harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G_{T}W_{\mathrm{e f f}; i}^{\ast}\frac{\pi_{\lambda}}{\pi_{i}}\left( 1 - \alpha_{B}\right) = G_{S}W_{\mathrm{e f f}; \lambda}^{\ast}\left( 1 - \alpha_{B}\right) + M_{i}^{2}+ \beta_{B}X_{B}
\Psi={\rm Tr}\int\left(\bar c\,\,d^\ast\!\!\,A+\bar b\wedge\,\,d^\ast\,\!B+\bar a\,\,d^\ast\!\, \Omega+\bar\theta\,\,d^\ast\,\!b-\,d^\ast\,\!\bar b\wedge\Xi\right)
c^{-(i_1+1)}=c^{-2}\cdot c^{-i_1+1}\geq c^{-2}\cdot c^{-i_0\alpha_2}\geq \left(c^{-2}\cdot c^{-i_0}\right)^{\alpha_2}\geq \left(c^{-2}\cdot c^{-i_0}\right)^{\alpha_2'}= \left(c^{-3}\cdot c^{-i_0+1}\right)^{\alpha_2'}\geq \left(\delta c^{-i_0+1}\right)^{\alpha_2'}> c^{-i_2},
\int_{B_r}div_{\mathbb{G}}F |\nabla_{H}v|^2 - 2\sum_{i=1}^{m}\int_{B_r}X_i v [X_i, F]v - 2 \int_{B_r}Fv \Delta_{H}v = \int_{\partial B_r}|\nabla_{H}v|^2 < F, \nu> -2\sum_{i=1}^{m}\int_{\partial B_r}Fv X_iv < X_i, \nu>.
K(x_1,x_2,x_3)=\frac{1}{\pi^2\det\Theta}\,\exp\left(-2i\bigl[x_1^{\alpha}(\Theta^{-1})_{\alpha\beta}x_2^{\beta}+x_2^{\alpha}(\Theta^{-1})_{\alpha\beta}x_3^{\beta}+x_3^{\alpha}(\Theta^{-1})_{\alpha\beta}x_1^{\beta}\bigr]\right),
\mathbf{Q}\left[:,:,0\right]=\left(\begin{array}{rr}\left(x^{3}+1\right)^{-\frac{1}{3}}\left(\frac{1}{x^{3}}+1\right)^{-\frac{1}{3}}-x 1\end{array}\right),\mathbf{Q}\left[:,:,1\right]=\left(\begin{array}{rr}1 1 \left(x^{3}+1\right)^{-\frac{1}{3}}\left(\frac{1}{x^{3}}+1\right)^{-\frac{1}{3}}\end{array}\right),
\hat{A}\geq \frac{\alpha+1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-2\vert L_{A\Omega_1}\vert -2\vert U_{A\Omega_1}\vert = \frac{\alpha+1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-2\vert B_{A\Omega_1}\vert:= T.
\zeta ( s , z ) = \frac{z^{1 - s}}{s - 1}+ \frac{z^{- s}}{2}+ \frac{s}{1 2}z^{- s - 1}- \frac{s ( s + 1 )}{2 !}\int_{0}^{\infty}\frac{\overline{{B}}_{2}( t )}{( z + t )^{s + 2}}d t .
\|I_2(D^{\beta_\perp}\mathcal{E}^{k_s}_\rho)\|_{L^2_{\gamma,\mu}(C(\Lambda[0,\frac{L}{2}],2\rho))}\le \|I_2(D^{\beta_\perp}\mathcal{E}^{k_s}_\rho)\|_{L^2_{\gamma,\mu}(\widetilde{\bf C}_0^{2\rho})}+ \|I_2(D^{\beta_\perp}\mathcal{E}^{k_s}_\rho)\|_{L^2_{\gamma,\mu}(\cup_{j=3}^{J-1}{\bf C}_j^{2\rho})}&=: I + II.
\mathcal{C}:= \left\{R_{(\omega_1 , \ldots , \omega_N)}\cap ( \pi^{ss}_\xi )^{-1}A : \omega_1 , \ldots , \omega_N \in \{0 , \ldots , \ell -1 \}, \, N \in \mathbb{N}\mbox{and}A \in \mathcal{B}( \mathbb{R}) \right\}
\pi R \cdot Re(dZ_{\gamma_m}\wedge d\overline{Z}_{\gamma_e}) = R[S_1 - \ln|c|]dc_1 \wedge dc_2 & \frac{1}{2\pi R}\cdot d\theta_{\gamma_m}\wedge d\theta_{\gamma_e}= i\partial \bar{\partial}\left[ \frac{t_1^2}{R(S_1 - \ln |c|)}\right] = i\partial \bar{\partial}\left[ \frac{S_1-\ln|c|}{4\pi^2 R}\cdot \theta^2_{\gamma_e}\right]
\|f_y\|_{L^4}^4 = - \sum\limits_{N_1\sim N_2}\int f_{N_1}f_{N_2,yy}(f_{\leq N_2,y})^2\,dxdy - 2\sum\limits_{N_1\sim N_2}\int f_{N_1}f_{N_2,y}f_{\leq N_2,y}f_{\leq N_2,yy}\,dxdy &\lesssim \|f_x\|_{L^\infty}\left(\|C_1[\partial_x^{-1}f_{yy},f_y,f_y]\|_{L^1}+ \|C_2[f_y,f_y,\partial_x^{-1}f_{yy}]\|_{L^1}\right),
d\frac{(1/f)^{(d-1)}}{(1/f)^{(d)}}=d\frac{(-1)^{d-1}(d-1)!c_{0}}{(-1)^{d}d!c_{0}}(x-a)(\frac{1+O((x-a)^{d})}{1+O((x-a)^{d+1})})
\limsup_{t\to\infty}\left(S(A|M)_{(\mathcal{N}(t)\otimes\mathbb{I}_M)(\hat{\rho}_{AM})}- n\ln t -n\right) \le \limsup_{t\to\infty}n\left(g(E+t) - \ln t - 1\right) &= 0\;.
