harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\zeta(x) \geq 2(1-\log x)^{\tau-1}(-\log x)^{\tau-1}- 2\tau(1-\log x)^{\tau-1}(-\log x)^{\tau-1}\qquad\mbox{}+ (\tau-1)(1-\log x)^{\tau}(-\log x)^{\tau-2}+ (\tau-1)(1-\log x)^{\tau-2}(-\log x)^{\tau}=(\tau-1)(1-\log x)^{\tau-2}(-\log x)^{\tau-2}\geq0,
{}~ \times \Bigg \{\sum_{\sigma \in Z_{N}}\frac{w ( x_{3}, x_{1 3}, x_{1}| d , h + \sigma ) w ( x_{4}, x_{2 4}, x_{2}| a , g + \sigma )}{w ( x_{4}, x_{1 4}, x_{1}| e , c + \sigma ) w ( x_{3}/ \omega , x_{2 3}, x_{2}| f , b + \sigma )}\Bigg \}_{0}.
\tilde{g}\tilde{g}\rightarrow(\overline{q}\tilde{q})(\overline{q}\tilde{q})\rightarrow(q\overline{q}\tilde{C}_{1}^{+})(q\overline{q}\tilde{C}_{1}^{+})\rightarrow(q\overline{q}W^{+})(q\overline{q}W^{+})\rightarrow
\mathbf U_k(z) = \sum_{m=1}^N (\mathbf P_m \cdot \mathbf T^{-1}\cdot \boldsymbol{\rho}_m \cdot \mathbf U_m) (z)+ \mathbf U^{ext}(z), \; z\in D_k \cup L_k \; (k=1,2,\ldots,N).
( \sigma_{\mu})_{\alpha \dot{\alpha}}J^{\mu \alpha}= J_{\alpha \dot{\alpha}}^{\alpha}= i \, \frac{T_{G}}{4 \pi^{2}}\, \mathrm{T r}\left[ \bar{G}_{\dot{\alpha}\dot{\beta}}\bar{\lambda}^{\dot{\beta}}\right] \, , \qquad \vartheta_{\mu}^{\mu}= \frac{T_{G}}{1 6 \pi^{2}}\, \mathrm{T r}\, \left[ G_{\rho \sigma}G^{\rho \sigma}\right] \, .
\frac{\partial f_j}{\partial \lambda_k}=-\frac{1}{m|g|}\sum_{h,i=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{j,k}[\tilde{v}_{i-h-1}\tilde{w}_{i-h-1}]_{k,j}\frac{\partial}{\partial\lambda_k}\frac{\lambda_k^{m-h-1}\lambda_j^h}{\lambda_k^m-\lambda_j^m}
\{A,D\}_{(3,1)}=-4d{\partial}_{1}a \cdot{\gamma}^{2}-4{\gamma}^{2}d \cdot{\sum_{k=1}^{\infty}}{\gamma}^{2k}{\partial}_{1}a_{2k}-2(1-4{\gamma}^{2}){\partial}_{1}d \cdot{\sum_{k=1}^{\infty}}k{\gamma}^{2k}a_{2k},
\varrho_{\Phi}= \left({\frac{1}{2}}{\frac{\dot{\Phi}^{2}}{c^{2}}}+ V \right){\frac{1}{c^{2}}}, \ \wp_{\Phi}={\frac{1}{2}}{\frac{\dot{\Phi}^{2}}{c^{2}}}- V , \I \equiv 1 + B{\frac{\dot{\Phi}^{2}}{c^{2}}}.
K_{\omega,\mathbf{c}}(U_0) =2L(U_0)+3N(U_0)+2\omega Q(U_0)+\mathbf{c}\cdot \mathbf{P}(U_0) &=2\omega Q(U_0)+\sqrt{\omega}\ \mathbf{c}_0\cdot \mathbf{P}(U_0)+2L(U_0)+3N(U_0)
V^D_\theta = V_0^D \equiv V^D \equiv\sum_n \mu^{\epsilon}{1\over 2}\int{d^{D-1}k\over (2\pi)^{D-1}}\log\left({k^2 + m_n^2(\varphi_i,D) \over\mu^2}\right),\quad \quad (D\neq{\rm integer}).
\nabla_x R(t,x,\xi) =\eta(t,0;x,\xi)=p(0,t;y(0,t;x,\xi),\xi) &=\xi+\int_0^t(\nabla_x V_\rho)(\tau,q(\tau,t;y(0,t;x,\xi),\xi))d\tau &=\xi+\int_0^t(\nabla_x V_\rho)(\tau,q(\tau,0;x,\eta(t,0;x,\xi)))d\tau .
\varepsilon_4 \delta_0^{-1}= C_2 \left(472.5+2B_3 (B_4-m_4) \sqrt{(\sqrt{2}+1) C}\right)^{-1}(\log{K})^{\delta_3}\geq C_2 \left(472.5+3.2 B_3(B_4-m_4) \sqrt{C}\right)^{-1}(\log{K})^{\delta_3}=:\varepsilon_5.
