harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\dot{\Upsilon}_{33}(t)= - \frac{\lambda}{\theta + \rho}a^{\alpha \beta}\Upsilon_{\alpha\beta}(t)- \frac{\lambda + 2\mu}{\theta + \rho}\Upsilon_{33}(t) - \frac{\theta}{\theta + \rho}a^{\alpha\beta}\dot{\Upsilon}_{\alpha\beta}(t).
p(x) = a_0\phi_0(x) + \sum_{k=1}^n \!\left[b_k[p](x) - (\alpha_kx+\beta_k)b_{k+1}[p](x) -\!\! \sum_{j=k+1}^{n-1}\gamma_{j,k+1}b_{j+1}[p](x)\right]\!\phi_k(x).
|\partial_{t}^{2}\left(\langle\left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}}(r))-1}{r^{2}}\right)v_{5}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r=R\lambda(t)}, \phi_{0}\rangle\right)| \leq \frac{C}{t^{4}\log^{3N-b-2}(t)}
\lim_{M\rightarrow\infty}\lim_{n\rightarrow\infty}II_1 \leq C \lim_{n\rightarrow\infty}\mathbb{E}\int_0^{T}(1+\Vert X_n(s)\Vert_H^{\lambda}) \Vert X_n(s) - X(s) \Vert_H^2 ds = 0 .
{\cal D}_{1}^{ab}= \left(\begin{array}{cc}0 \delta^{ab}\partial \noalign{\vspace{6pt}}\delta^{ab}\partial -f^{abc}J_{0}^{c}\end{array}\right)\;,\qquad{\cal D}_{2}^{ab}= \left(\begin{array}{cc}\delta^{ab}\partial 0 \noalign{\vspace{6pt}}0 \delta^{ab}\partial + f^{abc}J_{1}^{c}\end{array}\right)\;.
( 2 g )_{\infty}= \frac{1}{3 2 \pi^{2}( d - 4 )}\t r < - \ ( 2 l - 1 ) \ \partial_{\mu}\hat{S}^{\mu}\hat{W}+ ( 2 l - 2 ) \ \partial_{\mu}\hat{S}\ \hat{W}^{\mu}> .
a(x)=A_1x+a_2\,, r(x)=r_1^Tx+r_2\,, \alpha(x)=\alpha_1^Tx+\alpha_2\,, \theta(x)=\Theta_1 x+\theta_2\,,\intertext{and}b(x)=b,\;\beta(x)=\beta,\;\sigma(x)=\sigma\,,
\|F_{A(T)}\|^{2}_{L^{2}(M)}=-\int_{M}Tr(F_{A(T)}\wedge\ast_{M}F_{A(T)}) &=-\int_{M}Tr\big{(}F_{A(T)}\wedge(-\dot{A}(T)\wedge\ast_{M}P)\big{)}&\quad+\int_{M}Tr(F_{A(T)}\wedge F_{A(T)})\wedge\ast_{M}Q &=\|\dot{A}\|_{L^{2}(M)}^{2}-(n-3)\big{(}CS(A(T))-CS(A_{\infty})\big{)}
(g_{31}y_1+g_{32}y_2+g_{33}y_3)\{(g_{11}+g_{31})y_1 +(g_{12}+g_{32})y_2+(g_{13}+g_{23}+g_{33})y_3\}^{b-1}=g_{32}(g_{12}+g_{23})^{b-1}y_2(y_1+y_2)^{b-1}+\alpha y_3(y_1+y_2+y_3)^{a-1},
\sup_{z \in (\ell^{2},1]}\hat{U}_{3}(\ell^{1},\ell^{2},z)= \sup_{z \in [b,1]}\hat{U}_{3}(\ell^{1},\ell^{2},z)= \hat{U}_{3}(\ell^{1},\ell^{2},r).
S^{I}_{p,q}(N,z^{\beta}) \equiv -|z|^{\beta(N-1)}p^{-1}\; P \int_{0}^{\infty}\frac{t^{N+q/p-1}\, e^{-t^{1/p}}}{t-|z|^{-\beta}}\; dt +\frac{2 \pi i}{p}\; |z|^{-\beta q/p}& \times \;\;\sum_{j=1}^{M}e^{2ij q \pi /p}\exp \!\left( -|z|^{-\beta/p}e^{2ij \pi/p}\right) + \frac{\pi i}{p}\; |z|^{-\beta q/p}\, \exp \!\left( -|z|^{-\beta/p}\right) \;\;.
E_0(f,p)=\sum_{z\leq p}\frac{1}{p}\sum_{\substack{0\leq a< p \gcd(n,p-1)=1}}e^{\frac{i2\pi a(\tau^{n}-u)}{p}}\cdot \frac{1}{p}\sum_{\substack{0<\leq b< p gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{m}-v)}{p}}=0.
\psi\circ f\leq \phi ~\Rightarrow~ \psi\circ g\circ\eta_X\leq\phi ~\Rightarrow~ \widehat{\xi}\circ\eta_X\leq\phi ~\Rightarrow~ \forall x\in X, \xi(x)\leq\phi(x) ~\Rightarrow~ \widehat{\xi}(\phi)=0.
\sum_{m\ge0}(-1)^{m}g_{m}(z)w^{m}=\sum_{m\ge0}\sum_{n\ge m}\frac{(-1)^{n}}{n!}s(n,m)w^{m}z^{n}=\sum_{n\ge0}\frac{(-1)^{n}}{n!}z^{n}\sum_{m=0}^{n}s(n,m)w^{m}.
