harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2}\frac{W_{0}}{R^{\beta}}= \frac{W_{0}/2}{(\sqrt{2}|Q|^{1/2})^{\beta}}= 2^{-\beta/2}\left(\frac{a_{\beta}}{\Xi_{\beta}}\right)^{\beta-2}|Q|^{1-\beta/2}|Q|^{-1}\ge \frac{1}{2|Q|}\left(\delta\Xi_{\beta}\right)^{2-\beta},
p(u,w,\left\Vert \mathbf{x}-\mathbf{y}\right\Vert )=p(u)\p(w)\ \left( 1+\sum_{k=1}^{\infty}\gamma^{k}(\left\Vert \mathbf{x}-\mathbf{y}\right\Vert )\e_{k}(u)\e_{k}(w)\right)
\nabla_{e_i}\nabla_{e_i}(\frac{Y^{\perp}}{u}\partial_t) =\nabla_{e_i}[e_i(\frac{Y^{\perp}}{u})\partial_t+\frac{Y^{\perp}}{u}\nabla_{e_i}\partial_t] &=e_i(e_i(\frac{Y^{\perp}}{u}))\partial_t+2e_i(\frac{Y^{\perp}}{u})\nabla_{e_i}\partial_t+\frac{Y^{\perp}}{u}\nabla_{e_i}\nabla_{e_i}\partial_t.
\Omega(x_1,x_2) = (\iota^*\omega_2+\sqrt{-1}\iota^*\omega_3)(x_1,x_2) = (\omega_2+\sqrt{-1}\omega_3)(\iota_*x_1,\iota_*x_2) = tr(-a_2b_1+b_2a_1-i_2j_1+i_1j_2-a'_2b'_1+b'_2a'_1).
\phi\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes B\right )=\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes\varphi_{j_1,\ldots,j_{m-1},\mathrm{I}}(B)\hbox{for all\quad}B\in M_{n_m}.
\lim_{q \to 1}\left( P_q \mathcal{X}^{K\textnormal{th,eq}}\left(q,\varphi_{q,z}^{-1}(Q)\right) \right)_{0i}= Q^\frac{\lambda_i}{z}\sum_{d \geq 0}Q^d \prod_{r=1}^d \prod_{j=0}^N \frac{1}{(\lambda_i-\lambda_j+rz)}
\phi_{ij}(x)=\log\frac{\lambda e^x-\mu}{(\lambda-\epsilon)e^x-(\mu-\epsilon)},\phi_{k\ell}(x)=\log\frac{(\mu+\alpha-\epsilon)e^x-(\lambda+\alpha-\epsilon)}{(\mu+\alpha)e^x-(\lambda+\alpha)}, &\phi_{i\ell}(x)=\log\frac{1-\lambda(\mu+\alpha-\epsilon)e^x}{1-(\lambda-\epsilon)(\mu+\alpha)e^x}.
h(x) \le \Bigl(g_2(x)+\frac{2}{6^3}(2^3x-3)(12-3^3x)+\frac{2}{6^4}(3^4x-34)(7-2^4x)\Bigr)\Bigr|_{x=\frac{8}{19}}& + \frac1{10\cdot 6^4}- \eta &=-\frac{638551936297453983963801188142263}{3940960181626780198766689107612132000}<0,
\frac{S_{k+1}}{S_k}=\frac{\frac{1}{S_k}+2+r+\sqrt{\left(\frac{1}{S_k}+(2+r)\right)^2-4}}{2}\geq \frac{2+r+\sqrt{(2+r)^2-4}}{2}= \frac{2+r+\sqrt{r^2+4r}}{2}
I_\lambda(\omega) =I_\lambda(\omega)-\frac{1}{2}I'_\lambda(\omega)\omega &=\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx+\left(\frac{1}{2}-\frac{1}{2_\alpha^*}\right)\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx &\geq0,
\begin{aligned}p(z) = c \det\left( \begin{pmatrix}0 0 0 I-K \end{pmatrix}U^* \Delta(z) U + i\begin{pmatrix}2I 0 0 I+K \end{pmatrix}\right) &= c \det(I-K)\det\left( \begin{pmatrix}0 0 0 I \end{pmatrix}U^*\Delta(z) U + \begin{pmatrix}2iI 0 0 A \end{pmatrix}\right)\end{aligned}
P^1_2 = -2 r^2 K ( - 2 r^2 \hat{r}_2 + 6r^2 \hat{r}_6 + 8 r \hat{r}_2 \hat{r}_6+ 3 \hat{r}_2^2 \hat{r}_6 + \hat{r}_2 \hat{r}_6^2 ),
\hat{T}_{ij}^{(2)}=\frac{1}{2}(\hat{V}_{i}\hat{W}_{j}+\hat{V}_{j}\hat{W}_{i})-\frac{1}{3}\hat{V}_{k}\hat{W}_{k}\delta_{ij}=\hat{V}_{(i}\hat{W}_{j)}-T_{ij}^{(0)}
\|u(t,\cdot)\|\leq \beta( \|{u_0}\|,t)+\theta_1\bigg(\int_{0}^{t}\gamma_1(|d_1(s)|)\bigg) +\theta_2\bigg(\int_{0}^{t}\gamma_2(|d_2(s)|)\bigg),\ \forall t\geq 0.
\Vert (v_{s},v_{u})\Vert_{1}^{2}=\Vert v_{s}\Vert_{1}^{2}+\Vert v_{u}\Vert_{1}^{2}\leq (\frac{\lambda+\varepsilon}{\varepsilon}c)^{2}(\Vert v_{s}\Vert^{2}+\Vert v_{u}\Vert^{2}) \leq \frac{1}{\mu_{i}}(\frac{\lambda+\varepsilon}{\varepsilon}c)^{2}\Vert (v_{s},v_{u})\Vert^{2}.
