harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat{\pi}(y_t)=\frac{\exp\{-2 f(x)/(\tau_N^2 \gamma)\}}{\int \exp\{-2 f(y)/(\tau_N^2 \gamma)\}}=\frac{\exp\{-2N f(y)/(\sigma^2\tilde{\Gamma})\}}{\int \exp\{-2N f(y)/(\sigma^2\tilde{\Gamma})\}}
\#\left(\mathbf{k};\{p_{i,j}\},\{q_{i,j}\}\right) = (\mathbf{k}) \cdot (\{p_{i,j}\},\{q_{i,j}\}) \cdot \#(\mathbf{k}') = (\{p_{i,j}\},\{q_{i,j}\}) \cdot \prod_{1 \leq i<j \leq n}\frac{k_j-k_i}{j-i},
L^{2}({\cal A}_{\gamma}\times{\cal F}_{\gamma}) = \bigotimes_{e \in E}\bigl ( \bigoplus_{\rho \in \mathrm{R e p}( G )}\rho \otimes \rho^{*}\bigr ) \; \otimes \; \bigotimes_{v \in V}\bigl ( \bigoplus_{\rho \in S}\rho \bigr ) .
a_{4}^{1234}=a_{4}^{1243}=a_{4}^{1324}=a_{4}^{1342}=a_{4}^{1423}=a_{4}^{1432}= a_{4}^{2134}=a_{4}^{2143}=a_{4}^{2314}=a_{4}^{2341}=a_{4}^{2413}=a_{4}^{2431}= a_{4}^{3124}=a_{4}^{3142}=a_{4}^{3214}=a_{4}^{3241}=a_{4}^{3412}=a_{4}^{3421}= a_{4}^{4123}=a_{4}^{4132}=a_{4}^{4213}=a_{4}^{4231}=a_{4}^{4312}=a_{4}^{4321}
\ker(N_{N/M})=\prod_{\ell\mid R,\ \ell\equiv +1(5)}\,(\langle\mathfrak{L}^{1-\tau^2}\rangle/\langle\mathfrak{L}^{1-\tau^2}\rangle^5\times\langle\mathfrak{L}^{\tau-\tau^3}\rangle/\langle\mathfrak{L}^{\tau-\tau^3}\rangle^5)
I = \frac{1}{2}\int d^2 x \sqrt{- g}e^{- 2 \phi}\Big(R + 4 ( D \phi )^2 + 4 \Lambda - (D \psi )^2 - \frac{1}{4}(D \psi_1 )^2 - \frac{1}{4}e^{-\psi_1}F_2^2 - \frac{1}{4}e^{\psi_1}F^2 \Big)
(d_1 \circ s_0)(a \otimes b) = (d_1 \circ \varphi_1)(a \otimes z \otimes b) &= (\varphi_0 \circ \beta_1)(a \otimes z \otimes b) &= \varphi_0(az \otimes b) - \varphi(a \otimes zb) &= z\varphi_0(a \otimes b) - \varphi(a \otimes b)z .
\left\langle \bar{\psi}( x + \epsilon / 2 ) \gamma^{\mu}\gamma_{5}\psi ( x - \epsilon / 2 ) \right\rangle = 2 g \varepsilon^{\alpha \beta \mu \gamma}F_{\alpha \beta}( x ) ( \frac{- i}{8 \pi^{2}}\frac{\epsilon_{\gamma}}{\epsilon^{2}}) ,
\begin{array}[c]{rl}\left. \frac{\partial V}{\partial x}(s,x,W(s,x),W_{x}(s,x),\bar{u}(s))\right\vert_{x=\bar{X}^{t,x;\bar{u}}(s)}=K_{1}(s), \left. \frac{\partial^{2}V}{\partial x^{2}}(s,x,W(s,x),W_{x}(s,x),\bar{u}(s))\right\vert_{x=\bar{X}^{t,x;\bar{u}}(s)}=\tilde{K}_{2}(s),\end{array}
\begin{array}{rcl}\theta(\alpha \cdot f) (u \otimes n) = u (\alpha \cdot f)(n) = u (\alpha f(n) - f(\alpha \cdot n)) = \theta(f) (u \alpha \otimes n - u \otimes \alpha \cdot n) = \theta(f) ((u \otimes n) \cdot \alpha) = (\alpha \cdot \theta(f))(u \otimes n)\,. \end{array}
Z = \sum_{h_{i j}}\prod_{n i}\int_{- \infty}^{\infty}\! d A_{i}( n ) e x p \: ( - \frac{1}{4 \kappa^{2}}{\sum_{n i j}\: [ \bigtriangleup_{i}A_{j}( n ) - \bigtriangleup_{j}A_{i}( n ) + h_{i j}( n ) ]^{2}}) .
