harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_0 = \frac{1}{12}\int_{S^2}W_0^2 dS^2 -\frac{1}{60}\int_{S^2}|\alpha|^2 dS^2 + \frac{11}{45}\int_{S^2}|\beta|^2 dS^2 = \frac{1}{12}\int_{S^2}W_0^2 dS^2 + \frac{1}{9}\int_{S^2}|\beta|^2 dS^2.
S_{\mathrm{I}}= \frac{1}{1 6 \pi \alpha^{\prime 4}}\int d^{1 0}x \sqrt{| g_{1 0}|}\left[ e^{- \Phi}\left( R_{1 0}+ \left( \nabla \Phi \right)^{2}\right) - \frac{1}{1 2}H_{3}^{2}- \frac{1}{4}e^{- \Phi / 2}F_{2}^{2}\right]
f_{\phi, t}(g, o) = \left( \frac{mn}{d^2}\right)^t\sum_{o_1, \dots, o_{2t}}\left(\prod_{s=0}^{t-1}M_{\iota(o_{2s}), \iota(o_{2s+1})}M_{\iota(o_{2s+2}), \iota(o_{2s+1})}\right) \phi_{\iota(o_{2t})}
M_{k}:=\max_{\left\vert \sigma \right\vert \leq k,\sigma \in \mathcal{P}_{n}}\sup_{0\leq s<t\leq T}\sup_{D,D\subset \left[ s,t\right]}\left\Vert\sigma \left( \beta_{s,t}^{D}\right) \right\Vert \leq C_{n,\theta ,M,\omega\left( 0,T\right)}
W_{1}=\left\Vert \widetilde{p}_{0t}\right\Vert_{H^{2,1}\left( S_{T}\right)}+\left\Vert \widetilde{p}_{1t}\right\Vert_{H^{1,0}\left( S_{T}\right)}+\left\Vert \widetilde{q}_{0t}\right\Vert_{H^{2,1}\left( S_{T}\right)}+\left\Vert \widetilde{q}_{1t}\right\Vert_{H^{1,0}\left( S_{T}\right)}.
M_{n_1,n_2}^d(w_1,w_2) =k^{2(n_1+n_2-1)-d}\int_0^{\infty}dxx^{d\over{2}}\frac{\partial}{\partial{t}}\left\{\dot{P}^{2}(P+w_1)^{-n_1}(\tilde{P}+w_2)^{-n_2}\right\}
\prod_{j=1}^W(x_{j,1}-x_{j,2})^2(b_{j,1}+b_{j,2})^2 =\frac{(-a_+^2)^W}{M^W}(a_+-a_-)^{2W}\Big(1+O\Big{(}\frac{\Theta^{\frac32}}{\sqrt{M}}\Big{)}\Big) &\times\prod_{j=1}^W\Big{(}\mathring{x}_{j,1}-\mathring{x}_{j,2}+O\Big{(}\frac{\Theta}{\sqrt{M}}\Big{)}\Big{)}^2.
\begin{array}{c}{{\displaystyle \displaystyle \sum_{j = 1}^{M}\phi ( \Theta - \vartheta_{j}, 1 / 2 ) = \displaystyle \int_{a s y m p}^{\Theta}d x \, W ( x )}}\\{{\displaystyle \sum_{j = 1}^{M}\phi ( \Theta + \vartheta_{j}, 1 / 2 ) = - \displaystyle \sum_{j = 1}^{M}\phi ( - \Theta - \vartheta_{j}, 1 / 2 ) = - \displaystyle \int_{a s y m p}^{- \Theta}d x \, W ( x ) .}}\\ \end{array}
F_2(q^2)={Me^2\over 4\pi^2}\int_0^1d\alpha\,\int_0^1dx\,\,{m-xM\over\alpha (1-\alpha)\,{1-x\over x}\,{\vec q}_{\perp}^2-M^2+{m^2\over x}+{\lambda^2\over 1-x}}\ .
K(L,r) = 2^{n-2}\int_0^{\alpha_0(r)}(\cos\theta)^{n-2}\log\bigl(\dfrac{1}{1-r^2}\bigr)+\cdots = 2^{n-2}\int_0^{\pi/2}(\cos\theta)^{n-2}\log\bigl(\dfrac{1}{1-r^2}\bigr)+\cdots =2^{n-3}\sqrt{\pi}\; \frac{\Gamma(\frac{n-1}2)}{\Gamma(\frac n2)}\log\bigl(\dfrac{1}{1-r^2}\bigr)+\cdots
\textnormal{Res}_{k=\eta}T_{\eta,1}=\frac{3}{4\left(D_{0}^{-2}r^{-1}(\eta)-1\right)}\begin{pmatrix}\tilde{T}_{\eta,1}^{(11)}\tilde{T}_{\eta,1}^{(12)}\tilde{T}_{\eta,1}^{(21)}\tilde{T}_{\eta,1}^{(22)}\end{pmatrix}
\prod_{i=1}^{r-1}yt^{p_i}\prod_{\substack{j=1 j \neq l}}^r xt^{-q_j}\left(ht^{p_r - q_l}\,v\right) = (-1)^{r-1}\sum_{\pi^\prime \vdash r-1}C(\pi^\prime)\,\mathcal{H}(\pi^\prime; p^\prime, q^\prime) \,ht^{p_r - q_l}\,v.
|{\rm I}-{\rm I}'| \leq \sum_{\ell=0}^L\sum_{x\in E_n}\left(\frac{2+\lambda/N_n}{\lambda/N_n}\cdot \frac{1}{N_n^2}\right)\cdot \frac{2^\ell|Q^{(n),\ell}(x,x)-Q^{(\infty),\ell}|}{(\lambda/N_n+2)^{\ell+1}}&\leq \frac{1}{\lambda}\sum_{\ell=0}^L\int_{E_n}|Q^{(n),\ell}(x,x)-Q^{(\infty),\ell}| \pi^{(n)}(\d x)
\lim_{\beta\to0}\int_{0}^{\infty}\sum_{n=M_{j}}^{\infty}\mathcal{T}_{j,Y_{j}}\left(n,y,U,T_{1},T_{2},\beta\right)f_{Y_{j}}(y){\rm d}y=\int_{0}^{\infty}\sum_{n=M_{j}}^{\infty}\lim_{\beta\to0}\mathcal{T}_{j,Y_{j}}\left(n,y,U,T_{1},T_{2},\beta\right)f_{Y_{j}}(y){\rm d}y\;.
