harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left|\sum_{n=m}^\infty(-1)^na_n{\rm erfc}\left(\frac{x-b_n}{2\sqrt{s}}\right)\right| & = \left|\sum_{n=m}^\infty\left(\sum_{k=m}^n(-1)^ka_k\right)\left[{\rm erfc}\left(\frac{x-b_{n+1}}{2\sqrt{s}}\right)-{\rm erfc}\left(\frac{x-b_n}{2\sqrt{s}}\right)\right]\right| &\leq 2\sup_{N\geq m}\left|\sum_{k=m}^N(-1)^ka_k\right|,
V ( f_{1}, f_{2}) = \left| \frac{\mu}{2}\right|^{2}\frac{1}{f_{1}^{i}{\bar{f}}_{1 i}+ f_{2}^{i}{\bar{f}}_{2 i}}\left\{- | f_{1}^{i}{\bar{f}}_{2 i}- f_{2}^{i}{\bar{f}}_{1 i}|^{2}+ ( f_{1}^{i}{\bar{f}}_{1 i}+ f_{2}^{i}{\bar{f}}_{2 i})^{2}\right\}.
\langle e^{A}\rangle_{\beta}= \exp \left[ \frac{1}{2}\, \langle A^{2}\rangle_{\beta}\right] \ ; \ \langle \, : e^{A}: \, \rangle_{\beta}= \exp \left[ \frac{1}{2}\, \langle \, : A^{2}: \, \rangle_{\beta}\right] \, ,
X_A(x_{ij}) =(L_A)_*(X_{I_n})(x_{ij})=X_{I_n}(x_{ij}\circ L_A) =X_{I_n}\left(\sum_{k=1}^n a_{ik}x_{kj}\right)\quad\textrm{(the $(i,j)$-entry of $AB$ is $\sum_{k=1}^n a_{ik}b_{kj}$)}=\sum_{k=1}^n a_{ik}X_{I_n}(x_{kj})=\sum_{k=1}^nX_{I_n}(x_{kj})x_{ik}(A).
\delta^{(m,n)}_{\texttt{a};\lambda}(1)= \prod_{\substack{1\leq j<k\leq n \lambda_j-\lambda_k=0}}\frac{k-j}{1+k-j}\prod_{\substack{1\leq j<k\leq n \lambda_j-\lambda_k=m}}\frac{n-k+j}{n+1-k+j}.
\psi_{n+1}((ab^i)^{kp^n})= \big( ((a^{\alpha}b)^{ki}u_1)^{p^{n-1}}, ((a^{\alpha}b)^{ki}u_2)^{p^{n-1}}, \dots ((a^{\alpha}b)^{ki}u_p)^{p^{n-1}}\big),
\left(\langle\underline t,\partial_{\underline x}\rangle^2+\Delta_{\underline y}\right)\big(g(\theta,\rho)P_{\ell}(\underline y)\big)=\left(\partial_{\theta}^2g+\partial_{\rho}^2g+(2\ell+q-1)\frac{\partial_{\rho}g}{\rho}\right)P_{\ell},
\psi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)\equiv\phi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)+\frac{e}{g}\sum_{i=0}^{p}(-1)^{p\cdot i}\frac{\partial}{\partial X_{\mu_{i}}}B^{\mu_{i+1}\mu_{i+2}\ldots\mu_{p}\mu_{0}\mu_{1}\ldots\mu_{i-1}}(X).
\frac{1}{\ell^2-1}\Theta_2 \left(f_1^{\ell(\ell^2-1)}, R\left( (\ell^2-1)\ell^{n_2+1}\right) \right) = \left( \frac{1}{\ell^2-1}\Theta_1\left(f_1^{\ell(\ell^2-1)}\right), \left( \begin{matrix}0 \ell^{n_2+1}0 0 \end{matrix}\right) \right).
\begin{array}{r c l}{{i T}}&{{=}}&{{\displaystyle + \lambda v \int \left[ 3 D_{k}^{\sigma}+ D_{k}^{Z}+ 2 D_{k}^{W}\right]}}\\{{}}&{{}}&{{\displaystyle + \frac{n}{2}g^{2}v \int \left[ D_{k}^{W}+ D_{k}^{Z}\right]}}\\{{}}&{{}}&{{\displaystyle - \frac{1}{2}g^{2}v \int \left[ D_{k}^{W}+ D_{k}^{Z}\right] ,}}\\ \end{array}
\mathbf{W}_{n}=\frac{1}{n}\sum\limits_{i=1}^{n}\psi_{1}'\left( \frac{y_{i}-\mathbf{x}_{i,I}^{T}\boldsymbol{\theta}_{n}^{*}}{s_{n}}\right)\mathbf{x}_{i,I}\mathbf{x}_{i,I}^{T}
u(t,x,y+h\pi_{t})= \int_{0}^{\sigma_1}[D^{2}_{x}u(s,x,y)+f(s,x,y)]\,ds + u(\sigma_1,x,y+h)- u(\sigma_1,x,y)+\int_{\sigma_1}^{t}[D^{2}_{x}u(s,x,y+h)+f(s,x,y)]\,ds.
\sum_{i=1}^{N_n}E\left[\max_{1\leq k\leq d_n}\left|\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)W^{(i)}_j\right|^3\right]\leq 2d_n^{3/p}(p-1)^{3/2}\sup_{i\in\mathbb{N}}\|Y_i\|_p^3\sum_{i=1}^{N_n}\max_{1\leq k\leq d_n}\left(\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)^2\right)^{3/2}
G^M_N= \frac{6}{l^2}\delta^M_N + \kappa^2 \frac{\sqrt{-g_{brane}}}{\sqrt{-g}}T^M_N = \frac{6}{l^2}\delta^M_N + e^{-\beta}\kappa^2 T^M_N , \:\: (M,N =y,t,x^i).
