harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_{\mu_{1}\cdots \mu_{n}}( 0 ) \sim \mathcal{C}_{\left[ 1 + \frac{n}{2}\right]}+ p^{2}\mathcal{C}_{\left[ \frac{n}{2}\right]}+ \cdots + \frac{p^{2 \left[ \frac{n}{2}\right]}}{\left[ \frac{n}{2}\right] !}\mathcal{C}_{1}+ \mathrm{c o n s t .}
A_{M}= \left( \begin{array}{c}{a}\\{\psi}\\ \end{array}\right) , A_{M}^{\dagger}= \left( \begin{array}{c}{a^{\dagger}}\\{\psi^{\dagger}}\\ \end{array}\right) , \tilde{A}_{M}= \left( \begin{array}{c}{\tilde{a}}\\{\tilde{\psi}}\\ \end{array}\right) , \tilde{A}_{M}^{\dagger}= \left( \begin{array}{c}{\tilde{a}^{\dagger}}\\{\tilde{\psi}^{\dagger}}\\ \end{array}\right)
\sum\limits_{v=0}^{k-1}f^{\ast}(-b\left( \left[ R_{2}\right] -v\right))B_{1}\left( \frac{v+\left\{R_{2}\right\}}{k}\right) \sum\limits_{\mu=0}^{k-1}f(c(\mu+1+\left[ R_{1}\right] ))B_{1}\left( \frac{\mu+1-\left\{R_{1}\right\}}{k}\right) & =P_{1}\left( R_{2},B_{-b}\right) P_{1}\left( -R_{1},A_{-c}\right)+f(cR_{1})\lambda_{R_{1}}.
\Delta_{p,+r}^\alpha u(x) = \;_{-\infty}D_x^\alpha u(x) + a_2(r) h^2 \;_{-\infty}D_x^{2+\alpha}u(x) + O(h^{p}) & = \;_{-\infty}D_x^\alpha u(x) + a_2(r) h^2 D^2 \;_{-\infty}D_x^{\alpha}u(x) + O(h^{p}) & = P_x \;_{-\infty}D_x^\alpha u(x) + O(h^{p}),
A_n^{\rm pro}= \left( \begin{array}{cccc}1 0 \cdots 0 0 1 \cdots 0 \vdots \vdots \ddots \vdots 0 0 \cdots 1 0 0 \cdots 0\end{array}\right) B_n^{\rm pro}= \left( \begin{array}{cccc}0 0 \cdots 0 1 0 \cdots 0 0 1 \cdots 0 \vdots \vdots \ddots \vdots 0 0 \cdots 1\end{array}\right).
u_{1}^{\dot{a}}Q^{\dot{a}+}u_{2}^{a}Q^{b -}\mid B \rangle = u_{1}^{\dot{a}}\gamma_{\dot{a}b}^{I}u_{2}^{b}\oint ( \partial X^{I}+ M^{I J}\bar{\partial}X^{J}) \mid B \rangle
\int_A e^{-2 \pi i \langle n, x \rangle}\, \textrm{d}x = \sum_{\substack{(F_{d-1}, F_{d-2}, \dots, F_k) \textrm{relevant flags}}}C_n(F_{d-1}, F_{d-2}, \dots, F_k) \int_{\pi_k (F_k)}e^{-2 \pi i \langle \pi_k (n), x \rangle}\, \textrm{d}x
\max_{1 \le j \le m}\max_{s \in \mathcal{K}_j}\Biggl| \sum_{n=0}^M v_n(s,b(n);\alpha,\lambda_j) - h_j (s) \Biggr| < \frac{\varepsilon}{2}, &\max_{1 \le j \le m}\max_{s \in \mathcal{K}_j}\Biggl| \sum_{n>M}v_n(s,b(n);\alpha,\lambda_j) \Biggr| < \frac{\varepsilon}{2}
\frac{\varphi(\triangle u_\ell)}{\mathcal C_\ell}\leq\frac{c_2 h_\ell^{\beta}}{C_0}=\frac{c_2 h_0^{\beta}s^{-\beta\ell}}{C_0}\leq\frac{c_2 h_0^{\beta}}{C_0}.
\mu^*+\varkappa =\mu^{\rm N}+\varkappa^{\rm N},\mu^*=\mu^{\rm N}-\varkappa^{\rm N},\lambda =\lambda^{\rm N}, \beta =\gamma^{\rm N}-\beta^{\rm N}, \ \ \gamma =\gamma^{\rm N}+\beta^{\rm N}, \qquad\alpha =\alpha^{\rm N}\,,
M \leq\max\Big\{(2r_o)^{Q},{4\theta_b\over A_1A_3}\Big(1+\log^+\Big({2\over A_1}\Big)\Big), \Big({2\over A_5}\Big)^{Q\over Q+1}, \ \e^{Q}, \ \ \Big({\theta_bA_8\over A_{10}}\Big)^Q\Big\}=: C(Q,\theta_b).
