harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\bf A}= \frac{e^{\phi}}{2}\left( \begin{array}{cccc}\frac{1}{t-t^{'}}+\frac{1}{t^{'}-t^{''}}0 \frac{1}{t-t^{'}}0 0 \frac{1}{t-t^{'}}+\frac{1}{t^{'}-t^{''}}0 \frac{1}{t-t^{'}}\frac{1}{t-t^{'}}0 \frac{1}{t-t^{'}}+\frac{1}{t^{''}}0 0 \frac{1}{t-t^{'}}0 \frac{1}{t-t^{'}}+\frac{1}{t^{''}}\end{array}\right)
\frac{1}{\pi^{2}}\int_{0}^{\infty}d t \, \frac{\sinh ( 1 + \nu ) t}{\sinh t \sinh 2 \nu t}\left( \cos \pi \nu \Bigl ( \frac{t - \pi i}{\cosh \nu ( t - \pi i )}+ \frac{t + \pi i}{\cosh \nu ( t + \pi i )}\Bigr ) - \frac{2 t}{\cosh \nu t}\right) .
\hat{H}_{p h}^{F}\Phi_{n}^{F}= \left( \frac 1 2 \langle p_{a}, \hat{g}^{1 1}p_{a}\rangle + V_{q}[ a ] + \frac 1 2 \langle{\bf B},{\bf B}\rangle \right) \Phi_{n}^{F}= E_{n}\Phi_{n}^{F}\ .
V_{14}=-\varepsilon\bar{m}\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})\sum_{i=1}^{d}\sum_{j=1}^{d}\left[\Theta_{p_{i}p_{j}}(t,x,\mu^{2},Dw^{2})\left(\frac{\partial^{2}(w^{1}-w^{2})}{\partial x_{i}\partial x_{j}}\right)\right]\dx.
{E}^{0,i}\big[e^{- \alpha\underline{m}_{\mathbf{e}_q}}, J_{{m}_{\mathbf{e}_q}}= k\big] =\frac{\pi_k}{\pi_i}\hat{{E}}^{0,k}\big[e^{- \alpha\overline{m}_{\mathbf{e}_q}}, J_{\overline{m}_{\mathbf{e}_q}}= i\big],
\gamma^{N^F}(A^{N,F}_{\varepsilon,\theta}[z_1,\ldots,z_l])=\gamma^{N^F}(\mathcal{U}^{U}_1( A^{N,F}_{\varepsilon,\theta}[z_1,\ldots,z_l]))=\gamma^{N^F}(A^{N,F}_{\varepsilon,\theta}[Uz_1,\ldots,Uz_l]).
\frac{F(n+1)}{F(n)}= 2^n \cdot \frac{(4n+2)!}{(3n+2)!}\prod_{i=0}^{n-1}\frac{(n+2i+1)!}{(n+2i+2)!}= 2^n \cdot \frac{(4n+2)!}{(3n+2)!}\prod_{i=0}^{n-1}\frac{1}{n+2i+2}= \frac{2^n (4n+2)! n!!}{(3n+2)! (3n)!!}.
\begin{aligned}\dot{x}= -y + k_1 x^2 +k_2 x y -k_1 y^2 + x \left( 2 k_3 x y + k_1 k_2 y^2 \right), \dot{y}= x + 2 k_1 x y + k_2 y^2 + y \left( 2 k_3 x y + k_1 k_2 y^2 \right),\end{aligned}
A = \frac{1}{|z|}\Big| ( D_t \tilde h(z+t_0(X_1)) - D_t \tilde h(t_0(X_1))) b(X_1,z+t_0(X_1)) & - ( D_t \tilde h(z+t_0(X_2)) - D_t \tilde h(t_0(X_2))) b(X_2,z+t_0(X_2)) \Big| B = \frac{1}{|z|}\Big| D_t \tilde h(t_0(X_1))) ( D_t b(X_1,t_0(X_1)) z-b(X_1,t_0(X_1))) &-D_t \tilde h(t_0(X_2))) ( D_t b(X_2,t_0(X_2)) z-b(X_2,t_0(X_2))) \Big| .
F ( x^{-}, x^{+}) = \left|{\frac{L}{\pi}}\langle 0 | T^{+ +}( k ) | n \rangle \right|^{2}\left({\frac{x^{+}}{x^{-}}}\right)^{2}{\frac{M_{n}^{4}}{8 \pi^{2}k^{3}}}K_{4}\left( M_{n}\sqrt{2 x^{+}x^{-}}\right)
F_{B ( F )}\left( \beta , V , \frac{1}{m}\right) = - \frac{V}{2^{\frac{d}{2}+ 1}\pi^{\frac{d}{2}}}\int_{0}^{\infty}d s \; s^{- 1 - \frac{d}{2}}\left[ \theta_{3 ( 4 )}\left( 0 \left| \frac{i \beta^{2}}{2 \pi s}\right. \right) - 1 \right] e^{- \frac{m^{2}s}{2}}\; .
\Delta~ \partial_1 J^{(d)}_{1 1 1}=\frac12(d-3)(\partial_1 \Delta) J^{(d)}_{1 1 1}+(d-2)J^{(d)}_{0 1 1}+\frac{(d-2)}{4m_1^2}\left[(\partial_3 \Delta) J^{(d)}_{1 0 1}+(\partial_2 \Delta) J^{(d)}_{1 1 0}\right],
w^{(c)}_{q+1}= \lambda^{-1}_{q+1}\sum_{k\in \Lambda_v}{\nabla\big(a_{(v,k)}\phi_{{(\gamma,\tfrac{1}{2},k)}}g_{{(2, \sigma)}}\big)}\times F_{\bar{k}}+\lambda^{-1}_{q+1}\sum_{k\in \Lambda_b}{\nabla\big(a_{(b,k)}\phi_{{(\gamma,\tfrac{1}{2},k)}}g_{{(2, \sigma)}}\big)}\times F_{\bar{k}},
\mathrm{ch}(c) = \Gamma F[F,a_0]\prod_{k=1}^p [F,a_k] = \Gamma a_0\prod_{k=1}^p [F,a_k]-\Gamma Fa_0F\prod_{k=1}^p [F,a_k]. = \mathcal{W}_{\emptyset}(c)+(-1)^{p+1}\Gamma Fa_0\left(\prod_{k=1}^p [F,a_k]\right)F.
