harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_1^1 =YW H_1^2 =ZW H_2^1 =XH_1^1 H_2^2 =XH_1^2+Y^2W H_3^1 =X^2H_1^1+Z^2W H_3^2 =X^2H_1^2+XY^2W=XH_2^2 H_4^1 =X^3H_1^1+XZ^2W+Y^2ZW+W^3 H_4^2 =X^3H_1^2+X^2Y^2W+YZ^2W &=XH_3^1+Y^2ZW+W^3 =XH_3^2+YZ^2W.
S_pS_p^*S_qS_q^*= S_pf_{\Delta}S_p^*S_qf_{\Delta}S_q^* = f_{\Delta_p}f_{\Delta_q}= f_{\Delta_p\cap\Delta_q}= \begin{cases}f_{\Delta_r}pP\cap qP = rP 0 pP\cap qP = \emptyset\end{cases},
\sup_{R\in\mathcal{D}_{\mu}}\Big(\frac{1}{|R|}\int_R{\sum_{k=\max{(0,\mu-5)}}^{\infty}{2^{(m+s)kq}\big| \Pi^*_kf(x)\big|^q}}dx \Big)^{1/q}&\lesssim\sup_{0\leq k\leq \mu-1}{\big\Vert 2^{k(s+m)}\Pi_kf\big\Vert_{L^{\infty}}}+\sup_{R\in\mathcal{D}_{\mu}}\Big(\frac{1}{|R|}\int_R\sum_{k=\mu}^{\infty}{2^{(s+m)kq}\big|\Pi_kf(x) \big|^q}\Big)^{1/q}
{u_2}_x= \epsilon \left( b_1(u_1){u_1}_{xx}+c_1(u_1) ({u_1}_{x})^2 \right) +\epsilon^2 \left( b_2(u_1){u_1}_{xxx}+c_2(u_1){u_1}_{x}{u_1}_{xx}+d_2(u_1) ({u_1}_{x})^3 \right) + \dots \, ,
R_{n+1}^+(x) =2nxR_{n}^+(x)+2x(1-2x)\frac{d}{dx}R_{n}^+(x)+R_{n}(x), R_{n+1}^-(x) =(2n+1)xR_{n}^-(x)+2x(1-2x)\frac{d}{dx}R_{n}^-(x)+xR_{n}(x), R_{n+1}(x) =(1+(2n+1)x)R_{n}(x)+2x(1-2x)\frac{d}{dx}R_{n}(x)+xR_{n}^-(x),
\lfloor\frac{p}{2}\rfloor \binom{q}{\lfloor\frac{p}{2}\rfloor}\left(\lfloor\frac{p}{2}\rfloor \right)^{p}\leq \sqrt{k}e^kq^kk^k =\sqrt{\frac{p}{2}}e^{\frac{p}{2}}q^{\frac{p}{2}}\left( \frac{p}{2}\right)^{\frac{p}{2}}\leq \frac{p}{2}e^{\frac{p}{2}}q^{\frac{p}{2}}\left( \frac{p}{2}\right)^{\frac{p}{2}}.
\big\langle \mathrm{E}(\Sigma,c) , (V,w) \big\rangle= \big\langle (V_g,w_g),(V,w) \big\rangle= \int_\Sigma \big( V_g - u_g c H \big) V \, \mathrm{d}\mathcal{H}^d + \int_\Sigma u_g w \, \mathrm{d}\mathcal{H}^d
\mathrm{div}_f( X) =\overline{\mathrm{div}}(Y), \int_\Sigma |X|^2 e^{-f}d\mathrm{vol}_\Sigma \leq \int_{\overline{\Sigma}}|Y|^2 d\mathrm{vol}_{\overline{\Sigma}}\int_\Sigma (\mathrm{div}_f)_- (X) e^{-f}d\mathrm{vol}_\Sigma = \int_{\overline{\Sigma}}(\overline{\mathrm{div}})_-(Y) d\mathrm{vol}_{\overline{\Sigma}}.
\mathcal{X}_k'=\begin{cases}\mathcal{X}_k \mbox{if}k<v \emptyset \mbox{if}k=v \mathcal{X}_{k-1}\mbox{if}k>v\end{cases}, ~~~\mathcal{Y}_k'=\begin{cases}\mathcal{Y}_k \mbox{if}k<v \mathcal{Y}\mbox{if}k=v \mathcal{Y}_{k-1}\mbox{if}k>v\end{cases},
\frac{\sum_{j=1}^s a_j (2t_j)^{b_j}}{\prod_{j=1}^s (1+t_j)^{p(s)}}+\frac{\log C(s)+\sum_{j=1}^s \log (5(1+t_j))}{\prod_{j=1}^s (1+t_j)^{p(s)}\log\omega^{-1}}\geq &\qquad\qquad\qquad\qquad\qquad\ge \left(1-\frac{1}{\prod_{j=1}^s (1+t_j)}\right)^{p(s)}\frac{\log q^{-1}}{\log \omega^{-1}}M(s)^{-p(s)}.
\max_{1 \leq k \leq n}\Big\|\sum_{i=k}^n X_i \Big\|_H \leq \max_{1 \leq k \leq n-1}\Big \|\sum_{i=1}^{k}X_i\Big\|_H + \Big \|\sum_{i=1}^{n}X_i\Big\|_H \leq 2\max_{1 \leq k \leq n}\Big \|\sum_{i=1}^{k}X_i\Big\|_H
\tilde Z_n^N(i) := \Delta_n^{-\frac 12}\left((p_N\tilde{\Delta}_i^nY)^{\otimes 2}-\langle\langle p_N\tilde{\Delta}_i^nY\rangle\rangle\right) = \Delta_n^{-\frac 12}\left((p_N\tilde{\Delta}_i^nY)^{\otimes 2}-\int_{t_{i-1}}^{t_i}p_N\mathcal S(t_i-s)\Sigma_s\mathcal S(t_i-s)^* p_N ds\right).
