harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{A}=\left[0;a_0,a_1,a_2,\ldots, a_{2m}, \left( m +\tfrac{1}{\tfrac{1}{L_{A_{m}}-m}+\tfrac{1}{L_{A_{M+1}}-m}}\right) -m+1 \right] &=\left[0;a_0,a_1,a_2,\ldots, a_{2m},1, \tfrac{1}{L_{A_{m}}-m}+\tfrac{1}{L_{A_{m+1}-m}}\right] &=\left[0;a_0,a_1,a_2,\ldots, a_{2m},a_{2m+1}, a_{2m+2}, L_{A_{m+1}}-m \right]
F_{p}= - \frac{1}{4}\pi^{2}\tilde{V}_{3}N_{3}T^{4}\left[ 1 - 4 \left( \frac{M}{\pi T}\right) + 1 0 \left( \frac{M}{\pi T}\right)^{2}- 2 0 \left( \frac{M}{\pi T}\right)^{3}+ \cdots \right] .
dQ_\xi(\gamma)h-dQ_\xi(\gamma')h= -\frac{\alpha^2}{2\pi}\int_{-\infty}^{+\infty}(D_{\gamma,\xi}-iz)^{-1}V_{\gamma'-\gamma,\xi}(D_{\gamma',\xi}-iz)^{-1}V_{h,\xi}(D_{\gamma,\xi}-iz)^{-1}dz &-\frac{\alpha^2}{2\pi}\int_{-\infty}^{+\infty}(D_{\gamma',\xi}-iz)^{-1}V_{h,\xi}(D_{\gamma,\xi}-iz)^{-1}V_{\gamma'-\gamma,\xi}(D_{\gamma',\xi}-iz)^{-1}dz.
u_k (z) := U_k \left(\frac{z}{\left|z\right|^{2}}\right)+(m_1 +1)\beta_0\log\left(\frac{1}{\left|z\right|}\right) = u_0 \left(\frac{1}{\bar{z}^{m_1+1}}- \xi_k \right) - \beta_0\log|\bar{z}^{m_1+1}|
\lambda_k=\min\limits_{g\in V, <g, g_j>=0 \for \j=k+1, \cdots, n}\frac{<Ag, g>}{<g, g>}=\max\limits_{g\in V, <g, g_l>=0 \for \l=1, \cdots, k-1}\frac{<Ag, g>}{<g, g>}
(\omega(g)\phi, \phi)= ~\|\phi\|^2 (\xi\bar{\xi}')^{\mu_1+\frac{1}{2}}(\eta\bar{\eta}'\zeta\bar{\zeta}')^{\mu_2+\frac{1}{2}}(\gamma\bar{\gamma}')^{\alpha-\frac{1}{2}}(\cosh t)^{-\mu_2-\alpha-3}&\times (\cosh^{-1}t \cos \theta \cos \theta' + \zeta\bar{\zeta}'\sin \theta \sin \theta')^r.
\stackrel{\lbrack 2 ]}{S}= \int d^{D}x \left( \left( \varepsilon_{m n p}\eta_{n a}^{* ( m )}+ \bar{A}_{b}^{( p ) \alpha}\left( D_{\alpha}\right)_{a}^{b}\right) \pi_{p}^{a}+ \cdots \right) ,
\begin{aligned}\mu_{1,k}^T(u)= \|u\|_{L_T^\infty H^s}+\|D_x^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}+\|D_y^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}&+\|u\|_{L_T^{p_k}L_{xy}^\infty}+\|\partial_x u\|_{L_T^{\frac{12}{5}}L_{xy}^\infty}+\| u\|_{L_x^{4}L_{yT}^{\infty}},\end{aligned}
\mathrm{w i t h}{\cal{X}}^{\alpha}({\bf{r}}) ={\textstyle \frac{\partial_{i}}{\partial^{2}}}A_{i}^{\alpha}({\bf{r}}) \mathrm{a n d}\overline{{{\cal Y}^{\alpha}}}({\bf r}) ={\textstyle \frac{\partial_{j}}{\partial^{2}}\overline{{{\cal A}_{j}^{\alpha}}}({\bf r})}.
|\widetilde{\nabla}u|^2=(\widetilde{\nabla}u) \overline{u}=2\sum_{i,j}g^{i,j}\Bigg\{\frac{\partial u}{\partial \bar{z}_i}\overline{\frac{\partial u}{\partial \bar{z}_j}}+\overline{\frac{\partial u}{\partial z_i}}\frac{\partial u}{\partial z_j}\Bigg\}.
\begin{aligned}\mathbf{y}_{k}= \sum_{u=1}^{N}a_{u}[\mathbf{x}_{u}^{c}[k],\mathbf{x}_{u}^{c}[k+ K\cdot 1],\cdots, \mathbf{x}_{u}^{c}[k+K(L-1)] ]^{T}+ \mathbf{n}_{k}&= \mathbf{S}\mathbf{X}_{k}\mathbf{a}+ \mathbf{n}_{k},\end{aligned}
V_{1}^{[r,1]}(x) \coloneqq \sup\limits_{\tau_{1}\in \mathcal{T}}M^{x}_{1}(\tau_{1},D_{[r,1]}) = M^{x}_{1}(D_{[\ell^{1},\ell^{2}]},D_{[r,1]}),\forall x \in [0,1].
\widetilde{B}_{n}(x)-2nxD_{n-1}(x) = \sum_{i=0}^{n-2}\big((n+i-1)x+n+i+1\big)xT_{n-1,i}(x) +\sum_{i=0}^{n-2}\big((2n-i-1)x+i+1\big)T_{n-1,n+i-1}-\sum_{i=0}^{n-2}nx\big(T_{n-1,i}(x)+T_{n-1,n+i-1}(x)\big) = \sum_{i=0}^{n-2}\big( (n-i-1)x+i+1 \big) \big( xT_{n-1,i}(x)+T_{n-1,n+i-1}(x) \big),
\left(8\binom{z+1}{2}+1\right)^{k}+\left(8\binom{z-1}{2}+1\right)^{k}-2\left(8\binom{z}{2}+1\right)^{k}= 2{\sum_{1\leq j \leq{k}}\binom{2k}{2j}2^{2j}{\left(8\binom{z}{2}+1\right)}^{k-j}}.
