harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{e\mid d}\mu(d/e)f(x_1^{d/e},x_2^{d/e},\dots)^e =\sum_{e\mid d}\mu(d/e)\left(\sum_{i=1}^{n}g_i(x_1^{d/e},x_2^{d/e},\dots)\right)^e &=\sum_{e\mid d}\mu(d/e)\left(\sum_{i=1}^{n}g_i^{d/e}(x_1,x_2,\dots)\right)^e.
I ( T / \lambda ) = \int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{1}{e^{| k_{0}| / T}- 1}\delta ( k^{2}- \lambda^{2}) = \frac{1}{( 2 \pi )^{2}}\left[ \frac{\pi T}{2 \lambda}+ \frac{1}{2}\ln \frac{\lambda}{T}+ \cdots \right]
\begin{aligned}\lim_{r\to 0}\lim_{k\to+\infty}\left(\frac{r}{a}\int_{\partial B_{r}}\left|x\right|^{2\alpha_{k,1}}V_{k}(x)e^{au_k}d\sigma+r\int_{\partial B_{r}}\left|x\right|^{2\alpha_{k,2}}U_{k}(x)e^{u_k}d\sigma\right) &=-\pi\beta^{2}_{0}+\frac{4\pi(\alpha_{1}+1)}{a}\beta_{0,1}+4\pi(\alpha_2+1)\beta_{0,2},\end{aligned}
\frac{d}{dt}\eta_t^* (\alpha + t d h) = \eta_t^* \bigl( L_{X_t}(\alpha + t d h) + dh \bigr) = \eta_t^* \bigl( \imath_{X_t}d (\alpha + t dh)+ d \bigl((\alpha + t d h)(X_t)\bigr) + d h \bigr) = \eta^*_t (0 - dh + dh) = 0,
\|u(t_n)-u_h^{n}\|+\gamma^{\frac12}\Big{(}\Delta t\sum_{k=1}^n\|\sigma(t_{k-\theta})-\sigma_h^{k-\theta}\|^2\Big{)}^{\frac12}+\Big{(}\Delta t\sum_{k=1}^n\|\nabla (u(t_{k-\theta})-u_{h}^{k-\theta})\|^2\Big{)}^{\frac12}\leq C[\Delta t^2+h^{m+1}],
N^{\cal P}= \left( \frac{2}{k + 2}\right) \sum_{l = 1}^{k + 1}s i n^{2}\theta_{l}\sum_{m_{1}= - j_{1}}^{j_{1}}\cdots \sum_{m_{p}= - j_{p}}^{j_{p}}\exp \{2 i ( \sum_{n = 1}^{p}m_{n}) \theta_{l}\},
( R [ U_{2}] + R [ V_{1}] ) / ( R [ V_{1}] + R[V_{2}] ) \cong R [ U_{2}] / (R [ U_{2}] \cap R[ V_{1}\cup V_{2}]) &= R [ U_{2}] / R [ U_{2}\cap ( V_{1}\cup V_{2}) ] &= R [ U_{2}] / R [ V_{2}].
\widetilde{\Gamma}_{i}^{( 9 )}= \left( \begin{matrix}{0}&{t_{i}}\\{- t_{i}}&{0}\\ \end{matrix}\right) , \qquad \widetilde{\Gamma}_{i}^{( 0 )}= \left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) , \qquad \widetilde{\Gamma}_{i}^{( 8 )}= \left( \begin{matrix}{1}&{0}\\{0}&{- 1}\\ \end{matrix}\right) ,
I_{p}( 2 ) = \frac{1}{2 4}\sum_{n = 0}^{p - 1}\left( \begin{array}{c}{{2 p + 1}}\\{{n}}\\ \end{array}\right) ( 2 p + 1 - 2 n ) \{( 2 p + 1 - 2 n )^{2}- 1 \}\, .
\tilde{G}( x_{1}^{4}, \vec{x}_{1}, A_{1}, B_{1}; x_{2}^{4}, \vec{x}_{2}, A_{2}, B_{2}) = \tilde{G}( x_{1}^{4}, - \vec{x}_{1}, B_{1}, A_{1}; x_{2}^{4}, - \vec{x}_{2}, B_{2}, A_{2}) .
