harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\phi G\leq \inf_{C_{4}}\left\{\frac{n}{2(p-1)}(\frac{n}{8(p-1)}C_{4}^{2}p^{2}+\frac{2}{n}(p-1)^{2}L+p\lambda_{1}+2Kp )\right\}= \frac{n}{2(p-1)}\left(\frac{2}{n}(p-1)^{2}L+p\lambda_{1}+2Kp \right).\end{aligned}
h_{klij}= \h_{ijkl}-h_{ml}(h_{im}h_{kj}-h_{ij}h_{mk})-h_{mj}(h_{mi}h_{kl}-h_{il}h_{mk}) -h_{ml}(\delta_{ij}\delta_{km}-\delta_{ik}\delta_{jm}) -h_{mj}(\delta_{il}\delta_{km}-\delta_{ik}\delta_{lm}).
(-v_1^2 + v_1^4 + v_2^2 - 10 v_1^2 v_2^2 + 9 v_1^4 v_2^2 + v_2^4 - 9 v_1^2 v_2^4)V_1^2= (\tilde{c}-c)v_1^2(v_2^2+v_3^2)^2.
{\cal M}= \int \frac{d^{D}\! q}{( q - p )^{2}( q \! \cdot \! n )}\longrightarrow^{\! \! \! \! \! \! \! \! \! \! \! \! \! \! M L}\int \frac{d^{D}\! q \, ( q \! \cdot \! n^{*})}{( q - p )^{2}\left[ ( q \! \cdot \! n ) ( q \! \cdot \! n^{*}) + i \epsilon \right]}.
{\cal L}_{\mathrm{k i n}}^{( 0 )}= e^{- \phi}k^{-{\frac{1}{2}}}\sqrt{\dot{X}^{\mu}\dot{X}^{\nu}\bigl ( g_{\mu \nu}- k^{2}A_{\mu}^{( 2 )}A_{\nu}^{( 2 )}\bigr ) - k^{2}{\dot{S}}^{2}- 2 k^{2}\dot{S}{\dot{X}}^{\mu}A_{\mu}^{( 2 )}}\, .
\frac{|\left\langle g,g_\ast\right\rangle|^2}{\|g_\ast\|_{L^2}^2}=\frac{|\mathbb{S}^{n-1}|}{(2\pi)^n\lambda_0(w)}(P^{(0,1)})^2A_{0,1}(w)=\frac{A_{0,1}(w)}{(2\pi)^n\lambda_0(w)},
\begin{array}{cc}\displaystyle{\langle \psi^{i}(z_{1})\psi^{j}(z_{2})\rangle = C_{\psi}\frac{V^{ij}(z_{12})}{(\det{\cal X}_{12})^{\frac{1}{2}\eta}}}~~~~ ~~~~V^{ij}(z_{12})=V_{k}^{~j}(z_{12}){\cal E}^{ki}\end{array}
(-\frac{1}{2}\frac{d^2}{{(d x^1)}^2}+ \Lambda^3_1+\frac{1}{2}{(\Lambda^3_1)}^2 (x^1)^2+\frac{1}{(x^1)^2}( 8 (m^1)^2 - \frac{1}{8}))Y^1_1(x^1)=4E^1_{II}Y_1^1(x^1) ,
\left\langle \; \mathcal{C}^{( g )}( p ) \; \right\rangle = \left. \frac{\mathcal{C}^{( g )}\; \left( \frac{1}{i}\, \frac{\delta}{\delta J}\right) \; W_{0}( J )}{W_{0}( 0 )}\; \right\vert_{J = 0}= \; \bar{p}_{3}
\sum_{j = 1}^{L}|p_{j}(\xi_k)|^2 = \sum_{j = 1}^{\infty}|p_j(\xi_k)|^2 - \sum_{j = L + 1}^{\infty}|p_j(\xi_k)|^2 & \geqslant \delta^2 - \sum_{j = \ell_k + 1}^{\infty}|p_j(\xi_k)|^2 & \geqslant \delta^2 - \frac{\delta^2}{2}= \frac{\delta^2}{2}> 0.
\left[ \frac{r \Delta}{\sigma}(r \Delta)'\right]' -2(\frac{2r\Delta}{\sigma}\Delta'+ \frac{\Delta^2}{\sigma}) +\sigma = -2 r^2 \frac{\Delta^2}{\sigma}{\phi'}^2 + 2 r^2 \sigma e^{-2\phi}(f^2 + g^2)~~,
\frac{df_i}{dt}= f_i(-f_{i+1}f_{i+2}-f_{i+1}f_{i+4}-f_{i+3}f_{i+4}+f_{i+2}f_{i+3}+f_{i+2}f_{i+5}+f_{i+4}f_{i+5}) &-\frac{1}{2}(a_{i}+a_{i+2}+a_{i+4}-a_{i+1}-a_{i+3}-a_{i+5})f_i+a_i(f_i+f_{i+2}+f_{i+4})
\operatorname{Trace}(S_{x,\tau})=\sum_{j=1}^n\langle x_j,\tau_j \rangle=\sum_{j=1}^n\langle \tau_j,x_j \rangle; \operatorname{Trace}(S^2_{x,\tau}) =\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j,x_k\rangle \langle \tau_k,x_j\rangle=\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j,\tau_k\rangle \langle x_k,x_j\rangle .
P_z\left(\bar \tau_{B}=k, S_k=x\right) =P_z\left(S_k=x, S_1,S_2,\cdots, S_{k-1}\notin B\cup L_0\right) &=P_{x}\left(S_k=z, S_1,S_2,\cdots, S_{k-1}\notin B\cup L_0\right) &=P_{x}\left(S_k=z, \tau_{B\cup L_0}>k\right).
I_{f i e l d}= \langle I \rangle = \int d^{4}x \left[ - m^{2}\sqrt{-{\frac{1}{2}}W^{\mu \nu}( x ) W_{\mu \nu}( x )}-{\frac{1}{2}}W^{\mu \nu}( x ) \partial_{[ \mu}A_{\nu ]}\right] \ .