\sum_{m^{( 1 )}, \cdots , m^{( N )}}\exp \left[ - \pi \vec{m}^{T}M^{- 1}\vec{m}+ 2 \pi i \vec{m}^{T}\vec{x}\right] = \sum_{m^{( 1 )}, \cdots , m^{( N )}}( \det M )^{\frac{1}{2}}\, \exp \left[ - \pi ( \vec{m}+ \vec{x})^{T}M ( \vec{m}+ \vec{x}) \right] .
\frac{\partial k_{ia}}{\partial \lambda_{jb}}= [\delta_{ia,jb}- s_bF(\lambda_{jb}|\lambda_{ia})]2\pi \rho(\lambda_{jb}), \frac{\partial \lambda_{ia}}{\partial k_{jb}}= \frac{1}{2\pi \rho(\lambda_{ia})}[\delta_{ia,jb}- s_b F(\lambda_{ia}|\lambda_{jb})]
(x^\pm\otimes t^k)^{(r)}\cdot (v_1\otimes v_2)=\sum_{i=0}^{r}(x^\pm\otimes t^k)^{(i)}\cdot v_1\otimes (x^\pm\otimes t^k)^{(r-i)}\cdot v_2, \Lambda_r\cdot (v_1\otimes v_2)=\sum_{i=0}^{r}\Lambda_i\cdot v_1\otimes \Lambda_{r-i}\cdot v_2, \binom{h}{r}\cdot (v_1\otimes v_2)=\sum_{i=0}^{r}\binom{h}{i}\cdot v_1\otimes \binom{h}{r-i}\cdot v_2.
C_{m,n}(k) = \left(\begin{array}{cc}\frac{1}{c_{1,n}(k)}\frac{-m}{a_{n+1}(k)c_{1,n}(k)}\frac{-m}{a_{n}(k+1)c_{1,n}(k)}c_{2,n}(k) + \frac{m^2}{a_{n}(k+1)a_{n+1}(k)c_{1,n}(k)}\end{array}\right).
- \Omega_{u}( - u ) = \mathrm{i}\left( \begin{matrix}{- 1}&{2}\\{- 1}&{1}\\ \end{matrix}\right) \left( \begin{matrix}{f^{( 1 )}( u )}\\{f^{( 2 )}( u )}\\ \end{matrix}\right) = \mathrm{i}R \Omega_{u}( u )
\Gamma_{fb_m}^*\Gamma_{fb_m}(bw_n)= \Gamma_{fb_m}^*\Gamma_f(v_n)={\bf P}_+\Bigl(\overline{fb_m}\,\,\Gamma_{f}(v_n) \Bigr)={\bf P}_+\Bigl(\bar b_m{\bf P}_+\bigl(\bar f\Gamma_f(v_n)\bigr)\Bigr) ={\bf P}_+\left(\bar b_m\Gamma_f^*\Gamma_f(v_n)\right)=s^2_n(\Gamma_f){\bf P}_+\left(\bar b_mv_n\right)=s^2_n(\Gamma_f)bw_n.
Z(\vec{h};1) =\|\partial_t\vec{h}\|+\|\vec{h}\|_1 &\lesssim \|\partial_t\vec{h}\|+\|\mathfrak{A}\vec{h}\|+\|\vec{h}\| &=\|\partial_t\vec{h}\|+\|\partial_t\vec{h}-\vec{f}\|+\|\vec{h}\| &\lesssim \|\partial_t\vec{h}\|+\|\vec{h}\|+\|\vec{f}\|,
\sum_{n \geq 0}\frac{q^{n^2}(-q;q^2)_n}{(q^2;q^2)^2_n}= \frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}, \sum_{n \geq 0}\frac{q^{n^2+2n}(-q;q^2)_n}{(q^2;q^2)^2_n}= \frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}\sum_{n\geq 0}\frac{(-1)^n q^{n^2+n}}{(-q;q^2)_{n+1}}=\frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}\sum_{j\geq 0}q^{3j^2+2j}(1-q^{2j+1}).
\bigl\Vert \mathbb{E}_{0}(X_{k})\bigr\Vert_{2}^{2}=\mathbb{E}\bigl(X_{k}{\mathbb{E}}_{0}(X_{k})\bigr)\leq 2\int_{0}^{\tilde{\alpha}(k)}Q^{2}(u)\,\mathrm{d}u.
\Theta_{\{\Lambda\}}(v) = \sum_{n_i \in{\bf Z}}exp\bigl(3 \pi i (n_i +\frac{\lambda_i}{3})\tau_{ij}(n_j+\frac{\lambda_j}{3}) + 2\pi i\sqrt{3}(n_i+\frac{\lambda_i}{3})v_i \bigr)\ ,
\Omega_{j}^{-}:=\Omega_{j}, \Omega_{j}^{+}:=\mathbb{R}^{3}\backslash\overline{\Omega_{j}}, \mathbb{A}_{j}^{+}:=\left. \mathbb{A}_{j}^{\operatorname*{ext}}\right\vert_{\Omega_{j}^{+}}, p_{j}^{+}:=\left. p_{j}^{\operatorname*{ext}}\right\vert_{\Omega_{j}^{+}}.
