harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}h(w_i)z_ih(w_i)^{-1}= \begin{psmallmatrix}1 \sqrt{w_i}0 1\end{psmallmatrix}\begin{psmallmatrix}x_i -y_i y_i x_i\end{psmallmatrix}\begin{psmallmatrix}1 -\sqrt{w_i}0 1\end{psmallmatrix}&= \begin{psmallmatrix}x_i+y_i\sqrt{w_i}-y_i(1+w_i) y_i x_i-y_i\sqrt{w_i}\end{psmallmatrix}\end{aligned}
\lim_{s\to \infty}\limsup_{N\to \infty}d^{N,k}\left( (8\pi^2)^{-1}N^2\log k+sN^2 \right)=0, \lim_{s\to -\infty}\liminf_{N\to \infty}d^{N,k}\left( (8\pi^2)^{-1}N^2\log k+sN^2 \right)=1
\theta ( v ; \gamma , \delta ) = \sum_{n = - \infty}^{\infty}\exp \left\{\sum_{a , b = 1}^{g}\pi i ( n_{a}+ \delta_{a}) \tau_{a b}( n_{b}+ \delta_{b}) + \sum_{a = 1}^{g}\left[ v_{a}( n_{a}+ \delta_{a}) + 2 \pi i \gamma_{a}( n_{a}+ \delta_{a}) \right] \right\}\ ,
(1-z)^aP^{\lambda-1, -2\lambda-2a+1}_a\bigg(\frac{3+z}{1-z}\bigg)=(-1)^a\frac{\left(\lambda+ [\frac{a}{2}] \right)_{[\frac{a+1}{2}]}}{\left(\lambda - \frac{1}{2}\right)_{[\frac{a+1}{2}]}}C^{\lambda-\frac{1}{2}}_{a}(z).
\bar \eta_{K,M}^{(3)}\buildrel \Delta \over = \sum\limits_{j \in{\cal A}}{\mathop{\max}\limits_{1 \le{k_j}\le K}\sum\limits_{i ={k_j}}^K{\hat \phi_{ji}^{\left( 4 \right)}}}= \sum\limits_{j \in{\cal A}}{\hat \psi_j^{(K)}}= \sum\limits_{j \in{\cal A}}{\sum\limits_{l = 1}^{M+K}{\xi_j^{(l)}}}
\mathbf{U^{\pm}}=\chi(\tilde{v})A(\tilde{v})\pm\chi(\tilde{u})A(\tilde{u}) \mathbf{S}(\tilde{v}) =\chi(\tilde{v})A(\tilde{v})\chi(\tilde{v}) \mathbf{G}=\frac{1}{\lambda}\left[\chi(\tilde{v})A(\tilde{v})\tilde{\Gamma}\tilde{v}-\chi(\tilde{u})A(\tilde{u})\tilde{\Gamma}\tilde{u}\right].
\Phi(\eta_1,\eta_2,\xi)=p(\eta_1)+p(\eta_2)+p(\xi-\eta_1-\eta_2)-p(\xi)=-\frac{\eta_1}{\sqrt{1+\eta_1^2}}-\frac{\eta_2}{\sqrt{1+\eta_2^2}}-\frac{\xi-\eta_1-\eta_2}{\sqrt{1+(\xi-\eta_1-\eta_2)^2}}+\frac{\xi}{\sqrt{1+\xi^2}}.
g^1(N,C,D^2,\tau) =r[\displaystyle\frac{t}{2}+a_1v+c_1u+d_1rw^2+e_1rv^2+f_1rvu+g_1rvt+\cdots], g^2(N,C,D^2,\tau) =r[b_2t+a_2v+c_2u+d_2rw^2+e_2rv^2+f_2rvu+g_2rvt+\cdots],
|\tilde{w}_{xx}(t,L)|^2\leq \left(\frac{1}{L}+L\right)\Vert \tilde{w}_{xx}\Vert_{L^2(0,L)}^2 &+\left(2\lambda +\frac{1}{L}\right)\Vert \tilde w_x\Vert_{L^2(0,L)}^2 & +\frac{1}{L}\Vert \tilde{w}_t\Vert_{L^2(0,L)}^2
\lambda\left( \sum_{i =1}^{m}\sum_{j=J_{(i)}}^{k}d_{j}(\lambda_{i-1}) e_j + \sum_{j=q+1}^k d_j(\lambda_{m}) e_j + \sum_{i =m+1}^{n-1}\sum_{j=J_{(i)}}^{k}d_{j}(\lambda_{i}) e_j\, , ~ d(\lambda_0 \cdots \lambda_n) \right) \; .
