harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{F}{V}= - \sum_{\stackrel{n = 1}{n ~ \mathrm{o d d}}}^{\infty}\sum_{k = 1}^{\infty}\frac{2}{k}\left( \frac{1}{4 \pi^{2}\alpha^{\prime}\tau_{2}}\right)^{5}e^{- n^{2}\beta^{2}/ 4 \pi \alpha^{\prime}\tau_{2}}\sum_{M}e^{- \pi \alpha^{\prime}\tau_{2}M^{2}}
E_{\Phi}(\Gamma_0) = \frac{1}{n}\sum_{k=1}^n \sum_{j=1}^n \theta_\alpha \left(\ell+\frac{j-k}{n}\right) = \frac{1}{n}\sum_{k=1}^n \sum_{j=1}^n \theta_\alpha \left(\ell+\frac{j}{n}\right) = \sum_{j=1}^n \theta_\alpha \left(\ell+\frac{j}{n}\right),
\begin{array}{|c||c|c|c|c|}\hline\mbox{Matrix operators}J_{\ell}^{+ij}J_{\ell}^{+-i}J_{\ell}^{ijk}J_{\ell}^{-ij}\hline\hline\mbox{SUGRA modes}v^{\ell}_2 u^{\ell}\alpha^{\ell}v_1^{\ell}\hline\end{array}
p(\theta|Y, Z)\propto p(\theta) \prod_{t=1}^{T}{f(y_t|\theta,z_t=k,Y_{t-1})}= p(\theta)\prod_{t=1}^{T}{\frac{1}{\sqrt{2\pi H_{k,t}}}\exp(-\frac{y^2_t}{2H_{k,t}})},
\varphi \big( g^{i}_1 g_2 \big) = \varphi \big( g^{i}_1 \big) \varphi (g_2) \mbox{and}\varphi \big( g^{i}_1 g_3 \big) = \varphi (g_3) \varphi \big( g^{i}_1 \big) \,,
\begin{array}{l}e_{a}\circ_{X}(e_{b}\circ_{X}(e_{c}\circ_{X}e_{d})) = \pm e_{7}\mbox{if exactly one of}a,b,c,d= 0 \mbox{or}h=7, e_{a}(e_{b}(e_{c}e_{d}))=\pm 1\mbox{otherwise.}\end{array}
\sum_{k=1}^\lambda\frac{(-1)^{\lambda+k}(v+2k-1)!}{(2k-1)!(\lambda-k)!k!}= \frac{v!}{\lambda!}\sum_{k=0}^\lambda\binom{\lambda}{k}\binom{v-1}{v-\lambda-k}2^{\lambda-k}=2^\lambda \binom{v-1}{v-\lambda}\sum_{k=0}^{\lambda}\frac{\binom{\lambda}{k}\binom{v-\lambda}{k}}{\binom{\lambda+k-1}{k}}2^{-k},
2<\psi_\alpha(T)\psi_\beta(T)\psi_\mu(\tau_2) \psi_\nu(\tau_2)\psi_\kappa(\tau_1)\psi_\lambda(\tau_1)>_\psi={1\over 4}\delta_{\alpha\beta}[\delta_{\mu\lambda}\delta_{\nu\kappa}-\delta_{\nu\lambda}\delta_{\mu\kappa}][sign(\tau_1-\tau_2)]^2
\P\left(X_{t_{2r+1}}^{\left(n\right)}\right. \left.>0,\,\ldots, X_{t_{Tn-1}}^{\left(n\right)}>0,\, X_{T}^{\left(n\right)}=j|X_{t_{2r}}^{\left(n\right)}=0\right) &=2p \P\left(S_{2r+1}>0,\,\ldots, S_{Tn-1}>0,\, S_{Tn}=j|S_{2r}=0\right),
{{d^2y}\over{d\tilde{\lambda}^2}}-{1\over 2}{{{\cal W}^{\prime}}\over{\cal W}}\left({{dy}\over{d\tilde{\lambda}}}\right)^2-{1\over 2}\left[{\cal W}-\left({{dy}\over{d\tilde{\lambda}}}\right)^2\right]{\cal W}^{-1}\partial_y({\cal W}g_{\mu\nu}){{dx^{\mu}}\over{d\tilde{\lambda}}}{{dx^{\nu}}\over{d\tilde{\lambda}}}=0.
r^t(\mathbf{S}^*, \delta^*) \leq r^t(\mathbf{S}, \delta^*) \forall \,\mathbf{S}\in \mathbb{S}\;, r^r(\mathbf{S}^*, \delta^*) \leq r^r(\mathbf{S}^*, \delta) \forall \,\delta \in \Delta\;.
\lambda_1(t,x)=R_1(t,x,u(t,x),q(t.x)) \in X_0((0,T_1] \times D_{R_1})), b_1(t,x)=R_2(t,x,u(t,x),q(t.x)) \in X_0((0,T_1] \times D_{R_1}))
d^\star(j \circ u)(t) = A_0^{\odot \star}T_0^{\odot \star}(t-s)j\varphi + A_0^{\odot \star}\int_s^t T_0^{\odot \star}(t-\tau) [B(\tau)u(\tau) + f(\tau)] d\tau + B(t)u(t) + f(t) = A_0^{\odot \star}ju(t) + B(t)u(t) + f(t).
