harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[ N_{i},{\cal P}_{j}] = i \delta_{i j}\left( \kappa \sinh \frac{{\cal P}_{0}}\kappa - \frac{\vec{{\cal P}}{}^{2}}{2 \kappa}e^{{\cal P}_{0}/ \kappa}\right) , \quad [ N_{i},{\cal P}_{0}] = i{\cal P}_{i}e^{{\cal P}_{0}/ \kappa}.
d(m,n) = \sum_{r=1}^m u(m-r,n+r) + f(m-r,n+r), f(m,n) =\sum_{r=1}^m u(m-r,n) + d(m-r,n), u(m,n) = \sum_{r=1}^m f(m-r,n-r) + d(m-r,n-r),
\begin{aligned}\widetilde{U}(\widetilde{\mathcal{O}}f) = \widetilde{U}\Big[\Big(\frac{d^2}{dx^2}+2x\frac{d}{dx}\Big)f\Big] &= \Big(\frac{d}{dx}+x\Big)\Big(\frac{d}{dx}-x\Big)\widetilde{U}f &=- \mathcal{H}(\widetilde{U}f) - \widetilde{U}f.\end{aligned}
g(y_1) =\frac{1}{d(a_1,b_1)}\sum_{k=1}^{K}\big(g(u_{2k-2})-g(u_{2k-1})-g(u_{2k-1})+g(u_{2k})\big) =\frac{1}{d(a_1,b_1)}(2R+2(K-2)s)=1.
B^{s}_J(\bar{z}\oplus \bar{w}) := \prod_{\substack{(i_1, \ldots, i_k) \in [m_1] \times \ldots \times [m_k]}}\chi_{f^{\left[r_{s(\bar{i},l+1,[k]\setminus I)}, r_{s(\bar{i},l+1,[k] \setminus I)+1}\right)}}\left(\bar{z}\oplus \bar{0}^{k+1}_{w_{t,i_t}\to t, t \in [k] \setminus I}\right).
\int_{B_R}c_1^{-2}(y)\cdot f_1(y)\cdot |y|^\alpha\cdot \overline{Y_n^m}(\hat y)\dy = \int_{B_R}c_2^{-2}(y)\cdot f_2(y)\cdot |y|^\alpha \cdot \overline{Y_n^m}(\hat y)\dy,
\mathbf{P}\bigl(\Pi_{\varepsilon}=0,\,X_{t}(B)>0\bigr)\leq\mathbf{P}\biggl(\int_{t-\varepsilon}^{t}\mathrm{d}s\X_{s}(B)\frac{1}{(t-s)\log\bigl(\frac{1}{t-s}\bigr)}<\infty,X_{t}(B)>0\biggr).
0 \to E_0 \stackrel{i_0}{\to}E_1 \stackrel{f_1}{\to}S_1 \to 0 0 \to S_1 \stackrel{i_1}{\to}E_2 \stackrel{f_2}{\to}S_2 \to 0 \vdots 0 \to S_{n-2}\stackrel{i_{n-2}}{\to}E_{n-1}\stackrel{f_{n-1}}{\to}S_{n-1}\to 0 0 \to S_{n-1}\stackrel{i_{n-1}}{\to}E_n \stackrel{f_n}{\to}E_{n+1}\to 0.
U_{1 2}\, = \, V_{1 2}- \{P_{0}\! - \! S , \,{\frac{V_{1 2}}{2 ( P_{0}\! - \! h_{3})}}\, \}+{\frac{V_{1 2}\, ( P_{0}+ S - 2 h_{3}) \, V_{1 2}}{4 ( P_{0}- h_{3})^{2}}}
\begin{cases}\widehat{H}(x,P+\nabla \widetilde{u}_0 (x ) )=\ln \widetilde{m}_0 (x ) + \overline{H}, -\div\big (\widetilde{m}_0 (x )\widetilde{b}(x,P+\nabla \widetilde{u}_0 (x ) )\big )=0,\end{cases}
\beta_{r}( q_{1 3}^{0}, | \vec{q}_{1 3}| , q_{2 4}^{0}, | \vec{q}_{2 4}| ) = \frac{- ( q_{1 3}^{0}- q_{2 4}^{0})^{2}+ \vec{q}_{1 3}^{2}+ \vec{q}_{2 4}^{2}+ m_{r}^{2}- i \varepsilon}{2 | \vec{q}_{1 3}| | \vec{q}_{2 4}|}\, .
f_1 \approx(1.76,1.62,2.84,1.62,1,1)^T, f_2 =(0,-1,0,1,0,0)^T, f_3 =(0,0,0,0,-1,1)^T, f_4 \approx(-2.44,-0.62,1.51,-0.62,1,1)^T, f_5 \approx(0.82,-0.62,-0.51,-0.62,1,1)^T, f_6 \approx(-1.14,1.62,-1.84,1.62,1,1)^T.
-2\int_M|M|^2d\mu- \int_M\big(\frac{|\nabla R+R\nabla f+\frac{\alpha'}{2}R\nabla R|^2}{R}\big)d\mu= &-\int_M\big(|\nabla R|^2\big(\frac{\alpha'}{2}\sqrt{R}+\frac{1}{\sqrt{R}}\big)^2+\int_M\big(R+\frac{\alpha'}{4}R^2-r\big)^2d\mu.
|\chi J_1 + (1-\chi)J_2|_{g}\le \chi |J_1|_{g}+ (1-\chi) |J_2|_{g}&\le \chi |J_1|_{g_1}+ (1-\chi) |J_2|_{g_2}&+ \chi (1-\chi) |g_1 - g_2|_{g_1}|J_1|_{g_1}+ (1-\chi) \chi |g_1 - g_2|_{g_2}|J_2 |_{g_2}.