N_{f}\, \mathrm{T r}\, \sum_{n = 2}^{\infty}\frac{1}{( p - m + i \epsilon )^{n}}\, f_{n}( A ) = N_{f}\sum_{n = 2}^{\infty}\int \frac{d p}{2 \pi}\, \frac{1}{( p - m + i \epsilon )^{n}}\int d t \, f_{n}( A )
\begin{aligned}\left(x \frac{d}{dx}\right)^3 \log \left( \theta_3(x) \right) = \left(x \, \frac{d}{dx}\right)^2 \left(x \, \psi(x) \right) & = x \, \psi(x) + 3 x^2 \, \psi'(x) + x^3 \, \psi''(x).\end{aligned}
\sum(-1)^{f_\sigma}\cdots E_{ij}E_{kl}\cdots-\sum (-1)^{f_\sigma+(\bar i +\bar j)(\bar k +\bar l)}\cdots E_{kl}E_{ij}\cdots = \sum \delta_{jk}(-1)^{f_\sigma}\cdots E_{il}\cdots-\sum \delta_{il}(-1)^{f_\sigma+(\bar i +\bar j)(\bar k +\bar l)}\cdots E_{kj}\cdots
\frac{[x^{n-\Sigma_{\mathrm{k}}}]S(x)}{[x^n]S(x)}= z_n^{\Sigma_{\mathrm{k}}}\bigg(\exp\bigg\{-\frac{\big(A_1(z_n)-(n-\Sigma_{\mathrm{k}})\big)^2}{2A_2(z_n)}\bigg\}+ o(1) \bigg).
{\frac{u^{2}}{l^{2}}}\sin^{2}\alpha \rho^{1 2}+ \left({\frac{u^{2}}{l^{2}}}\cos^{2}\alpha -{\frac{u^{2}}{l^{2}}}\sin^{2}\alpha - 1 \right) \rho^{6}-{\frac{u^{2}}{l^{2}}}\cos^{2}\alpha = 0
\widehat{R}_{\mu \nu}\, \widehat{\xi}^{\mu}\, \widehat{\xi}^{\nu}= - \frac{d - 1}{d}\, \frac{d \widehat{K}}{d \tau}\ge 0 \mathrm{f o r ~ a n y ~ n u l l ~ v e c t o r}~ \widehat{\xi}^{\mu}.
P_k(x) ={}\frac{1}{2\pi i}\oint_{\Sigma}s^{-\nu}e^{s}p_k\left(\frac{x}{s}\right) \frac{ds}{s}, Q_k(y) ={}\int_{0}^{\infty}t^{\nu}e^{-t}q_k\left(\frac{y}{t}\right) \frac{dt}{t},
C^{a , b}( x_{1 2}, \partial_{2}) \, C^{a , b}( x_{3 4}, \partial_{4}) \frac{1}{x_{2 4}^{2 a}}= \frac{1}{x_{1 3}^{2 a_{+}}x_{2 4}^{2 a_{-}}}\,{\cal{E}}_{a , b}( u , v ) ,
\mathcal{R}^k:=\Gamma^k-\Gamma^0=\sum_{\ell = 1}^\aleph\left\langle[P_{D_\ell}]\cdot\sum_{j=1 \atop j \neq \ell}^\aleph\left(\Upsilon_{\ell j}^k-\Upsilon_{\ell j}^0\right)\cdot Q_j, \V_\ell\right\rangle.
\big|u_1^{s,N}(C_{0,k}) - u_1^{s,N}(B_{0,k})\big|^2 = \cos^2(\frac{2ks\pi}{N}) \big|u_1^{0,N}(C_{0,0}) - u_1(B_{0,0})\big|^2 \neq \big|u_1^{0,N}(C_{0,0}) - u_1(B_{0,0})\big|^2,
I = \frac{8 \beta \pi^{2}( 5 m \ell^{2}- 3 5 N^{4}r_{+}^{3}- 5 r_{+}^{7}+ 3 5 N^{6}r_{+}+ 2 1 N^{2}r_{+}^{5})}{5 \ell^{2}}
\|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}+H^{2m+2}].
\left( \begin{array}{c}{\hat{\xi}}^t{\hat{\xi}}^r{\hat{\xi}}^\varphi \end{array}\right)={\cal G}\left( \begin{array}{c}{\hat{\Sigma}}^t{\hat{\Sigma}}^r{\hat{\Sigma}}^\varphi \end{array}\right);\;\;\;\;\;\;{\cal G}=\mbox{Exp}\left( -\int^\tau{\cal D}(\tau')\; d\tau'\right),
\begin{array}{lll}f_1(x_1^k, \Psi_1^k) + \mathcal{Q}_2^{k-1}(x_1^k) = \mathfrak{Q}_1^{k-1}(x_0, \xi_1^k)=\mathfrak{Q}_1^{k-1}(x_0, \xi_1).\end{array}
\int_0^\infty e(u_2(y)) a(y) \, d y = \frac{a(y_0) e\Bigl(-x c^2 p^{-1}+ \dfrac{1}{8}\Bigr)}{\sqrt{u_2''(y_0)}}+ O( c^{\frac{7}{2}}T^4 N^{-\frac{11}{6}}m^{\frac{11}{3}}p^{-\frac{7}{4}}).
\begin{cases}a_{0,j}= h a_{0,j}+ k a_{1,j}+ t c_{0,j}j=0,1,2 a_{1,j}= h a_{1,j}+ k a_{2,j}+ t c_{1,j}j=0,1,2 a_{2,j}= h a_{2,j}+ k a_{3,j}+ t c_{2,j}j=0,1,2 \xi^2 b_0 = h b_0 + k b_1 + t d_0 \xi^2 b_1 = h b_1 + k b_2 + t d_1 \xi^2 b_2 = h b_2 + k b_3 + t d_2 \end{cases}
x_j^\tau \;=\; \max_{\lambda \in \Lambda}x_j^\lambda \;{\mbox{for}}j=2,\ldots, p \;\;\;\;{\mbox{and}}\;\;\;\;y_k^\tau \;=\; \max_{\lambda \in \Lambda}y^\lambda_k \;{\mbox{for}}k=2,\ldots, q.
t_{i}:=\min \{t\ge t_{i-1}+1:\exp \left(-\kappa_{i}(\bar{t})+1 \right)\le \frac{1}{i \beta(t_{i-1},i)\sqrt{L}},\forall \bar{t}\ge t \}, &i=2,\ldots,m\;{\rm for}\; m\ge 2;t_{1}:=t_{0}+\tau.
