harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_X\alpha =L^l_X\alpha+L^r_X\alpha\in\Omega^1_RA\otimes A\oplus A\otimes\Omega^1_RA, i_X\omega =i^l_X\omega+i^r_X\omega\in \Omega^1_RA\otimes A\oplus A\otimes\Omega^1_RA,
U_{a_{i}}=\begin{pmatrix}\frac{1}{\sqrt{1+a^{2}}}I_{p'-q}-\frac{a}{\sqrt{1+a^{2}}}I_{p'-q}\mbox{O}_{p'-q,1}\frac{1}{\sqrt{1+a^{2}}}I_{p'-q}\frac{a}{\sqrt{1+a^{2}}}I_{p'-q}\mbox{O}_{p'-q,1}\mbox{O}_{1,p'-q}\mbox{O}_{1,p'-q}1 \end{pmatrix},\quad\mbox{where $i=1,\dots,q$,}
0=c_{2}S(v)\begin{pmatrix}{\bf j}0 0{\bf j}\end{pmatrix}+c_{3}S(v)\begin{pmatrix}{\bf k}0 0{\bf k}\end{pmatrix}+d_{2}T(v)\begin{pmatrix}{\bf j}0 0{\bf j}\end{pmatrix}+d_{3}T(v)\begin{pmatrix}{\bf k}0 0{\bf k}\end{pmatrix},
\lim_{n\to\infty}\Lambda_{n,2}^{(1,1)}= \lim_{n\to\infty}\Lambda_{n,2}^{(1,3)}= \lim_{n\to\infty}\Lambda_{n,2}^{(3,3)}=1, &\lim_{n\to\infty}\Lambda_{n,2}^{(1,2)}= \lim_{n\to\infty}\Lambda_{n,2}^{(2,2)}= \lim_{n\to\infty}\Lambda_{n,2}^{(2,3)}=0.
\sum_{i=1}^{k}q^{d_{i}^{-}(x)}T_{i}^{(-)}(x).f(x-v_{i}) =T_{w_{x}}^{-1}\sum_{a=1}^{r}\sum_{l=1}^{c_{a}}q^{c_{a}-l}(T_{c_{1}+\cdots +c_{a-1}+l-1}^{-1}\cdots T_{c_{1}+\cdots +c_{a}-1}^{-1}).f(w_{x}x-v_{c_{1}+\cdots +c_{a}}).
\int_{\tilde{\gamma}}\widetilde{d}G_{\Omega}(\cdot,z') =\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,z')+\int_{\gamma_2\left(\gamma_1\right)^{-1}}\widetilde{d}G_{\Omega}(\cdot,z') &=\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,z') &=h(z').
{\cal T}_{\mu \nu}^{\alpha \beta}= 2{\it g_{1}}^{2}\epsilon_{~ \mu}^{\alpha ~ \rho \sigma}\epsilon_{~ \nu}^{\beta ~ \lambda \gamma}\bigg ( F_{\rho \sigma}F_{\lambda \gamma}+ \tilde{F}_{\rho \sigma}\tilde{F}_{\lambda \gamma}+{\frac{1}{\sqrt 2}}\hat{\theta}_{5}\big ( F_{\rho \sigma}\tilde{F}_{\lambda \gamma}+ \tilde{F}_{\rho \sigma}F_{\lambda \gamma}\big ) \bigg ) \, ,
\frac{1}{( q \! \cdot \! n )^{j}}= P V \left( \frac{1}{( q \! \cdot \! n )^{j}}\right) - i \pi \frac{( - 1 )^{j - 1}}{( j - 1 ) !}\delta^{( j - 1 )}( q \! \cdot \! n ) \; s g n ( q^{0}) ,
\langle A \rangle_{0}= \lim_{R \rightarrow \infty}\frac{1}{Z_{R}[ 0 ]}\mathrm{T r}\, ( e^{- \beta H_{R}}\, \delta ( Q_{R}) \, A ) = \lim_{R \rightarrow \infty}\frac{1}{Z_{R}[ 0 ]}\mathrm{T r}\, ( e^{- \beta H_{R}}\, P_{s}\, A ) \, ,
b_k^Ty= 0\ \k\in\{0,...,m\}\begin{bmatrix}b^0_1 \cdots b^0_n \vdots \ddots \vdots b^m_1 \cdots b^m_n\end{bmatrix}\begin{bmatrix}y_1 \vdots y_n\end{bmatrix}= \begin{bmatrix}0 \vdots 0\end{bmatrix}
\;(\|u_*\|_{D(A)}^2+\alpha^2 \|A^{1/2}u_*\|_{L^2}^2)(t) &\;+2\nu\int_0^t(\|A^{s/2}u_*\|^2_{D(A)}+\alpha^2\|A^{(1+s)/2}u_*\|^2_{L^2})(\tau)\,d\tau \leq \;\|u_0\|_{D(A)}^2+\alpha^2 \|A^{1/2}u_0\|_{L^2}^2 &\;+C\int_0^t (\|Au_*\|_{L^2}\| Au_*\|_{D(A^{s/2})}\|A^{1/2}u_*\|_{D(A^{s/2})})(\tau)\,d\tau.
II \le C \bigg(\sum_{j=1}^{J-1}(2^j\rho)^{2\mu}(II_A+II_B+II_C)^2\bigg)^\frac{1}{2}\le C\rho^{\gamma+\frac{3}{2}-k_\perp-\frac{1}{p}}|\log(\rho)| \|\sigma\|_{H^1(\Lambda)}\bigg(\sum_{j=1}^{J-1}(2^j\rho)^{2\mu+\frac{2}{p}}\bigg)^\frac{1}{2}.
e\Bigl( \frac{\pm n_2 \bar u}{m c n_1^{-1}}\Bigr) e\Bigl( \frac{-n_2 n_1^2}{c m p}\Bigr) = e \Bigl( \frac{\pm n_2 \bar u}{m c n_1^{-1}}- \frac{n_2 n_1^2}{c m p}\Bigr) = e \Bigl( \frac{\pm n_2 \bar u p - n_2 n_1}{m c n_1^{-1}p}\Bigr) &= e\Bigl( \frac{n_2 (\pm \bar u p - n_1)}{m c n_1^{-1}p}\Bigr).