|\partial_x^{(i)}p_t(x,y)| \lesssim \left(\frac{1}{\sqrt{1+x^2}}\right)^{i}\left(\frac{1}{\sqrt{1+y^2}}\right)\sum_{j=0}^{i}c(j,i,t)\exp \left(-\left(C_jt- \frac{(arsinh(x)-arsinh(y))^2}{C_j t}\right)^-\right) + \mathsf{l}.\mathsf{o}.\mathsf{t},
L^p(\mathcal I;L^r(\Omega_\eta)) :=\big\{v\in L^1(\mathcal I\times\Omega_\eta):\,\,v(t,\cdot)\in L^r(\Omega_{\eta(t)})\,\,t,\,\,\|v(t,\cdot)\|_{L^r(\Omega_{\eta(t)})}\in L^p(\mathcal I)\big\}, L^p(\mathcal I;W^{1,r}(\Omega_\eta)) :=\big\{v\in L^p(\mathcal I;L^r(\Omega_\eta)):\,\,\nabla v\in L^p(\mathcal I;L^r(\Omega_\eta))\big\}.
\tilde{H}= - i \vec{\alpha}\cdot \hat{e}_{j}\frac{\partial}{\partial \xi^{j}}- \frac{i}{2 \Delta}\vec{\alpha}\cdot \vec{\kappa}- \frac{i}{\Delta}\vec{\alpha}\cdot \hat{e}_{3}\Bigl ( \frac{\partial}{\partial s}+ \frac{1}{2 \Delta}\frac{\partial ( \vec{\kappa}\cdot \vec{\xi})}{\partial s}\Bigr ) + \beta m \ .
|e^{\int_0^tB_i(s)d\beta_i}f|_q=|e^{\theta_i\int_0^tB_i(s)ds}f|_q=|f|_q,\ \forall f\in L^q(\mathbb{R}^d),\s\in\mathbb{R},\t\geq0, i=1,2,...,N;
A+B+C-\omega = \alpha_{N-2,N+2,N,N}+ \alpha_{N-2,N,N-2,N}+ \alpha_{N+2,N,N+2,N}- \alpha_{N,N,N,N}& = \frac{\pi(2N-2)!}{(N-1)! (N-2)! 2^{2N+1}}\Big(\frac{2(2N-1)}{\sqrt{(N+2)(N+1)N(N-1)}}+\frac{(4N^2-2N-5)(N+1)}{(N+2)(N+1)N(N-1)}\Big).
\Hat{X}^i_t =\; \Hat{X}^i_0 + \Hat{E}^i_t - \Hat{S}^i(\mu^i \int_o^t B^i_{ns}\, d{s}) - \sum_{j: j\neq i}\Hat{S}^{ij}(\mu^{ij}\int_0^t B^{ij}_{ns}\, d{\bar{I}^j_s}) begin{align*}2mm] \; + \mu^i \Hat{I}^i_t - \sum_{j: j\neq i}\mu^{ij}\, \int_{0}^t B^{ij}_{ns}\, d{\Hat{I}^j_s},
\frac{\partial}{\partial t}\begin{pmatrix}<u_1(t)> <u_2(t)>\end{pmatrix}-\nu \frac{\partial^2}{\partial x_3^2}\begin{pmatrix}<u_1(t)> <u_2(t)>\end{pmatrix}=-\begin{pmatrix}\frac{p_1(t)}{\Pi_1}\frac{p_2(t)}{\Pi_2}\end{pmatrix}.
Q(\Psi_{\omega})=\omega^{1-\frac{d}{2}}Q(\Psi),\ \E(\Psi_{\omega})=\omega^{2-\frac{d}{2}}E(\Psi),\ \ \mathbf{P}(\Psi_{\omega})=\omega^{\frac{3-d}{2}}\mathbf{P}(\Psi).
p_0 = (t^{20}+ 228 t^{15}+ 494 t^{10}- 228 t^5 + 1)^3 p_{\infty}= 2^6 3^3 \bigl( t(t^{10}- 11 t^5 - 1) \bigr)^5 p_1 = (t^{30}- 522 t^{25}- 10005 t^{20}- 10005 t^{10}+ 522 t^5 + 1)^2 ,
R_{\alpha\beta3\alpha}= |\nabla u|^{-1}(\nabla_\alpha\nabla_\beta-\nabla_\beta\nabla_\alpha)\nabla_\alpha u = -|\nabla u|^{-1}\nabla_\alpha( k_{\alpha \beta}|\nabla u|)+|\nabla u|^{-1}\nabla_\beta( k_{\alpha \alpha}|\nabla u|) = -\nabla_\alpha k_{\alpha \beta}+\nabla_{\beta}k_{\alpha \beta}+k_{\alpha 3}k_{\alpha \beta}-k_{\beta 3}k_{\alpha\alpha},
\sum_{n \leq x}F_{\omega_0}(n)^{h_0}\prod_{i = 1}^{h_1}F_{\omega_{q_i}}(n)^2 \prod_{i = h_1 + 1}^{h_1 + h_2}F_{\omega_{q_i}}(n) \ll x (\log\log x)^{(3h_0 + 2h_1 + h_2 - 1)/2}.
\lim_{u\rightarrow \infty}\frac{1}{\overline{\mathbf{F}}\left( u\right)}\int_{u}^{\infty}g_{\mathbb{K}}\left( \frac{\overline{\mathbf{F}}\left(x\right)}{\overline{\mathbf{F}}\left( u\right)}\right) \log \frac{x}{u}d\mathbf{F}\left( x\right) =\gamma_{1},
\delta - \bar{\delta}= - \frac{4 i}{2 l + 1}\left( \frac{\alpha - 1}{8}\right)^{2 l + 1}\frac{\Gamma ( 1 + l - i \omega ) \Gamma ( 1 + l + i \omega )}{\Gamma ( l + \frac{1}{2})^{2}}\sinh \pi \omega .
