harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\phi(i_{\theta_j}(f)) \ge \phi\big(i_{\theta_j}\big(\sqrt{f}\big) s_{\theta_j}s_{\theta_j}^*i_{\theta_j}\big(\sqrt{f}\big)\big) = \phi\big(s_{\theta_j}^*i_{\theta_j}\big(\sqrt{f}\big)i_{\theta_j}\big(\sqrt{f}\big) s_{\theta_j}\big) = \phi(s_{\theta_j}^*i_{\theta_j}(f) s_{\theta_j}) = \phi(i_{\theta_j}(f \circ R_{-\theta_j}).
A_{r, k}^2(x) = \big(Q_{r-k-1}(x)\big)^2\cdot\frac{1-x^2}{2(2r-2k-1)}- \sum \limits_{n = r - k}^{r-1}{\big(Q_n^{(n+k-r)}(x)\big)^2\big(n + \frac 12\big)},
\bigg \|S^{n_k}\bigg(\sum_{i=s_{2k+1}}^\infty x_i e_i \bigg)\bigg\|\leq \bigg \| \sum_{i=s_{2k+1}}^\infty 2x_i e_i \bigg\| \leq \bigg \| \sum_{i=N}^\infty 2x_i e_i \bigg \|< \frac{\epsilon}{2}.
\mathcal Q_3^{\rm{low}}- \mathcal Q_3^{\rm{soft}}= \frac{1}{\vert \Lambda \vert}\sum_{\substack{k \in \mathcal P_H^c , k \neq 0 p \in \mathcal P_L}}\widehat g(k) \big( a_0^\dagger a_p^\dagger a_{p-k}a_k + h.c. \big) .
\int_{\gamma^*}^u G_j(v) \: dv = \int_{\gamma^*}^u \big( s(j -{\bar M}(v)) + s ({\bar M}(v) - M(v)) - \mu \big) \: dv &= \sum_{\ell = i}^{j + K - 1}\int_{\gamma_{\ell}\wedge u}^{\gamma_{\ell + 1}\wedge u}\big( s(j - \ell) + s ({\bar M}(v) - M(v)) - \mu \big) \: dv.
b_d(x,\beta)=-{7\pi^2 r_d \over 720\beta_l^4}+{r_d \over 48 \beta_l^2}\left[{1 \over 12}R+\frac 12 (\nabla w -w^2)+m^2\right]+{a_{d,2}(x) \over 16\pi^2}\ln{\mu\beta_l \over 2\pi}~~~,
g(p)=\begin{cases}\frac{2(1+\chi_D(p))}{p}+ O(1/p^2), \beta(p)=0, \frac{2}{p}+O(1/p^2), \beta(p)=1, \frac{3}{p}+O(1/p^2), \beta(p)=2k k\geq1, \frac{3}{p}+O(1/p^2), \beta(p)=2k+1 k\geq1,\end{cases}
\begin{array}{lcl}\Phi_{m+1}= \Phi_m+\Psi_m~,{\rm where}\Psi_m := (-1)^mq^{A_m}+(-1)^{m+1}D_{m+1}-(-1)^mD_m = (-1)^mq^{A_m}(q^{k+m-1/2}(1+q^{k+m+1/2})-q^{k+m}(1+q^{k+m-1/2}))/S_m = (-1)^mq^{A_m+k+m-1/2}(1-q^{1/2})(1-q^{k+m})/S_m~.\end{array}
\langle\lambda,\mu;l+\lambda,m-\mu|lm\rangle=(-1)^{\lambda+\mu}(\begin{matrix}l+\lambda-m+\mu\\ \lambda+\mu\end{matrix})^{1/2}(\begin{matrix}l+\lambda+m-\mu\\ \lambda-\mu\end{matrix})^{1/2}(\begin{matrix}2l+2\lambda+1\\ 2\lambda\end{matrix})^{-1/2}.
\mathsf{E}_k(x^*,k_0,f,g):=f\cdot\left\{\Vert x^0-x^*\Vert^2+\left[\mathsf{I}(x^*,k_0)L^2+B(x^*)^2\right]\mathsf{H}_\tau\mathsf{a}_0^k+g\mathsf{G}_\tau\mathsf{C}(x^*)^2\mathsf{b}_0^k\right\},
C(u)=\sum_{i=1}^{k}\frac{N_{i,n}(u)w_{i}}{\sum_{j=1}^{k}N_{j,n}(u)w_{j}}P_{i}=\frac{\sum_{i=1}^{k}N_{i,n}(u)w_{i}P_{i}}{\sum_{i=1}^{k}N_{i,n}(u)w_{i}}
\big<\Lambda g,\Lambda g\big>_{\xi}=\big|\nabla_{\xi}\cdot\big[\sigma\nabla_{\xi}g\big]\big|^{2}_{L^{2}_{\xi}}+|\psi(\xi)g|^{2}_{L^{2}_{\xi}}&+2\big<\psi(\xi),\left( \nabla_{\xi}g, \sigma\nabla_{\xi}g\right)\big>_{\xi}+2\big<g ,\left(\nabla_{\xi}\psi(\xi), \sigma\nabla_{\xi}g\right)\big>_{\xi}\,.
