harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\substack{nm\leqslant N (n,m)=1}}\frac{r(n)^2r(m)\omega(n)\omega'(m)}{\sqrt{m}}&\qquad=\big(1+O\big(N^{-1/(\log\log N)^3}\big)\big)\prod_p\left(1+r(p)^2\omega(p)+\frac{r(p)}{\sqrt{p}}\omega'(p)\right),
b(u,v,w) = (u \cdot \nabla v, w) + \frac{1}{2}((\nabla \cdot u)v, w), b(u,v,w) \leq C_{1}\| \nabla u \| \| \nabla v \| \| \nabla w \|, b(u,v,w) \leq C_{2}\sqrt{\| u \| \|\nabla u\|}\| \nabla v \| \| \nabla w \|.
\sum_{\gamma_{\chi}}\frac{\frac{1}{2}}{\frac{1}{4}+\gamma_{\chi}^2}\leq \left(\dfrac{1}{2}+\dfrac{2}{\log q -2}\right)\log{\dfrac{q}{\pi}}+2\left(1-\dfrac{2}{\log q}\right)^{-2}\log\log q\leq \dfrac{1}{2}\log{\dfrac{q}{\pi}}+ 2.6\log\log q.
p_{2}= \int_{0}^{\infty}\, \frac{d \omega}{2 \pi}\int_{0}^{\infty}\, \frac{d \omega^{\prime \prime}}{2 \pi}\vert \langle o u t \, \omega \omega^{\prime \prime}\vert i n \rangle \vert^{2}= \int_{0}^{\infty}\, \frac{d \omega}{2 \pi}p_{2 \omega}.
S_{\alpha}\geq \sum\limits_{|\gamma|=N}\frac{c_{\gamma+\alpha}^{2}}{c_{\gamma}^2}=\sum\limits_{k=0}^{N}\frac{c_{\alpha+(k,N-k)}^{2}}{c_{(k,N-k)}^2}=\sum\limits_{k=0}^{N}\frac{c_{(k+\alpha_1,N-k+\alpha_2)}^{2}}{c_{(k,N-k)}^2}=\sum\limits_{k=0}^{N}\frac{N-k+1}{N-k+\alpha_2+1}\int\limits_{0}^{1}\Phi_{2k+1,2(N-k)+2}(r)g_{\alpha}(r)dr.
\lim_{n\rightarrow \infty}\left( (\eta_0,w_0), S^*(\tau_n)(\varphi_{\tau_n},\psi_{\tau_n})\right)_{\overline{X}_2}= \left((\eta_0,w_0),S^*(\tau) (\varphi_{\tau},\psi_{\tau})\right)_{\overline{X}_2}.
S_3 \ll \left(\sum_{j=\lceil \log_{b}{N}\rceil}^{\infty}b^{j\left(r-\frac{1}{p}+1\right)q}\left( b^j\left(\frac{1}{b^{2j}}\right)^p\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}=\left(\sum_{j=\lceil \log_{b}{N}\rceil}^{\infty}b^{jq(r-1)}\right)^{\frac{1}{q}}&\ll b^{\log_{b}{N}(r-1)}=N^{r-1},
(\hat{B}^{(\alpha)})^{-1}f(x^{1/\alpha}) = \sum_{r=0}^{\infty}f_{r}(c_{z}x^{1/\alpha})^{r}\frac{1}{\Gamma(1+z)}\Big\vert_{z=0}= f(c_z x^{1/\alpha}) \frac{1}{\Gamma(1+z)}\Big\vert_{z=0}.
\overline{r}^{(k)}(x) :=\begin{pmatrix}\left( \frac{\mathrm{d}}{\mathrm{d}x}\right)^k p_0 (x) \left( \frac{\mathrm{d}}{\mathrm{d}x}\right)^k p_1 (x) \cdots \left( \frac{\mathrm{d}}{\mathrm{d}x}\right)^k p_{n-1}(x)\end{pmatrix}^T,\overline{r}_j^{(k)}:= \overline{r}^{(k)}(x_j),
\begin{minipage}{14cm}\begin{itemize}\item[(i)] $W_0(RF) = W_0(F).$\item[(ii)] $W_0(F)\to +\infty \mbox{as}\det F\to 0$.\item[(iii)] $W_0(\mbox{Id}) = 0$.\item[(iv)] $W_0(F) \geq c~\mbox{dist}^2(F, SO(3))$.\end{itemize}\end{minipage}
T_{j_{1}}\circ\dots\circ T_{j_{s}}(I)\cap T_{i_{1}}\circ\dots \circ T_{i_{\ell}}(I)=\emptyset, \quad\mbox{and}T_{j_{1}}\circ\dots\circ T_{j_{s}}(I) \cup T_{i_{1}}\circ\dots \circ T_{i_{\ell}}(I) \subset J.
\overline{Y}=\frac{1}{4}\cdot\frac{K_{I}[X]+2K_{II}[X]^{2}+3K_{III}[X]^{3}+4K_{IV}[X]^{4}}{1+K_{I}[X]+K_{II}[X]^{2}+K_{III}[X]^{3}+K_{IV}[X]^{4}}
{\rm grad}\,F(X) = \left( I_m - \frac{1}{2}XX^T \right) \nabla F(X) - \frac{1}{2}X\nabla F(X)^TX &= \left( I_m - \frac{1}{2}XX^T \right) \left( \nabla F(X) - X\nabla F(X)^TX \right).