( h ( M_{d}) - h ( M_{1 2}) ) F ( \eta_{3}) = - i ( 2 \pi )^{- 4}\Delta_{3}^{- 1}\int d^{4}q_{1 2}^{\prime}[ F ( \eta_{2}^{\prime}) \Delta_{1}^{- 1}+ ( 1 \leftrightarrow 2 ) ]
(\overline{A}_1)_j^{j'}(l):=-\frac{2 j\,c_2\,(j-j')^2\,\sqrt{\lvert j-j'\rvert \xi_{j-j'}}+6 j\,c_3\,\sqrt{\lvert j-j'\rvert \xi_{j-j'}}}{\overline{\omega}\cdot l+j'^3-j^3}
S_{I , \theta}= - \frac{\sigma}{2}\int \frac{d \nu}{2 \pi}\int_{\bar{R}_{1}}^{\bar{R}_{2}}d r \tilde{\theta}_{I}^{*}\left[ - \frac{\partial}{\partial r}\gamma^{- 1}r^{2}\frac{\partial}{\partial r}- ( \nu^{2}- \omega^{2}) \gamma r^{2}\right] \tilde{\theta}_{I}\, ,
\mathrm{E}_{\theta}\|\hat{\theta}-\theta\|^2 \le\mathrm{E}_{\theta}\hat{\mathrm{E}}(\|\vartheta-\theta\|^2|X)\le M_0r^2(\theta) + H_0 \int_0^{+\infty}\!\!\!\! e^{-m_0u/\sigma^2}du=M_0r^2(\theta) + \frac{H_0\sigma^2}{m_0}.
\int d x \sqrt{g}V [ \sigma ] = \int d x \sqrt{g}\frac{N}{2 \lambda_{B}}\sigma^{2}+ \frac{N}{2}\mathrm{T r}\int_{1 / \Lambda^{2}}^{\infty}\frac{d t}{t}e^{- t \sigma^{2}}e^{\not \nabla^{2}t}.
\star \equiv \exp \left({\frac{i\hbar}{2}}\int dx~ \left( \stackrel{\leftarrow}{\delta\over\delta\phi (x)}\stackrel{\rightarrow}{\delta\over\delta\pi (x)}-\stackrel{\leftarrow}{\delta\over\delta\pi (x)}\stackrel{\rightarrow}{\delta\over\delta\phi (x)}\right) \right) ,
2 w_{R}'' w_{R}= 2 \frac{w_{R}^{2}|w_{R}'|^{2}}{|w_{R}|^{2}}- \frac{m_{1}w_{R}^{2}}{|w_{R}|^{2}}+ |w_{R}|^{4}\nabla U_{1}( c_{1}+w_{R}^{2})
\begin{array}{ll}\sum\limits_{\ell=1}^{3}\|\partial_t^{\ell}\phi\|_{H^{3-\ell}(\Omega)}^2\leq c_{19}\|\xi_0\|_{H^4(\Omega)}^2\|\phi_0\|_{H^3(\Omega)}^2\left(\exp\{\|\xi_0\|_{H^4(\Omega)}\}\right)^{c_{18}}\exp\{-\beta_7 t\},\end{array}
(u_{\mp}-v_{\pm}-\frac{1}{2}h)(u_{\pm}-v_{\mp}+ \frac{1}{2}h)H_{i}^{\pm}(u)H_{i+1}^{\mp}(v) &=(u_{\mp}-v_{\pm}+\frac{1}{2}h)(u_{\pm}-v_{\mp}-\frac{1}{2}h)H_{i+1}^{\mp}(v)H_{i}^{\pm}(u).
\varepsilon_{k}(X) = -\frac{q^{1-g}h}{\zeta(k)(q-1)^2}+ \sum_{j=1}^{2g}\sum_{\ell=0}^{k-1}\frac{Z(\gamma_{j, \ell}^{-1})}{k\gamma_{j, \ell}^{1-k}Z'(\gamma_j^{-1})}\frac{\gamma_{j, \ell}}{\gamma_{j, \ell}-1}+ \frac{1}{2\pi i}\sum_{N=0}^{X-1}\oint_{C_\rho}\frac{1}{u^{N+1}}\frac{Z(u)}{Z(u^k)}\,du.
I_m = \int d^4x \sqrt{{g}}[-\frac{1}{2}D_{\mu}\phi(D^{\mu}\phi)^* - \frac{1}{2}D_{\mu}\Sigma (D^{\mu}\Sigma)^* - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{4}H_{\mu\nu}H^{\mu \nu}- V(|\phi|, |\Sigma |, \Lambda)],
\sum_{n=0}^{\infty}\textup{PDO}_\textup{t}(8n+6)q^n =16\dfrac{f_2^{16}f_{6}^{10}}{f_1^{17}f_3^3f_{12}^4}-8q\dfrac{f_2^{28}f_3f_{12}^{4}}{f_1^{21}f_4^8f_{6}^2}-128q^2\dfrac{f_2^{4}f_3f_4^8f_{12}^{4}}{f_1^{13}f_{6}^2},
\widehat{L}= \frac{1}{2}\partial_{\mu}\phi ( x ; \alpha ) \partial^{\mu}\phi ( x ; \alpha ) - \frac{m^{2}}{2}\phi ( x ; \alpha )^{2}- \frac{1}{2}\partial_{\mu}\widetilde{\phi}( x ; \alpha ) \partial^{\mu}\widetilde{\phi}( x ; \alpha ) + \frac{m^{2}}{2}\widetilde{\phi}( x ; \alpha )^{2},
V=\left(\begin{array}{l}X(u) = 2a(u)=\frac{2\sqrt{2}}{\pi}\int_{-1}^{1}\,\sqrt{{u-x}\over{1-x^2}}\, dx F_A (u) = 2\,a_D(u)=2i\frac{\sqrt{2}}{\pi}\int_{1}^{u}\,\sqrt{{u-x}\over{1-x^2}}\, dx\end{array}\right)\ .
