harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widetilde{I}= \int_{\mathbb{R}^n}e^{\vec{\eta}\vec{v}- g(\vec{v})}v_1^{i_1}\cdots v_n^{i_n}\, d\vec{v}= e^{h(\vec{v_0})}\int_{\mathbb{R}^n}e^{h(\vec{v})-h(\vec{v_0})}v_1^{i_1}\cdots v_n^{i_n}\, d\vec{v},
\tilde{G}^{(n)}_\sigma (\zeta)=(\sqrt{-1})^{-n}\sum_{k_0 , k_1 , \cdots k_{2n}}t^{k_0}_{k_1}(u_1 -u_0 ) \otimes \cdots \otimes t^{k_{2n-1}}_{k_{2n}}(u_{2n}-u_0 ) \tilde{F}^{(n)}_\sigma (\zeta)^{k_0 k_1 \cdots k_{2n}}.
\Delta = \frac{1}{r^2}\frac{\partial}{\partial r}\bigl( r^2\frac{\partial}{\partial r}\bigr) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial^2\phi}+ \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\bigl( \sin\theta\frac{\partial}{\partial\theta}\bigr).
R_{a b}= \frac{1}{6}F_{a c d e f}F_{b}^{c d e f}+ \frac{1}{8}\left( G_{a c d}G_{b}^{* c d}+ G_{a c d}^{*}G_{b}^{c d}- \frac{1}{6}g_{a b}G_{c d e}G^{* c d e}\right) ,
\begin{multlined}I_\varepsilon (t,\xi)-J_\varepsilon (u_0-W_{\delta_\varepsilon (t),\xi}) =\left\langle u_0-W_{\delta_\varepsilon (t),\xi}-i^\ast (f_\varepsilon(u_0-W_{\delta_\varepsilon (t),\xi})),\phi_{\delta_\varepsilon (t),\xi}\right\rangle_{P_g}+O(\left\|\phi_{\delta_\varepsilon (t),\xi}\right\|^2_{P_g})\end{multlined}
\sum_{k=1}^{k_0}c^2 (k-1)^2 N(c,k) - \tau_n c^2 k_0^2 N(c,k_0) &\leq \min_{S \subset [n]: |S| = \lceil \delta n \rceil}\|x_S\|_2^2 \leq \sum_{k=1}^{k_0}c^2 k^2 N(c,k) + \tau_n c^2 (k_0+1)^2 N(c, k_0+1).
{[k+1+\ell]\choose[q]}=\sum_{i=-1}^{q}{[k]\choose[i]}\cdot{[\ell]\choose[q-1-i]}={[k]\choose[q]}+{[\ell]\choose[q]}+\sum_{i=0}^{q-1}{[k]\choose[i]}\cdot{[\ell]\choose[q-1-i]},
\overline{M}^s(F) =\frac{1}{\bar{\mathbf{n}}\cdot\mathbf{n}(F)}\left(\begin{array}{cc}\bar{n}_yn_y(F)+\beta^s/\beta^{s'}\bar{n}_xv_x(F) -\bar{n}_xv_y(F)+\beta^s/\beta^{s'}\bar{n}_xv_y(F) -\bar{n}_yn_x(F)+\beta^s/\beta^{s'}\bar{n}_yv_x(F) \bar{n}_xv_x(F)+\beta^s/\beta^{s'}\bar{n}_yv_y(F)\end{array}\right),
\begin{array}{lll}d\left(\lambda x^-\right) = e^{-\lambda\sigma^-}\,d\left(\lambda\sigma^-\right)=-\lambda\left(x^-+\Delta_q\right)\,d\left(\lambda\sigma^-\right) &={N\over{48\lambda x^+}}\,d\left(\lambda\sigma^-\right)\;.\end{array}
M(x)=x^{-1/2}\left[C_1\> J_{\lambda}\left(\frac{\sqrt{4\alpha c_{\epsilon}}}{\epsilon \> x^{\epsilon \over 2}}\right)+C_2 \> J_{-{\lambda}}\left(\frac{\sqrt{4\alpha c_{\epsilon}}}{\epsilon \> x^{\epsilon \over 2}}\right)\right]\;\;\;,
\|w_{\varepsilon}(t,\cdot)\|^{2}_{\mathrm{H}_{\mathcal{H}_{\hbar,V}}^{1+s}}+\|\partial_{t}w_{\varepsilon}(t,\cdot)\|^{2}_{\mathrm{H}_{\mathcal{H}_{\hbar,V}}^{s}}\lesssim \varepsilon^{-2L_{1}-L_{2}-1}\varepsilon^{2L_{1}+L_{2}+1+q}=\varepsilon^{q},q\in\mathbb{N}_{0},
( c_{i}- c_{i+1}+ \varepsilon_1 )( c_{i}- c_{i+1}+ \varepsilon_2 ) x_{i,r}^+ x_{i+1,s}^+ = ( c_{i+1}- c_{i}+ \varepsilon_1 )( c_{i+1}- c_{i}+ \varepsilon_2 ) x_{i+1,s}^+ x_{i,r}^+.
u(t,x) =u_0(\overline{x})\exp\left(-\int_{0}^{t}\left(\operatorname{div}(\mathbf{B})-G\right)(s,X(s,\overline{x}))ds\right) &+\int_{0}^{t}F(\tau,X(\tau,\overline{x})) \exp\left(-\int_{\tau}^{t}\left(\operatorname{div}(\mathbf{B})-G\right)(s,X(s,\overline{x}))ds\right)d\tau,
\mathcal{F}(\Delta,\mathrm{Ind}_{H}^{\Delta}(\rho-\mathrm{dim}\rho \cdot 1_H)) =\mathcal{F}(H,\rho-\mathrm{dim}\rho\cdot1_H) \implies \frac{\mathcal{F}(\Delta,\mathrm{Ind}_{H}^{\Delta}\rho)}{\mathcal{F}(\Delta,\mathrm{Ind}_{H}^{\Delta}1_H)^{\mathrm{dim}\rho}}=\frac{\mathcal{F}(H,\rho)}{\mathcal{F}(H,1_H)^{\mathrm{dim}\rho}}.