0= (\frac{\mathrm{d}}{\mathrm{d}t}g_t^* \sigma^* (g_t^{-1})^* \omega_{\varphi_0})|_{t=0}= \sigma^* (\frac{\mathrm{d}}{\mathrm{d}t}(g_t^{-1})^* \omega_{\varphi_0})|_{t=0}+ (\frac{\mathrm{d}}{\mathrm{d}t}g_t^* \sigma^*\omega_{\varphi_0})|_{t=0}& = \sqrt{-1}\partial \bar{\partial}[( f+\bar{f}) - ( f+\bar{f})\circ \sigma]
\ell^2 :=\Big\{\{a_l\}_{l\in\mathbb{Z}}\in \mathbb{C}^{\mathbb{Z}}\Big|\sum_{l\in\mathbb{Z}}|a_l|^2< \infty\Big\}; \ell^2_k :=\Big\{\{a_l\}_{l\in\mathbb{Z}}\in \mathbb{C}^{\mathbb{Z}}\Big|\sum_{l\in\mathbb{Z}}(1+|l|)^{2k}|a_l|^2< \infty\Big\}.
S(a_j)S^{-1}(b_k) \otimes S^{-2}(b_i)b_j \otimes a_k a_i = a_jS^{-1}(b_k) \otimes S^{-1}\!\left(b_j S^{-1}(b_i)\right) \otimes a_k a_i = (\mathrm{id}\otimes S^{-1}\otimes \mathrm{id})(R_{12}R_{31}^{-1}R_{32}^{-1}) &= (\mathrm{id}\otimes S^{-1}\otimes \mathrm{id})( R_{32}^{-1}R_{31}^{-1}R_{12}) = S^{-1}(b_k)S(a_j) \otimes b_j S^{-2}(b_i) \otimes a_i a_k
\Phi~=~\sup_{\varphi\in\mathcal{G}}\varphi\quad\mathrm{with}\mathcal{G}~:=~\{\varphi:[0,+\infty)\to [0,+\infty)~|~\varphi~\mathrm{is~convex},~~\varphi(0)=0,~~\varphi\leq \mathfrak d\}.
Z_x = \left(1+u_0(t)\right)^{-1}+ \sum_{\substack{i \in K_{N_0}\beta \le i < N_0}}(Z_x)_i(t) x^i + O\left(x^{N_0}\right) \mbox{as}\, x \searrow 0 \, \mbox{and}\, t > 0,
\sum_{k=0}^n (-1)^k P^\gamma_{\tau(m),k}(\nu) = \sum_{k=0}^n \sum_{s \in W}(-1)^k \det(s) A(s\cdot \Lambda(\sigma_{\tau(m),k}+ \rho_M), \nu) B( s \cdot \Lambda(\sigma_{\tau(m),k}) + \rho_M).
R_n(z) \prod_{k=0}^{n-1}\frac{1 + \mathcal{R}e(\rho_{k}\alpha_{k-1})}{1 + \overline{\rho}_{k}\overline{\alpha}_{k-1}}= zS_{n-1}(z) - \rho_{n-1}S_{n-1}^{\ast}(z), n \geq 1,
{\tilde{\tilde{\Omega}}}_{c -}{}^{a b}= \Pi_{c}{}^{d}\;{\tilde{\Omega}}_{d -}{}^{a b}= \Pi_{c}{}^{d}\; \Pi_{d}{}^{e}\; \Omega_{e -}{}^{a b}= \Omega_{c -}{}^{a b}\ .
\sum_{n=1}^{\infty}\Big[ \prod_{k=1}^{n}\frac{\mathfrak{m}_k(0)}{1 - \mathfrak{m}_k(0)}\Big] = \sum_{n=1}^{\infty}\Big[ \prod_{k=1}^{n}\frac{\mathfrak{m}_k}{1 - \mathfrak{m}_k}\Big] = \sum_{n=1}^{\infty}\frac{(1)_n}{(2\lambda+3)_n}.
H^{-1}(z) = \left[ 1 - \frac{k d^2_L(z)}{a_0^2(1+z)^2}\right]^{-1/2}\frac{d}{dz}\left[ \frac{d_L(z)}{1+z}\right] \to \frac{d}{dz}\left[\frac{d_L(z)}{1+z}\right]
E_1(z) = \underset{:=\tilde{E}_1}{\underbrace{z_1^{n_1-1}z_2^{n_2}\overline{E_1(1/\bar{z}_1, 1/\bar{z}_2)}}}E_2(z) = \underset{:=\tilde{E}_2}{\underbrace{z_1^{n_1}z_2^{n_2-1}\overline{E_2(1/\bar{z}_1,1/\bar{z}_2)}}}.
\rho(i,i+1,t-s)=\begin{cases}\int_{-\infty}^{s}\gamma_{i}(s-u)\rho(i,i,t-u)du \int_{-\infty}^{s}\gamma_{i}(s-u)\rho(i,i,t-u)du+\gamma_{i}(s-t)\overline{\gamma}_{0}m_{i}\end{cases}.
S_{h}= \int \! \! d^{6}x \sqrt{-{\cal G}_{6}}e^{- 2 \phi_{6}^{( h )}}\left\{R + 4 \big ( \nabla \phi_{6}^{( h )}\big )^{2}-{\frac{1}{1 2}}( H_{6}^{( h )})^{2}-{\frac{1}{4}}{\cal F}_{6}^{2}\right\}
\int_0^\infty G(y)(y^2-1)\varphi(y)dy \leq G(y_0) \int_0^1 (y^2-1)\varphi(y)dy & + G(1)\int_1^\infty (y^2-1)\varphi(y)dy =(G(y_0)-G(1)) \int_0^1 (y^2-1)\varphi(y)dy<0\;.
