harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\{\tilde{\Theta}_{\alpha}( q , \phi ) , \tilde{\Theta}_{\beta}( q , \phi ) \}= 0 ~ ~ ; ~ ~ \tilde{\Theta}_{\alpha}( q , \phi ) = \Theta_{\alpha}( q ) + \Sigma_{n = 1}^{\infty}\tilde{\Theta}_{\alpha}^{( n )}( q , \phi ) ~ ~ ; ~ ~ \tilde{\Theta}^{( n )}\approx O ( \phi^{n})
\begin{aligned}\qquad\,\,\mbox{there are}\,\, \beta>2\,\, \mbox{and}\,\, r_0>0\,\, \mbox{such that}& tf(x,t)\geq \beta F(x,t)>0\,\, \mbox{for any}\,\, |t|\geq r_0\,\,\mbox{and every}\,\, x\in D,\end{aligned}
\frac{1}{2\pi}c^{(b)}: e^{\mp i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi +gN}}U_{1 b}\Phi^{(1)}(x)}:_{M^{(1)}}\; \prod_{I=2}^N: e^{\mp i 2 \sqrt{\pi}U_{I b}\Phi^{(I)}(x)}:_{M^{(I)}}e^{\pm i \frac{\theta}{N}}\; ,
u(x,t) = \sum_{k=1}^{\infty}E_{\alpha,1}(-\lambda_{k}t^{\alpha})u_{0,k}\phi_{k}(x) & + \sum_{k=1}^{\infty}\int_{0}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{k}(t-s)^{\alpha})f_{k}(s)ds\phi_{k}(x),
\sum_{t=0}^{p}r^t \begin{bmatrix}p+1 t+1 \end{bmatrix}=p! \left[ \binom{r+p+1}{p+1}- 1 \right] - \sum_{j=1}^{r-1}\sum_{t=0}^{p}j^t \begin{bmatrix}p+1 t+1 \end{bmatrix},
A_{ah}^{-3/2}\, \left( \begin{array}{cc}\cos \left( \frac{\sqrt{7}}{2}\ln (A_{ah}) \right) \sin \left( \frac{\sqrt{7}}{2}\ln (A_{ah}) \right) - \sin \left( \frac{\sqrt{7}}{2}\ln (A_{ah}) \right) \cos \left( \frac{\sqrt{7}}{2}\ln (A_{ah}) \right) \end{array}\right)\,{\cal A}^{-1}\,\left( \begin{array}{c}a b \end{array}\right)\, .
\|\{f_j\}_{j\in\mathbb Z}\|_{L\dot F_{pq}(E\times J)}:=\|\{f_j\}_{j\in\mathbb Z}\|_{L^p\ell^q(E\times J)}:=\|\{f_j\}_{j\in\mathbb Z}\|_{L^p(E;\ell^q(J))}:=\left\|\left(\sum_{j\in J}|f_j|^q\right)^{\frac{1}{q}}\right\|_{L^p(E)}
\begin{aligned}dE(u)v =\Bigl(\int_0^1v, \frac 12 \int_0^1\Bigl\{\Big\langle \int_0^s u, Av(s) \Big\rangle+ \Big\langle \int_0^s v,A u(s)\Big\rangle\Bigr\}ds\Bigr) & =\Bigl(\int_0^1v,\; \int_0^1\Big\langle A\Bigl(\frac x2 -\int_0^s u \Bigr), v(s)\Big\rangle ds \Bigr).\end{aligned}
I_{0}(u,\mu)={1\over \Gamma({1\over2})}\mu^{{1\over2}}\left\{\Gamma(-{1\over 4})^{2}F(-{1\over4},-{1\over4};{1\over2};{u^{2}\over 4\mu^{2}})-{u\over \mu}\Gamma({1\over 4})^{2}F({1\over4},{1\over4};{3\over2};{u^{2}\over 4\mu^{2}})\right\}.
R ( u , \lambda ) = \sum_{i = 1}^{n}E_{i , i}\otimes E_{i , i}+ \sum_{i \neq j}\alpha ( u , \lambda_{i j}) E_{i , i}\otimes E_{j , j}+ \sum_{i \neq j}\beta ( u , \lambda_{i j}) E_{i , j}\otimes E_{j , i},
\underbrace{\left[\begin{matrix}|b_1^1|^2 \cdots |b_1^n|^2 \vdots \vdots |b_q^1|^2 \cdots |b_q^n|^2\end{matrix}\right]}_B \left[\begin{matrix}d_1 \vdots d_n\end{matrix}\right] = \left[\begin{matrix}0 \vdots 0\end{matrix}\right]
\Delta_{r , s}= \frac{( P^{\prime}r - P s )^{2}- ( P^{\prime}- P )^{2}}{4 N P^{\prime}P}= \frac{[ ( N + N^{\prime}+ 2 ) r - ( N^{\prime}+ 2 ) s ]^{2}- N^{2}}{4 N ( N + N^{\prime}+ 2 ) ( N^{\prime}+ 2 )}.
\mathcal{M}_1=\sqrt{\frac{\pi}{2}}\,e^{\frac{\psi_{1}^2\tau_{\rm d}^2}{2N_{\rm a}^2}}\left[\mathrm{erf}\left(\frac{\kappa_{\rm t}+\frac{\psi_{1}\tau_{\rm d}}{N_{\rm a}}}{\sqrt{2}}\right)-\mathrm{erf}\left(\frac{\kappa_{\rm b}+\frac{\psi_{1}\tau_{\rm d}}{N_{\rm a}}}{\sqrt{2}}\right)\right].
