harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi_c \:C\:i^{|I_1|+\cdots+|I_k|}\:(-i)^{|K|}\;\gamma^I\;(\partial^{I_k}V^{(k)}_{J_k, c_k}(x)) \cdots(\partial^{I_1}V^{(1)}_{J_1, c_1}(x))\:(y-x)^K \;S^{\vee (h)}(x,y) \;\;\; ,
\begin{array}{c}max\Big\{min\Big[rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f(x,i,j);\qquad\qquad\qquad\qquad V^{ij}(x)-M^{ij}[V](x)\Big];V^{ij}(x)-N^{ij}[V](x)\Big\}=0,\end{array}
\chi=\sum_{\nu\in \mathcal{N}_h}\beta^{(\nu)}\phi^{(\nu)},{\rm where}\;\beta^{(\nu)}=\frac{1}{|\omega_\nu|}\sum_{x_K^{(j)}=\nu}\alpha_K^{(j)}{\rm if}\ \nu\in\mathcal{N}_h\{\rm and\}\nu\not\in\Gamma_1.
\sum_{j = 0}^k{2^j G_j^2}= \frac{{11}}{3}G_0^2 + 2G_1^2 - \frac{4}{3}G_2^2 - \frac{{11}}{3}2^{k + 1}G_{k + 1}^2 &\quad- 2^{k + 2}G_{k + 2}^2 + \frac{1}{3}2^{k + 3}G_{k + 3}^2
K_{p}( z ) \stackrel{( z \rightarrow 0 )}{\sim}\frac{1}{2}\, \left[ \Gamma ( p ) \left( \frac{z}{2}\right)^{- p}+ \Gamma ( - p ) \left( \frac{z}{2}\right)^{p}\right] \, \left[ 1 + O ( z^{2}) \right] \; .
\Theta_m = (2n+1)X_2 + (2n+1)2nX_1 + 2nY_2 + (2n+1)Y_1 \Theta_b = X_2 + 2nX_1 + Y_2 \Psi_r = (2n-1)X_2 + (2(n-1)(2n+1) + 4) X_1 + (2n+1)Y_2 + (2n-1)Y_1 \Psi_g = X_2 + (2n-1)X_1 + Y_2 + 2Y_1
\partial_t\mathcal{Z}^\alpha h-v^b\cdot\mathcal{Z}^\alpha\mathbf{N}-V^\alpha\cdot\mathbf{N}=-[\mathcal{Z}^\alpha,v{_y^b},\nabla_yh]+\frac{(\partial_zv)^b}{\partial_z\varphi}\mathcal{Z}^\alpha h\mathbf{N}:=\mathcal{C}^\alpha(h).
\frac{\dot{z}_{n}~z_{m}+\dot{z}_{m}~z_{n}}{z_{n}-z_{m}}=\frac{\dot{\theta}_{n}~\sin \theta_{m}~\cos \theta_{m}+\dot{\theta}_{m}~\sin \theta_{n}~\cos \theta_{n}}{\cos \theta_{n}~\cos \theta_{m}~\sin \left( \theta_{n}-\theta_{m}\right)}~,
\mathrm{Nm}_{F'/F}\left( c_{F'}\right) = \left( \prod_{\ell \in S' \setminus S}\left( P_\ell \left( \sigma^{-1}_\ell \right) \right) \right) \cdot c_F &= \left( \prod_{\ell \in S' \setminus S}\left( 1 - \overline{a_\ell(f)}\ell^{-1}\sigma^{-1}_\ell + \overline{\psi}(\ell)\ell^{-1}\sigma^{-2}_\ell \right) \right) \cdot c_F
\Theta^{\mu \nu}( x ) ={\frac{1}{2}}\int_{S}d^{2}S^{\mu \nu}( x ( \sigma ) ) \delta^{4}( x - x ( \sigma ) ) , \quad d^{2}S^{\mu \nu}: = d \sigma^{1}d \sigma^{2}{\frac{\partial ( x^{\mu}, x^{\nu})}{\partial ( \sigma^{1}, \sigma^{2})}}.
\frac{e^{-\gamma t}}{20\pi}[(8-3e^{-\gamma t}+2\gamma t+4\gamma^{2}t^{2})\frac{\delta(ct-r)}{r^{2}}+\frac{\gamma^{2}}{c}\Theta(ct-r)(\frac{1}{cu}I_{1}(\frac{\gamma u}{c})+\frac{4t}{u^{2}}I_{2}(\frac{\gamma u}{c}))]
\hat{\hat{K}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\; = \;{\textstyle \frac{1}{2}}\left( \hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{c}}\hat{\hat{b}}}\; + \; \hat{\hat{T}}_{\hat{\hat{b}}\hat{\hat{c}}\hat{\hat{a}}}\; - \; \hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\right) \; ,
\underline{u}'(z_1^-)=-\beta<0=\underline{u}'(z_1^+), begin{align*}0.2cm] \overline{v}'(0^+)=0<q(1)\lambda_1=\overline{v}'(0^-), begin{align*}0.2cm] \underline{v}'(z_2^-)=-q(1)\varepsilon e^{\lambda_1 z_2}<0=\underline{v}'(z_2^+).
(t+1)(f^{(t+1)}(\mathbf{x}^{(t+1)})-f^{(t)}(\mathbf{x}^{(t+1)})) =l^{(t+1)}(\mathbf{x}^{(t+1)}) -\frac{1}{t}\sum_{\tau=1}^{t}l^{(\tau)}(\mathbf{x}^{(t+1)}),
\beta_{k 0}( t \to \infty ) = \sqrt{\frac{\omega_{k}}{\tilde{\omega}}}\frac{( \tilde{\omega}- \omega_{k}) ( 2 \omega_{k}^{2}- 2 \tilde{\omega}^{2}+ i \pi g \omega_{k})}{\left[ 4 \left( \omega_{k}^{2}- \tilde{\omega}^{2}\right)^{2}+ \pi^{2}g^{2}\omega_{k}^{2}\right]}\sqrt{g \Delta \omega}\mathrm{e}^{i \omega_{k}t}.
