harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta_{\varepsilon ,\overline{\varepsilon}}^{(0)}=\left(1-\frac{\sigma^2}2\right) \cdot \frac{m-i\varepsilon\widehat{p}}{2m}\cdot \frac{m-i \overline{\varepsilon}\overline{p}}{2m}\cdot \left( 1-\sigma_p^2\right) =\Psi_{\varepsilon ,,\overline{\varepsilon}}^{(0)}\cdot \overline{\Psi}_{\varepsilon ,\overline{\varepsilon}}^{(0)},
G_{M N}d x^{M}d x^{N}= H_{p}^{-{\frac{4 ( n - 1 )}{( p + n ) \Delta_{p}}}}\left[ - d t^{2}+ d{\bf x}\cdot d{\bf x}\right] + H_{p}^{\frac{4 ( p + 1 )}{( p + n ) \Delta_{p}}}\left[ d \tilde{y}^{2}+ g_{i j}d y^{i}d y^{j}\right] ,
\mathbb{E}\bigg[\frac{\rho}{|\hat{h}_i|^2}\Big||\hat{h}_i|^2\ge\mu_i\bigg] =\frac{\rho}{\mathbb{P}(|\hat{h}_i|^2\ge \mu_i)}\int\limits_{\mu_i}^{\infty}\frac{e^{-x/\vartheta_i}}{\vartheta_i x}\mathrm{d}x=-\frac{\rho}{(1-\xi)\vartheta_i}\mathrm{li}(1-\xi),
\pi_g\Delta_gx =\pi_g\left( \sum_{h\in G}f_{gh}(x) \tau_{h}\right)=\sum_{h\in G}f_{gh}(x)\pi_g \tau_{h}=\sum_{h\in G}f_{gh}(x)\sum_{u\in G}f_u(\tau_h) \tau_{gu}=\sum_{h\in G}f_{gh}(x)\sum_{u\in G}f_{gu}(\tau_{gh}) \tau_{gu}=\sum_{h\in G}f_{gh}(x) \tau_{gh}=x,\forall x \in \mathcal{X}.
S_{f, \tau, \Lambda}x=\frac{o(\Lambda)}{o(G)}\sum_{\mu\in \Lambda^0}f(\pi(\mu)^{-1}\tau) \pi(\mu)x=\frac{o(\Lambda)}{o(G)}f(\tau)x+\frac{o(\Lambda)}{o(G)}\sum_{\mu\in \Lambda^0\setminus \{(0,0)\}}f(\pi(\mu)^{-1}\tau) \pi(\mu)x=x, \forall x \in \mathbb{C}^{o(G)}.
\prod_{i\in I}\prod_{j=1}^{v^i}\frac{\vartheta(z^{(i)}_j+\lambda_i)}{\vartheta(z^{(i)}_j)\vartheta(\lambda_i)}\cdot \prod_{i\in I}\prod_{j=1}^{v^i}\frac{\vartheta(z^{(i)}_j+\frac{c_{ki}}{2}\hbar)}{\vartheta(z^{(i)}_j-\frac{c_{ki}}{2}\hbar)}.
{{Q}}_1={\rm d}Z+2 L_xZ_x{\rm d}t+2\left(x+\varepsilon\right)^{2\gamma}L_yZ_y{\rm d}t+\tau L_{xx}Z{\rm d}t,{{Q}}_2= -L_t Z-Z_{xx}-\left(x+\varepsilon\right)^{2\gamma}Z_{yy}-L_x^2 Z-\left(x+\varepsilon\right)^{2\gamma}L_y^2 Z-\frac{\sigma}{(x+\varepsilon)^2}Z,{{Q}}= -(\tau-1) L_{xx}Z+\left(x+\varepsilon\right)^{2\gamma}L_{yy}Z.
\|r_3^{1}\| \leq 2 \int_{0}^{k}\|u_{tt}(\zeta)\|d\zeta, \|r_3^{n}\| \leq 4 \left( \int_{0}^{2k}\|u_{tt}(\zeta)\|d\zeta + k\int_{t_{n-1}}^{t_{n}}\|u_{ttt}(\zeta)\|d\zeta \right), n\geq 2.
W = (n-2-(n-3))\rho_G - \bigoplus_{i=1}^L \lambda(i) - \bigoplus_{i=1}^{\frac{1}{2}(m_1p-n)}\lambda(i) = \rho_G - \Big( \bigoplus_{i=1}^L \lambda(i) + \bigoplus_{i=1}^{\frac{1}{2}(m_1p-n)}\Big)
\pounds =\dot{x}\cdot p-\frac{1}{2}A^{22}p^{2}+\tilde{\theta}_{a}\gamma\cdot{p}\partial_{\tau}\theta^{a}\Rightarrow \frac{1}{2A^{22}}\left( \dot{x}^{\mu}+\tilde{\theta}_{a}\gamma^{\mu}\partial_{\tau}\theta^{a}\right)^{2}.
f(x)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f_{t}(x)dt=\frac{1}{2\pi}\int\limits_{E}f_{t}(x)dt=\frac{1}{2\pi}\int\limits_{E}\sum\limits_{k=-\infty}^{\infty}a_{k}(t)\Psi_{k,t}(x)dt
\prod_{k=0}^MA_k^+-\prod_{k=0}^M A_k^-=\sum_{\ell=0}^M \bigg(\prod_{k=0}^{\ell-1}A_k^-\bigg)\big(A_\ell^+-A_\ell^-\big)\bigg(\prod_{k=\ell+1}^M A_k^+\bigg),
e^{-t\mathfrak{D}_{{\rm left},a}}u(x) =\exp\bigg[-\int_{x-t}^{x}a(y)\,dy\bigg]u(x-t) e^{-t\mathfrak{D}_{{\rm right},a}}v(x) =\exp\bigg[-\int_{x}^{x+t}a(y)\,dy\bigg]v(x+t),
\Delta'(u,\psi) :=\min\{\Delta(u,\psi^u),\psi^u(V(u,\psi^u))\}, V'(u,\psi) :=(1-t)u+t(V(u,\psi^u)), \psi^u(v) :=\frac{\psi((1-t)u+tv)^2}{16d}.
