harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left|\sqrt{\widehat{C}(s)}-\frac{1}{\sqrt{2s}}\right| =\frac{1}{\sqrt{2s}}\left|\sqrt{\frac{2}{\pi}\int_0^{2\pi s}\frac{\sin y}{y}dy}-1\right|\leq \frac{1}{\sqrt{2s}}\frac{2}{\pi}\int_{2\pi s}^\infty \frac{\sin y}{y}dy=\mathcal{O}(s^{-3/2}),
b_R^{2,2}=-4(\mathbb{E}R)\mathbb{E}R^3 + 3(\mathbb{E}R^2)^2 + \mathbb{E}R^4 b_R^{2,1}= 6\sigma_R^2 a_R^{2,1}=4(\mathbb{E}R)^4 - 8(\mathbb{E}R)^2\mathbb{E}R^2+ \frac83(\mathbb{E}R)\mathbb{E}R^3 + 2(\mathbb{E}R^2)^2 - \frac23 \mathbb{E}R^4.
\mathcal{R}_{1,2}(t) = \int_0^t \int_0^\xi \mathcal{B}\left( F(v) \cdot e^{(\xi - s) \mathcal{A}}\mathcal{C}[G, \mathcal{A}] \left( e^{s \mathcal{A}}\overline v \right) \right) ds d\xi .
\mathcal L_{\mathrm{noise}}(\psi, \phi, x, t) = \mathcal L_{\mathrm{Dirac}}^{\mathrm{noise}}(\psi, x,t) + \mathcal L_{\mathrm{meson}}^{\mathrm{noise}}(\phi, x,t) = \psi^* \beta \psi \xi_1 + \frac 12 \phi^2 \xi_2 .
\sum_{n=0}^{\infty}\mathcal{E}_{n+m}^{\left( \alpha\right)}\left(\lambda\right) \frac{\left( 2i\pi\right)^{n}}{n!}=\mathcal{E}_{m}^{\left( \alpha\right)}\left( \lambda\right) , \sum_{n=1}^{\infty}\binom{n+m+l}{l}^{-1}\mathcal{B}_{n+m+l}^{\left( l\right)}\left( \lambda\right) \frac{\left( 2i\pi\right)^{n}}{n!}=0.
R_{2}^{1}= - 2 i \frac{d g_{1}\wedge d \overline{{g}}_{1}}{( 1 - g_{1}\overline{{g}}_{1})^{2}}, R_{4}^{3}= + 2 i \frac{d g_{2}\wedge d \overline{{g}}_{2}}{( 1 + g_{2}\overline{{g}}_{2})^{2}},
\exp [ f_{\nu}( z ) ] = \frac{\Gamma ( 2 + 4 \nu ) \Gamma ( 2 + 2 \nu ) \Gamma ( 2 \nu - z ) \Gamma ( - z )}{\Gamma ( 2 + 4 \nu + z ) \Gamma ( 2 + 2 \nu + z ) \Gamma ( 2 \nu )}( 4 a \nu^{4})^{z}.
\lambda_1(a) =\Lambda_1({\rm diag}(a,1,a,1))=\nu^{1/2}(a)\sigma_1(a)\omega_{\pi_1}(a), \lambda_2(a) =\Lambda_2({\rm diag}(a,1,a,1))=\nu^{-1/2}(a)\sigma_1(a).
\sum_{\substack{a< q (a,q)=1}}\sum_{\substack{|\eta|\le E/Y (\eta+a/q)Y\in\mathbb{Z}}}F_Y\Bigl(\frac{a}{q}+\eta\Bigr)\ll (d E)^{27/77}q^{1/21}+\frac{E^{5/6}d^{3/2}q}{Y^{10/21}}.
B_{8}= \left( \begin{array}{c c}{{P_{1}}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right) , B_{9}= \left( \begin{array}{c c}{{Q_{1}}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right) , B_{7}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{b I}}\\ \end{array}\right) .
\| w\cdot\delta_G- \delta_K\|^2 =\|\delta_G\|^2+\|\delta_K\|^2-2\,\langle w\cdot\delta_G,\delta_K\rangle &\geq \|\delta_G\|^2+\|\delta_K\|^2-2\,\langle w_{\emptyset}\cdot\delta_G,\delta_K\rangle=\|\beta_{\emptyset}\|^2\,.
\tilde{a}_{2n}= \frac{\tilde{Q}_{2n}\left (w_1, w_2, w_3, F, w_1', w_2', w_3', F', \dots,w_1^{(2n)}, w_2^{(2n)}, w_3^{(2n)}, F^{(2n)}\right )}{F^{2n}(w_1 w_2 w_3)^{3n-1}},
X_{j}=\frac{R_{j}}{T_{j}}=\frac{r_{j,j+1}+X_{j+1}e^{2ik_{j+1,z}d_{j}}}{1+r_{j,j+1}X_{j+1}e^{2ik_{j+1,z}d_{j}}}e^{-2ik_{j,z}d_{j}}
\left( \frac{\mathcal{\vartheta}_{1}\left( \nu \mid \theta \right)}{\nu \mathcal{\vartheta}_{1}^{^{\prime}}\left( 0\mid \theta \right)}\right)=\exp \left( -\sum\limits_{k=1}^{\infty}\frac{1}{2k}\nu^{2k}G_{2k}(q^{2})\right)
\log | \mathcal{L}(w,\chi_D) \cdot \ldots \cdot \mathcal{L}(wu^{l-1},\chi_D) | \leq \frac{2g}{N+1}\log 2 + \frac{2g}{N+1}\sum_{j=1}^{l-1}\log (1+q^{-(N+1) j \gamma}) & + \sum_{d(f) \leq N}\sum_{j=0}^{l-1}\frac{a_j(d(f)) \chi_D(f) \Lambda(f)}{\sqrt{|f|}}\cos((\theta+j \beta)d(f)) + O (\gamma^{-1}),
\left[-\frac{d^2}{dz^2}+\frac{l(l+1)}{(z+R)^2}+m^2+\frac{\lambda}{2}\phi_{\rm wall}^2(z)+\frac{\sigma}{4!}\phi_{\rm wall}^4(z)\right]\chi_{n,l}(z)=E_{\rm wall}^2 (n,l)\chi_{n,l}(z)\;.