\Delta v_1 =4\bigg(k_{11}e^{u_0'+v_1}+k_{12}e^{u_0''+v_2}-1\bigg)+g_0' \Delta v_2 =4\bigg(k_{21}e^{u_0'+v_1}+k_{22}e^{u_0''+v_2}-1\bigg)+g_0'',
\begin{array}{rcl}F^{(m)}(\zeta_1 , \cdots , \zeta_{2N}) =\displaystyle\sum_{\iota_1 , \cdots , \iota_{2N}}v_{\iota_1}\otimes \cdots \otimes v_{\iota_{2N}}F^{(m)}(\zeta_1 , \cdots , \zeta_{2N})_{\iota_1 , \cdots , \iota_{2N}}\end{array}
D^{\mu \nu}( x , y ) = i \int \frac{d^{4}k}{( 2 \pi )^{4}}\, \frac{e^{- i k_{\lambda}( x - y )^{\lambda}}}{k_{\lambda}k^{\lambda}+ i \epsilon}\left[ g^{\mu \nu}- \frac{\gamma k^{\mu}k^{\nu}}{k_{\lambda}k^{\lambda}+ i \epsilon}\right] ;
g(x_{L,\epsilon}(t))e^{c_1\int_0^t \frac{1}{\sigma(s)}\,ds}= x_1(\epsilon) &<\min_{T_7(\epsilon)\leq s\leq T_8(\epsilon)}g(x(s))e^{c_1\int_0^s\frac{1}{\sigma(u)}\,du}&\leq g(x(t))e^{c_1\int_0^t\frac{1}{\sigma(u)}\,du}.
\widetilde{\eta}_{t,1}(f)= \widetilde{\eta}_{0,1}(f)-\int_0^t\int_\mathbb{T}\widetilde{\eta}_{s,3}(\lambda_2)\lambda_1(v)\theta_s^S(v)f(v)dvds &-\int_0^t\int_{\mathbb{T}}\widetilde{\eta}_{s,1}(\lambda_1f)\lambda_2(v)\theta_s^I(v)dvds &-\int_0^t\mu_{s,1}\bigotimes\mu_{s,3}\left(\lambda(\cdot, \ast)(\tilde{G}(\cdot)-\tilde{F}(\cdot))\right)ds
\begin{aligned}Y^{m_k}_{N_k}(x) \sqrt{dV_g(x)}\sim Ai\left(-h_k^{-\frac{2}{3}}\rho(x) \right) \sum_{n = 0}^\infty u_{0,n}(x) h_k^{-{\frac{1}{6}}+ n}+ Ai'\left(h_k^{-\frac{2}{3}}\rho(x)\right)\sum_{n = 0}u_{1,n}(x) h_k^{\frac{1}{6}+n}\end{aligned}
\begin{cases}(\phi,\tilde{\Lambda})|_{\{t=0\}}= \varepsilon(\phi_0,\tilde{\Lambda}_0), (\phi_0,\tilde{\Lambda}_0)\in C_c^{\infty}(\mathbb{R}\times \mathbb{R}), (\partial_t\phi,\partial_t\tilde{\Lambda})|_{\{t=0\}}= \varepsilon(\phi_1,\tilde{\Lambda}_1), (\phi_1,\tilde{\Lambda}_1) \in C_c^{\infty}(\mathbb{R}\times \mathbb{R}). \end{cases}
y_{n+1}=P_Lx_{n+1}=P_L((1-\alpha_n)x_n +\alpha_n s_n) =(1-\alpha_n)P_Lx_n +\alpha_n P_Ls_n &\in (1-\alpha_n)y_n +\alpha_n P_B^{\mu_n}P_A^{\lambda_n}y_n.
u(1/2, 1/2, \tau, \sigma, -1/8) = - (1 + i) \frac{\Gamma(3/4; \tau, \sigma)}{\Gamma(1/4; \tau, \sigma)}\frac{1}{(\tau; \tau)(\tau+1/2; 2\tau)}\frac{1}{(\sigma; \sigma)(\sigma + 1/2; 2\sigma)}.
h(t) := \bigl( L(\sigma+i\tau;\alpha,\lambda_1), L'(\sigma+i\tau;\alpha,\lambda_1), \ldots, L^{(N-1)}(\sigma+i\tau;\alpha,\lambda_1), ,\ldots, L(\sigma+i\tau;\alpha,\lambda_m), L'(\sigma+i\tau;\alpha,\lambda_m), \ldots, L^{(N-1)}(\sigma+i\tau;\alpha,\lambda_m) \bigr) .
\begin{array}{c}{{{\tilde{\cal H}}= g_{A \bar{B}}V^{A}{\bar{V}}^{\bar{B}}+ i V_{, C}^{A}g_{A \bar{B}}{\bar{V}}_{, \bar{D}}^{\bar{B}}\theta^{C}{\bar{\theta}}^{\bar{D}}-}}\\{{- R_{A{\bar{B}}C{\bar{D}}}V_{, C}^{A}{\bar{V}}_{; \bar{D}}^{\bar{B}}\theta^{A}\theta^{C}{\bar{\theta}}^{\bar{B}}{\bar{\theta}}^{\bar{D}},}}\\ \end{array}
Z=(\begin{matrix}U_{1,1}&U_{1,2}\\U_{2,1}&U_{2,2}\end{matrix})(\begin{matrix}\Lambda_{1,1}&\Lambda_{1,2}\\ 0&\Lambda_{2,2}\end{matrix})(\begin{matrix}U_{1,1}^{T}&U_{2,1}^{T}\\U_{1,2}^{T}&U_{2,2}^{T}\end{matrix})
\frac{\delta\Gamma}{{\delta}A_{\mu}}=\langle\frac{{\delta}S}{\delta\hat A_{\mu}}\rangle,\qquad\frac{\delta\Gamma}{\delta{\bar\psi}^\alpha}=\langle\frac{{\delta}S}{\delta\hat{\bar\psi}^\alpha}\rangle,\qquad \frac{\delta\Gamma}{\delta{\psi}^\alpha}=\langle\frac{{\delta}S}{\delta\hat{\psi}^\alpha}\rangle.
