harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mu_{\mathbf{\Lambda}}\left(-r\right)=L\sum_{j=1}^{r}\sum_{i=1}^{m}\mathcal{D}\left(i,j\right)\frac{\left(-1\right)^{r-j}}{\left(r-j\right)!}\mathbf{b}_i^t\mathbf{\Psi}^{-1}\mathbf{D}_i\mathbf{a}_j .
\oint_{\partial M}\epsilon K|h|^{\frac{1}{2}}d^{3}y=\int_{t_{1}}^{t_{2}}dt\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}d\theta(\frac{2}{r_{0}})(r_{0}^{2}sin\theta)=8\pi r_{0}(t_{2}-t_{1})
B^s_{p,q}(\mathbb{R}^d) = \left\{f\in \mathcal{S}'(\mathbb{R}^d) \,\,|\,\, f \stackrel{\mathcal{S}'}{=}\sum_{j = 0}^{\infty}a_j(x);\,\, \mathrm{supp}F a_j \subset M_j;\right. &\left. \| \{a_j \}\|_{l^s_q(L_p)}= \left[ \sum_{j = 0}^{\infty}(2^{sj}\| a_j \|_{L_p(\mathbb{R}^d)})^q \right]^{1/q}< \infty \right\},\,\,\,\,\,\, q \in [1,\infty)
n!{\|h_n(.,t,x)\|}_{\mathcal{H}^{\otimes n}}^2 \leq\frac{1}{n!}C_1 e^{-(2-\delta)\mu_1 t}\int_{[0, t]^{2n}}\prod_{i=1}^n \gamma(t_i-s_i)\prod_{i=1}^n\Big(t_{\tau(i+1)}-t_{\tau(i)}\Big)^{-\beta/\alpha}d{\bf t}d{\bf s}.
\begin{array}{c}{G_{\rho \sigma \nu}^{a b c d}= \beta_{1}( G_{\rho \sigma \nu}^{( 1 )}+ G_{\rho \sigma \nu}^{( 2 )}) - \beta_{2}( G_{\rho \sigma \nu}^{( 1 )}+ G_{\rho \sigma \nu}^{( 3 )})}{+ \beta_{3}( G_{\rho \sigma \nu}^{( 3 )}- G_{\rho \sigma \nu}^{( 2 )}) = 0}\end{array}
\sum_{i}^{n}\! \left< g_{+ +}\! ( \xi_{1}) \! \dots \! \delta_{\! f}\! g_{+ +}\! ( \xi_{i}) \! \dots \! g_{+ +}\! ( \xi_{n}) \right> \! + \! \frac{d \! \! - \! \! 2 8 \! \! + \! \! \lambda^{D}}{i 4 8 \pi}\! \! \int \! \! d \xi^{2}\! \, \delta \! f \! ( \! \xi \! ) \left< \partial_{-}^{3}g_{+ +}\! ( \xi ) g_{+ +}\! ( \xi_{1}) \! \dots \! g_{+ +}\! ( \xi_{n}) \right> \! \! = \! \! 0 .
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m)/2}\cdot|M_v^{K_{\ell}}|^{-1}\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}\leq q_v^{-1/2}(q_v^{n_{v,\nu_v/2}}+1).
T_1=\left(\begin{array}{c|c|c|c|c|c}\Omega_{d_1}H W_1 \hline-W_1' 0 W_2 \hline -W_2' 0 W_3 \hline -W_3' \ddots \ddots \hline \ddots 0 W_{l}\hline -W_{l}' 0\end{array}\right)
R_{\{1 \}}(\Delta) = R_{\{2 \}}(\Delta) = \frac{1}{2}\log \frac{\sigma^2(1 + r_{\min}^2)}{{\Delta - \sigma^2(1 - r_{\min}^2 )}}, \sigma^2(1 - r_{\min}^2) \leq \Delta \leq 2 \sigma^2.
[T(a),T(b)]_{\alpha}= \alpha[T(a),T(b)]=\alpha\big(T(\varrho(T(a))b-\varrho(T(b))a\big), T\big(\tilde{\varrho}(T(a))b-\tilde{\varrho}(T(b))a\big) = T\big((\varrho(\alpha(T(a)))\beta(b)-\varrho(\alpha(T(b)))\beta(a)\big).
MASE=mean(\frac{|e_{j}|}{\frac{1}{T-1}\sum_{t=2}^{T}|Y_{t}-Y_{t-1}|})=\frac{\frac{1}{J}\sum_{j}|e_{j}|}{\frac{1}{T-1}\sum_{t=2}^{T}|Y_{t}-Y_{t-1}|}
\begin{aligned}u_\ell\ =\x_1\big(\textstyle\prod_{s=\ell}^{d-2}x_{n-st-1}\big)\big(\prod_{s=0}^{\ell-1}x_{n-st}\big) &=\x_1x_{n-(d-2)t-1}\cdots x_{n-\ell t-1}x_{n-(\ell-1)t}\cdots x_{n-t}x_n.\end{aligned}
i \frac{\partial}{\partial \tau_{2}}< m^{\prime}, \bar{m^{\prime}}; \tau_{2}| m , \bar{m}; \tau_{1}> = \hat{H}( m^{\prime}, \bar{m^{\prime}}) < m^{\prime}, \bar{m^{\prime}}; \tau_{2}| m , \bar{m}; \tau_{1}> .
\hskip -2mm \sum_{m=0}^{n-1}(8m^2+12m+5)g_m(-1) begin{align*}5pt] =\sum_{k=0}^{n-1}{2k\choose k}(-1)^k\sum_{i=0}^k \Bigg((8n^2-4n+1){n\choose k+i+1}-(16n-12){n\choose k+i+2}&\qquad{}+16{n\choose k+i+3}\Bigg){k+i\choose i}{k\choose i}.
