harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Vert g\Vert_{L_{\sigma}^{2}}^{2}=\int_{{\mathbb{R}^{3}}}|g|_{L_{\sigma}^{2}}^{2}dx\,, \quad\Vert g\Vert_{L_{\sigma}^{2}(m)}^{2}=\int_{{\mathbb{R}^{3}}}|g|_{L_{\sigma}^{2}(m)}^{2}dx\,.
Q_A V_k = U_{k+1}J_k, Q_A^T U_{k+1}= V_k J_k^T + \alpha_{k+1}v_{k+1}e_{k+1}^T, Q_B \widehat{V}_k = \widehat{U}_{k}\widehat{J}_k, Q_B^T \widehat{U}_{k}= \widehat{V}_k \widehat{J}_k^T + \hat{\beta}_{k}\hat{v}_{k+1}e_{k}^T,
g \lambda_r^{-1}\ = (f,(h_0,h_1))(-r\cdot (-\mathbf{e}_0),-\mathbf{e}_0) =\ (f- r \cdot (h_0,h_1)\cdot(- \mathbf{e}_0), \ (h_0,h_1) - \mathbf{e}_0) & = \ (f- r x^{h_0-1}(1+x)^{h_1}, (h_0-1,h_1)),
{\mathbf s}_{\textrm{dir}}[\widetilde{n}] = [s_{\textrm{dir}_1}[\widetilde{n}],\cdots,s_{\textrm{dir}_{N_r}}[\widetilde{n}]]^T =\sum_{k=1}^{K}\widetilde{\alpha}_{1,k}{\bf a}_r(\theta_t,f_k){\bf a}_t^T(\theta_t,f_k){\bf F}_k{\bf s}_ke^{\mathrm{j}2\pi{f_k}\frac{\widetilde{n}\triangle t}{K}},
F_{k;\epsilon_1\cdots\epsilon_k}(n_1,n_2;0)=\frac{(-1)^{n_0}}{k!}\frac{n_2!^2}{(2n_2\!-\!k)!}\;\frac{(2n_2\!-\!n_+\!-\!n_-)!}{\big(n_2\!-\!n_+\big)!\big(n_2\!-\!n_-\big)!}\; .
P= \frac{u^2}{2F}\varphi, Q^0= u^2\left\{\frac{1}{2\phi\Lambda}(\phi-z\phi_z)(\varphi_z-2\phi_{x^0})\Omega - (r^2-s^2)\left(z\frac{\phi_s}{\phi}U + \frac{\phi-z\phi_z}{\phi}V\right)\right\}, Q^i= -u^2\left\{sU+(r^2-s^2)(\frac{\phi_s}{\phi}U-\frac{\phi_z}{\phi}V) + \frac{\phi_z\Omega}{2\phi\Lambda}(\varphi_z-2\phi_{x^0})\right\}\frac{y^i}{u}+{u^2}Ux^i.
\begin{tabular}{r@{\, \, = \, \,}l}\large{$-\Gamma^{[r,0]}_{T}(k^{2}, \Lambda^{2})$}\Large{{$\frac{(k^{2})^{r+1}+ \zeta_{r,1}\Lambda^{2}(k^{2})^{r}+\ldots + \zeta_{r, r+1}(\Lambda^{2})^{r+1}}{(k^{2})^{r}+ \eta_{r,1}\Lambda^{2}(k^{2})^{r-1}+ \ldots+ \eta_{r,r}(\Lambda^{2})^{r}}\, ; $}}[4ex]$r$ $0,1,2,3, \ldots$ \, ;\end{tabular}
\operatorname*{sign}f^\prime(-1+h-a) = \operatorname*{sign}\left[ - \left(1 - \frac{b^2/a}{2 + a -h}\right)^2 + (1 - b^2/a^2)\right] = \operatorname*{sign}\left[ \frac{b^2/a^2}{(2+a-h)^2}\left( - (2+a-h)^2 + 2 a (2-h+a) - b^2\right)\right] \stackrel{(\ref{eq:lemma:inclusion-101})}{\leq}0.
\tilde\psi^G(t)=\textrm{e}_-^{-iH^\times t}|E_R+i\Gamma/2^+\rangle=\textrm{e}^{-iE_R t}\textrm{e}^{\Gamma t/2}|E_R+i\Gamma/2^+\rangle,\,\,\,\,\textup{only for}\,\,\,\,t\leq0.
\Lambda^{(1)}_{n,\delta}=\exp \left( i \mu Y^{n,1}_{t}+i \lambda n^{\alpha+\frac 12}\int_{t- \delta}^{\eta_n(t)}\psi_{n,1}(s,\eta_n(t)) dW_s \right) \times \left[\exp \left( i \lambda n^{\alpha+\frac 12}\int^{t- \delta}_0\psi_{n,1}(s,\eta_n(t)) dW_s \right) -1\right]
\gamma_{\scriptscriptstyle{l}}=\Phi/2 \pi - l +\frac{a^{2l}}{\int_0^a dr\,r^{2l-1}\,e^{2 \phi(r)}}-k^2\,a \chi_{\scriptscriptstyle{l}}^{\prime}(a) + \mbox{O}(k^4)\,.
(n+1)S_{n,l}S_{n,l}^{*}=t_{-1}(S_{n,l}S_{n+1,l}^{*}+S_{n,l}^{*}S_{n-1,l})(1-S_{n,l}S_{n,l}^{*})-l(1-S_{n,l}S_{n,l}^{*}),
\int\limits_0^T\int\limits_0^{\ell}\Big[B(x,t,v)\psi_t+v\psi_{xx}+f\psi\Big]\,dxdt + \int\limits_0^{\ell}B_0(x,0,\Phi(x))\psi(x,0)\,dx + \int\limits_0^Tp(t)\psi(\ell,t)\,dt-\int\limits_0^Tg(t)\psi(0,t)\,dt=0, \forall \psi\in W_2^{2,1}(D), \psi(x,T)=0, \psi_x(0,t)=\psi_x(\ell,t)=0.
