harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{m=1}^\ell X_{(m)}^{JSQ}(t) \leq_{st}\sum_{m=1}^\ell X_{(m)}^{\Pi}(t),\mbox{for}\ell=1,2,\ldots, N, \left\{L^{JSQ}(t)\right\}_{t\geq 0}\leq_{st}\left\{L^{\Pi}(t)\right\}_{t\geq 0},
g_{ij}=\begin{pmatrix}2u^3+\lambda (u^2)^2 -1-\lambda u^1 u^2 -u^1 -1-\lambda u^1 u^2 \lambda(u^1)^2 0 -u^1 0 \lambda\end{pmatrix}, w_{12}=\frac{\lambda}{\sqrt{-\det g}},w_{23}=0,w_{31}=\frac{\lambda u^1}{\sqrt{-\det g}},
- b (x) \cdot D\tilde \psi (x) \zeta_{2r}(x) \leq |- b (x) \cdot D\tilde \psi (x)| = |\bar f (x)| \leq \left( \log \frac{\kappa}{r}- \frac{\log 2}{2}\right)^{-1}\int_{\tau_-^\delta (y)}^{\tau_+^\delta (y)}|f (X(t, y))| \, dt,
|b_{22}^{(k)}-b_{44}^{(k)}|^{2}+|b_{11}^{(k)}-b_{33}^{(k)}|^{2}= 2|b_{24}^{(k)}|^{2}|\cot2\psi_{k+1}|^{2}+2|b_{13}^{(k)}|^{2}|\cot2\phi_{k+1}|^{2}& \leq2\cdot4.49^{2}\delta_{k}^{2}\delta_{k+1}^{2}S^{2}(B).
\begin{aligned}\int u^2(x,t)\phi_N(x,t)\,dx \leq c\bigl(\int u_0^2(x)\phi_N(x,0)\,dx\bigr)e^{\int_0^TM(t')\,dt'}&\leq c(a_0, \|e^{\frac12a_0^2x_+^{5/4}}u_0\|_2, \| u_0\|_{4,2},T)\int u_0^2(x)e^{a_0x_+^{5/4}}\,dx\,.\end{aligned}
p(x,A)=\inf\{p(x,a):a\in A\},\delta_p(A,B)=\sup\{p(a,B):a\in A\}, &\delta_p(B,A)=\sup\{p(b,A):b\in B\}H_p(A,B)=\max\{\delta_p(A,B),\delta_p(B,A)\}.
\tilde{J}^{\pm}(u)=\begin{pmatrix}1 0 f_{2}^{\pm}(u) 1\end{pmatrix}\begin{pmatrix}k_{2}^{\pm}(u) 0 0 k_{3}^{\pm}(u)\end{pmatrix}\begin{pmatrix}1 e_{2}^{\pm}(u) 0 1\end{pmatrix}
H^\bullet_\mathrm{abs}(X,F) = H^\bullet(X,F)\simeq H^\bullet\big(\Omega^\bullet(X,F),d^F\big), & H^\bullet_\mathrm{rel}(X,F) = H^\bullet(X,\partial X,F)\simeq H^\bullet\big(\Omega^\bullet_\mathrm{c}(X,F),d^F\big).
\sum_{n=-\infty}^{+\infty}\frac{1}{S_{0}^{2}+(2\pi n-\sigma)^{2}}= \frac{1}{4S_{0}}\left[ \coth\left(\frac{S_{0}}{2}+\frac{i\sigma}{2}\right) + \coth\left(\frac{S_{0}}{2}- \frac{i\sigma}{2}\right)\right]
\sum_{l=0}^{\infty}(2l+1)[s_l^{\prime}(\lambda r)e_l(\lambda\rho)]^2=\frac{1}{2r \rho}\int_{r-\rho}^{r+\rho}\left(\frac{1}{\lambda}\,\frac{\partial{\cal D}}{\partial r}\right)^2R\,dR\,.
H'_{\beta}\upharpoonright \,\bigotimes_{j=1}^{N-1}\mathcal{F}_j:=\Big(H_{Kitaev}+\beta \sum_{i=2}^{N-2}\tilde{V}_{I_{2;i-1}}+\beta \sum_{i=2}^{N-3}\tilde{V}_{I_{3;i-1}}+\dots +\beta\sum_{i=2}^{N-\bar{k}-1}\tilde{V}_{I_{\bar{k}+1;i-1}}\Big) \upharpoonright \,\bigotimes_{j=1}^{N-1}\mathcal{F}_j
I(d) := -\frac{N}{2(2\pi)^d}\int_{\mathbb{R}^d}\Big(\mathcal{A}_k -\sqrt{\mathcal{A}_k^2- \widehat{g}_k^2}- \frac{N+1}{\lvert \Lambda \rvert}G_d(k) \Big) \mathrm{d}k.
\partial_{x}(A_{1}(\textbf{w}+\bar{\textbf{u}})\partial_{x}\textbf{w}) \cdot \partial_{x}\textbf{w}= \frac{1}{2}\partial_{x}((A_{1}(\textbf{w}+\bar{\textbf{u}})\partial_{x}\textbf{w}) \cdot \partial_{x}\textbf{w}) + \frac{1}{2}(\partial_{x}A_{1}\partial_{x}\textbf{w}) \cdot \partial_{x}\textbf{w}.
2^{n-1}-1 = 1\cdot 2^{n-2}+ 1\cdot 2^{n-3}+ \dots + 1\cdot 2 + 1, 2^{n-1}-2 = 1\cdot 2^{n-2}+ 1\cdot 2^{n-3}+ \dots + 1\cdot 2 + 0.