{\displaystyle\sum\limits_{\overline{k}}}{\displaystyle\sum\limits_{i=1}^{M}}\left\langle D_{x}^{\overline{k}}u\right\rangle_{x_{i},\overline{\Omega}}^{(\widehat{d}_{i})}\leq C\left( \sum_{i=1}^{M}\left\langle u\right\rangle_{x_{i},\overline{\Omega}}^{(l_{i})}+|u|_{\overline{\Omega}}^{(0)}\right)
\int_{\delta(lz^{0}+Z^{\ast})}\vert v_{\varepsilon, k}(y) \vert^{2}dy=\int_{Z^{\ast}}\vert v_{\varepsilon\delta, kl}(z) \vert^{2}\delta^{n}dz\leq C(Z^{\ast})\delta^{2}\int_{Z^{\ast}}\vert\delta^{-1}\nabla_{z}v_{\varepsilon\delta, kl}(z) \vert^{2}\delta^{n}dz
( \partial_2 - a_s \partial_1^2 +1 )\partial_s w^{\tau}_{sT}= \partial_s ( g_{s \tau})_T + (\partial_s a_s \partial_1^2 w^{\tau}_{s})_T - [ a_s , (\cdot)_T ] \partial_1^2 \partial_s w^{\tau}_{s}\textrm{in}\mathbb{R}^2.
0 = \int_{B_\rho}\tilde{u}_{X_\circ,r_{j_\ell}}L_a \tilde{u}_{X_\circ,r_{j_\ell}}|y|^a \to \int_{B_\rho}\tilde u_{X_\circ,0}L_a \tilde u_{X_\circ,0}|y|^a \quad\ell \to \infty,
E_1 =E(e,d'(r),s,f), E_2 =E(e,r,d'(s),f), E_3 =E(e,d(e)r,s,f), E_4 =E(e,r,sd(f),f), E_5 =E(e,r,s,f), E_6 \mbox{being such that}\{(g,h)\in E\times E\mid ergd(g)hsf\ne 0\}\subseteq E_6\times E_6
\frac{1}{\lambda_{r}}+ \frac{D - 1}{( 4 \pi )^{\frac{D}{2}}}\Gamma ( 1 - \frac{D}{2}) \mathrm{t r}\mathrm{\boldmath ~ 1 ~}- \left( \frac{K_{c r}}{\sigma_{0}^{2}}\right)^{\frac{D - 2}{2}}\frac{1}{\sqrt{\pi}( 4 \pi )^{\frac{D}{2}}}\Gamma ( \frac{D - 1}{2}) \Gamma ( \frac{D}{2}) \Gamma ( 1 - \frac{D}{2}) \mathrm{t r}\mathrm{\boldmath ~ 1 ~}= 0 .
\begin{aligned}\sum_{n\leq \sqrt{x}}\left(\sum_{y< p< z}\frac{\left(\frac{f(n)}{p}\right)}{p}\right)^{2k}=\sum_{m}\frac{b_{2k}(m;y, z)}{m}\sum_{n\leq \sqrt{x}}\left(\frac{f(n)}{m}\right) \ll \sqrt{x}\sum_{m}\frac{b_{2k}(m;y, z)}{mm_0}, \end{aligned}
\|f\|_{{B_{rt}^s}(K)}= \|q^{sj}\Delta_j f \|_{\ell_t(L^r(K))}&= \Bigg\{\sum\limits_{j=0}^{\infty}\|q^{sj}\Delta_j f\|^t_{L^r(K)}\Bigg\}^{\frac{1}{t}}&= \Bigg\{\sum_{j=0}^{\infty}q^{sjt}\bigg\{\int_{K}|\Delta_j f|^r dx \bigg\}^{\frac{t}{r}}\Bigg\}^{\frac{1}{t}}.
\bigg[\left(D_{\theta}-\frac{n+2}{2}\right)\left(D_{\theta}+\frac{n+1}{2}\right)\left(D_{\theta}+\frac{j-n}{2}\right)\left(D_{\theta}+\frac{n+1-j}{2}\right) &+\frac{\theta^2}{4}\left(D_{\theta}+\frac{1}{2}\right)\left(D_{\theta}+1\right)\bigg]\hat{h}^n_{n-j}(\theta)=0,
A=[\begin{matrix}1&2\\ 3&4\end{matrix}]=\frac{5}{2}[\begin{matrix}1&0\\ 0&1\end{matrix}]+2[\begin{matrix}0&1\\ 0&0\end{matrix}]+\frac{3}{2}[\begin{matrix}-1&0\\ 0&1\end{matrix}]-3[\begin{matrix}0&0\\ -1&0\end{matrix}]=\frac{5}{2}I+2\sigma_{+1}+\frac{3}{2}\sigma_{0}-3\sigma_{-1}
\frac{d (a)_{2m}}{d a}= \frac{d}{d a}\left[2^{2 m}\left(\frac{a}{2}+\frac{1}{2}\right)_m \left(\frac{a}{2}\right)_m\right] = 2^{2 m}\frac{d}{d a}\left[\left(\frac{a}{2}+\frac{1}{2}\right)_m\right] \left(\frac{a}{2}\right)_m + 2^{2 m}\left(\frac{a}{2}+\frac{1}{2}\right)_m \frac{d}{d a}\left[\left(\frac{a}{2}\right)_m\right]
d(\lambda_{n-1}) =\sum_{i+j=n}\binom{i+j}{i}\lambda_{i-1}\lambda_{j-1}, d(\mu_{n-1}) =\sum_{i+j=n}\binom{i+j}{i}(\lambda_{i-1}\mu_{j-1}-\mu_{i-1}\lambda_{j-1}), d(\sigma\tau) =(-1)^{\deg \sigma}\sigma d(\tau)+d(\sigma)\tau.
