harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
K_{\beta}(\theta-\theta')=\frac{2^{1-\nu}}{D(\nu)}\Big(\sqrt 2\nu\sum_{k=1}^d|(\theta_k-\theta'_k)/\beta_k|^{2}\Big)^\nu k_\nu\Big(\sqrt 2\nu\sum_{k=1}^d|(\theta_k-\theta'_k)/\beta_k|^{2}\Big),~\nu\in(0,\infty),~\nu\notin\mathbb N,
\Delta_{i}^{KK}=\frac{{\overline{b}_{i}}}{2\pi}\sum_{(m_{1},m_{2})\not=(0,0)}^{{}}\ln \frac{M_{s}}{M_{m_1,m_2}}-\frac{{\overline{b}_{i}}}{4\pi}\sum_{(m_{1},m_{2})\not=(0,0)}^{{}}\int\limits_{-1/2}^{1/2}d\tau_{1}\int\limits_{0}^{\kappa_{\vec{m}}\sqrt{1-\tau_{1}^{2}}}\frac{dt}{t}(e^{-t}-1),
Q ( r ) = Q ( \bar{r}) + \left[{\frac{d Q}{d \mu^{2}}}\right]_{\bar{r}}\left( \mu^{2}( r ) - \mu^{2}( \bar{r}) \right) + \left[{\frac{d Q}{d ( \Gamma^{- 2})}}\right]_{\bar{r}}\left( \Gamma^{- 2}( r ) - \Gamma^{- 2}( \bar{r}) \right) + . . .
\operatorname{Vol}_{\tilde{g}((\tau)}(P) = \,\,\frac{1}{\tau^{(1+ \sum_{i=1}^m 2n_i)/2}}\operatorname{Vol}_{g (\tau)}(P) \sim \,\,\tau^{-(1+ \sum_{i=1}^m 2n_i)/2}\cdot \tau^{\sum_{i=1}^m n_i}\operatorname{Vol}_{g_0}(P) = \,\tau^{-1/2}\operatorname{Vol}_{g_0}(P) \rightarrow 0
\left(1-\frac{T_*}{n}\right)^2\frac{n}{T_*}=\left(\frac{n}{T_*}-1\right)+\left(\frac{T_*}{n}-1\right)\leq \frac{1}{T_*}\left|n-T_*\right|+\frac{1}{n}\left|T_*-n\right|\,.
\mathcal{K}_m' (\varphi) v (s) = \tilde{h}\sum_{j=1}^m \sum_{i=1}^\rho w_i \; \frac{\partial \kappa}{\partial u}(s, \zeta_i^j, \varphi (\zeta_i^j)) v (\zeta_i^j), \;\;\; s \in [a, b].
{\frac{\partial W_{0}}{\partial m^{2}}}= -{\frac{A T}{2}}\sum_{n = 1}^{\infty}\int{\frac{d k_{1}d k_{2}d k_{0}}{( 2 \pi )^{3}}}\int_{0}^{\infty}d s s^{\nu}e^{- i s ( - k_{0}^{2}+ k_{1}^{2}+ k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}+ m^{2})}\, .
\underset{t\to\infty}{\lim}\Delta_{\ell,\pi}^{av}(t)=\lim_{t\to\infty}\sum_{i=1}^{R_\ell^\pi(t)}\frac{\frac{(T_{\ell i}^\pi)^2}{2}+T_{\ell i}^\pi Z_{\ell i}^\pi}{t}.
\theta^{\ell_i-i-1}_t =\bigcirc_{\ell=0}^{i}(\theta_t+\gamma_{r-i+\ell})^{-1}\circ(\theta_t+\xi_{i,1})\circ \cdots \circ (\theta_t+\xi_{i,\ell_i}) &=\bigcirc_{\ell=0}^{j}(\theta_t+\gamma_{r-j+\ell})^{-1}\circ(\theta_t+\xi_{j,1})\circ \cdots \circ (\theta_t+\xi_{j,\ell_j})=\theta^{\ell_j-j-1}_t\enspace.
G_{E}( a_{3}, a_{4}, \phi ) = \mu^{4}\sqrt{a_{3}a_{4}}\pi \sum_{s = 0}^{\infty}\frac{( - 1 )^{s + 2}}{2 s + 2}g^{s + 1}\phi^{2 s + 2}\frac{\Gamma ( s )}{\Gamma ( s + 1 )}A_{2}^{c^{2}}( s , a_{3}, a_{4}) \, .
\sum_{k=1}^d ET(k) = 2\epsilon^2 \sum_{k=1}^d \sum_{i=k+1}^{d}\sum_{j=i+1}^{d}\frac{1}{((x_i-x_k)+\epsilon^2)((x_j-x_k)+\epsilon^2)((x_j-x_i)+\epsilon^2)}.
\Vert (B(\varepsilon ;\vartheta)-\widehat{\zeta}\varepsilon^2 Q_0)^{-1}-(B^0(\varepsilon ;\vartheta)-\widehat{\zeta}\varepsilon^2\overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant C_4 c(\phi )^2\varepsilon^{-1}.