P_0 =\;\frac{81\Big(4\sqrt{2}\xi({\xi}^2-81)-6{\xi}^2+2475\Big)}{4{\xi}^8}, P_1 =\;\frac{27\Big(\sqrt{2}\xi(93-2{\xi}^2)+3{\xi}^2-603\Big)}{2{\xi}^6}, P_2 =\;\frac{9\Big(12\sqrt{2}\xi(2{\xi}^2-43)+4{\xi}^4-36{\xi}^2+2781\Big)}{32{\xi}^4}, P_3 =\;\frac{9(6\sqrt{2}\xi+2{\xi}^2-27)}{4{\xi}^2}.
C^{\mu\nu\rho\sigma}_{\alpha\beta}=\eta^\nu_\alpha\eta^\rho_\beta\eta^{\mu\sigma}-\eta^\mu_\alpha\eta^\rho_\beta\eta^{\nu\sigma}-\eta^\nu_\alpha\eta^\sigma_\beta\eta^{\mu\rho}+\eta^\mu_\alpha\eta^\sigma_\beta\eta^{\nu\rho}.
\mathbb{E}_{x}\left[ S_{s,t}-\widetilde{S}_{s,t}\right]=\mathbb{E}_{x}\left[\mathbb{E}_{x}\left[S_{s,t}\,\big|\, \mathcal{F}_{t}\right]-\mathbb{E}_{x}\left[\widetilde{S}_{s,t}\,\big|\, \mathcal{F}_{t}\right]\right]\to 0,\mbox{as}t\to+\infty.
\langle D A^*u, A^*v\rangle =\langle A^*u, D^*A^*v\rangle=\langle A^*u, C^*v\rangle=\langle CA^*u, v\rangle &=\langle AC^*u, v\rangle=\langle C^*u, A^*v\rangle=\langle D^*A^*u, A^*v\rangle,
\frac{E_2^*(\tau_0)}{\omega_0^2}=\frac{(c\tau_0+d)^2E_2^*(\tau_0)}{\omega_1^2},\ \frac{E_4(\tau_0)}{\omega_0^4}=\frac{(c\tau_0+d)^4E_4(\tau_0)}{\omega_1^4},\ \frac{E_6(\tau_0)}{\omega_0^6}=\frac{(c\tau_0+d)^6E_6(\tau_0)}{\omega_1^6},
q\dim_{V_{\mathbb{Z}\beta}^{+}}V_{\mathbb{Z}\beta}^{T_{1},+}=q\dim_{V_{\mathbb{Z}\beta}^{+}}V_{\mathbb{Z}\beta}^{T_{1},-}=q\dim_{V_{\mathbb{Z}\beta}^{+}}V_{\mathbb{Z}\beta}^{T_{2},+}=q\dim_{V_{\mathbb{Z}\beta}^{+}}V_{\mathbb{Z}\beta}^{T_{2},-}=\sqrt{2k}.
\int_0^{t/\lambda}|u_{\lambda}|^{m-1}u_{\lambda}ds =\int_0^t \lambda^{1+\frac{1}{m-1}}|u(s',\lambda x)|^{m-1}u(s',\lambda x) \frac{ds'}{\lambda}=\lambda^{\frac{1}{m-1}}\int_0^t |u(s',\lambda x)|^{m-1}u(s',\lambda x) ds'.
T_{\mu}(z) \approx \frac{r_B^2}{N}\sum_{j=0}^{N-1}\frac{(J_{[a,b]}^{-})'(t(r_B \xi_N^j)) t'(r_B \xi_N^j) \xi_N^{2j}}{J_{[a,b]}^{-}(t(r_B \xi_N^j)) - z}.
\{f_{i,j}(\beta^{r^t})\}_{j\in[l]}=\{\beta^a:a_i=0\}\bigcup &\Big(\bigcup_{u=0}^{r-2}\{\beta^{l+a}:a_n=\dots=a_{t+2}=0,a_{t+1}=u,a_i=0\}\Big).
\|x\|\ =\ \left\|x-P_A(x) + \sum_{n\in A}e_n^*(x)e_n\right\| \ \leqslant\ \sup_{\delta}\left\|x-P_A(x) + 1_{\delta A}\right\| &\ \leqslant\ \mathbf C_{\lambda, ib}\left\|x-P_A(x) + 1_{\varepsilon B}\right\|.
\tilde{\hat{T}\hspace{. 0 3 5 i n}}= \hat{T}_{g_{\lambda}}- \hat{T}, \hspace{. 2 i n}\tilde{\hat{c}\hspace{. 0 1 8 i n}}= \hat{c}_{g_{\lambda}}- \hat{c}, \hspace{. 2 i n}\tilde{\hat{\Delta \hspace{. 0 1 i n}}}_{0}= \hat{\Delta}_{0}^{g_{\lambda}}- \hat{\Delta}_{0},
\frac{\#(S_n \cap A_\epsilon)}{\#S_n}\le C_2 \ \nu\left(\bigcup_{x \in S_n \cap A_\epsilon}O(x,R) \right) &\le C_2 \ \nu\left( \xi \in \partial \Gamma: \left| \frac{\log\|\rho(\xi_n)\|}{n}- \Lambda \right| > \frac{\epsilon}{2}\right).
{Y_i}{Z_i}{\left({\rho_i^*}\right)^2}- \left[{{X_i}\left({{W_i}+ 1}\right) +{Y_i}{Z_i}+{Z_i}}\right]\rho_i^* &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+{X_i}\left({W_i + 1}\right)= \ell_i\left(\rho_i^*\right) = 0.