D(\tau) = (\omega_{1r}+\omega_{1d})\wedge d\tau + \Big[ (rt_1+t_2) \d\omega_{1r}+ (r \omega_{2r}+\mu_{r}) + (r \omega_{2d}+\mu_{d})\Big]\wedge\tau D(\tau) = (\omega_{1r}+\omega_{1d})\wedge d\tau + \Big[ (rt_1+t_2) \d\omega_{1r}+ \nu_r + \nu_d\Big]\wedge\tau
T R_1 = H(M_1) = I(M_1; Y_0^T)+H(M_1|Y_0^T) \leq I(M_1; Y_0^T) + T \varepsilon_T \leq H(M_1)-H(M_1|Y_0^T) + T \varepsilon_T \leq H(M_1|M_2)-H(M_1|Y_0^T, M_2)+T \varepsilon_T = I(M_1; Y_0^T|M_2) + T \varepsilon_T.
\int_{\Omega}|\mathrm{E}|^2(\mathrm{y})d\mathrm{y}= \dfrac{1}{|1 - \alpha\lambda^{(3)}_{\mathrm{n}_{0}}|^2}\Big|\Big\langle \widehat{\nabla \mathrm{H}}^{\textbf{in}};e^{(3)}_{\mathrm{n}_{0}}\Big\rangle_{\mathbb{L}^{2}(\mathrm{B})}\Big|^2 \delta^2 + \mathcal{O}\Big(\delta^{2}\Big).
\int_X\mathrm{Tr}\, e^{-t\widetilde\Box^+_{b,m}}(x,x)dv_X(x) &\sim D_{1, m}\big((2\pi)^{-1}(2\pi t)^{-n}\mathrm{vol}(X)+t^{-n+1}S^+_{1, n-1}+t^{-n+2}S^+_{1, n-2}+\ldots\big) \qquad+(2\pi)^{-(n+1)}D_{2, m}\mathrm{vol}(X_{p_2})t^{-n+\frac{e_2}{2}}+O(t^{-n+\frac{e_2+1}{2}}).
I\!I_{12}= \int_{d(y,u)\geqslant \frac{1}{2}d(x,y)\atop\|u-y\|> t}\frac1{V(x,u,t+d(x,u))}\Big(\frac{t}{t+d(x,u)}\Big)^{\varepsilon_0}\times \frac1{V(u,y,s+d(u,y))}\bigg(\frac{s}{s+\|u-y\|}\bigg)^\varepsilon\bigg(\frac{s}{s+\|u-y\|}\bigg)^{\varepsilon_0-\varepsilon}d\omega(u).
\frac{1}{\varepsilon^2}\int_\Omega\left( v^\tau_\varepsilon-q\ln\frac{1}{\varepsilon}\right)^{p+1}_+ dx=\int_{\hat{B}_\tau(0)}q^{p+1}(\bar{x}+\varepsilon y)(\hat{U}(y)+\ln\tau)^{p+1}_+dy \rightrightarrows q^{p+1}(\bar{x})\int_{\hat{B}_\tau(0)}(\hat{U}(y)+\ln\tau)^{p+1}_+dy,
\left. e^{- \tilde{E}_{\beta}\, (a,q)}\leq e^{- E_{\beta}\, (a,q)}\ \ \ \ \ \ \mbox{with}\ \ \ \ \ \ \E_{\beta}\, (a,q) = q a - \frac{n_{b}}{\beta}\, c \, e^{2 \pi |a|}\ . \right.
|B_3| \prec \frac{1}{N^{d-m_0}}\Big( \frac{C_0}{N}\Big)^{m_0-1}\Psi^{m_2}\sum_{j_2,\ldots, j_d}\big|G_{j_q y_2}\big|\prod_{i=3}^{m_1}\big|G_{x_i y_i}\big|.
\delta S^{s p i n}= - \frac{i}{2}\omega_{a b}\sum_{i , j = 1}^{n}\int_{\Lambda}( D_{c}- \frac{4 x_{c}}{R^{2}}) [ \frac{\partial L}{\partial u_{i , c}}( \Gamma^{a b})_{j}^{i}u^{j}( x ) ] d \Omega ( x )
\sum_{k=0}^{\infty}\binom{s+k}{k}\left( r+k\beta\mid\alpha\right)_{n}x^{k}=\frac{1}{\left( 1-x\right)^{s+1}}w_{n}^{\left( s+1\right)}\left(\frac{x}{1-x};\alpha,\beta,r\right) ,
\mathcal{B}_{-}(\lambda^{-1})=-\frac{(\lambda^{2}q-1/\left( q\lambda^{2}\right) )}{(\lambda^{2}/q-q/\lambda^{2})}\mathcal{B}_{-}(\lambda ),\mathcal{C}_{-}(\lambda^{-1})=-\frac{(\lambda^{2}q-1/\left( q\lambda^{2}\right) )}{(\lambda^{2}/q-q/\lambda^{2})}\mathcal{C}_{-}(\lambda ).
