harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
=\varphi\Big(\prod_{i=1}^j h_i^{(-1)^{\dim \sigma_i}}\Big)\cdot \varphi\Big(\prod_{i=j+1}^k h_i^{(-1)^{\dim \sigma_i}}\Big)\cdot \tau\left(C_*^{\varphi}(M,\overline{W});\{\sigma_1,\dots,\sigma_j\},\mathfrak{h}'\right)\cdot \cdot\tau\left(C_*^{\varphi}(\overline{C},P^+);\{\sigma_{j+1},\dots,\sigma_k\},\mathfrak{h}''\right)=
f_w(w\nu_s(a)s) =f_w(wss\nu_s(a)s){}\stackrel{(i)}{=}f_w(wsh_s(a^{-1})\nu_s(a)\delta_ss\nu_s(a^{-1})){}=\theta(wsh_s(a^{-1})\nu_s(a)\delta_ssw^{-1})f_w(w\nu_s(a^{-1})){}\stackrel{(ii)}{=}\theta(wsh_s(a^{-1})\delta_ssw^{-1}){}=\kappa_{ws,s}\theta(wsh_s(a^{-1})sw^{-1}).
(\xi_1+\xi_2)d_i^{(2)}(z_{\ast})+\cdots+(-1)^{N+1}\xi_{N-1}d_i^{(N)}(z_{\ast})= &(\hat{\xi}_1+\hat{\xi}_2)d_i^{(2)}(z_{\ast})+\cdots+(-1)^{N+1}\hat{\xi}_{N-1}d_i^{(N)}(z_{\ast})
e_{n+1}: = v(t_{n+1})-v^{n+1}= (v(t_{n+1})-v_*(t_{n+1}))+(v_*(t_{n+1})-v^{n+1}) & = (v(t_{n+1})-v_*(t_{n+1}))+ (v(t_n) - v^n) & \quad-i \int_0^{\tau}S(t_n+r)^{-1}f \left( S(t_n+r)v(t_n)\right) d r + i \tau S(t_n+\tau\xi_n)^{-1}f\left( S(t_n+\tau\xi_n)v^n\right),
\begin{array}{r l}\hat{U}^{-1}(k_1,k_2) |A_{\bar{z}}(z,\bar{z})>= & = e^{i\pi kk_1k_2}\,e^{-\frac{1}{2}ik\left[(k_1\tau-k_2)\overline{A_{\bar{z}}^{++}(0,0)}+(k_1\bar{\tau}-k_2)A_{\bar{z}}^{++}(0,0)\right]}\|A_{\bar{z}}(z,\bar{z})-\frac{i\pi}{\tau_2}(\tau k_1-k_2)>\ \ \ ,\end{array}
a_1 = x_1x_2x_3^{2}x_4^{2}x_5^{4}, a_2 = x_1x_2x_3^{2}x_4^{4}x_5^{2}, a_3 = x_1x_2^{2}x_3x_4^{2}x_5^{4}, a_4 = x_1x_2^{2}x_3x_4^{4}x_5^{2}, a_5 = x_1x_2^{2}x_3^{4}x_4x_5^{2}
\ell(x) \geq \mu\, \min_{i=1,2,\ldots,k}\|x-x^{(i)}\|^2\textup{for all $x\in(\textup{ext}(\mathcal{K})\cap \mathcal{S}^{m-1}) \cap (\cup_{i=1}^{k}\mathcal{B}(x^{(i)},\epsilon))$}.
\left| \sum_{i=1}^N (f_i\circ \alpha)'_t g_i(\beta(t))\right|\leq \Phi(\alpha(t), (g_i(\beta(t)))|\alpha_t'|, t\not\in E_\alpha,\textrm{and}\left| \sum_{i=1}^N (g_i\circ \beta)'_t f_i(\alpha(t))\right|\leq \Psi(\beta(t), (f_i(\beta(t)))|\beta_t'|,t\not\in E_\beta.
\begin{cases}{\rm (a)}\i+j=k k \le n,{\rm (b)}\ (2n+1-i^-)+(2n+1-j^-)=2n+1-k^-, \k \le n \min\{i,j\}\le n,{\rm (c)}\i^-+j^-=k^-, \k \ge n+2 \max\{i,j \}\ge n+2,{\rm (d)}\ (2n+2-i)+(2n+2-j)=2n+2-k k \ge n+2.\end{cases}
\nu^{{\pi^{*,\infty}}}(0)=\frac{1+2^{\mu^s_0+\Delta{K}^{s,\infty}}}{1+2^{\mu^s_0+\mu^s_1+2\Delta{K}^{s,\infty}}+2^{\mu^s_0+1+\Delta{K}^{s,\infty}}},~\nu^{{\pi^{*,\infty}}}(1)=\frac{2^{\mu^s_0+\Delta{K}^{s,\infty}}(1+2^{\mu^s_1+\Delta{K}^{s,\infty}})}{1+2^{\mu^s_0+\mu^s_1+2\Delta{K}^{s,\infty}}+2^{\mu^s_0+1+\Delta{K}^{s,\infty}}}.
\int_{\mathbb{R}^3}\mathcal{R}_{ijk}\phi_{x_i, x_j, x_k, t}~\mbox{d}x =\left(\frac{\mbox{d}}{\mbox{d}t}\int_{\mathbb{R}^3}\mathcal{R}_{ijk}\phi_{x_i, x_j, x_k}~\mbox{d}x \right) -\int_{\mathbb{R}^3}(\mathcal{R}_{ijk})_t\phi_{x_i, x_j, x_k}~\mbox{d}x.
-{\frac{\partial^{4}\ln [ P_{\mathrm{v a c}}( x^{+}) ]}{\partial x^{+}\partial x^{-}\partial^{2}x_{\bot}}}={\frac{1}{4 \pi}}\sum_{n = 1}^{\infty}\left({\frac{e E ( x^{+})}{n \pi}}\right)^{2}\exp \left[ -{\frac{n \pi m^{2}}{\vert e E ( x^{+}) \vert}}\right] .
