harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_0^\ell\nu\|\nabla u(t+\zeta)\|_{L^2(\Omega)}^2+\lambda\gamma\|\nabla\Delta\phi(t+\zeta)\|_{L^2(\Omega)}^2\,d\zeta\leq \exp(\int_0^{t+\ell}\mathbb{L}(\zeta)\,d\zeta)(\|u(s)\|_{L^2(\Omega)}^2+\lambda\|\phi(s)\|_{H^1(\bar{\Omega},d\sigma)}^2) = \mathcal{M}_\ell(t)(\|u(s)\|_{L^2(\Omega)}^2+\lambda\|\phi(s)\|_{H^1(\bar{\Omega},d\sigma)}^2).
\nu(x_{2m+1,2}) =\nu{\left(\sum_{k=1}^\infty x_{2m,k}\beta_{k,2}\right)}=\min_{k\geq4}{\left\{\big(\nu(x_{2m,2})+\nu(\beta_{2,2})\big),\big(\nu(x_{2m,3})+\nu(\beta_{3,2})\big)\right.}, \qquad\qquad\left.\big(\nu(x_{2m,k})+\nu(\beta_{k,2})\big)\right\}\geq 2m+1.
L_\rhd(\{x, z\})[y, t] = \{x, z\}\rhd [y, t]= x\rhd[z\rhd y, t]- [z\rhd(x\rhd y), t] - [x\rhd(z\rhd t), y] + z\rhd [x\rhd t, y] &=L_\rhd(x)[L_\rhd(z)y, t]- [L_\rhd(z)L_\rhd(x)y, t] - [L_\rhd(x)L_\rhd(z)t, y]+ L_\rhd(z)[L_\rhd(x)t, y].
\exp_{\tau}(b, t) := \begin{cases}0, t < -\tau, 1 + \sum\limits_{k = 1}^{\left\lfloor \tfrac{t}{\tau}\right\rfloor + 1}\frac{(t - (k - 1) \tau)^{k}}{k!}b^{k}, t \geq -\tau. \end{cases}
\|\tilde{u}\|_{L_{[0,t]}^{\infty}L_x^{p_0}}\lesssim\big{\|}e^{-t\partial_x^3}\big{(}\tilde{u}(0)\big{)}\big{\|}_{L_{[0,t]}^{\infty}L_x^{p_0}}+\|\mathcal{E}\|_{L^{q_2'}_{[0,t]}L_x^{r_2'}}+\big{\|}\big{(}f(\tilde{u})\big{)}_x\big{\|}_{L^{q_2'}_{[0,t]}L_x^{r_2'}}&=I+II+III.
f_{n+1}\circ h_n(a_n) =f_{n+1}\circ(\ast_{1n}(\top_1,a_n))=f_1(\top_1)\ast_{1n}f_n(a_n) =\top_1\ast_{1n}f_n(a_n)=l_n(f_n(a_n))=l_n\circ f_n(a_n).
\partial_{G}(1)=0, \quad\partial_{G}(\Sigma \lambda_{i_1})=(1\otimes y_{i_1}-y_{i_1}\otimes 1), \forall \lambda_{i_1}\in \Lambda_1 &\partial_{G}(\Sigma^k\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{k-j}\Sigma^{k-1}\lambda_{i_1\cdots \hat{i_j}\cdots i_k}(1\otimes y_{i_j}+(-1)^ky_{i_j}\otimes 1),
\phi_{\Phi_k, \beta}(x, y) = \begin{dcases}|\{n \leq x; \ \ \frac{\phi(n)}{n^{\delta}}\leq y\}| , k |\{n \leq x; \ \ \frac{\phi(n) \alpha_{k}(n)}{n^{\delta}}\leq y\}| , k \end{dcases}
\begin{aligned}\nu_i\left(\mathbf{Q}_{i+1}^{{\bar\gamma}_{i+1}}\right)-\nu_i\left(Q_i^{\gamma_i-q}\left(\partial_{j_0}\mathbf{Q}_i^{{\bar\gamma}_i}\right)\prod\limits_{t=1}^q\left(\partial_{j_t}Q_i\right)\right)\le &\le\frac{j_0+j_1+\dots+j_q}{b_i}(\beta_i-\nu(\partial_{b_i}Q_i))=\frac b{b_i}(\beta_i-\nu(\partial_{b_i}Q_i)).\end{aligned}
\left[\mbox{Prod}\left(\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right),\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right)^{\top^{2}},\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right)^{\top}\right)\right]_{0,0,0}=
\left \Vert \begin{bmatrix}a_{i,0}\sqrt{\rho_{0}}a_{i,1}\sqrt{\rho_{1}}\!\ldots\! \underbrace{0}_{i}\!\ldots\! a_{i,N_{ue}}\sqrt{\rho_{N_{ue}}}\sigma_n\end{bmatrix}\right \Vert &\leq \frac{\sqrt{\rho_i}a_{i,i}}{\sqrt{\gamma_c}}, i=1, \ldots, N_{ue},
\big(\tau_{k,\ell}^{(\alpha,\beta)}\big)^{2}= \tau_{k, \ell}^{(\alpha + 1,\beta)}\tau_{k, \ell}^{(\alpha - 1,\beta)}+\tau_{k+1, \ell+1}^{(\alpha-1,\beta)}\tau_{k - 1, \ell-1}^{(\alpha + 1,\beta)}-\tau_{k+1,\ell}^{(\alpha-1,\beta)}\tau_{k - 1, \ell}^{(\alpha + 1,\beta)},
2n -2 \geq Rn - \sum_{i=1}^R(a_i+b_i) & \geq Rn + \sum_{i=1}^R(e_{i}-2)a_i + \sum_{q\in U_{T_0,f}}\bigl(e(q|t_0)-1\bigr) - (2n-2) - \sum_{i=1}^Rb_i.
