harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{1}^{\left( 0\right)}\,=\,-\frac{a}{2}A_{\mu}^{\left( 0\right)}\,A^{\mu\left( 0\right)}\,+\,\frac{b}{2}\left( F_{\mu \nu}^{\left( 0\right)}\right)^{2}\,+\,\theta \,\epsilon^{\mu \nu \rho}\,\,A_{\mu}^{\left(0\right)}\,\partial_{\nu}\,A_{\rho}^{\left( 0\right)}.
P =\sqrt{2p(1-p)}\; \exp{\left\{\frac{(1-2p)(y-b)}{2\sigma}- \frac{|y-b|}{2\sigma}\right\}}&= \sqrt{2p(1-p)}\; \exp{\left\{-\frac{1}{\sigma}\left[ p-I(y \le b)\right](y-b)\right\}}&=\sqrt{2p(1-p)}\; \exp{\left\{-\frac{1}{\sigma}\left[ p-I(y\le\mu+\sigma\alpha s)\right](y-\mu-\sigma\alpha s)\right\}}.
\phi = \frac{1}{4}P^{2}- \frac{1}{2}( m_{1}^{2}+ m_{2}^{2}) + \frac{( m_{1}^{2}- m_{2}^{2})^{2}}{4 P^{2}}+ q_{\bot}^{2}- \frac{\alpha \bigl ( P^{2}- ( m_{1}- \kappa m_{2})^{2}\bigr )}{\epsilon P \cdot r}= 0 ,
\lim_{n}J_{\rho_n}(v_n)=\lim_{n}c_{2, \rho_n}\leq c_{1, 0}+\left(\frac 1 p -\frac{1}{p^*_\alpha}\right)\frac{S_\alpha^{\frac{N-\alpha}{ps-\alpha}}}{\mu^{\frac{p}{p^*_\alpha-p}}}-\kappa,
F_{p}= \frac{V_{p - 2}\tilde{V}_{2}}{( 2 \pi )^{p}\tilde{g}\tilde{b}}\left( \frac{u}{R}\right)^{7 - p}\left[ \sqrt{\tilde{f}}- 1 + \frac{u_{0}^{7 - p}}{2 u^{7 - p}}\right] .
\begin{array}{c}{{W = X_{3}X_{8}X_{1 3}- X_{8}X_{9}X_{1 1}- X_{5}X_{6}X_{1 3}- X_{1}X_{3}X_{4}X_{1 0}X_{1 2}}}\\{{+ X_{7}X_{9}X_{1 2}+ X_{1}X_{2}X_{5}X_{1 0}X_{1 1}+ X_{4}X_{6}X_{1 4}- X_{2}X_{7}X_{1 4}.}}\\ \end{array}
\delta_1\phi'(t) =\delta_1\int_\Omega |u_t|^2 dx+\delta_1\int_\Omega uu_{tt}dx &=\delta_1\int_\Omega |u_t|^2dx-\delta_1\eta\int_\Omega|\Delta u|^2dx-a(t)\delta_1\int_\Omega \Delta \theta udx+\delta_1\int_\Omega f(t,u)udx &=\delta_1\int_\Omega |u_t|^2dx-\delta_1\eta\int_\Omega|\Delta u|^2dx-a(t)\delta_1\int_\Omega \theta \Delta udx+\delta_1\int_\Omega f(t,u)udx
\mathrm{lhs\of\ (\ref{3.1})}=\left(I_{0+}^{\alpha,\alpha',\beta,\beta',\gamma}\left(t^{\frac{\rho}{k}-1}\sum_{n=0}^{\infty}\frac{\prod_{i=1}^{p}\Gamma_k(a_i+n\alpha_i)}{\prod_{j=1}^{q}\Gamma_k(b_j+n\beta_j)}\frac{(at^{\frac{\mu}{k}})^n}{n!}\right)\right)(x).
H_2=\displaystyle \sum_{\beta \in F^{\times}}\sum_{\substack{X\in S(\beta) t=3}}\Theta \begin{bmatrix}m_1 X 0 m_2\end{bmatrix}\overline{\psi_A({m_1}^{-1}X})=\theta(a)q^8(1-q)(1-q^2).
\int^\infty_0x^{-3\alpha}\exp\left\{-(a\alpha+n)\frac{\theta}{x^2}\right\}\mathrm{d}x= \int^\infty_0u^{\frac{3(\alpha-1)}{2}}\exp\left\{-(a\alpha+j)u\right\}\mathrm{d}u = \left(\frac{1}{a\alpha+n}\right)^{\frac{3\alpha-1}{2}}\Gamma\left( \frac{3\alpha-1}{2}\right).
\int_{\mathbb{R}_{\geqslant0}}x^{\kappa}m_{\alpha,\beta}(\mathrm{d}x)\geqslant\left(\int_{\mathbb{R}_{\geqslant0}}xm_{\alpha,\beta}(\mathrm{d}x)\right)^{\kappa}=\left(\frac{\alpha}{\beta}\right)^{\kappa}.
B(k_1^*)C(k_2)f=P_+\left(k_1^*P_-(k_2f)\right)=P_+\left(k_1^*\left(\begin{matrix}P_-(d^*f_1-c^*f_2) 0\end{matrix}\right) \right) \hphantom{B(k_1^*)C(k_2)f}{}=P_+\left(\left(\begin{matrix}a^*P_-(d^*f_1-c^*f_2) 0\end{matrix}\right)\right)=0.