\delta \tilde{\lambda}( t , s , \{\vec{k}\}) = [ i \delta ( 0 ) ] ~ \frac{3}{2}a ( t )^{- 1}\tilde{\lambda}^{2}( t ) ( 2 \pi )^{- 3}\int d^{3}\vec{q}[ \vec{q}^{2}+ \tilde{m}^{2}( t ) - i \epsilon ]^{- 1}
\begin{matrix}\sum(x-m)^{2}&=&\sum(x^{2}&-&2xm&+&m^{2})\\ &=&\sum x^{2}&-&\sum2xm&+&\sum m^{2}\\ &=&\sum x^{2}&-&2m\sum x&+&Nm^{2}\\ &=&\sum x^{2}&-&2Nm^{2}&+&Nm^{2}\\ &=&\sum x^{2}&-&Nm^{2}.\end{matrix}
\begin{array}{c}{R_{i j}= R_{i}{}^{k}{}_{j k}, R_{i a}= P_{i}{}^{k}{}_{a k}= - P_{i a},}{R_{a i}= P_{a}{}^{b}{}_{i b}= P_{a i}, R_{a b}= S_{a}{}^{c}{}_{b c}= S_{a b}}\end{array}
\mu^{2 - 2 \omega}\int \frac{d^{2 \omega}k}{\left( 2 \pi \right)^{2 \omega}}\sum_{n = - \infty}^{\infty}\left( k^{2}+ \frac{1}{L^{2}}\left( 2 \pi n + \chi \right)^{2}\right)^{1 / 2}= - \frac{\pi^{2}}{4 5 L^{3}}\left[ 1 - \frac{1 5}{8}\left( \left( \frac{\chi}{\pi}- 1 \right)^{2}- 1 \right)^{2}\right]
\int_{\Omega}\langle TT_{\omega}x,TT_{\omega}x\rangle_{\mathcal{A}}d\mu(\omega)=\int_{\Omega}\langle T_{\omega}Tx,T_{\omega}Tx\rangle_{\mathcal{A}}d\mu(\omega) \leq B\langle Tx,Tx\rangle_{\mathcal{A}}&=B\|T\|^{2}\langle x,x\rangle_{\mathcal{A}}
\quad\mathbb{P}[\bigcap_{m=1,l=1}^{M,L}\{(U,S_m,T_l)\notin \mathcal{F}\}] &\le\mathbb{P}[\{(U,S,T)\notin\mathcal{F}\}\cup\{\imath_{S;T|U}(S;T|U)>\log(ML)-2\gamma\}] &\quad+\frac{\min\{M,L\}-1}{ML(\exp(-\gamma)-\exp(-2\gamma))}+e^{-\exp(\gamma)}.
w_{kl}=\max(|\mathrm{INR}_{kl}^{RR}-\mathrm{INR}_{kl}^{LR}|,|\mathrm{INR}_{kl}^{LL}-\mathrm{INR}_{kl}^{RL}|, |\mathrm{INR}_{lk}^{RR}-\mathrm{INR}_{lk}^{LR}|,|\mathrm{INR}_{lk}^{LL}-\mathrm{INR}_{lk}^{RL}|).
\Psi_{k_{1}, k_{1 z}; k_{2}, k_{2 z}; I , I_{z}}^{r_{1}, r_{2}}( \theta , \alpha , \beta , \gamma , \alpha_{I}, \beta_{I}, \gamma_{I}) = \sqrt{\frac{p !}{m_{1}! m_{2}! m_{3}! m_{4}!}}\Psi_{1}^{m_{1}}\Psi_{2}^{m_{2}}\Psi_{3}^{m_{3}}\Psi_{4}^{m_{4}},
\rho_1(c(x,a),(y,b)) =\rho_1((cx,ca),(y,b))=\rho(cx,y) &=(-1)^{\bar c\bar x}\rho(x,cy)=(-1)^{\bar c\bar x}\rho_1((x,a), (cy,cb))=(-1)^{\bar c\bar x}\rho_1((x,a), c(y,b)).
\begin{aligned}V\in L^{\infty}(0,T_{0};H^{s}), v_{2}\in L^{2}(0,T_{0};H^{s+1}), V_{t}\in L^{2}(0,T_{0};H^{s-1}), V(x,t)\in\mathcal{O}_{g_{2}}\quad\forall(x,t)\in Q_{T_{0}},\end{aligned}
x^5R^{(5)}(x) +10x^4R^{(4)}(x)+x^3(20x^2-20qx+22+10\tilde{p})R{'''}(x) + x^2(64x^2-76qx+4+44\tilde{p})R{''}(x) + 4x(16x^4-32qx^3+4(4q^2+4\tilde{p}+1)x^2-q(6+16\tilde{p})x+4\tilde{p}^2+7\tilde{p}-1)R{'}(x) + 8(-4qx^3+4(q^2+\tilde{p})x^2-q(6\tilde{p}-1)x+2\tilde{p}^2)R(x)=0.
\begin{aligned}g(t) \geq \frac{\alpha c_{0}\zeta}{2 B(\alpha)}E_{\alpha,\alpha}\big[-2\gamma \big] \int_{-2}^{t}h(s)ds \geq \frac{\alpha c_{0}\zeta}{2 B(\alpha)}E_{\alpha,\alpha}\big[-2c_{1}\big]c_{0}\mu & \geq \frac{\alpha}{2}E_{\alpha,\alpha}\big[-2c_{1}\big]c_{0}\mu.\end{aligned}
Y(z)=\begin{pmatrix}p_n(z) \frac{1}{2\pi i}\int_{-\infty}^{\infty}p_n(s)e^{-V(s)}\frac{ds}{s-z}-2\pi ih_{n-1}^{-1}p_{n-1}(z) -h_{n-1}^{-1}\int_{-\infty}^{\infty}p_{n-1}(s)e^{-V(s)}\frac{ds}{s-z}\end{pmatrix}.
