harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}I^{[k]}\ =\I_0^{[k]}+\sum_{\substack{p\in[s] \nu(I_p)\ge k-1}}u_pI_p^{[k-1]}, I^{[\ell]}\ =\I_0^{[\ell]}+\sum_{\substack{q\in[s] \nu(I_q)\ge\ell-1}}u_qI_q^{[\ell-1]}.\end{aligned}
\langle \frac{u}{q}\,,\,\frac{v}{q}\rangle=\langle \frac{z^Nu}{\widetilde{q}q}\,,\,v\rangle=\langle\frac{z^Nub}{\widetilde{q}}+Q_q\left(z^Nua\right)+\frac{R_q(z^Nua)}{q}\,,\,v\rangle=\langle Q_q\left(z^Nua\right)\,,\,v\rangle,
\begin{cases}\frac{\partial \mathbf{u}_0(\mathbf{x},t)}{\partial t}=-b(\mathbf{D})^*g^0b(\mathbf{D})\mathbf{u}_0(\mathbf{x},t),\mathbf{x}\in \mathcal{O},t>0, \mathbf{u}_0(\mathbf{x},0)=\boldsymbol{\varphi}(\mathbf{x}),\mathbf{x}\in \mathcal{O}, \mathbf{u}_0(\cdot ,t)\vert_{\partial \mathcal{O}}=0,t>0.\end{cases}
[\wedge^{i+1}E] \otimes [\alpha_* \omega_\alpha^0] = [\wedge^{i+1}E] \otimes [ \O + E^\vee] = [K_{i+1}(i+1)] + [\wedge^{i+1}E]\otimes[E^\vee],\ [\wedge^i E] \otimes [\alpha_* \omega_\alpha ] = [\wedge^i E]\otimes[\O + E] = K_{i+1}(i) + [\wedge^i E].
\overline{\mu_{11}}=1+\frac{1}{\pi^2}\int_{\mathbb C}d\sigma_{z_1}\int_{\mathbb C}d\sigma_{z_2}\frac{e^{i \Re(k\overline{z}_1)}}{\overline{z}-\overline{z_1}}\overline{Q}_{12}(z_1) \frac{e^{-i \Re(k\overline{z}_2)}}{z_1-z_2}{Q_{21}}(z_2) \overline{\mu_{11}}(z_2,k),
\frac{\,1\,}{\,n\,}\sum_{i=1}^{n}\mathbb E [ (\lvert X_{t+\delta, i}^{(u)}- X_{t, i}^{(u)}\rvert^{2}+ \lvert X_{t+\delta, i+1}^{(u)}- X_{t, i+1}^{(u)}\rvert^{2})^{1/2}\vert \mathcal F_{t}] \le \mathbb E \big [ \, \mathfrak f(\delta) \, \vert \, \mathcal F_{t}\big ] \, ; 0 \le t \le T-\delta \, ,
\big\{\sigma^1 > n \,,\, \sigma^2 > n \,,\, N^2_{[x,y]}> n \big\}\subseteq \big\{\sigma > n \,,\, \tau^1_{[x,y]}> n \,,\, \tau^2_{[x,y]}> n \big\}= \big\{\min \{\sigma, \tau^1_{[x,y]}, \tau^2_{[x,y]}\}> n \big\}\,.
h = \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) , \x = \left( \begin{array}{c c}{0}&{1}\\{0}&{0}\\ \end{array}\right) , \y = \left( \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\right)
\lim_{t\rightarrow 0^+}\omega_2(it)= \begin{cases}\infty, 0<|\lambda|\leq \lambda_1(\mu); i\frac{|\lambda|^2}{s(|\lambda|,0)}, |\lambda|\in (\lambda_1(\mu),\lambda_2(\mu)); 0, , |\lambda|\geq \lambda_2(\mu). \end{cases}
P_0Q(k)P_0-D_kP_0 =D_k\frac{\lvert k\lvert}{\omega(k)}P_0(\widehat{k}\cdot \nabla K(\xi-d\Gamma(m)))(1-P_0)Q(k)P_0 &+D_k\left(\frac{\lvert k\lvert}{\omega(k)}-C_\omega\right)\widehat{k}\cdot uP_0Q(k)P_0+ D_kP_0o_1(k)P_{0}.
r_{s-2}= q_s \left( r_{s-1}+ r_{s+1}\right) + r_s - q_s r_{s+1}&= q_s \left( 2 r_{s-1}- r_s \right) + r_s - q_s r_{s-1}+ q_s r_s &= q_s r_{s-1}+ r_s,
g_{\mathbb{H}}(\sigma_{\mathfrak{e}}^{-1}\eta z,\gamma_{i}^{n}\sigma_{\mathfrak{e}}^{-1}\eta z)=\log\bigg(1+\frac{1}{\sin^{2}(n\pi\slash m_{\mathfrak{e}})\sinh^{2}(\rho(\sigma_{\mathfrak{e}}^{-1}\eta z))}\bigg)\leq c_{X,\mathrm{ell}}\log\bigg(1+\frac{1}{\sinh^{2}(\rho(\sigma_{\mathfrak{e}}^{-1}\eta z))}\bigg).
\begin{array}{lll}d_v = 4\sum_{n=1}^{k-1}n + (2N-4k+1) k=2Nk-k(2k+1) , d_H = [\frac{1}{2}\times 2 \sum_{n=1}^{k-1}( (2n)^2+2n(2n+2) )+\frac{1}{2}( 2(2k)^2+ (2N-4k-2) 2k(2k+1) ) + 2k] - [2\sum_{n=1}^{k-1}(n(2n-1) +n(2n+1) ) + (2N-4k-1) k(2k+1)+2k(2k-1) ] = k.\end{array}
A = \frac{C_{l - 1}}{2 \pi}\int \frac{d t}{t}e^{- ( \frac{b^{2}t}{2 \pi \alpha^{\prime}})}( 8 \pi^{2}\alpha^{\prime}t )^{- ( \sharp N N - 1 ) / 2}\Theta_{3}( 0 , 8 i \pi^{2}\alpha^{\prime}t / L^{2}) B \times J .
