harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{p_{out,K}}= \Pr \left({{{\log}_2}{{\left({1 + \frac{{P}{{\left|{{h}}\right|}^2}}{\mathcal N_0}}\right)}^K}< \mathcal R}\right) = \frac{1}{{\Gamma \left( m \right)}}\Upsilon \left({m,\frac{{m\mathcal N_0\left({{2^{\mathcal R/K}}- 1}\right)}}{{{P}{\sigma}^2}}}\right).
\left\{Q_\alpha^a,Q_\beta^b\right\}=\delta^{ab}\gamma_{\alpha \beta}^\mu \,\,P_\mu +\sum_{p=0,1,\cdots}\gamma_{\alpha \beta}^{\mu_1\cdots\mu_p}\,\,\,Z_{\mu_1\cdots \mu_p}^{ab}.
\cdot\cdot\cdot+\frac{1}{3}\gamma_{txx}\frac{\partial^{3}A}{\partial x^{2}\partial t}+\frac{1}{3}\gamma_{tyy}\frac{\partial^{3}A}{\partial y^{2}\partial t}+\frac{1}{3}\gamma_{ttx}\frac{\partial^{3}A}{\partial t^{2}\partial x}+\cdot\cdot\cdot
L_{m,n}^{II,(\alpha)}(x)=\frac{(-1)^{n+1}-(m-\alpha-1)/(n-m)}{m!(n-m-1)!\prod_{1\leq i<j\leq n}\big(Y^{(\alpha)}_{j,m,n}-Y^{(\alpha)}_{i,m,n}\big)}\det\big(M_n^{II}\big).
F(\mathbf{m})= \begin{pmatrix}-qm_{2}+\sqrt{3}c_{1}m_{2}^{2}+c_{1}m_{1}m_{2}+A_{1}m_{1}^{3}+B_{1}m_{1}^{2}m_{2}+C_{1}m_{1}m_{2}^{2}+D_{1}m_{2}^{3}qm_{1}-c_{1}m_{1}^{2}-\sqrt{3}c_{1}m_{1}m_{2}+A_{2}m_{1}^{3}+B_{2}m_{1}^{2}m_{2}+C_{2}m_{1}m_{2}^{2}+D_{2}m_{2}^{3}\end{pmatrix}&+o(|\mathbf{m}|^3),
D_A^2=\lim_{N\to\infty}E\big(\xi_N^{(A)}(1)\big)^2=\sum_{s:q_{r_s}\in A}c_{i_A,r_i}D_{i,i}+2\sum_{i<j:q_{r_i},q_{r_j}\in A}c_{i_A,r_i}D_{i,j}.
\left( \begin{array}{c c}{{\left\{\bar{D}_{a}\, , D^{b}\right\}}}&{{\left\{\bar{D}_{a}\, , \bar{D}_{b}\right\}}}\\{{\left\{D^{a}\, , D^{b}\right\}}}&{{\left\{D^{a}\, , \bar{D}_{b}\right\}}}\\ \end{array}\right) = 2 i m^{2}\Delta \, .
S_k = \{x^* \in S_{k-1}: r_{k-1}\cos(\theta) = \langle \log_{y_{k-1}}(x^*), e_k^{(k-1)}\rangle = \langle \log_{y_{k-1}}(x^*), e_k \rangle\}&= \{x^* \in S_{k-1}: r_{k-1}\cos(\theta) = \frac{r_{k-1}}{\sinh(r_{k-1})}\langle x^*, e_k \rangle\}.
\sigma_j \sigma_1^{-1}\gamma_{1k}= \sigma_j \sigma_1^{-1}\gamma_{*1k}- i \sigma_j \Phi \Phi^* \sigma_k + i \sigma_j \sigma_1^{-1}\sigma_k \Phi \Phi^* \sigma_1 & \gamma_{1k}\sigma_1^{-1}\sigma_j = \sigma_j \sigma_1^{-1}\gamma_{*1k}- i \sigma_1 \Phi \Phi^* \sigma_k \sigma_1^{-1}\sigma_j + i \sigma_k \Phi \Phi^* \sigma_j
\sup_{x \in [X_{(1)},X_{(n)}]}\log \frac{1}{f_0(x)}\leq \sup_{x \in [F_0^{-1}(n^{-t/2}/\alpha),F_0^{-1}(1-n^{-t/2}/\alpha)]}\log \frac{1}{f_0(x)}&= \max\biggl\{\log \frac{1}{f_0\bigl(F_0^{-1}(n^{-t/2}/\alpha)\bigr)}\, , \, \log \frac{1}{f_0\bigl(F_0^{-1}(1 - n^{-t/2}/\alpha)\bigr)}\biggr\}&\leq \frac{t}{2}\log n,
\mathrm{\hat{H}= \frac{\k a p p a^{2}}{1 2}R^{p - 3}\frac{\partial}{\partial R}R^{- p + 2}\frac{\partial}{\partial R}- \frac{1}{2 R^{3}}\frac{\partial^{2}}{\partial \v a r p h i^{2}}+ \frac{\k a p p a^{1 2}}{R^{9}}V ( \v a r p h i ) ,}
G_{n}( x_{1}, \dots , x_{n}) = \left< \Omega \right| T \phi_{H}( x_{1}) \dots \phi_{H}( x_{n}) \left| \Omega \right> = \frac{\left< 0 \right| T \phi_{D}( x_{1}) \dots \phi_{D}( x_{n}) \;{\cal{S}}\left| 0 \right>}{\left< 0 \right|{\cal{S}}\left| 0 \right>}.