\sum_{i=1}^l |\alpha_i|^p + \sum_{j=1}^s \left| \sum_{i=1}^l a_{j\,i}\alpha_i + v_j \right|^p + \sum_{j=1}^m \left|\sum_{i=1}^l \bar a_{j\,i}\alpha_i + u_j \right|^p \to \min &\begin{cases}\alpha \in \mathbb{Z}^l \setminus \{0\}||\alpha||_1 \leq C. \end{cases}
\left< W ( C ) \right>_{\mathrm{l o w - e n e r g y}}= \int D h_{\mu \nu}\exp \left\{- \int d^{3}x \left[ \frac{1}{6 \zeta}H_{\mu \nu \lambda}^{2}+ g^{2}h_{\mu \nu}^{2}- 2 \pi i h_{\mu \nu}\Sigma_{\mu \nu}\right] \right\}.
\frac{\mu}{\hbar}\psi^{\dagger}(iI\delta_{ij}+\sigma_{k}\epsilon_{ijk})B_{i}\psi+\frac{\mu}{\hbar}\psi^{\dagger}(-iI\delta_{ij}+\sigma_{k}\epsilon_{ijk})B_{i}\psi=\frac{2\mu}{\hbar}(\psi^{\dagger}\sigma_{k}\psi)B_{i}\epsilon_{ijk}
Spl(f,\sigma)=Spl(f,\sigma\nu),\, Spl(f,\sigma,\{k_j\})=Spl(f,\sigma\nu,\{k_j'\}),\, Spl(f,\sigma,\{k_j\},L,\{R_j\})=Spl(f,\sigma\nu,\{k_j'\},L,\{R_j\}),
:{\bf X}^{\mu}({\bf z}_{1},\overline{\bf z}_{1}){\bf X}^{\nu}({\bf z}_{2},\overline{\bf z}_{2}):\, ={\cal R}{\bf X}^{\mu}({\bf z}_{1},\overline{\bf z}_{1}){\bf X}^{\nu}({\bf z}_{2},\overline{\bf z}_{2}){}-{\bf G}^{\mu\nu}\left({\bf z}_{1},\overline{\bf z}_{1}|{\bf z}_{2},\overline{\bf z}_{2}\right)~,
Q_k(x) = k! Y^a_k(x) x^a e^{-x}= \frac{1}{2\pi i}\int_{L}\frac{\Gamma( (w-1)\theta^{-1}+ 1)}{\Gamma( (w-1)\theta^{-1}+ 1-k)}\Gamma(w+a) x^{-w}dw .
\begin{cases}u_1^{\prime\prime}+ A u_1 = 0 \,, u_i^{\prime\prime}+ A u_i+ C_{i i-1}u_{i-1}= 0 \,, 2 \le i \le n \,, (u_i,u_i^{\prime})(0)=(u_i^0,u_i^1) \mbox{for}i=1, \ldots n\,,\end{cases}
{\cal S}^n\equiv{\cal S}\left( \varphi_n^\alpha ,\varphi_{n+1}^\alpha,\nabla \varphi_n^\alpha ,\nabla \varphi_{n+1}^\alpha ,T\right)\rightarrow{\cal S}^n\left( T_n\right) \equiv{\cal S}\left( \varphi_n^\alpha ,\varphi_{n+1}^\alpha ,\nabla \varphi_n^\alpha ,\nabla \varphi_{n+1}^\alpha ,T_n\right) ,
\nabla f(x^*)+\sum_{i=1}^m \lambda^*_i \nabla g_i(x^*) + \sum_{i=1}^p \mu^*_i \nabla h_i(x^*) + \sum_{i=1}^n \gamma^*_i e_i = 0, \lambda^*_i \geq 0,\lambda^*_i \cdot g_i(x^*) =0\forall i=1,\dots,m, \gamma^*_i =0\forall i \in I_{\pm0}(x^*,y^*).
h_0^{(1)}(k) = - k^2-m^2 - \int{d^4 l \over (2 \pi)^4}{R_{00}(k,l) \, (l^2+m^2) \over (l^2+m^2)^2 + [h_1^{(0)}(l)]^2 + [h_2^{(0)}(l)]^2 + [h_3^{(0)}(l)]^2}\, .
\times \exp \left( -2 e \sum_{b=1}^N \sum_{j=1}^{n_b}\Big( \frac{\varepsilon_{\mu \nu}\partial_\mu}{\triangle}a^{(b)}_\nu,\delta_n(x_j^{(b)}) - \delta_n(y_j^{(b)}) \Big) \right)
\mathcal{V}_\ast(t) := \{v \in C^2(\O({\Gamma(t)})) \mid \nabla v \cdot \nabla \phi = 0 \}\Vert v \Vert_{\mathcal{V}}:= \left(\Vert v \Vert_{H^1(\Gamma(t))}^2 + \Vert \nabla \phi \cdot \nabla v \Vert_{L^2(\O(\Gamma(t)))}^2\right)^{\frac12},
\alpha_i^* =\frac{c^*}{(N_c+1-i)!}\frac{d^{N_c+1-i}}{ds^{N_c+1-i}}\left[\frac{1}{\prod_{n=1}^{N-N_c}[s+(1+\frac{n}{N_c})\lambda_{0}]}\right] \bigg|_{s=-\lambda_{0}}&=\frac{c^*}{(N_c+1-i)!}\left[ \sum_{k=1}^{N-N_c}\frac{\theta_k^*(-1)^{N_c+1-i}(N_c+1-i)!}{( \frac{k}{N_c}\lambda_{0})^{N_c-i+2}}\right],
K(z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]=\tilde{p}(0,z) \exp(\int_0^T (b_0(s,z)\pi(s,z)-\frac{a_0(s,z)}{b_0(s,z)})dB(s) -\frac{1}{2}\int_0^T (b_0(t,z)\pi(s,z)-\frac{a_0(s,z)}{b_0(s,z)})^{2}ds).
