harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widehat{S}_{t}\psi(x)=\left( \frac{1}{2\pi i\hbar|t|}\right)^{1/2}i^{m}\int_{-\infty}^{\infty}\exp\left[ \frac{i}{\hbar}\frac{(x-x^{\prime})^{2}}{2t}\right] \psi(x^{\prime})dx^{\prime}
\psi_V([u,v,w]_{T'}) = \psi_V(\rho(T'u,T'v)w) = \rho(\psi_L(T'u),\psi_L(T'v))\psi_V(w) = \rho(T\psi_V(u),T\psi_V(v))\psi_V(w) = [\psi_V(u),\psi_V(v),\psi_V(w)]_T.
\Phi_{conv}:=\left\{\phi:X\to\mathbb{R}|\ \phi\left(x\right)=\left\langle u,x\right\rangle_X + c ,u\in X^{*}, c\in \mathbb{R}\right\}\Psi_{Q,b}:=\left\{\psi:Y\to\mathbb{R}|\ \psi\left(y\right)= b\left\Vert y\right\Vert^{2}_Y+\left\langle v,y\right\rangle_Y +d,v\in Y^{*},b,d\in\mathbb{R}\right\}.
f_{\mu\nu}^{\xi}(x) = \partial_\mu[n^A(x){\cal A}_\nu^A(x)] - \partial_\nu[n^A(x){\cal A}_\mu^A(x)] -{1 \over g}f^{ABC}n^A(x) \partial_\mu n^B(x) \partial_\nu n^C(x) ,
E^t_{GMS}(\alpha^* + dg_k) = \int_M \sqrt{t^{-2}+ |\alpha^* + dg_k|^2}dV_g & \leq \int_M (t^{-1}+ |\alpha^* + dg_k|) dV_g & = \frac{(M)}{t}+ TV_{\alpha^*}(dg_k) & < c - \delta
b_j(y) = \frac{1}{y^{3j+1}}g\Big( \frac{y^2 n_1^4}{N}\Big) \widehat{k^*}\Big( \frac{MTcm}{2\pi^2 y n_1^2}\Big) e\Big( -\frac{T^2cm}{4\pi^2 y n_1^2}\Big)
p(\lambda_1)(1+p(\eta)+p(\mu)) + p(x_1)(p(\lambda_1)+p(\mu)) = p(\lambda_1)(1+p(\eta)+p(\mu)) + (p(\lambda_1)+p(y_1))(p(\lambda_1)+p(\mu)) &= p(\lambda_1)(p(y_1)+p(\eta)) + p(y_1)p(\mu).
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( 1-z^n \right)^{\frac{m^3}{n^4}}= \sqrt[3]{1-z}\; \exp\left\{\frac{-1}{4}Li_2(z) - \frac{1}{4}\frac{z}{1-z}\right\}.
\sum_{k\ne i}M_{k}=\{m_{1}+\cdot\cdot\cdot+m_{i-1}+m_{i+1}+\cdot\cdot\cdot+m_{d}:m_{k}\in M_{k}forallk\}=span\bigcup_{k\in\{1,...,i-1,i+1,...,d\}}M_{k}.
w\partial_u H=\frac{w(u-u^*)}{f(u)q(u)}\leq\frac{w(u+u^*)}{q(0)f(u)}\leq\frac{2w}{q(0)f(u)}<\frac{2K}{q(0)}, begin{align*}0.2cm] y\partial_v H=\frac{y(v-v^*)}{sv}\leq\frac{1}{s}\left(\frac{c}{d}v-\frac{y}{v}v^*\right)\leq \frac{q(1)}{s}\left(\frac{c}{d}+\frac{2sv^*}{cq(0)}\right).
\Theta \cot \left( \Theta \left[ \ln \left( \frac{z}{2}\right) + \gamma \right] \right) \stackrel{( a \rightarrow 0 )}{=}\tilde{k}a \frac{J_{l}^{\prime}( \tilde{k}a )}{J_{l}( \tilde{k}a )}\left[ 1 + O \left( [ \kappa a ]^{2}\right) \right] .
