harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x=x_0-u_{10}\epsilon^2 \tau + \epsilon^3{{y}}\, , t=t_0 + \epsilon^2{\tau}\, ,{u_1}={u_1}_0 + \epsilon \upsilon_{11}+ \epsilon^2 \upsilon_{12}\, ,{u_2}={u_2}_0 + \epsilon^2 \upsilon_{22}\, ,
\mathrm{P}_{1,i}^o \geq \mathrm{P}\left( \frac{[\mathbf{R}_1]^2_{i,i}\alpha_i^2}{[\mathbf{R}_1]_{i,i}^2\beta_i^2 +\sum^{N}_{j=i+1}[\mathbf{R}_1]_{i,j}^2\beta_j^2 +\frac{1}{\rho}}<\epsilon_{1,i}\right) \geq \mathrm{P}\left( \frac{[\mathbf{R}_1]^2_{i,i}\alpha_i^2}{\sum^{N}_{j=i+1}[\mathbf{R}_1]_{i,j}^2\beta_j^2 +\frac{1}{\rho}}<\epsilon_{1,i}\right),
S=\begin{pmatrix}1 [3] [3] 1 [3] -1 -1 [3] [3] -1 -1 [3] 1 [3] [3] 1 \end{pmatrix}T =\begin{pmatrix}1 0 0 0 0 -i 0 0 0 0 i 0 0 0 0 -1 \end{pmatrix},
\int^T_0 \mathcal{E}(t) dt = \frac{3}{2\eta}\left( \mathcal{E}(0) - \mathcal{E}(T) \right) + (h(0)-h(T))\leq \left( \frac{3}{2\eta}\mathcal{E}(0) + h(0) \right)- h(T).
\frac{a(z\mathrm{e}^{it},\mathrm{e}^{if(z\mathrm{e}^{it})}w)-a(z,\mathrm{e}^{if(z)}w)}{t}&\qquad\qquad=\underbrace{\frac{a(z,\mathrm{e}^{if(z\mathrm{e}^{it})}w)-a(z,\mathrm{e}^{if(z)}w)}{t}}_{I_t(z,w)}+\underbrace{\frac{a(z\mathrm{e}^{it},\mathrm{e}^{if(z\mathrm{e}^{it})}w)-a(z,\mathrm{e}^{if(z\mathrm{e}^{it})}w)}{t}}_{II_t(z,w)}.
c_1-c_2 =\int_0^{\infty}\big(\cosh t-1\big)(\sinh t)^{-s-1}\,dt+\int_0^{\infty}\Big(\Big(\frac{t}{\sinh t}\Big)^{s+1}-1\Big)t^{-s-1}\,dt &=\int_0^{\infty}\big((\cosh t)(\sinh t)^{-s-1}-t^{-s-1}\big)\,dt.
J=(\begin{matrix}1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&1&0&0&0\\ 0&0&0&1&1&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&1\\ 0&0&0&0&0&0&1\end{matrix}),
\varphi_j(x)=\frac{b_j}{\Gamma(\alpha)}x^{\alpha-1}-\frac{1}{\Gamma(\alpha)}\underset{0}{\overset{x}{\int}}\frac{-k_j\varphi_j(s)+\left(D_{l_j-}^{\alpha}\varphi_j\right)(s)}{\left(x-s\right)^{1-\alpha}}ds,
\left(\frac{|A_{j+1}|}{|\Omega_{\tau}|}\right)^{2}=\left(\frac{|A_{j+1}|}{|A_{j}\setminus A_{j+1}|}\right)^{2}\left(\frac{|A_{j}\setminus A_{j+1}|}{|\Omega_{\tau}|}\right)^{2}\leq C_2 \frac{|A_{j}\setminus A_{j+1}|}{|\Omega_{\tau}|}.
U = R(1,4,\theta^{(0)})R(2,3,\theta^{(1)})R(1,3,\theta^{(2)})R(2,4,\theta^{(3)})R(1,2,\theta^{(4)})R(3,4,\theta^{(5)}), \mathsf{U}= R(1,3,\phi^{(0)})R(2,4,\phi^{(1)})R(1,4,\phi^{(2)})R(2,3,\phi^{(3)})R(1,2,\phi^{(4)})(3,4,\phi^{(5)}).
\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}\right)+\frac{\partial P}{\partial x}=\mu\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}\right),
\left\|\frac{\delta^2_h u}{|h|^{\frac{2s}{2-\theta}}}\right\|_p\leq \left[C\, [u]_{s,p}^{(1-\frac{p}{2})}\left(\|f\|_r\, \|u\|_t^{1-\theta}\right)^{\frac{1}{2}}\right]^{\sum_{i=0}^{+\infty}\frac{\theta^i}{2^i}},
0 = \gamma_1'''(c_0) (|\Omega|+|\partial \Omega|) + \lambda'''(c_0) \left\{\int_\Omega \left( m + (p-1)a c_0^{p-2}\right) + (q-1) c_0^{q-2}\int_{\partial \Omega}b \right\}&+ 3\lambda''(c_0) \left\{(p-1)(p-2)c_0^{p-3}\int_\Omega a w'(c_0) + (q-1)(q-2) c_0^{q-3}\int_{\partial \Omega}b w'(c_0) \right\}.
\tilde{\mathbf{f}}_{i}^{[i,j]}= \arg \max_{\{\mathbf{w}=\mathbf{c}_k: 1\le k \le N_f\}}\frac{\left|{\mathbf{f}_{i}^{[i,j]}}^{\mathsf{H}}\mathbf{w}\right|^2}{\left\|\mathbf{f}_{i}^{[i,j]}\right\|^2}.
g_{rr}= 1 + \left(\frac{\partial u}{\partial r}\right)^2 g_{rj}= \frac{\partial u}{\partial r}\frac{\partial u}{\partial\theta_j}g_{ir}= \frac{\partial u}{\partial\theta_i}\frac{\partial u}{\partial r}g_{ij}= r^2Z^2\lambda_i\delta_{ij}+ \frac{\partial u}{\partial\theta_i}\frac{\partial u}{\partial\theta_j}.
