harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
( \mathcal{Z}^{s})^{\ell}=\mathop{\textstyle \prod}\limits_{i=1}^{n}\mathcal{Z}_{i}^{\ell_{i}}=\mathop{\textstyle \prod}\limits_{i=1}^{n}\Big(\sum\limits_{j_{1},\cdots ,j_{\ell_{i}}=1}^{\infty}\mathop{\textstyle\prod}\limits_{k=1}^{\ell_{i}}\mathcal{Z}_{i,j_{k}}g_{j_{k}}\Big).
\zeta_{r}( s ) ={\frac{1}{3^{s}}}-{\frac{3}{\sqrt{\pi}}}{\frac{1}{2^{2 s + 1}}}{\frac{\Gamma \left( s +{\frac{1}{2}}\right)}{\Gamma ( s + 1 )}}-{\frac{3}{\sqrt{\pi}}}{\frac{1}{2^{2 s + 3}}}{\frac{\Gamma \left( s +{\frac{3}{2}}\right)}{\Gamma ( s + 2 )}}\F \left( 1 , s +{\frac{3}{2}}, s + 2 ,{\frac{3}{4}}\right)
\mathrm{T r}\ln \hat{\cal F}= \lim_{L \rightarrow \infty}\frac{i}{( 4 \pi )^{\frac{d}{2}}}\int_{1 / L^{2}}^{\infty}\frac{d s}{s^{\frac{d}{2}+ 1}}\int d^{d}x \mathrm{T r}\sum_{n = 0}^{\infty}( i s )^{n}\hat{a}_{n}( x ) ,
a_1^{x_1}a_2^{x_2}\cdots{}a_{m-1}^{x_{m-1}}a_m^{x_m}\cdots a_{n-1}^{x_{n-1}}a_n^{x_n}\cdot a_1^{-1}&=s_{1,2}^{x_{1,2}}s_{2,3}^{x_{23}}\cdots s_{m-1,m}^{x_{m-1,m}}s_{1,3}^{x_{1,3}}\cdots s_{1,m-1}^{x_{1,m-1}}s_{2,m}^{x_{2,m}}s_{1,m}^{x_{1,m}}\cdot s_{1,2}^{-1}.
{}^C D^{\alpha}_{a^+}f(t)= I_{a^+}^{m-\alpha}\frac{\mathrm d^m}{\mathrm dt^m}f(t)= \frac{1}{\Gamma(m-\alpha)}\int_a^{t}(t-s)^{m-1-\alpha}\frac{\mathrm d^m}{\mathrm ds^m}f(s) \, \mathrm ds.
\aligned \hat d_{2}^{i+1}\circ \frak H_{ab}^{i+1 i}-\frak H_{ab}^{i+1 i}\circ \hat d_{1}^{i}&= \frak h_b^{i+1 i}- \frak h_a^{i+1 i}+ \frak h_{ab}^{i+1}\circ \psi_1^{i+1 i}-\psi_2^{i+1 i}\circ \frak h_{ab}^{i}\endaligned
{\cal C}_{4}= \left( \begin{matrix}{{3 a_{1}^{4}+ 9 a_{1}^{2}[ a_{2}, a_{3}] + 3 [ a_{2}, a_{3}]^{2}}}\\{{3 a_{1}^{3}a_{2}+ 3 a_{1}a_{2}[ a_{2}, a_{3}]}}\\{{3 a_{1}^{3}a_{3}+ 3 a_{1}a_{3}[ a_{2}, a_{3}]}}\\ \end{matrix}\right)^{\prime}
\{F,G\}_{\rm new}=\{F,G\}_{\rm old}+\oint_{\partial\Sigma}d^{n-2}x\frac{\delta^\vee F}{\delta\phi_A}\delta_G\phi_A-\oint_{\partial\Sigma}d^{n-2}x\frac{\delta^\vee G}{\delta\phi_A}\delta_F\phi_A+?!,
s_X \circ f = s'_X \circ f' \iff \eta(s \otimes X) \circ f = \eta(s' \otimes X) \circ f' \iff \eta((s \otimes X) \circ f)) = \eta((s' \otimes X) \circ f') \iff (s \otimes X) \circ f =(s' \otimes X) \circ f'
\widetilde{WZ}^{(p+1)}=(2\pi\alpha^\prime)\left\{dc+{\textstyle\frac{1}{2\pi\alpha^\prime}}\left[C+{\textstyle\frac{m}{2}}\omega(e)e^{2\pi\alpha^\prime{\cal F}(e)}\right]e^{-2\pi\alpha^\prime{\cal F}(b)}\right\}\, .
\operatorname *{l i m s u p}_{n \to + \infty , \, \mathrm{t y p e \, 1}}\left| \frac{1 - q^{2 n}}{1 + q^{2 n}}\right|^{1 / ( 2 n )}\ \le \ \operatorname *{l i m s u p}_{n \to + \infty , \, \mathrm{t y p e \, 1}}1 \ = \ 1 \ .
\beta_{0,1}:=\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_{r}}\left|x\right|^{2\alpha_{k,1}}V_{k}e^{au_k(x)}=\frac{a\beta_0(\beta_0-4(\alpha_2+1))}{4((\alpha_1+1)-a(\alpha_2+1))},
G_{0}^{\mu \nu \alpha \beta}( x - x^{\prime}) = \int{\frac{d p^{4}}{( 2 \pi )^{4}}}{\frac{{\frac{1}{2}}( g^{\mu \alpha}g^{\nu \beta}+ g^{\mu \beta}g^{\nu \alpha}) -{\frac{1}{2}}g^{\mu \nu}g^{\alpha \beta}+{\cal O}( p )}{p^{2}- i \epsilon}}e^{- i p ( x - x^{\prime})}~ .
