harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Omega = - 2 V \int \frac{d^{3}p}{( 2 \pi )^{3}}\int_{C_{0}}\frac{d z}{2 \pi i}\left[ \mathrm{t a n h}( \frac 1 2 \beta z ) f ( z , \mu ) - \lim_{\beta \rightarrow \infty}\mathrm{t a n h}( \frac 1 2 \beta z ) f ( z , 0 ) \right] ,
\mathrm{~ \frac{1}{2}~}( S_{m}, S_{m})^{a}- V_{m}^{a}S = i \hbar \Delta^{a}S_{m}\qquad \Longleftrightarrow \qquad \bar{\Delta}_{m}^{a}\, \mathrm{e x p}\{( i / \hbar ) S_{m}\}= 0 ,
\mathfrak{C}_1=\mathop{\sum\sum}_{\substack{d_1, d_1'|q_1}}d_1d_1'\mathop{\mathop{\sideset{}{^\star}\sum}_{\substack{\alpha\bmod{q_1r/n_1}n_1\alpha\equiv -m\bmod{d_1}}}\;\mathop{\sideset{}{^\star}\sum}_{\substack{\alpha'\bmod{q_1r/n_1}n_1\alpha'\equiv -m'\bmod{d_1'}}}}_{\bar{\alpha}q_2'-\bar{\alpha}'q_2\equiv n_2\bmod{q_1r/n_1}}\; 1,
C= \sum_{i=1}^N \mathbb{E}[X_i] = \sum_{i=1}^N \int_0^T (1-F_i(x))\mathrm{d}x &= \sum_{i=1}^N \int_0^T \min\{(f(x)-i+1)_+, 1\}\mathrm{d}x=\int_0^T f (x) \mathrm{d}x \leq 1.
\frac{1}{p}\frac{d}{dt}\|\omega(t)\|_{L^{p}}^{p}= \int_{\mathbb{R}^{2}}(b\cdot\nabla j) \omega|\omega|^{p-2}\,dx \leq \|b\|_{L^{\infty}}\|\nabla j\|_{L^{p}}\|\omega\|_{L^{p}}^{p-1}.
\sum_{n=1}^{\infty}b_n \dfrac{x^n}{n}= \dfrac{1}{2}b_0 \left( \dfrac{a_1}{b_0\sqrt{p}}\ln \left( \dfrac{1- (a_1-b_0\sqrt{p}) x}{1-(a_1+b_0\sqrt{p})x}\right) - \ln \left( 1-2a_1x+(a_1^2-b_0^2p)x^2\right) \right).
B_{mw}B_{nw^{\prime}}^{\;\;\;\;\;\;\dagger}= \sum_{w^{\prime\prime}}(B_{mw^{\prime\prime}}f_{w^{\prime\prime},w,n,w^{\prime}}+ f^{*}_{w^{\prime\prime},w^{\prime},m,w}B_{nw^{\prime\prime}}^{\;\;\;\;\;\;\dagger}) + E_{mw;nw^{\prime}}(\phi)
\int_0^t S(t-\tau)\sin(kx - \omega \tau)d\tau = \int_0^t \sin\Big(kx - \frac{k}{1+k^2}(t-\tau) - \omega \tau\Big) d\tau = \left(\frac{k}{1+k^2}- \omega \right)^{-1}\left(\cos\Big(kx - \frac{k}{1+k^2}t \Big) - \cos \Big(kx - \omega t \Big)\right)
\begin{array}{rcl}-A^{\star}x = \bigl(P_1(\mathcal{H}_ex_e)'+P_0\mathcal{H}_ex_e\bigl)_{e\in E}\; \; x=(x_e)_{e\in E}begin{align*}9pt]D(-A^{\star}) = \bigl\{x\in X\:;\: (P_1(\mathcal{H}_ex_e)'+P_0\mathcal{H}_ex_e)_{e\in E}\in X\bigr\}.\end{array}
\lim_{n\rightarrow\infty}(n^{1-\alpha}L(n))^{-1}\int_{n_0}^n x^{-\alpha}L(x) dx = \int_{0}^1 y^{-\alpha}\lim_{n\rightarrow\infty}\frac{L(yn)}{L(n)}dy = \int_{0}^1 y^{-\alpha}dy = \frac{1}{1-\alpha}.
\frac{1}{2}\vert \left({}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \left[\hat X_{2,q},\hat P_{2,q}\right] \vert J_1,\gamma_1; J_2,\gamma_2\rangle_q\right)\vert = \frac{\hbar}{4(\lambda_1 +\lambda_2)^2}\vert \lambda_2K_1\left(1-(1-q^2)J_1\right) + \lambda_1K_2\left(1 -(1-q^2)J_2\right)\vert;
\left|\partial_{r}\hat{h}\left(r,\,\iota\right)\right|=\left|\partial_{z}\hat{w}\,\left(\frac{r}{\left(1-\frac{\iota}{2\sigma\tau_{*}}\right)^{\frac{1}{2}+\sigma}},\,\frac{\tau_{*}}{\left(1-\frac{\iota}{2\sigma\tau_{*}}\right)^{2\sigma}}\right)\right|\,\leq\,\frac{4}{3}
\begin{array}{@{}l@{}}{\displaystyle |\tau\sum_{k=1}^{m}V_{1}^{k}(\partial_{\tau}{\theta_{\Psi}^{k}})|\leq C\big(h^{2r}+\tau^{4}\big)+ C\|\theta_{\Psi}^{m}\|_{\mathcal{L}^2}^{2}+C\tau \sum_{k=1}^{m-1}{\|\theta_{\Psi}^{k}\|_{\mathcal{L}^2}^{2}}.}\end{array}
\bar{p}_{I}\to{\frac{3 \bar{p}_{I}+ \bar{s}}{3 + \bar{s}}}\forall \I < 8 \quad , \quad \bar{p}_{I}\to{\frac{3 \bar{p}_{I}- 2 \bar{s}}{3 + \bar{s}}}\forall \I \geq 8
R(x_n) - R(y_n) = \frac{1}{2\pi i}\oint_{|z|= r_n}\left(\frac{R(z)-\mathbb{I}}{z-x_n}- \frac{R(z)-\mathbb{I}}{z-y_n}\right) dz & = \frac{x_n-y_n}{2\pi i}\oint_{|z|=r_n}\frac{R(z) - \mathbb{I}}{(z-x_n)(z-y_n)}dz
\tilde{C}\cap \mathrm{H}_{A_{F_J}- V, \ge 0}\cap N_{M,\leq \delta}= ((\tilde{C}\cap \mathrm{H}_{A_{F_J}- V}\cap N_{M,\leq \delta}) + \mathbb{R}_{\ge 0}e^*_{F_J}) \cap N_{M,\leq \delta}.
