harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V^{a{\mu}_{1}{\cdots}{\mu}_{q + 1}}= \frac{( - 1 )^{p}}{( 2 p - q + 1 ) !}{\Gamma}^{a b}{\epsilon}^{{\mu}_{1}{\cdots}{\mu}_{q + 1}{\nu}_{1}{\cdots}{\nu}_{2 p - q + 1}}{\partial}_{[{\nu}_{1}}Z_{{\nu}_{2}{\cdots}{\nu}_{2 p - q + 1}]}^{b},
C_{4,3/4}=[\begin{matrix}c_{1}&c_{2}&c_{3}&0\\ -c_{2}^{*}&c_{1}^{*}&0&c_{3}\\ -c_{3}^{*}&0&c_{1}^{*}&-c_{2}\\ 0&-c_{3}^{*}&c_{2}^{*}&c_{1}\end{matrix}],
G_{\mu \nu}^{a b}( p ) = \frac{- i \delta^{a b}}{( 2 \pi )^{2 \omega}( p^{2}+ i \epsilon )}\Big [ g_{\mu \nu}- \frac{p_{\mu}n_{\nu}+ p_{\nu}n_{\mu}}{p \cdot n}+ \frac{n^{2}p_{\mu}p_{\nu}}{( p \cdot n )^{2}}\Big ] , \ \n^{2}> 0 , \, e p s i l o n > 0 ,
\alpha_{n}=(-1)^nq^{n(n-1)/2}\frac{(1-q^{2n+1})}{1-q}\sum_{i\ge0}(-1)^iq^{i(3i+1)/2}(1-q^{2i+1})\sum_{|j|\le i}(-1)^jq^{-j(j-1)/2+nj}.
\Psi_1\equiv \pi_a\pi^a-m^2 +{{2eF_{ab}\pi^a}\over{\pi^2}}\Theta_2^b\approx 0~~;~~\Psi_2 \equiv \pi .S-{{mj}\over 2}+{{ejF_{ab}\pi^a}\over{2m\pi^2}}\Theta_2^b\approx 0 ,
|(f_i\pm \varphi)(x) -(f_i\pm \varphi)(y)|=|f_i(x) \pm \varphi(y)| \leq |f_i(x)| +|\varphi(y)| = |f_i(x)-f_i(u)|+|\varphi(y)-\varphi(u)| \leq (1-\delta)d(x,u) +(1-\delta)d(u,y)\leq d(x,y).
t_f-t_i=\int_{r_i}^{r_f}\,du\,\frac{\pm 1}{\sqrt{2\left[E_r-V(u)-\frac{L^2}{2u^2}\right]}}\ \ ,\ \L=\frac{\varphi_f-\varphi_i}{g^2(t_f-t_i)+\int_{t_i}^{t_f}\,\frac{dt}{r^2(t)}}\ \ \ ,
|V(G)\cap \Delta| =|V(G)\cap \Delta_{0,1}|+|G_{\Delta_{1,k}}|+|V(G)\cap \Delta_{k,k+1}| &\le 2c(\partial_G\Delta_{0,1}+\partial_G\Delta_{k,k+1})+8c+4+f(\lceil 8c+4\rceil) &\le 2c\partial_G\Delta+8c^2+16c+4+f(\lceil 8c+4\rceil)\le g(\partial_G\Delta-2)-2,
0\le\Phi_{I,\lambda,P}(f)-\Phi_{I,\lambda,Q}(f) =|f(t_{j+1})-f(t_{j-1})|-\left[|f(t_{j+1})-f(t_{j})|+|f(t_{j})-f(t_{j-1})|\right]-\lambda <|f(t_{j+1})-f(t_{j-1})|-\left[|f(t_{j+1})-f(t_{j})|+|f(t_{j})-f(t_{j-1})|\right],
\begin{aligned}\bigg( \frac{f(x)}{\psi_\gamma(x)}- \frac{f(y)}{\psi_\gamma(y)}\bigg)(g(x)-g(y))\psi_\gamma(x)\psi_\gamma(y) &=(f(x)-f(y))\cdot (g(x)\psi_\gamma(x))-(f(x)-f(y))\cdot (g(y)\psi_\gamma(y)) +(f(x)g(x)-f(y)g(y))\cdot \psi_\gamma(y)-(f(x)g(x)-f(y)g(y))\cdot \psi_\gamma(x).\end{aligned}
\mathrm{c^{sm}}(\Omega_i) =\sum_{\substack{\gamma\in\mathcal{N}(\Delta) \mathcal{O}_\gamma\subset\Omega_i}}\mathrm{c^{sm}}(\mathcal{O}_\gamma) &=\sum_{\substack{\gamma\in\mathcal{N}(\Delta) \mathcal{O}_\gamma\subset\Omega_i}}h^{m-\dim(\gamma)}H^{\mathrm{dim}(\gamma)}.
\{g_1, \dots, g_m\}= \{\chi_{D_1}, \dots, \chi_{D_N}, x_1 \chi_{D_1}, \dots, x_1 \chi_{D_N}, x_2 \chi_{D_2}, \dots, x_2 \chi_{D_2}, \dots, x_n \chi_{D_N}, \dots, x_n \chi_{D_N}\}.
