harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mu_{Y}\left( \frac{1}{b_{T}(m)}{\textstyle\sum_{j=0}^{m-1}}\varphi_{Y}\circ T_{Y}^{j}\leq t\right) \Longrightarrow\Pr[\Gamma(1+\alpha)^{-1/\alpha}\,\mathcal{G}_{\alpha}\leq t]\text{as}k\rightarrow\infty
\ddot{\varrho}_t(x) = \frac{\kappa}{2}\partial_{xx}\dot{\varrho}_t(x) - \lambda \partial_x \dot{\Delta}_t(x)= \frac{\kappa}{2}\partial_{xx}\dot{\varrho}_t(x) - \lambda \partial_x\left( \frac{\kappa}{2}\partial_{xx}\Delta_t(x) - \lambda \partial_x\varrho_t(x) -2\gamma \Delta_t(x)\right).
\nu [ A ] \; = \; \lim_{T \rightarrow \infty}\lim_{V_{3}\rightarrow \infty}\frac{g^{2}}{8 \pi^{2}}\int_{V_{3}}\; d^{3}x \; \Big ( \; K_{4}\; \Big |_{x_{4}= T}\; - \; K_{4}\; \Big |_{x_{4}= - T}\; \Big ) \; = : \; \nu_{+}[ A ] \; - \; \nu_{-}[ A ] \; .
|x_n(t_n) - x_c| = |x_n(t_n) - x_n(T) + x_n(T) - x_0(T) + x_0(T) - x_c| &\leq |x_n(t_n) - x_n(T)| + |x_n(T) - x_0(T)| + |x_0(T) - x_c|.
|tA+(1-t)B\setminus S^1| =\int_{x\in\mathbb{R}^{d-1}}|S_x\setminus S^1| &\geq t^{d-1}\int_{x\in\mathbb{R}^{d-1}}|S_{tx}\setminus S^1| &\geq t^{d-1}\int_{x\in\mathbb{R}^{d-1}}|tA_x| &= t^{d}\int_{x\in\mathbb{R}^{d-1}}|A_x| &= t^{d}|A|.
w^{(\varepsilon)}-V = (w^{(\varepsilon)}-w_{n_0})+(w_{n_0}- V)\leq \frac{\varepsilon}{4}+ \frac{1}{n_0}\leq \frac{\varepsilon}{2}, \mbox{and}w^{(\varepsilon)}-V= w_{n_0}+ \frac{\varepsilon}{4}-V \geq \frac{\varepsilon}{4}.
q ( t , g ) = \frac{a}{2}[ \exp ( - i ( \Omega t + b ) ) + \frac{\lambda}{8}( 1 - \frac{2 1 \lambda}{8}) \exp ( - 3 i ( \Omega t + b ) ) + \frac{\lambda^{2}}{6 4}\exp ( - 5 i ( \Omega t + b ) ) ] + C . C . ,
\hat{Y_1}=\frac{Y_1}{\sqrt{A}}, \; \; \hat{Y_2}=\frac{Y_2}{\sqrt{B}}, \; \; \hat{Y_3}=\frac{Y_3}{\sqrt{C}}, \; \; \hat{Y_4}=\frac{Y_4}{\sqrt{D}},\hat{Y_5}=\frac{Y_5}{\sqrt{E}}.
U(x,t) = \frac{1}{\sqrt{8\,\pi^3\,K_{1/2}\,t^{1/2}}}\,G^{3,0}_{0,3}\left[ \left(\frac{x^2}{16\,K_{1/2}\,t^{1/2}}\right)^2 \left| \; \left( 0,\frac{1}{4},\frac{1}{2}\right) \right.\right],
\lim_{k\to\infty}H(Y_0^v|Y_{-k+1},\ldots,Y_{-1},Y_0^{<v}) =H(Y_0^v|Y_0^{<v},Y_{-1},Y_{-2},\ldots) =\lim_{k\to\infty}H(Y_0^v|Y_0^{<v},Y_{-1},\ldots;X_{-k}^{<v},X_{-k-1},\ldots),
\tilde{H}_{\mu \nu \rho}= \frac{1}{6 !}\epsilon_{\mu \nu \rho \sigma_{1}\sigma_{2}, \cdots , \sigma_{7}}\frac{1}{\sqrt{- X^{2}}}\frac{\delta}{\delta X_{\sigma_{1}}}B_{( 6 )}^{\sigma_{2}, \cdots , \sigma_{6}},
\sum_{j=0}^{m}\sum_{k=0}^{m}\left[(a+b)\tilde{\alpha}_{j+k+1}- \tilde{\alpha}_{j+k+2}- ab\tilde{\alpha}_{j+k}\right] \xi_{j}\xi_{k}= \int_a^b \cdots \int_a^b \left( \sum_{j=0}^{m}(x_1\cdots x_d)^{j}\xi_{j}\right)^2 (x_1 \cdots x_d)f(x_1\cdots x_d) p(\pmb x) d\mu \cdots d\mu,
\ \Xi_{\pm}v_{\mp 1 , r}= 0 , \; \Xi_{\pm}= \frac{1}{2}( 1 \pm \gamma^{0}\gamma^{D}) , \; \mathrm{r a n k}\; \Xi_{\pm}= J_{( d )}= 2^{[ \frac{d}{2}] - 1};
\frac{\partial Y}{\partial x}=\left( \frac{A_0^{(1)}}{x^2}+\frac{A_0^{(0)}}{x}+\frac{A_{t_1}}{x-t_1}+N \right)Y , \frac{\partial Y}{\partial t_1}=\left( N_1-\frac{A_{t_1}}{x-t_1}\right)Y ,\frac{\partial Y}{\partial t_2}=\left( N_2-\frac{\frac{1}{t_2}A_0^{(1)}}{x}\right)Y .
