harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\alpha_{\ell}= r_1 + \frac{(r_1 - r_2)\frac{\alpha_1 - r_1}{\alpha_1-r_2}q^{\ell-1}}{1 - \frac{\alpha_1 - r_1}{\alpha_1-r_2}q^{\ell-1}}, &= r_2 + \frac{r_2 - r_1}{\frac{\alpha_1 - r_1}{\alpha_1-r_2}q^{\ell-1}- 1}.
\int |\chi_R(x-z) \nabla u^{\xi}(t,x)|^2 dx - \frac{N\alpha}{2(\alpha+2)}\int |\chi_R(x-z) u(t,x)|^{\alpha+2}dx &= \|\nabla[\chi_R(\cdot-z) u^{\xi}(t)]\|^2_{L^2}- \frac{N\alpha}{2(\alpha+2)}\|\chi_R(\cdot-z) u^{\xi}(t)\|^{\alpha+2}_{L^{\alpha+2}}+ O(R^{-2}) &\geq \nu \|\nabla[\chi_R(\cdot-z) u^{\xi}(t)]\|^2_{L^2}+ O(R^{-2}).
T_{\Omega}u_h(x)=\min\{h,\max\{T_{\Omega}u(x),-h\}\}, &T_{\Omega^*\setminus\overline{\Omega}}\,v_h(x)=\min\{h,\max\{T_{\Omega^*\setminus\overline{\Omega}}\,v(x),-h\}\}.
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}})+f_{x}(x,y,t)
\partial_{t}(T(y_{1})-T(y_{2})) =-\int_{t}^{\infty}\cos((t-x)\sqrt{\omega})F_{2}(y_{1}-y_{2}) dx &+\int_{t}^{\infty}\cos((t-x)\sqrt{\omega}) \mathcal{F}(\sqrt{\cdot}(F_{3}(y_{1})-F_{3}(y_{2})(x,\cdot \lambda(x))))(\omega \lambda(x)^{2}) dx
\left\Vert \left( \mathbb{L}+i\mu\right) s_{n}-f\right\Vert =\left\Vert(I+R_{\mu})g_{n}-f\right\Vert & \leq||(I+R_{\mu})\left( g_{n}-g\right) || & \leq\left\Vert I+R_{\mu}\right\Vert_{op}||g_{n}-g||\rightarrow0\text{as}n\rightarrow\infty.
\left( \sum_{j=1}^{\infty}\vert x^*(T(x_j)) \vert^{p}\right)^{\frac{1}{p}}\leq C \cdot \left(\int_{B_{F^{^{**}}}}\vert \varphi(x^*)\vert^{p}d\mu(\varphi) \right)^{\frac{1}{p}},
\Delta (U_1^2+U_2^2-\xi) =\Delta (U_1^2+U_2^2)-\Delta\xi &\geq \left(1-\frac{\epsilon}{2}\right)\lambda_0(U_1^2+ U_2^2)-\left(\sigma^2\xi-\dfrac{\sigma\xi}{|x|}\right) &\geq \left(1-\frac{\epsilon}{2}\right)\lambda_0(U_1^2+ U_2^2)-\sigma^2\xi.
R\left( z\,\,\vline\begin{tabular}{ccc}$\omega_1$ $\dots$ $\omega_m$ $\varrho_1$ $\dots$ $\varrho_m$ \end{tabular}\right) =(-1)^{\sigma-1}\Gamma(\varrho_1)\dots\Gamma(\varrho_m)\sum^\infty_{l=0}b_l\frac{(\log(1-z))^{\sigma+l-1}}{\Gamma(\sigma+l)}&=\frac{\Gamma(\varrho_1)\dots\Gamma(\varrho_m)}{\Gamma(\sigma)}z^{\sigma-1}+\dots
K_2=\mathbb{E}\vert \int_{0}^{t}\int_{B_{M_{{\mathcal{P}}}(r)}}{c}(z,{X}^{{{{\mathcal{P}}}},{M_{{\mathcal{P}}}}}_{\tau^{{\{\ \mathcal{P}}}}(r)-})-{c}(z,{X}^{M_{{\mathcal{P}}}}_{r-}){N}(dz,dr)\vert.
\sum_{1\leq j_1\leq\cdots\leq j_{t+1}\leq n}{d_{j_1}+\cdots + d_{j_{t+1}}-1\choose n-1}=\begin{cases}\operatorname{rk}F_{n-t-1}= f_{n-t-1}t\geq n-m-1, 0 t<n-m-1.\end{cases}
\min_{u\in U}\varrho^{(N,J)}_K[u,X] \leq\varrho^{(N,J)}_K[u_{E},X] \leq \varrho^{(N)}_E[u_{E},X] + Lh_N^\alpha & = \min_{u\in U}\varrho^{(N)}_E[u,X] + Lh_N^\alpha\quad
C^{FB,A.2}_{X^n\rightarrow{Y^n}}=\sum_{y_{-1}\in\{0,1\},y_{-2}\in\{0,1\}}C_t(y^{-1}_{-2})\mu(y_{-2}^{-1}),~\mu(y_{-2}^{-1})~\mbox{is fixed}.