\vec{\cal K}U_{E,\hat q}=\left(\begin{array}{r}\vec{K}'\,u^{(1)}_{E,\hat q}E^{-1}\sqrt{V}\, \pi\vec{K}'\,u^{(1)}_{E,\hat q}\end{array}\right)\,,\quad\vec{K}'=\vec{K}+F\,\frac{\vec{\sigma}}{2}\,,
\int_{\mathbb R}\tilde\phi_2(x_1,x_2,x_3) dx_2=0\quad{\rm for\each\fixed\}(x_1,x_3)\{\rm in\}\mathbb R^2; &\int_{\mathbb R^2}\tilde\phi_2(x_1,x_2,x_3) dx_1dx_3=0 \quad{\rm for\each\fixed\}x_2\{\rm in\}\mathbb R.
\sum\limits_{k=1}^q\tau\sum\limits_{i=0}^{m-1}hv_{i\bar t}^2(k) +\sum\limits_{i=0}^{m-1}hv_{ix}^2(q)+ \sum\limits_{k=1}^q\tau^2 \sum\limits_{i=0}^{m-1}hv_{ix\bar t}^2(k) \leq \tilde C_{\infty}\Big(\Vert\Phi\Vert^2_{W_2^1(0,\ell)}+\Vert f\Vert_{L_\infty(D)}^2 +\Vert p\Vert_{W_2^1(0,T)}^2 +\Vert g^n\Vert_{W_2^1(0,T)}^2\Big)
M_{|g|}(t) \gg \frac{1}{\log t}N_{|g|}(t) = \frac{1}{y\log t}\int_{t-y}^{t}N_{|g|}(u) du +O\left(\frac{M_{|g|}(t)}{\log^3 t}\right)\geq \frac{t^2}{y\log t}\int_{t-y}^t \frac{N_{|g|}(u)}{u^2}du +O\left(\frac{M_{|g|}(t)}{\log^3 t}\right),
\sum_{n \geq 1}\frac{a_n q^{(m+1)n}}{(1-q^n)^{k+1}}= \frac{1}{(q; q)_{\infty}}\sum_{n \geq 1}\sum_{i=1}^{\lfloor n/(m+1) \rfloor}s_{n-m, i}\frac{a_i}{(1-q^i)^k}\cdot q^n,
\delta^{(s)}w_k+\frac{\sum_{ij}(\sum_l \Gamma^{k(s)}_{ij,l}c_l)v_i^{\phi^s}w_j^{\phi^s}}{(\sum_l r_l c_l)(\sum_l v_l^{\phi^s})}=0,\ \k\in \{1,\ldots,n\}.
\frac{1}{n}\sum_{i=2}^n A_i - m_{\frac{1}{2}}(\alpha_0)\delta_0^{1/2}= \frac{1}{n}\sum_{i=2}^n \left( A_i - \delta_0^{1/2}n^{1/ 2\alpha_0}|\Delta_i^n L - \Delta_{i-1}^n L|^{1/2}\right) + \delta_0^{1/2}\left( \frac{1}{n}\sum_{i=2}^n n^{1/ 2\alpha_0}|\Delta_i^n L - \Delta_{i-1}^n L|^{1/2}- m_{\frac{1}{2}}(\alpha_0) \right).
B(X,Y)= \langle \nabla_{X}Y,\nabla f\rangle\cdot\frac{\nabla f}{|\nabla f|^{2}}= [\nabla_{X}\langle Y,\nabla f\rangle-\langle Y,\nabla_{X}\nabla f\rangle]\cdot\frac{\nabla f}{|\nabla f|^{2}}= -{\rm Ric}(X,Y)\cdot\frac{\nabla f}{|\nabla f|^{2}},
A_1 := \begin{pmatrix}0 0 0 0 0 1 0 1 0\end{pmatrix}, A_2 := \begin{pmatrix}0 0 -1 0 0 0 -1 0 0\end{pmatrix}, B_\Gamma := \begin{pmatrix}0 1 0 0 -1 0 0 0 1 0 0 -1\end{pmatrix},
\begin{cases}{\displaystyle g_n := g_0 + \sum_{i=0}^{n-1}\frac{1}{d^{i+1}}(f^i)^*(\lambda)}(n \geqslant 1) begin{align*}1.5ex]{\displaystyle \varphi_0 = 1, \varphi_n = \prod_{i=0}^{n-1}(f^i)^*(\varphi)^{1/d^{i+1}}}(n \geqslant 1).\end{cases}
Si(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}=x-\frac{x^{3}}{3!\cdot3}+\frac{x^{5}}{5!\cdot5}-\frac{x^{7}}{7!\cdot7}\pm\cdot\cdot\cdot
\omega \; = \; \left( \begin{array}{c c}{{\gamma^{\mu}\, \omega_{1 \mu}}}&{{\begin{array}{c}{{\omega_{1}}}\\{{\omega_{2}}}\\ \end{array}}}\\{{\begin{array}{c c}{{\tilde{\omega}_{1}\,}}&{{\tilde{\omega}_{2}}}\\ \end{array}}}&{{\gamma^{\mu}\, \omega_{2 \mu}}}\\ \end{array}\right)
\begin{array}{l}{{\lambda_{0}= 2 e^{J_{1}}\cosh J + 2 e^{- J_{1}}\cos \varepsilon}}\\{{\lambda_{2}= 2 e^{J_{1}}\cosh J - 2 e^{- J_{1}}\cos \varepsilon}}\\{{\lambda_{1}= 2 e^{J_{1}}\sinh J + 2 e^{- J_{1}}\sin \varepsilon}}\\{{\lambda_{3}= 2 e^{J_{1}}\sinh J - 2 e^{- J_{1}}\sin \varepsilon \; .}}\\ \end{array}
\Big( \frac{1}{p_1^2 \tilde{t}^2}\frac{\vert w_1 \vert^2}{\vert \hat{w}\vert^2}- 1 \Big) \vert w_1 \vert^{2m-2}= - \frac{(m-1)^{m-1}}{m^m}\frac{\big( p_1^2\tilde{t}^2\vert \hat{w}\vert^2 \big)^{m-1}}{p_1^2}\big( \frac{1}{p_1^2}\big)^{m-1}
\begin{array}{l}{2 ( n - n_{1}) - 2 ( n_{2}+ \cdots + n_{j}) + \beta_{i - 1 j - 1}^{( k - 1 )}\geq a_{1}^{( j )}\geq \cdots \geq a_{n_{j}- n_{j + 1}}^{( j )}\geq 0 ,}\\{n - n_{1}- ( n_{2}+ \cdots + n_{k - 1}) \geq b_{1}\geq \cdots \geq b_{n_{k - 1}}\geq 0 ,}\\ \end{array}
A^2 A^* - \beta A A^* A + A^* A^2 - \gamma(AA^* + A^* A) - \varrho A^* = \gamma^* A^2 + \omega A + \eta I,{A^*}^2 A - \beta A^* A A^* + A{A^*}^2 - \gamma^* (A^*A + A A^*) - \varrho^* A = \gamma{A^*}^2 + \omega A^* + \eta^* I.