\Omega = \left( \begin{matrix}{{0}}&{{1}}&{{0}}&{{0}}&{{}}\\{{- q}}&{{0}}&{{0}}&{{0}}&{{}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{}}\\{{0}}&{{0}}&{{- q^{2}}}&{{0}}&{{}}\\ \end{matrix}\right) ,
-\frac{1}{2}\int_{\Omega}\langle\operatorname{adj}\nabla\varphi\cdot \varphi,\nabla \eta\rangle ~dx\geq \int_{\Omega}f\chi_{A_{\epsilon}}\eta~dx+\int_{\Omega}\frac{f_{\epsilon}}{1+\delta}\chi_{\Omega\setminus A_{\epsilon}}\eta~dx \geq \int_{\Omega}f\chi_{A_{\epsilon}}\eta~dx+\int_{\Omega}f\chi_{\Omega\setminus A_{\epsilon}}\eta~dx=\int_{\Omega}f\eta ~dx,
\rho_{M}( \lambda ) = \frac{1}{\pi}( 2 g \lambda^{2}+ 2 + m^{2}g ) \sqrt{m^{2}- \lambda^{2}}\ \ \mathrm{w i t h}\ \m^{2}= \frac{2}{3 g}( \sqrt{1 + 3 g}- 1 )
\begin{aligned}T_{f}\left(A^{{'E}_{1}}_{1}, h^{{'E}_{1}}\right)- T_{f}\left(A^{{'E}_{1}}_{1}, h^{{'E}_{1}}_{L^{2}}\right) &=\widetilde{f}\left(\nabla^{{'E}_{1}},h^{{'E}_{1}}_{L^{2}}, h^{{'E}_{1}}\right)-\widetilde{f}\left(\nabla^{{'E}_{2}},h^{{'E}_{2}}_{L^{2}}, h^{{'E}_{2}}\right),\end{aligned}
\Psi_\mu(E_{\chi_k^\mu}g_{1,2}) = \sum_{w \in{\langle s_1,s_2\rangle}}\Psi_\mu(E_{\chi_k^\mu}g_w) = \sum_{w \in{\langle s_1,s_2\rangle}}U_\mu\widetilde{\Psi}_\mu(E_{\chi_k^\mu}g_w)U_\mu^{-1}=\sum_{w \in{\langle s_1,s_2\rangle}}G_{\pi_{\mu,k}^{-1}w\pi_{\mu,k}}M_{k,k}.
\sup_{x, y \in D}\, \inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x,\,\varphi(T)=y}}\frac{1}{T}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t))\, \Bigl \vert \,\varphi(t) \in D \cup \partial D,\,\, \forall t \in [0, T] \biggr\}< r({\mathcal{K}}) + \varkappa,
-16632\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}=\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right. &\left.\cdot{\rm ch}(S^2\widetilde{TX}+\widetilde{TX}+\wedge^2\widetilde{L_R\otimes C}-\widetilde{L_R\otimes C}-\widetilde{TX}\otimes\widetilde{L_R\otimes C}\right\}^{(14)}.
h_0(\sigma) := \sigma^4 \ell^2 m_P m_S + \sigma^3\left( \gamma_P \left( m_P + m_S \right) + \ell^2 \gamma_S m_P \right)+ \sigma^2 \left( \gamma_S \gamma_P + K_s \ell^2 m_p \right)+ \sigma \gamma_P K_S +g_0 h_1(\sigma),
R(\varphi,\varphi')=\frac{\varphi-\pi}{2\pi k}\sum_{i}E_{ii}\otimes E_{ii}+\frac{i}{2k}\sum_{i\neq j}\frac{\cos{\frac{\pi\alpha_{ij}(X)}{k}}}{\sin{\frac{\pi\alpha_{ij}(X)}{k}}}e^{-i\frac{\alpha_{ij}(X)}{k}(\varphi-\varphi')}~E_{ij}\otimes E_{ij}
(\phi_\nu)'(z') = \gamma'\circ(\gamma^{-1}\circ\phi_\nu\circ\gamma)'\circ(\gamma^{-1})'(z) & =t_\nu^{1/2}AC_\nu A^{-1}z'-t_\nu^{1/2}AC_\nu A^{-1}\zeta'+\zeta' ~~\longrightarrow~~ \zeta'
\begin{array}{ll}\mbox{minimize}\langle C, X \rangle \mbox{subject to}\langle A_i, X \rangle = b_i \;\;\; \forall i \in [m] & X \in \mathbb{S}_{+}^{n_{1}}\times \cdots \times \mathbb{S}_{+}^{n_{r}}, \end{array}
\begin{array}{c c c c c c c c}{}&{{\cal M}}&{\tilde{\cal M}}&{{\cal D}}&{\tilde{\cal D}}&{{\cal Q}}&{\tilde{\cal Q}}&{T}\\{U ( 1 )}&{0}&{0}&{1}&{- 1}&{1}&{- 1}&{0}\\{U ( 1 )^{\prime}}&{1}&{- 1}&{1}&{- 1}&{0}&{0}&{0 ,}\\ \end{array}
B(0,y)= \begin{bmatrix}B_{0y}^{11}B_{0y}^{12}B_{0y}^{13}B_{0y}^{14}B_{0y}^{21}B_{0y}^{22}B_{0y}^{23}B_{0y}^{24}B_{0y}^{31}B_{0y}^{32}B_{0y}^{33}B_{0y}^{34}B_{0y}^{41}B_{0y}^{42}B_{0y}^{43}B_{0y}^{44}\end{bmatrix}.
q_d(k) :=\left[\prod_{j=1}^{d-1}\left( 1 + \gamma_j \, \omega\left( \left\{\frac{k Y_j z_j}{N}\right\}\right) \right)\right]\left[\prod_{j=d+1}^{s}\left( 1 + \gamma_j \, \omega\left( \left\{\frac{k z_j^0}{N}\right\}\right) \right)\right].
p_{1,k_1,k_2}(x,t)= \frac{(-1)^{k_2}\pi^{k_1+k_2}}{4^n (\pi\delta)^{n+k_1-\frac{m+1}{2}}\sqrt{2 \pi\kappa^m}}e^{-\frac{1}{4}d(x,t)^2}\left[1+O\left(\delta+\frac{1}{\kappa}\right) \right].
