harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
i \frac{\partial}{\partial t}\langle \vec q\,',t\mid \vec q,0 \rangle = -\frac{1}{2}\sum_{\nu=1}^3 \frac{\partial^2}{\partial q'^2_{\nu}}\langle \vec q\,',t\mid \vec q,0 \rangle + V(\vec q\,') \langle \vec q\,',t\mid \vec q,0 \rangle
- \frac{1}{k}\sum_{i=1}^k \lambda_i^T (g(x_i) + \delta_i) \le \frac{\| \lambda_1 \|_2^2}{2 \alpha k}+ \frac{\alpha \sigma_g^2}{2}+ \frac{\alpha}{2k}\sum_{i=1}^k \| \delta_i - \delta \|_2^2 & + \frac{\alpha}{k}\sum_{i=1}^k (\delta_i - \delta)^T (g(x_i) + \delta)
s_{\theta^{'}}(v,v)\ge s_{\theta}(v,v)-c\varepsilon \|v\|^2_s=(1-c\varepsilon)s_\theta(v,v)-c\varepsilon \big(1-\Lambda_1(S_\theta)\big)\|v\|^2_{L^2(\Omega_{\frac{\pi}{4}})}.
(S^{r}_\rho h)_n :=\frac{1}{2\rho+1}\sum\limits_{m=-\rho}^{\rho}(S^{r-1}_\rho h)_{n-m}=\frac{1}{2\rho+1}\sum\limits_{m=n-\rho}^{n+\rho}(S^{r-1}_\rho h)_{m},
\partial_{0}L_{0}- \partial_{x}L_{1}= i(\partial_{0}^2 \vec{n}\wedge \vec{n}- \partial_{x}^2 \vec{n}\wedge\vec{n}) \cdot\vec{\sigma}= i((\partial_{0}^2 \vec{n}- \partial_{x}^2 \vec{n})\wedge\vec{n}) \cdot \vec{\sigma}= 0
\delta S=\int_{t_{1}}^{t_{2}}[L(q+\epsilon,\dot{q}+\dot{\epsilon})-L(q,\dot{q})]dt=\int_{t_{1}}^{t_{2}}(\epsilon\cdot\frac{\partial L}{\partial q}+\dot{\epsilon}\cdot\frac{\partial L}{\partial\dot{q}})dt
\mathcal{D}(h,k):= \frac{1}{2}\sum_{\substack{1\leq q<\infty (q,hk)=1}}W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty (mn,q)=1 mh=nk}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right)\sum_{\substack{1\leq c\leq C, d \geq 1 cd=q d|mh\pm nk}}\phi(d)\mu(c),
\left\{\frac{4n+2k-2}{m}\right\}+3\left\{\frac{k}{m}\right\}+\left\{\frac{2n}{m}\right\}\leq 3\left\{\frac{2k}{m}\right\}+\left\{\frac{n}{m}\right\}+\left\{\frac{n-1}{m}\right\}&\quad+2\left\{\frac{n-k}{m}\right\}+\left\{\frac{2n+k-1}{m}\right\}
\begin{cases}\displaystyle \frac{\partial u\left(t,s\right)}{\partial t}+\frac{\partial u\left(t,s\right)}{\partial s}+\Psi\left(s, X(t) \right) u\left(t,s\right)=0, \displaystyle u\left(t,0\right)=\int_{s\in \mathbb{R}}\Psi\left(s, X(t) \right)u\left(t,s\right)ds,\end{cases}
(\mathcal M\rtimes\mathcal G)(a_1\dots a_n;a) \cong (\mathcal M\wr\widetilde{\mathbb G}_{\mathcal G})(a_1\dots a_n,a)_{\mu_n}&\xrightarrow[\cong]{H_{{\widetilde{\mathbb G}}_{\mathcal G}}}(\mathcal C\times_{\widetilde{\mathbb E}_{\mathcal G}}\widetilde{\mathbb G}_{\mathcal G})(H(a_1\dots a_n),H(a))_{\mu_n}\ .
\left(\begin{array}{c}u_1(-t+\bar{T}) u_2(-t+\bar{T}) u_3(-t+\bar{T}) \end{array}\right)=M(2p-1)\left(\begin{array}{c}u_1(t+\bar{T}) u_2(t+\bar{T}) u_3(t+\bar{T})\end{array}\right).
\int_{B_k\cap\Omega}G_{\Omega}(x, 1; z, t_k)G_{\Omega}(z, t_k; y, t_{k+1})\,d \mathrm{vol}_{g(t_k)}(z) &\le \int_{\Omega\setminus B_0(x,r_k)}G_{\Omega}(x, 1; z, t_k)G_{\Omega}(z, t_k; y, t_{k+1})\,d \mathrm{vol}_{g(t_k)}(z)\leq a_k.
\left[\begin{array}{cc}p_{n-1}p_{n}q_{n-1}q_{n}\end{array}\right]=\left[\begin{array}{cc}p_{n-m-2}p_{n-m-1}q_{n-m-2}q_{n-m-1}\end{array}\right]\left[\begin{array}{cc}0 1 1 a_{n-m}\end{array}\right]\cdots\left[\begin{array}{cc}0 1 1 a_{n}\end{array}\right]
C^{la}(G, L) = \prod_{g\in G/G_0}C^{la}(gG_0, L) = \prod_{g\in G/G_0}\varinjlim_{h>0}C^h(G_0,L) = \varinjlim_{\substack{g \in G / G_0 h_g>0}}\prod_{g \in G / G_0}C^{h_g}(G,L).
