harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}{{\bar R}_{{\rm{BF,upper}}}}={\log_2}\left( 1 + \frac{{E{N_{\rm{T}}}{N_{\rm{R}}}\kappa}}{{{\sigma^2}K{L_{\rm{R}}}\left({\kappa + 1}\right)}}\right.\times &\left.\left({{\rm{t}}{{\rm{r}}_1}\left({{{\bf{D}}_{K}}}\right) + \frac{\pi}{2}{\rm{t}}{{\rm{r}}_2}\left({{\bf{D}}_{K}^{1/2}}\right)}\right) \right).\end{aligned}
\lim_{y^- \rightarrow x^-}\lim_{L \rightarrow \infty}\Bigl\langle \Omega \Bigl\vert 2 \partial_+ J^+(x^+;x^-,y^-) - 2 i e m J_5(x^+,x^+;x^-,y^-) \Bigr\vert \Omega \Bigr\rangle ={e^2 \over \pi}E(x^+) \; .
i\left\{Y_{l_{1}}^{m_{1}},Y_{l_{2}}^{m_{2}}\right\}={\displaystyle\sum\limits_{_{\substack{l=|l_{1}-l_{2}|+1 l\equiv l_{1}+l_{2}-1}}}^{l_{1}+l_{2}-1}}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{2l+1}}\C_{0,0,0}^{l_{1}-1,l_{2},l}P_{m_{1},m_{2},m}^{l_{1}-1,l_{2},l}Y_{l}^{m}
[\begin{matrix}1&2\\ 0&3\end{matrix}][\begin{matrix}1&1\\ 0&1\end{matrix}]=[\begin{matrix}1&3\\ 0&3\end{matrix}]\ne[\begin{matrix}1&5\\ 0&3\end{matrix}]=[\begin{matrix}1&1\\ 0&1\end{matrix}][\begin{matrix}1&2\\ 0&3\end{matrix}].
D_{( n , j , m ) \, M^{\prime}}^{J}( h^{- 1}g ) = \sum_{m^{\prime \prime}}D_{( n , j , m ) \, ( n , j , m^{\prime \prime})}^{J}( h^{- 1}) \, D_{( n , j , m^{\prime \prime}) \, M^{\prime}}^{J}( g ) \; .
\iota(\eta, \xi)(x,y) = \Lambda\left( \begin{pmatrix}(\eta^{r_0}\circ\xi)^{r_1}\circ\eta(x) (\eta^{r_0}\circ\xi)^{r_1-1}\circ\eta(x) \end{pmatrix}, \begin{pmatrix}\eta^{r_0}\circ\xi(x) x \end{pmatrix}\right).
S = - \frac{1}{2}\int d^{d + 1}x [ 2 ( \nabla C ) . ( \nabla D ) + 2{m^{2}}_{1}C D +{m^{2}}_{2}C^{2}+ 2 ( \nabla \bar{\alpha}) . ( \nabla \alpha ) + 2{m^{2}}_{1}\bar{\alpha}\alpha ] .
|\nabla u(t)|^2_{2}+|\nabla \rho(t)|^2_{2}+|\nabla H(t)|^2_{2}+\int_0^t\Big( | u|^2_{D^2}+\frac{1}{\sigma}|H|^2_{D^2}\Big)t \leq C\exp \Big(\Big(1+\frac{1}{\sigma}\Big) \int_0^t(|\nabla u|^2_2+|\nabla \theta|^2_2+|D(u)|_\infty+1) s\Big) \leq C(\sigma).
\gamma = 8 \pi G \eta_{1}^{2}\ \ , \ \alpha = e^{2}/ \lambda_{1}\ \ , \ \q = \frac{\eta_{2}}{\eta_{1}}\ \ , \ \{\bar{\Lambda}}= \frac \Lambda{\lambda_{1}\eta_{1}^{2}}\ \ , \ \, b e t a_{i}^{2}= \frac{\lambda_{i}}{\lambda_{1}}\ \ , \ \i = 1 , 2 , 3 \ .
\sum_{\hat{\alpha}\in \hat{\mathcal{A}}}b_{\gamma \hat{\alpha}}\partial^{\beta}\left( \left[ S_{\hat{\alpha}}\odot_{0}\bar{P}_{\chi \left( \hat{\alpha}\right)}\right] \circ T\right) \left( 0\right) \cdot \delta^{\left\vert \beta \right\vert -\left\vert \hat{\alpha}\right\vert}=\delta_{\gamma \beta}\gamma ,\beta \in \hat{\mathcal{A}}
\mu_{i}\left(\mathcal{F}_{i}(x)\right)=\begin{cases}0 \mathcal{F}_{i}(x)\geq y_{i}^{0}, m_{i}(\mathcal{F}_i(x)) y_{i}^{0}\geq \mathcal{F}_{i}(x)\geq y_{i}^{1}, 1 \mathcal{F}_{i}(x)\leq y_{i}^{1},\end{cases}
\chi^{\mathcal{C}\pm}_q(\phi_q(\tau))=2^{\frac{q-1}{2}}\chi_\varnothing\bigg{(}\bigg{(}\frac{1}{2^{\frac{q-1}{2}}}(e_1+e_2)\dots(e_{q-1}+e_q)\bigg{)}^2\bigg{)}=\frac{1}{2^{\frac{q-1}{2}}}\chi_\varnothing(((e_1+e_2)\dots(e_{q-1}+e_q))^2),
\begin{cases}\frac{dx}{dt}= \frac{dx}{dz}\frac{dz}{dt}= -2\pi i x(1 + e^{6\pi it}x^2y) \frac{dy}{dt}= \frac{dy}{dz}\frac{dz}{dt}= -2\pi i y(1 - e^{6\pi it}x^2y)\end{cases}\, .