S_{a,c}^b(z) = z^{\frac{2(b-a)}{N}}q^{\frac{2(c-b)}{N}}p^{\frac{(b-a)(c-b)}{N}}\frac{\Theta_{p^N}(p^{N+c-a}q^2 z^2)}{\Theta_{p^N}(p^{N+c-b}z^2)\Theta_{p^N}(p^{N+b-a}q^2)}\;.
\Delta^n_{2t-1}= \rho_{n-t}(v_1,u_2,\ldots,v_{n-1},u_n,v_n), \nabla^n_{2t-1}= \rho_{n-t}(u_1,v_1,\ldots,u_{n-1},v_{n-1},u_n), \Delta^n_{2t}= \rho_{n-1-t}(v_1,u_2,\ldots,v_{n-1},u_n), \nabla^n_{2t}= \rho_{n-t}(u_1,v_1,\ldots,u_n,v_n).
\int_{\mathbb{R}^3}u^p \;dx = \int_{\mathbb{R}^3}u^{p\theta}u^{p(1-\theta)}(1+|x|)^m(1+|x|)^{-m}\;dx &\le \left(\int_{\mathbb{R}^3}u^{pp_1\theta}(1+|x|)^{p_1m}\;dx\right)^{\frac{1}{p_1}}\left(\int_{\mathbb{R}^3}u^{p(1-\theta)p_2}(1+|x|)^{-mp_2}\;dx \right)^{\frac{1}{p_2}},
\Theta^{\mu}_{\hskip1mm\mu}=\frac{2g^{\mu\nu}}{\alpha^2}\,\frac{\partial\alpha}{\partial g^{\mu\nu}}\,\frac{1}{16\pi}\,{\rm tr}\,F^2=\frac{1}{8\pi}\,\frac{\beta(\alpha)}{\alpha^2}\,{\rm tr}\,F^2
\frac{d^{2}\alpha}{( d \xi^{1})^{2}}\pm \left[ H^{2}\left( c_{2}^{2}- H^{2}c_{1}^{2}\right) \left( k_{1}^{2}- ( c_{1}+ H k_{2})^{2}\right) \right] e^{- \alpha}= 0 .
\| \rho(S_\alpha S_\alpha^*)e_i+\rho(U^k)^*\rho(S_\alpha S_\alpha^*)\rho(U)^k e_i\|^2= & = \| \rho(S_\alpha S_\alpha^*)e_i+\rho(U^k)^*\rho(S_\alpha S_\alpha^*)e_i\|^2= & = \| \rho(S_\alpha S_\alpha^*)e_i\|^2+\|\rho(U^k)^*\rho(S_\alpha S_\alpha^*)e_i\|^2=2
\dfrac{\partial f}{\partial \vec{z}^{*}}(\vec{z}^{*}(\bar{\vec{x}}),\bar{\vec{x}}) +[\boldsymbol{\lambda}^{*}(\bar{\vec{x}})]^{\top}\dfrac{\partial \vec{g}}{\partial \vec{z}^{*}}(\vec{z}^{*}(\bar{\vec{x}}),\bar{\vec{x}}) = \vec{0}_{s}^{\top}.
d{}^{*}{\hat{F}}^{( 4 )}+ d A^{( 3 )}\wedge{\hat{F}}^{( 4 )}= H^{( 3 )}\wedge{}^{*}J^{( 6 )}, \qquad d{}^{*}{\hat{F}}^{( 7 )}\equiv - d{\hat{F}}^{( 4 )}={~}^{*}J^{( 6 )}.
M^{6}r_{c}^{4}\sim \frac{M_{s}^{2}}{T_{2}}E_{2}( U , \bar{U}) = \frac{M_{s}^{2}}{T_{2}}\sum_{( m , n ) \in ( Z , Z ) - ( 0 , 0 )}\frac{U_{2}^{2}}{| m + n U |^{4}}.
\beta x = -p u = -\varepsilon p^0 \alpha x = \sqrt{\varepsilon^2 + \beta^2}\:x= \sqrt{-\varepsilon^2 p^2 + (p u)^2}&= \sqrt{-\varepsilon^2 \,\big( (p^0)^2 - |\vec{p}\,|^2 \big) + \varepsilon^2 (p^0)^2}= \sqrt{\varepsilon^2 \, |\vec{p}\,|^2}= \varepsilon\, |\vec{p}\,| \:.
+ i \{g g^{\mu \nu}\delta ( x - y ) +{\frac{1}{e^{2}}}N^{\mu \beta}( x ) N^{\nu \lambda}( y ) ({\frac{a}{a^{2}\partial^{2}+ 4 \alpha^{2}}}g_{\beta \lambda}\delta ( x - y ) +{\frac{2 \alpha}{\partial^{2}( a^{2}\partial^{2}+ 4 \alpha^{2})}}\epsilon_{\beta \lambda \sigma}\partial^{\sigma}\delta ( x - y ) ) \},
\sum_{|i-i_0| + |j-i_0| \leq N^{\nu}}\mathbb{E}[ O ( \rho (E) N (\mu_i (T) - E), \rho (E) N (\mu_j (T) - E )) | \{\lambda_k (0) \}_k] = \sum_{|i-i_0| + |j-i_0| \leq N^{\nu}}\mathbb{E}[ O ( \rho (E) N (\lambda_i (T) - E), \rho (E) N (\lambda_j (T) - E )) | \{\lambda_k (0) \}_k] + \O ( N^{- c_7})
\sum\limits_{\alpha}\langle \frac{\phi_{\alpha}}{-z-\bar{\psi}},\textbf{t}(\bar{\psi}),\ldots,\textbf{t}(\bar{\psi})\rangle^{\mathbb{T}}_{0,n+1,\mathcal{D}}\phi^{\alpha}_{\sigma,b}=\sum\limits_{\Gamma\in DT_{0,n+1}(\mathcal{P},\mathcal{D})}\mbox{C}(\Gamma)_{\sigma,b}.