\| U^{n}\|_{L^{p}(\Omega;H)}\leq \|U_{h}^{n}\|_{L^{p}(\Omega;H)}+ \Big \| \int_{0}^{T}\sum_{j=0}^{n-1}\chi_{[t_{j}, t_{j+1})}(s) B_{n-j}P_{h}\d W (s) \Big \|_{L^{p}(\Omega; H)}: ={\rm I}_{1}+{\rm I}_{2}.
Q(K + \sum_{j\in\mathcal{I}^-}L(A_j)\overline{\Psi}_j )Q^T = P^- +\left[\begin{array}{cc}(\hat{\lambda}+2\hat{\mu})g_n(F) \hat{\lambda}g_t(F) \hat{\mu}g_t(F) \hat{\mu}g_n(F)\end{array}\right]
\phi^*_{k}(x_{i,k}) = x_{i,k+1}, \phi^*_{k}(\tilde s_{i,k}) = \tilde s_{i,k+1}+ \xi_{k+1}\tilde s_{i+1,k+1}, \psi^*_{k}(x_{i,k+1}) = x_{i,k+1}+ \xi_{k+1}x_{i-1,k+1}, \psi^*_{k+1}(\tilde s_{i,k+1}) = \tilde s_{i,k+1}.
D_{a+}^{\alpha}f\left( t\right) =\frac{1}{\Gamma \left( 1-\alpha \right)}\left( \frac{f\left( t\right)}{\left( t-a\right)^{\alpha}}+\alpha\int_{a}^{t}\frac{f\left( t\right) -f\left( s\right)}{\left( t-s\right)^{\alpha +1}}ds\right)
- 2 \left[ 2 \zeta_{R}^{\prime}\left( - 1 ,{\frac{3}{2}}\right) - \sigma \psi \left({\frac{3}{2}}\right) + \sum_{k = 2}^{\infty}{\frac{\sigma^{k}}{k}}\zeta_{R}\left( 2 k - 1 ,{\frac{3}{2}}\right) \right]
[\begin{matrix}c_{3}c_{2}&-s_{3}c_{1}+c_{3}s_{2}s_{1}&s_{3}s_{1}+c_{3}s_{2}c_{1}\\s_{3}c_{2}&c_{3}c_{1}+s_{3}s_{2}s_{1}&-c_{3}s_{1}+s_{3}s_{2}c_{1}\\ -s_{2}&c_{2}s_{1}&c_{2}c_{1}\end{matrix}]
W(u_1,u_2,u_3)= (C_1D_1-D_2)W(u,v,\partial_\epsilon u)+D_1^2W(u,v,\partial_\epsilon v)+D_1W(u,\partial_\epsilon u, \partial_\epsilon v) &+\frac{D_1}{2}W(u,v,\partial_\epsilon^2 u)+\frac{1}{2}W(u,\partial_\epsilon u,\partial_\epsilon^2 u).
0 \in \partial g(y^{k+1})+\eta B^T(Ax^k+By^{k+1}-b-\frac{1}{\eta}\lambda^k)= \partial g(y^{k+1})+\eta B^T\frac{1}{\eta}(\lambda^k-\lambda^{k+1}-\lambda^k)=\partial g(y^{k+1})- B^T\lambda^{k+1}\ ,
A_{g r a v + s c a l}^{[ D ]}= \int \, d^{D}x \, \sqrt{- g}\, \left[ 2 \, R [ g ] + \alpha \, \frac{1}{2}g_{i j}( \phi ) \, \partial^{\mu}\, \phi^{i}\partial_{\mu}\phi^{j}- \mathcal{V}( \phi ) \right]
\bigg[ w_1 w_2 \cdots w_{n-1}y + w_1 w_2 \cdots w_n x + w_n \sum_{j=1}^{n-1}\big(\prod_{\substack{i=1 i \neq j}}^{n-1}w_i \big) k_j \bigg]_q &=[w_1 w_2 \cdots w_{n-1}]_q \Big[y + w_n x + \frac{w_n}{w_1}k_1+ \cdots + \frac{w_n}{w_{n-1}}k_{n-1}\Big]_{q^{w_1 w_2 \cdots w_{n-1}}}.
\left|N(f, r)-\sum_{j=r}^{d}{j \choose r}{q \choose j}(-1)^{j-r}q^{k-j}\right| &\leq \sum_{j=k+1}^d{j \choose r}{\frac{q}p+m\sqrt{q}+j \choose j}{m-1 \choose d-j}\sqrt{q}^{d-j}.
(q_0+q_2+...+q_{m-2}+q_m)(x) = -\frac{i}{2\pi}L^{\frac{n+3}{2}}(f_2 - f_1)(x)+\frac{i^{m+1}}{8\pi^2 m!}L^{\frac{n-m+1}{2}}(f_2-f_1)(y)|x-y|^{m-1}dy.
\begin{aligned}s^{\frac{3}{d+2}}\leq t^{\frac{3}{d+2}}\leq C\beta^{-\frac{1}{d+2}}N^{-\frac{1}{d+2}}+ s^{\frac{3}{d+2}}, s \leq C\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}. \end{aligned}
\mathfrak{L}(u):=\frac{1}{p}\int\limits_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u(x)|^{p}dx+\frac{1}{p}\int\limits_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u(x)|^{p}dx-\frac{1}{q}\int\limits_{\mathbb{G}}|u(x)|^{q}dx
\hat{\varphi}^{( 1 )}( x ) = \mathrm{i}\, \int d^{6}y \, \, \Big < \hat{\varphi}( x ) \, J ( y ) \, \hat{\varphi}( y ) \Big > = \frac{\kappa_{o r b}\, M}{2}\, \int \frac{d^{5}k}{( 2 \pi )^{5}}\frac{\mathrm{e}^{\mathrm{i}k \cdot x}}{k^{2}}~ ~ .