\iint_{\Sigma}((\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})dydz+(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})dzdx+(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy)=\oint_{\partial\Sigma}(Pdx+Qdy+Rdz),
\partial_{\underline{i}}\epsilon_{I}-{\textstyle \frac{i}{4}}e^{2 \phi}\bigl ( \partial_{\underline{i}}a \bigr ) \epsilon_{I}+{\textstyle \frac{1}{2}}\sqrt 2 e^{- U - \phi}\biggl ( F_{0 i}^{+ ( 1 )}\alpha_{I J}+ i F_{0 i}^{+ ( 2 )}\beta_{I J}\biggr ) \gamma^{0}\epsilon^{J}= 0 .
d_{\Lambda}EZ(t \otimes \sigma) = EZ( d_{\Lambda}t \otimes \sigma ) + (-1)^{|t|}EZ( t \otimes \partial' \sigma) & - \sum_{(\sigma)}[ (i,\sigma' ) | EZ (t \otimes \sigma'')] + \sum_{(\sigma)}(-1)^{|t| |\sigma'|}[EZ (t \otimes \sigma') | (j, \sigma'') ],
w(u\wedge v)=\sum_{i<j}(w_{i}e_{j}-w_{j}e_{i})|\begin{matrix}u_{i}&u_{j}\\v_{i}&v_{j}\end{matrix}|+\sum_{i<j<k}|\begin{matrix}w_{i}&w_{j}&w_{k}\\u_{i}&u_{j}&u_{k}\\v_{i}&v_{j}&v_{k}\end{matrix}|e_{i}\wedge e_{j}\wedge e_{k}
L^S(\tfrac{1}{2},BC(\pi(q))\otimes BC(\pi'(q)))=L^S(\tfrac{1}{2},\Pi^{{\sf v}}\otimes \Pi'^{{\sf v}}) =L^S(\tfrac{1}{2},\Pi^{c}\otimes \Pi'^{c}) = L^S(\tfrac{1}{2},\Pi\otimes \Pi').
A_4 =\||x|^{-\tau}|u|^{p-1}(I_\alpha\ast|\cdot|^{-\tau}|u|^{p-1}\nabla u)\|_{L_x^{\frac{2n}{n+2}}}&\lesssim \||x|^{-\tau}|u|^{p-1}\|_{L_x^{a_2}}\||x|^{-\tau}|u|^{p-1}\nabla u\|_{L_x^{b_2}}&\leq \||x|^{-\frac{\tau}{p-1}}|u|\|^{p-1}_{L_x^{(p-1)a_2}}\||x|^{-\frac{\tau}{p-1}}u\|^{p-1}_{L_x^{(p-1)b_4}}\|\nabla u\|_{L_x^r}&\lesssim \|\nabla u\|_{L_x^r}^{2p-1}
\lim_{{\vec{k}}\rightarrow 0}k^{\mu}{\cal F}_{\mu}( k ) ~ \equiv ~ \lim_{{\vec{k}}\rightarrow 0}{\bar{\epsilon}}\int d^{4}x \sqrt{- g}~ k^{\mu}p_{k}( x ) < 0_{-}|{\cal T}{\cal S}_{\mu}( x ){\cal O}_{(}y ) | 0_{-}> ~ ,
U \ll \frac{X}{\log X}\left(\frac{(\log X)^{1/2}}{\exp(M(f; X)/2)}+ 1\right) \int_{1/\log X}^{2/\log y}\frac{d\sigma'}{\sigma'^{3/2}}+ \frac{X}{(\log X)^{1/2}}&\ll \frac{X}{\exp(M(f; X)/2)}+ \frac{X}{(\log X)^{1/2}}.
e^{i W [ \vec{H}]}= \int{\cal D}\lambda{\cal D}{\vec{\Phi}}e^{i \int d^{d}x ( \frac{1}{2}( \partial_{\mu}{\vec{\Phi}}( x ) )^{2}- \frac{1}{2}\lambda ( x ) ({\vec{\Phi}}^{2}( x ) - C^{2}) + \frac{1}{4 U}\lambda^{2}( x ) +{\vec{H}}( x ) \cdot{\vec{\Phi}}( x )}
-\overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},\overline{\nabla}_{Z_{i}}Z_{j}) =g(\overline{\nabla}_{Z_{i}}\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{j})+ \overline{g}(\overline{\nabla}_{Z_{i}}\widehat{W},\overline{\nabla}_{\widehat{W}}Z_{j}) +\overline{g}(\widehat{W},\overline{\nabla}_{Z_{i}}\overline{\nabla}_{\widehat{W}}Z_{j}).
( \Omega \Omega^{*})_{N}( \vec{x}) \ = \e^{2 e_{o}\sigma_{N}( \vec{x})}\ + \ \frac{2 e_{o}^{2}}{N}e^{2 e_{o}\sigma_{N}( \vec{x})}( X_{i}\sigma_{N}( \vec{x}) )^{2}\ + \o ( e_{o}^{2}N^{- 2}) \ ,
D_j\left(\left(|v|^2+\varepsilon\right)^{\frac{p}{2}-1}\right) = \left(\frac{p}{2}-1\right)\left(|v|^2+\varepsilon\right)^{\frac{p}{2}-2}D_j\left(|v|^2+\varepsilon\right) = (p-2)\left(|v|^2+\varepsilon\right)^{\frac{p}{2}-2}\Re\left(\overline{D_j v}^T v\right),
\textbf{J}\leq \frac{6}{(1-s')^3}\frac{1}{(r^2 -1)^3}\sum_{n=0}^{\infty}(|z|^{-2}r_{s'})^{n}= \frac{6}{(1-s')^3}\frac{1}{(r^2 -1)^3}\frac{|z|^2}{|z|^2 - r_{s'}}.