\mathfrak{A}\in\mathcal{I}_{E/F}\Longleftrightarrow \mathfrak{A}=\prod_{\mathfrak{p}\in\mathbb{P}_F}\,(\prod_{i=1}^{g(\mathfrak{p})}\,\mathfrak{P}^{\frac{e_i}{e(\mathfrak{p})}})^{v(\mathfrak{p})},
\lim_{L\to\infty}\frac{e^{\ln(z_c)\sigma_{\mu,\nu}L^{2/3}\tilde y}}{e^{\ln(z_c)\sigma_{\mu,\nu}L^{2/3}\tilde x}}\cdot\sigma_{\mu,\nu}L^{1/3}\cdot\tilde K\left(x_cL-\sigma_{\mu,\nu}L^{1/3}\tilde x,x_cL-\sigma_{\mu,\nu}L^{1/3}\tilde y\right)=K_{Airy}(\tilde x,\tilde y),
A = \frac{1}{d^n}\exp\bigg( n \Big( s\, H(P) + s^2\, \frac{V(P)}{2}+ O(s^3) \Big)\bigg) &=\frac{1}{d^n}\exp\bigg( u n^{1-t}\, H(P) +u^2 n^{1-2t}\,\frac{V(P)}{2}+ O( n^{1-3t}) \bigg) ,
T[F_1(\zeta_1)F_2(\zeta_2)\dots F_k(\zeta_k)F_{k+1}\dots F_{k+n}](\zeta_{k+1},\dots,\zeta_{k+n}):=\sum_{\sigma\in\mathfrak{S}_{k+n}}\chi_{P'_{n,\sigma}}(\zeta_{k+1},\dots,\zeta_{k+n})F_{\sigma(1)}(\zeta_{\sigma(1)})\dots F_{\sigma(k+n)}(\zeta_{\sigma(k+n)}),
-b\sigma e^{h\sigma}k_{p}+be^{h\sigma}k_{i}= \sigma a_\sigma \;, be^{h\sigma}(1+h\sigma)k_{p}-hbe^{h\sigma}k_{i}= -a_{2\sigma}\;, -hbe^{h\sigma}(2+h\sigma)k_{p}+h^{2}be^{h\sigma}k_{i}= -2\;.
\vec{X}_{11}= \Bigg( 0, 1, - \frac{\lambda}{m-n}\bigg(\frac{1}{1+ B^{-1}}\bigg)\Bigg), \vec{X}_{12}= \bigg( 1\;,\;0\;,\; -\frac{\lambda}{m-n}\frac{\lambda c}{1+C^{-1}}\bigg).
S^{c}= \frac{e^{i \frac{\pi}{4}}}{2 ( 4 \pi )^{3 / 2}}\int_{0}^{\infty}d e_{0}\left( \det \frac{\sinh \frac{g e_{0}F}{2}}{g F}\right)^{- \frac{1}{2}}e^{i I [ x_{c l}, e_{0}]}\Phi [ x_{c l}, e_{0}] ,
-\frac{1}{2}\sum_{n_1}\, \log \frac{\cos [ \pi ( \varphi_2 + \tau_1(n_1 + \varphi_1) - i \tau_2 \mid n_1 +\varphi_1 \mid )]}{\cos [ \pi (\tau_1n_1 - i \tau_2 \mid n_1 \mid )]}\,.
\widetilde{\Phi}_{t}\left( \xi ,\zeta \right) =\widetilde{\Phi}_{0}\left( \frac{a\left( t\right) \xi +b\left( t\right)}{c\left( t\right) \xi +d\left(t\right)},\zeta \right) =\phi_{0}\left( \frac{a\left( t\right) \xi+b\left( t\right)}{c\left( t\right) \xi +d\left( t\right)}\zeta \right) ,
\left|\int_s^t\langle p_k^2+c_k^2, w_k\cdot\nabla\psi_L\rangle+\langle p_k^3+c_k^3, w_k\cdot\nabla\psi_L\rangle d\tau\right| &\leq C_T\|w_k\cdot\nabla\psi_L\|_{L^5(0,T;L^{15/8}(\Omega))}+C_T\|w_k\cdot\nabla\psi_L\|_{L^{q^\prime}(0,T;L^{(r_*)^\prime}(\Omega))}&\leq C_T\big(\|\nabla\psi_L\|_{30}+\|\nabla\psi_L\|_\sigma\big),
\frac{d}{dt}(p_{t}|(D\phi_{t})w)=(\dot{p}_{t}|(D\phi_{t})w)+(p_{t}|\frac{d}{dt}(D\phi_{t})w)=(\dot{p}_{t}|(D\phi_{t})w)+(p_{t}|(Dv_{t})_{|_{\phi_{t}}}(D\phi_{t})w)=0
E^0_{k,m}= \{n \in \mathbb{N}^k: 0 \le n \le m\}&E^1_{k,m}= \{n + v_i: n, n + e_i \in E^0_{k,m}\}&r(n+v_i) = n, s(n+v_i) = n + e_i, c(n + v_i) = c_i
\big\{h_\gamma(\lambda_2,\lambda_1),^2\!E_S[f] \big\}\!=\!\!\int_{x\in S}\!\!\!\!\!\!\mathrm{d}S^{bc}(\mathbf{x})f^k(\mathbf{x})v_{abc}(\mathbf{x})\!\int_{\lambda_1}^{\lambda_2}\!\!\!\mathrm{d}t\dot{\gamma}^a(t) \mathcal{E}^{(3)}_{\;\,\gamma(t)}(\mathbf{x})h_\gamma(\lambda_2,t)\frac{\tau_k}{2}h_\gamma(t,\lambda_1 )\,.
{\hat F[L,~f]}~=~ \alpha^{-\sigma[L,~f]}~\sum_{\{R_i\}}~~\mu_{R_1}~\mu_{R_2}~....~\mu_{R_r}~ V[L;~n_1,~n_2,~...~n_r;~R_1,~R_2,~....~R_r]
\begin{cases}v\left(t,0-\right)=v\left(t,0+\right) p\left(t,0-\right)=p\left(t,0+\right) \end{cases}\begin{cases}v\left(t,m-\right)=v\left(t,m+\right) p\left(t,m-\right)=p\left(t,m+\right) \end{cases}\mbox{for a.e.}t\geq 0 \,.