C r \sup_{x \geq t}|\lambda'''(x)| + C r \sup_{x \geq t}\left(|\lambda'''(x)|\lambda(x)^{\alpha-1}\left(\lambda(x)^{\alpha-1}-\lambda(t)^{\alpha-1}\right)\right)\lambda(t)^{2-2\alpha}&\leq \frac{Cr}{t^{3}\log^{b+1}(t)}+ \frac{Cr}{t \log^{b-b\alpha}(t)}\sup_{x \geq t}\left(\frac{|e'''(x)|x}{\lambda(x)^{1-\alpha}}\right)
u(x,t) = \frac{\Gamma(3/4)}{\pi \sqrt{2}}\int_0^{\infty}e^{-w}w^{-3/4}\frac{e^{-\frac{x^2}{4\sqrt[4]{tw}}}}{\sqrt{4\pi\sqrt[4]{tw}}}\mathrm{d}w =\frac{\sqrt{2}\Gamma(3/4)}{\pi\sqrt[4]{t}}\int_{0}^{\infty}\frac{e^{-\frac{x^2}{2w}}}{\sqrt{2\pi w}}\dot e^{-\frac{w^4}{2^4t}}dw,
(\tilde{S}\tilde{T}^{p}\tilde{S})_{\alpha\beta}=-e^{-\frac{i\pi}{4}p}\sqrt{\frac{i}{8Kp}}\sum_{n=0}^{p-1}\sum_{\mu_{1,2}=\pm1}\mu_{1}\mu_{2}\exp\left[-\frac{i\pi}{2Kp}(\mu_{1}\alpha+\mu_{2}\beta+2Kn)^{2}\right]
S_{cl}=-\frac{1}{2}\int d^{d}\vec{x}\sqrt{|h|}\int d\bar{\eta}d\eta n^{\mu}\Phi^{0}(x,\eta) \partial_{\mu}\Phi^{0}(x,\eta)+\int d^{d+1}x\sqrt{|g|}\int d\eta d\bar{\eta}\frac{\lambda_{n}(\eta)}{n!}(\Phi^{0}(x,\eta))^{n}
\underset{l\rightarrow \infty}{\lim}\left\{\frac{n^{d}}{\left\vert\Lambda_{l}\right\vert}\mathfrak{F}_{t,k,q}^{(\omega )}\left(lb_{j}\right) \right\}=\mathbb{E}\left[ \mathfrak{F}_{t,k,q}^{(\omega)}\left( \left\{0\right\}\right) \right] \ .
{\frac{1}{1 6 \pi G_{\mathrm{\scriptsize i n d}}}}\simeq 8 . 2 \cdot 1 0^{- 4}\mu^{2},{\frac{2 \Lambda_{\mathrm{\scriptsize i n d}}}{1 6 \pi G_{\mathrm{\scriptsize i n d}}}}\simeq 4 . 2 \cdot 1 0^{- 8}\mu^{4},
\Gamma_{\underline{\alpha}\underline{\beta}}^{a}= \left( \begin{array}{c c}{\gamma_{\alpha \beta}^{a}\delta_{q}^{r}}{0}{0}{\gamma^{a \alpha \beta}\delta_{q^{\prime}}^{r^{\prime}}}\end{array}\right) \, , \quad a = 0 , 1 , . . . , p \, ,
\omega =\sum\limits_{i=1}^n\frac{A_i(a)}{z-a_i}d(z-a_i) +\sum\limits_{k=1}^n \gamma_k(a)da_k+\sum\limits_{l=1}^n\sum\limits_{k=1}^n\sum\limits_{m=1}^{r_l}\frac{\gamma_{m, k, l}(a)}{(z-a_l)^m}da_k,
\prod_{i=1}^{k}n_i - \sum_{i=1}^{k-2}n_i + k - 2 \leq (2 \cdot n_k + n_{k-1}- 3) \cdot \left( \frac{h_l(n_1, n_2, \dots, n_k)}{3}- k + 2 \right) + 1.
\frac{1}{2\pi}\int_{0}^{1}[-\frac{1}{(1-t)^{2}}-\frac{1}{(2-t)^{2}}-\frac{1}{t^{2}}-\frac{1}{(1+t)^{2}}+\pi^{2}csc^{2}(\pi t)]sin(\pi t)dt\approx0.100466.
v_i({\gamma_*})^*B_i^*{u_i}({\gamma_*}) + \sum\limits_{p = 1}^{i - 1}\left({{{\bar{\gamma_*}}_{p,i - p}}}v_i({\gamma_*})^*{u_p}({\gamma_*})\right) = \alpha_k^* v_i({\gamma_*})^*{v_i}({\gamma_*}),
M^{0,0}_{\frac{2m-i-k+1}{2},\frac{i+k-3}{2}}M^{\frac{k-3}{2},\frac{k-3}{2}}_{\frac{i+k-3}{2},\frac{2m-i}{2}}=\frac{k-1}{2}M^{\frac{2m-k+1}{2},\frac{2m-i-k+1}{2}}_{\frac{2m-i-k+1}{2},\frac{2m-i}{2}}
\Big| \frac{s_1-\rho}{s_0-\rho}\Big| = \Big(1 - \frac{|s_0-\rho|^2-|s_1-\rho|^2}{|s_0-\rho|^2}\Big)^{\frac 12}\le \exp\Big( -2\lambda \frac{(1-\rho)}{|s_0-\rho|^2}\Big) = \exp\Big( -2\lambda \Big(\frac{1}{s_0-\rho}\Big) +\frac{2\lambda^2}{|s_0-\rho|^2}\Big) .