W(C)_{\rm mon}^{\cal N}=\frac{1}{N}{\,}{\rm tr}{\,}\exp\left(\frac{ig}{2}\vec H\int d^3x\vec F_{\mu\nu}^{\cal N}\Sigma_{\mu\nu}\right)=\frac{1}{N}\sum\limits_{\alpha=1}^{N}W_\alpha^{\cal N}.
S_0 = \sum_{n=X_1}^X \int_n^{n+1}\frac{dx}{\log x}e(qf(n)) + O \left( \frac{\pi(X)}{\log X}\right) = \sum_{n=X_1}^X \frac{1}{\log n}e(qf(n)) + O\left(\frac{\pi(X)}{\log X}\right).
A_\Gamma(q) = \left(1-q^{-1}\right)^{b(\Gamma)}\sum_{\Gamma'_1 \subsetneq \dots \subsetneq \Gamma'_{\beta-1}\subsetneq \Gamma'_{\beta}= \Gamma}\prod_{j=1}^{\beta-1}\frac{1}{q^{b(\Gamma)-b(\Gamma'_{j-1})}-1},
\Delta\nu =\nu_2\left(\frac{t_2}{\Delta t}\right)^{2/3}-\nu_1\left(\frac{t_1}{\Delta t}\right)^{2/3}, \Delta\eta =(\eta_2-\nu_2^2)\left(\frac{t_2}{\Delta t}\right)^{1/3}-(\eta_1-\nu_1^2)\left(\frac{t_1}{\Delta t}\right)^{1/3}+\Delta\nu^2.
\tilde{D}((X,d,x,f), (Y,d',y,g)) = \inf \bigg\{\epsilon>0: \phi:(X,d,x)\rightarrow (Y,d',y) &\psi:(Y,d',y)\rightarrow (X,d,x) &\epsilon &\sup_{B(x,1/\epsilon)}|f-g\circ\phi|<\epsilon \sup_{B(y,1/\epsilon)}|g-f\circ\psi|<\epsilon\bigg\}
\mathbf{J}_\nu(\, \underline{r}\,\underline{\omega}^\intercal ) = Q_1\mathbf{J}_{\nu,M}^{\rm ASY}(\, \underline{r}\,\underline{\omega}^\intercal )Q_1 + Q_1\mathbf{R}_{\nu,M}(\, \underline{r}\,\underline{\omega}^\intercal )Q_1 + \mathbf{J}_\nu^{\rm EVAL}(\, \underline{r}\,\underline{\omega}^\intercal ),
E|Y(N)|^2 = \kappa_{2,N}+ \kappa_{1,N}^2 = D_2 L(N) N^{2H}(1+o(1)), E|Y(N)|^4 = \kappa_{4,N}+ 3 \kappa_{2,N}^2 = D_4 L(N) N^{2H+2}(1+o(1)) + 3 D_2^2 L(N)^2 N^{4H}(1+o(1)).
\int_S|u|^2_{h_F\otimes h,\omega}dV_{\omega}\leq\int_S\langle B^{-1}_{h_F,\omega}f,f\rangle_{h_F\otimes h,\omega}dV_\omega=\int_X\langle B^{-1}_{h_F,\omega}f,f\rangle_{h_F\otimes h,\omega}dV_\omega,
y\left[ i \right] = \sum\limits_{t = 0}^{{n'_{{\rm{span}}}}}{\left({\sum\limits_{l' = 1}^{L'}{{\bf{g}}_{l'}^H\left[ t \right]{{\bf{\bar H}}_{l'}^\bot{\bf{\bar X}}_{l'}}}}\right){{\bf{d}}\left[{i - t}\right]}}+ z\left[{i}\right].
\sup_{t \in [0,T]}\| \tilde{X}_t - \tilde{O}_t \|_{\L^{p\max\{2,\varphi\}}( \P; H_{(\rho+\varrho)})}\leq \sup_{t \in [0,T]}\left( \int_0^t \| e^{(t-s)A}F( \tilde{X}_s ) \|_{\L^{p\max\{2,\varphi\}}( \P; H_{(\rho+\varrho)})}\, ds \right) < \infty .
\Lambda'g=(\hat g^{+,-})^{\frac{1}{2}}\Big[\Big(\frac{\partial u'}{\partial y_n}-iA_-'u'\Big)+\sum_{j=1}^{n-1}\frac{\hat g^{nj}}{\hat g^{+,-}}\Big(\frac{\partial u'}{\partial y_j}-iA_j'u'\Big)\Big]\Big|_{y_n=0},
\rho_N= \frac{r_{N+1}}{(2i\lambda)^N}+ \varphi_1\Big(\sum_{k=1}^{N}r_k r_{N+1-k}\Big)\frac{1}{(2i\lambda)^{N+1}}+\varphi_1\!\!\!\sum_{n=N+2}^{2N}\!\!\!\Big(\sum_{k=n-N}^{N}r_k r_{n-k}\Big)\frac{1}{(2i\lambda)^n}
\int\!\delta\xi\,ds Tr\left [{\bar\eta}_\mu(g^{\mu\nu}\delta_\rho\delta^\rho-\delta^\mu\delta^\nu) \eta_\nu\right ]=-\int\!\delta\xi\,ds Tr\left[(\delta_\nu{\bar\eta}_\mu-\delta_\mu{\bar\eta}_\nu)(\delta^\nu \eta^\mu-\delta^\mu \eta^\nu)\right ],
\mathcal{H}(g)=0,\qquad\forall g=wtG_k^{\triangle}U_kwB_n\ne G_k^{\triangle}U_k\omega^{-1}B_n, &\mathcal{H}(\omega^{-1}t)=0,\qquad\forall tG_k^{\triangle}U_k\omega^{-1}tB_{n,*}\ne G_k^{\triangle}U_k\omega^{-1}B_{n,*}.