\underset{s=-\ell}{\mathrm{Res}}\zeta_{F}\left( s,z\right) =\frac{1}{\log a}\sum_{j=0}^{\left\lfloor \frac{\mathrm{\ell}}{4}\right\rfloor}\binom{4j}{2j}\binom{\ell}{4j}5^{-2j}z^{\ell -4j}=\frac{1}{\log a}\sum_{j=0}^{\left\lfloor \frac{\mathrm{\ell}}{4}\right\rfloor}\binom{\ell}{2j,2j}5^{-2j}z^{\ell -4j},
\frac{f^{}_{n-1,p}\left(x\right)}{f^{}_{n,\frac{x+1}{n}}\left(x+1\right)}= \frac{f^{}_{n-1,p}\left(x\right)}{f^{}_{n-1,\frac{x+1}{n}}\left(x\right)}= \frac{\left(1+\frac{y}{n-x-1}\right)^{n-x-1}}{\left(1+\frac{y}{np}\right)^x}> \frac{\left(1+\frac{y}{n-x-1}\right)^{n-x-1}}{\left(1+\frac{y}{x}\right)^x}\geq\,\ 1,
\int \frac{d^3 \tilde{u}}{(2\pi)^3}\frac{\tilde{u}^2\tilde{u}_0^2}{[\tilde{u}^2+\tilde{u}_0^2 +\tilde{M}^2]^{\frac{3}{2}}}\sim \frac{\tilde{u}_0^2}{[\tilde{u}_0^2 + \tilde{M}^2]^{\frac{3}{2}}}
{\cal L}_0:\,\, \partial_z\partial_{\bar{z}}\chi_n=\widetilde{\mbox{Inh}}\left(\chi_0,\ldots,\chi_{n-1}\right)\,,\quad{\cal L}_1:\,\, \partial_z\partial_{\bar{z}}\chi_n+m^2\chi_n=\widetilde{\mbox{Inh}}\left(\chi_0,\ldots,\chi_{n-1}\right)\,.
\sum_{Q\in \Gamma\backslash \mathcal Q_D}\chi_d(Q) \int_{C_Q}i\partial_z\varphi(z) y^{s}\, dz = -2\pi i\sum_{n\neq 0}n a_\varphi(n) G(n,d) \int_0^\infty y^{s+\frac 12}K_{ir}(2\pi |n|y) \, dy
\stackrel{P}{\cal F}{}\!\!_{0,p}^{(\pm)}\left( \begin{array}{c}\!\!\alpha \!\! \!\!\beta \!\!\end{array}\right) = \; \sum_{s\in{\cal S}^{p-3}}\stackrel{P}{K}{}\!\!_{0,p}^{(\pm)}\left( \begin{array}{c}\!\!\alpha\!\! \!\!\beta,s\!\! \end{array}\right) \stackrel{\alpha}{\omega}_{0,p}\!\! \lbrack \beta+s \rbrack \;\;,
|\partial_x^\alpha \mathcal K(x+h,y)-\partial_x^\alpha\mathcal K(x,y)| =\left|\int_0^1h\cdot\nabla_x\partial_x^\alpha\mathcal K(x+th,y)\,dt \right| &\lesssim|h|\sup_{t\in(0,1)}|x+th-y|^{-n-|\alpha|-1}\sim|h|\,|x-y|^{-n-|\alpha|-1},
\Big|\mathcal{S}\Lambda_\iota+\mathcal{T}_\iota\Lambda_\iota^2+O(\Lambda_\iota^3)\Big|\prec \frac{\sqrt{(\Im m_{\mu_A\boxplus\mu_B}+\hat{\Lambda})(|\mathcal{S}|+\hat{\Lambda})}}{N\eta}+\frac{1}{(N\eta)^2},\iota=A, B\,.
a'(p')=\left.a(p)\right|_{p=\widehat{p}(p')}\,,\lambda_b^{(a')}(p')=\left.\sum_{c=1}^n\,J_{bc}(p)\,\lambda_c^{(a)}(p)\right|_{p=\widehat{p}(p')}\,,\forall\, b=1,\ldots,n\,.
\sum_{k=0}^n(-1)^{k}\frac{2n}{n+k}\binom{n+k}{n-k}m_k(t)[4(1-\epsilon)]^k = 2 \int_0^1 \left\{\sum_{k=0}^n(-2)^{k}\frac{n}{n+k}\binom{n+k}{n-k}[2x(1-\epsilon)]^k\right\}\nu_t(dx) & = 2 \int_0^1 T_n(1-2x(1-\epsilon)) \nu_t(dx).
Q(z) = \frac{1}{Z}\int_{\Sigma_z}e^{-\beta V(x)}\Big[\det(\nabla \xi\nabla \xi^T)(x)\Big]^{-\frac{1}{2}}\, d\nu_z(x) = \frac{1}{Z}\int_{\mathbb{R}^n}\delta(\xi(x) - z)\,e^{-\beta V(x)}\, dx
\begin{aligned}f_1(0,0;0) > 0, g_1(0,0;0) < 0, f_2(0,0;0) < 0, g_2(0,0;0) < 0, f_3(0,0;0) < 0, g_3(0,0;0) > 0, f_4(0,0;0) > 0, g_4(0,0;0) > 0,\end{aligned}
\left(r_s^{(K)}- r_{s-1}^{(K)}\right) \left( \bar{D}(r_{s}^{(K+1)}) - \bar{D}(r_s^{(K)}) \right)=\left(r_s^{(K)}- r_s^{(K+1)}\right) \left( \bar{D}(r_{s-1}^{(K)}) - \bar{D}(r_s^{(K)}) \right).
\sigma_q(F):=\left(\prod_{\substack{\gamma_q(p)\neq1}}(1-\gamma_q(p))\left(1+\frac{1}{p}\right)\prod_{\substack{\gamma_q(p)=1}}\left(1+\frac{1}{p}\right)\right)\prod_{p|q}\left(1-\frac{1}{p^2}\right)^{-1}.