\frac{1}{2}\varepsilon_{\mu \nu \rho \sigma}\gamma_5 (\{(\eta^2 +\Phi^2),\hat F_{\rho \sigma}\}+\hat D_{[\rho}\Phi \: \: \hat D_{\sigma]}\Phi) - (\{(\eta^2 +\Phi^2),\hat F_{\mu \nu}\}+\hat D_{[\mu}\Phi \: \: \hat D_{\nu]}\Phi) =0.
\partial_{a_{1}}\ldots \partial_{a_{n}}\delta_{n}\phi = \epsilon_{a_{1}}^{\bar{a}_{1}}\ldots \epsilon_{a_{n}}^{\bar{a}_{n}}\bar{\partial}_{\bar{a}_{1}}\ldots \bar{\partial}_{\bar{a}_{n}}( \rho_{0}^{\bar{c}}\bar{\partial}_{\bar{c}}\phi ) \; + \;{\cal O}( \phi^{2}) .
a^{-2\Delta}\langle \sigma(z^a_j) \rangle_{D_a}= a^{-\lambda/4}\exp[-2 \lambda\tilde \mu_{D_a}(\gamma: N_{\gamma}(z^a_j) )] & = a^{-\lambda/4}\big[ c_1 ( a^{-1}r(D_a,z^a_j) )^{1/4}[ 1 + O ( ( a^{-1}r(D_a,z^a_j) )^{-u}) ] \big]^{-\lambda}.
S_{n , l}^{+}= ( x + i t z )^{l}z^{n - l}\psi^{+}= \left( \frac{\alpha - 2 i D_{n}}{2 H_{n}}\right)^{l}S_{n , 0}^{+}, \quad S_{n , l}^{-}= ( S_{n , l}^{+})^{*},
\frac{\partial|H|^2}{\partial t}= \Delta |H|^2 -2|\nabla H|^2 +2nK|H|^2 + 2R_2 \frac{\partial|h|^2}{\partial t}= \Delta |h|^2 -2 |\nabla h|^2 + 2 R_1 + 4K|H|^2 -2nK|h|^2,
= ( 1 \frac{1}{2}- 1 \frac{1}{2}) + ( 2 \frac{1}{3}- 1 \frac{1}{3}) + ( 3 \frac{1}{4}- 1 \frac{1}{4}) + ( 4 \frac{1}{5}- 1 \frac{1}{5}) + ( 5 \frac{1}{8}- 1 \frac{1}{6})
m ={\textstyle \frac{1}{2}}\left( \begin{array}{c c}{m^{1}}&{m^{2}+ m^{3}}\\{}&{}\\{m^{2}- m^{3}}&{- m_{1}}\\ \end{array}\right) ={\cal Q}= \left( \begin{array}{c c}{\frac{\delta}{2}r}&{\beta q}\\{\gamma p}&{- \frac{\delta}{2}r}\\ \end{array}\right) .
\mathrm{Cov}(f,g) \geq c \sum_{i=1}^n \frac{I_i(f)}{\sqrt{\log \frac{e^3}{nI_i(f)^2}}}\frac{I_i(g)}{\sqrt{\log \frac{e^2}{I_i(g)}}}= \sum_{i=1}^n \psi_n(I_i(f)) \psi(I_i(g)),
\delta \sigma_{i}= \kappa \sqrt{\frac{\pi}{2 k}}\; e^{i ( 2 | p_{i}| + 1 ) \pi / 4}\; \frac{\sqrt{- k \eta}}{\bar{a}_{i}}\;{H_{| p_{i}|}}^{( 1 )}\left( - k \eta \right) \, .
|1-\alpha+\alpha\tfrac{1-\gamma \beta}{1+\gamma \beta}| \leq|1-\alpha|+|\alpha\tfrac{1-\gamma \beta}{1+\gamma \beta}| &=1-\alpha+\alpha\tfrac{1-\gamma \beta}{1+\gamma \beta}&\leq 1-\alpha+\alpha\tfrac{1-\gamma\sigma}{1+\gamma\sigma}=r_\sigma
\mathcal{L}_{\mathcal{I}_l}u_\epsilon(t)=U_0(t)\mathcal{L}_{\mathcal{I}_l}u_0+i\sum_{j=1}^N\left(S\mathcal{L}_{\mathcal{I}_l}V_\epsilon(x_j,\cdot)u_\epsilon\right)(t)-i\sum_{j<k}\left(S\mathcal{L}_{\mathcal{I}_l}W_\epsilon(x_j,x_k)u_\epsilon(\cdot)\right)(t).
r^{*,\pi}_n(x_n|y^{n-1}_{n-M},y_n)=\Big(\frac{q_n(y_n|y^{n-1}_{n-M},x_n)}{\nu^{\pi}_{n}(y_n|y^{n-1}_{n-J})}\Big){\pi}_{n}(x_n|y^{n-1}_{n-J}).