\lim_{\epsilon \rightarrow 0}\int_{- \infty}^{\infty}d p \frac{f ( p )}{p - k}\lim_{z \rightarrow 1}( 1 - z )^{2 \epsilon}\left\{( 1 - z )^{- i ( \sqrt{k^{2}- \mu^{2}}- \sqrt{p^{2}- \mu^{2}})}- ( 1 - z )^{i ( \sqrt{k^{2}- \mu^{2}}- \sqrt{p^{2}- \mu^{2}})}\right\}
\left\langle v_n-u^*-Tv_n+Tu^*,v_n-u^*\right\rangle =\Vert v_n-u^*\Vert^2-\left\langle Tv_n-Tu^*,v_n-u^*\right\rangle \ge\Vert v_n-u^*\Vert^2-\Vert Tv_n-Tu^*\Vert\cdot\Vert v_n-u^*\Vert \ge\Vert v_n-u^*\Vert^2-\Vert v_n-u^*\Vert^2=0,
p_{L,N+1}^{\ell_{1}}(x) =\sum\limits_{\ell=0}^N\frac{S_{\lambda\mu_{\ell}}\left(2\sum\limits_{j=0}^N\omega_j\tilde{\Phi}_{\ell}(x_j)f(x_j)\right)}{2}\tilde{\Phi}_{\ell}(x) &=\sum\limits_{\ell=0}^N\left(\sum_{j=0}^N\omega_j\tilde{\Phi}_{\ell}(x_j)f(x_j)\right)\tilde{\Phi}_{\ell}(x)+\sum_{\ell=0}^Nc_{\ell}\tilde{\Phi}_{\ell}(x),
\sum_{n=0}^{\infty}\frac{q^{n}}{(zq^{n+1};q)_{n+2}(zq^{2n+4};q^2)_{\infty}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}p_{\omega}(m,n)z^mq^n, begin{align*}6pt]\sum_{n=0}^{\infty}q^{n}(-zq^{n+1};q)_{n}(-zq^{2n+2};q^2)_{\infty}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}p_{\nu}(m,n)z^mq^n.
\hat{H}_{I}^{( q )}( \hat{x}, \hat{p}) = f^{2}\sum_{k = 1}^{\infty}\frac{V^{( k )}( 0 )}{k !}\biggl ( \sum_{i = 0}^{k - 1}\hat{x}^{( k - 1 ) - i}g ( \hat{x}, \hat{p}) \hat{x}^{i}\biggr ) ,
\mathbf P_{so}^{\,i}=1-\mathbb E\left[\frac{e^{-\frac{(2^{\frac{nr_s}{\beta}}-1) (I_B+1)}{\gamma_{A_i,B}}}}{\frac{\gamma_{A_i,E}}{\gamma_{A_i,B}}\frac{I_B+1}{I_E+1}2^{\frac{nr_s}{\beta}}+1}\right],
{\frac{\delta}{\delta \psi_{\alpha}^{a}( x )}}= \int d z \int d y{\frac{\delta \sigma_{\beta \rho}( y , z )}{\delta \psi_{\alpha}^{a}( x )}}{\frac{\delta}{\delta \sigma_{\beta \rho}( y , z )}}= - \int d y \bar{\psi}_{\beta}^{a}( y ){\frac{\delta}{\delta \sigma_{\beta \alpha}( y , x )}}
ds^2 = \left({\, d_{ABC}\,{\rm Im}Z^A \,{\rm Im}Z^B \,{\rm Im}Z^C\over \, [d_{ABC}\,{\rm Im}Z^A \,{\rm Im}Z^B \,{\rm Im}Z^C]_\infty}\right )^{-1}\, dt^2 -\left({\, d_{ABC}\,{\rm Im}Z^A \,{\rm Im}Z^B \,{\rm Im}Z^C \over\, [d_{ABC}\,{\rm Im}Z^A \,{\rm Im}Z^B \,{\rm Im}Z^C]_\infty}\right)\,d\vec x^2 \ .
(\Delta_C,L/K)\stackrel{?}{=}(-1)^{\#\big\{\textup{even-sized}G_k\textup{-orbits on}\mathcal{R}\cap K^{\textup{nr}}\big\}+\#\big\{G_k\textup{-orbits}O\subseteq T\textup{with}\prod_{\mathfrak{t}\in O}\gamma_{\mathfrak{t},L}(F)=-1\big\}}.
\mu_{z,X}(\phi)=\sum_{\substack{\beta \kappa_{\beta}=\kappa}}\int_{M}\int_{A}\int_{N_{P}}\int_{w\overline{N}_{P}w^{-1}\cap\overline{N}_{P}}a^{-\lambda+\rho_{P}}\Big(\sigma^{\vee}(m) c_{\beta},\partial^{\beta}\phi(man\overline{n})\Big)\,d\overline{n}\,dn\,da\,dm.