X_t - \Delta X + \lambda |\nabla X|^{\alpha}= 2MK_1 h'(t) [1-K_1(h(t)-r)] + 2MK_1^2 -\dfrac{N-1}{r^2}2MK_1[K_1(h(t)-r)-1] & +\lambda (2MK_1)^{\alpha}[1-K_1(h(t)-r)]^{\alpha}\geq 2MK_1^2 \geq 2M (h_0+1 - K_1^{-1})^{-\gamma}K_1^2(r+1)^{\gamma}\geq av^p = u_t - \Delta u +\lambda |\nabla u|^{\alpha}.
(H_1 + (X_0+Y_0) G_1) \circ F = \big(X_1^{a_1}\cdots X_n^{a_n}- u_1^{a_1}\cdots u_n^{a_n}X_0^{d-1}+ (X_0+Y_0)G_1\big)\circ F= & =\prod_{i=1}^n (a_i !) x_0 +G_1\circ \big(x_1^{a_1}\cdots x_n^{a_n}+ y_0^{b_0-1}y_1^{b_1}\ldots y_m^{b_m}\big) =0,
\begin{aligned}\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(q)\times Sp(1)]}= &\sum_{\substack{m=0 begin{align*}.2ex] 0 \le k \le m/2}}^\infty sum_{\substack{0\le e \le d begin{align*}.1ex] 0\le e'\le d' begin{align*}.2ex] d+e = m+\ell+4q begin{align*}.1ex] d'+e' = m-2k}}\sum_{j=0}^{\min(d-e,d'-e')}\pi^{Sp(p)}_{d,e}\otimes \pi^{Sp(q)}_{d',e'}\otimes \pi^{Sp(1)}_{d+d'-e-e'-2j}.\end{aligned}
0 = \langle Ah, g \rangle_{\mu_p}= \sum_{d=0}^n \left( -\frac{p}{1-p}\right)^d \langle h^{=d}, g \rangle \geq \mu_{p}(h) \mu_{p}(g) - \sum_{d=1}^n \left( \frac{p}{1-p}\right)^d | \langle h^{=d}, g \rangle | .
\int_{-1}^{1}\frac{2(y(1+\rho^{2})-2\rho x)\sqrt{1-y^{2}}dy}{\pi((1-\rho^{2})^{2}-4xy\rho(1+\rho^{2})+4\rho^{2}(x^{2}+y^{2}))}=-\rho x.
\sum_{l = 1}^{\infty}( 2 l + 1 ) s_{l}( y r ) e_{l}( y \rho ) + 1 = \frac{y r \rho}{| r - \rho |}e^{- y | r - \rho |}\equiv \overline{{{\cal D}}}( r , \rho , y ) \,{.}
(\Psi_{\Sigma}(f) \star_{H, \ell}\Psi_{\Sigma}(g))[\psi] = (\Psi_{\Sigma}(f) \cdot \Psi_{\Sigma}(g))[\psi] - \frac{\hbar}{4 \pi}\int_{\mathbb{R}^{2}}\frac{f(s) g(s')}{(s - s')^2}\,\mathrm{d}s\,\mathrm{d}s',
\partial_s^\beta \partial_\xi^\gamma \delta^\lambda a(x+h;s,\xi) - \partial_s^\beta \partial_\xi^\gamma \delta^\lambda a(x;s,\xi) = \sum_{1\leq j \leq d}h_j \int_0^1 \partial_{x_j}\partial_s^\beta \partial_\xi^\gamma \delta^\lambda a(x+th;s,\xi)dt.
\begin{cases}\partial_t^2 v(r,t;x)-\partial_r^2 v(r,t;x)+\frac{\mu}{1+t}\partial_t v(r,t;x)+\frac{\nu^2}{(1+t)^2}v(r,t;x) = \Omega_r[f](t,x), t>0, r>0, v(r,0;x)= \Omega_r[u_0](x), \ \ \, r>0, \partial_t v(r,0;x)= \Omega_r[u_1](x), r>0, v(0,t;x)=0, \ \, t \geqslant 0.\end{cases}
\Omega'\star d\Phi^c_\mathcal{E}(0,E) :=\,\frac{d{\Omega}(u)}{d u}\large|_{u=\Phi^c_\mathcal{E}(0,E)}\star \,\frac{d\Phi^c_{\mathcal{E}}(t,E)}{dt}\large|_{t=d_{\mathbb{K}min}}& =\,\frac{d{\Omega}(u)}{d u}\large|_{u=E}\star \,\frac{d\Phi^c_{\mathcal{E}}(t,E)}{dt}\large|_{t=d_{\mathbb{K}min}}.
\sum_{i=1}^{M}\frac{U_i}{\beta_{i,\xi}+\theta}= \sum_{i=1}^{M}\frac{H_ie^{\beta_{i,q}(b-y)}}{\theta+\beta_{i,q}}\left(1-\frac{\psi_{\xi}^+(\theta)}{\psi_{\xi}^+(-\beta_{i,q})}\right) &+\sum_{i=1}^{N}\frac{P_i}{\gamma_{i,q}-\theta}\left(\frac{\psi_{\xi}^+(\theta)}{\psi_{\xi}^+(\gamma_{i,q})}-1\right).