|\nabla g_Q(x)| \geq -\partial_1 g_Q(x) = c\int_{B(y_Q,r(V_Q))}\frac{x_1-y_1}{r(V_Q)\,|x-y|^{n+1}}\,d\mu(y) &\approx \frac{\mu(B(y_Q,r(V_Q))}{r(V_Q)^{n+1}}\approx \frac1{r(V_Q)}\quad\mbox{for all $x\in V_Q$.}
\Lambda=\left\{\lambda_{1}Z_{1}^{\ast}+\lambda_{2}Z_{2}^{\ast}+\beta_{2}X_{2}^{\ast}\in\Omega:\left( \lambda_{1},\lambda_{2},\beta_{2}\right)\in\mathbb{R}^{3}\right\}\simeq\mathbb{R}^{\ast}\times\mathbb{R}^{2}.
f_{T}(\cos(\alpha_{1}),...,\cos(\alpha_{n})|K_{n})\allowbreak= \allowbreak\frac{1}{2^{n}}\sum_{i_{1}\in\{-1,1\}}...\sum_{i_{n}\in\{-1,1\}}\frac{\sum_{k=0}^{n}(-1)^{k}\sum_{S_{k}\subseteq\mathcal{K}_{n}}^{\prime}\rho_{S_{k}}\cos(b_{S_{k}})}{\prod_{j=1}^{n}\prod_{m=j+1}^{n}(1-2\rho_{jm}\cos(\beta_{j,m}(i_{j},i_{m}))+\rho_{jm}^{2})},
\sup_{\mathcal{C}, x}\left| \frac{d f}{d s}\right| = \sup_{\mathcal{C}, x}\lim_{p \to x , \; p \in \mathcal{C}}\left| \frac{f ( x ) - f ( p )}{s ( x ) - s ( p )}\right| = \sup_{\mathcal{C}, x}\lim_{p \to x , \; p \in \mathcal{C}}\frac{| f ( x ) - f ( p ) |}{\mathrm{l e n g t h}( \widehat{x p})}~ .
\sum_{i = 0}^{4}a_{i}N_{i}= N_{0}( a_{0}- 1 0 a_{2}- 1 5 a_{3}- 6 a_{4}) + N_{1}( a_{1}+ 4 a_{2}+ 5 a_{3}+ 2 a_{4}) + \chi ( 1 0 a_{2}+ 1 5 a_{3}+ 6 a_{4}) .
u = \frac{x_{1 2}^{2}x_{3 4}^{2}}{x_{1 3}^{2}x_{2 4}^{2}}\, \, \, \, \, \, \mathrm{a n d}\, \, \, \, \, v = \frac{x_{1 2}^{2}x_{3 4}^{2}}{x_{1 4}^{2}x_{2 3}^{2}}.
ad(-a^2-b^2+5c^2-5d^2)+bc(5a^2-8b^2-5c^2+10d^2)=12D_1(q), &ad(2a^2-b^2-4c^2+d^2)+bc(-a^2+b^2-5c^2+7d^2)=24D_2(q), &ad(-a^2+2b^2-7c^2+10d^2)+bc(-a^2+4b^2+c^2-8d^2)=48D_3(q), &ad(a^2-8b^2-5c^2+20d^2)+bc(7a^2-10b^2+5c^2-10d^2)=48D_4(q).
S = \frac{\pi u}{\lambda^{2 / 3}}\left[ \frac{| m^{i}+ \tau n^{i}+ \omega p^{i}|^{2}}{\tau_{2}}+ \lambda^{2}\left( p^{i}\right)^{2}\right] - \pi u^{3}+ 2 \pi i C_{i j k}m^{i}n^{j}p^{k},
I\left(f_{\omega_{\pi_\nu},s}^\circ,v_{\pi_{\nu}}^\circ\right)= \frac{Z_{\mathrm{GJ}}\left(m(s+\frac{1}{2})-\frac{n-1}{2},c_{v_{\pi_{\nu}}^\circ,\check{v}_{\pi_{\nu}}^{\circ}},\Phi_0\right)}{L(m(s+\frac{1}{2}),\omega_{\pi_\nu})}v_{\pi_{\nu}}^\circ,
v^*_{\tilde T, j}= q^*_{\tilde T}\cdot y^*_j + (1-q^*_{\tilde T}) \cdot 0 = q^*_{\tilde T}\cdot v^3_{\tilde T,j}+ (1-q^*_{\tilde T}) \cdot v^2_{\tilde T,j}, & \bar v^*_{\tilde T, j}= q^*_{\tilde T}\cdot 0 + \bar q^*_{\tilde T}\cdot y^*_j = q^*_{\tilde T}\cdot \bar v^3_{\tilde T,j}+ (1-q^*_{\tilde T}) \cdot \bar v^2_{\tilde T,j},
G_{r-1}^C ( z | \underline \tau ) = \exp \left [ \frac{2\pi i}{(r+1)!}B_{r+1,r+1}^{\hat C}( z | \underline \tau,-1)\right ] \prod_{\rho \in\Delta^C_1}(SK_\rho)^* G_{r-1}(z|\underline \tau)
E(T,X(T)) = -(\nabla \phi \ast(G \ast \mathcal S + \mathcal S))(t,X(t)), \lim_{s\rightarrow \infty}\nabla \phi *\rho[h_{V(T)}])(s,0) = \lim_{R\rightarrow \infty}(\nabla \phi \ast(G \ast S_{R,X(T),V(T)}+ S_{R,X(T),V(T)})(R,X(T)).
