harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\uppercase\expandafter{\romannumeral1}\leq \exp\left({-n\lambda(\int_{\Omega}\log X dP + \frac{\delta}{2})}\right)E\left(\exp(\lambda\sum_{k=1}^n\log X_k)\right) = \exp\left({-n\lambda(\int_{\Omega}\log X dP + \frac{\delta}{2})}\right)E\left(X_1^{\lambda}X_2^{\lambda}\cdots X_n^{\lambda}\right).
\mathcal{I}(\delta_{A}) = \inf_{\lambda \in \mathbb{R}_{+}^L}\left[ \langle \lambda, \delta_{A}\rangle + \sup_{\pi \in \Pi(\mu_{1, L+1},\dotsc, \mu_{L,L+1})}\int_{\mathcal{V}}f_{\lambda, A}\, d \pi \right].
\dim_{\kappa(a')}\pi_i (M\otimes_{A}\widetilde{A}\otimes_{\widetilde{A}}\kappa(a')) = \dim_{\kappa(a')}\pi_i \left(M\otimes_A \kappa(a) \otimes_{\kappa(a)}\kappa(a')\right) & = \dim_{\kappa(a')}\pi_i \left(M\otimes_A \kappa(a)\right) \otimes_{\kappa(a)}\kappa(a') &= \dim_{\kappa(a)}\pi_i \left(M\otimes_{A}\kappa(a)\right),
\frac{4\pi v^{1-s}}{2s-1}\bigg(\delta_{p,q}y_{p}^{s}+\alpha_{p,q}(s)y_{p}^{1-s}+\sum_{n\not = 0}\alpha_{p,q}(n,s)W_{s}(n\sigma_{p}^{-1}z)\bigg)=\frac{4\pi v^{1-s}}{2s-1}\mathcal{E}_{\mathrm{par},q}(z,s).
v_i^\prime=u_{n-i+1}-\left(\frac{t_{i-1}}{\varphi_1}+\sum_{\ell=2}^{i-1}\frac{t_{\ell-1}^2t_{i-1}}{\varphi_{\ell}\varphi_{\ell-1}}\right)u_n-\sum_{r=1}^{i-2}\frac{t_{r}t_{i-1}}{\varphi_{r+1}}u_{n-r}+\sum_{r=1}^{i-3}t_ru_{n-r}\sum_{\ell=r+2}^{i-1}\frac{t_{\ell-1}^2t_{i-1}}{\varphi_{\ell}\varphi_{\ell-1}}.
\mathfrak{h}_{0}\Big(\varphi-\sum_{1\leq i\leq g}d^+_{i,l-2}(D^2\varphi)h_{i,2}-\sum_{1\leq i\leq g}d^+_{i,l-1}(D\varphi)h_{i,1}-\sum_{1\leq s<\gamma}d^0_{s,l-1}(D\varphi)c_{s,1}\Big) & = \sum_{1\leq i\leq g}d^+_{i,l}(\varphi)h_i+\sum_{1\leq s<\gamma}d^0_{s,l}(\varphi)c_s+\sum_{1\leq j\leq g}d^-_{-j,l}(\varphi)h_{-j}.
\langle X, \nabla(\frac{F}{\varphi}) \rangle =\langle X,\nabla W \rangle+\frac{\beta-1}{\beta}\sum_{m}\langle X,e_{m}\rangle^{2}&=\frac{\beta-1}{\beta}\sum_{i}\lambda_{i}^{-2}(\nabla_{i}F)^{2}.
f_2 ([x,x], x \partial^{2i-1}) - 2f_2([x, x\partial^{2i-1}], x) = 2 [ f_1(x), f_2(x, x \partial^{2i-1}) ] - [ f_1(x \partial^{2i-1}), f_2(x, x) ] + d f_3(x, x, x \partial^{2i-1}).
F=\left(\begin{array}{cccccc}0 \tilde{\nu}_1 0 0 0 0 -\tilde{\nu}_1 0 0 0 0 0 0 0 0 \tilde{\nu}_2 0 0 0 0 -\tilde{\nu}_2 0 0 0 0 0 0 0 0 \tilde{\nu}_3 0 0 0 0 -\tilde{\nu}_3 0\end{array}\right)
\frac{1}{r\!-\!1}\rho^{\frac{r}{r-1}}\!+\!\left(\frac{rm}{n}\right)^{\frac{r}{r-1}}\! \! =\frac{1}{r\!-\!1}\Bigg[\rho^{\frac{r}{r-1}}+\underbrace{\left(\frac{rm}{n}\right)^{\frac{r}{r-1}}+\cdots+\left(\frac{rm}{n}\right)^{\frac{r}{r-1}}}_{r-1}\Bigg] & \geq\frac{r^2m}{(r-1)n}\rho^{\frac{1}{r-1}}.
\int_{M}\left|\nabla \left( \sqrt{G(p,y)}\phi(y)\right)\right|^2\,dy \leq \frac{1}{2}\int_{\mathcal{L}_{p}(\frac{\delta \varepsilon}{2}, 2\varepsilon)}\frac{|\nabla G(p,y)|^2}{G(p,y)}\,dy + 2 \int_M G(p,y)|\nabla \phi|^2 \,dy &= C(-\log\delta+1) + 2 \int_M G(p,y)|\nabla \phi|^2 \,dy
\hat{G}(y_0,...,y_{n+1})=\delta\Big(\frac{y_0}{y_{n+1}}+\hat{f}\Big(\frac{y_1}{y_{n+1}},...,\frac{y_n}{y_{n+1}}\Big)\Big)\hat{g}\Big(\frac{y_1}{y_{n+1}},...,\frac{y_n}{y_{n+1}}\Big)y_{n+1}^{-\frac{n+2}{2}}
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-n+1)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=2}^{n}(q_v^i-(-1)^i)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-n)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=2}^{n+1}(q_v^i-(-1)^i)\Bigr\}^{-1}=\frac{q_v^{n_{v,\nu_v}}-(-1)^{n_{v,\nu_v}}}{q_v^{n+1}-(-1)^{n+1}}.