\frac{1}{2^{n}}\frac{1}{|D_{n}|}\frac{1}{\prod_{k<n}|D_{k}|}>\sum_{m=n+1}^{\infty}\frac{1}{2^{m}}\frac{|D_{m}|}{|D_{m}|}\frac{1}{\prod_{k<n}|D_{k}|}.
\int_{\mathcal{Y}^d}\widetilde{m}_{\Lambda_j}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{w}_{\Lambda_jy_i}dy+\int_{\mathcal{Y}^d}\widetilde{m}(\widetilde{\delta}_{ij}+\widetilde{w}_{y_i\Lambda_j})\widetilde{w}_{\Lambda_jy_i}dy=0.
\left( \frac{1}{T_i^{\infty}}+\frac{1}{\theta_i^2T_i^*(u)},\frac{Z_i^*(u)^2}{T_i^*(u)}+\frac{T_i^{\infty}Z_i^*(u)^2}{\theta_i^2T_i^*(u)^2}\right)_{i\in V}\overset{law}=\left(2\delta_i+\frac{1}{\theta_i^2T_i^*(u)},\frac{z_i^2}{\theta_i^2T_i^*(u)}+\frac{z_i^2}{2\delta_i(\theta_i^2T_i^*(u))^2}\right)_{i\in V}
\left(\pi_{\frac{n-m}{2}}(X\otimes Y) -\pi_{\frac{n-m}{2}}(X \circledcirc Y) - \frac{1}{2}\pi_{\frac{n-m}{2}}([X,Y]) - \lambda^c \pi_{\frac{n-m}{2}}(\langle X,Y \rangle)\right)=0.
R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}= \lambda_{s}^{2}\, ( \gamma_{2}+ \frac{\gamma_{1}^{2}}{2})^{2}+ \frac{2 \gamma_{1}^{2}\lambda_{s}^{2}}{r_{s}^{2}}+ \frac{4 \, ( 1 - \lambda_{s})^{2}}{r_{s}^{4}}+{\cal{O}}( r - r_{s})
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^2}{n^3}}= \sqrt{\frac{1}{1-z}}\; \exp\left\{\frac{1}{6}Li_2(z) + \frac{1}{3}\frac{z}{1-z}\right\},
\frac{1}{A^{\alpha}B^{\beta}}= \frac{\Gamma ( \alpha + \beta )}{\Gamma ( \alpha ) \Gamma ( \beta )}\int_{0}^{1}d x \frac{x^{\alpha - 1}( 1 - x )^{\beta - 1}}{[ A x + B ( 1 - x ) ]^{\alpha + \beta}},
\frac{p^{2}}{4}= \frac{p^{3}}{16 (p/4)}\leq \frac{p^{3}}{16 a_{1}^{*}}= a_{2}^{*}a_{3}^{*}a_{4}^{*}\leq \left(\frac{4}{3}\right)^{3}(a_{2}^{*})^{3}
\hat{\mathbb{E}}_{(\tau\wedge\sigma)+}[XI_{\{\tau\leq \sigma\}}] =\hat{\mathbb{E}}_{(\tau\wedge\sigma)+}[\sum_{i=1}^n\xi_iI_{A_i\cap\{\tau\leq \sigma\}}] &=\sum_{i=1}^n\hat{\mathbb{E}}_{(\tau\wedge\sigma)+}[\xi_i]I_{A_i\cap\{\tau\leq \sigma\}}&=\sum_{i=1}^n\hat{\mathbb{E}}_{(\tau\wedge\sigma)+}[\xi_i]I_{\{\tau\leq \sigma\}}I_{A_i}.
[y_{(1\,q)},y_{(2\,l)}]_c =[y_1,[y_{(1\,q)},y_{(2\,l)}]_c]_c+\chi_{(2\,l)}(g_{(2\,q)})[y_{(1\,l)},y_{(2\,q)}]_c &-\chi_{(2\,q)}(g_1)(1-\xi)y_{(2\,q)}y_{(1\,l)}&=[y_1,[y_{(1\,q)},y_{(2\,l)}]_c]_c-3\lambda_{122}\xi^2\chi_{(2\,l)}(g_{(3\,q)})y_{(3\,l)}y_{(3\,q)}&-\chi_{(2\,q)}(g_1)(1-\xi)y_{(2\,q)}y_{(1\, l)}
d s_{T^{1 , 1}}^{2}={\frac{1}{9}}\bigg ( d \psi + \sum_{i = 1}^{2}\cos \theta_{i}d \phi_{i}\bigg )^{2}+{\frac{1}{6}}\sum_{i = 1}^{2}\left( d \theta_{i}^{2}+ \mathrm{s i n}^{2}\theta_{i}d \phi_{i}^{2}\right) \ .
\begin{bmatrix}1 -q_1 0 \ldots 0 -a_0 1 -q_2 \ddots \vdots -a_1 \ddots \ddots 0 \vdots 1 -q_{m-1}-a_{m-2}1 -q_m -a_{m-1}1\end{bmatrix}s = \begin{bmatrix}0 0 \vdots 0 0 1\end{bmatrix}
\frac{1}{n}\Delta P = u^{\frac{\alpha+1}{\alpha}}(\frac{1}{n}|h|^{2}-H^{2})-\frac{1}{n}u^{\frac{1}{\alpha}}\overline{\mathrm{Ric}}(\nu,\lambda\partial_{r}^{T}) &+\frac{1}{\alpha}\bar{g}(\lambda\partial_{r},\nabla \log u)-\frac{1}{n\alpha}u^{\frac{\alpha+1}{\alpha}}|\nabla\log u|^{2}.