\mathcal{E}(q,\lambda(q,t))=\frac{-1+e^{\lambda(q,t)\cdot (1-q)}}{1-qe^{\lambda(q,t)\cdot(1-q)}}=\frac{e^{\lambda(q,t)}-e^{\lambda(q,t)\cdot q}}{e^{\lambda(q,t)\cdot q}-qe^{\lambda(q,t)}};
\sum_{k=rd}^{\infty}C_{r,d}\frac{k^{rd+1}}{rd+1}\exp\left(- C_{r,d}\frac{k^{rd+1}}{rd+1}\right) &\leq 2/e + \int_{0}^{\infty}C_{r,d}\frac{z^{rd+1}}{rd+1}\exp\left(- C_{r,d}\frac{z^{rd+1}}{rd+1}\right) dz.
v_i(\lambda) - v_{i+1}(\lambda) + \# A_{\lambda,i}^{l}- \# R_{\mu,i}^{l}&= \# \{b \mid 1 \leq b < a,\j_{b}= i \}+ \# \{b \mid a \leq b \leq Nd,\j_{b}= i+1 \}- d & \equiv \# \{b \mid 1 \leq b < a,\j_{b}= i \}+ \# \{b \mid a < b \leq Nd,\j_{b}= i+1 \}
\begin{array}{rl}\hat{\Gamma}(k_1,k_2,k_3) = \displaystyle (2\pi)^4\delta^4(\sum_ik_i)\sum_{\{\lambda_j\}}\prod_{j=1}^3\left[\eta_{\lambda_j}(k_j)\right] &\displaystyle\times\sum_{\pi_3}\exp[-i(k_{\pi(1)\lambda_{\pi(1)}},k_{\pi(2)\lambda_{\pi(2)}},k_{\pi(3)\lambda_{\pi(3)}})]. \end{array}
\|\tilde{K}_0\| = \big\||V|^{1/2}H_0^{-1}|V|^{1/2}\big\| \leq \big\||V|^{1/2}H_0^{-1/2}\big\| \big\|H_0^{-1/2}|V|^{1/2}\big\| \leq a \,.
\begin{array}{rclrcl}\hat{b}_{5}= v^{(0)}=0\, , \hat{b}_{i}= b_{i}\, ,\hspace{1cm}& \hat{a}^{(2)}{}_{i5}= b_{i}+v^{(1)}{}_{i}\, , \hat{a}^{(2)}{}_{ij}= a^{(2)\prime}{}_{ij}\, , \end{array}
|\varphi_Z(\theta)|= |\varphi_S(\theta) \varphi_Y(\theta)|\leq|\varphi_S(\theta)|= \prod_{j=1}^{b_t}|\varphi_{V_j}(\theta)|=\prod_{j=1}^{b_t}|\mathbb{E}(e^{i\theta V_j})|\leq (1-\eta)^{b_t}\leq e^{-\eta b_t}\leq \frac{1}{e\eta b_t},
2(-1)^{|a||b|}f(\{a,b\}) =(-1)^{|a||b|}g(a)f(b)+(-1)^{|a||b|}f(a)g(b)-g(b)f(a)-f(b)g(a) &=(-1)^{|a||b|}f(a)g(b)-g(b)f(a)+(-1)^{|a||b|}(-1)^{|a||b|}f(\{a,b\})
\tilde{Q}_n= \begin{bmatrix}\frac{1}{f_{1}-\alpha_n}e_M(\theta)1_{\{1\in J_w\}}+\hat{Z}_{1}\vdots \frac{1}{f_{y}-\alpha_n}e_M(\theta)1_{\{y\in J_w\}}+\hat{Z}_{y}\end{bmatrix},
C_{\ell(\lambda_1), \xi_1}= \sum \limits_{(\Lambda^{\prime}, A^{\prime}, \Xi^{\prime})}\left ( \prod_{j=2}^q m(\lambda_j) \sum \limits_{(d^j_k)_{k=1}^{\ell(\lambda_j)}}\prod \limits_{k=1}^{\ell(\lambda_j)}s(b_k^j , d_k^j) \right ).
K_{z}=K_{3}=i\frac{\partial\hat{B}(\varphi,\hat{e}_{z})}{\partial\varphi}|_{\varphi=0}=i(\begin{matrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{matrix}).
