harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
u_t(x)= P_t u_0(x) + \int_0^t P_{t-s}\big(\beta\,u_s(1-u_s)\big)(x)\,ds &+ \int_{[0,t]\times \Gamma}p(t-s,x,y)\,\ell(y)\sqrt{\gamma(y)\,u_s(y)\big(1-u_s(y)\big)}\,dW(s,y) &+\frac{1}{2}\int_0^t\sum_{v\in V}p(t-s,x,v)\,\ell(v)\,\hat{\beta}(v)\,u_s(v)(1-u_s(v))\,ds,
\begin{aligned}(\delta_P Q_C)^a F_a = P^\alpha (\delta^*_F Q_C)_\alpha + D_i \Psi^i(P,Q_C;F)= -P^\alpha (\delta^*_{Q_C}F)_\alpha + D_i \Psi^i(P,Q_C;F) &= -P^\alpha R_{Q_C}(F)_\alpha + D_i \Psi^i(P,Q_c;F)\end{aligned}
\pm \left[ ( + 1 , 0 , 0 , 0 , 0 , \, 0 , + 1 , 0 , 0 , 0 , \, 0 , \cdots , 0 ) \ominus ( 0 , + 1 , 0 , 0 , 0 , \, + 1 , 0 , 0 , 0 , 0 , \, 0 , \cdots , 0 ) \right]
=\mathbb P\{\sigma_{2}< T_2, \sigma_{3}> \overline T_2\}\sum_{l=0}^{\infty}\mathbb P\{\sigma_{2l+2}< T_2, \sigma_{2l+3}> \overline T_2\mid \eta_k=2l+1\}\mathbb P\{\eta_k=2l+1\}&=\mathbb P\{\sigma_{2}< T_2, \sigma_3> \overline T_2\}^2.
\log F(z\,|\,\overline\omega_1,\omega_2) = \log F(cz\,|\,c\overline\omega_1,c\omega_2) \sim \int_C \frac{e^{czs}}{e^{c\overline\omega_1 s}-1}\sum_{k\geq 0}\frac{B_k\cdot (c\omega_2)^{k-1}(s)^{k-2}}{k!}\, ds & = \sum_{k\geq 0}\frac{B_k\cdot \omega_2^{k-1}}{k!}\cdot \int_C \frac{e^{czs}\,(cs)^{k-2}}{e^{c\overline\omega_1 s}-1}\d(cs).
P_{{\bf b},d}^{(-1)}=\prod_{\substack{j=1 b_j\geq 2}}^{n}\Bigl\{(\theta-t)^{q^{d-2}}\mathbb{D}_{b_j-2}^{q^{d-b_j}}\Bigr\}P_{{\bf b},d}=\prod_{\substack{j=1 b_j\geq 2}}^{n}\biggl\{(\theta-t)^{q^{d-2}}\Bigl((\theta^q-t)^{q^{b_j-3}}\cdots(\theta^{q^{b_j-2}}-t)\Bigr)^{q^{d-b_j}}\biggr\}P_{{\bf b}, d}.
\mathbb E_{h}(\mathbf1_{D\setminus \bar V}u(X_{\tau_V})\mathbf1_{A\cap V}(X_{\tau_V-})) = 2\int_{\mathbb R^d\times\mathbb R^d}R^Vh(x)\mathbf1_{A\cap V}(x)\mathbf1_{D\setminus \bar V}u(y)\,J(dx,dy) & =\int_{A\cap V}\int_{D\setminus \bar V}R^Vh(x)u(y)j(|x-y|)\,dy\,dx &= \int_{A\cap V}\int_{D\setminus \bar V}\int_V G_V(x,z)h(z)\,dz \,u(y)j(|x-y|)\,dy\,dx.
U = \left( \begin{matrix}{{1}}&{{}}&{{}}&{{}}\\{{}}&{{e^{\frac{2 \pi i}{N}}}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{}}&{{e^{\frac{2 \pi i ( N - 1 )}{N}}}}\\ \end{matrix}\right)
\langle 0 | T ( a_{+}^{a *}( x ) \bar{a}_{-}^{b}( y ) ) | 0 \rangle = \langle 0 | T ( a_{-}^{a *}( x ) \bar{a}_{+}^{b}( y ) ) | 0 \rangle ={\cal F}\left[ \frac{( 2 i \delta^{a b})}{k}\right]
D_s\mathrm{(IV)}=D_s(R(\gamma_s,\gamma_t)\gamma_t) =D_sR(\gamma_s,\gamma_t)\gamma_t+R(D_s\gamma_s,\gamma_t)\gamma_t+R(\gamma_s,D_s\gamma_t)\gamma_t+R(\gamma_s,\gamma_t)D_s\gamma_t &=D_sR(\gamma_s,\gamma_t)\gamma_t+R(\gamma_s,D_t\gamma_s)\gamma_t+R(\gamma_s,\gamma_t)D_t\gamma_s-2\alpha R(\gamma_s,\gamma_t)\gamma_t
U^\mu=-\Gamma^\mu_{\rho\sigma}(\dot{\eta}^\rho\dot{\eta}^\sigma-\eta'^\rho\eta'^\sigma)-2\Gamma^\mu_{\rho\sigma,\lambda}\dot{q}^\rho\eta^\lambda\dot{\eta}^\sigma-\frac{1}{2}\Gamma^\mu_{\rho\sigma,\lambda\delta}\dot{q}^\rho\dot{q}^\sigma\eta^\lambda\eta^\delta.