\sum_{T\subseteq R}\sum_{\overline{T}_1=T}\sum_{\overline{T}_2=T}\ldots \sum_{\overline{T}_n=T}|\Lambda_T( (\psi_{T}- \psi_{T_k})v_{T_k}, ( \varphi_{T}- \varphi_{T_{k+1}})u_{T_{k+1}})|
S_{s}={\frac{1}{2 \kappa_{1 0}^{2}}}\int d^{1 0}x \sqrt{- G^{s}}\left[ e^{- 2 \Phi}\left\{{\cal R}_{G^{s}}+ 4 ( \partial \Phi )^{2}\right\}-{\frac{1}{2 \cdot ( p + 2 ) !}}F_{p + 2}^{2}\right] ,
\|F\|_\infty= \|F_R\|_\infty\leq C_{N,p,\lambda}\max\left\{1,\sup_{r\geq2}\frac{\|F_R\|_r}{r^\lambda}\right\}\log^\lambda(\|F_R\|_{W^{1,p}}+e) = C_{N,p,\lambda}\max\left\{1,\sup_{r\geq2}\frac{\|F\|_r}{r^\lambda R^{N/r}}\right\}\log^\lambda\left(e+\frac{\|F\|_p}{R^{N/p}}+\frac{\|\nabla F\|_p}{R^{N/p-1}}\right),
i \frac{\partial}{\partial t}\phi ( \vec{r}, t ) = \left[ - \frac{{\nabla_{r}}^{2}}{2 m}- \frac{m}{2}\left( \frac{\ddot{a}}{a}\right){\vec{r}}^{\ 2}+ V ( \vec{r}, t ) \right] \phi ( \vec{r}, t ) .
{\frac{g_{\tiny \mathrm{U V}}^{( a )}}{g_{\tiny \mathrm{I R}}^{( a )}}}={\frac{\sin{\frac{\pi a}{k + l + 2}}}{\sin{\frac{\pi}{k + l + 2}}}}{\frac{\sin{\frac{\pi}{l + 2}}}{\sin{\frac{\pi ( a - k )}{l + 2}}}}.
\Bigl|F'_{U}\Bigl(\sum_{i=1}^u\frac{t_i}{10^i}+\eta\Bigr)\Bigr| \ll U \prod_{i=1}^u\Bigl( G(t_i,\dots,t_{i+J})+O(10^i\eta)\Bigr) &\ll (U+O(U^2\eta))\prod_{i=1}^u G(t_i,\dots,t_{i+J}).
\sum_{i=1}^k Q_i^{\Pi_1}\geq \sum_{i=1}^k Q_i^{\Pi_2}\implies\sum_{i=1}^{k}(Q_i^{\Pi_1}-Q_i^{\Pi_2})^+\geq\sum_{i=1}^{k}(Q_i^{\Pi_2}-Q_i^{\Pi_1})^+.
\frac{\partial^\gamma}{\partial t^\gamma}u(x,t)|_{(x_j,t_{n+1})}=\frac{\partial^\gamma}{\partial t^\gamma}u_j^{n+1}=\frac{1}{\Delta t^\gamma \Gamma[2-\gamma]}\sum\limits_{l=0}^{n}w_l(u_j^{n-l+1}-u_j^{n-l}),
\theta = \nabla_{a}n^{a}={\frac{- 1}{\sqrt{g}}}{\frac{\partial}{\partial u}}{( \sqrt{g}g^{u v})}={\frac{1}{| g_{u v}| \sqrt{g_{x x}g_{y y}}}}{\frac{\partial}{\partial u}}{( \sqrt{g_{x x}g_{y y}})}\to - \infty
[\mathcal{U}(\mathbb{Z}[\zeta_{p_{i}^{n_{i}}}]): O_{(H_{i},K_{i})}]= [\mathcal{U}(\mathbb{Z}[\zeta_{p_{i}^{n_{i}}}]): P_{(H_{i},K_{i})}][ P_{(H_{i},K_{i})}: O_{(H_{i},K_{i})}]
{\overline{{{Q}}}}_{a}={\overline{{\lambda}}}_{a}^{\mu}P_{\mu}- 2 i G_{\mu \nu}{\overline{{\lambda}}}_{a}^{\mu}m^{\nu}+ i \Gamma_{\mu \nu \rho}{\overline{{\lambda}}}_{a}^{\mu}{\overline{{\lambda}}}^{\nu}\lambda^{\rho},
\widetilde{Y}_{\underline{s}}:= \{x\in \{0,1\}^\Lambda : |x \bmod L_\ell| = |\Lambda/ L_\ell|-s_\ell, \ell\geq 1\}, \widetilde{Y}_{\geq \underline{s}}:= \{x\in \{0,1\}^\Lambda : |x \bmod L_\ell| \leq |\Lambda/ L_\ell|-s_\ell, \ell\geq 1\}
t_\lambda t_\lambda^* 1 = t_\lambda ((\overline{f_\lambda}\circ \sigma_\lambda) \cdot (1 \circ \sigma_\lambda) \cdot \Phi_\lambda ) &= f_\lambda \cdot (\overline{f_\lambda}\circ \sigma_\lambda \circ \sigma^n) \cdot (1 \circ \sigma_\lambda \circ \sigma^n) \cdot (\Phi_\lambda \circ \sigma^n),
\text{LHS of (\ref{eqn:final})}= \sum_{m=1}^4 (-1)^m \binom{1+2m}{3}\binom{4}{m}= - \binom{3}{3}\binom{4}{1}+ \binom{5}{3}\binom{4}{2}- \binom{7}{3}\binom{4}{3}+ \binom{9}{3}\binom{4}{4}= 0
F(x_1,\ldots,x_r;s_1,\ldots,s_r;r) & =\frac{1}{\prod_{j=1}^{r}\Gamma(s_j)}\int_0^\infty\cdots\int_0^\infty \prod_{j=1}^{r}\left\{t_j^{s_j-1}\frac{e^{\sum_{\nu=j}^{r}x_\nu}}{1-e^{\sum_{\nu=j}^{r}x_\nu}(1-e^{\sum_{\nu=j}^{r}t_\nu})}\right\}\prod_{j=1}^{r}dt_j.