\acute{\Phi}|_{\mathcal{C}_{0}((0,1]) \otimes 1_{A}}= \Theta|_{\mathcal{C}_{0}((0,1])}\mbox{and}\grave{\Phi}|_{\mathcal{C}_{0}([0,1)) \otimes 1_{A}}= \Theta|_{\mathcal{C}_{0}([0,1))},
X_N =\frac{Z_0}{X_1\cdots X_{N-1}}, Y_{N-r+1}=X_{N-r+1}M_1\left(\frac{Y_1}{X_1},\dots,\frac{Y_{N-r}}{X_{N-r}},\frac{Z_1}{Z_0},\dots,\frac{Z_r}{Z_0}\right), &\vdots Y_N =\frac{Z_0}{X_1\cdots X_{N-1}}M_r\left(\frac{Y_1}{X_1},\dots,\frac{Y_{N-r}}{X_{N-r}},\frac{Z_1}{Z_0},\dots,\frac{Z_r}{Z_0}\right),
\begin{array}{lll}A \approx A_k:= C_A\M_A\R_A := AP\M_A\S^T\!A ~ , begin{align*}1mm] B \approx B_k:= C_B\M_B\R_B := BP_B\M_B\S^T\!B ~ , begin{align*}1mm] G \approx G_k:= C_G\M_G \R_G := GP \M_G \S_G^TG.\end{array}
P\big(S_{n+1}(d,p)=y\big|S_n(d,p)=x\big)=\frac{p}{2d}, &P\big(S_{n+1}(d,p)=\triangle\big|S_n(d,p)=x\big)=1-p, &P\big(S_{n+1}(d,p)=\triangle\big|S_n(d,p)=\triangle\big)=1
N_p(x) = \frac x{p\log x}\left(1 + O\left(\frac{(\log_2 p)^2}{\log_2 x}\right)\right) \sum_{k \le K \log_2 x}\frac{g(p, \mu)}{\Gamma(1+\mu)}\frac{(\log_2 x)^{k-1}}{(k-1)!}\sum_{\substack{P^+(A) \le p \Omega(A) = k}}\frac1A
\lim_{x\to\infty}\frac{f_s(x)}{f(x)}= \lim_{x\to\infty}\frac{f_{+a}(x)}{f(x)}\cdot \frac{f_{ar}(x)}{f_{+a}(x)}\cdot\frac{f_s(x)}{f_{ar}(x)}=(1-e^{-\lambda_s})^{-1}.
0=J^{'}(\widehat{u})+\lambda_+ \varphi_{w^+}^*+\lambda_- \varphi_{w^-}^*\ \ \ (W^{\alpha,G},(\mathbb{R}^d))^*,\ \ \varphi^*_{\widehat{w}^+}\in \partial\varphi_+(\widehat{w})\ \ \varphi^*_{\widehat{w}^-}\in \partial\varphi_-(\widehat{w}).
\prod_{i=0\atop i\neq l}^{j}\left ({\rm Coeff}(\varphi^*_{a}(F_i),t^{v_i})\right )^{\gamma_i}\cdot \left ({\rm Coeff}(\varphi^*_{a}(F_l),t^{v_l})\right )^{\gamma_{l}-1}\cdot{\rm Coeff}(\varphi^*_{a}(F_l),t^{k-I(F,H)+v_l}).
2a\cdot (C_n/P_k) - 2b\cdot \dim(C_n/P_k) \geq 2(2k-2)(2n-k+1) - k (4n-3k+1) = 4nk-k^2+7k-8n-4 =\frac{2nk}{3}-k^2 + \frac{10nk}{3}- 8n +7k-4 >0.
- i \bar{\psi}( g_{\rho}^{\prime}\rho_{\mu m}+ g_{\pi}^{\prime}{\gamma^{5}}\partial_{\mu}\pi_{m}) \psi_{\mu m}- i \bar{\psi}_{\mu m}( g_{\rho}^{\prime}\rho_{\mu m}+ g_{\pi}^{\prime}{\gamma^{5}}\partial_{\mu}\pi_{m}) \psi
\int\limits_{\mathbb{R}^{N}}\left\vert v\right\vert^{r}\frac{dx}{\left\Vert x\right\Vert^{s}}=d^{1+\frac{p-1}{p}r}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert T_{N,d,p}v\right\vert^{r}}{\left\Vert x\right\Vert^{N+sd-Nd}}dx
\partial_{\mu}^{x}U ( x , x^{\prime}) = i g \left[ A_{\mu}( x ) - \! \! \int_{0}^{1}\! \! d s s{\frac{d \omega^{\nu}}{d s}}\U ( x , \omega ) F_{\nu \mu}( \omega ) U ( \omega , x ) \right] U ( x , x^{\prime})
-\frac{\hbar^2}{2m}\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2}{\partial \phi^2}\right]\psi(r,\phi)-\frac{Ze^2}{r}\psi(r,\phi) = E\psi(r,\phi),
\hat{P}_{1,k+1}(x)\mathcal W(x) =-\begin{bmatrix}J_{k+1,k}J_{k+1,k+1}A \rho_{k+1}A^{\top}0 J_{k+1,k}\rho_{k+1}A^{\top}0\end{bmatrix}p_{k}(x)+\begin{bmatrix}I_{q}J_{k+1,k+1}A\rho_{k+1}0 \rho_{k+1}\end{bmatrix}p_{k+1}(x)+\begin{bmatrix}0 A 0 0\end{bmatrix}p_{k+2}(x),
d_{\gamma}^2 =\int_{\mathbb{B}^2}|z^{\gamma}|^2\mu(z)dV(z) &=4\pi^2\int_{\mathcal{R}}r_1^{2\gamma_1+1}r_2^{2\gamma_2+1}\exp \left( \frac{-1}{1-(r_1^2+r_2^2)}\right)dr_2dr_1
\widetilde{P}_{1,2,k}^{(0^{2})}(z; \lambda_1) = -4k ( k+4 ) P_{k}^{(1,2)}(z) \left( \lambda_{1}- \frac{1}{3}(z+1)^{3}+\frac{1}{8}(z+1)^{4}\right) \hphantom{\widetilde{P}_{1,2,k}^{(0^{2})}(z; \lambda_1) =}{}+(k+4) (1-z)^{2}(1+z)^{3}P_{k-1}^{(2,3)}(z), k\ne 0, \widetilde{P}_{1,2,0}^{(0^{2})}(z; \lambda_1) = 1.
