harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\textbf{w}_{j,k}= \frac{1}{\tau\lambda \sigma_{\hat{h}_{1,k}}^2}\textbf{h}_{j,k}\bigg(\sum_{k'\neq k}^{K}\big(\textbf{p}_{1,k}\textbf{s}_{j,k'}^{H}\big)\textbf{h}_{j,k'}^H+\sum_{i\neq j}^L\sum_{k'=1}^{K}\big(\textbf{p}_{1,k}\textbf{s}_{i,k'}^H \big)\textbf{h}_{i,k'}^H + \textbf{p}_{1,k}\textbf{N}_i^H+\textbf{h}_{1,k}\bigg)\hat{\textbf{h}}~\!\!_{1,k}^H.
\left( \Psi + n^{\mu}\, \nabla_{\mu}\right) \varphi \longrightarrow \Omega^{-{\frac{{\cal D}- 1}{2}}}\left( \Omega \overline{{{\Psi}}}\varphi + n^{\mu}\nabla_{\mu}\varphi -{\frac{{\cal D}- 2}{2}}{\frac{n^{\mu}\partial_{\mu}\Omega}{\Omega}}\varphi \right) ,
\int_0^{|x|^{-2}}\frac{y^{d/2-1}}{\phi(y)}\,dy\leq \frac{c_3}{\phi(|x|^{-2})}\int_0^{|x|^{-2}}y^{d/2-1}\left(\frac{|x|^{-2}}{y}\right)^{\delta}\,dy=\frac{c_4}{|x|^d\phi(|x|^{-2})}
(\sum_{n=0}^{N}a_n\square_{n})(k+\Delta')=\sum_{n=0}^{N}a_n\square_{n}(m-\square_{1}) &=\sum_{n=0}^{N}ma_n\square_{n}-\sum_{n=0}^{N}a_n(\square_{n+1}+A_n\square_{n}) &=ma_0+\sum_{n=1}^{N}\left((m-A_n)a_n-a_{n-1}\right)\square_{n}-a_N\square_{N+1}&=1-a_N\square_{N+1},
-P_g(r^{6-n})= \Big[\Delta^3+\Delta \delta T_2 d+\delta T_2 d \Delta+\frac{n-2}{2}\Delta(\sigma_1(A)\Delta)+\delta T_4 d-\frac{n-6}{2}Q_g\Big](r^{6-n}) := \sum_{k=1}^6 I_k.
\leq C \int_{\mathbb{R}^{n}}\left\{\sum_{j=1}^{k}\left(\frac{\lambda_{j}|B_j|^{\frac{\alpha}{n}}\chi_{B_j}(x)}{\| \chi_{B_j}\|_{p(.)}}\right)^{p_0}\right\}^{\frac{q_0}{p_0}}w(x) dx
P(S|E,A)=\prod_{i=1}^{I}\Phi_{i}!n(\Phi|e_{j})*\prod_{j=1}^{J}t(f_{j}|e_{a_{j}})*\prod_{j:a(j)\ne0}^{J}d(j|a_{j},I,J)(\begin{matrix}J-\Phi_{0}\\ \Phi_{0}\end{matrix})p_{0}^{\Phi_{0}}p_{1}^{J}
m^n_K(\xi)=-\frac{1}{\Delta t_n}\left[(v^{n+1/2}_K-\xi)^+-(v^n_K-\xi)^+\right]-\frac{1}{|K|}\sum_{L\in\mathcal{N}(K)}\int_\xi^{+\infty}a^n_{K\to L}(\zeta)d\zeta.
\iint_\Omega|\nabla\phi|^2~d\mathbf{x}dy=-\int_{\mathbb{R}^{n-1}}[\phi]_\Gamma(\mathbf{c}\cdot\nabla)\eta~d\mathbf{x}=\int_{\mathbb{R}^{n-1}}\eta\mathbf{c}\cdot([\nabla_\mathbf{x}\phi]_\Gamma+[\phi_y]_\Gamma\nabla\eta)~d\mathbf{x}.
p_\pi = \sum_\lambda \frac{\langle J_\lambda, p_\pi \rangle}{\langle J_\lambda, J_\lambda \rangle}J_\lambda = \langle p_\pi, p_\pi \rangle \sum_\lambda \frac{\theta_\pi(\lambda)}{\langle J_\lambda, J_\lambda \rangle}J_\lambda.
\displaystyle{\raisebox{1.1ex}{\scriptsize{$\diamond$}}}\mbox{\hspace{-0.33ex}}\tilde{S}_\tau^{(2 1)}(Q - K) \simeq i \, [\theta (q_0) - n_F (q)] \, \frac{\tilde{g}_\tau^{(i)}(q_0, q)}{[q_0 - \tau (q + m_f^2 / q) - k_0 + \tau{\bf k}\cdot \hat{{\bf q}}]^2 + [\tilde{g}_\tau^{(i)}(q_0, q) ]^2}\, ,
z_{\sf Aff}({\cal U}) \leq 2 \cdot z_{\sf Aff}( \tilde{\cal V}) &= 2 \cdot O \left( \frac{\log n}{\log \log n}\right) \cdot z_{\sf AR}( \tilde{\cal V}) &\leq 2 \cdot O \left( \frac{\log n}{\log \log n}\right) \cdot 4 \log L \cdot z_{\sf AR}({\cal U})= O\left( \frac{\log n}{\log \log n}\ \cdot \log L \right) \cdot z_{\sf AR}({\cal U}),
\lim_{n\to\infty}\frac{\textup{dim}(Y_n)}{2\lambda(n)}\leq\lim_{n\to\infty}\frac{n^2(n!)+n\big(\sum_{i=0}^{n-1}\sigma(i)\big)}{n^2(n!)}=1+\lim_{n\to\infty}\frac{1}{n}=1.