\int_{{\cal F}\times S^2}\omega \ (p,\zeta )\;\langle p^{\prime},\zeta^{\prime}|p,\zeta \rangle \langle p,\varsigma |p^{\prime \prime},\zeta^{\prime \prime}\rangle =c\langle p^{\prime},\zeta^{\prime}|p^{\prime \prime},\zeta^{\prime \prime}\rangle
\begin{array}{llll}a * a = a, a * b = b, a * c = d, a * d = d, b * a = a, b * b = b, b * c = c, b * d = a, c * a = b, c * b = b, c * c = c, c * d = d, d * a = a, d * b = c, d * c = c, d * d = d. \end{array}
\Psi_{\delta_{i}}^{R}\equiv \Psi_{l_{i}}^{R}, \Psi_{q_{i}}^{R}\qquad \Psi_{l_{i}}^{R}= \left( \begin{array}{c}{\psi_{\nu_{i}}^{R}}\\{\psi_{e_{i}}^{R}}\\ \end{array}\right) \qquad \Psi_{q_{i}}^{R}= \left( \begin{array}{c}{\psi_{u_{i}}^{R}}\\{\psi_{d_{i}}^{R}}\\ \end{array}\right)
\rho[\frac{\partial\overline{u_{i}}}{\partial t}+\overline{u_{j}}\frac{\partial\overline{u_{i}}}{\partial x_{j}}]=-\frac{\partial\overline{p}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}(\mu\frac{\partial\overline{u_{i}}}{\partial x_{j}}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}),
\bar{\psi}_i = \psi_i \mbox{if $i \notin \partial(Q,f)$}, \bar{\psi}_i = \psi_i + \psi_i^{(1)}+ \psi_i^{(2)}+ \psi_i^{(3)}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mbox{if $i \in \partial(Q,f)$}.
(1- \beta_k)^{c\ell}= (1- (1-\frac 1q)^k )^{c\ell}= (1- e^{\delta k})^{c\ell}&= (1 - e^{\delta \ln (n)})^{c\ell}= (1 - n^{\delta})^{c\ell}= \left((1- \frac 1{n^{-\delta}})^{n^{-\delta}}\right)^{\frac{cn^{1+ \delta}}{\ln(n)}}
\begin{array}{l}L^{(t)}(\mathbf{x}^{(t+1)})-L^{(t-1)}(\mathbf{x}^{(t)})=\smallskip L^{(t)}(\mathbf{x}^{(t+1)})-L^{(t)}(\mathbf{x}^{(t)})+L^{(t)}(\mathbf{x}^{(t)})-L^{(t-1)}(\mathbf{x}^{(t)}).\end{array}
\frac{2(2-x)}{(1-x)^2}\sum_{j=0}^\infty f_j \left(2-\frac{1}{x}, 2-\frac{1}{x}\right) = \frac{3}{1-x}+ \frac{1-x}{(1-2x)^2}\sum_{j=0}^\infty f_j \left(3-\frac{1}{x}, 3-\frac{1}{x}\right).
\frac{|\Phi_{U_{j},U_{A_j}}\Lambda_{\Psi^{k(\{j\}\cup A_j)}}\Phi_{U_{j},U_{A_j}}^t|}{|\Phi_{U_{A_j}}\Lambda_{\Psi^{k(A_j)}}\Phi_{U_{A_j}}^t|}=|\Lambda_{U''_j}|.
\big( F_h(x_1) - F_h(x_2), x_1 - x_2 \big) = |x_1 - x_2 |^2 - \beta_k h \big( f(x_1) - f(x_2), x_1 - x_2 \big) \ge \big(1 - \beta_k h_1 L \big) |x_1 - x_2 |^2.
K_{\mathrm{Fix}(c)/S}=R(f\circ c')^{!}\Lambda\to Ra_{\mathrm{Fix}(c)\ast}a_{\mathrm{Fix}(c)}^{\ast}R(f\circ c')^{!}\Lambda \;\overset{\beta_{a,f\circ c'}}{\longrightarrow}Ra_{\mathrm{Fix}(c)\ast}R(f_{T}\circ c'_{T})^{!}a^{\ast}\Lambda=Ra_{\mathrm{Fix}(c)\ast}K_{\mathrm{Fix}(c)_{T}/T}.