\left(\frac{\partial}{\partial t}- \mathcal L \right) \, u_{\mathbf{0}}(t, x) + \, q_{\mathbf{0}}(x) \, u_{\mathbf{0}}(t, x) = f_{\mathbf{0}}(t, x), u_{\mathbf{0}}(0, x) = g_{\mathbf{0}}(x).
y^{2}= \prod_{k = 1}^{n_{c}}( x - \phi_{k})^{2}+ 4 \Lambda^{2 n_{c}- n_{f}}\prod_{j = 1}^{n_{f}}( x + m_{j}) , \qquad S U ( n_{c}) , \, \, \, n_{f}\le 2 n_{c}- 2 ,
\mathbf{S}_{d,0}(s)\mathcal{K}_{t}u =\mathbf{S}_{d,0}(s)\mathbf{S}_{d,0}(t)|\cdot|^{\alpha}|u|^p+\mathbf{S}_{d,0}(s)\mathbf{S}_{d,0}(t)\zeta(\cdot)\mathbf{w}&=\mathbf{S}_{d,0}(t+s)|\cdot|^{\alpha}|u|^p+\mathbf{S}_{d,0}(t+s)\zeta(\cdot)\mathbf{w}&=\mathbf{S}_{d,-\alpha}(t+s)|u|^p+\mathbf{S}_{d,0}(t+s)\zeta(\cdot)\mathbf{w}&=\mathcal{K}_{t+s}u.
G_{m,3N+1}= G_{m,3N}+ bq^{3N+1}G_{m,3(N-1)}+ abq^{6N}G_{m,3(N-2)+1}, G_{m,3N}= G_{m,3(N-1)+2}+ q^{3N}G_{m,3(N-1)}, G_{m,3N+2}= G_{m,3N+1}+ aq^{3N+2}G_{3(N-1)+1}.
\begin{array}{l l}{\Gamma^{i}= \sigma^{i}\otimes \gamma^{( 7 )}}&{\mathrm{f o r}i = 1 , 2 , 3}\\{{}}&{{}}\\{\Gamma^{a}= 1 \otimes \gamma^{a}}&{\mathrm{f o r}a = 4 , 5 , 6 , 7 , 8 , 9 .}\\ \end{array}
Z(t,u) = 1 + \int_u^t A_1^\prime \big( X(s-r) \big) \, Z (s-r ,u) \, X(s) \, \mbox{d}W(s) + \int_u^t A_1 \big( X(s-r) \big)\, Z(s,u) \, \mbox{d}W(s) & = 1 + \int_u^t A_1 \big( X(s-r) \big) \, Z(s,u) \, \mbox{d}W(s)
\sum_{x,y \in \mathbb{Z}}(-1)^{x+y}q^{x^2+y^2}- \sum_{n=-\infty}^\infty (-1)^nq^{n^2}= \sum_{N\geq 1}(-1)^N r_2(N)q^N - 2\sum_{n\geq 1}(-1)^n q^{n^2}.
\begin{array}{r c l}{{\exp ({\cal K}_{0}( Q - 1 ) ) - 1}}&{{=}}&{{( f_{0}+ Q^{- 1 / 2}f_{1}) ( Q - 1 ) \; ,}}\\{{\exp ({\cal K}_{1}( Q - 2 ) ) - 1}}&{{=}}&{{Q^{- 1 / 2}f_{1}( Q - 2 ) \; .}}\\ \end{array}
x\in D(B)=\{x\in X\:;\: (x'_e)_{e\in E}\in X,\, x_{(\frac1{k+1},\frac1{k})}\bigl(\textstyle{\frac1{k+1}}\bigr)=x_{({\frac1{k+2}},\frac1{k+1})}\bigl({\frac1{k+2}}\bigr),\, x_{(\frac12,1)}(1)=0\}.
\mathsf{J}_{\mathsf{f}}:=\min\left\{\mathsf{J}\in\{\mathsf{J}_{\mathsf{f}-1}+1,\dots,\mathsf{B}-\mathsf{F}+\mathsf{f}\}:{\kappa_{\mathsf{J}_{\mathsf{f}-1}}^2-\kappa_{\mathsf{J}}^2}\geq\frac{1-\phi}{\mathsf{F}}\,\big[{\kappa_{\mathsf{A}}^2-\kappa^2_{\mathsf{B}}}\big]\right\}\,,\mathsf{J}_{0}:=\mathsf{A}\,.
d \Theta_{\underline{\mu}}^{2}= v_{\underline{\mu}q}^{+}\pi_{q}^{2 -}+ v_{\underline{\mu}\dot{q}}^{-}\pi_{\dot{q}}^{2 +}, = e^{- 2}\psi_{- 2 \dot{q}}^{2 +}v_{\underline{\mu}\dot{q}}^{-}+ e_{q}^{-}v_{\underline{\mu}q}^{+},
\left( R^{-2}\int_{B_R}|v_k - \overline{v_{k,R}}|^s \,dx \right)^{1/s}\leq C_s \left( \int_{B_R}|\nabla v_k|^2 \,dx \right)^{1/2}= C_s \left( \int_{B_R}|\nabla u_k|^2 \,dx \right)^{1/2}\leq C k^{1/2}.
y = \frac{1}{2}\left( \begin{array}{c c c c}{{0 1 0 1}}&{{0 1 0 1}}&{{0 1 0 1}}&{{0 1 0 1}}\\{{0 0 0 0}}&{{0 0 0 0}}&{{1 1 1 1}}&{{1 1 1 1}}\\{{0 0 0 0}}&{{1 1 1 1}}&{{0 0 0 0}}&{{1 1 1 1}}\\{{0 0 1 1}}&{{0 0 1 1}}&{{0 0 1 1}}&{{0 0 1 1}}\\ \end{array}\right) \ .