S^{I}( \tau ) = \int_{- \infty}^{\infty}\Bigl [ -{\frac{1}{2}}\varepsilon{\frac{x^{2}}{\tau^{2}}}\left({\partial_{x}\phi}\right)^{2}+{\frac{\lambda}{4}}{\frac{1}{1 +{\frac{x^{2}}{\tau^{2}}}( 1 - \varepsilon )}}\phi^{4}\Bigr ] d x .
\xi_{i,1}(t):= G(q(t),\psi_{i,1}),\ \xi_{i,2}(t):=G(q(t),\psi_{i,2})\ \textup{for}\i\in I_1, \xi_{i,3}(t):= G(q(t),\psi_{i,3}),\ \xi_{i,4}(t):=G(q(t),\psi_{i,4})\ \textup{for}\i\in I_2, \xi^\pm_{i}(t):= G(q(t),\psi^\pm_{i})\ \textup{for}\i\in I_3\cup I_4.
g_\varepsilon=\big(\sigma\big(u_\varepsilon,|\nabla\varphi_\varepsilon|\big)\nabla\varphi_\varepsilon-\sigma\big(u_\varepsilon,|\nabla\varphi|\big)\nabla\varphi\big)\cdot\nabla(\varphi_\varepsilon-\varphi)\quad\; Q_T.
-M_{im}\varphi_{xt^{k}}^{m}=M_{im}A_{2}^{ms}\partial_{x}^{-1}\frac{\delta\mathbf{H}_{k}}{\delta \varphi^{s}}=\frac{\delta \mathbf{H}_{k+1}}{\delta\varphi^{i}}, -M_{im}\varphi_{xt^{k+1}}^{m}=M_{im}A_{2}^{ms}\partial_{x}^{-1}\frac{\delta \mathbf{H}_{k+1}}{\delta \varphi^{s}}=\frac{\delta \mathbf{H}_{k+2}}{\delta \varphi^{i}},
\begin{array}{r c c c l c r c c c l}{{\left( \frac{G}{H}\right)_{7}}}&{{=}}&{{\frac{S O ( 8 )}{S O ( 7 )}}}&{{\equiv}}&{{S^{7}}}&{{;}}&{{\left( \frac{G}{H}\right)_{4}}}&{{=}}&{{\frac{S O ( 5 )}{S O ( 4 )}}}&{{\equiv}}&{{S^{4}}}\\ \end{array}
x \ast_{n, n}x = (y+z) \ast_{n,n}(y+z) = y\ast_{n,n}y + y\ast_{n,n}z + z\ast_{n,n}y + z\ast_{n,n}z = y\ast_{n,n}y + z\ast_{n,n}z= \top_n \ast_{n,n}y + \top_n \ast_{n,n}z = \top_n \ast_{n,n}(y+z) = \top_n \ast_{n,n}x
\begin{array}{ll}\delta F^{\nu}=F^{\nu}-F^0\ast^g_{sp}\Gamma^v_{\nu}=Q^S(F^{\nu},F^{\nu})\ast^g \Gamma^v_{\nu}=\sum_{j=1}^{2d}W^S_j(F^{\nu},F^{\nu})\ast^g \Gamma^v_{\nu}=\sum_{j=1}^dP^S_j(F^{\nu},F^{\nu})\ast^g \Gamma^{v,*}_{\nu,v_j}+\sum_{j=d+1}^{2d}W^S_j(F^{\nu},F^{\nu})\ast^g \Gamma^v_{\nu}\end{array}
\sum_{n=1}^{\infty}\frac{1}{n}\left\vert \mathrm{Tr}\, \left(\mathcal{L}_{s,\lambda}^{nN}\right) \right\vert = N \sum_{n=1}^{\infty}\frac{1}{Nn}\left\vert \mathrm{Tr}\, \left(\mathcal{L}_{s,\lambda}^{nN}\right) \right\vert \leq N \sum_{m=1}^{\infty}\frac{1}{m}\left\vert \mathrm{Tr}\, \left(\mathcal{L}_{s,\lambda}^{m}\right) \right\vert.
\lim_{x\to \infty}\chi(x)=\lim_{x\to \infty}\int\limits_{}^{}\frac{\sin(t)}{t}f(\frac{t}{x}) \,\mathrm{d}t\stackrel{}{=}\int\limits_{}^{}\frac{\sin(t)}{t}f(0) \,\mathrm{d}t =1,
i_{e}{}^{N_1 N_2 N_3 N_4}_{\alpha_1\alpha_2\alpha_3\alpha_4}{}^{(a)}= \sum_{\Lambda}\M_{e}{}^{(a)}_{\Lambda}\{C}^{N_1 N_2 N_3 N_4\,\Lambda}_{\alpha_1\alpha_2\alpha_3\alpha_4},
\int_\Omega \varphi_\epsilon(z)dx-\int_\Omega \varphi_\epsilon(\theta_{0x})dx \leq \iint_{Q_t}|b z_x||z\varphi_\epsilon''(z)|dxds + \iint_{Q_t}(|c z|+|c_x \theta|+|f_x|)dxds.
x(\varepsilon) =x^0+\frac{1}{2}\sum_{i=1}^n{[F_{1i}e_i,F_{2i}e_i](J(x^0))}\int\limits_0^\varepsilon\int\limits_0^\tau\big(u_{2i}(\tau)u_{1i}(\theta)-u_{1i}(\tau)u_{2i}(\theta)\big)d\theta d\tau+R(\varepsilon) =x^0{-}\varepsilon\sum_{i=1}^n\frac{\partial J(x^0)}{\partial x_i}F_{0i}(J(x^0))e_i+R(\varepsilon).