\lim_{M \rightarrow \infty}M^{n-\alpha}\mu^\alpha = \lim_{M \rightarrow \infty}M^{n-\alpha}\left(\frac{1}{M}\right)^{\alpha\beta(M-2n)^n/M^n}= \lim_{M \rightarrow \infty}M^{n-\alpha(1+ \beta)},
F_t(V^\leq (\epsilon)) =F_{t}(\bigcup_{t_i\geq 0}F_{t_i}(S_1(\epsilon))) =\bigcup_{t_i\geq 0}F_{t+t_i}(S_1(\epsilon)) = \bigcup_{t_i'\geq t}F_{t_i'}(S_1(\epsilon)) \subset \bigcup_{t'\geq 0}F_{t'}(S_1(\epsilon))=V^\leq (\epsilon),
\biggl [ G_{i j k l}( x ) \frac{\delta}{\delta h_{k l}( x )}\frac{\delta}{\delta h_{i j}( x )}- \frac{1}{( 1 6 \pi G )^{2}}\sqrt{h ( x )}\Bigl ( \! R ( x ) + 2 \Lambda \Bigr ) \biggr ] \Psi [ h ] = 0 \ ,
\dot{\sigma}_{ab}=-\frac{2\theta}{3}\sigma_{ab}-(\sigma_{am}{\sigma^{m}}_{b}+\omega_{am}{\omega^{m}}_{b})-{E[\vec{X}]}_{ab}+\frac{\sigma^{2}-\omega^{2}+{E[\vec{X}]^{m}}_{m}}{3}h_{ab}
H_k^{(s+2,t)}H_k^{(s,t)}=H_{k}^{(s+1,t)}H_{k}^{(s+1,t)}+H_{k+1}^{(s,t)}H_{k-1}^{(s+2,t)},s,t=0,1,\dots, &H_{k}^{(s,t+2)}H_{k}^{(s,t)}=H_{k}^{(s,t+1)}H_{k}^{(s,t+1)}+H_{k+1}^{(s,t)}H_{k-1}^{(s,t+2)},s,t=0,1,\dots.
g_{Y M}^{2}={\frac{1}{4 \pi^{2}}}{\frac{g_{s}}{\alpha^{\frac{3}{2}}}}= \mathrm{f i x e d}, \hspace{2 m m}\mu \alpha^{2}= \mathrm{f i x e d}, \hspace{2 m m}\alpha^{\prime}\to 0
J \epsilon_{a b 3}\epsilon_{i j}\int_{\tau_{0}}^{\tau_{0}- i \beta}d z^{0}\, \left[ S_{c}( z^{0}- x^{0}, \vec{p}) \right]_{\rho \mu}\left( \gamma_{5}C \right)_{\rho \sigma}\left[ S_{c}( z^{0}- y^{0}, - \vec{p}) \right]_{\sigma \nu}.
\frac{\partial}{\partial t'}[\omega_{a}^{QH'S}(t',t)]_{Q^{S}(t,z)}=z(\bar{C}(\Psi,t',t)_{a}^{b}- \Psi(0)\delta_{a}^{b})[\omega_{b}^{QH'S}(t',t)]_{Q^{S}(t,z)}
\setlength{\abovedisplayskip}{4 pt}\setlength{\belowdisplayskip}{4 pt}\boldsymbol{W}_j^{out}=\boldsymbol{s}_j{\boldsymbol{v}_j^{\rm T}}{\left({\boldsymbol{v}_j^{\rm T}}\boldsymbol{v}_j +{\lambda^2}\boldsymbol{\rm I}\right)^{- 1}},
\frac{1 6 \pi^{2}}{g_{\alpha ,{\scriptstyle \mathrm{b a r e}}}^{2}}= \frac{1 6 \pi^{2}}{g_{\alpha}^{2}( \mu )}- b_{\alpha}( 4 \pi )^{\epsilon}\int_{0}^{\infty}\frac{d t}{t^{1 - \epsilon}}e^{- t \frac{\mu^{2}}{M^{2}}}.
i_{e}{}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}}{}^{( a )}= \sum_{\Lambda}\M_{e}{}_{\Lambda}^{( a )}\{C}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}\, \Lambda},
\hat{r}_{\alpha}(m) = \sum_{k=0}^{\infty}(-1)^k (k+1) \Big[ \frac{1}{|m|+k(N+1)}\Big( q^{-(\alpha-\tfrac{1}{2})(|m|+k(N+1))}-q^{-2(|m|+k(N+1))}\Big) &-\frac{1}{(N+1)(k+2) - |m|}\Big( q^{(\alpha-\tfrac{1}{2})(|m|-(k+2)(N+1))}- q^{2(|m|-(k+2)(N+1))}\Big) \Big].
\widetilde{\mathcal{K}}^{}= \frac{1}{2}\sum_{k \neq 0}\widetilde{\mathcal{A}}_k \big( a_k^\dagger a_k + a_{-k}^\dagger a_{-k}\big) + \frac{1}{2}\sum_{k \neq 0}\mathcal{B}_k \big( a_k^\dagger a_{-k}^\dagger + a_k a_{-k}\big),
{\mit\Gamma}_{\rm cusp}(\theta; \alpha_s) = - \frac{\alpha_s N_c}{\pi}\frac{1}{\sqrt{1 + x}}\left\{\sqrt{1 + x}-\left(1 + \frac{x}{2}\right) \ln \frac{\sqrt{1 + x}+ 1}{\sqrt{1 + x}- 1}\right\}\, ,
P\int_0^{\infty{e}^{{i}\varphi}}\frac{t^{n-1}\,{e}^{at}}{z\,{e}^{t}-1\,}\,{d}t = z^{-1}\,(n-1)!\,\Phi(z^{-1},n,1-a)-{\rm sgn}(\varphi)\,{i}\,\pi\,(-\ln z)^{n-1}\,z^{-a},
\forall\,\mathbf{x}\in\displaystyle{\prod_{i=1}^{e^{'}}}\big{[}Y_{i}\big{]}^{r}\times\big{[}F|(\bigcup_{1\leq i<\theta}Y_{e^{'}+i})\big{]}^{r,f^{'}}\,\,\, M\models\sigma(\mathbf{x}).