- \frac{1}{2}\sum_{j = 1}^{N}D_{j}^{2}= - \frac{1}{2}\sum_{j = 1}^{N}\partial_{x_{j}}^{2}+ \sum_{1 \leq j < k \leq N}\frac{\beta ( \beta - K_{j k})}{( x_{j}- x_{k})^{2}}.
\mathcal{V}_{\phi}( \xi ; k_{M}) = : \exp \left( i \sum_{M = 0}^{p + 2}k_{M}X^{M}\right) ( \xi ) : = : \exp \left( i \sum_{\mu = 0}^{p}k_{\mu}X^{\mu}\right) ( \xi ) : V_{\phi}( \xi ; k_{m}) ~ ,
f(\phi)\otimes\max\{\psi_1, \psi_2\}=\phi\otimes(\max\{\psi_1, \psi_2\}\circ f) = \phi\otimes(\max\{\psi_1\circ f, \psi_2\circ f\}) =\max\{\phi\otimes(\psi_1\circ f),\phi\otimes(\psi_2\circ f)\}(\phi~{\rm is~flat}) =\max\{f(\phi)\otimes\psi_1,f(\phi)\otimes\psi_2\},
f^*(t)= \left(\frac{1}{3p-1}(s_{f}(p)+\frac{1-p}{2}\delta)+\frac{1}{2}(-f(t+p)+f(t))\right){\bf 1}_{[0,1-p]}(t) &+\frac{1}{2p-1}(s_f(p)-\frac{1-p}{3p-1}(s_{f}(p)-(2p-1)\delta)){\bf 1}_{(1-p,p)}(t) &+\left(\frac{1}{3p-1}(s_{f}(p)+\frac{1-p}{2}\delta)+\frac{1}{2}(f(t+p)-f(t))\right){\bf 1}_{[p,1]}(t), t\in[0,1].
\parallel \Omega(\xi)=(\Omega_{\textbf{j}}(\xi))_{{\textbf{j}}\in\mathbb{Z}^d_1}\parallel_{-2}:=\sup_{\textbf{j}\in\mathbb{Z}^d_1}|\Omega_{\textbf{j}}(\xi)|\textbf{j}|_2^2|;
\mathrm{P}^k_o = \mathrm{P}\left(\log\left(1+\frac{x_k \alpha_{1,k}^2}{x_k \alpha_{1,n}^2 + \frac{1}{\rho}}\right) <R_{1,k}\right) = \mathrm{P}\left(\log \sqrt{1+\rho x_k}<R_{1,k}\right).
\beta_{t-2}^k \beta_{t+k-1}= \begin{vmatrix}\gamma_{h-k}\cdots \gamma_{h-2}\gamma_{h-1}\gamma_{h}* \cdots * * * \vdots \ddots \vdots \vdots \vdots * \cdots * * *\end{vmatrix}\eqqcolon \Gamma_{k}, \forall k\in \{0,1,\ldots,h-1\},
\begin{array}{ll}\displaystyle\sum_{i=1}^k \|\mathcal{A}^*u^{i+1}+ \mathcal{B}^*v^{i+1}- c\|^2 \leq (1-\tau)^{-1}\sigma^{-1}\Phi_1(\bar{u}, \bar{v}, \bar{x}).\end{array}
|E_iM \setminus S_i| \geq |E_iM| - |S_i| \geq |E_i| - |S_i| \geq |F_{k_i}^{-M^2}| - |S_i| = \left(1-\frac{\varepsilon}{8}\right) |F_{k_i}| - (d-\frac{\varepsilon}{4}) |F_{k_i}| = \left( 1 - d + \frac{\varepsilon}{8}\right)|F_{k_i}|=\left( 1 - d + \frac{\varepsilon}{8}\right)|T_i|.
\begin{aligned}\mathcal{B}^n_t (\varphi) = \int_0^t ( \partial_s + A_n ) \mathcal{X}^n_s (\varphi ) ds & = \frac{1}{\sqrt{n}}\int_0^t \sum_{j \in \mathbb{Z}}\big[ n g_n (\eta^n_j (s) ) \nabla^n \varphi^n_j (s) - \frac{f_n}{n}(\eta^n_j (s) - \rho ) \partial_x \varphi^n_j (s) \big] ds . \end{aligned}
\begin{array}{l l}\Phi(L_1)=(\frac{n!}{2},0,0), \Phi(L_2)=(n!,0,0), begin{align*}15pt]\Phi(L_3)=(0,\frac{n!}{2},\frac{(n+1)!}{2}), \Phi(L_4)=(0,0,(n+1)!).\end{array}
\frac d{dt}\int \lambda |u|^2 dx= 2\Re \int \lambda u_t \bar u dx =2\Re \int \lambda (\frac i 2 u'' \bar u - i u \bar u|u|^\alpha)dx +2\Re \lambda(0) i |u(t,0)|^2=- \Re \int i \lambda' u' \bar u dx.
N^{\prime}(\eta_{\mu_{1}}(\alpha))=N^{(1)}(\eta_{\mu_{1}}(\alpha))=u_{\sigma}^{(1)}(a_{0})\cdot\frac{1}{(1-\eta_{\mu_{1}}(\alpha))^{2}}=\frac{b_{1}}{(1-\eta_{\mu_{1}}(\alpha))^{2}},
d_{2}(v_{1}\vee \dots \vee v_{k}) = \frac{1}{2}\sum_{\substack{i+j=k (\mu,\nu)\in \mathrm{Sh}_{i,j}}}\pm [v_{\mu(1)}\vee \dots \vee v_{\mu(i)}, v_{\nu(1)}\vee \dots \vee v_{\nu(j)}],
Q^{i,j}(A) := Q^{i,j}(A,\ldots,A) = \frac{(-1)^{i+j}}{n-1}(M^{i,j}(A)) = \frac{1}{n-1}(A)^{i,j}= \frac{1}{n-1}(A) (A^{-1})^{i,j}.