W\eta(k,x) =\sqrt{\frac{k}{N}}(A\delta_{a^{N-k}\beta(x)}+B\delta_{b^{N-k}\alpha\beta(x)})+\sqrt{\frac{N-k}{N}}(A\delta_{a^{-k}\beta(x)}+B\delta_{b^{-k}\alpha\beta(x)}) &=\sqrt{\frac{N-k}{N}}(A\delta_{a^{(N-k)-N}\beta(x)}+B\delta_{b^{-k}\alpha\beta(x)}) &+\sqrt{\frac{N-(N-k)}{N}}(A\delta_{a^{N-k}\beta(x)}+B\delta_{b^{N-k}\alpha\beta(x)}) &=\eta(N-k,\beta(x))
\mathcal{X}=^{-1}((\mathcal{X})) =^{-1}((\mathcal{X})^{\frac{1}{2}}(\mathcal{X})^{\frac{1}{2}}) =^{-1}((\mathcal{X})^{\frac{1}{2}})*^{-1}((\mathcal{X})^{\frac{1}{2}}) =\mathcal{X}^{\frac{1}{2}}*\mathcal{X}^{\frac{1}{2}},
\begin{array}{l}|w_j|^{p-2}w_j(w_j)_x|u_j|^q-|w_{j-1}|^{p-2}w_{j-1}(w_{j-1})_x|u_{j-1}|^q =(|w_j|^{p-2}w_j-|w_{j-1}|^{p-2}w_{j-1})(w_j)_x|u_j|^q \quad+|w_{j-1}|^{p-2}w_{j-1}((w_j)_x-(w_{j-1})_x)|u_j|^q \quad+|w_{j-1}|^{p-2}w_{j-1}(w_{j-1})_x(|u_j|^q-|u_{j-1}|^q).\end{array}
\sum_{\ell=1}^L n_\ell \le \frac{L-1}{-\varepsilon C}\left[\ln(\bar{\alpha})+\ln\left(\frac{2\beta\bar{\alpha}}{3}\right)+ \ln\left(\frac{\beta}{3}\right)+ \frac{1}{L-1}\ln\left( \frac{\varepsilon}{F(x^{1}) - \eta^1}\right)\right] + L.
{\frac{\partial \Gamma_{\Lambda}}{\partial \Lambda}}= -{\frac{1}{2}}\mathrm{t r}\left[{\frac{1}{\Delta_{I R}}}{\frac{\partial \Delta_{I R}}{\partial \Lambda}}\cdot \left( 1 + \Delta_{I R}\cdot{\frac{\delta^{2}\Gamma_{\Lambda}}{\delta \varphi^{c}\delta \varphi^{c}}}\right)^{- 1}\right] \quad .
u_{k}(\Psi_{\xi_{k}}^{\partial}(\bar{y},y_{n}))\chi_{R}(|\bar{y}|)\chi_{R}(y_{n})=u_{k}(\Psi_{\xi_{k}}^{\partial}(\varepsilon_{k}\bar{z},\varepsilon z_{n}))\chi_{R}(|\varepsilon_{k}\bar{z}|)\chi_{R}(\varepsilon z_{n})=w_{k}(\bar{z},z_{n}).
X_HG_X^{N_0,N_1}(x,\xi)=X_H(m_{X}^{N_0,N_1})(x,\xi)\ln(1+f(x,\xi))+m_{X}^{N_0,N_1}(x,\xi)\frac{X_Hf(x,\xi)}{1+f(x,\xi)}.
\frac{1}{\tau}\big\|(e_n-e_{n-1})_{n=k}^N\big\|_{L^p(W^{-1,q}(\varOmega))}+ \big\|(e_n )_{n=k}^N\big\|_{L^p(W^{1,q}(\varOmega))}\le C\tau^k , & \max_{k\le n\le N}\|e_n \|_{L^\infty(\varOmega)}\le C\tau^k ,
\sum_{\ell=0}^L h_\ell^{(\beta\lambda-d)/(\lambda+1)}\,\eqsim\, \sum_{\ell=0}^L 2^{\ell (d-\beta\lambda)/(\lambda+1)}\,\eqsim\, E_L \,:=\, \begin{cases}1 \mbox{when}\beta\lambda > d\,, L \mbox{when}\beta\lambda = d\,, 2^{L (d-\beta\lambda)/(\lambda+1)}\mbox{when}\beta\lambda < d\,. \end{cases}
(\mathbf{z}_t,A\mathbf{z})+\mu \|A\mathbf{z}\|_0^2+a_1(\mathbf{z},\mathbf{z},A\mathbf{z})+a_1(\bar{\mathbf{u}},\mathbf{z},A\mathbf{z})+a_1(\mathbf{z},\bar{\mathbf{u}},A\mathbf{z})+J(t;\mathbf{z},A\mathbf{z}) &\qquad=J_1(t;\bar{\mathbf{u}},A\mathbf{z})+(\mathbf{f}(t)-\bar{\mathbf{f}},A\mathbf{z}).
