset: IMG2LATEX

This dataset is a large-scale, cleaned, and unified collection of mathematical formula images and their corresponding LaTeX source code. It was created by combining and preprocessing several publicly available datasets.

The goal of this dataset is to provide a robust foundation for training Optical Character Recognition (OCR) models for mathematical equations, specifically for Image-to-LaTeX tasks. All images are preserved in their native resolution to avoid information loss from resizing or distortion.

Data Fields

The dataset contains two columns:

image: A PIL Image object containing the formula image in its original resolution and in grayscale.
formula: A string containing the ground-truth LaTeX formula.

Source Datasets

This dataset would not be possible without the amazing work from the creators of the original datasets. Please credit them if you use this work. The following datasets were used as sources:

Downloads last month: 11

Size of downloaded dataset files:

7.27 GB

Size of the auto-converted Parquet files:

7.27 GB

Number of rows:

1,848,682

image imagewidth (px) 15 6.82k	formula stringlengths 0 3.17k
	\begin{align}(K^{-1})^U_S = -\frac{z^{3}(1-A\|z\|^2)P'(y)} {e^{\tilde{K}/2}P''(y)},\mbox{ }(K^{-1})^U_T = \frac{(T+T^)z^{3}(1-A\|z\|^2)} {e^{\tilde{K}/2} (1-\frac{\bar{n}}{3}B(1-A\|z\|^2)\\|\Pi\\|)},\end{align*}
	\begin{align}[t_1v_1,t_2v_2]={}^{t_1}[v_1,t_2]\cdot {}^{t_1t_2}[v_1,v_2]\cdot{}^{t_2}[t_1,v_2]\in [^{t_1}v_1,t_2]\cdot[t_1,{}^{t_2}v_2]\,[N,N].\end{align}
	\begin{align}d(\phi\wedge \beta)=d\phi\wedge \beta-\phi\wedge d\beta=d\phi\wedge \beta=\alpha\wedge \beta.\end{align}
	\tau ^ { k } { } _ { B } ^ { A } \tau ^ { k } { } _ { D } ^ { C } = { \frac { 1 } { 4 } } ( \delta _ { D } ^ { A } \delta _ { B } ^ { C } - \epsilon ^ { A C } \epsilon _ { B D } ) ,
	L _ { a } = \frac { 1 } { 1 6 \pi } \mathrm { I m } \int d ^ { 2 } \theta W ^ { 2 } \tau ( z ^ { - 1 / 4 } ) ,
	\begin{align}D=-(v+A_2)(v^2-A_2^2+c)-8A_e^2,\end{align}
	{ \cal L } _ { p } = \sum _ { i = 0 } ^ { p } h ^ { a } \left( \phi _ { i } \right) \partial _ { a } ^ { ( i ) } .
	\Phi ^ { - } = - \frac { \partial ^ { I } } { \partial ^ { + } } \Phi ^ { I } + \frac { d - 3 } { \hat { \partial } ^ { + } } \Phi ^ { z } \, .
	F _ { t } ^ { 0 } ( s , u ) = Q _ { t } ( u + s ) \, , \; \; \; \; \; F _ { t } ^ { t } ( s , u ) = Q _ { t } ( u - s - t ) \, ,
	\begin{align}F^{}(t,x) = \int_{0}^{x}\int_{x-u}^{x_{}} u \beta(u,v)n(t, u) n(t, v) dv du, \end{align}
	1 \leq q , q ^ { \prime } \leq Q ^ { f }