\mathbf{y}_{k,l}=\mathbf{G}_k\mathbf{\Lambda}_k^{\frac{1}{2}}\mathbf{U}_k \sum^{K}_{i=1}\mathbf{P}_i \sum^{L}_{j=1}\mathbf{w}_{i,j}s_{i,j}+\mathbf{n}_{k,l},
\tilde{L}= \sum_i \tilde\alpha_i^{\prime2}- \tilde\alpha_i'\sum_j\tilde\alpha_j' +{1\over D-2}\varphi^{\prime2}+ \sum_a \alpha_a^{\prime2}-{Q^2 \over 2}\exp\left(-\tilde\Phi+2\sum_k \tilde\alpha_k \right) \ .
\mbox{$\cal{A}$}({\mathbf{u}}) =\int d^{1+n}x \frac{(1+{\mathbf{u}}^{\dag}{\mathbf{u}})\partial^{\mu}{\mathbf{u}}^{\dag}\partial_{\mu}{\mathbf{u}}- \partial^{\mu}{\mathbf{u}}^{\dag}{\mathbf{u}}{\mathbf{u}}^{\dag}\partial_{\mu}{\mathbf{u}}}{(1+{\mathbf{u}}^{\dag}{\mathbf{u}})^2},
|||u-v_\zeta|||^2 \leq \mathcal{M}^\oplus(\alpha,\beta;\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho}) := \mathcal{J}^\oplus(\alpha,\beta;\eta,\zeta,\boldsymbol{\tau}) - \mathcal{J}^\ominus(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho})
f(x)-\frac{c}{\Phi_0(\nu;x)}=\frac{c}{\Phi_0(\nu;x)}\left(\exp\left(E_0(x)\right)-1\right), &f(x)-\frac{c}{\Phi_0(\nu;x)+\mathrm{MC}_k(x)}=\frac{c}{\Phi_0(\nu;x)+\mathrm{MC}_k(x)}\left(\exp\left(E_k(x)\right)-1\right),k\ge 1.
\phi_t(t,r,v) = \frac{v(t)}{r^2(t)}r^2 -v\,=- \Big(v-v(t) -\, v(t)\frac{r^2-r(t)^2}{r(t)^2}\Big)\,, \phi(t,r,v) = \frac{r(t)^2-r^2}{r(t)^2}+ t\phi_t(t,r,v)\,.
{\cal L}= - \frac{1}{4}\left\{\phantom{|}^{\mu\nu\alpha\beta}_{abcd}\right\}[ 3 \Omega_\mu{}^{aef}\Omega_\nu{}^{bef}\delta_\alpha{}^c\delta_\beta{}^d - \frac{1}{3}\Omega_\mu{}^{abc}T_{\nu\alpha\beta}{}^d ]
A_i = \begin{pmatrix}1 \cdots 1 e_n^1 - z_{i, 1}\cdots e_n^1 - z_{i, n}e_n^2 - z_{i, 1}e_n^1 + z_{i, 1}^2 \cdots e_n^2 - z_{i, n}e_n^1 + z_{i, n}^2 \vdots \ddots \vdots \sum_{d=0}^{n - h - 2}(-z_{i, 1})^d e_n^{n - h - 2 - d}\cdots \sum_{d=0}^{n - h - 2}(-z_{i, n})^d e_n^{n - h - 2 - d}\end{pmatrix}.
\beta_q(k)=[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k.
V_{i}\cdot{\cal N}_{\overline{{{\alpha V}}}}\equiv{\vec{U}}_{i}\cdot{\vec{P}}_{\overline{{{\alpha V}}}}- \sum_{r}V_{i}^{r}N_{\overline{{{\alpha V}}}}^{r}+ \sum_{\ell : ~ r i g h t}T_{i}^{\ell}J_{\overline{{{\alpha V}}}}^{\ell}- \sum_{\ell : ~ l e f t}T_{i}^{\ell}J_{\overline{{{\alpha V}}}}^{\ell}~ .
-{\frac{g_{\Lambda}^{2}C_{F}\Lambda^{2}}{2 \pi^{2}P^{+}}}\ln \Biggl ({\frac{1}{\epsilon}}\Biggr ) \int{\frac{d Y d^{2}Q}{1 6 \pi^{3}}}\theta ( 1 - Y ) \theta ( Y ) | \phi ( Q , Y ) |^{2}.
