harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}I_3 \lesssim (\|u\|_{L^3}\|\nabla B\|_{L^2}+\|B\|_{L^3}\|\nabla u\|_{L^2})\|B\|_{L^6}+\|B\|_{H^2}\|\nabla B\|_{L^2}^2 &\lesssim (\|u\|_{H^1}+\|B\|_{H^1})(\|\nabla u\|_{L^2}^2+\|\nabla B\|_{L^2}^2) +\delta\|\nabla B\|_{L^2}^2 &\lesssim \delta(\|\nabla u\|_{L^2}^2+\|\nabla B\|_{L^2}^2).\end{aligned}
|| \sqrt{A_x}\, \partial_x w ||^2_{L^2_1}+ || \sqrt{A_z \over \varepsilon\left(z\right)}\, \partial_z w ||^2_{L^2_1}+|| \sqrt{A_x}\partial_x w\left( \cdot ,z_\iota\right)||^2_{L^2_2}=- \left( A_x \partial_x\xi_2^\prime, \partial_x w\left( \cdot ,z_\iota\right)\right)_{L^2_2}\,,
\int_{\Omega_\ell}A_\delta\nabla\phi_\ell.\nabla\phi_\ell = \int_{\Omega_{\ell_0}}A_\delta\nabla v_{\ell_0}.\nabla v_{\ell_0}+ \int_{\Omega_{\ell-\ell_0}}A_\delta \nabla \phi_\ell.\nabla\phi_\ell\,.
\begin{aligned}AG\geq \frac{2}{n}\phi G^{2}-C_{3}\phi G^{2}-\frac{4}{n}\phi G(p-1)\lambda_{1}\omega-\frac{C_{1}^{2}}{R^{2}}\frac{G}{C_{3}}-2KG\phi &-2\lambda_{1}G\phi+\lambda_{2}e^{b\omega}\phi(C_{3}G-bG+b(p-1)\lambda_{1}\omega) &+\lambda_{1}^2\phi(\frac{2}{n}(p-1)^{2}\omega^{2}+p\omega)+2Kp\lambda_{1}\omega \phi. \end{aligned}
d(u,x)+d(v,y) \ge (1-\varepsilon)\big(d(u,v)-d(x,y)\big)+\varepsilon \big(d(u,x)+d(v,y)\big) \ge(1-\varepsilon)\big(d(u,v)-d(x,y)\big)+2(1-\varepsilon)d(x,y) = (1-\varepsilon)\big(d(u,v)+d(x,y)\big).
(\mu*\gamma_{\sigma})(B_y) = \int_{B_y}f_{\mu}(z) dz \ge \inf_{z \in B_y}f_\mu(z) \times \int_{B_y}dz \ge c_3 e^{-c_2^2}f_{\mu}(0)e^{-2(1+c_1)(|x|^2+9)},
I_r \leq \prod_{k=2}^{r}(1+a_k)I_1+\sum_{l=2}^{r-1}b_l\prod_{j=l+1}^{r}(1+a_j)+b_r & \leq e^{\sum_{k\geq 1}a_k}I_1 + e^{\sum_{k\geq 1}a_k}\sum_{l\geq 1}b_l \leq (2K^{2n}+K^n 4^nn+n\sum_{l\geq 1}l^{-2})e^{K^n4^nn}.
W ( p_{1}, p_{2}, . . . p_{n}; q_{1}, q_{2}, . . . q_{n}) = V ( p_{1}, p_{2}, . . . , p_{n}) \exp \biggl [ - i \biggl ( \frac{1}{2}\sum_{i , j}I_{i j}q_{i}\times q_{j}\biggr ) \biggr ] ,
\begin{aligned}\mathbf{S}_\mathtt{s}(I):=\bigg\{\tilde{\mathtt{s}}: \tilde{\mathtt{s}}~ ~I, & \tilde{\mathtt{s}}(e)=0,~\tilde{\mathtt{s}}\ll \mathtt{s},~\frac{d\tilde{\mathtt{s}}}{d\mathtt{s}}=0~~1,~d\mathtt{s}\bigg\},\end{aligned}
\begin{aligned}\int_{D_{Rz/H}\times\{z\}}\Big|\frac{\partial W}{\partial z}\Big|^2 =\int_{D_{Rz/H}}\big|\nabla w(Hx/z)\cdot(Hx/z^2)\big|^2 &\leq\frac{1}{H^2}\int_{D_R}|\nabla w(y)|^2|y|^2\,dy\leq\Big(\frac{R}{H}\Big)^2 \int_{D_R}|\nabla w|^2\,.\end{aligned}
\sum_{m=2}^{\infty}\ell\left( m,0,\chi\right) \frac{\left( -s\right)^{m}}{m}=\log\Gamma^{\ast}\left( s,\chi\right) -\log\Gamma^{\ast}\left(0,\chi\right) -s\psi^{\ast}\left( 0,\chi\right) & =\log\Gamma^{\ast}\left( s,\chi\right) +s\ell\left( 1,\chi\right) .