\begin{aligned}\sum_{k=0}^{m}(-1)^k \binom{\zeta}{k}\binom{c-k}{c-m}=\sum_{k=0}^{m}(-1)^k \binom{\zeta}{k}\binom{c-k}{m-k}= (-1)^m \sum_{k=0}^{m}\binom{\zeta}{k}\binom{-(c-m+1)}{m-k}&= (-1)^m \binom{\zeta-c+m-1}{m-k}= \binom{c-\zeta}{m}.\end{aligned}
(\frac{1}{\alpha}-\Delta t \delta_{xx}) (\frac{1}{\alpha}-\Delta t \delta_{yy}) u^{k+1}_{i,j}= (\frac{1}{\alpha}+\Delta t \delta_{yy}) \frac{1}{\alpha}u^k_{i,j}- (\frac{1}{\alpha}-\Delta t \delta_{xx}) \Delta t \delta_{yy}u^k_{i,j}+ \frac{\Delta t}{\alpha^2}f^{k+1}_{i,j}.
0=\zeta_{l_0,t}-D_{l_0}\Delta \zeta_{l_0}=f_{l_0}(v)-f_{l_0}(u) &> f_{l_0}(\cdots,v_{j_0-1},u_{j_0},v_{j_0+1},\cdots)-f_{l_0}(u) &\geq f_{l_0}(\cdots,u_{l_0-1},v_{l_0},u_{l_0+1},\cdots)-f_{l_0}(u) &\geq-M\zeta_{l_0}=0\ \{\rm at}\ \ (x,t)=(x_0,t_0).
u =c_3x-c_4y-c_1+c_7xy+\frac{1}{6}(3y^2e^{-c_{10}}-x^2e^{3c_{10}}), v =c_2-c_6y+c_5x-c_8xy+\frac{1}{2}(x^2e^{c_{9}}+y^2e^{c_{9}-4c_{10}}).
M_{n,k}=\frac{1}{n}+\ldots+\frac{1}{n-k+1}&=\frac{1}{n}\left(1+\frac{1}{1-1/n}+\ldots+\frac{1}{1-(k-1)/n}\right) &=\frac{1}{n}\left(k+\frac{k(k-1)}{2n}+{\rm o}(1/n)\right)
\frac{c_{b}}{2}\int_{0}^{\infty}\sin(t \xi) \xi J_{0}(r \xi) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}d\xi = \frac{-b}{t^{2}\log^{b}(t)}+ E_{\partial_{r}v_{2},1}(t,r)
\Theta_{n-1}([q,(1,1)]_f) =[a,(b,c)]_g &=[q\xi_n(x),(b,c)]_g &=[q\xi_n(x),(b,c)\nabla^g_n(x)\nabla^g_n(x)\i]_g &=[q,(b,c)\nabla^g_n(x)]_g[\xi_n(x\i)\i,\nabla^g_n(x\i)]_g &=[q,(1,1)]_g[1,(bp_n(x),cg_n(x))]_g
\frac{1}{\kappa_K}\sum_{\mathrm{N}\mathfrak{n}< x}\frac{1}{\mathrm{N}\mathfrak{n}}\geq \log x - \sum_{j=1}^{n_K}\frac{1}{j}+ \gamma_K + O_{\epsilon}\left( x^{-\tfrac{\epsilon}{8}}\log x \right),
U_{1}(\phi_{0},a_{1},...,a_{d})=\mu^{D}\sqrt{a_{1}\cdots a_{d}}\sum_{s=1}^{\infty}f(D,d,s)g^{s}\phi_{0}^{2s}A_{d}^{c^{2}}(s-\frac{D-d}{2};a_{1},...,a_{d}),
{S_{L}}\left[ \varphi , \hat{g}\right] = -{\frac{d - 2 6}{4 8 \pi}}\int{d^{2}}\xi \sqrt{\hat{g}}\left({\frac{1}{2}}\varphi \hat{\Delta}\varphi + \hat{R}\varphi \right) +{\mu_{1}^{2}}\int{d^{2}}\sqrt{\hat{g}}{e^{\varphi}},
\begin{array}{l l}{{Z = \frac{1}{2}(}}&{{\mathrm{T r \thinspace ~}e^{- \beta ( H^{( + )}- \mu \hat{N})}+ \mathrm{T r \thinspace ~}e^{- \beta ( H^{( - )}- \mu \hat{N})}+}}\\{{}}&{{\mathrm{T r \thinspace ~}\tau e^{- \beta ( H^{( + )}- \mu \hat{N})}- \mathrm{T r \thinspace}\tau e^{- \beta ( H^{( - )}- \mu \hat{N})})}}\\ \end{array}
e^{- i \frac{\theta_{0}}{\hbar}J^{2}}J^{\pm}e^{i \frac{\theta_{0}}{\hbar}J^{2}}= e^{- i \frac{\theta_{0}}{\hbar}J^{2}}J^{1}e^{i \frac{\theta_{0}}{\hbar}J^{2}}\pm i J^{2}= \cos \theta_{0}\: J^{1}- \sin \theta_{0}\: J^{3}\pm i J^{2}. \nonumber
S_{1,}= q^{2g+1}\sqrt{q}\frac{1}{(2 \pi i)^3}\oint \oint \oint x^g \sum_{\substack{n=0 n}}^{4g}\frac{1}{q^n w^{n+1}x^{n/2}}\sum_{i=0}^{y}\frac{1}{q^{2i}u^{i+1}x^i}\sum_{\substack{r=0 r}}^{n-2g-2+2i}x^{r/2}\times \sum_{V_0 \in \mathcal{H}_r}\mathcal{F}(V_0;w,x,u) \, du \,dw \, dx,
R_{0}( \theta ) = \prod_{l = 1}^{\infty}\left[ \frac{\Gamma ( 4 l \lambda + 2 i \lambda \theta / \pi ) \Gamma ( 4 ( l - 1 ) \lambda + 1 + 2 i \lambda \theta / \pi )}{\Gamma ( ( 4 l - 3 ) \lambda + 2 i \lambda \theta / \pi ) \Gamma ( ( 4 l - 1 ) \lambda + 1 + 2 i \lambda \theta / \pi )}/ ( \theta \to - \theta ) \right]