P_{0, 4}(\lambda; \tau, \eta) S^+(\tau, \eta) P_{0, 4}(\lambda; -1/\tau, -\eta/\tau)^{-1}= -4 \sqrt{2}\pi i \tau \exp\Big(\pi i \frac{4 + 216 \eta^2 - 42 \eta(1 + \tau) - 3 \tau + 4\tau^2}{12\tau}\Big)
f_{3}= - \frac{i e^{2}}{2 \pi^{2}}\ln \frac{\Lambda}{\mu}\sum_{i}c_{i}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{M_{i}^{2}}{( k^{2}- M_{i}^{2})^{2}\Big ( ( k + p )^{2}- M_{i}^{2}\Big )}\qquad \quad
\nabla^{\mathrm{restr}}_{\partial z}(v_{1}) = (\frac{c +\alpha z}{z^{2}}-\frac{\lambda + 1}{2z}) v_{1}+\frac{\eta}{z^{2}}v_{2}& \nabla^{\mathrm{restr}}_{\partial z}(v_{2}) = \left( -\frac{\beta}{2}+\frac{\eta \gamma}{z}\right) v_{1}+( \frac{c+ \alpha z}{z^{2}}+ \frac{\lambda + 1}{2z}) v_{2}.
T_{j}^{(m+1)}-T_{j}^{n}+\alpha_{m+1}\lambda(u^{n}T^{(m)})|_{j-1/2}^{j+1/2}+\alpha_{m+1}\lambda(\gamma-2)T_{j}^{(m)}u^{n}|_{j-1/2}^{j+1/2}=0
\sum_{i=n}^{+\infty}q_{i}(x)R_G^i i^t \underset{n\rightarrow+\infty}{\sim}\frac{K_1Q^{\prime}(x+x_0)}{n^{m+2-t-\alpha}}, &\underset{n\rightarrow+\infty}{\sim}K_2 Q^{\prime}(x+x_0)\sum_{i=n}^{+\infty}p_iR_G^i i^{t-2},
[2]\sum_{k=0}^{n-1}\frac{(q,q^{-1};q^2)_kq^{2k}}{(q^2;q^2)_k^2}=\sum_{k=0}^{n-1}\frac{(q^{3},q^{-1};q^2)_kq^{2k}}{(q^2;q^2)_k^2}+q\sum_{k=0}^{n-1}\frac{(q;q^2)_k^2q^{2k}}{(q^2;q^2)_k^2}begin{align*}1.5mm] \equiv (-1)^{(n+1)/2}q^{(n^2+3)/4}+q(-1)^{(n-1)/2}q^{(n^2-1)/4},
y = p \, \sqrt{\frac{\lambda}{\lambda_{l}^{( \ast )}}}\, \left[ 1 + O \left( \epsilon \right) \right] = p \, \sqrt{1 + \frac{\Theta_{l}^{2}}{\lambda_{l}^{( \ast )}}}\, \left[ 1 + O \left( \epsilon \right) \right] \; ,
N = N_1 N_2 N_3 = \begin{bmatrix}A_1 0 C_1 B_1 \end{bmatrix}\begin{bmatrix}0 A_2 B_2 C_2 \end{bmatrix}\begin{bmatrix}B_3 C_3 0 A_3 \end{bmatrix}= \begin{bmatrix}0 A_1 A_2 A_3 B_1 B_2 B_3 C_1 A_2 A_3 + B_1 C_2 A_3 + B_1 B_2 C_3 \end{bmatrix}
\sum_{\substack{\delta_1,\ldots,\delta_K,j_1,\ldots,j_K, n_1,\ldots,n_K\in\mathcal{A}_{\{2,4,\ldots,K\}}, l_{k-1,1}=l_{k,1}\forall k\in\{2,4,\ldots,K\}, \Delta\in\mathcal{D}}}\frac{1}{2^K}\frac{1}{N^{K/2+1}}\prod_{k=1}^Ka'_{j_k}= |\mathcal{D}|\left(\sum_{\substack{j\in(\mathbb{Z}\backslash \{0\})^2, ||j||_{\infty}\leq N^K}}a'^2_j\right)^{K/2}.
\| m_{\mathbf{B},V}^\frac 12 (-i \nabla + \mathbf{A}) u\|^2=\langle (-i\nabla m_{\mathbf{B},V})(-i\nabla+\mathbf{A})u,u\rangle+\langle m_{\mathbf{B},V}(-i\nabla+\mathbf{A})^2u,u\rangle\,.
