harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S [ \varphi_{R}] = \int \,{\mathrm{d}}x \,{\mathrm{d}}y \, \left\{\theta ( x ) \, \left( \frac{1}{2}( \nabla \varphi_{R})^{2}+ \frac{m^{2}}{2}\varphi_{R}^{2}\right) + \frac{c}{2}\delta ( x ) \varphi_{R}^{2}\right\}
[T_k]_i^j(A_1,A_2,\cdots,A_k)=\dfrac{1}{k!}\delta_{ii_1i_2\cdots i_k}^{jj_1j_2\cdots j_k}(A_1)^{i_1}_{j_1}(A_2)^{i_2}_{j_2}\cdots (A_k)^{i_k}_{j_k}.
Y_{1,j}=\pi^{j-1}T_{n-j+1,n}\cdots T_{1,j}, j \ge 2, Y_{2,j}=T_{1,j-1}^{-1}\pi^{j-2}T_{n-j+2,n}\cdots T_{2,j}, , Y_{j-1,j}= T_{j-2,j-1}^{-1}\cdots T_{1,2}^{-1}\pi T_{n-1,n}\cdots T_{j-1,j}.
x_{\Delta}= \sqrt{\frac{A}{4 \pi}}\mathrm{w i t h}A = 2 \pi \int_{0}^{\pi}d \theta \sin \theta x_{h}^{2}\frac{\sqrt{l_{2}( x = x_{h}, \theta ) m_{2}( x = x_{h}, \theta )}}{f_{2}( x = x_{h}, \theta )}\ .
\begin{array}{c}{{g_{+}( - \infty ) = g_{0}( + \infty ) = 2 \chi_{\infty}( S - 2 S^{+}) + 2 \pi k_{W}^{+}}}\\{{g_{-}( + \infty ) = g_{0}( - \infty ) = - 2 \chi_{\infty}( S - 2 S^{-}) + 2 \pi k_{W}^{-}}}\\ \end{array}
\left| \sum_{n=X_1}^{X_2}a_n ( R(n)- R(n-1) ) \right| = \left| \sum_{n=X_1}^{X_2 -1}( a_n - a_{n+1}) R(n) + a_{X_2}R(X_2) - a_{X_1}R( X_1 - 1) \right| &\leq 2AR + R \sum_{n= X_1}^{X_2-1}|a_n - a_{n+1}|.
\mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S}{}^{\pm}\widetilde{K}_v(E_{p}/L_\infty))=\mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S_p}{}^{\pm}\widetilde{K}_v(E_{p}/L_\infty))=2.
\begin{aligned}\left(\lambda^2-1\right)c_{1,1}+(a_1+\lambda(\lambda-1))c_{1,0}=0 , \lambda(\lambda+2)c_{1,2}+(a_1+\lambda(\lambda+1))c_{1,1}-a_1^2c_{1,0}=0 , (\lambda+1)(\lambda+3)c_{1,3}+(a_1+(\lambda+1)(\lambda+2))c_{1,2}-a_1^2c_{1,1}+Ac_{1,0}=0 ,\end{aligned}
\begin{aligned}G_{1}= cx_{n+1}G_{2}= 2(d-1)cx_{n+1}x_0+ 2F_2 G_{3}= 3(d-1)(d-2)cx_{n+1}x^2_0+ 6(d-2)x_0F_2+6F_3 \dots G_{h-1}= (h-1)\frac{(d-1)!}{(d-h+1)!}cx_{n+1}x_0^{h-2}+\sum_{i=2}^{h-1}\frac{(h-1)!}{(h-i-1)!}\frac{(d-i)!}{(d-h+1)!}x_0^{h-i-1}F_i.\end{aligned}
H=\frac{P^{2}}{2M}+V(X)+\sum_{i}(\frac{p_{i}^{2}}{2m_{i}}+\frac{1}{2}m_{i}\omega_{i}^{2}q_{i}^{2})-X\sum_{i}C_{i}q_{i}+X^{2}\sum_{i}\frac{C_{i}^{2}}{2m_{i}\omega_{i}^{2}}
f_1\ast f^{\ast -1}_2(a) =\mathcal{B}_1(f\ast g)(a_1)\mathcal{B}_2(f\ast g)^{\ast -1}(a_2)=f\ast g(B_1(a_1))f\ast g(S(B_2(a_2))) &=f\ast g(\sigma(a))=f\ast g(a).
\left( x^{2}\sqrt{l}\Phi_{3}^{'}\right)^{'}= \frac{l^{3 / 2}x^{2}\Phi_{3}}{f}\left[ \beta_{2}^{2}\left( \Phi_{1}^{2}+ \Phi_{3}^{2}+ \Phi_{4}^{2}- 1 \right) + \beta_{3}^{2}\left( \frac{\Phi_{3}^{2}}{2}+ \frac{2}{5}\Phi_{4}^{2}- \frac{1}{\sqrt{5}}\Phi_{3}\Phi_{4}- \frac{7}{3 0}\right) \right]
| p_{( a_{1}, \cdots , a_{k})}|^{2}- | p_{( a_{1}, \cdots , a_{k}, a_{k + 1})}|^{2}- | p_{( a_{1}, \cdots , a_{k}, a_{k + 2})}|^{2}+ | p_{( a_{1}, \cdots , a_{k}, a_{k + 1}, a_{k + 2})}|^{2}= 0 .
F(x) = \int_{0}^{\infty}dA(t)\int_{0}^{x+t}d(u)\left[\bar{G}(u) \int_{t-u}^{x+t-u}dB(s) + G(u) + \bar{G}(u)\int_{0}^{t-u}V(x+t-u-s) dB(s) \right] + \int_{0}^{\infty}dA(t)\int_{0}^{t}dW(u)\left[ -G(u)\bar{V}(x+t-u) \right].