L_{E_i(T)}(S-R_{E_j(S)}) b = (S-R_{E_j(S)})L_{E_i(T)}b = (S-R_{E_j(S)})E_i(T) b, R_{E_j(S)}(T-L_{E_i(T)})b = (T-L_{E_i(T)}) R_{E_j(S)}b = (T-L_{E_i(T)})bE_j(S),
\frac{c^{-\nu}_{2r-2}}{c^{-\nu}_{2r}}=\frac{1}{\displaystyle h^{-2}[s^2-(-\nu+2r-2)^2]-\frac{1}{\displaystyle h^{-2}[s^2-(-\nu+2r-4)^2]-\frac{1}{\displaystyle h^{-2}[s^2-(-\nu+2r-6)^2]\cdot\cdot\cdot}}}
f_{\varLambda_i}(\varLambda_i)=\frac{1}{m_{c,i}}\mathrm{exp}\left(-\frac{\varLambda_i+m_{s,i}}{m_{c,i}}\right)I_0\left(\frac{2\sqrt{\varLambda_im_{s,i}}}{m_{c,i}}\right),
R_{(a_1,a_2, \dots , a_k )^{\downarrow}}^{n}= \binom{n - \sum_{j=1}^{k}a_{j}+ k - 1}{k-1}= \binom{n - \sum_{j=1}^{k}a_{j}+ k - 1}{n - \sum_{j=1}^{k}a_{j}}\,.
\frac{d}{dt}E(t)=-\nu\int_{\mathbb{R}^d}|\nabla u|^2\,dx-\Gamma \lambda\int_{\mathbb{R}^d}\textrm{tr}\left(L\Delta Q-aQ+b[Q^2-\frac{\textrm{tr}(Q^2)}{d}Id]-cQ\textrm{tr}(Q^2)\right)^2\,dx\le 0
F = \widehat{f}_{\mathcal{U}_0^+}= \left( \bigoplus_{U\in\mathcal{U}_0}\widehat{f}_U \right) \oplus \left( \mathrm{1}_X - \sum_{V \in \mathcal{U}_0}\widehat{f}_{V}\right) : Y \to |\mathcal{N}(\mathcal{U}_0^+) | \subset | C Z |,
\Gamma_{V D}[ \phi ] = S [ \phi ] - \frac{1}{2}\mathrm{T r \l n}\left[ G^{l i}\left( S_{, i j}- \left\{\begin{array}{c}{{k}}\\{{i \j}}\\ \end{array}\right\}S_{, k}- T_{i j}^{k}S_{, k}\right) \right] \ ,
(T\widetilde{f}-Tf)^2 = (T\widetilde{f}-Tf_1)^2 + [2T\widetilde{f}-Tf_1-Tf](Tf_1-Tf) & \leq (T\widetilde{f}-Tf_1)^2 + 4B_n|Tf_1-Tf| & \leq (\widetilde{f}-f_1)^2 + 4B_n|f_1-f|
x_1 = x_1(e^{\tau(m,x_1)}\cdot m) = \int_0^{\tau(m,x_1)}\frac{d}{dt}(x_1(e^t\cdot m))dt = \int_0^{\tau(m,x_1)}|T|^2(e^t\cdot m))dt \geq C^{-1}\int_0^{\tau(m,x_1)}\rho^2(e^t\cdot m)dt \geq C^{-1}\int_0^{\tau(m,x_1)}\rho^2(m)dt = C^{-1}\rho^2(m)\tau(m,x_1).
-B^{2}P_{i}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{\widehat{m}_{j}(P_{j}-\sigma(\langle P_{i},P_{j}\rangle+\sigma z^{2})P_{i})}{(\sigma-\sigma(\langle P_{i},P_{j}\rangle+\sigma z^{2})^{2})^{\frac{3}{2}}}-\sigma(\dot{p}_{i}\odot\dot{p}_{i})P_{i},\textrm{}i\in\{1,...,n\},
F \equiv ( F_{\mu \nu}) = \left( \begin{array}{c c c c}{0}&{E_{1}}&{E_{2}}&{E_{3}}\\{- E_{1}}&{0}&{B_{3}}&{- B_{2}}\\{- E_{2}}&{- B_{3}}&{0}&{B_{1}}\\{- E_{3}}&{B_{2}}&{- B_{1}}&{0}\\ \end{array}\right) .
\langle M^{n,f},M^{n,f}\rangle_{t}=\frac{\sigma_{n}^{2}}{n^{3}}\int_{0}^{t}\sum_{m=1}^{n}(f^{\prime}(\lambda_{m}^{(n)}(X_{s-}^{(n)})))^{2}ds & +\frac{1}{n}\int_{0}^{t}\int_{\mathbb{S}(\mathbb{C}_{n})}\int_{\mathbb{R}}\left(\mathrm{tr}\left[f(X_{s-}^{(n)}+rvv^{\ast})-f(X_{s-}^{(n)})\right] \right)^{2}\rho_{n}^{\alpha}(dr)\pi(dv)ds.
2(1+\eta) r =4\eta r+2(r-\eta r)< \sum_{j\le r}c^{(i)}_j=\sum_{j\le r}a_{i,j}+\sum_{i'<i}a_{i'}+ \sum_{j\le r}c_j\le \sum_{i'\le i}a_{i'}+ \sum_{j\le r}c_j
\partial_{1}\left( \frac{h_{2}h_{3}}{h_{1}}\right) = \partial_{2}\left( \frac{A_{1}h_{2}h_{3}}{h_{1}}\right) , \partial_{3}\left( \frac{h_{2}h_{1}}{h_{3}}\right) = \partial_{2}\left( \frac{A_{3}h_{2}h_{1}}{h_{3}}\right) .