\mathbb{E}(1+ e^{\lambda t_{e_k}'})^2 (F_m^{(k)}-F_m)_+^2 = \mathbb{E}(1+e^{\lambda t_{e_k}'})^2 \left( \frac{1}{\# B_m}\sum_{z \in B_m}(T_z^{(k)}-T_z) \right)_+^2\ .
{\cal P}( 2 \pi i z ) = - \frac{1}{4 \pi^{2}}\left( \frac{1}{z^{2}}+ \sum_{( m . n ) \neq ( 0 , 0 )}\left[ \frac{1}{( z - m - n \tau )^{2}}- \frac{1}{( m + n \tau )^{2}}\right] \right)
\left|\frac{k_1}{m_1}\theta-\frac{k_2}{m_2}-\frac{k}m\right| =\left|\frac{k_1}{m_1}\right|\cdot\left|\theta-\frac{m_1(k_2m+km_2)}{k_1m_2m}\right| >\frac{C}{k_1m_1m_2^2\cdot m^2}
\Omega_{\psi}( \omega , \eta ) = \frac{1}{\rho_{c}}\frac{d \rho_{\psi}}{d \operatorname{l o g}{\omega}}= g_{1}^{2}\Omega_{\gamma}( \eta ) \biggl [ \frac{\omega}{\omega_{1}}\biggr ]^{3 - 2 \mu}
\frac{\varphi_{g^{-1}s}(\pi(x))}{\varphi_{g^{-1}}(\pi(x))}=\frac{dg^{-1}s\nu}{dg^{-1}\nu}(\pi(x))=\frac{ds\nu}{d\nu}(g.\pi(x))=\frac{ds\nu}{d\nu}(\pi(g.x)).
\|e^{-\frac{1}{2}A_n^2(t-s)}(-A_n)^{\gamma}\|_{2}=\max_{1\le j\le n-1}e^{-\frac{1}{2}\lambda_{j,n}^2(t-s)}(-\lambda_{j,n})^\gamma\le C(t-s)^{-\frac{\gamma}{2}},\quad\gamma>0,
s(y_{3},\upsilon,\bar{\upsilon})= -2\ln \upsilon +\frac{4}{(e^{y_{3}}+e^{-y_{3}})^{2}}\cdot\frac{\upsilon{\bar \upsilon}-{\bar \upsilon^{2}}}{\upsilon^{2}+{\bar \upsilon}^{2}-\upsilon{\bar \upsilon}}+....
\begin{aligned}g_1 = 1 g_2 = \frac{x^3+x^2+x+1}{y^3}g_3 = g_2/y g_4 = \frac{4(x^3+x^2+x+1) - x^2y^4 - 2xy^4 - 3y^4}{4y^6}g_5 = g_5/y g_6 = g_6/y g_7 = \frac{16(x^3+x^2+x+1)-6x^3y^4 - 10x^2y^4 + xy^8 - 14xy^4 + 3y^8 - 18y^4}{6y^9}g_8 = \frac{32(x^3+x^2+x+1) - 3x^2y^8 - 8x^2y^4 - 4xy^8 - 16xy^4 - 3y^8 - 24y^4}{32y^{10}}\end{aligned}
m_{t}\coloneqq\mathbf{E}_{x_{0}}[X_{t}]=a_{t}+e^{\beta t}x_{0}, \Sigma_{t}\coloneqq\mathbf{E}_{x_{0}}[(X_{t}-m_{t})(X_{t}-m_{t})^{T}]=\sigma^{2}\int_{0}^{t}e^{\beta(t-s)}(e^{\beta(t-s)})^{T}\mathrm{d}s,
\begin{cases}^{C\!}D_{0+}^{\alpha}x_1(t)= f_1(t,x_1(t)),^{C\!}D_{0+}^{\alpha}x_2(t)= f_2(t,x_1(t),x_2(t)), \cdots \cdots^{C\!}D_{0+}^{\alpha}x_d(t)= f_d(t,x_1(t),x_2(t),\cdots,x_d(t)),\end{cases}
+3a_{2}^{1}c_{1}^{2}a_{2}^{2}+a_{1}^{0}c_{1}^{2}a_{1}^{1}+a_{1}^{1}c_{2}^{1}a_{2}^{3}+a_{1}^{0}c_{2}^{1}a_{2}^{4}-a_{2}^{0}c_{1}^{2}a_{2}^{3}+a_{2}^{0}c_{1}^{1}a_{2}^{4}).