\overline{\pi}_{1j}(u_1,u_2)>0~\forall j\geq 2,~~\sum_{j\in S}\overline{\pi}_{1j}(u_1,u_2)=0,~~\overline{\pi}_{1j}(u_1,u_2) \leq e^{-2\theta j}~\forall~j\geq 2,
{\mathcal{R}}_{0}= e x p ( - \sum_{{\bf{k}},{\bf{q}}, i}f_{{\bf{k}},{\bf{q}}}^{*}({\bf{x}}) f_{{\bf{k}},{\bf{q}}}({\bf{x}}) [ b_{i}({\bf{q}}) , a_{{\bf{k}}}^{\dagger}({\bf{q}}) ] [ a_{{\bf{k}}}({\bf{q}}) , b_{i}^{\dagger}({\bf{q}}) ] )
M_x[\alpha_{1, 2}]=M_x[a_{0, 2}]+M_x[a_{0, 1}\,(\beta_1)_x]=-2 c_6\,M_x[\overline{v}_x^2]-12 c_7 M_x[\overline{v}^2]+\frac{4}{3}c_2^2 M_x[\overline{v}_x^2]+4 c_2 c_3 M_x[\overline{v}^2], begin{align*}2mm] M_x[\alpha_{0, 2}]=M_x[a_{0, 1}\,(\beta_1)_{xx}]=-4 c_1 c_2 M_x[\overline{v}_{xx}^2]-12 c_1 c_3 M_x[\overline{v}_x^2].
0 = (I-Q) F_{\Psi\Psi}(\Psi^q,p_0)[\psi_p(0,p_0)p'(0),\psi_p(0,p_0)p'(0)] &+ (I-Q) F_{\Psi p}(\Psi^q,p_0)[\psi_p(0,p_0)p'(0), p'(0)] &+ (I-Q) F_\Psi(\Psi^q,p_0)\psi_{pp}(0,p_0)[p'(0), p'(0)] &+ (I-Q) F_{\Psi p}(\Psi^q,p_0)[\psi_p(0,p_0)p'(0), p'(0)] &+ (I-Q) F_{pp}(\Psi^q, p_0)[p'(0), p'(0)].
\left\langle \gamma \left( p , \varepsilon_{1}\right) , \gamma \left( p^{\prime}, \varepsilon_{2}\right) \mid \pi \left( q \right) \right\rangle = i \left( 2 \pi \right)^{4}T_{\mu \nu}\left( q ; p , p^{\prime}\right) \delta^{4}\left( q - p - p^{\prime}\right) \varepsilon_{1}^{\mu}\left( p \right) \varepsilon_{2}^{\nu}\left( p^{\prime}\right) .
K_{0}(p_{1j}^{2};m_{0},\epsilon) \equiv K_{00}(p_{1j}^{2};m_{0}^{2},\epsilon), \; \; \; \Delta K (p_{1j}^{2};m_{0},M,\epsilon)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}K_{lm}(p_{1j}^{2};M_{l}^{2},\epsilon) ,
d\frac{v_d^{(1)}}{v_d^{(0)}}-3(2d-1) \frac{v_{d-1}^{(1)}}{v_{d-1}^{(0)}}+(d-1)\frac{v_{d-2}^{(1)}}{v_{d-2}^{(0)}}=2d(d-1),(d\ge 2) \frac{v_0^{(1)}}{v_0^{(0)}}= \frac{v_1^{(1)}}{v_1^{(0)}}= 0.
\left(\begin{array}{cccc}0 a_{1,2}\cdots a_{1,n}a_{2,1}0 \cdots a_{2,n}\vdots \vdots \ddots \vdots a_{n,1}a_{n,2}\cdots 0 \end{array}\right)\left(\begin{array}{cccc}c_1 c_2 \vdots c_n\end{array}\right)=\left(\begin{array}{cccc}x_1 x_2 \vdots x_n\end{array}\right).
{\bf e}_{N+1}\equiv{\bf p}^\bot_N/c_N,\quad{\bf e}_{2N+1-a}=-{1\over \sqrt{\Delta_{a}\Delta_{a+1}}}\left | \begin{array}{ccc}{\Phi}_{N.N}\cdots{\Phi}_{N.a}\vdots \vdots{\Phi}_{a+1 .N}\cdots{\Phi}_{a+1.a}{{\bf p}^\bot}_{N}\cdots{{\bf p}^\bot}_{a},\end{array}\right |,
\begin{array}{l}\mathbb{E}[(f(Z_t)-f(Z_s))^4]= \mathbb{E}[F_{f,Z_s}^4(Z_t)]= \ \ (by \Lemma \ 4.4. \ \cite{DGP12}) \int\limits_s^t\mathbb{E}[F_{f,Z_s}^2(Z_u)(6GF_{f,Z_s}^2(Z_u)-8F_{f,Z_s}(Z_u)GF_{f,Z_s}(Z_u))]du\leq C_2\int\limits_s^t \mathbb{E}[F_{f,Z_s}^2(Z_u)]du\leq C_1C_2\int\limits_s^t(u-s)du=\frac{C_1C_2(t-s)^2}{2}.\end{array}
A_{K,M}^{(1,2)}\buildrel \Delta \over = \max \left\{{\frac{1}{{\left|{1 - \frac{1}{{MN + KN}}\left({{\lambda_{\cal M}}+{\lambda_{\cal N}}}\right)}\right|}},\frac{1}{{\left|{1 - \frac{1}{{MN + KN}}\left({\hat \lambda_{{\cal M},j}^{(K)}+ \hat \lambda_{{\cal N},j}^{(K)}}\right)}\right|}}}\right\},
\Psi^{\mu}( z ) \Psi^{\mu}( w ) = \frac{\alpha^{\prime}}{2}\frac{1}{z - w}+ \cdots \, , \quad \tilde{\Psi}^{\mu}( z ) \tilde{\Psi}^{\mu}( w ) = \frac{\alpha^{\prime}}{2}\frac{1}{\bar{z}- \bar{w}}+ \cdots \, ,