\lambda_{\ast,m}\ = \ \ (i-\ast)^{-1}\lambda'_{\ast,m}\lambda_{\ast,m-1}\ = \ \ (i-\ast)^{-1}(\lambda'_{\ast,m-1}- \lambda_{\ast, m}) \vdots \ \ \ \lambda_{\ast,1}\ = \ \ (i-\ast)^{-1}(\lambda'_{\ast,1}- \lambda_{\ast, 2}).
\widetilde{\mathcal{H}}_\pm = \bigoplus_{M=0}^N \widetilde{\mathcal{H}}_{\pm,M}\, , \widetilde{\mathcal{H}}_{\pm,M}\equiv \ker\bigl[S^z - \tfrac12(N-2M) \bigr] \subset \widetilde{\mathcal{H}}_\pm \, .
M_2+M_3\alpha^2 +M_4\alpha^4 +M_5 \alpha^6 +M_6\alpha^{8}+M_7\alpha^{10}+M_8\alpha^{12}=0, &M_{10}+M_{11}\alpha^2 +M_{12}\alpha^4 +M_{13}\alpha^6 +M_{14}\alpha^{8}+M_{15}\alpha^{10}+M_{16}\alpha^{12}=0.
\rho\mathbf{u}=\sqrt{\rho}\cdot\sqrt{\rho}\mathbf{u}\in L^\infty_tL^2_x, & \rho\nabla\mathbf{u}\in L^\infty_tL^2_x, & \nabla\rho\bar{x}^a\in L^\infty_tL^q_x\ \ \q>2.
T= \begin{pmatrix}0 1 0 \ldots 0 0 0 1 0 \ldots 0 \vdots \vdots \ddots \ddots \ldots 0 \vdots \vdots . 0 \ddots 0 . . . . 0 1 0 0 . . \ldots 0 \end{pmatrix},
\Delta M=\frac{1}{2}\sum_i\omega_i+\frac{\left<V\right>}{4\pi}+\frac{1}{2}\int_{-\infty}^{\infty}\frac{dk}{2\pi}\omega(k)\frac{d}{dk}\delta(k)-\frac{\left<V\right >}{4}\int_{-\infty}^\infty\frac{dk}{2\pi}\frac{1}{\sqrt{k^2+m^2}}\;.
\delta\Psi=\epsilon^2(1+i\Psi\partial_\tau\Psi), \quad\delta\Phi=-i\epsilon^2(2\eta-\Psi\partial_\tau\Phi)=-i\epsilon^2(2\eta-L),\quad L=\Psi\partial_\tau\Phi={{\partial_\tau\Phi D\Phi}\over{1+\sqrt{1-(\partial_\tau\Phi)^2}}},
\frac{d^2u_{n\ell}(z)}{dz^2}+\frac{(1-z)}{z(1-z)}\frac{du_{n\ell}(z)}{dz}+\left\{\frac{z^2(-\xi_{n\ell}^2-\beta_1-\beta_2)+z(2\xi_{n\ell}^2+\beta_1)+(-\xi_{n\ell}^2)}{z^2(1-z)^2}\right\}u_{n\ell}(z)=0.
\rho=\frac{1}{2}|R\rangle\langle R|+\frac{1}{2}|L\rangle\langle L|=\frac{1}{2}|H\rangle\langle H|+\frac{1}{2}|V\rangle\langle V|=\frac{1}{2}(\begin{matrix}1&0\\ 0&1\end{matrix}).
\displaystyle\sum_{n=1}^\infty \ln\left|1-\frac{z^2}{|\lambda_n|^2}\right| \leq \displaystyle\sum_{n=1}^\infty \ln\left(1+2\frac{|z|^2}{|\lambda_n|^2}\right)\leq \displaystyle\sum_{n=1}^\infty \ln\left(1+2\frac{|z|^2}{n^2}\right)\leq \int_{0}^{\infty}\ln\left(1+2\frac{|z|^2}{x^2}\right)dx, =\sqrt{2}\pi|z|,
\int_{\Omega\times\Omega}|\langle \tau_\alpha, \tau_\beta\rangle|^{2m}\, d(\mu\times\mu)(\alpha,\beta)=\int_{\Omega}\int_{\Omega}|\langle \tau_\alpha, \tau_\beta\rangle|^{2m}\, d \mu(\alpha)\, d \mu(\beta)\geq \frac{\mu(\Omega)^2}{{d+m-1\choose m}}, \forall m \in \mathbb{N}.
SINR_{i,k}= \frac{|\mathbf{v}_{i,k}^H\mathbf{H}_{i,k}\mathbf{p}_i |^2}{\sum_{m=1, m\neq i}^{M}|\mathbf{v}_{i,k}^H\mathbf{H}_{i,k}\mathbf{p}_{m}|^2 +|\mathbf{v}_{i,k}|^2\frac{1}{\rho}}.