\mu_1^{(\dagger)}(\ell, \omega) = \inf\{q^{(\dagger)}(\ell, \omega)[g] : g \in \tilde H^{1/2}_0(\Gamma) \cap L_{2,\dagger}(\Gamma)\; \wedge \; \| g\|_{L_2(\Gamma)}= 1 \}\ge \frac{\mu_*}\ell - C_1 ,
T = \left( \begin{array}{c c c c c}{\mathrm{r o t}\theta_{1}}&{}&{}&{}&{}\\{}&{\mathrm{r o t}\theta_{2}}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{\mathrm{r o t}\theta_{N}}&{}\\{}&{}&{}&{}&{{\large 1}}\\ \end{array}\right)
G^2(x,\mathbf{y}) = \left( FP^2_1 \right) y^1 +( FP^2_2 )y^2+ \left( L^2_1\right) \left(y^1 \right)^2 + L^2_2 \left(y^2 \right)^2+(Q_0^2)y^1y^2 &+ \left( \frac{A^2_0}{F}\right) \left(y^1 \right)^3+ \left( \frac{B^2_0}{F}\right) \left(y^1 \right)^2 y^2 + \left( \frac{D^2_0}{F}\right) y^1 \left(y^2 \right)^2+F^2R_0^2,
\inf_{B_{\frac{r}{2}}(p)}v \ge c(n)(\frac{1}{\omega_nr^n}\int_{B_r(p)}v^{p_0})^{\frac{1}{p_0}}&\ge c(n)(\frac{1}{\omega_nr^n}\int_{B_r(p)\backslash E}v^{p_0})^{\frac{1}{p_0}}&=c(n)\sup_{B_r(p)}u(\frac{1}{|B_r(p)|}|E^c\cap B_r(p)|)^{\frac{1}{p_0}}&\ge c(n)\sup_{B_r(p)}u.
Q_{\cal M}=\left(\begin{array}{ccccccccc}1 1 -3 1 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 1 -2 1\end{array}\right).
F_2^{(2,3)}(z) = \frac{1+z_2+z_2^2+z_1z_2^2+z_1z_2^3+z_1z_2^4}{(1-z_1^2z_2^3)(1-z_2^3)}= \frac{1+z_1z_2^2}{(1-z_1^2z_2^3)(1-z_2)}
I(t) := \frac{\Gamma(2t - 1/4; \tau, \sigma)}{\Gamma(2t + 1/4; \tau, \sigma)}\frac{\theta_0(2t; \tau)}{\theta_0(2t + 1/4; \tau)}\frac{\theta_0(2t; \sigma)}{\theta_0(2t + 1/4; \sigma)}
U_t+u\cdot\nabla U-b\cdot\nabla B+U\cdot\nabla u^\eta-B\cdot\nabla b^\eta+\nabla \pi =\nu\Delta U, B_t+u\cdot\nabla B-b\cdot\nabla U+U\cdot\nabla b^\eta-B\cdot\nabla u^\eta-\eta\nabla\times ((\nabla\times b^\eta)\times b^\eta) =\mu\Delta B, \nabla \cdot U=0, \ \ \ \nabla\cdot B =0.
\tilde{p}_0 (s) = \frac{1}{s}-\frac{1}{s}\int_0^{\infty}dw\, \lambda e^{-\lambda w}e^{-\int_0^w s^{\alpha (\tau)}d\tau}, \tilde{p}_{k}(s) =\int_{0}^{\infty}dx e^{-\int_{0}^{x}s^{\alpha (\tau)}d\tau}\frac{\lambda^{k}x^{k}e^{-\lambda x}(kx^{-1}-\lambda )}{s\Gamma(k+1)}, k\geq 1.
\Sigma_N = \sum_{d \mid v}\mu(d) \sum_{(a,w)=1}e \left( \frac{ad \overline{m\ell_1\ell_3 v}}{w}\right) \frac{N}{dw}\sum_{|h| \leq W^{1+\epsilon}d/N}e \left(\frac{ah}{w}\right) \widehat{G}\left(\frac{hN}{dw}\right) + O_\epsilon \left(Q^{-100}\right),
\Delta((f+100)^{-8}) + \langle X,\nabla ((f+100)^{-8}) \rangle = -8 \, (f+100)^{-9}\, (\Delta f + \langle X,\nabla f \rangle) + 72 \, (f+100)^{-10}\, |\nabla f|^2 \leq -8 \, (f+100)^{-9}+ 72 \, (f+100)^{-10}\leq -(f+100)^{-9}.
\hat{A}\geq \frac{1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-2\vert L_{A\Omega_1}\vert -2\vert U_{A\Omega_1}\vert \geq \frac{1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-2\vert B_{A\Omega_1}\vert:= T.