\alpha_{\chi}^k = \sum_{1 \leq i_1 < i_2 <...<i_l \leq n}\epsilon_{i_1}...\epsilon_{i_l}\cdot d_{k,r}\left(\langle e_{i_1}\cdot...e_{i_l}\cdot \chi, \chi \rangle_{\Delta_{r,s}}\right) e^{\flat}_{i_1}\wedge...\wedge e^{\flat}_{i_l}.
\mathcal{S}_{n}^{\prime}(z) =\frac{1}{\pi^{2n}}\operatorname{Im}[Li_{2n}(-\exp(-2\pi iz))],n>0, \mathcal{C}_{n}^{\prime}(z) =\frac{1}{\pi^{2n+1}}\operatorname{Re}[Li_{2n+1}(-\exp(-2\pi iz))],
\bigcup_{0\leq j < N_k}B\big(j\theta,{1\over N_k^\tau}\big) \subset \bigcup_{i=0}^{q_k-1}\bigcup_{j=0}^{\lceil N_k/q_k\rceil}B\Big((i+jq_k)\theta,{1\over N_k^\tau}\Big).
K_{X/Y}+\Delta^{=1}+(l-1)L-\lfloor (l-1)^2 \{\Delta\}\rfloor &=l(K_{X/Y}+\Delta^{=1})+l(l-1)\{\Delta\}-\lfloor (l^2-2l+1)\{\Delta\}\rfloor +(l-1)(A+S) &=l(K_{X/Y}+\Delta)+(l-1)(A+S).
\partial_{\mu}^{z}V_{i}^{\mu}( z , u , v ) = \frac{i g_{1}}{2}\left[ \delta^{4}\left( z - v \right) - \delta^{4}( z - u ) \right] \left( Y_{R}^{i}L + Y_{L}^{i}R \right) \Sigma_{1}^{i}( u - v ) ,
\sigma^2(t) =2\int_{0}^{t}\int_{0}^{x}e^{-x-y}\sqrt{y/x}\dydx &=2\int_{0}^{1}\frac{\sqrt{z}(1-e^{-t(1+z)})}{(1+z)^2}dz - 2\int_{0}^{1}\frac{t\sqrt{z}e^{-t(1+z)}}{1+z}dz &=\frac{2}{3}t^2+\sum_{k=3}^{\infty}(-1)^{k}\frac{2(k-1)}{k!}t^{k}\int_{0}^{1}(1+z)^{k-2}\sqrt{z}dz.
\chi_{1 / 2}^{1 , 2}| \Omega{\rangle}_{L}\otimes{\Psi}_{1 / 2}^{a}{\Psi}_{1 / 2}^{i *}; \, \,{\Psi}_{1 / 2}^{a *}{\Psi}_{1 / 2}^{i}| \Omega{\rangle}_{R}\ \ \mathrm{a n d ~ e x c h a n g e}\ \Psi \longrightarrow \Phi ,
- \beta F^{\mathrm{{\tiny ~ S t r i n g}}}( \beta ) = - \frac{N^{2}}{8}\int_{0}^{\infty}\frac{d t}{2}\, t^{- 1}\left[ \theta_{3}\left( 0 , \frac{2 \pi i t}{\beta^{2}}\right) - \theta_{2}\left( 0 , \frac{2 \pi i t}{\beta^{2}}\right) \right] f ( t )
\begin{array}{c}{\{\quad , \quad \}= i{\tilde{g}}^{A \bar{B}}\nabla_{A}\wedge{\bar{\nabla}}_{\bar{B}}+ g^{A \bar{B}}\frac{\partial}{\partial \theta^{A}}\wedge \frac{\partial}{\partial{\bar{\theta}}^{\bar{B}}}}\\ \end{array}
Q_p \sum_{n = - \infty}^\infty \frac{1}{\mid \vec{y}- 2\pi n a \hat{z}\mid^{7 - p}}\rightarrow \frac{Q_p}{2\pi a}\int_{- \infty}^\infty \frac{dz}{(r^2 + z^2)^{(7 - p)/2}}= \frac{\tilde{Q}_p}{r^{6 - p}},
{\cal U}_{\Delta}^{\dagger}B_{i}^{3}{\cal U}_{\Delta}= \frac{1}{2}\epsilon_{i j k}\left\{\left( e^{- i g \Delta}F_{j k}e^{i g \Delta}\right)^{3}+ i g{\cal U}_{\Delta}^{\dagger}\left( [ A_{j}, A_{k}] \right)^{3}{\cal U}_{\Delta}\right\}\ .
