harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
< \Phi^{n}( t_{2}) \left[ \Phi^{*}( t_{1}) \right]^{n}> ={\frac{1}{Z}}\int [ d \phi^{*}d \phi ] \Phi^{n}( t_{2}) \left[ \Phi^{*}( t_{1}) \right]^{n}e^{- S}\ .
\lambda \omega^p_{\mathbb{B}^n}(Q_s) \geq F_1^*(\omega^p_{\mathbb{B}^{N_1}})(Q_s) \geq F_1^*\ \left(\frac{\sum_{k=1}^{N_1}d w_k \wedge d \overline{w_k}}{1-|w|^2}\right)^p (Q_s) &= \frac{1}{(1-|F_1(Q_s)|^2)^{p}}\sum_{k_1 < \cdots < k_p}C_{k_1, \cdots, k_p}\Theta_{k_1, \cdots, k_p}(Q_s),
\frak R_{j,k}(x_1, x_2, x_3,v_1,v_2,v_3) &:=\sum_{R=I\times J\times S\in \mathcal R^N_{\frak z}(j,k)}\int_{R}\psi_{j,k}(x_1-y_1, x_2-y_2, x_3-y_3) &\times(\psi_{j,k}(y_1-v_1, y_2-v_2, y_3-v_3) -\psi_{j,k}(x_I-v_1, x_J-v_2, x_S-v_3)) dy_1dy_2dy_3,
\Delta = \alpha_1 \cdots \alpha_{\mu-1}\beta_{\nu}\cdots \beta_1 = \frac{\alpha_1 \cdots \alpha_{\mu-1}\beta_1 \cdots \beta_{t-2}\Gamma_{h-1}\cdots \Gamma_0}{\beta_{t-2}^{1+2+\cdots + (h-1)}}.
D_{1}= \begin{pmatrix}-kc_{k}* * 0 \langle\Psi_{2}^{0}, \mathcal{A}_{1}[\phi]\Phi_{2}^{0}\rangle \langle\Psi_{2}^{0}, \mathcal{A}_{1}[\phi]\Phi_{3}^{0}\rangle 0 \langle\Psi_{3}^{0}, \mathcal{A}_{1}[\phi]\Phi_{2}^{0}\rangle \langle\Psi_{3}^{0}, \mathcal{A}_{1}[\phi]\Phi_{3}^{0}\rangle\end{pmatrix},
f_{\overline{Y}}(y) = \sqrt{\frac{n}{2\pi \kappa''\left(\hat{\gamma}\right)}}e^{n\left[ \kappa\left(\hat{\gamma}\right) - y \hat{\gamma}\right]}\left[ 1 + \frac{1}{n}\left( \frac{\hat{\rho}_4}{8}- \frac{5 \hat{\rho}_3^2}{24}\right) + O\left(n^{-2}\right) \right]
=c\sum_{i=1}^{q}\int_{B_{4\Lambda}}\left\vert X_{i}\widetilde{u}\left(x\right) -X_{i}\widetilde{u}_{B_{4\Lambda}}\right\vert dx & \leq c\sum_{i,j=1}^{q}\int_{B_{4\Lambda^{2}}}\left\vert X_{j}X_{i}\widetilde{u}\left( x\right) \right\vert dx & =c\sum_{i,j=1}^{q}\int_{B_{4\Lambda^{2}}}\left\vert X_{j}X_{i}u\left(x\right) \right\vert dx.
\sigma_{ac|Bz}=\sigma_{ac|B}-\frac{(\Sigma_{aB}\Sigma^{-1}_{BB}\Sigma_{Bz})(\Sigma_{cB}\Sigma^{-1}_{BB}\Sigma_{Bz})}{\sigma_{zz|B}}&=\sigma_{ac|B}-\frac{(\sigma_{ax}\sigma_{xz}Q_1)(\Sigma_{cB}\Sigma^{-1}_{BB}\Sigma_{Bx}\sigma_{xz})}{\sigma_{zz|B}}=\sigma_{ac|B}\left(1+\sigma^2_{zx}Q_1\sigma^{-1}_{zz|B}\right).
c_n :=2(4-n)\frac{\Gamma(\frac{n+1}{4})^2}{\Gamma (\frac{n-1}{4})^2}; c_{2,k}:=\begin{cases}c_2^{-1}k\in\mathfrak{T}_2^{-}, c_2k\in\mathfrak{T}_2^{+};\end{cases}c_{3,(l,m,s)}:=c_3^{2s}.
||\partial_2(\dot{H}^+_2\partial_1\hat{\Phi}^{+})||^2_{s-1,\ast,t}\leq||\dot{H}^+_2\partial_1\hat{\Phi}^{+}||^2_{s,\ast,t}\leq C(K)\Big(||{\mathbf V}||^2_{s,\ast,t}+||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s,\ast,t}\Big).
\frac{d}{dt}\int_0^L\Big(\frac{\rho u^2}{2}+\frac{\epsilon}{2}\chi_x^2+\Phi(\rho)+\frac{\rho(\chi^2-1)^2}{4\epsilon}\Big)dx+\int_0^L\Big(\mu^2+\nu u^2_x\Big)dxd\tau=0.