\sum_{R\in G}^{6}\Gamma(R)_{11}^{*}\Gamma(R)_{11}=1^{2}+1^{2}+(-\frac{1}{2})^{2}+(-\frac{1}{2})^{2}+(-\frac{1}{2})^{2}+(-\frac{1}{2})^{2}=3.
\begin{pmatrix}1 0 \cr - \hat{x}_{n}^{-1}1 \end{pmatrix}L_n(t \mid \hat{y}_n, \hat{Y}_n) \begin{pmatrix}1 0 \cr \hat{x}_{n-1}^{-1}1 \end{pmatrix}= \begin{pmatrix}A_n B_n \cr 0 D_n \end{pmatrix}\;.
\frac{\overline{(\Delta N)^2}}{\overline{N}^2}\sim \beta \sim \frac{M_s}{M}\ , \qquad \frac{\overline{(\Delta \,{\cal R})^2}}{\overline{\cal R}^2}\sim \frac{\beta}{\delta}\sim \frac{(R^2 / \ell_s^2)}{(M / M_s)}\, .
\begin{array}{rcl}k_iX^+_jk_i^{-1}=q_i^{1/2}X^+_j~, begin{align*}2mm] X^+_iX^-_j-q^{a_{ij}/2}_iX^-_jX^+_i=\delta_{ij}\displaystyle\frac{1-k_i^{-2}}{1-q^{-1}_i}~.\end{array}
\left(e_{i(k-1)}e_{-i(k-1)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}x_-\right)\left(e_{i(k)}e_{-i(k)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}x_-\right)^{-1}=e_{-i(k)}e_{i(k)}^{-1}\in B_-s_{i(k)}B_-,
\left< \phi^{2}( x ) \right> ={\frac{a^{2 - N}}{2^{N}\pi^{\frac N 2 - 1}\Gamma \left( \frac N 2 \right)}}\sum_{n = 1}^{\rho_{N}}( - 1 )^{n}c_{2 n}^{N}( a^{2}b )^{n - \frac 1 2}.
\sqrt{\tfrac{2\pi}{z_{\pi,t'}}}= \sqrt{\tfrac{2\pi}{z_{\pi_x\delta_x,t'}}}\sqrt{\tfrac{\pi_xe^{t_x'}}{\sum_{i\in V}\pi_ie^{t_i'}}}\le \sqrt{\tfrac{2\pi}{z_{\pi_x\delta_x,t'}}}.
P^t = l_{0,0}, \mathcal{W}^t =\min\left( \{l_{0,0}-l_{1,1}, l_{a,b}\}\cup\left(\bigcup_{\substack{1\leq i\leq a-1 1\leq j\leq b}}\{l_{i,j}-l_{i+1,j}\}\right)\cup\left( \bigcup_{\substack{1\leq i\leq a 1\leq j\leq b-1}}\{l_{i,j}-l_{i,j+1}\}\right)\right).
\begin{aligned}u(t)=\gamma_{1}(t)H_{1}(\alpha_{1}[u])+\delta_1(t)L_{1}(\beta_{1}[v])+G_1(u)(t)+F_1(u,v)(t), v(t)= \gamma_{2}(t)H_{2}(\alpha_{2}[v])+\delta_2(t)L_{2}(\beta_{2}[u])+G_2(v)(t)+F_2(u,v)(t),\end{aligned}
\partial_y \left(\frac{{\partial^2_{x}{\partial^n_x u}}}{u^s_y +{u}_y}\right) = \partial^2_xg_n + 2\partial_x g_n \eta_1 + 2\partial_x \partial_y^{-1}g_n \partial_y \eta_1 - 2g_n\eta_1^2 -{4 \partial_y^{-1}g_n \eta_1 \partial_y \eta_1}+\partial_y\bigg(\partial_y^{-1}g_n\,\frac{{u}_{xxy}}{u^s_y +{u}_y}\bigg)\, .
\int_G\int_G\mu^{-1/2}(t)\mu^{-1/2}(s)\langle T^{1/2}s(e),T^{1/2}(t(e))\rangle_{H^i(E)}\cdot\langle s(h),t(h)\rangle_H\,ds\,dt &=\int_G\bigg\langle e,\left(\int_Gs(T)\,ds\right)(e)\bigg\rangle_{\mathcal{E}^i}(t)\cdot\langle h,t(h)\rangle_H\,dt.
\left(\begin{array}{c}\dot{x}_{1}\dot{x}_{2}\end{array}\right)=\left(\begin{array}{cc}0 1 -2 -2\end{array}\right)\left(\begin{array}{c}x_{1}x_{2}\end{array}\right)+\left(\begin{array}{c}0 1\end{array}\right)u,\,\,\,\,\,|u|\leq 1, \,\,\,\,\,x_2 - u \leq 0
\sum\limits_{n=1}^{\infty}\frac{\chi\left( n\right)}{n\left( e^{n\gamma/2}+e^{-n\gamma/2}\right)}+\sum\limits_{n=1}^{\infty}\frac{\chi\left(n\right)}{n\left( e^{n\theta/2}+e^{-n\theta/2}\right)}=\frac{\pi}{8},
\partial_v H_\phi (x,r) = - 2 \int |z|^{-1}\phi' \left({\textstyle\frac{|z|}{r}}\right)\sum_i \partial_v u_i (x+z) \cdot u_i (x +z)\, dz \partial_r H_\phi (x,r) = \frac{m-1}{r}H_\phi (x,r) - \frac{2}{r^{m-1}}\int |\zeta|^{-1}\phi' (|\zeta|) \sum_i \partial_\zeta u_i (x+r\zeta)\cdot u_i (x+r\zeta)\, d\zeta\, \, .