a_{D}= \partial_{a}{\cal F}={\frac{\sqrt 2}{\pi}}\int_{\Lambda^{2}}^{u}{\frac{d x \sqrt{x - u}}{\sqrt{x^{2}-{\Lambda^{4}}}}}, \qquad a ={\frac{\sqrt 2}{\pi}}\int_{-{\Lambda^{2}}}^{\Lambda^{2}}{\frac{d x \sqrt{x - u}}{\sqrt{x^{2}-{\Lambda^{4}}}}},
\left[\begin{array}{c}\dot{x}_1(t) \dot{x}_2(t)\end{array}\right]=\left[\begin{array}{cc}0 I -L -L\end{array}\right]\left[\begin{array}{c}x_1(t) x_2(t)\end{array}\right]+\left[\begin{array}{c}0 I\end{array}\right]w(t)\,,
Y_{i,j}= T_{i-1,j-1}^{-1}T_{i-2,j-2}^{-1}\cdots T_{1,j-i+1}^{-1}\pi^{j-i}~T_{n-j+i,n}\cdots T_{i+1,j+1}T_{i,j}, 1 \le i < j \le n,
H_l(x^{B(l)}| \bar{x}^{B(l)}) = \sum_{i \in B(l)}\psi (x_i) + \frac{1}{2}\sum_{i, j \in B(l)}M_{ij}x_i x_j + \sum_{\substack{i \in B(l) k \notin B(l)}}M_{ij}x_i x_k
[D_x (\frac{v}{u})]_{H^{\alpha}( G \cap Q_{r/4}( \frac{r}{2}e_n,0))}= [D_x (\frac{\tilde v}{\tilde u})]_{H^{\alpha}( \tilde G \cap Q_{1/4}( \frac{1}{2}e_n,0))}
(\begin{matrix}\Delta OD_{\lambda_{1}}\\ \Delta OD_{\lambda_{2}}\end{matrix})=(\begin{matrix}\epsilon_{\lambda_{1}}^{Hb}d&\epsilon_{\lambda_{1}}^{HbO_{2}}d\\ \epsilon_{\lambda_{2}}^{Hb}d&\epsilon_{\lambda_{2}}^{HbO_{2}}d\end{matrix})(\begin{matrix}\Delta[X]^{Hb}\\ \Delta[X]^{HbO_{2}}\end{matrix})
\widetilde{G}(\mathbf{x}) = \sum\limits_{\substack{\mathbf{k}\in\mathbb{Z}^m \Vert \mathbf{k}\Vert_\infty \leqslant X}}b_X(\mathbf{k}) e(\mathbf{k}\cdot \mathbf{x}) + O_{c,C,\varepsilon}\Big(\frac{\log X}{\sigma_G X}\Big),
\Big(<\nabla g_j(g^{-1}(v_-)),\int_{-\infty}^{+\infty}F(. v_-+x_-)(\tau)d\tau>\Big)_{j=1\ldots n}=\sqrt{1-{|v_-|^2\over c^2}}\int_{-\infty}^{+\infty}F(. v_-+x_-)(\tau)d\tau.
\Delta(M_a)= M_{\epsilon}\otimes M_{445132}+M_{{\rm Park}(445)}\otimes M_{{\rm Park}(132)}begin{align*}5pt] +M_{{\rm Park}(132)}\otimes M_{{\rm Park}(445)}+M_{445132}\otimes M_{\epsilon}begin{align*}5pt]= 1\otimes M_{445132}+M_{112}\otimes M_{132}+M_{132}\otimes M_{112}+M_{445132}\otimes 1.
X_{a - 1 , a + 1}^{a}( u^{'}) X_{a + 1 , a - 1}^{a}( u ) = X_{a - 1 , a + 1}^{a}( u ) X_{a + 1 , a - 1}^{a}( u^{'}) \; ; 2 \leq a \leq p - 2
{\frac{1}{m}}{\frac{d m}{d \tilde{\lambda}}}= g_{\mu \nu}A^{\mu}{\frac{d x^{\nu}}{d \tilde{\lambda}}}= \sqrt{h}\left[ 2 - h \left({\frac{d t}{d \tilde{\lambda}}}\right)^{2}+ y^{2}\sum_{i}\left({\frac{d x^{i}}{d \tilde{\lambda}}}\right)^{2}\right] \left[{\frac{h^{\prime}}{2}}\left({\frac{d t}{d \tilde{\lambda}}}\right)^{2}- y \sum_{i}\left({\frac{d x^{i}}{d \tilde{\lambda}}}\right)^{2}\right] .
\mathcal{R}(R,L) :=-2\sum_{i}\beta_i\otimes R(L(\beta_i)), \mathcal{T}(R,L) :=-\sum_i\big(L^*(\alpha_i)\otimes1+1\otimes \alpha_i\big)\circ\big(L^*(R(\alpha_i))\otimes 1+1\otimes R(\alpha_i)\big),
{\cal L}= e \left({\frac{1}{2}}R + g^{2}V -{\frac{1}{4}}G_{I J}F_{\mu \nu}{}^{I}F^{\mu \nu J}-{\frac{1}{2}}{\cal G}_{i j}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{- 1}}{4 8}}\epsilon^{\mu \nu \rho \sigma \lambda}C_{I J K}F_{\mu \nu}^{I}F_{\rho \sigma}^{J}A_{\lambda}^{K}\, \right) ,
l = \int_{0}^{\infty}e^{-\tau 2^{1/p}}E_p^2(-\tau^p) d\tau, m = \int_{0}^{\infty}e^{-\tau 2^{1/p}}~(\tau 2^{\frac{1}{p}}-1 + p)~ E_p^2(-\tau^p)~d\tau.
W(10^l|0^k1) = \sum_{j=1}^l \left[ A(0^{j+k}1...) - A(0^{j+\alpha}1...) \right] + A (10^{l+k}1...) - A(10^{j+\alpha}1...) &= \sum_{j=1}^l (a_{j+k}- a_{j+\alpha}) + d_{l+k}- d_{l+\alpha}\ .