\frac{R}{\sqrt{1+(\alpha-\epsilon)^{2}}}-\frac{R}{\sqrt{1+(\alpha+\epsilon)^{2}}}=\frac{R}{\sqrt{1+\alpha^{2}}}\left(1+O(\epsilon)\right)-\frac{R}{\sqrt{1+\alpha^{2}}}\left(1+O(\epsilon)\right) &=O(R \epsilon).
p_{0}= 1 , \quad p_{1}= w_{1}, \quad p_{2}= \frac 1{2 !}( w_{1}^{2}+ w_{2}) , \quad p_{3}= \frac 1{3 !}( w_{1}^{3}+ 3 w_{1}w_{2}+ 2 w_{3}) , \quad \ldots
(-1)^{k}\left( \bigg( \sum_{n=1}^\infty u_m(n)\, q^n \bigg) \bigg( \sum_{n=-k}^{k}(-1)^{n}\, q^{n(3n-1)/2}\bigg) -\sum_{n=0}^\infty C_m(n)\,q^n\right) & = \left( \sum_{n=0}^\infty C_m(n)\,q^n \right) \left( \sum_{n=0}^\infty \widetilde{P}_k(n)\,q^n\right),
\pi_n = \inf_{|h| \leq \epsilon a_n}\P_h\left( \left|\tfrac{S_{r_n}}{a_n}\right| \leq \epsilon, \tfrac{S_j}{a_n}\in I, \xi_j\leq a_n, j \in [r_n]\right) \bar{\pi}_n = \inf_{|h| \leq \epsilon a_n}\P_h\left( \tfrac{S_{\Delta_n}}{a_n}\in [c,d], \tfrac{S_j}{a_n}\in I, \xi_j \leq a_n, j \in [\Delta_n] \right)
\frac12\int_{M}(|\partial_tu(t,\cdot)|^2+|\nabla u(t,\cdot)|^2) & = E(u(t,\cdot)) +\rho_1\log\int_{M}e^{u(t,\cdot)-\overline{u}(t)}+\rho_2\log\int_{M}e^{-u(t,\cdot)+\overline{u}(t)}& \leq E(u(t,\cdot)) +(\rho_2-\rho_1)\overline{u}(t) +C &= E(u(0,\cdot)) +(\rho_2-\rho_1)\overline{u}(0) +C & \leq C \mbox{for all}t\in[0,T_0),
|(j_{2}{}_{3}j_{1})jm\rangle=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}\sum_{m_{3}=-j_{3}}^{j_{3}}C_{m_{2},m_{3},m_{23}}^{j_{2},j_{3},j_{23}}C_{m_{23},m_{1},m}^{j_{23},j_{1},j}|j_{1}m_{1}j_{2}m_{2}j_{3}m_{3}\rangle
+Y_1(z)^6\cdot{}_2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{Y_1(z)^{12}}{27}\right)=\frac{9\sqrt{3}\Gamma\left(\frac{1}{3}\right)^3}{4\pi \sqrt[3]{2}}.
D_{ij}^{1}=\frac{1}{2}M_{kkij}^{(2)}-\frac{1}{4}M_{kkll}^{(2)}\delta_{ij}D_{ij}^{2}=\frac{1}{2}M_{ijkk}^{(2)}-\frac{1}{4}M_{kkll}^{(2)}\delta_{ij}
\textsf{R}'(z) = \rho_N(z^2)\left[\sum_i e_{i,i}\otimes e_{i,i}+ \frac{q(1-z^2)}{1-q^2z^2}\left(\sum_{i<j}q^{(2j-2i-N)/N}+\sum_{i>j}q^{(2j-2i+N)/N}\right)e_{i,i}\otimes e_{j,j}\right. &\left. + \frac{z(1-q^2)}{1-q^2z^2}\left(\sum_{i<j}z^{(2j-2i-N)/N}+\sum_{i>j}z^{(2j-2i+N)/N}\right)e_{i,j}\otimes e_{j,i}\right] \,.
\widetilde{V}^{(0^{2})}\big(x;\omega ,1,- \Vert \psi_{0}\Vert^{2}\big) =\frac{\omega^{2}}{4}x^{2}+\frac{3}{4x^{2}}+\frac{4\omega}{\omega x^{2}+2}-\frac{16\omega}{( \omega x^{2}+2)^{2}}= V^{( 1,- )}(x;\omega ,2)+2\omega
\mathrm{\boldmath ~ \Psi ~ \unboldmath}= \left( \begin{array}{c}{\Psi_{+ 1}({\bf x})}{\Psi_{- 1}({\bf x})}\end{array}\right) \; , \; \; \Psi_{\zeta}({\bf x}) = \left( \begin{array}{c}{\chi_{\zeta}({\bf x})}{\varphi_{\zeta}({\bf x})}\end{array}\right) \; ,
\Delta_2(w) =\Delta_2(s_{\alpha_1})\Delta_2(s_{\alpha_2})\cdots\Delta_2(s_{\alpha_k}) &=\left({s''_{\alpha_1}}^{(1)}{s''_{\alpha_1}}^{(2)}\right)\left({s''_{\alpha_2}}^{(1)}{s''_{\alpha_2}}^{(2)}\right)\cdots\left({s''_{\alpha_k}}^{(1)}{s''_{\alpha_k}}^{(2)}\right)
\int_{\Omega}f_{|\tau}(x) I_{\tau<\infty}\,\mu(dx)=\int_{\Omega}f(x) I_{\tau<\infty}\,\mu(dx), begin{align*}10pt] \int_{\Omega}f_{|\tau}(x) \,\mu(dx)=\int_{\Omega}f(x) \,\mu(dx).
\begin{cases}u_t+u\cdot\nabla u-\mu\Delta u+\nabla P=-\lambda\nabla\cdot\big(\nabla d\odot\nabla d-\frac12{|\nabla d|^2}\mathbb{I}_3\big), \nabla \cdot u=0, d_t+u\cdot\nabla d=\gamma\big(\Delta d+|\nabla d|^2d\big),\end{cases}
\!\!\!\!\Pi_{\rm SIC}(v,n) = \Pi_{\rm NC}(v) &\!\!\!\!+ \sum_{i=1}^n \left(\prod_{j=0}^{i-1}\left(1-\Pi_{\rm C}(v,j)\right)\right) \left(\prod_{j=0}^{i}\Pi_{\rm D}(v,j)\right) \Pi_{\rm C}(v,i).