S = \frac{1}{\pi}\int d^{2}x \left[ e^{- 2 \phi}( - 2 \partial_{+}\partial_{-}\rho + 4 \partial_{+}\phi \partial_{-}\phi + \lambda^{2}e^{2 \rho}) + \frac{1}{2}\partial_{+}f \partial_{-}f + \mu^{2}( \cos f - 1 ) e^{2 \rho - 2 \phi}\right] ,
\begin{array}{c}-\frac{1}{2}(k-1)k(k+1)<O_{k-2}O_{k_{1}}... O_{k_{p-1}}P>_{g-1}+4t_{0}(k+1)<O_{k}O_{k_{1}}... O_{k_{p-1}}P>_{g}+4t_{2}(k+2)<O_{k+2}O_{k_{1}}... O_{k_{p-1}}P>_{g}+2 \sum_{i=1}^{p-1}(2k +k_{i}+2) <O_{k+k_{i}}O_{k_{1}}...\hat{O}_{k_{i}}...O_{k_{p-1}}P>_{g}=0, \end{array}
m=\prod_{\rho=1}^n\sum_{j_\rho=1}^N\sum_{k_\rho=1}^Na_{j_\rho}(t_\rho)a_{k_\rho}(\tau_\rho) a_{j_{\rho}}c_{j_{\rho}+k_\rho}\phi(N \xi_{\rho}-j_{\rho})\phi(N \eta_\rho-k_\rho),
\displaystyle\lim_{n\rightarrow \infty}\nu_{x_n-x, z}(t) =\displaystyle\lim_{n\rightarrow \infty}\frac{t}{t+\|x_n-x,z\|}&= \frac{t}{t + \displaystyle\lim_{n \rightarrow \infty}\|x_n-x,z\|}=\frac{t}{t + 0}= 1, i.e.x_n\rightarrow x\textrm{for the 2-P norm}.
0=DJ(\theta^* \vee l)_{\sigma(t,l)}=k_{\sigma(t,l)}+p_{\sigma(t,l)}^{\theta^*\vee l}\leq u(\sigma(t,l),\theta^*_{\sigma(t,l)}\vee l)+p_{\sigma(t,c)}^{\theta^*\vee l}&\leq u(\sigma(t,l),\theta^*_t\vee l)+p_{\sigma(t,c)}^{\theta^*\vee l}
\left\|\mathcal H^{m,n}_{\psi,\overrightarrow{s}}\right\|_{L^{p_1}_{\omega_1}\left(\mathbb R^d\right)\times\cdots\times L^{p_m}_{\omega_m}\left(\mathbb R^d\right)\to L^{p}_{\omega}\left(\mathbb R^d\right)}={\mathcal A}_\star.
\rho_1=f_1\cdot\mathcal{L}^N:=\frac{|\phi|^{q-2}\,\phi_+}{\displaystyle\int_\Omega |\phi|^{q-2}\,\phi_+\,dx}\cdot\mathcal{L}^N \mbox{and}\rho_0=f_0\cdot\mathcal{L}^N=\frac{|\phi|^{q-2}\,\phi_-}{\displaystyle\int_\Omega|\phi|^{q-2}\,\phi_-\,dx}\cdot\mathcal{L}^N.
F_{p r^{2}_{ii}|\frac{p}{N_{0}}\rightarrow \infty}(x)= &\frac{m!(m-i+1)^{1-i}\left(\frac{N_{0}x}{p}\right)^{n-i+1}}{(i-1)!(m-i)!(n-i+1)!}+o\left(\left(\frac{p}{N_{0}}\right)^{-(n-i+1)}\right).
[\psi]^p_{W^{\alpha,p}(\mathbb{R}^N)}=[\psi]_{W^{\alpha,p}(B_R)}^p+2\,\int_{B_R}\int_{\mathbb{R}^N\setminus B_R}\frac{|\psi(x)|^p}{|x-y|^{N+\alpha\,p}}\,dx\,dy.
I_{m,p}(s,x):=\frac{1}{2\pi i}\,\intop_{\mu-i\infty}^{\mu+i\infty}\left(2z,2\pi pm\right)_{m}\,\Gamma\left(z\right)\,\Gamma\left(s+z-\frac{1}{2}\right)\,(\pi\,x\,m)^{-2z}\,dz.
{P ( \Delta s ) V [ \phi ] P ( \Delta s )}={g}{\frac{\mu^{\epsilon}}{4 !}}\sum_{n = 0}^{4}\left({4 \atop n}\right) \int \! d^{d - 1}x \, \langle 0 | \phi_{>}( x )^{4 - n}| 0 \rangle \, \phi_{<}( x )^{n}\quad .
X^2_-(t) =\sum_{i:\gamma^+_i<\gamma_*}|\xi_i^+(t)|^2+\sum_{i:\gamma^-_i<\gamma_*}|\xi_i^-(t)|^2+\sum_{i\in I_1}|\xi_{i,1}(t)|^2+|\xi_{i,2}(t)|^2+\sum_{i\in I_2}|\xi_{i,3}(t)|^2+\varepsilon_1^2 |\xi_{i,4}(t)|^2.
\int_0^1 dt \delta_{\vec{u}}(t) <{\rm Tr}(H_{12}(\gamma)H_{34}(\gamma))> = -{i \pi \over k}{\rm Tr}(H_{12}(\gamma) H_{34}(\gamma))+{2i \pi \over k}{\rm Tr}(H_{12}(\gamma)){\rm Tr}(H_{34}(\gamma))
\psi_1(x, x,u)=0,\ \d_x\psi_1(x, x,u)=d_x\psi(x,x,u),\ \d_y\psi_1(x, x,u)=d_y\psi(x, x,u), &{\rm Im\,}\psi_1(x,y,u)\geq0,\ \ \forall (x,y,u)\in D\times D\times(-M,M).