\begin{aligned}f_i = a_{i0}z_{i0}+ \sum_{j = 1}^{n_i}(a_{ij}- a_{ij-1})z_{ij}, z_i \in \Delta_i, z_{ij}\geq \delta_{ij}\geq z_{ij+1}j \in \{1, \ldots,n_i-1\}, \delta_{i}\in \{0,1\}^{n_i-1}, \end{aligned}
S_{\mu\nu}:= \left( \begin{array}{c}G_{\mu\nu}E_{\mu\nu}\end{array}\right)=\left( \begin{array}{c}g_{\mu\nu}H_{\mu\nu}\end{array}\right)+i\left( \begin{array}{c}B_{\mu\nu}M_{\mu\nu}\end{array}\right)\,,
\mathbf{\sigma}_{\mathrm{p}}\left( \alpha \right) =-\frac{\alpha^{2}}{2}D_{\left\{0\right\}}+\int\nolimits_{\mathbb{R}\backslash \left\{0\right\}}\left( \mathrm{e}^{i\alpha \nu}-1-i\alpha \nu \mathbf{1}\left[ \left\vert\nu \right\vert <1\right] \right) \mathfrak{m}_{\mathrm{AC}}\left( \mathrm{d}\nu \right) \ ,
(\rho_0c_0+\alpha_1) u_0^{\mathcal{L}}-i\omega\sqrt{d_1}u_1^{\mathcal{L}}=x(\rho_0u_0^{\mathcal{L}}+\sqrt{d_1}u_1^{\mathcal{L}}), &i\omega\sqrt{d_{k}}u_{k-1}^{\mathcal{L}}-f_{k}^{-1}s_{k}u_{k}^{\mathcal{L}}-i\omega\sqrt{d_{k+1}}u_{k+1}^{\mathcal{L}}=x(\sqrt{d_{k}}u_{k-1}^{\mathcal{L}}+\rho_ku_{k}^{\mathcal{L}}+\sqrt{d_{k+1}}u_{k+1}^{\mathcal{L}}).
\overline{\mathcal{M}}_{0,[k]}(X,1):= \underbrace{\overline{\mathcal{M}}_{0,1}(X,1)\times_{\overline{\mathcal{M}}_{0,0}(X,1)}\times\cdots\times_{\overline{\mathcal{M}}_{0,0}(X,1)}\overline{\mathcal{M}}_{0,1}(X,1)}_{k\ \mbox{factors}}.
(\lambda C_\lambda)^{(p-1-\alpha)(p-1-k_2)-k_1k_3}=\left( \frac{c_1}{\mu^{k_1}B_\lambda}\right)^{p-1-k_2}\left( \frac{c_2}{\mu^{k_2}B_\mu}\right)^{k_1}, (\mu C_\mu)^{(p-1-\alpha)(p-1-k_2)-k_1k_3}=\left( \frac{c_2}{\mu^{k_2}B_\mu}\right)^{p-1-\alpha}\left( \frac{c_1}{\mu^{k_1}B_\lambda}\right)^{k_3}.
\xi=W^{\frac{1}{2}}~\psi =exp(\frac{2i}{F_\pi}~\gamma_5 \mbox{\boldmath$\tau$}\cdot \mbox{\boldmath$\phi$}+ \frac{g_\omega}{2M_\omega^2}~\sigma_{\alpha\beta}\omega^{\alpha\beta}+\frac{g_\rho}{2M_\rho^2}~\sigma_{\alpha\beta}\mbox{\boldmath$\tau$}\cdot \mbox{\boldmath$\rho$}^{\alpha\beta})~\psi
\int_{B_{R_{1}}}K\left( \left| x\right| \right) \left| u\right|^{q_{1}}dx=\left\| u\right\|^{q_{1}}\int_{B_{R_{1}}}K\left( \left| x\right|\right) \frac{\left| u\right|^{q_{1}}}{\left\| u\right\|^{q_{1}}}dx\leq\left\| u\right\|^{q_{1}}\mathcal{S}_{0}\left( q_{1},R_{1}\right)
\mathcal{M}\left( p^{2}+ p_{\perp}^{2}+ M^{2}\right) G ( p , p_{\perp}) + \frac{p^{2}}{2 \pi}\int_{- \infty}^{\infty}d p_{\perp}^{\prime}G ( p , p_{\perp}^{\prime}) = 1 ,
{\left( \gamma^a \right)_\alpha}^\beta = \left[ \left( \begin{array}{cc}i 0 0 -i \end{array}\right) , \, \left(\begin{array}{cc}0 i i 0 \end{array}\right) , \, \left( \begin{array}{cc}0 i -i 0 \end{array}\right) \right] \, .
\int_{-\infty}^{\infty}\mathcal{L}_s^{-1}f(z,w)e^{i\lambda w}\,dw=(2\pi)^{-n}|\lambda|^n\sum_{k=0}^{\infty}(2|\lambda|)^{-s}\frac{\Gamma\big(\frac{2k+n}{2}+\frac{1-s}{2}\big)}{\Gamma\big(\frac{2k+n}{2}+\frac{1+s}{2}\big)}f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z).