\|f\|_{\vec{p}}:=\left(\int_\mathbb{R}\cdots\left(\int_\mathbb{R}\left(\int_\mathbb{R}|f(x_1,\dots,x_d)|^{p_1}dx_1\right)^{\frac{p_2}{p_1}}dx_2\right)^{\frac{p_3}{p_2}}\cdots dx_d\right)^{\frac{1}{p_d}}<\infty.
C_2 :=\sum_{p>23}b_p f(p) = \frac{\pi}{4}e^\gamma \sum_{p> 23}\frac{M_p}{m_p^2}{\rm d}({\rm L}_p) \le \frac{\pi}{4}\frac{Me^\gamma}{\mu_{23}^2}\prod_{p\le 23}\big(1-\tfrac{1}{p}\big) = 0.251135\cdots,
{{\mathbf{P}}_B}^{(q)}={\mkern 1mu}\frac{1}{{(q - 1)!}}\left({\left({{\mathbf{x}}_n^{(\alpha )}\otimes{{\mathbf{J}}_{1,n + 1}}}\right) - \left({{{\left({{\mathbf{x}}_n^{(\alpha )}}\right)}^T}\otimes{{\mathbf{J}}_{n + 1,1}}}\right)}\right)_{(q-1)}\circ{{\mathbf{P}}_B}^{(1)},
\|u_{\epsilon}\|_{q}^{q}=R^{q}\omega\epsilon^{\frac{N}{2}}\int_{0}^{2}\frac{(\varphi^{q}(r)-1)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{N}{2}}}dr+R^{q}\omega\epsilon^{\frac{N}{2}}\int_{0}^{2}\frac{r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{N}{2}}}dr =I_{1}(\epsilon)+I_{2}(\epsilon).
\begin{array}{l l}{{\cal L}_{\chi}}&{= ( 2 \pi R ){\cal L}_{5},_{\chi}= \tilde{g}^{M N}( \partial_{M}\chi )^{*}( \partial_{N}\chi ) - V}\\{}&{= e^{\sqrt{\frac 2 3}\kappa \varphi}g^{\mu \nu}( \partial_{\mu}\chi )^{*}( \partial_{\nu}\chi ) - V}\\ \end{array}
(u , \chi)_{L/K}= \left( \frac{\tau(\pi_L)}{\pi_L},\chi \right)_{LK'/K'}+\displaystyle\sum_{i=1}^r \left( \frac{\sigma(\beta_i)}{\beta_i}, \chi \right)_{LK'/K'}&= \left( \frac{\tau(\pi_L)}{\pi_L}, \chi \right)_{LK'/K'}.
\begin{bmatrix}\lambda 1 0 0 \dots 0 0 \lambda 1 0 \dots 0 \vdots \ddots \ddots \ddots \ddots \vdots 0 0 \dots \lambda 1 0 0 0 \dots 0 \lambda 1 0 0 \dots 0 0 \lambda\end{bmatrix}.
\int_0^\infty\Phi(K^{\sigma^2}_t f(x))\nu_{\sigma^2}(x) dx =\int_0^\infty \frac{1}{\sigma^2}\Phi\left(K_t d_{\sigma^2}f\left(\frac{x}{\sigma^2}\right)\right)\nu\left(\frac{x}{\sigma^2}\right)dx =\int_0^\infty \Phi(K_t d_{\sigma^2}f(x))\nu(x) dx.
\begin{array}{l}R(\Delta )= I(\Delta )=\Delta^{N-1}(\Delta^{-1}\Omega \Delta^{-1})^{N-1}, R(\Omega )=\Delta^{N-2}\Omega (\Delta^{-1}\Omega \Delta^{-1})^{N-1}, I(\Omega )= \Delta^{N-1}(\Delta^{-1}\Omega \Delta^{-1})^{N-1}\Omega\Delta^{-1}.\end{array}
B ( p^{2}) = m_{0}+ \frac{\alpha}{2 \pi^{2}}\int \frac{d^{2}q B ( q^{2})}{q^{2}+ m_{t o t}^{2}}\int_{0}^{\infty}\frac{d x \exp ( - x l^{2}/ 2 )}{x + ( q - p )^{2}}.
\sup_{|z-c|\leq R}\sum_{k=n}^\infty |a_kz^k| \leq \sum_{k=n}^\infty C_2(2R)^k\exp\left(-\gamma_2 k^{\frac{d}{d-1}}\right) &\leq \sum_{k=n}^\infty C_3 \exp\left(-\gamma_3 k^{\frac{d}{d-1}}\right) \leq C_4 \exp\left(-\gamma_4 n^{\frac{d}{d-1}}\right)
S_{n}\left(e^{2\xi}q^{-1}\right)=\frac{q^{\binom{n+1}{2}}\left(-e^{\xi}t\right)^{n}h_{n}\left(\sinh\left(\xi+\log q^{\left(n-1\right)/2}\right)\vert q\right)}{\left(q;q\right)_{n}},
\begin{aligned}\frac{\partial}{\partial y}\int dx \xc(x,y,t) = \left(\frac{2}{\pi}\right)^{1/2}\biggl[\left(4y^{2}-1\right)e^{-2y^{2}}-4y\left(\frac{4}{3}y^{3}-y\right)e^{-2y^{2}}\biggr] = \left(\frac{2}{\pi}\right)^{1/2}\left(-\frac{16}{3}y^{4}+8y^{2}-1\right)e^{-2y^{2}}\end{aligned}
\Sigma_1=\left(\begin{array}{cccc}0 0 0 0 0 i 0 -i 0 \end{array}\right)\; , \Sigma_2=\left(\begin{array}{cccc}0 0 -i 0 0 0 i 0 0 \end{array}\right)\; , \Sigma_3=\left(\begin{array}{cccc}0 i 0 -i 0 0 0 0 0 \end{array}\right)\; \, ,
\int_{\Omega_T}(u_n\cdot\nabla\phi_n-u\cdot\nabla\phi)\psi\,dx=-\int_{\Omega_T}\phi(u_n-u)\cdot\nabla\psi\,dx-\int_{\Omega_T}(\phi_n-\phi)u_n\cdot\nabla\psi\,dx
Z\left( T_{1},\ldots,T_{d},\mathcal{O}\right) :={\textstyle\sum\nolimits_{\underline{n}\in S}}\left[ \mathcal{I}_{\underline{n}}\right] \mathbb{L}^{-\left\Vert\underline{n}\right\Vert}T^{\underline{n}}\in\mathcal{M}_{k}[\![T_{1},\ldots,T_{d}]\!] ,
\frac{1}{N}\sum_{i=1}^N \nabla^T f(y_{i,t}) e_{i,t}=\frac{1}{N}\sum_{i=1}^N\left(\nabla^T f(y_{i,t})-\nabla^T f(\overline{x}_t)\right) (y_{i,t}-\overline{x}_t)\leq \frac{1}{N}\sum_{i=1}^N \mu \|e_{i,t}\|^2= 2\mu\overline{V}_t.