\prod_{i=1}^{m_0(\mu)}(1-t^i)P_{\lambda}(x_1,x_2,x_3;t)= &(1-t)(1-t^2)(1-t^3)\Big( x_1^3 x_2 x_3 + (1-t) x_1^2 x_2^2 x_3 + x_1 x_2^3 x_3 + (1-t) x_1 x_2^2 x_3^2 + x_1 x_2 x_3^3 + (1-t) x_1^2 x_2 x_3^2 \Big).
\mathbb{P}_\varphi (X_{t+1}=x_{t+1}| \mu^{(T)}=m, X^{(t)}=x^{(t)}) & = \mathbb{P}_\varphi(\Psi(t,x_{t},m_t,u_t,\xi_{t+1})=x_{t+1}| \mu^{(T)}=m, X^{(t)}=x^{(t)}) = \sum_{\gamma \in \Gamma}\mathbb{P}_\varphi(\Psi(t,x_{t},m_t,\gamma,\xi_{t+1})=x_{t+1})\mathbb{P}_\varphi(u_t=\gamma | \mu^{(T)}=m, X^{(t)}=x^{(t)}) >0.
J=\left(\begin{array}{ccc}\sqrt{3}x_{4}-x_{0}-2\cos t \sqrt{3}x_{3}-\sqrt{3}\bar{x}_{2}\sqrt{3}\bar{x}_{3}-\sqrt{3}x_{4}-x_{0}-2\cos t -\sqrt{3}x_{1}\sqrt{3}x_{2}\sqrt{3}\bar{x}_{1}2x_{0}-2\cos t\end{array}\right) \ ,
\textrm{Tr}_{\mathbb{F}}\big[\mathbf{U}_t(P_\psi\otimes\Phi\Phi^\star)\mathbf{U}_t^\star\big]=\Phi^\star\widetilde{\mathbf{U}}_t \big(P_\psi\otimes{\mathbf{I}}^\otimes\big)\widetilde{\mathbf{U}}_t^\star\Phi,
\phi_{\ell(x)-\alpha x^2}(t) =\exp\left\{{\alpha x^2}\frac{(e^{it}-1)}{2x-(2x-1)e^{it}}-\alpha x^2 it\right\}=\exp\left\{{\alpha x^2}\left(\frac{(e^{it}-1)}{2x-(2x-1)e^{it}}- it\right)\right\}.
\frac{1}{N^{2}}\int_{G_{N}}\sum_{i=1}^{N}\left|\nabla_{i}f(x)\right|M_{\epsilon,i}(x)dx\le C\left[1+\left(\frac{\mathcal{D}_{N}(f)}{N}\right)^{\frac{3}{4}}\right]\epsilon^{\frac{1}{2}}.
\gamma_{[-d,f|_S,M]}(x)=\inf_{\begin{array}{c}s\in S, n\in\mathbb{N},\x_1=s, \{(x_i,y_i)\}_{i=1}^n\subset G(M) \end{array}}f(s)+\sum_{i=1}^{n-1}[d(x_i,y_{i+1})-d(x_{i+1},y_{i+1})]+d(x_n,x)
F(A,B,A',B'):= \Pi(A,B,A')-\Pi(A',B',A)-\Pi(B,A,B')+\Pi(B',A',B) +\Pi(A',0,A)-\Pi(B',0,B)+\Pi(B,A,0)-\Pi(B',A',0)-\Pi(0,A',B)+\Pi(0,B',A) +\Pi(0,0,B)-\Pi(0,0,A)-\Pi(0,0,B')+\Pi(0,0,A').
(2\phi_{n-1-k}+\frac{1}{2\phi_{n-1-k}})^2 =4\phi_{n-1-k}^2+\frac{1}{4\phi_{n-1-k}^2}+2 &=4\phi_{k}^2+\frac{1}{4\phi_{k}^2}+2 &=(2\phi_{k}+\frac{1}{2\phi_{k}})^2.
\frac{(m+1)!}{a_{1}!\cdots a_{i}!(m+1-s-r)!s!}=\binom{m+1}{m+1-a_{1}}\binom{m+1-a_{1}}{m+1-a_{1}-a_{2}}\cdots\binom{m+1-a_{1}-\cdots-a_{i}}{m+1-a_{1}-\cdots-a_{i}-a_{i+1}}
\left[ D(\ell^{n_2}),R(\ell^{n_2}) \right]=R\left(\ell^{2n_2}(2+\ell^{n_2}) \right), \left[ D(\ell^{n_2}),L(\ell^{n_2}) \right]=L\left(-\ell^{2n_2}\frac{2+\ell^{n_2}}{(1+\ell^{n_2})^2}\right)
-3F(2\pi/3)+\sum_{i=1}^{3}F(\theta_{i})\leq-\frac{2.44}{\pi^{2}}\lambda_{1,n}^{r,s}\leq-\frac{7.32}{2\pi^{2}}\lambda_{1,n}^{r,s}\sum_{i=1}^{3}\Big(\frac{\theta_{i}}{2\pi}-1/3\Big)^{2}.
g_{2+3\tau}(x)=\, (x-1)\sum_{\substack{0\le i_0\le 2\cr 0\le i_1\le 3}}(-1)^{i_0+i_1}\binom 2{i_0}\binom 3{i_1}\frac{y^{i_0+i_1}-1}{y-1}\cr \cdot\Bigl[\sum_{l=0}^{r-1}\binom{i_0+2+(l-3)k}2\binom{i_1+2-lk}3x^l\cr -\sum_{l=0}^2\binom{i_0+2+(l-3)k}2\binom{i_1+2-lk}2x^l\Bigr]-1.
g = -[G:H] + 1 + \frac{[G:H]}{2}\sum_{i=1}^{3}\frac{|I_i|-1}{|I_i|}- \frac{[N_G(H):H]}{2}\sum_{i=1}^{3}\frac{|N_G(I_i)|}{|N_H(I_i)|}\frac{|H\cap I_i|-1}{|H\cap I_i|}.