\quad\quad\quad\int_{0}^{T}\mathcal{E}(u,u^{-})dt = \int_{0}^{T}\mathcal{E}(u^{+},u^{-})dt - \int_{0}^{T}\mathcal{E}(u^{-},u^{-})dt, &\int_{0}^{T}\mathcal{E}(u^{-},u^{-})dt = \int_{0}^{T}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}(u^{-}(x,t) - u^{-}(y,t))^{2}k(x,y)dxdydt > 0,
s_{2B}(k)=\sum_{j=0}^{k}(\begin{matrix}2j\\j\end{matrix})(\begin{matrix}2j\\j\end{matrix})(\begin{matrix}4j\\ 2j\end{matrix})(\begin{matrix}k+j\\k-j\end{matrix})(-64)^{k-j}=1,-40,2008,-109120,6173656,...
\int_0^t\int_\mathbb{R}g_x^2 w_p^2 dxd\tau \leqslant O(1)\bigg(\|w_p(0)\|^2+ \int_0^t \| w_{px}\|^2 d\tau+ \int_0^t <w_{p\tau}, w_p g^2>_{H^{-1}\times H^1}d\tau\bigg).
(e^{-t[H_{j, A}^m-m+V]}g)(x) = E^{\lambda^0}\bigg[e^{-S_{j,A,V}^m(x,t; \Phi_m(X))}g\big(x+\Phi_m(X)(t)\big)\bigg] j=1,2, (e^{-t[H_{3, A}^m-m+V]}g)(x) = E^{\mu \times \nu^0}\bigg[e^{-S_{3,A,V}(x,t; B, \Psi_m(T))}g\Big(x+B\big(\Psi_m(T)(t)\big)\Big)\bigg].
\mathbb{P}\{\bar{z}'_{a^*,s}\geq Z(a^*)/\beta K+c_{k,s}\}\leq e^{-4\ln k}=k^{-4}, &\mathbb{P}\{\bar{z}'_{n,s_n}\leq Z(n)/\beta K -c_{k,s_n}\}\leq k^{-4}.
\sum_{{\bf{k}}}\Lambda_{{\bf{k}}}({\bf{q}})\frac{-i\langle T a^{\dagger n}_{{\bf{k}}}(-{\bf{q}})a^{\dagger}_{{\bf{k}}^{'}}({\bf{q}}^{'}) \rangle}{\langle T 1 \rangle}= G_{2}({\bf{q}},{\bf{k}}^{'},{\bf{q}}^{'}; n)
H\left( a_{\hbar},a_{\hbar}^{\dagger}\right) =\sum_{k=0}^{d}\sum_{l=0}^{k}\left( \hbar\right)^{k/2}g_{l,k}\left( x\right) \partial_{x}^{l}=\sum_{l=0}^{d}\hbar^{\frac{l}{2}}\left( \sum_{k=l}^{d}\hbar^{\frac{k-l}{2}}g_{l,k}\left( x\right) \right) \partial_{x}^{l}.
p_{j,\mathbb{R}}(x,t;y)=p_{j,\mathbb{R}}(|x-y|,t) =\frac{1}{2\pi}\int_{\mathbb{R}}e^{-t|k-\textrm{sign}(k)\frac{\pi g(s)}{a_j}|^{2s}+i(k-\textrm{sign}(k)\frac{\pi g(s)}{a_j})|x-y|}dk, \,\, g(s)=\frac{1-s}{2}
V_F/D(V_F) \cong \prod_{i=1}^{d+1}\mathbb{Z}/\epsilon_i \mathbb{Z}\otimes_\mathbb{Z}\mathbb{Z}/2 \mathbb{Z}\cong \begin{cases}(\mathbb{Z}/2 \mathbb{Z})^{l+1}d^{(i)}(\mathbb{Z}/2 \mathbb{Z})^{l}.\end{cases}
\beta\left(R(r+s)\right) =\int^{T_2}_{T_1}G(T_2,\tau)f_\uparrow\left(r(\tau)+s(\tau)\right)d\tau &\le f_\uparrow\left(b+4^k c\right)\int^{T_2}_{T_1}G(T_2,\tau)\;d\tau &<b.
\exp \bigg( \int_{\tau_{j+1}}^{\tau_{j+\ell+1}}G_{j+\ell}(v) \: dv \bigg) \geq \bigg( \frac{s}{C_6 \mu}\bigg)^{\ell}\exp \bigg( \sum_{m=1}^{\ell}s (\ell - m) \cdot \frac{a_N}{3 k_N}\bigg) &= \bigg( \frac{s}{C_6 \mu}\bigg)^{\ell}\exp \bigg( \frac{s a_N \ell (\ell - 1)}{6 k_N}\bigg).
P_{k}^{T}= \left. \frac{1 2 8}{3}G^{2}\varepsilon \right|_{k = a H}\approx \left. \frac{1 2 8}{3}G^{2}V \right|_{k = a H}= \left. \frac{1 6}{\pi}G H^{2}\right|_{k = a H},
m(0,z) = - k - \int_0^1 V(t) e^{-2kt}\,dt + \frac 1k \int_0^1 \int_0^{t_1}e^{-2kt_1}(1 -e^{-2kt_2}) V(t_1) V(t_2)\,dt_2\,dt_1 + O(\lvert k\rvert^{-2}).
