harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi_V= E_V-\sum_{i=1}^n(\dot{x}_i-\mu_i)\sigma_i\partial_y\widetilde{R}_i+\sum_{i=1}^n\bigg(\dot{\mu}_i+\sum^n_{\substack{j=1, j\not=i}}\frac{a_{ij}}{x^3_{ij}}+\sum^n_{\substack{k,j=1, j\not=i}}\frac{b_{ijk}\mu_k}{x^3_{ij}}\bigg)\frac{\sigma_i\Lambda\widetilde{R}_i}{1+\mu_i},
H_{n}(a,b)=\frac{\zeta^{1-n}}{2\pi}exp(-\frac{a^{2}+b^{2}}{2})\int_{0}^{2\pi}\frac{cos(n-1)\theta-\zeta cosn\theta}{1-2\zeta cos\theta+\zeta^{2}}exp(abcos\theta)d\theta,
\mathcal{L}= \frac{r^{2}\partial_{r}W \partial_{r}W^{*}+ \partial_{\theta}W \partial_{\theta}W^{*}- r^{2}\dot{\lambda}^{2}\partial_{\lambda}W \partial_{\lambda}W^{*}}{4 r^{2}( 1 + | W |^{2})^{2}}.
P_{F}( q )_{{\bf s}^{\prime}{\bf s}}^{{\hat{\mu}}{\hat{\nu}}}= \sum_{N_{X}= 0}^{\infty}d ( N_{X}) \cdot{\bar{\psi}}_{{\bf s}^{\prime}}q_{\hat{\rho}}\Gamma^{\hat{\rho}}\psi_{\bf s}{\frac{\eta^{{\hat{\mu}}{\hat{\nu}}}}{q^{2}+ M_{N_{X}}^{2}}}.
\frac{\partial x_0}{\partial z}(u,\tau)=2\pi i x_0\left( - n \sum_{m \geq 1}\frac{q^{nm}u^{-n}}{1-q^{nm}u^{-n}}-n \sum_{m \geq 1}\frac{q^{nm}u^{n}}{1-q^{nm}u^{n}}\right) +\frac{2 \pi n x_0}{1-u^{-n}}.
\mathrm{P}_{t,i}=e^{-\lambda_c\pi \left( \frac{\epsilon_1}{\rho}+\frac{(1+\epsilon_1)}{\rho \phi_i}\right)^{-\frac{2}{\alpha}}}\sum^{t-1}_{k=0}\frac{(\lambda_c\pi)^{k}\left( \frac{\epsilon_1}{\rho}+\frac{(1+\epsilon_1)}{\rho \phi_i}\right)^{-\frac{2k}{\alpha}}}{k!}.
h_{0}= \frac{l_{p}^{\prime 9}}{R_{6}^{\prime}R_{7}^{\prime}R_{8}^{\prime}R_{9}^{\prime}R_{1 1}^{\prime}}\left( \frac{Q_{0}}{R_{1 1}^{\prime}}\right) \frac{1}{r^{3}}= \frac{Q_{0}l_{p}^{3}}{r^{3}}
\left( \begin{matrix}{Z_{A B}}\\{Z_{I}}\\ \end{matrix}\right) = \int \left( \begin{matrix}{T_{A B}}\\{T_{I}}\\ \end{matrix}\right) = \int \left( \begin{matrix}{T_{A B}^{-}}\\{T_{I}^{-}}\\ \end{matrix}\right) = \left( \begin{matrix}{f_{A B}^{\Lambda}}\\{f_{I}^{\Lambda}}\\ \end{matrix}\right) e_{\Lambda}- \left( \begin{matrix}{h_{\Lambda A B}}\\{h_{\Lambda I}}\\ \end{matrix}\right) g^{\Lambda}
\begin{array}{c}{K = h^{\mu \nu}\nabla_{\mu}u_{\nu}= - h^{\mu \nu}\Gamma_{\mu \nu}^{\lambda}u_{\lambda}= - \frac{1}{2 \sqrt{f}}h^{\mu \nu}\left( \partial_{\nu}h_{\mu \tau}+ \partial_{\mu}h_{\nu \tau}- \partial_{\tau}h_{\mu \nu}\right)}\\{= \frac{1}{2 \sqrt{f}}h^{\mu \nu}\partial_{\tau}h_{\mu \nu}}\\ \end{array}
\lim_{N\to\infty}\Bigg|\frac1N\sum_{n=1}^N \prod_{i=1}^k\prod_{j=1}^\ell \tilde{f}_i(R_X^{in}x)\tilde{g}_j(R_Y^{jn}y) - \Bigg(\frac1N\sum_{n=1}^N \prod_{i=1}^k\tilde{f}_i(R_X^{in}x)\Bigg) \Bigg(\frac1N\sum_{n=1}^N \prod_{j=1}^\ell \tilde{g}_j(R_Y^{jn}y) \Bigg)\Bigg|=0.
\forall\xi \in\mathbb{R}^{n}\backslash\{0\}\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1 2 \leq\left\vert j-j^{\prime}\right\vert \Rightarrow\mathrm{Supp}(\varphi(2^{-j}\cdot))\cap\mathrm{Supp}(\varphi(2^{-j^{\prime}}\cdot))=\emptyset.
Q(t)=[\begin{matrix}t-i&0\\ \frac{2t}{t+i}&\frac{t^{2}-1}{t+i}\\ \end{matrix}]UU(t)=\frac{1}{\sqrt{2}}[\begin{matrix}t+i&t-i\\i(t-i)&-i(t+i)\\ \end{matrix}].
