harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int\gamma D(F)(m)d\nu_j = \int_{T_1}\gamma D(F)(m)d\nu_j+\int_{T_2}\gamma D(F)(m)d\nu_j & = \gamma_c\int_{T_1}D(F)(m)d\nu_j-\gamma_c\int_{T_2}D(F)(m)d\nu_j & = \gamma_c\int_{L_1}D(F)(m)d\mu_j-\gamma_c\int_{L_2}D(F)(m)d\mu_j & = e^{-i\theta}\int \gamma D(F)(m)d\mu_j
\widetilde{W}_{N_1}=\widetilde{W}( :, [1\, 3 \, 4 \, 2 \, 5 \, 6]),\widetilde{W}_{N_2}=\widetilde{W}( :, [2\, 3 \, 4 \, 1 \, 5 \, 6]), \widetilde{W}_{N_3}=\widetilde{W}( :, [1\, 5 \, 6 \, 2 \, 3 \, 4]),\widetilde{W}_{N_4}=\widetilde{W}( :, [2\, 5 \, 6 \, 1 \, 3 \, 4]).
I_{m}= \int d^{4}x \sqrt{{g}}[ - \frac{1}{2}D_{\mu}\phi ( D^{\mu}\phi )^{*}- \frac{1}{2}D_{\mu}\Sigma ( D^{\mu}\Sigma )^{*}- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{4}H_{\mu \nu}H^{\mu \nu}- V ( | \phi | , | \Sigma | , \Lambda ) ] ,
h(x)= \left\lbrace\begin{array}{cl}|x-\frac{1}{2}|, 0\leq x\leq 1 |x-\frac{3}{2}|, 1<x\leq 2 \end{array}\right.K(x)=\left\lbrace\begin{array}{cl}[-\frac{3}{2}x+\frac{3}{2},2], 0\leq x\leq 1 \left[0,-\frac{3}{2}x+\frac{7}{2}\right], 1<x\leq 2 \end{array}\right.
m \int_0^t \sum_x \psi_x^{K_n}(\eta_s^n) f' \big( \tfrac{x}{n}\big) ds,\footnote{We actually proved this with $m$ replaced by $\sum_{y=1}^{K_n-1}y a(y)$, but this last sum converges to $m$ as $n \to \infty$.}
\sum_{k=0}^{\infty}\frac{1}{(2k)!}=\frac{1}{0!}+\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+\frac{1}{8!}+\cdot\cdot\cdot=\frac{1}{2}(e+\frac{1}{e})=cosh1
\sum_{1\leq i,j\leq n}\|\mathbf x_i-\mathbf y_j\|^2 = \sum_{1\leq i<j\leq n}\|\mathbf x_i - \mathbf x_j\|^2 + \sum_{1\leq i< j\leq n}\|\mathbf y_i- \mathbf y_j\|^2 + n^2\|\mathbf x -\mathbf y\|^2,
(S^{i,j}f(y))^{2}= \sum_{\substack{R_1 \in \mathcal{D}_1 R_2 \in \mathcal{D}_2}}\bigg(\sum_{\substack{P_1 \in (R_1)_{i_1}P_2 \in (R_2)_{i_2}}}|\hat{f}(P)| \bigg)^{2}\bigg(\sum_{\substack{Q_1 \in (R_1)_{j_1}Q_2 \in (R_2)_{j_2}}}\frac{1_{Q_1}}{|Q_1|}\otimes \frac{1_{Q_2}}{|Q_2|}(y) \bigg).
{{f}_{{{I}_{x}}}}\left({{I}_{x}}\right)=\frac{{{\gamma}_{1}}I_{x}^{{{m}_{1}}{{\gamma}_{1}}-1}}{{{\left({{{\Omega}_{1}}}/{{{m}_{1}}}\; \right)}^{{{m}_{1}}}}\Gamma \left({{m}_{1}}\right)}\exp \left( -\frac{{{m}_{1}}}{{{\Omega}_{1}}}I_{x}^{{{\gamma}_{1}}}\right)
\sum_{k\in S_{m,N}}|x_k|^{p_1}\leq \left(\sum_{k\in S_{m,N}}|x_k|^{p_2}\right)^\frac{p_1}{p_2}\left(\sum_{k\in S_{m,N}}1\right)^{1-\frac{p_1}{p_2}}&= |S_{m,N}|^{1-\frac{p_1}{p_2}}\left(\sum_{k\in S_{m,N}}|x_k|^{p_2}\right)^\frac{p_1}{p_2}&= |S_{m,N}|\left(\frac{1}{|S_{m,N}|}\sum_{k\in S_{m,N}}|x_k|^{p_2}\right)^\frac{p_1}{p_2}.
\Big| \sum_{j=0}^{n-1}\phi \circ F_j - \sum_{j=0}^{n-1}\phi \circ f^j \circ h^{-1}\Big| \le \sum_{j=0}^{n-1}\| \phi_j -\phi\|_{C^0}\le |\phi|_\alpha L^\alpha \sum_{j=0}^{n-1}a_j^\alpha
\widehat{E}_{C_2}= E_{C_2}[1/\tau] = \left( E_{C_2}\overset{\tau}{\longrightarrow}\Sigma^{\sigma-|\sigma|}E_{C_2}\overset{\tau}{\longrightarrow}\Sigma^{2\sigma - |2\sigma|}E_{C_2}\overset{\tau}{\longrightarrow}\cdots \right)
\frac{m^{L/m}}{\Gamma(L)^{1/m}}\leq \frac{(CL)^{1/C}}{\Gamma(L)^{1/(CL)}}=\frac{C^{1/C}L^{1/(2CL)}e^{1/C}}{(2\pi)^{1/(2CL)}}[1+\textrm{o}(1)]\leq (eC)^{\frac 1C}K^{\frac 1{2C}},
L(s, \operatorname{sym}^2 f)= \zeta(2s) \sum_{n \geq 1}\frac{\lambda(n^2)}{n^s}=\prod_{p}\left( 1-\frac{\lambda(p^2)}{p^s}+\frac{\lambda(p^2)}{p^{2s}}-\frac{1}{p^{3s}}\right)^{-1}.