\|A(T) - A(\tau)\|^2_{H^k(\Omega_r)}\leq C_{\ref{omega}}\left( \|F(\tau - \tau_0)\|^2 - \|F(T)\|^2 \right) \left( T-\tau \right) & \cdot \left(1 + \sup_{\tau \leq t < T}\|A(t) \|^{2k}_{H^{k-1}\left( \Omega \right)}\right)
L_a = \frac{1}{32\pi^2}\mbox{Im}\,\mbox{tr}\int d^2\theta\W^2 g(z)= \frac{1}{32\pi^2}\mbox{Im}\,\mbox{tr}\int d^2\theta\W^2 \Big(-i\ln z+\sum\limits_{n=1}^\infty c_n z^n\Big).
\sigma^{1}= \frac{\varphi + \bar{\varphi}}{1 + \varphi \bar{\varphi}}; \sigma^{2}= \frac{i ( \bar{\varphi}- \varphi )}{1 + \varphi \bar{\varphi}}; \sigma^{3}= \frac{1 - \varphi \bar{\varphi}}{1 + \varphi \bar{\varphi}}
{\cal H}^{M}\approx \frac{1}{8}{\Pi}_{a i}^{+ ( 0 )}{\Pi}_{a i}^{+ ( 0 )}\approx \frac{1}{2}( D_{a 1}^{( 0 )}D_{a 1}^{( 0 )}+ B_{a 1}^{( 0 )}B_{a 1}^{( 0 )})
I_d(tu_1,\ldots,tu_{d-1},t\theta w) =\frac{1}{4}\Big(\sum_{i=1}^{d-1}\|t u_i\|_{\lambda_i}^2 +\theta^2\|t\theta w\|_{\lambda_d}^2\Big) &=\frac{t^2}{4}\Big(1+C_1\theta^2\Big)\sum_{i=1}^{d-1}\|u_i\|_{\lambda_i}^2.
p_{\hat{S}_1(t),\hat{S}_2(t)|S(t)}(\hat{s}_1,\hat{s}_2|s)=\sum\limits_{\substack{s^m:s(t)=s \hat{s}^m_i:\hat{s}_i(t)=\hat{s}_i,\mbox{}i=1,2}}p_{\hat{S}^m_1,\hat{S}^m_2|S^m}(\hat{s}^m_1,\hat{s}^m_2|s^m)\prod\limits_{t':t'\neq t}p_S(s(t')).
((\tilde \gamma+2\gamma I)^{-1})^i_j=\left( \begin{array}{ccc}3\gamma/(9\gamma^2+B_0^2) -B_0/(9\gamma^2+B_0^2) 0 B_0/(9\gamma^2+B_0^2) 3\gamma/(9\gamma^2+B_0^2) 0 0 0 1/(3\gamma) \end{array}\right).
{f_{\left. T \right|S}}(\left. t \right|s) = \int{d{\lambda_{ex}}{f_{\left.{{\Lambda_{ex}}}\right|S}}\left({\left.{{\lambda_{ex}}}\right|s}\right)}{f_{\left. T \right|{\Lambda_{ex}},S}}\left({\left. t \right|{\lambda_{ex}},s}\right).
\frac{2m\!+\!1}{4(m\!+\!1)}\sqrt{\frac{a}{2}}\!\sum_{K_1\ni j=k}^{\infty}\|x^{j+1}\!-\!x^j\|\le c_1\sum_{j=k}^{\infty}\!\Xi_{j}+c_2\sum_{j=k}^{\infty}\!\Xi_{j}^{\frac{1-\theta}{\theta}}+a_1(m\!+\!1)(\Delta_{k-m}-\Delta_{k}).
r_{i}= \frac{\det(B_{i})}{\det(B)}= -\frac{\prod_{1 \leq l \leq t+1}(x_{t+2}y_{l}- x_{l}y_{t+2})}{\prod_{\substack{1 \leq l \leq t+2 l \neq i}}(x_{i}y_{l}- x_{l}y_{i})}.
\varphi(\mathrm{v},\mathrm{y},\mathrm{t}, \tau)-\Phi^{\textbf{e}}(\xi,\mathrm{t};\mathrm{z},\tau) = \mathcal{O}\Bigg(\sqrt{\alpha}|\mathrm{y}-\mathrm{v}| \ \Vert \partial_\mathrm{s}\Phi^{\textbf{e}}(\xi,\mathrm{t};\mathrm{z},\cdot)\Vert_{\mathrm{H}^{-\frac{1}{4}}(0,t)}\Bigg),
WK(u) = \lim_{h \to 0}\frac{K(uW^h) - K(u)}{h}= t^{-\alpha}\lim_{h \to 0}\frac{K(s_t(u) W^{th}) - K(s_t(u))}{h}= t^{-\alpha+1}\lim_{h \to 0}\frac{K(s_t(u) W^{th}) - K(s_t(u))}{th}= t^{-\alpha+1}WK(s_t(u)).