\left| n^{-2}\sum_{k=1}^{K_n}\sum_{\ell=0}^{L_n-1}c^{(1)}_{\ell}c^{(2)}_{k,\ell}\right| \leq \left[ n^{-2}\sum_{k=1}^{K_n}\sum_{\ell=0}^{L_n-1}\left( c^{(1)}_{\ell}\right)^2 \right]^{1/2}\left[ n^{-2}\sum_{k=1}^{K_n}\sum_{\ell=0}^{L_n-1}\left( c^{(2)}_{k,\ell}\right)^2 \right]^{1/2}\to 0
\widetilde{\Psi}_{R}^{c}( x ; k ) = \left( \begin{array}{c}{\frac{\Psi_{R}^{1}( x )}{\frac{\pi}{L}( k + \frac{1}{2}) - g q}}\\{\frac{\Psi_{R}^{2}( x )}{\frac{\pi}{L}( k + \frac{1}{2}) + g q}}\\ \end{array}\right)^{c},
\mathop{\max}_{i=1,\cdots,q}f_{i}(\mathcal{X}^{t})\geq \delta_{\infty}^{2}(\mathcal{Q},\boldsymbol{\mathcal{S}})\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}, \mathbf{E}_{i\sim \mathbf{p}}[f_{i}(\mathcal{X}^{t})]\geq \delta_{\mathbf{p}}^{2}(\mathcal{Q},\boldsymbol{\mathcal{S}})\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}.
\mathcal C(k,d)= \left\{A\in\binom{[n]}{k}:[d-1]\subseteq A,A\cap[d,k]\neq\emptyset\right\}& \cup \left\{A\in\binom{[n]}{k}:|A\cap[d-1]|=d-2,[d,k]\subseteq A \right\},
S = \int \mathrm{d}x \left\{- \frac{1}{1 2}F_{\mu \nu \rho}F^{\mu \nu \rho}+ B^{* \mu \nu}\left( \partial_{\mu}C_{\nu}- \partial_{\nu}C_{\mu}\right) + C^{* \mu}\partial_{\mu}C \right\},
( e ) = -{\frac{\lambda^{2}}{3 2 ( 2 \pi )}}{\tilde{G}}_{\beta}^{2}( 0 ) \int_{0}^{\frac{\beta}{2}}d \tau \int_{- \infty}^{+ \infty}{\frac{d k}{g^{2}( k )}}\left[ e^{- g ( k ) \tau}+{\frac{e^{g ( k ) \tau}+ e^{- g ( k ) \tau}}{e^{\beta g ( k )}- 1}}\right]^{2},
C(h):= \displaystyle \sum_{n=1}^{+\infty}\mathbb P[S_n=h, \min_{1\leq i\leq n}S_i >0], C'(-h):= \displaystyle \sum_{n=1}^{+\infty}\mathbb P[S_n'=-h, \min_{1\leq j\leq n}S_j' >0].
r_{-}(\eta) := \frac{\sqrt{3}}{4}-\frac{1}{2}\sqrt{\frac{3}{4}- \frac{16\sqrt{3}M}{K}-4\eta}, r_{+}(\eta) := \frac{\sqrt{3}}{4}+ \frac{1}{2}\sqrt{\frac{3}{4}- \frac{16\sqrt{3}M}{K}-4\eta}.
X_{t}^{\varepsilon}[\phi ]=x_{0}[\phi ]+\int_{0}^{t}b(X_{s}^{\varepsilon})[\phi ]ds+\varepsilon \int_{\mathbb{X}\times \lbrack0,t]}G(s,X_{s-}^{\varepsilon},y)[\phi ]{\mathbf{n}}_{\varepsilon}(dy,ds).
v_{t}\left( \theta,t\right) =\alpha v^{p}\left( \theta,t\right) \left[v_{\theta\theta}\left( \theta,t\right) +v\left( \theta,t\right)-\lambda\left( t\right) \right] ,\ \ \p=1+\frac{1}{\alpha}>1,\ \ \ \left(\theta,t\right) \in S^{1}\times\lbrack0,\infty)
{\lim_{\longleftarrow}}_{\mathcal{A}}F={\lim_{\longleftarrow}}_{B\in\mathcal{A}}FB=\{(f_B)_{B\in\dot{\mathcal{A}}}\in\prod_{B\in\dot{\mathcal{A}}}FB,\forall (B_1\supset C, B_2\supset C)\in \overrightarrow{\mathcal{A}^{op}},{f_{B_1}}_{\vert C}={f_{B_2}}_{\vert C}\}
P(\eta_{\alpha}>\eta+ \delta) &= \int_{\mathbb R^d \times \mathcal M}P(\eta_{\alpha}>\eta+ \delta: \alpha_{\eta}=i, X_{\eta}=x )P((X_{\eta}, \alpha_{\eta})\in (dx,di)) &\geq 1- \frac{1}{4}P(B).
m_{\tilde G}={1\over3}\langle\,|M|\,\rangle = \frac{\, 1}{\, 4\,}\,\langle\,|\sum_ab'_au_a - 4e^{K/2}W|\,\rangle= \frac{\, 1}{\, 4\,}\,\langle\,|\sum_ab_au_a|\,\rangle\,\approx\,{1\over 4}b_+\langle\,\rho_{+}\,\rangle.
\zeta_{P{\cal K}( c )}( s ) = \frac{1}{\Gamma ( s )}\int_{0}^{\infty}\d \beta \beta^{s - 1}\mathrm{T r}e^{- \beta P{\cal K}( c )}\qquad , \qquad \zeta_{{\cal V}_{1}}( s ) = \frac{1}{\Gamma ( s )}\int_{0}^{\infty}\d \beta \beta^{s - 1}\mathrm{T r}e^{- \beta{\cal V}_{1}}\quad .
F(n)=A(n)\sum_{d\mid n}f(d) G(n)=B(n)\sum_{d\mid n}g(d), \intertext{where}\sum_{d=1}^{\infty}\frac{|f(d)|}{d}<+\infty, \sum_{d=1}^{\infty}\frac{|g(d)|}{d}<+\infty.
Q_{\alpha}(r) = \lim_{n\to\infty}\left( \frac{1}{4n^{2}}\right)^{2}\int_{0}^{\infty}\int_{rs}^{\infty}p_{n}\left( \frac{s}{4n^{2}},\frac{s_{2}}{4n^{2}}\right)ds_{2}ds = \int_{0}^{\infty}\int_{rs}^{\infty}p(s,s_{2})ds_{2}ds.