\left\Vert \partial_{t}^{s}\widetilde{u}\right\Vert_{H^{2,1}\left(Q_{\varepsilon ,T}\right)},\left\Vert \partial_{t}^{s}\widetilde{m}\right\Vert_{H^{2,1}\left( Q_{\varepsilon ,T}\right)},\left\Vert \widetilde{k}\right\Vert_{L_{2}\left( \Omega \right)}\leq C\delta^{1-\rho},\forall \delta \in \left( 0,\delta_{0}\right) ,s=0,1,2.
T_{0}^{\mu \nu}= - \left( \frac{g_{s t r}\alpha^{\prime}}{2}\right) \prod_{u}\frac{\delta ( \sqrt{\alpha^{\prime}}k_{u})}{\sqrt{\tau_{2}}}\frac{\eta^{\mu \nu}}{2 \pi \alpha^{\prime}}\left( \prod_{o}\frac{\delta ( \sqrt{\alpha^{\prime}}k_{o})}{\sqrt{\tau_{2}}}X_{1 , 1}+ 2^{- ( d + 1 )}X_{1 , 0}\right)
\mathcal{I}_{\mathrm{D}}(0) = \sup_{\pi \in \Pi(\mu_1, \mu_2)}\inf_{\lambda \in \mathbb{R}^2_+}\int g_\lambda \, d \pi =\inf_{\lambda \in \mathbb{R}^2_+}\sup_{\pi \in \Pi(\mu_1, \mu_2)}\int g_\lambda \, d \pi = \sup_{\pi \in \Pi(\mu_1, \mu_2)}\int g \, d \pi.
=(G^{T}WG)^{-1}G^{T}W\Omega^{1/2}(I-\Omega^{-1/2}G(G^{T}\Omega^{-1}G)^{-1}G^{T}\Omega^{-1/2})\Omega^{1/2}WG(G^{T}WG)^{-1}
\sum\limits_{A=2}^m\phi_A^af_{1t}^A=-\sum\limits_{A=2}^m[\phi_{At}^af_1^A+\phi_{A1}^af_t^A+\sum\limits_{B=2}^m(\phi_{AB}^af_t^B)f_1^A]-\phi_{1t}^a,\;\;\;a=1,2,...,m-1.
{\cal I}_{\cal K}^{2}={\cal I}_{\cal K}^{3}= \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) \; , \qquad{\cal I}_{\cal K}^{4}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) .
H(e_i)(b) = H(e_{t_0})(b) = H(e_{t_0})(H(u_{p_1})(a)) = H(e_{t_0}\cdot u_{p_1})(a) = \mathstrut = H(e_{t_1}\cdot v_{q_1})(a) = H(e_{t_1})(H(v_{q_1})(a)) = H(e_{t_1})(b),
\begin{array}{l}a_0(f, D) = \int_M dV ~Tr\{f\}a_2(f, D) = \frac{1}{6}\int_M dV \,Tr\{f(6V + R)\}a_4(f, D) = \frac{1}{360}\int_M dV \, Tr\{f(60\Delta V + 60RV + 180V^2 + 12\Delta R+ 5R^2 -2 R^{ij}R_{ij}+2 R^{ijkl}R_{ijkl}) \}\end{array}
\begin{array}{lcc}\mbox{matrix norm}\mbox{Lipschitz constant}\mbox{Bregman Condat--V\~u}\|A\|_{1,2}=\sqrt 2 L_1=\max_{i,j}|(C^TC)_{ij}| \mbox{Bregman PD3O}\|A\|_2 \leq 2 L_2=\|C\|_2^2.\end{array}
De = -z^{-2}Dz\langle \phi_0,F(z\phi_0+h)\rangle + z^{-1}\langle \phi_0, D[F(z\phi_0+h)]\rangle, Dh = (H+q^2/2)^{-1}\{-P_c D[F(z\phi_0+h)]+[De]h + e[Dh]\}.
K = \left( \begin{array}{l l}{- \sigma_{1}\bar{\beta}_{1 1}}&{- \sigma_{1}\bar{\beta}_{1 2}}\\{- \sigma_{2}\bar{\beta}_{1 2}}&{- \sigma_{2}\bar{\beta}_{2 2}}\\ \end{array}\right) \equiv \left( \begin{array}{l l}{K_{1 1}}&{K_{1 2}}\\{K_{2 1}}&{K_{2 2}}\\ \end{array}\right) .
\begin{cases}|\nabla u_+|=\alpha_+>0\quad\partial\{u>0\}\setminus \partial\{u<0\}\cap D, |\nabla u_-|=\alpha_->0\quad\partial\{u<0\}\setminus \partial\{u>0\}\cap D, |\nabla u_\pm|\ge \alpha_\pm\quad|\nabla u_+|^2-|\nabla u_-|^2=\alpha_+^2-\alpha_-^2\quad\partial\{u>0\}\cap \partial\{u<0\}\cap D.\end{cases}
\frac{1}{N+1}{L^{1}_{N}}(x)=\frac{1}{2\pi i}\int_{{\cal C}_0}\frac{dz}{z^{N+1}}\exp\Biggl[\sum_{k=1}^{+\infty}L^{0}_{k}(\frac{kx}{N+1})\frac{z^k}{k}\Biggr],
B = \frac{2}{\pi}\int f^{\prime}\sin^{2}f ~ \mathrm{d}r ~ \frac{1}{4 \pi}\int \frac{( | R_{z}|^{2}- | R_{{\bar{z}}}|^{2}) ( 1 + | z |^{2})^{2}}{( 1 + | R |^{2})^{2}}\frac{2 i ~ \mathrm{d}z \mathrm{d}{\bar{z}}}{( 1 + | z |^{2})^{2}}.
