harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_a(x) = \frac12\,\left(1 + \frac{4a^2}{1+2a}\,x^{\frac1{2a}}\right) + \frac{1+\frac{2a}{1+2a}\log(1+x^{\frac1{2a}})}{2(1+x^{\frac1{2a}})^{2a}}R_a(x) = \frac{\pi_{\frac1{2a}}}4 + \frac{1+2a \log(1+x^{\frac1{2a}})}{2(1+x^{\frac1{2a}})^{2a}},
\Xi_{{m}_{{1}}\ldots{m}_{{5}}}^{{( n )}}\equiv S p ( v^{{T}}\tilde{\sigma}_{{m}_{{1}}{\bf . . .}{m}_{{5}}}v \sigma^{{( n )}}) \equiv v_{\alpha}^{{a}}( \tilde{\sigma}_{{m}_{{1}}{\bf . . .}{m}_{{5}}})^{\alpha \beta}v_{\beta}^{{b}}( \sigma^{{( n )}})_{{a b}}= 0
(Au^s)_r + (Au^r)_s = 2(Au^s)_r + \sum_{i \in I}u^r_i(Au^s)_i \geq \sum_{i \in I}u^r_i(Au^s)_i = \sum_{i \in I}u^r_i(Au^s)_i + \sum_{i \in I}u^s_i(Au^r)_i.
Q_{2+k+1}(a) = \frac{\xi_{2+k+1}'(a)}{(T_a^{2+k+1})'(x)}= Q_2(a) \prod_{j=0}^k \biggl(1+\frac{\partial_a T_a(\xi_{2+j}(a))}{T_a'(\xi_{2+j}(a)) \xi_{2+j}'(a)}\biggr).
\mu(\Omega;p):=\inf_{\phi\in \mathrm{W}^{1,p}(\Omega)\setminus\{0\}}\left\{\frac{\displaystyle\int_\Omega |\nabla \phi|^p\,dx}{\displaystyle\int_\Omega |\phi|^p\,dx}\,:\,\int_\Omega |\phi|^{p-2}\,\phi\,dx=0\right\}.
(\mathfrak{I}_3+\varepsilon)^{s_c}\ge{}H(\Psi_\lambda)^{s_c}M(\Psi_\lambda)^{1-s_c}={}(\tfrac{d}{2(1-s_c)})^{s_c-1}\tfrac{d}2(\tfrac{2s_c}{d}H(\Psi_\lambda) + M(\Psi_\lambda)) \ge{}(\tfrac{d}{2(1-s_c)})^{s_c-1}\tfrac{d}2 \mathfrak{I}_2(1) ={}(\tfrac{d}{2(1-s_c)})^{s_c}\tfrac12(-g_{\min})^{s_c-\frac{d}2}\|Q\|_{L^2}^2,
S_{B u l k}= \frac{1}{4 \pi}\int d^{2}z \left( \frac{2}{\alpha^{\prime}}\partial X^{\mu}\partial X_{\mu}+ \psi^{\mu}\overline{{\partial}}\psi_{\mu}+ \tilde{\psi}^{\mu}\partial \tilde{\psi}_{\mu}\right) \ .
\mbox{$H' \neq 0$ at $s$}\; \; \Longrightarrow \;\;\mbox{$\lambda_2$ does not change sign at $s$}, \mbox{$H'=0$ \& $H''\ne 0$ \& $V'\ne 0$ at $s$}\; \; \Longrightarrow \;\;\mbox{$\lambda_2=0$ \& $\lambda_2$ changes sign at $s$}.