\mathbf{C}_{\mathbf{y}}=E\left[\left(\mathbf{H}\mathbf{x}+\mathbf{n}\right)\left(\mathbf{H}\mathbf{x}+\mathbf{n}\right)^H\right]=\sigma_x^2\mathbf{HH}^H+\sigma_n^2\mathbf{I}_M.
\begin{array}{ll}\sigma^0=-{\bf 1}_{2\times 2}={\bar\sigma}^0, \sigma^1=\tau_z=-{\bar\sigma}^1, \sigma^2=\tau_x=-{\bar\sigma}^2, \sigma^{3}=\tau_y=-{\bar\sigma}^{3},\quad ({\rm or}\quad\sigma^{y}=\tau_y=-{\bar\sigma}^{y}),\end{array}
\begin{array}{c}A_\mu =\left( \frac 12\left( l_\mu^1+l_\mu^2\right) ,\frac 12\left( l_\mu^1+l_\mu^2\right) \right) , B_\mu =\left( \frac 12\left( l_\mu^1-l_\mu^2\right) ,-\frac 12\left( l_\mu^1-l_\mu^2\right) \right) ,\end{array}
\mbox{$H_{nm}^{\{{\bf R}\}}(y_1,y_2)$}=R_{12}^m\sum_{k=o}^{\mu_{mn}}\left(\frac{R_{11}}2\right)^{\frac{n-k}2}\frac{(-1)^kn!m!y_1^{m-k}}{(n-k)!(m-k)!k!}H_{n-k}\left(\frac{R_{11}y_1+R_{12}y_2}{\sqrt{2R_{11}}}\right).
\bigg\|\sum_{k=0}^n a_k\sum_{i\in F_{\sigma|k}}d_iy^{**}_{\xi_{\sigma|k}}(e_i^*)e_i\bigg\| \leq\max_{0\leq k\leq n}|a_k|\bigg\|\sum_{k=0}^n \sum_{i\in F_{\sigma|k}}y^{**}_{\xi_{\sigma|k}}(e_i^*)e_i\bigg\|
I_{5}=-\frac{(n-2)^{2}}{2(n-1)}\int_{\mathbb{R}^{n-1}}\left(U(0,z)\chi(0,\delta z)\right)^{\frac{2(n-1)}{n-2}}|g(0,\delta z)|^{1/2}dz =-\frac{(n-2)^{2}}{2(n-1)}\int_{\mathbb{R}^{n-1}}U(0,z)^{\frac{2(n-1)}{n-2}}\left(1-\frac{\delta^{2}}{6}\bar{R}_{ij}(q)z_{i}z_{j}\right)dz+o(\delta^{2});
+ \frac{4 \lambda_{2}}{r}h ( 3 f^{2}- 1 ) \left( ( h^{2}- \eta^{2}) ( f^{2}- 1 ) + 2 h^{2}f^{2}\right) + 2 \lambda_{4}r^{3}( h^{2}- \eta^{2})^{3}h
\theta( (F(t_2) - F(t_1))\,t_1^{k_1}\cdots t_n^{k_n}) = \begin{cases}-2q_{n-2}(t_1, \dots, t_n) k_2 = n-2 \ , -q_{k_2}(t_1, \dots, t_n) k_2 \neq n-2\ .\end{cases}
v_i v_j =( c_i-4 s_{i}-4z_{i})( c_j-4 s_{j}-4z_{j}) &= c_i c_j-4\left({c_i(s_{j}+z_i)+ c_j(s_{i}+ z_{i})}\right) &\phantom{{}={}}+ 16(s_{i}s_{j}+s_i z_{j}+s_j z_{i}+z_{i}z_{j}) &= -3c_{i,j}{-6t_{i,j}}+ c_i c_j+ 16(s_i z_{j}+s_j z_{i}+z_{i}z_{j})
\begin{aligned}f =f(\mathbf 0)+\delta f,\vert \delta f \vert \leq \|f\|_{C^\alpha}\vert \mathbf x \vert^\alpha, v =v(\mathbf 0)+\delta v,\vert \delta v \vert \leq \|v\|_{C^\alpha}\vert \mathbf x \vert^\alpha, \eta =\eta(\mathbf 0)+\delta \eta, \vert \delta \eta\vert \leq \|\eta\|_{C^\alpha}\vert \mathbf x\vert^\alpha.\end{aligned}
2\ell\left( -s,a,\chi\right) =\sum\limits_{j=0}^{l}\frac{(s)_{j}}{j!}E_{j,\overline{\chi}}\left( 0\right) a^{s-j}+\frac{(s)_{l+1}}{l!}\int\limits_{0}^{\infty}\frac{\overline{E}_{l,\overline{\chi}}\left(-t\right)}{\left( t+a\right)^{l+1-s}}dt,
\Sigma_{aB}=\Sigma_{a(xc)}\Sigma^{-1}_{(xc)(xc)}\Sigma_{(xc)B}=[\sigma_{ax},0]\Sigma^{-1}_{(xc)(xc)}\Sigma_{(xc)B}=\sigma_{ax}[1,0]\Sigma^{-1}_{(xc)(xc)}\Sigma_{(xc)B}=\sigma_{ax}\mathcal{Q}_{cxB}.
{\rm acc}(G[A];A_1,A_2) \geq 2\ell - |X \cap A_1| - 2|X \cap A_2| \geq 2\ell - \min \lbrace a_1,\ell-a_2 \rbrace - 2a_2 &= \ell - a_1 - 2 a_2 + \max \lbrace \ell,a_1+a_2 \rbrace \geq \ell-a_2.
\bar{A}= T_{\rm l}^\top \hat{A}T_{\rm r}, \bar{B}= T_{\rm l}^\top \hat{B}, \bar{C}= \hat{C}T_{\rm r}, \bar{E}= T_{\rm l}^\top \hat{A}T_{\rm r}, \bar{F}(\bar{v}) = T_{\rm l}^\top \hat{F}( T_{\rm r}\bar{v}) .
0 =f(0) 0 =\Delta f(0)=f(1) &\ \,\vdots 0 =\Delta^{\eta-1}f(0)=\Delta^{\eta-2}f(1)=\cdots f(\eta-1) 0 =f(0)=\Delta^{\eta}f(0)=\Delta^{\eta-1}f(1)=\cdots=f(\eta) &\ \,\vdots
Q_2 = \mathrm{P}\left( x_n \left( \frac{x_k- \frac{\epsilon_{1,k}}{\rho}}{x_k}\right)<\tilde{\epsilon}_{1,n}, x_k> \frac{\epsilon_{1,k}}{\rho}\right) = \mathcal{E}_{x_k}\left\{\mathrm{P}\left( x_k< x_n <\frac{\tilde{\epsilon}_{1,n}x_k}{x_k- \frac{\epsilon_{1,k}}{\rho}}\right)\right\}.