X_{\rm Anderson}= \left( \begin{array}{rrrrr}-0.0590 0.1700 0.0030 0.6650 0.6552 0 -0.1730 0.0300 0 0 0 1.1800 -1.3160 0.0080 0 0 0.8010 0.4950 -1.1780 1.3570 0 1.0400 2.7560 0 -0.1830 \end{array}\right)
2(p_n|q_n)_o = d_{\Omega}(o,p_n)+d_{\Omega}(o, q_n) - d_{\Omega}(p_n,q_n) & \geq 2T - 2R-2\log(2) & +d_{\Omega}(\sigma_{\xi_n^+}(T), p_n)+ d_{\Omega}(\sigma_{\xi_n^-}(T),q_n)- d_{\Omega}(p_n,q_n) & \geq 2T-2R-2\log(2) - d_{\Omega}(\sigma_{\xi_n^-}(T),\sigma_{\xi_n^+}(T)).
g''(m) = q [(n+h)^2 f''(mn + mh) - n^2 f(mn)] = \frac{q}{m^2}[t_1^2 f''(t_1) - t^2 f''(t)] &= \frac{q}{m^2}\cdot mh \cdot \frac{d}{dt}(t^2 f''(t)) \big|_{t= t_0}= \frac{qh}{m}(2 t_0 f''(t_0) + t_0^2 f'''(t_0)).
= - \left[ (5\,h_{0}+ 1)\, h_{0}\,\kappa\left(\frac{2}{3}h + \frac{2}{15}- \frac{22 + 5c}{240}\right)+ 18\, q\, q_{0}\,(h_{0}+ 1) \right]\; \zeta^{-1}\;\left\langle \phi',\left( \phi \otimes \phi_{0}\right) \right\rangle,
\begin{cases}b_2 = a_1^2 + 4 a_2 b_4 = a_1 a_3 + 2 a_4 b_6 = a_3^2 + 4 a_6 b_8 = a_1^2 a_6 - a_1 a_3 a_4 + 4 a_2 a_6 + a_2 a_3^2 - a_4^2 c_4 = b_2^2 - 24 b_4 c_6 = -b_2^3 + 36 b_2 b_4 - 216 b_6 \Delta = -b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6 j ={c_4^3}/{\Delta}\end{cases}
\left.{z_1 dz_2- z_2 dz_1\over z_0^*}\right|_{S}=\left.{z_0 dz_2- z_2 dz_0\over z_1^*}\right|_{S}=\left.{z_1 dz_0- z_0 dz_1\over z_2^*}\right|_{S}
\frac{d\pi_\varkappa^\theta}{d\pi_\varkappa}(\gamma)=\exp\left[\left\langle\log\left(\frac{d\varkappa^\theta}{d\varkappa}\right),\gamma\right\rangle+\int_{\hat X}\left(1-\frac{d\varkappa^\theta}{d\varkappa}\right)d\varkappa\right],
e^{-i \zeta\cdot x}L_{\sigma,\mu}(e^{i \zeta\cdot x}r) = -\Delta_\zeta r - \nabla_\zeta(\nabla\log\mu\cdot r) - \nabla\log\sigma\times \nabla_\zeta\times r - i\omega (\sigma\mu - \sigma_0 \mu_0) r.
\begin{array}{l}F=u_0+v=\alpha +\bar \beta , \quad quad \leftrightarrow \quad v=W=\alpha +\bar \beta -u_0, \bar F=u+v_0=\bar \alpha +\beta , \quad \quad \leftrightarrow \quad u=\bar W=\bar \alpha +\beta -v_0.\end{array}
(Q_{2p})_{ii}= \begin{cases}1, \lceil \alpha\beta^{-p}N^{\frac{1}{2}}\rceil \leq i\leq N, 0, , \end{cases}\, (Q_{2p+1})_{ii}= \begin{cases}1, \lceil\alpha\beta^{p}N^{\frac{1}{2}}\rceil \leq i\leq \lfloor \alpha\beta^{p+1}N^{\frac{1}{2}}\!\rfloor , 0, . \end{cases}
{\cal{L}}_{\psi}= \frac{1}{2}\overline{{\Psi}}^{a}\left[ \sqrt{- \left( D^{2}\right)^{a c}+ \left( F_{\rho \sigma}^{b}S^{\rho \sigma}\right) \left( t^{b}\right)^{a c}}\right] \Psi^{c}
L_{D}^{\ast}=\int_{\cal M}d^{3}x\varepsilon^{ijk}\bigg \{-{\frac{3}{4a}}(a\partial_{i}B_{jAB}+bG_{ijAB})[{\bf R}^{-1}]_{kn}^{ABCD}\varepsilon^{lmn}(a\partial_{l}B_{mCD}+bG_{lmCD})+c\alpha_{i}^{AB}G_{jkAB}\bigg\},
|(\zeta-\zeta_0)^2+\tau^2| \geqslant \big|(u-u_0)^2-\big(a(u)-a(u_0)\big)^2 +\tau^2\big| = (u-u_0)^2-\big(a(u)-a(u_0)\big)^2 +\tau^2 > \frac{3\tau^2}{4}.
\delta U_{n,\lambda}(T):= \|t^{\gamma_1}\delta u_{n,\lambda}\|_{L^{2r}(0,T;L^{p_3}_x)}+\|t^{\gamma_2}\delta u_{n,\lambda}\|_{L^{\infty}(0,T;L^{p_3}_x)}+\|t^{\beta}\nabla \delta u_{n,\lambda}\|_{L^{2r}(0,T;L^{p_2}_x)},
\begin{array}{rlll}\Delta \mathcal{D}_0 = -\lambda\,\alpha_3\,\Bigl(\frac{1}{(1+\vert z\vert^2)^{1/2}}-\frac{1}{\vert z\vert}\Bigr) \mathbb{R}^3, \mathcal{D}_0 \rightarrow 0 \vert z\vert\rightarrow \infty.\end{array}
(C_1) = \left(\int_{C_1}\omega\right) = \left(\lim_{\varepsilon \rightarrow 0}\int_{-\pi}^{\pi}\int_\varepsilon^{1}\frac{1}{h}dh\wedge d\theta\right) = (\lim_{\varepsilon \rightarrow 0}-2\pi \log|\varepsilon|)
|F^{1,2}_2| \le \frac{\epsilon}4\int (\partial_x^2 w)^2\partial_x\phi_N\,dx+\frac{4}{\epsilon}\int u^2_1 w^2 \partial_x\phi_N\,dx &\le \frac{\epsilon}4\int (\partial_x^2 w)^2\partial_x\phi_N\,dx+\frac{c}{\epsilon}\|u_1\langle x\rangle^{1/8}\|_{\infty}^2\int w^2\phi_N\,dx
\iota(\mathbf{h}) \odot \iota(\mathbf{h}') = \left(x,\omega,\tau+\tfrac{1}{2}x \cdot \omega\right) \odot \left(x',\omega',\tau'+\tfrac{1}{2}x' \cdot \omega'\right) & = \left(x+x',\omega+\omega',\tau+\tau'+\tfrac{1}{2}(x \cdot \omega + x' \cdot \omega') + x' \cdot \omega \right) & = \iota(\mathbf{h}\bullet \mathbf{h}') .
