harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[f(s|D|),x] - \frac{s}{2}\{f'(s|D|),\delta(x)\}= -s^2\int_{-\infty}^\infty \int_0^1 \widehat{h''}(u)(1-v)e^{iu(1-v)s|D|}h(s|D|)\delta^2(x)e^{iuvs|D|}\,dvdu -s^2\int_{-\infty}^\infty \int_0^1 \widehat{h''}(u)(1-v)e^{iuvs|D|}\delta^2(x)h(s|D|)e^{iu(1-v)s|D|}\,dvdu.
||v_{n+2\,m}^1||^2= -\frac{c_{n+2\,m}}{\overline{b_{n+1\,m+3}}}||v_{n+1\,m+3}^1||^2= \frac{c_{n+2\,m}}{\overline{b_{n+1\,m+3}}}\cdot \frac{\overline{d_{n+1\,m+3}}}{a_{nm}}||v_{nm}^1||^2.
\Big\|(a_{i,j})_{0\leq i\leq N,j\in I_n}\Big\|=\max_{j\in I_n}\bigg(|a_{0,j}|+t_j\sum_{1\leq i_1<i_2<\ldots<i_j\leq N}\sum_{s=1}^j|a_{i_s,j}|\bigg).
F_{a}\circ F_{b}=\begin{cases}(u_1\cdot(w_1+m),\,\ldots,\,u_r\cdot(w_r+m),\, w_{r+1}+m,\,\ldots,\,w_{s}+m), \textrm{if}r\leq s; begin{align*}5pt](u_1\cdot (w_1+m),\,\ldots,\,u_s\cdot (w_s+m),\,u_{s+1},\,\ldots,\,u_{r}), \textrm{if}r>s.\end{cases}
{\delta_{1}^{( 0 )}L_{1}^{( 3 )}= - i{\det e}\left( \bar{\epsilon}^{*}\gamma^{\mu}\psi_{\mu}^{*}\right) \lambda^{1 6}\left( 8 f^{( 1 2 , - 1 2 )}+ 6 \cdot 1 4 4 D_{1 1}f^{( 1 1 , - 1 1 )}\right) ,}
e x p \bigg ( + i \tilde{L}_{D}\bigg ) = \int{\cal D}G{\cal D}B \e x p \bigg ( + i \int_{X}d^{3}x \varepsilon^{i j k}( a B_{i}^{A B}H_{j k A B}+ b A_{i}^{A B}G_{j k A B}+ c B_{i}^{A B}G_{j k A B}) \bigg ) .
\begin{aligned}\frac{d}{dt}Q(M_t) = ~|M_t|\int_{M_t}H^{1-\alpha}|\mathring{A}|^2d\mu_t-\int_{M_t}H^{1-\alpha}\int_{M_t}|\mathring{A}|^2d\mu_t +\alpha|M_t|\int_{M_t}H^{-1-\alpha}|\nabla H|^2d\mu_t.\end{aligned}
\cos(\frac{k_1^{\prime}\wedge k_2^{\prime}}{2s^2})\cos(\frac{k_3^{\prime}\wedge k_4^{\prime}}{2s^2})=\cos(\frac{k_1^{\prime}\wedge k_2^{\prime}}{2}-tk_1^{\prime}\wedge k_2^{\prime})\cos(\frac{k_3^{\prime}\wedge k_4^{\prime}}{2}-tk_3^{\prime}\wedge k_4^{\prime})
LV( x ,1 ) ={\rm sign}x \cdot( -x^3 -x ) + ( \frac{1}{3}+\frac{1}{4}\cos x )( 2- 1 )|x| & \leq -|x| + \frac{7}{12}|x| = -\frac{5}{12}|x|= -\frac{5}{12}V( x ,1 ), LV(x,2 ) = 2{\rm sign}x \times (-\frac{1}{2}x) + ( \frac{7}{3}+\frac{1}{2}\sin x )(1- 2 )|x| & \leq - |x|-\frac{11}{6}|x|=-\frac{17}{12}V(x,2 ).
\mathbb{E}(|C|) \leq (2\epsilon+o(1)) g (d-1)^{(3 + \log_{p}(4+o(1)) - 2+ o(1)) \frac{g}{4}}= (2\epsilon+o(1)) g (d-1)^{(1+o(1))\frac{g}{4}}= o(g (d-1)^{(1+o(1))\frac{g}{4}})
\begin{array}{lll}V =\frac{1}{2}\int_{M}d^{2}x \Omega \left[ \frac{1}{2}F_{ij}F^{ij}+D_{i}\phi\overline{D^{i}\phi}\right. \left. + \frac{1}{4}(|\phi|^{2}-1)^{2}\right]\end{array}
\begin{array}{l}3(3,2)_{\frac{1}{6}}+3(\overline{3},1)_{\frac{1}{3}}+3(\overline{3},1)_{-\frac{2}{3}}+12(1,2)_{-\frac{1}{2}}+9(1,2)_{\frac{1}{2}}\ \ \ +12(1,1)_1+9(1,1)_0+9(1,1)_{-1}.\end{array}
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_{IV}^-}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,s,k}T R^d \theta^{d-1}\end{aligned}
\mathbb{P}[^{+}_{\omega \ge \omega_1}X^{h}(\omega) \in d\omega_2, \max\limits_{\omega \ge \omega_1}X^{h}(\omega) \in dx^{*}]= \int_{-\infty}^{x^{*}}\mathbb{P}[X^{h}(\omega^{*}) \in dy,^{+}_{\omega \ge \omega_1}X^{h}(\omega) \in d\omega_2, \max\limits_{\omega \ge \omega_1}X^{h}(\omega) \in dx^{*}]
\varPsi_2(u):=\sqrt{u^2 (4-u) ( 16-u)}\det\begin{pmatrix}D^0\mu^1_{2,2}(u) D^0\mu^1_{2,3}(u) D^1\mu^1_{2,2}(u) D^1\mu^1_{2,3}(u) \end{pmatrix}
K(\beta) = (g(x_0)- \underline{g}) \!\left[ \min \! \left( c_1 \tau_1 \min(\underline{\alpha_1}, \underline{t_1}) \varepsilon_1^2, c_2 \tau_2^2\min \! \left( \min \! \left(\alpha_{02}, \dfrac{3|2c_2-1| \varepsilon_2}{M_g}\right)\!, \underline{t_2}\right)^2 \! \varepsilon_2 \right) \!\right]^{-1}\!.
