harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{P_c}:=\{\lambda \in P\mid 0\leq \langle \lambda,\beta\rangle\leq c,\ \forall\beta\in{\hat R_0^+}\}, \hat P_c:={\{\mu\in \hat P\mid 0\leq \langle \mu ,\alpha\rangle \leq c,\, \forall \alpha\in R_0^+\}},
(f)_{i,j}\in \begin{cases}\mathbb{Z}, i \leq n-1 j \geq i p\mathbb{Z}, i \leq n-1 i \geq j+1 \mathbb{Z}, i \geq n j \geq i+1 p\mathbb{Z}, i \geq n i+1 \geq j+1.\end{cases}
\Gamma_{\{\ell+1,\ell+2\}}^{2}+\Gamma_{[\ell+2]\setminus \{\ell+1\}}^2=\Gamma_{[\ell+2]}^2-\Gamma_{[\ell+1]}^2+\Gamma_{[\ell]}^2+\Gamma_{\{\ell+2\}}^2+\Gamma_{\{\ell+1\}}^2-1/4.
\gamma \le s \frac{\binom{n/2}{s/2}^2}{\binom{n}{n/2}}\le s \frac{1}{2\pi}\frac{n/2}{s/2 (n-s)/2}2^{n h_2(s/n)}\left( \sqrt{\frac{n}{8 (n/2)^2}}2^n \right)^{-1}\le \sqrt{\frac{2n}{\pi^2}}2^{- n( 1- h_2(s/n) )},
E_1 :=\{(\lambda, x,t)\in (0,4r)\times 2 \Delta:\ \eta\lambda/32\leq\delta(x,t)\leq \eta\lambda/8\}, E_2 :=\{(\lambda,x,t) \in (2r,4r)\times 2\Delta: \delta(x,t) \leq \eta \lambda/8\}, E_3 :=\{(\lambda,x,t) \in (\epsilon,2\epsilon)\times 2\Delta: \delta(x,t) \leq \eta \lambda/8\}.
0 = \nabla_{( \alpha}\overline{{\xi}}_{\overline{{\beta}}) a}= \partial_{( \alpha}\overline{{\xi}}_{\overline{{\beta}}) a}- H_{\alpha \overline{{\beta}}}^{\, \, \, \overline{{\gamma}}}\overline{{\xi}}_{\overline{{\gamma}}a}+ H_{\alpha \overline{{\beta}}}^{\, \, \, \gamma}\xi_{\gamma}.
sB_{\mu \nu}^a=\varepsilon_{\mu \nu \lambda \rho}\left(D^\lambda \right)_{\;\;b}^a\eta^{\rho b},\;sA_\mu^a=0,\;s\varphi^{(1)a}=\left( D_\mu \right)_{\;\;b}^a\eta^{\mu b},
R^{SS}_N(z) = \frac{2\,(-1)^{N+1}\,z}{(2\pi z)^{2N-2}}\sum_{n=1}^{\infty}\frac{1}{n^{2N-2}}\int_0^{\infty}dy \; \frac{y^{2N-2}\, e^{-y}}{\left( y^2 + 4 \pi^2 n^2 z^2 \right)}\;,
{\Pr}\bigg({x_R}= 1\bigg)=\Pr \bigg(x_T= 1 \bigg){\Pr}\bigg({{x}_R}= 1 ~\bigg|~{x_T}= 1\bigg)+ \Pr \bigg(x_T= 0 \bigg){\Pr}\bigg({{x}_R}= 1 ~\bigg|~{x_T}= 0\bigg).