\tau_{1}^{-k-2}W''(f)(\tau_{1}^{-1}, \tau_{2}) = - \frac{1}{2\pi i}\theta_{2}W(f)(\tau_{1}, \tau_{2}) + W''(f)(\tau_{1}, \tau_{2}), W''(f)(\tau_{1}, \tau_{2}) = W''(f)(\tau_{2}, \tau_{1}).
LC_{t}=S\Phi(a_{1}(S,m))-me^{-r\tau}\Phi(a_{2}(S,m))-\frac{S\sigma^{2}}{2r}(\Phi(-a_{1}(S,m))-e^{-r\tau}(m/S)^{\frac{2r}{\sigma^{2}}}\Phi(-a_{3}(S,m))),
\bigg(\frac{1}{2}-\Pr(U_{\bar{S},1}=1|{\bar{Y}}_2^n=\omega) \bigg)^2 &\ \ \ \ \ \ -\left(\frac{1}{2}-\Pr(U_{S,1}=1|Y_2^n=\omega) \right)^2 &=\left(\frac{1}{2}(1-a_\omega) \right)^2-\left(\frac{1}{2}(1-a_\omega)-b_\omega \right)^2 &=b_\omega(1-a_\omega)-b_\omega^2=b_{\omega}c_\omega\geq 0,
\hat{\cal W}_{\mu}\to U_{\alpha}U_{\beta}\hat{\cal W}_{\mu}U_{\beta}^{-1}U_{\alpha}^{-1}+ i (\partial_{\mu}U_{\alpha}) U_{\alpha}^{-1}+ i U_{\alpha}(\partial_{\mu}U_{\beta}) U_{\beta}^{-1}U_{\alpha}^{-1}.
l A_{1}+ n B_{1}= ( U_{l}^{n}) \sum_{q = 1}^{l}S t r \left( \ldots \phi^{i_{2 k - 1}^{\prime}}\ldots \underbrace{\partial_{a_{q}}\phi^{j_{q}}\phi^{i_{2 k}^{\prime}}}\ldots \right) + ( U_{l}^{n}) \sum_{s = 1}^{n}S t r \left( \ldots \phi^{i_{2 k - 1}^{\prime}}\ldots \underbrace{\phi^{i_{s}}\phi^{i_{2 k}^{\prime}}}\ldots \right)
\mathbf{P}^n\equiv \int d^3\mathbf{x}\left( \nabla \varphi_n^\alpha \right)\left\{\frac{\partial \mathcal{F}^n}{\partial \varphi_n^\alpha}-\nabla \mathbf{\cdot}\left( \frac{\partial \mathcal{F}^n}{\partial \nabla \varphi_n^\alpha}\right) \right\}.
E_5^{'}-E_1 = \left(f_2^*+f_3^* \right)\frac{2^{R_{51}^{'}+R_{52}^{'}}-1}{g_{51}}-f_2^*\frac{2^{R_2^*}-1}{g_{51}}+f_3^*\frac{2^{R_3^*}-1}{g_{51}}= \frac{\left(f_2^*+f_3^* \right)2^{\frac{f_2^*R_{2}^{*}+f_3^*R_{3}^{*}}{f_2^*+f_3^*}}-f_2^*2^{R_2^*}-f_3^*2^{R_3^*}}{g_{51}}<0
=i_{0}e^{-\frac{R}{L}t}+\frac{E\omega}{L[(\frac{R}{L})^{2}+\omega^{2}]}e^{-\frac{R}{L}t}+\frac{E}{L[(\frac{R}{L})^{2}+\omega^{2}]}[\frac{R}{L}sin(\omega t)-\omega cos(\omega t)]
\left( \left[{\alpha}_{m-n}^{\mu},{\alpha}_{-s}^{\sigma}\right]\left[{\alpha}_n^{\nu},{\alpha}_{s-m}^{\rho}\right] +\left[{\alpha}_{m-n}^{\nu},{\alpha}_{s-m}^{\rho}\right]\left[{\alpha}_n^{\nu},{\alpha}_{-s}^{\sigma}\right] \right)
\delta G^{A B}( \partial_{A}\partial_{B}\varphi - \overline{{{\Gamma}}}_{A B}^{C}\partial_{C}\varphi ) + \overline{{{G}}}^{A B}( \partial_{A}\partial_{B}\chi - \overline{{{\Gamma}}}_{A B}^{C}\partial_{C}\chi - \delta \Gamma_{A B}^{C}\partial_{C}\varphi ) + \frac{\partial V}{\partial \varphi}\chi = 0 ,
\prod_{a\in A}(\sqrt{2\pi m})^{1/(m-D)}= \sqrt{2\pi m}\cdot (\sqrt{2 \pi m})^{D/(m-d)}\leq \sqrt{2\pi m}\cdot e^{O(D\log{m}/(m-D))}\leq \sqrt{2\pi m}\cdot e^{O(D\log{m}/m)}.
\sigma_{5,10,2}=[2,3,5,6,8,9,11,12,14,15,1,4,7,10,13] \in \mathbb{S}_{15}, \mathfrak{G}_{5}(\beta)= (1,20,210,1420,7085,27636,87430,230240,516375,997790,1676587,2466840, \hphantom{\mathfrak{G}_{5}(\beta)= (}{}3204065,3695650,3778095,3371612,2569795,1610910,782175,262200,{\bf 45885}).