g(\lambda,\delta,\mu)=\max\mathcal{L}_1^{\rm NOMA}(\{p_k(\nu)\},\{p_{\bar k}(\nu)\},\lambda,\delta,\mu), &\mathtt{s.t.}~~p_k(\nu)\ge 0, p_{\bar k}(\nu)\ge 0, p_k(\nu)+p_{\bar k}(\nu)\le\hat P, \; \forall\nu.
\overline{g}(\mathrm{div}^{\nabla'}(T_{r}), Y) =(-1)^{r}\overline{g}(\nabla'S_{r},Y)+\mathrm{tr}(T_{r-1}(\nabla'_{Y}\mathcal{A}_{\widehat{W}})) &+\overline{g}(\mathrm{div}^{\nabla'}(T_{r-1}),Y) + \overline{g}(Y, \sum_{i=1}^{n}\epsilon_{i}\overline{R}(\widehat{W},T_{r-1}Z_{i})Z_{i}).
\zeta_{\pm}^{2}( z ) = N_{2}\left( \begin{array}{c}{{z^{\frac{1 - M}{2}}( 1 - z )^{- \frac{M}{2}}F ( \frac{3}{2}- M , - \frac{1}{2}- M , \frac{3}{2}- M ; z )}}\\{{\mp i ( \frac{1}{2}- M ) z^{- \frac{M}{2}}( 1 - z )^{\frac{M}{2}}F ( - 1 , 1 , \frac{1}{2}- M ; z )}}\\ \end{array}\right)
\sum_{k_i}\langle\gamma_0^{k_0}, \ldots, \widehat{\gamma_i^{k_i}}, \ldots, \gamma_m^{k_m}\rangle_{g,\beta}^X \Big(\int_\beta \gamma_i\Big)^{k_i}\frac{y_i^{k_i}}{k_i!}= \langle\gamma_0^{k_0}, \ldots, \widehat{\gamma_i^{k_i}}, \ldots, \gamma_m^{k_m}\rangle_{g,\beta}^X e^{y_i\int_\beta \gamma_i}
\frac{B(p^{2})}{A(p^{2})}=p^{-\frac{1}{2}}\sin\left(\frac{1}{2}\left[ \frac{128}{3\pi^{2}N}-1\right]^{\frac{1}{2}}\left\{\ln \left[\frac{pA(0)}{B(0)}+\delta\right]\right\}\right),
\left. \phantom{{\frac{A}{B}}}{\epsilon^{i_{1}\ldots i_{n}}\phi_{i_{1}}d \phi_{i_{2}}\cdots d \phi_{i_{n}}}\right|_{\Sigma}= \left.{\frac{\overline{{{\Delta}}}}{\Delta}}{\epsilon^{j_{1}\ldots j_{n}}\overline{{{\phi}}}_{j_{1}}d \overline{{{\phi}}}_{j_{2}}\cdots d \overline{{{\phi}}}_{j_{n}}}\right|_{\Sigma}
\begin{cases}\nabla\times(\epsilon^{-1}\nabla\times\widetilde{H_{\eta,\ell}})-k^{2}\mu\widetilde{H_{\eta,\ell}}=-\nabla\times((\epsilon^{-1}(x)-\epsilon_{0}^{-1}(x))\nabla\times H_{\eta,\ell})\mbox{in}\Omega, \nu\times\widetilde{H_{\eta,\ell}}=0\mbox{on}\partial\Omega.\end{cases}
{\cal F}^i_\epsilon(\vec{x},t) = \frac{1}{2\pi}\int d^2\vec{p}\;{\rm e}^{i\vec{p}\vec{x}}\, \tilde{\cal F}^i_\epsilon(\vec{p},t),\qquad\overline{\tilde{\cal F}}{}^i_\epsilon(\vec{p},t) = \tilde{\cal F}^i_\epsilon(-\vec{p},t),
|N\ast (\eta_{\varepsilon}b_2) (x)| \le C \begin{cases}1+ \log \langle x \rangle (n=2) \langle x \rangle^{2-n}(n=1, n\ge 3) \end{cases}, |\nabla N\ast (\eta_{\varepsilon}b_2) (x)| \le C \langle x \rangle^{1-n}
\mathcal{A}^\pm:=\{\theta=\sum_{j\geq 0}\theta_jr^j| \theta_{2j}\in\Gamma\big(S^{+\partial_r}(X,\bar{g})\big), \theta_{2j+1}\in\Gamma\big(S^{-\partial_r}(X,\bar{g})\big), \; \nabla^{\bar{g},S}_{\partial_r}\theta_j=0 \},
\int_0^1 \int_X \dot{\phi}_t \omega_{\phi_t}^n dt = - \int_0^1 C_H'\int_{\chi+\Delta^t\cap\bar{C}^+}\dot{u}_t^*(2\chi-2q)P_{DH}(q)dq & = -C_H' \int_{\chi+\Delta^t\cap\bar{C}^+}(u^*-u_{\mathrm{ref}}^*)(2\chi-2q)P_{DH}(q)dq & = - (2\pi)^n n! \int_{\Delta^+}(u^*-u_{\mathrm{ref}}^*)(2\chi-2q)P_{DH}(q)dq
\mathcal{L}_{I_{m}}\big(s_{c}\big) = e^{-2\pi \psi_0 \, p(r) (s_{c})^{\frac{2}{\alpha}}\mathbb{E}[{p^{\frac{2}{\alpha}}_{i}}] \int^{\infty}_{R_0}\big(\frac{u}{1+u^{\alpha}}\big) du}.
\begin{bmatrix}v_0^* v_1^* v_2^* v_3^* v_4^* \end{bmatrix}=\left[\begin{array}{rrrrrrrrr}-2 2 -1 -2 0 0 1 0 2 0 2 3 -2 3 0 -3 0 -3 -3 7 0 -7 -3 1 0 -1 6 9 -14 0 14 -3 2 0 -2 -6 0 6 0 -6 -6 0 0 0 6 \end{array}\right].