\Psi^{(s)}_{i}(\mathbf{x})=e^{i\mathbf{p}_{i}.\mathbf{x}}u_{i}=\frac{e^{ip^{\bot}\rho \cos \varphi}}{\sqrt{4\pi}\sqrt{2s}}\left(\begin{array}{c}\sqrt{E+sm}\sqrt{s+1}\epsilon_{3}\sqrt{E-sm}\sqrt{s-1}\epsilon_{3}\sqrt{E+sm}\sqrt{s-1}\sqrt{E-sm}\sqrt{s+1}\end{array}\right) ,
\frac{d}{d\alpha}\varphi(\alpha)=\int_{0}^{1}\frac{\partial}{\partial\alpha}(\frac{\alpha}{x^{2}+\alpha^{2}})dx=\int_{0}^{1}\frac{x^{2}-\alpha^{2}}{(x^{2}+\alpha^{2})^{2}}dx=-\frac{x}{x^{2}+\alpha^{2}}|_{0}^{1}=-\frac{1}{1+\alpha^{2}},
F(t) = \prod_{j=1}^{\frac{n+1}{2}}(t - \lambda_{2j+1}) + \frac{4}{n + 3}\sum_{j=1}^{\frac{n+1}{2}}\sin^{2}\left[\frac{(2j + 1) \pi}{n + 3}\right] (\lambda_{2j+1}- \lambda_{1}) \prod_{\substack{m=1 m \neq j}}^{\frac{n+1}{2}}(t - \lambda_{2m+1})
\prod_{i=1}^k v_i \leq \prod_{i=1}^k u_i = \left( \prod_{i=1}^{\ell-2}u_i \right) \left( u'_{\ell-1}u'_{\ell}\right) \left( \prod_{i= \ell+1}^k u_i \right) = \prod_{i=1}^k u'_i.
(\mathbb{V}, \phi)= \bigoplus (\mathcal{F}_i, \phi)^{f_i}\oplus \bigoplus ((\mathcal{E}_i, \phi)\oplus (\mathcal{E}^*_i, \phi))^{e_i}\oplus \bigoplus ((\mathcal{S}_i, \phi)\oplus (\mathcal{S}^*_i, \phi))^{s_i}
0 =\theta_t^i u_i'(x_t^i)-\lambda_t, t=0,1,\ldots, 0 =\theta_t^i\delta^i-\mu_t\theta_{t+1}^i, t=0,1,\ldots, 0 =z_t^i - V_{\theta^i}(k_{t+1},\theta_{t+1}), t=0,1,\ldots, 0 =-\lambda_t + \mu_t V_k(k_{t+1},\theta_{t+1}), t=0,1,\ldots,
DS_k (C,X) = (\partial_{c_1}S_{k},\partial_{x_1^{(1)}}S_{k},...,\partial_{x_1^{(n)}}S_{k},\partial_{c_2}S_{k},...,\partial_{x_k^{(n)}}S_{k}) &= (s(x_1), c_1\partial_1 s|_{x=x_1},...,c_1\partial_d s|_{x=x_1},s(x_2),...,c_1 \partial_n s|_{x=x_k}).
\frac{1}{2}\beta = \sum_{q\geq 0}d_{2q+1}^2 (2q+1)! 2^{-1-2q}\sum_{i\in\mathbb{Z}}\widetilde{\rho}(i)^{2q+1}= \sum_{q\geq 0}\ell_{2q+1}^2 (2q+1)! \sum_{i\in\mathbb{Z}}\widetilde{\rho}(i)^{2q+1}\,.
\vert\mathcal J_{1,2}\vert\le \varepsilon I_{1,n}(t)+\frac{C_2}{\varepsilon}\left\{\Vert{\mathbf F}\Vert_{L^2(\Omega_t)}^2+\Vert\varphi(t)\Vert_{L^2(\mathbb R)}^2+\int_0^t\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)}ds\right\}&+\frac{C_3}{\varepsilon}\int_0^t (I_{1,\ast}+I_{1,n})(s)ds
\begin{aligned}\mathcal{L}\phi \leq \, U_{11}\sum_{i \geq 2}F^{ii}A^{ii}_{p_1 p_1}- \psi_{p_{1}p_{1}}U_{11}+ E + \frac{C}{U_{11}}F^{ii}U_{ii}^2 + C \sum F^{ii}+ C\end{aligned}
Q\left( z \right)=\Gamma_{0}^{+}\left({\begin{array}{*{20}c}0 0 0 W \end{array}}\right)\Gamma_{0}=\Gamma_{0}^{+}\left\{P\left( A-zI \right)P-PA(I-P)\left( \tilde{A}-z \right)^{-1}(I-P)AP \right\}^{-1}\Gamma_{0}.