J_{\ast,t}(V) :=\int_{0}^{t}\int_{\lbrace z\leqslant1\rbrace}e^{cs}\left(V(X_{s-}+z)-V(X_{s-})\right)\widetilde{N}(\mathrm{d}s,\mathrm{d}z),t\geqslant0, J_{t}^{\ast}(V) :=\int_{0}^{t}\int_{\lbrace z>1\rbrace}e^{cs}\left(V(X_{s-}+z)-V(X_{s-})\right)\widetilde{N}(\mathrm{d}s,\mathrm{d}z),t\geqslant0.
\int_{0}^{r_{s}}\left[ \frac{3 2 r \sqrt{1 - r^{2}}}{3 \pi}\frac{P_{y}^{2}}{y}+ \frac{3 \pi r}{2 \sqrt{1 - r^{2}}}( y - y_{0}) \right] \Psi [ y ] d r = 0
\bar{\mathbf{b}}+\bar{\mathbf{c}}=(\bar{\beta}_1+\bar{\gamma}_1,\bar{\beta}_2+\bar{\gamma}_2,b_1+\bar{\gamma}_3,b_2+c_1,b_3+c_2,\cdots,b_{2k-1}+c_{2k-2},\bar{\beta}_3+c_{2k-1},\bar{\beta}_4+\bar{\gamma}_4).
\nu(\mathcal{B}_n)= \prod_{i\in\mathcal{I}_b}p^n_U\big(\mathbf{u}(i)\big) \prod_{\big(\hat{i},j\big)\in\mathcal{I}_n\times\mathcal{J}_n}p^n_{V|U}\Big(\mathbf{v}\big(\hat{i},j\big)\Big|\mathbf{u}(\hat{i})\Big).
\rho_a(z) =\sum_{k=0}^\infty \left( \sum_{j=0}^k a_jb_{k-j}^{2^j}a^{2^j}\right)z^{2^k}&=\sum_{k=0}^\infty \left( \sum_{j=0}^k \frac{a^{2^j}}{d_j\ell_{k-j}^{2^j}}\right)z^{2^k}.
h_1 = -a A f_1 + \rho_1f_2 - a A^\frac12 f_3\in H^{-1}, h_2 = -a A^\frac12f_1 - (a + bA) f_3 + \rho_2f_4\in H^{-1}, h_3 = \rho_3f_5\in H.
10_{F}=(\begin{matrix}0&u_{3}^{c}&-u_{2}^{c}&u_{1}&d_{1}\\ -u_{3}^{c}&0&u_{1}^{c}&u_{2}&d_{2}\\u_{2}^{c}&-u_{1}^{c}&0&u_{3}&d_{3}\\ -u_{1}&-u_{2}&-u_{3}&0&e_{R}\\ -d_{1}&-d_{2}&-d_{3}&-e_{R}&0\end{matrix})
f_{\mathrm{I}_{ij}}(\mathrm{I}) = \frac{\left({\alpha_{ij}}/{b_{0_{ij}}}\right)^{\alpha_{ij}}}{\Gamma(\alpha_{ij})}& \times \int_0^{\infty}b^{\alpha_{ij}-2}\exp\left(-\frac{\alpha_{ij}b}{b_{0_{ij}}}-\frac{\mathrm{I}+{A_{ij}}^2}{b}\right)I_0\left(\frac{2{A_{ij}}\sqrt{\mathrm{I}}}{b}\right)\mathrm{d}b
\mu_1\left((a,b)\right) =\mu_1\left((a,b)\right), \int_a^b x^k \mu_1(dx) =\int_a^b x^k \mu_2(dx),for \:k=1,\ldots ,n, \int_a^b \bigl(t-x\bigr)^n_+\mu_1(dt) =\int_a^b \bigl(t-x\bigr)^n_+\mu_2(dt),for \:x\in (a,b),
\mathbf{I}\leq \frac{C_2}{t}\int^\infty_{\sqrt{t}}\left(\frac{r}{\sqrt{t}}\right)^n \exp\left(-\frac{r^2}{C_1t}\right)\cdot \frac{k(x_0,r)}{r}\,dr\leq \frac{C_3}{t}\int^\infty_{\sqrt{t}}\frac{k(x_0,r)}{r}\,dr.
I ( p^{2}, m_{2}^{2}) = ( 1 - \frac{d}{2}) N_{d}( - \frac{1}{\epsilon}+ O ( \epsilon ) ) \sum_{k = 0}^{\infty}C_{\frac{d}{2}- 2}^{k}\frac{p^{2 k}}{m_{2}^{2 k}}\int_{0}^{1}d t ( t ( 1 - t ) )^{k}.
\begin{cases}\partial_{t}u+\Lambda^{\alpha}u+\nabla\cdot(u\nabla \psi)=0 \mbox{in}\ \ \mathbb{R}^n\times(0,\infty), -\Delta \psi=u \mbox{in}\ \ \mathbb{R}^n\times(0,\infty), u(x,0)=u_0(x), \ \ \mbox{in}\ \ \mathbb{R}^n.\end{cases}
{\cal L}_{\mathrm{G N J L}}= -{\frac{1}{4}}F_{\mu \nu}^{2}+ \bar{\psi}( i \gamma^{\mu}D_{\mu}- m_{0}) \psi +{\frac{G_{0}}{2}}[ ( \bar{\psi}\psi )^{2}+ ( \bar{\psi}i \gamma_{5}\psi )^{2}] ,
W_m=\left[w_m(x_i-x_j)\cdot \frac{1}{n}\right]_{i,\j=0}^{n-1}=\left[\frac{w(g_{m}^{-1}(x_i-x_j))}{\sum_{z_r=-1}^{1}w(z_r) |g'_{m}(z_r)| \Delta z_r}\cdot \frac{1}{n}\right]_{i,\j=0}^{n-1}
d s^{2}= \left[ - d t^{2}+{\frac{2 a \sin^{2}{\theta}}{\Xi}}d t d \phi + l^{2}{\frac{d \theta^{2}}{\Delta_{\theta}}}+ l^{2}{\frac{\sin^{2}{\theta}}{\Xi}}d \phi^{2}+ l^{2}\cos^{2}{\theta}d{\Omega_{2}}^{2}\right] \ .