U:=\sum_{\substack{\delta X<n_1^k,n_2^k,n_3^k,n_4^k\le X \vert n_1^k+n_2^k-n_3^k-n_4^k\vert \le 1/\tau}}1,\textrm{and}V:=\sum_{\substack{\delta X<n_1^k,n_2^k,n_3^k,n_4^k\le X \vert n_1^k+n_2^k-n_3^k-n_4^k\vert > 1/\tau}}\frac{1}{\vert n_1^k+n_2^k-n_3^k-n_4^k\vert},
\partial_{\mu}P ( Z ) = U \left( \begin{array}{c c}{I}&{Z^{\dag}}\\{- Z}&{I^{\prime}}\\ \end{array}\right)^{- 1}\left( \begin{array}{c c}{}&{\partial_{\mu}Z^{\dag}}\\{\partial_{\mu}Z}&{}\\ \end{array}\right) \left( \begin{array}{c c}{I}&{- Z^{\dag}}\\{Z}&{I^{\prime}}\\ \end{array}\right)^{- 1}U^{\dag}
\lim_{N \to \infty}\frac{1}{R_{\sigma_N}(\theta_N)}=\lim_{N \to \infty}\frac{\theta_N^2}{\sigma_NR_{\sigma_N}(\theta_N)}=\infty, \lim_{N \to \infty}(1+R_{\sigma_N}(\theta_N))^{1/2R_{\sigma_N}(\theta_N)}=e^{1/2}<e^1=\lim_{N \to \infty}\left(1+R_{\sigma_N}(\theta_N)\frac{\sigma_N}{\theta_N^2}\right)^{{\theta_N^2}/{\sigma_N}R_{\sigma_N}(\theta_N)}.
\begin{aligned}\lVert \widetilde{M}_{k}(\alpha,\theta,E,0)\rVert \leq \lVert M_{k}(\alpha,\theta,E,0)\rVert+\sum_{i=0}^{k-1}|g(i)|\lVert M_{k-i-1}(\alpha,\theta,E,i+1)\rVert\lVert\widetilde{M}_{i}(\alpha,\theta,E,0)\rVert. \end{aligned}
F_1 (t_1,\dots,t_k)=\prod_{i=1}^k \frac{\psi(t_i/U_k)}{1+T_kt_i}, F_2 (t_1,\dots,t_k)=\sum_{1\le j\le k}\Bigl(\frac{\psi(t_j/2)}{1+T_kt_j}\prod_{\substack{1\le i\le k i\ne j}}\frac{\psi(t_i/U_k)}{1+T_kt_i}\Bigr).
\times\left(\frac{1}{1-te^{-2x}}\right)^{r'-g_{Y}}\cdot\prod_{i=1}^{g_{Y}}\left(\frac{1}{1-te^{4\sigma_{i}-2x}}\right)\cdot \left(\frac{1}{1-te^{-2x}}\right)^{r''-g_{Y}}\cdot\prod_{i=1}^{g_{Y}}\left(\frac{1}{1-te^{-4\sigma_{i}-2x}}\right).
G^D( x, y)=\int_0^\infty p^D(t,x,y)dt =\int_0^s p^D(t,x,y)dt+ \int_0^\infty p^D(t+s,x,y)dt &=\int_0^s p^D(t,x,y)dt+ P^D_s\int_0^\infty p^D(t,\cdot,y) (x)dt=\int_0^s p^D(t,x,y)dt+P_s^DG^D (\cdot,y)(x).
o_n(1)=I'_\lambda(\omega_n)\Psi_n =(I'_\lambda(\omega_n)-I'_\lambda(\omega_0))\Psi_n=\|\Psi_n\|_X^2-\displaystyle\int_{\mathbb{R}^N}b(x)|\Psi_n(x,0)|^{2_\alpha^*}dx.
\frac{1}{k^{2}}\rightarrow \lim_{\alpha \rightarrow 1 + i \epsilon}\frac{1}{\alpha^{2}k_{0}^{2}- \vec{k}^{2}}= \frac{1}{k^{2}+ 2 i \epsilon k_{0}^{2}}, \hspace{. 5 c m}\epsilon \rightarrow 0^{+}.
Q(M,P_i,\tau)=\left\{e^{\frac{1}{24}E_2(\tau)A_1}\left(\prod_{j=1}^{5}\frac{x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\right)\frac{\sqrt{-1}\theta(u,\tau)}{\theta_1(0,\tau)\theta_2(0,\tau)\theta_3(0,\tau)}\right. &\left.\frac{1}{2}\left(\prod_{l=1}^8\theta_1(y_l^i,\tau)+\prod_{l=1}^8\theta_2(y_l^i,\tau)+\prod_{l=1}^8\theta_3(y_l^i,\tau)\right)\right\}^{(10)}.
[D_{x^\prime}D^{m-1}u]_{\frac{\gamma}{2m},\gamma, Q_1^+(X_0)}+\lambda^{\frac{1}{2}}[u]_{\frac{\gamma}{2m},\gamma, Q_1^+(X_0)}\le N\sum_{k=0}^m\lambda^{\frac{1}{2}-\frac{k}{2m}}\|D^k u\|_{L_2(Q_4^+(X_0))}.