\sup_{k \leq M-1}2^{-\frac{k}{p}}\bigg( \int_{x_k}^{\infty}u \bigg)^{\frac{1}{q}}V_r(0,x_k) \approx \sup_{k \leq M-1}2^{-\frac{k}{p}}\bigg( \int_{x_k}^{\infty}u \bigg)^{\frac{1}{q}}V_r(x_{k-1},x_k) = A_1.
\left\vert u^{-}\right\vert_{g,c,2^{\ast}}=\min_{v\in\mathcal{P}^{\Gamma}}\left\vert u-v\right\vert_{g,c,2^{\ast}}\leq C\min_{v\in\mathcal{P}^{\Gamma}}\left\Vert u-v\right\Vert_{g,a,A}=C\,_{A}(u,\mathcal{P}^{\Gamma})
\Sigma_{\mathrm{g f}}= - s \displaystyle{\int}d^{3}x e g^{\mu \nu}\partial_{\mu}\bar{c}_{a}A_{\nu}^{a}= - \displaystyle{\int}d^{3}x e g^{\mu \nu}\left( \partial_{\mu}b_{a}A_{\nu}^{a}+ \partial_{\mu}\bar{c}_{a}D_{\nu}c^{a}\right) ,
F=\sum_{0\leq T\in\Lambda_n}a(F,T)\,q^T =\sum_{t_i}\left(\sum_{t_{ij}}a(F,T)\prod_{i<j}q_{ij}^{2t_{ij}}\right) \prod_{i=1}^nq_i^{t_{ij}}\in\mathbb{C}[q_{ij}^{-1},q_{ij}][\![q_1,\ldots,q_n]\!].
\frac{1}{2}\frac{W_{0}}{R^{\beta}}= \frac{W_{0}/2}{(\sqrt{3}|Q|^{1/3})^{\beta}}= 3^{-\beta/2}\left(\frac{a_{\beta}}{\Lambda_{\beta}}\right)^{\beta-2}|Q|^{1-\beta/3}|Q|^{-1}\ge \frac{a_{\beta}}{|Q|}\frac{3^{-3/2}}{\Lambda_{\beta}}\left(\delta\Lambda_{\beta}\right)^{3-\beta},
N_{\pm}( X ; E_{p}, \pm \hat{\bf p}) = \int \frac{d^{\, 3}q}{2 | p_{0}|}\, e^{- i{\bf p}\cdot{\bf q}X_{0}/ p_{0}+ i{\bf q}\cdot{\bf X}}F_{\pm}( \pm{\bf p}-{\bf q}/ 2 ; \pm{\bf p}+{\bf q}/ 2 ) .
Z(s,f,\chi)=\frac{q^{-8-18s}(1-q^{-1})^2}{(1-q^{-3-6s})(1-q^{-5-12s})}+\frac{q^{-3-6s}(1-q^{-1})^2}{(1-q^{-3-6s})}+\frac{q^{-4-6s}(1-q^{-1})}{(1-q^{-3-6s})}+q^{-1}(1-q^{-1}).
\Gamma_{\varphi}= \frac{\varphi_0^3 + \|\nabla\varphi\|_\infty^2}{r^{5d}\varphi_0^2}\,, N_\varphi = \frac{\varphi_0^4}{r^{6d}}+ \frac{\|\nabla\varphi\|_\infty^8}{(r^{5d}\varphi_0^2)^4}\,,
w_h(t_n) - W_h^n= \frac{1}{2\pi\mathrm{i}}\int_{\Gamma_{\theta,\delta}\backslash\Gamma_\tau}e^{zt_n}K_1^s(z)A_hv_hdz &+\frac{1}{2\pi\mathrm{i}}\int_{\Gamma_\tau}e^{zt_n}(K^s_1(z)-e^{-z\tau}K^s_2(z))A_hv_hdz = I+II.
(q - 1)(q - 2)^3 (pq - 1) + (p - 1)(p-2)^3q(pq - 1) > (q - 1)^2(q - 2)^3 + (p - 1)^2(p - 2)^3q^2 + q(q - 1)(p - 1)(p-2)(q-2)(p+q-4) = \Big{(}(q-1)(q-2)+q(p-1)(p-2)\Big{)}\Big{(}(q-1)(q-2)^{2}+q(p-1)(p-2)^{2}\Big{)}
\phi^1=du_{xy}-a_1dx-a_2dy,\phi^2=du_{yy}-a_2dx-a_1u_{yy}^4dy, \phi^3=dv_{xy}-a_5dx-a_6dy, \phi^4:=dv_{yy}-a_6dx-(4a_1v_{yy}u_{yy}^3-a_5u_{yy}^4)dy.
S^{m}=\left(\begin{array}[c]{ccccc}\left( 2j\right)^{m}0 \cdots 0 0 0 \left( 2j-2\right)^{m}\cdots 0 0 \vdots \vdots \ddots \vdots \vdots 0 0 \cdots \left( -2j+2\right)^{m}0 0 0 \cdots 0 \left( -2j\right)^{m}\end{array}\right) \ .
q_F=\sum_{C \!\hbox{\tiny{child of}}\! F}\Big(\frac{\mu(C)}{\mu(F)}-\frac{\nu(C)}{\nu(F)}\Big)_+=\sum_{C \!\hbox{\tiny{child of}}\! F}\Big(\frac{\nu(C)}{\nu(F)}-\frac{\mu(C)}{\mu(F)}\Big)_+.
