harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{yy}(t)\otimes (\delta(-t) - \alpha \theta(-t-\tau))\otimes (\delta(t) - \alpha \theta(t-\tau)) = R_{n_{\rm R}n_{\rm R}}(t) \otimes h_{\rm RD}(-t) \otimes \theta(-t)\otimes h_{\rm RD}(t) \otimes \theta(t) & + R_{n_{\rm D}n_{\rm D}}(t) \otimes(\delta(-t) - \alpha \theta(-t-\tau))\otimes (\delta(t) - \alpha \theta(t-\tau)).
|f_{I}-f_{J}| \lesssim \frac{1}{|I|}\int_{I}|f(e^{is})-f_{J}|ds &\lesssim\Big(\frac{1}{|I|}\int_{I}|f(e^{is})-f_{J}|^{2}ds\Big)^{1/2}&\lesssim|J|^{2-2\beta}\|f\|_{BMO^{\beta}(\mathbb{T})}.
M_3(G) = M_3(T) +M_3(CQ) + \nu_2M_2(T)+(\nu_1-1)M_2(CQ) &= \frac{(\nu_1 + \nu_2)^3 - 12(\nu_1 + \nu_2)^2 + 35(\nu_1 + \nu_2)+ 6(x+y)}{6}&= f(n) + x+y.
F_{N}(a,b;t)=(1- t q^{N})\sum_{n=0}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(a q)_n(q)_n\left(\frac{at q}{b}\right)_n(a t q^2)_{N-1}(tb)^nq^{n^2}(1-at q^{2n+1})}{(b q)_n(t)_{n+1}(a t q^2)_{N+n}}.
H ( r_{0}) = ( 3 - p ) \, \frac{q_{1}^{2}}{r_{0}^{7 - 2 p}}\left( \frac{h_{2}^{\prime}}{h_{2}}- \frac{h_{1}^{\prime}}{h_{1}}\right)^{- 1}\Bigg \vert_{r = r_{0}}\, .
f^{( 2 )}= \left( \begin{array}{c c c c}{{0}}&{{- \delta_{i j}}}&{{2 a_{i}}}&{{\pi_{i}}}\\{{\delta_{i j}}}&{{0}}&{{0}}&{{a_{i}}}\\{{- 2 a_{i}}}&{{0}}&{{0}}&{{0}}\\{{- \pi_{i}}}&{{- a_{i}}}&{{0}}&{{0}}\\ \end{array}\right) ,
[ p_{i}, x_{j}] = \delta_{i j}e^{- 2 p_{0}/ \kappa}- \frac{1}{\kappa^{2}}\left( \vec{p}{}^{2}\delta_{i j}- 2 p_{i}p_{j}\right) , \quad [ p_{0}, x_{i}] = - \frac{2}\kappa p_{i}
X_{+}(x) = X_{-}(x) \begin{pmatrix}1 \sqrt{2}x^{\beta}e^{-nV(x)}0 0 1 0 0 0 1\end{pmatrix}, x > 0, X_{+}(x) = X_{-}(x) \begin{pmatrix}1 0 0 0 0 (-1)^{n+1}0 (-1)^n 0\end{pmatrix}, x < 0,
\int_0^x e^{-\beta_0/y}dy=xe^{-\beta_0/x}-\Gamma\left(0,\frac{\beta_0}{x}\right)=xe^{-\beta_0/x}-\left(1,\frac{\beta_0}{x}\right) \hphantom{\int_0^x e^{-\beta_0/y}dy}{}=x-\beta_0 \log (x)+\beta_0(\log (\beta_0)+\gamma-1)-\frac12\beta_0^2 x^{-1}+\cdots.
{\mathsf W}_{- +}^{( 4 5 )}= \frac{1}{2 !}h_{a b}^{^{( 4 5 )}}< \widehat{\Psi}_{( - ) a}^{*}| B \Sigma_{\mu \nu}| \widehat{\Psi}_{( + ) b}> \Phi_{\mu \nu}
2^{-p}\int_M |u-v|^p \omega_{(\frac{u+v}{2})}^n\wedge\eta= \int_M \left|u-\frac{u+v}{2}\right|^p \omega_{(\frac{u+v}{2})}^n\wedge\eta\leq d_p\left(u, \frac{u+v}{2}\right)^p
[u,e_1]+\lambda J[u,e_1] + J[u,Xu+Ye_1+Ze_2]-\lambda [u,Xu+Ye_1+Ze_2] = -e_1-\lambda Je_1 -YJe_1-Z\theta Je_2 + \lambda Ye_1+\lambda Z\theta e_2 = (-1-\lambda Y-Y^2-Z\theta B+\lambda Y)e_1+(-\lambda Z-YZ-Z\theta C+\lambda Z\theta)e_2+ &+ (-\lambda X-XY-Z\theta A)u
w(q) = \phi(-q^3)\frac{\eta_2^2\eta_3}{\eta_1\eta_6}&= \phi(-q^3)\left(\phi(-q^9)+\frac{\eta_3\eta_{18}^2}{\eta_6\eta_9}\right) &= \phi(-q^3)\phi(-q^9)+\frac{\varphi_3^2}{\varphi_6}q\frac{\varphi_3\varphi_{18}^2}{\varphi_6\varphi_9}&=\phi(-q^3)\phi(-q^9)+qx(q^3).