\mathbf{F}^{k}=\mathbf{B}^{k}\mathbf{G}^{k}, \mathbf{B}^{k}=\operatorname{diag}(b^k_{1,1}, b^k_{2,1}, \cdots, b^k_{M,1}, b^k_{1,2}, \cdots, b^k_{M,N}),
T^*\Lambda_k^*g=T^*\Big(\sum_{i\in\mathbb{N}}\Lambda_i^*\Theta_i\Lambda_k^*g\Big) =\sum_{i\in\mathbb{N}}T^*\Lambda_i^*\Theta_i\Lambda_k^*g &=\sum_{i\in\mathbb{N}}\Lambda_{i+1}^*\Theta_i\Lambda_k^*g =\sum_{i\in\mathbb{N}}\big(\Lambda_k \Theta_i^*\Lambda_{i+1}\big)^*g,
A(C_n,q,T)= (q-1)\frac{TA_n(T)}{(1-T)^n(1-qT)}+n\frac{TA_{n-1}(T)}{(1-T)^n}= \frac{T\sum_{j=0}^{n-2}A(n-1,j)\left[q(j+1)+n-1-j\right]T^j}{(1-T)^{n-1}(1-qT)},
J_{4}\le \sup_{j\in \mathbb{Z}}\sum_{|k-j|\le4}\|\Delta_{j}(\Delta_{k}\delta B\cdot\nabla S_{k-1}u_{2})\|_{L^2}+\sup_{j\in \mathbb{Z}}\sum_{|k-j|\le4}\|\Delta_{j}(S_{k-1}\delta B\cdot\nabla \Delta_{k}u_{2})\|_{L^2}&+\sup_{j\in \mathbb{Z}}\sum_{k\ge j-3}\|\Delta_{j}(\Delta_{k}\delta B\cdot\nabla\tilde{\Delta}_{k}u_{2})\|_{L^2}= J_{41}+J_{42}+J_{43},
-g^2T^2\sum_{n,l}\int \frac{d^5p}{(2\pi)^5}\,\frac{d^5k}{(2\pi)^5}frac{e^{i(\theta_{12}(p_1\, k_2-p_2\, k_1)+\theta_{34}(p_3\, k_4-p_4\, k_3))}}{(p^2+\frac{4\pi^2 n^2}{\beta^2})(k^2+\frac{4\pi^2 l^2}{\beta^2})( (p+k)^2+\frac{4\pi^2 (n+l)^2}{\beta^2})}.
c(T^*)=e(T^{\prime})+f_3(T)+r =3(k^2-2k-3)(t-2)+2(t-2)+r &=(3k^2-6k-7)(t-2)+r &=(3k^2-6k-7)\frac{n-r-2}{k^2-k-2}+r &=\left(\frac{k-3}{k-2}+\frac{2}{3(k+1)(k-2)}\right)(3n-6)-\frac{2k^2-5k-5}{k^2-k-2}r,
- \Delta \Lambda_{n}= 4 \pi G_{n}( n - 2 ) ( \sigma - p ) \frac{\Delta [ \epsilon h \dot{t}]}{Z}+ 4 \pi G_{n}( \sigma + \rho ) \Delta \left[ \frac{\ddot{Z}+ \frac{1}{2}h^{\prime}}{\epsilon h \dot{t}}\right] .
P=\{0,\epsilon_{1},\frac{1}{N}-\epsilon_{2},\frac{1}{N}+\epsilon_{2},\frac{1}{N-1}-\epsilon_{2},\frac{1}{N-1}+\epsilon_{2},...,\frac{1}{3}-\epsilon_{2},\frac{1}{3}+\epsilon_{2},\frac{1}{2}-\epsilon_{2},\frac{1}{2}+\epsilon_{2},1\}
\langle \psi, -\mathrm{i}\boldsymbol{\sigma}\cdot \nabla \psi \rangle =\langle \mathcal{F}_2\psi,\boldsymbol{\sigma}\cdot \boldsymbol{p}\mathcal{F}_2\psi \rangle , \langle \zeta, -\mathrm{i}\boldsymbol{\alpha}\cdot \nabla \zeta \rangle =\langle \mathcal{F}_3 \zeta,\boldsymbol{\alpha}\cdot \boldsymbol{p}\mathcal{F}_3 \zeta \rangle .
\theta_1=-\frac{1}{t^5}+\frac{3}{t^8}-\frac{k_1}{t^{11}}, \theta_2=t+\frac{1}{t^5}-\frac{2}{t^8}+\frac{k_2}{t^{11}}\ \ \theta_3=t^4+3t-\frac{1}{t^8}+\frac{k_3}{t^{11}}
\mathbb B=\big\{(x_1,\dots, x_n)\in\mathbb R^{n}\ \big | \ \sum_{i_1}^nx_i^2\leq 1\big\}\quad &\mathbb S=\big\{(x_1,\dots, x_n)\in\mathbb R^{n}\ \big | \ \sum_{i_1}^nx_i^2=1\big\}.
E_{1 \, n o n - d i s p .}= ( \varepsilon - 1 )^{2}\Bigl ( - \frac{2 3}{1 2 8}\frac{V}{\pi^{2}\lambda^{4}}+ \frac{2 3}{3 8 4}\frac{S}{\pi^{2}\lambda^{3}}+ \frac{2 3}{1 5 3 6}\frac{1}{\pi a}\Bigr ) \, \, + \, \mathrm{c o n s t}
K_{\delta,q}=\left\langle Z_{\delta,q}^{1},\dots,Z_{\delta,q}^{n}\right\rangle K_{\delta,q}^{\bot}=\left\{\varphi\in H^{1}(M)\ :\ \left\langle \left\langle \varphi,Z_{\delta,q}^{b}\right\rangle \right\rangle_{H}=0,\b=1,\dots,n\right\}
R_j(t) := \Vert f_j(t)\Vert_{L^2}+\Vert g_j(t)\Vert_{L^2}+ \Vert \widetilde{g}_j(t)\Vert_{L^2}+ \Vert \nabla \dot S_{j-1}u(t)\Vert_{L^{\infty}}X_j(t).