\sum_{n=0}^{\infty}B_{n, p}(x)\frac{t^{n}}{n!}= \frac{(p+1)(t-H_{p})e^{(x+p)t}}{(e^{t}-1)^{p+1}}+(p+1)e^{xt}\sum_{k=1}^{p}\binom{p}{k}\frac{H_{k}}{(e^{t}-1)^{k+1}}.
q^{H}a_{1}^{\varepsilon}= a_{1}^{\varepsilon}q^{H + 1}\, , \, \, q^{H}a_{2}^{\varepsilon}= a_{2}^{\varepsilon}q^{H - 1}\, , \, \, q^{\bar{H}}{\bar{a}}_{\varepsilon}^{1}={\bar{a}}_{\varepsilon}^{1}q^{{\bar{H}}- 1}\, , \, \, q^{\bar{H}}{\bar{a}}_{\varepsilon}^{2}={\bar{a}}_{\varepsilon}^{2}q^{{\bar{H}}+ 1}\, ,
\pi_1(u,v) =\sum_{j=0}^\infty\, (u_0+\dots+u_{j-2})\cdot v_j, \pi_2(u,v) =\sum_{j=0}^\infty\, (u_{j-1}\cdot v_j+u_j\cdot v_j +u_j\cdot v_{j-1}), \pi_3(u,v) =\sum_{j=0}^\infty\, u_j\cdot (v_0+\dots+v_{j-2}) =\pi_1(v,u).
v_h^{{B}}+\sum_{i=1}^{h-1}\ell^{{B}}_i(v_{h-i}^{{B}})^{q^{if}}= \ell^{{B}}_h = \gamma(\ell^{{A}}_{hf}) = \gamma\left(\sum_{i=0}^{hf-1}\ell^{{A}}_i(v_{hf-i}^{{A}})^{q^i}\right) = \sum_{i=0}^{h-1}\ell^{{B}}_i\gamma(v_{(h-i)f}^{{A}})^{q^{if}}= \gamma(v_{hf}^{{A}})+ \sum_{i=1}^{h-1}\ell^{{B}}_i (v_{h-i}^{{B}})^{q^{if}},
T_{pq}=\frac{1}{16\pi}\left[ F_{p}G_{q}^{\ast}+F_{p}^{\ast}G_{q}+F_{q}G_{p}^{\ast}+F_{q}^{\ast}G_{p}\right. & \qquad\quad-\left. \delta_{pq}\left( \mathbf{F}\cdot\mathbf{G}^{\ast}+\mathbf{F}^{\ast}\cdot\mathbf{G}\right) \right] =T_{qp}\qquad\left(p,q=1,2,3\right)
\sum_{\ell'=0}^L (\mathcal C_{\ell'}^{\mu_2}\|\triangle u_{\ell'}\|)^{\frac{1}{\mu_2+1}}\leq \max_{i=1,2,3}\{\tilde d_i\}\varepsilon^{-\max\{1,\frac{\gamma\mu_2-\beta}{\alpha(\mu_2+1)}\}}.
\int_{B_{1}}b^{ij,kl}f_{ij}\tau^{4}f_{kl}dx=-\int_{B_{1}}b^{ij,kl}f_{ij}\left( (\tau^{4})_{kl}f+ (\tau^{4})_{l}f_{k}+ (\tau^{4})_{k}f_{l}\right) dx.
\left|\frac{\lambda_{+}}{\lambda_{+}- \lambda_{-}}+ \frac{1}{\nu^{2}}|\xi|^{2(1-2 \sigma)}\right| & = \frac{1}{\nu^{2}}|\xi|^{2(1-2 \sigma)}\left(\frac{-2}{\left(1+ \sqrt{1-\frac{4}{\nu^{2}}|\xi|^{2(1-2\sigma)}}\right)\sqrt{1-\frac{4}{\nu^{2}}|\xi|^{2(1-2\sigma)}}}+1\right)
E |\mu^N(t) - \widetilde{\mu}_N(t)| + \frac1N \sum_{i=2}^N E|X^N_i(t) - \widetilde{X}^N_i(t)| &\leq\frac dN + 2 K_1 \frac1N \sum_{i=2}^N \int_0^t \left[ E|X^N_i(s) - \widetilde{X}^N_i(s)| + E |\mu^N(s)- \widetilde{\mu}_N(s)|\right] ds
J =(W_t-W_r-\sigma)\beta e^{-\beta t}+f(W)-f(W+\sigma e^{-\beta t}) =(W_t-W_r-\sigma)\beta e^{-\beta t}-f'(W+\theta )\sigma e^{-\beta t}= e^{-\beta t}\Big\{\big[-f'(W+\theta)-\beta\big]\sigma+(W_t-W_r)\beta\Big\},
\left\langle \Theta_{2 2}( \vec{x}) \right\rangle = - \frac{1}{1 6 \pi^{2}}D_{2}( \vec{x}) + \frac{1}{4 8 \pi^{2}}D_{1}( \vec{x}) - \frac{1}{2 4 \pi^{2}}D_{2 2}( \vec{x}) + \frac{1}{6 4 \pi^{2}}F_{2}( \vec{x}) - \frac{1}{1 9 2 \pi^{2}}F_{1}( \vec{x})
V ( \overline{{\rho}}) = \frac{3 e^{4}}{6 4 \pi^{2}}\overline{{\rho}}^{4}\left( \ln \frac{\overline{{\rho}}^{2}}{\sigma^{2}}- \frac{1}{2}\right) + \frac{e^{2}}{8}T^{2}\overline{{\rho}}^{2}- \frac{e^{3}}{4 \pi}T | \overline{{\rho}}|^{3}
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_X=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{X};
_{2}F_{1}\left({\frac{1}{3}},{\frac{5}{6}};1;{-\frac{2}{25}}\right)=\frac{\sqrt[3]{5}}{\sqrt{3}}{\,}_{2}F_{1}\left({\frac{1}{6}},{\frac{2}{3}};1;{-\frac{2}{25}}\right),
\frac{\partial^{n+1}t}{\partial u^n\partial v}=\frac{\partial^{n-1}}{\partial u^{n-1}}(L\mu-K\nu) &=F_1+F_2\frac{\partial^nt}{\partial u^n}+F_3\frac{\partial^n\alpha}{\partial u^n}+F_4\frac{\partial^n\beta}{\partial u^n},
[D^*D,D^k]\psi= \sum_{l=0}^{k-1}D^{l}R^M\star D^{k-l}\psi+\sum_{l=0}^{k}D^{l}R^{SM}\star D^{k-l}\psi &\quad+\sum_{\sum l_i+\sum{m_j}=k}{}^{G}\nabla^{l_1}R^N\star\underbrace{D^{m_1+1}\phi\star\ldots\star D^{m_{l_1}+1}\phi}_{l_1-}\star D^{l_2+1}\phi\star D^{l_3+1}\phi\star D^{l_4}\psi.