Z_{t}^{i, N}= Z^{i, N}_{0}+ \int_{0}^t b\left(Z_{s}^{i, N}- M_{s}^{i, N}\right) ds + \frac{\alpha}{N}\sum_{j = 1}^N M_{t}^{j, N}+ W_{t}^i, &M_{t}^{i, N}= \bigl \lfloor \bigl(\sup_{s \in [0,t]}Z_{s}^{i, N}\bigr)_{+}\bigr\rfloor =\sup_{s \in [0,t]}\bigl \lfloor \bigl(Z_{s}^{i, N}\bigr)_{+}\bigr\rfloor,
\operatorname*{tr}\left( I+\mathbb{A}\right)^{-1}\partial\mathbb{A}=-\operatorname*{tr}\left\langle \cdot,a\right\rangle \left( I+\mathbb{A}\right)^{-1}a & =-\left\langle \left( I+\mathbb{A}\right)^{-1}a,a\right\rangle .
|R_{k}(z)|\le\sum_{j=k+1}^{\infty}\frac{M_{r}|z-c|^{j}}{r^{j}}=\frac{M_{r}}{r^{k+1}}\frac{|z-c|^{k+1}}{1-\frac{|z-c|}{r}}\le\frac{M_{r}\beta^{k+1}}{1-\beta},\frac{|z-c|}{r}\le\beta<1.
\langle \alpha |\beta \rangle =det_{\mathsf{N}}||\mathcal{M}_{a,b}^{\left(\alpha ,\beta \right)}||\mathcal{M}_{a,b}^{\left( \alpha,\beta \right)}\equiv \sum_{h=0}^{p-1}\alpha_{a}^{(h)}\beta_{a}^{(h)}(X_{a}^{(h)})^{(b-1)}.
L(x_1,y_1,\ldots,x_K,y_K,\dot x_1, \dot y_1,\ldots,\dot x_K,\dot y_K) = \frac{1}{2}\sum_{i=1}^K \Gamma_i (x_i \dot y_i - y_i \dot x_i) - H(x_1,y_1,\ldots,x_K,y_K)
P_{(r)}\big(t^{N-1}, \dots, t, 1; q, t\big) = \prod_{j=2}^N{\frac{\big(q^{r}t^{j-1}; q\big)_{\infty}(t^j; q)_{\infty}}{(q^rt^j; q)_{\infty}(t^{j-1}; q)_{\infty}}}={\frac{(q^rt; q)_{\infty}(t^N; q)_{\infty}}{(q^rt^N; q)_{\infty}(t; q)_{\infty}}}= \frac{\big(t^N; q\big)_r}{(t; q)_r}.
P_1^1=-\frac{1}{4m}V^{\prime \prime}\left(x^1\right),P^1_2=0,P^2_1=-\frac{1}{2}V^{\prime \prime}\left(x^1\right)y^1,P_2^2=-\frac{1}{4m}V^{\prime\prime}\left(x^1\right).
[y_{(1\,l)},y_{(2\,p)}]_c=\begin{cases}\lambda_{122}(1-\xi^2)\chi_2(g_{(3\,l)})y_{(3\,l)}, p=2 \lambda_{122}(1-\xi^2)^2\chi_2(g_{(3\,l)})y_{(3\,l)}y_{(3\,p)}, p>2. \end{cases}
\mathcal{P}(x,y;t)=\hat{\phi}_0(x) \sum_{n=0}^{\infty}\Bigl( e^{-\mathcal{E}(n)t}\hat{\phi}_n(x)\hat{\phi}_n(y) +e^{-\mathcal{E}^{\prime}(n)t}\hat{\phi}^{(-)}_n(x)\hat{\phi}^{(-)}_n(y) \Bigr)\hat{\phi}_0(y)^{-1}\ \ (t>0),
T^+u(z,\xi, w)=\sum_{i,j\geq0}\sum_{0\leq \alpha\leq j/2}u_{ij}\binom{j}{2\alpha}z^i(w+\frac{1}{4}\xi^2)^\alpha(-\frac{1}{2}\xi)^{j-2\alpha}, T^-u(z,\xi,w)=\sum_{i\geq0,j>0}\sum_{0\leq \alpha<j/2}u_{ij}\binom{j}{2\alpha+1}z^i(w+\frac{1}{4}\xi^2)^\alpha(-\frac{1}{2}\xi)^{j-2\alpha-1}.
M_{2}(\Gamma_{0}(20))=\mbox{span}_{\mathbb{C}}\left\{2P(q^{2})-P(q),\,4P(q^4)-P(q),\,5P(q^5)-P(q),\atop10P(q^{10})-P(q),\,20P(q^{20})-P(q),\,z\right\}.
\tilde{g}_{\mu \nu}= \left( \begin{array}{l c}{{e^{- U}}}&{{e^{- U}b_{j}}}\\{{e^{- U}b_{i}}}&{{e^{- U}b_{i}b_{j}+ e^{- U}\tilde{\gamma}_{i j}}}\\ \end{array}\right) \ , B_{\mu \nu}= \left( \begin{array}{l c}{{0}}&{{\omega_{j}}}\\{{- \omega_{i}}}&{{b_{i j}}}\\ \end{array}\right) \ .
\int_{- 1}^{{x_k}}{{P_{k,m}}f(x)\,dx}= \frac{{{x_k}+ 1}}{2}\sum\limits_{i = 0}^m{f_{m,k,i}^{(\alpha_k^*)}\,\int_{- 1}^1{{\mathcal{L}}_{OB,m,i}^{(\alpha_k^*)}(t; - 1,{x_k})\,}\,dt}.