\Sigma_{\psi}^{a}= - i \int \frac{d^{D}k}{( 2 \pi )^{D}}\frac{( \not \! \! k + \not \! \! p + M ) \cos^{2}( k \wedge p )}{[ ( k + p )^{2}- M^{2}] [ k^{2}- 4 M^{2}]}\Delta_{\lambda}( k )
|\int_{0}^{\infty}d\xi \chi_{\leq 1}(r \xi) \sin(t\xi) J_{1}(r \xi) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}| \leq C r \int_{0}^{\infty}d\xi \chi_{\leq 1}(r \xi) \xi \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}&\leq \frac{C}{r \log^{b-1}(r)}, r \geq \frac{t}{2}> 4
h_{H}(f^{n}(P))= -\sum_{i=0}^{n-1}\delta^{n-1-i}c_{1}h_{Z_{1}}(p^{-1}(f^{i}(P)))+\sum_{i=0}^{n-1}\delta^{n-1-i}h_{E}(p^{-1}(f^{i}(P))) &+\sum_{i=0}^{n-1}\delta^{n-1-i}c_{1}h_{E_{1}'}(f^{i}(P))+\sum_{i=0}^{n-1}\delta^{n-i}h_{N}(f^{i}(P))+\delta^{n}h_{H}(P).
\tilde{\mathcal Q}_A:=\sum_{k\in A}\omega^{(1)}(\xi)\cdot L(k)|z'_k|^2+\sum_{\ell\in E_b(A)}c_q(\ell)z'_h \bar z'_k + \sum_{\ell\in E_r(A)}c_q(\ell)[ z'_h z'_k + \bar z'_h\bar z'_k]
\sum_{n =1}^{\infty}\frac{(b/a)_n a^n}{(1- c q^n) (b)_n}= \sum_{m=0}^{\infty}\frac{(b/c)_mc^m}{(b)_m}\left(\frac{aq^m}{1-aq^m}-\frac{bq^m}{1-bq^m}\right)
\begin{array}{l}{\rm Tr}~ D = \lambda_1^2+\lambda_2^2+\lambda_3^2 \frac{1}{2}({\rm Tr}~ D)^2 - \frac{1}{2}{\rm Tr}~ D^2 =\lambda_1^2\lambda_2^2+\lambda_2^2\lambda_3^2+\lambda_1^2\lambda_3^2 \det D = \lambda_1^2 \lambda_2^2 \lambda_3^2.\end{array}
(Wf)_{i_1 \dots i_m j_1 \dots j_m}= 2^m\sigma(i_1 \dots i_m) \sigma(j_1 \dots j_m) (Rf)_{i_1 j_1 \dots i_m j_m}(Rf)_{i_1 j_1 \dots i_m j_m}= \frac{1}{(m+1)}\alpha(i_1 j_1) \dots \alpha(i_m j_m) (Wf)_{i_1 \dots i_m j_1 \dots j_m}.
A_{2\mu+2}= \frac{9^{2\mu+2}(6\mu+\nu)}{3\cdot(\mu+1)!}\frac{d^\mu}{dx^\mu}\left.\left\{\frac{(1+x/3)^{5-\nu}}{(x-1)^{2\mu+2}(x-9)^{2\mu+2}}\right\}\right|_{x=0}.
m={\textstyle\frac{1}{2}}\left(\begin{array}{cc}m^{1}m^{2}+m^{3}& m^{2}-m^{3}-m_{1}\end{array}\right)={\cal Q}= \left(\begin{array}{cc}\frac{\delta}{2}r \beta q \gamma p -\frac{\delta}{2}r \end{array}\right)\, .
\frac{\partial}{\partial r}\varphi(t,r) =\lim_{\xi\to 0}\frac{f(t,y(r)+\xi G(x)h)-f(t,y(r))}{\xi}&=\lim_{\xi\to 0}\frac{f(t,y(r)+\xi G(y(r))G(y(r))^{-1}G(x)h)-f(t,y(r))}{\xi}&=\nabla^G f(t,y(r))(G(y(r))^{-1}G(x)h).
+ \frac{1}{2}\partial_{+}f_{2}\partial_{-}f_{1}+ \frac{5}{3}e^{- 2 \rho}\Omega^{2 / 3}( e^{- \sqrt{2}f_{1}}F_{1 + -}^{2}+ e^{- \sqrt{2}f_{1}}F_{2 + -}^{2}+ e^{\sqrt{2}( f_{1}+ f_{2})}F_{+ -}^{2}) = 0
|f^2_{\alpha a}|\lesssim \sum\limits_{\tiny\begin{matrix}b+c=a \beta+\gamma=\alpha\end{matrix}}|\nabla^\perp_j H^{(\beta,b)}\cdot\omega^\perp\nabla_j V^{(\gamma,c)}| &+\sum\limits_{\tiny\begin{matrix}b+c=a \beta+\gamma=\alpha\end{matrix}}|(\nabla^\perp_j H^{(\beta,b)}\cdot\omega+\nabla^\perp_j V^{(\beta,b)})\nabla_j V^{(\gamma,c)}\omega|,
\kappa_m\left(\sum_{i=1}^n c_i Z_{\gamma_1,\gamma_2}(t_i)\right) \sim\left(\frac{\gamma_1+\gamma_2+3/2}{\gamma_2+1}\right)^{m/2}(m-1)! \left[\sum_{i,j=1}^n c_ic_j~ \frac{1}{2}\left( |t_i| +|t_j| -|t_i-t_j| \right)\right]^{m/2}&\rightarrow \rho^{m/2}(m-1)! \left[\sum_{i,j=1}^n c_ic_j~ \frac{1}{2}\left( |t_i| +|t_j| -|t_i-t_j| \right)\right]^{m/2},
\sum\limits_{\sigma \in \mathcal{A}_3}\omega([\eta_{\sigma(1)}, \eta_{\sigma(2)}] , \eta_{\sigma(3)}) = \kappa ([\eta_1 , \eta_2 ], D\eta_3) + \kappa([\eta_2 , \eta_3] , D\eta_1) &+ \kappa ([\eta_3 , \eta_1] , D\eta_2).