\big\|e^{t\Delta}u_0\big\|_{\mathcal{L}^{\tilde p, 1}}= \big\|e^{-t\left| \xi \right|^2}\hat u_0(\xi)\big\|_{L^{\tilde p', 1}_\xi}= \big\|e^{-t\left| \xi \right|^2}|\xi|^{1-\frac{d}{p}}|\xi|^{\frac{d}{p}- 1}\hat u_0(\xi)\big\|_{L^{\tilde p', 1}_\xi}.
- \nabla^{\mu}\nabla_{\mu}V_{\nu}= \left[ - \nabla^{K}\nabla_{K}+ \left({\frac{\partial}{\partial r}}+{\frac{d - 1}{r}}\right) \left({\frac{\partial}{\partial r}}-{\frac{1}{r}}\right) -{\frac{1}{r^{2}}}\right] V_{\nu}~ ~ ~
\tilde{Z}^{( n )}( \{q_{N_{s}}\}, L ) = \tilde{Z}_{0}^{( n )}( \{q_{N_{s}}\}, L ) \Upsilon_{m}^{( n )}( \{q_{N_{s}}\}, L ) \Upsilon_{g h}^{( n )}( \{q_{N_{s}}\}, L )
\widehat{y}_1 (x)= \lim_{K \to -\infty \atop{L \to +\infty}}\widehat{y}_1^{\,[K,L]}(x) = (q-1)\alpha_1\sum_{n=-\infty}^{+\infty}q^n\xi\frac{\big(q^{\lambda+n+1}\xi / x, q^{n+1}\xi\alpha_1, q^n \xi\alpha_2 ; q\big)_{\infty}}{\big(q^{n+1}\xi / x, q^n \xi\beta_1, q^n \xi\beta_2 ; q\big)_{\infty}}.
y_{\mu}^\lambda=\begin{cases}\sum_{i=a_{k-1}+1}^{a_k}q^{a_k-i}T_{i,l}E^f x_\mu \hbox{if $\mu=\lambda\setminus p$,}E^fx_\lambda T_{l,2f}^{-1}T_{b_k,2f-1}^{-1}\sum_{i=b_{k-1}+1}^{b_k}q^{b_k-i}T_{b_k,i}\hbox{if $\mu=\lambda\cup p $.}\end{cases}
\frac{\partial}{\partial{p}}g(p,y)=-f'_{p}/f'_x, begin{align*}5pt] \frac{\partial^2}{\partial{p^2}}g(p,y)=\left(f'_x\right)^{-2}\left\{2f'_{p}f''_{xp}-f'_xf''_{pp}-\left(f'_{p}\right)^2f''_{xx}/f'_x\right\}, begin{align*}5pt] \frac{\partial^2}{\partial{p^2}}\log{g(p,y)}=(xf'_x)^{-2}\left\{2xf'_{p}f''_{p{x}}-xf'_xf''_{pp}-x(f'_{p})^2f''_{xx}/f'_x-(f'_{p})^2\right\},
(e,z)(g,1)\cdot \tilde{x}_0 = (e,z)(g,1)\tilde{H}_0 = (g,e^{\tau})\tilde{H}_0 = (g\lambda(e^{-\tau/m}),1)(\lambda(e^{\tau/m}),e^{\tau})\tilde{H}_0 = (g\lambda(e^{-\tau/m}),1)\tilde{H}_0 = (g\lambda(e^{-\tau/m}),1)\cdot \tilde{x}_0.
H^{(n)}_{i_1}:= \bigcup_{|u| \le \sqrt{n'}\log n'}H^{(n')}_{i_1}(u) \,, H_{i_1,i_2}^{(n)}:=\bigcup_{|u|,|v| \le \sqrt{n'}\log n'}H^{(n')}_{i_1,i_2}(u,v), H^{(n')}_{i_1}(u) := \{S_{i_1}-S_{i_1-n'}=u\}\,, H^{(n')}_{i_1,i_2}(u,v) := H^{(n')}_{i_1}(u) \cap \{S_{i_2}-S_{i_2-n'}+S_{{i_1}+n'}-S_{i_1}=v\}\,.
C = \sum_{i=1}^k d(\beta(x_{i-1}),\beta(x_i)) \leq \sum_{i=1}^N d(\beta(t_{i-1}),\beta(t_i)) \leq L\sum_{i=1}^N d(f(\beta(t_{i-1})),f(\beta(t_i))) \leq L \ell(f \circ \beta).
\int_{\mathbb R^2}(\Delta a)(y) y_1 \partial_1 F(y) dy =0, \int_{\mathbb R^2}(\Delta a)(y) y_2\partial_2 F(y) dy =0, \int_{\mathbb R^2}(\Delta a)(y) ( y_1 \partial_2 F(y) + y_2 \partial_1 F(y)) dy =0.
\Delta{K}^s_{n+1}=0 \Delta{K}^s_{t}=\big(\mu^s_1(t)(\beta_t-1)-\mu^s_0(t)(\alpha_t-1)\big)+H(\alpha_t)-H(\beta_t) &\qquad\qquad+\log\Big(\frac{1+2^{\mu^s_1(t)+\Delta{K}^s_{t+1}}}{1+2^{\mu^s_0(t)+\Delta{K}^s_{t+1}}}\Big)+s,~t\in\{n,\ldots,0\}.
\left( \begin{array}{c c c}{( \kappa - \ln \tilde{q}) \eta ( \tilde{\tau})^{2}}&{- \Lambda_{1 , 2}( \tilde{\tau})}&{- \eta ( \tilde{\tau})^{2}}\\{- \Lambda_{1 , 2}( \tilde{\tau})}&{0}&{0}\\{- \eta ( \tilde{\tau})^{2}}&{0}&{0}\\ \end{array}\right)
R_1 \le I(X_1;Y_k) + I(X_2;Y_k|X_1) + I(X_3;Y_k|X_1,X_2,W_1), R_2 \le I(W_1;Y_k|X_1,X_2) + I(W_2;Y_k|X_1,X_2,X_3,W_1).