L^{( - )}= \left( \begin{array}{c c}{( - 1 )^{F}q^{- \left( \begin{array}{c c}{1}&{0}\\{0}&{0}\\ \end{array}\right)}}&{0}\\{( q^{- 1}- q ) ( - 1 )^{F}Y^{-}}&{( - 1 )^{F}q^{- \left( \begin{array}{c c}{0}&{0}\\{0}&{1}\\ \end{array}\right)}}\\ \end{array}\right)
- \int_r^R \pi t\partial_t (G_t(\eta,\xi_k,\dots, \xi_n)) \frac{dt}{t}& = \int_r^R 2\pi^2 t^2 \Big(\alpha^2\eta^2 + \sum_{j=k}^n \alpha_j^2(\xi_j^2 + (\xi_j+\eta)^2) \Big) \,G_t(\eta,\xi_k,\dots, \xi_n) \frac{dt}{t}.
c_{d}( f_{1}, f_{2}) = d \left\{m_{1}\lambda_{2}- m_{2}\lambda_{1}+ \frac{1}{4 \pi}\int_{0}^{2 \pi}\biggl ( \frac{d \tilde{f}_{1}}{d \sigma}\tilde{f}_{2}- \frac{d \tilde{f}_{2}}{d \sigma}\tilde{f}_{1}\biggr ) d \sigma \right\}.
\begin{cases}-dY_{t}=f(t,Y_{t},Z_{t},Y_{t+\delta_{t}},Z_{t+\zeta_{t}}) dt - Z_{t}dW_{t}, \ \ \ 0\leq t\leq T; Y_{t}=\xi_{t}, \ \Z_{t}=\eta_{t}, \ \ \T\leq t\leq T+K, \end{cases}
\pi_{\phi}(x)=\sum_{\substack{\textup{$\mathfrak{p}$ unramified in $L$}\mathrm{N}_{F/\mathbb{Q}}~\mathfrak{p}\leq x}}\phi(\mathrm{Frob}_{\mathfrak{p}}),\widetilde{\pi}_{\phi}(x)=\sum_{\substack{\textup{$\mathfrak{p}$ unramified in $L$}\mathrm{N}_{F/\mathbb{Q}}~\mathfrak{p}^m\leq x}}\frac{1}{m}\phi(\mathrm{Frob}_{\mathfrak{p}}^m).
|r^{2}\partial_{r}^{2}v_{4}(t,r)| \leq \frac{C r}{t^{2}\log^{3b+2N}(t)}+ \frac{C r^{2}}{t^{4}\log^{3b-2+N}(t)}+ \frac{C r}{t^{2}\log^{3b+2N-1}(t)}+ \frac{C r}{t^{2}\log^{3b+2N-1}(t)}&\leq \frac{C r}{t^{2}\log^{3b+2N-1}(t)}, r \leq \frac{t}{2}
\sum_{\substack{a=1 (a,q)=1}}^q\int_{\beta\in\mathcal{B}(q,a)}\Big(\cdots\Big)d\beta = \int_{|\beta|\le\frac{1}{qP}}\sum_{|v|\le P}\sigma(v;\beta,q)\sum_{\substack{a=1 (a,q)=1}}^q\Big(\cdots\Big)e\Big(-\frac{\overline{a}\,v}{q}\Big)d\beta
\sum_{i,k=r+1}^{d}(\lambda_i - \widetilde \lambda_k)^2 \langle p_i, q_k\rangle^2 \leq \sum_{i,k=r+1}^{d}(\lambda_i^2 + \widetilde \lambda_k^2) \langle p_i, q_k\rangle^2 & \leq 2 \sum_{i=r+1}^{d}\lambda_i^2.
\sum_{k=1}^4\log[(1-|\zeta_k|^2)|f'(\zeta_k)|]+\sum_{k=1}^4\sum_{{\substack{l=1 l\neq k}}}^4\delta_k\delta_l\log\left|\frac{1-\overline{\zeta}_k\zeta_l}{\zeta_k-\zeta_l}\right|\le 2\log \frac{4\rho^2}{1-\rho^2}.
\begin{aligned}\|e^{u(s,\cdot)}\|_{L^2}^2=~ \int_{M}e^{2u(s,\cdot)}=\int_{M}e^{2(u(s,\cdot)-\overline{u})}e^{2I}\leq~ C\exp\left(\frac{1}{4\pi}\int_{M}|\nabla u(s,\cdot)|^2\right)e^{2I}\leq Ce^{2I}e^{\frac{1}{4\pi}R^2},\end{aligned}
R_1(x;p) = \int_x^p (u-x)r''(u)du \leq \int_x^p (u-x) 2u^{2\alpha-2}du \leq 2 \int_x^p u^{2\alpha-1}du \leq 2 \int_0^p u^{2\alpha-1}du = \frac{p^{2\alpha}}{\alpha}.
\sum_{I_{1}I_{2}\cdots I_{n}= ( I )}X^{I_{1}, I_{2}, \ldots , I_{n}}\equiv \sum_{I_{1}, I_{2}, \ldots , I_{n}}\delta_{( I )}^{I_{1}I_{2}\cdots I_{n}}X^{I_{1}, I_{2}, \ldots , I_{n}}.
\det(J_{n}-zI_{n})=\left(\prod_{k=1}^{n}(\lambda_{k}-z)\right)\mathfrak{F}\!\left(\frac{\gamma_{1}^{\,2}}{\lambda_{1}-z},\frac{\gamma_{2}^{\,2}}{\lambda_{2}-z},\dots,\frac{\gamma_{n}^{\,2}}{\lambda_{n}-z}\right)\!.
\int\limits_{-b}^{-a}\hat\Gamma(z) e\left(-\beta z^3-\alpha z\right) \dz - \hat\Gamma\left(-\frac{|\alpha|^{1/2}}{(3\beta)^{1/2}}\right) \cdot \frac{1}{2^{1/2}(3|\alpha|\beta)^{1/4}}\cdot e\left(-\frac{1}{8}+\frac{2|\alpha|^{3/2}}{3^{3/2}\beta^{1/2}}\right) = O\left(\Delta\log(2+\beta)\cdot |\alpha|^{-1}\right).