V_{\lambda,\nu}(t) := \prod_{\substack{\beta\in{\hat R_0^+}\langle \lambda,\beta\rangle=0 \langle \nu,\beta\rangle > 0}}\frac{1-t_{\beta}\hat e_t (\beta)}{1-\hat e_t (\beta)}\prod_{\substack{\beta\in{\hat R_0^+}\langle \lambda,\beta\rangle=c \langle \nu,\beta\rangle < 0}}\frac{1-t_{\beta}\hat h_t \hat e_t (-\beta)}{1-\hat h_t \hat e_t (-\beta)}
S_{n,2}(\omega)=\mathrm{ord}(\omega)^4 a_{\frac{n}{\mathrm{ord}(\omega)},\frac{y^2(x-y)^2}{2}-\frac{(x-y)^2}{6}}+ \mathrm{ord}(\omega)^2 a_{\frac{n}{\mathrm{ord}(\omega)},\frac{(x-y)^2}{6}}.
\partial_{w_{bb}}\left(\partial_{w_{aa}}\left(\frac{1}{I_{q'^{2}}+R\left(W'\right)}\right)\cdot G\left(W,Z\right)\right)=\partial^{2}_{w_{bb}w_{aa}}\left(\frac{1}{I_{q'^{2}}+R\left(W'\right)}\right) \cdot G\left(W,Z\right)+\partial_{w_{aa}}\left(\frac{1}{I_{q'^{2}}+R\left(W'\right)}\right)\cdot\partial_{w_{bb}}\left( G\left(W,Z\right)\right),
Q(x) = \div X_0(x) + \frac12 \sum_{i = 1}^r X_i \div X_i(x) \, , \tilde Q(x, v) = \div \tilde X_0(x, v) + \frac12 \sum_{i =1}^r \tilde X_i \div \tilde X_i(x, v) \,.
\hat X = \sum_{n=0}^\infty \sum_{k=0}^n \alpha_{n-k}\alpha_k \left( -\partial_\mu \partial^\mu -m^2 \right) \left( -\partial_\mu \partial^\mu \right)^{n}= \sum_{k=0}^\infty \beta_k \left( -\partial_\mu \partial^\mu \right)^k
\tilde{Z}_{A B}= i \sigma_{2}\left( \begin{matrix}{{z_{1}}}&{{0}}\\{{0}}&{{z_{2}}}\\ \end{matrix}\right) , \qquad \tilde{Z}_{A B}= i \sigma_{2}\left( \begin{matrix}{{z_{1}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{z_{2}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{z_{3}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{z_{4}}}\\ \end{matrix}\right) ,
\begin{aligned}\liminf_{n\to\infty}\frac 1n \log P^{\pi,\omega}_0 \big(\mathfrak L_n \in \mathcal G\big) \geq - \inf_{\mu\in \mathcal G}\mathfrak I(\mu) &=- \inf_{\mu\in \mathcal G}\mathfrak I^{\star\star}(\mu).\end{aligned}
\begin{matrix}\begin{array}{l}M'=(2R)^{m^2}, c'=(\acute b R\nu'^{-1}\delta^{-1}\tilde M)^{-rm^3(m+6)}c, \zeta'=10m\nu',\end{array}\begin{array}{l}\delta'=\acute b^{-1}((R^2\tilde M)^{-1}\nu')^{3m}\delta, r'=r, \tilde{M}'=\acute b (\nu'^{-1}\tilde{M})^{m^2(m+2)},\end{array}\end{matrix}
\mathcal{Q}_U :=|\partial_x|DN_2+\frac{i}{2}|\partial_x|^{1/2}\big[\phi_x^2-(|\partial_x|\phi)^2\big], \mathcal{C}_U :=|\partial_x|DN_3+\frac{3i}{2}|\partial_x|^{1/2}(h_{xx}h_x^2)-i|\partial_x|^{1/2}\big[|\partial_x|\phi\cdot(DN_2+h_x\phi_x)\big],
K_{\hbox{direct}}(r,\phi ; r,\phi ; t) ={1\over{8\pi^2\alpha t}}\int_{A+B}{{d\delta}\over{1-e^{-{{i\delta}\over\alpha}}}}e^{-{{r^2}\over{2t}}(1-\cos \delta)}\quad - (\hbox{the pole at $\delta=0$})
{}_\alpha\phi_\beta\!\left[\begin{matrix}a_1,\ldots,a_\alpha b_1,\ldots,b_\beta\end{matrix};q,z\right]:=\sum_{k=0}^\infty\frac{(a_1,\ldots,a_\alpha;q)_k}{(q,b_1,\ldots,b_\beta;q)_k}\left((-1)^kq^{\binom k2}\right)^{1+\beta-\alpha}z^k.
\tilde{E}_{n \delta}^{( i )}= \sum_{p_{1}+ 2 p_{2}+ \cdots + n p_{n}= n}\frac{\left( q^{( \alpha_{i}, \alpha_{i})}- q^{- ( \alpha_{i}, \alpha_{i})}\right)^{\sum_{i}p_{i}- 1}}{p_{1}! \; . . . \; p_{n}!}( E_{\delta}^{( i )})^{p_{1}}( E_{2 \delta}^{( i )})^{p_{2}}. . . ( E_{n \delta}^{( i )})^{p_{n}}
c_+^{(ba^{k-1},1)}(p,s)= \,\mathrm{pexp}\left(\frac{Q^8-Q^{-8}}{(1+s^{-1})(1+p^{-1}s^{2-k})(1+ps^{k-1})}\right)=c_+^{(ba^k,1)}(p/s,s), c_-^{(b^{-1}a^{k-1},1)}(p,s)= \,\mathrm{pexp}\left(-\frac{Q^8-Q^{-8}}{(1+s^{-1})(1+p^{-1}s^{1-k})(1+ps^k)}\right)=c_-^{(b^{-1}a^k,1)}(p/s,s),
\theta_k(a,b,c,d) begin{align*}1mm] \:=\frac{2k(1+2a-b-c-d+3k)}{a}+\frac{(a+k)(1+a-b-c+k)}{a(1+a-b+2k)}begin{align*}1mm] \quad\times\frac{(1+a-b-d+k)(1+a-c-d+k)(2+a-b-d+3k)}{(1+a-d+2k)(2+a-b-c-d+2k)}begin{align*}1mm] \:+\frac{(a+k)(c+k)(1-b+k)(1-d+k)}{a(1+2k)(1+a-b+2k)(1+a-c+2k)}begin{align*}1mm] \quad\times\frac{(1+a-b-c+k)(1+a-b-d+k)(1+a-c-d+k)}{(1+a-d+2k)(2+a-b-c-d+2k)}.