(d^{[{\sf d}]}_{11},\cdots,d_{KN}^{[{\sf d}]},d^{[{\sf u}]}_{11},\cdots,d_{KN}^{[{\sf u}]})=\lim_{P\to\infty}\left(\frac{R^{[{\sf d}]}_{11}}{\frac{1}{2}\log P},\cdots,\frac{R^{[{\sf d}]}_{KN}}{\frac{1}{2}\log P},\frac{R^{[{\sf u}]}_{11}}{\frac{1}{2}\log P},\cdots,\frac{R^{[{\sf u}]}_{KN}}{\frac{1}{2}\log P}\right)
\|\mathcal E_\lambda^\gamma f\|_{L^q(d\mu)}\le \sum_{j=0}^{n-1}\| \mathcal E_\lambda^\gamma f\chi_{[jh, jh+h ]}\|_{L^q(d\mu)}= \sum_{j=0}^{n-1}(C_{\gamma,j,h})^{\frac1q}\| \mathcal E_\lambda^{\gamma_j}f_j \|_{L^q(d\mu_j)},
\frac{\partial}{\partial \lambda_{1}}\cdots \frac{\partial}{\partial \lambda_{p + q}}\int_{- \infty}^{\infty}\: d s \: \eta \left. T_{\bullet}^{( a_{1})}\cdots T_{\bullet}^{( a_{p})}\: \overline{{{T_{\bullet}^{( b_{1})}}}}\cdots \overline{{{T_{\bullet}^{( b_{q})}}}}\right._{| \lambda_{1}= \cdots = \lambda_{p + q}= 0}\; .
\lambda ( p^{\mu}) \equiv \left( \begin{array}{c}{\left( \zeta_{\lambda}\Theta_{[ j ]}\right) \phi_{_L}^{\ast}( p^{\mu})}\\{\phi_{_L}( p^{\mu})}\\ \end{array}\right) , \quad \rho ( p^{\mu}) \equiv \left( \begin{array}{c}{\phi_{_R}( p^{\mu})}\\{\left( \zeta_{\rho}\Theta_{[ j ]}\right)^{\ast}\phi_{_R}^{\ast}( p^{\mu})}\\ \end{array}\right) .
\begin{array}{c}{\omega_{a f t e r}( A \mathcal{) =}\omega_{b e f o r e}( P A P ) + \omega_{b e f o r e}( \left( 1 - P \right) A \left( 1 - P \right) ) ,}\\{A \mathcal{\in A}, \mathrm{a l g e b r a o f o b s e r v a b l e s}}\\ \end{array}
\langle Pu,\phi \rangle = (u,P\phi)_{L^{2}(\mathbb{R}^{n})}\le \|u\|_{L^{2}(\mathbb{R}^{n})}\|P\phi\|_{L^{2}(\mathbb{R}^{n})}\le C\|u\|_{L^{2}(\mathbb{R}^{n})}\|\phi\|_{H^{2}(\mathbb{R}^{n})}
{\cal L}_{1 / m^{3}}^{( 2 + 1 ) s p i n}= \frac{\alpha}{7 2 0 m^{3}}\left[ 1 8 F^{\nu \lambda}F_{\nu \lambda , \mu}^{\mu}+ 7 F^{\nu \lambda , \mu}F_{\nu \lambda , \mu}- 2 F_{\lambda}^{\nu \lambda ,}F_{\nu \mu ,}^{\mu}\right] ,
u_{s2}(x,t) = \frac{4c_{1}c_{2}\left( c_{1}\lambda_{1}^{2}+ c_{2}\lambda_{2}^{2}+ (c_{1}+c_{2})^{2}c_{1}^{-1}c_{2}^{-1}(c_{1}-c_{2})^{-2}\right)}{\left( c_{1}c_{2}\lambda_{1}\lambda_{2}-(c_{1}+c_{2})^{2}(c_{1}-c_{2})^{-2}\right)^{2}+ \left( c_{1}\lambda_{1}+ c_{2}\lambda_{2}\right)^{2}},
w^{n+1}_i=w^n_i+\frac{\Delta t}{N}\sum_{j=1}^{N}P^n_{ij}(w^n_j-w^n_i)+\Delta t u^n,w^0_i= w_{0i}, & u^n = -\frac{\Delta t}{\nu N}\sum_{j=1}^N(w_j^{n+1}-w_d).
\langle \psi_{j_1,k_1},\psi_{j_2,k_2}\rangle = \langle \tilde{D_c}\psi_{j_1-1,k_1},\psi_{j_2,k_2}\rangle &=\langle \psi_{j_1-1,k_1},\tilde{D_c}^\dagger \psi_{j_2,k_2}\rangle &=-c\C(j_2,k_2) \langle \psi_{j_1-1,k_1}, \psi_{j_2-1,k_2}\rangle .
\|\nabla (\phi_{m}\circ \ldots \circ \phi_1)-\nabla (\phi_n \circ \ldots \circ \phi_1)\|_{L^{q}}\leq \sum_{k=n}^{m-1}\|\nabla (\phi_{k+1}\circ \ldots \circ \phi_1)-\nabla (\phi_k \circ \ldots \circ \phi_1)\|_{L^{q}}\leq C\tau_0\|2f\|_{L^p}^{p/(2q)}\sum_{k=n}^{m-1}(1+\tau_0)^{k[(1+\lambda)(q-1)-(p-1)/2]/q}
\begin{array}{c c}{{\alpha^{\prime}M^{2}= 4 \left( N_{R}- 1 \right) + \alpha^{\prime}Q_{+}^{2}= 4 \left( N_{L}- 1 \right) + \alpha^{\prime}Q_{-}^{2},}}&{{N_{R}- N_{L}= m_{0}w_{0},}}\\ \end{array}
\widehat{Q}_k^{(11)}=\dfrac{\gamma I_n}{r^k}, \widehat{Q}_k^{(22)}= J(I_n+\mathcal{C}_{J}(\gamma I_{n})^{r^{k-1}})(I_n-\mathcal{C}_{J}(\gamma I_{n})^{r^{k-1}})^{-1}.