G' = \left\{g \in \operatorname{SL}_2(\mathbb{Z}_\ell)^2 \bigm\vert \Theta_2(g) \in [\mathcal{L}(G),\mathcal{L}(G)], \; \operatorname{tr}(g) - 2 \in C \right\}\supseteq \mathcal{B}_\ell(2k,2k).
|\widehat{f}_c(k) - \widehat{h}(k) | = \frac{1}{2m}\left| \left( \sqrt{\frac{1}{c^2}(k+1)^2+m^2}+ \sqrt{\frac{1}{c^2}(k-1)^2+m^2}-2m\right) \widehat{h}(k) \right| &\leq \frac{1}{2m}\frac{2k^2+2}{mc^2}|\widehat{h}(k)|.
\begin{aligned}u^n(s,{X}_s)=\ \Phi({X}_T)- w^n(s,{X}_s) &+\int^T_sf(r,{X}_r,u^n(r,{X}_r),\nabla u^n(r,{X}_r))\d r &-\int^T_s\nabla w^n(r,{X}_r)\d{W}_r-\int^T_s\nabla u^n(r,{X}_r)\d{W}_r.\end{aligned}
\begin{aligned}u^0_e\partial_x u^1_e +v^1_e\partial_Yu^0_e-h^0_e\partial_x h^1_e-g^1_e\partial_Y h^0_e+\partial_x p^1_e &=-\int_Y^\infty E_b(x,\theta)\mathrm{d}\theta-\sqrt{\epsilon}h^0_e\partial_x\rho-\epsilon\int_0^y\partial_x \rho(x,s)\mathrm{d}s\cdot\partial_Yh^0_e,\end{aligned}
\dot{\Psi}^{R}= i [ H^{R}, \Psi^{R}] = \gamma_{0}( - i \gamma_{i}\partial_{i}- e \gamma_{i}V_{i}^{R}+ m ) \Psi^{R}+{\frac{e}{2}}\left\{\Psi^{R},{\frac{1}{\vec{\partial}^{2}- M^{2}}}J_{0}\right\}~ ,
E_0^{(r)}T_a=\begin{cases}2\phi_1(a)\int_0^ady\thinspace\phi_1^{-2}(y)\int_{-\infty}^y\phi_1(z)dz,\a>0; \frac{2\phi_1(a)}{\int_0^{\infty}\phi_1^{-2}(x)dx}\int_0^{\infty}dx\thinspace\phi_1^{-2}(x)\int_a^0dy\thinspace\phi_1^{-2}(y)\int_y^x\thinspace\phi_1(z)dz,\a<0.\end{cases}
\textup{L.H.S. of (\ref{eqn591})}\lesssim\sum_{\tau=p, N_1}\sum_{\mu, \nu, \kappa\in\{+,-\}}|\mathfrak{G}_{\mu, \nu}^{\kappa,\tau}| +\big| \sum_{\tau=p,N_1}\sum_{\mu, \nu, \mu',\nu' \in\{+,-\}}\mathfrak{G}_{\mu, \nu;\mu',\nu'}^{\tau}+ \frac{d}{d t}E_{modi}^{S, low2}(t)|
\mathbf{b}_{s,i}-\mathbf{b}_{i}=s\Biggl[\log \left(\theta_{n-m+i}\right),\theta_{n-m+i}\log \left(\theta_{n-m+i}\right), \cdots,\theta_{n-m+i}^{n-m-1}\log \left(\theta_{n-m+i}\right) \Biggr]^t +o\left(s\right) =s\log\left(\theta_{n-m+i}\right)\mathbf{b}_i+o\left(s\right).
q \equiv - \frac{1}{1 6 \pi^{2}}\int d^{4}x \mathrm{~}t r G_{\mu \nu}\tilde{G}_{\mu \nu}= \frac{1}{2}\int d \sigma_{1}d \sigma_{2}d \tau_{1}d \tau_{2}\epsilon_{\mu \nu \alpha \beta}\dot{z}_{1 \mu}z \prime_{1 \nu}\dot{z}_{2 \alpha}z \prime_{2 \beta}\delta ( z_{1}- z_{2})
||\delta{{\mathbf V}_i}||_{s,\ast,T}+||\delta\psi_i||_{H^s(\Gamma_T)}&\leq C\Big(||{f}_i||_{s+2,\ast,T}+||{g}_i||_{H^{s+2}(\Gamma_T)}&\quad+(||{f}_i||_{8,\ast,T}+||{g}_i||_{H^{8}(\Gamma_T)})||(\tilde{\mathbf U}^a+{\mathbf V}_{i+\frac{1}{2}}, \nabla(\Psi^a+\Psi_{i+\frac{1}{2}}))||_{s+4,\ast,T}\Big).
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m+1)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m-1}}(2^{2i}-1)}{|M_2^{K_{\ell}}|}\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^L|}\Bigr\}^{-1}\leq 2\cdot\frac{2^{n_{2,\nu_2}}-1}{2^{n+1}-1}.
\begin{aligned}\partial_{k}\left(\frac{\pi^2}{8}\omega(k)\right) = \frac{(2k^2 - 1) K (\mathcal{E}+ (k^2 - 1) K)}{k(1 - k^2)}+ kK^2 = \frac{k^2(2k^2 - 1) K \Theta + k^2(1 - k^2) K^2}{k(1 - k^2)}. \end{aligned}
N_{2}= i \sin \varphi x \left( \begin{array}{cc}0 - \frac{q (p - q)^{-}(p + q)^{+}}{(p^2 - q^2)}\frac{q (p - q)^{+}(p + q)^{-}}{(p^2 - q^2)}0\end{array}\right).