P^{-}= -{\frac{g^{2}}{2 ( 2 \pi )^{2}}}{\frac{d^{2}}{d z^{2}}}- g^{2}\int_{- L}^{L}d x^{-}T r \left( J^{+}{\frac{1}{D_{-}^{2}}}J^{+}\right) .
( D_{x_{1}}^{2}+ \bar{D}_{x_{1}}^{2}) \gamma^{r}[ \theta , p ] = \sum_{i}( D_{x_{1}}^{2}+ \bar{D}_{x_{1}}^{2}) B_{i}( \theta , p ) \Big ( F_{i}( p ) - P_{i}( p ) \Big ) .
{\cal G}( z , z^{\prime}) \simeq{\frac{1}{4 \pi}}e^{f ( z , z^{\prime})}\int_{\delta}^{\infty}{\frac{d s}{s}}\langle z | e^{- s O_{\Sigma}}| z^{\prime}\rangle = -{\frac{1}{4 \pi}}e^{f ( z , z^{\prime})}\langle z | \ln O_{\Sigma}| z^{\prime}\rangle .
F^{\mu \nu}= \left( \begin{array}{c c c c}{0}&{- l_{x}}&{- l_{y}}&{- l_{z}}\\{l_{x}}&{0}&{- w_{z}}&{w_{y}}\\{l_{y}}&{w_{z}}&{0}&{- w_{x}}\\{l_{z}}&{- w_{y}}&{w_{x}}&{0}\\ \end{array}\right) .
K_{1}(w,\lambda(t))=\int_{0}^{\infty}\frac{r dr}{\lambda(t)^{2}(1+\frac{r^{2}}{\lambda(t)^{2}})^{3}}\int_{0}^{w}\frac{\rho d\rho}{w}\left(1+\frac{r^{2}-1-\rho^{2}}{\sqrt{(r^{2}-1-\rho^{2})^{2}+4r^{2}}}\right)
p(M+1 - \delta) = 2(M+1)\cdot\frac{1-\frac{\delta}{M+1}}{1-\frac{1}{2}\cdot\frac{\delta}{M+1}}\equiv 2(M+1)\delta_1, &q = \frac{2}{1+\frac{1}{2}\cdot\frac{\delta}{M+1}}\equiv 2\delta_2
e^{c \phi}e^{f ( q^{a})}- e^{- 2 c \phi}e^{- 2 f ( q^{a})}= e^{\phi}[ \, c + c_{1}+ Y_{0}X_{0}f ( q^{a}) \, ] - e^{- 2 \phi}[ \, c + c_{1}- Y_{1}X_{0}f ( q^{a}) \, ] .
\lambda = \lambda_{\max}\left(\mathbf{F}^H\mathbf{P}^H\mathbf{P}\mathbf{F}\right) & = \lambda_{\max}\left(\mathbf{P}^H\mathbf{P}\right) & = N_c\left\|\mathbf{F}^H\left(\mathbf{S}\check{\mathbf{F}}\hat{\mathbf{h}}^{(i)}\right)\right\|_\infty^2,
P \biggl\{\int\limits_{\mathbb{T}}\bigl| Y(t)\bigr|^pd\mu(t)>\varepsilon \biggr\}\leq2\cdot\bigl(2^{\frac{p}{2}}\bigr)^lL_0(s)\bigl(p^p\bigr)^l\bigl(\exp\{-p\}\bigr)^lC_p^l\bigl(s^{-p}\bigr)^l\cdot\frac{(l^p)^l}{\varepsilon^l}& =2L_0(s) a^l\bigl(l^p\bigr)^l,
S = \frac{\pi u}{\lambda^{2 / 3}}\left[ \frac{| m^{i}+ \tau n^{i}+ \omega p^{i}|^{2}}{\tau_{2}}+ \lambda^{2}\left( p^{i}\right)^{2}\right] - \pi u^{3}+ 2 \pi i C_{i j k}m^{i}n^{j}p^{k}\, ,
\mathcal{V}_s^\eta = \left\{(Z_s,Z_s^\prime) \in \overline{\mathcal{D}_s}\times\overline{\mathcal{D}_s}\left|\begin{aligned}\inf_{1 \leq i \neq j \leq s}|v_i-v_j^\prime| > \eta & \textnormal{and}& \inf_{1 \leq i \leq s \; : \; |v_i - v_i^\prime|\neq 0}|v_i - v_i^\prime|>\eta\end{aligned}\right. \right\}
{\tilde{G}}[{\tilde{x}},{\tilde{x}^{\prime}}] = \sum_{i}\left[ u_{i}({\tilde{x}}) u_{i}^{*}({\tilde{x}^{\prime}}) \right] \sum_{j}\left[ v_{j}^{*}({\tilde{x}}) v_{j}({\tilde{x}^{\prime}}) \right] .
\mathcal{I}-d_-(x,A)+\mathcal{I}-d_-(x,B) \leq \mathcal{I}-\liminf \{a_n+b_n\}\leq \mathcal{I}-\liminf \{1+p_n\}= 1+ \mathcal{I}-\liminf p_n = 1+\mathcal{I}-d_-(x,A \cap B)
\left|A(z,\zeta)\right|^{1+\mu_{0}}\delta_{\Omega}^{-\gamma\mu_{0}-\varepsilon}(\zeta)\lesssim\left(2^{i}\delta_{\Omega}(z)\right)^{-\mu_{0}(\gamma+n)-\varepsilon}\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,2^{i}\delta_{\Omega}(z)\right)
f(z)=\cdot\cdot\cdot+(\frac{1}{3!})z^{-3}+(\frac{1}{2!})z^{-2}+2z^{-1}+2+(\frac{1}{2!})z+(\frac{1}{3!})z^{2}+(\frac{1}{4!})z^{3}+\cdot\cdot\cdot.
