harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta_{[ N + 2 k ] \times [ 2 k ]}\rightarrow \left( \! \begin{array}{c c}{{1_{[ N ] \times [ N ]}}}&{{0_{[ 2 k ] \times [ N ]}}}\\{{0_{[ N ] \times [ 2 k ]}}}&{{\bar{\Lambda}_{[ 2 k ] \times [ 2 k ]}}}\\ \end{array}\! \right) \Delta_{[ N + 2 k ] \times [ 2 k ]}\Lambda_{[ 2 k ] \times [ 2 k ]},
N_{1 2}^{b}( x , a , b , d ) |_{m = 0}= - \frac{1}{a^{d - 3}b}\frac{h ( d )}{2^{d}}\Gamma ( d - 3 ) \left( \zeta ( d - 3 , \frac{x}{a}+ 1 ) + \zeta ( d - 3 , - \frac{x}{a}+ 1 ) \right) .
\bar{J}_{|m}y^m-J_ia^{ik}b_{k|m}y^m-J_l\frac{\partial(G^l-\bar{G}^l)}{\partial y^i}b^i-2\frac{\partial \bar{J}}{\partial y^l}(G^l-\bar{G}^l)+\boldsymbol{K}\alpha^2 \phi^2 I_ib^i=0.
a_j' = \frac{\lambda_j^\ell{\textsl{\footnotesize R}}_j(0)}{d_j}= \frac{\lambda_j^\ell \sin(\theta_j)}{d_j}, a_j = \frac{\lambda_j^\ell{\textsl{\footnotesize R}}_j(0)}{d_j}\left(\frac{\ell}{\lambda_j}- \frac{d_j'}{d_j}\right) = \frac{\lambda_j^\ell\sin(\theta_j)}{d_j}\left(\frac{\ell}{\lambda_j}- \frac{d_j'}{d_j}\right),
\begin{aligned}\|D\vec v\|_{L^q(B_{\rho})}+\frac{1}{r-\rho}\|\vec v\|_{L^q(B_\rho)}&\le \frac{C}{r-\rho}\left(\|b-b_0\|_{L^{nq/(n+q)}(B_r)}+\|D\vec v\|_{L^{nq/(n+q)}(B_{r})}+\frac{1}{r-\rho}\|\vec v\|_{L^{nq/(n+q)}(B_{r})}\right).\end{aligned}
w_2 (x_1,x_2,t_1,t_2) & = \int\limits_0^{t_1}d\tau_1 \int\limits_0^{t_2}d\tau_2 ~q(t_1-\tau_1) ~q(t_2-\tau_2)~ g(x_2,\tau_1 + \tau_2|x_1) .
\frac{1}{n}\log d(\textbf{\textit{f}}_{0}^{-n}(q),\textbf{\textit{f}}_{0}^{-n}(p)) \leq \frac{1}{n}\log\frac{2}{\Delta_{0}}+ \frac{\log d(q,p)}{n}+\frac{1}{n}\sum_{k=-1}^{-n}\log\tau_{k}.
{\frac{1}{2 j + 1}}\left({\frac{D_{\mathrm{e f f}\s}^{j}}{D_{s}^{j}}}\right)^{2}= 4 \pi \sqrt{2}( T_{D 2}\alpha^{\prime})^{2}( k + 2 )^{3 / 2}\left[{\frac{\sin x_{j}}{x_{j}}}\right]
v_{n-1}\tfrac{\|z^{n-1}\|_{2}}{\|z^{n}\|_{2}}(- \mu_{2}+ n) + u_{n-1}\tfrac{\|z^{n-1}\|_{1}}{\|z^{n}\|_{1}}(-\mu_{1}+ n)= (\mu_{2}- \mu_{1})(\lambda + \mu_{2}+ \mu_{1}- 1) + (\lambda + 2n - 1), n \in \mathbb{Z}.
\lim_{t\to\infty}t^{\frac{N+A}{2}}v\left(t^{\frac{1}{2}}y,t\right)=c_dm(\varphi^{0,1})|y|^Ae^{-\frac{|y|^2}{4}}\,\,\,\mbox{in}\,\,\,L^2({\bf R}^N,e^{|y|^2/4}\,dy)\,\cap\,L^\infty(K), \lim_{t\to\infty}\,t^{\frac{N+2A}{2}}\frac{v(x,t)}{U(|x|)}=\frac{c_d}{c_*}m(\varphi^{0,1})\,\,\,\mbox{in}\,\,\,L^\infty(B(0,R)).
\mu_{p}(\mathcal{S})=1-p^{-nm}- p^{-(n-1)(m-1)}\cdot \frac{\left( 1-p^{-n}\right) \left( 1-p^{-m}\right)}{1-p^{-1}}=1 -\sum_{(n-1)(m-1)}^{(n-1)m}p^{-i}+\sum_{n(m-1)+1}^{nm-1}p^{-i}\,.
(b_n-2)(n-2) =\frac{(b_n-2)(2n-4)}{2}&<u_n=2n-4+(b_n-2)(2n-4)\sum_{i=b_n+1}^n\frac{1}{i-2}&\le 2n-4+\frac{(b_n-2)(2n-4)}{2}=b_n(n-2),
V_1(V_2V_1)^*\phi(t^{-1}D)f(V_2V_1)V_1^* =V_1V_1^*V_2^*\phi(t^{-1}D)V_2fV_1V_1^* &=V_1V_1^*V_2^*\phi(t^{-1}D)V_2f &\varpropto (-1)^{\partial \phi\cdot\partial f}V_1V_1^*fV_2^*\phi(t^{-1}D)V_2 &=(-1)^{\partial \phi\cdot\partial f}fV_2^*\phi(t^{-1}D)V_2 &\varpropto V_2^*\phi(t^{-1}D)fV_2\,.
