harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|\phi_1(q_0)-\phi_1(q)\|_1^2+ \int_\Omega q_0(\phi_1(q_0)-\phi_1(q))^2\,dx-\lambda_1(q_0)\int_\Omega (\phi_1(q_0)-\phi_1(q))^2\,dx + &\int_\Omega(q_0-q)\phi_1(q)(\phi_1(q_0)-\phi_1(q))\,dx-(\lambda_1(q_0)-\lambda_1(q))\int_\Omega \phi_1(q)(\phi_1(q_0)-\phi_1(q))\,dx=0.
\frac{1}{2}\|\nabla V^{j_0}\|^2_{L^2}- \frac{1}{\alpha+2}\|V^{j_0}\|^{\alpha+2}_{L^{\alpha+2}}-\frac{c}{2}\limsup_{n\rightarrow \infty}\int |x|^{-2}|V^{j_0}(x-x^{j_0}_n)|^2 dx = d_M.
D_{2N-1}^{(p\to p)}(n-\lambda+1)\circ P^p(\lambda)= (\lambda-n+p-1)(\lambda-p)D_{2N}^{(p\to p)}(n-\lambda), D_{2N}^{(p\to p)}(n-\lambda+1)\circ P^p(\lambda)= -(\lambda-n+p-1)(\lambda-p)(2N+1)\times \times(2\lambda-n-2N-2)D_{2N+1}^{(p\to p)}(n-\lambda).
H_{\Lambda_L}= S \sum_{\langle x,y\rangle \subset \Lambda_L}\left( -a^*_x\sqrt{1-\frac{n_x}{2S}}\sqrt{1-\frac{n_y}{2S}}a_y - a^*_y \sqrt{1-\frac{n_y}{2S}}\sqrt{1-\frac{n_x}{2S}}a_x + n_x + n_y - \frac{1}{S}n_x n_y\right).
T^{(-)}=\frac{M}{2}\left(\left(\frac{dr}{dt}\right)^2+R^2{\rm sh}^2\left(\frac{r}{R}\right)\left(\frac{d\phi}{dt}\right)^{2}\right)-\frac{I}{2}\left(\frac{d\psi}{dt}+{\rm ch}\left(\frac{r}{R}\right)\frac{d\psi}{dt}\right)^2
\mathrm{Re}( H_N(l)) &=\frac{1}{4}\Big(\left[\left(G_N(l)-G_N(l-1)\big)-\big(G_N(l-1)-G_N(l-2)\right)\right] &\ \ \ \ \ -\left[\left(G_N(2)-G_N(1)\right)-\left(G_N(1)-G_N(0)\right)\right]\Big) &=\frac{1}{4}\left[\left(K_N(l)-K_N(l-1)\right)-\left(K_N(2)-K_N(1)\right)\right]
R_1 \le \min\{I(X_1;Y_1) + I(X_2,X_3;Y_1|X_1,W),\,I(X_1,X_2;Y_2) + I(X_3;Y_2|X_1,X_2,W)\}, R_2 \le \min\{I(W;Y_1|X_1),\,I(W;Y_2|X_1,X_2) \}.
k =1 d_1^{-2^{1-1}}=d_1^{-1}\equiv d_8P_{64}(1)=d_8d_4d_2d_1= & =d_8P_{64}(2)d_1\pmod{2}, k =2 d_1^{-2^{2-1}}=d_1^{-2}\equiv d_8P_{64}(2)=d_8d_4d_2= & =d_8P_{64}(3)d_2\pmod{2}, k =3 d_1^{-2^{3-1}}=d_1^{-4}\equiv d_8P_{64}(3)=d_8d_4\pmod{2}.
0=[f,P_0]_1|_a^b=[f,1]_1|_a^b =\langle \ell[f],1\rangle_{L^2[(a,b),w]}-\langle f,\ell[1]\rangle_{L^2[(a,b),w]}&=\int_a^b\left(\dfrac{1}{w(x)}[p(x)f'(x)]'\right)w(x)dx-0 &=\lim_{x\to b^-}(p(x)f'(x))-\lim_{x\to a^+}(p(x)f'(x)).
\mathbb{P}(\widehat{Q}_0 < \widehat{Q}_1 < \cdots \,|\, \widehat{Q}_0 = m ) \stackrel{1}{=}\frac{m}{m+1}\prod_{j={m+1}}^\infty \frac{j (j+2)}{(j+1)^2}= \frac{m}{m + 2}.
\sum_{n=1}^{\infty}\frac{a(n)}{n}\left(\{nx\}^{N}+N!\sum_{k=0}^{N-1}\frac{(-1)^k\zeta(-k)}{(N-k)!k!}\right)=\sum_{n=1}^{\infty}a_n\sin(2 \pi n x)+\sum_{n=1}^{\infty}b_n\cos(2 \pi n x),
(0,\pm\varphi,\pm\frac{1}{\varphi})\frac{\sqrt{75+6\sqrt{5}}}{11\sqrt{3}},(\pm\frac{1}{\varphi},0,\pm\varphi)\frac{\sqrt{75+6\sqrt{5}}}{11\sqrt{3}},(\pm\varphi,\pm\frac{1}{\varphi},0)\frac{\sqrt{75+6\sqrt{5}}}{11\sqrt{3}}.
(\kappa)={1\over\sqrt{1+\vert\nu_B\vert^2}}\,\left(\begin{array}{c c}\kappa_B \kappa_B\,\nu_B \nu_B\,\kappa_B \nu_B^c\,\kappa_B\,\nu_B\end{array}\right)\,{1\over\sqrt{1+\vert\nu_B\vert^2}}\;,
\langle f, E_2^{*}P_{k, m}\rangle = \frac{1}{\mu_{\Gamma}}\int_{\Gamma\setminus\mathcal{H}}f(z) \overline{E_2^{*}(z) P_{k,m}}~y^{k+2}~\frac{dx dy}{y^2}&= \frac{1}{\mu_{\Gamma}}\int_{\Gamma\setminus\mathcal{H}}f(z) \overline{E_2^{*}(z)}~\overline{\sum_{\gamma\in\Gamma_{\infty}\setminus \Gamma}e^{2\pi imz}\mid_k\gamma}~y^{k+2}~\frac{dx dy}{y^2}.