\frac{2p}{2-p}\log(\frac{\left \|\sqrt{n_{\pi}}a_{\pi}\right \|_p}{\left \|\sqrt{n_{\pi}}\widehat{a_{\pi}}\right \|_{p'}}) =\frac{2p}{2-p}\log(\frac{\left \|\sqrt{n_{\pi}}a_{\pi}\right \|_p}{n_{\pi}^{\frac{1}{2}-\frac{1}{p}}}) &\leq \frac{2p}{2-p}\log(B_1^{\frac{2-p}{p}})=\log(B_1^2),
\Delta_1 = g_3g_1^{-1}g_3g_1^{-1}g_3x_0(C_2), \Delta_2 = g_3g_1^{-1}g_3g_1^{-1}g_3g_2x_0(C_1,) \Delta_3 = g_3g_1^{-1}g_3g_2x_0(C_1), \Delta_4 = g_3x_0(C_3), \Delta_5 = x_0(C_2)\cup x_0(C_3).
f_1(x_{-2},x_{-1},x_1,x_2) =x_{-1}(a_m-x_1)\le 0 f_2(x_{-2},x_{-1},x_1,x_2) =x_{-2}(a_m-x_1)+x_1(a_m-x_{-2})\le 0 f_3(x_{-2},x_{-1},x_1,x_2) =x_{-1}(a_m-x_{2})+x_2(a_m-x_{-1})\le 0 f_4(x_{-2},x_{-1},x_1,x_2) =x_1(a_m-x_{-1})\le 0.
Z_r= \,\sum_{j=1}^{\ell_r}\frac{1}{j}\sum_{t=1}^jY_t=\sum_{t=1}^{\ell_r}Y_t\sum_{j=t}^{\ell_r}\frac{1}{j}= \sum_{t=1}^{\ell_{r-1}}Y_t\sum_{j=t}^{\ell_r}\frac{1}{j}+\sum_{t=\ell_{r-1}+1}^{\ell_r}Y_t\sum_{j=t}^{\ell_r}\frac{1}{j}=:Z_{r,1}+Z_{r,2}.
\frac{\phi}{\chi^{q_{1}/ q_{2}}}= C \frac{\prod_{r = 1}^{n_{1}}( z - Z_{r})}{\prod_{r = 1}^{n_{2}}( z - Z_{r}^{\prime})^{q_{1}/ q_{2}}}, \mathrm{( C : a c o m p l e x n u m b e r )}.
G_{( i a ) ( j b )}( x , y ; \mu ) = \left( 1 + e_{R}^{2}\,{\frac{1 1 N}{1 2}}\,{\frac{1}{2 \pi^{2}}}\{\log}( \mu / m_{R}) \right) \, G_{( i a ) ( j b )}^{0}( x , y ) +{\cal O}( e_{R}^{3})
A^{p,n}_{E\otimes F,h_\nu\otimes h_F,\omega}=[i\Theta_{E,h_\nu}\otimes\mathrm{id}_F,\Lambda_\omega]+[i\Theta_{F,h_F}\otimes\mathrm{id}_E,\Lambda_\omega] \geq[i\Theta_{F,h_F}\otimes\mathrm{id}_E,\Lambda_\omega]=B_{h_F,\omega}>0,
{\cal Z}(x)\> = \> 1 \> + \> \frac{\alpha}{4\pi}\frac{(4\pi)^\epsilon}{\Gamma(2-\epsilon)}\> \xi \> \left[{\epsilon \over 2 - \epsilon}\int_0^x dy \>{y^{1-\epsilon}\over x^2}{\cal Z}(y) \> - \> \int_x^\infty dy \> y^{-\epsilon-1}{\cal Z}(y) \right]\;\;\;.
\begin{aligned}H = H(n_1,n_2,n_3) := (n_1+n_2+n_3)^5 - n_1^5 - n_2^5 - n_3^5 = \frac52(n_1+n_2)(n_1+n_2)(n_2+n_3)(n_1^2+n_2^2+n_3^2+n^2).\end{aligned}
\varepsilon_{j}( \theta ) = \delta_{j , 0}\, m R \, \cosh \theta + \sum_{k = 0}^{p - 2}\int_{- \infty}^{\infty}\Phi_{j , k}( \theta - \theta^{\prime}) \log \big ( 1 + e^{- \varepsilon_{k}( \theta^{\prime})}\, \big ) \, \, d \theta^{\prime}, \qquad j = 0 , 1 , \ldots , p - 2 .
\frac{d(x,y)}{\sqrt{d(x,p)d(y,p)}}\leq\frac{d(x,z)}{\sqrt{d(x,p)d(z,p)}}+ &+\frac{d(z,y)}{\sqrt{d(z,p)d(y,p)}}+\frac{d(x,z)d(z,y)}{d(z,p)\sqrt{d(x,p)d(y,p)}}.