(0^{(k_1)}, 1^{(t_1)}, \ldots,0^{(k_s)}, 1^{(t_s)}):=(\underbrace{0, \ldots, 0}_{k_1},\underbrace{1, \ldots, 1}_{t_1}, \ldots, \underbrace{0, \ldots, 0}_{k_s}, \underbrace{1, \ldots, 1}_{t_s}),
\begin{array}{rcl}\hat{Q}_8^{(iv)}(x)\!\! \!=\! \!\! 215040 x^4-322560\pi x^3+(172800\pi^2-1036800) x^2 begin{align*}1.0 ex]\!\! \! \! \!\! +(-38400\pi^3+714240\pi) x+2880\pi^4-114048\pi^2+96768.\end{array}
S_0=\int d^Dx\left[\pi^i\dot A_i-{1\over2}{\pi^i}^2-{1\over4}F_{ij}^2+i\bar\psi\gamma^\mu D_\mu\psi+\lambda^1(J^0+\partial_i\pi^i)+\lambda^2(\partial_i A^i)\right]\,,
c^{\chi}_{i_1,\ldots,i_k}=\prod_{i_p\in \{n,\ldots,1\}}q^{2i_p-1}s_1 \prod_{\overline{i_p}\in\{\bar 1,\ldots,\bar n\}}q^{-2i_p+1}s_1^{-1}\prod_{j=1}^r \frac{q^{-j+1}s_1-q^{j-1}s_1^{-1}}{q^j-q^{-j}}\,.
\begin{array}{l}\dot{x}= -y+k_1 x^2+k_2 x y+ x \left((k_3+k_4)x^2+(k_3-k_4)y^2+2 k_5xy \right), \dot{y}= x+k_1 xy+k_2 y^2+ y \left((k_3+k_4)x^2+(k_3-k_4)y^2+2 k_5xy \right).\end{array}
\alpha(q'_{j,i},q'_{j+1,i}) - \alpha(q'_{j,i},q'_{j+2,i-1})- \alpha(q'_{j+1,i-1},q'_{j+1,i}) + \alpha(q'_{j+1,i-1},q'_{j+2,i-1}) &=\alpha(q_{j,i},q_{j+1,i}) - \alpha(q_{j,i},q_{j+2,i-1})- \alpha(q_{j+1,i-1},q_{j+1,i}) + \alpha(q_{j+1,i-1},q_{j+2,i-1})
U=\frac{-Q_{1}}{2\pi\epsilon}\int d\rho\rho\lambda(\rho,\theta)ln\rho+\frac{1}{2\pi\epsilon}\sum_{k=1}^{\infty}\int d\theta d\rho[C_{1k}\frac{cosk\theta}{\rho^{k-1}}+S_{1k}\frac{sink\theta}{\rho^{k-1}}]\lambda(\rho,\theta)
\begin{aligned}|\nabla B|^2 \leq 4B\sum_{i,k=1}^nh_{iik}^2 &=4B\bigl(\dfrac{n}{n+2}\sum_{k=1}^nh_{kkk}^2+\dfrac{2}{n+2}\sum_{k=1}^nh_{kkk}^2+\sum_{i\neq k}h_{iik}^2\bigl) &\leq \dfrac{4n}{n+2}B\bigl(\sum_{k=1}^nh_{kkk}^2+3\sum_{i\neq k}h_{iik}^2\bigl).\end{aligned}
\Sigma^{-1}C(\tilde{k}^1) \stackrel{\tilde{K}}{\longrightarrow}\Sigma J(\widetilde{W}^3_n) \longrightarrow C( \tilde{K}), J(W^1_n) \stackrel{k}{\longrightarrow}\Sigma J(W^2_n) \longrightarrow C(k).
\mathcal{H}_\varepsilon^\alpha\left(E\right) \leq (2nC_1)^\alpha \sum_{j=m}^\infty \left( M^{n-\alpha}(2\mu)^\alpha \right)^m = (2nC_1)^\alpha \frac{\left( M^{n-\alpha}(2\mu)^\alpha \right)^m}{1- M^{n-\alpha}(2\mu)^\alpha}.
\Vert z_i - a_i \Vert \le \Big \Vert z_i - \frac{y_i}{1+14\rho}\Big \Vert + \Big \Vert \frac{y_i}{1+14\rho}- y_i \Big \Vert + \Vert y_i -a_i \Vert & < \Bigl( 1- \frac{1}{1+14\rho}\Bigr) + \Vert y_i\Vert \Bigl( 1- \frac{1}{1+14\rho}\Bigr) +86\rho & < 28\rho +86\rho = 114\rho <\varepsilon.
\begin{aligned}f(x) \,\to\,\min\limits_{x,y,z}g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 j\in\mathcal P G_l(x)-y_l \,\leq\,0 l\in\mathcal Q H_l(x)-z_l \,\leq\,0 l\in\mathcal Q 0\,\leq\,y_l\,\perp\,z_l \,\geq\,0 l\in\mathcal Q \end{aligned}
\frac{d}{dt}\left(t^5 \left(z_{k}^- \right)^2\right) =2 t^5 z_{k}^- \frac{d}{dt}z_{k}^-+ 5 t^4 \left(z_{k}^- \right)^2 & \leq - 2 t^5 c_k \left(z_{k}^- \right)^2 + \frac{C C^*}{t^{\frac 12}}\leq -2 \overline c t^5 \left(z_{k}^- \right)^2 + \frac{CC^*}{t^{\frac 12}}.
\sum_{j=-N}^{N}\int_{\mathbb{R}^6}F(y,x')F(x,x')F(y,y')F(x,y') begin{align*}-1ex] |\Phi|_{2^j}(x'-p,y'-p) \omega_{2^j}(x-q)\omega_{2^j}(y-q) \,dx dy dx' dy' dp dq.
\left|{\bf p}\right|^{v^\prime v^{\prime\prime}}=\vert{\bf F}\vert\alpha_{0v}\int\tilde{\psi}^\ast_{v^\prime}\tilde{\psi}_{v^{\prime\prime}}dx+\vert{\bf F}\vert\alpha_{v}^1\int x\tilde{\psi}^\ast_{v^\prime}\tilde{\psi}_{v^{\prime\prime}}dx~.
