harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}H_n''(t) \leqslant \mathbb{E}_{n-1}\Big[ 5\lambda^2 |Z_n|^2 \cosh{\big(\lambda |g_{n-1}+tZ_n|}\big)\Big] \leqslant 5 \lambda^2 S_n^2\mathbb{E}_{n-1}\Big[ \cosh{\big(\lambda |g_{n-1}+tZ_n|}\big)\Big] = 5\lambda^2 S_n^2 H_n(t).\end{aligned}
\int_{\mathbb{R}^N}(I_\mu\ast|u_n|^{2^*_\mu})|u_n|^{2^*_\mu}=\int_{\mathbb{R}^N}(I_\mu\ast|u|^{2^*_\mu})|u|^{2^*_\mu}+\int_{\mathbb{R}^N}(I_\mu\ast|\tilde{u}_n|^{2^*_\mu})|\tilde{u}_n|^{2^*_\mu}+o_n(1),
\frac{d}{dt}\|\tilde{\phi}(t)\|_3\leq C\left(\|v\|_3\|\tilde{\phi}\|_3+ (|\psi|_\infty+\|\nabla\psi\|_2)\|v\|_3+|\psi \nabla^4 v|_2\right) \leq C\left(c_3\|\tilde{\phi}\|_3+c^2_3+|\psi \nabla^4 v|_2\right).
\frac{1}{N}\underset{i=1}{\overset{N}{\sum}}\mathbb{E}\Big[\sup_{0\leq s\leq t}|x_s^i-x_s^{*,i}|^2\Big] \leq\frac{C}{N^2}\Big(1+\frac{1}{N^2}\sum_{i,j=1}^N\mathbb E\big[|\xi^i-\xi^j|^2\big]\Big).
m(B_{t+\delta}\setminus B_{t-\delta}) =m(B_{t+\delta})-m(B_t) \leq m(B_{t+\delta})-\frac{F(t-\delta)}{F(t)}m(B_t) =m(B_{t+\delta}\setminus B_t)+\frac{F(t)-F(t-\delta)}{F(t)}m(B_t),
\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 ,
\begin{aligned}\frac{1}{2Z}|\sqrt{Z'}-\sqrt{Z}|^2 =\frac{1}{2Z}\frac{|Z-Z'|^2}{|\sqrt{Z}+\sqrt{Z'}|^2}&\leqslant K_1|Z-Z'|^2 &=K_1|\int\exp(-\varrho(x))\rho_0(x)dx- \int\exp(-\varrho'(x))\rho_0(x)dx|^2 &\leqslant K_1\int|\exp(-\varrho(x))-\exp(-\varrho'(x))|^2\rho_0(x)dx\end{aligned}
\det \left( A_{0}\tilde{E}_{0}\right) \otimes \left( \det \left( \tilde{E}_{0}\right) \right)^{- 1}, \quad \tilde{E}_{0}: = E_{0}\left( A_{1}, A_{0}\right) = A_{0}^{- 1}K_{- 1}\left( S_{1}\right) ,
\begin{array}{ll}\displaystyle \lim_{k\to\infty}\|\mathcal{A}^*u^{k+1}+ \mathcal{B}^*v^{k}- c\|= 0, begin{align*}8pt]\displaystyle\lim_{k\to\infty}(1-\tau)\|\mathcal{A}^*u^{k+1}+ \mathcal{B}^*v^{k+1}- c\|= 0, begin{align*}8pt]\displaystyle \lim_{k\to\infty}\|w^{k+1}- w^k\|_{\mathcal{Q}+\,(\mathcal{S}, \mathcal{T})}= 0. begin{align*}8pt]\end{array}
a_{A}( k ) a_{A}^{\dagger}( k^{\prime}) + q^{- 2}\sum_{l , l^{\prime}\in{\cal I}}{\hat{\cal R}}\left( k^{\prime}, l ; k , l^{\prime}\right) a_{A}^{\dagger}( l^{\prime}) a_{A}( l ) = \delta_{k k^{\prime}},
\left( \Delta_{r \theta}- \frac{{\hat{J}}^{2}}{r^{2}\sin^{2}\theta / 2}- \frac{{\hat{L}}^{2}}{r^{2}\cos^{2}\theta / 2}\right) \psi + \frac{2 M}{\hbar^{2}}\left( \epsilon + \frac{\alpha}{r}\right) \psi = 0
\|\lambda_y\|^2= \int_{\mathbb{R}^3_+}\mathrm{d}k\, \frac{\chi^2_{\Lambda}(k)}{2 \pi^2|k|}\left\{(1-\hat{k}_1^2) (1 + \cos(2k_1 y)) + (2-\hat{k}_2^2 - \hat{k}_3^2) (1 - \cos(2k_1y)) \right\}.
h_{1}^{2}v^{2}\rightarrow h_{2}^{2}v^{2}+ ( 2 h_{2}h_{3}+ h_{3}^{2}\phi^{2}) ( v^{j}\phi^{j})^{2}\equiv g_{1}v^{2}+ g_{2}( v^{i}\phi^{i})^{2},
N^{+}( [ 1 , n ] ) = \sum_{k = 0}^{n}( - 1 )^{k}\sum_{\begin{array}{c}{\scriptstyle j_{0}+ \dots + j_{k}= n + 1}\\{\scriptstyle j_{i}> 0}\\ \end{array}}c_{j_{0}}\cdots c_{j_{k}}.