\Gamma[\bar{\Phi},\Phi] = \int d^8z{\cal L}_{eff}(\Phi,D_A\Phi,D_A D_B\Phi;\bar{\Phi},D_A\bar{\Phi},D_A D_B\bar{\Phi})+ (\int d^6z{\cal L}^{(c)}_{eff}(\Phi) + h.c.) +\ldots
\sum_{d_1 \dots d_H \leq y}= \sum_{d_1, d_2 \dots, d_{H-1}\leq X}\sum_{d_H \leq y/(d_1 d_2 \dots d_{H-1})}+ \sum_{i=1}^{H-1}\sum_{d_1, \dots, d_{i-1}\leq X}\sum_{\substack{d_{i+1}, \dots, d_H: d_1 \dots \hat{d_i}\dots d_{i-1}\leq y/X}}\sum_{X \leq d_i \leq y/(d_1 \dots \hat d_i \dots d_H)}.
{\mathbb E}_{\mu}\left ( \sup_{w \in K^{\circ}\cap L}\langle J(O)v,w \rangle \right) = R(K^{\circ})\int\limits_{x \in S^{2n-2}}\sup_{w \in K^{\circ}\cap L}\langle x,w \rangle d \sigma_{2n-2}= R(K^{\circ}) M^*(K^{\circ}\cap L).
\det C_{n}\left( H\right)=H_{k,1}N^{n-1}+H_{k,1}M^{n-2}\sum\limits_{i=1}^{n-1}\left( -\frac{H_{k,2}H_{k,i+1}}{H_{k,1}}+H_{k,i+2}\right) \left( \frac{N}{M}\right)^{i-1},
\begin{array}{l}f(\vec{z}+\vec{x}\tau+\vec{y};\tau)=(-1)^{\vec{x}^tQ\vec{x}+\vec{y}^tQ\vec{y}}e(-\vec{x}^t Q (\vec{z}+\vec{x}\tau/2))f(\vec{z};\tau), begin{align*}.5ex]f(\vec{z}/(c\tau+d);(a\tau+b)/(c\tau+d))=(c\tau+d)^w e(c\vec{z}^{\,t}Q\vec{z}/2(c\tau+d))f(\vec{z};\tau),\end{array}
h_{n|1}=-\left(P_{n|1}\circ\log_2 P_{n|1}\right)1_{n}= -\left[\left(\tilde{I}_nP_{n|0}\tilde{I}_n\right)\circ\log_2 \left(\tilde{I}_nP_{n|0}\tilde{I}_n\right)\right]1_{n}= -\tilde{I}_n\left(P_{n|0}\circ\log_2 P_{n|0}\right)\tilde{I}_n1_{n}= \tilde{I}_n h_{n|0}
\begin{aligned}|\nu_2(A \cap B(x,r))- \nu_2(A \cap B(y,r))| \leq \nu_2(B(x,r) \triangle B(y,r)) & \leq \nu_2( B(x,r+d(x,y)) \setminus B(x,r) ) & = h_2(r+d(x,y)) - h_2(r).\end{aligned}
A_{\rm planar}= -{2e^{2}p^{2}\over (4\pi)^{2}}\,\left[\left({13\over6}-{\xi\over 2}\right) \left({1\over \epsilon}- \log{-p^{2}\over4\pi \mu^{2}}- \gamma\right) +{31\over 9}- (1-\xi) +{(1-\xi)^{2}\over 4}\right]
\theta(n_s(\lambda))T_{n_s}- T_{n_s}\theta(\lambda) = \sum_{k = 0}^{\langle \nu(\lambda),\alpha\rangle - 1}\theta(n_s(\lambda)\mu_{n_s}(k))c_{n_s,k}& = \sum_{k = 1}^{\langle \nu(\lambda),\alpha\rangle}c_{n_s,k}\theta(\mu_{n_s}(-k)\lambda).
\varphi(v) = \begin{cases}u_1 \ \ \textrm{if}\ \v = v_{k_2+2}u_2 \ \ \textrm{if}\ \v \in \{v_3,v_4, \dots,v_{d-k_1},v_{k_2+3}, \dots,v_d \}u_3 \ \ \textrm{if}\ \v \in \{v_1, v_2 \}. \end{cases}
\left.\frac{d}{dt}\mbox{Cap}_{\mathcal{A}}(E_t) \right|_{t = 0}\, = (p-1) \int_{\mathbb{S}^{n-1}}(h_1(\xi)-h_0(\xi)) d\mu(\xi) &=(n-p) [\mbox{Cap}_{\mathcal{A}}(E_1)-\mbox{Cap}_{\mathcal{A}}(E_0)].
L_D=\int_{\cal M}{1\over 4\pi}{\rm Tr}\bigg[\widetilde{d}\bigg({k\over{2}}\widetilde{d}g^{-1}d_0g-{k\over{2}}\widetilde{d}h^{-1}d_0h+\widetilde{d}\alpha d_0 h h^{-1}\bigg)-{k\over{6}}(h^{-1}dh)^3+{k\over{6}}(g^{-1}dg)^3 \bigg].
y_{i}(t+1) =\alpha(t) (y_{i}(t)- \beta z_{i}(t))+(1-\alpha(t)) ((1-\eta) y_{i}(t) &\quad{}\quad{}+\eta \sum_{j \in \mathcal{N}_{i}^{in}(\omega^{*}(t)) \cup \{i \}}\mathcal{W}_{ij}(\omega^{*}(t)) y_{j}(t)), z_{i}(t) =x_{i}(t)-R_{i}, x_{i}(t) =\arg\min_{q_{i}}(f_{i}(q_{i})-y^{T}_{i}(t) q_{i}),
\left \{\begin{matrix}p 1 a_{21}p+a_{22}\end{matrix}\right \}=\left [\begin{matrix}p 1 a_{21}a_{22}+p\end{matrix}\right ]+v^{- a_{22}-1}\ \overline{[p+1]}\left [\begin{matrix}p+1 0 a_{21}-1 a_{22}+p +1\end{matrix}\right ].