Q_{B_n}P_{B_n}(f) =V_n^*(0,P_{B_n}(f))=1_{B_n}P_{B_n}(f)=1_{B_n}f Q_{B_n^c}P_{B_n^c}(f) =V_n^*(P_{B_n^c}(f),0)=1_{B_n^c}P_{B_n^c}(f)=1_{B_n^c}f
W_{\mathbf h,\xi}(1) = e^{-2\pi\xi}2^{-\nu(N)}c(\mathfrak d_{\xi})\chi(\mathfrak f_{\xi})\mathfrak f_{\xi}^{k-1/2}\prod_p \Psi_p(\xi;\alpha_p) = 2^{-\nu(N)}c(\mathfrak d_{\xi})\mathfrak f_{\xi}^{k-1/2}\prod_v W_{v,\xi}(1),
\mathcal{F}\{DN_3[h,h,\phi]\}=\frac{1}{4\pi^2}\int_{\mathbb{R}}\widehat{h}(\xi-\eta)\widehat{h}(\eta-\sigma)\widehat{\phi}(\sigma)n_3(\xi,\eta,\sigma)\,d\eta, &n_3(\xi,\eta,\sigma)=\frac{|\xi||\sigma|}{2}\big(|\eta|+|\xi+\sigma-\eta|-|\xi|-|\sigma|\big),
\left|\frac{\partial^j}{\partial\tilde{\mathbf{x}}^j}(\tilde{g}_{t}(\tilde{\mathbf{x}})_{ab}-\delta_{ab})\right| \leq \frac{\epsilon_t^{-2}}{100}|\tilde{\mathbf{x}}|^{2-j}\,\;{\rm for}\;\, |\tilde{\mathbf{x}}| \leq 2\epsilon_t\;\,{\rm and}\;\,j=0,1.
\frac{e_{t+1}(L_{t}\cup M, F_{t+1})}{|L_{t}\cup M|}= \frac{e_t(L_t, F_t) \left ( \binom{r-2}{2}+2(r-2) \right )}{|L_t|+(r-2)e_t(L_t, F_t)}= \frac{e_t(L_t, F_t) \left ( \binom{r}{2}-1 \right )}{|L_t|+(r-2)e_t(L_t, F_t)}.
\Lambda_{n}= \left[ \begin{matrix}{0}&{1}&{0}&{\cdots}&{0}\\{\vdots}&{0}&{1}&{\ddots}&{\vdots}\\{\vdots}&{{}}&{\ddots}&{\ddots}&{0}\\{0}&{{}}&{{}}&{\ddots}&{1}\\{\lambda}&{0}&{\cdots}&{\cdots}&{0}\\ \end{matrix}\right]
\frac{1}{z^{n_0}}\prod_{j=1}^N \prod_{\substack{0\leq b<\langle q_j r \rangle+\sum_i\langle q_j(\vec l_0)_i \rangle \langle b \rangle = \langle q_j r +\sum_i q_j(\vec l_0)_i \rangle}}(b+q_j)z
a_{12}^{1234}=a_{12}^{1243}=a_{12}^{1324}=a_{12}^{1342}=a_{12}^{1423}=a_{12}^{1432}= a_{12}^{3124}=a_{12}^{3142}=a_{12}^{3412}=a_{12}^{4123}=a_{12}^{4132}=a_{12}^{4312}
\prod_{j = 1}^{J + m}( y^{\prime}q^{2 J^{\prime}}+ y q^{- ( \varpi - 2 J + 2 j - 1 )}) \prod_{\ell = 1}^{J^{\prime}+ m^{\prime}}( y^{\prime}+ y q^{\varpi - 2 J^{\prime}+ 2 m + 2 \ell - 1}) =
\frac{\tau^{-1}}{\Gamma(1-\alpha)}\mathcal{D}_t^{(\alpha)}( \mu_1 \delta_t U_i^{n-\frac{1}{2}}+\mu_2 U_i^{n-\frac{1}{2}})+\psi(U_i^{n-\frac{1}{2}},U_i^{n-\frac{1}{2}})-\frac{h^2}{2}\psi(W_i^{n-\frac{1}{2}},U_i^{n-\frac{1}{2}}) =\lambda W_i^{n-\frac{1}{2}}+P_i^{n-\frac{1}{2}},1\leq i \leq M-1,1\leq n\leq N,
\mathcal{L}^{\mathrm{Optimal}}(\overline{\pmb{\theta}}, \overline{\pmb{\omega}}) = \varpi \mathcal{L}^{\mathrm{Optimal}}_{1}(\overline{\pmb{\theta}}, \overline{\pmb{\omega}}) + (1-\varpi) \mathcal{L}^{\mathrm{Optimal}}_{2}(\overline{\pmb{\theta}}, \overline{\pmb{\omega}}),
g = \lambda^2 (dx^2+dy^2) +\Bigl(- \frac{2\tau}{\kappa}\frac{\lambda_y}{\lambda}dx +\frac{2\tau}{\kappa}\frac{\lambda_x}{\lambda}dy +dt\Bigr)^2 &= \lambda^2 (dx^2+dy^2) + \Bigl( \lambda \tau \big(ydx -xdy \big) + dt\Bigr)^2.
\int\mid \alpha, \lambda ><\alpha, \lambda \mid{1\over 2\pi}K_{\lambda +\frac{1}{2}}(\mid \alpha \mid )I_{\lambda +\frac{1}{2}}(\mid \alpha \mid )d^2\alpha =\sum^{\infty}_{m=0}\mid m, \lambda><m, \lambda\mid ={\bf 1},
\sum_{k = 0}^{s - 1}\lambda_{1, k}^{j - 1}c^{I}_{1, b(1, k)}+ \sum_{i = 2}^{i = n}\lambda_{i, b_{i}}^{j - 1}\sum_{k = 0}^{s - 1}c^{I}_{i, b(1, k)}= 0 j \in [r].