\frac{1}{\left (k_2 + |\tau-k_2| \right )^{2 (1-b')}\left ( k_3 + |\tau-k_4| \right )^{2b}}= \frac{1}{[f(\tau)]^{2(1-b')}}\le \frac{1}{[f(k_4)]^{2(1-b')}}= \frac{1}{k_3^{2b}(k_3+|k_2-k_4|)^{2 (1-b')}}
e^{-2\hat \phi}=e^{-2\hat \phi_0}+{2\tilde m_k\over r}\ ,\qquad A_u = (A_u)_0 +{2\hat m_k\over r}\ , \qquad A_i = (A_i)_0 +{2 (q_k)_i \over r}\ ,\qquad i=4, \dots 9.
\begin{aligned}{}\|\tilde{u}-L\|_{L^{\infty}(\tilde{\Omega}\cap B(\sigma))}=\sup_{p\in\tilde{\Omega}\cap B(\sigma)}\Big|\frac{(u-L_{\nu})(\delta_{\sigma^{\nu}}(p))}{\sigma^{\nu}\omega(\sigma^{\nu})}-L(p)\Big| &=\frac{1}{\sigma^{\nu}\omega(\sigma^{\nu})}\|u-L_{\nu+1}\|_{L^{\infty}(\Omega\cap B(\sigma^{\nu+1}))},\end{aligned}
\begin{array}{rcl}S ={1\over \kappa}\int d^{11}x\sqrt{|g|}[R] & & -{T_{KK11}\over 2}\int d^{7}\xi \sqrt{|\gamma|}\left[k^{4/7}\gamma^{ij}\partial_{i}X^{\mu}\partial_{j}X^{\nu}\Pi_{\mu\nu}-5\right]\, , \end{array}
\overline{e}_{f}(k,i+1)=\frac{\overline{e}_{f}(k,i)-\overline{\delta}(k,i)\overline{e}_{b}(k-1,i)}{\sqrt{(1-\overline{\delta}^{2}(k,i))(1-\overline{e}_{b}^{2}(k-1,i))}}
\mathcal{B}_{G,h,p_{1},...,p_{m}}^{\mathcal{T}}(F_{1},...F_{m},f,k) =\sup\{\int_{0}^{k}G[(M_{\mathcal{T}}\phi)^{\ast}(t)]h(t)dt:\phi\geq0\text{measurable on}X \left\Vert \phi\right\Vert_{1}=f,\left\Vert\phi\right\Vert_{p_{1},\infty}\leq F_{1},...,\left\Vert \phi\right\Vert_{p_{m},\infty}\leq F_{m}\}
A(n+1)={}\frac{1}{2^{4(n-1)}}\sum_{m=1}^{n}\sum_{\ell=1}^{m}\sum_{k=1}^{\ell}\frac{[(2 n-2m)!]^3}{[(n-m)!]^4}\frac{[(2 m-2\ell)!]^3}{[(m-\ell)!]^4}\frac{[(2 \ell-2k)!]^3}{[( \ell-k)!]^4}\frac{[(2k-2)!]^3}{[(k-1)!]^4}.
G_{m, n}(x)=\mathbb{P}\left(\sum_{i=m+1}^{n}\frac{X_{i}-\mu}{\sigma \sqrt{n}}\leq x\right) =G_{n-m}\left(\frac{x \sqrt{n}}{\sqrt{n-m}}\right)=G_{n-m}\Big(x\left(1+R_{n}\right)\Big),
{w_j}- \sum\limits_{i = 0}^n{\left({p_{B,j,i}^{(1)}+ \sum\limits_{k = 0}^n{p_{B,n + 1,i}^{(1)}p_{B,j,k}^{(1)}{e^{x_{n,k}^{(\alpha )}x_{n,i}^{(\alpha )}}}}}\right){\mkern 1mu}{w_i}- \sum\limits_{i = 0}^m{p_{OB,j,i}^{(1)}\;{r_{j,i}}}- 1 = 0},j = 0, \ldots ,n,
{}^{\Psi}\Gamma_{p}^{(\alpha,\beta)}(x){}^{\Psi}\Gamma_{p}^{(\alpha,\beta)}(y) =4\int_{0}^{\infty}\int_{0}^{\infty}\eta^{2x-1}\xi^{2y-1}{}_{0}\Psi_{1}\left( \alpha,\beta;-\eta^{2}-\frac{p}{\eta^{2}}\right) &~~~\times{}_{0}\Psi_{1}\left( \alpha,\beta;-\xi^{2}-\frac{p}{\xi^{2}}\right)d\eta d\xi.
\kappa\int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left(1-\frac{x_{j}}{x_{1}}\right) \left\{\frac{x_{2}}{x_{1}}\tau (1,2) \mathrm{d}\mu_{1,2}(x) +\mathrm{d}\mu_{1,2}(x) \tau(1,2) \right\}{}=2\kappa\tau(1,2) \int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left( 1-\frac{x_{j}}{x_{1}}\right) \mathrm{d}\mu_{1,2}(x) ,
(\rho^{u})\mathtt{C}\mathtt{S}(\phi) = (\mu_{S_1e}\ast \rho(e,u,f))\mathtt{C}\mathtt{S}(\phi) &= \mu_{v\mathtt{C}(\phi)(S_1e)}\ast \mathtt{C}(\phi)(\rho(e,u,f)) &= \mu_{v\Phi(S_1e)}\ast \Phi(\rho(e,u,f)) &=\mu_{S_2e\phi}\ast\rho(e\phi,u\phi,f\phi) &=\rho^{e\phi}\ast\rho(e\phi,u\phi,f\phi) &=\rho^{e\phi u\phi}=\rho^{e u\phi}= \rho^{u\phi}.