\sum\limits_{i,j=1}^{n+1}[d_G(y,y_0)^{-1-2\alpha+\epsilon}Y_i Y_j \tilde{v}]_{C^{0,\epsilon}_\ast(\mathcal{B}_1^+(y_0))}&+ \sum\limits_{i,j=1}^{n+1}\|d_G(y,y_0)^{-1-2\alpha}Y_i Y_j \tilde{v}\|_{L^{\infty}(\mathcal{B}_1^+(y_0))}\leq C\|f\|_{Y_{\alpha,\epsilon}}.
\begin{aligned}I_i = \Big[\frac{1}{v}+\frac{(\frac{|B|^2+\beta^2}{v}-c^2)\frac{1}{v}+\frac{2c^2}{v}}{\sqrt{\Big( \frac{|B|^2+\beta^2}{v}-c^2 \Big)^2+4\frac{|B|^2c^2}{v}}}\Big]B_i^2 &>\frac{\frac{2c^2}{v}B_i^2}{\sqrt{\Big( \frac{|B|^2+\beta^2}{v}-c^2 \Big)^2+4\frac{|B|^2c^2}{v}}}\ge0,\end{aligned}
R\left(e_{Q,\omega}w_{\tau,T}\right)=\int_{\mathbb{R}^{nd}}R\left(e_{Q,\omega}\tilde{e}_{Q,T,y}\right)\widehat{\zeta}_{\tau}\left(y\right)dy=\int_{\mathbb{R}^{nd}}\left(\theta_{T,y}\left(\omega\right)-\vartheta_{T,y}\left(\omega\right)\right)\widehat{\zeta}_{\tau}\left(y\right)dy.
\phi_{i}( g_{1}, g_{2}, g_{3}, g_{4}) ={\frac{1}{4 !}}\sum_{\sigma \in \Sigma_{4}}\chi_{i}( \sigma ) \phi_{i}( g_{\sigma ( 1 )}, g_{\sigma ( 2 )}, g_{\sigma ( 3 )}, g_{\sigma ( 4 )}) .
p^{10}=\left[\begin{array}{ccccc}1 1 1 2 2 2 2 2 1 3 3 3 3 3 1 \end{array}\right],p^{11}=\left[\begin{array}{ccccc}1 1 1 2 3 2 2 2 1 1 3 3 3 3 2 \end{array}\right],p^{12}=\left[\begin{array}{ccccc}1 1 1 3 2 2 2 2 2 3 3 3 3 1 1 \end{array}\right],
|[ (Q\widetilde{T})^* (z^*) ] (e_k \otimes y^*) - \langle u^* (y^*), Q^* (z^*) \rangle | = |\langle (T(e_k))^* (y^*) - u^*(y^*) , Q^* (z^*) \rangle | &\leq \| (T(e_k))^* - u^* \| \| y^* \| \| Q^* (z^*) \| \longrightarrow 0
N_0 = \prod_{\substack{p\mid N, -v_{p}(\zeta_{p})\leq 0}}p^{n_{p}}, N_1 = \prod_{\substack{p\mid N, 0<-v_{p}(\zeta_{p})<n_{p}}}p^{n_{p}}N_2= \prod_{\substack{p\mid N, n_{p}\leq -v_{p}(\zeta_{p})}}p^{n_{p}}.
a(k)= N_{11}(0,0,k)=\bar N_{22}(0,0,\bar k)=\frac{\varkappa(k)+\varkappa^{-1}(k)}{2}\frac{F^0_1(k)}{F^0_1(\infty)}, b(k)= N_{21}(0,0,k)=-\bar N_{12}(0,0,\bar k)=\frac{\varkappa(k)-\varkappa^{-1}(k)}{2}\frac{F^0_2(k)}{H^0_2(\infty)},
\mathrm{P}^{OMA}_{m,1}= \mathrm{P}\left( z_t>\frac{\epsilon_1}{\rho}, z_m<\frac{\epsilon_1}{\rho}\right)+ \mathrm{P}\left( z_t<\frac{\epsilon_1}{\rho}, z_m<\frac{\epsilon_1}{\rho}\right) =\mathrm{P}_{m,1}.
e^{{\cal{G}}_{\rm ac}\{{\phi}\}}= \frac{\int{\cal{D}}\{\phi^{\prime}\}e^{- S_0 \{\phi^{\prime}\}- S_{\rm int}\{\phi^{\prime}+{\phi}\}}}{\int{\cal{D}}\{\phi^{\prime}\}e^{- S_0 \{\phi^{\prime}\}}}\; .
\chi((b,k))=L_{(0,i_1)}^{p_1}\ldots L_{(0,i_s)}^{p_s}((b,k))=(b+\sum_{\ell=1}^{s}p_\ell\cdot c_{{i_\ell},k},k)=(b+\sum_{\ell=1}^{s}p_\ell\cdot (c_{i_\ell-k}- c_{-k}),k).
g_{a a}= - \frac{1}{4}\frac{\Lambda^{\prime}( \lambda_{a})}{A ( \lambda_{a})}= - \frac{1}{4}\frac{\displaystyle \prod_{b \neq a , b = 1}^{N}( \lambda_{a}- \lambda_{b})}{\displaystyle \prod_{b = 1}^{N}( \lambda_{a}- r_{b})}, \g_{a b}= 0 , \forall a \neq b
aa_1af(\zeta_1,\ldots ,\zeta_n)^2 +aa_2af(\zeta_1,\ldots ,\zeta_n)aa_3af(\zeta_1,\ldots ,\zeta_n) +f(\zeta_1,\ldots ,\zeta_n)aa_5af(\zeta_1,\ldots ,\zeta_n)aa_6a-aa_5af(\zeta_1,\ldots ,\zeta_n)^2aa_6a +f(\zeta_1,\ldots ,\zeta_n)aa_4af(\zeta_1,\ldots ,\zeta_n) =0, \forall \ \zeta_1,\ldots ,\zeta_n\in \mathcal{R}.