\mbox{Tr}[ (D^2 + B + m^2)^{-1}- (P^2 + m^2)^{-1}]=\int_{0}^{\infty}\, dt e^{- t m^2}\int\,d^2r <\!{\bf r}\,|\,e^{-(D^2 + B)t}- e^{-P^2 t}\,|\,{\bf r}\!>\,.
\sum_{q \geq 2}\frac{\Lambda(q)}{\phi(q)^2}= \sum_{p}\sum_{j = 1}^\infty \frac{\Lambda(p^j)}{\phi(p^j)^2}= \sum_{p}\frac{\log p}{(p-1)^2}\sum_{j = 1}^\infty \frac{1}{(p^{j-1})^2}= \sum_{p}\frac{\log p}{(p-1)^2}\frac{p^2}{p^2-1}= 4A_0,
\left\|g_u^{-1}(x) - g_w^{-1}(x)\right\| = \left\| g_u^{-1}( g_u(z) ) - g_v^{-1}( g_u(z) ) \right\|= \left\| g_v^{-1}( g_v(z) ) - g_v^{-1}( g_u(z) ) \right\| &\leq \Lambda \| g_v(z) - g_u(z) \|\leq \Lambda \sqrt{d}\|g_v - g_u\|_{L_{\infty}([0, 1]^d)}
s(f - xy^n) = s x y^2(x-y)\left(\sum_{j=0}^{m-1}x^{n-3-2j}y^{2j}\right) &\equiv xy(x^2 -y^2)\left(\sum_{j=0}^{m-1}x^{n-3-2j}y^{2j}\right) &= xy(x^{2m}- y^{2m}) = x^{n}y - x y^{n}.
\partial_y \log \widetilde Z_{n,\alpha}(y;r) = \frac{n}{4\pi i y}\int^\infty_{yr}x \left(\mathrm{Tr}\left({Y}^{-1}_{+}(x){Y}^{\prime}_{+}(x)\sigma_3 \right) - \mathrm{Tr}\left({Y}^{-1}_{-}(x){Y}^{\prime}_{-}(x) \sigma_3 \right)\right) dx.
\begin{array}{l}{{\dot{J}_{1}= [ J_{1}, H ]_{G P B}= \displaystyle \frac{I - I_{3}}{I I_{3}}J_{2}J_{3}~ ,}}\\{{\dot{J}_{2}= [ J_{2}, H ]_{G P B}= \displaystyle \frac{I_{3}- I}{I I_{3}}J_{1}J_{3}~ ,}}\\{{\dot{J}_{3}= [ J_{3}, H ]_{G P B}= 0 ~ .}}\\ \end{array}
E_{m}=\frac{RT}{F}ln(\frac{P_{K}[K^{+}]_{out}+P_{Na}[Na^{+}]_{out}+P_{Cl}[Cl^{-}]_{in}}{P_{K}[K^{+}]_{in}+P_{Na}[Na^{+}]_{in}+P_{Cl}[Cl^{-}]_{out}})
\sigma_{2}(G) \le d_{G}(v_{1}) + d_{G}(u_{2}) = d_{G}(v_{1}) + \big( d_{G_{2}}(u_{2}) + d_{G_{1}}(u_{2}) \big) &\le (s_{1}+ s_{2}- 2) + \big( (s_{2}- 1) + (s_{1}+ 1) \big) = 2(s_{1}+ s_{2}) - 2,
\frac{d}{dT}\left[ T \left( L_{{\rm eff}}(\omega)+ E + \frac{\omega}{2}\right) \right] = L_{{\rm eff}}(\omega)+ E + \frac{\omega}{2}+ T \frac{d\omega}{dT}\left( \frac{dL_{{\rm eff}}}{d\omega}+ \frac{1}{2}\right) = 0 \,.
\hat{\Delta}^{(m,n)}_{\texttt{a};\lambda}:= \Biggl(\sum_{\mu\in \Lambda^{(m,n)}_{\texttt{a}}}\left| P_{\texttt{a};\mu}\bigl(\boldsymbol{\xi}^{(m,n)}_{\texttt{a};\lambda};q\bigr) \right|^2 \delta^{(m,n)}_{\texttt{a};\mu}(q) \Biggr)^{-1}.
f_{k}( x ) ={\frac{1}{\sqrt{4 \pi \omega}}}\left\{\Theta ( - x ){\frac{1}{\sqrt{n}}}T e^{i ( n k x - \omega t )}+ \Theta ( x ) \left[ e^{i ( k x - \omega t )}- R e^{- i ( k x + \omega t )}\right] \right\}.
x_{1}^{\left(t_{0}+1\right)}=\gamma_{1}x_{1}^{\left(t_{0}\right)}y^{\left(t_{0}\right)},x_{2}^{\left(t_{0}+1\right)}=\gamma_{2}x_{1}^{\left(t_{0}\right)}y^{\left(t_{0}\right)},y^{\left(t_{0}+1\right)}=\gamma x_{1}^{\left(t_{0}\right)}y^{\left(t_{}\right)}.
