harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\omega \in Z\left(\zeta_L\right) \atop \omega \neq \omega_0}\frac{\Re s-1}{|s-\omega|^2}\leq \sum_{\omega \in Z\left(\zeta_L\right)}\Re \frac{1}{s-\omega}= \frac{1}{2}\log d_L + \Re \left(\frac{1}{s}+\frac{1}{s-1}\right) + \Re \frac{\gamma_L^{\prime}}{\gamma_L}(s) + \Re \frac{\zeta_L^{\prime}}{\zeta_L}(s).
\widehat{F}\big(\Psi(z)\big)= \sum_{j=1}^{n-1}\left[(x_1,\ldots,x_{n-1}) X^{T}\widehat{Q}_{j}X\left(\begin{array}{c}x_1 \vdots x_{n-1}\end{array}\right)\right]x_n^{j-1}=F(z).
\Vert \eta -\hat{\eta}\Vert_{L^2 \left( \partial K \right)}\lesssim_{n_1,\ldots,n_{J},k_{0}}\dfrac{C_k}{c_k}\left\{\sum_{j=1}^{4}\left( \log k \right)^{1/2}\left| \eta^{\rm slow}\circ \phi \right|_{n_j,\mathcal{I}_j}d_j^{-n_j}+ \sum_{j=5}^{J}\left| \eta^{\rm slow}\right|_{n_j,\mathcal{I}_j}d_j^{-n_j}\right\}.
\langle \tau_p(g)\breve{\mathbf g}_p, \breve{\mathbf g}_p \rangle = \int_{K_{00}}\mathbf g_p(h g \varpi_p) \overline{\mathbf g(h\varpi_p)}dh = p^{1/2}\xi_1(p)^2 \int_{K_{00}}\mathbf g_p(h g \varpi_p)\underline{\chi}_p^{-1}(h) dh.
\hat{q}_{\beta}(n,t)=\begin{cases*}E_{\beta, 1}\left(-\lambda t^{\beta}\right),\n=0, \displaystyle\sum_{\Omega_{n}}\prod_{j=1}^{n}\frac{\left((1-p)^{j}/j\right)^{x_{j}}}{x_{j}!}z_{n}!\left(\frac{-\lambda t^{\beta}}{\ln p}\right)^{z_{n}}E_{\beta, z_{n}\beta +1}^{z_{n}+1}\left(-\lambda t^{\beta}\right),\n\ge1,\end{cases*}
\sup_{1\le m\le (\log(1/\epsilon))^{1+\delta}}\left\| B_1(\epsilon)\cdots B_m(\epsilon) - \left(\tilde{B}^m + \epsilon \sum_{i=1}^{m-1}i \tilde{B}^i \tilde{B}^{(1)}\tilde{B}^{m-i-1}\right) \right\| = O(\epsilon^2(\log(1/\epsilon))^v)
\Omega_{\Sigma}^*= \{(z_1,\dots,z_{n}) \in \mathbb D^n \,:(z_1,\dots, z_{n}) \in \sigma_T(P_1^{\perp}U_1^*+zP_1U_1^*, \dots, P_n^{\perp}U_n^*+zP_nU_n^*)\}.
\frac{\sum_{j=1}^s a_j (2t_j)^{b_j}}{\prod_{j=1}^s (1+t_j)^{p(s)}}=t^{1-p(s)\sum_{j=1}^s b_j^{-1}}\frac{\sum_{j=1}^s a_j (2\left\lceil t^{1/b_j}\right\rceil/t^{1/b_j})^{b_j}}{\prod_{j=1}^s (\left\lceil t^{1/b_j}\right\rceil /t^{1/b_j}+t^{-1/b_j})^{p(s)}}.
{\cal I}= \frac{1}{4 \pi}\int \left( \frac{( | R_{z}|^{2}- | R_{{\bar{z}}}|^{2}) ( 1 + | z |^{2})^{2}}{( 1 + | R |^{2})^{2}}\right)^{2}\frac{2 i \mathrm{d}z \mathrm{d}{\bar{z}}}{( 1 + | z |^{2})^{2}}.
\|u_k\|^2 = \pi \Omega_\beta' \mu_k^\frac{2-\beta}{2\beta}(1+o(1))= \pi \Omega_\beta' \left(\frac{\pi}{\Omega_\beta}k\right)^\frac{2-\beta}{2+\beta}(1+o(1)), k \to \infty,
\lim_{N\to \infty}\frac{1}{N}\sum_{i=1}^N1_{(\mathcal{L}(i-1)=\mathcal{L}(i)=0)}=\mathbb{P}_p(\mathcal{L}(0)=\mathcal{L}(1)=0) > 1-\epsilon,\ \ \mathbb{P}_p-
(-1)^{t-1}\prod_{s=1}^{t-1}g(z_{\ell(s)}, z_{\ell(s+1)})+\frac{g(w, z_{\ell(t)})}{g(z_{\ell(t)}, z_{\ell(1)})}\prod_{s=1}^{t-1}g(z_{\ell(s+1)}, z_{\ell(s)}) =\frac{g(w, z_{\ell(t)})}{g(w, z_{\ell(1)})}\prod_{s=1}^{t-1}g(z_{\ell(s+1)}, z_{\ell(s)}).
\begin{aligned}g(t) \geq \frac{\alpha}{2}E_{\alpha,\alpha}\big[-2c_{1}\big]c_{0}\big|\big\{x:u(x,t)\leq 0 \big\}\cap B_{1}\times [-2,-1] \big| & \geq \frac{\alpha}{2}E_{\alpha,\alpha}\big[-2c_{1}\big]c_{0}\mu,\end{aligned}
{\rm (i)}\,\,\,\,\, \tau_{A}(x, \xi)=v_{\xi}^{-1}(x)(Av_{\xi})(x);{\rm (ii)}\,\,\,\,\, \tau_{A}(x,\xi)=v_{\xi}^{-1}(x)\int_{\Omega}\widetilde{K}_{A}(x,y)v_{\xi}(y)dy.