\frac{3}{2}F^{\prime \prime}- \frac{1 5}{2}\frac{K^{\prime}}{K}F^{\prime}+ 6 \bigg [ 2 \bigg ( \frac{K^{\prime}}{K}\bigg )^{2}- \frac{K^{\prime \prime}}{K}\bigg ] F + \bigg [ \sum_{i}\frac{\Lambda_{i}}{K}\delta ( z - z_{i}) + ( \phi_{0}^{\prime})^{2}\bigg ] F + \phi_{0}^{\prime}\varphi^{\prime}= 0 ,
\Phi^{\ell}( t , \bar{t}, \psi ) = \sum_{n = 0}^{\infty}a_{n}t_{0 n}^{\ell}( t , \bar{t}, \psi ) = \frac{e^{- i \ell \psi}}{( 1 - | t |^{2})^{\ell}}\sum_{n = 0}^{\infty}b_{n}t^{n}\equiv e^{- i \ell \psi}\varphi ( t , \bar{t}) \,
\phi(-q^{1/2}x_3) \phi(-q^{1/2}x_1) \phi(-q^{1/2}x_2) =\sum \frac{q^{\frac{\ell^2}{2}}(x_{3})^{\ell}}{(q)_{\ell}}\frac{q^{\frac{m^2}{2}}(x_{1})^{m}}{(q)_{m}}\frac{q^{\frac{n^2}{2}}(x_{2})^{n}}{(q)_{n}}&=\sum \frac{q^{\frac{n^2}{2}+\frac{\ell^2}{2}+\frac{m^2}{2}-mn-n\ell}(x_{2})^{n}(x_{3})^{\ell}(x_{1})^{m}}{(q)_{n}(q)_{\ell}(q)_{m}}.
\overline{\beta_{ij}}\cdot\omega\,=\, \sum_{i\leq k\leq j-1}\overline{\alpha_{ik}}\cdot\alpha_{k+1,j}0\,=\,\sum_{i\leq k\leq j-1}(\overline{\alpha_{ik}}\cdot\beta_{k+1,j}-\overline{\beta_{ik}}\cdot\overline{\alpha_{k+1,j}})\, .
{\bf A}_+^{(I)}(\lambda)|_{\lambda\to\infty}=-(-\lambda)^l\phi^{2M}q^{l-IM}({\boldsymbol I}+O(\lambda^{-2})), &{\bf A}_-^{(I)}(\lambda)|_{\lambda\to\infty}=\phantom{-}(-\lambda)^{I M-l}q^{l-IM}({\boldsymbol I}+O(\lambda^{-2})).
\pi_r(\boldsymbol{K}):= \frac{\prod\limits_{\stackrel{i=1}{i\neq r}}^n (\boldsymbol{K}-\lambda_i\boldsymbol{I})}{\prod\limits_{\stackrel{i=1}{i\neq r}}^n (\lambda_r-\lambda_i)}= \frac{\partial}{\partial q_r}\otimes \mathrm{d}q_r+ \frac{\partial}{\partial p_r}\otimes \mathrm{d}p_r, r=1,\ldots,n ,
\begin{array}{r l}{}&{G ( x_{1}, x_{2}, x_{3}, x_{4})}\\{=}&{\displaystyle \int d^{4}x_{5}\int d^{4}x_{6}< 0 | T \left( \pi_{1}\pi_{2}\chi_{3}\chi_{4}( \pi \star \sigma \star \pi )_{5}( \chi \star \sigma \star \chi )_{6}\right) | 0 > ,}\\ \end{array}
\frac{\Sigma_n}{a^{m+1}}\,g_a^{m+1}=D_{\lambda+m+2}\left[1-(2\lambda+2m+1)r^2+2(\lambda+m)r\cos\theta\right]\sum_{k=0}^mR_k^m(r)\cos^k\theta &+D_{\lambda+m+2}\left[1+r^2-2r\cos\theta\right]r\sum_{k=0}^mR_k^{m\,\prime}(r)\cos^k\theta.
{\cal F}= \sum_{n_{1}\geq n_{2}\geq \cdots \geq n_{N}\geq 0}^{\infty}\det \left[ d_{n_{j}+ N - j , i}\right] \det \left[ t_{i}^{n_{j}+ N - j + \frac{1}{2}}- t_{i}^{- ( n_{j}+ N - j + \frac{1}{2})}\right] .
\sigma_{-2}(N^\omega_{\partial M})(t,\xi') = -\frac{\chi(\xi')}{\sigma_1(\Lambda_{g,F}^\omega)(t,\xi')}\left(p_{-1}(t,\xi')\sigma_0(\Lambda_{g,F}^\omega)(t,\xi')+\nabla_{\xi'}p_{-1}\cdot D_{t'}\sigma_1(\Lambda_{g,F}^\omega)\right).
\delta_{c_1}\circ \delta_{c_4}=h_0 \delta_{c_0}+ h_1 \delta_{c_1}+ h_2 \delta_{c_2}+ h_2 \delta_{c_3}+ h_1 \delta_{c_4}, \delta_{c_4}\circ \delta_{c_1}= h_0 \delta_{c_0}+ i_1 \delta_{c_1}+ i_2 \delta_{c_2}+ i_2 \delta_{c_3}+ i_1 \delta_{c_4},
X_{0}=v,X_{1}=J_{z_1}(v),X_{2}=J_{z_2}(v),X_{3}=J_{z_3}(v),X_{4}=J_{z_4}(v), &X_{5}=J_{z_5}(v),X_{6}=J_{z_6}(v),X_{7}=J_{z_7}(v), &X_{8}= w, X_{9}=\tilde{J}_{z_1}(w),X_{10}=\tilde{J}_{z_2}(w),X_{11}=\tilde{J}_{z_3}(w),X_{12}=\tilde{J}_{z_4}(w), &X_{13}=\tilde{J}_{z_5}(w),X_{14}=\tilde{J}_{z_6}(w),X_{15}=\tilde{J}_{z_7}(w).