\delta^2(a_{m-1}a_m) = \delta^2(a_{m-1})a_m+2\delta(a_{m-1})\delta(a_m)+a_{m-1}\delta^2(a_m), [F,\delta(a_{m-1}a_m)] = [F,\delta(a_{m-1})]a_m+[F,a_{m-1}]\delta(a_m)+\delta(a_{m-1})[F,a_m]+a_{m-1}[F,\delta(a_m)].
\alpha_0(i,j,k,l) = b_0([1+2i,1+2j,2k,2l]), \beta_0(i,j,k,l) = b_0([2i,1+2j,1+2k,2l]), \gamma_0(i,j,k,l) = b_0([21,2j,1+2k,1+2l]).
|\nabla u|^2 = \frac{|\nabla f|^2}{f^2}=\frac{1}{|x|^2 f^2}, |\nabla^2 u|^2 \leq C \left( \frac{|\nabla f|^4}{f^4}+ \frac{|\nabla^2 f|^2}{f^2}\right)\leq C \left( \frac{1}{|x|^4 f^4}+ \frac{1}{|x|^4 f^2}\right),
S = 0 \longrightarrow D \longrightarrow E^i \longrightarrow \cdots \longrightarrow E^1 \longrightarrow C \longrightarrow 0 \mbox{and}\hat{S}= 0 \longrightarrow D \longrightarrow \hat{E}^i \longrightarrow \cdots \longrightarrow \hat{E}^1 \longrightarrow C \longrightarrow 0
\begin{cases}- \Delta u^{(j)}+ \nabla p^{(j)}= \lambda_j u^{(j)}\div u^{(j)}=0 u^{(j)}=0 \ \Gamma_0 u^{(j)}=\eta^{(j)}e_2 \ \Gamma_1 -\partial_{x_1}^2 \eta^{(j)}- p^{(j)}=\lambda_j \eta^{(j)}\ \mathcal{I}\end{cases}
\frac{\partial L_{d}F\left(p\|q\right)}{\partial q_{j}}=\left[\frac{a-1}{a-b}p_{j}\left(Z_{j}\right)^{a-2}-\frac{b-1}{a-b}p_{j}\left(Z_{j}\right)^{b-2}\right]\frac{\partial Z_{j}}{\partial q_{j}}+\left(1-\alpha\right)
\Gamma(u)=\left[ u \; A u \; A^2 u \dots A^{n-1}u\right], \Gamma_1(u,v)=\left[ v \; u \; A u \dots A^{n-2}u\right],\;\; \Gamma_2(u,v)=\left[A v \; u \;A u \dots A^{n-2}u\right].
\pounds_{12}=\varphi^{\prime}\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+M^2\right) \varphi +\chi^{\prime \alpha}\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+M^2\right) \chi_\alpha +f^{\prime}\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+M^2\right) f+h.c
A_1:=\{y| |y-2^{|j|+2N}e_1|\le2^{2N-1}\}, &A_2:=\{y| |y-2^{|j|+2N}e_1|\ge2^{2N-1}, |y-2^{|k|+2N}e_1|\ge2^{2N-2}\}, &A_3:=\{y| |y-2^{|j|+2N}e_1|\ge2^{2N-1}, |y-2^{|k|+2N}e_1|\le 2^{2N-2}\},
g =\displaystyle \sum_{s \in L(\sigma(b))}(s\rho(bwa))^{\ast}\left(\rho^{k}\left(m_{1}\hdots m_{l}\Delta(r) \Diamond z\right)\right)s &=\displaystyle \sum_{s \in L(\sigma(b))}(s\rho(bwa))^{\ast}\left(r\rho(cyc(\bar{e_{k}}zm_{1}\hdots m_{l}))-\rho(cyc(\bar{e_{k}}zm_{1}\hdots m_{l}))r\right)s
K_{i \omega}( \mu \rho ) \simeq{\frac{i \pi}{2 \sinh \pi \omega}}\left[{\frac{1}{\Gamma ( i \omega + 1 )}}\left({\frac{\mu \rho}{2}}\right)^{i \omega}-{\frac{1}{\Gamma ( - i \omega + 1 )}}\left({\frac{\mu \rho}{2}}\right)^{- i \omega}\right] .
R^h =\frac{1}{a^2+b^2}\bigl\{\cos{2t}[a(\pi_1-\pi_2)-b\pi_3]-\sin{2t}[-a\pi_3-b(\pi_1-\pi_2)]\bigr\}begin{align*}4pt] =\frac1{a^2+b^2}\bigl\{(a\cos{2t}+b\sin{2t})(\pi_1-\pi_2)-(b\cos{2t}-a\sin{2t})\pi_3\bigr\}.
- \partial_{z}^{\mu}[ \delta^{( 4 )}\left( x - z \right) \delta^{( 4 )}( y - z ) \gamma_{\mu}] = i \delta^{( 4 )}( y - z ) \overline{{{S}}}^{- 1}( x , y ) - i \delta^{( 4 )}\left( x - z \right) \overline{{{S}}}^{- 1}( x , y )
d ~ = ~ \left( \begin{matrix}{{0}}&{{1}}&{{- 1}}&{{0}}&{{0}}&{{1}}&{{- 1}}&{{0}}\\{{1}}&{{- 1}}&{{0}}&{{0}}&{{1}}&{{- 1}}&{{0}}&{{0}}\\{{- 1}}&{{0}}&{{0}}&{{1}}&{{- 1}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{1}}&{{- 1}}&{{0}}&{{0}}&{{1}}&{{- 1}}\\ \end{matrix}\right)
\langle M \rangle \approx \frac{2}{3 ( q^{- 2}+ 1 )^{3 / 2}}\lambda ( 2 \mu )^{3 / 2}+ \frac{0 . 6 4 \lambda}{( q^{- 2}+ 1 )^{3 / 2}\beta^{2}\sqrt{2 \mu}}.