{\tau}^{p \to n}_{\rm Rindler}= \frac{\pi^2 a e^{\pi \frac{\Delta m}{a}}}{G_F^2 m_e}\left[ \int_{-\infty}^{+\infty}d{\omega_R}\frac{K_{i \frac{{\omega_R}}{a}+ \frac{1}{2}}( m_e/a) K_{i \frac{{\omega_R}}{a}- \frac{1}{2}}( m_e/a)}{\cosh[\pi ({\omega_R}-{\Delta m})/a ]}\right]^{-1}\, .
[\mathcal{X}_{|n|+2},\mathcal{Y}_{|n|+2}] [t_{\delta_1}^{-4n}t_{\delta_2}^{-8n}\cdots t_{\delta_{k-1}}^{-4(k-1)n}, r_k] = [\mathcal{X}_{|n|+2}t_{\delta_1}^{-4n}t_{\delta_2}^{-8n}\cdots t_{\delta_{k-1}}^{-4(k-1)n}, \mathcal{Y}_{|n|+2}r_k],
\frac{d E ( X^{0})}{d l}= 2 N \; \frac{\mid \cos ( H X^{0}) \mid [ ( 1 + H^{2}r^{2}( \sigma ) )^{2}+ H^{2}L^{2}\tan^{2}( H X^{0}) ]}{( 1 + H^{2}r^{2}( \sigma ) )^{3 / 2}\sqrt{H^{2}r^{2}( \sigma ) + \cos^{2}( H X^{0})}},
\begin{aligned}\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2 =-S(1-\dfrac12S)+(S-\sup H^2)^2-\dfrac12(\sup H^2)^2+\bar H^2(\bar \lambda_1^2+\bar\lambda_2^2) &=(\frac56S-1)S.\end{aligned}
\aligned\|u_n^+\|_{E_a}^2= \langle\Phi'_{j_n}(u_n),u_n^+\rangle+\int(I_\alpha\ast|u_n|^p)|u_n|^{p-2}u_nu_n^+dx \leq 1\cdot\|u_n^+\|_{E_a}+\|u_n\|_{\mathcal Q^{\alpha,p}}^{2p-1}\|u_n^+\|_{\mathcal Q^{\alpha,p}}\leq \|u_n^+\|_{E_a}+[C(1+\|u_n\|_{E_a})]^{1-\frac{1}{2p}}\|u_n^+\|_{\mathcal Q^{\alpha,p}}\endaligned
\det \left(\frac{\partial x}{\partial y}\right) = \frac{|x|^n}{|y|^n}\det \left( \begin{array}{cccc}a\frac{y_1^2}{|y|^2}+1 a\frac{y_1 y_2}{|y|^2}\cdots a\frac{y_1y_n}{|y|^2}a\frac{y_2y_1}{|y|^2}a\frac{y_2^2}{|y|^2}+1 \cdots \vdots \vdots \vdots \ddots \vdots a\frac{y_ny_1}{|y|^2}\cdots \cdots a\frac{y_n^2}{|y|^2}+1 \end{array}\right).
\| (I_{N}^{\lambda}u-u)'\|_{L^2(\mathbb R)}\le c\big(\|I_N^G \breve U-\breve U\|_{L^2_{\omega_\lambda}(I)}+ \| \sqrt{1-t^2}(I_N^G \breve U-\breve U)'\|_{L^2_{\omega_\lambda}(I)}\big).
I_{\frac{5}{2}}\geqslant L(1)\tau \int_{\tau^2}^{1}u^{1/2}e^{uy}du =L(1)\tau\bigg[\frac{\tau}{y}\big(e^{y/2}- e^{\tau^2y}\big)+\frac{1}{\sqrt{2}y}\big(e^{y}-e^{y/2}\big) \bigg], \mbox{for}\tau < \frac{1}{\sqrt{2}}
\begin{aligned}\big[2 - h(1+\kappa)\big]\big(f_{0}-f_{1}\big){}\geq{}\frac{h}{2L}\big[\kappa h^2 - 2h(1+\kappa) + 3\big] \big\|g_{0}\big\|^2 + \frac{h}{2L}\|g_1\|^2 \end{aligned}
\sigma(X)=\sum_{i<j\le n}w_{ij}(d_{ij}(X)-\delta_{ij})^{2}=\sum_{i<j}w_{ij}\delta_{ij}^{2}+\sum_{i<j}w_{ij}d_{ij}^{2}(X)-2\sum_{i<j}w_{ij}\delta_{ij}d_{ij}(X)
\omega = -{\frac{e^{2}}{2 \theta}}\sum_{a , b = 1}^{N}\int_{0}^{T}d t \int_{0}^{t}d \tau{\dot{x}}_{a}^{i}( t ) \epsilon_{i j}{\dot{x}}_{b}^{j}( \tau ) \delta ({\bf x}_{a}( t ) -{\bf x}_{b}( \tau ) ) .
\nu^{2}{\frac{\theta_{i j}}{( z_{i j})^{2}}}\xi_{i}\epsilon_{j}K_{i j}^{\mathrm{t o t}}, \quad \nu^{2}{\frac{\theta_{i j}}{( z_{i j})^{2}}}\xi_{i}\epsilon_{j}K_{i j}^{\mathrm{t o t}}K_{j k}^{\mathrm{t o t}}.
\left( \begin{matrix}{{1}}&{{0}}&{{- \sqrt{2}\xi}}&{{\sqrt{2}\xi}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{\sqrt{2}\xi}}&{{0}}&{{1 - \xi^{2}}}&{{\xi^{2}}}\\{{\sqrt{2}\xi}}&{{0}}&{{- \xi^{2}}}&{{1 + \xi^{2}}}\\ \end{matrix}\right) .