{\cal{L}}= \int d^{2}\theta d^{2}{\bar{\theta}}\sum_{i = 1}^{3}{\bar{\Phi}}_{i}\mathrm{e}^{2 g V}\Phi_{i}+ \frac{1}{8 \pi}I m \left[ \int d^{2}\theta \, \tau W^{\alpha}W_{\alpha}\right]
| A \cap I_{u,n}| = \sum_{j=1}^{q+1}\left| A \cap I_{u,n}^{(j)}\right| \leq \sum_{j=1}^q \left( \left| I_{u,n}^{(j)}\right| - 1 \right) + \left| I_{u,n}^{(q+1)}\right| & = \sum_{j=1}^{q+1}\left| I_{u,n}^{(j)}\right| - q = |I_{u,n}| - q = n - q < n - \frac{n}{d}+1 = \left(1 - \frac{1}{d}\right)n + 1.
\phi^{\lambda}\left(d + \sum_{i=0}^\ell \psi_i\check f_idt + bdt\right) \phi^{-\lambda}\coloneqq d + \sum_{i=0}^\ell \phi^{-\langle\lambda,\check\alpha_i\rangle}\psi_i\check f_i dt - \lambda \phi^{-1}d\phi + \phi^{\lambda}b\phi^{-\lambda}dt,
(\tilde{\varphi}_a^i \smile\tilde{\psi}_a^j)(1) = \tilde{\varphi}_a^i(1) \otimes \tilde{\psi}_a^j(1) = [\varphi_a^i]\otimes[\psi_a^j], (\varphi_a^i \smile \psi_a^j)(1) = \varphi_a^i(1)\otimes \psi_a^j(1) = [u_a^i]\otimes [v_a^j], (u_a^i\smile v_a^j)(1) = u_a^i(1)\otimes v_a^j(1) \mapsto \frac{a^2}{\delta_i\gcd(\delta_i, c_a^i-1)},
\langle \tilde{\pi}_p(\alpha_n)\mathbf h_p, \mathbf h_p\rangle = \int_{\mathcal A_1(n)}\mathbf h_p(h\alpha_n)\overline{\mathbf h_p(h)}dh + \int_{\mathcal A_2(n)}\mathbf h_p(h\alpha_n)\overline{\mathbf h_p(h)}dh =: A_1(n) + A_2(n).
\Lambda = \left( \begin{array}{c c c c c c}{{0}}&{{0}}&{{0}}&{{\ldots}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{\ldots}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{\ldots}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{\ldots}}&{{0}}&{{0}}\\{{.}}&{{.}}&{{.}}&{{.}}&{{.}}&{{.}}\\{{0}}&{{0}}&{{0}}&{{\ldots}}&{{1}}&{{0}}\\ \end{array}\right)
I_n(f_n)=n!\int_0^T\int_{\mathbb{R}_0}\int_0^{t_n}\int_{\mathbb{R}_0}...\int_0^{t_2}\int_{\mathbb{R}_0}f_n(t_1,\zeta_1,...,t_n,\zeta_n)\tilde{N}(dt_1, d\zeta_1)
H_{B}^{\mathrm{b a r e}}= - \frac{1}{2}\sum_{k = 1}^{\infty}\Bigl ( \sigma_{k + 1}^{x}\sigma_{k}^{x}+ \sigma_{k + 1}^{y}\sigma_{k}^{y}+ \Delta \sigma_{k + 1}^{z}\sigma_{k}^{z}\Bigr ) + h \sigma_{1}^{z}.
|z|^k \sum_{|c| \geq 2}\sum_{|d| \geq 1}|cz+d|^{-k}\ll \Big(\frac{|z|^2}{4y^2}\Big)^{k/2}\leq \Big(\frac{\frac{9}{25}+ y^2}{4y^2}\Big)^{k/2}= (\tfrac{100}{43})^{-k/2},
D_i(y^0_i, y_i) := J_i(y_i) - [J_i(y^0_i) + \langle J_i'(y^0_i), y_i - y^0_i \rangle], \Pr_{{\cal Y}_i}(-\tilde x, y_i^0, \tau) := \arg\min\limits_{y_i \in{\cal Y}_i}\left\{\langle - \tilde x, y \rangle + J_i(y_i)+ \tau D_i(y^0_i, y_i)\right\},
qL_{X\wedge Y}i_Z\omega =(-1)^{l}\nabla_X\omega(Y\wedge Z)+(-1)^{kl}\omega(Y\wedge \nabla_X Z) &+(-1)^{l}\omega((\nabla_X Y)\wedge Z)+(-1)^{k(l-1)}\nabla_Y\omega(X\wedge Z) &+\omega(X\wedge \nabla_YZ)+(-1)^{k(l-1)}\omega((\nabla_Y X)\wedge Z) &-(-1)^l\omega([X,Y]\wedge Z).
s([n,n+1]) = s_1(s_3s_1)^{n-1}, s([n+1,1]) = s_3(s_1s_3)^{n-1}, s([1,1]) = s_2, \beta([n,n+1]) = n \alpha_1+ (n-1) \alpha_3 , \beta([n+1,1]) = (n-1) \alpha_1+ n \alpha_3 , \beta([1,1]) = \alpha_2 .