{\cal P}({\cal E}) = \sum_{\vert \Psi \rangle}\vert{\cal A}({\cal E}) \vert^{2}= \int_{- \infty}^{\infty}d \tau \int_{- \infty}^{\infty}d \tau^{\prime}e^{- i{\cal E}( \tau - \tau^{\prime})}{\tilde{G}}\left[{\tilde{x}}( \tau ) ,{\tilde{x}}( \tau^{\prime}) \right] ,
\beta_{ni}(\tau) = \frac{\exp\left(-\frac{\sum\limits_{j=-\tau}^{-1}(x_{n+j}- x_{(i-1)l+j + \tau + 1})^2}{2h^2}\right)}{\sum\limits_{k=1}^{N}\exp\left(-\frac{\sum\limits_{j=-\tau}^{-1}(x_{n+j}- x_{(k-1)l+j + \tau + 1})^2}{2h^2}\right)},
\|W(t)\|_1^2 + \|W_t(t)\|^2+\int_0^t \int_\mathbb{R}\bigg(\sum_{i,j = 1\atop j \neq i}^n |u^j_x| w_i^2+\sum_{i=1 \atop i \neq p}^n |u^i_x| w_i^2+ |u^p_x|^2w_p^2\bigg)dxd\tau &+ \int_0^t(\|W_x\|^2 + \|W_\tau\|^2)d\tau\leqslant O(1)\big( \|W(0)\|_1^2 + \|W_t(0)\|^2\big)+ O(1) \varepsilon_0 \delta_0^{{1}/{2}},
T_2= (n-2) \sigma_1(A_g)g-8A_g=-\frac{8}{n-2}{\rm Ric}_g+\frac{n^2-4n+12}{2(n-1)(n-2)}R_gg; T_4= -\frac{3n^2-12n-4}{4}\sigma_1(A_g)^2 g+4(n-4)|A|_g^2 g+8(n-2)\sigma_1(A_g)A_g &+(n-6)\Delta_g \sigma_1(A_g) g-48A_g^2-\frac{16}{n-4}B_g; v_6= -\frac{1}{8}\sigma_3(A_g)-\frac{1}{24(n-4)}\langle B,A\rangle_g.
T_{1}=(2p^{2})^{-1}[ f^{(2)}(\tau_{1})E\{(\|X_{1}-X_{2}\|_{2}^{2}-p\tau_{1})^{2}\}+f^{(2)}(\tau_{2})E\{(\|Y_{1}-Y_{2}\|_{2}^{2}-p\tau_{2})^{2}\}&-2f^{(2)}(\tau_{3})E\{(\|X_{1}-Y_{1}\|_{2}^{2}-p\tau_{3})^{2}\}]
{\cal L}_0(\phi,\partial_\mu\phi)=\frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-\frac{1}{2}m^2\phi^2=\frac{1}{2}\left(\partial_\mu\phi\right)^2- \frac{1}{2}m^2\phi^2\ .
\begin{aligned}\Phi^{P}_{g}(f_0, f_1, \dots, f_m) := \int_{h \in M/Z_M(x)}\int_{K N}\int_{G^{\times m}}C\big( H(g_1...g_mk), H(g_2...g_mk ),\cdots, H(g_mk)\big) &f_0 \big (kh g h^{-1}nk^{-1}(g_1\dots g_m)^{-1}\big)f_1(g_1) \dots f_m(g_m)dg_1\cdots dg_m dk dn dh, \end{aligned}
V_{i \bar{j}}\Big [ \begin{array}{c c}{b_{3}}&{b_{4}}\\{\emptyset}&{\emptyset}\\ \end{array}\Big ] = a_{2 3}^{i}\bar{a}_{4 5}^{j}( a_{1 3 4 5 2 6}\bar{a}_{1 6 4 3 2 5}) | ( 1 2 4 | 3 5 6 )^{2}( 1 3 5 | 2 4 6 ) | ,
-\frac{i}{2}\left(q_tr+qr_t\right)-\frac{i}{2}\left(r^2 qq_x+q^2rr_x\right)-\frac{1}{2}\left(rq_{xx}-qr_{xx}\right) =0 q_t -iq_{xx}+q^2 r_x -\frac{i}{2}r^2q^3 =0 r_t +ir_{xx}+r^2 q_x +\frac{i}{2}r^3 q^2 =0 \frac{i}{2}\left(q_tr+qr_t\right)+\frac{i}{2}\left(r^2 qq_x+q^2rr_x\right)+\frac{1}{2}\left(rq_{xx}-qr_{xx}\right) =0
I_{4}= \underset{\mathcal{A}^{c}}{\int}\frac{u_{a_{j}}( a_{i})}{(1+r^{2})^{\frac{n+2}{2}}}(\frac{1}{\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\gamma_{n}G_{a_{j}}^{\frac{2}{2-n}}(\exp_{g_{a_{i}}}\frac{x}{\lambda_{i}})})^{\frac{n-2}{2}}.
[\begin{matrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&1\\ \end{matrix}]
\underline{\underline{A_{\epsilon}}}=[\begin{matrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&-1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&1\end{matrix}]
\sum_{k=0}^{n}q_1^k P_k(q_2)*\sum_{p=0}^{m-1}q_1^p H_p(q_2)+\sum_{k=0}^{m}q_1^kQ_k(q_2)*\sum_{p=0}^{n-1}q_1^p K_p(q_2)\equiv 0
M(\lambda):= \left\{\left( \phi_{2}(\lambda) - \phi_{1}(\lambda), V \right) \mid V \geq 0 \right\}; & M_{\Phi}(\lambda):= \left\{\left( \phi_{2}(\lambda) - \phi_{1}(\lambda), \Phi \left( \phi_{2}(\lambda) - \phi_{1}(\lambda); \lambda, \alpha \right) \right) \mid \alpha \geq 0 \right\}.