A = V \operatorname{diag}(\Lambda_{-}, \Lambda_{+}) V^{*}= \underbrace{V \operatorname{diag}(-I_{\nu}, I_{n-\nu}) V^{*}}_{=:U}\cdot \underbrace{V\operatorname{diag}(\vert \Lambda_{-}\vert, \vert \Lambda_{+}\vert) V^{*}}_{=:H}
\bullet \textrm{For}p=0 , \vert \lambda_0 \vert\leq C \Vert \psi \Vert_{X_r}&\bullet \textrm{For}p\in\mathbb{Z}^*, \vert \lambda_p \vert\leq \dfrac{C}{p^4}\Vert \psi \Vert_{X_r}
\hat{H}_{AF}=-\sum_{k,\lambda}\hbar(g_{k,\lambda}\hat{\sigma}_{+}\hat{a}_{k,\lambda}-g_{k,\lambda}^{*}\hat{\sigma}_{-}\hat{a}_{k,\lambda}^{\dagger}-g_{k,\lambda}\hat{\sigma}_{+}\hat{a}_{k,\lambda}^{\dagger}+g_{k,\lambda}^{*}\hat{\sigma}_{-}\hat{a}_{k,\lambda}),
0=\lambda_{\boldsymbol{x}}(\epsilon)-\lambda_{\boldsymbol{y}}(\epsilon)=-2\sum_{j=1}^n(x_j-y_j)+2\epsilon\sum_{j=1}^n\mu_j(x_j-y_j)-\epsilon^2\sum_{j=1}^n\mu_j^2(x_j-y_j)+O\left(\epsilon^4\right)
\mathrm{t r i r e s}\left( \Phi_{A}^{* ( 2 )}\right) = \left( g h_{1}\left( \Phi^{A}\right) , g h_{2}\left( \Phi^{A}\right) + 1 , g h_{3}\left( \Phi^{A}\right) \right) = - \mathrm{t r i g h}\left( \Phi_{A}^{* ( 2 )}\right) ,
[g(n),\mathcal{Y}(w_1,x_2)] =\mathrm{Res}_{x_0}\, (x_2+x_0)^n\mathcal{Y}(g(x_0)w_1,x_2)=\sum_{i\geq 0}\binom{n}{i}x_2^{n-i}\mathcal{Y}(g(i)w_1,x_2).
\overline{\sigma}^{\mu}\frac{\partial}{\partial x^{\mu}}\psi_{L}(x)\mapsto\overline{\sigma}^{\mu}\frac{\partial}{\partial x^{\prime\mu}}\psi_{L}(x^{\prime})=S\overline{\sigma}^{\mu}\frac{\partial}{\partial x^{\mu}}\psi_{L}(x)
L^{\tilde{\pi}}_{r}(t) = L^{\pi^{i,j^*}}_{r}(t), R^{\tilde{\pi}}_{r}(t) = \beta(x - x_{j^*}) 1_{\lbrace t =0 \rbrace}+ R^{\pi^{i,j^*}}_{r}(t) 1_{\lbrace t> 0 \rbrace},
(\partial_t-\Delta_{\omega(t)})tr_{\omega(t)}\hat\omega =g(t)^{\bar ji}g(t)^{\bar lk}\hat R_{i\bar jk\bar l}-g(t)^{\bar ji}g(t)^{\bar lk}\hat g^{\bar ba}\nabla^{g(t)}_i\hat g_{k\bar b}\nabla^{g(t)}_{\bar j}\hat g_{a\bar q}&\le g(t)^{\bar ji}g(t)^{\bar lk}\hat R_{i\bar jk\bar l}&\le A(tr_{\omega(t)}\hat\omega)^2,
[\begin{matrix}t\\x\\y\\z\end{matrix}]\rightarrow[\begin{matrix}t\\x\\y\\z\end{matrix}]+Re(\alpha)[\begin{matrix}x\\t-z\\ 0\\x\end{matrix}]+Im(\alpha)[\begin{matrix}y\\ 0\\z-t\\y\end{matrix}]+\frac{|\alpha|^{2}}{2}[\begin{matrix}t-z\\ 0\\ 0\\t-z\end{matrix}].
i =\alpha^{2^{n-2}}=\alpha^{l(l_1\cdot2^{n-2-s})+2^n(n_1\cdot2^{n-2-s})}=(\alpha^l)^{l_1\cdot2^{n-2-s}}(\alpha^{2^n})^{n_1\cdot2^{n-2-s}}= &=(\alpha^l)^{l_1\cdot2^{n-2-s}}
{\rm L}^{p,q}_v (G) = \left\{F: G \to \mathbb{C}~:~\int_H \left(\int_{\mathbb{R}^d}|F(x,h)|^p v(x,h)^p dx \right)^{q/p}\frac{dh}{|{\rm det}(h)|}< \infty\right\}~,
\frac{\phi'_\omega(t)}{t}=\|\omega\|_X^2-\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx &~~~-\lambda\ln t\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx-t^{2_\alpha^*-2}\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx.
\mathcal{S}\left( J_{3}^{\mathbb{O}}\right) \supset \mathcal{S}\left(\mathbb{R}\oplus J_{2}^{\mathbb{O}}\right) \cong \left. so(1,q)\right\vert_{q=8}\supset \left. so(q-1)\right\vert_{q=8}\oplus so(1,1).
K_s(1,w)= \frac{\Gamma(n+2-s)}{4^n\pi^{n+1}}\int_0^{\infty}\frac{\cos \Big((n-s+2)\arctan\Big(\frac{4w}{\coth t}\Big)\Big)}{\Big(w^2+\frac{1}{16}\coth^2t\Big)^{\frac{n-s+2}{2}}}(\cosh t)(\sinh t)^{-n-2+s}\,dt.
\sum_{\substack{d\leq D (d, q_1) = 1}}\sum_{\substack{\nu \bmod d \nu^2+1\equiv 0 \bmod d}}\left|\sum_{n\leq N}\alpha_n e\left(\frac{\nu' n}{dq_1}\right)\right|\leq C D^{\frac{1}{2}}(D+N)^{\frac{1}{2}}\|\alpha\|.
X(i,j) =\left\{\left( [x_1: \dots :x_i:0: \dots :0], [0: \dots 0: y_j : \dots : y_n] \right) \in \mathbb{P}^{n-1}\times \mathbb{P}^{n-1}\biggr\rvert \sum x_k y_k =0 \right\}.