\eta(q_na_{t_n})\leq d(q_na_{t_n})^2=d(q_n)^2\epsilon^{\frac2{2-\gamma}}d(q_n)^{-\frac4{2-\gamma}}=\epsilon^{\frac2{2-\gamma}}d(q_n)^{-\frac{2\gamma}{2-\gamma}}= \epsilon e^{-\gamma t_n}.
\mathbb{E}_0\left[ h\left({\bf R}^{(12)}_{{\bf i}^1}\right) h\left({\bf R}^{(12)}_{{\bf i}^2}\right) h\left({\bf R}^{(12)}_{{\bf j}^1}\right) h\left({\bf R}^{(12)}_{{\bf j}^2}\right)\right] = (\zeta_d^h)^2.
\int_{B_{\theta R}\setminus B_{R_0}}\frac{h(x, y)}{|x-y|^{N+s\,p}}\, dy \leq \frac{C\,(\varepsilon+\varepsilon_0)}{(1-\theta)^{N+s\,p}\,|x|^{N+s\,p}}\int_{B_{\theta R}\setminus B_{R_0}}\frac{1}{|y|^{N-s\,p}}\, dy &\leq C\,\frac{(\varepsilon+\varepsilon_0)}{(1-\theta)^{N+s\,p}}\,\frac{(\theta\,R)^{s\,p}}{|x|^{N+s\,p}},
\sum_{(p,2)=1}(\log p)\chi_{8p}(m\ell )V \left( \frac{m}{p}\right) \Phi\left(\frac{p}{X}\right) = \sum^{\infty}_{n=1}\chi_{8n}(m\ell ) \Lambda(n) V \left( \frac{m}{n}\right) \Phi \left( \frac{n}X \right) +O \left(\sum_{\substack{p^j \leq X^{1+\varepsilon}, j \geq 2}}(\log p)\Phi \left( \frac{p^j}X \right) \right ).
{\widehat \Gamma}_{\Lambda_{1}\dots \Lambda_{n}}^{( n )}( k_{1}\dots k_{n}) = 2^{1 - n}P_{\Lambda_{1}\alpha_{1}}( k_{1}^{0}) \dots P_{\Lambda_{n}\alpha_{n}}( k_{n}^{0}) \Gamma_{\alpha_{1}\dots \alpha_{n}}^{( n )}( k_{1}\dots k_{n}) ,
[ Q , \bar{C}^{a}]_{P . B .}\stackrel{d e f .}{=}\tilde{K}_{a}= K_{a}+ f_{a b}^{c}\bar{C}_{c}C^{b}~ , ~ [ \tilde{K}_{a}, \tilde{K}_{b}]_{P . B .}= f_{a b}^{c}\tilde{K}_{c}~ .
\tilde \xi_t(y)= \xi(y) - q_{\xi,t}(y), \;\;\; y \in B_{R_t^\ast}\setminus \{0\}\tilde\sigma_t(y)=\begin{cases}\frac{\sigma(y)}{q_{\sigma,t}}\mbox{if}y=0, \sigma(y) \mbox{otherwise}.\end{cases}
{\displaystyle\sum\limits_{n\geq0}}\frac{u_{n+k}(z)}{t^{2n+1}}={\displaystyle\sum\limits_{n\geq k}}\frac{u_{n}(z)}{t^{2n-2k+1}}=t^{2k}\left( S-{\displaystyle\sum\limits_{n=0}^{k-1}}\frac{u_{n}(z)}{t^{2n+1}}\right) .
[ e_{\{m \}, i}^{( n )}\otimes^{n}1_{N}] [ e_{\{m^{\prime}\}, j}^{( n )}\otimes^{n}1_{N}] = \delta_{i , j}\delta ( \{m \}, \{m^{\prime}\}) \frac{n !}{f_{\{m \}}^{( n )}}e_{\{m \}, i}^{( n )},
R = \begin{bmatrix}1 -q^-_2 0 0 \ldots 0 0 -q^+_2 1 -q^-_3 0 \ldots 0 0 0 -q^+_3 1 -q^-_4 \ldots 0 0 \vdots \vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 0 \ldots 1 -q^-_{N-1}0 0 0 0 \ldots -q^+_{N-1}1\end{bmatrix}
{L}_{\xi}f ( x ) : = \lim_{\lambda \rightarrow 0}\frac{1}{\lambda}\{\phi_{\lambda}^{*}f ( e x p ( \lambda \xi ) x ) - f ( x ) \}= \frac{d}{d \lambda}|_{\lambda = 0}( \phi_{\lambda}^{*}f ( x^{\prime}) ) , \quad x^{\prime}= e x p ( \lambda \xi ) x .
I_{( M 2 )}= -{\frac{1}{4 \pi^{2}}}\int d^{3}\xi \bigg \{\sqrt{- \det ( \partial_{\alpha}{\bar{X}}^{M}\partial_{\beta}{\bar{X}}_{M})}+{\frac{1}{6}}\epsilon^{\alpha \beta \gamma}A_{M N R}\partial_{\alpha}{\bar{X}}^{M}\partial_{\beta}{\bar{X}}^{N}\partial_{\gamma}{\bar{X}}^{R}\bigg \},
\frac1\rho\partial_x f(z_-,z_-) = - \frac{z_+-z_-}{z_+-y_-}\left( z_--y_-\right)^{\rho-1}+ \frac{z_--y_-}{z_+-y_-}\left( z_+-z_-\right)^{\rho-1}&=\frac{z_+-z_-}{z_+-y_-}\left( z_--y_-\right)^{\rho-1}\left(\left(\frac{z_--y_-}{z_+-z_-}\right)^{2-\rho}- 1\right).