\sigma^{(\mathcal{I},red)}(H_{\omega,\Lambda_L(x_0)}):= \left\{E\in\sigma^{(\mathcal{I})}(H_{\omega,\Lambda_L(x_0)}):\left(E,\sigma^{(\mathcal{I})}(H_{\omega,\Lambda_{L_n}(x_0)}\right)\leq 2e^{-\frac{\hat{m}}{K}L_n},n=1,\cdots,n_1 \right\}
S =w_1GD_1(p-2+s)+w_2GD_2(p-2+s)+\epsilon w_3GD_1(p-4+s)+\epsilon w_4GD_2(p-4+s) =w_1GD_1(0)+w_2GD_2(0)+w_3\epsilon GD_1(-2)+w_4\epsilon GD_2(-2),
[y_1, y_1]=\alpha_2y_5, [y_1, y_2]=\frac{\alpha_3}{\beta_1}y_4+\theta_1y_5, [y_2, y_1]=-\frac{\alpha_3}{\beta_1}y_4+\theta_2y_5, [y_2, y_2]=y_4, [y_1, y_3]=\beta_4y_5, [y_3, y_2]=\gamma_4y_5.
\begin{aligned}\dot{x}= -y+a_{20}x^2-2 b_{20}x y+a_{20}b_{20}x^3+r_{11}x^2 y:=P_4(x,y), \dot{y}= x+a_{20}x y+b_{20}x^2-b_{20}y^2+a_{20}b_{20}x^2 y+r_{11}x y^2:=Q_4(x,y).\end{aligned}
\varphi(r)(T+z^2)^{-d}(p)\sim_{|z|\to\infty}&(4\pi)^{-\frac{m}{2}}\varphi(r)\sum_{j=0}^{\infty}z^{-2d+m-2j}u_{j}(p)\frac{\Gamma(-\frac{m}{2}+d+j)}{(d-1)!}.
\int_{-\infty}^x |u(y)I(y)|~dy \le C\int_{-\infty}^x w^{-s-r}(y)|w^su(y)|~dy &= C\int_{-\infty}^x w^{-r-\frac12(s-\frac12)}(y)w^{-\frac12(s+\frac12)}(y)|w^su(y)|~dy &\le Cw^{-r-\frac12(s-\frac12)}(x) \int_{\real}w^{-\frac12(s+\frac12)}(y)|w^su(y)|~dy &\le Cw^{-(r+\frac12(s-\frac12))}(x) \|w^{-\frac12(s+\frac12)}\|_2\|u\|_{L^2_s}
\begin{aligned}\varphi_{A}(z, w) = [w_j z^{u_i^{j}}]_{0 \leq i \leq N_j, 0 \leq j \leq l}= [ w_0z^{u^0_{0}}: \cdots : w_0 z^{u^0_{N_0}}: w_1 z^{u^1_{0}}: \cdots : w_1 z^{u^1_{N_1}}: \cdots: w_l z^{u^l_{0}}: \cdots : w_l z^{u^l_{N_l}}] \end{aligned}
C(\beta) = 2\,\int_0^\infty \varrho^{N-1}\, \left|1-\varrho^{-\beta}\right|^{p-2}\, (1-\varrho^{-\beta})\, \Phi(\varrho)\,d\varrho &=-2\,\int_0^1 \varrho^{N-1}\,\left|1-\varrho^{-\beta}\right|^{p-1}\, \Phi(\varrho)\,d\varrho &+2\,\int_1^\infty \varrho^{N-1}\,\left|1-\varrho^{-\beta}\right|^{p-1}\, \Phi(\varrho)\,d\varrho.
\operatorname{Re}\left( c\left( s\right) \left(\mbox{\boldmath$ \alpha$},\mbox{\boldmath$ \alpha$}\right) -\frac{1}{2}\left\langle\mbox{\boldmath$ \alpha$},\overline{\mbox{\boldmath$ \alpha$}}\right\rangle_{\mathbb{X}}\right) =\operatorname{Re}c\left( s\right)\left(\mbox{\boldmath$ \alpha$},\mbox{\boldmath$ \alpha$}\right)
\frac{d}{dr}\left(\frac{1}{4\pi r^2}\int_{\partial B(z,r)}f(s)\, ds\right) =\frac{1}{4\pi}\frac{d}{dr}\int_{\mathbb{S}^2}f(z+r\omega)d\omega =\frac{1}{4\pi r^2}\int_{\partial B(z,r)}\frac{\partial f(s)}{\partial n}\,ds=\frac{1}{4\pi r^2}\int_{B(z,r)}\Delta_x f(x)\,dx
\hat{T}{\bf f}= 0 , \hat{T}= \frac{\imath s m}{2 \hat{\omega}}\hat{\zeta}\Sigma^{3}\left[ \hat{\zeta}\hat{\omega}\Sigma^{3}+ \imath \partial_{1}( \imath s \Sigma^{2}) + \imath \partial_{2}( - \imath s \Sigma^{1}) - s m \right] .
\left( f^{-1}\right)_n = \frac 1{n!}\sum_{k=1}^{n-1}\frac{1}{f_1^{n+k}}B_{n-1+k, k}\left( 0, -2! f_2,- 3! f_3 , \ldots \right), \left( f^{-1}\right)_1=\frac{1}{f_1}.
\mathbb{G}(x,\, y\, |\, z):=\mathbb{G}_\lambda(x,\, y\, | \, z)=\sum_{n=0}^\infty p^{(n)}(x,\, y)z^n,~ x,~y\in V(G),~z\in \mathbb{C},\ |z| < R_{\mathbb{G}}\, ,
m(-\frac{z}{qy},q^2,qy) -m(-\frac{z}{qy},q^2,-y)=\frac{yq}{z}\frac{J_2^3j(-q;q^2)j(yz;q^2)}{j(qy;q^2)j(-y;q^2)j(-z;q^2)j(qz;q^2)}.
9996 =58^2+14^2+6^2+80^2\ \ 58+3\times14=10^2, 99999999 =139^2+19^2+6866^2+7269^2\ \ 139+3\times19=14^2, 3999999999 =2347^2+18^2+12671^2+15295^2\ \ 2347+3\times18=49^2.