2T_{2p q}(x)X'_n(x) =2T_{2p q}(x)T_{N(n)}(x) &=T_{pqn+1+2pq}(x)+T_{pqn+1-2pq}(x) &=T_{pq(n+2)+1}(x)+T_{pq(n-2)+1}(x) &=T_{N(n+2)}(x)+T_{N(n-2)}(x) &=X'_{n+2}(x)+X'_{n-2}(x).
\sum_{\beta}\n_{[ l , l^{\prime}] \, \alpha}{}^{\beta}n_{[ j , j^{\prime}] \, \beta}{}^{\gamma}\ = \ \sum_{[ k , k^{\prime}]}\N_{[ l , l^{\prime}] [ j , j^{\prime}]}{}^{[ k , k^{\prime}]}n_{[ k , k^{\prime}] \, \alpha}{}^{\gamma}\ \ .
\sum_{n=0}^{\infty}\frac{q^{n}}{(zq^{n+1};q)_{n+2}(zq^{2n+4};q^2)_{\infty}}=\sum_{n=0}^{\infty}\frac{z^nq^{2n^2+2n}}{(q;q^2)_{n+1}(zq;q^2)_{n+1}}.
\mathcal{H}[ \psi ( x ) , \bar{\psi}( x ) ] = \psi_{a}^{\dag \alpha}( x ) \ \! ( \mathbf{a}. \mathbf{p})_{\alpha}^{\phantom{\alpha}\beta}\ \! \psi ( x )_{\beta a}^{\phantom{\beta}}+ V ( \psi^{\dag}( x ) \gamma_{0}\psi ( x ) )
\frac{\overline{IA}\cdot\overline{IA}}{\overline{CA}\cdot\overline{AB}}+\frac{\overline{IB}\cdot\overline{IB}}{\overline{AB}\cdot\overline{BC}}+\frac{\overline{IC}\cdot\overline{IC}}{\overline{BC}\cdot\overline{CA}}=1.
\{x_{3},x_{2},x_{1},y_{1}\}=\{(\begin{matrix}0\\ 0\\ 1\\ 0\end{matrix})(\begin{matrix}-2\\ 2\\ 0\\ 0\end{matrix})(\begin{matrix}2\\ 0\\ 0\\ 0\end{matrix})(\begin{matrix}-14\\ 4\\ -3\\ 1\end{matrix})\}.
\Delta_{h}(r,z,t,x) = u(r,z + h) v^{(r,z + h)}(t,x)-u(r,z)v^{(r,z)}(t,x) = \big(u(r,z + h)-u(r,z))v^{(r,z)}(t,x)+u(r,z + h)\big(v^{(r,z + h)}(t,x)-v^{(r,z)}(t,x) \big).
\frac{d^2}{dt^2}\lambda(h,0) \leq \int_M \langle \frac{1}{2}\triangle h+^* h,h \rangle- \langle\mathring{R}h,h \rangle-R_{ij}h_{ik}h_{jk}dV_g &= -\frac{1}{2}\|Dh\|_{L^2}^2+2\| h\|_{L^2}^2-2\int_M R_{ij}h_{ik}h_{jk}dV_g\leq 0 \quad
(s_4 s_3 s_2 s_3 s_4)s_3 s_2 s_3 s_2 = (s_4 s_{\alpha_2 + 2\alpha_3}s_4) s_3 s_2 s_3 s_2 = s_{\alpha_2 + 2 \alpha_3 + 2\alpha_4}(s_3 s_2 s_3) s_2 = s_{\alpha_2 + 2 \alpha_3 + 2\alpha_4}s_{\alpha_2 + 2 \alpha_3}s_2.
X(t) = \exp\Big(W(t) - \frac{t}{2}\Big) \Big[ 1 - \int_0^t\int_0^r \exp\big(\frac{r}{2}- W(r)\big) g(u)du dr &+ \int_0^t \int_0^r \exp\big(\frac{r}{2}- W(r)\big) g(u)dW(u) dr \Big].
-{1\over 8}f_1 f_2{(-G)\over (-G_5)^2}\epsilon_{\rho'\mu\nu\lambda\sigma}\partial_\rho \xi G_5^{\rho\sigma}\tilde{H}^{\rho'\mu}\tilde{H}^{\nu\lambda}(G\tilde{H}G)_{\eta\zeta}\tilde{H}^{\zeta\eta}= - z_1 f_1 f_2 W.
\liminf_{\lambda \rightarrow \infty}\frac{N_{sc}[\mathfrak{e},\lambda V]}{\mathcal{L}_{V}[\mathrm{e}^{-\ell_{\lambda}}]}\ \geq \ 1-c_{1}+\frac{d\,c_{1}}{2}\int_{0}^{\infty}\exp \left( [g_{-}(V)-1]\,\frac{d}{2}r\right)\,\mathrm{d}r\ =\ \infty .
\frac{\langle s^{\prime}| e^{- \hat{H}_{s}\tau_{f}}\hat{O}_{1}^{s}( \tau_{1}) \hat{O}_{2}^{s}( \tau_{2}) \ldots \hat{O}_{n}^{s}( \tau_{n}) e^{\hat{H}_{s}\tau_{i}}| s \rangle}{\langle s^{\prime}| e^{- \hat{H}_{s}( \tau_{f}- \tau_{i})}| s \rangle}\, .
E_{k}^{2}= \frac{\cos ( 2 \pi w_{r}/ H ) + \cos ( \pi ( 2 j - 1 ) / H )}{\cos ( \pi / H ) + \cos ( \pi ( 2 j - 1 ) / H )}; ~ E_{r}= \frac{\sin ( \pi w_{r}/ H )}{\sin ( \pi w_{r}) \, \sin ( \pi / 2 H )}.