\sum_{\ell = 0}^{\infty}\frac{( 2 q + 2 \ell ) \Gamma ( 2 q + \ell )}{\Gamma ( \ell + 1 )}J_{q + \ell}( z )^{2}= \frac{\Gamma ( 2 q + 1 )}{\Gamma ( q + 1 )^{2}}\left( \frac{z}{2}\right)^{2 q}
\begin{cases}\Box u + u = P(u, \partial_t\partial_xu, \partial^2_xu\,; \partial_tu,\partial_xu) & u(1,x)=\varepsilon u_0(x) & \partial_tu(1,x)=\varepsilon u_1(x)\end{cases}t\ge 1 \, , x \in \mathbb{R}
\mathop{{\rm Res}}_{s=1}\,\xi_{n,1}(s) & =\pi^{-n}\cdot 2^{-3-2n}\cdot 3^{-1}\frac{\left(\tfrac{n-5}{4}\right)!\left(\tfrac{n+3}{2}\right)!(n+1)!(n-1)!}{\left(\tfrac{n-1}{4}\right)!\left(\tfrac{n-3}{2}\right)! \left\{\left(\tfrac{n-1}{2}\right)!\right\}^2}\frac{B_{\frac{n-1}{2}}}{B_{\frac{n+3}{2}}B_{n+1}B_{n-1}}.
S = \int d^{4}x \; \mathrm{T r}\; \left\{-{\frac{1}{8}}\nabla^{\alpha \alpha^{\prime}}\phi^{a b}\nabla_{\alpha \alpha^{\prime}}\phi_{a b}+{\frac{1}{2}}\chi_{\alpha^{\prime}}^{a}\nabla^{\alpha \alpha^{\prime}}\chi_{\alpha a}+{\frac{1}{6}}G^{\alpha^{\prime}\beta^{\prime}}F_{\alpha^{\prime}\beta^{\prime}}- \phi^{a b}\chi_{a}^{\alpha}\chi_{\alpha b}\right\}\ .
{\mathcal B}= \{a:\; \;\mbox{the sequence}\; \{(h_{2r}^{11}(a))^{-2}\sum_{q = 1}^{2r}(h_{2r}^{1 q}(a))^3, \;\; r = 1, 2, 3, \dots \}\; \mbox{is constant in}\; r\}.
\Vert D_{p}g^{n}(v)\Vert \leq 2\Vert D_{p}g^{n}(v)\Vert^{\star}\leq 2\eta^{n}(\Vert v_{s}\Vert^{\star}+\Vert v_{u}\Vert^{\star}) \leq 2\eta^{n}(1+\alpha)\Vert v_{s}\Vert^{\star}& \leq 2\eta^{n}(1+\alpha)\frac{\lambda+\varepsilon}{\varepsilon}c (1-\alpha \frac{\lambda+\varepsilon}{\varepsilon}c)^{-1}\Vert v\Vert= c^{\prime}\eta^{n}\Vert v\Vert,
a^{2413}=a^{2143},a_{1}^{2413}=a_{1}^{2143},a_{2}^{2413}=a_{2}^{2143},a_{3}^{2413}=a_{3}^{2143},a_{4}^{2413}=a_{4}^{2143}, a_{12}^{2413}=a_{12}^{2143},a_{13}^{2413}=a_{13}^{2143},a_{14}^{2413}=a_{14}^{2143},a_{24}^{2413}=a_{24}^{2143},a_{34}^{2413}=a_{34}^{2143}, a_{124}^{2413}=a_{124}^{2143},a_{134}^{2413}=a_{134}^{2143}
\|R f\|_{\ell^1(h^1_E)}= \sum_{X,U}\big\|R_{X,U}\sum_\nu c_\nu a_{P,\nu}\big\|_{h^1_E}\le C\sum_{X,U}\sum_{\substack{\nu: |x_{P_\nu}-X|\le C_d |u_{P_\nu}-U|\le C_d}}|c_\nu| \le C' \sum_\nu |c_\nu|.
A = \left( \begin{array}{ccccc}0.7 0.2 0.1 0.5 1.0 0.3 0.6 0.2 0.8 0.3 0.5 0.7 0.9 1.0 0.5 0.1 0.1 0.3 0.8 0.3 0.8 0.2 0.9 0.3 0.2\end{array}\right), \mbox{with}\rho(A)=2.4031.