\!\!\!\!\!\!\!\!\!\! -{1\over 2}\sin{\phi_3}\sin{\phi_2}\sin{\phi_1}\, t\,\hat{\rho}(-\theta_3-\theta_2) - \sin^2{\phi_3\over 2}\sin{\phi_2}\cos{\phi_1}\, t\,\hat{\rho}(2\theta_3 -\theta_2)
\frac{\sum_{j=1, j \neq \ell}^{K}\gamma^2_{(j,\ell)}P_{(j,\ell)}+\gamma^2_{(\ell,\ell)}P_{\ell,\ell}}{\gamma^2_{(\ell,\ell)}P_{\ell,\ell}}=\sum_{j=1 \atop j \neq \ell}^{K}\frac{\gamma^2_{(j,\ell)}+\gamma^2_{(\ell,\ell)}}{\gamma^2_{(\ell,\ell)}}
(1-b^2)^{\frac{3}{2}}r+\lambda\left(\frac{r}{\sqrt{r^2+\dot{r}^2}}+b(\left( -\dot{r}\sin(\theta-t)+r\cos(\theta-t))\theta_{r}+\sin(\theta-t)\right) \right) -\lambda\frac{d}{dt}\left(\frac{\dot{r}}{\sqrt{r^2+\dot{r}^2}}+b\cos(\theta-t)\right) =0.
E_2 = n^{-1/3}\int_{n^{1/3}\mathcal{C}_{0\not\in}^{\rm local}}\int_{n^{1/3}\mathcal{C}_{0\not\in}^{\rm local}}\frac{(\tilde z-\tilde w)e^{\frac{\sigma^3}{3}\tilde z^3 + \frac{\sigma^3}{3}\tilde w^3 -\sigma x\tilde z -\sigma y\tilde w}}{4\tilde z\tilde w(\tilde z+\tilde w)}\frac{4(\tilde z +\tilde w)}{2\alpha-1}\mathrm{d}\tilde z\mathrm{d}\tilde w.
\int\limits_GF(g)\,d'g =2^n\int\limits_A\int\limits_{N_1}\int\limits_{N_2}\int\limits_KF(an_1n_2k)\,dk\,dn_2\,dn_1\,da =2^n\int\limits_A\int\limits_{N_1}\int\limits_{N_2}F(n_2an_1)\det(a)^{-(n+1)}\,dn_2\,dn_1\,da.
\sum_{n \leq x}\prod_{j=1}^k F_{g_j}(n) = \sum_{n \leq x}\prod_{j=1}^k \sum_{p\leq z}g_j(p)f_p(n) &= \sum_{p_1,\dots,p_k \leq z}g_1(p_1) \cdots g_k(p_k) \sum_{n \leq x}f_{p_1\cdots p_k}(n) &= \sum_{p_1,\dots,p_k \leq z}g_1(p_1) \cdots g_k(p_k) \big( H(p_1\cdots p_k)x + O\big( 2^{\omega(p_1\cdots p_k)}\big) \big),
\int_{f^{-1}(B_\frac\rho2(x))}|A|^4 + |\nabla A|^2\,d\mu\le \int_{f^{-1}(B_\rho(x))}|A|^4 + |\nabla A|^2\,d\mu\bigg|_{t=0}+ c_0t\rho^{-4}\varepsilon_1 \,,t\in[0,t_0)\,.
K_{\alpha,\mathcal{F},z}^{s}=\{(v,w)\in \tilde{E}_{z}^{s}\oplus \tilde{E}_{z}^{u}: \Vert w\Vert_{\star}< \alpha \Vert v \Vert_{\star}\}\cup\{(0,0)\}, K_{\alpha,\mathcal{F},z}^{u}=\{(v,w)\in \tilde{E}_{z}^{s}\oplus \tilde{E}_{z}^{u}: \Vert v\Vert_{\star}< \alpha \Vert w\Vert_{\star}\}\cup\{(0,0)\}.
\begin{multlined}\gamma_{\mathcal M\rtimes\mathcal G}\bigl((f,x);(f_1,x_1),\dots,(f_n,x_n)\bigr) = \left(\gamma_{\mathcal M}(f;f_{x^{-1}(1)},\dots,f_{x^{-1}(n)}),\gamma_{\mathcal G}(x;x_1,\dots,x_n)\right)\ .\end{multlined}
N_6[w] = \{\dot{\sf M}+ f(u) - f_2(u) \}(1-\chi_4) = \underbrace{\{\dot{\sf M}+ f(u) - f_2(u) \}(1-\chi_{4,{\sf a}})}_{={\sf t}_5}+ \underbrace{\{\dot{\sf M}+ f(u) - f_2(u) \}(\chi_{4,{\sf a}}-\chi_4)}_{={\sf t}_6}.
\lim_{n\rightarrow\infty}\left|\frac{(n!)^2}{((n+1)!)^2}\right|\left|\frac{\left|\gamma(2n+1,-\log( \mathrm{\mathsf{x}}))\right|}{\left|\gamma(2n-1,-\log( \mathrm{\mathsf{x}}))\right|}\right|=\lim_{n\rightarrow\infty}\left|\frac{1}{(n+1)^2}\right|\log( \mathrm{\mathsf{x}})^2=0\;.