	\begin{align}(\{\lambda_{j\|v}(s)\},\{\Gamma_{jk\|v}(s)\})=\sum_{u=1}^m h_{u\|v}(s)(\{\Lambda_{uj}'\},\{\theta_{ujk}\})-\delta_{HP}(\{g_{j\|v}(s)\})\end{align}
	\langle \Psi ^ { F _ { \pm } } \| \Psi ^ { F _ { \pm } } \rangle = \Bigl [ \mathrm { d e t } ( ( 1 - M ) ( 1 + M / 2 ) ^ { 2 } ) \Bigr ] ^ { 5 } .
	m \, \frac { d ^ { 2 } \mathbf { x } } { d t ^ { 2 } } = \frac { q \, g } { 4 \pi } \, \mathbf { v } \times \frac { \mathbf { r } } { r ^ { 3 } } .
	\begin{align} \epsilon \leq \langle e_i, g \cdot e_j \rangle= \frac{ \langle e_i, g e_j \rangle }{\\|g e_j\\|} = \frac{g^{i,j}}{ \sum_{i=1}^d \langle e_i, g e_j \rangle }= \frac{g^{i,j}}{ \sum_{i=1}^d g^{i,j} }.\end{align}
	\delta \dot { \mathbf { x } } ( t ) = \mathbf { J } ( { \bf x } , t ) \delta \mathbf { x } ( t ) \, ,
	\begin{align}E = \prod_{i=1}^{s}\left[ \dfrac{a_{i}}{b^{d_{i}}},\dfrac{a_{i+1}}{b^{d_{i}}} \right]\end{align}
	\begin{gather}\left\\|\left(f(s) - \sum_{n=0}^{M} \frac{1}{(n+c)^s}\right)- \sum_{M<n \leq N_0} \frac{\beta_n}{(n+c)^s} \right\\|< \frac{\epsilon}{3}\end{gather}
	\displaystyle M _ { f } ^ { r } ( k ) ^ { \top } Q ( k { + } 1 ) M _ { f } ^ { r } ( k ) { - } M _ { f } ^ { r } ( k { - } 1 ) ^ { \top } Q ( k ) M _ { f } ^ { r } ( k { - } 1 )
	\displaystyle \frac { d N } { d t } =
	\begin{align}\varepsilon _{0}^{\beta }=\frac{\|e\|H}{(2\pi )^{2}} \sum_{n=0}^{\infty} \int_{-\infty }^{\infty} dp \;\epsilon _{n} \left\{ \frac{-2+{\rm e}^{\beta \epsilon _{n}}}{1-{\rm e}^{\beta \epsilon _{n}}}\right\} .\end{align}
	\begin{align}-\frac{b_1}{\varpi^{a_1}}\log_F(1+(y+\varpi^{\kappa} x)\varpi^{a_1-l}) = \sum_{j\geq 1} \frac{(-1)^{j} b_1}{j}(y+\varpi^{\kappa}x)^j\varpi^{(j-1)a_1-jl}. \end{align}
	\begin{align} r(p) \ =\ 2^{\gamma} 3^{\beta}\ \geq\ 2^{10^m-8}\cdot 3\ \geq\ 10^{m+1}\end{align}
	\sqrt { - G } T ^ { A B } ( X ) = \frac { 1 } { 2 \pi \alpha ^ { \prime } } \int d \sigma d \tau \, ( \dot { X } ^ { A } \dot { X } ^ { B } - X ^ { A } X ^ { B } ) \, \delta ^ { ( 3 ) } ( X - X ( \tau , \sigma ) ) .
	\begin{align}A_{i}^{j}=B_{i}^{j}\end{align}
	\begin{align} e^a_\mu (x)\, \frac{\partial x^\mu}{\partial q^\alpha}\, e^\alpha_b (q) \, V^\dagger (q) \, \sigma^b\, V (q) = \sigma^a.\end{align}
	\displaystyle d _ { h h ^ { \prime } ; b + }
	\begin{align} \left. \begin{array}{l l}\mathcal{E}_i= \{ x,y,z \ \| \ H_i > 0 \} \end{array} \right. \end{align}
	\begin{align} \mathcal F_{\delta} = \overline{\mathcal T}_{\delta}, \mathcal F = \bigcup_{0 < \delta < \delta_0} \mathcal F_{\delta},\end{align}