U(x_3)=a_1\left(1-\frac{\cosh\left((x_3-\frac{x_3^{(u)}+x_3^{(l)}}{2})/{\alpha}\right)}{\cosh h/(2\alpha)}\right)+a_2\left(1-\frac{(x_3-\frac{x_3^{(u)}+x_3^{(l)}}{2})^2}{(h/2)^2}\right),
\left(e^{2\lambda r_c}- \frac{M_0\pi}{\lambda^2}\right)\left(e^{2\lambda r_c}- \frac{M_2\pi}{\lambda^2}\right)=\left(e^{2\lambda r_c}- \frac{M_1\pi}{\lambda^2}\right)\left(e^{2\lambda r_c}- \frac{\tilde{M}_1\pi}{\lambda^2}\right)
\tilde{\partial}_i(1_\sigma) =\begin{cases}\partial_i(1_\sigma) i\notin \{j+1,j\}, \sum_{\tau\in \Delta_j\setminus\{\alpha\}}[\sigma:\tau]\cdot 1_\tau i=j+1, \sum_{\tau\in \Delta_{j-1}\setminus\{\beta\}}\left([\sigma:\tau]-\frac{[\sigma:\beta][\alpha:\tau]}{[\alpha:\beta]}\right)\cdot 1_\tau i=j\end{cases}
H_\epsilon'' = \frac{v_\epsilon^{(4)}}{v_\epsilon''}- \frac{(v_\epsilon''')^2}{(v_\epsilon'')^2}- \frac{v_\epsilon'''}{v_\epsilon'-a_t}+ \frac{(v_\epsilon'')^2}{(v_\epsilon' - a_t)^2}+ \frac{v_\epsilon'''}{b_t - v_\epsilon'}+ \frac{(v_\epsilon'')^2}{(b_t - v_\epsilon')^2},
\delta Y_t^{i,r}(m)\leq\bar{Y}_t^r =\frac{C_vC_{\eta}}{C_{\eta}-C_v}\frac{e^{\rho t}(e^{-(\rho t+C_{\eta}(t-r))}-e^{-(\rho m+C_{\eta}(m-r))})}{\rho+C_{\eta}}|v-\bar{v}| &\leq\frac{C_v}{C_{\eta}-C_v}|v-\bar{v}|
-{1\over 4}{\cal A}_{\hat{\ell}\hat{\ell}}\varepsilon_{\ell,u}\pm{1\over 6}{\bf P}_{\hat{i}\hat{j}}\Gamma^{\hat{i}\hat{j}}\varepsilon_{u,\ell}-{i\over 6}{\bf Q}_{\hat{i}\hat{k}}\Gamma^{\hat{i}\hat{k}}\varepsilon_{u,\ell}= 0 \ \ \ \ \ (\hat{\ell}\neq \hat{i},\hat{j}, \hat{k}),
V=V_{1}U_{2},\, X=U_{1}V^{T}_{1}, &\|V\|_{S_{\widehat{p}_{2}}}=\|U_{2}\|_{S_{q}}\|V_{1}\|_{S_{p_{2}}}, &\|U_{1}\|_{S_{p_{1}}}\leq\|U\|_{S_{\widehat{p}_{1}}}\|U_{2}\|_{S_{q}}
0 =\varepsilon(\chi^{i})1_{H}\otimes\chi_{i}=(m\otimes\mathrm{Id})(S\otimes\mathrm{Id}\otimes\mathrm{Id})(\Delta\otimes\mathrm{Id})(\chi) &\overset{\eqref{eq:CC4'2}}=(m\otimes\mathrm{Id})(S\otimes\mathrm{Id}\otimes\mathrm{Id})(1_{H}\otimes\chi+(\mathrm{Id}\otimes\rho^{l})(\chi)) &=\chi+(m\otimes\mathrm{Id})(S\otimes\rho^{l})(\chi).
{\mathbb E}\,{\rm e}^{-\alpha \bar S_{N_{\beta,\omega}}- \gamma G_{N_{\beta,\omega}}}&= \exp\left[-\int_0^\infty \int_{(0,\infty)}\frac{1}{t}{\rm e}^{-\beta t}(1-{\rm e}^{-\omega t})(1-{\rm e}^{-\gamma t-\alpha x})\,{\mathbb P}(Y(t)\in{\rm d}x)\,{\rm d}t\right];
\hat{H}= p^1u_1 + p^2u_2 + p^{(2,0)}u_{(2,0)}+ p^{(1,1)}u_{(1,1)}+ p^{(0,2)}u_{(0,2)}&\quad{}- \frac{1}{2}u_{(2,0)}^2 - u_{(1,1)}^2 - \frac{1}{2}u_{(0,2)}^2 + qu \, ,
\sum_{k=0}^m[3k+1]\,\frac{(aq;q^2)_k(q/a;q^2)_k(b;q)_k(q/b;q)_k(q;q^2)_k\,q^k}{(aq;q)_k(q/a;q)_k(bq^2;q^2)_k(q^3/b;q^2)_k(q^2;q^2)_k}\equiv \frac{(bq;q^2)_{(n-1)/2}(q^2/b;q^2)_{(n-1)/2}}{(bq^2;q^2)_{(n-1)/2}(q^3/b;q^2)_{(n-1)/2}}[n].
H = \frac{\sqrt{\alpha^\prime}}{2}g^{\mu \nu}(P_\mu - Q_e A_\mu) (P_\nu - Q_e A_\nu) + (\frac{m^2 R^2}{2 \sqrt{\alpha^\prime}\alpha^\prime}+ \frac{\sqrt{\alpha^\prime}Q_e^2}{2 k^2}- \frac{Q_e \lambda R}{k}) ,
Q_i^{\epsilon}:= (V_k \cdots V_{i+2}V_{i+1}) P_i^{\epsilon_i}(V_k \cdots V_{i+2}V_{i+1})^{-1}&= (V_k \cdots V_{i+2}V_{i+1}V_i) t_{b_i}^{-\epsilon_i}(V_k \cdots V_{i+2}V_{i+1})^{-1}.
\| f_T \| = \| (\partial_1^2 f^1 + \partial_2 f^2 + f^3)_T \| \lesssim (T^{\frac{1}{4}})^{\alpha-2}([f^1]_{\alpha}+ [f^2]_\alpha + \|f^3\| ) \lesssim (T^{\frac{1}{4}})^{\alpha-2}[f]_{\alpha-2},
\left\| \sum_{j\in \mathbb{S}}\Psi_j^*A_jh\right\|^2 = \left\langle \sum_{j\in \mathbb{S}}c_jA_j^*A_jh, \sum_{k\in \mathbb{S}}c_kA_k^*A_kh\right\rangle=\sum_{j\in \mathbb{S}}c_j^2 \|A_jh\|^2 \leq \left(\sup\{c_j^2\}_{j \in \mathbb{J}}\right)\sum\limits_{j \in \mathbb{S}}\| A_jh\|^2, ~
A = \left[ \begin{array}{c ccccccc}1 1 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 1 \end{array}\right].