W^{g e n}\kappa \partial_{\kappa}\Gamma = [ W^{g e n}, \kappa \partial_{\kappa}] \Gamma + \kappa \partial_{\kappa}W^{g e n}\Gamma = - ( \kappa \partial_{\kappa}W^{g e n}) \Gamma + \kappa \partial_{\kappa}( \chi \Delta_{b r}+ \chi_{A}\Delta_{b r_{A}})
\eta_{+}^{A}= \epsilon_{\pi}( y ) e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{\hat{\eta}_{R}}\\{- \hat{\eta}_{L}}\\ \end{array}\right)^{A}\ , \qquad \eta_{-}^{A}= \epsilon_{0}( y ) e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{\hat{\eta}_{L}}\\{\hat{\eta}_{R}}\\ \end{array}\right)^{A}\ ,
\frac{1}{4}\xi_{m-1}^*+\frac{3}{2}\xi_m^*+\frac{1}{4}\xi_{m-1}^*+ &\left(\frac{1}{4}\xi_{m-1}+\frac{3}{2}\xi_m+\frac{1}{4}\xi_{m-1}\right) &\left(\frac{-1}{h}\xi_{m-1}+\frac{1}{h}\xi_{m+1}\right) =\nu\left(\frac{2}{h^2}\xi_{m-1}-\frac{4}{h^2}\xi_m+\frac{2}{h^2}xi_{m+1}\right)
\hat{g}_m(\zeta_*) = -\frac{1}{4}\log |q| \left(1-\frac{2}{m^2-1}\cdot \frac{m^2+m-2}{2}\right) +\frac{2}{m^2-1}\cdot \frac{m^2-2m+1}{16}\log|q| &=-\frac{1}{4}\log|q| \left(\frac{1-m}{m^2-1}\right) + \frac{1}{8}\log|q| \cdot \frac{m^2-2m+1}{m^2-1}& = \frac{1}{8}\log|q| \left(\frac{2m-2+m^2-2m+1}{m^2-1}\right) = \frac{1}{8}\log|q|\ .
(K_r\circ K_{\bar r})(x,\{x\})= \sum_{z\in X:\> \pi(x,z)=r}\frac{\omega_X(\{z\})}{\omega_D(\{r\})}\cdot \frac{\omega_X(\{x\})}{\omega_D(\{\bar r\})}\> \delta_{\bar r,\pi(z,x)}= \frac{\omega_X(\{x\})}{\omega_D(\{\bar r\})}.
\sum_{i=1}^{\ell}\int_{Q_T}c_i\omega(p) f'(u)\Big(\partial_n G_i(p,n) (\nabla p\cdot \nabla n-\partial_t n) +\partial_p G_i(p,n) pu-G_i(p,n)\Delta p \Big)-c_iG_i(p,n)\nabla p\cdot \nabla (\omega(p)f'(u))
\Lambda_{+ A Y^{\prime}}= u_{A}^{+}\Lambda_{+ Y^{\prime}}^{-}+ u_{A}^{-}\Lambda_{+ Y^{\prime}}^{+}\; , \Lambda_{+ Y^{\prime}}^{-}\equiv - u^{- A}\Lambda_{+ A Y^{\prime}}\; , \; \; \Lambda_{+ Y^{\prime}}^{+}\equiv u^{+ A}\Lambda_{+ A Y^{\prime}}\; .
\int_x^{x+1}\sup_{N\ge 1}\Big|\sum_{n=1}^N \alpha_n{\rm e}^{it \lambda_n\mu_k}\Big|^2dt = \frac1{\mu_k}\int_{\mu_k x}^{\mu_k(x+1)}\sup_{N\ge 1}\Big|\sum_{n=1}^N\alpha_n{\rm e}^{it \lambda_n}\Big|^2dt \le \frac{[\mu_k]+1}{\mu_k}\Big\| \sup_{N\ge 1}\big|\sum_{n=1}^N \alpha_n{\rm e}^{it \lambda_n}\big|\, \Big\|_{\S^2}^2\, ,
\left(\begin{array}{c}\varphi^{*}X_{1}\varphi^{*}X_{2}\end{array}\right) = \left( \begin{array}{c}X_{1}X_{2}\end{array}\right) + \left(\begin{array}{c}\sum_{j=1}^{n-2}g_{j}' Y_{j}\sum_{j=1}^{n-2}h_{j}' Y_{j}\end{array}\right)
\left|m_{N}^{N^\delta}(f)-T_{p}.m_{N}^{N^\delta}(f) \right| \leq \frac{1}{N^\delta}\left|\sum_{i=0}^{N^\delta-1}(T_{p}^{i}).m_{N}(f)-(T_{p}^{i+1}).m_{N}(f) \right| = \frac{1}{N^\delta}\left|m_{N}(f)-T_{p}^{N}.m_{N}(f) \right| \leq \frac{2\|f\|_{\infty}}{N^\delta}\xrightarrow{N\to\infty}0.
\big(y_{0}^{o}(t)\big)^{T}D_{2}y_{0}^{o}(t)=\big(P_{10}x_{0}^{o}(t)+h_{10}(t)\big)^{T}A_{2}D_{2}^{-1}A_{2}^{T}\big(P_{10}x_{0}^{o}(t)+h_{10}(t)\big).
\det(A_{V_H, S \smallsetminus \{i_k \},{R}}) y^H \Big(\prod_{i \notin S \smallsetminus \{i_k \}}y^{A_i}\Big) = -\sum_{\substack{H \subset G |G| = s}}\det(A_{V_{H,G}, S,{R}\cup \{i_k\}}) y^G \Big(\prod_{i \notin S}y^{A_i}\Big).
\begin{aligned}||p-\Pi_hp||_K\leq Ch^s||p||_{s,K},0\leq s\leq k+1, &||\nabla(p-\Pi_hp)||_K\leq Ch^s||p||_{s+1,K},0\leq s\leq k, &|p-\Pi_hp|_e+|p-\Pi_ep|_e\leq Ch^{s+\frac{1}{2}}||p||_{s+1,K},0\leq s\leq k,\end{aligned}
y_{g_2,2}(\lambda,x_{g_2,2})=\frac{\Delta_{B_{g_2-1,2}}^D(\lambda)}{\Delta_{2}(\lambda)}c_{g_2,2}(\lambda,x_{g_2,2})+\sum\limits_{k=g-d(w)+1}^{g-1}\frac{\Delta_{A_{k,2}}^D(\lambda)}{\Delta_{2}(\lambda)}s_{g_2,2}(\lambda,x_{g_2,2}),
\varphi^{[B;m]}(\frac{\tau}{M},\frac{u}{M},\frac{v}{M},0)=\!\!\!\sum\limits_{0 \leq a < M \atop b \in I}\!\!(-1)^{a+b}e^{\frac{2\pi i(2m+1)}{M}(au+bv)}q^{\frac{2m+1}{2M}(a^2-b^2)}\varphi^{[B;m]}(M\tau,u+a\tau,v-b\tau,0).