\begin{array}{lll}x_i = \prod_{j=1}^r\overline x_j^{a_{ij}}(\overline x_{r+\overline m}+\overline \alpha)^{a_{i,r+1}}\mbox{for}1\le i\le r\mbox{and}x_{r+\overline m}= \prod_{j=1}^r\overline x_j^{a_{r+1,j}}(\overline x_{r+\overline m}+\overline \alpha)^{a_{r+1,r+1}}\end{array}
w_{ij}:= \begin{bmatrix}v_{ij1}\vdots v_{ijr}\end{bmatrix}, \ \v_{ijk}:= \sum_{\substack{s\ge i, t\ge j (s,t)\neq(i,j)}}\left(( A_k )_{is}( X_k )_{st}( B_k )_{tj}- ( C_k )_{is}( X_{k+1})_{st}( D_k )_{tj}\right);
\varOmega_{2k-1}(u)=\frac{(-1)^{\frac{(k-1)(k-2)}{2}}k[\Gamma(k/2)]^{2}}{u^{k(2k-1)/2}(2k+1)}\frac{(\det\mathbf N_{k-1})^2}{2^{(k-1)(2k-1)+1}}\prod_{j=1}^k\left[\frac{(2j)^2}{(2j)^2-u}\right]^{k-\frac{1}{2}},\forall u\in(0,4).
d_i^{ne}=\frac{B\left(|\mathcal{S}|-1\right)}{w_i\sum_{j \in \mathcal{S}}\frac{c_j+\lambda_j}{w_j}}\left(1-\frac{c_i+\lambda_i}{w_i}\frac{\left(|\mathcal{S}|-1\right)}{\sum_{j \in \mathcal{S}}\frac{c_j+\lambda_j}{w_j}}\right).
\omega_l^{(1)}=\sqrt{(\omega_l^{(0)})^2+C_{p+1}^{p/2}2^{-p}a_l^p\delta}&=\omega_l^{(0)}+\frac{C_{p+1}^{p/2}2^{-p}a_l^p\delta}{\sqrt{(\omega_l^{(0)})^2+C_{p+1}^{p/2}2^{-p}a_l^p\delta}+\omega_l^{(0)}},
(\gamma_{4})^{T}=\alpha(\begin{matrix}0.7308\\ 0.2692\end{matrix})\circ(\begin{matrix}0.6273\\ 0.3727\end{matrix})=\alpha(\begin{matrix}0.4584\\ 0.1003\end{matrix})=(\begin{matrix}0.8204\\ 0.1796\end{matrix})
\iint_{\mathbb{R}^3\!\times\mathbb{R}^3}\left[(\bar{f}\varphi)(t)-(\bar{f}\varphi)(0) \right] dy'dw' = \int_0^t\!\iint_{\mathbb{R}^3\!\times\mathbb{R}^3}\Big\{\bar{f}\big[(\partial_s+w'\!\cdot\!\nabla_{\!y'})\varphi-\mathbb{B}\cdot\nabla_{\!w'}\varphi\big] - \nabla_{\!w'}\bar{f}\cdot(\mathbb{A}\nabla_{\!w'}\varphi) \Big\}dy'dw'ds
\theta_{\{b_{i,j}\}}^{\{c_{i,j}\}}(\underline{\lambda}\times\underline{\lambda}') = e( u_1\lambda_1+ u_2\lambda_2+\cdots+ u_{n-1}\lambda_{n-1}+ p^{-b_{12}}c_{12}\lambda_1'+ p^{-b_{23}}c_{23}\lambda_2'+\cdots+ p^{-b_{n-1,n}}c_{n-1,n}\lambda_{n-1}').
|\nu_{m_i}(\sigma_{i+1})-\nu_{m_{i+1}}(\sigma_{i+1})| \leq|\nu_{m_i}(\sigma_{M'})-\nu_{m_{i+1}}(\sigma_{M'})| &\ \ \ \ +|\nu_{m_i}(\sigma_{i+1}\bigtriangleup\sigma_{M'})|+|\nu_{m_{i+1}}(\sigma_{i+1}\bigtriangleup\sigma_{M'})| &<1/8E_\flat+1/16E_\flat+1/16E_\flat &=1/4E_\flat.
c^T(\mathcal{E}- \mathcal{V}_{k+1}- \mathcal{U}_{l+1}) = \frac{\prod_{i=1}^n(1 - t_i^2u^2)}{\prod_{i=k+1}^{n}(1+\pmb{z}_iu) \prod_{i=l+1}^n(1-t_iu)}= \prod_{i=1}^{n}\frac{1+ t_i u}{1 + \pmb{z}_i u}\prod_{i=1}^k (1+\pmb{z}_i u) \prod_{i=1}^{l}(1-t_i u),
\sigma_n\int_{\mathcal{A}_n^2}y^{1-2\alpha}|w_n|^{2}\,dxdy+\int_{\mathcal{A}_n^2\times\{y=0\}}|w_n|^{2^*_\alpha}\,dx +\int_{\mathcal{A}_n^2\times\{y=0\}}|w_n|^{2}\,dx \le C\sigma_n\sigma_n^{-\frac{N+2-2\alpha}2}+ C\sigma_n^{-\frac{N}{2}}\le C\sigma_n^{-\frac{N-2\alpha}2}.
