harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C^*C =Q^*B^{\beta}Q=B^{\beta}, C^{*2}C^2 =Q^*B^\frac{\beta}{2}C^*CB^\frac{\beta}{2}Q=Q^*B^{2\beta}Q=B^{2\beta}=(C^*C)^2, C^{*3}C^3 =C^{*}(C^{*2}C^2)C=Q^*B^\frac{\beta}{2}B^{2\beta}B^\frac{\beta}{2}Q &=Q^*B^{3\beta}Q=B^{3\beta}=(C^*C)^3.
\begin{array}{r c l}{d s^{2}}&{=}&{d s^{2}( \mathrm{A d S}_{2}\times \mathrm{S}^{2}) + d x^{2}}\\{C_{m n x}}&{=}&{C_{m n}( \mathrm{A d S}_{2}\times \mathrm{S}^{2})}\\ \end{array}
\gamma_{g,9}:|D_{\rm inv}\rangle \rightarrow{1\over\sqrt{D}}\sum_\alpha \sum_{\beta,\gamma}c_{\alpha\beta}{\overline c}_{\gamma\alpha}|d_\beta\otimes{\overline d}_\gamma \rangle={1\over \sqrt{D}}\sum_\alpha |d_\alpha\otimes{\overline d}_\alpha\rangle=|D_{\rm inv}\rangle~,
P_{out}(R_{E},\phi_{2}) \!=\!Pr(\rho_e|h_{e}|^{2}\!<\!\rho_1|h_{1}|^{2})P_{out,1}(R_{E},\phi_{2}) &\!+\!Pr(\rho_e|h_{e}|^{2}\!\geq\!\rho_1|h_{1}|^{2})P_{out,2}(R_{E},\phi_{2}),
Z_j (\varphi , \Psi_j ; (\Psi_k)_{k<j}| \Lambda ) = e^{-E_{j}|\Lambda| + e_j}\sum_{X\in \mathcal{P}_j (\Lambda)}e^{U_j (\Lambda \backslash X, \varphi)}\prod_{Y \in \textnormal{Comp}_j (X)}\big( K_j (Y, \varphi ; (\Psi_k)_{k < j}) + \Psi_j (Y, \varphi) \big) ,
S_{c}\omega=\int_{c}^{t}\omega_{B}(\tau)d\tau;dS_{c}\omega=d\left(\int_{c}^{t}\omega_{B}(\tau)d\tau\right)=dt\wedge\omega_{B}(t)+\int_{c}^{t}d_{N}\omega_{B}(\tau)d\tau, S_{c}d\omega=\int_{c}^{t}d\omega=\int_{c}^{t}\left(\frac{\partial\omega_{A}}{\partial t}(\tau)\right)d\tau-\int_{c}^{t}d_{N}\omega_{B}(\tau)d\tau.
\mathcal{S}(J^{-1})(w)=\varphi_{ww}(w)-\frac{1}{2}\varphi_{w}(w)^{2}=\sum_{i=1}^{n-1}\left(\frac{1}{2(w-w_{i})^{2}}+\frac{c_{i}}{w-w_{i}}\right),
w_{01}^1 \ = \l_{11}+l_{12}, w_{21}^1 \ = \ \frac{l_{11}+l_{12}}{l_{21}}, w_{12}^2 \ = \ \frac{w_{01}^2+1}{l_{12}}, w_{22}^2 \ = \ \frac{w_{01}^2+1}{l_{22}}.
h_{l,p,q}(\zeta)=\sum_{n=1}^{\infty}\left[\int_{\mbox{arccosh}(y)}^{\infty}\frac{udu}{\sqrt{\cosh u-y}}\frac{l^2+pe^{-u}}{\sinh^q(u/2)}\right]_{y=1+n^2\zeta^2/2},
K(n, k^2) \equiv \frac{1}{2 n}\int_{0}^{2 n \pi}d \theta_2 \int_{0}^{2 n \pi}d \theta_1~{\cal F}(\theta_1)^{k^2-1}{\cal F}(\theta_2)^{k^2-1}\left| 2 n \sin \frac{\theta_2-\theta_1}{2 n}\right|^{-2 k^2},
\mathcal{H}(\rho,m)= \frac{h}{2\tau}\sum_{i=1}^s\sum_{j=1}^N \frac{\hat m^2_{i,j}}{\rho_{i,j}}D^{-1}_{i,j}+ h\sum_{i=1}^s\sum_{j=1}^{N}\rho_{i,j}\log \rho_{i,j}, \mathcal{I}(\phi)= \frac{1}{8h}\sum_{j=1}^{N}\epsilon_j(\phi_{j+1}-\phi_{j-1})^2 +\frac{1}{8\beta_a}(\phi_0+\phi_1)^2+\frac{1}{8\beta_b}(\phi_N+\phi_{N+1})^2.
P_h\delta_t e_i^{m+1/2}- \Delta_{2}e_i^{m+1/2}= R_i^m, 1\le i \le N-1 , 0\le m\le M-1 e_0^m = 0, e_M^m = 0 e_i^0 = 0 , 0 \le i \le N.
\left[\begin{array}{cc}\!\!I_{M}, \!\!\!\!I_{M}\!\!\! \end{array}\right]P_{\operatorname{Box}[\boldsymbol{0}, \boldsymbol{1}]}\biggl(\boldsymbol{\rho}\left(i+\frac{1}{2}\right)-\begin{bmatrix}\boldsymbol{\lambda}^{\star}\boldsymbol{\lambda}^{\star}\end{bmatrix}\biggl)=\boldsymbol{1}.