H=\int d^{d}x[\frac{1}{2}E^{2}+\frac{1}{4}B_{ij}B_{ij}-\pi\nabla\cdot\vec{A}+\vec{E}\cdot\nabla\phi+\frac{m^{2}}{2}A^{2}-\frac{m^{2}}{2}\phi^{2}]
\frac{fl}{k_{B}T}=L^{-1}(\frac{R}{Nl})\approx3\frac{R}{Nl}+\frac{1}{5}(\frac{R}{Nl})^{2}sin(\frac{7R}{2Nl})+\frac{(\frac{R}{Nl})^{3}}{1-\frac{R}{Nl}},
1 = \frac{\left|\bar v_{1,n}(0)-\bar v_{1,n}\left(\frac{y_n-x_n}{r_n}\right)\right|}{\left|\frac{y_n-x_n}{r_n}\right|^{\alpha}}= \frac{\left|\bar v_{1,n}(0)-\bar v_{1,n}\left(\frac{y_n-x_n}{r_n}\right)\right|}{\left|\frac{y_n-x_n}{r_n}\right|}\left|\frac{y_n-x_n}{r_n}\right|^{1-\alpha}\le C \left|\frac{y_n-x_n}{r_n}\right|^{1-\alpha}\to 0,
\sum_{k=1}^{n}k^{\ell-1}S(k) =\sum_{k=1}^{n}k^{\ell-1}|S\cap [1,k]|=\sum_{k=1}^{n}k^{\ell-1}\left(\frac{k}{n}t+E_k\right) &=\frac{t}{n(\ell+1)}\left(B_{\ell+1}(n+1)-B_{\ell+1}\right)+\sum_{k=1}^{n}k^{\ell-1}E_k,
\begin{cases}t_1 := \min \bigl\{t \in [0,1] \mid H_i^{\pi}(t) = m_i^{\pi}\bigr\}, begin{align*}1.5mm]t_0 := \max \bigl\{t \in [0,t_1] \mid H_i^{\pi}(t) = m_i^{\pi}+ 1 \bigr\};\end{cases}
(\hat{\xi}_{0,k}^{i},\hat{\xi}_{1,k}^{i},\cdot\cdot\cdot,\hat{\xi}_{k-1,k}^{i},\hat{\xi}_{k,k}^{i}),(\xi_{0,k}^{i},\xi_{1,k}^{i},\cdot\cdot\cdot,\xi_{k-1,k}^{i},\xi_{k,k}^{i})
\mathrm{Mab}_{\Theta}(u) = \sum_Y \Lambda_Y \int_{\tilde{\Delta}^+_Y}(nu^*(p)-u^*(p)\sum \frac{\chi(\alpha^{\vee})}{q(\alpha^{\vee})}+d_pu^*(p))P_{DH}(q)dq & +\int_{\Delta^+}u^*(p)(\sum \frac{\chi^{ac}(\alpha^{\vee})}{q(\alpha^{\vee})}-\bar{S}_{\Theta})P_{DH}(q)dq -\int_{\Delta^+}I_H(d_pu^*)P_{DH}(q)dq &-\int_{\Delta^+}\ln\det (d^2_pu^*)P_{DH}(q)dq
\prod_{k=1}^{\infty}\left( \frac{z- \mu_k^2}{z-\lambda_k^2}\right) = 1-\sum\limits_{k=1}^{\infty}a_{k}\frac{(\lambda_{k}^2-z)+z}{\lambda_{k}^2(\lambda_{k}^2-z)}&= 1-\sum\limits_{k \geq 1}\frac{a_k}{\lambda_k^2}- \sum\limits_{k \geq 1}\frac{a_k z}{\lambda_k^2 (\lambda_k^2 -z)},
T_3 = -\sum_{k=1}^d \sum_{i=k+1}^d \sum_{j=i+1}^d \frac{(X_i-X_k)(X_j-X_k)(X_j-X_i)}{(X_i-X_k+\epsilon^2)(X_j-X_k+\epsilon^2)(X_j-X_i+\epsilon^2)}.
\psi (\prod_{k=1}^p a_k ) - \pi_{IY}(\prod_{k=l}^p a_l) =\sum_{k=1}^p ( \prod_{n=1}^{k-1}\psi (a_n) ) ( \psi (a_k) - \pi_{IY}(a_k) ) ( \prod_{m=k+1}^p \pi_{IY}(a_m) ).
Z = \int \prod_{x , i , a}d \bar{E}_{x , i}^{a}d \mu ( U_{x , i}) \exp \left[ -{\frac{\bar{H}_{\mathrm{e l}}+ \bar{H}_{\mathrm{m a g}}}{T_{\mathrm{G}}}}\right] \delta ( D_{x , i}^{a b}\bar{E}_{x , i}^{b}) \delta ( F ( U_{x , i}) ) ,
\left( Q_{2}^{\dagger}\right)^{3 1}u_{B}^{-}({\bf r}) \propto \sum_{n = 0}^{\infty}{\frac{( - 1 )^{n}}{( 2 n ) ! ! ( 2 n + 2 \left( 2 \ell + 1 - \lambda \right) ) ! !}}\left( \sqrt{E}r \right)^{2 n + \left( 2 \ell + 1 - \lambda \right)}e^{- i \left( 2 \ell + 1 \right) \theta}
\sum_{A \in \mathcal{F}}\left(\frac{q}{q+1}\right)^{|A|}\leqslant\sum_{A\in\mathcal{F}}2^{-|A|/q}\leqslant\binom{n}{\lceil n/2\rceil}2^{-\binom{\lceil n/2\rceil}{k}/q}&\leqslant\left(\frac{en}{\lceil n/2\rceil}\right)^{\lceil n/2\rceil}2^{-(1+1/\ln 2)n}<\left(\frac{e}{2^{1/\ln 2}}\right)^n=1,
\widehat{u}(t,\xi) = \dfrac{1}{e^{i2\pi\xi c_{j0}}- 1}\int_{0}^{2\pi}e^{i\xi \widetilde{H}(t_j, \tau)}\widehat{f}_j(t_1,\ldots,t_j+\tau,\ldots, t_n,\xi) d\tau,
f(\chi_1,\ldots,\chi_n) b^{\prime}f(\mathcal{Y}_1,\ldots,\mathcal{Y}_{n-1},\mathcal{V}_n)- b^{\prime}f(\mathcal{Y}_1,\ldots,\mathcal{Y}_{n-1},\mathcal{V}_n)f(\chi_1,\ldots,\chi_n)=0
\displaystyle \sum_{w \in \mathsf{sib}_{i_j}(v_G | u_{i_j})}\varPhi(f_1 \otimes \cdots \otimes f_k)(w) = 0, v_G \in \mathsf{sib}_{F, G}(u),~ G = F \setminus \{i_j \}, ~j=1, \ldots, k.