J_{\alpha}^+(x) := \frac{\pi}{2}\frac{J_{-\alpha}(2x) + J_{\alpha}(2x)}{\cos(\pi \alpha/2)}, J_{\alpha}^-(x) := \frac{\pi}{2}\frac{J_{-\alpha}(2x) - J_{\alpha}(2x)}{\sin(\pi \alpha/2)}, \tilde{K}_{\alpha}(x) = 2\cos\left(\frac{\pi}{2}\alpha\right) K_{\alpha}(2x),
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{xx}g_{xx}dxdy=\int_{-\infty}^{\infty}f_{x}g_{xx}|_{-\infty}^{\infty}dy-\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{x}g_{xxx}dxdy=-\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{x}g_{xxx}dxdy
\sum_{k_1 \leq M}\frac{\mu^2(k_1)\chi(k_1)}{k_1 \tau(k_1)}\prod_{p\mid k_1}\left(1 + \frac{1}{p}\right)^{-1}= \sum_{k_1 \leq M}\frac{\mu^2(k_1)\chi(k_1)}{k_1 \tau(k_1)}\sum_{d \mid k}\mu(d)f(d),
\displaystyle\prod_{\mu =1}^{m}\oint_{C^{(N)}}{dx_{\mu}\over 2\pi i}=\displaystyle \prod_{\mu =1}^{m-1}\oint_{C'{}^{(N)}}{dx_{\mu}\over 2\pi i}\left( \oint_{C'{}^{(N)}}{dx_m\over 2\pi i}+m{\rm Res}_{x_m=z_N\tau^-1}\right),
\bigoplus_{k=0}^{\frac{n-1}{2}}f^{k}_{0}\otimes \dots \otimes f^{k}_{2k+1}\mapsto \sum_{k=0}^{\frac{n-1}{2}}\frac{b^{\frac{n+1}{2}+k}}{(2k+1)!}f^{k}_{0}df^{k}_{1}\wedge \dots \wedge df^{k}_{2k+1}\ .
J_{3}= \sum_{r = 1}^{m}\left[ 2 \left( \partial \theta_{r}\right)^{2}\left( \partial \theta_{r + 1}- \partial \theta_{r - 1}\right) + \left( \partial \theta_{r}\right) \left( \partial^{2}\theta_{r + 1}- \partial^{2}\theta_{r - 1}\right) \right] ,
\Delta_{\nu}(x)>\frac{1}{\sqrt{\pi}}\frac{\Gamma\left(\nu+\frac{1}{2}\right)x^{2\nu}}{\Gamma(\nu+2)\Gamma(2\nu+1)}+\frac{1}{\sqrt{\pi}}\frac{\Gamma\left(\nu+\frac{3}{2}\right)x^{2\nu+2}}{\Gamma(\nu+3)\Gamma(2\nu+2)}=\frac{\left(\frac{1}{4}x^{2}\right)^{\nu}}{\Gamma(\nu+1)\Gamma(\nu+2)}+\frac{2\left(\frac{x}{2}\right)^{2\nu+2}}{\Gamma(\nu+1)\Gamma(\nu+3)},
\int_{B_R}\int_{B_R}\, \left(\frac{|\delta_h u(x)|^{p}}{|h|^{\gamma+t}}+\frac{|\delta_h u(y)|^{p}}{|h|^{\gamma+t}}\right)\, \frac{|\eta(x)-\eta(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy &\le \frac{C}{(R-r)^{s\,p}}\,\left(\frac{R}{R-r}\right)^p\,\frac{1}{s\,(1-s)}\,\int_{B_R}\left|\frac{\delta_h u}{|h|^{\gamma}}\right|^p\,dx,
\left|\frac{ds_{i}(t)}{dt}\right| \leq\left|p-2\right|\sqrt{n}\cdot\begin{cases}1 |s_{i}(t)|\leq t ~, (p-1)^{-\frac{1}{2}}\left(t/|s_{i}(t)|\right)^{\frac{p-2}{2}}\end{cases}
{\cal B}^{(\varphi,t)}_{WZW}= \frac{\ell}{8 \pi G}\int_{\cal V}Tr [ \partial_\varphi (Q_2^{-1}\partial_{[r}Q_2 Q^{-1}\partial_{t]}Q) + \partial_t (Q_2^{-1}\partial_{[\varphi}Q_2 Q^{-1}\partial_{r]}Q)] dr\, d\varphi\, dt \quad .
x\in\, \mathfrak{g}[1]\longmapsto \Phi_0(x)={\mathcal{L}}_{\varrho(x)}\in\mathfrak{X}_0(E)[1] x\wedge y\in\wedge^2 \mathfrak g[1]\longmapsto \Phi_1(x,y)=\iota_{\chi(x,y)}\in\mathfrak{X}_{-1}(E)[1]
Q_{2323}=\frac{3e^{2 q_3+q_2-3q_1}+2e^{2 q_3-2 q_1}-2e^{2q_3-q_2-q_1}-2 e^{2q_3-2q_2}-e^{q_3+q_2-2q_1}+e^{q_3-q_1}+e^{q_3-q_2}}{(1+e^{q_3-q_2})\Delta_3},
S_0 \equiv \int_{-\infty}^{\infty}dt \int_{-\infty}^{\infty}dx_1[ \bar{\psi}(x_1,t) ({\gamma}^{\mu}i{\hbar}c{\partial}_{\mu}- mc^2){\psi}(x_1,t) - e\bar{\psi}(x_1,t){\gamma}^{\mu}{\psi}(x_1,t) \cdot \bar{A}_{\mu}^{ext}
\left[ N \frac{\partial}{\partial N}+ 2 \right] F ( N , 0 , 1 , g , 0 , 0 , \ldots ) = \sum_{( n_{1}, \ldots , n_{k})}\beta_{n_{1}\cdots n_{k}}( g ) \frac{\partial F}{\partial g_{n_{1}\cdots n_{k}}}( N , 0 , 1 , g , 0 , 0 , \ldots ) .
