harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E^*_{i+k-1}A_1E^*_{i+k-2}A_1E^*_{i+k-3}\cdots E^*_{i+1}A_1E^*_{i}=\big((\frac{k-2}{2})!\big)^2\frac{k}{2}M^{\frac{k-2}{2},\frac{k-2}{2}}_{\frac{i+k-2}{2},\frac{2m-i}{2}}.
\eta_{+}^{A}= e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{\hat{\eta}_{R}}\\{- \hat{\eta}_{L}}\\ \end{array}\right)^{A}\ , \qquad \eta_{-}^{A}= \epsilon ( y ) e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{\hat{\eta}_{L}}\\{\hat{\eta}_{R}}\\ \end{array}\right)^{A}\ ,
\vert\widetilde{T}_{F,G,N}^L(\widetilde{f_1},\dots,\widetilde{f_d}) \vert \ll_G \frac{1}{N^{d-m}}\int\limits_{\substack{\mathbf{x}\in \mathbb{R}^d L\mathbf\mathbf{x}= \mathbf{0}}}\Big(\prod\limits_{j=1}^d \widetilde{f_j}(x_j)\Big) F(\mathbf{x}) \, d \mu(\mathbf{x}),
\P\left(\rho(X_N)\geqslant \left(\frac{1-\delta}{1+\delta}\right)^{1/s}\frac1{n}\right) \geqslant f\left(N, \frac{\mathcal{H}_s(K)n^s}{(1-\delta)\upsilon_s}, \tau_n n^s\right) = f\left(N, \frac{\mathcal{H}_s(K)n^s}{(1-\delta)\upsilon_s}, \kappa_n \cdot\frac{\mathcal{H}_s(K)n^s}{(1-\delta)\upsilon_s}\right),
W(B_{1}(n,d,x)) =\frac{n^{3}}{3}+(-\frac{3d}{2}-\frac{5}{4})n^{2}+(4d^{2}+10d+\frac{13}{3}-2x)n+ & +2x^{2}-\frac{8}{3}d^{3}-12d^{2}-\frac{46d}{3}-7.
\delta ( \kappa_{f}- \kappa_{i}) \delta ( \kappa_{f}- \kappa_{i}^{\prime}) \simeq \delta ( \kappa_{p_{f}}- \kappa_{p_{\mathrm{T}}}- \kappa_{p_{\mathrm{B}}}) \delta ( \kappa_{\rho_{\mathrm{T}}}+ \kappa_{\rho_{\mathrm{B}}}- \kappa_{\rho_{\mathrm{T}}^{\prime}}- \kappa_{\rho_{\mathrm{B}}^{\prime}}) .
\log 4 \mathbf{e}_1+\log 4\mathbf{e}_2+\log 4\mathbf{e}_3+\log \frac{3}{2}\mathbf{e}_{12}+\log \frac{3}{2}\mathbf{e}_{123'}&=[\log 9,\log 9,\log 6,\log 54,\log 54,\log 54,\log 216]^\intercal
\begin{array}{rcl}{\cal N}_{\rm bulk}\approx \displaystyle\sum_{i=1}^N\frac{(q^{2i}l_p)^2}{(q^{2i}l_p-q^{2(i-1)}l_p)^2}~,~~~~~~~~q^{2N}=R~, &= \displaystyle\frac{q^4\ln\left(\frac{R}{l_p}\right)}{2(q^2-1)^2\ln q}~.\end{array}
R_2^{(n)}: =\sup_{0\le k,l\le\lfloor n^\mu\rfloor}\sup_{\{t:|t-t_k^{(\mu)}|\le T/n^\mu\}}\sup_{\{x:|x-x_l^{(\mu)}|\le \pi/n^\mu\}}|u^n(t_k^{(\mu)},x_l^{(\mu)})-u^n(t,x)|^{2p}, R_3^{(n)}: =\sup_{0\le k,l\le\lfloor n^\mu\rfloor}\sup_{\{t:|t-t_k^{(\mu)}|\le T/n^\mu\}}\sup_{\{x:|x-x_l^{(\mu)}|\le \pi/n^\mu\}}|u(t_k^{(\mu)},x_l^{(\mu)})-u(t,x)|^{2p}.
M = 4 \pi \int_{0}^{\infty}d r ~ r^{2}\{\frac{F_{\pi}^{2}}{8}[ F^{2}+ 2 \frac{s i n^{2}F}{r^{2}}] + \frac{1}{2 e^{2}}~ \frac{s i n^{2}F}{r^{2}}~ [ \frac{s i n^{2}F}{r^{2}}+ 2 F^{2}] \}
\begin{array}{c}{L = \left[ 1 + \frac 1 2 h \right] i \bar{\psi}\gamma^{a}\left[ \eta_{a}^{\mu}+ h_{a}^{\mu}\right] \left[ \partial_{\mu}+ \Omega_{\mu}\right] \psi + + \left[ 1 + \frac 1 2 h \right] m \bar{\psi}\psi}\\ \end{array}
u_j(x)=a_j+\sum_{n\in \mathbb{Z}_{\star}^3}\big(b_{n,j}\cos(n\cdot x)+ c_{n,j}\sin(n\cdot x) \big),\ \ \j=0,1,\ \ \ \mathbb{Z}_{\star}^3 = \mathbb{Z}^3\backslash \{0\}
\sum_{z\in L}\lambda_1 \mathbf{v}_z = \sum_{z\in L}\sum_{y\sim z}\mathbf{v}_y = \sum_{z\in L}\left( \sum_{\substack{y\sim z y\in S}}\mathbf{v}_y + \sum_{\substack{y\sim z y\in L}}\mathbf{v}_y \right) \leq \epsilon e(S,L) + 2e(L) \leq \epsilon (2n-4) + \frac{18\sqrt{2n-4}}{\epsilon}.