\Phi_3(x)=\sum_{j,m_j,k_j\in\mathcal{A}_3}\frac{1}{2\sqrt{\pi}}\left(\begin{array}{cc}F_{j,m_j,k_j}(r)\Xi^+_{j,m_j,k_j}(\theta,\phi) G_{j,m_j,k_j}(r)\Xi^-_{j,m_j,k_j}(\theta,\phi)\end{array}\right)
\gamma(M,K,d,\varepsilon):= \gamma -2\left(|g|_\alpha+\frac{{\rm diam}(D)^2|f|_\alpha+2{\rm diam}(D)|f|_\infty}{d}\right)\frac{{\sf diam}(D)}{K}&-\left(\frac{|\Delta g|_{\infty}}{2}+ |f|_\infty \right)|v|_{\infty}(\varepsilon) - (8|g|_\infty+ \frac{2}{d}{\sf diam}(D)^2|f|_\infty)e^{-\frac{\beta^2\varepsilon^2}{4{\sf diam}(D)^2}M}.
\delta_{H}^2 \simeq \frac{512\pi}{75}\, \frac{A}{q^3{M_P}^6}\, \frac{\phi^{2+q}}{\phi_{0}^q}\left [ 1+\xi (3 q -\frac{3\kappa^2\phi^2}{2}+\frac{\kappa^2 \phi^2}{q})\right ]^2
\begin{array}{r c c c r c c}{{{\bf a}}}&{{\mapsto}}&{{g \,{\bf b}}}&{{,}}&{{a}}&{{\mapsto}}&{{g \b ,}}\\{{{\bf b}}}&{{\mapsto}}&{{- g^{- 1}{\bf a}}}&{{,}}&{{b}}&{{\mapsto}}&{{- g^{- 1}a .}}\\ \end{array}
\check{R}(u)=P_{j=k}+\frac{y^{2}-q^{2k}}{1-y^{2}q^{2k}}P_{j=k-1}+ \ldots +\frac{y^{2}-q^{2k}}{1-y^{2}q^{2k}}\frac{y^{2}-q^{2(k-1)}}{1-y^{2}q^{2(k-1)}}\ldots \frac{y^{2}-q^{2}}{1-y^{2}q^{2}}P_{j=0}\;,
\gamma_{i}( \beta \sqrt{-\triangle}) = \int_0^\infty\!\!{\mathrm d}t \,\, g_{i}(t)\left[ \frac{2\pi}{\beta{\sqrt{-\triangle}}\, t}\, \frac{1}{{\mathrm sh}(2\pi t/(\beta{\sqrt{-\triangle}}))}- \frac{1}{t^2}\right],
Z'(\gamma) \, Z\Big( \gamma+\frac{2 \pi \kappa}{\log T}\Big)=-i\,\chi(\rho)^{1/2}\, \chi\Big(1-\rho -\frac{2\pi i \kappa}{\log T}\Big)^{1/2}\, \zeta'(1-\rho) \, \zeta\Big(\rho+\frac{2\pi i \kappa}{\log T}\Big).
\mathbf{y}_k=\mathbf{H}_k\sum_{i\in \mathcal{M}_k}a_i s_i\mathbf{p}_i+\mathbf{H}_k\sum\limits_{\substack{l=1 l \neq k}}^{K}\sum\limits_{j\in \mathcal{M}_l}a_j s_j\mathbf{p}_j+\mathbf{n}_k
{\rm ob}_\infty(x,y) = \inf_{R > R_0}\sup_{|zp| > R/2}\max \left\{\angle (\Uparrow_x^y, \uparrow_x^z), \angle (\Uparrow_y^x, \uparrow_y^z) \right\}- \pi/2 &\ge \inf_{R > R_0}\sup_{R/2 < |zp| \le R}\max \left\{\angle (\Uparrow_x^y, \uparrow_x^z), \angle (\Uparrow_y^x, \uparrow_y^z) \right\}- \pi/2 &\ge \epsilon_n(v).
\iota_{1\leq 2}^{(2)}[24]=(S^{(2)}_{0,2})^{12}\iota_{1\leq 2}^{(2)}(S^{(2)}_{0,1})^{-12}=\Sigma^8\iota_{1\leq 2}^{(2)}\Sigma^{-6}=\Sigma^2\iota_{1\leq 2}^{(2)}
{{\bf{H}}_{{\rm{T}},k}}= \sum\nolimits_{l \in{\mathcal L}_{{\rm T},k}}{{\alpha_{{\rm{T}},k,l}}{{\bf{a}}_{{\rm{S}},k}}\left({\Theta_{{\rm{T}},k,l}^{\rm{A}}}\right){\bf{a}}_{\rm{T}}^H\left({\Theta_{{\rm{T}},k,l}^{\rm{D}}}\right)},
Y_j^{(1,i)}= \begin{cases}0 \mbox{if}j=0 [X_j,X_i] \mbox{if}1\leq j\leq m \end{cases}\qquad\mbox{and}\tilde Y_j^{(1,i)}= \begin{cases}[X_0,X_i]+\frac{1}{2}\sum_{l=1}^m[X_l,[X_l,X_i]] \mbox{if}j=0 0 \mbox{otherwise}\end{cases}\;.
\log \mathbb{E}\left[ \exp\left(\lambda E_{k,j}\right)\middle| D_{k+1},W\right]= \frac{D_{k+1}\lambda}{n^2}(\sqrt{n}f_{1,k}+f_{2,k}+f_{3,k}) &-D_{k+1}\log\left(1-\frac{q_k}{p_k}\left(\exp\left(-(\sqrt{n}f_{1,k}+f_{3,k})\lambda/n^2\right)-1\right)\right).
\frac{1}{8\pi}\int_{\partial{\cal M}}Tr{\cal K}=\frac{1}{8\pi}\frac{\partial}{\partial{\bf n}}\, ({\rm Vol}\,\partial{\cal M}) = \frac{1}{8\pi}\frac{\partial}{\partial\, t}\, \left[\left(\frac{3}{2}\,t\right)^{1/3}\,{\cal V}\right]\ .