(\overline{\bf 4},{\bf 1},{\bf 2})(-1) =(\overline{\bf 3},{\bf 1})(\textstyle{-{1 \over 3},{1 \over 2},-1}) \oplus(\overline{\bf 3},{\bf 1})(\textstyle{-{1 \over 3},-{1\over 2},-1}) \oplus({\bf 1},{\bf 1})(\textstyle{1,{1 \over 2},-1}) \oplus({\bf 1},{\bf 1})(\textstyle{1,-{1 \over 2},-1})
\Pi^{( 1 )}( k^{2}, m ) = \frac{g^{2}}{2 \pi}\left[ \frac{\sqrt{m^{2}}}{2}- \left( \frac{m^{2}}{2}+ \frac{k^{2}}{8}\right) \frac{1}{\sqrt{k^{2}}}l n \left( \frac{2 \sqrt{m^{2}}+ \sqrt{k^{2}}}{2 \sqrt{m^{2}}- \sqrt{k^{2}}}\right) \right] \ ,
\begin{aligned}Y \le c\cdot \,\int_s^{s+\delta}|X_u-X_s|^{p-2}\cdot \bigl( 1+X_u^2 + |X_s|\cdot(1 + |X_u|^{\ell_\mu+1})\bigr)\, du & + \sup_{s\le t\le s+\delta}\Bigl|\int_s^t p \cdot (X_u-X_s)\cdot |X_u-X_s|^{p-2}\cdot \sigma(X_u)\, dW_u\Bigr|\end{aligned}
\mathop{\mathtt{Min.}}\limits_{\{\mathbf{v}^{}_i\},\{p^{}_i\}}~~ \sum_{i=1}^{K}p^{}_i \mathtt{s.t.}~~^{}_i \geq \gamma^{}_i, \forall i \in \mathcal{K}& ~~ 0 \leq p^{}_i \leq P^{}_{i,}, \forall i \in \mathcal{K}.
t_1^{a_2^k}(a_1^{x_1}a_2^{x_2}a_3^{x_3}) =x_1=t_1 t_2^{a_2^k}(a_1^{x_1}a_2^{x_2}a_3^{x_3}) =x_2-1=t_2-k t_3^{a_2^k}(a_1^{x_1}a_2^{x_2}a_3^{x_3}) =x_3=t_3 t_1^{a_3^k}(a_1^{x_1}a_2^{x_2}a_3^{x_3}) =x_1=t_1 t_2^{a_3^k}(a_1^{x_1}a_2^{x_2}a_3^{x_3}) =x_2=t_2 t_3^{a_3^k}(a_1^{x_1}a_2^{x_2}a_3^{x_3}) =x_3-1=t_3-k
\Delta \phi = \Delta (u^2+|\nabla u|^2 ) =2\left( u \Delta u+ |\nabla u|^2 + |\nabla^2 u|^2 + \nabla \Delta u \cdot \nabla u+ Ric(\nabla u, \nabla u)\right) \geq 2\left( -nu^2 +|\nabla u|^2+ \frac{(\Delta u)^2}{n}-n |\nabla u|^2+ (n-1) |\nabla u|^2\right) =0.
\left\langle f,g \right\rangle_{\mathcal{H}_K}=\sum_{k=0}^M \beta_k^2\sum_{j=1}^{2k+1}\left\langle Y_{k,j},f\right\rangle_{L^2(\mathbb{S}^2)}\left\langle Y_{k,j},g\right\rangle_{L^2(\mathbb{S}^2)}.
\partial_{\alpha}f^{j+1}=\left( -1\right)^{j+1}{\displaystyle\sum\limits_{k=0}^{j}}\frac{\left( -1\right)^{k}F_{\alpha}^{k}}{F_{y}^{k}}{\displaystyle\prod\limits_{i>k}^{j}}\frac{F_{x}^{i}}{F_{y}^{i}}& =-\frac{F_{x}^{j}\partial_{\alpha}f^{j}+F_{\alpha}^{j}}{F_{y}^{j}},
\partial_{t}\varphi + \frac{1}{2 m}({\mathrm{\boldmath ~ \nabla ~}}\varphi )^{2}+ \frac{\hbar^{2}}{4 m}\left[ \frac{1}{2}\left( \frac{{\mathrm{\boldmath ~ \nabla ~}}\rho}{\rho}\right)^{2}- \frac{\triangle \rho}{\rho}\right] + U = 0
\mathcal{H}'_{\tilde{x}^{(0)}}(\tilde{x}(s),\tilde{\lambda}(s))\left[\begin{array}{c}\frac{d\tilde{x}}{ds}\frac{d\tilde{\lambda}}{ds}\end{array}\right]=0, |( \frac{d\tilde{x}}{ds},\frac{d\tilde{\lambda}}{ds})\|_2=1, \ \tilde{x}(0)=\tilde{x}^{(0)}, \ \tilde{\lambda}(0)=e.