{\mathbb P}^S_{n}(t_0):=\left[\begin{array}{ccc}S_1 \cdots S_{n+1}\vdots \vdots S_{n+1}\cdots S_{2n+1}\end{array}\right]{\bf \Psi}_n(t_0) \left[\begin{array}{ccc}\overline{S}_0 \ldots \overline{S}_n \ddots \vdots 0 \overline{S}_0 \end{array}\right]
r_1(\rho) =\mathrm{E}\left[\max I(X;Y|F) \right]=\mathrm{E}\left[\max I(X;Y|H) \right] &=\int\limits_0^{\infty}\log_2\left(1+z\rho \right)e^{-z}dz=-\frac{1}{\mathrm{ln}2}\mathrm{Ei}\left(-\frac{1}{\rho}\right)\exp(\frac{1}{\rho}),
a_{\mu}^{\frac 13 - \frac{1}{3q}}\mu^{-\frac14}+ a_{\mu}^{\frac12 - \frac 1q}\mu^{-\frac 12 + \frac 1q}= a_{\mu}^{\frac 13 - \frac{1}{3q}}\mu^{-\frac14}\left(1 + (a_\mu^{\frac 16}\mu^{-\frac 14})^\frac{q-4}{q}\right)
(2k-1)\,\left(\sum\limits^{2k}_{i=1}\,|\lambda_i|\right)^2 \geq 4k\,\sum\limits_{i<j}\,|\lambda_i|\,|\lambda_j|, &=2k\,\left[\left(\sum\limits^{2k}_{i=1}\,|\lambda_i|\right)^2-\sum\limits^{2k}_{i=1}\,\lambda^2_i\right],
\tau(J_{2,1}) =\begin{cases}2 \tau(J) +1 \epsilon(J) = -1 0 \epsilon(J)=0 2 \tau(J) \epsilon(J) = 1.\end{cases}\tau(J_{2,-1}) =\begin{cases}2 \tau(J) \epsilon(J) = -1 0 \epsilon(J)=0 2 \tau(J) -1 \epsilon(J) = 1.\end{cases}
v^{B}(x) := \frac{v(x)}{\chi*\chi(x/\ell_{})}, w_{B_u}(x,y) := \chi_{B_u}(x) v^B(x-y)\chi_{B_u}(y), v_1^{B}(x) := \frac{g(x)}{\chi*\chi(x/\ell_{})}, w_{1,B_u}(x,y) := \chi_{B_u}(x) v_1^B(x-y)\chi_{B_u}(y), v_2^{B}(x) := \frac{g(x)(1+\omega(x))}{\chi*\chi(x/\ell_{})}, w_{2,B_u}(x,y) := \chi_{B_u}(x) v_2^B(x-y)\chi_{B_u}(y),
\begin{array}{c c c}{( \rho_{1}\rho_{2}\rho_{3})^{*}}&{\Longrightarrow}&{\rho_{3}^{*}\circ ( \rho_{1}\rho_{2})^{*}}\\{\Downarrow}&{}&{\Downarrow}\\{( \rho_{2}\rho_{3})^{*}\circ \rho_{1}^{*}}&{\Longrightarrow}&{\rho_{3}^{*}\circ \rho_{2}^{*}\circ \rho_{1}^{*}}\\ \end{array}
\langle{\cal O}[\overline{\psi},\psi,A,h] \rangle \; = \;\frac{1}{Z}\sum_{n=0}^\infty\frac{(-1)^n}{n!}\langle{\cal O}[\overline{\psi},\psi,A,h]\Big(S_M[\overline{\psi},\psi]\Big)^n \rangle_0 \; ,
H_N^\mathrm{eff}=\sum_{j=1}^N-\Delta_{x_j}+2\sum_{j<k}^N \nabla_{x_j}\left( u_\ell(x_j-x_k) \nabla_{x_j}\right) + 2\lambda_\ell \sum_{j<k}^N (\chi_\ell f_\ell^2)(x_j-x_k).
4^rP+e_{r-1}= 4^r \frac{2^kB-1}{3}+ e_{r-1}& = \frac{4^r \times 2^kB - 4^r + 3e_{r-1}}{3}&= \frac{4^r \times 2^kB - 4^r + (3e_{r-1}+1) -1}{3}\mbox{(adding and subtracting $1$)}= \frac{4^r \times 2^kB - 4^r + 4^r -1}{3}\mbox{(since $3e_{r-1}+1 = 4^r$)}& = \frac{4^r \times 2^kB -1}{3}&= A.
\partial_x \phi(x,y)=\begin{pmatrix}1+2\tau \frac{-1+|M|^{1/2}}{x^2 |M|^{1/2}}y[1-(1+x)\partial_y \phi_2(\ldots)]+\frac{1}{|M|^{1/2}}\phi_2(\ldots)\end{pmatrix}
f_{\zeta}= \frac{p_{\zeta}+\zeta^n \widetilde{p_{\zeta}}}{p_{\zeta}-\zeta^n \widetilde{p_{\zeta}}}=\frac{\bar{\mu}p_{\zeta}+\mu \widetilde{p_{\zeta}}}{\bar{\mu}p_{\zeta}-\mu \widetilde{p_{\zeta}}}.
{\lambda^{i}_{\epsilon}}-\lambda^{i}=\epsilon^{2}(\lambda^{i}+\theta(1))\nabla_{x}\upsilon^{ij}(0).{M}\nabla_{x}\upsilon^{ij}(0)+\theta(\epsilon^{\frac{5}{2}}(\lambda^{i}+\theta(\epsilon^{2-\alpha}))).
\begin{array}{c}{D ( \beta , R ) - i \mu \gamma^{0}= e^{\mu \beta t}D ( \beta , R ) \e^{- \mu \beta t},}\\{K_{b c}( t , \vec{x}; t^{\prime}, \vec{x^{\prime}}) = e^{\mu \beta t}k ( t , \vec{x}; t^{\prime}, \vec{x^{\prime}}) \e^{- \mu \beta t \prime},}\\ \end{array}
\pi ( x ) ={\frac{\partial{\cal L}_{\sigma}}{\partial \partial_{+}\psi ( x )}}= i \bar{\psi}\gamma^{+}= \left( \begin{matrix}{{\psi_{R}^{*}}}&{{\psi_{L}^{*}}}\\ \end{matrix}\right) \left( \begin{matrix}{{i \sqrt{1 + s}}}&{{0}}\\{{0}}&{{i \sqrt{1 - s}}}\\ \end{matrix}\right) .
e^{-c_{N,s}t}\hat{p}_t^{(N+1),s}(x,y)\frac{\left(\hat{m}_s^{(N+1)}\right)^{-1}(y)}{\left(\hat{m}_s^{(N+1)}\right)^{-1}(x)}=p_t^{(N),s}(x,y).
\|X_r\circ Y_r-\mu_{r}I\|_F =\left\|\begin{pmatrix}D_{X_{r}}O O E_{X_{r}}\end{pmatrix}\circ\tilde{Y}_r-\mu_r I\right\|_F &\ge \sqrt{\sum_{i=p+1}^{m}(e^r_{i}\tilde{y}_{ii}^r-\mu_r)^2},
\Psi(d):=\frac{n-2}{2n}\left(\frac{n-2}{2n}\mathfrak{A}-\mathfrak{D}\right)+\frac{(n-2)^2}{4n}\mathfrak{A}\log d - \left(\frac{n-2}{2}\mathfrak{C}+\frac{\mathfrak{B}}{n}\right) d.