G_{1}\equiv \sum_{m = 0}^{\infty}m^{2 k + 4}\alpha_{m}^{- 2 k - 9}, \; G_{2}\equiv \sum_{m = 0}^{\infty}m^{2 k + 3}\alpha_{m}^{- 2 k - 9}, \; k = 0 , 1 , 2 . . . ,
2 \int_{0}^{\infty}a^{d}( w ) q ( w ) d w = 2 \int_{0}^{\infty}a^{d}( w ) \biggl [ 1 - 2 \epsilon \frac{{\cal H}^{2}}{a^{2}}( d - 2 ) ( d - 1 ) \biggr ] d w ,
({\bar p_j}\cup p_j^{\vee}) = ({\bar m_j}\cup m_j^{\vee}) \cup ({\bar x}^+ \cup (x^{+})^\vee) \setminus ({\bar x}^- \cup x \cup (x^{-})^\vee ) ({\bar m_j}\cup m_j^{\vee}) = ({\bar p_j}\cup p_j^{\vee}) \cup ({\bar x}^- \cup x \cup (x^{-})^\vee) \setminus ({\bar x}^+ \cup (x^{+})^\vee)\rlap{;}
\ell \, \neq \, 5 \quad \, S p D_{5 6}\left( E^{{\vec{\alpha}}_{\ell}}\right) = \left( \begin{matrix}{{A [{{\vec{\alpha}}_{\ell}}]}}&{{{\bf 0}}}\\{{{\bf 0}}}&{{- A^{T}[{{\vec{\alpha}}_{\ell}}]}}\\ \end{matrix}\right)
I_{2}\; : = \; \frac{1}{N}\left[ \ln ( 2 L ) \; + \; \ln \left( \frac{e}{2}\sqrt{\frac{N}{\pi}}\right) \; + \; \mathrm{K}_{0}\left( 2 L e \sqrt{\frac{N}{\pi}}\right) \; + \; \gamma \right] \; .
\beta_v(f_{1,v},f_{2,v}):=\frac{L(1,\eta_{T,v})\cdot L(1,\Pi_v,{\rm ad})}{\zeta_{v}(2)\cdot L(1/2,\Pi_v,\chi_v)}\int_{T(F_v)}\chi_v(t_v)\frac{\langle\pi_v(t_v)f_{1,v},f_{2,v}\rangle_v}{\langle f_{1,v},f_{2,v}\rangle_v}d^\times t_v
\mathrm{R}\left(c_0~\Phi_0(\nu;x)-U(x)c_0~\Phi_0(\nu;x+1)-V(x)\right) = \max_{c, \Phi(\nu;x)}\mathrm{R}\left(c~\Phi(\nu;x)-U(x)c~\Phi(\nu;x+1)-V(x)\right).
T_{\ \mu \rho \sigma}^{\mu}= - p^{2}T_{1}\Biggl [ \left( p_{\rho}p_{\sigma}- p^{2}g_{\rho \sigma}\right) \mp \frac{1}{4}\left( \varepsilon_{\rho \lambda}p^{\lambda}p_{\sigma}+ \varepsilon_{\sigma \lambda}p^{\lambda}p_{\rho}\right) \Biggl ] .
\phi\big(\exp(H+\sum_{j=1}^mp_j\log X_j)\big) = \ \phi\Big(\exp\big(H+r\sum_{j=1}^m\frac{p_j}{r}(\log X_j-\log C_j)+ \sum_{j=1}^mp_j\log C_j\big)\Big) = \ \phi\Big(\exp\big(L+r\sum_{j=1}^m\frac{p_j}{r}(\log X_j-\log C_j)\big)\Big) \leq \ \sum_{j=1}^m\frac{p_j}{r}\phi\big(\exp(L+r\log X_j-r\log C_j)\big),X_j=A_j,B_j.
\bar{\mathbf{a}}+\bar{\mathbf{b}}=(\bar{\alpha}_1+\bar{\beta}_1,\bar{\alpha}_2+\bar{\beta}_2,\bar{\alpha}_3+\bar{\beta}_3,\bar{\alpha}_4+b_1,a_1+b_2,\cdots,a_{2k-3}+b_{2k-2},a_{2k-2}+b_{2k-1},a_{2k-1}+\bar{\beta}_4).
\|(-P)^{s}u\|_{L^{2}(\mathbb{R}^{n})}^{2}= ((-P)^{2s}u,u)_{L^{2}(\mathbb{R}^{n})}\le \|(-P)^{2s}u\|_{H^{-2s}(\mathbb{R}^{n})}\|u\|_{H^{2s}(\mathbb{R}^{n})}\le C\|u\|_{H^{2s}(\mathbb{R}^{n})}^{2}.
\stackrel{*}{\chi^a_b}(z)=\Phi^a_b(0,z), \, \chi^a_b(z)=\Psi^a_b(z,0),\, \lambda^a_b(z)=\Phi^a_b(z,z^{-1}), \, \stackrel{*}{\lambda^a_b}(z)=\Psi^a_b(z,z^{-1}),
\sup_{x \in \mathbb{R}^d}\max_{r=1,\ldots,m}\frac{\|f^{(r)}(x)\|}{f(x)}\leq d^{m/2}\sup_{x \in \mathbb{R}^d}\max_{r=1,\ldots,m}\frac{q_r(\|x\|)}{(1+\|x\|^2/\rho)^r}=: A_{d,m,\rho}^{(2)},
\psi_{ij}(x^{-},0) = \frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}dk^{+}\left (b_{ij}(k^{+}){\rm e}^{-ik^{+}x^{-}}+b_{ji}^{\dag}(k^{+}){\rm e}^{ik^{+}x^{-}}\right ).
z_{1}={\frac{1}{2 \pi}}\left({\frac{m_{W}}{( e^{2}/ 4 \pi )}}\right)^{3 / 2}A ( \lambda / e^{2}) \exp \left[ -{\frac{m_{W}}{e^{2}/ 4 \pi}}C ( \lambda / e^{2}) \right] .
uv={T}_{u}v+{T}_{v}u+{R}(u,v)\quad{T}_{u}v=\sum_{j\in\mathbb{Z}}{S}_{j-1}u{\Delta}_jv \quad\mbox{and}\quad{R}(u,v)=\sum_{j\in\mathbb{Z}}{\Delta}_ju\tilde{\Delta}_jv.