-\log\,\det\left(\Phi_n^*\mathcal{T}_{n+1}^*\mathcal{T}_{n+1}\Phi_n\right)=\log\,\det\left(\mathcal{T}_{n+1}^{-1}(\mathcal{T}_{n+1}^*)^{-1}\right)-\log\,\det\left((\Phi_n^{\perp})^*\mathcal{T}_{n+1}^{-1}(\mathcal{T}_{n+1}^*)^{-1}\Phi_n^{\perp}\right)
a_{h,2}(\ell,k)=\frac 1{(2\pi i)^4}\int_{\gamma_{r_2}}dz\int_{\Gamma_{D_2}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1,\mu_1,\xi_1}(z)G_{\Delta n,\Delta\mu,\Delta\xi}(w)\left(\frac 1{z-w}-h\right)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(z-\zeta)(w-\omega)}
K_{s,e}=2\int_0^\eta d\rho\;\rho^{N-2}\int_0^{f(\rho e)}\frac{dh}{\left(\rho^2+h^2\right)^{s+N/2}}+2\int_\eta^{+\infty}d\rho\;\rho^{N-2}\int_0^{f(\rho e)}\frac{dh}{\left(\rho^2+h^2\right)^{s+N/2}}.
x_3 p_3 + x_4 p_4 = 8f + 4a ~~,~~~~x_3 q_3 + x_4 q_4 = 12 g~~,~~~~x_3 r_3 + x_4 r_4 = 6 k~~,~~~~x_3 s_3 + x_4 s_4 = 4 m
M_{\mathrm{i n t}}^{2}( x_{1}, x_{2}, s_{1}, s_{2}) \equiv \frac{s_{1}^{2}+ r_{n}}{x_{1}}+ \frac{s_{2}^{2}+ r_{n}}{x_{2}}+ \frac{( s_{1}+ s_{2})^{2}+ r_{n}}{1 - x_{1}- x_{2}}. \nonumber
g =\frac{rP}{2R_{+}R_{-}}dr\otimes dr+\frac{2R_{+}R_{-}P}{r}\theta^{1}\otimes\theta^{1}&+\frac{R_{+}Q}{R_{-}}\left(\theta^{2}\otimes\theta^{2}+\theta^{3}\otimes\theta^{3}+\theta^{4}\otimes\theta^{4}\right) &+\frac{R_{-}Q}{R_{+}}\left(\theta^{5}\otimes\theta^{5}+\theta^{6}\otimes\theta^{6}+\theta^{7}\otimes\theta^{7}\right).
\left[\begin{array}{c|cccc}B_k(\lambda) -I_m B_{k-1}(\lambda) \lambda^\ell I_m -I_m B_{k-2}(\lambda) \lambda^\ell I_m \ddots \vdots \ddots -I_m B_1(\lambda) \lambda^\ell I_m\end{array}\right]= \left[ \begin{array}{c|c}M_2^\ell(\lambda) L_{k-1}(\lambda^\ell)^T\otimes I_m\end{array}\right],
\det(B_{ij})^{n+1}_{i,j=1}= 2\det(x^{n-j+1}_i+x^{-(n-j+1)}_i+x^{n-j}_i+x^{-(n-j)}_i)^{n}_{i,j=1}= 2(x^{\frac{1}{2}}_i+x^{-\frac{1}{2}}_i)\det(x^{n-j+\frac{1}{2}}_i+x^{-(n-j+\frac{1}{2})}_i)^{n}_{i,j=1}.
\displaystyle \rho_{\infty}= \int_{\Omega\times{\mathbb R^d}}\, f(v)\, dxdv, u_{\infty}= \frac{1}{\rho_{\infty}}\, \int_{\Omega\times{\mathbb R^d}}\, vf(v)\, dxdv, begin{align*}2pt] \displaystyle T_{\infty}= \frac{1}{N\rho_{\infty}}\int_{\Omega\times{\mathbb R^d}}\, |u_{\infty}-v|^2\, f(v)\, dxdv,
\left( \sum_{j=1}^{k}\vert x^*(T(x_j)) \vert^{p}\right)^{\frac{1}{p}}\leq C \cdot \left(\int_{B_{F^{^{**}}}}\vert \varphi(x^*)\vert^{p}d\mu(\varphi) \right)^{\frac{1}{p}},
\begin{aligned}\|W_{< m,g,\xi}\|_{\mathfrak{S}_{\infty,1}}\leq \|W_{< m,g,\xi}\|_{\mathfrak{S}_{2,1}}=\|W_{< m,g,\xi}\|_{L^1(Q_\ell^*;L^2(Q_\ell\times Q_\ell))}&\leq \|W_{<m,\ell}(\xi,x-y)\|_{L^1(Q_\ell^*;L^\infty(Q_\ell\times Q_\ell))}\|g_{\xi'}(x,y)\|_{L^1(Q_\ell^*;L^2(Q_\ell\times Q_\ell))}.\end{aligned}
[ M_{\hat{a}\hat{b}}, M_{\hat{c}\hat{d}}] = \eta_{\hat{b}\hat{c}}M_{\hat{a}\hat{d}}- \eta_{\hat{b}\hat{d}}M_{\hat{a}\hat{c}}+ \eta_{\hat{a}\hat{d}}M_{\hat{b}\hat{c}}- \eta_{\hat{a}\hat{c}}M_{\hat{b}\hat{d}}.
T_{n+i}=\frac{1}{\sqrt{mt_i+1}}\exp\biggl(m(n+i)t_i-(n+i)+\frac{n+i}{mt_i}+2ct_i-{1+c\over m}\biggr)\cdot\biggl(1+O\Bigl(\frac{\log^{7/2}{n}}{\sqrt{n}}\Bigr)\biggr).
\frac{1}{(1-2X)^2(1-X)^2}\sum_{n=0}^{\infty}\lambda_{n+1}\left( \frac{X}{X-1}\right)^n= \sum_{n=0}^{\infty}\left[ \sum_{k=1}^{n+1}k\, 2^{k-1}\sum_\rho \rho^{-(n-k+2)}\right] X^n.