\hat{\gamma}_{ij}d\hat{x}^id\hat{x}^j=\left(1+{\hat{k}\over 4}\delta_{mn}\hat{x}^m\hat{x}^n\right)\delta_{ij}d\hat{x}^id\hat{x}^j={{d\hat{r}^2}\over{1-\hat{k}\hat{r}^2}}+\hat{r}^2\left(d\hat{\theta}^2+\sin^2\hat{\theta}d\hat{\phi}^2\right).
M_{\mu,\nu}(y) = \frac{\Gamma(1+2\nu)}{\Gamma(\nu-\mu+\frac{1}{2})}e^{\pi i\mu}\mathcal{M}^+_{\mu,\nu}(y) + \frac{\Gamma(1+2\nu)}{\Gamma(\nu+\mu+\frac{1}{2})}e^{-\pi i (\nu-\mu+\frac{1}{2})}W_{\mu,\nu}(y), 2\nu \not\in \mathbb{Z}_{<0},
\left(\mathrm{Log}\,\mathrm{Tr}\,\mathrm{A}\right)^{-(\alpha+1)}_\lambda:=\frac{d}{d\alpha}\mathcal{M}_\lambda\left[\mathrm{Tr}\,\mathrm{E}^{-\mathrm{A}};\alpha\right]=\zeta'_{A_\lambda}(\alpha)\;,
(\widetilde f_2\otimes\widetilde f'_2)(\omega_{Z,Z'})= 2\,\rho_{Z^{(2)}}\otimes\left[\rho_{\binom{3}{2}}+\rho_{\binom{2}{3}}\right]= 2\sqrt 2\,\left[\rho^{(3)}_{Z^{(2)}}\otimes\rho^{(3)}_{Z'^{(2)}}\right].
|\frac{16}{\lambda(t)^{2}}\int_{t}^{\infty}dx \lambda'''(x) \left(K_{1}(x-t,\lambda(t)) -\frac{\lambda(t)^{2}}{4(1+x-t)}\right)| &\leq \frac{C}{\lambda(t)^{2}}\sup_{x \geq t}(|\lambda'''(x)|) \int_{t}^{\infty}|K_{1}(x-t,\lambda(t))-\frac{\lambda(t)^{2}}{4(1+x-t)}| dx &\leq C \sup_{x \geq t}|\lambda'''(x)|
M\int_{T_3}^t \tilde{I}_3(s)\,ds \leq K_1 \int_{T_3}^t \frac{x'(s)}{f^2(x(s))}\cdot \frac{Mf(x(s))}{x'(s)}\,ds &\leq 2K_1 \int_{T_3}^t \frac{x'(s)}{f^2(x(s))}\,ds=2K_1 \int_{x(T_3)}^{x(t)}\frac{1}{f^2(u)}\,du.
\begin{array}{r c l}{}&{}&{( \epsilon \otimes \mathrm{i d}) \Delta ( H ) = \epsilon ( H ) 1 + H = H ,}\\{}&{}&{( \epsilon \otimes \mathrm{i d}) \Delta ( X^{+}) = \epsilon ( X^{+}) q^{- H}+ \epsilon ( q^{H}) X^{+}= X^{+}.}\\ \end{array}
(\sqrt{a}-\sqrt{b})^2-a\nabla_{\nu}b =b\nabla_{\nu}a-2\sqrt{ab}&\geq-a\sharp_{\nu}b+\sum_{k=0}^{\infty}r_{k}\big[\big(a^{\frac{m_k}{2^k}}b^{1-\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{\frac{m_k+1}{2^k}}b^{1-\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2}.
x_{n+1}= t_nTx_n + (1-t_n)x_n &= T_{t_n}x_n &= T_{t'_n \cdot \frac{1-k}{d}}x_n &= T_{t'_n}(T_{\frac{1-k}{d}}x_n) &= t'_n T_{\frac{1-k}{d}}x_n + (1-t'_n) x_n .
\L_{1}(E_\infty) \geq \limsup_{Q\to \infty}\frac{(\sum_{n=N}^{Q}\L_{1}(E_n))^2}{\sum_{n,m=N}^{Q}\L_{1}(E_n\cap E_m)}&\geq \limsup_{Q\to \infty}\frac{(\frac{rd_{\lambda}(x')}{4})^2\left(\sum_{n=N}^{Q}h(n)\right)^2}{rd_{\lambda}(x')\left(\sum_{n=N}^{Q}h(n)+\left(\sum_{n=N}^{Q}h(n)\right)^2\right)}&\gg rd_{\lambda}(x').
\int_\Omega u^{\beta+1}=\lVert \sqrt{u}\rVert^{2(\beta+1)}_{L^{2(\beta+1)}(\Omega)}\leq 2^{2\beta+1}C_{GN}^{2(\beta+1)}\Big[\frac{m}{4^\beta}\Big(\int_\Omega \frac{\lvert \nabla u \rvert^2}{u}\Big)^\beta+m^{\beta+1}\Big]
\widetilde{\mathcal{Z}}_{T}= \widetilde{T}{(\widetilde{I}_{\mathcal{H}^{Jm}_{+}}+\widetilde{T}^{*}\widetilde{T})}^{-\frac{1}{2}}&= \widetilde{T}{(I_{\mathcal{H}}+ \widetilde{T}^{*}\widetilde{T})}^{-\frac{1}{2}}&= \mathcal{Z}_{\widetilde{T}}.