K_{S_{2}}( z , z^{\prime}, s ) ={\frac{1}{l_{1}^{2}}}{\frac{\sqrt{2}e^{-{\bar{s}/ 4}}}{( 4 \pi \bar{s})^{3 / 2}}}\sum_{n = - \infty}^{\infty}( - 1 )^{n}\int_{\sigma}^{\pi}{\frac{d \varphi ( \varphi + 2 \pi n ) e^{-{\frac{( \varphi + 2 \pi n )^{2}}{4 \bar{s}}}}}{\sqrt{\cos \sigma - \cos \varphi}}}~ ~ ,
\oint_{C}d z \omega ( z ) = - ( - z_{\infty})^{- ( 1 - \vartheta_{3})}4 \sin \pi \vartheta_{2}\sin \pi \vartheta_{1}e^{\pi i \vartheta_{1}}\frac{\Gamma ( \vartheta_{1}) \Gamma ( \vartheta_{2})}{\Gamma ( 1 - \vartheta_{3})},
\rho_{ij}= \frac{\sum_{k=1}^n(x_{ki}-\bar{x}_i)(x_{kj}-\bar{x}_j)}{\sqrt{\sum_{k=1}^n(x_{ki}-\bar{x}_i)^2}\cdot\sqrt{\sum_{k=1}^n(x_{kj}-\bar{x}_j)^2}},1\leq i, j \leq p,
\mathfrak{M}= [\mathfrak{B}_{1}, \mathfrak{C}_{1}], \mathfrak{C}_{1}= [\mathfrak{B}_{2}, \mathfrak{C}_{2}],\ \dots,\ \mathfrak{C}_{l-1}= [\mathfrak{B}_{l}, \mathfrak{C}_{l}]
\L = \left( \begin{array}{c c}{{\bf 1}_{d_{\cal H}}}&{0}\\{0}&{-{\bf 1}_{d_{\cal K}}}\\ \end{array}\right) , \mathrm{s u c h t h a t}f^{A B C}= \L^{A A^{\prime}}\L^{B B^{\prime}}\L^{C C^{\prime}}f^{A^{\prime}B^{\prime}C^{\prime}},
A=\Lambda_\mathbb{Z}( ((x_6)_b\gamma_i,\;(x_6)_by_i,\;(x_2)_m(x_6)_by_1y_2,\;(x_2)_m(x_6)_b(y_1\gamma_2-y_2\gamma_1), & (x_2)_m(x_6)_by_2\gamma_1^3,\;(x_2)_m(x_6)_b(y_1\gamma_1+y_2\gamma_2),\; (x_2)_m(x_6)_b(\gamma_1^2+\gamma_2^2), (x_2)_m(x_6)_b\gamma_1^4,\;(x_2)_m(x_6)_b\gamma_1^3\gamma_2)
\int_\Omega\Phi(f(t))\, \mathrm{d}x-\int_\Omega\Phi(f(t_0))\, \mathrm{d}x=\int_{t_0}^t\Big\langle\partial_t f(\tau),\Phi'(f(\tau))\Big\rangle_{(H^1)',H^1}\mathrm{d}\tau\,.
\begin{aligned}0 \leqslant \frac{1}{2}\ddot{z}(t_{0}) \cos^{2}t_{0}- \dot{z}(t_{0}) \cos t_{0}\sin t_{0}- z(t_{0}) + 1 - 2 \delta \cos^{2}t_{0}& - \frac{\dot{z}(t_{0})}{4 z(t_{0})}\cos t_{0}\{\dot{z}(t_{0}) \cos t_{0}- 2 z(t_{0}) \sin t_{0}+ 2 \sin t_{0}\}.\end{aligned}
\rho ={\bf \nabla}.{\bf E}= - \left({\frac{\partial^{2}\phi}{\partial x^{2}}}+{\frac{\partial^{2}\phi}{\partial y^{2}}}\right) \qquad ; \qquad{\bf j}={\bf \nabla}\times{\bf B}= - \left({\frac{\partial^{2}\phi}{\partial x^{2}}}+{\frac{\partial^{2}\phi}{\partial y^{2}}}\right) \,{\hat{\bf z}},
\tilde{P}^{\lambda}= \int\,d^n\mathbf{p}\, \left(\vert\vec{p}\vert^2+m^2\right)^{\lambda}\exp({-i\,p_k z^k}) &= \frac{2^{\lambda +1}(2\pi)^{\frac{1}{2}n}}{\Gamma(-\lambda)}\left(\frac{m}{\vert \vec{z}\vert}\right)^{\frac{1}{2}n+\lambda}K_{\frac{1}{2}n+\lambda}(m\vert \vec{z}\vert ),
H_{+}\phi ( r , \theta ) = \left( \begin{array}{c c}{{m}}&{{i e^{- i \theta}\left( \partial_{r}- \frac{i}{r}\partial_{\theta}- \frac{e B}{2}r \right)}}\\{{i e^{i \theta}\left( \partial_{r}+ \frac{i}{r}\partial_{\theta}+ \frac{e B}{2}r \right)}}&{{- m}}\\ \end{array}\right) \phi ( r , \theta ) = E \phi ( r , \theta ) .
a_k = \frac{b_k -\alpha_k b_{-k}^\dagger}{\sqrt{1- \alpha_k^2 \vphantom{\alpha_{p-k}^2}}}, a_{p-k}= \frac{b_{p-k}-\alpha_{p-k}b_{k-p}^\dagger}{\sqrt{1- \alpha_{p-k}^2}}.