\sum_{n=1}^{\infty}A_{\pi}(m,n) e\left(\frac{an}{q}\right) \psi(n) =q \sum_{\pm}\sum_{n_{1}|qm}\sum_{n_{2}=1}^{\infty}\frac{A_{\pi}(n_{2},n_{1})}{n_{1}n_{2}}S\left(m \bar{a}, \pm n_{2}; mq/n_{1}\right) \, G_{\pm}\left(\frac{n_{1}^2 n_{2}}{q^3 m}\right)
\Psi(0)=-\frac{\Gamma_p(\frac12)\Gamma_p(\frac12+\beta)\Gamma_p(\beta)}{\Gamma_p(\frac12-\frac12a)\Gamma_p(1+\frac12a)\Gamma_p(\beta-\frac12a)\Gamma_p(\frac12+\beta+\frac12a)}=-\Phi(0).
{\lim\limits_{M,N \to +\infty}}{C}_S^{\min}(\boldsymbol{\psi}) = \mathop{\lim}\limits_{M,N \to + \infty}C_S(\boldsymbol{\psi},\textbf{W}^*) \overset{(g)}{\ge}\mathop{\lim}\limits_{M,N \to + \infty}C_S(\boldsymbol{\psi},\widetilde{\textbf{W}}_S^*),
E_{2}(\mbox{\mathversion{bold}$\varphi$}(t))=E_{2}(\mbox{\mathversion{bold}$\varphi$}_{0})+R_{2}(\mbox{\mathversion{bold}$v$}^{R},\mbox{\mathversion{bold}$\varphi$}_{0})-R_{2}(\mbox{\mathversion{bold}$v$}^{R},\mbox{\mathversion{bold}$\varphi$}(t)).
\gamma^{i}\left( \begin{matrix}{c}&{0}\\{0}&{d}\\ \end{matrix}\right) \gamma^{j}= - \left( \begin{matrix}{( \delta^{i j}+ i \epsilon^{i j}) d}&{0}\\{0}&{( \delta^{i j}- i \epsilon^{i j}) c}\\ \end{matrix}\right)
H_{\rm par}= \frac{1}{2}\big ( (\vec{p}\cdot \vec{u}) (\vec{p}\cdot \vec{r}) -(\vec{r}\cdot\vec{u})(\vec{p}\cdot\vec{p})\big)-(\vec{r}\cdot\vec{u})V_{\rm Cal}\hphantom{H_{\rm par}}{}= \frac{1}{2}\big ( (p_x+p_y+p_z)(xp_x+yp_y+zp_z)-(x+y+z)\big(p_x^2+p_y^2+p_z^2\big)\big)-(x+y+z)V_{\rm Cal}
\|K_W\|^2_{\mathbb{D},{H^2_\beta(\mathbb{U}_{\mathbb{B}\mathbb{C}})}}= \|\sum_{n=0}^{\infty}\frac{\left(W^{\dagger_3}\right)^n}{\beta(n)^2}Z^n\|^2_{\mathbb{D},{H^2_\beta(\mathbb{U}_{\mathbb{B}\mathbb{C}})}}&=\sum_{n=0}^{\infty}\left|\frac{\left(W^{\dagger_3}\right)^n}{\beta(n)^2}\right|_k^2\beta(n)^2=\sum_{n=0}^{\infty}\frac{|W|_k^{2n}}{\beta(n)^2}=T(|W|^2_k).
\mathcal{J}_{N}(x,s):=\intop_{0}^{\infty}\,\frac{y^{s-\frac{1}{2}}\left\{J_{s-\frac{1}{2}}(2\pi xy)-\sum_{k=0}^{N}\frac{\left(-1\right)^{k}\left(\pi xy\right)^{s+2k-\frac{1}{2}}}{k!\Gamma\left(s+k+\frac{1}{2}\right)}\right\}}{\sigma\left(y\right)e^{2\pi y}-1}\,dy
LHS \leq \frac{B_{1}+1}{(4\pi)^{N/2}}\,|\Gamma_1|^{\alpha}\int_{0}^{t}\tau^{\frac{1-(N-1)\alpha}{2}-1}\,d\tau &= \frac{2\,(B_1+1)}{(4\pi)^{N/2}\,\big[1-(N-1)\alpha\big]}\,|\Gamma_1|^{\alpha}\,t^{\frac{1-(N-1)\alpha}{2}},
|\dot{x}_i(t)-\mu_i(t)|\lesssim\frac{1}{t^{5/4-\delta_0}}, &\bigg|\dot{\mu}_i(t)+\sum^n_{\substack{j=1, j\not=i}}\frac{a_{ij}}{x^3_{ij}(t)}+\sum^n_{\substack{k,j=1, j\not=i}}\frac{b_{ijk}\mu_k(t)}{x^3_{ij}(t)}\bigg|\lesssim \frac{1}{t^{9/4-\delta_0}}.
T^t_t = T^z_z = \varphi'^2 e^{-2gM}+\frac{A^2\varphi^2}{r^2}e^{-2gM}+{\cal P}M'^2 +\left(\frac{A'}{r h}\right)^2 +{\cal R}\left(\frac{G}{gr}\right)^2 + U.
\|dE(u_n)\|_*=\sup_{\psi \in T_{u_n}S(c), \|\psi\|_{H^{s_1}}\leq 1}|dE(u_n)[\psi]|=\sup_{\psi \in T_{u_n}S(c), \|\psi\|_{H^{s_1}}\leq 1}|dE(u_n)[(\psi_{\frac{1}{t_n}})_{t_n}]|.
