harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lim_{s\downarrow1}\frac{1}{\zeta(s)}\sum_{m=m_0}^\infty\frac{1}{m^s}\left(f(m)^{-s+1}- g(m)^{-s+1}\right) &=\lim_{s\downarrow1}\frac{1}{\zeta(s)}\sum_{m=m_0}^\infty\frac{1}{m^s}\left(u(m)^{-s+1}- v(m)^{-s+1}\right).
F_{i}( \zeta , \upsilon + \chi ) \equiv \frac{( \upsilon + \chi ) f_{i}}{2 \sqrt{2}( \zeta + k )^{2}( \zeta + b )^{1 / 2}}[ \zeta^{2}+ ( 3 h_{i}- k ) \zeta + 2 b ( h_{i}- k ) + k h_{i}] , \qquad i = E^{\prime}, U^{\prime}, D^{\prime}.
G_{N}(x^{N}(\eta))-G_{N}(x^{N}(0))+\frac{a\eta(N-1)}{2}-\frac{2a}{N}\sum_{i\neq j}A_{ij}^{N}(\eta) =\frac{a}{N}\sum_{i=1}^{N}\int_{0}^{\eta}\left[\sum_{j:j\neq i}g'(x_{i}^{N}(s)-x_{j}^{N}(s))\right]dx_{i}^{N}(s)
Z(t) = \frac{1}{C(H)}\int_{\mathbb{R}}\bigg((t - s)^{H - \frac{1}{2}}_+ - (-s)^{H- \frac{1}{2}}_+ \bigg) dB(s) &= \frac{1}{C(H)}\Bigg( \int^0_{- \infty}((t - s)^{H - \frac{1}{2}}- (-s)^{H- \frac{1}{2}}) dB(s) + \int^{t}_0 ((t - s)^{H - \frac{1}{2}}dB(s) \Bigg)
\lim_{r\rightarrow \infty}F(r\omega )=\sum_{0\leq p\leq m}\frac{\Gamma(m-p+1)\Gamma (n+2p+2\nu )}{(-1)^{p-m}\Gamma (m+n+p+2\nu )}\sum_{j=1}^{d(n,p,p+2\nu )}a_{j}^{\nu ,p}h_{p,p+2\nu}^{j}(\omega ,\bar{\omega})
\begin{array}{lll}\lambda_2(1-\lambda_2)b_1+2\lambda_2(1-\lambda_2)a_1+\lambda_2\lambda_3b_1a_1-\lambda_2\lambda_3b_1a_2+\lambda_2\lambda_3a_1^2-\lambda_2\lambda_3a_1a_2\ & \ \ =(\lambda_2-1)(1-\lambda_3),\end{array}
G_{ml}=\mu^2\sum_{p\in\mathbb{N}}{\cal{R}}_{mp}\lambda_p{\cal{R}}^\dag_{pl},\ \sum_{p\in\mathbb{N}}{\cal{R}}_{mp}{\cal{R}}^\dag_{pl}=\sum_{p\in\mathbb{N}}{\cal{R}}^\dag_{mp}{\cal{R}}_{pl}=\delta_{ml},\{\cal{R}}^\dag_{mn}={\cal{R}}_{nm}.
\psi_{l,k}:=\left[\begin{array}{cccc}\psi_{l,k+1}(\sqrt{\bar{\alpha}}\mathcal{A}_{1,k}+\sqrt{\bar{\alpha}}\mathcal{A}_{2,k}) \psi_{l,k+1}\sqrt{1-\bar{\alpha}}\mathcal{A}_{1,k}\psi_{l,k+1}(\sqrt{\bar{\alpha}}\mathcal{C}_{1}+\sqrt{\bar{\alpha}}\mathcal{C}_{2,k}) \psi_{l,k+1}\sqrt{1-\bar{\alpha}}\mathcal{C}_{1}\end{array}\right],
I_{+}= \frac{2\pi i}{(2\pi)^{6}}\frac{z^{4}}{(1-z^{2})^{4}}\frac{\Theta(\Delta x^{+})}{2}\int_{0}^{\infty}dk_{-}k_{-}^{3}e^{-ik_{-}Y^{2}}e^{i\frac{m^{2}\Delta x^{+}}{k_{-}}}
y_{u,G}(\mathbf{X}_{u})={\displaystyle \sum_{{\textstyle{\mathbf{j}_{|u|}\in\mathbb{N}_{0}^{|u|}\atop j_{1},\cdots,j_{|u|}\neq0}}}}C_{u\mathbf{j}_{|u|}}\psi_{u\mathbf{j}_{|u|}}(\mathbf{X}_{u})
\begin{aligned}M_{\delta,+,1}(x ; z)=e_1+\mathcal{P}^+\left(r_{\delta,-,2}e^{2iz x}M_{\delta,-,2}(x; \cdot)\right) , &M_{\delta,-,2}(x ; z)=e_2+\mathcal{P}^{-}\left(\bar{r}_{\delta,-,1}e^{-2iz x}M_{\delta,+,1}(x; \cdot)\right) .\end{aligned}
y_{\ast}^{2}={\frac{( 4 \pi )^{2}\alpha}{2 a \alpha_{C}}}, ~ ~{\frac{\lambda_{\ast}}{y_{\ast}^{4}}}={\frac{2 a}{( 4 \pi )^{2}}}~{\frac{\alpha_{C}}{\alpha}}, ~ ~ \xi_{\ast}={\frac{1}{6}}.