\forall\, g \in H, \: \forall \, \psi \in \mathcal{O}(H), \:\:\:\: \widetilde{g}\widetilde{h}= \widetilde{gh}, \:\:\: g\widetilde{h}= \widetilde{h}g, \:\:\: \widetilde{h}\psi = \psi\!\left(S^{-1}(h'')?\right)h'.
d\left[ K^{\left( 2q+2\right)}\left( S\right) -C^{\left(2q-1\right)}K^{\left( 3\right)}\left( \Gamma_{11}\right) \right]=dK^{\left( 2q+2\right)}\left( S\right) +K^{\left( 2q\right)}\left(S\Gamma_{11}\right) K^{\left( 3\right)}\left( \Gamma_{11}\right) \quad ,
u(x)=\inf \left\{\left.I[\gamma]: =\int_0^\infty e^{-s}\left( \frac{\left( \dot{\gamma}(s) \right)^2}{2}+ f(\gamma (s)) \right)ds \right| \gamma \in \mathrm{AC}([0, \infty); [-1,1]), \gamma(0)=x \right\}.
\begin{array}{r c l}{{\Phi}}&{{=}}&{{\phi + \phi_{0},}}\\{{\phi}}&{{=}}&{{\left( \pi_{1}, \dots , \pi_{N - 1}, ( \sigma + i \pi_{0}) / \sqrt{2}\right)^{T},}}\\{{\phi_{0}}}&{{=}}&{{( 0 , \dots , 0 , v / \sqrt{2})^{T},}}\\ \end{array}
\theta_s\theta_{t_i}\cong\begin{cases}\theta_{s_2}, i=1; \theta_{s_{i+1}}\oplus \theta_{s_{i-1}}, i>1;\end{cases}\theta_t\theta_{s_i}\cong\begin{cases}\theta_{t_2}, i=1; \theta_{t_{i+1}}\oplus \theta_{t_{i-1}}, i>1;\end{cases}
(\blacksquare)\geq\sqrt{\cos^{2}\varphi\,\left(1+\frac{\left\Vert v_{2}\right\Vert^{2}}{\left\Vert v_{1}\right\Vert^{2}}\right)-\frac{\left\Vert v_{2}\right\Vert^{2}}{\left\Vert v_{1}\right\Vert^{2}}}\overset{(\ref{eq:useful_identity})}{=}\sqrt{\frac{\cos^{2}\varphi-\cos^{2}\angle\left(v,V^{\bot}\right)}{1-\cos^{2}\angle\left(v,V^{\bot}\right)}}>0\mbox{.}
C_1 := \{(x,y,z)^\top \in \mathbb{R}^3 \colon 0 \leq x, 0\leq y, 0\leq z\leq x+y\}, C_2 := \{(x,y,z)^\top \in \mathbb{R}^3 \colon 0 \leq x,0\geq y, y\leq z\leq x\}, C_3 := \{(0,0,z)^\top \in \mathbb{R}^3 \colon 0\leq z\}
\tilde{f}_{i_1}^{a_1}\cdots \tilde{f}_{i_l}^{a_l}\left( b_\lambda \otimes b \right) = \tilde{f}_{i_1}^{c_1}\cdots \tilde{f}_{i_l}^{c_l}(b_\lambda) \otimes \tilde{f}_{i_1}^{d_1}\cdots \tilde{f}_{i_l}^{d_l}(b),
{\cal A}_{f i}= \int d{\bf x \,}\left[ - \frac e{w_{{\bf p}}}{\bf A}({\bf x}) \,{\bf \cdot \, p}+ V ({\bf x}) \right] \, \mathrm{e}^{- \, i \, ({\bf p}^{\prime}-{\bf p}) \,{\bf \cdot \, x}}= - \frac e{w_{{\bf p}}}\,{\bf \tilde{A}( q}) \,{\bf \cdot \, p}+ \tilde{V}({\bf q}) \; \; \;
\lim_{n\to \infty}\Bigg( \frac{1}{n}\sum_{i=1}^{n}\sum_{p = 1}^{\lfloor (n-i)/2 \rfloor}\binom{n}{i}\binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)}\Bigg) \le 1,
f_{0}( x , p ) ={\frac{1}{2 \pi}}\int \! \! d y ~{\frac{e^{- i y p}}{2 \cosh ( x / \hbar - y / 2 ) \cosh ( x / \hbar + y / 2 )}}={\frac{1}{\pi}}\! \int_{0}^{\infty}\! \! \! \! d y ~{\frac{\cos ( y p )}{\cosh ( 2 x / \hbar ) + \cosh ( y )}}
\begin{aligned}(\lambda-2\eta)\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}|D^2u|^2\phi^2dxdt \leq \frac{C}{\eta}\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}|Du|^2|D\phi|^2dxdt +C\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p+s-\gamma}{2}}|\phi_t|\phi dxdt,\end{aligned}
k_1\pi^{(0)}_{0,k_1+k_2}+\frac{\mu_1}{\mu_2}\sum_{h=1}^{k_1-1}(k_1-h)\pi^{(0)}_{1,h}= \frac{1}{k_2}\left[k_1\left(1-\frac{\lambda_1}{\mu_1}\right)-k_2\frac{\lambda_1}{\mu_2}\right].