\left[w_{1}w_{2}x+w_{2}\left(j_{1}+\cdots+j_{r}\right)+w_{1}\left(y_{1}+\cdots+y_{r}\right)\right]_{q}= \left[w_{1}\right]_{q}\left[w_{2}x+\frac{w_{2}}{w_{1}}\left(j_{1}+\cdots+j_{r}\right)+\left(y_{1}+\cdots+y_{r}\right)\right]_{q^{w_{1}}}.
\hat{E}(\rho_1,\hat{\phi},\partial_{\rho_1}\hat{\phi},\partial^2_{\rho_1}\hat{\phi};{\cal S}_0\times{\cal S}_1)\equiv(k_{1,0}^2-{\bf k_1}^2)(\rho_1^2\partial^2_{\rho_1}\hat{\phi}+\rho_1\partial_{\rho_1}\hat{\phi})+m^2\hat{\phi}+\lambda\hat{\phi}^3=0.
\left( \begin{matrix}{{\omega_{1}^{\prime}}}\\{{\omega_{3}^{\prime}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{a_{1 1}}}&{{a_{1 2}}}\\{{a_{2 1}}}&{{a_{2 2}}}\\ \end{matrix}\right) \left( \begin{matrix}{{\omega_{1}}}\\{{\omega_{3}}}\\ \end{matrix}\right) , \quad a_{i j}\in{\bf Z}, \quad i , j = 1 , 2 , \quad a_{1 1}a_{2 2}- a_{1 2}a_{2 1}= 1 .
\mathrm{sf}(\hat{H}^\sharp|_{\overline{\alpha}_y}) = \mathrm{sf}(\hat{H}^\sharp|_{\gamma}) = \sum_{j = 1}^\ell \epsilon(f_j) I(f_j, \gamma)+ \sum_{j = 1}^{\ell^c}\epsilon(f^c_j) I(f^c_j, \gamma) &= \sum_{j = 1}^\ell \epsilon(f_j) I(f_j, \overline{\alpha}_y)+ \sum_{j = 1}^{\ell^c}\epsilon(f^c_j) I(f^c_j, \overline{\alpha}_y).
\lim_{T\to \infty}\frac{1}{T}\log \sum_{\gamma\in\Gamma:|\gamma|\le T}1= P_{top}(T_f^t,0)=h(f)\intertext{and}\lim_{T\to \infty}\frac{1}{T}\log \sum_{\gamma\in\Gamma:|\gamma|\le T}E_\gamma^{-1}= P_{top}(T_f^t,-\log \det DT_f^t) =0.
\|\Delta(|A(\eta_1+\eta_2)|) +\eta_1+\eta_2\|_X =\|\Delta(|A(\eta_2-\eta_1)|)+(\eta_2-\eta_1)+2\eta_1\|_X \geq2\|\eta_1\|_X-\|\Delta(|A(\eta_2-\eta_1)|)+(\eta_2-\eta_1)\|_X.
\underset{T\rightarrow\infty}{\lim}\int_{{\varLambda}^{k-1}}G(u_{1}-v_{1},\ldots,u_{k-1}-v_{k-1})\varPhi_{k,T}^{h}(u_{1},\ldots,u_{k-1})\,du_{1}\ldots du_{k-1}&\quad{}=G(v_{1},\ldots,v_{k-1}),
X_t=X_0 exp\Big(\int_{0}^t \Big\{\alpha_s-c_s-\frac{1}{2}\rho^2_s\Big\}ds+\int_{0}^t\rho_sdW_s\Big)exp\Big(-\int_{0}^t \mu_s \lambda_s ds \Big) \Big(1+\mu_\tau\mathbf{1}_{\tau\leq t}\Big).
\partial_t (u^2_t, m_t) = (u_t^2, \Delta m_t) - 2( |\nabla u_t|^2, m_t) - 2( F(u_t), \nabla(u_t m_t)) & - (u_t^2, \mathrm{div}( \beta_j m_t) ) \dot{Z}_t^j + (u_t^2, \partial_t m_t) & = (u_t^2, \Delta m_t+ \partial_t m_t - \mathrm{div}( \beta_j m_t) \dot{Z}_t^j ) - 2( | \nabla u_t|^2, m_t) - 2( F(u_t), \nabla(u_t m_t)).
\pi_S(b)\xi(x) = S_\mu S_\mu^*(\xi)(x) - \rho(\Lambda)^{(m-n)/2}S_\nu S_\mu^*(\xi)(x) &= \chi_{Z(\mu)}(x) \rho(\Lambda)^{m/2}S_\mu^*\xi (\sigma^m(x)) - \chi_{Z(\nu)}(x)\rho(\Lambda)^{m/2}S_\mu^*\xi(\sigma^n(x)).
\boldsymbol u_\nu(0)=\boldsymbol u_0,\quad \theta_\nu(0)=\theta_0,\qquad\Omega, \boldsymbol u_\nu\cdot \boldsymbol n=0,\quad\frac{\partial(\boldsymbol u_\nu\cdot\boldsymbol\tau)}{\partial\boldsymbol n}=0,\quad \theta_\nu=0,\partial\Omega.
\sum_{i=1}^d\sum_{j_i\geq J_{n,i}(\boldsymbol{\alpha})}2^{-j_i\mu_i}\prod_{l\neq i}^d\sum_{j_l<J_{n,l}(\boldsymbol{\alpha})}2^{-j_l\mu_l}\lesssim\sum_{i=1}^d2^{-J_{n,i}(\boldsymbol{\alpha})/2},
\mathfrak{S}(n) = \sum_{q=1}^\infty \frac{1}{\phi^4(q)}\sum_{\substack{a=1 (a,q)=1}}^q \bigg( \sum_{\substack{r=1 (r,q)=1}}^q e_q\big( ar^2 \big) \bigg)^4 e_q(-an), \mathfrak{I}(t) = \int_{-\infty}^\infty \bigg(\int_{1/3}^{2/3}e(x^2\gamma) \, dx \bigg)^4 e\big(-t\gamma\big) \, d\gamma.