G_{1 2 , 2 3}^{\delta}( p_{3 0}^{\prime}, p_{1 0}) \, = \,{\frac{- 2 i \pi}{( P_{0}- S ) - ( p_{3 0}^{\prime}- h_{3}) + i \epsilon}}\, \beta_{2}\beta_{3}\, \delta ( p_{1 0}- h_{1})
\begin{cases}0 \textrm{if}\deg g - \deg h \neq d_2-d_1 q^{- \frac{d_1+d_2}{2}}A_{d_2-\deg g}\textrm{if}\deg g - \deg h = d_2-d_1\textrm{and}d_1 + d_2 \leq N q^{\frac{d_1+d_2}{2}-N}A_{N-d_1-\deg g}\textrm{if}\deg g - \deg h = d_2-d_1\textrm{and}d_1 + d_2 > N \end{cases}
= \ - \sum_{i = 1}^{N}\left[ d_{i}^{2}+ \nu \sum_{j \neq i}\frac 2{x_{i}- x_{j}}d_{i}- \nu \sum_{j \neq i}\frac 1{( x_{i}- x_{j})^{2}}( 1 - K_{i j}) \right] + \sum_{i = 1}^{N}x_{i}^{2}\ =
E_{i}E_{j}- E_{j}E_{i}= 0, \a_{ij}= 0, E_{i}^{2}E_{j}- (q + q^{-1}) E_{i}E_{j}E_{i}+ E_{j}E_{i}^{2}= 0, \a_{ij}= - 1, F_{i}F_{j}- F_{j}F_{i}= 0, \a_{ij}= 0, F_{i}^{2}F_{j}- (q + q^{-1}) F_{i}F_{j}F_{i}+ F_{j}F_{i}^{2}= 0, \a_{ij}= - 1.
s_{z_o, 0}= s^+_0 = \nu_{z_o}^*L_0^+, \,\, s_{z_0, i}= s^+_{1,i}= \nu_{z_o}^*L^+_{1,i},\, (i =1, 2, 3) \,\, s_{z_o,4}= s^+_{2,1}= \nu_{z_o}^*L_{2,1}^+.
\hat{\pi}_{h, n}(x) := \frac{1}{n \Delta_n}\frac{1}{\prod_{l = 1}^d h_l}\sum_{i = 0}^{n-1}\prod_{l = 1}^d K(\frac{x_l - X_{t_i}^l}{h_l}) (t_{i + 1}- t_i) &= \frac{1}{n}\sum_{i = 0}^{n-1}\mathbb{K}_h(x - X_{t_i}),
q_H(z) = \lim_{\varepsilon\to0}q_{H_\varepsilon}(z)= i\frac{2\kappa}{\kappa_2}\cdot\frac{|\Gamma(1)|^2}{(\Gamma(1))^2}\cdot\frac{1}{2}\Bigl(1-i\frac{\kappa_3}{\kappa}\Bigr)\Bigl(\frac{z}{i}\Bigr)^0= i\frac{\kappa-i\kappa_3}{\kappa_2}\,.
\kappa_0=\frac{1}{27},(\kappa_1,\kappa_2,\kappa_3,\ldots)=\frac{1}{54}\left(\frac{2^3}{1},\frac{4^3}{3},\frac{5^3}{3},\frac{7^3}{5},\frac{8^3}{5},\frac{10^3}{7},\frac{11^3}{7},\frac{13^3}{9},\frac{14^3}{9},\frac{16^3}{11},\frac{17^3}{11},\ldots\right).
v_{A(1)}= -\frac{1}{2}E^{\gamma_{A(1)}}\Omega_\lambda-\frac{1}{2}E^{\gamma_{A(2)}}\Omega_\lambda-\frac{1}{12}E^{\gamma_{A(3)}}\Omega_\lambda v_{A(2)}= -\frac{1}{3}E^{\gamma_{A(2)}}\Omega_\lambda-\frac{1}{12}E^{\gamma_{A(3)}}\Omega_\lambda v_{A(3)}= -\frac{1}{12}E^{\gamma_{A(3)}}\Omega_\lambda.
K_n(x,y) = \frac{1}{(2\pi i)^2}\int_C ds \int_{\Sigma_n}dt \prod_{j=0}^m \frac{\Gamma(s+1+\nu_j) \Gamma(t+1+\ell_j-n)}{\Gamma(t+1+\nu_j) \Gamma(s+1+\ell_j-n)}\frac{x^t y^{-s-1}}{s-t}.
B(a_1,a_2,\cdots,a_{n+1}) :=\int_{\Delta^n}t_1^{a_1-1}\cdots t_{n+1}^{a_{n+1}-1}dt_1\wedge\cdots\wedge dt_{n}&=\frac{\Gamma(a_1)\cdots\Gamma(a_{n+1})}{\Gamma(a_1+\cdots+a_{n+1})}
\left \vert \left\langle \left (M^{old}-\hat M\right )_{\bar \Omega},\hat M-M_0\right\rangle\right \vert \leq \left \Vert \left (M^{old}-\hat M\right )_{\bar \Omega}\right \Vert \left \Vert\hat M-M_0\right \Vert_{*}&\hskip 0.5 cm\leq \lambda/3\,\left \Vert \hat M-M_0\right \Vert_{*}.
\alpha_{2}^{( 1 )}- \alpha_{2}^{( 2 )}\tau \rho + i \alpha_{1}^{( 2 )}\rho - i \alpha_{1}^{( 1 )}= 0 , \ \beta_{2}^{( 1 )}- \beta_{2}^{( 2 )}\tau \rho + i \beta_{1}^{( 2 )}\rho - i \beta_{1}^{( 1 )}= 0 .
\sum_{m=0}^{p_{2}-1}\sum_{j=0}^{p_{1}}\binom{p_{1}}{j}m^{p_{1}-j}S\left(j,l-m\right) S\left( p_{2},m\right) =\sum_{k=1}^{p_{2}-1}\left( -1\right)^{p_{2}+k+1}s\left( p_{2},k\right) S\left( p_{1}+k,l\right) .