(H\varphi)(x) =\sum_{j=1}^{2n}\frac{t_{\rm s}^{(j)}}{2i}[\varphi(x+e^{(j)})-\varphi(x-e^{(j)})]\gamma^{(j)}&+\Big\{m_0+\sum_{j=1}^{2n}\frac{t_{\rm c}^{(j)}}{2}[\varphi(x+e^{(j)})+\varphi(x-e^{(j)})]\Big\}\gamma^{(2n+1)},
\beta_1 \oplus \beta_2 = \begin{cases}\emptyset i_1 = j_1 i_2 < j_1-1 j_2 \leq i_2, \{(\alpha_{i_1, j_2}, \alpha_{j_1, i_2})\}j_1 -1 \leq i_2 < j_2. \end{cases}
\eta^N(t) \leq C \int_0^t dt_1\left[ N^{3\alpha+2}\delta^N(t_1)+e^{-\frac{\lambda}{2}N^2}\right]\leq &CN^{3\alpha+2}\int_0^t dt_1\int_0^{t_1}dt_2\,\eta^N(t_2)+\int_0^tdt_1e^{-\frac{\lambda}{4}N^2},
\phi_{p_1, q_1, \cdots, p_n, q_n}:= \prod_{i=1}^n (t_{d_{2i-1}}^{q_i}\circ t_{c_{2i}}^{p_i}\circ t_{a_{2i}}^{-1}\circ t_{b_{2i}}^{-1}\circ t_{a_{2i-1}}\circ t_{b_{2i-1}}).
\underline{y}_k^c = ( y_{k,i}^c)_{i=1,\dots,n}, \,\underline{y}_k^s = ( y_{k,i}^s)_{i=1,\dots,n}, \, \underline{p}_k^c = ( p_{k,i}^c)_{i=1,\dots,n}, \,\underline{p}_k^s = ( p_{k,i}^s)_{i=1,\dots,n}\in \mathbb{R}^n
0 = \sum_{x\in\Gamma}G(o,x) = \sum_{\underset{G(o,x)\geq 0}{x\in \Gamma}}G(o,x) + \sum_{\underset{0>G(o,x)> -\tfrac{C_1C_3}{2}\tfrac{D^2}{n}}{x\in \Gamma}}G(o,x) + \sum_{\underset{-\tfrac{C_1C_3}{2}\tfrac{D^2}{n}\geq G(o,x)}{x\in \Gamma}}G(o,x) =: A_1 + A_2 + A_3.
A = -L(-1)(\beta_0L(1)-L_\xi(1))^\top-L_\xi(-1)L^\top(1), B = 2 \int_{-1}^1 L_\xi L_\xi^\top d\xi +L(-1)(\beta_0L(-1)+L_\xi(-1))^\top{+}L(1)(\beta_0L(1)-L_\xi(1))^\top +L_\xi(-1)L^\top(-1)-L_\xi(1)L^\top(1), C = -L(1)(\beta_0L(-1)+L_\xi(-1))^\top+L_\xi(1)L^\top(-1), K = \frac{h}{2}\int_{-1}^1 L(\xi)L^\top(\xi)d\xi.
\frac{\partial^{2}\psi}{\partial x^{2}}\rightarrow\frac{L_{\mathbb{Z}}\psi(j\Delta x,t)}{\Delta x^{2}}\equiv\frac{\psi((j+1)\Delta x,t)-2\psi(j\Delta x,t)+\psi((j-1)\Delta x,t)}{\Delta x^{2}}
+ 3 a_{2}^{1}c_{1}^{2}a_{2}^{2}+ a_{1}^{0}c_{1}^{2}a_{1}^{1}+ a_{1}^{1}c_{2}^{1}a_{2}^{3}+ a_{1}^{0}c_{2}^{1}a_{2}^{4}- a_{2}^{0}c_{1}^{2}a_{2}^{3}+ a_{2}^{0}c_{1}^{1}a_{2}^{4}) .
\mathbb E\bigg[\exp\Big(-16\varepsilon^2\kappa(d,N)^2\sum_{j\in \sigma(I_x)\cap J_r}\|\xi_j\|^2\Big)\bigg]\le C\exp\Big(-c'\kappa(d,N)^2\sum_{j=r+1}^d\|\xi_j\|^2\Big),
\ll \frac{T_k\phi(a_m W B)^2\log{R}}{k^2 a_m^2B^2W^2}\sum_{\substack{\mathbf{r}\in\mathcal{D}'_k r_m=1}}\frac{Y_{\mathbf{r}}^2\prod_{p|r}(\phi_L(p)-1)}{\phi_\omega(r)^2}\sum_{A|r}\frac{\log{A}}{\prod_{p|A}(\phi_L(p)-1)}\sum_{\substack{\mathbf{s}\in\mathcal{D}'_k A(\mathbf{r},\mathbf{s})=A}}(\log\log{R})^2.
\left( \begin{array}{cc}a_1 b_1 c_1 d_1 \end{array}\right) \ast\left(\begin{array}{cc}a_2 b_2 c_2 d_2 \end{array}\right)=\left( \begin{array}{cccc}a_1 0 b_1 0 0 a_2 0 b_2 c_1 0 d_1 0 0 c_2 0 d_2 \end{array}\right),
I_n(\lambda)_k=\frac{\dot{c}(X_{{i_k}/{n}}^{\lambda},\lambda_k)^2}{nD^{n, \lambda_k,k}( X_{{i_k}/{n}}^{\lambda})},N_n(\lambda)_k=\frac{\sqrt{n}(X_{({i_k+1})/{n}}^{\lambda}-X_{i_k/n}^{\lambda}-c(X_{{i_k}/{n}}^{\lambda},\lambda_k))}{\sqrt{nD^{n, \lambda_k,k}( X_{{i_k}/{n}}^{\lambda})}}.