\limsup_{N \to \infty}\bigg|V_S^N(\nu^1,\dots,\nu^N)-V_S \bigg( \frac{1}{N}\sum_{i=1}^{N}\nu^i \bigg) \bigg|=\lim_{m \to \infty}\bigg|V_S^{N_m}(\nu^1,\dots,\nu^{N_m})-V_S \bigg( \frac{1}{N}\sum_{i=1}^{N_m}\nu^i \bigg) \bigg|.
\sum_{j=1}^m\Big(\sum_{i=1}^k |\Delta_i\Delta_j \nabla_x F^n_{t_i}(^{x_j}X_{t_i})|^a\Big)^{\frac{b}{a}}\le \int_0^2\sum_{j=1}^m\Big(\sum_{i=1}^k|\Delta_i\Delta_j(\nabla^w F_{t_i})(^{x_j - \frac{z}{n}}X_{t_i})|^a \Big)^{\frac{b}{a}}\rho(z)dz
J_{T^*T^*G}^{-1}(0) = \{((g, \pi), ( 0, \omega)) \in (G\times \mathfrak{g}^*)\times (\mathfrak{g}^*\times\mathfrak{g})\}\cong (G\times \mathfrak{g}^*)\times \mathfrak{g}, J^{-1}_{TT^*G}(0) = \{((g, \pi), (\omega, ad^*_\omega\pi)) \in ( G\times \mathfrak{g}^*) \times( \mathfrak{g}\times \mathfrak{g}^*)\}\cong G\times \mathfrak{g}^*\times \mathfrak{g},
\sum_{n_1=0}^\infty\sum_{m_1=0}^{n_1}\cdots\sum_{n_N=0}^\infty\sum_{m_N=0}^{n_N}\langle\eta_{m_1}^+(x_1),\eta_{n_1-m_1}^-( y_1)\rangle\cdots\langle\eta_{m_N}^+(x_N),\eta_{n_N-m_N}^-(y_N)\rangle &\langle Be_{(m_1,...,m_N)},Ae_{(n_1-m_1,...,n_N-m_N)}\rangle.
m^{-}(L'')=\pi^{-4}\cdot 2\cdot \frac{1}{6}\cdot 2\cdot \frac{3^\frac{13}{2}}{2^4}\cdot (1-2^{-4})\cdot (1-3^{-4})\cdot \zeta(4)=\frac{3^{1/2}\cdot 5}{2^4}\approx 0.5412
L_0g \leq [-\eta\beta(1-\eta)+w^2C]c_0e^{\frac{8}{3}\eta-\frac{8}{3}}e^{-\beta \xi}-\varepsilon &\leq [-\frac{1}{4}\beta(1-\eta)+C^2(1-\eta)^{\frac{3}{2}}]c_0e^{\frac{8}{3}\eta-\frac{8}{3}}e^{-\beta \xi}-\varepsilon &<0 in \{\eta> \frac{1}{4}\}\cap D_{T^*},
{\rm ~e}^{-S_{\rm eff}[A_0]}\equiv \sum_{B_M}\int [d\vec a] <\vec a\vert{\rm ~e}^{-H/T}{\rm ~e}^{i\frac{\kappa}{T}\int A_0(B_M+B/2)}\vert\vec a+\vec \nabla A_0/T>
p^{-1}(\Gamma ga_s)\cap \Omega =\{\Gamma c_1 ga_s u_{p_1},\ldots, \Gamma c_n ga_{s}u_{p_n}\}&=\{\Gamma c_1 g u_{p_1}a_s u_{b_1},\ldots, \Gamma c_n g u_{p_n}a_su_{b_n}\},
\int f(x) h(x/q) \Big[ \frac{1}{q}\frac{D_{q^{-1}}D_q h(x)}{h(x/q)}+ p(x) \frac{D_{q^{-1}}h(x)}{h(x/q)}+ \frac{r(x) h(x)}{h(x/q)}\Big]y(x) d_qx = f(x/q) h(x/q) \Big[y(x/q) \frac{D_{q^{-1}}h(x)}{h(x/q)}- D_{q^{-1}}y(x)\Big].
f(x) =\int_{-\infty}^{x}g(x)q(z)dz+\int_{x}^{\infty}g(z)q(z)dz f'(x) =g(x)q(x)+\int_{-\infty}^{x}g'(x)q(z)dz-g(x)q(x) & =g'(x)Q(Z\leq x)
{\cal{J}}_{1}= \frac{x^{2}}{\varrho}\, ; \quad{\cal{J}}_{2}= n \varrho \, ; \quad{\cal{J}}_{3}= 2 \alpha x^{2}- v^{2}\varrho + \frac{x v}{2}\varrho_{z}\, ; \quad \varrho = \left( 1 - 2 \alpha z^{2}\right) \, .