= \; 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}^{2n_b}{2 n_b \choose q_b}E^\mu(\{q,n\}) \; ,
\lVert \widetilde\phi_{l}-\phi_{l}\lVert^{2}\lesssim \int_{0}^{\widetilde Ke^{-\omega_{l}s_{0}}}y^{d-2\gamma-1}\,dy+\int_{e^{\widetilde\sigma s_{0}}}^{\infty}y^{4\lambda_{l}+d-1}e^{-\frac{y^{2}}{4}}\,dy \lesssim (Ke^{-\omega_{l}s_{0}})^{2+\omega}=e^{-(2+\omega)(1-\widetilde k)\omega_{l}s_{0}}
u_t = \Delta u x\in \Omega, \quad 0<t<T v_t =\Delta_{M}v+u^2 v^2 x\in M,\quad 0<t<T \frac{\partial u}{\partial \eta}=-u^2 v^2 x\in M, \quad 0<t<T u =u_0 x\in\Omega ,\quad t=0 v =v_0 x\in M , t=0
P^{}_{,\chi}=\mathrm{Pr}\big\{{R}^{}_\chi< \Tilde{R}_\chi\big\}= \mathrm{Pr}\Big\{|H^\chi|^2 < \frac{(2^{2\Tilde{R}_\chi}-1)\sigma^2_0}{2c^2_\chi P_{BS}}\Big\},
\hat{R}^{*}_{\beta}(t) ={R}^{*}_{\beta}+ \frac{\partial{R}_{\beta}}{\partial \lambda_{\beta}}\Big|_{\lambda_{\beta}}\left( \hat{\lambda}_{\beta}- \lambda_{\beta}\right) + O\left( \left( \hat{\lambda}_{\beta}- \lambda_{\beta}\right)^2 \right).
c(W):=\dim_{\mathrm{k}}(\mathrm{k}\langle WV \rangle)-(1-\frac{\varepsilon}{2})\dim_{\mathrm{k}}(W)\leq(2-\varepsilon)\dim_{\mathrm{k}}(V)-(1-\frac{\varepsilon}{2})\dim_{\mathrm{k}}(V)=(1-\frac{\varepsilon}{2})\dim_{\mathrm{k}}(V).
\alpha_{\widetilde{\sigma}}= \left(\begin{array}{ccc}-6 -2 \theta^2 + 2 \theta^2 + 11 2 \theta^2 + 2 -10 3 \theta^2 + 1 -\theta^2 + 11 -3 \theta^2 + 1 2 \end{array}\right) ,
0<\mu<\mu_0\Rightarrow\frac{\frac{1}{1+\mu}\left(1+\left\Vert\sum_{i=1}^n \lambda_iy_i\right\Vert-\mu \right)-\mu}{1+\mu}>1+\left\Vert\sum_{i=1}^n\lambda_i y_i\right\Vert-\varepsilon
\frac32{\rm Cl}_2(\theta_7)-\frac32{\rm Cl}_2(2\theta_7)+\frac12{\rm Cl}_2(3\theta_7)=\frac74\left\{{\rm Cl}_2\left(\frac{2\pi}{7}\right)+{\rm Cl}_2\left(\frac{4\pi}{7}\right)-{\rm Cl}_2\left(\frac{6\pi}{7}\right)\right\}
w^{( n )}= W^{2}g_{n}\left( \frac{v^{3}}{W^{2}}\right) \left( \frac{\Lambda}{v}\right)^{n ( 3 N_{c}- N_{f})}= S g_{n}\left( \frac{v^{3}}{S}\right) \left( \frac{\Lambda}{v}\right)^{n ( 3 N_{c}- N_{f})}
\Sigma_{\mathrm{B F Y M}}= \int d^{n}x \, \, \left\{\varepsilon^{\mu_{1}\ldots \mu_{n}}B_{\mu_{1}\ldots \mu_{n - 2}}^{a}F_{\mu_{n - 1}\mu_{n}}^{a}+ \left( B_{\mu_{1}\ldots \mu_{n - 2}}^{a}+ D_{[ \mu_{1}}\eta_{\mu_{2}\ldots \mu_{n - 2}]}^{a}\right)^{2}\right\}\, \, ,
K_{w}(t_{1},t_{2},\cdots,t_{l})=\prod_{j=1}^{l}\Gamma(-\beta_{j})^{-1}\times\{\exp(-t_{l}F_{m_{l}}) \cdots\exp(-t_{1}F_{m_{1}})\prod_{j=1}^{l}t_{j}^{-\beta_{j}-1}1\},
W_{1 2}\, \psi \, = \, ( h_{1}+ h_{2}+ \Lambda_{1 2}^{+ +}\, V_{1 2}\, \Lambda_{1 2}^{+ +}+ \Lambda_{2}^{+}\, V_{2}^{+}\, \Lambda_{2}^{+}+ \Lambda_{1}^{+}\, V_{1}^{+}\, \Lambda_{1}^{+}) \, \psi
S(\mathbf{P}_{\!\sigma}) = S(\mathbf{p}_{\sigma_d};\ldots;\mathbf{p}_{\sigma_1}) = -\sum_{n=1}^{d}C_{n}^{(d), \sigma}(\mathbf{P}_{\!\sigma}) \cdot \int\! \log \boldsymbol{\mu}_n^{\sigma}\,\mathrm{d}\mathbf{p}_{\sigma_n}.
