harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M_{j,m,n}(\xi,\eta):= m_j(\xi,\eta)\hat{\Phi}\left(\frac{\xi}{2^{aj+m}}\right)\hat{\Phi}\left(\frac{\eta}{2^{bj+n}}\right) = \int \rho(t)e^{-2\pi i\left(\frac{\xi}{2^{aj}}(t^a+\epsilon_P(t))+\frac{\eta}{2^{bj}}(t^b+\epsilon_Q(t))\right)}\, dt\hat{\Phi}\left(\frac{\xi}{2^{aj+m}}\right)\hat{\Phi}\left(\frac{\eta}{2^{bj+n}}\right)
|e_{n}^{(k,i)}| \le \sum_{m=0}^{k-1}|e_{m}^{(k,i)}|\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|s_{i,m}^{(k,i)}| +(\Delta t)^{\alpha}\sum_{j=0}^{n-k}g_{n-k-j}^{(-\alpha)}\sum_{i=0}^{j}|r_{j-i}^{(k,i)}|\left(L^{(k,i)}|e_{i+k}^{(k,i)}|+|\tau_{i+k}^{(k,i)}|\right).
\frac{d}{dt}\Big|_{t=s}\varphi_t^* \o_t = \varphi_s^*\left(\mathcal{L}_{X_s}\o_s\right) + \varphi_s^* (d \mu) = \varphi_s^* \left(d(\iota_{X_s}\o_s) - \iota_{X_s}d \o_s \right) + \varphi_s^* d \mu = -\varphi_s^* (d\mu) + \varphi_s^* (d \mu) =0.
\prod_{j, i_j}A_{j, i_j}< \left( \prod_{j, i_j}\left( | a_j | + 1 \right)^L \right) \prod_{j, i_j}\left( 1 + \frac{1}{| a_j |}\right)^{i_j}&= \left( \prod_j \left( | a_j | + 1 \right)^{\nu_j L}\right) \prod_j \left( 1 + \frac{1}{| a_j |}\right)^{\frac{\nu_j (\nu_j + 1)}{2}}\leq f^{ML}g^{\frac{M^2}{2}}.
\widetilde{\mathrm{S}}_{M}(A\mid\alpha):=\frac{\mathrm{S}_{M}(A\mid\alpha)}{\mathrm{S}_{M}(\{\theta:\|\theta-\theta_{0}\|^{2}/\varepsilon^{2}\leq 2(B/\varepsilon)^{2/(2\alpha_{0}+1)}\}\mid\alpha)}
\left( I_{2}\right)_{\mu}=-\frac{1}{2}(k_{1}+k_{2})_{\alpha}\left[ \Delta_{\alpha \mu}\right] -\frac{1}{2}(k_{1}+k_{2})_{\mu}\left\{I_{\log}(m^{2})-\left( \frac{i}{(4\pi )^{2}}\right)Z_{0}(m^{2},m^{2},(k_{1}-k_{2})^{2};m^{2})\right\}.
1 =\langle z|z\rangle=\overline{A}_{0}\sum_{m=0}^{\infty}\frac{\overline{z}^{m}}{\sqrt{m!}}A_{0}\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!}}\langle m|n\rangle &=|A_{0}|^{2}\sum_{n=0}^{\infty}\frac{|z|^{2n}}{n!}=|A_{0}|^{2}e^{|z|^{2}}.
F\Big(\{x_{i,r}\}\Big)\Big|_{(x_{i,1},x_{i,2},x_{i,3}, \dots, x_{i,1-c_{ij}}) \mapsto (w,w q_i^2, w q_i^4, \dots, w q_i^{-2c_{ij}}),\, x_{j,1}\mapsto w q_i^{-c_{ij}}}=\, 0
P_1+P_2 =\prod{\tilde{\lambda}_j}\int_{\mathbb{R}^n}{\frac{u(y)-u(0)}{(\tilde{\lambda}_1^2y_1^2+...+\tilde{\lambda}_n^2y_n^2)^{(n+2s)/2}}dy}\geq \frac{\eta_0}{1-s}>0,
S:= 1+\sum\limits_{i=1}^n\big((q-1)E_{ii}+(q^{-1}-1)E_{-i,-i}\big)\otimes (E_{i,i}+E_{-i,-i}) &+(q-q^{-1})\sum\limits_{1\leqslant|j|<|i|\leqslant n}(-1)^{[j]}\big(E_{ij}-(-1)^{[i]([j]+1)}E_{-j,-i}\big)\otimes E_{ji}&+(q-q^{-1})\sum\limits_{i=-n}^{-1}E_{i,-i}\otimes E_{-i,i}\in\mathrm{End}(V_q)^{\otimes 2}.
\left\|u(\cdot, t)-t^{-\frac{1}{2}}f_{M}\left((\cdot)t^{-\frac{1}{2}}\right)\right\|_{L^{\infty}}=\left(\left\|\widetilde{f}_{M}(\cdot)\right\|_{L^{\infty}}+o(1)\right)t^{-1}\log t as \ \t\to \infty.