\underset{t\to\infty}{\lim}\Delta_{\ell,sr}^{av}(t)=\lim_{t\to\infty}\frac{\sum_{i=1}^{R_\ell^m(t)}\left(\frac{(T_{\ell i}^m)^2}{2}+T_{\ell i}^m Z_{\ell i}^m\right)}{t},
C_3=\left(\begin{tabular}{ccc}-0.8365 0.4830 0 0.4830 0.8365 0.2241 0.1294 0.2241 -0.8365\end{tabular}\right),\;C_4=\left(\begin{tabular}{ccc}0.8365 0.2241 -0.1294 0.2241 -0.8365 0.4830 0 0.5000 0.8660\end{tabular}\right),
\left| D_k\right|\leq C\int\limits_{B_{C_1\Lambda_n}(M_{k,n-2})\setminus\widetilde{\Omega}_{n-2}}\,\frac{\omega(\Lambda_{n})}{\Lambda^{2}_{n}}\cdot\frac{1}{\Lambda_n (k-k_0)^2}\, dm_3(M)\leq &\leq C\frac{\omega (2^{-n})}{(k-k_0)^2}.
{_1 F}_{1}\left( - n + l + 1 , 2 l + \delta_{1}+ \delta_{2}+ 2 ; \varepsilon r \right) = \sum_{s = 0}^{n - l - 1}\frac{( - n + l + 1 )_{s}}{( 2 l + \delta_{1}+ \delta_{2}+ 2 )_{s}}\frac{( \varepsilon r )^{s}}{s !}
=lim_{t\rightarrow\infty}(\frac{1}{c+exp(x)}\frac{1}{2}\sqrt{\pi}erf(x))|_{0}^{t}-lim_{t\rightarrow\infty}\int_{0}^{t}\frac{1}{2}\sqrt{\pi}erf(x)-(c+exp(x))^{-2}(exp(x))dx
{\cal O}_{1}{}^{i}{}_{j}\left[ \begin{array}{l}{\bar{\psi}_{n}^{j \dot{1}}}{\psi_{n 2}^{j}}\end{array}\right] = - i m_{n}^{( 1 )}\left[ \begin{array}{l}{\psi_{n 1}^{i}}{\bar{\psi}_{n}^{i \dot{2}}}\end{array}\right]
[y_1, y_2]=y_4+\theta_1y_5, [y_2, y_1]=\frac{\alpha_5}{\alpha_3}y_4+\theta_2y_5, [y_2, y_2]=\frac{\beta_2}{\gamma_6}y_5, [y_1, y_3]=\frac{\beta_4\gamma_1}{\alpha_3\gamma_6}y_5, [y_3, y_1]=\frac{\beta_6\gamma_1}{\alpha_3\gamma_6}y_5, [y_2, y_3]=y_4, [y_3, y_2]=\frac{\gamma_4}{\gamma_6}y_5, [y_3, y_3]=y_5.
H^{n-1}(\{dist(\cdot, \partial C)=r\}) \le\Sigma H^{n-1}(B_{2r}(p_j)\cap \{dist(\cdot, \partial C)=r\}) &\le\Sigma 2^{2n+1}n\omega_nr^{n-1}&\le 2^{2n+1}n\omega_n\Sigma \frac{1}{\gamma}H^{n-1}(\partial C\cap B_r(p_j)) &\le \frac{\Gamma}{\gamma}H^{n-1}(\partial C).
\int_{{\cal M}_{\beta}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}=\alpha\int_{{\cal M}_{\beta_H}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}+8\pi(1-\alpha)\int_{\Sigma}R_{\mu\nu\lambda\rho}n^{\mu}_in^{\lambda}_in^{\nu}_j n^{\rho}_j+O((1-\alpha)^2)~~~,
\begin{cases}\partial_t \tilde{u}(t,x) -D\Delta \tilde{u}(t,x) = \left(\lambda(S,N,F) + \mu\right)\tilde{u}(t,x) \le 0\ (0,T)\times \Omega, \partial_\nu \tilde{u}(t,x) = 0 (0,T)\times\partial\Omega,\end{cases}
\sum_{k=1}^{n+1}S_{2,\lambda}(n+1,k)(k-1)!x^{k}=\sum_{k=1}^{n+1}\big\{S_{2,\lambda}(n,k-1)+(k-n\lambda)S_{2,\lambda}(n,k)\big\}(k-1)!x^{k}&=\sum_{k=1}^{n}k!S_{2,\lambda}(n,k)x^{k}+x\sum_{k=1}^{n}S_{2,\lambda}(n,k)k!x^{k}-n\lambda \sum_{k=1}^{n}S_{2,\lambda}(n,k)(k-1)!x^{k}&=(1+x)F_{n,\lambda}(x)-n\lambda\int_{0}^{x}\frac{F_{n,\lambda}(t)}{t}dt,(n\ge 1).
e^{lin}_{n,T}= \Delta[\chi_n(\frac{x-x_n}{\lambda_n})]e^{i(t+\lambda_n^2t_n)\Delta}g_n[P_n\phi] &+2\nabla[\chi_n(\frac{x-x_n}{\lambda_n})]e^{i(t+\lambda_n^2t_n)\Delta}\nabla g_n[P_n\phi]
- \frac{1}{\pi \, d}\int_{-\alpha_{N}}^{\alpha_{N}}\frac{x\, w'(x)}{w(x)}\, \mathrm{d}x- \frac{4(\pi \alpha_{N}+ 2d)}{\pi^{3}}\frac{w'(\alpha_{N})}{w(\alpha_{N})}= 2(N+1).
