harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varepsilon_{16}^N=\int_0^T \frac{1}{N\gamma_N}\sum_{i=1}^N\sum_{j=1}^N\lambda_1\left(\frac{i}{N}\right)\lambda_2\left(\frac{j}{N}\right)\times &\left(\mathbb{E}\left(S_s^N(i)I_s^N(j)\right)-\mathbb{E}\left(S_s^N(i)\right)\mathbb{E}\left(I_s^N(j)\right)\right)\left(G_s\left(\frac{i}{N}\right)-F_s\left(\frac{i}{N}\right)\right)ds,
\Im_{f}=\left\{\frac{N}{K}\subseteq \frac{M}{K}\left|\;\forall \bar{U}\in \frac{N}{K}\; \exists\; \, n\in{\mathbb Z}^{+}\, \,{\rm such\; that\; \;}\bar{U}+\frac{M_{n}+K}{K}\subseteq \frac{N}{K}\right. \right\}.
\bigg(1 - \frac{3\delta}{4}\bigg)\bigg\| \sum_{j=1}^3 L_j^2f \bigg\|_{L^2(\mathbf{R}^3)}^2\leq\bigg(\frac{3}{4\delta}- \frac{9}{16}\bigg)\bigg\| \frac{f}{|x|^2}\bigg\|^2_{L^2(\mathbf{R}^3)}+\|\Delta f\|^{2}_{L^2(\mathbf{R}^3)}.
m(\mu) =2^{l}\binom{d}{l}u(\mu) \ \ \ \u(\mu)=\binom{l}{m_1(\mu),\ldots,m_h(\mu)}, X^{(d)}_{m,h}(n;\mu) =2(d-1)\delta_{h,0}+\frac{2n\bigl(-(2d-1)\bigr)^{\frac{n-mh}{2}}}{n+mh}\binom{mh}{m\mu_1,\ldots,m\mu_l}P^{(m\mu,-1)}_{d,\frac{n-mh}{2}}\Bigl(\frac{2d-3}{2d-1}\Bigr).
\Psi(E_j(x, a)) = \begin{cases}2 \sqrt{a}\left( \cot\left( \dfrac{\pi}{2j+2}\right) - 1 \right) , 2 \nmid j , 2 \sqrt{a}\left( \csc\left( \dfrac{\pi}{2j+2}\right) - 1 \right) , 2 \mid j . \end{cases}
\|U_F^{\mathcal{N}}\|\leq \bar{C}(T)\exp(\mathcal{N}\tau_F)\left(\|U_F^{0}\| +\sum^{\mathcal{N}}_{n=1}\tau_F\|b^{n}\|+\sum^{\mathcal{N}}_{n=1}\tau_F +\sum^{\mathcal{N}}_{n=1}\tau_F\|U_C^n\|\right)\leq \bar{C}.
{\bf E}\,\exp \left\{\lambda |\xi_{n}|\right\}\le 4\,\exp \left\{\sum_{m=2}^{\infty}\left( \frac{720e\lambda \kappa}{\log m}\right)^{m}\left( \Omega^{m/2}(n,\overline{C}_{n})+\frac{1}{n^{m/2-1}}\,\Omega (n,\overline{C}_{n})\right) \right\},
\varLambda(-b^*-^*a,-d^*-^*c) =\varLambda(b^*+a^*,d^*+c^*)-\varLambda(b^*,c^*)-\varLambda(a^*,d^*)+\varLambda(b^*,^*c)+\varLambda(^*a,d^*) =\varLambda(b^*+a^*,d^*+c^*)+ \langle b,c \rangle - \langle d,a \rangle .
\begin{aligned}\int_{\Sigma_s}(\partial_t \phi) (\partial_t \psi) + (\partial^i \phi) (\partial_i \psi) d x = \int_{\Sigma_0}(\partial_t \phi) (\partial_t \psi) + (\partial^i \phi) (\partial_i \psi) d x - \int_0^s \int_{\Sigma_t}(\Box \phi) (\partial_t \psi) d x d t, \end{aligned}
\left( P e^{- i g \int_{x_{0}}^{x_{0}+ \beta}d t^{\prime}A_{0}( t^{\prime}, \vec{x})}\right) \rightarrow U^{- 1}( x_{0}, \vec{x}) \left( P e^{- i g \int_{x_{0}}^{x_{0}+ \beta}d t^{\prime}A_{0}( t^{\prime}, \vec{x})}\right) U ( x_{0}, \vec{x})
\sum_{|{\alpha_1}|\le k}\left\|\|\nabla^{|\alpha_1|}f\|_{L^2_{p,\mathcal{I}}}\|\nabla^{k-|\alpha_1|}f\|_{L^2_{p,\mathcal{I}}}\right\|_{L^2_x}^2\le C\delta^2\biggl(\|\nabla^{k+1}f\|^2_{L^2_{x,p,\mathcal{I}}}+ \|\nabla^k \{I-P\}f\|^2_{L^2_{x,p,\mathcal{I}}}\biggl)
F = \left[\begin{array}{ccccc}0 I 0 \cdots 0 0 0 I \cdots 0 \vdots \vdots \vdots \ddots \vdots 0 0 0 \cdots I \end{array}\right], G = \left[\begin{array}{ccccc}I 0 \cdots 0 0 0 I \cdots 0 0 \vdots \vdots \ddots \vdots \vdots 0 0 \cdots I 0 \end{array}\right],
\tilde{\Gamma}_{5}( p , p^{\prime}) \to \gamma_{5}\left( 1 - \frac{\alpha}{2 \pi}\right) - \gamma_{5}\frac{\alpha}{2 \pi}P^{2}\int_{4 m^{2}}^{\infty}\frac{d M^{2}}{M^{2}}\frac{a ( M^{2})}{M^{2}+ P^{2}},
\frac{2\,(2n-1)^3\,\pi^3\,e^{2(2n-1)\pi y}}{(e^{(2n-1)\pi y}-1)^3}-\frac{(2n-1)^3\,\pi^3\,e^{(2n-1)\pi y}}{(e^{(2n-1)\pi y}-1)^2}=\frac{(2n-1)^3\,\pi^3\,e^{(2n-1)\pi y}}{(e^{(2n-1)\pi y}-1)^3}\left(e^{(2n-1)\pi y}+1\right).