T_{m}^{( r +{\frac{\epsilon - 1}{2}})}( u ) = \sum_{( \zeta^{( 1 )}, \ldots , \zeta^{( m )}) \in S p i n_{m}^{\epsilon}}\prod_{i = 1}^{m}s p ( \zeta^{( i )}; u - m + 2 i - 1 ) .
g^0 = a_1^+i\alpha_{-\frac{\pi}{2}, 1}^- - a_1^- i\alpha_{-\frac{\pi}{2}, 1}^+ + b_1 W + c_1 (-i\Lambda W) + a_2^+(\lambda^0)^2i\alpha_{0, \lambda^0}^- - a_2^-(\lambda^0)^2i\alpha_{0, \lambda^0}^+ + b_2 iW_{\lambda^0}+ c_2 \Lambda W_{\lambda^0},
\int_0^\epsilon \exp\left(-ca_k K\tilde{H}(x) \right) dx \lesssim \sum_{\ell=N_0}^\infty \exp \left(-\ell\cdot \beta\left( 1- \frac{ca_k}{\pi}\right) \log 2 \right) \lesssim |\epsilon|^{\beta'}< +\infty,
\begin{aligned}\frac{1}{1 + \big(\gamma^{-\alpha}p\big)^{-\alpha}}= \sum_{k=1}^{\infty}\frac{d^{\alpha k-2}t^{\alpha k-1}}{\Gamma(\alpha k)}= \frac{d}{dt}E_{\alpha}\big[\gamma^{-2\alpha}t^{\alpha}\big] = d^{-2\alpha}t^{\alpha-1}E_{\alpha,\alpha}\big[\gamma^{-2\alpha}t^{\alpha-1}\big]\end{aligned}
\aligned \int_{((X_1,\widehat{\mathcal U_1}) \times_{R\times P}(X_2,\widehat{\mathcal U_2}),(\widehat{\frak S_1}\times_{R\times P}\widehat{\frak S_2})^{\epsilon})}h_1 \wedge h_2 &=\int_{((X_2,\widehat{\mathcal U_2}),\widehat{\frak S_2})}(\pi_{R}\circ \widehat f_2)^*(\pi_{R}\circ \widehat f_1)!(h_1;(\widehat{\frak S_1})^{\epsilon})\wedge h_2.\endaligned
x_{\sigma (j)}^{\rho + \Delta}x_{\sigma (k)}^{\rho - \Delta}+ x_{\sigma (j)}^{\rho - \Delta}x_{\sigma (k)}^{\rho + \Delta}- x_{\sigma (j)}^{\rho + \delta}x_{\sigma (k)}^{\rho - \delta}- x_{\sigma (j)}^{\rho - \delta}x_{\sigma (k)}^{\rho + \delta}= 0.
V^{(2n+1)}= \widetilde A_0+ z\widetilde A_1+\ldots+ z^{n+1}\widetilde A_{n+1}+z^{n+2}\widetilde A_{n+2}, \m=2n+1, V^{(2n)}= \widetilde B_0+ z\widetilde B_1+\ldots+ z^{n}\widetilde B_{n}+z^{n+1}\widetilde B_{n+1}, \m=2n.
L_{f}^{m}= q^{1 / 2}p_{\mu}\dot{x}^{\mu}+ \frac{i}{2}( \psi_{\mu}\; \dot{\psi}^{\mu}- \psi_{5}\dot{\psi}_{5}) - \; \frac{e}{1 + q^{2}}( p^{2}- m^{2}) + i \chi ( \psi_{\mu}p^{\mu}- \psi_{5}m ) ,
G(\varepsilon ;\zeta):=\widetilde{g}^\varepsilon S_\varepsilon b(\mathbf{D})(B^0-\zeta \overline{Q_0})^{-1}+g^\varepsilon \bigl(b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon S_\varepsilon (B^0-\zeta \overline{Q_0})^{-1}.
p^* (Y_{1,k})= 1+p^* (X_{1,k})= 1+\frac{\Delta_{I(1,k+1)}\Delta_{I(2,k)}}{\Delta_{I(1,k-1)}\Delta_{I(2,k+1)}}= \frac{\Delta_{I(1,k)}\Delta_{J(2,k)}}{\Delta_{I(1,k-1)}\Delta_{I(2,k+1)}}.
h_\epsilon(x):=\begin{cases}h^+_n(x), x\in [e_n,b_n+\varepsilon_n),\; 1\leq n\leq M, h^-_n(x), x\in (a_n-\varepsilon'_n, e_n],\; 1\leq n\leq M, 0, x=b_n=a_j , h(x), .\end{cases}
f_1(x)=1+g_1^{\prime}x+\cdots+g_{s-1}^{\prime}x^{s-1}+x^s, &f_2(x)=1+g_1^{\prime\prime}x+\cdots+g_{r-1}^{\prime\prime}x^{r-1}+x^r, & ~~~~ f_{14}(x)=1+g_1^{\prime\prime\prime}x+\cdots+g_{t-1}^{\prime\prime\prime}x^{t-1}+x^t ,
c(p_1,p_2)\ge\max\bigg( \sum\limits_{k=1}^{p_1}(-1)^{k-1}c(p_1-k,p_2){a_2-cp_2+k-1\choose k}, \sum\limits_{j =1}^{p_2}(-1)^{j -1}c(p_1,p_2- j ){a_1-bp_1+ j -1\choose j}\bigg).