(x_{1},x_{2};x_{3},x_{4})=\frac{x_{3}-x_{1}}{x_{3}-x_{2}}/\frac{x_{4}-x_{1}}{x_{4}-x_{2}}=\frac{(x_{3}-x_{1})(x_{4}-x_{2})}{(x_{3}-x_{2})(x_{4}-x_{1})}.
\|\partial_{x_i}\partial_{x_k}P_{2t}\phi\|_{\infty}= \|\partial_{x_i}\partial_{x_k}P_{t}P_{t}\phi\|_{\infty}= \|\partial_{x_i}P_{t}(\partial_{x_k}P_{t}\phi)\|_{\infty}\leq c(t) \|\partial_{x_k}P_{t}\phi\|_{\infty}&\leq c(t)^2 \|\phi\|_{\infty}.
\Phi(M_t)=\frac{1}{p}\int_{\mathbb{S}^n}f(x)h^p(x,t)dx-\frac{1}{(\sqrt{2\pi})^{n+1}}\int_{\mathbb{S}^n}d\xi\int_0^{r(\xi,t)}e^{-\frac{s^2}{2}}s^nds, \ \p\neq0,
\int_0^\infty e^{\beta t}\int_{B_R(0)\times B_R(0)}p_D(t,x_1,y_1)p_D(t,x_2,y_2)f(y_1,y_1) \d t\,\d y_1\,\d y_2 =\int_0^{t_0}e^{\beta t}\int_{B_R(0)\times B_R(0)}p_D(t,x_1,y_1)p_D(t,x_2,y_2) f(y_1,y_1) \d t\,\d y_1\,\d y_2 +\int_{t_0}^\infty e^{\beta t}\int_{B_R(0)\times B_R(0)}p_D(t,x_1,y_1)p_D(t,x_2,y_2) f(y_1,y_1) \d t\,\d y_1\,\d y_2 :=I_1+I_2.
M_1(\varepsilon)=\left( \begin{matrix}0 \varepsilon 1 0 \end{matrix}\right), \; M_2(\varepsilon)=M_2= \left( \begin{matrix}1 0 0 -1 \end{matrix}\right), \; M_3(\varepsilon)=\left( \begin{matrix}0 -\varepsilon 1 0 \end{matrix}\right).
\|X\|_{S_{p}}=\min_{U\in\mathbb{R}^{m\times d},\widehat{V}\in\mathbb{R}^{n\times d}:X=U\widehat{V}^{T}}\|U\|_{S_{\widehat{p}_{1}}}\|\widehat{V}\|_{S_{\widehat{p}_{2}}}.
H(t,x,y,\varphi,u,z,p,q,r)=H(t,x,y,\varphi,u,z,p,q,r,\omega) &=\mathbb{E}[\delta_Z(z)|\mathcal{F}_t] h(t,x,y,u,z)+[A_u(\varphi)+a(t,x,y,u,z)]p & + b(t,x,y,u,z)q+\int_{\mathbb{R}}c(t,x,y,u,z,\zeta)r(\zeta)\nu(d\zeta).
(-1)^{|y||z|}\Big((x\circ z)\circ \alpha(y)\Big) = (-1)^{|y||z|}\Big((P(x)\ast P(z))\ast \alpha(y)+(P(x)\ast z)\ast \alpha(P(y))+(x\ast P(z))\ast \alpha(P(y)) &+\lambda (x*z)\ast \alpha(P(y))+\lambda (P(x)\ast z)\ast \alpha(y)+\lambda (x\ast P(z))\ast \alpha(y)+\lambda^{2}(x*z)\ast \alpha(y)\Big).
= \; \int_{x = 0}^{1}d x \int \frac{d^{2 \omega}r}{( 2 \pi )^{2 \omega}}\frac{\mathrm{T r}( \gamma_{\mu}\gamma_{\alpha}\gamma_{\nu}\gamma_{\beta}) ( r_{\alpha}- x k_{\alpha}) ( r_{\beta}+ ( 1 - x ) k_{\beta}) - m^{2}2 \delta_{\mu \nu}}{[ r^{2}+ m^{2}+ k^{2}x ( 1 - x ) ]^{2}}
{\cal B}(P_{XY}) := \left\{P^\prime_{XY}\in{\cal P}({\cal X}\times{\cal Y}) : \frac{1}{2}\| P_{XY}- P^\prime_{XY}\|_1 \le \varepsilon \right\}, \overline{{\cal B}}(P_{XY}) := \left\{P^\prime_{XY}\in \overline{{\cal P}}({\cal X}\times{\cal Y}) : \frac{1}{2}\| P_{XY}- P^\prime_{XY}\|_1 \le \varepsilon \right\}.
|\int_{0}^{s-t}\rho d\rho \frac{\lambda''(s)}{r}\partial_{r}\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(-1-\rho^{2}+r^{2})^{2}+4r^{2}}}\right)| \leq C(s-t)^{2}|\lambda''(s)|, s-t \leq \frac{1}{2}
\forall x \in \mathbb{Z}, \overleftarrow{\eta}^{\ell}(x) - \overleftarrow{\eta}^{\varepsilon n}(x) = \frac{1}{m \ell}\sum_{j=1}^{m-1}\sum_{z = x-\ell}^{x-1}[ \eta(z) - \eta(z - j \ell) ].