A_{w,1}f(v_{\ell,\ell,\ell})=q^3f(v_{\ell-1,\ell-1,\ell-1})+(q^2+q+1)f(v_{\ell+1,\ell,\ell}) &A_{w,2}f(v_{\ell,\ell,\ell})=(q^4+q^3+q^2)f(v_{\ell,\ell-1,\ell-1})+(q^2+q+1)f(v_{\ell+1,\ell+1,\ell}) &A_{w,3}f(v_{\ell,\ell,\ell})=(q^3+q^2+q)f(v_{\ell,\ell,\ell-1})+f(v_{\ell+1,\ell+1,\ell+1}).
\|(z_1,z_2)\|_{\Omega^*}=\sup_{(w_1,w_2)\in\Omega}|z_1\overline{w_1}+z_2\overline{w_2}| &=\sup_{(w_1,w_2)\in\Omega}||z_1|\overline{e^{-iz_1}w_1}+|z_2|\overline{e^{-iz_2}w_2}|.
\tilde{g}_{\mu\nu}= \left (\begin{array}{lc}e^{U+a}e^{U+a/2}\omega_j begin{align*}9pt]e^{U+a/2}\omega_i e^U\omega_i \omega_j + e^{-U}\tilde{\gamma}_{ij}\end{array}\right) \ , \hspace{30pt}B_{\mu\nu}=\left (\begin{array}{lc}0 e^{a/2}b_j begin{align*}9pt] - e^{a/2}b_i b_{ij}+\omega_{[i}b_{j]}\end{array}\right) \ .
\int_{[0,t]^n}A^2_{t,x}(u)du\le \frac{C^n}{n!}\sum_{m=0}^n \binom{n}{m}\frac{t^{m(\alpha_2-1)+n}}{\Gamma(m(\alpha_2-1)+n+1)}\le \frac{(2C)^n}{n!}\frac{T^n}{\Gamma(n\alpha_2+1)}.
L^t_{\theta}(u_1[f_1]\cdots u_n[f_n] \cdot v )= u_1[f_1]\cdots u_n[f_n]\cdot L^t_\theta(v) + \sum_{i=1}^n \left(u_1[f_1]\cdots u_i[\theta(f_i)]\cdots u_n[f_n]\cdot v \right)
\tau ( r_{i}) = \frac{1}{2 \kappa^{2}}\frac{H^{\prime}}{H}= \frac{1}{2 \kappa^{2}}\frac{( p - 7 ) + \frac{Q^{2}}{( 3 - p ) \rho r_{i}^{3 - p}}}{1 - \frac{Q^{2}}{\rho ( 3 - p ) ( 1 0 - 2 p ) r_{i}^{3 - p}}}\frac{1}{r_{i}}\, s i m \, \frac{1}{2 \kappa^{2}}\Bigg ( \frac{p - 7}{r_{i}}+ \frac{Q^{2}}{( 1 0 - 2 p ) \rho}\frac{1}{r_{i}^{4 - p}}\Bigg )
x\cdot x = 0, y\cdot x = (1-\alpha^{-1})z, z\cdot x = (\alpha+1)x, x\cdot y = -(1-\alpha^{-1})z, y\cdot y = 0, z\cdot y = (\alpha-1)y, x\cdot z = \alpha x, y\cdot z = \alpha z, z\cdot z = \alpha z,
\sup_{\lambda/n\leq s<t}\frac{n^{\eta}\left\vert \widetilde{\beta}_{n}\left(s;t\right) -\widetilde{B}_{n}\left( s\right) \right\vert}{s^{1/2-\eta}}=O_{\mathbb{P}}\left( 1\right) =\sup_{\lambda/n\leq s<t}\frac{n^{\nu}\left\vert \widetilde{\alpha}_{n}\left( s;t\right) -\widetilde{B}_{n}\left( s\right) \right\vert}{s^{1/2-\nu}}.
\sum_{n=0}^\infty q_n^b(t,x,y)\geq \frac{1}{2}p_0(t,x,y) \hbox{for}0<t \leq 1\wedge( A_0/\|b\|_\infty)^{2/(2 -\beta)}\hbox{and}|x-y|\leq 3 t^{1/2}.
\sum \limits_{m=1}^k \int_\Omega \left(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}\right)u_m\,dx \geq \sum_{m=1}^k \frac{q}{q+1}\int_\Omega \left(|u_m|^{q+1}-|u_{m-1}|^{q+1}\right)\,dx begin{align*}2mm] = \frac{q}{q+1}\int_\Omega \left(|u_k|^{q+1}-|u_{0}|^{q+1}\right) \,dx.
\Delta w=w_{jj}=\frac{2u_{ij}^2}{u^2}+\frac{2u_iu_{ijj}}{u^2}-\frac{4u_iu_{ij}u_j}{u^3}-\frac{4u_iu_{ij}u_j}{u^3}-\frac{2u_i^2u_{jj}}{u^3}+\frac{6u_i^2u_j^2}{u^4}.
= \sum_{j=1}^{|n\oplus m/2|}\sum_{j^\prime = 1}^{|n|}c_{2j-1}c_{2j^\prime - 1}\left ( \frac{1}{2}\{\gamma_{2j-1}, \gamma_{2j^\prime - 1}\}+ \frac{1}{2}[ \gamma_{2j - 1}, \gamma_{2j^\prime - 1}] \right )
{x_u(z)}= \frac{1}{N}\sum\limits_{n=1}^N{\frac{{{q}}}{{\frac{1}{N}\sum\nolimits_{i=1}^U{\frac{{{q}}}{{1 +{x_i(z)}}}}+ \sum\nolimits_{k=1}^K{{p_{k,n}}g_{k,n}}+{\sigma^2}}}}.
