harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{_rT_{ij}}\{T_\infty}\ -\{_rT_{ik}}\{_rT_{jk}}\ +\{_rT_{il}}\{_rT_{jk}}; 0\le i<j\le n_r,k=n_r+1,\l= n_r+2,{_rT_{ij}}\{_rT_{kl}}\ -\{_rT_{ik}}\{_rT_{jk}}\ +\{_rT_{il}}\{_rT_{jk}}; 0\le i<j< k<l\le n+2,k\le n_r,
\frac{(n-2)(10-n)}{4}\int_{B_{\rho}}r^{2-n}u_r^2 \d x & \leq C \int_{B_{2\rho}\setminus B_{\rho}}r^{2-n}|\nabla u|^2 \d x +C \varepsilon \int_{B_{2\rho}}r^{4-n}|D^2 u| |\nabla u| \d x & + C \varepsilon \int_{B_{2\rho}}\big( r^{3-n}+ \varepsilon r^{4-n}\big) |\nabla u|^2 \d x.
\begin{array}{ll}{1\over8\pi}\,\delta A =\left\{r_++M\left(1+{\left(r_+-M\right)}^{-1}\left(M^2+a^2\over M\right)\right)\right\}\delta M-\frac{a}{\left(r_+-M\right)}\,\delta J &=\frac{r_+^2+a^2}{r_+-M}\,\delta M-\frac{a}{r_+-M}\,\delta J={1\over\kappa}\,\delta M-\frac{\Omega_H}{\kappa}\,\delta J\;,\end{array}
\int_a^b w(x) f'(x)^2 dx \leq \int_a^b w(x) f'(x)^2 dx-\frac{1}{2}\int_a^b w''(x) f(x)^2 dx =\int_a^b \bigl( \sqrt{w(x)}f(x) \bigr) \cdot \bigl(- \sqrt{w(x)}f''(x) \bigr) dx.
p_2(x,y,z+\phi_1 + \phi_2) = xy \frac{a_{000}\left( a_{011}a_{101}- a_{001}a_{111}\right)}{a_{001}^2}+ x^2 \left( \frac{a_{000}a_{101}^2}{a_{001}^2}+ y \frac{a_{000}a_{101}a_{111}}{a_{001}^2}\right)+ z\left( a_{001}+ a_{011}y + a_{101}x + a_{111}xy\right)
\gamma ( s_{p j}, s_{q j}^{\prime}, t ) = \exp \left( \frac{t}{2}\left( \frac{\partial^{2}}{\partial s_{p j}^{2}}+ \frac{1}{s_{p j}}\frac{\partial}{\partial s_{p j}}\right) \right) \frac{\delta ( s_{p j}- s_{q j}^{\prime})}{\sqrt{s_{p j}s_{q j}^{\prime}}}
\tilde{b}_{~ [ l_{1}\cdots l_{F + 4 + k}]}^{~ [ A_{1}\cdots A_{F - k}]}= \epsilon_{i_{1}\cdots i_{2 F + 8}}x_{i_{1}}^{A_{1}}\cdots x_{i_{F - k}}^{A_{F - k}}\bar{C}_{l_{1}i_{F + 1 - k}}\cdots \bar{C}_{l_{k + F + 4}i_{2 F + 4}}W_{i_{2 F + 5}i_{2 F + 6}}W_{i_{2 F + 7}i_{2 F + 8}}.
[x_{0}^{+}(z),x_{0}^{-}(w)] &=-\big(:\beta(z)\beta^{*}(z):+:X(-\varepsilon_{1},z)X(\varepsilon_{1},z):\big)\delta(z-w)-\partial_{w}\delta(z-w) &=-\big(\alpha_{0}(z)\delta(z-w)+\partial_{w}\delta(z-w)\cdot 1\big),
\begin{array}{l}\displaystyle{f_n\Big[n-\frac12\Big]=\frac{n-\frac32}{2(n-1)}A Q^{-1}f_{n-1}\Big[n-\frac32\Big]\,,}\displaystyle{g_n\Big[n+\frac12\Big]=-\Big(n-\frac12\Big)\frac{\gamma}{\epsilon}Q'Q^{-1}f_n\Big[n-\frac12\Big]\,.}\end{array}
\begin{array}{l}\left[ u(i+1,j)-2u(i,j)+u(i-1,j)\right] +\left[ u(i,j+1)-2u(i,j)+u(i,j-1)\right] \bigskip +\lambda f((i,j),u(i,j))=0,\bigskip i\in \{1,..,m\},j\in \{1,...,n\}\bigskip u(i,0)=u(i,n+1)=0i\in \{1,..,m\}\bigskip u(0,j)=u(m+1,j)=0j\in \{1,...,n\}\end{array}
J_{m n}= \frac{1}{1 6 \pi^{2}}\frac{1}{A^{4}}\eta_{n m}\Gamma ( \frac{\epsilon}{2}) \int \frac{d x d y d z \delta ( 1 - x - y - z )}{{( a_{2}x + a_{3}( y + z ) - p^{2}{( x + y )}^{2})}^{\epsilon / 2}}
\int_{-\infty}^{\infty}q^{\alpha^{2}}\left(-iq^{-n/2}\right)^{\nu-\alpha}S_{n}\left(-q^{\nu+\alpha};q\right)J_{\nu-\alpha}^{(2)}\left(2iq^{n/2};q\right)d\alpha=\frac{1}{\log q^{-1}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q}\right)}{\left(q;q\right)_{\infty}\left(q;q\right)_{n}}dx,
\|f_1f_2\|_{L^2}^2= \sum\limits_{k\in\mathbb{Z}}\|f_1f_{2,k}\|_{L^2}^2\leq \sum\limits_{k\in\mathbb{Z}}C\big(\|f_1\|_{L^2}^2+\|\partial_zf_1\|_{L^2}^2\big)\|f_{2,k}\|_{H^{1}}^2 \leq C(\|f_1\|_{L^2}^2+\|\partial_zf_1\|_{L^2}^2)\|f_{2}\|_{H^{1}}^2,
[\begin{matrix}c_{2}c_{1}&-s_{2}&c_{2}s_{1}\\c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}c_{2}&c_{3}s_{2}s_{1}-s_{3}c_{1}\\s_{3}s_{2}c_{1}-c_{3}s_{1}&s_{3}c_{2}&s_{3}s_{2}s_{1}+c_{3}c_{1}\end{matrix}]
f_\mathbf{U}(\mathbf{u}|\alpha, \beta) = \left(\frac{\alpha}{2\pi\beta^2\Gamma(2/\alpha)}\right)^{M_\mathrm{F}}\prod_{l=0}^{M_\mathrm{F}-1}\exp\left(-\left(\frac{|u_l|}{\beta}\right)^\alpha\right)
D:=\min\Big\{\frac{|\chi_2\mu_2-\chi_1\mu_1|}{2\sqrt{\lambda_2}}+\frac{\chi_1\mu_1|\sqrt{\lambda_1}-\sqrt{\lambda_2}|}{2\sqrt{\lambda_1\lambda_2}}, \frac{|\chi_1\mu_1-\chi_2\mu_2|}{2\sqrt{\lambda_1}}+\frac{\chi_2\mu_2|\sqrt{\lambda_2}-\sqrt{\lambda_1}|}{2\sqrt{\lambda_1\lambda_2}}\Big\}.