I_2(\langle s,t\rangle)=\{s^2,t,st,s^2t,s^3t\},I_2(N\alpha)=\{s^ju^k\alpha\mid j,k\in\{0,1,2,3\}\}, &I_2(N\beta)=\{s^jtu^jv\beta, s^ju^k\beta\mid j,k\in \{0,1,2,3\},\, j+k\equiv0\pmod{4}\}, &I_2(N\alpha\beta)=\{s^ju^j\alpha\beta\mid j\in \{0,1,2,3\}\}\cup\{(stv\alpha\beta)^{\beta^k},(s^2tu^3v\alpha\beta)^{\beta^k}\mid k\in\{0,1\}\}.
\frac{H_{a+1,b,c}(u,v;m)}{H_{a,b,c}(u,v;m)}= \frac{\tilde d_m}{a+b+c}\frac{F_1(\tilde d_m+1;c,b;a+b+c+1;u,v)}{F_1(\tilde d_m;c,b;a+b+c;u,v)}.
q_{n}(x) =q\left( x\right) \chi_{\left( -n,\infty\right)}\left(x\right) \rightarrow q\left( x\right) & \Rightarrow\begin{array}[c]{cccc}S_{n}\rightarrow S_{q}q_{n}\left( x,t\right) \rightarrow q\left( x,t\right) \text{locallyuniformly for}t>0\end{array}
\eta(2^k i)=\dfrac{\pi^{1/4}}{\Gamma(3/4)\cdot 2^{\frac{k+1}{2}}}\cdot e^{-\frac{\pi(2^{k}-1)^2}{12\cdot 2^k}-\frac{1}{2\pi}\sum_{\ell=1}^k 2^\ell L_\ell},
||f-p||_1 =\int |f(x)-p(x)|\, dx &=\int |f(x)-\frac{1}{2}(p_1(x)+p_2(x))|\, dx &\leq\int \frac{1}{2}|f(x)-p_1(x)|+\frac{1}{2}|f(x)-p_2(x)|\, dx &=\frac{1}{2}||f-p_1||_1+\frac{1}{2}||f-p_2||_1.
\Delta =\vee^2\{2\}= \vee^1\{\{2\},\{1,2,3\},\{1,2,4\},\{1,2,5\},\{2,3,4\},\{2,3,5\},\{2,4,5\}\}= \{\{2\},\{1,2,3\},\{1,2,4\},\{1,2,5\},\{2,3,4\},\{2,3,5\},\{2,4,5\},\{1,2,3,4,5\}\}.
\int_{0}^{\infty}\frac{x^{\alpha - 1}\cos{b x}}{x \sinh{c x}}d x = \frac{\Gamma ( \alpha )}{( 2 c )^{\alpha}}\left\{\zeta \left[ \alpha , \frac{1}{2}\left( 1 + \frac{i b}{c}\right) \right] + \zeta \left[ \alpha , \frac{1}{2}\left( 1 - \frac{i b}{c}\right) \right] \right\}\, ,
k_{m-1}^{(r)}+ k_m^{(r-1)}= \sum_{i=0}^{m}\left[\binom{r+i-2}{i-1}+ \binom{r+i-2}{i}\right] h_{m-i}= \sum_{i=0}^m \binom{r+i-1}{i}h_{m-i}= k_m^{(r)}\,.
h(x) \le \Bigl(g_3(x)+\frac{2}{6^4}(3^4x-33)(7-2^4x)+\frac{2}{6^5}(2^5x-13)(101-3^5x) &+\frac{2}{6^6}(3^6x-302)(27-2^6 x)\Bigr)\Bigr|_{x=\frac{276}{665}}+ \frac1{10\cdot 6^6}- \eta &=-\frac{9623392693609973991877849224521309}{425623699615692261466802423622110256000}<0,
: \underset{\{x[i],y[i]\}}{\max}\underset{i=1}{\overset{N}{\sum}}\underset{j=i}{\overset{N}{\sum}}~w[i][j]f_{3,}(x[i],x[j],y[i],y[j]) \mathrm{s.t.}\; (\ref{maxV1}), (\ref{maxV2}), (\ref{maxV3}). ~~~~~~~~~~~~~~~~~~~~~~~~
z{\cal H}_{k-1}^{(s+1,t)}(z)={\cal H}_{k}^{(s+1,t)}(z)+(q_{k}^{(s+1,t)}+e_{k-1}^{(s+1,t)}){\cal H}_{k-1}^{(s+1,t)}(z)+q_{k-1}^{(s+1,t)}e_{k-1}^{(s+1,t)}{\cal H}_{k-2}^{(s+1,t)}(z).
\begin{gathered}|(F-F_k)(s,x+z)-(F-F_k)(s,x)| \leq c_k \min\{1,|z|\}|(F-F_k)(s,x+z)-(F-F_k)(s,x)-\nabla_x (F-F_k)(s,x) \cdot z| \leq C \min\{1, |z|^{\gamma}\}\end{gathered}
\biggl[ \biggl\{\frac{\partial F_1( t, \theta )}{\partial t}\frac{\partial t^*_{\alpha'}(\theta)}{\partial\theta}+\frac{\partial F_1( t, \theta )}{\partial\theta}\biggr\}\bigg|_{t=t^*_{\alpha'}(\theta)}\biggr] \bigg|_{\theta=\theta_0(\alpha')}=0.
\lambda_N = \frac{3a}{N\ell^3}+ \frac{O(1)}{N^{2}\ell^4}, 0 \leqslant 1-f \leqslant \frac{C\mathbf{1}(|x|\leqslant \ell)}{N|x|}, |\nabla f| \leqslant \frac{C\mathbf{1}(|x|\leqslant \ell)}{N|x|^2}.
\frac{d}{ds}\Big|_{s=0}\frac1h \int \left( u^n_s-u^{n-1}\right) \cdot G_h\ast\left( u^n_s-u^{n-1}\right) dx = -2\int \frac{u^n-u^{n-1}}{h}\cdot G_h \ast \left( \xi \cdot \nabla u^n\right)dx.