\bar{u}(t,x)=\sup_{h \in{\mathcal{H}}(T)}\inf_{\gamma \in \Gamma(T)}I(f,x,h,\gamma,t,T) \underline{u}(t,x)=\inf_{\gamma \in \Gamma(T)}\sup_{h \in{\mathcal{H}}(T)}I(f,x,h,\gamma,t,T) u(t,x)=\bar{u}(t,x)=\underline{u}(t,x)
\left\|B\vec{t}\right\|_{\dot a^{0,\tau}_{p,q}(W)}^r \lesssim\sum_{k\in\mathbb{Z}}\sum_{l=0}^\infty(1+k_-+l) &\quad\times\left[2^{-(E-\frac{n}{2})k_-}2^{-k_+(F+\frac{n}{2}-\frac{n}{a})}2^{-(D-\frac{n}{a})l}2^{(k_-+l)n\widehat\tau}\left\|\vec{t}\right\|_{\dot a^{0,\tau}_{p,q}(W)}\right]^r,
{^{( + )}\xi}^{i}= \left( - \beta \partial_{\phi}{^{( + )}\xi}^{\phi}( x^{+}) ,{^{( + )}\xi}^{\phi}( x^{+}) \right) , \qquad{^{( - )}\xi}^{i}= \left( - \beta \partial_{\phi}{^{( - )}\xi}^{\phi}( x^{-}) ,{^{( - )}\xi}^{\phi}( x^{-}) \right)
\sum_{r,s=1}^n \dot{h_{rs}}\frac{\partial}{\partial z_r}\wedge \frac{\partial}{\partial z_s}=\sum_{r,s,a,b=1}^n \dot{h_{rs}}\frac{\partial \xi_j^a (z,0)}{\partial z_r}\frac{\partial \xi_j^b(z,0)}{\partial z_s}\frac{\partial}{\partial \xi_j^a}\wedge \frac{\partial}{\partial \xi_j^b}
\partial_{x^{2}}f^{k}=\partial_{x}f^{k}{\displaystyle\sum\limits_{j=0}^{k-1}}\frac{F_{x^{2}}^{j}-2F_{xy}^{j}\nu_{j}+F_{y^{2}}^{j}\nu_{j}^{2}}{F_{x}^{j}}\partial_{x}f^{j}.
G_{E}(a_{3},a_{4},\phi)=\mu^{4}\sqrt{a_{3}a_{4}}\pi\sum^{\infty}_{s=0}\frac{(-1)^{s+2}}{2s+2}g^{s+1}\phi^{2s+2}\frac{\Gamma(s)}{\Gamma(s+1)}A^{c^{2}}_{2}(s,a_{3},a_{4})\, .
_h^w (\overline{q}_0^N (X/\sqrt{\varepsilon})) [_h^w(p_0),_h^w (q_0^N (X/\sqrt{\varepsilon}))] &=\frac{1}{\varepsilon^{2N}}_h^w (\overline{q}_0^N (X)) [_h^w(r),_h^w (q_0^N (X))].
\begin{array}{c}lim \varepsilon \rightarrow 0\end{array}\int D\phi \exp \{i\int d^{4}x[\frac{1}{2}\phi (\alpha \partial^{2}_{0}-\nabla^{2}-m^{2}+i\varepsilon )\phi +\frac{\lambda}{4}\phi^{4}]+i\int d^{4}xJ\phi \}\bullet \left( \exp [i\delta \alpha \int d^{4}x\phi \partial^{2}_{0}\phi ]-1\right)
\Big [{\frac{1}{4 \, v^{2}}}-{\frac{f^{\prime}}{v f}}+{\frac{b}{v}}-{\frac{f^{\prime \prime}}{f}}+{\frac{2 b f^{\prime}}{f}}- b^{2}-{\frac{1}{4 \, v^{2}}}- m^{2}\Big ] \varphi^{( 0 )}= 0 ~ .
\begin{array}{rcl}x^ix^j = qx^jx^i~,~~~~~{\rm for}~i<j~ ~{\rm and}~~i\not=-j~, x^ix^{-i}= x^{-i}x^i+\lambda q^{i-2}L_{i-1}= q^{-2}x^{-i}x^i+\lambda q^{i-2}L_i~,~~~~~{\rm for}~i>0~,\end{array}
\varphi(\Theta)={}\mathrm{itr}(\Theta \Psi_{n}){}-{}\frac{1}{2}\mathrm{tr}\left( \Theta \mathcal{A}_{n}\Theta \right){}+{}\int_{\mathbb{H}_{n}}\left( \mathrm{e}^{\mathrm{itr}(\Theta \xi)}{}-1{}-\mathrm{i}\frac{\mathrm{tr}(\Theta \xi)}{1+\left \Vert \xi \right \Vert^{2}}{}\right)\nu_{n}(\mathrm{d}\xi),\ \Theta \in \mathbb{H}_{n},
E(m^2) = \sum_{j=1}^\infty \sum_{|x|\leq L^j}|x|^p O(L^{-j (d - 1)}L^{-2k(j-j_m)_+}) &= \sum_{j=1}^\infty O(L^{(p+1)j}L^{-2k(j-j_m)_+}).
\lambda_{a|b|c|d}^{e|f|g|h}= \sum_{(p,q)\in [a,b]\times [c,d]}\lambda_{a|p|c|q}^{e|e|g|g}\lambda_{p|b|q|d}^{e|f|g|h}&= \sum_{_{(p,q)\in [a,b)\times [c,d]}}\lambda_{a|p|c|q}^{e|e|g|g}\lambda_{p|b|q|d}^{e|f|g|h}+ \sum_{q \in [c,d]}\lambda_{a|b|c|q}^{e|e|g|g}\lambda_{b|b|q|d}^{e|f|g|h}&=0
\left\langle \frac 1Nv^{-1}\sum_{k=1}^N(\sigma\circ w)^k\mu,\alpha\right\rangle = \frac 1N\sum_{k=1}^N \langle \mu,(\sigma\circ w)^k v\alpha\rangle = \frac 1N\sum_{k=1}^N \langle \mu,(\sigma\circ w)^k v\alpha\rangle + \Phi^+((\sigma\circ w)^k v\alpha) - \Phi^+((\sigma\circ w)^{k+1}v\alpha) = \frac 1N\sum_{k=1}^N \ell(x,(\sigma\circ w)^kv\alpha).