B_{0,4}= \left(\pi^G_{3}\right)^{-1}\left( G_{0,3}\right), B_{0,n+1}= \left(\pi^G_{n}\right)^{-1}\left(B_{0,n}\cup \bigcup_{i=1}^{n-2}\left(\textrm{Im}\left(\sigma^G_i\right) \cdot B_{0,n}\right)\right), \mbox{for}n\geq 4.
\begin{array}[c]{l}\left\langle J_{ab},P_{c}\right\rangle =\alpha_{0}\,\varepsilon_{abc}, \left\langle J_{ab},Z_{c}\right\rangle =\alpha_{1}\,\varepsilon_{abc}, \left\langle Z_{ab},P_{c}\right\rangle =\alpha_{1}\,\varepsilon_{abc}, \left\langle Z_{ab},Z_{c}\right\rangle =\alpha_{2}\,\varepsilon_{abc}.\end{array}
{\cal M}= \frac{i}{\sqrt{{\mathrm{\boldmath ~ \beta ~}}^{2}}}\left( \begin{array}{l l}{E_{\mathrm{\boldmath ~ \beta ~}}}{- E_{-{\mathrm{\boldmath ~ \mu ~}}}}{E_{\mathrm{\boldmath ~ \mu ~}}}{\, \, \, E_{-{\mathrm{\boldmath ~ \beta ~}}}}\end{array}\right) \, \, .
\lim\limits_{H^2 \rightarrow +\infty}\frac{r}{k-r}w_2 =\ H^2\left(\frac{k}{r}-\frac{1}{2}\right)+ \frac{H^2r}{2(k-r)}\left(\frac{k}{r}-\epsilon \right)\left(-\frac{k}{r}-\epsilon + 1 \right)
P_2 \left( n \right) = \delta^{4n - 8}\left( \frac{1}{n - 3}\bar B_1 \left( \frac{1}{\delta^2},n - 1,n - 2 \right) \right. \,\,\,+ \frac{1}{n - 1}\bar B_1 \left( \frac{1}{\delta^2},n - 3,n \right) \,\,\, \left. - \frac{2}{n - 2}\bar B_1 \left( \frac{1}{\delta^2},n - 1,n - 2 \right) \right).
\begin{cases}y_i^{\prime\prime}+ A y_i +C_{i+1 i}^{\ast}y_{i+1}= 0 \,, 1 \le i \le n-1\,, y_{n}^{\prime\prime}+ A y_{n}= B_n v_n\,, (y_i,y_i^{\prime})(0)=(y_i^0,y_i^1) \mbox{for}i=1, \ldots, n \,,\end{cases}
\langle(-\triangle_{g})^{\alpha}u,v\rangle=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g\bigg{(}\frac{u(x)-u(y)}{|x-y|^{\alpha}}\bigg{)}\frac{v(x)-v(y)}{|x-y|^{\alpha}}\frac{dxdy}{|x-y|^{d}},
\Big|1-\sum_{h\in H_{r(k)}}\mu(h)\Big| =\Big|1-\sum_{h\in H_{r(k)}}\phi_0(s(h))\mu_0(h)+\phi_1(s(h))\mu_1(h)\Big| & = \phi_0(r(k))\Big|1-\sum_{h\in (H_0)_{r(k)}}\mu_0(h)\Big|+\phi_1(r(k))\Big|1-\sum_{h\in (H_1)_{r(k)}}\mu_1(h)\Big| <\phi_0(r(k))\epsilon+\phi_1(r(k))\epsilon=\epsilon.
z=\frac{\Lambda^{3N_c-N_f}S^{N_f-N_c}}{\mbox{det}M - (\tilde B^{A_1 A_2\ldots A_{N_f-N_c}}M_{A_1}{}^{B_1}M_{A_2}{}^{B_2}\ldots M_{A_{N_f-N_c}}{}^{B_{N_f-N_c}}B_{B_1 B_2\ldots B_{N_f-N_c}})}.
\begin{array}{ll}\frac{1}{\tau}\int\limits^{t_{n}}_{t_{n-1}}\Delta u(\cdot,t)dt= \begin{cases}\Delta u^{1}+(R1)^{1}, \Delta u^{n-\frac{1}{2}}+(R1)^{n}, 2\leq n\leq \mathcal{N}, \end{cases}\end{array}
\int I_{( 1 )}^{g h}= \int \prod_{i < j}^{2}d^{2}\tau_{i j}( \det{\mathrm{I m \,}\tau})^{- 1}( \overline{{{\Delta_{( 2 )}}}})^{- 2}\, \overline{{{\Theta ( \Lambda_{2 4})}}}\, Z_{R}^{g h}( \tau ) \, ,
{\frac{\xi_{0}}{\lambda}}= 2 i \xi_{0}\left({\frac{1}{2 L}}\sum_{n = \pm \frac{1}{2}, \pm \frac{3}{2}, \cdots}\right) \int_{- \infty}^{\infty}{\frac{d p^{-}}{2 \pi}}{\frac{1}{2 p_{n}^{+}p^{-}- \xi_{0}^{2}+ i \epsilon}},
\lim_{n \to \infty}\int_{G}^{}f_{(n)}\circ ( \phi_1 * \phi_2 ) (g) dg = \int_{G}^{}f \circ ( \phi_1 * \phi_2 ) (g) dg, \lim_{n \to \infty}\int_{\mathbb{R}}^{}f_{(n)}\circ ( \phi_1^* * \phi_2^* ) (x) dx = \int_{\mathbb{R}}^{}f \circ ( \phi_1^* * \phi_2^* ) (x) dx
E_{1, 1}\cdot \Lambda_{a, b, c}= c\Lambda_{a, b, c}, E_{2, 2}\cdot \Lambda_{a, b, c}= (b-c)\Lambda_{a, b, c}, E_{3, 3}\cdot \Lambda_{a, b, c}= (a-b)\Lambda_{a, b, c}, E_{4, 4}\cdot \Lambda_{a, b, c}= (n-a)\Lambda_{a, b, c},
\phi_1 := \phi_{1A}(\tau, z) = 2 \zeta^{\pm 1}+ 20 + (20 \zeta^{\pm 2}- 128 \zeta^{\pm 1}+ 216)q + (2 \zeta^{\pm 3}+ 216 \zeta^{\pm 2}- 1026 \zeta^{\pm 1}+ 1616)q^2 + O(q^3).