\lim_{\beta \rightarrow \infty}\lim_{m \rightarrow 0^{-}}\langle{\bar{\psi}}\psi \rangle_{\beta}= \lim_{\beta \rightarrow \infty}\lim_{m \rightarrow 0^{+}}\langle \bar{\psi}\psi \rangle_{\beta}= \frac{e B}{4 \pi}.
\begin{array}{rl}h_= (\lambda_1\begin{bmatrix}2x_1 -1\end{bmatrix}+\lambda_2\begin{bmatrix}-2x_1 4\end{bmatrix}+\lambda_3\begin{bmatrix}0 -1\end{bmatrix}, &\lambda_1(x_1^2-x_2),\lambda_2 (-x_1^2+4x_2),\lambda_3(-x_2+1),1-\lambda_0^2-\lambda_1^2-\lambda_2^2-\lambda_3^2)\,.\end{array}
\sum_{k=0}^{a}\frac{(-a-p)_{p+k}^2(\beta)_{p+k}}{(1)_{p+k}^2(1-\beta-a-p)_{p+k}}\cdot z^{k}=\frac{(\beta)_p(-\beta-p-a+1)_a}{(-\beta-p-a+1)_p(\beta)_a}\sum_{k=0}^{a}\frac{(-a-p)_k^2(\beta)_{k}}{(1)_k^2(1-\beta-a-p)_{k}}\cdot z^{a-k},
\rho_x(\{y: f_{q}(x)\leq y \leq f_{q}(x)+s\}=\frac{1}{\sqrt{2\pi}\sigma}\int_{f_\rho(x)}^{f_\rho(x)+s}\exp\{-\frac{(y-u_x)^2}{2\sigma^2}\}dy =\frac{1}{\sqrt{2\pi}\sigma}\int_{0}^{s}\exp\{-\frac{y^2}{2\sigma^2}\}dy\geq\frac{1}{\sqrt{2\pi}\sigma}\int_{0}^{s}\exp\{-\frac{s^2}{2\sigma^2}\}dy\geq\frac{e^{-\frac{1}{2}}}{\sqrt{2\pi}\sigma}s.
\frac{\log((n p_n b_c^{(n)'})/a_c^{(n)})}{n p_n}=\frac{\log(n p_n)}{n p_n}+\frac{\log b_c^{(n)'}}{n p_n}-\frac{\log a_c^{(n)}}{n p_n}&\sim_e -1-\frac{\log a_c^{(n)}}{n p_n}\sim_e -1,
K_a = \frac{\pi}{2}\int_0^{2\pi}\frac{a}{(a^2\cos^2\theta + \sin^2\theta)^{1/2}}d\theta & A = \frac{4\pi^2a}{K_a}|u_j(x^*)|^2e^{\phi(x^*)}, & B = \frac{4\pi^3 a^2 |u_j(x^*)|^2e^{\phi(x^*)}}{K_a^2}(H(x^*)-\partial_{\nu}\phi(x^*))
\begin{array}{l l}{{l = ( l_{1 2}+ l_{1 3}+ l_{2 3}) / ( 2 N ) ,}}&{{l^{\prime}= ( l_{1 4}+ l_{1 5}+ l_{4 5}) / ( 2 N )}}\\{{l^{\prime \prime}= ( l_{2 4}+ l_{2 6}+ l_{4 6}) / ( 2 N ) ,}}&{{l^{\prime \prime \prime}= ( l_{3 5}+ l_{3 6}+ l_{5 6}) / ( 2 N )}}\\ \end{array}
\left\langle u,\mathsf{L}_{j}\left( s\right) \overline{v}\right\rangle_{\mathbb{R}^{3}}=\left\langle \psi,\gamma_{\operatorname*{N};j}^{\operatorname*{ext}}\left( s\right) \overline{v}\right\rangle_{\Gamma_{j}}\quad\forall v\in H^{1}\left( \mathbb{R}^{3},\mathbb{A}_{j}^{\operatorname*{ext}}\right) .
||h||_{H_{x,v}^{s,r}L_z^{\infty}}=\sup_{z\in I_z}\, ||h||_{H_{x,v}^{s,r}}\,, ||h||_{H_{x,v}^s L_z^{\infty}}=\sup_{z\in I_z}\, ||h||_{H_{x,v}^s}\,.
\varphi^{d+1}g_{e_j}=b\varrho^{-1}t^{n_{e_j}+1-p^{d+1}}g_{e_j}, \gamma(x)g_{e_j}-x^{s_{e_j}}g_{e_j}\in t\cdot k_{\mathcal E}^+\cdot g_{e_j}.
K(u,w)=\frac{p(u,w,\left\Vert \mathbf{x}-\mathbf{y}\right\Vert )}{p(u)\p(w)}=\sum_{k=0}^{\infty}r_{k}(\left\Vert \mathbf{x}-\mathbf{y}\right\Vert )e_{k}(u)e_{k}(w),
\{f(\phi_n)\}_{n=1}^{\infty}\;\textit{is uniformly bounded in}\;L^\infty(0,T;L^{\frac{p}{p-1}}(\Omega)), &\{g(\phi_n)\}_{n=1}^{\infty}\;\textit{is uniformly bounded in}\;L^\infty(0,T;L^{\frac{q}{q-1}}(\Gamma)).
(s_{j+3}-s_{j+2}) - (s_{j+1}-s_j) \leq (s_{j'+1}-s_{j'}) - (s_{j+1}-s_j) &= (s_{i+3}-s_{i+1})-(s_{i+2}-s_{i}) & = (s_{i+3}-s_{i+2})-(s_{i+1}-s_{i}) .
v_i^S = v_i^{S \cup \{j\}}+ c^{S}_j \, x_{ij}^{1, S \cup \{j\}}, j \in N^{\{0, 1\}}\setminus S v_i^{S \setminus \{j\}}= v_i^S + c^{S}_j \, x_{ij}^{0, S \setminus \{j\}}, j \in S.
{\nabla}_{\mathfrak{U}^1}^p(s_{U^1})=\theta_p\theta_{p-1}\cdots \theta_1g{\nabla}_{\mathfrak{U}^2}^p(s_{U^2})+\sum_{j=0}^{p-1}P_j^p{\nabla}_{\mathfrak{U}^2}^j(s_{U^2}).