(\underline{X},\underline{A})^{\emptyset}=(\underline{X},\underline{A})^{\emptyset,c}=\prod^m_{j=1}A_j \qquad\mbox{and}(\underline{Y},\underline{B})^{\emptyset}=(\underline{Y},\underline{B})^{\emptyset,c}=\prod^m_{j=1}B_j,
f_1(\mu)=\hat{\lambda}_{\max}^{\mu\geq\mu_0}\log_2(1+\mu)=n\sum_{i=0}^{n-1}\binom{n-1}{i}r_i \left(1-\hat{q}_0\right)^{n-1-i}\hat{q}_0^{i+1}\cdot \log_2(1+\mu)
\frac{u_{-}-v_{+}+h}{u_{-}-v_{+}}k_{1}^{+}(u)k_{1}^{-}(v)e_{1}^{-}(v)=k_{1}^{-}(v)e_{1}^{-}(v)k_{1}^{+}(u)+\frac{h}{u_{+}-v_{-}}k_{1}^{-}(v)k_{1}^{+}(u)e_{1}^{+}(u)
\mu(x) = \frac{1}{2\pi \i}(R(x-\i 0)-R(x + \i 0))=|\alpha_x\rangle\langle\alpha_x| |\alpha_x\rangle = (1+\pi^2)^{-1/2}\left(\frac{\mathcal{P}}{x-\Omega}|E\rangle+|\delta_x\rangle\right)
S_{m-1}^{m,1}=a_0\dots a_{m-2},\; m\ge 2;\;S_{r+1}^{1,1}=a_0\dots a_{r-1};S_{n-1}^{1,n}=1,\;S_{n+q}^{1,n}=\sum_{l=0}^{n-1}b_l;
\int_X e^{ik\cdot h_n}v_{-k}\,w_k\circ f^n\,d\mu=\int_\Delta e^{ik\cdot \hat h_n}\hat v_{-k}\,\hat w_k\circ \hat f^n\,d\mu_\Delta=\int_X 1_{\hat Y}L_k^n1_{\hat Y}\hat v_{-k}\,\hat w_k\,d\mu_\Delta.
{\cal K}_{i}=\sum_{a}\sum_{s=0}^{\infty}D^{s}[-\varphi_{a}\delta^{j}_{i}+v^{j}_{a}(x_{ai}-\varphi_{a}\varphi_{ai})\varphi^{-1}_{at}]\frac{\partial}{\partial x_{a}^{j(s)}}.
\lambda(p^\mu)\,\equiv \left( \begin{array}{c}\left(\zeta_\lambda\,\Theta_{[j]}\right)\,\phi^\ast_{_L}(p^\mu) \phi_{_L}(p^\mu) \end{array}\right)\,\,,\quad\rho(p^\mu)\,\equiv\left(\begin{array}{c}\phi_{_R}(p^\mu) \left(\zeta_\rho\,\Theta_{[j]}\right)^\ast\,\phi^\ast_{_R}(p^\mu)\end{array}\right)\,\,.
\lefteqn{\sum_i \left(\lambda_{A}^i(x_A) - \sum_{y_a}\lambda_{A}^i(x_{A \setminus a}, y_a)\right) \lambda_{B}^i(x_B)}&= \sum_i \lambda_{A}^i(x_A) \lambda_{B}^i(x_B) - \sum_{y_a}\sum_i \lambda_{A}^i(x_{A \setminus a}, y_a) \lambda_{B}^i(x_B) &= \lambda(x_A, x_B) - \sum_{y_a}\lambda(y_a, x_{A \setminus a}, x_B) &= \lambda(x_A, x_B).
\int_{-\pi}^{\pi}\left(P(h;re^{i\theta})-S_{\lambda}h(re^{i\theta})\right)\overline{f(e^{i\theta})}dm_{\lambda}(\theta) &=\int_{-\pi}^{\pi}h(e^{i\varphi})re^{i\varphi}\,\overline{\int_{-\pi}^{\pi}e^{i\theta}C(e^{i\theta},re^{i\varphi})f(e^{i\theta})dm_{\lambda}(\theta)}\,dm_{\lambda}(\varphi) &=0.
\langle \partial_s u(s,\cdot), v(s,\cdot) \rangle-\langle A^*(s,\cdot)\nabla u(s,\cdot),\nabla v(s,\cdot) \rangle = 0, \langle u(s,\cdot),\partial_s v(s,\cdot) \rangle+\langle\nabla u(s,\cdot), A(s,\cdot)\nabla v(s,\cdot) \rangle = 0.
\begin{array}{l l}{{D + \alpha \tilde{D}+ \frac{e}{2}E_{I}| \sigma_{I}|^{2}+ k = 0 ,}}\\{{\tilde{D}+ \alpha D + \frac{q}{2}Q_{i}| \phi_{i}|^{2}+ \eta = 0 ,}}\\{{\bar{G}_{i}= - \frac{\partial W}{\partial \bar{\phi}_{i}}= - m \phi_{i},}}\\{{\bar{F}_{I}= - \frac{\partial W}{\partial \bar{\sigma}_{I}}= - \tilde{m}\sigma_{I}.}}\\ \end{array}
\|v_t\|_{L^2(\mathrm{M}_0)}^2 =(u|\partial_\nu w_t)_{L^2(\partial \mathrm{M}_0)}-(\partial_\nu u|w_t)_{L^2(\partial \mathrm{M}_0)}&\le \|u\|_{L^2(\partial \mathrm{M}_0)}\|\partial_\nu w_t\|_{L^2(\partial \mathrm{M}_0)}+\|\partial_\nu u\|_{L^2(\partial \mathrm{M}_0)}\|w_t\|_{L^2(\partial \mathrm{M}_0)}.
(1,-1) \cdot (x_1, x_2, x_3, x_4, x_5, x_6, x_7) = (x_1, x_2, x_3, -x_4, -x_5, -x_6, -x_7) (-1,1) \cdot (x_1, x_2, x_3, x_4, x_5, x_6, x_7) = (x_1, -x_2, -x_3, x_4, x_5, -x_6, -x_7).