A_\nu(x_1,\ldots,x_d) \coloneqq B_\nu\Big(p_1/N + \sum_{j=1}^d x_j q_{1,j}/N,\, \ldots,\, p_s/N + \sum_{j=1}^d x_jq_{s,j}/N \Big) \prod_{i=1}^s \gamma_{\nu,i}^{p_i}.
\begin{aligned}\max_{\boldsymbol{\Theta},\:\mathbf{w}}\biggl|\Bigl(\mathbf{g}^T \boldsymbol{\Theta}\mathbf{H}+\mathbf{h}_d^T\Bigr)\mathbf{w}\biggr|^2 \textrm{s.t.}\|\mathbf{w}\|^2\leqslant 1 \phi_n\in [0,2\pi), \: \forall n=1,2,\ldots,N.\end{aligned}
={\pi (\Gamma (\ell +1))^2\over 2^{2\ell +2}(\Gamma (\ell +{3\over 2}))^2 (\Gamma (2\ell +1))^2}\;k^{2\ell -1}r_0^{2\ell +1}e^{-2\pi kr_0}|\Gamma (\ell + 1+2ikr_0)|^2
[ Q_{0}( \Phi_{0}) , \Phi_{0}] = [ [ Q_{0}, R ] , R ]^{L}\mathcal{I}= [ [ Q_{0}, R ]^{L}, R^{L}] \mathcal{I}= [ [ Q_{0}^{L}, R^{L}] , R^{L}] \mathcal{I}\ .
R_{1}(a,b,d)=-\frac{1}{\left( 2\pi \right)^{d-2}ab}\sum_{n^{\prime}=1}^{\infty}\int d^{d-2}p\frac{1}{(\vec{p}^{\,2}+m^{2}+(\frac{n^{\prime}\pi}{b})^{2})}
x \displaystyle\frac{d}{dx}\ln \left( \displaystyle\frac{I_l}{K_{l +N}}\right) = x \displaystyle\frac{I'_l}{I_l}+ l + N + 0 \left(\displaystyle\frac{1}{N + l}\right) > l + N + 0 \left(\displaystyle\frac{1}{N + l}\right),
r_j = A^{\frac{-\lambda}{n-\lambda(n-2m)}}\frac{|x_j|^{1+b}}{\psi(|x_j|)^{\frac{\lambda-1}{n-\lambda(n-2m)}}}&\le A^{\frac{-\lambda}{n-\lambda(n-2m)}}|x_j|^{1+b/2}.
\mathbf{Pr}\left(\hat{x}_{i,min,\gamma}> x_i + \epsilon\right)=\left[1-\frac{1}{K}\left( \gamma K \right)E\left\{\frac{2}{\pi}\tan^{-1}\left(\left(\frac{\sqrt{\epsilon}}{\eta_{ij}}\right)\right)\right\}\right]^M = \left[1-\frac{1}{K}H(\gamma, K; \epsilon,0.5)\right]^M
\begin{bmatrix}n_{11}n_{12}n_{13}n_{21}n_{22}n_{23}n_{31}n_{32}n_{33}\end{bmatrix}\begin{bmatrix}2 -1 -1 -1 2 -1 -1 -1 2\end{bmatrix}= \begin{bmatrix}-16 8 8 8 -16 8 8 8 -16\end{bmatrix}
S = \sum_{d | q}d \sum_{n' = 1}^{q/d - 1}\Big| \sum_{b = B+1}^{B+L}e\Bigl(-\frac{n' b}{q/d}\Bigr) \Big| &\ll \sum_{d | q}d \sum_{n' = 1}^{q/d - 1}\min\Bigl(L, \frac{1}{\|\frac{n'}{q/d}\|}\Bigr) &\ll \sum_{d | q}d \Bigl[ \sum_{n' \le q/(Ld)}L + \sum_{q/(Ld) < n' \le q/(2d)}\frac{q/d}{n'}\Bigr] \ll \sum_{d | q}d \frac{q}{d}\log q \ll d(q) q \log q.
\frac{\partial}{\partial t}(\frac{g}{f}\nabla^{2}Z)+V\cdot\nabla\eta-f\frac{\partial\omega}{\partial p}=(\xi\frac{\partial\omega}{\partial p}-\omega\frac{\partial\xi}{\partial p})+(\frac{\partial\omega}{\partial y}\frac{\partial u}{\partial p}-\frac{\partial\omega}{\partial x}\frac{\partial v}{\partial p})
d\boldsymbol{X}_t=g(\boldsymbol{X}_t)d\boldsymbol{W}_th(\boldsymbol{X}_t)+h(\boldsymbol{X}_t)d\boldsymbol{W}_t^*g(\boldsymbol{X}_t)+\left(b(\boldsymbol{X}_t)+\alpha Tr\left(\boldsymbol{X}_t\right)\boldsymbol{I}\right)dt,
{\cal T}_\alpha=[\bar{C}_\alpha,\Omega]=T_\alpha+C^{\dagger\beta}U^\gamma_{\beta\alpha}\bar{C}_\gamma(-1)^{\varepsilon_\alpha}+\bar{U}^\gamma_{\alpha\beta}\bar{C}_\gamma C^\beta(-1)^{\varepsilon_\beta}+\bar{C}^\dagger_\gamma\bar{U}^{\dagger\gamma}_{\beta\alpha}C^\beta(-1)^{\varepsilon_\alpha},
Rf(z, t) = \frac{1}{\chi}\int_{[0,1]}\int_{-t}^\infty f(y,u)\,du d\nu(y) + \lim_{s\rightarrow 0^+}\frac{1}{2\pi}\int e^{-it \theta}U(s-i\theta)\hat{f}(z,\theta)\,d\theta,
\bar{P}_{\pi}^{( \pm )}= 2 ( 1 - x^{2}) ( 1 -{\frac{\omega}{v}}) + \omega^{2}x^{2}( 1 - v^{2}x^{2}) \mp s \sqrt{1 - v^{2}x^{2}}\; ( 2 - 2 x^{2}- \omega v x^{2}){\frac{M \omega}{p_{\perp}}}\, .