\frac{d}{dq(\epsilon)}\vartheta_{00}[q(\epsilon)]=\sqrt{2}\pi^{-5/2}q(\epsilon)^{-1}K(\epsilon)^{3/2}(1+\sqrt{1-\epsilon^{2}})[E(\frac{1-\sqrt{1-\epsilon^{2}}}{1+\sqrt{1-\epsilon^{2}}})-\sqrt{1-\epsilon^{2}}K(\epsilon)]
[ \delta_{2}, \delta_{1}] H_{\lambda \mu \nu}= \delta_{5 \mathrm{D}}H_{\lambda \mu \nu}+ \Lambda^{\tau \kappa}x_{\tau}\partial_{\kappa}H_{\lambda \mu \nu}- 3 \Lambda_{\kappa [ \lambda}\partial^{\kappa}B_{\mu \nu ]}\, .
R=\left(\frac12R_{\alpha\beta\gamma}^{\;\;\;\;\;\;\delta}\, z^\alpha\wedge z^\beta +R_{\alpha\bar\beta\gamma}^{\;\;\;\;\;\;\delta}\, z^\alpha\wedge \bar{z}^{\bar{\beta}}+\frac12R_{\bar\alpha\bar\beta\gamma}^{\;\;\;\;\;\;\delta}\, \bar{z}^{\bar\alpha}\wedge \bar{z}^{\bar\beta}\right)\otimes z^\gamma \otimes z_\delta.
S_{\mathrm{e f f}}\left[ A_{\mu}\right] = S_{\mathrm{Y M}}\left[ A_{\mu}, A_{3}= 0 \right] + S_{\mathrm{g f}}\Bigl [ \int_{0}^{L}d z \, A_{\mu}^{3}\Bigr ] + M^{2}\int d^{4}x \, \, \Phi_{\mu}^{\dagger}( x ) \Phi^{\mu}( x ) .
\begin{aligned}L^\Phi(\theta)[\cdot]:= -\big({\rm div}_{\! z}+ 2i\pi \theta \big)\big(A{( \Phi^{-1}(z,\omega),\omega)}{\big( \nabla_{\!\! z}+ 2i\pi \theta \big)}[\cdot] \big) +V{\big(\Phi^{-1}\left(z,\omega\right),\omega\big)}[\cdot].\end{aligned}
M_{x}f_{m}=\frac{1}{\sqrt{2}}\left( \bar{a}+a^{\ast}\right)f_{m}\rightarrow\frac{1}{\sqrt{2}}\left( \bar{a}+a^{\ast}\right) f\text{and}\partial_{x}f_{m}=\frac{1}{\sqrt{2}}\left( \bar{a}-a^{\ast}\right)f_{m}\rightarrow\frac{1}{\sqrt{2}}\left( \bar{a}-a^{\ast}\right) f.
m_{1}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=0,m_{2}=\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}=-\frac{2}{n}a_{2},m_{3}=\frac{1}{n}\sum_{i=1}^{n}x_{i}^{3}=-\frac{3}{n}a_{3}
{\mathbb D}^{(A}{}_{(B}{\mathbb D}^{C)}{}_{D)}= \, \mathrm{tf}\big({\mathbb D}^{(A}{}_{(B}{\mathbb D}^{C)}{}_{D)}\big)+\tilde{A}\delta^{(A}{}_{(B}{\big({\mathbb D}_o^2\big)}^{C)}{}_{D)}& +\tilde{B}\delta^{(A}{}_{(B}\delta^{C)}{}_{D)}\langle{\mathbb D},{\mathbb D}\rangle
\left[ \sum_{i = 1}^{N}\frac{\partial^{2}}{\partial{x_{i}}^{2}}+ 2 ( a + \frac{1}{2}) \sum_{i < j}^{N}\frac{1}{( x_{i}- x_{j})}( \frac{\partial}{\partial x_{i}}- \frac{\partial}{\partial x_{j}}) \right] P_{k}( x ) = 0 .
m_I^{\Sigma}(\xi) = \prod_{k=1}^3m_k^{I,\Sigma}(\xi_k) : = \prod_{k=1}^3m_k^I(\xi_k)\Sigma(\xi_k)^{-1}\,, \widetilde{m}_I^{\Sigma}(\xi) = \prod_{k=1}^3\widetilde{m}_k^{I,\Sigma}(\xi_k):= \prod_{k=1}^3\widetilde{m}_k^I(\xi_k)\Sigma(\xi_k)^{-1}\, .
\dot{x}_{2}=l(\dot{\theta}_{1}cos\theta_{1}+\frac{1}{2}\dot{\theta}_{2}cos\theta_{2})\rightarrow\dot{x}_{2}^{2}=l^{2}(\dot{\theta}_{1}^{2}cos^{2}\theta_{1}+\dot{\theta}_{1}\dot{\theta}_{2}cos\theta_{1}cos\theta_{2}+\frac{1}{4}\dot{\theta}_{2}^{2}cos^{2}\theta_{2})
\sup_{x=(z,y) \in B_{\frac{1}{4}}\times Y}\sup_{x' \in B^{g_{z,t}}(x,\frac{1}{8})}\frac{|{g}_t^\bullet(x) - \mathbf{P}^{g_{z,t}}_{x'x}({g}_t^\bullet(x'))|_{g_{z,t}(x)}}{d^{g_{z,t}}(x,x')^\alpha}\leq C'.