{\mathcal{K}}( \hat{M}_{f}, \hat{M}_{b}) = \mathrm{c o n s t}\; \int{\mathcal{D}}{\mathcal{F}}{\mathcal{D}}{\mathcal{B}}\exp \left\{-{\mathcal{W_{F}}}-{\mathcal{W_{B}}}-{\mathcal{W_{F B}}}\right\}
2 \mathrm{Re}\langle u_{\alpha}\otimes 2 \alpha^{3/2}\eta \Phi_*^1\,|\, H_y R_y \rangle = 2 \mathrm{Re}\langle u_{\alpha}\otimes 2 \alpha^{3/2}\eta \tilde{\Phi}_*^1\,|\, \alpha V_y \Phi^y_{\#}\rangle &+ 2 \mathrm{Re}\langle u_{\alpha}\otimes 2 \alpha^{3/2}\eta \tilde{\Phi}_*^1\,|\, \alpha V_y R^{\#}_y \rangle,
\Phi ( \omega ; \lambda ) \simeq{\frac{2 \omega^{D - 2}}{( 4 \pi )^{( D - 1 ) / 2}}}\sum_{n = 0}^{\infty}\left[{\frac{a_{n}( \lambda )}{\Gamma \left({\frac{D - 1}{2}}- n \right)}}\omega^{- 2 n}+{\frac{b_{n}( \lambda )}{\Gamma \left({\frac{D - 2}{2}}- n \right)}}\omega^{- ( 2 n + 1 )}\right] .
R_{nl}(r)=\sqrt{{(\frac{2Z}{na_{\mu}})}^{3}\frac{(n-l-1)!}{2n(n+l)!}}e^{-Zr/na_{\mu}}(\frac{2Zr}{na_{\mu}})^{l}L_{n-l-1}^{(2l+1)}(\frac{2Zr}{na_{\mu}})
\pi ( u ) = \frac{2 k - 1}{4 \pi}\int_{\tau = 0}^{\infty}d \tau \; \sinh \tau \int_{0}^{2 \pi}d \theta \left[ \cosh \frac{\sigma}{2}- i \hat{n}. \hat{r}\sinh \frac{\sigma}{2}\right]^{- 2 k}
Kurt(Z)= &\qquad\frac{\sigma_1^4}{\sigma^4}\frac{\Gamma(\frac{1}{\alpha})\Gamma(\frac{5}{\alpha})}{\Gamma(\frac{3}{\alpha})^2}+\frac{\sigma_2^4}{\sigma^4}\frac{\Gamma(\frac{1}{\beta})\Gamma(\frac{5}{\beta})}{\Gamma(\frac{3}{\beta})^2}+6\frac{\sigma_1^2\sigma_2^2}{\sigma^4}-3
\frac{1}{2}\sum_{x\in \mathcal{S}, g \in \rm{DA}}\left(h(x.g)-h(x)\right)^2\mu(g)=\sum_{n=0}^\infty \left(\frac{a}{q^n}\right)^2\frac{q^n}{4}+\left(\frac{a}{2q^{n+1}}\right)^2\frac{q^n}{2q}\frac{q(q-1)}{2}<\infty.
[W,e^{\hat{K}}]=\left[W,\sum_{N=0}^{\infty}\frac{1}{N!}\hat{K}^N\right]=\sum_{N=0}^{\infty}\frac{1}{N!}\hat{K}^N\left([W,\hat{K}]\right)=e^{\hat{K}}([W,\hat{K}]) \ .
2^{n(d-k)}\left|\Delta f_n^{(k)}( \gamma)-\frac 1{2^n}f_n^{(k+1)}( \gamma) + \sum_{\ell=2}^{d-k}t_{k,k+\ell}\frac 1{2^{n\ell}}f_n^{(k + \ell)}( \gamma)\right|\le \varepsilon_n,
\ell \left( \frac{x}{\delta}\right) =\frac{x}{\delta}\log \left( \frac{x}{\delta}\right) -\frac{x}{\delta}+1 & =\frac{1}{\delta}\ell (x)+\frac{\delta -1}{\delta}-\frac{(x-1)}{\delta}\log \delta -\frac{\log \delta}{\delta}& \leq \frac{1}{\delta}\ell (x)+|x-1|\frac{|\log \delta |}{\delta}+\frac{\delta +|\log \delta |}{\delta}.
\sum_{\substack{n\in\mathcal{A}(x) n\equiv v_0 \pmod{W}}}w_n=\Bigl(1+O\Bigl(\frac{1}{(\log{x})^{1/10}}\Bigr)\Bigr)\frac{W^{k-1}B^k \mathfrak{S}_{WB}(\mathcal{L})\#\mathcal{A}(x)}{\phi(WB)^k}(\log{R})^kI_k(F).
= \sum_{B\in{\mathcal B}_{\lambda,\mu}^\nu}(-1)^{|\mu| - \#}\left(\prod_{}\!\!\!\!\!\!\!\!{\tt edgefactor}(\ell)\right)\sum_{A\in \Phi^{-1}(B)}\!\! (-1)^{}\!\!\!\!\!\!\!\!\!\!\!\!\!\prod_{}\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\tt factor}_K(\ell).
u'(x) = \int_{x_0}^x u''(y) \mathrm{d}y \leq \int_{x_0}^x \lvert u''(y) \rvert \mathrm{d}y \leq \frac{2 c_0}{\lvert x_0 \rvert}\int_{x_0}^x u'(y) \mathrm{d}y
\mathcal{A}=\left(\begin{array}{ccc}\frac 3r \frac{\cos \theta}{\sin \theta}-\frac{6y}{p(y)}0 \frac 1{3\sqrt{3}}\frac 1{\sin \theta}0 0 -\frac 16\frac{\cos \theta}{\sin \theta}\frac 1{p(y)}\end{array}\right) \,.