\begin{aligned}U_R(x)=\left[a+b\left\{\frac{1}{|x|^{\frac{n-p}{p-1}}}-\frac{1}{R^{\frac{n-p}{p-1}}}\right\}^{-\frac{p}{n-p}}\right]^{1-\frac{n}{p}}\end{aligned}
{\cal I}= \int_{{\cal F}}\frac{d^2 \tau}{Im \tau}\sum_{m_1,m_2\atop n_1,n_2}\left( \sum_{b=even}q^{\frac{1}{2}p_L^2}\bar q^{\frac{1}{2}p_R^2}h_0(\tau)+\sum_{b=odd}q^{\frac{1}{2}p_L^2}\bar q^{\frac{1}{2}p_R^2}h_1(\tau)-c(0)\right)
\begin{aligned}w^* =\left(1-\alpha P(w,v)\right)w+\alpha P(w,v)v-\frac{\beta}{2}\left((v-w_d)+(w-w_d)\right) &\qquad-\alpha\frac{\beta}{2}((P(w,v)-P(v,w))(w-v))+\Theta_1 D(w), v^* =\left(1-\alpha P(v,w)\right)v+\alpha P(v,w)w-\frac{\beta}{2}\left((v-w_d)+(w-w_d)\right) &\qquad-\alpha\frac{\beta}{2}((P(v,w)-P(w,v))(v-w))+\Theta_2 D(w), \end{aligned}
\begin{array}{c}{{< j_{1}- m_{1}\j_{2}- m_{2}| j_{1}+ j_{2}\ - m_{1}- m_{2}>_{q}=}}\\{{= < j_{2}m_{2}\j_{1}m_{1}| j_{1}+ j_{2}\m_{1}+ m_{2}>_{q}=}}\\{{= < j_{1}m_{1}\j_{2}m_{2}| j_{1}+ j_{2}\m_{1}+ m_{2}>_{q^{- 1}}}}\\ \end{array}
\mathrm{F}_1 = \dfrac{\tanh{(\sqrt{\mu}|\mathrm{D}|)}}{\sqrt{\mu}|\mathrm{D}|},\mathrm{F}_2 = \frac{3}{\mu |\mathrm{D}|^2}(1- \mathrm{F}_1), \mathrm{F}_3 =\mathrm{sech}(\sqrt{\mu}|D|), \mathrm{F}_4 = \frac{2}{\mu |\mathrm{D}|^2}(1- \mathrm{F}_3).
\begin{array}{c}{g_{+}( - \infty ) = g_{0}( + \infty ) = 2 \chi_{\infty}( S - 2 S^{+}) + 2 \pi k_{W}^{+}}\\{g_{-}( + \infty ) = g_{0}( - \infty ) = - 2 \chi_{\infty}( S - 2 S^{-}) + 2 \pi k_{W}^{-}}\\ \end{array}
\left[\epsilon^0q^{d}\right] \frac{(-1)^{(N-1)d}}{\det(z_i^{N-j+1})}\begin{vmatrix}(z_1-y)z_1^{d+N-1}\cdots (z_1-y)z_1^{d}(z_2-y)z_2^{d+N-1}\cdots (z_2-y)z_2^{d}\vdots \cdots \vdots (z_N-y)z_N^{d+N-1}\cdots (z_N-y)z_N^{d}\end{vmatrix}\prod_{i\in [N]}\bigg( \frac{\alpha_i-z_{N+1}}{\alpha_i-y}\bigg)^{b_i}.
\mathcal{H}=\left\lbrace f:\mathbb{R}^{d}\rightarrow \mathbb{R}: \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}f(\mathbf{u}) f( \mathbf{v}) \vert \mathbf{u}- \mathbf{v}\vert^{\mathbf{2H-2}}d\mathbf{u}d\mathbf{v}< + \infty \right\rbrace
\hat{C}_{(2n)}^{+}\;=\; C_{(2n)}^{+}\,-\, C^{+}_{(2n-1)}\left( d\underline{y}+A^{(2)}\right) \,+\, \underline{y}\G_{(0)}^{+}\textstyle{\frac{1}{n!}}\hat{B}^{n}\; ,
F_{j_{1}m_{1}, j_{2}m_{2}}^{+ + ~ j_{3}m_{3}}\langle{\cal O}_{j_{3}m_{3}}^{r_{1}+ r_{2}~ +}{\cal O}_{j_{3}~{- m_{3}}}^{r_{3}~ -}\rangle = F_{j_{2}m_{2}, j_{3}~{- m_{3}}}^{+ - ~ j_{1}{- m_{1}}}\langle{\cal O}_{j_{1}m_{1}}^{r_{1}+}{\cal O}_{j_{1}~{- m_{1}}}^{r_{2}+ r_{3}-}\rangle \nonumber
\underset{p_k, q_k, \tau_k}{\max}\mathcal{U}=\sum\limits_{k=1}^K R^{u}_{k}& L\sum\limits_{i=1}^l q_i \leq \sum\limits_{i=1}^l e^{r}_{i}, \forall l \in [1:K], & p_k \leq p, ~\mbox{and}p_k \geq 0, ~ \forall k \in [1:K], & \tau_k\in[0,T), ~\mbox{and}~q_k\geq0,~~\forall k \in [1:K],
\left\langle T_{\ell}^{i j}( p ) \, T_{\ell^{\prime}}^{k m}( - p ) \right\rangle = -{\frac{3}{4 \kappa_{1 0}^{2}{\alpha^{\prime}}^{3}}}\, F_{\ell}^{i j ; k m}( p ) \, \delta_{\ell \ell^{\prime}},
\mathcal{H}[U\cup \{u_1\}] = \mathcal{G}[U\cup \{u_1\}] \cong K_{t+2}^{3}\quad\mathcal{H}[U\cup \{v\}] = \mathcal{G}[U\cup \{v\}] \cong K_{t+2}^{3}.
\begin{aligned}\tilde{H}_{iv_i}(t)= \tilde{H}_{i{v_i}}(t,x,y,z,z_1,z_2,u_1,u_2;q_i,k_i,k_{1i},k_{1i},p_i,Q_{1i},Q_{2i}) = H_{iv_i}(t)-\sum_{j=1}^2q_i(t)\sigma_j(t,x(t),u_1(t),u_2(t))h_{jv_i}(t,x(t),u_1(t),u_2(t)),\end{aligned}
R_{\mathsf{u}}> N\sum_{j=1}^{L}\binom{L}{j}\left(\frac{1}{N}\right)^{j}\left(1-\frac{1}{N}\right)^{L-j}&=N\sum_{j=0}^{L}\binom{L}{j}\left(\frac{1}{N}\right)^{j}\left(1-\frac{1}{N}\right)^{L-j}- N\left(1-\frac{1}{N}\right)^{L}&=N(1-(1-1/N)^L).