\left|{\bf E\,}\left| \frac{\sum_{i=1}^{n}\omega_{i}}{\sqrt{n}}\cdot \frac{\sum_{i=1}^{n}\zeta_{i}}{\sqrt{n}}\right| -{\bf E\,}|Z_{1}^{*}Z_{2}^{*}|\right| \leq \frac{A}{\left( 1-\rho^{2}\right)^{3/2}\sqrt{n}}\left({\bf E\,}|\omega |^{3}+{\bf E\,}|\zeta |^{3}\right)
\Gamma^{( 1 )}= \frac{g_{\varepsilon}^{2}\lambda^{2}\hbar}{4 \pi c \tau_{D}\tau_{S}}\, \frac{\operatorname{c o s}( | \vec{x}_{D}- \vec{x}_{S}| / \lambda )}{| \vec{x}_{D}- \vec{x}_{S}|}\, .
k\delta_k^{\textrm{Ric}}\leq 2k(k-1)c + \frac{1}{2}\sum_{r}\big\{(\sum_{i=1}^{2k}h_{ii}^r)^2-\frac{1}{k}(\sum_{i=1}^{2k}h_{ii}^r)^2\big\}&=2k(k-1)c+\frac{k-1}{2k}(\sum_{i=1}^{2k}h_{ii}^r)^2 &=2k(k-1)c+\frac{k-1}{2k}4k^2 H^2 &=2k(k-1)(c+H^2),
\phi_{F'}(s) = \int_0^\infty f_{F'}(t)e^{st}\mathrm{d}t &= \int_0^\infty \frac{1}{1-p}\lambda e^{-t\lambda}\mathbb{P}(G>t)e^{st}\mathrm{d}t &= \frac{1}{1-p}\left(\frac{\lambda}{\lambda-s}-\lambda\int_0^\infty \mathbb{P}(G<t)e^{st}\mathrm{d}t\right)
e^{-\gamma (t_2-t_1)}X_-(t_2)\leq X_-(t_1)+\int_{t_1}^{t_2}e^{-\gamma (s-t_1)}Y_-(s)\, ds \leq X_-(t_1)+\int_{t_1}^{\infty}e^{-\gamma (s-t_1)}|Y_-(s)|\, ds.
\int_{\mathbb R^2}( \rho_{(1),\infty}^{\rm OCP}(\mathbf r; Q) - 1) e^{i \mathbf k \cdot \mathbf r}\, d \mathbf r = \sum_{j=0}^\infty{(-1)^j \over (j!)^2}\Big ({|\mathbf k |^2 \over 4}\Big )^j \int_{\mathbb R^2}( \rho_{(1),\infty}^{\rm OCP}(\mathbf r; Q) - 1) | \mathbf r |^{2j}\, d \mathbf r,
\Gamma \lbrack A \rbrack = \int_{0}^{\infty}{\frac{d T}{T}}\, \mathrm{e}^{- m^{2}T}\int{\cal D}x \, \mathrm{e x p}\left[ - \int_{0}^{T}\! \! \! d \tau \left({\frac{1}{4}}{\dot{x}}^{2}+ i e A_{\mu}\dot{x}^{\mu}\right) \right]
\sum_{j\in\mathbb{N}}\frac{w_r(Q_j)}{w_r(Q)}\leq \frac{w_r\left(\bigcup_{j\in\mathbb{N}}Q_j\right)}{w_r(Q)}\leq \left(\frac{\mu\left(\bigcup_{j\in\mathbb{N}}Q_j\right)}{\mu(Q)}\right)^{1/r'},
a_1(J_{q,p}^t-J_p^rJ_q^sJ_{s,r}^t)+a_2(J_{p,q}^t-J_q^rJ_p^sJ_{s,r}^t)+a_3(J_p^rJ_{q,r}^t+J_q^sJ_{s,p}^t)+a_4(J_q^rJ_{p,r}^t+J_p^sJ_{s,q}^t)a=0.
\ll x e^{O(k)}\sum_{\alpha \in \mathcal{B}_j}\sum_{\substack{t_j^k<n_1, n_2 \leq t_{j+1}^k P^{-}(n_1n_2)> y n_1n_2= \square}}\frac{|g_{t_j, t_{j+1}, k}(n_1, \alpha)g_{t_j, t_{j+1}, k}(n_2, \alpha)|}{n_1n_2}\ll e^{O(k)}4^j x \sum_{n\geq t_j^{k}}\frac{d_{2k}(n^2)}{n^2},
\begin{array}{lcl}P_{0}= -\theta^4-(r+2-e_1-e_2-e_3)\theta^3-(r^2+(2-e_1-e_2-e_3)r-e_1 -e_2-e_3+e_1e_2+e_1e_3+e_2e_3)\theta^2 -(r^3+(2-e_1-e_2-e_3)r^2 -(e_1+e_2+e_3-e_1e_2-e_1e_3-e_2e_3)r-2+e_1 +e_2+e_3-e_1e_2e_3)\theta-(r-1)(r-e_1+1)(r+1-e_2)(r+1-e_3).\end{array}
U_m^-(x,t) =\inf_{S\in \mathcal{S}_m}\sup_{A\in \mathcal{AC}_m}\mathbb E \Big[e^{-r(T-t)}g\big(X(T)\big)\Big], U_m^+(x,t) =\sup_{S\in \mathcal{S}_m}\inf_{A\in \mathcal{AC}_m}\mathbb E \Big[e^{-r(T-t)}g\big(X(T)\big)\Big]
\prod_{j\in J}\cos(2\pi \xi_j) =\prod_{j\in I}\bigg(\frac{1+\cos (2\pi \xi_j)}{2}+\varepsilon(J)_j\frac{1-\cos (2\pi \xi_j)}{2}\bigg) &=\prod_{j\in I}\big(\cos^2 (\pi \xi_j)+\varepsilon(J)_j\sin^2(\pi \xi_j)\big) &=\sum_{S\subseteq I}a_S(\xi)w_S(\varepsilon(J)),
z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right)\Omega_q = \Omega_q
G(n_1,\ldots,n_r)=\sum_{d_1+\cdots+d_r=d}\frac{1}{d_1!\cdots d_r!}e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)n_1^{d_1}\cdots n_r^{d_r}.
m_{3}^{eq}=\alpha \, \rho \,,\, m_{4}^{eq}=\beta \, \rho \,,\,m_{5}^{eq}=-\frac{j_{x}}{\lambda}\,,\, m_{6}^{eq}=-\frac{j_{y}}{\lambda}\,,\, m_{7}^{eq}=0 \,,\, m_{8}^{eq}=0 \, .