{\cal S}[ e , \psi ] = \int d^{4}x \sqrt{g}\left\{\frac{i}{2}[ \overline{{\psi}}\gamma^{\hat{\alpha}}D_{\hat{\alpha}}\psi - ( \overline{{D_{\hat{\alpha}}\psi}}) \gamma^{\hat{\alpha}}\psi ] - m \overline{{\psi}}\psi \right\}
i \int_0^t\Delta I_h z(s) \d s= I_h z(t)-I_h z_0+\int_0^t I_h(i\nu z(r) +\epsilon (\gamma z(r) -\mu \overline{z}(r)))\d r +\tfrac{1}{2}\int_0^t I_h(z(r)F_\Phi )\d r -i \kappa \int_0^t I_h(\theta_R(|z|_{X^{\mathfrak{s}}_r})|z(r)|^2z(r)) \d r +i \int_0^t I_h(z(r)\cdot) \d W(r)
\widetilde{P}_{n,m}(r)=\left[\omega rL^{-\alpha+1}_n(y)L^{\alpha}_m(-y)+\frac{2}{r}(m-l-\frac{3}{2})L^{\alpha-1}_m(-y)L^{\alpha}_n(y)\right]_{y=\frac{1}{2}\omega r^2},
\int \phi_m(x) q_2(x) dP_0^m(x) = \int \int \phi_m(x) q\left(\frac{x- w}{\sqrt{2}}\right)q\left(\frac{x + w}{\sqrt{2}}\right) dP_0^m(x) \, dP_0^m(w) = \bar{G}\int \Pi_{j=1}^{m}\phi_1\left( x^j\right) q_2(x) dP_0^m(x).
[ \hat{G}_{1}^{\prime}, \hat{G}_{2}^{\prime}] = 2 i{\bf 1}, [ \hat{G}_{3}^{\prime}, \hat{G}_{4}^{\prime}] = - 2 i{\bf 1}, [ \hat{N}_{1}^{\prime}, \hat{N}_{2}^{\prime}] = 2 i{\bf 1},
\min\limits_{\sum_{i=1}^2\alpha_k=1}\sum_{i=1}^2\frac{(2^{\frac{\bar R_i}{\alpha_i}}-1)\alpha_i}{g_i}\ge \frac{2^{\bar R_2}-1}{g_2}+\frac{2^{\bar R_2}(2^{\bar R_1}-1)}{g_1},
|f_{\nu,L}(X_1,\dots, X_n)-f_{\nu,L}(X_1',\dots, X_n')|\leq &4\|C_r\|_{\infty}^k \|P_r\|_1 (k+2)(3\delta)^{k-1}\frac{\|\Sigma\|_{\infty}^{1/2}+\sqrt{2\delta}}{\sqrt{n}}\biggl(\sum_{j=1}^n \|X_j-X_j'\|^2\biggr)^{1/2},
{y_i^T\over i}={y_{km+n}^T\over km+n}\le{y_{km}^T+ y_n^T+ 1\over km+n}\le{k y_{m}^T+ y_n^T+ k\over km+n}={k y_{m}^T\over km+n}+{y_n^T\over km+n}+{k\over km+n}\,.