{\cal H}_{\mathrm{T C S}}( s , w , E_{\mathrm{c u t}}) = \left\{\left| \Psi \right\rangle : \: \left( L_{0}- \bar{L}_{0}\right) \left| \Psi \right\rangle = s \left| \Psi \right\rangle , \: Q_{k}\left| \Psi \right\rangle = w \left| \Psi \right\rangle , H_{\mathrm{C F T}}\left| \Psi \right\rangle \leq E_{\mathrm{c u t}}\left| \Psi \right\rangle \right\}\: ,
-\mathcal{A}_1^T\phi+\mathcal{B}^T\psi+\mathcal{C}^Tb=0 &\left(\mathcal{B}^T\right)^*\phi+\mathcal{A}_2^T\psi-\mathcal{D}^Tb=0 &\left(\mathcal{C}^T\right)^*\phi-\left(\mathcal{D}^T\right)^*\psi-P\left(T^{-2}-l^T\right)b=0
\theta^{2}\Delta E_{n}^{( 1 )}( e v e n ) = \frac{\theta^{2}V_{0}}{6}[ 1 - 4 m a^{2}V_{0}( V_{0}- n \pi - \frac{\pi}{2}) ( \frac{1}{3}- \frac{1}{2 ( n + \frac{1}{2})^{2}\pi^{2}}) ] .
\sum_{k=0}^{n-1}[2dk+1]\frac{(aq;q^d)_k (q/a;q^d)_k (bq;q^d)_k (q/b;q^d)_k}{(aq^d;q^d)_k(q^d/a;q^d)_k (bq^d;q^d)_k (q^d/b;q^d)_k}q^{(d-2)k}\equiv 0\pmod{[n]}\intertext{and, for $d\ne2$,}\sum_{k=0}^{n-1}[2dk+1]\frac{(aq;q^d)_k (q/a;q^d)_k (q;q^d)_k^2}{(aq^d;q^d)_k(q^d/a;q^d)_k (q^d;q^d)_k^2}q^{(d-2)k}\equiv 0\pmod{[n]\Phi_n(q)}.
\langle T_{i}^{k}\rangle_{{\mathrm{s u b}}}^{( R )}= - \frac{\delta_{i}^{k}\xi^{- d - 1}}{2^{d - 1}\pi^{d / 2}\Gamma ( d / 2 )}\int_{0}^{\infty}\frac{\omega^{d}g^{( i )}( \omega ) d \omega}{e^{2 \pi \omega}+ ( - 1 )^{d}}
Q_{\ell_1}^\delta+\sum\limits_{j=0}^{\delta-1}a_{j,\ell_1}Q_{\ell_1}^j=\left(Q_i-w_i\right)^\delta+\sum\limits_{j=0}^{\delta-1}a_{j,\ell_1}\left(Q_i-w_i\right)^j
\lim_{\lambda \to \infty}\frac{E_{\lambda}( \ell )}{L^{D - 1}}= - \frac{\Gamma ( D + 1 ) \zeta ( D + 1 )}{2^{2 D + 1}\pi^{D / 2}\Gamma \left( \frac{D}{2}+ 1 \right)}\ell^{- D}.
{\cal S}^*=\left\{(x^*,\lambda^*)\in \Re^{2n}:\begin{array}{l}\nabla f(x^*)+\bigg[\displaystyle \sum_{i=1}^p[y^I]^*_j\nabla^2 c_j(x^*)-A^TA-{\cal J}c_{\gamma}(x^*)^T{\cal J}c_{\gamma}(x^*) begin{align*}4pt] -{\cal J}c_{\beta}(x^*)^T{\rm Diag}([v_b]_{\beta}){\cal J}c_{\beta}(x^*)\bigg]\lambda^*=0 \end{array}\right\}.
\det (I_{n+2M}-u{\cal A})=(1+2( \beta +2 \delta )u+4 \delta ( \delta +1) u^2 )^{M-N}\ \ \ \times \det ( \{1+(2 \beta +5 \delta )u+(2 \delta ( \beta +4 \delta +2)+d \beta{}^2 ) u^2 +2 \delta (2 \delta ( \delta +1)+d \beta{}^2 ) u^3 \}I_N & \ \ \ - \beta (1+2 \delta u)(1+( \beta +2 \delta )u){\bf A}(G) ) .