S_{\mathrm{b o u n d a r y}}^{G L S M}= \int d x^{0}\left\{i{\frac{\Theta}{2 \pi r}}\sum_{i}( \phi_{i}\widetilde{D}_{0}\overline{{{\phi}}}_{i}- \overline{{{\phi}}}_{i}\widetilde{D}_{0}\phi_{i}) \right\}
\begin{array}{rcl}\hat{P}_6(x)\!\! \!=\! \!\! 64 x^6-192\pi x^5+\left(240\pi^2-1344 \right) x^4+\left(-160\pi^3+2688\pi \right) x^3 [1.0 ex]\!\! \! \! \!\! \!+\!\left(60\pi^4\!-\!2016\pi^2\right)x^2\!+\!\left(\!-\!12\pi^5\!+\!672\pi^3\right)x\!+\!\pi^6\!-\!84\pi^4\!+\!7560,\end{array}
T_{\hat M}^{~\hat N}={\rm diag}\left (\frac{n(n+1)}{2\ell^2}-\frac{{\cal Q}^2}{a^{2n}}, \frac{n(n+1)}{2\ell^2}+\frac{{\cal Q}^2}{a^{2n}}, \cdots, \frac{n(n+1)}{2\ell^2}+\frac{{\cal Q}^2}{a^{2n}}, \frac{n(n+1)}{2\ell^2}-\frac{{\cal Q}^2}{a^{2n}}\right).
G_2^{\mathrm{reg}}(\xi,\sigma_1,\sigma_2,\alpha_1,\alpha_2,\beta_1,\beta_2)=\phi_2(\sigma_1,\sigma_2,\alpha_1,\alpha_2,\beta_1,\beta_2) &+H_2(\xi,\sigma_1,\sigma_2,\alpha_1-\tau\beta_1,\alpha_2-\tau\beta_2,\beta_1+\beta_2,\beta_2)+\phi_1(\sigma_1,\alpha_1,\beta_1)H_1(\xi,\sigma_2,\alpha_2-\tau\beta_2,\beta_2)\,.
\Pi\Bigl(\sum_{i=1}^N(\kappa_if_i- \kappa_if_{0,i})^2 \leq \epsilon^2/2\Bigr) \geq \exp\Bigl(-\Bigl(1+2\alpha+2p+\frac{\log 2}{2}\Bigr)N\Bigr) \exp\Bigl(-N^{(1+2\alpha-2\beta)\wedge 0}\|f_0\|_\beta^2\Bigr) \times{}\Pr\Bigl(\sum_{i=1}^N V_i^2 \leq 2\delta^2N^{1+2\alpha+2p}\Bigr).
(ab)^i{}_{jkl}=a^i{}_{\alpha\beta\gamma}b^\alpha{}_jb^\beta{}_kb^\gamma{}_l+a^i{}_{\alpha\beta}b^\alpha{}_{jl}b^\gamma{}_k+a^i{}_{\alpha\beta}b^\alpha{}_jb^\gamma{}_{kl}+a^i{}_{\alpha\beta}b^\beta{}_lb^\gamma{}_{jk}+a^i{}_\alpha b^\alpha{}_{jkl}
\nabla G(U,x,t)^T\nabla A(U,x,t) =\nabla A(U,x,t)^T\nabla G(U,x,t) \nabla f_\alpha(U,x,t)^T\nabla A(U,x,t) =\nabla f_\alpha(U,x,t)^T\nabla G(U,x,t)\qquad\alpha=1,\dots,d,
g^{ij}\, \Pi \left( F_{sij}\right) = g^{ij}\, \nabla_{F_i}^{\perp}\nabla_{F_j}^{\perp}V - g^{ij}\, \nabla_{F_i}^{\perp}\left( F_{\ell}\, g^{\ell k}\langle F_{jk}, V \rangle \right) = g^{ij}\, \nabla_{F_i}^{\perp}\nabla_{F_j}^{\perp}V - g^{ij}\, A_{i\ell}\, g^{\ell k}A^V_{jk}\, .
\mathrm{d}\equiv \mathrm{d}z^{{\cal A}}\frac{\partial ~}{\partial z^{{\cal A}}}= e^{{\cal A}}D_{{\cal A}}= e^{A}\partial_{A}+ \mathrm{d}\theta^{i \alpha}D_{i \alpha}+ \mathrm{d}\tilde{\theta}_{\alpha}^{\tilde{\imath}}{\tilde{D}}_{\tilde{\imath}}^{\alpha}
sp_\lambda(\mathbf{x}^{\pm};\mathbf{y}^{\pm};z)=\sum_{\substack{\mu l(\mu)=m}}sp_\mu(\mathbf{y}^{\pm})sp_{\lambda/\mu}(\mathbf{x}^{\pm};z), &o_\lambda(\mathbf{x}^{\pm};\mathbf{y}^{\pm};z)=\sum_{\substack{\mu l(\mu)=m}}o_\mu(\mathbf{y}^{\pm})o_{\lambda/\mu}(\mathbf{x}^{\pm};z).
T ( z ) = g_{a b}( z ) \frac{\delta S}{\delta g_{a b}( z )}= \frac{\varepsilon \mu^{- \varepsilon}}{8 \pi \alpha^{\prime}}g^{a b}( z ) \partial_{a}x^{\mu}\partial_{b}x^{\nu}G_{\mu \nu}- \frac{\mu^{- 1 - \varepsilon}}{4 \pi \alpha^{\prime}}H_{\mu \nu}\dot{x}^{\mu}\dot{x}^{\nu}\delta_{\partial M}( z )
S = \frac{1}{g^{2}}\mathrm{T r}\Bigl ( \frac{1}{4}F_{j k}F^{j k}+ \frac{1}{2}\sum_{\alpha}[ \nabla_{j}, \Phi^{\alpha}] [ \nabla^{j}, \Phi^{\alpha}] + V ( \Phi ) \Bigr )
\frac{1}{N^{h-m}}\int\limits_{\mathbf{x}\in \mathbb{R}^h}F(\sum\limits_{i=1}^{h}x_i\mathbf{v^{(i)}})G(L(\sum\limits_{i=1}^{h}x_i\mathbf{v^{(i)}}))(\prod\limits_{j=1}^{d}g_j(\xi_j(\Phi(\mathbf{x_1^{h-m}})+\sum\limits_{i=h-m+1}^{h}x_i \mathbf{v}^{\mathbf{(i)}})+ \widetilde{\mathbf{r}}_j)) \, d\mathbf{x},
\sum_{n=0}^\infty \frac{1}{n!}\langle \phi_a\,\psi^m,\mathbf{t},\dots,\mathbf{t}\rangle_{g,n+1}(t)= \frac{1}{4}\sum_{i=1}^N{\rm Res}_{\lambda=u_i}\,\frac{(I_{\beta_i}^{(-m-1)}(t,\lambda),\phi_a)}{(I^{(-1)}_{\beta_i}(t,\lambda),1)}\ \Omega^{(g)}_{\beta_i,\beta_i}(t,\lambda;\mathbf{t})\d\lambda,
W_I = \lambda_{1}\epsilon_{abc}{\mbox{Tr}}(P_a P_b Q_c) + \lambda_{2}{\mbox{Tr}}(Q_1 R_2 \Phi_3 + Q_2 R_3 \Phi_1 + Q_3 R_1 \Phi_2) + \lambda_{3}{\mbox{Tr}}(R_1 R_2 R_3) + \cdots ~.