\chi = \begin{bmatrix}1 0 \cdots 0 1 0 \cdots 0 0 1 \cdots 0 1 1 \cdots 0 \vdots \vdots \ddots \vdots \vdots \vdots \ddots 0 0 0 \cdots 1 1 0 \cdots 1\end{bmatrix}.
ab\equiv a\otimes b\equiv ab^{T}=(\begin{matrix}a_{1}\\a_{2}\\a_{3}\end{matrix})(\begin{matrix}b_{1}&b_{2}&b_{3}\end{matrix})=(\begin{matrix}a_{1}b_{1}&a_{1}b_{2}&a_{1}b_{3}\\a_{2}b_{1}&a_{2}b_{2}&a_{2}b_{3}\\a_{3}b_{1}&a_{3}b_{2}&a_{3}b_{3}\end{matrix}).
\gamma_{\sigma_1, \sigma_2}^{[i,j]}(n_{\sigma_1}\otimes n_{\sigma_2}) \,\,=\,\, \sum_\nu \bigl( \,\, \cdots \otimes n_{pj}s''_\nu \otimes n_{jk}\otimes \cdots \otimes n_{hi}s'_\nu \otimes n_{iq}\otimes \cdots\bigr).
B_{6}^{+}=\frac{det(\begin{matrix}1&2&0&0&0&0\\ 1&3&3&0&0&0\\ 1&4&6&4&0&0\\ 1&5&10&10&5&0\\ 1&6&15&20&15&6\\ 1&7&21&35&35&21\end{matrix})}{7!}=\frac{120}{5040}=\frac{1}{42}
\Big(\frac{R}{K}\Big)_{prop}=\Big({\frac{1 - \frac{1}{q}}{2}}\Big)^z \;\; z \in \{2,3,\dots,m\}\geq \frac{1 - \frac{z}{q}}{2^z}\; \;z<q \;\; z \in \{2,3,\dots,m\}
\|\mathcal{A}(\mathcal{M}_f^{})^\nu\|_{\mathcal{L}\times \mathcal{L}}\le \eta(\nu-1) \, \|\mathcal{D}_f\|_{\mathcal{L}\times \mathcal{L}}\mathcal{D}_f = \begin{bmatrix}\hat{A}- A 0 0 \hat{S}- (C + B \hat{A}^{-1}B^\top) \end{bmatrix}.
\left| u_{\beta}( \phi \psi_{\beta}) \right| \leq C \sup_{\stackrel{{| p | \leq m + n}}{z \in K}}\left| D^{p}( \phi \psi_{\beta}) ( z ) \right| , \quad \phi \psi_{\beta}\in G_{0}^{\infty}( \widetilde{X}_{\beta}) \,
\frac{x}{p}= \frac{1-\frac{m^{1/2}}{c n^{1/2}}}{1 - \frac{cm^{1/2}}{2n^{1/2}}+O(m n^{-1})}= 1 - \frac{m^{1/2}}{c n^{1/2}}+ \frac{cm^{1/2}}{2n^{1/2}}+O(m n^{-1})
|B_{\epsilon/\log n}(S_n)| \le (2p)^{\frac{\epsilon n}{\log(n)}}&= (2\rho n)^{\frac{\epsilon n}{\log(n)}}&= (e^{\log(2\rho n)})^{\frac{\epsilon n}{\log(n)}}= e^{\epsilon \log(2\rho)\frac{n}{\log(n)}}e^{\epsilon n}&\le e^{2\epsilon n}.
\bar{F}(x) = \rho_{2}\bar{V}^{r}(x)+\rho_{1}f^{*}(\gamma)\bar{F}_{\gamma}(x) & + \rho_{1}f^{*}(\gamma)\left\lbrace \int_{0^-}^{\Delta}\bar{B}^{r}(x-u)dF_{\gamma}(u) + \int_{\Delta}^{x-\Delta}\bar{B}^{r}(x-u)dF_{\gamma}(u) + \int_{x-\Delta}^{x}\bar{B}^{r}(x-u)dF_{\gamma}(u)\right\rbrace.
K_{i k}\, K_{k j}~ = ~ K_{i j}~ , ~ t_{0}^{k l}\, K_{i j}~ = ~ e^{\,{\frac{i}{2}}\, ( \, \alpha_{i k}+ \alpha_{k j}+ \alpha_{j l}+ \alpha_{l i}\, ) \,}~ K_{i j}\, t_{0}^{k l}~ .
\begin{aligned}2n^{-2}(| \mathrm{supp}G_j | + 1 ) \sum_{\substack{j, k \in \mathbb{Z}, k \in \mathrm{supp}G_j \cup (\mathrm{supp}G_j+1 )}}E_{\nu^n_\rho}[ g(\eta_k ) G_j (\eta )^2 ](\nabla^n \varphi^n_j)^2 + \frac{n^2}{2}\sum_{k \in \mathbb{Z}}E_{\nu^n_\rho}[ g(\eta_k ) ( \nabla_{k,k- 1}f (\eta) )^2 ] .\end{aligned}
\lambda_{((a_{i_1}a_{i_2})\cdots) a_{i_n}}:= \lambda_{((a_{i_1}a_{i_2})\cdots )a_{i_{n-1}}}\lambda_{a_{i_n}}+ [\lambda_{((a_{i_1}a_{i_2})\cdots )a_{i_{n-1}}},\rho_{a_{i_n}}]\cr\rho_{((a_{i_1}a_{i_2})\cdots) a_{i_n}}:= \rho_{a_{i_n}}\rho_{((a_{i_1}a_{i_2})\cdots )a_{i_{n-1}}}+ [\lambda_{a_{i_n}}, \rho_{((a_{i_1}a_{i_2})\cdots )a_{i_{n-1}}}]
T_{n k}( x | g ) = \left( \begin{array}{c c}{{\Lambda_{\mu \nu}( x | g )}}&{{0}}\\{{0}}&{{- \varepsilon}}\\ \end{array}\right) \; , \; \; \Lambda_{\mu \nu}( x | g ) = \eta_{\mu \nu}\varepsilon - \frac{e_{0}}{2}\varepsilon g F_{\mu \nu}( x ) \varepsilon \; .