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^+_I}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_d \left(s+k-1\right) R^d \eta^{-1}y\end{aligned}
\phi \ : \X = \begin{bmatrix}X_{11}\dots X_{1h}\cr \vdots \ddots \vdots \cr X_{k1}\ldots X_{kh}\end{bmatrix}, \hbox{}X_{ij}\in\mathbb{F}^{n\times n}\mapsto F(x,y) = \sum_{i=0}^{k-1}\sum_{j=0}^{h-1}X_{k-i,h-j}\phi_i(y)\phi_j(x).
(\chi_{\{m'\}},T(t)\chi_{\{m\}})\, = \, \exp(-\frac{1}{2}\lambda_{\{m\}}g_0^2 L_x t)\delta_{\{m'\},\{m\}}\, \stackrel{!}{=}\, (\chi_{\{m'\}}, \exp(-t H)\chi_{\{m\}})
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j_{b,m}(X,E) &=r(\sum^{p}_{s=1}e^{\frac{2\pi(s-1)}{p}mi})\frac{m^{n}}{n!(2\pi)^{n+1}}\int_X(-d\omega_0)^n\wedge\omega_0+O(m^{n-1}),
a = \left( \begin{array}{c c c c c c c}{0}&{1}\\{1}&{0}&{- 1}\\{}&{- 1}&{0}&{1}\\{}&{}&{1}&{0}\\{}&{}&{}&{}&{\ddots}\\{}&{}&{}&{}&{}&{0}&{1}\\{}&{}&{}&{}&{}&{1}&{0}\\ \end{array}\right)
\sum_{i_1=0}^{r-1}\sum_{i_2=0}^{s-1}\sum_{i_3=0}^{t-1}\binom{i_1+i_2+i_3}{i_1,i_2,i_3}^2\sum_{m_1+m_2+m_3<p}\binom{m_1+m_2+m_3}{m_1,m_2,m_3}^2 &\qquad\cdot\left(1+2p((i_1+i_2+i_3)H_{m_1+m_2+m_3}-i_1H_{m_1}-i_2H_{m_2}-i_3H_{m_3})\right).
S_4 = (2 \pi)^4 \int \left[ \prod_{j = 1}^2 \frac{d^{\, 4}P_j}{(2 \pi)^4}\frac{d^{\, 4}Q_j}{(2 \pi)^4}\right] G_4 \, H \, \delta^4 \left( \sum_l P_l + \sum_{j = 1}^2 (Q_j - P_j) \right).
d\bar{\tilde{v}}_{n}^{1}\wedge d\tilde{v}_{n}^{1}=-d\bar{\tilde{v}}_{n}^{3}\wedge d\tilde{v}_{n}^{3}, \ \ \d\bar{\tilde{v}}_{n}^{2}\wedge d\tilde{v}_{n}^{2}=0,
\Delta(k)=\frac{1}{2}\left( \begin{array}{cc}\chi_{R,L}(k)+\chi_{R,L}^{-1}(k) i\left(\chi_{R,L}(k)-\chi_{R,L}^{-1}(k)\right) -i\left(\chi_{R,L}(k)-\chi_{R,L}^{-1}(k)\right) \chi_{R,L}(k)+\chi_{R,L}^{-1}(k) \end{array}\right),
\mathrm{SLNR}_{n,k}=\frac{\big|\mathbf{h}_{n,k}\mathbf{w}_{n,k}\big|^{2}}{\sigma_{k}^{2}+\sum\limits_{\substack{i=1 i\neq{}k}}^{k_{n}}\big|\mathbf{h}_{n,i}\mathbf{w}_{n,k}\big|^{2}+\sum\limits_{\substack{j=1 j\neq{}n}}^{N}\sum\limits_{q=1}^{k_{j}}\big|\mathbf{g}_{n,q}\mathbf{w}_{n,k}\big|^{2}}.
\lim_{n \to \infty}\| P_k ( \alpha_{\pi(t)}\phi_n(T) \alpha_{\pi(t^{-1})}- \phi_n( \alpha_{\pi(t) \oplus \pi(t)}T \alpha_{\pi(t^{-1}) \oplus \pi(t^{-1})}) ) P_k \| = 0.
\lambda_1(A_{\alpha}(K_m\vee H)) -\lambda_1(A_{\alpha}(K_m\vee P_n)) &\geq2(1-\alpha)( x_{i-1}x_{i+k+1}+x_{i}x_{i+k}-x_{i-1}x_i-x_{i+k}x_{i+k+1}) &=2(1-\alpha)(x_{i+k}-x_{i-1})(x_{i}-x_{i+k+1}).
E^\mathrm{Bog}=\; -\frac{1}{2}\mathrm{Tr}_{\perp,r}\left[ D_r+\lambda Q_r K_{11}Q_r-\sqrt{D_r^2+2\lambda D_r^{1/2}Q_rK_{11}Q_rD_r^{1/2}}\right] &-\frac{1}{2}\mathrm{Tr}_{\perp,\ell}\left[ D_\ell+\lambda Q_\ell K_{22}Q_\ell-\sqrt{D_\ell^2+2\lambda D_\ell^{1/2}Q_\ell K_{22}Q_\ell D_\ell^{1/2}}\right].