\gamma_{\operatorname*{D},j}\left( s\right) \mathsf{E}_{j}\left( s\right)\varphi=\varphi\quad\left\Vert \mathsf{E}_{j}\left( s\right)\varphi\right\Vert_{H^{1}\left( \mathbb{R}^{3}\right) ;s}\leq C\left\Vert\varphi\right\Vert_{H^{1/2}\left( \Gamma_{j}\right)}.
\begin{array}{r c l r c l}{{B^{( 1 ) \prime}}}&{{=}}&{{- B^{( 2 )}\, ,}}&{{B^{( 2 ) \prime}}}&{{=}}&{{- B^{( 1 )}\, ,}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{g_{\mu \nu}^{\prime}}}&{{=}}&{{\left( \ell^{2}+ k e^{- 2 \phi}\right)^{\frac{1}{2}}g_{\mu \nu}\, , \hspace{1 c m}}}&{{C^{\prime}}}&{{=}}&{{- C \, .}}\\ \end{array}
L_{4}^{g}= 4 8 \kappa^{2}\left[ 2 B_{\mu \nu \alpha \beta}B^{\mu \alpha \nu \beta}- B_{\mu \nu \alpha \beta}B^{\mu \nu \alpha \beta}+ 6 B_{\mu \rho \alpha}^{\rho}B_{\ \ \ \, s i g m a}^{\mu \sigma \alpha}- \frac{1 5}{4 9}\: ( B_{\ \ ~ \mu \nu}^{\mu \nu})^{2}+ P_{\mu \nu \alpha \beta}P^{\mu \nu \alpha \beta}\right] .
\sum_{j=1}^3\left(\|\theta_{\epsilon}^{\frac{1}{2}}|\nabla \varphi_j|\|_{L^2}^2+\|\theta_{\epsilon}^{\frac{1}{2}}|\varphi_j|\|_{L^2}^2\right)\le 2(3+p)C\theta_{\epsilon}(R)\prod_{j=1}^3\|\varphi_j\|_{H^1}.
[Tu,Tv,T_1(w)]+[Tu,T_1(v),Tw]+[T_1(u),Tv,Tw] &=T\Big(D(Tu,T_1(v))w+D(T_1(u),Tv)w+\theta(Tv,T_1(w))u+\theta(T_1(v),Tw)u &-\theta(Tu,T_1(w))v-\theta(T_1(u),Tw)v\Big)+T_1\Big(D(Tu,Tv)w+\theta(Tv,Tw)u-\theta(Tu,Tw)v\Big),
V_{9}\leq\varepsilon \mathcal{G}_{1}(K)\sum_{i=1}^{d}\sum_{j=1}^{d}\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})\frac{\partial^{2}(w^{1}-w^{2})}{\partial_{x_{i}}\partial_{x_{j}}}\dx.
G\left(x,\bar x;y,\bar y\right)=\int\limits_0^{+\infty}ds\int\limits_0^{+\infty}d\bar s{\rm e}^{-m^2\left(s+\bar s\right)}\int Dz\int D\bar z\exp\left(-\frac14\int\limits_0^s\dot z^2 d\lambda-\frac14\int\limits_0^{\bar s}\dot{\bar z}^2d\lambda\right)\left<W(C)\right>.
\partial_{\underline{i}}\epsilon_{I}+{\textstyle \frac{1}{4}}\biggl ({\frac{\partial_{\underline{i}}{\overline{{\cal H}}_{2}}}{\overline{{\cal H}}_{2}}}-{\frac{\partial_{\underline{i}}{\cal H}_{2}}{{\cal H}_{2}}}\biggr ) \epsilon_{I}-{\textstyle \frac{1}{2}}\left( \partial_{\underline{i}}U \right) \epsilon_{I}= 0 .
{\cal{Z}}^{P F}( X_{1}, \cdots , X_{M}; p ) ={\frac{\det ( X_{j}^{2 M + p + 1 - i}- X_{j}^{i})}{\det ( X_{j}^{2 M + 1 - i}- X_{j}^{i})}}~ ~ ; ~ ~ 1 \le i , j \le M .
(e_nx^2+f_nx)\mathcal{L}_{n+1}(x) =(p_nx^3+q_nx^2+r_nx)\mathcal{L}_n(x) +(t_nx^4+u_nx^3+\omega^2t_nx^2+\omega^2u_nx)\mathcal{L}_{n-1}(x), n\geq 1.
|M_n(t,k,\xi,\eta)|\leq C^n\int_{\zeta_1+\cdots+\zeta_n=\xi-\eta}\prod_{j=1}^n(|\widehat{h_1}(\zeta_j)|+|\widehat{h_2}(\zeta_j)|)\,\mathrm{d}\zeta_1\cdots\mathrm{d}\zeta_{n-1}
w_1(x,t)=\min\{\rho^{-N-2+\kappa-\kappa_1},\max\{|x-x_0|,\sqrt{2|t-t_0|}\}^{-N-2+\kappa-\kappa_1}\}, & w_2(x,t)=\min\{\rho^{-N+\vartheta-\vartheta_1},|x-x_0|^{-N+\vartheta-\vartheta_1}\},
\psi_{n}\ \longrightarrow \ \psi_{n}^{\prime}\ = \, p s i_{n}+{\textstyle{\frac 1 2}}( p_{\psi n}- \psi_{n}) \qquad \mathrm{s o \t h a t}\qquad \{\psi_{n}^{\prime}, \psi_{m}^{\prime}\}\ = \{\textstyle{\frac 1 2}}\eta_{n m}\quad ,
F_\varepsilon(t,x,w,p,X)=\min\{F(t,x',w,p,X)\mid x'\in\overline{B(x,M\varepsilon^{1/2})}\},\quad &F^\varepsilon(t,x,w,p,X)=\max\{F(t,x',w,p,X)\mid x'\in\overline{B(x,M\varepsilon^{1/2})}\}.