\int_T^{2T}\left(\frac{h}{k}\right)^{it}|\zeta(\sigma+it)|^2\,dt =\int_T^{2T}\left(\zeta(2\sigma)\left(\frac{(h,k)^2}{hk}\right)^\sigma+\left(\frac{t}{2\pi}\right)^{1-2\sigma}\zeta(2-2\sigma)\left(\frac{(h,k)^2}{hk}\right)^{1-\sigma}\right)dt &+O(T^{1-\sigma+\epsilon}\min(h,k)).
|\mathcal{BC}_{(s,s+d,s+2d)}|=|\mathcal{CS}_{(s,s+d,s+2d)}| &~~=\sum_{i=0}^{(s-1)/2}\binom{(d-1)/2+i}{\lfloor i/2 \rfloor}\left( \binom{(s+d-2)/2}{(d-1)/2+i}+ \binom{(s+d-4)/2}{(d-1)/2+i}\right).
K_{L , n}^{( 2 )}= \frac 3 4 \frac{\sqrt{\pi}\Gamma ( L + \frac 1 2 ) \Gamma ( N + L + 1 )}{\Gamma ( n + 1 ) \Gamma ( L + 1 ) \Gamma ( 2 L + 2 )}\, \frac{1 - 4 N^{2}}{( 2 L - 1 ) ( 2 L + 3 )}
d ( d \omega ) = \frac{1}{r !}\left( \frac{\partial^{2}}{\partial x^{\rho}x^{\nu}}\omega_{\mu_{1}\mu_{2}\cdots \mu_{r}}\right) d x^{\rho}\wedge d x^{\nu}\wedge d x^{\mu_{1}}\wedge \cdots \wedge d x^{\mu_{r}},
{Q_1}(M,P_i,P_j,\tau)= \left\{e^{\frac{1}{24}E_2(\tau)A_2}\widehat{L}(TM,\nabla^{TM}){\rm ch}\left[\bigotimes_{n=1}^{\infty}S_{q^n}(\widetilde{T_CM})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^m}(\widetilde{T_CM})\right]\right. &\left.\cdot\varphi(\tau)^{16}{\rm ch}(\mathcal{V}_i){\rm ch}(\mathcal{V}_j)\right\}^{(12)},
m_{G}^{2}\; = \; - 2 e^{2}\int_{{\mathcal R}}\frac{d^{2}p}{( 2 \pi )^{2}}\, \frac{\partial}{\partial p_{\mu}}( \frac{p_{\mu}}{p^{2}}) + 2 e^{2}\int_{{\mathcal R}}\frac{d^{2}p}{( 2 \pi )^{2}}\, \frac{\partial}{\partial p_{\mu}}( \frac{p_{\mu}}{p^{2}+ \Lambda^{2}}) \; ,
A=\sum_{r_{1}=1}^{R_{1}}\sum_{r_{2}=1}^{R_{2}}\cdot\cdot\cdot\sum_{r_{M}=1}^{R_{M}}s_{r_{1},r_{2},...,r_{M}}u_{r_{1}}\otimes u_{r_{2}}\otimes\cdot\cdot\cdot\otimes u_{r_{M}},
\big[\#\textrm{of ``$-$'' under}(i,j)\big] \ge \sum_{(k,\ell) \,\in\, \mathcal E}\big[\#{\textrm{of ``$-$'' under}(k,\ell)}\big] \ge \sum_{(k,\ell)\, \in\, \mathcal E}\tfrac12(\ell-k+1) = \tfrac12(j-i-1),
-\frac{1}{4}\left\langle Q \left| \cal F^{\cal A}_{\mu \nu}\cal F^{\cal A \mu \nu}\right | Q \right\rangle = -\frac{1}{4}\left\langle \cal F^{\cal A}_{\mu \nu}\cal F^{\cal A \mu \nu}\right\rangle = -\frac{1}{4}\left\langle{\cal F}^3_{\mu \nu}{\cal F}^3_{\mu \nu}+{\cal F}^m_{\mu \nu}{\cal F}^m_{\mu \nu}\right\rangle
u_n\coloneqq\sum_{i=1}^{n\wedge k}m_i\delta_{x_i}+\sum_{i=1}^{(n2^n)^N}u^d_+(Q^n_i)\delta_{x^n_i}-\sum_{i=1}^{(n2^n)^N}u^d_-(Q^n_i)\delta_{y^n_i},
\underset{t\rightarrow\infty}{\lim}\dfrac{\mathbb{U}_{\mathbf{F}}\left(tx\right) /\mathbb{U}_{\mathbf{F}}\left( t\right) -x^{\gamma_{1}}}{\mathbf{A}_{\mathbf{F}}\left( t\right)}=x^{\gamma_{1}}\dfrac{x^{\rho_{1}}-1}{\rho_{1}},
\left\|T(t)v-v\right\|_{L^p_{\theta}}\leqslant \left\|\left[G(t,0)v\right](e^{tS}\cdot)-v(e^{tS}\cdot)\right\|_{L^p_{\theta}}+\left\|v(e^{tS}\cdot)-v(\cdot)\right\|_{L^p_{\theta}}=: \left\|v_1(\cdot,t)\right\|_{L^p_{\theta}}+ \left\|v_2(\cdot,t)\right\|_{L^p_{\theta}}.
|b_s+c_pb_c\tilde{b}|=O(b_c)\bigg{(}b_c^{2}+\mathcal{N}^{\frac{1}{2}}+\bigg{|}\frac{\lambda_s}{\lambda}+b\bigg{|}+\bigg{|}\frac{x_s}{\lambda}-1\bigg{|}\bigg{)},
(\Psi_{N,t},e^{i\left\langle \gamma-\gamma_{1}+t,x\right\rangle})=\dfrac{(q\Psi_{N,t},e^{i\left\langle \gamma-\gamma_{1}+t,x\right\rangle})}{\Lambda_{N}(t)-\mid\gamma-\gamma_{1}+t\mid^{2}}.