Y_n(\frac{L_{t_0}}{L_t},t)=\sqrt{\frac{L_{t_0}}{L_t}}\frac{1}{\sigma_t\sqrt{N_4(t,h_n)}}\sum_{k=0}^{\lfloor N_4(t,h_n) \rfloor-1}\big(\phi(f_t^k(c)) -\int \phi \d\mu_t\big).
\hat{\Omega}= i\hbar\Delta_\psi+{\cal J}_i\frac{\delta}{\delta\varphi^{*}_i}+L_m\sigma^m_{,A}\bigg(\frac{\delta{\cal W}}{\delta{\cal J}}+\frac{\hbar}{i}\frac{\delta}{\delta{\cal J}},\psi\bigg) \frac{\delta}{\delta\phi^{*}_A}.
E_{\frac{n-m}{2}}^{(n)}(Z,s)= F_{n,m}(s)\cdot E_{\frac{n-m}{2}}^{(n)}\left(Z,\tfrac{m+1}{2}-s\right), F_{n,m}(s)=\gamma_{n,m}(s)\cdot\xi_{n,m}(s).
{\cal C}_k =\Big( \displaystyle\sum_{\sigma \subset U \atop |\sigma|=k/2}\prod_{u_i \in \sigma}(d_i)(d_i-1) \Big)\Big( \displaystyle\sum_{\sigma \subset W \atop |\sigma|=k/2}\prod_{w_i\in \sigma}(d_i')(d_i'-1) \Big)\Big( \dfrac{(\frac{k}{2})!(\frac{k}{2})!}{k}\Big)\:,
g(\pi) = \sum_{i=1}^{n}g_i(\pi) = \sum_{i=1}^{k-t+2}g_i(\pi) + \sum_{i=k-t+3}^{n}g_i(\pi) = \sum_{i=1}^{k-t+2}\bigl( h_i(\pi_1 \dotsm \pi_k) - h'_i(\pi_1 \dotsm \pi_k) \bigr) + \sum_{i=k-t+3}^{n}g_i(\pi)
\begin{aligned}\big(\sigma \partial_t^{1/2}\boldsymbol{y},\partial_t^{1/2}\boldsymbol{v}\big) = \big(\sigma \partial_t \boldsymbol{y},\boldsymbol{v}^{\perp}\big) \mbox{and}\big(\sigma \partial_t^{1/2}\boldsymbol{y},\partial_t^{1/2}\boldsymbol{v}^{\perp}\big) = \big(\sigma \partial_t \boldsymbol{y},\boldsymbol{v}\big) \end{aligned}
\alpha_{n, M}(1)=\sum_{\ell \equiv n \bmod M}\frac{a(\ell)}{\ell}V\left(\frac{X}{2\pi \ell}\right) + \mathcal{O}\bigg(M^{\frac{1}{2}+2\epsilon+\epsilon'}\sigma_0(M)^2 X^{-\frac{1}{2}-\epsilon}q^{\frac{1}{2}+\epsilon}\prod_{\substack{p\mid \gcd(q, M) p^2\mid q}}p\bigg).
K_1 = \left\lceil \frac{1}{\alpha}\left(\frac{1}{t \epsilon}\left( \frac{1}{2}\|x_0-x_*\|_v^2 + \xi_k \right) -1\right) \right\rceil, K_2 = \left\lceil \frac{2 c}{\alpha\epsilon}\log\left(\frac{t}{\rho}\right)\right\rceil.
\forall n^0,x^X,y^X\exists z^X\preceq_X L1_X\exists w^X\preceq_XL2^{\alpha_n+1}1_X &\qquad\qquad\qquad\left(y\in A(\tilde{x}^L)\rightarrow (w\in Az\land \tilde{x}^L=_Xz+_X\gamma_n w)\right)
\Lambda_{N_j}^{\alpha,\beta}(f_{jk}(z_k,s),s)=\sum_{a,b=1}^n\Lambda_{N_k}^{a,b}(z_k,s)\frac{\partial f_{jk}^{\alpha}}{\partial z_k^a}\frac{\partial f_{jk}^{\beta}}{\partial z_k^b}
D^{2r+1}\Bigl(\frac{L_{r}(\lambda)}{H_\lambda}\Bigr) =0, (r\geq 0) D^{2r}\Bigl(\frac{L_{r}(\lambda)}{H_\lambda}\Bigr) = \frac{(2r-1)!!}{H_\lambda}, (r\geq 1) D^{2r-1}\Bigl(\frac{L_{r}(\lambda)}{H_\lambda}\Bigr) = \frac{(2r-1)!!}{H_\lambda}|\lambda|. (r\geq 1)
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}.
s_n(\nu|\mu) := \sum_{\eta \in \Omega_{\Lambda_n}}\sum_{\Delta \subset \Tilde{\Lambda}_n}\sum_{\xi_{\Delta}\neq \eta_{\Delta}}f(\nu, \Lambda_n, \eta, \xi_{\Delta})\hat{c}^{B_{n-1}(x(\Delta))}_{\Delta}(\xi_{\Delta}\eta_{\Delta^c},\eta_{\Delta}),
- e_{A}E_{t}\frac{\kappa}{r}+ \frac{e_{A}}{r}\, ( \vec{\Theta}\gamma^{t s}\vec{\Theta}) E_{s}+ \frac{E_{t}}{2 r}\, \left( \vec{\Theta}_{\hat{\alpha}A}( \vec{\Theta}_{\hat{\alpha}}\cdot \vec{e}) - ( \vec{\Theta}_{\hat{\alpha}}\cdot \vec{e}) \vec{\Theta}_{\hat{\alpha}A}\right) \, .