\lim_{\ell \longrightarrow \infty}\frac{\mathcal{N}_{q,D_{V_\omega}(b)}^\pm \big( \vert \omega \vert \gamma_\ell,\vert \omega \vert r_0 \big)}{\textup{Tr}\,\textbf{\textup{1}}_{[\gamma_{\ell},\infty)}\big({\bf P}_q^\pm \vert W \vert{\bf P}_q^\pm \big)}= 1.
J=\left( \begin{array}{cc}\alpha^{t}\tilde{f}_{2}f_{1}\tilde{A}\end{array}\right) \left( \begin{array}{ll}0 0 0 L\end{array}\right) \left( \begin{array}{cc}\alpha \tilde{f}_{1}f_{2}A\end{array}\right) =\left( \begin{array}{ll}\tilde{f}_{2}Lf_{2}\tilde{f}_{2}LA \tilde{A}Lf_{2}\tilde{A}LA\end{array}\right)
\begin{aligned}d\eta(g_0,g_1) =(g_0+g_1)e^{g_0}(1+g_1)\,dg_0+(g_0+2g_1)e^{g_0}\,dg_1 &=(g_0+g_1)(\partial_{g_0}\beta\,dg_0+\partial_{g_1}\beta\,dg_1)+g_1e^{g_0}\,dg_1 &=(g_0+g_1)\,d\beta+g_1e^{g_0}\,dg_1\end{aligned}
[\begin{matrix}y_{1}-x_{0}\\y_{2}-x_{0}\\y_{3}-x_{0}\\y_{4}-x_{0}\end{matrix}]=[\begin{matrix}w&w^{2}&w^{3}&w^{4}\\w^{2}&w^{4}&w&w^{3}\\w^{3}&w&w^{4}&w^{2}\\w^{4}&w^{3}&w^{2}&w\end{matrix}][\begin{matrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{matrix}]
\mathbb{P}\big( X_t \in \mathrm{d}x , L_t \in \mathrm{d}l \big) = \frac{|x| + l}{\sqrt{2 \pi t^3}}\exp \Big\{- \frac{(|x| + l)^2}{2 t}\Big\}\mathrm{d}x \mathrm{d}l x \in \mathbb{R}l > 0
\frac{1}{n}\int_{-m^*}^{m^*}\frac{1}{|\phi_X(u)|^{2(K-1)}}\d u\leq 2 a C_2 n^{-\frac{2\beta -1}{2\beta K}}=o \big((\log n)^{-\frac{2\beta - 1}{\rho}}\big).
K(x,u) = (x-1)({u}^{4}-3{u}^{3}x+{u}^{2}{x}^{2}+{u}^{3}-u{x}^{2}-{u}^{2}+2xu-{x}^{2}) R_1(x,u) = -x ({u}^{3}x-{u}^{2}{x}^{2}-{u}^{3}+2{u}^{2}x+u{x}^{2}-{u}^{2}-xu+{x}^{2}+u-x) R_2(x,u) = xu( x-1 )({u}^{3}-{u}^{2}x+{u}^{2}-u+x ) R_3(x,u) = - xu(xu-u+x).
C_\infty^{-p}(a+bC_\infty^{-p}\|u_k\|_\infty^p)^{p-1}\|u_k\|_\infty^p \leq(a+b\|u_k\|_{E^{\alpha,p}}^p)^{p-1}\|u_k\|_{E^{\alpha,p}}^p &=\int_0^Tf(t,u_k(t))u_k(t)dt &\leq\varepsilon\int_0^T|u_k(t)|^pdt &\leq\varepsilon T\|u_k\|_\infty^p,
\psi\otimes\overline{\psi}=(\psi,\psi)+(\psi, e_a.\psi)e^a+(\psi, e_{ba}.\psi)e^{ab}+...+(\psi, e_{a_p...a_2a_1}.\psi)e^{a_1a_2...a_p}+...+(-1)^{\lfloor{n/2}\rfloor}(\psi, z.\psi)z
\langle E_l^{\mathbf{1}, \alpha}E_{k-l}^{\overline{\mathbf{1}, \alpha_N}}, f_i\rangle = \frac{(k-l-1)!(k-2)!N^{k-l}}{(-2\pi i)^{k-l}(4\pi)^{k-1}L(\alpha_N,k-l)^2 G(\alpha)}L(f_i, k-1) L((f_i)_{\alpha}, k-l).