\| c \| = \| \langle \zeta, g \zeta \rangle \| \leq \| g \| \| \langle \zeta, \zeta \rangle \| = \| g \| = \| \langle \xi, T \xi \rangle \| &\leq \| T \| \, \| \langle \zeta, \zeta \rangle \| = \| T \| = \| y_0 \| < \| y_0 \| + \epsilon.
\mathfrak{d}^{\rm (out)}_{n,d}= \mathfrak{d}^{\rm (out)}_{n-2,d+1}+\mathfrak{d}^{\rm (out)}_{n-1,d-1}+ \mathfrak{d}^{\rm (out)}_{n-3,d}+ \sum\limits_{i=0}^{n-3}\mathfrak{d}^{\rm (out)}_{i,1}\cdot \mathfrak{d}^{\rm (out)}_{n-i-3,d-1}+ \sum\limits_{i=0}^{n-2}\sum\limits_{j=0}^{d-2}\tilde{s}_{i,j}\cdot \mathfrak{d}^{\rm (out)}_{n-2-i,d-2-j}.
\tilde{\phi}(p_{r(0^{(n)}0^{(n)})}) = \tilde{\phi}(p_{r(1^{(n)}1^{(n)})}) = 1/ (6 \cdot 2^{n}) \hbox{, and}\tilde{\phi}(p_{r(0^{(n)}1^{(n)})}) = \tilde{\phi}(p_{r(1^{(n)}0^{(n)})}) = 1/ (3 \cdot 2^{n}).
\vartheta_{m}^{i+1}= - \frac{\gamma}{(\gamma-1)\Delta x}\big( \widetilde{m}_{i+1}^{\gamma-1}- \widetilde{m}_{i}^{\gamma-1}\big) +\frac12 \chi\left(\frac{(c_x)^i}{1+\widetilde{m}_{i}}+\frac{(c_x)^{i+1}}{1+\widetilde{m}_{i+1}}\right),
\left|\varphi_0\left(\frac{e_n-x}{\delta}\right)\right| \leq \frac{C}{(1+\frac{1}{2}\delta^{-1})^{n-3}}\leq C2^{n-3}\delta^{n-3},U_{\delta,-e_n}^p= \left(\frac{\delta}{\delta^2+|x+e_n|^2}\right)^\frac{n+2}{2}\leq 2^{n+2}\delta^\frac{n+2}{2}
\mathbb{P}_{\!x}\{\tau_{ab}\in\mathrm{d}t,X_{\tau_{ab}}\in\mathrm{d}z\}/\mathrm{d}t\,\mathrm{d}z =\sum_{k=0}^{N-1}(-1)^k I_k^-(t;x)\,\delta_a^{(k)}(z)+\sum_{k=0}^{N-1}(-1)^k I_k^+(t;x)\,\delta_b^{(k)}(z)
\mu_{\mathrm{w k b}}= 2^{- 1 / 2}\biggl ( \frac{n \epsilon}{2 \pi}\biggr )^{- 1 / 2}n^{- 1 / 2}b_{1}^{- 1 / 4}\vert \eta \vert^{- 1 / 2}\exp \biggl [ \mp i b_{1}^{1 / 2}\biggl ( \frac{n \epsilon}{4 \pi}\biggr ) n \biggl ( \vert \eta \vert^{2}- \vert \eta_{\mathrm{i}}\vert^{2}\biggr ) \biggr ] .
(b-a)\limsup_{n\to+\infty}\mathbb{E}\chi^{(n)}((b,+\infty))\leq\lim_{n\to+\infty}\mathbb{E}\sum_{k=1}^n((\tau_k-a)_+-(\tau_k-b)_+) =(\pi/2)e^{c_0-a}-(\pi/2)e^{c_0-b}\leq(b-a)\liminf_{n\to+\infty}\mathbb{E}\chi^{(n)}((a,+\infty)).
P_Y(z,w) = \left(\frac{1}{1-qz}\right)^w\, \exp\left(wR(q^{-1})+\sum_{k\geq 2}\frac{I(q^{-k})\,(1-(1-w)^k)}{k}\right)\,(1+o(1)) P_Z(z,w) = \left(\frac{1}{1-qz}\right)^w\,\exp\left(wR(q^{-1})+\sum_{k\geq 2}\frac{I(q^{-k})\,w^k}{k}\right)\,(1+o(1))
\mathcal{J}-j \leq \frac{\log(1/\alpha_{j})}{\log 20}, \mbox{and}\sum_{X^{\alpha_{j}}< p \leq X^{\alpha_{j+1}}}\frac{1}{p}= \log \alpha_{j+1}- \log \alpha_{j}+ o(1) = \log 20 + o(1) \leq 10.
g_W(R^W(\sigma_1,\sigma_2)\sigma_3, \sigma_4)= \sigma_1( g_W(\nabla_{\sigma_2}^W\sigma_3, \sigma_4))-g_W(\nabla_{\sigma_2}^W\sigma_3, \nabla_{\sigma_1}^W\sigma_4) - \sigma_2( g_W(\nabla_{\sigma_1}^W\sigma_3, \sigma_4))+g_W(\nabla_{\sigma_1}^W\sigma_3, \nabla_{\sigma_2}^W\sigma_4).