\mathcal{I}_2 = \exp\left({- \frac{\theta_{\mathrm{S}}N_0}{\Omega_{\mathrm{ST-SD}}P_{\mathrm{ST}}}}\right)\sum_{n=1}^N n{N \choose n}(-1)^{n+1}& \times \left( \mathcal{I}_{2,1,n}+ \mathcal{I}_{2,2,n}\right),
\begin{aligned}\left[c_{p,r}\left(\bigcap_{i=1}^{N}\left\{a_{i}<X_{i}<b_{i}\right\}\right)\right]^{p}\leq \left[\sum_{l=0}^{r}N^{lp}C_{H}^{lp/2}\left(\frac{M_{r}}{c}\right)^{lp}\right]P\left(\bigcap_{i=1}^{N}\left\{a_{i}-c_{i}<X_{i}<b_{i}+c_{i}\right\}\right)\end{aligned}
\frac{\lambda}{4}\int_A J(x_1,y) dy = \frac{\lambda}{4}\int_A \frac{f(|y-x_1|^{-1})}{|y-x_1|^d}dy \ge \frac{\lambda}{4}\int_A \frac{f((2r)^{-1})}{2^d r^d}dy &\ge \frac{\lambda}{2^{d+2}}a_2^{-1}2^{-\delta_2}f(r^{-1})r^{-d}|A| \ge C_3 f(r^{-1})
\frac{E_{k-2}'(\tau)}{k-2}\left\{\binom{k-3}{s-1}(-1)^r-\frac12\binom{k-2}{s-1}(-1)^{r}+ \binom{k-3}{r-1}(-1)^{s}-\frac12 \binom{k-2}{r-1}(-1)^{s}\right\},
SNR_{ins}= \frac{\sigma^2_s}{n_{eq}[i]}Tr({\boldsymbol R}_{SDA}[i]{\boldsymbol A}^\emph{H}_S[i]) = \frac{\sigma^2_s}{n_{eq}[i]}Tr(\sum_{k=1}^{n_r}{\boldsymbol R}_{{\boldsymbol G}_{eq_k}}[i]{\boldsymbol A}^\emph{H}_k[i]),
\mathrm{P}\left(\alpha_{1,n}^2 >0,\log(1+SINR^k_{1,n})>R_{1,k},SINR^n_{1,n}<\epsilon_{1,n}\right) =\mathrm{P}\left(\alpha_{1,n}^2 >0, SINR^n_{1,n}<\epsilon_{1,n}\right).
\frac{q[q^{2},-q^3,q^{4};q^{6}]_\infty(q^{6};q^{6})^4_\infty}{[q,-q,-q^2,-q^{2},-q^4;q^{6}]_\infty}=\frac{(q^2;q^2)_\infty^2}{(-q^2;q^2)_\infty^2}\sum_{n=-\infty}^\infty \frac{(-1)^nq^{n(3n+3)+1}}{1-q^{6n+3}}.
u^{(n)}(t,x) = \int_t^T \left(A_x u^{(n)}(s,x)+b(s,x) \cdot \nabla_x u^{(n)}(s,x) + g(s,x)\right) \, ds\intertext{we get}u(t,x) = \int_t^T \left(A_x u(s,x) + b(s,x) \cdot \nabla_x u(s,x) + g(s,x)\right) \, ds.
\delta_{\mathsf{P-IA}}= \max\left\{\frac{(1-\mu M)K}{Mr}~,~ \frac{(1-\mu M)K}{\min\{M,K\}}+ \mu(M+K-1)\right\}, = \frac{(1-\mu M)K}{Mr}, ~~~~~~~~~~~~~~\mu \leq \mu_1.
\begin{array}{c}A_{1111}= 0.2883, A_{1112}= -0.0031, A_{1113}=0.1973, A_{1122}= -0.2485, A_{1123}= -0.2939, A_{1133}=0.3847, A_{1222}=0.2972, A_{1223}=0.1862, A_{1233}=0.0919, A_{1333}=-0.3619, A_{2222}=0.1241, A_{2223}=-0.3420, A_{2233}=0.2127, A_{2333}=0.2727, A_{3333}=-0.3054.\end{array}
\pounds_{1}= \partial_{\mu}\tilde{\partial}_{+}\bar{\varphi}\partial^{\mu}\tilde{\partial}_{-}\varphi + \bar{\chi}_{\dot{\alpha}}\bar{\sigma}_{\mu}^{\dot{\alpha}\beta}{}\partial_{\mu}\tilde{\partial}_{-}\chi_{\beta}- s \tilde{\partial}_{+}\bar{f}\tilde{\partial}_{-}f
A_{\sigma(UV)}A_{\sigma(UW)}^* = A_I\pi(\sigma(UV))^{-1}( A_I\pi(\sigma(UW))^{-1})^* &=A_I(\overline{f(UV)}\pi(U)\pi(V))^{-1}\overline{f(UW)}\pi(U)\pi(W)A^*_I &=f(UV)\overline{f(UW)}A_I\pi(V)^{-1}(A_I\pi(W)^{-1})^* &=f(UV)\overline{f(UW)}A_VA_W^*,
\int d \mu_{C^{1}- C^{2}}[ \theta_{1}] \int_{\Lambda}d^{2}x t ( x ) : \cos \Big [ 2 \sqrt{\pi}\Big ( \theta_{1}^{( b )}( x ) + \theta_{2}^{( b )}( x ) \Big ) \Big ] :_{C^{1}- C^{2}, C^{2}}
|1+z|^p \ge \begin{cases}1 + p\Re(z) + \tfrac12 p(p-1) (\Re(z))^2 + \tfrac{p}2(\Im(z))^2 - C\eta |z|^2 |z|\le\eta 1+p\Re(z)+c \eta^{2-p}|z|^p |z|>\eta. \end{cases}
e'_k=\int^1_0\mathbb{L}''({\mathbf U}^a+{\mathbf V}_k+\tau\delta{\mathbf V}_k,\Psi^a+\Psi_k+\tau\delta \Psi_k)((\delta{\mathbf V}_k,\delta\Psi_k),(\delta{\mathbf V}_k,\delta\Psi_k))(1-\tau)d\tau,
|B_{3/4}\setminus E| \le |B_{1}| - |E| \le \pi - \frac{\pi}{(1+\delta)^{2}}= \frac{2+\delta}{(1+\delta)^{2}}\pi\delta \le \frac{2+\delta}{1+\delta}\pi\delta \le 2\pi\delta \le \frac{\pi (3/4)^{2}}{36}\,,
I_{(b, \tau)}=R_{k, \beta}\left(\frac{1}{(1-\delta)}\right)y^{\frac{1}{(1-\delta)}}+O_{\delta}\left(x \exp \left( -4 \exp\left( \kappa (c_k \alpha x^\delta)^{\frac{1}{4}}\right)^{-1}\right) \right).