F(t,x,v) = \begin{cases}x/v, \mbox{for}x/v \leq t, t, \mbox{for}x/v >0,\, v >0 t, \mbox{for}v=0 t, \mbox{for}(1-x)/|v| >t,\, v <0 (1-x)/|v| \mbox{for}-(1-x)/v \leq t. \end{cases}
\hat{A}_{n m}( \omega ) ={\frac{2 \coth ({\frac{\omega}{2}}) \sinh \Bigl ( ( \nu - \max ( n , m ) ){\frac{\omega}{2}}\Bigr ) \sinh \Bigl ( ( \min ( n , m ) ){\frac{\omega}{2}}\Bigr )}{\sinh ({\frac{\nu \omega}{2}})}}
\leq 2^{4d+2}C \sum\limits_{n=1}^{\infty}\sum\limits_{r=0}^{\infty}2^{4n+2dr}e^{-\alpha\delta^2(n+r)}= 4^{2d+1}C \sum\limits_{n=0}^{\infty}\sum\limits_{r=0}^{\infty}2^{4n+2dr}e^{-\alpha\delta^2(1+n+r)},
E'(k) := \min \{\ell(k_i) : 1\leqslant i \leqslant d \}, \ell(k_i) := \begin{cases}2-2\cos(k_i/2), \ \k_i \in (0,2\pi/3] 2+2\cos(k_i), \ \k_i \in (2\pi/3, \pi) \cup (\pi, 4\pi/3] 2+2\cos(k_i/2), \ \k_i \in (4\pi/3,2\pi). \end{cases}
\Phi^{\mathrm{H}}_{\tau, \, 0}u_0(z) :=2\int_{0}^\tau \frac{\partial K}{\partial y}(\tau - s, z , 0) u_0(s)\, ds &= \frac{2z}{\sqrt{2 \pi}}\int_0^\tau \frac{e^{-\coth(2(\tau-s))z^2/2}}{\sinh(2(\tau-s))^{3/2}}u_0(s) ds &= \frac{z}{2 \sqrt{\pi}}\int_0^{\frac{\tanh(2\tau)}{2}}\frac{e^{-\frac{z^2}{4x}}}{x^{3/2}}(1-4x^2)^{-1/4}u_0(\alpha^{-1}(x)) dx.
\Delta_2\leq -2(X-Y)\cdot(|X|^\gamma X-|Y|^\gamma Y) +2|X|^{2+\gamma}+2|Y|^{2+\gamma}-4(|X||Y|)^{\gamma/2}(X\cdot Y) = 2(X\cdot Y)(|X|^{\gamma/2}-|Y|^{\gamma/2})^2.
\||D|^{-m_1}x|D|^{m_2}e^{-s^2D^2}\|_1 \leq \Big(\sum_{l=0}^{\infty}2^{l(p-m_2)}\cdot e^{-2^{2l}s^2}\Big)\cdot O(1) = O(s^{m_2-p}).
\psi_{\alpha}={\frac{1}{2 \pi i}}\int_{C_{+}}d t e^{- i x \cosh t}{\frac{e^{( 1 - \alpha ) t}}{e^{t}+ e^{i \theta}}}+{\frac{1}{2 \pi i}}\int_{C_{-}}d t e^{- i x \cosh t}{\frac{e^{( 1 - \alpha ) t}}{e^{t}+ e^{i \theta}}},
f*g=fg+\frac{i\hbar}{2}\sum_{i,j}\Pi^{ij}(\partial_{i}f)(\partial_{j}g)-\frac{\hbar^{2}}{8}\sum_{i,j,k,m}\Pi^{ij}\Pi^{km}(\partial_{i}\partial_{k}f)(\partial_{j}\partial_{m}g)+...,
\frac{x_{n+1}}{x_0}-f\Big(\frac{x_1}{x_0},...,\frac{x_n}{x_0}\Big)=0,~~~~~~~\frac{y_0}{y_{n+1}}+\hat{f}\Big(\frac{y_1}{y_{n+1}},...,\frac{y_n}{y_{n+1}}\Big)=0.
f'(x)-h'(x)=(H'_y(x,h(x)))^{-1}H'_x(x,h(x))-(F'_y(x,f(x)))^{-1}F'_x(x,f(x))= =(H'_y(x,h(x)))^{-1}H'_x(x,h(x))-(F'_y(x,h(x)))^{-1}F'_x(x,h(x))+ +(F'_y(x,h(x)))^{-1}F'_x(x,h(x))-(F'_y(x,f(x)))^{-1}F'_x(x,f(x)).