\begin{array}{lll}X^{*}= (I-J^{(r)}\hat{C}C\hat{C}^{-1}J^{(r)})^{-1}\cdot \Big[J^{(r)}\Big((\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}(Dp - U(J^{(r)}X^{*}) - \mathbf{1}\frac{\beta}{2})\Big)\Big]. \end{array}
\begin{array}{c c c}{{}}&{{\sigma}}&{{}}\\{{\bar{\chi}_{+}\, \,}}&{{}}&{{\, \, \chi_{-}}}\\{{}}&{{v_{\mu}}}&{{}}\\{{\bar{\chi}_{-}\, \,}}&{{}}&{{\, \, \chi_{+}}}\\{{}}&{{\bar{\sigma}}}&{{}}\\ \end{array}
\begin{array}{ccc}\{h_\mathbf{v}(\tau)\}_\mathbf{v}+\{k_\mathbf{v}(\tau)\}_\mathbf{v}= \{(h_\mathbf{v}+k_\mathbf{v})(\tau)\}_\mathbf{v}, \{h_\mathbf{v}(\tau)\}_\mathbf{v}\cdot\{k_\mathbf{v}(\tau)\}_\mathbf{v}= \{(h_\mathbf{v}k_\mathbf{v})(\tau)\}_\mathbf{v}.\end{array}
\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}\chi (\tilde{F}_{\alpha\beta}\tilde{F}^{\alpha\beta})g^{\mu\gamma}g^{\nu\delta}\tilde{F}_{\gamma\delta}]-\frac{\tilde{e}^{2}m^{4}}{8}\varphi^{2}\chi g^{\alpha\nu}\tilde{A}_{\alpha}=0
T_i(h_{j,1}) = T_i\big(\big[x_{j}^+,x_{j,1}^-\big]\big) = \big[\big[x_i^+,x_{j}^+\big], -\big[x_i^-,x_{j,1}^-\big]\big] \hphantom{T_i(h_{j,1})}{}= - \big[\big[\big[x_i^+,x_j^+\big],x_i^-\big],x_{j,1}^-\big]-\big[x_i^-,\big[\big[x_i^+,x_j^+\big],x_{j,1}^-\big]\big].
f(X)= -\sum_{j=1}^{r+s}\sum_{\substack{i=1 \lambda_i (X) <0}}^n |a_j^i|^2(X) \lambda_i(X)= - \sum_{j=1}^{r+s}\sum_{\lambda (X) <0}\|p_{\lambda(X)}(\vec{x}_j)\|^2_{z_0}\lambda(X).
\sup_{R >0}\frac{1}{R^{2\alpha}}\displaystyle\inf_{l \in \textrm{Span}\{1, x_1 \}}\| u - \sigma_i(x_0) v_{i}(\cdot, a(x_0)) - l \|_{B_R(x_0)}\leq (\epsilon^{2 - 2\alpha}+ \epsilon^{2 \alpha}) M + ( \epsilon^{\alpha-2}+ \epsilon^{-(4-2\alpha)}+ \epsilon^{-4}+ 1) \tilde{N}.
I(q\,|\,a, \tau) =\log\frac{G(1+a+\tau\,|\,\tau)}{G(1-q+a+\tau\,|\,\tau)}-q\log\Bigl[\Gamma\bigl(1+\frac{a}{\tau}\bigr)\Bigr]+\frac{(q^2-q)}{2\tau}\psi\bigl(1+\frac{a}{\tau}\bigr).
L_{1}=\left(\begin{array}[c]{ccc}0 0 0 0 0 -1 0 1 0\end{array}\right) ,L_{2}=\left(\begin{array}[c]{ccc}0 0 1 0 0 0 -1 0 0\end{array}\right) ,L_{3}=\left(\begin{array}[c]{ccc}0 -1 0 1 0 0 0 0 0\end{array}\right)
( \bar{C}_{\alpha},{\cal T}_{\beta}^{\dagger})_{\Omega}= \frac{1}{2}[{\cal T}_{\alpha},{\cal T}_{\beta}^{\dagger}] , \quad ({\cal T}_{\beta}, \bar{C}_{\alpha}^{\dagger})_{\Omega}= \frac{1}{2}[{\cal T}_{\beta},{\cal T}_{\alpha}^{\dagger}] ,
\|u_{n}\|_{L^{p}(B)}\le \|u_{n}-\psi_n\|_{L^{p}(B)}+\|\psi_n\|_{L^{p}(B)}\le C\, \big(\|D(u_{n}-\psi_n)\|_{L^{p}(B)}+\|\psi_n\|_{L^{p}(B)}\big) \le C\, \big(\|Du_{n}\|_{L^{p}(B)}+\|\psi_n\|_{W^{1,p}(B)}\big),
Q^{\left( \alpha\right)}\left( t;\mathbf{F}\right) :=\frac{M^{\left(\alpha\right)}\left( t;\mathbf{F}\right) -\Gamma\left( \alpha+1\right)\left( M^{\left( 1\right)}\left( t;\mathbf{F}\right) \right)^{\alpha}}{M^{\left( 2\right)}\left( t;\mathbf{F}\right) -2\left( M^{\left(1\right)}\left( t;\mathbf{F}\right) \right)^{2}},
m_\pi = \dfrac{q^{(1/2)(n(n-1))}- q^{(1/2)(n-1)(n-2)}}{2}&=(1/2)\cdot q^{(1/2)(n-1)(n-2)}(q^{n-1}-1) &=\dfrac{q-1}{2}\cdot q^{(1/2)(n-1)(n-2)}\cdot (1+q+q^2+ \cdots + q^{n-2}).