\begin{array}{l}X_1(q)=-2qhxix^{-1}h^{-1}, X_2(q)=q\left(-X_3(\beta) hxix^{-1}h^{-1}-(1-\sqrt{3}e^{-2\beta})\hxjx^{-1}h^{-1}\right), X_3(q)=q\left( X_2(\beta)\hxix^{-1}h^{-1}-(1+\sqrt{3}e^{-2\beta}) \hxkx^{-1}h^{-1}\right).\end{array}
\begin{array}{c}{s \left( \lambda , \mu \right) = \frac t{\lambda - \mu}\left( \phi \left( \lambda \right)^{- 1}- \phi \left( \mu \right)^{- 1}\right) ,}{a \left( \lambda , \mu \right) = \frac t{\lambda - \mu}\left( \phi \left( \lambda \right)^{- 1}- \phi \left( \mu \right)^{- 1}\right) .}\end{array}
J_1= \int_{|z'|\leq\left(\frac{1}{3M_0}\right)^{\frac{1}{1+\alpha}}}dz' \int_{-M_0\left(\frac{h}{\rho_0}\right)^\alpha|z'|^{1+\alpha}}^{M_0\left(\frac{h}{\rho_0}\right)^\alpha|z'|^{1+\alpha}}\frac{1}{D(z)}dz_3,
\mathcal{O}_{\nu+1}=\mathcal{O}_{\nu}\setminus\big(\bigcup_{k\in\mathbb{Z}^n\setminus\{0\},|l|\leq2\atop{l\neq e_{-j}-e_j}}\mathcal{R}_{kl}^{\nu}(\alpha_{1,\nu})\cup\bigcup_{k\in\mathbb{Z}^n,\pm j\in\mathbb{Z}_*\atop{|j|\leq \Pi_{\nu}}}\mathcal{R}_{k(-j)j}^{\nu}(\alpha_{2,\nu})\big),
Equ > p^s+1-(\tau+1)p^t-\left( (p-\tau+1)p^{s-t-1}-1 \right)\times \left( 1+\frac{\tau p^t-1}{(p-\tau)p^{t-1}+1}\right) = p^s+1-(\tau+1)p^t-\left( (p-\tau+1)p^{s-2}-p^{t-1}\right)\times \frac{p(\tau-1)-\tau}{(p-\tau)p^{t-1}+1}.
\begin{array}{c}{c_{j}^{( i )}= \frac{1}{( \frac{d \varphi_{i}}{d N_{i}})_{0}}\delta_{i j}}\\{c_{i j}^{( i )}= - \frac{1}{2}( c_{i j}+ c_{j i}^{*}) c_{i}^{( i )}c_{j}^{( j )}}\\{c_{j k}^{( i )}= 0 , \quad j \neq i , \quad k \neq i}\\ \end{array}
\begin{array}{l}{{u_{+}^{2}= u_{+}; \ \u_{+}u_{-}= 0 ; \ \v_{+}^{( n )}u_{+}= u_{-}v_{+}^{( n )}}}\\{{u_{+}v_{+}^{( n )}= v_{+}^{( n )}u_{-}= 0 ; \ \v_{+}^{( \ell )}v_{-}^{( m )}= - \delta_{\ell m}u_{0}; \ \v_{+}^{( \ell )}v_{+}^{( m )}= \epsilon_{\ell m n}v_{-}^{( n )}.}}\\ \end{array}
\begin{array}{l}R(x,y,z,w) = g(y,z)\rho(x,w) - g(x,z)\rho(y,w) begin{align*}4pt]\phantom{R(x,y,z,w) =}+ \rho(y,z)g(x,w) - \rho(x,z)g(y,w) begin{align*}4pt]\phantom{R(x,y,z,w) =}-\dfrac{\tau}{2}\{g(y,z)g(x,w) - g(x,z)g(y,w)\}.\end{array}
U_{( 0 , \, r - 1 )}^{2}+ \Lambda_{1}^{4}\Omega_{( 2 , \, r - 1 )}+ \Lambda_{r}^{4}\Omega_{( 1 , \, r - 2 )}- M_{0}\Omega_{( 1 , \, r - 1 )}- \Lambda_{( 1 , \, r )}^{2}= 0 .
H_{\Lambda}= \frac{1}{2}\sum_{x \in [N, M [}J_x ( \sigma_x^{(1)}\sigma_{x + 1}^{(1)}+ \sigma_x^{(2)}\sigma_{x + 1}^{(2)}) + \frac{1}{2}\sum_{x \in \Lambda}\lambda_x \sigma_x^{(3)}.
{\rm Var}\,G_{N_{\beta,\omega}}= \frac{1}{\beta^2}- \frac{1}{(\beta+\omega)^2}+ \frac{\psi''(\beta)}{\psi(\beta)}- \frac{\psi''(\beta+\omega)}{\psi(\beta+\omega)}- \left(\frac{\psi'(\beta)}{\psi(\beta)}\right)^2 + \left(\frac{\psi'(\beta+\omega)}{\psi(\beta+\omega)}\right)^2 .
\rho = \rho_{X 0}a^{- 3 w_{X}}\left[ \frac{\rho_{m 0}}{\rho_{X 0}}a^{3 ( w_{X}- 1 )}+ 1 \right] = \rho_{m 0}a^{- 3}+ \rho_{X 0}a^{- 3 w_{X}}.
( \partial_{\sigma_{c}}^{2}- \partial_{\tau}^{2}) \delta x_{\perp}^{( 2 )}= U_{\perp \perp}^{( 2 )}\delta x_{\perp}^{( 0 )}+ U_{\perp \perp}^{( 1 )}\delta x_{\perp}^{( 1 )},
W_{0, \nu}(y) = \big( \frac{y}{\pi}\big)^{\frac{1}{2}}K_\nu\big( \frac{y}{2}\big), K_\nu(y) = \frac{1}{2}\int_0^\infty e^{-\frac{1}{2}y(u + u^{-1})}u^\nu \frac{du}{u}.