\left(\begin{array}{cc}0 - B B 0 \end{array}\right)\left(\begin{array}{c}\dot V_{{\rm even}}\dot V_{{\rm odd}}\end{array}\right)=\lambda\left(\begin{array}{cc}-{1 \over 2}B^2 0 0 -{1 \over 2}B^2 \end{array}\right)\left(\begin{array}{c}\ddot V_{{\rm even}}\ddot V_{{\rm odd}}\end{array}\right),
\frac{\partial}{\partial y}\int dx \xa(x,y,t)=\frac{\partial}{\partial y}(yp_{0}(y,t))+\frac{1}{2}\frac{\partial^{2}}{\partial y^{2}}\int dx \x^{2}\varphi(x,y,t)-\frac{\partial}{\partial t}p_{0}(y,t)
\begin{aligned}u(s,{X}_s)=\ \Phi({X}_T)- w(s,{X}_s) &+\int^T_sf(r,{X}_r,u(r,{X}_r),\nabla u(r,{X}_r))\d r &-\int^T_s\nabla w(r,{X}_r)\d{W}_r-\int^T_s\nabla u(r,{X}_r)\d{W}_r.\end{aligned}
\left(\prod_{k=1}^{K_n}|\alpha_{k,n}|\right)^{1/n}\le \left(S_m(|\alpha_{1,n}|,\ldots,|\alpha_{K_n,n}|)\right)^{K_n/(mn)}\le \left(\binom{n}{m}/ \binom{K_n}{m}\right)^{K_n/(mn)}\left(S_m(|\alpha_{1,n}|,\ldots,|\alpha_{n,n}|)\right)^{K_n/(mn)}.
(M_r{V})_T= &\sum_{a,b}\left [\binom{a}{2}+\binom{b}{2}\right ]\left [\binom{|S|-3}{a-2}\binom{r-|S|-1}{b-1}+\binom{|S|-3}{a-1}\binom{r-|S|-1}{b-2}\right . &\left . -\binom{|S|-3}{a-1}\binom{r-|S|-1}{b-2}-\binom{|S|-3}{a-2}\binom{r-|S|-1}{b-1}\right ]=0.
\tilde{f}\mapsto\sum_{\alpha=1}^{N}\frac{\partial f^{\alpha}}{\partial u^{i}}\frac{\partial\tilde{f}^{\beta}}{\partial u^{j}}+\sum_{\alpha=1}^{N}\frac{\partial\tilde{f}^{\alpha}}{\partial u^{i}}\frac{\partial f^{\beta}}{\partial u^{j}}.
0=\lim_{i\rightarrow\infty}\frac{dL}{dt}\left( t_{i}\right) =-\int_{0}^{2\pi}w\left( \theta\right) d\theta+\dfrac{2\pi}{{L\!}\left(\infty\right)}\int_{0}^{2\pi}w^{1-\frac{1}{\alpha}}\left( \theta\right)d\theta,\ \ \ 1-\frac{1}{\alpha}>0.
\Pi_{\mathrm{T}}^{( 1 ) ( \beta )}( 0 ) = \frac{C ( D ) e^{2}}{2^{D - 2}\pi^{\frac{D - 1}{2}}\Gamma ( \frac{D - 1}{2})}\int_{m}^{\infty}d \omega_{k}\, \frac{\partial}{\partial \omega_{k}}\left( ( \omega_{k}^{2}- m^{2})^{\frac{D - 1}{2}}\, \frac{n_{F}( \omega_{k})}{\omega_{k}}\right) = 0 , \qquad D \geq 2
\frac{\partial \delta v(t,x)}{\partial t}=\sum_{ij}\left(a^{v^2}_{ij}-a^{v^1}_{ij}\right)\frac{\partial v^1}{\partial x_i\partial x_j}+\sum_{ij}a^{v^2}_{ij}\frac{\partial^2 \delta v}{\partial x_i\partial x_j},
\{\underline{{Q}}_{\underline{{\alpha}}}, \underline{{Q}}_{\underline{{\beta}}}\}= 2 i P_{\underline{{a}}}( C \Gamma^{\underline{{a}}})_{\underline{{\alpha}}\underline{{\beta}}}, \quad [ \underline{{Q}}_{\underline{{\alpha}}}, P_{\underline{{a}}}] = 0 .
\begin{array}{l}{\ell_{3}^{+}= \ell^{+}, \ell^{-}= 0}\\{\hat{h}= \frac{k^{-}}{k}\ell^{+}= \frac{\hat{q}}{2}}\\{\hat{h}= \hat{h}^{\prime}, \hat{q}= \hat{q}^{\prime}}\\ \end{array}
B_8=\left(\begin{array}{cc}P 0 0 0 \end{array}\right),B_9=\left(\begin{array}{cc}Q 0 0 0 \end{array}\right),B_7=\left(\begin{array}{cc}bI 0 0 0 \end{array}\right),B_1=\left(\begin{array}{cc}Ivt 0 0 0 \end{array}\right).