\times \prod_{I=2}^N \;\Bigg\langle\prod_{b=1}^N \; \prod_{j=1}^{n_b}: e^{- i 2 \sqrt{\pi}U_{I b}\Phi^{(I)}(x^{(b)}_j)}:_{M^{(I)}}\; \prod_{l=1}^{m_b}: e^{+ i 2 \sqrt{\pi}U_{I b}\Phi^{(I)}(y^{(b)}_l)}:_{M^{(I)}}\Bigg\rangle_{Q^{(I)}}
\psi (z^1,...,z^{k+1}) = \frac{1}{\pi}\int_{|\zeta| < 2r}\chi\left (\frac{\zeta}{r}\right )\omega_{k+1\overline{k+1}}(z^1, z^2,...,z^k, \zeta) \log |z^{k+1}-\zeta |^2 dA(\zeta),
\kappa(\vec{\theta})=\frac{1}{2}\nabla_{\vec{\theta}}^{2}\psi(\vec{\theta})=\frac{4\pi GD_{ds}D_{d}}{c^{2}D_{s}}\int dz\rho(D_{d}\vec{\theta},z)=\frac{\Sigma}{\Sigma_{cr}}=\sum_{i}\frac{4\pi GM_{i}D_{is}}{c^{2}D_{i}D_{s}}\delta(\vec{\theta}-\vec{\theta}_{i})
q_{i}+ e_{i - 1}- \lambda = \hat{q}_{i}^{( 0 )}+ \hat{e}_{i - 1}^{( 0 )}, i = 1, \dots , N , q_{i}e_{i}= \hat{q}_{i}^{( 0 )}\hat{e}_{i}^{( 0 )}, i = 1, \dots , N - 1,
=\frac{z^{-\mu}4^{\mu-1}}{\pi}G_{{0},{4}}^{{4},{0}}\left( \frac{z^{4}}{256}\bigg{|}\begin{array}{lll}\frac{\mu+\nu}{4}~,\frac{2+\mu+\nu}{4},~\frac{\mu-\nu}{4},\frac{2+\mu-\nu}{4}\end{array}\right),
\langle{\mathcal L}Z_y,\phi\rangle \!=\!\int_U\! \left( (a_{ij}-a_{ij}(y))\partial_i\partial_j Z_y\phi-a_{ij}(y)\,\partial_iZ_y\,\partial_j\phi+b_k\partial_kZ_y\, \phi\right)dx.
\left| \smash{\prod_{j=1}^n}\vphantom{\prod}(\partial^{\gamma_j}|\bullet|^2) (\tau) \right| \leq 2^n \cdot \prod_{j=1}^n |\tau|^{2 - |\gamma_j|}\leq 2^{|\alpha|}\cdot |\tau|^{2n - |\alpha|}.
\sum_{k=N}^\infty\frac{(\alpha t)^k}{k!}k^\beta \gtrsim(\alpha t)^N\sum_{k=0}^\infty\frac{k!}{(1+N-\beta)(2+N-\beta)\cdots(k+N-\beta)}\frac{(\alpha t)^k}{k!}&=(\alpha t)^N\sum_{k=0}^\infty\frac{(1)_k}{(1+N-\beta)_k}\frac{(\alpha t)^k}{k!}
g_{AB}= \left(\begin{array}{ccc}-n^2(1+2A) a^2 B_{,i}nA_y a^2 B_{,j}a^2\left[ (1+2{\cal{R}})\delta_{ij}+ 2E_{,ij}\right] a^2 B_{y,i}nA_y a^2 B_{y,i}b^2(1+2A_{yy}) \end{array}\right) \, .
\min_{|\Delta g_{ps}^i|\leq \varepsilon}\sum_{i=1}^{N}\beta_{i}\log_{2}\left ( 1+\frac{\left | g_{ss}^{i}\right |^{2}p_s^{i}}{\sigma_n^2+\left(\left| \hat{g}_{ps}^{i}+\Delta g^i_{ps}\right|^{2}\right)p_{p}}\right )
{\cal L}=\frac{|\partial_t W|^{2}-2 |\partial_z W|^{2}}{(1+|W|^{2})^{2}}+8\theta_{1}\frac{|\partial_z W|^{2}}{(1+|W|^{2})^{4}}(|\partial_{t}W|^{2}-|\partial_{z}W|^{2}),
v_{0}=\sqrt{\varphi_{1}^{2}+g_{0}\theta_{1}^{2}}v_{1}=\frac{1}{v_{0}}[2\varphi_{1}\varphi_{2}+\frac{f_{0}}{2}\varphi_{1}\theta_{1}^{2}+2g_{0}\theta_{1}\theta_{2}],
G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|\frac{z}{w})=w^{1-a_{1}}\sum_{h=0}^{\infty}\frac{(1-w)^{h}}{h!}G_{p,q}^{m,n}(\begin{matrix}a_{1}-h,a_{2},...,a_{p}\\b_{q}\end{matrix}|z),n\ge1,
\mathrm{T r}e^{- \beta \, P \bar{\cal K}}= \left[ \frac{m L}{\sqrt{4 \pi \beta}}+ \frac{1}{\sqrt{\pi}}\sum_{n = 1}^{\infty}\frac{2^{n}\, \beta^{n - \frac{1}{2}}}{1 \cdot 3 \cdot 5 \cdot . . . \cdot ( 2 n - 1 )}\right] e^{- \beta}- 1
q^{(j)}_{j,e}\!\cdot\! f^{(j)[e]}_{k-i}x^{[k+e-i]}\! =\! -\!\Big(q^{(j)}_{0,d}x^{[d]}+ \sum_{\ell=1}^{j-1}q^{(j)}_{\ell,e}\cdot f^{(\ell)[e]}_{k-i}x^{[k+e-i]}\Big)\!\!
\rho(h) = \frac{1}{2}\left[e^{h H}+ e^{-h H}-\left(e^{h/2}-e^{-h/2}\right)^{2 H}\right] = \frac{1}{2}\left[e^{-h H}+\sum_{k=1}^{\infty}{2H\choose k}(-1)^{k-1}e^{-h(k-H)}\right].