\begin{aligned}\mathcal{R}f: =(\mathcal{R}_{1}f,\mathcal{R}_{2}f,\ldots ,\mathcal{R}_{k}f)^{T}, \mathcal{Q}f: =(\mathcal{Q}_{1}f,\mathcal{Q}_{2}f,\ldots,\mathcal{Q}_{k}f)^{T}, \end{aligned}
\mathcal{T}=\frac{1}{2}\Gamma^{\mu\nu}p_{\mu}p_{\nu}, \Gamma^{\mu\alpha}\Gamma_{\alpha\nu}=\delta^{\mu}{}_{\nu}, p_{\mu}=\frac{\partial T}{\partial \dot{q}^{\mu}}=\Gamma_{\mu\nu}\frac{dq^{\nu}}{dt}.
\rho|v_t|^2+\nu\frac{d}{dt}\|\nabla\times v\|^2 =(-\Delta F,v_t)-(\rho u\cdot\nabla v,v_t) &\leq C(\|F\|_2^2+\|\Psi\|^2)+\frac{\underline{\rho}}{4}\|\nabla\times v_t\|^2,
\Delta = ( \partial_{\bar{z}}A_{0}^{\bar{z}}) ( \partial_{z}A_{0}^{z}) - ( \partial_{z}A_{0}^{\bar{z}}) ( \partial_{\bar{z}}A_{0}^{z}) ={\cal L}|_{\varphi = \varphi_{0}}={\cal L}_{0}= \partial_{z}\varphi_{0}\partial_{\bar{z}}\varphi_{0}- V ( \varphi_{0})
\sum_{n=1}^{\infty}|<f_{n},x>| =\sum_{n=1}^{\infty}|\sum_{k=p_{n}}^{q_{n}}x(k)\alpha_{n}(k)| &\leq \sum_{n=1}^{\infty}\sum_{k=p_{n}}^{q_{n}}|x(k)||\alpha_{n}(k)| &\leq \sum_{n=1}^{\infty}\sum_{k=p_{n}}^{q_{n}}|x(k)|\leq \|x\|.
\| v_2 \| = \sqrt{\sum_{\stackrel{k :}{y_k > x_1+ \delta/2}}| \langle \delta_{x_1}, \rho(s_1,t_k) \delta_{y_k}\rangle |^2}\leq \sum_{\stackrel{k :}{y_k > x_1 + \delta/2}}\hat \rho(x_1,y_k) \, .
|\chi_S ( \mathbf{d}(c) - \tilde{B}_{\mathbf{d}- \mathbf{e}_a - \mathbf{e}_b, \mathcal{A}}(ab) ) | = |\chi_S ( \tilde{B}_{\mathbf{d}- \mathbf{e}_a - \mathbf{e}_b, \mathcal{A}}(ab) ) | \leq C_5 \frac{d^2}{n}(24 d)^{-|S|}(4 |S|)! .
\delta\left(\sum_{i=1}^r b^*_{i,1}\otimes b^*_{i,2}\otimes a_{i,3}\right)=\sum_{i=1}^r j^*(b^*_{i,1})\otimes j^*(b^*_{i,2})\otimes a_{i,3}\in \mathcal{A}^*\otimes \mathcal{A}^*\otimes \mathcal{A},
(S^U) R (\underline{x}) ({}^tS^W) = \sum_{\ell =1}^p x_{\ell}\left [ (S^U) R^\ell ({}^tS^W) \right ]=\sum_{\ell =1}^p x_\ell \left( \sum_{i=1}^p s_{\ell i}R^i \right) =\sum_{i=1}^p \left(\sum_{\ell =1}^p s_{\ell i}x_\ell \right) R^i = \sum_{i=1}^p \widehat{x}_i R^i = R(\underline{\widehat{x}}),
K(u;x,y,p,q)F(u) &=\frac{xy}{1-yu}+\frac{x}{1-u}\left(\frac{2yu}{1-yu}+p+q-\frac{1}{1-u}\right)F(1)+\frac{x}{1-u}\frac{\partial}{\partial u}F(u)\mid_{u=1},
\prod_{\substack{m,n \geq 1 m \leq n ; \, \gcd(m,n)=1}}\left( \frac{1}{1- y^m z^n}\right)^{\frac{1}{m^a n^b}}= \exp\left\{\sum_{n=1}^{\infty}\left( \sum_{m=1}^{n}\frac{y^m}{m^a}\right) \frac{z^n}{n^b}\right\}.
\lim_{\beta\rightarrow\infty}U[v_{j}\left(t,\beta\right) ]=0,1\leq j\leq N_{R}; \lim_{\beta\rightarrow\infty}U[v_{j}\left(t,\beta\right) ]=\frac{1}{2},N_{R}+1\leq j\leq2N.