p\left({{d_{p2}}\leq \delta}\right) = \frac{{M{\delta^{2nk -{k^2}}}}}{{k!}}\int_{{{\mathbf{\Omega}}_2}}\prod\limits_{i = 1}^k{{x_i}^{n - 2k}}\,\,\, \times \prod\limits_{1 \leq i < j \leq k}{{{\left({{x_i}-{x_j}}\right)}^2}}d{x_1}\cdots d{x_k}
\sum_{\substack{r_1,r_2,r_3 \geq 0}}y_1^{r_1}y_2^{2r_3+r_2}\frac{q^{r_1^2+(r_2+r_3)^2+r_3^2-r_1(r_2+2 r_3)}}{(q)_{r_1}(q)_{r_2}(q)_{r_3}}= \sum_{\substack{n_1,n_2,n_3,n_4,n_{5}\geq 0}}y_1^{n_1+n_2+n_4}y_2^{2n_1+2n_3+n_4+n_5}\frac{q^{n_1^2+n_2^2+(n_3+n_5)^2+n_4^2+n_3^2+(2 n_3+n_5)(n_1)+n_4(n_1+n_2)+n_3 n_4}}{(q)_{n_1}(q)_{n_2}(q)_{n_3}(q)_{n_4}(q)_{n_5}}
\frac{n}{B_n}\left(\|\hat P^{(n)}- P^{(n)}\|_2^2 - \mathbb E\|\hat P^{(n)}- P^{(n)}\|_2^2\right) = \frac{n}{B_n}\langle \Xi^{(n)},u\rangle + o_{\mathbb P}(1),\; u = (0,0,1,0,0).
P^{-1}AP=[\begin{matrix}1&0&1\\ 1&2&0\\ 0&1&-1\end{matrix}]^{-1}[\begin{matrix}0&1&-2\\ 0&1&0\\ 1&-1&3\end{matrix}][\begin{matrix}1&0&1\\ 1&2&0\\ 0&1&-1\end{matrix}]=[\begin{matrix}1&0&0\\ 0&1&0\\ 0&0&2\end{matrix}]=D.
\zeta_{2}\left(-n_{1},-n_{2}\right)=\frac{1}{n_{2}+1}\sum_{k_{2}=0}^{n_{2}+1}\sum_{l_{1}=0}^{n_{1}+n_{2}+2-k_{2}}\sum_{l_{2}=0}^{k_{2}}\frac{\binom{n_{2}+1}{k_{2}}\binom{n_{1}+n_{2}+2-k_{2}}{l_{1}}\binom{k_{2}}{l_{2}}}{n_{1}+n_{2}+2-k_{2}}\mathcal{B}^{l_{1}}\mathcal{B}^{l_{2}}.
\times\qquad\left\vert\begin{array}{cccc}(x_1-x_{r+1})_1 (x_2-x_{r+1})_1 \ldots (x_r-x_{r+1})_1 (x_1-x_{r+1})_2 (x_2-x_{r+1})_2 \ldots (x_r-x_{r+1})_2 \vdots \vdots \ddots \vdots (x_1-x_{r+1})_r (x_2-x_{r+1})_r \ldots (x_r-x_{r+1})_r\end{array}\right\vert\qquad=\qquad
\begin{array}{l}u_{tt}+A_1u+CC^*u_t=\nabla G(u)+kBB^*u_t(t-\tau),t>0, u(0)=u_0,\u_t(0)=u_1, \B^*u_t(t-\tau)=g(t),\t\in (0,\tau),\end{array}
\Gamma_q(\theta N)\prod_{i=1}^N{\frac{\Gamma_q(\lambda_i + \theta(N - i) - z)}{\Gamma_q(\lambda_i + \theta(N - i + 1) - z)}}= (1 - q)\frac{(q; q)_{\infty}}{(q^{\theta N}; q)_{\infty}}\prod_{i=1}^N{\frac{\big(q^{\lambda_i + \theta(N - i + 1) - z}; q\big)_{\infty}}{\big(q^{\lambda_i + \theta (N - i) - z}; q\big)_{\infty}}}
\|\mathcal{A}\, (\mathcal{M}_s^{})^\nu\|_{\mathcal{L}\times \mathcal{L}}\le \bar{c}\, \eta(\nu) \, \|\mathcal{D}_s^{}\|_{\mathcal{L}\times \mathcal{L}}\mathcal{D}_s = \begin{bmatrix}\hat{A}_s 0 0 \hat{S}\end{bmatrix}.
(\phi^2+\phi*\phi)(a) = c^2(1+J'(\alpha^2,\alpha^2))\alpha^4(a)+\bar c^2(1+J'(\bar\alpha^2,\bar\alpha^2))\bar\alpha^4(a) +2(1+J(\alpha^2,\bar\alpha^2)) = c^2(1+J'(\alpha^2,\alpha^2))\bar\alpha^4(a)+\bar c^2(1+J'(\bar\alpha^2,\bar\alpha^2))\alpha^4(a) = 2\mathrm{Re}(c^2(1+J'(\alpha^2,\alpha^2)))\alpha^4(a).