\frac{(2n!-2^{\lfloor log_{2}n!\rfloor+1})\cdot\lceil log_{2}n!\rceil+(2^{\lfloor log_{2}n!\rfloor+1}-n!)\cdot\lfloor log_{2}n!\rfloor}{n!}=\lceil log_{2}n!\rceil+1-\frac{2^{\lceil log_{2}n!\rceil}}{n!}
+c_1\lambda^2\,(\overline{\psi}\Gamma^{\mu}\partial_{\mu}\psi)^2+c_2 \lambda^2\, \overline{\psi}\Gamma^{\mu}\partial^{\nu}\psi\overline{\psi}\Gamma_{\mu}\partial_{\nu}\psi+c_3\lambda^2 \overline{\psi}\Gamma^{\mu}\partial^{\nu}\psi\overline{\psi}\Gamma_{\nu}\partial_{\mu}\psi
\sum_{i=1}^dx_ia_i(x) =\sum_{i=1}^dx_i\sum_{l;l\neq i}\phi_{il}(x_i-x_l) &=\sum_{k,l;k>l}\{x_k\phi_{kl}(x_k-x_l)+x_l\phi_{lk}(x_l-x_k)\}&=\sum_{k,l;k>l}(x_k-x_l)\phi_{kl}(x_k-x_l).
F(x^{(\ell-1)}) - \eta^{k(\ell)-1}= \biggl[F(x^{(\ell-1)}) - \widetilde{F}^{k(\ell)-1}(x^{(\ell)})\biggr] - \frac{\rho}{2}\big\|x^{(\ell)}- x^{(\ell-1)}\big\|^2 \leq \frac{1}{\beta}\big(F(x^{(\ell-1)}) - F(x^{(\ell)})\big) - \frac{\rho}{2}\big\|x^{(\ell)}- x^{(\ell-1)}\big\|^2.
P \int_{0}^{\infty}d k \frac{k^{n}}{2 \mu E - k^{2}}= - \int_{0}^{\infty}d k k^{n - 2}+ 2 \mu E P \int_{0}^{\infty}d k \frac{k^{n - 2}}{2 \mu E - k^{2}},
u \doteqdot (4\pi\epsilon)^{-n/2}e^{-f}; 1 =\int_M (4\pi\epsilon)^{-n/2}e^{-f}d\mu (\tau_0); |\nabla f|^2 =|\nabla f_1|^2=|\nabla \frac{|x|^2}{4\epsilon}|^2=\frac{|x|^2}{4\epsilon}, \textrm{for $|x|<\rho$}.
A_{M_1,\ldots, M_d;S_1,\ldots,S_d}^{n_1,\ldots, n_d}f(x)\coloneqq \frac{1}{M_1\cdots M_d}\sum_{n_1=1}^{M_1}\ldots \sum_{n_d=1}^{M_d}f(S_1^{n_1}\cdots S_d^{n_d}x), x \in X,
g(z)= \begin{cases}0, \quad\quad\ \mbox{if}z \in (-\infty, a) \lambda( r_{\ell}(z) - r_{\ell}(a)), \mbox{if}z \in (a, b) \lambda( r_{\ell}(b) - r_{\ell}(a)), \mbox{if}z \in (b, + \infty). \end{cases}.
g=\frac{4\pi^2 R}{e^2}\,\left(C_{AB}d\vec r_A\cdot d\vec r_B + (C^{-1})_{AB}(d\psi_A+\vec \omega_{AC}\cdot d\vec r_C)(d\psi_B+\vec \omega_{BD}\cdot d\vec r_D)\right) .
\gamma_1=\sum_{I\in \mathcal{I}}\omega_Iu_1^{\epsilon_1}v_1^{(p-1)i_1-\epsilon_1}\cdots u_s^{\epsilon_s}v_s^{(p-1)i_s-\epsilon_s}S_s(\Sigma^{-(s-1)}x_sy_s^{-1}\otimes m)\in(\Gamma^+\Sigma^{-(s-1)}\widehat{P}\otimes M)_s.
s_{\lambda}(x|a)=s_{\lambda}(x)+\sum_{\mu \varsubsetneq \lambda}(-1)^{|\lambda /\mu |}\det (e_{\lambda_{i}-\mu_{j}-i+j}(a_{1},\ldots,a_{n+\lambda_{i}-i}))_{1\leq i,j\leq n}~s_{\mu}(x)\;.
|E(Tx ) - E (x)| = |\epsilon (x) -\epsilon (Tx) + \langle s_n (T x) , v_n \rangle - \langle s_n (x) , v_n \rangle | \leq \frac 14 + |\langle e , v_n \rangle | \leq \frac 14 + ||v_n|| \leq \frac 12 .
{\bf Z ({\bf x})}= \frac{1}{{\bf x}}+ \sum_{q \neq 0}\{\frac{1}{{\bf x - q}}+ \frac{1}{{\bf q}}+ \frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}+ \frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}+ \frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}\},
\left\langle Db(x) u, u \right\rangle = -2 \left| \left\langle x, u \right\rangle \right|^2 + \left( 1 - |x|^2 \right) | u |^2 &\leq \left( 1 - |x|^2 \right) |u|^2 \leq |u|^2
\begin{array}{l l}{z^{\prime}= z + \varepsilon \zeta \theta ;}&{\theta^{\prime}= \theta + \varepsilon \zeta}\\{\bar{z}^{\prime}= \bar{z}- \bar{\varepsilon}\bar{\zeta}\bar{\theta};}&{\bar{\theta}^{\prime}= \bar{\theta}+ \bar{\varepsilon}\bar{\zeta}.}\\ \end{array}
\frac{1}{(z^{2}+\epsilon^{2})^{3/2}}-\frac{1}{((z+1)^{2}z^{2}+\epsilon^{2})^{3/2}}<\frac{1}{(z^{2})^{3/2}}-\frac{1}{((z+1)^{2}z^{2})^{3/2}}
\lim_{k\to\infty}\frac{1}{k}\log\|S(k)(D^l\mathfrak{r}_{a,n}(\varphi))\|_{\sup}=-\lambda_1(n-l-a)0\leq l<n \lim_{k\to\infty}\frac{1}{k}\log\big(\|S(k)(D^l\mathfrak{r}_{a,n}(\varphi))\|_{L^1(I^{(k)})}/|I^{(k)}|\big)=-\lambda_1(n-l-a)0\leq l\leq n.