0 = \int d^{5}x \left( i e \frac{\delta \Gamma}{\delta \Phi_{c}^{*}}\Phi_{c}^{*}w_{c}- i e \frac{\delta \Gamma}{\delta \Phi_{c}}w_{c}\Phi_{c}+ D_{\alpha}\frac{\delta \Gamma}{\delta \Gamma_{\alpha}^{c}}w_{c}+ \frac{1}{\xi}\frac{\delta \Gamma}{\delta u_{c}}D^{\alpha}\Gamma_{\alpha}^{c}\right) .
M ( p , p^{\prime}) = \int d^{4}x d^{4}x^{\prime}\bar{E}_{p}( x ) M ( x , x^{\prime}) E_{p^{\prime}}( x^{\prime}) = \tilde{\Sigma}_{A}( \bar{p}) ( 2 \pi )^{4}\hat{\delta}^{( 4 )}( p - p^{\prime}) ,
\Delta_{2^{p_0}+\cdots+2^{p_s}}= 2^{p_0-p_1}(2^{p_1-p_0}-2)\left[4\Delta_{2^{p_1}+\cdots+2^{p_s}}+(2^{p_1-p_0}-2)(1+2^{p_2-p_1}+\cdots+2^{p_s-p_1})\right] &+2^{p_0-p_1}R_1(p_1,\cdots,p_s),
size_{j}((\langle f,\Phi^{j}_{P_{j}}\rangle)_{P\in\mathbb{P}}):=\sup_{T\subseteq\mathbb{P}}(\frac{1}{|I_{T}|}\sum_{P\in T}|\langle f,\Phi^{j}_{P_{j}}\rangle|^{2})^{\frac{1}{2}},
\langle M_{1,2}\psi, M_{2,3}\psi\rangle_{L^2(\Omega_T)}= -\frac{1}{2}\int_{\Omega_T}\partial_t \left(\left(\frac{\tau}{2}-s\alpha\right) (\partial_t \log \gamma)\right)\, \psi^2 \,\d x\,\d t
\begin{aligned}\ \sum_{r+s+t=n}(-1)^{\deg x_1+\dots+\deg x_r+r}\overline{f}_{r+t+1}(x_1,\dots, x_r,\overline{m}_s(x_{r+1},\dots,x_{r+s}),x_{r+s+1},\dots,x_n)= &\sum_{i_1+\dots+i_r=n}\overline{m}_r(\overline{f}_{i_1}(x_1,\dots,x_{i_r}),\overline{f}_{i_2}(x_{i_1+1},\dots,x_{i_1+i_2}),\dots,\overline{f}_{i_r}(x_{i_1+\dots+i_{r-1}+1},\dots,x_n))\end{aligned}
\sigma_{1}^{j - 1}\cdot H_{j, a_{1, 1}}\cdot c^{1}_{1}+ \sum_{q_i \in Q}\sigma_{q_i}^{j - 1}\cdot H_{j, a_{q_i, 1}}\cdot c^{q_i}_{1}+ \sum_{v_i \in V}\sigma_{v_i}^{j - 1}\cdot H_{j, a_{v_i, 1}}\cdot c^{v_i}_{1}= 0 j \in [r].
R_g = e^{-\beta(t)g(r)}\left(\left(1 + r^2 \beta(t) g'(r^2)\right)\frac{\partial}{\partial t}-\frac{r\beta'(t)g(r^2)}{2}\frac{\partial}{\partial r}- 2\beta(t)g'(r^2)\frac{\partial}{\partial \theta}\right).
\|\mathcal{L}_{X_r}\mathcal{L}_{Y_r}\mathcal{L}_{X_r}^{-1}-\mathcal{L}_{Y_r}\|_2 \le\|\mathcal{L}_{Y_r}\mathcal{L}_{X_r}-\mathcal{L}_{X_r}\mathcal{L}_{Y_r}\|_2\|\mathcal{L}_{X_r}^{-1}\|_2 &\le \mu_r\|\mathcal{L}_{X_r}^{-1}\|_2\frac{\|X_r\circ Y_r-\mu_rI\|_F}{\mu_r}&=\frac{\mu_r}{\lambda_r}\frac{\|X_r\circ Y_r-\mu_rI\|_F}{\mu_r},
\kappa[ a(\mathbf{k}_1,-1); a(\mathbf{k}_2,\sigma); a(\mathbf{k}_3,1); a(\mathbf{k}', \sigma') ] = \delta(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}') \widehat{F}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}',\sigma, \sigma')
\widetilde{V}^{(0^{2})}(x;1,2,\lambda_{1}) = \frac{3}{4\sin^2 x}+ \frac{15}{4\cos^2 x}- 16 + 32 \frac{(1-4\cos 2x) \sin^2 x \cos^4 x}{\lambda_1 - \frac{8}{3}\cos^6 x + 2\cos^8 x}\hphantom{\widetilde{V}^{(0^{2})}(x;1,2,\lambda_{1}) =}{}+ 512 \frac{\sin^6 x \cos^{10}x}{\big(\lambda_1 - \frac{8}{3}\cos^6 x + 2\cos^8 x\big)^2}
\Omega_2:=\left[1+\sqrt{\frac{\gamma_n}{4n}}+\frac{C_n}{\sqrt{4 n \gamma_n}},~1+\frac{n^{\tau}}{\sqrt{n}}\right]\times \left[-\frac{n^{\tau/2}}{n^{1/4}},~\frac{n^{\tau/2}}{n^{1/4}}\right],
\int_{\Omega_{\delta}}\big\{{-}( x_{i}\mathcal{D}_{i}f(x))^{\ast}K(x) g(x) +f(x)^{\ast}K(x) x_{i}\mathcal{D}_{i}g(x)\big\}\mathrm{d}m(x){}=\int_{\Omega_{\delta}}x_{i}\partial_{i}\{f(x)^{\ast}K(x) g(x)\}\mathrm{d}m(x).