H = \frac{1}{2 m}\overrightarrow{\nabla}^{2}- \frac{i e}{2 m}\overrightarrow{\nabla}. \overrightarrow{A}- \frac{i e}{m}\overrightarrow{A}. \overrightarrow{\nabla}- \frac{e^{2}}{2 m}\left| \overrightarrow{A}\right|^{2}- e \phi .
t_{j}=\frac{1}{2\cos^{2}(\frac{\pi j}{2H})}\prod_{i=0}^{j-1}\frac{\tan (\frac{\pi (j-1-2i+w_{+})}{2H})\tan (\frac{\pi (j-1-2i+w_{-})}{2H})}{\tan^{2}(\frac{\pi (i+1)}{2H})}.
\Vert u(t)\Vert_{\infty}\leq \begin{cases}K \, t^{-\frac{1}{m_2-1}}\forall t \in \left(0,t^\ast \right] , 1 \forall t \in \left( t^\ast , 2T \right) , K \, t^{-\frac{1}{m_1-1}}\forall t \ge 2T \, .\end{cases}
\psi_{, T T}^{z}- \psi_{, \sigma \sigma}^{z}+ \frac{4 \alpha}{1 + \alpha}( T + \tilde{\Lambda})^{- 1}\psi_{, T}^{z}- \frac{2 \alpha ( 1 - \alpha}{( 1 + \alpha )^{2}}( T + \tilde{\Lambda})^{- 2}\psi^{z}= 0 ,
H^{(n,m)}_{\texttt{a};j,k}(\boldsymbol{\xi)}:=\partial_{\xi_j}\partial_{\xi_k}V^{(n,m)}_{\texttt{a};\lambda}(\boldsymbol{\xi}) &=\begin{cases}m + \sum_{\substack{1\leq l\leq n l\neq j}}u_q(\xi_j-\xi_l) -u_q(\xi_j-\xi_k) \end{cases},
L = \frac{1}{2}\Gamma_i \Lambda^{ij}\dot{\Gamma}_j - \rho(\Gamma ) = q_i \dot{x}_i + \frac{B}{2}\epsilon_{ij}x_i \dot{x}_j - \frac{1}{2}\left( \frac{q^2_i}{m}+ kx^2_i \right)
\int_{|x|\geq \tilde C(\eta)\sqrt{t}}\bigl||x|^{|s_c|}e^{-it\Delta}\Psi(t)\bigr|^2 \,dx = \lim_{n\to\infty}\int_{|x|\geq \tilde C(\eta)\sqrt{t}}\bigl||x|^{|s_c|}e^{-it\Delta}v^{[t_n]}(t)\bigr|^2\,dx & = \lim_{n\to\infty}\int_{|x|\geq\tilde C(\eta)\sqrt{t_nt}}\bigl||x|^{|s_c|}e^{-it_nt\Delta}v(t_nt)\bigr|^2\,dx \leq\eta
g_{n_k}=a_{g_{n_k}}^{\gamma_{g_{n_k}}}b_{g_{n_k}}^{\delta_{g_{n_k}}}c_{g_{n_k}}^{\varepsilon_{g_{n_k}}}a_{g_{n_k}}^{-\gamma_{g_{n_k}}}, \ (b_{g_{n_k}}, c_{g_{n_k}})\in V_{n_k}a_{g_{n_k}}\in W_{n_k}.