{}\, =(\psi\otimes\eta)\bigl((c^*\otimes d^*)z(1\otimes b)\bigr) &=(\psi\otimes\eta\otimes\varphi)\bigl((c^*\otimes d^*\otimes1)\Delta_{13}(r^*y)\Delta_{23}(s)(1\otimes b\otimes1)\bigr) &=(\psi\otimes\eta\otimes\varphi)\bigl((c^*\otimes d^*\otimes1)\Delta_{13}(r^*y)(1\otimes E)\Delta_{23}(s)(1\otimes b\otimes1)\bigr),
\int_{\partial B_R(0) \cap \Omega}x \times u\, dS = \int_{\partial B_R(0) \cap \{x_n < 0\}}x \times \nabla \left(\frac{p\cdot x}{|x|^n}\right)\, dS + O\left( \frac{1}{R^{1+\varepsilon}}\right).
A_{2}=\left[\begin{array}{ccc|ccc|cc}1 0 0 1 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 \hline0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 \hline1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1\end{array}\right]
A_{C_{5}}=\begin{cases}\#M_{p}^{-}(a,c) (p^{e-2d-1}+p^{m-d-1}), \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=-1, \#M_{p}^{+}(a,c) (p^{e-2d-1}+p^{m-d-1}), \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=1.\end{cases}
\begin{array}{c}\mathbb{C}T^*X = T^{*1,0}X \oplus T^{*0,1}X \oplus \left\{\lambda \omega_0: \lambda \in \mathbb{C}\right\}, \mathbb{C}TX = T^{1,0}X \oplus T^{0,1}X \oplus \left\{\lambda T: \lambda \in \mathbb{C}\right\}.\end{array}
\mathrm{S o l v}\left( \frac{\mathrm{U S p}( 3 , 3 )}{\mathrm{S U}( 3 ) \times{U}( 1 )}\right) \subset \mathrm{S o l v}\left( \frac{\mathrm{S U}( 3 , 3 )}{\mathrm{S U}( 3 ) \times \mathrm{S U}( 3 ) \times \mathrm{U}( 1 )}\right) .
\underset{k\rightarrow\infty}{\lim\sup}F_{w}^{-}\left( u_{k}\right)=\underset{k\rightarrow\infty}{\lim\sup}F_{w}\left( u_{k},v_{k}\right)\leq\underset{k\rightarrow\infty}{\lim\sup}F_{w}\left( u_{k},v_{0}\right)\leq F_{w}\left( u_{0},v_{0}\right) =F_{w}^{-}\left( u_{0}\right) .
|\Lambda(f_1, \ldots, f_{n+1}) | \lesssim \|\mathrm{M}(f_1,\ldots, f_{n+1})\|_{1}, \mathrm{M}(f_1,\ldots, f_{n+1})(x) = \sup_{x\in Q}\prod_{j=1}^{n+1}\langle |f_j| \rangle_Q.
\limsup_{k\to\infty}\frac{Y_{nk}}{\log{k}}\leq \frac{1}{\log 1/p}, \limsup_{k\to\infty}\frac{\max_{1\leq i\leq nk}Y_i}{\log{k}}\leq \frac{1}{\log 1/p}, \limsup_{k \to \infty}\frac{\sum_{i=1}^{nk}Y_i}{k \log k}\leq \frac{2n}{\log 1/p}.
X_1\times X_2\times\cdots\times X_s=\sum\limits_{n=0}^{\infty}\begin{pmatrix}\sum\limits_{n_1+\cdots+n_s= n}\frac{n!}{n_1!\times n_2!\cdots n_s!}x_{n_1}^{1}\times x_{n_2}^{2}\cdots x_{n_s}^{s}\end{pmatrix}\frac{m^n}{n!}
D\left( G\left( \mathcal{N}\right) \right) :=\left\{u\in L^{2}\left(m\right) :\sum_{n=0}^{\infty}\left\vert G\left( n\right) \right\vert^{2}\left\vert \left\langle u,\Omega_{n}\right\rangle \right\vert^{2}<\infty\right\}
\|H(f)\|_{\dot{B}^{\alpha,q}_{p,\mathcal F}}\lesssim \bigg\{\sum_{k\in \Bbb Z}\Big(2^{k\alpha}\sum_{k'\in \Bbb Z}2^{-|k-k'|\varepsilon'}\|D_{k'}T_N^{-1}(f)\|_{L^p_\mu}\Big)^q\bigg\}^{1/q}&\lesssim \bigg\{\sum_{k'\in \Bbb Z}\Big(2^{k'\alpha}\|D_{k'}T_N^{-1}(f)\|_{L^p_\mu}\Big)^q\bigg\}^{1/q}.