\frac{d}{dt}v_t^n = A_n v_t^n + F(v^n_s + z_s, u_s), \frac{d}{dt}v_t^{n,\varepsilon}= A_n v_t^{n,\varepsilon}+ F(v^{n,\varepsilon}+ z_s, u^\varepsilon_s)
\left( Q_{2}^\dagger \right)^{31}Q_{2}^{13}\u_A ({r}) ={1 \over m}\left[ -{\partial^2 \over \partial r^2}-{1 \over r}{\partial \over \partial r}+{\left( 2 \ell + 1 - \lambda \right)^2 \over r^2}+ \lambda{\delta (r) \over r}\right] u_A = E u_A ({r})
W_{2}(\mu_{1}(t),\mu_{2}(t)) =W_{2}(\mu(t,-n,\mu_{1}(-n)),\mu(t,-n,\mu_{2}(-n))) & \leq \left(\mathbb E\|u(t,-n,\zeta_{n,1}) -u(t,-n,\zeta_{n,2})\|^{2}\right)^{1/2}& \leq{\rm{e}}^{-(\lambda_{*}-\lambda_{f}-\frac{L_{g}^{2}}{2})(t+n)}\left(\mathbb E\|\zeta_{n,1}-\zeta_{n,2}\|^{2}\right)^{1/2}\rightarrow0{\rm{as}}~n\rightarrow\infty.
f_{\mathrm{CPF}}^* = \frac{L^*}{R_{\mathrm{CPF}}(P_{sm})}f_i^* = \frac{L^*}{R_{i}(P_{r_i})}L^* = \frac{1}{\frac{1}{R_{\mathrm{CPF}}(P_{sm})}+\sum_{i=1}^{M}\frac{1}{R_{i}(P_{r_i})}}
A_1(t,x):= \alpha(t+\theta)^{-\alpha-1}[\Phi(x-\underline h(t))+\Phi(-x-\underline h(t))-\mathbf{u}^*], A_2(t,x):= - (1-\epsilon) \delta'(t)[\Phi'(x-\underline h(t))+\Phi'(-x-\underline h(t))] &+(1-\epsilon)[F(\Phi(x-\underline h(t)))+ F(\Phi(-x-\underline h(t)))]- F(\underline U(t,x)).
E_{t-1}(H_{m,t}|z_t) \leq a_{0m}+|a_{2m}-a_{1m}|M^2 &+\sum_{z_{t-1}=1}^{K}{{[a_{1m}+|a_{2m}-a_{1m}|(\delta+\frac{1}{2})]p(z_{t-1}|z_t)}E_{t-2}[H_{z_{t-1},t-1}|z_{t-1}]}&+\sum_{z_{t-1}=1}^{K}{b_mp(z_{t-1}|z_t)E_{t-2}(H_{m,t-1}|z_{t-1})},
g(A) =\sum_{r=1}^{a-1}\left(\sum_{i=1}^{k}x_i\right)_r+\frac{(a-1)(a+d-1)}{2}&=\sum_{i=1}^{m-1}i(2^k-1)+m(2^{k-1}-1)+\sum_{r=1}^{a-1}\left(\sum_{i=1}^{k-1}x_i\right)_r+\frac{(a-1)(a+d-1)}{2}.
\|f_n-f_m\|_{\alpha}= \left( 2\|f_n-f_m\|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{q_1}(f_n-f_m)\|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{q_2}(f_n-f_m)\|_{L^2 (\mathbb{R}^2)}^2 \right)^{1/2}< \epsilon
\alpha\circ P_{d}(L_{m}) =\alpha( P_{d}(L_{m}))=\alpha(f_1(m+d)L_{m+d}+g_1(m+d)W_{m+d}) &=f_1(m+d)(q^{m+d}+q^{-(m+d)})L_{m+d}+g_1(m+d)(q^{m+d}+q^{-(m+d)})W_{m+d}, P_{d}\circ\alpha(L_{m}) =P_{d}(\alpha(L_{m}))=P_{d}((q^{m}+q^{-m})L_{m}) &=f_1(m+d)(q^{m}+q^{-m})L_{m}+g_1(m+d)(q^{m}+q^{-m})W_{m}.