L_{\rm S}= \mbox{}- \frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}- \frac{m^{2}}{2}A_{\mu}A^{\mu}- \frac{1}{2}\partial_{\mu}B\partial^{\mu}B - \frac{m^{2}}{2}B^{2},
\lim\limits_{\Delta\rightarrow 0}\frac{1}{\Delta^{\frac{n-1+p}{p}}}|H_p^n(\Delta)|_n= \frac{1}{n}\left(\frac{p}{p-1}\right)^{\frac{n-1+p}{p}}|B_p^{n-1}|_{n-1}-\frac{1}{n-1+p}\left(\frac{p}{p-1}\right)^{\frac{n-1+p}{p}}|B_p^{n-1}|_{n-1}= \frac{p-1}{n(n-1+p)}\left(\frac{p}{p-1}\right)^{\frac{n-1+p}{p}}|B_p^{n-1}|_{n-1}.
C_D = \tanh(\beta_1(a))\Re(z_1\mu_{\beta_1}) +\coth(\beta_1(a))\Im(z_1\mu_{\beta_1}) & +\tanh(\beta_2(a))\Re(z_2\mu_{\beta_2}) +\coth(\beta_2(a))\Im(z_2\mu_{\beta_2}).
C ( n , \tau ) = \left( \theta_{3}^{4}( 0 , \tau ) - \theta_{4}^{4}( 0 , \tau ) - e^{i \pi n}\theta_{2}^{4}( 0 , \tau ) \right) \left( \bar{\theta}_{3}^{4}( 0 , \tau ) - \bar{\theta}_{4}^{4}( 0 , \tau ) - e^{i \pi n}\bar{\theta}_{2}^{4}( 0 , \tau ) \right)
c\sum\limits_{k=0}^{\infty}2^{(n-d+a)k}\left\Vert \frac{1}{\left\Vert \chi_{P}\right\Vert_{p(\cdot )}^{r}}\sum\limits_{h\in \Omega_{k}}\delta^{-\frac{r}{q\left( \cdot \right)}}2^{v(\alpha \left( \cdot \right)+n/2)r}\left\vert \lambda_{v,h}\right\vert^{r}\chi_{v,h}\chi_{Q(c_{P},2^{k-v_{P}+4})}\right\Vert_{p(\cdot )/r}
{}(v_n^{a_n}\otimes \cdots \otimes v_1^{a_1}) \mathbf{G}_{\sigma}= ( v_n^{a_n}\otimes \cdots \otimes v_1^{a_1}) \sigma = v_{i_n}^{a_{i_n}}\otimes \cdots \otimes v_{i_1}^{a_{i_1}}
\sum_{v \in V(D_j)}\frac{\mathbf{Q}(L_w[0,j]=\gamma[0,j], L_w(j+1)=v)}{\mathbf{Q}(L_{v_0}[0,j]=\gamma[0,j])}=\frac{\mathbf{Q}(L_w[0,j]=\gamma[0,j])}{\mathbf{Q}(L_{v_0}[0,j]=\gamma[0,j])}=M_j.
{\tau}_{\mathrm{I n e r t}}^{p \to n}( a ) = \frac{2 \pi^{3 / 2}e^{\pi \frac{{\Delta m}}{a}}}{G_{F}^{2}m_{e}}\left[ G_{1 \; 3}^{3 \; 0}\left( \frac{m_{e}^{2}}{a^{2}}\left| \begin{array}{l}{{1}}\\{{- \frac{1}{2}\; , \; \frac{1}{2}+ i \frac{\Delta m}{a}\; , \; \frac{1}{2}- i \frac{\Delta m}{a}}}\\ \end{array}\right. \right) \right]^{- 1}\; ,
\sum_{n\ne0}\widehat{\Delta}(an)^{2}\le\sum_{n \in \mathbb{Z}}\widehat{\Delta}(an) - 1 =\frac{1}{a}\sum_{m\in\mathbb{Z}}\Delta\left(\frac{m}{a}\right)-1=\frac{1}{a}-1<\frac{1}{a}.
\mathrm{Z}_{\mathrm{R}}^{I}( r ) = \frac{1}{2}\mathrm{T r}_{\mathrm{R}}\left( r^{2 L_{0}}\right) = \mathrm{T r}_{\mathrm{R}}\left( r^{2 L_{0}}{\frac{1 + \Omega}{2}}\right) ,
\frac{h(s)-h(s_0)}{s-s_0}= \frac{\int_0^1 |\phi_{s_0}'|^2 \frac{P_s-P_{s_0}}{s-s_0}- \lambda_{1,2}(\sigma_{s_0}) |\phi_{s_0}|^2 \frac{Q_s-Q_{s_0}}{s-s_0}\,dt}{\int_0^1 |\phi_{s_0}|^2 Q_s \,dt}
\vec{s}(t) = \sum_{k = -\infty}^{\infty}\left( \vphantom{\frac 12}\vec{x}[k] \, h\left(2 (t - kT)\right) \right. &+ \left. \vec{SI}\left(\vec{x}[k], \vec{x}[k+1], \frac 12\right) \, h\left(2 \left( t - \left(k + \frac 1 2 \right)T\right)\right) \right)
f(-q) :=f(-q,-q^2)=\sum_{n=-\infty}^{\infty}(-1)^nq^{n(3n-1)/2}=(q;q)_{\infty}, \varphi(-q) :=f(-q,-q)=\sum_{n=-\infty}^{\infty}(-1)^nq^{n^2}=\dfrac{(q;q)_{\infty}}{(-q;q)_{\infty}}, \psi(-q) :=f(-q,-q^3)=\sum_{n=0}^{\infty}(-q)^{n(n+1)/2}=\dfrac{(q^2;q^2)_{\infty}}{(-q;q^2)_{\infty}}.