\int_{\mathbb{R}^{n+1}_+\setminus B^+_{r+\bar{l}+1}}{y^a|\nabla W_k|^2\,dX}+\int_{\mathbb{R}^n\setminus\{B_{r+\bar{l}+1}\cap\{y=0\}\}}{(W_k)_+^{2^*_s}(x,0)\,dx}&=\int_{\mathbb{R}^{n+1}_+\setminus B^+_{r+\bar{l}+1}}{y^a|\nabla U_k|^2\,dX}+\int_{\mathbb{R}^n\setminus\{B_{r+\bar{l}+1}\cap\{y=0\}\}}{(U_k)_+^{2^*_s}(x,0)\,dx}\geq \eta_0,
I_{m,p}^{\star(1)}(s,x)=(-1)^{m}\sqrt{\pi}2^{\frac{1}{2}-s}x^{s+\frac{1}{2}}\,\intop_{0}^{\infty}y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(xy)\,e^{-my}dy
\chi^0=\chi_0:=\frac{\int_{\mathbb{R}^d}\left(\frac{1}{2}|\nabla\phi_0|^2+V|\phi_0|^2+\beta|\phi_0|^4-\Omega\overline{\phi}_0L_z\phi_0\right)\mathrm{d}\mathbf{x}}{\|\phi_0\|^2}+\sigma R^\prime(\|\phi_0\|^2-1).
| \phi_{n}\rangle_{g h}^{x}= \hat{\Gamma}^{n}| 0 \rangle_{g h}^{x}= \hat{Q}\hat{\Phi}\hat{\Gamma}^{n - 1}| 0 \rangle_{g h}^{x}= \hat{Q}^{\dagger}\hat{\Phi}^{\prime}\hat{\Gamma}^{n - 1}| 0 \rangle_{g h}^{x}
H(\theta_{0}):=\bigg{\{}X:\log \int \frac{\mathrm{d}P_{\theta,\varepsilon^{2}}}{\mathrm{d}P_{\theta_{0},\varepsilon^{2}}}(X)\mathrm{d}\widetilde{\mathrm{S}}_{M}(\theta\mid\alpha) \geq -2 ( B / \varepsilon)^{2/(2\alpha_{0}+1)}\bigg{\}}.
|\int_{t}^{\infty}ds \frac{-\lambda''(s)}{r^{2}}\int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\left(1+\frac{r^{2}-1-\rho^{2}}{\sqrt{(r^{2}-1-\rho^{2})^{2}+4r^{2}}}\right)| \leq \frac{C}{r^{2}}\int_{t}^{\infty}ds |\lambda''(s)|(s-t)
\tilde{G}_{\mu \nu}^{0 A}- \tilde{G}_{\mu \nu}^{0 L}= -{\frac{1}{k^{2}}}k_{\mu}k_{\nu}\biggl ({\frac{( \lambda k^{2}+ \eta^{2})}{( \eta \cdot k )^{2}}}+{\frac{( 1 - \lambda )}{k^{2}}}\biggr ) +{\frac{k_{[ \mu}\eta_{\nu ]_{+}}}{k^{2}\eta \cdot k}}.
\mathcal{E}^{(q)}(B_{r})=\frac{m\cos (\frac{\pi}{4}-\frac{\pi}{2h})}{4\sin(\frac{\pi x}{2h})\cos (\frac{\pi (1-x)}{2h})}+\frac{m(\cos (\frac{\pi w_{r}}{h})-\cos (\frac{\pi x}{2h}))}{2\sqrt{2}\sin (\frac{\pi x}{2h})\,\sin (\frac{\pi (1-x)}{h})};
\left(\partial_{0,\nu}M_{0}\left(\mathrm{m}_{0}\right)+M_{1}\left(\mathrm{m}_{0}\right)+A\right)^{*}=\left(\overline{M_{0}\left(\mathrm{m}_{0}\right)\partial_{0,\nu}^{*}+M_{1}\left(\mathrm{m}_{0}\right)^{*}-A}\right).
\beta^{(j)}_{i}\leq a^{(j)}_{i}p^{(j)}_{i}\cdots p^{(j)}_{r_j}\beta_{+}+ \frac{q^{(j)}_{i+1}}{a^{(j)}_{i+1}}(\mu_{\mathfrak{f}^{(j)}_i}-1)+\sum_{i'=i+1}^{r_j}\frac{a^{(j)}_{i}p^{(j)}_{i}\cdots p^{(j)}_{i'-1}q^{(j)}_{i'+1}}{a^{(j)}_{i'}a^{(j)}_{i'+1}}(\mu_{\mathfrak{f}^{(j)}_{i'}}-1).
\frac{\theta}{4}\sqrt{\frac{i \pi}{2}}\, e^{- i \frac{\Phi}{2}}\, \, k^{3 / 2}\, \, \left( \pi \, \delta [ \varphi ] -{\cal P}\left[ \frac{1}{1 - e^{- i \varphi}}\right] - \frac{\Phi}{2 \pi}\log [ 1 - e^{- i ( - i \epsilon + \varphi )}] + \dots \right) .