\left\Vert u\right\Vert_{H^{1}\left( \omega,\mathbb{B}\right)}:=\left(\left\Vert u\right\Vert_{H^{1}\left( \omega\right)}^{2}+\left\Vert\operatorname{div}\left( \mathbb{B}\nabla u\right) \right\Vert_{L^{2}\left( \omega\right)}^{2}\right)^{1/2}.
F'=\overline{W}_0 T^{\varepsilon_1}_{\alpha_1}{\overline{W}\,}^{-1}_1 \overline{W}_1 T^{\varepsilon_2}_{\alpha_2}{\overline{W}\,}^{-1}_2 \cdots{\overline{W}\,}^{-1}_{m - 1}\overline{W}_{m - 1}T^{\varepsilon_m}_{\alpha_m}{\overline{W}\,}^{-1}_m.
\beta_k u_k u_k^TA_k^{-1}g_k = (\frac1{\delta_k}-1 ) g_k + \frac{g_k^T g_k}{p_{k-1}^T g_{k-1}}g_{k-1}=- \frac{g_k^T g_k}{p_{k-1}^T g_{k-1}}(\gamma_{k}g_k-g_{k-1}),
\begin{aligned}\max\limits_{B \leq 2f_{\rm c}\frac{1-\cos \theta_{\rm c}}{1+\cos \theta_{\rm c}}}R=B \log_2 \left(1 + \frac{P_{\rm R}}{BN_0 |\cos \theta_{\rm c}|\log\sqrt{\frac{f_{\rm c}+B/2}{f_{\rm c}- B/2}}}\right). \end{aligned}
\nabla^i \nabla_i f_t + t^2|\beta_t|^2 \nabla^i \nabla_i f_t - t^2(\beta_t)_i (\beta_t)_j \nabla^i \nabla_j f_t & = -\nabla^i (\alpha^*)_i - t^2|\beta_t|^2 \nabla^i (\alpha^*)_i + t^2 (\beta_t)_i (\beta_t)_j \nabla^i (\alpha^*)_j.
{\partial}^{-}a_{+}= n_{-}{\pi}^{+}- \frac{{\partial}_{+}}{{\partial}_{\bot}^{2}}( \sum_{i = 1}^{2}{\partial}_{i}{\pi}^{i}) + \frac{n_{+}^{2}}{{\partial}_{\bot}^{2}+ n^{2}{\partial}_{+}^{2}}{\partial}_{+}B ,
P^{0,1}\mu_1P^{1,0}{\rm ad}\bar\nu_j^T\Delta^{-1}_0\star{\rm ad}\bar\nu_i^T\star P^{1,0}\bar\mu_2P^{0,1}\nu_i =i P^{0,1}\mu_1P^{1,0}{\rm ad}(\Delta^{-1}_0\star{\rm ad}\bar\nu_i^T\star P^{1,0}\bar\mu_2\nu_i)\star\nu_j
(-1)^{k+l}2^{m+2}[y^{-m-2}]\left(\partial_y^{k+2}v\right)\left(\partial_y^{l+2}v\right) = \sum_{i=0}^{k-m+l+1}(-1)^{l+1-i}\frac{(2k-2i+1)!!}{(2k-2m-2i-1)!!}C_iC_{k-m+l+1-i}.
\frac{d}{dt}x(t) = - 2 \nu \left\|u(t)\right\|_{m+\alpha/2}^2-2\left\langle u^{(m)}(t), (f(u(t)))^{(m+1)}\right\rangle \indent \leq - 2 \nu \left\|u(t)\right\|_{m+\alpha/2}^2 + C \left\| u(t) \right\|_{m+\alpha/2}^{4m/(2m+\alpha-1)}. \indent = (- 2 \nu y(t)^{1/\beta(2m+\alpha-1)}+C) y(t)^{2m/(2m+\alpha-1)}.
\mu =t^{2}\lambda_{n}\left[ z_{2n}\frac{c_{2n}}{2}+ \frac{c_{n}^{2}}{8}\right] +O ( t^{3}) =t^{2}\lambda_{n}\frac{c_{n}^{2}}{8}\frac{2 \gamma -1}{\gamma -1}+O ( t^{3})
\begin{array}{| c | c | c | c |}\hline{{~}}&{{[ e ]}}&{{[ r^{k}]}}&{{[ s ]}}\\ \hline{{J_{0}}}&{{1}}&{{1}}&{{1}}\\{{J_{1}}}&{{1}}&{{1}}&{{- 1}}\\{{\alpha_{j}}}&{{2}}&{{q^{j k}+ q^{- j k}}}&{{0}}\\ \hline{{}}\\ \end{array}
W_n(\lambda)=\frac{1}{n d_n}\sum\limits_{i=1}^{\lfloor n\lambda\rfloor}\sum\limits_{j=\lfloor n\lambda\rfloor+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right), \ 0\leq \lambda\leq 1,
R^\mathrm{RLN}_\mathsf{A}(Q_{U,V,X|S})=\min\Big\{I(V;Y,S_1|U)-I(V;Z,S_2|U),I (U,V;Y,S_1)-I(U,V;S) &,I(U,V;Y,S_1)-I(U;S)-I(V;Z,S_2|U)\Big\}
Q^{\mu i}(\xi_i)=-\sum_{\scriptstyle n=-\infty \atop \scriptstyle n\neq 0}^\infty\frac{1}{\sqrt{n}}\alpha^{\mu i}_n(\xi_i)^{-n}+\alpha^{\mu i}_0\ln \xi_i+\frac{\partial \!\!\!\!{}^{{}^\leftarrow}}{\partial \alpha^i_{0\mu}}\ \ .