\Psi^{[k](\alpha)}(z)=\frac{(-1)^k}{\tau_{k}^{(\alpha)}}\begin{bmatrix}z^{k}0 0 z^{-k}\end{bmatrix}\begin{bmatrix}S^{+}(z) 0 0 S^{-}(z)\end{bmatrix}\begin{bmatrix}\tau_{k}^{(\alpha)}\tau_{k-1}^{(\alpha)}/z \tau_{k+1}^{(\alpha)}/z \tau_{k}^{(\alpha)}\end{bmatrix},
\Gamma_{H}~ = ~ \frac 1 4 \pi^{2}r_{0}^{2}( a + b ) [ \frac 1 4 + \frac{9}{1 6}( a + b )^{2}] ~ \left( e^{\frac{\pi}{2}( a + b )}- 1 \right)^{- 1}\left( e^{\frac 3 2 \pi ( a + b )}+ 1 \right)^{- 1}~ \frac{d^{3}k}{( 2 \pi )^{3}}
A^{Abelian}_{Total}=16\frac{G^2_{0}}{2\alpha^{'}}[\frac{K}{st}cos(\frac{p_1\theta p_2-p_3\theta p_4}{2})+ \frac{K}{su}cos(\frac{p_1\theta p_2+p_3\theta p_4}{2})+ \frac{K}{ut}cos(\frac{p_1\theta p_4+p_2\theta p_3}{2})]
K_n(s,f,\chi) = \zeta^{(N)}(2s)\sum_{m=1}^\infty\frac{f_m}{m^s}\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\sum_{g\in H_k(N,\chi)}\rho_g(n)\overline{\rho_g(m)}= \frac{\Gamma(k-1)}{( 4 \pi)^{k-1}}\sum_{g\in H_k(N,\chi)}\rho_g(n)L(s,f\times\bar{g}),
( \partial X^{j}( z ) )^{2}\cdot \partial_{n}X^{i}( w ) \sim G^{i i}\frac{1}{( z - w )^{2}}\partial_{n}X^{i}\sim \frac{t^{- 2 p}}{( z - w )^{2}}\partial_{n}X^{i}, \qquad ( \mathrm{n o}~ ~ \mathrm{s u m}~ ~ \mathrm{o v e r}~ ~ i )
\rho_{0,2}(\alpha) := \pi \sum_{k=1}^\infty \binom{1/2}{k}\left( - \frac{\pi^2}{\alpha^2}\right)^k X_k(0) = \frac1{2}\sum_{k=2}^\infty \binom{1/2}{k}\left(- \frac{\pi^2}{\alpha^2}\right)^k \frac{1}{k-1}.
\int_0^T\eta(t)\int_\Omega \big(\frac12|\nabla d|^2\nabla\cdot X-\nabla d\odot\nabla d:\nabla X\big)=\int_0^T\eta(t)\int_\Omega \big\langle \partial_t d+u\cdot\nabla d, X\cdot\nabla d\big\rangle.
h_c(G) =h_c(G \backslash v) + f_c(b)-f_c(0)+\sum_{i=1}^b \Delta_c(d_i) \le h_c(G \backslash v)+f_c(b)-f_c(0)+b \cdot \Delta_c\left( \frac{m+\binom{b}{2}}{b}\right) < h_c(K_{1,m-b})+f_c(m)-f_c(m-b)+b \Delta_c\left(1\right) =h_c(K_{1,m}).
\alpha_{2\mu}(12\mu) = \Y^{2\mu}\ \ (1-Y^2)^{-4\mu-1}(1+2Y+Y^2) &= \Z^{\mu}\ \ (1-Z)^{-4\mu-1}+ \Z^{\mu-1}\ \ (1-Z)^{-4\mu-1}={{5\mu}\choose{\mu}}+{{5\mu-1}\choose{\mu-1}}= 6{{5\mu-1}\choose{\mu-1}}.
\begin{aligned}q^{10}L_{11}(q^{96}) = \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{(24n+4)^2 - 6(8j+1)^2}+ q^{(24n + 20)^2 - 6(8j+1)^2}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{(24n+32)^2 - 6(8j+3)^2}+ q^{(24n+16)^2 - 6(8j+3)^2}.\end{aligned}
\begin{cases}\frac{\partial \omega_{\varphi_{\epsilon}}}{\partial t}=-{\rm Ric}(\omega_{\varphi_{\epsilon}})+\gamma\omega_{\varphi_{\epsilon}}+(1-\beta)\eta_{\epsilon}+L_X \omega_{\varphi_{\epsilon}}\omega_{\varphi_{\epsilon}}|_{t=0}=\omega_{\epsilon},\end{cases}
\left| \overline{GS^{r_i}_i}S_i{\overline{GS^{r_i+1}_i}\,}^{-1}\right|_\mathfrak{U}=\left| \overline{G'S^{r'_i}_i}S_i{\overline{G'S^{r'_i+1}_i}\,}^{-1}\right|_\mathfrak{U}
|\partial_{r}N_{2}(f_{v_{5}})(t,r)| \leq \begin{cases}\frac{C \log(r)}{r^{3}(t-r)^{2}}, t-\sqrt{t}\leq r \leq t-t^{1/4}t+t^{1/4}\leq r \leq t+\sqrt{t}\frac{C}{r^{7/2}}, t-t^{1/4}\leq r \leq t+t^{1/4}\end{cases}
\lambda(\theta\alpha v) - \Delta(\theta\alpha v) = - \theta\alpha\nabla_H\pi - 2\nabla(\theta\alpha)\cdot\nabla v - (\Delta(\theta\alpha)) v \quad\Omega', \partial_z(\theta\alpha v)|_{\Gamma_u' \cup \Gamma_b'}= 0, \theta\alpha v \Gamma_l'.