\Psi ( t , \theta , \phi ) = e^{- i E t}\exp{i ( n + \frac{1}{2}- \frac{1}{2}\sigma_{_{3}}) \phi}\left( \begin{array}{c}{A J_{_{| \frac{1}{\beta}( n + \frac{1 - \beta}{2}) |}}( \alpha c r )}\\{B J_{_{| \frac{1}{\beta}( n + \frac{1 - \beta}{2}) + 1 |}}( \alpha c r )}\\ \end{array}\right) ,
S_{\nu \mu_1}= zm(z) S_{\mu_1 \mu_1}^{(\nu)}\sum_{p,q}S_{pq}^{(\nu \mu_1)}(Y_{\gamma-1}^*)_{\nu p}(Y_{\gamma-1})_{q\mu_1}+z(S_{\nu \nu}-m(z)) S_{\mu_1 \mu_1}^{(\nu)}\sum_{p,q}S_{pq}^{(\nu \mu_1)}(Y_{\gamma-1}^*)_{\nu p}(Y_{\gamma-1})_{q\mu_1}.
\oplus_{\mu\colon \rho(\mu)=\lambda}\Delta_{\mu}^{m_\mu}\twoheadrightarrow \imath_{\leq \lambda}\imath^{*}_{\leq \lambda}M / \imath_{<\lambda}\imath^{*}_{<\lambda}M = (M)_{\not<\lambda}/ (M)_{\not\leq \lambda},
f_{u_1}=(f_1)_{u_1}\partial_{x_1}+(f_2)_{u_1}\partial_{x_2}+(f_3)_{u_1}\partial_{x_3}=\frac{(f_1)_{u_1}}{f_1}X_1+(f_3)_{u_1}X_2+\sqrt{L}\left[\frac{(f_2)_{u_1}}{f_1}-(f_3)_{u_1}\right]\widetilde{X_3},
\tilde{\Phi}(\epsilon\eta^{m-1}\phi(|\nabla \overline{u}||\nabla \overline{u}|)) \le \max\{(\epsilon\eta^{m-1})^{\ell'},(\epsilon\eta^{m-1})^{m'}\}\tilde{\Phi}(|\nabla \overline{u}||\nabla \overline{u}|) = g_2(x,\epsilon) \tilde\Phi(\phi(|\nabla \overline{u}|)|\nabla \overline{u}|),
\begin{array}{c c c c c}{{\pi_{\tau \tau}\approx 0 ,}}&{{\pi_{\tau \sigma}\approx 0 ,}}&{{\pi_{\sigma \sigma}\approx 0 ,}}&{{\chi_{1}\left( P , X^{\prime}\right) \approx 0 ,}}&{{\chi_{2}\left( P , X^{\prime}\right) \approx 0 .}}\\ \end{array}
(2)_{t}^{2}(3)_{t}\cdot(6)_{t}\cdot(6)_{t}\cdot(2)_{t^{2}}^{2}(3)_{t^{2}}\cdot(6)_{t^{2}}\cdot(2)_{t^{3}}(6)_{t^{3}}\cdot(2)_{t^{3}}^{2}(3)_{t^{3}}
\limsup_{n \to \infty}n^{-1-\alpha-\beta}\sum_{k=1}^n k^\alpha (n-k+1)^\beta =\limsup_{n \to \infty}n^{-1-\alpha-\beta}\sum_{k=1}^n k^\alpha (n-k)^\beta &=\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^n \left(\frac{k}{n}\right)^\alpha \left(1-\frac{k}{n}\right)^\beta &=\int_0^1x^\alpha(1-x)^\beta dx \in (0,1)
h_{\alpha \beta}^{( \beta )}( t ; x , y ) = \sum_{m = - \infty}^{\infty}\left( \sum_{n = 1}^{\infty}\sum_{\sigma = \pm}e^{- 2 n E t}{\psi_{n , m}^{( \sigma )}( x )_{\alpha}{\psi}_{n , m}^{( \sigma )}( y )_{\beta}^{*}}+ \phi_{m}^{( + )}( x )_{\alpha}\phi_{m}^{( + )}( y )_{\beta}^{*}\right)
L\big(\tau^i,x^\mu,x^\mu_i\big) = -T\sqrt{-D}= -T \sqrt{-\det \big(g_{\mu\nu}x^\mu_i x^\nu_j \big)}= -T\sqrt{(g_{\alpha\beta}g_{\mu\nu}-g_{\alpha\mu}g_{\beta\nu}) x^\alpha_0x^\mu_0x^\beta_1x^\nu_1}.