F^{m}{}_{\mu \nu}= 2 \partial_{[ \mu}A^{m}{}_{\nu ]}- f_{n p}{}^{p}A^{n}{}_{\mu}A^{p}{}_{\nu}= 2 \partial_{[ \mu}A^{m}{}_{\nu ]}- Q^{m q}\epsilon_{n p q}A^{n}{}_{\mu}A^{p}{}_{\nu}.
[ \mathrm{P}( p ) , \mathrm{K}( q , m ) ]_{-}= \hbar \frac{\pi}{\pi}{L}p \{\mathrm{K}( q , p ) \cdot ( \delta_{m , - p - q}+ \delta_{m , p}) - ( p \rightarrow - p ) \}, \hspace{5 m m}q \neq \pm p ,
u_{(1)}(y) =-\frac{1}{54\hat{T}^{1/3}}\frac{5(\hat{z}_{+}(y)^{2/3}+\hat{z}_{-}(y)^{2/3})+11\hat{T}^{1/3}}{(\hat{z}_{+}(y)^{2/3}+\hat{z}_{-}(y)^{2/3}+\hat{T}^{1/3})^5}.
{\cal L}_{1}^{m}{}[ \Lambda ] = -{\frac{1}{4}}m^{2}( \partial_{\mu}\Lambda_{\nu}- \partial_{\nu}\Lambda_{\mu})^{2}+{\frac{\alpha_{1}m^{2}}{2}}( \Lambda_{\mu})^{2}+{\frac{\alpha_{1}m^{2}}{2}}( \partial^{\mu}\Lambda_{\mu}) \Delta^{- 1}( \partial^{\nu}\Lambda_{\nu}) .
[\begin{matrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{matrix}]=\frac{E}{1-\nu^{2}}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&\frac{1-\nu}{2}\end{matrix}][\begin{matrix}\epsilon_{11}\\ \epsilon_{22}\\ 2\epsilon_{12}\end{matrix}].
\quad\sum_{h_2, k}\theta(g_j, g_k)(\pi(\xi)|_{g_jX_k\cap X_j}) &=\sum_{h_2, k}\theta(g_j, g_k)(\pi(\xi)|_{g_jX_{g_k, h_2}\cap X_j})~(\mbox{as $X_k=X_{g_k, h_2}$ by convention}) &=\sum_{g}\sum_{\substack{h_2 h_2\in c(g, X)}}\theta(g_j, g)(\pi(\xi)|_{g_jX_{g, h_2}\cap X_j}) &=\sum_g\theta(g_j, g)(\pi(\xi)|_{X_j}).
\begin{gathered}g(t)=e^{-tB}g(0), f(t)=e^{tbB}f(0)+\int_{0}^te^{(t-s) bB}g( s) ds, u(t)=e^{taB}u(0)+\int_{0}^te^{(t-s) aB}f( s) ds.\end{gathered}
\omega_\mu^{\ab}=\frac{1}{2}e^{a\nu}e^{b\rho}\left[e_{\mu c}(\partial_\nu e_\rho^{\c}-\partial_\rho e_\nu^{\c})-e_{\nu c}(\partial_\rho e_\mu^{\c}-\partial_\mu e_\rho^{\c})-e_{\rho c}(\partial_\mu e_\nu^{\c}-\partial_\nu e_\mu^{\c})\right].
\begin{array}{r c l}{{\frac{d f}{d t}}}&{{=}}&{{\frac{\partial f}{\partial t}+ \{f , H_{T}\}}}\\{{}}&{{}}&{{}}\\{{}}&{{=}}&{{\frac{\partial f}{\partial t}+ \{f , H_{T}\}_{D}+ \{f , \chi_{s}\}\left( \Delta^{- 1}\right)^{s s^{\prime}}\{\chi_{s^{\prime}}, H_{T}\}\approx \frac{\partial f}{\partial t}+ \{f , H_{T}\}_{D}\ .}}\\ \end{array}
H_{X}(R_{X}) = \sum_{[P]}\sum_{n \geq 2}\frac{1}{n}R_{X}^{n \ell(P)}\leq \sum_{[P]}\sum_{n \geq 2}R_{X}^{n \ell(P)}&= \sum_{[P]}\frac{R_{X}^{2 \ell(P)}}{1- R_{X}^{\ell(P)}}\leq \frac{1}{1-R_{X}}\sum_{[P]}R_{X}^{2 \ell(P)}= \frac{P_{X}(R_{X}^{2})}{1-R_{X}}< +\infty,
P(\underline{T}\leq\underline{t}) = P(T_1\leq t_1,\cdots, T_m \leq t_m) &=\Phi_m\bigg[\frac{1}{\alpha_1}\bigg(\bigg(\frac{t_1}{\beta_1}\bigg)^\nu-\bigg(\frac{\beta_1}{t_1}\bigg)^\nu\bigg),\cdots,\frac{1}{\alpha_m}\bigg(\bigg(\frac{t_m}{\beta_m}\bigg)^\nu-\bigg(\frac{\beta_m}{t_m}\bigg)^\nu\bigg);\Gamma\bigg]
\sqrt{\delta_1}=\||b| (\lambda - \Delta)^{-\frac{1}{2}}\|_{2 \rightarrow 2}\leq\|b\|_{d,w}\Omega_d^{-\frac{1}{d}}\||x|^{-1}(\lambda - \Delta)^{-\frac{1}{2}}\|_{2 \rightarrow 2}\leq \|b\|_{d,w}\Omega_d^{-\frac{1}{d}}\frac{2}{d-2},