(\begin{matrix}\alpha&\beta\\ \gamma&\delta\end{matrix})=(\begin{matrix}\alpha&\beta\\ \gamma&\delta\end{matrix})^{\dagger},\beta\alpha=q^{2}\alpha\beta,[\alpha,\delta]=0,[\beta,\gamma]=(1-q^{-2})\alpha(\delta-\alpha),[\delta,\beta]=(1-q^{-2})\alpha\beta
P^{\textsc{idle}}\left( \pmb{\rho}^{(N)}\right) = \prod_{u=1}^V \left(1-\rho_u^{(N)}p_u^{(N)}\right)^{N_u^{(N)}}&= \prod_{u=1}^V \left(1-\frac{\rho_u^{(N)}p_u^{(N)}N}{N}\right)^{\frac{N_u^{(N)}}{N}N}
\mbox{exp}\left[\frac{2i\pi}{k}\tau e(\tilde{n}-1/2)+\frac{2i\pi}{k}e(\widetilde{n_{0}}z_{1}+(1-\widetilde{n_{0}})z_{2}-w)+\frac{2i\pi}{k}(\tilde{n}-1/2)e_{0}(z_{1}-z_{2})\right] dw
\int_{G}|\langle\lambda_g(F),F\rangle|^2\;g = \int_{G^3}F(g^{-1}x)\overline{F(x)}\,\overline{F(g^{-1}y)}F(y)\;g\;x\;y & = \int_{G^3}\int_{H}F(g^{-1}xt)\overline{F(x)}\,\overline{F(g^{-1}yt)}F(y)\;g\;x\;y\;t
C_{T}=c_{0}^{-1}(1+\left\|a\right\|_{L^{\infty}})e^{c_{0}^{-1}\left(1+\|a^{\prime}\|_{L^{\infty}}+\|q\|_{L^{\infty}}+2\left\|a\right\|_{L^{\infty}}\right)T},
d s_{1 1}^{2}= \frac{1}{H_{M 5}( \rho )^{1 / 3}}\left[ - d t^{2}+ d x^{i}d x^{i}\right] + H_{M 5}( \rho )^{2 / 3}\left[ d \rho^{2}+ \rho^{2}d \Omega_{4}^{2}\right] ,
C^2 \left\|Q_N^{1}x\right\|_1^{2}\, \ge\, \left\|Q_N^{1}x\right\|_2^{2}\, = \, \left\|\left(P_N^{2}x-P_N^{1}x\right)+Q_N^{2}x\right\|_2^{2}
\sum_{n=1}^{[N/\Delta]}\frac{1}{n}- \gamma = \log \left[ \frac{N}{\Delta}\right]+ \frac{1}{2 [N/\Delta]}- \sum_{s=1}^{k}\frac{B_{2s}}{2s [N/\Delta]^{2s}}+ O \left( \frac{1}{N^{k+1}}\right) & = \log \left[ \frac{N}{\Delta}\right]- \sum_{s=1}^{2k}\frac{B_{s}}{s [N/\Delta]^{s}}+ O \left( \frac{1}{N^{k+1}}\right),
\frac{dq_1}{dt}=\frac{\partial I_i}{\partial p_1}, \frac{dp_1}{dt}=- \frac{\partial I_i}{\partial q_1}\frac{dq_2}{dt}=\frac{\partial I_i}{\partial p_2}, \frac{dp_2}{dt}=- \frac{\partial I_i}{\partial q_2}
U^{2a}_{2k,2a}(x;q) &=(xq)^{2a+1}\sum_{h=1}^{k-a}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q)+(xq)^{2a+1}\sum_{h=0}^{k-a-1}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q) &+(xq)^{2a-1}\sum_{h=1}^{k-a+1}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q)+(xq)^{2a-1}\sum_{h=0}^{k-a}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q).
\int\limits_{|y|<2|x|}dy\, |y|^{-\sigma}|y-x|^{2-d-\alpha}= |x|^{2-\sigma-\alpha}\int\limits_{|y|<2}dy\, |y|^{-\sigma}|y-\omega_x|^{2-d-\alpha}\lesssim |x|^{-\sigma}
S_{k}=\left\{\left( \bar{u},\bar{v}\right) \in\mathbb{H}_{0}^{\alpha/2}:\varphi_{k}\left( \bar{u},\bar{v}\right) =\max\limits_{u}\min\limits_{v}\varphi_{k}\left( u,v\right) =\min\limits_{v}\max\limits_{u}\varphi_{k}\left( u,v\right) \right\}.
U'_{\mathcal X_1\to y_1}=\{u'\in \mathcal P_{\mathcal X_1}: \{u', \alpha^{1}_{\mathcal X_2}\}\frac{\rho}{2}\prod_{j\in [k]\setminus \{1,2\}}|\mathcal P_{\mathcal X_j}| \mathcal A_{\mathcal Y}\}.