Q/Q(2) \downarrow X = 3 \otimes 1^{[1]}+ 1 \otimes 1^{[1]}+ 0, Q(2)/Q(3) \downarrow X = 3 \otimes 1^{[1]}+ 1 \otimes 1^{[1]}, Q(3)/Q(4) \downarrow X = 4 + 2 + 2 \otimes 2^{[1]}+ 0, Q(4)/Q(5) \downarrow X = 1 \otimes 1^{[1]}, Q(5)/Q(6) \downarrow X = 1 \otimes 1^{[1]}, Q(6)/Q(7) \downarrow X = 2.
+ \binom{k-1}{i+1}\frac{m!}{(m - i - 1)!}e^x D^{k-1-i-1}\left( e^{-x}x^n\right)y^{m-i-1}+ \cdots + \frac{m!}{(m-k+1)!}x^n y^{m-k+1}\biggr)\biggr).
y_{1}= \rho_{0}^{2}\mathrm{e x p}\left( 1 -{\frac{A}{B}}\right) , y_{\chi}= \sqrt{e}y_{1}, \chi = 1 +{\frac{B}{2 \rho_{0}^{4}}}\mathrm{e x p}\left({\frac{2 A}{B}}- 3 \right) .
F_n^{(2m)}\sim -(1-2\alpha) \frac{m(m-1)}{\Gamma(2\alpha)}n^{-m(1-2\alpha)}\sum^{n-1}_{j=1}j^{-2(1-\alpha)+m(1-2\alpha)}&\sim -\frac{m}{\Gamma(2\alpha)}n^{-(1-2\alpha)}
J(x):=\sup_{\tau_1,\tau_2\in\mathcal{T},\atop\tau_1\leq\tau_2}\mathbb{E}^x\left[\int_{\tau_1+\delta}^{\tau_2+\delta}f(X(t))\mathrm{d}t+g_1(X(\tau_1+\delta))+g_2(X(\tau_2+\delta))\right],
\langle xP_n,P_n\rangle=\langle x^{n+1},P_n\rangle+a_{n,n-1}\langle x^n, P_n\rangle=\left|\begin{array}{cc}\Lambda_{n-1}(\theta_n^n)^\top \theta_{n+1}^n \boxed{m_{2n+1}}\end{array}\right|+\left|\begin{array}{cc}\Lambda_{n-1}e_{n-1}^\top \theta_n^n \boxed{0}\end{array}\right|H_n,
\Sigma(s,r_1,0) \le c_5 D(t) \Big[\Sigma(s,r_1-1,0)+ \Sigma(s-1,r_1,0)\Big], \Sigma(0,r_1,r_2) \le c_5 D(t) \Big[\Sigma(0,r_1,r_2-1)+ \Sigma(0,r_1-1,r_2)\Big],
\sigma_1(u_1, u_2, e_1, e_2) := \frac{1}{2}\left\langle \begin{bmatrix}u_1 u_2 e_1 e_2\end{bmatrix}, \begin{bmatrix}0 -I 0 0 -I 0 I 0 0 I 0 0 0 0 0 0\end{bmatrix}\begin{bmatrix}u_1 u_2 e_1 e_2\end{bmatrix}\right\rangle.
4 g_{\mu \upsilon}= 2{\frac{\partial \psi^{\beta}}{\partial \omega_{1}}}{\frac{\partial \psi^{\beta}}{\partial \omega_{1}}}{\frac{M_{\mu}M_{\upsilon}}{( \omega_{1})^{2}}}+{\frac{4 ( S_{1})_{\mu \upsilon}}{( \omega_{1})^{2}}}\sin^{2}\left({\frac{\theta_{1}+ 2 \theta_{2}}{2}}\right)
\sum_{j\ne i}\frac{2}{z_i-z_j}+ \frac{-12b_3z_i^4 - 4(2b_2-3b_3)z_i^3 - 4(b_1-2b_2)z_i^2 + (4a+4b_1+3)z_i - 4a+2l+d-3}{2z_i(z_i-1)}= 0, i=1, 2, \ldots, n.
C_{\lambda}:= \frac{1}{\lambda}D|_T - \frac{1-\lambda}{\lambda}\left( \frac{1}{2}T|_S \right) = \frac{2 \gamma - 1 + \lambda}{2 \lambda}\Gamma + \frac{2 \delta - 1 + \lambda}{2 \lambda}\Delta + \frac{1}{\lambda}\Xi.
J_{x_i,t}^{N,i}(v_i; \{v^*_j\}_{j\neq i}) = J_{x_i,t}^{mfg}(v_i;\bar X_i^*,\bar v^*_i) + o(1) &\geq J_{x_i,t}^{mfg}(v^*_i;\bar X_i^*,\bar v^*_i) + o(1) &= J_{x_i,t}^{N,i}(v^*_i; \{v^*_j\}_{j\neq i}) + o(1),
\mathbf{R}=\left[\begin{array}{cccc|cccc}0 0 0 0 -1 0 0 0 0 1 0 0 -1 0 0 -1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 \hline 1 0 0 0 -1 -1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 1 -1 -1\end{array}\right]
\mathbb{E}[A(j+1)] = \frac{2p\cdot |\textbf{F}_u(j)|}{\sum_{v\in [n]}|\textbf{F}_v(j)|}&= \frac{2c[f(t_{j})\pm g_f(t_{j})]}{n^{1+s}[f(t_{j})\pm g_f(t_{j})]}&\subseteq \frac{2c}{n^{1+s}}\Bigl[1\pm \frac{3g_f(t_{j})}{f(t_{j})}\Bigr] &\subseteq \frac{1}{n^{1+s}}\Bigl[2c\pm \frac K2g_d(t_{j})\Bigr],
\frac{d}{dt}\Bigg( \int_0^t (t-\xi)^{\alpha-1}E_{\alpha, \alpha}[-\mu_{m,n}(t-\xi)^{\alpha}]F_{m,n}(\xi)d\xi\Bigg)_{t=0}= (t - t)^{\alpha - 1}E_{\alpha, \alpha}(-\mu_{m,n}(t -t)^\alpha)F_{m,n}(t)\Big|_{t=0}+ \Bigg(\int_0^t \frac{\partial}{\partial t}(t-\xi)^{\alpha -1}E_{\alpha, \alpha}(-\mu_{m,n}(t - \xi)^{\alpha})F(\xi)d\xi\Bigg)_{t=0}= 0.