\left( T_{1}+T_{2}\right)^{2m-1}\left( T_{1}-T_{2}\right) h_{2m}^{-}& =2\left( N\kappa+m+1\right) \left( T_{1}+T_{2}\right)^{2m-1}h_{2m-1}^{+}& =2^{m+1}\left( N\kappa+m+1\right) \left( N\kappa-\kappa+1\right)_{m-1}\left( N\kappa+1\right)_{m}& =2^{m+1}\left( N\kappa-\kappa+1\right)_{m-1}\left( N\kappa+1\right)_{m+1},
(T\widetilde{f}-Tf)(2y-T\widetilde{f}-Tf) \leq 2|T\widetilde{f}-Tf||y| + 2B_n|T\widetilde{f}-Tf| & = 2|T\widetilde{f}-Tf|(|y|-B_n) + 4B_n|T\widetilde{f}-Tf| & \leq 4B_n(|y|-B_n) + 4B_n|\widetilde{f}-f|.
\log \alpha+\log (1-\alpha)+\log (-\chi')+\log\left(1+\sum_{k=0}^\infty\frac{b_k'}{(-\chi') s^{k+1}}\right)+\sum_{k=0}^\infty \frac{b_k}{s^k}=-\sum_{k=0}^\infty \frac{\omega_{k+1}}{s^{k}}.
\left[ \Omega_{1},\Omega_{1}\right]^{*}=\frac{1}{3}f_{\;a\left[ d\right.}^{e}f_{\;\left. bc\right]}^{a}\int d^{3}x\left( \epsilon^{0ijk}A_{i}^{d}A_{j}^{b}A_{k}^{c}\eta_{e}\right) .
\psi''_{\gamma}(\tau)=a(t(N))\cdot \psi''_{\gamma_{t(N)}}(a(t(N))\cdot \tau)-\sum_{i=0}^{N-1}a(t(i+1))\,\varphi''_{p(i),t(i+1)-t(i)}(a(t(i+1))\cdot \tau).
\Theta_{n+1}= \Theta_n + \frac{\sigma_n}{\gamma}\int_{t_n}^{t_{n+1}}(AX_t^\dagger)^{\rm T}{\rm d}X_t^{\dagger}- \frac{1}{2}K_n A X_{t_n}^\dagger \left( \Theta_n + \pi_n[\theta] \right) \Delta t ,
\delta \eta_{( p ) \mu_{1}\ldots \mu_{p - 2 k - 1}}^{*}= - \left( \left( p - 2 k \right) \partial^{\mu}A_{( p ) \mu \mu_{1}\ldots \mu_{p - 2 k - 1}}^{*}+ \partial_{\left[ \mu_{1}\right.}A_{( p ) \left. \mu_{2}\ldots \mu_{p - 2 k - 1}\right]}^{*}\right) .
\lVert a(k_1,\dots,k_n)\psi_{g,\eta}\lVert \leq \sum_{i=1}^{n}\frac{\omega(k_i)}{\omega(k_1)+\cdots+\omega(k_n)}g^{n}\frac{\lvert v(k_1)\lvert\dots\lvert v(k_n)\lvert}{\omega(k_1)\dots\omega(k_n)}&=g^{n}\frac{\lvert v(k_1)\lvert\dots \lvert v(k_n)\lvert}{\omega(k_1)\dots\omega(k_n)}.
h(x) =\frac{\left[\Phi\left(\frac{x-\mu}{\sigma}\right)\right]^{\alpha-1}\,\left[1-\Phi\left(\frac{x-\mu}{\sigma}\right)\right]^{\beta-1}\,\phi\left(\frac{x-\mu}{\sigma}\right)}{\sigma\,\mathrm{B}(\alpha,\beta)\,\left[1-I_{\Phi(\frac{x-\mu}{\sigma})}(\alpha,\beta)\right]},
\bar{\cal L}_c =-\frac{1-\rho^2}4\left(\bar f_{\mu\nu}^i\right)^2 -\frac14\left(\bar B_{\mu\nu}^i\right)^2 +\frac i2\rho\bar B_{\mu\nu}^i{}^\ast\!\bar f^{\mu\nu i}-\frac1{2\beta}\left(\partial^\mu\bar A_\mu^i\right)^2 .
T(\lambda)=\begin{pmatrix}1 \frac{i}{4}(\eta_1^2+2\tau_m\eta_1)r^{3/2}0 1\end{pmatrix}r^{-\frac{\sigma_3}{4}}\Psi(r\lambda+r\eta_1;x=r\tau_m,r\vec{\eta},\vec{s}).