[ W^{gen}, \underline{m}\partial_{\underline{m}}] = z \int \left\{\hat{\xi}_A \frac{m}{e}\frac{\delta}{\delta \varphi_2}+ \xi_A \frac{m}{e}\frac{\delta}{\delta \hat{\varphi}_2}\right\}+ (\chi \partial_{\xi}+ \chi_A \partial_{\xi_A}) z \xi_A \frac{m}{e}\int \frac{\delta}{\delta q_2}
\sum_{n=1}^{\infty}n^2\Vert f_n^d(t) \Vert_{L^1(S_2)}\Vert I_{1n}\Vert_{L^1(S_2)}\leq \Big(\sum_{n=1}^{\infty}\sum_{q=1}^{\infty}\dfrac{\vert a_{qn}\vert^2}{q^4}\Big)^{1/2}\Big(\sum_{n=1}^{\infty}n^4\Vert f_n^d \Vert_{L^1(S_2)}^2\Big)
x^2y^4 + xy^3z^3 = \left(xy^2 + \frac{1}{2}yz^3\right)^2 - \frac{1}{4}y^2z^6 x^4y^4 + x^2y^4z^3 = \left(x^2y^2 + \frac{1}{2}y^2z^3\right)^2 - \frac{1}{4}y^4z^6.
{\cal L}_{W Z}^{( 1 )}\, = \, - \,{\frac{1}{2}}\, ( \, \dot{\theta}^{2}\, + \, 3 \,{\theta^{\prime}}^{2}\, ) \, - \, \lambda \, ( \, \dot{\theta}\, + \, \theta^{\prime}\, ) \, - \,{\frac{1}{2}}\, \lambda^{2}\; \; ,
\tilde s_{k}- \tilde s_{k-1}= \sum_{i = 0}^{p-1}\mathbf{h}_i \cdot \mathbf{e}_{{k-i}}- \sum_{i = 0}^{p-1}\mathbf{h}_i \cdot \mathbf{e}_{{k-1-i}}&= \sum_{i = 0}^{p-1}(\mathbf{h}_{i}- \mathbf{h}_{i-1}) \cdot \mathbf{e}_{{k - i}}= \sum_{i = 0}^{p-1}\mathbf{n}_{i-1}\cdot \mathbf{e}_{{k-i}}= s_{k-1},
I_{2,2}(h) =\sup_{t\in \lbrack 0,t_{0}]}\Vert f(t,0)\Vert\int_{0}^{t_{0}}\left( (t_{0}-\tau )^{\alpha -1}-(t_{0}+h-\tau )^{\alpha-1}\right) d\tau & \leq \sup_{t\in \lbrack 0,t_{0}]}\Vert f(t,0)\Vert \times \frac{h^{\alpha}}{\alpha}.
\sum_{j,\vec{k},\vec{l}}c_{j\vec{k}\vec{l}}z^{j}p_{1}^{k_{1}}p_{2}^{k_{2}}\cdot\cdot\cdot p_{n}^{k_{n}}q_{1}^{l_{1}}q_{2}^{l_{2}}\cdot\cdot\cdot q_{n}^{l_{n}},
F(T) = \lim_{m\to\infty}\sum_{n=0}^{[2^m T]-1}\mathbb E\Bigl[F\Bigl(\frac{n+1}{2^m}\Bigr) - F\Bigl(\frac{n}{2^m}\Bigr)\Big|\mathcal F_{\frac{n}{2^m}}\Bigr],
f=\frac{(1-b^2)^{\frac{3}{2}}}{2}(x^{1}\dot{x}^{2}-x^{2}\dot{x}^{1}), g=\sqrt{(\dot{x}^{1})^2+(\dot{x}^{2})^2}+b(\cos \theta(x^1,x^2) \dot{x}^1 +\sin \theta(x^1,x^2) \dot{x}^2).
\omega^\xi_\eta(\rho)=w^\xi_\eta(\rho):=\sum_{\{\alpha,\beta;\epsilon\}\in\vec{\mathcal{\triangle}}^\xi_\eta}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho)+\ell_{\beta}(\rho))}+(-1)^{\alpha\cdot\beta}\right)^{-1}.
\nu_\alpha^{-n}:=u_\alpha^q(y_\alpha)+\left\vert\frac{\nabla u_\alpha(y_\alpha)}{u_\alpha(y_\alpha)}\right\vert^{n}+\left\vert\frac{\nabla^2u_\alpha(y_\alpha)}{u_\alpha(y_\alpha)}\right\vert^\frac{n}{2}+\vert\mathcal{L}_gW_\alpha(y_\alpha)\vert.
|u_*(|x|,t)|\le Ct^{-\gamma-\frac{d}{2}}, |F_N^0(|x|,t)|\le Ct^{-\gamma-\frac{d}{2}-1}|x|^2\le C\epsilon^2t^{-\gamma-\frac{d}{2}}, |(\partial_r F_N^0)(|x|,t)|\le Ct^{-\gamma-\frac{d}{2}-1}|x|\le C\epsilon t^{-\gamma-\frac{d}{2}-\frac{1}{2}},
[N(x),N(y)] = N(x) \cdot N(y) - N(y) \cdot N(x) &= N\big(N(x) \cdot y + x \cdot N(y) - N(x\cdot y) - N(y) \cdot x - y \cdot N(x) + N(y \cdot x)\big) &= N\big([N(x), y] + [x,N(y)] - N[x, y]\big).