(i) \, h^0(X, T_X) = 0 \mbox{(cf. \cite[\S\,1.2.4, p.\,13]{Hu})}(ii) \,h^0(X, S^mT_X) =0, \;\; \forall \;\; m \geq 1 \mbox{(cf. \cite[Cor.\,9.4.13, p.\,183]{Hu})}(iii) \, T_X \, \mbox{is}\, \mu_H-\mbox{stable}\mbox{(cf. \cite[Prop.\,9.4.5, p.\,180]{Hu})}
\bar E_n\Bigl |_{n\to \infty}\sim \frac{c \xi^2}{4\pi a^2}\left[\frac{10-3 \xi^2}{960n^2}-\frac{28224-7344\xi^2+720\xi^4}{15482880n^4}+O\left(\frac{1}{n^6}\right)\right] \,{.}
T_{n}( x_{1}, \ldots , x_{n}; \hbar ) = \sum_{\pi}T_{1}( x_{\pi ( 1 )}) \ldots T_{1}( x_{\pi ( n )}) \Theta ( x_{\pi ( 1 )}^{0}- x_{\pi ( 2 )}^{0}) \ldots \Theta ( x_{\pi ( n - 1 )}^{0}- x_{\pi ( n )}^{0}) ,
\begin{aligned}\mathcal B'(\delta \b u)=\begin{pmatrix}\delta \b u|_{\partial\Omega}\delta \b u|_{\Omega\times\{t=0\}}\partial_t \delta \b u|_{\Omega\times\{t=0\}}\delta p|_{\partial\Omega}\delta \mu|_{\partial\Omega}\end{pmatrix}= 0 .\end{aligned}
z_\mu \chi_\lambda(g_\mu) = \chi_\lambda(C_\mu) = \sum_\nu a_{\mu,\nu}\chi_\lambda(\tilde{e}_\nu) = a_{\mu,\lambda}\dim L_\lambda \implies a_{\mu,\lambda}= \frac{z_{\mu}}{\dim L_\lambda}\chi_\lambda(g_\mu).
\varphi'(\Gamma_i)\frac{\partial |G_{ii}|^2}{\partial g_{ik}}= \varphi'(\Gamma_i)\overline{G_{ii}}\frac{\partial G_{ii}}{\partial g_{ik}}+ \varphi'(\Gamma_i) G_{ii}\frac{\partial \overline{G_{ii}}}{\partial g_{ik}}\,.
\{\lambda'_i(x)\}_{i\in S_n}=Eig^{\neq 0}(\bar{A}_n(x))=\{\lambda\in Eig(\bar{A}_n(x)):\lambda\neq 0\}&=Eig^{\neq 0}(\bar{B}_n(x))=\{\lambda\in Eig(\bar{B}_n(x)):\lambda\neq 0\}
d(\varphi_i(x_0),\mathcal{Y}(x_0)g)\leq d(\varphi_i(x_0),\mathcal{Y}_0(x_0h_n))+d(g_n,g) = d(\varphi_i(x_0),\mathcal{Y}(x_0)h_n)+d(g_n,g) \leq d(\varphi_i(x_0),\varphi_i(x_0)h_n)+d(g_n, g) \to 0 \,\,n\to \infty.
(f_h-f_H,v_h)_{L^2(\Omega)}=(f_h-f_H,v_h-\mathcal{S}_H v_h)_{L^2(\Omega)}=\sum\limits_{K\in\mathcal{T}_H\backslash\mathcal{T}_h}(f_h-f_H,v_h-\mathcal{S}_H v_h)_{L^2(K)}
R_{| n_{1}, n_{2}, \dots , n_{k}\rangle}^{( n )}( \eta , \vartheta , u ) = R^{( n )}( \eta , \vartheta , u ) \prod_{i \textrm{}\mathrm{o d d}}b_{n_{i}}^{n}( \eta , u ) \prod_{i \textrm{}\mathrm{e v e n}}b_{n_{i}}^{n}( \bar{\eta}, u ) \, \, \,
\chi = m^{2}\frac{\operatorname{s i n}\left( \beta \sqrt{\phi^{2}+ \frac{1}{4 \pi}\operatorname{l n}[ \frac{m^{2}}{\chi}]}\right)}{\beta \sqrt{\phi^{2}+ \frac{1}{4 \pi}\operatorname{l n}[ \frac{m^{2}}{\chi}]}}
\mbox{for $k\geq 0$,}\sqrt{2\pi}\mathcal{K}(x,k) = \chi_+(x)T(k)\psi_+(x,k)+\chi_-(x)\left[\psi_-(x,-k) + R_-(k) \psi_-(x,k) \right], \mbox{for $k<0$,}\sqrt{2\pi}\mathcal{K}(x,k) =\chi_-(x)T(-k) \psi_-(x,-k)+\chi_+(x)\left[ \psi_+(x,k) + R_+(-k)\psi_+(x,-k) \right].
{_{n+1}}F_{n}\left(\begin{array}{cccc}A_1, A_2, \ldots, A_{n+1}B_1, \ldots, B_n \end{array}\mid t\right):=\frac{p}{p-1}\sum_{\chi\in\widehat{\mathbb{F}_p^\times}}{A_1\chi\choose\chi}\cdots{A_{n+1}\chi\choose B_n\chi}\chi(t).