\left(\boldsymbol{\sigma}\cdot\boldsymbol{a}\right) \left(\boldsymbol{\sigma}\cdot\boldsymbol{b}\right)=\left(\boldsymbol{a}\cdot\boldsymbol{b}\right)I_{2\times 2}+i\, \boldsymbol{\sigma}\cdot\left( \boldsymbol{a}\times\boldsymbol{b}\right),
\sup_{k \in \{1,\ldots,k^*\}}\frac{k^{2/d}}{n^{2/d}}\sup_{f \in \mathcal{F}_{d,\theta}}\int_{\mathcal{X}_n^c}\frac{\Delta f(x)}{f(x)^{2/d}}\,dx = O \biggl( \frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\biggr)
(A_C(m^\phi|_C))_{r(e)}\leq (A_C(m^\phi|_C))_{r(e)}+A(r(e),s(e))m^\phi_{s(e)}&\leq((Am^\phi)|_C)_{r(e)}\leq e^\beta m^\phi_{r(e)}=\rho(A_C)(m^\phi|_C)_{r(e)},
\left[ e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi+g}}\big( \delta(x), C_{m_d}\alpha \big) + i\theta}\; + \;e^{- 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi+g}}\big( \delta(x), C_{m_d}\alpha \big) - i\theta}\right] \; .
\begin{array}{l}I_3=\frac{2}{a}I_2-\frac{1}{a}I_1, I_4=(\frac{88}{21a^2}-\frac{8}{7a})I_2-\frac{44}{21a^2}I_1+\frac{4}{7a}hI_0, I_5=(\frac{572}{63a^3}-\frac{209}{42a^2})I_2-(\frac{286}{63a^3}-\frac{5}{4a^2}-\frac{1}{a}h)I_1+\frac{26}{21a^2}hI_0.\end{array}
\hat{\Pi}^{\underline{{m}}}= d \hat{X}^{\underline{{m}}}- i d \hat{\Theta}^{1}{\sigma}^{\underline{{m}}}\hat{\Theta}^{1}- i d \hat{\Theta}^{2}\tilde{\sigma}^{\underline{{m}}}\hat{\Theta}^{2}= d \xi^{m}\hat{\Pi}_{m}^{\underline{{m}}}( \xi )
\Sigma_{\theta}^2 =\sup_{0\le a \le 1}\sigma_{\theta}^2(a)=\sup_{0\le a \le 1}\biggl(V(a ,a)+2\sum_{k=1}^\infty\frac{V(\langle p^k a \rangle, \langle q^k a \rangle)}{p^kq^k}\biggr),
[\begin{matrix}V_{a}\\V_{b}\\V_{c}\end{matrix}]=[\begin{matrix}V_{a,0}\\V_{b,0}\\V_{c,0}\end{matrix}]+[\begin{matrix}V_{a,1}\\V_{b,1}\\V_{c,1}\end{matrix}]+[\begin{matrix}V_{a,2}\\V_{b,2}\\V_{c,2}\end{matrix}]
\tau_{0}= \frac 1 2 \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \, , \quad \tau_{1}= \frac 1 2 \left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) \, , \quad \tau_{2}= - \frac 1 2 \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) \, .
S_{+}= \int d^{4}x \sqrt{- g^{0}}\frac{1}{2}V_{0}^{+}\Phi \Bigr |_{y = 0}+ \int d^{4}x \sqrt{- g^{\pi r_{c}}}\frac{1}{2}V_{\pi}^{+}\Phi \Bigr |_{y = \pi r_{c}}\ ,
{\frac{\ddot{a}_{0}}{a_{0}}}= -{\frac{1 6 \pi^{2}G_{5}^{2}}{9 c^{6}}}( 2 \varrho^{2}c^{4}+ \sigma \varrho c^{2}+ 3 \sigma \wp + 3 \wp \varrho c^{2}) -{\frac{{\cal C}c^{2}}{a_{0}^{4}}}+{\frac{4 \pi G_{5}}{3 c^{2}}}\left( \Lambda +{\frac{4 \pi G_{5}}{3 c^{4}}}\sigma^{2}\right) ,
\begin{aligned}F_\tau(n;q) = \frac{1}{2\pi\mathrm{i}}\int_{\Gamma_{\theta,\sigma}^\tau}e^{zt_n}{e^{-z\tau}}\delta_\tau(e^{-z\tau})^{\alpha-1}({\delta_\tau(e^{-z\tau})^\alpha}+A(q))^{-1}\,\d z, E_\tau(n;q) = \frac{1}{2\pi\mathrm{i}}\int_{\Gamma_{\theta,\sigma}^\tau}e^{zt_n}e^{-z\tau}({\delta_\tau(e^{-z\tau})^\alpha}+A(q))^{-1}\,\d z,\end{aligned}
G = \left(\underbrace{\begin{array}{ccc}1 0 0 0 1 0 0 0 1\end{array}}_{I}\underbrace{\begin{array}{ccc}0 0 1 0 1 0 1 0 0\end{array}}_{C}\underbrace{\begin{array}{ccc}0 0 1 0 1 0 1 0 0\end{array}}_{C}\underbrace{\begin{array}{ccc}1 0 1 0 1 0 1 0 1\end{array}}_{I+C-C_1}\right)
\nabla_{\alpha\beta}u-\mathcal{A}_{\alpha\beta}(x,\nabla u,-D_\gamma u) =D_\gamma(u-\varphi)\mathfrak{C}_{\alpha\beta}+\nabla_{\alpha\beta}\varphi-\mathcal{A}_{\alpha\beta}(x,\nabla \varphi,-D_\gamma u), \ \ \alpha,\beta=1,\cdots,n-1,
\left| \int_{\mathbb{R}^2}(F(y) - F(x))\partial_1^2 G(y) \psi_T(x-y) \, \textrm{d}y \right|\lesssim C(G) [F]_{\alpha}\int_{\mathbb{R}^2}|x_2 |^{\frac{\alpha -2}{2}}d^{\alpha}(x,y) | \psi_T(x - y)| \, \textrm{d}y \lesssim C(G) \left[ F \right]_{\alpha}(T^{\frac{1}{4}})^{2\alpha-2}.