\mathcal{I}_0=\{i\in\mathcal{V}\mid[\hat{p}_{i0}^v~\hat{p}_{i1}^v]^T=[0~0]^T\}, &\mathcal{I}_1=\{i\in\mathcal{V}\mid[\hat{p}_{i0}^v~\hat{p}_{i1}^v]^T=[1~1]^T\}.
R_{\alpha\beta\rho}^{\gamma}:=\frac{\partial}{\partial X^{\rho}}[\Gamma_{\alpha\beta}^{\gamma}]-\frac{\partial}{\partial X^{\beta}}[\Gamma_{\alpha\rho}^{\gamma}]+\Gamma_{\mu\rho}^{\gamma}\Gamma_{\alpha\beta}^{\mu}-\Gamma_{\mu\beta}^{\gamma}\Gamma_{\alpha\rho}^{\mu}=0
\begin{aligned}\mu : =\mu_{\{b_k\},\{D_k\}}&=\delta_{b_{1}^{-1}D_1}\ast\delta_{b_{1}^{-1}b_{2}^{-1}D_2}\ast\delta_{b_{1}^{-1}b_{2}^{-1}b_{3}^{-1}D_3}\ast\cdots\ast\delta_{b_{1}^{-1}b_{2}^{-1}\cdots b_{k}^{-1}D_k}\ast\delta_{b_{1}^{-1}b_{2}^{-1}\cdots b_{k+1}^{-1}D_{k+1}}\ast \cdots &=\mu_{k}\ast\nu_{>k}(b_{1}b_{2}\cdots b_{k}\cdot),\end{aligned}
\Phi(x_1, x_2; \rho)^{(2l-1)}_\alpha := \left.\frac{\partial^{2l-1}\Phi}{\partial \varphi^{2l-1}}\right|_{\varphi = \alpha}= 0 \mbox{\rm for all}\; \alpha \in \Pi\; \mbox{\rm and}\;1 \le l \le \mu_\alpha.
\int_{0}^{\infty}d x \, \left( x^{2}+ u^{2}\right)^{\nu - 1}e^{- p x}= \frac{\sqrt{\pi}}{2}\left( \frac{2 u}{p}\right)^{\nu - \frac{1}{2}}\Gamma \left( \nu \right) \left[ H_{\nu - \frac{1}{2}}\left( u p \right) - N_{\nu - \frac{1}{2}}\left( u p \right) \right] ,
\int_\Omega v_j(x,t)\cdot D\zeta(x,t) dx =-\int_\Omega u_j(x,t)\zeta(x,t)\,dx, \|(v_j)_t\|_{L^\infty(\Omega_T)}\le \frac{1}{2}, \int_0^T\int_\Omega |(v_j)_t-\sigma(Du_j)|\,dxdt \leq\frac{1}{j}|\Omega_T|.
\sum_{\substack{S\subseteq B_1, |S|=k T\subseteq \{v_1\}, |T|=\ell}}p_{H_1,S\cup T}(x)=\sum_{\substack{S\subseteq B_2, |S|=k T\subseteq \{v_2\}, |T|=\ell}}p_{H_2,S\cup T}(x).
|\Phi(x_n)-\Phi(y_n)|=|\Phi(x_n)-\Phi_{\varepsilon_n}(x_n)+\Phi_{\varepsilon_n}(y_n)-\Phi(y_n)|=|(\Phi-\Phi_{\varepsilon_n})(x_n)-(\Phi-\Phi_{\varepsilon_n})(y_n)|.
\begin{aligned}\int_{\tau}^{\tau+2}\int_{\Omega}(\varphi_\tau(t)(u_1(x,t)+u_3(x,t)))\theta(x,t) dxdt &\leq \|\theta\|_{L^{p_0'}(\Omega_{\tau,\tau+2})}\left(C\|u_1+u_3\|_{L^{p_0}(\Omega_{\tau,\tau+1})}^{1-\alpha}+\frac{|d_1-d_3|}{d_1+d_3}C_{\rm{mr},p_0'}\|\varphi_\tau(u_1+u_3)\|_{L^{p_0}(\Omega_{\tau,\tau+2})}\right)\end{aligned}
\begin{array}{l}{\rho ( \theta ) \rho ( - \theta ) ( \sin^{2}\frac{n \gamma}{1 6}+ \sinh^{2}\frac{\theta}{2}) \cosh^{2}\frac{\theta}{2}= 1 \vspace{3 m m}}\\{\rho ( \theta ) = \rho ( i \pi - \theta )}\\ \end{array}.