= 2^{N}\prod_{a = 1}^{N}\sum_{n_{a}= 0}^{\infty}\frac{( \beta^{( a )})^{2 n_{a}}}{( 2 n_{a}) !}\prod_{b = 1}^{N}\sum_{q_{b}= 0}^{2 n_{b}}{\binom{2 n_{b}}{q_{b}}}E^{\mu}( \{q , n \}) ,
\sum_{n_{i}= 0}^{\infty}\left[ \begin{array}{c}{{n_{i}+ \tilde{n}_{i}+ \tilde{V}_{i , r}}}\\{{n_{i}}}\\ \end{array}\right] \tilde{x}^{n_{i}}= \frac{1}{( 1 - \tilde{x})^{\tilde{n}_{i}+ 1 + \tilde{V}_{i}, r}}.
J(-\tau,x)=\begin{bmatrix}\frac{\partial\Phi^1_\tau}{\partial x_1}\dotsb \frac{\partial\Phi^1_\tau}{\partial x_d}\vdots \vdots \frac{\partial\Phi^d_\tau}{\partial x_1}\dotsb \frac{\partial\Phi^d_\tau}{\partial x_d}\end{bmatrix}=\left(\frac{\partial\Phi^i_\tau}{\partial x_j}\right)_{i,j=1}^d.
\begin{array}{l}\Phi_3 \times \Phi_{\overline 3}=\Phi_1+\Phi_8\ ,\ \Phi_6\times\Phi_{\overline6}=\Phi_1+ \Phi_8\ ,\ \Phi_{10}\times \Phi_{\overline{10}}=\Phi_1 \Phi_8\times\Phi_8=\Phi_1+2\Phi_8+\Phi_{10}+\Phi_{\overline{10}}\ ,\ \Phi_{15}\times\Phi_{\overline{15}}=\Phi_1+\Phi_8\end{array}
R^d_{c, ab}=\frac{\partial L^d_{cb}}{\partial z^a}-\frac{\partial L^d_{ca}}{\partial z^b}+L^f_{cb}L^d_{fa}-L^f_{ca}L^d_{fb}\,,\,R^{\overline{d}}_{\overline{c}, \overline{a}\,\overline{b}}=\overline{R^d_{c, ab}},
M_{b c a}^{( s , t , r ) \d}\equiv \hat{G}_{a b}\delta_{c}^{d}\delta_{r + s , 0 \hspace{. 0 5 i n}m o d \hspace{. 0 4 i n}\lambda}+ \hat{G}_{a c}\delta_{b}^{d}\delta_{r + t , 0 \hspace{. 0 5 i n}m o d \hspace{. 0 4 i n}\lambda}- f_{a b}^{\ \e}f_{e c}^{\ \d}.
F_{1}= - \, \frac{1}{2 4}\log \Bigl \{\biggl ( N_{1}+ 2 \varphi \frac{1}{y_{+}^{3 / 2}\, y_{-}^{1 / 2}}\biggr ) \biggl ( K_{1}+ 2 \varphi \frac{1}{y_{+}^{1 / 2}\, y_{-}^{3 / 2}}\biggr ) ( y_{+}- y_{-})^{4}\Bigr \}.
g_{t,l,v}= \left(\begin{matrix}\varpi_{p}^t 0 0 1\end{matrix}\right)w\left(\begin{matrix}1 v\varpi_{p}^{-l}0 1\end{matrix}\right) = \left(\begin{matrix}0 \varpi_{p}^t -1 -v\varpi_{p}^{-l}\end{matrix}\right).
\frac{\sqrt{2}}{2}= \sum_{n=0}^\infty \frac{3 + 184n -336n^2}{(4n-1)(4n-3)}\frac{\binom{4n}{2n}}{2^{10n}}, \frac{\pi^2}{2}= \sum_{n=1}^\infty \frac{(14n-3)(3n-1)}{n^3(2n-1)}\frac{16^n}{\binom{4n}{2n}^2\binom{2n}{n}}.
u(1/2, 1/2, \tau, \sigma, 1/8) = - (1 - i) \frac{\Gamma(3/4; \tau, \sigma)}{\Gamma(1/4; \tau, \sigma)}\frac{1}{(\tau; \tau)(\tau+1/2; 2\tau)}\frac{1}{(\sigma; \sigma)(\sigma + 1/2; 2\sigma)}.
\mathbf{u}^{k+1}= (2 - \tau^2 (M^{L})^{-1}K)\mathbf{u}^k - \tau^2 (M^{L})^{-1}G \mathbf{u}^k -\mathbf{u}^{k-1}, \boldsymbol{\lambda}^{k-1}= - \tau^2 (M^{L})^{-1}G \boldsymbol{\lambda}^k + (2 - \tau^2 (M^{L})^{-1}K) \boldsymbol{\lambda}^k -\boldsymbol{\lambda}^{k+1}.
\psi_k(\xi) =\psi(\xi/2^k)-\psi(\xi/2^{k-1}), \psi_{\leq k}(\xi)=\psi(\xi/2^k),\psi_{\geq k}(\xi)=1-\psi(\xi/2^{k-1}), \tilde\psi_k(\xi) =\psi_{k-1}(\xi)+\psi_k(\xi)+\psi_{k+1}(\xi).
H_{0}= \int d^{3}x \biggl ( \psi^{\dagger}( i \mathrm{\boldmath \alpha}\cdot{\bf D}+ \gamma_{0}m ) \psi +{\frac{1}{2}}\mathrm{\boldmath \pi}^{2}+{\frac{1}{4}}F^{i j}F_{i j}\biggr ) .