q_L(\theta) =\frac{\rho_{0,L}}{2}a_0+\sum_{\ell=1}^L\rho_{\ell,L}a_{\ell}\cos\ell\theta &=\frac{\rho_{0,L}}{2}a_0+\sum_{\ell=1}^L\rho_{\ell,L}a_{\ell}T_{\ell}(x) =\frac{\sqrt{\pi/2}\rho_{0,L}}{2}a_0\tilde{T}_0+\sum_{\ell=1}^L\sqrt{\pi}\rho_{\ell,L}a_{\ell}\tilde{T}_{\ell}(x),
H=\frac{1}{\sqrt{2n}}\begin{pmatrix}\mathcal{N}(0, 2) \chi_{(n-1) \beta}\chi_{(n-1) \beta}\mathcal{N}(0, 2) \chi_{(n-2) \beta}\ddots \ddots \ddots \chi_{2\beta}\mathcal{N}(0,2) \chi_{\beta}\chi_{\beta}\mathcal{N}(0,2) \end{pmatrix}
\prod_{m=0}^{(p-1)/2}\prod_{n_1=-\infty}^{\infty \prime}\left [{{\omega^{(m)}_{n_10}}\over{2\pi\alpha^{\prime}}}\right ]^{-1}= \prod_{m=0}^{(p-1)/2}\prod_{n_1=1}^{\infty}\left [{{\omega^{(m)}_{n_10}}\over{2\pi\alpha^{\prime}}}\right ]^{-2}= (2\pi \alpha^{\prime}l^2 )^{-(p+1)/2}{\rm det}({\bf 1}+{\bf B}) \quad ,
\int_{\mathcal{Y}^d}\widetilde{m}_{x_j}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{w}_{x_jy_i}dy+\int_{\mathcal{Y}^d}\widetilde{m}\widetilde{w}_{y_ix_j}\widetilde{w}_{x_jy_i}dy=0.
^{P}\mathcal{X}(\boldsymbol{\Sigma}/\sigma) =[(P\times_{(\mathbb{C}^{\times})^{n+m}}(\mathbb{C}^{\times})^{l+m}\times_{(\mathbb{C}^{\times})^{l+m}}\mathcal{U}^{S}(\sigma))/G^{S}(\sigma)] & =[(P\times_{(\mathbb{C}^{\times})^{n+m}}\mathcal{U}^{S}(\sigma))/G^{S}(\sigma)]
J_{x,y}(\theta_1,\theta_2) = 8 \pi^2 (\cos(2 \pi (\theta_1 + 2\theta_2)) + \cos(2 \pi (2\theta_1 - \theta_2)) - \cos(2 \pi (2\theta_1 + \theta_2)) & - \cos(2 \pi (\theta_1 - 2\theta_2)).
f_{-1}(x)= \oint_{\Sigma_{-\infty}}\frac{dt}{2 \pi i}\Gamma(t) e^{-\frac{\gamma}{2}t^{2}+ x t}, g_{-1}(x)= \int^{1 + i\infty}_{1 - i\infty}\frac{ds}{2\pi i}\frac{1}{\Gamma(s)}e^{\frac{\gamma}{2}s^{2}-x s}
\left ( \begin{array}{c}(\mathcal{F}f_{1})(\xi) \vdots (\mathcal{F}f_{N})(\xi) \end{array}\right ) = \frac{1}{2}\sum_{k = k_{0}}^{k_{1}}A_{k}e^{- i k \frac{\xi}{2}}\left ( \begin{array}{c}(\mathcal{F}f_{1})(\frac{\xi}{2}) \vdots (\mathcal{F}f_{N})(\frac{\xi}{2}) \end{array}\right ) .
\begin{array}{l}W_A= \{(d_1,d_2) \,|\, a_i\leq d_i\}Z_A=\left\lbrace\begin{array}{c}(d_1,d_2)\,|\, \forall j=1,\dots, m \exists u,v \mbox{with}u+v=1 \mbox{such that}a_1\leq d_1\leq a_1+b_{2j}-1+v \quad\mbox{and}a_2\leq d_2\leq a_2+b_{1j}-1+u\end{array}\right\rbrace.\end{array}
\int_{|k|<\frac{\Lambda}{s}}\frac{d^D k_1}{(2\pi)^D}\int_{|k|<\frac{\Lambda}{s}}\frac{d^D k_2}{(2\pi)^D}\int_{\frac{\Lambda}{s}<|k|<\Lambda}\frac{d^D k_3}{(2\pi)^D}\int_{\frac{\Lambda}{s}<|k|<\Lambda}\frac{d^D k_4}{(2\pi)^D}\phi_{4f}(k_4)\phi_{3f}(k_3)\phi_{2s}(k_2)\phi_{1s}(k_1).
xJ = \left[ \begin{array}{cc}0 x -x 0 \end{array}\right] = \left[ \begin{array}{cc}u 0 0 1 \end{array}\right] \left[ \begin{array}{cc}0 \varpi^{-k}-\varpi^{-k}0 \end{array}\right] \left[ \begin{array}{cc}u 0 0 1 \end{array}\right].
u(x,t)=v(t,r,\theta) &=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu(k)}(r\rho)e^{it\rho^2}b^0_{k,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho &=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\mathcal{H}_{\nu(k)}\big[e^{it\rho^2}b^0_{k,\ell}(\rho)\big](r).