I:=\int_{B(x, R/8)^c}\frac{h(y)+h_x(y)}{|y-x|^{\alpha+d}}dy, II:=\int_A \frac{h(y)+h_x(y)}{|y-x|^{\alpha+d}}dy,III:=\int_E \frac{|h(y)-h_x(y)|}{|y-x|^{\alpha+d}}dy.
\begin{cases}\underline u_t-\underline u_{rr}-\frac{N-1}{r}\underline u_r\leq f(\underline u) \mbox{for}r\in [ct, \tilde c_k t], \underline u(\tilde c_kt,t)\leq u(\tilde c_k t, t) \mbox{for}t\geq T, \underline u(ct,t)\leq u(c t, t) \mbox{for}t\geq T, \underline u(r, T)\leq u(r, T) \mbox{for}r\in [cT, \tilde c_k T].\end{cases}
c(2k)=\left(\frac{-\lambda}{1-p^{-2}}\right)^{k-1}\frac{c(2)p^{k(k-1)}(p^2-1)^{k-2}}{(p^4-1)^2(p^6-1)^2\ldots(p^{2k-2}-1)^2 (p^{2k}-1)}\ .
\phi\left( x,t\right) =\cos\left( t\sqrt{m^{2}-\nabla_{x}^{2}}\right)\phi\left( x,0\right) +\sin\left( t\sqrt{m^{2}-\nabla_{x}^{2}}\right)\frac{1}{\sqrt{m^{2}-\nabla_{x}^{2}}}~ \pi\left( x,0\right)
\int_0^T \int_N h(\nabla \phi(x,t),u_t(x)) d\alpha_t\,dt = \int_0^T (\int_Nh(\nabla\phi(x,t),v_t(x)) \lambda d\mu_t + h(\nabla\phi(x,t),w_t(x))(1-\lambda) d\nu_t)\,dt
\widehat{\phi_{N(k,n)}}(\alpha,y, \xi) = \frac{2^{\lambda-(n-k)/2}\sqrt{\pi}}{\Gamma(\lambda) k^{|\alpha|}}y^{\lambda+|\alpha|-1-(n-k)/2}e^{-2y}e^{-|\xi|^2/8y}\frac{1}{\alpha_1!\cdots \alpha_k!},
T_{\mathcal I}f := \sum_{I \in \mathcal I}T_I f , j =1,2,3. T_{\ast , \mathcal I}f := \sup_{l \geq k_0}\Bigl\lvert \sum_{I \in \mathcal I \,:\, \lvert I\rvert \geq 2^{l}}T_I f \Bigr\rvert,
H=\frac{1}{2}\sum_{k,\mu=\pm1}\hbar\omega({a^{\dagger}}^{(\mu)}(k)a^{(\mu)}(k)+a^{(\mu)}(k){a^{\dagger}}^{(\mu)}(k))=\sum_{k,\mu}\hbar\omega({a^{\dagger}}^{(\mu)}(k)a^{(\mu)}(k)+\frac{1}{2})
{\cal M}_{1 - l o o p}( \theta ) = \frac{i g^{4}}{8 \pi m \Theta^{2}}[ \varepsilon^{c a b}T_{c}\otimes T_{b}T_{a}] = - \frac{i g^{4}}{8 \pi m \Theta^{2}}[ T^{a}\otimes T_{a}] \; .
\left(\begin{matrix}x=0 x=\infty \, \left( \frac12 \right) \overbrace{\begin{matrix}0 0 t_2 \theta^0\end{matrix}}\overbrace{\begin{matrix}1 0 -t_1/2 \theta^\infty_1/2 -1 0 t_1/2 \theta^\infty_1/2\end{matrix}}\end{matrix}\right) ,
E(s;\mathbf{t})=\sum_{n\sim E}\frac{e_j(n;\mathbf{t})\chi(n)}{n^s}\,F(s;\mathbf{t})=\sum_{n\sim F}\frac{f_j(n;\mathbf{t})\chi(n)}{n^s},P(s;\mathbf{t})=\sum_{p\sim P}\frac{\chi(p)}{p^{s+i(t_1+ t_2 +t_3)}}.
\nu_i^{S \cup \{j\}}=\frac{\nu_i^S w_i^S - \nu_j^S \frac{w_j^S}{a_{jj}^S}a_{ij}^S}{w_i^S - \frac{w_j^S}{a_{jj}^S}a_{ij}^S}= \nu_j^S - \frac{w_i^S}{w_i^S - \frac{w_j^S}{a_{jj}^S}a_{ij}^S}(\nu_j^S - \nu_i^S) = \nu_j^S - \frac{w_i^S}{w_i^{S \cup \{j\}}}(\nu_j^S - \nu_i^S),
\frac{9 \|f\|_{\mathrm{Lip}}^2}{\epsilon^2 N}\mathbb{E}\left(\mathrm{Tr}[(X_N-\widetilde{X}_N)^2]\right)=\frac{9 \|f\|_{\mathrm{Lip}}^2}{\epsilon^2 N}\sum_{1\leq i,j \leq N}\mathbb{E}\left(([X_N]_{ij}-[\widetilde{X}_N]_{ij})^2\right)=\frac{9 \|f\|_{\mathrm{Lip}}^2}{\epsilon^2}\,t
g(z) = \sum_{n=0}^{\infty}\int_{-1}^{1}P_n^{(1,1)}(x)z^nf(x) \mathrm{d}x = \int_{-1}^{1}\sum_{n=0}^{\infty}P_n^{(1,1)}(x)z^nf(x) \mathrm{d}x &=\int_{-1}^{1}\frac{4}{R(1-z+R)(1+z+R)}f(x)\mathrm{d}x\ .