\dot{c}_{m}(t)+(\frac{i}{\hbar}E_{m}(t)+\langle m(t)|\dot{m}(t)\rangle)c_{m}(t)=\sum_{n\ne m}\frac{\langle m(t)|\dot{H}|n(t)\rangle}{E_{m}(t)-E_{n}(t)}c_{n}(t).
\begin{aligned}f_1(x_1)-f_1(x^{k+1}_1)+(x_1-x_1^{k+1})^T[-A_1^T\lambda^k &+\beta^kA_1^T(A_1{x}_1^{k+1}+A_2\hat{x}_2^k-b)]\geq0,~\forall x_1.\end{aligned}
T_j \big( \prod_{i=1}^{n}(x_i+ \hat{x}_i) (y_i+ \hat{y}_i)(z_i+ \hat{z}_i) (w_i+ \hat{w}_i) \big) = \ (x_j+\hat{x}_j+ z_j) \prod_{k\neq j}(x_k+\hat{x}_k) \times \prod_{i=1}^{n}(y_i+ \hat{y}_i)(z_i+ \hat{z}_i) (w_i+ \hat{w}_i)
\det\limits_{x,y \in \mathbb{T}_{m_1,m_2}}K^{\alpha,\beta,\gamma}_{m_1,m_2,\theta}(x,y) = \prod_{j \in \mathbb{T}_{m_1,m_2}}\left( \alpha + \beta \exp \left\{2 \pi i \frac{j_1 + \theta_1 /2}{m_1}\right\}+ \gamma \exp \left\{2 \pi i \frac{j_2 + \theta_2 /2}{m_2}\right\}\right).
| B_{m n}^{( + )}| \leq ( n + m )^{- k}\sqrt{{\frac{n}{m}}}s u p \{|{\frac{d^{k + 1}\varphi_{t}}{d y^{k + 1}}}| : 0 \leq y \leq 2 \pi , 0 \leq t \leq 1 \}.
a(x^\perp,t)=\left(\frac{\sinh(\sqrt{3\Lambda}t/2m_3)}{\sinh(\sqrt{3\Lambda}t_0/2m_3)}\right)^{\frac{2}{3}}\left[1+\sqrt{\frac{\Lambda}{3}}\coth\!\left(\frac{\sqrt{3\Lambda}}{2m_3}t\right)\frac{x^\perp}{m_3}\right],
L_{1}^{\ast}=\frac{c_{1}\left[\xi_{p}^{2}\overline{G}_{(1)}\left(\frac{1}{2}\xi_{p}^{2}\right)-\xi_{q}^{2}\overline{G}_{(1)}\left(\frac{1}{2}\xi_{q}^{2}\right)\right]}{F_{Y}(\xi_{p},\xi_{q})},
g = \int_{S_{\infty}^{2}}d \vec{\sigma}\cdot \vec{B}= \frac{1}{2 e v^{3}}\int_{S_{\infty}^{2}}d^{2}\sigma^{i}\varepsilon_{i j k}\varepsilon^{a b c}\Phi_{a}\partial^{j}\Phi_{b}\partial^{k}\Phi_{c}= \frac{4 \pi n}{e},
F[x,y,z]=\int_{a}^{t}\sqrt[3]{z^{\prime\prime\prime}(x^{\prime}y^{\prime\prime}-y^{\prime}x^{\prime\prime})+z^{\prime\prime}(x^{\prime\prime\prime}y^{\prime}-x^{\prime}y^{\prime\prime\prime})+z^{\prime}(x^{\prime\prime}y^{\prime\prime\prime}-x^{\prime\prime\prime}y^{\prime\prime})}dt
\tilde{G}( x ) = \tilde{S}( x ) - 4 \pi \alpha \hspace{- 1 m m}\int \hspace{- 1 m m}d^{4}x_{1}d^{4}y_{1}e^{i x A ( x_{1}) + i x_{1}A ( y_{1})}\tilde{S}( x - x_{1}) \gamma^{\mu}\tilde{G}( x_{1}- y_{1}) \gamma^{\nu}\tilde{G}( y_{1}){\cal D}_{\mu \nu}( x_{1}- y_{1}) ,
J^{-{\frac{1}{n}}}\equiv f_{n}= \left| \begin{matrix}{s_{1}^{'}}&{\cdots}&{s_{1}^{( n - 1 )}}\\{\vdots}&{\ddots}&{\vdots}\\{s_{n - 1}^{'}}&{\cdots}&{s_{n - 1}^{( n - 1 )}}\\ \end{matrix}\right|^{-{\frac{1}{n}}}
A_pu_n(x)=(1+|x|^\alpha)\Delta (\eta_n u)(x)-|x|^\beta \eta_n(x) u(x) &=\eta_n(x)A_pu(x)+2(1+|x|^\alpha) \nabla \eta_n(x)\nabla u(x)+(1+|x|^\alpha)\Delta \eta_n (x) u(x) &=\eta_n(x)A_pu(x)+\frac{2}{n}(1+|x|^\alpha) \nabla \eta\left(\frac{x}{n}\right)\nabla u(x) +\frac{1}{n^2}(1+|x|^\alpha)\Delta \eta \left(\frac{x}{n}\right) u(x)\;,
\sum_{1-n \leq 2r \leq n}\frac{1}{4r^2 - 1}\left(1 + \frac{r}{j n}\right)^{-s-1}= -1 + \sum_{k=1}^{n/2}\frac{2}{4k^2 - 1}\sum_{q=0}^{\infty}\frac{(s+1)_{2q}}{(2q)!}\left( \frac{k}{jn}\right)^{2q}~~~~~~~~~~~ - \frac{1}{n^2 - 1}\left( 1 - \frac{1}{2j}\right)^{-s-1}.
\begin{aligned}a^{\textnormal{hom}}_{ijrs}(\theta) := \sum_{k=1}^n \sum_{l=1}^d \int_Y a_{ijkl}\big( \partial_{l}N^{(rs)}_{\theta k}(y) +e^{\i \theta \cdot y}\delta_{kr}\delta_{ls}\big) e^{-\i \theta \cdot y}\,{\textnormal{d}}y ({i,r \in \{1,\ldots,n\}, j,s \in \{1,\ldots,d\}}),\end{aligned}
\sum_{i=1}^r \left(\sum_{j=1}^i \gamma_j -\sum_{j=1}^{i-1}\gamma_j\right) \lambda_i=\sum_{i=1}^r \left(\sum_{j=1}^i \gamma_j\right) (\lambda_i-\lambda_{i+1}) = \sum_{i=1}^{\lambda_1}m_i \leqslant 0.