D^{2}\Phi\left( T,T\right) =TT\Phi-\left( \nabla_{T}T\right) \Phi & =TT\Phi-\left( \nabla_{T}^{\Sigma}T\right) \Phi+\Pi\left( T,T\right)\chi & =D^{2}\Phi\left( T,T\right) +\Pi\left( T,T\right) \chi.
\begin{array}[c]{lll}S\cdot1=S S\cdot T=ST S\cdot S^{2}T=(-S-1)T S\cdot S=S^{2}S\cdot T^{2}=ST^{2}S\cdot ST^{2}=S^{2}T^{2}S\cdot S^{2}=-S-1 S\cdot ST=S^{2}T S\cdot S^{2}T^{2}=(-S-1)T^{2}\end{array}
S_{i n t}= \frac{k}{2}\int d^{4}x [ G^{a}V^{a}]_{D}= \frac{k}{2}\int d^{4}x \; T r \left\{C^{a}D^{a}- A_{\mu}^{a}v^{a \mu}- \bar{\lambda}{}^{a}\chi^{a}\right\}.
\tau^{\ast}= \lim\limits_{P \to \infty}\lim\limits_{F \to \infty}\frac{T}{F/\log P}\geq \max\limits_{s\in[3]}\max\left\{\frac{1}{N}(1-s\mu_R),\frac{s}{3M}(1-s\mu_R)\right\}&=\max\left\{\frac{1}{N}(1-\mu_R),\max\limits_{s\in[3]}\frac{s}{3M}(1-s\mu_R)\right\},
R = 2n \frac{a^2}{b^4}- \frac{2n(2n+2)}{b^2}+ 4n \frac{a'b'}{ab}+ 2n(2n-1)\left(\frac{b'}{b}\right)^2 -2f'\left(\frac{a'}{a}+ 2n \frac{b'}{b}\right).
P(E)= \mathcal{H}^{n-1}( \partial^{*}F_{\ell}\cap \{z < \bar{z}\}) + \mathcal{H}^{n-1}( \partial^{*}E \cap \{z = \bar{z}\}) + \mathcal{H}^{n-1}( \partial^{*}F_{\ell}\cap \{z > \bar{z}\}).
\Vert (B^0+\lambda_0 \overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow H^2(\mathbb{R}^d)}=\sup \limits_{\boldsymbol{\xi}\in\mathbb{R}^d}(\vert \boldsymbol{\xi}\vert^2+1)\vert L(\boldsymbol{\xi})^{-1}\vert \leqslant \check{c}_*^{-1}.
\mathcal{I}:=\mathcal{I}_{\Delta}=\mathbf{1}\{\max_{1 \leq j \leq p}|\mathbf{Z}_j-\widetilde{\mathbf{Z}}_j|<\Delta, |\max_{1 \leq j \leq p}|\mathbf{U}_j-\widetilde{\mathbf{U}}_j|<\Delta\},
s_{\mkern-1mu+}:= \min\left\{k \mkern-1mu\in\mkern-1mu{\mathbb{Z}}_{>0}\mkern1mu|\mkern3mu y_1^{\textsc{d}}k, \ldots, \mkern-1mu y_{n_{{\textsc{d}}}}^{\textsc{d}}\mkern-2mu k,\mkern1muy_{1\mkern-.5mu +}^{\textsc{g}}\mkern-.5mu k, \ldots,\mkern-1mu y_{n_{{\textsc{g}}}\mkern-.5mu +}^{\textsc{g}}\mkern-.5mu k \in{\mathbb{Z}}\right\}
\frac{\partial^2 g\circ f}{\partial x_l^2}= \sum_{i,j=1}^3\frac{\partial^2 g}{\partial f_i \partial f_j}\left(\frac{\partial f_i}{\partial x_l}\frac{\partial f_j}{\partial x_l}\right) +\sum_{i=1}^3\frac{\partial g}{\partial f_i}\frac{\partial^2 f_i}{\partial x_l^2}
R_c( R_p) \geq \ell \left(\frac{\Delta e^{R_p}}{2\Delta e^{R_p}-1+\rho}\right) &=\frac{1}{2}\log^+ \frac{N^2(X,Y)}{(1-\rho) \left( 2 \Delta e^{R_p}-1+\rho \right)},
\frac{d}{dt}\Big[(3+t)E(u;t)\Big] =\|u'\|^2+\|A^{1/2}u\|^2-2(3+t)\|u'\|^2, \frac{d}{dt}E_*(u;t) =2\|u'\|^2+2(u,u''+u') &=2\|u'\|^2-2\|A^{1/2}u\|^2.