\mathcal{L}^0(h,k) = - \sum_{\substack{1\leq c\le C (c,hk)=1}}\mu(c)\sum_{\substack{1\leq m,n<\infty (mn,c)=1}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right)\sum_{\substack{1\le d < \infty (d,mhnk)=1}}W\left( \frac{cd}{Q}\right).
-G(z,n)^{-1}=z-q_n+b_n^2m_n^+(z)+b_{n-1}^2m_n^-(z) =z-q_n+b_n^2\sum_{k\in\widetilde{M}}\frac{\tau_k}{c_k-z}+b_{n-1}^2\sum_{j=1}^{n-1}\frac{\kappa_j}{d_j-z}\,,
\langle D_1J_1(f;v^1+sw^1,v^2)- D_1 J_1(f;v^1,v^2),w^2 \rangle = \iint_{\mathcal{O}_{1}\times (0,T)}(\phi^s-\phi) w^2 dxdt + s \mu_1 \iint_{\mathcal{O}_1 \times (0,T)}w^1 w^2 dxdt.
\nabla^L_{X_j}X_j=0,~~~1\leq j\leq 3,~~~\nabla^L_{X_1}X_2=\frac{L-1}{2L}X_3,~~~ \nabla^L_{X_2}X_1=\frac{-L-1}{2L}X_3, &\nabla^L_{X_1}X_3=\frac{1-L}{2}X_2,~~\nabla^L_{X_3}X_1=\frac{-1-L}{2}X_2,\nabla^L_{X_2}X_3=\nabla^L_{X_3}X_2=\frac{1+L}{2}X_1.
g^{(1)}_{\mu\nu}= -{l^2 \over d-2}\left({1\over a^2}-1 \right) \left( R_{\mu\nu}-{1\over 2(d-1)}h_{\mu\nu}R \right) -{2l \over d}\left({1 \over a^d}- 1 \right) \chi_{\mu\nu}\ ,
v(t,x)-v(t,y) =\mathbb{\hat{E}}[\mathbf{1}_{B_t+x\leq 0}]-\mathbb{\hat{E}}[\mathbf{1}_{B_t+y\leq 0}]\geq -\mathbb{\hat{E}}[-\mathbf{1}_{-y<B_t\leq -x}]>0.
\begin{array}{l}\|L(|w_j|^{p-1}|w_j-w_{j-1}|)|u_j|^q\|_1 =\|L\left(\left||w_j|^{(p-1)/p}|w_j-w_{j-1}|^{1/p}\right|^p|u_j|^q\right)\|_1 \le C(T+R)\||w_j|^{(p-1)/p}|w_j-w_{j-1}|^{1/p}\|_2^p\|u_j\|_1^q \le C(T+R)\|w_j\|_2^{p-1}\|w_j-w_{j-1}\|_2\|u_j\|_1^q\end{array}
K_{i j}( x_{0}, x ; y_{0}, y ; t ) = \frac{1}{\beta ( 2 \pi )^{3}}\sum_{r = - \infty}^{\infty}\int d^{3}k \, e^{i ( k ( x - y ) + 2 \pi r ( x_{0}- y_{0}) / \beta )}e^{- t k^{2}}e^{- t ( 2 \pi r / \beta - a_{0}( y ) )^{2}}\left( e^{- ( A + B )}\right)_{i j}
\sum_{n=1}^\infty \rho\left(\frac{1}{n^{\frac{1}{p}}}\right)\lambda\left(\frac{1}{n}\right)<\infty~~\hbox{and}~~\sum_{n=1}^\infty \sigma\left(\frac{1}{n^{\frac{1}{q}}}\right)\mu\left(\frac{1}{n}\right)<\infty.
\begin{cases}\lambda_1' = \sqrt{\frac{s_{-}(a)s_{+}(c)}{s_{+}(b)}}= \sqrt{\frac{s_+(c)}{s_+(a)s_+(b)}}= \frac{1}{\lambda_2}\lambda_2' = \sqrt{\frac{s_{-}(a)s_{+}(b)}{s_{+}(c)}}= \sqrt{\frac{s_+(b)}{s_+(a)s_+(c)}}= \frac{1}{\lambda_1}\lambda_3' = \sqrt{\frac{s_{+}(b)s_{+}(c)}{s_{-}(a)}}= \sqrt{s_+(a)s_+(b)s_+(c)}= \frac{1}{\lambda_4}.\end{cases}
\bar{D}^{\dot{\alpha}}J^{0}_{\alpha\dot{\alpha}}= \frac{1}{3}D_{\alpha}\left\{\left[3{\cal W}- \sum_i S_i \,\frac{\partial{\cal W}}{\partial S_i}\right] - \left[\frac{b}{16\pi^2}{\rm Tr}\, W^2 + \frac{1}{8}\sum_i\gamma_i Z_i \bar{D}^2\left({S_i}^\dagger e^{V}S_i\right) \right] \right\}
\begin{cases}\frac{d}{dt}y(t)=-\frac{1}{12}\frac{y(t)^4+z(t)^4}{y(t)^5 z(t)^8}, \, \,\, \frac{d}{dt}z(t) = \frac{1}{12}\frac{z(t)^2-y(t)^2}{y(t)^4 z(t)^7}, y(0)=1, \, \,\, z(0)=1.\end{cases}
(1-\pi_{11})(1-\pi_{22}) > (1 - \bar\alpha_1)(1 - \bar\beta_1)(1 - \bar\alpha_2)(1 - \bar\beta_2) &= (1 - \bar\alpha_1)(1 - \bar\beta_2)(1 - \bar\alpha_2)(1 - \bar\beta_1) = (1-\pi_{12})(1-\pi_{21}).