\kappa^n = (0,\,\kappa^{n-1}_{01}) = (0,\,\,(-1)^n(n-1)!\, \check{\chi}_{H_{n}\cup H_-}\, \bar{\partial}\varphi_+ \wedge \bar{\partial}\varphi_1 \wedge \cdots \wedge \bar{\partial}\varphi_{n-2}),
\sum_{i=1}^{m}H_d(F(y_i),F(y_{i-1})) = \sum_{i=1}^{m}H_d(\lim_{n \to \infty}F_n(y_i),\lim_{n \to \infty}F_n(y_{i-1})) = \lim_{n \to \infty}\sum_{i=1}^{m}H_d(F_n(y_i),F_n(y_{i-1})) \le \liminf_{n \to \infty}\sup_{P}\sum_{i=1}^{m}H_d(F_n(y_i),F_n(y_{i-1})),
\underline{\theta}:=\widetilde{g}_{01}{_{|\mathcal{I}^1}},\;\underline{\Theta}_{ab}:= \widetilde{g}_{ab}{_{|\mathcal{I}^1}},\;\underline{\phi}:= \Phi_{|\mathcal{I}^1},\;\textbf{\underline{f}}:=\rho_{|\mathcal{I}^1},\;\underline{\psi}_{\mu\nu}:= \frac{\partial\widetilde{g}_{\mu\nu}}{\partial y^1}.
\begin{multlined}[t][12.5cm]\widetilde{\Pi}_{\nu_{G},k\nu_{T}}\left(x_{1k},y_{2k}\right) =\frac{d_{\nu_{G}}}{(2\pi)^{d_{T}}}\int_{G}\int_{T}\chi_{\nu_{G}}(g^{-1})t^{-k\nu_{T}}\cdot\Pi\left(\tilde{\mu}^{P}_{p_{0}^{-1}}\circ\tilde{\mu}^{G}_{g^{-1}}\circ\tilde{\mu}^{T}_{t^{-1}}\left(x\right), x_{2k}\right)dtdg.\end{multlined}
\begin{array}{c}\epsilon_{ab}=\left (\begin{array}{cc}0 q^{\frac{1}{2}}-q^{-\frac{1}{2}}0 \end{array}\right ) \epsilon_{ab}\epsilon_{bc}=-\delta_{ac}(\epsilon_{\overline{a}\overline{b}})_{q}=-(\epsilon_{a b})_{q^{-1}}\end{array}
\langle e^{\frac{-ikx^{2}}{2z}}|e^{\frac{-ikx^{2}}{2z}}\rangle=e^{\frac{-ikx^{2}}{2z}}(e^{\frac{-ikx^{2}}{2z}})^{*}=e^{\frac{-ikx^{2}}{2z}}e^{\frac{+ikx^{2}}{2z}}=e^{0}=1
D(A) = \biggl\{x = \sum_{m=1}^{\infty}a_m f_m: \sum_{m=1}^{\infty}2^m a_m f_m \biggr\}A \biggl(\sum_{m=1}^{\infty}a_m f_m \biggr) = \sum_{m=1}^{\infty}2^m a_m f_m.
E_{1}^{\ast}=E_{1},E_{3}^{\ast}=E_{2},E_{4}^{\ast}=E_{4},F_{1}^{\ast}=F_{1},F_{2}^{\ast}=F_{3},F_{4}^{\ast}=F_{4}, &E_{2}E_{3}+E_{4}^{2}=E_{4},F_{2}F_{3}+F_{4}^{2}=F_{4}.
\sup_{\pi\in\Pi}\left| E_{\mu}^\pi\left(\sum_{n=1}^N \beta^{n-1}r_i(s_n,a_n)\right)-E_{\mu}^\pi\left(\sum_{n=1}^N \beta^{n-1}r_i^{l}(s_n,a_n)\right)\right| \to 0 \quad\mbox{as}l\to\infty.
+ \frac{1}{N}\sum_{(b_1,\vec{b}_\ast):\;\mathcal{U}_{b_1,\vec{b}_\ast}\neq 0, \; \mathcal{U}_{b_1-1,\vec{b}_\ast}= 0}\mathcal{U}_{b_1,\vec{b}_\ast}\delta_{b_1,\vec{b}_\ast}\;\;
\lim_{\lambda\to\infty}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^{r,\tau}(\mathbb{R}^n)}=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^{r,\tau}(\mathbb{R}^n)};
\begin{array}{ll}\pi \circ f_B - f_{B_1}\circ\pi =c^a_n v^n_b + c^k_i B^i_j v^j_k \delta^a_b + c^d_n B^n_j v^j_d \delta^a_b + c^d_i B^i_n v^n_d \delta^a_b &= c^k_i B^i_n v^n_k \delta^a_b v^j_k = \delta^j_n &=0.\end{array}
\iint_{Q_{\rho}}\textbf{A}(x,t,u,Du) \cdot D\varphi \,dx\,dt \leq -s C_0 \iint_{Q_{\rho}}G(|Du|) G^{r-1}\left(\frac{\zeta \bar{u}}{\rho}\right)\bar{u}^{s}\zeta^{q}\,dx\,dt + \{(r-1)g_0 + q\}C_1 \iint_{Q_{\rho}}g(|Du|)|D\zeta|G^{r-1}\left(\frac{\zeta \bar{u}}{\rho}\right)\bar{u}^{s+1}\zeta^{q-1}\,dx\,dt .
{{\cal F}_1}\left( \gamma \right) = \exp \left({- \frac{\gamma}{{{{\bar \gamma}_0}}}}\right)\sum\limits_{j}{{\phi_{T_1,j}}}\frac{{{{\bar \gamma}_0}}}{{\gamma +{{{{\bar \gamma}_0}}\mathord{\left/{\vphantom{{{{\bar \gamma}_0}}{{\xi_{{T_1},j}}}}}\right.\kern-\nulldelimiterspace}{{\xi_{{T_1},j}}}}}}
0 \leq\lim_{\mu\to+\infty}f_{s,\mu,+\infty}\ast\rho_\nu(z)=\lim_{\mu\to+\infty}\int f_{s,\mu,+\infty}(z-w)\rho_\nu(w)dV_w \leq\limsup_{\mu\to+\infty}\int f_{s,\mu,+\infty}(z-w)\rho_\nu(w)dV_w\leq\int\limsup_{\mu\to+\infty}f_{s,\mu,+\infty}(z-w)\rho_\nu(w)dV_w=0,
j :=-\frac{1}{2}\left(\frac{S_1\mu}{R_1}\left(\mu \frac{f_*^2}{2}+\mu f_*g_*+\frac{g_*^2}{2}\right)\sigma^\prime(\Gamma_*)-z\left(S_1 \frac{g_*^2}{2}+S_1\mu\left(\frac{f_*^2}{2}+f_*g_*\right)\right)\Gamma_*\right),
\left\langle (\varphi\otimes\bar{u}),\phi(\psi\otimes\bar{v}; z,\bar{z})\;\phi(\chi\otimes\bar{w}; \zeta,\bar{\zeta})\;\left(\Omega \otimes \overline{\Omega}\right) \right\rangle =(\bar{z}- \bar{\zeta})^{\bar{\Delta}_{l}- \bar{\Delta}_{\bar{v}}-\bar{\Delta}_{\bar{w}}}\; F_{\psi \chi}(z,\zeta).