{A_4^{YM}= \langle T_{\mu_1\nu_1}(1)T_{\mu_2\nu_2}(2) T_{\mu_3\nu_3}(3) T_{\mu_4\nu_4}(4) \rangle = N^2 A_4^{(1)}+ \tilde k N^{1/2}f^{(0,0)}(S,\bar S) A_4^{(2)}+ \dots ,}
F(x,y)=\begin{cases}F_1(x,y) x\in(0,x_1), \frac{\partial^2F_1(x,y)}{\partial x\partial y}>0, F_2(x,y) x\in(x_1,x_2),\frac{\partial^2F_2(x,y)}{\partial x\partial y}<0, F_3(x,y) x\in(x_2,1),\frac{\partial^2F_3(x,y)}{\partial x\partial y}>0.\end{cases}
(x \circ y)^{\gamma(z)^{-1}}= (x^{\gamma(y)}y)^{\gamma(z)^{-1}}&= x^{\gamma(y) \gamma(z)^{-1}}y^{\gamma(z)^{-1}}&= (x^{\gamma(z)^{-1}})^{\gamma(y^{\gamma(z)^{-1}})}y^{\gamma(z)^{-1}}&= x^{\gamma(z)^{-1}}\circ y^{\gamma(z)^{-1}},
p_x =\sum_{k=3}^n\frac1n\times \frac{\binom{k-1}{2}}{\binom{n-1}{2}}\times \frac{k-2}{(k-1)(k-2)}\frac{n-k}{n-3}=\frac{1}{n(n-1)(n-2)(n-3)}\sum_{k=3}^n(k-2)(n-k) =\frac{1}{6n}
G_{ij}= \frac{{2w_k w_{kij}+ 2w_{ki}w_{kj}}}{{\left|{Dw}\right|^2}}- \frac{{4w_k w_{ki}w_l w_{lj}}}{{| Dw |^4}}+ ah_{ij}= \frac{{2w_{1ij}}}{{w_1}}+ \frac{{2w_{ki}w_{kj}}}{{w_1^2}}- a^2h_i h_j + ah_{ij}.
curlF=\nabla\times F=(\frac{\partial F_{3}}{\partial y}-\frac{\partial F_{2}}{\partial z})e_{1}-(\frac{\partial F_{3}}{\partial x}-\frac{\partial F_{1}}{\partial z})e_{2}+(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y})e_{3}.
\ell:\mathcal{C}\longrightarrow \mathcal{A}\otimes \mathcal{A},\sum_{\mu}a_{\mu}\otimes r_{\mu}(c):=\ell(c)=:\ell\left( c\right)^{\langle 1\rangle}\otimes \ell\left(c\right)^{\langle 2\rangle}
\left( 1- \frac{2}{n}\right)^{p+1}\left( \frac{n+1}{n}\right)^{p \beta}+ \left( C_{1}' + C_{2}'c_{\gamma}^{2}\right) 2^{(p+2)\alpha}\frac{1}{(n+1)^{(p+2) \alpha - p \beta}}= 1- \left( 2p+2 -p\beta \right)\frac{1}{n}+ o \left( \frac{1}{n}\right) .
A = \frac{2 \gamma a (1-e^{-2\gamma n a})(1-e^{-\gamma n a})^2}{(1-e^{-2 \gamma a})(1-e^{-4\gamma n a})}\, e^{-2 a \gamma}B = \frac{2 \gamma a (1-e^{-2\gamma n a})(1+e^{-\gamma n a})^2}{(1-e^{-2 \gamma a})(1-e^{-4\gamma n a})}.
S = \int_M d^6 x \sqrt{G}G^{\lambda \rho}G^{\mu \sigma}G^{\nu \tau}\left( - \frac{3}{4}D_{[\lambda}B_{\mu \nu]}D_\rho B_{\sigma \tau}- \frac{\alpha}{2}D_\lambda B_{\rho \mu}D_\nu B_{\tau \sigma}\right)
\sqrt{\alpha}\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\exp\left({-\left(\frac{\alpha}{n}\right)^2}\right) - \sqrt{\beta}\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\exp\left({-\left(\frac{\beta}{n}\right)^2}\right) = -\frac{1}{2\sqrt{\beta}}\sum_{\rho}\frac{\Gamma\left(\frac{1-\rho}{2}\right) \beta^\rho}{\zeta'(\rho)}.
c_{X,n}:= \min\left( \frac{1}{8 \gamma_{X,n}}, \frac{\tau_{X,n}-1}{4 \gamma_{X,n}(\tau_{X,n}+1)}, \frac{1-\mu_X}{72 \gamma_{X,n}d_{n}}, \frac{1}{4\beta_{X,n}}\right).
E =\Big(y,x_1C^{(1)},\ldots, x_{2^{n-3}-1}C^{(2^{n-3}-1)},zA,x_{2^{n-3}-1}C^{(2^{n-3}-1)}_{\overline{R}}, \ldots, x_1C^{(1)}_{\overline{R}}\Big), F =\Big(y,x_1D^{(1)},\ldots, x_{2^{n-3}-1}D^{(2^{n-3}-1)},zB,x_{2^{n-3}-1}D^{(2^{n-3}-1)}_{\overline{R}}, \ldots, x_1D^{(1)}_{\overline{R}}\Big),
\langle P_h^k(\cdot, s), E_\infty^k(z, \tfrac{k+1}{2}) + E_0^k(z, \tfrac{k+1}{2})\rangle = \frac{\overline{\rho_\infty^k(h, \frac{k+1}{2}) + \rho_0^k(h, \frac{k+1}{2})}}{(4\pi h)^{s-1}}\frac{\Gamma(s + \frac{k}{2}- \frac{1}{2}) \Gamma(s -\frac{k}{2}- \frac{1}{2})}{\Gamma(s - \frac{k}{2})}.
p_t(x,y)~=~\frac{\sqrt{\pi}}{(2 \pi)^{\frac{d+1}{2}}}\cdot\frac{\exp\left(-b_d \cdot t\right)}{\sqrt{t}}\cdot \left( \frac{-1}{\sinh(r)}\partial_r \right)^{\frac{d-1}{2}}\exp\left( \frac{-r^2}{4t}\right)\,,
{\frac{\omega^{\prime}}{T_{L}^{\prime}}}={\frac{\omega + k_{4}}{T_{L}}}\ , \quad{\frac{\omega^{\prime}}{T_{R}^{\prime}}}={\frac{\omega - k_{4}}{T_{R}}}\ , \quad{\frac{\omega^{\prime}}{T_{H}^{\prime}}}={\frac{\omega - \phi k_{4}}{T_{H}}}\ ,
\left| \frac{\partial ( a_{1}^{1}, a_{1}^{2}, a_{1}^{3}, a_{2}^{1}, a_{2}^{2}, a_{2}^{3})}{\partial ( r , \Theta , \Phi , \varphi^{1}, \varphi^{2}, \varphi^{3})}\right| = \frac{1}{8}\: r^{5}\sin ( 4 \Theta ) .