\bar{\beta}_{\mathrm{x}}(\delta) =\min_{i,x}\{\hat{B}_{\mathrm{x}}(x)\vert \{C_{\mathrm{x}}^ix=d_{\mathrm{x}}^i}\}, i=1,\dots,q_{\mathrm{x}}\, , \bar{\beta}_{\mathrm{u}}(\delta) =\min_{u,j}\{\hat{B}_{\mathrm{u}}(u) \vert \{C_{\mathrm{u}}^ju=d_{\mathrm{u}}^j}\}, j=1,\dots,q_{\mathrm{u}}\, .
\lim_{s\to \infty}\mathbb{P}_x(A \,| \tau^{\{0\}}>t+s)=\lim_{s\to\infty}\mathbb{E}_x\left[\mathbf{1}(A, \, t<\tau^{\{0\}})\frac{\mathbb{P}_{X_t}(\tau^{\{0\}}>s)}{\mathbb{P}_x(\tau^{\{0\}}>t+s)}\right].
K_\infty( x,b,c,d,\nu) :=\inf\left\{\int_{Q_{\nu}}f^{\infty}(x,w( y),\eta( y) ,\nabla w(y))dy:w\in\mathcal{A}( c,d,\nu), \eta\in L^{\infty}( Q_{\nu};\mathbb{R}^{m}), \int_{Q_{\nu}}\eta\,dy=b\right\},
( 2 i )^{N_{+}}\kappa ( a ) = \prod_{\alpha > 0}^{}\left( e^{i \pi ( a , \alpha ) / a_{0}}- e^{- i \pi ( a , \alpha ) / a_{0}}\right) = \sum_{\hat{s}\in W}^{}\det \hat{s}\exp \left( \frac{2 \pi i}{a_{0}}( \hat{s}\rho , a ) \right)
\Big|\sum_{m,n\geq0 \atop m+n>0}\frac{a_mb_n}{m+n}\Big| \leq \pi\Big(\sum_{m=0}^\infty |a_m|^2\Big)^\frac{1}{2}\Big(\sum_{n=0}^\infty |b_n|^2\Big)^\frac{1}{2}.
\frac{\tau}{\sqrt{a n\log n}}= O\left( \left( \sqrt{a}-\sqrt{b}\right) \sqrt{\frac{n}{\log^5 n}}+ \frac{1}{\log n}\right) =O\left( \frac{1}{\log n}\right) \frac{\tau}{a n}= O\left(\frac{1}{\log^2 n}+ \frac{1}{\sqrt{a n \log n}}\right) =O\left( \frac{1}{\sqrt{\log n}}\right) ,
\frac{1}{\sqrt{-g}}{\cal L}=\frac{1}{2\kappa}R-\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\cdot\partial_\nu\phi-\frac{1}{2}\exp(-2\frac{\phi}{f_\phi})g^{\mu\nu}\partial_\mu a\cdot\partial_\nu a
\Biggl[\prod_{j=1}^{N-1}\left(1-R\,\kappa^{+}_j\right)\Biggr]^{-\frac12}- \Biggl[\prod_{j=1}^{N-1}\left(1+R\,\kappa^{-}_j\right)\Biggr]^{-\frac12}= \Biggl[\prod_{j=1}^{N-1}\left(1-R\,\kappa_j(Q)\right)\Biggr]^{-\frac12}.
0\le \sum_{n \le X}\Bigl(1- \frac{\Phi(n)}n \Bigr) \le \sum_{n \le X}\sum_{\substack{d \mid n: d > T}}\frac1d = \sum_{T < d \le X}\frac1d \sum_{\substack{n \le X: d\mid n}}1 & \le X \sum_{T < d \le X}d^{-2}\ll \frac{X}{T}.
\int_X\psi\omega^n = n! 2^{|\Phi_s^+|-r}C_H\int_{\chi-(-\Delta^t)\cap \bar{C}^-}\psi(d_{2\chi-2q}u^*)\prod_{\alpha\in\Phi_{Q^u}\cup \Phi_s^+}q(\alpha^{\vee}) dq.
\nu_0:=\frac{1}{\gamma_p+\gamma_q}\frac{S^{\frac{(2-\gamma_p-\gamma_q)2^*_\mu}{22^*_\mu-2}}_{H,L}}{C_{N,p,q}}\frac{(22^*_\mu-2)(2-\gamma_p-\gamma_q)^{\frac{2-\gamma_p-\gamma_q}{22^*_\mu-2}}}{(22^*_\mu-\gamma_p-\gamma_q)^{\frac{22^*_\mu-\gamma_p-\gamma_q}{22^*_\mu-2}}(a^2+b^2)^{\frac{p+q-\gamma_p-\gamma_q}{2}}},
\; p_k^{1/2}|f^{(\ell)}_k(x)\;|=\; \big| (-i)^\ell\big( p_k^{-ix}\log^{\ell}p_k-p_{r_m}^{-ix}\log^{\ell}p_{r_m}\big)\big| \leq \; |p_k^{-ix}-p_{r_m}^{-ix}|\log^{\ell}p_k+ |\log^{\ell}p_k -\log^{\ell}p_{r_m}|\lesssim \; \log^{\ell}(p_{r_m})(\log p_k-\log p_{r_m}) \lesssim \; \log^{\ell}(p_{r_m})\frac{p_k-p_{r_m}}{p_{r_m}}.