\lim_{u\to\infty}\limsup_{N\to\infty}\sup_{0\leq t\leq T}|\frac1N\sum_{n=1}^{[Nt]}E(W_{N,i,n}^{(u)})E(W_{N,i',n}^{(u)})-\frac t{c^{(k)}}D_{i,i'}|=0
\zeta_{n}^{*}(k_{1},...,k_{n-1},k_{n})=\sum_{0<m_{1}<m_{2}<\cdot\cdot\cdot<m_{n}}\frac{1}{m_{1}^{k_{1}}\cdot\cdot\cdot m_{n-1}^{k_{n-1}}m_{n}^{k_{n}}}.
|X\chi_R| =2 \frac{|(x^2+y^2)x + yz|}{||(x,y,z)||^4_{\mathbb H}\ln R}\ \ \ \ |Y\chi_R| = 2 \frac{|(x^2+y^2)y - yz|}{||(x,y,z)||^4_{\mathbb H}\ln R},
\sum_{k=0}^n \binom{p+k}{p}\binom{q+n-k}{q}\binom{k}{c}\binom{n-k}{d}k = \binom{n+p+q+1}{n-c-d}\binom{p+c}{c}\binom{q+d}{d}\cdot \frac{n(p+c+1)+(q+1)c-(p+1)d}{p+q+c+d+2}
\frac{1}{2}=\lim_{\beta\rightarrow\infty}U\left[ v_{j}\left( t,\beta\right)\right] =\lim_{\beta\rightarrow\infty}\mathcal{V}\left( \dot{Q}_{j}\left(t\,,\beta\right) ,Q_{j}\left( t\,,\beta\right) \right) , \lim_{\beta\rightarrow\infty}\mathcal{T}\left( \dot{Q}_{j}\left(t\,,\beta\right) ,\dot{Q}_{j}\left( t\,,\beta\right) \right) =0.
c_1 = \int_{\partial B_1}\bar u^+ = \int_{\partial B_1}v^+=\int_{\partial B_1}v^+ x\cdot\nu=\int_{B_1}{\rm div}\, (v^+ x)= \int_{B_1}(2v^+ +\nabla v^+\cdot x) \qquad\qquad\le 2{\mathcal{L}}^2 (B_1\cap \{v>0\})+\int_{B_1}|\nabla v^+|.
\begin{aligned}\hat{F}(1,\phi,\psi) =-\hat{F}(0,\phi,\psi), \hat{G}(1,\phi,\psi) =-\hat{H}(0,\phi,\psi), \hat{H}(1,\phi,\psi) =\hat{G}(0,\phi,\psi)\end{aligned}
d s^{2}= e^{\pm 2 \eta}\left( e^{2 \varphi_{0}}+ 2 \alpha^{\prime}v \frac{2 \rho^{2}+ e^{\pm 2 \eta}}{( \rho^{2}+ e^{\pm 2 \eta})^{2}}\right) \left( - d \eta^{2}+ d \Omega_{- 1}^{2}\right) .
\int_{0}^\delta \frac{\sin^2{(\pi T \|k \| s)}}{\pi^2 || k||^2 s^2}(\sqrt{1-s^2})^{d-3}\, ds \leq \frac{T}{\pi ||k||\sqrt{1-\delta^2}}\int_{0}^{\pi T ||k|| \delta}\frac{\sin^2{u}}{u^2}\, du & \leq \frac{T}{\pi ||k||\sqrt{1-\delta^2}}\int_{0}^{\infty}\frac{\sin^2{u}}{u^2}\, du &=\frac{T}{2||k||\sqrt{1-\delta^2}}.
M_{\pm \gamma}=\left( M_{{ij}_{\pm \gamma}}\right) M_{{ij}_{\pm \gamma}}=\begin{pmatrix}\frac{s_{ij}\pm \gamma c_{ij}}{2}\frac{s_{ij}\mp \gamma c_{ij}}{2}\frac{s_{ij}\mp \gamma c_{ij}}{2}\frac{s_{ij}\pm \gamma c_{ij}}{2}\end{pmatrix},\ 1\leq i,j\leq n
\sum_j\int_{S_j(x_1)}\frac{1}{\mu(Q_{j})}\int_{Q_{j}}f(x_1,y_2)\,d\mu(y_2)\,\chi_{S_j}(x)\,d\mu(x_2) &=\sum_{j\colon S_j(x_1)\neq \emptyset}\int_{Q_j(x_1)}\frac{1}{\mu(Q_{j}(x_1))}\int_{Q_{j}(x_1)}f(x_1,y_2)\,d\mu(y_2)\,\,d\mu(x_2)\leq \int_Y f(x)\,d\mu(x_2)
\sum_{Q_1\in \mathcal{P}}\varphi_{Q_1}= \sum_{Q_1\in \mathcal{P}\setminus\mathcal{P}_1}\varphi_{Q_1}+ \sum_{Q_1\in \mathcal{P}_1}(\varphi_{Q_1}- \varphi'_{Q_1}) + \sum_{Q_1\in \mathcal{P}_1}\varphi'_{Q_1}& \in \sum_{Q_1\in \mathcal{P}\setminus\mathcal{P}_1}I_{Q_1,A}+ \sum_{Q_1\in \mathcal{P}}I_{Q_1,A'}+ \sum_{Q_1\in \mathcal{P}_1}\varphi'_{Q_1}.