\frac{a}{a-\mathrm{{i}}z}= \exp \biggl[ \int_0^\infty\bigl(\mathrm{e}^{\mathrm{i}zx}-1 \bigr) x^{-1}\mathrm{e}^{-a x}\,\mathrm{d}x \biggr], a>0 , z \in{\mathbb{R}}
f_{4,4,3}(q^3,q^2;q) =J_{1}\psi(q) f_{4,4,3}(-q^3,-q^2;q) =\frac{1}{4}\overline{J}_{0,3}\mu(q^3) -\frac{1}{2}\overline{J}_{1,3}\phi(q)+\Theta(q),
\begin{aligned}\int_S g(x) p_{S}(x)\,dU(x) = \int_S \int_S g(x) 1_{B_y}(x)\,dU(y)\,dU(x) &= \int_S \left(\frac{1}{U(B_y)}\int_{B_y}g(x)\,dx\right)\,dU(y) > \int_S g(y)dU(y) = 0,\end{aligned}
1-\langle \hat{u}_j^{(n)},u_j^{(n)}\rangle=1-|\langle \hat{u}_j^{(n)},u_j^{(n)}\rangle|\leq 1-\langle \hat{u}_j^{(n)},u_j^{(n)}\rangle^2=\frac{1}{2}\|\hat P_j^{(n)}-P_j^{(n)}\|_2^2\xrightarrow{\P}0.
0\rightarrow\mathbb{Z}^{(\begin{matrix}n\\n\end{matrix})}\rightarrow\mathbb{Z}^{(\begin{matrix}n\\n-1\end{matrix})}\rightarrow\cdot\cdot\cdot\rightarrow\mathbb{Z}^{(\begin{matrix}n\\ 1\end{matrix})}\rightarrow\mathbb{Z}^{(\begin{matrix}n\\ 0\end{matrix})}\rightarrow0
f_{r_m,r_t}(x,y)= \frac{4(\lambda_c\pi)^{t}}{(t-m-1)!(m-1)!}e^{-\lambda_c \pi y^2}\sum^{t-m-1}_{p=0}(-1)^p \times{t-m-1\choose p}y^{2(t-m-1)-2p+1}x^{2m+2p-1}.
|r^{1/2}u_t - r^{1/2}u_r|_{(r,t)=(r_0,t_0)}\leq |(\sqrt{r}u)_t - (\sqrt{r}u)_r|_{(r,t)=(r_0,t_0)}+ (1/2) r_0^{-1/2}|u(r_0,t_0)| \lesssim_\delta 1.
S_{i,j}= X \sum_{\substack{\alpha \leq Z (\alpha, 2)=1}}\frac{\mu(\alpha)}{2\alpha^2}\sum_{m \geq 1}\sum_{\substack{(p,2\alpha)=1}}\frac{a_{\pi}(p)\log p}{p}\hat{\phi}\left( \frac{\log p}{M\log X}\right) \left( \frac{2jm}{p}\right) \widetilde{W}_{i}\left(\frac{mX}{\alpha^2p}\right).
u_j=\sum_{l=1}^{j-1}C_{j-l}\frac{\partial_{\epsilon}^{l-1}u}{(l-1)!}+\sum_{l=1}^{j-1}D_{j-l}\frac{\partial_{\epsilon}^{l-1}v}{(l-1)!}+\frac{\partial_\epsilon^{j-1}u}{(j-1)!},
\det\begin{pmatrix}3 s_1(a) s_1(b) s_1(a) s_1(a^2) s_1(ab) s_1(b) s_1(ab) s_1(b^2) \end{pmatrix}= \begin{array}{l}3s_1(a^2)s_1(b^2) - 3s_1(ab)^2 \ -\s_1(a)^2s_1(b^2) - s_1(a^2)s_1(b)^2 \ +\ 2s_1(a)s_1(ab)s_1(b). \end{array}
B_{11}:= P_5, &A_{11}:= P_4-(B_{12}+B_{12}^T), &B_{22}:= P_3-(B_{13}+B_{31}^T)-(A_{12}+A_{12}^T), &A_{22}:= P_2-(B_{23}+B_{23}^T) - (A_{13}+A_{13}^T), &B_{33}:= P_1-(A_{23}+A_{23}^T), \mbox{and}&A_{33}:= P_0..
( \alpha_{o p t}^{2})^{( e v )}= \sqrt{\frac{2 \mid \beta \mid^{2}t a n h \mid \beta \mid^{2}+ 2 \mid \beta \mid^{2}c o s ~ 2 \theta + 1}{2 \mid \beta \mid^{2}t a n h \mid \beta \mid^{2}- 2 \mid \beta \mid^{2}c o s ~ 2 \theta + 1}}( \alpha_{o p t}^{2})^{( o )}\nonumber \,
\partial_L f(x) = \{l\in L: \forall y\in X, f(y) \geq f(x) + l(y)-l(x)\}= \{l\in L: \forall y\in X, f(y) \geq f(x) + l_x(y)-l_x(x)\}= \{l\in L: l-l(x) \in \partial_{L_x}f(x)\}
[\psi_1,\cdots,\psi_N](x)=\prod_{j>l}^N(k_j-k_l)\cdot e^{\sum_{j=1}^Nk_jx}u_N(x), &U_N(x)=-2\partial_x^2\log[\psi_1,\cdots,\psi_N](x)=-2\partial_x^2\log u_N(x).
\widetilde{u}_n \in \left( Q_1(\alpha-1,A_n,u_n + a_nf_{q(n)}) \right)^{(\alpha)}\subseteq \left( \bigcap_{j=1}^\infty K(\alpha-1,A_j,u_j + a_jf_{q(j)}) \right)^{(\alpha)}= K^{(\alpha)}.