\log \left\{(q^a:q^{a+b})_{\infty}(q^b:q^{a+b})_{\infty}\right\}= \frac{\pi^{2}}{3(a+b)\log q}+ \log \left\{2 \sin \left( \frac{a}{a+b}\pi \right) \right\}+ O(\log q).
A_{N}^{\ast}(w_{1,r}) = A_{N}^{\ast}(a_{1},\{1\}_{b_{1}-1},w_{2,r}) & = \sum_{i_1=1}^{N}\frac{1}{i_1}\dots\sum_{i_{k-1}=1}^{i_{k-2}}\frac1{i_{k-1}}\sum_{i_{k}=1}^{i_{k-1}}\frac1{i_{k}^{b_1+1}}A_{i_k}^{\ast}(a_2,\{1\}_{b_2-1},w_{3,r}).
\frac{1}{2}\int_{\varepsilon^2}^{\infty}\frac{dt}{t^{3}}\sum_{k=-\infty}^{\infty}\left[e^{-k^2 \pi t/R^2}-e^{-(k+1/2)^2 \pi t/R^2}\right]=\frac{1}{R^4}\frac{31}{16}\frac{3}{4\pi^2}\zeta(5)
a_{\frac{1}{t}}n_{-u}\cdot f_{-}= \frac{1}{\sqrt{2}}\left(0, t, \frac{-2u}{t}, \frac{1+u^2}{t^3}\right), a_{\frac{1}{t}}n_{-u}\cdot f_{+}= \frac{1}{(108)^{\frac{1}{4}}}\left( 0, 3t, \frac{-6u}{t}, \frac{-1+3u^2}{t^3}\right).
h_{\mu \nu}^{\pm}= \gamma_{\mu \nu}^{\pm}+ \varphi^{\pm}\eta_{\mu \nu}+ f_{\mu , \nu}+ f_{\nu , \mu}\, , \, \, \,{\gamma_{\mu \nu}}^{, \, \nu}= \eta^{\mu \nu}\gamma_{\mu \nu}= 0 .
\alpha(R_{n+1},K_{n+1}) \, = \, \alpha\left(R_{n}\setminus A_{i_{n+1}}C_{n+1},K_{n+1}\right) &\leq \, \frac{\alpha(R_{n},K_{n+1})+\alpha(A_{i_{n+1}}C_{n+1},K_{n+1})}{\epsilon}&\leq \, 2\epsilon^{2(N-n)}\, \leq \, \epsilon^{2(N-n)-1}\, = \, \epsilon^{2(N-(n+1))+1},
G_{6 6}\sim 1 + \frac{2 0}{3 !}g \int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{d^{3}p}{( 2 \pi )^{3}}\frac{1}{k^{2}}\frac{1}{p^{2}}\frac{1}{( p + k )^{2}},
\begin{array}{c c c c c}{{E G}}&{{\leftarrow}}&{{E G \times M}}&{{\rightarrow}}&{{M}}\\{{\downarrow}}&{{}}&{{\downarrow}}&{{}}&{{\downarrow}}\\{{B G}}&{{\leftarrow}}&{{E G \times_{G}M}}&{{\rightarrow}}&{{M / G .}}\\{{}}&{{\pi}}&{{}}&{{\sigma}}&{{}}\\ \end{array}
\sum_{m=-\infty}^{\infty}f_0(a_0m)f_1(a_1m)\cdot\ldots\cdot f_n(a_nm)e^{-im\pi}= \int_{-\infty}^{\infty}2\cos(\pi t)f_0(a_0t)f_1(a_1t)\cdot\ldots\cdot f_n(a_nt)\, dt
m_{3 + \nu}= \left( \frac{2 ( \nu + 1 ) m_{3}^{2}}{( \nu + 2 ){\ell^{\perp}}^{\nu}}\right)^{\frac{1}{\nu + 2}}> 1 0^{( 3 7 - 1 2 \nu ) / ( \nu + 2 )}\, \mathrm{G e V}.
\rho^\vee(\mu_{\underline{m}'})(\kappa(\alpha,\beta))=\mu_{\underline{m}'}\left(\left|\begin{array}{cc}X Y -\bar\beta \bar\alpha\end{array}\right|^{\underline{k}-2}\right)=(-\bar\beta)^{\frac{k_{\rm id}-2}{2}-m_{\rm id}'}\bar\alpha^{\frac{k_{\rm id}-2}{2}+m_{\rm id}'}(-\beta)^{\frac{k_{c}-2}{2}-m_{c}'}\alpha^{\frac{k_c-2}{2}+m_{c}'}.
\begin{array}{ll}\int_{\mathbb R}\zeta_k \nu_{n_k}|D|coth(\sqrt{\mu_2}|D|)(\varphi_k \nu_{n_k})dx =\int_{\mathbb R}\zeta_k \nu_{n_k}\Big[|D|\coth(\sqrt{\mu_2}|D|)-\frac{1}{\sqrt{\mu_2}}\Big](\varphi_k\nu_{n_k})dx &= \int_{\mathbb R}\zeta_k \nu_{n_k}\mathcal T|D|(\varphi_k\nu_{n_k})dx\equiv L.\end{array}
\nabla^{2}\ln | \Psi_{p}|^{2}= \sigma_{p}\frac{2}{\hbar c}\left\{\sum_{p^{\prime}}\beta_{p p^{\prime}}| \Psi_{p^{\prime}}|^{2}- e_{p}B^{\mathrm{e x}}\right\}+ 4 \pi \sum_{r = 1}^{n_{p}}\delta^{2}({\bf r}-{\bf R}_{r}^{( p )}) \, , \hspace{3 m m}( p = 1 , \ldots , M )
A\times(B\vec{r})-C\vec{r}=[\begin{matrix}2&3\\ 3&4\end{matrix}]([\begin{matrix}1&0\\ 1&2\end{matrix}][\begin{matrix}1\\ 0\end{matrix}])-[\begin{matrix}6&5\\ 8&7\end{matrix}][\begin{matrix}1\\ 0\end{matrix}]=[\begin{matrix}-1\\ -1\end{matrix}].