C_n(y_{n-1})=\sum_{y_n\in\{0,1\}}\log\Big(\frac{q_n(y_n|x_n,y_{n-1})}{\nu^{\pi^*}_n(y_n|y_{n-1})}\Big)q_n(y_n|x_n,y_{n-1}),~\forall{x_n}.
\partial^{\beta_0}_xP_t\varphi(x)=\mathbb{E}[\partial^{\beta_0}_x(\varphi(X_t(x))]=\sum_{\vert\alpha_0\vert\leq\vert\beta_0\vert}\mathbb{E}[(\partial^{\alpha_0}\varphi)(X_t(x))\mathbf{P}_{\alpha_0}(x)],
\int_{B_{r\tau}'}u^2(x)dx =r^n\int_{B_{\tau}'}u^2(rx)dx &=r^n\frac{H(r)}{r^{n+1-2s}}\int_{B'_\tau}\frac{u^2(rx)}{H(r)/r^{n+1-2s}}dx &=r^n\frac{H(r)}{r^{n+1-2s}}\int_{B'_\tau}u^2_r(x)dx &=r^n\frac{H(r)}{r^{n+1-2s}}\tau^{n+2\gamma}\int_{B_1'}u^2_r(\tau x)/\tau^{2\gamma}dx &\ge r^n\frac{H(r)}{r^{n+1-2s}}\tau^{n+2\gamma}\frac{1}{2}c.
S_{4d-2d}=\frac{1}{2\hbar\pi}\int_{\mathbb{R}^{2}\times C}\omega\wedge CS(A)+\sum_{\alpha=1}^{n}\frac{1}{\hbar}\int_{\mathbb{R}^{2}\times z_{\alpha}}L_{\alpha}(\phi_{\alpha};A_{w}|_{z_{\alpha}},A_{\overline{w}}|_{z_{\alpha}}),
{\cal A}_{\alpha}= \left( \begin{matrix}{0}&{0}&{0}&{0}\\{\noalign{\vskip 1 m m}0}&{0}&{0}&{C_{\alpha \beta \gamma}}\\{\noalign{\vskip 1 m m}0}&{\delta_{\alpha}^{\gamma}}&{0}&{0}\\{\noalign{\vskip 1 m m}\delta_{\alpha}^{\beta}}&{0}&{0}&{0}\\ \end{matrix}\right) \ .
Y^{n,m}_t(x,v) := f^n_t(x,v) \sqrt{1+v^{2m}}\,, Z^{n,m}_t(x,v) := (D f^n_t(x,v))^\top \sqrt{1+v^{2m}}\,, H^{n,m}_t(x,v) := D^2f^n_t(x,v) \sqrt{1+v^{2m}}\,.
\left|\frac{p}{q}\right|\leq\left|\frac{\alpha_3}{\alpha_1}\right|+\frac{1}{qH}<\frac{1}{\alpha_1}+1\leq\tau\Rightarrow|p|<\tau q \Rightarrow|a|^2=q^2v^2+q^2u^2+p^2v^2<\tau^2q^2+\tau^2q^2+\tau^4q^2<3\tau^4q^2
\bar{H}_{\mathrm{g}}= H_{\mathrm{g}}( q^{r}, p^{r}) + \lambda_{a}^{( 1 ) r_{a}}\pi_{a}^{r_{a}}+ \lambda_{a}^{( 2 ) r_{a}}\psi_{a}^{r_{a}}+ \lambda_{a}^{( 3 )}\theta_{a}+ \lambda_{a}^{( 4 )}\gamma_{a},
G_{1}(x) = \displaystyle \ln \cos x - \ln\!\left(\frac{\pi}{2}\!-\!x\right)-\omega_{1}(x) \ln \! \left( \frac{2}{\pi}+ \frac{\pi\!-\!2}{\pi^{3}}\left(\pi^{2}\!-\!4\left(\frac{\pi}{2}\!-\!x\right)^{2}\right) \right)\!,
Z\subseteq\bigcap_{n\geq 1}\bigcup_{j=1}^k \overline{B_{10 m_n^{-1/2}}(x_j)}= \bigcup_{j=1}^k \bigcap_{n\geq 1}\overline{B_{10 m_n^{-1/2}}(x_j)}= \bigcup_{j=1}^k \lbrace x_j \rbrace.
\limsup_{h\rightarrow 0}\| C_{\mathrm{ext}}(t+h) z -C_{\mathrm{ext}}(t) z \|^2 = \ \limsup_{h\rightarrow 0}\sum_{n=1}^{N}\| C (nt+nh) z_n -C (nt) z_n \|^2 =0,
\langle \widetilde{u}(\epsilon_m,...,\epsilon_1), \widetilde{u}(\delta_m,...,\delta_1) \rangle = \begin{cases}0 (\epsilon_m,...,\epsilon_1) \neq (\delta_m,...,\delta_1) \epsilon_1 (\epsilon_m,...,\epsilon_1) = (\delta_m,...,\delta_1) \end{cases}
\Phi_{q}(q,t)=\int_{\Omega}a\left( x\right) \left[ (t\phi_{1}+w)^{q}\left\{\log(t\phi_{1}+w)+\frac{qw_{q}}{t\phi_{1}+w}\right\}-w_{q}\right] \phi_{1}. & \Phi_{t}(q,t)=\int_{\Omega}a(x)\left\{q(t\phi_{1}+w)^{q-1}(\phi_{1}+w_{t})-(\phi_{1}+w_{t})\right\}\phi_{1}.