\Lambda_{BHT;\mathbb{P}}(f_1,f_2,f_3) \lesssim \prod_{j=1}^{3}\left( (\langle f_j, \phi_{P_j}^j \rangle_P) \right)^{\theta_j}\left( (\langle f_j, \phi_{P_j}^j \rangle_P)\right)^{1-\theta_j}
w_1(M) = (a-3)t_1-t_2-\dots-t_{b-1}+ (b-3)s_1-s_2-\dots -s_{a-1}= (a-3)\beta_1 + \dots + (a-b-1)\beta_{b-1}+\dots + (a-b-1)\beta_{n-2}+ \frac{1}{2}\left(a-b-1\right)\left(\beta_{n-1}+\beta_n\right) + (b-3)\gamma_1+\dots + (b-a-1)\gamma_{a-1}+\dots + (b-a-1)\gamma_{n-2}+ \frac{1}{2}(b-a-1)(\gamma_{n-1}+\gamma_n).
\phi (x)=\langle IN|\tilde T(e^{-i\int^td^4x^{\prime}~J(x^{\prime})\Phi (x^{\prime})})\Phi (x)T(e^{i\int d^4x^{\prime \prime}~J(x^{\prime \prime})\Phi (x^{\prime \prime})})|IN\rangle |
d(f(x),f(y))<\lambda d(x,y) \;\;\;\mbox{for}\;\;\;y\in W^s_{\delta_1}(x), &d(f^{-1}(x),f^{-1}(y))<\lambda d(x,y) \;\;\;\mbox{for}\;\;\;y\in W^u_{\delta_1}(x).
S_\omega(u^\gamma, v^\gamma) = \frac{1}{2}K(u^\gamma,v^\gamma) + \frac{\omega}{2}M(u^\gamma,v^\gamma) - P(u^\gamma,v^\gamma) &=\frac{\gamma^2}{2}K(u,v) + \frac{\omega}{2}M(u,v) - \gamma^{\frac{5}{2}}P(u,v).
I = \frac{1}{16\pi G_N}\int d^4x\sqrt{-g}\left[ R + a_1R^2 + a_2(3R^{\mu}_{\nu}R_{\mu}^{\nu}-R^2) + a_3(R^{\mu\nu}_{\ \ \rho\sigma}R^{\rho\sigma}_{\ \ \mu\nu}-4R^{\mu}_{\nu}R_{\mu}^{\nu}+R^2) \right],
f_{1}= - \frac{i}{2}\Bigg ( \int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{1}{k^{2}( k + p )^{2}}- \sum_{i}c_{i}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{1}{( k^{2}- M_{i}^{2}) \Big ( ( k + p )^{2}- M_{i}^{2}\Big )}\Bigg )
\sum\limits_{n = -\infty}^{+\infty}\exp\left( -\gamma\,(n+1/2)^2 \right) \,\left[ \gamma \, (n+1/2)^2 \right]^p = \left({\pi \over \gamma}\right)^{1/2}\,{(2p-1)!! \over 2^p}\left[ 1 + 2 \sum\limits_{n=1}^\infty (-1)^n \, \exp(-\pi^2 n^2/\gamma)\sum\limits_{l=0}^p \left({-\pi^2 n^2 \over \gamma}\right)^l C_l^p \right]
ds^2 = - 4 M^2\,{{\left( 1 - \sin (u) \right)}^2}\,du\,dv +{{\cos^2 (u)}\over{(1 - \sin (u)})^2}\,dx^2+{{{\cos^2 (u)}}\,{\left( 1 - \sin (u) \right)}^2}\,dy^2.
\sum_{p=1}^{r_i}\left(E_{\Lambda^n}\left[\frac{h(\textbf{A}_{p,i,n}\textbf{X}^n+\textbf{B}_{p,i,n}|\Omega^n,\Lambda^n=\lambda^n,W_{i+1},\ldots,W_K)}{r_ir_{i+1}}\right.\right. &\left.\left.-\frac{h(\textbf{C}_{i,n}\textbf{X}^n+\textbf{D}_{i,n}|\Omega^n,\Lambda^n=\lambda^n,W_{i+1},\ldots,W_K)}{r_ir_{i+1}}\right]\right)
\cdots \stackrel{a ( l - 1 )^{\dagger}}{\longrightarrow}{\cal E}( l - 1 ) \stackrel{a ( l )^{\dagger}}{\longrightarrow}{\cal E}( l ) \stackrel{a ( l + 1 )^{\dagger}}{\longrightarrow}{\cal E}( l + 1 ) \stackrel{a ( l + 2 )^{\dagger}}{\longrightarrow}\cdots
( (W^{(1)}\mapsto \ldots \mapsto W^{(k_1)}= U^{(k_2)}), &((a_h^{(1)}, a_j^{(1)}), \ldots, (a_h^{(k_1 - 1)}, a_j^{(k_1 - 1)})), &(d^{(1)}, \ldots, d^{(k_1)}= b^{(k_2)})
\sum_{k}\int w\left( e_{\pm}\right) \left\vert \phi_{k}^{\pm}(I_{\pm})\right\vert^{2}dI_{\pm}=\int\int w\left( e_{\pm}\right) \left\vert\phi\left( x\right) \right\vert^{2}dxdv & \leq\sup_{x}\int w\left( e_{\pm}\right) dv\left\Vert \phi\right\Vert_{L^{2}}^{2}\lesssim\left\Vert \phi\right\Vert_{L^{2}}^{2}.
-\langle\nabla_{X_{2}}\nabla_{X_{1}}X_{2},X_{1}\rangle=\displaystyle\frac{-c\lambda_{2}(\nu^{2}-b_{2}^{2})}{\lambda_{2}-\lambda_{1}}+\displaystyle\frac{-\nu^{2}b_{1}^{2}}{(\lambda_{2}-\lambda_{1})^{2}}.