\frac{\partial\Psi}{\partial\lambda}=\left(\begin{array}{cc}\frac{x}{2}+\frac{v+\frac{\Theta}{2}}{\lambda}-\frac{v+\frac{\Theta}{2}}{\lambda-1}~~-\frac{u(v+\Theta)}{\lambda}+\frac{yuv}{\lambda-1}\frac{v}{u\lambda}-\frac{v+\Theta}{uy(\lambda-1)}~~-\frac{x}{2}-\frac{v+\frac{\Theta}{2}}{\lambda}+\frac{v+\frac{\Theta}{2}}{\lambda-1}\end{array}\right)\Psi
M_{n}=\frac1{(\sqrt{2})^{n}}M_n^{(\sqrt{2},2)}=\frac1{(\sqrt{2})^{n}}M_n^{(3+\sqrt{2}-3,2)}=\frac1{(\sqrt{2})^{n}}\sum_{k=0}^n\binom{n}{k}M_k^{(3,2)}(\sqrt{2}-3)^{n-k}=\frac1{(\sqrt{2})^{n+2}}\sum_{k=0}^n(\sqrt{2}-3)^{n-k}\binom{n}{k}S_{k+1}.
\int_{0}^{2 \pi}\frac{d ( e^{\pm i N \chi}e^{\pm i N^{\prime}( \cos^{- 1}( p ) \pm \cos^{- 1}( q ) )})}{\pm i N \mp i N^{\prime}( \frac{1}{\sqrt{1 - p^{2}}}\frac{d p}{d \chi}\pm \frac{1}{\sqrt{1 - q^{2}}}\frac{d q}{d \chi})}F ( N , N^{\prime}, \chi )
\Pi_{\epsilon}^{( \pm 1 )}= \frac 1{2 m_{2}}\sigma_{k}\left( \sigma_{k}\pm 1 \right) \left( m_{2}- B \right) \Lambda_{\epsilon}= \Psi_{\epsilon}^{( \pm )}\cdot \overline{{{\Psi}}}_{\epsilon}^{( \pm )}.
\dot{\Pi}^{\varepsilon}_{\vec{\mathbb{P}}}(f_{1},f_{2})(x)=\int_{0}^{1}\int_{0}^{1}\sum_{\vec{P}\in\vec{\mathbb{P}}}\frac{C^{\varepsilon}_{Q_{\vec{P}}}}{|I_{\vec{P}}|^{\frac{1}{2}}}\langle f_{1},\Phi^{1,\nu}_{\vec{P}_{1}}\rangle\langle f_{2},\Phi^{2,\nu}_{\vec{P}_{2}}\rangle\Phi^{3,\nu}_{\vec{P}_{3}}(x)d\nu.
\begin{aligned}JO_{n+1}^{(3)}+JO_{n}^{(3)}+2JO_{n-1}^{(3)}=\sum_{s=0}^{7}(J_{n+s+1}^{(3)}+J_{n+s}^{(3)}+2J_{n+s-1}^{(3)})e_{s}&=J_{n+2}^{(3)}+\sum_{s=1}^{7}J_{n+s+2}^{(3)}e_{s}&=JO_{n+2}^{(3)}\end{aligned}
\Phi_{[p]}(([p],[q,v])+([p],[q',v'])) =\lfloor p,gv+g'v'\rfloor, \Phi_{[p]}([p],[q,v])+\Phi_{[p]}([p],[q',v'])=\lfloor p,gv\rfloor+\lfloor p,g'v'\rfloor =\lfloor p,gv+g'v'\rfloor.
\big|u_2^{s,N}(C_{0,k}) - u_2^{s,N}(B_{0,k})\big|^2 = \sin^2(\frac{2ks\pi}{N}) \big|u_1^{s,N}(C_{0,0}) - u_1^{s,N}(B_{0,0})\big|^2,
\begin{aligned}H=H(x,D_x)=H(x,D)=H_0+V, &H_0=H_0(D_x)=H_0(D),\V=V(x),\ \ (D_x=D=-i\partial_x), &H(x,\xi)=H_0(\xi)+V(x),\ \H_0(\xi)=\sum_{i=1}^n\frac{1}{2\mu_i}|\xi_i|^2, \ \V(x)=\sum_{1\le i<j\le N}V_{ij}(x_{ij}).\end{aligned}
[\tilde{\cal R}_\mu,\tilde{\cal R}_\nu]=-i\epsilon_{\mu\nu\lambda}\tilde{\cal R}{}^\lambda,\quad[\tilde{\cal R}_\mu,\tilde{\cal R}_\nu]_{{}_+}=-\frac{1}{2}\Pi_-\eta_{\mu\nu},\quad[\tilde{\cal R}_\mu,\Pi_-]=0,
\psi_1((b^{\mathbf n})^d) = ((b^{\mathbf n})^{d|_0}, (a^{\mathbf n \cdot \mathbf{e}_{{1}}})^{d|_0}, \dots, (a^{\mathbf n \cdot \mathbf{e}_{{p-1}}})^{d|_0})^{d_0}&= ((b^{\mathbf n})^{d|_0}, a^{j_1 \cdot \mathbf n \cdot \mathbf{e}_{1.d_0}}, \dots, a^{j_1 \cdot \mathbf n \cdot \mathbf{e}_{(p-1).d_0}}).
\sum_{p \leq X}\frac{|1-f(p)p^{-it_1}g_{\mathcal{J}}(p)|}{p}= \sum_{\substack{p \leq X p \in \cup_{j \in \mathcal{J}}(P_j, Q_j]}}\frac{1}{p}+ \sum_{\substack{p \leq X p \not \in \cup_{j \in \mathcal{J}}(P_j, Q_j]}}\frac{|1-f(p)p^{-it_1}|}{p}
\tilde{V}(t) = \int_0^T \left\{\frac{\ 1 \}{t}\int_0^t f^\prime (X(s)) \, Z(s,u) \, A_1 (u, X_u) \, \mathbb{I}_{(u \le s)}\, \mbox{d}s \right\}^2 \mbox{d}u, & \tilde{V}_N (t_1, \dots, t_N) = \int_0^T \left\{\frac{\ 1 \}{N}\sum_{k=1}^N f^\prime (X(t_k)) \, Z(t_k, u) \, A_1 (u, X_u) \, \mathbb{I}_{(u \le t_k)}\right\}^2 \mbox{d}u.