\mathrm{Z}_{\mathrm{o r b .}}( \alpha^{\prime 1 / 2}) = \mathrm{Z}_{\mathrm{c i r c .}}({\frac{\alpha^{\prime 1 / 2}}{2}}) = \mathrm{Z}_{\mathrm{c i r c .}}( 2 \alpha^{\prime 1 / 2}) \equiv \mathrm{Z}_{K T}\quad .
\begin{aligned}U(a,z,\alpha)\cdot v(\chi,w) =U(a(1,0,\alpha)(1,z,0)(1,0,\chi))w &=U(a\chi(z)(1,0,\chi+\alpha)(1,z,0))w &=a^d\chi(z)^dv(\chi+\alpha,w).\end{aligned}
0=1-g_c + \frac{(D-1)}{2N_c}\left[ 1-\frac{1}{\tilde{g}_{sc}}\log(1+\tilde{g}_{sc}) \right] - \frac{(D-1)^2(4-D)}{2N_c(D-2)}\left[ 1-\frac{1}{g_{Vc}}\log(1+g_{Vc}) \right]\ ,
\frac{\partial^{2}\ln I ( \phi , \chi )}{\partial \phi_{a}\partial \phi_{b}}\stackrel{a \neq b}{=}\oint \frac{d z_{1}}{2 \pi \imath}\oint \frac{d z_{2}}{2 \pi \imath}\frac{1}{( z_{1}- z_{2})^{2}( p ( z_{1}, 1 ) - \phi_{a}) ( p ( z_{2}, 1 ) - \phi_{b})}
(v, Ku) = -4\pi \int_0^\infty (\overline{{\mathcal F}^\ast N_v)(t)}({\mathcal F}^\ast N_u)(t) dt & = -4\pi \big ({\mathcal F}^\ast N_v ,{\mathcal R}{\mathcal F}^\ast N_u \big )= - 4\pi \big ( N_v,{\mathcal F}{\mathcal R}{\mathcal F}^\ast N_u \big ).
\prod_{1\le i \le m}^{\curvearrowleft}(f(w, z_{i})Y_{i-1}(w, z_{i}))u(b, a^{m})=q^{m}u(a^{m}, b)+\sum_{\ell=1}^{m}g(w, z_{\ell})Z_{\ell}^{[1, m]}(\vec{z}) u(b, a^{m})
\begin{array}{r c l r c l}{{\ell^{\prime}}}&{{=}}&{{- \ell \, , \hspace{1 . 5 c m}}}&{{A^{( 1 ) \prime}}}&{{=}}&{{- A^{( 1 )}\, ,}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{B^{( 2 ) \prime}}}&{{=}}&{{- B^{( 2 )}\, ,}}&{{C^{\prime}}}&{{=}}&{{- C \, ,}}\\ \end{array}
S_{W}\cong \int \mathcal{A}\star Q_{N S}\mathcal{A}+ \frac 2 3 \int \mathrm{X}\mathcal{A}\star \mathcal{A}\star \mathcal{A}+ \int Y ( i ) \Psi \star Q_{R}\Psi + 2 \int \Psi \star \mathcal{A}\star \Psi .
\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(2i-3)(2j-3)(i+j+1)}=\frac{1}{4}\left(\sum_{i=0}^{p-1}\frac{p(-1)^{i}}{2i-3}\right)^2-\frac{1}{4}\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(2i-3)(i+j+1)}.
\tau(e_n a e_n + e_n b e_n)+2\tau(e_{n+1}) =\tau(e_n a e_n + e_{n+1}+ e_nbe_n + e_{n+1}) &=\tau(e_nae_n+e_{n+1})+\tau(e_nbe_n+e_{n+1}) &=\tau(e_n a e_n)+\tau(e_n b e_n)+2\tau(e_{n+1}).
\frac{1}{2}(c_{L}- C_{2}h)\|z_{m}\|_{H}^2 + h\sum_{n=0}^{m-1}\|B_{1}^{1/2}z_{n+1}\|_{H}^2 + \frac{1}{2}(\omega_{2, 1}- C_{2}h)\|v_{m}\|_{V}^2 + \frac{\omega_{2, 1}}{2}h^2\sum_{n=0}^{m-1}\|z_{n+1}\|_{V}^2 \leq C_{3}+ \frac{C_{2}}{2}h\sum_{j=0}^{m-1}\|v_{j}\|_{V}^2 + \frac{C_{2}}{2}h\sum_{j=0}^{m-1}\|z_{j}\|_{H}^2
S_n(z_1, z_2) = \underbrace{(2n^2)^{2m -s -1}\sum_{j = 0}^{\lfloor n \varepsilon \rfloor -1}\cdots}_{= : \, I_n(\varepsilon)}\, + \, \underbrace{(2n^2)^{2m -s -1}\sum_{j = \lfloor n \varepsilon \rfloor}^{n-1}\cdots}_{= : \, II_n(\varepsilon)}.
\sup_{n \geq 1}\frac{J_2(n,b)}{\left(n^{\frac{2}{\eta}}p_n\bar{F}(b)\right)^2}= O \left( \frac{b^{2r}\bar{F}^{\frac{1-\gamma}{\gamma q}}(b)}{\bar{F}^2(b)}\right), b \rightarrow \infty.