\widetilde{F}_n = \sum\limits_{r\geq 0}\frac{1}{r!}\sum\limits_{\sum\limits_{i=0}^rk_i=n}\widetilde{P}_{k_{0}}\widetilde{F}_{k_{1}}\cdots \widetilde{F}_{k_{r}}{n \choose k_0,k_1,\hdots,k_r}(k_0)^r.
A(z) = B(z) + \big[C_\Sigma A_-\big(v_A v_B^{-1}- I\big)B_-^{-1}\big](z)B(z) \hphantom{A(z)}{}= B(z) + \big[C_\Sigma B_-\big(v_A v_B^{-1}- I\big)B_-^{-1}\big](z)B(z) \hphantom{A(z)=}{}+ \big[C_\Sigma (A_--B_-)\big(v_A v_B^{-1}- I\big)B_-^{-1}\big](z)B(z).
\|\sqrt\rho u\|_{L^\infty(0,T;L^2(\Omega_R))}\leq c, \left\|\frac{\rho-\varrho}{\varepsilon}\right\|_{L^\infty(0,T;L^2(\Omega_R))}\leq c, \nu^\frac{1}{2}\|\nabla u\|_{L^2(0,T;L^2(\Omega_R))}\leq c
\mathcal S^{\frac{1}{2}}(\partial\Omega)=\mathcal S^{\frac{1}{2}}_{\mu}(\partial\Omega):=\left\{f\in L^2(\partial\Omega): f=\sum_{j=1}^{\infty}\hat b_j\hat v_{j,\mu}{\rm\with\}\left(\sqrt{\lambda_j(\mu)}\hat b_j\right)_{j=1}^{\infty}\in l^2\right\}.
d s^{2}= ( 1 - ( \tau_{0}/ \tau )^{D - 2}) d \chi^{2}- \frac{d \tau^{2}}{1 - ( \tau_{0}/ \tau )^{D - 2}}+ \tau^{2}[ d \theta^{2}+ s i n h^{2}\theta d \Omega_{D - 2}^{2}]
S_\ell=\Big\{x\in \mathbb{R}^{d}: |x_{1}|\lesssim\lambda^{1-\frac{1}{d-\ell}}, |x_{2}|\lesssim\lambda^{1-\frac{2}{d-\ell}}, \cdots, |x_{d-\ell}|\lesssim\lambda^{1-\frac{d-\ell}{d-\ell}}=1,\cdots, |x_{d}|\lesssim 1\Big\}.
s_{b+1}- s_b = \left(\sum_{k=1}^2 (s_{b+1,k}- s_{b,k}) - \sum_{k=4}^\infty s_{b,k}\right) + \left(\sum_{k=3}^{\infty}s_{b+1,k}- s_{b,3}\right) > \frac{1}{b}- \frac{6\log b}{b(b-1)}- \frac{5}{b^2}.
cdf_{MJ}(\gamma)=\frac{2\sqrt{2}}{3}\frac{e^{-1/\theta}}{\theta K_{2}(\frac{1}{\theta})}\sqrt{\gamma-1}^{3}+\frac{1}{5\sqrt{2}}\frac{(5\theta-4)e^{-1/\theta}}{\theta^{2}K_{2}(\frac{1}{\theta})}\sqrt{\gamma-1}^{5}+O(\sqrt{\gamma-1}^{7})
\begin{aligned}\int_G \partial_j \zeta(0,\omega) \; \overline{f(0,\omega)}d\mu(\omega) = \int_{G^\wedge}2 \pi i \; y_j(\xi)\widehat{\zeta(0,\xi)}\; \overline{\widehat{f(0,\xi)}}d\nu(\xi) begin{align*}5pt] = - \int_{G^\wedge}\widehat{\zeta(0,\xi)}\; \overline{2 \pi i \; y_j(\xi) \widehat{f(0,\xi)}}d\nu(\xi).\end{aligned}
\mu_{n}~\exp \left[ \left( N+1\right) ~i~\theta_{n}\right] ~\dot{\theta}_{n}=-\eta_{n}~\exp \left[ \left( N+1\right) ~i~\theta_{n}\right]+\sum_{m=1}^{N}\left[ h_{m}~\exp \left( 2~m~i~\theta_{n}\right) \right] ~,
g_{n,k}(\tau):=\frac{|h_{n+1,k}(\tau)|}{|h_{n,k}(\tau)|}\frac{|P_{n,k-1}(\tau)|}{|P_{n+1,k-1}(\tau)|}\,\frac{|P_{n,k+1}(\tau)|}{|P_{n+1,k+1}(\tau)|}.