{\bf 5 6}\rightarrow{\bf ( 8_{v}, 2 , 1 , 1 ) \, \oplus \, ( 8_{s}, 1 , 2 , 1 ) \, \oplus \, ( 8_{s^{\prime}}, 1 , 1 , 2 )}\, \oplus \,{\bf ( 1 , 2 , 2 , 2 )}
\Gamma^1_{11}=\frac{GE_x-2FF_x+FE_y}{2(EG-F^2)}, \Gamma^2_{11}=\frac{2EF_x-EE_y-FE_x}{2(EG-F^2)}, & \Gamma^1_{12}=\Gamma^1_{21}=\frac{GE_y-FG_x}{2(EG-F^2)}, \Gamma^2_{12}=\Gamma^2_{21}=\frac{EG_x-FE_y}{2(EG-F^2)}, & \Gamma^1_{22}=\frac{2GF_y-GG_x-FG_x}{2(EG-F^2)}, \Gamma^2_{22}=\frac{EG_y-2FF_y+FG_x}{2(EG-F^2)}.
{\beta^*_{1}}=t_{3,3}^*\, \left( = id \right);{\beta^*_{2}}=t_{3,2}^* + t_{2,2}^*\, \left( = t_{3,2}^* + id \right);{\beta^*_{3}}=t_{3,1}^* + t_{2,1}^* + t_{1,1}^*\, \left( = t_{3,1}^* + t_{2,1}^* + id \right).
S_{P}[ \eta , A ] = t \sum_{x}\sum_{\stackrel{\scriptstyle \mu , \nu = 1}{\mu < \nu}}^{3}A_{\mu \nu}( x ) \Big [ \eta_{+ \mu}( x , \nu ) \eta_{- \mu}( x + \hat{\mu}, \nu ) + \eta_{+ \nu}( x , \mu ) \eta_{- \nu}( x + \hat{\nu}, \mu ) \Big ] ,
\langle \omega_0, \xi_M\rangle T\mathcal T^{k-1}(\chi f)=-\xi_M \mathcal T^{k-1}(\chi f)+\sum_{j=1}^{n-1}a_jL_j\mathcal T^{k-1}(\chi f)+\sum_{j=1}^{n-1}b_j\overline L_j\mathcal T^{k-1}(\chi f)+O(|z|)D\mathcal T^{k-1}(\chi f).
H_{1}(t)=C\|\nabla K(t)\|_{L_{t}^{1}L^{2}}\| j_{0}\|_{L^{2}};H_{3}(t)= C\|b\|_{L_{t}^{\infty}L^{\infty}}\|\nabla^{2}K(t)\|_{L_{t}^{1}L^{1}};
\nu^{S_1}_j = \frac{c_j}{w^{S_1}_j}, j \in S_1 = J \nu^{S_{k}}_j = \nu^{S_{k-1}}_j + \left(\frac{w^{S_{k-1}}_j}{w^{S_k}_j}- 1\right) \, \left[\nu^{S_{k-1}}_j - \nu^{S_{k-1}}_{\pi_{k-1}}\right], \, j \in S_{k}, 2 \leq k \leq n,
T_{\mu\nu}(k)=i\int \frac{d^2p}{(2\pi )^2}\frac{{\rm tr}\{\gamma_{\mu}\left[ (p\!\!\!\slash +k\!\!\!\slash) + m\right]\gamma_{\nu}(p\!\!\!\slash +m)\}}{\left[ (p+k)^2 -m^2\right] (p^2 - m^2 )}.
\hat{\tilde{r}}_\mathrm{upa}=\Big(1-\frac{\tau_\mathrm{a}+\tau_\mathrm{p}}{\tau_\mathrm{f}}\Big)\frac{1}{N+\mu}\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}\log_2(1+\hat{\tilde{\gamma}}_{k,n}),
G = \begin{cases}(\bigoplus_{p\in P_1}\mathbb{Z}_{p^{m_p}}^{n_p}) \oplus (\bigoplus_{p\in P_2}\mathbb{Z}[p^{\infty}]^{n_p}) \mbox{if}G (\bigoplus_{p\in \mathbb{P}}\mathbb{Z}[p^{\infty}]^{n_p}) \oplus (\mathbb{Q}^{n}) \mbox{otherwise,}\end{cases}
\mathcal{D}_{1}=\sup_{R > 0}\left(\int_{|z| \geq R}\left( G\left(\frac{z}{2}\right) \right)^{q}|z|^{-\beta q}dz\right)^{\frac{1}{q}}\left(\int_{|z| \leq R}|z|^{\alpha p(1-p')}dz\right)^{\frac{1}{p'}}<\infty.
\nabla' \psi (\xi)= s(s+2)(s+4)(y_0'\cdot v)^2 \cdot \frac{y_0'-\xi'}{|y_0-\xi|^{s+6}}-s(s+2)\frac{y_0(d+1)}{\xi(d+1)}\frac{y_0'-\xi'}{|y_0-\xi|^{s+4}},
\frac{1}{\phi(Q)|Q|}\int_Q\sum_{j \in J_1}\lambda_j a_j(y)\,dy =\frac{1}{\phi(Q)|Q|}\int_Q\sum_{j \in J_1}\lambda_j\left(\frac{1}{|Q_j|}\int_{Q_j}a_j(y)\,dy\right)\chi_{Q_j}(z)\,dz &\le\frac{1}{\phi(Q)|Q|}\int_Q\sum_{j \in J_1}\lambda_j\chi_{Q_j}(z)\,dz &\le\left\|\sum_{j=1}^\infty\lambda_j\chi_{Q_j}\right\|_{{\mathcal M}_{1,\phi}}.