[V_\alpha,V_\beta]=\epsilon_{\alpha\beta}^{~~~\!\gamma}V_\gamma, \;\;\; [V_\alpha,V_m^a]=V_m^b(\sigma_\alpha)_b^{~a}, \;\;\; \{V_m^a,V_m^b\}=-m^2(\sigma^\alpha)^{ab}V_\alpha,
\frac{(\alpha_2-\lambda_i)}{(\alpha_2-\lambda_1)}\cdot \mathbf M_{ij}=\frac{(\lambda_{n+2}-\lambda_i)(\lambda_{n+3}-\lambda_i)}{(\lambda_{n+2}-\lambda_1)(\lambda_{n+3}-\lambda_1)}\cdot\mathbf M_{1j}
\sum_{n=1}^{[N/\Delta]}\frac{1}{n}= \log N - \log \Delta+ \gamma & - \sum_{s=1}^{k}\left( a^{s}+ B_{s}\Delta^{s}\sum_{m=0}^{k-s}\binom{s+m-1}{m}\left( \frac{a}{N}\right)^{m}\right) \frac{1}{s N^{s}}+ O \left( \frac{1}{N^{k+1}}\right),
Q_{1}^{K}=-D\frac{\partial}{\partial x_{1}}(\nabla^{2}w^{K}),Q_{2}^{K}=-D\frac{\partial}{\partial x_{2}}(\nabla^{2}w^{K}),\Omega:=\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{\partial x_{1}}.
\underset{t\rightarrow\infty}{\lim}\dfrac{\mathbb{U}_{\mathbf{G}}\left(tx\right) /\mathbb{U}_{\mathbf{G}}\left( t\right) -x^{\gamma_{2}}}{\mathbf{A}_{\mathbf{G}}\left( t\right)}=x^{\gamma_{2}}\dfrac{x^{\rho_{2}}-1}{\rho_{2}},
\langle \partial X^{\mu}( z ) \bar{\partial}X^{\nu}( \bar{w}) \rangle = - \delta^{\mu \nu}\left[ \pi \delta^{( 2 )}( z - w ) -{\frac{\pi}{2}}\omega ( z ) \cdot ( \mathrm{I m}\tau )^{- 1}\cdot \bar{\omega}( \bar{w}) \right] ,
\sum_{j=1}^r\frac{\alpha^j\left(1+\cos(t j)\right)-2}{j}\geq \sum_{j=1}^{r}\frac{\cos(t j)-1}{j}+\frac{2}{r}\frac{\alpha^{r+1}}{\alpha-1}\left(\frac{\pi^{-2}}{1+(r\alpha-r)^2}-\alpha^{-r/2}\right)_+-4.
\frac{1}{r^2-1}\frac{2}{i}\int_{|w|=1}\frac{1}{w^{\tilde{k}}}\frac{1}{w^{2}-2\frac{1+r^2}{1-r^{2}}w+1}dw= \frac{1}{r^2-1}\frac{2}{i}\frac{1}{2\sqrt{\left(\frac{1+r^2}{1-r^{2}}\right)^{2}-1}}\int_{|w|=1}\frac{1}{w^{\tilde{k}}}\left(\frac{1}{w-w_{+}}- \frac{1}{w-w_{-}}\right) dw.
e^{z L_{- 1}+ x J_{0}^{-}}( L_{0}- h_{1}- z \partial_{z}- h_{0}) e^{- z L_{- 1}- x J_{0}^{-}}\; \; e^{z L_{- 1}+ x J_{0}^{-}}\Phi_{\jmath_{0}}( - z , - x ) | \jmath_{1}, t \rangle = 0
\int_{0}^{\infty}T_{-1}(s)A_{-1}x u(s)\,ds = \int_{0}^{\infty}e^{-(i\omega +\varepsilon) s}T_{-1}(s)A_{-1}x \,ds = (i\omega+\varepsilon -A_{-1})^{-1}A_{-1}x \in X.
\frac{\dot{z}_{n}+\dot{z}_{m}}{z_{n}-z_{m}}=\frac{\dot{\theta}_{n}~\cos^{2}\theta_{m}+\dot{\theta}_{m}~\cos^{2}\theta_{n}}{\cos \theta_{n}~\cos\theta_{m}~\sin \left( \theta_{n}-\theta_{m}\right)}~,
\lim_{s\downarrow 0}\frac{1}{s}\int_0^T\iint\limits_{\mathbb{R}^N\times \mathbb{R}^N}\frac{H(x,y,(u+s\psi)(x,t)-(u+s\psi)(y,t))-H(x,y,u(x,t)-u(y,t))}{|x-y|^N}\,dx\,dy\,dt =\int_0^T\iint\limits_{\mathbb{R}^N\times \mathbb{R}^N}\int_0^1 \frac{D_\xi H(x,y,u(x,t)-u(y,t))\cdot(\psi(x,t)-\psi(y,t))}{|x-y|^N}\,d\sigma\,dx\,dy\,dt
L_\#=\{\xi\in L^0_+: \exists \lambda\in{\mathbb R}, \lambda>0 \xi\geqslant \lambda \}; & L_\#[0,1]=\{\xi\in L_\#: \xi\leqslant 1\}; & L_\#[0,2]=\{\xi\in L_\#: \xi\leqslant 2\}.
\textrm{SIR}= \frac{\frac{1}{M}g_{0}r_0^{-\alpha}}{\sum_{i\in \Phi_{a}\setminus \{0\}}\mathbf{1}\{\tilde{M}_{i}\geq 1\}\frac{1}{\tilde{M}_{i}}g_{i}r_i^{-\alpha}},
H^{-1}\circ B \circ A^{r_0}\circ H(x,y) = (a_{x}^{r_0}(b_{0}(x) + O(\varepsilon)), x) = (a_{0}^{r_0}(b_{0}(x)) + O(\varepsilon), x) = (\eta^{r_0}\circ \xi(x) + O(\varepsilon), x).