f ( \zeta ) = \frac{( - q \zeta^{- 1}; p )_{\infty}( - p q^{- 1}\zeta ; p )_{\infty}}{( - q \zeta ; p )_{\infty}( - p q^{- 1}\zeta^{- 1}; p )_{\infty}}g ( \zeta^{2}) .
R_{\mu_{1}\nu_{1}\cdots \mu_{s}\nu_{s}}= \partial_{\alpha_{1}\cdots \alpha_{s}}h_{s}^{\beta_{1}\cdots \beta_{s}}\prod_{i = 1}^{s}( \delta_{\mu_{i}\alpha_{i}}^{{}}\delta_{\nu_{i}\beta_{i}-}\delta_{\mu_{i}\beta_{i}}^{{}}\delta_{\nu_{i}\alpha_{i}}) .
\frac{1}{W_f(t)}= 1 - \frac{3}{t^{-1}+1}+ \sum_{ef}\frac{1}{[2,m(e)](t^{-1})}- \frac{1}{W_f(t^{-1})}= &= 1 - \frac{3}{t^{-1}+1}+ 2A(f) + \frac{1}{W_f(t^{-1})}.
H_0^1(0,1):=\Big\{f\in L^2[0,1]\;\Big|\;\frac{df}{dx}\in L^2[0,1],\f(0)=f(1)=0\Big\}, &H^2(0,1):=\Big\{f\in L^2[0,1]\;\Big|\;\frac{df}{dx},\frac{d^2f}{dx^2}\in L^2[0,1]\Big\}.
\int_M r(y)^\gamma \,u(y)^2\,dy = \int_M r(y)^\gamma \,\left(\sum_k \varphi_k (y) u(y)\right)^2\,dy &\leq 2\sum_k \int_M r(y)^\gamma \,\varphi_k (y)^2 u(y)^2\,dy &\leq C\sum_k (k-1)^\gamma \int_M \varphi_k (y)^2 u(y)^2\,dy &\leq C\sum_k \int_M |\nabla\left(\varphi_k (y) u(y)\right)|^2\,dy,
\prod_{i=1}^{N}\left(\bigg(\frac{z_i(\alpha_i+y)}{\alpha_i(z_i+y)}\, \frac{R(z_i)}{\prod_{j=1}^{N}(z_j-\alpha_i)}\bigg)^{b_i}\prod_{p=1}^{r}\bigg(\frac{1+\alpha_ix_p}{1+z_ix_p}\bigg)^{a_i+m_p+1}\right)=(-1)^{(N-1)\sum b_i}f(z_{N+1}).
\partial^{\mu}\partial_{\nu}\partial_{\lambda}\sigma^{2}+ \delta^{\mu}{}_{\nu}\partial_{\lambda}[ \sigma \partial_{\alpha}\partial^{\alpha}\sigma - 2 ( \partial_{\alpha}\sigma ) ( \partial^{\alpha}\sigma ) ] - [ \delta^{\mu}{}_{\nu}\partial_{\lambda}+ \delta^{\mu}{}_{\lambda}\partial_{\nu}+ \delta_{\lambda \nu}\partial^{\mu}] ( \partial_{\alpha}\sigma ) ( \partial^{\alpha}\sigma ) = 0 .
V_\nu(x) := -\frac{2}{\sqrt{\pi}}\frac{\Gamma(\tfrac{1}{2}+\nu)}{\Gamma(\nu)}(1+x^2)^{-1/2}{}_2F_1(1, \tfrac{1}{2}+\nu,\tfrac{1}{2},-x^2) u_\nu(x) := \frac{1}{(1+x^2)^{\nu}}.