{\frac{d^{2}t}{d \tilde{\lambda}^{2}}}= \sqrt{h}\left[{\frac{h^{\prime}}{2}}\left({\frac{d t}{d \tilde{\lambda}}}\right)^{2}- y \sum_{j}\left({\frac{d x^{j}}{d \tilde{\lambda}}}\right)^{2}-{\frac{h^{\prime}}{h}}\right]{\frac{d t}{d \tilde{\lambda}}},
\int^{T}_{0}\int_{\Omega}\big(\rho u\cdot \partial_t\phi+\rho u\otimes u:\nabla_x\phi+\rho^{\gamma}{\rm{div}}_x\phi-\mathbb{S}(\nabla_x u):\nabla_x\phi &+(j-nu)\cdot\phi\big)\,dxdt+\int_{\Omega}\rho_0u_0\cdot\phi(0,x)\,dx=0,
(1_{X_4}* W_{321}) \odot W_{4(32)1}\odot (W_{432}*1_{X_1}) = e_{1_{X_4}* W_{321}}\odot e_{W_{4(32)1}}\odot e_{W_{432}*1_{X_1}}& = e_{(1_{X_4}* W_{321}) \odot W_{4(32)1}\odot (W_{432}*1_{X_1})},
g_\infty = - \: 16 du^2 + u^2 d\theta^2 + 4 c_\lambda^{-1}\left[ e^{P_\infty}(d\widehat{\sigma}+ Q_\infty d\widehat{\delta})^2 + e^{- \: P_\infty}d\widehat{\delta}^2 \right].
f(g_s, s\xi, \hat{\psi})=\big(\mathcal{T}_{a^+,a^-}\circ B(s\mathfrak{k}_{t_0}), P_{\ker(D|_{L^2_1})}(D_{pert,\eta_s}(\psi+\hat{\psi}+s\mathfrak{k}_s)\big)
\begin{aligned}\left| \frac{\bar{\nu}(B(x,r)\cap A)}{h(r)}- \frac{\bar{\nu}(B(y,r)\cap A)}{h(r)}\right| = \frac{1}{h(r)}|\bar{\nu}(A \cap B(x,r))- \bar{\nu}(A \cap B(y,r))| & \leq \frac{1}{h(r)}\nu(B(x,r) \triangle B(y,r)) & \leq \frac{1}{h(r)}\nu( B(x,r+d(x,y)) \setminus B(y,r) ) & = \frac{h(r+d(x,y)) - h(r)}{h(r)}.\end{aligned}
\int_{t_{1}}^{t_{2}}\, \left( \frac{\partial L}{\partial x}\; \delta x + \frac{\partial L}{\partial{\cal V}}\; \left[ \, \frac{1 - i}{2}\; \frac{d}{d t_{+}}+ \frac{1 + i}{2}\; \frac{d}{d t_{-}}\, \right] \, \delta x \right) \, d t = 0 ,
\mathcal R_j(\tilde\Lambda )=\cup_{|k|=0}^{\tilde N_j}\cup_{\bar\Lambda^{(j)}\in \mathcal C(\Lambda^{(j)})}\cup_{i\in J_{j,k}(\bar\Lambda^{(j)})}I_{k,i}( \bar \Lambda^{(j)})=:\cup_{i\in J_j(\tilde\Lambda)}I_i^{(j)},\ \ \Lambda^{(j+1)}=\Lambda^{(j)}\backslash\mathcal R_j(\tilde\Lambda).
d s^{2}= M d{\bf X}^{2}+{\frac{1 6 \pi^{2}}{M}}d \chi^{2}+ k \left[{\frac{d a^{2}}{a}}+ a \left( d \alpha^{2}+ \sin^{2}\alpha d \beta^{2}+ ( d \gamma + \cos \alpha d \beta )^{2}\right) \right]
K_{n}^{m}\left({\gamma^{2}}\right) =\frac{\left({-1}\right)^{m}}{2^{m/2}m!}\sum\limits_{k=-k^{-}}^{\infty}{\left({-1}\right)^{k}\frac{\left({n+2k+m}\right) !}{\left({n+2k-m}\right) !}a_{n,k}^{m}\left({\gamma^{2}}\right)}.
0=_{q/2}\Bigl(1+\frac 1{1+a+b}\Bigr)=_{q/2}\Bigl(1+\frac 1{1+a^2+b^2}\Bigr)=_{q/2}\Bigl(1+\frac 1{1+b}\Bigr)=_{q/2}\Bigl(\frac b{1+b}\Bigr).
\rho_n^2k_n^{1+2\alpha}= n^{-\frac{2(\alpha \wedge \beta)}{1+2\alpha+2p}}\cdot n^{\frac{1+2\alpha}{1+2\alpha+2p}}= n^{\frac{1+2\alpha - 2(\alpha \wedge \beta)}{1+2\alpha+2p}}= n\epsilon_n^2,
\begin{array}{c}{R ( u / v ) ( K_{l}( u ) \otimes{\bf 1}) R ( u v q^{- 1}) ({\bf 1}\otimes K_{l}( v ) =}\\{= ({\bf 1}\otimes K_{l}( v ) ) R ( u v q^{- 1}) ( K_{l}( u ) \otimes{\bf 1}) R ( u / v ) ,}\\ \end{array}
\mathbf{f}= \log 3 \mathbf{e}_1+\log 3\mathbf{e}_2+\log 3\mathbf{e}_3+\log \frac{4}{3}\mathbf{e}_{123'}= [\log 4,\log 4,\log 4,\log 16,\log 16,\log 16,\log 48]^\intercal
\mathcal{L}_{\mathcal{I}^{({\rm CCU})}_{\rm d \rightarrow d}}(s|r_o)=\mathcal{L}_{\mathcal{I}^{({\rm CEU})}_{\rm d \rightarrow d}}(s|r_o) =\exp \left( -\pi \lambda \sqrt{s P_{\rm{d}}}\arctan \left( \sqrt{s P_{\rm{d}}}r_o^{-2}\right) \right),
r^{\frac{(p-2)n}{p}}\|u\|^2_{L^{p}(\Omega)}=r^{2\left(\frac{(p-2)}{p}-s'\right)}\,r^{2s'}\,\|u\|^2_{L^{p}(\Omega)}& \le C'(\Omega,p) \,D_0^{2\left(\frac{(p-2)}{p}-s'\right)}\,r^{2s'}\|\nabla u\|_{L^2(\Omega)}^2\,.