I [ L , X ] = I [ L ] + i \int d^{2}\xi d^{2}\theta \left[ \langle L^{- 1}D_{+}L , T^{a}\rangle F_{a \beta}^{R}D_{-}X^{\beta}+ D_{+}X^{\alpha}\hat{F}_{\alpha \beta}D_{-}X^{\beta}\right] ,
\frac{d^2R(s)}{ds^2}+\frac{3}{2s}\frac{dR(s)}{ds}+\left[\frac{\left(-\frac{m^2\omega^2}{4\hbar^2}\right)s^2+\left(\frac{mE}{2\hbar^2}\right)s-\frac{\ell}{4}(\ell+1)}{s^2}\right]R(s)=0.
\int_{{\cal C}_{0}}\left( B_{( 2 )}\, + \, 2 \pi \alpha^{\prime}{\cal F}\right) = - 4 \pi^{2}\alpha^{\prime}\, \left( \frac{1}{2}+ \frac{1}{4 \pi^{2}\alpha^{\prime}}\, \tilde{b}_{1}\right) + 4 \pi^{2}\alpha^{\prime}= 4 \pi^{2}\alpha^{\prime}\, \left( \frac{1}{2}- \frac{1}{4 \pi^{2}\alpha^{\prime}}\, \tilde{b}_{1}\right)
(\tau\xi)^{-1}\circ\nu(x) =\widetilde{\nu}(x) &=y_1^ny_2^mxy_2^{-m}y_1^{-n}&=y_1^ny_2^{m-1}\xi(x)y_2^{1-m}y_1^{-n}&=\cdots &=y_1^n\xi^m(x)y_1^{-n}&=\cdots &=\tau^n\xi^m(x).
\sigma_2^-+\sigma_3^-={\Gamma(1+x)\over \Gamma(1-x)}{\Gamma(1/2-x)\over \Gamma(1/2+x)}{\Gamma(1/2-\Delta+x)\over \Gamma(1/2-\Delta-x)}{\Gamma(1-\Delta-x)\over \Gamma(1-\Delta+x)}
{\cal Y}(T,U) \stackrel{{\Gamma^o(3)}_T}{\rightarrow}(i \gamma T + \delta )\;{\cal Y}(T,U) - i \gamma\{(\partial_{\frac{U}{3}}\eta^{-2}(\frac{U}{3}) \;\eta^{-2}(T) + (\partial_U \eta^{-2}(U)) \;\eta^{-2}(\frac{T}{3})) \}.
H:=\{x\in\mathbb{R}^2\, | \ (\beta_1v_{11}+\beta_2v_{12})x_1+(\beta_1v_{21}+\beta_2v_{22})x_2 \geq\beta_1\alpha_1+\beta_2\alpha_2, (\beta_1v_{11}-\beta_2v_{12})x_1+(\beta_1v_{21}-\beta_2v_{22})x_2 \geq\beta_1\alpha_1-\beta_2\alpha_2\}.
\begin{array}{rcl}\displaystyle\frac{\partial^{k}\lambda_{i}^{t}}{\partial t^{k}}= u_{k,0}c^{\frac{k+1}{2}}\|T\|^{k+1}+u_{k,2}c^{\frac{k-1}{2}}\|T\|^{k-1}(\lambda_{i}^{t})^{2}+u_{k,4}c^{\frac{k-3}{2}}\|T\|^{k-3}(\lambda_{i}^{t})^{4}+\ldots+u_{k,k+1}(\lambda_{i}^{t})^{k+1},\end{array}
\mu_s(\Phi_s(U) \cap \Phi_s(V)) = \mu_s(\Phi_s(U)) \cdot \mu_s(\Phi_s(U)) \intertext{and}\mu_q(\Phi_q(U) \cap \Phi_q(V)) = \mu_q(\Phi_q(U)) \cdot \mu_q(\Phi_q(V)),
\frac{d\mu_{E_\infty}}{dx}(x) = G(x) = \frac{2\arcsin(1/r)}{\pi^2 \sqrt{r^2-x^2}}+ \frac{1}{\pi^2 x}\log\left|\frac{(x+1)(r^2-x+\sqrt{r^2-x^2}\sqrt{r^2-1})}{(x-1)(r^2+x+\sqrt{r^2-x^2}\sqrt{r^2-1})}\right|.
y =\left(a_{s-\ell}^\tau k^s+\sum_{r=\ell}^{s-1}a_{r-\ell}k^r\right)t+ik^\ell+\sum_{r=0}^{\ell-1}a_{r+s+1-\ell}k^r &=a_{s-\ell}^\tau n+\left(\sum_{r=\ell}^{s-1}a_{r-\ell}k^r\right)t+ik^\ell+\sum_{r=0}^{\ell-1}a_{r+s+1-\ell}k^r.
P_q^{\perp}(X_*) = (I - U_1 U^{\top}_1) X_* - (I - U_1 U^{\top}_1) X_* V^{(q)}_0 (V^{(q)}_0)^{\top}= = (I - U_1 U^{\top}_1) X_* + (I - U_1 U^{\top}_1) H V^{(q)}_0 (V^{(q)}_0)^{\top}.