\lim_{a\to\infty}\frac{Z^{(q)}(a)}{W^{(q)}(a)}=\frac{q}{\Phi(q)}, \quad\lim_{a\to\infty}\frac{\mathbb{W}^{(q)}(a-z)}{\mathbb{W}^{(q)}(a)}=e^{-\varphi(q)z}, \lim_{a\to\infty}\frac{W^{(q)}(a-z)}{\mathbb{W}^{(q)}(a)}=\lim_{a\to\infty}\frac{Z^{(q)}(a-z)}{\mathbb{W}^{(q)}(a)}=0.
f_{\pm}( r ) = 1 - \frac{\omega_{n}M}{r^{n - 1}}+ \frac{n \omega_{n}^{2}Q^{2}}{8 ( n - 1 ) r^{2 ( n - 1 )}}- \frac{2 \Lambda_{\pm}r^{2}}{n ( n + 1 )}.
f_{r_t|r_m}(y)= 2y\lambda_c\pi e^{-\lambda_c \pi (y^2-x^2)}\left(\sum^{t-m-1}_{n=0}S_n - \sum^{t-m-1}_{n=1}S_{n-1}\right) = 2y(\lambda_c\pi)^{t-m}e^{-\lambda_c \pi (y^2-x^2)}\frac{(y^2-x^2)^{t-m-1}}{(t-m-1)!},
u_t (t,x) -\frac{1}{\,2\,}\sup_{a\in \mathcal{A}(x)}\left( \sum_{i,j}x_i x_j a_{ij}D^2_{ij}u (t,x) \right) = 0\,,\ \ \ \ (t,x) \in (0, \infty)\times \mathbb{R}_{+}^n
tH_{\mathrm{Suz}}^{2+\frac{3}{2}}\left({-\theta^0_1,\, \theta^0_2-\theta^0_1 \atop \theta^\infty_2}; t;{q_1,p_1 \atop q_2,p_2}\right)=tH_{\mathrm{III}(D_7)}\big({-}\theta^0_1; t; q_1,p_1\big) \qquad{}+tH_{\mathrm{III}(D_7)}\big(\theta^0_2-\theta^0_1; t; q_2,p_2\big)+p_2q_1\big(p_1(q_1+q_2)+\theta^\infty_2\big)-q_1.
{\rm ch}[W(\Lambda_0)] \ \_{=}^{?}\sum_{n_1,n_2,n_3,n_4,n_5 \geq 0}\frac{q^{n_1^2+n_2^2+(n_3+n_5)^2+n_4^2+n_3^2+(2 n_3+n_5)(n_1)+n_4(n_1+n_2)+n_3 n_4}}{(q)_{n_1}(q)_{n_2}(q)_{n_3}(q)_{n_4}(q)_{n_5}}
\langle \sigma_{I}| = \operatorname*{l i m}_{\widetilde{\xi}^{1}\rightarrow 0}\left( \frac{1}{\widetilde{\xi}^{1}}\right)^{2 h_{\sigma_{I}}}\, \langle 0 | ~ \sigma_{I}^{-}\left( - \frac{1}{\widetilde{\xi}^{1}}\right) ~ ,
r_{\langle\rangle}(\mathcal C_tP) =(\exists_z,\mathbb L_{\vartheta(t+\Omega^{o_{\langle\rangle}(n(P,0))})}^u,\forall_{x\in a}\exists_{y\in z}\theta), n(\mathcal C_tP,a) =\mathcal B_{\exists,\forall_{x\in a}\exists_y\theta}^{\vartheta(t+\Omega^{o_{\langle\rangle}(n(P,0))})}\mathcal C_tn(P,0).
\lim_{p\rightarrow0^{+}}\mathcal{K}_{\nu,p}(x) =-\intop_{0}^{\infty}\,\frac{y^{-\nu-\frac{1}{2}}(y+1)^{-\nu-\frac{1}{2}}}{e^{(2y+1)x}+1}dy=\sum_{n=1}^{\infty}(-1)^{n}\intop_{0}^{\infty}y^{-\nu-\frac{1}{2}}(y+1)^{-\nu-\frac{1}{2}}\,e^{-(2y+1)xn}dy =\frac{\Gamma\left(\frac{1}{2}-\nu\right)(2x)^{\nu}}{\sqrt{\pi}}\sum_{n=1}^{\infty}(-1)^{n}\,n^{\nu}\,K_{\nu}(xn).
\frac{\sigma_{m_n}- n}{\sqrt{m_n \log(m_n)}}\stackrel{d.}{\longrightarrow}Z_{2, B_1}, \mbox{as $n \rightarrow \infty$.}& \frac{\sum_{i=1}^{m}Q_i - m \mu_Q(m)}{m}\stackrel{d.}{\longrightarrow}Z_{1, B_2, C}, \mbox{as $m \rightarrow \infty$.}
\sum_{l=0}^{\lfloor\lambda\rfloor}\frac{\lambda^l}{l!}e^{-\lambda}+\sum_{l=\lfloor\lambda\rfloor +2}^{k-1}\frac{\lambda^l}{l!}e^{-\lambda}<U\leq \sum_{l=0}^{\lfloor\lambda\rfloor}\frac{\lambda^l}{l!}e^{-\lambda}+\sum_{l=\lfloor\lambda\rfloor +2}^{k}\frac{\lambda^l}{l!}e^{-\lambda}
C=e^{\pi\sqrt{-1}[(n-1)(n+2)+2(n-1)^2 \tau ]/2n}\displaystyle\frac{\prod_{i=0}^{n-1}(\sum_{j=0}^{n-1}\omega^{ij}\theta^{(j)}(1/2))}{h(n/2-(n-1)\tau /2)\prod_{k=1}^{n-1}h(k\tau /n)^{n-k}},
\hat u'' + \frac{1}{r}\hat u' = \Big(c_1 + O(|\hat u| + |\hat v| + r^{-2}) \Big)\,\hat u+ \Big(c_2 + O(|\hat u| + |\hat v|) \Big)\,\hat v + \frac{k^2\,s_+}{\sqrt{2}r^2}, \hat v'' + \frac{1}{r}\hat v' = \Big(c_2 + O(|\hat u| + |\hat v|) \Big)\hat u+ \Big(c_3 + O(|\hat u| + |\hat v|) \Big) \hat v.