\mathbb{E}(\widetilde{N}_0 /n) - \frac{1-\varepsilon}{2}\{P_0(A)+P_1(A)\}- \frac{\varepsilon}{2}= P(A) - \frac{1-\varepsilon}{2}\{P_0(A)+P_1(A)\}- \frac{\varepsilon}{2}\geq (1-\varepsilon)P_0(A) - \frac{1-\varepsilon}{2}\{P_0(A)+P_1(A)\}- \frac{\varepsilon}{2}= \frac{1-\varepsilon}{2}\mathrm{TV}(P_0,P_1) - \frac{\varepsilon}{2}.
\lambda_{2}( a ) = \lambda_{2}^{( 0 )}( a ) + k^{- 2}\lambda_{2}^{( 2 )}( a ) g_{a}^{\mu \nu}\partial_{\mu}\partial_{\nu}+ k^{- 4}\lambda_{2}^{( 4 )}( a ) g_{a}^{\mu \nu}g_{a}^{\rho \sigma}\partial_{\mu}\partial_{\nu}\partial_{\rho}\partial_{\sigma}+ \ldots ,
\Phi(r) = \Psi\left( \varphi \left( \frac{a + b\omega_K}c\right)\right)=\Psi\left(1 - \frac ac, \frac ac - \frac bc, \frac bc, 0\right) &=\frac{((c-a)c, (a-b)c, bc, a^2 - ab + b^2 - ac)}{a^2 - ab + b^2 + c^2 - ac}&=\frac{(|\beta|^2 - a, a-b, b, |\alpha|^2 - a)}{|\alpha|^2 + |\beta|^2 - a}.
P_\gamma^\pm P_\delta^\pm=\frac{1}{2}P_\gamma^\pm+\frac{1}{2}P_\delta^\pm+\frac{\cos[\pi(\gamma-\delta)]}{4}R_\gamma R_\delta,P_\gamma^- P_\delta^+ =-\frac{e^{i\pi(\delta-\gamma)}}{4}R_\gamma R_\delta.
m ={\textstyle \frac{1}{2}}\left( \begin{array}{c c}{{m^{1}}}&{{m^{2}+ m^{3}}}\\{{}}&{{}}\\{{m^{2}- m^{3}}}&{{- m_{1}}}\\ \end{array}\right) ={\cal Q}= \left( \begin{array}{c c}{{\frac{\delta}{2}r}}&{{\beta q}}\\{{\gamma p}}&{{- \frac{\delta}{2}r}}\\ \end{array}\right) \, .
e^{2t}| T_u v(t) - u(t)| = \left| \int_t^\infty e^{2t}\Phi(t, s) f(v(s)) ds \right| & \le C_1K_1C_0^2 e^{-2t}+ C_1 K_2 C_0^3 e^{-4t}& \le C_1 K_1 C_0^2 + C_1 K_2 C_0^3 & = 4C_1 K_1 r^2 + 8C_1 K_2 r^3
\mathbb{E}^{SC}_{\sigma_{0},\tilde{\sigma}_{0}}\big[\,\Vert S_{t+1}-\tilde{S}_{t+1}\Vert_{1}-\Vert S_{t}-\tilde{S}_{t}\Vert_{1}\,\vert\,\mathcal{F}_{t}\,\big]\leq-\frac{1}{n}\,\big(\tilde{S}_{t}^{2}-S_{t}^{2}\big)-\frac{1}{n}\,\big(\tilde{S}_{t}^{3}-S_{t}^{3}\big)=-\frac{1}{2n}\,\Vert S_{t}-\tilde{S}_{t}\Vert_{1}.
f\in C^0([-1,1])\cap C^1(-1,1), &\lim\limits_{s\rightarrow -1}f^\prime(s) = -\infty,\;\;\;\;\lim\limits_{s\rightarrow 1}f^\prime(s)= +\infty,\;\;\;\;f^{\prime\prime}(s) \geq \alpha > 0.
-\tilde{A}_{1}\hat{Q}^{-1}\hat{A}_{12}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}P_{N(\check{A}_{p})\cap S}\frac{\partial z_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}=0.
Q_{j}(\rho)=\big\{\omega\in \Omega: \exists \psi\in\mathbb{C}^{[-L,L]}, (\psi)\subset \widetilde{I}_{j},\|\psi\|=1,\big\|(H_{\omega}\upharpoonright \widetilde{I}_{j}-E)\psi\upharpoonright{\widetilde{I}_{j}}\big\|,\big|\psi\upharpoonright{\partial \widetilde{I}_{j}}\big|<\rho\big\}.
\mathrm{Sub}(A)=\sigma_A^{(1)}+\frac{\mathrm{i}}{2}\cdot\frac{\partial^2\sigma_A^{(0)}}{\partial x^k\partial\xi_k}+\frac{\mathrm{i}}{2}\cdot \frac{\partial\sigma_A^{(0)}}{\partial{\xi_k}}\cdot\frac{\partial (\log\sqrt{G})}{\partial x^k},
\alpha(X) = \P(O_S = X){\widehat \P}(X \rightarrow a), \beta(X) = \P(O_S = X){\widehat \P}(X \rightarrow b), \gamma(X) = \P(O_S = X){\widehat \P}(X \rightarrow a \cap X \rightarrow b), \delta(X) = \P(O_S = X).
A_{t,11}=-W_t^TY_t-X_t^T[A_{13}Y_t-A_{11}Z_t]+W_t^T[A_{11}^TY_t+A_{21}Z_t], &A_{t,13}=X_t^TW_t+X_t^T[A_{13}X_t-A_{11}W_t]-W_t^T[A_{11}^TX_t+A_{21}W_t], &A_{t,21}=-Z_t^TY_t-Y_t^T[A_{13}Y_t-A_{11}Z_t]+Z_t^T[A_{11}^TY_t+A_{21}Z_t].