L = \frac{1}{8}\, T_{( \mu \nu ) \sigma}T^{( \mu \nu ) \sigma}+ \frac{1}{4}T_{( \mu \nu ) \sigma}T^{( \mu \sigma ) \nu}+ \frac{1}{2}T_{( \mu \nu ) \sigma}\epsilon^{\mu \nu \alpha \beta}\partial_{\alpha}h_{\; \beta}^{\sigma}+ h^{\alpha \beta}\Theta_{\alpha \beta}\ ,
\liminf_{n \to \infty}\frac{1}{n}\sum_{k = 0}^n V_k \geq \liminf_{n \to \infty}\frac{i_n + 1}{\sigma_{i_n + 1}}\cdot \left[ \frac{1}{i_n + 1}\sum_{j = 1}^{i_n}Q_j \right] &= \liminf_{i \to \infty}\frac{i+1}{\sigma_{i+1}}\cdot \left[ \frac{1}{i+1}\sum_{j = 1}^{i}Q_j \right]= \frac{\mu_{Q}}{\mu_{\sigma}}~,~ P_{0,\pi}^V ~ \mbox{a.s.}
{}^{M\!L\!\!}{B}_{p}(x,1-y)=\int_{0}^{1}t^{x-1}\left[ \sum_{n=0}^{\infty}\frac{(y)_n}{n!}t^n\right]E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right) dt.
\int_{0}^{\lambda(t-s)}\bigl(K_{0}(z) \bigr)^{k}\,dz =\int_{0}^{\eta}\bigl(K_{0}(z) \bigr)^{k}\,dz+\int_{\eta}^{\lambda(t-s)}\bigl(K_{0}(z) \bigr)^{k}\,dz &:=I_{1}+I_{2}.
A_{x_{| |}}^{1 1}= \frac{1}{\sqrt{\rho}}\int_{\tilde{T}^{2}}d^{2}z \, K_{-}( \omega , z ; x ) e^{\phi ( z )}\left( D_{x}^{\dagger}( \hat{A}) \partial_{x_{| |}}D_{x}( \hat{A}) \right)^{1 1}\frac{e^{\phi ( z )}}{\sqrt{\rho}}K_{-}( z , \omega ; x ) .
\exp\left\{-\Gamma_{\rm W}[\tilde{A}\,]\right\}\propto \, \prod_{k}\, \frac{\lambda_k}{\lambda_k + i\Lambda_0}\, \left(\, \prod_{l=1}^{\infty}\frac{\lambda_k^2 + M_l^2}{\lambda_k^2 + (M_l+\Lambda)^2}\,\right)\; ,
\nu[A] \;=\; \frac{e}{4 \pi \hbar}\int d^2 x \epsilon_{\mu\nu}F^{\mu\nu}\;=\;\frac{e}{2 \pi \hbar}\int \limits_{-\frac{1}{2}T}^{\frac{1}{2}T}dt\int \limits_{-\frac{1}{2}{R}}^{\frac{1}{2}{R}}dx(\dot A_1 -\partial A_0)
\Phi(f) \geq e^{-\frac{xa_u}{\tau|v_1|}}f_{L}(v)1_{v_1>0}+e^{-\frac{(1-x)a_u}{\tau|v_1|}}f_{R}(v)1_{v_1<0}\cr \geq e^{-\frac{a_u}{\tau|v_1|}}f_{L}(v)1_{v_1>0}+e^{-\frac{a_u}{\tau|v_1|}}f_{R}(v)1_{v_1<0}\cr =e^{-\frac{a_u}{\tau|v_1|}}f_{LR}.
\left \|\sum\limits_{j=1}^{\infty}\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}+\left \|\sum\limits_{j=1}^{\infty}\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}\leq C\left \|f\right \|_{h_{\omega}^{p}}
~R_T =\mathbb{E}\left[\sum_{t=1}^{T}x_t(u(t,n)-\mu^*)\right] &=u_m\mathbb{E}\left[\sum_{t=1}^{T}x_t\left\{\left(1-\frac{\mu^*}{u_m}\right)-\left(1-\frac{u(t,n)}{u_m}\right)\right\}\right].