\{a^{p^{-1}}| p \in P_a\}= \{a^{g{g^{-1}}{p^{-1}}}| p \in P_a\}= \{a^{g{{(pg)}^{-1}}}| p \in P_a\}= \{a^{g{q^{-1}}}| q g^{-1}\in P_a\}\textrm{(where}q = pg ) = \{a^{g{q^{-1}}}| q \in P_a*g\}= \{b^{q^{-1}}| q \in P_b\}
(\delta u)^{p+1,q}_{i_0\ldots i_{p+1}}=\sum_{k=1}^p(-1)^k u^{p,q}_{i_0\ldots \widehat{i_k}\ldots i_{p+1}}|_{U_{i_0\ldots i_{p+1}}}\, u^{p,q}\in C^p(\mathcal{U},^q_{\mathfrak{F}}(E,F))
f_{N}^{\prime}\left[ \phi ( x ) \right] ={\frac{1}{Z_{0}}}\sum_{n = 0}^{N}{\frac{\left( - 1 \right)^{n}}{n !}}e^{- S_{0}}\left({\frac{\lambda}{4}}\int \mathrm{d^{d}}x \phi^{4}\right)^{n}\mathrm{C h}( \Omega_{N}, \left\{\phi ( x ) \right\})
\begin{aligned}II_2\leq C\epsilon\int_{t_1}^{t_2}\left\|u(\tau)\right\|_{L^{\infty}}^{q-m-1}\left\|\nabla u^{m}(\tau)\right\|^2_{L^2}\,d\tau+\frac{C}{\epsilon}\int_{t_1}^{t_2}\left(\left\|u(\tau)\right\|_{L^{\infty}}^{q-m}+\left\|\hat{u}(\tau)\right\|_{L^{\infty}}^{q-m}\right)\left\|\left(u-\hat{u}\right)(\tau)\right\|_{L^{1}}\,d\tau\end{aligned}
y_1=\frac{1}{\sqrt{\mu}}x_3, \, y_2= \frac{1}{\sqrt{1-\eta}}(x_1+x_2),\, y_3= x_1; &[y_1,y_2]=\frac{1-w}{\sqrt{\mu}}y_2+\frac{2w}{\sqrt{\mu(1-\eta)}}y_3,\,[y_2,y_3]= 0, \, [y_3,y_1]=-\frac{1+w}{\sqrt{\mu}}y_3.
c_{\psi}= \sin^2(\alpha) \int_0^{2\pi}\frac{\psi_0 \cos \theta + \psi_1 \sin \theta}{\left( \dfrac{\cos \theta}{r_0^2}- \dfrac{\sin \theta \cos \alpha}{r_0r_1}\right) \psi_0 +\left(\dfrac{\sin \theta}{r^2_1}- \dfrac{\cos \theta \cos \alpha}{r_0r_1}\right) \psi_1}d\theta
H'(u(t))\frac{d}{dt}(k*u)(t) = \frac{d}{dt}(k*H(u))(t) + (-H(u(t))+H'(u(t))u(t))k(t) & + \int_{0}^{t}(H(u(t-s))-H(u(t))-H'(u(t))[u(t-s)-u(t)])\left(-\frac{dk(s)}{ds}\right)ds,
\limsup_{i \rightarrow 0}u_i(q_i)^{-1}\int_{B_g(p_i,r_1)}L_{g}u_i\,dx \leq \limsup_{i \rightarrow 0}u_i(q_i)^{-1}\int_{B_g(p_i,r_1)}\Delta_{g}u_i\,dx &= \int_{\partial B_g(p_\infty,r_1)}\partial_\nu (a G_{p_\infty})\,dS + O(r_1^{n-1}) < 0
R_2 = H(Y_2 | Y_1,{\bf S}) &= P_{Y_1}(1) H(Y_2 | Y_1 = 1,{\bf S}) + P_{Y_1}(0) H(Y_2 | Y_1 = 0,{\bf S}) = \frac{B_1}{M}H( \frac{B_2^1}{B_1}) + (1 - \frac{B_1}{M}) H( \frac{B_2^0}{M - B_1}).
\begin{array}{ll}\delta'(h_\lambda)=0, h_\lambda\in\frak h, \delta'(k_\lambda)=0, k_\lambda\in\frak k, \delta'(x_\beta)= 2^{-1}x_\beta\wedge (h_\beta+k_\beta), \beta\in\bf R^+, \delta'(x_{-\beta})=2^{-1}x_{-\beta}\wedge (h_{\beta}+k_{\beta}), \beta\in\bf R^+,\end{array}
\frac{1}{\prod_{k=1}^{n}(1+ic_k)}R_n(z) = \frac{1+ic_{n+1}}{2(1-\mathfrak{m}_n)}\, \Big[\hat{S}_n(z) + \tau_{n+1}\,\hat{S}_n^{\ast}(z)\Big], n \geq 0.
\begin{aligned}\mathbf{A}=A_{i_1i_2\cdots i_n}\vec e_{i_1}\vec e_{i_2}\cdots\vec e_{i_n}=A'_{j_1j_2\cdots j_n}\vec e'_{j_1}\vec e'_{j_2}\cdots\vec e'_{j_n}. \end{aligned}
- \frac{L'}{L}(s,\pi) = \frac12\log\left(\frac{N}{\pi^m}\right)+ \frac12 \, \sum_{j=1}^m \, \frac{\Gamma'}{\Gamma}\!\left( \frac{s + \mu_j}{2}\right) - B_\pi + \frac{r(\pi)}{s}+ \frac{r(\pi)}{s-1}- \sum_{\rho_{\pi}}\left( \frac{1}{s-\rho_{\pi}}+ \frac{1}{\rho_{\pi}}\right)
\widetilde{T_n}: = I\otimes E_n + \sum_{j=1}^n \frac{1}{j!}(i(D+I))^j \otimes B_n^j Q_n : = \{\widetilde{T}_n (\xi \otimes e_n) : \xi \in \mathrm{dom}(D^n) \}=\{\xi \otimes e_n + \sum_{j=1}^n \frac{1}{j!}(i(D+I))^j \xi \otimes e_{n-j}\, : \, \xi \in \mathrm{dom}((D+I)^n) \,\}
\frac{\lambda(x,x^{1+})}{\lambda(x^{1+},x)}=e^{U(x)-U(y)+F_\varepsilon(x,x^{1+})}=\frac{\lambda_0(x,x^{1+})}{\lambda_0(x^{1+},x)}e^{F_\varepsilon(x,x^{1+})}.