\begin{array}{rcl}H[n,x(n,m,\xi)] = \sum\limits_{k=-\infty}^{+\infty}G(n,k+1)g(k,H[k,x(k,m,\xi)]) & -\sum\limits_{k=-\infty}^{+\infty}G(n,k+1)f(k,x(k,m,\xi))+x(n,m,\xi).\end{array}
\widehat\Lambda_n -\Lambda_n= \left(\frac{1}{N}\sum_{i=1}^nR_{n,i}X_{n,i}Z_{n,i}^{\prime}\right)\left(\frac{1}{N}\sum_{i=1}^nR_{n,i}Z_{n,i}Z_{n,i}^{\prime}\right)^{-1}
\begin{aligned}I_{1}''+I_{1}'''+I_{4}= \delta^{2}\left(-\frac12\int_{\mathbb{R}_{+}^{n}}|\nabla v_{q}|^{2}dzdt+\frac n2\int_{\partial\mathbb{R}_{+}^{n}}U^{\frac{2}{n-2}}v_{q}^{2}dz\right) +o(\delta^{2}) &= \frac12\delta^{2}\int_{\mathbb{R}_{+}^{n}}\Delta v_{q}v_qdzdt+o(\delta^{2})\end{aligned}
R(t) = \begin{bmatrix}R_1(t)_{k \times k}R_2(t)_{k \times (n-k)}R_3(t)_{(n-k) \times k}R_4(t)_{(n-k) \times (n-k)}\end{bmatrix}, G(t) = \begin{bmatrix}G_1(t)_{k \times k}G_2(t)_{k \times (n-k)}G_3(t)_{(n-k) \times k}G_4(t)_{(n-k) \times (n-k)}\end{bmatrix},
W_{\pi}(g_{t,l,v}) = \delta_{\Delta \in F^{2\times}}q^{\frac{\Delta}{4}}\sum_{\pm}\gamma_F(-1\pm \sqrt{\Delta},\rho)\gamma_F(\Delta \pm \sqrt{\Delta},l-\frac{v(\Delta)}{2})\chi^2(-\frac{1}{2v}\pm\frac{\sqrt{\Delta}}{2v}) \psi(\varpi^{-\frac{n}{2}}(\frac{\Delta-3}{4v})),
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{\binom kj\binom{k-j}s\left({\frac{{f_2}}{{f_1}}}\right)^j \left({\frac{{f_3}}{{f_1}}}\right)^s X_{m - ak - (b - a)j - (c - a)s}}}= \frac{{X_m}}{{f_1^k}}\,,
m\epsilon_{\mu\alpha\beta}\partial^{\alpha}\hat A^{\beta}\approx \epsilon_{\mu\alpha\beta}\partial^{\alpha}(\hat f^{\beta}-m\hat a^{\beta})-\epsilon_{\nu\alpha\beta}(m\partial^{\alpha}\hat A^{\beta}-\partial^{\alpha}\hat f^{\beta})\frac{\delta \hat a^{\nu}}{\delta \hat A^{\mu}}.
h_k(\underbrace{x_A,\l,x_A}_{\hbox{$m$ times,}}, \underbrace{x_B,\l,x_B}_{\hbox{$m$ times}})= \sum_{r=0}^k{r+m-1 \choose r}{k-r+m-1\choose k-r}x_1^r x_2^{k-r}.
g(y, \alpha, \sigma) \sim \frac{|S^{d-2}|}{(2\pi \sigma^{1/\alpha})^{d}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}a_{k}\left(\frac{|y|}{\sigma^{1/\alpha}}\right)^{2k},
\begin{aligned}\int_{\Omega}{\int_{\Omega}{\frac{\overline{u}(x) -\overline{u}(y)}{|x-y|^{n+2s}}(\overline{u}(x)-\hat{u}(x) -\overline{u}(y)+\hat{u}(y))\,dx\,dy}}& =\int_{\Omega}{\int_{\Omega}{\frac{|\overline{u}(x) -\overline{u}(y)|^2}{|x-y|^{n+2s}}\,dx\,dy}}\ge 0\end{aligned}
U_k^N(s,x) =U(0,x)-\alpha\int_0^{s}\int_0^1 A(x,y)U_{k-1}(u\wedge \tau_N,y) 1_{(0,\tau_N)}(u) \mathrm{d}y\mathrm{d}u +\alpha\beta\int_0^s 1_{(0,\tau_N)}(u)\mathrm{d}\xi_2(u\wedge \tau_N,x)+\alpha\zeta\int_0^s 1_{(0,\tau_N)}(u)\mathrm{d}\xi_3(u\wedge \tau_N,x).
|X^{0,u}_x - X^0_x|_{C([0,T_0]:E)}\leq \kappa_3 |Y^{0,u}_x - Y^0_x|_{C([0,T_0]:E)}= \kappa_3|Y^{0,u}_x|_{C([0,T_0]:E)}\leq \kappa_4 T_0^{\frac{1}{p}}|u|_{L^2([0,T_0]:H)}
{\mathcal O}_{\widehat{\mathfrak Y}}(k)|_{\widehat{\mathfrak U}_{\{Z_{\gamma_n}\}}}=\widehat{\pi}^{kn}{\mathcal O}_{\widehat{{\mathfrak U}}_{\{Z_{\gamma_n}\}}}\quad\mbox{inside}\quad{\mathcal O}_{\widehat{{\mathfrak U}}_{\{Z_{\gamma_n}\}}}\otimes_{{\mathcal O}_{\widehat{K}}}\widehat{K}.
\frac{\partial_v^2}{\partial_v^2-k^2}f(v)=f(v)+\int_{\mathbb{R}}c_1|k|e^{-|k||v-w|}f(w)\,\mathrm{d}w,\frac{\partial_v}{\partial_v^2-k^2}f(v)=\int_{\mathbb{R}}c_2\mathrm{sgn}(v-w)\cdot e^{-|k||v-w|}f(w)\,\mathrm{d}w,
y_i = C_1\frac{\theta_p\big(a x_i^{\pm 1}\big)}{\theta_p\big(b x_i^{\pm 1}\big)},z_i = C_2\frac{\theta_{p^{1/3}}\big(c x_i^{\pm 1}\big)}{\theta_{p^{1/3}}\big(d x_i^{\pm 1}\big)},
\alpha + \omega^{l+1}\cdot(1+\mu_1)+\rho ~ = ~ \alpha_i+\omega^{l+1}\cdot\mu_0+\sigma+ \omega^{l+1}+ \omega^{l+1}\cdot\mu_1+\rho = ~ \alpha_i+\omega^{l+1}\cdot(\mu_0+ 1 + \mu_1)+\rho ~ = ~ \alpha_{i+1},
H_1^1 =Y^3-Z^3 H_1^2 =Y^2Z H_2^1 =XH_1^1+YZ^3 H_2^2 =XH_1^2 H_3^1 =XH_2^1-Y^2Z^3 H_3^2 =XH_2^2 H_4^1 =XH_3^1+Y^3Z^3-4Z^6 H_4^2 =XH_3^2 H_5^1 =XH_4^1+Y^7-Y^4Z^3+4YZ^6 H_5^2 =XH_4^2.