\left( X^{\mu}\right) ( \tau , \sigma^{1}, \sigma^{2}) = \left( \begin{array}{c}{{\tau}}\\{{X^{1}( \tau , \sigma^{1}, \sigma^{2})}}\\{{X^{2}( \tau , \sigma^{1}, \sigma^{2})}}\\{{X^{3}( \tau , \sigma^{1}, \sigma^{2})}}\\ \end{array}\right) ,
R_w(\nu',\nu) = \sum_{\alpha}''' w\left(\frac{N(\alpha)}{X}\right) \frac{L\left(\chi_{\alpha},\frac{1}{2}+ \nu'\right)}{L\left(\chi_{\alpha},\frac{1}{2}+ \nu\right)}= W^*(X) \left(\frac{\zeta\left(\frac{3}{2}+3\nu'\right)}{\zeta\left(\frac{3}{2}+2\nu'+\nu\right)}A(\nu';\nu) \right) + O\left(X^{1/2+\epsilon}\right),
\lim_{k\rightarrow+\infty}\left[\left(\frac{\pi}{k\,\nu}\right)^d \cdot H(X)_{k\boldsymbol{\nu}}\right]= \int_M \,\mathrm{d}V_M(m)\left[\left(\frac{1}{2\,\lambda (m)}\right)^{d+1}\right].
E[e^{(U_1+U_2+\cdots+U_k)(e^t-1)}] = \sum_{m=0}^\infty E[(U_1+\cdots+U_k)^m] \frac{(e^t-1)^m}{m!}&= \sum_{m=0}^\infty E[(U_1+\cdots+U_k)^m] \sum_{n=m}^\infty S_2(n,m) \frac{t^n}{n!}&= \sum_{n=0}^\infty \left( \sum_{m=0}^n S_2(n,m) E[(U_1+\cdots+U_k)^m] \right) \frac{t^n}{n!}.
\frac{\int_{\bar{D}_\delta^{N-1}}\exp\left(-\frac{\beta}{2}\sum_{j=2}^N \lambda^s_j a_j^2 \right) da_2\dots da_N}{\int_{\mathbb{R}^{N-1}}\exp\left( -\frac{\beta}{2}\sum_{j=2}^N \lambda^s_j a_j^2\right) da_2\dots da_N}> 1- \hat{\delta}
E[U_n(\theta)U_n(\theta')]-E[U^*_n(\theta)U^*_n(\theta')|\mathcal{F}^X] &=\sum_{I,J}\{E[X^1(I)^2X^2(J)^2]-X^1(I)^2X^2(J)^2\}K(I,J_{-\theta})K(I,J_{-\theta'})\in\overline{\mathcal{P}}_4(\mathbb{H})
-\frac{1}{2}\bar{\varphi}(r_i)r_i^{-1/2}\bar{\psi}^{(i)}(r_i,0) = \bar{\Psi}(r_i,\pi) - \bar{\Psi}(r_i,0) = \ \int_0^{\pi}\partial_{\theta_i}\bar{\Psi}(r_i, \theta_i)\,d\theta_i = \ \int_0^{\pi}r_i^{-1}\left(\partial_{\theta_i}^2\bar{S}_i \bar{\psi}^{(i)}+ \partial_{\theta_i}\bar{S}_i \partial_{\theta_i}\bar{\psi}^{(i)}\right)\,d\theta_i
\|u\|_{(\kappa_1+1)p_*,\partial\Omega}=\|u^{\kappa_1+1}\|_{p_*,\partial\Omega}^{\frac{1}{\kappa_1+1}}\leq M_{15}(\kappa_1, u) \left [\|u\|_{p_*,\partial\Omega}^{p_*}+1\right]^{\frac{1}{(\kappa_1+1)p}}<\infty.
\lim_{K \to \infty}K^{-\delta}\limsup_{N \to \infty}\int_{D}\mathbb{E}(|V_{N}(\tau+i\eta)|^{1+\delta})\chi_{N}(\eta)|\overline{\partial}\Psi_{f}(\tau,\eta)|^{1+\delta}\,d\tau\,d\eta &\leq \lim_{K \to \infty}K^{-\delta}\int_{D}c_1\eta^{-1-\delta}|\overline{\partial}\Psi_f(\tau,\eta)|^{1+\delta}\,d\tau\,d\eta = 0
\begin{array}{ll}{\mathcal E}(V^+_{n+1}, \Phi^+_{n+1}) =\sum\limits_{k=0}^n({\mathbb E}g_{k,1}+h_k^++\overline{e}_{+,k}) begin{align*}2mm] ={\mathbb E}\left(({\mathcal B}(U^{a,+}+V^+_{n+1}, U^{a,-}+V^{-}_{n+1}, \psi^a+\phi_{n+1}))_1-\sum\limits_{k=0}^n\tilde{e}_{k,1}\right)+\sum\limits_{k=0}^n(h_k^++\overline{e}_{+,k})\end{array}
+ i \hbar ( V_{A B}^{C E}V_{E}^{D}( - 1 )^{\varepsilon_{C}}- ( C \leftrightarrow D ) ( - 1 )^{\varepsilon_{C}\varepsilon_{D}}) - [ V_{A B}^{C D}, \Omega ] ( - 1 )^{\varepsilon_{C}+ \varepsilon_{D}}- \frac{1}{2}( i \hbar )^{2}V_{A B}^{E F}V_{F E}^{C D}.