L_0^{\delta}q_1^{\delta}(0, \cdot) \equiv 0\,, L_0^{\delta}q_2^{\delta}(0, \cdot) =E_{12}^{\delta}\delta^2\q_1^{\delta}(0, \cdot)\,, L_0^{\delta}q_3^{\delta}(0, \cdot) =E_{13}^{\delta}\delta\q_1^{\delta}(0,\cdot)\,,
\tilde{J}_{\mu}^{( \pm )}\equiv \int d^{3}x j_{\mu}^{( \pm )}( x ) = \sum_{{\bf k}}\frac{k_{\mu}}{\omega_{{\bf k}}}( a_{{\bf k}}^{\dagger}a_{{\bf k}}\pm b_{{\bf k}}^{\dagger}b_{{\bf k}}) .
\langle \!\langle G, Q_{\alpha_2\alpha_1}(t) k_0\rangle \!\rangle -\langle \!\langle G, Q^\sigma_{\alpha_2\alpha_1}(t) k_0\rangle\!\rangle = \langle \!\langle G, \Upsilon_1(t,\sigma) \rangle\!\rangle + \langle \!\langle G, \Upsilon_2(t,\sigma) \rangle\!\rangle.
A_L = \begin{bmatrix}2 r_L \cos \left( 2 \pi \omega_L \right) 1 -r_L^2 0\end{bmatrix}\;, A_R = \begin{bmatrix}\frac{2}{s_R}\cos \left( 2 \pi \omega_R \right) 1 -\frac{1}{s_R^2}0\end{bmatrix}\;, B = \begin{bmatrix}1 0\end{bmatrix}\;,
\|{Q}_{k+1}-\sqrt{S}\| ={\|({Q}_k-\sqrt{S})(I_n-\mathcal{C}_{\sqrt{S}}(Q_1)^k)}\mathcal{C}_{\sqrt{S}}(Q_1)(I_n-\mathcal{C}_{\sqrt{S}}(Q_1)^{k+1})^{-1}\| &\leq \|\mathcal{C}_{\sqrt{S}}(Q_1)\|\dfrac{1+ \|\mathcal{C}_{\sqrt{S}}(Q_1)\|^{k}}{1-\|\mathcal{C}_{\sqrt{S}}(Q_1)\|^{k+1}}\|Q_k-\sqrt{S}\|.
\sigma_{~ ~ ; \alpha}^{; \alpha}= 2 - \frac{1}{6}g_{\gamma \delta}R \sigma^{; \gamma}\sigma^{; \delta}+ \frac{1}{2 4}g_{\gamma \delta}R_{; \rho}\sigma^{; \gamma}\sigma^{; \delta}\sigma^{; \rho}+ O ( \sigma^{2}) ~ .
( \widetilde \psi , \widetilde \psi ) \ = \ - \eta \int d^{3}x^{T}4 P^{2}\ \overline{{\widetilde \Psi}}\ \frac{\partial}{\partial s}[ \widetilde g_{0}^{- 1}+ \widetilde V ] \ \widetilde \psi \ = \ 2 P_{L}\ ,
\Xi \equiv \left( \begin{array}{l l}{\xi_{a}( x )}\\{\bar{\xi}^{\dot{a}}( x )}\\ \end{array}\right) , X \equiv \left( \begin{array}{l l}{\chi_{a}( x )}\\{\bar{\chi}^{\dot{a}}( x )}\\ \end{array}\right) , \Lambda \equiv \left( \begin{array}{l l}{\lambda_{a}( x )}\\{\bar{\lambda}^{\dot{a}}( x )}\\ \end{array}\right) ;
\rho^m\Delta_\phi W = \left((1-m)(2-m)A\rho^{-m}+ c\Phi \rho^{-m}+ o(\rho^{-m})\right)\left(e^{-Bs}-e^{-Bs_0}\right) & + \left(\rho + o(\rho)\right)\left(-\frac{nB}{s}+B^2+o(1)\right)e^{-Bs}.
\frac{\det D ( A )}{\det D ( 0 )}\; = \; e^{- \frac{i}{2 \pi}A_{1}A_{2}}\; q^{\frac{1}{8 \pi^{2}}A_{1}^{2}}\; \frac{\vartheta ( \alpha , \tau )}{\vartheta ( 0 , \tau )}\; ,
\begin{matrix}\ell^{(1)}=(-1,\;\;\,0,\;\;\,0,\;\;\,1,\;\;\,0,\;\;\,0,\;\;\,0,\;\;\,1,-1), \ell^{(2)}=(\;\;\,0,-1,\;\;\,0,\;\;\,1,-1,\;\;\,1,\;\;\,0,\;\;\,0,\;\;\,0), \ell^{(3)}=(\;\;\,0,\;\;\,0,-1,\;\;\,0,\;\;\,0,\;\;\,1,-1,\;\;\,1,\;\;\,0), \ell^{(4)}=(\;\;\,0,\;\;\,0,\;\;\,0,-1,\;\;\,1,-1,\;\;\,1,-1,\;\;\,1).\end{matrix}
{\hat \Pi}= D{\hat X}^{\hat \mu}D{\hat X}^{\hat \nu}{\hat g}_{{\hat \mu}{\hat \nu}}= \partial{\hat X}^{\hat \mu}\partial{\hat X}^{\hat \nu}\left({\hat g}_{{\hat \mu}{\hat \nu}}+ |{\hat k}|^{-2}{\hat k}_{\hat \mu}{\hat k}_{\hat \nu}\right) \, ,
g^{*n}-(P_m)^{*n}=\big((g-P_m)+P_m\big)^{*n}-(P_m)^{*n}&=(g-P_m)^{*n}+\sum_{\nu=1}^{n-1}\binom{n}{\nu}(g-P_m)^{*(n-\nu)}*(P_m)^{*\nu}.