\frac{d\mathcal{K}_{p,q}}{dk}=\frac{q}{p^*}\int_0^{\frac{\pi_{p,q}}{2}}\dfrac{k^{q-1}\sin_{p,q}^q{\theta}}{(1-k^q\sin_{p,q}^q{\theta})^{2-\frac1p}}\,d\theta.
g f_{i,j}=g u_i \otimes g^{n-i+1}w_j &=(u_i+u_{i-1}) \otimes g^{n-i}(w_j+w_{j-1}) &=u_i \otimes g^{n-i}w_j + u_i \otimes g^{n-i}w_{j-1}+u_{i-1}\otimes g^{n-i}(w_j+w_{j-1}) &=f_{i,j}+f_{i,j-1}+u_{i-1}\otimes g^{n-(i-1)}w_j &=f_{i,j}+f_{i,j-1}+f_{i-1,j}
\frac{1}{2}\zeta^2-\frac{1}{2}\alpha^2\ln(2\zeta)-\frac{1}{4}\alpha^2+\frac{1}{2}\alpha^2\ln(\alpha)+O\left (\alpha^4\zeta^{-2}\right )=\int_{\lambda_2}^\lambda Q^{1/2}(\eta,x)d \eta.
W_1 = \langle x \, \omega_{1,0}, \dots, x^{d_1-1}\, \omega_{1,0}\rangle = \langle v_1 ,\dots, v_{d_1-1}\rangle, W_3 = \langle x \, \omega_{3,0},\dots, x^{d_3-1}\, \omega_{3,0}\rangle = \langle w_1 ,\dots, w_{d_3-1}\rangle,
\left| D^n_s \left( \eta^{\rm slow}\circ \phi \right) \right|\lesssim_{n}\sum_{\left\{m_{\ell}\right\}}\left| (D^{m}_s \eta^{\rm slow}) (\phi) \right|\prod_{\ell=1}^{n}\left| D^{\ell}_s \phi \right|^{m_{\ell}}
d_1 = a_1, \:\:\: d_i = a_{i-1}b_{i}a_{i}^{-1}b_{i}^{-1}\:\:\: 2 \leq i \leq g, & e_1 = a_1, \:\:\: e_i = b_1 a_1^{-1}b_1^{-1}a_1 \ldots b_{i-1}a_{i-1}^{-1}b_{i-1}^{-1}a_{i-1}b_i a_i^{-1}b_i^{-1}\:\:\: 2 \leq i \leq g.
\bigg\|\frac{T^{*2K}x}{x_{2K}w_{2K-1}\cdots w_{0}}-e_{0}\bigg\|^{2}\leq \frac{\mu^{2}\delta}{w_{0}^{2}w_{1}^{2}}\sum_{i=0}^{\infty}w_{i+2K}^{2}~\bigg|\frac{x_{i+2K+1}}{x_{2K}}\bigg|^{2}\leq \frac{\mu^{2}\delta}{w_{0}^{2}w_{1}^{2}}\sum_{i=0}^{\infty}w_{i+2K}^{2}< C\epsilon,
\left[ \frac{\Gamma(1/p) \Gamma(1/q)}{\Gamma(1)}\right]^{1/q}\left[ \frac{\Gamma(1/q) \Gamma(1/p)}{\Gamma(1)}\right]^{1/p}= \Gamma\left(\frac{1}{p}\right) \Gamma\left(1 - \frac{1}{p}\right) = \frac{\pi}{\sin(\pi / p)}
\Delta =\,\Delta E_G =\,-4\,\pi\,G\,\int_{M_3(1)}\,\int_{M_3(1)}\,d^3x\,d^3 y \, \sqrt{\det \,g_3(x)}\sqrt{\det\,g_3(y)}\cdot & \cdot\frac{\left(\rho_1(x)-\rho_2(x)\right)\,\left(\rho_1(y)-\rho_2(y)\right)}{|x-y|}.
d s_{T^{1 , 1}}^{2}={\frac{1}{9}}\bigg ( d \psi + \sum_{i = 1}^{2}\cos \theta_{i}d \phi_{i}\bigg )^{2}+{\frac{1}{6}}\sum_{i = 1}^{2}\left( d \theta_{i}^{2}+ \mathrm{s i n}^{2}\theta_{i}d \phi_{i}^{2}\right) \
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int|\nabla \bar{d}^{k+1}|^2 dx+\int |\Delta \bar{d}^{k+1}|^2dx &=\int \left[\nabla \bar{d}^{k+1}: (\nabla d^k+\nabla d^{k-1})d+|\nabla d^{k-1}|^2 \bar{d}^k -(\bar{u}^k \cdot)d^k-(u^{k-1 \cdot \nabla})\bar{d}^k\right]\cdot (-\Delta \bar{d}^{k+1})dx &=\sum_{i=1}^4 I_{9i}.\end{aligned}
\langle f, g \rangle_{k,l,m}= \left( \int_{-1}^{-q^{k+1}}+ \int_{zq^{m+1}}^{zq^{-l}}\right) f(x)\overline{g(x)}w(x) d_qx &= (1-q)\sum_{n=0}^k f(-q^n)\overline{g(-q^n)}w(-q^n)q^n & + (1-q)\sum_{n=-l}^m f(zq^n) \overline{g(zq^n)}w(zq^n) zq^n.