\sup_{Q'\in \mathcal{D}_\nu}\sum_{Q\in \mathcal{D}_\nu}(1+2^\nu{\rm dist}(Q,Q'))^{N}\|1_Q e^{-tL}1_{Q'}\|_{2\to 2}\leq C e^{M_{0}t}, \ \ \ \N=\lfloor n/2\rfloor+1.
\sup_{f \in \mathcal{F}_{d,\theta}}\biggl| \mathbb{E}_f(\hat{H}_n) - H + \frac{\Gamma(k+2/d)}{2(d+2)V_d^{2/d}\Gamma(k)n^{2/d}}\int_\mathcal{X}\frac{\Delta f(x)}{f(x)^{2/d}}\,dx\biggr| = o\Bigl(\frac{k^{2/d}}{n^{2/d}}\Bigr).
\prod_{\substack{k\in M k\ne k_0,k_0\pm 1}}\left(1-\frac{z}{\mu_k}\right) \left(1-\frac{z}{\lambda_k}\right)^{-1}=\prod_{\substack{k\in M k\ne k_0,k_0\pm 1}}\frac{\lambda_k}{\mu_k}\prod_{\substack{k\in M k\ne k_0,k_0\pm 1}}\frac{z-\mu_k}{z-\lambda_k}\,.
\left(\frac{\sqrt{p}}{2\pi}\right)^{k+t}\Gamma(k+t)\, L(f, k+t)=i^{-2k}\epsilon_f\left(\frac{\sqrt{p}}{2\pi}\right)^{k-t}\Gamma(k-t)\, L(f, k-t) \implies \left(\frac{p}{4 \pi^2}\right)^t \Gamma(k+t)^2 L(f, k+t)^2= \left(\frac{p}{4 \pi^2}\right)^{-t}\Gamma(k-t)^2 L(f, k-t)^.
\sigma( \left(z_1,z_2, \dots, z_k\right) ) = \left(z_1 +1, z_2, \dots, z_k\right) s_i(\left(z_1, z_2, \dots, z_k\right)) = \left(z_{\tau_i(1)},z_{\tau_i(2)}, \dots, z_{\tau_i(k)}\right)
F(a,b,c,z)= ~\sum^\infty_{n=0}\frac{a(a+1)\cdots (a+n-1) b(b+1)\cdots (b+n-1)}{n!c(c+1)\cdots (c+n-1)}z^n, = ~ \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum^\infty_{n=0}\frac{\Gamma(a+n)\Gamma(b+n)}{n!\Gamma(c+n)}z^n,
\frac{1}{8}\mathrm{T r}\left[ \left( M^{- 1}\dot{M}\right)^{2}\right] = \frac{1}{4 d}\left[ \mathrm{T r}\left( G^{- 1}\dot{G}\right) \right]^{2}+ \frac{1}{4}\left\{\mathrm{T r}\left[ \left( G^{- 1}\dot{G}\right)_{t}^{2}\right] - \mathrm{T r}\left[ \left( G^{- 1}\dot{B}\right)^{2}\right] \right\},
( \Delta F )_{2 2}= \frac{g}{2 \pi}\int_{L}d k e^{i k x}[ \frac{M_{0}^{( - )}( k ) \hat{R}_{1}^{( + )}- M_{0}^{( + )}\hat{R}_{1}^{( - )}]}{M_{0}^{( - )}( k ) M^{( - )}( \nu , k )};
\| f \|_{\dot B^{s}_{p,q}}:= \begin{cases}\Bigl(\sum_{N>0}N^{sq}\| P_N f \|_{L^p(\mathbb R^d)}^q \Bigr)^{\frac 1q}, \sup_{N>0}N^{s}\| P_N f \|_{L^p(\mathbb R^d)}, \end{cases}
\begin{aligned}\sigma_{i}^{(1)}={\rm Cone}\left\{\rho_{1},\rho_{2},\rho_{3},\rho_{6-i}\right\}\;(i=1,2) \sigma_{i}^{(2)}{\rm =Cone}\left\{\rho_{j},\rho_{k},\rho_{4},\rho_{5}\right\}\;\;(\{i,j,k\}=\{1,2,3\}).\end{aligned}
\int_{0}^{+\infty}\int_{-\infty}^{+\infty}\triangle_{{\alpha},q}f(x,y)g(x,y) y^{2\alpha+1}d_{q}x d_{q}y=\int_{0}^{+\infty}\int_{-\infty}^{+\infty}\triangle_{{\alpha},q}g(x,y)f(x,y) y^{2\alpha+1}d_{q}x d_{q}y.
\rho^2(x,y,z,\lambda)\begin{bmatrix}\frac{\partial G_1}{\partial \lambda}\frac{\partial G_2}{\partial \lambda}\frac{\partial G_3}{\partial \lambda}\frac{\partial G_4}{\partial \lambda}\end{bmatrix}=M.\begin{bmatrix}G_1 G_2 G_3 J(F)\end{bmatrix},
\exp\left(\sum_{g=0}^\infty x^{2g-2}\mathcal{F}_g\right)=e^{-\frac{D(0,0,0)}{2}\zeta(1)}\prod_{(\ell,n,\gamma,j)>0}(1-p^\ell q^n\zeta^\gamma y^j)^{D(\ell n,\gamma,j)}\,.
\begin{array}{ll}v_{k}=E_{1+k,1}-E_{1,1+k}+E_{1+l+k,1}-E_{1,1+l+k}, 1\leq k \leq l, w_{ij}=E_{1+i,1+j}-E_{1+j,1+i}+E_{1+l+i,1+l+j}-E_{1+l+j,1+l+i}, u_{ij}=E_{1+l+i,1+j}-E_{1+l+j,1+i}+E_{1+i,1+l+j}-E_{1+j,1+l+i}, 1\leq j<i\leq l, \end{array}
\frac{d^2}{dt^2}\frac{\partial L}{\partial \ddot h}+ \nabla^2\frac{\partial L}{\partial\nabla^2h}- \frac{d}{dt}\nabla^2\frac{\partial L}{\partial\nabla^2\dot h}- \frac{d}{dt}\frac{\partial L}{\partial\dot h}- \nabla\frac{\partial L}{\partial\nabla h}+ \frac{\partial L}{\partial h}= 0 \quad .