\partial_z q^i(z)+\langle Z, \alpha_i\rangle q^i(z)=L_i(z)\Big[{p^i(z)}\Big]^{2}\prod_{j\neq i, j\in I_w\cup I_g}p^j(z)^{a_{ji}}\prod_{j\neq i, j\in I_b}p^j(z)^{a_{ji}/2}.
\mathcal{G}^{4s}_{|\cdot|}(\vec{a};\vec{b}) =|\langle \vec{a},\vec{b}\rangle|\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}dx_1dx_2\cdots dx_n &=|\langle \vec{a},\vec{b}\rangle|\times \bigg|\prod_{i=1}^{n}|b_i|-|a_i|\bigg|.
\int \frac{d^2 \vec k_{T}}{(2 \pi)^2}e^{i \vec k_{T}\cdot \vec b_T}e^{- \frac{q^2}2 \tau}e^{- \frac{k^2}2 l^\prime}=\frac 1{2\pi l}e^{\frac 1{2l}(\vec p_T \tau - i \vec b_T)^2}e^{- \frac{p^{(1)2}\tau + p^{(2)2}l^\prime}{2 \sinh^2 (v_1 - v_2)}}\;.
T^{+ 2}= - 2 i \pi_{q}^{1 +}\pi_{q}^{1 +}\! - \! 2 i \pi_{q}^{2 +}\pi_{q}^{2 +}, \quad T^{- 2}= - 2 i \pi_{\dot{q}}^{1 -}\pi_{\dot{q}}^{1 -}\! - \! 2 i \pi_{\dot{q}}^{2 -}\pi_{\dot{q}}^{2 -}
\ell(\operatorname{Ad}_\psi u+w,\psi\cdot\rho)=\frac12\int_M\big(\big(\psi^{-1}\big)^*\rho\big)\big(\operatorname{div}(\psi^{-1})^*u\big)^2\mu=\frac12\int_M\rho(\operatorname{div}u)^2\psi^*\mu=\ell(u,\rho),
\begin{aligned}\left(u_{k,0},\;\rho_{k,0},\;r_{k,0}^2(u_{k,0})_x,\;r^2_{k,0}(\rho_{k,0})_x,\;r_{k,0}^2(\rho_{k,0}(r^2_{k,0}u_{k,0})_x)_x\right) \rightarrow \left(u_0,\;\rho_0,\;r_0^2(u_0)_x,\;r_0^2(\rho_0)_x,\;r_0^2(\rho_0(r_0^2u_0)_x)_x\right)L^2,k\rightarrow+\infty.\end{aligned}
\begin{cases}L^{1}=L^{\mathfrak{p}_{2}}=L^{2}=L^{2\mathfrak{p}_{2}}=L^{\mathfrak{p}_{5a}}=L^{\mathfrak{p}_{5b}}=L^{\mathfrak{p}_{2}\mathfrak{p}_{5a}}=L^{\mathfrak{p}_{2}\mathfrak{p}_{5b}}, L^{\mathfrak{m}}=L^{\mathfrak{p}_{2}\mathfrak{m}},\mathfrak{p}_{2}\nmid\mathfrak{m}.\end{cases}
\frac{\left<e_j, u_1 \right>}{\left<u_1, u_1 \right>}u_1 = \frac{-(k'_2+\ldots+k'_{\nu})}{k_1 (k'_2+\ldots+k'_{\nu})^2 + k_2 k'^2_2+\ldots+k_{\nu}k'^2_{\nu}}u_1.
(d_n\theta)(a_1\otimes\cdots\otimes a_n) =a_1\theta(a_2\otimes\cdots\otimes a_n) &+ \sum_{i=1}^{n-1}(-1)^i \theta(a_1\otimes \cdots \otimes a_{i-1}\otimes a_ia_{i+1}\otimes a_{i+2}\otimes\cdots\otimes a_n).
\mathcal{R}(\psi) = \left( \begin{array}{lll}m_{00}m_{01}0 m_{10}m_{11}0 E F 1\end{array}\right), \ \left( \begin{array}{ll}m_{00}m_{01}m_{10}m_{11}\end{array}\right) = M_\psi\ \ \ \E,F \in \mathbb{N}.
||\delta^{\ell}(a)||_{\infty,0}\le C||\delta^{\ell}(a)||_{s-|\ell|}&= C||\sum_m m^{\ell}a_m\prod_{j=1}^nU_j^{m_j}||_{s-|\ell|}&< C||\sum_m(1+|m_1|^2+\cdots+|m_n|^2)^{|\ell|}a_m\prod_{j=1}^nU_j^{m_j}||_{s-|\ell|}&= C||P_{\lambda^{|\ell|}}(a)||_{s-|\ell|}&= C||a||_s
I_{m,p}^{\star}(s,x):=\frac{1}{2\pi i}\,\intop_{\sigma-i\infty}^{\sigma+i\infty}\Gamma(z)\,\Gamma(s-z)\,(2z,2\pi pm)_{m}\,\left(\frac{x}{m}\right)^{2z}\,dz=I_{m,p}^{\star(1)}(s,x)+I_{m,p}^{\star(2)}(s,x),
{\cal L}_{b r a n e}= - 2 g \left( \delta ( x^{5}) - \delta ( x^{5}- \tilde{x}^{5}) \right) \left( \sqrt g_{( 4 )}3 W ( \phi ) +{\frac{1}{4 !}}\varepsilon^{\mu \nu \rho \sigma}A_{\mu \nu \rho \sigma}\right) .