a(S) =C_SN_1^{-3/2}\,\det(S)^{\frac k2}\prod\limits_{p | N_1}\sum\limits_{i=0}^{b_p}p^{\frac i2}p^{-(a_p-i)(2k-3)/2}\Big(c(p^{a_p-i})- \delta_p p^{-\frac 12}p^{(2k-3)/2}c(p^{a_p-i-1})\Big) =C_SN_1^{-k}\det(S)^{\frac k2}\prod\limits_{p | N_1}\sum\limits_{i=0}^{b_p}p^{i(k-1)}\Big(c(p^{a_p-i})- \delta_p p^{k-2}c(p^{a_p-i-1})\Big).
\|{w}^{k+\frac{1}{2}}+\Delta_{1}{w}^k-{w}^{\ast}\| \le \|{w}^{k+\frac{1}{2}}-w^{\ast}\|+\|\Delta_{1}{w}^k\| &=\|{w}^{k+\frac{1}{2}}-w^{\ast}\|+O(\|{w}^{k+\frac{1}{2}}-{w}^{\ast}\|^{1+\tilde{c}\alpha}) &=O(\|{w}^{k+\frac{1}{2}}-{w}^{\ast}\|),
\partial_t^\alpha (u^{n+1}-u^n) =L_{\lambda_0}(u^{n+1}-u^n)+(\lambda-\lambda_0)(L_1-L_0)(u^n-u^{n-1}) &~~~ +\partial_{t}^{\beta}\int_{0}^{t}\Lambda^k_{\lambda_0}(u^{n+1}-u^n)+(\lambda-\lambda_0)(\Lambda_{1}-\Lambda_0)(u^n-u^{n-1}) dw_{s}^{k}.
F'(x) = \frac{1}{x \ln(x)}, & F''(x) = - \frac{1}{x^2 \ln(x)}- \frac{1}{x^2 \ln^2(x)}, & F'''(x) = \frac{2}{x^3 \ln(x)}+ \frac{3}{x^3 \ln^2(x)}+ \frac{2}{x^3 \ln^3(x)}.
\begin{aligned}\int_{0}^{\infty}d_0^2(s,f)ds =\int_{0}^{T-t}\left\{\frac{1}{b}m(T-t-s) + f(t+s)\right\}^2ds &=\frac{1}{b^2}\int_{0}^{T-t}\left\{m(s) + e^{-bs}- 1\right\}^2ds.\end{aligned}
\aligned\overset{\circ}S_A([0,1)^k) =\{(t_1,\dots,t_k) \in [0,1)^{k}\mid i\in A \Rightarrow t_i = 0,\,\, i \notin A \Rightarrow t_i > 0\}, \overset{\circ}S_A(V_{\frak r}) =V_{\frak r}\cap (\overline V_{\frak r}\times \overset{\circ}S_A([0,1)^k)), \Gamma_{\frak r}^A =\{\gamma \in \Gamma_{\frak r}\mid \gamma A = A\}.\endaligned
\tilde{u}\left( x,t\right) =\left( \sqrt{\alpha \beta}\rho^{-1}\left(t\right) \right)^{\frac{n}{2}}u\left( \sqrt{\alpha \beta}x\rho^{-1}\left(t\right) ,\beta t\rho^{-1}\left( t\right) \right) e^{\varphi}.
g_M(\bar{X},\bar{Y}) =e^{-2\phi/3}g_N(X,Y)+e^{4\phi/3}C_1(X)C_1(Y) g_M(\bar{X},\bar{S}) =-e^{4\phi/3}C_1(X)\alpha^{10}(S) g_M(\bar{S},\bar{S}) =e^{4\phi/3}g_{S^1}(S,S).
\begin{array}{r c l}{{\overline{{{D_{m}( k_{1}+ p )}}}}}&{{=}}&{{\displaystyle D_{m}( k_{1}) \left[ - 2 k_{1}\cdot p D_{m}( k_{1}) \right.}}\\{{}}&{{}}&{{\displaystyle \left. - p^{2}D_{m}( k_{1}) + 4 ( k_{1}\cdot p )^{2}D_{m}^{2}( k_{1}) + O ( p^{3}) \right] .}}\\ \end{array}
\xi_p \left \{\begin{matrix}0 1 a_{21}a_{22}\end{matrix}\right \}= \left [\begin{matrix}p 1 a_{21}a_{22}+p\end{matrix}\right ]+ ( v^{- a_{22}-3}\ \overline{[p]}- v^{a_{22}+1})\left \{\begin{matrix}p+1 0 a_{21}-1 a_{22}+p+1\end{matrix}\right \},
\hat{H}_{1}= \phi_{( \tilde{u}+ 2 , \bar{\tilde{u}})}^{7_{3}7_{3}}\phi_{( \tilde{u}, \overline{{\tilde{u}+ 1}})}^{7_{3}7_{3}}\phi_{( 1 , \bar{\tilde{u}})}^{3 7_{3}}\phi_{( \tilde{u}, \bar{2})}^{7_{3}3}M_{s}^{- 3}
[\epsilon^0 q^d]\frac{(-1)^{(N-1)d}}{\det(z_i^{N-j+1})}\,\begin{vmatrix}z_1^{d+N}z_1^{d+N-1}\cdots z_1^{d+1}f(z_{1}) z_2^{d+N}z_2^{d+N-1}\cdots z_2^{d+1}f(z_{2}) \vdots \vdots \cdots \vdots \vdots z_{N+1}^{d+N}z_{N+1}^{d+N-1}\cdots z_{N+1}^{d+1}f(z_{N+1})\end{vmatrix}.
