harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{gathered}n_{\sigma_1}= \,\,\cdots \otimes n_{hi}\otimes n_{ij}\otimes n_{jk}\otimes \cdots \,\, \, \in \,\, N_{\sigma_1}, n_{\sigma_2}= \,\, \cdots \otimes n_{pj}\otimes n_{ji}\otimes n_{iq}\otimes \cdots \,\,\, \in \,\, N_{\sigma_2},\end{gathered}
V ( \overline{{{\rho}}}) = \frac{3}{2}\int \frac{d^{4}p}{( 2 \pi )^{4}}\ln ( p^{2}+ e^{2}\overline{{{\rho}}}^{2}) + \frac{1}{2}m^{2}\overline{{{\rho}}}^{2}+ \frac{\lambda}{2 4}\overline{{{\rho}}}^{4}.
{f^{[2]}\left(z_0,z_{1},z_2 \right)}:=\begin{cases}\frac{f^{[1]}(z_0,z_1) - f^{[1]}(z_1,z_2)}{z_0 - z_2}, \text{if~$z_0\neq z_2$}, \frac{d}{dz}f^{[1]}(z,z_1)_{\vert z=z_0}, \end{cases}, z_0, z_1,z_2\in\mathbb{T}.
Q^{0}_{P'}=\omega^{0}_{P'_{1}}\times\omega^{0}_{P'_{2}}\times\omega^{0}_{P'_{3}}:=\{(\xi^{1}_{1},\xi^{1}_{2},\xi^{1}_{3})\in\mathbb{R}^{3}: (2^{k'}\xi^{1}_{1},2^{k'}\xi^{1}_{2},2^{k'}\xi^{1}_{3})\in Q_{P'}\},
\|u_\tau(t) - \bar u_\tau(t)\|_{L^2(\Omega)}= \left|\dfrac{t_k-t}{\tau_k}\right|\left\| u_k - u_{k-1}\right\|_{L^2(\Omega)}&\leq \left\| \dfrac{u_k - u_{k-1}}{\tau_k}\right\|_{L^2(\Omega)}\tau_k &\stackrel{\eqref{ei-dsc}}\leq C |\tau|^{1/2}\mbox{for all}\t \in (t_{k-1}, t_k], \k = 1,2,\ldots,m,
\Delta m^2 \sim -\frac{\lambda}{2\pi L}\, \int_0^\infty dx \,\ln \Bigl(1 - e^{-2\pi L \sqrt{x^2 +m^2/4\pi^2}}\Bigr) =\frac{\lambda}{4\pi^2}m^2 \int_1^\infty{{(t^2-1)^{{1\over 2}}dt}\over{{\rm e}^{mL t}-1}}\, .
S(x,y)=\int_{-1}^{1}\,(x-v)\mathrm{sgn}(y-v)w_r^{(m)}(x)w_r^{(m)}(v)\sum_{j=0}^{N-2}\bigg(\prod_{i=1}^m\frac{(L_i+j)!}{L_i!j!}\bigg)(xv)^j\mathrm{d}v,
g = d \vec x_a \Phi_{a b}\cdot d \vec x_{b}+ \left(\frac{d \phi_a}{4 \pi}+ \vec\Omega_{ac}\cdot d \vec x_{c}\right) (\Phi^{-1})_{ab}\left(\frac{d \phi_b}{4 \pi}+ \vec\Omega_{bd}\cdot d \vec x_{d}\right),
\limsup_{R \to \infty}\limsup_{k\to +\infty}\bigg| \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\big[ \mathcal{A}(u_k(x)-u_{k}(y))u_k(x)\big](\Phi_{R}(x)-\Phi_{R}(y))K(x,y)\,dy\,dx + \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\big[ \mathcal{A}(v_k(x)-v_{k}(y))v_k(x)\big](\Phi_{R}(x)-\Phi_{R}(y))K(x,y)\,dy\,dx \bigg|=0.
\int \frac{g(|y|)}{|x-y|}\,dy \leq C\left[\int_{|x-y|\leq 1}\frac{g(|y|)}{|x-y|}\,dy+\sum_{i\in{\mathbb{Z}}^3:|i|\geq 1}\frac{1}{|i|}\int_{| x+i-y|\leq 1}g(|y|)\,dy\right] &\leq C\left[1+\sum_{\substack{i\in{\mathbb{Z}}^3:|i|>1 |x+i|\geq 1}}\frac{1}{|i||x+i|^{2+\epsilon}}\right].
\Upsilon_{\partial B_i}- \Upsilon_{\partial B_{i-1}}= \sum_{\substack{C \partial B_i C \phi_i^{-1}(x_i)}}u_{\rho(C)}\, \, \, \, \, - \sum_{\substack{C \partial B_{i-1}C x_i}}u_{\rho(C)}.
\langle J^{a i}( x ) \rangle_{\tilde{A}, \phi}= - \frac{\delta S}{\delta \tilde{A}_{i}^{a}( r_{b}, x )}, \hspace{1 e m}\langle{\cal O}^{I}( x ) \rangle_{\tilde{A}, \phi}= - \frac{\delta S}{\delta \phi^{I}( r_{b}, x )}
\kappa(h) =\prod\limits_{\alpha >0}\left[\frac{\pi (h,\alpha )}{a_0}\prod\limits_{n=1}^\infty\left(1-\frac{(h,\alpha )^2}{a_0^2n^2}\right)\right] =\prod\limits_{\alpha >0}\sin\frac{\pi (h,\alpha )}{a_0}\ .