\frac{1}{C_{1}\xi \log^{2}(\xi)}\leq \rho(\xi) \leq \frac{C_{1}}{\xi \log^{2}(\xi)}, 0 < \xi < \frac{1}{2e^{2}}&\frac{\xi}{C_{1}}\leq \rho(\xi) \leq C_{1}\xi, \xi > \frac{1}{2e^{2}}
J_2=C_{2}\int_{\Omega \cap B_{r_{0}}^c(x_1)}\frac{v^p (y)}{|x_1-y|^2}\mathrm{d}y\leq C_{2}r_{0}^{-2}\int_{\Omega}v^p(y)\mathrm{d}y\leq C_{5}N^{\frac{2p}{n}}.
Q_1:= \alpha F^{-\alpha-2}\ddot{F}^{kl,mn}\nabla_1h_{kl}\nabla_1h_{mn}-\alpha(\alpha+1)F^{-\alpha-3}(\nabla_1F)^2 &+\alpha(\alpha+1)F^{-\alpha-4}\dot{F}^{kl}\nabla_kF\nabla_lF\kappa_1 &\quad+2\alpha F^{-\alpha-1}\sup_{\Lambda}\dot{F}^{kl}\left(2\Lambda_k^p\nabla_lG_{1p}-\Lambda_k^p\Lambda_l^qG_{pq}\right)~\geq~0
p^\ell(p^\ell+1)(k-1) =\frac{e_0p^a(p^{2a}-1)}{2(p^{\ell}-1)}-p^\ell (p^{\ell}+1) =\frac{e_0(p^a-1)}{2(p^{\ell}-1)}(p^{2a}+p^a+2)-\frac{e_0(p^a-1)}{p^{\ell}-1}-p^\ell(p^\ell+1),
n(\beta, \mu)=\frac{\sum_{l=0}^{n-1}\sum_{p=0}^{n-1}q^{-pl}\sum_{k^{m}_{l}}f_{q^{p}}(\beta(\epsilon^{l}(k^{m}_{l})-\mu))\prod_{k^{m}_{l}}(1 + q^{p}e^{\beta (\mu - \epsilon^{l}(k^{m}_{l}))})}{M \sum_{l=0}^{n-1}\sum_{p=0}^{n-1}q^{-pl}\prod_{k^{m}_{l}}(1 + q^{p}e^{\beta (\mu - \epsilon^{l}(k^{m}_{l}))})}\ .
Z_\phi (\sqrt{E})= &2^{N}\displaystyle \int_{\sum_{i=1}^{N-1}y_i^2 \leq E}\frac{1}{\left |\phi'(\phi^{-1}(E-\sum_{i=1}^{N-1}y_i^2 ))\right |}\prod_{i=1}^{N-1}\frac{|y_i|}{|\phi'(\phi^{-1}(y_i^2))|}dy_1 \dots dy_{N-1}.
\chi_{t o p}= \int \langle q ( x ) q ( 0 ) \rangle d^{2}x = \frac{e^{2}}{( 2 \pi )^{2}}\int \langle F_{1 2}( x ) F_{1 2}( 0 ) \rangle d^{2}x = \frac{e^{2}}{2 \pi}\hat{G}_{F F}( 0 ) \; ,
n = ( n_{-}+ n_{+}) = \frac{< N >}{V \Lambda^{2}}= \Bigg [ ( 1 + \beta^{\prime})^{\frac{1 + \beta^{\prime}}{2}}( 1 - \beta^{\prime})^{\frac{1 - \beta^{\prime}}{2}}4 \pi \beta \frac{e^{2 \gamma_{E}}}{8}\frac{N_{c}}{N_{c}- 1}2^{\frac{8 ( N_{c}- 1 )^{2}}{\beta N_{c}^{2}}}\Bigg ]^{\frac{\beta N_{c}^{2}}{8 ( N_{c}- 1 )^{2}- \beta N_{c}^{2}}}\; .
d s^{2}= \left( \frac{l}{u}\right)^{2}d u^{2}+ \left( \frac{u}{l}\right)^{2}d x^{+}d x^{-}+ \frac{l^{2}}{4}( d \theta^{2}+ \sin^{2}\theta d \phi^{2}) + d s^{2}( M^{6}) ,
\limsup_{t\to+\infty}\|\theta(t)\|_{L^1(B^c_{A\sqrt t})}\le \kappa A^{-1}, &\limsup_{t\to+\infty}\|u(t)\|_{L^3(B^c_{A\sqrt t})}\le \kappa A^{-2}, \qquad &\limsup_{t\to+\infty}\sqrt t\, \|u(t)\|_{L^\infty(B_{A\sqrt t}^c)}\le \kappa A^{-3}.
\prod_{j=1}^{s-1}\left(1 + \gamma_j^{\frac{1}{\alpha - 2 \delta}}2 \zeta\left(\frac{\alpha}{\alpha - 2 \delta}\right) \right) =\exp\left(\log \left(\prod_{j=1}^{s-1}\left(1 + \gamma_j^{\frac{1}{\alpha - 2 \delta}}2 \zeta\left(\frac{\alpha}{\alpha - 2 \delta}\right) \right)\right)\right),
\textrm{conv}(g)(x)=\inf\left\lbrace \sum_{j=1}^{n+1}\lambda_{j}g(x_j) \, : \, \lambda_j\geq 0,\sum_{j=1}^{n+1}\lambda_j =1, \, x=\sum_{j=1}^{n+1}\lambda_j x_j \right\rbrace
\setlength{\nulldelimiterspace}{0pt}\left[ \begin{array}{c}\gamma_{1}\gamma_{2}. . . \gamma_{T}\end{array}\right]_{2T\times N}\cdot\;\;\;\;\; x_{N \times 1}\;\; = \left[ \begin{array}{c}y_{1}y_{2}. . . y_{T}\end{array}\right]_{2T\times 1}.