\hat{u}(\xi) = \frac{\widehat{Vu}(\xi)}{|\xi|- \lambda}= \chi(|\xi| < \lambda /2) \frac{\widehat{Vu}(\xi)}{|\xi|- \lambda}+\chi(|\xi| \geq \lambda /2) \frac{\widehat{Vu}(\xi)}{|\xi |-\lambda}
W=-\sum_{k\geq 1}\frac{B_{2k}u^{2(2k-1)}}{2k(2k-1)}(\frac{\partial}{\partial t_{2k}}-\sum_{i\geq 0}t_i\frac{\partial}{\partial t_{i+2k-1}}+\frac{1}{2}\sum_{i+j=2k-2}(-1)^i\frac{\partial^2}{\partial t_i\partial t_j}).
h_{a^*\varpi}(r^*\varpi^{4-j},s;\psi_{\varpi^{j+1}}\chi) =\sum_{(n,a^*\varpi)=1}\frac{\psi_{\varpi^{j+1}}(n)\chi(n)g_6(a\varpi^{4-j},n)}{N(n)^s}=\sum_{(n,a^*)=1}\frac{\psi_{\varpi^{j+1}}(n)\chi(n)g_6(a\varpi^{4-j},n)}{N(n)^s}-\sum_{\substack{(n,a^*)=1 \varpi | n}}\frac{\psi_{\varpi^{j+1}}(n)\chi(n)g_6(a\varpi^{4-j},n)}{N(n)^s}.
| Z(\mathbb{F}_q)| = \sum_{M \in \mathbb{G}}\Bigl( |((L_1\cap H) \cup (L_2^{(M)}\cap H) \cup \dots \cup (L_d^{(M)}\cap H))(\mathbb{F}_q)| -|(L_1\cap H)(\mathbb{F}_q)| \Bigr) + |(L_1\cap H)(\mathbb{F}_q)|.
z_{a}=\frac{p}{2}\frac{(1+\frac{p^{2}}{4\pi^{2}})(1+\frac{p^{2}}{16.\pi^{2}})(1+\frac{p^{2}}{36\pi^{2}})\cdot\cdot\cdot}{(1+\frac{p^{2}}{\pi^{2}})(1+\frac{p^{2}}{9\pi^{2}})(1+\frac{p^{2}}{25\pi^{2}})\cdot\cdot\cdot}
||U(t,0)G(x)-G(x)||_{L^{\infty}(0,\tau_1;L^2)}= ||(U(t,0)-I)e^{(\alpha-1)W}g(x)||_{L^{\infty}(0,\tau_1;L^2)}\le ||U(t,0)-I||||e^{(\alpha-1)W}||_{L^{\infty}(0,\tau_1;L^{\infty})}||g(x)||_{L^2}\lesssim ||g(x)||_{L^2}\lesssim ||x||_{H^2}^{\alpha}.
\widetilde{\nu}(\widetilde{N}-1\pm\widetilde{\Phi}) =\widetilde{\nu}(\alpha_{i}f_{i}-1\pm\left(\alpha_{i}\varphi_{i}+\beta_{i})\right)=\alpha_{i}f_{i}\left(f_{i},_{r}\pm\varphi_{i},_{r}\right)>0
\Phi(\mathrm{x},t;\mathrm{y},\tau):=\ \begin{cases}\frac{\alpha}{4\pi(t-\tau)}\textbf{exp}\Bigg(-\frac{\alpha |\mathrm{x}-\mathrm{y}|^2}{4(t-\tau)}\Bigg), \ \ \t > \tau 0 ,\end{cases}
\Phi\left( \theta,t\right) =\frac{1}{k\left( \theta,t\right)}-\frac{L\left( t\right)}{2\pi}-\frac{1}{2\pi}\int_{0}^{t}\int_{0}^{2\pi}k^{\alpha}\left( \theta,\sigma\right) d\theta d\sigma,\ \ \ \left(\theta,t\right) \in S^{1}\times\lbrack0,T),
\langle{\cal O}[ \overline{{{\psi}}}, \psi , A , h ] \rangle \; = \; \frac{1}{Z}\sum_{n = 0}^{\infty}\frac{( - 1 )^{n}}{n !}\langle{\cal O}[ \overline{{{\psi}}}, \psi , A , h ] \Big ( S_{M}[ \overline{{{\psi}}}, \psi ] \Big )^{n}\rangle_{0}\; ,
\begin{array}{c c}{{\bar{\theta}_{i}= \theta^{i \dagger}\tilde{\gamma}^{0}= ( \theta^{j})^{t}\bar{\cal E}_{j i}~}}&{{~ \bar{\tilde{\theta}}_{i}= \tilde{\theta}^{i \dagger}\gamma^{0}= ( \tilde{\theta}^{j})^{t}\bar{\cal E}_{j i}}}\\ \end{array}
\bar{I}_{K_S}=-H\frac{\Gamma(1+\frac H z)^4\Gamma(1-\frac H z)}{\Gamma(1+\frac{3H}{z})}\sum_{m\geq 0}(-1)^m\frac{(2\pi i)e^{\frac{2\pi im}{3}}}{e^{-2\pi i \frac{H}{z}}-e^{\frac{2\pi im}{3}}}\frac{q^{-\frac m 3}}{\Gamma(m)\Gamma(1-\frac m 3)^4\Gamma(1+\frac m 3)}.
\hat{\delta}_2(\hat{B},Q)^2 =\min_{\pi}\|\hat{B}_{\pi}-Q\|_2^2 \geq \frac{1}{2}\|\hat{B}_{\pi}\|_2^2-\|Q\|_2^2 = \frac{1}{2}\|\hat{B}\|_2^2-\|Q\|^2.
x_1^3x_2^{4}x_3^{4}x_4^{5}x_5^{7}= Sq^1(x_1^3x_2x_3^{2}x_4^{9}x_5^{7}+ x_1^3x_2x_3^{2}x_4^{5}x_5^{11}) + Sq^2(x_1^5x_2^{2}x_3^{2}x_4^{5}x_5^{7}+ x_1^5x_2x_3^{2}x_4^{6}x_5^{7}) \quad+ Sq^4(x_1^3x_2^{2}x_3^{2}x_4^{5}x_5^{7}+ x_1^3x_2x_3^{2}x_4^{6}x_5^{7}) \ (P_5^-(B_5(\omega_{(4)}))).