\sum_{k \geq 2}I(q^{-k}) = \sum_{\substack{k\geq 2 m\geq 1}}\frac{\mu(m)}{m}\log\left(\frac{1}{1-q^{1-km}}\right) = \sum_{D=km \geq 2}\log\left(\frac{1}{1-q^{1-D}}\right)\,\sum_{\substack{m \mathrel{|}D m \neq D}}\frac{\mu(m)}{m}&= \sum_{D \geq 2}\frac{\varphi(D)-\mu(D)}{D}\,\log\left(\frac{1}{1-q^{1-D}}\right) = S(q^{-1}) - R(q^{-1}).
\displaystyle{m^{2}}_{D 0 + \tilde{D}4}= \frac{1}{\lambda}\left({\frac{N}{2}}\left( 2 - \lambda + 2{\tilde{b}}g{\tilde{b}}\right) - q \right)^{2}+ 2 N^{2}{\tilde{b}}g{\tilde{b}},
+\sum_{j=1}^{N-1}\sum_{i=j+1}^{N}f_{ij}\{\epsilon_{ij}[(\frac{r_{ij}^{0}}{r_{ij}})^{12}-2(\frac{r_{ij}^{0}}{r_{ij}})^{6}]+\frac{q_{i}q_{j}}{4\pi\epsilon_{0}r_{ij}}\}
(1+T) \! \int_0^T \! \! e_1(W_{n+q+1}) dt \! \ge \! Te_1(W_{n+q+1})(0) - C(T) \! \int_0^T \! \sum_{l=0}^{q}\! \| \mathcal{\mathbf{B}}_{n+l}^{\ast}U_{n+l}\|_{G_{n+l}}^2 .
ed'(r)sf =\sum_{g\in E'}ed'(r)gsf =\sum_{g\in E'}(ed(erg)g-ed(e)rg-erd(g)g)sf, erd'(s)f =\sum_{g\in E'}ergd'(s)f =\sum_{g\in E'}er(gd(gsf)f-gd(g)sf-gsd(f)f),
\big\|\partial^{1/2}_t v^\perp\big\|^2 = \frac{T}{2}\sum_{k=1}^\infty k \omega \|\boldsymbol{v}_k^\perp\|_{\Omega}^2 = \frac{T}{2}\sum_{k=1}^\infty k \omega \|\boldsymbol{v}_k\|_{\Omega}^2 = \big\|\partial^{1/2}_t v\big\|^2 \forall \, v \in H^{0,\frac{1}{2}}_{per}(Q_T).
d_{k[S]}:k[S]^*\otimes k[S]\to k,d_{k[S]}(f\otimes x) = f(x), b_{k[S]}:k\to k[S]\otimes k[S]^*,b_V(1) = \sum_{g\in S}g\otimes g^*, d_{k[S]^*}:k[S]\otimes k[S]^*\to k,d_{k[S]^*}(x\otimes f) = f(x), b_{k[S]^*}:k\to k[S]^*\otimes k[S],b_{k[S]^*}(1) = \sum_{g\in S}g^*\otimes g,
\langle q^{n(x_N(t)+N)}\rangle =\frac{(-1)^nq^{\frac12 n(n-1)}}{(2\pi i)^n}\int \prod_{1\leq j<k\leq n}\frac{z_j-z_k}{z_j-qz_k}\prod_{j=1}^n \left(\prod_{m=1}^N\frac{a_m}{a_m-z_j}\right) e^{(q-1)tz_j}\frac{dz_j}{z_j},
\Phi^1_n:=\sqrt{q^2+p^2}\,T_n\left(q(q^2+p^2)^{-\frac{1}{2}}\right)-1\quad,\quad \Phi^2_n:=p\,U_{n-1}\left(q(q^2+p^2)^{-\frac{1}{2}}\right)
( u_{1}, u_{2}) = \left({\frac{\phi_{2}}{\phi_{1}}},{\frac{\phi_{3}}{\phi_{1}}}\right) , \quad ( v_{1}, v_{2}) = \left({\frac{\phi_{1}}{\phi_{2}}},{\frac{\phi_{3}}{\phi_{2}}}\right) , \quad ( w_{1}, w_{2}) = \left({\frac{\phi_{1}}{\phi_{3}}},{\frac{\phi_{2}}{\phi_{3}}}\right) .
\mathrm{Re}\left(e^z+(-1)^k\right)^{-1}=\mathrm{Re}\left(e^{\bar{z}}+(-1)^k\right)^{-1}=\tfrac{1}{2}\left(\left(e^z+(-1)^k\right)^{-1}+\left(e^{\bar{z}}+(-1)^k\right)^{-1}\right) &=\frac{e^x\cos|y|+(-1)^k}{e^{2x}+2(-1)^k e^x\cos|y|+1}z=x+yi.
\lim_{n\to\infty}\{\|\nabla u_n\|_{L^2}^2-\sum_{j=1}^J\|\nabla \phi^j\|_{L^2}^2-\|\nabla w_n^J\|_{L^2}^2\}=0 \lim_{n\to\infty}\{P(u_n)-\sum_{j=1}^JP(g_n^j[e^{it_n^j\Delta}\phi^j])-P(w_n^J)\}=0.
g\in Lin(f) \Longleftrightarrow\begin{cases}\forall i,j\in [n], \: i\geq j\mbox{and}f(i)\leq f(j)\Longrightarrow g(i)\leq g(j), \forall i,j\in [n], \: g(i)=g(j)\Longrightarrow f(i)=f(j)\end{cases}&\Longleftrightarrow f\leq g.
Y_{1}Z_{2}X_{3}=[\begin{matrix}c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}\\s_{2}&c_{2}c_{3}&-c_{2}s_{3}\\ -c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}\end{matrix}]
w_j=\left( \begin{array}{c}w_j^1 w_j^2 \vdots w_j^r \end{array}\right),\f_{kj, \alpha}=\left( \begin{array}{c}f_{kj, \alpha}^1 f_{kj, \alpha}^2 \vdots f_{kj, \alpha}^r \end{array}\right).