u(0,x_0) = \mathbb E\bigg[ \Phi(W_T+x_0) & + \int_0^T f\left ( r,W_r +x_0, u(r, W_r+x_0), \nabla u(r, W_r+x_0){\mathrm dr}\right) \mathrm dr & + A_T^{W ,W}( (\nabla u^* b)(x_0 + \cdot ) )\bigg],
S_{e f f}= \int d^{2}x \sqrt{g}[{\bar{\psi}}( i{D \! \! \! /}-{A \! \! /}-{j \! \! /}) \psi + L_{g}] + \int_{\Sigma}( \frac{i}{2 \pi}d B{\wedge}A -{\^{*}J}{\wedge}A )
=e^{-\frac{i}{2R}\frac{\partial}{\partial t_{0}}}\sum_{n}\frac{1}{(\frac{1}{2\sqrt{\alpha '}}(2n+1)i -t_{0})^{2}}=e^{-\frac{i}{2R}\frac{\partial}{\partial t_{0}}}\frac{\partial}{\partial t_{0}}\sum_{n}\frac{1}{\frac{1}{2 \sqrt{\alpha '}}(2n+1)i -t_{0}}.
\nu_{x_0}(n) := \mathbb{E}_{\rho_{x_{0}}}\left(\displaystyle \sum_{j = 0}^{C_1 - 1}\mathbb{1}_{n}(X_j^{(0)}) \right),\,\,\,\,\,\,\,\,\, n \in \mathcal{I}(r_{x_0})
\phi =\left[ \begin{array}{cccc}a_{11}a_{12}0 0 a_{21}a_{22}0 0 a_{31}a_{32}a_{11}^{2}a_{12}^{2}a_{41}a_{42}a_{21}^{2}a_{22}^{2}\end{array}\right] :a_{11}a_{22}\neq 0,a_{12}=a_{21}=0\mbox{or}a_{12}a_{21}\neq0,a_{11}=a_{22}=0.
\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\phi_{4}(x_{4})\rangle=f\left(\frac{x_{12}x_{34}}{x_{13}x_{24}}, \frac{x_{12}x_{34}}{x_{23}x_{14}}\right)\prod_{i<j}^{4}x_{ij}^{\Delta/3-\Delta_{i}-\Delta_{j}},
\sum_{n\leq x}\left(\frac{f(n)}{q}\right) = \sum_{a=1}^q\sum_{\substack{n\leq x n\equiv a \bmod q}}\left(\frac{f(n)}{q}\right)= \sum_{a=1}^q\left(\frac{f(a)}{q}\right)\sum_{\substack{n\leq x n\equiv a \bmod q}}1= \frac{x}{q}\sum_{a=1}^q\left(\frac{f(a)}{q}\right) + O(q).
f_1 =\eta^{2k_0}\cdot\sum_{m \geq 0}\frac{a_m}{n^mm!}q^{m+\frac{r}{n}+\frac{e-1}{3}}, f_2 =\eta^{2k_0}\cdot\sum_{m \geq 0}\frac{b_m}{n^{2m}(2m)!}q^{m+\frac{r}{n}+\frac{e}{3}-1}, f_3 =\eta^{2k_0}\cdot\sum_{m \geq 0}\frac{c_m}{n^{2m}(2m+1)!}q^{m+\frac{r}{n}+\frac{e}{3}-1}.
-{\biggl (}\gamma^2-\frac{\varpi^2}{\alpha^2}{\biggr )}\alpha^2 \rho^2 dt^2+ \frac{d \rho^2}{\alpha^2 \rho^2}+{\biggl (}\gamma^2-\frac{\varpi^2}{\alpha^2}{\biggr )}\rho^2 d \varphi^2 \:,
\begin{aligned}J_1(v_1(\cdot),u_2(\cdot))- J_1(u_1(\cdot),u_2(\cdot))\geq \mathbb{E}\int_0^T\bigg[(H_1^{v_1}(t)-H_1(t))-(x^{v_1}(t)-x(t))H_{1x}(t) - (y^{v_1}(t)-y(t))H_{1y}(t)-(z^{v_1}(t)-z(t))H_{1z}(t)-\sum_{j=1}^2(z_j^{v_1}(t)-z_j(t))H_{1z_j}(t)\bigg]dt.\end{aligned}
\Delta^{(+)}(x) =\frac{1}{(2\pi)^3}\int_{0}^{\infty}\frac{dp^+}{2p^+}e^{-ip^-x^+ -ip^+x^- +ip^{\bot}x^{\bot}}=\frac{m}{4\pi^2\sqrt{-x^2}}K_1 (m\sqrt{-x^2}),
\frac{\pi^2}{D^2}\leq \frac{\int_0^{\frac{D}{2}}(y')^2}{\int_0^{\frac{D}{2}}y^2}= -\frac{\int_0^{\frac{D}{2}}yy''}{\int_0^{\frac{D}{2}}y^2}&= \frac{(n-1)\int_0^{\frac{D}{2}}\cot(x)yy'+\lambda_1\int_0^{\frac{D}{2}}y^2}{\int_0^{\frac{D}{2}}y^2}&\leq \lambda_1,
\operatorname{rk}(\mathcal{D}_N)= &=\operatorname{rk}\left[\begin{array}{ccccc}w_1-w_2 D^{(m-1)}_1 D^{(m-1)}_2 w_3-w_2 D^{(m-1)}_2 D^{(m-1)}_3 \vdots \ddots \ddots (-1)^N(w_{N-1}-w_N) D^{(m-1)}_{N-1}D^{(m-1)}_N\end{array}\right]+N-1,
\begin{aligned}\mathcal{H}^-_k(N; \eta, m) \leq \sum_{f\in \mathcal{H}^-_k(N; \eta, m)}N(1-\eta, 100(kN)^\eta,{\rm sym}^m f) & \leq \sum_{f\in \mathcal{H}^*_k(N)}N(1-\eta, 100(kN)^\eta,{\rm sym}^m f)\ll_{\eta}(kN)^{65\eta}.\end{aligned}
f_A(x) =A_{1,1}+ A_{1,2}x + A_{2,1}x^2+ A_{2,2}x^3 + R x^4, \mbox{and}f_B(x) =B_{1,1}+ B_{2,1}x^{-1}+ B_{1,2}x^{-5}+ B_{2,2}x^{-6}+ S x^{-10}