\delta_{v}j^{i_{1}\ldots i_{s}}= [ D - 2 + s ] ( v \cdot x ) j^{i_{1}\ldots i_{s}}+ \sum_{p = 1}^{s}( v^{i_{p}}x_{k}- x^{i_{p}}v_{k}) j^{i_{1}\ldots i_{p - 1}k i_{p + 1}\ldots i_{s}}
\left( \begin{array}{c c}{{A}}&{{B}}\\{{C}}&{{D}}\\ \end{array}\right)^{- 1}= \left( \begin{array}{c c}{{( A - B D^{- 1}C )^{- 1}}}&{{- A^{- 1}B ( D - C A^{- 1}B )^{- 1}}}\\{{- D^{- 1}C ( A - B D^{- 1}C )^{- 1}}}&{{( D - C A^{- 1}B )^{- 1}}}\\ \end{array}\right) .
\begin{aligned}\mathbb{E}\bigg[\Big(\int_0^T[\delta f_x(t,v(t),p^0\delta\sigma(t))\delta^1X(t)+\delta f_{x'}(t,v(t),p^0\delta\sigma(t))\delta^1\widehat{X}(t)+\delta f_{y}(t,v(t),p^0\delta\sigma(t))\delta^1Y(t)]\mathbf{1}_{E_\varepsilon}(t)dt\Big)^2\bigg] &\leq\varepsilon^2\rho(\varepsilon).\end{aligned}
f_{11}= \left(\frac{1}{\sqrt{5}}- \frac{1}{4}\right)E_1^{\textbf{1},\phi}E_1^{\textbf{1},\overline{\phi}}-\left(\frac{1}{\sqrt{5}}+ \frac{1}{4}\right)E_1^{\textbf{1},\phi^3}E_1^{\textbf{1},\overline{\phi}^3}.
\frac{d}{dt}\int_M u d\mu=\int_M(\square u - Ru) d\mu \leq \int_M u d\mu, \Rightarrow \left. \int_M u d\mu \right|_{t}\leq \left. e^{t+K^{-2}r^2}\int_M ud\mu \right|_{s_0}< 2 e^{t+K^{-2}r^2}.
p_{\tau_{1}}\circ \xi_{1}(t) =p_{\tau_{1}}\circ f(\tau(g,\xi(t)))=\overline{f}\circ p_{\tau}(\tau(g,\xi(t)))=\overline{f}\circ p_{\tau}(\xi(t))= p_{\tau_{1}}(f \circ \xi(t))= p_{\tau_{1}}(\tau_1 (g,f \circ \xi(t))) &= p_{\tau_{1}}\circ \xi_{2}(t).
\frac{f_{zz}}{f_z-ig_z}=-\frac{f^*_{z^*z^*}}{f^*_{z^*}+ig^*_{z^*}}=i\lambda^1,\frac{g_{zz}}{f_z-ig_z}=-\frac{g^*_{z^*z^*}}{f^*_{z^*}+ig^*_{z^*}}=i\lambda^2,
k_{r}^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2},k_{x}=\frac{m\pi}{l},k_{y}=\frac{n\pi}{w},k_{z}=\frac{p\pi}{h}k_{xy}^{2}=k_{x}^{2}+k_{y}^{2}
V(\overline{\rho})=-\frac{1}{12\pi}[|M_{+}(\overline{\rho}^{2})|^{3}+|M_{-}(\overline{\rho}^{2})|^{3}]+\frac{\lambda}{4!}\overline{\rho}^{4}+\frac{\eta}{6!}\overline{\rho}^{6}.
\widetilde \phi ( x ) = K^{\partial_{x}^{2}}\phi ( x ) = t_{0}+ t_{1}K^{- \frac{1}{{R^{2}}}}\cos \left({\frac{x}{R}}\right) + t_{2}K^{- \frac{4}{{R^{2}}}}\cos \left({\frac{{2 x}}{R}}\right) + \cdots ,
\textstyle C_1= C_2=FH e_j, C_{12}=FH e'~~\mbox{where}~ e'=\sum_{\epsilon\in E-\{e_j,\overline e_j\}}\epsilon; \widehat C_{12}=\{(c,c)\,|\,c\in C_{12}\}(\mbox{i.e.,~ take $g=e'$ in Theorem \ref{Goursat C}}).
\frac12\Delta_B|\phi|^2 = \langle \nabla_T^*\nabla_T\phi,\phi\rangle + \langle \phi,\bar\nabla_T^*\bar\nabla_T\phi\rangle -\sum_{a=1}^n\{|\nabla_{\bar V_a}\phi|^2 + |\nabla_{V_a}\phi|^2\}&+\frac 12 (H^{1,0}-H^{0,1})|\phi|^2,
\phi_{Wk}=\phi_{wk}-(\frac{\delta x_{i}}{2})(\frac{\partial\phi}{\partial x})_{wk}+\frac{1}{2!}(\frac{\delta x_{i}}{2})^{2}(\frac{\partial^{2}\phi}{\partial x^{2}})_{wk}+\cdot\cdot\cdot.
(i) \,\,\, \mu_\alpha\sim\mu_{inv}, (ii) \,\,\, S^L_{11}(\mu_\alpha)<\infty, (iii) \,\,\,{\bf 1}\not\in \langle \left(C(T_{1n})-p\right){\bf 1}\mid n\geq 2\rangle, (iv) \,\,\, \Delta_{1}:=\lim_{n\to\infty}\prod_{k=2}^np^{-1}C(T_{1k}).
\sum_{n=[q_{1}^{j_{1}},\ldots,q_{\lambda}^{j_{\lambda}}]}\log q_{1}\cdots \log q_{\lambda}= \sum_{\lambda_1 + \cdots + \lambda_s = \lambda}\frac{\lambda !}{\lambda_1 ! \cdots \lambda_s !}\prod_{i=1}^s (h_i^{\lambda_i}- (h_i-1)^{\lambda_i})(\log q_i)^{\lambda_i}= \rho_{\lambda}(n).