A_{x_{||}}^{11}=\frac{1}{\sqrt{\rho}}\int_{\tilde T^2}d^2z\,K_-(\omega,z;x)e^{\phi(z)}\left(D^\dagger_x(\hat A)\partial_{x_{||}}D_x(\hat A)\right)^{11}\frac{e^{\phi(z)}}{\sqrt{\rho}}K_-(z,\omega;x).
\sum_{k=0}^{m}[6k+1]\frac{(aq;q^2)_k (q/a;q^2)_k (q;q^2)_k (b;q^4)_k q^{k^2+2k}}{(aq^4;q^4)_k(q^4/a;q^4)_k(q^4;q^4)_k (q^3/b;q^2)_k b^k}\equiv \frac{(q^rb;q^4)_{(n-r)/4}}{(q^{4+r}/b;q^4)_{(n-r)/4}}\, b^{-(n-r)/4}(-q)^{(1-r)/2}[n].
\begin{array}{rl}\displaystyle{\frac{\partial p_i(t)}{\partial t}= \sum_q\sum_j (\pi_{qij}\,a_{qj}\,p_j(t)-\pi_{qji}\,a_{qi}\,p_i(t))= \sum_q\sum_j (\pi_{qij}\,a_{qj}\,p_j(t))-a_{qi}\,p_i(t)}.\end{array}
u(t) \leq \frac{u(x_n + b_n) - u(x_n - a_n)}{b_n + a_n}(t- (x_n + b_n)) + u(x_n + b_n) & = \frac{3 \theta_n g (b_n) - 3 \theta_n g (a_n)}{b_n + a_n}(t- (x_n + b_n)) + w(x_n) + 3\theta_n g (b_n) & \leq w(x_n) + 3\theta_n g (b_n).
\left( \Omega^{g_1 g_2, g_3}_{\alpha \beta}\right) \,\left( g_3^* \Omega^{g_1, g_2}_{\alpha \beta}\right) \: = \:\left( \Omega^{g_1, g_2 g_3}_{\alpha \beta}\right) \,\left( \Omega^{g_2, g_3}_{\alpha \beta}\right) \,\left( \omega^{g_1, g_2, g_3}_{\alpha}\right) \,\left( \omega^{g_1, g_2, g_3}_{\beta}\right)^{-1}
{\mathrm R e}\partial_{p^{2}}\Gamma_{+ -}( p^{2}) \big |_{p^{2}= \mu_{W}^{2}}= 1 \quad{\mathrm R e}\partial_{p^{2}}\Gamma_{H H}( p^{2}) \big |_{p^{2}= \mu_{H}^{2}}= 1 \quad{\mathrm R e}\partial_{p^{2}}\Gamma_{\chi \chi}( p^{2}) \big |_{p^{2}= \mu_{\chi}^{2}}= 1
\int_{t_2}^1y(t_1)(t_2-t_1)dt_1 =t_2\int_{t_2}^1y(t_1)dt_1-\int_{t_2}^1t_1y(t_1)dt_1 =t_2\int_{t_2}^1y(t_1)dt_1-\left[s\int_{s}^1y(t_1)dt_1\right]_{s=t_2}+\int_{t_2}^1\int_{s}^1y(t_1)dt_1ds =\int_{t_2}^1\int_{s}^1y(t_1)dt_1ds.
\nu_{-}= \left( \begin{array}{c c}{{I_{n}}}&{{0}}\\{{\nu_{- 2 1}}}&{{I_{k}}}\\ \end{array}\right) , \qquad \nu_{+}= \left( \begin{array}{c c}{{I_{n}}}&{{\nu_{+ 1 2}}}\\{{0}}&{{I_{k}}}\\ \end{array}\right) , \qquad \eta = \left( \begin{array}{c c}{{\eta_{1 1}}}&{{0}}\\{{0}}&{{\eta_{2 2}}}\\ \end{array}\right) ,
\sum_{1\leq j,k\leq n}H^{(m,n)}_{\texttt{b};j,k}x_j x_k = \sum_{1\leq j\leq n}\Bigl( 2(m+1)+ u_{q_0}(\xi_j) + u_{q_1}(\xi_j)\Bigr) x_j^2 &+ \sum_{1\leq j<k\leq n}\Bigl( u_q(\xi_j +\xi_k)(x_j+x_k)^2+ u_q(\xi_j -\xi_k)(x_j-x_k)^2\Bigr) \ge 2(m+1) \sum_{1\leq j\leq n}x_j^2.
\widehat{\mathcal E}_\mu(u) :=\frac{1}{2}\left\|u\right\|^2 -\frac{\mu}{2^*}\int_\Omega (u^+)^{2^*}dx, \widetilde{\mathcal E}_{\lambda,\mu}(u) :=\frac{\mu}{2^*}\int_\Omega (u^+)^{2^*}dx +\frac{\lambda}{r}\int_\Omega (u^+)^r dx +\lambda\int_\Omega G(u^+) dx
C_{2}=\frac{1}{2}\tilde{A}^{\nu}_{\mu}\tilde{A}^{\mu}_{\nu}=\frac{P(N+P)(N-1)}{2N},\; \tilde{A}^{\nu}_{\mu}-\frac{\delta_{\mu}^{\nu}}{N}\sum_{\lambda}A^{\lambda}_{\lambda}\;.
E(\phi^{\epsilon},u^{\epsilon},\eta^{\epsilon},\tau^{\epsilon})=\|\phi^{\epsilon}\|_{H^3_{P'(\rho^{\epsilon})}}^2+\|u^{\epsilon}\|_{H^3_{\rho^{\epsilon}}}^2+[\beta(L-1)+2\bar{\mathfrak{z}}]\|(\eta^{\epsilon}-1)\|_{H^3}^2+\frac{\beta}{2k^2}\|\tau^{\epsilon}\|_{H^3}^2,
T(a(1+q), b(1+q)) = \frac{\Delta}{2}\Bigl(\frac1{\Delta}-(\alpha-\beta)-(\alpha-\beta)q - \sqrt{1 - 2q(\alpha+\beta)+q^2(\alpha-\beta)^2}\,\Bigr) =\frac{\Delta(\frac1\Delta -1-(\alpha-\beta))}2 + \Delta T(\alpha q, \beta q).