D_0=\frac{1}{2}\left(\begin{array}{cc}1 0 0 -1\end{array}\right) \; ,\;\;\;D_+=\left(\begin{array}{cc}0 \sigma_1 0 0\end{array}\right) \; ,\;\;\;D_-=\left(\begin{array}{cc}0 0 \sigma_1 0\end{array}\right);
-u(x,t)+\frac{1}{2\delta}\left[|x-y_{2}|^{2}+(t-s_{2})^{2}\right]+p_{2}\cdot x &=\tilde{u}(x,t)+\frac{1}{2\delta}\left[|x-y_{2}|^{2}+(t-s_{2})^{2}-|x-y_{1}|^{2}-(t-s_{1})^{2}\right]+p_{2}\cdot x &=\tilde{u}(x,t)+\left(\frac{1}{\delta}(-y_{2}+y_{1})+p_{2}\right)\cdot x+\frac{1}{2\delta}\left[(t-s_{2})^{2}-(t-s_{1})^{2}+|y_{2}^{2}|-|y_{1}|^{2}\right],
\widetilde{E}_\eta(n)= \frac{HH'}{\Lambda}\sum_{c}\frac{w_0(c/C)}{c^2}\sum_{m_1}\sum_{m_2}\lambda_{1}(m_1)\lambda_{2}(m_2)S(m_1+m_2,2n;c) &\cdot W^{\star}_{\eta H}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right)V^{\star}_{-\eta H'}\left(\frac{m_2n}{c^2},\frac{m_2H'}{c^2}\right).
p^* (L_{i,b})= p^* (X_{i,b}L_{i+1,b}) =\left(\lambda_{i-a}\frac{A_{i+1,b}A_{i-1,b-1}}{A_{i,b}A_{i,b-1}}\right) \left(\frac{A_{i,b-1}}{A_{i+1,b}}\prod_{\substack{k= i-a+1}}^{n}\lambda_k\right) & =\frac{A_{i-1,b-1}}{A_{i,b}}\prod_{k=i-a}^{n}\lambda_k.
=\frac{3}{2\epsilon}\int_{1}^{1/\epsilon}(\frac{x^{2}}{x^{2}+1})^{3/2}\frac{2+\epsilon x}{(1+\epsilon x)^{2}x^{2}+1}dx\le\frac{3}{2\epsilon}\int_{1}^{\infty}\frac{3}{x^{2}}dx=\frac{9}{2\epsilon}
\sum_{j\geq -n}\int_{\mathcal L_{2j+1}}\mathbf h_p(h) dh = -p \sum_{j\geq -n}\mathrm{vol}(\mathcal L_{2j+1}) = -p(1-p^{-1})^2 \sum_{j\geq -n}p^{-2j-1}= -p^{2n}\frac{(1-p^{-1})^2}{1-p^{-2}}= -p^{2n}\frac{p-1}{p+1}.
M_1(t) = \int_0^t \langle \nabla V(X_s,\alpha_s), \sigma(s,X_s,\mathcal{L}_{X_s},\alpha_s)dW_s\rangle, M_2(t) = \int_0^t \int_{\mathbb R}V(X_s,\alpha_0+ h(X_s, \alpha_{s-}, z))- V(X_s,\alpha_s) \mu(ds,dz),
H = \sum_{\bf n}( \frac{\lambda}{4}( \psi_{+}({\bf n}) + \psi_{-}({\bf n}) ) ( \psi_{+}({\bf n + 1}) - \psi_{-}({\bf n + 1}) ) + [ \psi_{+}({\bf n}) , \psi_{-}({\bf n}) ] )
\max \mathrm{tr}\left[ \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{R}\mathbf{W}^{\mathrm{RF}}\right] = \max \mathrm{tr}\left[ \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{P}\mathbf{M}\mathbf{P}^\mathrm{H}\mathbf{W}^{\mathrm{RF}}\right] = \max \sum_{m=1}^M c_m \mu_m,
\dot{x}= n(x)-h(x,v), \dot{y}= \int_0^{\infty}f_1( \tau )h(x(t- \tau),v(t- \tau)) d \tau-ag_1(y)-pw(y,z), \dot{v}= k \int_0^{\infty}f_2(\tau) g_1(y(t- \tau)) d \tau, \dot{z}= c \int_0^{\infty}f_3(\tau) w(y(t-\tau),z(t-\tau)) d\tau -bg_3(z).
Y(n)_{s,t}=(1,\int_{s<s_{1}<t}dY(n)_{s_{1}},...,\int_{s<s_{1}<...<s_{\lfloor p\rfloor}<t}dY(n)_{s_{1}}\otimes\cdot\cdot\cdot\otimes dY(n)_{s_{\lfloor p\rfloor}}).
a_n(r,s)\equiv \prod_{i=0}^{m}a_{n_i}(r,s)+ 3\sum_{\nu=1}^m\left(\vphantom{\sum_{k_{\nu-1}=0}^2}\right.\underset{i\ne \nu-1,\nu}{\prod_{i=0}^m}a_{n_i}(r,s)\left.\vphantom{\sum_{k_{\nu-1}=0}^2}\right)f(n_{\nu-1},n_{\nu};r,s)(),
\lambda^r((a_0,a_1,a_2,\dots)) = \left(\lambda^r(a_0), \sum_{i=0}^{r-1}\lambda^i(a_0)\lambda^{r-i}(a_1), \sum_{i=0}^{r-1}\lambda^i(a_0)\lambda^{r-i}(a_2), \dots \right).