\dot{\mathbf{y}}=\mathbf{V}^{-1}\mathbf{A}\mathbf{V}\mathbf{y}+\mathbf{V}^{-1}\mathbf{G}_{nlin}( \mathbf{V}\mathbf{y})=\mathbf{\Lambda}\mathbf{y}+\mathbf{G}(\mathbf{y}).
M= \mu\left( \begin{array}{cccccc}0 -a_1 0 0 0 0 a_1 0 0 0 0 0 0 0 0 -a_2 0 0 0 0 a_2 0 0 0 0 0 0 0 0 -a_3 0 0 0 0 a_3 0\end{array}\right)
\Gamma\lbrack A\rbrack = \int_0^{\infty}{dT\over T}\,{\rm e}^{-m^2T}\int{\cal D}x\,{\rm exp}\left[ - \int_0^T \!\!\! d\tau \left({1\over 4}{\dot x}^2 + ieA_{\mu}\dot x^{\mu}\right) \right]
\Psi_{\bf C}\rightarrow \left( \begin{array}{c c}{e^{\frac{\mathrm{i}}{2}\mathrm{{\bf \sigma \cdot}}( \mathrm{{\bf \theta}}- \mathrm{i}\mathrm{{\bf \phi}})}}&{0}\\{0}&{e^{\frac{\mathrm{i}}{2}\mathrm{{\bf \sigma \cdot}}( \mathrm{{\bf \theta}}+ \mathrm{i}\mathrm{{\bf \phi}})}}\\ \end{array}\right) \cdot \Psi_{\bf C},
(\,g_j\,|\,{\rm P\,}f_j\,) =(\,g_j-u^0\,|\,{\rm P\,}f_j\,)+(\,u^0\,|\,{\rm P\,}f_j\,) &=(\,g_j-u^0\,|\,{\rm P\,}f_j\,)+(\,u^0\,|\,{\rm P\,}v^0\,)+(\,u^0\,|\,{\rm P\,}(f_j-v^0)\,).
- \Delta v_{k,R}+ \frac{5}{3}v_{k,R}^{7/3}+ \left( m_{k}- v_{k,R}^{2}\cdot \chi_{B_{R}(0)}- u_{k}^{2}\cdot \chi_{B_{R}(0)^{\rm c}}\right) v_{k,R}= 0, & v_{k,R}= u_{k}\partial B_{R}(0).
\frac{(A_{k+1}- A_k)^{\frac{\tilde p}{\tilde p-1}}}{A_{k+1}}\leq (\epsilon\sigma)^{\frac{1}{\tilde p-1}}\frac{\tilde p}{\tilde p-1}\frac{(N^2 -1)^{\frac{\tilde p-1}{2\tilde p-2}}}{2 N}:= C_{\epsilon, \sigma,\tilde p,N}.
\widetilde{S}^{m}_t=X_0 +\int_0^t b\big(r, \widetilde{S}^{m},\overline{\beta}^{m}_r,\widetilde{\gamma}^m_r\big) \mathrm{d}r+\int_0^t \sigma \big(r, \widetilde{S}^{m},\overline{\beta}^{m}_r,\widetilde{\gamma}^m_r\big)\mathrm{d}W_r+ \int_0^t \sigma_0\big(r, \widetilde{S}^{m},\overline{\beta}^{m}_r,\widetilde{\gamma}^m_r\big) \mathrm{d}B_r,
\sum_{a\in S}a^{\ell+1}=\begin{cases}\frac{1}{\ell+2}tn^{\ell+1}+O_\ell\left(n^{\ell+11/8}\right),~ ~t\geqslant n^{1/2}-n^{1/4}\frac{1}{\ell+2}tn^{\ell+1}+O_\ell\left(n^{\ell+5/4}\sqrt{n^{1/2}-t}\right),~ t<n^{1/2}-n^{1/4}.\end{cases}
{( - 1)^{{\textstyle{L \over 2}}}}{T_{{\textstyle{L \over 2}}}}(C,1) = -{(\omega{\omega^{' - 1}})^{{\textstyle{L \over 2}}}}-{(\omega{\omega^{' - 1}})^{-{\textstyle{L \over 2}}}}-{( - 1)^{{\textstyle{L \over 2}}}}{s^{-{\textstyle{L \over 2}}}}{\eta^L}{(\omega{\omega^{' - 1}})^{-{\textstyle{L \over 2}}}}{f^L}{e^L}.
\sum_{m=0}^{n}\genfrac{[}{]}{0pt}{}{n}{m}_{q}h_{m}(\cos(\theta+\varphi)|q)h_{n-m}(\cos(\theta-\varphi)|q)=\sum_{j=0}^{\left\lfloor n/2\right\rfloor}\frac{(q)_{n}}{(q)_{j}(q)_{n-2j}}h_{n-2j}(\cos\theta|q)h_{n-2j}(\cos\varphi|q).
K_{i j}^{0}( p_{i j 0}^{\prime}, p_{i j 0}, P_{i j 0}) = \beta_{i}\beta_{j}\Lambda_{i j}^{+ +}\beta_{i}\beta_{j}K_{i j}^{T}( s_{i j}, s_{i j}, P_{0}- h_{k}) \Lambda_{i j}^{+ +}
\Lambda = v^{\mu}\langle \partial_{\mu}\gamma_{21}\psi^{-1}\rangle_0 -v^{\mu}(\partial_{\mu}\ln{\rho})\langle \psi\gamma_{21}\psi^{-1}\rangle_0 - v^{\mu}\partial_{\mu}\beta\langle \gamma_5 \psi \gamma_{21}\psi^{-1}\rangle_{0}.