\tau_{0}= \frac 1 2 \left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) \, , \quad \tau_{1}= - \frac 1 2 \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) \, , \quad \tau_{2}= \frac 1 2 \left( \begin{array}{c c}{{- 1}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right) \,
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_1-2e_3+\delta e_5,[e_{i},e_{n+1}]=(i-4)e_{i},(3\leq i\leq n),[e_{n+1},e_1]=-e_1+2e_3-\delta e_5, \displaystyle[e_{n+1},e_3]=e_3,[e_{n+1},e_4]=e_2,[e_{n+1},e_j]=(4-j)e_j, (\delta=\pm1,5\leq j\leq n,n\geq5). \end{array}
A_{r,j}\leq \frac{q_r}{q_r - 2}\leq \frac{1000}{998}< 2 A_{r,j}=\lfloor \sqrt{n_{r+1}- n_r}\rfloor < 2 \sqrt{n_{r+1}- n_r}< 3 n_{r+1}- n_r < 9 2^{n_{r+1}- n_r}< 512.
\mathcal{L}[\beta](k,x)\!\! = \!\! \frac{-4\sigma L^2}{|k|^2}\int_0^{\infty}\!\! \left[v(|p|) - \frac{Lx}{2\pi |k|}\arctan\left(\frac{2\pi |k|}{Lx}v(|p|)\right)\right](1+|p|^2) (-f_0'(|p|))d|p|.
p_{k}=n\frac{\dot{x}_{k}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}}=n\frac{dx_{k}}{\sqrt{dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}=n\frac{dx_{k}}{ds}
\begin{array}{l}{{\partial_{-}\partial_{0}= q^{- 1}\partial_{0}\partial_{-}~ , ~ ~ \partial_{0}\partial_{+}= q^{- 1}\partial_{+}\partial_{0}~ ,}}\\{{\partial_{-}\partial_{+}- \partial_{+}\partial_{-}= ( q^{1 / 2}- q^{- 1 / 2}) \partial_{0}\partial_{0}~ .}}\\ \end{array}
m(a_{\gamma})= \inf_{(x,y)\in \Gamma(f)}{\{\langle a_{\gamma}, (x, y)\rangle \}}= \inf_{(x,y)\in \Gamma(f)}\{j_0x+i_0y\}= \inf_{(x,y)\in \Gamma(f)}\Big\{i_0j_0\Big(\frac{x}{i_0}+\frac{y}{j_0}\Big)\Big\}= i_0j_0\inf_{(x,y)\in \Gamma(f)}\Big\{\frac{x}{i_0}+\frac{y}{j_0}\Big\}= i_0j_0,
f^{\left( 1\right)}(\theta )=\sum_{l}\left( 2\pi ik\right)^{-\frac{1}{2}}e^{il\theta}\left[ -i\pi\alpha \mathrm{sgn}\left( l\right) \left( 1-\delta_{l,0}\right) -i\pi \alpha \delta_{l,0}\cos\beta\right],
\frac{1-t_i}{1-t_i/t_{p+1}}-Q_{i}^N=\frac{1-t_i}{1-t_i/t_{p+1}}\cdot \frac{t_i^{N+1}}{t_{p+1}^{N+1}}\cdot \frac{1-t_{p+1}}{1-t_i}.
a_{\mu}^{\frac 13 - \frac{1}{3q}}\mu^{-\frac14}+ a_{\mu}^{\frac12 - \frac 1q}\mu^{-\frac 12 + \frac 1q}= a_{\mu}^{\frac12 - \frac 1q}\mu^{-\frac 12 + \frac 1q}\left(1 + (a_\mu^{\frac 16}\mu^{-\frac 14})^\frac{4-q}{q}\right)
\eta_{H}( 0 ) = - 1 - 2 \mu ( \gamma ) , \ \zeta_{H}( 0 ) = - 1 , z e t a_{H}^{\prime}( 0 ) = 2 \ln \left| \frac{\sqrt{c}\ell [ \gamma ]}{2 \sin ( \sqrt{c}\ell [ \gamma ] )}\right| ,
\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}}{45 L^{3}}\left[ 1-\frac{15}{8}\left( \left(\frac{\chi}{\pi}-1\right)^{2}-1\right)^2\right]
\int_{K \backslash G /H}Q(f) (\ddot{n}) d\mu_1 (\ddot{n}) = \int_N f(n) \rho_1 (n)dn &=\int_N f(n) \big(\frac{\rho_1(n)}{\rho_2(n)}\big) \rho_2 (n) dn &=\int_{K \backslash G /H}Q(f) (\ddot{n})\varphi(\ddot{n})d\mu_2(\ddot{n}).
\sum_{j=0}^{m}\frac{d u_j(t)}{d t}J_{L,j}^{\beta,\gamma}(x_q)=d\sum_{j=0}^{m}\sum_{k=0}^{j}\sum_{i=0}^{k}u_j(t)\times \Psi_{i,j,k}^{\alpha,\beta,\gamma}\big{|}_{x=x_q}\times J_{L,j}^{\beta,\gamma}(x_q)-e\sum_{j=0}^{m}u_j(t)\frac{d}{dx}J_{L,j}^{\beta,\gamma}(x)\big{|}_{x=x_q}+s(x_q,t).
\zeta_0(t) \doteq 0, \zeta_k(t) \doteq p_k(f_1(t))^k, k \in \mathbb{N}, \psi(t) \doteq -2 \int_0^t r_0(\boldsymbol{\zeta}(s))\,ds + \sum_{k=1}^\infty (k-2) (p_k- \zeta_k(t)).