\min_{\mu \in \mathcal{P}(\mathbb{T}^{d})}\mathcal{E}_{V}(\mu) = \lim_{N \to \infty}\min_{X_{N}\in \mathbb{T}^{d \times N}}\left( \frac{1}{N^{2}}\mathcal{H}_{N}(X_{N}) \right),
\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}\|Z^m(t)-Z^{n}(t)\|_{L^p(\Omega;H^\beta)}= \lim_{n\rightarrow\infty}\sup_{t\in[0,T]}\|P^m(I-P_n) Z(t)\|_{L^p(\Omega;H^\beta)}\leq C\lim_{n\rightarrow\infty}\lambda_{n+1}^{-\frac{\gamma-\beta}2}\sup_{t\in[0,T]}\|Z(t)\|_{L^p(\Omega;H^\gamma)}.
i \alpha^{\prime}\int_{0}^{2 \pi}d \theta ~ \mathrm{T r}~{\cal P}\left( U [ A ] ~ \delta_{1}A_{a}\partial_{\theta}\overline{{{X}}}^{a}\right) + i \alpha^{2}\int_{0}^{2 \pi}d \theta ~ \mathrm{T r}~{\cal P}\left( U [ A ] ~ \delta_{2}A_{a}\partial_{\theta}\overline{{{X}}}^{a}\right)
N^{\cal P}~ = ~ \left( \frac{2}{k + 2}\right) ~ \sum_{l = 1}^{k + 1}s i n^{2}\theta_{l}~ \sum_{m_{1}= - j_{1}}^{j_{1}}\cdots \sum_{m_{p}= - j_{p}}^{j_{p}}\exp \{2 i ( \sum_{n = 1}^{p}m_{n}) ~ \theta_{l}\}~ ,
\begin{array}{rcll}-\Delta_{\xi}\Z^{(0)}_p(\xi) = 0, \xi \in \Pi_0 , \partial_{\xi_2}Z^{(0)}_p(\xi_1,0) = 0, \xi_1 \in (0,1)\setminus \big( \frac12 - \frac{h_0}{2}, \frac12 - \frac{h_0}{2}\big), \partial_{\xi_1}Z^{(0)}_p(\xi) = - \delta_{p,1}, \xi \in \partial\Pi^{-}_{h_1}\cap \{\xi: \ \xi_2 <0\}, p=1, 2.\end{array}
\forall z\in D,\forall v,w\in \mathbb{C}^2\setminus\{0\},Bis_{z^{(\nu)}}^D\left( \left(\partial\Lambda^{(\nu)}_{z^{(\nu)}}\right)^{-1}(v),\left(\partial\Lambda^{(\nu)}_{z^{(\nu)}}\right)^{-1}(w)\right)=Bis_{0}^{\Omega_\nu}\left(v,w\right),
\mathrm{Sh}_n(\Omega(C))=[\mathrm{Sh}_{n-1}(\sigma(\mathrm H_C^{\mathrm{dR}}))|\Omega(C)]+\sum_{k=0}^{n-1}[\mathrm{Sh}_{k-1}(\Omega(C))|d\mathcal O(C)|\mathrm{Sh}_{n-k}(\Omega(C))].
\phi(W''_0) =_H w_0w_1^{-1}a_2^{\prime\,-1}(a_4^{-1}t_1)^2 a'_2 w_3 &= w_0 w_1^{-1}a_2^{\prime\,-1}(a_4^{-1}a_2 w_1a'_1)^2 a'_2 w_3 &= w_0 a'_1 a_4^{-1}a_2 w_1 a'_1 a'_2 w_3 &= \theta(W).
\begin{aligned}x_{e,0}- x_{e,0}= 0 x_{e,e}- x_{e,e}= 0 x_{e,e}+ x_{e,0}= 0 x_{0,e}+ x_{e,e}= 0 x_{0,e}- x_{e,0}= 0 x_{0,e}- x_{0,e}= 0 \end{aligned}
\begin{cases}Y_t+u_0 \cdot \nabla Y=0 (0,+\infty)\times \mathbb{R}^3, begin{align*}10pt]Z_t- Y^2\triangle Z=0 \ (0,+\infty)\times \mathbb{R}^3 , begin{align*}10pt](Y,Z)|_{t=0}=(\phi_0,u_0) \mathbb{R}^3, begin{align*}10pt](Y,Z)\rightarrow (\overline{\phi},0) |x|\rightarrow +\infty,t> 0,\end{cases}
\begin{array}{l l l l l l}{\displaystyle D}&{= 3}&{: \quad}&{S p ( 4 ; R )}&{\to}&{O S p ( N ; 1 | R )}\\{\displaystyle D}&{= 4}&{: \quad}&{S U ( 2 , 2 )}&{\to}&{S U ( 2 , 2 | N )}\\{\displaystyle D}&{= 6}&{: \quad}&{U_{\alpha}( 4 ; H )}&{\to}&{U U_{\alpha}( N ; 4 | R )}\\ \end{array}
2 \alpha^{5/2}\mathrm{Re}\langle u_{\alpha}\tilde{\Phi}_*^1\,|\, (\|\lambda_y\|^2 + A^+_y A^-_y) \Phi_{\#}^y \rangle = 0, 8 \alpha^2 \mathrm{Re}\langle u_{\alpha}\tilde{\Phi}_*^1\,|\, P u_{\alpha}A^+_y \Omega \rangle = 0
E(\varphi_{\varepsilon_{0},k};k,R) = E(\varepsilon_{0}\psi;k,R) + E( \xi u_{k};k,R) + \varepsilon_{0}^{2}\int_{B_{R}(0)}\left( (\xi u_{k})^{2}\cdot \chi_{B_{R}(0)}* Y_{a_{k}}\right) \psi^{2}&\leq E(\varepsilon_{0}\psi;k,R) + \varepsilon_{0}^{4}\leq - C_{1}< 0,
C_{\beta,\delta}:= \frac{1}{\Lambda_{\beta}}\min\left\{\frac{1}{3^{3/2}}\left(\delta\Lambda_{\beta}\right)^{3-\beta},\frac{1}{\sqrt{3}}(\delta\Lambda_{\beta})\left(1- \left(\delta\Lambda_{\beta}\right)^{\frac{\beta-2}{2}}\tanh\left(\left(\delta\Lambda_{\beta}\right)^{-\frac{\beta-2}{2}}\right)\right)\right\}.