-\int_{B_r(p)}\nabla \eta \cdot (\cos R^{u}\cos R^{v}\nabla F)\, d\mu_g =-\lim_{\epsilon \to 0}\int_{B_r(p)}\phi_{\epsilon}\nabla \eta \cdot (\cos R^{u}\cos R^{v}\nabla F) \, d\mu_g \ge 0,
\begin{aligned}\left| \left\langle x-Ax,y-Ay \right\rangle \right| =\left| \left\langle x,y \right\rangle -\left\langle \left( 2A-A^2 \right)x,y \right\rangle \right| \ge \left| \left\langle x,y \right\rangle \right|-\left| \left\langle \left( 2A-A^2 \right)x,y \right\rangle \right|. \end{aligned}
\mu_{\lambda}[s_n \ldots s_0]_l = \lambda_{s_n}p_{s_n, s_{n-1}}\ldots p_{s_1, s_0}&= p_{s_{n-1}, s_n}\lambda_{s_{n-1}}p_{s_{n-1},s_{n-2}}\ldots p_{s_1, s_0}&= p_{s_{n-2},s_{n-1}}p_{s_{n-1}, s_n}\lambda_{s_{n-2}}\ldots p_{s_1, s_0}&= \ldots = p_{s_0, s_1}\ldots p_{s_{n-1},s_n}\lambda_{s_0}&= \mu_{\lambda}[s_0 \ldots s_n]_k
\textbf{e}_1 +\textbf{e}_2 =1,\textbf{e}_1 . \textbf{e}_2 =\textbf{e}_2 . \textbf{e}_1 =0,\textbf{e}_1^2 =\textbf{e}_1 ,\textbf{e}_2^2 =\textbf{e}_2.
d^h_0(p,g_1,\ldots,g_m) =(g_1^{-1}p,g_2,\ldots,g_m) d^h_i(p,g_1,\ldots,g_m) =(p,g_1,\ldots,g_i\cdot g_{i+1},\ldots,g_n) s^h_0(p,g_1,\ldots,g_m) =(p,\sigma f(g_1),g_1,\ldots,g_n) s^h_i(p,g_1,\ldots,g_m) =(p,g_1,\ldots,g_i,\sigma f(g_i),g_{i+1},\ldots,g_n).
\mbox{\bf Term 2}= \left (F_i(x_i^k,N \hat v_i^k) - F_i(x_i^k,Ny^k)\right)^T(x_i^k-x^*_i) +\left(F_i(x^k_i,Ny^k) - F_i(x^*_i,\bar{x}^*)\right)^T(x_i^k-x^*_i).
\mathcal{A}=\begin{pmatrix}A_{11}A_{12}0_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}A_{23}A_{24}0_{d\times d}0_{d\times d}A_{33}A_{34}A_{41}A_{42}0_{d\times d}0_{d\times d}\end{pmatrix};
\frac{1}{\rho}k(\rho\mu)^{2}= \left. \frac{1}{\rho}\frac{a(\rho) c_{1}(\rho\mu)^{2}+ b(\rho) c_{2}(\rho\mu)^{-2\gamma}}{a(\rho) c_{3}+ c_{4}b(\rho) (\rho\mu)^{-2\gamma}}\right.
\frac{1}{L_{1}\cdots L_{n}}=\frac{(-1)^n}{a}I(C)=\frac{(-1)^n}{a}\sum_{\mu\in J}I(D_\mu)=\sum_{\mu\in J}\frac{b_\mu}{a}\frac{1}{M_{\mu 1}\cdots M_{\mu n}}.
q = L^n q_{2d}\/, X_a = \exp\left( L \sigma_a \right) = 1 + L \sigma_a + \frac{L^2}{2}{\sigma_a}^2 + \cdots, T_i = \exp\left( L m_i \right) = 1 + L m_i + \frac{L^2}{2}{m_i}^2 + \cdots
\mathbb{E}_{k}(\delta^{k,1}_{N}(z_{1})) =N^{-1}\sum_{i<k,j<k}\mathbb{E}_{k}(G^{(1)}_{k})_{ij}\overline{W_{ik}}W_{jk}-N^{-1}\sum_{j<k}\mathbb{E}_{k}(G^{(1)}_{k})_{jj}&= N^{-1}\sum_{\substack{i<k,j<k i \neq j}}\mathbb{E}_{k}(G^{(1)}_{k})_{ij}\overline{W_{ik}}W_{jk}+N^{-1}\sum_{j<k}\mathbb{E}_{k-1}(G^{(1)}_{k})_{jj}(|W_{jk}|^{2}-1)
\begin{array}{@{}l@{}}{\displaystyle |\sum_{k=1}^{m}R_5^{k,1}| + |\sum_{k=1}^{m}R_5^{k,2}| \leq C\big\{h^{2r}+(\Delta t)^{4}\big\}+C \|\theta_{\psi}^{m}\|_{\mathcal{L}^2}^{2}}begin{align*}2mm]{\displaystyle \quad+ \frac{1}{16}\|\nabla\theta_{\psi}^{m}\|_{\mathbf{L}^2}^{2}+C\Delta t \sum_{k=1}^{m}\|\nabla\theta_{\psi}^{k}\|_{\mathbf{L}^2}^{2}.}\end{array}
\sum_{n=0}^{\infty}\frac{\left(a,b;q\right)_{n}z^{n}}{\left(q,c;q\right)_{n}}=\frac{\left(c/b,bz;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(abz/c,b;q\right)_{n}}{\left(q,bz;q\right)_{n}}\left(\frac{c}{b}\right)^{n}
v_{l j}= a_{l j}{\frac{| \lambda | -{\frac{i E \gamma}{k}}}{- \kappa +{\frac{i M{\gamma}^{\prime}}{k}}}}{\rho}^{| \lambda |}\Phi ( | \lambda | -{\frac{i E \gamma}{k}}+ 1 , 2 | \lambda | + 1 , - 2 i \rho ) .