\frac{1}{2}\frac{d}{dt}|f_{\lambda}(t)|^{2}= -\lambda \cdot \coth \rho_{\lambda}(t) \bigg( \big<(I + \theta(t)\theta(t)^{T})f_{\lambda}(t), f_{\lambda}(t)\big> &+ \big< f_{\lambda}(t) \theta(t)^{T}f_{\lambda}(t), f_{\lambda}(t)\big> \bigg) - \big< \theta'(t), f_{\lambda}(t) \big>.
n(R_2-\epsilon_n) \leq I(X_1^n;Y_2^n|W_1,X_2^n) = h(X_1^n+\tfrac{1}{\sqrt{a}}Z_2^n|W_1) - h(Z_2^n) +\tfrac{n}{2}\log a \leq \tfrac{n}{2}\log[2\pi e(\tfrac{1}{a}+\alpha P_1)] - \tfrac{n}{2}\log(2\pi e) +\tfrac{n}{2}\log a =n\mathcal{C}(a\alpha P_1),
=\frac{\sigma_{n-1}}2\int_0^{\infty}\frac{t^{n/2-1}}{(1+t)^{n/2+(1-\beta)/2}}\,dt=\frac{\sigma_{n-1}}2\frac{\Gamma\left(\frac n2\right)\Gamma\left(\frac{1-\beta}2\right)}{\Gamma\left(\frac n2+\frac{1-\beta}2\right)}=\frac{\pi^{n/2}\Gamma\left(\frac{1-\beta}2\right)}{\Gamma\left(\frac n2+\frac{1-\beta}2\right)}.
\int \limits_{0}^{t}\int\limits_{\Omega}2~\frac{1}{c^2}\partial_{tt}u ~\partial_t u~ dxd\tau - \int \limits_{0}^{t}\int\limits_{\Omega}2 \nabla \cdot (a \nabla u)~\partial_t u~ dxd\tau =2\int\limits_{0}^{t}\int\limits_{\Omega}g~\partial_t u~dxd\tau.
\sum_{k\in{E}}P_{R_{j,k}}=\sum_{k\in\tilde{E}}P_{R_{j,k}}-\sum_{k\in\{\pm 1\}^d}P_{R_{j,k}}=P_{I_{j}^1}\otimes P_{I_{j}^2}\otimes\cdots\otimes P_{I_{j}^n}-P_{I_{j-1}^1}\otimes P_{I_{j-1}^2}\otimes\cdots\otimes P_{I_{j-1}^d}.
\int_{Q_t}|\partial_t\chi_{\varepsilon}|^2 + \ell \int_{Q_t}\partial_t \varphi_{\varepsilon}\partial_t \chi_{\varepsilon}+ \frac{1}{2}\int_{\Omega}|\nabla \chi_{\varepsilon}(t)| + \int_{Q_t}\zeta_{\varepsilon}\partial_t \chi_{\varepsilon}= \int_{Q_t}f_{\varepsilon}\partial_t \chi_{\varepsilon}+ \frac{1}{2}\int_{\Omega}|\nabla \chi_{0}| .
o(1)= \langle u_k-u,DE_f[u_k]\rangle=\langle u_k-u, DE_f[u_k]-DE_f[u]\rangle = \int_M |\nabla \Delta (u-u_k)|_g^2d\mu_g-\int_M f|u-u_k|^{2^\sharp}d\mu_g+o(1).
\begin{array}[c]{l}Y_{s}^{1,u}=Y_{s}^{u}-\varphi\left( s,X_{s}^{u}\right) ,\Z_{s}^{1,u}{=Z_{s}^{u}-{D\varphi\left( s,X_{s}^{u}\right)}^{\intercal}\sigma\left(s,X_{s}^{u},Y_{s}^{u},Z_{s}^{u},u_{s}\right) .}\end{array}
A_{z z^{\prime}}= \ln \frac{( a_{z}- a_{z^{\prime}}) ( b_{z}- b_{z^{\prime}})}{( a_{z}+ b_{z^{\prime}}) ( b_{z}+ a_{z^{\prime}})}, \qquad \theta ( z ) = \theta^{( 0 )}( z ) + \sum_{p = 1}^{\infty}( a_{z}^{p}- ( - b_{z})^{p}) t_{p},
\mathbb{T}\Big[\nabla \mathrm{H}\Big](\mathrm{x}) = \int_{\Omega}\nabla \nabla\cdot \Big(\mathbb{G}^{(\mathrm{k})}-\mathbb{G}^{(0)}\Big)(\mathrm{x},\mathrm{y})\nabla \mathrm{H}(\mathrm{y})d\mathrm{y}.
|\{x\in \mathbb{R}^{n}: |T_{\alpha}(\vec{f})(x)|> \lambda\}| \le \sum_{s=1}^{2^{m}}|E_{s}| + C |\Omega^{*}| \le \sum_{s=2}^{2^{m}}|E_{s}| + C B^{q}\lambda^{-\frac{n}{mn-\alpha}}\gamma^{q-\frac{n}{mn-\alpha}}+C (\lambda\gamma)^{\frac{-n}{mn-\alpha}}.
T_{2k+1,2a+1}(x;q)-T_{2k+1,2a-1}(x;q) &=(-xq;q^2)_\infty[\overline{Q}_{k+\frac{1}{2},a+\frac{1}{2}}(x^2;q^2)-\overline{Q}_{k+\frac{1}{2},a-\frac{1}{2}}(x^2;q^2)] &=(-xq;q^2)_\infty[(xq)^{2a+1}\overline{Q}_{k+\frac{1}{2},k-a}(x^2q^2;q^2)+(xq)^{2a-1}\overline{Q}_{k+\frac{1}{2},k-a+1}(x^2q^2;q^2)].
- ( l_{1}+ l_{0}+ 2 ) v ( l_{0}+ 1 , l_{1}; l_{0}+ 1 , l_{1}+ 1 ) w ( l_{0}+ 1 , l_{1}+ 1 ; l_{0}+ 1 , l_{1}) v ( l_{0}+ 1 , l_{1}; l_{0}, l_{1}) .
F_{+}^{( 0 , 2 )}( A ) = 0 \quad \Longleftrightarrow \quad - \frac{1}{2}\bar{\Omega}_{\bar{i}\bar{j}\bar{k}\bar{l}}F^{( 2 , 0 ) \bar{k}\bar{l}}= 4 F_{\bar{i}\bar{j}}^{( 0 , 2 )}.