\frac{G_{2}}{S O ( 4 )}, \, \, \, \frac{F_{4}}{S p ( 3 ) \times S p ( 1 )}, \, \, \, \frac{E_{6}}{S U ( 6 ) \times S p ( 1 )}, \, \, \, \frac{E_{7}}{S p i n ( 1 2 ) \times S p ( 1 )}, \, \, \, \frac{E_{8}}{E_{7}\times S p ( 1 )}\ ,
f\big(\frac{a}{M}+iy\big) = \xi_R(-M) \overline{\xi_{q/R}(a)}i^{k}(MR^{\frac{1}{2}}y)^{-k}\tilde{f}_R\big(-\frac{\overline{Ra}}{M}+ i\frac{1}{M^2 Ry}\big).
\dim_K(F_{red}^{i+1}) =\dim_K(H^{d}(X_{\Gamma},{\mathcal{M}}^{*,{\Gamma}}\otimes_K\Omega^{\bullet}_{X_{\Gamma}}))-\dim_K(\check{F}_{red}^{d-i}){}\le\dim_K(\check{F}^{i+1}_{\Gamma})=\dim_K(F^{i+1}_{\Gamma}).
\lbrack K ]_{q \bar{q}}= \left( \begin{array}{l l}{{q \bar{q}+ q^{- 1}\bar{q}^{- 1}}}&{{- 1}}\\{{- ( 1 + \bar{q}^{2}+ \bar{q}^{- 2})}}&{{q \bar{q}^{3}+ q^{- 1}\bar{q}^{- 3}}}\\ \end{array}\right)
\Sigma_{1}^{*}\Sigma_{1}+\Sigma_{2}^{*}\Sigma_{2}=[\begin{matrix}\Sigma_{1}\\ \Sigma_{2}\end{matrix}]^{*}[\begin{matrix}\Sigma_{1}\\ \Sigma_{2}\end{matrix}]=W^{*}P_{1}^{*}[\begin{matrix}U_{1}&0\\ 0&U_{2}\end{matrix}][\begin{matrix}U_{1}^{*}&0\\ 0&U_{2}^{*}\end{matrix}]P_{1}W=I.
\Delta \triangleq \sum^{M}_{i=1}\left[\log \left(1+SNR_{i,k}\right) +\log \left(1+SNR_{i,n}\right)\right] - \frac{1}{2}\sum^{M}_{i=1}\left[ \log \left(1+\rho|\mathbf{v}_{i,k}^H\mathbf{h}_{i,ik}|^2 \right) +\log \left(1+|\mathbf{v}_{i,n}^H\mathbf{h}_{i,in}|^2 \right)\right].
P(T, x) = \prod_{j = 1}^{13}\mathcal{G}(u_j, x) = x^9 \ \dfrac{x^2-2}{x}\ \dfrac{(x^2-1)(x^2-4)}{x(x^2-2)}\ \dfrac{x^2-4}{x}\ \dfrac{x^6 - 8x^4 + 12x^2 - 4}{x(x^2-1)(x^2-4)}= x^5 (x^2-4)(x^6 - 8x^4 + 12x^2 - 4) .
\sigma_{3}^{(2)}: T^{*}M|_{\Xi,\Psi}\times T^{*}M|_{Q,P}\rightarrow T^{*}M ; (\sigma_{3}^{*}\Xi^{a},\sigma_{3}^{*}\Psi_{a},\sigma_{3}^{*}\Theta^{\alpha}:=\epsilon^{\alpha},\sigma_{3}^{*}\Theta_{\alpha}:=\epsilon_{\alpha},\sigma_{3}^{*}Q^{i},\sigma_{3}^{*}P_{i})\mapsto (\Xi^{a},\Psi_{a},\Theta^{\alpha},\Theta_{\alpha},Q^{i},P_{i}).
F_k^{(\ell)}(q) := q^k \sum_{n_1, \dots, n_k\geq 0}(q)_{n_k}\, q^{n_1^{2}+ \cdots + n_{k-1}^{2}+ n_{\ell}+ \cdots + n_{k-1}}\, \prod_{j=1}^{k-1}\begin{bmatrix}n_{j+1}+ \delta_{j,\ell-1}n_j \end{bmatrix},
r^{6}p/\left( p-1\right) <r^{5}\left( r-1\right) p/\left( p-1\right)+5r+8rp/\left( p-1\right) & =\left( r^{6}-r^{5}+5r\left( p-1\right) /p+8r\right) p/\left(p-1\right) & <\left( r^{6}-r^{5}+13r\right) p/\left( p-1\right) ,
x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n+\lambda_n\theta_n(z-x_n) =x_n-\lambda_n\left(x_n-Tx_n+\theta_n(x_n-z)\right) =x_n-\lambda_n\left(Ux_n+\theta_n(x_n-z)\right).
h(fg) =\psi(fg)=\sum_{\alpha\in\Lambda}\psi_{\alpha}(fg)y_{\alpha}=\sum_{\alpha\in\Lambda}f\psi_{\alpha}(g)y_{\alpha}+\sum_{\alpha\in\Lambda}(-1)^{|f||g|}g\psi_{\alpha}(f)y_{\alpha}&=f\psi(g)+(-1)^{|f||g|}g\psi(f)=m(f)h(g)+(-1)^{|f||g|}m(g)h(f).