\sum_{k=1}^{n}f(k)=\int_{0}^{n}f(x)dx+\frac{f(n)+f(0)}{2}+\frac{B_{2}}{2}(f^{\prime}(n)-f^{\prime}(0))-\frac{1}{2}\int_{0}^{n}f^{\prime\prime}(x)P_{2}(x)dx.
g ( (x_{-(l+r)}\ldots x_{-1}x_0 x_1 \ldots x_{l+r})) = (f (x_{-(l+r)}\ldots x_{-1}x_0 x_1 \ldots x_{l+r})) = (y_{-l}\ldots y_{-1}y_0 y_1 \ldots y_{l}) .
i\,\lim_{r\rightarrow 0}\left(m r\right)^{1-a}g_{\ell}^{\pm}(r)\sin\left(\frac{\pi}{4}+\frac{\Theta^{\pm}}{2}\right)=\lim_{r\rightarrow 0}\left(m r\right)^{a}f_{\ell}^{\pm}(r)\cos\left(\frac{\pi}{4}+\frac{\Theta^{\pm}}{2}\right)\,.
E_{(x,T)}\int_0^{\sigma}\! \langle \dot{z}_\tau, N_\tau\rangle_{g_T}\, d\tau=E_{(x,T)}[\langle z_\sigma, N_\sigma\rangle_{g_T}-\int_0^{\sigma}\! \langle{z}_\tau, dN_\tau\rangle_{g_T}]= E_{(x,T)}\langle z_\sigma, N_\sigma\rangle_{g_T},
\begin{array}{c}\mbox{for every}m,\ell\in\mathbb{N},\mbox{and}j_1,j_2\in\{1,\dots, r(\sigma) \}, \sigma^m(x_{j_1})=\sigma^{\ell}(x_{j_2})\mbox{if and only if}j_1=j_2\mbox{and}|x_{j_1}^{\langle\sigma\rangle}| \mbox{divides}\ell-m.\end{array}
\beta_m = \min \left\{\sum_{\omega \in \Omega_m'}|\omega| x_{\omega}q_{\omega}\, \,:\, \, x_{\omega}\geq 0 \,\, \forall \omega \in \Omega_m', \,\sum_{\omega \in \Omega_m'}|\omega| x_{\omega}= 1, \, \sum_{\omega \in \Omega_m'}x_{\omega}A_{\omega}\succeq 0 \right\}
\Gamma(k) = \begin{bmatrix}\Gamma(k)_{1,1}\Gamma(k)_{2,1}\vdots \Gamma(k)_{n,1}\end{bmatrix}\otimes \begin{bmatrix}\Gamma(k)_{1,1}\Gamma(k)_{1,2}\ldots \Gamma(k)_{1,n}\end{bmatrix}.
S_{11}^{A_{2}^{(1)}}\left( \theta \right) =S_{22}^{A_{2}^{(1)}}\left( \theta\right) =\left\{1\right\}_{\theta}\qquad S_{12}^{A_{2}^{(1)}}\left(\theta \right) =\left\{2\right\}_{\theta}\qquad S^{A_{2}^{(2)}}\left(\theta \right) =\left\{1\right\}_{\theta}\left\{2\right\}_{\theta}\,.
\sup_{B}\big|P \bigl(\bigl(X_{n:n}^n,\dots,X_{n-k+1:n}^n\bigr)\in B \bigr)-H_k(B)\big| &\quad\le \bigl(D^{**}n^{-2\delta}k^{2\delta+1}\bigr)^{1/2}+ck/n=D\bigl((k/n)^\delta k^{1/2}+k/n\bigr),
\partial_\mu E = \left\{x\in G:\, \min\left\{\frac{\mu(E\cap B_d(x,r))}{\mu(B_d(x,r))}, \frac{\mu(E^c \cap B_d(x,r))}{\mu(B_d(x,r))}\right\}>\epsilon \, \,\forall r>0\right\}.
g_{1}(\nabla^{^{M_1}}_{U}V,X) =g_{1}(V,\hat{\nabla}_{U}\phi\mathcal{B}X)+g_{1}(V,\mathcal{T}_{U}\omega\mathcal{B}X)+g_{2}((\nabla\pi_{\ast})(U,\varphi V),\pi_{\ast}\mathcal{C}X) &+g_{1}(V,\varphi U)\eta(X),
=-n^{n-2}(n+1)^{n-1}\times\left|\begin{array}{cccccc}-1 1 0 \cdots 0 0 0 -1 1 \cdots 0 0 0 0 -1 \cdots 0 0 \cdots \cdots \cdots \cdots \cdots \cdots 0 0 0 \cdots -1 1 -1 -2 -3 \cdots -(n-2) n^{2}-n+1 \end{array}\right|
\partial J=\dfrac{\partial J}{\partial{\mathcal F}_{ij}}\partial{\mathcal F}_{ij}= J{\mathcal{A}}_{ij}\partial{\mathcal F}_{ij}, &\partial{\mathcal{A}}_{ij}= -{\mathcal{A}}_{i\ell}\partial{\mathcal F}_{m\ell}{\mathcal{A}}_{mj},
{l_j}={g_{ij}}\frac{{{y^i}}}{F}={(\frac{1}{2}{F^2})_{{.i}{.j}}}\frac{{{y^i}}}{F}= \frac{{{{(\frac{1}{2}{F^2})}_{{.j}}}}}{F}= \frac{{F{F_{{y^j}}}}}{F}={F_{{.j}}},
2\langle \hat{\Pi}_{J_0}v-\hat{\Pi}_{J}v,f-v \rangle_n &\leq (1/3)\|\hat{\Pi}_{J_0}v-\hat{\Pi}_{J}v\|_n^2+3\|f-v\|_n^2 &= (1/3)\left(\Vert\hat{\Pi}_{J_0}v\Vert_n^2-\Vert\hat{\Pi}_{J}v\Vert_n^2\right)+3\|f-v\|_n^2,
\frac{d}{dx}f(x) = \frac{\theta^{2}}{B(a,b)\,x^6}\exp\left(-\frac{a\theta}{x^2}\right)\left[1-\exp\left(-\frac{\theta}{x^2}\right)\right]^{b-1}&\times\left[4a-\frac{6x^2}{\theta}+\frac{4(b-1)}{1-\exp\left(\frac{\theta}{x^2}\right)}\right],x>0.