0\leq\nabla_{\tilde E_n}\nabla_{\tilde E_n}Q(x_0,y_0,X_0,t_0)=\frac{E_{f_{nn}}(x_0,y_0,t_0)}{\psi(r_0/2,t_0)}-\frac{m\psi_{ss}}{\psi}(r_0/2,t_0)
\mathrm{d i m .}{\cal M}( \Sigma ) = h^{1}= - 2 \chi + 7 \mid \tau \mid + \mathrm{d i m . ~ o f{}~ t h e ~ I s o m e t r y}+ 2 b_{3}( K^{\ast}) + b_{3}- 2 b_{4}( K^{\ast}) - b_{4}\ .
T_{n}+T_{n-1}=(\frac{n^{2}}{2}+\frac{n}{2})+(\frac{(n-1)^{2}}{2}+\frac{n-1}{2})=(\frac{n^{2}}{2}+\frac{n}{2})+(\frac{n^{2}}{2}-\frac{n}{2})=n^{2}=(T_{n}-T_{n-1})^{2}.
C_{n}\doteq \frac{\iota_n}{n}\sum\limits_{j=1}^{p}\frac{\vert \widehat{\beta}_{A,j}\vert^{t}}{\vert \widehat{\beta}_{2,j}\vert^{\varsigma}}- \frac{\vert \beta_{0,j}\vert^{t}}{\vert \widehat{\beta}_{2,j}\vert^{\varsigma}}.
\mathcal{A(O}^{\prime})\subseteq\mathcal{A(O})^{\prime}\mathcal{\}Einsteincausality,\EC & \mathcal{A(O)}\mathcal{=A(O}^{\prime\prime}\mathcal{)}causal\text{}completionproperty,\CC & \mathcal{A(O}^{\prime})=\mathcal{A(O})^{\prime},\Haag\duality,\HD
G ( \epsilon ) = \exp \left[ \frac{V}{( 2 \pi )^{( n - 1 )}}\sum_{\sigma}\int d^{n - 1}p \; \epsilon ( p ) [ d_{p}^{( \sigma )}\bar{d}_{\tilde{p}}^{( - \sigma )}- d_{p}^{( \sigma ) \, \dagger}\bar{d}_{\tilde{p}}^{( - \sigma ) \, \dagger}] \right] \,{,}
\Big\|\widehat{\mathcal{L}^{\mathbb{R}^n}_\varepsilon c}+ \widehat{\Delta c}\Big\|_{L^2(\mathbb{R}^n)}^2 = \frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}\big|\big(-\mathcal{F}(J_\varepsilon)(\xi) + \mathcal{F}(J_\varepsilon)(0) - |\xi|^2\big)\mathcal{F}(c)(\xi)\big|^2\xi,
P\left( \lambda_{\mathrm{max}}(\mathbf{G}) < x \right) = \frac{\mathcal{C}\Gamma_K \left( K \right)}{\mathcal{C}\Gamma_K \left( M + K \right)}x^{KM}\times{}_1F_1 \left(M;M + K; - x{\mathbf{I}}\right),
\omega_1 \equiv \dfrac{i}{2}\,\lambda_3+\dfrac{i}{2\sqrt{3}}\,\lambda_8 = \dfrac{i}{3}\small{\begin{pmatrix}2 0 0 0 -1 0 0 0 -1 \end{pmatrix}}\ \ , \ \ \ \omega_2 = \dfrac{i}{\sqrt{3}}\,\lambda_8 = \dfrac{i}{3}\small{\begin{pmatrix}1 0 0 0 1 0 0 0 -2 \end{pmatrix}}\ .
(1/2 + 1/2^{n+1})s_l - 3k - (1/2 + 1/2^{n+2})(s_l + k) > 1/2^{n+2}N_n - 4k > 1/2^{n+2}K_n^2 - 8K_n - 160 = K_n(1/2^{n+2}K_n - 8) - 160 > 0.
2\int_{\Sigma_{\epsilon,\beta,O}}|\nabla \psi|^2\,\mathrm d\mathcal H^2_g\geq &\int_{\Sigma_{\epsilon,\beta,O}}\left(R(g)-R_{\epsilon,\beta,O}+H_{\epsilon,\beta,O}^2+|A_{\epsilon,\beta,O}|^2+2\partial_\nu(h_{\epsilon,\beta}\circ \rho)\right)\psi^2\,\mathrm d\mathcal H^2_g
{\cal L}=-\frac{1}{4}F_{ij}^aF_{ij}^a+\frac{1}{2}F_{i0}^aF_{i0}^a+j_i^aA_i^a-j_0^aA_0^a+{\cal L}_{\mbox{gauge}}-{\bar \psi}\left(m-i{\gamma}\cdot\partial\right)\psi
\begin{cases}\tau u_{ttt}+ \alpha u_{tt}-c^2 \Delta u -b \Delta u_t =0 \textrm{on $(0,T)\times\Omega$}\frac{\partial u}{\partial \nu}=g \textrm{on $(0,T)\times\Gamma_0$}\frac{\partial u}{\partial \nu}+\frac{1}{c}u_t=0 \textrm{on $(0,T)\times\Gamma_1$}\end{cases}
\sum_{v\in \Omega^\prime|_N}\left|\pi_N^{-1}(v)\right|^w \leq \sum_{v\in \Omega^\prime|_N}\left(\sum_{k: V_k|_N = \{v\}}t_k\right)^w \leq \sum_{v\in \Omega^\prime|_N}\left( \sum_{k: V_k|_N = \{v\}}t_k^w\right) = \#^w\left(\pi, \Omega, d_N,d^\prime_N, \varepsilon\right).