\Gamma = - \frac{T^{2}}{8 \pi^{2}}\int d^{4}x \, M^{2}\left[ \frac{\, \pi^{2}}{3}+ \frac{1}{8}\frac{M^{2}}{T^{2}}\ln \frac{M^{2}}{T^{2}}+ \frac{\xi^{( 1 )}( - 2 ) M^{4}}{2 4 T^{4}}+ \cdots \right] \, ,
d \Theta_{\underline{\mu}}^{2}= v_{\underline{\mu}}^{( \underline{\alpha})}\pi_{( \underline{\alpha})}^{2}= v_{\underline{\mu}q}^{+}\pi_{q}^{2 -}+ v_{\underline{\mu}\dot{q}}^{-}\pi_{\dot{q}}^{2 +},
\log \det ( K + X^{-1}) \equiv \min_{Z \succ 0}\, \log \det (Z^{-1}) & \ \mbox{\rm s.t.}\ \left[ \begin{array}{@{}c@{\,\,}c@{}}X - Z X K^{1/2}K^{1/2}X I + K^{1/2}X K^{1/2}\end{array}\right] \succeq 0 \,.
\|\sum_{j=2}^\infty\tilde S^{j-2}f \tilde S_j g \|_{B^s_{p, \infty}(\real^{d_s})}\le C \sup_{k\ge 2}2^{ks}\sum_{\ell=-1}^{+1}\|\tilde S^{k+\ell-2}f \tilde S_{k+\ell}g \|_{L_p(\real^{d_s})}\, .
(x_{i-1,j-1}+ x_{m+2-i,n+2-j}) + (x_{i-1,j}+ x_{m+2-i,n+1-j}) + (x_{i,j-1}+ x_{m+1-i,n+2-j}) + (x_{i,j}+ x_{m+1-i,n+1-j}) = 2S.
g:(\Gamma_1^{1,9},\Gamma_2^{1,9},\Gamma_1^{1,1},\Gamma_2^{1,1},\Gamma^{2,2})\mapsto (\Gamma_2^{1,9},\Gamma_1^{1,9},\Gamma_1^{1,1},-\Gamma_2^{1,1},-\Gamma^{2,2}).
\nu_i\left(\mathbf{Q}_i^{{\bar\gamma}_i}\right)-\nu_i\left(\partial_{j_0}\mathbf{Q}_i^{{\bar\gamma}_i}\right)=\nu_{i_0}\left(\mathbf{Q}_i^{{\bar\gamma}_i}\right)-\nu_{i_0}\left(\partial_{j_0}\mathbf{Q}_i^{{\bar\gamma}_i}\right)\le\frac{j_0}{b_i}(\beta_i-\nu(\partial_{b_i}Q_i)),
\langle\Psi\rangle_{R}\rightarrow\frac{1}{2}(\begin{matrix}\psi_{11}+\psi_{11}^{*}&\psi_{12}+\psi_{21}^{*}\\ \psi_{21}+\psi_{12}^{*}&\psi_{22}+\psi_{22}^{*}\end{matrix})
S_l = \sum_{k=1}^{l}2^{-k}\sum_{j=1}^{2^{k-1}}E_A\left(2^{-k}(2j-1),2^{-k}(2j)\right] = \sum_{m=1}^{2^l-1}m2^{-l}E_A\left(2^{-l}m,2^{-l}(m+1)\right].
P={\textstyle\sum\limits_{l=0}^{n}}P_{l}={\textstyle\sum\limits_{l=0}^{n}}{\textstyle\sum\limits_{m=-l}^{l}}a_{lm}\mathbf{e}(l,m)K={\displaystyle\sum\limits_{l=0}^{n}}K_{l}=\sum_{l=0}^{n}\gamma_{l}\mathbf{e}(l,0)
\hat{G}( x_{0}, \vec{k}) = \frac{1}{2}\int_{- \infty}^{+ \infty}d k_{0}~ \left[ ( - k^{2}+ m^{2}- i 0 )^{- \alpha}+ ( - k^{2}+ m^{2}+ i 0 )^{- \alpha}\right] e^{i k_{0}x_{0}}
\int_\Omega \theta_h\cdot\tau_h +\int_\Omega u_h{\rm div}( \tau_h)+ \langle \lambda_h,\tau_h\cdot n\rangle_{\partial{\cal T}_h}=0, -\varepsilon\int_\Omega{\rm div}(\theta_h) v_h+\int_\Omega cu_hv_h =(f,v), \langle \theta_h\cdot n,\mu_h\rangle_{\partial{\cal T}_h}=0.
x^i(q(x))^j =x^i\Big(\sum_{0\leq i_1,\ldots,i_j\leq \ell}q_{i_1}q_{i_2}\cdots q_{i_j}x^{i_1+\ldots+i_j}\Big) &=\sum_{p=0}^{j\ell}\Big(\sum_{\substack{0\leq i_1,\ldots,i_j\leq \ell, i_1+\ldots+i_j=p}}q_{i_1}q_{i_2}\cdots q_{i_j}\Big)x^{i+p}&=\sum_{s=i}^{i+j\ell}q_{i,j,s}x^s,
S_{i n t}= \frac{1}{g^{2}}\frac{p^{2}}{p - 1}\left[ \frac{2 \pi p \ln p}{p^{2}- 1}\right]^{\frac{1}{2}}\int d^{D - 1}y \sum_{\ell = 3}^{p + 1}\frac{p \ !}{\ell \ ! ( p + 1 - \ell ) !}\sum_{n_{1}}\cdots \sum_{n_{\ell}}A_{n_{1}n_{2}\dots n_{\ell}}\prod_{i = 1}^{\ell}\psi_{n_{i}}( y ) ,
\sum_{x,y=0}^n(\delta_{x,y}-M_{x,y}(m))f_y^\star f_x = \sum_{j,j'=0}^n \left(1- \Theta_m(\mu_j,\mu_{j'})\right) \left|\sum_{x=0}^n \psi_j(x)f_x\psi_{j'}(x)\right|^2.