\|\nabla_j^{l_1}\nabla_k^{l_2}\mathcal{K}_{j,k}v\|_{L^{\widetilde{p},2}_{{\mathcal{I}_l},j,k}}\lesssim_{\widetilde{p}}\|\mathcal{K}_{\mathcal{I}_l}v\|_{L^{\widetilde{p},2}_{j,k}},\,\,\,l_1,l_2=0,1.
\frac{p_{\beta_{1},\lambda}}{p_{0}}\frac{p_{\beta_{2},\lambda}}{p_{0}}=(1+\lambda^{2})\exp^{\frac{y^{2}}{1+\lambda^{2}}-\frac{1}{2}((y-\frac{\lambda}{\sqrt{2}}(u+v))^{2}+(y-\frac{\lambda}{\sqrt{2}}(u-v))^{2})}&=(1+\lambda^{2})\exp^{\frac{y^{2}}{1+\lambda^{2}}-(y-\frac{\lambda}{\sqrt{2}}u)^{2}-\frac{\lambda^{2}}{2}v^{2}}
I^m_\gamma(c) \leq \inf \left\{E_\gamma(f) \ : \f \in S(c) \cap B_\gamma(m/4)\right\}&< \inf\left\{E_\gamma(f) \ : \f \in S(c) \cap \left( B_\gamma(m) \backslash B_\gamma(m/2)\right)\right\}&\leq E_\gamma(\phi) = I^m_\gamma(c)
\omega_{2k}(u)=(-1)^{\frac{k(k-1)}{2}}\frac{(2k+1)(\det \mathbf M_k)^{2}}{2^{(2k-1)k+1}u^{k^{2}}(k+1)}\prod_{j=1}^{k+1}\left[\frac{(2j-1)^{2}}{(2j-1)^2-u}\right]^{k},\forall u\in(0,1).
d_\infty((u_{1},v_1,w_2),(u_{1},v_1,w_2)) \leq \sum \limits_{l=1}^{T-1}\frac{1}{|\bar{l}|}|v_1w_1^* - v_2w_2^* |_{S_2}+ \left| \sum \limits_{l=1}^{T-1}\frac{1}{|\bar{l}|}(e^{i2 \pi l (u_1-z)}- e^{i2 \pi l (u_2-z)}) \right| \lesssim \log(T) |v_1 - v_2 |_2+ \log(T) |w_1 - w_2 |_2 + T |u_1-u_2|.
\frac{1}{2}\tilde \Delta ( \tilde \Delta + 2 ) Y_0^{(3)}= \tilde \nabla^a (\alpha_H^{(2)})_a + \tilde \nabla^a ( f^{(1)}\tilde \nabla_a C_i\tilde X^i )+ \frac{1}{2}\tilde \Delta (f^{(1)}C_i\tilde X^i ).
\mathbb{T}^{T(a,b)}_{\mu}\left(\rho_{u,\omega_1}\right) = \frac{\mathbb{T}^{T(a,b)}_{\lambda}\left(\rho_{u,\pm}\right)}{d\,v(u)/d\,u}= \frac{2a+1}{2\left(\omega_1-\omega_1^{-1}\right)^2}
\psi_{1}\left( \rho , \varphi \right) = \psi_{n - 1 , l - 1}^{\left( j \right)}\left( \rho , \varphi \right) , \psi_{2}\left( \rho , \varphi \right) = - i \psi_{n , l}^{\left( j \right)}\left( \rho , \varphi \right) .
\displaystyle \sum_{n=1}^{\infty}S_n^{\alpha}\frac{|A_{n}-A_{n-1}|^{2}}{n}\geq \frac{(\alpha-1)^{2}}{4}\displaystyle \sum_{n=1}^{\infty}\frac{n}{S_n^{2-\alpha}}|A_{n}|^{2},
\sum_{n \leq X}|A_f(n)|^2 = \sum_{n\leq X}\sum_{d^2 | n}\mu_N(d)C_f(n/d^2) = \sum_{d^2 \leq X}\mu_N(d) \sum_{v \leq X/d^2}C_f(v) = \sum_{d \leq \sqrt{X}}\mu_N(d) \left(R_1 \frac{X}{d^2}+O\left(\frac{X^{3/5}}{d^{6/5}}\right) \right).
\mathcal{E}(\vec{v})=\frac{2k+1}{2}\mathrm{vol}(K)+\frac{1}{2}\int_{K}[\sum\limits_{i,j=1}^{2k}(h_{ij}(\vec{v}))^{2}+\sum\limits_{i=1}^{2k}(\left\langle\nabla_{\vec{v}}\vec{v},e_{i}\right\rangle)^{2}]
Q_t = \begin{pmatrix}Q_{t_1}F_{t_1}Q_{t_2}F_{t_2}\end{pmatrix}= \begin{pmatrix}Q_{t_1}Q_{t_2}\end{pmatrix}\widehat Q_t, \widehat Q_t := \begin{pmatrix}F_{t_1}F_{t_2}\end{pmatrix}.
