harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal F}\Big ( \{x , y \}\Big ) \; : = \; \prod_{a = 1}^{N}\Bigg | \frac{\prod_{i , j = 1}^{n_{a}}\big ( \hat{x}_{i}^{( a )}- \tilde{y}_{j}^{( a )}\big ) \big ( \tilde{x}_{i}^{( a )}- \hat{y}_{j}^{( a )}\big )}{\prod_{i , j = 1}^{n_{a}}\big ( \hat{x}_{i}^{( a )}- \tilde{x}_{j}^{( a )}\big ) \big ( \tilde{y}_{i}^{( a )}- \hat{y}_{j}^{( a )}\big )}\Bigg |^{[ 1 - \frac{g}{\pi + g N}]}
E^p[H_1](\tau_2)+C_*E^{p}[H_0]^{\leq 1}(\tau_2)+\int_{\tau_1}^{\tau_2}E^{p-1,deg}[H_1](\tau)d\tau \leq C\Big(E^p[H_1](\tau_1)+C_*E^{p}[H_0]^{\leq 1}(\tau_1)\Big).
\begin{aligned}\frac{d}{dt}\int |\nabla d|^4dx+4\int |\nabla d|^2 |\nabla^2 d|^2dx+2\int |\nabla(|\nabla d|^2)|^2 dx &=2\int_{\partial \Omega}|\nabla d|^2<\nabla (|\nabla d|^2), \nu>d\sigma+4\int \nabla(|\nabla d|^2 d):|\nabla d|^2 \nabla d \dx &-4\int \nabla (u \cdot \nabla d):|\nabla d|^2 \nabla d \dx=\sum_{i=1}^3{I\!I}_{1i},\end{aligned}
\sum_{d>x}\frac{|f(d)|}{d}=\sum_{x<d\le 2x}\frac{|f(d)|}{d}+\sum_{2x<d\le 4x}\frac{|f(d)|}{d}+\sum_{4x<d\le 8x}\frac{|f(d)|}{d}+\cdots &\le \frac{E_2(2x)}{x}+\frac{E_2(4x)}{2x}+\frac{E_2(8x)}{4x}+\cdots &\le \frac{E_2(x)}{x}(c+c^2/2+c^3/4+c^4/8+\cdots) &\le \frac{2c}{2-c}\frac{E_2(x)}{x}.
\{E_2\}=\sum_{u=1}^{m-1}\sum_{v=2}^{n-2}\{E_2^{u,v}\}\leq (m-2)(n-4)\{E_2^{1,2}\}\leq n^2\ln n O(n^{-3}\ln n) =O(n^{-1}(\ln n)^2).
\beta_{\l}+ \sigma_{\l-1}\lambda_{\l}\left(1-\frac{1}{\l}\right)^2 = (2\l+d-2) \left(1-\frac{1}{\l}\right) + \frac{2 \delta}{\l-1}\lambda_{\l}\left(1-\frac{1}{\l}\right)^2 = 2(1+\delta)\l + O(1)
\sum_{k=1}^\infty (-1)^{k+1}\sum_i 2^{L-i}\binom{L}{i}\binom{N}{n+K-i}= \sum_i 2^{L-i}\binom{L}{i}\sum_{k=1}^\infty (-1)^{k+1}\binom{N}{n+k-i}&= \sum_i 2^{L-i}\binom{L}{i}\binom{N-1}{n-i},
= log(\rho_{0}) + log(1 + \frac{1}{N}\sum_{{\bf{q}}\neq 0}(e^{i{\bf{q}}.({\bf{x}}-{\bf{x}}^{'})}- 1)\theta(k_{f}- |{\bf{q}}|)) \approx log(\rho_{0}) + \frac{1}{N}\sum_{{\bf{q}}\neq 0}(e^{i{\bf{q}}.({\bf{x}}-{\bf{x}}^{'})}- 1)\theta(k_{f}- |{\bf{q}}|)
\left (\frac{I_{\mathcal{B}_{i}}}{I_{\mathcal{B}_{i-1}}}\right)^2=\frac{I_{\mathcal{B}_{i}}}{I_{\mathcal{B}_{i-1}}}\frac{I_{\mathcal{B}_{i}}}{I_{\mathcal{B}_{i-1}}}=\frac{I_{\mathcal{B}_{i}}^2+I_{\mathcal{B}_{i-1}}}{I_{\mathcal{B}_{i-1}}}=\frac{I_{\mathcal{B}_{i}}}{I_{\mathcal{B}_{i-1}}}
\tau_{i}= 1/s_{i},\tau = \left(A_{\alpha}\,\sum\limits_{i=1}^N \frac{\tilde{\mu}_i}{s_i}\right)^{-1/\alpha},\mu_{i}= \frac{\tilde{\mu}_i}{s_i}\,\left(\sum\limits_{i=1}^N \frac{\tilde{\mu}_i}{s_i}\right)^{-1},
(q^{a_i-a_j+n_i-n_j};q)_{n_j+1}(q^{a_i-a_j-n_j};q)_{n_i}=(q^{a_i-a_j-n_j};q)_{n_i+n_j}=(q^{a_i-a_j-n_j};q)_{n_j+1}(q^{a_i-a_j+1};q)_{n_i}.