\mathcal{A}_{h}:=\left\{x\in\Omega:h\left( y\right) =0\text{}a.e.y\in\left( a,x\right) \right\}, \mathcal{B}_{h}:=\left\{x\in\Omega:h\left( y\right) =0\text{}a.e.y\in\left( x,b\right) \right\},
\sum_{n=0}^{\infty}(n+1)f_{n+1}(t)x^{n}=t\left(\sum_{n=0}^{\infty}(1-2^{\nu_{2}(n+1)+1})x^{n}\right)\left(\sum_{n=0}^{\infty}f_{n}(t)x^{n}\right) =t\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}(1-2^{\nu_{2}(n-k+1)+1})f_{k}(t)\right)x^{n}.
\mathcal{L}(\mathcal{X}) =\lim_{t \to \infty}\frac{1}{t}\log \Vert \mathbf{X}_{i_1}\cdots \mathbf{X}_{i_t}\Vert=\lim_{t \to \infty}\frac{1}{t}\log [ \mathbf{X}_{i_1}\cdots \mathbf{X}_{i_t}]_{jk}\,,
\partial_y \partial_\xi^{n}V_{<k;\xi}- [V_{<k;y}, \partial_\xi^{n}V_{<k;\xi}]= \partial_\xi^{n+1}O_{<k;x}+ \sum_{j=1}^{n}[\partial_\xi^j O_{<k;x}, \partial_\xi^{n-j}V_{<k;\xi}].
\eta_{E}^{-}= + i ( \eta_{E}^{+})_{c}, \quad \eta_{E}^{+}= - i ( \eta_{E}^{-})_{c}, \quad \eta_{M}^{-}= - i ( \eta_{M}^{+})_{c}, \quad \eta_{M}^{+}= + i ( \eta_{M}^{-})_{c},
\frac{1}{x}\sum_{\substack{n\le x n\in{\rm L}_{A\cap (y,\infty)}}}1 \le \frac{1}{x}\sum_{a\in A,a> y}\left\lfloor \frac{x}{a}\right\rfloor \le \sum_{a\in A,a> y}\frac{1}{a}= o_y(1).
\lambda_i(T) \rho_j(S) b &= \lambda_i(T) \left(b E_j(S) \oplus (S-R_{E_j(S)}) b \right) &= E_{i}(T)bE_j(S) \oplus (T-L_{E_i(T)})bE_j(S) \oplus L_{E_i(T)}(S-R_{E_j(S)}) b \oplus \left([(T-L_{E_{i}(T)})1_B] \otimes [(S-R_{E_j(S)}) b] \right),
M_{i j}= \left( \begin{array}{c c}{{{\frac{\partial \beta_{( 2 i )}^{\lambda}}{\partial \lambda_{( 2 j )}}}}}&{{{\frac{\partial \beta_{( 2 i )}^{\lambda}}{\partial \xi_{( 2 j )}}}}}\\{{{\frac{\partial \beta_{( 2 i )}^{\xi}}{\partial \lambda_{( 2 j )}}}}}&{{{\frac{\partial \beta_{( 2 i )}^{\xi}}{\partial \xi_{( 2 j )}}}}}\\ \end{array}\right) .
e_i \pi := ( \nu_1,\,\ldots,\,\nu_p,\,r_i \nu_p,\,r_i \nu_{p+1},\,\ldots,\, r_i \nu_q,\,\nu_{q+1},\,\ldots,\,\nu_s ; & a_0,\,\ldots,\,a_{p-1},\,t_0,\,a_p,\,\ldots,\,a_{q}=t_1,\,\ldots,\,a_s);
\int_{\Omega}|\nabla w^i (\cdot,t) | \, dx \leq \int_{\Omega}\sigma^{-1}\Big( \frac{\varepsilon_i |\nabla \varphi^i |^2}{2}+\frac{W (\varphi^i)}{\varepsilon_i}\Big) \, dx = \mu_t^{\varepsilon_i}(\Omega) \leq D_1
\langle \;{\cal B}({\{x \}}) \; \rangle_{0}^{\theta}\; \; : = \; \; e^{i \theta \frac{Q_{5}({\scriptstyle{\cal B}})}{2 N}}\lim_{\tau \rightarrow \infty}\langle{\cal U}_{\tau}({\scriptstyle{\cal B}}) \;{\cal B}({\scriptstyle \{x \}}) \rangle_{0}\; .
w'(NEP) = \frac{(n - 0 + (m+1) - 1)}{2}+ w'(P) = \frac{n+m}{2}+ v_{n-1,m}= \frac{n+m}{2}- \sum_{k=D_{n-1,m}}^{D_{n-1,m}+m-1}k = \frac{n+m}{2}- \sum_{k=D_{n,m+1}+1}^{D_{n,m+1}+m}k = -\sum_{k=D_{n,m+1}}^{D_{n,m+1}+(m+1)-1}k = v_{n,m+1}.
\tau=\begin{cases}t \alpha+\delta+\beta=1, \frac{\log((1-\alpha-\delta-\beta)\eta t+1)}{(1-\alpha-\delta-\beta)\eta}\alpha+\delta+\beta<1, \frac{\log(\frac{T^*-t}{T^*})}{(1-\alpha-\delta-\beta)\eta}\alpha+\delta+\beta>1.\end{cases}
\left( \begin{array}{c c}{{\varphi}_{o u t}^{\prime}}{{{\varphi}^{\prime}}_{o u t}^{*}}\end{array}\right) = \left( \begin{array}{c c}{- 1}{0}{0}{- 1}\end{array}\right) \left( \begin{array}{c c}{\varphi_{i n}}{\varphi_{i n}^{*}}\end{array}\right) ,
q_{\infty}^{*}=f_{1,K_1^c+F_1^{b*},\infty}\sum_{n\in\mathcal F_1^{c*}\cup \mathcal F_1^{b*}}a_n+\sum_{n\in\mathcal F_2^{c*}}a_nf_{2,K_2^c,\infty}(T_n^{*}).