a_{24}^{4213}=a_{24}^{4231}=a_{24}^{4123}=a_{24}^{4132}=a_{24}^{4321}=a_{24}^{4312}= a_{24}^{1423}=a_{24}^{1432}=a_{24}^{1342}=a_{24}^{3421}=a_{24}^{3412}=a_{24}^{3142}
\begin{array}{l}\bar{N}_{x}=\{N_{x},+,\times_{x},<_{x}, 0_{x},1_{x}\}\bar{I}_{x}=\{I_{x},\pm_{x},\times_{x},<_{x}, 0_{x},1_{x}\}\overline{Ra}_{x}=\{Ra_{x},\pm_{x},\times_{x},\div_{x},<_{x}, 0_{x},1_{x}\}\bar{R}_{x}=\{R_{x},\pm_{x},\times_{x},\div_{x},<_{x}, 0_{x},1_{x}\}\bar{C}_{x}=\{C_{x},\pm_{x},\times_{x},\div_{x}, 0_{x},1_{x}\}\end{array}
e(G') =e(S)+e(\bar{S})+e(\bar{S},S)\leq \binom{s}{2}+\frac{1}{2}\sum_{x\in \bar{S}}\left(d_S(x)+d_{G'}(x)\right)\leq \binom{s}{2}+\frac{1}{2}(s-1+k)(n-s).
\sum_{n=0}^{\infty}{\frac{a}{a+bn}\binom{a+bn}{n}z^{n}}= x^{a}= x^{a_{1}}x^{a_{2}}\cdots x^{a_{r}}& = \prod_{i=1}^{r}{\left(\sum_{n=0}^{\infty}{\frac{a_{i}}{a_{i}+bn}\binom{a_{i}+bn}{n}z^{n}}\right)}& = \sum_{n=0}^{\infty}{\left(\sum_{n_{1}+n_{2}+\cdots+n_{r}=n}\prod_{i=1}^{r}{\frac{a_{i}}{a_{i}+bn_{i}}\binom{a_{i}+bn_{i}}{n_{i}}}\right)}z^{n}.
A_{i}= \frac{i}{2}[ \frac{\phi_{1}}{s}\: \sigma_{i}\: \: + \frac{1}{s^{2}}( A_{1}- \frac{\phi_{1}}{s}) \: x_{i}\vec{x}. \vec{\sigma}\: \: + ( \frac{\phi_{2}+ 1}{s^{2}}) \: \varepsilon_{i j k}\sigma_{j}x_{k}]
A \, \left\|\,f,\, a_{\,2},\, \cdots,\, a_{\,n}\,\right\|^{\,2}\,\leq\, \sum\limits^{\infty}_{i \,=\, 1}\,\left|\,\left<\,f,\, f_{\,i}\,|\, a_{\,2},\, \cdots,\, a_{\,n}\,\right>\,\right|^{\,2}\,\leq\, B\, \left\|\,f,\, a_{\,2},\, \cdots,\, a_{\,n}\,\right\|^{\,2}
\bigl| \bigl( (1-\varepsilon )\partial_t u_{\lambda, \varepsilon}, 0 \bigr) \bigr|_{L^2(0,T;\boldsymbol{V}^*)}= \bigl| (1-\varepsilon ) \partial_t u_{\lambda, \varepsilon}\bigr|_{L^2(0,T;H)}& \le (1-\varepsilon ) C^* M_2 & \le C^*M_2.
b(a_0\otimes a_1\otimes \cdots \otimes a_n) = a_0a_1\otimes a_2\otimes \cdots\otimes a_n + \sum_{k=1}^{n-1}(-1)^k a_0\otimes \cdots \otimes a_ka_{k+1}\otimes\cdots \otimes a_n + (-1)^na_na_0\otimes a_1\cdots\otimes a_{n-1}.
\parallel\ \dfrac{2q-n+1}{3(n+1)}T_g(r) \le \sum_{i=1}^qN_{(g,a_i)}^{[n]}(r)+o(T_g(r)+T_f(r)) &\le n\sum_{i=1}^qN_{(g,a_i)}^{[1]}(r)+o(T_g(r)+T_f(r)) &= \sum_{i=1}^qnN_{(f,a_i)}^{[1]}(r)+o(T_g(r)+T_f(r)) &\le qn T_f(r)+o(T_g(r)+T_f(r)).
0 = d(m_0, m_0) \longleftarrow \, d(m_0, \exp(m_0, (\Upsilon_{g_i})'X_i)) &= d( \Upsilon_{g_i}p_i, \Upsilon_{g_i}\exp(p_i, X_i)) = d(p_i, \exp(p_i, X_i)).
\widetilde{c}_{{X^{\ast}}}(g_1,g_2) =m_{X^{\ast}}(g_1g_2) m_{X^{\ast}}(g_1)^{-1}m_{X^{\ast}}(g_2)^{-1}\overline{c}_{X^{\ast}}(g_1, g_2) &=[\overline{\beta_1}(g_1g_2)m_{X^{\ast}}(g_1g_2)] [\overline{\beta_1}(g_1)m_{X^{\ast}}(g_1)]^{-1}[\overline{\beta_1}(g_2)m_{X^{\ast}}(g_2)]^{-1}.
\sigma_{l}= \frac{2 \pi^{\frac{d - 1}{2}}\Gamma \left( l + d - 2 \right) \left( l + d / 2 - 1 \right)}{\omega^{d - 1}\Gamma \left( d / 2 - 1 / 2 \right) \Gamma \left( l + 1 \right)}P ( l ) ,
\mathcal{Q}:= \{Q\in\mathfrak{P}(\Omega_T):\Q\lll\mathcal{P},\E_Q[|\Delta S_t|\ |\mathcal{F}_t]<\infty\Q t=0,\dotso,T-1, &H\cdot SQ\ \forall H\in\mathcal{H}\}.