g_\mathcal{M}(\tau) = \sum_{k\in\mathcal{M}}e(k_0+k-k_0,\tau) = e(k_0,\tau) \sum_{k\in\mathcal{M}}e(k-k_0,\tau) & = e(k_0,\tau) \sum_{m\in \mathcal{M}-k_0=\mathcal{M}}e(m,\tau) = \underbrace{e(k_0,\tau)}_{\neq 1}g_\mathcal{M}(\tau),
\begin{array}{c}{t_{i}e_{j}t_{i}^{- 1}= q^{a_{i j}}e_{j}}\\{t_{i}f_{j}t_{i}^{- 1}= q^{- a_{i j}}f_{j}}\\{{}[ e_{i}, f_{j}] = \delta_{i j}\frac{t_{i}- t_{i}^{- 1}}{q - q^{- 1}}}\\ \end{array}
\begin{aligned}\ \overline{c}_1\circ \overline{c}_2(x_1,\dots,x_{m+n-1})= &\sum_{0\le i\le m-1}(-1)^{d_2(\deg x_1+\dots+\deg x_i+i)}\overline{c}_1(x_1,\dots,x_i,\overline{c}_2(x_{i+1},\dots,x_{i+n}),x_{i+n+1},\dots,x_{m+n-1})\end{aligned}
H=\Big{\langle}x_1,x_2,\ldots, x_n \colon [x_i,x_j,x_k]=1, [x_i,x_j]^p=1, x_i^p=\prod_{j<k}[x_j,x_k]^{c_{ijk}}, \prod_{j<k}[x_j,x_k]^{d_{ljk}}=1, l=1,2,\ldots,t\Big{\rangle},
Y_{N L M}( \theta_{1}, \theta_{2}, \phi ) = 2^{L + 1}\Gamma ( L + 1 ) \sqrt{\frac{n ! \, N}{2 \pi \Gamma ( N + L + 1 )}}\, ( 1 - x^{2})^{\frac L 2}C_{n}^{L + 1}( x ) Y_{L M}( \theta_{2}, \phi )
f_1 =xc, f_2 =xa, f_3 =xp+ab, f_4 =yz+qc,f_5=pq, f_6 =yc, f_7 =ya, f_8 =yp+bb, f_9 =yy+qb, f_{10}=zc, f_{11}=za, f_{12}=zp+cb, f_{13}=yx+qa, f_{14}=xb, f_{15}=xq+ac, f_{16}=xz+pc, f_{17}=zz+qq+ca, f_{18}=yb, f_{19}=yq+bc, f_{20}=xy+pb, f_{21}=zy+qp+ba, f_{22}=zb, f_{23}=zq+cc, f_{24}=xx+pa, f_{25}=zx+pp+aa.
\begin{aligned}\tau=j-i=(j_0-i_0)2^0+(j_1-i_1)2^1+\cdots &+(j_{\pi(m-3)}-i_{\pi(m-3)})2^{\pi(m-3}+\cdots+(j_{m-3}-i_{m-3})2^{m-3}& \leq 2^0+2^1+\cdots+2^{m-3}-2^{\pi(m-3)}\leq 2^{m-2}-2^{\pi(m-3)}-1, \end{aligned}
{{\alpha}_{n}}={{\gamma}_{2,n}}p_{n}^{3/2}q_{n}^{1/2}{{\left( 2\pi \right)}^{1-\left({{p}_{n}}+{{q}_{n}}\right)/2\}}&{{\omega}_{n}}={{\left({{\Omega}_{2,n}}{{p}_{n}}\right)}^{{{p}_{n}}}}{{\left({{q}_{n}}{{\Omega}_{1,n}}\right)}^{{{q}_{n}}}}
\boldsymbol{\omega}_k(i) =\sum_{l\in \mathcal{N}_k}c_{kl}\boldsymbol\Lambda_l^2\boldsymbol{\omega}_l(i-1)\big(\boldsymbol{\omega}_l^H(i-1)\boldsymbol\Lambda_l^2\boldsymbol{\omega}_l(i-1)\big)^{-1}& \times\boldsymbol\omega_l^H(i-1)\boldsymbol\omega_l(i-1).
\begin{aligned}\left(4D^{2}_{1}e^{4tD^{2}_{1}}\right)(x, x')=\left(4\overline{D}^{2}_{1}e^{4t\overline{D}^{2}_{1}}\right) ( x, x') &+(-1)^{{\rm \mathbf{bd}}}\phi^{*}_{\phi^{-1}(x)}\left(4\overline{D}^{2}_{1}e^{4t\overline{D}^{2}_{1}}\right)(\phi^{-1}(x), x').\end{aligned}
\Phi_{R}\left( p^{\mu}\right) = \Lambda_{R}\left( \dot{p}^{\mu}\to p^{\mu}\right) \Phi_{R}\left( \dot{p}^{\mu}\right) = e^{i \vec{J}\cdot ( \vec{\vartheta}- i \vec{\varphi})}~ \Phi_{R}\left( \dot{p}^{\mu}\right) ,
\widehat{K(g,f)}(\eta) = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}b_1(\theta) \left[ \hat{g}(\eta^-)\hat{f}(\eta^+) - \hat{g}(0) \hat{f}(\eta)\right] \,\mathrm{d}\theta, \eta^+ = \eta \cos\theta, \eta^- = \eta \sin\theta.
(Q_{m-1}) \int_0^T |u_{n-1}|^2 dt \le C\Big(\sum_{i=1}^{m-1}e_{1-i}(U_{n-i})(0) + \sum_{p=1}^{m-1}|A^{-p/2}w_{n-m}^{m-p}(0)|^2 &+\int_0^T |w_{n-m}^{m-1}|^2 \,dt\Big) \,.
\mathbb{E}_{N}^{eq}\exp\left\{\sum_{i=1}^{N}\left[\alpha_{i}\left(\widetilde{A}_{i}^{N}(t_{i})-\widetilde{A}_{i}^{N}(s_{i})\right)-\frac{\alpha_{i}^{2}}{\lambda}\left(A_{i}^{N}(t_{i})-A_{i}^{N}(s_{i})\right)\right]\right\}\le1.
[z_{1}\otimes\alpha_{1},[z_{2}\otimes\alpha_{2},z_{3}\otimes\alpha_{3}]]=[[z_{1}\otimes\alpha_{1},z_{2}\otimes\alpha_{2}],z_{3}\otimes\alpha_{3}]+[z_{2}\otimes\alpha_{2},[z_{1}\otimes\alpha_{1},z_{3}\otimes\alpha_{3}]].