A_{-\alpha}(\lambda) = -\frac{\pi}{2}\, \Big( v'_{-\alpha}(\lambda,1)\, Y_0(\sqrt{\lambda}) +\sqrt{\lambda}\v_{-\alpha}(\lambda,1)\, Y_1(\sqrt{\lambda}) \Big ) B_{-\alpha}(\lambda) = \frac{\pi}{2}\, \Big( v'_{-\alpha}(\lambda,1) \, J_0(\sqrt{\lambda}) +\sqrt{\lambda}\v_{-\alpha}(\lambda,1) \, J_1(\sqrt{\lambda}) \Big ) ,
B_\eta \left( \left(\overline{x}_{n+\i,\widehat{n+\eta}}\right)_{\i=1}^{\infty},\left(\overline{x}_{n+\i+1,\widehat{n+\eta+1}}\right)_{\i=1}^{\infty}\right) & = d\left(\overline{x}_{n+\eta},\overline{x}_{n+\eta+1}\right)
g_n(x) := \exp\big\{{-}n(W_1(z_1^* + \tfrac{x}{\sqrt{n}}) - W_1(z_1^*))\big\}(z_1^* + \tfrac{x}{\sqrt{n}})^{-(2i - 1)}1_{\left\{z_1^* + \tfrac{x}{\sqrt{n}}> 0\right\}}
\Vert f\Vert_{\mathcal{H}}^{2}= \int_{\mathbb{R}^{d\cdot q}}\left( J(f) (\mathbf{y}_{1}, \ldots ,\mathbf{y}_{q}) \right)^{2}d\mathbf{y}_{1}\ldots d\mathbf{y}_{q}.
g_2(t) = \int_{0}^t e^{is\Lambda^{1/2}}R\cdot V_\omega(s)\,ds, g_3(t) = \int_0^t e^{is\Lambda^{1/2}}\Lambda^{1/2}\big( (R\cdot V_\omega)(\Lambda^{1/2}u)\big) - \nabla \cdot \big( u V_\omega\big) + \Lambda( u R\cdot V_\omega), g_4(t) = \int_0^t e^{is\Lambda^{1/2}}\Lambda^{1/2}(R\cdot V_\omega)^2\, ds.
\left( L^{\frac{3}{2}}\right)_{+}= \left( L L^{\frac{1}{2}}\right)_{+}= \partial^{3}+{\frac{1}{4}}u \partial +{\frac{1}{8}}u_{x}= \partial^{3}+{\frac{1}{8}}( \partial u + u \partial ) ={\frac{1}{4}}B
y_1 =-93506(\zeta^5+\zeta^2)-152738(\zeta^4+\zeta^3)-147903, y_2 =-9177(\zeta^5+\zeta^2)-13557(\zeta^4+\zeta^3)-58289, y_3 =56798(\zeta^5+\zeta^2)+98510(\zeta^4+\zeta^3)-85253, y_4 =75152(\zeta^5+\zeta^2)+125624(\zeta^4+\zeta^3)+31325,
\mathop{{\mathrm{maximize}}}\limits_{{\bf w}_p,{\bf\Phi}_m,{\bf F}_k}{\quad}\min_{p}\{\mathrm{SINR}_{R_p}\}\textrm{subject to}{\quad}\|{\bf F}_k\|_F^2 \leq \mathcal{P}_k, \forall k,{\quad}\mathrm{SINR}_{C_u}\geq \xi, \forall u,{\quad}|{\bf\Phi}_m(i,i)|=1, \forall i,\forall m,
\displaystyle\frac{1}{k}(u,\bar{u}_h)+ (\nabla u, \nabla\bar{u}_h) +(\widetilde{u}{\boldsymbol \sigma},\nabla\bar{u}_h)=\frac{1}{k}(u^{n-1}_h,\bar{u}_h), \ \forall \bar{u}_h\in U_h,
\int_0^{x_{k}}\bigg(\int_0^t w \bigg)^{-\frac{p}{p-r}}w(t) V_r(0,t)^{\frac{pr}{p-r}}dt = \sum_{i= -\infty}^k \int_{x_{i-1}}^{x_i}\bigg(\int_0^t w\bigg)^{-\frac{p}{p-r}}w(t) V_r(0,t)^{\frac{pr}{p-r}}dt \lesssim \sum_{i= -\infty}^k 2^{-i\frac{r}{p-r}}V_r(x_{i-1}, x_i)^{\frac{pr}{p-r}}.
\lambda_1= \inf_{\varphi\in C_0^{\infty}(\Sigma),\int \varphi^2e^{\frac{|x|^2}{4}}d\sigma\neq 0}\dfrac{\int|\nabla \varphi|^2e^{\frac{|x|^2}{4}}d\sigma}{\int \varphi^2e^{\frac{|x|^2}{4}}d\sigma}\geq \dfrac{n}{2}+\inf_{x\in \Sigma}|{\bf H}||^2\geq \frac n2.
\phantom{=}\left\| |\varphi |^{p_{r;s}^{\uparrow *}}\int_{\mathbb{R}^N\setminus B_R(0)}K(x-y)|v_n(y)|^{p_{r;s}^{\uparrow *}}dy\right\|_{L^\sigma_x(\operatorname{supp}\varphi)}& \leq \|K\wedge C_{\varphi,R}\|_{\tilde{r}}\| |v_n|^{p_{r;s}^{\uparrow *}}\|_{\ell_r}\| |\varphi |^{p_{r;s}^{\uparrow *}}\|_{q_\varphi}\leq C_3
\|\sigma^0\|_{\mathcal{C}_h}= \|u_h^0 - \mathcal{P}_h \, u(0)\|_{\mathcal{C}_h}= \|u_{h,0}- \mathcal{P}_h \, u_0\|_{\mathcal{C}_h}\leq C \, h^2 \, |u_0|_{H^2(\Omega)}.