\lim_{K \rightarrow \infty}\frac{1}{\psi_2(N,K,s)}\leq\frac{N}{\lambda}\sum_{i=0}^{\infty}\left(\frac{(1-\lambda)(N-1)}{\lambda}\right)^i =\frac{N}{\lambda}\cdot\frac{1}{1-\frac{(1-\lambda)(N-1)}{\lambda}}&=\frac{N}{N\lambda-(N-1)}.
\begin{aligned}\langle\nabla_U V,A\rangle{}=\alpha-\frac{a}{b(\alpha-\gamma)}\bigl(A\alpha-\frac{3abc}{4}\bigr), \langle\nabla_A U,V\rangle{}=\gamma-\frac{Ab}{a}, \langle\nabla_V U,A\rangle{}=-\Bigl(\beta+\frac{b}{a(\beta-\gamma)}\bigl(A\beta+\frac{3abc}{4}\bigr)\Bigr).\end{aligned}
\bar{q}_{\beta}(n,t) =\sum_{k=1}^{n}\mathrm{Pr}\{X_{1}+X_{2}+\dots+X_{k}=n\}\mathrm{Pr}\{N_{\beta}(t)=k\}&=\sum_{k=1}^{n}\sum_{\Theta_{n}^{k}}k!\prod_{j=1}^{n}\frac{\left(\rho^{j}\binom{r+j-1}{j}\right)^{x_{j}}}{x_{j}!}\left(\frac{\lambda t^{\beta}(1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{k}E_{\beta,k\beta+1}^{k+1}(-\lambda t^{\beta}),
A(r)\equiv\frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\times F(r^{\prime})}{|r-r^{\prime}|}dV^{\prime}-\frac{1}{4\pi}\oint_{S}\hat{n}^{\prime}\times\frac{F(r^{\prime})}{|r-r^{\prime}|}dS^{\prime}
\frac{\partial^{2}A_{\varphi}}{\partial \rho^{2}}+ \frac{1}{\rho}\frac{\partial A_{\varphi}}{\partial \rho}- \frac{A_{\varphi}}{\rho^{2}}- \gamma C_{2}\bigg ( A_{\varphi}+ \frac{1}{\rho^{2}}\int_{0}^{\rho}A_{\varphi}( \rho^{\prime}) \rho^{\prime}d \rho^{\prime}\bigg ) = m_{C S}\gamma e \lambda K_{1}( \lambda \rho ) .
\frac{x^{\frac{k-2}{2}-m}y^{\frac{k-2}{2}+m}}{2^{2-k}}=\sum_{\frac{2-k}{2}\leq n\leq\frac{k-2}{2}}\binom{k-2}{n+\frac{k-2}{2}}\mu_m\left(P_{n+\frac{k-2}{2}}\right)z^{n+\frac{k-2}{2}}\bar z^{\frac{k-2}{2}-n}.
\lim_{r\rightarrow 0}{\frac{\sigma(B(x_0s,r))}{r^n}}={\frac{\mbox{vol}(B_n)}{\mbox{vol}(S^n)}}={\frac 1{2\sqrt{\pi}}}{\frac{\Gamma\begin{pmatrix}{\frac{n+1}2}\end{pmatrix}}{\Gamma\begin{pmatrix}{\frac n2}+1\end{pmatrix}}}
E_1(f,p)=\sum_{z\leq p}\frac{1}{p}\sum_{\substack{0< b< p gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{m}-v)}{p}}\cdot \frac{1}{p}\sum_{\substack{0\leq b< p gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{em}-v)}{p}}=0.
M_{W^{+}}^{2}={\frac{v^{2}}{4}}g^{2}, M_{W^{-}}^{2}={\frac{v^{2}}{4}}g^{2}, M_{Z}^{2}={\frac{v^{2}}{4}}( g^{\prime 2}+ g^{2}) , M_{A}^{2}= 0 \ .
[(I_{L}^{\mathfrak{m}}\cap L^{\times}):(I_{L}^{\mathfrak{n}}\cap L^{\times})]=[I_{L}^{\mathfrak{m}}:I_{L}^{\mathfrak{n}}]=\prod_{\mathfrak{p}}\left[U_{\mathfrak{p}}^{(v_{\mathfrak{p}}(\mathfrak{m}))}:U_{\mathfrak{p}}^{(v_{\mathfrak{p}}(\mathfrak{n}))}\right].
G((p+1)/2,j) =-\frac{2p}{4^{p-1}}\binom{p-1}{(p-1)/2}^2\frac{\binom{p-1+2j}{(p-1)/2+j}\binom{p-1+2j}{2j}}{\binom{2j}{j}}&=-\frac{2p^3}{4^{p-1}}\cdot\frac{\binom{p-1}{(p-1)/2}^2}{\binom{2j}{j}}\cdot\frac{\binom{p-1+2j}{(p-1)/2+j}}{p}\cdot\frac{\binom{p-1+2j}{2j}}{p}.
\tilde{z}(\tilde{t}_1) = \tilde{z}(s) - \int_{\tilde{t}_1}^{s}\tilde{z}'(t) \,dt < w(x_n) -2 \theta_n g_n (s) - \int_{\tilde{t}_1}^{s}(-2 \theta_n g_n'(t)) \,dt & = w(x_n) -2 \theta_n g_n (\tilde{t}_1) & = w(x_n) + 2 \theta_n |g_n(\tilde{t}_1)|,
\int_{a\exp\{-\int_0^tI(y_2,s)ds\}}^{b\exp\{-\int_0^tI(y_2,s)ds\}}\frac{1}{((x_2-y_2)^2+y_1^2)^2}dy_1\sim_{a,b}\frac{e^{-G}}{((x_2-y_2)^2+e^{-2G})^2}.