U_{\hbar}\left( \alpha\right)^{\ast}P\left( a_{\hbar},a_{\hbar}^{\dag}\right) U_{\hbar}\left( \alpha\right) =P\left( a_{\hbar}+\alpha,a_{\hbar}^{\dag}+\bar{\alpha}\right) =\sum_{k=0}^{d}P_{k}\left( \alpha:a_{\hbar},a_{\hbar}^{\dag}\right)
\begin{aligned}\alpha'' =\varphi^{-1}\frac{\partial\varphi}{\partial v}=\frac{\partial x^0}{\partial v}E_0+\frac{\partial x^1}{\partial v}e^{-\mu_1x^0}E_1+\frac{\partial x^2}{\partial v}e^{-\mu_2x^0}E_2. \end{aligned}
\left.\frac{\partial}{\partial \nu}\,J_\nu(z)\right|_{\nu=n}= \left(\ln (z/2)+\gamma\right)\,J_n(z) - (z/2)^n\sum_{m=0}^\infty \frac{(-z^2/4)^m}{m!\,(m\!+\!n)!}\,\hat{H}_{m+n}^{(1)}\,,
h \propto ( C \Gamma_{\mu_{1}\dots \mu_{5}})_{\alpha \beta}\Pi^{\mu_{1}}\dots \Pi^{\mu_{5}}\Pi^{\alpha}\Pi^{\beta}- \frac{1 5}{2}( C \Gamma_{\mu_{1}\mu_{2}})_{\alpha \beta}\Pi^{\mu_{1}}\Pi^{\mu_{2}}\Pi^{\alpha}\Pi^{\beta}H
\sum_{j=1}^m \sum_{l=k_{j-1}}^{k_j-1}\int_{\mathbb{R}^6}F(y,x') G(x,x')F(y,y')G(x,y') & \ \vartheta_{2^{k_j}}(x'-p-q) \psi_{2^l}(y'-p-q)\omega_{2^l}(x-p) \omega_{2^l}(y-q) \,dx dy dx' dy' dp dq.
v^{(n)}(x)(v(x))^{n-1}= -\frac{f^{(n+1)}(x)}{(f^\prime(x))^{n+1}}+Q_n\left(\frac{f^{\prime \prime}(x)}{(f^\prime(x))^{2}}, \ldots,\frac{f^{(n)}(x)}{(f^\prime(x))^{n}}\right),n\geq 1,
I^{(\alpha)}(\Omega) = \sum_{n \in \mathbb{Z}}\frac{\left|\left\langle w^\infty_{\alpha\alpha}, \varphi^{(n)}_{\Omega, \alpha}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{\Omega, \alpha}\right|}< + \infty
\sum_{m \in \Delta(n)}\frac{2}{m^{d_k}(1 - m^2)}= -2 \sum_{j= 0}^{\infty}\sum_{m \in \Delta(n)}\frac{1}{m^{d_k+2j+2}}& = \frac{2}{n^{d_k}(n^2 - 1)}- \sum_{j= 0}^{\infty}\frac{1}{2^{d_k+2j+1}}\zeta\left(d_k+2j+2, \frac{n}{2}\right) ,
\tilde{S}:=\Bigl\{(x,s,\delta) \Bigm| \bigl(Z(s),\delta\bigr) \in(\ref{eq:Inc-1}),\ (x,s_{\cdot n}) \in W,\s_{ij}= \min \bigl\{b_{ij}(\delta),s_{in}\bigr\}\; \forall i \; \forall j \Bigr\}.
\displaystyle\frac{c\mu(\nu^{2}-b_{2}^{2})}{\gamma-\mu}-\displaystyle\frac{\nu^{2}b_{2}^{2}}{(\gamma-\mu)^{2}}+\displaystyle\frac{\nu^{2}b_{1}^{2}}{(\gamma-\mu)(\gamma-\lambda)}-\mu\gamma-c(1-b_{2}^{2})=0.
\delta \mathrm{A}= \mathrm{A}^{\prime}\xi_{\mathrm{B}}+ \xi_{\mathrm{B}}^{\prime}\mathrm{A}- [ \Lambda_{\mathrm{B}}\, \mathrm{A}- \mathrm{A \,}\Lambda_{\mathrm{B}}] + k \, \mu \, \Lambda_{\mathrm{B}}^{\prime}= 0 ,
Q_{jk}= \dfrac{K^2 \left| \left\langle L_{F'}e_k',e_j'\right\rangle \right|^2}{\xi^2 \, \left|\Re\left(\left\langle L_{F'}e_j', e_j' \right\rangle\right)\right|\left|\Re\left(\left\langle L_{F'}e_k', e_k' \right\rangle\right)\right|}j<k.
\partial_t (| \nabla f|^2) = \partial_t(g^{ij}\partial_i f \partial_j f) \displaystyle = (\partial_t g^{ij}) \partial_i f \partial_j f + 2 g^{ij}\partial_i f \partial_j f_t \displaystyle = 2 h^{ij}\partial_i f \partial_j f + 2 g^{ij}\partial_i f \partial_j f_t \displaystyle = 2 h_{ij}f_i f_j + 2 f_i f_{t, j},
C ( l )_{n , n^{\prime}}^{m , m^{\prime}}= \left( \begin{array}{c c}{n}\\{l}\\ \end{array}\right) \left[ \begin{array}{c}{m^{\prime}+ n^{\prime}}\\ \end{array}\right]_{l}- \left( \begin{array}{c c}{n^{\prime}}\\{l}\\ \end{array}\right) \left[ \begin{array}{c}{m + n}\\ \end{array}\right]_{l}
\begin{aligned}\langle x^*,x_k-x\rangle\leq \|y^*\|_{Y^*}\|y_k-y\|_Y+(\varepsilon+\gamma)(\|x_k-x\|_X+\|y_k-y\|_Y) \leq \big(\kappa\|y^*\|_{Y^*}+(\varepsilon+\gamma)(1+\kappa)\big)\|x_k-x\|_X\;\mbox{when}\;k>k_0.\end{aligned}
\log BF_{kl}=\log \frac{f_{k}(\mathbf{y}|\boldsymbol{\theta}_{k})}{f_{l}(\mathbf{y}| \boldsymbol{\theta}_{l})}+\log \frac{\pi_{k}(\boldsymbol{\theta}_{k})}{\pi_{l}(\boldsymbol{\theta}_{l})}-\log \frac{\pi(\boldsymbol{\theta}_{k}| \mathbf{y})}{\pi(\boldsymbol{\theta}_{l}| \mathbf{y})}.