\int_{\mathbb{R}}x^k\d\mu^{v_N}_{A_N+U_NB_NU_N^*}(x)=\tau^{v_N}_{A_N,U_NB_NU_N^*}((X+Y)^k)=\varphi((X+Y)^k)+o(1) &=\int_{\mathbb{R}}x^k\d\nu_1\prescript{}{\mu_1}{\boxplus}^{}_{\mu_2}\mu_2(x).
u_t = D\Delta u+H(u^{+}) x\in \Omega, \quad 0<t<T v_t =\tilde D\Delta_{M}v+F(u^{+},v^{+}) x\in M,\quad 0<t<T D\frac{\partial u}{\partial \eta}=G(u^{+},v^{+}) x\in M, \quad 0<t<T u =u_0 x\in\Omega ,\quad t=0 v =v_0 x\in M , t=0
\Gamma_a^0(x,y) =\int_0^\infty O(at)^T G(O(at)x-y, t)\, dt, \Gamma_a^{11}(x,y) =\int_0^\infty\!\!\! O(at)^T\!\!\! \int_t^\infty\!\!\! G(O(at)x-y, s)\,\frac{(O(at)x-y)\otimes (O(at)x-y)}{4s^2}\,ds\, dt, \Gamma_a^{12}(x,y) =\int_0^\infty O(at)^T \int_t^\infty G(O(at)x-y, s)\,\frac{-1}{2s}\,ds\, dt.
I_{k}:= \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{(|\tilde{u}_k(y)|^{p(x,y)}+|\tilde{v}_k(y)|^{p(x,y)})| \phi (x)- \phi (y)|^{p(x,y)}}{\vartheta_{k}^{p(x,y)}|x-y|^{N+sp(x,y)}}\,dy\,dx.
\tan \mathrm{\boldmath \Large \left( \right.}\! \! \! \phi_{l}^{( D )}( k ; a ) \mathrm{\boldmath \Large \left. \right)}\! \! = \tanh \left( \frac{\pi \Theta_{l}}{2}\right) \frac{1 -{\mathcal T}_{l}( k ; a ) \varrho_{l}}{{\mathcal T}_{l}( k ; a ) + \varrho_{l}},
\sum_{k=1}^{n+1}S_{2,\lambda}(n+1,k)k!x^{k-1}=\frac{d}{dx}\sum_{k=1}^{n+1}S_{2,\lambda}(n+1,k)(k-1)!x^{k}&=-\frac{n\lambda}{x}\sum_{k=1}^{n}S_{2,\lambda}(n,k)k!x^{k}+F_{n,\lambda}(x)+(1+x)F_{n,\lambda}^{\prime}(x),
0\leq F_\varphi(t)-G_\varphi(t) = 2t^2\left(\int_X A\varphi |Df|^2 \,d\mu -2\int_X \varphi Df\cdot DAf \,d\mu -2K \int_X \varphi |Df|\,d\mu \right)+o(t^2)
\gamma_2 = \left[\frac{\lambda\, l^{-\epsilon}|g|^2}{P\left(q^{-\epsilon}|u|^2 + r^{-\epsilon}|v|^2\right) d^{-\epsilon}|f|^2}-1\right]^{\dag}= \left[\frac{c_2}{T}-1\right]^{\dag},
\begin{array}{c}{\operatorname{d i m}H^{1}( M , T ) = \operatorname{d i m}H^{1}( \tilde{M}, T^{\star}) ~ ,}{\operatorname{d i m}H^{1}( M , T^{\star}) = \operatorname{d i m}H^{1}( \tilde{M}, T ) ~ .}\end{array}
\dot{z}_j(t)= (v-\frac{\lambda_j i}{\overline{z-z_j(t)}})\Big|_{z=z_j(t)}=\sum_{1\leq k\leq N, k\neq j}\frac{\lambda_k i}{2\pi(\overline{z_j(t)-z_k(t)})}+\bar{F}(z_j(t)).
iw_t + w_{xx}+ (\lambda p/ 2) |u|^{p-2}(w + z) + (\lambda (p-2)/ 2 ) |u|^{p-4}u^2 \overline{(w+ z)}= 0\, , & w ( x, 0 ) = \phi_1 (x) , w ( 0 , t ) = 0 \, .
\frac{e_{t+1}(L_t \cup M, F_{t+1})}{\ell + |M|}\geq \frac{\left(\binom{r}{2}-1 \right) e_t(L_t, F_t)}{\ell + (r-2) e_t(L_t, F_t)}= \frac{\frac{(r-2)(r+1)}{2}e_t(L_t, F_t)}{\ell + (r-2) e_t(L_t, F_t)}.
\int_{M\setminus B(p,\varepsilon)}\eta r^{2-n}\Delta_gu\,dv^g-\int_{M\setminus B(p,\varepsilon)}u\Delta_g(\eta r^{2-n})\,dv^g &=\int_{\partial B(p,\varepsilon)}\eta r^{2-n}\partial_{\nu}u\,ds^g-\int_{\partial B(p,\varepsilon)}u\partial_{\nu}(\eta r^{2-n})\,ds^g.
W: \begin{cases}x_{1}' =\;\bigl(\gamma_{1}x_{1}+\delta_{1}x_{2}\bigr)y\medskip x_{2}' =\;\bigl(\gamma_{2}x_{1}+\delta_{2}x_{2}\bigr)y\medskip y' =\;\bigl(\gamma x_{1}+\delta x_{2}\bigr)y.\end{cases}
\Lambda(z)=e^{Q_\lambda}z^{\lambda_0}(s_1z)^{-\bar\lambda_0}:\exp\Bigl(\sum_{r\neq0}\lambda_{r}z^{-r}\Bigr)\,, &\bar\Lambda(z)=e^{Q_{\bar\lambda}}z^{\bar\lambda_0}(s_1z)^{-\lambda_0}:\exp\Bigl(\sum_{r\neq0}\bar\lambda_{r}z^{-r}\Bigr)\,.