[{\mathbb D},{\mathbb D}]_A{}^{B}=(\mathrm{tr}({\mathbb D}\wedge{\mathbb D}))_A{}^{B}={\mathbb D}^{B}{}_{R}{\mathbb D}^{R}{}_{A}- \,{\mathbb D}^{R}{}_{A}{\mathbb D}^{B}{}_{R},
{\hat{\pi}}_{a}^{i}( 0 ) \simeq m{\frac{\Delta{\hat{x}}_{a}^{i}}{\varepsilon}}-{\frac{e}{2}}\sum_{b = 1}^{N}\epsilon^{i j}[{\hat{B}}({\hat{\bf x}}_{b}( \varepsilon ) ) \delta_{a b}+{\frac{e}{\theta}}\delta ({\hat{\bf x}}_{a}( \varepsilon ) -{\hat{\bf x}}_{b}( \varepsilon ) ) ] \Delta{\hat{x}}_{b}^{j}\, ,
F = \left( \begin{array}{c c c c c}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{f}&{0}&{0}\\{0}&{- f}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\ \end{array}\right)
W_T[\phi;G(k)] = V_{tree}(\phi)+{1 \over 2}\, \hbox{$\sum$}\!\!\!\!\!\!\!\int^{(T)}_k \, Tr [\ln G^{-1}(k) + D_{tree}^{-1}(\phi;k) G(k) -1]+ W^*_T[\phi;G(k)]~,
m_{ij}= c'_{\lambda_i}\,c'_{\lambda_j}+ \sum_{m>0}\beta^m\, c'_{\lambda_i+m}\,c'_{\lambda_j}+ \sum_{\ell>0}\sum_{m\geq 0}(-1)^\ell \left( \binom{\ell+m-1}{m}+ \binom{\ell+m}{m}\right) \beta^m \, c'_{\lambda_i+\ell+m}\, c'_{\lambda_j-\ell},
\varkappa(m,p_1,p_2):=\sum_{s=0}^{n}\sum_{\nu\in{I}_s}k^{-\frac{sr}{4}}(\ln k)^{2sr}\prod_{\nu_j>k}[\nu_j^{-\frac{r}{4}}(\ln \nu_j)^{2r}]\cdot |m(\nu)|^r<\infty,
\vartheta_{1}= \omega_{1}\vartheta_{2}= \omega_{2}+ \frac{\alpha_{3}}{2}\omega_{1}\vartheta_{3}= \omega_{3}-\frac{\alpha_{6}}{2}\omega_{2}-\left(\frac{\alpha_{3}\alpha_{6}}{4}+ \frac{\alpha_{5}}{2}\right)\omega_{1}
\left|t_{N, N / 2 + (N/3)^{1/2}}- \frac{N + 2 (N/3)^{1/2}}{2 N}\right| \ge\frac{(N/3)^{1/2}}{N}=\frac{1}{(3 N)^{1/2}}\gg \frac{K}{N}.
P^{(1)}L_{\phi}= L_{\phi}L_{\phi}^{*}L_{\phi}+ ( Q^{- 1})^{*}d_{\phi , h}^{*}M d_{\phi , h}Q^{-1}Q d_{\phi , h}a_{h}&= L_{\phi}L_{\phi}^{*}L_{\phi}+ ( Q^{- 1})^{*}d_{\phi , h}^{*}M d_{\phi , h}d_{\phi , h}a_{h}&= L_{\phi}L_{\phi}^{*}L_{\phi}= L_{\phi}P^{(0)}.
\dot{\beta}(t, u^{1}, \dotsc, u^{m-d-1}) ={}a_{1}X_{1}(t) + \dotsb + a_{m-1}X_{m-1}(t) &- u^{m-d}\dot{X}_{m-d}(t) - \dotsb - u^{m-1}\dot{X}_{m-1}(t).
S = - \frac{1}{g}\int_{D 6}( 1 + \frac{1}{4}) \left( F_{\mu \nu}F^{\mu \nu}+ 2 i \overline{{{\theta}}}\Gamma^{\mu}\partial_{\mu}\theta + 2 \delta_{i , j}\partial_{\mu}X^{i}\partial^{\mu}X^{j}\right)
\int_{0}^{T}- \psi'(t) (\tilde{\textbf{u}}^{\varepsilon}, \phi)_{0}~ dt + \sum_{j=1}^{d}\int_{0}^{T}\psi(t) (\textbf{P}J_{\varepsilon}A_{j}(J_{\varepsilon}(\tilde{\textbf{u}}^{\varepsilon}+\bar{\textbf{u}}))\partial_{x_{j}}J_{\varepsilon}\tilde{\textbf{u}}^{\varepsilon}, \phi)_{0}~ dt = 0.
f(q,p)\star g(q,p)={\rm exp}\Big\{\frac{i \hbar}{2}\Pi_{ij}\Big(\frac{\partial}{\partial q_i}\frac{\partial}{\partial p_j'}-\frac{\partial}{\partial p_i}\frac{\partial}{\partial q_j'}\Big)\Big\}f(q,p)g(q',p')|_{q=q',p=p'}.