w_+=\begin{cases}w \mbox{a.e.~in}\Omega_+, 0 \mbox{a.e.~in}\Omega\setminus \Omega_+, \end{cases}\nabla w_+=\begin{cases}\nabla w \mbox{a.e.~in}\Omega_+, 0 \mbox{a.e.~in}\Omega\setminus \Omega_+. \end{cases}
\begin{aligned}T(k_1,\ldots,k_n;(\alpha,\beta))-T(k_n,k_1,\ldots,k_{n-1};(\alpha,\beta)) = Z_{II}(k_n,k_1,\ldots,k_{n-2},k_{n-1}+1;(\alpha,\beta)) &-\sum_{j=0}^{k_n-2}Z_{I}(j+1,k_1,\ldots,k_{n-1},k_n-j;(\alpha,\beta)),\end{aligned}
=\hat{a}_{j}^{\dagger}(\hat{a}_{l}^{\dagger}\hat{a}_{i}+\delta_{il})\hat{a}_{k}+\delta_{kl}\hat{a}_{j}^{\dagger}\hat{a}_{i}+\delta_{ij}\hat{a}_{l}^{\dagger}\hat{a}_{k}+\delta_{ij}\delta_{kl}
{\Omega}_{i j k}^{u}= \frac{{\partial}{\omega}_{j k}^{u}}{{\partial}{\xi}^{i}}+ \frac{{\partial}{\omega}_{k i}^{u}}{{\partial}{\xi}^{j}}+ \frac{{\partial}{\omega}_{i j}^{u}}{{\partial}{\xi}^{k}}.
\vert f(z)-P_{m}(z)\vert =\left\vert \sum_{h=1}^{\infty}c_{m+h}z^{m+h}\right\vert\leq \sum_{h=1}^{\infty}\vert c_{m+h}\vert T_{m}^{m+h}< \frac{1}{2}\tau_{m}^{-m-1}
\Theta_{n+1}= \Theta_n + K_n \left\{(X_{t_{n+1}}^\dagger - X_{t_n}^\dagger) - \frac{1}{2}\left( f(X_{t_n}^\dagger,\Theta_n ) + \pi_n[f(X^\dagger_{t_n},\theta)] \right)\Delta t \right\}
\sum_{\mathfrak I \in \Sigma}\tilde \chi^k(\mathfrak I)N(\mathfrak I)^{-s}=\frac 1{1-(-2)^k 4^{-s}}\sum_{\mathfrak I \in S_2}\tilde \chi^k(\mathfrak I)N(\mathfrak I)^{-s}= L\big(\chi^k, s - k/2\big).
A_{2,3+p}(r)^2 = \bigg( \sum_{j\ge 1}\sum_{k\ge 1}j^{p/2+3/2}k^{3/2}\sqrt{c_k r^{jk}}\cdot j^{p/2+1/2}\sqrt{kc_kr^{jk}}\bigg)^2 \le A_{3,4+p}(r)A_{1,2+p}(r) \quad A_{1,2+p}(r)^3 = \bigg(\sum_{k\ge 1}j^{p/3+1}k (c_kr^{jk})^{1/3}\cdot j^{2p/3}(c_kr^{jk})^{2/3}\bigg)^3 \le A_{3,4+p}(r) A_{0,1+p}(r)^2.
\frac{1}{\rho}\frac{d(P-P_{e})}{dX}+\frac{1}{2}[\frac{f}{D}+\frac{f_{e}}{D_{e}}(\frac{F}{F_{e}})^{2}]W^{2}+[(2-\beta)-(2-\beta_{e})(\frac{F}{F_{e}})^{2}]W\frac{dW}{dX}=0
v(\psi_{n}(P)) =\widehat{\lambda}(nP) +\frac{n^2-1}{12}v(\Delta)- n^2 \widehat{\lambda}(P) &=\frac{v(\Delta)}{12}-\frac{c_v(nP)}2+\frac{(nP.O)_v}2 +\frac{n^2-1}{12}v(\Delta)- n^2\frac{v(\Delta)}{12}+n^2\frac{c_v(P)}2-\frac{n^2(P.O)_v}2 &=\frac{(nP.O)_v}2-\frac{n^2(P.O)_v}2+\frac{1}{2}n^2 c_v(P)-\frac{1}{2}c_v(nP).
F_{H_{k}}\wedge F_{H_{k}}\wedge\omega^{k-1}&=-\frac{a_{k}^{2}(k+1)!(1+\tau)^{k}-b_{k}^{2}(k+1)!(1+\tau)^{k-2}}{(2\pi)^{2}}\phi(\tau)\omega_{\Sigma_{1}}\wedge\dots\wedge\omega_{\Sigma_{k}}\wedge\frac{\sqrt{-1}dw\wedge d\bar{w}}{\lvert w\rvert^{2}}
\big|y(x)-\hat{y}(x)\big| \leq|{\epsilon}|(x-a)^{\gamma-1}\sum_{j=0}^{\infty}\frac{A^j(x-a)^{\alpha{j}}}{\Gamma(\alpha{j}+\gamma)}&=|{\epsilon}|(x-a)^{\gamma-1}E_{\alpha,\gamma}(A(x-a)^{\alpha}),
M_X=\begin{pmatrix}0 \infty 3 \infty 2 0 \infty 5 \infty 3 0 \infty \infty \infty 6 0\end{pmatrix}M_X^2=\begin{pmatrix}0 6 3 \infty 2 0 5 5 5 3 0 8 \infty 9 6 0\end{pmatrix}M_X^3=M_X^4=\begin{pmatrix}0 6 3 11 2 0 5 5 5 3 0 8 11 9 6 0\end{pmatrix}
\tilde{d e g}T \equiv d e g [ ( J_{3}^{1})^{n_{1 3}}( J_{2}^{1})^{n_{1 2}}( J_{3}^{2})^{n_{2 3}}( J_{2}^{3})^{n_{3 2}}( J_{d})^{n_{d}}({\tilde{J}}_{d})^{n_{\tilde{d}}}( J_{1}^{3})^{n_{3 1}}( J_{1}^{2})^{n_{2 1}}] \ =
\prod_{r=s+1}^t \Big(1-\frac{b}{r}\Big) =\exp\Big[b\ln\Big(\frac{s}{t}\Big)+O\Big(\frac{t-s}{st}\Big)\Big] &=\Big(\frac{s}{t}\Big)^b\Big(1+O\Big(\frac{t-s}{st}\Big)\Big).