M_{4}= \left( \begin{array}{c c c c}{{c_{1 1}}}&{{c_{1 3}}}&{{0}}&{{d_{1 3}}}\\{{c_{1 3}}}&{{c_{3 3}}}&{{- d_{1 3}}}&{{0}}\\{{0}}&{{- d_{1 3}}}&{{c_{1 1}}}&{{c_{1 3}}}\\{{d_{1 3}}}&{{0}}&{{c_{1 3}}}&{{c_{3 3}}}\\ \end{array}\right)
R_{r,v}^{\natural}(W_{v},\phi_{v}\chi'_{v}, \psi_{v}): ={|d_{v}|^{-2}|D_{v}|^{-1/2}}{\zeta_{F, v}(2)L(1, \eta_{v})\over L(1/2, \sigma_{E,v}\otimes \chi'_{v})}R_{r,v}^{\circ}( W_{v},\phi_{v}\chi_{v}', \psi_{v}).
B_1\!\!=\!B_{11}\!=\!\left\{\frac{5}{12},\frac{7}{12},\frac{11}{12},\frac{13}{12}\right\},B_2\!\!=\!B_{21}\!=\!\left\{\left.\frac{10+q+2k}{2(12+q)}\right|0\leq k\leq 11+q, k\neq 1\right\}.
\overline{\rho}\partial_{t}\overline{u}_{i}+\overline{\rho}\overline{u}_{j}\partial_{j}\overline{u}_{i}=-\partial\overline{p}_{i}+\eta\partial_{j}^{2}\overline{u}_{i}-\partial_{j}(\overline{\rho v_{i}v_{j}}/\partial x_{j}).
a_{S}( k ) \, a_{S}^{\dagger}( k^{\prime}) - \sum_{l , l^{\prime}\in{\cal I}}{\hat{\cal R}}\left( k^{\prime}, l ; k , l^{\prime}\right) a_{S}^{\dagger}( l^{\prime}) \, a_{S}( l ) = \delta_{k \; k^{\prime}}\; ,
\sum_{\mu\in M}x_\mu \prod_{j=1}^{s}p_j^{\sum_i \lambda^{j}_i \mu^{j}_i}=\sum_{\mu\in M}y_\mu \prod_{j=1}^{s}p_j^{\sum_i \lambda^{j}_i \mu^{j}_i}=C_\lambda.
\Gamma_{\nu}( p ) = - \int \frac{d^{3}q}{( 2 \pi )^{3}}D ( p - q ) \, \gamma_{\mu}\, S \left[ - \left( m + \frac{\delta}{2}\right) i w + q \right] \Gamma_{\nu}( q ) \, S \left[ \left( m + \frac{\delta}{2}\right) i w + q \right] \gamma_{\mu}.
\sum_{M\geq0}g(M)\,z^M =\sum_{M\geq0}z^M\sum_{0\leq 2n\leq M-1}(M-2n)\binom{M-n-1}n &=\sum_{M,n\geq0}z^{M+2n+1}(M+1)\binom{M+n}n & =\sum_{M\geq0}{z^{M+1}(M+1)}{(1-z^2)^{-(M+1)}}=\sum_{M\geq1}{z^MM}{(1-z^2)^{-M}}&=\frac{\frac{z}{1-z^2}}{\left(1-\frac{z}{1-z^2}\right)^2}=\frac{z-z^3}{(1-z-z^2)^2}.
\tilde{N}_p^{t-q}[q] :=\tilde{N}_p^{t-q}\tilde{S}_{t-q}[q]\cap\tilde{N}_p^t, \tilde{S}_{t-q}[q] :=\tilde{N}_p^{t-q}\tilde{S}_{t-q}[q]\cap\tilde{S}_t.
\int_{|z|\geq R}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle \rho, z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p'}|z|^{-\alpha p'}dz \leq Ce^{-\xi p'\frac{R}{2}},
1_{N_{B}}( x_{\eta}^{- 1}x g_{A}x^{- 1}x_{\zeta}) \beta_{i , j}( x_{\eta}^{- 1}x g_{A}x^{- 1}x_{\zeta}) \phi ( x ) = 1_{N}( g_{A}) 1_{N_{B}}( x_{\eta}^{- 1}x_{\zeta}) \beta_{i j}( x_{\eta}^{- 1}x_{\zeta}) \phi ( x ) .
(B_1(a_1)\otimes a_2\otimes B_2(a_3))\cdot_{H^{\otimes 3}}(B_1(b_1)\otimes b_2\otimes B_2(b_3)) = B_1(a_1)B_1(b_1)\otimes \epsilon(a_3)B_1(a_2)b_2S(B_2(a_5))\otimes B_2(a_4)B_2(b_3) = B_1(a_1)B_1(b_1)\otimes B_1(a_2)b_2S(B_2(a_3))\otimes B_2(a_4)B_2(b_3) = B_1(a_1\circ b_1)\otimes a_2\circ b_2\otimes B_2(a_3\circ b_3),
V_{\mathrm{n p}}= \frac{A^{2}N_{f}^{2}\Sigma_{0}^{3}}{3 2}\left[ -{\frac{7}{3}}+ \frac{3}{4 \nu^{2}}-{\frac{1}{\nu}}\sin{2 \theta}- \left( 1 + \frac{3}{4 \nu^{2}}\right) \cos{2 \theta}\right] .
\sum\limits_{i\in I}\left(n_{i,{j}}-n_{i+1,{j}}\right)^{2}=2n_{{j},{j}}. \footnote{Here $\sigma_{\ell}=\sigma_i$ if $\ell\equiv i \ (\mathrm{mod}\n+1)$. Therefore, $\sigma_{n+2}=\sigma_1$ and $\sigma_{0}=\sigma_{n+1}$, $n_{n+2,j}=n_{1,j}$ and $n_{0,j}=n_{n+1,j}$ for any $j\in I$.}
q\left( x,t\right) =-2\partial_{x}^{2}\log\det\left( \delta_{mn}+c_{n}^{2}\dfrac{\mathrm{e}^{-\left( \kappa_{m}+\kappa_{n}\right) x+8\left(\kappa_{m}^{3}+\kappa_{n}^{3}\right) t}}{\kappa_{m}+\kappa_{n}}\right) .