\left|\tau_s(g f)\right| = \left|\tau_s\left(g \prod_{u=1}^k f_u\right)\right| \leq \|g\|_{Y( \{X_{j}^s:j\in \mathcal J\})}\prod_{u=1}^k \|f_u\|_{X_{j_u}^s}, \|f_u\|_{X_{j_u}^s}= 1, u=1,\ldots, k,
F((\bar{a},\bar{c},0,0,0),(\bar{b},\bar{c},0,0,1)) =(\bar{z},\bar{c},0,0,0)\in X_{\phi}, F((\bar{b},\bar{c},0,0,1),(\bar{a},\bar{c},0,0,0)) =(\bar{z},\bar{c},0,0,1)\in X_{\phi},
\alpha^{\pm\pm}(n_1,n_2)=\sqrt{\frac{1}{f^{\pm\pm}(n_1,n_2)}\frac{N_k}{4\hbar\tau_2}}Big[-i\tau \hat{A}_1^{\pm\pm}(n_1,n_2)+i\hat{A}_2^{\pm\pm}(n_1,n_2)\Big]\ \ \ ,
\delta S_{\psi \psi \lambda \lambda}= \int d^{2}\sigma \left[ \frac{1}{2}v^{\rho}\hat{D}^{( + )}{}_{\rho}F_{\mu \nu a b}- 2 i F_{\mu \nu c b}\Sigma^{c}{}_{a}\right] \psi^{\mu}\psi^{\nu}\lambda^{a}\lambda^{b}\ .
\sum_{n}\sum_{\sigma\subset \mu_\ell}c_{\ell}x_{\ell,n}\pi^{n-\omega(\mu_\ell)}+ c_{\kappa}\pi^{-\omega(\kappa)}=0 \sigma \subset \kappa \sum_{n}\sum_{\sigma\subset \mu_\ell}c_{\ell}x_{\ell,n}\pi^{n-\omega(\mu_\ell)}=0 \sigma \not \subset \kappa.
\frac{t_{k-1}+ 9 + 7\eta_{k-1}}{15+11\eta_{k-1}}\le \frac{\varphi+8 + (\varphi + 6)\eta_{k-1}}{15+11\eta_{k-1}}\le \frac{\varphi+8 + (\varphi + 6)(\varphi -1)}{15+11(\varphi-1)}& = \frac{7\varphi + 3}{11 \varphi + 4}= \frac{5\sqrt{5}+69}{122}= 0.6572\ldots
\sum_{k}\varphi_k(d\phi_{ik}+\Omega_{ij}) =\sum_k\varphi_kd\phi_{ik}+\sum_l\varphi_l\phi_{il}\wedge\Big(\sum_k\varphi_k\phi_{kl}\Big)=\sum_k\varphi_kd\phi_{ik}+\sum_l\varphi_l\phi_{il}\wedge(2\varphi_l\phi_{ll}) =\sum_k\varphi_kd\phi_{ik}-\sum_{l}\phi_{il}\wedge d\varphi_l=d\Big(\sum_k\varphi_k\phi_{ik}\Big)=0.
\begin{aligned}R_1 := \tau_1+\tau_2+\tau_3, R_2 := \tau_1 \tau_2+\tau_2 \tau_3+ \tau_3 \tau_1, R_3 := \tau_1 \tau_2 \tau_3, \end{aligned}\begin{aligned}S_1 := \tau_4+\tau_5, S_2 := \tau_4 \tau_5,\end{aligned}
F(x)=\frac{3\pi\mu_{0}}{2}M^{2}R^{4}L^{2}\frac{1}{x^{4}}=\frac{3\mu_{0}}{2\pi}M^{2}V^{2}\frac{1}{x^{4}}=\frac{3\mu_{0}}{2\pi}m_{1}m_{2}\frac{1}{x^{4}}
\begin{aligned}\|\nabla^2 w\|_{L^2(\Sigma_{p_2\varepsilon})}\leq C\varepsilon^{-\frac{1}{2}}\big\|(\nabla w)^*\big\|_{L^2(\partial\Omega)}+ C\big\|\nabla w\big\|_{L^2(\Omega)}&\leq C\varepsilon^{-\frac{1}{2}}\Big\{\|F\|_{L^q(\Omega)}+\|h\|_{W^{1,q}(\Omega)}+\|g\|_{H^1(\partial\Omega)}\Big\}.\end{aligned}
\begin{aligned}S_K(u,v) :=\; h_K^{-1}\;\sum_{F\subset \partial K}\Bigg[(Q_K u-Q_F u,Q_K v-Q_F v)_{F}&+\epsilon_F h_F\sum_{e\subset \partial F}( (u-Q_F u), (v-Q_F v))_e\Bigg],\end{aligned}
{\cal N}( \alpha ) = \int_{x ( 0 ) = x ( T ) = 0}{\cal D}x ( \tau ) e x p \left[ -{\frac{1}{4}}\int_{0}^{T}d \tau \dot{x}^{2}- \sum_{i = 1}^{N}\alpha_{i}x^{2}( \tau_{i}) \right] ={\frac{1}{( 4 \pi T )^{2}}}d e t^{- 2 N}( 1 + 4 \hat{\Delta}) .