\psi_p\left(\begin{pmatrix}1 u_1 * \cdots * 1 u_2 \cdots * \cdots \cdots \cdots 1 u_{n-1}1\end{pmatrix}\right) = \xi( \nu_1 u_1+ \nu_2 u_2+ \nu_3 u_3+\cdots+ \nu_{n-1}u_{n-1}),
\rho_{a,\omega,n}=(\rho_a\cdot \varrho_{a,\omega,n})\circ g_{a,\omega,n}^{-1}\circ b_{a,\omega,n}^{-1},\tilde{\rho}_{a,\omega,n}=(\tilde{\rho}_a\cdot \tilde{\varrho}_{a,\omega,n})\circ g_{a,\omega,n}^{-1}\circ b_{a,\omega,n}^{-1}
\frac{(n-2)(10-n)}{4}\int_{B_{\rho}}r^{2-n}u_r^2 \d x & \leq C \int_{B_{2\rho}\setminus B_{\rho}}r^{2-n}|\nabla u|^2 \d x + C \varepsilon \int_{B_{4\rho}}r^{3-n}|\nabla u|^2 \d x, \rho \leq 1/4 \varepsilon \leq \varepsilon_0,
\partial^{\mu}\widetilde{A}_{\mu}(x) = -4\pi\int\delta\xi ds\, \epsilon_{\mu\nu\rho\sigma}\, \delta^{\mu}(s)\left[\Omega_\xi(s,0) L^{\rho\sigma}[\xi|s] \Omega_{\xi}^{-1}(s,0)\right] \dot{\xi}^\nu(s) \delta^4(x-\xi(s)).
{P_1}\left( n \right) = \frac{1}{{n - 3}}\bar B\left({\frac{1}{2},n - 1,n - 2}\right) \,\,\, + \frac{1}{{n - 1}}\bar B\left({\frac{1}{2},n - 3,n}\right) \,\,\, - \frac{2}{{n - 2}}\bar B\left({\frac{1}{2},n - 2,n - 1}\right)
\mu^{n-\left(i_{j}+\epsilon\right)}\underset{=+1}{\underbrace{\left(-1\right)^{k+1+i_{j}+\epsilon+\epsilon}}}\det\left(C_{[n]\setminus\{i_{1},i_{1}+1,\dots,i_{\frac{m}{2}},i_{\frac{m}{2}}+1\}}\right)+o\left(\mu^{n-\left(i_{j}+\epsilon\right)}\right),
\sqrt{z^2-4xy^2}= z\sqrt{1-\frac{4xy^2}{z^2}}&=z-z\sum_{k\ge1}\frac{(2k-2)!}{2^{2k-1}k!(k-1)!}\frac{(4xy^2)^k}{z^{2k}}&=z-\frac{2xy^2}z-z\sum_{k\ge 2}\frac{(2k-2)!}{2^{2k-1}k!(k-1)!}\frac{(4xy^2)^k}{z^{2k}}&=z-\frac{2xy^2}{z}-2\sum_{k\ge0}\frac{(2k+2)!}{(k+2)!(k+1)!}\frac{x^{k+2}y^{2k+4}}{z^{2k+3}}.
\mu(C\cap T^{-\gamma}D) = \mu(T^{a}(C\cap T^{-\gamma}D))=\mu(T^{a}C \cap T^{a-\gamma}D), \int f\cdot g\circ T^{\gamma}\,d\mu = \int f\circ T^{-\gamma}\cdot g \,d\mu,
0= \left\| p/q \right\| - \left\| p/q' \right\| =\sum_{j=1}^{m}w_j ( ( p c_j - q b_j ) - ( q' b_j - p c_j ) ) &=\sum_{j=1}^{m}w_j ( 2 p c_j - ( q + q' ) b_j ) &= 2p\sum_{j=1}^{m}w_j c_j -(q + q')\sum_{j=1}^{m}w_j b_j \;.
e_1\otimes e_1 + e_1\otimes \alpha(e_1) = e_1\otimes e_2, e_2\otimes e_1 + e_1\otimes \alpha(e_2) = e_1\otimes e_1+e_2\otimes e_1 , e_1\otimes e_2 + e_2\otimes \alpha(e_1) = e_1\otimes e_2 + e_2\otimes e_1 + e_2\otimes e_2, e_2\otimes e_2 + e_2\otimes \alpha(e_2) = e_2\otimes e_1 + e_2\otimes e_2.
\sigma_{g^{-1}}(y^*)_{(\vec{0},s^{-1})}= (y^*)_{((i,j),h)(\vec{0},s^{-1})}= (y^*)_{((i,j),hs^{-1})}& = (y^{sh^{-1}})_{\varphi_{(sh^{-1})}(i,j)}& = (\sigma_{-(\varphi_{(sh^{-1})}(i,j))}(y^{sh^{-1}}))_{\vec{0}}.
\zeta(s+1)^{-1}Z_{r_1}(d_2(f), s+1) =f(0)q^{As}+\zeta(1)^{-1}\sum_{j=-A}^{A-1}f(0,0,0,\varpi^j,0,0)q^{js}(1-\zeta(-1)\zeta(-s-1)^{-1}) =f(0)+s(\log q)A f(0)-s(\log q)\sum_{j=-A}^{A-1}f(0,0,0,\varpi^j,0,0)+O_f(s^2).