\sum_{i=1}^n\{\tau^{-1/2}e_1 \lambda_i\}^*e_2\{\tau^{-1/2}e_1\lambda_i\}={\tau}^{-1}\sum_{i=1}^n{\lambda_i}^*e_1e_2e_1\lambda_i ={\tau}^{-1}\tau\sum_{i=1}^n{\lambda_i}^*e_1\lambda_i =1~~~ (\textrm{since}~~\{\lambda_i\}~~ \textrm{is a basis}).
\tilde{t}^{2}- \tilde{w}^{2}- \tilde{x}^{2}- \tilde{y}^{2}- \tilde{z}^{2}= \left( \frac{\tilde{t}- \tilde{w}}{\lambda}\right) k_{1}\left[ h^{- 1}\left( \frac{\tilde{t}- \tilde{w}}{\lambda}\right) \right] .
\sqrt{\langle \rho^{2}\rangle}\sim \sqrt{N_{4}g_{s}l_{s}}\sqrt{l_{s}^{2}/ r_{\perp 0}}= ( g_{s}N_{4})^{1 / 3}l_{s}\sqrt{\frac{( g_{s}N_{4})^{1 / 3}l_{s}}{r_{\perp 0}}}.
\mathsf{F}(\cdot)=\mathring{\mathbb{G}}(\cdot)\mathring{\mathbb{F}}(\cdot)\sim \frac{1}{(N\eta)^{n+2}}\Big(1+\frac{M\eta}{\sqrt{M}}\sum_{j=1}^W \mathsf{l}_j(\mathring{x}_{j,1},\mathring{x}_{j,2},\mathring{v}_j)+\cdots\Big) &\times \Big(1+\frac{M\eta}{\sqrt{M}}\sum_{j=1}^W \mathsf{l}'_j(\mathring{b}_{j,1},\mathring{b}_{j,2}, \mathring{t}_j)+\cdots\Big),
H_\mu(m,Y^{\mu,t}(b,z)m') =H_\mu(Y^{\mu,s}(A(s_0,z)\prod_{n=1}^{\infty}e^{\frac{-2(t-s)}{n}(-z)^{-n}a_n}b,z^{-1})m,m') &=H_\mu(Y^{\mu,s}(e^{zL(s_0)_1}z^{-2L(s_0)_0}g\prod_{n=1}^{\infty}e^{\frac{-2(t-s)}{n}(-z)^{-n}a_n}b,z^{-1})m,m') &=H_\mu(Y^{\mu,s}(e^{zL(s_0)_1}\prod_{n=1}^{\infty}e^{\frac{-2(t-s)}{n}z^{n}a_n}z^{-2L(s_0)_0}g(b),z^{-1})m,m').
\{\; \stackrel{1}{U}( x , \lambda ) \; , \; \stackrel{2}{U}( y , \mu ) \; \}= \,{[}\; r ( \lambda , \mu ) \; , \; \stackrel{1}{U}( x , \lambda ) \; + \stackrel{2}{U}( x , \mu ) \;{]}\; \delta ( x - y )
\begin{array}{c}{\dot{x}^{A}= \mu p^{A}\ , \quad \dot{p}_{A}= 0 \ ,}\\{\dot{Z}_{i}^{A}= 0 \ , \quad \dot{\psi}_{a}= 0 \ , \quad ( m o d u l o \t r a n s f o r m a t i o n s \ ( \ref{e q r e l}) )}\\ \end{array}
\frac{dy}{dx}=-\frac{1}{\sqrt{1-(\frac{1}{x})^{2}}}\cdot(-\frac{1}{x^{2}})=\frac{1}{x^{2}\sqrt{1-\frac{1}{x^{2}}}}=\frac{1}{x^{2}\frac{\sqrt{x^{2}-1}}{\sqrt{x^{2}}}}=\frac{1}{\sqrt{x^{2}}\sqrt{x^{2}-1}}=\frac{1}{|x|\sqrt{x^{2}-1}}
\mathcal{L}_{\Pi\alpha}\nu^\prime = \mathcal{L}_{\Pi\alpha}f\nu &= f\mathcal{L}_{\Pi\alpha}\nu+ ((a\circ \Pi)(\alpha)f)\nu &= f\big(\iota_\alpha\widetilde{\mathcal{M}}_\Pi^\nu+ (-1)^{n-1}i_{d_{A}\alpha}\Pi \big)\nu+ ((a\circ \Pi)(\alpha)f)\nu
{}_2F_1\bigg[\begin{matrix}\alpha \alpha &1\end{matrix}\bigg|\,z\bigg]=(1+z)^{-\alpha}{}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12+\frac12\alpha &1\end{matrix}\bigg|\,\frac{4z}{(1+z)^2}\bigg].
t_1-t_0={1\over mn}-{1\over m^2 n t_0}+O\Bigl({1\over n\log^2{n}}\Bigr), begin{align*}5pt] t_2-t_1={1\over mn}-{1\over m^2 n t_0}+O\Bigl({1\over n\log^2{n}}\Bigr), begin{align*}5pt] 2t_1-t_0-t_2={1\over m n^2}+O\Bigl({1\over n^2\log{n}}\Bigr), begin{align*}5pt]{1\over t_0}+{1\over t_2}-{2\over t_1}=O\Bigl({1\over n^2\log^2{n}}\Bigr).