\widehat{C}_{0,T}(f,u) =C^{/T}_0(f,u)-\frac{|\partial T|_1}{\pi |T|}C^{/T}_1(f,u)+\left(\frac{1}{2\pi}\left(\frac{|\partial T|_1}{|T|}\right)^2-\frac{1}{|T|}\right)C^{/T}_2(f,u), \widehat{C}_{1,T}(f,u) =C^{/T}_1(f,u)-\frac{|\partial T|_1}{2 |T|}C^{/T}_2(f,u), \widehat{C}_{2,T}(f,u) =C^{/T}_2(f,u) .
P^{(j + \sigma)}= \eta_j I - \sigma\Big(\frac{\gamma^{(j + \sigma)}}{2h}s(Q) + \frac{D^{(j + \sigma)}_{+}}{h^{\beta}}s(W_{\beta}) + \frac{D^{(j + \sigma)}_{-}}{h^{\beta}}s(W^{T}_{\beta})\Big),
[g] = \underbrace{\begin{bmatrix}1 0 0 r^2Z^2\lambda_i\delta_{ij}\end{bmatrix}}_{=: M}+ \underbrace{\begin{bmatrix}\frac{\partial u}{\partial r}\frac{\partial u}{\partial\theta_i}\end{bmatrix}}_{=:\eta}\begin{bmatrix}\frac{\partial u}{\partial r}\frac{\partial u}{\partial\theta_j}\end{bmatrix}.
e_{\bot}(x_{\bot}) = g \nabla_{\bot}\int dy^{-}dy_{\bot}d(x_{\bot}- y_{\bot})\left \{f^{3ab}A^a_{\bot}(y_{\bot}, y^-) \Pi^b_{\bot}(y_{\bot}, y^-) + \rho^3_m(y_{\bot}, y^-) \right \}\frac{\tau^3}{2}\, .
Y_n (T)-Y_n(T-1) &=\mathbb E_0\bigg[\exp\big\{H_{\beta,T}(W,B)\big\}I_n(T,W_T) -\exp\big\{H_{\beta,T-1}(W,B)\big\}I_n(T,W_T) &\qquad+\exp\big\{H_{\beta,{T-1}}(W,B)\big\}\Big(I_n(T,W_T)-I_n(T-1,W_{T-1})\Big)\bigg] &=\mathbb E_0\bigg[\exp\big\{H_{\beta,T-1}(W,B)\big\}\,\Big(\exp\big\{H_{\beta,T-1,T}(W,B)\big\}-1\Big)\,I_n(T,W_T)\bigg],
\frac12\frac{d}{dt}(\|\nabla w\|_{L^2(\Omega)}^2+\|w\|_{L^2(\Omega)}^2)+\int_\Omega e^{-v}u^2dx=\int_\Omega e^{-v}uwdx\leq \frac12\int_\Omega e^{-v}u^2dx+\frac12\int_\Omega e^{-v}w^2dx.
g^{\otimes n}=(g-I_N+I_N)^{\otimes n}&=I_N^{\otimes n}+\displaystyle\sum_{\substack{1\leq m \leq n 1\leq k_1<\ldots<k_m \leq n}}\iota_{k_1,\ldots,k_m}((g-I_N)^{\otimes m}),
d \bar{s}^{2}\equiv \mathrm{e}^{2 \phi}d s^{2}= \Gamma \left[ \frac{( d t + 2 N \cos \theta d \varphi )^{2}}{( r + 2 M )^{2}}- \frac{d r^{2}}{r^{2}}- d \Omega^{2}\right]
\log Z_{2n+1,k,\alpha}^{pGUE}=\sum_{j=0}^{n}\log \widehat h_{2j}(2n+1,k,\alpha)+\sum_{j=0}^{n-1}\log \widehat h_{2j+1}(2n+1,k,\alpha) &=\sum_{j=0}^{n}\log h_j(n+\frac{1}{2},k,\alpha-\frac{1}{2})+\sum_{j=0}^{n-1}\log h_j(n+\frac{1}{2},k,\alpha+\frac{1}{2}) &=\log A(n+1,n+\frac{1}{2},k,\alpha-\frac{1}{2})+\log A(n,n+\frac{1}{2},k,\alpha+\frac{1}{2}),
A = 2 \pi^{2}( r_{0})^{3}={\frac{\pi^{2}}{3}}\left\{( C^{I J}q_{I}q_{J})_{\partial i Z = 0}\right\}^{3 / 4}\qquad \Longrightarrow \qquad 8 \pi \sqrt{\frac{Q_{H}( Q_{F})^{2}}{2}}\ ,
\operatorname{supp}\hat{f}\subset\lambda\mathcal{C}\ \ \Rightarrow\ \C^{-1-k}\lambda^{k}\|f\|_{L^{a}}\leq \sup_{|\alpha|=k}\|\partial^{\alpha}f\|_{L^{a}}\leq C^{1+k}\lambda^{k}\|f\|_{L^{a}}.
\Big\| ( ax, by ) - \frac{1}{m}\sum_{i=1}^{m}\big(a x_i, b y_i \big) \Big\|_N = N\Big( a \Big\| x - \frac{1}{m}\sum_{i=1}^{m}x_i \Big\|, b \Big\| y - \frac{1}{m}\sum_{i=1}^{m}y_i \Big\| \Big) \leq N(\gamma a, \gamma b) = \gamma N(a, b) = \gamma,
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=(2^{(\dim M_2^{K_{\ell}}-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^{K_{\ell}}|})(2^{(\dim M_2^L-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^L|})^{-1}\leq 2^{\frac{1}{2}}(2^{n_{2,\nu_2}}-1)
\left[q^d\right] \frac{(-1)^{(N-1)d}}{\mathsf V_{N+1}}\,\begin{vmatrix}z_1^{d+N}z_1^{d+N-1}\cdots z_1^{d+1}z_2^{d+N}z_2^{d+N-1}\cdots z_2^{d+1}\vdots \vdots \cdots \vdots z_N^{d+N}z_N^{d+N-1}\cdots z_N^{d+1}\end{vmatrix}\prod_{i=1}^{N}\bigg( \frac{\alpha_i-z_{N+1}}{\alpha_i}\bigg)^{b_i}\bigg{|}_{\epsilon=0}.