\lim_{r\to \infty}w_{0}(x_{0},x_{1},x_{2},r,\Phi_{1}(b_{1})) = \frac{x_{0}+x_{1}+x_{2}}{3}\lim_{r\to \infty}z_{0}(x_{0},x_{1},x_{2},r,\Phi_{1}(b_{1})) = \frac{x_{0}+x_{1}+x_{2}}{3}\lim_{r\to \infty}y_{0,1}(x_{0},x_{1},x_{2},r,\Phi_{1}(b_{1})) = \frac{x_{0}+x_{1}+x_{2}}{3}
\lim_{t \to \infty}e^{-\lambda t}M_j(t) = \frac{\nu q Y}{\lambda}\int_0^1 (1-py)^{-1}(1-y) \sum_{k=j}^{\infty}y^{k-1}dy = \frac{\nu q Y}{\lambda}\int_0^1 (1-py)^{-1}y^{j-1}dy.
J_r^{(n)}(t;\{q\};L)\approx\frac{\hat S_r^{(n)}(\{q\};L)+\hat\Sigma_r^{(n)}(\{q\};L)\vartheta}{z-z_0}\equiv \frac{\hat J_r^{(n)}(\{q\};L;\vartheta)}{z-z_0}\,.
\frac{z+w}{z-w}\prod_{k = 1}^{m}\frac{z-\pi_{k}}{z+ \pi_{k}}\frac{w+\pi_{k}}{w-\pi_{k}}=\frac{z+w}{z-w}+\sum_{p=1}^{m}\Big(\frac{w+\pi_{p}}{w-\pi_{p}}-\frac{z-\pi_{p}}{z+\pi_{p}}\Big) \prod_{k = 1}^{p-1}\frac{z-\pi_{k}}{z+ \pi_{k}}\frac{w+\pi_{k}}{w-\pi_{k}},
(\omega \wedge \psi )_{h_{k}...h_{1}s_{m}...s_{1},j}^{i}\overset{def}{=}\left[ \omega_{h_{k}...h_{1},a}^{i}\psi_{s_{m}...s_{1},j}^{a}\right]_{[h_{k}...h_{1}s_{m}...s_{1}]}
\lim_{h\to 0}\int_0^T\iint\limits_{\mathbb{R}^N\times\mathbb{R}^N}\frac{H\left(x,y,[v]_h(x,t)-[v]_h(y,t)\right)}{|x-y|^N}\,dx\,dy\,dt=\int_0^T\iint\limits_{\mathbb{R}^N\times\mathbb{R}^N}\frac{H\left(x,y,v(x,t)-v(y,t)\right)}{|x-y|^N}\,dx\,dy\,dt
\mathbb{S}_q(\ell_{\infty})(x_{\delta^\vee}) = \prod_{m\geq 1}\prod_{k=0}^{m-1}\big( (1-(q^{\frac{1}{2}})^{2-m+2k}x_{m \delta}) (1-(q^{\frac{1}{2}})^{-m+2k}x_{m \delta}) \big)x_{\delta^\vee},
P_{0}^{i}\, = \,{\frac{\Pi_{0}^{i}}{\epsilon}}\, \, \, , \, \, \, P_{1}^{i}\, = \, \Pi_{1}^{i}\, \, \, \, , . . . \, \, \, , P_{n}^{i}\, = \, \Pi_{n}^{i}\, \, \, \, , \, . . . \, \, , P_{N - 1}^{i}\, = \, \Pi_{N - 1}^{i}\, \, , \, \, P_{N}^{i}\, = \,{\frac{\Pi_{N}^{i}}{\epsilon}}\, \, .
\langle A + | A - \rangle \; = \; \det_{\lambda , \lambda^{\prime}}\, \langle v_{+}( \lambda ) | v_{-}( \lambda^{\prime}) \rangle \; \; , \; \; \langle \pm | A \pm \rangle \; = \; \det_{\lambda , \lambda^{\prime}}\, \langle v_{\pm}^{0}( \lambda ) | v_{\pm}( \lambda^{\prime}) \rangle \; ,
\delta U=\int_{\Omega^{0}}[-N_{\alpha\beta,\alpha}\delta u_{\beta}^{0}+M_{\alpha\beta,\beta}\delta w_{,\alpha}^{0}]d\Omega+\int_{\Gamma^{0}}[n_{\alpha}N_{\alpha\beta}\delta u_{\beta}^{0}-n_{\beta}M_{\alpha\beta}\delta w_{,\alpha}^{0}]d\Gamma
\begin{cases}\begin{aligned}\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2=0, \ \ \bar \lambda=0, \ \ \bar\lambda_1\bar\lambda_2\neq 0, & S<\sup H^2=\bar H^2 \leq 3S-2 \ \ \ \S<\sup H^2<\dfrac43S\end{aligned}\end{cases}
h_1 =\lambda_1+\lambda_{12}+\lambda_{123'}, h_2 =\lambda_2+\lambda_{12}+\lambda_{123'}, h_3 =\lambda_3+\lambda_{123'}, h_{12}=\lambda_1+\lambda_2+\lambda_{12}+2\lambda_{123'}, h_{13}=\lambda_1+\lambda_3+\lambda_{12}+2\lambda_{123'}, h_{23}=\lambda_2+\lambda_3+\lambda_{12}+2\lambda_{123'}, h_{123}=\lambda_1+\lambda_2+\lambda_3+\lambda_{12}+2\lambda_{123'}.
|(\phi_h^n, 1)_{\mathcal{T}_h}| =\frac{1}{\epsilon}|\big(f^n(u_h^n), 1\big)_{\mathcal{T}_h}| \leq \frac{1}{\epsilon}(||u_h^n||_{L^3}^3+||u_h^n||_{L^1})\leq \frac{C}{\epsilon},
\Pr(\gamma_{F,j}\geq\Gamma_F\big{|}x_F)=\int^{\frac{x_F}{\Gamma_F}-\mu_F}_{0}\frac{1}{\lambda_F}e^{-y/\lambda_F}dy= 1-e^{\frac{-x_F+\mu_F\Gamma_F}{\Gamma_F \lambda_F}}.