\dot{\bar{\alpha}}_{1}+\left[(2a+f_{2}^{2})-b_{1}^{2}m_{1}^{-1}\alpha\right]\bar{\alpha}_{1}+l_{1}=0\hbox{on}\; (0, T], \dot{\bar{\alpha}}_{1}(0)=-r_{1}, \dot{\bar{\alpha}}_{2}+\left[(2a+f_{2}^{2})-b_{2}^{2}m_{2}^{-1}\alpha\right]\bar{\alpha}_{2}+l_{2}=0 \hbox{on}\; (0, T], \dot{\bar{\alpha}}_{2}(0)=-r_{2},
\phi_0(E,\zeta) = C\left(2+\frac{K}{\rho L}\zeta\right) \frac{E}{\exp\left(\frac{\zeta^2 E^2}{4L^2}\right) - 1}+\sqrt{2L\log\left(\frac{1}{1 - \exp\left(-\frac{\zeta^2 E^2}{4L^2}\right)}\right)}.
\sum_{i=0}^n\sum_{j=0}^i((s+1)(i-j)j(i-2) + (n-i)(j(j-1) + (i-j)(i-j-1)))\frac{x_{i-j}}{(i-j)!}\frac{x_j}{j!}\sum_{\ell=0}^n (s+\ell)!\frac{B_{n-i, \ell}(x_1, x_2, \ldots)}{(n-i)!}= 0
\int\frac{\big|\widetilde{U}(t)\big|^2}{2h(t)}+\int^{t}_{0}e^{r(t-s)}\left[\int\frac{\widetilde{W}^{{\rm T}}Q_r(w^*)\widetilde{W}}{2h}+ \widetilde{W}\cdot\mathrm{L}(w^*)\right]{\rm d}s = e^{rt}\int\frac{\big|\widetilde{U}(0)\big|^2}{2h(0)}.
g^{\mu \rho}M^{\nu \sigma}- g^{\nu \rho}M^{\mu \sigma}+ g^{\mu \sigma}M^{\rho \nu}- g^{\nu \sigma}M^{\rho \mu}= q ( F^{\nu \sigma}x^{\mu}x^{\rho}- F^{\mu \sigma}x^{\upsilon}x^{\rho}+ F^{\rho \mu}x^{\mu}x^{\sigma}- F^{\rho \mu}x^{\nu}x^{\sigma}) ,
Var(\hat{\pi}_{h,n}(x)) \le \frac{c}{T_n}\frac{\sum_{l = 1}^d |\log(h^*_l)|}{\prod_{l \ge 3}h^*_l}+ \frac{c}{T_n}\frac{1}{\prod_{l =1}^d h_l^*}\Delta_n \le \frac{c}{T_n}\frac{\sum_{l = 1}^d |\log(h^*_l)|}{\prod_{l \ge 3}h^*_l}.
i\hbar\frac{\partial}{\partial t}\tilde{\Psi}(p,t)=\frac{p^{2}}{2m}\tilde{\Psi}(p,t)+(2\pi\hbar)^{-3/2}\int d^{3}p^{\prime}\tilde{V}(p-p^{\prime})\tilde{\Psi}(p^{\prime},t).
\left(l_{j, 1}, \;\; l_{j, 2}\right) = \left(a_{i, 1}, \;\; a_{i, r}\right) \left(\begin{array}{cc}a_{1,1}a_{r,1}a_{r,1}a_{r,r}\end{array}\right)^{-1}&= \frac{1}{a_{1,1}\, a_{r, r}- a_{r,1}^2}\left(a_{i, 1}\, a_{r, r}- a_{r, 1}\, a_{i, r}, \;\; a_{1,1}\, a_{i, r}- a_{r, 1}\, a_{i, 1}\right).