\rho (x) = \frac{1}{x^{2}}+\sum_{\{q \}\neq 0}\{\frac{1}{(x-q)^{2}}-\frac{1}{q^{2}}-\frac{2 q \cdot x}{q^{4}}-\frac{1}{q^{6}}(4(q \cdot x)^{2}- q^{2}x^{2}) \}
\begin{gathered}\mathcal{I}^u_k:L^2(\mathbb{R}_+\times \mathcal{C}^n,\mathcal{P},dt\times \mu^u(d\omega))\to L^2(\mathcal{C}^n,\mathcal{B}^n,\mu^u), \mathcal{I}^u_k h=\int^\infty_0 h(t)dw_k(t).\end{gathered}
E_{j(t)}:\;y^2 - x^2y =(40960000000000 - 14580000000 j^5 + 531441 j^{10})\; x^6 - (86400000000 j^2 - 39366000 j^7 )\;x^5 - (1/4 + 48600000 j^4 + 177147 j^9 /4)\;x^4 + (320000000 j + 291600 j^6 )\;x^3 - 1350000 j^3 \;x^2 &+ (1600000 + 243 j^5 )\;x-500 j^2,
\begin{aligned}D_{f_p}(z, x) = \frac{1}{q}\|x\|^{p}+ \frac{1}{p}\|z\|^{p}-\langle J_{p}(x),z\rangle, = \frac{1}{q}\|J_p(x)\|^{q}- \frac{1}{q}\|J_p (z)\|^{q}-\langle J_{p}(x)-J_p(z),z\rangle = D_{f_q}(J_p(x), J_p(z)) \end{aligned}
\|I^{+}f\|_{\mathcal{B}_{\alpha}^{2}}=\sum_{k\in\mathbb{Z}^{n}\setminus\{0\}}|k|^{4}\sup_{t\in[0,\infty)}e^{\alpha t |k|}\left|\int_{0}^{t}e^{-|k|^{4}(t-s)}\hat{f}(s,k)\ds\right|.
(-1)^{(n-n/d)(a-1)}\mu(d)\binom{2(a,n/d)-1}{(a,n/d)-1}&\kern2cm+(-1)^{(n-n/pd)(a-1)}\mu(pd)\binom{2(a,n/pd)-1}{(a,n/pd)-1}&\kern-10pt=\binom{2(a,n/d)-1}{(a,n/d)-1}\mu(d)\left((-1)^{(a-1)(n-n/d)}-(-1)^{(a-1)(n-n/pd)}\right) &\kern-10pt=0.
\prod_{j=1}^J\left(\sum_{p_1 < \cdots < p_h \in P_j}\frac{1}{p_1\cdots p_h}\right)\cdot \left(\sum_{p_1 < \cdots < p_{k-Jh}\in P_{J+1}}\frac{1}{p_1\cdots p_{k-Jh}}\right).
\frac{h_{\overline{z}}}{h_{z}}=\frac{(a^{2}+b^{2})E(E-G+2iF)}{(a^{2}+b^{2})E(E+G+2\sqrt{EG-F^{2}})}=\frac{E-G+2iF}{E+G+2\sqrt{EG-F^{2}}}=\mu(z).
N^{\eta}_{\overline{T}_K}=\sum^{\infty}_{q=0}\sum^{\left\lfloor q/2\right\rfloor}_{\ell=0}b^{\eta}_{q-2\ell}a_{2\ell}\int_{[0,K\pi]_{-\alpha}}H_{q-2\ell}\left(\overline{T}_K(s)\right)H_{2\ell}\left(\mathcal{T}^{\prime}_K(s)\right)v_K(s)ds.
\langle \varphi^{2}( x ) \rangle_{b}\approx - \frac{\Gamma^{- 2}( \nu_{0}+ 1 )}{2^{2 \nu_{0}}\pi a^{D - 1}S_{D}\sigma^{D - 1}}\left( \frac{r}{a}\right)^{2 \nu_{0}- n}\int_{m a}^{\infty}d z \frac{z^{2 \nu_{0}+ 1}}{\sqrt{z^{2}- m^{2}a^{2}}}\frac{\bar{K}_{\nu_{0}}( z )}{\bar{I}_{\nu_{0}}( z )}, \quad r \to 0 ,
\mathcal{M}_q[N]=\left\{\prod_{\substack{\mathcal{R},\bar{\mathcal{R}}_i,\mathcal{T}: \mathcal{R}\cup\bar{\mathcal{R}}_i\not\ni q}}\left[\alpha_{{\mathcal{R}},{\mathcal{T}}}^{\bar{\mathcal{R}}_i}\tilde{h}_{q,\mathcal{T}}^{\bar{\mathcal{R}}_i}\right]^{s_{{\mathcal{R}},{\mathcal{T}}}^{i}}: 1\le s_{{\mathcal{R}},{\mathcal{T}}}^{i}\le N\right\},
\Phi = V^{-2}\circ t_{\delta_3}\circ t_{\delta_2}\circ t_{\delta_1}\circ V^{-1}= V^{-2}\circ (t_{\alpha_3}\circ t_{\alpha_2}\circ t_{\alpha_1}\circ t_{\beta})^3\circ V^{-1}= (V^{-1}\circ t_\beta\circ V)\circ t_{\beta}\circ (V\circ t_\beta \circ V^{-1}) =t_{\gamma_{-1}}\circ t_{\beta}\circ t_{\gamma_{1}}.
w(r_1) =\sum_{i=1}^k2^ix_i^{\prime}=\sum_{i=1}^{k-1}2^ix_i^{\prime}+2^kx_k^{\prime}\leq 2^k+2^kx_k^{\prime}=2^k(1+x_k^{\prime})\leq 2^k x_k\leq w(r_2).