a ( a^{\dagger})^{2 n}= 2 n ( a^{\dagger})^{2 n - 1}+ ( a^{\dagger})^{2 n}a , b ( b^{\dagger})^{2 n}= 2 n ( b^{\dagger})^{2 n - 1}+ ( b^{\dagger})^{2 n}b
1 + \sum_{n\geq 1}\frac{t\alpha_n(t,q)}{(1-t)^n}\frac{x^n}{2^{n-1}n!}= \sum_{v\geq 0}\left(\frac{2t}{1-t}\right)^v\left[\sum_{u \geq 0}\left(\frac{q-t}{1-t}\right)^u\left(e^{x/2}-1\right)^u\right]^v
I_{1 1 d}= -{\frac{1}{2 \kappa_{( 1 1 )}^{\ 2}}}\; \int d^{1 1}x \ \left[ \sqrt{- G}\; \left( R +{\frac{1}{4 8}}( d A )^{2}\right) +{\frac{1}{6}}A \wedge d A \wedge d A \right] \ ,
\begin{array}{l}{{w_{2 i + 1}= x_{i}^{- 1}x_{i + 1}, ~ ~ i = 0 , 1 , 2 , \cdots , N - 1 ,}}\\{{w_{2 j}= y_{j}, ~ ~ j = 1 , 2 , \cdots , N - 1 ,}}\\{{w_{2 N}= \prod_{l = 1}^{N - 1}y_{l}^{- 1},}}\\ \end{array}
\lim_{N\to\infty}\sup_{t\in[0,T]}\sup_{n\in\mathbb N}\sum_{m,k\geq N}\mathbb E\left[ \langle \tilde Z_t^{2,n}, e_k\otimes e_m\rangle_{\mathcal H}^2\right] = \lim_{N\to\infty}\sup_{t\in[0,T]}\sup_{n\in\mathbb N}\mathbb E\left[\|p_N\tilde Z_t^{2,n}\|_{\mathcal H}^2\right]=0,
0 = \vert \nabla f + \ell_{0}\nabla \ell_{0}\vert^{2}- \vert \nabla f \vert^{2}&= \vert \nabla f \vert^{2}+ 2 \ell_{0}\nabla f \cdot \nabla \ell_{0}+ \ell_{0}^{2}\vert \nabla \ell_{0}\vert^{2}- \vert \nabla f \vert^{2}&= \ell_{0}\big( 2 \nabla f \cdot \nabla \ell_{0}+ \vert \nabla \ell_{0}\vert^{2}\ell_{0}\big) .
\sum_{n=q_{k-1}}^{q_k-1}\min(\psi(n),\|q_{k-1}\theta\|)+\sum_{n=q_k}^{q_{k+1}-1}\min(\psi(n),\|q_k\theta\|)\geq(q_{k+1}-q_{k-1})\|q_k\theta\|\geq\frac{q_{k+1}}{2}\|q_k\theta\|\geq\frac{1}{4}.
\liminf_{u \to \infty}\Pr[ \theta(X) \in G \mid \rho(X) > u ] = \liminf_{u \to \infty}\frac{\Pr[ u^{-1}X \in T^{-1}( (1, \infty) \times G ) ]}{\Pr[ \rho(X) > u ]}\ge \mu \circ T^{-1}( (1, \infty) \times G ) = H(G).
\mathcal{M}_{( n )}^{( \mathcal{N}= 4 )}= \frac{\mathrm{S U}( 1 , 1 )}{\mathrm{U}( 1 )}\times \frac{\mathrm{S O}( 6 , n )}{\mathrm{S O}( 6 ) \times \mathrm{S O}( n )}.
\begin{array}{l l}{{n \left( \frac{1 - k}{1 + k}\right) = 2 G \left( U - T - J^{2}- \nu S^{2}\right)}}\\{{B^{2}= 1 - 8 G \left( U + \frac{( J^{2}+ \rho S^{2})}{2}\right)}}\\{{m^{2}= 4 G ( J^{2}+ \nu S^{2}) .}}\\ \end{array}
\Vert u(t) \Vert_{s}^2 \rbrace \lesssim \int_0^1 \frac{1}{\ell^{(2s+1)}}\Big( \int_{S^1}{|u(t,x+\ell)-u(t,x)|^2 dx}\Big) d \ell = \int_{0}^{1}{\frac{S_2(\ell)}{\ell^{(2s+1)}}d\ell}=\int_{J_1}{\frac{S_2(\ell)}{\ell^{(2s+1)}}d\ell}+\int_{J_2}{\frac{S_2(\ell)}{\ell^{(2s+1)}}d\ell}+\int_{J_3}{\frac{S_2(\ell)}{\ell^{(2s+1)}}d\ell}.
\mathbb P\left [\underset{1\leq r\leq n_1n_2}{\max}\;\underset{0\leq k\leq \binom{n_1n_2}{r}}{\max}\left \{\Vert \Pi_{r,k}(E)\Vert^{2}_{2}-K_1 r\log\left (\dfrac{e\,n_1\,n_2}{r}\right )\right \}\geq \Delta\right ]\leq c_1\exp\left \{-\frac{c_2\,\Delta^{2}}{K^{2}}\right \},
N_{\mathrm{c}\,m+1}=N_{m+1}-N_{\mathrm{d}\,m+1}-\sum_{n=1}^{m}\sum_{r=1}^{n}\binom{m+1}{m-n+1}N_{\mathrm{d}\,m-n+1}\mathcal{C}_{r}^{n}\left(N_{r}-N_{\mathrm{d}r}\right).
{\mathcal L}_G^{(1)}|=\frac{1}{\gamma^2}\,\, \frac{\partial_zv_1 \partial_{\bar z}v_1 +\partial_zv_2\partial_{\bar z}v_2}{1+v_1^2 + v_2^2},~~~~~~~~{\mathcal L}_G^{(2)}|=\frac{1}{\gamma^2}\,\, \frac{\partial_zc \partial_{\bar z}c +\partial_zv_0\partial_{\bar z}v_0}{c^2 + v_0^2-1}.