R_{\boldsymbol{k}}(\tau):=\int_0^{i\infty}\int_{w_1}^{i\infty}\frac{\Theta_1(2p,k_1,2p;w_1)\Theta_0(6p,k_2,6p;w_2)}{\sqrt{-i(w_1+\tau)}(-i(w_2+\tau))^\frac32}dw_2dw_1.
\begin{aligned}\partial_t\eta+\langle\nabla_x\Phi,\nabla_x\eta\rangle-\partial_z\Phi=0 \ @ \z=\eta(x,t)\cr \partial_t\Phi+{1\over2}|\nabla\Phi|^2+g\eta+{p\over\rho}=0 \ @ \z=\eta(x,t)\cr\end{aligned}
\left(\begin{array}{cccc}x=0: 0 e_1 e_2 x=1: 0 e_3 e_4 x=\infty: e_5 e_6 e_7\end{array}\right)\to\left(\begin{array}{ccc}0 -e_1 -e_2 0 -e_3 -e_4 2-e_5 2-e_6 2-e_7\end{array}\right),
\left( \frac{1}{c^{2}}\left( \frac{\partial}{\partial t}\right)^{2}- \sum_{j = 1}^{3}\left( \frac{\partial}{\partial x^{j}}\right)^{2}\right) A_{0}( x ; y ) = 4 \pi m_{y}G \delta ({\bf x}-{\bf y}( t ) ) ,
f^{\star}(p) = \mathcal{L}[f (x); p] = \int_{0}^{\infty}e^{-p x}f (x) dx, f (x) = \mathcal{L}^{-1}[f^{\star}(p); x] = \frac{1}{2\pi i}\int_{c-i \infty}^{c+i \infty}e^{p x}f^{\star}(p) dp,
\int d^{8}x \vert_{\theta_{j}^{\alpha}= \bar{\theta}_{j}^{\bar{\alpha}}= 0}\left[ ( D_{+})^{8}( D_{-})^{8}f_{V}( W ) + \int d \zeta ( D_{-})^{8}( \bar{D}^{+})^{8}f_{T}( \tilde{L}, \zeta ) \right] + c . c .
M^{(3,1)}\otimes M^{(2,2)}= M^{(2,1,1)}\oplus M^{(2,1,1)}\displaybreak[1] M^{(3,1)}\otimes M^{(3,1)}= M^{(3,1)}\oplus M^{(2,1,1)}\displaybreak[1] M^{(3,1)}\otimes M^{(4)}= M^{(3,1)}
S= -\frac{1}{2\pi}\int d^2\sigma ~e~\left [ h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu}-i\bar \Psi^{\mu}\rho^{\alpha}\nabla_{\alpha}\Psi_{\mu}+ 2\bar{\chi}_{\alpha}\rho^{\beta}\rho^{\alpha}\Psi^{\mu}\partial_{\beta}X_{\mu}+\frac{1}{2}\bar{\Psi}^{\mu}\Psi_{\mu}\bar{\chi}_{\beta}\rho^{\beta}\rho^{\alpha}\chi_{\alpha}.\right ].
\begin{cases}\varphi_{\sigma}(x)=-\frac{1}{2}\varphi_{\sigma}^{\prime\prime}(x)+\varphi_{\sigma}^{2\sigma+1}(x),0< x<+\infty, \varphi_{\sigma}(0)=0,\lim\limits_{x\to+\infty}\varphi_{\sigma}(x)=1.\end{cases}
\frac{\partial}{\partial a}\mathrm{B}_{z}\left( a,b\right) =z^{a}\left\{\ln z\sum_{k=0}^{\infty}\frac{\left( 1-b\right)_{k}\,z^{z}}{k!\left(a+k\right)}-\sum_{k=0}^{\infty}\frac{\left( 1-b\right)_{k}}{k!\left(a+k\right)^{2}}\right\}.
\underset{n\to\infty}{\lim}\frac{\P\left(|\hat\tau(Y_{1:n},Z_{1:n})-\tau(Y_1,Z_1)|\ge \gamma\sqrt{\frac{\log p}{n}}\right)}{2\left(1-\Phi\left(\frac{3}{2}\gamma\sqrt{\log p}\right)\right)}=1.
i D_{T_{\mu \nu}}^{\underline{{{a}}}\underline{{{b}}}}( k ) = \frac{i \delta_{\underline{{{a}}}\underline{{{b}}}}}{k^{2}- m_{G}^{2}}\left[ - g_{\mu \nu}+ \frac{k_{\mu}k_{\nu}}{k^{2}}\right] \; .
A_m = \frac{1}{N}\left[ \prod_{i=1}^{d}(m_i!)^{-1}\sum_{i=1}^{N}z_k^{(i)}J_{m-e_k}(y^{i}) - \sum_{i=1}^{N}r_{ki}m_k^{-1}A_{m-e_k-e_f}\right], k = 1,\ldots,d
[\begin{matrix}Z_{11}^{\prime}&Z_{12}^{\prime}\\Z_{21}^{\prime}&Z_{22}^{\prime}\end{matrix}]=[\begin{matrix}n_{11}&n_{12}\\n_{21}&n_{22}\end{matrix}][\begin{matrix}Z_{11}&Z_{12}\\Z_{21}&Z_{22}\end{matrix}][\begin{matrix}n_{11}&n_{12}\\n_{21}&n_{22}\end{matrix}]
f_{m,p}(x) =xf_{m-1,p-1}(x)+x\sum_{i=p}^{m-1}f_{m,i}(x) &+\frac{x^{p+3}}{(1-x)^2}\left(\sum_{i=0}^{m-1-p}x^iF_{m-1-p-i}(x;0)+\sum_{i=0}^{m-2-p}x^iF_{m-1-p-i}(x;1)\right) &+\frac{x^{p+3}}{(1-x)^2}\left(\sum_{i=0}^{m-2-p}\frac{x^i}{1-x}F_{m-1-p-i}\left(x;\frac{1}{1-x}\right)-\sum_{i=0}^{m-2-p}x^iF_{m-1-p-i}(x;1)\right).