H_{M}(q)=\frac{\eta_{W}(q^{2})^{2}}{\eta_{W}(q)^{2}}[\frac{\vartheta_{01}(q^{5})}{\vartheta_{01}(q)}]^{1/2}[\frac{5\vartheta_{01}(q^{5})^{2}}{4\vartheta_{01}(q)^{2}}-\frac{1}{4}]^{-1/2}R(q)^{1/2}
\Delta_{\bf d}(32) = -467775 \Delta_{\overline{\bf d}}(32) = 245025 \Delta_{\bf d}(64) = -448046589375 \Delta_{\overline{\bf d}}(64) = 218813450625 \Delta_{\bf d}(128) = -396822986774382287109375 \Delta_{\overline{\bf d}}(128) = 200745746250569862890625,
\left[ X_{1},Y_{1}\right] =Z_{1},\left[ X_{1},Y_{2}\right]=Z_{2},\left[ X_{1},Y_{3}\right] =Z_{3}\left[ X_{2},Y_{1}\right] =Z_{2},\left[ X_{2},Y_{2}\right]=Z_{3},\left[ X_{2},Y_{3}\right] =Z_{2}\left[ X_{3},Y_{1}\right] =Z_{3},\left[ X_{3},Y_{2}\right]=Z_{1},\left[ X_{3},Y_{3}\right] =Z_{1}.
Q_{\mathrm{I}}= \frac{1}{\sqrt{2}}( e^{i \pi / 4}Q_{2}+ e^{- i \pi / 4}\overline{{{Q}}}_{\dot{2}}) , \qquad Q_{\mathrm{I I}}= \frac{1}{\sqrt{2}}( e^{- i \pi / 4}Q_{1}+ e^{i \pi / 4}\overline{{{Q}}}_{\dot{1}}) ,
S_{\rm cl}={\rm tr}\left( \hat{\bar \sigma}^{\dot \alpha}\bar D_{\dot \alpha}D_\alpha \hat\sigma^\alpha \right) +{\rm tr}\left({\hat{\bar \sigma}}^{\dot \alpha}\bar D_{\dot \alpha}\left( e^{-V}-1 \right) D_\alpha{\hat\sigma}^\alpha \right)\,.
\lim_{n \uparrow \infty}\mathbb E\left[e^{\phi \underline{X}_{e(q_n)}}\right]=\mathbb E\left[e^{\phi \underline{X}_{e(q)}}\right], \ \lim_{n \uparrow \infty}\mathbb E\left[e^{-\phi \overline{X}_{e(p+q_n)}}\right]=\mathbb E\left[e^{-\phi \overline{X}_{e(p+q)}}\right].
lim_{x\rightarrow0}\frac{e^{x}-1}{x}=lim_{x\rightarrow0}\frac{x+\frac{x^{2}}{2}+\frac{e^{\xi_{x}}}{6}x^{3}}{x}=lim_{x\rightarrow0}(1+\frac{x}{2}+\frac{e^{\xi_{x}}}{6}x^{2})
\lim_{R\rightarrow 0^{+}}\mathcal{S}_{0}\left( q_{1},R\right)=\lim_{R\rightarrow +\infty}\mathcal{S}_{\infty}\left( q_{2},R\right) =0, \tag*{$\left({\cal S}_{q_{1},q_{2}}^{\prime \prime}\right) $}
\times \theta({k^+\Delta \over p^+}-k_\perp^2)\bigl[{{1}\over{(k_\perp^2+{m^2k^{+2}\over p^{+2}}})}\bigr] \biggl[ 1 -{{m^2k{+2}\over p^{+2}}\over{k_\perp^2 +{m^2k^{+2}\over p^{+2}}}}\biggr] \;.
|(w_{>1},r_{> 1})(\alpha)| \lesssim \sum_{2^\ell=1}^\infty 2^{-\frac{\ell}2}|\alpha|^{-1}(\|(w,r)\|_{L^2}+ \|(g,k)\|_{L^2}) \approx |\alpha|^{-1}(\|(w,r)\|_{L^2}+ \|(g,k)\|_{L^2}) ,
h^{\mu \nu}( z , w ) = \frac{1}{2}\sum_{n , m = 1}^{\infty}( - w )^{n - 1}( - z )^{m - 1}\sqrt{n m}S^{n m}\eta^{\mu \nu}- \frac{1}{2}\eta^{\mu \nu}\frac{1}{( z - w )^{2}}.