F_{\mu \nu \alpha \beta}^{- 1 \sigma \rho}F_{\rho \omega}^{\alpha \beta \eta \kappa}= \frac{1}{2}\delta_{\omega}^{\sigma}\left( \delta_{\mu}^{\eta}\delta_{\nu}^{\kappa}- \delta_{\mu}^{\kappa}\delta_{\nu}^{\eta}\right)
\int_0^{\frac{\pi}{2}}\cos\left( 2xz \right) \log\left( \tan x \right) \d x = \frac12 \int_0^{\frac{\pi}{2}}\left[e^{2izx}+e^{-2ixz}\right] \log\left( \tan x \right) \d x = -\sin\left( \pi z \right) \sum_{n=1}^{\infty}\left( 1 - \frac{1}{2^{2n+1}}\right)\zeta(2n+1) z^{2n-1},
(b,y)\in B_I \Longleftrightarrow Z_{B(y)}(b)\cap D_n=D_I &\Longleftrightarrow Z_{B(ty)}({}^tb)\cap{}^tD_n={}^tD_I &\Longleftrightarrow Z_{B(ty)}({}^tb)\cap D_n=D_I &\Longleftrightarrow ({}^tb,ty)\in B_I\,,
\Big[\sum_{i=0}^{+\infty}x_it^{i},\sum_{j=0}^{+\infty}y_jt^{j},\sum_{k=0}^{+\infty}z_kt^{k}\Big]=\sum_{s=0}^{+\infty}\sum_{i+j+k=s}[x_i,y_j,z_k]t^{s},\;\forall x_i,y_j,z_k\in L.
\sum_{x^n:c(x^n)>0}\frac{c(x^n)}{c(f_n,x^n)}&= \sum_{m=1}^{2^{nR}}\sum_{x^n \in f_n^{-1}(m), c(x^n)>0}\frac{c(x^n)}{\sum_{\tilde{x}^n \in f_n^{-1}(m)}c(\tilde{x}^n)}&\leq 2^{nR},
\int \delta w^1_h(x,y)\frac{(2-\sigma)\langle A^h(x)y,y\rangle\rho(x+he,y)}{|y|^{n+\sigma+2}}dy=\frac{f(x+he)-f(x)}{|h|^\beta}+k_h(x)|h|^{1-\beta}-I^1_h(x)-I^2_h(x)
\frac{Z_t}{|Z_t|}= \frac{Z_u}{|Z_u|}+ \int_u^t \Big(2\frac{Z^\perp_s}{|Z_s|}\frac{d\gamma_s}{|Z_s|^{1/3}}-\frac{2Z_s}{|Z_s|^{5/3}}ds\Big)=\frac{Z_u}{|Z_u|}+ \int_u^t \Big(\frac{Z^\perp_s}{|Z_s|}\frac{d\gamma_s}{\sqrt{R_s}}-\frac{Z_s}{|Z_s|}\frac{ds}{2R_s}\Big).
\begin{array}{c l}{{}}&{{\mathrm{R e s}_{z_{2}= z_{1}\tau^{2}}\cdots \mathrm{R e s}_{z_{n}= z_{n - 1}\tau^{2}}w_{\alpha_{1}\cdots \alpha_{n}}( z_{1}, \cdots , z_{n})}}\\{{=}}&{{\delta_{\alpha_{1}n}\delta_{\alpha_{2}n - 1}\cdots \delta_{\alpha_{n}1}z_{1}^{n - 1}( \tau^{2}- 1 )^{n - 1}\tau^{( n - 1 )^{2}}\left[ n - 1 \right] ! ~ ~ v^{( n \cdots 2 1 )},}}\\ \end{array}
\frac{1}{2}\langle\Psi_{4,3}|Id(a,b)|\Psi_{2,1}\rangle=\langle\Psi_{4,3}|\mathrm{B}(a,b)|\Psi_{2,1}\rangle+\langle\Psi_{4,2}|\mathrm{B}(a,c)|\Psi_{3,1}\rangle+\langle\Psi_{3,4}|\mathrm{B}(b,c)|\Psi_{2,1}\rangle\;.
(e^{At}\xi)^{(i)}=e^{\lambda t}\sum_{j=0}^{d-i}\frac{t^j}{j!}\xi^{(i+j)},(e^{-As}\xi)^{(i)}=e^{-\lambda s}\sum_{j=0}^{d-i}\frac{(- s)^j}{j!}\xi^{(i+j)}.
\sup_{i=1,\dots,d}\sup_{\substack{x \neq y x,y \in \overline{\Omega_n'}}}\frac{|\bar v_{i,n}(x)-\bar v_{i,n}(y)|}{|x-y|^{\alpha}}= \frac{|\bar v_{1,n}(0)-\bar v_{1,n}\left(\frac{y_n-x_n}{r_n}\right)|}{\left|\frac{y_n-x_n}{r_n}\right|^{\alpha}}= 1
2\Lambda_1 = \lim_{m\rightarrow \infty}\left[\frac{u_{2m}(\mathrm{i}\pi s/2)^{m-1/2}}{\Gamma(m-1/2)}-\frac{u_{2m+2}(\mathrm{i}\pi s/2)^{m+1/2}}{\Gamma(m+1/2)}\right], 2\mathrm{i}\Lambda_2 = \lim_{m\rightarrow \infty}\left[ \frac{u_{2m}(\mathrm{i}\pi s/2)^{m-1/2}}{\Gamma(m-1/2)}+\frac{u_{2m+2}(\mathrm{i}\pi s/2)^{m+1/2}}{\Gamma(m+1/2)}\right].