(w_1,\dots, w_n; w_{n+1},\dots, w_{2n}; w_{2n+1}) =\left(\frac{u_{11}}{v_{11}},\dots ,\frac{u_{nn}}{v_{nn}};x_1,\frac{v_{21}}{u_{21}},\dots, \frac{v_{n,n-1}}{u_{n,n-1}};t_{n+1}\right).
(\eqref{eq_generalized_order_1}) &=\frac{1}{X+1}\sum_{l=1}^n\prod_{j=1}^l\Bigg(\sum_{m_j=1}^n\Bigg)1_{\sum_{j=1}^lm_j=n}c^{(n)}(l,m_1,m_2,\cdots,m_l)(-1)^l &\quad\cdot \prod_{k=1}^l\Bigg(\frac{1}{1-f(0)}\left(\frac{d}{dx}\right)^{m_k}(1-f(x))\Big|_{x=0}\Bigg)\frac{1}{1-f(0)}.
\tilde{f}\left(\frac{\theta}{\gamma}\right)-\frac{1}{\gamma}\tilde{f}(\theta) = \frac{1}{\gamma}\theta\ln(\theta)-\frac{1}{\gamma}\theta\ln(\gamma)+\frac{1}{e}\geq -\frac{1}{\gamma}\theta\ln(\gamma)+\frac{1}{e}\frac{\gamma-1}{\gamma}= -\frac{1}{\gamma}g(\theta,\gamma)\ln(\gamma)
P_{\sigma \nu}= -\frac{1}{n-1}\left(\Omega^0_\nu(e_\sigma,e_0) + \Omega^\rho_{\mu}(e_\sigma,e_\rho)-\frac{1}{n+1}\Omega^0_0(e_\sigma,e_\nu)-\frac{1}{n+1}\Omega^\rho_\rho(e_\sigma,e_\nu) \right), \tilde{P}_{\sigma \nu}= -\frac{1}{n-2}\left(\Omega^\rho_\mu(e_\sigma,e_\rho)-\frac{1}{n}\Omega^{\rho}_\rho(e_\sigma,e_\nu)\right).
\mbox{RHS of (\ref{6phi5even})}={(ax;q)_\infty\over (qx/a;q)_\infty}\sum_{l\geq 0}\sum_{j\geq 0}\sum_{m\geq 0}A(l+2j+2m,j,m) &=\sum_{j\geq 0}\sum_{m\geq 0}A(2j+2m,j,m)=\Phi(x;s|a,-a,c,-c|q)=\mbox{LHS of (\ref{6phi5even})}.
\sum_{m=0}^{p_{2}+q_{2}}S_{a,b}^{a,b+1,q_{2}}\left( p_{2},m\right)S_{a,am+b}^{a,am+b+1,q_{1}}\left( p_{1},l-m\right)=S_{a,b}^{a,b+1,q_{1}+q_{2}}\left( p_{1}+p_{2},l\right) .
F\left( x_{i}, \dots, x_{i-m+1}\right) = F_1\left( x_{i}\right) + \ldots + F_m\left( x_{i-m+1}\right) , F_j\left(x_{i-j+1}\right) = U\left[ j, \frac{x_\mathrm{max}- n x_\mathrm{min}+ \left(n-1\right)x_{i-j+1}}{x_\mathrm{max}- x_\mathrm{min}}\right] ,
\lim_{k\to+\infty}\|\nabla u(t_k,\cdot)\|_{L^2}=+\infty, \lim_{k\to+\infty}\max\left\{\int_{M}e^{u(t_k,\cdot)}, \int_{M}e^{-u(t_k,\cdot)}\right\}=+\infty.
G = \frac{\binom{n}{i}}{\binom{k}{i}}(k-i+1)K_{i-1}(x,k-y,r-1) H_j(y,n-i,k-i) +\frac{\binom{n}{i}}{\binom{k}{i}}K_{i-1}(x,k-y,r-1) \quad\Big(A_j(n-i,k-i)H_{j+1}(y,n-i,k-i) +B_j(n-i,k-i)H_j(y,n-i,k-i) & +C_j(n-i,k-i)H_{j-1}(y,n-i,k-i)\Big).
w_{=n}(\zeta)<w_{n}(\zeta), w_{=n}^{\ast}(\zeta)<w_{n}^{\ast}(\zeta), \widehat{w}_{=n}(\zeta)<\widehat{w}_{n}(\zeta), \widehat{w}_{=n}^{\ast}(\zeta)<\widehat{w}_{n}^{\ast}(\zeta).
Q_{\ell +1}=Q_\ell^{\alpha_{\ell +1}}+\sum\limits_{j=0}^{\alpha_{\ell +1}-1}\left(\sum\limits_{{\bar\gamma}_\ell}c_{\ell +1,j{\bar\gamma}_\ell}\mathbf{Q}_\ell^{{\bar\gamma}_\ell}\right)Q_\ell^j.
\delta>0\quad\frac{1}{M}=\gamma^{\delta/q}, \beta_*>2(2K)^{1-\alpha}\beta\Bigl(1-\bigl(1+\frac{1}{M}\bigr)^{-\alpha/\delta}\Bigr)^{-1}>2\beta(2K)^{1-\alpha}, k_0\in\mathbb{N},k_0>\frac{q}{\log \gamma}\log (c_1(2^{\frac{1}{1-\alpha}}-1)),
(a)_n=(a;q)_n :=\prod_{k=0}^{n-1}(1-aq^{k}), (a)_\infty=(a;q)_{\infty}:=\prod_{k\ge 0}(1-aq^{k}), (a_1,a_2,\cdots,a_m;q)_\infty :=(a_1;q)_\infty(a_2;q)_\infty\cdots(a_m;q)_\infty.