\mathcal{M}_1 =\lbrace \hat{\boldsymbol{x}}^{\boldsymbol{\omega}}, \: \boldsymbol{\omega}\in \Omega \rbrace, \hat{\mathcal{D}}_{\hat{\boldsymbol{x}}^{\boldsymbol{\omega}}}= \cap_{k \in \mathbb{Z}}\hat{\mathcal{D}}_{\omega_k,k}, \mathcal{M}_2 = \cup_{\boldsymbol{\omega}\in \Omega}\hat{\mathcal{D}}_{\hat{\boldsymbol{x}}^{\boldsymbol{\omega}}}.
L(x_1,x_2)=(L_1(x_1,x_2), L_2(x_1,x_2))=\left(x_1+\frac{1}{p}L_2(x_1^{p^{h_1}},x_2^{p^{h_1}}), x_2+ \frac{1}{p}L_1(x_1^{p^{h_2}},x_2^{p^{h_2}}) \right),
R^{(1)}_{m}\big(i(i-1)\big)= (\tfrac{\beta}{2}-m+1)_{2m}(\tfrac{\beta}{2}+i)(\tfrac{\beta}{2}-i+1) \sum_{j=1}^m\tfrac{(-m-i+1)_{j-1}(-m+i)_{j-1}}{(\tfrac{\beta}{2}-m+1)_j(-\tfrac{\beta}{2}-m)_j}.
\Pr{B_k}{M_k}= \sum_{i=k+1}^\infty\Pr{s_i=M_k}{M_k}\prod_{n=1}^{i-(k+1)}\Pr{s_{k+n}<M_k}{M_k}= \sum_{i=k+1}^\infty\frac{c}{M_k^{5/4}}\Pr{s_{k+1}<M_k}{M_k}^{i-(k+1)}.
\langle Q | \hat{\psi}_{s p}( x , t ) | Q + 1 \rangle = \int \frac{d \omega \, d k}{( 2 \pi )^{2}}\frac{{\cal A}( k )}{k^{2}+ m^{2}\sin^{2}\left( \frac{\lambda^{2}\bar{Q}}{4}\right)}\delta ( \omega - M ( Q + 1 ) + M ( Q ) ) ,
B_n[(\alpha, g)] = [1, (\beta, 1_G), (\beta, 1_G)][(\alpha, g), (\alpha, g), (\alpha,g)] &=[(\alpha,g), (\beta\alpha, g), (\beta\alpha,g)] & = [(\alpha, g)][(\beta\alpha, g)]^*[(\beta\alpha, g)] &= [(\alpha, g)]B_{n+|\alpha|}
p_{i,j}= p_{i,j}(\xi) := \frac{\varphi(x_i+\xi-x_j)}{\sum_k \varphi(x_i+\xi-x_k)}= \frac{\varphi(x_i+\xi-x_j)}{N \varrho_N^\varphi(x_i+\xi)}\,, q_{i,j}= q_{i,j}(\xi) := \frac{\varphi(y_i+\xi-y_j)}{\sum_k \varphi(y_i+\xi-y_k)}= \frac{\tilde \varphi(y_i+\xi-y_j)}{N \varrho_N^\varphi(y_i+\xi)}\,.
K_{U^{\alpha}_\eta}(z,z^{\prime},w;\bar{\zeta},\bar{\zeta^{\prime}},\bar{\eta})=(1-|\eta|^2)^{-\alpha\cdot \mathbf 1}K_{\Omega}\big(f_{\alpha}(z,\eta),z^{\prime};\overline{f_{\alpha}(\zeta,\eta)},\bar{\zeta^{\prime}}\big).
\hat{J}_+\psi- \frac{\Delta(VJ_+)+\frac{1}{2}\langle[\hat{V},\hat{J}_+]\rangle}{(\Delta V)^2}\hat{V}\psi+ \left(\frac{\Delta(VJ_+)+\frac{1}{2}\langle[\hat{V},\hat{J}_+]\rangle}{(\Delta V)^2}V-J_+\right) \psi=0\,.
\oint_{C}Ldx+Mdy=\oint_{\partial D}\omega=\int_{D}d\omega=\int_{D}\frac{\partial L}{\partial y}dy\wedge dx+\frac{\partial M}{\partial x}dx\wedge dy=\iint_{D}(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y})dxdy.