\partial_t^j \partial_x^n \partial_y^m J_1(t,x,y;\nu) = \sum\limits_{l=1}^{+\infty}\frac1{2\pi}\int_{\mathbb R}(i\theta)^j \frac{r_1^n e^{r_1x}-r_2^ne^{r_2x}}{r_1-r_2}\widehat\nu(\theta,l)\,d\theta\, \psi_l^{(m)}(y)
\frac{dx_j}{ds}=\frac{\partial L_0(x(s),\xi(s))}{\partial \xi_j},\ \x_j(0)=y_j, \ \ 0\leq j\leq n, &\frac{d\xi_j}{ds}=-\frac{\partial L_0(x(s),\xi(s))}{\partial x_j},\ \ \xi_j(0)=\eta_j, \ \ 0\leq j\leq n,
A_{n}={{\frac{1}{4}}\sum_{m}}\Biggl [{\frac{1}{( n - m - z )^{2}}}+{\frac{1}{( n - m + z )^{2}}}-{\frac{1}{( n - m + z )^{2}}}-{\frac{1}{( n + m - z )^{2}}}\Biggr ]
\left| \sum_{d \mid n, \, d \neq 1}\frac{n}{d}\sum_{f \in \mathcal{P}_{\frac{n}{d}}}\chi^d(f) \right| \le \sum_{d \mid n, \, d \neq 1}\frac{n}{d}|\mathcal{P}_{n/d}| \le \sum_{d \mid n, \, d \neq 1}q^{n/d}\le 2q^{\frac{n}{2}},
e^{2(\frac{\gamma}{2}+J+\frac{1}{2}{\sf c}_1)\tau}\|w_2^\bot\|_\rho^2 - \underbrace{e^{2(\frac{\gamma}{2}+J+\frac{1}{2}{\sf c}_1)\tau}\|w_2^\bot\|_\rho^2 |_{\tau=-\log T}}_{=T^{-2(\frac{\gamma}{2}+J+\frac{1}{2}{\sf c}_1)}\|w_0^\bot\|_\rho^2}\lesssim \int_0^t (T-t)^{-1+{\sf c}_1}dt \lesssim T^{{\sf c}_1}.
< \dot{\omega}\left( q \right) ~ \omega \left( f_{1}\right) \cdots \omega \left( f_{n}\right) > = c o n s t L^{2 + \lambda}~ \left\{q_{+}< \Delta | L_{1}\omega \left( q \right) \cdots \omega \left( f_{n}\right) | > - c . c . \right\}~ + ~ O \left( L^{\lambda}\right)
\eta^{\mu}( \sigma ) = \overline{{\eta}}^{\mu}+ \widetilde \eta^{\mu}( \sigma ) \ , \quad \overline{{\eta}}^{\mu}\equiv{\frac{1}{\int_{\Sigma}d^{2}\sigma \sqrt g}}\int_{\Sigma}d^{2}\sigma \sqrt g \, \eta^{\mu}( \sigma )
f( t_1, t_2)= t_1^{\lambda_1}t_2^{\lambda_2}(1-\bar t_1/\bar t_2)=\det\begin{pmatrix}t_1^{\lambda_1}t_1^{\lambda_1}\bar t_1 t_2^{\lambda_1}\bar t_2^{-1}t_2^{\lambda_2}\end{pmatrix}.
\hat{q}_{\beta}(n,t) =\sum_{k=1}^{n}\mathrm{Pr}\{X_{1}+X_{2}+\dots+X_{k}=n\}\mathrm{Pr}\{N_{\beta}(t)=k\}&=\sum_{k=1}^{n}\sum_{\Theta_{n}^{k}}k!\prod_{j=1}^{n}\frac{((1-p)^{j}/j)^{x_{j}}}{x_{j}!}\left(\frac{-\lambda t^{\beta}}{\ln p}\right)^{k}E_{\beta,k\beta+1}^{k+1}(-\lambda t^{\beta}),
\zeta_{0,0}(s,c)=\frac{\pi}{\sqrt{c}}\,\frac{1}{s-1}+\frac{\pi}{\sqrt{c}}\left(2\gamma-\log\left(\frac{c}{4}\right)-8\,\sum_{n=1}^{\infty}\frac{1}{2n-1}\cdot\frac{1}{e^{(2n-1)\pi\sqrt{c}}+1}\right)+O(s-1).
|a_n\alpha^{n+1}| \leq a_0 + (a_1 - a_0)|\alpha| + \cdots + (a_n - a_{n-1})|\alpha|^{n}& < a_0|\alpha|^n + (a_1 - a_0)|\alpha|^{n}+ \cdots + (a_n - a_{n-1})|\alpha|^{n}= |a_n\alpha^n|
\Big\|\frac{u^{\theta}}{r}\Big\|_{4}\leq ||u^{\theta}||_{4}=\Big\|(ru^{\theta})^{\frac{1}{2}}\Big(\frac{u^{\theta}}{r}\Big)^{\frac{1}{2}}\Big\|_{4}\leq ||ru^{\theta}||_{\infty}^{\frac{1}{2}}\Big\|\frac{u^{\theta}}{r}\Big\|_{2}^{\frac{1}{2}}.
\hat{A}_{1}V_{c}=-\frac{1}{\lambda_{c}}\hat{A}_{1}\hat{Q}_{1}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]-\frac{1}{\lambda_{c}}\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\mu_{c}.
3D_{i\alpha}(W(\theta)\tilde{\gamma}^{A})^{[\beta\gamma ]}=\delta_{\alpha}^{~\beta}D_{i\delta}(W(\theta)\tilde{\gamma}^{A})^{[\delta\gamma]}-\delta_{\alpha}^{~\gamma}D_{i\delta}(W(\theta)\tilde{\gamma}^{A})^{[\delta\beta]}
\sum_{\ell(\lambda)\leq K\atop \ell(\mu)\leq K}{\mathcal B}_{\lambda, \mu}(\vec a;\sigma)t^{|\lambda|+|\mu|}=\frac{Q(\vec a, K, \sigma)}{t^{K(K-1)}}\times &\times \frac{1}{(2K)!}\frac{1}{2\pi i}\oint_{|q|=1}\left[\underbrace{\int\!\cdots\! \int}_{2K}\prod_{1\leq i<j\leq 2K}(x_i-x_j)^{2}t^{\sum_{i=1}^{2K}{x_i}}\prod_{i=1}^{2K}d\nu(x_i)\right]\frac{dq}{q}
\begin{array}{rcl}\nu = \nu_o+\frac{\displaystyle\left(E_{q-{\rm rot}}(J+2)-E_{q-{\rm rot}}(J)\right)}{\displaystyle hc}~, begin{align*}2mm] = \nu_0+B\left([J+3]_{q}[J+2]_{q}-[J+1]_{q}[J]_{q}\right)~,\end{array}
B_1(x_1,x_2,x_3; y_1,y_2,y_3) = x_2 y_1 - x_1 y_2, B_2(x_1,x_2,x_3; y_1,y_2,y_3) = x_3 y_1 + a_1 x_2 y_1 + a_3 x_1 y_1 - x_2 y_3, B_3(x_1,x_2,x_3; y_1,y_2,y_3) = x_2 y_2 + a_2 x_2 y_1 + a_4 x_1 y_1 - x_3 y_3.