f_n^*(x)=\frac{\rho_nh(x)}{\|\tilde{u}_n^{(1)}-\tilde{u}_n^{(2)}\|_{L^{\infty}(M)}}\left(e^{\tilde{u}_n^{(1)}(x)}-e^{\tilde{u}_n^{(2)}(x)}\right), \;x\in M, \ \ \textrm{and}
\begin{array}{l}\check{X}_{k+1}-Y_{k+1}= (1-h)\mathbf{L}\check{X}_k-\alpha_{k+1}h \left(\nabla U_v(X_k)-\frac{1}{m}\sum_{i=1}^m \mathbf{1}\otimes \nabla u_i(X^{(i)}_k)\right)\end{array}
u:= S(x_{n_{1}}\otimes \cdots \otimes x_{n_{r+1}}\otimes x_{n_{r}}\otimes \cdots \otimes x_{n_{i+1}}) + S(\mu^{i-1}\alpha_{T}(x_{n_{1}}\otimes \cdots\otimes x_{n_{r-1}})\otimes [x_{n_{r}},x_{n_{r+1}}]_{\mathfrak{g}}\otimes \alpha_{T}(x_{n_{r+2}}\otimes \cdots \otimes x_{n_{i+1}})),
c_f(r) =\begin{dcases}\sqrt{s(\alpha,a)}, a_f(r)>1-\frac{4\alpha}{(1-\alpha)(1+6\alpha+\alpha^2)}, a_f(r)-1+\dfrac{1}{1-\alpha}, 1-\frac{4\alpha}{(1-\alpha)(1+6\alpha+\alpha^2)}\geq a_f(r),\end{dcases}
\beta_i=(\mathbf 1_{A_{\alpha^*}}\bar{m}_{\tilde\xi_i}'',u)_{L^2(d\nu)}-(\bar{m}_{\tilde\xi_i}',\alpha b_1)_{L^2(d\nu)}+c' c\|u\|_{L^2(d\nu)}^2+|A_{\alpha^*}^c|+R_{\max},
\!\!\!\!\!\!\!\!\!\! y_{\chi}(t)= A \sqrt{P_{r_o}h_o}s_{\chi}(t) + \sum_{k \in \tilde{\Psi}_{\chi}}i_{k}^{(\chi)}(t)+ \sum_{j \in \tilde{\Psi}_{\bar{\chi}}}i_{j}^{(\bar{\chi})}(t) + i_{\rm s}(t) + n_{o}(t).
S(\xi(g)) = \{(\alpha, \beta) \in G^2\colon \alpha \xi(g) = \xi(g) \beta\}= \{(\alpha, \beta) \in G^2\colon \xi (\alpha g) = \xi(g\beta)\}= \{(\alpha, \beta) \in G^2\colon \alpha g = g\beta\}= S(g).
N(c,H) ={\mathcal{Q}}(G) +{\mathcal{C}}(G) = c^2 \mathcal{Q}(G_0) +\mathcal{Q}(H) + 2 c \mathcal{Q}(G_0,H) + c^3 \mathcal C (G_0) +\mathcal{C}(H) + 3 c^2 \mathcal{C}(G_0,G_0,H) + 3 c \mathcal{C}(G_0,H,H).
A^1_n=-[\partial^n_x,\, (u^s +{u})] \partial_x u=-\sum_{i=1}^{n}C^i_n\partial_x^i{u}\, \partial^{n + 1 -i}_x{u}, \,\, A^2_n=-[\partial^n_x,\, (u^s_y + \partial_y{u})]{v}= -\sum_{i=1}^{n}C^i_n\partial_x^i{w}\,\partial^{n -i}_x{v}.
X = \sum_{b \in \mathcal{B}}\sum_{i=1}^{n+1}(s_i \cdots s_n \check{b}_{n+1}\otimes b_{n+1}s_n \cdots s_i \otimes g + s_i \cdots s_n c_{n+1}\check{b}_{n+1}\otimes b_{n+1}c_{n+1}s_n \cdots s_i \otimes g).
\displaystyle{S_{1}(a)}\, \equiv \,\sum_{k=1}^{\infty}{\frac{\Gamma \left( k+1-a \right) \Gamma \left( a+k \right)\left( \Psi \left( k+1-a \right)+\Psi(a+k)-2\Psi(k)\right)}{k^3\, \Gamma \left( k \right)^{2}\mbox{}}}\,.
f(\mathbf{x}) = G(\mathbf{x}) - \frac{\mathbf{c}(3,d)'}{3!}G^{(3)}(\mathbf{x}) + \frac{\mathbf{c}(4,d)'}{4!}G^{(4)}(\mathbf{x}) - \frac{\mathbf{c}(5,d)'}{5!}G^{(5)}(\mathbf{x}) & + \frac{\mathbf{c}(6,d)' + 10 \mathbf{c}(3,d)^{\otimes 2'}}{6!}G^{(6)}(\mathbf{x}) + \ldots
\sqrt{\ell}\sum_{\nu\geq\ell/k}\frac{1}{\sqrt{\nu}(\nu+\ell)}\cdot\frac{1}{\nu\frac{k}{\ell}}= \frac{\ell^{3/2}}{k}\sum_{\nu\geq \ell/k}\frac{1}{\nu^{3/2}(\nu+\ell)}=\mathcal{O}\left(\frac{1}{\sqrt{k}}\right)
\gamma_i =\frac{|h_{i,i}|^2p_i}{\sum_{j\ne i}|h_{i,j}|^2p_j+\sigma_i^2}, \tilde{\gamma}_k = \frac{|\tilde{h}_{k,k}|^2p_k}{\sum_{j=1}^{L}|\tilde{h}_{k,j}|^2p_j+\tilde{\sigma}^2_k}.