\tau_1 < \min \biggl( \frac{2 \alpha}{5\alpha+3d}\, , \, \frac{\alpha-d}{2\alpha}\, , \, \frac{4\beta^*}{4\beta^*+3d}\biggr), \tau_2 := \min\biggl(1-\frac{d/4}{1+\lfloor d/4 \rfloor},1-\frac{d}{2\beta}\biggr)
J' (\Phi\Psi)^{-1}\mathbb{H}_0 = (\Phi\Psi)^{-1}\mathbb{H}_0 \iff J' \Psi^{-1}\Phi^{-1}\mathbb{H}_0 = \Psi^{-1}\Phi^{-1}\mathbb{H}_0 \iff \Psi^{-1}J \Phi^{-1}\mathbb{H}_0 = \Psi^{-1}\Phi^{-1}\mathbb{H}_0 \iff J \Phi^{-1}\mathbb{H}_0 = \Phi^{-1}\mathbb{H}_0.
\left\langle F_y\psi,\psi \right\rangle = \left\langle \left(e^{iHt_0}\pi(e_y)e^{-iHt_0}\right)\psi,\psi \right\rangle = \tau\left(e^{iht_0}\pi(e_y)e^{-iht_0}\right) = \tau(e_y)= t.
e(G,N)=\sum_{i=1}^{s}\left\lvert \mathrm{Orb}(N_{i}) \right \rvert = \sum_{i=1}^{s}\dfrac{\left\lvert \mathrm{Aut}(G)\right\rvert}{\left\lvert\mathrm{Stab}(N_{i})\right\rvert}= \sum_{G_{N}}\dfrac{\left\lvert \mathrm{Aut}(G)\right\rvert}{\left\lvert\mathrm{Aut}_{\mathcal{B}r}(G_{N})\right\rvert}.
\mathrm{P}_{k,q,p}= \int_{\rho\xi^*_{k,q,p}}^{\infty}\pi_p^Pf(x) \left(F(x) \right)^{P-p}\left(1-F (x) \right)^{p-1}dx = \sum^{p-1}_{i=0}{p-1 \choose i}(-1)^i \int_{\rho\xi^*_{k,q,p}}^{\infty}\pi_p^Pf(x) \left(F(x) \right)^{P-p+i}dx.
\mathbf{\Delta}\mathbf{S}_{b}\mathbf{\Delta}+\mathbf{S}_{a}^{-1}\mathbf{S}_{ab}\mathbf{\Delta}+\mathbf{\Delta}\mathbf{S}_{bc}\mathbf{S}_{c}^{-1}+\mathbf{S}_{a}^{-1}\mathbf{S}_{abc}\mathbf{S}_{c}^{-1}=\mathbf{S}_{b},
, \ldots , \left( \begin{array}{c c c c c c c c c c c}{0}&{\cdots}&{0}&{1}&{0}&{\cdots}&{0}&{0}&{0}&{\cdots}&{0}\\{0}&{\cdots}&{0}&{0}&{0}&{\cdots}&{0}&{1}&{0}&{\cdots}&{0}\\ \end{array}\right)
\frac{1}{2}m\omega^{2}\sum_{j}(x_{j}-x_{j+1})^{2}=\frac{1}{2}m\omega^{2}\sum_{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})=\frac{1}{2}m\sum_{k}{\omega_{k}}^{2}Q_{k}Q_{-k},
B^n_t=B_{\frac{k+2}{2^n}}+2^n(t-\frac{k+1}{2^n})(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}), \mbox{for}t\in [\frac{k}{2^n}, \frac{k+1}{2^n}].
\displaystyle\begin{array}{cc}F(x)^T \cdot x=\Big(c_r(h_r)-\pi_{ab}h_r-d_r(\pi_{ab})\Big)\cdot \left(\begin{array}{cc}h_r \pi_{ab}\end{array}\right)=0, x=\left(\begin{array}{cc}h_r \pi_{ab}\end{array}\right)\geq 0, F(x)=\left(\begin{array}{cc}c_r(h_r)-\pi_{ab}h_r-d_r(\pi_{ab})\end{array}\right)\geq0,\end{array}
(H_\lambda-z)^{-1}=(H-z)^{-1}-\lambda (H-z)^{-1}C_1(H_\lambda-z)^{-1}&=(H-z)^{-1}-\lambda (H-z)^{-1}C_1(H-z)^{-1}&\qquad+\lambda^2 (H-z)^{-1}C_1(H_\lambda-z)^{-1}C_1(H_\lambda-z)^{-1},
e_5 =-24 p^4\mu^2+24 a\mu p^3+204 \mu^2 p^3-4p^2a^2-192 a\mu p^2-12 \mu^2 p^2+24 \mu p^3+45 pa^2-16 p^2a -192 p^2\mu+3 a^2+90 ap-4 p^2+6a+45p+3 e_6 =360 p (a+1-2\mu p)^2\ge 0.
\phi_k^{(n)}\circ \phi_{\infty, k+1}^{(n)}= (id_n \otimes \phi_k) \circ (id_n \otimes \phi_{\infty,k+1}) &= id_n \otimes ( \phi_k \circ \phi_{\infty, k+1}) = id_n \otimes \phi_{\infty, k}= \phi_{\infty, k}^{(n)},
\eta(k;l) =\sum_{\substack{m>0 0<a_1\leq\cdots\leq a_{k-1}\leq m 0<b_1\leq\cdots\leq b_{l-1}\leq m}}\frac{1}{m^2a_1\cdots a_{k-1}b_1\cdots b_{l-1}}&=\zeta\bigl((\underbrace{1,\ldots,1}_k)^\star\circledast(\underbrace{1,\ldots,1}_l)^\star\bigr)
g_s^{1/2}\,ds^2 = H_2^{-5/8}H_6^{-1/8}\eta_{\mu\nu}dx^\mu dx^\nu +H_2^{3/8}H_6^{7/8}(dr^2 + r^2 d\Omega) + V^{1/2}H_2^{3/8}H_6^{-1/8}ds^2_{K3}
\psi^{\tau}(x,t):= \psi_i(k),\psi_x^{\tau}(x,t):= \psi_{ix}(k)\psi_t^{\tau}(x,t):= \psi_{it}(k), x_i\leq x<x_{i+1},t_{k-1}< t\leq t_{k},i =\overline{0,m},~k =\overline{0,n}.