\mathfrak{F}^3_g(X)=\sum_{\rho\in \mathcal{I}}\sum_{\rho'\in \mathcal{I}}c_{\lambda\lambda}^\rho\frac{|C_\rho|}{|C_\lambda||C_\lambda|}c_{\rho\lambda}^{\rho'}\frac{|C_{\rho'}|}{|C_\rho||C_\lambda|}\mathfrak{F}_{g_{\rho'}}(X),
\begin{array}{c c}{{\displaystyle{\langle V^{A}( z_{1}) V^{B}( z_{2}) \rangle = C_{V}\frac{I^{A B}( \hat{{\cal X}}_{1 2})}{( \det{\cal X}_{1 2})^{\frac{1}{2}\eta}}}~}}&{{~ I^{A B}( \hat{{\cal X}}_{1 2}) = \textstyle{\frac{1}{4}}\mathrm{t r}( \gamma^{A}\hat{{\cal X}}_{1 2}\gamma^{B}\hat{{\cal X}}_{2 1})}}\\ \end{array}
\Delta m =(m\otimes m)(B\otimes c\otimes B)(\Delta \otimes \Delta ), \Delta u =(u\otimes u)\Delta_{\mathbf{1}}, \varepsilon m =m_{\mathbf{1}}\left( \varepsilon \otimes \varepsilon \right), \varepsilon u =\mathrm{Id}_{\mathbf{1}}.
S(q)\approx\frac{S(q_{P})}{1+\frac{r}{(1-r)^{2}}\frac{\Delta q^{2}a^{2}}{2}}\approx\frac{S(q_{P})}{1+\frac{\Delta q^{2}}{[q_{P}^{2}\sigma_{2}^{2}/a]^{2}/2}}
\mathrm{tr}\;P=\frac{1}{4}\left( f_{1}^{2}+2f_{2}g_{1}+g_{2}^{2}\right) =\frac{1}{4}\left(f_{1}^{2}+2f_{1}g_{2}+g_{2}^{2}-2f_{1}g_{2}+2f_{2}g_{1}\right) =\frac{1}{4}\left[ \left( \mathrm{tr}\;A\right)^{2}-2\det A\right] ,
\varphi_3(\varepsilon,b,\eta,n,\varphi) :=\min\{\varphi_1(\varepsilon/3,b,n,\varphi),\varphi_1(\eta(\min\{\varepsilon/3b,2\}) \varepsilon/4,b,n,\varphi), &\qquad\qquad\qquad\varphi_1(\varepsilon/4,b,n,\varphi),\lambda_0/2\}
\Omega_2^j(x) := \Big\{\eta \in \Omega: \overleftarrow{\eta}^{\ell}(x)\geq \frac{2}{\ell}\Big\}\cup \Big\{\eta \in \Omega: \overleftarrow{\eta}^{\ell}(x- j \ell)\geq \frac{2}{\ell}\Big\}.
\frac{d}{dt}\| \bar{w}\|_{H^{m-2, m-3}_{k+\ell}(\mathbb{R}^2_+)}^2+ \|\partial_y\bar{w}\|_{H^{m-2, m-3}_{k+\ell}(\mathbb{R}^2_+)}^2\le \bar C_1\| \bar{w}\|_{H^{m-2}_{k+\ell}}^2,
e^{s(q_j + C_9)(\xi_j^- - \tau_j)}= \frac{e^{-b}}{sq_j}\exp \bigg( \frac{C_9}{q_j}\log \bigg( \frac{1}{sq_j}\bigg) - \frac{C_9 b}{q_j}\bigg) \leq \frac{2 e^{-b}}{sq_j}.
~\begin{aligned}J_2 =\iint_{[-\infty,\infty]\times[y_{0}+\epsilon,\infty]}\biggl[\frac1{(a+xw_1+yw_2)^k}&\quad-k\frac{w_1}{(a+xw_1+yw_2)^{k+1}}P_{1}(x) &\quad-k\frac{w_2}{(a+xw_1+yw_2)^{k+1}}P_{1}(y) &\quad+k(k+1)\frac{w_1w_2}{(a+xw_1+yw_2)^{k+2}}P_{1}(x)P_{1}(y)\biggl]dxdy,\end{aligned}
\phi_{a}( x_{0}, \psi_{0}, \bar{\psi}_{0}, \tau_{a}) = \int_{x_{0}, \psi_{0}, \bar{\psi}_{0}}{\cal D}x_{d}^{I}{\cal D}\psi_{\alpha}{\cal D}\bar{\psi}_{\dot{\alpha}}V_{a}e^{- S ( x_{d}^{I}, \psi_{\alpha}, \bar{\psi}_{\dot{\alpha}})}
\mathfrak{J}:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}),\ \mathfrak{J}:=\sum_{n,k}u^n_kP^n_k,\ (u^n_k)^2=1,\ \forall k\in\{0,1,...,n-1\},\n\in\mathbb{N}^*,
T_{0 0}= K ( -{\cal B}( x^{0}) +{\cal A}_{0 0}( x^{0}) ) = -{\cal T}_{p}~ , ~ T_{i j}= K ({\cal B}( x^{0}) +{\cal A}_{i j}( x^{0}) ) = \delta_{i j}{\cal T}_{p}f ( x^{0}) ~ .
\mathbf{A}_{B}^{A}=\omega_{B}^{A}-\Omega_{B}^{A}\mathbf{m}\text{;}\mathbf{A}_{B^{\prime}}^{A^{\prime}}=\omega_{B^{\prime}}^{A^{\prime}}-\Omega_{B^{\prime}}^{A^{\prime}}\overline{\mathbf{m}}
\begin{array}{c}\bar{N}= \{N,+,\times,0,1\},\mbox{the natural numbers}\bar{R}=\{R,\pm,\times,(-)^{-1},<,0,1\}, \mbox{the real numbers}\bar{C}=\{C,\pm,\times,(-)^{-1},^{*},0,1\}\mbox{the complex numbers}.\end{array}
F\left( \beta \right) =E_{0}+\widetilde{F}\left( \beta \right) =-\frac{\pi^{2}a}{6\beta^{2}}-\frac{1}{2\beta}\log \left( \frac{\beta}{2\pi a}\right) -\frac{1}{\beta}\sum^{\infty}_{p,\, n=1}\frac{1}{n}\, e^{-4\pi^{2}\, n\, p\, a/\beta}.