X = \left[\begin{array}{cccccc}1 \cdots 0 0 \cdots 0 \vdots \ddots \vdots \vdots \vdots 0 \cdots 1 0 \cdots 0 0 \cdots 0 X_{11}\cdots 0 \vdots \vdots \vdots \ddots \vdots 0 \cdots 0 0 \cdots X_{rr}\end{array}\right]
|\langle b_{j},\varphi \rangle|\le C_{\varphi,\theta}\left(\sum_{l=1}^\infty\left(2^{-\frac{n\theta}{p}}\left\|{\mathcal M}f \cdot \chi_{{\mathcal O}_j}\right\|_{L^{p}([-2^l,2^l]^n)}\right)^{\frac{1}{\kappa}}\right)^{\kappa}
\psi^{\hbar}_{n - e_{j}}= \frac{1}{\sqrt{n_{j}}}\, \hat{a}_{j}\psi^{\hbar}_{n}, \psi^{\hbar}_{n + e_{j}}= \frac{1}{\sqrt{n_{j}+ 1}}\, \hat{a}^{*}_{j}\psi^{\hbar}_{n},
A_{1}=\left[\begin{array}{cccccc|cc}1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 \hline1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1\end{array}\right]
H_{2n}= 2^{4n^2-1}\frac{(2n+2)!}{(4n+1)!}\prod_{k=1}^{2n-1}\frac{k!^2}{(2k+1)!^2}, H_{2n+1}= 2^{4n(n+1)}\frac{(2n+1)(2n+2)!}{(4n+3)!}\prod_{k=1}^{2n}\frac{k!^2}{(2k+1)!^2}.
\frac{d}{dt}(I_{\hat\omega}(\varphi_t)-J_{\hat{\omega}}(\varphi_t)) =-\frac{1}{V}\int_M\varphi_t{\Delta}_t\dot{\varphi_t}{\omega}_t^n =\frac{1}{V}\int_M({\Delta}_t\dot{\varphi_t}+t\dot{\varphi_t}){\Delta}_t\dot{\varphi_t}{\omega}_t^n.
\int_{0}^{\infty}d k \hspace{. 2 c m}Q_{\ell}({\frac{k^{2}+ q^{2})}{2 k q}}){\frac{q^{\ell + d_{1}}}{( c^{2}+ k^{2})^{d_{2}}}}B_{j}( k ) = q^{\ell + 1}\sum_{j = 0}^{3}a_{j}( q - p )^{j}.
f(H^E_b) = \tilde{\eta}_L f(H_b)\eta_L +\tilde{\eta}_0 f(H_b^E)\eta_0 +\pi^{-1}\int_{\mathcal{D}}\overline{\partial}f_N(z)(H^E_b-z)^{-1}W_L(z) \d z_1 \d z_2.
J_\phi (t) = I_\phi (t) - A(n,p,\ell,\phi ), \widetilde J_\phi (0) = 2^{-\frac{1}{p-1}}\|\phi^\ell\|_{L^1(\mathbb R^n)}^{-1}J_\phi(0), A_1 = \frac{J_\phi '(0)}{J_\phi (0)}= \frac{I_\phi '(0)}{I_\phi (0) - A(n,p,\ell,\phi )}.
\Psi_{(3,3,2)}=\prod_{i<j}(z_i^{(1)}-z_j^{(1)})^{3}\prod_{i<j}(z_i^{(2)}-z_j^{(2)})^{3}\prod_{i,j}(z_i^{(1)}-z_j^{(2)})^{2}\; \Psi_{0}.
<\alpha|T_1|\gamma>=\frac{2}{\left(2\pi\right)^{2\left(\nu-1\right)}}\delta\left(p_1-q_1\right)\frac{e^{-iq_1^{o}x}}{\sqrt{2{\omega}_1\left(q_1^{o}\right)}}\delta\left(p_2-q_2\right)\;\;\times
\begin{matrix}T_{A}=&0&:&\frac{a-b+c}{b}&:&\frac{a+b-c}{c}\\T_{B}=&\frac{-a+b+c}{a}&:&0&:&\frac{a+b-c}{c}\\T_{C}=&\frac{-a+b+c}{a}&:&\frac{a-b+c}{b}&:&0\end{matrix}
S o l v \left( E_{7 ( 7 )}/ S U ( 8 ) \right) = S o l v \left( S L ( 2 , R ) / O ( 2 ) \right) \, \oplus \, S o l v \left( \frac{S O ( 6 , 6 )}{S O ( 6 ) \times S O ( 6 )}\right) \, \oplus \,{\cal W}_{7}
\mathcal{S(}\Sigma \mathcal{)=}\int d^4x\frac{\delta \Sigma}{\delta \varphi^i}\frac{\delta \Sigma}{\delta \varphi^{*i}}\;+\;\frac12C_{\;AB}^C\varepsilon^A\varepsilon^B\frac{\partial \Sigma}{\partial\varepsilon^C}\;,
\int_{t= \frac{\sqrt{\beta +1}-\sqrt{\beta -1}}{\sqrt{2}}}^{\frac{\sqrt{\beta -1}+\sqrt{\beta +1}}{\sqrt{2}}}t^{-n-m-3}b_mb_n(t^2+1) \sqrt{\frac{t^4-2 \beta t^2+1}{1-\beta}}dt = \begin{cases}0 \mbox{if}n\not=m, \frac{\pi}{2}(\beta+1)\sqrt{\beta-1}\mbox{if}n =m. \end{cases}
[\boldsymbol{I}]_{\mathcal{B}_{\rm cycl}}^{\mathcal{B}_{\rm int}}=\boldsymbol{V}(\boldsymbol{q})=\begin{bmatrix}1 \lambda_1^{2}\ldots \lambda_1^{n-1}1 \lambda_2^{2}\ldots \lambda_2^{n-1}\ldots \ldots \ldots \ldots 1 \lambda_n^{2}\ldots \lambda_n^{n-1}\end{bmatrix}.