V = \frac{1}{2}\int_{M}d^{2}x \left( \Omega^{- 1}F_{1 2}^{2}+ D_{1}\phi \overline{{{D_{1}\phi}}}+ D_{2}\phi \overline{{{D_{2}\phi}}}+ \frac{\Omega}{4}( | \phi |^{2}- 1 )^{2}\right)
\sum_{|X'|}\binom{n}{|X'|}\binom{n}{|Y^*|}\binom{n}{|Z'|}2^{3p|X'||Y^*|/2}\beta^{p|X'||Z'|/2}\le \sum_{|X'|}\big(\tfrac{en}{|X'|}\big)^{6|X|'}2^{3p|X'|m/2}\beta^{p|X'|m/8}\le \sum_{|X'|}\exp\big(12|X'|\log\tfrac{en}{|X'|}+\tfrac32C'|X'|\log\tfrac{en}{|X'|}-12.5 C'|X'|\log\tfrac{en}{|X'|}\big) \le \sum_{|X'|}e^{-|X'|}
I_6=C\left| \int^T_0 (\nabla v_3|\nabla (v\cdot\nabla \psi))_{H^1}dt\right| \leq C\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}^2\|\nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))},
\mathcal Q_1^n(\zeta) =\zeta-(n+1)(a_{2,1}+na_{3,1}), \mathcal Q_2^n(\zeta) =\zeta^2-(2a_{3,1}(1+n)^2+a_{2,1}(2 n+3))\zeta+(n+2)\left[(1+n) (a_{2,1}+a_{3,1}n) (a_{2,1}+a_{3,1}+a_{3,1}n)-a_{2,0}(a_{2,2}+a_{3,2}+a_{3,2}n)\right].
E\left[\exp\left(\alpha\sum_{m, n = 1}^N\int_0^T\!\!\!\int_0^t\!\!\int_{k \in \mathbb{R}^3}\frac{1}{\omega(k)}e^{ik(X_t^m - X_s^n)}e^{-\omega(k)(t - s)}\chi_{\Lambda}(k)\,dk\,ds\,dt\right)\right]
\Psi[A_i]=\int Ds_iD\phi\exp\left\{i\kappa[s_i\epsilon_{ij}A_j-\partial_i\phi\epsilon_{ij}A_j+\epsilon_{ij}\partial_i\phi s_j]-\frac{1}{2}A_i^{\phi,s}G^{-1}A_i^{\phi,s}\right\}
\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}|\xi|^{s}|\widehat{f}(\xi)|^2d\xi =\frac{2^{1-s}\Gamma(1-s/2)}{\Gamma ({s}/{2})}\int_{\mathbb{R}^{n+1}_{+}}|\nabla_{(x,t)}u(x,t)|^2t^{1-s}dxdt,
\tilde{B}_{J_X}= \Big\{E\in \mathrm{FE}(\Lambda^i\setminus \Lambda^i H_{J_X}):E\subseteq \bigcup_{j=1}^k\Lambda^{e_j}, E\cup F\in \mathrm{FE}(\Lambda) F\subseteq r(E)\Lambda^{e_i}\Big\}.
\int_{0}^{\infty}k(t) \Big|\sum_{n=n_0}^{\infty}R_{n}e^{r_n t}\Big|^2\d t\le\varepsilon\ \pi T\sum_{n= n_0}^\infty\frac{|C_{n}|^2+|d_nD_{n}|^2}{\pi^2+T^2r_{n}^2}\,.
Z_{\Phi_{y,p}}(p, \chi_{}, 0, f_j) +_{t^{m-1}}\Big(\frac{(t-p)Z_{\Phi_{y,p}}(p, \chi_{}, s, f_j)}{(p-1)(1-t)}\Big) &+ \sum_{\substack{\chi \neq \chi_{}, c(\chi)=1}}g_{\chi^{-1}}\chi(u)_{t^{m-1}}(Z_{\Phi_{y,p}}(p, \chi, s, f_j)).
f_4 =1728^{1/14}\eta^{12}K^{-1/14}\_{3}F_2(-1/14, 11/42, 25/42; 5/7, 4/7; K) = f_1^3 + 3f_2f_3^2, f_5 = 1728^{-3/14}\eta^{12}K^{3/14}\ \_{3}F_2(3/14, 23/42, 37/42; 9/7, 6/7; K) = f_2^2f_1 - (1/3)f_3^3, f_6 =1728^{-5/14}\eta^{12}K^{5/14}\ \_{3}F_2(5/14, 29/42, 43/42; 10/7, 8/7; K)= (3/2)f_1^2f_3 - (1/2)f_2^3.