\left| \Omega \right\rangle \equiv \left| \overline{{\Psi}}\left({\bf y}\right) \Psi \left({\bf 0}\right) \right\rangle = \overline{{\psi}}\left({\bf y}\right) \exp \left({i g \int_{\bf 0}^{\bf y}{d z^{i}A_{i}\left( z \right)}}\right) \psi \left({\bf 0}\right) \left| 0 \right\rangle ,
\|x_m^A-x_m^B\| = t_m \|v_m^A-v_m^B\| \le \frac{\|x_m^A-x_0\|}{1 - \|v_m^A-v^A\|}\left( \|v_m^A-v^A\|+ \|v^A-v^B\| + \|v^B-v_m^B\|\right) \, .
\exp\left(\sum_{n=1}^N\Big(\int_0^x\varphi_1r_n\,dt\Big)\frac{1}{(2i\lambda)^n}\right)=1- \frac{1}{2i\lambda}\int_0^x \varphi_1\varphi_2\,dt+ O\Big(\frac{1}{\lambda^2}\Big)
\langle P_h(\cdot,s),\mu_j \rangle = \frac{\overline{\rho_j(h)}\sqrt{\pi}}{(4\pi h)^{s-\frac{1}{2}}}\frac{\Gamma(s-\frac{1}{2}+it_j)\Gamma(s-\frac{1}{2}-it_j)}{\Gamma(s)}.
\int_\Omega|\nabla\psi|^2\,dx+\int_\Omega V\,\psi^2\,dx \le\int_\Omega|V-V_0|\,|v_0|\,|\psi|\,dx &\le\left(\int_\Omega|V-V_0|^p\,dx\right)^{1/p}\left(\int_\Omega \big(|v_0|\,|\psi|\big)^{p/(p-1)}\,dx\right)^{(p-1)/p}&\le\|V-V_0\|_{L^p(\Omega)}\,\|v_0\|_{L^{2p/(p-1)}(\Omega)}\,\|\psi\|_{L^{2p/(p-1)}(\Omega)}.
\overline{P}_{m,m'}(N,Q) \le \overline{\prod_{n=1}^N Z_{m,m'}(n)}& \stackrel{(1)}{=}\prod_{n=1}^N \overline{Z}_{m,m'}(n) \stackrel{(2)}{=}\left[\,\overline{Z}_{m,m'}(1)\right]^N
\sum_{k=2}^r\frac{\tilde\lambda_k}{\tilde\lambda_1-\tilde\lambda_k}+\frac{\tilde\lambda_1}{\tilde\lambda_1-\tilde\lambda_2}\leq 1+2\sum_{k=1}^r\frac{\lambda_k}{\tilde\lambda_1-\lambda_k}, \frac{\tilde\lambda_1}{\tilde\lambda_1-\tilde\lambda_2}\leq 1+\frac{\lambda_1}{\tilde\lambda_1-\lambda_1}.
\mathcal{H}_K^{\infty}(M) = \{\varphi \in \mathcal{H}^{\infty}(M)| \varphi = \varphi \circ \sigma \sigma \in K\}; \mathcal{H}_K^{k,\alpha}(M) = \{\varphi \in \mathcal{H}^{k, \alpha}(M)| \varphi = \varphi \circ \sigma \sigma \in K\};
\theta_{3}(q) = \sum_{n= - \infty}^{\infty}q^{n^{2}}, \theta_{4}(q) = \sum_{n = - \infty}^{\infty}(-1)^{n}q^{n^{2}}, \theta_{2}(q) = \sum_{n=-\infty}^{\infty}q^{(n + 1/2)^{2}}.
B_{\delta,m}= F(z+\pi i \theta_1,0) - \int_{-1/2}^{1/2}F(z+\pi i \theta_1 + 2 \pi i u) + O(1/m_0) + O(m_0^2/m) &= F(z+\pi i \theta_1,0) - \int_{-1/2}^{1/2}F(z+\pi i \theta_1 + 2 \pi i u) \mathrm{d}u + O(m^{-1/3}).
h=\left(\begin{array}{cc}1 0 X 1 \end{array}\right) \left(\begin{array}{cc}e^{-\frac{1}{2}\tilde{\Phi}}0 0 e^{\frac{1}{2}\tilde{\Phi}}\end{array}\right) \left(\begin{array}{cc}1 Y 0 1 \end{array}\right).
\Psi_1(v_1\otimes \cdots \otimes v_8)(X,Y) = D(v_2,v_5,v_7)D(v_4,v_6,v_8)(v_1\times v_3)\times (X\times Y), \Psi_2(v_1\otimes \cdots \otimes v_8)(X,Y) = D(v_2,v_5,v_7)D(v_4,v_6,v_8) (D(v_1,v_3,X)Y+D(v_1,v_3,Y)X).