V_{e}={\frac{\lambda^{2}}{R}}\Delta ( 0 , \pi R ) ( ( \phi_{2}^{*}+ \bar{\phi}_{2}) T^{a}( \phi_{2}+ \bar{\phi}_{2}^{*}) ) ( ( \phi_{1}^{*}+ \bar{\phi}_{1}) T^{a}( \phi_{1}+ \bar{\phi}_{1}^{*}) )
_{r}F_{s}\left( \begin{array}{c|c}a_{1},\ldots ,a_{r}x b_{1},\ldots ,b_{s}\end{array}\right) =\sum_{k\geq 0}\frac{\left( a_{1},\ldots ,a_{r}\right)_{k}}{\left(b_{1},\ldots ,b_{s}\right)_{k}}\frac{x^{k}}{k!},
\left( \begin{array}{c c c}{{\psi_{R},}}&{{\psi_{G},}}&{{\psi_{B}}}\\ \end{array}\right) \rightarrow \left( \begin{array}{c c c}{{\psi_{R},}}&{{\psi_{G},}}&{{\psi_{B}}}\\ \end{array}\right) \exp ( - i \theta_{a}\lambda_{a}^{*}/ 2 ) ,
\langle q2',t'\mid q,t \rangle = \left( \frac{-i}{2\pi\Delta t}\right)^{1/2}\exp \left\{i \frac{(q'-q)^2}{2\Delta t}\right\}\left\{1- \sum_{n=1}^{\infty}(i \Delta t)^n Y_n(q',q) \right\}
-\frac{1}{n}\log{{E}}e^{-nh(\boldsymbol{X}^{n},Y^{n})}=\inf_{\boldsymbol{\varphi}^{n}=(\varphi_{k}^{n})_{k\in \mathbb{N}_{0}}\in \bar{\mathcal{A}}_{b}}{{E}}\left\{\sum_{k=0}^{\infty}\int_{[0,T]\times \lbrack 0,1]}\ell(\varphi_{k}^{n}(s,y))\,ds\,dy+h(\bar{\boldsymbol{X}}^{n},{\bar{Y}}^{n})\right\}.
v o l ( G ) = \int_{G}\D g = \int_{\raise 1 p t \mathrm{\scriptstyle G}/ \lower 1 p t \mathrm{\scriptstyle H_x}}D [ g ] v o l ( H_{x}) = v o l ( H_{x}) v o l \bigl ( \! \! \raise 4 p t \mathrm{\G}\! / \! \! \! \lower 4 p t \mathrm{\H_x}\bigr )
- \mathrm{T r}_{2 + 1}\ln ( -{\hat{\partial}}^{2}- \partial_{2}^{2}+ \kappa^{2}) + \frac{1}{2}\mathrm{T r}_{2 + 1}\ln ( -{\hat{\partial}}^{2}- \partial_{2}^{2}+ 4 \kappa^{2}) ,
\Big| \int_r^R \int_{\mathbb{R}^{n+1}}\prod_{i=0}^{n}F_i(x_0,\dots,x_{i-1},x_{i+1},\dots,x_n) & h_t(x_0+\dots + x_n)dx_0 \dots dx_n \frac{dt}{t}\Big|\leq C \Big(\log \frac{R}{r}\Big)^{1-2^{-n+1}},
x\overline{\mathbf{d}}z =\inf\{\epsilon>0:\forall r\in(0,\infty)\z^\bullet_r\subseteq x^\bullet_{r+\epsilon}\}. z\underline{\mathbf{d}}y =\inf\{\epsilon>0:\forall r\in(0,\infty)\z^r_\bullet\subseteq y^{r+\epsilon}_\bullet\}.
\ln\zeta(s) = \sum_{n=2}^\infty \underbrace{\pi(n)}_{}\int_{n}^{n+1}\frac{s}{x(x^s-1)}dx &= \sum_{n=2}^\infty \int_n^{n+1}\frac{s\pi(x)}{x(x^s-1)}dxn: 2\rightarrow 3,3\rightarrow4,\dots \ln\zeta(s) = \int_2^\infty \frac{s\pi(x)}{x(x^s-1)}dx\intertext{or}\frac{\ln\zeta(s)}{s}= \int_2^\infty \frac{\pi(x)}{x(x^s-1)}dx.