\underline{{{\Theta}}}^{0}= \ell{\frac{d r}{r}}\quad , \quad \underline{{{\Theta}}}^{a}={\frac{r}{\ell}}\underline{{{\theta}}}^{a}+{\frac{\ell}{r}}\underline{{{\sigma}}}^{a}+ O ( r^{- 3}) \qquad ,
\sum_{j=1,2}(\|\partial_{t}^{\ell}\nabla_{x}^{k}K_{jM}(t)g\|_{2}+\|\partial_{t}^{\ell}\nabla_{x}^{k}K_{jH}(t)g\|_{2}) & \le C e^{-ct}t^{-\frac{n}{2(1- \sigma)}(\frac{1}{r}-\frac{1}{2})-\frac{k-\tilde{k}}{2(1- \sigma)}}\| \nabla^{\tilde{k}+ 2(1-\sigma) \ell}_{x}g\|_{r},
f_{\psi_3}^L(z) =\frac{2z^2-(1+\sqrt{2})z+\sqrt{2}}{\sqrt{2}z^2-(1+\sqrt{2})z+2},f_{\psi_3}^R(z) =\frac{2^{\frac{3}{2}}z^3 - 2z^{2}- (\sqrt{2}-1)z + 2}{2z^3-(\sqrt{2}-1)z^2-2z+2^{\frac{3}{2}}}.
\partial_{t}^{2}\partial_{i}(\bar q(R)) = \bar q'(R) \partial_{t}^{2}\partial_{i}R + 2\bar q''(R) \partial_{t}\partial_{i}R \partial_{t}R + \bar q''(R) \partial_{i}R\partial_{t}^2 R + \bar q'''(R) (\partial_{t}R)^2 \partial_{i}R
\langle H_y \rangle_{u_{\alpha}\otimes 2 \alpha^{3/2}\Phi_*^1}= 4\alpha^3 \Big( e_{\alpha}+ \alpha \|\lambda_y \|^2 + \|(H_f^+)^{1/2}\tilde{\Phi}_*^1\|^2 + 2 \alpha \|A^-_y \tilde{\Phi}_*^1\|^2\Big) &= 4 \alpha^3 \|\tilde{\Phi}^1_*\|^2_* + O(\alpha^4) = 4 \alpha^3 \|\Phi^1_*\|^2_* + O(\alpha^4),
\begin{cases}u_{tt}-\Delta u +\frac{\mu}{1+t}u_t+\frac{\nu^2}{(1+t)^2}u=f(t,x), x\in \mathbb{R}^n, \t>0, u(0,x)=u_0(x), x\in \mathbb{R}^n, u_t(0,x)=u_1(x), x\in \mathbb{R}^n,\end{cases}
H_{q,n}(f):=\begin{vmatrix}a_{n}a_{n+1}\cdots a_{n+q-1}a_{n+1}a_{n+2}\cdots a_{n+q}\vdots \vdots \vdots \vdots a_{n+q-1}a_{n+q}\cdots a_{n+2(q-1)}\end{vmatrix}.
\nabla_{\mathbf{x}}\mathcal{L}_k(\mathbf{x},\mathbf{y})=(H+ \rho I )\mathbf{x}-A^T \mathbf{y}+\mathbf{g}-\rho{\mathbf{x}_k}- \begin{bmatrix}0 \frac{\mu}{x_{\bar{d}+1}}\vdots \frac{\mu}{x_{d}}\end{bmatrix}=0 ; -\nabla_{\mathbf{y}}\mathcal{L}_k(\mathbf{x}, \mathbf{y}) = (A\mathbf{x}+\delta (\mathbf{y}-\mathbf{y}_k) -\mathbf{b})=0 .
\frac{|H^0(G_{\Sigma},M)| \cdot |H^2(G_{\Sigma},M)|}{|H^1(G_{\Sigma},M)|}=\prod_{v:\ \mathrm{infinite}}|H^0(G_v,M)| \cdot p^{-(r_1+r_2) \cdot v_p(|M|)}.
\langle T,T^{\prime}\rangle_{0}:=\delta_{T,T^{\prime}}\times \prod_{\substack{1\leq i<j\leq N, c(i,T) \leq c(j,T) -2}}\left( 1-\frac{1}{( c(i,T) -c(j,T))^{2}}\right) , T,T^{\prime}\in\mathcal{Y}(\tau) .
H(x)\equiv \nabla^2 f(x) &=\frac{m(m-1)\mathcal{A}x^{m-2}}{\mathcal{B}x^m}-\frac{m(m-1)\mathcal{A}x^m\mathcal{B}x^{m-2}+m^2(\mathcal{A}x^{m-1}\circledcirc\mathcal{B}x^{m-1})}{(\mathcal{B}x^m)^2}&+\frac{m^2\mathcal{A}x^m(\mathcal{B}x^{m-1}\circledcirc\mathcal{B}x^{m-1})}{(\mathcal{B}x^m)^3},
\Pi_S^{\perp}[\partial_{xx}(a_1 \partial_x)+\partial_x (a_0 \cdot)]\Pi_S (\mathcal{A}-\mathrm{I}) \Pi_S^{\perp}=\varepsilon^2 \Pi_S^{\perp}[\partial_{xx}(a_{1, 1}\partial_{xx}\Pi_S[\beta_1 \,\cdot])+\partial_x (a_{0, 1}\,\partial_x\Pi_S[\beta_1\,\cdot])]+o(\varepsilon^2).
ch(Sym^t(\nu))=\frac{(a+1)^{t+1}-(b+1)^{t+1}}{a-b}= =(t+1) +{t+1 \choose 2}(b+a) +{t+1 \choose 3}\left((a+b)^2 - ab\right) + +{t+1 \choose 4}(a+b)\left((a+b)^2 - 2ab\right) +{t+1 \choose 5}\left(a^2b^2 + (a+b)^2\left((a+b)^2 - 3ab\right)\right).