J^{\alpha \beta}( x ) ={\frac{1}{2}}\left[ \psi_{A}^{\alpha \dagger}( x ) , \psi_{A}^{\beta}( x ) \right] = \psi_{A}^{\alpha \dagger}( x ) \psi_{A}^{\beta}( x ) - \delta^{\alpha \beta}{\frac{{\cal N}_{C}}{2}}= ( J^{\beta \alpha}( x ) )^{\dagger}
\Psi_{i}^{( s )}( \mathbf{x}) = e^{i \mathbf{p}_{i}. \mathbf{x}}u_{i}= \frac{e^{i p^{\bot}\rho \cos \varphi}}{\sqrt{4 \pi}\sqrt{2 s}}\left( \begin{array}{c}{{\sqrt{E + s m}\sqrt{s + 1}}}\\{{\epsilon_{3}\sqrt{E - s m}\sqrt{s - 1}}}\\{{\epsilon_{3}\sqrt{E + s m}\sqrt{s - 1}}}\\{{\sqrt{E - s m}\sqrt{s + 1}}}\\ \end{array}\right) ,
\frac{P_{\lambda}\big(q^rt^{N-1}, t^{N-2}, \dots, t, 1; q, t\big)}{P_{\lambda}\big(t^{N-1}, \dots, t, 1; q, t\big)}= \frac{P_{(r)}\big(q^{\lambda_1}t^{N-1}, \dots, q^{\lambda_{N-1}}t, q^{\lambda_N}; q, t\big)}{P_{(r)}\big(t^{N-1}, \dots, t, 1; q, t\big)}.
\delta \eta_{\alpha_{2k+1}}^{*}=Z_{\;\;\alpha_{2k+1}}^{\alpha_{2k}}\Phi_{\alpha_{2k}}^{*}+A_{\alpha_{2k+1}}^{\;\;\alpha_{2k+2}}\Phi_{\alpha_{2k+2}}^{*},\;k=0,\cdots ,b,
{\frac{M_{s}^{- 4}\pi^{4}2}{3}}\int d \tau_{2}\tau_{2}d^{4}\eta \delta ( 1 - \sum_{i}\eta_{i}) e^{{\frac{2 \pi \tau_{2}}{M_{s}^{2}}}( s \eta_{1}\eta_{3}+ t \eta_{2}\eta_{4})}e^{-{\frac{2 \pi \tau_{2}}{M_{s}^{2}}}\sum_{i}m_{i}^{2}\eta_{i}}\equiv{\frac{\pi^{2}}{6}}B^{\prime}\ .
w_0 = \frac{1}{\Delta s}\int_{s_0}^{s_1}(s_1-s)w(s)ds = \Psi_2(0), w_K = \frac{1}{\Delta s}\int_{s_{K-1}}^{s_K}(s-s_{K-1})w(s)ds = \Psi_1(K-1), w_k = \frac{1}{\Delta s}\int_{s_{k-1}}^{s_k}(s-s_{k-1})w(s)ds + \frac{1}{\Delta s}\int_{s_k}^{s_{k+1}}(s_{k+1}-s)w(s)ds = \Psi_1(k-1) + \Psi_2(k).
\prod_{\substack{(a,b,c,d)=1 a,b,c,d \geq 1}}\left( \frac{1}{1-w^a x^b y^c z^d}\right)^{\frac{1}{a^r b^s c^t d^u}}= \exp\left\{\left( \sum_{h=1}^{\infty}\frac{w^h}{h^r}\right) \left( \sum_{i=1}^{\infty}\frac{x^i}{i^s}\right) \left( \sum_{j=1}^{\infty}\frac{y^j}{j^t}\right) \left( \sum_{k=1}^{\infty}\frac{z^k}{k^u}\right)\right\}.
\mathcal{L}[\widehat{\rho^h_{\cdot}}(k)](s) = \mathcal{L}[\alpha(k,\cdot)](s) + \frac{\mathcal{L}[\beta(k,\cdot)](s)}{1-\mathcal{L}[\beta(k,\cdot)](s)}\mathcal{L}[\alpha(k,\cdot)](s).
^{(0)}_{\psi,T,R}\langle Dp,f,\eta_1|Dp,f,\eta_2 \rangle^{(0)}_{\psi,T,R}= \lim_{x \to 1}\;^{(0)}_{\psi,T,R}\langle Dp,f,\eta_1|x^{2(F_0 + G_0)}|Dp,f,\eta_2 \rangle^{(0)}_{\psi,T,R}
D_{3,b}=\frac{1}{|z|}\big[ Q(X_1,z+t_0(X_1),D_t\tilde h(X_1,t_0(X_1)))- D_t Q(X_1t_0(X_1),D_t\tilde h(X_1,t_0(X_1))) & -\big( Q(X_2,z+t_0(X_2),D_t\tilde h(X_2,t_0(X_2)))- D_t Q(X_2,t_0(X_2),D_t\tilde h(X_2,t_0(X_2)))\big)\big] -
\nabla u=\frac{\widetilde{\nabla}u}{f(u)^2-|\widetilde{\nabla}u|_{\widetilde{g}}^2}, |\nabla u|_g^2=\frac{|\widetilde{\nabla}u|_{\widetilde{g}}^2}{f(u)^2-|\widetilde{\nabla}u|_{\widetilde{g}}^2},
\tilde{P}_{V, t}(Y) = \int_{\mathbb{R}^{p_2\times r}}\frac{1}{(2\pi)^{p_1p_2/2}}\exp(-\|Y - 2t WV^{\intercal}\|_F^2/2 ) & \cdot (\frac{p_1}{2\pi})^{p_1r/2}\exp(-p_1\|W\|_F^2/2) dW.