\|\psi^{p}(t)\|^2 \geq 2^{J}\sum_{k=0}^{2^{J}-1}\left\{\sum_{m\in \mathbb{Z}}\psi(2^{J}t + 2^{J}m - k) \right\}^{2}= 2^{J}\sum_{k=0}^{2^{J}-1}\sum_{m\in\mathbb{Z}}\{\psi(2^{J}t + 2^{J}m - k )\}^{2}
c(d+1,d) = \frac{2}{d}c(d,d-1) = \frac{2}{d}\frac{2}{d-1}c(d-1,d-2) = \cdots = \frac{2}{d}\frac{2}{d-1}\cdots \frac{2}{2}\frac{2}{1}c(1,0) = \frac{2^d}{d\,!}
(\omega(a_tk)\phi, \omega(k')\phi) = \int_{\mathbb{C}^9}\omega(a_tk)\phi\cdot\overline{\omega(k')\phi}\cdot e^{-\pi |z|^2}dz = ~(\xi\bar{\xi}')^{\mu_1+\frac{3}{2}}(\eta\bar{\eta}'\zeta\bar{\zeta}')^{\mu_2+\frac{3}{2}}(\bar{\gamma}\gamma')^{\nu+\frac{3}{2}}(\cosh t)^{-\mu_2-\nu-3}J_1 J_2,
S_{k}(c;\xi)={\sum_{a\,\hbox{\tiny mod}\, k}}^{\!\!\!*}\sum_{{\tiny \begin{tabular}{c}$z\,\hbox{\tiny mod}\, k\ell$ $z=\xi(\ell)$\end{tabular}}}e_{k\ell}\big(a\ell F_h(z) +c\cdot z\big) ={\sum_{a\,\hbox{\tiny mod}\, k}}^{\!\!\!*}\;\sum_{z'\,\hbox{\tiny mod}\, k}e_{k\ell}\big(a\ell F_h(\xi+\ell z') +c\cdot (\xi+\ell z')\big).
U_{1}(n)=\{g :\, \textrm{$g$ square-free, $\deg g =n$}\}, & U_{2}(n)=\{x^{2}g :\, \textrm{$g$ square-free, $\deg g =n-2, x\nmid g$}\}, U_{3}(n)=\{(x+1)^{2}g :\, \textrm{$g$ square-free, $\deg g= n-2, x+1\nmid g$}\}, U_{4}(n)=\{x^{2}(x+1)^{2}g :\, \textrm{$g$ square-free, $\deg g= n-4, x\nmid g, x+1\nmid g$}\}.
S^*_{n}(qm_1+m_2) =\sum_{\substack{k=1, (k,p)=1}}^{qm_1+m_2}k^n = \sum_{\substack{k=1, (k,p)=1}}^{qm_1}k^n+\sum_{\substack{j=1, (j,p)=1}}^{m_2}(qm_1+j)^n & \equiv q S^*_{n}(m_1)+S^*_{n}(m_2) \pmod{p^d},
\begin{array}{c}E_{1,2}^2=p_{\bot}^2+m^2+ \frac{g^2}4\left( \tau_1+\tau_2\right) \pm g\sqrt{\tau_1p_1^2+\tau_2p^2+ \frac{\left( H_z^3\right)^2}4}E_3^2=p_{\bot}^2+m^2,\ \ \left( p_{\bot}^2=p_1^2+p_2^2\right) . \end{array}
I_{\mathrm{g}}( \tau ) = ( \det P^{+}P )^{1 / 2}( \det \Delta_{\mathrm{g}})^{- 1 3}= e^{c ( 2 \mathrm{g}- 2 )}\left( \frac{d}{d s}Z ( s ) |_{s = 1}\right)^{- 1 3}Z ( 2 ) \, ,
G_n=\left| \begin{array}{llcl}p_1p_1 p_1p_2 \cdots p_1p_n p_2p_1 p_2p_2 \cdots p_2p_n \multicolumn{4}{c}\dotfill p_np_1 p_np_2 \cdots p_np_n\end{array}\right| ,
R_{\Delta_k}= k^{j_m}K_{\Delta_k}(\mathbf y ;m) (\nabla^2 k) \cdot g^{-1}+ k^{j_m-1}H_{\Delta_k}(\mathbf y_1, \mathbf y_2 ;m) (\nabla k \nabla k) \cdot g^{-1},
\Delta_{\mu \nu \pm}( k ) = - \frac 1{k^{2}}\Big ( g_{\mu \nu}- ( 1 - \alpha ) \frac{k_{\mu}k_{\nu}}{k^{2}}\mp 2 \beta \frac{\tilde{k}_{\mu}\tilde{k}_{\nu}}{( \tilde{k}^{2})^{2}( k^{2}\pm \frac{2 \beta}{\tilde{k}^{2}})}\Big ) .