\nabla E(\bar{x}) = 2k \, [d_{x_k}x_1 \, \nabla d_{x_k}x_1 +d_{x_2}x_1 \, \nabla d_{x_2}x_1 , \, d_{x_1}x_2 \, \nabla d_{x_1}x_2 +d_{x_3}x_2 \, \nabla d_{x_3}x_2, & \ldots ,d_{x_{k-1}}x_k \, \nabla d_{x_{k-1}}x_k +d_{x_1}x_k \, \nabla d_{x_1}x_k ]
2m(t_2-t_1) ={2\over n}+O\Bigl({1\over n\log{n}}\Bigr), begin{align*}5pt]-(mn+2c)(2t_1-t_0-t_2) =-{1\over n}+O\Bigl({1\over n\log{n}}\Bigr), begin{align*}5pt]{2\over m}\Bigl({1\over t_2}-{1\over t_1}\Bigr) =O\Bigl({1\over n\log^2{n}}\Bigr), begin{align*}5pt]{n\over m}\Bigl({1\over t_0}+{1\over t_2}-{2\over t_1}\Bigr) =O\Bigl({1\over n\log^2{n}}\Bigr).
\begin{aligned}d\big(\Psi_k(x),\Psi_k(y)\big)\ \le\ s_1 d(\big(T_2 \circ T_3 \circ \dots \circ T_k(x),T_2 \circ T_3 \circ \dots \circ T_k(y)\big)) & \le s_1s_2d(\big(T_3 \circ T_4 \circ \dots \circ T_k(x),T_3 \circ T_4 \circ \dots \circ T_k(y)\big)) ... & \le \big(\prod_{i=1}^k s_i\big) d(x,y),\end{aligned}
\begin{cases}\overline{V}^{(n)}:=\overline{\Psi}(\overline{V}^{n-1}), \,\, \overline{V}^{0}=V^{(0)}, \underline{V}^{(n)}:=\underline{\Psi}(\underline{V}^{n-1}), \,\, \underline{V}^{0}=V^{(0)}. \end{cases}
\begin{array}{c}{{{\displaystyle K_{a b}^{-}~ = ~ \frac{\epsilon}{4}\, ( E_{a a}- E_{b b}+ E_{\theta ( a ) \, \theta ( a )}- E_{\theta ( b ) \, \theta ( b )}) ~ ,}}}\\{{{\displaystyle M_{a b}^{-}~ = ~ \frac{\epsilon}{4}\, ( E_{a a}- E_{b b}- E_{\theta ( a ) \, \theta ( a )}+ E_{\theta ( b ) \, \theta ( b )}) ~ ,}}}\\ \end{array}
exp\psi(x+\frac{1}{2})\sim x+\frac{1}{4!\cdot x}-\frac{37}{8\cdot6!\cdot x^{3}}+\frac{10313}{72\cdot8!\cdot x^{5}}-\frac{5509121}{384\cdot10!\cdot x^{7}}+\cdot\cdot\cdot
A^{(2)}_{\mu\nu\eta\sigma\rho}(x)=\frac{e^2}{2}(A_\mu A_\eta \partial_\sigma F^0_{\nu\rho}-\partial_\nu A_\rho\partial_\eta A_\mu A_\sigma+ A_\mu F^0_{\nu\eta}F^0_{\sigma\rho}) \,\,\, ,
|N(x) \cap S_4| = \frac{1}{|S_0 \setminus (S \cup T)|}|S_4| |N(y) \cap S_0 \setminus (S \cup T)| = \frac{1}{|S_0 \setminus (S \cup T)|}|S_4| \cdot \frac12|S_0 \setminus (S \cup T)| = \frac12 |S_4|.
T(u) T_1(v) + T_1(u) T(v) =~ T_1 ( u \cdot T(v) + T(u) \cdot v + H (Tu, Tv)) &+ T \big( u \cdot T_1(v) + T_1(u) \cdot v + H (T(u), T_1(v)) + H ( T_1(u), T(v)) \big), T_1(u) T_1(v) =~ T ( H (T_1(u), T_1(v)) ) + T_1 \big( u \cdot T_1(v) + T_1(u) \cdot v + H (T(u), T_1(v)) + H (T_1(u), T(v)) \big), 0 =~ T_1 ( H (T_1(u), T_1(v) ) ).
\lambda_{n,m_1,m_2}= B(2n+1) + \big(\textstyle{\frac{\pi m_1}{d_1}}\big)^2 + \big(\textstyle{\frac{\pi m_2}{d_2}}\big)^2,n\in\mathbb{N}_0,\; m_1,m_2\in\mathbb{N}
[\begin{matrix}\epsilon_{xx}&\epsilon_{xy}&\epsilon_{xz}\\ \epsilon_{yx}&\epsilon_{yy}&\epsilon_{yz}\\ \epsilon_{zx}&\epsilon_{zy}&\epsilon_{zz}\\ \end{matrix}]=[\begin{matrix}\epsilon_{xx}&\gamma_{xy}/2&\gamma_{xz}/2\\ \gamma_{yx}/2&\epsilon_{yy}&\gamma_{yz}/2\\ \gamma_{zx}/2&\gamma_{zy}/2&\epsilon_{zz}\\ \end{matrix}]
I:= \int_0^1\int_0^1 (1-t)^{2c-1}(1-ts)^{-1/2-d}\Big(\frac{s}{1-s}\Big)^c\Big(\frac{s(1-s)}{1-ts}\Big)^n dsdt\sim \frac{n^{d-3c}}{\sqrt n}\Gamma(2c)\Gamma\Big(\frac{1}{2}-d+c\Big)
A\left( \begin{array}{c}f\left( x\right) R^{\prime}\left( f\right)\end{array}\right) =\left( \begin{array}{c}\tau \left( f\right) -R\left( f\right)\end{array}\right) ,\left( \begin{array}{c}f\left( x\right) R^{\prime}\left( f\right)\end{array}\right) \in D\left( A\right) .