\Delta \mathcal{O}^2 ={}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \mathcal{O}^2\vert J_1,\gamma_1; J_2,\gamma_2\rangle_q -\left({}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \mathcal{O}\vert J_1,\gamma_1; J_2,\gamma_2\rangle_q\right)^2.
u^n_h =\sum_{j=1}^m U^n_{j}\varphi_j, \eta_{n,{\kappa}}=\frac{\lambda_{n,{\kappa}}+B_{\kappa}u^n_h}{x^{\bar{p}_{\kappa}-2}+r}, \lambda^{n+1}_h =\lambda^n_h +r(B u^n_h- \eta^n_h).
\mathbb{E}(\nu_i)= \mathbb{E}\left(\sum_{\substack{1\leq j\leq t \psi(E_N^{(j,M)})\in\mathcal{P}_i}}1\right)= t \cdot \mathbb{E}\left(\sum_{\substack{G\in\Phi_M \psi(G)\in\mathcal{P}_i}}1\right)=t\pi_i.
T(\overline{x}(t)+h)- T(\overline{x}(t)) - \langle \overline{p}(t),h \rangle =T(x_h(t))-T(\overline{x}(t))- \langle \overline{p}(t),h \rangle &\geq T(x_h(0))- T(x_0)- \langle \overline{p}(t),h \rangle .
\begin{aligned}\Big(\tilde{\Omega}^{(1)}_{\mathcal{P}}(t_*) \tilde{v}, \tilde{v}\Big)_{\mathbb{C}^n}= 2 \Big( (X_1 (t_*)^t Y_1' (t_*) - Y_1^t (t_*) X_1' (t_*)) M_1 (t_*)^2 \Big{\{}X_1 (t_*)^t v_1 + Y_1 (t_*)^t v_2 \Big{\}}, & M_1 (t_*)^2 \Big{\{}X_1 (t_*)^t v_1 + Y_1 (t_*)^t v_2 \Big{\}}\Big)_{\mathbb{C}^n},\end{aligned}
\Omega_{1}( E ) = \frac{V \pi^{- 5 1 / 2}}{\Gamma \left( \frac{2 5}{2}\right) 2^{3 8}}\sum_{r = 1}^{\infty}\sum_{k = 0}^{\infty}r^{- 2 6}a_{k}^{2}E \left( E^{2}- \frac{4 r^{2}k}{\alpha^{\prime}}\right)^{2 3 / 2}{\cal H}\left( E^{2}- \frac{4 r^{2}k}{\alpha^{\prime}}\right) \vspace{4 m m}
h(Y^n_2|\mathbf{V}^n_2) \geq\frac{n}{2}\log\Big(e^{\frac{2}{n}h(Y^n_1|\mathbf{V}^n_2)}+e^{\frac{2}{n}h(Z^n)}\Big) &=\frac{n}{2}\log\Big(e^{\frac{2}{n}h(Y^n_1|\mathbf{V}^n_2)}+2\pi e(N_2-N_1)\Big),
u(t,x) =U_L+(U_M-U_L)(H(x-c_1 t)-H(x-ct)) &+(-U_M-U_L)(H(x-c t)-H(x-c_2t))+(U_R-U_L)H(x-c_2t) v(t,x) =V_L+(V_M-V_L) (H(x-c_1 t)-H(x-ct)) &+(V_M-V_L)(H(x-c t)-H(x-c_2t))+(V_R-V_L)H(x-c_2t) &+\beta_1(t) \delta(x-c_1t)+\beta_2(t) \delta(x-c_2t),
Q_n(s)=\! \begin{bmatrix}e_M(\theta)1_{\{1\in J_r^{[s]}\}}\!+\!(f_{g((s-1)\ell_r^*+1)}-\alpha_n)\tilde{Z}_{s,1}\vdots e_M(\theta)1_{\{\ell_r^*\in J_r^{[s]}\}}\!+\!(f_{g(s\ell_r^*)}-\alpha_n)\tilde{Z}_{s,\ell_r^*}\end{bmatrix},
a =\left(1, x_1, 0, \tfrac{117}{4}- x_1, \tfrac{117}{4}, \tfrac{117}{4}+ x_1, \tfrac{117}{2}, \tfrac{117}{2},\tfrac{117}{2}, \tfrac{351}{4}-x_1\right), aQ =(352, 0, 0, 0, 0, 128 x_1,0,0,0, 32 (117 - 4 x_1)).