\!=\![x_{1k}x_{2k}\!-\!x_{12}x_{2k}\!-\!2x_{12}(x_{1k}x_{2k}-x_{12}x_{2k})]\tau_{1n}\!=\![x_{1k}x_{2k}\!-\!x_{12}x_{2k}(2x_{1k}\!-\!I)]\tau_{1n}.
\gamma_{12}^*(\alpha \boxtimes \gamma ) =s_{12}^*\left(\left(g_{12}^*(st_1\times st_2)^*\alpha\right)\boxtimes \gamma\right) =s_{12}^*s^*_3\left(\left((g_{1}\times g_2)^*(st_1\times st_2)^*\alpha\right)\boxtimes \gamma\right),
{\cal S}_{\psi}= i \int d^{4}x \int d \phi ~ [{\it d e t}~ V ] ~ \bar{\psi}~ [ \gamma^{a}v_{a}^{\mu}~ ( \partial_{\mu}- \frac{i}{2}G_{L N}\sigma^{a b}v_{a}^{\nu}\partial_{\mu}v_{b}^{\lambda}\delta_{\nu}^{N}\delta_{\lambda}^{L}- G_{A D}\sigma^{a b}v_{a}^{\beta}v_{b}^{\delta}{\bar{\Gamma}_{M B}}^{A}\delta_{\mu}^{M}\delta_{\beta}^{B}\delta_{\delta}^{D}) ] ~ \psi ~ \delta ( \phi - \pi )
\Gamma^{\prime}_{\mathit{eff}}[\phi ]=i \int \frac{d^4k}{(2 \pi)^4}\mathrm{tr}\langle k| \g \ \phi (\hat{x}) \S(\hat{p}) + \frac{1}{2}\g phi (\hat{x}) \S(\hat{p}) \g \ \phi (\hat{x}) \S(\hat{p}) +... |k \rangle.
\tilde{A}_{1}^{( 1 )}( x ) \; = \; - \; \frac{1}{e}\; \frac{x_{2}}{x_{1}^{2}+ x_{2}^{2}+ \lambda^{2}}\; \left[ 1 + \frac{x_{1}^{2}}{( x_{1}^{2}+ \lambda^{2})^{\frac{3}{2}}}\right]
\mathbb{S}_{j}={\textstyle\bigcup\nolimits_{v=1}^{m_{j}}}\mathbb{S}(v,j),\mathbb{S}={\textstyle\bigcup\nolimits_{j=1}^{m}}\mathbb{S}_{j},\mathbb{T}_{j}={\textstyle\bigcup\nolimits_{v=1}^{s_{j}}}\mathbb{T}(v,j),\mathbb{T}={\textstyle\bigcup\nolimits_{j=1}^{s}}\mathbb{T}_{j},
W_{\bf a}= -{\frac{1}{2}}\int_{\frac{4 \pi \epsilon}{\bf a}}^{+ \infty}{\frac{d \tau}{\tau}}e^{\tau^{2}}( 1 - \Phi ( \tau ) ) \simeq{\frac{1}{2}}\ln ({\frac{4 \pi \epsilon}{\bf a}}) .
F_{i j}^{a}= \partial_{j}A_{i}^{a}- \partial_{i}A_{j}^{a}- g{\epsilon}^{a b c}A_{i}^{b}A_{j}^{c}, F_{i 0}^{a}= \partial_{0}A_{i}^{a}+ \partial_{i}A_{0}^{a}+ g{\epsilon}^{a b c}A_{i}^{b}A_{0}^{c},
\eta = \frac{3}{6 4 k_{*}v}\left[ 6 v^{1 / 3}( v - 2 ) - 8 + ( v - 2 ) \log \frac{( 1 - v^{1 / 3})^{3}}{1 - v}+ 2 \sqrt{3}( 2 - v ) \tan^{- 1}\frac{2 v^{1 / 3}+ 1}{\sqrt{3}}\right]
j = \left( \begin{array}{c c c c c c}{{\theta_{1}}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{\theta_{q + 1}}}&{{}}&{{}}&{{b}}\\{{}}&{{}}&{{}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{\theta_{p}}}&{{}}\\{{}}&{{}}&{{c^{t}}}&{{}}&{{}}&{{d}}\\ \end{array}\right) .
K (z_0; z) := \frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0; z) \rangle}{|\nabla_{(x,t)}\Gamma(z_0;z)|}, M(z_0; z) := \frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0; z) \rangle}{\Gamma(z_0; z)^2}.
p\left({d_{pF}\leq \delta}\right) = p\left({d_{pF}^2 \leq \delta^2}\right) = M\iint\limits_\Omega \sin^{n - 4}\theta_1 \sin^{n - 4}\theta_2 \, \, \, \times \left({\sin^2 \theta_i - \sin^2 \theta_j}\right)^2 d\sin^2 \theta_1 d\sin^2 \theta_2
\int_{0}^{\infty}\mathrm{d}r \, \left[ u_{1 \epsilon \kappa}^{2}\left( r \right) + u_{2 \epsilon \kappa}^{2}\left( r \right) - v_{1 \epsilon \kappa}^{2}\left( r \right) - v_{2 \epsilon \kappa}^{2}\left( r \right) \right] = 2 \eta_{\epsilon \kappa}^{\prime}\left( k \right) + \left( - 1 \right)^{l}2 \pi \delta \left( k \right) \sin^{2}\eta_{\epsilon \kappa}\left( k \right) \, ,
L_1 =\left(i J_1+\mu_2\frac{s_3}{s_2}R_2-\mu_3 \frac{s_2}{s_3}R_3\right)R_2+\mu_2R_3+\mu_3 R_2+\frac{1}{2}R_2R_3, L_3 =\left(i J_3+\mu_1\frac{s_2}{s_1}R_1-\mu_2 \frac{s_1}{s_2}R_2\right)R_1+\mu_1R_2+\mu_2 R_1+\frac{1}{2}R_1R_2.