	\begin{align}\frac{\theta_\Gamma (c_{\mathbf{O}_{1m}})^2\theta_\Gamma(c_{\mathbf{O}_{2h}})^2}{\theta_\Gamma (c_{\mathbf{O}_{2m}})^2\theta_\Gamma(c_{\mathbf{O}_{1h}})^2}=\frac{\lambda_h-\lambda_m}{\lambda_h-\lambda_l}\frac{\lambda_k-\lambda_m}{\lambda_k-\lambda_l}\end{align}
	\begin{align}V(t,x_t,\dot x_t)=V_P(x(t))+V_U(t,\dot x_t)+V_X(t,x_t),\end{align}
	\begin{align}G(z,t)= e^{-t (c_{1}\{(\lambda_{1}+\mu(1-z))^{\alpha_{1}}-{\lambda_{1}}^{\alpha_{1}}\} +c_{2}\{(\lambda_{2}+\mu(1-z))^{\alpha_{2}}-{\lambda_{2}}^{\alpha_{2}}\})}, \; \|z\|\leq 1, \;\; \mu \leq \frac{\lambda_{i}}{2},\; i=1,2.\end{align}
	\begin{align} \langle f, g \rangle_{\mathrm{pet}}= \int_{\Gamma\backslash\mathbb{H}} f(z) \overline{g(z)}\, (\mathrm{Im}(z))^{k}\, \mu_{\mathrm{hyp}}(z)\,\ \,\ f, g\in S_{k}(\Gamma).\end{align}
	S _ { L 1 } = \sum _ { n = 1 } ^ { \infty } f _ { L q } ( n ) \, \left( { \frac { v ^ { 2 } } { R ^ { 2 } M _ { p } ^ { 9 } V _ { T } ~ b ^ { 7 - p } } } \right) ^ { 2 n } ~ \left( \frac { v ^ { 2 } } { R ^ { 2 } M _ { p } ^ { 6 } b ^ { 4 } } \right) ^ { - 2 n } = \sum _ { n = 1 } ^ { \infty } f _ { L q } ( n ) \, \left( { \frac { 1 } { M _ { P } ^ { 3 } V _ { T } ~ b ^ { 3 - p } } } \right) ^ { 2 n }
	\begin{align}\psi_1(h):[\textbf{q},\textbf{p}] \mapsto [\textbf{Q},\textbf{P}]:= [\textbf{q}+h \textbf{p}/m, \textbf{p}].\end{align}
	\begin{align}w(T)=\prod_{b \in T}w_T(b),\end{align}
	Y _ { L } ^ { \alpha } ( { \tilde { x } _ { L } } , { \tilde { \theta } } ) = \frac { 1 } { \sqrt { 2 } } ( { { \tilde { \theta } } ^ { \alpha } } + i \psi ^ { \alpha } ( { \tilde { x } _ { L } } , { \tilde { \theta } } ) ) .
	\begin{align} I(a,b) = \int_{\o} \psi(ax^2 + bx) \, dx\end{align}
	\rho _ { \mathrm { N } } ^ { \mathrm { t o t } } \equiv \rho _ { \mathrm { t o t } , + } + \mathrm { N } \cdot { \rho } _ { \mathrm { t o t } , - } ,
	{ \cal L } _ { \mathrm { B , c } } = - { \frac { 3 } { 2 } } \left[ S _ { 0 } \overline { S } _ { 0 } \, e ^ { - K / 3 } \right] _ { D } ,
	\begin{align}y \odot \lambda_a(x)=y\odot \lambda_b(x)\end{align}
	\exp ( x T r P ) = \sum _ { n _ { 1 } \geq n _ { 2 } \geq \cdots \geq n _ { N } \geq 0 } ^ { \infty } \det \left[ d _ { n _ { j } + N - j + 1 , i } \right] \chi _ { ( n _ { 1 } , n _ { 2 } , \ldots , n _ { N } ) } ( P ) ,
	\begin{align}\delta\int_{t_1}^{t_2}L_{\gamma_0}(\chi(t),\dot\chi(t))dt=0,\end{align}
	[ \{ - \partial _ { x } ^ { 2 } + ( q B x + k _ { y } ) ^ { 2 } \} + \{ - \partial _ { z } ^ { 2 } - ( \omega + q E z ) ^ { 2 } \} + m ^ { 2 } ] f _ { \omega k _ { y } } ( x , z ) = 0 .
	\begin{align}\omega(t,r,\theta) = A(r,\theta) a^{-3}(t).\end{align}