\nu_2(B,h_r)= \:{1}+{2r^2}(a^{i})^2 \Bigl(\frac{1}{3(N+2)}R_{ii}-b_i \Bigr) +o(r^2) = \:{1}+ \frac{2r^2}{3N(N+2)}S(y_0) +o(r^2).
r ( n , - 2 t ) \sim{\frac{1 ! 2 ! \cdots ( n - 1 ) !}{\pi^{n / 2}2^{( 5 n^{2}+ n ) / 2}}}t^{- 3 n^{2}/ 4}\exp \left({\frac{8}{3}}n t^{3 / 2}\right) .
\frac{1}{\displaystyle \max_{\{I_j\}\in \mathcal C_n(I)}\min_{1\leq j\leq n}f_j(I_j)}=\min_{\{I_j\}\in \mathcal C_n(I)}\max_{1\leq j\leq n}\frac{1}{f_j(I_j)},
\int^1_{-1}p_n(x)^2w(x)dx =c^2_n\int^\pi_0 \Biggl( z^{-2n}\frac{\phi^2_{2n+1}(z)}{(z-1)^2}& +\frac{z^{2n}\phi_{2n+1}(1/z)^2}{(1/z-1)^2}+\frac{2\phi_{2n+1}(z)\phi_{2n+1}(1/z)}{(z-1)(1/z-1)}\Biggr)(1-\cos\theta)f_3(\theta)d\theta.
\begin{aligned}\mathcal{A}(E) =2^n\cdot\lim\limits_{t\rightarrow+\infty}e^{-nt}Vol(\partial B_E^{g^+}(t),g^+) &=2^n\cdot\lim\limits_{t\rightarrow+\infty}Vol(\partial B_E^{g^+}(t),g^E|_{\partial B_E(t)}) &=2^n\cdot Vol(\partial X,\hat{g}^E)\end{aligned}
q = \int_{c_1 + \sqrt{n_1}G_m^{-1}(p) + o(1)}^\infty [ 1 - \Phi( c_2 - z + (\sqrt{n_1}+\sqrt{n_2}) G_m^{-1}(p) + o(1) ) ] \, d \Phi(z) + o(1) & = \int_{c_1 + \sqrt{n_1}G_m^{-1}(p) + o(1)}^\infty [ 1 - \Phi( c_2 - z + (\sqrt{n_1}+\sqrt{n_2}) G_m^{-1}(p)) ] \, d \Phi(z) + o(1),
\varphi_{n_{k+1}, \infty}(i_{n_k, n_{k+1}}(\theta_{n_k, \mu})) = \varphi_{n_k, \infty}\Big(\sum_{\lambda \in E^{<n_{k+1}}, [\lambda]_{n_k}= \mu}\theta_{n_{k+1}, \lambda}\Big) = \psi\Big(\sum_{\lambda \in E^{<n_{k+1}}, [\lambda]_{n_k}= \mu}\chi_{Z(\lambda,n_{k+1})}\Big) = \psi(\chi_{Z(\mu,k)}) = \varphi_{n_k, \infty}(\theta_{n_k, \mu}).
e(G) \le \begin{cases}3k-6 +ex_{_\mathcal{P}}(n-k,\Theta_k) \, |U|=0 2k-3+ex_{_\mathcal{P}}(n-k+1,\Theta_k) \, |U|=1 k+ex_{_\mathcal{P}}(n-k+2,\Theta_k) \,|U|=2. \end{cases}\medskip
X_{n}(t, s) =\int_{0}^{t}\int_{0}^{s}(t-u)_{+}^{\frac{D}{2}-\frac{I}{2}}(s-v)_{+}^{\frac{D}{2}-\frac{I}{2}}B_{n}(du, dv) &=n^{2}\sum_{i=1}^{\lfloor nt\rfloor}\sum_{j=1}^{\lfloor ns\rfloor}\eta_{i, j}^{(n)}\int_{\frac{i-1}{n}}^{\frac{i}{n}}\int_{\frac{j-1}{n}}^{\frac{j}{n}}(t-u)_{+}^{\frac{D}{2}-\frac{I}{2}}(s-v)_{+}^{\frac{D}{2}-\frac{I}{2}}dudv.
\hat{\lambda}_j-\lambda_j-\lambda_j\bar{\eta}_{jj}=\langle \hat u_j,\hat\Sigma\hat u_j\rangle-\lambda_j\langle \hat u_j,\hat u_j\rangle-\langle u_j,Eu_j\rangle &=\langle \hat u_j,E\hat u_j\rangle-\langle u_j,E u_j\rangle+\langle \hat u_j,(\Sigma-\lambda_jI)\hat u_j\rangle
p_{b \perp}\sim - \varepsilon^{b}\sim \frac{D a p_{b}}{( D - 1 ) ( a - r )}\sim \frac{D \Gamma ( ( D + 1 ) / 2 ) ( \xi - \xi_{D})}{2^{D}\pi^{( D + 1 ) / 2}( a - r )^{D + 1}}\left( 1 - 2 \delta_{B 0}\right) .
\limsup\limits_{x \rightarrow\infty}\frac{\mathcal{K}_2(x)}{\mathbb{P}(S_\eta>x)}\leqslant\limsup\limits_{x \rightarrow\infty}\frac{c_7}{\mathbb{P}(S_\eta> x)}\sum\limits_{1 \leqslant k \leqslant\frac{x-1}{c_5-1}}\mathbb{P}(S_{\kappa+k}> x)\mathbb{P}(\eta=\kappa+k) &\leqslant c_7.