\left( e^{s \bar{D}^{2}}\right)_{A x , B y}= \int d^{4}z_{1}\dots \int d^{4}z_{n}\left( e^{\frac{s \bar{D}^{2}}{n + 1}}\right)_{A x , C_{1}z_{1}}\left( e^{\frac{s \bar{D}^{2}}{n + 1}}\right)_{C_{1}z_{1}, C_{2}z_{2}}\dots \left( e^{\frac{s \bar{D}^{2}}{n + 1}}\right)_{C_{n}z_{n}, B y}
f^{i_1\ldots i_{m_1}j_1\ldots j_{m_2}\ldots k_1\ldots k_{m_{N-1}}}\; u^1_{i_1}\ldots u^1_{i_{m_1}}u^2_{j_1}\ldots u^2_{j_{m_2}}\ldots u^{N-1}_{k_1}\ldots u^{N-1}_{k_{m_{N-1}}}\;.
s + \frac{3}{2}x = \frac{5 - \mu l^{2}e^{- \psi_{2 0}}}{2 \mu l^{2}e^{- \psi_{2 0}}- 2}\left( x - \frac{2}{3}\psi_{0}\right) + \frac{\epsilon_{x}c_{1}}{2}( 3 - \mu l^{2}e^{- \psi_{2 0}}) \left| x - \frac{2}{3}\psi_{0}\right|^{2 - \mu l^{2}\exp ( - \psi_{2 0})},
\left| \log q - L(r) \cdot g \right| \le \left| \log q - \log q_n \right| + \left| \log q_n - n \cdot g\right| + (L(r)-n)\cdot g &\le \left| \log (m/q_n)\right| + \delta \cdot n+ O(\delta \log m) &= O(\delta \cdot n) = O(\delta \cdot L(r)).
\psi ( R ) = - \left( \frac{\bar{\mu}\mu_{P}}{2 g}+ \frac{g}{2}\bar{q}\epsilon_{P}\right) \sum_{j}\ln \left| \frac{R - r_{j}}{a}\right| + \frac{1}{4}\left[ \frac{\mu_{P}\bar{m}}{2 g}+ \frac{g}{2}\epsilon_{P}\bar{q}\right] R^{2}
b_m =\sum_{1\leqslant i_1<i_2<\cdots <i_m\leqslant d}i_1(i_2-1)(i_3-2)\cdots (i_m-m+1) &\qquad\qquad{}\times(d+1)^{i_1-1}d^{i_2-i_1-1}(d-1)^{i_3-i_2-1}\cdots (d-m+1)^{d-i_m}&=\sum_{1\leqslant k_1\leqslant k_2\leqslant\cdots\leqslant k_m\leqslant d-m+1}\prod_{i=0}^{m}k_{i}(d+1-i)^{k_{i+1}-k_{i}},
\sum_{r\geq0}\nu^r\sum_{u+v=r\atop u,v\geq0}\rho_u(\rho_v(FG)H)=\sum_{r\geq0}\nu^r\sum_{u+v=r\atop u,v\geq0}\rho_u(F\rho_v(GH)),\quad\forall F,G,H\in N[\nu].
\mathbf{F}_{\frac{n+1}{2}}:= \left[\frac{(\lambda_{2k+1}- \lambda_{1}) \sin\left[\frac{(2k+1)\pi}{n + 3}\right]}{\lambda_{2k+1}- \phi_{\ell}}\right]_{k,\ell}\mathbf{G}_{\frac{n-1}{2}}:= \left[\frac{(\lambda_{2k}- \lambda_{n+1}) \sin\left(\frac{2k\pi}{n + 3}\right)}{\lambda_{2k}- \psi_{\ell}}\right]_{k,\ell}
|\varepsilon^n| \ll \big(\prod_{\mathfrak{p}|u_n v_n}N(\mathfrak{p})^{v_\mathfrak{p}(p)}\big)^{1+\delta}= \bigg(\prod_{\mathfrak{p}|u_n}N(\mathfrak{p})^{v_\mathfrak{p}(p)}\prod_{\mathfrak{p}|v_n}N(\mathfrak{p})^{v_\mathfrak{p}(p)}\bigg)^{1+\delta}
(\begin{matrix}n\\t\end{matrix})+(\begin{matrix}n\\t+s\end{matrix})+(\begin{matrix}n\\t+2s\end{matrix})+...=\frac{1}{s}\sum_{j=0}^{s-1}(2cos\frac{\pi j}{s})^{n}cos\frac{\pi(n-2t)j}{s}.
S_{0}={\frac{1}{\pi}}\int d^{2}x [ e^{- 2 \phi}( 2 \partial_{+}\partial_{-}\rho - 4 \partial_{+}\phi \partial_{-}\phi + \lambda^{2}e^{2 \rho}) +{\frac{1}{2}}\sum_{i = 1}^{N}\partial_{+}f_{i}\partial_{-}f_{i}] .
\big|p\,\int_{\Omega}\phi^{p-1}u_n\,\mathcal{A}(x,\nabla u_n)\cdot\nabla\phi\,dx\big| \leq \epsilon\,\int_{\Omega}\,w|\phi|^p|\nabla u_n|^p\,dx + c_\epsilon\,\int_{\Omega}\,w\,|u_n|^p\,|\nabla\phi|^p\,dx.