\tilde{S}_{\mathrm{I F}}[ \phi , \phi^{\prime}; \xi_{1}, \xi_{2}, \xi_{3}] = \mathrm{R e}\left( S_{\mathrm{I F}}[ \phi , \phi^{\prime}] \right) + \sum_{N = 1}^{3}{\frac{1}{N}}\int_{t_{i}}^{t}d^{4}x R_{N}^{-}( x ) \xi_{N}( x ) .
\begin{aligned}\Vert \langle y\rangle^{l+k}D^{\tilde{\alpha}}u\cdot D^{\alpha-\tilde{\alpha}}\partial_x u\Vert_{L^2_y}=\Vert\langle y\rangle^{l+k}\partial_x^{\tilde{\beta}-1}(\partial_x u)\cdot D^{\alpha-\tilde{\alpha}}(\partial_x u)\Vert_{L^2_y}&\leq C\Vert u\Vert_{H^{m}_l}^2 &\lesssim E_u^2,\end{aligned}
\begin{gathered}\sum_{M=0}^\infty M^{\#S-1-k}e^{-M}\sum_{N=0}^{\lfloor w\rfloor+M}N^k \leq \sum_{M=0}^\infty M^{\#S-1-k}e^{-M}(w +M)^{k+1}\lesssim \sum_{M=0}^\infty M^{\#S}e^{-M}\end{gathered}
A_1 = \sum_{j = 0}^{\infty}2^{-2j}(|(-\Delta_x)^{1/6}u_0|^p)^{1/p}_{Q_{\nu r, 2^j \nu r}(z_0)}, A_2 = \lambda^{1/2}(|u_0|^p)^{1/p}_{Q_{\nu r}(z_0)}+ (|D_v u_0|^p)^{1/p}_{Q_{\nu r}(z_0)}.
d s^{2}= - \left( r^{2}/ l^{2}- \frac{\mu}{r^{d - 3}}- 1 \right) d t^{2}+ \frac{d r^{2}}{\left( r^{2}/ l^{2}- \frac{\mu}{r^{d - 3}}- 1 \right)}+ r^{2}d \sigma_{- 1}\; ,
{J}\pi(a){J}^{-1}= p_+{J}\pi_0(f){J}^{-1}p_+ + p_-{J}\pi_0(g){J}^{-1}p_- &= p_+ \pi_0(\bar f) + p_-\pi_0(\bar g) = \pi(\bar f, \bar g) = \pi(a^*).
\phi_{m}( z , \bar{z}) = \frac{\bar{z}^{m}}{\sqrt{\pi \Gamma ( m + 1 )}}\left( \frac{e B}{2}\right)^{( m + 1 ) / 2}\exp \left( - \frac{e B}{4}z \bar{z}\right) \left( \begin{matrix}{0}\\{1}\\ \end{matrix}\right) ,
\tilde{T}_{a}\Psi^{( 0 )}= \left( \begin{array}{c}{{e^{i \chi ( a ; s^{k})}\phi^{( 0 )}( s^{i}+ a^{i})}}\\{{e^{- i \chi ( a ; s^{k})}\phi^{\ast ( 0 )}( s^{i}+ a^{i})}}\\{{A_{k}^{( 0 )}( s^{i}+ a^{i}) - \frac{1}{q}\partial_{k}\chi ( a ; s^{i})}}\\ \end{array}\right) ,
I_{\tilde{a}}[tu] \leq C_1' t^p\|u\|^p-C_2' t^{2\cdot p^\sharp}\|u\|^{2\cdot p^\sharp}-C_3' t^{2\cdot p^\flat}\|u\|^{2\cdot p^\flat}\phantom{=}-C_4' t^{p^\flat + p^\sharp}\|u\|^{p^\flat+p^\sharp}-C_5' t^{p_g}\|u\|^{p_g}.
w(\boldsymbol{\varphi_{x}}) =\sum_{i=1}^{k}\sum_{\substack{(y,z)\in S_{i}x\not\neq y,z}}\frac{1+b_{k}-b_{i}}{4}w(\boldsymbol{\varphi_{\{x,y\}}}) =2\cdot\frac{1+b_{k}-b_{i}}{4}i(x,y)\in S_{i}
A(x) = 2\left( \frac{\rho-(1-\rho)c_N}2 \right)\sqrt{\left( \frac{\rho-(1-\rho)c_N}2 \right)^2x^2+(1-\rho)x}B(x) = 1-\rho + 2 \left( \frac{\rho-(1-\rho)c_N}2 \right)^2x.