\! \! \! \! \! \!\! \! \! \! \! \! \mathcal{E}_{UL}(\alpha,\theta)= \left( B_{\rm{u}}+\alpha B \right) \log_2 (1+\theta) \exp \left(- U(\theta) - \frac{\pi^2 \lambda}{2}\sqrt{\frac{\theta}{\rho}}\sqrt{P_{\rm{d}}|\mathcal{\tilde{C}}_{\rm{u}}(\alpha)|^2}-\theta \frac{{\beta}P_{\rm{d}}|\mathcal{\tilde{C}}_{\rm{u}}(\alpha)|^2}{\rho}\right),
\begin{aligned}\Delta_n^{s,t}\leq K_1 K_2 \sum_{k=1}^{\infty}\{S_{1,2k-1}(n,n-s,t-1) + S_{2,2k-1}(n,n-s,t-1)\}& \qquad+K_1 K_2 \sum_{k=1}^{\infty}\{S_{1,2k}(n,s-1,t-1) + S_{2,2k}(n,s-1,t-1)\}.\end{aligned}
\begin{array}{l}(a^1+b^1)(x_1^0)^2+[(a^0+a^1)+\nu(a^1+b^1)](x_1^1)^2+b^1x_1^0x_1^1+m_1^1x_1^0+(m_1^0+m_1^1)x_1^1 +\ldots+(a^1+b^1)(x_{r-1}^0)^2+[(a^0+a^1)+\nu(a^1+b^1)](x_{r-1}^1)^2+b^1x_{r-1}^0x_{r-1}^1 +m_{r-1}^1x_{r-1}^0+(m_{r-1}^0+m_{r-1}^1)x_{r-1}^1+d^1=0. \end{array}
\left[ \prod_{i \in S}B_{i}\right] = \sum_{P \in \mathbf{P}( S )}( - 1 )^{( \mathbf{N}( P ) - 1 )}( \mathbf{N}( P ) - 1 ) ! \prod_{i \in P}\left\langle \prod_{j \in i}B_{j}\right\rangle
\begin{cases}\mathbf{S}= \{(x_1, x_2, x_3) \in \mathbb{R}^3 \mid x_1^2 + x_2^2 + x_3^2 = 2 \}, \mathbf{n}= (0, 1, 1), \mathbf{P}= \{(x_1, x_2, x_3) \in \mathbb{R}^3 \mid x_2 + x_3 = 1 \},\end{cases}
\delta_{B}( f^{( 0 )}u^{4}) + \delta_{F}( f^{( 2 )}u^{3}\left[ \psi^{2}\right] ) + u^{i}u^{2}\epsilon \dot{N}^{i ( 0 )}\psi + \frac{i}{2}u^{3}\psi \dot{M}^{( 0 )}\epsilon = 0 ~ ,
\alpha\leq \inf_{x}f\left(x\right)+\psi\left(x\right)=\begin{cases}c-10 a=3,b=3,c\leq-9 -\frac{\left(b-3\right)^{2}}{4\left(3-a\right)}-10+c 2<a<3,c\leq-\frac{\left(1-b\right)^{2}}{4\left(a-2\right)}-8 -\infty a>3.\end{cases}
\sum_{\ell=0}^{\infty}c_q(\ell d)\sum_{k=0}^{d-1}\frac{c_q(\ell d+k)}{c_q(\ell d)}=\frac{(q;q^2)_\infty(q^6;q^6)_\infty(aq^3;q^6)_\infty(q^3/a;q^6)_\infty}{(1-q)\,(q^2;q^2)_\infty(q^3;q^6)_\infty(aq^6;q^6)_\infty(q^6/a;q^6)_\infty}.
\begin{array}{lll}(\pi_J)_! (c_{\omega'}((\prod_{i=1}^{k}S^{2j_i+1})\times_{T^k}TM)) = f_J^*(\pi_!c_{\omega'}(ET^k\times_{T^k}TM)) &= f_J^*((\sum n_{J'}^{\omega'})x^{J'}) &= \sum n_{J'}^{\omega'}x^{J'}, \end{array}
S_{h}= \int\!\! d^6x \sqrt{-{\cal G}_6}e^{-2\phi^{(h)}_{6}}\left \{R + 4 \big( \nabla \phi^{(h)}_6 \big)^2 -{{1}\over{12}}(H^{(h)}_{6})^2-{{1}\over{4}}{\cal F}_{6}^2 \right \}
\hat S(t,x) :=Im\left(\sqrt{\frac{A^{-1}(r(t))\left(x-r(t)\right)\cdot\gamma'(s(t))}{c_{A,\gamma'}(t)\sqrt{1-|c_{A,\gamma'}(t)|^2|\dot s(t)|^2}}+i\frac{\left(x-r(t)\right)\cdot n(s(t))}{c_{A,n}(t)}}\right)\,,
L^{(a)}(z)\begin{pmatrix}f_+ f_-\end{pmatrix}=\begin{pmatrix}L^{(a)}_{11}(z) L^{(a)}_{12}(z) L^{(a)}_{21}(z) L^{(a)}_{22}(z)\end{pmatrix}\begin{pmatrix}f_+ f_-\end{pmatrix}=\lambda \begin{pmatrix}f_+ f_-\end{pmatrix}.
\lim_{n\to+\infty}\sum_{i=0}^{n-1}\int_{\{z\in M:-\Psi(z)\in I_{i,n}\backslash U_k\}}|\tilde F|^2_h \le \frac{G(T_1)}{\int_{T_1}^{+\infty}c(s)e^{-s}ds}\limsup_{n\to+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{i,n}\backslash U_k}c(t)}\int_{I_{i,n}\backslash U_k}c(s)e^{-s}ds.