I:=\int_{\mathbb{S}^{2}}\frac{f^{2}}{\theta^{2}\,\phi\left(\theta\right)}d\sigma-\int_{\mathbb{S}^{2}}\frac{f^{2}}{\theta\,\phi\left(\theta\right)}\frac{\cos{\theta}}{\sin{\theta}}d\sigma=\int_{\mathbb{S}^{2}}\frac{1}{\theta\,\phi\left(\theta\right)}\left(\frac{1}{\theta}-\frac{\cos{\theta}}{\sin{\theta}}\right)f^{2}d\sigma
J(\Omega_t) = J(\Omega) + t \int_{\Gamma}\left(\nabla u \cdot{\nu}+ \lambda \right) \left(-\nabla u \cdot{\nu}+ \lambda\right) V_{n}\operatorname{d}\!s + O(t^2) &= J(\Omega) + t \int_{\Gamma}\underbrace{\left(\nabla u \cdot{\nu}+ \lambda \right)}_{<\ 0}|V_{n}|^2 \operatorname{d}\!s + O(t^2)< J(\Omega).
|H^\chi|<\sqrt{\frac{\tau_\chi}{P_{BS}}}\ \Rightarrow \ \Big(|h^\chi_d|<\sqrt{\frac{\tau_\chi}{P_{BS}}}\Big)\land \Big(|g_1||h^\chi_1|<\sqrt{\frac{\tau_\chi}{P_{BS}}}\Big)\land\cdots \land\Big(|g_M||h^\chi_M|<\sqrt{\frac{\tau_\chi}{P_{BS}}}\Big),
{\cal S}\supset \int d^{5}x \int d^{4}\theta \left( c_{1}{\frac{\Sigma^{\dagger}\Sigma B B^{\dagger}}{M_{P l}^{2}}}\delta ( x_{5}- r_{c}) + c_{2}{\frac{Q Q^{\dagger}B B^{\dagger}}{M_{P l}^{2}}}\delta ( x_{5}) \right)
\left(\hat{q}^a_i\right)^\dagger=\hat{q}^a_i\ \ ,\ \ \left(\hat{p}^a_i\right)^\dagger=\hat{p}^a_i\ \ ,\ \ \left[\hat{q}^a_i,\hat{p}^b_j\right]=i\hbar\delta^{ab}\delta_{ij}\ \ ,
(\begin{matrix}e_{x}\\e_{y}\\e_{z}\end{matrix})=U^{-1}(\begin{matrix}e_{+}\\e_{-}\\e_{0}\end{matrix}),U^{-1}=(\begin{matrix}-\frac{1}{\sqrt{2}}&+\frac{1}{\sqrt{2}}&0\\ +\frac{i}{\sqrt{2}}&+\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{matrix}).
\frac{1}{\lambda_{MFP}}=\frac{1}{\lambda_{el-el}}+\frac{1}{\lambda_{ap}}+\frac{1}{\lambda_{op,ems}}+\frac{1}{\lambda_{op,abs}}+\frac{1}{\lambda_{impurity}}+\frac{1}{\lambda_{defect}}+\frac{1}{\lambda_{boundary}}
x_k\sum_{i=1}^{n}{\frac{1}{x_k-w_i}}-\frac{1}{2}\sum_{i=1}^{n}\frac{R_n^{\prime \prime}(w_i)}{R_n^{\prime}(w_i)(x_k-w_i)}= 0, k=1,\,2,\, \ldots, n.
\varkappa_1=\begin{cases}(N_1,N_1,N_2,N_2)(N_3,N_1,N_2,N_2),G\in\mathcal{T}(\langle 729,49\rangle), (N_2,N_2,N_3,N_4)(N_2,N_3,N_3,N_4),G\in\mathcal{T}(\langle 729,54\rangle).\end{cases}
\sum_{i=1}^{J}\sum_{j\ne i}\frac{\lambda_{i}}{\lambda_{j}}p_{ij}\mu_{j}(x_{j})=\sum_{j=1}^{J}[\sum_{i\ne j}\frac{\lambda_{i}}{\lambda_{j}}p_{ij}]\mu_{j}(x_{j})=\sum_{j=1}^{J}[1-p_{jj}-\frac{\alpha p_{0j}}{\lambda_{j}}]\mu_{j}(x_{j}).
\begin{aligned}\frac{d}{dt}E^{s}(t) \lesssim (\|\varphi_x\|_{C^{1, \delta}}+\|\varphi_{tx}\|_{L^\infty})\|\varphi_x\|_{C^{1, \delta}}E^{s}(t) + \|\mathcal{R}\|_{H^s}\|v\|_{H^s}.\end{aligned}
\left|\mathbb E\left[\langle B_1,B_2\rangle\sum_{i=1}^n \int_{(i-1)\Delta_n}^{i\Delta_n}\Delta_n^{-(\mathfrak H+\epsilon)}\langle (\mathcal S(i\Delta_n-s)-I) B_1,B_2\rangle ds\right]\right|>c,
Q_{n}^{(N,M)}(1) =-\frac{1}{2}\int_{-1}^{1}d\zeta (1-\zeta)^{N}(1+\zeta)^{M}\big( P_{n}^{( N,M)}(\zeta) \big)^{2}\hphantom{Q_{n}^{(N,M)}(1)}{}=-\frac{2^{N+M}}{2n+N+M+1}\frac{(n+N) !(n+M) !}{n!( n+N+M) !}.
\sum_{r\ge0}\bigl(\partial_\theta^r\theta^v\bigm|_{\theta=0}\bigr)[w^r](1-\hbar w/2)^{v-1}\bigl(\tfrac{1+\hbar w/2}{1-\hbar w/2}\bigr)^{\theta/\hbar}=v![w^v](1-\hbar w/2)^{v-1}\bigl(\tfrac{1+\hbar w/2}{1-\hbar w/2}\bigr)^{\theta/\hbar}.