\|f(t)\|_{\mathcal{E}^\alpha_{w}}: =\langle t\rangle^{-p_0}\|f(t)\|_{\dot{H}^{N_0+\alpha,p_1}}+\langle t\rangle^{1/2}\big\|f(t)\big\|_{\dot{W}^{N_2+\alpha,2/5}}+\langle t\rangle^{-4p_0}\|Sf(t)\|_{\dot{H}^{N_1+\alpha,p_1}}.
\int_{\mathbb{T}^d}\prod_{l=1, l\neq j}^d K_n(y_l)|f(x-y)|\,dy=\int_{\mathbb{T}^d}(\prod_{l=1, l\neq j}^d K_n)(x-y)|f(y)|\,dy &=\int_{\mathbb{T}^{d-1}}(\prod_{l=1, l\neq j}^d K_n)(\tilde{x}_j-\tilde{y}_j)[\int_\mathbb{T}|f(y_1,\cdots, y_j,\cdots, y_d)|dy_j]\,d\tilde{y}_j &=\int_{\mathbb{T}^{d-1}}K_n^{d-1}(y)|f|_j(\tilde{x}_j-y)|\,dy.
R_{1}(l)^{'}=\frac{1}{25}\sum_{s=1}^{l}(-1)^{s+1}5^{s}\frac{2l}{l+s}\binom{l+s}{l-s}\Bigg\lbrack \sqrt{5}-5^{1-s}\hbox{res}_w\Bigg(\frac{(1+w)^{2s}}{(w){^s}((1+w)^2-5w)}\bigg)\Bigg\rbrack.
\hat I(t) = \frac{1}{2}\left[ (mf^{-1}\dot\rho\hat x_1 -\rho\hat p_1)^2 + \frac{\nu^2}{\rho^2}\hat x_1^2 + (mf^{-1}\dot\rho\hat x_2 -\rho\hat p_2)^2 + \frac{\nu^2}{\rho^2}\hat x_2^2 \right],
k_{P_H}= \frac{\left( \frac{(1+e^{ab})^{k_1-1}}{\left(1+e^{ab}\left(1-\zeta\right)\right)^{k_1}}-\frac{1}{1+e^{ab}}\right)^2}{\frac{(1+e^{ab})^{k_1-2}}{\left(1+e^{ab}\left(1-\zeta\right)\right)^{k_1}}-\frac{(1+e^{ab})^{2k_1-2}}{\left(1+e^{ab}\left(1-\zeta\right)\right)^{2k_1}}},
\sum_{j \in N_{i'}}\max \left\{a_{i'j}, b - \sum_{k \in M_C - i'}a_{k t_k}\right\}x_{i'j}& + \sum_{i \in M_C - i'}\left(\sum_{j =1}^{t_i}a_{i t_i}\max \left\{1, \frac{a_{ij}}{b - \sum_{k \in M_C -{i, i'}}a_{k t_k}- a_{i' n_{i'}}}\right\}x_{ij}+ \sum_{j=t_i+1}^{n_i}a_{ij}x_{ij}\right) > b?
d s_{1 1}^{2}= \frac{1}{H_{M 2}( \rho )^{2 / 3}}\left[ - d t^{2}+ d x^{i}d x^{i}\right] + H_{M 2}( \rho )^{1 / 3}\left[ d \rho^{2}+ \rho^{2}d \Omega_{7}^{2}\right] ,
l_{0}^{o}+ l_{1}^{o}+ l_{2}^{o}+ l_{3}^{o}+1 \geq0, l_{0}^{o}+ l_{1}^{o}- l_{2}^{o}- l_{3}^{o}-1 \geq0 , & l_{0}^{o}- l_{1}^{o}+ l_{2}^{o}- l_{3}^{o}-1 \geq0 , l_{3}^{o}=(l_{0}-l_{1}-l_{2}-1)/2 \geq-1,
\frac{d^{\ell+1}}{dz^{\ell+1}}\left[P_\ell(z) \ln\left(\frac{1+z}{1-z}\right)\right] = \sum_{m=1}^{\ell+1}\frac{(\ell+1)!}{m! (\ell+1-m)!}\frac{d^{\ell+1-m}P_{\ell}(z)}{dz^{\ell+1-m}}\frac{d^m}{dz^m}\ln\left(\frac{1+z}{1-z}\right),
240\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right\}^{(8)}=\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{V_C}-\widetilde{V_C}\otimes \widetilde{V_C}+\widetilde{V_C})\right\}^{(8)}.
|I_{2}|\leq \frac{\mu}{8}\sum_{g\geq -1}\lambda_q^{2s+2\alpha}\|b_q\|_2^2+\frac{\nu}{16}\sum_{g\geq -1}\lambda_q^{2s+2}\|u_q\|_2^2 &+C_{\nu,\mu}\left(\sum_{q\geq -1}\lambda_q^{2s}\|b_q\|_2^2\right)^{\gamma_1}+C_{\nu,\mu}\left(\sum_{q\geq -1}\lambda_q^{2s}\|u_q\|_2^2\right)^{\gamma_2},
[\begin{matrix}1&3&1&9\\ 1&1&-1&1\\ 3&11&5&35\end{matrix}]\rightarrow[\begin{matrix}1&3&1&9\\ 0&-2&-2&-8\\ 0&2&2&8\end{matrix}]\rightarrow[\begin{matrix}1&3&1&9\\ 0&-2&-2&-8\\ 0&0&0&0\end{matrix}]\rightarrow[\begin{matrix}1&0&-2&-3\\ 0&1&1&4\\ 0&0&0&0\end{matrix}]
(X^{k+1}A^{k})A(X^{k+1}A^{k})= X^{k+1}A^{k+1}X^{k+1}A^{k}=X^{k}A^{k}(X^{k+1}A^{k}) = X^{k}X^{k+1}A^{k}A^{k}=X^{k+1}X^{k}A^{2k}=X^{k+1}A^{k}.