\tilde{\eta}= < ( e^{\frac{\alpha G}{2}})_{, A}> \phi^{A}+ < ( e^{\frac{\alpha G}{2}})_{, \bar{A}}> \chi^{\bar{A}}, \qquad \qquad \bar{\tilde{\eta}}= < ( e^{\frac{\alpha G}{2}})_{, A}> \chi^{A}+ < ( e^{\frac{\alpha G}{2}})_{, \bar{A}}> \phi^{\bar{A}}.
(M_{(P^\perp+zP)U}R_q,R_{\overline q}M_{U^*(P+zP^\perp)})=(V_1'\tau_{\rm BCL}\mathfrak r_q\tau_{\rm BCL}^*,\tau_{\rm BCL}\mathfrak r_{\overline q}\tau_{\rm BCL}^*V_2')=\tau_{\rm BCL}(V_1\mathfrak r_q,\mathfrak r_{\overline q}V_2)\tau_{\rm BCL}^*.
D(0,0) =1,D(j_1,j_2)=D(j_1,0)D(0,j_2),\;\; j_1,j_2\ge1 D(j,0) =\prod_{s=1}^{j}\big(2k+p-(2s-1)\big),D(0,j)=\prod_{s=1}^{j}\big(2\ell+q-(2s-1)\big),j\ge1.
x_{i}^{t}+\delta x_{i\rightarrow a}^{t}=\frac{1}{\sigma_{i}^{2}}\eta_{t}\left(\sum_{b\in[m]}A_{bi}\left(r_{b}^{t}+\delta r_{b\rightarrow i}^{t}\right)\right)+ \quad\frac{1}{\sigma_{i}^{2}}A_{ai}r_{a}^{t}\eta'_{t}\left(\sum_{b\in[m]}A_{bi}\left(r_{b}^{t}+\delta r_{b\rightarrow i}^{t}\right)\right),
\langle | J^{s a}( Z_{1}) K^{s b}( Z_{2}) K^{s c}( Z_{3}) | \rangle = \frac{- i \epsilon^{a b c}\theta_{1 2 3}}{Z_{1 2}^{1 / 2}Z_{3 1}^{1 / 2}Z_{2 3}^{3 / 2}}
F_1(a+) = a^{-1}(a+x\partial_x + y\partial_y)F_1 F_1(b+) = b^{-1}( b+x\partial_x )F_1 F_1(b'+) = b'^{-1}( b'+y\partial_y )F_1 F_1 = c^{-1}(c+ x \partial_x + y\partial_y )F_1(c+) .
F_{scat}(r)=\frac{k^{4}\alpha^{2}}{6\pi cn_{0}^{3}\epsilon_{0}^{2}}I(r)\hat{z}=\frac{8\pi n_{0}k^{4}a^{6}}{3c}(\frac{m^{2}-1}{m^{2}+2})^{2}I(r)\hat{z}.
\phi_{+}= f - f ( \Gamma ) + \frac{\ell_{0 , x}^{2}}{2}\ell_{0 , x}( y , z ) = \Big( 2 \int_{0}^{1}( 1 - t ) \partial^{2}_{z , z}g ( y , t z ) \, d t \Big)^{1 / 2}z.
\begin{cases}-\Delta \psi_1'(0) = m \psi_1(0) + (p-1)aU_{j,0}^{p-2}\psi_1(0) + \gamma_1'(0) \psi_1(0) \mbox{in $\Omega$}, \frac{\partial \psi_1'(0)}{\partial \mathbf{n}}= (q-1)bU_{j,0}^{q-2}\psi_1(0) + \gamma_1'(0) \psi_1(0) \mbox{on $\partial \Omega$}. \end{cases}
\begin{aligned}\frac{\partial^\alpha u(x,y,t)}{\partial t^\alpha}+\kappa_x\frac{\partial u(x,y,t)}{\partial x}+\kappa_y\frac{\partial u(x,y,t)}{\partial y}&-\varepsilon_x\frac{\partial^2 u(x,y,t)}{\partial x^2}-\varepsilon_y\frac{\partial^2 u(x,y,t)}{\partial y^2}=f(x,y,t),\end{aligned}
\lambda_{1}^{2}( 1 + 9 \lambda_{2}^{2}\operatorname{c o s}^{4}f ) ( \rho^{2}f_{\rho})_{\rho}+ 8 \lambda_{2}^{2}\operatorname{s i n}^{2}f \: ( a^{2}f_{\rho})_{\rho}
f(\mathbf{x})=\frac{q}{p}\left(\sum_{i=1}^{m-1}a_{i}x_{\pi(i)}x_{\pi(i+1)}+\sum_{k=1}^{m}d_{k}x_{k}^{2}\right) + \mathcal{L}_{c}(x) = \frac{q}{p}\mathbf{x}^{T}\mathbf{A}\mathbf{x}+ \mathcal{L}_{c}(\mathbf{x})
P'(w')=L^{\beta k}\;\sum_{\left[ n_{i'}\right]^{k}_{i'=0}}\;\sum_{\left[ X_{i}\right]^{n}_{i=0}}\int \prod^{n}_{i=0}dt_{i}\;\prod^{k}_{j'=0}\;\delta (\sum^{m_{j'}}_{{i}=m_{j'-1}+1}\;\;t_{i}-L^{\beta}t'_{j'})\times
\bar\nabla_{J}^k \dot\gamma = \bar\nabla_{\dot\gamma}\bar\nabla^{k-1}_{J}J +\sum_{i=0}^{k-2}\bar\nabla_J^i (\bar R (\dot\gamma, J) \bar\nabla_{J}^{k-2-i}J) &= \bar\nabla_{\dot\gamma}\bar\nabla^{k-1}_{J}J + \sum (\bar\nabla^{i_r}\bar R)^{i_r'}* (\dot\gamma)^j * (\bar \nabla_{J}^{k_p}J)^{k_p'}* \big(\bar\nabla_{\dot\gamma}\bar\nabla_{J}^{l}{J}\big)^{l'},
L_{i}((x_{j}^{k})_{k,j},(\lambda_{k})_{k},(\mu_{j})_{j})=f^{i}(x^{i})+\sum_{k=2}^{m}\lambda_{k}(z_{k}-f^{k}(x^{k}))+\sum_{j=1}^{n}\mu_{j}(b_{j}-\sum_{k=1}^{m}x_{j}^{k})
\{\eta_{M N}\}= \left( \begin{array}{c c}{{g_{\alpha \beta}+ g_{\rho \sigma}A_{\alpha}^{\rho}A_{\beta}^{\sigma}}}&{{g_{\mu \rho}A_{\beta}^{\rho}}}\\{{A_{\alpha}^{\rho}g_{\rho \nu}}}&{{g_{\mu \nu}}}\\ \end{array}\right) \, ,
\langle u, v \rangle_{\mathcal{D}_g}:= \langle (L_{g^{-1}})_{\ast}u, (L_{g^{-1}})_{\ast}v\rangle_{\mathcal{D}_e}&= \langle (L_{g^{-1}})_{\ast}u,(L_{g^{-1}})_{\ast}v\rangle_{\mathcal{H}}u, v \in\mathcal{D}_g.