Q^{N_p/2}=Q^{N_{n-h}/2}Q^{N_h/2}=\sum_{\{\epsilon_i=\pm1\}}\cos\left[\frac{\pi}{2}e_0\sum_{i}\epsilon_i\right]\sum_{\{\epsilon_j=\pm1\}}\exp\left[4i\lambda\sum_j \epsilon_j\right]
C \left(\frac{a_{N}^{2}}{\left\vert \Lambda_{N}\right\vert^{4}}\sum_{i, j\in \Lambda_{N}}\Psi(x_{i}, x_{j})^{2}\right)^{2}= C \left(\frac{a_{N}^{2}}{\left\vert \Lambda_{N}\right\vert^{4}}\sum_{i, j\in \Lambda_{N}}d(x_{i}, x_{j})^{-2\alpha}\right)^{2}.
\lambda =\frac{2k(k-1)(k-2)(m-1)!(m-2)!}{(m+1)(m^2-2) |G_\Delta|}&=\frac{72\cdot 35\cdot 34\cdot 10!\cdot 9!}{12\cdot 119\cdot (24)^2}&= \frac{(10!)^2}{96}=137,168,640,000.
p_{\varepsilon}(\rho)=\begin{cases}p(\rho) \mbox{for}\rho \in [0,\overline{\rho}-\varepsilon] p(\overline{\rho}-\varepsilon) + |(\rho - \overline{\rho}+\varepsilon)^{+}|^{\gamma}\mbox{for}\rho \in (\overline{\rho}-\varepsilon,\infty),\end{cases}
\mathbb{E}(t):=\Xi(t)+\beta\Big[\alpha\Xi_1^{(0)}(t)+\Xi_2^{(0)}(t)\Big]+\beta\Big[\Xi_3(t)+\beta\Xi_4(t)\Big]+\beta^3\Big[\alpha\Xi_1^{(1)}(t)+\Xi_2^{(1)}(t)\Big],
F_{\beta_{s}}F_{\beta_{l+2}}-q^{(\beta_s,\beta_{l+2})}F_{\beta_{l+2}}F_{\beta_{s}}=\sum_{n_{s+1},\ldots,n_{l+1}\geq 0}c(n_{s+1},\ldots,n_{l+1})F_{\beta_{s+1}}^{n_{s+1}}\ldots F_{\beta_{l+1}}^{n_{l+1}}.
| \Psi_{( n , 0 )}\rangle = \int_{0}^{P^{+}}{\frac{d q_{1}\dots d q_{n}}{\sqrt{q_{1}. . . q_{n}}}}\delta ( q_{1}+ \cdots + q_{n}- P^{+}) \hspace{1 m m}f^{( n , 0 )}( q_{1}, \dots , q_{n}) \mathrm{t r}[ a^{\dagger}( q_{1}) \dots a^{\dagger}( q_{n}) ] | 0 \rangle .
E(Iu(T)) = \lambda^k E(Iu_\lambda(\lambda^k T)) = \lambda^k E(I u_\lambda(C_1 N^{\gamma_0(k)-}\delta)) &\sim \lambda^k \leq N^{\frac{k(k/2-\gamma)}{\gamma}}\sim T^{\frac{k(k-2\gamma)}{2(\gamma_0(k)+k)\gamma-k^2}+}.
(i)~R_{\omega}= \int_{\mathbb T}\varrho(\theta, \omega, t) \cos(\theta - \phi_{\omega}) ~ d\theta, 0 = \int_{\mathbb T}\varrho(\theta, \omega, t) \sin(\theta - \phi_{\omega}) ~ d\theta. \cr (ii)~R = \int_{\mathbb R}g(\omega) R_\omega \cos(\phi_\omega - \phi) ~ d\omega, 0 = \int_{\mathbb R}g(\omega) R_\omega \sin(\phi_\omega - \phi) ~ d\omega.
\mathbf{0}=\left(M^{n+k}+c_{k-1}M^{n+k-1}+c_{k-2}M^{n+k-2}+\cdots+c_0M^n\right)\mathbf{B}(0) =\mathbf{B}(n+k)+c_{k-1}\mathbf{B}(n+k-1)+c_{k-2}\mathbf{B}(n+k-2)+\cdots+c_0\mathbf{B}(n).
e_{(l+1)}=\Big(\frac{q-q^{-1}}{z-z^{-1}}\Big)^{l-2}e_{(l-1)}H_{2l-2}H_{2l-1}H_{2l}H_{2l+1}H_{2l-3}^{-1}H_{2l-2}^{-1}H_{2l-1}^{-1}H_{2l}^{-1}e_{(l)}.
Q_{0}= - \sum_{\kappa = - \infty}^{\infty}\vert \kappa \vert \left\{N_{\kappa}^{+}- N_{\kappa}^{--}n_{l}^{+}+ n_{\overline{{l}}}^{-}+ \frac{2}{\pi}\int_{0}^{\infty}\mathrm{d}r \; \varphi \left( r \right) \right\}\, ,
a_{10}^{1}(0) = a_{01}^{2}(0) = 1, \,\, a_{01}^{1}(0) = a_{10}^{2}(0) = 0, \,\, a_{kl}^{j}(0) = 0 \,\,\, (j = 1, \, 2, \, k+l \geq 2).