w\left( \begin{matrix}1 \zeta_{p}0 1 \end{matrix}\right) = \left( \begin{matrix}1 0 -\zeta_{p}1 \end{matrix}\right)w = \left( \begin{matrix}1 -\zeta_{p}^{-1}0 1 \end{matrix}\right)\left( \begin{matrix}\zeta_{p}^{-1}0 0 -\zeta_{p}\end{matrix}\right)\left( \begin{matrix}0 1 1 -\zeta_{p}^{-1}\end{matrix}\right)w
\limsup_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}J^{n}_{i}\left(\boldsymbol{u}^{n}\right) \leq \limsup_{n\to\infty}J^{n}_{1}\left(\boldsymbol{u}^{n}\right) \leq \limsup_{n\to\infty}J^{n}_{1}\left(\boldsymbol{\tilde{u}}^{n}\right).
A_1(x) =\begin{bmatrix}-\delta_+x+\delta_- \delta_-x-\delta_+ -\delta_-x+\delta_+ \delta_+x-\delta_-\end{bmatrix}, \Lambda_n =\begin{bmatrix}A_+-\delta_+\Big(n+\frac{1}{2}\Big) A_-+\delta_-\Big(n+\frac{1}{2}\Big) A_--\delta_-\Big(n+\frac{1}{2}\Big) A_++\delta_+\Big(n+\frac{1}{2}\Big)\end{bmatrix},
S_{\mathrm{m a s s i v e \D 0 - b r a n e}}= T \int d \tau \Big ( e^{- 3 \hat{\phi}/ 4}\sqrt{- \dot{X}^{\hat{\mu}}\dot{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}}}+ \dot{X}^{\hat{\mu}}\hat{A}_{\hat{\mu}}+ m \hat{V}_{t}\Big ) ,
{\cal E}= \frac{1}{2}\, \mu^{1 - 2 \nu}\sum_{n = 1}^{\infty}\int \frac{d^{2}p}{( 2 \pi )^{2}}\left\{{\bf p}^{2}+ \frac{\pi^{2}}{\ell^{2}}\left[ ( n + s )^{2}+ \frac{g}{3}\right] \right\}^{\nu}\Bigg |_{\nu = 1 / 2}.
\int_{m^{1+\varepsilon}}^{x}\frac{\pi(t;m,1)}{t^2}dt =\int_{m^{1+\varepsilon}}^{x}\frac{1}{\varphi(m)t\log t}dt+O\left(\int_{m^{1+\varepsilon}}^{x}\frac{1}{\varphi(m)t\log^2 t}dt\right) &=\frac{\log\log x-\log\log m}{\varphi(m)}+O\left(\frac{1}{\varphi(m)\log m}+\frac{\varepsilon}{\varphi(m)}\right)
\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}=\left(1-\frac{c_x}{2}\right) \sqrt{n}\langle x, x \rangle^\frac{1}{2}=\sqrt{n}\langle x, x \rangle^\frac{1}{2}\left(1-\frac{c_x}{2}\right).
\zeta_{D}^{c}( 2 , x^{2}) = - \frac{\alpha R}{8 x^{3}}+ \frac{3 \alpha}{5 1 2 R x^{5}}+ \frac{5 5 5}{2^{1 7}}\frac{\alpha}{R^{3}x^{7}}+{\cal O}( x^{- 9}){.}
\begin{array}{l}{{\{n^{a}( x ) , n^{b}( y ) \}= 0 \; \; , \{{\pi}^{a}( x ) , n^{b}( y ) \}= ( \delta^{a b}- n^{a}( x ) n^{b}( x ) ) \delta ( x - y )}}\\{{\{{\pi}^{a}( x ) ,{\pi}^{b}( y ) \}= ({\pi}^{b}( x ) n^{a}( x ) -{\pi}^{a}( x ) n^{b}( x ) ) \; \delta ( x - y )}}\\ \end{array}
\frac{du_\varepsilon(t,\varepsilon)}{m}= -\frac{\Lambda(dt)}{m}+\frac{\kappa_c}{2}\left\{\ell\, \frac{\nabla u_\varepsilon(t,\varepsilon)}{\varepsilon}+m\mu_c(u_\varepsilon(t,\varepsilon)+u_\varepsilon(t,2\varepsilon))+2mp(N-u_\varepsilon(t,\varepsilon)) \right\}dt
\lambda_{k}\left( n\right) =\max_{v\left( G\right) =n}\lambda_{k}\left( G\right) , \lambda_{-k}\left( n\right) =\max_{v\left( G\right) =n}\text{}\left\vert \lambda_{n-k+1}\left( G\right) \right\vert , \lambda_{k}^{\ast}\left( n\right) =\max_{v\left( G\right) =n}\lambda_{k}^{\ast}\left( G\right) .