C_{1}^{J}=\frac{1}{k_{1}}\frac{q_{1}q_{2}q_{3}}{d};C_{2}^{J}=\frac{1}{k_{2}}\frac{q_{2}q_{3}}{d};C_{3}^{J}=\frac{1}{k_{3}}\frac{q_{3}}{d}
(Z_{1/\Omega}\circ H)g(t+pT,\nu) & = \Omega\,\sum_{q=0}^{L-1}c_{-q}\,\sum_{\ell=0}^{L-1}\eta^{QP}_H(t+(q+p)T,\nu+m\Omega) & \hskip.5in e^{-2\pi i\nu qT}\,e^{-2\pi i\nu mq/L}.
\tilde{W}_1^{BC}(\beta,\alpha,y,\eta) =U(\beta,\alpha,y) +{(\alpha - 1)^2 \over 12 \alpha}\ln\left[{\eta \beta\over 2 \pi \alpha\mu}\right] +{\alpha -1 \over 24}\ln \alpha -{\alpha -5\over4}+ C ~~~,
[ \bar{C}_{\beta},{\cal T}_{\alpha}^{\dagger}] = \bar{C}_{\gamma}^{\dagger}\bar{U}_{\alpha \beta}^{\dagger \gamma}( - 1 )^{\varepsilon_{\beta}}+ \bar{U}_{\beta \alpha}^{\gamma}\bar{C}_{\gamma}( - 1 )^{\varepsilon_{\alpha}}.
\bar{I}^{A}{}_{R}I^{R}{}_{B}= \,\,{\mathbb Y}_{r}{}^A{}_B\big(v^s(\nabla_s\bar{v}^r)_o +\tfrac{1}{6}v^r(\nabla_t\bar{v}^t)-\tfrac{1}{3}v^r(\nabla_t\bar{v}^t)\big) & + \mathrm{"lower\,\, adjoint\,\, tractor \,\, slots"}
\pi_hA^\Delta_\pi v = \int_\Delta \pi_{hg}v\,d\eta(g)= \int_{h\Delta}\pi_{g}v\,d\eta(g) &= \sum_{\gamma \in \Gamma}\int_{h\Delta\cap \Delta \gamma}\pi_{g}v\,d\eta(g)=\sum_{\gamma\in\Gamma}\int_{h\Delta\gamma^{-1}\cap \Delta}\pi_{g\gamma}v d\eta(g)
\sigma \epsilon^{p - 3}= \Phi \! \! \left( \! \! \begin{array}{c c}{2}&{1}\\{1}&{3}\\ \end{array}\! \! \right) = \Phi \! \! \left( \! \! \begin{array}{c c}{p - 3}&{p - 3}\\{2}&{1}\\ \end{array}\! \! \right) = \sigma^{p - 4}\bar{\sigma}^{p - 2}
\begin{aligned}{U}_{F}^+. v =\span\bigg\{\prod_{h+1\leq b\leq m+n}E_{h,b}^{(A_{h,b})}\,\prod_{{\epsilon_a-\epsilon_b\in\Phi^+}\atop a\neq h}E_{a,b}^{(A_{a,b})}\, .v\;\bigg| \;A\in P(m|n)\bigg\}& =\span\bigg\{\bigg(\prod_{h+1\leq b\leq m+n}E_{h,b}^{(A_{h,b})}\bigg) \;v\bigg|\; A_{h,b}\in\mathbb N\bigg\}.\end{aligned}
\mathbb{G}_{j}( \mathcal{P}_{1},\ \mathcal{P}_{2},\ \mathsf{Q}) = \mathbb{G}_{j}(-\tilde{K}^{-1}(2s_{0}M+D),\ -\tilde{K}^{-1}M,\ \tilde{K}^{-1}F), = \mathbb{G}_{j}(-\tilde{K}^{-1}((2s_{0}+\alpha)M+\beta K),\ -\tilde{K}^{-1}M,\ \tilde{K}^{-1}F), & = \mathbb{K}_{j}( \mathcal{P}_1,\ \mathsf{Q}),
\widetilde{\Xi}_1(z)=f_\mathrm{Pr}\left(r\right)e^{- z{{P_S}G_\mathrm{M}^{S}G_\mathrm{M}\beta \left({\max{{\{r,d\}}}}\right)^{-{\alpha_\mathrm{LoS}}}}}+(1-f_\mathrm{Pr}\left(r\right))e^{- z{{P_S}G_\mathrm{M}^{S}G_\mathrm{M}\beta \left({\max{{\{r,d\}}}}\right)^{-{\alpha_\mathrm{NLoS}}}}}
\int_{\prod_{k=1}^rx_{d-r+k}\leq (nb)^{1/m},x_{d-r+k}\geq1}\left(b\prod_{k=1}^rx_{d-r+k}^m - n^{-1}\prod_{k=1}^rx_{d-r+k}^{2m}\right) & \quad\quad\quad\quad\quad\quad\quad\times \left(1+\sum_{j=d-r+1}^px_j^2\right)^{-1}dx_{d-r+1}\cdots dx_{d}\asymp 1.
\begin{array}{ll}Ric(U, X) = (m-n)g(\nabla_U H, X) -\sum\limits_{i=n+1}^{m}g((\nabla_{U_i}T)_U U_i, X) &+ \sum\limits_{j=1}^{n}g((\nabla_X A)_{X_j}X_j, U) - \sum\limits_{j=1}^{n}g((\nabla_{X_j}A)_X X_j, U) &- \sum\limits_{j=1}^{n}g(T_U X_j, \nu [X, X_j]),\end{array}
{\cal A}( \mu_{k}; e ) ={\frac{\epsilon}{2 4 \pi}}\int_{\Sigma^{+}}{\frac{e}{v}}\partial_{z}\bigg ( v \partial_{z}\bigg ( v \partial_{z}\partial_{\bar{z}}\bigg ({\frac{e_{k}}{v}}\bigg ) \bigg ) \bigg ) ,
r \geq |N_{F-E(F_{0})}(z^{-}_{1})\cap V(D_{2})| \geq |N_{F_{0}}[z_{2}]|-|N_{F-E(F_{0})}(z_{1})\cap N_{F_{0}}[z_{2}]| &\geq k'+1-\bigg(\bigg\lceil\frac{k'}{2}\bigg\rceil-1\bigg) &=\bigg\lfloor\frac{k'}{2}\bigg\rfloor+2=\bigg\lceil\frac{k'}{2}\bigg\rceil+1.