	\begin{align}\bar H := H \cup \{ (v_i, w_i) \}^m_{i=1}.\end{align}
	\sum _ { n , n ^ { \prime } } \delta _ { n , n ^ { \prime } } ( \rho _ { n n } ^ { L } + \rho _ { n n } ^ { R } ) ( \rho _ { n ^ { \prime } n ^ { \prime } } ^ { L } + \rho _ { n ^ { \prime } n ^ { \prime } } ^ { R } ) ,
	\: \sigma ^ { \pm } = z ^ { \pm } ( \tau ) . \:
	\begin{align} y_0 = x_0^2,~~ y_1 = x_0x_1,~~ y_2 = x_1^2,~~ y_3 = x_2 \end{align}
	\begin{align}([x,y],z,w)=0 \mbox{ if } \mathsf{char}(F)\neq 2, 3.\end{align}
	- \triangle \ = \ \nabla _ { i } ^ { * } \; g ^ { i j } \; \nabla _ { j } \; , \quad \nabla _ { j } \ = \ \partial _ { j } \ - \ \Gamma _ { j l } ^ { k } \; a ^ { * l } \; a _ { k } \ ,
	\begin{align}\frac{\partial e_j}{\partial [\tilde{v}_i]_{j,\alpha}}=\frac{[\tilde{w}_i]_{\alpha,j}}{m\|g\|\lambda_j}\bigg[c_{i-1}+\frac{i}{m}\|g\|-[\tilde{v}_i\tilde{w}_i]_{j,j}-\sum_{s=0}^{m-1}\frac{m-s}{m}[\tilde{v}_s\tilde{w}_s]_{j,j}\bigg].\end{align}
	\begin{align}u_{yy}+u_{xt}+u_x u_{xy}-u_y u_{xx}=0,\end{align}
	\begin{align} S_{a,1}:\,(r_1,y_1,z,\epsilon_1) = (r_1,\left(\frac{2(\vert \beta \vert+\mathcal O(z))}{\phi^{[k]}(\vert b\vert + \mathcal O(z))(-z) } \right)^{1/k} + \mathcal O(r_1),z,0).\end{align}
	\begin{align}<p_{1},...,p_{n}\|S\|q_{1},....,q_{m}>= lim_{T\rightarrow \infty}\frac{1}{T} \int_{0}^{T}d\tau E\Big[\prod_{j}\prod_{k}( \overline{f_{p_{j}}},\partial_{\tau}\phi_{\tau})( f_{q_{k}},\partial_{\tau}\phi_{\tau})\Big]\end{align}
	\begin{align} L(\gamma,Y)=-Y\|\eta\|+H^{1/q}&\bigg(\left(\frac{\gamma}{\gamma+1}\right)^{p-2}(1+Y)^{p-1}\left(Y-\frac{1}{p-1}\right)\\&+\frac{(2-p)\gamma^{p-1}+\gamma^{p-2}}{p-1}\frac{Z}{\|\zeta\|^p}\bigg)^{1/p},\end{align}
	q = e ^ { - \frac { \omega \hbar } { 4 m c ^ { 2 } } }
	\begin{align} G(F(z)) - G(F(0)) = \lambda G(z),\quad\|z\|<1.\end{align}
	x ( { \mathbf t } ) = x ( { \mathbf 0 } ) \cdot g ( { \mathbf t } ) .
	\begin{align} \{x^i,x^j\}=c^{ij}_kx^k.\end{align}
	\psi \to e ^ { i \omega _ { i j } \sigma _ { i j } / 4 } \psi , \qquad \sigma _ { i j } = \frac { i } { 2 } [ \gamma _ { i } , \gamma _ { j } ] .
	\begin{align} y\left[ i \right] = \sum\limits_{t = 0}^{{n'_{{\rm{span}}}}} {\left( {\sum\limits_{l' = 1}^{L'} {{\bf{g}}_{l'}^H\left[ t \right]{{\bf{\bar H}}_{l'}^ \bot {\bf{\bar X}}_{l'} }} } \right){{\bf{d}}\left[ {i - t } \right]}}+ z\left[ {i} \right]. \end{align}
	\begin{align} \varphi = -dr^1 \wedge dr^2 \wedge dr^3 + dr^1 \wedge \omega + dr^2 \wedge \Re (\Omega) + dr^3 \wedge \Im (\Omega).\end{align}
	\begin{align} AX(PX^T\!AXP)^\dagger X^T\!A &= AX(UU^TX^T\!AXUU^T)^\dagger X^T\!A \\ &= AXU(U^TX^T\!AXU)^\dagger U^TX^T\!A.\end{align}