A_{j + 1}= 2^{-1}\{\Pi_r(\theta_j - \eta_jZ_j) - \theta\}^2 = 2^{-1}\{\Pi_r(\theta_j - \eta_jZ_j) - \Pi_r(\theta)\}^2 \leq 2^{-1}(\theta_j - \eta_jZ_j - \theta)^2 = A_j + 2^{-1}\eta_j^2 Z_j^2 - \eta_j (\theta_j - \theta)Z_j.
\begin{array}{r c l}E_0=2\hbar g\omega\ : \ \ \ |E_0,p=0>=|0>\ \ \ , E_1=4\hbar g\omega\ : \ \ \ |E_1,p=\pm 1>={\alpha^+_\pm}^\dagger\,{\alpha^-_\pm}^\dagger\,|0>\ \ \ , & \ \ \ |E_1,p=0,\sigma=\pm 1>=\frac{1}{\sqrt{2}}\,\left({\alpha^+_+}^\dagger\,{\alpha^-_-}^\dagger\,\pm\,{\alpha^+_-}^\dagger\,{\alpha^-_+}^\dagger\right)\,|0>\ \ \ ,\end{array}
\alpha + \sum_{i = 1}^{3}\alpha_{i}= 0 \, , \; \; \alpha \mu^{2}+ \sum_{i = 1}^{3}\alpha_{i}\mu_{i}^{2}= 0 \, , \; \; \sum_{i = 1}^{3}\alpha_{i}\mu_{i}^{2}\ln ( \mu_{i}^{2}/ \mu^{2}) = 0 \, ,
\phantom{=}\lim_{R\to\infty}\limsup_{n\to\infty}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u_n(y)|^p|\eta_R (x)-\eta_R (y)|^p}{|x-y|^{N+sp}}dxdy & = \lim_{R\to\infty}\limsup_{n\to\infty}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|(1-\eta_R (x))-(1-\eta_R (y))|^p}{|x-y|^{N+sp}}|u_n(y)|^p dxdy =0,
U_t^{(m)}(x;q):=q^{-t}\sum_{\substack{k_t\ge \cdots \ge k_1\ge 1 k_m\ge 1}}(-xq)_{k_t-1}(-q/x)_{k_t-1}q^{k_t}\cdot &\cdot \prod_{i=1}^{t-1}q^{k_i^2}\Big [ \begin{array}{c}k_{i+1}-k_i-i+\sum_{j=1}^{i}(2k_j+\chi(m>j)) k_{i+1}-k_i \end{array}\Big ]_q,
K = \left( \begin{array}{l l}{{- \sigma_{1}\bar{\beta}_{1 1}}}&{{\ \ - \sigma_{1}\bar{\beta}_{1 2}}}\\{{- \sigma_{2}\bar{\beta}_{1 2}}}&{{\ \ - \sigma_{2}\bar{\beta}_{2 2}}}\\ \end{array}\right) \equiv \left( \begin{array}{l l}{{K_{1 1}}}&{{\ \K_{1 2}}}\\{{K_{2 1}}}&{{\ \K_{2 2}}}\\ \end{array}\right) .
\int_{\Sigma_{p_2\varepsilon}}|\nabla^2 w|^2 dx= \int_{\Sigma_{p_2\varepsilon}\setminus\Sigma_{c_0}}|\nabla^2 w|^2 dx+\int_{\Sigma_{c_0}}|\nabla^2 w|^2 dx\leq \int_{\Sigma_{p_2\varepsilon}\setminus\Sigma_{c_0}}|\nabla^2 w|^2 dx+ C\|\nabla w\|_{L^2(\Omega)}^2.
G_{\tilde{\lambda}}( l_{1}, l_{2}; t = 1 ) = \langle l_{2}| \hat{T}| l_{1}\rangle = \mathrm{e}^{- \tilde{\lambda}( l_{1}+ l_{2})}\sum_{T : l_{1}\rightarrow l_{2}}1 \equiv \mathrm{e}^{- \tilde{\lambda}( l_{1}+ l_{2})}\frac{1}{l_{1}+ l_{2}}\Biggl ({l_{1}+ l_{2}\atop l_{1}}\Biggr ) .
( q^{3 / 2}- q^{- 1 / 2}) a_{1}^{\pm}a_{2}^{\mp}= q^{H}- q^{1 \mp N}, \ ( q^{1 / 2}- q^{- 3 / 2}) a_{2}^{\pm}a_{1}^{\mp}= q^{H}- q^{\pm N - 1}
w_1 = (s_{\alpha_3 + \alpha_4}s_{\alpha_2 + \alpha_3 + \alpha_4 + \alpha_5}s_{\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_5 + \alpha_6})_{i \rightarrow i+2}s_{\alpha_5 + \alpha_6}s_{\alpha_4 + \alpha_5 + \alpha_6 + \alpha_7}s_{\alpha_3 + \alpha_4 + \alpha_5 + \alpha_6 + \alpha_7 + \alpha_8}.
0\longrightarrow\left(\ker L|_{F_b^p(M)},d_b\right)=\left(\mathcal{A}_b^p(M),d_b\right)\stackrel{i_b}{\longrightarrow}\left(F_b^p(M),d_b\right)\stackrel{L}{\longrightarrow}\left({\rm Im}_b^{p+2}L,d_b\right)\longrightarrow0.
\mathop{{\rm Res}}_{s=1}\,F_{n,1}(s)=\lim_{s\to 1}\gamma_{n,1}(s)\cdot \mathop{{\rm Res}}_{s=1}\,\xi_{n,1}(s) & =\pi^{-n}\cdot (-1)^{\frac{n-1}{4}}2^{-\frac{7n+17}{4}}\cdot 3^{-1}\frac{\left(\tfrac{n+3}{2}\right)!(n+1)!}{\left(\tfrac{n-1}{2}\right)!}\frac{B_{\frac{n-1}{2}}}{B_{\frac{n+3}{2}}B_{n+1}B_{n-1}}.