\chi^{\epsilon,L}_t \leq \frac{1}{1+\frac{1}{t}}+ p_{d,t,\mu,m^2=1}(\lambda),\int_{0}^{t_0}\frac{p_{d,t,\mu,m^2=1}(\lambda)}{t^2}\, dt \leq c_d(\lambda,\mu,m^2=1),
n_{o}p_{o}=(N_{C}e^{-\frac{E_{C}-E_{F}}{k_{B}T}})(N_{V}e^{-\frac{E_{F}-E_{V}}{k_{B}T}})=N_{C}N_{V}e^{-\frac{E_{g}}{k_{B}T}}=n_{i}^{2}
(\tilde{Q}_i)_{j,j}=([\tilde{Q}]_{i+1,i})_{j,j}=\phi_j+\lambda_j^{-1}\bigg(c_i+\sum_{r=0}^i[\tilde{v}_r \tilde{w}_r]_{j,j}-\sum_{s=0}^{m-1}\frac{m-s}{m}[\tilde{v}_s\tilde{w}_s]_{j,j}\bigg)
H_k =\langle x^iy^j\mid i\geq k\rangle, A=H_0>H_1>\cdots>H_r=0, V_k =\langle x^iy^j\mid j\geq k\rangle, A=V_0>V_1>\cdots>V_s=0, D_k =\langle x^iy^j\mid i+j\geq k\rangle, A=D_0>D_1>\cdots>D_{r+s-1}=0.
F_k(\alpha) = \max_{\tilde{p}\!, p_1\!, p_2\!}\alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}}{\sigma^2_n}\!\Big),
\det\left[\begin{array}{cc}\left[{\cal K}_t(x,n-i;x,n-j)\right]_{i,j=0}^{\ell-1}\left[-{\cal K}_t(x,n-i;x+1,n-\ell)\right]_{i=0}^{\ell-1}\left[{\cal K}_t(x+1,n-\ell;x,n-j)\right]_{j=0}^{\ell-1}1-{\cal K}_t(x+1,n-\ell;x+1,n-\ell)\end{array}\right].
H^{(0)}_{m,n}=E\big(F_\ell(\xi^{(1)}_m,\xi_{2m}^{(2)},..., \xi^{(\ell)}_{\ell m})F_\ell(\xi^{(1)}_n,\xi_{2n}^{(2)},..., \xi^{(\ell)}_{\ell n})\big)
d\tilde W^n(t) = \left(-e(\tilde W^n(t))+(\mu-r)\pi^*(\tilde W^n(t))+\sigma\pi^*(\tilde W^n(t))\dot\psi^n(t)\right)dt+\frac{1}{\sqrt{n}}\sigma\pi^*(\tilde W^n(t))dB^n(t),
H ( x_{1}, x_{2}) = \left( \begin{array}{c c}{{\bar{q}_{D}( x_{1}) q_{C}( x_{2})}}&{{D_{A B}( x_{1}) q_{C}( x_{2})}}\\{{\bar{q}_{D}( x_{1}) \bar{D}_{E F}( x_{2})}}&{{D_{A B}( x_{1}) \bar{D}_{E F}( x_{2})}}\\ \end{array}\right)
(\ln f)^{\ast}(1) =\lim_{v\rightarrow1+}(\ln f)^{\ast}(v)=\lim_{y\rightarrow-\infty}\left[ y\varphi(y)-\ln f(y)\right] & =\lim_{y\rightarrow-\infty}\bigg[y\frac{1+\sum_{n\geq2}n^{2}e^{(n^{2}-1)y}}{1+\sum_{n\geq2}e^{(n^{2}-1)y}}-y-\ln\bigg(1+\sum_{n\geq2}e^{(n^{2}-1)y}\bigg)\bigg]=0.
{\cal S}( \hat{\Gamma}) = \int \left\{\partial_{\mu}c \frac{\delta \hat{\Gamma}}{\delta A_{\mu}}+ B \frac{\delta \hat{\Gamma}}{\delta \bar{c}}+ \frac{\delta \hat{\Gamma}}{\delta \underline{{Y}}}\frac{\delta \hat{\Gamma}}{\delta \underline{{\varphi}}}+ \underline{{q}}\frac{\delta \hat{\Gamma}}{\delta \underline{{\hat{\varphi}}}}\right\}= 0
p^{\alpha_{m+1}}(m+1,t) =(-1)^{m+1}\sum_{k=m+1}^{\infty}(-\lambda)^k\underset{\Theta^{k}_{m+1}}{\sum}\frac{t^{\sum_{j=0}^{m+1}k_j\alpha_j}}{\Gamma\left(\sum_{j=0}^{m+1}k_j\alpha_j+1\right)},
W_{a}^{\prescript{w}{}{\chi}}\circ \overline{\mathcal{A}}_{w}(\pi(\varpi^{\lambda}) \phi_{K}^{\chi}) = \tau_{\nu, \nu}^1 W_{a}^{\chi}(\pi(\varpi^{\lambda}) \phi_{K}^{\chi}) + \tau_{w \cdot \nu, \nu}^2 W_{w \cdot a}^{\chi}(\pi(\varpi^{\lambda}) \phi_{K}^{\chi})
A_{4}= \lambda^{6}\left[{\frac{\sin^{2}\theta^{\prime}}{1 6 \pi^{2}}}\left({\frac{1}{\sqrt{3}}}\ln{( \sqrt{3}+ 2 )}+ \cos \theta^{\prime}\right) + \sin^{4}\theta^{\prime}\left( c_{4}+ d_{4}\cos \theta^{\prime}\right) \right] ,
V_i \cdot{\cal N}_{\overline{\alpha V}}\equiv{\vec U}_i \cdot{\vec P}_{\overline{\alpha V}}- \sum_{r}V^r_i N^r_{\overline{\alpha V}}+ \sum_{\ell:{\mbox{right}}}T^\ell_i J^\ell_{\overline{\alpha V}}- \sum_{\ell:{\mbox{left}}}T^\ell_i J^\ell_{\overline{\alpha V}}~.