{\cal J}_{i}^{\mu \nu}[ \xi | s ] = 4 \pi \int d^{4}x \epsilon^{\mu \nu \rho \sigma}\left[ \Omega_{\xi}( s , 0 ) T_{i}\Omega_{\xi}^{- 1}( s , 0 ) \right]^{j}\dot{\xi}_{\sigma}( s ) \widetilde{j}_{\rho}^{j}( x ) \delta^{4}( x - \xi ( s ) ) .
\mathop{\max}\limits_{{F_{\left. T \right|S}}\left({\left. t \right|s}\right)}h\left({\left. T \right|s}\right), s.t. &{F_{\left. T \right|S}}\left({\left. t \right|s}\right) = \Pr \left({\left.{T < t}\right|s}\right), &E\left({\left. T \right|s}\right) ={g_0}, &E\left({\left.{{{\log}_e}T}\right|s}\right) ={g_1},
T\left( x,0\right) =T_{0}\left( x\right) ,\;\;q\left( x,0\right) =0,\;\;x\in \mathbb{R}, \lim_{x\rightarrow \pm \infty}T\left( x,t\right) =0,\;\;\lim_{x\rightarrow\pm \infty}q\left( x,t\right) =0,\;\;t>0.
p(r)=(\int_{-\sqrt{r^2+\rho^2}}^{\sqrt{r^2+\rho^2}}(1-\frac{t^2}{r^2+\rho^2})^{\frac{n-3}{2}}dt)^{-1}\int_{\rho}^{\sqrt{r^2+\rho^2}}(1-\frac{t^2}{r^2+\rho^2})^{\frac{n-3}{2}}dt.
U_{\mu}^{-}(x):= \max\{0, \varphi_{\mu}(x)-d\varphi_{\tilde{\mu}}(x)\}=\begin{cases}0\ \x\leq \underline{a}_{\mu,\tilde{\mu},d}\varphi_{\mu}(x)-d\varphi_{\tilde{\mu}}(x)\x\geq \underline{a}_{\mu,\tilde{\mu},d}.\end{cases}
\langle 0 | e^{k g \varphi ( 0 )}| 0 \rangle = \sum_{j = 0}^{\infty}\, \frac{k^{j}\, g^{j}}{j !}\langle 0 | : \! \varphi^{j}( 0 ) \! : | 0 \rangle = 1 + o ( k^{2}) \, .
\begin{aligned}\sum_{2\le|a|\le N}{\tilde P}^a_{M_a}e^{-itH}f=e^{-itH}f=\phi(H)e^{-itH}f=\sum_{2\le|a|\le N}\phi(H){\tilde P}^a_{M_a}e^{-itH}f.\end{aligned}
\beta = \frac{\chi}{1+ \chi}, \gamma \leq \min \left \{\frac{\tilde{\lambda}\chi(1-\chi)\xi}{8(1+\chi)}, \frac{1+\chi}{2(2+\chi)R_0 \|\nabla_x M\|_\infty}\right \}, \mbox{with}\xi:=\begin{cases}\tilde C^{k-2}, \mbox{if}k<2, 1, \mbox{if}k=2, \|\nabla_x M\|_\infty^{k-2}, \mbox{if}k>2.\end{cases}
\frac{\partial}{\partial t}\left( \begin{array}{c}{a ({\bf k})}\\{a^{*}({\bf k})}\\ \end{array}\right) = \mathrm{\cal f i n i t e t e r m s}+ \lambda^{2}\left( \begin{array}{c c}{A ( 0 ; t )}&{A^{*}( 0 ; t )}\\{A ( 0 ; t )}&{A^{*}( 0 ; t )}\\ \end{array}\right) \ \left( \begin{array}{c}{a ({\bf k})}\\{a^{*}({\bf k})}\\ \end{array}\right)
0\leq \langle I_{\sigma}^{\prime}(U),\left(U_{\epsilon}-U \right) \rangle=\epsilon \langle I_{\sigma}^{\prime}(U),\left(\varphi,\psi \right) \rangle -\langle I_{\sigma}^{\prime}(U),\left(w^{\epsilon},z^{\epsilon}\right) \rangle +\langle I_{\sigma}^{\prime}(U),\left(w_{\epsilon},z_{\epsilon}\right) \rangle,
\exp_2\,\phi_*(\xi_1)=\phi\,\exp_1(\xi_1)=\phi(z_1)=z_2=\exp_2(\xi_2)\;, d_2(\phi(c_1),c_2)\le d_2(\phi(c_1),z_2)+d_2(z_2,c_2)=d_1(c_1,z_1)+d_2(z_2,c_2)<2\epsilon\;.
E_{n,l}= \lambda \Bigl[2a(4n+2l+d) + 2n(2n+1) + 2b_1\Bigl(4a+4n-2l-d+3-4\sum_i z_i\Bigr){}- 16b_2\Bigl(\sum_i z_i^2 - \sum_i z_i\Bigr) - l(l+d-1)\Bigr],
\|R_0(\lambda\pm i0)f\|_{L^2(|V|)}=\sup_{\|\widetilde{g}\|_{L^2}\leq1}|\langle|V|^{1/2}R_0(\lambda\pm i0)f,\widetilde{g}\rangle| &=\sup_{\|\widetilde{g}\|_{L^2}\leq1}|\langle R_0(\lambda\pm i0)f,|V|^{1/2}\widetilde{g}\rangle| &=\sup_{\|g\|_{L^2(|V|^{-1})}\leq1}|\langle R_0(\lambda\pm i0)f,g\rangle|,
e^{(0)}_\mu = \frac{p_\mu}{\sqrt{-p^2}}, \quad e^{(\alpha)}_\mu \eta_{\alpha\beta}e^{(\beta)}_\nu = \eta_{\mu\nu},\quad\varepsilon_{\mu\nu\lambda}e^{(0)\mu}e^{(i)\nu}e^{(j)\lambda}=\varepsilon^{0ij}.