{\frac{d^{2}x^{\rho}}{d \tilde{\lambda}^{2}}}+ \tilde{\Gamma}_{\mu \nu}^{\rho}{\frac{d x^{\mu}}{d \tilde{\lambda}}}{\frac{d x^{\nu}}{d \tilde{\lambda}}}= -{\frac{1}{2}}{\frac{{\cal W}^{\prime}}{\cal W}}{\frac{d y}{d \tilde{\lambda}}}{\frac{d x^{\rho}}{d \tilde{\lambda}}}.
f(z) = \sum_{n \in \mathbb{Z}}\sum_{j=0}^{r-1}\biggl(c_{n,j}^- u_{k,n}^{[j],-}(y)e^{2\pi i n x}+ c_{n,j}^+ u_{k,n}^{[j],+}(y)e^{2\pi i n x}\biggr),
\nabla \kappa_{\varepsilon}(t) \cdot \nabla \sigma_{\varepsilon}(t) = \frac{|\nabla \kappa_{\varepsilon}(t)|^2}{\max{\{\varepsilon, \| \kappa_{\varepsilon}(t) \|_H \}}}\geq 0 \ \ \ \textrm{a.e. in $\Omega$, for a.e. $t \in (0,T)$},
(-i\epsilon^a_{~bcd}e^\mu_a\omega_\mu^{bc}+i\kappa e^\mu_a R^{bc}_{\mu\nu}\epsilon^{da}_{~~bc}(\bar{\beta}^\nu-\bar{\alpha}^{\nu\rho}p_\rho){\cal S}_1= \frac{1}{3}\kappa\epsilon^{da}_{~~bc}e^\mu_a(\nabla_\rho R_{\mu\nu}^{bc})\alpha^{\nu\rho}{\cal S}_0
\mathcal{I}\left(c,\theta\right) =\int_{0}^{\infty}Q_z(c\sqrt{x},\theta)f_\gamma(x)\, dx =\frac{1}{\pi}\int_0^\theta \int_{0}^{\infty}\mbox{exp}\Big(-\frac{c^2 x}{2\sin^2\phi}\Big) f_\gamma(x)\, dx\, d\phi =\frac{1}{\pi}\int_0^\theta\mathcal{G}_{\gamma}\Big(\frac{c^2}{2\sin^2\phi}\Big)d\phi.
3(\Psi_{11})_t+(\Psi_{11})_{xx}+\big(t-x^2\big)(\Psi_{11})_x{}= x^2\left(r_2+\frac{t}{2}\right)\Psi_{11}+x(3b-2e_1)\Psi_{21}+x \left(r_1-\frac{1}{2}-\frac{q_2}{2}\right)\Psi_{11}{}+(e_2+3c)\Psi_{21}+\left(\frac{U}{2}+ \frac{3}{2}q_2(b-e_1) + q_1 + r_0\right)\Psi_{11}.
={\frac{1}{( 4 \pi s )^{2}}}\left( \Omega_{4}+ \Sigma_{2}{\frac{i s}{4}}\int_{\Gamma}{\frac{\cot ( \pi \alpha^{- 1}w )}{\sin^{2}w / 2}}\exp \left( -{\frac{r_{0}^{2}\sin^{2}w / 2}{s}}\right) d w \right) .
\frac{(z_0')^2-(z_1')^2-(z_2')^2}{2z_1'z_2'}&=\frac{\big(z_0(z_0-z_2)-z_1^2\big)\big(z_0(z_0+z_2)-z_1^2\big)-(z_0^2-z_2^2)z_2^2}{2z_1^2z_2^2}&=\frac{z_0^4+z_1^4+z_2^4-2z_0^2z_1^2-2z_0^2z_2^2}{2z_1^2z_2^2}&=\frac{(z_0^2-z_1^2-z_2^2)^2-2z_1^2z_2^2}{2z_1^2z_2^2}&=2\big(\frac{z_0^2-z_1^2-z_2^2}{2z_1z_2}\big)^2-1.
\|f_1f_2\|_{L^2}^2= \sum\limits_{k\in\mathbb{Z}}\|f_1f_{2,k}\|_{L^2}^2,\ \ \|f_2\|_{L^2}^2=\sum\limits_{k\in\mathbb{Z}}\|f_{2,k}\|_{L^2}^2,\ \ \|\partial_zf_2\|_{L^2}^2=\sum\limits_{k\in\mathbb{Z}}\|\partial_zf_{2,k}\|_{L^2}^2,
L_{e f f}^{( 0 )}= \int \; d^{2}x \; [ ( \partial_{t}\mid \phi \mid )^{2}+ ( \partial_{t}\mid \chi \mid )^{2}+ \frac{\kappa}{q_{1}}B_{2}\partial_{t}\omega_{1}+ \frac{\kappa}{q_{2}}B_{1}\partial_{t}\omega_{2}] \; \; .
E_{\xi_{i}}E_{\xi_{i}}^{*}- q^{-2}E_{\xi_{i}}^{*}E_{\xi_{i}}\sim -q^{-1}(q - q^{-1}) \sum_{k = i + 1}^{N}q^{3(i - k)}E_{\xi_{k}}^{*}E_{\xi_{k}}.
{\mathbf T}^{(a)}_m(\theta+\frac{\pi}{3}i ){\mathbf T}^{(a)}_m(\theta-\frac{\pi}{3}i )={\mathbf T}^{(a+1)}_m(\theta){\mathbf T}^{(a-1)}_m(\theta) +{\mathbf T}^{(a)}_{m+1}(\theta){\mathbf T}^{(a)}_{m-1}(\theta)
\begin{aligned}\frac{dx_i}{dt}= 0 \Rightarrow y_{i}^* =\frac{r_i(K_i-x_i^*)(1+x_i^*)}{a_iK_i}\frac{dy_j}{dt}= 0 \Rightarrow x_{i}^* = \frac{-\rho_jsy_i^*+\rho_jsy_j^*+d_jy_j^*}{a_i\rho_jsy_i^*y_j^*-a_i\rho_jy_i^*y_j^*+\rho_jsy_i^*-\rho_jsy_j^*-d_jy_j^*}\end{aligned}
\exists \ \gamma_1>0,\,\gamma_2>0: \quad|f_1'(r)|\le \gamma_1|f_{\Gamma 1}'(r)|+\gamma_2 \quad\forall\,r\in (-1,1), begin{align*}1mm] \lim_{r\searrow -1}f_1'(r)=\lim_{r\searrow -1}f_{\Gamma 1}'(r)=-\infty, \lim_{r\nearrow +1}f_1'(r)=\lim_{r\nearrow +1}f_{\Gamma 1}'(r)=+\infty.