\| \alpha_k (\beta) (x) \|_2 = \sqrt{\sum_\sigma \left( \int_0^1 t^{k - 1}f_\sigma (t x) dt \right)^2}& \le \frac{1}{k}\sqrt{\sum_\sigma \max_{0 \le t \le 1}f_\sigma^2 (t x)}& \le \frac{1}{k}\sqrt{m \choose k}\max_\sigma \max_{0 \le t \le 1}| f_\sigma (t x) | & \le \frac{1}{k}\sqrt{m \choose k}\max_{0 \le t \le 1}\| \beta (t x) \|_2,
V(\tilde{x}(T))-V(x^*(T)) = 2\sum_{p=1}^r (a_p-u_p^*(\tau))m_p(\tau) + o(\varepsilon) = 2(a_i-u_i^*(\tau))m_i(\tau) + 2(a_k-u_k^*(\tau))m_k(\tau) + o(\varepsilon) = -2zm_i(\tau)+2zm_k(\tau) + o(\varepsilon) = 2z(m_k(\tau)-m_i(\tau)) + o(\varepsilon),
e(G') \leq \max\left\{\binom{k}{2},\binom{\frac{k}{2}+1}{2}+\left(\frac{k}{2}-1\right)\left(n-\frac{k}{2}-1\right)\right\}&=\max\left\{\binom{k}{2},\binom{\frac{k}{2}-1}{2}+\left(\frac{k}{2}-1\right)\left(n-\frac{k}{2}+1\right)+1\right\}.
\alpha_1 = \frac{1}{r^2}\, (x_1 \, dx_2 - x_2 \, dx_1 + x_3 \, dx_4 - x_4 \, dx_3), \alpha_2 = \frac{1}{r^2}\, (x_1 \, dx_3 - x_3 \, dx_1 + x_4 \, dx_2 - x_2 \, dx_4), \alpha_3 = \frac{1}{r^2}\, (x_1 \, dx_4 - x_4 \, dx_1 + x_2 \, dx_3 - x_3 \, dx_2).
\delta(s(e)) = \tilde{\mathbf{Q}}\left( \delta\big( e^{-\pi i( \frac{k_L}{3}+ \frac{1}{2})}\tilde{\alpha}_0^{-1}\big) g_p\!\left( \tilde{\chi}_{\sigma_L}^{-1}\circ \tilde{\chi}_{\sigma_R}\right) \delta\big( e^{\pi i \frac{k_R}{3}}\tilde{\alpha}_0 \big) \right).
\begin{array}{l}\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\lambda f((x,y),u(x,y))=0\bigskip u(x,0)=u(x,n+1)=0,x\in (0,m+1)\bigskip u(0,y)=u(m+1,y)=0y\in (0,n+1)\end{array}
\|q_{\ell}(f)\|_2 = \Big\|\sum_{\substack{k_i\geq\ell_i i=1,\cdots,d}}q_{\ell}(\delta_k(f))\Big\|_2\leq \sum_{\substack{k_i\geq\ell_i i=1,\cdots,d}}\|q_{\ell}(\delta_k(f))\|_2.
\Lambda_{i}^{(\alpha,\lambda)}(\{Q_{j}\}_{j=1}^{i}) =\sum_{x_{i},y_{i},u_{i},v_{i}}P^{(\alpha,\lambda)}(x_{i},y_{i},u_{i},v_{i})g_{Q_{i},P_{i}}^{(\alpha,\lambda)}(x_{i},y_{i}|u_{i},v_{i}).
L^{(2 )}_{\ell}(x_i;f):= \frac{1}{x_i}\sum_{\substack{j=1 x_i \geqslant y_{j-1}}}^J \int_{y_{j}}^{y_j(1+1/X)}X \sum_{\max \{t/(1+1/X),y_{j-1}\}< p \leqslant y_j}\frac{1}{p}\big|\Psi_f^{\prime}(x_i/t,p)\big|^2{\rm d}t.
\left\langle \left\langle \cdot ,\cdot \right\rangle \right\rangle:\mathbb{C}l_{(n+m)}\times \mathbb{C}l_{(n+m)}\rightarrow \mathbb{C}l_{(n+m)}(\xi_{1},\xi_{2}) \mapsto \left\langle \left\langle \xi_{1}, \xi_{2}\right\rangle \right\rangle =\tau (\xi_{2})\xi_{1}.
G=4\pi log(\frac{G(\frac{3}{8})G(\frac{7}{8})}{G(\frac{1}{8})G(\frac{5}{8})})+4\pi log(\frac{\Gamma(\frac{3}{8})}{\Gamma(\frac{1}{8})})+\frac{\pi}{2}log(\frac{1+\sqrt{2}}{2(2-\sqrt{2})}).