\|(\partial^{\alpha_1}h_1)\cdot\dots\cdot(\partial^{\alpha_m}h_m)\|_{L^2}\leq \Big(\sum^m_{i=1}\|h_i\|_{H^s}\Big)\Big(\prod_{j\neq i}\|\hat h_j\|_{L^1}\Big)\leq C\Big(\sum^m_{i=1}\|h_i\|_{H^s}\Big)\Big(\prod_{j\neq i}\|h_j\|_{H^\kappa}\Big).
\widetilde{x}_n^{(M)}(t)-\zeta_{n-1}= \max\big\{\zeta_n-\zeta_{n-1}+B_n(t),\sup_{0\leq s \leq t}(\widetilde{x}_{n-1}^{(M)}(s)-\zeta_{n-1}+B_n(t)-B_n(s))\big\},
{\frac{1}{3 2 \omega^{2}}}\left( -{\frac{3 2 \omega z_{0}^{2}}{l^{2}}}+ 9 + 3 \sqrt{- 6 4 \omega{\frac{z_{0}^{2}}{l^{2}}}+ 9 - 6 4{\frac{z_{0}^{2}}{l^{2}}}\omega^{2}}\right) < z_{0}^{2}\varepsilon^{2}<{\frac{1}{8 \omega + 3}}\left( - 3 -{\frac{5 z_{0}^{2}}{l^{2}}}\right) \, .
\overline{A}=\sum_{\alpha=1}^{D}|c_{\alpha}|^{2}A_{\alpha\alpha}+i\hbar lim_{\tau\rightarrow\infty}[\sum_{\alpha\ne\beta}^{D}\frac{c_{\alpha}^{*}c_{\beta}A_{\alpha\beta}}{E_{\beta}-E_{\alpha}}(\frac{e^{-i(E_{\beta}-E_{\alpha})\tau/\hbar}-1}{\tau})].
\mathrm{T r}d \Phi d \Phi^{\dagger}= \sum_{i}d \lambda_{i}^{2}+ \sum_{\alpha , \beta}d t_{\alpha}d t_{\beta}\left( \sum_{i}\lambda_{i}\alpha_{i}\right) \left( \sum_{j}\lambda_{j}\beta_{j}\right) \mathrm{T r}E_{\alpha}E_{\beta}.
p_{X_{T^c,k}|W_{\mathcal{I}}, X_{\mathcal{I}}^{k-1}, Y_{T^c}^{k-1}}\stackrel{}{=}p_{X_{T^c,k}|W_{T^c}, X_{T^c}^{k-1}, Y_{T^c}^{k-1}}\stackrel{}{=}s_{X_{T^c,k}|W_{T^c},X_{T^c}^{k-1},Y_{T^c}^{k-1}}
f_1\otimes \bar f_2 ((\exp(r^k Z), \exp(s^k Z))(x_1, x_2))=f_1(\exp (r^k Z) x_1)\cdot \bar f_2(\exp(s^k Z) x_2) = \chi_1(\exp(r^k Z)) \overline{\chi_2(\exp(s^k Z))}\cdot f_1\otimes \bar f_2(x_1, x_2) = (\chi_1^{r^k}\cdot\chi_2^{-s^k})(\exp(Z))\cdot f_1\otimes \bar f_2(x_1, x_2).
S_A = -\frac{1}{4}\int d^6x \sqrt{^6g}g^{AB}g^{MN}F_{AM}F_{BN}\simeq -\frac{1}{4}\epsilon^2 \int_1^a d\phi \int d^4x\sqrt{-\eta}\eta^{\mu\nu}\eta^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}.
L_{\mathrm{AdS}}^{\left( 5\right)}=\kappa\left( \frac{1}{5l^{5}}\epsilon_{a_{1}\cdots a_{5}}e^{a_{1}}\cdots e^{a_{5}}+\frac{2}{3l^{3}}\epsilon_{a_{1}\cdots a_{5}}R^{a_{1}a_{2}}e^{a_{3}}\cdots e^{a_{5}}+\frac{1}{l}\epsilon_{a_{1}\cdots a_{5}}R^{a_{1}a_{2}}R^{a_{3}a_{4}}e^{a_{5}}\right) ,
J_{n n^{\prime}}( r ) \equiv \sum_{m = 0}^{\mathrm{m i n}( n , n^{\prime})}\frac{n ! n^{\prime}!}{m ! ( n - m ) ! ( n^{\prime}- m ) !}[ i ~ \mathrm{s g n}( e H ) r ]^{n + n^{\prime}- 2 m}.
\hat{q}=i\hbar(-1)^{\varepsilon_\alpha}\bigg( \frac{\delta_l}{\delta\psi^\alpha}- (-1)^{\varepsilon_i\;\varepsilon_\alpha}\;\cdot\;\; (\Gamma^{''-1})^{ij}\;\, \bigg(\frac{\delta_l}{\delta\varphi^j}\frac{\delta_l\Gamma}{\delta\psi^\alpha}\bigg) \frac{\delta_l}{\delta\varphi^i}\bigg)\cdot \rule{2.1cm}{0cm}
\left. + \left( b_{1}\left( \kappa_{s}+ \kappa_{t}\right) + b_{2}\kappa_{s}+ \frac{\Lambda}{3}b_{3}\right) \dot{\phi}^{2}+ \frac{\Lambda}{3}b_{4}\ddot{\phi}+ c_{1}\dot{\phi}^{4}+ c_{2}\dot{\phi}^{2}\ddot{\phi}+ c_{3}\ddot{\phi}^{2}\right] + ( \mathrm{s . t .}) \; ,
\left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right)_{\xi \eta}= \left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right)_{\xi}\otimes \left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right)_{\eta}
\mathbf{v}_{i,k}^H\underset{\tilde{\mathbf{H}}_{i,k}}{\underbrace{\begin{bmatrix}\mathbf{h}_{1,ik}\cdots \mathbf{h}_{i-1,ik}\mathbf{h}_{i+1,ik}\cdots \mathbf{h}_{M,ik}\end{bmatrix}}}=0.