\frac{d^2x^1}{dt^2}=\frac{1}{m}\frac{dx^2}{dt}=\frac{1}{m}y^2, \frac{d^2x^2}{dt^2}=-V^{\prime \prime}\left(x^1\right)\frac{dx^1}{dt}=-V^{\prime \prime}\left(x^1\right)y^1.
\begin{array}[c]{rl}\mathbb{E}\left[{\int}_{t}^{T}\left( |\tilde{Y}_{s}^{m}-Y_{s}^{m}|^{2}+|\tilde{Z}_{s}^{m}-Z_{s}^{m}|^{2}\right) ds\right] \leq C^{2}\sum\limits_{i=0}^{m-1}\mathbb{E}\left[ \left\vert W(t_{i+1}^{m},X_{t_{i+1}^{m}}^{m})-\tilde{Y}_{t_{i+1}^{m}}^{m}\right\vert^{2}\right] & \leq\frac{C^{2}}{m}.\end{array}
G(\omega, \mbox{\boldmath $k$}, \rho \Lambda ; v, e^{2}) = \exp{\left\{\int^{\rho}\frac{d \rho '}{\rho '}\; \gamma\right\}}\; \; G (\omega, \mbox{\boldmath $k$}, \Lambda ; v_{eff}(\rho), e^{2}_{eff}(\rho) )
\int_{\mathbb{R}^{N}}[\mathrm{H}_{u}(u_k,v_k)\varphi+\mathrm{H}_{v}(u_k,v_k)\psi]\,dx \longrightarrow \int_{\mathbb{R}^{N}}[\mathrm{H}_{u}(u,v)\varphi+\mathrm{H}_{v}(u,v)\psi]\,dx.
< \Psi_{n p}^{j}, \Psi_{n^{\prime}p^{\prime}}^{j^{\prime}}> = \int_{H_{3}^{+}}d h \left( \Psi_{n p}^{j}( h ) \right)^{\ast}\Psi_{n^{\prime}p^{\prime}}^{j^{\prime}}( h ) = ( 2 \pi )^{3}\delta_{n , n^{\prime}}\delta ( p - p^{\prime}) i \delta ( j - j^{\prime}) ,
{\cal L}_{{BI}}^{{SUSY}}= \frac{1}{4e^2}\left[ \int\! d^2\theta ~W^2+\int\! d^2\bar \theta ~\bar W^2\right] + \sum_{s,t=0}^\infty a_{st}^{(\beta)}\int\! d^4\theta~ W^2\bar W^2~ X^s Y^t
\begin{aligned}H_n(t,\zeta)+o(1)>0 \zeta\le Z_n(t)-Z_*(D), & H_n(t,\zeta)+o(1)<0 \zeta\ge Z_n(t)+Z_*(D), & F(0)|H_n(t,\zeta)+o(1)|\le 2 h_n Z_*(D) \zeta\in[Z_n(t)-Z_*(D),Z_n(t)+Z_*(D)], \end{aligned}
I_H^{BH}={\beta\over{r_+}}\left[-r_+^4 +r_+^2(1+\sum_{i<j}^4q_iq_j) +r_+(\sum_{i=1}^4q_i+2\sum_{i<j<k}^4q_iq_jq_k)+3\prod_{i=1}^4q_i\right]\ ,
\alpha , \, \alpha + \delta , \, \cdots , \, \alpha + n \delta , \, \cdots , \, \delta , \, 2 \delta , \, \cdots , \, m \delta , \, \cdots \, , \, \cdots \, , \, ( \delta - \alpha ) + l \delta , \, \cdots \, , \, \delta - \alpha \, ,
s_{1,j}= \beta_1'w_j = \frac{w_1'(I_d-P_B)w_j}{\|(I_d-P_B)w_1\|}=\frac{w_1'w_j - w_1'BB'w_j}{\left(w_1'w_1 - w_1'BB'w_1\right)^{1/2}}=\frac{w_1'w_j - \|x\|^2}{\left(w_1'w_1 - \|x\|^2\right)^{1/2}}.
\frac{\phi(\alpha)}{\phi(\beta)}=\frac{\Gamma(1+\beta)\,\Gamma(1+r-\beta)}{\Gamma(1+\alpha)\,\Gamma(1+r-\alpha)}=\lim_{n\rightarrow\infty}\prod_{i=1}^n \,\delta_i(\alpha,\beta),\,\mbox{where}
t_{t_{c_1}(z_1)}^{-1}t_{a_1}t_{z_1}^{-1}t_{a_1}= [t_{t_{c_1}(z_1)}^{-1}t_{a_1}, \phi_1], t_{t_{c_1}^{-2m-1}(z_1)}^{-1}t_{a_1}^{2m+1}t_{c_1}^{-2m-1}t_{a_1}= [t_{t_{c_1}^{-2m-1}(z_1)}^{-1}t_{a_1}^{2m+1}, \phi_2], t_{c_1}^{-2m}t_{a_1}^{2m}= [t_{c_1}^{-2m}, \phi_3].
\Lambda_\mu(p,q;\xi) = \frac{-1}{16\pi^2}\gamma_\alpha \gamma_\mu \gamma_\beta \frac{\partial^2}{\partial p_\alpha \partial q_\beta}\int d^3x\,d^3y\, e^{i(p\cdot x - q\cdot y)}e^{-e^2\xi\left|x-y\right| / 8\pi}\frac{1}{|x|^3 |y|^3}.
\begin{array}{c}\medskip$$G_1E = q E G_1, G_2 E = q^{-1}EG_2, G_1 F = q^{-1}F G_1, G_2 F = q F G_2, G_1 G_2 = G_2 G_1, G_1G_1^{-1}= G_1^{-1}G_1 = 1, G_2G_2^{-1}=G_2^{-1}G_2 =1, EF - FE = \displaystyle \frac{G_1G_2^{-1}- G_2G_1^{-1}}{q - q^{-1}}.\end{array}
|h'_i(y)| = |h'_i(y)-h'_i(x)| &= \left| \int_0^{d_{L_0}(x,y)}(dh'_i)_{\gamma(t)}\left(\dot{\gamma}(t)\right)dt\right| &\leq ||dh'_i|| \int_0^{d_{L_0}(x,y)}|\dot{\gamma}(t)|dt &\leq \mathrm{Diam}(L_0)||dh'_i||,
E_{n\ell}\approx \hbar\omega\left[ \left( n + \frac{D}{2}\right) + \frac{1}{2}\left\{(k^2 +{k'}^2 )\left( n + \frac{D}{2}\right)^2 + (k^2 -{k'}^2 )\left(L^2 + \frac{D^2}{4}\right) +{k'}^2\frac{D}{2}\right\}\right]\;.