X^\epsilon(t_{n+1})=\varphi\bigl(X^\epsilon(t_n),\zeta^\epsilon(t_{n+1})-\zeta^\epsilon(t_n)\bigr)=\varphi\bigl(X^\epsilon(t_n),\frac{1}{\epsilon}\int_{t_n}^{t_{n+1}}m^\epsilon(s)ds\bigr)
-\int_0^{\frac{\lfloor sT \rfloor}{T}}\Theta( u ,x) f'(u) \mathrm{d}u+ f(s){\Theta}(\frac{\lfloor sT\rfloor}{T},x)-f(0){\Theta}(0,x) +o(1) =-\eta d T\big( \int_0^{\frac{\lfloor sT \rfloor}{T}}\int_0^1 f(u)A(x,y)\Theta(u,y)\mathrm{d}y\mathrm{d}u +o(1) \big) +\sigma^2 \eta^2 T \big( \int_0^s f(u)\mathrm{d}\xi_1(u,x)+o(1)\big).
\begin{array}{rcl}L_{ijk}^{(s)}:= \sum_m L_{ij}^{m(s)}(q_{mk}^{(s)})^{p^s}, \ \ \ \Gamma_{ijk}^{(s)}:= \sum_m \Gamma_{ij}^{m(s)}(q_{mk}^{(s)})^{p^s}.\end{array}
{}_{a}^{C}\mathrm{D}_{t}^{\alpha}u(t)=\frac{1}{\Gamma (1-\alpha )}\int_{a}^{t}\frac{u^{\prime}(\theta )}{(t-\theta )^{\alpha}}\mathrm{d}\theta ,\;\;\;\;{}_{t}^{C}\mathrm{D}_{b}^{\alpha}u(t)=-\frac{1}{\Gamma (1-\alpha )}\int_{t}^{b}\frac{u^{\prime}(\theta )}{(\theta-t)^{\alpha}}\mathrm{d}\theta ,
\exp \left( a T r U - b T r U^{2}\right) = \sum_{r}b^{( n_{1}+ n_{2}+ \cdots + n_{N}) / 2}\det \left( \frac{H_{n_{j}+ i - j}( a / 2 \sqrt{b})}{( n_{j}+ i - j ) !}\right) \chi_{r}( U ) .
U_{2m}^{(2)}=U_{m}^{2}&=U_{m}\left( pU_{m-1}-qU_{m-2}\right) &=pU_{m}U_{m-1}-q\left( pU_{m-1}-qU_{m-2}\right) U_{m-2}&=pU_{2m-1}^{(2)}-pqU_{2m-3}^{(2)}+q^{2}U_{2m-4}^{(2)}.
\operatorname{div}_{\Gamma}\mathbf{E}^{\operatorname*{curl}}=0\left( \mathbf{E}^{\operatorname*{curl}},\mathbf{E}^{\nabla}\right)_{\Gamma}=0=b_{k}\left( \mathbf{E}^{\operatorname*{curl}},\mathbf{E}^{\nabla}\right) =b_{k}\left( \mathbf{E}^{\nabla},\mathbf{E}^{\operatorname*{curl}}\right) .
\frac{c}{4}(-b_1^2-b_2^2+2a^2){}=\langle\bar{R}(U_1,U_2)\xi,\eta\rangle=\langle S_\xi U_1,S_\eta U_2\rangle-\langle S_\eta U_1,S_\xi U_2\rangle &{}=(\lambda_1-\lambda_2)\langle S_\eta U_1,U_2\rangle.
\mathcal{E}(\mathbf{t}, \mathbf{f}) = \det\begin{bmatrix}f_0 t_0 t_1 \ldots t_{\frac{N+1}{2}-1}f_1 t_1 t_2 \ldots t_{\frac{N}{2}}\vdots \vdots \vdots \vdots \vdots f_{\frac{N+1}{2}}t_{\frac{N+1}{2}}t_{\frac{N+1}{2}+1}\ldots t_{N}\end{bmatrix}
k=\frac{\sqrt{\sqrt{c^{4}+1}+1}-c}{\sqrt{2c^{2}+2+2\sqrt{c^{4}+1}}},k^{\prime}=\frac{\sqrt{\sqrt{c^{4}+1}+1}+c}{\sqrt{2c^{2}+2+2\sqrt{c^{4}+1}}}.
\varrho_N(x_1,\dots,x_N) = \frac{1}{N!}\det_{N\times N}\big[ K(x_i,x_j) \big] & = \frac{1}{N!}\det_{N\times N}\big[ P^N_{j}(x_i) \big] \det_{N\times N}\big[ Q^N_{j}(x_i) \big] ,
\left(\mathcal{A}_\varepsilon - zI \right)^{-1}= \mathcal{G}_\varepsilon^{-1}\left( \int_{Y'}^\oplus \left(\frac{1}{\varepsilon^2}\mathcal{A}_{\chi,\varepsilon}- zI \right)^{-1}d\chi \right) \mathcal{G}_\varepsilon, z \in \rho(\mathcal{A}_\varepsilon),
x_{\ell}=\sum_{((\ell_1,\ldots, \ell_j))}\frac{1}{\prod_{m=1}^{\kappa(\ell_1, \ldots, \ell_j)}g_m!}\prod_{k=1}^j \frac{\prod_{u=0}^{\ell_k-1}(\ell-\sum_{i=1}^{k-1}\ell_i-u)^2}{2 \ell_k}
\int^\infty_0\!\!e^{-tx}t^mdt= \begin{cases}m!x^{-m-1}, m\in\mathbb Z,\ \m\geq 0, \frac{(-1)^m}{(-m-1)!}x^{-m-1}(\log x+c-\sum^{-m-1}_1\frac{1}{j}), m\in\mathbb Z,\ \m<0.\end{cases}
u_+(x) = x\exp\Bigl(-\frac{d}{\gamma}x^\gamma\Bigr)M\biggl(\frac{\gamma+1-\frac{c}{d}}{2\gamma},\frac{\gamma+1}{\gamma},\frac{2d}{\gamma}x^\gamma\biggr), begin{align*}1ex]u_-(x) = \exp\Bigl(-\frac{d}{\gamma}x^\gamma\Bigr)M\biggl(\frac{\gamma-1-\frac{c}{d}}{2\gamma},\frac{\gamma-1}{\gamma},\frac{2d}{\gamma}x^\gamma\biggr),
0=\sum_{\substack{r\geqslant 0 s<0}}v_r w_s u (z_0 +z_2)^{-r-1}z_2^{-s-1}=\sum_{\substack{r\geqslant 0 s<0}}\sum_{\ell\geqslant 0}\binom{-r-1}{\ell}v_r w_s u z_0^{-r-\ell-1}z_2^{\ell-s-1}.