D_0(z) = \begin{pmatrix}e^{\frac{2}{3}(2\varphi_{1}(z)+\varphi_{2}(z))}0 0 0 e^{\frac{2}{3}(\varphi_{2}(z)-\varphi_{1}(z))}0 0 0 e^{-\frac{2}{3}(\varphi_{1}(z)+2\varphi_{2}(z))}\end{pmatrix}.
\left( \mathcal{N}+1\right)^{\beta}\mathcal{\bar{A}}f=\mathcal{\bar{A}}\left( \left( \mathcal{N}+l\right)_{+}+1\right)^{\beta}f\text{for all}f\in D\left( \mathcal{N}^{\beta+\frac{k}{2}}\right) .
\rho^{(1)}_{\mu_1 \dots \mu_{p-1}\alpha_1}=(C \Gamma_{\nu \mu_1 \dots\mu_{p-1}})_{\beta \alpha_1}\Pi^\nu \Pi^\beta\ , \ \rho^{(2)}_{\mu_1 \dots \mu_{p-1}\alpha_1}= (C \Gamma^\nu)_{\beta\alpha_1}\Pi_{\nu \mu_1 \dots \mu_{p-1}}\Pi^\beta\quad .
H_{\ell , \ell + 1}^{D}= g_{\ell + 1}^{2}H_{\ell , \ell + 1}^{A}g_{\ell + 1}^{- 2}= ( t^{- 1}S_{\ell}^{+}S_{\ell + 1}^{-}+ t S_{\ell}^{-}S_{\ell + 1}^{+}+ 2 \cos \gamma S_{\ell}^{3}S_{\ell + 1}^{3}) \ ,
| \Psi_{2}^{1 1}( p ) \rangle = \int [ 1 2 3 ] \frac{\tilde{\delta}( p - p_{1 2 3}) f_{1 2}f_{( 1 2 ) 3}}{2 p_{1 2}^{+}( p^{-}- p_{1 2 3}^{-}) ( p^{-}- p_{( 1 2 ) 3}^{-})}| 1 2 3 \rangle \; .
[\Pi_{n-1}\alpha, \Pi_{n-1}\beta]_{A}= [\Pi\alpha, \Pi]_{A}+ \Pi(\mathcal{L}_{\Pi\alpha}\beta) = [\Pi\alpha,\Pi]_{A}\beta+ \Pi(\iota_{\Pi\beta}d_{A}\alpha)+ \Pi(\mathcal{L}_{\Pi\alpha}\beta-\iota_{\Pi\beta}d_{A}\alpha) = \Pi(\mathcal{L}_{\Pi\alpha}\beta- \iota_{\Pi\beta}d_{A}\alpha) = \Pi_{n-1}(\ll \alpha,\beta\gg_\ast).
\int_{1 0}\hat{F}_{( 5 )}{}^{\star}\hat{F}_{( 5 )}= \int_{9}\left[ K^{- \frac{3}{4}}F_{( 5 )}{}^{\star}F_{( 5 )}- K^{\frac{3}{4}}F_{( 4 )}{}^{\star}F_{( 4 )}\right] ,