\left\{j_\alpha^0\left( t,\sigma \right) ,j_m^0\left( t,\sigma^{\prime}\right) \right\}\approx \left( emb^{*}S_{\alpha m}\left( \Delta\right) \right)_r\cdot \Lambda^r\cdot \delta \left( \sigma -\sigma^{\prime}\right) \quad ,
\frac{L(g,1)}{F_1(1)}=-\frac18, \frac{L(g,2)}{F_2(1)}=\frac1{32}, \frac{L(g,3)}{F_3(1)}=-\frac3{448}, \frac{L(g,4)}{F_4(1)}=\frac1{640}\qquad\frac{L(g,5)}{F_5(1)}=-\frac5{12032}.
\frac{1}{2}\frac{d}{dt}\mu( (v_N^-)^2 ) = - \mu( v_N^- \, \partial_t v_N) = - \mu(v_N^- L v_N) + \mu( a_N(t) v_N^+ v_N^-) - \mu(\sum_{i=1}^{N-1}\frac{a_i}{Z}v_i^+ \, v_N^- ).
D_{\mu \nu}= \frac{1}{p^{2}+ M^{2}}( \epsilon_{\mu \nu \lambda}p_{\lambda}+ \frac{p_{\mu}p_{\nu}}{M}+ M \delta_{\mu \nu}) ,{}D_{\mu \nu}^{- 1}= - \epsilon_{\mu \nu \lambda}p_{\lambda}+ M \delta_{\mu \nu}.
\lim_{\varepsilon\rightarrow0^{+}}\frac{1}{\varepsilon}\left\{\frac{1}{\mathcal{\varepsilon}^{N}}\int_{Q\left( x_{0},\varepsilon\right)}\left\vert u(x)-u(x_{0})-\nabla u\left( x_{0}\right) \cdot(x-x_{0})\right\vert^{\frac{N}{N-1}}dx\right\}^{\frac{N-1}{N}}=0.
|D|^{-1}[D^2,a]|D|^{-p}= \Big(F\cdot\partial(a)|D|^{-p}+\partial(a)|D|^{-p}\cdot F\Big) + \Big(-\partial(\delta(a))|D|^{-p-1}\cdot F+|D|^{-1}\cdot\partial(\delta^2(a))|D|^{-p-1}\cdot F\Big).
S_{GF+FP}= - \int d^4x i \delta_B \left[ \bar C^a \left\{D_\mu[a]A^\mu +{\alpha \over 2}B \right\}^a- i{\zeta \over 2}g f^{abi}\bar C^a \bar C^b C^i- i{\zeta \over 4}g f^{abc}C^a \bar C^b \bar C^c \right] ,
p_{j}^{*}\big(a(D)[u]\big) = p_{j}^{*}\Big( \big[a(D)u \big] \Big) = \left (\int_{|\xi|\leqslant j}|a(\xi)|^{2}\, \big| \widehat{[u]}(\xi) \big|^{2}\, d\xi \right)^{1/2}\leqslant \| a \|_{L^{\infty}(B(0,j))}\, \big\| \widehat{[u]}\big\|_{L^{2}(B(0,j))}= \| a \|_{L^{\infty}(B(0,j))}\, p_{j}^{*}\big([u]\big),
C_3\ \int^t_0\|\tau^{k}\partial^{k}_\tau f\|_{L^2}\| f\|_{L^2_A}\|\tau^k\partial^k_\tau f\|_{L^2_A}d\tau &\le \frac{1}{8}\|\tau^k\partial^k_\tau f\|^2_{L^2([0,t];L^2_A)}+2C^2_3 B^2_0\epsilon^2 |\tau^{k}\partial^{k}_\tau f\|^2_{L^\infty([0, t]; L^2)},
\sum_{p=n_{*}{\scriptscriptstyle (K)}+1}^\infty \frac{1}{\mu_{p}}\leq I\quad\textup{and}\sum_{j=n_{*}{\scriptscriptstyle (K)}}^\infty\frac{1}{\lambda_{j}\pi_j}\sum_{p=j+1}^\infty \pi_{p}\ \le \C\,K.
\zeta_{0}= - \sqrt \tau +{\frac{3}{2}}\sqrt \tau \coth^{2}\left( \sqrt{\frac{3}{2}}\tau^{1 / 4}D -{\frac{1}{2}}\log{\frac{\sqrt{\zeta + \sqrt \tau}- \sqrt{\frac{3}{2}}\tau^{1 / 4}}{\sqrt{\zeta + \sqrt \tau}+ \sqrt{\frac{3}{2}}\tau^{1 / 4}}}\right)
\nabla^{2k + 1}_{\partial_n}\partial_i = \nabla_{\partial_n}^{2k - 1}(\widehat{R}(\partial_i)) = \sum_{l = 0}^{2k - 1}\binom{2k - 1}{l}(\nabla_{\partial_n}^l \widehat{R})(\nabla_{\partial_n}^{2k - 1 - l}\partial_i)
(D_t\tilde{Y})_{u_{t_1}}=\frac{d}{dt}|_{t_1}\tilde{Y}(u_t)=\frac{d}{dt}|_{t_1}u_t^{-1}Y_{\pi u_t}= u_{t_1}^{-1}\frac{d}{ds}|_0 P_{t_1,t_1+s}^{-1}Y_{(x,t_1+s)}=u_{t_1}^{-1}(\nabla_t Y)_{(x,t_1)}=(\widetilde{\nabla_t Y})_{u_1}.
u= a_3\times (10^3-1)+a_2\times 10 \times (10^1-1) - a_1\times 10 \times (10^1-1)-a_0\times(10^3-1) = a_3\times (999)+a_2\times 10 \times (9) - a_1\times 10 \times (9)-a_0\times(999).