v^{i}=g^{ik}v_{k}\Rightarrow v^{1}=g^{11}v_{1}=\frac{v_{1}}{h_{1}^{2}};v^{2}=g^{22}v_{2}=\frac{v_{2}}{h_{2}^{2}};v^{3}=g^{33}v_{3}=\frac{v_{3}}{h_{3}^{2}}
L^{\otimes 5}\cong\omega_{C}\left(\sum_{i=1}^{n+1}5[q_i]\right)\Longrightarrow \left( L\otimes\O\left(-[q_{n+1}]\right)\right)^{\otimes 5}\cong \omega_{C}\left(\sum_{i=1}^{n}5[q_i]\right).
\hphantom{=}\left\| \lambda(a) - \sum \theta_{\xi_i, \eta_i}\right\| \leq \|\lambda(a) - \lambda(ae) \| + \left\| \lambda(ae) - \sum \theta_{\xi_i, e \eta_i}\right\| + \sum \| \theta_{\xi_i, e \eta_i}-\theta_{\xi_i, \eta_i}\|.
\widetilde{H}_1 = z + \frac{k}2(c_3\,x-b_3\, y)^2 - \frac{k h^2}{8}\big( A_2\,x - A_1\,y \big)^2, \widetilde{H}_2 = H_2 + \frac{h^2}{24}\bigg(A_3(-c_1 x^2-2c_2xy-2c_3xz+2b_3yz+b_2y^2)+(A_1c_3-A_2b_3)z^2 +k(-2c_3A_2^2 x^3+2A_2(b_3A_2+2c_3A_1)x^2 y -2A_1(c_3A_1+2b_3A_2)xy^2+2b_3A_1^2 y^3)\bigg),
w^{[h]}=u^{[h]}-b_1u_{xx}^{[h]}+bu_{xxxx}^{[h]}\in L^{\infty}(0,T';L^2(\mathbb{T})) w_{t}^{[h]}+\frac{a}{b}w_{x}^{[h]}=[(\frac{a}{b}-q)u_x]^{[h]}-[(\frac{ab_1}{b}+p)u_{xxx}]^{[h]}-[ru_{xx}]^{[h]}\in L^{\infty}(0,T';L^2(\mathbb{T})).
\mathcal{E}Q^{(\pm)}_x(\mathcal{E}) =-B^{(\pm)}(x)Q^{(\pm)}_{x+1}(\mathcal{E}) +\bigl(B^{(\pm)}(x)+D^{(\pm)}(x)\bigr)Q^{(\pm)}_x(\mathcal{E}) -D^{(\pm)}(x)Q^{(\pm)}_{x-1}(\mathcal{E}),
{\cal B}^r(t)\sim \frac{1}{(\Omega!)^2}\left[\Phi^*(t)\right]^r\sum_{s=0}^r{r\choose s}(\Omega-s)! \left[\Omega-(r-s)\right]! x^{2(r-s)}\Phi^s(t) \Phi^{r-s}(t-1) \;.
\sum_{k=1}^K \omega_k v_k (\zeta\gamma)_{kj}= \sum_{k=1}^K \sum_{i=1}^{K-1}\omega_k v_k \left(\frac{1}{1-v_k\, \lambda_i}- \frac{1}{1+v_k\, \lambda_i}\right) \gamma_{ij}= 0.
M(\lambda,L,m,d)=\lambda^{2}\int d^{d-1}r \int d^{d-1}r^{\prime}\int_{0}^{L}dz \int_{0}^{L}dz^{\prime}F(\vec{r},\vec{r}',z,z')(G_{0}^{(2)}(\vec{r}-\vec{{r}^{\prime}},z,z^{\prime}))^2,
Z(q\gamma_j^{-1}) = \frac{\prod_{m = 1}^{2g}(1 - q\gamma_j^{-1}\gamma_m)}{(1 - q\gamma_j^{-1})(1 - q^2\gamma_j^{-1})}= q^{2g - 3}\gamma_j^{-2g + 2}\prod_{m = 1}^{2g}\gamma_m \cdot \left(\frac{\prod_{m = 1}^{2g}(1 - q^{-1}\gamma_j\gamma_m^{-1})}{(1 - q^{-1}\gamma_j)(1 - q^{-2}\gamma_j)}\right).
P_a=2\sqrt{\beta}\left(\Pi_a+\frac{1}{4}C^{cc'}_{\;\cdot a}M_{cc'}\right)\;\;\;\;\leftrightarrow\;\;\;\;P_0=2\sqrt{\beta}\Pi_0,\;P_j=2\sqrt{\beta}\left(\Pi_j-\frac{1}{2}J_j\right),
\begin{array}{l}h\Delta_{g_B}h + (r-1)|grad_{g_B}h|^2 + \rho h^2 = h\Big\{\varphi^2\varepsilon_{i_0}\left[h''-(n-2)\frac{\varphi'h'}{\varphi}\right] + m\varepsilon_{i_0}\varphi^2\frac{h'f'}{f}\Big\}+ (r-1)\varepsilon_{i_0}\varphi^2(h')^2 + \rho h^2.\end{array}
\hat{\beta}=(\frac{1}{n}X^{\prime}X)^{-1}\frac{1}{n}X^{\prime}y=\beta+(\frac{1}{n}X^{\prime}X)^{-1}\frac{1}{n}X^{\prime}\epsilon=\beta+(\frac{1}{n}\sum_{i=1}^{n}x_{i}x_{i}^{\prime})^{-1}(\frac{1}{n}\sum_{i=1}^{n}x_{i}\epsilon_{i})
Z(A_{n})=\sum_{j_{1}+2j_{2}+3j_{3}+\cdot\cdot\cdot+nj_{n}=n}\frac{1+(-1)^{j_{2}+j_{4}+\cdot\cdot\cdot}}{\prod_{k=1}^{n}k^{j_{k}}j_{k}!}\prod_{k=1}^{n}a_{k}^{j_{k}}.