(0,x+\mathbf{i}y,z_1) \cdot (\theta,0,0)= (\theta, e^{-\mathbf{i}\theta}(x+\mathbf{i}y) , z )=(\theta,0,0)\cdot(0,e^{-\mathbf{i}\theta}(x+\mathbf{i}y),z)
H_{\phi}\Phi_{\nu}(\phi) = \epsilon_{\nu}^2 \Phi_{\nu}(\phi), H_{\phi}= - \partial_{\phi}^2 + \frac{\alpha^2 - 1/4}{\cos^2\phi}+ \frac{\beta^2 - 1/4}{\sin^2\phi}, \epsilon_{\nu}= \alpha + \beta + 1 + 2\nu, \nu=0, 1, 2, \ldots,
G_{m n}^{\prime}= G_{m n}- \frac{G_{m 1}G_{1 n}+ B_{m 1}B_{1 n}}{G_{1 1}}\; , B_{m n}^{\prime}= B_{m n}- \frac{G_{m 1}B_{1 n}+ B_{m 1}G_{1 n}}{G_{1 1}}\; ,
\overline{z}_{m}=\frac{\hat{\eta}_{\alpha}^{*}z_{m}-\hat{\eta}_{\beta}z_{m}^{*}}{|\hat{\eta}_{\alpha}|^{2}-|\hat{\eta}_{\beta}|^{2}}=\frac{\hat{\eta}_{\alpha}^{*}}{|\hat{\eta}_{\alpha}|^{2}-|\hat{\eta}_{\beta}^{*}|^{2}}(z_{m}-\frac{\hat{\eta}_{\beta}}{\hat{\eta}_{\alpha}^{*}}z_{m}^{*})
\frac{\langle \Phi \mid T \left\{\hat{A}_{I}\left( x_{1}\right) \hat{A}_{I}\left( x_{2}\right) \hat{A}_{I}\left( x_{3}\right) . . . \exp \left( - \int_{- \infty}^{\infty}H_{i}\left( t \right) d t \right) \right\}\mid \Phi \rangle}{\langle \Phi \mid T \left\{\exp \left( - \int_{- \infty}^{\infty}H_{i}\left( t \right) d t \right) \right\}\mid \Phi \rangle},
\int_{\mathcal{V}\times \mathcal{V}}g \, d \pi = \lim_{n \rightarrow \infty}\int_{\mathcal{V}\times \mathcal{V}}g \, d \pi_n \leq \sup_{\gamma \in \mathcal{P}_{\mathrm{D}}}\int_{\mathcal{V}}g \, d \gamma.
\frac{(q)_{\infty}(q^2;q^2)_{\infty}}{(-q)_{\infty}(xq^2;q^2)_{\infty}}=\sum_{n\ge0}q^{n^2}(-x)^n(1-q^{2n+1})\frac{(q^2/x;q^2)_n}{(q^2x;q^2)_n}\sum_{0\le j\le n}\frac{(1+q^{j})(x;q^2)_j}{q^{j(j-1)/2}x^j(q^2/x;q^2)_j},
\begin{array}{rcl}B = -\eta_{33}{\cal C}_{11}{\cal C}_{44}-\eta_{11}\left({\cal C}_{11}{\cal C}_{33}-2{\cal C}_{13}{\cal C}_{44}-{\cal C}_{13}^2\right)-{\cal C}_{44}e_{15}^2-2{\cal C}_{11}e_{15}e_{33}+2\left({\cal C}_{13}+{\cal C}_{44}\right)e_{15}\left(e_{31}+e_{15}\right)-{\cal C}_{44}\left(e_{31}+e_{15}\right)^2\end{array}
\arraycolsep=1.4pt\def\arraystretch{1.7}\begin{array}{l}\displaystyle\hat{y}_1^h = x_1 + h^2\left(u^h - x_2\frac{R^h_{21}}{h}- x_3\frac{R^h_{31}}{h}\right) + h^2\beta_1^h\,, \hat{y}_i^h = hx_i + hv_i^h + h^2w^h x_i^\bot+ h^2\beta_i^h\,,\i=2,3\,,\end{array}
\begin{aligned}\frac{e^{-t}u\left(\xi(t)\right)-e^{-(t+h)}u(\xi(t+h))}{h}\leq \frac{e^{-t}\phi\left(\xi(t)\right)-e^{-(t+h)}\phi(\xi(t+h))}{h}\Longrightarrow \frac{1}{h}\int_t^{t+h}e^{-s}C_p\left|\dot{\xi}(s)\right|^{q}ds \leq \frac{e^{-t}\phi\left(\xi(t)\right)-e^{-(t+h)}\phi(\xi(t+h))}{h}\end{aligned}
\sum_{p \leq N}e(h g(p)) = \sum_{N_0 < n \leq N}\Lambda_1(n) e(h g(n)) + O \left( \frac{\pi(N)}{\log N}\right) \left( N_0 = \frac{N}{\log N}\right) = \sum_{N_0 < n \leq N}\frac{1}{\log n}e (h g(n)) + O \left( \frac{\pi(N)}{\log N}\right) = \sum_{n \leq N}\frac{1}{\log n}e (h g(n)) + O \left( \frac{\pi(N)}{\log N}\right).
(x+y)^{p^{m+n+1}}= (x^{p^m}+y^{p^m})^{p^{n+1}}+\sum_{i=1}^p \binom{p}{i}(x^{p^m}+y^{p^{m}})^{p^{n}(p-i)}(p^{n+1}L)^i &=(x^{p^m}+y^{p^m})^{p^{n+1}}\mod p^{n+2}.
\nu_{R}\left( B\right) :=\int 1_{B}\left( y/R\right) \nu \left( dy\right) ,B\in \mathcal{B}\left( \mathbf{R}_{0}^{d}\right) ,~\tilde{\nu}_{R}\left( dy\right):=w\left( R\right) \nu_{R}\left( dy\right) .
z_0^2 \varepsilon^2 <{1 \over{32}}\left( -32 \omega z_0^2 + 9l^2- 3 \sqrt{-64\omega z_0^2 l^2 + 9l^4 - 64 l^2 z_0^2 \omega^2}\right) / (\omega^2 l^2) \, .