K(g) = -\frac{{-a^2+b^2+c^2}}{2bc}\, \zeta^1 \odot \bar \zeta^1 - \frac{a^2-b^2+c^2}{2ac}\, \zeta^2 \odot \bar \zeta^2 -\frac{a^2+b^2-c^2}{2ab}\, \zeta^3\odot \bar \zeta^3\,.
[ | u_i | = d_U ] = \Lambda_{d_U}\approx{M \beta \choose d_U}\left( \frac{1}{N}\right)^{d_U}\left( 1 - \frac{1}{N}\right)^{M \beta - d_U}& = \frac{\left( \frac{M}{N}\beta \right)^{d_U}}{d_U !}e^{-\frac{M}{N}\beta}, \; 1 \leq d_U \leq M,
S_5=\int \!\!d^3x~\left( \alpha_{abcde}\varphi_a\varphi_b\varphi_c\varphi_d\varphi_e\right) ,\quad S_6=\int\!\!d^3x~\left( \alpha_{abcdef}\varphi_a\varphi_b\varphi_c\varphi_d\varphi_e\varphi_f\right) ,
\mathcal{J}^{+}_1(N_0\xi,m,q) = \int_0^\infty U(y) \,y^{-it}\, e\left(\frac{2\sqrt{mNy}}{p_1q \sqrt{p_2}}+\frac{3(NN_0 \xi (y+w))^{1/3}}{p_1qr^{1/3}}\right) dy.
G_{k,l}- G_{k+1,l}=\sum^{n}_{j=l}\frac{{{j-1}\choose l-1}}{{{n-1}\choose l-1}}\sum^{m}_{i=k}\frac{(i-1)(i-2)...(i-k+1)}{(m-1)(m-2)...(m-k+1)}{\bigg (}1-\frac{i-k}{m-k}{\bigg )}P(S< i, T<j).
\begin{aligned}\int_{\Gamma_h^{\pm}}e^{\rho\cdot \mathbf x}\mathrm d \sigma =\frac{\Gamma(1)}{\tau}\frac{1}{(\mathbf d+\mathrm i\mathbf d^{\perp})\cdot \mathbf{\hat x_1}}-I_{R_1}+\frac{\Gamma(1)}{\tau}\frac{1}{(\mathbf d+\mathrm i\mathbf d^{\perp})\cdot \mathbf{\hat x_2}}-I_{R_2}, \end{aligned}
\sum_{t=1}^{T}\sum_{i=1}^n(G_1+G_2\|q_{i,t+1}\|)\|\epsilon^z_{i,t}\|\le\sum_{t=1}^{T}\sum_{i=1}^n\Big(\frac{2G_1^2\alpha_{t}}{\sigma}+\frac{2G_2^2\alpha_{t}\|q_{i,t+1}\|^2}{\sigma}+\frac{\sigma\|\epsilon^z_{i,t}\|^2}{4\alpha_{t}}\Big).
{}_3\phi_2\begin{bmatrix}q^{-N}, c/b, t ;q,q at, q^{1-N}/b\end{bmatrix}=\frac{(c,t;q)_N}{(b, at; q)_N}{}_3\phi_2\begin{bmatrix}q^{-N}, a, b ;q,q c, q^{1-N}/t\end{bmatrix}.
[\Delta k(a\rightarrow j)]=\int_{0}^{\infty}\Delta k(a\rightarrow j)d\omega=\frac{\pi^{2}}{\hbar}(N_{a}-N_{j})(\frac{\alpha^{2}}{n})(|\langle a|m_{-}|j\rangle|^{2}-|\langle a|m_{+}|j\rangle|^{2})
\mu_i(\overline{B}(x,\gamma_i))=e^{-m_i(x)s}\leq\sum_{y\in E_{i+1}(x)}e^{-m_{i+1}(y)s+f_{m_{i+1}}(y)}\leq(1+2^{-i-1})\mu_i(\overline{B}(x,\gamma_i)),
\vert g(0)-g(s_b)\vert /\vert s_b\vert \le \left\vert \sum^{+\infty}_{n=1}\frac{d^n g}{d s^n}(0) s^{n-1}_b/n!\right\vert\le C_{R_0}\sum_{n=1}^{+\infty}\left(\sigma^{(1)}_{\theta_{\hat{x}}}\right)^{-n}\vert s_b\vert^{n-1}\le C_{R_0}.
c^{\dagger}_{{\bf{k+q/2}}}c_{{\bf{k-q/2}}}= O({\bf{k}})\mbox{}\delta_{{\bf{q}},{\bf{0}}}+[\sqrt{n_{{\bf{k}}+{\bf{q}}/2}}A_{{\bf{k}}}(-{\bf{q}}) + A^{\dagger}_{{\bf{k}}}({\bf{q}})\sqrt{n_{{\bf{k}}-{\bf{q}}/2}}]
\sigma A^{2}r^{2}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\left(1-\sigma(q_{i}\odot_{k}q_{j})\right)}{(\sigma-\sigma(q_{i}\odot_{k}q_{j})^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\}.