S(v)=\begin{pmatrix}s_{11}s_{12}s_{21}s_{22}\end{pmatrix}=\begin{pmatrix}1 \frac{v-|v|^2\overline{v}}{2|v|^2}\frac{{v}-|v|^2\overline{v}}{2|v|^2}-1\end{pmatrix},~~T(v)=\begin{pmatrix}t_{11}t_{12}t_{21}t_{22}\end{pmatrix}.
i_{*h^p}h^q= \binom{2n-p-q}{n-p}p!q!(\det h)\frac{h^{p+q-n}}{(p+q-n)!}, (*h^p)(*h^q)= \binom{2n-p-q}{n-p}p!q!(\det h)(*\frac{h^{p+q-n}}{(p+q-n)!}).
\int_M |\nabla_gu(x)|^2{\rm d}v_g -\mu\int_M\frac{u^2(x)}{d^2_g(x_0,x)}{\rm d}v_g+\int_Mu^2(x){\rm d}v_g\leq |\lambda|C_0 \|\alpha\|_{L^\infty}\int_Mu^2(x){\rm d}v_g.
C_{m_{1},m_{2},m_{3}}^{l_{1},l_{2},l_{3}}\ =\ \delta(m_{1}+m_{2},m_{3})\sqrt{2l_{3}+1}\ \Delta(l_{1},l_{2},l_{3})\S_{m_{1},m_{2},m_{3}}^{\l_{1},\ \l_{2},\ \l_{3}}& \cdot\displaystyle{\sum_{z}}\frac{(-1)^{z}}{z!(l_{1}+l_{2}-l_{3}-z)!(l_{1}-m_{1}-z)!(l_{2}+m_{2}-z)!(l_{3}-l_{2}+m_{1}+z)!(l_{3}-l_{1}-m_{2}+z)!}
A_{p}^{(\ell)}(z) = \sum_{j=1}^{p}(-1)^{j-1}\sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor}\binom{n-p+j+2t}{t}\sum_{\beta=0}^{p-j-2t}2^{p-j-2t-\beta}\binom{n-\ell}{\beta}\binom{\ell}{p-j-2t-\beta}\sum_{\alpha=0}^\beta \binom{\beta}{\alpha}\sum_{i=0}^{j-1}z^{p-2(j+t+\alpha-i)}.
\operatorname{N}:=\operatorname{N}_{\operatorname{M}_{\Lambda''}|\operatorname{M}_{\Lambda'}}(g_{\sigma_1})=\prod\limits_{\sigma \in \operatorname{Gal}(\operatorname{M}_{\Lambda''}|\operatorname{M}_{\Lambda'})}\sigma(g_{\sigma_1}) \in \operatorname{M}_{\Lambda'}^{\times}.
({\rm Char}^{\cal R}(j_!{\cal F}),T^*_XX)_{T^*X}=(2-2g)\cdot ({\rm Char}^{\cal R}(j_!{\cal F}|_Y),T^*_YY)_{T^*Y}+\sum_{x\in Z}({\rm Char}^{\cal R}(j_!{\cal F}),[df])_{T^*X,x}.
\sum_{2\leq |\gamma|< n}\sum_{|\beta|\geq 1}A_\gamma R^{|\beta|}\cdot X^\gamma\cdot \prod_{\lambda=1}^r\left(X^\lambda+A(X)\right)^{\beta_\lambda}=A(X)\cdot\left(-1+\prod_{\lambda=1}^r\frac{1}{1-R(X^\lambda+A(X))}\right).
dH_r(\xi, \sigma_1,\ldots ,\sigma_r,\alpha_1,\ldots ,\alpha_r,\beta_1,\ldots ,\beta_r)\,= &\textbf{e}((\xi-\sigma_r)\beta_r)\hat{F}(\sigma_r-\xi,\alpha_r)H_{r-1}(\xi,\sigma_1,\ldots ,\sigma_{r-1},\alpha_1,\ldots ,\alpha_{r-1},\beta_1-\beta_2,\ldots ,\beta_{r-1}-\beta_{r})\,.
c_{t,l}(\mu_p) = c_{p}(\pi_{p},l,t,\mu_p)\zeta_{p}(1)p^{-\frac{l+t+a(\mu_p\pi_p)}{2}}\lambda_{\chi_{\mu_p}\pi}(p^{t+a(\mu_p\pi_{p})+\delta_{\mu_p\pi_{p}}}),
(\gamma_{0})^{T}=\alpha(\begin{matrix}0.5000\\ 0.5000\end{matrix})\circ(\begin{matrix}0.6469\\ 0.3531\end{matrix})=\alpha(\begin{matrix}0.3235\\ 0.1765\end{matrix})=(\begin{matrix}0.6469\\ 0.3531\end{matrix})
a(x, y, t)\in S^{1}_{{\rm cl\,}}(X\times X\times]0, \infty[), &\mbox{$a(x, y, t)\sim\sum^\infty_{j=0}a_j(x, y)t^{1-j}$ in $S^1_{1,0}(X\times X\times]0,\infty[)$}, &a_j(x,y)\in C^\infty(X\times X),\ \j=0,1,\ldots, &a_0(x,x)=\frac{1}{2}\pi^{-n},\ \ \forall x\in X.