T_{S \circ\psi}\cong T_S\quad,P_{F\circ \Psi}\cong P_F\quad,T_{\Phi\circ S}\cong \Phi\circ T_S\circ \Phi^{-1}\quad,P_{\Phi\circ F}=\Phi\circ P_F\circ\Phi^{-1}\,;
\P (X_{n+1}=j+1|X_{n}=j) =\frac{f(j)}{n}\times \frac{n}{n+1}=\frac{f(j)}{n+1}, \P (X_{n+1}=j|X_{n}=j) =(1-\frac{f(j)}{n})\times \frac{n}{n+1}=\frac{n-f(j)}{n+1}, \P (X_{n+1}=0|X_{n}=j) =\frac{1}{n+1}.
\left\|\frac{R}{ks}- \mathrm{I}_{nk}\right\| = \left\|\frac{W_A \otimes \mathrm{I_k}}{ks}-\frac{W_{AB}}{ks}- \mathrm{I}_{nk}\right\| &\leq \frac{1}{k}\left\|\frac{W_{AB}}{s}\right\| + \left\|\left(\frac{W_A}{ks}- \mathrm{I}_{n}\right) \otimes \mathrm{I_k}\right\| &\leq \frac{1}{k}\left\|\frac{W_{AB}}{s}\right\| + \left\|\frac{W_A}{ks}- \mathrm{I}_{n}\right\|,
\begin{array}{r l r l}{{\Delta ( z^{\, i}{}_{j})}}&{{z^{\, i}{}_{k}\otimes z^{\, k}{}_{j}}}&{{\Delta ({\bf 1})}}&{{{\bf 1}\otimes{\bf 1}}}\\{{\varepsilon ( z^{\, i}{}_{j})}}&{{\delta^{\, i}{}_{j}}}&{{\varepsilon ({\bf 1})}}&{{1}}\\{{S ( z^{\, i}{}_{j})}}&{{( S ( Z ) )^{\, i}{}_{j}}}&{{S ({\bf 1})}}&{{{\bf 1}}}\\ \end{array}
L_{soft}= \frac{1}{2}\sum_{\alpha}M_{\alpha}{\tilde \lambda}{\tilde \lambda}- \sum_{i}m_{i}^{2}|{\tilde \phi}|^{2}-(A_{ijk}{\tilde Y}_{ijk}{\tilde \phi}_{i}{\tilde \phi}_{j}{\tilde \phi}_{k}+ B_{ij}{\tilde \mu}_{ij}{\tilde \phi}_{i}{\tilde \phi}_{j}+ h.c),
U_n(s)= \sum_{i\in J_w^{[s]}}\tilde{\Delta}_{\theta,i}^{[s]}\prod_{j\in J_w^{[s]}, j\neq i}(f_{g((s-1)\ell_w^*+j)}-\alpha_n) \quad+\prod_{j\in J_w^{[s]}}(f_{g((s-1)\ell_w^*+j)}-\alpha_n)z,
S = \int d^{2}x \left[ -{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}+{\frac{1}{2}}\partial_{\mu}\phi \partial^{\mu}\phi + e A_{\nu}( \eta^{\mu \nu}- \epsilon^{\mu \nu}) \partial_{\mu}\phi +{\frac{1}{2}}a e^{2}A_{\mu}A^{\mu}\right]
\frac{1+1/\tau'}{w_{i}+1}- \sum_{j=1}^{i-1}\frac{w_{j}-1/\tau'}{w_{i}+1}\cdot \frac{\log q_{k_j}}{\log q_{k_i}}\ge \frac{1+1/\tau'}{w_{i}+1}- \sum_{j=1}^{s}\frac{w'_{j}-1/\tau'}{w_{i}+1}\cdot \frac{\log q_{k'_j}}{\log q_{k_i}}.
\hat{z}_{i+1}= \scriptstyle{\cal \prod}\left( \hat{z_{i}}- \eta \cdot \frac{\partial ||y-z.G||^2}{\partial z}\vert_{z=\hat{z}_{i}}\right) = \scriptstyle{\cal \prod}\left( \hat{z_{i}}- 2 \eta \cdot yG^{T}+ \eta \cdot \hat{z}_{i}G G^{T}\right),
\tilde{F}_1(i,v_1,\hat{v}_2(i))= \bigg[\sum_{j\in S}{\psi}_{1}^{\hat{v}_2}(j)\pi_{ij}(v_1,\hat{v}_2(i))+c_1(i,v_1,\hat{v}_2(i)){\psi}_{1}^{\hat{v}_2}(i)\bigg],
\widetilde{J}(p):=\sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p \left[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t\right. &\quad\left.-\exp(-r\tau_I)(k_1P(\tau_I)+k_0)-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\right],
I_0(s) =\sum_{k=0}^{+\infty}(-1)^k\dbinom{\frac{d-2}{2}}{k}\int_a^{+\infty}e^{-su}u^{d-2(k+1)}\, d u &=\sum_{k=0}^{+\infty}(-1)^k\dbinom{\frac{d-2}{2}}{k}s^{2k+1-d}g_{d-2(k+1)}(sa).