L^{\prime}= -{\frac{1}{4 g^{4}}}t r F_{\mu \nu}F^{\mu \nu}+ L_{M A T}-{\frac{1}{2 \alpha}}t r ( \partial_{\mu}A^{\mu})^{2}+{\frac{1}{\alpha}}t r c M \overline{{c}}+ t r ( \rho^{\mu}\Delta A_{\mu}+ U \Delta \overline{{c}}) + Y^{a}\Delta \phi_{a}
\bar{\psi}^T\Omega^{-1}\bar{\psi}+ \frac{2}{3}\frac{1}{n}\sum_{i=1}^n \left( \bar{\psi}^T \Omega^{-1}\psi_i\right)^3 + \Pi^T \Omega^{-1}\Pi - \frac{1}{2}\frac{1}{n}\sum_{i=1}^n \left( \bar{\psi}^T \Omega^{-1}\psi_i \right)^4 + O_p(n^{-5/2}) \,.
\left\{\left[ \begin{array}{c}g_k(\underbar{y}^{(CR)}_k) R_k(\underbar{y}^{(CR)}_k) \end{array}\right] : \underbar{y}^{(CR)}_k\in \mathbb{R}^{N^{(CR)}_k}\right\}= \left\{\left[ \begin{array}{c}\underbar{b}_k \widehat{R}_k(\underbar{a}^{(CR)}_k) \end{array}\right] : \underbar{b}_k\in \mathcal{B}_k \right\}.
\mathbb{E}\left[ \left| f'(W_1)\right|^2\right] \leq 2\left(\int_{\sqrt{2a}}^{+\infty}(-a x^{-a-1}+\frac12 x^{1-a})^2 dx + \int_{-\infty}^{\sqrt{2a}}|g'(x)|^2 \phi(x) dx\right) & = C'\left( \int_{\sqrt{2a}}^{+\infty}(x^{-2a-2}+ x^{-2a+2}+ x^{-2a})dx +1 \right)<+\infty, \; \textrm{since}a> \frac32.
\frac{p\binom{n}{r}-i+1}{q\binom{n}{r}-i+1}\ge \frac{p\binom{n}{r}-n+1}{q\binom{n}{r}-n+1}& \ge \frac{pn/2-1}{qn/2-1}& = \frac{p}{q}\left(1-\frac{q/p-1}{qn/2-1}\right) & > \frac{p}{q}\left(1-\frac{2/p}{n}\right)\;.
|G_0(j,j; E)| + \sum_{r = 1}^n 2^{r-1}|p_r| |g_{r-1}(j; E)| \left(\sum_{d(j,k) \le r}\! w(k) |g_r(k; E)|^2 \right)^{1/2}\le |G_0(j,j; E)| + \sum_{r = 1}^n |p_r| 2^{\mu r}2^{r-1}|g_{r-1}(j; E)| \left(\sum_{d(j,k) \le r}\! |g_r(k; E)|^2 \right)^{1/2}.
X^{\alpha \beta}= - \partial_\mu \partial_\mu \delta^{\alpha \beta}+ r_0 \delta^{\alpha \beta}- 2 g_0 A_\mu^{\alpha \beta}\partial_\mu + v_0 T^{\alpha \beta}+ w_0 S \delta^{\alpha \beta}+ g_0^2 A_\mu^{\beta \gamma}A_\mu^{\alpha \gamma}.