\widetilde{\psi_x}(a,t) = \left(\dots, s_i u_i(a_i) s_i, u_{i+1}(a_{i+1})s_{i+1}, \dots \right) & = \left(\dots, u_i(-a_i^{-1})s_i u_i(-a_i) a_i^{\bar \alpha_{i}^\vee}, u_{i+1}(a_{i+1})s_{i+1}, \dots \right) & \simeq \left(\dots, u_i(-a_i^{-1})s_i u_i(-a_i), a_i^{\bar \alpha_{i}^\vee}u_{i+1}(a_{i+1}) s_{i+1}, \dots \right).
X_{N}=\left(\begin{array}{cccc}X_{N}(1,1) X_{N}(1,2) \ldots X_{N}(1,N) X_{N}(2,1) X_{N}(2,2) \ldots X_{N}(2,N) \vdots \vdots \vdots X_{N}(N,1) X_{N}(N,2) \ldots X_{N}(N,N)\end{array}\right)
\Gamma(t):=e^{\sum_{i=1}^N\left[\int_0^tB_i(s)d\beta_i+\theta_i\beta_i-\frac{1}{2}\int_0^tB_i^2(s)ds-\frac{1}{2}\theta_i^2t-\theta_i\int_0^tB_i(s)ds\right]}
\boldsymbol{w}_{k+1}= \boldsymbol{w}_k + \mu e_k \boldsymbol{x}_k- \rho_{\rm r}\frac{(\boldsymbol{\Psi}\boldsymbol{w}_k) \boldsymbol{\Psi}}{\epsilon_{\rm r}+ |\boldsymbol{\Psi}\boldsymbol{w}_{k-1}|}.
\begin{array}{l c l}{\phi_{a}( b )}&{=}&{\int a^{- n / 2}\bar{\psi}\left( \frac{x - b}{a}\right) \phi ( x ) d^{n}x ,}\\{\phi ( x )}&{=}&{C_{\psi}^{- 1}\int a^{- n / 2}\psi \left( \frac{x - b}{a}\right) \phi_{a}( b ) \frac{d a d b}{a^{n + 1}},}\\ \end{array}
<{\cal J}_{\mu_{1}\cdots \mu_{s}}^{( s )}( x ) ~{\cal J}_{\nu_{1}\cdots \nu_{s}}^{( s )}( 0 ) > ={\prod}_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}\left({\frac{c_{s}( t )}{x^{4}}}\right) + \sum_{s^{\prime}= 0}^{s - 2}
\beta_{0 0}= \sum_{\rho = 0}^{N}\frac{\left( \eta_{0 \rho}\right)^{2}}{4}\left[ \left( \frac{\tilde{\omega}}{\Omega_{\rho}}- \frac{\Omega_{\rho}}{\omega}\right) \mathrm{e}^{i \Omega_{\rho}t}+ \left( \frac{\Omega_{\rho}}{\tilde{\omega}}- \frac{\tilde{\omega}}{\Omega_{\rho}}\right) \mathrm{e}^{- i \Omega_{\rho}t}\right] \; .
H\left(q^{i-5}\dfrac{E_{5}^{6i-1}}{E_{1}^{6i}}\right) =\sum_{j=1}^{\infty}m(6i,i+j)q^{5j-5}\dfrac{E_{25}^{6j}}{E_{5}^{6j+1}}, H\left(q^{i-4}\dfrac{E_{5}^{6i}}{E_{1}^{6i+1}}\right) =\sum_{j=1}^{\infty}m(6i+1,i+j)q^{5j-5}\dfrac{E_{25}^{6j-1}}{E_{5}^{6j}}.
\tau^2\sup\limits_{\rho \leq \bar{r}\leq r}\left\| \nabla a^{ij}\right\|_{L^{n+1}(A_{\bar{r}, e \bar{r}}^+)}\leq \delta \mbox{and}\tau^2\sup\limits_{\rho \leq \bar{r}\leq r}|a^{ij}(x)-\delta^{ij}| \leq \delta.
m^{\varepsilon}_{\widetilde{Q'},Q''}(\xi_{1},\xi_{2},\xi_{3}):=m^{\varepsilon}(\xi_{1},\xi_{2})\cdot (\widetilde{\phi}_{\widetilde{Q'}}\otimes(\phi_{Q''_{1}\times Q''_{2}}\cdot\widetilde{\phi}_{Q''_{3},3}))(\xi_{1},\xi_{2},\xi_{3}),
\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \prod_{i=1}^k \prod_{j=1}^\ell T^{in}f_i \; S^{jn}g_j=\left(\lim_{N\to\infty}\frac1N\sum_{n=1}^N\prod_{i=1}^k T^{in}f_i\right) \left( \lim_{N\to\infty}\frac1N\sum_{n=1}^N\prod_{j=1}^\ell S^{jn}g_j\right)
S^{BH}(G_{\tiny\mbox{ren}},c^i_{\tiny\mbox{ren}})={1 \over 4 G_{\tiny\mbox{ren}}}{\cal A}-\int_{\Sigma}\sqrt{\sigma}d^2\theta~(8\pi c^1_{\tiny\mbox{ren}}R+4\pi c^2_{\tiny\mbox{ren}}{\cal Q}+ 8\pi c^3_{\tiny\mbox{ren}}{\cal R})~~~.
\frac{1}{2Q\cdot k_i+k_i^2}=\frac{1}{2Q\cdot k_i}- \frac{k_i^2}{\left(2Q\cdot k_i\right)^2}+\frac{\left(k_i^2\right)^2}{\left(2Q\cdot k_i\right)^3}+\cdots .
\int_{\mathfrak{m}}\big|S(\alpha)\big|d\alpha\le \int_{\mathfrak{m}}\int_{[0,1]^3}\Big|\mathcal{F}(\alpha,\boldsymbol{\beta})\prod_{j=1}^{n-3}f_j(\alpha,\boldsymbol{\beta})\Big|d\boldsymbol{\beta}d\alpha.