\frac{K_{0}}{c_{3}^{2}\left( cf_{1}f_{1}^{\prime \prime}-\left(f_{1}^{\prime}\right)^{2}\right)}=\frac{f_{2}^{2c}}{\left( af_{1}^{\prime}f_{2}+c_{3}f_{1}f_{2}^{c}\right)^{4}}.
\left[{dn_d(\omega,D) \over d\omega}\right]_{\mbox{div}}=r_d{\Gamma\left(1-\frac D2\right) \over (4\pi)^{D/2}}{m^{D-4}\over \kappa}\int_{\Sigma}\left[2\left(m^2+{R\over 12}-{R_{ii}\over 8}\right)-{\omega^2 \over \kappa^2}P\right]~~~.
\;\; - \frac{1}{N}\sum_{(b_1,\vec{b}_\ast):\;\mathcal{U}_{b_1,\vec{b}_\ast}= 0, \; \mathcal{U}_{b_1-1,\vec{b}_\ast}\neq 0}\mathcal{U}_{b_1-1,\vec{b}_\ast}\delta_{b_1,\vec{b}_\ast}.
F_{\mu \nu}= i F_{\mu \nu}^{1}+ F_{\mu \nu}^{5}\Gamma_{5}+ F_{\mu \nu}^{a}\Gamma_{a}+ F_{\mu \nu}^{a 5}\Gamma_{a}\Gamma_{5}+ \frac{1}{4}F_{\mu \nu}^{a b}\Gamma_{a b},
log\Gamma_{b}(w)=\int_{0}^{\infty}\frac{dt}{t}[\frac{e^{-wt}-e^{-\frac{Q}{2}t}}{(1-e^{-bt})(1-e^{-b^{-1}t})}-\frac{(\frac{Q}{2}-w)^{2}}{2}e^{-t}-\frac{\frac{Q}{2}-w}{t}].
N_R^0 = \sum_{n=1}^{\infty}n(a_n^\alpha{}^\dagger a_n^\alpha +s_n^A{}^\dagger s_n^A), ~~~~~ N_L^0 = \sum_{n=1}^{\infty}n(\tilde{a}_n^\alpha{}^\dagger \tilde{a}_n^\alpha + \tilde{s}_n^A{}^\dagger \tilde{s}_n^A).
\begin{aligned}(\int_t^{+\infty}[A(s)]^{\frac{1}{1-p}}ds)^{1-p}\leq\int_t^{+\infty}[(\mathcal{A}(q)+\varepsilon)(\frac{e^s}{2})^n]^{\frac{1}{1-p}}ds)^{1-p}=(\mathcal{A}(q)+\varepsilon)\frac{1}{2^n}(\int_t^{+\infty}e^{\frac{ns}{1-p}}ds)^{1-p}=\frac{1}{2^n}(\frac{n}{p-1})^{p-1}(\mathcal{A}(q)+\varepsilon)e^{nt}\end{aligned}
K = *{\frac{( \kappa^{2})}{2}}={\frac{i}{2}}\left[ f_{\mu_{1}\bar{\nu}_{1}}f_{\mu_{2}\bar{\nu}_{2}}\right] \epsilon_{\alpha}^{\mu_{1}\mu_{2}}\epsilon_{\bar{\beta}}^{\bar{\nu}_{1}\bar{\nu}_{2}}d z^{\alpha}\wedge d \bar{z}^{\beta}
E [ f^{(0)}+ f^{(1)}] = E [ f^{(0)}] + \frac{\delta E}{\delta f}[ f^{(0)}] \cdot f^{(1)}+ \frac{1}{2}\frac{\delta^2 E}{\delta f^2}[ f^{(0)}] \cdot (f^{(1)})^2 + \cdots,
\nabla_{2k+2}=\sum_{s=0}^{k+1}(-1)^{s-1}\nabla_s\Delta_{2k+2-s}+\sum_{\substack{t=k+3 t}}^{2k+1}\nabla_t\Delta_{2k+2-t}-\sum_{\substack{s=k+2 s}}^{2k}\nabla_s\Delta_{2k+2-s}=\sum_{s=0}^{2k+1}(-1)^{s-1}\nabla_s\Delta_{2k+2-s}.
\Theta_{\Lambda}\left[\begin{array}{c}a b\end{array}\right](v^{I}+\tau p^{I}+q^{I}|\tau)=e^{-i\pi\tau q^{2}-2\pi i\,p\cdot(v+b)+2\pi i a\cdot q}\Theta_{\Lambda}\left[\begin{array}{c}a b\end{array}\right](v^{I}|\tau).
\widetilde{W}=\left( \begin{array}{ll}A-\widehat{\theta}\Phi A A\theta -\widehat{\theta}(\Phi A\theta+(A^{t})^{-1}) \Phi A \Phi A\theta +(A^{t})^{-1}\end{array}\right) =\left( \begin{array}{ll}M N R S\end{array}\right)
\tilde{\eta}(\omega)(m) =\sigma_{\eta(T)}(m)=<\eta(T)m,m> & =\int_{\mathcal{M}_{j}}\sigma_{T}(m^{\prime})<m,m^{\prime}><m^{\prime},m>dm^{\prime}& =\int_{\mathcal{M}_{j}}\omega(m^{\prime})[<m,m^{\prime}><m^{\prime},m>]dm^{\prime}& =\int_{\mathcal{M}_{j}}N^{\prime}(m,m^{\prime})\omega(m^{\prime})dm^{\prime}
a_{\beta}^R = a_{\beta}\delta\exp\left(-(\delta\Xi_{\beta})^{\frac{\beta-2}{2}}\frac{\mathcal{I}_0\left((\delta\Xi_{\beta})^{-\frac{\beta-2}{2}}\right)}{\mathcal{I}_1\left((\delta\Xi_{\beta})^{-\frac{\beta-2}{2}}\right)}\right).