H = F(z,\tau)^2 I = 2\, G(z,\tau) T = 8 \sum_{k \geq 1}\frac{1}{k^3}y^k + 12 \sum_{k,n \geq 1}\frac{1}{k^3}q^{kn}+ 8 \sum_{k,n \geq 1}\frac{1}{k^3}(y^k + y^{-k}) q^{kn}+ 2 \sum_{k,n \geq 1}\frac{1}{k^3}(y^{2k}+ y^{-2k}) q^{(2n-1) k}.
L_{e f f}^{i i}= \frac{1}{2 \pi}\left( \hat{\Phi}^{i}~ \hat{\eta}^{i}\right) \left( \begin{array}{c c}{{A^{i i}}}&{{C^{i i}/ 2}}\\{{C^{i i}/ 2}}&{{B^{i i}}}\\ \end{array}\right) ~ \left( \begin{array}{c}{{\hat{\Phi}^{i}}}\\{{\hat{\eta}^{i}}}\\ \end{array}\right) ,
{\cal L}(J) = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi + \frac{J}{2}\phi^2 + \frac{\lambda \mu^{2\varepsilon}}{4!}\phi^4 +{\cal L}_{\rm CT}(J)-\mu^{-2\varepsilon}\frac{\zeta}{2}\;J^2.
p^*\left(\frac{\prod\limits_{i=1}^aL_{i,b-k+i}}{\prod\limits_{i=1}^{a}L_{i,b-k+i+1}}\right)=\frac{\Delta_{\{k-a,\dots, k-1\}}}{\Delta_{\{k-a+1,\dots, k\}}}\prod_{i=k-a}^{k-1}\lambda_i=p^* (M_k ).
\bar{\gamma}_{ki}^{s}=\frac{\rho_{kii}^{2}\bar{\upsilon}_{ki}^{s}}{\rho_{kii}{\displaystyle \sum_{j=1}^{L}\alpha_{j}\mathbb{E}\left[\mathsf{\Gamma}_{ji}\right]+\bar{\upsilon}_{ki}^{s}\sum_{j\neq i}^{L}\rho_{kji}^{2}}}
h_{\mu \nu}( x , y ) \rightarrow h_{\mu \nu}^{\prime}( x , y ) = h_{\mu \nu}( x , y ) - \frac{1}{k}\partial_{\mu}\partial_{\nu}\xi^{5}( x ) + e^{- 2 k y}[ 2 \partial_{( \mu}\hat{\xi}_{\nu )}( x ) - 2 k \xi^{5}( x ) \eta_{\mu \nu}] .
\mathcal{I}_{\nu}(t) = \frac{e^{t}}{\sqrt{2\pi t}}\left(1 + \mathcal{O}(t^{-1})\right), K_{\nu}(t) = \sqrt{\frac{\pi}{2t}}e^{-t}\left(1 + \mathcal{O}(t^{-1})\right),
g^{(h)}({([c_k]S^{\ell_1p^k}T^{\ell_2p^k})_{k=0}^\infty}) =\sum_{k=0}^h p^k([c_k]S^{\ell_1p^k}T^{\ell_2p^k})^{p^{h-k}}&=(\sum_{k=0}^h p^k[c_k]^{p^{h-k}})S^{\ell_1p^h}T^{\ell_2p^h}.
\widetilde G_n(x,z)dz :=\P_x(S_n \in dz; \rho > n) & = \prod_{j=1}^d \lambda_j e^{-\sum_{j=1}^d \lambda_j(z_j - x_i)}\det(q_n(z_j - x_i))_{i, j = 1}^d dz.
\Omega = G_{A_{0}}( \stackrel{( 1 , 0 )}{\eta^{A_{0}}}+ \stackrel{( 0 , 1 )}{\eta^{A_{0}}}) + ({\raise . 5 p t \mathrm{~ \stackrel{( 0 , - 1 )}{{\cal ~ P}_{A_{0}}}~}}-{\raise . 5 p t \mathrm{~ \stackrel{( - 1 , 0 )}{{\cal ~ P}_{A_{0}}}~}}) \stackrel{( 1 , 1 )}{\pi^{A_{0}}}+ \cdots
\lim_{M\to\infty}\sum_{m=1}^MS_m^{(1)}=\frac{1}{\pi^2}\int_0^4 d\lambda_1\, \delta(\lambda_1)\,\mathcal{P}\!\!\,\int_0^4 d\lambda_2\, \frac{\delta'(\lambda_2)}{\lambda_2-\lambda_1}.
A_{[n,0]}=\sum_{\gamma\in\mathcal{D}_{[n,0]}}w(\gamma)=\sum_{j=0}^{p}\sum_{\gamma\in\mathcal{L}_{[n,j]}}w(\gamma)=\sum_{j=0}^{p}a_{0}^{(j)}A_{[n-1,j]},n\geq 1.
q_1 =\left(\begin{array}{cccc}0 0 \frac{i u_4}{a(t)}-u_1 \frac{i u_2}{a(t)}+\frac{u_3}{a(t)}0 0 \frac{i u_2}{a(t)}-\frac{u_3}{a(t)}-u_1-\frac{i u_4}{a(t)}-u_1-\frac{i u_4}{a(t)}-\frac{i u_2}{a(t)}-\frac{u_3}{a(t)}0 0 \frac{u_3}{a(t)}-\frac{i u_2}{a(t)}\frac{i u_4}{a(t)}-u_1 0 0 \end{array}\right),
\frac{d m_i}{d z_r}(z_1^*, z_1^*, \ldots) = (z_1^*)^{i - 1}\frac{d m_i}{d z_r}(1, 1, \ldots) = (z_1^*)^{i - 1}\left(\binom{2i}{i - r}- \binom{2i}{i - r - 1}\right).