Q=s+\omega^{+}\delta^{+}+\omega^{-}\delta^{-}+\varepsilon^{+\mu}\delta_\mu^{+}+\varepsilon^{-\mu}\delta_\mu^{-}+v^\mu \partial_\mu -(\omega^{+}\varepsilon^{+\mu}+\omega^{-}\varepsilon^{-\mu})\frac \partial{\partial v^\mu}\,.
\sum_{i<j}\left\{{\lambda'_i}(t)\lambda_j(t)+\lambda_i(t){\lambda'_j}(t)\right\}f^i\wedge J_0 f^i\wedge f^j\wedge J_0 f^j = &-\frac{1}{\varepsilon^2_t}\sum_{i<j}\left[\left(\frac{n_1}{\lambda_1(t)}+\frac{n_2}{\lambda_2(t)}+\frac{n_3}{\lambda_3(t)}\right)\left(n_i\lambda_j(t)+n_j\lambda_i(t))\right)-2n_in_j\right]f^i\wedge J_0 f^i\wedge f^j\wedge J_0 f^j.
E_{\varphi_m}[\mu] = \left(\iint_{F_m \times F_m}+ \iint_{(X \times X) \setminus (F_m \times F_m)}\right) [G(p,q) + \frac12 (\varphi_m(p) + \varphi_m(q))] d \mu(p) d \mu(q) &\geq \mu(F_m)^2 E_{\varphi_m}[\tilde\mu] + (M+1)(1-\mu(F_m)^2).
I^0_{\Omega_1}(p)|\det J_{\mathbb{C}}f(p)|^2 =I^0_{\Omega_2}(f(p)), I^1_{\Omega_1}(p;X)|\det J_{\mathbb{C}}f(p)|^2 =I^1_{\Omega_2}(f(p);df(X)), I^2_{\Omega_1}(p;X,Y)|\det J_{\mathbb{C}}f(p)|^2 =I^2_{\Omega_2}(f(p);df(X),df(Y)),
f^{[2]}(z_0,1,z_2) =\frac{f^{[1]}(z_0,1)-f^{[1]}(1,z_2)}{z_0-z_2}=\frac{\frac{f(z_0)-f(1)}{z_0-1}-\frac{f(1)-f(z_2)}{1-z_2}}{z_0-z_2}& =\frac{g(z_0)-g(z_2)}{z_0-z_2}=g^{[1]}(z_0,z_2).
\int_Y Sf \; (\mu \times \lambda_G) = \int_X \int_G f(Tx, \phi(x)g) \; \lambda_G(g) \; \mu(x) &= \int_G \int_X f(Tx, g) \; \mu(x) \; \lambda_G(g)= \int_Y f \; (\mu \times \lambda_G).
S^{I}_{2,-1}\left( N, (1/2n \pi z)^2 \right) \equiv \frac{(-1)^N}{(2n\pi z)^{2N-2}}\int_0^{\infty}\frac{y^{2N-2}\,e^{-y}}{y^2+4 n^2 \pi^2 z^2}\; dy - \frac{1}{2 n z}& \times \;\; \sum_{j=1}^M (-1)^{M-j}\, \exp\Bigl( -2(-1)^{M-j}n i \pi z \Bigr) \;.
\begin{aligned}j^{(1)}\geq j^{(2)}\geq j^{(4)}\,,j^{(0)}\geq j^{(4)}\,, j^{(i)}\geq 0\,,i=0,1,...,4\,, &j^{(1)}+j^{(2)}+j^{(0)}+j^{(4)}\geq j^{(3)}\geq j^{(1)}+j^{(2)}+j^{(0)}-j^{(4)}\,. \end{aligned}
\log v_n+I_n =\log v_n-\log\sqrt{(u_n-x)^2+v_n^2}+\log\sqrt{(u_n-x+\delta)^2+v_n^2}&=-\log\sqrt{1+(u_n-x)^2/v_n^2}+\log\sqrt{(u_n-x+\delta)^2+v_n^2}
\int_{(P_{k+1},\dots,P_{g})\in X^{g-k}}\Phi^{*}\nu^{g}(P_{1},\dots,P_{g}) = (g-k)!\left(\tfrac{i}{2}\right)^{k}g!\sum_{1\le j_{1}<\dots<j_{k}\le g}\sum_{\rho,\tau\in\mathrm{Sym}(k)}\bigwedge_{m=1}^{k}\psi_{j_{m}}(P_{\rho(m)})\wedge\bar{\psi}_{j_{m}}(P_{\tau(m)}),
\begin{aligned}H^{s}(I;H) := [L^2(I;H),H^1(I;H)]_{{s}}\;, s\in (0,1)\;, H^{s}(I_>;H) := [L^2(I_>;H),H^1(I_>;H)]_{{s}}\;, s\in (0,1)\;, H^{s}_{0,\{0\}}(I_>;H) := [L^2(I_>;H),H^1_{0\{0\}}(I_>;H)]_{s}\;, s\in (0,1)\setminus\{\tfrac12\}\;, H^{\frac12}_{00,\{0\}}(I_>;H) := [L^2(I_>;H),H^1_{0,\{0\}}(I_>;H)]_{\frac12}\;. \ \end{aligned}
\sum_{m,v \in \mathcal{M}'}\sum_{p \in P}\frac{r(m)r(v)}{p^{\sigma}}\Phi\left( T\log\frac{mp}{v}\right) \ge \sum_{p \in P}\sum_{\substack{m,v \in \mathcal{M}mp=v}}\frac{f(m)f(v)}{p^{\sigma}}= \sum_{v \in \mathcal{M}}f(v)^2 \sum_{p|v}\frac{1}{f(p)p^{\sigma}}.
\begin{aligned}\mathbf{R}_{k,i}=\mathbf{I}+\sum\limits_{\pi_{k',i'}>\pi_{k,i}}\mathbf{W}_{k,i}\mathbf{H}_{k'}\mathbf{P}_{k',i'}\mathbf{P}_{k',i'}^H\mathbf{H}_{k'}^H\mathbf{W}_{k,i}^H.\end{aligned}
\sum_{i=1}^{n_1}\int_{B_{\rho}(y_i)}u^2(x,0)dx \le e^{M_1(1-\beta)}\sum_{i=1}^{n_1}\left(\int_{B_{\rho/2}(y_i)}u^2(x,0)dx\right)^{\beta}\left(N\int_{Q_{4}}u^2( x, t)\right)^{1-\beta}.