\begin{aligned}\Vert q_{1}-q_{2}\Vert_{H^{-1}(\Omega)}^{2}\leq \frac{C}{h^{\frac{2n}{n+2}}}e^{\frac{12R}{h}}\Vert u_{1}-u_{2}\Vert_{\mathcal{C}}^{2}+C h^{\frac{4}{n+2}}\leq C e^{\frac{14R}{h}}\Vert u_{1}-u_{2}\Vert_{\mathcal{C}}^{2}+ C h^{\frac{4}{n+2}}. \end{aligned}
\left(\prod_{i=1}^3[\mathrm{cok}(\delta(\mathcal{F}_i)_{tor})]^{(-1)^{i+1}}\right) = \frac{\nu(\mathcal{H}^0)_{\mathbb{R}}\nu(\mathcal{H}_S/\mathcal{H}^0)}{\nu(\mathcal{H}_S)_{\mathbb{R}}}.
\begin{array}{ccc}g^{n,h}_f =\left(\begin{array}{ccc}x 1 0 1 y_f 0 0 0 y_e\end{array}\right) \hspace{.5in}g^{n,h}_e =\left(\begin{array}{ccc}y_f 0 0 0 y_e 1 0 1 x\end{array}\right)\end{array}{}.
R_{i j k l}= q^{2}[ g_{j l}\widetilde{M_{i k}}- g_{j k}\widetilde{M_{i l}}+ g_{i k}\widetilde{M_{j l}}- g_{i l}\widetilde{M_{k j}}] +{\frac{q^{2}( c_{0}^{2}- 1 )}{c_{0}^{2}}}F_{i j}F_{m l}
\frac{\partial\gamma_1}{\partial\omega_k}=\frac{{u}_2^{(0)}}{\kappa_1({U}_2^{(0)})},\frac{\partial\gamma_1}{\partial\alpha_i}=\frac{r_{i}^{(2)}({U}_2^{(0)})}{\kappa_1({U}_2^{(0)})},i=2,3,5,
\delta L_{T}=\left[\dot{Q}^{a}-\frac{\partial H_{T}}{\partial P_{a}}\right]\delta P_{a}+\left[-\dot{P}_{a}-\frac{\partial H_{T}}{\partial Q^{a}}\right]\delta Q^{a}+\frac{d}{dt}\left[P_{a}\delta Q^{a}\right]
c_l=c_l(\delta):=\begin{cases}0 l=0, \displaystyle\frac{1+\delta \sum_{m=0}^{l-1}c_m^2}{\sqrt{1-\delta^2 \sum_{m=0}^{l-1}c_m^2}}1\leq l \leq n_1(\delta).\end{cases}
\int_0^1\sum_{N\ge 1}y^N(1-t)^N [z^N] \,\frac{(1-z)^{-x}}{1-tz}\,dt= \,\int_0^1\sum_{N\ge 1}[z^N]\,\frac{\bigl(1-(1-t)yz\bigr)^{-x}}{1-t(1-t)yz}\,dt = \,\int_0^1\frac{\bigl(1-(1-t)y\bigr)^{-x}}{1-t(1-t)y}\,dt -1.
\left. \frac{d \mu}{dt}\right|_q \lesssim C_\mu \left[ \sum_{j_1,\ldots,j_Q=1}^d \left| \sum_{i_1,\ldots,i_Q=1}^d{\mathcal A}_{q}(M_{j_1 i_1}\partial_{t_{i_1}},\ldots,M_{j_Q i_Q}\partial_{t_{i_Q}}) \right|^2 \right]^{\frac{d}{2Q}}
\begin{array}{l c r}{{q_{n - 1}^{n - 1}= q_{n}^{n}= - 2 , \quad q_{n - 1}^{n - 2}= q_{n}^{n - 2}= 1}}\\{{q_{n - 2}^{n - 1}= q_{n - 2}^{n}= q_{n + 3}^{n - 1}= q_{n + 1}^{n}= q_{n + 2}^{n}= 1}}\\{{q_{n}^{n - 1}= q_{n - 1}^{n}= q_{n + 1}^{n - 1}= q_{n + 1}^{n}= q_{n + 2}^{n - 1}= 0 .}}\\ \end{array}
v_{t}(x, t) = a_{1}^{2}v_{xx}(x, t) + a_{2}^{2}v_{xx}(x, t - \tau) + b_{1}v_{x}(x, t) + b_{2}v_{x}(x, t - \tau) + d_{1}v(x, t) + d_{2}v(x, t - \tau) + g(x, t),
L^\pm(k,\chi) =\frac{\partial_u F(\chi,u_0(k,\chi))}{\partial_u \alpha(\chi,u_0(k,\chi))}, , \Psi^\pm(k,\chi,y) = u_0(k,\chi)+ y \frac{\partial_u F(\chi,u_0(k,\chi))}{\partial_u \alpha(\chi,u_0(k,\chi))}, .
\sum_{i=0}^{n-2}\epsilon_i 2^i \leq t+t\cdot2+t\cdot2^2+\cdots+t\cdot2^{n-k-3}+2^{n-k-2}+2^{n-k-1}+\cdots+2^{n-2}&<t\cdot2^{n-k-2}+2^{n-1}-1 &\leq2^{k+1}\cdot2^{n-k-2}+2^{n-1}-1 &=2^n-1 &<2^n.
\rho_{L,\varepsilon}(u, v) =\rho\left(\mathcal{A}_{L,\varepsilon}(u)\bigcap\mathcal{A}_{L,\varepsilon}(v)\right)= \begin{cases}\log(L/|u-v|) \, , 1+\log(L/\varepsilon)-|u-v|/\varepsilon \, \text{if$|u-v|<\varepsilon$},\end{cases}
\Gamma_{\rm b}\left(a=-a_l\right)=\eta_{11}\left(a_4-a_l\right)\left[\left({\cal C}_{66}-{\cal C}_{11}\right)\left({\cal C}_{33}-a_l{\cal C}_{44}\right)+\left({\cal C}_{13}+{\cal C}_{44}\right)^2\right]
d_N=\left(\prod_{k=2}^Nk^{p^{N+1-k}}\right)K^{(1+p+\cdots+p^{N-2})}c_2,m_N\triangleq \left(\prod_{k=2}^Nk^{p^{N+1-k}}\right)K^{1+p+\cdots+p^{N-1}}M^{qp^N}.