\mathfrak{L}^{k,t}= \sum_{i=1}^{k+1}\sum_{\alpha\in I_{i,k}}\biggl\{\nabla^i X|_{\Phi^t}\circ \nabla^{\alpha}\Phi^t_* \circ \mathfrak{S}_{\alpha}+ \sum_{j=0}^{i-2}((\nabla^j_\bullet R)(\nabla^{i-2-j}_\bullet X, \bullet)\bullet)|_{\Phi^t}\circ \nabla^{\alpha}\Phi^t_* \circ \mathfrak{S}_{\alpha,j}\biggr\}.
\underset{\textbf{W}_k}{\max}\left\{\!\underset{\hat{\textbf{x}}_k}{\max}\sum\limits_{l=1}^k\left[ \, p_{DI}\left(\textbf{y}_l \big|{\textbf{x}},\textbf{W}_{l}\right)\bigg|_{\textbf{x}=\hat{\textbf{x}}_k}\right]\right\}~ \eqref{eq_observation_vector},\eqref{eq_Fc}, \eqref{eq_sv}, \eqref{eq_FeDI},\eqref{eq_constrant1DI}.
e^{i k_{1}\cdot X}e^{i k_{2}\cdot X}\sim \left( \tau - \tau^{\prime}\right)^{2 \alpha^{\prime}G^{\mu \nu}k_{1 \mu}\cdot k_{2 \nu}}\times \left[ e^{- \frac{i}{2}\Theta^{\mu \nu}k_{1 \mu}\cdot k_{2 \nu}}\right] \times e^{i \left( k_{1}+ k_{2}\right) \cdot X}+ \cdots ,
\|\gamma_{i_{k}}\| = \frac{1}{\|A_{i_{k}}\|}|(r-y^{k})_{i_{k}}|\leq \Big(1-\frac{\delta^{2}\alpha (2 - \alpha)}{n}\Big)^{k/2}\frac{\|b\|}{\min_{i \in [m]}\|A_{i}\|}=: \gamma^{k}M,
\mathrm{R e}\Pi^{\mathrm{( n o n \mathrm{-}p l a n a r )}}= \frac{2 \pi^{\frac{n + 1}{2}}}{( 2 \pi )^{n}}\left( | \bar{\theta}| \right)^{\frac{3 - n}{2}}\int_{0}^{\infty}d z \frac{1}{z^{2}- p^{2}}z^{\frac{n - 3}{2}}J_{\frac{n - 3}{2}}( \frac{| \bar{\theta}| z}{2})
\psi_{\alpha, \varepsilon}(x) = \frac{2\sqrt{\alpha^2-1}-\alpha \varepsilon x}{2\alpha\pi \sqrt{1-x^2}}+ \frac{1}{\alpha \pi}\arctan\left( \frac{\sqrt{1-x^2}}{\sqrt{\alpha^2-1}}\right),x \in (-1,1).
r''_{11}=d_1;r''_{22}=d^{*}_{22}&r''_{1i}=r''_{k1}=0(i,k\ne 1);r''_{2i}=r''_{k2}=0(i,k\ne 2) &r''_{i+2,k+2}=r^{*}_{ik}(i,k>1).
\left[{\frac{d^{2}}{d x^{2}}}+{\frac{3}{x}}{\frac{d}{d x}}-{\frac{2 s^{2}( x_{+}^{2}- x_{-}^{2})}{x^{4}( \cosh 2 \delta + \cosh 2 \gamma )}}-{\frac{m^{2}l^{2}}{x^{2}}}\right] \Phi_{s}^{\mathrm{f a r}}= 0 .
\begin{cases}\begin{gathered}-\operatorname{div}\big({\mathcal{A}(x,\nabla u)}\big) = \frac{f_n(x)}{\big(u+\frac{1}{n}\big)^\delta}\;\;\mbox{in}\;\;\Omega, u>0\Omega,\,\,u = 0\;\mbox{on}\;\partial\Omega.\end{gathered}\end{cases}
\nabla_{e_0}e_{0}=-\mu_1e_2,\ \nabla_{e_0}e_{1}=0,\ \nabla_{e_0}e_{2}=-\mu_1e_0, \nabla_{e_1}e_{0}=0,\ \nabla_{e_1}e_{1}=\mu_2e_2,\ \nabla_{e_1}e_{2}=-\mu_2e_1, \nabla_{e_2}e_{0}=-\mu_1e_0,\ \nabla_{e_2}e_{1}=-\mu_2e_1, nabla_{e_2}e_{2}=0.