\frac{k-t+q-2}{|B|-2}\leq 1+\frac{q-2}{(|B|-1)(|B|-2)}+\frac{q}{|B|-2}\leq 1+\frac{q-2}{12}+\frac{q}{3}= 1+\frac{5q-2}{12}\leq \frac{5(k-t)}{12},
\{W|P|Q\}(\alpha,\alpha^{*},t)=\frac{1}{\pi[\kappa+\langle n\rangle(1-e^{-2\gamma t})]}exp(-\frac{|\alpha-\alpha_{0}e^{-(\gamma+i\omega_{0})t}|^{2}}{\kappa+\langle n\rangle(1-e^{-2\gamma t})}).
\bar{R}_{i j k l}=\frac{1}{2(n-2)r}\left(g_{i k}-\partial_i f \partial_k f\right)\left(g_{j l}-\partial_j f \partial_l f\right) -\frac{1}{2(n-2)r}\left(g_{i l}-\partial_i f \partial_l f\right)\left(g_{j k}-\partial_j f \partial_k f\right) +O(r^{-\frac{3}{2}-\delta}).
\sum_{\substack{z_1<q\le p\le z_2 z_3\le p q<z_5 z_6\le q^2 p\le X}}S_{p q}(\min(q,(X/p q)^{1/2}))=\sum_{\substack{z_1<q\le p\le z_2 z_3\le p q<z_5 z_6\le q^2 p\le X}}S_{p q}((X/p q)^{1/2})+\sum_{\substack{z_1<q\le p\le z_2 z_3\le p q<z_5 z_6\le q^2 p\le X q<r\le (X/p q)^{1/2}}}S_{p q r}(r).
\frac{1}{n}P_n^{(m)}= \frac{(-2)^m}{m!}\sum_{k=0}^{n-1}\frac{(n-1)!}{(n-k-1)!}\frac{\Gamma(k+(m+2)/2)\Gamma(k+(m+3)/2)}{\Gamma(k+2)\Gamma(k+3/2)}\frac{(-\epsilon)^{k+1}}{\Gamma(k+1)}& = [(-1)^{m-1}2^m\epsilon]\frac{\Gamma((m+2)/2)\Gamma((m+3)/2)}{\Gamma(m+1)\Gamma(3/2)}\sum_{k=0}^{n-1}\frac{(1-n)_k}{k!}\frac{((m+2)/2)_k((m+3)/2)_k}{(2)_k(3/2)_k}(\epsilon)^{k}.
C_wC_s=\begin{cases}-\left(q^{\frac{1}{2}}+q^{-\frac{1}{2}}\right)C_w ~s\in\mathcal{R}(w) \sum_{\substack{|\ell(w)-\ell(y)|=1 ys<y}}C_y \textit{if}~s\not\in\mathcal{R}(w)\end{cases}.
\mathbf{\Theta}(\gamma)+\mathbf{\Sigma}_{\mathbf{U}_{1,\epsilon},U_{2,\epsilon}}&=\mbox{diag}(\mathbf{\Theta}_{1}(\gamma)+\mathbf{\Sigma}_{\mathbf{U}_{1,\epsilon}}, \theta_{2}(\gamma)+\sigma^2_{U_{2,\epsilon}}).
\prod_{a=1}^{n}f(z+\frac{1}{2}hk-u_{a})\prod_{a=1,\cdots,n}^{\rightarrow}\bar{R}_{0a}(z+\frac{1}{2}hk-u_{a})A_{n}L_0^{+}(z)qdetL^{-}(u) &=qdetL^{-}(u)L_0^{+}(z)A_{n}\prod_{a=1,\cdots,n}^{\leftarrow}\bar{R}_{0a}(z-\frac{1}{2}hk-u_{a})\prod_{a=1}^{n}f(z-\frac{1}{2}hk-u_{a}).
\mathcal{C}_{r,s}(\varepsilon ;\tau) = F_{2p-s-pr, 2p}\left( \frac{-i\varepsilon}{\sqrt{2p}};\frac{\tau}{4p}\right) -F_{2p+s-pr, 2p}\left( \frac{-i\varepsilon}{\sqrt{2p}};\frac{\tau}{4p}\right) .
P(l) = \frac{2\pi \lambda}{l!^2(l+1)!^2}\left( \frac{\omega r_0}{2}\right)^{2l+2}\left| \frac{\Gamma^2 \left( \frac{l+2}{2}- \frac{i\lambda}{2}\right)}{\Gamma \left( 1-i\lambda\right)}\right|^2\,.
0=\frac{dS}{d\beta}(\hat{\beta})=\frac{d}{d\beta}(y^{T}y-\beta^{T}X^{T}y-y^{T}X\beta+\beta^{T}X^{T}X\beta)|_{\beta=\hat{\beta}}=-2X^{T}y+2X^{T}X\hat{\beta}
\frac{1}{\sqrt{- g}}{\cal H}= \frac{1}{2}g^{i j}\partial_{i}\phi \cdot \partial_{j}\phi + \frac{1}{4}\exp ( \sqrt{2 \kappa}\phi ) ( V_{i j}{}^{k}+ \sqrt{2 \kappa}A^{k}f_{i j}) ( V^{i j}{}_{k}+ \sqrt{2 \kappa}A_{k}f^{i j})
s(\xi_j - \tau_j) = \frac{1}{q_j}\bigg( \log \bigg( \frac{1}{s q_j}\bigg) + b \bigg) \leq \frac{1}{(1 - 2 \delta) k_N}\bigg( \log \bigg( \frac{1}{s}\bigg) + b \bigg) \rightarrow 0
\limsup_{m\to \infty}|v_{n(m)}|\,\cdot\, 2^{n(m)}(\Delta b_{n(m)})^m\left(\prod_{j=1}^{m-1}\frac{|v_{n(j)}|}{\Delta b_{n(j)}}\, \prod_{\nu=b_{n(j)}+1}^{b_{n(j)+1}-1}|w_{\nu}|\right)<\infty.