S_{PV}(\Phi,\Phi^*,\underline \Phi_i)=\frac{1}{2}\sum_{i=1}^{N}{\underline\Phi}^A_i S_{AB}{\underline \Phi}^B_i - \frac{1}{2}M^2_i{\underline\Phi}^A_i T_{AB}{\underline \Phi}^B_i,
\mathrm{~ \n u_{r s}\equiv \n u ~ i s ~ i n d e p e n d e n t ~ o f ~ r , s ~ , ~ w h e r e}~ ~ \nu_{r s}\equiv{\frac{{\widetilde b}_{r}-{\widetilde b}_{s}}{b_{r}^{*}- b_{s}^{*}}}~ ~ \mathrm{f o r ~ r \not = s ~ .}
|{u}_{\iota}x_0-{u}_0x_0| \geqslant \sum_{l=0}^{\iota-1}|{u}_lx_0-{u}_{l+1}x_0|- 48\iota \delta & \geqslant \iota \; |{u}_jx_0-{u}_{j+1}x_0| -48 \iota \delta.
\begin{cases}x_j^{k+1}= \mathrm{prox}_{\frac{\alpha_j}{\beta}f_j}(x_j^k-\alpha_j A_j^\top(\sum_{i=1}^{j-1}A_i(x_i^{k+1}+\theta(x_i^{k+1}-x_i^k))+A_jx_j^k +\sum_{i=j+1}^s A_i((2+\theta)x_i^k-\theta x_i^{k-1})-b)-\frac{\alpha_j}{\beta}A_j^\top y^k), ~~ j\in\mathbb{N}_s, y^{k+1}= y^k+\beta(\sum_{i=1}^s A_ix_i^{k+1}-b). \end{cases}
\omega ( \partial_u K,\partial_{\theta_n^0}K) = \omega ( \partial_u K,\partial_{\varphi_i}K) =\omega ( \partial_{\theta_n^0}K,\partial_{\varphi_i}K) = \omega ( \partial_{\varphi_j}K,\partial_{\varphi_i}K) = 0,
\int_{N_d(K)}H(x, y, t)d\mathrm{vol}(y)\lesssim\sum_{\rho=1}^\infty (1+\rho^n)e^{-\frac{\rho^2}{4t}-\frac{(n-1)^2}{4}t-\frac{n-1}{2}\rho}\mathrm{vol}(\mathrm{An}(\rho)\cap N_d(K)),
f^{(k)}(x)=\sum_{\left\{m_i\right\}}\frac{k!}{m_1!m_2!\cdots m_k!}g^{(m_1+m_2+\cdots m_k)}(h(x))\prod_{j=1}^k\left[\frac{h^{(j)}(x)}{j!}\right]^{m_j}
m_{u,p}= \bigg\{\lambda=\{\lambda_k\}_{k\in \mathbb{Z}^d}\subset \mathbb{C}: \|\lambda| m_{u,p}\| = \sup_{j\in\mathbb{N}_0; m\in \mathbb{Z}^d}|Q_{-j,m}|^{\frac{1}{u}-\frac{1}{p}}\Big(\sum_{k:\, Q_{0,k}\subset Q_{-j,m}}\!\!\!|\lambda_k|^p\Big)^\frac{1}{p}< \infty \bigg\},
\| u_{x_1, r}- p_{x_1}\|_{L^1(\partial B_1)}+ \| u_{x_2, r}- p_{x_2}\|_{L^1(\partial B_1)}\leq C \, (-\log(r))^{-\frac{1-\gamma}{2\gamma}}= C \big(-\log |x_1-x_2|\big)^{-\frac{1-\gamma}{2\gamma}}
\frac{d b(p)}{d p}\bigg{|}_{p=5}=\frac{\|\mathcal{Q}\|_{L^2}^2}{\|\mathcal{Q}\|_{L^1}^2}>0, \frac{\partial\gamma}{\partial b}\bigg{|}_{b=b(p)}=-\frac{\|\mathcal{Q}_p\|_{L^1}^2}{8\|\mathcal{Q}_p\|_{L^2}^2}+O(|p-5|)<0, \frac{1}{2}\int|v_y(b(p),p,y))|^2dy-\frac{1}{p+1}\int|v(b(p),p,y)|^{p+1}dy=0.
\left(\int_{D}|u(x)|^2 d\mu \right)^{1/2}= \left(\sum_{x\in D}\mu (x) |u(x)|^2 \right)^{1/2}\leq \frac{1}{\sqrt{\lambda_1}}\left( \int_{D}|\nabla u|^2(x) d\mu \right)^{1/2},
\rho_{\frac{s}{2}}\left( x \right) = \sinh \frac{i}{2}\frac{d}{d x}\rho \left( x \right) \qquad \qquad \rho_{\frac{c}{2}}\left( x \right) = \cosh \frac{i}{2}\frac{d}{d x}\rho \left( x \right)
\Delta^2 S_2 = \Delta^2 \sum\limits_{j=1}^{\frac{t_A}{\Delta}- 1}\left( \frac{t_A}{\Delta}+ \frac{t_B}{\Delta}- 1 - j\right) = \left( t_A - \Delta \right) \left( t_A + t_B - \Delta \right) - \Delta^2 \sum\limits_{j=1}^{\frac{t_A}{\Delta}- 1}j &= \left( t_A - \Delta \right) \left( t_A + t_B - \Delta \right) - \frac{1}{2}t_A\left( t_A - \Delta \right).
\rho(x,y,z,w,\lambda)\begin{bmatrix}\frac{\partial F_1}{\partial \lambda}\frac{\partial F_2}{\partial \lambda}\end{bmatrix}=X.\begin{bmatrix}F_1 F_2\end{bmatrix}+\begin{bmatrix}A_{11}A_{12}A_{21}A_{22}\end{bmatrix}\begin{bmatrix}F_1 F_2\end{bmatrix}
\big{\langle}R\big{(}\alpha_{v_q}\big{)}, W_{\mathsf{v}^*(\alpha_{v_q})}\big{\rangle}= - \big{\langle}\alpha_{v_q}, \bar W_{v_q}\big{\rangle}+ \big{\langle}W_{\mathsf{v}^*(\alpha_{v_q})}, \bar W_{v_q}\big{\rangle}^{T},
\int_X \alpha(\pi(x,y))\> \overline{\beta(\pi(z,y))}\> d\omega_X(y)=\int_D \alpha(h)\> \overline{\beta(\pi(z,x)*h)}\>d\omega_D(h) &=\int_D \alpha(h)\>\overline{\beta(h)}\>d\omega_D(h)\cdot \overline{\beta(\pi(z,x))}.