\int_{\Omega}\varphi^{\prime}\overline{w}\; \d x + \int_{\Omega}\nabla \varphi \cdot \overline{\nabla w}\; \d x = - \int_{\Omega}u \cdot \nabla \varphi \overline{w}\; \d x + 2 \int_{\Omega}\lvert \nabla d \rvert^2 \varphi \overline{w}\; \d x
\bar{R}_{~ \lambda \mu \nu}^{\sigma}( \bar{\Gamma}) = \bar{R}_{~ b \mu \nu}^{a}( \bar{\Omega}) e_{a}^{~ \sigma}e_{~ \lambda}^{b}= ( \partial_{\mu}\bar{\Omega}_{~ b \nu}^{a}- \partial_{\nu}\bar{\Omega}_{~ b \mu}^{a}+ \bar{\Omega}_{~ \alpha \mu}^{a}\bar{\Omega}_{~ b \nu}^{\alpha}- \bar{\Omega}_{~ \alpha \nu}^{a}\bar{\Omega}_{~ b \mu}^{\alpha}) e_{a}^{~ \sigma}e_{~ \lambda}^{b}
\begin{cases}\dot{\theta}_j=j\,\,\dfrac{\partial}{\partial I_j}\mathcal{H}_{\leq 5}(\theta, I, 0), j\in S^+, begin{align*}3mm]\dot{I}_j=-\dfrac{\partial}{\partial \theta_j}\mathcal{H}_{\leq 5}(\theta, I, 0), j\in S^+.\end{cases}
\big(x-q^{\lambda+1}\beta^{-1}\big)h(x/q) + \big(q^{2\lambda-\mu}\alpha\beta^{-1}x-q^{2\lambda-\mu+1}\beta^{-1}\big)h(qx) &{}-\big\{\big(q^{\lambda-\mu}\alpha\beta^{-1}+q^{\lambda}\big)x-q^{\lambda+1}\beta^{-1}- q^{2\lambda-\mu+1}\beta^{-1}\big\}h(x)=0.
A=(\begin{matrix}0&0&0&0&0&0&0\\ 0&0&0&1&0&1&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1\\ 0&0&0&0&1&0&0\\ 0&0&1&1&0&0&0\\ 0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0\\ \end{matrix})
D_{\mu \nu}= \left[ \begin{matrix}{{p^{2}- p_{0}^{2}+ \frac{1}{\alpha \beta^{2}}}}&{{- p_{0}p_{1}}}&{{- p_{0}p_{2}}}&{{- p_{0}p_{3}}}\\{{- p_{0}p_{1}}}&{{p^{2}- p_{1}^{2}}}&{{- p_{1}p_{2}}}&{{- p_{1}p_{3}}}\\{{- p_{0}p_{2}}}&{{- p_{1}p_{2}}}&{{p^{2}- p_{2}^{2}}}&{{- p_{2}p_{3}}}\\{{- p_{0}p_{3}}}&{{- p_{1}p_{3}}}&{{- p_{2}p_{3}}}&{{p^{2}- p_{3}^{2}}}\\ \end{matrix}\right] \beta^{2}.
R_{12}(\lambda /\mu )\,\mathcal{U}_{1}(\lambda )\,R_{12}(\lambda \mu /q)\,\mathcal{U}_{2}(\mu )=\mathcal{U}_{2}(\mu )\,R_{12}(\lambda \mu /q)\,\mathcal{U}_{1}(\lambda )\,R_{12}(\lambda /\mu ).
\mathsf G_v=\left(\prod_{e:\,v=t(e)}\mathsf T_e \right)\left(\prod_{e:\,v=s(e)}\mathsf T_e^3 \right)\prod_{f:\, v=b(f)}\left(\mathsf T_f^2\frac{1-\mathsf U_f^2}{2}+\frac{1+\mathsf U_f^2}{2}\right).
Z_{1 \mathrm{i n s t}}= \frac{( - )^{N}}{\sqrt{2 \pi N \frac{1 6 \pi^{4}m^{4}}{A^{2}}\sqrt{\frac{A}{\pi^{2}m^{2}}- 1}}}( \mathrm{e}^{i \frac{3}{4}\pi - \frac{2 \pi^{2}m^{2}N}{A}\gamma ( \frac{A}{\pi^{2}m^{2}})}+ \mathrm{e}^{- i \frac{3}{4}\pi + \frac{2 \pi^{2}m^{2}N}{A}\gamma ( \frac{A}{\pi^{2}m^{2}})}) .
\widetilde{\Delta^\Phi_K}(t)= (8+4(\xi^3+\xi^4))(t^6+1)-(81+48(\xi^3+\xi^4)-16(\xi^2+\xi^5))(t^5+t) &+ (287+189(\xi^3+\xi^4)-45(\xi^2+\xi^5)+27(\xi+\xi^6))(t^4+t^2) &- (300+160(\xi^3+\xi^4)-188(\xi^2+\xi^5)-75(\xi+\xi^6))t^3.
d(z)\!\!=\!\! \left (\!\! q^{2j-1}x_{+}z\! -\! 2x_0skq^{j}\phantom{\frac{q^j}{s}}\!\!\!\!\!\!-\! x_{-}k^2 z^{-1}\! \right )\!\! \left ( -q^{2j}y_{+}z\! +\! \frac{2y_0}{tk}q^{j}\!\! +qy_{-}k^{-2}z^{-1}\! \right ),
a_n\big((m_1^{(n)}, \ldots, m_k^{(n)}) - (m_1^*, \ldots, m_k^*)\big) = a_n\big(\varphi_k^{[a,b]}(p_1^{(n)}, \ldots, p_k^{(n)}) - \varphi_k^{[a,b]}(\vec{y}^*)\big)
P(\eta^N\in O) \geq\exp\left(-\frac{\gamma_N^2}{N}(J_{ini}(W^m_0)+\mathcal{J}_1(W^m, F^m, G^m, H^m)+4\epsilon)\right) &\times P(D_{5,\epsilon}\cap D_{4,\epsilon}\cap\{\eta^N\in O\}) &=\exp\left(-\frac{\gamma_N^2}{N}(J_{ini}(W^m_0)+J_{dyn}(W^m)+4\epsilon)\right)(1+o(1)).