\sum_{j_{1 2}}( - 1 )^{2 ( 2 ( j_{1}+ j_{2}) ( j_{1 2}+ j_{1 2}^{\prime}) + j_{1 2})}\left\{\begin{array}{c c c}{j_{1}}&{j_{2}}&{j_{1 2}}\\{j_{1}}&{j_{2}}&{j_{1 2}^{\prime}}\\ \end{array}\right\}^{S}= 1 .
S ( z_{1}, z_{1}^{\prime}; z_{2}, z_{2}^{\prime}) L ( z_{1}, z_{1}^{\prime}) \otimes L ( z_{2}, z_{2}^{\prime}) = L ( z_{2}, z_{2}^{\prime}) \otimes L ( z_{1}, z_{1}^{\prime}) S ( z_{1}, z_{1}^{\prime}; z_{2}, z_{2}^{\prime}) .
II\lesssim \Bigg{(}\int_0^t\bigg{(}\frac{b_c^{1+\frac{1}{p_0}-\delta}}{\lambda(\tau)^{2+\frac{2}{p-1}+\frac{1}{r_2}}}\bigg{)}^{q_2'}d\tau\Bigg{)}^{\frac{1}{q_2'}}\lesssim \frac{b_c^{\frac{p_0+1}{3p_0}-\frac{2\delta}{3}}}{\lambda(t)^{\frac{2}{p-1}-\frac{1}{p_0}}}= \frac{b_c^{\frac{7}{15}-\frac{2\delta}{3}}}{\lambda(t)^{\frac{2}{p-1}-\frac{1}{p_0}}}.
\left| L_{1}\right| \lesssim \left\vert \eta \right\vert \, \left\vert{\bf{P}}_{1}\right\vert \, \aleph \overset{(\ref{DefP1S})}{=}\mathcal{O}\left( \left\vert \eta \right\vert \, a^{5}\, \aleph \right)=\mathcal{O}\left(a^h\right),
\ddot{z}_j(t)=-\sum_{k:k\neq j}\frac{\lambda_k i\overline{\dot{z}_j(t)-\dot{z}_k(t)}}{2\pi(\overline{z_j(t)-z_k(t)})^2}+\bar{F}_z(z_j(t),t)\dot{z}_j(t)+\bar{F}_t(z_j(t),t)
\int_{\Sigma}\sqrt{W}[ - \overline{{\Psi}}\Gamma_{-}\Gamma_{r}\mathcal{D}_{r}\Psi + \overline{{\Psi}}\Gamma_{-}\Gamma_{m}\{X^{m}, \Psi \}+ \Lambda \{\overline{{\Psi}}\Gamma_{-}, \Psi \}] ,
L\Phi(x)= \sum_{i=3}^{N-2}m \eta_i (c_{i-1}+c_{i+1}-2c_i)+ m c_2(\eta_1-2\eta_2)+m c_{N-1}(\eta_{N}-2\eta_{N-1}) &+ m c_3\eta_2 + m c_{N-2}\eta_{N-1}& +c_1(bm+(b-d-m)\eta_1+m\eta_2)+c_N(bm+(b-d-m)\eta_N+m\eta_{N-1}).
\frac{\beta}{2}\sum_{i<j}^N \frac{z_i + z_j}{z_i - z_j}\, \bigl(z_i \, \partial_{z_i}- z_j \, \partial_{z_j}\bigr) = \frac{\beta}{2}\sum_{j=1}^N \Biggl( \, \sum_{i (\neq j)}^N \frac{z_i + z_j}{z_i - z_j}\Biggr) z_j \, \partial_{z_j}\, ,
\infty>i(c)\,(=i(c, g))\,\ge 2, d_c\,(=d_{c,g, m, I}=d_{c,g, m, \tau(I)}=d_{c,g, m,\tau(\Omega)}) &\qquad\qquad\qquad\qquad\quad\,\,\,:=(\sqrt{\pi})^{e_{i(c)(\gamma_{i(c)})}}\sum_{h\in\mathbb{N}, h<c, (h, c)=1}\frac{e^{-i\frac{2\pi h}{c}{m}}}{\prod_{r=1}^{i_f}\left\vert \lambda_r-1\right\vert^{m_r}}
\begin{aligned}\lambda_D =(b_i-d_i)^{-1}(c_i-d_i), \i=1,2, \lambda_E =(c_i-e_i)^{-1}(a_i-e_i), \i=1,2, \lambda_F =(a_i-f_i)^{-1}(b_i-f_i), \i=1,2.\end{aligned}
{\sigma^{2}_{v}}^{(l)}=\sigma^{2}_{0}+\mathrm{var}\bigg(\sum_{i=1}^{d_{v}-1}\mu_{i}^{(l-1)}\bigg)+2\mathrm{cov}\bigg(u_{0},\sum_{i=1}^{d_{v}-1}\mu_{i}^{(l-1)}\bigg),
q_{k,n,D_{2,\ell_0,0}}\left({\bf p},d\right) &={\rm E}_{I_{2,n},I_{2,-n},I_{1}}\left[{\rm Pr}\left[\left|h_{2,\ell_0,0}\right|^{2}\ge \left(2^{\frac{k\tau}{W}}-1\right)D_{2,\ell_0,0}^{\alpha_2}\left(I_{2,n}+I_{2,-n}+I_{1}\frac{P_1}{P_2}+\frac{N_{0}}{P_2}\right)\Big|D_{2,\ell_0,0}=d\right]\right]
\sigma_{p}i\infty = p \mathrm{and}\sigma_{p}^{-1}\Gamma_{p}\sigma_{p}= \langle\gamma_{\infty}\rangle,\mathrm{where}\,\,\, \gamma_{\infty}=\left(\begin{array}{ccc}1 1 0 1 \end{array}\right)\quad \mathrm{and}\quad\Gamma_{p}=\langle\gamma_{p}\rangle
{\cal Q}={\frac{1}{1 6 \pi G_{D}}}\int_{S^{\bar{d}- 1}}* F_{q + 2}= ( d - 1 ) \Omega_{d}\tilde{Q}, \qquad \tilde{Q}={\frac{1}{1 6 \pi G_{D}}}\sqrt{\frac{2 ( D - 2 )}{\Delta}}Q ,
-{\rm dim\tot}_u\phi(p_L,j_!{\cal F}) =({\rm Char}_{\cal M}(\bar j_!{\cal F}),[dp_L])_{T^*X,u}, -{\rm dim\tot}_{u'}\phi(p_{L'},j'_!{\cal F}') =((T^*f)^*{\rm Char}_{\cal M}(\bar j_!{\cal F}),[dp_{L'}])_{T^*X',u'}.