{\cal M}( a , b ) = \int_{{\mathrm i}\, \eta}^{\infty +{\mathrm i}\, \eta}{\mathrm d}t \, \exp ( - t ) \, \left( \frac{1}{1 -{\mathrm i}\, a \, t}+ \frac{1}{1 +{\mathrm i}\, a \, t}+ \frac{1}{1 - b \, t}+ \frac{1}{1 + b \, t}\right) ,
{\partial_r}g(\Delta u(t, p_0),e^2) =\{\partial_r (-\frac{1}{s^2}u_\theta+\frac cs \partial_r u_\theta +\partial_r^2 u_\theta+\frac{c}{s^2}\partial_\theta u_r-\frac 1s \partial_r\partial_\theta u_r)\}|_{(t,p_0)}=\{-\frac 2{s^2}\partial_r u_\theta+\frac cs \partial^2_r u_\theta +\partial_r^3 u_\theta-\frac 1s \partial^2_r\partial_\theta u_r\}|_{(t,p_0)}.
g(x) = \begin{cases}a_1 x\in E_1=\bigcup_{k=1}^{M_1}Q(x_k,\tau_1), a_2 x\in E_2=\bigcup_{k=M_1+1}^{M_2}Q(x_k,\tau_2), \vdots a_m x\in E_m=\bigcup_{k=M_{m-1}+1}^{M_m}Q(x_k,\tau_m), 0 . \end{cases}
\gamma_{G_1}(t) = \gamma_{G_2}(t), \_0\gamma_{G_1}(t) ={_0\gamma_{G_2}(t)}, \^0\gamma_{G_1}(t)={^0\gamma_{G_2}(t)}\ \textrm{and}\^0_0\gamma_{G_1}(t) ={^0_0\gamma_{G_2}(t)}.
\Gamma(s)G(z,s;A,B) =\left( -2\pi i/k\right)^{s}k\left( A\left(z,s;A,\widehat{B}_{-1}\right) +e(s/2)A\left( z,s;A_{-1},\widehat{B}\right)\right) & \quad+\Gamma(s)f(0)\left( L(s;B)+e(s/2)L(s;B_{-1})\right)
d s^{2}= \left[ \left( 1 - \frac{q^{2}}{r_{1}^{2}}\right) \left( 1 - \frac{p^{2}}{r_{2}^{2}}\right) \right]^{- \frac{1}{2}}d t^{2}- \left[ \left( 1 - \frac{q^{2}}{r_{1}^{2}}\right) \left( 1 - \frac{p^{2}}{r_{2}^{2}}\right) \right]^{\frac{1}{2}}d \vec{x}^{2}\, .
\sin(\zeta(s)) = 2P(s)Q(s) \cos\theta_{\hat x}+ (2P^2(s)-1) \sin\theta_{\hat x}, &\cos(\zeta(s)+\theta_y) ={(2P^2(s)-1)\cos(\theta_{\hat x}+ \theta_{\hat y})- 2P(s)Q(s) \sin(\theta_{\hat x}+ \theta_{\hat y})}.
x=(\begin{matrix}0&1&0\\ 0&0&0\\ 0&0&0\end{matrix}),y=(\begin{matrix}0&0&0\\ 0&0&1\\ 0&0&0\end{matrix}),z=(\begin{matrix}0&0&1\\ 0&0&0\\ 0&0&0\end{matrix}).
1= K_1e_\lambda(I-R(\lambda,A)Le_\lambda)^{-1}e^{-\int_0^\cdot(\lambda+\mu(s))ds}= K_1e_\lambda f = \int_0^{+\infty}\int_{-r}^0\beta_1(\sigma,a)e^{\lambda \sigma}e^{-\int_0^a(\lambda+\mu(s)+e^{-\lambda r}\alpha(s))ds}d\sigma da,
\ \lim_{k\to \omega}\| \tilde{\alpha}_n (x_k) -\theta_t(x_k)\|_{\varphi \circ \theta_s}^\sharp =\mathrm{weak}\lim_{k \to \omega}\frac{1}{2}( |\tilde{\alpha}_n (x_k) -\theta_t(x_k )|^2 +|(\tilde{\alpha}_n (x_k)-\theta_t(x_k))^*|^2 ) & =2\delta >0
S_{s o u r c e}^{( 3 )}= \int d^{4}x \left( \sqrt{\det ( G_{E}+ e^{- \phi / 2}{\cal F})}+{\frac{i}{4}}\chi{\cal F}*{\cal F}+{\frac{i}{2}}B^{R}*{\cal F}+{\frac{i}{2 4}}\epsilon^{\alpha \beta \gamma \delta}A_{\alpha \beta \gamma \delta}^{( 4 )}\right) ,
H_{\bar{l}}(\bar{x}^{B(l)}) = H(x) - H_l(x^{B(l)}| \bar{x}^{B(l)}) & = \sum_{i \notin B(l)}\psi (x_i) + \frac{1}{2}\sum_{i, j \notin B(l)}M_{ij}x_i x_j.