d s^{2}= d \tilde{x}^{\tilde{m}}\tilde{g}_{\tilde{m}\tilde{n}}d \tilde{x}^{\tilde{n}}= e^{m}\eta_{m n}e^{n}+ e^{\rho}e^{\rho}= \rho^{2}d x^{2}+ R^{2}\, \left( \frac{d \rho}\rho \right)^{2}\, ,
[ \hat{B}^{0 i}, \hat{Q}^{a}] = i \varepsilon^{a b}\hat{\pi}_{( C ) b}^{i}, \; [ \hat{\pi}_{( B ) 0 i}, \hat{Q}^{a}] = - i \varepsilon^{a b}( \partial_{i}\hat{\pi}_{( C ) 0 b}+ \hat{A}_{i}\wedge \pi_{( C ) 0 b}) ,
|\{(cl(i),cl(j)) \mid -j \in C_i, 1\le i,j\le l\}| &= \begin{cases}0 \mbox{if $1 \le l \le q^{m/2}-2$}, 1 \mbox{if $q^{m/2}-1\le l\le 2q^{m/2}-3$}, 2 \mbox{if $l=2q^{m/2}-2$}, 4 \mbox{if $2q^{m/2}-1 \le l \le 2q^{m/2}$}.\end{cases}
h'(r) = \int_{\varphi_{0}(r)}^{1-\varphi_{0}(r)}\frac{2(r-\cos(2\pi\varphi))}{1+r^{2}-2r\cos(2\pi\varphi)}\, d\varphi &\ge \int_{\varphi_{0}(r)}^{1-\varphi_{0}(r)}\frac{r}{1+r^{2}-2r\cos(2\pi\varphi)}\, d\varphi \ge 0
\begin{aligned}\left(\frac{\partial e_f}{\partial t},\psi_h\right) + a_h^0(e_f,\psi_h) + \mathcal{N}(u;f,\psi_h) - \mathcal{N}^h(u_h;f_h,\psi_h) = 0\quad\forall\, \psi_h \in \mathcal{Z}_h , \end{aligned}
\epsilon_{0}^{>}= - \frac{8 k^{2}}{\ell_{0}^{2}}B_{1}^{<}, \quad \epsilon_{2}^{>}= k^{2}B_{2}^{<}- \frac{3 w^{( 2 ) >}k^{2}}{\ell_{0}^{2}}B_{1}^{<}.
\int_{\sigma}^{\infty}{\frac{d \sigma^{\prime}}{\sqrt{1 + 4{\sigma^{\prime}}^{3}}}}={\frac{1}{3}}\int_{0}^{\infty}{\frac{X^{- 2 / 3}d X}{\sqrt{1 + 4 X}}}-{\frac{1}{3}}\int_{0}^{\sigma^{3}}{\frac{X^{- 2 / 3}d X}{\sqrt{1 + 4 X}}}= i g_{3}^{\frac{1}{6}}( \eta - \eta_{\infty}) ,
\left(\int_{\Omega}|f|^2\delta^{-2s}\right)^{1/2}\le\left(\int_{\Omega}|f|^{r}\right)^{1/r}\left(\int_{\Omega}\delta^{-2s\cdot\frac{r}{r-2}}\right)^{\frac{r-2}{2r}}=C\|f\|_{L^{r}},
\sum_{j_{1 2}}( - 1 )^{I_{1 2}( I_{1 2 3}+ 1 )}\left\{\begin{array}{c c c}{{j_{1}}}&{{j_{2}}}&{{j_{1 2}}}\\{{j_{3}}}&{{j}}&{{j_{2 3}}}\\ \end{array}\right\}^{S}~ \left\{\begin{array}{c c c}{{j_{1}}}&{{j_{2}}}&{{j_{1 2}}}\\{{j_{3}}}&{{j}}&{{j_{2 3}^{\prime}}}\\ \end{array}\right\}^{S}= ( - 1 )^{I_{2 3}( I_{1 2 3}+ 1 )}\delta_{j_{2 3}\, j_{2 3}^{'}}~ .
\mathcal{H}_{ij,\alpha}(x,y)_p=\partial_i\mathcal{G}_{\alpha,j}(x,y)_p+\partial_j\mathcal{G}_{\alpha,i}(x,y)_p-\frac{2}{n}\xi_{ij}\sum_{k=1}^n\partial_k\mathcal{G}_{\alpha,k}(x,y)_p.
\limsup_{\varepsilon \rightarrow 0^{+}}\limsup_{n \rightarrow \infty}\mathbb{E}_{\mu_n}\Big[ \Big| \int_{0}^{t}\frac{1}{n}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \overleftarrow{\eta}_{s}^{\ell}(x) [\eta_{s}(x+1)-\overrightarrow{\eta}_{s}^{\varepsilon n}(x+1)]ds \Big| \Big] =0.
\Phi_\delta^{\gamma,\varepsilon}(t)=\frac{1}{2}\int_{D}\log\left(1+\frac{|X_{1t}(x)-X_{2t}(x)|^2+\gamma \langle \eta_t^\varepsilon(X_{1t}(x)),X_{1t}(x)-X_{2t}(x)\rangle^2}{\delta^2}\right)dx.
\lim_{k\rightarrow \infty}\mathcal{L}_k(\mu_1,\overline{D}_1)-\mathcal{L}_k(\mu_0,\overline{D}_0)=-\frac{1}{\mathrm{vol}(D)}\int_{\Delta_D}(\check{g}_{\overline{D}_1}-\check{g}_{\overline{D}_0})dx.