{\bf 1}= \int | z )_{q}d \mu_{q}( z )_{q}( z | \hspace{1 . 5 c m}\mathrm{w i t h}\hspace{1 . 5 c m}d \mu_{q}( z ) = \frac{[ 2 j + 1 ]}{_q ( z | z )_{q}^{2}}d_{q}^{2}z \ ,
(\begin{matrix}&h&\\v&&v^{\prime}\\ &h^{\prime}&\end{matrix})\circ_{2}(\begin{matrix}&k&\\v^{\prime}&&v^{\prime\prime}\\ &k^{\prime}&\end{matrix})=(\begin{matrix}&hk&\\v&&v^{\prime\prime}\\ &h^{\prime}k^{\prime}&\end{matrix})
\bar{q}_{\beta}(n,t)=\begin{cases*}E_{\beta, 1}\left(-\lambda t^{\beta}\right),\n=0, \displaystyle\rho^{n}\sum_{j=1}^{n}\sum_{m=1}^{j}(-1)^{m}\binom{j}{m}\left(\frac{-\lambda t^{\beta}}{(1-\rho)^{-r}-1}\right)^{j}\binom{rm+n-1}{n}E_{\beta, j\beta +1}^{j+1}\left(-\lambda t^{\beta}\right),\n\ge1.\end{cases*}
\hat{U}^{- 1}( \theta_{0}) | A_{\bar{z}}( z , \bar{z}) > = e^{- \frac{i k}{4 \pi}\int_{T_{2}}d z \wedge d \bar{z}\left[ \partial_{\bar{z}}\chi \partial_{z}\theta_{0}- \partial_{\bar{z}}\theta_{0}\overline{{{\partial_{\bar{z}}\chi}}}\, \right]}\| A_{\bar{z}}( z , \bar{z}) + \partial_{\bar{z}}\theta_{0}( z , \bar{z}) > \ \ \ ,
B^{a}= \left( \begin{array}{c c c c}{c_{1}^{a}}&{\ldots}&{0}&{0}\\{\vdots}&{\ddots}&{\vdots}&{0}\\{0}&{\ldots}&{c_{l}^{a}}&{0}\\{0}&{0}&{0}&{Y_{n}^{a}}\\ \end{array}\right) .
\hskip -2mm128(n-1)(n+2)(2n+1)(2n+3)(7n+15)X_{n}&-16(n+1)(7n^4-6n^3-121n^2-210n-90)X_{n+1}&-4(n+1)( n+2)^2( 56n^2+127n+57 )X_{n+2}&+(n+1)(n+2)(n+3)^2(7n+8)X_{n+3}=0.
p_1:=\frac{\binom{k+l-3}{k-1}}{2^{k+l-3}}\hskip 1truecm \hskip 1truecm p_2:=\frac{2^{k+l-3}-\binom{k+l-3}{k-1}-\binom{k+l-3}{k-2}}{2^{k+l-3}}.
-{\textstyle \frac{1}{2 \sqrt{2}}}e^{- \phi}\sigma^{a b}F_{a b}^{+}\alpha_{I J}\gamma_{\hat{\imath}}\epsilon_{J}= \gamma^{j}[ -{\textstyle \frac{1}{\sqrt{2}}}\omega_{\hat{\imath}}F_{0 j}^{+}+{\textstyle \frac{1}{\sqrt{2}}}e^{- 2 \phi}F_{0 j}^{+}\gamma^{i}\gamma^{0}] \alpha_{I J}\epsilon_{J}\, .
||\phi^\ast(\omega)||^2_f =\int_{M}||\phi^\ast(\omega)||^2_{p}f(p) dV_M &\leq \int_{M}\frac{a(p)}{b(p)}||\omega||^2_{\phi(p)}g(\phi(p)) \phi^\ast(dV_N) &\leq K \int_{M}||\omega||^2_{\phi(p)}g(\phi(p)) \phi^\ast(dV_N) &= K\int_N ||\omega||^2_x g(x)dV_N=K ||\omega||^2_g
\P(Y_{k_j}^* \ge u_j) \ge (1-2\theta_j)\frac{n}{\sqrt{n}+ n}=: 1- z_j, z_j = \left(1-\frac{n}{\sqrt{n}+ n}\right) + 2 \theta_j \frac{n}{\sqrt{n}+ n}= \frac{1}{\sqrt{n}+ 1}+ 2 \theta_j \frac{n}{\sqrt{n}+ n}.
\lim_{\lambda\in\Lambda}\varphi \left( f,h_\lambda,\dotsc,h_\lambda \right)=\sum_{[\pi]\in\mathcal{F}}\varphi \left( d_\pi f\ast\chi_\pi,d_\pi \chi_\pi,\dotsc,d_\pi \chi_\pi \right)=\Phi_0(f).
d_{C_1Cn}(M, \, Y_q(\Omega)) \underset{\mathfrak{Z}_0}{\lesssim}\sum \limits_{t=0}^{\hat t(n)}\sum \limits_{m=0}^\infty d_{k_{t,m}}(W_{t,m}, \, l_q^{\nu_{t,m}}) + \sup_{f\in M}\|f\|_{Y_q(\tilde \Omega_{\hat t(n)+1})}.
\int_{-\infty}^{\infty}\epsilon\left[\phi_1, \phi_2\right] dz \geq\int_{-\infty}^{\infty}dz \,\frac{d}{dz}\left({\phi_1}-\frac{2}{3}\,\phi_1^3 -2 \phi_2^2 \phi_1 \right) = \frac{\sqrt 2}{3},
\mathbb{E}\Big[ r_{s}(X_n^x) \psi_s \big( \log |G_n x| - n(q_t+l) \big) \Big] = \frac{r_t(x)}{\nu_t(r_t)}\nu_{t}(r_s) \frac{e^{-n \Lambda^*(q_t+l)}}{(t -s)\sigma_{t}\sqrt{2 \pi n}}\big[ 1 + o(1) \big].
i^{0,0}(H^3_c(Y;\mathbb{Q})) = 1, i^{1,1}(H^3_c(Y;\mathbb{Q})) = ph(Y,w) - 3, i^{2,1}(H^3_c(Y;\mathbb{Q})) = i^{1,2}(H^3_c(Y;\mathbb{Q}))=h^{1,2}(Z).