d'_{r,t}\triangleq\max\limits_{1\le t'\le t}\left\{\frac{\binom{N_R-1}{r}\binom{N_T}{t'}\binom{N_R-r-1}{t'-1}t'}{\binom{N_R-1}{r}\binom{N_T}{t'}\binom{N_R-r-1}{t'-1}t'+\binom{N_R-1}{r+1}\binom{N_R-r-2}{t'-1}\binom{N_T}{t'-1}}\right\}
\Lambda^{*}_f(s):=\left(\frac{\sqrt{|D|}}{i}\right)^k\int^{+\infty}_{0}f(it)t^{s-1}dt=G(s)\left(\frac{\sqrt{|D|}}{i}\right)^k\sum^{\infty}_{n=1}\frac{a_f(n)}{n^s},
\|\left(\Phi(X^*AY)\Phi(Y^*AY)^{-1}\Phi(Y^*AX)\right)^{\frac{p}{2}}\Phi(X^*AX)^{-\frac{p}{2}}\| &\le \frac{(M-m)^{p}(M+m)^{\frac{p}{2}}}{2^{2+\frac{p}{2}}M^{\frac{3p}{4}}m^{\frac{3p}{4}}},~~~p\ge 2
A_r = \begin{pmatrix}1.8965 0.0237 0.7778 0.0237 3.1554 1.8009 0.7778 1.8009 3.1784\end{pmatrix}, B_r = \begin{pmatrix}-0.2677 1.1820 1.5124 0.2049 -0.7759 1.2155\end{pmatrix}, C_r = \begin{pmatrix}0.8726 0.1503 -0.0630 0.3321 0.0680 1.3121\end{pmatrix}.
\sum_{i}e^{i}(\nabla_{e_{i}}(\nabla_{Z}Z + \nabla_{Y}Y))= \frac{{\rm div}(\rho_{\phi}[\iota] (\pi_{L}\phi ([Z, Y]) + 2 f Y))}{\rho_{\phi}[\iota]}&+g(- \pi_{L}[Z, Y] + 2 \eta (Y) Y, H_{\phi}) &+ \eta (\pi_{L}[Z, Y]) -2 \eta (Y)^{2}.
U_{p - a}( u ) = - D_{a}( u ) \; , \hspace{1 c m}X_{p - a - 1 , p - a + 1}^{p - a}X_{p - a + 1 , p - a - 1}^{p - a}= X_{a - 1 , a + 1}^{a}X_{a + 1 , a - 1}^{a}\; .
AB^m=\langle a^m,b^{m+1}\rangle CD^m=\langle c^m,d^m\rangle F_\pm=\langle F_+,F_-\rangle G_\pm=\langle G_+,G_-\rangle FG_+=\langle F_+,G_+\rangle FG_-=\langle F_-,G_-\rangle.
z(x,t) =\left(\frac{2\beta^2\cosh\theta+2\mathrm{i}\beta\sqrt{2-\beta^2}\sinh\theta}{2\cosh\theta-\sqrt{4-2\beta^2}\cos(\sqrt\omega\beta x)}-1\right)\sqrt\omega\mathrm{e}^{\mathrm{i}\omega t},\theta=\omega\beta\sqrt{2-\beta^2}t,\quad\beta<\sqrt{2}, u(x,t) =\operatorname{Re}(z),v(x,t)=\operatorname{Im}(z).
C_d P_{\lambda_0}( C_{\ll d}u_{\lambda_1}C_{\ll d}v_{\lambda_2}) ={}C_d P_{\lambda_0}( C^+_{\ll d}u_{\lambda_1}C^+_{\ll d}v_{\lambda_2}) + C_d P_{\lambda_0}( C^-_{\ll d}u_{\lambda_1}C^-_{\ll d}v_{\lambda_2}).
\sum_{v=0}^{\ell^m-1}f(z)\mid_{-1}\alpha_0\sigma_{w_v,\ell^m}=\ell^{\frac{3}{2}}\sum_{\substack{n=n_0 n\equiv 0\pmod{\ell^m}}}^{\infty}a_0(n)q_{24\ell^m}^{n}.
K_{n}^{(3)}K_{m}^{(3)}+K_{n+1}^{(3)}K_{m+1}^{(3)}+K_{n+2}^{(3)}K_{m+2}^{(3)}=\left\lbrace \begin{array}{c}21\cdot 2^{n+m}+2^{n}\left(M_{m+1}^{(2)}+3M_{m+2}^{(2)}\right) +2^{m}\left(M_{n+1}^{(2)}+3M_{n+2}^{(2)}\right) +3(\omega_{1}^{n}\omega_{2}^{m}+\omega_{1}^{m}\omega_{2}^{n})\end{array}\right\rbrace ,
\int{\cal D}\varphi \quad e^{+{\frac{1}{2}}( d \varphi , d \varphi )}\quad \prod_{k = 1}^{i}e^{2 i \sqrt{\frac{\pi}{\xi}}\varphi ( y_{k})}\prod_{k = 1}^{i}e^{- 2 i \sqrt{\frac{\pi}{\xi}}\varphi ( x_{k})}\propto ( \Lambda_{U V})^{2 i}K_{i}^{- 1}( x , y ) .