	P ( \rho ) = \rho ^ { 2 } \alpha ^ { \, \prime } ( \rho ) .
	\mathcal { E } ^ { ( c l ) } ( C _ { r - 1 } ^ { \vee } ) = \mathcal { E } ^ { ( c l ) } ( B _ { r } ) = \frac { r m \cos ( \pi / 4 - \pi / 4 r ) } { \pi b ^ { 2 } \cos ( \pi / 4 r ) } ; \quad w _ { 0 } = w _ { r } = 0
	\begin{align}\displaystyle X_V=\Xi-X_W+c(z_r,w)+\sum_{i=0}^r \lambda_i c(z_i,\eta z_i)+ \sum_{i=0}^{r-1} \mu_i c(z_i,z_{i+1})\end{align}
	\int _ { \mathbf { y } } ^ { y \prime } d z _ { i } \delta ^ { ( 3 ) } \left( \mathbf { x - z } \right) = (
	\begin{align}J(X)=\left\{ \delta_{ij}:1\leq i<j\leq n\right\} \cup\left\{ A_{1rs}:1<r<s\leq n\right\} . \end{align}
	K _ { \mu \nu } = - \frac { 1 } { 2 } g _ { \mu \nu , y } = \Sigma _ { \mu \nu } + { \frac { 1 } { 4 } } g _ { \mu \nu } K \ .
	\begin{align} u(\phi(\alpha(i_1,i_2)))=c(i_2+i_1+1)-(-1)=c(i_1+i_2)+c+1=u(\alpha(i_1,i_2))+c+1 \, . \end{align}
	\displaystyle \frac { \partial \sigma _ { i j } ^ { \mathrm { N } } } { \partial q ^ { 0 } }
	\begin{align} \operatorname{lev}\nolimits_{\varphi_{A,k},=}(t) = tk+ \operatorname{bd}A \forall\, t \in \mathbb{R}. \end{align}
	S U ( n ) \simeq S ^ { 2 n - 1 } \otimes S ^ { 2 n - 3 } \otimes \cdots \otimes S ^ { 3 } \; .
	R _ { 1 2 } R _ { 1 3 } R _ { 2 3 } = R _ { 2 3 } R _ { 1 3 } R _ { 1 2 } .
	\begin{align}\sum_{\lambda\in\{\lambda\|\lambda\vdash m, \ell(\lambda)\geq 1\}}\frac{(-1)^{\ell(\lambda)-1}}{\ell(\lambda)}\frac{\ell(\lambda)!}{\lambda_{1}!\cdot\cdot\lambda_{\ell(\lambda)}!\|Aut(\lambda)\|}=0\end{align}
	\begin{align} R_n^{(1)}(z) = zS_{n-1}(z) + S_{n-1}^{\ast}(z) \mbox{and} (z-1)R_n^{(2)}(z) = zS_{n}(z) - S_{n}^{\ast}(z), n\geq 1,\end{align}
	\begin{align}\widehat{A}&=a\cdot (y^{-1}x^2y)+ b \cdot (y^{-1})+c \cdot (x^2 y)+d\cdot (1)\\&=a \cdot (x^2)+ b \cdot (y) +c \cdot (x^2 y)+d \cdot (1),\\\widehat{B}&=e\cdot ((x^2y)^{-1} x^{-1}y)+ f \cdot ((x^2 y)^{-1} )+ g\cdot (x^{-1} y)+ h\cdot (1)\\&=e \cdot (x^3) + f\cdot (x^2 y) + g\cdot (x^3y)+ h\cdot (1).\end{align}
	\begin{align} \mathcal{K}^{\mu}(x,y):=\frac{G^{\mu}(x^{-1}y)}{G^{\mu}(y)}. \end{align}
	\begin{align}\bigcup_{n=1}^{\infty} S_{\mu}^n = \Gamma\,,\end{align}
	{ \cal I } _ { I J } \, \equiv \, { \hat { c } } _ { I } \, \cap \, { \hat { c } } _ { J } \ .
	\begin{align}K(N):&=\prod_{p\nmid N}\left( 1-\frac{(\frac{N-1}{p})^2p+1}{(p-1)^2(p+1)}\right)\prod_{{p\mid N}}\left(1-\frac{1}{p^{\nu_p(N)}(p-1)}\right) \end{align}
	\begin{align}b ( x ) ( \Delta_\lambda ( x ) - \Delta_\lambda ( q^{-1} x ) + c ) = 0 \end{align}