\left[ \phi\partial_{x}-\left( n+1-\frac{\tau_{n}}{\gamma_{n}\gamma_{n-1}}\right) x\right] P_{n+1}=\left[ \lambda_{n}+\tau_{n}\frac{x^{2}-\gamma_{n}}{\gamma_{n}\gamma_{n-1}}-\left( n+1\right) \gamma_{n+1}\right] P_{n}.
h_{s-i}=\displaystyle{\sum_{k=0}^{a_1-1}h'_{s'-(i-k)}}&=\displaystyle{\sum_{k=0}^{a_1-1}\binom{c-2+i-k}{c-2}}+\displaystyle{\sum_{k=0}^{a_1-1}\binom{c-2+i-k-a_2}{c-2}}&=\binom{c-1+i}{c-2}-\binom{c-1+i-a_1}{c-1}.
\exp \left\{\frac 1 2 C_{\sigma}\left( \left| \vec{p}\right| \right) \left( A_{\vec{p}, \sigma}^{a +}+ i \int d^{4}x J^{\mu , a}\left( x \right) f_{p , \mu}^{\sigma}\left( x \right) \right)^{2}\right\}\mid 0 \rangle \
\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}- \mu_{k_{1}}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) \stackrel{d}{\to}N_{l}(0, I_{l}).
e_1 = \frac{1}{\sqrt{2}}(\partial_1 - \sqrt{-1}\partial_2), \e_2 = \frac{1}{\sqrt{2}}(\partial_3 - \sqrt{-1}\partial_4), \ldots, e_n = \frac{1}{\sqrt{2}}(\partial_{2n-1}- \sqrt{-1}\partial_{2n}),
L_R= \sum_i \int_V d^3{\bf x}e^{3Ht}\frac{1}{2}\left[ (\partial_0 \eta_i({\bf x},t))^2-(e^{-Ht}{\bf \nabla}\eta_i({\bf x},t))^2 - \mu_i^2 \eta_i^2({\bf x},t)\right].
\begin{aligned}||e^{\mathcal Rt}\Delta^h_k(\partial_tu)_\phi||^2_{L^2}=e^{2\mathcal Rt}(\Delta^h_k\partial^2_yu_\phi, \Delta^h_k(\partial_yu)_\phi)_{L^2}&-e^{2\mathcal Rt}(\Delta^h_k(U\partial_xu)_\phi, \Delta^h_k(\partial_yu)_\phi)_{L^2}-e^{2\mathcal Rt}(\Delta^h_k(v\partial_yU)_\phi, \Delta^h_k(\partial_yu)_\phi)_{L^2}\end{aligned}
[z^j]\Big(\frac{z}{f(z)}\Big)^j =\frac{1}{2\pi i}\int_{|w|=\delta_1}\frac{d w}{w^{j+1}}-\frac{1}{2\pi i}\int_{|w|=\delta_1}\frac{\Phi'}{\Phi}(w) \frac{dw}{w^j}&=-[w^{j-1}]\Big(\frac{\Phi'}{\Phi}(w)\Big).
[-\infty, y] := \{\infty\}\cup \left<-\infty, y\right], \left[-\infty, y\right> := \{\infty\}\cup \left<-\infty, y\right>, & [y, +\infty] := \left[y, +\infty\right> \cup \{\infty\}, \left<y, +\infty\right] := \left<y, +\infty\right> \cup \{\infty\}.
\left[ E_{ij},[B_{A(1,l)}^+,\dots,B_{A(\lambda_l',l)}^+]\right]=\sum_{k=1}^{\lambda_l'}\delta_{j,A(k,l)}[B_{A(1,l)}^+,\dots,B_{A(k-1,l)}^+,B_i^+,B_{A(k+1,l)}^+,\dots,B_{A(\lambda_l',l)}^+],
( x_{\alpha \dot{\alpha}}) = \left( \begin{array}{c c}{{z_{2}}}&{{z_{1}}}\\{{- \bar{z}_{1}}}&{{\bar{z}_{2}}}\\ \end{array}\right) , \qquad ( \bar{x}^{\dot{\alpha}\alpha}) = \left( \begin{array}{c c}{{\bar{z}_{2}}}&{{- z_{1}}}\\{{\bar{z}_{1}}}&{{z_{2}}}\\ \end{array}\right) .
\bigg|\int_0^t (t - \xi)^{\alpha - 1}E_{\alpha, \alpha}(-\mu_{m,n}(t- \xi)^\alpha)F_{m,n}(\xi)d\xi\bigg| = \frac{1}{\mu_{m,n}}\bigg|\int_0^t(t - \xi)^{-1}E_{\alpha, 0}(-\mu_{m,n}(t-\xi)^\alpha) F_{m,n}(\xi)d\xi\bigg| &\leq \frac{||f||_\infty}{\mu_{m,n}}\int_0^t\bigg|(t - \xi)^{-1}E_{\alpha, 0}(-\mu_{m,n}(t-\xi)^\alpha)\bigg|d\xi
-iq_{0}^{-1}\sin\psi\cdot\left(\frac{F[\frac{1}{q_{0}}](i\mu_{0})w_{2}[\frac{1}{q_{0}}](i\mu_{0})w_{3}[\frac{1}{q_{0}}](i\mu_{0})}{w_{1}[\frac{1}{q_{0}}](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{1}\partial_{\eta}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\mathbb{E}(\mathrm{y},t) := \int_{\partial\Omega}\frac{(\mathrm{y}-\mathrm{v})\cdot\nu_{\mathrm{v}}}{2\pi|\mathrm{y}-\mathrm{v}|^2}\Big[\varphi(\mathrm{v},\mathrm{y},\mathrm{t},\tau)-\Phi^{\textbf{e}}(\xi,t;z,\tau)\Big]d\sigma_\mathrm{v}.