\| |D|^{-2}P_k [A_{j,k_1}, \partial_0 A_{j,k_2}]\|_{L^1 L^\infty}^2 \lesssim \ 2^{-4k}2^{2k_1}\left( \sum_{\mathcal C_k}\| P_{\mathcal C_k}A_{j,k_1}\|_{L^2 L^\infty}^2 \right) \left( \sum_{\mathcal C_k}\| P_{\mathcal C_k}A_{j,k_2}\|_{L^2 L^\infty}^2 \right) \lesssim \ \| A_{j,k_1}\|_{S^1}^2 \| A_{j,k_2}\|_{S^1}^2
y_n =\frac{1}{n+1}\sum_{k=1}^{n}\binom{n+k}{k}\frac{k!}{n!}B_{n,k}(0,\dots,(q+1)!c_{q+1},(q+2)!c_{q+2},\dots) =\frac{1}{n+1}\sum_{1\le k< n/q}\binom{n+k}{k}\frac{k!}{(n-qk)!}B_{n-qk,k}(1!c_{q+1},2!c_{q+2}\dots).
\begin{aligned}U^{j}\overset{\ast}{\rightharpoonup}U^{0}~\mbox{in}~ L^{\infty}(0,T_{0};H^{s}), v^{j}\rightharpoonup v^{0}~\mbox{in}~ L^{2}(0,T_{0};H^{s+1}), U_{t}^{j}\rightharpoonup U_{1}~\mbox{in}~ L^{2}(0,T_{0};H^{s-1}).\end{aligned}
C_2^{S^{\star}}=\begin{cases}\,\,\int_{-\beta^{\star}\eta^{\star}}^{\eta^{\star}-\beta^{\star}\eta^{\star}}{\gamma^2 p(\gamma)d\gamma}, \eta^{\star}> 0, \,\,\int_{\eta^{\star}-\beta^{\star}\eta^{\star}}^{-\beta^{\star}\eta^{\star}}{\gamma^2 p(\gamma)d\gamma}, \eta^{\star}< 0.\end{cases}
\sup\limits_{x\in D}\left | u(x)-\mathbb{U}_M^{N}(\cdot,x)\right| =|u-\widetilde{u}_M^N|_\infty\leq |u-u_M^N|_\infty +|u_M^N-\widehat{u}_M^N|+|\widehat{u}_M^N-\widetilde{u}_M^N| \leq |u-u_M^N|_\infty+\epsilon_g+\frac{M{\rm diam}(D)^2\epsilon_f}{d}+\gamma/6 \leq |u-u_M^N|_\infty+\gamma/2,
\partial_\mu E = \left\{x\in G:\, \min\left\{\frac{\mu(E\cap B_d(x,r))}{\mu(B_d(x,r))}, \frac{\mu(E^c \cap B_d(x,r))}{\mu(B_d(x,r))}\right\}>\epsilon \, \,\forall r>0\right\},
\begin{array}{c}{\displaystyle{R e s \zeta_{\nu}( s , x ) |_{s = 1}= \frac{1}{\pi},}}\\{\displaystyle{R e s \zeta_{\nu}( s , x ) |_{s = 0}= 0 ,}}\\{\displaystyle{R e s \zeta_{\nu}( s , x ) |_{s = - 1}= \frac{1 - 4{\nu}^{2}}{8 \pi},}}\\ \end{array}
\sum_{i = 0}^{s}\frac{(q^{-s})_{i}}{(q)_{i}}q^{i}\sum_{i_{1}= 0}^{n_{1}}\dotsm \sum_{i_{r}= 0}^{n_{r}}\sum_{j = 0}^{s - 1 - I_{r}}\frac{(q^{1 - s})_{I_{r}+ j}(a q^{-i})_{j}(b_{1}q^{i})_{i_{1}}\dotsm (b_{r}q^{i})_{i_{r}}}{(c q^{i})_{1 - s + I_{r}+ j}(q)_{j}(q)_{i_{1}}\dotsm (q)_{i_{r}}}(x q^{i})^{j}x_{1}^{i_{1}}\dotsm x_{r}^{i_{r}}= 0,
r(0, 2, 2, 2; n)=\frac1{(q)_n}\frac{\left(-q^{2+n}; q^2\right)_{\frac{n-1}{2}}}{\left(-q^2; q^2\right)_{\frac{n-1}{2}}}=\frac{\left(-q; q^2\right)_n}{(q)_n\left(-q^2; q^2\right)_{\frac{n-1}{2}}\left(-q; q^2\right)_{\frac{n+1}{2}}}=\frac{\left(-q; q^2\right)_n}{\left(q^2; q^2\right)_n}.
\frac{dU^{\nu}}{d\tau}+\tilde{\Gamma}_{\alpha\beta}^{\mu}U^{\alpha}U^{\beta}+2\tilde{\Gamma}_{5\alpha}^{\mu}U^{\alpha}U^{5}+\tilde{\Gamma}_{55}^{\mu}(U^{5})^{2}+U^{\mu}\frac{d}{d\tau}ln\frac{cd\tau}{ds}=0.
G^{\rm (TM)}_{(1+2)}(\tau)= 360\tau^5(e^{\Omega}-1)\int_0^{\infty}d\Omega' \int_0^{\Omega'}dK\,\frac{K\,\left[K^2-\Omega'\left(\Omega+\Omega'\right)\right]^2}{\sqrt{(\Omega'^2-K^2)\left[(\Omega+\Omega')^2-K^2\right]}}\,\frac{e^{\Omega'}}{\left(e^{\Omega'}-1\right)\left(e^{\Omega+\Omega'}-1\right)},
\Delta f = \frac{1}{\sqrt \rho}\left( \frac{1}{\epsilon}\nabla^{a}\frac{\partial^{L}}{\partial \theta^{a}}+ \frac{1}{\bar{\epsilon}}{\overline{{\nabla^{a}}}}\frac{\partial^{L}}{\partial{\bar{\theta}}^{a}}\right) ( \sqrt \rho f ) ,
\|\Theta[n+1]\|_\mathrm{HS}\le g N^{3\beta}\|\upsilon\|_{L^\infty}\|\phi_{n+1}\|_{L^2}^2 + \frac{1}{N}\|(\Theta^\mathrm{pair}[n+1])\|_\mathrm{HS}&\le g\|\upsilon\|_{L^\infty}\Big\{N^{3\beta}+ \frac{C^2}{2c}\Big\},
\Psi^{n,4}_s = 2\sigma(X_s) \left( \int_0^s \psi_{n,1}(u,s) dW_u \right)\left( \int^s_{\eta_n(s)}\left(s-\eta_n(u)\right)^{\alpha}\sigma(X_{\eta_n(u)})\, dW_u \right),