{\frac{1}{2}}( h_{1}- h_{2})^{\prime}= \left[{\frac{d^{\prime}}{d}}+{\frac{1 2 / r^{2}- ( 2{f^{\prime}}_{1}^{2}/ f_{1}^{2}+{f^{\prime}}_{K}^{2}/ f_{K}^{2})}{f^{\prime}/ f + 6 / r}}-{\frac{4}{r^{6}d}}{\frac{Q_{K}^{2}/ f_{K}^{2}- Q_{5}^{2}/ f_{1}^{2}}{f_{1}^{\prime}/ f_{1}- f_{K}^{\prime}/ f_{K}}}\right] h_{2}.
\nu_j = \frac{1}{\mu}\sum_{i=1}^{j+1}\Delta h_i \, \left(1 + \cdots + \rho^{i-1}\right) = \begin{cases}\displaystyle \frac{1}{\mu}\sum_{i=1}^{j+1}\Delta h_i \, \frac{\rho^i -1}{\rho-1}\rho \neq 1 \displaystyle \frac{1}{\mu}\sum_{i=1}^{j+1}i \, \Delta h_i \rho = 1.\end{cases}
J_{i,k}^{(s)}(l_1,l_2)= \frac{1}{[2\Gamma(1/s)]^{k+1}}\sum_{j=0}^k\binom{k}{j}[\Gamma(1/s)]^{k-j}&\times\int_{l_1}^{l_2}y^{\frac{i+1}{s}-1}\exp(-y)\left[y^{1/s}\sum_{m=0}^\infty \frac{(-y)^m}{(1/s+m)m!}\right]^j\mathrm{d}y.
v\left(y,\,s_{0}\right)=e^{-\lambda_{2}s_{0}}\left(\frac{1}{c_{2}}\varphi_{2}\left(y\right)+\frac{a_{1}}{c_{1}}\varphi_{1}\left(y\right)+\frac{a_{0}}{c_{0}}\varphi_{0}\left(y\right)\right)
|f(z)| \leq \left(\sum_{n=-\infty}^{\infty}|a_{n}|^{2}I_{n}(r) \right)^{1/2}\left(\sum_{n=-\infty}^{\infty}\frac{|z|^{2n}}{I_{n}(r)}\right)^{1/2}=\frac{1}{\sqrt{2\pi}}\left( \iint_{\mathbb{A}_r}(1-|z|^2)^2 |f(z)|^2 \,dA \right)^{1/2}\left(\sum_{n=-\infty}^{\infty}\frac{|z|^{2n}}{I_{n}(r)}\right)^{\frac{1}{2}}.
\begin{aligned}S_{1,2,\dots,N+2}^{[0]}(i,j) =\mathcal X^{[i-1]\,T}_{j-1}=-\mathcal X^{[j-1]}_{i-1},S_{1,2,\dots,N+2}^{[0]}(j,i)=S_{1,2,\dots,N+2}^{[0]\,T}(i,j),\;j\ge i,i,j=1,2,\dots,N+2, \end{aligned}
T_{f_{+}}{\Gamma^{*}_{\overline{g_{+}}}}+{\Gamma^{*}_{\overline{f_{+}}}}{S_{g_{+}}}=T_{g_{+}}{\Gamma^{*}_{\overline{f_{+}}}}+{\Gamma^{*}_{\overline{g_{+}}}}{S_{f_{+}}}, \Gamma_{f_{-}}T_{g_{-}}+S_{f_{-}}{\Gamma_{g_{-}}}=\Gamma_{g_{-}}T_{f_{-}}+S_{g_{-}}{\Gamma_{f_{-}}}.
\sum_{k=3}^5 C_k D_k E_k \leq \Big( n \sum_{i=1}^{2n}(Z_{2n;i}(y) - Z_{2n;i}(x))^2 \Big)^2 \cdot \mathcal{O}(1) \leq \Big( n \sum_{i=1}^{2n}(Z_{2n;i}(y) - Z_{2n;i}(x))^2 \Big)^{3/2}\cdot \mathcal{O}_P(1),
\sum_i{|O_i| \cdot (-1)}\leq \sum_i{|O_i| \cdot \frac{1}{|O_i|}f(\sum_{\zeta \in O_i}{\zeta}}, \sum_{\zeta \in O_i}{\zeta^2}) = -M(b\phi(c) + d\phi(c) + 2a + b) \Rightarrow \sum_i{|O_i|}\geq M(b\phi(c) + d\phi(c) + 2a + b).
cos(n\theta)=\sum_{keven}(-1)^{\frac{k}{2}}(\begin{matrix}n\\k\end{matrix})cos^{n-k}\theta sin^{k}\theta=\sum_{i=0}^{n/2}\sum_{j=0}^{i}(-1)^{i-j}(\begin{matrix}n\\ 2i\end{matrix})(\begin{matrix}i\\j\end{matrix})cos^{n-2(i-j)}\theta
R_n= &\frac{2(1+\delta)\varphi^2d\Vert f_1\Vert^2(\|f-\Pi_{V_2}f\|^2+\sigma^2)}{(1-\delta)^2(1-\rho_0^2)k_n^2}+2(\Vert f_1\Vert+k_n)^2d \exp\left(-\kappa\frac{\delta^2n}{\varphi^2d}\right),
a^+_i = c_i^+, i = 2, \dots, m+1, a_{m+2}^+a_j^- = c_{m+2}^+c_j^-, j = 2, \dots, n-m, b_i^+ = d_i^+, i = 1, \dots, m+1, b_j^- = d_j^-, j = 2, \dots, n-m.