\sum_{\alpha=1}^\infty\frac{1-e^{-2\lambda_\alpha t}}{2\lambda_\alpha}\leq \frac{t}{\sqrt{2\pi}}\sum_{\alpha=1}^\infty\frac{2\sqrt{\lambda_\alpha}t-\sin(2\sqrt{\lambda_\alpha}t)}{2\lambda_\alpha^{3/2}}\leq \sum_{\alpha=1}^\infty \frac{t}{\lambda_\alpha}\leq \frac{t}{6},
0 = G_i = \frac{{(|D w|^2 )_i}}{{|D w|^2}}+ h'u_i + \alpha_0 d_i, i = 1, 2, \cdots, n, 0 \geq G_t = \frac{{(|D w|^2 )_t}}{{|D w|^2}}+ h'u_t.
\beta_a \leftarrow \zeta_b\cdot \zeta_a^{-1}= -\zeta^a \frac{\sin(3\pi a /2^n)}{\sin(\pi a/2^n)}\b = 3a\bmod 2^{n}, 3a<2^{n-1}. \beta_a \leftarrow \zeta_b\cdot \zeta_a^{-1}= -\zeta^{-a}\frac{\sin(3\pi a /2^n)}{\sin(\pi a/2^n)}\b = -3a\bmod 2^{n}, 3a\geq2^{n-1}.
\int_C \big( \mathbf{e}^{\frac{1}{\mathbf{l}(\mathbf{z})}}\big)^{d\mathbf{z}}=\frac{\mathbf{z}_2}{\mathbf{z}_1}=\frac{(a,\pi )}{(a,-\pi )}=(1,2\pi )=\mathbf{e}^{2\pi i}.
\dot\Delta(\xi_n)= -\dot \theta_N(\xi_n)2 \sin \theta_N(\xi_n)+ \frac{2i(-1)^{n+1}}{(2i\xi_n)^{N+1}}\int_0^1\varphi_1(t)r_{N+1}(t)\,dt+\frac{1}{n^{N+1}}\ell^2_n
r(e_{\ell-2i-1}e_{\ell-2i}) =r(e_{\ell-2i-1}e_{\ell-2i}e_{\ell-2i+1})r(e_{\ell-2i+1}) &=r(e_{\ell-2i+2}^\psi)r(e_{\ell-2i+1})=\psi^{-1}r(e_{\ell-2i+2})\psi r(e_{\ell-2i+1})
y_{n}(t)=\int_{-\infty}^{+\infty}\cdot\cdot\cdot\int_{-\infty}^{+\infty}h_{n}(\tau_{1},\tau_{2},\cdot\cdot\cdot,\tau_{n})\prod_{i=1}^{n}x(t-\tau_{i})d\tau_{i}
\max\left\{|\cot2\phi_{k}|,|\cot2\psi_{k}|\right\}= \frac{1-t^2}{2t}=\frac{1}{2}\left|\frac{1}{t}-t \right| = \gamma + \frac{1}{2}\frac{\gamma^2}{1-\gamma}& \leq \left(1+\frac{1}{2}\frac{4.444\cdot 0.0045}{1-4.444\cdot 0.0045}\right)4.444\delta_k \leq 4.48935\delta_k.
H(U(V)+U(B)) =\log(p_\mu e^{U_\mu(V)}e^{U_\mu(B)}) &=\log\left(p_\mu \frac{A_{\mu\lambda}V_\lambda}{\alpha_\sigma V_\sigma}\frac{A_{\mu\nu}B_\nu}{\alpha_\epsilon B_\epsilon}\right)
\left[\frac{\delta_h (u_R\,\eta_i)}{|h|^\frac{\gamma_i+t}{p}}\right]^p_{W^{s,p}(B_{r_i})}\le \frac{C}{(1-s)\,s}\,i_0^{2\,p}\,\left\|\frac{\delta_h u_R}{|h|^{\gamma_i}}\right\|^p_{L^p(B_{r_i})}+C\, i_0^{4\,(N+p)}\,\mathcal{A}_1(u_R,f_R),i=0,\dots,i_0-1,
z_n(t) =\frac{\widehat{\sigma}^2_n(t)-\sigma^2(t)}{\sigma^2(t)\mathfrak{s}_n(t)},t\in[0,T], e_n =\sup_{t\in[a_n,T-a_n]}\left|\frac{M_n(t)}{\sigma^2(t)\mathfrak{s}_n(t)}-F_n(t)\right|, v_n =\sqrt{nh^{1+2\gamma}}+(nh)^{-1/2}.
g(x) = 8x^6 + 74x^5 + 271x^4 + 452x^3 + 325x^2 + 110x + 64, h(x) = 8x^2(x+3)^2, P_0(x) = - x^6 - 10x^5 - 46x^4 - 104x^3 - 95x^2 - 24x - 9, P_1(x) = x^3 + 6x^2 + 3x - 9.
\eta_{1}( \beta \sqrt{- \triangle}) = \int_{0}^{\infty}{\frac{{\mathrm{d}}s}{s}}\left[ \theta_{3}\Big ( 0 ,{\mathrm e}^{-{\frac{\beta^{2}}{4 s}}}\Big ) - 1 \right] \frac{f ( - s \triangle ) - 1}{s \triangle}.
\left(\frac{b-a}{a+b\omega}\right)=\left(\frac{-1}{a+b\omega}\right)\left(\frac{a-b}{a+b\omega}\right) =\left(\frac{-1}{a+b\omega}\right)(-1)^{((N(a-b)-1)/2)((N(a+b\omega)-1)/2)}\left(\frac{a+b\omega}{a-b}\right).