\psi_1(u,v,w)= u(\alpha u+(1-\alpha)u^2+\beta w), \psi_2(u,v,w)= v(\alpha v+(1-\alpha)v^2+\beta w), \psi_3(u,v,w)= u(\alpha u+(1-\alpha)v^2+\beta w)\shortintertext{and}\psi_4(v,w)= v(\alpha v+(1-\alpha)u^2+\beta w).
\begin{aligned}\int_{t}^{T_{x, \xi}}C_pq\left| \dot{\xi}(s)\right|^qds= -C_p(q-1)p^q\left(f(\xi(t))-u(\xi(t))\right)+f(\xi(t)) \Longrightarrow \int_{t}^{T_{x, \xi}}p\left(f(\xi(s))-u(\xi(s))\right)ds=u(\xi(t)).\end{aligned}
\mathcal{W}^{\mathrm{U(1)}}=\partial_\mu \frac \delta{\delta A_\mu}+\varepsilon^{ab}\left( A_\mu^a\frac \delta{\delta A_\mu^b}+c^a\frac\delta{\delta c^b}+b^a\frac \delta{\delta b^b}+\overline{c}^a\frac \delta{\delta \overline{c}^b}+A_\mu^{a*}\frac \delta{\delta A_\mu^{b*}}+c^{a*}\frac \delta{\delta c^{b*}}\right) \;.
\begin{aligned}\sqrt{5+x_2^2}- \sqrt{5}= \sqrt{5+s_{21}(x)}- \sqrt{5}+ \sqrt{5+s_{22}(x)}- \sqrt{5+s_{21}(x)}&\geq \frac{\sqrt{8}- \sqrt{5}}{3}s_{21}(x) + \frac{\sqrt{9}- \sqrt{8}}{1}\bigl(s_{22}(x) - s_{21}(x)\bigr),\end{aligned}
W_{\mu}^{T}= W_{\mu}^{F}+ W_{\mu}^{G}= \int d^{4}\! x t r \big ( F_{\mu \nu}( x ){\frac{\delta}{\delta A_{\nu}( x )}}+ D_{\nu}A_{\mu}( x ){\frac{\delta}{\delta A_{\nu}( x )}}\big ) ,
(u \delta_e) (a_g \delta_g) (u \delta_e) (b \delta_e) (u \delta_e) =(u \delta_e) (b \delta_e) (u \delta_e) (a_g \delta_g) (u \delta_e) \Longleftrightarrow(u a_g \delta_g) (ubu \delta_e) = (ubu \delta_e) ( \alpha_g( \alpha_{g^{-1}}(u a_g) u ) \delta_g ) \Longleftrightarrow\alpha_g( \alpha_{g^{-1}}(u a_g) ubu ) \delta_g =ubu a_g \alpha_g(u) \delta_g.
A_{1}\otimes A_{2}=(sym(A_{1})\oplus skew(A_{1}))\otimes(sym(A_{2})\oplus skew(A_{2}))=sym(A_{1})\otimes sym(A_{2}) \oplus skew(A_{1})\otimes skew(A_{2})\oplus skew(A_{1})\otimes sym(A_{2})\oplus sym(A_{1})\otimes skew(A_{2}).
\mid \alpha > = \left( \begin{array}{c}{{\alpha_{0}}}\\{{\alpha_{1}}}\\{{\alpha_{2}}}\\{{\alpha_{3}}}\\{{\vdots}}\\ \end{array}\right) , ~ \mid 0 > = \left( \begin{array}{c}{{1}}\\{{0}}\\{{0}}\\{{0}}\\{{\vdots}}\\ \end{array}\right)
\bar{I}_2=\lim \limits_{P_0/N_0\rightarrow \infty}I_2=\frac{1-\alpha_0}{2\sigma_2^2}\bar{\tilde{B}}_3\exp\left(\frac{\bar{\tilde{B}}_2}{\sigma_2^2}\right)E_1\left(\frac{\bar{\tilde{B}}_2}{\sigma_2^2}\right)
\binom{m+1}{m+1-(s+r)}\binom{s+r}{s+r-a_{1}}\cdots\binom{s+r-a_{1}-\cdots-a_{i-1}}{s+r-a_{1}-\cdots-a_{i-1}-a_{i}}=\frac{(m+1)!}{(m+1-s-r)!a_{1}!\cdots a_{i}!s!}.
\begin{aligned}\|C - CZ_{j}Z_{j}^{T}\|_F = \|C - C(\hat{V}_j - \bar{E})(\hat{V}_j - \bar{E})^{T}\|_F & \leq \|C - C\hat{V}_{j}\hat{V}^{T}_{j}\|_F + \|C\bar{E}\hat{V}^{T}_{j}\|_F + \|C \hat{V}_{j}\bar{E}^{T}\|_F + \|C\bar{E}\bar{E}^{T}\|_F & < \|C - C\hat{V}_j \hat{V}_j^{T}\|_F + (2\xi \sqrt{1+\xi}+ \xi^2) \|C\|_F.\end{aligned}
\begin{array}{lll}\overline{\mathrm{Ric}}(x,y) = \mathrm{Ric}(x,y)+[\beta^2-\beta+(1-2n)\alpha^2]g(x,y) +[(2n+1)\beta-\beta^2+(2n-1)\alpha^2]\eta(x)\eta(y) +[(\alpha\beta-\alpha)(2-2n)+\alpha] g(x,\varphi y) \end{array}
\prod_{(i,j)\in \lambda}P(\vec a; i-j+\sigma) =\frac{G(\vec a;K+1+\sigma)}{G(\vec a;1+\sigma)}\prod_{i=1}^{K}\frac{1}{\Gamma(\vec{a};K-(L_i-\sigma))}\prod_{(i,j)\in \mu}P(\vec a; i-j-\sigma) =\frac{G(\vec a;K+1-\sigma)}{G(\vec a;1-\sigma)}\prod_{i=1}^{K}\frac{1}{\Gamma(\vec{a};K-(M_i+\sigma))}\ .