\begin{aligned}a' \le\binom{|V|}{2}^{-1}\bigg(|W|\binom{2p|V|}{2}+2(c'p)^2|W|\binom{|V|}{2}\bigg) \le \big(4+2(c')^2\big)p^2|W|\le 5p^2|W|\,. \end{aligned}
\partial_{\mu}J{^{5 , \mu}_{3}}= \partial_{+}J{_3^{R}}- \partial_{-}J{_3^{L}}={\frac{g}{\pi}}( \partial_{-}A_{3}- \partial_{+}v ) ={\frac{g}{2 \pi}}\epsilon^{\mu \nu}F_{\mu \nu}^{3},
D^{2}\eta(x)(v)^{2}=D^{2}\psi(x,x_0)(v,0)^{2}\leq 2\frac{\sqrt{K_0}d(x,x_0) \cosh\left(\sqrt{K_0}d(x,x_0)\right)}{\sinh\left(\sqrt{K_0}d(x,x_0)\right)}\|v\|_{x}^{2},
f_if_{i'}=0, \ \f_ih_{i'}=0, \ \h_ih_{i'}=0, \ \h_i^2=f_i^3, &h_{S,j}h_{S,j'}=\sum_{i\in S}a_{ij}a_{ij'}f_i, &f_ih_{S,j}=a_{ij}h_i, &h_ih_{S,j}=a_{ij}f_i^2,
A ~ = ~{\frac{1}{2}}\; ( \; X_{1}^{\dagger}~ X_{1}~ + ~ X_{2}^{\dagger}~ X_{2}\; ) , ~ ~ B ~ = ~{\frac{1}{3}}~ \sqrt{X_{3}^{\dagger}~ X_{3}}, ~ ~ x ~ = ~ 4 ~ \sqrt{B}\; \mathrm{c o s}\left( ~{\frac{1}{3}}\; \mathrm{A r c c o s}~{\frac{~ A ~}{B^{3 / 2}}}~ \right) \; .
\Xi_1(z)=f_\mathrm{Pr}\left(r\right)e^{- z{{P_t}G_{}^2\beta \left({\max{{\{r,d\}}}}\right)^{-{\alpha_\mathrm{LoS}}}}}+(1-f_\mathrm{Pr}\left(r\right))e^{- z{{P_t}G_{}^2\beta \left({\max{{\{r,d\}}}}\right)^{-{\alpha_\mathrm{NLoS}}}}}
\gamma_{mk}(L)=(k)_{q^{-2}}!(m-k)_{q^{-2}}!q^{k(k+1)+\frac{m(m-3)}{2}}\sum_{\tau}q^{\sum_{i=1}^{k}l_{m\tau(i)}-\sum_{i=k+1}^{m}l_{m\tau(i)}}
r=2,s=2, t=4, u=3,k=1, &w=1,x=q, &a_R=(qs/ac,-qs/ac),b_S=(q^{1/2}s/a,-q^{1/2}s/a), &c_T=(q^{-n},q^{n+1}s^2/a^2,s,-qs/a^2),d_U=(qs/a,-qs/a,q^2s^2/a^2c^2), &e_K=q^2s^2/a^2c^2,\gamma=s^2/a^2.
U^{-1}=\left[ \begin{array}{cccccc}\ast \ast \cdots \ast \ast a_{n-1}0 \ast \cdots \ast \ast a_{n-2}\vdots \vdots \ddots \vdots \vdots \vdots 0 0 \cdots \ast \ast a_{2}0 0 \cdots 0 \ast a_{1}0 0 \cdots 0 0 a_{0}\end{array}\right]
\Psi_2 \left( \rho_1(y_1, y_1^\prime)^{p}, \boldsymbol{d}_{\mathcal{S}_2}(s_2, s_2^\prime)^p \right) = \Psi_2 \circ \psi_2 \left( \boldsymbol{d}_{\mathcal{Y}_1}(y_1, y_1^\prime)^{p}, \boldsymbol{d}_{\mathcal{Y}_2}(y_2, y_2^\prime )^p , \boldsymbol{d}_{\mathcal{X}}(x, x^\prime)^p \right).
\tilde \xi_{n, i}= \log(q_{2n, 2i}p_{2n, 2i}) + \log(q_{2n, 2i - 1}p_{2n, 2i - 1}), \xi_{n, i}(t) = \frac{2}{\sqrt{n}}(\lfloor nt \rfloor - i + 1) \big(\tilde\xi_{n, i}- \mathbb{E}[\tilde\xi_{n, i}]\big),
\begin{aligned}\operatorname{Div}R_1+\operatorname{Div}^{\vartriangle}R_2 =Q^{\alpha}F_{\alpha}^{\ast}+Q^{\alpha}_{\ast}F_{\alpha}&=Q^{\alpha}F_{\alpha}^{\ast}-v^{\alpha}\bold{pr}X(F_{\alpha})+\operatorname{Div}P_1+\operatorname{Div}^{\vartriangle}P_2.\end{aligned}
+ \left[ \partial_{\mu}C^{a}{}_{c}{}^{\nu}- C^{a}{}_{d \mu}C^{d}{}_{c}{}^{\nu}\right] A^{\prime c \mu}+ A^{\prime b}{}_{\mu}R^{\prime a}{}_{b}{}^{\mu \nu}+ \left[ C^{a}{}_{c \mu}+ f^{a}{}_{c b}A^{\prime b}{}_{\mu}\right] \partial^{\nu}A^{\prime c \mu}
C_\nu(t) \coloneqq \begin{cases}C e^{-td^\nu}(t\ge 1) C t^{-\lambda_\nu}(0<t\le 1), \end{cases}\lambda_\nu \coloneqq \max\Big\{\frac{d}{\nu}\Big( \frac{1}{p_2}-\frac{1}{p_1}\Big),0 \Big\},
|f(x)-f_{Q,\mu}| =f(x)-f_{Q,\mu}\geq f(x)-\log \beta_\mu=\log\left(\frac{v(x)}{\beta_\mu}\right) &= \frac{1}{2}\log\left[\frac{M(w\chi_Q)(x)}{\beta_\mu^2 w_{Q,\mu}}\right]\geq \frac{1}{2}\log\left[\frac{w(x)\chi_Q(x)}{\beta_\mu^2 w_{Q,\mu}}\right].