\delta E_{k}( \rho_{k}, n_{k}) = \frac{n_{k}^{2}}{2 \rho_{k}}\frac{\Gamma \left( 2 -{\frac{1}{2 n_{k}}}\right)}{\Gamma \left({\frac{1}{2 n_{k}}}\right)}+ \frac{n_{k - 1}^{2}}{2 \rho_{k}}\left( \frac{\rho_{k}}{\rho_{k - 1}}\right)^{2 n_{k - 1}}\frac{\Gamma \left({\frac{4 n_{k - 1}- 1}{2 n_{k}}}\right)}{\Gamma \left({\frac{1}{2 n_{k}}}\right)}
\begin{array}{lll}\check{\alpha}_0^{(1)}=h_{q_1}+h_{p_1}-e_{1,2,3,4}, \check{\alpha}_1^{(1)}=h_{q_1}-e_{5,6}, \check{\alpha}_2^{(1)}=h_{p_1}-e_{7,8}, \check{\alpha}_0^{(2)}=h_{q_2}+h_{p_2}-e_{9,10,11,12}, \check{\alpha}_1^{(2)}=h_{q_2}-e_{13,14}, \check{\alpha}_2^{(2)}=h_{p_2}-e_{15,16}\end{array}.
\omega_{\exp(x)H}= \sum_{1\leq j_1,j_2 \leq r}\frac{1}{4}\frac{\partial^2 u}{\partial x_{j_1}\partial x_{j_2}}(x)\gamma_{j_1}\wedge \bar{\gamma}_{j_2}+ \sum_{\alpha \in \Phi_{P^u}}\left< \alpha, \nabla u(x)/2-t_{\chi}\right>\gamma_{\alpha}\wedge \bar{\gamma}_{\alpha}.
m^2_{phys.}= m^2 + g^2 \delta m^2_0 + \frac{g^2}{32\pi^2}\int_0^1dx \int_{{\cal M}^2_0}^\infty d{\cal M}^2\frac{f^2_{\lambda_0}}{{\cal M}^2 - m^2}= m^2 + g^2 \delta\tilde{m}^2_0 \; .
\int \mathrm{d}^{n}x \frac{\delta^{R}\Gamma}{\delta \Phi_{c}^{a}( x )}\frac{\delta^{L}\Gamma}{\delta K_{a}( x )}+ \sum_{r \geq 2}\frac 1{r !}\frac{\partial^{R}\Gamma}{\partial \xi^{B}}f_{A_{1}\cdots A_{r}}^{B}\xi^{A_{r}}\cdots \xi^{A_{1}}= 0 .
{\frac{d}{d t}}( L^{n})_{\mu \nu}= i [{\cal H}, ( L^{n})_{\mu \nu}] = [ L^{n}, M ]_{\mu \nu}\nonumber \, = \sum_{\lambda \in{\cal R}}\left( \phantom{\mathrm{\huge ~ H}}( L^{n})_{\mu \lambda}M_{\lambda \nu}- M_{\mu \lambda}( L^{n})_{\lambda \nu}\right) , \quad n = 1 , \ldots .
d=-\frac{\left\{\phi_{3}^{G},\Phi_{1}\right\}}{\left\{\phi_{2}^{G},\Phi_{1}\right\}}\,,\;\;v=-\frac{\left\{\phi_{3}^{G},\Phi_{6}\right\}}{\left\{\Phi_{7},\Phi_{6}\right\}}\,,\;\;u=-\frac{\left\{\phi_{3}^{G}+v\phi_{1}^{\left( 1\right)},\phi_{4}^{G}\right\}}{\left\{\phi_{2}^{(1)},\phi_{4}^{G}\right\}}\;.
0\to H^0(U,V)\to \bigoplus_{s\in S}(j_*V)_s\to H^1_c(U,j_*V)\to H^1(\P^1,j_*V)\to 0 0\to H^1(\P^1,j_*V)\to H^1(U,V)\to \bigoplus_{s\in S}(R^1j_*V)_s\to H^2(\P^1,j_*V)\to 0
W_{\infty}^{3,D}={1\over 16}(1-\gamma)^{-1}\sum_{k=0}^{\infty}(-1)^{k}\beta^{k}(k+1)(k+3)(k+5)\sum_{m=0}^{\infty}(m^{2+k}-m^{1+k})\alpha_{m}^{-k-7}.
U_x=\;v_y\circ\phi^{-1}\cdot J_{\phi^{-1}}\ \ \ ,\ \ \U_y= -u_y\circ \phi^{-1}\cdot J_{\phi^{-1}}\ \ , V_x= -v_x\circ\phi^{-1}\cdot J_{\phi^{-1}}\ \ \ ,\ \ \V_y= \;u_x\circ \phi^{-1}\cdot J_{\phi^{-1}}\ \ .
\sup_{k\in \{0, \ldots, N \}}\tfrac{1}{M^{(N-k)}k!}=\frac{1}{M^{N}}\sup_{k\in\{0, \ldots, N \}}\tfrac{M^{k}}{k!}\leq \frac{1}{M^{N}}\sum_{k=0}^\infty\tfrac{M^{k}}{k!}=\frac{e^{M}}{M^{N}}.
\sum_{k=0}^{\infty}\frac{q^{2k^{2}-\left(2n-1\right)k}}{\left(q^{2},q^{3};q^{2}\right)_{k}}=\frac{\left(-1\right)^{n}\left(-q;q\right)_{n}}{\left(q^{3};q^{2}\right)_{\infty}q^{\binom{n}{2}}}.