R^{*}(D, \epsilon, \delta) := \inf \{R : \mbox{$\exists${\rm a}$(D, R, \epsilon, \delta)${\rm code}}\}, \tilde{R}(D, \epsilon, \delta) := \inf \{R : \mbox{$\exists${\rm a deterministic}$(D, R, \epsilon, \delta)${\rm code}}\}.
\stackrel{.}{a_{1}}= \Gamma^{\rho}\sum_{i , j}T_{i}X_{0 , \rho}T_{j}\, e^{\left( m_{i}^{2}- m_{j}^{2}\right) t}- \sum_{i , j}T_{i}X_{1}T_{j}\, e^{\left( m_{i}^{2}- m_{j}^{2}\right) t}
R^{(n-1)}_{\triangleleft h}(h) =\sum_{k=n}^\infty\bigg( \sum_{i_1+\dotsm+i_n=k}(i_1+1)(i_1+i_2+1)\dotsm (i_1+\dotsb+i_{n-1}+1)h_{i_1}\dotsm h_{i_n}\bigg)x^k.
S_\alpha(z) \,S_A(z)=\left.\widetilde S_\alpha(\bar z) \,\widetilde S_A(\bar z)\right|_{z=\bar z}~~~,~~~~S^{\dot\alpha}(z) \,S^A(z)=\left.\widetilde S^{\dot \alpha}(\bar z) \,\widetilde S^A(\bar z)\right|_{z=\bar z}~~,
\mathcal{Y}=L^{2}\left(\Omega,\mathbb{C}^{3}\right)\oplus\left(L^{2}\left(\Omega,\mathrm{sym}\left[\mathbb{C}^{3\times3}\right]\right)\oplus\tilde{Y}\right)\oplus L^{2}\left(\Omega,\mathbb{C}^{3}\right)\oplus\left(L^{2}\left(\Omega,\left[\mathbb{C}^{3}\right]\right)\oplus Y\right)
\begin{aligned}\Delta\left(\frac{f}{H+\varepsilon}\right) = \left \langle - \frac{1}{2}F - \frac{2}{H+\varepsilon}\nabla (H+\varepsilon), \nabla\left(\frac{f}{H+\varepsilon}\right)\right\rangle + \frac{f}{(H+\varepsilon)^2}\left(\left(\lambda + \frac{1}{2}\right)H - \varepsilon \left( |A|^2 - \lambda\right)\right).\end{aligned}
\inf_{k \in [m]_0}\left\{\inf_{\mathcal{Q}\subseteq [m]}\left\{\inf_{x \in \mathcal{X_\mathcal{Q}}}\left\{\Gamma \Delta u_k^T l_k(x) + \sum_{i \in [m]}\overline{u}_i^Tl_i(x) + \sum_{q \in \mathcal{Q}}\Delta u_q^T l_q(x) - \Delta u_k^T l_k(x) \right\}\right\}\right\},
\dot{\varrho}+3{\dot{a}_0\over a_0}\left(\varrho+{\wp\over c^2}\right)=-{\dot{G}_4\over G_4}\varrho+{{3kc\dot{c}}\over{4\pi G_4a^2_0}}-{{3{\cal C}c\dot{c}}\over{4\pi G_4a^4_0}}.
(\alpha_0|\alpha_0)=2,\;\; (\alpha_1|\alpha_1)=0,\;\; (\alpha_2|\alpha_2)=-\frac{1}{2},\;\; (\alpha_0|\alpha_1)=-1,\;\; (\alpha_1|\alpha_2)=\frac{1}{2},\;\; (\alpha_0|\alpha_2)=0.
a(\tilde{u}_{\Lambda,X}) H_0^{-1}a^*(u_{\Lambda,X}) - a(\tilde{u}_{X}) H_0^{-1}a^*(u_X) \ \ \ = a(\tilde{u}_{\Lambda,X}- \tilde{u}_X) H_0^{-1}a^*(u_{\Lambda,X}) - a(\tilde{u}_X) H_0^{-1}a^*(u_X - u_{\Lambda,X}).
y_{c}^{\prime}= \stackrel{-}{( + )}\frac{{r_{0}}^{p - 1}}{\sqrt{r^{2 p - 2}-{r_{0}}^{2 p - 2}}}, \sqrt{1 +{y_{c}^{\prime}}^{2}}= \stackrel{-}{( + )}\frac{r^{p - 1}}{\sqrt{r^{2 p - 2}- r_{0}^{2 p - 2}}}
\begin{array}{rcl}S = \displaystyle\int d^{8}z E^{-1}\left[\bar{Q}Q+\bar{Q}^{c}Q^{c}+G(\bar{Q}^{c}Q)(\bar{Q}Q^{c}) +\bar{\xi}_{1}(\bar{Q}Q) +\bar{\xi}_{2}(\bar{Q}^{c}Q^{c})\right],\end{array}
\partial_{t}v_{4}(t,r) = \frac{-1}{2\pi}\int_{0}^{\infty}dw \int_{0}^{w}\frac{\rho d\rho}{\sqrt{w^{2}-\rho^{2}}}\int_{0}^{2\pi}d\theta \frac{\partial_{1}v_{4,c}(t+w,\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)})}{\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)}}(r+\rho \cos(\theta))
\int \phi P\phi\dv = \int (\Delta \phi)^2\dv - 4 \int A(\nabla \phi,\nabla \phi)\dv + 3 \int \sigma_1(A)|\nabla \phi|^2\dv + \frac{1}{2}\int Q \phi^2\dv &\geq \frac{3}{7}\int (\Delta \phi)^2\dv + \int \sigma_1(A)|\nabla \phi|^2\dv + \frac{1}{2}\int Q \phi^2\dv,
\frac{f^{2}}{6}\, E \int_{- \infty}^{\infty}d x \, \psi_{n}^{( 0 ) *}( x ) \bigl \{1 - 4 \mu x^{2}[ V ( x ) - E ] -{\textstyle \frac{2}{3}}x^{3}\mu V^{\prime}( x ) \bigr \}\psi_{n}^{( 0 )}( x ) = 0 .