\{\Phi(g'_c, \Phi (g_n, D_1)),\ldots ,\Phi( (g'_c)^2g_cg_n,s)^{-1}\Phi \Big( g'_c, \Phi\Big(\displaystyle{\prod_{i=1}^{n}}g_i , s\Big)\Big), \dots , \Phi(g'_c, \Phi (g_n, D_{n-1})), \Phi(g'_c, \Phi (g_n, D_n)) \},
Z [ A_{+}, \eta ,{\bar{\eta}}] = \int \! D \bar{\psi}D \psi \exp \{i \int \! d^{2}x \, [ \bar{\psi}i \gamma^{\mu}\partial_{\mu}\psi + \psi_{-}^{\dagger}A_{+}\psi_{-}+ \eta \psi_{+}^{\dagger}\psi_{-}+{\ \bar{\eta}}\psi_{-}^{\dagger}\psi_{+}] \}\quad .
S = \int \mathrm{d}x \; \left\{- \frac{1}{1 2}F_{\mu \nu \rho}F^{\mu \nu \rho}+ B^{* \mu \nu}\left( \partial_{\mu}C_{\nu}- \partial_{\nu}C_{\mu}\right) + C^{* \mu}\partial_{\mu}C \right\}\; ,
\gamma_z((s_\lambda s_\mu^*)\cdot_{\sigma}(s_\nu s_\tau^*)) =\sigma(g,h)(z^{d(\lambda)-d(\mu)}s_\lambda s_\mu^*)(z^{d(\nu)-d(\tau)}s_\nu s_\tau^*) &=\gamma_z(s_\lambda s_\mu^*)\cdot_\sigma\gamma_z(s_\nu s_\tau^*).
\int d^{3}\zeta - \sqrt{- g ( Z ( \zeta ) )}= \int - h^{- 1}\sqrt{1 - h v^{2}}\Big [ 1 + \frac{1}{2}h^{2 / 3}\big [ 2 h^{- 1 / 3}\bar{\theta}\Gamma^{\tilde{1}}\partial_{1}\theta + 2 h^{- 1 / 3}\bar{\theta}\Gamma^{\tilde{2}}\partial_{2}\theta
\rho \ddot{u}_i = \big(A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}- B_{iJKj}\dot{u}_{j})_{,K}\big)_{,J}- E_{ij}\dot{u}_{j}.
\vert \cos\theta_{\hat x}-\cos \theta_c\vert = \left\vert 2\sin \left(({\theta_c-\theta_{\hat x}})/2\right)\sin \left(({\theta_c+\theta_{\hat x}})/2\right)\right\vert\le k^{-1/2}_+ \vert x \vert^{-1/2},
\left( \begin{array}{c c c}{{A_{3}+ A_{8}/ \sqrt{3}}}&{{R \bar{G}}}&{{R \bar{B}}}\\{{G \bar{R}}}&{{- A_{3}+ A_{8}/ \sqrt{3}}}&{{G \bar{B}}}\\{{B \bar{R}}}&{{B \bar{G}}}&{{- 2 A_{8}/ \sqrt{3}}}\\ \end{array}\right)
h_{T_0}^2(t) \ = \t(v_1^2+v_2^2+v_3^2)+t^2((v_1+v_2)^2+(v_2+v_3)^2+(v_3+v_1)^2-v_1^2-v_2^2-v_3^2) h_{T_1}^2(t) \ = \tv_1^2+t^2((v_1+v_2)^2 + (v_1+v_3)^2 - v_1^2)+t^3(v_2+v_3)^2 h_{T_2}^2(t) \ = \t^2(v_2+v_3)^2+t^3((v_1+v_2)^2 + (v_1+v_3)^2 - v_1^2)+t^4v_1^2
\bullet \forall \, p \geqslant r b - f'(p) = \int_p^{+\infty}f''(x) \, dx \geqslant c_- \int_p^{+\infty}x^{-\beta_-}dx = - \frac{c_-}{1 - \beta_-}\, p^{1-\beta_-}\; , & \bullet \forall \, p \leqslant -r f'(p) - a = \int_{-\infty}^p f''(x) \, dx \geqslant c_- \int_{-\infty}^p (-x)^{-\beta_-}dx = - \frac{c_-}{1 - \beta_-}\, (-p)^{1-\beta_-}\; .
S_1 = \sup_{i\in I_1}-\frac{Q_{i}^{(K_i-1)}(0)}{Q_{i}^{(K_i)}(0)}S_2 = \sup_{i\in I_2}\inf\left\{-\frac{(Q_{i,s_{i,-\kappa-2}})^{(K_i-1)}(0)}{Q_{i}^{(K_i)}(0)}\!: s_{i,-\kappa-2}\in (t_{\infty}(\mathbf{s}_{i}),\infty)\right\}.
\begin{array}{l}\displaystyle \beta_1^*(x_*,\mu) = \frac{(3x_*+\mu-1)(x_*+\mu)^3}{2(1-\mu)}, [10pt]\displaystyle \beta_2^*(x_*,\mu) = \frac{(3x_*+\mu)(x_*+\mu-1)^3}{2\mu}, \end{array}{}
g\cdot m = \frac{\alpha\xi(\frac{\|m\|^2}{2}+v^Tm+\frac{\|v\|^2}{2})+A(m+v)}{\frac{\alpha}{2}\|\xi\|^2(\frac{\|m\|^2}{2}+v^Tm+\frac{\|v\|^2}{2})+\xi^TA(m+v)+\alpha^{-1}}.