\sum_{n \ge 1}\frac{|\lambda_{u_j}(n)|^2}{\sqrt{n}}\cdot e^{-n/X}= \frac{1}{2\pi i}\int_{(2)}\frac{L(\tfrac 12 + s, u_j \otimes u_j)}{\zeta(2s+1)}\cdot \Gamma(s) X^{s}ds.
t_{2n}=\frac{\lambda}{n}\left( t_{2n-1}+\frac{2n-1-2\kappa_{0}}{2\lambda}t_{2n-1}\right) \delta_{0}^{-1}=\frac{2\lambda+2n-1-2\kappa_{0}}{2n}t_{2n-1}\delta_{0}^{-1}, t_{2n+1}=\frac{2\lambda}{2n+1-2\kappa_{0}}\left( t_{2n}+\frac{n}{\lambda}t_{2n}\right) \delta_{0}^{-1}=\frac{2\lambda+2n}{2n+1-2\kappa_{0}}t_{2n}\delta_{0}^{-1}.
({\cal M})_{i_{1}}^{j_{1}}({\cal M})_{i_{2}}^{j_{2}}\cdots ({\cal M})_{i_{F - n}}^{j_{F - n}}\epsilon^{i_{1}i_{2}\cdots i_{F - n}i_{F - n + 1}\cdots i_{F}}\epsilon^{j_{1}j_{2}\cdots j_{F - n}j_{F - n + 1}\cdots j_{F}}= B^{i_{F - n + 1}\cdots i_{F}}\bar{B}_{j_{F - n + 1}\cdots j_{F}}.
{\cal R}_{a b c}^{d}= \partial_{b}\Omega_{a c}^{d}- \partial_{c}\Omega_{a b}^{d}+ \Omega_{0 b}^{d}\Omega_{a c}^{0}+ \Omega_{e b}^{d}\Omega_{a c}^{e}- \Omega_{0 c}^{d}\Omega_{a b}^{0}- \Omega_{e c}^{d}\Omega_{a b}^{e}.
SA=(\begin{matrix}0&0&0&0&0\\ 1&1&1&1&1\\ 1&2&2&2&1\\ 1&2&3&2&1\\ 1&2&2&2&1\end{matrix});AS=(\begin{matrix}1&1&1&1&0\\ 2&2&2&1&0\\ 2&3&2&1&0\\ 2&2&2&1&0\\ 1&1&1&1&0\end{matrix}).
\mathbf{A}\cdot\left( \nabla\times\left( f\nabla\times\mathbf{B}\right)\right) -\mathbf{B}\cdot\left( \nabla\times\left( f\nabla\times\mathbf{A}\right) \right) & \quad=\nabla\cdot\left( f\left( \mathbf{B}\times\left( \nabla\times\mathbf{A}\right) -\mathbf{A}\times\left( \nabla\times\mathbf{B}\right) \right) \right) .
S_{\sigma}={\frac{1}{4 \pi \alpha^{\prime}}}\int d^{2}\sigma [ \sqrt{\gamma}\gamma^{\alpha \beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\hat{G}_{M N}+{\epsilon}^{\alpha \beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\hat{B}_{M N}-{\frac{1}{2}}\sqrt{\gamma}\hat{\Phi}{\cal R}^{( 2 )}] .
u ={\frac{2 c_{_\mathrm{P}}^{2}v \cos \theta \pm c_{_\mathrm{P}}\sqrt{\big ( c_{_\mathrm{P}}^{2}- v^{2}\big ) \big ( 3 c_{_\mathrm{P}}^{2}- ( 1 + 2 \cos^{2}\theta ) v^{2}\big )}}{3 c_{_\mathrm{P}}^{2}- v^{2}}}+{\cal O}\{\Theta^{4}\}\ .
K_{23}(t,x,y)|_{t=\log\left(\frac{1+s}{1-s}\right)}\lesssim\frac{(1-s)^n|(1+s)y-(1-s)x|}{s^{(n-1)/2}}\exp\Big(-\frac{|(1+s)y-(1-s)x|^2}{8s}\Big)e^{|y|^2-|x|^2}&\lesssim\frac{(1-s)^n}{s^{n/2}}\exp\Big(-\frac{|(1+s)y-(1-s)x|^2}{16s}\Big)e^{|y|^2-|x|^2}, (x,y)\in N^c.