\begin{array}{rcl}p_q = a p_{rr}+ a p_{qq}+ 2d p_{qr}p_{qq}= \beta + a p_{qrr}+ a p_{qqq}+ 2d p_{qqr}p_r = b p_{rr}+ b p_{qq}+ 2ep_{qr}p_{qr}= \gamma + b p_{qrr}+ b p_{qqq}+ 2e p_{qqr}p_{qr}= \gamma + 2d p_{qrr}+ a p_{rrr}+ a p_{qqr}p_{rr}= \alpha + 2e p_{qrr}+ b p_{rrr}+ b p_{qqr}\ .\end{array}
\int_M\eta|\nabla u_\alpha|_g^2dv_g + \frac{1}{2}\int_M\Delta_g(\eta)u_\alpha^2dv_g + \alpha\int_M\eta u_\alpha^2dv_g = \lambda_\alpha\int_M\frac{\eta u_\alpha^{2^*(s)}}{d_g(x,x_0)^s}dv_g.
\begin{cases}\frac{|\widetilde{\Lambda}+\nabla_y \widetilde{u}_1 (x,y ) |^2}{2}+V (x,y )=\ln \widetilde{m}_1 (x,y )+\widehat{H}(x,\widetilde{\Lambda}), -\div_y \big(\widetilde{m}_1 (x,y ) (\widetilde{\Lambda}+\nabla_y \widetilde{u}_1 (x,y ) )\big )=0.\end{cases}
E_\pi(T_1, \ldots, T_n)= E\left(T'_{q_1}T'_{q_2}\cdots T'_{q_{p-1}}T_{q_p}L_{E_{\pi|_{V_0}}((T_1, \ldots, T_n)|_{V_0})}\right)= E\left(T'_{q_1}T'_{q_2}\cdots T'_{q_{p-1}}T_{q_p}R_{E_{\pi|_{V_0}}((T_1, \ldots, T_n)|_{V_0})}\right).
\begin{aligned}c\bigl(\iota_j\bigl(f\bigr)(0)\bigr) = \operatorname{Ad}(v_0^j) \Bigl( \operatorname{diag}\bigl[ \overbrace{c(f(0)), \dots, c(f(0))}^{a'}, &f\bigl(t_{n_0^0 + q'}^j(0)\bigr), f\bigl(t_{n_0^0 + 2q'}^j(0)\bigr), \dots, f\bigl(t_{k-n_0^1}^j(0)\bigr), &\qquad\qquad\qquad\qquad\quad\underbrace{c(f(1)), \dots, c(f(1))}_{b'}\bigr] \Bigr) \end{aligned}
\begin{array}{r l}{{\bar{\phi}_{a b}( z ) : =}}&{{( \bar{\phi}_{a b}^{( 0 )}( z ) , \cdots , \bar{\phi}_{a b}^{( n - 1 )}( z ) ) ,}}\\{{\bar{\phi}_{a a + \bar{\epsilon}_{j}}^{( i )}( z ) : =}}&{{( \tilde{\Phi}( z ) )_{j}^{i}/ C h ( z - ( n - 1 ) / 2 ) ,}}\\ \end{array}
(a+u) \succ (b+v) :=\hat T(a+u) \cdot (b+v)=T(u)\cdot (b+v)=T(u)\cdot b+l(T(u)) v, (a+u) \prec (b+v) := (a+u) \cdot \hat T(b+v)=(a+u)\cdot T(v)=a\cdot T(v)+ r(T(v)) u,
m_2(\xi,\eta) := \widetilde{\chi}(\xi-\eta,\eta)[\xi(\xi-\eta)-|\xi||\xi-\eta|]-\chi(\xi-\eta,\eta)|\xi||\xi-\eta|\varphi_{\leq 0}(\eta), q_2(\xi,\eta) := \widetilde{\chi}(\xi-\eta,\eta)[\eta(\xi-\eta)+|\eta||\xi-\eta|]/2+\chi(\xi-\eta,\eta)|\eta||\xi-\eta|\varphi_{\leq 0}(\eta),
\iint_D f(a_1, a_2, b_1, b_2) \cdot \phi_{n-2}(-a) \, dab = \iint_D f(a_1,a_2,b_1,b_2) \cdot \left(1 + \frac{\alpha_1}{n}+ \frac{\alpha_2 a}{n}- \frac{a^2}{2n}\right) \, dab + o(n^{-1}) \; ;
R_{l}^{\rm WKB}(r) \propto \frac{1}{\sqrt{\kappa(r)}}\left\{Ae^{- \int_{r_{2}}^{r}dr' \kappa(r')}+ B e^{\int_{r_{2}}^{r}dr' \kappa(r')}\right\}, \;\;\; \kappa(r) \equiv \sqrt{2\mu (V_{\it eff}(r) - E )}
\begin{aligned}3^{l_1+1}x^{r + 18}+ 3a(r) x^{33}+ 3b(r) x^{51}+ 3 a(r)^2 x^{39}+ 2 \cdot 3a(r) b(r) x^{57}+ 3 b(r)^2 x^{75}+ 3 \cdot 3^{2l_1}x^{2 r+9}+ 2 \cdot 3^{l_1+1}a(r) x^{24+r}+ 2 \cdot 3^{l_1+1}b(r) x^{42 + r}\in \langle F \rangle. \end{aligned}
\nabla g(a)^\perp = \left\langle\begin{bmatrix}4 4 0 0 \end{bmatrix}\right\rangle^\perp= \left\langle \begin{bmatrix}1 -1 0 0\end{bmatrix}, \begin{bmatrix}0 0 1 0\end{bmatrix}, \begin{bmatrix}0 0 0 1\end{bmatrix}\right\rangle,