\mathcal{L}_{os}=\frac{1}{2}\left( \partial_{\mu}\eta \partial^{\mu}\eta+2\mu^{2}\eta^{2}\right) +\frac{1}{2}\left( \partial_{\mu}\xi \partial^{\mu}\xi \right) -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+qvA_{\mu}\left(\partial_{\mu}\xi \right) +\frac{q^{2}v^{2}}{2}A_{\mu}A^{\mu}~.
-{d^3 \over d \tau^3}\Big ( (\tau^3 - 4 \tau) \hat{r}(\tau) \Big ) + 4{d^2 \over d \tau^2}(\tau^2 \hat{r}(\tau)) - 2 (1 - 2 p^2){d \over d \tau}( \tau \hat{r}(\tau)) - 4 p^2 \hat{r}(\tau) = 0,
H(t, x) = \int_{-\infty}^{\infty}v_{\alpha_{1}, \beta_{1}}(t, y) v_{\alpha_{2}, \beta_{2}}(t, x-y) dy = \int_{-\infty}^{\infty}v_{\alpha_{1}, \beta_{1}}(t, x-y) v_{\alpha_{2}, \beta_{2}}(t, y) dy,
\Phi_{v a c}= \left( \begin{array}{c c c c}{{0}}&{{V_{v a c}}}&{{0}}&{{0}}\\{{V_{v a c}^{*}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
\int_{0}^{\infty}U(r)\ln(r/a)r\,dr = \frac{1}{|Q|\ln\left(\sqrt{2}|Q|^{1/2}/a\right)}\int_{a}^{\sqrt{2}|Q|^{1/2}}\ln(r/a)r\,dr &\leq \frac{1}{|Q|}\int_{a}^{\sqrt{2}|Q|^{1/2}}r\,dr \leq 1.
SINR^1_{1,1}= \frac{|\mathbf{v}_{1,1}^H\mathbf{H}_{1,1}\mathbf{p}_1 |^2 \alpha_{1,1}^2}{\sum_{m=2}^{M}|\mathbf{v}_{1,1}^H\mathbf{H}_{1,1}\mathbf{p}_{m}|^2 +|\mathbf{v}_{1,1}|^2\frac{1}{\rho}}.
\lVert \psi_q(a) + \Phi_p(a) (1 - h^+_q) \rVert = \lVert \psi_q(a) + \Phi(a) (1 - h^+_q) \rVert &\approx_{\frac{\gamma}{36M}}\lVert \psi_q(a) + \Phi_q(a) (1 - h^+_q) \rVert < \frac{3}{2}\lVert a \rVert,
P_{0,x,y}(p) = 0,P_{1,x,y}(p) = - \frac{p}{xy},P_{2,x,y}(p) = \frac{3 p \left(p^2-x^2-y^2\right)}{2 x^2 y^2}P_{3,x,y}(p) = -\frac{3 p \left(5 \left(p^2-x^2\right)^2+2 y^2 \left(3 x^2-5 p^2\right)+5 y^4\right)}{8 x^3 y^3}
{\cal R}_{abc}^d=\partial_b\Omega_{ac}^d-\partial_c\Omega_{ab}^d+\Omega_{0b}^d\Omega_{ac}^0+\Omega_{eb}^d\Omega_{ac}^e-\Omega_{0c}^d\Omega_{ab}^0-\Omega_{ec}^d\Omega_{ab}^e.
{}_2\phi_1 \biggl(\!\! \begin{array}{c}a,b c \end{array}\! ;q,z \biggr) = \frac{(b,az;q)_{\infty}}{(c,z;q)_{\infty}}\,_2\phi_1 \biggl(\!\! \begin{array}{c}c/b,z az \end{array}\! ;q, b \biggr)
f\left(x\right) =e^{-x}\int_{0}^{x}\varphi\left(y\right)e^{y}dy+e^{x}\int_{x}^{1}e^{-y}\varphi\left(y\right)dy f'\left(x\right) =-e^{-x}\int_{0}^{x}\varphi\left(y\right)e^{y}dy+e^{x}\int_{x}^{1}\varphi\left(y\right)e^{-y}dy f''\left(x\right) =f\left(x\right)-2\varphi\left(x\right);
\rho_{1,0,n}=\left(n-\frac{1}{2}\right)\pi+\frac{(-1)^{n+1}q_1(1)}{n\pi}\cos \left(n-\frac{1}{2}\right)\pi a+\frac{\kappa_{1,0,n}}{n},\, \, \{\kappa_{1,0,n}\}\in l_2, n\in\mathbb{Z}.