N_R = \sum_{n=1}^{\infty}n(a_n^i{}^\dagger a_n^i +s_n^a{}^\dagger s_n^a), ~~~~~ N_L = \sum_{n=1}^{\infty}n(\tilde{a}_n^i{}^\dagger \tilde{a}_n^i + \tilde{s}_n^a{}^\dagger \tilde{s}_n^a),
S_{1}'(\ell_{*}) = \psi_{1}'(\psi_{2}(\ell_{*}))\psi_{2}'(\ell_{*}) = \psi_{1}'(r_{*})\psi_{2}'(\ell_{*}) & = \left(\frac{g_{1}'(r_{*}) - f_{1}'(\ell_{*})}{f_{1}''(\ell_{*})(r_{*}- \ell_{*})}\right)\left(\frac{f_{2}'(r_{*}) - g_{2}'(\ell_{*})}{f_{2}''(r_{*})(r_{*}- \ell_{*})}\right) & = \psi_{2}'(\psi_{1}(r_{*}))\psi_{1}'(r_{*}) & = S_{2}'(r_{*}).
|e_{\sigma}(\ell - \rho)| = |D_{\sigma}u_{0}(\ell - \rho)| \leq |u_{0}(\ell - \rho + \sigma)| + |u_{0}(\ell - \rho)| \leq C \b( \log(2 + |\ell - \rho|) + |\sigma| \, \b) \leq C\b( \log(1 + |\rho|) + |\sigma| \, \b),
0 < q_0 \leq \mathcal{F}_{g_0}[u] = V^{-\frac{n-4}{n}}\int u P_{g_0}u\dv_0 = V^{-\frac{n-4}{n}}\int u_0 P_{g_0}u_0 \dv_0 = \frac{n-4}{2}V^{-\frac{n-4}{n}}\int Q_{g_0}\dv_0,
\Sigma_0 = \left(S_0, L_{T_0}, I_{T_0}, R_{T_0}, I_{H_0}, A_0, L_{TH_0},I_{TH_0}, R_{TH_0}, A_{T_0}\right) = \left(\frac{\Lambda}{\mu},0, 0, 0, 0, 0, 0, 0, 0, 0 \right) \, .
\frac{\partial}{\partial t}\left( \frac{\partial \bar{\textbf{g}}}{\partial \textbf{x}}\right) =\frac{\partial}{\partial \textbf{x}}\left( \frac{\partial \bar{\textbf{g}}}{\partial t}\right) =-\frac{e}{m}\textbf{A}' \frac{\partial \bar{\textbf{g}}}{\partial \textbf{x}}.
\varphi(x_k)-\varphi(x_{k+1}) =\varphi(x_k)-\min \varphi-(\varphi(x_{k+1})-\min \varphi) \leq (\frac{1}{\nu}-\frac{1}{2L})\|\mathcal{R}_{1/L}(x_{k-1})\|^2-\frac{1}{2L}\sum_{i=1}^\infty\|\mathcal{R}_{1/L}(x_{k+i})\|^2, ~k\geq 1.
Z_{\rm F}[A ] = \int \! D{\bar \psi}D \psi e^{-\int \! d^2 x\,[ \psi_{1}^{\dagger}i \partial \psi_{1}+\psi_{2}^{\dagger}i{\bar \partial}\psi_{2}-\psi_{1}^{\dagger}A\psi_{1}- \psi_{2}^{\dagger}{\bar A}\psi_{2}]}\; ,
\rho \ddot{u}_i = \big(A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L})_{,K}\big)_{,J}-E_{ij}\dot{u}_j, a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i,K}+ m_{IJ}\dot{\tau}_{,IJ}+ M_{jLKI}u_{j,LKI}+K_{IJ}\tau_{,IJ}
|H(z)-H(z')| \le \sum_{\ell=0}^{\varphi(a)-1}|h(f^\ell z)-h(f^\ell z')|\le |h|_{C^\eta}\sum_{\ell=0}^{\varphi(a)-1}d(f^\ell z,f^\ell z')^\eta \le C_1' |h|_{C^\eta}\varphi(a)d(Gz,Gz')^\eta\le C_1' |h|_{C^\eta}\varphi(a)d_\theta(Gz,Gz')= C_1' \theta^{-1}|h|_{C^\eta}\varphi(a)d_\theta(z,z'),
\varphi_4'(\varepsilon,b,c,\eta,n,\omega,\varphi) :=\min\{\varphi_1(\varepsilon/3,b,n,\varphi),\varphi_2(\varepsilon/3,b,n,\omega,\varphi), &\qquad\varphi_1(\eta(\min\{\varepsilon,2\})c/4,b,n,\varphi),\sqrt{\varphi_2(c/2,b,n,\omega,\varphi)}, &\qquad\qquad\eta(\min\{\varepsilon,2\})c/8b,\sqrt{\varphi_2(c/2,b,n,\omega,\varphi)},1,\lambda_0/2\}
\rho \ddot{u}_i = \left( A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L})_{,K}\right)_{,J}- E(|\dot{u}|) \dot{u}_i, a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i,K}+ m_{IJ}q_{I,J}+ M_{jLKI}u_{j,LKI}+ K_{IJ}\tau_{,IJ}, \kappa \dot{q}_i = \dot{\tau}_{,i}- q_{i}
\mathbb Q_s^x \big( \log|G_n x| \geq n( q_t+l ) \big) & = \frac{\nu_t(r_s)}{\nu_t(r_t)}\frac{r_{t}(x)}{r_{s}(x)}\frac{\exp \{-n(\Lambda^*(q_t + l) - \Lambda^*(q_s) - s(q_t -q_s +l)) \}}{(t -s)\sigma_{t}\sqrt{2\pi n}}[ 1 + o(1)].