q = P( \overline{X}_1^* - \sqrt{n_1}G_m^{-1}(p) + o(1) > c_1, \overline{X}_1^* + \overline{X}_2^* - (\sqrt{n_1}+\sqrt{n_2}) G_m^{-1}(p) + o(1) > c_2 ) = P( Z_1 - \sqrt{n_1}G_m^{-1}(p) + o(1) > c_1, Z_1 + Z_2 - (\sqrt{n_1}+\sqrt{n_2}) G_m^{-1}(p) + o(1) > c_2 )
\iint G(p,q) d\mu(p) d\mu(q) \leq \iint \log \frac{1}{|z_{\infty}(p)^{-1}- z_{\infty}(q)^{-1}|}d\mu(p) d\mu(q) + C_1 & = \iint \log \frac{1}{|z^{-1}- w^{-1}|}d\omega(z) d\omega(w) + C_1 & = \iint \log \frac{|z w|}{|z-w|}d\omega(z) d\omega(w) + C_1 & \leq - \log \frac{\delta}{4}+ 2 \int \log |z| d\omega(z)+ C_1.
\delta E\!\simeq\!{M|{|\delta\omega|\over2\pi}\over r\!+\!{|\delta x|\over2}}\!-\!{M{|\delta\omega|\over2\pi}\over r\!-\!{|\delta x|\over2}}\!\simeq\!-\!{M|\delta\omega|\over 2\pi r^2}|\delta x|\!\simeq\!-\!{M\over 2\pi r^2},
|\sqrt{1+\eta_1}-1| = \Bigl|\eta_1\sum_{n=1}^\infty \binom{1/2}{n}\eta_1^{(n-1)}\Bigr|\le0.585787\times 2^{-d'}\Bigl|\frac{1}{\sqrt{1+\eta_1}}-1\Bigr| =\Bigl|\eta_1\sum_{n=1}^\infty \binom{-1/2}{n}\eta_1^{(n-1)}\Bigr|\le 0.828428\times 2^{-d'}
\begin{aligned}{}[K_1,K_2]=K_3 \qquad\quad\ [K_2,K_3] ={K_2}^2+\{K_1,K_2\}+dK_2+e_1 \quad\ [K_3,K_1] ={K_1}^2+\{K_1,K_2\}+dK_1+e_2\end{aligned}
\langle gE_{k-l}^{\textbf{1},\psi}(\cdot, s), f \rangle = \frac{\Gamma(s + k - 1)(k-l-1)!N^{k-l}}{2(4 \pi)^{s + k- 1}(-2\pi i)^{k-l}L(\overline{\psi},k-l+2s)G(\overline{\psi_0})}\sum_{n \geq 1}\frac{\overline{a_n}b_n}{n^{s + k -1}},
\nabla_{\!x}\cdot \Big(\big\{\beta_\varepsilon(v) + [v\!-\!\beta_\varepsilon(v)] \eta_\varepsilon(x) \big\}\psi \Big) =0, begin{align*}5pt] \nabla_{\!v}\cdot\big(\mathbf{B}\psi \big) = \mathbf{B}\cdot\nabla_{\!v}\psi + (\nabla_{\!v}\cdot\mathbf{B}) \psi = 0,
[{\cal J}^{\mu \nu},{\cal J}^{\rho \sigma}] = \eta^{\mu \rho}{\cal J}^{\nu \sigma}- \eta^{\nu \sigma}{\cal J}^{\mu \rho}+ \eta^{\nu \rho}{\cal J}^{\mu \sigma}- \eta^{\mu \rho}{\cal J}^{\nu \sigma}\,,
\mu_{2n+1}(q^{2})-\mu_{j}(q^{2})-\mu_{2n-j}(q^{2})-\mu_{j}(q^{2})\mu_{2n-j}(q^{2})(q^{2}-1) =\ q^{2n-j}(\theta_{j}(q^{2})-\mu_{j}(q^{2}))(q^{2n-j+1}-(-1)^{2n-j+1})
D_2(m_1,\ell) := \frac{1}{m_1+\ell}, &D_k(m_1,\ell) := \int_{0}^{\infty}dm_2 \cdots \int_{0}^{\infty}dm_{k-1}\frac{1}{\{\prod_{i=1}^{k-2}(m_{i}+m_{i+1})\}(m_{k-1}+ \ell)},\ \k\ge 3.
T \le e^{t(1-q)^2}qs + 2 \left(1- tq(1-q) + \frac{1}{2}t^2 q^2(1-q)^2 \right) q(1-s) + \left(1+ tq^2 + t^2 q^4\right) (1-2q+qs) = \alpha_0 + \alpha_1 t + \alpha_2 t^2 \, ,
\rho\left(n,\theta \frac{a_n}{n}\right) = \P\left( \exists |u| = n : \forall j \leq n, V(u_j) \geq -\theta a_n \frac{j}{n}\right) \leq \P\left( \exists |u| \leq n : V(u) \geq a_n g(|u|/n), V(u_j) \in I^{(n)}_j, j < |u| \right).