C(l,h)=\begin{cases}\displaystyle\frac{-(-1)^{n-l}}{(n-1)!}{n-2 \choose l-1}h=0, begin{align*}4mm]\displaystyle\frac{(-1)^{n-l}}{(n-1)!}{n-2 \choose l-1}h=1, begin{align*}4mm]0 h\ge2.\end{cases}
\begin{array}{rcl}(-c^2\Delta_{x,y}+m^2c^4)U(x,y) = 0 \mathbb{R}^{n+1}_+ -c\frac{\partial U}{\partial y}(x,0) = (mc^2-\mu)u(x) + |u|^{p-2}u(x) \partial\mathbb{R}^{n+1}_+ = \mathbb{R}^n\end{array}
L_iL_j u - L_jL_i u = \bigg[(a\nabla\xi_i)_l \frac{\partial(a\nabla\xi_j)_{l'}}{\partial x_l}- (a\nabla\xi_j)_l\frac{\partial(a\nabla\xi_i)_{l'}}{\partial x_l}\bigg] \frac{\partial u}{\partial x_{l'}}= B_{ij, l'}\frac{\partial u}{\partial x_{l'}}\,,
\Psi(Z) = q^A \zeta^B s^C \prod_{(n,r, m) > 0}(1 - q^n \zeta^{r}s^{tm})^{\mathrm{mult}(n,r,m)}= q^A \zeta^B s^C \prod_{d | N}\prod_{(n, r, m) > 0}(1 - q^{dn}\zeta^{dr}s^{dtm})^{\mathrm{mult}_d(nm, r)}
\mathbf{a}_{\mathrm{h}}\left(\phi,\theta\right) = \frac{1}{\sqrt{M_{\mathrm{h}}}}\left[1, \,\, e^{j2\pi \frac{d_{\mathrm{h}}}{\lambda}\cos\phi\sin\theta}, \,\, \cdots, \,\, e^{j2\pi \left(M_{\mathrm{h}}-1\right)\frac{d_{\mathrm{h}}}{\lambda}\cos\phi\sin\theta}\right]^{\mathrm{T}},
\frac{q}{4}\sum_{k=3}^{n-3}\sum_{l=3}^{n-k}kl{n \choose k}{n-k \choose l}\frac{(k-1)!}{2}\frac{(l-1)!}{2}&=\frac{q}{4}\sum_{k=3}^{n-3}\sum_{l=3}^{n-k}{n \choose k}{n-k \choose l}\frac{k!}{2}\frac{l!}{2}&=\frac{q}{16}\sum_{k=3}^{n-3}\sum_{l=3}^{n-k}\frac{n!}{(n-k-l)!}.
\tilde{q}_{c}^{( a b )}( r ) = \frac{1}{4 \pi^{2}b^{4}}\sum_{m = 0}^{\infty}{'}\int_{0}^{\infty}{d z \sum_{\lambda = 0}^{1}z^{3}\tilde{\Omega}_{\lambda m}^{c}( \sigma , z ) F_{\lambda m}^{( q )}( z , z y )}
\Sigma_{(f)}= M^2 \{{{\hat{\lambda}}\over 2\epsilon}+{\hat{\lambda}}^2 ({1\over 2\epsilon^2}-{1\over 4\epsilon}) \}+{{\hat{\lambda}}^2\over 24\epsilon}p^2 +{{\hat{\lambda}}^2\over 2\epsilon}M^2{\partial\over \partial{M^2}}G +{3\hat{\lambda}\over 2\epsilon}G.
Z_{3}^{- 1}= 1 + \frac{\alpha_{0}}{3 \pi}\ln \frac 1{i \gamma m^{2}s_{0}}+ \frac{\alpha_{0}^{2}}{4 \pi^{2}}\left( \ln \frac 1{i \gamma m^{2}s_{0}}+ \frac{1 0}{3}- \frac 5 3 \ln 2 \right) .
E_{r,n}(x^{-r}\ln x;[cn^{-2},1]) \ge \min\left\{\frac{C_1}{M_1}-2\sqrt{c}, 2\left(\frac{C_2}{2rM_1}- \sqrt{c}\right)\right\}\cdot m^r &= \min\left\{\frac{C_1}{M_1}-2\sqrt{c}, 2\left(\frac{C_2}{2rM_1}- \sqrt{c}\right)\right\}\cdot (\frac{n}{\sqrt{c}})^r.
\langle g\left(\cdot,t\right), \phi_p \rangle= \begin{cases}\dfrac{1}{n}\sum_{k=1}^n g(x_k,t) - \widetilde{H}_{n0}(t), p=0, \dfrac{\pi}{n}\sum_{k=1}^n g(x_k,t) \phi_p(x_k) - \widetilde{H}_{np}(t), 1 \le p \le n-1.\end{cases}
4r^2 \| (I-\eta H)^{\tau+1}H (I-\eta H)^{\tau+1}\|_2 + 4 \eta r (L_2 12^2(\mathcal P \cdot \hat c)^2) \sum_{k=0}^{\tau}\|(I-\eta H)^{\tau+1}H(I-\eta H)^k\|_2 &\leq 2\mathcal P^2 \gamma \hat c\log^{-1}(d\kappa /\delta).
\Lambda_0=\left( a_0(u)+\frac{a_1(u)}{u_1}+\frac{a_2(u)}{u_1^2}+\left(a_1(u)+2a_2(u)\frac{1}{u_1}\right) \xi_1+ a_2(u)\xi_1^2\right)\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}
\varphi(\theta^{k},\tilde{\theta}^{k},\zeta^k)=\langle\lambda^{k}-\tilde{\lambda}^{k},\tilde{y}^{k}-y^{k}\rangle_{\mathcal{L}^2}+\langle\theta^{k}-\tilde\theta^{k}, Gd(\theta^k,\tilde{\theta}^{k},\zeta^k)\rangle_{\mathcal{L}^2},
\pi^*_{1,j}=-\infty, ~ \mathrm{and}~\pi^*_{1,j+R_1}=0, \varphi_j(\hat{\xi}_{n,j}(\theta'_n))=0, ~ \mathrm{and}~\varphi_{j+R_1}(\hat{\xi}_{n,j+R_1}(\theta'_n))=0,
E[S_n'(\theta_j^{(n)})]=n\pi_n(\theta_j^{(n)}) \geq\mathrm{e}^{-3}(1-r^{-1})^{r-1}\frac{K^{r-1}}{r}\theta_j^{(n)}>K^{1/\lceil K\rceil}\theta_j^{(n)}=\theta_{j+1}^{(n)}.