L= \frac{64 k^2}{a^8 \xi}\ \ \ \ \ \ \ \M=\frac{D-1}{4\xi}\ \ \ \ \ \ \ \N=\frac{4}{3a^2 \xi}\left( 1+\frac{D-1}{2}-\frac{k^2}{a^2}\right)-\frac{4}{3a^4}
\sqrt{T}E^{\Pi^{D_T}}[\max_{\lambda \leq J,k,j}|Z_{\lambda,k,j}| |X^T] &\lesssim \sqrt{2 \log 2 (V_J^{\otimes d})}(C_T'+1) \max_{\lambda \leq J,k,j}(\|\tilde{\Phi}_{\lambda,k,j}\|_{\mu_0}^2 + c_T') =O_{P_{b_0}}( \sqrt{J})
h''(u)= \frac{2 \log(e/(2u))}{u^2}+ \frac{2 \log(e/(2 - 2u))}{(1 - u)^2}\leq \frac{2 \log(e/(2 D_{\min}))}{D_{\min}^2}+ \frac{2 \log(e/(2 - 2D_{\max}))}{(1 - D_{\max})^2}.
I_A= -2\int \Big(\partial^{\zeta^2+\zeta^3}_x \nabla \phi \phi\Big) \cdot Q(v) \cdot \partial^{\zeta+\zeta^1}_xu \leq C|\nabla^3 \phi|_2|\nabla v|_\infty|\phi\nabla^4 u|_2\leq Cc^4_3+\frac{\alpha}{20}|\sqrt{\phi^2+\eta^2}\nabla^4 u|^2_2.
\sum_{p=1}^{r^{(1)}_{i}}\sum_{q=1}^{r^{(1)}_{i-1}}\mathrm{min}\{n_{p,i},n_{q,{i-1}}\}=\sum_{s=1}^{k}r^{(s)}_{i-1}r_{i}^{(s)}, \sum_{p=1}^{r_{i}^{(1)}}(\sum_{p>p'>0}2\mathrm{min}\{n_{p,i},n_{p',i}\}+n_{p,i}) = \sum_{s=1}^{k}r^{(s)^{2}}_{i},
\phi(x_{[k]}) = \sum_{i \in [s]}\tilde{\beta}_i(x_I) \tilde{\gamma}_i(x_{[k] \setminus I}) + \sum_{i \in [r_2]}\beta^{(2)}_{i, 1}(x_{I^{(2)}_{i, 1}}) \dots \beta^{(2)}_{i, d^{(2)}_i}(x_{I^{(2)}_{i, d^{(2)}_i}}) + \sum_{i \in [r_3]}\beta^{(3)}_{i, 1}(x_{I^{(3)}_{i, 1}}) \dots \beta^{(3)}_{i, d^{(3)}_i}(x_{I^{(3)}_{i, d^{(3)}_i}})
\big\|B_l^{(j+1)}\big\|_l\leq \sum_{q=2}^k\sum_{(p_1,\dots,p_q)\in\mathbb{I}_{2k}^q,\,\deg(S_{p_1}) +\cdots+\deg(S_{p_q})=l}|\alpha_{p_1,\dots,p_q}|2^{p_1\wedge\cdots\wedge p_q}\prod_{i=1}^q\tilde{Q}_{p_i}\big(\nu_1,\dots,\sqrt[k]{\nu_k}\big),
P_{(n-1)/2}^{\left(\frac{1}{2}, -\frac{\beta}{\lambda}-\frac{1}{2}\right)}(1+2\lambda r^2) \propto \frac{C_n^{(-\beta/\lambda)}\left(\sqrt{-\lambda r^2}\right)}{\sqrt{-\lambda r^2}}\text{for $p=1$, $n=2n_r+1$},
(\Gamma_{q^{2}}(x)\Gamma_{q^{2}}(1-x))^{-1}=\frac{q^{2x(1-x)}}{(q^{-2};q^{-2})_{\infty}^{3}(q^{2}-1)}\theta_{4}(\frac{1}{2i}(1-2x)logq,\frac{1}{q}).
p_{cov}^{c}= \frac{e^{-\frac{\pi^2 R^2 \psi_0 p(r)}{2}\sqrt{\frac{\gamma p_i}{p_c}}}-1}{-\frac{\pi^2 R^2 \psi_0 p(r)}{2}\sqrt{\frac{\gamma p_i}{p_c}}}.
d_{P}^{2}(h_{(j,N)}^{-},h_{(j,N)}^{+}) =\int_{A_{j}^{(N)}}\left(g_{+}-g_{-}+\mathbb{E}(g_{+}-g_{-}\;|\;A_{j}^{(N)})\right)^{2}dP & =P(1_{A_{j}^{(N)}}(g_{+}-g_{-})^{2})+P(A_{j}^{(N)})\mathbb{E}(g_{+}-g_{-}\;|\;A_{j}^{(N)})^{2}& +2\mathbb{E}(g_{+}-g_{-}\;|\;A_{j}^{(N)})P(1_{A_{j}^{(N)}}(g_{+}-g_{-})).