\left\langle \prod_{b=1}^N2 \cosh\left( \; 2 \omega\beta^{(b)}\int_{\Lambda}d^2x \;t(x)\; : \cos \left( 2 \sqrt{\pi}\varphi^{(b)}(x)\right) :_{Q^{M,S}}\; \right) \;\right\rangle_{Q^{M,S}}\; .
\{g\star m\}_{\alpha,\lambda}(x) \equiv g_{1-\alpha,0}(x)\star m_{\alpha,\lambda}(x)/\lambda \mathrm{and}\phi_{\alpha,\lambda}(t) \equiv \frac{\alpha}{\pi}{\rm Im}\left\{\frac{(e^{-i\pi}t)^{\alpha-1}}{\lambda+e^{-i\pi\alpha}\,t^\alpha}\right\}= \frac{\alpha}{\pi}\frac{\lambda\,t^{\alpha-1}\sin\pi\alpha}{\lambda^2+2\lambda\,t^\alpha\cos\pi\alpha+t^{2\alpha}}
S = \frac{1}{g_{n}^{2}}\int d^{3}y \epsilon^{\mu \nu \rho}T r_{U ( n )}\lbrack \partial_{\mu}A_{\nu}A_{\rho}- \frac{i}{3}A_{\mu}\{A_{\nu}, A_{\rho}\}\rbrack + 0 ( 2 ) .
\int_0^\infty \int_0^\infty u(a) v(b) \big(\max\{u(a), v(b)\}\big)^{\lambda /n}da db \geqslant C \bigg(\int_0^\infty p u(a) a^{p-1}\int_0^{a^{p/r}}r v(b)b^{r-1}db da\bigg)^{1+\lambda/n}&+ C \bigg(\int_0^\infty p u(a) a^{p-1}\int_{a^{p/r}}^\infty r v(b)b^{r-1}db da\bigg)^{1+\lambda/n}\geqslant 2^{-\lambda/n}C .
\mathfrak W_{h_1,h_2,0}(x_1,x_2,x_3,t) =\mathcal Z_{h_1}(x_1,t)+ \mathcal Z_{h_2}(x_2,t) \intertext{and}\mathfrak W_{h_3,0,h_4}(x_1,x_2,x_3,t) =\mathcal Z_{h_3}(x_1,t) + \mathcal Z_{h_4}(x_3,t) ,
{\cal K}^{( 2 )}= 2 \partial \omega^{( 1 )}+{\frac{1}{2 \pi \alpha^{\prime}}}( i_{k}C^{( 3 )}) - 2{\cal K}^{( 1 )}\left( C^{( 1 )}+ ( i_{k}C^{( 1 )}) A^{( 1 )}\right) .
\sigma^0_\Gamma(\alpha)(\mathcal{F}_q(\delta_\gamma\otimes x)) = \sigma_{\alpha}(\sigma_\gamma(q)x)=\sigma_{\alpha\gamma}(q)x = \mathcal{F}_q (\delta_{\alpha\gamma}\otimes x) &=\mathcal{F}_q ((\lambda_\Gamma \otimes id)(\alpha) (\delta_\gamma\otimes x))
\limsup_{j\to \infty}\,\int_{\mathbb{R}^s}\,\phi_2(\xi)-H(\xi)\,\chi_F(\xi)\,d\tau_j(\xi)\le\lim_{j\to \infty}\,\int_{\mathbb{R}^s}\,\phi_2(\xi)-\phi_1(\xi)\,d\tau_j(\xi) &=\int_{\mathbb{R}^s}\,G(\xi)\,\left(\phi_2(\xi)-\phi_1(\xi)\right)\,d\xi\le B\,\int_{\mathbb{R}^s}\,\left(\phi_2(\xi)-\phi_1(\xi)\right)\,d\xi\le B\,\epsilon
A^{ij}(-r^{-2}(d^{2})_{i;j}+C\hat{u}_{i;j})= -r^{-2}A^{ij}(d^{2})_{i;j}& + C\left( nWH+\dfrac{f+W^{2}}{W^{2}}\left\langle \Pi_{\ast}\bar{\nabla}_{D_{0}}D_{0},Du \right\rangle +A^{ij}\gamma_{ij}\right) ,
E(y_{n\ell}^4)=\sigma_{np_n}^{-4}\sum_{1\le j_1, j_2, j_3, j_4\le \ell-1}E(\hat{r}_{\ell j_1}\hat{r}_{\ell j_2}\hat{r}_{\ell j_3}\hat{r}_{\ell j_4})=\sigma_{np_n}^{-4}O\left(\frac{\ell^2}{n^4}\right)=O\left(\frac{\ell^2}{p_n^4}\right)
0 = - \int d^{N^{2}}X d^{N^{2}}Y \, \sum_{\alpha = 1}^{N^{2}}\left[ \frac{\partial}{\partial X_{\alpha}}\mathrm{e}^{- S}\frac{\partial}{\partial X_{\alpha}}+ \frac{\partial}{\partial Y_{\alpha}}\mathrm{e}^{- S}\frac{\partial}{\partial Y_{\alpha}}\right] \mathrm{e}^{K \cdot \Psi}.
M^{-(a_1+a_2+a_4)l+ra_3l}\sum_{m=2}^N c_0^{\frac{m}{2}}M^{\frac{a_1 l}{2}m}\alpha^m\sum_{n=2}^{\infty}\|T_m^{l,(n)}\|_{l,r}&\le c M^{-2a_1+a_2+2a_4-ra_3+a_4l}\alpha^{-2}.