K [ H_{n}, 1 ] = \left( \frac{m \omega \sqrt{\xi \xi_{0}}}{i \hbar \sin \omega t}\right) \left( \frac{m \omega \sqrt{\eta \eta_{0}}}{i \hbar \sin \omega t}\right) J_{n}\left( \frac{m \omega \sqrt{\xi \xi_{0}}}{i \hbar \sin \omega t}\right) J_{n}\left( \frac{m \omega \sqrt{\eta \eta_{0}}}{i \hbar \sin \omega t}\right) \nonumber
m_{1,5}m_{7,9}+m_{1,6}m_{8,9}=0, m_{1,5}=0, m_{1,6}m_{7,9}=0, 1-m_{1,2}m_{7,9}-m_{1,3}m_{8,9}=0, m_{1,2}-m_{7,9}=0, -m_{1,3}m_{7,9}-m_{8,9}=0, 1+m_{4,5}m_{7,9}+m_{4,6}m_{8,9}=0, -m_{4,5}+m_{7,9}=0, m_{4,6}m_{7,9}+m_{8,9}=0.
\mu\big(\begin{smallmatrix}b_{11}0 b_{21}b_{22}\end{smallmatrix}\big)^\ast f (x) = \mu_2(1/(\sqrt{L}a_{11}))\, \mu_1(-a_{21}/a_{11})f(x) = (\sqrt{L}a_{11})^{1/2}\,e^{-\pi i a_{21}/a_{11}(\sqrt{L}a_{11}x)^2}\, f(\sqrt{L}a_{11}x) & =(\sqrt{L}a_{11})^{1/2}\,e^{-\pi i L a_{21}a_{11}x^2}\, f(\sqrt{L}a_{11}x)
|q_k\nu-e-p_k|\leq\Delta(p_k,q_k)^{|P_k|L_{k}}, \Delta(p_k,q_k)=\min\left\{|q\nu-re-p|\colon 0<|r|+|q|<3(q_k+1),\right. \qquad\qquad\left. (p,q,r)\neq 0, \pm( p_k,q_k, 1), \pm2(p_k,q_k,1)\right\}.
\varphi_{\beta,\varepsilon}'(s) = - \frac{\beta}{\gamma_{\varepsilon}}e^{-s}M(\gamma_{\varepsilon}-\beta, \gamma_{\varepsilon}+ 1; s), \varphi_{\beta,\varepsilon}''(s) = \frac{\beta(\beta+1)}{\gamma_{\varepsilon}(\gamma_{\varepsilon}+1)}e^{-s}M(\gamma_{\varepsilon}-\beta, \gamma_{\varepsilon}+ 2; s).
( 3 , 1 ) : \quad \left( \begin{array}{c}{{u_{L}^{i}}}\\{{d_{L}^{i}}}\\{{b_{L}^{i}}}\\ \end{array}\right) , ~ \quad ( 1 , \bar{3}) : \quad \left( \begin{array}{c}{{\hat{u}_{R}^{i}}}\\{{\hat{d}_{R}^{i}}}\\{{\hat{b}_{R}^{i}}}\\ \end{array}\right)
\sum_{k=0}^{n-1}\frac{(a^{d-1}q^{d-1},a^{d-3}q^{d-1},\cdots,a^{3-d}q^{d-1},a^{1-d}q^{d-1};q^d)_kq^{dk}}{(q^d,a^{d-2}q^d,a^{d-4}q^d,\cdots,a^{4-d}q^d,a^{2-d}q^d;q^d)_k}\equiv B_q(d,n,d-1)
{\hat{q'}}_{l}(\zeta)=f(\theta'_{+l}(\zeta),\theta_{+k}(\zeta)){\hat{q}}_{k}(\zeta),\quad{\bar{\hat{q}}}'_{l}(\zeta)={\bar{\hat{q}}}_{k}(\zeta)f^{*}(\theta'_{-l}(\zeta),\theta_{-k}(\zeta)),
\hat{\mathbf{t}}_{l}=\left[\sum_{\mu=\pi_{c}\left(j\right)}^{L}\hat{\mathbf{t}}_{l,\mu}\right]\bmod\Lambda_{s}=\left[\sum_{\mu=\pi_{c}\left(l\right)}^{L}\mathbf{t}_{l,\mu}\right]\bmod\Lambda_{s}=\mathbf{t}_{l}
b(p_r)=4\left.\frac{[(\tilde{k}_r+\tilde{h})r-(k_r+h)\tilde{r}]\sinh\frac{\beta|p^0|}{2}}{(k_r+h)^2+r^2\sinh^2\frac{\beta|p^0|}{2}}\right|_{p^2=p^2_r=m^2_{\phi_S^0}}.
g \, t_{i_1 \ldots i_n \Phi^2}+U \, t_{i_1 \ldots i_n \Phi}+c\, (t_{(i_1+1) i_2 \ldots (i_n-1) \Phi}+ t_{(i_1-1) i_2 \ldots (i_n+1) \Phi})- \sum_{j=1}^{n-1}t_{i_1 \ldots i_j}\, t_{i_{j+1}\ldots i_n}=0
\sum_{i\in Z}(\begin{matrix}m\\i\end{matrix})(u_{q+i}(v))_{m+n-i}(w)=\sum_{i\in Z}(-1)^{i}(\begin{matrix}q\\i\end{matrix})(u_{m+q-i}(v_{n+i}(w))-(-1)^{q}v_{n+q-i}(u_{m+i}(w)))
C_{I J K}h^{I}h^{J}h^{K}= 3 \left\{h^{1}\big ( h^{2}\big )^{2}- h^{1}\big ( h^{\mu}\big )^{2}- h^{2}\big ( h^{i}\big )^{2}+ \gamma_{\mu i j}h^{\mu}h^{i}h^{j}\right\}.