\frac{E_{2n-1}(x)}{(2n-1)!}= \frac{4 (-1)^n}{\pi^{2n}}\times \sum_{k \geq 0}\frac{\cos\left((2k+1) \pi x\right)}{(2k+1)^{2n}}\frac{E_{2n}(x)}{(2n)!}= \frac{4 (-1)^n}{\pi^{2n+1}}\times \sum_{k \geq 0}\frac{\sin\left((2k+1) \pi x\right)}{(2k+1)^{2n+1}}.
d(\mu_{n})=\sum_{\substack{p+q=n+1 p,q\geq 2}}\mu_{p}\{\mu_{q}\}={}\sum_{\substack{p+q=n+1 2\leq p<q}}[\mu_{p},\mu_{q}] &+\mu_{\frac{n+1}{2}}\{\mu_{\frac{n+1}{2}}\}.
\begin{aligned}\int_{\mathbb{H}^n}(v^{1/m}-\Tilde{v}^{1/m})p_n(v-\Tilde{v})+\mathcal{E}_J(v-\Tilde{v},p_n(v-\Tilde{v}))=\int_{\mathbb{H}^n}(g-\Tilde{g})p_n(v-\Tilde{v}). \end{aligned}
\frac{d}{dt}\int_0^t \pi_{i,j}(s) \, \bar{\nu}(t-s,i) ds \, - \delta_{ij}\bar{\nu}(t,i) \, = \, -\theta_i \pi_{i,j}(t) + \sum_{k \in \mathcal{S}}\theta_i h_{ik}\pi_{k,j}(t).
\begin{aligned}\underset{0 \le t \le T}{\sup}(\|\rho_t\|_{L^q}+\|u\|_{H^2}+\|P\|_{H^1}+\|\nabla d_t\|_{L^2}) &+\int_0^T \left(\|u\|_{W^{2,r}}^2+\|P\|_{W^{1,r}}^2+\|d_{tt}\|_{L^2}^2+\|\nabla^4 d\|_{L^2}^2\right)dt\le C.\end{aligned}
< p_{1}, . . . , p_{n}| S | q_{1}, . . . . , q_{m}> = l i m_{T \rightarrow \infty}\frac{1}{T}\int_{0}^{T}d \tau E \Big [ \prod_{j}\prod_{k}( \overline{{{f_{p_{j}}}}}, \partial_{\tau}\phi_{\tau}) ( f_{q_{k}}, \partial_{\tau}\phi_{\tau}) \Big ]
\frac{1}{r-k}\left(-kH^2 + \frac{r}{2}H^2+s-r\right)\overset{\eqref{bound for s}}{>}\ \frac{1}{r-k}\left(-kH^2 + \frac{r}{2}H^2+\frac{k^2}{2r}H^2-r -1+\frac{1}{r}\right) = \ \frac{r-k}{2r}H^2 + \frac{1}{r-k}\left(-r -1+\frac{1}{r}\right) > \ 1\,.
\frac{1}{2\omega(B)|x-x_B|^n}\int_B|b(y)-\langle b\rangle_B|dy &\qquad\quad=\frac{1}{|x-x_B|^n}\int_B(b(y)-\langle b\rangle_B)a(y)dy\lesssim M_{\tilde{\phi}}((b-\langle b\rangle_B)a)(x).
\begin{tabular}{||c|c|c||}\hline$\Gamma = \gamma^0$ $M>0$ $M=0$ \hline$b>0$ Theorem \ref{gwp-hartree4}Theorem \ref{gwp-hartree2}\hline$b =0$ Theorem \ref{nonscatter-thm}Theorem \ref{gwp-hartree2}\hline\end{tabular}
p_{3Dc}\left( xy|\mu_{x}\sigma_{x}\sigma_{y};\rho\right) =\dfrac{1}{2\pi\sigma_{x}\sigma_{y}\sqrt{1-\rho^{2}}}\exp\left[\frac{-1}{2\left( 1-\rho^{2}\right)}\left( \frac{\left( x-\mu_{x}\right)^{2}}{\sigma_{x}^{2}}+\frac{y^{2}}{\sigma_{y}^{2}}-\frac{2\rho\left(x-\mu_{x}\right) y}{\sigma_{x}\sigma_{y}}\right) \right]
\eqref{term-3}= \sum_{j,k}\iint |u(t,y)|^2 \psi_R(x-y) (x_j-y_j) \partial_j \partial^2_k(|u(t,x)|^2) dxdy &=-\sum_{j,k}\iint |u(t,y)|^2 \partial^x_j(\psi_R(x-y)(x_j-y_j)) \partial^2_k(|u(t,x)|^2) dxdy &=\sum_{k}\iint |u(t,y)|^2 \partial^x_k(3\phi_R(x-y) + 2(\psi_R-\phi_R)(x-y)) \partial_k(|u(t,x)|^2) dxdy.
A_*:=\left(\frac{\displaystyle{(N+1)\int_0^{\infty}\left(1-B_0r^{p/(p-1)}\right)_+^{(p-1)/(p-2)}r^{N-1}\,dr}}{\displaystyle{\eta^{q/(p-1)}\int_0^{\infty}\left(1-B_0r^{p/(p-1)}\right)_+^{q/(p-2)}r^{N-1+q/(p-1)}\,dr}}\right)^{p(p-2)(N+1)\eta/(p-1)}.
(f_{(u,v),(u',v')},\Phi_k)\cr=\, \bigl(-X^u(X^{u'-u}-1)(X^v-1),\Phi_k\bigr)\cr=\, \pm\prod_{l\in\Bbb Z_k^\times}(\zeta_k^{(u'-u)l}-1)(\zeta_k^{vl}-1)\cr=\, \pm\Phi_{k/(k,u'-u)}(1)^{(k,u'-u)}\Phi_{k/(k,v)}(1)^{(k,v)}
( n + k ) d_{k l}^{n}\pm ( k + 1 ) d_{k + 1 , l}^{n}= \left( \frac{( k + 1 ) ( k + 2 )}{2}- g \right) d_{k + 2 , l}^{n - 1}- g \sum_{m = k + n - 1}^{\infty}s_{m}d_{k - m , l}^{n - 1}.