\Delta \lambda_{n}^{c r i t}= - 2 \theta a \left( \frac{\lambda_{n}^{c r i t}}{2 ( n + 1 )}\right)^{2}\left( c_{n + 1}+ \left( \frac{\lambda_{n + 1}^{c r i t}}{2 ( n + 2 )}\right)^{2}\left( c_{n + 2}+ \cdots \right) \right) .
\begin{aligned}\sum_{2\le|a|\le N}\frac{1}{t}\int_0^t (f,e^{isH}({\tilde P}^a)^*i[H^a,A^a]{\tilde P}^a e^{-isH}f) ds &\sim_{\epsilon_M}\sum_{2\le|a|\le N}\frac{1}{t}\int_0^t (f,e^{i(s-t(s))H}({\tilde P}^a)^*e^{it(s)H_a}i[H^a,A^a]{\tilde P}^a e^{-isH}f) ds.\end{aligned}
A^{2}= \left( \begin{matrix}{{- \omega_{1}^{2}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- \omega_{1}^{2}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- \omega_{2}^{2}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- \omega_{2}^{2}}}\\ \end{matrix}\right) ,
\delta L = \frac{d}{d \tau}\left( \rho \left( s m + e^{- 1}\dot{x}J - J^{2}e^{- 1}\left( \dot{x}J + ( \dot{x}\xi ) ( \xi J ) \right) \cdot ( J^{2}+ ( J \xi )^{2})^{- 1}\right) \right) ,
\begin{aligned}\frac{p_{n}}{M(n)}= \frac{1+\frac{\beta}{M(n)}}{n\left(1+\frac{\beta}{(2+\beta)n}\right)}g\left(\frac{M(n)+\beta}{n+\beta/(2+\beta)}\right) &=\frac{1}{n}g\left(\frac{M(n)+\beta}{n+\beta/(2+\beta)}\right)\left(1+O\left(\frac{1}{M(n)}\right)\right).\end{aligned}
\mathbb Z_{\geq 0}(\Phi^\vee)^{\diamond, +}\, = \, \mathbb Z(\Phi^\vee)^\diamond \cap \mathbb R_{\geq 0}\breve{\Phi}^{\vee +}\, = \,\mathbb Z\breve{\Sigma}^\vee \cap \mathbb R_{\geq 0}\breve{\Phi}^{\vee +}\, = \, \mathbb Z_{\geq 0}\breve{\Sigma}^{\vee +}.
E [ p ] = \frac{L g^{2}}{2 N}s_{\theta}[ n ] = \frac{L g^{2}}{2 N}\left[ \sum_{j = 1}^{N}\left( p_{j}- \frac{\theta}{2 \pi}\right)^{2}- \sum_{j = 1}^{N}\left( \frac{N + 1}{2}- j - \frac{\theta}{2 \pi}\right)^{2}\right]
\inf\left\{R:D_{1+s}(P_{X^{n}Y^{n}|U_{n}}\|\pi_{X^{n}Y^{n}}|P_{U_{n}})\rightarrow0\right\}\leq\min_{Q_{XYW}:\,Q_{XY}=\pi_{XY},\,X-W-Y}I_{Q}(XY;W) =C_{\mathsf{Wyner}}(X;Y).
Y_{k,r}:=\left\{\begin{array}{cc}\omega \in \wedge^{k+\ell}V_d, \; \omega=\sum \alpha_i \wedge \beta_i \mbox{such that}\beta_i \in \wedge^\ell V_d \mbox{and}\alpha_i \in \wedge^k U \mbox{for some}U \subset V_d \mbox{of dimension}r\end{array}\right\},
\int_{\frac{-1}{2\log T_k}}^{\frac{1}{2\log T_k}}p^{-it}\, dt = \frac{2 \sin \bigl( \frac{\log p}{2\log T_k}\bigr)}{\log p}, \, \,\int_{\frac{-1}{2\log T_k}}^{\frac{1}{2\log T_k}}p^{-2it}\, dt = \frac{1}{\log T_k}+ O \biggl( \frac{(\log p)^2}{(\log T_k)^3}\biggr).
R(r)=Be^{-\eta{r}}r^{(\frac{1+\tau}{2})}\Big[e^{2\eta{r}}r^{-(1+\tau+\mathrm{b})}\Big]_{-(1+\mathrm{b}E^{-1})}\quad\Bigg(\mathrm{b}=\Big(\frac{\beta-\eta(1+\tau)}{2\eta}\Big)\Bigg),
E_{\gamma}(A, \mathcal{V}_{N}\setminus A)= \frac{\Vert \gamma. F - F \Vert^{2}}{\vert \mathcal{V}_{N}\vert^{2}}\geq \frac{c_{0}^{2}\Vert F\Vert^{2}}{\vert \mathcal{V}_{N}\vert^{2}}= c_{0}^{2}\left( 1-\frac{\vert A\vert}{\vert \mathcal{V}_{N}\vert}\right) \vert A\vert.
y^{\nu_{r-1}(a_s)}R_{r-1}(a_s) +y^{\nu_{r-1}(a'_s)}R_{r-1}(a'_s) = \begin{cases}0, \mbox{if}v_{r-1}(a_s+a'_s)>v_{r-1}(a_s), y^{\nu_{r-1}(a_s+a'_s)}R_r(a_s+a'_s), \mbox{if}v_{r-1}(a_s+a'_s)=v_{r-1}(a_s).\end{cases}
\min\limits_{\partial B(p, r_i)}\sum\limits_{s}|g_s|^2=\min\limits_{\partial B(p_i, 1)}\sum\limits_{s}|g_s|^2=\min\limits_{\partial B(p_i, 1)}\sum\limits_{s}|g^i_s|^2 \geq c(n, v)r_i^2>0.