\frac{2}{\sqrt{-1}}\tilde{S}_{l}(\varphi) = \iint_{D\setminus\cup_{i=1}^{n}D_{i}}\omega(\varphi)+\sum_{i=1}^{n}\int_{l_{i}}\left(\frac{2\log|w-w_i|}{\bar{w}-\bar{w}_i}+\frac{2\log\left(\log|w-w_i|\right)^2}{\bar{w}-\bar{w}_i}\right)d\bar{w}.
-\frac 12 \int_{\partial_e \mathcal B_n}y^{1-2\alpha}|\nabla v_n|^2(X-z_0,\nu)\,dS &+\int_{\partial_e \mathcal B_n}y^{1-2\alpha}(\nabla v_n, X-z_0)\frac{\partial v_n}{\partial\nu}\,dS. = \frac 12 \int_{\partial_e \mathcal B_n}y^{1-2\alpha}|\nabla v_n|^2(X-z_0,\nu)\,dS\le0.
\mathbf{A}\otimes\mathbf{B}= \left[\begin{array}[]{c c c c}a_{11}\mathbf{B}a_{12}\mathbf{B}\cdots a_{1q}\mathbf{B}a_{21}\mathbf{B}a_{22}\mathbf{B}\cdots a_{2q}\mathbf{B}\vdots \vdots \ddots \vdots a_{p1}\mathbf{B}a_{p2}\mathbf{B}\cdots a_{pq}\mathbf{B}\end{array}\right]
\overset{\longrightarrow}{iL}_{-1}\phi_+=i+\rho_0\big(1+2u_1 z+3u_2z^2+\cdots\big)\left(\frac{i(u_1+u_1^*)}{\rho_0}z-\frac{i}{\rho_0}-\frac{i}{\rho_0}z^2\right).
\int_{-\infty}^\infty x^n\,h_j^{(s)}\mathrm{d}x =\sigma\,\int_{-\infty}^\infty(\sigma z+\mu)^n\,[\Phi_s(z)]^{j+\alpha-1}\,\phi_s(z)\mathrm{d}z &=\sigma\mu^n\,\sum_{i=0}^n\,\binom{n}{i}\,\left(\frac{\sigma}{\mu}\right)^i\,\int_{-\infty}^\infty z^i\,[\Phi_s(z)]^{j+\alpha-1}\,\phi_s(z)\mathrm{d}z.
m_3(6)=\frac{{}_2F_{1}\left(\frac{1}{6},\frac{5}{6};1;\beta_{9r}\right)}{{}_2F_{1}\left(\frac{1}{6},\frac{5}{6};1;\beta_r\right)}=\sqrt{3r}\cdot\sqrt[4]{\frac{1+8\alpha'_{3r}}{1+8\alpha_{3r}}}
L = -{\frac{1}{4}}T r{\bf F}_{\mu \nu}({\bf A}){\bf F}^{\mu \nu}({\bf A}) -{\frac{1}{2}}T r \left[ m \left({\bf A}_{\mu}-{\bf \Phi}_{\mu}\right) - D_{\mu}({\bf \Phi}){\bf C}\right]^{2}
\begin{array}{lll}h^0(H_{S^{[2]}}) = \frac{1}{8}\left((H\cdot H)+4\right)\left((H\cdot H)+6\right), h^0(H_{Z}) = \frac{1}{2}\left(h^0(H)\right)^2+\frac{1}{16}h_{\Sigma_{S\times S}}, h^0(H_{Y}) = \frac{1}{2}h^0(H_Z)+\frac{1}{16}h_{\Sigma_{Z}}+1=\frac{1}{4}\left(h^0(H)\right)^2+\frac{1}{32}h_{\Sigma_{S\times S}}+\frac{1}{16}h_{\Sigma_{Z}}+1.\end{array}
\left\| \frac{z-w_i}{\|z-w_i\|}- \frac{z}{\|z\|}\right\| \le \frac{\|w_i\|}{\|z-w_i\|}+ \|z\| \left| \frac{1}{\|z-w_i\|}- \frac{1}{\|z\|}\right| \le \frac{R_0}{\|z\|-R_0}+ \frac{| \|z\| - \|z-w_i\| |}{\|z-w_i\|}&\le \frac{R_0}{\|z\|-R_0}+ \frac{R_0}{\|z\|-R_0}&< \frac{2R_0}{(\frac{2+\delta}{\delta})R_0-R_0}= \delta.
R_{II}(E,\Delta T,\xi) = \Delta T \sum_{\mu}|Z|^{2}|f_{\mu}(\xi)|^{2}\left\{\frac{\sin\left[(E-\omega_{\mu})\Delta T\right]}{(E-\omega_{\mu})\Delta T}+\frac{\sin\left[(E+\omega_{\mu})\Delta T\right]}{(E+\omega_{\mu})\Delta T}\right\}.