\nabla_1 P^\Sigma_{22}= D_1 P_{22 \nu \nu \nu}-2 h_{12}P_{\nu 2 \nu \nu \nu}+h_{11}(P_{221\nu\nu}+P_{22\nu 1 \nu}+P_{22\nu \nu 1}) & +h_{12}(P_{222\nu \nu}+P_{22\nu 2 \nu}+P_{22 \nu \nu 2})
n \: C \; \alpha_{1}^{1 - p}\: \beta_{1}^{1 - q - \gamma}\left( \prod_{j = 2}^{p}\; \: \alpha_{1}^{a_{j}- \alpha_{j}}\: \alpha_{j}^{- a_{j}}\right) \beta_{i_{0}}^{\gamma}\left( \prod_{k = 2}^{q}\beta_{1}^{b_{k}- \beta_{k}}\: \beta_{k}^{- b_{k}}\right) ,
\theta_{2}(q)=\sum_{n=-\infty}^{\infty}q^{(n+\frac{1}{2})^{2}}\theta_{3}(q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}\theta_{4}(q)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n^{2}}.
\|\rho_3 f\|_{L^2(0,T;H_0^1(I))}^2 + \|(\rho_3)_t f\|_{L^2(Q)}^2 \leq C\b((\hat{u},\hat{z}^1,\hat{z}^2),(\hat{u},\hat{z}^1,\hat{z}^2)) := C \left( \iint_Q \rho_0^2 |\hat{y}|^2 dxdt + \sum_{i=1}^2 \iint_Q \rho_0^2 |\hat{p}^i|^2 dxdt \right. &+ \left. \iint_{\mathcal{O}\times (0,T)}\rho_1^2 |\hat{f}|^2 dxdt \right)
\int_{\Omega}\vert \nabla{y_1}_*\vert^{p - 2}\nabla{y_1}_*\cdot \nabla hdz + \int_{\Omega}\eta\vert \nabla{y_1}_*\vert^{q - 2}\nabla{y_1}_*\cdot \nabla hdz \geq \int_{\Omega}a_1{y_1}_*^{- \nu}hdz + \lambda(\kappa_1 + 1)\int_{\Omega}\vert{y_1}_*\vert^{\kappa_1}\vert{y_2}_*\vert^{\kappa_2 + 1}h dz,
\langle \lambda^{\prime}| \langle \mu | \left\{[ B_{1}^{-}( \zeta ) , [ B_{1}^{+}( \zeta ) , \widetilde M^{-}( - z ) ] ] + [ A_{1}( \zeta ) , \widetilde M^{-}( - z ) ] + M^{+}( - z ) \right\}= \langle \lambda^{\prime}| \langle \mu | i S ( \lambda , \mu ) \, ,
K(t,{\bf x}, s\,\vert\, t,{\bf x}, 0) = \big\langle t,{\bf x}\big\vert \exp -i\left[\left({\partial_{t}}^2 -{\nabla}^2 +2iq\phi (\partial_t + \partial_{z})\right)s\right]\,\big\vert t,{\bf x}\big\rangle.
\sum_{n=0}^{\infty}\frac{z^nq^{2n^2+2n}}{(q;q^2)_{n+1}(zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}b_\omega^1(m,n)z^m q^n =\sum_{n=0}^{\infty}\frac{q^n}{(zq;q^2)_{n+1}}
Cx=C_{1}x\oplus\cdots\oplus C_{n}x=\left(\begin{array}{c}C_{1}x \vdots C_{n}x\end{array}\right)=\left(\begin{array}{c}C_{1}\vdots C_{n}\end{array}\right)x\in\left(\begin{array}{c}Y_{1}\vdots Y_{n}\end{array}\right)=Y
(h^{-1}\,\partial_{n^\alpha}A^{01}\,h)^\bot=\sum_{\gamma,\beta}(h^{-1}\phi_\gamma\, h)^\dagger\ (\Delta^{-1})_{\gamma\beta}\ \big\langle (h^{-1}\phi_\beta\, h)^\dagger\,\big|\,h^{-1}\,\partial_{n^\alpha}A^{01}\,h\big\rangle
f_{m,p;i}(x) =xf_{m,p+i-1}+\frac{x^{p+1}}{(1-x)^2}\sum_{j=-1}^{i-2}x^{i-j}f_{m-i+j+1-p;j}(x) &+\frac{x^{3+p}}{(1-x)^3}\sum_{k=0}^{i-1}\sum_{j=k}^{m-i-1+k-p}\frac{x^{i-k}}{(1-x)^{j-k}}f_{m-i+k-p,j}(x)\,.