I_{m-1}(t) = \frac{\alpha}{m-\alpha}t^{m-\alpha}\int_0^\infty L_1(t/w)\left(1- e^{-w}\right)^{m-1}w^{\alpha-m}dw = \frac{\alpha}{m-\alpha}t^{m-\alpha}\int_0^\infty L_1(t z) \left(1- e^{-\frac{1}{z}}\right)^{m-1}z^{m-\alpha-2}dz .
T_3 \le \Gamma(2m)\frac{\Gamma(m+5/2)\Gamma(m-5/2)}{\Gamma(2m)}\int_{0}^1y^{4}(1-y)^{m-3/2}\,dy &=\frac{\Gamma(m+5/2)\Gamma(m-5/2)}{\Gamma(m-1/2)^2}\frac{\Gamma(5)\Gamma(m-1/2)}{\Gamma(m+9/2)}.
h'' \begin{pmatrix}v w\end{pmatrix}=0 \Leftrightarrow \begin{cases}v+Ew=0, E^*v-w=0\end{cases}\Leftrightarrow \begin{cases}v+EE^*v=0, w=E^*v\end{cases}\Leftrightarrow \begin{cases}v=0, w=0.\end{cases}
\frac{1}{D}\frac{d}{dD}\left(D^2(\bar{\lambda}_2(n,D,K)-\bar{\lambda}_1(n,D,K))\right) =\frac{d\lambda_2(\tilde{L}_c)}{dc}\biggr|_{c=1}- \frac{d\lambda_1(\tilde{L}_c)}{dc}\biggr|_{c=1}.
{\cal Z}= \int \prod_{i = 1}^{3}[ \mathrm{d}{\phi}^{i}( \tau , x ) ] \prod_{i = 1}^{3}[ \mathrm{d}{\Pi}^{i}( \tau , x ) ] \delta \left( | \vec{\phi}| - 1 \right) e^{- S_{e}[ \vec{\phi}, \vec{\Pi}]}.
\frac{1}{12}\frac{(q;q)_{\infty}^6}{(q^3;q^3)_{\infty}^3}=\frac{1}{12}\left(\sum_{n\ge0}(-1)^n(2n+1)q^{n(n+1)/2}\right)^2\left(\sum_{n\ge0}p(n)q^{3n}\right)^3.
{\cal O}_{1}{}^{i}{}_{j}\left[ \begin{matrix}{{\bar{\psi}_{n}^{j \dot{1}}}}\\{{\psi_{n 2}^{j}}}\\ \end{matrix}\right] = - i m_{n}^{( 1 )}\left[ \begin{matrix}{{\psi_{n 1}^{i}}}\\{{\bar{\psi}_{n}^{i \dot{2}}}}\\ \end{matrix}\right]
M=\alpha_1^{a_1}\alpha_2^{a_2}\alpha_3^{a_3}\alpha_4^{a_4}\alpha_5^{a_5}\alpha_6^{a_6}\beta_1^{b_1}\beta_2^{b_2}\beta_3^{b_3}\beta_4^{b_4}\beta_5^{b_5}\beta_6^{b_6}.
\frac{\partial^{2}z}{\partial t^{2}}=[\frac{\partial^{2}}{\partial x^{2}}(\frac{\partial x}{\partial t})^{2}+\frac{\partial^{2}}{\partial x\partial y}\frac{\partial x\partial y}{\partial t^{2}}+\frac{\partial^{2}}{\partial y\partial x}\frac{\partial y\partial x}{\partial t^{2}}+\frac{\partial^{2}}{\partial y^{2}}(\frac{\partial y}{\partial t})^{2}]z
\begin{cases}\mathcal{Q}\,^{C}_{t}D_{T}^{q}\Psi(x,t)= A^{*}\mathcal{Q}\Psi(x,t) + C^{*}C \mathcal{Q}\varphi(x,t) \hbox{in}Q_{T}\Psi(\xi,t) = 0 \hbox{on}\Sigma_{T}\Psi(x,T) = 0 \hbox{in}\Omega.\end{cases}
S_{eff}= \int d^{3}x\, \left\{\frac{|m|}{16\pi}\, \mbox{tr}\,(\partial_{\mu}\hat{n})^{2}+i\frac{\mbox{sgn}(m)}{4\pi}\, \epsilon^{\mu\nu\lambda}\, a_{\mu}\,\partial_{\nu}a_{\lambda}\right\},
\begin{aligned}(u_1)_t-\mu_1\Delta u_1 = a_1(t) u_1 - b_1(t) u_1^2 -\alpha_1(t) u_1u_2, x \in B,\t>0, (u_2)_t-\mu_2\Delta u_2 = a_2(t) u_2 - b_2(t) u_2^2 -\alpha_2(t) u_1u_2, x\in B,\t>0, \frac{\partial u_i}{\partial\nu}=0 \text{on $\partial B \times (0,\infty)$,}u_i(x,0) =u_{0,i}(x)\geq 0 \end{aligned}
\mu_0(f^{-1}(C)) = \pi((f^{-1}(C)\times X)\cap (\Gamma_{u}\setminus\{x=y\})) &= \pi ( X\times f^{-1}(C))\cap (\Gamma_{u}\setminus\{x=y\}) &= \mu_1(f^{-1}(C));