\mathcal F(\lambda, Z) = \left[\begin{array}{c}(\lambda I - A)^{-1}B I \end{array}\right]^H\left[\begin{array}{cc}M_{11}+ Z_{11}M_{12}+ Z_{12}M_{21}+Z_{21}M_{22}+ Z_{22}\end{array}\right] \left[\begin{array}{c}(\lambda I - A)^{-1}B I \end{array}\right]
\Gamma_4 = \frac{i \lambda^2}{2}\cos \frac{{\tilde p}_1 p_2}{2}\cos \frac{{\tilde q}_1 q_2}{2}\int \frac{d^2 k d \omega}{(2 \pi)^3}\frac{\mbox{cos}^2 \frac{{\tilde P}k}{2}}{(\omega - \frac{k^2}{2}+ i \epsilon)(E -\omega - \frac{(k-P)^2}{2}+ i \epsilon)},
\sum_{n_k\ge n_{k-1}\ge \dots \ge n_1 \ge 1}\frac{q^{n_1+n_2+\dots+n_k}}{(1-q^{n_k})^2(1-q^{n_{k-1}})^2\dots(1-q^{n_1})^2}=\sum_{n=1}^\infty \frac{(-1)^{n+1}q^{n(n-1)/2+kn}(1+q^{n})}{(1-q^{n})^{2k}}.
\Omega_{3}=2\int_{\wp (\omega_{3}/2)}^{+\infty}\frac{d\mathbf{u}}{\sqrt{4\mathbf{u}^{3}-g_{2}\mathbf{u}-g_{3}}}= \int_{a_{4}}^{X_{\pm}(b_{4})}\frac{dx}{\sqrt{D(x)}},
\hat{S}= \int d^{10}\hat{x}\sqrt{|\hat{g}|}\,\left\{e^{-2\hat{\phi}}\left[ \hat{R}-4(\partial\hat{\phi})^{2}\right] +{\textstyle\frac{(-1)^{7-p}}{2\cdot (8-p)!}}\left(\hat{G}^{(8-p)}\right)^{2}\right\}\, .
H_{n}(t,\theta) \leq e^{\theta+\int_{0}^{t}(e^{x(\theta)}-1)\gamma_{n}(s)ds}&\leq e^{\theta+(e^{x(\theta)}-1)\int_{0}^{t}\gamma_{n}(s)ds}&\leq e^{\theta+(e^{x(\theta)}-1)\Vert\gamma_{n}\Vert_{L^{1}}}&\leq e^{\theta+(e^{x(\theta)}-1)\rho}&=x(\theta),
\int_{-\pi}^{\pi}e^{i\theta k}\left[\left( A(e^{i\theta}) -{h}(\theta) \right)^{<\alpha - 1>}f(\theta) - \left({h}(\theta) \right)^{<\alpha - 1>}g(\theta)\right] d\theta = 0, \, k = -1, -2, \dots.
\int_{K_2}\frac{|\sin u-\sin v|^2}{|u-v|^{2-2H}}dudv &\le 4\int_0^{\sqrt{\lambda_\alpha}}\bigg[\int_0^{v-1}(v-u)^{2H-2}du+\int_{v+1}^{\sqrt{\lambda_\alpha}}(u-v)^{2H-2}du\bigg] dv &=\frac{4}{H(1-2H)}\lambda_\alpha^H.
R_\mathsf{A}^\mathsf{Alt}\left(Q_{U,V,X|S}\right)\triangleq\min\left\{\begin{aligned}I(V;Y|U) -I(V;Z|U), I(U,V;Y) -I(U,V;S), I(U,V;Y) -I(U;S)-I(V;Z|U)\end{aligned}\right\},
{\rm Hom}_{{\rm Pro}({\rm Qcoh}(X))}(i_!(F),G) =\varprojlim_{i\in\mathcal{I}}{\rm Hom}_{{\rm Pro}({\rm Qcoh}(X))}(i_!(F),G(i)) &\cong\varprojlim_{i\in\mathcal{I}}{\rm Hom}_{{\rm Qcoh}(U)}(F,i^*G(i)) &={\rm Hom}_{{\rm Pro}({\rm Qcoh}(U))}(J(F),{\rm Pro}(i^*)(G)),
w_2(\rho_0,\rho_1)^2=\inf \left\{\int_0^1 \int_\Omega |v|^2\, \mu_t\,dx\,dt\, :\, \begin{array}{c}\partial_t \mu_t+\mathrm{div}(v_t\, \mu_t)=0 \mu_0=\rho_0\ \mbox{and}\ \mu_1=\rho_1\end{array}\right\}.
W^T \alpha L(u^a)Q\leqslant O(1) \bigg(\sum_{i,j = 1\atop j \neq i}^n |u^j_x| w_i^2+ \delta^{{1}/{2}}\sum_{i=1 \atop i \neq p}^n |u^i_x| w_i^2+ \delta^{-{1}/2}|u^p_x|^2 w_p^2+ \varepsilon_0 |W_x|^2\bigg).
\dfrac{1+\psi(1-\lambda)-\psi(\lambda)}{2}f(\sigma(t)) =\dfrac{1}{\int_a^b \nu(t)\Delta t}\int_a^b K(s,t)f^{\Delta}(\sigma(s))\Delta s + \dfrac{1}{\int_a^b \nu(t)\Delta t}\int_a^b \nu(s)f(\sigma^2(s))\Delta s & - \dfrac{\psi(\lambda)f(\sigma(a))+\left(1-\psi(1-\lambda)\right)f(\sigma(b))}{2}.
(p_{1},\ldots,p_{n-3},f_{n-2},f_{n-1},f_{n}) = & \Big[p_{1},\ldots,p_{n-2},f_{n-1}\circ (p_{1},\ldots,\widetilde{f}_{n-2},p_{n-1},p_{n}),f_{n}\circ (p_{1},\ldots,\widetilde{f}_{n-2},p_{n-1},p_{n}) \Big] \circ & (p_{1},\ldots,f_{n-2},p_{n-1},p_{n}).
\lefteqn{f_T(y)-f_T(x)-(y-x)_1\partial_1f_T(x)}&=\int_0^1(y_2-x_2)\partial_2f_T(y_1,sy_2+(1-s)x_2)ds &+\int_0^1(y_1-x_1)^2\partial_1^2f_T(sy_1+(1-s)x_1,x_2)\, (1-s) ds,