S(x^i y^{n+2-i}, g) = - s x^{i-1}y^{n+3-i}- x^{i+1}y^{n+1-i}+ x^{i-1}y^{n+3-i}&= (-s+1) x^{i-1}y^{n+3-i}+ (-1)x^{i+1}y^{n+1-i}.
\widehat{\mu}_{\min}^{\mathrm{asy}}(\overline L)= \inf_{Y\in \Theta_X}\widehat{\mu}_{\max}^{\mathrm{asy}}(\overline L|_Y) = \inf_{Y\in \Theta_X}\frac{h_{\overline L}(Y)}{(\dim(Y)+1)\deg_L(Y)}.
T_4{}:=X_4, T_3{}:=X_3, T_2{}:=X_2-\frac{q^4}{(q^2+1)(q+q^{-1})}X_3^2 X_4^{-1}, T_1{}:= X_1-\frac{q^2(q+q^{-1})}{q^2-1}X_2 X_3^{-1}.
\begin{aligned}s (\rho, q) = \rho F_\rho (\rho, q) + q \cdot F_q (\rho, q) - F (\rho, q)\, , r (\rho, q) = \rho F_q (\rho, q)\, , H(\rho , q) = q \otimes F_q (\rho, q) \, .\end{aligned}
I = \int d x [ \Omega^{\prime}\rho^{\prime}- \frac{\mu}{8}e^{2 \rho}\Omega^{1 - \lambda - \gamma / 4}- \frac{1}{4}\Omega f^{\prime 2}+ \frac{1}{4}e^{\chi f - 2 \rho}\Omega^{1 + \gamma / 4 - \epsilon / 2}A^{\prime 2}] .
Var(\hat{\pi}_{h,n}(x)) \le \frac{c}{T_n}\big[ \frac{1}{\prod_{l =1}^d h_l}(\delta_1 + \Delta_n) + \frac{1}{\prod_{l \ge k_0 + 1}h_l}\delta_1^{1 - \frac{k_0}{2}}+ \delta_2^{1 - \frac{d}{2}}+ D + \frac{1}{(\prod_{l = 1}^d h_l)^2}e^{- \rho D}\big].
A_1(n):=\{(\lambda,(a^b))\in A(n) \mid a+(2r-1)(b-1) \not\in \lambda \}; \ A_2(n) :={\small{\{(\lambda,(a^b))\in A(n) \mid a+(2r-1)(b-1) \in \lambda \}};}\ A_3(n):=\{(\lambda,(a^b))\in A(n) \mid \lambda =(a+(2r-1)(b-1)) \}.
\zeta_{p,\infty}(s,c)=\sum_{m,n\neq0}\frac{p^{2}+\lambda_{n}^{2}}{p(p+\frac{1}{\pi})+\lambda_{n}^{2}}\,\frac{1}{\left(\lambda_{m}^{2}+cn^{2}\right)^{s}},\,\,\,\,\,(s)>1.
A =\overline{\rm{conv}}\left\{(0,0,1),(1/2,0,\sqrt{3}/2),(0,1/2,\sqrt{3}/2)\in \mathbb{S}^2 \right\}, B =\overline{\rm{conv}}\left\{(1/3,0,2\sqrt{2}/3),(2/3,1/3,2/3),(2/3,2/3,1/3)\in \mathbb{S}^2 \right\}
\begin{cases}\rho_\epsilon\rightharpoonup \rho \{\rm{weak^*\in}}\L^\infty([0,T], L^\gamma(\Omega)), u_\epsilon\rightharpoonup u \{\rm{in}}\L^2([0,T], H^1_0(\Omega)), d_\epsilon\rightharpoonup d \{\rm{weak^*\in}}\L^\infty([0,T], H^1(\Omega)),\end{cases}
u(z) =1-\alpha e^{\lambda_1z}-\beta ze^{\lambda_2z}+h.o.t., begin{align*}0.2cm] w(z) =0-\alpha \lambda_1 e^{\lambda_1z}-\beta \left[\lambda_2 z+f(1)/(2\lambda_2-c)\right]e^{\lambda_2z}+h.o.t., begin{align*}0.2cm] v(z) =0-\alpha \psi\left(\lambda_{1}\right) e^{\lambda_1z}-\beta e^{\lambda_2z}+h.o.t..
( f )_{\mu}^{\sim}({\bf l}) = \int_{0}^{2 \pi R}d \theta_{1}\cdots \int_{0}^{2 \pi R}d \theta_{d}\exp \bigl [ i R^{- 1}\sum_{k = 1}^{d}l_{k}\theta_{k}\bigr ] ( f )_{\mu}( \mathrm{\boldmath{~ \t h e t a ~}}) .
\Sigma_{a}( p ) = - i \frac{\lambda_{1}}{2}\int \frac{d^{3}k}{( 2 \pi )^{3}}\Delta ( k ) = \frac{\lambda_{1}}{2}\int^{\Lambda_{0}}\frac{d^{3}k}{( 2 \pi )^{3}}\frac{1}{k^{2}- m^{2}+ i \epsilon}\; .
w ( l , \mu ) \: = \; \int_{- \infty}^{+ \infty}\! \! d p \int_{\mu}^{\infty}\! \! d x \; \mathrm{e}^{\textstyle - l p^{2}- \sqrt{x}\, l}\; = \; \frac{2 \sqrt{\pi}}{l^{5 / 2}}\; \mathrm{e}^{\textstyle - \sqrt{\mu}\, l}\; \left[ 1 + \sqrt{\mu}\, l \right] ,
\frac{T_{m , m - 1}}{T_{m + 1 , m}}= \frac{T_{m , m}}{T_{m + 1 , m}}= \frac{T_{m + 1 , m}}{T_{m + 2 , m}}= \frac{T_{m + 1 , m}}{T_{m + 2 , m + 1}( 1 - T_{m + 1 , m})^{2}}.