- \int_M (\nabla_{ij}A_{kl}\nabla^jA^{kl}) (\nabla^i|\nabla A|^2) \,\gamma^s\,d\mu = - 2\int_M (\nabla_{ij}A_{kl}\nabla^jA^{kl}) (\nabla^i\nabla_pA_{qr}\nabla^pA^{qr}) \,\gamma^s\,d\mu = -\frac12\int_M \big|\nabla|\nabla A|^2\big|^2\,\gamma^s\,d\mu\,.
\Delta(p)=\sum_{n=1}^{N}\frac{Z_{n}}{p^{2}-m_{n}^{2}+i\epsilon}+\int_{4m_{N}^{2}}^{\infty}d\mu^{2}\rho_{c}(\mu^{2})\frac{1}{p^{2}-\mu^{2}+i\epsilon}
< 0 |{\cal T}\left[ \varphi ( y , t ) \varphi ( y_{0}, t_{0}) \right] | 0 >_{E}= \int_{0}^{\infty}\frac{\mathrm{d}\theta}{2 \pi}e^{- m | x - x_{0}| \cosh \theta + i m ( y - y_{0}) \sinh \theta},
S^{m''}_{1,A}(x,t; X) -S^{0}_{1,A}(x,t; X) &= \sum_{s\leq t}\mathbf{1}_{|X(s)-X(s-)|\geq 1}\bigg[ \big( A(x+\Phi_{m''}(X)(s-) + \tfrac{1}{2}\phi_{m''}(X(s)-X(s-))) & - A(x+X(s-)+\tfrac{1}{2}(X(s)-X(s-)))\big)\cdot \phi_{m''}(X(s)-X(s-)) +A(x+\tfrac{1}{2}(X(s)+ X(s-)))\cdot \big(\phi_{m''}(X(s)-X(s-))- (X(s)-X(s-))\big) \bigg],
({\frac{\delta \tilde{\Gamma}}{\delta A}},{\frac{\delta \tilde{\Gamma}}{\delta K}}) + ({\frac{\delta \tilde{\Gamma}}{\delta c}},{\frac{\delta \tilde{\Gamma}}{\delta L}}) +{\frac{\delta \tilde{\Gamma}}{\delta V}}\cdot{\frac{\delta \tilde{\Gamma}}{\delta M}}= 0 .
A_{K,m, \xi}(t) = \tilde V_m( K\cap \{\xi^\perp + t\xi\}) = \frac{1}{n-1}\int_{S^{n-1}\cap \xi^\perp}\rho_{K-t\xi}^m(\theta)\, d\theta & = \frac{\kappa_{n-1}}{\kappa_m}\int_{G(\xi^\perp, m)}V_m ((K-t\xi) \cap H ) \, dH,
\mathcal{G}^i \triangleq \left\{\theta^i \mathbf{G}^i: \theta^i>0, \mathbf{G}^i \in \mathbf{R}^{n_i\times n_i}, (\mathbf{G}^i)^T \mathbf{Q}^i \mathbf{G}^i = \mathbf{Q}^i \right\},
\left\langle \mathbf P^{\bot}_{M_0}(W),M_0-\hat M\right\rangle=-\left\langle W ,\mathbf P^{\bot}_{M_0}\left (\hat M\right )\right\rangle=-\left \Vert \mathbf P^{\bot}_{M_0}\left (\hat M\right )\right \Vert_{*}.
I_{5}=-\frac{(n-2)^{2}}{2(n-1)}\omega_{n-1}\left(I_{n-1}^{n-2}-\frac{\delta^{2}}{6(n-1)}\bar{R}_{ii}(q)\omega_{n-1}I_{n-1}^{n}\right) =-\frac{(n-2)^{2}(n-3)}{2(n-1)^{2}}\omega_{n-1}I_{n-1}^{n}+\delta^{2}\frac{(n-2)^{2}}{12(n-1)^{2}}\bar{R}_{ii}(q)\omega_{n-1}I_{n-1}^{n}.
t_{s_{1,1}}^{2m}= \prod_{i=1}^{m}{}_{t_{c_1}^{-2i}}(t_{t_{c_1}(z_1)}^{-1}t_{a_1}t_{z_1}^{-1}t_{a_1}) \cdot t_{c_1}^{-2m}t_{a_1}^{2m}\cdot t_{d_{g-1}}^{2m}t_{d_1}^{2m}
\left[{\frac{d^{2}}{d t^{2}}}+{\frac{1}{2}}\sum_{k}{\frac{1}{t - a_{k}}}{\frac{d}{d t}}-{\frac{1}{4 \hbar^{2}}}\sum_{k}{\frac{f_{k}}{t - a_{k}}}\right] \Psi ( t ) = 0
s_3\leq (25n)^{\frac{1}{2}(u+1)}\left(1+\sum_{i=1}^{3u+3}p_i(\kappa')/n^i\right) e^{-2\kappa'}+ (25n)^{\frac{1}{2}(u+1)}(q(\kappa')/n^{3u+4})e^{2\kappa'}.
\beta_{2,\infty}:=\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_r}\frac{1}{\left|x\right|^{2(N+2)-\beta_k}}e^{\hat{u}_{k}}=\frac{\beta_{\infty}\left[4+a\beta_{\infty}-2a\beta\right]}{4\left[1-a(N+1)\right]}.