2i\ |z_{13}|^{-3/2}|z_{14}|^{-3/2}|z_{15}|^{-1}|z_{23}|^{-3/2}|z_{24}|^{-3/2}|z_{25}|^{-1}|z_{35}|^{-1}|z_{45}|^{-1}\cdot\biggl(\frac{z_{14}z_{23}- z_{13}z_{24}}{z_{12}z_{34}}\biggr)
\mathbf{h}_{\rm v}^{\rm s}\!=\!\mathbf{P}^{\rm T}\mathbf{h}_{\rm v}\!\triangleq\!\big[\mathbf{e}_{\pi(1)}, \mathbf{e}_{\pi(2)}, \cdots, \mathbf{e}_{\pi(N_{\rm v})}\big]^{\rm T}\mathbf{h}_{\rm v}.
\partial_i\partial_j\bar{\eta}(x) = \sum_{k=1}^n\partial_iR_{jk}(x)\overline{\underline{D}_k\eta}(x)+\sum_{k,l=1}^nR_{jk}(x)R_{il}(x)\overline{\underline{D}_l\underline{D}_k\eta}(x).
\frac{W_+(z)W_+(z')}{e^{i\hbar \gamma^2 h^+(z-z')}}-\frac{W_+(z')W_+(z)}{e^{-i\hbar \gamma^2 h^-(z-z')}}=\frac{i\hbar}{2\gamma^2}\left(\eta^2-1+\frac{\gamma^2\hbar}{\pi}\right):\! e^{2i\gamma\varphi(z)}e^{2i\gamma\varphi(z')}\!:\!\partial_z\delta_{2\pi}(z-z'),
P^{(m)}_{}(\gamma_{}) =\exp\left(-\frac{N_{0}\gamma_{}/p}{\left(1-\left(\kappa^{2}_{T}(\omega+1)+\omega\right)\gamma_{}\right)}\right) &\times \sum^{\infty}_{k=n}\frac{\left(\frac{N_{0}\gamma_{}/p}{\left(1-\left(\kappa^{2}_{T}(\omega+1)+\omega\right)\gamma_{}\right)}\right)^{k}}{k!}.
P = \frac{(T - \lambda_{1}I) (T - \lambda_2I)}{(1 - \lambda_1)(1 - \lambda_2)}, Q = \frac{(T - I)(T - \lambda_2I)}{(\lambda_1 - 1)(\lambda_1 - \lambda_2)}, R = \frac{(T - I)(T - \lambda_1 I)}{(\lambda_2 - 1)(\lambda_2 - \lambda_1)}
g(n) \overline{g(n+1)}= \mu^2(n) \mu^2(n+1) \left( \frac{n+1}{n}\right)^{i s_{m+1}}+ O\left( \frac{1}{t_m}\right) &= \mu^2(n) \mu^2(n+1) + O\left( \frac{s_{m+1}}{t_{m+1}^{3/4}}\right) + O\left( \frac{1}{t_m}\right) &= \mu^2(n) \mu^2(n+1) + O\left( \frac{1}{t_m}\right),
v'_1 = \frac{v_2\Delta (v_1)}{\omega v_1\Delta (v_2)},~~\Delta (v'_1)=\frac{\Delta (v_2/(\omega v_1))}{\Delta (v_2)},~~v'_2=\frac{\Delta (v_1)}{\Delta (v_2)},~~\Delta(v'_2)=\frac{\omega v_1\Delta(v_2/(\omega v_1))}{\Delta(v_2)};
[\hat{x}_{1}^{\prime},\hat{x}_{2}^{\prime}]=i\alpha^{2}, \hspace{0.4cm}[\hat{p}_{1}^{\prime},\hat{p}_{2}^{\prime}]=i\alpha^{-2}, \hspace{0.4cm}[\hat{x}_{i}^{\prime},\hat{p}_{j}^{\prime}]=i\delta_{ij}.
T_{\mu \nu}^{brane,i}= V_i\, \omega(x,y) \,\delta (\vec{n}_i \cdot \vec{x}) \left( \begin{array}{cccccc}1 &-1 & -1 & -1 & -\cos^2 \varphi_i -\sin \varphi_i \cos \varphi_i & -\sin \varphi_i \cos \varphi_i -\sin^2 \varphi_i \end{array}\right ).
a_3 - a_1 a_2 + \tfrac{1}{3}a_1^{\, 3}= \tfrac{1}{15^3}\left( -25 + 50 \lambda + 15^2 (\lambda-2) \omega \right) = \tfrac{1}{135}\left( -1 - 18\omega + 2\omega^5 + 9\omega^6 \right)
{\frac{1}{B^{2}}}\left({\frac{1}{A^{2}}}-{\frac{\ell^{2}}{\tilde{B}^{2}r^{2}}}\right) = 0 , \qquad{\frac{1}{B^{2}}}\left({\frac{( A^{2})^{\prime}}{A^{4}}}-{\frac{( \tilde{B}^{2}r^{2})^{\prime}\ell^{2}}{\tilde{B}^{4}r^{4}}}\right) = 0 .
\sup_{\pi\in\Pi}E_{\mu}^\pi\left(\sum_{n=N+1}^\infty \beta^{n-1}r_i^{l}(s_n,a_n)\right)\le\sup_{\pi\in\Pi}E_{\mu}^\pi\left(\sum_{n=N+1}^\infty \beta^{n-1}w(s_n)\right).