\dot{\cal G}_{B}( \tau_{i}, \tau_{i}) \rightarrow \dot{\cal G}_{B}( \tau_{i}, \tau_{i}) -{\cal G}_{F}( \tau_{i}, \tau_{i}) ={\frac{i}{\mathrm{s i n}( e F T ) \mathrm{c o s}( e F T )}}-{\frac{i}{e F T}}\quad .
(D^{\alpha,0}_{0+}\, f)(x) = (I_{0+}^{0}\, \frac{d}{dt}\, I_{0+}^{1-\alpha}\, f)(t)\, = \, \frac{d}{dt}\, (I_{0+}^{1-\alpha}\, f)(t)\, = \,(D^\alpha_{0+}\, f)(t),\ 0 \le \alpha < 1,
\mathbb{E}\exp\left(N\sum_{k=1}^3\mu^N_{0,k}(f_k)\right) =\mathbb{E}\left(\exp\left(N\sum_{k=1}^3\mu^N_{0,k}(f_k)\right)\mathcal{U}^N_{F,G,H}\left(T, \xi^N\right)\right) &\geq \mathbb{E}\left(\exp\left(N\sum_{k=1}^3\mu^N_{0,k}(f_k)\right)\mathcal{U}^N_{F,G,H}\left(T, \xi^N\right)1_{\{\mu^N\in C\}}\right).
\binom{5n}{4n}=\frac{(5n)!}{(4n)!(n!)}&>\frac{l(5n)}{u(4n)u(n)}&=\sqrt{\frac{5}{8\pi n}}{\left(\frac{3125}{256}\right)}^{n}e^{\frac{1}{60n+1}-\frac{1}{48n}-\frac{1}{12n}}&>0.446024n^{-\frac{1}{2}}{\left(\frac{3125}{256}\right)}^{n},
F^N(x,v)=\begin{cases}\displaystyle V'(|x|,v)\frac{x}{|x|}\mathcal{H}(x,v), |x| \ge N^{-\theta}, begin{align*}5mm]\displaystyle N^{\theta}V'(|x|,v)x \mathcal{H}(x,v), |x| < N^{-\theta}. \end{cases}
\chi_{\tau}(t,r)=-\int_0^r \frac{(B_r-B_u)}{\pi|B_r-B_u|^2}\Big( e^{-\frac{|B_r-B_u|^2}{2(i\tau +t-r)}}-e^{-\frac{|B_r-B_u|^2}{2i\tau}}\Big) du,
= \left( \delta_{\lambda \nu}\delta_{\mu \rho}- \delta_{\nu \rho}\delta_{\mu \lambda}\right){\cal G}( x ) + \left( \delta_{\mu \lambda}\partial_{\rho}\partial_{\nu}+ \delta_{\nu \rho}\partial_{\mu}\partial_{\lambda}- \delta_{\mu \rho}\partial_{\lambda}\partial_{\nu}- \delta_{\lambda \nu}\partial_{\mu}\partial_{\rho}\right) g ( x ) ,
\textrm{P}_\textrm{so}^{u}(\beta_e) = 1 - \exp \!\! \left[ - \frac{\pi^{\frac{3}{2}}\lambda_e}{2}\sqrt{\frac{\rho}{\beta_e w}}\exp \!\! \left(\! \frac{\theta^2}{4 w}\! \right) \mathrm{Erfc}\! \left( \! \frac{\theta}{2 \sqrt{w}}\! \right ) \! \right]
\rho(a_1,a_2,\ldots,a_{n_0 -1},a_{n_0}) = (a_{n_0}, a_{n_0 -1},\ldots,a_2,a_1) \mbox{\ \ \ \and}\sigma(a_1,a_2,\ldots,a_{n_0 -1},a_{n_0}) = (S- a_1,S- a_2,\ldots,S- a_{n_0 -1},S- a_{n_0}).
\begin{aligned}\nu \partial_y D^\alpha u|_{y=0}= D^\gamma \big[\big((u+u_b)\partial_x +v\partial_y\big)u-\big(h\partial_x+g\partial_y\big)h \big]|_{y=0}= \bigg[D^\gamma \left( (u+u_b)\partial_x u-h\partial_x h\right)+D^\gamma \left(v \partial_y u-g\partial_y h\right)\bigg]|_{y=0}.\end{aligned}
\sum_{k=0}^{9}c_{k}t^{k}<10^{7}(-29t+30t^{2}+20t^{3}+25t^{4}-30t^{5}-40t^{6}-35t^{7}-10t^{8}-3t^{9}) & =10^{7}t(-29+30t+20t^{2}+25t^{3}-30t^{4}-40t^{5}-35t^{6}-10t^{7}-3t^{8}) & =10^{7}t((-4+20t^{2}-30t^{4})+(-5+25t^{3}-35t^{6})-10t^{7}-3t^{8}-\eta (t)) &=10^{7}t\left(-A^{2}-5B^{2}-\frac{101}{84}-10t^{7}-3t^{8}-\eta(t)\right),
e=e_{1}^{1}e_{1}^{2}\cdot\cdot\cdot e_{1}^{n_{T}}e_{2}^{1}e_{2}^{2}\cdot\cdot\cdot e_{2}^{n_{T}}\cdot\cdot\cdot e_{T}^{1}e_{T}^{2}\cdot\cdot\cdot e_{T}^{n_{T}}.