\big[V^A\big]_g:=\begin{pmatrix}v^+ begin{align*}1mm]v_a\, begin{align*}2mm]v^-\end{pmatrix}\mapsto\begin{pmatrix}v^+ begin{align*}1mm]v_a-\hat n_aH v^+ begin{align*}2mm]v^-+\frac12 H^2 v^+\end{pmatrix}=\big[U^A{}_B\big]_{\bar g}^g\, \big[ V^B\big]_g=:\big[\bar V^A\big]_{\bar g}\, ,
\mathcal Q_k(t,s,\xi) = \mathrm I + \sum_{j=1}^\infty \i^j \int_s^t \mathcal R_k(t_1,s,\xi)\int_{s}^{t_1}\cdots \int_s^{t_{j-1}}\mathcal R_k(t_j,s,\xi)\, \d t_j \cdots \d t_2 \d t_1.
E_{\pm N}= \Big \lbrace \mu^{2}+ M_{N}^{2}\pm \Big \lbrack 4 \mu^{2}M_{N}^{2}+ | \bar{\Phi}|^{4}\Big ( V_{t r e e}^{\prime \prime}\Big )^{2}\Big \rbrack^{1 / 2}\Big \rbrace^{1 / 2},
\langle x(t, 0, u), p_{k}\rangle_2^2 \leq \lambda_{k}\left[\int_0^t x^T(s)(A^T P^{-1}+P^{-1}A+ \sum_{i=1}^m N_i^T P^{-1}N_i+k^2 P^{-1}) x(s)\right. ds \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.+2\int_0^t x^T(s) P^{-1}B u(s) ds\right].
\frac{n}{2}\left(1 - \frac{a_{n/2+p}}{\binom{n}{n/2}g(1)}\right) = \frac{g_0}{g(1)}p^2 + \frac{g_1}{g(1)}(p-1)^2 + \ldots + \frac{g_p}{g(1)}0^2 + \frac{g_{p+1}}{g(1)}1^2 + \ldots + \frac{g_\ell}{g(1)}(\ell-p)^2 +\Theta(n^{-1}).
H_j(T)= \|u_j\|_{C_t^0\dot{H}_x^\gamma(S_T)}+ \sup_{q_0^\ast \le \tau \le \frac{\mu_\ast\left(\kappa-1\right)}2}\bigl\| |D_x|^{\frac{(m+2)n+2}{\tau(m+2)}-\frac 4{(m+2)(\kappa-1)}}u_j\bigr\|_{L^\tau(S_T)}
\tilde{\mathbf{E}}^{+}_{h}\ = \prod_{(\beta,s)\in\Delta^{+}\times\mathbb{Z}}\limits^{\rightarrow}\tilde{\mathbf{E}}_{\beta,s}^{+,(h(\beta,s))}, \tilde{\mathbf{E}}^{-}_{h}\ = \prod_{(\beta,s)\in\Delta^{+}\times\mathbb{Z}}\limits^{\rightarrow}\tilde{\mathbf{E}}_{\beta,s}^{-,(h(\beta,s))}.
\left\langle \rho, u\alpha_{1}, v\alpha_{3}\right\rangle \cong M_{1}\A=\begin{pmatrix}u_{2}v_{2}u_{3}v_{3}\end{pmatrix}\in \mathrm{GL}_{2}(\mathbb{F}_{p}) \ \v_{2}-u_{3}-\mathrm{det}(A)\not\equiv 0 \ \mathrm{mod}\p.
\begin{cases}D_{A_1}\psi_1 =0 F_{A_1}^{+}= \frac{r}{2}(q(\psi_1,\psi_1)- i\Omega_X) + i\mu_1 *d*b_1 -2^{-\frac{1}{2}}r^{\frac{1}{2}}(\eta_1^* \psi_I-\psi_I^* \eta_1)=0, \end{cases}
\langle f,g\rangle :=\langle -\ln r\sum\limits_{j=-\infty}^\infty\sum\limits_{Q\in Q^j}w(Q)\psi_Q(\cdot,x_{Q})q_{Q}h(x_{Q}), g\rangle =-\ln r\sum\limits_{j=-\infty}^\infty\sum\limits_{Q\in Q^j}w(Q)\psi_Qg(x_{Q})q_{Q}h(x_{Q}),
V[\rho]=\sum\limits_{n=-\infty}^{+\infty}\int d^3x\left\{\rho\left[\ln\left[\frac{\rho}{2\zeta}+\sqrt{1+\left(\frac{\rho}{2\zeta}\right)^2}\right]+2\pi in\right]- 2\zeta\sqrt{1+\left(\frac{\rho}{2\zeta}\right)^2}\right\},
d_{1,\mathrm{rel}}^G=\sum_{n>0}x_{i,-n}\frac{\partial}{\partial \varphi_{i,-n}}-\frac{1}{2}\sum_{p,q>0}(-1)^{\bar{x}_i\bar{x}_k+\bar{x}_i+\bar{x}_j}c_{i,j}^k \varphi_{k,-p-q}\frac{\partial}{\partial \varphi_{i,-p}}\frac{\partial}{\partial \varphi_{j,-q}}.
Z = \int{\cal D}B e x p \bigg ( -{\frac{4 \pi}{g}}\varepsilon^{l m n}( B_{l}^{A B}\partial_{m}B_{n A B}-{\frac{4 \pi}{g}}f_{A B C D E F}B_{l}^{A B}B_{m}^{C D}B_{n}^{E F}) \bigg ) .