\partial_t \big( \tfrac{1}{2}\rho |u|^2 + \varepsilon \tfrac{1}{2}n |v|^2 + h_1(\rho) + h_2(n) + \delta \tfrac{1}{2}|\nabla \phi|^2 \big) + \nabla \cdot \mathcal{F}= 0 \ ,
\mathrm{Vol}(S^{m-1}) = \int_{S^{m-1}}d\mathbf{s}= \frac{2 \pi^{m/2}}{\Gamma(m/2)}, \int_{S^{m-1}}\xi_j \xi_l d\mathbf{s}= \frac{\pi^{m/2}}{\Gamma(1+m/2)}\mathbf{1}_{jl}= \mathrm{Vol}(S^{m-1}) \frac1m \mathbf 1_{jl}.
F^{\mu \nu}= - \frac{1}{2}\epsilon^{\mu \nu \rho \sigma}\hspace{0.8mm}{^*}\hspace{-0.8mm}F_{\rho \sigma}= \left( \begin{array}{cccc}0 -E_1/c -E_2/c -E_3/c E_1/c 0 -B_3 B_2 E_2/c B_3 0 -B_1 E_3/c -B_2 B_1 0 \end{array}\right) .
(\mathcal{N}_1,\mathcal{N}_1'), \mathcal C_1=\left(\begin{matrix}\mathfrak{s}^2 \rho \rho \mathfrak{s}^2 \end{matrix}\right), &(\mathcal{N}_2,\mathcal{N}_2'), \mathcal C_2=\left(\begin{matrix}\mathfrak{s}^2+|\rho| 0 0 \mathfrak{s}^2+|\rho| \end{matrix}\right).
TS =P|^{-1}_{Y_1}Q^{-1}_{X1}Q(P+(I-Q))+(I-Q)|^{-1}_{X_2}(I-P)|^{-1}_{Y_2}(I-P)(P+(I-Q)) &=P|^{-1}_{Y_1}Q^{-1}_{X1}QP+(I-Q)|^{-1}_{X_2}(I-P)|^{-1}_{Y_2}(I-P)(I-Q) &=P^{-1}|_{Y_1}P+(I-Q)|^{-1}_{X_2}(I-Q)=I
\mathbb{E}_0\left[ h\left({\bf R}^{(12)}_{{\bf i}^1}\right) h\left({\bf R}^{(12)}_{{\bf i}^2}\right) h\left({\bf R}^{(12)}_{{\bf j}^1}\right) h\left({\bf R}^{(12)}_{{\bf j}^2}\right)\right] = \eta^h.
\int_{\mathcal{X}_n}\|\chi(\vartheta_n)\|^2\mathrm{d}\vartheta_n:=\underset{\inf{\Delta}<t_1<\ldots< t_n<\sup{\Delta}}{\int\cdots\int}\|\chi(t_1,t_2,\ldots,t_n)\|^2\mathrm{d}t_1\mathrm{d}t_2\cdots\mathrm{d}t_n
\frac{d\phi}{d\tau}= \frac{r}{R}\cos \phi \left( -\frac{1}{2}+ \frac{2(n-1)}{r^2-s^2}\right) + \frac{\lambda}{R}+ \frac{1}{R}\sin \phi \left( \frac{s}{2}+ \frac{2s(n-1)}{r^2-s^2}\right).
-\int_{Q_4^+}u \phi_t \,dx\,dt+\sum_{j=1}^{d-1}\int_{Q_4^+}D_j^muD_j^m\phi\,dx\,dt+\int_{Q_4^+}A^{\hat{\alpha}\hat{\alpha}}(t)D^m_du D^m_d\phi \,dx\,dt=0,
\begin{matrix}\dot A_0=\frac{1}{2}(A_1B_0A_0-A_0B_0A_1) \dot A_1=\frac{1}{2}(A_1B_1A_0-A_0B_1A_1) \dot B_0=\frac{1}{2}(B_1A_0B_0-B_0A_0B_1) \dot B_1=\frac{1}{2}(B_1A_1B_0-B_0A_1B_1). \end{matrix}
D^2 = (\tilde{C}+ \sum_i E_i)^2 = \tilde{C}^2 - \sum_p (\omega_p -2r_p) p_{a}(D) = p_{a}(\tilde{C}) + \sum_p p_a(E_p) - s + \tilde{C}.E= p_{a}(\tilde{C}) +\sum_p (r_p-1) e(\tilde{C}) = e(C) + \sum_p (r_p -1) (K_S+D).D = 2p_{a}(D) -2 = 2p_{a}(\tilde{C}) -2 + 2 \sum_p (r_p - 1) = -e(C) + \sum_p(r_p -1) K_S^2 - \tilde{C}^2 = K_X^2 -C^2 + \sum_i (m_i^2-1).
\|\tilde{v}^l\|_{L^{\alpha+2}}\leq \limsup_{n\rightarrow \infty}\|\tilde{v}^l_n\|_{L^{\alpha+2}}= \limsup_{n\rightarrow \infty}\|v^l_n\|_{L^{\alpha+2}}\rightarrow 0, l \rightarrow \infty.
\delta^{-n}\sigma_4^{-1}s_2 \underbrace{\delta^n r_1 \delta^{-n}s_2 \delta^n \dots}_{= (\delta^n r_1 \delta^{-n}s_2)^{n-1}}r_1 \delta^{-n}s_2 \delta^{n}r_1 \sigma_4
\begin{array}{l}C=\sum_{i=1}^{N}\sum_{j=1}^{N}q_{i,j}\Omega_{i,j}+((\eta +\kappa )P_{I}^{(th)}{\rm +}\sum_{j=1}^{N}\tau_{j}{\rm \; \; \; \;}+\sum_{i=1}^{N}\mu_{i}(\beta -p_{f}(\lambda_{i})) +\sum_{i=1}^{N}\delta_{i}(p_{d}(\lambda_{i})-1+\alpha ), \end{array}
[a_{(1\,p)},a_{(1\,l)}]_c=-3\xi^2 \lambda_{112}\chi_{(1\,l)}(g_{(2\,p)})a_{(3\,l)}a_{(2\,p)}+3\lambda_{112}\chi_{1}(g_{(3\,p)})a_{(3\,p)}a_{(2\,l)}.