\frac{\big(1-\alpha_N^{0}(\mu_n)\big)\big(1-\beta_N^{0}(\mu_n)\big)}{1-\alpha_N^{0}(\mu_n)\beta_N^{0}(\mu_n)}2 i \sin \theta_N(\mu_n) = & \; i(-1)^{n+1}\big(\hat\varphi_1(-n)+\hat\varphi_2(n)\big)+ \frac{2(-1)^{n}}{(2i\pi n)^{N+1}}\int_0^1\varphi_1r_{N+1}\,dt + \frac{\ell^2_n}{n^{N+1}}.
s(e_{x_1}e_{x_2}\cdots e_{x_{n-1}}e_{x_n}) = (-1)^{n \choose 2}(- e_{x_n}x_n^{-1})(- e_{x_{n-1}}x_{n-1}^{-1}) \cdots (-e_{x_2}x_2^{-1})(-e_{x_1}x_1^{-1}) &=(-1)^{\frac{n(n+1)}{2}}\, e_{x_n}\, e_{x_{n-1}^{x_n}}\cdots e_{x_2^{x_3\cdots x_n}}\, e_{x_1^{x_2\cdots x_n}}\, x_n^{-1}\cdots x_1^{-1}
\frac{1}{|B_{R_\gamma}|}\sum_{y \in B_{R_\gamma}(x)}\sigma(y) =\frac{1}{|B_{R_\gamma}|}\sum_{y \in B_{R_\gamma}(x)}(\sum_{z \in \mathbb{Z}^d}J_\gamma(z,y)-s(\gamma)) \sigma(y) &=\frac{1}{|B_{R_\gamma}|}\sum_{z \in B_{R_\gamma}(x)}I_z^\gamma(\sigma) + O(\gamma^{-1}/R_\gamma) + s(\gamma)O(1)\,,
\int_{W^d}(f_i(x_j))_{i, j = 1}^d (g_i(x_j))_{i, j = 1}^d \prod_{i=1}^d \nu(dx_i) = \left(\int_{\mathbb{R}}f_i(x) g_j(x) \nu(dx)\right)_{i, j =1}^d.
\zeta ( z , q ) = \frac{1}{2 q^{z}}+ \frac{q^{1 - z}}{z - 1}+ 2 \int_{0}^{\infty}( q^{2}+ y^{2})^{\frac{- z}{2}}\sin ( z \arctan \frac{y}{q}) \frac{1}{e^{2 \pi y}- 1}d y .
c_{a b}= \left( \begin{array}{c c c c c}{{0}}&{{- \omega_{1}}}&{{}}&{{}}&{{}}\\{{\omega_{1}}}&{{0}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{0}}&{{- \omega_{\frac{p + 1}{2}}}}\\{{}}&{{}}&{{}}&{{\omega_{\frac{p + 1}{2}}}}&{{0}}\\ \end{array}\right) .
X(\mathcal{A},\mathcal{B}) = \bigcup_{(\tilde{\mathcal{A}},\tilde{\mathcal{B}}) \in \sigma^{-1}(\mathcal{A},\mathcal{B})}X((\tilde{\mathcal{A}},\tilde{\mathcal{B}})) C(\mathcal{A},\mathcal{B}) = \bigcup_{(\tilde{\mathcal{A}},\tilde{\mathcal{B}}) \in \sigma^{-1}(\mathcal{A},\mathcal{B})}C((\tilde{\mathcal{A}},\tilde{\mathcal{B}}))
\overline{\cal C}=\left[\begin{array}{cccc}\overline{c}_{11}E^{|a_{1}||a_{1}|}\cdots \overline{c}_{1r}E^{|a_{1}||a_{r}|}0 \vdots \ddots \vdots \vdots \overline{c}_{r1}E^{|a_{r}||a_{1}|}\cdots \overline{c}_{rr}E^{|a_{r}||a_{r}|}0 0 \cdots 0 0\end{array}\right]\,,
\begin{aligned}k^{-1}\lVert \widetilde{M}_{k}(\alpha,\theta,E,0)\rVert \leq C'+C'\sum_{i=0}^{k-1}i|g(i)| i^{-1}\lVert\widetilde{M}_{i}(\alpha,\theta,E,0)\rVert. \end{aligned}
-2\sum_{x\in X_n}\omega(x) \int_{-\infty}^\infty K_t(x,y)W(y)\mathrm{d}y = -2t^{-1}\sum_{x\in X_n}\omega(x) W(x) & = -2 t^{-1}\int_{-\infty}^\infty W^2(x)dx=-\frac{2}{\sqrt{2}t}.