\frac{d}{d t}S_{W Z}( t ) = i^{[ \frac{p + 1}{2}]}\frac{2 i}{p !}\int \phi^{*}\Pi^{a_{1}}( t ) \wedge \cdots \wedge \Pi^{a_{p}}( t ) \wedge \left( \bar{\nu}( t ) \Gamma_{a_{1}}\cdots \Gamma_{a_{p}}t d \theta \right)
\dot{R}^{2}+ k = \frac{\mu}{R^{2}}+ \Bigl ( \frac{\tilde{\rho}^{2}}{\ell^{2}\sigma^{2}}- q^{2}\Bigr ) \frac{1}{R^{4}}+ \frac{2 \tilde{\rho}}{\ell^{2}\sigma}\frac{1}{R}+ \frac{\ell^{2}( \Delta \mu )^{2}}{1 6 ( R^{3}+ \tilde{\rho}/ \sigma )^{2}}.
e_i b = b_L \otimes \cdots \otimes b_{j_++1}\otimes e_ib_{j_+}\otimes b_{j_+-1}\otimes \cdots \otimes b_1, f_i b = b_L \otimes \cdots \otimes b_{j_-+1}\otimes f_ib_{j_-}\otimes b_{j_--1}\otimes \cdots \otimes b_1.
\int_{B(0,R)}|\nabla u_j|^2\, dZ =\int_{B(q_j, Rr_j)}\frac{r_j^{n-2}}{(\omega(B(q_j,r_j)))^2}|\nabla G(X_0,Y)|^2\, dY &\leq C\frac{r_j^{n-4}}{(\omega(B(q_j,r_j)))^2}\int_{B(q_j, 2Rr_j)}G(X_0,Y)^2 dY.
\varepsilon^{ijk}{^{\pm}}A_i^{AB}\partial_j{^{\pm}}A_{kAB}=\varepsilon^{ijk}\big({\frac{1}{2}}A_i^{AB}\partial_j A_{kAB}\mp{\frac{i}{2}}\varepsilon^{ABCD}A_{iAB}\partial_j A_{kCD}\big),
\mathbb{E}(\tau) = \sum_{n=1}^{\infty}i b_i = \sum_{j=1}^{\infty}\left\lceil{\frac{j}{k}}\right\rceil a_j' \leq \frac{1}{k}\left(k+ \sum_{j=1}^{\infty}j a_j' \right) = 1+ \sum_{n=1}^{\infty}n a_n,
\frac{c_{2}^{- \nu}}{c_{0}^{- \nu}}= \frac{1}{\displaystyle h^{- 2}[ s^{2}- ( - \nu + 2 )^{2}] - \frac{1}{\displaystyle h^{- 2}[ s^{2}- ( - \nu + 4 )^{2}] - \frac{1}{\displaystyle h^{- 2}[ s^{2}- ( - \nu + 6 )^{2}] \cdot \cdot \cdot}}}
\partial(\rho_i, -w_i)(\chi^{w_k}) = \langle w_k, \rho_i \rangle \chi^{w_k - w_i}= \delta_{ik}i=1, \dots, n, \partial(\rho_j,0)(\chi^{w_k}) = \langle w_k, \rho_j \rangle \chi^{w_k}= - a_{kj}y_k j=n+1, \dots, n+r,
\mathfrak H^s(\Gamma) = \begin{cases}H^s(\Gamma) \{u \in H^s(\Gamma) \colon u^\prime(0) = u^\prime(1) = 0\}\{u \in H^s(\Gamma) \colon u^\prime(0) = u^\prime(1) = u^{\prime\prime\prime}(0) = u^{\prime\prime\prime}(1) = 0\}\end{cases}
G_{r}(z,v_{I},v_{L}) =\Phi^{r}(T(zv_{I},tv_{L}))\vert_{t=z}= \frac{1-u^{2^{r+1}}}{(1-u^{2^{r+1}-1})(1+u)}T\Bigg(\frac{u(1-u^{2^{r+1}-1})^{2}}{(1-u^{2^{r+1}})^{2}}v_{I},\frac{u^{2^{r+1}-1}(1-u)^{2}}{(1-u^{2^{r+1}})^{2}}v_{L}\bigg),
\frac{|S^n|}{|G|}\ge\frac{|S|}{|G|}\ge\frac{1}{\gamma^{\varepsilon^{-1}}}=\left(\frac{\gamma^{1/2}}{\gamma}\right)^{2\varepsilon^{-1}}\ge\left(\frac{n}{\gamma}\right)^{2\varepsilon^{-1}}.
\partial_y\left(\frac{{\partial^2_{y}{\partial^n_x u}}}{u^s_y +{u}_y}\right) =\partial^2_y g_n + 2 (\partial_y g_n) \eta_2 + 2g_n \partial_y \eta_2 - 4g_n\eta_2^2 - 8 \partial_y^{-1}g_n \eta_2 \partial_y \eta_2 + \partial_y\bigg(\partial_y^{-1}g_n\, \frac{u_{yyy}^s +{u}_{yyy}}{u^s_y +{u}_y}\bigg).
m(\pi_{\rm ss}, \bar\sigma_{s,a}) = m(\pi_{\rm ss}, I(\tau, \bar\sigma_{s,a})) = \frac{1}{2}\left(m(R^G_{T, s}, R^G_{S, s'_a}) + m(R^G_{T_{\rm a},s}, R^G_{S, s'_a})\right).
P_{t-s}( h \Gamma(f_s) ) \leq \frac{2\rho q(s)}{2\rho\theta-1+e^{-2\rho(t-s)}}\log \left( P_{t}\left( e^{q(0) (\Gamma(f))}\right) \right)^{1/q(0)}& = \frac{2\rho q(s)/q(0)}{2\rho\theta-1+e^{-2\rho(t-s)}}\log P_{t}\left( e^{\alpha (\Gamma(f))}\right) .