g_{2}((\nabla\pi_{\ast})(X,\mathcal{C}Y)-(\nabla\pi_{\ast})(Y,\mathcal{C}X),\pi_{\ast}\omega W) =g_{1}(\mathcal{A}_{X}\mathcal{B}Y+\mathcal{A}_{Y}\mathcal{B}X,\omega W) &+\eta(Y)g_1(X,\omega W)-\eta(X)g_1(Y,\omega W)
V((r,t),(s,u)) = \frac{\sigma(r,s)}{(3-d(r,s))(2-d(r,s))}[\overline{c(s,r)}t^{3-d(r,s)}+c(r,s) u^{3-d(r,s)}-C(r,s;t-u) \vert t-u \vert^{3-d(r,s)}] ,
\begin{aligned}\lim\limits_{\stackrel{x \to x_{0}}{x \in K^{'}}}\big\langle \nabla_{x}(S\varphi)(x,t),n(x_{0})\big\rangle=-\frac{1}{2}\varphi(x_{0},t) &+\int_{0}^{t}\int_{\partial\Omega}\frac{\partial\varepsilon_{n, a}(x_{0}-\xi,b(t,\tau))}{\partial n(x_{0})}\varphi(\xi,\tau)a(\tau)dS_{\xi}d\tau,\end{aligned}
g'(x) =-\frac{{\rm sech}^{p+2}(x)}{(1-{\rm sech}^p(x))^2}(1+p\cosh^2(x)-p-\cosh^p(x)) &<-\frac{{\rm sech}^{p+2}(x)}{(1-{\rm sech}^p(x))^2}(1+p\cosh^2(x)-p-\cosh^2(x)) &<-\frac{{\rm sech}^{p+2}(x)}{(1-{\rm sech}^p(x))^2}((p-1)(\cosh^2(x)-1))<0.
\zeta_{\pm}^{1}( z ) = N_{1}\left( \begin{array}{c}{{z^{\frac{M}{2}}( 1 - z )^{- \frac{M}{2}}F ( 1 , - 1 , \frac{1}{2}+ M ; z )}}\\{{\pm \frac{i}{\frac{1}{2}+ M}z^{\frac{1 + M}{2}}( 1 - z )^{\frac{M}{2}}F ( - \frac{1}{2}+ M , \frac{3}{2}+ M , \frac{3}{2}+ M ; z )}}\\ \end{array}\right)
\gamma^{\mathcal O\rtimes E\mathcal M}\big((o,u_\bullet); (f^{(1)},v^{(1)}_\bullet),\dots,(f^{(r)},v^{(r)}_\bullet)\big) &\quad{}=\big(\gamma^{\mathcal O}(o, u_1.f^{(1)},\dots,u_r.f^{(r)});\gamma^{E\mathcal M^\bullet}(u_\bullet;v_\bullet^{(1)},\dots,v_\bullet^{(r)})\big)
\begin{aligned}\mathbf{S}_{\infty m}(q,1) = \big\{(q_{ij})_{\substack{i \geq 1 0 \leq j \leq m}}\big| \, q < q_{ij}< 1 \,, q_{ij}< q_{i,j-1}(\forall i \geq 1, 0 \leq j \leq m) \,, q_{i+1,0}< q_{im}(\forall i \geq 1) \big\}\,.\end{aligned}
\bigl[\mathcal{H},[\mathcal{H},\eta^{}]\bigr] =\eta^{}R^{}_0(\mathcal{H}) +[\mathcal{H},\eta^{}]R^{}_1(\mathcal{H}) +R^{}_{-1}(\mathcal{H}), R^{}_1(z)=R_1(z), R^{}_0(z)=R_0(z), R^{}_{-1}(z)=aR_{-1}(z)-bR_0(z).
\sum_{k=0}^{p-1}c_{k,2p-1}(-1)^{p+k}q^{4(p+k)-3}\leqslant A_{q,1}^{(2p-1)}(r)\leqslant \sum_{k=0}^{p-1}d_{k,2p-1}(-1)^{p+k}q^{4(p+k)-3}; \sum_{k=0}^{p}c_{k,2p}(-1)^{p+k}q^{4(p+k)-3}\leqslant A_{q,1}^{(2p)}(r)\leqslant \sum_{k=0}^{p}d_{k,2p}(-1)^{p+k}q^{4(p+k)-3}.
\begin{array}{rcl}\hat{Q}_{10}^{(vi)}(x)\!\! \!\!\!=\!\!\! \!\!151200(-128\pi^2+1280)x^4+60480(320\pi^3-3072\pi)x^3 begin{align*}1.0 ex]\!\! \!\!\! \!\!\! \!\!\!+20160(-320\pi^4+4224\pi^2-13440)x^2+5040(160\pi^5-4640\pi^3 begin{align*}1.0 ex]\!\! \!\!\! \!\!\! \!\!\!+32256\pi)x-28800\pi^6+2592000\pi^4-26611200\pi^2+48384000.\end{array}
M\mapsto{\bf D}(\Theta_*{\mathcal V}_M){\bf D}(\Theta_*{\mathcal V}_M)=H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)\otimes_{{\mathcal O}_{\mathcal E}^+}{\mathcal O}_{\mathcal E}=H_0({\mathfrak X}_+,\Theta_*{\mathcal V}_M)\otimes_{{\mathcal O}_{\mathcal E}^+}{\mathcal O}_{\mathcal E}
W_\mathcal{A}(Op_{w,2d}(a)f,g)=Op_{w,4d}(b)W_\mathcal{A}(f,g), W_\mathcal{A}(f,Op_{w,2d}(a)g)=Op_{w,4d}(\tilde b)W_\mathcal{A}(f,g), W_\mathcal{A}(Op_{w,2d}(a)f)=Op_{w,4d}(c)W_\mathcal{A}(f).