\frac{\partial}{\partial \tilde{\zeta}_{n}}\, \Big \{\ \sum_{i = 1}^{5 6}\, \big ( \sum_{k}M_{i k}\tilde{\zeta}_{k}- b_{i}\big )^{2}\ \Big \}\ = \ 0 \qquad ( \n \, = \, 1 \ldots 1 7 \ )
\left(\widehat{i\ 0}\right)=V_{\mathbb{Z}\beta+\frac{i}{4k}\beta}\otimes(V_{\mathbb{Z}\beta}^{T_{1}})^{+}+V_{\mathbb{Z}\beta+\frac{2k+i}{4k}\beta}\otimes(V_{\mathbb{Z}\beta}^{T_{1}})^{+},
\mathcal{F}_{1,\theta}[\Sigma_t]=-\int_\Sigma H_2(t)\left<\xi_t,\eta_t\right>\,d\mu_{\Sigma_t}+\int_{\partial\Sigma}\left<\xi_t,(P_1\nu-\vert{P_1\nu}\vert\cos\theta\,\overline\nu)_t\right>\,d\mu_{\partial\Sigma_t},
L = \int d^{4}\theta \left( \sum_{i = 1}^{N + 1}\overline{{{\Phi}}}_{i}e^{2 \sum_{a = 1}^{N - 1}Q_{a i}V_{a}}\Phi_{i}- \sum_{a = 1}^{N - 1}\frac{1}{2 e_{a}^{2}}\overline{{{\Sigma}}}_{a}\Sigma_{a}\right) - \frac{1}{2}\left( \int d^{2}\widetilde{\theta}\sum_{a = 1}^{N - 1}t_{a}\Sigma_{a}+ \mathrm{c . c .}\right) ,
S'=\Big<s,s_1, \ldots, s_{2^n-4}\colon\, s^2=1, \s_{\alpha}^2=-1, ss_{\alpha}=-s_{\alpha}s, \s_{\alpha}s_{\beta}=-s_{\beta}s_{\alpha}, \ \ 1\le \alpha\neq \beta\le 2^n-4\Big>.
\hat{\Pi}_{V_1}-\sum_{j=1}^k(\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^j(1-\hat{\Pi}_{V_1}) &=\sum_{j=0}^{k-1}\hat{\Pi}_{V_1}(\hat{\Pi}_{V_2}\hat{\Pi}_{V_1})^j(1-\hat{\Pi}_{V_2})+(\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^k\hat{\Pi}_{V_1},
\begin{array}{rcl}({\bf 1})_{abcd}= S_{ac}S_{bd}(Q-1)(P_{0})_{abcd}= S_{ab}S_{cd}{\rm and \;\;}(Q-2)(P_{1})_{abcd}= Q\sum_{\alpha}S_{a\alpha}S_{b\alpha}S_{\alpha c}S_{\alpha d}-S_{ab}S_{cd}\;. \end{array}
\times \exp \Biggl \{\int d^{4}x \Biggl [ - \frac 1 4{\bf F}_{\mu \nu}^{2}- \eta^{2}\sum_{a = 1}^{3}\left( \partial_{\mu}\theta_{a}- g_{m}{\bf e}_{a}{\bf B}_{\mu}\right)^{2}+ i k \sum_{a = 1}^{3}\theta_{a}^{\mathrm{r e g .}}\Biggr ] \Biggr \}.
\sigma_n\tau_n =(\rho_n,(k_1,\ldots,k_j+k_{n+1},\ldots,k_n,0)) ((j,n+1),(0,\ldots,k_j,\ldots,-k_j)) &=(\rho_n \circ (j,n+1),(k_1,\ldots,0,\ldots,k_n,k_j+k_{n+1})+(0,\ldots,k_j,\ldots,0,-k_j)) &=(\rho_{n+1},(k_1,\ldots,k_{n+1})) = \sigma_{n+1}.
{\cal Z}_{\mathrm{g r a n d}}^{\mathrm{3 D}}= \int{\cal D}{\bf j}\exp \left\{- \left[ \frac{\pi \eta^{2}}{2}\int d^{3}x d^{3}y j_{\mu}({\bf x}) \frac{\mathrm{e}^{- m |{\bf x}-{\bf y}|}}{|{\bf x}-{\bf y}|}j_{\mu}({\bf y}) + V [ 2 \pi{\bf j}] \right] \right\}.
\|u\|_{L^{2}_{\tau}((a, b), X)}^{2}= \sum_{k = i}^{j}\|u_{k}\|_{L^{2}((0, \tau), X)}^{2}, \|u\|_{H^{1}_{\tau}((a, b), X)}^{2}= \sum_{k = i}^{j}\|u_{k}\|_{H^{1}((0, \tau), X)}^{2}.
\int_{\mathcal{B}^c_{\epsilon , k}}\frac{\overline{\widehat{\varphi}(kp)}\widehat{\eta}(kp)}{(p^2 +1 )^2}\mathrm{d}p= k^2\int_{\mathcal{B}^c_{\epsilon , 1}}\frac{\overline{\widehat{\varphi}(t)}\widehat{\eta}(t)}{(t^2 +k^2)^2}\mathrm{d}t= \mathcal{O}(k^2 )\,.