|f(Y)-f(X)| =\Big|\sum_{j\in[q]}h_j(1)-\sum_{j\in[q]}h_j(0)\Big| &\le\sum_{j\in[q]}|h_j(1)-h_j(0)| &=\sum_{j\in[q]}|h_j'(\tau_j)| &\le\sum_{j\in[q]}\sqrt{2}\|a_j\|_2\|Y-X\|_* &=\Big(\sqrt{2}\sum_{j\in[q]}\|a_j\|_2\Big)\|Y-X\|_*.
q_{\rho}\left( x,t\right) =2\left[{\displaystyle\sum\limits_{n=1}^{N}}c_{n}^{2}\mathrm{e}^{8\kappa_{n}^{3}t-\kappa_{n}x}\psi_{\rho}\left(x,t;\kappa_{n}\right) \right]^{2}& -4{\displaystyle\sum\limits_{n=1}^{N}}\kappa_{n}c_{n}^{2}\mathrm{e}^{8\kappa_{n}^{3}t-\kappa_{n}x}\psi_{\rho}\left(x,t;\kappa_{n}\right) .
n^2 \sum\limits_{i=0}^k \P(A_{r, r_1, k, i}) =n^2\sum\limits_{i=0}^k{k\choose i}p^{i}q^{k-i}f^{\left(i\right)}_{2r_1}f^{\left(k-i\right)}_{2(r-r_1)}\to \frac{2pql^2}{\pi(x(t-x))^{3/2}}e^{-\frac{l^2}{2}\left(\frac{p^2}{x}+\frac{q^2}{t-x}\right)},
c = \frac{\delta(n+1)}{u H_{Y}(u)}\left(\frac{H_{Y}(u)(H_{Y}(u)-1)}{2}+\epsilon'\right) &\le \frac{\delta(n+1)H_{Y}(u)}{2u}=\frac{\delta\binom{n+u}{u}\epsilon}{2d(2n+1)}&\le \delta e^n(2n+4)^{n-1}q^nd^{n^2+n-1}\deg(V)^n\epsilon^{-n+1}.
\lambda\left(B\cap\bigcap_{n\le k}H_n^c\right) =\sum_{j=1}^I\lambda\left(A_j\cap K_j\cap\bigcap_{n\le k}H_n^c\right) &=\lim_r\sum_{j=1}^I\lambda\left(A_j\cap K_j\cap\bigcap_{n\le k}H_n^c\cap\bigcup_{n\le r}H_n\right) &\le\lim_r\lambda\left(\bigcap_{n\le k}H_n^c\cap\bigcup_{n\le r}H_n\right)
\Psi^\star_L(\mu, \nabla \tilde G_L) = 2\sum_{i\in\mathbb T_L^d}\sum_{k=1}^d \hat a_{i,i+e_k}(\mu) L^2\Bigl[\cosh\bigl(\tfrac 12 \nabla^{i,i+e_k}G(\cdot/L)\bigr)-1\Bigr]
\gamma(\nabla_X\alpha)\phi+\gamma(\alpha)\widetilde{\nabla}_X\phi =\gamma((X\alpha_k)X_k+\alpha_k\nabla_XX_k)\phi+\gamma(\alpha)\widehat{\nabla}_X\phi+\gamma(\alpha)L(X)\phi &=(X\alpha_k)(R_k\phi)+\alpha_k\gamma(\nabla_XX_k)\phi+\gamma(\alpha)\widehat{\nabla}_X\phi+\gamma(\alpha)L(X)\phi.
\sum_{f=1}^{F}\sum_{k=1}^{K}\mathbb{E}\{\mid\tilde{\boldsymbol{H}}(f,k)\mid^2\}=\sum_{f=1}^{F}\sum_{k=1}^{K}\mathbb{E}\{\mid\hat{\boldsymbol{H}}(f,k)\mid^2\}&\stackrel{(\star)}{=}2FK\hat{\sigma}^2,
HP(\varphi: (F, w_F)\circlearrowleft , \Gamma) \simeq HM(M_\varphi,\mathfrak{s}_\Gamma) \text{if \(d_\Gamma:=\langle P.D.[F],\Gamma\rangle<g-1\), and}HP(\varphi: (F, w_F)\circlearrowleft , \Gamma) \simeq HM(M_\varphi, \mathfrak{s}_\Gamma, c_-)
\int_0^{\infty}u e^{-a u^2}ds =\int_0^{\infty}\frac{1}{2a}e^{-z}dz = \left. -\frac{1}{2a}e^{-z}\right|_0^{\infty}= \frac{1}{2a}, \int_0^{\infty}u e^{-au}du = \int_0^{\infty}\frac{z}{a^2}e^{-z}dz = \frac{\Gamma(2)}{a^2}= \frac{1}{a^2}.
| (\Phi^{b,\psi}_L)_* \mu(A_p) - (\Phi^{a,\phi}_L)_*\mu(A_p) | = |\mu(\tilde{D}_{\jmath_p}) - \mu(B_{\jmath_p})| \leq |\mu(\tilde{D}_{\jmath_p}) - \mu(D_{\jmath_p})| + |\mu(D_{\jmath_p}) - \mu(B_{\jmath_p})| < (|F_p| k^{|F_p|}) ( 4 \delta k^{2 |F|}) + 2 \delta k^{2 |F|}.
\{\theta_{1}, \theta_{2}\}_{\mathrm{c . b .}}= \beta_{1}\left( P - \frac{1}{N}I \right) + \left\{\theta_{1}[ \theta_{2}, r ] \right\}+ \beta_{2}\left( \theta_{1}P \theta_{2}+ \theta_{2}P \theta_{1}- \frac{2}{N}\theta_{1}\theta_{1}- \frac{2}{N}\theta_{2}\theta_{2}\right) .
\chi_{\min}(f) =\lim_{t\to\infty}\frac{1}{t}\log\left(\min_{z\in X_{f}}E(z,t;f)\right)\intertext{and}\chi_{\max}(f) =\lim_{t\to\infty}\frac{1}{t}\log\left(\max_{z\in X_{f}}E(z,t;f)\right).