{}_{p}\Psi_q(z)={}_{p}\Psi_q\Bigg[\left.\begin{matrix}(a_i,\alpha_i)_{1,p}(b_j,\beta_j)_{1,q}\end{matrix}\right|z\Bigg]:=\sum_{n=0}^{\infty}\frac{\prod_{i=1}^{p}\Gamma(a_i+n\alpha_i)}{\prod_{j=1}^{q}\Gamma(b_j+n\beta_j)}\frac{z^n}{n!},
\{w_i,w_j\}= \frac{1}{t-1}(W_iW_j - W_jW_i) \mod (t-1) &= \frac{1}{t-1}(t^{a_{ij}}-1)W_jW_i \mod (t-1) &= (1 + t + \dots + t^{a_{ij}-1})W_iW_j \mod (t-1) &= a_{ij}w_iw_j.
D_{\mu,n}(f)=\frac{1}{n!(n-1)!}\displaystyle\int_{\mathbb D}\big|f^{(n)}(z)\big|^2 P_{\!\mu}(z)(1-|z|^2)^{n-1}dA(z)=\int_{\mathbb \mathbb T}D_{\lambda,n}(f) d\mu(\lambda).
(-1)^{n+m_{3}}\sqrt{\frac{(2j+1)(2l_{3}+1)}{(2l_{1}+1)(2l_{2}+1)}}\left[\begin{array}[c]{ccc}l_{1}l_{2}l_{3}m_{1}m_{2}-m_{3}\end{array}\right] \!\!{[j]}& =C_{m_{1},m_{2},m_{3}}^{l_{1},l_{2},l_{3}}\Phi(l_{1,}l_{2},l_{3};n+1)
r_{e}^{3 - p}\, = \, 2 \sqrt{\frac{2 V}{V_{*}}}\frac{q_{2}}{q_{1}}\left( r_{2}^{3 - p}- r_{1}^{3 - p}\right) + r_{1}^{3 - p}\, = \, r_{2}^{3 - p}\, ,
\langle\hat{T}\rangle=\langle\psi|\sum_{i=1}^{N}\frac{-\hbar^{2}}{2m_{e}}\nabla_{i}^{2}|\psi\rangle=-\frac{\hbar^{2}}{2m_{e}}\sum_{i=1}^{N}\langle\psi|\nabla_{i}^{2}|\psi\rangle
a_j^{BW}({\mathsf s})= \frac{R_Z}{{\mathsf s}-{\mathsf s}_R}= \frac{R_Z}{{\mathsf s}-\bar M^2_Z+i\bar M_Z\bar\Gamma_Z}= \frac{R_Z}{{\mathsf s}-(M_R-i\frac{\Gamma_R}{2})^2}\, , \quad m_0^2\leq{\mathsf s}<\infty \, .
\|\varphi\|^p_{1,2,p; W^{2, p}(\overline{D})}\ = \ \| \varphi \|^p_{p; W^{2,p}(\overline{D})}+ \| \frac{\partial \varphi}{\partial t}\|^p_{p; L^p(\overline{D})}, \ 1 \leq p < \infty.
\begin{aligned}2, \,\, 5, \,\, 11, \,\, 17, \,\, \, 31, \,\, 47, \,\, 53, \,\, 3^2 \cdot 7, \,\, 3 \cdot 13, \,\, 3 \cdot 19, \,\, 3 \cdot 23, \,\, 3 \cdot 29, \,\, 3 \cdot 37, \,\, 3 \cdot 41, \,\, 3 \cdot 43, \,\, 3 \cdot 59, \,\, 3^2 \cdot 61, \,\, 3 \cdot 113.\end{aligned}
\begin{aligned}AG\geq \frac{2}{n}\phi G^{2}-C_{3}\phi G^{2}-\frac{4}{n}\phi G(p-1)\lambda_{1}\omega-\frac{C_{1}^{2}}{R^{2}}\frac{G}{C_{3}}-2KG\phi+2Kp\lambda_{1}\omega \phi = (\frac{2}{n}-C_{3})G^{2}\phi-\frac{C_{1}^{2}}{R^{2}}\frac{G}{C_{3}}-2KG\phi+\lambda_{1}\omega \phi(-\frac{4}{n}G(p-1)+2Kp).\end{aligned}
m_{d+1,1^0}(x_1,x_2,\ldots,x_i) = \begin{cases}m_{d,1^0}(x_1,x_2,\ldots,x_i) x_{i+1}(d\ge2), m_{1^{1}}(x_1,x_2,\ldots,x_i) x_{i+1}(d=1) \end{cases}
\int_{B_2}|f_j|^2\leq2 H_j(0,2)\leq 2^{2+2I_j(0,2)}H_{j}(0,1) = \frac{2^{2+2I_j(0,2)}}{I_j(0,1)}\leq Q2^{2+2I_j(0,2)}\leq C(Q,\Delta_0)\,,
V(P, I) = \sum_{\{n_{i, k}\}}\sum_{j \geq 0}\widetilde{C}(\{n_{i, k}\}, j) \hbar^{p(\{n_{i, k}\}, j)}\left(\sum_l \partial_{P_l^{(1)}}\partial_{P_l^{(2)}}\right)^j\prod_{i, k \geq 0}(S^kI_{i, k})^{n_{i, k}}.
C(x_V \mid x_{[k]-V}, p_{[k+1..\ell]}) \stackrel{(a)}{\leq}C(x_V \mid x_{[k]-V}, x_{k+1}, p_{[k+2..\ell]}) + c \log n &\stackrel{}{=}C(x_V \mid x_{[k+1]-V}, p_{[k+2..\ell]}) + c \log n & \stackrel{(b)}{\leq}\sum_{j \in V}n_j + O(\ell-k-1) \cdot \log n + c \log n & = \sum_{j \in V}n_j + O(\ell-k) \cdot \log n.