q((T'\circ T) \otimes \mathrm{id}_{A})\left(|\Omega\rangle\langle\Omega|\right) &=\sum_{k,l}((T'\circ T) \otimes \mathrm{id}_{A})\left(|e_k\rangle|e_k\rangle\langle e_l|\langle e_l|\right) &=\sum_{k,l}(T'\circ T) \left(|e_k\rangle\langle e_l|\right)\otimes |e_k\rangle\langle e_l|
u_H(r,s) =\big\langle U_Y(r,s)F(s),B^*\zeta\big\rangle_V,\ 0\leq s\leq r\leq t, u(r,s) =\big(\mathcal{K}^*_H u_H(\,.\,,s)\big)(r),\ 0\leq s\leq r\leq t,
\begin{aligned}\int_{B_R\times\left\{t\right\}}\psi^2\left(u\right)\xi^2\,dx\leq \left[s_2^2\left(\log2\right)^2\left(\frac{1-\rho_0}{1-\frac{\rho_0}{2}}\right)+C\left(\frac{s_2\log 2}{\nu^2}\left(\frac{\Lambda}{\theta_0^{\,\beta}}\right)^{m-1}+\left(\Lambda^{m+1}+\Lambda^{2(b-1)}\right)4^{s_2+1}R^{2-2\epsilon}s_2\log 2\right)\right]\left|B_R\right|\end{aligned}
S^{2}= \Delta \left( Y^{\prime}- \frac{2 C_{2}+ \frac{1}{\Delta}( a C_{3}+ \frac{\tilde{d}C_{4}}{D - 2}) r^{\tilde{d}}}{r ( C_{2}+ C_{1}r^{2 \tilde{d}})}\right)^{2}+ \frac{K}{( C_{2}+ C_{1}r^{2 \tilde{d}})^{2}}r^{2 \tilde{d}- 2},
\begin{cases}z_{tt}-iaz_{\alpha}=-i \frac{d}{dt}z_j(t)=(v-\frac{\lambda_j i}{2\pi(\overline{z-z_j})})\Big |_{z=z_j}(I-\mathfrak{H})\Big(\bar{z}_t+\sum_{j=1}^N \frac{\lambda_j i}{2\pi(z(\alpha,t)-z_j(t))}\Big)=0.\end{cases}
G(x,y)=\sum_{j=0}^\infty g_j(x,y)=\sum_{j=0}^{r_0}g_j(x,y)=k\log 2\left( (r_0+1)-\frac1\pi \sum_{j=0}^{r_0}\big( 2\theta_{j,d}+\sin(2\theta_{j,d}) \big)\right)\,.
u_{l , m , n}( x ) = \frac 1{\sqrt{2 \pi R^{2}L \omega_{l , m , n}}J_{l + 1}\left( \alpha_{m}^{( l )}\right)}J_{l}\left( \frac{\alpha_{m}^{( l )}}R r \right) e^{- i \omega_{l , m , n}t + i l \theta + i \frac{n \pi}{2 L}z}
b_{2}=-\zeta \cdot\frac{4y_{2}(y_{1}y_{2}+(y_{1}^{2}+y_{2}^{2})\arctan{y_{2}/y_{1}})}{y_{1}(y_{1}^{2}+y_{2}^{2})}+\zeta \cdot f(y_{1}) y_{2},
F_1^0 (e_r, e_r) = \sum_{\{\beta \in Q_1: \t(\beta)=r \}}(\beta , \beta) - \sum_{\{\beta \in Q_1: \s(\beta)=r \}}(\beta, \beta ), F_1^1 (e_r, \gamma) = \sum_{\{\beta \in Q_1: \t(\beta)=r \}}(\beta , \beta \gamma) - \sum_{\{\beta \in Q_1: \s(\beta)=r \}}(\beta, \gamma \beta ),
\mathcal{Y}(s)=\mathcal{Y}_{f,a_1,b_1,a_2,b_2}(s):=\int_0^1 \int_0^1 f(x,y)^s x^{a_1s+b_1}y^{a_2s+b_2}\frac{dx}{x}\frac{dy}{y}.
\frac{dC}{da^{L}}\cdot\frac{da^{L}}{dz^{L}}\cdot\frac{dz^{L}}{da^{L-1}}\cdot\frac{da^{L-1}}{dz^{L-1}}\cdot\frac{dz^{L-1}}{da^{L-2}}\cdot...\cdot\frac{da^{1}}{dz^{1}}\cdot\frac{\partial z^{1}}{\partial x},
\phi^{NL}(x,f)=-\int_{\mathcal{P}(E)}c(x,\pi)\pi(f)G(x,{\rm d}\pi)-\int_{\mathcal{P}(E)}\int_{(0,+\infty)}\left(1-\mathrm{e}^{-\theta\pi(f)}\right)\Pi^{NL}(x,\pi,{\rm d}\theta)G(x,{\rm d}\pi),
d_{g(t)}(x_0, y_0) \leq 40^{N}\left\{d_{g(0)}(x_0, y_0) +C \right\}\leq C 40^{N}d_{g(0)}(x_0,y_0)=C 40^{\frac{t-t_0}{\epsilon}}d_{g(0)}(x_0,y_0).
\vert \partial V_t^+(\vec x)\vert = \sum_{i< j}w_{ij}(\chi_{x_i \le t<x_j}+ \chi_{x_j\le t<x_i}), \vert \partial V_t^-(\vec x)\vert = \sum_{i< j}w_{ij}(\chi_{x_i<-t\le x_j}+\chi_{x_j<-t\le x_i})