\Delta A_\mu = (\bar\epsilon_2 \zeta^{(9)}- \bar\epsilon_1) \Gamma_\mu\lambda + ({1\over 3}\bar\epsilon_1 - \bar\epsilon_2 \zeta^{(9)}) \Gamma_{\rho}\lambda \bar\lambda \Gamma^\rho \partial_\mu \lambda+\xi^\rho \partial_\rho A_\mu + \partial_\mu \xi^\rho A_\rho,
w(r) =\begin{cases}A \, r^{1-N/2}J_{N/2-1}\left(\sqrt{\tilde{\lambda}_0 \overline{m}}r \right) 0\le r <1 \; , B \, r^{1-N/2}K_{N/2-1}\left(\sqrt{\tilde{\lambda}_0 \underline{m}}r \right) r \ge 1 \; , \end{cases}
|\partial_\theta x_{s_{k+1}}| \leq |\partial x^{(0)}_0| \epsilon(m)^{k+1}+ \|\partial_\theta c\| \sum \limits_{j=1}^{k+1}\epsilon(m)^j + \lambda^{-1/4}\sum \limits_{j=0}^{k}\epsilon(m)^j \leq &\leq \left( |\partial x^{(0)}_0| + \|\partial_\theta c\| + \lambda^{-1/4}\right) \cdot \epsilon(m) + \lambda^{-1/4}.
\Gamma={2G_F^2\alpha E_1^5\over 135(2\pi)^4}\sin^6\theta\left[{1\over 50}({B\over B_c})^6-\left({8\over 105}-{1\over 49}{E_1^2\sin^2\theta\over m^2}\right)({B\over B_c})^8+\cdots \right] .
\Omega([000]) =\{(3210)\}\Omega([100]) =\{(2310), (2130), (2103) \}\Omega([010]) =\{(3120), (1320), (1302), (3102), (1032)\}\Omega([001]) =\{(3201), (3021), (0321)\}\Omega([110]) =\{(1203), (1023), (1230) \}\Omega([101]) =\{(2013), (2031), (0213), (0231), (2301) \}\Omega([011]) =\{(3012), (0312), (0132) \}\Omega([111]) =\{(0123)\}
< 0 | T_{\mu \nu}( x ) | 0 > = \mathop{{\sum}^{\prime}}_{\gamma}{\frac{\Delta_{\gamma}( x , x )^{1 / 2}}{\pi^{2}s_{\gamma}( x , x )^{4}}}t_{\mu \nu}( x ; \gamma ) + O ( s_{\gamma}( x , x )^{- 3}) .
\theta_I=\frac{\partial F_{I}}{\partial x^{i}}\rho^{i}_{\beta}y^{\beta}+\frac{\partial F_{I}}{\partial y^{\beta}}\Gamma^{\beta}+C^{\gamma}_{I\beta}F_{\gamma}y^{\beta}=0, I=1,\ldots,d=\mbox{dim}(\mbox{Ker}(\rho)) \leq n.
F=\begin{pmatrix}2\overline{M'}(M+\overline{M'})^{-1}M -i(M-\overline{M'})(M+\overline{M'})^{-1}-i(M+\overline{M'})^{-1}(M-\overline{M'}) 2(M+\overline{M'})^{-1}\end{pmatrix}.
\begin{aligned}\int_0^T\sum_{x\in G}\,V(x)\, \partial_t v_+(x,t) v_+(x,t) \,\eta^2(x)e^{\xi(x,t)}\mu(x)\,dt &\le \int_0^T\sum_{x\in G}\Delta(v_+(x,t))\,v_+(x,t)\,\eta^2(x)e^{\xi(x,t)}\mu(x)\,dt.\end{aligned}
\triangledown^2P(a,b) =\frac{1}{2}\left(\begin{array}{cc}\frac{\gamma(1+a)}{(1-a)^3}-\partial_{xx}\xi(a,b) -\partial_{xy}\xi(a,b) -\partial_{xy}\xi(a,b) \frac{(1-\gamma)(1+b)}{(1-b)^3}-\partial_{yy}\xi(a,b)\end{array}\right),
[ a_{\lambda \lambda^{\prime}}( \xi , \nu ) , a_{\lambda^{\prime \prime}\lambda^{\prime \prime \prime}}^{\dag}( \xi^{\prime}, \nu ) ] = c_{\nu}\delta_{\lambda \lambda^{\prime \prime}}\delta_{\lambda^{\prime}\lambda^{\prime \prime \prime}}\frac{\xi^{0}}{| \xi^{4}|}\delta^{3}( \vec{\xi}- \vec{\xi}^{\prime}) .
\begin{aligned}\|y_i^{k+1}-{y'_i}^{k+1}\|_1 \leq (1-\alpha^k-\gamma^k_2|C_{ii}|)\|y_i^k-{y'_i}^k\|_1 + \|g_i^{k+1}-{g'_i}^{k+1}\|_1+(1-\alpha^k)\|g_i^k-{g'_i}^k\|_1 \leq (1-\alpha^k-\gamma^k_2|C_{ii}|)\|y_i^k-{y'_i}^k\|_1 + (2-\alpha^k)2C, \end{aligned}
_{a}\mathcal{I}_{x}^{\beta}f(x) &:=\frac{1}{\Gamma^{\alpha}_{F}(\beta)}\int_{S_{F}^{\alpha}(a)}^{S_{F}^{\alpha}(x)}\frac{f(t)}{(S_{F}^{\alpha}(x)-S_{F}^{\alpha}(t))^{\alpha-\beta}}d_{F}^{\alpha}t.