\tau \cdot v_1^2 = (\tau \cdot v_1)^2 = (t_1 v_1)^2 = t_1^2 v_1^2, \tau \cdot v_1 v_2 = (\tau \cdot v_1) (\tau \cdot v_2) = (t_1 v_1) (t_2 v_2) = t_1 t_2 v_1 v_2, \tau \cdot v_2^2 = (\tau \cdot v_2)^2 = (t_2 v_2)^2 = t_2^2 v_2^2.
\lim_{n \to \infty}n^{d(r,s)-3}\Big(\{nu\}E(S_{\lfloor nt \rfloor}(r)\overline{X_{\lfloor nu \rfloor+1}(s)})+\{nt\}E(S_{\lfloor nu \rfloor}(s)\overline{X_{\lfloor nt \rfloor+1}(r)})+ &\{nu\}\{nt\}E(X_{\lfloor nu \rfloor+1}(r)\overline{X_{\lfloor nt \rfloor+1}(s)})\Big)=0.
\left( \frac{1 + i x}{1 - i x}\right)^{1 / 3}= \sum_{e \geq 0}A_{e}x^{e}+ i \sum_{o > 0}A_{o}x^{o}\, , ~ \left( \frac{1 + i x}{1 - i x}\right)^{2 / 3}= \sum_{e \geq 0}B_{e}x^{e}+ i \sum_{o > 0}B_{o}x^{o}\, .
\Sigma_{( f )}= M^{2}\{{\frac{\hat{\lambda}}{2 \epsilon}}+{\hat{\lambda}}^{2}({\frac{1}{2 \epsilon^{2}}}-{\frac{1}{4 \epsilon}}) \}+{\frac{{\hat{\lambda}}^{2}}{2 4 \epsilon}}p^{2}+{\frac{{\hat{\lambda}}^{2}}{2 \epsilon}}M^{2}{\frac{\partial}{\partial{M^{2}}}}G +{\frac{3 \hat{\lambda}}{2 \epsilon}}G .
V^{i^{\prime}}= \frac{n^{\prime}}{n^{\prime}+ \frac{n^{\prime}}{( e{\cal P}^{\prime})}( e Q )}\left[ 1 + \frac{n^{\prime}}{( e{\cal P}^{\prime})}( e Q ) \right] \left[ Q^{i^{\prime}}-{\cal P}_{L}^{i^{\prime}}\frac{( e Q )}{( e{\cal P}^{\prime})}\right]
\Phi_u(X) = (X-e^{2i\pi /u})(X-e^{2i\pi (u-1)/u})(X-e^{2i\pi a/u})(X-e^{2i\pi (u-a)/u}) &(X-e^{2i\pi b/u})(X-e^{2i\pi (u-b)/u})(X-e^{2i\pi c/u})(X-e^{2i\pi (u-c)/u})
\Phi_t(m\rq_{1,t}(x,y))=m_{1,t}(\Phi_t(x),\Phi_t(y)), & \Phi_t(m\rq_{2,t}(x,y))=m_{2,t}(\Phi_t(x),\Phi_t(y)),\,\,\,~ x,y\in \mathfrak{g}.
g(x_{1},x_{2},x_{3})=\sum_{m_{1}=-\infty}^{\infty}\sum_{m_{2}=-\infty}^{\infty}h^{two}(m_{1},m_{2},x_{3})\cdot e^{i2\pi\frac{m_{1}}{a_{1}}x_{1}}\cdot e^{i2\pi\frac{m_{2}}{a_{2}}x_{2}}
f_{n}=(-1)^{n}\alpha^{-n}q^{\frac{1}{2}n(n+1)}\left[1+o(1)\right]\!, g_{n}=(-1)^{n}\alpha^{n}q^{-\frac{1}{2}n(n-1)}\left[\theta_{q}\!\left(z^{-1}\alpha^{-1}\right)+o(1)\right]\!,
\tilde j_{n,mn}(\gamma^n_z(s_{n,(e,\mu)}) = \tilde j_{n,mn}(z s_{n,(e,\mu)}) = \sum_{\tau \in E^{<mn}, [\tau]_n=\mu}z s_{mn,(e,\tau)}= \gamma^{mn}_z (\tilde j_{n,mn}(s_{n,(e,\mu)})),
\nu [ A ] = \frac{e}{4 \pi \hbar}\int d^{2}x \epsilon_{\mu \nu}F^{\mu \nu}= \frac{e}{2 \pi \hbar}\int_{- \frac{1}{2}T}^{\frac{1}{2}T}d t \int_{- \frac{1}{2}{R}}^{\frac{1}{2}{R}}d x ( \dot{A}_{1}- \partial A_{0})
\inf_{\Vert \tilde{D}\Sigma^{1/2}\Vert_F=1}\: Q_{2\xi}(\tilde{D}) \ge 1-\sup_{\Vert \tilde{D}\Vert_F\ge \sigma_{\max}\left(\Sigma^{1/2}\right)^{-1}}\: \frac{4 \xi}{\sqrt{\pi}}\left(\frac{e}{2}\right)^{\frac{1}4}\frac1{t}\sum_{s=0}^{t-1}\frac{1}{\sqrt{\sum_{r=0}^{t-1}\tilde{D}_{s,r}^2}}.