W_\phi(w_1, w_2) = \frac{2^{w_2-3}}{\pi^{\frac{1+w_1+w_2}{2}}}\Gamma\left(1-w_2\right)\cos\left(\frac{\pi}{2}(1-w_2)\right) &\times \Gamma\left(\frac{-1+w_1+3w_2 +2it_\phi}{4}\right)\Gamma\left(\frac{-1+w_1+3w_2 -2it_\phi}{4}\right)
\Psi(\Phi(\{u,v\}_1)) = \Psi ([u^r,v]_1 [u^{\frac{b}{h}},v]_2) & = \{u^{\frac{ra}{h}},v\}_1 \{u^{-\frac{br}{h}},v\}_2 \{u^{\frac{bs}{h}},v\}_1 \{u^{\frac{br}{h}},v\}_2 & = \{u^{\frac{ra+bs}{h}},v\}\:\:\mbox{(By (A1) and (iii))}& = \{u,v\}_1
\begin{aligned}\sum\limits_{i=1}^{n}a^{ii}\left(-\frac{1}{2}\int_{\partial B_{\varepsilon}(0)}|\nabla u^{k}_{i}|^2(x\cdot\mathbf{n})\mathrm{d}S+\int_{\partial B_{\varepsilon}(0)}\big(x\cdot\nabla u^k_i\big)\big(\mathbf{n}\cdot\nabla u^k_i\big)\mathrm{d}S\right)=-4\pi\sum\limits_{i=1}^{n}a^{ii}\alpha_{i}^{2}+o(1).\end{aligned}
\begin{aligned}E_{\nu^n_\rho}[ f ( \nabla_{j -1 , j}\overrightarrow{W}^\ell_j ) W_{j - 1}] = \ell^{-1}E_{\nu^n_\rho}[W_j^2 (\nabla_{j,j-1}f)] + \ell^{-1}E_{\nu^n_\rho}[W_j (W_j -W_{j-1})f] . \end{aligned}
(u-v\mp a) x_i^{\pm}(u)x_j^{\pm}(v)=(u-v\pm a) x_j^{\pm}(v) x_i^{\pm}(u)+ \hbar\Big( [x_{i,0}^{\pm}, x_j^{\pm}(v)]-[x_i^{\pm}(u), x_{j, 0}^{\pm}]\Big)
||\sqrt{\omega}\lambda(x)\left([\xi \partial_{\xi},K](y(x,\frac{\cdot}{\lambda(x)^{2}}))\right)(\omega \lambda(x)^{2})||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}&\leq C\left(||y(x)||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}+||\sqrt{\omega}\lambda(x) y(x) ||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}\right)
{\frac{\delta{\cal E}( h )}{h^{2}}}= \kappa_{0}\ln{\frac{h}{m}}+ \kappa_{1}\ln \ln{\frac{h}{m}}+ \kappa_{2}+{\cal O}{\mathrm{\Large (}}\ln \ln{\frac{h}{m}}{\mathrm{\Large /}}\ln{\frac{h}{m}}{\mathrm{\Large )}},
\left\lbrace\begin{tabular}{l l l l}$u(0,t) = 0$, $u(L,t) = 0$, $u_{x}(L,t) = h_2(t)$ in $(0,T)$, $v(0,t) =g_0(t)$, $v(L,t) = g_1(t)$, $v_{x}(L,t) = g_2(t)$ in $(0,T)$,\end{tabular}\right.
A_k=\frac{2^{2s}}{b_k}\quad\mbox{with}b_k=\begin{cases}l \mbox{if}A_{0}A_{1}\cdots A_{k-1}\in\left[\frac{1}{2},1\right), l+1 \mbox{if}A_{0}A_{1}\cdots A_{k-1}\in[1,2]. \end{cases}
C_{\Omega}=2^{10/3}3^{-1}\pi^{13/6}\kappa^{-5/3}N^{1/2}m^{1/2}r_+ \frac{15r_+^4 +3l^2r_+^2 -2l^4}{3r_+^4 +4l^2r_+^2 +l^4}\frac{3r_+^2 +l^2}{3r_+^2-l^2}.
\|f\ast g\|_{\tau(n)}=\|f\ast g\|_{1}+\sum_{k=1}^{n}\|(f\ast g)\tau^{k}\|_{1}&\leq \|f\|_{1}\|g\|_{1}+\sum_{k=1}^{n}\|f\tau^{k}\|_{1}\|g\tau^{k}\|_{1}&\leq \|f\|_{\tau(n)}\|g\|_{\tau(n)}.
\frac{\partial U^{\epsilon}}{\partial t}(t,x) = \frac{\partial^2 U^{\epsilon}}{\partial x^2}(t,x) +\sqrt{\epsilon}\sigma(t, x, U^\epsilon(t,x))\frac{\partial^2W}{\partial t \partial x}(t,x) & +\frac{\partial}{\partial x}g(t,x,U^{\epsilon}(t,x))+ f(t,x,U^\epsilon(t,x)),
\langle\prod_{i=1}^{4}O_{i}(x_{i})\rangle=\frac{(\frac{|x_{24}|}{|x_{14}|})^{\Delta_{1}-\Delta_{2}}(\frac{|x_{14}|}{|x_{13}|})^{\Delta_{3}-\Delta_{4}}}{|x_{12}|^{\Delta_{1}+\Delta_{2}}|x_{34}|^{\Delta_{3}+\Delta_{4}}}g(u,v).