|\nabla|^{2\sigma}e^{i(t-s)\sqrt{1-\Delta}}P_k^2 F_{\rho_1}(\cdot,s) &=\int_{\mathbb{R}^n}\bigg(\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi+i(t-s)\sqrt{1+|\xi|^2}}|\xi|^{2\sigma}\phi(2^{-k}|\xi|)^2 d\xi\bigg)F_{\rho_1}(y,s) dy &:=\int_{\mathbb{R}^n}I_k(x-y,t-s)F_{\rho_1}(y,s) dy
\lim_{n\rightarrow\infty}h(\tilde{p}_n)^{\frac{1}{n}}\leq \lim_{n\rightarrow\infty}\left(-2(1-\tilde{p}_n)\log(1-\tilde{p}_n)\right)^{\frac{1}{n}}=\left(\frac{p_{12}p_{21}}{p_{11}p_{22}}\right)^{\frac 12}
P^{\textnormal{SIC}}_c\geq \frac{\eta}{1-\eta}\sum_{k=1}^K \frac{1}{(1+\theta)^{\frac{\beta}{2}k(k-1)}\Gamma(1+k\beta)}\left(\frac{1-\eta}{\theta^\beta\Gamma(1-\beta)}\right)^k,\; \forall K\geq 1,
\mathcal F(s,z)= (1+s)^{n-1}\bigg(1+\frac{|z|^2}{2(1+s)^2}\Biggr)+O\big(|z|^4\big)=1+(n-1)s+\frac{(n-1)(n-2)}{2}s^2 +\frac{|z|^2}{2}+O\big(|s|^3+|z|^3\big),
0\longrightarrow{\cal O}_{{\cal F}_L}\longrightarrow\oplus_{l\in L}{\cal O}_{{\cal G}_{(\emptyset,\{l\})}}\longrightarrow\oplus_{\stackrel{L'\subset L}{|L'|=2}}{\cal O}_{{\cal G}_{(\emptyset,L')}}\longrightarrow\ldots\longrightarrow{\cal O}_{{\cal G}_{(\emptyset,L)}}\longrightarrow 0
Z[J]=e^{i\int d^{d}x(L[B(x)]+J(x)B(x))}\int D\eta e^{\frac{i}{2}\int d^{d}xd^{d}y\frac{\delta^{2}L}{\delta\phi(x)\delta\phi(y)}[B]\eta(x)\eta(y)+\cdot\cdot\cdot}.
\alpha_{i,\varpi_p}(\xi,m)=\eta \Leftrightarrow\vee_{j\in J}(\phi_{ij}(\xi)\wedge\eta_{ij}(\eta,m)); \beta_{i,\varpi_p}(\xi,m)=\nu \Leftrightarrow\vee_{j\in J}(\theta_{ij}(\xi)\wedge\nu_{ij}(\nu,m)); (\xi,m,\varsigma)\in V_{i,\varpi_{p}}\Leftrightarrow\vee_{j\in J}(\varsigma_{ij}(\varsigma,\xi)\wedge\tau_{ij}(m)).
|h(t)-h(s)|^{2}\leq C_{2}(\gamma^\prime)(t-s)^{2\gamma^\prime-1}|h|_{W^{\gamma^\prime,2}}^{2}= C_{2}(\gamma^{\prime})(t-s)^{2\gamma^\prime-1}\int\int_{[s,t]^2}\frac{|h(u)-h(v)|^2}{|u-v|^{1+2\gamma^\prime}}\, \mathrm{d}u \, \mathrm{d}v.
{\mathbf[ \prod_{r=3}^{5}2 \sin ( \pi kv_r ) ] \sum_{i=3}^{5}\frac{Tr\gamma_{k,5_i}}{2\sin ( \pi kv_i)}- \sum_{i=3}^{5}[ 2\sin (\pi kv_i) ] Tr\gamma_{k,7_i}=0},
A_{k}\left( \mathbf{v},\widehat{\mathcal{N}}(\mathbf{w},\mathbf{h})\right)=\left(\kern-.1em\left(\kern-.1em\mathbf{v},\mathbf{w}\kern-.1em\right)\kern-.1em\right) -\operatorname*{i}kb_{k}\left( \mathbf{v}^{\operatorname*{curl}},\mathbf{h}^{\operatorname*{curl}}\right) \qquad\forall\mathbf{v}\in\mathbf{X}.
(\partial^{p,q}_{{\rm{ver}}}\sigma)(i_0, \dots, i_p, j_0, \dots, j_{q+1}) = \sum_{r=0}^{q+1}(-1)^r\sigma(i_0, \dots, i_p, j_0, \dots, \widehat{j_r}, \dots, j_{q+1})|_{U_{i_0\dots i_pj_0\dots j_{q+1}}}.
H_{c}^{r}= \int \left[ ( \pi_{a}\pi_{a}) ( \phi^{b}\phi^{b}) - ( \phi^{a}\pi_{a}) ( \phi^{b}\pi_{b}) - \phi^{a}\partial_{i}\partial_{i}\phi^{a}{\frac{1}{\phi^{b}\phi^{b}}}\right] d x .
\beta\gamma =(\rho\tau)(m,d(\eta)\vee m)\,\lambda(d(\eta)\vee m-d(\eta),d(\lambda)) &=(\eta\lambda)(m,d(\eta)\vee m)\,\lambda(d(\eta)\vee m-d(\eta),d(\lambda)) &=(\eta\lambda)(m,d(\eta)\vee m)\,(\eta\lambda)(d(\eta)\vee m,d(\eta\lambda)) &=(\eta\lambda)(m,d(\eta\lambda)).