\bar{g}_1(x) = \frac{(1-q)\beta_1}{\alpha_1}x^{\lambda}\frac{( \beta_1 x, q, q \beta_1 / \beta_2 ; q)_{\infty}}{\big( q^{-\lambda}x \beta_1, \beta_1 / \alpha_1,q \beta_1 / \alpha_2 ; q\big)_{\infty}}\,{}_3\phi_2 \biggl(\!\! \begin{array}{c}q^{-\lambda}\beta_1 x, \beta_1/\alpha_1, q\beta_1/\alpha_2 \beta_1 x, q\beta_1/\beta_2 \end{array}\! ;q,q \biggr)
\stackrel{(-1,0)}{\Phi}_{\alpha_{L-2}}^{*(1)}Z_{\;\;\alpha_{L-1}}^{\alpha_{L-2}}=\delta_1\left( \stackrel{(-2,0)}{\eta}_{\alpha_{L-1}}^{*(11)}-\stackrel{(-2,0)}{\gamma}_{\alpha_L}A_{\alpha_{L-1}}^{\;\;\alpha_L}\right) .
W({x}^{i_{\nu}}; \tilde \varepsilon) \le W({x}^{\nu}; \tilde \varepsilon) -\omega\sum_{t=\nu}^{i_{\nu}-1}\gamma^{t}\|d({x}^{t})\|^{2}\le W({x}^{\nu}; \tilde \varepsilon) - \omega\frac{\delta^{2}}{16}\sum_{t=\nu}^{i_{\nu}-1}\gamma^{t}.
\big((\tau_\theta)^{(p)}\times(\tau_\theta)^{(q)}\big)_g(x_1H,h_1,x_2H,h_2)= \qquad \big(g^px_1H,\theta(g^p,x_1H)h_1,g^qx_2H, \theta(g^q,x_2H)h_2\big)
\bigl\| \sum_i \varrho(a_i)Vx_i \bigr\|^2 \ =\ \sum_{i,j}\bigl< V^* \varrho(a_j^*a_i) Vx_i, x_j\bigr> &\ =\ \sum_{i,j}\bigl< \varphi(a_j^* a_i)x_i, x_j \bigr> &\ =\ \bigl\| \sum_i \varrho'(a_i) V'x_i \bigr\|^2.
Z ( \beta ; R ( S_{\nabla}^{{\bf C}}) , \langle \cdot , \cdot \rangle_{H}, \langle \cdot , \cdot \rangle ) = \pi^{\zeta}e^{- \frac{i \pi}{2}( ( \frac{2 \theta}{\pi}\mp 1 ) \zeta \pm \eta )}| \beta |^{- \zeta}\tau ( M , \chi , \langle \cdot , \cdot \rangle_{H})^{- 1}
\mathcal{J}(w)(t) =\Phi_{M_n,n}(\widetilde u_T )(x,t)-\sum_{p=0}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}F_p(w)(s)ds \Bigg]\phi_p(x) &+\sum_{p=M_n+1}^{\infty}\Bigg[ \int_0^t e^{(s-t)p^{2\beta}}F_p(w)(s)ds \Bigg]\phi_p(x),
\left[ \hat{x}^{\mu}, \hat{\pi}_{\nu}\right]_{-}= i \left\{x^{\mu}, \pi_{\nu}\right\}_{D ( \Phi_{3}^{( 1 )})}= i \delta_{\nu}^{\mu}\; , \; \; \left[ \hat{\psi}^{\mu}, \hat{\psi}^{\nu}\right]_{+}= i \left\{\psi^{\mu}, \psi^{\nu}\right\}_{D ( \Phi_{3}^{( 1 )})}= - \frac{1}{2}g^{\mu \nu}\; .
(\widetilde{\Delta}_{2}) =p^{-4}\times(P_{1}+P_{2}+P_{3})=\frac{2}{p^{4}}\times\bigg\{\frac{1}{n(n-1)}+\frac{1}{m(m-1)}+\frac{2}{nm}\bigg\}(Q_{2}-2Q_{1}) \le\frac{6}{p^{4}}\times\left\{\frac{1}{n(n-1)}+\frac{1}{m(m-1)}+\frac{2}{nm}\right\}Q_{2},
H^{\bullet}(G, b, \mu )[\rho]=\sum\limits_{(M_S, \mu_S) \in \mathcal{I}^{G, \mu}_{M_S, b_S}}(\mathrm{Ind}^G_{P_S}H^{\bullet}(M_S, b_S, \mu_S)[\rho])\otimes [1][|\cdot|^{\langle \rho_G, \mu_S \rangle - \langle \rho_G, \mu \rangle}],
Re \frac{d}{dt}\int i \partial_{\alpha}D^k u \overline{D^k u}= Re~ i\int \partial_{\alpha}\partial_t D^k u \overline{D^k u}+\partial_{\alpha}D^k u \overline{\partial_t D^k u}= 2Re\int i \partial_{\alpha}D^k u\overline{\partial_t D^k u}.
I_{a+}^{\alpha ;\psi}f\left( x\right) :=\frac{1}{\Gamma \left( \alpha\right)}\int_{a}^{x}\psi^{\prime}\left( t\right) \left( \psi \left(x\right) -\psi \left( t\right) \right)^{\alpha -1}f\left( t\right) dt
\|\phi_j f\|_b^2 =\|a_1\widetilde{f}\phi_j +\phi_j p\|^2_b & \le 2(\|a_1\widetilde{f}\phi_j\|^2_b +\|\phi_j p\|^2_b ) &=2(\|\widetilde{f}\phi_j\|^2_{H^2}+\|\phi_j p\|^2_b ).