{\mathcal M}\big(e^{\psi_m}\big)(z) =\dfrac 1{\Omega_{2n}\delta^{2n}}\int_{B(z,\delta)}(1+|\zeta|^2)^{a_m}e^{2mV(\zeta)}\, d\lambda(\zeta) &\leq (1+(|z|+\delta)^2)e^{2m(V(z)+\varepsilon)},
A_n = \frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}C_n = \frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}.
\begin{cases}\widetilde{H}(x,P+\nabla u_0 (x ) )=\ln (m_0 (x) )+\overline{H}(P),\ \ \mathbb{T}^d, -\div\big(m_0 (x )D_\Lambda\widetilde{H}(x, P+\nabla u_0 (x)\big)=0,\ \ \mathbb{T}^d, \int_{\mathbb{T}^d}m_0 dx=1. \end{cases}
\frac{(x-1)! \cdot 2^{x+1}}{q}\leq \frac{1}{4}, \frac{3x}{q^{n-2k+1}}\leq \frac{3x}{q^3}\leq \frac{3x}{2^{3(2x+1)}}\leq \frac{3}{1024}, x^2 \cdot 2^{x}\cdot q^{-\delta}\leq \frac{x^2 \cdot 2^x}{2^{2x+1}}\leq \frac{x^2}{2^{x+1}}\leq \frac{9}{16}, \frac{q}{\binom{x}{2}}\geq \frac{(x-1)! \cdot 2^{2x+2}}{x(x-1)}\geq 32.
{\bf E}[M^{-n}_{(\tau, \lambda_1, \lambda_2)}] = (2\pi)^{-n}\prod\limits_{j=0}^{n-1}\frac{\Gamma(1+\lambda_1+\frac{(j+1)}{\tau}) \,\Gamma(1+\lambda_2+\frac{(j+1)}{\tau})\Gamma(1-\frac{1}{\tau})}{\Gamma(1+\lambda_1+\lambda_2+\frac{(j+1)}{\tau})\,\Gamma(1+\frac{j}{\tau})}.
\|F_{il}(x_{il},u_l)-F_{il}(x_{il},z_l)\| = \sqrt{|c_{il}'(g_{il}) - c_{il}'(g_{il})|^2+|p_l(u_l) + p'_l(u_l)s_{il}-p_l(z_l)-p'_l(z_l)s_{il}|^2}\cr = \sqrt{|(p_l(u_l) - p_l(z_l))+ (p'_l(u_l) - p'_l(z_l))s_{il}|^2}\cr \le \sqrt{2}\sqrt{|p_l(u_l) - p_l(z_l)|^2+ |p'_l(u_l) - p'_l(z_l)|^2s_{il}^2},
\begin{aligned}V_1(x) = x_3 R_{f_2}(x) - x_2 R_{f_3}(x), V_2(x) = x_1 R_{f_3}(x) - x_3 R_{f_1}(x), V_3(x) = x_2 R_{f_1}(x) - x_1 R_{f_2}(x).\end{aligned}
S=[\begin{matrix}-\beta_{1}&\beta_{1}&0&0&0&0\\ 0&-\beta_{1}&\beta_{1}&0&0&0\\ 0&0&-\beta_{1}&0&0&0\\ 0&0&0&-\beta_{2}&\beta_{2}&0\\ 0&0&0&0&-\beta_{2}&\beta_{2}\\ 0&0&0&0&0&-\beta_{2}\\ \end{matrix}].
\tilde{A}( p ) = 3 g^{2}N ( N^{2}- 1 ) \delta_{a b}\int \frac{d^{4}k}{( 2 \pi )^{4}}d^{4}\theta \left\{\frac{m^{2}\left[ V^{a}( p , \theta ,{\overline{\theta}}) V^{b}( - p , \theta ,{\overline{\theta}}) \right]}{( k^{2}+ m^{2}) [ ( p - k )^{2}+ m^{2}]}\right\}.
{\bf h}_{u,k}= \sum_{l=1}^{L}\sum_{k=1}^{K}{\alpha_{l}}e^{-\mathrm{j}\frac{2\pi kd}{K}}{\bf a}_t(\phi_{l},f_k)r(kT_s-\tau_l),\;\in\mathbb{C}^{{N_t}\times1},
\int_S2^{-m\frac{\gamma^2}{2}}\mathbb{E}\left(e^{\gamma \Gamma^U(\rho_{x,2^{-m}})}\right) \, \nu(dx)=\int_S R(x,U)^{\frac{\gamma^2}{2}}\, \nu(dx) \le 16^{\frac{\gamma^2}{2}}\cdot 2^{-l\frac{\gamma^2}{2}}\nu(S),
\psi({g})=\left(\begin{array}[c]{ccc}(x_{1}^{2}-x_{2}^{2}-y_{1}^{2}+y_{2}^{2}) 2(x_{1}y_{1}-x_{2}y_{2}) x -2(x_{1}y_{1}+x_{2}y_{2}) (x_{1}^{2}+x_{2}^{2}-y_{1}^{2}-y_{2}^{2}) y -2(x_{1}x_{2}-y_{1}y_{2}) -2(x_{1}y_{2}+x_{2}y_{1}) z\end{array}\right)
S_{l o c a l}[ g ] ={\frac{1}{1 6 \pi G}}\int \, d^{4}\, x \, \sqrt{- g}\, ( R - 2 \Lambda ) + S_{l o c a l}^{( 4 )}+ \sum_{n = 3}^{\infty}S_{l o c a l}^{( 2 n )}\, .