\beta_i:=\alpha_i \ (1\leq i\leq p-1), \beta_p:=\alpha_p+\alpha_{p+1}+\cdots+\alpha_{m+n-q}, &\beta_i:= \alpha_{i+m+n-p-q}\ (p+1\leq i\leq p+q-1), &\gamma_i:=\alpha_{p+i}\ (1\leq i\leq m+n-p-q-1).
\begin{aligned}\sup_{B_{r_{k+1}}}v\leq\sup_{B_{\tau_2r_{k}}}v \leq (1-\mu)\left(\hat{C}M r_{k}^{\beta}-M r_{k}^{\alpha}\right)+M r_{k}^{\alpha}&\leq \hat{C}M r_{k+1}^{\beta}\cdot \frac{1-\mu}{\tau_1^{\beta}}+\mu M r_{k}^{\beta}&\leq \hat{C}M r_{k+1}^{\beta}\left(\frac{1-\mu}{\tau_1^{\beta}}+\frac{\mu}{\hat{C}\tau_1^{\beta}}\right).\end{aligned}
K_l(u)=\left ( \begin{array}{ll}2x_{0}(q^{-1}s^{-1}u-qsu^{-1}) \; x_{+}(q^{-1}u^2 -qu^{-2}) x_{-}(q^{-1}u^2 -qu^{-2}) \; -2x_{0}(su-s^{-1}u^{-1})\end{array}\right )\, ,
\mathcal{R}^+_1 = \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}g\Bigl( \frac{m^2 n}{N}\Bigr) \sum_{c \geq \frac{C_1}{m}}\frac{S(n, p; c)}{c}H_{m, n}^+ \Bigl( \frac{4 \pi \sqrt{n p}}{c}\Bigr),
\hat{K}_{( 1 / 2 )}^{\beta}( s \mid \tau , \mathrm{\boldmath x}; \tau^{\prime}, \mathrm{\boldmath x '}) = \sum_{n = - \infty}^{\infty}( - 1 )^{n}\hat{K}_{( 1 / 2 )}( s \mid \tau , \mathrm{\boldmath x}; \tau^{\prime}+ \beta n , \mathrm{\boldmath x '})
\sum_{t=0}^{n-1}\{S_{1,k}(n,s,t) + S_{2,k}(n,s,t) \}\leq\frac{\pi K_3 \{r\sin(\pi d)\}^{k-1}n^{(1/2)-d}}{(1-4d^2)^{1/2}(s+1)^{(1/2)+d}}.
(k+m)^{n-1}(k+l)^{g}T_G\left(\frac{kx+kw+lw_*+m}{k+m},\frac{ky+kz+l+mz_*}{k+l}\right) = \sum_{\O}k^{|\O^{o}|}l^{|\O^{u}|}m^{|\O^{b}|}x^{| I(\O^{+}) |}w^{| I(\O^{-}) |}{w_*}^{| I(\O^{u}) |}y^{| L( \O^{+}) |}z^{| L(\O^{-}) |}{z_*}^{| L(\O^{b}) |}
k_{(i)\mu\nu}= -{\mu\over 4}(f^Y_{~\mu\lambda}f^{(i)\lambda}_{~~\nu}+f^Y_{~\nu\lambda}f^{(i)\lambda}_{~~\mu})+{1\over 2\mu}(k_{(5)\mu}k_{(i)\nu}+k_{(5)\nu}k_{(i)\mu}).
\dot{r}= p_{r}\ \ \ , \ \ \ \dot{p}_{r}= \frac{1}{r^{3}}p_{\theta}^{2}\, - \, \omega^{2}r \ \ \ ; \ \ \ \dot{\theta}= \left( \frac{1}{r^{2}}+ g^{2}\right) p_{\theta}\ \ \ , \ \ \, d o t{p}_{\theta}= 0 \ \ \ ,
{\bf Y}_2 =\xi^2\chi_{1}(g_{(2\,l)})^2\chi_{(1\,l)}(g_{(q+1\,l)})y_1y_{(1\,q)}y_{(1\,l)}+\xi^2\chi_{1}(g_{(2\,l)})^2y_1y_{(1\,l)}y_{(1\,q)}&+\xi\chi_{1}(g_{(2\,l)})y_{(1\,l)}y_1y_{(1\,q)}+\chi_{112}(g_{(q+1\,l)})\chi_{(3\,l)}(g_{(q+1\,l)})y_{(1\,q)}y_{(1\,l)}y_1 &+\xi\chi_{1}(g_{(2\,l)})\chi_{112}(g_{(q+1\,l)})\chi_{(3\,l)}(g_{(q+1\,l)})y_{(1\,q)}y_1y_{(1\,l)}
CP_1(T,r)=E_{h_i}\left[e^{-T r^\alpha I-T r^{\alpha}\frac{\sigma^2}{P}}\right] = \underset{i \in \psi}\prod E_{h_i}\Big[e^{-T r^\alpha h_i d_i^{-\alpha}}\Big]e^{-T r^{\alpha}\frac{\sigma^2}{P}}=\underset{i \in \psi}\prod \frac{1}{1+T r^\alpha d_i^{-\alpha}}e^{-T r^{\alpha}\frac{\sigma^2}{P}},
\begin{array}{c c c c c c c}{{{\cal L}_{\mathbf{I}}K^{x}}}&{{=}}&{{\epsilon^{x y z}K^{y}W_{\mathbf{I}}^{z}}}&{{;}}&{{{\cal L}_{\mathbf{I}}\omega^{x}}}&{{=}}&{{\nabla W_{\mathbf{I}}^{x}}}\\ \end{array}
\Phi_{1}^{( 2 )}= \pi^{2}\; , \; \; \Phi_{2}^{( 2 )}= \pi_{\mu}\psi^{\mu}\; , \; \; \Phi_{3 \mu}^{( 2 )}= \frac{1}{8 !}\epsilon_{\mu \nu \rho_{2}\ldots \rho_{9}}\pi^{\nu}\psi^{\rho_{2}}\ldots \psi^{\rho_{9}}+ \frac{\alpha}{2^{8}}\pi_{\mu}\; .