\mathbf{x}_{k+1}=\frac{\mathbf{y}_{k+1}}{\left\Vert \mathbf{y}_{k+1}\right\Vert}=\frac{\mathbf{x}_{k}+\Delta \mathbf{y}_{k}}{\left\Vert\mathbf{x}_{k}+\Delta \mathbf{y}_{k}\right\Vert}.
c_w w + c_{rw}rw = c_w w + c_{w_1^{-1}s_i w_1w}w_1^{-1}s_i w_1w = w_1^{-1}\big({}^{w_1}c_w +{}^{w_1}c_{w_1^{-1}s_i w_1w}s_i\big) w_1 w,
P_l^{(\lambda)}(\xi)= \frac{1}{l!}\left\{\frac{d^{l}}{dr^{l}}\left[(1-2r\xi+r^{2})^{-\lambda}\right]\right\}_{r=0}= \sum_{j=0}^{[\frac{l}{2}]}(-1)^j\frac{\Gamma(l-j+\lambda)}{\Gamma(\lambda)j!(l-2j)!}(2\xi)^{l-2j}
U=\left( \begin{smallmatrix}Hom(B,B) Hom(C,B) Hom(B,C) Hom(C,C) \end{smallmatrix}\right)andW_U=\left( \begin{smallmatrix}\mathfrak{J}(EndB) Hom(C,B) Hom(B,C) \mathfrak{J}(EndC) \end{smallmatrix}\right).
\begin{aligned}\uppercase\expandafter{\romannumeral1}=\sum\limits_{j=1}^{\infty}\lambda_{j}\vert g(a_{j})(x)\vert\chi_{2Q_{j}}(x)+\sum\limits_{j=1}^{\infty}\lambda_{j}\vert g(a_{j})(x)\vert\chi_{(2Q_{j})^{c}}(x) &:=\uppercase\expandafter{\romannumeral1}_{1}+\uppercase\expandafter{\romannumeral1}_{2}\end{aligned}
\sqrt{N^{2}+d}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n)!d^{n}}{(1-2n)n!^{2}4^{n}N^{2n-1}}=N+\frac{d}{2N}-\frac{d^{2}}{8N^{3}}+\frac{d^{3}}{16N^{5}}-\frac{5d^{4}}{128N^{7}}+\cdot\cdot\cdot
{a - k + 2 \choose r}n^{r^{k-1}}\prod_{i \leq k-2}{|U_i| \choose r}={a - k + 2 \choose r}|U_{k-1}|^r\prod_{i \leq k-2}{|U_i| \choose r}& \geq r!{a - k + 2 \choose r}\prod_{i \leq k-1}{|U_i| \choose r}\geq (a - k + 2)\prod_{i \leq k-1}{|U_i| \choose r},
\begin{array}{l c l}{{x_{a}\star x_{b}}}&{{=}}&{{\sum_{\alpha = 0}^{\infty}( x_{a}\star x_{b})_{\alpha}}}\\{{}}&{{=}}&{{x_{a}x_{b}+ \sum_{{\alpha}= 1}^{\infty}\frac{1}{\alpha !}( \frac{i}{2})^{\alpha}\prod_{i = 1}^{\alpha}\theta^{\mu_{i}\nu_{i}}( \prod_{j = 1}^{\alpha}\partial_{\mu_{j}}x_{a}) ( \prod_{k = 1}^{\alpha}\partial_{\nu_{k}}x_{b})}}\\ \end{array}
g ( u ) = \int_{1}^{u}d z \eta ( z ) ={\frac{1}{\pi^{2}}}\int_{1}^{u}d x \int_{- 1}^{1}d y{\frac{y - x}{\sqrt{( x^{2}- 1 ) ( y^{2}- 1 )}}}\log \left[{\frac{2 u - x - y + 2 \sqrt{( u - x ) ( u - y )}}{x - y}}\right] .