W ={\frac{i}{4}}u_{i}^{+}u_{j}^{+}\bar{D}_{\dot{\alpha}}^{i}\bar{D}^{j \dot{\alpha}}\sum_{n = 1}^{\infty}\int d u_{1}\ldots d u_{n}\;{\frac{( - i g )^{n}V^{+ +}( u_{1}) \ldots V^{+ +}( u_{n})}{( u^{+}u_{1}^{+}) ( u_{1}^{+}u_{2}^{+}) \ldots ( u_{n}^{+}u^{+})}}
\sup_{0<|h|<h_0}\left\|\frac{\delta_{h}(u\,\eta)}{|h|^\tau}\right\|^p_{L^p(\mathbb{R}^N)}\le \frac{C}{(1-\tau)^p}\,\left(\frac{R}{r}\right)^N\,\left(\frac{R}{h_0}\right)^{1+p}\times \left[(1-s)\,\mathcal{M}_\gamma+\left(h_0^{-\tau\,p}+h_0^{-p}+1\right)\,\left\|u\right\|^p_{L^p(B_{R+h_0})}\right],
|p_{n + 1}^y(x, y)| \le |p_{n}^y(x, y_{i - 1})|+ \frac{\alpha}{2}\max_{k}\beta^i_k &= |p_{n}^y(x, y_{i - 1})| + \frac{\delta}{2}\max_{k}(p_n^x(x_k, y_{i - 1}) - p_n^x(x_k, y_i)) &\le \delta (2 - 2^{-(n - 1)}) + \delta 2^{-n}= \delta (2 - 2^{-n}),
\int_{M}\alpha A ={\frac{1}{2}}\langle \int_{M}\alpha ({\tilde{\overline{{\Psi}}}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\overline{{\Psi}}}}_{L}}}+{\tilde{\Psi}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L}}}) \rangle .
\vert V \rangle = \pi^{- 1 / 4}\sqrt{\frac{2}{3}}\exp \Biggl [{\frac{1}{6}}[ ( a_{1}^{\dagger})^{2}+ ( a_{2}^{\dagger})^{2}+ ( a_{3}^{\dagger})^{2}] -{\frac{2}{3}}[ a_{1}^{\dagger}a_{2}^{\dagger}+ a_{1}^{\dagger}a_{3}^{\dagger}+ a_{2}^{\dagger}a_{3}^{\dagger}] \Biggr ] \vert 0 \rangle_{1}\otimes \vert 0 \rangle_{2}\otimes \vert 0 \rangle_{3}.
\mathbf{k}=\frac{v-v_*}{|v-v_*|},\ \mathbf{i}=\frac{v\times v_*}{|v\times v_*|},\ \mathbf{h}=\mathbf{i}\times \mathbf{k}= \frac{\big((v-v_*)\cdot v\big) v_* -\big((v-v_*)\cdot v_*\big) v}{|v-v_*| |v \times v_*|}.
\frac{1}{z}e^{\frac{1}{4}\pi i}(1-(\begin{matrix}1/4\\ 1\end{matrix})3z+(\begin{matrix}1/4\\ 2\end{matrix})3^{2}z^{2}-(\begin{matrix}1/4\\ 3\end{matrix})3^{3}z^{3}+\cdot\cdot\cdot).
<\tau\vert g_0^{-1}M_+^{-1}M_-\vert\tau >^{(1)}= <\tau\vert(g_0^{-1}N_-g_0)N_+^{-1}\vert\tau>^{(1)}=<\tau\vert N_+^{-1}\vert\tau >^{(1)},
{\bf B}^{S}(\nu ;\Delta)=\prod_{\substack{p|f p\not\in S}}\left\{\frac{\zeta_p(-\nu_p)}{L_p\left(\frac{-\nu_p+1}{2},\chi_{D}\right)}|f|_p^{\frac{\nu_p-1}{2}}+\frac{\zeta_p(\nu_p)}{L_p\left(\frac{\nu_p+1}{2},\chi_{D}\right)}|f|_p^{\frac{-\nu_p-1}{2}}\right\},
A(r)-(1 + \gamma)B(r) \geq w'\Big[ E+(\varphi')^2\big(1+2\mathcal{W}\Phi - 2(1 + \gamma) \mathcal{W}|\Phi|^2 \min \big( \mathcal{W}, \frac{h}{4 \varphi'}\big)\big) &-2(1 + \gamma)h^{-2}\mathcal{W}|V_S|^2 \min \big( \mathcal{W}, \frac{h}{4 \varphi'}\big)-V_L-\mathcal{W}\big(V'_L + \frac{h^2q}{4r^2}\big)\Big].
\left|\sum\limits_{i=0}^{k}{k\choose i}p^{i}q^{k-i}f^{\left(i\right)}_{2r_1}f^{\left(k-i\right)}_{2(r-r_1)}-\frac{2}{\pi}\sum\limits_{i=0}^{k}\frac{{k\choose i}p^{i}q^{k-i}i(k-i)e^{-\frac{i^2}{2(2r_1-i)}-\frac{(k-i)^2}{2(2(r-r_1)-k+i)}}}{(2r_1-i)^{3/2}(2(r-r_1)-k+i)^{3/2}}\right|\to 0,
b_2 =(4\pi)^{-\frac{m}{2}}\frac{1}{2}\frac{\Gamma'(d)}{\Gamma(d)}\bigg(r^{m}\int_Nu_{\frac{m}{2}}(p)dvol_N\bigg)|_{r=0}=-\frac12\frac{\Gamma'(d)}{\Gamma(d)}Res_1\zeta^{\frac{n-1}{2}}_N(-1/2).