S_{\mathrm{g f}}+ S_{\mathrm{F P}}= \int_{M}g^{- 2}( \varphi_{\mathrm{o}}) \left( \frac{1}{2}\nabla^{\mu}A_{\mu}\nabla^{\nu}A_{\nu}+ \partial_{\mu}\bar{\theta}\partial^{\mu}\theta \right) \sqrt{h}\, d^{3}x ,
\frac{\rho}{h}\int_Q \frac{\eta_k^i-\eta_{k-1}^i}{\tau}dx = \int_{\partial Q}d \sigma^i_k + \frac{\rho}{h}\int_Q \zeta_k dx \frac{\rho}{h}\int_Q \partial_t \eta(t) dx = \int_{\partial Q}d \sigma_i + \frac{\rho}{h}\int_Q \partial_t \eta(t-h) dx \frac{d}{dt}\rho \int_Q \partial_t \eta dx = \int_{\partial Q}d \sigma_i \in L^2(I)
\sum_{Q\in S^-(\alpha)}{\left(\frac{|Z(Q)|}{q^k}\right)}^{m-k}\leq \leq{\left(\frac{2q-1}{q^2}\right)}^{\frac{k(k-1)}{2}}q^{(-\frac{\alpha^2}{2}+\alpha)k^2+o(k^2)}=q^{\mu(\alpha)k^2+o(k^2)},
\mathcal J_2(m) = \int_{\mathcal B_2^-(m)}(c,-d)_p \chi_{\psi}(d) \chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh + \int_{\mathcal B_2^+(m)}\chi_{\psi}(d) \chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh.
X_1(s) =\sqrt{2}(1+sf)(ie^{i\theta_1},0)+\sqrt{2}s\textstyle\frac{\partial f}{\partial\theta_1}(e^{i\theta_1},e^{i\theta_2}) X_2(s) =\sqrt{2}(1+sf)(0,ie^{i\theta_2})+\sqrt{2}s\textstyle\frac{\partial f}{\partial\theta_2}(e^{i\theta_1},e^{i\theta_2}).
\tau_J^\circ \cap N_{M, \leq \delta}^\circ = (\sigma_{A_{F_J}}^{\circ}\cap \tau_J^\circ \cap N_{M, \leq \delta}^\circ) \cup \bigcup_{\substack{C \in \Sigma \cap \Sigma_{\mathcal{Z}}C^\circ \subset \tau_J^\circ C \cap N_{M, \leq \delta}^\circ \not= \emptyset C \not \subset \sigma_{A_{F_J}}}}((C \cap \sigma_{A_{F_J}})^\circ \cup C^\circ) \cap N_{M, \leq \delta}^\circ.
(1-q^{N})\sum_{n=1}^{\infty}\textup{ssptd}_{o}(n, N)q^n= \frac{1}{2}\left\{\sum_{m=1}^{\infty}\frac{q^{m}}{1-q^{m}}\frac{(-1)_m}{(-q^N)_m}- \sum_{m=N}^{\infty}\frac{2q^m}{1-q^{2m}}+ \frac{q^N}{1-q^N}(-1)_N\right\}.
s(f,\Psi,\mathcal{P}^1_n) = \sum_{k}f(x_k^{n,1})\, |I^{n,1}_k|_\Psi = \sum_{k}\int_{I_k^{n,1}}f(x_k^{n,1}) (\psi-\psi(x_k^{n,1}))+ S(f\psi,\Lambda,\mathcal{P}_n),
\sum_{n=0}^{\infty}\frac{1}{(N+1)^{s+1}}(\int_{0}^{\infty}u^{s}e^{-u}du)=(\int_{0}^{\infty}u^{s}e^{-u}du)(\sum_{n=0}^{\infty}\frac{1}{(N+1)^{s+1}})=
K_n^{(j)}(x,y)=\frac{1}{4\pi^2}\int_{\gamma}\d u\int_{\tilde\gamma}\d v\, \frac{F_n^{(j)}(u)}{F_n^{(j)}(v)}\frac{x^{-u}y^{v-1}}{u-v}, j=1,2,3
\frac{Mn^2\|R\|_{\mathcal{L}^1}}{\lambda}\leq\frac{Mn^2}{\lambda}\max\{1,\lambda^{n-1}\}\sum_{i=1}^{n}\frac{1}{\lambda^{n-i}}\prod_{j=i}^{n}(1+\frac{\sigma_j^2}{2j\lambda})<1
\noindent \begin{cases}p_0=0,q_0=0, p_m=\frac{(k_0+m)^{-\alpha}}{\zeta_{H}(\alpha)}, m\in\mathbb{N}, \alpha\in\mathbb{R}, \alpha>1, q_n=\frac{(k_0+n)^{-\beta}}{\zeta_{H}(\beta)}, n\in\mathbb{N}, \beta\in\mathbb{R}, \beta>1,\end{cases}
\hat{H}_{\mathcal M}^{2 \! \times \! 2}= - \frac{g^{2}}{2 r^{3}}\frac{\partial}{\partial r}r^{3}\frac{\partial}{\partial r}+ \frac{2 g^{2}}{r^{2}}\hat{L}^{2}+ \frac{r^{4}}{8 g^{2}}\sin^{2}\! \vartheta \: \sin^{2}\! \psi \: .
\rho_s\partial_{tt}{b}- \beta \partial_{xx}{b}+ \alpha \, \partial_{xxxx}{b}- \gamma \, \partial_{xxt}{b}= \; q , \mbox{on}(0, L), \partial_{t}{b}= \; \partial_x [{b}^3 \partial_{x}q], \mbox{on}(0, L) \,,
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(n)q^n =\dfrac{1}{2}\left(\dfrac{f_3^5}{f_1^3f_6^2}-\dfrac{f_6}{f_1f_2f_3}\right) \intertext{and}\sum_{n=0}^{\infty}\textup{PDO}_\textup{t}(n)q^n =\dfrac{qf_2f_3^2f_{12}^2}{f_1^2f_6},
\left[\overline{\mathcal{W}}_{3,1}\right] \cdot \left[\overline{\Theta}_{3,1}\right] |_{T_3}= 27, \left[\overline{\mathcal{H}yp}_{3,1}\right] |_{T_3}= 0, \left[\overline{\mathcal{W}}_2 \right] |_{T_3}= -3, \gamma_1 |_{T_3}=12, \gamma_2 |_{T_3}=0.