R_{(c,d)}(n) = 12\sigma(\frac{n}{c}) - 36\sigma(\frac{n}{3c}) + 12\sigma(\frac{n}{d}) -36\sigma(\frac{n}{3d}) + 144\, W_{(c,d)}(n) + 1296\, W_{(c,d)}(\frac{n}{3}) - 432\, \biggl( W_{(3c,d)}(n) + W_{(c,3d)}(n) \biggr).
\Theta_{l_1,l_2}(t_4(\theta_1,\theta_2))=\frac{-e^{il_1\theta_1+il_2\theta_2}+e^{il_2\theta_1+il_1\theta_2}}{(e^{i\theta_1}-e^{-i\theta_1})(e^{i\theta_2}-e^{-i\theta_2})( 1 - e^{i\theta_1+i\theta_2})(e^{-i\theta_1}-e^{-i\theta_2})}.
\sigma_{r , 3}= \sigma_{r , 1}^{*}; \, \, \sigma_{r , 4 s + 3}= \sigma_{r , 4 s + 1}^{*}; \quad | \sigma_{r , 4 s + 1}| > | \sigma_{r , 1}| \quad ( s = 1 \ldots \frac{r - 1}{2}) \, .
N_{q}\left( D_{q}^{n}f\left( t\right) \right) \left( u;v\right) =\frac{v^{n}}{u^{n}}\left( N_{q}\left( f\right) \right) \left( u,v\right)-\sum_{i=0}^{n-1}\left( \frac{u}{v}\right)^{n-1-i}D_{q}^{i}f\left( 0\right).
f_{ug}(\tau_{uh}) =f_{ug}(\theta_\tau\delta_{uh})=\zeta_{ug}(\theta_f\theta_\tau\delta_{uh})=\zeta_{ug}\left(\sum_{v\in G}\eta(v)\rho_v \delta_{uh}\right) &=\sum_{v\in G}\eta(v)\zeta_{ug}\rho_v \delta_{uh}=\sum_{v\in G}\eta(v)\zeta_{ug}\delta_{uhv^{-1}}=\eta(g^{-1}h)=f_g(\tau_h), \forall u,g,h \in G.
Q^{\boldsymbol{f}}(\boldsymbol{x}_0,\boldsymbol{x},\boldsymbol{y})=P_{H^{\boldsymbol{f}}(\boldsymbol{x}_0,\boldsymbol{x})\cap H^{\boldsymbol{f}}(\boldsymbol{x},\boldsymbol{y})}^{\boldsymbol{f}}\boldsymbol{x}_0.
\sum_{j=1}^\infty \left(2\kappa_j-k_n -\frac{1}{\ell}\right)Y_{j,0}Y_{j,n}=2\sum_{j=1}^\infty \kappa_j Y_{j,0}Y_{j,n}-\left(k_n +\frac{1}{\ell}\right)\sum_{j=1}^\infty Y_{j,0}Y_{j,n}=2\sum_{j=1}^\infty \kappa_j Y_{j,0}Y_{j,n}.
H_{min}\left( (E_1[t]\times E_1[t]^{\vee})\otimes F \right) =H_{min}\left( E_1[t] \otimes F \times E_1[t]^{\vee}\otimes F \right) &=\min \left( H_{min}(E_1[t] \otimes F), H_{min}(E_1[t]^{\vee}\otimes F)\right) & \leq H_{min}(E_1[t] \otimes F)= tH_{min}(E_1\otimes F) \leq t H_{min}(E_1)H_{min}(F)
\lambda_1 =q\sqrt{q}(z_1+z_2+z_3+z_4) \lambda_2 =q^2(z_1z_2+z_1z_3+z_1z_4+z_2z_3+z_2z_4+z_3z_4) \lambda_3 =q\sqrt{q}\left(\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}+\frac{1}{z_4}\right)
L_3^{-1}=(1-T) (1+T) (9T^4 + 6T^3 + 4T^2 + 2T + 1), L_7^{-1}=(1 + T^3) (7T^2 + 3T + 1), L_{31}^{-1}=(1 - T) (31^3T^6 + 1581T^4 + 36T^3 + 51T^2 + 1), L_{53}^{-1}=(1+T) (53^3T^6 + 8427T^5 +1537T^4 + 670T^3 + 29T^2 + 3T + 1),
G(E_{r};\vec{x}_{b},\vec{x}_{a}) = i\int_{0}^{\infty}d\tau_{f}\int{\cal D}\vec{p}{\cal D}\vec{x}exp\{i\int_{0}^{\tau_{f}}d\tau [ \vec{p}\dot{\vec{x}}- f(\vec{x}(\tau))H ]\}
B(1,Y) = 1/2 + \sum_{j=d}^{N-d}b_j Y^j + 1/2 Y^{N}= (1-6Y + Y^2)(1+Y)^{N-2}\cdot \sum_{i=0}^{\lfloor \frac{N-2}{4}\rfloor}e_i (Y(1-Y)^2(1+Y)^{-4})^i.
[A,K] = \sum_{i=1}^{L+1}\alpha_i\{A^i,C\}+ 2\epsilon C + \sum_{i=1}^{L+1}\frac12 m_i\{A^i,C\}+\beta\eta C + \sum_{i=1}^{L}\beta\omega_{i}\{A^{i},C\}+\ell_1 C.
\partial_{x_1}P_t(x,y)= \frac1{\pi^{(n+1)/2}}\int_0^\infty \frac{t\,e^{-\frac{t^2}{4s}}}{s^{3/2}}\,\frac{e^{-s}}{\sqrt{1-e^{-2s}}}\,\frac{y_1-e^{-s}x_1}{\sqrt{1-e^{-2s}}}\frac{e^{-\frac{|y-e^{-s}x|^2}{1-e^{-2s}}}}{(1-e^{-2s})^{n/2}}\,ds.