\Psi_{\gamma+t}(x) =e^{i\left\langle \gamma+t,x\right\rangle}+\sum\limits_{\delta\in\Gamma(k+)}c(\gamma,\delta)e^{i\left\langle\gamma+\delta+t,x\right\rangle}= & e^{i\left\langle \gamma+t,x\right\rangle}+\sum_{p\in\mathbb{N}}\sum\limits_{\delta\in\Gamma(k,p)}c(\gamma,\delta)e^{i\left\langle\gamma+\delta+t,x\right\rangle},
h_{1,j'-1}(\theta^{q^N})\biggl(\prod_{\substack{l=1 l\neq m_a}}^{d-1}(\theta^{q^{l+N-m_u}}- \theta^{q^N})^{c(l)q^{d-l}}\alpha_1^{q^{n+d}}\biggr)+\cdots & \cdots+h_{r-j'+1,j'-1}(\theta^{q^N})\biggl(\prod_{\substack{l=1 l\neq m_u}}^{d-1}(\theta^{q^{l+N-m_u}}-\theta^{q^N})^{c(l)q^{d-l}}\alpha_{r-j'+1}^{q^{n+d}}\biggr)=0.
\begin{aligned}\frac{d}{dt}\int |\nabla^2 B|^2dx+\int |\nabla^3 B|^2 dx &\lesssim \|\nabla (u,B)\|_{H^1}^2\|\nabla^2 (u,B)\|_{L^2}^2 +\|\nabla^2 \varrho\|_{L^2}^2\|\nabla B\|_{H^1}^2 &\lesssim (1+t)^{-\frac{5}{2}}(1+t)^{-\frac{5}{2}}+(1+t)^{-\frac{5}{2}}(1+t)^{-\frac{5}{2}}&\lesssim (1+t)^{-5}.\end{aligned}
d_{j}^-(\lambda)=\frac{(\lambda_j+1)j(\lambda_j+n-j+[\lambda_j]_2(p-n))}{(-1)^{\sum_{\alpha=j}^n(\alpha+1)(\lambda_\alpha-\lambda_{\alpha+1})}}\bigg(\prod_{\ell=j+1}^n\frac{\lambda_j-\lambda_\ell-j+\ell-1}{\lambda_j-\lambda_\ell-j+\ell-[\lambda_j-\lambda_\ell]_2}\bigg).
S^{- 1}\left( p \right) = Z_{2}\left( p / - m_{0}\right) - i Z_{1}e^{2}\int^{\Lambda}\frac{d^{4}k}{\left( 2 \pi \right)^{4}}\gamma^{\mu}S \left( k \right) \gamma^{\nu}D_{\mu \nu}\left( k - p \right)
V \Big [ \begin{array}{c}{{\alpha + n b_{3}+{m}b_{4}}}\\{{\beta + \tilde{n}b_{3}+ \tilde{m}b_{4}}}\\ \end{array}\Big ] = V \Big [ \begin{array}{c}{{\alpha + n b_{3}+{m}b_{4}+ n_{1}b_{1}+ m_{1}b_{2}}}\\{{\beta + \tilde{n}b_{3}+ \tilde{m}b_{4}+ n_{2}b_{1}+ m_{2}b_{2}}}\\ \end{array}\Big ] ,
\begin{aligned}\partial_t L_\tau U_{\tau,h}(t_{n,0}) + \mathcal A_h U_{\tau,h}(t_{n,0}) - \mathcal P_h I_\tau^{\operatorname{GL}}F(t_{n,0}) &= \partial_t L_\tau U_{\tau,h|I_{n-1}}(t_{n-1}) + \mathcal A_h U_{\tau,h|I_{n-1}}(t_{n-1}) - \mathcal P_h I_\tau^{\operatorname{GL}}F(t_{n-1}) = 0 \,.\end{aligned}
\sigma^+_1 - \sigma^+_{-1}= \sigma_{1}^- - \sigma^-_{-1}+ 2\left[ g(J^+_*) - g(J^-_*) \right]\delta &= \sigma_{1}^- - \sigma^-_{-1}+ 2 g'(s) \left[ J^+_* - J^-_* \right]\delta
C_{\mu}^{ab}= 2ig \, \overline{\psi_A^a}\, g^{AB}\psi_C^b \,{A_{\mu B}}^C + \left( \, \overline{\psi_A^a}\, g^{AB}\, (\partial_{\mu}\psi_B^b) + \psi_B^b \, g^{AB}\, (\partial_{\mu}\overline{\psi_A^a}) \, \right),
\hat{H}_{I}^{( q )}( \hat{x}, \hat{p}) = \frac{f^{2}\hat{x}^{2}}{6}\bigl \{\hat{x}V^{\prime}( \hat{x}) \partial_{\hat{x}}^{2}+ \bigl [ \hat{x}V^{'^{\prime}}( \hat{x}) + 3 V^{'}( \hat{x}) \bigr ] \partial_{\hat{x}}+{\textstyle \frac{1}{3}}\hat{x}V^{'^{\prime \prime}}( \hat{x}) +{\textstyle \frac{3}{2}}V^{'^{\prime}}( \hat{x}) \bigr \}.
\varpi_{j*}\circ \tilde{\tau}_j^s(1_{Z_j})= \begin{cases}\phi_j\left( t_j^{\lambda_j+s}\dfrac{\prod_{i=1}^{j-1}( 1-\bar t_i/\bar t_j)}{\prod_{i=1}^{\gamma_j}(1 - t_i/\bar t_j)}\right) (\chi_j<0), \phi_j\left( \dfrac{t_j^{\lambda_j+s}}{2+\beta t_j}\dfrac{\prod_{i=1}^{j-1}( 1-\bar t_i/\bar t_j)}{\prod_{i=1}^{\gamma_j}(1 - t_i/\bar t_j)}\right) (0\leq \chi_j),\end{cases}
\gamma_{n-1}\left( B_{n}B_{n-1}-A_{n-1}\partial_{x}B_{n}\right)+xB_{n}+1 & =\frac{\left( n-2\gamma_{n}\right) \left[ 2x^{2}\phi-\left( \phi+nx^{2}-2x^{2}\phi-2x^{2}\gamma_{n}\right) C_{n}\right]}{4\gamma_{n}C_{n-1}C_{n}^{2}}+1.