\partial \bar{\partial}\log K (z,z) \geq \sup\Big \{\tfrac{|f^\prime(z)|^2}{(1 - |f(z)|^2)^2}: f\in{\rm Rat}(\Omega), \|f\|_{\Omega, \infty}\leq 1\Big \}= \mathbb S_\Omega(z,z)^2,
e(\lambda_{k}(t))f=\left( \overline{\alpha_{k}(t)}\right)^{-1}(f,\Psi_{k,t}^{\ast})\Psi_{k,t},\left\Vert e(\lambda_{k}(t))\right\Vert=\left\vert \left( \alpha_{k}(t)\right)^{-1}\right\vert
\frac{\partial^{j}A_{1,n}}{\partial x_2^j}(\cdot,\cdot,0)=\frac{\partial^j B_{1,n}}{\partial x_2^j}(\cdot,\cdot,0)=\frac{\partial^j B_{2,n}}{\partial x_2^j}(\cdot,\cdot,0)=0, &j=1,...,L-1;\,\,n=2,...,L-j+1;\,\,
|T'_1\cap T'_2| =|T'_1|+|T'_2|-|T'_1\cup T'_2| &\ge |T'_1|+|T'_2|+|T_3| &\quad-(|A_{123}|+|A_{12}|+|A_{13}|+|A_{23}|) &\ge N/2-|A_{12}|-|A_{13}|-|A_{23}|-|T'_3|+|T_3| & \ge N/8>b'.
\overrightarrow{\not\!\!D}_{R}\equiv e^{\mu}_{a}\gamma^{a}\left(\vec{\partial}_{\mu}+\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd}+ig^{\prime}B_{\mu}\right); \quad\overleftarrow{\not\!\!D}_{R}\equiv\left(\overleftarrow{\partial}_{\mu}-\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd}-ig^{\prime}B_{\mu}\right)\gamma^{a}e_{a}^{\mu};
r_{s,p}=\left(\frac{\sqrt{|\alpha'(\eta_s)|}\sqrt{|\beta'(\zeta_1)|}}{\sqrt{|\alpha'(\eta_p)|}\sqrt{|\beta'(\zeta_s)|}}\frac{\eta_p}{\eta_s}\frac{\eta_1-\zeta_s}{\eta_p-\zeta_s}r_{1,s}+\frac{\sqrt{|\beta'(\zeta_1)|}}{\sqrt{|\beta'(\zeta_s)|}}\frac{\eta_p-\zeta_1}{\eta_p-\zeta_s}r_{1,p}\right)
C\|\Sigma\|_{\infty}\left( \sqrt{\frac{t}{n}}\bigvee \frac{t}{n}\right) > \frac{\bar g_r}{24}- \mathbb{E}\|\hat\Sigma - \Sigma\|_{\infty}\geq \frac{\bar g_r}{24}- \frac{\bar g_r}{48}=\frac{\bar g_r}{48}.
\Big(\sum_{j=1}^n x_j^qz_j\Big)^{\frac1{q}}\leq\sum_{j=1}^n x_jy_j\max_{1\leq k\leq n}\Big(\sum_{j=1}^k z_j\Big)^{\frac1{q}}\Big(\sum_{j=1}^k y_j\Big)^{-1},
s_i (2 \rho) = 2 \rho - 2 \alpha_i s_{i+1}(2 \rho) = 2 \rho- 2\alpha_{i+1}, s_{i+1}s_{i}( 2\rho) = 2 \rho - 2\alpha_{i}- 4 \alpha_{i+1}, s_i s_{i+1}( 2\rho) = 2 \rho - 4 \alpha_{i}- 2\alpha_{i+1}, s_i s_{i+1}s_i (2 \rho) = 2 \rho - 4 \alpha_i - 4\alpha_{i+1}.
\lambda_{1,5}= \frac{2}{\pi}\left(2\int y_{1,2}y_{1,3}- 2\int (y_{1,2}')(y_{1,3}')+\frac{(n-1)(n-3)}{4}\frac{62}{315}\int x^8\cos^2(x) \right) &= \frac{(n-1)^2(n-3)^2}{24^2}\frac{(-30240+4410\pi^2-147\pi^4+\pi^6)}{70}+\frac{(n-1)(n-3)}{2\pi}\frac{62}{315}\int x^8\cos^2(x) &\approx \frac{(n-1)^2(n-3)^2}{24^2}(-1.039) + \frac{(n-1)(n-3)}{2\pi}(0.10734).
p ( \alpha ) ~ = ~ \frac{a ( \alpha - \mu )}{\pi^{2 \mu}a ( \alpha )}~ ~ , ~ ~ r ( \alpha ) ~ = ~ \frac{\alpha p ( \alpha )}{( \mu - \alpha )}~ ~ , ~ ~ m ( \alpha ) ~ = ~ \frac{[ 4 ( \mu - \alpha )^{2}- 1 ] p ( \alpha )}{4 ( \mu - \alpha )^{2}}
M(k) =\exp\left(\sum_{i=1}^k\frac{1}{q_0\kappa^i}\log\left(\frac{\rho_{i-1}}{\rho_i}\right) \right)\leq \sum_{i=1}^k\frac{1}{q_0\kappa}\left(\log \rho_{i-1}-\log \rho_i\right) begin{align*}2mm] \leq \exp\left(\frac{1}{\gamma}\log\left(\frac{1}{\sigma}\right)\right)\leq \left(\frac{1}{\sigma_0}\right)^{\frac{1}{\gamma}}.