e_{i,j}=\begin{cases}e_t ~~{\rm if}~~i<2t-1,~~i ~~{\rm is ~ odd}~~{\rm and}~~j=2t-1, e_t ~~{\rm if}~~i<2t,~~i ~~{\rm is ~ even}~~{\rm and}~~j=2t, e_t ~~{\rm if}~~i=2t-1, ~ j>2t-1~{\rm and}~~j ~~{\rm is ~ even}, e_t ~~{\rm if}~~i=2t,~ j>2t~{\rm and}~~j ~~{\rm is ~ odd}. \end{cases}
\prod_{i=1}^3\left( \frac{[H^0_{et}(U,\mathcal{F}_{i})_{tor}][Ext^1_{U}(\mathcal{F}_i,\mathbb{G}_m)]}{[Hom_{U}(\mathcal{F}_i,\mathbb{G}_m)_{tor}][H^0_{et}(U,\mathcal{F}_{i,B})_{tor}]}\right)^{(-1)^{i+1}}= \frac{\nu(\mathcal{H}^0)_{\mathbb{R}}[Q_1][R_2]}{\nu(\mathcal{H}om)_{\mathbb{R}}\nu(\mathcal{H}_S)_{\mathbb{R}}[P_1]}.
h_1(0;y) =(N_1+\nu_1N_1^{2/3}+yN^{1/3})\log(1-d_2N_1^{-1/3}) &+\frac 12(N_1-\nu_1N_1^{2/3})(1-d_2N_1^{-1/3})^2-(2N_1+\lambda_1N^{1/3})(1-d_2N_1^{-1/3}).
D(y)= \sum_{i,j=1}^n\Big(\nabla \Phi(\Psi(y))\,\nabla\omega(\Psi(y))\,\nabla \Psi(y)\Big)_{ij}^2,F(y)=\sum_{i,j=1}^n\left(\sum_{k=1}^nS_{ij}^k(\Phi,y)\omega_k(\Psi(y))\right)^2 \smallskip E(y)= 2\sum_{i,j,k=1}^nS_{ij}^k(\Phi,y)\omega_k(\Psi(y))\left(\nabla \Phi(\Psi(y))\,\nabla\omega(\Psi(y))\,\nabla \Psi(y)\right)_{ij}
(\Lambda_{L,{I_{ij}}}, \Lambda_{R,{I_{ij}}},\Lambda_{D,{I_{ij}}}, \Lambda_{U,{I_{ij}}}) =(\min(B^{1,1,1,0}_1, B^{1,1,0,1}_1,B^{1,0,1,1}_1), \min(B^{1,1,1,0}_2, B^{1,1,0,1}_2, B^{0,1,1,1}_2),\min(B^{1,1,1,0}_3, B^{1,0,1,1}_3, B^{0,1,1,1}_3), \min(B^{1,1,0,1}_4, B^{1,0,1,1}_4,B^{0,1,1,1}_4)).
\frac{1}{2\pi i}\oint_{C^{\mathfrak{p}}_{g,n+2}}\phi(p)\cdot d_p\left(\frac{\eta^p(p_0)}{2\omega_{0,1}(p)}\omega_{g,n+1}(p,J)\right)=-\frac12\omega_{g,n+1}(p_0,J),
T =\frac{\partial}{\partial t},\ \Omega_1=\frac{\partial}{\partial \phi}, \Omega_2 =\cos\phi\frac{\partial}{\partial \theta}-\cot\theta\sin\phi\frac{\partial}{\partial \phi}, \Omega_3 =\sin\phi\frac{\partial}{\partial \theta}+\cot\theta\cos\phi\frac{\partial}{\partial \phi}.
u^{(n+1)}(t, x) =\int_{\mathbb R}{p}(t, x-y)u_0(y)\,dy &\quad{}+\int_0^t \,ds \int_{\mathbb R}{p}(t-s,x-y)f\bigl(s, y, u^{(n)}(s, y) \bigr)\,dy &\quad{}+\int_{(0,t]}\,d\mu(s) \int_{\mathbb R}{p}(t-s,x-y)\sigma(s, y)\,dy .
F_n={}_{0}F_{2}\!\left(\left.\begin{matrix}- \lambda+n,1-\mu-n\end{matrix}\right|x\right): &-F_n'=\frac{(n+\mu)}{(n+\lambda)(2n+\lambda+\mu)(2n+\lambda+\mu-1)}F_{n+1}+{\frac{2}{(2n+\lambda+\mu)(2n+\lambda+\mu-2)}}F_n &+\frac{(n+\lambda-1)}{(n+\mu-1)(2n+\lambda+\mu-1)(2n+\lambda+\mu-2)}F_{n-1};
\begin{array}{lll}D^r \, \phi^{(n,0)}= \nabla_{2^{-1}}^r \, \phi^{(n-r,0)}, r\le n-1\,, D^r \, \phi^{(n,m)}= \nabla_{2^{-m}}^r \, \phi^{(n-r,m)}, r\le n-1\,, m>0\,.\end{array}
V_{(1,2)}+ 1 - \bigoplus_{i=1}^{p-1}\lambda(i) - \lambda(1) = (n-2)\rho_G -\bigoplus_{i=1}^{\ell}\lambda(i) - \bigoplus_{i=1}^{p-1}\lambda(i) - \lambda(1) & = (p^k-2)\rho_G - \bigoplus_{i=1}^{\ell+p-1}\lambda(i) - \lambda(1)
\phi ( x ) \sim{\frac{\alpha_{0}}{B}}\int_{\partial M}d^{d}x^{\prime}\left( \frac{z}{z^{2}+ | \vec{x}- \vec{x}^{\prime}|^{2}}\right)^{\Delta}\int_{0}^{\epsilon}d x_{0}^{\prime}\,{z}^{\Delta - d - 1}J ( x^{\prime}) .
\lim\limits_{r\rightarrow 0}G_{i}(r)\sim \frac{q_{i}}{\pi r}\frac{1}{1+\exp\varepsilon_{i}(0,r,0)}\left. \frac{d\varepsilon_{i}(0,r,\mu_{i}/2)}{d\mu_{i}}\right\vert_{\mu_{i}=0}\,.