\begin{aligned}d_1+d_{11122}>2d_{112},\ \d_{1112}+ d_{11122}> 3d_{112},\ \d_{1112}+ d_{12}> 2d_{112}, d_{1112}+ d_{2}> d_{112}+ d_{12},\ \d_{112}+ d_{2}> 2d_{12},\ \d_{11122}+ d_{2}> 3 d_{12}.\end{aligned}
S_{B H}={\frac{A_{H}}{4 G_{D}}}={\frac{m^{\frac{D - p - 2}{D - p - 3}}V_{S^{D - p - 2}}\cosh^{\frac{4 ( p + 1 ) ( D - p - 2 )}{( D - 2 ) \Delta_{p}}}\alpha_{p}}{4 G_{D}}}.
\frac{d}{dt}R_{N+1}u_{N+1}(t) =\frac{d}{dt}\left(e^{-(N+1)t}p_{N+1,N+1}(t)\right) &=-(N+1)e^{-(N+1)t}p_{N+1,N+1}(t)-e^{-(N+1)t}\sum_{m+j=N+1}R_{N+1}B(q_m(t),q_j(t) &=-AR_{N+1}u_{N+1}(t)-\sum_{m+j=N+1}R_{N+1}B(u_m(t),u_j(t)).
I_{\epsilon}^{( 1 )}( a^{2}) ={\frac{\pi}{2}}{\frac{1}{c o s{\frac{\pi}{2}}\epsilon}}\left\{{\frac{1}{a^{1 + \epsilon}}}+ 2 \sum_{n = 1}^{\infty}{\frac{1}{( 2 \pi n )^{1 + \epsilon}}}+ 2 \sum_{n = 1}^{\infty}{\frac{1}{( 2 \pi n )^{1 + \epsilon}}}\; [{\frac{1}{( 1 +{\frac{a^{2}}{4 \pi^{2}n^{2}}})^{\frac{1 + \epsilon}{2}}}}- 1 ] \right\}\; .
\int_{\mathbb{R}^m}|k|^{-1}e^{-|k||v-w_1|}\prod_{j=1}^{m+1}(h_1(w_j))^{q_j-1}\prod_{j=1}^m(|k|h_1(w_j)+h_2(w_j)\mathrm{sgn}(w_j-w_{j+1}))e^{-|k||w_j-w_{j+1}|}\,\mathrm{d}w_1\cdots\mathrm{d}w_m,
\frac{\gamma_{k+1}(\Gamma_{n,c})}{\gamma_{k+2}(\Gamma_{n,c})}=\left< \overline{a}^{(p_1^{y_1}-1)m^{k-1}}\gamma_{k+2}(\Gamma_{n,c}),...,\overline{a}^{(p_r^{y_r}-1)m^{k-1}}\gamma_{k+2}(\Gamma_{n,c}) \right>=\left< \overline{a}^\beta \gamma_{k+2}(\Gamma_{n,c}) \right>,
\left. \tilde{S}_{\mathrm{\scriptsize r e d}}( \hat{\xi}( p ) , \hat{\eta}( p ) ) = S_{\mathrm{\scriptsize e f f}}( \hat{\xi}, \hat{\eta}) \right|_{\hat{\xi}( p ) = \hat{\eta}( p ) = 0 \atop{| p | > \Lambda}}.
\sum^{N}_{k=1}\sum^4_{\mu=1}\beta^\mu_{kx}(u)\,p_{kx\mu}\big|_{{\bf D}_0}+\,\sum^{N}_{k=1}\sum^4_{\mu=1}\, \beta^\mu_{ky}(u)\,p_{ky\mu}\big|_{{\bf D}_0}=0.
(yW_{h})(yW_{h})^{*}yW_{gh}=yE_{h}y^{2}W_{gh}= y^{3}E_{h}W_{gh}\textrm{~~(since $E_{h}$ commutes with $y$)}= x W_{h}W_{g}\textrm{~~(since $W_{h}W_{g}=E_{h}W_{hg}$)}.
\sum_{\mu,\nu=1}^n \frac{\partial^2 \log |g|^2}{\partial z_\mu \partial \bar{z}_\nu}\xi_\mu \bar{\xi}_\nu \ge |g|^{-4}\sum_{j<k}\left|\sum_{\mu=1}^n \left(g_j\frac{\partial g_k}{\partial z_\mu}-g_k\frac{\partial g_j}{\partial z_\mu}\right) \xi_\mu \right|^2
J[f]=\int_{B}F(x,f,\partial^{\alpha_{1}}f,\partial^{\alpha_{2}}f,...,\partial^{\alpha_{n}}f)dx=\int_{B}[\sum_{i=1}^{N}(f(x)-f_{i})^{2}\delta(x-c_{i})+\lambda|\nabla^{m}f|^{2}]dx.