B_{i}^{+}(k,s) = \frac{F_{i}(D^{+}U(k,s + 1),D^{-}U(k,s+1)) - F_{i}(D^{+}U(k,s),D^{-}U(k,s+1))}{D^{+}U(k,s + 1) - D^{+}U(k,s)}, B_{i}^{-}(k,s) = \frac{F_{i}(D^{+}U(k,s),D^{-}U(k,s+1)) - F_{i}(D^{+}U(k,s),D^{-}U(k,s))}{D^{-}U(k,s+1) - D^{-}U(k,s)}, \Gamma(k,s) = f_{i}(s \Delta t, -k\Delta x).
h(\lambda) := \phi_\lambda^{-1}\circ \mathfrak{a}_{\underline{n}}(\lambda) = \left(\phi_{\lambda,1}^{-1}\left(f_\lambda^{n_1}(a_1(\lambda))\right),\ldots,\phi_{\lambda,m}^{-1}\left(f_\lambda^{n_m}(a_m(\lambda))\right)\right).
{\cal T}_{\mu \nu}^{\alpha \beta}\sim \epsilon_{~ \mu}^{\alpha ~ \rho \sigma}\epsilon_{~ \nu}^{\beta ~ \lambda \gamma}\big ( A F_{\rho \sigma}F_{\lambda \gamma}+ B F_{\rho \sigma}\tilde{F}_{\lambda \gamma}+ C \tilde{F}_{\rho \sigma}\tilde{F}_{\lambda \gamma}\big ) \, ,
h_a(l)=\frac{1}{2}\sigma^2l(l-1)+l\Gamma-\delta+\int_\mathbb{R}\Big\{(1+\gamma(z))^l-1-l\gamma(z)\Big\}\nu(dz) h_b(x)=\epsilon \ln{x}-\delta(b\ln{x}+c)+b\Gamma-\frac{1}{2}\sigma^2b+\int_\mathbb{R}\big\{b\ln(1+\gamma(z))-b\gamma(z)\big\}\nu(dz),
\frac{\partial}{\partial a}\mathrm{B}_{z}\left( a,b\right) =z^{a}\left\{\frac{\ln z}{a}\,_{2}F_{1}\left( \left.\begin{array}{c}1-b,a a+1\end{array}\right\vert z\right) -\frac{1}{a^{2}}\,_{3}F_{2}\left( \left.\begin{array}{c}1-b,a,a a+1,a+1\end{array}\right\vert z\right) \right\}.
E[| L^{n,2}_\tau +L^{n,3}_\tau|^2] = n^{4\alpha+ 2}\int_{[0,\tau]^2}\gamma_{s_1}\gamma_{s_2}\mathbf{1}_{|s_1-s_2|\le\frac 1n}(s_1-\eta_n(s_1))^\alpha(s_2-\eta_n(s_2))^\alpha & \times E[(\sigma'\sigma)^2(X_{\eta_n(s_1)})\Lambda^n_{s_1}(\sigma'\sigma)^2(X_{\eta_n(s_2)})\Lambda^n_{s_2}] ds_1ds_2
D_{x_{N}}^{m-j}u(x,t)=\frac{b_{j}a}{x_{N}^{n-j}}+o(x_{N}^{-(n-j)})=D_{x_{N}}^{m-j}\widetilde{Q}_{u}(x_{N})+o(x_{N}^{-(n-j)}),\,(x,t)\rightarrow(0,0),
(1+q-q^{2n})(q^2;q^2)_{n-1}- q(1-q^{2n-2})(q^2;q^2)_{n-2}&=(q^2;q^2)_n +q(q^2;q^2)_{n-1}-q(q^2;q^2)_{n-1}&=(q^2;q^2)_n.
\sum_{n=0}^{\infty}w_{n}^{\left( s\right)}\left( x;\alpha,\beta,r\right)\frac{t^{n}}{n!}=\left( \frac{1}{1-x\left( \left( 1+\alpha t\right)^{\beta/\alpha}-1\right)}\right)^{s}\left( 1+\alpha t\right)^{r/\alpha},
\langle E_{i}\rangle=\int dX_{i}\alpha_{i}X_{i}^{2}p_{i}(X_{i})=\frac{\int dX_{i}\alpha_{i}X_{i}^{2}e^{-\frac{\alpha_{i}X_{i}^{2}}{k_{B}T}}}{\int dX_{i}e^{-\frac{\alpha_{i}X_{i}^{2}}{k_{B}T}}}
\Big{(}\mathcal{D}_tu_h^{\frac12},v_h\Big{)}-\gamma(\nabla \sigma_h^{\frac12},\nabla v_h)+(\nabla u_h^{\frac12},\nabla v_h)+(f^{\frac12}(u_h),v_h)= (g^{\frac12},v_h),~\forall v_h\in L_h,
\begin{array}{rl}M_2^*(R) = \!\! \!\!\displaystyle \sup\limits_{B_{\frac{\delta R}{16}}(\bar x)}(d_x^2 |D^2u(x)|) \le (\frac{17}{16}\delta R)^2\sup\limits_{B_{\frac{\delta R}{16}}(\bar x)}|D^2u| \le \!\! \!\!\displaystyle (\frac{17}{16}\delta R)^2 C (1+\frac{1}{(\delta R/ 8)^2}),\end{array}