Z_{N}( x ) \equiv \frac{N}{4}\left[ \phi \left( x + \Theta , \frac{\pi \beta^{2}}{2}\right) + \phi \left( x - \Theta , \frac{\pi \beta^{2}}{2}\right) \right] + \sum_{r = 1}^{l}\phi ( x - \alpha_{r}, \pi \beta^{2}) + 2 \pi \beta^{2}\varepsilon l \, .
a_1^{56}=y_1^{26}=:b_1, a_2^{56}=y_1^{36}=:b_2, &a_3^{57}=y_2^{27}=:c_3, a_4^{57}=y_2^{37}=:c_4, &v^{25}+y_2^{26}=a_3^{56}=a_1^{57}+u_2^{67}=:b_3, &v^{35}+y_2^{36}=a_4^{56}=a_2^{57}-u_1^{67}=:b_4, &v^{25}+a_1^{57}=y_1^{27}=y_2^{26}+u_2^{67}, &v^{35}+a_2^{57}=y_1^{37}=y_2^{36}-u_1^{67}
\left\vert I_{2}\right\vert \leq\frac{1}{\varepsilon}\Vert Z_{\varepsilon,\xi}^{h}-i_{\varepsilon}^{\ast}[b(x)f^{\prime}(W_{\varepsilon,\xi(y)})Z_{\varepsilon,\xi}^{h}\Vert_{\varepsilon}\Vert\phi_{\varepsilon,\xi(y)}\Vert_{\varepsilon}=O(\varepsilon).
\Psi_L(x_1^{K+1}, \cdots, x_N^{K+1}, \lambda^{K+1}, x_N^K) - \Psi_L(x_1^1, \cdots, x_N^1, \lambda^1, x_N^0) \leq -\tau\sum_{k=1}^{K}\sum_{i=1}^N \left\| x_i^k - x_i^{k+1}\right\|^2,
|\langle 0, \mu_1,\epsilon_1, j, \mu_2,\epsilon_2 | 0,\mu_{12},\epsilon_{12}\rangle|^2 =\begin{cases}\displaystyle \frac{(\frac{j}{2}+1+\mu_1+\mu_2)_{j/2}}{2^{j/2}(\frac{1}{2}+\mu_1)_{j/2}}j ,{}\displaystyle \frac{(\frac{j+1}{2}+\mu_1+\mu_2)_{(j+1)/2}}{2^{(j+1)/2}(\frac{1}{2}+\mu_1)_{(j+1)/2}}j .\end{cases}
=\sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}[I^G_{M_S}(\rho)]\left[ \bigoplus_{(M_S, \mu_S) \in \mathrm{Rel}^{M_b, \mu_b}_{M_S,b'}}r_{-\mu_S}\circ LL(I^{G}_M(\rho))|_{W_{E_{\{\mu_S\}_{M_S}}}}| \cdot|^{-\langle \rho_G, \mu \rangle}\right].
R = \left( \begin{matrix}{{q^{2}}}&{{0}}&{{0}}&{{0}}&{{}}\\{{0}}&{{q^{2}- q}}&{{1}}&{{0}}&{{}}\\{{0}}&{{1}}&{{0}}&{{1}}&{{}}\\{{0}}&{{0}}&{{0}}&{{q^{2}}}&{{}}\\ \end{matrix}\right) .
\|\nabla u\|_{L^\gamma}\leq \sum_{i=0}^{+\infty}\|\nabla u_i\|_{L^\gamma}\leq C\, \|f\|_{L^1}^{\frac{1}{p}}\sum_{i=0}^{+\infty}2^{iN(1-\frac{\gamma}{p})}\, 2^{-i\frac{N-p}{p(p-1)}},
(\hat{m}_k^A, \hat{s}^A_k) = \arg\min_{{m_k^A\in \{0,\cdots, M_\mathrm{F}-1\}}\atop{s_k\in \{s_0, \cdots, s_{Q-1}\}}}\sum_{l=0}^{M_\mathrm{F}-1}\left|\Omega_{k,l}-H^A_{k,l}s^A_k\delta_{m_k^A,l}\right|^2.
Z ={\textstyle\sum\limits_{k=n\vee m,n\neq m}}TP_{N\geq ke}e={\textstyle\sum\limits_{m=2}^{\infty}}2(m-1)TP_{N\geq ke}e & =2{\textstyle\sum\limits_{k>1}}\left({\textstyle\sum\limits_{m=1}^{k-1}}m\right) P_{N=ke}e={\textstyle\sum\limits_{k=1}}\left( k^{2}-k\right) P_{N=ke}e=TN^{2}-TN.
\vert\mathcal J_{2}\vert\le \varepsilon\left\{I(t)+I_{1,n}(t)\right\}+\frac{C_2}{\varepsilon}\Vert\varphi(t)\Vert^2_{L^2(\mathbb R)}&+C_3\left\{\int_0^t(I(s)+I_{1,n}(s))ds+\int_0^t\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)}ds\right\}\,.
\mbox{\boldmath$ \gamma$}_{\operatorname*{C};j}^{\mathbb{B}}:H^{1}\left( \mathbb{R}^{3},\mathbb{B}\right) \rightarrow H^{1/2}\left( \Gamma_{j}\right) \times H^{-1/2}\left(\Gamma_{j}\right) ,\quad\mbox{\boldmath$ \gamma$}_{\operatorname*{C};j}^{\mathbb{B}}u:=\left( \gamma_{\operatorname*{D};j}u,\gamma_{\operatorname*{N};j}^{\mathbb{B}}u\right) .