T r \, e^{- i s H}={\frac{T \, 2 a \ell^{2}}{4 \pi^{2}i}}{\frac{e^{- i s m^{2}}}{s^{2}}}[ 1 + 2 \sum_{n = 1}^{\infty}( - 1 )^{n}e^{i ( a n )^{2}/ s}] [ 1 + i s e B \, L ( i s e B ) ] \, ,
\left. \exp \left\{- \frac{i}{\hbar}\sum_{j = 1}^{N}\epsilon \, H_{\Delta}\left( \frac{\hbar}{i \epsilon}\frac{\partial}{\partial K_{j}}\, , \frac{\hbar}{i \epsilon}\frac{\partial}{\partial J_{j}}\right) \right\}W_{f i}[ K , J ] \right|_{K , J = 0}\, .
B_n^{\overbrace{\scriptstyle{(0,\ldots,0}}^{r-1},-1)}=\displaystyle \sum_{l=1}^{1}{\alpha}_l^{\overbrace{\scriptstyle{(0,\ldots,0}}^{r-1},1)}(l+r)^n &={\alpha}_1^{\overbrace{\scriptstyle{(0,\ldots,0}}^{r-1},1)}(1+r)^n.
K_{m,n}(k) = \left(\prod_{i=0}^{k-1}C_{m,n}(i)\right)K_{m,n}(0) = \left(\prod_{i=0}^{k-1}C_{m,n}(i)\right) \left(\begin{array}{c}K_{m,n}^{(1)}(0) K_{m,n}^{(2)}(0)\end{array}\right),
\begin{cases}-\operatorname{div}(y^a\varphi(x)\nabla_{x,y}w)=0, \hbox{in}~\mathcal{C}_{\Omega}, w=0, \hbox{on}~\partial_{L}\mathcal{C}_{\Omega}, -\underset{y\rightarrow0^{+}}{\lim}y^aw_y=f, \hbox{on}~\Omega.\end{cases}
8 a^2 \phi_x + \left(a_x - 2 b\right) \left(\tau a_t+\xi a_x-a \dot{\tau}\right) - a [2 \xi a_{xx}+2 \tau(a_{xt}-2 b_t) -4 \xi_t-4 \xi b_x ]=0, &c \tau_t+\tau c_t-\phi_t+\xi c_x+b \phi_x+a \phi_{xx}=0,
\frac{d}{dt}\int (\partial_x^ku)^2\psi \; dx + 3 \int (\partial_x^{k+1}u)^2\psi' \; dx &\qquad= \int (\partial_x^ku)^2\psi''' \; dx+ \int \partial_x(\psi u)(\partial_x^ku)^2 \; dx+ \int \partial_x^ku [\partial_x^k;u]\partial_xu\psi \; dx.
\left\Vert Bq^{m}\frac{a_{k}}{b_{k}}\alpha^{k}\right\Vert=\left\Vert d_{k}a_{k}q^{m}\alpha^{k}\right\Vert \leq \vert a_{k}\vert d_{k}q^{m-k}\left\Vert q^{k}\alpha^{k}\right\Vert\leq ABq^{m-k}\left\Vert q^{k}\alpha^{k}\right\Vert.
h_p^{t+1}= A_p^* m_p^t - q_p^t,q_p^t = f_t(h_p^t,x_p) &b_p^t = A_p q_p^t-\lambda_p^t m_p^{t-1},m_p^t = b_p^t +\sum_{u\neq p}b_u^{\theta(t)}-w
Q_{1,\alpha}=~~\mu_\alpha^2 \left(\frac{n-2}{4(n-1)}S_g(\bar{x}_0) \int_{\mathbb{R}^n}u_0^2 dx+\tilde{K}\int_{B_0(\sqrt{n(n-2)})}\Psi_0(x) dx \right) ~~+o(\mu_\alpha^2)+o(\mu_\alpha^{n-2}r_\alpha^{2-n}) \geq \eta \mu_\alpha^2 +o(\mu_\alpha^2)+o(\mu_\alpha^{n-2}r_\alpha^{2-n})
r \phi * (a+b) = r \phi * r \phi_d^{-1}* r\phi_d * (a+b) = r \phi * r \phi_d^{-1}*(r\phi_d * a + r\phi_d*b) = r \phi * r \phi_d^{-1}*(r\phi_d * r*a + r\phi_d*r*b) = r \phi * r \phi_d^{-1}*r\phi_d * (r*a + r*b) = r \phi * (r*a + r*b)
d s^{2}= \left[ H ( r ) \right]^{- \frac{\widetilde{d}}{D - 2}}\, d x^{\mu}\otimes d x^{\nu}\, \eta_{\mu \nu}+ \left[ H ( r ) \right]^{\frac{d}{D - 2}}\, \left( d r^{2}+ r^{2}d s_{S^{D - p - 2}}^{2}\right)
|\hat{\mathcal{H}}_d -H g_0^*|_{S_2}=|H\hat{g}-H g_0^* - \hat{\mathcal{H}}_{\bar{d}}|_{S_2}\leq |H\Delta g|_{S_2}+ |\hat{\mathcal{H}}_{\bar{d}}|_{S_2}\leq 2|H\Delta g^*|_{S_2}\leq \frac{10\sqrt{2}}{3\sigma_u^2}d_0^{1/2}\lambda T,
\rho^q\rho_q(\rho_i\Psi_{jk}-\rho_j\Psi_{ik}) =2\rho_k\rho^q(\rho_i\Psi_{jq}-\rho_j\Psi_{iq}), \rho^q\rho_q\rho^k\Psi_{ik}=2\rho_i(\rho^j\rho^k\Psi_{jk})
\int_0^\infty xP_{n+k}(x) Q_{n}(x) dx = \frac{1}{2\pi i}\oint_{\Sigma_n}\prod_{l=0}^m (z+\eta_l+1) \frac{\Gamma(z-n-k)}{\Gamma(z+2-n)}dz \alpha_{n+k,-k}= \frac{1}{2\pi i}\oint_{\Sigma_n}\frac{\prod_{l=0}^m (z+\eta_l+1)}{(z-n+1)\cdots(z-n-k)}dz .