\int_{0}^{\pi}\exp\left(-\frac{\beta^{\frac{1}{2}}N^{\frac{1}{2}}s^{\frac{3}{4}}t^{\frac{3}{4}}\theta^2}{2t}\right)d\theta \le C\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}s^{-\frac{3}{8}}t^{\frac{1}{8}}&\le C\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}t^{-\frac{1}{4}}.
\begin{cases}\Psi''-\Delta \Psi+ \overline A_p^T\Psi=0 \hbox{in}(0, +\infty)\times \Omega, \Psi=0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu \Psi+\overline B_p^T\Psi=0 \hbox{on}(0, +\infty)\times\Gamma_1\end{cases}
x^-_{\theta}(1)=x_{n\,1}^-(1)=[\,x_1^-(0),\ldots,x_{n}^-(1)\,]_{(r^{-2},\,\ldots,\, r^{-2})}, &x_{\theta}^+(-1)=x_{n\,1}^+(-1)=[\,x_{n}^+(-1),\,\ldots,\,x_1^+(0)\,]_{\langle s^{-2},\,\ldots,\, s^{-2}\rangle}.
\Phi_{n}(x,t) := \sum_{i=1}^{n}\gamma_{n,i}(t) w_{i}(x), \Xi_{n}(x,t) := \sum_{i=1}^{n}\delta_{n,i}(t)w_{i}(x), \Sigma_{n}(x,t) := \sum_{i=1}^{n}\eta_{n,i}(t)w_{i}(x)
F(A) =\left(m(2^k-1)+2^{k-1}-1+d-2+m+2\right)\cdot (m(2^k-1)+2^{k-1}-1)-d &=\left((2m+1)2^{k-1}-1\right)^2+(d-m)\left((2m+1)2^{k-1}-1\right)-md-d.
S \, = \, S_{0}+ \phi_{A}^{\ast \, a}\, s_{a}\phi^{A}+{\frac{1}{2}}{\bar{\phi}}_{A}s_{2}s_{1}\phi^{A}\, + \,{\frac{1}{2}}\epsilon^{a b}\phi_{A \, a}^{\ast}\, \pi_{b}^{A},
(a)\ -\bigg\langle \nabla V(x),f(x) + \sum_{i = 1}^m g_i(x)u_i(x)\bigg\rangle + \sum_{j = 1}^Ja_j(x)\sigma_j(x) + \sum_{\ell = 1}^L b_\ell(x)\rho_\ell(x)\ \mathrm{is\SOS}, (b)\ \sigma_j(x)\ \mathrm{is\SOS},
\partial^{\mu}{\cal{G}}_{\mu \mu_{2 n}. . . \mu_{s}}^{\{n - 1 \}}={\frac{( s + 1 - 2 n )}{( s + 2 - 2 n )}}\partial_{( \mu_{2 n}}{\cal{G}}_{\mu_{2 n + 1}. . . \mu_{s})}^{\{n \}}
\begin{aligned}|u-Q_F u_I|_{1,F}\lesssim |u-\Pi_F u_I|_{1,F}+|\Pi_F u_I-Q_F u_I|_{1,F}& \lesssim |u-\Pi_F u_I|_{1,F}+\epsilon_F^{-1}h_F^{-1}\|\Pi_F u_I-Q_F u_I\|_{0,F}& \leq |u-\Pi_F u_I|_{1,F}+\epsilon_F^{-1}h_F^{-1}\|\Pi_F u_I- u_I\|_{0,F}& + \epsilon_F^{-1}h_F^{-1}\|u_I-Q_F u_I\|_{0,F}\end{aligned}
e^{(4)}= e^0\wedge e^1\wedge e^2\wedge e^3, e_a^{(3)}= \frac{1}{3!}\epsilon_{abcd}e^{b}\wedge e^{c}\wedge e^{d}\hbox{and}e_{ab}^{(2)}= \frac{1}{2!}\epsilon_{abcd}e^{c}\wedge e^{d}
L_{2}^{\ast\ast}= \frac{\Gamma(t-2)\left[\xi_{p}\left(1+\xi_{p}^{2}\right)^{-(t-2)}-\xi_{q}\left(1+\xi_{q}^{2}\right)^{-(t-2)}\right]}{4\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})}+\frac{1}{(2t-5)(2t-3)}\frac{F_{Y_{(2)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},~t>\frac{5}{2},
V_2(\rm x) = \frac{1}{32}\rm x^6 - \frac{\omega}{16\rm g}\rm x^4 +(\frac{\omega^2}{32\rm g^2}+ \frac{3j}{4}+ \frac{3}{8}) \rm x^2 -\frac{\omega}{4\rm g}(5 j+\frac{1}{2})
D_{k,i}^{\alpha}u_{n}=\frac{(\Delta t)^{-\alpha}}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha}\frac{\mathrm{d}P_{i}^{k}(t_{j-1}+s\Delta t)}{\mathrm{d}s}\mathrm{d}s &=(\Delta t)^{-\alpha}\sum_{j=0}^{k-1}w_{n,j}^{(k,i)}u_{j}+(\Delta t)^{-\alpha}\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}u_{j},
(T_n^*)^r \leq \frac{r}{r-1}\sum_{i=1}^n T_i^*\left ( (T_i^*)^{r-1}-(T_{i-1}^*)^{r-1}\right ) = \frac{r}{r-1}\sum_{i=1}^n T_i\left ( (T_i^*)^{r-1}-(T_{i-1}^*)^{r-1}\right ) \, ,
\begin{aligned}D_{f,l}=\{s_{2f,i_f}s_{2f-1,j_f}\cdots s_{2,i_1}s_{1,j_1}\mid i_1<\ldots<i_f, k\leq j_k<i_k\leq l, 1\leq k\leq f \}, \tilde D_{f,l}= \{s_{2f,i_f}s_{2f-1,j_f}\cdots s_{2,i_1}s_{1,j_1}\mid k\leq j_k<i_k\leq l, 1\leq k\leq f \}, \end{aligned}
\sum_j (-1)^{1+\dots+p^t-j}\binom{j(p-1)}{1+\dots+p^t-j}Q_{(2r+ps-2pj)(p-1)}Q_{2j(p-1)}&=q^{\tfrac{p-1}{2}}\sum_i (-1)^{r+p^{t+1}-1-i}\binom{i(p-1)}{r+p^{t+1}-1-i}Q_{(2r+ps-2pi)(p-1)}Q_{2i(p-1)}.