F_2(z_2,\bar{z}_1z_2 + \bar{z}_1^2)= \frac{{\bigl(\bar{F}_1(-\bar{z}_1-z_2)\bigr)}^2-{\bigl(\bar{F}_1(\bar{z}_1)\bigr)}^2}{\bar{F}_1(\bar{z}_1)-\bar{F}_1(-\bar{z}_1-z_2)}=\bar{F}_1(\bar{z}_1)+ \bar{F}_1(-\bar{z}_1-z_2) .
\begin{pmatrix}a_1 a_2 a_1 \end{pmatrix}\begin{pmatrix}d_1 d_2 d_1 \end{pmatrix}-\begin{pmatrix}b_1 b_2 b_1 \end{pmatrix}\begin{pmatrix}c_1 c_2 c_1 \end{pmatrix}= \begin{pmatrix}1 1 \end{pmatrix}
\begin{matrix}{{\bar{C}_{z z}^{\prime \prime}= - b_{z z},}}&{{C^{+ \prime}= c^{z},}}\\{{\bar{\Gamma}_{+ z}^{\prime \prime}= - i \beta_{+ z},}}&{{\Gamma^{+ \prime}= - i \gamma^{+},}}\\{{\bar{\Gamma}_{- z}^{\prime \prime}= \beta_{- z},}}&{{\Gamma^{- \prime}= - \gamma^{-},}}\\ \end{matrix}
\beta^{\mathbb{Z}_p}_{i+a}(J^{h})-\beta^{\mathbb Q}_{i+a}(J^{h}) \geq \sum_{\ell=1}^h \left[\beta^{\mathbb{Z}_p}_{i+a}(T^{\ell})-\beta^{\mathbb Q}_{i+a}(T^{\ell}) \right] \geq \sum_{\ell=1}^h \binom{\ell+r-1}{r}=\binom{h+r}{r+1}.
\left( \begin{array}{c}{\phi}\\{\psi}\\ \end{array}\right) ( \theta ) | \frac{1}{2}\rangle = \left( \begin{array}{l l}{\mathcal{A}_{+}}&{\mathcal{B}}\\{\tilde{\mathcal{B}}}&{\mathcal{A}_{-}}\\ \end{array}\right) \left( \begin{array}{c}{\phi}\\{\psi}\\ \end{array}\right) ( - \theta ) | \frac{1}{2}\rangle
+ \nabla_m(\pi^{ij}N^m) -\pi^{jm}\nabla_m N^i -\pi^{im}\nabla_m N^j+N \sum_n \delta^2(x - q_n)\frac{P_{na}P_{nb}g^{ai}g^{bj}}{2\sqrt{m_n^2+P_{na}P_{nb}g^{ab}}}
T_{\eqref{e_37_aux_approx}}\overset{\eqref{d_aux_cg_ham}}{=}\int_0^T \int_{\mathbb{T}^1}\bar{H}_K (\bar{\eta}^{\nu}) \beta(t) d\theta dt & = \int_0^T \int_{\mathbb{T}^1}\varphi (\bar{\eta}^{\nu}) \beta(t) d\theta dt + \int_0^T \int_{\mathbb{T}^1}\left(\bar{H}_K (\bar{\eta}^{\nu}) - \varphi (\bar{\eta}^{\nu}) \right) \beta(t) d\theta dt
r' (h, h_{i_{1}}\cdots h_{i_{t}}) = r' \big( h_{(1)}, h_{i_{t}}\big) \cdots r' \big( h_{(t)}, h_{i_{1}}\big) & = r'' \big( h_{(1)}, h_{i_{t}}\big) \cdots r'' \big( h_{(t)}, h_{i_{1}}\big) & = r'' (h, h_{i_{1}}\cdots h_{i_{t}})
\sum_{k=0}^\infty\frac{q^{2k^2+(2\mu_a-1) k}}{(q)_{2k+\mu_a}}=\frac{Q(q^2)}{Q(q)}=q^{-1/24}\chi_{1,2}^{(4)}(q)\equiv q^{-1/24}\chi_{1/16}(q).
\tau=\frac{1}{2}\begin{pmatrix}\sqrt{-3}\eta^{-1}-1 -1 \sqrt{-3}\eta \end{pmatrix}, &\zeta=\frac{1}{2}\left((\dfrac{z_1}{1-\omega}-\dfrac{z_2}{1-\omega^2})\eta^{-1},\dfrac{z_1}{1-\omega}+\dfrac{z_2}{1-\omega^2}\right).
G(z_1,\ldots,z_n,w) =\overline{F_1}(\bar{z}_1,\ldots,\bar{z}_n,\bar{w})F_2(z_1,\ldots,z_n,w) +{\bigl( \overline{F_1}(\bar{z}_1,\ldots,\bar{z}_n,\bar{w}) \bigr)}^2 .
E^{( 0 )}( \lambda ) = \frac{4 L^{2}\alpha \lambda^{2}\bigl [ ( 1 0 + 1 8 \lambda^{2}- 1 8 \lambda^{4}- 1 0 \lambda^{6}) + ( 3 + 2 1 \lambda^{2}+ 2 1 \lambda^{4}+ 3 \lambda^{6}) \log ( \lambda^{2}) \bigr ]}{3 ( \lambda^{2}- 1 )^{5}},
(V_2 * U_2) \odot (V_1 * U_1) = C_{P_{012}}\big(C_{P_0\times_{X_{012}}P_1}(U_1\times V_1)\times C_{P_1\times_{X_{012}}P_2}(U_2\times V_2)\big) \cong C_{P_{012}\bullet ( (P_0\times_{X_{012}}P_1) \times (P_1\times_{X_{012}}P_2) )}\big(U_1\times U_2 \times V_1 \times V_2 \big),
\tilde{\Phi}(\xi' ,\eta') := \Bigl| \xi_1'{\xi'}^2 + \eta_1'{\eta'}^2 - \frac{{\xi_1'}^3 +{\eta_1'}^3}{4}\Bigr|\leq 2^5 A^{-\frac{3}{2}}d^{-1}N_1^3 , \tilde{F}(\xi' ,\eta') := \Bigl| \frac{3}{2}\, \xi_1' \, \eta_1' + 2 \, \xi' \, \eta' \Bigr|\leq 2^5 A^{-1}d^{-1}N_1^2,
\Delta(j^{\min\{p, q\}}) \prod_{i = 1}^{\min\{p, q\}}(j^{\min\{p, q\}}_i)^{p - \min\{p, q\}}(1 - j^{\min\{p, q\}}_i)^{q - \min\{p, q\}}\prod_{n = 1}^{\min\{p, q\}}\prod_{i = 1}^n dj^n_i.