I = \frac{1}{1 6 \pi G_{N}}\int d^{4}x \sqrt{- g}\left[ R + \tilde{a}_{1}R^{2}+ \tilde{a}_{2}R_{\nu}^{\mu}R_{\mu}^{\nu}+ \tilde{a}_{3}R_{\rho \sigma}^{\mu \nu}R_{\mu \nu}^{\rho \sigma}\right] ,
-\mathcal{L}_{\epsilon,0}^{\gamma}\bigl(\cdot\bigr) \bigl( x \bigr) = \Bigl \langle \triangledown \bigl(\cdot\bigr),\, \bigl( A + \sum\nolimits_{i=1}^n B_i \gamma_i\bigr)x \Bigr \rangle + \frac{\epsilon}{2}\operatorname{tr}\Bigl \{\sigma(x) \sigma^T(x) \triangledown^2 \bigl(\cdot\bigr) \Bigr\},
res(A,B)=a_{0}^{e}b_{0}^{d}\prod_{\begin{matrix}1\le i\le d\\ 1\le j\le e\end{matrix}}(\lambda_{i}-\mu_{j})=a_{0}^{e}\prod_{i=1}^{d}B(\lambda_{i})=(-1)^{de}b_{0}^{d}\prod_{j=1}^{e}A(\mu_{j}),
\|\widetilde{c}_{jl}\| = \left(\begin{array}{cccc}1 \widetilde{c}' \widetilde{c}' \widetilde{c}' \widetilde{c}' 1 1/2 1/2 \widetilde{c}' 1/2 1 1/2 \widetilde{c}' 1/2 1/2 1 \end{array}\right) ,\quad\widetilde{c}' = - \sqrt{\frac{1+3c}{2(1+2c)}}.
\begin{aligned}\phi(f^L) + \sum_{i=1}^d\Biggl[ \phi\Biggl( \sum_{j=1}^{i-1}e_{\omega_j}f_{\omega_j}^U + e_{\omega_i}f_{\omega_i}+ \sum_{j=i+1}^de_{\omega_j}f_{\omega_j}^L \Biggr) & - \phi\Biggl( \sum_{j=1}^{i-1}e_{\omega_j}f_{\omega_j}^U + \sum_{j=i}^de_{\omega_j}f_{\omega_j}^L \Biggr) \Biggr].\end{aligned}
\begin{aligned}\hat{H}- H(f) = H(f_h) - H(f) + \hat{H}- H(f_h) &= H(f_h) - H(f) + \left(\hat{H}_{\mathsf{discrete}}+ \sum_{i=1}^{S}p_i\ln p_i\right) &=h^d\sum_{i=1}^{S}(-\frac{p_i}{h^d}\ln \frac{p_i}{h^d}+\frac{1}{h^d}\int_{I_i}f(t)\ln f(t)dt) + \left(\hat{H}_{\mathsf{discrete}}+ \sum_{i=1}^{S}p_i\ln p_i\right). \end{aligned}
\chi: T^*Y \setminus 0 \to T^*X\setminus 0 (y', y_n, \eta', \eta_n) \mapsto (x'(y', y_n, \eta', \eta_n), x_n(y', y_n, \eta', \eta_n), \xi'(y', y_n, \eta', \eta_n), \xi_n(y', y_n, \eta', \eta_n)),
\overline{e}_{b}(k,i+1)=\frac{\overline{e}_{b}(k-1,i)-\overline{\delta}(k,i)\overline{e}_{f}(k,i)}{\sqrt{(1-\overline{\delta}^{2}(k,i))(1-\overline{e}_{f}^{2}(k,i))}}
{}_2F_{1}\left(\frac{1}{2},\frac{1}{2};1;x^2\right)=\sum^{\infty}_{n=0}\frac{\left(\frac{1}{2}\right)^2_n}{(n!)^2}x^{2n}=\frac{2}{\pi}K(x)=\frac{2}{\pi}\int^{\pi/2}_{0}\frac{d\phi}{\sqrt{1-x^2\sin^2(\phi)}}
1-G_1\big((\alpha^*_nx+\beta^*_n)^{\frac{1}{p}}\big)=n^{-1}e^{-x}\left\{1-\frac{(1-p)x^2}{2\log \frac{n}{2}}+\frac{(1-p)[3(1-p)x-4(1-2p)]x^{3}}{24(\log \frac{n}{2})^{2}}+o(\frac{1}{(\log \frac{n}{2})^{2}})\right\}.
(q^3-1) \Big(q^2+(q^2-1)q(q+q^2+3q^3+q^4) + (q^2-1)(q-1)(q^3+3q^4+4q^5+4q^6+3q^7)\Big) &=(q^3-1)\Big(q^6+(q^2-1)q^3(3q^3+q^4)+(q^2-1)(q-1)(3q^6+3q^7)\Big).
(\bar{\xi}\circ \pi)(\mathfrak{a})= (\bar{\xi}\circ \pi)(\mathfrak{a}'\odot a_{i+1})=\bar{\xi}(\pi(\mathfrak{a}'))\cdot_{A}\bar{\xi}(\pi( a_{i+1})) = \bar{\xi}'(\pi(\mathfrak{a}'))\cdot_{A}\bar{\xi}'(\pi( a_{i+1}))=(\bar{\xi}'\circ \pi)(\mathfrak{a}'\odot a_{i+1})=(\bar{\xi}'\circ \pi)(\mathfrak{a}).
\sqrt{g}g^{ij}\delta^{kl}W_{ik}W_{jl}={\mbox{Trace}}(O^{T}DOWW^{T}) & ={\mbox{Trace}}(OO^{T}DOWW^{T}O^{T}) & ={\mbox{Trace}}(D\left( OW\right) (OW)^{T}) & \geq\min_{i}D_{ii}\cdot\mbox{Trace}\left( (OW\right) (OW)^{T}) & =\min_{i}D_{ii}\left\Vert OW\right\Vert_{HS}^{2}& =\min_{i}D_{ii}\left\Vert W\right\Vert_{HS}^{2},
a_1(k)=\frac{1}{k}(v_1\bar{s}_1-\bar{v}_1s_1-v_2\bar{s}_2+\bar{v}_2s_2)|_{x=0,t=0}+O(1), k\rightarrow 0, a_2(k)=k\frac{2i}{A}(v_1\bar{h}_1+\bar{v}_1h_1-v_2\bar{h}_2-\bar{v}_2h_2)|_{x=0,t=0}+O(k^2), k\rightarrow 0, b(k)=v_1\bar{h}_1-v_2\bar{h}_2+\frac{2i}{A}(v_1\bar{s}_1-v_2\bar{s}_2)|_{x=0,t=0}+O(k), k\rightarrow 0.