(\nabla_X K)(Y;t) =\nabla^U_X(K_Y t)-K_{\nabla_X Y}t-K_Y(\nabla^V_X t) &=\nabla^U_X(K_Y t)-K_{\tilde{\nabla}_X Y-(\nabla_X df)(Y)}t-K_Y(\nabla^V_X t) &=K_{(\nabla_X df)(Y)}t,
\begin{aligned}V^{-1}(s)_{i2}\leq \delta_{i1}, V^{-1}(s)_{i0}+ V^{-1}(s)_{i4}\leq 1- \delta_{i1}\forall i \in \{1, 2\}V^{-1}(s)_{i0}+ V^{-1}(s)_{i1}\leq \delta_{i2}, V^{-1}(s)_{i3}+ V^{-1}(s)_{i4}\leq 1 - \delta_{i2}\forall i \in \{1, 2\}.\end{aligned}
\delta \psi_{A \mu}= \mathcal{D}_{\mu}\epsilon_{A}- \frac{1}{3}\mathcal{T}_{A B}^{\rho \sigma}\left( g_{\mu \rho}\gamma_{\sigma}- \frac{1}{8}\epsilon_{\mu \rho \sigma \lambda \nu}\gamma^{\lambda \nu}\right) \epsilon^{B}
Q_1(x,y,q,k) :=\left(l_1x^{l_1}+l_2x^{l_2}\right)\left(y-x-q+b(1+k)[\ln{x}-\ln{y}]\right) &\qquad-(x-b(1+k))(y^{l_1}-x^{l_1}+y^{l_2}-x^{l_2}), Q_2(x,y,q,k) :=\left(l_1y^{l_1}+l_2y^{l_2}\right)\left(y-x-q+b(1+k)[\ln{x}-\ln{y}]\right) &\qquad-(y-b(1+k))(y^{l_1}-x^{l_1}+y^{l_2}-x^{l_2}).
\bigl[(\gamma_C\otimes\gamma_B)(\sigma E)\bigr](b\otimes c) :=(\gamma_C\otimes\gamma_B)\bigl((\gamma_C^{-1}(b)\otimes\gamma_B^{-1}(c))(\sigma E)\bigr)=E(b\otimes c), (b\otimes c)\bigl[(\gamma_C\otimes\gamma_B)(\sigma E)\bigr] :=(\gamma_C\otimes\gamma_B)\bigl((\sigma E)(\gamma_C^{-1}(b)\otimes\gamma_B^{-1}(c))\bigr)=(b\otimes c)E.
\hat{R}_{\mu}{}^{\nu}( z ; g ) = \Omega ( z ; g )^{- 1}R_{\mu}{}^{\nu}( z ; g ) = \textstyle{\frac{1}{2}}\mathrm{t r}( \gamma_{\mu}L ( z ; g ) \gamma^{\nu}L^{- 1}( z ; g ) ) \in \mathrm{S O}( 1 , 2 ) \, .
\int_{\mathbb{B}^{N}}a(x)\left|u_{n}\right|^{p+1}\mathrm{~d}V_{\mathbb{B}^{N}}(x) -\int_{\mathbb{B}^{N}}a(x)|u|^{p+1}\mathrm{~d}V_{\mathbb{B}^{N}}(x) &=\int_{\mathbb{B}^{N}}a(x)\left|u_{n}-u\right|^{p+1}\mathrm{~d}V_{\mathbb{B}^{N}}(x)+o(1).
\langle \varepsilon_t - \varepsilon_{t+1}, a_{i,j}(\varepsilon_i - \delta_j) \rangle = \delta_{t, i}a_{i,j}- \delta_{t+1, i}a_{i,j}\langle \varepsilon_t - \varepsilon_{t+1}, b_{i,j}(\delta_i - \varepsilon_j) \rangle = -\delta_{t, j}b_{i,j}+ \delta_{t+1,j}b_{i,j}
\sigma_0(u_1, u_2, e_1, e_2) := \left\langle \begin{bmatrix}u_1 u_2 e_1 e_2\end{bmatrix},\begin{bmatrix}I 0 0 0 0 I 0 0 0 0 -\gamma^2 I 0 0 0 0 -\gamma^2 I\end{bmatrix}\begin{bmatrix}u_1 u_2 e_1 e_2\end{bmatrix}\right\rangle
S=\frac{1}{2}\int d^{4}x \epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}\bigg[ ({^+}\tau -{^-}\tau) F^{ab}_{\mu\nu}F^{cd}_{\alpha\beta}+ ({^+}\tau +{^-}\tau) F^{ab}_{\mu\nu}\tilde F^{cd}_{\alpha\beta}\bigg].
\sqrt{u}\acute \nu_{k,j}^\ell(u)=\begin{cases}-\frac{2k+1}{2(k+1)}\mu^\ell_{k-1,1}+o(1), j=1 O(u), j\in\mathbb Z\cap[2,k] -\mu^{\ell}_{k-1,j-k-1}+o(1), j\in\mathbb Z\cap[k+2,2k] \end{cases}
b_{n}J_{n - 1 +{\frac{D}{2}}}^{\prime}( | \omega | r^{\prime}) + c_{n}J_{- n + 1 -{\frac{D}{2}}}^{\prime}( | \omega | r^{\prime}) - a_{n}J_{n - 1 +{\frac{D}{2}}}^{\prime}( | \omega | r^{\prime}) ={\frac{( 2 n + D - 2 ) \Gamma \left({\frac{D - 2}{2}}\right)}{4 ( \pi r^{\prime})^{\frac{D}{2}}| \omega |}}
\Gamma_k:=\frac{1}{2}{{\displaystyle\sum_{i=1}^n}\Delta_{\frac{1}{2}}x_i^k\left(\mathcal{L}_{{F}(x^k)}\mathcal{L}_{{V}_k}\mathcal{L}_{{F}(x^k)}^{-1}-\mathcal{L}_{V_k}\right)F_i}.
\bar{x}_{i,j}= \bar{x}_{i-1,j-1}+\beta \bar{x}_{i,j-1}+\bar{y}_{i,j-1}, \bar{y}_{i,j}= \delta \bar{x}_{i,j-1}+2\beta \bar{x}_{i-1,j-1}+\bar{y}_{i-1,j-1},
a_n=\Big (\prod_{j=1}^{n-1}\sigma_j^z \Big )\sigma^{+}_n\ , a_n^{\dag}=\Big (\prod_{j=1}^{n-1}\sigma_j^z \Big )\sigma^{-}_n \ ,\sigma_n^z=1-2a_n^{\dagger}a_n \ ,\sigma_n^{\pm}=\tfrac{1}{2}(\sigma_n^x\pm \mathrm{i}\sigma_n^y)\ .