t^{N(M-N)}\prod_{j=1}^N z_j^{j-1-N}(1+tz_j^2)\prod_{1 \le j<k \le N}(1+tz_j z_k)(1+tz_j z_k^{-1})sp_{\overline{\lambda}}( \{tz \}_N ) = \sum_{\{\overline{x^1}\},\dots,\{\overline{x^{N-1}}\}}\prod_{k=1}^N(\alpha(z_k,\{\overline{x^{k-1}}\},\{\overline{x^{k}}\})+\beta(z_k,\{\overline{x^{k-1}}\},\{\overline{x^{k}}\})),
D\mathfrak{u}=\sum_{l,k}e^{ilt}\left( \begin{array}{c}e^{i(k-\frac{1}{2})\theta}(2lU^+_{k,l}+\frac{d}{dr}U^-_{k,l}+\frac{(k+\frac{1}{2})}{r}U^-_{k,l}) e^{i(k+\frac{1}{2})\theta}(-2lU^-_{k,l}-\frac{d}{dr}U^{+}_{k,l}+\frac{(k-\frac{1}{2})}{r}U^{+}_{k,l})\end{array}\right)=0
\begin{aligned}A_{2}=\frac{1}{q^{4}(q+1)(q-1)^2}(q^7 + q^{2 m}- q^{1 + m}- q^{5 + m}- q^{6 + m}+ q^{7 + m}+ q^{2 + 2 m}- q^{5 + 2 m}- q^{6 + 2 m}+ q^{7 + 2 m}) =\frac{1}{q^{4}(q+1)(q-1)^2}(q^7+ q^{m+1}(q^{6}-q^5-q^4-1) + q^{2 m}(q^{7}-q^6-q^5+q^2+1))>0 \end{aligned}
\mathbb E_{P_\sigma}\left[e^{iX_\varphi^{(\sigma)}}\right]=\int_{\mathcal S^\prime}e^{i\langle \varphi,\xi\rangle}dP_\sigma(\xi)=e^{-\frac{1}{2}\int_{\mathbb R}|\widehat{\varphi}(u)|^2d\sigma(u)}.
f=\sum_{j=1}^{4s+1}\left(\delta_{b_j}^{(\alpha-\beta)/2}+\delta_{b_j}^{-(\alpha-\beta)/2}-\delta_{b_j}^{n-(\alpha+\beta)/2}-\delta_{b_j}^{-(n-(\alpha+\beta)/2))}\right)^{s}\ast f_j.
\exists A>0,\ \left\Vert E^{l}u\right\Vert_{2,\left( \tilde{P}_{0}\right)^{l}}\leq A^{l}\left\Vert u\right\Vert_{L^{2}},\forall u\in C_{0}^{\infty}\left( \Omega \right) ,\l=0,1,...,
H = N \left[ - \frac{1}{2 4}\epsilon_{i j k}( \hat{\theta}^{i}- A^{i}) ( \hat{\theta}^{j}- A^{j}) ( \hat{\theta}^{k}- A^{k}) + \frac{1}{4}F^{i}( \hat{\theta}^{i}- A^{i}) \right] + h \, ,
p^{\alpha_3}_{k-1}(3,t)=-(-\lambda)^{k-1}\underset{\Theta^{k-1}_{3}}{\sum}\frac{t^{k_0\alpha_0+k_1\alpha_1+k_2\alpha_2+k_3\alpha_3}}{\Gamma\left(k_0\alpha_0+k_1\alpha_1+k_2\alpha_2+k_3\alpha_3+1\right)}.
\vert \pi_{P_{F(w)}}(DF(z)a-DF(w)a)\vert =\vert \pi_{P_{F(w)}}(Df(z)a-Df(w)a)\vert =\vert (\pi_{P_{F(w)}}-\pi_{P_x})(Df(z)a-Df(w)a)\vert \leq C_2(m) \tilde \delta_{x,k}\vert Df(z)a-Df(w)a\vert.
\mathcal{L}^0(h,k) := \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}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right)\sum_{\substack{c>C, d \geq 1 cd=q}}\mu(c)
\frac{du_\varepsilon(t,\ell)}{m}= -\frac{\kappa_c}{2}\left\{\ell\, \frac{\nabla u_\varepsilon(t,\ell-\varepsilon)}{\varepsilon}+m\mu_c(u_\varepsilon(t,\ell-\varepsilon) +u_\varepsilon(t,\ell)) +2mp(N-u_\varepsilon(t,\ell-\varepsilon))\right\}dt
\psi(u) =\mathbb{P}\left(\bigcup_{t\geqslant 0}\left\{W(t)<0\right\}\right) &=\mathbb{P}\left(\inf_{n\geqslant1}\left\{u+c T_n-\sum_{i=1}^{n}X_n\right\}<0\right) &=\mathbb{P}\left(\sup_{n\geqslant1}\sum_{k=1}^n(X_k-c \theta_k)>u\right).
0 = \left({\int}_{- L}^{L}d x^{-}\left[{\Theta}_{\mu}^{\; \; +}( x ) \right]_{x^{+}= - T}^{x^{+}= T}+{\int}_{- T}^{T}d x^{+}\left[{\Theta}_{\mu}^{\; \; -}( x ) \right]_{x^{-}= - L}^{x^{-}= L}\right) ,
\sum_{m=1}^{n}\frac{1}{m!}\sum_{\substack{n_0+\ldots+n_m=n n_0\ge0 n_1,\ldots,n_m\ge1}}\{\{\ldots\{\{f_{n_0},G^0_{n_1}\},G^0_{n_2}\}\ldots\},G^0_{n_m}\}
\ln \left[{\cal L}^{\mathrm{I},{\cal R}}( \nu , x ) \right] \sim \ln \left[ 1 + \sum_{k \geq 1}{\frac{v_{k}( t ( x ) )}{\nu^{k}}}+{\alpha}{\frac{t ( x )}{\nu}}\left( 1 + \sum_{k \geq 1}{\frac{u_{k}( t ( x ) )}{\nu^{k}}}\right) \right] = \sum_{n \geq 1}{\frac{{\cal U}_{n}^{\mathrm{I},{\cal R}}( t ( x ) )}{\nu^{n}}}
\widetilde{H}'_c = \frac{1}{4}(B_{ij}-Q_{ij})^2-\frac{1}{2}(A_i+\partial_i\theta)^2+\frac{1}{4m^2}F_{ij}F^{ij}-\frac{1}{2\cdot3!m^2}H_{ijk}H^{ijk}.