{_{*}X}_{{_{*}\phi}^{(s+1)}_{\alpha}}={_{*}X}_{\{{_{*}\phi}^{(s)}_{\alpha},H_{T}\}}=[{_{*}X}_{T},{_{*}Y}^{(s)}_{\alpha}]={_{*}Y}^{(s+1)}_{\alpha}, &{_{*}X}_{{_{*}\phi}^{(m_{\alpha}+1)}_{\alpha}}={_{*}C}_{\alpha s}^{\beta}{_{*}Y}^{(s)}_{\alpha},
:= \; \left\langle \prod_{b=1}^N2 \left( \sum_{n_b=0}^\infty \frac{1}{(2 n_b)!}\Big(\beta^{(b)}\Big)^{2n_b}\;\left[\chi^{(b)}_+[Q^W] + \chi^{(b)}_-[Q^W]\right]^{2n_b}\right)\right\rangle_{Q^\mu}
\int_{0}^{1}t^{q(\frac{1}{p}-1)}(\int_{0}^{t}g(u)du)^{q-1}g(t)dt=\frac{1}{q}\int_{0}^{1}t^{q(\frac{1}{p}-1)}[(\int_{0}^{t}g(u)du)^{q}]^{\prime}dt= =\frac{1}{q}(\int_{0}^{1}g(u)du)^{q}+(1-\frac{1}{p})\int_{0}^{1}(t^{\frac{1}{p}-1}\int_{0}^{t}g(u)du)^{q}\frac{dt}{t}=\frac{f^{q}}{q}+\frac{1}{p^{\prime}}\Delta(g)
\gamma_{1 , 1}^{6}= 7 2 0 , \gamma_{2 , 1}^{6}= 6 9 8 4 , \gamma_{3 , 1}^{6}= 2 3 3 2 8 , \gamma_{4 , 1}^{6}= 3 9 6 7 2 , \gamma_{5 , 1}^{6}= 4 5 9 3 6 \nonumber
\lim_{L\to\infty}\frac 1{L^d}\int_0^T\Psi_L\bigl(\mu^L_t,\jmath^L_t\bigr)\;\!\mathrm dt = \frac 12\int_0^T\|\dot\rho_t\|_{-1,\chi(\rho_t)}^2\;\!\mathrm dt
E(u_{0,n}, u_{1,n}) = \sum_{j = 1}^J E \left ( U^j_L( -t_{j,n}/\lambda_{j,n}), \partial_t U^j_L( -t_{j,n}/ \lambda_{j,n}) \right ) + E ( w_{0,n}^J, w_{1,n}^J ) + o_n(1).
({\bf a}_{i}\cdot{\bf a}_{j}) = ({\bf b}_{i}\cdot{\bf b}_{j}) = ({\bf c}_{i}\cdot{\bf c}_{j}) = \delta_{i j}, \quad ({\bf a}_{i}\cdot{\bf b}_{j}) = - \frac{1}{2}, \quad ({\bf a}_{i}\cdot{\bf c}_{j}) = \frac{1}{2}, \quad
f \star g = \sum_{s = 0}^{\infty}\frac{\kappa^{s}}{s !}\sum_{t = 0}^{s}( - 1 )^{t}\left( \begin{matrix}{s}\\{t}\\ \end{matrix}\right) [ \partial_{x}^{s - t}\partial_{p}^{t}f ] [ \partial_{x}^{t}\partial_{p}^{s - t}g ] .
\textstyle\pi_{\eta}'\colon X'\to Y', (B_{l,1},B_{l,2},i_{l})_{1\le l\le e}\mapsto (\bigoplus_{l=1}^{e}B_{l,1},B_{2},\bigoplus_{l=1}^{e}i_{l},Z_{1},Z_{2},...,Z_{e}),
Z_{\Lambda}[J,\phi^{\phantom{a}}_{UV}]=\int\!\! \mathcal{D}\phi^{\phantom{a}}_{IR}\e^{-\frac{1}{2}\phi^{\phantom{-1}}_{IR}.\Delta_{IR}^{-1}.\phi^{\phantom{-1}}_{IR}- S^{\phantom{-1}}_{\Lambda_{o}}[\phi^{\phantom{-1}}_{IR}+ \phi^{\phantom{-1}}_{UV}] + J.(\phi^{\phantom{-1}}_{IR}+\phi^{\phantom{-1}}_{UV})}.
1_{N S}{\frac{\buildrel T_{\alpha}}{\longrightarrow}}( 0_{w}| 1_{N S}) , 5_{N S}{\frac{\buildrel T_{\alpha}}{\longrightarrow}}( 0_{m}| 5_{N S}) , ( p + 1 )_{R}{\frac{\buildrel T_{\alpha}}{\longrightarrow}}( p | p_{R}, ( p + 2 )_{R}) ,
\sum_{j = 0}^{\infty}r_{j}^{2-n}\int_{B_{r_{j+1}}\setminus B_{r_{j+2}}}|\nabla u|^2 \d x & \leq C \int_{B_{1}}r^{2-n}u_r^2 \d x + C \varepsilon^2 \int_{B_{1}}r^{3-n}|\nabla u|^2 \d x \varepsilon \leq \varepsilon_0.
\mathcal{R}_1' = \mathcal{R}' \cap \mathcal{G}_{l_1}= \mathcal{R}_1 \sqcup \Delta\mathcal{R}_1 \mathcal{R}_2' = \mathcal{R}' \cap \mathcal{G}_{l_2}= \mathcal{R}_2 \setminus \Delta\mathcal{R}_2
\sum_{k=K+1}^\infty \frac{1}{(1+\theta)^{\frac{\beta}{2}k(k-1)}\Gamma(1+k\beta)}\left(\frac{1-\eta}{\theta^\beta\Gamma(1-\beta)}\right)^k\leq \frac{1}{\Gamma(1+(K+1)\beta)}\frac{C_2^{K+1}}{1-C_2},
a_0(r,s) =1, a_1(r,s) =1+2^s\equiv\begin{cases}2\pmod 9, s\equiv 0, 3\pmod 9, s\equiv 1, 5\pmod 9, s\equiv 2, 0\pmod 9, s\equiv 3, 8\pmod 9, s\equiv 4, 6\pmod 9, s\equiv 5,\end{cases}
\lbrack v]_{x_{N},t,\overline{Q}}^{(1,\frac{\gamma}{m})}=\sup_{\substack{\theta>0,\tau>0 (x,t)\in\overline{Q}}}\frac{|\Delta_{\theta,x_{N}}^{2}\Delta_{\tau,t}v(x,t)|}{\theta\tau^{\frac{\gamma}{m}}}.