D^{\alpha-1}\left((P_t^{\nu_\alpha})^*(f)\right)(x) + \frac{1}{p_\alpha(x)}R^\alpha \left(p_\alpha, (P_t^{\nu_\alpha})^*(f) \right)(x) = \dfrac{- x e^{-\alpha t}}{(1-e^{-\alpha t})}(P^{\nu_\alpha}_t)^{*}(f)(x) & + \dfrac{e^{-t}}{\left(1-e^{-\alpha t}\right)}(P^{\nu_\alpha}_t)^{*}(hf)(x),
\tau^r_\varphi (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)\varphi (e,g_1,g_1g_2,\ldots,g_1\cdots g_k)dg_1\cdots dg_k
\widetilde{\Pi}_n^{(n)}( z_1, z_2) = \frac{(z_1 z_2 )^{\frac{s}{2}}}{\prod_{i = 1}^m ( v_i+ z_1 )^{\frac{1}{2}}(v_i + z_2)^{\frac{1}{2}}}\Sigma_n (z_1, z_2 ),
\int_{\Omega_j \cap \Omega}| \widehat{\nabla}_x f|^2 dx \geq C \int_{\Omega_j \cap \Omega}| \nabla_x f|^2 dx = \int_{B_1^+}|\nabla_y \widetilde{f}L|^2 \iota^{-1}dy \geq C' \int_{B_1^+}|\nabla_y \widetilde{f}|^2 dy,
\widetilde{\Gamma}_{kp}\ =\ \frac{1}{\mu_k{-}\bar{\mu}_p}\,T_k^\dagger\,T_p\quad, \qquad\textrm{i.e.}\qquad\sum_{k=1}^m\,\Gamma^{\ell k}\,\widetilde{\Gamma}_{kp}\ =\ \delta_{\p}^\ell \quad.
D^{(1)}=\sum_{a=1}^r \left(\sum_{i=1}^n y_{i,a}^2\right)\left(\sum_{j=1}^n \partial_{j,a}^2\right)+\sum_{1\leq a\neq b\leq r}\left(\sum_{i=1}^n y_{i,a}y_{i,b}\right)\left(\sum_{j=1}^n \partial_{j,a}\partial_{j,b}\right).
B[v,\varpi]=\int_{\mathcal{Y}^d}e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+\lambda V}\big( (\Lambda+\nabla_y\widetilde{w})^T\nabla_yv (\Lambda+\nabla_y\widetilde{w})+\nabla_y v )\big)^T\nabla_y \varpi dy.
\delta_{w^\ast}(x_{n}^\ast) :=\sup\left\{\mathrm{diam}\,\mathrm{clust}(x_{n}^\ast(x)):\;x\in B_{X}\right\}\delta_{w}(x_{n}^\ast) :=\sup\left\{\mathrm{diam}\,\mathrm{clust}(x^{\ast\ast}(x_{n}^\ast)):\;x^{\ast\ast}\in B_{X^{\ast\ast}}\right\}
-3F(2\pi/3)+\sum_{i=1}^{3}F(\theta_{i})\leq-\frac{1}{\pi^{2}}\frac{13}{72}\lambda_{1,n}^{r,s}.\leq-\frac{1}{\pi^{2}}\frac{13}{9}\lambda_{1,n}^{r,s}\sum_{i=1}^{3}\Big(\frac{\theta_{i}}{2\pi}-1/3\Big)^{2}.
e(G_1-C)+e(G_1-C,C) =e(H)+e(H,C)-e(B_1)-e(B_1-b_1, C) &>\left(\left\lfloor\frac{c}{2}\right\rfloor-1\right)\cdot (n-r-c)=\left(\left\lfloor\frac{c}{2}\right\rfloor-1\right)\cdot (|V(G_1)|-c).
c_{4}^{\{3 , L \}}\approx - 1 . 7 4 2 0 4 + 0 . 5 9 2 1 3 \, L^{- 1}- 0 . 0 9 9 3 3 \, L^{- 2}- 0 . 0 1 3 4 6 \, L^{- 3}\, .
\begin{array}{c}0=\left[-(\Gamma^0+\cos\alpha\Gamma^{\phi})\Gamma(e^{\psi})+\sin\alpha\right]\psi= =\left[\Gamma(e^{\psi})(\Gamma^0+\cos\alpha\Gamma(e^{\phi}))+\sin\alpha\right]\psi= =\left[\Gamma(e^{\psi})(\Gamma(e^{||})+\cos\alpha\Gamma(e^{\phi}))+\sin\alpha\right]\psi\end{array}
\begin{array}{r c l}{I_{2}( \rho^{2}, m )}&{=}&{\displaystyle \mu^{4 - n}\int \frac{d^{n}k}{( 2 \pi )^{n}}[ ( k + p )^{2}+ m^{2}]^{- 2}\cos ( k \cdot \rho )}\\{}&{=}&{\displaystyle I_{1}( \rho^{2}, m ) \cos ( p \cdot \rho ) .}\\ \end{array}
E_q(z):= \sum_{j=0}^\infty q^{j(j-1)/2}\,\frac{z^j}{[j]_q!}; \, z\in \mathbb{C}\, \mbox{and}\, e_q(z):= \sum_{j=0}^\infty \frac{z^j}{[j]_q!}; \, |z|< 1.
a_E(m) = \beta_{\infty}(m) \beta_2(m) \beta_7{m}\beta_{13}(m) \left( \prod_{2,7,13\neq p \mid m}\beta_p(m)\right) \left( \prod_{2,7,13 \neq q \nmid m}\beta_q(m) \right) &= \dfrac{182m}{213}\beta_2(m)\beta_7(m) \beta_{13}(m) \left(\prod_{2,7,13\neq p \mid m}\dfrac{\beta_p(m)p^2}{p^2-\chi_Q(p)}\right) \; .
\widehat{\mathbf{R}}_k = \epsilon_k \mathbf{D}\circ \left(\widetilde{\mathbf{V}}_k^H\widetilde{\mathbf{V}}_k^H\widetilde{\boldsymbol{\Sigma}}_k + \widetilde{\boldsymbol{\Sigma}}_k\widetilde{\mathbf{V}}_k^H \widetilde{\mathbf{V}}_k\right),
- \log \left[ ( x )_{i \frac{\pi}{2}+ \theta}( x )_{i \frac{\pi}{2}- \theta}\right] \, \sim \, - 4 \sin \pi x \, \mathrm{e}^{- \left| \theta \right|}+ O \left( \mathrm{e}^{- 2 \left| \theta \right|}\right) \ .