D_{n,k}(y+\alpha y^{-1},\alpha)=\frac{y^{2n}+k\alpha y^{2n-2}+\cdot\cdot\cdot+k\alpha^{n-1}y^{2}+\alpha^{n}}{y^{n}}=\frac{y^{2n}+\alpha^{n}}{y^{n}}+(\frac{k\alpha}{y^{n}})\frac{y^{2n}-\alpha^{n-1}y^{2}}{y^{2}-\alpha},
\sqrt{2}\pi^{-s-1/4}\zeta(s+1) \Gamma(s/2+1/4+ir)\Gamma(s/2+1/4-ir)k^{s/2}\sum_{n\geq 1}\frac{\rho(n)}{n^{s-1/2}}&=k^{s/2}\pi^{-s-1/2}\Gamma(\frac{s+1/2+2ir}{2})\Gamma(\frac{s+1/2-2ir}{2}) \sum_{n=1}^{\infty}\frac{a_{u}(n)}{n^{s+1/2}}.
M_{1, 1}(\delta, \nu) \leq (C_{1}\nu^{-O(1)})^{2}M_{2^{N - 1}, 2^{N}}(\delta, \nu)^{1/2^N}\prod_{j = 0}^{N-1}D(\frac{\delta}{\nu^{2^j}})^{1/2^{j + 1}}.
\kappa_1=\frac{\sqrt{20}}{3}\frac{\sigma_2}{\sigma_1^2}, \hskip .1in \kappa_2=\frac{\sqrt{105}}{2}\frac{\sigma_3}{\sigma_1 \sigma_2}, \hskip .1in \kappa_3=\frac{\sqrt{336}}{5}\frac{\sigma_4}{\sigma_1 \sigma_3}, \hskip .1in \kappa_4=\dfrac{\sqrt{825}}{4}\frac{\sigma_5}{\sigma_1 \sigma_4}.
\overline{\Pi}_{1}\overline{\overline{N}}(a)=-\theta_{1}\overline{\pi}_{1}\overline{p}\xi_{e_{k}a}&\overline{\Pi}_{2}\overline{\overline{N}}(a)=-\theta_{2}^{-1}\overline{\gamma}' \xi_{e_{k}}a &\overline{\Pi}_{3}\overline{\overline{N}}(a) = \overline{\Pi}_{4}\overline{\overline{N}}(a) = 0
{\frac{\chi^{\prime \prime}}{A}}+{\frac{1}{A}}\left({\frac{B^{\prime}}{2 B}}-{\frac{A^{\prime}}{2 A}}+{\frac{2}{r}}\right) \chi^{\prime}-{\frac{\operatorname{s i n}2 \chi}{r^{2}}}= 0 \, .
\begin{aligned}\widetilde{u^{\alpha}_{J_1+\bold{1};J_2}}=S_{J_2}\left(D_{\widetilde{x}}\widetilde{u^{\alpha}_{J_1;\bold{0}}}\right)=(S_{J_2}D_{\widetilde{x}})\widetilde{u^{\alpha}_{J_1;J_2}}&=\left(D_x\widetilde{u^{\alpha}_{J_1;J_2}}\right)S_{J_2}(D_x\widetilde{x})^{-1}.\end{aligned}
\int_{s_1}^{s_2}\big| \varphi''(s) \big| \, ds = \left| \int_{p_1 + \delta}^{p_0-\delta}\left( \frac{1}{\psi'}\right)'(p) \, dp \right| = \left| \frac{1}{\psi'(p_0 - \delta)}- \frac{1}{\psi'(p_1 + \delta)}\right| \leqslant \delta^{1-\rho}m^{-1}\; ,
M(w_1,\dots,w_{\chi(1)}) =\sum_{k,l}m_{k,l}v_{l,k}(w_1,\dots,w_{\chi(1)}) &=\sum_{k,l}m_{k,l}(w_1,\dots,w_{\chi(1)})E_{k,l}&=(w_1,\dots,w_{\chi(1)})(m_{i,j}),
\Xi^{\lambda}(K_{a}\otimes F_{a}\otimes1\otimes E_{a})=\sum_{i,j,n,o,p}(2c_{j}^{i}-\delta_{j}^{i})c_{n}^{j}\pi(K_{a}F_{a})_{o}^{n}c_{p}^{o}\pi(E_{a}K_{\lambda})_{i}^{p}.
K(\psi,A) = \textrm{Var}[V_2 |\, F_{1}(V_1) \in A], K'(\psi,A) =\frac{\textrm{Var}[V_2 |\, F_{1}(V_1) \in A]}{\textrm{Var}[V_1 |\, F_{1}(V_1) \in A]},
S ={\frac{1}{2 \kappa_{\scriptscriptstyle{D}}^{2}}}\int d^{\scriptscriptstyle{D}}x \sqrt{- g_{\scriptscriptstyle{D}}}e^{- \varphi}\left\{R_{\scriptscriptstyle{D}}+ ( \nabla \varphi )^{2}-{\frac{1}{1 2}}H^{2}\right\},
L(\tilde\sigma) =L(\sigma)+m, \ell(\tilde\sigma)=\ell(\sigma)+n-1 L(\hat\sigma) =L(\sigma)+m, \ell(\hat\sigma)=\ell(\sigma)+n-1 L(\check\sigma) =L(\sigma)+2m,\:\: \ell(\check\sigma)=\ell(\sigma)+2(n-1),