T_N(\varphi):= \left(\begin{array}{cccc}\varphi_0 \varphi_{-1}\ldots \varphi_{-N}& \varphi_1 \varphi_0 \ldots \varphi_{-N+1}& \vdots \vdots \ddots \vdots & \varphi_N \varphi_{N-1}\ldots \varphi_0\end{array}\right),
(U_{\lambda}(0), D_0^2u(0))=(D_0^2U_{\lambda}(T),u(T)) +(D_0U_{\lambda}(T), D_0u(T))+(U_{\lambda}(T),D_0^2(T)) -((D_1^2+x_1^2D_n^2)U_{\lambda}(T),u(T)).
D_{S, x}T_\rho f = \sum_{T \supseteq S}\rho^{|S|}T_\rho D_{S, x}[f^{=T}] = \rho^{|S|}\sum_{T \subseteq [n]}T_\rho D_{S, x}[f^{=T}] = \rho^{|S|}T_\rho D_{S, x}f
|\mathcal{F}_2| =\sum_{i=0}^{(s-2)/2}\binom{(s+d-3)/2}{\left\lfloor i/2 \right\rfloor}\binom{(s+d-3)/2-\left\lfloor i/2 \right\rfloor}{(s-2)/2-i}, |\mathcal{F}_1|+|\mathcal{F}_3| =\sum_{i=0}^{(s-2)/2}\binom{(s+d-5)/2}{\left\lfloor i/2 \right\rfloor}\binom{(s+d-1)/2-\left\lfloor i/2 \right\rfloor}{(s-2)/2-i},
C = \left\{\lambda_1\varpi_1 + \lambda_2\varpi_2 +\lambda_3\varpi_3\in P^+\ \middle\vert\ \lambda_1+2\lambda_2 +\lambda_3< 3\right\}= \{0,\varpi_1,\varpi_2,\varpi_3,2\varpi_1,2\varpi_3,\varpi_1+\varpi_3\}.
\phi ( x , t ) = \Phi_{1}( x - \alpha \t + \int_{- \infty}^{\sigma ( x , t )}\frac{{\Phi^{\prime}}_{2}^{2}- p \Phi_{2}^{\prime}}{a + q \Phi_{2}^{\prime}}\d \tilde{\sigma}) \ + \Phi_{2}( x - \beta \t - \int_{\rho ( x , t )}^{\infty}\frac{{\Phi^{\prime}}_{1}^{2}+ q \Phi_{1}^{\prime}}{a - p \Phi_{1}^{\prime}}\d \tilde{\rho}) .
[v(\cdot, S^1), S^2]_t =[M^v, M^2]_t &=\left[ \int_0^\cdot \nabla v^*(r, S^1_r) \mathrm d M^1_r, M^2\right] &=\int_0^t \nabla v^* (r, S^1_r) \mathrm d[M^1, M^2 ]_r,
j^{k}( x^{0},{\vec{x}}) \; = \; e \, \int d t \, \sum_{a = 1}^{N}\, \frac{d x_{a}^{k}( t )}{d t}\, \delta ( x^{0}- t ) \, \delta^{( 2 )}({\vec{x}}-{\vec{x}}_{a}( t ) )
Y_t^i = \xi^i + \int_t^T \Big(f^i(s, 0, 0)+f^\prime(s, \sum_{j=0}^{i}Y_s^j, \sum_{j=0}^{i}Z_s^j) - f^\prime(s, \sum_{j=0}^{i-1}Y_s^j, \sum_{j=0}^{i-1}Z_s^j) \Big)dA_s + \int_t^T g_s d \langle \widetilde{N}^{i}\rangle_s -\int_t^T \big(Z_s^i dM_s +d\widetilde{N}_s^{i}\big),
E_{C}( a , T ) = \frac{5 \, \xi^{2}}{3 2 a \pi}+ \frac{8 a^{3}\pi^{3}\xi^{2}{T^{4}}}{4 5}+ \frac{5 1 2 a^{5}\pi^{5}\, \xi^{2}\,{T^{6}}}{9 4 5}+{\cal O}( T^{7})
\limsup_{\varepsilon \to 0}E[\mu_{\varepsilon,h}- \mu] = \limsup_{\varepsilon \to 0}\left( E[\mu_{\varepsilon,h}] + E[\mu]- 2 \iint G(p,q) d\mu(p) d\mu_{\varepsilon,h}(q) \right) \leq E[\mu] + E[\mu] - 2 E[\mu] = 0,
L_{1 1}^{( \pm )}= q^{\pm \frac{1}{2}}q^{\pm J}= q^{\pm \left( \begin{array}{c c}{1}&{0}\\{0}&{0}\\ \end{array}\right)}, L_{2 2}^{( \pm )}= q^{\pm \frac{1}{2}}q^{\mp J}= q^{\pm \left( \begin{array}{c c}{0}&{0}\\{0}&{1}\\ \end{array}\right)},
\mathbf{1}\left[{B_{T_0}^{r\not \to m}}\right](\omega) =\mathbf{1}\left[{B_{T_0}^{(r,0)\not\to m}}\right](\omega) &= \prod_{j=1}^k \mathbf{1}\left[{B_{T_j}^{(x_j,t_j(\omega))\not\to m-1}}\right](\omega) \leq \prod_{j=1}^k \mathbf{1}\left[{B_{T_j}^{x_j \not \to m-1}}\right](\omega).