(\lambda+c\tau)^{-n-1}\vec{P}_{n+1}\cdot\vec{\mu}=( \lambda+c\tau)^{-n}\vartheta\vec{P}_{n}\cdot\vec{\mu}+\vec{P}_{n}\cdot(\lambda+c\tau)^{-n}\vartheta \vec{\mu}-n\tau c(\lambda+c\tau)^{-n-1}\vec{P}_{n}\cdot\vec{\mu}\hphantom{(\lambda+c\tau)^{-n-1}\vec{P}_{n+1}\cdot \vec{\mu}}{}=\vartheta\big[ (\lambda+c\tau)^{-n}\vec{P}_{n}\cdot\vec{\mu}\big] .
\int_{0}^{T}\left\langle \partial_{t}u\left( t\right) ,u\left( t\right)\right\rangle_{H^{-1}(\Omega ),H_{0}^{1}(\Omega )}dt +\int_{\Omega_{T}}\left( \int_{\mathcal{Y}_{n,m}}a_{0}\left(x,t,y^{n},s^{m}\right) dy^{n}ds^{m}\right) \cdot \nabla u\left( x,t\right)dxdt =\int_{\Omega_{T}}f\left( x,t\right) u\left( x,t\right) dxdt
0 = H(\emptyset) 0 \le H(A) = H(A|\emptyset) 0 \le H(A|B) 0 \le I(A;B) H(A,B|C) \le H(A|C) + H(B|C) H(A|B,C) \le H(A|B) \leq H(A,C|B) I(A;B,C) = I(A;B|C) + I(A;C).
M_{1, 1}(\delta, \nu) \leq C_{1}^{2}\nu^{-O(1)}D(\frac{\delta}{\nu^{2^{N - 1}}})^{\frac{1}{3 \cdot 2^N}}D(\frac{\delta}{\nu^{2^N}})^{\frac{2}{3 \cdot 2^N}}\prod_{j = 0}^{N - 1}D(\frac{\delta}{\nu^{2^j}})^{\frac{1}{2^{j + 1}}}
( \kappa_{4})^{4}U_{O M}= \frac{3 | h |^{2}}{4 d J^{2}}\Bigg \{\beta \Big [ e^{- J \beta x}- e^{- J \beta ( 1 - x )}\Big ]^{2}+ \frac{2}{3}\frac{J}{V}\beta^{2}\Big [ e^{- J \beta x}x + e^{- J \beta ( 1 - x )}( 1 - x ) \Big ]^{2}\Bigg \}\; .
\left \|\log \|A_n x\|-\log \|A_n y\| \right \|_r^r \leq D \sum_{k=1}^n \left ( \|X_{k, \overline x}-X_{k, \overline y}\|_r^r + \||X_{k, \overline x}-X_{k, \overline y}|^{r-1}\|_{1}\right ) \, ,
{\Gamma^{i}}_{kl}=\frac{1}{2}g^{im}(\frac{\partial g_{mk}}{\partial x^{l}}+\frac{\partial g_{ml}}{\partial x^{k}}-\frac{\partial g_{kl}}{\partial x^{m}})=\frac{1}{2}g^{im}(g_{mk,l}+g_{ml,k}-g_{kl,m}),
\eta_{(j_1-1)2^{{r\over 2}-1}+k}= \theta_{j_1}+ \lambda_k^+, \eta_{m_12^{{r\over 2}-1}+(j_2-1)2^{{r\over 2}-1}+k}= \theta_{j_2}' + \lambda_k^-, \eta_{m_12^{{r\over 2}-1}+[{m_2\over2}]2^{{r\over 2}-1}+l}= \lambda_l^- ,
R_N^{k+1}= \gamma \left(\sum\limits_{i=1}^{N-1}A_i x_i^{k+1}+ x_N^{k+1}- b\right), R_{\lambda}^{k+1}= b - \sum\limits_{i=1}^{N-1}A_i x_i^{k+1}- x_N^{k+1}
\epsilon^{\mu \nu}={\frac{1}{2}}e ( e^{\mu{[ + 2 ]}}e^{\nu{[ - 2 ]}}- e^{\mu{[ - 2 ]}}e^{\nu{[ + 2 ]}}) , \qquad \left( \epsilon^{{0 1}}= - \epsilon_{{0 1}}= 1 \right) ,
\sum_{j \geq 0}V_{1,j}=1 , V_{n,k}=\sum_{j=0}^{\infty}\binom{k+j}{j}\left(\left(b\right)_{n\uparrow\tau}\right)^{j}\left(a+b+\tau\left(n-1\right)\right)^{k}V_{n+1,k+j}.
\mathfrak{i}(q)~ :=(2\pi\hbar)^{n/2}\int_{T_{q}^{\ast}M}e^{ig^{jk}(q)p_{j}p_{k}/2\sigma\hbar~}\psi(\pi\circ\exp_{x}(p)) & \qquad\qquad\qquad\qquad\times\left( \sqrt{\frac{\det(g_{jk}(\exp_{x}(p))}{\det(g_{jk}(x))}}\frac{d^{n}x\wedge d^{n}(\exp_{x}(p))}{i^{n}d^{n}x\wedge d^{n}p}\right)^{1/2}d^{n}p.
H_1 \leq C\Big( |\phi|^p_{\frac{2n}{n+2}}+|U_{\mu_i,\xi_i}^{p-2}\phi^2|_{\frac{2n}{n+2}}\Big) \leq C \Big( |\phi|^{p-2}_{\frac{2n}{n-2}}+|U_{\mu_i,\xi_i}|_{p-2}^{p-2}\Big)|\phi|^2_{\frac{2n}{n-2}}= C \Big( |\phi|^{p-2}+1\Big)\|\phi\|^2.