B^{\mathrm{T}}( l_{3}, l_{2}, l_{1}) \; = \; e^{\pi i ( \Delta_{l_{3}}- \Delta_{l_{2}}- \Delta_{l_{1}})}, \qquad \Delta_{l}= \frac{Q^{2}}{4}+ \Bigl ( \frac{l}{4 \pi b}\Bigr )^{2}.
\begin{array}{ll}w^1 = \frac{1}{\sqrt{\pi}}\Big(\int\limits_{-\frac{y}{2\sqrt{t}}}^{+\infty}e^{-\xi^2}w_0^1(2\sqrt{t}\xi + y) \,\mathrm{d}\xi- \int\limits_{\frac{y}{2\sqrt{t}}}^{+\infty}e^{-\xi^2}w_0^1(2\sqrt{t}\xi - y) \,\mathrm{d}\xi\Big).\end{array}
J_L(g,g\cdot\xi_1,\dots,g\cdot\xi_k)=\mu\iff \pmb{F}\ell(\xi_1,\dots,\xi_k)=\big({\rm Ad}^*_{g}\big)^k\mu=({\rm Ad}^*_{g}\mu_1,\dots,{\rm Ad}^*_{g}\mu_k).
[D(2,3s+r+2p):D(3,3s+r)]_q =q^{4p}[D(2,3(s-1)+r+2p):D(3,3(s-1)+r)]_q &+q^{2(3s+r+2p-1)}[D(2,3s+r+2(p-1)):D(3,3s+r)]_q &+q^{3s+r+6p-3}[D(2,3(s-1)+r+2(p-1)+1):D(3,3(s-1)+r)]_q.
\aligned\frac{d}{dt}E(w(t)) = \int ((w(t,x))^{3}- (u(t,x))^{3}) w_{t}(t,x) dx = - \int w_{t}(t,x) [(v(t,x))^{3}+ 3 v(t,x)^{2}w(t,x) + 3 v(t,x) w(t,x)^{2}] dx.\endaligned
B_{11}= P_5, &A_{11}= P_4-(B_{12}-B_{12}^T), &B_{22}= P_3-(B_{13}-B_{31}^T)-(A_{12}-A_{12}^T), &A_{22}= P_2-(B_{23}-B_{23}^T) - (A_{13}-A_{13}^T), &B_{33}= P_1-(A_{23}-A_{23}^T), \mbox{and}&A_{33}= P_0,
\widetilde{J^\textnormal{coh}}(z,Q) = Q^{\frac{H}{z}}\sum_{d \geq 0}\frac{Q^d}{\prod_{r=1}^d \left( H + rz\right)^{N+1}}\in H\left( \mathbb{P}^N \right) \otimes \mathbb{C}[z,z^{-1}][\![Q]\!]
g_{M N}= \left( \begin{matrix}{{e^{- \gamma}e^{- 2 \sigma}g_{\mu \nu}}}&{{0}}&{{0}}\\{{0}}&{{e^{2 \gamma}e^{- 2 \sigma}}}&{{0}}\\{{0}}&{{0}}&{{e^{\sigma}\delta_{i \overline{{j}}}}}\\ \end{matrix}\right) .
\frac{1}{(2\pi )^{n-1}n^{1/2}}\int_{\mathbb{A}^{(n)}_{\texttt{a}}}f(\boldsymbol{\xi}) | C_{\texttt{a}}(\boldsymbol{\xi};q) |^{-2}\boldsymbol{\xi}=\sum_{\lambda\in\Lambda^{(m,n)}_{\texttt{a}}}f \bigl( \boldsymbol{\xi}^{(m,n)}_{\texttt{a};\lambda}\bigr) \hat{\Delta}^{(m,n)}_{\texttt{a};\lambda},
V = q\left({{\cal A}_0 \left( y \right) -{\cal A}_0 \left({y^\prime}\right)}\right) = \frac{{q^2 \sqrt \pi}}{{2g}}\sqrt{\frac{a - 1}{{a}}}\left({1 - e^{- \frac{g}{{\sqrt \pi}}\sqrt{\frac{a}{{a - 1}}}|y - y^\prime |}}\right).
a_{q}< a_{q_0}= \sqrt{d^{\frac2q_0 - 1}(2q_0 - 1)}\leq d^{\frac{1}{\ln d}- \frac12}\sqrt{4\ln d - 1}=\frac{e}{\sqrt{d}}\sqrt{4\ln d - 1}\leq d^{\frac1q - \frac12}\sqrt{32\ln d - 8},
[ J_{0}^{k}, \phi^{c}( w ) ] | \Psi^{a b}\rangle ~ + ~ \phi^{c}( w ) J_{0}^{k}| \Psi^{a b}\rangle = w^{- \Delta_{\phi}}C_{d}^{a b , c}f_{e}^{k d}| \phi^{e}\rangle ~ + ~ . . .
\begin{array}{rcl}S_{{\rm twisted}}\sim \int d^6 y\sqrt{|{\rm det}g|}\left(1+({\tilde{\cal F}}^{(4)})^2+\dots\right)-\int d^6 y\sqrt{|{\rm det}g|}= & &= \int d^6 y\sqrt{|{\rm det}g|}({\tilde{\cal F}}^{(4)})^2+\dots\, , \end{array}
(-1)^{l+m-1}n^{l+2m-2}2^{l+2m-2}P_{l+m-2}(m) \frac{(l+2m-2)!}{(l+2m-3)!}\alpha_p(1)^{l+2m-3}\alpha_p(3) = (-1)^{l+m-1}n^{l+2m-2}2^{l+2m-2}P_{l+m-2}(m)(l+2m-2)\alpha_p(3) + \cdots.
C_6^{\frac{pq}{p-q}}\lesssim \sum_{k=-\infty}^{M-1}\bigg(\int_{x_k}^{x_{k+1}}u(t)\bigg(\int_t^{\infty}u \bigg)^{\frac{q}{p-q}}dt\bigg) \Phi(0,x_k) &+ \sum_{k=-\infty}^{M-1}\sup_{y \in (x_k, x_{k+1})}\bigg(\int_y^{x_{k+1}}u(t)\bigg(\int_t^{\infty}u \bigg)^{\frac{q}{p-q}}dt\bigg) \Phi(x_k,y) & = C_{6,1}+ C_{6,2}.
a_{k,n}^{\left[ j\right]}=\left( -4\right)^{n}\left( -1\right)^{k/2}k!\sum_{m=1}^{j+1-n}\left( V^{-1}\left[ j\right] \right)_{k+1,m}\frac{\left( j+1-m\right)}{n}\binom{j+n-m}{2n-1}\ .