\pi_{b}^{a}\pi_{a}^{b}= \frac{2 P_{z}P_{\bar{z}}}{4 \pi^{2}}\frac{( z_{1}- z_{2}) ( \bar{z}_{1}- \bar{z}_{2})}{( z - z_{1}) ( \bar{z}- \bar{z}_{1}) ( z - z_{2}) ( \bar{z}- \bar{z}_{2})}
\mathcal{A}\left(\mathbb{H}_{\mathbf{f}}\left([x_{\textup{f}}],\partial_0(d_{\widetilde\delta})\right)\right)(\kappa,s) =-\mathcal{A}\left(\mathbb{H}_{\mathbf{f}}\left(\partial_0(d_{\widetilde\delta}),[x_{\textup{f}}]\right)\right)(\kappa,\kappa-s-k) &=-\left\langle\Psi_2,[x_p^+] \right\rangle\cdot(\kappa-k-s)\,.
\mathsf{E}\left[g(X)X^n\right]=\sum_{\ell=0}^n\binom{n}{\ell}\mu^{n-\ell}\sum_{k=0}^{\lfloor\ell/2\rfloor}H_{\ell,k}\sigma^{2(\ell-k)}\mathsf{E}\left[g^{(\ell-2k)}(X)\right], \ \n\in\mathbb{N}.
p^{(2n)}_\lambda(o,o) \sim \begin{cases}\frac{(d-1-\lambda)^2}{16(\pi\lambda)^{1/2}(d-1)^{3/2}}\rho_{\mathbb{T}_d}(\lambda)^{2n}n^{-3/2}\hbox{if}\ \lambda\in (0, \, d-1), \frac{1}{\sqrt{\pi n}}\hbox{if}\ \lambda=d-1. \end{cases}
\tau^+{:=}\sum_{\alpha{\in}X}c_{\alpha}e^{\alpha}, \tau^-{:=}\sum_{\alpha{\in}X}c_{\alpha}\varepsilon(h, \alpha)e^{\alpha-h}, \overline{\tau}^+{:=}\sum_{\alpha\in X}d_{h{-}\alpha}e^{h-\alpha}, \overline{\tau}^-{:=}-\sum_{\alpha\in X}d_{h-\alpha}\varepsilon(h, \alpha)e^{-\alpha}.
\partial_2 A_{11}= \partial_2 (\nabla_3k_{11}-\nabla_1k_{13}) = \nabla_2(\nabla_3k_{11}-\nabla_1k_{13})-k_{2}^\alpha\nabla_\alpha k_{11}+2k_{21}\nabla_3 k_{31}&-k_{21}\nabla_3k_{13}-k_{21}\nabla_1 k_{33}+k_{2}^\alpha\nabla_1 k_{1\alpha}= \nabla_2(\nabla_3k_{11}-\nabla_1k_{13}).
\int_{0}^{1/2}\widehat \chi_{\left[-s, s\right]}(k)^2 ds = \int_{0}^{1/2}\widehat \chi_{\left[-s, s\right]}(k)^2 ds &=\int_{0}^{1/2}{\left( \frac{\sin{(2 k \pi s)}}{k \pi}\right)^2 ds}= \frac{1}{k^2 \pi^2}\int_{0}^{1/2}{\sin{(2 k \pi s)^2}ds}= \frac{1}{4 k^2 \pi^2}.
U_{z}=\int_{\mathrm{T}_{d}}x_{1}^{z_{1}}\cdots x_{d}^{z_{d}}\mathrm{d}\mathrm{P}\left( x\right) =\int_{\Theta_{d}}\mathrm{e}^{i\theta \cdot z}\mathrm{d}\left( \mathrm{F}_{\ast}\mathrm{P}\right) \left( \theta \right) \,z\in \mathbb{Z}^{d}\ ,
a_{2}( x ) = \frac{1 - \alpha^{2}}{r^{4}}\left[ ( \alpha^{2}- 1 ) \left( \frac 1{9 0}+ 2 \left( \frac 1 6 - \xi \right)^{2}\right) + \frac 2 3 \left( \frac 1 5 - \xi \right) \alpha^{2}\right] + \frac{q^{2}}{6 r^{4}}\ .
P_{g.s.}=\sum_{\alpha=1}^{\frac{N}{2}}p_{\alpha}=N\int_{-\infty}^{\infty}p(\lambda)\rho(\lambda)d\lambda=-\frac{N}{2}\int_{-\infty}^{\infty}d\lambda\frac{\pi}{\cosh\pi\lambda}=\frac{N}{2}\pi\quad ({\rm mod}\quad 2\pi)\quad .
\left(\prod_{i=1}^{q}w_{i}^{m_{i}}\delta\right)^{\ell}=\prod_{j=0}^{\ell-1}\delta^{j}\left( \prod_{i=1}^{q}w_{i}^{m_{i}}\right) \delta^{-j}= \prod_{i=1}^{q}(w_{1}\cdots w_{q})^{m_{i}\ell/q}= (w_{1}\cdots w_{q})^{\frac{\ell}{q}\sum_{i=1}^{q}m_{i}}.