{\bf A_{2}A_{1}}f ( \bar{z}_{2}) = \int d \bar{z}_{0}d z_{0}e x p_{q}\{- q^{k}\bar{z}_{0}z_{0}\}\left[ \int d \bar{z}_{1}d z_{1}e x p_{q}\{- \bar{z}_{1}z_{1}\}A_{2}( \bar{z}_{2}, z_{1}) A_{1}( \bar{z}_{1}, z_{0}) \right] f ( \bar{z}_{0}) \ .
J_{1lin}= \frac{m(T_o)T_o}{8\pi^2}\left(H e^{Ht_o}\right)^4\int^{\infty}_{1}\frac{dz}{z}\frac{\sqrt{z^2-1}}{\left[\frac{\lambda_R T^2_c}{24}+m^2(T)z\right]^2}
\nu_{(I,J)}^q(x):=\frac{Z^{\beta, u, q}_{IN}\left(\Omega^{(I,J)}(\Gamma^{}_{(0,0)\to(I,J)}) \cap \{s_{IN}=x\}\right)}{Z^{\beta, u, q}_{IN}\left(\Omega^{(I,J)}(\Gamma^{}_{(0,0)\to(I,J)})\right)}, x \in R(I,J),
(1)\Phi^{n-1}([\varDelta]^{(n)})=\varDelta,(2)E_{12}^{(n)}+7\cdot [\varDelta]^{(n)}\equiv \vartheta_L^{(n)}\pmod{23}& (3)\quad[\varDelta]^{(n)}\equiv \mathcal{G}_{12}^{(n)}\pmod{691}.
\Pi_x^{(\phi)}[(\phi^{-1}f)(\xi_t)]= \phi(x)^{-1}\Pi_x[f(\xi_t) e^{- \int_0^t \beta(\xi_s) ds}]= \phi(x)^{-1}P^\beta_t f(x)\xrightarrow[t\to \infty]{}1,x\in E.
\left[\phi_{ij}(x^-,x_\perp),\partial_-\phi_{kl}(y^-,y_\perp)\right]=\left\{\psi_{ij}(x^-,x_\perp),\psi_{kl}(y^-,y_\perp)\right\}=\frac{1}{2}\delta(x^- -y^-)\delta(x_\perp -y_\perp)\delta_{il}\delta_{jk}.
[ M_{j0}+x^{0}P_{j},X_k] =\frac{i}{2}\big( d\Gamma(X_jV_k )+d\Gamma( V_kX_j)\big) +x^{0}[ P_{j},X_k] &= i\eta_{jk}\, x^{0}N+\frac{i}{2}\big( d\Gamma(X_jV_k )+d\Gamma( V_kX_j)\big),
\dim \mathcal O_k = \dim\left(J_0^{k+1}(\mathbb R^2, \mathbb R^4) \times J_0^k(\mathbb R^2,\mathbb R)\right) &= 4\dim \left(J_0^{k+1}(\mathbb R^2, \mathbb R)\right)+\dim\left( J_0^k(\mathbb R^2,\mathbb R)\right) &=4 \binom{k+3}{2}+\binom{k+2}{2}= \frac{(k+2)(5k+13)}{2}.
\sum_{d\in \mathcal{F}(x)}\Bigg|\sum_{\substack{1< n\leq N^{k}P^{-}(n)>y}}\frac{\chi_d(n)g_{1, N, k}(n, \alpha)}{n}\Bigg|^{2}\ll x \sum_{\substack{1<n_1, n_2 \leq N^{k}P^{-}(n_1n_2)> y n_1n_2= \square}}\frac{d_k(n_1)d_k(n_2)}{n_1n_2}\leq x \sum_{\substack{n>1 P^{-}(n)> y}}\frac{d_{2k}(n^2)}{n^2}.
\Phi_{v^\iota_n}(z^\iota_n, x^\iota_n) = \frac{\sqrt{v^\iota_n}}{2}(\Phi_{\widetilde{F}}(z^\iota_n, x^\iota_n) + o(1)) = \frac{k_n}{2n}+o\left(\frac{k_n}{n}\right).
M_{11}=\left(\begin{array}{ccc}1 0 0 1 0 0 0 0 0\end{array}\right),\,\,M_{12}=\left(\begin{array}{ccc}0 1 1 0 1 1 0 0 0\end{array}\right), M_{21}=\left(\begin{array}{ccc}0 0 0 0 0 0 1 0 0\end{array}\right),\,\,M_{22}=\left(\begin{array}{ccc}0 0 0 0 0 0 0 1 1\end{array}\right).
d(x) =\prod_{d\in S}\delta(d)\phi_d(x) &=\prod_{d\in S}\prod_{d'|d}\left(1-x^{d'}\right)^{\mu(\frac{d}{d'})}&\equiv \prod_{1\leq d' \leq r}\prod_{\substack{d\equiv 0 \mod d' d\in S}}\left(1-x^{d'}\right)^{\mu(\frac{d}{d'})}\mod x^{r+1}&\equiv \prod_{j=1}^{r}(1-x^j)^{l_j}\mod x^{r+1},
\sum_{l=0}^{k-2}\binom{k-1}{l+1}\frac{\partial^{k-1}f}{\partial u^{k-1-l}\partial v^l}=(k-1)v^2\frac{d^{k-2}A}{dv^{k-2}}+(k-1)(k-2)v\frac{d^{k-3}A}{dv^{k-3}}&+\frac{(k-1)(k-2)(k-3)}{3}\frac{d^{k-4}A}{dv^{k-4}}+P_{n-k+2}.