Pr(K=3)=\frac{(\begin{matrix}3\\ 3\end{matrix})}{\sum_{K=0}^{7}(\begin{matrix}K\\ 3\end{matrix})}=\frac{(\begin{matrix}3\\ 3\end{matrix})}{(\begin{matrix}8\\ 4\end{matrix})}=\frac{1}{70}.
g_{\alpha}(\alpha_{b_i},c_i)=-\tilde{c}_2'(\alpha_{b_i})\alpha_{b_i}\tau N+\frac{-\tilde{c}_2'(\alpha_{b_i})NN_s}{h_ic_i}=\frac{1}{\ln(2)\sum_{j=i}^{N}\left(\lambda_j\right)}
\beta^\frac{1}{q}\,[u]_{W^{\beta,q}(\mathbb{R}^N)}\le C\,\left(\frac{\alpha}{\alpha-\beta}\right)^\frac{1}{q}\,\|u\|_{L^p(\mathbb{R}^N)}^{(1-\theta)\,\left(1-\frac{\beta}{\alpha}\right)}\,\|u\|^\theta_{L^r(\mathbb{R}^N)}\,\|\nabla u\|_{L^p(\mathbb{R}^N)}^{\frac{\beta}{\alpha}\,(1-\theta)},
\tau^r_c(A_0,\ldots,A_k):=\int_{G^{\times k}}{}^b{\rm Tr}_S\left(\Phi_{A_0}((g_1\cdots g_k)^{-1})\circ \Phi_{A_1}(g_1)\circ \ldots\circ \Phi_{A_k}(g_k)\right)c(e,g_1,g_1g_2,\ldots,g_1\cdots g_k)dg_1\cdots dg_k
I_{M}= < - \frac{1}{3 !}m^{2}a_{m n p}a^{m n p}- \frac{1}{2 !}m^{2}\psi^{2}+ \frac{1}{4 !}m \epsilon^{m n p q}\psi G_{m n p q}+ \frac{1}{3 !}m \epsilon^{m n p q}( a_{m n p}- C_{m n p}) \partial_{q}\phi > .
S_n= \begin{bmatrix}\frac{1}{f_{1}-\alpha_n}\begin{bmatrix}W_{1,1}\vdots W_{M,1}\end{bmatrix}+\sum_{j=0}^{\lfloor\frac{N}{2}\rfloor-1}\alpha_n^j I_{1,j}\vdots \frac{1}{f_{y}-\alpha_n}\begin{bmatrix}W_{1,y}\vdots W_{M,y}\end{bmatrix}+\sum_{j=0}^{\lfloor\frac{N}{2}\rfloor-1}\alpha_n^j I_{y,j}\end{bmatrix},
C = \biggl(n_{\alpha}\biggl\lceil \dfrac{B(\alpha^{\vee},\mu)}{n_{\alpha}Q(\alpha^{\vee})}\biggr\rceil - \dfrac{B(\alpha^{\vee},\mu)}{Q(\alpha^{\vee})}\biggr) \alpha^{\vee}\, ,D = g(B(\alpha^{\vee},\mu) - Q(\alpha^{\vee})),
A^{(1)}={\pi^2 \over 2}{\rm tr}(T^{a_1}T^{a_2}T^{a_3}T^{a_4}){u(\sin \pi{s\over 2}+ \sin \pi{t\over 2}+ \sin \pi{u\over 2})\over \Gamma({s\over2}+1) \Gamma({t\over 2}+1) \Gamma({u\over 2}+1)\sin(\pi{s\over 2})\sin(\pi{t\over 2})\sin(\pi{u\over 2})}
- \frac{1}{2}\sum_{m , n = - \infty}^{\infty}( m - n ) : ( \tilde{c}_{m}\tilde{c}_{n}\tilde{b}_{m + n}- \tilde{c}_{m}\tilde{c}_{n}b_{- ( m + n )}) : | B ; p , 0 \rangle = 0 ,
\begin{aligned}\mbox{Case (i)}:~v^k_{r}(T\gamma_g (2\tau+1))\leq\frac{\bar{v}^k (T\gamma_g (2\tau+1))+\underline{v}^k(T\gamma_g(2\tau+1))}{2}. & \mbox{Case (ii)}:~v^k_{r}(T\gamma_g (2\tau+1))\geq\frac{\bar{v}^k (T\gamma_g (2\tau+1))+\underline{v}^k(T\gamma_g(2\tau+1))}{2}.\end{aligned}
| H_{t}(x + if_{t}(x)) - H_{r}(y + if_{t}(y)) | &\leq \left| H_{t}(x + if_{t}(x)) - H_{r}(x + if_{t}(x)) \right| + \left| H_{r}(x + if_{t}(x)) - H_{r}(y + if_{r}(y)) \right| &\leq (t-r)\left[ |x + if_{t}(x)| + |F_{\mu}(x + if_{t}(x))|\right] + 2 |(x,f_{t}(x)) - (y,f_{r}(y))| &= O\left(\min\left\{(t-r) , |(x,f_{t}(x)) - (y, f_{r}(y))|\right\}\right)
\frac{m^2}{t_h}+ \frac{h^{2b + 1}- (h - t_h)^{2b + 1}}{2b + 1}= \frac{m^2}{\tau_h}+ \frac{h^{2b + 1}- (h - \min\{h,\tau_h\})^{2b + 1}}{2b + 1},
\begin{array}{rclcrcl}|e_R^c\rangle = \displaystyle\bigg|1, \frac{1}{2}, \frac{1}{2}\bigg\rangle, \quad -|\nu_R^c\rangle = \displaystyle\bigg|1, \frac{1}{2}, -\frac{1}{2}\bigg\rangle, |e_L^c\rangle = |2,0,0\rangle, \quad |\nu_L^c\rangle = |0,0,0\rangle,\end{array}
\mathbb{E}\big(|\widetilde{\nu}_{n+1}(A)-\widetilde{\nu}_{n}(A)|^p\big)\le N^{p-1}K_p\sum_{j=1}^N \Bigg[\Bigg(\sum_{S\in \mathcal{S}_n^j(A)}\mathbb{E}\bigg(\Big| \int_{S\cap A}U_S(x)V_S(x)\, \nu(dx)\Big|^2\bigg)\Bigg)^{p/2}&+\sum_{S\in \mathcal{S}_n^j(A)}\mathbb{E}\Big(\Big| \int_{S\cap A}U_S(x)V_S(x)\, \nu(dx)\Big|^p \Big)\Bigg].