\sum_{z\in \mathbb{Z}^{d}}\left\vert f\left( z\right) \right\vert=\left\vert f\left( 0\right) \right\vert +\sum_{n=1}^{d}\sum_{\mathcal{I}\in2^{\left\{1,\dots ,d\right\}}:\left\vert \mathcal{I}\right\vert =n}sum_{z\in \mathbb{Z}^{n}\backslash \left\{0\right\}}\left\vert f\circ \xi_{\mathcal{I}}\left( z\right) \right\vert \ .
\|\tilde x_i-\tilde y_i\| \le K_2\mu (\mu +\tilde g(\mu))\frac{c^i-1}{c-1}\le \frac{K_2}{K_1}(\mu +\tilde g(\mu)) \mathrm{e}^{i\,K_1\mu}\le \frac{K_2}{K_1}(\mu +\tilde g(\mu))\mathrm{e}^{K_1}.
\int_{t_{0}}^{\infty}r_{1}^{-\frac{1}{\alpha}}(v)\left(\int_{t_{0}}^{v}r_{2}^{-\frac{1}{\beta}}(u)\left(\int_{t_{0}}^{u}q(s)s\right)^{\frac{1}{\beta}}u\right)^{\frac{1}{\alpha}}v = \infty ,
\Vert 1_A - P_t 1_A \Vert_1 = \int_A (1-P_t 1_A) d\mu + \int_{A^c}P_t(1_A) d\mu = \int_A (1-P_t 1_A) d\mu + \int_A (P_t 1_{A^c}) d\mu = 2 \left( \mu(A)- \int_A P_t (1_A) d\mu \right) = 2 \left( \mu(A)- \Vert P_\frac{t}{2}(1_A) \Vert_2^2 \right).
D_{T}^{[ r , 0 ]}( k^{2}) = \frac{k^{2}+ u_{r , 2}\Lambda^{2}}{( k^{2}+ \sigma_{r , 1}\Lambda^{2}) ( k^{2}+ \sigma_{r , 3}\Lambda^{2})}\prod_{s = 1}^{( r - 1 ) / 2}\frac{( k^{2}+ u_{r , 4 s}\Lambda^{2}) ( k^{2}+ u_{r , 4 s + 2}\Lambda^{2})}{( k^{2}+ \sigma_{r , 4 s + 1}\Lambda^{2}) ( k^{2}+ \sigma_{r , 4 s + 3}\Lambda^{2})}\, ,
b(a_0 \otimes \cdots \otimes a_n) = \sum_{i=0}^{n-1}(-1)^i a_0 \otimes \cdots \otimes a_i a_{i+1}\otimes \cdots \otimes a_n + (-1)^n a_n a_0 \otimes a_1 \otimes \cdots \otimes a_{n-1},
\frac{d}{d t}\Bigl ( \dot{q}_{i}+ \lambda^{2}q_{i}\Bigr ) = - \biggl [ \lambda^{2}+ \frac{2 \tilde{T}_{2}( 2 \tilde{T}_{1}+ 1 )}{( q^{2}+ 2 \eta )^{2}}\biggr ] ( \dot{q}_{i}+ \lambda^{2}q_{i}) - \frac{1}{q^{2}}( \dot{q}_{k}+ \lambda^{2}q_{k})^{2}\, q_{i}\, .
k_{ij}(C,\tilde C)=\min \{\dfrac{m_{j}.q_{1}\ldots q_{i}+l_{i}.n_{1}.\ldots n_{j}}{2.l_{i}.n_{1}.\ldots n_{j}},\dfrac{m_{j}.q_{1}\ldots q_{i}+l_{i}.n_{1}.\ldots n_{j}}{2.m_{j}.q_{1}\ldots q_{i}}\}.
\bar{F}_{p + 1}= \sum_{i_{k}, j_{k}}\frac{d}{d x^{+}}\bar{A}_{i_{1}\cdots i_{p}}( - 4 x^{+}) Q_{j_{1}}^{i_{1}}\cdots Q_{j_{p}}^{i_{p}}d x^{+}\wedge d x^{j_{1}}\wedge \cdots \wedge d x^{j_{p}}.
\mathbf E[Y_n(T)|\mathcal{F}_S] =\mathbb E_0\Big[\mathbf E\big[\exp\big\{H_{\beta,T}(W,B)\big\}\big|\mathcal{F}_S\big]I_n(T,W_T)\Big] &=\mathbb E_0\Big[\exp\big\{H_{\beta,S}(W,B)\big\}\mathbb E_0\big[I_n(T,W_T)\big|\mathcal{G}_S\big]\Big] &=Y_n(S).
t^{-\frac12}\tau^\vee_i = t^{-\frac12}T_i + \frac{(1-t^{-1})Y_iY^{-1}_{i+1}}{1-Y_iY^{-1}_{i+1}}\qquad\hbox{and}t^{\frac12}\tau^\vee_i = t^{\frac12}T^{-1}_i + \frac{(1-t)Y^{-1}_iY_{i+1}}{1-Y^{-1}_iY_{i+1}}.