C_{i_1...i_s}^g = \int_{M_{g,s+1}}\langle\oint_{C_{z_1}}\! \! G^-{\bar G}^- \phi_{i_1}... \oint_{C_{z_s}}\! \! G^-{\bar G}^- \phi_{i_s}\prod_{j,{\bar j}=1}^{3g-3}G^- (\chi_j){\bar G}^- ({\bar \chi}_{\bar j}) \rangle
\begin{aligned}\left\|\frac{\Delta_\sigma^{k_1}V_{t,s}(y)}{|t-s|^\frac{1}{p}}\right\|_{L^{2a}}\leq C_1^V(\sigma\,;y), \qquad\textrm{and}\left\|\frac{\Delta_\sigma^{k_1-1}W_{t,s}(y)}{|t-s|^\frac{2}{p}}\right\|_{L^a}\leq C_1^W(\sigma\,;y),\end{aligned}
q(x) = \frac{1}{1+25x^2}+ \frac{\alpha \left(1 + \sinh(1+x)\right)}{(1+x)^2}, \,\,\, \alpha=\frac{1}4,\frac{3}4,\frac{5}4,\begin{array}{l}y(-1)=0, y(1) = 2y'(1). \end{array}
[\begin{matrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{matrix}]=\frac{E}{1-\nu^{2}}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{12}\end{matrix}].
\int_0^x\mathrm{e}^{-\gamma t}t^{\nu}\mathbf{L}_{\nu+n}(t)\,\mathrm{d}t =\int_0^x\mathrm{e}^{-\gamma t}\frac{1}{t^{n+1}}t^{\nu+n+1}\mathbf{L}_{\nu+n}(t)\,\mathrm{d}t & >\frac{\mathrm{e}^{-\gamma x}}{x^{n+1}}\int_0^xt^{\nu+n+1}\mathbf{L}_{\nu+n}(t)\,\mathrm{d}t =\mathrm{e}^{-\gamma x}x^{\nu}\mathbf{L}_{\nu+n+1}(x),
\varepsilon_X V(\rho)V(\gamma){\varepsilon_S}^{-1}= \varepsilon_X V(\rho\gamma){\varepsilon_S}^{-1}& = \varepsilon_X VU(\alpha){\varepsilon_S}^{-1}& = \alpha \varepsilon_S{\varepsilon_S}^{-1}& = \alpha,
\dfrac{I(t)-I(0)}{t}=\frac{1}{t}\int_0^t(\left( \beta I(s)S(s)-(\rho+\phi+\mu)I(s)+\alpha A(s)+\omega C(s)\right)ds &+\frac{1}{t}\int_0^t \sigma ISdW_s+\frac{1}{t}\int_0^t \int_U J(u)IS \check{N}(ds,du) & \geq -(\rho+\phi+\mu)<I(t)>+\frac{1}{t}\int_0^t \sigma ISdW_s+\frac{1}{t}\int_0^t \int_U J(u)IS \check{N}(ds,du).
(i) \ \ \Vert u - (\bar{A}_j \, \cdot + \bar{c}_j)\Vert^2_{L^2(\Omega_u)}\le C C_\rho^2\varrho^2 (\alpha + \epsilon \Vert W \Vert_*), (iii) \ \ \Vert (\bar{A}_{j_1}\, \cdot + \bar{c}_{j_1}) - (\bar{A}_{j_2}\, \cdot + \bar{c}_{j_2})\Vert^2_{L^2(\Omega_u^H )}\le CC_\rho^2 \varrho^2 (\alpha + \epsilon \Vert W \Vert_*)
W_{z}\left( \begin{array}{c c}{a + N \bar{\epsilon}_{j}}&{b + N \bar{\epsilon}_{j}}\\{d + N \bar{\epsilon}_{j}}&{c + N \bar{\epsilon}_{j}}\\ \end{array}\right) = W_{z}\left( \begin{array}{c c}{a}&{b}\\{d}&{c}\\ \end{array}\right) .
H = \frac{p^{2}}2 + \frac 1 2 W^{2}( x ) + v + W^{\prime}( x ) \left[ \alpha + R ( W ( x ) ) \right] N , \quad Q^{+}= z^{\alpha}e^{i \int_{\varphi_{0}}^{\varphi}R ( \rho \cos \lambda ) \, d \lambda}\theta^{+},
\mathcal{M}^0 = \{u \in \mathcal{M}: \langle J''(u)u,u \rangle = 0\}, \mathcal{M}^+ = \{u \in \mathcal{M}: \langle J''(u)u,u \rangle > 0\}, \mathcal{M}^- = \{u \in \mathcal{M}: \langle J''(u)u,u \rangle < 0\}.
\int_{- 1}^{{x_{n,j}^{(\alpha )}}}{f(x) dx}= \sum\limits_{i = 0}^n{p_{B,j,i}^{(1)}}{f_{n,i}^{(\alpha )}}+ E_n^{(\alpha )}\left({x_{n,j}^{(\alpha )}},{\zeta_{n,j}^{(\alpha)}}\right)\, \forall x_{n,j}^{(\alpha )}\in \mathbb{S}_n^{(\alpha )},
\cdot:\Gamma_g\times \mathcal{C}_g \longrightarrow \mathcal{C}_g \cr(\gamma,m) \longmapsto \gamma \cdot\begin{bmatrix}m' m'' \end{bmatrix}:= \left [ \begin{pmatrix}d -c - b a \end{pmatrix}\begin{pmatrix}m' m'' \end{pmatrix}+ \begin{pmatrix}diag(c{}^t d ) diag( a{}^t b) \end{pmatrix}\right ] \, 2.