\int \left( \sum_{1 \le n \le N}\frac{w_n \mathbf{1}_{B^{cl}(x_{n}, r_{n})}}{\mu B^{cl}(x_{n}, r_{n})}\right) h \d\mu =\sum_{1 \le n \le N}\frac{w_n}{\mu B^{cl}(x_{n}, r_{n})}\int h \mathbf{1}_{B^{cl}(x_{n}, r_{n})}\d\mu
\langle \eta_{S}^{mic,Q_{H}}(h\sigma), (P(\xi),\sigma_{P(\xi)})\rangle =(-1)^{\dim S}\, \zeta \ \mathrm{tr}\left( \tau_{S}^{mic}(P(\xi'))(h)\right) & = (-1)^{\dim S}\, \mathrm{tr}\left( \tau_{S}^{mic}(P(\xi'))(h\sigma_{P(\xi)})\right)
\Delta_{ij}\to\tilde \Delta_{ij}:=m_i\tilde n^{(0)}_j-m_j\tilde n^{(0)}_i=m_i(n^{(0)}_j+km_j)-m_j(n^{(0)}_i+km_i)=m_in^{(0)}_j-m_jn^{(0)}_i=\Delta_{ij}\;,
i\alpha' \int_0^{2\pi}d\theta ~{\rm Tr}~{\cal P}\left( U[A]~\delta_1 A_a \partial_{\theta}\overline{X}^a \right) + i\alpha'^2\int_0^{2\pi}d\theta ~{\rm Tr}~{\cal P}\left( U[A] ~\delta_2 A_a\partial_{\theta}\overline{X}^a \right)
= \sum_{k=0}^{\infty}\frac{1}{(k+1)^{s-3}}\sum_{r_1=0}^{n}\sum_{r_2=0}^{n}\sum_{r_3=0}^{n}\frac{a_{r_1}b_{r_2}c_{r_3}}{(r_1+k+1)(r_2+k+1)(r_3+k+1)}.
\begin{array}{c c c}{{\epsilon \tilde{w}{}^{t}\epsilon^{- 1}= - \tilde{w}\, , ~ ~}}&{{~ \bar{\epsilon}^{- 1}w^{t}\bar{\epsilon}= - w \, , ~}}&{{\tilde{w}{}^{\dagger}= - w \, .}}\\ \end{array}
I_N = 1 + \sum_{d=1}^\infty z^d \sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}s_\lambda(a_1,\dots,a_N) s_\lambda(b_1,\dots,b_N) \Omega_{\frac{q}{N}}^{-1}(\lambda) \Omega_{\frac{1}{N}}^{-1}(\lambda),
\# \left\{n < N : \varepsilon_i(A_1n) = a,\, \varepsilon_j(A_2n) = b\right\}= d_{0,\Delta}(a)d_{0,\Delta}(b) N + O\left( N (\log N)^{-C}\right) &= \frac{N}{q^2}+ O\left( N (\log N)^{-C}\right)
\lesssim \| \mathfrak{m}(\cdot, \zeta)\|_{\mathcal{S}^\infty}2^{m/4+ k/2 + n+\max\{-j, p, n+\epsilon M_t\}/2-p/2 + \max\{l,p\}+ 8\epsilon M_t}\big(2^{\min\{n,p\}+ j}\mathbf{1}_{l=-j, l> n+\epsilon M_t}
0 =\sum_{j=0}^{\ell-1}a_j(x)\widetilde{b_j(x)}=\sum_{j=0}^{\ell-1}\left(\sum_{i=0}^{n-1}a_{ij}x^i\right)\left(\sum_{k=0}^{n-1}b_{kj}^qx^{-k}\right)
\vert C_{a}\rangle = \left[{\frac{2^{d ( J_{a})}V ( J_{a})}{V ( R J_{a})}}\right]^{\frac{1}{2}}\sum_{{\widetilde p}\in 2 \Lambda ( J_{a})}\sum_{{\widetilde w}\in{\widetilde \Lambda}( R J_{a})}\sum_{\bf n}\zeta_{\bf n}V_{L}({\bf n}) V_{R}({\bf n}) \vert{\widetilde w},{\widetilde p}\rangle ~ .
| \mathrm{\boldmath{~ \e t a ~}}; M ; D_{r}\rangle ={\frac{1}{\sqrt{M !}}}\left( \sum_{j = 1}^{r}\eta_{j}a_{j}^{\dagger}+ \sum_{j = 1}^{r}\eta_{- j}b_{j}^{\dagger}\right)^{M}| 0 \rangle ,
e^{{1\over 2}\epsilon\varphi_{\alpha}-(p+1)A}=H_{\alpha},\ \ds^2=\prod^N_{\alpha=1}H^{-{{\tilde{p}+1}\over{D-2}}}_{\alpha}dx^{\mu}dx^{\nu}\eta_{\mu\nu}+\prod^N_{\alpha=1}H^{{p+1}\over{D-2}}_{\alpha}dy^mdy^m,
P^u_{k^1,\ldots,k^n}f (t^1,\ldots,t^n)=\mathcal{R}^{t^2_{|k^2|},\ldots,t^{n-1}_{|k^{n-1}|};u}_{k^n}(P^{p^{n-1}(\omega^n)}_{k^1,\ldots,k^{n-1}}(f(\cdot,\omega^n))),
\ell^{-1}\sum_{k=1}^{\ell}\xi^{a_{j+k-1}-a_{i+k}}={\xi^{a_{j-1}-a_i}}{\ell}^{-1}\sum_{k=1}^{\ell}\xi^{k(j-1-i)}=\delta_{i,j-1}\xi^{a_{j-1}-a_i}=\delta_{i,j-1}.
=\frac{\left\langle \Psi \left| H^{Schw.}-H^{Flat}\right| \Psi \right\rangle}{\left\langle \Psi |\Psi \right\rangle}+\frac{\left\langle \Psi \left|H_{quasilocal}\right| \Psi \right\rangle}{\left\langle \Psi |\Psi\right\rangle}.
\langle \, : (\Delta \, P_+)^2 : \, \rangle_{\beta , \gamma}\simeq\langle \, : (\Delta \, P_-)^2 : \, \rangle_{\beta , \gamma}\simeq 4 \, \langle N \rangle_{\beta , \gamma}\simeq 4 \, \alpha' \, M^2 \, .
|1-\phi_{\varepsilon}(\lambda)|=|\mathbb{E}(1-\cos(\lambda \varepsilon))-\iota \mathbb{E}\sin(\lambda \varepsilon) | \leq c_1\, ( |\lambda|^{\alpha'}\mathbb{E}|\varepsilon|^{\alpha'})\wedge 1\leq c_2 (|\lambda|^{\alpha'}\wedge 1),