\int_0^{\infty}e^{-\mu t}\sinh^\nu\beta t\,dt=\frac{1}{2^{\nu+1}}\frac{\Gamma\Big(\frac{\mu}{2\beta}-\frac{\nu}{2}\Big)\Gamma(\nu+1)}{\Gamma\Big(\frac{\mu}{2\beta}+\frac{\nu}{2}+1\Big)},
\begin{cases}h_1(c,x):= x^2(4-c^2)^2(2c^2+(36-13c^2)x)+2c^2x^2; h_2(c,x):= 8cx(4-c^2)^2(1+x)(1-x^2); h_3(c,x):= 8(4-c^2)^2(8+x^2)(1-x^2); h_4(c,x):= 72x(4-c^2)^2(1-x^2).\end{cases}
\sum_{k=0}^n \binom{a}{k}\binom{b}{n+d-k}\binom{k}{c}\binom{n+d-k}{d}H_{b-d-n+k}={}\binom{a+b-c-d}{n-c}\binom{a}{c}\binom{b}{d}(H_{a+b-n-d}-H_{a+b-c-d}+H_{b-d})
( d s )^{2}= a^{2}( d X^{0})^{2}+ \frac{K^{2}( X^{0})}{( 1 + \frac{1}{4}\epsilon r^{2})^{2}}\left[ ( d X^{1})^{2}+ ( d X^{2})^{2}+ . . . + ( d X^{D})^{2}\right] \ ,
\begin{aligned}\int_0^T \sum_{x\in G}\, V(x)\,\partial_t v_+(x,t)\,v_+(x,t)\eta^2(x)e^{\xi(x,t)}\mu(x)\,dt &\le \int_0^T\frac1{4}\sum_{x,y\in G}v_+^2(x,t)\left[1-e^{\xi(y,t)-\xi(x,t)}\right]^2\eta^2(x)e^{\xi(x,t)}\,\omega(x,y)\,dt &+ \int_0^T\sum_{x,y\in G}v_+^2(x,t)[\eta(y)-\eta(x)]^2e^{\xi(y,t)}\omega(x,y)\,dt.\end{aligned}
\begin{array}{ccccc}F_{4(4)}\rightarrow SO(5,4) \rightarrow SO(5)\times SO(4){\bf 26}{\bf 9}+{\bf 16}+{\bf 1}[({\bf 5};{\bf 1},{\bf 1})+({\bf 1};{\bf 2},{\bf 2})]\,+ & [({\bf 4};{\bf 1},{\bf 2})+({\bf 4};{\bf 2},{\bf 1})]+ ({\bf 1};{\bf 1},{\bf 1})\ . \end{array}
\Pi_{1}( m_{0}) = \frac{\lambda}{2}~ T \sum_{n}\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{1}{\omega_{n}^{2}+ \omega_{k}^{2}}= \frac{\lambda}{2}\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{1}{\omega_{k}( m_{0})}\left[ n_{B}( \omega ) + \frac{1}{2}\right] ,
|S(n,\chi)| = q^{n/2}|[u^n]L(u/\sqrt{q},\chi)| \le q^{n/2}[u^n] \exp\bigg( \sum_{k \ge 1}\frac{u^k}{k}\min\{q^{k/2}, d(\chi)\}\bigg) &\le q^{n/2}\exp\bigg( \sum_{k\ge 1}\frac{R^k}{k}\min\{q^{k/2}, d(\chi)\}\bigg) R^{-n}
\begin{aligned}g_1\left(x_1,x_2,x_3,y_1 \right) = 0, & \frac{\partial g_1\left(x_1,x_2,x_3,y_1 \right)}{\partial y_1}= 0, & \frac{\partial g_1}{\partial x_1}f_1 + \frac{\partial g_1}{\partial x_2}f_2 + \frac{\partial g_1}{\partial x_3}f_3= 0.\end{aligned}
e^{- \delta S_{e f f}}\; = \; \int{\cal D}\vec{a}\exp \left\{- \sum_{n = - \infty}^{+ \infty}\frac{1}{4}\int \frac{d^{2}k}{( 2 \pi )^{2}}{\tilde{F}}_{j k}( - n , - k ) [ 1 + \Pi ( n , k ) ]{\tilde{F}}_{j k}( n , k ) \right\}\; .
({\rm Char}_{\cal L}(\bar j_!{\cal F}),[dp_L])_{T^*X,u}=({\rm Char}_{\cal M}(\bar j_!{\cal F}),[dp_L])_{T^*X,u}, ({\rm Char}_{{\cal L}'}(\bar j_!{\cal F}),[dp_{L'}])_{T^*X',u}=((T^*f)^*{\rm Char}_{\cal M}(\bar j_!{\cal F}),[dp_{L'}])_{T^*X',u'}.
L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\theta\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu}+\overline{\psi}(i\gamma^{\mu}D_{\mu}-me^{i\theta^{\prime}\gamma_{5}})\psi.
3\Big(\mu_{1}(\frac{\pi}{3})-\mu_{1}(\frac{\pi}{2})\Big)=4\Big(-R_{2}+\frac{1}{2}R_{4}-\frac{1}{4}R_{5}-R_{6}\Big), &3\Big(\mu_{1}(\frac{2\pi}{3})-\mu_{1}(\frac{\pi}{2})\Big)=4\Big(R_{2}+\frac{1}{2}R_{4}-\frac{1}{4}R_{5}+R_{6}\Big).
\text{LHS of (\ref{eqn:final})}= \sum_{m=2}^4 (-1)^m \binom{1+m}{3}\binom{4}{m}= \binom{3}{3}\binom{4}{2}- \binom{4}{3}\binom{4}{3}+ \binom{5}{3}\binom{4}{4}= \binom{4}{2}- 4^2 + \binom{5}{2}= 0
T=diag\{\tilde q^{(p-1)^2/4-p/8},\tilde q^{(p-2)^2/4-p/8},\tilde q^{(p-2)^2/4-p/8},\ldots \tilde q^{2^2/4-p/8},\tilde q^{2^2/4-p/8},\tilde q^{1^2/4-p/8}\}
A = \frac{1}{1 6 \pi}\int_{0}^{\beta}d t \int_{S_{\perp}}d^{2}x_{\perp}\int_{x_{1}}^{x_{2}}d x \sqrt{f}R = \frac{\beta S_{\perp}}{1 6 \pi}\left[ \frac{f^{\prime}}{\sqrt{f}}\right]_{x_{1}}^{x_{2}}
T(v_j,0) =U(v_j,0)=s_{w_{je}}U(v_j,i) =s_{w_{je-i}}s_{w_{je+i}}T(v_j,i) =U(v_j,0)\cdots U(v_j,i)=T(v_j,i-1)U(v_j,i) S(v_j,i) =T(v_j,i-1)\cdots T(v_j,0)=T(v_j,i-1)S(v_j,i-1).
\Omega_n=\sqrt{1+\frac{m^2\alpha'^2}{n^2 l^2}},\;\;\;\;a^R_n=(a^R_{-n})^{\dag},\;\;\;\;\tilde{a}^R_n=(\tilde{a}^R_{-n})^{\dag},\;\;\;\tilde{A}^R_0=(A^R_0)^{\dag}.