\left\{\mathcal{H}_{x}^{t}\longleftrightarrow\mathcal{H}_{y}^{t}\right\}:=\bigcup_{\left( x^{\prime},y^{\prime}\right) \in\mathcal{H}_{x}^{t,-}\cap\mathbb{Z}^{d}\times\mathcal{H}_{y}^{t,+}\cap\mathbb{Z}^{d}}\left\{\left\vert \mathbf{C}_{\left\{x^{\prime},y^{\prime}\right\}}\right\vert>0\right\}\ ,
{\cal T}^{\alpha\beta}_{\mu\nu}\sim\epsilon^{\alpha~\rho\sigma}_{~\mu}\epsilon^{\beta~\lambda\gamma}_{~\nu}\big( A F_{\rho\sigma}F_{\lambda\gamma}+ B F_{\rho\sigma}\tilde F_{\lambda\gamma}+ C \tilde F_{\rho\sigma}\tilde F_{\lambda\gamma}\big)\,,
\mathcal H\bigl(P_L|P_L^{\tilde V}\bigr) = \mathcal H\bigl(\mu^L_0|(X_0)_\# P_L^{\tilde V}\bigr) + \frac 12 \int_0^T \Phi_L\bigl(\mu^L_t,\jmath^L_t,F^{\tilde V_t}(\mu^L_t)\bigr)\;\!\mathrm dt .
X_r^{(t+1)}=X_{r+1}^{(t)}+X_{r-1}^{(t)}+X_r^{(t-1)}&=k_1W_1^{(t-r)}+k_MW_M^{(t+1-(M-r))}+k_1W_1^{(t-(r-2))}&+k_MW_M^{(t-1-(M-r))}+k_1W_1^{(t-r)}+k_MW_M^{(t-1-(M-r))}&=k_1W_1^{(t+1-(r-1))}+k_MW_M^{(t+1-(M-r))},
\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g\left( \frac{u_n(x)-u_n(y)}{\vert x-y\vert^{\alpha}}\right)\frac{u_n(x)-u_n(y)}{\vert x-y\vert^{\alpha+d}}dxdy + \int_{\mathbb{R}^{d}}g(u_n)u_ndx=\int_{\mathbb{R}^{d}}K(x)f(x,u_n)u_n dx,\ \n\in\mathbb{N}.
I_-^\mu \big\{(1+x)^{-\mu}\hat h_j(x)\big\}= (1+x) \sum_{l=0}^{N-2}\frac{\Gamma(l+2-\mu)}{(l+1)!}\hat \varrho_{l+1,j}\, P_l^{(0,1)}(x), &I_-^\mu \big\{(1+x)^{-\mu}\hat h_j(x)\big\}\big|_{x=1}= 2\sum_{l=0}^{N-2}\frac{\Gamma(l+2-\mu)}{(l+1)!}\hat \varrho_{l+1,j}\,,
((a,u)(b,v))(c,w) = (ab,f_3(a,b)u+f_4(a,b)v)(c,w) &= (abc,f_3(ab,c)(f_3(a,b)u+f_4(a,b)v)+f_4(ab,c)w), (a,u)((b,v)(c,w)) = (a,u)(bc,f_3(b,c)v+f_4(b,c)w) &= (abc,f_3(a,bc)u+f_4(a,bc)(f_3(b,c)v+f_4(b,c)w)).
\Delta \hat{E}_{n}^{( q )}= \frac{f^{2}}{6}\int_{- \infty}^{\infty}d \hat{x}_{0}\psi_{n}^{( 0 ) *}( \hat{x}_{0}) \Bigl \{V ( \hat{x}_{0}) \Bigl [ 1 - 4 \mu \hat{x}_{0}^{2}\Bigl ( V ( \hat{x}_{0}) - E_{n}^{\mathrm{( u n )}}\Bigr ) \Bigr ] -{\textstyle \frac{2}{3}}\mu E_{n}^{\mathrm{( u n )}}\hat{x}_{0}^{3}V^{\prime}( \hat{x}_{0}) \Bigr \}\psi_{n}^{( 0 )}( \hat{x}_{0}) .
A=\begin{bmatrix}3 1 1 1 1 -2 0 -1 -1 -1 -2 -1 0 -1 -1 -2 -1 -1 0 -1 -2 -1 -1 -1 0 \end{bmatrix},B=\begin{bmatrix}3 2 2 2 2 -1 0 -1 -1 -1 -1 -1 0 -1 -1 -1 -1 -1 0 -1 -1 -1 -1 -1 0 \end{bmatrix}
p_j\frac{\partial H}{\partial t}+\frac{\partial H}{\partial q^j}+\sum_{i=1}^n \left(p_i\frac{\partial H}{\partial p_i}-H\right)\frac{\partial \gamma^j}{\partial t}+\sum_{i=1}^n \frac{\partial H}{\partial p_i}\frac{\partial \gamma^j}{\partial q^i}=0
\frac{zd_{\nu}'(z)}{d_{\nu}(z)}=\nu-2\sum_{n\geq 1}\frac{z^2}{\alpha_{\nu,n}^2-z^2}&=\nu-2\sum_{n\geq 1}\sum_{m\geq1}\left(\frac{z^2}{\alpha_{\nu,n}^2}\right)^m &=\nu-2\sum_{m\geq 1}\left(\sum_{n\geq1}\frac{1}{\alpha_{\nu,n}^{2m}}\right)z^{2m},
\hat{P}_{\mu}= ({P}_{\mu}^{( 1 )}+{P}_{\mu}^{( 2 )}) M^{- 1}, M^{2}= ( P_{\mu}^{( 1 )}+ P_{\mu}^{( 2 )}) ( P^{( 1 ) \mu}+ P^{( 2 ) \mu})
\tilde{A}_{0,k_{i}}^{(e)}= \frac{\sqrt[4]{\pi}\sqrt{(2k_{i})!}}{2^{k_{i}-1/2}k_{i}!}\sim \frac{1}{2^{k_{i}}}\frac{(2k_{i})^{k_{i}+1/4}\exp(-k_{i})}{k_{i}^{k_{i}+1/2}\exp(-k_{i})}\sim k_{i}^{-1/4}
E_{2}( a , \lambda ) = \int_{m}^{\infty}\frac{d t}{2 \pi}\frac{1}{\sqrt{t^{2}- m^{2}}}\left\{t \ln \left[ 1 + \frac{\lambda}{t}+ \frac{\lambda^{2}}{4 t^{2}}\left( 1 - e^{- 4 t a}\right) \right] - \lambda \right\}.
p=\frac{NkT}{V-Nb^{\prime}}-\frac{N^{2}a^{\prime}}{V^{2}}\Rightarrow(p+\frac{N^{2}a^{\prime}}{V^{2}})(V-Nb^{\prime})=NkT\Rightarrow(p+\frac{n^{2}a}{V^{2}})(V-nb)=nRT.