\frac{1}{1-\alpha}log(\int_{\mathbb{R}}|f(x)|^{2\alpha}dx)+\frac{1}{1-\beta}log(\int_{\mathbb{R}}|g(y)|^{2\beta}dy)\ge\frac{\alpha}{2(\alpha-1)}log\frac{(2\alpha)^{1/\alpha}}{(2\beta)^{1/\beta}};
\begin{aligned}\boldsymbol{K}_\ell\left(\mu_\ell,\gamma^{\prime \prime}_{\ell,3}\right) \leq \lambda \boldsymbol{K}_\ell( \mu_\ell, \gamma_\ell) + (1-\lambda) \boldsymbol{K}_\ell \left( \mu_\ell,\gamma_\ell^\prime \right) \leq \lambda \delta_\ell + (1- \lambda) \delta_\ell^\prime.\end{aligned}
\Delta\check\S^{3124}=1 \otimes \check\S^{3124}+ \check\S^{1}\otimes \check\S^{123}+ \check\S^{1}\otimes \check\S^{213}+\check\S^{1 2}\otimes \check\S^{1 2}+ \check\S^{21}\otimes \check\S^{12}+ \check\S^{312}\otimes \check\S^{1}+ \check\S^{3124}\otimes 1,
f_{\mathbf{M^{\ast\ast}}_{-\xi_{\boldsymbol{s}k}}}(\boldsymbol{v})=c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast\ast}\exp\left\{-\frac{1}{2}\boldsymbol{v}^{T}\boldsymbol{v}-\frac{1}{2}\xi_{\boldsymbol{s}k}^{2}\right\}=\phi_{n-1}(\boldsymbol{v}),~\boldsymbol{v}\in \mathbb{R}^{n-1},
\begin{aligned}0 \geq \sum_l X_{l\bar li\bar i}- \frac{w_i w_{\bar i}}{w}- w A \varphi_{t,i\bar i}&= \sum_l \left(X_{i\bar il\bar l}- R_{l\bar l i\bar i}X_{i\bar i}+ R_{i\bar il\bar l}X_{l\bar l}+ G_{i\bar il\bar l}\right) - \frac{w_i w_{\bar i}}{w}- w A \varphi_{t,i\bar i}.\end{aligned}
S^{BH}={\pi\over 6}\sum_{i=1}^{N}\int_{\Sigma}d\sigma \left[2\langle \hat{\phi}_{i}^2 \rangle+{1 \over m_{d,i}}\langle \hat{\bar{\psi}}_{i}\hat{\psi}_{i}\rangle-2\langle \hat{V}^2_{i}\rangle\right]~~~.
\phi_T(X) = TX + gX^q - h^{q-1}X^{q^2}, \psi_T(X) = TX + h^{q-1}X^{q-1}, \lambda = \lambda_T\,h^{-1}\in \psi[T], \tilde{\jmath}(\phi,\lambda) = g^{(q+1)/2}/ h^{(q-1)/2}
\left|U_1U_2-\epsilon U^2\right|^2 =(U_1U_2-\epsilon U^2)(\overline{U}_1\overline{U}_2-\epsilon \overline{U}^2) &=\left| U_1\right|^4+2\epsilon\left(\left| U\right|\left| U_1\right|\right)^2+\left| U\right|^4 &=(\left| U_1\right|^2+\epsilon \left| U\right|^2)^2.
\frac{\partial}{\partial \theta}\begin{array}{c}lim \varepsilon \rightarrow 0\end{array}I(p,m,\theta ,\varepsilon )\mid_{_{\theta^{-}_{0}}}\neq \frac{\partial}{\partial \theta}\begin{array}{c}lim \varepsilon \rightarrow 0\end{array}I(p,m,\theta ,\varepsilon )\mid_{_{_{\theta_{0}^{+}}}}
S_{p}^{( 0 )}= - \tau_{p}\int d^{p + 1}\sigma e^{\Phi ( - 1 - \gamma ( p + 1 ) / 2 )}\sqrt{- d e t [ \tilde{G}_{a b}+ e^{\gamma \Phi}( \tilde{B}_{a b}+ 2 \pi \alpha^{\prime}F_{a b}) ]},
\begin{aligned}{R_{{\rm{DB}}}}&\approx \frac{1}{{{M_{\rm{R}}}}}{\log_2}\left({1 + \frac{{E{\sum\limits_{m = 1}^{{M_{\rm{R}}}}{\sum\limits_{k = 1}^K{\sum\limits_{n = 1}^K{{\left|{\left[{{{\bf{\Xi}}_{{\rm{DB}},m}}}\right]_{k,k}^*{{\left[{{{\bf{\Xi}}_{{\rm{DB}},m}}}\right]}_{n,n}}}\right|}}}}}}}{{K{\sigma^2}}}}\right)\end{aligned}
G_{MN}dx^Mdx^N=H^{-{{4(n-1)}\over{(p+n)\Delta_p}}}_p\left[-dt^2+d{\bf x}\cdot d{\bf x}\right]+H^{{4(p+1)}\over{(p+n)\Delta_p}}_p\left[d\tilde{y}^2+g_{ij}dy^idy^j\right],
\sum_{n=1}^{\infty}n^{s-\frac{1}{2}}K_{s-\frac{1}{2}}(2\pi n\,x)+\frac{(\pi x)^{\frac{1}{2}-s}}{4}\Gamma\left(s-\frac{1}{2}\right)-\frac{\pi^{-s}x^{-s-\frac{1}{2}}\Gamma(s)}{4}.