R_{\mu \nu \lambda}{}^{\rho}=\widetilde{R}_{\mu \nu \lambda}{}^{\rho}+R_{\mu \nu \lambda}^{{\footnotesize (T)}}{}^{\rho}+2(T_{[\mu \mid \tau\mid}{}^{\rho}\widetilde{\Gamma}_{\nu ]\lambda}{}^{\tau}+\widetilde{\Gamma}_{[\mu \mid \tau \mid}{}^{\rho}T_{\nu ]\lambda}{}^{\tau}),
\frac{\partial}{\partial \xi}F = S ( \frac{\partial}{\partial \xi}f ) S^{- 1}+ \frac{\bar{\xi}- \xi}{2}\; \frac{\partial S}{\partial \xi}S^{- 1}, \quad \frac{\partial}{\partial \bar{\xi}}F = S ( \frac{\partial}{\partial \bar{\xi}}f ) S^{- 1}- \frac{\bar{\xi}- \xi}{2}\; \frac{\partial S}{\partial \bar{\xi}}S^{- 1}.
= 2 ( N_{c}^{2}- 1 ) V \int \! \frac{d^{3}k}{( 2 \pi )^{3}}\frac{| k |}{2}+ \begin{array}{c}{{\mathrm{p e r t u r b a t i v e}}}\\{{\mathrm{c o r r e c t i o n s}}}\\ \end{array}
y_{2i-1,2n'_0 -1}= n(i-1)+n'_0 \mbox{\for all}1 \leqslant i \leqslant m^{+}_0, \mbox{and}y_{2i,2n'_0 -1}= MN - ni + n'_0 \mbox{\for all}1 \leqslant i \leqslant m_0.
z = \frac{(a_1 + a_2 + a_3 + a_4) (a_1 a_2 a_3 a_4 + 1) + 2 (a_1 a_2 a_3 + a_1 a_2 a_4 + a_1 a_3 a_4 + a_2 a_3 a_4) \pm 2 s}{(a_1 a_2 a_3 a_4 - 1)^2}\,,
\rho'_i(t)=\xi_{b_{i+1}}'(t)-\xi_{b_i}'(t)=\frac{u_t(\xi_{b_i}(t),t)}{u_r(\xi_{b_i}(t),t)}-\frac{u_t(\xi_{b_{i+1}}(t),t)}{u_r(\xi_{b_{i+1}}(t),t)},
\ |\Sigma_1|+|\Sigma_3|\leq\sum_{k\leq k_0-2 \atop k\ge k_0+3}\, |A_k|+\sum_{k\leq k_0-2 \atop k\ge k_0+2}\, |D_k|\leq \ \leq C\omega(2^{-n})\sum_{\nu =1}^{\infty}\frac{1}{\nu^{2}}\leq C\omega(2^{-n}).
(u,v)_s :=\left(1+|x|^2)^{\frac{s}{2}}\widehat{u}(x),(1+|x|^2)^{\frac{s}{2}}\widehat{v}(x)\right)_{L^2(\mathbf{R}^n,\mathbf{C}^N)}\|u\|_s :=\sqrt{(u,u)_s}.
p^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{1}}\frac{N_{n_{s}}}{N_{n_{1}}}+p^{n_{2}^{\prime}-n_{s}^{\prime}}q^{n_{s}^{\prime}-n_{1}^{\prime}}a_{n_{s}}=q^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{2}}\frac{N_{n_{s}}}{N_{n_{2}}}.
\tilde u =h_{\beta}(z,w)u+g_{\beta}(z,w)m \ , \cr\tilde v =g_{\alpha}(u,v)p+h_{\alpha}(u,v)z \ , \cr\tilde c =(h_\beta (z,w)-g_\beta(z,w)l_\alpha(u,v) m) c +(h_\alpha(u,v)-g_\alpha(u,v) l_\beta(z,w) p)d+g_\alpha(u,v)g_\beta(z,w) e \ . \cr
\min_{x\in \mathbb{R}^2}\max_{y\in \mathbb{R}^2}F(x, y)=\tfrac{1}{2}x^T\begin{pmatrix}L 0 0 0\end{pmatrix}x+x^T\begin{pmatrix}0 1 1 0 \end{pmatrix}y-\tfrac{1}{2}y^T\begin{pmatrix}L 0 0 \mu_y\end{pmatrix}y.
\sum_{K_1 \sim K_2}\Big| \int (f_{N_1,K_1,L_1}* g_{N_2,K_2,L_2}) h_{N_3,\lesssim K_1,L_3}\Big| \lesssim \sum_{K_1 \sim K_2 \gtrsim N_1}C(N_1,N_2,N_3) \| f_{N_1,K_1,L_1}\|_{L^2}\| g_{N_2,K_2,L_2}\|_{L^2}\| h_{N_3,\lesssim K_1,L_3}\|_{L^2}
1_{\bigcup_{k}A_{k}}=1-\sum_{F\subseteq\{1,2,...,n\}}(-1)^{|F|}1_{\bigcap_{F}A_{k}}=\sum_{\emptyset\ne F\subseteq\{1,2,...,n\}}(-1)^{|F|+1}1_{\bigcap_{F}A_{k}}
\xi_{n,m,\boldsymbol{K}}(s) = \rho_{n,m,\boldsymbol{K}}(s)\cdot \xi(2m-2s), \rho_{n,m,\boldsymbol{K}}(s)= \frac{\prod_{\substack{0\leq j\leq n-1 j\ne n-m}}\xi(n+m-2s-j;\chi_{\boldsymbol{K}}^j)}{\prod_{j=0}^{n-1}\xi(2s+n-m-j;\chi_{\boldsymbol{K}}^j)}.