Q = \begin{bmatrix}-\beta_0 \beta_0 0 0 \delta_1 -(\beta_1+\delta_1) \beta_1 0 \vdots 0 \delta_2 -(\beta_2+\delta_2) \beta_2 \vdots 0 0 \delta_3 -(\beta_3+\delta_3) \cdots \cdots \ddots\end{bmatrix}
\zeta_{\cal M}^{\mathrm{I I},{\cal D}}( s ) ={\frac{1}{a}}\left[ -{\frac{2}{3 1 5 \pi}}\left({\frac{1}{s + 1}}+ \ln a \right) - 0 . 0 0 3 2 6 + O ( s + 1 ) \right] .
\begin{cases}\partial_{\bar{t}}u+u\partial_{\bar{x}}u-Ah\partial_{\bar{x}}\tilde{h}+(1-A)h\partial_{\bar{y}}\tilde{h}\partial_{\bar{y}}u-Ah^2\partial_{\bar{y}}^2u=0, \partial_{\bar{t}}\tilde{h}+u\partial_{\bar{x}}\tilde{h}-Bh\partial_{\bar{x}}u+(1-B)h(\partial_{\bar{y}}\tilde{h})^2-Bh^2\partial_{\bar{y}}^2\tilde{h}=0.\end{cases}
I_{\textbf{s}\rightarrow\textbf{i}}(l) = \frac{\sigma_{\mathbf{d}_\mathbf{s}}^2}{K^2}\sum\limits_{\tau=-KT+\frac{T}{2}}^{\frac{T}{2}}\Bigg| \sum\limits_{k=-K+1}^{K-1}G_{|k|}e^{j\pi\frac{k}{K}\tau}&\times (\pi(\frac{k}{K}+l)) \Bigg|^2
{\cal F}_{\mathrm{v a c}}={\cal F}_{\mathrm{v a c}}^{+}+{\cal F}_{\mathrm{v a c}}^{-}= ( 1 - \mathrm{N}^{2}) \frac{e_{+}^{2}}{2 \pi}( \frac{1}{2}\epsilon ( x - y ) - \frac{1}{1}{L}( x - y ) )
p_{a \gamma}( f \star g ) \leq \sum_{b \leq a}\sum_{\epsilon \leq \gamma}\left( \begin{array}{c}{a}\\{b}\\ \end{array}\right) \left( \begin{array}{c}{\gamma}\\{\epsilon}\\ \end{array}\right) q_{a - b , \gamma - \epsilon}( f ) q_{0 , \eta - \epsilon}( g )
\sup_{t\in[0,1]}\max_{i,i'}\bigl|(X_{1,t})_i^k(\hat\lambda)-(X_{1,t})_{i'}^k(\hat \lambda)\bigr|\le C\,\log N\,N^{1/(\sigma-15)}|\lambda_i^k-\lambda_{i'}^{k}|\forall\,k=1,\dots,d,
\partial_+ \left( \Gamma^{(1)-1}\partial_- \Gamma^{(1)}\right) = - \Gamma^{(1)-1}\Gamma^{(2)}, \quad\partial_+ \left( \Gamma^{(2)-1}\partial_- \Gamma^{(2)}\right) = \Gamma^{(1)-1}\Gamma^{(2)}.
\hat R(X,Y)Z= c\{g( Y,Z) X-g( X,Z) Y+g( JY,Z) JX &-g( JX,Z) JY-2g( JX,Y) JZ\}&+c\sum_{a=1}^3\{g( J_aY,Z) J_aX-g( J_aX,Z) J_aY-2g( J_aX,Y) J_aZ &+g( JJ_aY,Z) JJ_aX-g( JJ_aX,Z) JJ_aY\}.
d\bar{x}_i(s) = \frac{\partial (G^{-1})_i}{\partial \phi_j}\mathcal{L}\phi_j\,ds + \frac{1}{\beta}(\nabla \phi a \nabla\phi^T)_{jl}\frac{\partial^2 (G^{-1})_{i}}{\partial \phi_j\partial \phi_l}ds + \sqrt{2\beta^{-1}}\frac{\partial (G^{-1})_i}{\partial \phi_j}(\nabla \phi\,\sigma)_{jl}\, dw_l(s)\,,
\left\{\alpha_{{\mathcal{R}},{[N_T]}}^{\bar{\mathcal{R}}_i}\sum_{p=1}^{N_T}h_{qp}c_p\right\}_{\mathcal{R},i:\mathcal{R}\ni q,|\mathcal{R}|=r+1,i\in[\rho]}\cup\left\{\alpha_{{\mathcal{R}},{[N_T]}}^{\bar{\mathcal{R}}_i}\sum_{p=1}^{N_T}h_{qp}c_p\right\}_{\mathcal{R},\bar{\mathcal{R}}_i:\mathcal{R}\cup\bar{\mathcal{R}}_i\not\ni q}.
\begin{array}{r}\nabla_{K}(z)=\bigl(b_{1}b_{2}+\varepsilon b_{3}(b_{3}+\delta)\bigr)z^{4}+(b_{1}+b_{2}-\varepsilon\delta)z^{2}+1~\medskip (b_{1}, b_{2}, b_{3}\in \mathbb{Z},\ \varepsilon\in\{\pm 1\},\ \delta\in\{0,1\}).\end{array}
\mathcal{T}_{00}= \int_1^T \int \langle D \rangle^N Q_0^2 \langle D \rangle^N \left( g \partial_Y Q^2_0\right) \,dV\,dt + \int \langle D \rangle^N Q_0^2 \langle D \rangle^N \left( U^3_0 \partial_Z Q^2_0\right) \,dV\,dt = \mathcal{T}_{00}^2 + \mathcal{T}_{00}^3.