\lambda_1^{-1/2}= \frac{t}{\sqrt{2}}+O \left({t}^{3}\right) \lambda_2^{-1/2}= \sqrt{2}+O \left({t}^{2}\right) \lambda_3^{-1/2}= t+O \left({t}^{3}\right) \lambda_4^{-1/2}= \frac{1}{\sqrt{12}}t^2 + O \left({t}^{3}\right).
S_{1}'(\ell_{*}) = S_{2}'(r_{*}) = \left(\frac{g_{1}'(r_{*}) - f_{1}'(\ell_{*})}{f_{1}''(\ell_{*})(r_{*}- \ell_{*})}\right)\left( \frac{f_{2}'(r_{*}) - g_{2}'(\ell_{*})}{f_{2}''(r_{*})(r_{*}- \ell_{*})}\right).
\binom{n}{k}>\sum_{i=1}^{k-1}\left(\frac{j-1}{2}\right)^i\binom{n}{k-i}>\sum_{i=1}^{k-r-1}\frac{j-1}{2}\binom{n}{k-i}+\sum_{i=k-r}^{k-1}j\binom{n}{k-i}.
\min_j \frac{[\mathbf{D}\mathbf{v}_{\min}]_j}{[\mathbf{v}_{\min}]_j}=\frac{[\mathbf{D}\mathbf{v}_{\min}]_i}{[\mathbf{v}_{\min}]_i}< \rho(\mathbf{D}_{\min}) = \max_j \frac{[\mathbf{D}\mathbf{v}_{\min}]_j}{[\mathbf{v}_{\min}]_j}\,.
\fbox{\begin{array}{c l}{A{\circ_{X , Y}}B}&{= ( A X ) ( Y^{\dagger}B )}\\{}&{= A{\circ_{X}}( ( X Y^{\dagger}){\circ_{X}}B )}\\{}&{= ( A{\circ_{Y}}( X Y^{\dagger}) ){\circ_{Y}}B ,}\\ \end{array}}
\Sigma_{a_{1}, \ldots , a_{p}; \; b_{1}, \ldots , b_{q}; c}^{\; i_{1}, \ldots , i_{p}; \; j_{1}, \ldots , j_{q}}( u ) \equiv \sum_{j = 1}^{\infty}\frac{1}{\left( 2 j \atop j \right)}\frac{u^{j}}{j^{c}}[ S_{a_{1}}( j \! - \! 1 ) ]^{i_{1}}\ldots [ S_{a_{p}}( j \! - \! 1 ) ]^{i_{p}}\; [ S_{b_{1}}( 2 j \! - \! 1 ) ]^{j_{1}}\ldots [ S_{b_{q}}( 2 j \! - \! 1 ) ]^{j_{q}},
e^{C_h t}\bar Y_t^{1n}= e^{C_h T}g^1(X_T^{t,x})-\int_t^T C_he^{C_h r}(1+|\varphi_n(X_r^{t,x})|^\gamma)dr+\int_t^T e^{C_h r}\bar Z_r^{1n}dB_r^n.
\langle F_{\mu}^{i}[ \xi | s ] F_{\mu^{\prime}}^{i^{\prime}}[ \xi^{\prime}| s^{\prime}] \rangle = -{\frac{3 i}{4 a_{\xi^{\prime}}( s^{\prime})}}\delta^{i i^{\prime}}g_{\mu \mu^{\prime}}\delta ( s - s^{\prime}) \prod_{\bar{s}= 0}^{2 \pi}\delta^{4}( \xi ( \bar{s}) - \xi^{\prime}( \bar{s}) ) ,
\tau_{D'}^{-1}(T^{*}_{X}X(\log D')) =\bigcup_{I''\subset I'-\{r\}}T^{*}_{D_{I''}}X\cup\bigcup_{\substack{I''\subset I' r\in I''}}T^{*}_{D_{I''}}X &=\tau_{E'}^{-1}(T^{*}_{X}X(\log E'))\cup i_{r\circ}\tau_{i_{r}^{*}E'}^{-1}(T^{*}_{D_{r}}D_{r}(\log i_{r}^{*}E')).
|f(X_1,\dots, X_n)-f(X_1',\dots, X_n')| \leq D_{\gamma}\frac{\delta}{\bar g_r^2}\frac{\|\Sigma\|_{\infty}^{1/2}+\delta^{1/2}}{\sqrt{n}}\biggl(\sum_{j=1}^n \|X_j-X_j'\|^2\biggr)^{1/2}.
\hat{T}_{\mu\nu}= -\frac{1}{2}\left(\hat{F}_{\mu\lambda}\star\hat{F}_{\nu}^{\,\lambda}+ \hat{F}_{\nu\lambda}\star\hat{F}_{\mu}^{\,\lambda}- \frac{1}{2}\,\eta_{\mu\nu}\,\hat{F}_{\lambda\rho}\star\hat{F}^{\lambda\rho}\right)
\Delta(a\circ b) =\Delta(B_1(a_1)bS(B_2(a_2)))=B_1(a_1)b_1S(B_2(a_4))\otimes B_1(a_2)b_2S(B_2(a_3)) &=B_1(a_1)b_1S(B_2(a_2))\otimes B_1(a_3)b_2S(B_2(a_4))=a_1\circ b_1\otimes a_2\circ b_2