\chi( \alpha_{fN,\bar{d}_m}) = \prod_{\ell \mid \frac{\bar{d}_m}{(\bar{d}_m, fN)}}(\ell - \chi(\sigma_{\ell})) \left[ \sum_{t \mid (\bar{d}_m,fN)}\mu(t) \cdot \frac{(\bar{d}_m,fN)}{t}\cdot \chi\big( \alpha_{fN}(t) \big) \right] ,
\mathbf{v}+\mathbf{v}_{\ast}=\mathbf{v}^{\prime}+\mathbf{v}_{\ast}^{\prime}, \mathbf{v}^{2}+\mathbf{v}_{\ast}^{2}=\mathbf{v}^{\prime 2}+ \mathbf{v}_{\ast}^{\prime 2}.
f'(y^n;\mu,\Sigma)\pi(\mu,\Sigma;\nu,\sigma^2, \rho^2) = f(y^n;A^{-1}\mu+\mu_0,A^{-1}\Sigma (A^\top)^{-1}) \pi(\mu,\Sigma;\nu,\sigma^2, \rho^2) &= f(y^n;\mu',\Sigma') \pi(A(\mu'-\mu_0), A\Sigma' A^\top; \nu, \sigma^2, \rho^2) &= f(y^n;\mu',\Sigma')\pi(\mu',\Sigma';\nu,\mu_0, \Sigma_0, \rho^2),
\sup_{s\in [0,t]}p\left(\imath\left(J'_s - \sum_{n=1}^N J'([0,s]\times C_n)\right)\right) \leq c_{p}\sup_{s\in [0,t]}\left\|J'_s- \sum_{n=1}^N J'([0,s]\times C_n)\right\|_K
\int_{Q}\left(\sigma \partial_t^{1/2}\boldsymbol{y}\cdot\partial_t^{1/2}\boldsymbol{v}^{\perp}+ \nu \, \textbf{curl}\, \boldsymbol{y}\cdot \textbf{curl}\, \boldsymbol{v}\right)\, d\boldsymbol{x}\,dt= \int_{Q}\boldsymbol{u}\cdot\boldsymbol{v}\, d\boldsymbol{x}\,dt
\omega = \partial_{z}\log{\frac{E ( z , P_{+})}{E ( z , P_{-})}}- 2 \pi i \sum_{j , k = 1}^{h}\left( \mathrm{I m}\int_{P_{-}}^{P_{+}}\omega_{j}\right){\Omega_{j k}^{( 2 )}}^{- 1}\omega_{k}( z ) ,
\begin{aligned}\{a,b,c,\infty\}, \{a,a_1,a_2,\infty\}, \,\, \{b,b_1,b_2,\infty\}, \,\, \{c, c_1,c_2,\infty\}, \{b,c,a_1,a_2\}, \{a,b,c_1,c_2\}, \, \, \{a,c,b_1,b_2\}; \end{aligned}
\begin{array}{l}\vert 0, z\rangle_0=\displaystyle\exp_{q^{-1}}(-\frac{1}{2}|qz|)\sum_{m=0}^\infty \frac{(-qz)^m}{[m]_{q^{-1}}!}({\cal D}_0)^m\vert 0 \rangle_0~,~~~~~{\cal X}^0\vert 0\rangle_0=0~,{\cal X}^{0}\vert 0,z\rangle_{0}=z\vert 0,z\rangle_{0}~.\end{array}
T^{(n)}[[\xi_{i,j}]_{i,j=1}^n] :=\sum_{i,j=1}^n [T\xi_{i,j}]_{i,j=1}^n[\xi_{i,j}]_{i,j=1}^n\in L^2 (\mathbb{M}_n (N), \o\otimes{\rm tr\,}_n )\, ,
\mathrm{DTE}_{(p,q)}(X)=\mu+\sigma\frac{\Gamma(t-1)\left[\left(1+\xi_{p}^{2}\right)^{-(t-1)}-\left(1+\xi_{q}^{2}\right)^{-(t-1)}\right]}{2\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},
A_{n m}^{l}= \frac{2}{2 l + \delta_{1}+ \delta_{2}}\left[ \frac{( l - | m | ) ( l + | m | + \delta_{1}+ \delta_{2}) ( l + \delta_{1}) ( l + \delta_{2}) ( n - l ) ( n + l + \delta_{1}+ \delta_{2})}{( 2 l + \delta_{1}+ \delta_{2}- 1 ) ( 2 l + \delta_{1}+ \delta_{2}+ 1 )}\right]^{1 / 2}.
\sigma_k(\lambda(D^2\underline{u})) =(\underline{u}')^k+\underline{u}''(\underline{u}')^{k-1}\Sigma_{i=1}^n\sigma_{k-1;i}(a)(a_ix_i^2) &\geq(\underline{u}')^k+2h_k(a)\underline{u}''(\underline{u}')^{k-1}\tau &=\overline{f}\quad\ \mathbb{R}^n\setminus\overline{D}_{s_0}.
\frac{L [ C ]}{\epsilon}+ \mathcal{A}[ C ] \mathop{=}_{\epsilon , \epsilon^{\prime}\to 0}\frac{L [ f ( C ) ]}{\epsilon^{\prime}}+ \mathcal{A}[ f ( C ) ] + \oint_{C}\left| \frac{d f ( x ( s ) )}{d s}\right| d s \int_{\epsilon / x ( s )^{2}}^{\epsilon^{\prime}}\frac{d y}{y^{2}}\; .
H ({\lambda}) = - \frac{1}{s i n \gamma \; \; d_{0}^{2}d_{1}^{2}}\sum_{k = 1}^{L}\frac{{\lambda}q +{\lambda}^{- 1}q^{- 1}}{2}( S_{k}^{X}S_{k + 1}^{X}+ S_{k}^{Y}S_{k + 1}^{Y}) - \frac{1}{2}S_{k}^{Z}S_{k + 1}^{Z}