\sum_{n=1}^{+\infty}I_n\big(z_n(t,\,.\,)\big) =Z_t=\int_0^t S_A(t-r)BZ_r\d B^H_r=\int_0^t\sum_{n=0}^{+\infty}I_n\big(S_A(t-r)B z_n(r,\,.\,)\big)\d B^H_r &=\sum_{n=0}^{+\infty}I_{n+1}\big(\textrm{Sym}\big(S_A(t-\,.\,)B z_n(\,.\,)\big)\big)=\sum_{n=1}^{+\infty}I_n\big(\textrm{Sym}\big(S_A(t-\,.\,)B z_{n-1}(\,.\,)\big)\big),
{I_R}= Q\left({{d}/{{\left\|{\left[{- 1/k,1}\right]\sqrt{\textbf{R}}}\right\|}}}\right),{I_L}= Q\left({{d}/{{\left\|{\left[{1/k,1}\right]\sqrt{\textbf{R}}}\right\|}}}\right),{I_B}= Q\left({{d}/{{\left\|{\left[{0,1}\right]\sqrt{\textbf{R}}}\right\|}}}\right)
\frac{1}{2}\int\int\mathrm{d}w\mathrm{d}x\left(\left(\partial_\mu\tilde{y}\right)\left(\partial_\mu\tilde{y}\right)+\left(\partial_\mu\tilde{z}\right)\left(\partial_\mu\tilde{z}\right)\right)=\frac{1}{2}\int\mathrm{d}t\left(\tilde{y}\left(n.\partial\tilde{y}\right)+\tilde{z}\left(n.\partial\tilde{z}\right)\right)
\frac{Var_Q\Big( Z^{\beta, u, q}_{N, \tilde{\nu}^q_{(I,0)}}(\Omega_N^{},\theta_{IN,0}(V))\ \big|\ \mathcal{F}_{I}\Big)}{\Big( E^Q\Big(Z^{\beta, u, q}_{N, \tilde{\nu}^q_{(I,0)}}(\Omega_N^{},\theta_{IN,0}(V))\ \big|\ \mathcal{F}_{I}\Big)\Big)^2}\leq \frac{\epsilon}{8},.
\begin{aligned}\eta^{\kappa}(x,y \mid t) = (1-\kappa)\eta^{0}(x,y \mid t) + \kappa \pi^{*}_x G(\pi^*)_{x,y}&= (1-\kappa)^2\eta^{0}(x,y \mid t) + (2\kappa(1-\kappa)+\kappa^2) \tau^{\kappa}(x,y \mid t).\end{aligned}
r^*(X)+r^*(Y) = r(E-X)+||X||_r-r(E)+r(E-Y)+||Y||_r-r(E) \geq r(E-(X\cup Y))+r(E-(X\cap Y))+||X\cup Y||_r+||X\cap Y||_r-2r(E) = r^*(X\cup Y)+r^*(X\cap Y).
r_d(s,t)=\frac{\omega(d)}{2}\sum_{i=0}^{\min \{s-1,t-1\}}\frac{(s+t-i-2)!}{i! (s-i-1)! (t-i-1)!}(2\omega(d)-1)^i(1-\omega(d))^{s+t-2i-2}
\sum_{x\in (D,f)\times_S{\bar \eta}}{\rm dim\tot}_x(j_!{\cal F}|_{(U,f)\times_S{\bar \eta}}) =\sum_{x\in (D,g)\times_S{\bar \eta}}{\rm dim\tot}_x(j_!{\cal F}|_{(U,g)\times_S{\bar \eta}}),{\rm dim\tot}_u(j_!{\cal F}|_{(U,f)\times_S{\bar s}}) ={\rm dim\tot}_u(j_!{\cal F}|_{(U,g)\times_S{\bar s}})
\chi^{(U,U')}(w) = \sum_{\substack{\{C \; | \; w \cdot C = C \}s \in C}}\chi^{U \boxtimes U'}(s^{-1}ws) = \sum_{\substack{\{C \; | \; w \cdot C = C \}s \in C}}\chi^{U}(p_m(s^{-1}ws)) \cdot \chi^{U'}(p_{n-m}(s^{-1}ws))
C^{\infty}L^{p}(M,\Lambda^{k}):=C^{\infty}(M,\Lambda^{k})\cap L^{p}(M,\Lambda^{k}); C^{\infty}\Omega_{q,p}^{k}(M):=C^{\infty}(M,\Lambda^{k})\cap\Omega_{q,p}^{k}(M);
\Big( \sum_{\tau}\|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{p/2}= \Big( \sum_{\tau}\|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{(p-3)/2}\Big( \sum_{\tau}\|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{3/2}&\lesssim R^{C\delta}R^{\frac{3}{4}- \frac{p}{4}}\Big( \sum_{\tau}\|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{3/2}.
e^{- H_{\infty}^\uparrow(X^n|Y^n)}= \sum_{y^n}P_{Y^n}(y^n) \max_{x^n}P_{X^n|Y^n}(x^n|y^n) = P_{Y_1}(y_1) \max_{x_1}P_{X_1|Y_1}(x_1|y_1) \prod_{i=2}^n W_Y(y_i|y_{i-1}) T(y_i | y_{i-1}).