M_{NC}(n,d) \le H_t(n,d)2^d + \left(\binom{n}{d}-H_t(n,d)\right)\frac{2^{d+1}}{t+1}= \left(h_t(n,d) + (1-h_t(n,d))\frac{2}{t+1}\right) \cdot 2^d \binom{n}{d}\enspace .
(S^*)^{N/(ps)}(t_0^*)^{p-1}-\frac{b}{p^\sharp}C_U(S^*)^{\frac{N+\alpha}{ps}}(t_0^*)^{2\cdot p^\sharp-1}-\varepsilon_g (S^*)^{N/(ps)}(t_0^*)^{p_s^*-1}=0.
|\varphi_k\dot\varphi_{k'}-\dot\varphi_k\varphi_{k'}|=|\dot\varphi_k|^2|\varphi_{k'}|^2+|\dot\varphi_k|^2|\varphi_{k'}|^2+\frac12(\dot\varphi_k\varphi_k^*+\varphi_k\dot\varphi_k^*)(\dot\varphi_{k'}\varphi_{k'}^*+\varphi_{k'}\dot\varphi_{k'}^*)+2
T^\epsilon_m(g, w)(0) = \sum^m_{n=1}\| g^\epsilon_n(0)\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2+\| \tilde{w}_\epsilon(0)\|_{H^{m, m -1}_{k+\ell}(\mathbb{R}^2_+)}^2 &\le C\| \tilde{u}_0\|_{H^{m+1}_{k + \ell'-1}(\mathbb{R}^2_+)}^2,
\left[{\frac{g_{B}}{\pi}}( B - 1 ) + B -{\frac{2 \sqrt{2}}{\pi^{2}}}\right] X^{2}+ \left({\frac{g_{B}}{\pi}}- B +{\frac{4 \sqrt{2}}{\pi^{2}}}\right) X -{\frac{2 \sqrt{2}}{\pi^{2}}}= 0 .
\left< \phi^{2}( x ) \right> = \left.{\frac{d}{d s}}\right|_{s = 0}{\frac{s}{\mu^{2}}}\zeta ( s + 1 , x | L_{b}/ \mu^{2}) = \lim_{s \rightarrow 0}\left[ ( 1 + s \ln \mu^{2}) \zeta ( s + 1 , x | L_{b}) + s \zeta^{\prime}( s + 1 | L_{b}) \right] ,
( C^{- 1})^{\alpha \beta}( C^{- 1})^{\gamma \delta}- ( \Gamma_{\dot{a}}C^{- 1})^{\alpha \beta}( \Gamma^{\dot{a}}C^{- 1})^{\gamma \delta}\; , \; C_{\alpha \beta}C_{\gamma \delta}- ( C \Gamma_{\dot{a}})_{\alpha \beta}( C \Gamma^{\dot{a}})_{\gamma \delta}
\frac{\mu^{s+1}}{2}\left(\frac{2\pi}{L}\right)^{-s}\sum_{n=-\infty}^{\infty}\frac{1}{\Gamma\left(\frac{s}{2}\right)}\int_0^{\infty}dt\,t^{(\frac{s}{2}-1)}e^{-(n^2+(\frac{mL}{2\pi})^2)t}\rfloor_{s=-1}\,=
P_{\infty}(R_s) = \Pr\left(\gamma_E^{\infty}\!>\! \frac{\gamma_B \!+\! 1}{2^{R_s}}\!-\! 1\right) = 1 \!-\! F_{\gamma_E^{\infty}}\left(\frac{\gamma_B \!+\! 1}{2^{R_s}}\!-\! 1\right),
\lim_{n\rightarrow\infty}\frac{1}{n}\log\left(\prod_{i=0}^{n-1}\sup_{t\in D^+}\left[\frac{m_i(pt)}{m_i(t)^p}\right]\right)=\mathbb E\log \sup_{t\in D^+}\left[\frac{m_0(pt)}{m_0(t)^p}\right]<0
p\sum_{k=0}^{n-1}\frac{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\equiv \frac p2\sum_{k=0}^{n-1}\frac{\binom{2k}k^2H_{k}}{(2k+1)16^k}+\sum_{k=0}^{n-1}\frac{\binom{2k}k^2}{(2k+1)16^k}-\sum_{k=0}^{n-1}\frac{\binom nk\binom{2k}k}{(2k+1)(-4)^k}\pmod{p^2}
I_{\beta}( D , s , m ) = \frac{m^{D - 2 s}}{( 2 \pi^{1 / 2})^{D}\Gamma ( s )}\left[ \Gamma ( s - \frac{D}{2}) + 4 \sum_{n = 1}^{\infty}\left( \frac{2}{m n \beta}\right)^{D / 2 - s}K_{D / 2 - s}( m n \beta ) \right]
\widehat{N}_{1}^{(r)}= \frac{N_{(0,rx_{1})}-\mu_{\alpha}(rx_{1})}{\sqrt{\sigma^{2}(rx_{1})}}, \widehat{N}_{j}^{(r)}= \frac{N_{(rx_{j-1},rx_{j})}-\big(\mu_{\alpha}(rx_{j}) - \mu_{\alpha}(rx_{j-1})\big)}{\sqrt{\sigma^{2}(rx_{j})+\sigma^{2}(rx_{j-1})}}, j=2,\ldots,m.
\sum_{k=0}^{k_0-1}u_{d-km}\ge\frac{1}{m}\sum_{k=1}^{k_0-1}\sum_{j\in U_{k-1}}u_j \ge\frac{1}{m}\Big(\bar{m}-\sum_{j\in U_{k_0}}u_j-\sum_{j\in U_{k_0-1}}u_j\Big)\ge\frac{1}{m}\big(m-(1-\delta_0)m\big)\ge\delta_0.