\partial_\xi m_1(\xi)= \frac{-2h\xi}{(1+h \xi^2)^2}, \partial_\xi m_2(\xi)=i\frac{1-h\xi^2}{(1+h \xi^2)^2}, \partial_\xi m_3(\xi)=\frac{-2\xi}{(1+h \xi^2)^2},
\int_{0}^{T}\int_{\Omega}\int_{0}^{1}\int_{Y^{\ast}}u_{1}(x,t,y,s)v_{1}(x)c_{1}(t)c_{2}(s)\nabla_{y}\cdot (\nabla_{y}\rho(y))dydsdxdt =\int_{0}^{T}\int_{\Omega}\int_{0}^{1}\int_{Y^{\ast}}u_{1}(x,t,y,s)v_{1}(x)c_{1}(t)c_{2}(s)v_{2}(y)dydsdxdt
\|\rho_n(a)\|^2 =\Big\|\sum_{\lambda,\mu}\sum_{\lambda',\mu'}T_\lambda\rho_0(a_{\lambda,\mu})T_{\mu}^*T_{\mu'}\rho_0(a_{\lambda',\mu'})^*T_{\lambda'}^*\Big\| &=\Big\|\sum_{\lambda,\mu}\sum_{\lambda'}T_\lambda\rho_0(a_{\lambda,\mu}q_{s(\mu)}a_{\lambda',\mu}^*)T_{\lambda'}^*\Big\|,
\frac{7!3!}{2}(C\Gamma^{[\nu_1})_{\alpha{'}\beta{'}}(C\Gamma^{\nu_2\nu_3]})_{\delta{'}\epsilon{'}}+ \frac{6!4!7}{4!}(C\Gamma^{\mu_7\nu_1\nu_2\nu_3}\Gamma_{11})_{\alpha{'}\beta{'}}(C\Gamma_{\mu_7}\Gamma_{11})_{\delta{'}\epsilon{'}}=0\quad,
\begin{aligned}\sqrt{n+1}\,C_{m,n+1}(i,k)= \left(\frac{i}{\alpha}\right) \sin \theta\,C_{m,n}(i-1,k) &+\left(\frac{k}{\beta}\right) \cos \theta\,C_{m,n}(i,k-1)-(\alpha \sin \theta+\beta \cos \theta) C_{m,n}(i,k).\end{aligned}
wt(C_{3}) =\biggl(p^{e-2}-\biggl(\frac{-1}{p}\biggr)p^{m+d-1}\biggr)+\sum_{i=1}^{p-1}\biggl(p^{e-2}-\biggl(\frac{i^{2}-c^{2}}{p}\biggr)p^{m+d-1}\biggr)-p^{e-2}&=(p-1)p^{e-2}-\biggl(\biggl(\frac{-1}{p}\biggr)+L_{p}(c)\biggr)p^{m+d-1}&=(p-1)p^{e-2}+p^{m+d-1}.
\begin{aligned}g(t)^{-1}\biggl( \frac{\sqrt{\pi}(\sqrt{2}-1)}{2}\sum_{k\geq M}l_k-\frac{1}{4\sqrt{t}}\sum_{k\geq M}l_k^2+\frac{E_M(t)}{\sqrt{t}}\biggr) \geq g(t)^{-1}\biggl( \frac{\sqrt{\pi}(\sqrt{2}-1)}{2}\sum_{k\geq M}l_k-\frac{1}{4\sqrt{t}}\sum_{k\geq M}l_k^2-(M+4|\Omega_M|-1)\sqrt{t}\biggr)\,.\end{aligned}
H_{k}^{p}(C,t):=\begin{vmatrix}c_pt c_{p+1}\cdots c_{p+k-1}c_{p+1}c_{p+2}t \cdots c_{p+k}\vdots \vdots \ddots \vdots c_{p+k-1}c_{p+k}\cdots c_{p+2k-2}t\end{vmatrix}.
\mathcal{C}_{\mbox{\tiny FP}}c = \bar c \; , \;\; \mathcal{C}_{\mbox{\tiny FP}}\bar c = - c \; , \;\; \mathcal{C}_{\mbox{\tiny FP}}B = B + \widetilde Z_1 (1-\alpha ) (\bar c \times c) \; , \;\; \mathcal{C}_{\mbox{\tiny FP}}A = A \;.
|\bar\nabla\bar\Delta(f\nu)|^2-2m|\bar\Delta (f\nu)|^2= |\nabla\Delta f|^2+4m^2|\nabla f|^2+4m\nabla\Delta f\nabla f+8|\nabla^2 f|^2+16|\nabla f|^2 &+8\nabla\Delta f\nabla f+8|\Delta f|^2+16mf\Delta f.