\rho^{-n-4}\int_{N \cap B_{\rho}\cap \{r>4\rho\tau\}}\beta^2 + \sum_{i=1}^k\rho^{-n-2}\int_{N \cap B_{\rho}\cap \{r>4\rho\tau\}}y_i^2 \leq C\tau^{-2}\|\underline{f}\|^2_{L^2}(2\rho,\tau)
\Xi_{n , m}( - \frac{1}{\tau}) = \widetilde{\Xi}_{n , m}( \tau ) = \sqrt{\frac{- i \tau}{2 m}}\sum_{n^{\prime}\mathrm{m o d}2 m}\sin ( - 2 \pi \frac{n n^{\prime}}{2 m}) \Xi_{n^{\prime}, m}( \tau ) ,
K_{m}=-(1+\frac{\beta}{\alpha})m +\frac{1}{\beta}\sum_{r=1}^{m}(-\delta)^{r-1}[r-1]!\,q^{-(m-1)r}e_{r}(1, q, \ldots , q^{m-1})\prod_{p=1}^{r}\frac{\alpha+\beta q^{p-1}}{1+\beta\gamma[p-1]}.
[X_r, X_j] = \sum_i f_i \bigg[ \frac{\partial}{\partial x^i}, X_j \bigg] + \sum_\rho \varphi_\rho \bigg[ \frac{\partial}{\partial \xi^\rho}, X_j \bigg] - \sum_i X_jf_i \frac{\partial}{\partial x^i}- \sum_\rho X_j \varphi_\rho \frac{\partial}{\partial \xi^\rho}= 0.
D_{mk}=\begin{bmatrix}1 1 \ldots 1 1 1 \ldots 1 \hdotsfor{4}1 1 \ldots 1\end{bmatrix},C_{kk}=\begin{bmatrix}1 1 \ldots 1 0 1 \ldots 1 \hdotsfor{4}0 0 \ldots 1\end{bmatrix}.
f_{X_{i}}(x_{i})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\cdot\cdot\cdot\int_{-\infty}^{\infty}f(x_{1},x_{2},...,x_{n})dx_{1}dx_{2}\cdot\cdot\cdot dx_{i-1}dx_{i+1}\cdot\cdot\cdot dx_{n}.
\left\langle f,|x-y|^{-1}f \right\rangle \leq \lim_{n\to\infty}\left\langle f_n,|x-y|^{-1}f_n \right\rangle & \leq 2 \lim_{n\to\infty}\langle f_n,(-\Delta_{x})^{\frac{1}{4}}(-\Delta_{y})^{\frac{1}{4}}f_n \rangle = 2\langle f,(-\Delta_{x})^{\frac{1}{4}}(-\Delta_{y})^{\frac{1}{4}}f \rangle.
(sR+dL) \cap [0,k] \ =\ (sn-s(A \cap [n-k,n]) + d(A \cap [0,k]) ) \cap [0,k] &\ \cong\ (s(A \cap [n-k,n]) - d(A \cap [0,k]) ) \cap [sn-k, sn] &\ = \ (sA-dA) \cap [sn-k, sn].
I_{\mathrm{o n e - l o o p}}= ( e^{i W})_{\mathrm{o n e - l o o p}}= \prod_{l = 0}^{\infty}{}\left\{{\frac{( \det{\cal D}_{3 / 2})^{2}}{\det{\cal D}_{2}\det{\cal D}_{1}}}\right\}= \prod_{l = 0}^{\infty}{}\left\{{\frac{\det{\cal D}_{3 / 2}}{\det{\cal D}_{1}}}\right\}^{2}= 1 .
W(p^\Lambda ,q_\Lambda) =-{(p\cdot q)}^2+4\bigl ((p^1q_1)(p^2q_2)+(p^1q_1)(p^3q_3)+(p^3q_3)(p^2q_2)\bigr ) - 4 p^0 q_1 q_2 q_3 + 4q_0 p^1 p^2 p^3 \ .
\mathfrak{G}_{{\sigma_{k}}^{(n)}}^{(\beta)}(x_j=1,\, \forall\, j)={n+k-1 \choose k}\left(\sum_{j=0}^{k-1}{k \over n+j}{k-1 \choose j}\beta^{j}\right) = \sum_{j=0}^{k-1}{n+j-1 \choose j}(1+\beta)^{j}.
-\frac{b_1}{\varpi^{a_1}}\log_F(1+(y+\varpi^{\kappa}x)\varpi^{a_1-l}) = \sum_{j\geq 1}\frac{(-1)^{j}b_1}{j}(y+\varpi^{\kappa}x)^j\varpi^{(j-1)a_1-jl}.
f(22) = f(2)\,f(11) = 2f(11) &= f(1^2+1^2+1^2+1^2+3^2+3^2) = 4+2f(9) = 22 f(26) = f(2)\,f(13) = 2f(13) &= f(1^2+4^2+2^2+2^2+2^2+3^2) = 1+4f(4)+f(9) = 26 f(38) = f(2)\,f(19) = 2f(19) &= f(1^2+1^2 + 3^2+3^2+3^2+3^2) = 2+4f(9) = 38.