[a,[b,c]]=[[a,b],c]+[b,[a,c]]. & \pi[a,b]=[\pi(a),\pi(b)]. & [a,fb]=f[a,b]+\pi(a)fb. & \pi(a)\langle b,c\rangle=\langle [a,b],c \rangle+\langle a,[b,c] \rangle. &[a,b]+[b,a]=\mathcal{D}\langle a,b \rangle
A_{k+1}f(x_{k+1}) \leq \sum_{i=0}^k \alpha_i f_i(x^\ast) + A_0 (\phi_0(x^\ast) - \tilde \varepsilon_0) + \tilde \varepsilon_{k+1}\overset{\eqref{Eq:Under}}{\leq}\sum_{i=0}^k \alpha_i f(x^\ast) + A_0\phi_0(x^\ast) + \tilde \varepsilon_{k+1}= A_{k+1}f(x^\ast) + A_0\phi_0(x^\ast) + \tilde \varepsilon_{k+1}.
t^2|\sqrt{\rho}u_{tt}(t)|^2_2 +\int_{0}^{t}s^2|\nabla u_{tt}|^2_2 s\leq C\int_{\tau}^{t}(c^6_3+c^2_3|v_t|^2_{D^1_0}) s^2|\sqrt{\rho}u_{tt}|^2_2s +Cc^{30}_3.
\Sigma^{A B}={\frac{i}{4}}\left[ \Gamma^{A}, \Gamma^{B}\right] \to{\frac{i}{4}}\left[ \omega \left( \Gamma^{A}\right) , \omega \left( \Gamma^{B}\right) \right] \equiv \Omega \, \Sigma^{A B}\, \Omega^{- 1}\, .
\mathcal{K}_{t, \mathcal{Q}^{\flat}}=\bigoplus_{m\in \mathbb{B}^{\xi, \flat}}\mathbb{Z}[t^{\pm 1/2}]E_t(m)=\bigoplus_{m\in \mathbb{B}^{\xi, \flat}}\mathbb{Z}[t^{\pm 1/2}]L_t(m)
d s^{2}= \left({\frac{r}{R}}\right)^{\frac{2}{\omega}}\, d x^{m}\, \eta_{m n}\, d x^{n}+ \left({\frac{R}{r}}\right)^{2}d x^{m^{\prime}}\delta_{m^{\prime}n^{\prime}}d x^{n^{\prime}}\ .
|\nabla f|^2=\frac{S(g_b)-2(\frac{2n-1}{n})S_C(g_b)}{2(2n-1)(n-1)}=\frac{(\frac{1-n}{n})S_C(g_b)-\frac{1}{4}|T(g_b)|^2}{(2n-1)(n-1)},
[ \Gamma^{( 2 )}, \Gamma^{( 6 )}] = \Gamma^{( 6 )}, \qquad [ \Gamma^{( 3 )}, \Gamma^{( 6 )}] = \Gamma^{( 7 )}+ \Gamma^{( 3 )}, \qquad [ \Gamma^{( 6 )}, \Gamma^{( 6 )}] = \Gamma^{( 1 0 )}+ \Gamma^{( 6 )}+ \Gamma^{( 2 )}.
I_{12}=\frac{|F_s(t)|}{|t|^{1-\frac1{2s}}}\approx\frac{t^2}{t^{2-\frac1{2s}}(1+t^{\frac1s})^{\frac{N+2s}2}}=\frac{t^{\frac1{2s}}}{(1+t^{\frac1s})^{\frac{N+2s}2}}\lesssim\min\left(1,\frac1{t}\right).
{\mathbf P}(X_2(n+1)=l\,|\, X_1(n+1)=k,X_1(n)=i, X_2(n)=j) =\frac{{\mathbf P}(X_1(n+1)=k,X_2(n+1)=l\,|\, X_1(n)=i, X_2(n)=j)}{{\mathbf P}(X_1(n+1)=k\,|\, X_1(n)=i,X_2(n)=j)}= \frac{P_{(i,j),(k,l)}}{P_{i,k}}.
\| ( t_{0}, t_{1}, t_{2}, t_{3}) \|^{2}= c \sum_{0}^{3}\| t_{i}( + \infty ) \|^{2}+ \int_{0}^{+ \infty}\sum_{0}^{3}\left( \| t_{i}( s ) \|^{2}- \| t_{i}( + \infty ) \|^{2}\right) d s .
\boldsymbol{T}_{k}\, \left(E^{T}_{j_{0}}\right)(x) - \eta \, \sum_{j=1 \atop j \neq j_{0}}^{\aleph}\int_{D_{j}}\Upsilon_{k}(x,y) \cdot E^{T}_j(y)\, dy = E^{Inc}_{j_{0}}(x),
\nabla(|\nabla_b\varphi |^2) = (\varphi_1\varphi_{\bar{1}})_1Z_{\bar{1}}+(\varphi_1\varphi_{\bar{1}})_{\bar{1}}Z_1 = (\mu-W)h_{\bar{1}1}\varphi_1Z_{\bar{1}}+ (\mu-W)h_{1\bar{1}}\varphi_{\bar{1}}Z_1 = (\mu -W)\nabla \varphi.