\lim_{n \to \infty}\sqrt{n}\, a_n = \lim_{n \to \infty}\sqrt{n}\int_{- \sqrt{12 / n}}^0 F_n( x ) \, dx = \lim_{n \to \infty}\sqrt{n}\, \sqrt{12 / n}\ \phi( 0 ) = \sqrt{12}/ \sqrt{2\pi}= \sqrt{6 / \pi}.
t^k B_k^{\varphi}\left(\frac{1}{t}\right)=t^k\sum_{j=0}^k \binom{k}{j}\left(1-\frac{1}{t}\right)^j\gamma_{k-j}\frac{1}{t^{k-j}}=\sum_{j=0}^k \binom{k}{j}(t-1)^j\gamma_{k-j},
\gamma (ht +\tau) - \gamma(\tau) = \gamma'(\tau) ht + \gamma''(\tau) \frac{(ht)^2}{2!}+ \cdots + \gamma^{(k)}(\tau) \frac{(ht)^k}{k!}+ \mathbf e (\tau, h, t) = M_\tau^{\gamma, k}D_h^k \gamma_\circ^k(t) + \mathbf e (\tau, h, t),
\Theta^{[2]}(\mathcal{M}) = \Theta^{[2]}\begin{pmatrix}\rho \kappa_u \kappa_s \end{pmatrix}=\begin{pmatrix*}[l]A_c Dk_c + Dg_c(K{}{}) \kappa - Dk_c\left(R\right) P_\rho -A_u^{-1}Dg_u(K{}{}) \kappa + A_u^{-1}\kappa_u(R) P_\rho A_s \kappa_s(T) Q_\rho + Dg_s(K{}{}\circ T) \kappa(T) Q_\rho\end{pmatrix*},
\langle v, (T_{y, x})^* w\rangle =\langle T_{y, x}v, w\rangle &=\langle(\pi_y\circ T\circ\iota_x)(v), w\rangle &=\langle v, (\pi_y\circ T\circ\iota_x)^*(w)\rangle &=\langle v, (\iota_x^*\circ T^*\circ\pi_y^*)(w)\rangle &=\langle v, (\pi_x\circ T^*\circ\iota_y)(w)\rangle &=\langle v, (T^*)_{x, y}(w)\rangle.
f ( \kappa ,{\textstyle \frac{3}{2}}) \simeq - \int \frac{d^{3}p}{( 2 \pi )^{3}}\log [ 1 8 ( 3 - 2 \sqrt{2}) \{s^{2}{\bf p}^{2}+ \frac{s^{4}}{2}+ \frac{1}{4}( p_{2}^{2}p_{3}^{2}+ p_{3}^{2}p_{1}^{2}+ p_{1}^{2}p_{2}^{2}) + O ( p^{6}) \}] .
1_A \hat{v}(r, X^{s,x; \alpha,v}_r)= 1_A w(r, X^{s,x; \alpha,v}_r)\leq \mathbb{E}[1_Aw(t, X^{s,x; \alpha[r,\xi],v}_t)|\mathcal{F}_r] + +1_A\times (t-r)\varphi_w (t-r)\leq \mathbb{E}[1_A \hat{v}(t, X^{s,x; \alpha[r, \hat{\xi}],v}_t)|\mathcal{F}_r] +1_A\times (t-r)\varphi_w (t-r).
\gamma^{(r),(i)}_{j,k}(x)+\Big[(\hat P_{k-N+1}^{[2]}(x))^\top,\dots,(\hat P_{k}^{[2]}(x))^\top\Big](\hat H_{(k,N)})^{-1}\omega_{(k,N)}\begin{bmatrix}\pi^{(r),(i)}_{j,k+1}\vdots \pi^{(r),(i)}_{j,k+N}\end{bmatrix}=0
\mathbf u = \sum_{l=1}^{L}a_{l}\mathbf x_l =\sum_{l=1}^{L}a_{l}\mathbf H^{-1}(\mathbf c_l -\mathbf k_l) = \mathbf H^{-1}\sum_{l=1}^{L}a_{l}(\mathbf c_l -\mathbf k_l).
\frac{u_{n+1}}{u_n}=Al^{\alpha+k+1-\gamma}\frac{\lim_{m\to\infty}\frac{m^{(n+3)k+(n+2)(\alpha+1-\gamma)+1}m!}{((n+3)k+(n+2)(\alpha+1-\gamma)+1)\cdots((n+3)k+(n+2)(\alpha+1-\gamma)+m+1)}}{\lim_{m\to\infty}\frac{m^{(n+3)(k+\alpha)+(n+2)(1-\gamma)+1}m!}{((n+3)(k+\alpha)+(n+2)(1-\gamma)+1)\cdots((n+3)(k+\alpha)+(n+2)(1-\gamma)+m+1)}}
\frac{1}{N}\sum_{i=1}^{N}f(T,\,x_{i}^{N}(T))-\frac{1}{N}\sum_{i=1}^{N}f(0,\,x_{i}^{N}(0)) =\frac{1}{N}\int_{0}^{T}\sum_{i=1}^{N}\left\{f_{t}+b_{c(i)}f_{x}+\frac{1}{2}f_{xx}\right\}(t,\,x_{i}^{N}(t))dt+\frac{1}{N}\int_{0}^{T}\sum_{i=1}^{N}f_{x}(t,\,x_{i}^{N}(t))d\beta_{i}(t)
d s^{2}= H_{p}^{- 1 / 2}[ - d t^{2}+ \sum_{i = 1}^{p - 1}( d x^{i})^{2}+ H_{5}^{N S}( d x^{p})^{2}] + H_{p}^{1 / 2}[ \sum_{i = p + 1}^{6}( d x^{i})^{2}+ H_{5}^{N S}( d r^{2}+ r^{2}d \Omega_{2}^{2}) ]