\mathcal{C}:=\{{v}\in\mathcal{S}_{sym}(a^-,a^+)\;:\; \mathfrak{E}_{W}(v)\leq C_0 \mbox{and}\forall t\geq 2S,\, |{v}(t)-a^+|=|{v}(-t)-a^-|\leq E(t)\}.
X_{i j k l}={\frac{1}{\Delta_{i}\Delta_{j}\Delta_{k}\Delta_{l}}}\left\{d_{i j k l}-{\frac{1}{2}}\left( \sum_{m}c_{i j m}{\frac{1}{\Delta_{m}}}c_{m k l}+ \mathrm{c r o s s i n g ~ t e r m s}\right) \right\}^{2}\F \ .
(\bar{u}_\ell^{(\nu+1)},\bar{p}_\ell^{(\nu+1)}) - (u_\ell,p_\ell) = [I - P_{\ell}^{\ell-1}] [(u_\ell^{(\nu)}, p_\ell^{(\nu)}) - (u_\ell,p_\ell)],
\int_0^t \| e^{( t - s ) A}F( Z_s ) \|_{L^p( \P; H_r )}ds \leq\sup\nolimits_{s\in[0,T]}\| F(Z_s)\|_H\int_0^t \| (-A)^{r}e^{(t-s)A}\|_{L(H)}\, ds &\leq\sup\nolimits_{s\in[0,T]}\| F(Z_s)\|_H\int_0^t (t-s)^{-r}\, ds=\sup\nolimits_{s\in[0,T]}\| F(Z_s)\|_H\tfrac{t^{1-r}}{1-r}<\infty
\lim_{y\to0+}\sqrt{(x+iy-\alpha)^2 - 4\beta}= \begin{cases}-\sqrt{(x-\alpha)^2 - 4\beta}, x < \alpha - 2\sqrt{\beta}i \sqrt{4\beta - (x-\alpha)^2}, x\in [\alpha - 2\sqrt{\beta}, \alpha + 2\sqrt{\beta}] \sqrt{(x-\alpha)^2 - 4\beta}, x >\alpha + 2\sqrt{\beta}\end{cases}.
P^{+}=\int_{\frac{Ag(0)}{L-1}}^{\infty}\frac{1}{\sqrt{2\pi}\sigma_{N}}e^{-\frac{x^{2}}{2\sigma_{N}^{2}}}dx=\frac{1}{2}erfc(\frac{Ag(0)}{\sqrt{2}(L-1)\sigma_{N}})
(f,g)~~~\leftrightsquigarrow ~~~G(x_0,...,x_{n+1})=\delta\Big(\frac{x_{n+1}}{x_0}-f\Big(\frac{x_1}{x_0},...,\frac{x_n}{x_0}\Big)\Big)~g\Big(\frac{x_1}{x_0},...,\frac{x_n}{x_0}\Big)~x_0^{-\frac{n+2}{2}}
\underline{\Psi}_{n}(\xi, \tau) = \left(\begin{array}{cc}e^{i \tau (1 + \mu_{0}\xi)}0 0 e^{-i \tau (1 - \mu_{0}\xi)}\end{array}\right) &\times\left(\begin{array}{cc}\cosh(\tau \partial_{\xi}) \sinh(\tau \partial_{\xi}) \sinh(\tau \partial_{\xi}) \cosh(\tau \partial_{\xi})\end{array}\right)\underline{\Psi}_{n-1}(\xi, \tau),
L^{(3)}w_{n3}=0\ \ \mbox{in}\ \D_1^{(3)},\ \ w_{n3}\big|_{\partial'D_1^{(3)}\setminus(\Gamma_0\times(t_1,t_2))}=0,\ \w_{n3}\big|_{\Gamma_0\times(t_1,t_2)}=f_n, w_{n3}\big|_{\partial_-D_1^{(3)}}=\frac{\partial w_{n3}}{\partial x_0}\Big|_{\partial_-D_1^{(3)}}=0.
\sup_{R \leq 1}\frac{1}{R^{2\alpha}}\displaystyle\inf_{l \in \textrm{Span}\{1, x_1 \}}\| u - \sigma_i(x_0) v_{i}(\cdot, a(x_0)) - l \|_{B_R(x_0)}& \lesssim (\epsilon^{2 - 2\alpha}+ \epsilon^{2 \alpha}) M + (\epsilon^{-(4-2\alpha)}+ \epsilon^{-4}+1)\tilde{N}.
\int{\cal D}A_{\mu}^{a}{\cal D}B^{a}{\cal D}\bar{c}^{a}{\cal D}c^{a}\exp \{- S_{Y M}( A_{\mu}^{a}) + \int d x [ - i B^{a}( \partial^{\mu}A_{\mu}^{a}) + \bar{c}^{a}( - \partial_{\mu}( D^{\mu}c )^{a}] \}
(\mathbb{F}_{p}\otimes_{\mathbb{Z}}^{L}\mathbb{F}_{p})\langle x\rangle=\frac{(\mathbb{F}_{p}\otimes_{\mathbb{Z}}^{L}\mathbb{F}_{p})[x_{1},x_{2},...]}{x_{i}x_{j}=(\begin{matrix}i+j\\i\end{matrix})x_{i+j}}