{D}^{( 1 / 2 )}({\bf p},{\bf n}) = \frac{p n + m - i{\vec{\sigma}}\cdot ({\bf p}{\times}{\bf n})}{\sqrt{2 ( p_{0}+ m ) ( p n )}}, \quad{D}^{\dag ( 1 / 2 )}({\bf p},{\bf n}){D}^{( 1 / 2 )}({\bf p},{\bf n}) = 1 ,
\mathbb{E}_{\rho,\phi}\left[W_p(\Theta_t)\right] = \mathbb{E}_{\rho,\phi}\left[W_p(\Theta_t){\bf 1}_{\{T_0 > t\}}\right] + \mathbb{E}_{\rho,\phi}\left[W_p(\Theta_t){\bf 1}_{\{T_0 < t\}}\right] & = \mathbb{P}_{\rho,\phi}(T_0 > t) + \mathbb{E}_{\rho,\phi}\left[W_p(\Theta_t){\bf 1}_{\{T_0 < t\}}\right]
{\hat{Q}}_{\pm}^{B C}= \int d x^{1}\sum_{a , b}\left[ f^{B}\right]^{a b}\left[ ( \partial_{0}\phi^{a}\mp \partial_{1}\phi^{a}) \psi_{\pm}^{b}\pm \sum_{c}\left[ f^{C}\right]^{b c}\frac{\partial W^{C}}{\partial \phi^{c}}\psi_{\mp}^{a}\right]
w'(\theta)=\left.\frac{V^{\tfrac{1}{m_1}-1}(x)}{m_1}\left(\frac{1-m_1}{m_1V(x)}\Big(\sum\limits_{i=1}^n\frac{\partial V(x)}{\partial x_i}y_i\Big)^2+\sum_{i,j=1}^n\frac{\partial^2 V(x)}{\partial x_i\partial x_j}y_iy_j\right)\right|_{x=x^0+\theta y}.
0 = \frac{1}{2\pi}\varepsilon L_a\tilde{v}_\varepsilon + a\varepsilon^2\frac{\tilde{u}_j(t')}{\lambda_j - \lambda_{j,\varepsilon}}\int_{\mathbb{D}}\tilde{u}_j(s')\tilde{v}_\varepsilon(s')e^{\tilde{\phi}(s')}ds' + \mathcal{R}_\varepsilon \tilde{v}_\varepsilon + \mathcal{R}^{\lambda_j,\lambda_{j,\varepsilon}}\tilde{v}.
\Phi(z,\vec x,t)\,=\,\sum_{p=1}^\infty \,\int{d \vec k \over (2\pi)^{n-1}}\,{z^{n/2}\,J_\nu (u_p z ) \over R w_p(\vec k ) \,J_{\nu\,+\,1}(u_p R )}\lbrace{{\bf a}_p(\vec k )\}e^{-iw_p(\vec k ) t +i\vec k \cdot \vec x}\,\,+\,\,c.c.\rbrace\,,
P(\boldsymbol{\varphi}) = \max_{\sigma \in \mathcal{A}}\, \sup_{\mathbf{P}_{\!\sigma}\in \mathcal{Q}^{\sigma}}\, \sum_{n=1}^{d}C_{n}^{(d), \sigma}(\mathbf{P}_{\!\sigma}) \cdot\bigg(H(\mathbf{p}_{\sigma_{n}}) + \int\! \varphi_n^{\sigma}\,\mathrm{d}\mathbf{p}_{\sigma_n}\bigg).
Op_h(a)(v_1, \dotsc, v_n) = \frac{1}{(2\pi h)^n}\int_{\mathbb{R}^n}e^{\frac{i}{h}x(\xi_1 + \dots + \xi_n)}a(x, \xi_1, \dotsc, \xi_n)\prod_{l=1}^n \hat{v}_j(\xi_l)\, d\xi_1 \dots d\xi_n \, .
\mathcal{I}_1^*(h,k) = J_1+J_2+J_3 + O\big( X^{-\frac{1}{2}+\varepsilon}Q^{\frac{5}{2}}(hk)^{\varepsilon}\big) + O\big( X^{\varepsilon}Q^{\frac{3}{2}+\varepsilon}(hk)^{\varepsilon}\big).
Z_{\mu}= \frac{( x - z )_{\mu}}{( x - z )^{2}}- \frac{( y - z )_{\mu}}{( y - z )^{2}}\; \mathrm{,}\hspace{0 . 8 c m}Z^{2}= \frac{( x - y )^{2}}{( x - z )^{2}( y - z )^{2}}\; \mathrm{.}
\phi_-(\Gamma)=- \frac{dh}{h}\wedge \left( \xi_\Gamma +\sum_{N\geq 1}(-1)^N\sum_{\gamma_N\subset \cdots \subset \gamma_1 \subset \gamma_0=\Gamma}\xi_{\gamma_N}\wedge \bigwedge_{j=1}^N \eta_{\gamma_{j-1}/\gamma_j}\right),
I(\gamma_{u,v}(t))= \frac{t^{2\alpha}}{2}\big(\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}\big)+\frac{t^{2(\alpha-1)}}{2}\big(\|u\|_{2}^{2}+\omega\|v\|_{2}^{2}\big) &+\frac{t^{6\alpha-4}}{2}\Big(B(u)+B(v)\Big)-\frac{t^{2p\alpha-2}}{2p}F(u,v).
\frac{\Phi_{\pmb{\phi}_p}(\alpha_n)}{||\pmb{\phi}_p||^2}= \chi_{\overline{\psi}_p}(p^n)p^{-|n|/2}, \frac{\Phi_{\pmb{\phi}_p}(\beta_m)}{||\pmb{\phi}_p||^2}= \chi_{\overline{\psi}_p}(p^m)p^{-|m|/2}.
p(z) = (z-b)(a-b)^{-1}f(a) +(z-a)(b-a)^{-1}f(b) \cr = \left(b(b-a)^{-1}f(a) a(a-b)^{-1}f(b)\right)+ z \left((a-b)^{-1}f(a) -a(b-a)^{-1}f(b)\right).
E_{j}\triangleright (E_{\xi_{i}}^{*}E_{\xi_{i}}) = \delta_{j, i}q^{-2}E_{\xi_{i + 1}}^{*}E_{\xi_{i}}- \delta_{j,i - 1}(K_{j}^{-1}\triangleright E_{\xi_{i}})^{*}E_{\xi_{i-1}}.