\frac{1+\sum_{i=0}^hk_1(i)}{k_1(0)+r}a^{\widehat{Q}^{+r}}+\frac{1+\sum_{i=0}^hk_2(i)}{k_2(0)+s}b^{\widehat{Q}^{+s}}= 1 + \sum_{\substack{j=1 2 \notin I_j}}^h\left(\frac{k_1(j)}{k_1(0)+k_1(j)}a^{\widehat{Q}_j}+\frac{k_2(j)}{k_2(0)+k_2(j)}b^{\widehat{Q}_j}\right).
\Im{(\sigma_{p}^{-1}z)}^{1-s}=1 - (s-1)\log\big(\Im{(\sigma_{p}^{-1}z)}\big)+ O\big((s-1)^{2}\big); \,\,\,\,\frac{1}{2s-1}= 1- 2(s-1) + O\big((s-1)^{2}\big).
\int_0^1 \log(\tilde f_1(x)f_2(x))\,dx = \int_0^1 \log\tilde f_1(x)\,dx+\int_0^1 \log f_2(x)\,dx \geq \frac{1}{2}\int_0^1 \log(f_1(x)) + \log(f_1(1-x))\,dx + \int_0^1 \log f_2(x)\,dx = \int_0^1 \log f_1(x)\,dx + \log f_2(x)\,dx = \int_0^1 \log(f_1(x)f_2(x))\,dx,
\sup \limits_{\nu \geqslant c_\flat}\frac{\nu}{\vert \nu -\zeta^* \vert^2}\leqslant \begin{cases}(c_\flat +1)c(\psi)^2\vert \zeta -c_\flat \vert^{-2}, \vert \zeta -c_\flat \vert <1, (c_\flat +1)c(\psi)^2\vert \zeta -c_\flat \vert^{-1}, \vert \zeta -c_\flat\vert \geqslant 1.\end{cases}
I =\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\Phi(u_{n}(x)-u_{n}(y))(u_{n}(x)-u_{n}(y))\xi_{j,\delta}(x)K(x,y)dx dy &\geq \Lambda^{-2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+ps}}\xi_{j,\delta}(x)dx dy &=\Lambda^{-2}\int_{\mathbb{R}^{N}}|D^{s}u_{n}|^{p}\xi_{j,\delta}(x)dx
\epsilon=\frac{1}{N(a-2)^N a^{N-1}2^{a-1}}\leq \frac{1}{N(a-2)^N \lfloor \frac{a}{\mathsf{a}(n,\mathbf{d})}\rfloor^{N-1}2^{a-2}}\cdot\frac{1}{2}.
G_{ik}=\frac{1}{4\pi\mu r}[\begin{matrix}1-\frac{1}{2b}\frac{z^{2}}{r^{2}}&0&\frac{1}{2b}\frac{\rho z}{r^{2}}\\ 0&1-\frac{1}{2b}&0\\ \frac{1}{2b}\frac{z\rho}{r^{2}}&0&1-\frac{1}{2b}\frac{\rho^{2}}{r^{2}}\end{matrix}]
\tilde\Pi_{R{(2)}}^{(\beta)}(\lambda,p) = -\frac{8 e^4}{p^2}m^2 I(T/\lambda)\int d^{2}q\,n_{F}(|q^{0}|) \left(q^{2}+m^2\right)\delta^{\prime\prime}(q^2-m^2).
a ( 1 , 2 , 3 ) = x_{1 2}^{2}g ( 1 , 2 , 3 ) + x_{1 2}^{2}x_{1 3}^{2}2 \partial_{2}\cdot \partial_{3}f ( 1 , 2 , 1 , 3 ) + \mathrm{c y c l e}\ .
\mathcal{G}_{\{n-k+1,\dots,n\}}=\frac{\det\left( x_j^{k-i}\prod_{m=1}^{n-k+i-1}(1-x_j/y_m) \right)_{1 \leq i,j \leq k}}{\prod_{1 \leq i<j \leq k}(x_i-x_j)}.
W=\lambda_U UQ_i\phi_{2i}+\lambda_{U'}U'Q_i\phi_{1i}+\epsilon_{ijk}(\lambda_DD_i\bar\phi_{2j}Q_k+\lambda_E E_i\bar\phi_{2j}L_k+ \lambda_{D'}\bar\phi_{1i}D_jD'_k +\lambda_{E'}\bar\phi_{1i}E_jE'_{k})
\sup_{0<r\le \lambda_i^{-1}}\frac1{r}\int_{-r}^{r}\int_{B^2_{\lambda_i^{-1}}(0)}|\widehat\tau_i|^2(x,t)\,dxdt=\lambda_i^2 \frac1{\lambda_ir}\int_{-\lambda_i r}^{\lambda_i r}\int_{B_1^2}|\widetilde\tau_i|^2(x,t)\,dxdt\le CL_2^2\lambda_i^2\rightarrow 0,
\gamma \triangleq \frac{\rho_d\sigma_{\widehat{h}_{ab}}^2}{\sigma_b^2+\rho_d\sigma_{\widetilde{h}_{ab}}^2}= \frac{\rho_d\lambda_{ab}^2n_p\rho_p}{\sigma_b^4+\rho_d\lambda_{ab}\sigma_b^2+\sigma_b^2\lambda_{ab}n_p\rho_p}.
\max\limits_{j \neq l}| \langle \phi_j, \phi_l \rangle |^2 = \frac{1}{2}+ \frac{1}{2}\max\limits_{j \neq l}\langle y_j, y_l \rangle &\geq \frac{1}{2}+ \frac{1}{2}\left[ \frac{1}{2}\csc^2\left(\frac{N \pi}{6(N-2)}\right) - 1 \right].
J_\nu (z) = \frac{H_\nu^{(1)}(z) + H_\nu^{(2)}(z)}2, \hskip 30 pt J_{-\nu}(z) = \frac{e^{\pi i \nu}H_\nu^{(1)}(z) + e^{-\pi i \nu}H_\nu^{(2)}(z)}2.