\begin{cases}\mathcal{E}x_{k+1}=[A_1+A_2+B(K_1+K_2) \ \ \ \ \ \ \ \ \ \ \ +\bar{\alpha}B(\Delta K_{1,k}+\Delta K_{2,k})]\mathcal{E}x_k, \mathcal{E}x_{0}=x_0=\xi,\k\in{\mathcal N}_{T-1}.\end{cases}
M^{\otimes_{w^*h}\eta}=[Y_1\otimes_{h M'}\cdots \otimes_{h M'}Y_n]^{\natural M' normal}\cap [Z_1\otimes_{h M'}\cdots \otimes_{h M'}Z_n]^{\natural M' normal}&\simeq Y_1^{\natural}\otimes_{eh M}\cdots \otimes_{eh M}Y_n^{\natural}\cap Z_1^{\natural}\otimes_{eh M}\cdots \otimes_{eh M}Z_n^{\natural}
2^{2n}\widehat{A}_{2n+1}(x) =\prod_{i=1}^n((1+x)^2+a_i(1-x)^2), 2^{2n+1}\widehat{A}_{2n+2}(x) =(1+x)\prod_{i=1}^n((1+x)^2+b_i(1-x)^2).
|\partial_{t}\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right) v_{3}\vert_{r = R\lambda(t)},\phi_{0}\rangle| \leq \frac{C \log(\log(t))}{t^{3}\log^{2}(t)}+ \frac{C}{t^{2}\log^{-b +1 +b\alpha}(t)}
\Sigma_{T}= 3 2 g^{2}\int{\frac{d^{3}k}{( 2 \pi )^{3}}}{\frac{\sin^{2}{\frac{\tilde{p}\cdot k}{2}}}{k}}\left( n_{B}( k ) + n_{F}( k ) \right) + 4 g^{2}P^{2}\bar{\Sigma},
\sum_{n=0}^{\frac{p-1}{2}}F(n,0)= \sum_{n=0}^{\frac{p-1}{2}}F(n,p) + \sum_{k=1}^{p}G\Bigl(\frac{p+1}{2}, k\Bigr)= F\Bigl(\frac{p-1}{2}, p\Bigr) + \sum_{k=1}^{p}G\Bigl(\frac{p+1}{2}, k\Bigr),
I=\int_{C}d\tau \; \eta_{ab}\;[p^{a}\partial_{\tau}q^{b}+t^{\mu}(p^{a}e^{b}_{\;\mu}+j^{a}\omega^{b}_{\;\mu})]+\lambda_{1}(p^{2}-m^{2})+\lambda_{2}(W^{2}-m^{2}s^{2})
\|x\|_{l_p^d}:=\begin{cases}\Bigl(\sum\limits_{i=1}^d\vert x_i\vert^p\Bigr)^{\frac1p}\ 0<p<\infty, \#\{i\mid x_i\neq0\}\p=0, \max\limits_{i=i,\ldots,d}\vert x_i\vert \p=\infty, \end{cases}
\phi_{\gamma(n)}(t) = \prod_{i=1}^{n}\phi_{\tilde{\gamma}_i}(t) = \left[ \frac{2(-\frac{jst}{\bar{L}})^{\frac{K}{2}}}{(K-1)!}K_{K}\left( \sqrt{-\frac{4jst}{\bar{L}}}\right) \right]^n,
\int d^{d}\theta \; \theta^{0}\theta^{1}\theta^{2}\theta^{3}\theta^{5}. . . \theta^{d}= 1 , \quad d^{d}\theta = d \theta^{d}. . . d \theta^{5}d \theta^{3}d \theta^{2}d \theta^{1}d \theta^{0}.
H(v,v^{(0)},\mu)=\left[\begin{array}{c}(1-\mu)(w-z_1+(m-1)(\hat{\mathcal{A}}x^{m-2})^T(x-z_2))+\mu(x-x^{(0)}) Z_1x-\mu Z_1^{(0)}x^{(0)}Z_2w-\mu Z_2^{(0)}w^{(0)}+(1-\mu)Xw w-(1-\mu)(\mathcal{A}x^{m-1}+ q)-\mu w^{(0)}\end{array}\right]=0
0 =P_M[R_o^c]\ge P_M\left[\cap_{i=1}^M\{o(\eta,S)^{(i)}\}\right] &=E_M\left[P_M\left[\cap_{i=1}^M\{o(\eta,S)^{(i)}\}\big|\,{\cal G}\right]\right] &=E_M\left[\left(P_M\left[o(\eta,S)^{(1)}\big|\,{\cal G}\right]\right)^M\right].