\Delta^{+}( \omega ) = \ln \left( \frac{\Lambda^{2}}{m^{2}}\right) - \int_{- \Lambda}^{+ \Lambda}\frac{d{\bf k}}{[{\bf k}^{2}+ m_{\mathrm{\scriptsize ~ e f f}}^{2}]^{1 / 2}}\frac{{\bf k}^{2}}{[{\bf k}^{2}+ m_{\mathrm{\scriptsize ~ e f f}}^{2}- \omega^{2}/ 4 + i \epsilon ]},
\aligned\partial_t v_1^i= \nu \Delta v_1^{i}+\sum_{i_1=1}^3P^{ii_1}\xi^{i_1},\textrm{div}v_1=0, \partial_t v^i= \nu \Delta v^{i}-\sum_{i_1,j=1}^3P^{ii_1}\frac{1}{2}D_j [(v^{i_1}+v_1^{i_1}) (v^j+v_1^j)],\textrm{div}v=0.\endaligned
U ( t , t^{\prime}) = \prod_{n = 1}^{N}\exp \left( - \frac{i}{\hbar}\int_{t^{\prime}}^{t}V ( \widehat{x}+ c t^{\prime \prime}- x_{n}) d t^{\prime \prime}\left[ \sigma_{+}^{( n )}\exp \left( - i \frac{\omega}{c}\widehat{x}\right) + \mathrm{h . c .}\right] \right) .
\#\{\beta \in o\mid \varphi(v'\beta)<0\}= \#\{\beta \in -s_{\alpha}(o)\mid \varphi(v'\beta)<0\}= \#\{\beta \in o\mid \varphi(-v\beta)<0\}\leq \#\{\beta\in o\mid \varphi(v\beta)\geq 0\}= 2-\#\{\beta \in o\mid \varphi(v\beta)<0\}.
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int( |\nabla^k \varrho|^2+\!|\nabla^k u|^2) dx+\int( \mu |\nabla^{k+1}u|^2+(\mu+\nu)|\nabla^k{\rm div}u|^2) dx &=\int \nabla^k S_1 \cdot \nabla^k \varrho \dx+\int \nabla^k S_2 \cdot \nabla^k u \dx.\end{aligned}
\frac{(G_{D_1}(x, x_0)\wedge C_2) (G_{D_1}(y, x_0)\wedge C_2)}{(G_{D_1}(A, x_0)\wedge C_2)^2}= \frac{G_{D_1}(x, x_0)G_{D_1}(y, x_0)}{G_{D_1}(A, x_0)^2}\asymp \frac{g(x) g(y)}{g(A)^2}.
f(u_n)-f(v_n) =\min_{x,y\in M_{n-1}}\big(f(x)+(1-1/2^n)d(x,u_n)-f(y)+(1-1/2^n)d(y,v_n)\big) \ge\min_{x,y\in M_{n-1}}\big((1-1/2^n)\big(d(x,u_n)+d(y,v_n)\big)-(1-1/2^{n-1})d(x,y)\big) \ge(1-1/2^n)\min_{x,y\in M_{n-1}}\big(d(x,u_n)+d(y,v_n)-(1-1/2^{n+1})d(x,y)\big) \ge (1-1/2^n)(1-1/2^{n+1})d(u_n,v_n) \ge (1-1/2^{n-1})d(u_n,v_n),
+ 2 ( n + 1 )^{2}G_{2 n}c_{- k + 2 - 2 n}^{( 3 )}+ ( n + 1 )^{3}G_{3 n}c_{- k - 3 n}^{( 3 )}+ c_{- k + 2}^{( 1 )}+ c_{- k - n}^{( 1 )}G_{n}+ c_{- k}^{( 0 )}= 0
\{c_{f,w}, c_{g,v}\}=[(\Phi_+\eta|\rho)-(\Phi_+\beta|\gamma)]c_{f,w}c_{g,v}+2\sum_{\nu\in\bf R^+}(c_{f,x_\nu w}c_{g,x_{-\nu}v}-c_{fx_\nu,w}c_{gx_{-\nu},v})
E(x^{1},x^{2},\dot{x}^1,\dot{x}^{2},u^{1},u^{2})=h(x^{1},x^{2},u^{1},u^{2})-h(x^{1},x^{2},\dot{x}^1,\dot{x}^{2})-\sum\limits_{j=1}^{2}(u^{j}-\dot{x}^{j})\frac{\partial h(x^{1},x^{2},\dot{x}^1,\dot{x}^{2})}{\partial \dot{x}^j}.
\varphi_{i}\mathbf{=}\frac{l_{i}^{x}}{l_{1}^{x}}\frac{\sum\limits_{j=1}^n{\mathellipsis \sum\limits_{k=1}^p{e_{1jk}y_{j}\mathellipsis z_{k}}}}{\sum\limits_{j=1}^n{\mathellipsis \sum\limits_{k=1}^p{e_{ijk}y_{j}\mathellipsis z_{k}}}},\thinspace i=1,\thinspace \mathellipsis ,m
\sum_{\sigma \in D_n^{\{0,\,1\}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\substack{\{\sigma \in D_n^{\{0,\,1\}}: | \sigma(3) |=n\}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\substack{\{\sigma \in D_n^{\{0,\,1\}}: | \sigma(n) |=n\}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}.
\mathcal G(t,r_1,r_2,l) =\frac{F(r_1+t)-F(r_2)}{l\,f^n(r_1+t)}&+\frac{1}{l\,f^n(r_1+t)}\int_0^l\int_s^lf'(F^{-1}(\bar U(s'))\,ds'\,ds.
\gamma_{\textrm{NDF}}^{\textrm{NOMA}}=\frac{\left(1-\beta_{\textrm{N}}\right)P_\textrm{F}}{\left(1-\beta_{\textrm{N}}\right)P_\textrm{N}+{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^2}/{|h_{\textrm{BN}}|^2}},