\zeta ( s , r ) = \frac{1}{2 \pi \alpha r^{3 - 2 s}}\frac{\Gamma ( s - 1 )}{\Gamma ( s )}\left[ G_{2 \alpha}( s - 1 / 2 ) - G_{2 \pi}( s - 1 / 2 ) \right] = \frac{1}{2 \sqrt \pi \alpha r^{3 - 2 s}}\frac{I_{2 \alpha}( s - 1 / 2 )}{\Gamma ( s )}{,}
(-\Delta)^{s}\Big\{\frac{x}{\left(1+x^2\right)^{\gamma}}\Big\}= (2s+1)A_s^\gamma\,\frac{x}{(1+x^2)^{s+\gamma}}\,{}_{2}F_{1}\Big(\!-s, s+\gamma;\frac{3}{2};\frac{x^2}{1+x^2}\Big),
F_{\lambda}\hat{\lambda}=-J_{3}\frac{1}{r^{5}}\frac{3}{2}(5sin^{2}\lambda-1)cos\lambda\hat{\lambda}=-J_{3}\frac{1}{r^{5}}\frac{3}{2}(5sin^{2}\lambda-1)(sinicosu\hat{t}+cosi\hat{z})
\displaystyle{\int_Q f(\xi_{2k+1}, v_{2k+1},\nabla \xi_{2k+1})dx =\int_{\Sigma}\int_{-\frac{1}{2}}^{-\frac{1}{2(2k+1)}}f(c,v(x),0)dx+\int_{\Sigma}\int_{\frac{1}{2(2k+1)}}^\frac{1}{2}f(d, v(x),0)dx}&\displaystyle{+ \int_{\Sigma}\int_{|x_N|<\frac{1}{2(2k+1)}}f(\xi((2k+1)x), \varphi((2k+1)x), (2k+1)\nabla \xi((2k+1)x)dx.}
S_{\mathrm{m a s s i v e \D 0 - b r a n e}}= T \int d \tau \, \Big ( e^{- 3 \hat{\phi}/ 4}\sqrt{- \dot{X}^{\hat{\mu}}\dot{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}}}+ \dot{X}^{\hat{\mu}}\hat{A}_{\hat{\mu}}+ m \, \hat{V}_{t}\Big ) \, ,
S=\int d^{10}x \sqrt{-g_{10}}\left\{e^{-\varsigma}\left[ R_{10}+\left( \nabla \varsigma \right)^2 \right] -\frac{1}{4}F_{AB}F^{AB}- \frac{1}{48}F_{ABCD}F^{ABCD}-\frac{1}{2}m^2 \right\},
\hat{T}= \sum_{\{\Delta_0 S_\gamma \}}\prod_{p \in \Delta_0 S_\gamma}\hat{W}_p^{n_p}\exp{[ \, -\frac{1}{2\beta_0}\sum_\ell \hat{E}_\ell^2 -\frac{1}{2\beta_s}\sum_{p_s \in (S_{t+a_0}-S_t)}n_p^2 \, ] \,},
a_n(p^1_n(x)p^2_{n-1}(y)-p^1_{n-1}(x)p^2_n(y)) =(x-y)\sum_{i=0}^{n-1}p^1_i(x)p^2_i(y) & +a_0(p^1_0(x) p^2_{-1}(y)-p^1_{-1}(x)p^2_0(y)).
_{n,m}^{2}=\frac{1}{n(n-1)}\sum_{i_{1}\neq i_{2}}k(X_{i_{1}},X_{i_{2}})+\frac{1}{m(m-1)}\sum_{j_{1}\neq j_{2}}k(Y_{j_{1}},Y_{j_{2}})-\frac{2}{nm}\sum_{i,j}k(X_{i},Y_{j}),
G(c_1,\dots,c_n,0) &=(H_1(c_1,\dots,c_n,0),\dots,H_n(c_1,\dots,c_n,0)) =(p_1(c_1,\dots,c_n),\dots,p_1(c_1,\dots,c_n)) &=(c_1,\dots,c_1) =\Delta_np_1(c_1,\dots,c_n),
\rho\frac{Du}{Dt}=\rho(\frac{\partial u}{\partial t}+(u\cdot\nabla)u)=-\nabla p+\nabla\cdot\{\mu[\nabla u+(\nabla u)^{T}-\frac{2}{3}(\nabla\cdot u)I]+\zeta(\nabla\cdot u)I\}+\rho g.
c_n =\frac{(-1)^{n(n+1)/2}x^{n^2/2}}{(x^2 ; x^2 )_\infty^{n(n-3)/2}(x^{2r}; x^{2r})_\infty^{3n(n-1)/2}}\frac{\{x^2 \}_\infty^n \{x^6 \}_\infty^n\{x^{2r+2}\}_\infty^n \{x^{2r+6}\}_\infty^n}{\{x^4 \}_\infty^n \{x^8 \}_\infty^n\{x^{2r}\}_\infty^n \{x^{2r+4}\}_\infty^n},
{\cal Z}_N (J) \,=\, \int{\cal D}\theta \, \exp [ - \frac{t}{2}\partial_{\mu}\theta \partial_{\mu}\theta - \frac{t}{2 g^2}J^2 + \theta ( - \frac{t}{g}\epsilon_{\mu \nu}\partial_{\mu}J_{\nu}+ i g \rho_N ) ] \;.