F_{m}( 0 ) [ l ] = \frac{1}{k}\sum_{n = 0}^{k - 1}A_{m}( n ) [ l ] = \frac{l^{2 \Delta_{a}}}{( 2 \pi )^{2 \Delta_{a}}}g ( a ) N ( l ) , \quad a = \frac{m \beta}{k},
\tilde\psi\left(\begin{pmatrix}x_0 y_0\end{pmatrix},\ldots,\begin{pmatrix}x_{q-1}y_{q-1}\end{pmatrix}\right)=\sum_{t=0}^{\max(y_j)-1}(h_t^*\psi_1)\left(\begin{pmatrix}x_0 y_0\end{pmatrix},\ldots,\begin{pmatrix}x_{q-1}y_{q-1}\end{pmatrix}\right).
\sum_{i}B_{ij}^{(u)}w_{i}^{(u)}\geq\sum_{i\in I_{>}}B_{ij}^{(u)}w_{i}^{(u)}=\sum_{i\in I_{>}}B_{ij}^{(u)}\rho_{u}\geq B_{i^{*}j}^{(u)}\rho_{u}\geq\sigma\rho_{u}& \geq a_{j}^{(u)}+\sum_{u=v+1}^{N}(t_{v}-t_{v-1})K_{ij}^{(v,u)}w_{i}^{(v)}\mbox{(by (\ref{copteq248}) and $\rho_{v}\geq w_{i}^{(v)}\geq 0$)}.
N(A,B) =\sum_{x \in \mathbb{F}_r}\Big( \dfrac{1}{p}\sum_{y\in \mathbb{F}_p}\zeta_p^{y(\mathrm{Tr}(x )-B)}\Big) \Big( \dfrac{1}{p}\sum_{z\in \mathbb{F}_p}\zeta_p^{z(\mathrm{Tr}(x^2)-A)}\Big) &= p^{m-2}+p^{-2}(S_1+S_2+S_3 ),
\frac{e^{2}}{( 2 \pi )^{2}}\int \langle F_{1 2}( x ) F_{1 2}( 0 ) \rangle e^{- i p x}d x = P^{0}( 0 ) - \int_{0}^{\infty}\frac{d \rho ( \mu^{2})}{p^{2}+ \mu^{2}}.
\begin{cases}-\Delta f_\ell(x)=\lambda_\ell f_\ell(x) \mathfrak{a}/N \le |x| \le \ell f_\ell(x)=0 |x|\le \mathfrak{a}/N \partial_{|x|}f_\ell(x)=0 |x|=\ell f_\ell(x)=1 |x|\ge \ell,\end{cases}
S_{1^j}(z)=\sum_{{\scriptstyle k_1+2k_2+ \cdots +mk_m=j}\atop{\scriptstyle k_1 \geq 0,\cdots ,k_m \geq 0}}(-1)^{k_2+k_4+ \cdots +k_{2 \left[\frac m 2\right]}}\frac1{k_1! \cdots k_m!}z_1^{k_1}\cdots z_m^{k_m}
d(\xi,y) \leq \tau(d(\xi,x) + d(x,y)) &<\tfrac{1}{1+\gamma}d(x,z)+\tau d(x,y) &\leq \tfrac{\tau}{1+\gamma}d(x,y)+\tfrac{\tau}{1+\gamma}d(y,z)+\tau d(x,y) &< \tau\bigl(\tfrac{1}{1+\gamma}+\tfrac{\gamma}{1+\gamma}+1\bigr)d(x,y)= 2\tau d(x,y).
A_{I_1,J_1}= \{0\}^{I_1}\times \{1\}^{J_1}\times \{0,1\}^{T\backslash(I_1\cup J_1)}\times \{0,1\}^{T^c}A_{I_2,J_2}= \{0\}^{I_2}\times \{1\}^{J_2}\times \{0,1\}^{T\backslash(I_2\cup J_2)}\times \{0,1\}^{T^c}
P_{{\underline X}_1}({\underline x}_1) = \begin{cases}r_0 \{\underline x_1}= \underline{0}r_{c}\alpha_{c, \ell}{{\underline x}_1}={\underline c}_{\ell}, \forall \ell \in \left[{M \choose c}\right], c \in[ B_],\end{cases}
\int_{\mathbb{T}_\alpha}K_2^{k}(x,y)dx =\int_{\mathbb{T}_\alpha}\frac{R_+^2(\alpha k-1,y,\lambda(x,y,E),E)r(\lambda(x,y,E))}{\sum_{i=0}^{\alpha-1}f_iu_+^2(\alpha k+i,y,\lambda(x,y,E),E)}dx.
a_{jk}^* (z) = \sum_{\ell=j}^d \frac{p_{j{\ell}}^* (z)}{(z^{-1}- 1)^{\ell-j}}\left( (z^{-1}+ 1 ) \widetilde b_{{\ell}k}^* (z) - \sum_{r=0}^{k-1}w_{k,r+1}\frac{\widetilde b_{{\ell}r}^* (z)}{z^{-1}- 1}\right), j,k=0,\dots,d.
\mbox{Res}\Big(\frac{J_{R}^{(1)}(s)}{s},s = 0\Big) = \frac{1}{2}\begin{pmatrix}-(d_{0}+d_{1}d_{0}^{2}) -id_{0}^{2}-i \big( \alpha^{2}+ d_{0}^{2}d_{1}^{2}+2d_{0}d_{1}+\frac{3}{4}\big) d_{0}+ d_{1}d_{0}^{2}\end{pmatrix}.
\nu(s-m-1,n-m-1) =\nu(s-1,n-1)-\nu(s-m-1,n+2r_n-m-1), \nu(s+m-2s_0,n) =\nu(s+2m-2s_0,n+m)-\nu(s+m-2s_0,n+2r_n), \nu(s-2s_0+j-1,n-m-1) =\nu(s+m-2s_0+j-1,n-1)-\nu(s-2s_0+j-1,n+2r_n-m-1).