K_{M}= \left[ \frac{2 q^{2}+ 3}{3}- \frac{r_{-}}{r_{+}}\right] \left\{\frac{1}{r_{+}}\left[ \frac{2 q^{2}+ 3}{3}+ \frac{r_{-}}{r_{+}}\right] + \frac{2 q^{2}+ 6}{2 q^{2}+ 3}\frac{r_{-}^{2}}{r_{+}^{3}}\left[ 1 - \frac{r_{-}}{r_{+}}\right]^{- 1}\right\}^{- 1}.
{\mathrm{~ c_{\b e t a - 1}~ a n d ~ c_{\b e t a}~}}\left\{\! \! \! \begin{array}{c}{{{\mathrm{c o i n c i d e}}}}\\{{{\mathrm{a r e ~ o p p o s i t e}}}}\\ \end{array}\! \! \right\}{\mathrm{i f ~ W^{( \b e t a - 1 )}~ c o n t a i n s ~ a n}}\left\{\! \! \! \begin{array}{c}{{{\mathrm{e v e n}}}}\\{{{\mathrm{o d d}}}}\\ \end{array}\! \! \right\}{\mathrm{n u m b e r ~ o f ~ f a c t o r s ~ Y ~ ,}}
\mathbf{B}(i,j)=\mathbf{A}(i,j)^{\mathsf{T}}\circ \mathbf{A}(i,j)=0, \and \mathbf{B}(j,i)=\mathbf{A}(j,i)^{\mathsf{T}}\circ \mathbf{A}(j,i)=0.
-Ric(\nabla_{\nu}e_2,e_2)+Ric(\nabla_{\nu}e_1,e_1)=h_{22}R_{22}-h_{11}R_{11}-\frac{4L_{22}}{|\nabla f|^{3}}=HS|\nabla f|-\frac{4L_{22}}{|\nabla f|^{3}}.
t^{2k-2}t_1t_2t_{2n+1}t_{2n+2}t_{2n+3}t_{2n+4}=t^{k-2}t_3t_4t_{2n+1}t_{2n+2}t_{2n+3}t_{2n+4}&=\cdots=t^{k-2}t_{2n-1}t_{2n}t_{2n+1}t_{2n+2}t_{2n+3}t_{2n+4}=pq.
N\,=\,\left(\begin{array}{ll}-\omega e^{\frac{\beta}{2}}- \frac{g}{2}\overline\omega e^{-\frac{\beta}{2}}\omega e^{-\frac{\beta}{2}}- \frac{g}{2}\overline\omega e^{\frac{\beta}{2}}-\omega e^{\frac{\beta}{2}}- \frac{2}{g}\overline\omega e^{-\frac{\beta}{2}}-\omega e^{-\frac{\beta}{2}}+ \frac{2}{g}\overline\omega e^{\frac{\beta}{2}}\end{array}\right)\,\,\, .
D_m= \sum_{r\geq 0}\sum_{p\in P_{m,r}}r \sum_{j=1}^{m}\left( p(2j-1)+\begin{cases}p(2j) p(2j)-1 \end{cases}\right) = \sum_{r\geq 0}\sum_{p\in P_{m,r}}r \sum_{j=1}^{m}2p(2j-1) = 4m \sum_{p\in \widetilde{P}_m}\sum_{j=1}^m p(2j-1),
(f,A_{\Omega,2m}(a,b,q) f)_{L^2({\Omega})}= \big\|A_{\Omega,2}(a,b,q)^{\ell}f\big\|^2_{L^2({\Omega})}\geq \varepsilon_{\ell}^2 \|f\|^2_{L^2({\Omega})}= \varepsilon_{m}\|f\|^2_{L^2({\Omega})},
\boldsymbol{\pi}=\sigma(y_1,\ldots,y_n)\Big(\mathfrak{m}_1\mapsto y_1\mapsto\cdots\mapsto T^{k_1-1}y_1\Big)\cdot\ldots\cdot\Big(\mathfrak{m}_n\mapsto y_n\mapsto \cdots\mapsto T^{k_n-1}y_n\Big).
\mathcal{H}(u,v)= -(u+u^*+e_2)(u-u^*)^2+[\varrho s-a(u+u^*+e_2)](u-u^*)(v-v^*)-\varrho s(v-v^*)^2 begin{align*}0.2cm] = -\left[\sqrt{(u+u^*+e_2)}(u-u^*)-\frac{\varrho s-a(u+u^*+e_2)}{2\sqrt{(u+u^*+e_2)}}(v-v^*)\right]^2 begin{align*}0.2cm] \quad-\left(\varrho s-\frac{[\varrho s-a(u+u^*+e_2)]^2}{4(u+u^*+e_2)}\right)(v-v^*)^2.
\mathbf{Q}_5=\left(\begin{array}{cc}\underline{a}_1 0 \underline{a}_2 0 0 \underline{b}_1 0 \underline{b}_2\end{array}\right),\mathbf{Q}_6=\left(\begin{array}{cc}a_1 +\underline{a}_1 0 a_2 + \underline{a}_2 0 0 b_1+\underline{b}_1 0 b_2+\underline{b}_2\end{array}\right).