\{z_i , z_k \}= \begin{cases}\langle \gamma^i(\alpha_i), \, \gamma^k(\alpha_k) \rangle z_iz_k, \mbox{if}\gamma_i=e -\langle \gamma^i(\alpha_i), \, \gamma^k(\alpha_k) \rangle z_iz_k -\langle \alpha_i,\alpha_i \rangle \sigma_i(z_k) \mbox{if}\gamma_i=s_i\end{cases},1\leq i < k \leq n,
\sum_{j=1}^{d_n}(g_j-\varepsilon)t^j = \sum_{j=1}^{d_n}g_j t^j - \varepsilon \sum_{j=1}^{d_n}t^j \ge s -\varepsilon \sum_{j=1}^{d_n}t_0^j > s- \frac{(s+c_n)(t_0-1)}{t_0(t_0^{d_n}-1)}\sum_{j=1}^{d_n}t_0^j = s- (s+c_n) = -c_n.
K(k)=\frac{\pi}{2}(1+(\frac{1}{2})^{2}k^{2}+(\frac{1\cdot3}{2\cdot4})^{2}k^{4}+\cdot\cdot\cdot+(\frac{(2n-1)!!}{(2n)!!})^{2}k^{2n}+\cdot\cdot\cdot),
\Psi_\theta\Psi_\theta^+=\frac{1}{X_\theta^2+Y_\theta^2+Z_\theta^2}\begin{pmatrix}X_\theta^2 X_\theta Y_\theta X_\theta Z_\theta Y_\theta X_\theta Y_\theta^2 Y_\theta Z_\theta Z_\theta X_\theta Z_\theta Y_\theta Z_\theta^2 \end{pmatrix},
=E_{0}\frac{r_{0}}{x}(1+\frac{2x^{2}}{n^{2}+x^{2}}-\frac{2x^{2}}{m^{2}+x^{2}}+\frac{2x(n+x)}{(n+x)^{2}+m^{2}}-\frac{2x(n-x)}{(n-x)^{2}+m^{2}})
X^c = \left(\bigcap_{i \in \mathbb{N}}X_i\right)^c \subseteq \left(\bigcap_{i \in \mathbb{N}}X_{i,m_{i,n}^{\infty}}\right)^c = \bigcup_{\sigma \in \mathbf{C_n^{\infty}}}[\mathbf{I_{\sigma,n}^{\infty}}]
\tilde{E}^{(i)}_{n\delta}=\sum_{\begin{array}{c}k_1p_1+...+k_mp_m=n 0<k_1<...<k_m\end{array}}\frac{\left ( q^{(\alpha_i,\alpha_i)}-q^{-(\alpha_i,\alpha_i)}\right )^{\sum_ip_i-1}}{p_1!\;...\;p_m!}(E^{(i)}_{k_1\delta})^{p_1}...(E^{(i)}_{k_m\delta})^{p_m}
\delta_{\bf v}{\cal L}_{0}= \mu \dot{\bf x}\cdot \delta_{\bf v}\dot{\bf x}+ \frac{\mu}{R^{2}}{\bf x}\cdot \delta_{\bf v}{\bf x}= \mu{\bf v}\cdot \frac{d}{d t}\left[{\bf x}\cosh \left( \frac{t}{R}\right) \right] ,
K(T,R) &:=\left\{v \in C^1([0,T) ; L^2(\mathbb{R}^n)) \cap C ([0,T) ; H^1(\mathbb{R}^n))\,;\, \sup_{t\in [0,T)}\| (v, v_t )(t) \|_{H^1 \times L^2}\le 3R \right\}.
\Delta\big(\frac{\partial f}{\partial t}-\big(1+\frac{\alpha'}{2}R\big)\Delta f-rf-\frac{\alpha'}{4}R^2f\big)=-\alpha'(\frac{1}{2}R\nabla R\nabla f+\frac{1}{4}f\Delta R^2+\nabla R\nabla\Delta f+\Delta R\Delta f)-\frac{\partial r}{\partial t}.
\lim_{m\rightarrow+\infty}\frac{y^{(t)}}{\varpi\circ W\left(z^{(t)}\right)}=\begin{cases}\qquad\gamma \mbox{if}\gamma_{2}=0,\delta_{1}\neq0, \frac{\gamma x_1^{(t_0)}+\delta x_{2}^{(t_{0})}}{x_{1}^{\left(t_{0}\right)}+x_{2}^{\left(t_{0}\right)}}\mbox{if}\gamma_{2}=\delta_{1}=0, \qquad\delta \mbox{if}\gamma_{2}\neq0,\delta_{1}=0.\end{cases}
d(T^n x, T^{n+l}x) \le \sum_{i=0}^{l-1}d(T^{n+i}x,T^{n+i+1}x) \le \sum_{i=0}^{l-1}k^{n+i}d(x,Tx) \le \frac{k^n}{1-k}d(x,Tx).
\mu \,:\, A\otimes A \to A ~~ , \begin{pmatrix}x_0 x_1\end{pmatrix}\otimes\begin{pmatrix}y_0 y_1\end{pmatrix}\mapsto \begin{pmatrix}x_0y_0+x_1y_1 x_0y_1+x_1y_0 \end{pmatrix}
|F_{n+1}|\lesssim \mathbf{M}^{1-},|\nabla_pF_{n+1}|\lesssim \mathbf{M}^{1-}, &|\nabla_xF_{n+1}|\lesssim \mathbf{M}^{1-},|\nabla_x\nabla_pF_{n+1}|\lesssim \mathbf{M}^{1-}, &|\nabla_x^2F_{n+1}|\lesssim \mathbf{M}^{1-},|\nabla_x^2\nabla_pF_{n+1}|\lesssim \mathbf{M}^{1-}.