\phi_{\kappa}( \zeta ) = F \left( \begin{array}{c c c c}{{d_{1 , \kappa},}}&{{\dots ,}}&{{d_{n - 1 , \kappa},}}&{{d_{n , \kappa}}}\\{{e_{1 , \kappa},}}&{{\dots ,}}&{{e_{n - 1 , \kappa},}}&{{\zeta}}\\ \end{array}\right)
\frac{1}{N^{\frac{n}{2}+ 1}}\sum^N_{j_1, \ldots, j_n = 1}\varphi_q\left(z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right)z_2\left(j_{s^{-1}_\chi(2)}, j_{s^{-1}_\chi(2) + 1}; k_2\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right) \right)
\mathcal{Z}[N,\lambda] = \int_{(\mathbb{C}^{N})^{\otimes D}}dT d\bar{T}\exp\Biggl(-N^{D-1}\biggl(\frac{1}{2}(\bar{T}\cdot T) +\frac{\lambda}{4}\sum_c (\bar{T}\cdot_{\hat{c}}T)\cdot_c (\bar{T}\cdot_{\hat{c}}T)\biggr)\Biggr)
\{\underline{a}\colon \underline{a}/\varpi^k \in L(\varpi^m\underline{c})\}= \{a\underline{c}^\bot+\varpi^{k-m}\underline{d}\colon |a|<|\varpi|^{k-m}, (a,\varpi)=1, |\underline{d}|<|\varpi|^m\}\setminus &\{a\underline{c}^\bot +\varpi^{k-m+1}\underline{d}\colon |a|<|\varpi|^{k-m+1},|\underline{d}|<|\varpi|^{m-1}, (a,\varpi)=1\}
\begin{array}{l l}{{{\sf I}_{{\sf p}, \mathrm{a b}}^{\sf i}= \langle \underline{{{\sf p}}}, \mathrm{a}|{\sf I}^{\sf i}| \underline{{{\sf p}}}, \mathrm{b}\rangle}}&{{{\sf i}= 1 , 2 , 3}}\\{{{\sf C}_{{\sf p}, \mathrm{a b}}^{\sf c}= \langle \underline{{{\sf p}}}, \mathrm{a}|{\sf C}^{\sf c}| \underline{{{\sf p}}}, \mathrm{b}\rangle}}&{{{\sf c}= 1 , 2 , . . . , 8}}\\ \end{array}\hspace{7 c m}
|E_{int,1}| \leq \frac{(b+1)}{t \log^{b+2}(t) \sqrt{\log(\log(t))}}+ \frac{1}{2 t \log^{b+2}(t) (\log(\log(t)))^{3/2}}&\leq \frac{1}{100}\frac{1}{t \log^{b+1}(t) \sqrt{\log(\log(t))}}, t \geq T_{0}
a_1(y_1,y_2) := 1 - \frac{1}{2}\sin(2\pi(y_1+y_2)) + \frac{1}{4}\sin(4\pi y_2), a_2(y_1,y_2) := 1 + \frac{1}{2}\sin(2\pi(y_1+y_2)) + \frac{1}{4}\cos(2\pi y_1)
\mathbb{E}_1=\frac{\sqrt{8}}{8}(e_1\otimes e_1\otimes e_1\otimes e_1+e_2\otimes e_2\otimes e_2\otimes e_2-e_1\otimes e_1\otimes e_2\otimes e_2-e_1\otimes e_2\otimes e_1\otimes e_2 -e_2\otimes e_1\otimes e_1\otimes e_2-e_2\otimes e_1\otimes e_2\otimes e_1-e_1\otimes e_2\otimes e_2\otimes e_1-e_2\otimes e_2\otimes e_1\otimes e_1)
P\Big(\frac{2}{k+1+2/\theta_0}\Big) \geq \frac{4}{(k+1+2/\theta_0)^2}+ \Big(\frac{2}{k+1+2/\theta_0}- 1\Big) \frac{4}{(k+2/\theta_0)^2}= \frac{4}{(k+1+2/\theta_0)^2 (k+2/\theta_0)^2}\geq 0.
\int_r^R \Big( 1+ \alpha^{-2}\sum_{j=k}^{n}\alpha_j^2 \Big) 2\pi^2 t^2 \alpha^2 \eta^2 \,G_t(\eta,\xi_k,\dots, \xi_n) \frac{dt}{t}& + \int_r^R \Big( \sum_{j=k}^n 4\pi^2 t^2 \alpha_j^2 \xi_j (\xi_j+\eta) \Big) \,G_t(\eta,\xi_k,\dots, \xi_n) \frac{dt}{t}.
D(a, b)^{-2}A(\nu)^2 = \big(a^{\nu+1}+a^{-\nu-1}-b^{\nu+1}-b^{-\nu-1}\big)^2 & = 4 + a^{2(\nu+1)}+ a^{-2(\nu+1)}+ b^{2(\nu+1)}+ b^{-2(\nu+1)}& - 2(ab)^{\nu+1}- 2(ab)^{-\nu-1}- 2(ab^{-1})^{\nu+1}- 2(a^{-1}b)^{\nu+1}.
\Delta_1(\alpha \star \beta; a\ (qr)) := \sum_{n = a\ (qr)}(\alpha \star \beta)(n) - \frac{1}{\phi(q)}\sum_{\substack{(n,q)=1 n = a\ (r)}}(\alpha \star \beta)(n), \Delta_2(\alpha \star \beta; a\ (qr)) := \frac{1}{\phi(q)}\sum_{\substack{(n,q)=1 n = a\ (r)}}(\alpha \star \beta)(n) - \frac{1}{\phi(qr)}\sum_{(n,qr)=1}(\alpha \star \beta)(n).
\begin{array}{lll}L_b=\log_2 \prod_{i=1}^m\left(1+\rho_k\alpha_i+\eta_k\beta_i\right) U_b=\log_2 \prod_{i=1}^m\left(1+\rho_k\alpha_i+\eta_k\beta_{m+1-i}\right)\end{array}
\mathbf{D\mathbf{e}}^{a}=d\mathbf{\mathbf{e}}^{a}+\mathbf{A}_{b}^{a}\mathbf{\mathbf{e}}^{b}=0, \mathbf{F}_{b}^{a}=d\mathbf{A}_{b}^{a}+\mathbf{A}_{c}^{a}\mathbf{A}_{b}^{c},
\begin{aligned}\Vert q_{1}-q_{2}\Vert^{2}_{H^{-1}(\Omega)}\leq C \Vert q_{1}-q_{2}\Vert^{2}_{L^{\infty}(\Omega)}\leq \frac{4CM^{2}}{\delta}\delta \leq \frac{4CM^{2}}{\delta}\Vert \mathcal{N}_{q_{1}}-\mathcal{N}_{q_{2}}\Vert\end{aligned}
\mathbb{L}[\varphi_0]\varphi=\varphi_{tt}-\left(\mu-2\phi_{0,x}\right)\varphi_{xx}-2\left[\mathbb H\,;\phi_{0,x}\right]\phi_{xx}-2\mathbb H[\phi_{0,xx}\phi_x]+\left(\left[\phi;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\phi_{xx}\right)_x\,.