\frac{d^m\delta_0}{dv^m}=\frac{\lambda}{6\kappa}\left\{v^2\frac{d^{m-1}y}{dv^{m-1}}+2(m-1)v\frac{d^{m-2}y}{dv^{m-2}}+(m-1)(m-2)\frac{d^{m-3}y}{dv^{m-3}}\right\}+\bar{P},
\int_{Y_1}|\psi(u,v)|^2 \,\frac{du\,dv}{u^2v^2}\,=\int_{Y_1}|\psi(-u,-v)|^2 \,\frac{du\,dv}{u^2v^2}\,=1,\, \int_{Y_1}\psi(-u,-v)\overline{\psi(u,v)}\,\frac{du\,dv}{u^2v^2}\,= 0,
S = \int d^{6}x \sqrt{g}\left[{\frac{1}{2}}( \partial \phi )^{2}+{\frac{1}{2}}m^{2}\phi^{2}+{\frac{g_{4}}{4 !}}\phi^{4}+ \cdots \right] + \int d^{4}x \sum_{n = 0}^{\infty}{\frac{\lambda_{2 n}}{( 2 n ) !}}\phi^{2 n}
T_{n}(x)=\left(\begin{array}{c}\frac{\underline{\alpha}\alpha^{n+1}(x)}{(\alpha(x)-\omega_{1}(x))(\alpha(x)-\omega_{2}(x))}-\frac{\underline{\omega_{1}}\omega_{1}^{n+1}(x)}{(\alpha(x)-\omega_{1}(x))(\omega_{1}(x)-\omega_{2}(x))}+\frac{\underline{\omega_{2}}\omega_{2}^{n+1}(x)}{(\alpha(x)-\omega_{2}(x))(\omega_{1}(x)-\omega_{2}(x))}\end{array}\right)
N ( x ) = \frac{- 1}{\sqrt{\Lambda}}[ 1 + A ( t ) \sin ( \phi_{\xi}^{\theta}) \cos ( \phi_{\xi}^{\varphi}) + B ( t ) \sin ( \phi_{\xi}^{\theta}) \sin ( \phi_{\xi}^{\varphi}) + C ( t ) \cos ( \phi_{\xi}^{\theta}) ] .
\begin{array}{l}\Gamma_1(u,n)(t)=\eta(t) W_0^t\big(u_0^e, g \big) -i \eta(t) \int_0^t e^{i(t- t^\prime)\Delta}F(u,n) \d t^\prime +i\eta(t) W_0^t\big(0, q \big), \Gamma_2(u,n)(t)= \eta(t) V_0^t\big(\phi_{\pm}, h \big) + \frac12\eta(t) (n_+ + n_-) - \frac12\eta(t) V_0^t(0,z), \end{array}
\phi_{2,i}= \frac{n-2}{2}u_{a_{i}}\frac{\lambda_{i}^{2}r^{2}-1}{\lambda_{i}^{2}r^{2}+1}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n-2}{2}}& +O(r^{n-2}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n-2}{2}})
\begin{aligned}R_{k}=\left(1-\frac{\tau_k}{T}\right) \log_2e \left( e^{\rho_k}\sum_{l=0}^{M-1}E_{l+1}\left(\rho_k\right) \right. - & \left. \int\limits_{\nu_k=0}^{1}\left(1-\left(1-\nu_k\right)^{M-1}\right)^{N_k}\frac{M}{\nu_k}e^{\left(\frac{\rho_k}{\nu_k}\right)}E_{M+1}\left(\frac{\rho_k}{\nu_k}\right) d\nu_k \right)\end{aligned}
\sum_{k=0}^{n}{n\choose k}6^k (\sqrt{5})^{n-k}\big ({1-3^{n-k-1}}\big )F_{k}B_{n-k}= n 2^{n-1}L_{2n-2}+ \sum_{m=1}^{n}{n\choose m}m4^{m-1}L_{m-1}.
\gamma(t)^{(N)}= e^{-it H_{N-n}}e^{-it H_n}\gamma e^{it H_n}e^{it H_{N-n}}-i \int_0^t e^{-i(t-s) H_{N-n}}e^{-i(t-s) H_n}\left[ W_n, \gamma(s)^{(N)}\right]e^{i(t-s) H_{N-n}}e^{i(t-s) H_n}ds
\delta_tV^{n-\frac{1}{2}}+ AV^{n-\frac{1}{2}}+ \sum\limits_{j=1}^m I^{(\alpha_j)}B_jV^{n-\frac{1}{2}}- \kappa V^{n-\frac{1}{2}}= g^{n-\frac{1}{2}}, 2\leq n \leq N,
\partial_tu+(v\cdot\nabla_H)u-\left(\int_{-h}^z\nabla_H\cdot v(x,y,\xi,t)d\xi\right)\partial_zu-\Delta_Hu -\varepsilon\partial_z^2u+(u\cdot\nabla_H)v-(\nabla_H\cdot v)u+f_0k\times u-\nabla_HT=0.
K_{N}=\frac{g_{11}(L_{12}^{1}L_{22}^{2}-L_{12}^{2}L_{22}^{1})-g_{12}(L_{11}^{1}L_{22}^{2}-L_{11}^{2}L_{22}^{1})+g_{22}(L_{11}^{1}L_{12}^{2}-L_{11}^{2}L_{12}^{1})}{W^{3}}.
K_{U^{\alpha}}=\frac{(m+n)!}{\pi^{m+n+1}}\frac{(1-w\bar{\eta})^m(n+1-(n+1)\langle z^{\prime},\zeta^{\prime}\rangle+m\frac{\langle z,\zeta\rangle}{1-w\bar{\eta}})}{(1-w\bar{\eta}-\langle z,\zeta\rangle-\langle z^{\prime},\zeta^{\prime}\rangle+w\bar{\eta}\langle z^{\prime},\zeta^{\prime}\rangle)^{m+n+2}}.
\mathcal{A}f(x) = (\gamma_U x + \gamma_L) \alpha x^{-\alpha -1}-\frac{1}{2}( x^2 \sigma_U^2 + 2 x \sigma_{UL}+ \sigma_L^2) \alpha (\alpha+1)x^{-\alpha-2}& =\alpha x^{-\alpha}\left( \gamma_U - \frac{(\alpha + 1) \sigma_U^2}{2}\right)+ O(x^{-\alpha -1}).