\langle \, 0 \, | \, \mathcal{O}( x ) \, | \, p_{1}, \dots , p_{n}\, \rangle^{i n}= e^{- i x ( p_{1}+ \dots + p_{n})}\, \mathcal{O}_{n}( \theta_{1}, \dots , \theta_{n}) ~ , \mathrm{f o r}~ ~ \theta_{1}> \dots > \theta_{n}
\hat{\pi}(t_0) =\mathbb{P}[\mathrm{Bin}(t_0,p)\geq r]=\sum_{j=r}^{t_0}\binom{t_0}{j}p^{j}(1-p)^{t_0-j}\stackrel{\eqref{t0small}}{=}(1+o(1))\frac{t_0^r p^r}{r!}&=(1+o(1))\frac{t_0^{r-1}p^r}{r!}t_0=(1+o(1))\frac{1}{r}\frac{t_0}{n}\stackrel{np=\omega(1)}{=}o\left(t_0 p\right)\stackrel{\eqref{t0small}}{=}o(1).
\delta \Psi_{\hat{a}}= \partial_{\hat{a}}\epsilon +{1\over 8}H^{-1}(\partial_{\hat{a}}H) \epsilon +{1\over 8}W H^{{m-n \over 4}}\gamma^{\underline{uxy}}\gamma_{\underline{\hat{a}}}(i\sigma_2) \epsilon = 0 \, .
Mf_{3}(y) =e^{-(1-2\sigma)\lambda_{l}s}\frac{\int_{y}^{\infty}r^{2l+1+\omega}e^{-r^{2}/4}dr}{\int_{y}^{\infty}r^{1+\omega}e^{-r^{2}/4}dr}\lesssim y^{2l}e^{-(1-2\sigma)\lambda_{l}s}\le y^{2l}e^{-(1-2\sigma) \lambda_{l}s_{0}}\le e^{2l\sigma s}e^{-(1-2\sigma) \lambda_{l}s_{0}}.
\int_{0}^{1}d r_{0}\, f_{1}( r_{0}) = - \frac{\zeta ( 3 )}{1 2 8 \, \pi^{4}}- \frac{4 5 \, \zeta ( 3 )}{5 1 2 \, \pi^{6}}+ \frac{1 5 \, \zeta ( 5 )}{1 2 8 \, \pi^{6}}
\left( \begin{array}{c c}{a^{\prime \prime}}{b^{\prime \prime}}{c^{\prime \prime}}{d^{\prime \prime}}\end{array}\right) = \left( \begin{array}{c c}{a}{b}{c}{d}\end{array}\right) \left( \begin{array}{c c}{a^{\prime}}{b^{\prime}}{c^{\prime}}{d^{\prime}}\end{array}\right)
\sum_{i, j}\mathbb{E}[ O ( \rho_T (E) N (\mu_i (T) - E), \rho_T (E) N (\mu_j (T) - E )) | \{\lambda_k (0) \}_k] = \sum_{|i-i_0| + |j-i_0| \leq N^{\nu}}\mathbb{E}[ O ( \rho_T (E) N (\mu_i (T) - E), \rho_T (E) N (\mu_j (T) - E )) | \{\lambda_k (0) \}_k] + N^{-10}
2^{k\alpha}=2^{(k-k_0)\alpha}2^{(k_0-\ell_0)\alpha}2^{(\ell_0-k_1)\alpha}\cdots2^{(\ell_{m-1}-k_m)\alpha}2^{(k_m-\ell_m)\alpha}2^{(\ell_m-k'-j)\alpha}2^{(k'+j)\alpha}
C^*_2:= \min_{\partial \mathcal{D}_{\delta}}\, ({\Phi}_* - \varphi )(t,x) > 0(\mathrm{by}\ \eqref{eq:min})\,, C^*_3:=\frac{C^*_2\, e^{-C_1}(||x_0||_1-n\delta)}{2(||x_0||_1 + n\delta)}> 0 (\mathrm{by}\ \eqref{eq:||x||_1})
s_{\widetilde{\lambda}_h+1}s_{\widetilde{\lambda}_h+2}\cdots s_{\widetilde{\lambda}_h+a}d_M=s_{\widetilde{\lambda}^+_h-1}s_{\widetilde{\lambda}^+_h-2}\cdots s_{\widetilde{\lambda}^+_h-b}d_{M^+_{h,k}}.
\int_{\frac{i}{k}}^{\frac{i+1}{k}}r |u_{\varepsilon,i}|^p \, dr = k^{-2}\int_0^1 (s+i) |v_\varepsilon|^p \, ds> k^{-2}\int_\varepsilon^{1-\varepsilon}(s+i)\, ds = \frac{k^{-2}}{2}(2i+1)(1-2 \varepsilon).
P = \sum_a (-1)^a \left [\binom{|S|-|U'|}{2}\binom{|S|-|U'|-2}{a-|U'|-2}\right . \left . + (a-|U'|)|U'|\binom{|S|-|U'|}{a-|U'|}+\binom{|U'|}{2}\binom{|S|-|U'|}{a-|U'|}\right ].
D_m \le \frac{1}{1-\delta}C_0\max_{M=m,m+1,\cdots,m+m_0}\max_{n=1,\cdots,N}|F_{n,2^{M}}\nabla(\phi,\sigma)| &+\frac{\delta}{1-\delta}+\delta^{m_0+1}D_{m+m_0+1}.
\frac{\partial}{\partial x^i}\Delta^{\gamma/2}p(s,\tau, z)=(s-\tau)^{-d/\gamma}(s-\tau)^{-1-1/\gamma}\frac{\partial}{\partial x^i}\Delta^{\gamma/2}q_1(s,\tau, (s-\tau)^{-1/\gamma}z).
I ( \phi ) \; = \; \frac{1}{2}\int_{\cal G}d^{d + 1}y \, \partial_{\mu}\, ( \sqrt{g}\, g^{\mu \nu}\, \phi \, \partial_{\nu}\phi ) \; = \; \lim_{\epsilon \rightarrow 0}\; \frac{1}{2}\int_{\bar{\cal G}\cap T_{\epsilon}}d^{d}{\bf y}\, \sqrt{h}\, \phi \,{\bf n}\cdot \nabla \, \phi