\left[a+b\right] =\max(a,b)=\max(b,a)=\left[b+a\right], \left[ab\right] =a+b=b+a=\left[ba\right], \left[a(b+c)\right] =a+\max(b,c)=\max(a+b,a+c)=\left[ab+ac\right].
ds^2_{(10)}=e^{\frac23 \phi}\left[dx^a \, dx^b \,\eta_{ab}+{1 \over\sqrt{2 \kappa |u|^2 + u_0}}\left(du^i + \epsilon^{ijk}\tilde{A}^ju^k\right)^2 + \sqrt{2 \kappa |u|^2 + u_0}\;d\tilde{s}_3\right]
-A_p(X) = \overline{\nabla}_X \eta = \sum_{i=1}^{n+2}\overline{\nabla}_X(a_i \widetilde{e}_i) = \sum_{i=1}^{n+2}\left[ a_i(p) \overline{\nabla}_X \widetilde{e}_i + X(a_i) \widetilde{e}_i (p) \right] = \overline{\nabla}_X \widetilde{e}_{n+2}+ \sum_{i=1}^{n+2}X(a_i) e_i = \alpha_p(X) + \Gamma_p^{-1}(D \gamma(p) \cdot X),
\ell\left(y_i \right) \triangleq \ell\left( c_{i,k}\vert y_i \right) = \ln{\frac{\sum_{\hat{x}\in\mathcal{K}_{k,1}}p(y_i\vert x=\hat{x})}{\sum_{\hat{x}\in\mathcal{K}_{k,0}}p(y_i\vert x=\hat{x})}},
\Sigma_{}:= \left\{x \in \Sigma: \theta^n_{\{u>0\}}(x) =0\right\}\Sigma_{}:= \left\{x \in \Sigma : \theta^n_{\{u>0\}}(x) \in (0, 1/2)\right\}\Sigma_{}:= \left\{x \in \Sigma : \theta^n_{\{u>0\}}(x)=1/2\right\}.
\dfrac{(f^{\tilde{\phi}^{-n}}\circ \rho_{\eta}(X))'}{f^{\tilde{\phi}^{-n}}\circ \rho_{\eta}(X)}= \eta \dfrac{(f^{\tilde{\phi}^{-n}})'}{f^{\tilde{\phi}^{-n}}}\circ \rho_{\eta}(X).
\left|S^{(8 )}(m,d,D,\mathfrak{C}_1\times\mathfrak{C}_2)\right|\ll \frac{(qm)^3}{(dD)^2}\max_{i=1,2}\sum_{\overrightarrow{b}\mod d^*qmdD}\left|S\left(d^*qmdD,\overrightarrow{b},\mathfrak{C}_i\right)\right|^2.
[z^n] (P_1+P_2 + \cdots + P_k) = [z^n] \Big( 1 - \frac{z F_{k,1}(A)}{A}\Big) = - [z^{n-1}] A^{-1}&= \frac{1}{n-1}[t^{n-2}] t^{-2}(1-t)^{-k(n-1)}= \frac{1}{n-1}[t^{n}] (1-t)^{-k(n-1)}&= \frac{1}{n-1}\binom{(k+1)(n-1)}{n}= \frac{k}{n}\binom{(k+1)(n-1)}{n-1}.
\frac{\partial}{\partial t}\rho + \div(\rho u) =0 \frac{\partial}{\partial t}(\rho u) + \div (\rho u \otimes u) + \nabla (p(\rho) + C_\kappa \frac{\alpha}{2}\rho^2) = C_\kappa\alpha \rho \nabla c c - \frac{1}{\alpha}\triangle c = \rho\, ,
\hat X_g(f) =\Bigl((R_g)_*(\hat X_e)\Bigr)(f)=\hat X_e(f\circ R_g) =v(f\circ R_g)=X_e(f\circ R_g) =\left.\frac{d\}{dt}\right|_{t=0}(f\circ R_g)(\exp(tX)) = \left.\frac{d\}{dt}\right|_{t=0}f(\exp(tX)g).
\mathcal{L}= \partial_{\mu}\phi^{*}\,\partial^{\mu}\phi -\phi^{*}m^{2}\phi +\sum_{i=1}^{2}\left[\frac{1}{2}\,\partial_{\mu}\varphi_{i}\,\partial^{\mu}\varphi_{i}- \phi^{*}U_{i}(\varphi_{i})\phi\right],
J_n: = \int_{(n-1)/n}^{1}(Q(u) -Q(u_{n,n}) )^2 du\le \int_{(n-1)/n}^{1}(Q(u) -q )^2 du & = \int_{(n-1)/n}^{1}Q(u)^2 du -2 q \int_{(n-1)/n}^{1}Q(u) du+q^2 n^{-1}.
\mathrm{tr}(Y) = \sum_{m=1}^x \sum_{q=1}^Q S(m,m;q) = \sum_{m=1}^x \sum_{q=1}^Q \sum_{\substack{k=1 \gcd(k,q)=1}}^q \exp (\frac{2 \pi i}{q}(mk+mk^*)) = \sum_{q=1}^Q \sum_{\substack{k=1 \gcd(k,q)=1}}^q \sum_{m=1}^x \exp (\frac{2 \pi i}{q}(k+k^*)m) ,
\tau(\widetilde K_n)\ge [\Lambda(g,2A,M)]^{-1}(1+\varepsilon) \geq [\Lambda(g,A,M)]^{-2}(1+\varepsilon) > \left[ \frac{1+\varepsilon}{1+\frac{\varepsilon}{2}}\right]^{-1}(1+\varepsilon) = 1+\frac{\varepsilon}{2},
P_{1}= - \frac{1}{2}\mathrm{T r}R^{2}\hspace{5 m m}\mathrm{a n d}\hspace{5 m m}P_{2}= - \frac{1}{4}\mathrm{T r}R^{4}+ \frac{1}{8}( \mathrm{T r}R^{2})^{2},
\operatorname{Re}\psi_7(i\rho,i\tau,\xi+i\eta)=-(1-\alpha)\tau^2-\beta\tau\eta\leqslant 0,\shortintertext{and}\operatorname{Re}\psi_8(i\rho,i\tau, \xi+i\eta)=-(1-\alpha)\tau^2-\beta\rho\eta \leqslant 0,