\psi(\bar\rho)-\psi(R) =\psi'(R)R\rho+\psi''(R(1+\theta\rho))\frac{R^2\rho^2}{2}&=\phi'(R)R^2\rho+\bigl(\phi'(R(1+\theta\rho))+R(1+\theta\rho)\phi''(R(1+\theta\rho))\bigr)\frac{R^2\rho^2}{2}.
\kappa ( u ) = \frac{\sin ( u - 2 \lambda )}{\sin \lambda}\prod_{k = 1}^{L / 2 - 1}\frac{\sin ( u + ( 2 k - 1 ) \lambda )}{\sin ( u + 2 k \lambda )}\frac{\sin \frac{L + 1}{L - 1}( u + 2 k \lambda )}{\sin \frac{L + 1}{L - 1}( u + ( 2 k - 1 ) \lambda )}.
\begin{cases}\partial_{t}V_{4}= D_{4}\partial_{xx}^{2}V_{4}-U \partial_{x}V_{4}+ (g - \delta - \frac{gM}{K_{B}}) V_{4}\triangleq D_{4}\partial_{xx}^{2}V_{4}- U \partial_{x}V_{4}+ \tilde{F}_{4}, D_{4}\partial_{x}V_{4}(x,t) - U V_{4}(x,t) \rvert_{x=0}= \partial_{x}V_{4}(x,t) \rvert_{x=1}= 0, \end{cases}
\nu_{\Phi^{\alpha}_{ij}}(z) \geqslant\min\{\nu_{F_1^{ij}}(z)-1,\nu_{F_2^{ij}}(z)-1,\nu_{F_3^{ij}}(z)-1\}\geqslant -\sum_{u=1}^3\nu^\infty_{F_u^{ij}}(z)-1\geqslant P(z).
|\widehat{\partial_xp^n_{ih}}|^2\big|_{x_{j+\frac12}}= \left|\{\partial_xp_{ih}^n\}+\beta_0\frac{[p^n_{ih}]}{h}+\beta_1h[\partial_x^2p^n_{ih}]\right|^2 &\leq 2 \left(\beta_0^2\frac{[p^n_{ih}]^2}{h^2}+\left(\{\partial_xp^n_{ih}\}+\beta_1h[\partial_x^2p^n_{ih}] \right)^2 \right).
\frac{\partial \tilde D_{k+1}^{\ell}(\Gamma_{k+1}^{\ell})}{\partial x_{k+1}^{\ell,j_{\ell}}}= \sum_{i \in I_k}\Psi_{j_{\ell}}' (x_k^{i}) \, p_k^i = \mathbb E \Psi_{j_{\ell}}' (\hat X_k)
H^{\psi_1}=(t_{a_g}t_{a_g^\prime}\cdot t_{\psi_1(A_{2g})}\cdots t_{\psi_1(A_2)}t_{a_2}t_{a_2}t_{\psi_1(A_2)}\cdots t_{\psi_1(A_{2g})})^2 \cdot t_{a_{g+1}}^{-1}.
\frac{V}{N}= - \frac{\sigma}{g_{0}}+ \frac{1}{4 \pi \rho^{2}}\sum_{l = 0}^{I}\int_{| k | < \Lambda_{1}}\frac{d k}{2 \pi}( 2 l + 1 ) \ln ( 1 + \frac{\sigma}{k^{2}+ \frac{l ( l + 1 )}{\rho^{2}}+ \frac{2 \xi}{\rho^{2}}+ i \epsilon}) .
O\biggl(\frac{1}{N}\,\int_{G_\vartheta}\sum_{(i_1, \ldots, i_m) \in J_{\hat\lambda,\tilde E}}|(X_{2,1}^N)_{i_j}^k|\,dP_{\beta}^{N,0}\biggr)=O\bigl(N^{(m+2)\vartheta-1}\bigr).
\hat{C}(k) \leq \frac{1}{(1+|k_1|)^{\lambda_1}(\sqrt{1+|k_2|})^{\lambda_2}}, k =(k_1,k_2) \in (2\pi \mathbb{Z})^2, \lambda_1 + \lambda_2 = -1 + 2 \alpha' \lambda_1, \frac{\lambda_2}{2}< 1.
x^m A_h = \sum_{k+|\alpha|\leq m}h^m a_{m k \alpha}(h,x,y)(x D_x)^k D_y^\alpha + \sum_{k+|\alpha|\leq j\leq m-1}x^{m-j}h^j a_{j k\alpha}(h,x,y)(x D_x)^k D_y^\alpha.
E_{i}={\frac{1}{\kappa \left( 1 -{\frac{e^{2}}{\kappa^{2}}}| \Phi |^{2}\right)}}\left( -{\frac{i e}{2}}\right) \epsilon_{i j}\left[ \Phi^{\ast}\nabla_{j}\Phi - \Phi \left( \nabla_{j}\Phi \right)^{\ast}\right] \, .
\nu_{p}(a_{n}) = \nu_{p}(a_{\rho p^{u}m}) & = \nu_{p}(a_{\rho})+\nu_{p}\left(\prod_{d\mid p^um}b_{\rho d}\right) & = \alpha+\nu_{p}\left(\prod_{d\mid p^u}b_{\rho d}\right)+\nu_{p}\left(\prod_{\substack{d\mid p^u e\mid m e>1}}b_{\rho de}\right)
|\mathcal{E}_{q, p}|\leq \sum_{l=1}^{\lfloor \frac{p}{2}\rfloor}\binom{q}{l}l^{p}\leq \lfloor \frac{p}{2}\rfloor \binom{q}{\lfloor \frac{p}{2}\rfloor}\left( \lfloor \frac{p}{2}\rfloor\right)^{p}\leq \frac{p}{2}e^{\frac{p}{2}}q^{\frac{p}{2}}\left( \frac{p}{2}\right)^{\frac{p}{2}}.