\left(B^{\frac{1}{p}}\,\widetilde{\boldsymbol\tau},D^\star{\boldsymbol\psi}\right)=\left(B \,{\boldsymbol\tau}\left(D^\star{\boldsymbol u}\right),D^\star{\boldsymbol\psi}\right) \mbox{for all}\{\boldsymbol\psi}\in \boldsymbol V_B^{p}.
w(\epsilon)\,=\,\frac{1}{2}\,\sum_{n=1}^{\infty}\,\sqrt{n^2\,+\,\epsilon^{-1}}\,=\,\frac{1}{4}\,\left(\sum_{n=-\infty}^{+\infty}\,\sqrt{n^2\,+\,\epsilon^{-1}}\,-\,\frac{1}{\sqrt{\epsilon}}\right)\,{,}
\overset{5}{R}(X,Y)Z = \overset{g}{R}(X,Y)Z - \frac{1}{2}\overset{1}{D}(X,Y) A Z - \frac{1}{2}\overset{3}{D}(X,Z) A Y + \frac{1}{2}\overset{2}{D}(Y,Z) A X - \frac{1}{2}\pi (Y) (\pi (X) Z - \pi (Z) X ).
\beta_{i\bar\jmath}=-\frac1{2\pi}R_{i\bar\jmath}-\frac{4\zeta(3)}{3(4\pi)^4}\partial_i\partial_{\bar\jmath}\left[R_{k\bar lm\bar n}R^{p\bar lq\bar n}R_p{}^k{}_q{}^m-R_{k\bar lm\bar n}R^{m\bar np\bar q}R_{p\bar q}{}^{k\bar l}\right]+\ldots
\Vert f \Vert_{\infty , \mathcal{A}}= \sup \lbrace \Vert f(x) \Vert_{\mathcal{A}}:\, x \in X \rbrace &= \sup \lbrace \Vert \overline{f}(\delta_{x}) \Vert_{\mathcal{A}}:\, x \in X \rbrace &\leq \Vert \overline{f}\Vert_{\infty ,\mathcal{A}}.
\int_M u dd^c_B v\wedge \omega_\phi^{k}\wedge (\omega^T)^{n-k-1}\wedge \eta= \int_M v dd^c_B u \wedge \omega_\phi^{k}\wedge (\omega^T)^{n-k-1}\wedge \eta = -\int_M d u\wedge d^c_B v \wedge \omega_\phi^{k}\wedge (\omega^T)^{n-k-1}\wedge \eta
\int_{FG}e^{sx}\overline{G}(s,t)ds=-\int_{r}^{-r+\lambda_2-\lambda_1}e^{-x\lambda_1}e^{-wx}e^{t(c_1\lambda_1^{\alpha_1}+c_2\lambda_2^{\alpha_2})}e^{-t[c_1(w^{\alpha_1}e^{-i\pi \alpha_1})+c_2(\lambda_2-\lambda_1+we^{-i\pi})^{\alpha_2}]}dw.
\tilde{h}= \frac{1}{1 + ( a u )^{7 - p}}, \ \ \a^{7 - p}= \tilde{b}^{2}/ R^{7 - p}, \ \ \R^{7 - p}= \frac{1}{2}( 2 \pi )^{6 - p}\pi^{- ( 7 - p ) / 2}\Gamma [ ( 7 - p ) / 2 ] \hat{g}_{s}N .
\int\limits_{(\overline \Delta \times M^k) \setminus J^{-1}(0)}d(d\ln \frac{J}{\overline J}\wedge \Phi^*\omega)=\int\limits_{(S^1 \times M^k) \setminus J^{-1}(0)}d\ln \frac{J}{\overline J}\wedge \Phi^*\omega + 4\pi i \int\limits_{J^{-1}(0)}\Phi^*\omega.
\psi(u_t T) =\varphi(e_M u_t T e_M)=\varphi(e_M (u_t Ju_tJ) T(Ju_tJ)^* e_M)=\varphi(\alpha_t^0 e_M T (Ju_tJ)^* e_M) &=\varphi(e_M T (Ju_tJ)^* e_M\alpha_t^0)=\varphi(e_M T (Ju_tJ)^* u_t Ju_tJ e_M)=\psi(T u_t).
\int_{B_t}|\nabla u_k(x)|^2\,|x_{n+1}|^a\,dx \leq \frac{D_{u_k}(1)}{2(1-t)}= \frac{I_{u_k}(1)\,H_{u_k}(1)}{2(1-t)}\leq \frac{M}{2(1-t)},
\varphi(x)=\alpha_1\, \phantom{}_0 F_2(; \frac{3}{4},\frac{1}{2}|\frac{x^4}{4^3})+\alpha_2 x\, \phantom{}_0 F_2(; \frac{3}{4},\frac{5}{4}|\frac{x^4}{4^3})+ \alpha_3 x^2\, \phantom{}_0 F_2(; \frac{5}{4},\frac{3}{2}|\frac{x^4}{4^3}).