F'(t)= \gamma'(t) \left\{\int_{\Sigma_t}|\nabla u|^{-2}|\nabla_{\Sigma_t}|\nabla u||^2 + \frac{1}{2}R_{M}+\frac12 |\overset{\circ}{h}|^2 + (2 a + 1) \left( \frac{H}{2}- \eta(u) |\nabla u| \right)^2 \right\}&\gamma'(t)\left(4\pi-\int_{\Sigma_t}K_{\Sigma_t}\right).
W_{j}(z) :=\sum_{n=0}^{\infty}\frac{W_{[n,j]}}{z^{n+1}}=\sum_{n\in\mathbb{Z}}\frac{W_{[n,j]}}{z^{n+1}}A_{j}(z) :=\sum_{n=0}^{\infty}\frac{A_{[n,j]}}{z^{n+1}}=\sum_{n\in\mathbb{Z}}\frac{A_{[n,j]}}{z^{n+1}}B_{j}(z) :=\sum_{n=0}^{\infty}\frac{B_{[n,j]}}{z^{n+1}}=\sum_{n\in\mathbb{Z}}\frac{B_{[n,j]}}{z^{n+1}}
\beta^{u_1+t}\alpha_{i_1}^{u_1+p_{i_1}-1}= \beta^{u_1+t}\alpha_{i_1}^{u_1+t+ \lfloor \frac{p_{i_1}-1-t}{s_1}\rfloor s_1}\alpha_{i_1}^{p_{i_1}-1-t - \lfloor \frac{p_{i_1}-1-t}{s_1}\rfloor s_1}\in W_{i_1}\alpha_{i_1}^{p_{i_1}-1-t - \lfloor \frac{p_{i_1}-1-t}{s_1}\rfloor s_1}\subseteq K
s_k = t_{k-1}+ \varepsilon x_{k-1},y_k = f_p(s_k),t_k = s_k + y_k,x_k = f_p(t_k), I_k =(s_k,t_k],\tilde I_k = \frac{I_k}{x_k}= (\tilde s_k, \tilde t_k],|\tilde I_k| = \frac{y_k}{x_k}.
\| A^{r-\frac{1}{2}}e^{\tau A}\partial_t \widetilde{v}\| \leq C_r \Big(\|e^{\tau A}\overline{v}\|_{H^{r+\frac{1}{2}}}^2 + \|e^{\tau A}\widetilde{v}\|_{H^{r+\frac{1}{2}}}^2 + |\Omega|\|A^{r}e^{\tau A}\widetilde{v}\| \Big) .
\frac{{\partial_y{\partial^n_x u}}}{u^s_y +{u}_y}\frac{u^s_{yy}+{u}_{yy}}{{u^s_y +{u}_y}}= \left(\frac{{{\partial^n_xu}}}{u^s_y +{u}_y}\right)_y \frac{u^s_{yy}+{u}_{yy}}{{u^s_y +{u}_y}}- \frac{{{\partial^n_xu}}}{u^s_y +{u}_y}\left(\frac{u_{yy}^s +{u}_{yy}}{(u^s_y +{u}_y)}\right)^2.
\phi^{\prime}_{n,\lambda}(x) =\frac{d}{dx}\phi_{n,\lambda}(x)=\sum_{k=0}^{n}\binom{n}{k}(1)_{n-k,\lambda}\phi_{k,\lambda}(x)-\phi_{n,\lambda}(x) &=\sum_{k=0}^{n-1}\binom{n}{k}(1)_{n-k,\lambda}\phi_{k,\lambda}(x),(n\ge 1).
\sum_{p=1}^n u^{p\bar{p}}\frac{\beta_{p\bar{p}}}{\beta}\leq \sum_{p=1}^n u^{p\bar{p}}\left ( \vert A u_p + B\bar z_{p}\vert^2 - u u_{p \bar p}- \vert u_p\vert^2 - B\right)
c_{\Phi}(\xi,\tau)+C_1 \lambda \frac{\left|p_{\Phi}(x_0,\xi,\tau)\right|^2}{|\xi|^2+\tau^2}= \lambda \left( c_{\Psi}(\xi,\lambda \tau)+2\lambda \left|\{p_{\Psi},\Psi\}(x_0,\xi,\lambda \tau)\right|^2 +C_1 \frac{\left|p_{\Psi}(x_0,\xi,\lambda \tau)\right|^2}{|\xi|^2+\tau^2}\right) .
U_t = D\Delta U+H( u+u_0) x\in \Omega, 0<t<T V_t =\tilde D\Delta_{M}V+F( u+u_0, v+v_0) x\in M, 0<t<T D\frac{\partial U}{\partial \eta}=G( u+u_0, v+v_0) x\in M, 0<t<T U =u_0 x\in\Omega , t=0 V =v_0 x\in M ,\quad t=0
\{f \in C(\Omega,\mathbb{D}) : \Sigma_\mathrm{ac}(\lambda f) = \emptyset 0 < \lambda \le 1 \}& = \bigcap_{j=1}^4 \{f \in C(\Omega,\mathbb{D}) : M(\lambda f,I_j) = 0 0 < \lambda \le 1 \}.
<{\cal O}_{1}\cdots{\cal O}_{k}\mathrm{e x p}\left\{\sum_{k = 0}^{4}\sum_{p = 4 - k}^{4 n}\sum_{\beta_{k}= 1}^{b^{k}({\cal M})}\sum_{\alpha_{p}= 1}^{b^{p}({\cal N})}s_{\alpha_{p}, \beta_{k}}^{p , k}{\cal D}_{p , k}^{\alpha_{p}, \beta_{k}}\right\}> .