	\begin{align}\frac{a_y}{\sqrt B}\sin\phi+\sigma\cos\phi+\tilde\sigma=0.\end{align}
	\begin{align}\left\{\begin{array}{l}x_0=0\\begin{align}2mm]\displaystyle\sum_{i=1}^r d_i x_i+2x_{r+1}=0.\\\end{array}\right.\end{align*}
	\begin{align}\omega_\alpha( \textit{\textbf{p}}_\lambda, \textit{\textbf{p}}_\mu)=\max\Big\{\frac{s_\lambda}{s_\mu},\frac{s_\mu}{s_\lambda}\Big\}(1+d_\alpha(\textit{\textbf{p}}_\lambda,\textit{\textbf{p}}_\mu))\,,\end{align}
	\begin{align}\{ \ ( \sum_{(\lambda,\mu)} \phi \mid_{k,m} [\lambda,\mu] )\ \mid U_l , \psi \} &= \{ \phi, \sum_{(\lambda,\mu)} \psi \mid A_l \mid_{k,m} [-\lambda,-\mu] \}\end{align}
	\forall a , b \in { \cal G } \equiv s l ( N , { \mathbb C } ) \; , \; \mathrm { T r } _ { { \cal G } \otimes { \cal G } } \; { \cal C } ^ { s l ( N ) } \: a \otimes b = \mathrm { T r } _ { \cal G } \; a b
	\begin{align}\left[\mathcal{J}_n^T[\mathbf{u}^{\mathcal{L}}(x_j^{(n)})]^T\right]^T \mathbf{u}^{\mathcal{R}}(x_k^{(n)})=[ \mathbf{u}^{\mathcal{R}}(x_k^{(n)})]^T \mathcal{J}_n^T [\mathbf{u}^{\mathcal{L}}(x_j^{(n)})]^T = \omega_{n,j,k}^{-1}\delta_{j,k}.\end{align}
	\begin{align}[L_{-2}, \, X_{2, \, j+2, \, 13}] = 0.\end{align}
	\displaystyle q ^ { 2 } \left[ \Pi _ { A } ( q ^ { 2 } ) - \Pi _ { V } ( q ^ { 2 } ) \right]
	\begin{align} \lim_{\tau \downarrow 0}\sup_{\|x\| > \rho \zeta_{\tau}} h(x,\tau) = 0.\end{align}
	\begin{align}L_3=\{(\gamma_1, \gamma_2, \gamma_3)\in T^(PM)^3 \mid \gamma_1* \gamma_2 \sim \gamma_3 \}.\end{align*}
	\begin{align} \langle \omega_{n-1}\otimes\omega_{n-2}\otimes\dots\otimes\omega_{0},\tilde\omega_{n-1}\otimes\tilde\omega_{n-2}\otimes\dots\otimes\tilde\omega_{0}\rangle_{F^{\otimes n}}=\prod \langle \omega_{a},\tilde \omega_{a}\rangle_{F}.\end{align}
	\overrightarrow { J } = - i \overrightarrow { P } \times \overrightarrow { \bigtriangledown } + ( \overrightarrow { S } + \overrightarrow \Sigma )
	\delta ( \partial _ { \mu } \phi + A _ { \mu } \phi + A _ { \mu } \zeta ) = \epsilon ( \partial _ { \mu } \phi + A _ { \mu } \phi + A _ { \mu } \zeta ) .
	\phi _ { \pm } ^ { \mu , k } ( \sigma , \tau ) = \frac { 1 } { \sqrt 2 } \sum _ { r \in Z + \frac { 1 } { 2 } } b _ { r } ^ { \prime \, \mu , \; k } e ^ { - i r ( \tau \pm \sigma ) }
	\begin{align}\\|u\\|_{X^{s,b}_{\|\tau\|=\|\xi\|}}=\\|\left<\xi\right>^s\left<\|\tau\|-\|\xi\|\right>^{b}\hat{u}(\tau,\xi)\\|_{L_{\tau,\xi}^2},\end{align}
	c h _ { G } ^ { g } ( E ) = \sum _ { i = 1 } ^ { s } \lambda _ { i } c h ( E ^ { i } )
	\rho = \frac { r } { r _ { 0 } } \ll 1 \qquad \frac { 1 } { 2 } N ^ { 2 } \rho ^ { D } \gg 1 ,