\int_{0}^{1 / l_{0}}. . . ={\frac{8 g_{Y M}^{4}}{M_{s}^{4}}}[{\frac{1}{1 2 l_{0}^{2}}}+{\frac{\pi ( s + t )}{1 8 0 l_{0}^{3}M_{s}^{2}}}] + \cdots \ .
\|u\|_{H^t}\lesssim \|L^* u\|_{H^{t-2}}+ \|b . \nabla u + div(b) u\|_{H^{t-2}}& \lesssim \|L^*u\|_{H^{t-2}}+ \|b\|_{B^{t-2}_{\infty \infty}}\|u\|_{H^{t-1}}+ \|b\|_{B^{t-1}_{\infty \infty}}\|u\|_{H^{t-2}}
F \epsilon^{ij}={1\over 2}E^{{\alpha}i}\wedge E^{{\beta}j}\epsilon_{{\alpha}{\beta}}+{1\over 2}\bar{E}^{\dot{\alpha}i}\wedge \bar{E}^{\dot{\beta}j}\epsilon_{\dot{\alpha}\dot{\beta}}+{1\over 2}E^a \wedge E^b f_{ab}\epsilon^{ij}\; , \quad dF\equiv 0 \; .
U_{1}(x,m,t)=\partial_{m_{1}}V_{1}(m,t)(x) \, ,U_{-1}(x,m,t)=\partial_{m_{2}}V_{1}(m,t)(x) \, , & U_{2}(x,m,t)=\partial_{m_{2}}V_{2}(m,t)(x) \, ,U_{-2}(x,m,t)=\partial_{m_{1}}V_{2}(m,t)(x) \, ,
W_{LaL\underline L}= r \beta_a +r^2 D\beta_a +\frac{1}{2}r^3 D^2\beta_a+O(r^4) W_{L\underline LL\underline L}=\rho + r D \rho+r^2 [ \frac{1}{2}D^2 \rho -\frac{1}{3}|\beta|^2]+O(r^3).
H_t = \{r : 0 < r_2 \leq b_2-a_2 \textrm{and}\phi(a) \leq \phi(r)-t < \phi(a)+\phi(d)-\phi(b)\}&\cup \{r : b_2-a_2 < r_2 \leq c_2-a_2 \textrm{and}\phi(b) \leq \phi(r)-t < \phi(a)+\phi(d)-\phi(b)\}&\cup \{r : c_2-a_2 < r_2 \leq m \textrm{and}\phi(d) \leq \phi(r)-t < \phi(a)+\phi(d)-\phi(b)\}.
\tilde{T}_{mn}= \tilde{T}_m\circ \tilde{T}_n, (m,n) = 1\ \mbox{\ \ \and}\tilde{T}_{p^{r+2}}(x) = \tilde{T}_p\left(\tilde{T}_{p^{r+1}}(x)\right) - \frac{1}{p}\Psi^p\tilde{T}_{p^r}(x) p\in P,\r\geq 0,\x\in E^*(X).
E\|X_1- X_2\|^{2l}_2+E\|Y_1-Y_2\|^{2l}_2-2E\|X_1- Y_1\|^{2l}_2 &=\sum_{0\leq a_1+a_2\leq l}\frac{(-2)^{l-a_1-a_2}l!}{a_1!a_2!(l-a_1-a_2)!}\Big[E\{\|X_1\|_{2}^{2a_1}\|X_2\|_{2}^{2a_2}(X_1^\top X_2)^{l-a_1-a_2}\}&\quad+E\{\|Y_1\|_{2}^{2a_1}\|Y_2\|_{2}^{2a_2}(Y_1^\top Y_2)^{l-a_1-a_2}\}-2E\{\|X_1\|_{2}^{2a_1}\|Y_1\|_{2}^{2a_2}(X_1^\top Y_1)^{l-a_1-a_2}\}\Big].
u'=\begin{cases}\frac{1}{d_0}\left( -1+ (\frac{d_0+d_1}{d_1}x)^{1/n}\right), 0\leq x\leq \frac{d_1}{d_0+d_1}, \frac{1}{d_1}\left( 1- (\frac{d_0+d_1}{d_0}(1-x))^{1/n}\right), \frac{d_1}{d_0+d_1}\leq x\leq 1.\end{cases}
\begin{array}{rl}\underset{\mathbf{u}}{}\mathbf{u}^H\mathrm{Diag}(\mathbf{y})^H\mathbf{V}\mathrm{Diag}(\mathbf{y})\mathbf{u}|u_n| = 1,n=0,1,\dots,N_c-1.\end{array}
u(\vec{p})\,\,=\,\, \left( \frac{w_p+m}{2\,w_p}\right)^{\frac{1}{2}}\,\left[ \begin{array}{c}1 \frac{p^{2}\,-\,ip^{1}}{w_p\,+\,m}\end{array}\right] \,\,\quad v(\vec{p})\,\,=\,\, \left( \frac{w_p+m}{2\,w_p}\right)^{\frac{1}{2}}\,\left[ \begin{array}{c}\frac{p^{2}+i p^{1}}{w_p\,+\,m}1\end{array}\right] \,\,.