A^{a, m}_{i,\sqrt{T}}= \left({\sum_{j, i \in j}\left(\frac{a(j)}{\hat a(i)}m^{*\sqrt{T}}\right)_{\sharp -e_{j^c}}}\right)^{*\frac{1}{\sqrt{T}}}, \, \, \, \, \, A^{a, m}_{-i,\sqrt{T}}=\left( \sum_{j, i \notin j}\left(\frac{a(j)}{\hat a(-i)}m^{*\sqrt{T}}\right)_{\sharp e_{j}}\right)^{*\frac{1}{\sqrt{T}}}.
(\sum_{\atop{l(\tau)=N^{-1/4}}}\|E_{\tau}g_i\|_{L^{p/2}(w_{B})}^2)^{\frac1{2}}\le (\sum_{\atop{l(\tau)=N^{-1/4}}}\|E_{\tau}g_i\|_{L^{2}(w_{B})}^2)^{\frac{1-\kappa_p}{2}}(\sum_{\atop{l(\tau)=N^{-1/4}}}\|E_{\tau}g_i\|_{L^{p}(w_{B})}^2)^{\frac{\kappa_p}{2}}.
\sigma(x) =\sup_{}\left\{t\geqslant 0\,|\,x+t\Vec{u}\in C \right\}&=\sup_{}\left\{t\geqslant 0\,|\,x+t\tau_C(x,y)(y-x)\in C \right\}&=\frac{1}{\tau_C(x,y)}\sup_{}\left\{t'\geqslant 0\,|\,x+t'(y-x)\in C \right\}&=\frac{1}{\tau_C(x,y)}\tau_C(x,y) &=1.
X_0(j+1,k) =\|\partial_t^{j+1}\vec{h}|_{t=0}\|_k &=\|-\partial_t^j\mathfrak{A}\vec{h}+\partial_t^j\vec{f}|_{t=0}\|k &=\|-\mathfrak{A}(\partial_t^j\vec{h})+[\partial_t^j,\mathfrak{A}]\vec{h}+\partial_t^j\vec{f}|_{t=0}\|_k &\leq X_0(j,k+1)+W_0(\vec{h};n)+W_0(\vec{f};n),
\tilde{T_{a}}\Theta_{k}\cong \Theta_{k}+ \frac{1}{M}C^{l}\left[ \begin{array}{c c c}{D_{l}^{( 0 )}}&{0}&{0}\\{0}&{D_{l}^{\ast ( 0 )}}&{0}\\{0}&{0}&{\frac{\partial}{\partial s^{l}}}\\ \end{array}\right] \Theta_{k},
\eta^{1/2}(\breve T_2-T_2)-{\textstyle{1\over 6}}(\breve \kappa_3-\kappa_3)\eta T_1^2 =O_p(n^{-1}), (\breve \kappa_1-\kappa_1)-{\textstyle{1\over 6}}(\breve \kappa_3-\kappa_3) =O(n^{-1}).
\begin{array}{rclrcl}T_{1}= M_{1}\, ,\hspace{1cm}\tilde{T}_{1}= R_{2}P_{3}\, , & T_{2}= R_{2}P_{1}\, ,\hspace{1cm}\tilde{T}_{2}= R_{2}P_{2}\, , & T_{3}= R_{2}P_{0}\, ,\hspace{1cm}\tilde{T}_{3}= M_{2}\, , \end{array}
\sum_{j=1}^{\left\lfloor (k+1)/2\right\rfloor}w_{k+2-2j,k+1}- \sum_{j=1}^{\left\lfloor k/2\right\rfloor}w_{k+1-2j,k+1}=\sum_{j=1}^{\left\lfloor (k+1)/2\right\rfloor}(w_{k+2-2j,k+1}- w_{k+1-2j,k+1}) &\geq 0,
\{\lambda\}=\{-(\frac{\epsilon_{0}}{2}E^{2}+\frac{1}{2\mu_{0}}B^{2}),\pm\sqrt{(\frac{\epsilon_{0}}{2}E^{2}-\frac{1}{2\mu_{0}}B^{2})^{2}+\frac{\epsilon_{0}}{\mu_{0}}(E\cdot B)^{2}}\}
S_{1,k}(n,s,t) :=\sum_{\ell = 0}^{\infty}\frac{1}{(s + \ell +2)^{1 + d}(t+1)^d}\Vert \tilde{b}^k_{n,1,\ell}\Vert, S_{2,k}(n,s,t) := \sum_{\ell = 0}^{\infty}\sum_{u=0}^{t-1}\frac{1}{(s + \ell +2)^{1+d}(u + 1)^d}\Vert \tilde{b}^k_{n,t-u,\ell}- \tilde{b}^k_{n,t+1-u,\ell}\Vert.
(x+fy)(u+fv) = \big(x_0u_0 + c \sigma(x_1)u_1 + \tau(y_0) v_0 d+ c \sigma \tau(y_1)v_1 d\big) &+ e \big( x_1u_0 + \sigma(x_0)u_1 + \tau(y_1)v_0 d+ \sigma \tau(y_0)v_1d \big) &+ f \big( y_0 u_0 + c \sigma(y_1)u_1 +\tau(x_0)v_0 + c \sigma\tau(x_1)v_1 \big) &+ fe \big( y_1u_0 + \sigma(y_0)u_1 +\tau(x_1)v_0 + \sigma\tau(x_0) v_1 \big).
\frac{d}{dt}\int_M |\dot\phi^\epsilon_t|^2 (\omega_{\phi^\epsilon_t})^n\wedge \eta= 2\int_M \dot \phi^\epsilon_t (\ddot \phi^\epsilon_t-|\nabla \dot\phi^\epsilon_t|_{\phi_t^\epsilon}) (\omega_{\phi^\epsilon_t})^n\wedge \eta = 2\epsilon \int_M \dot\phi^\epsilon_t (\omega^T)^n\wedge \eta
\tau_1 := \inf\{t \geq 0: s_1, s_2\in [t, t+1], (B_1(s_1) - B_2(s_1))(B_1(s_2) - B_2(s_2)) \geq 0 \}\tau_2 := \inf\{t > \tau_1 : B_1(t) = B_2(t)\}.