\begin{aligned}\Phi^{-1}u \in W^{1 , s}(0 , T ; W^{-1 , \frac{p}{2}}_{\sigma}(\Omega)) \cap L^s (0 , T ; \Phi^{-1}W^{1 , \frac{p}{2}}_{0 , \sigma}(\Omega)), d^{\prime}, B_{\frac{p}{2}}d \in L^s (0 , T ; L^{\frac{p}{2}}(\Omega)^3)\end{aligned}
e^{S_{w}p}=\prod_{l=1}^{k-1}p_{w_{l}w_{l+1}}\cdot\sup_{s\in S}p_{w_{k}s}=\frac{\sup_{s\in S}p_{w_{k}s}}{\mu\left(\left[w_{1}\right]\right)}\cdot\mu\left(\left[w\right]\right),
H(x,y) := \{-\Delta_x + V_\mathrm{trap}(x) + (\upsilon_N\ast\rho^\mathrm{pair})(x)-\mu\}\delta(x-y)+ (\upsilon_N n)(x,y), B(x) := \big(-\Delta_x + V_\mathrm{trap}(x) + \frac{1}{N}(\upsilon_N\ast \rho)(x) - \mu\big)\phi(x) &+ \int{dy\{(\upsilon_N n)(x,y)\phi(y)+(\upsilon_N m)(x,y)\overline{\phi(y)}\}},
f(0,x,v)=f_{0}(x,v),\rho(0,x)=\rho_0(x),u(0,x)=u_0(x), &\gamma^-f(t,x,v)\big|_{(0,T)\times\Sigma^-}=g(t,x,v),\Phi(t,x)\big|_{(0,T)\times\partial\Omega}=0, &\rho(t,x)\big|_{(0,T)\times\Gamma_{\rm{in}}}=\rho_B(x),u(t,x)\big|_{(0,T)\times\partial\Omega}=u_B(x),
\begin{array}{l}\displaystyle \psi^{(n,0)}= \sum_{\alpha = -n}^{n+1}\, (-1)^\alpha \, g_{\alpha-1}^{(n,0)}\ \phi^{(n,1)}(\cdot-2^{-1}\alpha)\,, \displaystyle \psi^{(n,m)}= \sum_{\alpha =-2n}^{n+1}\, (-1)^\alpha \, g_{\alpha-1}^{(n,m)}\ \phi^{(n,m+1)}(\cdot-2^{-(m+1)}\alpha)\,, m>0\,,\end{array}
<\chi|H_{p}(\frac{1}{E_{0}-H_{C}}H_{p})^{N-1}|\psi>=-\frac{N^{2}}{2}\frac{|t|^{N}}{2^{N}}\frac{4^{N-1}}{e_{L}^{2(N-1)}}\frac{N!}{((\frac{N}{2})!)^{2}}
P_i'(x)= 2^{-i}F'(2^{-i}(1+x)) +(-1)^i 2^{-i}F'(2^{-i}(1-x)) = & 2^{1-i}(F(2^{1-i}(1+x))-(-1)^{i-1}F(2^{1-i}(1-x)) = 2^{1-i}P_{i-1}(x)
\boldsymbol{K}^{(k)}_e \begin{bmatrix}\boldsymbol{K}^{(k)}_{e,II}\boldsymbol{K}^{(k)}_{e,IB_e}\boldsymbol{K}^{(k)}_{e,B_e I}\boldsymbol{K}^{(k)}_{e,B_e B_e}. \end{bmatrix}
\begin{aligned}\{Q_tf(n) \}_{t\ge0}= \{D L^{-1/2}P_t f(n) \}_{t\ge0}&= \Big\{\frac1{\sqrt{\pi}}\int_0^\infty e^{-t^2/(4v)}DW_vf(n) \frac{dv}{v^{1/2}}\Big\}_{t\ge0}.\end{aligned}
U_{j}( \underline{{{p}}}, \underline{{{q}}}) \equiv \frac{g m_{j}}{\sqrt{1 6 \pi^{3}}}\frac{1}{p^{+}\sqrt{q^{+}}}\, , \; \; V_{2 s}( \underline{{{p}}}, \underline{{{q}}}) \equiv \frac{g}{\sqrt{8 \pi^{3}}}\frac{\vec{\epsilon}_{2 s}^{\, *}\cdot \vec{p}_{\perp}}{p^{+}\sqrt{q^{+}}}\, ,
\int_{G_{N}}\left\langle \nabla f(x),\,\mathbf{V}(x)\right\rangle dx= -\int_{G_{N}}f(x)\left(\nabla\cdot\mathbf{V}\right)(x)dx -\frac{h(1)-h(0)}{2}\sum_{i\neq j}\int_{F_{ij}}\left[f_{ij}(x)+f_{ji}(x)\right]dS_{ij}(x).
\frac{d^{n}}{dz^{n}}\left\lbrace{}^{\Psi}F_{p}^{(\alpha,\beta)}(a,b;c;z)\right\rbrace =\frac{(a)_{n}(b)_{n}}{(c)_{n}}\left[{}^{\Psi}F_{p}^{(\alpha,\beta)}(a+n,b+n;c+n;z)\right],
J(f,g)^2 = \begin{cases}(2\pi)^{-2}(1-e^{-2t}) \frac{a^2 b^2}{4(1+ab)(1-e^{-2t}) - 3 + 2(2e^{-2t}-1) (a+b)}begin{align*}1ex] +\infty \end{cases}
ag_4^2-bg_3g_4+cg_3^2 = \det (g) a, -ag_4g_2+b(g_1g_4+g_2g_3)/2-cg_3g_1 = \det(g) b/2, ag_2^2 - b g_2g_1 +cg_1^2 = \det (g) c.
\eqref{eqDI1}=\bigg(\prod_{l=1}^N\frac{w_l}{w_l+\lambda}\bigg)\det_{1\leq k,l\leq N}\left[w_l^{k-1}\right]=\det_{1\leq k,l\leq N}\bigg[\frac{w_l^k}{w_l+\lambda}\bigg].