B(\omega_0, \omega_1, \dots, \omega_{n-1}, \omega)(t, u_0, \dots, u_n) = \widetilde{\omega}\biggl(t (|\omega_0| + |\omega_1| + \cdots + |\omega_{n-1}| + |\omega|) + u_0 (|\omega_0| + \cdots + |\omega_{n-1}|) + u_1 (|\omega_1| + \cdots + |\omega_{n-1}|) \cdots + u_{n-1}(|\omega_{n-1}|)\biggr)
\alpha''(s) = -\frac{\lambda}{s^2}\int_{0}^{s}t\alpha(t)dt + \lambda\alpha(s) - \frac{1}{s^2}\int_{0}^{s}tF(t)dt + F(s) = \frac{\lambda}{2}\alpha(s) + \frac{1}{2}F(s) + \frac{1}{2s^2}\int_0^{s}t^2(\lambda\alpha'(t)+ F'(t))dt.
a^{2314}=a^{3214},a_{1}^{2314}=a_{1}^{3214},a_{2}^{2314}=a_{2}^{3214},a_{3}^{2314}=a_{3}^{3214},a_{4}^{2314}=a_{4}^{3214}, a_{12}^{2314}=a_{12}^{3214},a_{13}^{2314}=a_{13}^{3214},a_{23}^{2314}=a_{23}^{3214},a_{24}^{2314}=a_{24}^{3214},a_{34}^{2314}=a_{34}^{3214}, a_{123}^{2314}=a_{123}^{3214},a_{234}^{2314}=a_{234}^{3214}
r^{\gamma}= \limsup_{T \rightarrow \infty}\,\inf_{\varphi(t) \in \Psi}\frac{1}{T}\biggl \{I_{T}^{\gamma}(\varphi(t)) \, \Bigl \vert \, \varphi(t) \in D \cup \partial D,\,\, t \in [0, T] \biggr\}
m(f_1)+m(f_2)=\lim_{\epsilon\rightarrow 0}\,\Bigl\{\int_\epsilon^1f_1(x)\,dx+\int_0^{1-\epsilon}f_2(x)\,dx\Bigr\}=\lim_{\epsilon\rightarrow 0}\int_0^{1-\epsilon}\bigl\{f_1(x+\epsilon)+f_2(x)\bigr\}\,dx\le\lim_{\epsilon\rightarrow 0}\int_0^{1-\epsilon}c(x+\epsilon,x)\,dx=0.
f(0)={}g(J_1(0),J_2(0))=0, f'(0)={}g(J'_1(0),J_2(0))+g(J_1(0),J_2'(0))=g(\gamma'_1(0),W_2'(0))=0, f''(0)={}2g(J_1'(0),J_2'(0))=0{}.
\langle 1 \cdots M|\mathcal{B}^\prime(z)|\overline{x}\rangle=\frac{(1+t^{-1}z^2)}{tz}\frac{z^{\overline{\lambda}+1}-z^{-\overline{\lambda}-1}}{z-z^{-1}},\ \ \ \overline{\lambda}=\overline{x}-1.
\begin{array}{llllll}a_{11}^{1}=a \ \a_{11}^{2}=b a_{11}^{3}=c \ \a_{12}^{1}=k \ \a_{12}^{2}=m \ \a_{12}^{3}=n a_{22}^{1}=d \ \a_{22}^{2}=e a_{22}^{3}=f \ \a_{13}^{1}=p \ \a_{13}^{2}=q \ \a_{13}^{3}=r a_{33}^{1}=g \ \a_{33}^{2}=h \ \a_{33}^{3}=j \ \a_{23}^{1}=s a_{23}^{2}=t \ \a_{23}^{3}=v.\end{array}
\varphi_1(\rho)=\begin{cases}1, \rho\in[0,\delta], 0, \rho\in[\delta_1,\infty), \end{cases}\varphi_2(\rho)=\begin{cases}0, \rho\in[0,\delta], 1, \rho\in[\delta_1,1/2], \varphi(\rho) \rho\in[1/2,\delta_2], 0, \rho\in[\delta_2,\infty), \end{cases}
\hat{\mathbb{J}}((\phi_d)_{d | N}) = \frac{1}{N}\sum_{a, b, c \in \mathbb{Z}/N\mathbb{Z}}e^{-2\pi i a c / N}\sum_{n \in \mathbb{Q}}\sum_{r \in \mathbb{Z}/2t\mathbb{Z}}c_{a, b}(n, r) q^{n - r^2 / 4t}\mathfrak{e}_{(c/N, r/2t, b/N)},
\mathbb{P}(L_n^{*}= \ell \,|\, K_n = k) = \sum_{j=1}^k \mathbb{P}(\mbox{pick}L_{n,j}\mbox{and}L_{n,j}= \ell \,|\, K_n = k) &= k \mathbb{P}( \mbox{pick}L_{n,1}\mbox{and}L_{n,1}= \ell \,|\, K_n = k ) &= k \frac{\ell}{n}\mathbb{P}( L_{n,1}= \ell \,|\, K_n = k),
T_{(A,j+n+i\ \left( \operatorname{mod}2n\right) )}=-T_{(A,j+i\ \left(\operatorname{mod}2n\right) )}, T_{(A,(j+i)+n\ \left( \operatorname{mod}2n\right) )}=-T_{(A,j+i\ \left( \operatorname{mod}2n\right) )}.
\frac{\Gamma\left(b,xw^{b^{i}}\right)}{\Gamma\left(b\right)}=1-\frac{x^bw^{b^{i+1}}}{\Gamma\left(b+1\right)}+\frac{x^{b+1}w^{b^{i+1}+b^{i}}}{\left(b+1\right)\Gamma\left(b\right)}+\dots
M_{t}(x,y)=\frac{1}{(1-e^{-2t})^{n/2}}\exp\bigg(-\frac{e^{-2t}|x|^{2}-2e^{-t}\langle x,y\rangle+e^{-2t}|y|^{2}}{2(1-e^{-2t})}\bigg).