C_{i_1,\ldots,i_k}(1) = \sum_{r=0}^{i_1-1}C_{r,i_1-1-r,i_2,\ldots,i_k}(1) + C_{i_2,\ldots,i_k}\left(C_{i_1}(1) - \sum_{r=0}^{i_1-1}C_{r,i_1-1-r}(1)\right) &+ 2\sum_{r=2}^k i_rC_{i_1+i_r-1,i_2,\ldots,\widehat{i_r},\ldots,i_k}
\lim_{m \to \infty}\lim_{l \to \infty}\inf_{v}\left[ b(x,v)\cdot \nabla u^{\kappa}_{\alpha,m}+ \theta r(x,v)u^{\kappa}_{\alpha,m}\right]\ = \ \inf_{v}\left[ b(x,v)\cdot \nabla u^{\kappa}_{\alpha}+ \theta r(x,v)u^{\kappa}_{\alpha}\right] \mbox{in}[\kappa,1] \times \overline{C}.
y\left( x\right) =y_{0}^{\ast}\left( x\right) +\frac{1}{\Gamma \left(\alpha \right)}\int_{x_{1}}^{x}\psi^{\prime}\left( t\right) \left( \psi\left( x\right) -\psi \left( t\right) \right)^{\alpha -1}f\left( t,y\left(t\right) \right) dt,
\int_{\mathbb{Z}_p}(1-4t)^{\tfrac{x}{2}}d\mu_{-1}(x) = \frac{2}{1+\sqrt{1-4t}}= \sum_{n=0}^\infty \frac{1}{n+1}{2n \choose n}t^n = \sum_{n=0}^\infty C_n t^n.
\mathcal{H}(V_A^{(n)},A(x))-\mathcal{H}(V_A^{(n-1)},A(x))= \mathcal{H}_e(V_A^{(n-1)},V_{A,0},A(x),A(0),Z_A^{(n-1)}) &-\mathcal{H}_e(V_A^{(n-2)},V_{A,0},A(x),A(0),Z_A^{(n-2)}).
\begin{aligned}\int_{C^{j-1}}^{C^{j-1}+\zeta(j)}\frac{1}{u}\mathrm{d}u = \log\left(\frac{C^{j-1}+\zeta(j)}{C^{j-1}}\right) & = \log(1 + (C-1)\tau_A(C,j))\end{aligned}
\sum_{j = 0}^k{( - 1)^{rj}\binom kj\left({\frac{{F_r}}{{F_n}}}\right)^j G_{(n + r)j}}= ( - 1)^{rk}\left({\frac{{F_{n + r}}}{{F_n}}}\right)^k G_{rk},n\ne 0\,,
\Longrightarrow \Big( I^{H - \frac{1}{2}}_{-}{\bf{1}}_{[0,t]}\Big) (s) = \frac{1}{\Gamma(H - 1/2)}\int^{\infty}_{x}{\bf{1}}_{[0,t]}(u) (u-s)^{H- 3/2}du & = \frac{1}{\Gamma(H - 1/2)}\int^{t}_{0}(u-s)^{H- 3/2}du = \frac{1}{\Gamma(H + 1/2)}\Big[ (t - s)^{H - 1/2}- (-s)^{H - 1/2}\Big]
\langle u, v \rangle = \langle u_1 + u_0 + u_{\frac{1}{4}}+ u_{\frac{1}{32}}, v_1 + v_0 + v_{\frac{1}{4}}+ v_{\frac{1}{32}}\rangle = \langle u_1, v_1 \rangle + \langle u_0, v_0 \rangle + \langle u_{\frac{1}{4}}, v_{\frac{1}{4}}\rangle + \langle u_{\frac{1}{32}}, v_{\frac{1}{32}}\rangle.
E^*_{i+k}A_1E^*_{i+k-1}A_1E^*_{i+k-2}\cdots E^*_{i+1}A_1E^*_{i}=\big((\frac{k}{2})!\big)^2 M^{\frac{2m-k}{2},\frac{2m-i-k}{2}}_{\frac{2m-i-k}{2},\frac{2m-i}{2}}.
f_\epsilon^{'}\Big(V_m(\mu_m^iy+\xi_m)\Big) = f_\epsilon^{'}\Big( PU_{\mu_{in},\xi_{in}}(\mu_m^iy+\xi_m)+ \sum\limits_{i=1, j\neq i}^kPU_{\mu_{jm},\xi_{jm}}(\mu_m^iy+\xi_m)\Big) = f_\epsilon^{'}\Big((\mu_m^{i})^{-\frac{n-2}{2}}U (y-\sigma_{in})+ U_{\mu_{jm},\xi_{jm}}(\mu_m^iy+\xi_m)+o(1)\Big),
\tau =\lim_{P\to\infty}\lim_{F\to\infty}\frac{T\log P}{F}&\ge\frac{1}{l}\left\{(s_1+s_2)-(N_T-l)s_2\mu_T-\left(\frac{2s_2+s_1+1}{2}\cdot s_1+s_2^2\right)\mu_R\right\}.
-te_{\lambda}^{-r}(t) =-t\sum_{n=0}^{\infty}\frac{(-r)_{n,\lambda}}{n!}t^{n}=t\sum_{n=0}^{\infty}(-1)^{n-1}\frac{\langle r\rangle_{n,\lambda}}{n!}t^{n}&=\sum_{n=1}^{\infty}(-1)^{n}\langle r\rangle_{n-1,\lambda}\frac{t^{n}}{(n-1)!}=\sum_{n=1}^{\infty}(-1)^{n}\langle r\rangle_{n-1,\lambda}n\frac{t^{n}}{n!},