\sum_{j=0}^{a-1}\left(\binom{a-1}{j}-1\right)\sum_{i=1}^j\sum_{r=0}^{b-2-a}\binom{b-1-a}{r}=\left(2^{b-1-a}-1\right)\sum_{j=0}^{a-1}\left(\binom{a-1}{j}-1\right)j &=\left(2^{b-1-a}-1\right)\left((a-1)2^{a-2}-\binom{a}{2}\right)
d V \epsilon_{\alpha \beta \gamma \delta}\Phi_{\alpha}\frac{\partial{( \Phi_{\beta}, \Phi_{\gamma}, \Phi_{\delta})}}{\partial ( x , y , z )}= 3 d \eta \hspace{0 . 5 m m}d \xi \hspace{0 . 5 m m}d \phi \hspace{0 . 5 m m}m n \frac{d}{d \eta}\biggl ( \frac{1}{1 + f^{2}( \eta )}\biggr ) ,
2 \sum_{2 \leq q_1 \leq X}\sum_{q_2 > X}\frac{\Lambda(q_1)\Lambda(q_2)}{\phi(q_1)\phi(q_2)^2}= 2 \bigg( \sum_{2 \leq q_1 \leq X}\frac{\Lambda(q_1)}{\phi(q_1)}\bigg) \bigg( \sum_{q_2 > X}\frac{\Lambda(q_2)}{\phi(q_2)^2}\bigg) \ll{\log X}\cdot \frac1{X}.
P_{0, 4}(\lambda; \tau, \eta) = - 2\pi \frac{\Gamma(-6\eta; \tau, -8\eta)}{\Gamma(-2\eta; \tau, -8\eta)}\frac{1}{\theta_0(4\eta; \tau)(\tau;\tau)^3}\frac{(-4\eta; -4\eta)}{(-2\eta; -4\eta)}.
\begin{gathered}\sum_{M=0}^\infty\sum_{N=\lfloor w \rfloor+M}^\infty N^k e^{w -N}\leq \sum_{M=0}^\infty (M+1)(w + M)^k e^{-M}\lesssim \sum_{M=0}^\infty M^{k+1}e^{-M}\end{gathered}
\langle d_jf,g\rangle_{X^{j+1}}= \frac12 \sum_{s\in X^{j+1}}m(s)df(s)\overline{g(s)}= \frac12 \sum_{s\in X^{j+1}}m(s)\overline{g(s)}\sum_{r\subset s}f(r) = \frac12 \sum_{r\in X^{j}}m(r)f(r) \overline{\sum_{r\subset s}\frac{m(s)}{m(r)}g(s)}\ .
\begin{aligned}y^3 = \, \, x^{4}+ 2 \cdot 7 \cdot 67 \cdot 179 \, x^{2}+ 2^3 \cdot 3^3 \cdot 5 \cdot 67 \cdot 137 \, x + 5^2 \cdot 7 \cdot 67^2 \cdot 71 \cdot 89 begin{align*}8pt]y^{3}= \, \, x^{4}+ \left(12222 v^{2}- 263088 v - 1290744\right) x^{2}+ \big(-19721880 v^{2}+ 232016400 v & + 1277237160\big) x + 11453819175 v^{2}- 62791404525 v - 447679991475 \,.\end{aligned}
V_1(\mu,\eta_1,\eta_2,t)= \exp\left\{-(\eta_1^2+\eta_2^2) \right\}\sum\limits_{r^2_s+q^2_s\neq 0}a'_s \eta_1^{m_s}\eta_2^{n_s}t^{l_s}\mu^{r_s}\ln^{q_s}\mu,
\mathbf{F}_{B}^{A}\mathbf{e}^{BA^{\prime}}+\mathbf{F}_{B^{\prime}}^{A^{\prime}}\mathbf{e}^{AB^{\prime}}=0, \mathbf{DF}_{B}^{A}=\mathbf{DF}_{B^{\prime}}^{A^{\prime}}=0,
\left[ k\right]_{q}\left[ k+1\right]_{q}...\left[ k+m-1\right]_{q}={\displaystyle\prod\limits_{s=0}^{m-1}}\left( q^{s}\left[ k\right]_{q}+\left[ s\right]_{q}\right) =\sum_{s=1}^{m}S_{q}\left( m,s\right) \left[ k\right]_{q}^{s},