p_1\,=\,-\,\frac{\partial L}{\partial \dot x}\,-\, \dot p_2,\quad p_2\,=\,-\,\frac{\partial L}{\partial \ddot x}\,-\,\dot p_3,\quad p_3\,=\,-\,\frac{\partial L}{\partial\stackrel{\ldots}{x}}\,{,}
J=\begin{pmatrix}b_0 a_0 0 0 \cdots a_0 b_1 a_1 0 \cdots 0 a_1 b_2 a_2 \cdots 0 0 a_2 b_3 \cdots \vdots \vdots \vdots \vdots \ddots \end{pmatrix},
v = \frac{z}{R_T}+ \sum_{n=1}^\infty \sin\left( n \frac{z}{R_T}\right)\left[ \frac{1}{n}f_d \left(n\frac{r}{R_T}\right) - \frac{1}{d-3}\frac{r}{R_T}f_d'\left(n \frac{r}{R_T}\right) \right]
a^\tau :=\tau \tilde{a}+(1-\tau)a+\sum_{i=- +}\chi_{u_i}\mu_{u_i}(\tau)[u-{u_i}(x)] f^\tau(x,u,u_x) :=\tau \tilde{f}+(1-\tau)f+\sum_{i=- +}\chi_{u_i}\mu_{u_i}(\tau)[u-{u_i}(x)]
\psi ({\xi}) = \sqrt{\frac{2 \sqrt{\beta}}{\pi}}\int_{- \infty}^{+ \infty}\frac{d p}{( 1 + \beta p^{2})^{3 / 2}}e^{\frac{i{\xi}\tan^{- 1}( \sqrt{\beta}p )}{\hbar \sqrt{\beta}}}\psi ( p )
0 <\int_{a}^{x_{o}}\left[v-u\right]\; dx \le\int_{a}^{b}\left[v-u\right]\; dx= \lambda\int_{a}^{b}\left[f(x,u)-f(x,v)\right]_{x}\; dx =\lambda\left[f\left(b-,u(b-)\right)-f\left(b-,v(b-)\right)\right]- \lambda\left[f\left(a+,u(a+)\right)-f\left(a+,v(a+)\right)\right].
\partial_{\mu}( \sqrt{g}R^{\mu}) = - \frac{1}{3 g^{3}}\widetilde{\beta}( g ) ( F \widetilde{F}) - \frac{\tilde{b}( g )}{4 8 \pi^{2}}( B \widetilde{B}) + \frac{\tilde{c}( g ) - a ( g )}{2 4 \pi^{2}}R \widetilde{R}+ \frac{5 a ( g ) - 3 \tilde{c}( g )}{9 \pi^{2}}( V \widetilde{V})
\sum_{n=1}^{\infty}n^2 \, |d_{n(j+k-1)}|^2 \le \frac{1}{4(j+k-1)^2}\left(\frac{|A-B|}{B}\right)^2\sum_{n=1}^{\infty}B^{2n}\mbox{for}B\neq -1 & = \frac{1}{4(j+k-1)^2}\frac{|A-B|^2}{1-B^2}\mbox{for}B\neq -1,
\begin{aligned}h(x,y) \geq h_0(x,y)-\int_0^x \Vert \partial_x h\Vert_{L^\infty}\geq h_0-Cx\cdot \sup\limits_{x\in[0,L]}\Vert h\Vert_{H^3_l}\geq h_0-Cx\left(F_0+\int_0^x F\right)^{1/2}\left\{1-2C\vartheta_0^{-8}\left(F_0+\int_0^x F\right)^2x\right\}^{-1/4}\end{aligned}
\bigg(w(e)^{p-r}\prod_{v\in e}B(v,e)\bigg)^{1/p}\!\geq\! \bigg(w_1(e)^{\xi(p-r)}w_2(e)^{(1-\xi)(p-r)}\prod_{v\in e}(B_1(v,e))^{\eta}(B_2(v,e))^{1-\eta}\bigg)^{1/p}& \!=\! \bigg(\!w_1(e)^{p_1-r}\prod_{v\in e}B_1(v,e)\!\bigg)^{\mu/p_1}\! \bigg(\!w_2(e)^{p_2-r}\prod_{v\in e}B_2(v,e)\!\bigg)^{(1-\mu)/p_2}& =\alpha_1^{\mu/p_1}\alpha_2^{(1-\mu)/p_2}.