\left|\left[F-(_{x}L\circ_{y}M)F\right](x,y) \right| \leq \displaystyle \sum_{\rho=0}^{r}\Gamma_{\rho,L}(x)\omega_{\rho}(F;\Lambda_{\rho,L}(x),0) &+\|L\|\sum_{\sigma=0}^s\Gamma_{\sigma,M}(y)\omega_{\sigma}(F;0,\Lambda_{\sigma,M}(y)),
\left[\begin{array}{ll}\partial_t u(t,x) = \partial_{xx}u(t,x) + f(u(t,x)) \dot{W}(t,x), (t,x) \in [0, T] \times [0, \pi], u(t,0) = u(t, \pi) = 0, \textrm{for all}\,\, t \in [0,T], u(0,x) = 0, \textrm{for all}\,\, x \in [0, \pi].\end{array}\right.
f_N \left[ \phi (x) \right]={1 \over Z_0}\sum_{n=0}^{N}{\left( -1 \right)^n \over n!}\left({\lambda \over 4}\int{\rm d^d}x \phi^4 \right)^n e^{- \int{\rm d^d}x \left[{1 \over 2}\left( \partial_{\mu}\phi \right)^2 +{1 \over 2}m^2 \phi^2 \right]}
A_{\mu}^{a}~ = ~ \left( \begin{matrix}{{V^{M}{}_{\mu}}}\\{{B_{\mu M}}}\\{{A^{I}{}_{\mu}}}\\ \end{matrix}\right){}~ F_{\mu \nu}^{a}~ = ~ \left( \begin{matrix}{{V^{M}{}_{\mu \nu}}}\\{{H_{\mu \nu M}}}\\{{F^{I}{}_{\mu \nu}}}\\ \end{matrix}\right)
\nu \Big(\prod_{i=1}^{k}a_{p_i}^*\prod_{j=1}^{k'}a_{q_j}\Big) \, = \, \delta_{k , k'}(-1)^\frac{k (k-1)}{2}\det \big[ \nu(a_{p_i}^* a_{q_j}) \big]_{1 \leq i,j \leq k}\ .
y = \frac{y\wedge x_2\wedge x_3\wedge \cdots \wedge x_n}{x_1\wedge x_2\wedge \cdots \wedge x_n}x_1 + \frac{x_1\wedge y\wedge x_3\wedge \cdots \wedge x_n}{x_1\wedge x_2\wedge \cdots \wedge x_n}x_2+\cdots + \frac{x_1\wedge x_2\wedge \cdots \wedge x_{n-1}\wedge y}{x_1\wedge x_2\wedge \cdots \wedge x_n}x_n.
d^2\!f(\overline{x}|\overline{v})(w) =\inf_{z\in\mathbb{X}}\Big\{d^2\vartheta(F(\overline{x}))(F'(\overline{x})w|D^2F(\overline{x})(w,w)\!+\!F'(\overline{x})z) -\langle \overline{v},z\rangle\Big\}, = d^2\vartheta(F(\overline{x}))(F'(\overline{x})(w))+\sup_{\xi\in \Lambda_{\overline{x},\overline{v}}}\langle \xi,D^2 F(\overline{x})(w,w)\rangle
X_{(P_{\ell(\alpha)},\alpha)}X_{(P_{\ell(\beta)},\beta)}=X_{(P_{\ell(\alpha)}\mid P_{\ell(\beta)}, \alpha\cdot\beta)}=X_{(P_{\ell(\alpha\cdot\beta)},\alpha\cdot\beta)}+X_{(P_{\ell(\alpha\odot\beta)},\alpha\odot\beta)}.
t^2|u_{t}|^2_{D^2}\leq Ct^2\big(|(\rho u_t+\rho v\cdot \nabla v)_t|^2_{2}+|\nabla P_t|^2_{2}+|(H\times H)_t|^2_{2}+|u_t|^2_{D^1_0}\big) \leq Cc^{32}_3, t^2|u|_{D^{3,q}}\leq Ct^2(c^4_1+c^2_1|u_t|_\infty+c^2_1|\nabla u_t|_q+Cc^3_1|v|_{D^{2,q}})\leq Cc^{34}_3.
\hat y_t = \frac{\exp \! \left(-\eta_t \sum_{s=1}^{t-1}\ell'(\hat y_s,y_s) \right)}{1 + \exp \! \left(-\eta_t \sum_{s=1}^{t-1}\ell'(\hat y_s,y_s) \right)}\in [0,1]\,,
\mathop{\min}\limits_{\theta, W, Z}\F(\theta, W, Z)\doteq \sum \limits_{i=1}^l \frac{\log(1+ \theta_i/\tau)}{\log(1+1/\tau)}+ \gamma (||Z||_{*}-||Z||_{F_{2n+m}}){\rm s.t.}\ Z=\left[\begin{array}{cc}0 M N I_{2n+m}\end{array}\right] & (\ref{const2})-(\ref{const5})
ST =(P+(I-Q))P|^{-1}_{Y_1}Q|^{-1}_{X1}Q+(P+(I-Q))(I-Q)|^{-1}_{X_2}(I-P)|^{-1}_{Y_2}(I-P) &=PP|^{-1}_{Y_1}Q|^{-1}_{X1}Q+(I-Q)(I-Q)|^{-1}_{X_2}(I-P)|^{-1}_{Y_2}(I-P) &=Q|^{-1}_{X_1}Q+(I-P)|^{-1}_{Y_2}(I-P)=I.
S = \int d^4 x \sqrt{g}\left({{m_s^8 v_1 v_2 v_3}\over (2 \pi)^7 \lambda^2}R +{1 \over 4}{{m_s^4 v_2 v_3}\over{(2 \pi)^5 \lambda}}F^2+{1 \over 4}{{m_s^4 v_1 v_3}\over{(2 \pi)^5 \lambda}}\tilde{F}^2 +\dots \right)
(u,1,u,1) \left( \begin{smallmatrix}1 0 0 0 \sqrt{bf}+1 1 \sqrt{be}\sqrt{cf}\sqrt{bc}0 b c \sqrt{ef}0 e f\end{smallmatrix}\right) = \left( (\sqrt{bc}+1)u + \sqrt{bf}+1 + \sqrt{ef}, 1, \sqrt{be}+ e + bu, f + \sqrt{cf}+ cu \right).