V_{\mu}\otimes V_{\nu}=\bigoplus_{\substack{0\leq k_1\leq\cdots\leq k_n begin{align*}1pt] \mu+\nu-\sum_{i\in I}k_i\alpha_i\in P_{+}}}V_{\mu+\nu-\sum_{i=1}^n k_i\alpha_i}.
P\big( W^- > (1 + 2 \delta) x e^{(\lambda_j^- - \nu_j^-)u}\big) = \bigg( \frac{\lambda_j^- - \nu_j^-}{\lambda_j^-}\bigg) e^{-(1 + 2 \delta) x e^{(\lambda_j^- - \nu_j^-) u}(\lambda_j^- - \nu_j^-)/\lambda_j^-}&\geq (1 - \delta) e^{-(1 + 3 \delta) sq_j x e^{sq_j u}}.
{\cal S K}_{n + 1}= \frac{S U ( 1 , 1 )}{U ( 1 )}\otimes \frac{S O ( 2 , n )}{S O ( 2 ) \otimes S O ( n )}\quad \quad ; \quad \quad{\cal Q M}_{m}= \frac{S O ( 4 , m )}{S O ( 4 ) \otimes S O ( m )}
\mathcal{L}= - m \sqrt{- \Delta^{1 / 2}( g_{\mu \nu}\dot{x^{\mu}}\dot{x^{\nu}}) - \Delta^{- 1 / 2}\sum_{i}X_{i}^{- 1}\mu_{i}^{2}( L \dot{\phi_{i}}+ A_{\mu}^{i}\dot{x^{\mu}})^{2}}
\begin{cases}t_0 := \max \bigl\{t \in [0,1] \mid H_i^{\pi}(t) = m_i^{\pi}\bigr\}, begin{align*}1.5mm]t_1 := \min \bigl\{t \in [t_0,1] \mid H_i^{\pi}(t) = m_i^{\pi}+ 1 \bigr\};\end{cases}
\nu^{S_{k}}_j = \begin{cases}\frac{\displaystyle c_j}{\displaystyle w^{S_1}_j}, \nu_{\pi_{k-1}}+ \frac{\displaystyle c_j - \nu_{\pi_1}\, w^{S_1}_j - \sum_{l=2}^{k-1}\left(\nu_{\pi_{l}}- \nu_{\pi_{l-1}}\right) \, w^{S_{l}}_j}{\displaystyle w^{S_{k}}_j}, \text{if $k \geq 2$}; \end{cases}
J ( p_{1}, p_{2}) = \int \frac{d^{4}k_{1}d^{4}k_{2}}{( 2 \pi )^{8}}\frac{k_{1}^{2}p_{1}^{2}+ k_{2}^{2}p_{2}^{2}- 2 k_{1}k_{2}p_{1}p_{2}}{\Omega ( k , p )}
Z_t = X_t + e^{-\alpha_t}\dot X_t \frac{d}{dt}\nabla h(Z_t) = \dot \beta_t\nabla h(X_t) - \dot \beta_t \nabla h(Z_t) - \frac{e^{\alpha_t}}{\mu}(\nabla \varphi(X_t)+ \nabla \psi(Z_t)),
\sigma_k = \frac{1}{k!}\det \begin{pmatrix}p_1 1 0 \dots 0 p_2 p_1 2 \dots 0 \vdots \vdots \ddots \ddots \vdots p_{k-1}p_{k-2}\dots p_1 k-1 p_k p_{k-1}\dots \dots p_1 \end{pmatrix}.
{1\over 2}(1-\gamma)^{-3}\left(\sum_{k=0}^{\infty}(-1)^{k}{(k+1)(k+2)\over 2}\beta^{k}\right)\left(-\sum_{r=0}^{3}\sigma_{3,2r}\right)=0 \;{\rm from}\; W_{\infty}^{5,D},
g_{2}((\nabla \pi_{\ast})(U,X),\pi_{\ast}Y) =g_{1}(\mathcal{T}_{U}\phi BX,Y)+g_{2}((\nabla\pi_*)(U,\omega\mathcal{B}X),\pi_*Y)-g_{1}(\mathcal{T}_{U}\mathcal{C}X,\mathcal{B}Y) &+g_{2}((\nabla\pi_*)(U,\mathcal{C}X),\pi_*\mathcal{C}Y)+\eta(X)g_{1}(QU,\varphi Y) &-U\eta(X)\eta(Y)-g_{1}(X,\omega U)\eta(Y).
G ( x^{\prime}, t^{\prime}; x^{\prime}, t^{\prime}) = \frac{1}{2}\int \frac{d k^{\prime \prime}}{2 \pi}\frac{1}{\sqrt{k^{2}+ 4}}\frac{i k^{\prime \prime}+ 2 \sigma_{1}}{i k^{\prime \prime}- 2 \sigma_{1}}e^{- 2 i k^{\prime \prime}x^{\prime}}.
\frac{1}{4}\omega_{\hat{\mu}}^{\hat{i}\hat{j}}\gamma_{\hat{i}\hat{j}}\, \varepsilon_{a}\, = \, - \frac{1}{2}\bar{g}A_{\hat{\mu}}^{I}\left( \Gamma_{I 4 5}\right)_{a}^{\; b}\, \varepsilon_{b},
\sigma(\dot{q}_{i}\odot_{k}\dot{q}_{i})=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\left(1-\sigma(q_{i}\odot_{k}q_{j})\right)}{(\sigma-\sigma(q_{i}\odot_{k}q_{j})^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\}.
\Re G^{R}( k ) ={\frac{N^{2}\epsilon^{2 ( \Delta - 4 )}}{8 \pi^{2}( \Delta - 3 ) !^{2}2^{2 \Delta - 5}}}| k^{2}|^{\Delta - 2}\ln{| k^{2}|}, \Delta = 2 , 3 , 4 , \ldots