\frac{dE_-(t)}{dt}=\int_{q(t,a)}^{q(t,b)}e^{-y}m_t(t,y)dy =\int_{-\infty}^{\infty}e^{-y}m_t(t,y)dy =-\int_{-\infty}^{\infty}\frac{1}{2}[(uv-u_yv_y)m]_ye^{-y}dy +\int_{-\infty}^{\infty}\frac{1}{2}(uv_y-vu_y)me^{-y}dy =\int_{-\infty}^{\infty}\frac{1}{2}(u+u_y)(v_y-v)me^{-y}ydy\leq 0,
\begin{array}{l}\partial_t a_i-\nabla\cdot(D_i\nabla a_i)=Q_i(a),i\in\{1,2,3,4\},\t\geq 0,\x\in\mathbb R^N, begin{align*}.3cm]Q_i(a)=(-1)^{i+1}(a_2a_4-a_1a_3),\end{array}
\begin{aligned}-\begin{bmatrix}1-F_{j,j-1}0 -\Gamma_{j}-G_{j-1,j-1}-\widetilde{t}_{j}\rho^2 0 1-F_{j,j-1}^{T}x -M\rho^2\widetilde{t}_{j}^2 1-f_{j,j}\end{bmatrix}\end{aligned}
V_{e}= - 2 i \pi \hat{\theta}_{e}\int d p_{0}^{\prime}d p_{0}\delta ( p_{0}^{\prime}\! - \! \mu_{e}) \beta_{1}\beta_{2}K_{e}^{T}( p_{0}^{\prime}, p_{0}) \delta ( p_{0}\! - \! \mu_{e}) \hat{\theta}_{e},
S_{i}=\sum_{k=1}^{N-1}\left\{\sum_{l=1}^{k}\left( 1-r_{i}\left( l^{\ast}\right) \right) \right\}\left( \frac{1}{\frac{\lambda_{\left(k+1\right)}^{2}(A)}{d_{i}}+1}-\frac{1}{\frac{\lambda_{\left( k\right)}^{2}(A)}{d_{i}}+1}\right)
\hat{B}_{n m}( \omega ) = -{\frac{2 \operatorname{c o t h}({\frac{\omega}{2}}) \operatorname{s i n h}\Bigl ({\frac{n \omega}{2}}\Bigr ) \operatorname{s i n h}\Bigl ({\frac{m \omega}{2}}\Bigr )}{\operatorname{s i n h}({\frac{\nu \omega}{2}})}}.
\rho(t) =\rho_{0}+\rho_{1}t+\rho_{2}t^{2}+.....+ \varphi(t) =\varphi_{0}+\varphi_{1}t+\varphi_{2}t^{2}+.....+ \theta(t) =\theta_{0}+\theta_{1}t+\theta_{2}t^{2}+.....+
\mathcal{S}_{\alpha\beta}\bigg\|(\phi \tilde{u})^{\frac{\alpha(\cdot)}{q(\cdot)}}(\phi \tilde{v})^{\frac{\beta(\cdot)}{q(\cdot)}}\bigg\|_{L^{q(\cdot)}(\mathbb{R}^{N})}\leqslant \|(\phi \tilde{u}_k,\phi \tilde{v}_k)\|= \|\phi \tilde{u}_k\|+ \|\phi \tilde{v}_k\|.
\Gamma_{\nu}( p , P ) = - e^{2}\int \mathrm{\frac{d^3 q}{( 2 \pi )^3}}D ( p - q ) \gamma_{\mu}S ( \mathrm{\frac{1}{2}}P + q ) \Gamma_{\nu}( q , P ) S ( - \mathrm{\frac{1}{2}}P + q ) \gamma_{\mu},
\delta_{\xi}\widetilde \Psi\ =\ -\frac{i}{2P_L}ie^2\delta\xi H_0\left \{(\Delta G_0T)+(\widetilde \Delta \widetilde g_0\widetilde T)\right \}(1-\widetilde g_0\widetilde T)^{-1}\widetilde \Psi -\frac{i}{2P_L}ie^2\delta\xi H_0\widetilde \Delta \widetilde \Psi\ .
I_{W}( z ) K_{W}( z ) = \frac{1}{2 W}\left[ 1 + \frac{z^{2}/ 2}{1 - W^{2}}- \frac{\pi ( z / 2 )^{2 W}}{W \Gamma^{2}( W ) \sin \pi W}+ O ( z^{4}, z^{2 W + 2}) \right] .
T\!\left(\overline{l},\overline{\alpha}\right) := \alpha_1^{\binom{l_1}{2}}\cdots \alpha_m^{\binom{l_m}{2}}\prod_{1 \le i < j \le m}(\alpha_i - \alpha_j)^{\min \left\{l_i^2,\, l_j^2 \right\}}
I(p)= \langle a_{d}b_{d-1}, a_{d}b_{d-2}, \dots, a_{d}b_{j}, \dots, a_{i+1}b_{j}\rangle: \langle a_{i}b_{j}\rangle =\langle a_d, \dots , a_{i+1}, b_{i-1}, \dots , b_{j+1}\rangle.
\begin{array}{rl}\mbox{Bekenstein-Verlinde bound}: S_{\rm BV}=\frac{2\pi}{n}ER, \mbox{Bekenstein-Hawking bound}: S_{\rm BH}=(n-1)\frac{V}{4G_nR}, \mbox{Hubble bound}: S_{\rm H}=(n-1)\frac{HV}{4G_n}.\end{array}
\langle \pi_q(g) \phi'_q, \phi'_q \rangle = 1, g \in K_q(q,q) \langle \pi_q(g) \phi'_q, \phi'_q \rangle = \chi(-1), g \in wK_q(q,q) \langle \pi_q(g) \phi'_q, \phi'_q \rangle \ll q^{-1/2}, g \notin W K_q(q,q)
\widehat{G}(0)\frac{N}{w}\sum_{d \mid v}\frac{\mu(d)}{d}\sum_{(a,w)=1}e \left( \frac{ad \overline{m\ell_1\ell_3 v}}{w}\right) = \widehat{G}(0)\mu(w)\frac{N}{w}\frac{\varphi(v)}{v},