( b \eta_{k,1,t}+ \eta_{k-1,1,t}b) c = \, c \, - \, \Psi_{k,1,t}c \, + \, \Psi_{k,1,t}s_{k-1,0}b_{k,0}c , begin{align*}2mm] ( b \eta_{k,k+1,t}+ \eta_{k-1,k+1,t}b ) c = = \Psi_{k,k,t}(-1)^k c \, + \, \Psi_{k,k,t}\sum_{j=0}^{k-1}\, (-1)^j \, s_{k-1,k-1}b_{k,j}c \, + \, (-1)^{k+1}\Psi_{k,t}\, c .
\Lambda(X) =\tfrac{1}{(g-1)!g!}\int_{X^{g-1}}\log\|\eta\|(P_1+\dots+P_{g-1})\Phi_{\Theta}^*\nu^{g-1}&=\tfrac{1}{(g!)^2}\int_{X^g}\log\|\eta\|(P_1+\dots+P_{g-1})\Phi^*\nu^g,
d s^{2}= \frac{- d \tau^{2}+ e^{2 k \tau}\delta_{i j}d x^{i}d x^{j}+ b_{0}^{2}d{\tilde{y}}^{2}}{\operatorname{s i n h}^{2}( k b_{0}| \tilde{y}| + \tilde{c}_{0})}.
\lim_{n\to\infty}\int_0^{1-t_n}\left(f'(Q(\frac{r}{\lambda_n}))\left(\frac{1}{\lambda_n}\partial_rQ(\frac{r}{\lambda_n})\right)^2+\frac{f^2(Q(\frac{r}{\lambda_n}))}{r^2}\right)\,rdr=0.
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int \rho |u_t|^2dx+\frac{1}{C}\int |\nabla u_t|^2 dx & \le C\int \left(|v| |\nabla \rho| |\nabla u| |\nabla u_t|+\rho |v| |u_t| |\nabla u_t| + |v|^2 |\nabla \rho| |\nabla u| |u_t|\right. &\left. +\rho |v_t| |\nabla u| |u_t|+|f_t||\nabla u_t|\right)dx=\sum_{i=1}^5 I_{2i}.\end{aligned}
H^1(\langle g \rangle, Cl_K)\simeq H^1(\langle \sigma \rangle, Cl_K)=\hat{H}^1(\langle \sigma \rangle, Cl_K)=\hat{H}_0(\langle \sigma \rangle, Cl_K)=\frac{\ker({\rm N}_{\sigma, 1})}{D(Cl_K)},
\sum\nolimits_{v \in M_1}x_v = \sum\nolimits_{v \in M_2}x_v \sum\nolimits_{v \in M_1}x_v = \sum\nolimits_{v \in M_3}x_v \vdots \sum\nolimits_{v \in M_1}x_v = \sum\nolimits_{v \in M_n}x_v
\vec{V}(z)=\Biggl(z+{1\over 4}{({g^2\over 2\pi})^2\over 1+{g^2\over 2\pi}}(z-\bar{z}),-{{g^2\over 2\pi}(2+{g^2\over 2\pi})\over 1+{g^2\over 2\pi}}(z-\bar{z})\Biggr).
\frac{1}{(2\pi H_{2,T}(0))^{k}}\biggl\{\int f(\gamma)H_{1,T}(\gamma -\lambda)H_{1,T}(-\gamma+\lambda)\,d\gamma \biggr\}^{k}&\quad{}= \biggl\{\int f(\gamma)\varPhi_{2,T}^{h}(\gamma-\lambda )\,d\gamma \biggr\}^{k}.
\begin{aligned}\bold{pr}X =\xi^i\left(\partial_{x^i}+\sum_I\left(D_{i;I}-\partial_{x^i}\right)\right)+\sum_{\alpha,J_1,J_2}(D_{J_1}S_{J_2}Q^{\alpha})\partial_{u^{\alpha}_{J_1;J_2}}&=\xi^iD_i+\sum_{\alpha,J_1,J_2}\left(D_{J_1}S_{J_2}Q^{\alpha}\right)\partial_{u_{J_1;J_2}^{\alpha}},\end{aligned}
G[F_q] = V_{z^d,G}(\sigma_{\!q}^*G[z^d]) = V_{z^d,G}\circ z^{(n-1)d}(\sigma_{\!q}^*G[z^{nd}]) = V_{z^d,G}^n\circ F_{q,G}^{n-1}(\sigma_{\!q}^*G[z^{nd}]) \subset V_{z^d,G}^n(\sigma_{\!q^n}^*G[z^{nd}]).
A_{a}^{i}=\frac{1}{2}{\epsilon^{i}}_{jk}A_{a}^{jk}=\frac{1}{2}{\epsilon^{i}}_{jk}(\omega_{a}^{jk}-i\frac{1}{2}({\epsilon^{jk}}_{m0}\omega_{a}^{m0}+{\epsilon^{jk}}_{0m}\omega_{a}^{0m}))=\Gamma_{a}^{i}-i\omega_{a}^{0i}
V_{i}\cdot V_{j}={\vec{U}}_{i}\cdot{\vec{U}}_{j}- \sum_{r}V_{i}^{r}V_{j}^{r}+ \sum_{\ell :{\mathrm{r i g h t}}}T_{i}^{\ell}T_{j}^{\ell}- \sum_{\ell :{\mathrm{l e f t}}}T_{i}^{\ell}T_{j}^{\ell}.