\log\mathfrak{F}(t_{1},\ldots,t_{n})=-\sum_{\ell=1}^{n-1}\,\sum_{m\in\mathbb{N}^{\ell}}\,\alpha(m)\sum_{k=1}^{n-\ell}\,\prod_{j=1}^{\ell}\left(t_{k+j-1}t_{k+j}\right)^{m_{j}}.
U^{(pq)}_h = \frac{1}{{n \choose k}}\sum_{\mathbf{i}\in\mathcal{P}(n,k)}h\left(\mathbf{X}^{(pq)}_{\mathbf{i}, 1}, \dots, \mathbf{X}^{(pq)}_{\mathbf{i}, k}\right) = \frac{1}{{n \choose k}}\sum_{\mathbf{i}\in\mathcal{P}(n,k)}h\left(\mathbf{X}^{(pq)}_{\mathbf{i}}\right).
\tilde{Y}_{1}\hat{R}d K_{1}\hat{R}= \hat{R}d K_{1}\hat{R}\tilde{Y}_{1}\quad \Longleftrightarrow \quad \hat{R}d K_{1}^{\dagger}\hat{R}Y_{1}= Y_{1}\hat{R}d K_{1}^{\dagger}\hat{R};
\int_M \sum_a V_a\langle \nabla_{\bar V_a}\phi,\psi\rangle =\frac12\int_M{\rm div}_\nabla Y =\int_M\langle\nabla_{H^{0,1}}\phi,\psi\rangle, \int_M \sum_a \bar V_a\langle \phi,\nabla_{\bar V_a}\psi\rangle =\frac12\int_M{\rm div}_\nabla Z =\int_M\langle\phi,\nabla_{H^{0,1}}\psi\rangle.
\mathcal{A}(i)= \big\{\mu_1 C_{S1}(i), \mu_2 C_{S2}(i),...,\mu_M C_{SM}(i), & (1-\mu_1) C_{1D}(i),(1-\mu_2) C_{2D}(i),..., (1-\mu_M) C_{MD}(i) \big\}.
\int_{x_i}^{x_{i+1}}\hat{m}(y_i)C_{t}(y,x')_{ij}dy_i =-A_t(x,x')_{i+1j}+A_t(x,x')_{ij}, \int_{x_i}^{x_{i+1}}\hat{m}(y_i)D_t(y,y')_{ij}dy_i =-B_t(x,y')_{i+1j}+B_t(x,y')_{ij}+\hat{m}(y_j')\textbf{1}\left(j=i\right),
P\left( \mathcal{E}^{\tau_{z,m}}(x) = j \mid \mathcal{E}^{\tau_{z,m}}(x-1)=i \right) = P\left( \zeta_1 = j \mid \zeta_0 = i \right),\ \z<x<0, \ \ P\left( \mathcal{E}^{\tau_{z,m}}(x) = j \mid \mathcal{E}^{\tau_{z,m}}(x-1)=i \right) = P\left( \tilde\zeta_1 = j \mid \tilde\zeta_0 = i \right), \ \x > z\vee 0.
b_i^{2}= [2] b_i, \mspace{40mu}b_ib_j =b_jb_i \mspace{10mu}\mspace{10mu}\vert i-j\vert>1, \mspace{40mu}b_ib_{i+1}b_i + b_{i+1}= b_{i+1}b_ib_{i+1}+ b_{i}, \rho\rho^{-1}=1=\rho^{-1}\rho,\mspace{40mu}\rho b_i \rho^{-1}= b_{i+1},
{\it \Pi}_i(t,\omega)\,\equiv \,{\mu}_i(t,\omega)\,:=\, \frac{\omega_i(t)}{\,\omega_1(t) + \cdots + \omega_n(t)\,}\,, \forall \,\,\,\, i=1, \cdots, n\,, \,\,\,0 \le t<\infty\,,
|\nabla f_0(x)|^2 \le \frac{1}{f_0(x)^2}\bigg|\sum_{y\in\pi^{-1}(x)}f(y)\pi_*(\nabla f(y)) \bigg|^2 \le \frac{1}{f_0(x)^2}\sum_{y\in\pi^{-1}(x)}f(y)^2\sum_{y\in\pi^{-1}(x)}|\nabla f(y)|^2 = \sum_{y\in\pi^{-1}(x)}|\nabla f(y)|^2.
P_{co}(l_k) =1-\exp\left[-2\pi \lambda_J \int_{0}^{\infty}\left( \frac{1}{1+\frac{P_{S_k}/d_{S_k,D_k}^\alpha}{\gamma_C P_{\bar{J}}/r^\alpha}}\right)rdr\right] &=1- \exp\left(-A_{co}d_{S_k, D_k}^2 P_{S_k}^{-\frac{2}{\alpha}}\right).