\begin{array}{c}\mu(x, y)=0.24, \mathbf{b}(x, y)=[5, \sin(6x)]^T, \sigma(x, y) = 0.1, begin{align*}2mm]f(x,y)=10\chi_{E_1}(x, y)+10\chi_{E_2}(x, y), \end{array}
\begin{aligned}\inf_{q\in Q}\sup_{\vec{v}\in V}\frac{(\operatorname{div}\vec{v},q)}{\|\varepsilon(\vec{v})\|\|q\|}=\inf_{\vec{v}\in W^{\perp}}\sup_{q\in Q}\frac{(\operatorname{div}\vec{v},q)}{\|\varepsilon(\vec{v})\|\|q\|}\ge \beta>0.\end{aligned}
C_{N}(z_{1},z_{2}) = \tilde{C}_{N}(z_{1},z_{2}) + \sum_{k=1}^{N}\mathbb{E}_{k-1}(Y_{k}(z_{1})-\tilde{Y}_{k}(z_{1}))\tilde{Y}_{k}(z_{2})+\sum_{k=1}^{N}\mathbb{E}_{k-1}(Y_{k}(z_{2})-\tilde{Y}_{k}(z_{2}))\tilde{Y}_{k}(z_{1}) &+\sum_{k=1}^{N}\mathbb{E}_{k-1}(Y_{k}(z_1)-\tilde{Y}_{k}(z_1))(Y_{k}(z_2)-\tilde{Y}_{k}(z_2)) + O_{L^1}(1/d_N^2)
24\theta'=(E_2-\theta^4+80\Delta_4^4)\theta\,, E_4=\theta^8+224\theta^4\Delta_4^4+256\Delta_4^8\,, &24\Delta_4'=(E_2+5\theta^4-16\Delta_4^4)\Delta_4\,, E_6=(\theta^4+16\Delta_4^4)(\theta^8-544\theta^4\Delta_4^4+256\Delta_4^8)\,.
\begin{array}{l}F_2(s,r)= \mathbb{E}[\{f(\zeta^1(s),\zeta^{2,t}(r))g(r,\zeta^{2,t}(r))+\mathcal{G}^{2,t}(r)f(\zeta^1(s),\zeta^{2,t}(r))\}. \exp\{\int\limits_0^r g(\tilde{u},\zeta^{2,t}(\tilde{u}))d\tilde{u}\}].\end{array}
e_{1}{\phi}_{( 1 )}^{\mathrm{s e l f}}+ e_{2}{\phi}_{( 2 )}^{\mathrm{s e l f}}= \frac{1}{8 \mathrm{L}}( e_{1}^{2}+ e_{2}^{2}) ( x_{+}^{2}+ x_{-}^{2}) + \frac{1}{4 \mathrm{L}}( e_{1}^{2}- e_{2}^{2}) x_{+}x_{-}+ \frac{\mathrm{L}}{4}( e_{1}^{2}+ e_{2}^{2}) ,
L^{\xi}= p \cdot \hat{H}+ X^{\xi}, \quad X^{\xi}= i \sum_{\rho \in \Delta_{+}}g_{| \rho |}( \rho \cdot \hat{H}) \left( x ( \rho \cdot q ) - x ( \rho^{\vee}\! \cdot \hat{H}\xi ) \right) \hat{s}_{\rho},
p_\zeta^{(1)}(t,r,s) = \frac{2\Gamma(\zeta+1)}{\sqrt\pi}\cdot \frac{t}{\left(r^2+s^2+t^2\right)^{\zeta+1}}\times\,_2\tilde{F}_1\left(\zeta+1,\zeta+2;\zeta+\frac{1}{2};\frac{4r^2s^2}{\left(r^2+s^2+t^2\right)^2}\right).
\theta_\rho(t) = \int_{-\infty}^t \theta_\rho'(s) ds \theta_\rho'(t) = \begin{cases}\frac{1}{\rho}\mid t - t_2 \mid < \rho \frac{-1}{\rho}\mid t - t_1 \mid < \rho 0 ,\end{cases}
R_{s,1}=\max_{\alpha_1,\gamma_1\in[0,1]}\min\{\textrm{rhs}(\ref{rs2b})+\textrm{rhs}(\ref{rs2d}),\ \textrm{rhs}(\ref{rs2b})+\textrm{rhs}(\ref{rs2e}),\ \textrm{rhs}(\ref{rs2f}),\ \textrm{rhs}(\ref{rs2g})\}, R_{s,2}=\mathcal{C}(P_1)+\mathcal{C}(P_2),
Q_{p - 1}\wedge E^{b}\wedge E^{a}={\frac{1}{( p + 1 ) !}}E^{a_{0}}\wedge E^{a_{1}}\wedge . . . \wedge E^{a_{p}}\epsilon_{a_{0}a_{1}. . . a_{p}}e^{-{\frac{p - 3}{2}}\phi}{\frac{\partial}{\partial F_{a b}}}( \sqrt{- d e t ( \eta + F )})
\partial_{t}\mathbf{u}- \mathrm{div}(\mathbf{H}\nabla \mathbf{u}) = \mathbf{0}(0, \infty) \times G, \tau \partial_{t}\mathbf{H}+ \mathbf{H}- \mathbf{F}(\nabla \mathbf{u}) = \mathbf{0}(0, \infty) \times G.