\xi_1=\xi_2=\cdots=\xi_{r_1}=a_1 &\xi_{r_1+1}=\xi_{r_1+2}=\cdots=\xi_{r_1+r_2}=a_2 &\xi_{r_1+r_2+1}=\xi_{r_1+r_2+2}=\cdots=\xi_{r_1+r_2+r_3}=a_3 &\mbox{and so on.}
\tau^{Y,r}_\varphi (A_0,\ldots,A_k):= \int_{G^{\times k}}{}^b{\rm Tr}_Z\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) &\qquad\varphi (e,g_1,g_1g_2,\ldots,g_1\cdots g_k)dg_1\cdots dg_k.
\Bigl(~_{s+1}F_s(1, m, \ldots, m; 1+m, \ldots, 1+m)(z^{q^{-m+1}})\Bigr)^{q^m}&=\biggl(\sum_{i\geq 0}\frac{D_i(m)_i\cdots(m)_i}{(1+m)_i\cdots(1+m)_iD_i}z^{q^{i-m+1}}\biggr)^{q^m}=\biggl(\sum_{i\geq 0}\frac{1}{[i+m]^{nq^{-m}}}z^{q^{i-m+1}}\biggr)^{q^m}=\sum_{i\geq 0}\frac{z^{q^{i+1}}}{[i+m]^{s}}.
\tilde{u}(t)= e^{-A(t-t_1)}u(t_1) -\chi\int_{t_1}^t e^{-A(t-s)}\nabla\cdot ( \tilde{u}(s) \nabla A^{-1}u(s))ds &+\int_{t_1}^t e^{-A(t-s)}\tilde{u}(s)\Big(1+ a_0(s,\cdot)-a_1(s,\cdot)u(s)-a_2(s,\cdot)\int_{\Omega}u(s)\Big)ds.
\| P_Ng(t_k, \cdot)-P_Ng(t, \cdot)\|_{H^s_r(\mathbb R^d)}= \| P_N (g(t_k, \cdot)-g(t, \cdot))\|_{H^s_r(\mathbb R^d)}& \leq \| g(t_k, \cdot)-g(t, \cdot)\|_{H^s_r(\mathbb R^d)},
\|F_c \phi\|_{L^p_x}=\|e^{ix^2/4t}F_c \phi\|_{L^p_x}\lesssim \|P e^{ix^2/4t}F_c \phi\|_{L^2_x}^a\|e^{ix^2/4t}F_c \phi\|_{L^2_x}^{1-a}&\lesssim \| (1/t) |2Pt-x| F_c \phi\|_{L^2_x}^a \|F_c \phi\|_{L^2_x}^{{1-a}}\lesssim t^{(-1+\alpha)a}\|F_c \phi\|_{L^2_x}=t^{(-1+\alpha)a}\|F(\frac{|x|}{t^{\alpha}}\leq 1)e^{i\Delta t}\phi\|_{L^2_x}
e_1\;=\;\begin{pmatrix}0 \vdots 0 0 1 \end{pmatrix}\;,e_2\;=\;\begin{pmatrix}0 \vdots 0 1 0\end{pmatrix}\;,\ldots\;,e_{\mathsf{L}}\;=\;\begin{pmatrix}1 0 \vdots 0 0\end{pmatrix}\;.
{\cal L}_{s f i e l d}= \frac 1 8 \int d^{2}\theta ~ \mathrm{T r}\overline{{{D{\bf G}}}}D{\bf G}= \frac 1 2 \left( Z^{2}+ J_{\mu}\cdot J^{\mu}+ i \overline{{{\chi}}}\cdot \partial \! \! \! / \chi + i \varepsilon^{j k l}\overline{{{\chi}}}^{j}J \! \! \! \! /^{k}\chi^{l}+ \frac 1 4 ( \overline{{{\chi}}}\cdot \chi )^{2}\right) .
\langle\nabla_{e_{j}}^{*}\langle\bullet, \bar{s}^{+}\otimes \bar{f}\rangle_{h},\star\rangle=\langle\langle\bullet, \bar{s}^{+}\otimes \bar{f}\rangle_{h},\nabla_{e_{j}}\star\rangle=\langle\nabla_{e_{j}}\star,\bar{s}^{+}\otimes \bar{f}\rangle_{h}.
\gamma_{\mu AB}^{{\footnotesize (T)}\prime}{}=\Delta_{{\small \Lambda}}\gamma_{\mu AB}^{{\footnotesize (T)}}{},\Gamma_{\mu AB}^{{\footnotesize (T)}\prime}{}=(\Delta_{{\small \Lambda}})^{-1}\Lambda_{A}{}^{C}\Lambda_{B}{}^{D}\Gamma_{\mu CD}^{{\footnotesize (T)}}{}=\Gamma_{\mu AB}^{{\footnotesize (T)}}{}.
\frac{1}{2}\frac{d}{dt}|\big< \theta(t), f_{\lambda}(t) \big>|^{2}= \big< \theta(t), f_{\lambda}(t) \big> \bigg[ - \lambda \cdot \coth \rho_{\lambda}(t) \bigg( 2 \big< \theta(t), f_{\lambda}(t) \big> &+ \big< \theta(t), f_{\lambda}(t) \big>^{2}\bigg) + \big< \theta'(t), f_{\lambda}(t) \big> \bigg].
\partial_1^{k_1+1}\partial_h^2 \left[ (\psi_{\pi/2}-P_{2,0}\psi_{\pi/2})a_{k_1,k_2,\pi/2}\right](0)=\frac{2}{\pi}(-1)^{k_1}{i^{k_2-n}\left(i\frac{\pi}{2}\right)^{n+k_1+k_2}(k_1+1)!}.