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^3}{n^4}}= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{n^2}{4}+ \frac{n^3}{3}+ \frac{n^4}{4}\right) \frac{z^n}{n^4}\right\}
\left[ \frac{d}{d \eta}( 1 - \eta^{2}) \frac{d}{d \eta}- \frac{( | m | + \delta_{1})^{2}}{2 ( 1 + \eta )}- \frac{( | m | + \delta_{2})^{2}}{2 ( 1 - \eta )}+ 2 \alpha R \eta + \frac{E R^{2}}{2}( 1 - \eta^{2}) \right] \psi_{2}= - \lambda ( R ) \psi_{2},
( M_{0 , \pi})_{A}^{\; B}= \frac{1}{2}\alpha_{0 , \pi}( \sigma_{1})_{A}^{\; B}\ , \quad ( \bar{M}_{0 , \pi})_{A}^{\; B}= \frac{1}{2}\alpha_{0 , \pi}\mathrm{i}( \sigma_{2})_{A}^{\; B}\ ,
\frac{p(r|c_i)}{p(r|c_j)}= \frac{p(c_i|r)}{p(c_j|r)}p(c_i|r) = \prod_{i=1}^{n}q_i q_i = \begin{cases}p_i r_i \neq c_i 1 - p_i r_i = c_i \end{cases}.
b_{nk}=\frac{1}{n+k}\left[\frac{(k+1)(k+2)}{2}b_{n-1, k+2}+ \frac{\lambda (\lambda -1)}{2}\sum_{m=0}^k (-1)^{m+1}\frac{m+1}{q^{m+2}}b_{n-1, k-m}\right].
{\cal H}= \frac{1}{2}( \vec{E}^{a}\vec{E}^{a}+ \vec{B}^{a}\vec{B}^{a}) + \pi^{\dagger}\pi + ( \vec{D}\varphi )^{\dagger}( \vec{D}\varphi ) + V ( \varphi^{\dagger}\varphi ) \ .
\|u(a-\epsilon)_+u^*-b\| =\|v(x^*x-\delta)_+v^*-b\|=\|(xx^*-\delta)_+-b\|\le \|(xx^*-\delta)_+-xx^*\|+\|xx^*-b\| &\le \delta+4(\epsilon')^\frac{1}{2}\le 5\epsilon^\frac{1}{2}.
\omega_{\tilde\sigma_+(0),\tilde\sigma_-(0),\infty}^{-1}(2\tilde b_\infty)= \min_{f_+,f_-\in\mathcal{F}_{RDT,p}(C)}\sqrt{\frac{p_{X,+}(0)}{\tilde\sigma_+^2(0)}\int_0^\infty f_+(u)^2\, du+\frac{p_{X,-}(0)}{\tilde\sigma_-^2(0)}\int_{-\infty}^0 f_-(u)^2\, du}&f_+(0)+f_-(0)\ge \tilde b_\infty.
V_1 =(a_{s+1}a_1)^*(a_{s+1}a_2)^*\dotsm(a_{s+1}a_{s-1})^*, V_2 =(a_{s+2}a_1)^*(a_{s+2}a_2)^*\dotsm(a_{s+2}a_{s-2})^*, V_3 =(a_{s+3}a_1)^*(a_{s+3}a_2)^*\dotsm(a_{s+3}a_{s-3})^*, & \vdotswithin{=}V_{t-1}=(a_{s+t-1}a_1)^*(a_{s+t-1}a_2)^*\dotsm(a_{s+t-1}a_{s-t+1})^*.
\eta_t =\min\left(\frac{1}{8\kappa \bar{L}d},\, d^{-\frac{2(\beta-1)}{2\beta-1}}T^{-\frac{\beta}{2\beta-1}}\right) \qquad h_t = \left(\frac{d^2}{T}\right)^{\frac{1}{2(2\beta-1)}}\enspace,
u(t,x) =\int_0^\ell u_0(y) p_{\ell,0}(t,y,x)\,dy-\int_0^t p_{\ell,0}(t-s,0,x)\,\Lambda(ds) &+ \rho N(1-\frac{2x}{\ell})+ 4\rho \sum_{n=1}^\infty \cos(\frac{n\pi x}{\ell}) \frac{((-1)^n -1)}{n^2\pi^2}e^{-2\kappa(n^2\pi^2)t}
-E(l_rl_sl_t) =\lambda_{rs,t}+\lambda_{rt,s}+\lambda_{st,r}+\lambda_{rst}, E(l_rl_sl_{tu}l_v) =-\lambda_{rs}\lambda_{tu,v}-\lambda_{rv}\lambda_{tu,s}-\lambda_{sv}\lambda_{tu,r}+O(n^{3/2}), E(l_rl_sl_tl_u) =\lambda_{rs}\lambda_{tu}+\lambda_{rt}\lambda_{su}+\lambda_{ru}\lambda_{st}+O(n^{3/2}).
\tilde{C}_{N}(z_{1},z_{2}) := \frac{1}{d_{N}^{2}}\frac{\partial^{2}}{\partial z_{1}\partial z_{2}}\left[s(z_{1})s(z_{2})\sum_{k=1}^{N}\mathbb{E}_{k-1}[\mathbb{E}_{k}\delta^{k,1}_{N}(z_{1})\mathbb{E}_{k}\delta^{k,1}_{N}(z_{2})]\right]
0=\nabla f(\bar x)+\sum\limits_{i\in I^g(\bar x)}\lambda_i\nabla g_i(\bar x)+\sum\limits_{j\in\mathcal P}\rho_j\nabla h_j(\bar x) &+\sum\limits_{l\in I^{0+}(\bar x)\cup I^{00}(\bar x)}\mu_l\nabla G_l(\bar x)+\sum\limits_{l\in I^{+0}(\bar x)\cup I^{00}(\bar x)}\nu_l\nabla H_l(\bar x),