\frac{\rho_R}{k}p_{k-2}=\sum\limits_{i=0}^{k-2}\rho_V^{i}\frac{\rho_R^{k-i-1}}{(k-2-i)!k}&<\sum\limits_{i=0}^{k-2}\rho_V^{i}\frac{\rho_R^{k-i-1}}{(k-1-i)!}<p_{k-1},
S(f;a_i) = 2N |x-a_i|^2 -4N (x-x_0)\cdot (x-a_i) -2N (r^2- |x-x_0|^2) = -2Nr^2 +2N |(x-a_i)-(x-x_0)|^2 = -2N r^2 +2N |a_i-x_0|^2 = 2N (|a_i-x_0|^2-r^2) =0,
B_*(t) = \begin{cases}\displaystyle \frac{n^{-1/2}\zeta(\max \{t, \sigma \})}{1 - 4 \, n^{-1/2}\zeta( \max \{t, \sigma \})}\bigl[ 6 + (\kappa-1)^{-1}\bigr] \zeta( \max \{t, \sigma \}) \leq \sqrt{n}+ \infty \end{cases}
\mu\left(\{x\in B:\ |b(x)-b_B|> t \}\right) \leq \mu\left(\{x\in B:\ |b(x)-b_B|> Ck\alpha \}\right) &\leq \sum_j \mu(B^{(k)}_j) \leq{1\over \alpha^k}\mu(B^{(0)})=\mu(B)e^{-k\log \alpha}&\leq \mu(B) \alpha e^{-{t\log (\alpha)\over C\alpha}},
H^{1,0}(Q_T) = \{v \in L^2(Q_T) : \nabla v \in [L^2(Q_T)]^d \}, H^{0,1}(Q_T) = \{v \in L^2(Q_T) : \partial_t v \in L^2(Q_T) \}, H^{1,1}(Q_T) = \{v \in L^2(Q_T) : \nabla v \in [L^2(Q_T)]^d, \partial_t v \in L^2(Q_T) \},
f_{\scriptscriptstyle{L , L}}( a ) = \frac{1}{2 \pi \, a^{L}}\int_{0}^{a}d r \, r^{L}\int_{0}^{2 \pi}d \theta \, e^{i ( 1 - L ) \theta}\, e^{2 \phi ( r , \theta )}\sum_{m = 0}^{\infty}b_{\scriptscriptstyle{m , L}}\, r^{m}\, e^{i m \theta}\, .
p_{cov}^{c}= \int^{R}_{R0}e^{-\frac{2\pi^2 \psi_0 p(r)\,r^{2}_{c}}{\alpha \sin(\frac{2 \pi}{\alpha})}\big(\frac{\gamma}{p_{c}}\big)^{\frac{2}{\alpha}}\mathbb{E}[{p^{\frac{2}{\alpha}}_{i}}]}\,\frac{2 r_{c}}{R^{2}}dr_{c}.
R(a) \ast_R R(b) =~ R^2 (a) \cdot R(b) + R(a) \cdot R^2 (b) =~ R (R(a) \ast_R b) + \kappa ~ \! R(a) \cdot b + R (a \ast_R R(b)) + \kappa ~ \! a \cdot R(b) =~ R \big( R(a) \ast_R b + a \ast_R R(b) \big) + \kappa ~ \! a \ast_R b.
\sup_{E\in[-\mathcal{K}, E_--N^{-\frac23+6\varepsilon}]}\Im m_H(E+\mathrm{i}\eta)= \sup_{E\in [-\mathcal{K}, E_--N^{-\frac23+6\varepsilon}]}\frac{1}{N}\sum_{i=1}^N \frac{\eta}{(\lambda_i-E)^2+\eta^2}\geq \frac{1}{N\eta}\,,
\dfrac{\partial i}{\partial r}= \dfrac{1}{1+\xi_2'(i) (g(t)-g_0)}:=\sqrt{A_2(g(t),i)}, -\dfrac{1}{g'(t)}\dfrac{\partial i}{\partial t}= \dfrac{\xi_2(i)}{1+\xi_2'(i) (g(t)-g_0)}:=C_2(g(t),i), \dfrac{\partial^2 i}{\partial r^2}= -\dfrac{\xi_2''(i)(g(t)-g_0)}{[1+\xi_2'(i) (g(t)-g_0)]^3}:=B_2(g(t),i),
\langle \:{\hat \phi}_{\bar t}\prod_{i \in S}\phi_i \:\rangle_{g, \:{\Delta_{2}}}\; = \; \frac{1}{2}\:\sum_{r=0}^{g}\:\sum_{X \cup Y = S}{\bar C}_{\bar t}^{\alpha \beta}\: \langle \phi_{\alpha}\: \prod_{j \in X}\phi_j \rangle_{r}\:\langle \phi_{\beta}\: \prod_{k \in Y}\phi_k \rangle_{g-r}
s(\Gamma)\equiv\int(\frac{\delta\Gamma}{\delta\varrho}\frac{\delta\Gamma}{\delta A}+B \frac{\delta\Gamma}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\sigma}\frac{\delta\Gamma}{\delta c}+\frac{\delta\Gamma}{\delta\bar{Y}}\frac{\delta\Gamma}{\delta\psi}+\frac{\delta\Gamma}{\delta\bar{\psi}}\frac{\delta\Gamma}{\delta Y}=0
\Rightarrow\int_{t_{1}}^{t_{2}}[\frac{1}{\rho}\frac{dp}{dt}]dt+\int_{t_{1}}^{t_{2}}[\frac{1}{2}\frac{dv^{2}}{dt}]dt+\int_{t_{1}}^{t_{2}}[g\frac{dz}{dt}]dt=0
\psi_{-}( \vec{x}) = \frac{1}{\sqrt{2}}\frac{1}{i \partial_{-}- e V_{-}}\ast \xi ( \vec{x}) - \frac{e}{\sqrt{2}}\frac{1}{\Delta_{\perp}}\ast ( J_{p z m}^{i}( x_{\perp}) - q_{i}) \alpha^{i}\frac{1}{i \partial_{-}- e V_{-}}\ast \psi_{+}( \vec{x}) \; ,
\|\varphi\otimes\psi\|_{C^{k}_{{\rm L}}(\overline{\Omega}\times \overline{\Omega})}:=\max_{j+l\leq k}\|{\rm L}^{j}\varphi\|_{L^2(\Omega)}\|{\rm L}^{l}\psi\|_{L^2(\Omega)}, k\in\mathbb N_0,\; \varphi, \psi\in C_{{\rm L}}^{\infty}(\overline{\Omega})