|P|(x) \ge \sup\Bigl\{\, \sum_{i}|A(u^1_{i},x,\dots,x)|: u^1 \in \Pi(x)\Bigr\}= \sup\Bigl\{\, \sum_{i}\Bigl|\int_K u^1_i x^{n-1}\,d\mu\Bigr|: u^1 \in \Pi(x)\Bigr\}= \int_K x^n\,d|\mu|\,,
\frac{\partial g_{ij}}{\partial t}=-2H\cdot h_{ij}begin{align*}5pt]\frac{\partial H}{\partial t}= \Delta H + H \cdot h_{pq}h_{pq}begin{align*}5pt]\frac{\partial |h|^2}{\partial t}= \Delta |h|^2-2|\nabla h|^2 +2R_1 begin{align*}5pt]\frac{\partial |H|^2}{\partial t}=\Delta |H|^2-2|\nabla H|^2+2R_2
\lim_{u\to D}\frac{g^{-1}(D-(D-u)x^{1/\gamma})}{g^{-1}(u)}=x \lim_{u\to D}\frac{g(g^{-1}(u)x)-u}{q(u)}=\lim_{u\to D}\frac{g\left(g^{-1}(u)\frac{g^{-1}(D-(D-u)x^{1/\gamma})}{g^{-1}(u)}\right)-u}{D-u}=1-x^{1/\gamma}
u_{- 1}= - 5 u_{1}+ v_{- 1}+ 2 w_{1}\, a^{- \gamma}+ ( \mathrm{~ \frac{9}{2}~}u_{1}^{2}- \mathrm{~ \frac{1}{2}~}v_{- 1}^{2}- 3 u_{1}v_{- 1}+ w_{2}) a^{- 2 \gamma}
\sum_{\rho}\frac{1+\eta-\beta}{|1+\eta+it-\rho|^2}\leq \frac{1}{5\eta}\#\{\rho\colon |\rho-(1+it)|\leq \eta,~L(\rho,\pi\otimes\chi)=0\}&\leq \frac{1}{2}\log q_{\pi\otimes\chi}+\frac{1}{2}\sum_{j=1}^{m}\mathrm{Re}\frac{\Gamma'}{\Gamma}\Big(\frac{1+\eta+it+\mu_{\pi\otimes\chi}(j)}{2}\Big)+\sum_{n=1}^{\infty}\frac{|\Lambda_{\pi\otimes\chi}(n)|}{n^{1+\eta}}
S_{P}^4 = \int d^4x\Bigl[{1 \over 4}\varepsilon^{\mu\nu\lambda\rho}B_{\mu\nu}F_{\lambda\rho}-{1 \over 4}B_{\mu\nu}B^{\mu\nu}-{\mu^2 \over 2}A_{\mu}A^{\mu}\Bigl]\;,
-\frac{\partial L_{d}D_{\beta}I(p\|q)}{\partial q_{j}}=Z\;\left[\left(\frac{\sum_{i}p_{i}q^{\beta-1}_{i}}{\sum_{i}q^{\beta}_{i}}\right)^{\beta-1}p_{j}q^{\beta-2}_{j}-\left(\frac{\sum_{i}p_{i}q^{\beta-1}_{i}}{\sum_{i}q^{\beta}_{i}}\right)^{\beta}q^{\beta-1}_{j}\right]
n_z(u)z=z,n_z(u)x=-(u,x)z+x,n_z(u)z'=-q(u)z+u+z', &m_z(a,g_1)z=az,m_z(a,g_1)x=g_1x,m_z(a,g_1)z'=a^{-1}z'+(a-a^{-1})q(z')z
A_3 =\frac{\beta}{2}\int_{-\infty}^0x|u|^2(\bar{u}u_x+u\bar{u}_x)dx =\frac{\beta}{2}\int_{-\infty}^0x\frac{1}{2}(|u|^4)_xdx =-\frac{\beta}{4}\int_{-\infty}^0|u|^4dx.
\begin{aligned}K =\int_0^1\mathrm{d}t\left((1-t^2)(1-(\alpha t/\beta)^2)\right)^{-1/2}\;, K' =\int_0^{\infty}\mathrm{d}t\left((1+t^2)(1+(\alpha t/\beta)^2)\right)^{-1/2}\ .\end{aligned}
\begin{array}{rll}\lambda (\chi E\bar v) - \Delta_H(\chi E\bar v) + \nabla_H(\chi E\pi) = \chi E\bar f - \partial_{z}(\chi Ev)|_{\Gamma_{b2}}- 2\nabla_H\chi\cdot\nabla_H(E\bar v) - (\Delta_H\chi)E\bar v - (\nabla_H\chi)E\pi \mathrm{div}_H(\chi E\bar v) = \nabla_H\chi\cdot E\bar v \end{array}
\lim_{T \to 0}\sup_{(0,T] \times D_R}|b_1(t,x)| = o(R) \mbox{(as $R \longrightarrow +0$)}, \lim_{T \to 0}\sup_{(0,T] \times D_R}|(\partial b_1/\partial x)(t,x)| = o(1) \mbox{(as $R \longrightarrow +0$)}.
{\cal Z}_{o p e n}^{9 - 9}= \int_{0}^{\infty}\frac{d s}{s}\left[ \mathrm{T r}_{\mathrm{N S}}\left({{\frac{1 + ( - 1 )^{F}}{2}}}{\frac{1 + \Omega}{2}}\, q^{2 L_{0}- 1}\right) - \mathrm{T r}_{\mathrm{R}}\left({{\frac{1 + ( - 1 )^{F}}{2}}}{\frac{1 + \Omega}{2}}\, q^{2 L_{0}}\right) \right] ~ ~ ,
\gamma_{\scriptscriptstyle{l}}= \Phi / 2 \pi - l + \frac{a^{2 l}}{\int_{0}^{a}d r \, r^{2 l - 1}\, e^{2 \phi ( r )}}- k^{2}\, a \chi_{\scriptscriptstyle{l}}^{\prime}( a ) + \mathrm{O}( k^{4}) \, .