{\mathbb S}_k^{\gamma,u_h^1,s_h}(\gamma)=c_n(k),\quad\forall\, s_h\in \big\{{\mathbb S}_k(u^1_h,\,\cdot\,)>c_n(k)-\lambda_h\big\}\cap (u_h^1)^{-1}(\mathcal W_\gamma),
\Gamma \sim \frac{\kappa_{5}^2 L_{eff}\cos^2 \psi}{T_{eff}^2 \omega}\int^{\infty}_{-\infty}dp_{1}dp_{2}dp_{3}\delta(p_{1}+ p_{2}+ p_{3}- \frac{\omega}{2}) \prod_{i=1,3}\frac{p_{i}}{1 - e^{-\frac{p_{i}}{T_{L}}}}\times \frac{\omega}{2( 1 - e^{- \frac{\omega}{2 T_{R}}})},
\textrm{E}\left[ \|I_2^n (\theta',\theta)\|^2 \right] \leq \left(\frac{\theta-\theta'}{\theta \theta'}\right)^{2\gamma}\theta^{2\left(H-\frac{1}{2}-\gamma\right)}\left(\sum_{m=1}^\infty d^{m+1}C^m \|b_n\|_{L_{\infty}^1}^m A_m^{\gamma}(H,d,|T|)^{1/4}\right)^2.
\lim_{\epsilon \rightarrow 0}\left[{\frac{\hbar^{2}}{2}}\int_{0}^{1}d s{\frac{\delta^{2}}{\delta \sigma^{\mu \nu}( s - \epsilon / 2 ) \delta \sigma_{\mu \nu}( s + \epsilon / 2 )}}- m^{4}\right] \Psi [ C ] = 0 \ ,
f ( \hat{Z}_{-}) = \Bigl ( 1 -{\frac{Z_{-}^{2}}{2 N}}\Bigr ) \rho +{\cal O}( 1 / N^{2}) \, , \quad f ( \hat{Z}_{1}) = \Bigl ( 1 -{\frac{Z_{1}^{2}}{2 N}}\Bigr ) \rho +{\cal O}( 1 / N^{2}) \, .
R_{\alpha_i,\alpha_j}=\left(1+\frac{\alpha_i}{\hbar}u_{\sigma_i}\right)\left(1+\frac{\sigma_i\alpha_j}{\hbar}u_{\sigma_j}\right) &=1+\frac{\alpha_i}{\hbar}u_{\sigma_i}+\frac{\alpha_j}{\hbar}u_{\sigma_j}+\frac{\alpha_i\alpha_j}{\hbar^2}u_{\sigma_i\sigma_j}&=R_{\alpha_j,\alpha_i}.
A_{4 F}= \int_{z_{4}}^{z_{2}}d z_{3}< O_{- \frac{1}{2}}^{( 1 )}( z_{1}) O_{- \frac{1}{2}}^{( 2 )}( z_{2}) V_{- \frac{1}{2}}^{( 3 )}( z_{3}) O_{- \frac{1}{2}}^{( 4 )}( z_{4}) >
\hat{H}_{JC}=\hbar\omega_{c}(\hat{a}_{c}^{\dagger}\hat{a}_{c}+\frac{1}{2})+\hbar\omega_{eg}|e\rangle\langle e|+\hbar g_{c}(\hat{\sigma}_{+}\hat{a}_{c}+\hat{\sigma}_{-}\hat{a}_{c}^{\dagger})
\int \frac{d^{3}k}{( 2 \pi )^{3}}\left\{D_{\mu \nu}^{\mathrm{T}}( k - \ell ) \mathrm{t r}\left[ \gamma_{\mu}S ( k ) \Gamma_{\nu}^{\mathrm{B C}}( k , \ell ) S ( \ell ) \right] + \frac{1}{( k - \ell )^{2}}F ( k , \ell ) \right\}= 0 ,
T_{\vec{v}}^{-n}\phi_u\widetilde{B}= \phi_u(T^{(0,1)}\times id)T_{\vec{v}}^{-n}(\widetilde{B}\cap(X\times A_1^n)) &\cup\phi_uT_{\vec{v}}^{-n}(\widetilde{B}\cap(X\times A_2^n))\cup\phi_u(T^{-(0,1)}\times id)T_{\vec{v}}^{-n}(\widetilde{B}\cap(X\times A_3^n)).
\bigg(\frac{1}{|Q|}\iint_{T_Q}|\Theta f(x', t)|^2 \frac{dx' dt}{t}\bigg)^{\frac12}= \bigg(\frac{1}{|Q|}\iint_{T_Q}|\Theta (f-f_Q)(x', t)|^2 \frac{dx' dt}{t}\bigg)^{\frac12}\le{\rm I}+{\rm II},
A(\tau_y(x)) + \sum_{n=1}^N \left[ \hat{A}\circ \hat{\sigma}^{-(n+1)}(y|x) - \hat{A}\circ \hat{\sigma}^{-n}(\sigma(y)|x') - \hat{A}\circ \hat{\sigma}^{-n}(y|x) + \hat{A}\circ \hat{\sigma}^{-n}(y|x') \right] = A(\tau_{y,N+1}(x)) + \sum_{n=1}^N \left[ \hat{A}\circ \hat{\sigma}^{-n}(y|x') - \hat{A}\circ \hat{\sigma}^{-n}(\sigma(y)|x') \right] \ .
\mathcal{E}_{\varrho,s}:C(Y) \to\tilde{L}^{\infty}\left(Y\right), f \mapsto\mathcal{E}_{\varrho,s}f,\mbox{where}\left(\mathcal{E}_{\varrho,s}f\right)(\bar{u},\bar{v},\cdot)=\mathbb{E}_{\lambda}\left[\tilde{f}\left(\bar{u},\cdot\right)|\mathcal{F}^{s}\right],
\sup_{f\in M}\|f\|_{Y_q(\tilde \Omega_{[\hat t(n)]})}\stackrel{(\ref{nu2}), (\ref{emb_nu})}{\underset{\mathfrak{Z}_0}{\lesssim}}2^{((1‐\lambda)\mu_*‐\lambda \alpha_*)k_*\hat t(n)}\stackrel{(\ref{hat_mt})}{=}n^{((1‐\lambda)\mu_*‐\lambda \alpha_*)/\gamma_*}= n^{‐\theta_2}