J_{22}=\frac{4}{mn\pi^{2}}\int_{h_{1}}^{\pi}\int_{0}^{h_{2}}H_{x,z_{1},y,z_{2}}(t_{1},t_{2})\frac{S(t_{1},t_{2})}{t_{1}^{2}\left( 2\sin \frac{t_{2}}{2}\right)^{2}}dt_{1}dt_{2}
\left[\begin{array}{c}p_{n-1}q_{n-1}\end{array}\right] =\left[\begin{array}{cc}0 1 1 a_1\end{array}\right]\left[\begin{array}{cc}0 1 1 a_2\end{array}\right]\cdots\left[\begin{array}{cc}0 1 1 a_{n-1}\end{array}\right]\left[\begin{array}{c}0 1\end{array}\right]
g_{1}\in \Gamma \vert{i}{I}{J}\ \ \mathrm{, ~}g_{2}\in \Gamma \vert{j}{J}{K}\ \ \mathrm{, ~}\ \g_{3}\in \Gamma \vert{k}{K}{L}\ \ \mathrm{, ~}g_{4}\in \Gamma \vert{l}{L}{I}\ \
k_{\mathbb{A}}(w,\xi;w'',\xi'';\eta) & =\langle \eta\rangle \int d\tilde{w}\; \exp(i(\xi (\Lambda^{-1}(\tilde{w}+w''))-\xi'' \tilde{w}-\eta \varpi \cdot \Lambda^{-1}(\tilde{w}+w'')-\eta\cdot \sigma(\tilde{w}+w''))) &\qquad\qquad\qquad\cdot \exp(-\langle \eta\rangle \|\Lambda^{-1}(\tilde{w}+w'')-w\|^2/2-\langle \eta\rangle \|\tilde{w}\|^2/2).
\partial_{y}\Big [ \big ( \partial_{y}+ 2 \, \ell^{- 1}\big ) \, h + \frac{( 1 - \gamma )^{- 1}\, \ell}{3 \, M_{*}^{3}}\, e^{- 2 y / \ell}\, T_{5 5}( y ) \Big ] = \frac{( 1 - \gamma )^{- 1}\, \ell}{3 \, M_{*}^{3}}\partial_{\mu}T^{\mu 5}\, .
\left|\zeta_{v,t}^{\mathcal{M}}(x,a)- \zeta_{v,t}^{\mathcal{M}}(y,b) \right| = \left|\int_{-\infty}^{\infty}[v(x + A_{\cdot, t+1}w)-v(y + A_{\cdot, t+1}w)] \phi(w)dw \right| & \le L_{v}\int_{-\infty}^{\infty}\left|(x + A_{\cdot,t+1}w)-(y + A_{\cdot,t+1}w) \right|\phi(w)dw & = L_{v}|x-y|
\int\limits_{-\infty}^{-\epsilon}\frac{\cos(xt)}{t}h\left(\frac{t}{2\pi}\right)\,\mathrm{d}t =-\int\limits_{\epsilon}^{\infty}\frac{\cos(xt)}{t}h\left(\frac{t}{2\pi}\right)\,\mathrm{d}t .
{\cal L}_{ct}=\gamma \left[ \left( -\frac{d-1}{\ell}+\frac{\Theta \left(d-3\right) d\left( d-1\right)}{2\ell (d-2)\cosh^{2}\left( \tau /\ell\right)}\right) +\frac{\Theta \left( d-4\right) d(d-1)}{8\ell (d-4)\cosh^{4}\left( \tau /\ell \right)}+\frac{\Theta \left( d-5\right) \left(d-1\right) d}{16\ell (d-6)\cosh^{6}\left( \tau /\ell \right)}\right]
{\Gamma}_{{t_j}}\ast g_{t_{n+j}}(x) = 2^{{d}{t_n}/p}2^{{d}{t_j}/p}\int_{|y|\leq 2^{-M}}{{\Gamma}(2^{{t_j}}x-y)g(2^{{t_n}}y)}dy &\geq 2^{{d}t_n/{p}}2^{{d}t_j/{p}}2^{-{t_n}d}\Vert g \Vert_{L^1}
F^{-1}\mathop{=}\limits_{\zeta\rightarrow 0}\zeta^{n+1}(F_{-1}+F_{0}\zeta+O(\zeta^{2}))^{-1}\mathop{=}\limits_{\zeta\rightarrow 0}\zeta^{n+1}(u^{-1}-u^{-2}F_{0}\zeta+O(\zeta^{2})),~~as~~P\rightarrow P_{\infty\pm},
\int_M|\nabla \Delta u_k|_g^2 d\mu_g= \int_M |\nabla \Delta u|_g^2 d\mu_g+\int_M |\nabla \Delta (u-u_k)|_g^2 d\mu_g+o(1), \int_M f|u_k|^{2^\sharp}d\mu_g= \int_M f|u|^{2^\sharp}d\mu_g+\int_M f|u-u_k|^{2^\sharp}d\mu_g+o(1),
[\mspace{1.5mu}u\mspace{1.5mu}]^2_{\frac{1}{2}}\!=2\pi^2\mspace{-4.5mu}\int_{\mathbb{R}^3}\mspace{-7.5mu}d\vec{k}\:\lvert\vec{k}\rvert\mspace{1.5mu}\lvert\hat{u}(\vec{k})\rvert^2.