\begin{array}{l l}{d e t_{q}L ( u ) =}&{a ( u + i \kappa / 2 ) d ( u - i \kappa / 2 ) - b ( u + i \kappa / 2 ) c ( u - i \kappa / 2 )}\\{}&{= Q_{1}u^{2}+ Q_{2}u - Q_{3}+ \kappa^{2}/ 4 Q_{1}\quad ,}\\ \end{array}
\dot Y_n (t) =\begin{cases}\cfrac{b(Y_n (t))}{\varepsilon}- (-1)^i \nu \, \cfrac{DH(Y_n (t))}{|DH(Y_n (t))|}\ \ \ t \in (0, s_n), \cfrac{b(Y_n (t))}{\varepsilon}+ \alpha_n (t - s_n) \ \ \ t \in (s_n, t_n),\end{cases}
\sum_{k=0}^n \binom{n}{k}\frac{2^k}{k+1}\left((-1)^k\frac{F_{j(k+1)}}{L^k_j}\Bigl( \frac{L_j}{\sqrt5F_j}\Bigr)^n - (1+(-1)^n) \frac{2^{k+3}-2}{k+2}F_j B_{k+2}\right)=0\,.
\|K(t)\|_{L^{\infty}}\leq C \|K(t)\|_{L^{2}}^{\frac{1}{2}}\|K(t)\|_{\dot{H}^{2}}^{\frac{1}{2}}\leq Ct^{-\frac{1}{4}}g(A_{t})^{-\frac{1}{4}}t^{-\frac{3}{4}}g(A_{t})^{-\frac{3}{4}}= Ct^{-1}g(A_{t})^{-1}.
\mathbf{x}^{(k)}=\mathbf{r}^{(k)}+\mathcal{B}^{(k)}\left[\widetilde{\mathcal{H}}^{(k)}\left[\mbox{soft}\left[\mathcal{H}^{(k)}\left(\mathcal{M}^{(k)}\left(\mathbf{r}^{(k)}\right)\right), \theta^{(k)}\right]\right]\right]
u(t) \,\,\, \leq \int_{0}^{t}\bigg(2\sum_{i=1}^{\infty}\sum_{j=1}^{i}j V_{i,j}|u_j|\,\psi_i + \sum_{i=1}^{\infty}\sum_{j=i}^{\infty}V_{i,j}|u_j|\, \psi_i \bigg)(h) dh.
\bigl[ s [y_i, x_j], y_{i+1}\dots y_{i' - 1}\bigr] = \sum_{m=i+1}^{i'-1}y_{i+1}\dots y_{m-1}\bigl[ s [y_i, x_j], y_m \bigr] y_{m+1}\dots y_{i'-1}= \ \sum_{m=i+1}^{i'-1}y_{i+1}\dots y_{m-1}\bigl( [s, y_m] [y_i, x_j] + s[y_i, x_j, y_m] \bigr) y_{m+1}\dots y_{i'-1}
p_{B,n + 1,i}^{(1)}= \sum\limits_{k = 0}^N{\;\varpi_{N,k}^{(0.5)}{\mkern 1mu}{\mathcal{L}}_{B,n,i}^{(\alpha )}\left({x_{N,k}^{(0.5)}; - 1,x_{n,n + 1}^{(\alpha )}}\right)},i = 0, \ldots ,n,
\mathbf{C}_{k,q,p}= \tilde{\mathbf{H}}_{k,q,p}^{\dagger}(\tilde{\mathbf{H}}_{k,q,p}^{\dagger})^H = ( \mathbf{P}_k^H\mathbf{U}_k^H\mathbf{\Lambda}_k^{\frac{1}{2}}\mathbf{G}_{k,q,p}^H \mathbf{G}_{k,q,p}\mathbf{\Lambda}_k^{\frac{1}{2}}\mathbf{U}_k \mathbf{P}_k)^{-1}= ( \mathbf{P}_k^H \mathbf{R}_k \mathbf{P}_k)^{-1}.
P(\sum_{j=1}^n X_j > x) \ge \sum_{k=1}^nP(X_k > A, \sum_{j=1}^n X_j > x)-\sum_{1 \le k < l \le n}P(X_k > A, X_l > A,\sum_{j=1}^n X_j > x) =nJ_1(x) + nJ_2(x)-2^{-1}n(n-1)P(X_1 > A, X_2 > A,\sum_{j=1}^n X_j > x) \ge nJ_1(x) -2^{-1}n(n-3)J_2(x)-2^{-1}n(n-1)J_3(x).
\frac{1}{G}_{i\overline{j}}^{g}g^{i\overline{q}}g^{p\overline{j}}h_{p\overline{q}}\ = \ \frac{1}{G}\langle^{g},h\rangle_{g}\ = \ \frac{1}{G}\langle-g+th-\frac{\alpha}{2\beta}dd^c \varphi_t, h\rangle_{g}&\ = \ \frac{1}{G}\langle-g+th, h\rangle_{g}-\frac{1}{G}\frac{\alpha}{2\beta}\langle \varphi_t, h\rangle_{g}.
\Lambda_{k+m+1}\le M{\textstyle\sum_{j=k}^{k+m}}\,\Xi_j^{\frac{1-\theta}{\theta}}\le M(m\!+\!1)^{\frac{2\theta-1}{\theta}}\big[{\textstyle\sum_{j=k}^{k+m}}\,\Xi_j\big]^{\frac{1-\theta}{\theta}}\le M(m\!+\!1)^{\frac{2\theta-1}{\theta}}\big(\Lambda_{k}\!-\!\Lambda_{k+m+1}\big)^{\frac{1-\theta}{\theta}},