\frac{da}{ds}\Big|_{s=0}= \frac{1}{p(r)}\frac{da}{dr}\Big|_{r=r_b}= \frac{1}{2L}\frac{da^2}{dr}\Big|_{r=r_b}= -\frac{(n+1)L}{r_b}b|_{s=0}= (L^2 - r_b^2).
\mathrm{M . S}: \mathrm{\bf T r}_{{\cal H}_{o}}\left[ \frac{\Omega}{2}( - 1 )^{F_{s}}e^{- 2 \pi t \mathrm{H}_{o}}\right] - \mathrm{\bf T r}_{{\cal H}_{o}}\left[ \frac{\Omega}{2}e^{- 2 \pi t \mathrm{H}_{o}}\right]
S(\Phi) = \int_{-\infty}^{0}dx \int_{-\infty}^{\infty}dt\left ( \frac{1}{2}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a}-\frac{m^{2}}{\beta^{2}}\sum_{i=0}^{r}n_{i}e^{\beta \alpha_{i}\cdot \Phi}\right )
0 =\langle\ast([F_{A},\eta]\wedge\phi^{(3)}),\eta\rangle=-\langle\ast([F_{A}\wedge\phi^{(3)},\eta]),\eta\rangle &=-\langle\ast([2\ast F^{7}_{A}-\ast F^{14}_{A},\eta]),\eta\rangle &=-2\langle F^{7}_{A},[\eta,\eta]\rangle+\langle F^{14}_{A},[\eta,\eta]\rangle.
|\mathring{Ric}|\triangle|\mathring{Ric}| \geq\frac{2}{n}|\nabla |\mathring{Ric}||^2-\sqrt{\frac{n-2}{2(n-1)}}\left(|W|^2+\frac{2n}{n-2}|\mathring{Ric}|^2\right)^{\frac 12}|\mathring{Ric}|^2+\frac{R}{n-1}|\mathring{Ric}|^2
H_2(t,x,y)=2^{\sigma_2 l+k_2}\int_{n_2 2^{-(\sigma_2 l+k_2)}+P_2(2^{-l}t)}^{(n_2+1)2^{-(\sigma_2l+k_2)}+P_2(2^{-l}t)}|f_2|\left(x,y+v\right) \,\mathrm{d}v.
\max_{\rho :\rho \in \mathcal{G},\mathrm{Tr}\rho F\leq E}H\left( \Phi \left[\rho \right] \right) =\max_{\boldsymbol{\nu}:\mathrm{tr}\boldsymbol{\nu\epsilon}\leq E}\,\mathrm{tr\,}g(\mathbf{K}^{\ast}\boldsymbol{\nu}\mathbf{K}+\boldsymbol{\mu}+\left( \mathbf{K}^{\ast}\mathbf{K}-\mathbf{I}_{B}\right) /2)].
{C(x,u^2/x) \over u^{\kappa}}=r(x)^{\frac{\gamma_0\gamma_1}{\tilde{\gamma}_1+\gamma_0}}\bigg ( r(x)^{-\frac{\gamma_0}{\tilde{\gamma}_1+\gamma_0}}+r(x)^{\frac{\tilde{\gamma}_1/\gamma_0}{\tilde{\gamma}_1+\gamma_0}}-u^{\frac{2}{\tilde{\gamma}_1+\gamma_0}}\bigg)^{-\gamma_0}\to c \in (0,\infty)
X^{m,n}=\left(\begin{smallmatrix}1 x_{12}x_{13}\dots x_{1m}\dots x_{1n}0 1 x_{23}\dots x_{2m}\dots x_{2n}0 0 1 \dots x_{3m}\dots x_{3n}\dots \dots 0 0 \dots x_{m-1m}\dots x_{m-1n}0 0 \dots 1 \dots x_{mn}\end{smallmatrix}\right).
f_{\rho,\tau}(\theta) = \cos (\theta- \tilde{\theta}) f_{\rho,\tau}(\tilde{\theta})+ \sin (\theta - \tilde{\theta})f_{\rho,\tau}' (\tilde{\theta})-3 \int_{\tilde{\theta}}^\theta \sin (\theta - \tau) \sin 2\tau d\tau
V_{0}+ V_{v^{2}}\rightarrow - \frac{4 \alpha_{S}}{3 r}\left[ 1 - \frac{1}{2}\left( \bf v_{\mathrm{1}}\cdot v_{\mathrm{2}}+ v_{\mathrm{1}}\cdot \hat{r}v_{\mathrm{2}}\cdot \hat{r}\right) \right] .