c_{12}c_{23}t_{12}=t_{23}c_{12}c_{23}c_{23}t_{12}c_{23}^{-1}=c_{12}^{-1}t_{23}c_{12}t_{12}c_{23}c_{12}=c_{23}c_{12}t_{23}c_{23}^{-1}t_{12}c_{23}=c_{12}t_{23}c_{12}^{-1}
h_n^+(x):=\begin{cases}0, e_n\leq x\leq b_n-\delta, h(b_n)\cdot \displaystyle{\frac{\ss_n(x)-\ss_n(b_n-\delta)}{\ss_n(b_n-\delta')-\ss_n(b_n-\delta)}}, b_n-\delta<x<b_n-\delta', h(b_n), b_n-\delta'\leq x\leq b_n.\end{cases}
\upsilon^{\textrm{F}}_{i,(k,m)}=\prod\limits_{p\in\varphi\setminus\{i\}}\Big(\sum\limits_{(q,u)\in\mathcal{G}^{\textrm{F}}_{p,(k,m)}}\pi_{p,(q,u)}\Big), ~\textrm{if}~k \geq 0,~m < M,
\liminf_{n\to\infty}\frac{1}{n}D_{1+s}(P_{X^{n}Y^{n}}\|\pi_{X^{n}Y^{n}}) \geq\liminf_{n\to\infty}\left\{\min\left\{1,\frac{1-s}{s}\right\}\delta_{n}-\frac{1}{ns}\log2\right\}=\min\left\{1,\frac{1-s}{s}\right\}\liminf_{n\to\infty}\delta_{n}>0.
\nu(a_t^c)\big[(z_{11}\cos\theta -z_{21}\zeta\sin\theta )^r(z_{11}z_{22}-z_{21}z_{12})^{\mu_2}z_{33}^\nu\big] = \int_{\mathbb{C}^9}K_{a_t^c}(z,\bar{w})(w_{11}\cos\theta -w_{21}\zeta\sin\theta )^r(w_{11}w_{22}-w_{21}w_{12})^{\mu_2}w_{33}^\nu e^{-\pi|w|^2}dw.
{\cal A}_L^{(1)\,{\rm lead}}=\frac m{8\pi}\left( \frac{\lambda^2}{16m^4}- \frac{4e^4}{m^2\Theta^2}\right) \left[ \ln \left( \frac{\Lambda^2}{{\bf p}^2}\right) +i\pi \right] \hspace{0.3cm}.
\left(A\xi_{\lambda}\right)\left(x\right)=-\left(D_{F}^{*}\xi_{\lambda}\right)\left(x\right)=\xi'_{\lambda}\left(x\right)\overset{\left(\ref{eq:Dadj9}\right)}{=}i\lambda\xi_{\lambda}\left(x\right),\;\forall x\in\left(0,a\right)
\tilde{A}_{1}(p) = - g^{2}N(N^{2}-1)\delta^{a}_{b}\delta_{J}^{I}\int \frac{d^{4}k}{(2\pi)^{4}}d^{4}\theta \, \frac{1}{k^{2}[(p-k)^{2}+m^{2}]}\left\{\Phi^{\dagger}_{aI}(p,\theta,{\overline \theta}) \Phi^{bJ}(-p,\theta,{\overline \theta}) \right\}.
\begin{array}{lll}H_{m_s}(n_1,\ldots,n_r) = \frac{1}{m_s^d}\left(\lim_{m\rightarrow\infty}\frac{\ell_R(R/I(mn_1F_{m_s,1})\cdots I(mn_rF_{m_s,r}))}{m^d}\right) = \frac{-1}{m_s^dd!}((-n_1F_{m_s,1}-\cdots-n_rF_{m_s,r})^d) \end{array}
\infty>\int_{x^{\ast}}^{1}\left\vert f^{\prime\prime\prime}(x)\right\vert^{2}(1-x)(1+x)^{\beta+3}dx\geq\frac{c^{2}}{4}\int_{x^{\ast}}^{1}\frac{(1+x)^{\beta+3}}{1-x}dx=\infty.
Z_\pm^\varepsilon(x,t) = \pm e^{-(c_5+1)t}\zeta^\varepsilon(x,t)-\left(\sup_{\overline{\Omega_\varepsilon(0)}}|e^{-c_4\sigma_\varepsilon(\cdot,0)}\chi_0^\varepsilon|+\sup_{Q_{\varepsilon,T}}|e^{-c_4\sigma_\varepsilon}f^\varepsilon|+\frac{1}{\varepsilon}\sup_{\partial_\ell Q_{\varepsilon,T}}|\psi^\varepsilon|\right)
\sum_{i=1}^L \exp\left(\frac{- k_22^{i-1}n^{\min\{\delta^2,1/25\}/3}p_d^2}{8}\right) \le L\exp\left(\frac{- k_2n^{\min\{\delta^2,1/25\}/3}p_d^2}{8}\right) & \le \exp\left(- n^{\min\{\delta^2,1/25\}/4}\right)