\frac{d}{ds}\sigma^{\infty}( X^* ( \tilde{x_1},\tilde{x_2}, \tilde{x_3}, \tilde{m}; s)) = - \sigma^{\infty}(X^* (\tilde{x_1}, \tilde{x_2}, \tilde{x_3}, \tilde{m}; s)) \times
S^{(m)}_2=\frac{\varphi(VP_f)^kX(\log R)^{k+1}}{V(VP_f)^k\log X}J^{(m)}_k(F)+O\left(\frac{F_{max}^2\varphi(VP_f)^kX(\log X)^k}{V(VP_f)^kD_0}\right),
\lim_{N\rightarrow+\infty}\phi\Big(\big(K_N^*A^\frac{r}{N}K_N\big)^{N}\Big) = \ \lim_{N\rightarrow+\infty}\phi\left(\Big(\exp\big(\frac{1}{2N}H\big)\exp\big(\frac{r}{N}\log A\big)\exp\big(\frac{1}{2N}H\big)\Big)^N\right) = \ \phi\big(\exp(H+r\log A)\big).
h_{n}= - \frac{1}{2}\frac{1}{N^{2}}\frac{\partial^{2}}{\partial \lambda_{n}^{2}}+ \frac{1}{2}\{g^{2}\lambda_{n}^{4}- 2 g \lambda_{n}^{3}+ \lambda_{n}^{2}+ 2 g \lambda_{n}- 1 \}
u_{m}^{( i )}= \frac 1 8 ( v_{\alpha A}^{+}\tilde{\sigma}_{m}^{\alpha \beta}v_{\beta \dot{A}}^{-}) \gamma_{A \dot{A}}^{i}= - \frac 1 8 ( v_{A}^{\alpha -}\sigma_{m \alpha \beta}v_{\dot{A}}^{\beta +}) \gamma_{A \dot{A}}^{i}.
D(-\lambda)=W(\hat c_0,c_0)+\sum_{n=1}\lambda^n \int_{0\leq \xi_1\leq \xi_2\cdots \xi_n\leq 1}\hat c_0(\xi_n)\rho(\xi_n)\left(\prod_{j=1}^{n-1}(\xi_{j+1}-\xi_j) \rho(\xi_j)\right)c_0(\xi_1) d\xi_1d\xi_2\dots d\xi_n,
QR = \begin{pmatrix}0 \frac{-\pi_2}{\pi_1}\frac{-\pi_3}{\pi_1}\cdots \frac{-\pi_D}{\pi_1}0 1 0 \cdots 0 0 0 1 0 \cdots \vdots \vdots \vdots \ddots \ddots 0 0 \cdots \cdots 1 \end{pmatrix}.
E_{c}= \frac{2 n r_{\mathrm{B H}}^{n - 1}\Omega_{n}}{1 6 \pi G_{n + 2}}, \ 2 E - E_{c}= - \frac{2 n r_{\mathrm{B H}}^{n + 1}\Omega_{n}}{1 6 \pi G_{n + 2}l_{n + 2}^{2}}.
\cos \theta = \frac{\dot{\chi}}{\sqrt{\dot{\chi}^{2}+ \dot{\varphi}^{2}e^{- \alpha \chi}}}, \sin \theta = \frac{\dot{\varphi}e^{- \frac{\alpha \chi}{2}}}{\sqrt{\dot{\chi}^{2}+ \dot{\varphi}^{2}e^{- \alpha \chi}}}.
(i\xi)^{-3}P(-i\xi X,\xi;k,A)=\det\left(\frac{1}{i\xi}M_\xi(k,A)+X\left(\widetilde\Pi_\xi\Pi_\xi\right)(k,A)\right) &\qquad\qquad=-d_3(\xi;k,A)X^3+d_2(\xi;k,A)X^2+d_1(\xi;k,A)X-d_0(\xi;k,A);
(q',10)=(g_1',10)=(b_2',d_0d_1q' g_1')=1, &X(b_2'/d_0 d_1 q' g_1'+\nu_2)\in\mathbb{Z}, &\exists\, b_1',g_2\Bigl|\frac{a_1}{X}-\frac{b_1'}{q' d_0g_2}\Bigr|\le \frac{E_0}{X},\,(b_1',d_0q' g_2)=1,\,g_2\sim G_2.
\frac{\pi}{2q}=\pi\frac{1}{q}-\frac{\pi}{2q}\le\pi\mathcal{I}-\frac{\pi}{2q}\le\pi\left(\frac{1}{p}+\Re\gamma\right)+\frac{\theta\omega(t)\Im\gamma}{2}\pm\frac{\theta\omega(t)x}{2}&\le\pi\mathcal{S}+\frac{\pi}{2q}\le\pi\left(1-\frac{1}{q}\right)+\frac{\pi}{2q}=\pi-\frac{\pi}{2q}
\mathfrak{A}(n;q_1,q_2,q_2')=\sideset{}{^\star}\sum_{a,b\bmod{q_1}}\;\sideset{}{^\star}\sum_{c,d\bmod{q_1}}e_{q_1}(a+b+c+d) &\times\sum_{\alpha\bmod{q_1}}e_{q_1}(\alpha_1\ell\bar{q}_2^3\bar{a}\bar{b}-\alpha_1\ell\bar{q_2'}^3\bar{c}\bar{d}+\alpha_1 \bar{q}_2\bar{q}_2' n).
\eta'_{\tau}(\rho)=\frac{\rho^{-3}r^2}{1+\rho^{-2}r^2}-\frac{\tau}{\rho}=\frac{(1-\tau)r^2-\tau\rho^2}{\rho(\rho^2+r^2)}\begin{cases}>0 1<\rho<\rho_0, =0 \rho=\rho_0, <0 \rho>\rho_0.\end{cases}
\hat{L}^{-}{}_{\alpha}| \mathrm{p h y s}\rangle = 0 , \qquad ( \Gamma^{n}\hat{p}_{n}\hat{L})_{\alpha}| \mathrm{p h y s}\rangle = 0 , \qquad \hat{p}^{2}| \mathrm{p h y s}\rangle = 0 . \qquad
L_{i}(\beta)=\frac{\lambda(Y_{i}|X_{i})}{\sum_{j:Y_{j}\ge Y_{i}}\lambda(Y_{i}|X_{j})}=\frac{\lambda_{0}(Y_{i})\theta_{i}}{\sum_{j:Y_{j}\ge Y_{i}}\lambda_{0}(Y_{i})\theta_{j}}=\frac{\theta_{i}}{\sum_{j:Y_{j}\ge Y_{i}}\theta_{j}},