\eta = \frac{z_{1 2}z_{3 4}}{z_{1 3}z_{2 4}}\qquad 1 - \eta = \frac{z_{1 4}z_{2 3}}{z_{1 3}z_{2 4}}\qquad \frac{\eta}{1 - \eta}= \frac{z_{1 2}z_{3 4}}{z_{2 3}z_{1 4}}
\Gamma_{\rho_{\bullet}}:= \pi (\mathcal{G}_{\rho_{\bullet}}) \subseteq \Gamma, \mathcal{E}_{\rho_{\bullet}}:=i^{-1}(\mathcal{G}_{\rho_{\bullet}}) = i^{-1}(\ker(\pi|_{\mathcal{G}_{\rho_{\bullet}}}))
\Phi'=\left(\bigoplus_{l=0}^{r-1}h^l \circ{}^t\Gamma_{\pi}\right) \oplus\gamma'\circ{}^t\Gamma_{\pi}: \left( \bigoplus_{l=0}^{r-1}\mathfrak{h}(Y)(l)\right)\oplus \mathfrak{h}(Y)(r) \longrightarrow \mathfrak{h}(E)
S_{\mathrm{g r a v i p h o t o n}}= \int d^{4}x \sqrt{- g}\left[ - \frac{1}{8}g^{\mu \rho}g^{\nu \sigma}F_{\mu \nu}^{( \alpha )}F_{\rho \sigma}^{( \alpha )}\right] \int_{- \infty}^{\infty}d z e^{- 2 k | z |},
-\langle\frac{dE}{dx}\rangle=\frac{4\pi}{m_{e}c^{2}}\cdot\frac{nz^{2}}{\beta^{2}}\cdot(\frac{e^{2}}{4\pi\epsilon_{0}})^{2}\cdot[ln(\frac{2m_{e}c^{2}\beta^{2}}{I\cdot(1-\beta^{2})})-\beta^{2}]
g_n = \sum_{z\in\Lambda_n}\delta A^n_z\varphi^n_z, \delta\xi_n = \sum_{z\in\Lambda_n}\Big(\sum_{u\in\Lambda_{n+1}}A^{n+1}_u\langle\varphi^{n+1}_u,\psi^n_z\rangle\Big)\,\psi^n_z.
V^{\prime}( x ) = n^{2}-{n ( n - 1 )}/{\cosh^{2}({\frac{\sqrt{2 m}x}{\hbar}})}\qquad \mathrm{~ t o ~ i t s ~ c o n t i g u o u s}\qquad V ( x ) = n^{2}-{n ( n + 1 )}/{\cosh^{2}({\frac{\sqrt{2 m}x}{\hbar}})},
\aligned |F(X_{k+1},(k+1)\Delta t)|^2 =[2\langle X_k,f(X_k,k\Delta t)\rangle+|g(X_{k},k\Delta t)|^2+(1-2\theta)|f(X_{k},k\Delta t)|^2\Delta t]\Delta t &\quad+|F(X_{k},k\Delta t)|^2+M_k,\endaligned
\mathrm{mod}_p(\rho)= \mathrm{Im}\sum_{\{\alpha,\beta;\epsilon\}\in\triangle^+(\rho)}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho)+\ell_{\beta}(\rho))}+(-1)^{\alpha\cdot\beta}\right)^{-1}=- \mathrm{Im}\sum_{\{\alpha,\beta;\epsilon\}\in\triangle^-(\rho)}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho)+\ell_{\beta}(\rho))}+(-1)^{\alpha\cdot\beta}\right)^{-1}.
\begin{matrix}{{T_{c}/ m ( 0 ) = 1 / 2 \ln 2 ,}}&{{\mathrm{i f}\}}&{{\mu_{c}= 0}}\\{{\mu_{c}/ m ( 0 ) = 1 ,}}&{{\mathrm{i f}\}}&{{T_{c}= 0}}\\ \end{matrix}
(A\sharp B)(t,x,y):=\int^t_0\int_XA(t-s,x,z)B(s,z,y)dv_X(z)ds &(\in C^\infty(\mathbb{R}_+\times X\times X, (T^{*0,+}X\otimes E)\boxtimes(T^{*0,+}X\otimes E)^*)).
\begin{cases}\lambda_1 = \sqrt{\frac{s_{\pm}(a)s_{\pm}(c)}{s_{\pm}(b)}}\lambda_2 = \sqrt{\frac{s_{\pm}(a)s_{\pm}(b)}{s_{\pm}(c)}}\lambda_3 = \sqrt{\frac{s_{\pm}(b)s_{\pm}(c)}{s_{\pm}(a)}}\end{cases}
|F_{4}(\lambda(s)^{1-\alpha},s-t)-F_{4}(\lambda(t)^{1-\alpha},s-t)| \leq C \frac{\lambda(t)^{1-\alpha}}{(s-t)^{4}}\cdot |\lambda(t)^{1-\alpha}-\lambda(s)^{1-\alpha}| &\leq C \frac{\lambda(t)^{1-2\alpha}}{(s-t)^{3}}\cdot \sup_{x \geq t}\left(\frac{|\lambda'(x)| \lambda(t)^{\alpha}}{\lambda(x)^{\alpha}}\right)
\mathrm{d}X(t)=\left[\begin{array}{c}1 \mu P(t)\end{array}\right]\mathrm{d}t+\left[\begin{array}{c}0 \sigma P(t)\end{array}\right]\mathrm{d}B(t),\;\,X(0)=\left[\begin{array}{c}s p\end{array}\right].
\prod_{j=1}^{N}(x_{j,1}B^{m}+x_{j,0})=\sum_{i=1}^{2^{N+1}-1}\prod_{j=1}^{N}x_{j,c(i,j)}B^{m\sum_{j=1}^{N}c(i,j)}=\sum_{j=0}^{N}z_{j}B^{jm}
\left ( \prod_{i=1}^mY_i^{- \frac{2n_i}{n}}\right )^{-1}\frac{d \bar{\lambda}(Y(u))}{du}\leq - \left ( \frac{1}{\sqrt{n}}\left ( \sum_{i=1}^m n_iY_i^2 (2 p_i - q_i^2Y_i)^2 \right )^{1/2}-\sqrt{\frac{n-1}{2n}}E(Y) \right )^2 \leq 0.
\left\langle N^{(\alpha_1)}_{\nu_{\alpha_1}},\cdots,N^{(\alpha_n)}_{\nu_{\alpha_n}}\right\vert \left.N^{(\alpha_1)}_{\nu_{\alpha_1}},\dots,N^{(\alpha_n)}_{\nu_{\alpha_n}}\right\rangle={\cal A}^{\{{\cal N}\}}\,.