Df =\int_{0}^{1}\kappa(r)\,\partial_r [r^{\mathcal{L}}-e^{\frac{1}{4}(1-r^2)\Delta}]\,r^{A}f\,dr, \tilde{T}f =\int_{0}^{1}\kappa(r)\,\partial_r e^{\frac{1}{4}(1-r^2)\Delta}\, r^{A}f\,dr,
\mathrm{div}_T\Big(h\,\tfrac{1}{2}\nabla_\xi H^2(\nabla w) \Big) \,h\,\tfrac{1}{2}\nabla_\xi H^2(\nabla w)\cdot \nu -h^2\, \nabla_T\Big[ h\,\tfrac{1}{2}\nabla_\xi H^2(\nabla w)\Big] \,\Big[\tfrac{1}{2}\nabla_\xi H^2(\nabla w)\Big]_T\cdot \nu=h\,H(\nu)\,H^2(\nabla w)\,\mathrm{tr}\,\mathcal{B}^H
(Df)(B) =\frac{1}{|8B|}\int_{8B}|f-f_B|\,d\mu \le \frac{1}{|8B|}\int_{8B}|u(B)-T_nu|\, d\mu+ \frac{1}{|8B|}\sum_{k=n}^\infty \int_{8B}|T_{k+1}u-T_ku|\, d\mu & \lesssim \frac{1}{|B|}\sum_{B'\in\bigcup_{k=n}^\infty V_k,\; B'\cap 8B\not=\emptyset}|B'| \tilde d u(B') \le \mathcal{M}(\tilde du)(B),
\nabla^2K(x)=\nabla^2 H(\pi_{p_0}^{-1}(x))[\nabla \pi_{p_0}^{-1}(x),\nabla \pi_{p_0}^{-1}(x)]+\nabla H(\pi_{p_0}^{-1})[\nabla^2\pi_{p_0}^{-1}(x)]
\mathbf{S}_{f}:=\left(\begin{array}{ccccc}f\left(1\right) 0 0 \cdots 0 f\left(2\right) f\left(2\right) 0 \cdots 0 \vdots \vdots \vdots \ddots \vdots f\left(N\right) f\left(N\right) f\left(N\right) \cdots f\left(N\right)\end{array}\right)
\tilde{m}\sim N_{f}( q + \tilde{q}){\frac{\Lambda^{3}}{\xi^{2}}}\left[{\frac{m}{\Lambda}}\left({\frac{\xi}{M_{P}}}\right)^{q + \tilde{q}}\right]^{N_{f}/ N_{c}}= N_{f}( q + \tilde{q}){\frac{< \lambda \lambda >}{\xi^{2}}},
F^{- 1}= \left( \begin{array}{c | c | c}{{f^{- 1}- \sum_{i = 1}^{2}f^{- 1}A_{i}C_{i}^{- 1}\tilde{A}_{i}f^{- 1}}}&{{- f^{- 1}A_{1}C_{1}^{- 1}}}&{{- f^{- 1}A_{2}C_{2}^{- 1}}}\\ \hline{{C_{1}^{- 1}\tilde{A}_{1}f^{- 1}}}&{{C_{1}^{- 1}}}&{{0}}\\ \hline{{C_{2}^{- 1}\tilde{A}_{2}f^{- 1}}}&{{0}}&{{C_{2}^{- 1}}}\\ \end{array}\right) .
\sum_{\substack{r=1 (r,n)=1}}^{\lfloor n/6\rfloor}\frac{1}{r}\equiv -2q_2(n)-\frac{3}{2}q_3(n)+nq_2^2(n)+\frac{3}{4}nq_3^2(n)+\frac{1}{6}J_{6}(n)n^{\phi(n)-1}\phi_{J_{6}}^{(2-\phi(n))}(n)\frac{B_{\phi(n)-1}(\frac{1}{6})}{\phi(n)-1}\pmod{n^2}.
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_X=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{X}.
\Theta_n^{W}:=\big\{(j,k,e)\in\Theta^W\, \big|\, j=0,\ldots,J-1\big\}, &\Theta_n:=\big\{(j,\tilde{\theta})\in\Theta\, \big|\, j=0,\ldots,\widetilde{J}-1\big\},
\frac1{2\pi i}\int_{1/2-i\infty}^{1/2+i\infty}\biggl(\frac\pi{\sin\pi t}\biggr)^2\frac{(t-1)(t-2)\dotsb(t-\ell)}{\ell!}\,\d t=\frac{(-1)^\ell}{\ell+1}.
\frac{dL}{dt}=\frac{\partial L}{\partial t}+\sum_{i=1}^{n}\frac{\partial L}{\partial x_{i}}\frac{dx_{i}}{dt}=(\frac{\partial}{\partial t}+\sum_{i=1}^{n}\frac{dx_{i}}{dt}\frac{\partial}{\partial x_{i}})(L).
P_s=\sum\limits_{i=1}^N \frac{w_i e^{-\rho_I \delta_0(-1+{}_2F_1(-\frac{1}{\alpha_I}, k, 1-\frac{1}{\alpha_I}, -\frac{T R^{2 \alpha_d}\gamma_0 \theta}{c_0 \overline{g_p}t_i \delta_0^{\alpha_I}}))}}{e^{\frac{T \sigma^2 R^{2 \alpha_d}}{c_0 \overline{g_p}t_i}}}
K_1 := \frac14+\Delta t\Bigg[ |\boldsymbol{u}(t_1)|_{W^{1,\infty}}+ C h^{2z}|\boldsymbol{u}(t_1)|_{W^{1,\infty}}^2 + C\frac{h^2}{\gamma}| \boldsymbol{u}(t_1)|^2_{W^{1,\infty}}&\qquad\qquad+ C \frac{h^{2z-2}}{\gamma}\|\boldsymbol{u}(t_1)\|_{\infty}^2 + C h^{2z-2}\|\widetilde{\boldsymbol{u}}_h^1\|_{\infty}^2\Bigg]<1.
I_1(\ell_1,\ell_2):=P(W^k_{\ell_1\Delta,t}(j)=1,W^k_{\ell_2\Delta,t}(j)=1), & I_2(\ell_1,\ell_2):=\nu\Delta p^k_{j,+}(t-\Delta\ell_2)E[Z_0(\Delta\ell_2);W^k_{\ell_1\Delta,t}(j)=1].