\dot{\hat{R}}_{k}(t)=-2\beta(\hat{R}_{k}(t)+\frac{1}{2})-(\omega_{0}/\hbar)\{[\hat{b}(t)+\hat{b}^{\dagger}(t)]\vec{\mu}\cdot\hat{\vec{A}}_{free}^{(+)}(\vec{r},t)+H.c.\}
K_{m}^{(j)}(X,Q)= D_{m}^{(j)}(X,Q)-\mathrm{S. P.}[D_{m}^{(j)}](X,Q)=\mathrm{I. P.}[D_{m}^{(j)}](X,Q) = (x_{j}-v_{j})\sum_{k=2m}^{\infty}[C_{m,-k}(X,Q)+\widetilde{C}_{m,-k}(X,Q)\log|Q|]\frac{1}{(1+|Q|^{2})^{\frac{k}{2}-m+\frac{n+3}{2}}},
G_{\Psi}^{a}= \sum_{k = 1}^{n}( \Psi_{k})_{i}^{\dagger}T_{i j}^{a}( \Psi_{k})_{j}- \sum_{k = n + 1}^{N}( \Phi_{k})_{i}^{\dagger}T_{j i}^{a}( \Phi_{k})_{j}.
{\Lambda_n}= \frac{1}{{{e^{\mu M\kappa}}{{\ln}^n}2}}\sum\limits_{i = 0}^\infty \frac{{{{\left({\mu \kappa M}\right)}^i}}}{{i!\Gamma \left({\mu M + i}\right)}}{{\left({\frac{{\mu \left({1 + \kappa}\right)}}{\Omega}}\right)}^{\mu M + i}}&\times J\left({\mu M + i,\frac{{\mu \left({1 + \kappa}\right)}}{\Omega},n}\right) ,
|E( h^{(2)}(X_i, X_j) h^{(2)}(X_k, X_l))| & \le C(Cov(X_j, X_i)^{\delta}+ Cov(X_j, X_k)^{\delta}+ Cov(X_j, X_l)^{\delta}), & |E( h^{(2)}(X_i, X_j) h^{(2)}(X_k, X_l))| & \le C(Cov(X_k, X_j)^{\delta}+ Cov(X_k, X_i)^{\delta}+ Cov(X_k, X_l)^{\delta}), .
\rho^{(-1)}_t= (a^{-1})_bD^bF, \rho^{( 1)}_t= 2P^{(1)}u_{b+1}D^{b+1}F+P^{(1)}_bu_{b+1}^2D^bF, \rho^{( 3)}_t= 2P^{(3)}u_{b+2}D^{b+2}F +4Q^{(3)}u_{b+1}^3D^{b+1}F &\quad\quad+(P^{(3)}_bu_{b+2}^2+Q^{(3)}_bu_{b+1}^4)D^{b}F .
{F}_{\mathrm{r e n}}^{\infty}= - \frac 1{8 \pi^{2}}\int \!{\mathrm d}^{3}x{g}^{1 / 2}{\mathrm t r}\Big \{R_{i j}\gamma_{1}( - \triangle ) R^{i j}+ R \gamma_{2}( - \triangle ) R +{\mathrm{O}}[ \Re^{3}] \Big \}.
vol (D^2_x(1)):=\frac{vol (B^2(1))b^2A^{\gamma\tau}\left[\cos^2\theta z^{1}_{\gamma}z^{1}_{\tau}+\sin^2\theta z^{2}_{\gamma}z^{2}_{\tau}+2\sin\theta \cos\theta z^{1}_{\gamma}z^{2}_{\tau}\right]}{2\sqrt{\det A}}.
A_{\mu_{1}\dots \mu_{n}}(x,e)=A_{\mu_{1}\dots \mu_{n}}^{\rm P}(x)+\partial_{\mu_{1}}\phi_{\mu_{2}\dots \mu_{n}}+\partial_{\mu_{1}}\partial_{\mu_{2}}\phi_{\mu_{3}\dots \mu_{n}}+\cdots +\partial_{\mu_{1}}\cdots\partial_{\mu_{n_{n}}}\phi.
\sum_{\substack{m\in \mathbb{Z}3m^2+m\leq 2n}}\left(6m+1\right)c\left(f; n-\frac{3}{2}m^2-\frac{1}{2}m\right)=-4\sigma(n)-16\sigma\left(\frac{n}{2}\right)+4\sum_{\substack{a,b\in \mathbb{Z}^+ ab=2n b> 3a a\not\equiv b \pmod{2}}}(3a+b).
M_1:= \{A^{26},A^{27}, A^{36}, A^{37}\}, M_2:= \{A^{56},A^{57}, y^{26}, y^{36}, y^{27}, y^{37}, v^{25}, v^{35}, u^{67}\}, M_3:= \{A^{25},A^{35}, u^{27}, u^{37}, u^{26}, u^{36}\}, M_4:= \{A^{67}, u^{25}, u^{35}, v^{26}, v^{27}, v^{36}, v^{37}\}, M_5:= \{u^{56}, u^{57}, y^{25}, y^{35}, v^{67}\}.
E^{2}( R ) = 2{\frac{j ( j + 1 )}{k + 2}}-{\frac{m^{2}+ \bar{m}^{2}}{k}}+{\frac{1}{2 k}}\left[ ( m - \bar{m}+ k M )^{2}R^{2}+{\frac{( m + \bar{m}+ k N )^{2}}{R^{2}}}\right] + N^{\prime}+ \bar{N}^{\prime}
\lim_{\lambda\to\infty}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^{\vec{r}}(\mathbb{R}^n)}=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^{\vec{r}}(\mathbb{R}^n)};