\displaystyle\frac{\partial J_\varepsilon}{\partial \lambda_j}= B_j \langle\frac{\partial^2U_{y^j, \lambda_j}}{\partial \lambda_j^2}, v\rangle + \sum_{l = 1}^NC_{jl}\langle\frac{\partial^2 U_{y^j, \lambda_j}}{\partial y^j_l\partial \lambda_j}, v\rangle,j = 1,2,
\mathbf{d}(\mathbf{G}^{(t)}) =\sum_{n=1}^{N}\mathbf{d}(\mathbf{G}_{n}^{(t)})\in\mathbb{R}^{K}, \mathbf{G}^{(t)}\mathbf{x}^{(t)}-\mathbf{b}^{(t)}=\sum_{n=1}^{N}\left(\mathbf{G}_{n}^{(t)}\mathbf{x}^{(t)}-\mathbf{b}_{n}^{(t)}\right)\in\mathbb{R}^{K}.
\sum_{n\ge 1, n\neq k}\frac{1}{\alpha^2_{\nu,n}-\alpha^2_{\nu,k}}=\frac{1}{4\alpha^2_{\nu,k}}\left[2\nu+1-\frac{\alpha^2_{\nu,k}+2\nu-1}{\alpha^2_{\nu,k}-2\nu+1}\right],
{\rm WF}_h'(e^{-it_0h^{-1}\widetilde{P}_h(z)}\widetilde{R}_h(z))\cap S^*(X\times X) \subset \kappa(\{(x,\xi,y,\eta)\,:\,(e^{-t_0H_p}(x,\xi),y,\eta) \in \Delta(T^*X)\cup \Omega_+\cup(E^*_u\times E^*_s)\setminus\{0\}\xi=0,\eta\neq 0\}).
B_x ={a_x^2}- \frac{\alpha_x+\beta_x R_x}{2 x^{2}},\ \B_x^{\dagger}= (a_x^{\dagger})^{2}- \frac{\alpha_x+\beta_x R_x}{2 x^{2}}, & B_y ={a_y^2}- \frac{\alpha_y+\beta_y R_y}{2 y^{2}},\ \B_y^{\dagger}={(a_y^{\dagger})^{2}}- \frac{\alpha_y+\beta_y R_y}{2 y^{2}},
|\hat\lambda_j^{(n)}-\lambda_j^{(n)}|/g_j^{(n)}\geq \frac{\hat\lambda_j^{(n)}-\lambda_j^{(n)}}{\lambda_{j-1}^{(n)}-\lambda_j^{(n)}}= 1-\frac{\lambda_{j-1}^{(n)}-\hat\lambda_j^{(n)}}{\lambda_{j-1}^{(n)}-\lambda_j^{(n)}},
\sup_{|x|\geq 6R}|x||v_\varepsilon(x)| \leq C(1+|a|^{-1})\Big[\int_{\mathbb R^2}(1+|x|) \left|\left\{g_\varepsilon+(1-\psi)f\right\}(x)\right|\,dx &\qquad+\sup_{|x|\geq 3R}|x|^3(\log |x|)|f(x)|\Big]
c_{\Psi}(\xi,\tau)=\frac{1}{i\tau}\{\overline{p_{\Psi}},p_{\Psi}\}(x_0,\xi,\tau), \tau >0 c_{\Psi}(\xi,0)= 2\{p_2,\{p_2 ,\Psi\}\}(x_0, \xi) ,
q_{\rm SVC}(\mathbf p) = \sum_{n\in\mathcal N}a_{n}\sum_{\ell\in\mathcal L}b_{n,\ell}\sum_{k\in\mathcal K_{{\rm SVC},n,\ell}}\Pr[K_{{\rm SVC},n,\ell}=k]\Pr[{\rm SIR}_{{\rm SVC},n,\ell}\geq\tau_k],
\left( \prod_{1 \leq i<j\leq k}d_B(C_i,C_j)-\eta \binom k2 \right) \prod_{i=1}^k |C_i| \geq \left( \delta^{\binom k2}-\eta \binom k2 \right) \alpha^k N^k \geq \frac 12 \delta^{\binom{k}{2}}\alpha^k N^k
\left(1-x+\frac{xv}{1-v}\right)^2D(x,v)=x\left(1-x+\frac{xv}{1-v}\right)B(x,v) &\qquad+\frac{x}{v}\left(\frac{xv^3}{1-v}C(x,1)+\frac{x^3v^3(2-x-xv)}{(1-x)^2(1-xv)^2}\right)+\frac{xv^2}{1-v}\left(1-x+\frac{xv}{1-v}\right)D(x,1).
\frac{1}{4}\int_{s=1^-}^{\infty}\exp\left(-\frac{8\pi^2 t\|\lambda^*\|_2^2s^2}{2k+1}\right)d\left(\left|\Lambda^\vee \cap B_2(0,s\|\lambda^*\|) \right| \right) &= \frac{4\pi^2 t \|\lambda^*\|_2^2}{2k+1}\int_{1-}^\infty s\exp\left(-\frac{8\pi^2 t\|\lambda^*\|_2^2 s^2}{2k+1}\right) \left|\Lambda^\vee \cap B_2(0,s\|\lambda^*\|) \right|ds.
\begin{aligned}\|\nabla w_\varepsilon\|_{L^p(\Omega)}+ \|p_\varepsilon\|_{L^p(\Omega)/\mathbb{R}}\leq C\Big\{\|F\|_{W^{-1,p}(\Omega)}+ \|\nabla G\|_{L^p(\Omega)}+ \|\nabla v\|_{L^p(\Omega)}\Big\}\leq C\Big\{\|F\|_{W^{-1,p}(\Omega)}+ \|h\|_{L^p(\Omega)}+ \|g\|_{B^{1-1/p,p}(\partial\Omega)}\Big\}.\end{aligned}