S \equiv S_{1}= ( S_{1 , i j})_{i , j = - 1 , 0 , 1}= \left( \begin{array}{l l l}{{1 ,}}&{{0 ,}}&{{0}}\\{{0 ,}}&{{- q ( q^{2}+ 1 )^{- 1},}}&{{0}}\\{{0 ,}}&{{0 ,}}&{{1}}\\ \end{array}\right) .
S_{\mathrm{main}}= \sum_{d_1,d_2<D}\frac{1}{\sqrt{d_1d_2}}\sum_{Id_1, Jd_2<D}\alpha_{Id_1}\bar{\alpha}_{Jd_2}\sum_{L, M\geq 1}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}\,\ \delta_{IL\, JM}
f=8\pi \tau_2^2h(t) +16\pi\left(\frac{\chi}{192\pi}-c_{\ell}\right)+ \frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{|m\tau+n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\,,
\int_A\L^{-\ell}d\tilde\mu^{G_i}=\sum_{n}[A_n]\L^{-(N+1)d-n}=\sum_{n,e}[A_{n,e}]\L^{-(N+1)d-n}&=\sum_{n,e}[\tilde{A}_{n,e}]\L^{-(N+1)d-(n+e)}=\sum_{m}\left(\sum_{n+e=m}[\tilde{A}_{n,e}]\right)\L^{-(N+1)d-m}&\qquad=\sum_{m}\left[\pi_N\left(\tilde{\ell}^{-1}(m)\right)\right]\L^{-(N+1)d-m}=\int_{h^{-1}(A)}\L^{-\tilde{\ell}}d\tilde\mu^{G_i},
\frac{d\varphi}{dt}=\left( \frac{\delta \bar{H}}{\delta \mu}\right)_{\varphi},\ \ \frac{d\mu}{dt}=-\mathcal{L}_{\frac{\delta \bar{H}}{\delta \mu}}\mu -\left( div_{d^{n}q}\frac{\delta \bar{H}}{\delta \mu}\right) \mu -\left( \frac{\delta \bar{H}}{\delta \varphi}\right)\circ \varphi^{-1},
\dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in I_1\right )_{\mathcal R}\ge k+1-(n+1-\sharp I_1)=k-n+\sharp I_1, \dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in A_2\right )_{\mathcal R}\ge k+1-(n+1-\sharp A_2)=k-n+\sharp A_2.
F^{a{\mu}_{1}{\cdots}{\mu}_{p + 1}}= \frac{( - 1 )^{p}}{( p + 1 ) !}{\Gamma}^{a b}{\epsilon}^{{\mu}_{1}{\cdots}{\mu}_{p + 1}{\nu}_{1}{\cdots}{\nu}_{p + 1}}G_{{\nu}_{1}{\cdots}{\nu}_{p + 1}}^{b}.
=\prod_{i=1}^{m}(\chi(a_{i})^{\sum_{j}\sigma((p^{s_{j}}-1)r_{ji})}\pi^{\sum_{j}\sigma((p^{s_{j}}-1)r_{ji})})\cdot\prod_{j}(-\prod_{i=1}^{m}\prod_{t=0}^{s_{j}-1}\Gamma_{p}(\{p^{t}r_{ji}\})).
Y_n(p,\ell)\, \uparrow\, [p,1]\sum_{i=0}^{\ell}\frac{p^{-i^2}}{[p,i]^2}=:Y(p,\ell) \ \{\mathrm{as}}\ \n\to \infty\,,Y(p,\ell) \, \uparrow\, \frac{[p,1]}{C_p}\ \{\mathrm{as}}\ \ \ell\to \infty\,,
a_{k, r}(i)= \dfrac{(k+r)!}{(r+i)!(k-i)!}\cdot\dfrac{(i+r-1)!}{(r-1)!i!}= \dfrac{r}{r+i}\cdot\dfrac{k!}{(k-i)!i!}\cdot\dfrac{(k+r)!}{r!k!}= \dfrac{r}{r+i}\dbinom{k}{i}\dbinom{k+r}{r}.
\dot{\theta_{l}}\approx \frac{\left( p_{\alpha}( l ) - p_{\alpha}( l + 1 ) \right)}{\left( p_{\mu}( l + 1 ) - p_{\mu}( l ) \right)^{2}}\int_{\bar{\theta}_{l}}^{\bar{\theta}_{l + 1}}d \theta \dot{P}_{\alpha}( \theta )
\epsilon_{X,(B')}\circ (u\times B')\circ (A\times f) = \epsilon_{X,(B)}\circ (X^f\circ u)\times B \epsilon_{X,(B')}\circ (X^{B'}\times f)\circ (u\times B) = \epsilon_{X,(B)}\circ (X^f\times B)\circ (u\times B)
\frac{s_{1}^{n_{1}}}{n_{1}}=\frac{s_{1}^{n_{1}}(\rho_{1})}{n_{1}}=\int_{0}^{\rho_{1}}t_{1}^{n_{1}-1-\frac{\gamma_{1}}{p-1}}dt=\frac{p-1}{n_{1}(p-1)-\gamma_{1}}\rho_{1}^{\frac{n_{1}(p-1)-\gamma_{1}}{p-1}},
\begin{aligned}(C_{aj}-C_j-C_a)\,[C_{ik},C_{k\ell}] +(C_{ai}-C_i-C_a)[C_{k\ell},C_{jk}] +(C_{a\ell}-C_\ell-C_a) [C_{ij},C_{jk}] & \qquad+(C_{ak}-C_k-C_a)[C_{i\ell},C_{j\ell}] =0\,.\end{aligned}
\begin{aligned}\mathfrak{S}_{\delta}^{\tau_k}(x, D) = - \frac{1}{2\pi}\sum_{H'}\sum_{M}\sum_{N}\big(\mathfrak{S}_{\delta, \flat}^{\tau_k}(H', M, N) + \overline{\mathfrak{S}_{\delta, \flat}^{\tau_k}(H', M, N)}\big) & \hskip 22mm+ \sum_{M}\sum_{N}\mathfrak{S}_{\delta, \dagger}^{\tau_k}(M, N),\end{aligned}
P_{\alpha}[\overline{L_{f}}](s,t)=\frac{1}{\alpha^{2}f^{2}}\iint\overline{L}_{f}(u(1-1/\alpha)+s/\alpha,v(1-1/\alpha)+t/\alpha,u,v)dudv\equiv\frac{1}{\alpha^{2}f^{2}}I_{2}^{4}\circ B_{\alpha}[L_{f}]
\|g(\tau)\|^2_{\ell^2}=\sum_{n=1}^{\infty}\chi^2_{[e^{-(n+1)}, e^{-n})}(\tau)=\sum_{n=1}^{\infty}\chi_{[e^{-(n+1)}, e^{-n})}(\tau)=\chi_{(0,e^{-1})}(\tau)\quad\mbox{for any}\tau\geq 0.