\sum_{\nu=2}^{(N-t)/2}\min_{u\ge 1}\frac{\sum_{\boldsymbol\mu}S_{\boldsymbol\nu,\boldsymbol\mu}}{S_{N,M}}=O\bigl(N^t\cdot N^{-2(k_1-1)}\bigr)=O\bigl(N^{-(k_1-1)}\bigr).
y=\frac{1}{4}(A_{y}-\sqrt{A_{x}^{2}+A_{y}^{2}})[1-(1-\frac{x}{A_{x}})^{2}]-\frac{1}{2}(A_{y}+\sqrt{A_{x}^{2}+A_{y}^{2}})ln(1-\frac{x}{A_{x}})
L_{0}^{\beta}\left( N \right) = \frac{1}{2}\beta_{0}^{2}+ \sum_{e \geq 2}^{2 N}\beta_{- e}^{x}\cdot \beta_{e}^{x}+ \sum_{o \geq 1}^{2 N - 1}\beta_{- o}^{p}\cdot \beta_{o}^{p}.
u(x)=U(t(x)),\breve u(x)=\frac{u(x)}{s(x)}=\frac{U(t)}{S(t)}=\breve U(t),\;\;\;{\rm where}s(x):=\frac{1}{(1+x^2)^{(\lambda+1)/2}}= (1-t^2)^{(\lambda+1)/2}:=S(t).
\mu(B)^\ast Hg(\sqrt{L}(a_{22}t- (q-k)/L)) & = (\sqrt{L}a_{11})^{1/2}\,e^{-\pi i L a_{21}a_{11}(\sqrt{L}(a_{22}t- (q-k)/L))^2}& \hskip.5in H g(\sqrt{L}a_{11}\sqrt{L}(a_{22}t- (q-k)/L)) & = (\sqrt{L}a_{11})^{1/2}\, e^{-\pi i a_{21}a_{11}(La_{22}t- (q-k))^2}& \hskip.5in H g(t- a_{11}(q-k))
p_{0\mu}= p_{\mu}- \frac{1}{2}\tilde{S}^{h_1 h_2}\;\omega_{h_1 h_2 \mu}- \frac{1}{2}\tilde{S}^{k_1 k_2}\;\omega_{k_1 k_2 \mu}- \frac{1}{2}\tilde{S}^{h_1 k_1}\;\omega_{h_1 k_1 \mu},
F_1(\mathbf u^{(l)})=-\sum_{k=1}^{n}\Big( (\alpha_k)^{l}\sum_{j=1}^{n}\big(\frac{(\lambda_{n+1}-\lambda_j)(\lambda_{n+2}-\lambda_j)(\lambda_{n+3}-\lambda_j)}{(\lambda_j-\alpha_1)(\lambda_j-\alpha_k)(\lambda_{n+1}-\alpha_k)}x_j^2\big)\Big).
\mu \left( \Biggl\{\omega \in \Omega : \frac{1}{n}\sum_{i=1}^n h(Z_i(\omega)) \geq \sqrt{\frac{8 (\log n)^{\frac{2}{\gamma}}\sigma^2 \tau}{n}}+ \frac{8 (\log n)^{\frac{2}{\gamma}}B \tau}{3 n}\Biggr\}\right) \leq 2 e^{- \tau}.
T_{\mu\nu}(x)=\sum_{a=1}^3F_{\mu\nu}^a(x)\widehat{\phi}^a(x)-\frac{1}{\lambda}\sum_{a,b,c=1}^3\epsilon^{abc}\widehat{\phi}^a(x)(D_\mu\widehat{\phi}(x))^b(D_\nu\widehat{\phi}(x))^c,
\begin{cases}dY(t,x,z) = [A_uY(t,x,z)+a(t,x,Y(t,x,z),u(t,x,z),z)]dt + b(t,x,Y(t,x,z),u(t,x,z),z)dB(t) + \int_{\mathbb{R}}c(t,x,Y(t,x,z),u(t,x,z),z,\zeta)\tilde{N}(dt,d\zeta); (t,x)\in (0,T)\times D Y(0,x,z)=\xi(x,z); x\in D \int_{\mathbb{R}}Y(t,x,z)\delta_Z(z)dz=\theta(t,x); (t,x)\in[0,T]\times \partial D.\end{cases}
\int d^4x \sqrt{g_{{}_E\pm}}{\cal K}_{{}_E\pm}^n =\int d^4x \left({z_\pm\over z_0}\right)^{(4-n) \beta}{1\over z_\pm^{n}}\propto\int d^4x \; r_\pm^{4+(4 /3) D_2 - (n/3) D_2},
\int_{B_{x_\alpha}(r_\alpha)\backslash B_{x_\alpha}(\mu_\alpha)}(qv_\alpha-1)^2 \Psi_\alpha dv_g = \mu_\alpha^2 \int_{B_0(r_\alpha/\mu_\alpha)\backslash B_0(1)}(q\hat{v}_\alpha-1)^2 \hat{\Psi}_\alpha dv_{\hat{g}_\alpha}= o(\mu_\alpha^2).
(\nabla_{\xi}\Phi)(\xi,e_k) =g(\xi,\nabla_\xi(\xi\times e_k))+g(\nabla_\xi e_k,\xi\times\xi) &=g(\xi,\nabla_\xi \xi\times e_k)+g(\xi,\xi\times\nabla_\xi e_k) &=-g(e_k,(\nabla_\xi \xi)\times\xi).
f((\ominus\,\mathbf{x})\oplus\mathbf{x}) =f(\mathbf{x}\oplus(\ominus\,\mathbf{x}))=f(0), f((\bar{\ominus}\,\mathbf{x})\,\bar{\oplus}\,\mathbf{x}) =f(\mathbf{x}\,\bar{\oplus}\,(\bar{\ominus}\,\mathbf{x}))=f(0),
x=\left[\begin{array}{ccc}a_1 \frac{b_1}{2}+b_2\zeta^{-1}a_2\end{array}\right],y=\left[\begin{array}{ccc}-c_1 -\frac{b_1}{2}+b_2\zeta^{-1}c_2\end{array}\right],x\phi_1 x^{\ast}=y\phi_2 y^{\ast}.