\widetilde{A}^{0}(U)=\left(\begin{array}{cccc}1 &\rho\mathbb{I}_{3}& \rho e_{\theta}& \tau\mathbb{I}_{3}\end{array}\right),\widetilde{D}(U)=\left(\begin{array}{cc}\mathbb{O}_{5\times 5}&\mathbb{I}_{3}\end{array}\right),
\partial_{t}u = \mathrm{div}\, \big(\mathbf{g}(\nabla u) \nabla u\big) (0, \infty) \times G, \mathbf{g}(\nabla u) \cdot \mathbf{n}= 0 (0, \infty) \times \Gamma, u(0, \cdot) = \tilde{u}^{0}\Omega,
(\Gamma_N(b,a))^*=a^*(N(b^*))^*-(N(b^*a))^*+(N(a))^*b \hphantom{(\Gamma_N(b,a))^*}{}=a^*N^\sharp(b)-N^\sharp(a^*b)+N^\sharp(a^*)b=\Gamma_{N^\sharp}(a, b),
|\zeta-\zeta_0|^2+\tau^2 \leqslant |\zeta-\zeta_0|^2+\frac49|\zeta-\zeta_0|^2 = \frac{13}{9}|\zeta-\zeta_0|^2, |(\zeta-\zeta_0)^2+\tau^2| \geqslant |\zeta-\zeta_0|^2-\tau^2 \geqslant \frac59|\zeta-\zeta_0|^2,
P_1 =\langle t_{13}\rangle, P_2 =\langle t_{8}-t_{12}, t_{6}-t_{10}, t_{4}+t_{5}-t_{10}-t_{11}, t_{3}-t_{11}, t_{2}+t_{7}-t_{9}-t_{12}, t_{1}-t_{9}, t_{5}t_{7}-t_{5}t_{9}-t_{7}t_{11}+t_{9}t_{10}+t_{9}t_{11}-t_{9}t_{12}-t_{10}t_{11}+t_{11}t_{12}\rangle.
\delta^{*}(t)=\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)^{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)ds
\left| \nabla_{y'}^\alpha q_\lambda(y',y_d,z_d)\right|+\left|\partial_{y_d}^\alpha q_\lambda(y',y_d,z_d)\right| \leq \frac{Ce^{-c|\lambda|^{\frac 12}z_d}}{(y_d + z_d +|y'|)^{d-1+\alpha}},
\widehat{T}[W_t u_0, v, \cdots, v]_k = \sum_{\ell\,\,odd}\widehat{T_\ell}[v, \cdots, v, \underset{\ell'th}{\underbrace{W_t u_0}}, v, \cdots, v]_k = \frac{p+1}{2}\widehat{T_1}[W_tu_0, v, \cdots, v]_k,
\max \mathrm{tr}\left[ \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{R}\mathbf{W}^{\mathrm{RF}}\right], & \,\, \mathrm{s.t.}\,\, \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{W}^{\mathrm{RF}}= \mathbf{I}_R, \left| w^{\mathrm{RF}}_{mr}\right| = \frac{1}{\sqrt{M}}.
|Q_{2l-1,2l-1}|-\sum_{\{j\,:\,2j-1\in[1,N], j\neq l\}}|Q_{2j-1,2l-1}|\ge |Q_{2l-1,2l-1}|-\sum_{j=1,j\neq l}^\infty|\tilde{M}_{2j-1,2l-1}| &\ge 1-\sum_{k=1}^\infty\alpha^k \left(\Big(\frac{1}{kR_0-1}\Big)^{2l-1}-\Big(\frac{1}{kR_0+1}\Big)^{2l-1}\right)\ge C(\alpha,R_0)>0,
\Delta(x)=\begin{cases}a_{2,k}\otimes1+b_{2,k-1}\otimes b_{2,k-1}+1\otimes a_{2,k}, \textrm{if}x=a_{2,k}, n=3, k\geq 1, a_{2,0}\otimes1+b_1\otimes b_1+1\otimes a_{2,0}, \textrm{if}x=a_{2,0}, n=3, x\otimes 1+1\otimes x, \textrm{otherwise.}\end{cases}
\frac{P_{m,n}(u,v)}{h_n(v)}=\frac{1}{h_n(v)}\!\!\sum_{k=0}^{h_n(v)-1}\!\!\!{\bf 1}_{B_m(u)}(T^{-k}y^{(n)})\xrightarrow[n\to+\infty]{}\mu_m(u).
{\cal L}_{\mathrm{e f f}}/ N ={\frac{1}{2}}\tilde{A}_{\mu}^{( 1 )}\left\{( k^{2}-{\frac{e^{2}}{\pi}}) \left( - g^{\mu \nu}+{\frac{k^{\mu}k^{\nu}}{k^{2}}}\right) + 2 e^{2}\xi_{0}^{2}U ( k^{2}){\frac{k^{\mu}k^{\nu}}{k^{2}}}\right\}\tilde{A}_{\nu}^{( 1 )}~ .
\{X^{i}, X^{j}\}_{M B}= \lim_{\stackrel{\alpha^{\prime}\to \alpha}{\beta^{\prime}\to \beta}}\sin \lambda ( \partial_{\alpha}\partial_{\beta^{\prime}}- \partial_{\alpha^{\prime}}\partial_{\beta}) X^{i}( \alpha , \beta , \sigma ) X^{j}( \alpha^{\prime}, \beta^{\prime}, \sigma ) .
\|e^{it^n A}\tau_{x^n}W_n\|_{L^p}^p=\|v_n\|_{L^p}^p- \|\tilde\psi\|_{L^p}^p+ o(1) = \|v_n\|_{L^p}^p- \|\tau_{-x^n}e^{it^n A}\tau_{x^n}\psi\|_{L^p}+ o(1) = \|v_n\|_{L^p}^p- \|e^{it^n A}\tau_{x^n}\psi\|_{L^p}+ o(1).
\mbox{var}\left( \#\mathcal{P}_{N}\cap [0, \textbf{x}] \right) = \mbox{var}\left( X_1 + \dots + X_N \right) &= \mbox{var}\left( X_1 \right) + \dots + \mbox{var}\left( X_N \right) &= N \mbox{var}\left( X \right) &= N |[0, \textbf{x}]| \left(1 - |[0, \textbf{x}]| \right).