[\widetilde J_{(\omega,x,y)}\psi](n) = \overline{\widetilde p( \widetilde T^{n-1}(\omega,x,y))}\psi(n-1) + \widetilde q(\widetilde T^n(\omega,x,y))\psi(n) \qquad\qquad + \widetilde p(\widetilde T^n(\omega,x,y))\psi(n+1).
F_\alpha(\lambda m)(d)=(\lambda m)(\psi_{t_\alpha}(d))=m(\lambda^\#\psi_{t_\alpha}(d))= =m(\psi_{t_\alpha}(\varphi_{t_\alpha}(\lambda^\#)d))=F_\alpha(m)(\varphi_{t_\alpha}(\lambda)^\#d)=(\varphi_{t_\alpha}(\lambda)F_\alpha(m))(d)\ .
\left< \bar v\cdot\nu_{\partial G}, \phi \right>_{W^{1-1/p',p'}(\partial G)}= \sum_{i=1}^2 \left[ \left< \bar v\cdot\nu_{\partial G}, \phi \right>_{W^{1-1/p',p'}(\partial G \cap\{x_i ={0}\})}+ \left< \bar v\cdot\nu_{\partial G}, \phi \right>_{W^{1-1/p',p'}(\partial G \cap\{x_i ={1}\})}\right]
\left(T^{\lambda,r_1}\circ(T^{\lambda,r_1})^*(F)\right)(r_2,\theta_2) = \int_{-\infty}^\infty \left(T^{\lambda,r_1}_{r_2}\circ(T^{\lambda,r_1}_{\tilde{r}_2})^*(F(\tilde{r}_2,\cdot))\right)(\theta_2) \tilde{r}_2d\tilde{r}_2.
\widetilde{p_D(t-s,x,y)h_n(., s,y)}= \frac{1}{n+1}\Big[ p_D(t-s,x,y)h_n(s_1,x_1, s_2,x_2,\cdots, s_n,x_n,s,y) + \sum_{i=1}^n p_D(t-s_j,x,y_j)h_n(s_1,x_1, s_2,x_2,\cdots,s_{j-1},x_{j-1}, s,y, s_{j+1},x_{j+1}, \cdots, s_n,x_n, s_j,x_j) \Big].
\big\lVert \omega \big\rVert_{Z_R}^2 =\big\lVert \omega \big\rVert_{Z_{1,0}\cup Z_{2,0}}^2 +\big\lVert \omega^\mathrm{zm}\big\rVert_{Y_{[-R,R]}}^2 +\big\lVert \omega^\mathrm{nz}\big\rVert_{Y_{[-R,R]}}^2.
\nu(P_n(z))=\sum\nu(z-z_i)-\nu([\tfrac{n-1}2]!)+\begin{cases}\nu([\tfrac{n+1}2]) n\equiv0,3\ (4)n+z \min(15,2\nu(z-148)) n=21 \min(9,2\nu(z-19)) n=29 \min(8,2\nu(z-11)) n=45 \min(10,2\nu(z-3) n=61 0 \end{cases}
\psi_{\vec{\mu}}^{a}( \theta + 2 \pi ) = e x p ( i \vec{a}. \vec{\mu}) \psi_{\vec{\mu}}^{a}( \theta ) \hspace{. 6 i n}\zeta_{\vec{\mu}}^{a}( \theta + 2 \pi ) = e x p ( i \vec{a}. \vec{\mu}) \zeta_{\vec{\mu}}^{a}( \theta )
{\bf E}^{\tau p}={\frac{1}{2}}\left( \begin{array}{c c}{{{a_{1}^{\tau p}}^{\dagger}}}&{{{a_{2}^{\tau p}}^{\dagger}}}\\ \end{array}\right){\bf \sigma}\left( \begin{array}{c}{{a_{1}^{\tau p}}}\\{{a_{2}^{\tau p}}}\\ \end{array}\right)
\begin{pmatrix}0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \end{pmatrix},\begin{pmatrix}\epsilon^2 -\epsilon^5 \epsilon^6 -\epsilon^4 0 -\epsilon \epsilon^4 -\epsilon^5 \epsilon^3 0 \epsilon^{-2}-\epsilon \epsilon^2 -1 0 -1 \epsilon^3 -\epsilon^4 \epsilon^2 0 -1 \epsilon^3 -\epsilon^4 \epsilon^2 0 \end{pmatrix}.
\bar{\bf N}=-\sqrt{1-\bar U^2}\frac{~\partial}{\partial U}-\frac{\bar U}{\sqrt{1-\bar U^2}}\left( T\frac{~\partial}{\partial T}+V\frac{~\partial}{\partial V}+X\frac{~\partial}{\partial X}+Y\frac{~\partial}{\partial Y}+Z\frac{~\partial}{\partial Z}\right)
P_h = x_h A \prod_{i=1}^{h}F_{k,i}(A)^{-1}\prod_{i=1}^{h-1}G_{k,i}(A), P_{k+1-h}= x_{k+1-h}z \prod_{i=1}^{h}F_{k,i}(A) \prod_{i=1}^h G_{k,i}(A)^{-1}.
S^{(1)}\big(L_i(x)\big) = \frac{\alpha_i}{a_i}S^{(1)}(x) + \frac{T_i \theta^4 + S_i \theta^3 (1-\theta) + U_i \theta^2 (1-\theta)^2 + V_i \theta (1-\theta)^3 + W_i (1-\theta)^4}{[ 1 + (r_i -3 ) \theta (1-\theta)]^2},
\gamma_{2 a}\equiv - \left( \left( \mathrm{D}_{i}\right)_{a}^{\; \; b}\pi_{b}^{i}- \mathrm{i g}\bar{\psi}_{\alpha}^{\; B}\left( \gamma^{0}\right)_{\; \; \beta}^{\alpha}T_{a \; \; B}^{A}\psi_{\; A}^{\beta}\right) \approx 0 ,