\sum_{n=0}^\infty \frac{(q;q^2)_nq^n}{(q^2;q^2)_n}= \frac{(q;q^2)_{\infty}}{(q^2;q^2)_{\infty}}\sum_{n=0}^\infty\sum_{|m|\leq n}(-1)^{n+m}q^{n^2+n-m^2}&= \frac{(q;q^2)_{\infty}}{(q^2;q^2)_{\infty}}f_{0,1,0}(-q,-q;q^4) &=\frac{J_2^2}{J_1},
\tilde{\phi}= \phi -{\frac{R_{y}}{\ell^{2}}}a_{2}\varphi -{\frac{R_{y}}{\ell^{3}}}a_{1}\tau , \quad \tilde{\psi}= \psi -{\frac{R_{y}}{\ell^{2}}}a_{1}\varphi -{\frac{R_{y}}{\ell^{3}}}a_{2}\tau ,
|u(r)|^p =\int_r^\infty\left(|u(s)|^p\right)'\mathrm ds=\int_r^\infty p|u(s)|^{p-2}u(s)u'(s)\mathrm ds &\leq p\int_r^\infty|u(s)|^{p-1}|u'(s)|\left(\dfrac{s}{r}\right)^{\frac{\alpha_0(p-1)+\alpha_1}{p}}\mathrm ds &\leq p\|u\|_{L^p_{\alpha_0}}^{\frac{p-1}{p}}\|u'\|_{L^p_{\alpha_1}}^{\frac{1}{p}}r^{-\frac{\alpha_0(p-1)+\alpha_1}{p}}.
T_i = C_{s_i}-t^{-\frac12}= t^{\frac12}s_i + \frac{t^{\frac12}-t^{-\frac12}}{1- X_iX^{-1}_{i+1}}(1-s_i),\qquad\hbox{for $i\in \{1, \ldots, n-1\}$.}
f_X(x)=\frac{2\lambda}{\pi}\frac{1}{(\lambda^2+x^2)}=\frac{1}{\lambda\pi}I^{1,1}_{1,1}\Bigg[\frac{x}{\lambda}\left|\begin{matrix}\left(0,\frac{1}{2},1\right) \left(0,\frac{1}{2},1\right) \end{matrix}\right.\Bigg],\ \x>0.
\left(\begin{array}{cc}0 f -f 0\end{array}\right) \star \left(\begin{array}{cc}0 g -g 0\end{array}\right) -\left(\begin{array}{cc}0 g -g 0\end{array}\right) \star \left(\begin{array}{cc}0 f -f 0\end{array}\right) =\left(\begin{array}{cc}\left[ g,f\right]_{\star}0 0 \left[ g,f\right]_{\star}\end{array}\right)
\begin{array}[c]{ll}\left[ \left( \mathsf{D}_{j}\left( s\right) \psi\right) \right]_{\operatorname*{D};j}\left( s\right) =\psi, \left[ \left(\mathsf{D}_{j}\left( s\right) \psi\right) \right]_{\operatorname*{N};j}^{\operatorname*{ext}}\left( s\right) =0.\end{array}
Y_t^{b,n}=\xi^b-\int_{t}^{T}g^b(s,Y^{b,n}_s,Z_s^{b,n})-\tilde\lambda_s^{b,n}\xi_s^a +\tilde\lambda_s^n Y_s^{b,n}ds-\int_{t}^{T}Z_s^{b,n}\cdot dW_s, ~t\in[0, T],
e^{izH_0}U_{n+1}(z',z'') e^{-izH_0}\Psi = -i \int_{z''}^{z'}d\zeta e^{izH_0}H_1(\zeta ) U_n (\zeta , z'') e^{-izH_0}\Psi & = -i \int_{z''}^{z'}d\zeta H_1(\zeta + z) U_n (\zeta + z , z'' +z) \Psi & = -i \int_{z''+z}^{z'+z}d\zeta H_1(\zeta ) U_n (\zeta , z'' +z) \Psi & = U_{n+1}(z' +z , z'' +z) \Psi ,
\frac{1}{2}\sum_{n\in\Psi_M}|f_j(n)-f(n)|^2 \leq\sum_{n\in\Psi_M}|f_j(n)-f_M(n)|^2+\sum_{n\in\zeta_M}|f_M(n)-f(n)|^2 &\leq\frac{2|\Psi_M|}{j}+\sum_{i=1}^{M-1}\sum_{n\in\Xi_i}|f_M(n)-f_i(n)|^2 &\leq\frac{2|\Psi_M|}{j}+\frac{|\Psi_M|}{M}
= \frac{\hbar}{8 \pi^{2}}\alpha^{4}\phi_{c}^{4}\sum_{N = 1}^{m}\frac{K_{2}\left( 2 N \alpha a \phi_{c}\right)}{\left( N \alpha a \phi_{c}\right)^{2}}+ \frac{\hbar}{8 \pi^{2}}\alpha^{4}\phi_{c}^{4}\sum_{N = m + 1}^{\infty}\frac{K_{2}\left( 2 N \alpha a \phi_{c}\right)}{\left( N \alpha a \phi_{c}\right)^{2}}.
\begin{cases}\cot(\pi\theta(n+1)-\pi k)=\cot\pi\theta(n)-\frac{V(n)}{\sin\pi k}, \cot(\pi\tilde{\theta}(n+1)-\pi \tilde{k})=\cot\pi\tilde{\theta}(n)-\frac{V(n)}{\sin\pi \tilde{k}},\end{cases}
\sum_{(\log Z)^9 < p \leq Z^{\frac{2}{d}}(\log Z)^{-9}}\frac{1}{p^k}= O \left(\sum_{(\log Z)^9 < p}\frac{1}{p^k}\right) & = O \left(\frac{1}{(\log Z)^{9(k-1)}}\right) & = O \left(\frac{1}{(\log Z)^9}\right).