\frac{1}{2}\frac{d}{dt}\int_{\Omega}(\mu|\nabla u|^2+(\mu+\lambda)|u|^2-(P+\frac{1}{2}|H|^2) u H\cdot \nabla u\cdot H\Big)x+C|\nabla^2 u|^2_2 \leq C(|\nabla u|^2_2+|\nabla H|^2_2+1)(|\nabla u|^2_2+|D(u)|_\infty+1).
|I_{t_1}(y_1)-I_{t_2}(y_2)| =\Big|\int_{t_2}^{t_1}v^n_{s-}(x)p^n_{t_1-s}(x,y_1)ds + \int_0^{t_2}v^n_{s-}(x)[p^n_{t_1-s}(x,y_1)-p^n_{t_2-s}(x,y_2)]ds\Big| &\leq \int_{t_2}^{t_1}p^n_{t_1-s}(x,y_1)\,ds + \int_0^{t_2}\big|p^n_{t_1-s}(x,y_1)-p^n_{t_2-s}(x,y_2)\big|\,ds.
\begin{aligned}|N_{G}(v)\cap (V(T')\backslash \{u_{1},\cdots,u_{s}\})| =|N_{G}(v)\cap (X\backslash \{u_{1},\cdots,u_{s}\})| &\leq |X|-\lfloor\frac{s}{2}\rfloor &\leq t-\lfloor\frac{s}{2}\rfloor\end{aligned}
\dot \omega(t)=g\big(g^{-1}(\tilde a)-\int_t^{+\infty}F(z_-+y_-)(\tau)d\tau\big)-g\big(g^{-1}(\tilde a)-\int_t^{+\infty}F^l(z_+(\tilde a,x_-,.))(\tau)d\tau\big),
{C}_{\alpha}{\mathcal F}_{V_B,\epsilon}^{0}\cap{\rm Im}[C_{\alpha}{\mathcal F}_{V_B,\epsilon'}^{1}\to C_{\alpha}{\mathcal F}_{V_B}^{0}]={\rm Im}[{C}_{\alpha}{\mathcal F}_{V_B,\epsilon}^{1}\to C_{\alpha}{\mathcal F}_{V_B}^{0}]
\begin{array}{l c r}{\eta_{j}^{+}= u_{\alpha}^{+}\eta_{j}^{\alpha}; \qquad \overline{{\eta}}_{j}^{+}= u_{\alpha}^{+}\overline{{\eta}}_{j}^{\alpha}}\\{\eta_{j}^{\alpha}\overline{{\eta}}_{\alpha j}= 1 ; \qquad \eta_{j}^{\alpha}{\eta}_{\alpha j}= \overline{{\eta}}_{j}^{\alpha}\overline{{\eta}}_{\alpha j}= 0}\\ \end{array}
S_{-1,\sigma}(t,\pi) \geq -\frac{\log C(t,\pi)}{\pi}\left(\frac{e^{-2 \pi \beta \Delta}}{1 + e^{-2\pi \beta \Delta}}\right) - \dfrac{2\,d\,\beta\,e^{(1-2\beta)\pi\Delta}}{\pi(\frac{1}{4}-\beta^2)\big(1+ e^{-2\pi\beta\Delta}\big)^2}+ O_c\left(\dfrac{d\,\beta\,e^{(1-2\beta)\pi\Delta}}{(\frac{1}{2}-\beta)^2\Delta}\right)+O(d).
\displaystyle{\left( \delta^{I J}- \frac{( x_{a}^{I}- x_{b}^{I}) ( x_{a}^{J}- x_{b}^{J})}{( x_{a}- x_{b})^{2}}\right)}( \textstyle{\frac{d \,}{d t}x_{a}^{J}- \frac{d \,}{d t}x_{b}^{J}}) = 0 \, ,
\begin{aligned}\mathcal{L}_{\infty}(\omega):=\Phi_{\infty}^{-1}(\omega) \circ \mathcal{L}_5 \circ \Phi_{\infty}(\omega)=\omega\cdot \partial_{\varphi}+\mathcal{D}_{\infty}(\omega), & \mathcal{D}_{\infty}(\omega):=\mbox{diag}_{j\in S^c}\{\mu_j^{\infty}(\omega) \}.\end{aligned}
h_{\frac{l}{k}, \frac{p}{q}}(t, x) = \sum_{r=0}^{\infty}\sum_{n=0}^{m_{1}-1}\frac{(-t)^{n+r}}{n!\, r!}\, \frac{\Gamma(1 + Mn + mr)}{x^{1+ M n + mr}}\sin(\pi M n + \pi mr) & \times{_{1+lp}F_{m_{1}}}\left({1, \Delta(lp, 1 + M n + mr) \atop \Delta(m_{1}, 1+n)}; (-1)^{m_{1}- lp}\kappa\right),
k_1 = \left( \begin{array}{c}p p \end{array}\right) \;\;\;k_2 = \left( \begin{array}{c}e -p \end{array}\right) \;\;\;k_3 = \left( \begin{array}{c}-p -p \end{array}\right) \;\;\;k_4 = \left( \begin{array}{c}-e p \end{array}\right).
\sum_{k = 0}^{\frac{r + s}{2}- \frac{\mid r - s \mid}{2}}\frac{( C_{k + \mid \frac{r - s}{2}\mid}^{\frac{r + s}{2}}) ( C_{k}^{\frac{r + s}{2}}) \mid 2 k + \mid \frac{r - s}{2}\mid - \frac{r + s}{2}\mid}{C_{\frac{r + s}{2}}^{r + s}}.
\tilde{p}(t,z)=\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(s,z)\pi(s,z)-\frac{a_0(s,z)}{b_0(s,z)})^{2}ds),