{(\overline{F}\cdot \mu_A)_{c, d, e}}_{(f, g, y)}={\overline{F}_{c, e, fg}}_{(y)},{((\overline{F}\cdot Ta) \circ (b \cdot T\overline{F}))_{c, d, e}}_{(f, g, y)}= (\overline{F}_{c, d, f}\circ A(g))_y \circ (B(f) \circ \overline{F}_{d, e, g})_y.
\phi = \phi_{\rm reg}+ \sum_{e\in E}\phi^e + \sum_{v\in V}\phi^v + \sum_{v\in V}\sum_{e\in E(v)}\phi^{ev}, u = u_{\rm reg}+ \sum_{e\in E}u^e + \sum_{v\in V}u^v + \sum_{v\in V}\sum_{e\in E(v)}u^{ev},
{\cal A}= ( - 1 ) ( \tilde{F}_{1}\tilde{F}_{1}^{*}+ \tilde{F}_{7}\tilde{F}_{7}^{*}) + 2 ( \tilde{F}_{2}\tilde{F}_{2}^{*}+ \tilde{F}_{6}\tilde{F}_{6}^{*}) + ( - 1 ) ( \tilde{F}_{3}\tilde{F}_{3}^{*}+ \tilde{F}_{5}\tilde{F}_{5}^{*}) + 8 \tilde{F}_{4}\tilde{F}_{4},
c_n :={s_n}^{-1}\cdot\sup_{(t,x) \in [0,1] \times [j_n,j_{n+1}]}|f(\varphi (s_n+t,x))-f(\varphi (s_n,x))|, d_n := \max\{M_n,1\}\cdot \max_{n-1 \leq i \leq n +1}\left\{|a_{i}|, |a_{i}-a_n|, c_i \right\}
\mathcal{T}^* := \,\, \sum_{n_1\sim N_1}\cdots\sum_{n_6\sim N_6}(\log n_1)\mu(n_4)\mu(n_5)\mu(n_6) & \,\, \qquad\qquad\times \mathbf{e}\big(jd^{-1}(n_1n_2\cdots n_6)^\gamma+m_1(n_1n_2\cdots n_6)\big),
\mathbb{EI}_{\ell-1}(\theta^{(\ell)}) \le (p'\theta^{(\ell)}-p'\theta^{*,\ell-1})_+\bigl(1-\Phi(-R/\varsigma)\bigr) &= (p'\theta^*-p'\theta^{*,\ell-1}-(p'\theta^*-p'\theta^{(\ell)}))_+\bigl(1-\Phi(-R/\varsigma)\bigr) &\le (p'\theta^*-p'\theta^{*,\ell-1})\bigl(1-\Phi(-R/\varsigma)\bigr)+|p'\theta^*-p'\theta^{(\ell)}|,
A^{W,W}_\cdot(l) = \lim_{n\to\infty}A^{W,W}_\cdot (l_n) & =\lim_{n\to\infty}\left( \phi_n(\cdot, W) - \phi_n(0, W_0) - \int_0^\cdot \nabla \phi_n^* (r, W_r) \mathrm d W_r \right).
\Theta=\dfrac{\prod_{i=1}^m\alpha^r_i\prod_{s=0}^{r-1}a_{0,s}^m}{(n-1)!^{r^2m}}\cdot{\rm{det}}\left({\rm{Eval}}_{\alpha_i}\bigcirc_{{{w}}=0}^s(\theta_t+\gamma_{r-s+{{w}}})^{-1}(t^nH_{\ell}(t))\right)_{\substack{0\le \ell \le rm-1 1\le i \le m, 0\le s \le r-1}}.
r_{W_{\mathcal{I}},X_{T^c}^n, Y_{T^c}^n}= p_{W_{\mathcal{I}}}\prod_{k=1}^n \left( p_{X_{T^c,k}|W_{T^c}, X_{T^c}^{k-1}, Y_{T^c}^{k-1}}s_{Y_{T^c,k}|X_{T^c,k}}^{(\lambda, T)}\right).
G({\bf x}_{n+1},{\bf x}_n;\varepsilon\vert{\hat{\bf A}})\simeq\biggl({m\over{2\pi i\varepsilon}}\biggr)^Nexp\biggl[ i\sum_{a=1}^N \biggl({m(\Delta{\bf x}^a_{n})^2\over2\varepsilon}+ e\Delta{\bf x}^a_{n}\cdot{\hat{\bf A}}({\bar{\bf x}}^a)\biggr)\biggr]\,.
\begin{aligned}Q_a^{\varepsilon}(t)^*Q_a^{\varepsilon}(t)f(x) =(2\pi)^{-\nu n}\iint e^{i(x-y)\xi}q_a^{\varepsilon}(t,x,\xi)^2f(y)dyd\xi &+(2\pi)^{-\nu n}\iint e^{i(x-y)\xi}D_\xi\tilde\nabla_xq_a^{\varepsilon}(t,x,\xi,y)f(y)dyd\xi.\end{aligned}
Y^{(0)}(t,x) =Y^{(0,m,\beta)}(t,x)=Y_<(t,x), Y^{(0,\beta)}_<(t,x) =Y^{(0,\beta)}(t,x)=1,u^{(0,\beta)}_<(s,y;t,x)=g(t-s,x-y)