\int_M | W_{\sigma_i}(u) - L_{\sigma_i}(u) |^2\: d{\mu}_{i,u}\le \int_M | W_{\sigma_i}(u) - \nabla (\sigma_i \circ F_{i,u}^{-1}) |_g^2\: d{\mu}_{i,u}= \int_M | (dF_{i,u})^{-1}_* W_{\sigma_i}(u) - \nabla \sigma_i |_{F_{i,u}^* g}^2\: d{\mu}_{i,0}.
\begin{aligned}\lambda q(\eta,\eta) = \int_X \eta\wedge\eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n-1}- -\frac{n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right)\end{aligned}
E\left[X^r\right] = E\left[\int_0^{\infty}\frac{\xi^{S_N + r-1}}{(S_N-1)!}\, e^{-\xi}d\xi\right]= E\left[\frac{\Gamma(S_N + r)}{(S_N-1)!}\right] = E\left[S_N^{(r)}\right].
\int_{\Omega}\mathbf{E}_{\overline{\mathbf{P}}_{N}^{x}}[{\rm Num}(\square_{1})] \, dx = \int_{\Omega}\left| \theta_{N^{\frac{1}{d}}x}X_{N}' \big|_{\square_{1}}\right| \, dx \geq \int_{\Omega}{\rm emp}_{N}\, dx.
\bar{D}^{2}e^{- 2 V}D^{2}e^{2 V}\rightarrow \bar{D}^{2}e^{i \Lambda}e^{- 2 V}e^{- i \bar{\Lambda}}D^{2}e^{i \bar{\Lambda}}e^{2 V}e^{- i \Lambda}= e^{i \Lambda}\left( \bar{D}^{2}e^{- 2 V}D^{2}e^{2 V}\right) e^{- i \Lambda}.
T_1(q) = 2\sum_{k,\ell, \, \mathrm{ord}(\omega) \mid \ell}{n \brack k}_q^2 q^{f(n,k,\ell)}q^2{n \brack \ell}'_q{k \brack \ell}'_q{k+\ell \brack n}_q, T_2(q) = 2\sum_{k,\ell, \, \mathrm{ord}(\omega) \nmid \ell}{n \brack k}_q^2 q^{f(n,k,\ell)}q^2{n \brack \ell}'_q{k \brack \ell}'_q{k+\ell \brack n}_q.
|\mathcal{B}(s_1,s_2)| \ll \prod_{p^{\alpha}\mid \mid h}\sum_{a=0}^{\alpha}\frac{\tau_{k}(p^a) \tau_{\ell}(p^a) j(p^a)}{(p^a)^{\sigma_1+\sigma_2+1}}\ll \prod_{p \mid h}\Big( 1 + \frac{k \ell}{p^{\sigma_1+\sigma_2+1}}\Big) = \Theta(\sigma_1+\sigma_2+1, h),
X_{jk}=X_{{\epsilon}_j-{\epsilon}_k}=G_{jk}^{+},\; 1\leq j\neq k \leq n, Y_{jk}=X_{{\epsilon}_j+{\epsilon}_k}=G_{kj}^{-},\; 1\leq j< k \leq n, Z_{jk}=X_{-({\epsilon}_j+{\epsilon}_k)}=G_{jk}^{-},\; 1\leq j< k \leq n.
Z_m(e^{\varphi i}\zeta,e^{\psi i}\eta)=Z_m(e^{-\psi i}\eta,e^{-\varphi i}\zeta)\quad\textrm{and}\overline{Z_m(e^{\varphi i}\zeta,e^{\psi i}\eta)}=Z_m(e^{\psi i}\eta,e^{\varphi i}\zeta).
\tilde g = \frac{(R_s M)^{\frac{3-d}{4}}}{\prod r_i}\ ,\quad\tilde \alpha' = \frac{R_s^{1/2}}{M^{3/2}}\ ,\quad\tilde R_i = \frac{1}{r_i M}\ ,\quad R_s\rightarrow 0\ ;\M,r_i \mbox{~fix{\'e}s}\ .
_{\Gamma}\left(\sum_{\gamma\in\Gamma}n_{\gamma}\gamma(\mathcal{G_A})\right)=\left\{\gamma_1\cdot\gamma_2\cdot\gamma_3^{-1}|\ \gamma_2\in_{\Gamma}(\mathcal{G_A}),\ \n_{\gamma_1},n_{\gamma_3}>0\right\}.
C^{k_i+1}(n_i) =\frac{(2(2a+1)+1) \cdot 3^{k_i}-1}{2}=\frac{4a\cdot 3^{k_i}+3 \cdot 3^{k_i}-1}{2}\quad =2a\cdot 3^{k_i}+ \frac{3^{k_i+1}-1}{2}
\|f\|_{\partial D_1\times M_1,\rho}=\frac{1}{2\pi}\lim_{r\rightarrow1-0}\int_{M_{1,r}}\int_{\partial D_{1,r}}|f^*(w_1,\hat w_1)|^2\rho_1(w_1)|dw_1|\lambda_1(\hat w_1)d\mu_1(\hat w_1)