Z(C(a_d,a_{d-1},\ldots,a_3,1,a_3,\ldots,a_{d-1},a_d)) = Z(C(a_d,a_{d-1},\ldots,a_3,1))\cdot Z(C(a_3,\ldots,a_{d-1},a_d)) &+Z(C(a_d,a_{d-1},\ldots,a_3))\cdot Z(C(a_4,\ldots,a_{d-1},a_d)) = Z(C(1,a_3,\ldots,a_d))\cdot (m-n)+(m-n)\cdot Z(C(a_4,\ldots,a_d)) = (m-n)(Z(C(a_d,a_{d-1},\ldots,a_3,1))+ Z(C(a_4,\ldots,a_{d-1},a_d)),
\lambda_{-k}(f(A_{\cal S})) = \min_{\{h\in{\cal F}\,:\,\Vert h\Vert=1\}}\langle h, f(A_{\cal S})h\rangle = \min_{\{h\in{\cal F}\,:\,\Vert h\Vert=1\}}f(\langle h, A_{\cal S}h\rangle) \le \inf_{\{h\in{\cal F}\,:\,\Vert h\Vert=1\}}\langle h, f(A)h\rangle \le \lambda_{-k}(f(A)_{\cal S})
\frac{sa_r-t(a_1+1)}{a_1+1-s}(a_r-s)_{q}=\frac{(s\hat\alpha\hat{x}-q\hat{x}-qs-t)(\hat{x}+\hat\alpha\hat{x})_q}{\hat{x}+1}=H_1(\hat{x},\hat\alpha)\cdot p(\hat{x}), &\frac{ta_r-s(a_1+1)}{a_1+1-t}(a_1-s)_{q}=\frac{(t\hat\alpha \hat{x}+q\hat{x}+qs-s)(\hat{x})_q}{\hat{x}+1-q}=H_2(\hat{x},\hat\alpha)\cdot p(\hat{x}).
Z \sim \int d{\hat{y}}_{i}d{\hat{p}}_{i}\frac{1}{g_{s}\ln \delta}e^{- \frac{({\hat{q}}_{m}- q_{m}) G^{m n}({\hat{q}}_{n}- q_{n})}{g_{s}\ln \delta}}\int D X e^{- S^{*}- \int{\hat{q}}_{m}V_{m}( X )}
d\mathbf{Y}_t = \mathbf{F}^{\ast}d\tilde{\mathbf{A}}_t^{\ast}+\mathbf{F}d\tilde{\mathbf{A}}_t+ \mathbf{G}d\tilde{\mathbf{\lambda}}_t, \mathbf{F}= \begin{pmatrix}1 0 0 0\end{pmatrix},\; \mathbf{G}= \begin{pmatrix}0 0 0 1\end{pmatrix}.
\psi_{i,\ell}^{(n)}(x) \coloneqq \begin{cases}\sqrt{\frac{2^n}{\binom{n}{\ell}}}\cdot \psi_i^{(n,\ell)}(x) \textnormal{if}x \in \binom{V(G_n)}{\ell}\cr 0 \textnormal{otherwise.}\end{cases}
h_f g h_f^{-1}A = h_f g(h_f^{-1}Ah_f)h_f^{-1}&= h_f gf h_f^{-1}& = h_f fg h_f^{-1}& = (h_f f h_f^{-1})h_f g h_f^{-1}& = A h_f g h_f^{-1}.
J_n \approx \begin{pmatrix}\bar a_1 \bar b_1 \bar b_1 \bar a_2 \bar b_2 &\ddots \ddots \ddots\end{pmatrix}+\frac{1}{n^r}\begin{pmatrix}\eta_1 \zeta_1 \zeta_1 \eta_2 \zeta_2 &\ddots \ddots \ddots\end{pmatrix},
\textrm{Re}\, \big( p(t(\varphi)) - p(t_0)\big) = \log \left|\frac{z}{2}\, e^{i\varphi}\left( \frac{z}{2}\, e^{i\varphi}+ z\right)\right| - \log \left( \frac{z}{2}\right)^2 *[0.3cm] = \log |e^{i\varphi}+ 2| = \frac{1}{2}\, \log \big( 1 + 4 (1 + \cos\varphi)\big) \geq 0\, ,
\begin{pmatrix}x_2^{k+1}\lambda^{k+1}\end{pmatrix}=\begin{pmatrix}x_2^{k}\lambda^{k}\end{pmatrix}-\begin{pmatrix}I_{n_2}0 -s\beta A_2 (r+s)I_l \end{pmatrix}\begin{pmatrix}x_2^{k}-\widetilde{x}_2^k \lambda^{k}-\widetilde{\lambda}^k \end{pmatrix}.
r_i = \left[\begin{array}{rl}r_i^{(1)}r_i^{(2)}\end{array}\right] = \left[\begin{array}{rl}\sum^2_{j=1}R g_{ji}h_{ji}P_t T_s s_j^{(1)}+ n_i^{(1)}~~~~~~~ \sum^2_{j=1}R g_{ji}h_{ji}P_t T_s (1-s_j^{(1)}) + n_i^{(2)}\end{array}\right],
\begin{cases}\,dX/dt \!\!=\, Y^{k-1}v \alpha w + Y^{k-2}v \alpha w Y + \ldots + v \alpha w Y^{k-1}\,dY/dt \!\!=\, 0 \,dv/dt \!\!=\, Y^k v \alpha \,dw/dt \!\!=\, - \, \alpha w Y^k \ .\end{cases}
g^{(p)}_{nm}(t)=\sqrt{\frac{P_p}{R}}\sum_{r=1}^{R}\sqrt{\exp(j\Phi_{rp}).G_R(\theta_{rp}).\sigma_{rp}.\mathcal{P}_{rp}.G_T(\phi_{rp})}. \exp{(\frac{j2\pi}{\lambda}(d_s\sin(\theta_{rp})+d_m\sin(\phi_{rp}+\Upsilon_{rp})))}. \exp{(\frac{-j2\pi V_m}{\lambda}\cos(\vartheta_v-\theta_{rp}))}