\exp_{q}(-tF)x_{1}^{\alpha_{1}}\cdots x_{m}^{\alpha_{m}}=x_{1}^{\alpha_{1}}\cdots x_{m}^{\alpha_{m}}\prod_{i=0}^{m-1}\frac{(q^{2s_{i}}t/x_{i+1})_{\infty}}{(q^{2r_{i}}t/x_{i+1})_{\infty}},
\mu (\Gamma )=2\pi \left(\varepsilon h-2+k+\sum_{i=1}^{r}{\left(1-{\frac{1}{{m_{i}}}}\right)}+{\frac{1}{2}}\sum_{i=1}^{k}\sum_{j=1}^{s_{i}}{\left(1-{\frac{1}{{n_{ij}}}}\right)}\right),
T_{L}=T_{1}+T_{2}=\frac{L_{L}/\gamma(v)}{c-v}+\frac{L_{L}/\gamma(v)}{c+v}=\frac{2L_{L}/\gamma(v)}{c}\frac{1}{1-\frac{v^{2}}{c^{2}}}=\frac{2L_{L}\gamma(v)}{c}
\delta\varepsilon T_{\delta}(T_{\varepsilon}\left( \operatorname{div}u_{\varepsilon\delta}\right) )=\delta T_{\delta}(\varepsilon T_{\varepsilon}(\operatorname{div}u_{\varepsilon\delta}))=\delta T_{\delta}(\operatorname{div}_{y}T_{\varepsilon}(u_{\varepsilon\delta})=\operatorname{div}_{z}T_{\delta}(T_{\varepsilon}\left( u_{\varepsilon\delta}\right) )
\frac{\partial^{n+1}f^A}{\partial z^{n+1}}=\sum_{\alpha =1}^n W_{\alpha}\frac{\partial^{\alpha}f^A}{\partial z^{\alpha}},\quad\frac{\partial^{n+1}\bar{f}^A}{\partial \bar{z}^{n+1}}=\sum_{\alpha =1}^n \bar{W}_{\alpha}\frac{\partial^{\alpha}\bar{f}^A}{\partial \bar{z}^{\alpha}}
F\big(3^{-1/3}x, 3^{-2/3}t;\beta=6\big) = (\Psi(x,t))_{11}, \Psi(x,t) = \kappa e^{\frac{x^3}{6}-\frac{xt}{2}}R(x,t)e^{-\frac{i\pi}{2}\sigma_3}u^{-\frac{1}{2}\sigma_3}\Psi_{0}(x,t),
H \left( t \right) = \omega \left( a^\dagger a +\sigma^{3}/ 2 \right) + \mathrm{i}\left( \alpha \left( t \right) a^\dagger - \overline{\alpha \left( t \right)}a + \overline{\beta \left( t \right)}\sigma^{-}- \beta \left( t \right) \sigma^+ \right),
\Delta n \sum_{l = 1}^{\infty}( 2 l + 1 ) A_{l}= \left. y \Delta n \left( \frac{\partial^{2}}{\partial r^{2}}- 2 \frac{\partial^{2}}{\partial r \partial \rho}+ \frac{\partial^{2}}{\partial \rho^{2}}\right) \overline{{\cal D}}( r , \rho , y ) \right|_{r = \rho = 1}+ 1{.}
\left. -2\sqrt{2}\left[2 F_{\alpha}^{\mu\,A}C^{[\alpha}\partial^{\nu]}D -\frac{g^{\mu\nu}}{2}F_{\alpha_1\alpha_{2}}^{\,A}C^{[\alpha_1}\partial^{\alpha_{2}]}D \right] +2\left[2C_{[\alpha}\partial^{\mu]}DC^{[\alpha}\partial^{\nu]}D \right. \right.
P(D) =p_e^{n_e}(1-p_e)^{{n\choose 2}-n_e}(1-p_d)^{n_{as}}(2p_d-1)^{n_s}&=p_e^{n_e}(1-p_e)^{(n(n-1)-2n_e)}\frac{p_a^{n_{as}}(1-p_a)^{n_{as}}}{p_e^{n_{as}}}\frac{p_a^{2n_s}}{p_e^{n_s}}&=p_e^{n_e-n_{as}-n_s}p_a^{n_{as}+2n_s}(1-p_a)^{n(n-1)-2n_e+n_{as}}&=p_a^{n_a}(1-p_a)^{n(n-1)-n_a},
H_\nu(\alpha\Vert\beta):=\mathbb E^U_{\nu,\alpha}\mathbb E^V_{\nu,\beta}\log\frac{1-\beta UV}{1-\beta V}-\mathbb E^U_{\nu,1-\alpha}\mathbb E^V_{-\nu-1,\beta}\log\left[ 1-\frac{1-\alpha}{\alpha}\frac{\beta}{1-\beta}(1-U)(1-V)\right]
L\left(\theta_{1,1}, \cdots, \theta_{K,M}, \upsilon\right) &=t\left(P_c+\frac{1}{\omega}\sum_{k=1}^K\sum_{m=1}^M \frac{\sigma^2}{\left(\mathbf{\Lambda}_k\right)_{m}}\left(2^{\theta_{k,m}}-1\right)+\kappa\phi\left(\frac{1}{t}\right)\right) &+\upsilon \left(1-tB\sum_{k=1}^K\sum_{m=1}^M \theta_{k,m}\right).
\bigg\vert \int_{x-2\vert u_n-x\vert}^{x+2\vert u_n-x\vert}\frac{(u_n-t)\varphi(t)dt}{(u_n-t)^2+v_n^2}\bigg\vert \leq \frac{k}{v_n}\int_{x-2kv_n}^{x+2kv_n}\vert \varphi(t) \vert dt=o(1).