\psi(dydy)(z)={\frac{5}{12}}\,{z}^{-2}+\frac{1}{6}\,{z}^{-1}-\frac{1}{36}+{\frac{1}{216}}\,{z}^{2}-{\frac{1}{120}}\,{z}^{3}-{\frac{19}{9072}}\,{z}^{4}+O\!\left({z}^{5}\right),
g_{ab}^j(f_{jk}^1(z_k,t),...,f_{jk}^n(z_k,t),t_1,...,t_m)=\sum_{r,s=1}^n g_{rs}^k \frac{\partial f_{jk}^a}{\partial z_k^r}\frac{\partial f_{jk}^b}{\partial z_k^s}\,\,\,\,\,\,\,\,\,\mathcal{U}_j \cap \mathcal{U}_k
\Psi ( x ) \, = \, \sum_{\sigma = + j}^{- j}\int{\frac{d^{3}p}{( 2 \pi )^{3}}}{\frac{1}{2 \, \omega_{\vec{p}}}}\Big [ u_{\sigma}( \vec{p}\, ) \, a_{\sigma}( \vec{p}\, ) \, e^{- i p \cdot x}+ v_{\sigma}( \vec{p}\, ) \, b_{\sigma}^{\dagger}( \vec{p}\, ) \, e^{i p \cdot x}\Bigr ] \quad ,
H_{T}=\frac{1}{2}(P_{1})^{2}+\frac{1}{2}(Q^{1})^{2}+\Psi_{1}P_{1}+\zeta^{1}\Psi_{1}+f(\Xi^{1},\Psi_{2},\Theta^{1})+g(\Psi_{1},\Psi_{2},\Theta^{1},\Theta_{1}),
\begin{cases}\mathcal{L}({\mathbf V},\Psi):=\mathbb{L}({\mathbf U}^a+{\mathbf V},\Psi^a+\Psi)-\mathbb{L}({\mathbf U}^a,\Psi^a)=\mathcal{F}^a, \Omega_T, \mathcal{B}({\mathbf V}|_{x_1=0},\psi):=\mathbb{B}({\mathbf U}^a|_{x_1=0}+{\mathbf V}|_{x_1=0},\varphi^a+\psi)=0, \Gamma_T, ({\mathbf V},\psi)=0, t<0.\end{cases}
\frac{1+\frac{1}{2}z+\frac{3}{28}z^{2}+\frac{1}{84}z^{3}+\frac{1}{1680}z^{4}}{1-\frac{1}{2}z+\frac{3}{28}z^{2}-\frac{1}{84}z^{3}+\frac{1}{1680}z^{4}}
A_{\pm}A_{\mp}\psi_{n_{\pm}, m_{\pm}}T_{n_{\pm}, m_{\pm}}= \left( \begin{array}{c}{\scriptstyle n_{+}}\\{\scriptstyle{n_{-}+ 1}}\\ \end{array}\right) \psi_{n_{\pm}, m_{\pm}}T_{\kappa_{\pm}; n_{\pm}, m_{\pm}}.
h(XY+UV)+YV+h^2XU-(1-h\lambda)XV=0, & \alpha (XY+UV)+XV+\alpha^2YU-(1-\alpha\lambda)XU=0, & (h-\alpha)(XY+UV)+(1-\alpha\lambda)YU-(1-h\lambda)YV & \qquad+h^2(1-\alpha\lambda)XV-\alpha^2(1-h\lambda)XU=0.
\begin{aligned}C(x,\rho)= \cos \rho (x-a)+\frac{1}{\rho}\left(q_{1}(x)-q_{1}(a)\cos \rho(x-a)\right) &-\frac{1}{\rho}\int\limits_{a}^{x}q_{1}^{\prime}(t)\cos \rho (x-t)dt+\frac{1}{\rho}\int\limits_{a}^{x}q_{0}(t)\sin \rho (x-t)dt.\end{aligned}
\psi_2(\bar z)+h_2^*\psi_1(\bar z) =\sum_{i=0}^{q-1}(-1)^i\varphi(h_0(z_0),\ldots,h_0(z_i),h_2(z_i),\ldots,h_2(z_{q-1})) +\sum_{i=0}^{q-1}(-1)^i\varphi(h_2(z_0),\ldots,h_2(z_i),h_3(z_i),\ldots,h_3(z_{q-1})) =\sum_{i=0}^{q-1}(-1)^i\varphi(h_0(z_0),\ldots,h_0(z_i),h_3(z_i),\ldots,h_3(z_{q-1}))+d_{q-2}\chi_2(\bar z) =\psi_3(\bar z)+d_{q-2}\chi_2(\bar z).
V(\mathbf{j}''') =3j'''_{n-2}-s(j'''_{n-2})+2j'''_{n-1}-s(j'''_{n-1})-c(j'''_n,-\alpha_{\mathbf{j}'''}(n)-1) & = 3(0)-s(0)+2(16q+6)-s(16q+6)-c(5,8q) & = 32q+12-s(q)-s(6)-c(5,8q) & = 32q+10-s(q).
{}_{{}_x}\mathfrak{J}^L = \mathfrak{J}_L L_{{}_{h_{{}_1}(x)}}\mathfrak{J}_L L_{{}_{{h_{{}_1}(x)}^{-1}}}\,\,\, \textrm{and}\,\,\,{}_{{}_y}\mathfrak{J}^M = \mathfrak{J}_M M_{{}_{h_{{}_2}(y)}}\mathfrak{J}_M M_{{}_{{h_{{}_2}(y)}^{-1}}}
g_{aa}=-\frac{1}{4}\frac{\Lambda'(\lambda_a)}{A(\lambda_a)}=-\frac{1}{4}\frac{\displaystyle \prod_{b\neq a,b=1}^N(\lambda_a-\lambda_b)}{\displaystyle \prod_{b=1}^N(\lambda_a-r_b)},\g_{ab}=0, \forall a\neq b
D_0^{(e)}=1,\mbox{}D_1^{(e)}=0,\mbox{}D_0^{(o)}=0,\mbox{}D_1^{(o)}=0, D_n^{(e)}=(n-1)(D_{n-2}^{(o)}+D_{n-1}^{(o)}),\mbox{}D_n^{(o)}=(n-1)(D_{n-2}^{(e)}+D_{n-1}^{(e)}), n\geq 2.
\psi ( x , t ) = \left( \begin{array}{c}{\psi_{1}( x , t )}\\{\psi_{2}( x , t )}\\{\psi_{3}( x , t )}\\ \end{array}\right) = \left( \begin{array}{c}{f_{1}( t ) \phi_{1}( x )}\\{f_{2}( t ) \phi_{2}( x )}\\{f_{3}( t ) \phi_{3}( x )}\\ \end{array}\right)