M_{X_{i}}=\frac{1}{b}\left( -a+\sqrt{a^{2}+2b\left[ -\ln \left( 1-\left( \frac{1}{2}\right)^{(\frac{1}{\theta_{i}+\theta_{3}})}\right) \right]^{\frac{1}{\alpha}}}\right) ,\ \ \ \i=1,2.
\begin{array}{rcl}Z_0.d\phi_t+Z_{th}.d\phi_1 = (1-t)Z_0.d\phi_0+t[Z_0.d\phi_1+Z_h.d\phi_1] \geq (1-t)\delta (|Z_0|^2+|d\phi_0|^2)+t\delta' (|Z_0+Z_h|^2+|d\phi_1|^2) \geq min(\delta,\delta')[(1-t)|Z_0|^2+t|Z_0+Z_h|^2+(1-t)|d\phi_0|^2+t|d\phi_1|^2]\end{array}
\frac{1}{R(t)\log t}=\frac{6\cdot 0.04962}{\log t}\left(\frac{1-a_1(t)}{1+a_2(t)}\right) =\frac{6\cdot 0.04962}{\log t}\left(1-\frac{a_1(t)+a_2(t)}{1+a_2(t)}\right) \geq\frac{1}{3.359\log t}\left(1-\frac{8.02\log\log t}{\log t}\right)
\begin{aligned}k^{y}_{j+1}(T) \leq k^{y}_{0}(T)+ C C_{2}(T) B \big(1-3 \big(\tfrac{1}{p}-\tfrac{1}{q}\big),1-\tfrac{3}{q}\big) \Big[k^{u}_{j}(T) k^{\nabla y}_{j}(T) + k^{\nabla y}_{j}(T)^2 (k^{x}_{j}(T)+k^{y}_{j}(T)+ \lvert \overline{b}\rvert ) \Big].\end{aligned}
\mathcal{H}^{(k)}_e : W^{(k)}\to V_{h,e}^{(k)}: &\begin{cases}\mathcal{H}^{(k)}_e{u_{B_e}}\in V_{h,e}^{(k)}: a^{(k)}_e(\mathcal{H}^{(k)}_e{u_{B_e}},u^{(k)})=0 \forall u^{(k)}\in V_{I,h}^{(k)}, \mathcal{H}^{(k)}_e{u_{B_e}}_{|\Gamma^{(k)}}={u_{B_e}}_{|\Gamma^{(k)}}, \end{cases}
H =s''b'r'r''a'C_{r''a',ra_{1}}z(b_{1}ra_{1})m_{1}\hdots m_{l-1}s'n &=\displaystyle \sum_{w_{1}\in L(k)}s''b'w_{1}^{\ast}(r'r'')w_{1}a'C_{r''a',ra_{1}}z(b_{1}ra_{1})m_{1}\hdots m_{l-1}s'n
\|\nabla (dist_N(u)-dist_N(v))\|^2_{L^2(B)}\le C\int_B(|w|^2|\nabla u|^2+(|\nabla u|+|\nabla v|)|\nabla w||w|)dz \le\varepsilon\|\nabla w\|^2_{L^2(B)}+C(\varepsilon)\|w\|^2_{L^4(B)}(\|\nabla u\|^2_{L^4(B)}+\|\nabla v\|^2_{L^4(B)}).
\| \sum_{k > 0}u_k v_k\|_{L^1(Q)}\ \lesssim \ \| u_k v_k\|_{\ell^1_k L^1(Q)}\lesssim \| u_k\|_{\ell^2_k L^2(Q)}\| v_k\|_{\ell^2_k L^2(Q)}\lesssim \|S_{>0}(u)\|_{L^2(Q)}\|S_{>0}(v)\|_{L^2(Q)}\ \lesssim \|u\|_{BMO}\|v\|_{BMO}.
c_{\mu}^{( s )}( \rho ) = \frac{i \sqrt{\sinh \pi \mu}}{2 \pi}\phi_{M}( 0 , \rho ) \left\{\Gamma ( - i \mu ) \left( \frac{m \rho}2 \right)^{i \mu}- \Gamma ( i \mu ) \left( \frac{m \rho}2 \right)^{- i \mu}\right\},
\int\limits_0^T\int\limits_0^{\ell}\Big[-b_n(\tilde v)\psi_t^{\tau}+v_x^{\tau}\psi_x^{\tau}-f\psi^{\tau}\Big]\,dx\,dt -\int\limits_0^{\ell}b_n(\tilde\Phi)\psi^{\tau}(x,\tau)\,dx -\int\limits_0^Tp(t)\psi^{\tau}(\ell,t)\,dt +\int\limits_0^Tg^n(t)\psi^{\tau}(0,t)\,dt + \int\limits_{T-\tau}^T\int\limits_0^{\ell}b_n(\tilde v)\psi_t^{\tau}\,dx\,dt = 0 .
c_{k-2}= - n(n-1) a_k - n b_{k-1}, c_l = - n(n-1) a_{l+2}- n b_{l+1}- 2 \sum_{m=2}^{k-1-l}a_{l+m+1}\biggl[(n-1) m_{(m-1,\dot{0})}+ \sum_{p=1}^{[(m-1)/2]}m_{(m-1-p,p,\dot{0})}\biggr] - \sum_{m=1}^{k-2-l}b_{l+m+1}m_{(m,\dot{0})}, l=0, 1, \ldots, k-3.
\begin{cases}I^-:=[\ell q_n+b_n, \ell q_n+m_n-1], I^+:=[\ell q_n+m_n+1, (\ell+1)q_n-b_n], |\phi(x_0^-)|:=\max_{y\in I^-}|\phi(y)|, |\phi(x_0^+)|:=\max_{y\in I^+}|\phi(y)|, R_{\ell}:=[\ell q_n-b_n, \ell q_n+b_n], r_{\ell}:=\max_{k\in R_{\ell}}|\phi(k)|.\end{cases}
{\bf T}^{(a)}_{m}(\theta) =1 + h t^{(a)}_{1,m}(\theta) +h^2 t^{(a)}_{2,m}(\theta)+ O(h^3) m=4k,4k+1,{\bf T}^{(a)}_{m}(\theta) = h t^{(a)}_{1, m}(\theta)+ h^2 t^{(a)}_{2,m}(\theta)+ O(h^3) m=4k+2,4k+3.