\begin{cases}\begin{array}{ll}v_m(.,t)\in SPSH(\Omega) \dot{v}_m=\log\det (v_m)_{\alpha\bar{\beta}}-Av_m+g(z,t)\;\;\; \mbox{on}\;\Omega\times (0,T), v_m\searrow\psi \mbox{on}\;\partial\Omega\times [0,T), v_m\searrow v_0 \mbox{on}\;\bar{\Omega}\times\{0\}. \end{array}\end{cases}
\bar{y}(P)\notin T_{P}\Phi,\bar{y}_{/i}(P)\in T_{P}\Phi (i=1,\ldots,n)~~\footnote{Partial derivatives of a function $f$ are denoted by$f_{/i}:=\frac{\partial f}{\partial u^{i}},~f_{/ij}:=\frac{\partial^{2}f}{\partial u^{i}\partial u^{j}}\medskip$}
F\left( \begin{array}{c}z x \end{array}\right) = \left( \begin{array}{c}z + q_1(z+ \delta x) \delta x - q_1(z+ \delta x) \end{array}\right) \; \;\mathrm{and}\; \;G\left( \begin{array}{c}w y \end{array}\right) = \left( \begin{array}{c}w + q_2(w+ \delta y) \delta y - q_2(w+ \delta y)\end{array}\right).
D_{x,2}= \frac{1}{2L_{xy}}\left(\sqrt{n\sigma_{G,r}^2}+ L_{y}\sqrt{n}M_y+\|\nabla G(Y^*)\| \right) &+\frac{1}{2\mu_{x}}\left(\sqrt{n\sigma_{F,r}^2}+ L_{x}\sqrt{n}M_x+\|\nabla F(X^*)\| \right),
\widehat{\rho}= \sqrt{\frac{n_1}{n_2}}\frac{\widehat{\gamma}}{\widehat{\sigma}_1 \widehat{\sigma}_2}\ \ \ \ \widehat{\sigma}_j^2 = \frac{1}{n}\sum_{i=1}^n (X_{ji}- \overline{X}_i)^2, \j = 1,2,
{\cal L}= \int_{0}^{2 \pi R}\! \! \! d y \left[ - \frac{1}{4 g^{2}}F^{M N}F_{M N}+ \bar{\psi}i \Gamma^{M}\big ( \partial_{M}- i A_{M}\big ) \psi - M \eta ( y ) \bar{\psi}\psi \right] .
\lambda^{2n}H,E_\alpha, \lambda^{2n}E_{\pm\alpha}, F_{\alpha_1}, \lambda^{2n}F_{\pm\alpha_1}, F_{\alpha_2}, \lambda^{2n}F_{\alpha_2},\lambda^{2n}F_{-\alpha_2}, \lambda H',\lambda^{2n+1}H',~~n>0
\tilde \gamma_u^{a({\rm{MF}})}= \frac{{{q}{{\left|{{\lambda_u}}\right|}^2}}}{{\frac{1}{N}\sum\nolimits_{i=1, i\ne u}^U{{q}{{\left|{{\lambda_i}}\right|}^2}}+ \frac{1}{N}\sum\nolimits_{n=1}^N{\sum\nolimits_{k=1}^K{{g_{k,n}}{{p_{k,n}}}}}+{\sigma^2}}}.
\hat{\Phi}(\vec{x},t)=\sum\limits_{\vec{p}}\frac{1}{\sqrt{2\omega_{\vec{p}}V}}[\hat{a}_{\vec{p}}e^{\frac{i}{\hbar}(\vec{p}\cdot\vec{x}-Et)}+{\hat{b}}^{\dag}_{\vec{p}}e^{-\frac{i}{\hbar}(\vec{p}\cdot\vec{x}-Et)}]
\mathbb{E}(T_{i-1}Y_{i-1}) = \frac{k\theta}{\lambda}(1+k\lambda\theta)+q^k\left(\frac{1-k\lambda\theta(2k+1)}{\lambda^2}\right) +q^{k+1}\left(\frac{k\theta(1+k\lambda\theta+2k)}{\lambda}\right)-\frac{1}{\lambda^2}q^{2k}-\frac{k\theta}{\lambda}q^{2k+1}