\frac{1}{6}\frac{2 M}{\eta}K_{1}( 2 \eta M ) - 2 M \sum_{k = 1}^{\infty}\frac{1}{( k \pi )^{2}}\frac{K_{1}( 2 M \sqrt{\eta^{2}+ k^{2}\pi^{2}S^{2}})}{\sqrt{\eta^{2}+ k^{2}\pi^{2}S^{2}}}\hspace{0 . 5 c m}.
\theta_{T,y}\left(\omega\right)=\sum_{v\in\mathbb{Z}^{nd}}\exp\left(\overline{Q}_{T,y}\left(\omega,v\right)\right)\quad\textrm{and}\quad\vartheta_{T,y}\left(\omega\right)=\int_{\mathbb{R}^{nd}}\exp\left(\overline{Q}_{T,y}\left(\omega,v\right)\right)dv,
-y(t) =\int_{t}^{T}\dot{y}\left( s\right) ds & \leq\left\vert y(0)\right\vert \int_{t}^{+\infty}e^{-\Gamma(s)}ds+\int_{t}^{+\infty}e^{-\Gamma(s)}\int_{0}^{s}e^{\Gamma(\tau)}\omega(\tau)d\tau ds & :=\varpi(t).
I = \int_{t_{0}}^{t_{1}}\int_{0}^{1}\Big [ x ( y^{\prime}\dot{z}- \dot{y}z^{\prime}) - H_{1}\Big ( \frac{\partial H_{2}}{\partial x}x^{\prime}+ \frac{\partial H_{2}}{\partial y}y^{\prime}+ \frac{\partial H_{2}}{\partial z}z^{\prime}\Big ) \Big ] d s d t .
d s_{P}^{2}={\frac{\left|{\partial_{a}^{3}{\cal F}}\right|^{2}}{\left( \mathrm{I m}\tau \right)^{2}}}| d a |^{2}={\frac{\left| \partial_{u}\tau \right|^{2}}{\left( \mathrm{I m}\tau \right)^{2}}}| d u |^{2}= e^{\varphi}| d u |^{2},
\frac{w(\upsilon+\tau)}{w(\upsilon)}= \det \left([A(\upsilon)]^{-1}A(\upsilon+\tau) \right) \leq \big\| [A(\upsilon)]^{-1}A(\upsilon+\tau) \big\|^d = \big\| [\phi_{\upsilon}(\tau)]^T \big\|^d \leq [v_0(\tau)]^d.
\left( \left({\begin{array}{*{20}c}f_{1}f_{2}\end{array}}\right),\left({\begin{array}{*{20}c}z^{-1}+z^{-2}z^{-1}z^{-1}z^{-1}\end{array}}\right)\left({\begin{array}{*{20}c}h_{1}h_{2}\end{array}}\right) \right)=0,\thinspace \forall h=\left({\begin{array}{*{20}c}h_{1}h_{2}\end{array}}\right)\in \mathbf{C}^{2}
f_i(q) = \frac{1}{i!n^i}\left( (q-1)^{i}G_{g_{i},n}(q) - \sum_{m_1=0}^{i-1}\sum_{m_2=m_1}^{i}\binom{i}{m_2}f_{m_1}^{(i-m_2)}(q)R_{n,m_1,m_2}(q)(q-1)^{i-m_2}q^{i-m_2}\right),
x_J^0 = x_S x_S = x_J^0 x_J^1 = x_J x_J = x_J^1 x_J^2 = 2 x_J + x_K + 2 x_{\emptyset}x_K = -\tfrac{14}{5}x_J^1 + \tfrac{8}{5}x_J^2 - \tfrac{1}{10}x_J^3 x_J^3 = 4x_J + 6 x_K + 32 x_{\emptyset}x_{\emptyset}= \tfrac{2}{5}x_J^1 - \tfrac{3}{10}x_J^2 + \tfrac{1}{20}x_J^3
\frac{k^{3}-7k^{2}+10k-2(b+1)k-2-\sqrt{D(b+1,k)}}{4(k-1)}-\frac{k^{3}-7k^{2}+10k-2bk-2-\sqrt{D(b,k)}}{4(k-1)}= \frac{\sqrt{D(b,k)}-\sqrt{D(b+1,k)}-2k}{4(k-1)}\;.
G ' := \left\{( \xi , x ) \in G : \begin{array}{l}\mu \left( B_r \left( x, W(x) \right) \cap G_\xi \times \mathbb{R}\right)\geqslant r^{\gamma_1 + \gamma_2 + \delta}\mbox{and}\nu_{\left( \xi, q_\xi (x) \right)}\left( B_r (x) \right) \leqslant r^{\gamma_1 - \delta}\end{array}\mbox{for}\forall r \in ( 0 , \varepsilon_1 ) \right\}
\begin{aligned}\Big(\frac{d}{dx}+x\Big)^\alpha u(x) = e^{-x^2/2}\Big(\frac{d}{dx}\Big)^\alpha\big[e^{(\cdot)^2/2}u(\cdot)\big](x) \Big(-\frac{d}{dx}+x\Big)^\alpha v(x) = e^{x^2/2}\Big(-\frac{d}{dx}\Big)^\alpha\big[e^{-(\cdot)^2/2}v(\cdot)\big](x). \end{aligned}
D_{q_{1}q_{2}}^{j}( Z ) \: = \: \left[ \frac{( j + q_{1}) ! ( j - q_{1}) !}{( j + q_{2}) ! ( j - q_{2}) !}\right]^{1 / 2}\quad \sum_{s = \max ( 0 , q_{1}+ q_{2})}^{s = \min ( j - q_{2}, j + q_{2})}{\binom{j + q_{2}}{s}}\times