\int_{U \cap Y}|f|_{he^{-\lambda \varphi}}^2 d\mu=\int_{U \cap Y}|f|_h^2(-\log |t|^2)^{2\lambda}\prod_{i=3}^l(-\log |t_i|^2)^{\lambda}\prod_{i=l+1}^n(1-|t_i|^2)^{\lambda}d\mu< \infty
B_d \,\,= \, \left(\begin{array}{cccccc}y z 0 0 \cdots 0 x y z 0 \cdots 0 0 2x y z \cdots 0 0 0 3x y \cdots 0 \vdots \vdots \vdots \vdots \vdots 0 0 \cdots (d-1)x y z \end{array}\right).
\left(1+\frac{1}{2}\sqrt{r}\right)^{2}= \frac{2+r/2+\sqrt{4r}}{2}\leq \frac{2+r+\sqrt{r^2+4r}}{2}\leq \frac{2+2r+\sqrt{4r^2+4r}}{2}=\left(1-\sqrt{\frac{r}{r+1}}\right)^{-1}.
\int_0^\infty p_\zeta^{(\alpha)}(t,r,s) s^{2\zeta}\,ds = 1, \int_0^\infty p_\zeta^{(\alpha)}(t,r,z) p_\zeta^{(\alpha)}(t',z,s) z^{2\zeta}\,dz = p_\zeta^{(\alpha)}(t+t',r,s), p_\zeta^{(\alpha)}(t,r,s) = t^{-\frac{2\zeta+1}{\alpha}}p_\zeta^{(\alpha)}\left(1,\frac{r}{t^{1/\alpha}},\frac{s}{t^{1/\alpha}}\right).
\sup\limits_{z\in\mathbb{C}^2}|z_q|^2e^{-\frac{1}{2}\Big(|e^{tM}z|^2-|z|^2\Big)}=\sup\limits_{s\in\mathbb{R}_{+}}|l(e'_q)|^2e^{-\frac{s Q_t(e'_q)}{2}}=\frac{2|l(e'_q)|^2}{Q_t(e'_q)}\sup\limits_{\sigma\in \mathbb{R}_{+}}\sigma e^{-\sigma}=c_0\frac{2|l(e'_q)|^2}{Q_t(e'_q)}=c_0\frac{2}{Q_t(e'_q)}
\hat{\Phi}_{\rm{st},\beta}(t) = \frac{S_\beta}{\pi E_{\rm{b}}}\bigg(\frac{P_{\rm{cu}}}{\hat{R}^{*2}_{\beta}}+ \lambda_{\beta}\left( 2^{C/W}- 1 \right)P_{\rm{tr},1}(h_{\beta,1}^*)\hat{R}^{*2}_{\beta}\bigg),
2\nabla V\cdot\nabla\partial_xf-2(\partial_y+\kappa\partial_z) (\partial_yV\partial_xf) &=2\partial_yV(\partial_y+\kappa\partial_z)\partial_x f-2(\partial_y+\kappa\partial_z)(\partial_yV\partial_xf) &=-2(\partial_y^2V+\kappa\partial_z\partial_yV)\partial_xf.
A=\left[\begin{array}{ccc}4 \ -2 \ 2 -1 \ 3 \ 1 1 \ -1 \ 5 \end{array}\right] V=\left[\begin{array}{ccc}| | | v^1 v^2 v^3 | | | \end{array}\right]=\left[\begin{array}{ccc}1 0 1 1 1 0 0 1 1 \end{array}\right].
a^{\dagger}( k , \rho ) \vert 1 \colon p_{i}\rangle = -{\frac{e}{( 2 \pi )^{3 / 2}}}{\frac{1}{\sqrt{2 k^{+}}}}f^{*}( k , \rho : p_{i}) \vert 1 \colon p_{i}\rangle + \vert 2 \colon p_{i}, k_{i}\rangle .
c =\tilde{c},\ \alpha = \tilde{\alpha},\c_{0}= \tilde{c}_{0}; & \frac{\tau_{2}^{(0)}}{2}+ \tau_{1}^{(0)}( c_{1}- \tilde{c}_{1}) =0; (n-1) \tau_{1}^{(n-1)}+ \sum_{l=2}^{n}\tau_{2}^{(n-l)}(c_{l-1}-\tilde{c}_{l-1}) =0; & (n-\frac{1}{2}) \tau_{2}^{(n-1)}+ \sum_{l=1}^{n}\tau_{1}^{(n-l)}(c_{l}- \tilde{c}_{l}) =0,
|\lambda|_a = 1-2+3-\cdots +(-1)^{\lambda_1-1}\lambda_1+(-1)^{\lambda_1}\lambda_2+\cdots+(-1)^{\lambda_1+k-2}\lambda_k &= (-1)^{\lambda_1-1}(\lceil\frac{\lambda_1}{2}\rceil+\sum_{i=2}^{k}(-1)^i\lambda_i) &= (-1)^{l(\lambda^T)}\lfloor \frac{e(\lambda^T)-o(\lambda^T)}{2}\rfloor ,
S^{(2)}= \frac{1}{2\pi}\int d^{2}x \sqrt{-g}\left[e^{-2\phi}\left(R^{(2)}+ 4{(\nabla \phi )}^{2}+ 4 \lambda^{2}\right)- \frac{1}{2}\sum_{i=1}^{N}{(\nabla f_{i})}^{2}\right]
b_{s}( x , \beta ) = -{\frac{\pi^{2}}{9 0 \beta_{l}^{4}}}-{\frac{1}{2 4 \beta_{l}^{2}}}\left[ \left( \frac 1 6 - \xi \right) R - m^{2}\right] -{\frac{a_{s , 2}( x )}{1 6 \pi^{2}}}\ln{\frac{\mu \beta_{l}}{2 \pi}},
\ln \mathrm{d e t}_{{S c h}}= \quad - ( \mathrm{n o . ~ b o u n d ~ s t a t e s}) X \quad \left[ \ln \left( \displaystyle \frac{\mathrm{n o . ~ b o u n d ~ s t a t e s}}{m a}\right) + 0 ( 1 ) \right] .