y_- \mkern1mu=\mkern1mu{{\textstyle{-\mkern-1.5mu\sum\limits_{i=1}^{n_{{\textsc{d}}}}}y_i^{\textsc{d}}}}-{{\textstyle{\sum\limits_{i=1}^{n_{{\textsc{g}}}}}y_{i+}^{\textsc{g}}}}\mkern1.5mu+\mkern4mu \max_{k>0}\mkern1.5mu y'_-(k).
\langle 0 p_{1}p_{2}p_{3}p_{4}\rangle = \xi_{1 2 3}^{2}\xi_{1 2 4}^{2}F^{p_{1}- 4 \vert p_{2}- 4 \vert p_{3}- 2 \vert p_{4}- 2}( x , u ) + O ( \theta^{5}\bar{\theta}) .
\begin{aligned}|g'(z_0)| =|h'(0)|\cdot |\phi'(z_0)| \leq 2|h(0) \left(\big{|}\log|h(0)|\big{|}+A\right)\frac{\pi}{4|x_0|}&=\frac{\pi}{2|x_0|}|g(z_0)|\left(\big{|}\log|g(z_0)|\big{|}+A\right).\end{aligned}
\hat{h}_{\mu \nu}^{\mathrm{o d d}}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{h_{0}( r )}}\\{{0}}&{{0}}&{{0}}&{{h_{1}( r )}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{h_{0}( r )}}&{{h_{1}( r )}}&{{0}}&{{0}}\\ \end{array}\right) e^{- i \omega t}\sin \theta{\frac{d P_{l}( \theta )}{d \theta}}
\langle g^l \rangle^{(m)}=\omega^{l(m+1)}\displaystyle\frac{(q^2 ; q^{2})_{\infty}^{2}}{(t^2 ; t^{2})_{\infty}^{2}}\displaystyle\frac{(t^2 \omega^{l}; t^{2})_{\infty}(t^{2}\omega^{-l}; t^{2})_{\infty}}{(q^2 \omega^{l}; q^{2})_{\infty}(q^{2}\omega^{-l}; q^{2})_{\infty}}.
S(x)=S(x;N,h)=\frac{(-1)^{h+1}}{N}\left(\frac{2\pi}{x}\right)^{\frac{N-2h+1}{N}}\sum_{n=1}^{\infty}\frac{1}{n^{\frac{2h-1}{N}}}\sum_{j=1}^{\frac{N}{2}}f_{2j-1}(x;n,N,h)
\lambda_* q(Z_\infty) = Z_\infty\cdot \nabla_{x}q(Z_\infty) = -Z_\infty\cdot \nabla_{x}q_\infty(0) = - \lim_{\ell \to \infty}\frac{r_\ell}{h_{r_\ell}}(Z_\infty \cdot \nabla_{x}p_*(X_\ell)) = 0.
{\cal C}_{kk}\sim\Big( \displaystyle\sum_{\sigma \subset U \atop |\sigma|=k/2}\prod_{u_i \in \sigma}(d_i)(d_i-1) \Big)^2\Big( \displaystyle\sum_{\sigma \subset W \atop |\sigma|=k/2}\prod_{w_i\in \sigma}(d_i')(d_i'-1) \Big)^2\Big( \dfrac{(k/2)!(k/2)!}{k}\Big)^2\:.
\frac{\partial^2{\rm dist}_N^2}{\partial p_i\partial p_j}(p)=2\sum_{k=1}^L\left(\left(\delta_{ik}-\frac{\partial P_N^k}{\partial p_i}(p)\right)\left(\delta_{jk}-\frac{\partial P_N^k}{\partial p_j}(p)\right)-\left(p_k-P_N^k(p)\right)\frac{\partial^2 P_N^k}{\partial p_i\partial p_j}(p)\right).
\psi_{x_1}=\psi_{x_5}=\psi_{x_9}=(x_1x_9x_5)(x_4x_{12}x_8), \psi_{x_2}=\psi_{x_6}=\psi_{x_{10}}=(x_2x_{10}x_6)(x_4x_{12}x_8), \psi_{x_3}=\psi_{x_7}=\psi_{x_{11}}=(x_3x_{11}x_7)(x_4x_{12}x_8), \psi_{x_4}=\psi_{x_8}=\psi_{x_{12}}=(x_2x_3x_6x_7x_{10}x_{11})(x_4x_{12}x_8).
I\!I_1 = \int_{\|u-y\|> t}\frac1{V(x,u,t+d(x,u))}\Big(\frac{t}{t+d(x,u)}\Big)^{\varepsilon_0}\frac1{V(u,y,s+d(u,y))}\Big(\frac{s}{s+\|u-y\|}\Big)^{\varepsilon_0}d\omega(u)