D_{\alpha\beta}(p\|q)= \frac{1}{(\beta-1)(\alpha+\beta-1)}\sum_{i}p^{\alpha+\beta-1}_{i}&+\frac{1}{\alpha(\alpha+\beta-1)}\sum_{i}q^{\alpha+\beta-1}_{i}&-\frac{1}{\alpha(\beta-1)}\sum_{i}p^{\alpha}_{i}q^{\beta-1}_{i}
\dot w(t) = \sum\limits_{i=1}^n \frac{\partial}{\partial \bar q^c_i}U(\bar q^c(t))\dot{\bar{q^c}}_i(t) & \geq \sum\limits_{i=1}^n \left[U(\dot{\bar{q^c}}_i(t) + \bar q^c_i(t),\bar q^c_{-i}(t)) - U(\bar q^c(t))\right] & = \sum\limits_{i=1}^n \left[\max_{\alpha_i \in \Delta_i}U(\alpha_i,\bar q^c_{-i}(t)) - U(\bar q^c(t)) \right]\geq 0,
\Vert (v_{s},v_{u})\Vert_{i}= \begin{cases}\sqrt{a^{2i}\Vert v_{s}\Vert^{2}+b^{2i}\Vert v_{u}\Vert^{2}}\mbox{if}i\geq0 \Vert (v_{s},v_{u})\Vert \mbox{if}i<0, \end{cases}
R_{k+1}\leq \: \left( \bigcup_{j=k+1}^{\infty}A_{j}\right)=\sum_{j=1}^{\infty}R_{k+j}\leq \frac{\alpha}{1-2\gamma}\sum_{j=1}^{\infty}d_{k+j}<d_{k}\gamma^{k}=\rho_{k}<R_{k}.
{\frac{1}{{\widetilde g}_{\mathrm{{\small Y M}}}^{2}( p^{2})}}\equiv{\frac{1}{g_{\mathrm{{\small Y M}}}^{2}}}+{\frac{\beta_{0}}{1 6 \pi^{2}}}\ln \left({\frac{\Lambda^{2}}{p^{2}}}\right) ,
f_2(f_1(x)) &=\sum_{m=0}^n\frac{1}{m!}\left(\frac{d}{d y}\right)^{m}f_2(y)\Big|_{y=f_1(x_0)}(f_1(x)-f_1(x_0))^m &+\frac{1}{n!}\int_{f_1(x_0)}^{f_1(x)}(f_1(x)-y)^n\left(\frac{d}{d y}\right)^{n+1}f_2(y)dy.
\tilde{U}( \phi_{\mathrm{k i n k}}( x ) ) = \frac{\tilde{m}^{2}}{2 \phi_{0}( T )^{2}}( 3 \phi_{\mathrm{k i n k}}( x )^{2}- \phi_{0}( T )^{2}) = \tilde{m}^{2}\left( - \frac{3}{2}\mathrm{s e c h}^{2}\frac{\tilde{m}x}{2}+ 1 \right) \, .
\left\langle i \right|F(p,q,m)\left| i \right\rangle =\left\langle i \right|Y_0 \left| i \right\rangle - \frac{e^2}{\pi}\left\langle i \right|\beta m \left( \frac 32\ln(A_0)+\frac 1{4A_0^2}-\frac 14 \right)\left|{i}\right\rangle,
\mathsf{d_h}:{\mathsf{\Omega}}^{r,s}\left({\sf J}^kY\right)\rightarrow{\mathsf{\Omega}}^{r+1,s}\left({\sf J}^{k+1}Y\right),\mathsf{d_h}F:=\sum_{i=1}^n\frac{\mathsf{d}}{\mathsf{d}x_i}^{(k+1)}(F)\,dx_i.
X^{(0)}(s_{0}) = \ \mathsf{Q}, X^{(1)}(s_{0}) = \ \mathcal{P}_{1}X^{(0)}(s_{0}), \mathrm{and}X^{(j)}(s_{0})= \ \mathcal{P}_{1}X^{(j-1)}(s_{0}) + \mathcal{P}_{2}X^{(j-2)}(s_{0})
C(\beta)=2\,\int_{\mathbb{R}^N}\frac{\left||x|^{-\beta}-|y|^{-\beta}\right|^{p-2}\,\left(|x|^{-\beta}-|y|^{-\beta}\right)}{|x-y|^{N+s\,p}}\, dy,x\in\mathbb{S}^{N-1},
\Pr(G \mid \sigma = \tau) = p^{|E(G) \cap A_\tau|}q^{|E(G) \cap B_\tau|}(1-p)^{|A_\tau \setminus E(G)|}(1-q)^{|B_\tau \setminus E(G)|}= (1-p)^{|A_\tau|}(1-q)^{|B_\tau|}\left(\frac{p}{1-p}\right)^{|E(G) \cap A_\tau|}\left(\frac{q}{1-q}\right)^{|E(G) \cap B_\tau|}.
\begin{array}{rcl}R_{\mu\nu}+\frac{1}{2}\partial_{\mu}\varphi\partial_{\nu}\varphi+\frac{1}{12}{\cal V}_{0}e^{-\varphi}g_{\mu\nu}= 0\, , & \nabla^{2}\varphi+\frac{1}{2}{\cal V}_{0}e^{-\varphi}= 0\, . \end{array}
((L_{0},\varphi_0^{(n)})\cdots (L_d,\varphi_d^{(n)}))_S\geqslant \sum_{i=0}^d\delta_i\frac{\widehat{\mu}_{\min}(E_{i,n},\xi_{i,n})}{n}&-\frac 32\nu(\Omega_\infty)\sum_{i=0}^d\frac{\delta_i}{n}\ln(r_{i,n}+1)-\nu(\Omega_\infty)\sum_{i=0}^d\frac{\delta_i}{n}\ln(n^d\delta_i).