w^{ii}D_{p_kp_l}A_{ii}D_k\eta D_l \eta = \sum_{k,l \neq i}D_{p_kp_l}A_{ii}D_k\eta D_l \eta + 2\sum_{l \neq i}D_{p_ip_l}A_{ii}D_i\eta D_l \eta \quad\quad+ D_{p_ip_i}A_{ii}D_i\eta D_i \eta
H^{\prime}= \frac{1}{2 m}\sum_{n}\left[ \vec{p}_{n}- \frac{e}{c}\vec{A}( x_{n}) - \frac{e}{c}\; \vec{a}( x_{n}) \right]^{2}+ \sum_{n}e A_{0}( x_{n}) + \sum_{n < n^{\prime}}V ( x_{n}- x_{n^{\prime}})
\begin{array}{ll}\sum \limits_{m'= \bar{m}}^{m}\int_{|x| \geq m'}\frac{m'}{|x|}\left( |\partial_{r}u(t)|^{2}+ \frac{|u(t)|^{2}}{|x|^{2}}\right) \; dx \lesssim 1 \cdot\end{array}
E_2^{ij}=H^i(A,R^ja_*((\Omega_X^1\otimes P)\otimes a^*(F^{e,*}(\widehat{L}^{\vee})\otimes Q^{\vee}))) &=H^i(A,R^ja_*(\Omega_X^1\otimes P)\otimes F^{e,*}(\widehat{L}^{\vee})\otimes Q^{\vee}) &\Rightarrow H^{i+j}(X,\Omega_X^1\otimes P\otimes a^*(F^{e,*}(\widehat{L}^{\vee})\otimes Q^{\vee}))
\langle \,S_{(\lambda,\mu)},S_{(\lambda,\mu)}\,\rangle_{\beta}=\frac{m!(N-m)!}{f_{\mu}}\;\frac{d'_{(\lambda^R, \mu)}d_{(\lambda^R, \mu^R)}}{d'_{(\lambda^R,\mu^R)}d_{(\lambda,\mu^R)}}\langle\, E_{(\lambda^R,\mu^R)},E_{(\lambda^R, \mu^R)}\, \rangle_{\beta}\, ,
\frac{1}{({M_{i}}^{2}-{p_{1}}^{2}) ({M_{i}}^{2}-{p_{2}}^{2})}= \int_{0}^{1}d \, \xi \frac{1}{{[{M_{i}}^{2}-{p_{1}}^{2}- ({p_{2}}^{2}-{p_{1}}^{2}) \xi ]}^{2}}
x\wedge y=\sum_{j}(x_j e_j)\wedge\sum_k (y_k e_k)= \sum_{1\leq j<k\leq m}(x_jy_k-x_ky_j)e_j\wedge e_k =:\sum_{1\leq j<k\leq m}(x\wedge y)_{jk}e_j\wedge e_k.
\sum_{w \in \{I, w_4, w_5\}}K_{w_6}^{--}(y; w(\mu)) = \frac{1}{48\pi^2}\sum_{w \in \{I, w_4, w_5\}}\Bigl(4 \mathcal{J}_1^{-}(y; w(\mu)) +2 \mathcal{J}_1^+(y; w(\mu)) \Bigr).
G= \{(x,r):x\in \mathbb{R}^k, 0<r<d(x,K),z_1,z_2\in K &d(x,z_1),d(x,z_2)\leq d(x,K)+\epsilon r & |\theta|>\cos^{-1}\left(2\left(\frac{1-(2\delta +\epsilon)}{1+2\delta}\right)^2-1\right)\}
E_N:= \sqrt{\pi}e^{\frac{\pi i}{6}}\begin{pmatrix}1 -1 -i -i\end{pmatrix}\begin{pmatrix}f_N^{1/4}a^{-1}0 0 f_N^{-1/4}a\end{pmatrix}\mbox{and}\mathbb{T}:= E_N \mathbb{T}_0 \mbox{in III}^+ ,
f_k(Q_n\cdot(an,1)) =i^{nk}\bigg(\prod_{j=1}^n\frac1{a_j+a_j^{-1}}\bigg)^{(2n+1)(s+\frac12)}\!f_k(\left[\begin{smallmatrix}C S &C S &-S C -S C\end{smallmatrix}\right]\!\!\left[\begin{smallmatrix}U &I_n &^tU^{-1}& I_n\end{smallmatrix}\right]\!\!\left[\begin{smallmatrix}I_n -X &I_n & I_n & I_n\end{smallmatrix}\right]).
Q_{i+1}(g)=Q_i^{(1)}(\pi_i(g))\times\frac{\mu_{i+1}(\sigma_1)}{\mu_i(\pi_i(\sigma_1))}\frac{\mu_{i+1}(\sigma_2)}{\mu_{i+1}(\partial_+\sigma_1)}\times\cdots\times\frac{\mu_{i+1}(\sigma_{m^{i+1}})}{\mu_{i+1}(\partial_+\sigma_{m^{i+1}-1})}.
L_{g h}^{f}( x_{f}) = J_{\psi}L_{g h}( x ) = J_{\psi}< ( e^{\mu}\partial_{\mu}C )^{+}, e^{\nu}\partial_{\nu}C > = J_{\psi}S ( B_{f})^{2}( D_{l}C_{f})^{+}D_{l}C_{f}.
r((t,1))( (1, \gamma_i)v ) =(t,\gamma_i)v =(1, \gamma_i)(\gamma^{-1}_i(t),1)v =(1, \gamma_i)r_{-\mu_S}((\gamma^{-1}_i(t),1))(v) =(1,\gamma_i) \mu'(\gamma^{-1}_i(t))v =\gamma_i(\mu')(t) (1, \gamma_i)v.
\begin{aligned}{{\bar R}_{{\rm{SM,upper}}}}= \sum\nolimits_{n \in{\mathcal K}_{\rm SM}}{\log_2}\left(1+\frac{1}{c_n}\right) = \sum\nolimits_{n \in{\mathcal K}_{\rm SM}}{\log_2}\left(1+\frac{{{E{N_{\rm{T}}}N_{{\rm{S}},n}^2\rho_n^2\kappa}}}{{{{\sigma^2}L_{\rm R}\left({\kappa + 1}\right)}}}\right).\end{aligned}
F'(U,V)(H,K) = \langle R(U,V),H\Sigma V^{T}+ U\Sigma K^{T}- P(H\Sigma V^{T}+ U\Sigma K^{T})\rangle = \langle R(U,V),H\Sigma V^{T}+ U\Sigma K^{T}\rangle = \langle R(U,V)V\Sigma^{T},H\rangle + \langle \Sigma^{T}U^{T}R(U,V),K^{T}\rangle.
\langle e^{- I_{\mathrm{b n d y}}}\cdots \rangle = \langle P \exp [ - \int d t d \theta \Gamma T ({\bf X}) ]_{\Gamma - \mathrm{e v e n}}\cdots \rangle = \langle P \cosh [ \int d t d \theta T ({\bf X}) ] \cdots \rangle ~ ,