\frac{D_n({\rm e}^V{\rm d}\mu_{FH})}{D_n({\rm d}\mu_{FH})}={\rm e}^{n V_0 + \sum_{k=1}^\infty k V_k V_{-k}}\prod_{j=1}^r{\rm e}^{-(\alpha_j+\beta_j) \sum_{k>0}V_k z_j^k-(\alpha_j-\beta_j) \sum_{k>0}V_{-k}z_j^{-k}}\times (1+o(1)),
\Big\|\frac{\partial_t^j\hat m(t,\xi,\eta)}{(|\xi|+|\eta|)^{3/2+\epsilon'}}\Big\|_2\lesssim_{\epsilon'}\sum_{l=0}^j\|\partial_t^l\phi\|_{4-2\epsilon''}\|\partial_t^{j-l}\phi\|_{4-2\epsilon''}.
\begin{aligned}\frac{1}{sp}+\int_{0}^{1}\frac{\left[1-(1-k)^s\right]^{p-1}-\left[(1+k)^s-1\right]^{p-1}}{k^{1+sp}}\mathrm{d}k-\int_{1}^{\infty}\frac{\left[(1+k)^s-1\right]^{p-1}}{k^{1+sp}}\mathrm{d}k=0.\end{aligned}
x=[\frac{(1-4k^{2}k^{\prime2})^{2}}{k^{2}k^{\prime2}}]^{\frac{1}{4}}\theta^{-1},c=\frac{1}{2}[\frac{(1-4k^{2}k^{\prime2})^{2}}{k^{2}k^{\prime2}}]^{\frac{1}{4}}
e^{x}=c_{0}+c_{1}x+(c_{2}-\frac{c_{0}}{2})x^{2}+(c_{3}-\frac{c_{1}}{2})x^{3}+(c_{4}-\frac{c_{2}}{2}+\frac{c_{0}}{4!})x^{4}+\cdot\cdot\cdot
\Pr \left\{\xi_1 \wedge \xi_2 < \left(a_{_{J_n}}^\eta \wedge \gamma\right) b \ \left|\frac{}{}\right. \J_n \right\}\leq \prod_{i=1, i \neq J_n}^n F\left( \frac{i^ka_{_{J_n}}^\eta b}{\tilde{c}_k}\right) \leq F(1)^{\left(\frac{\tilde{c}_k}{a_{_{J_n}}^\eta b}\right)^{\frac{1}{k}}-2},
P(S_n(\kappa_n(x))\leq\lfloor\kappa_n(x)\rfloor-a_n) =P(S_n(\kappa_n(x))\leq ((1-r^{-1})^{-1}x+o(1))a_c^{(n)}) &=P\left(\frac{S_n(\kappa_n(x))}{h(x)a_c^{(n)}}\leq\frac{((1-r^{-1})^{-1}x+o(1))}{h(x)}\right) &=P\left(\frac{S_n(\kappa_n(x))}{(1-r^{-1})^{-1}h(x)a_c^{(n)}}\leq\frac{(x+o(1))}{h(x)}\right).
\mathcal{I}(\delta) = \inf_{\lambda \in \mathbb{R}^2_+}\left\{\langle \lambda, \delta \rangle + \mathcal{I}^{\star}(\lambda) \right\}= \inf_{\lambda \in \mathbb{R}^2_+}\left\{\langle \lambda, \delta \rangle + \sup_{\gamma \in \Pi\left(\mu_1, \mu_2\right)}\int_{\mathcal{V}}f_\lambda(v) \, d \gamma(v) \right\},
\left. \begin{array}{l l}\mathcal{T}_p:= \! \big\{\textbf{x}\ | \ \| \textbf{x}_1\!-\!\textbf{x}_E \|_2 =0 \big\}\cup \big\{\textbf{x}\ | \ \| \textbf{x}_2\!-\!\textbf{x}_E \|_2 =0 \big\}\cup \big\{\textbf{x}\ | \ \| \textbf{x}_3\!-\!\textbf{x}_E \|_2 =0 \big\}\end{array}\right.
F_5(a,b,c,d,e) = \int\limits_0^{\infty}\!dt \,t\!\int\limits_0^{\infty}\!\! dx x J_0(cx) J_0(dx) J_0(ex) J_0(tx) &\times \int\limits_0^{\infty}\!dy y J_0(ay) J_0(by) J_0(ty) \;.
L_{b_{2}( k )}^{a_{2}}( \mu ) L_{b_{1}( j )}^{a_{1}}( \lambda ) = e^{- i \theta ( \hat{a}_{1}- \hat{b}_{1}) ( \hat{a}_{2}- \hat{b}_{2})}L_{b_{1}( j )}^{a_{1}}( \lambda ) L_{b_{2}( k )}^{a_{2}}( \mu ) , \ll{b r a n y}
\rho_s(x) = \left( 1 + O\left( x^{-1}\right) \right) (x/e)^{\sigma}\left( 1 + \sigma/x \right)^{x + \sigma}& \times \exp \left( \tfrac12 (x+\sigma) \log \big( 1 + \left(\tfrac{t}{x + \sigma}\right)^2 \big) -t \arctan \left( \tfrac{t}{x + \sigma}\right) \right) & \times \exp \left( i \phi_1(t) + i \phi_2(t) + it(-1 + \log x) \right)
Z = \sum_{n}{\frac{\left( z_{1}\right)^{n}}{n !}}\int \prod_{k = 1}^{n}d{\bf R}_{k}\exp \left( -{\frac{\pi m_{W}}{2 e^{2}}}\sum_{i \not = j}{\frac{q_{i}q_{j}}{\vert{\bf R}_{i}-{\bf R}_{j}\vert}}\right)