d_Y(f_{n,n}(x),f(x))\le \sum\limits_{k=m}^{n-1}d_Y(f_{n,k+1}(x),f_{n,k}(x))+d_Y(f_{n,m}(x),f_n(x))< <\sum\limits_{k=m}^{n-1}\frac{1}{2^k}+\frac{1}{2^m}<\frac{1}{2^{m-2}}+\frac{1}{2^m}<\frac{1}{2^{m-3}}<\varepsilon
\begin{aligned}2 \bigg\langle \sqrt{n}\sum_{j \in \mathbb{Z}}(\overline{W}_{j - 1}- \overline{W}_j ) \nabla^n \varphi^n_j , f \bigg\rangle_{L^2 (\nu^n_\rho )}= - 2 \sqrt{n}E_{\nu^n_\rho}\bigg[ \sum_{j\in \mathbb{Z}}W_{j - 1}\nabla^n \varphi^n_j \nabla_{j-1 , j}f \bigg] .\end{aligned}
{\bf E}\bigl[M_{(\tau=1,\lambda_1,\lambda_2)}^q\bigr] = \frac{G(2+\lambda_1)}{G(2-q+\lambda_1)}\frac{G(2+\lambda_2)}{G(2-q+\lambda_2)}\frac{G(1)}{G(-q+1)}\frac{G(4-2q+\lambda_1+\lambda_2)}{G(4-q+\lambda_1+\lambda_2}.
\hat{Q}_{2}\left( z \right):=\left( \Gamma_{0}^{+}\Gamma_{0}\right)^{-1}\Gamma_{0}^{+}A(I-P)\left( \tilde{A}-z \right)^{-1}(I-P)A\Gamma_{0}\left( \Gamma_{0}^{+}\Gamma_{0}\right)^{-1}
a_n = \frac{1}{2}- \frac{1}{8n^2}- \frac{2C}{(n\log n)^2}+ o\left(\frac{1}{n^2(\log n)^2}\right), b_n = \frac{1}{4}- \frac{1}{32n^2}+ \frac{C}{(n \log n)^2}+ o\left(\frac{1}{n^2(\log n)^2}\right),
( \partial_{\bar{l}}A_{0}^{z}) ( \partial_{l}A_{0}^{\bar{z}}) - ( \partial_{\bar{l}}A_{0}^{\bar{z}}) ( \partial_{l}A_{0}^{z}) = \frac{1}{2}{\cal L}_{0}= \frac{4}{\cosh^{2}( l )}.
AG \geq \frac{2}{n}\phi G^{2}+\bigg(-\frac{4}{n}(\lambda -1)aH^{p}+(\lambda -2)paH^{p-1}\bigg)\phi G-2\left\vert \nabla\phi \right\vert G^{\frac{3}{2}}-\left(2K+\frac{12\left\vert b \right\vert}{n}\right)\phi G.
\frac{\nu^{1/2}}{c}\log\frac{\tfrac{\nu^{1/2}}{c}}{h^*}= \,\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{h_k}-k+O(1), \frac{\nu^{1/2}}{c}\log\frac{\tfrac{\nu^{1/2}}{c}}{w^*}= \,\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{w_k}-k +O(1).
F^\alpha_j{\frac{\partial C^i_{kl}}{\partial u^\alpha}}+F^\alpha_k{\frac{\partial C^i_{lj}}{\partial u^\alpha}}+F^\alpha_l{\frac{\partial C^i_{jk}}{\partial u^\alpha}}=\bigl(C^i_{mj}C^m_{kl}+C^i_{mk}C^m_{lj}+C^i_{ml}C^m_{jk}\bigr)
\mu_{-}= \left( \begin{array}{c c}{{I_{m_{1}}}}&{{0}}\\{{\mu_{- 2 1}}}&{{I_{m_{2}}}}\\ \end{array}\right) , \qquad \mu_{+}= \left( \begin{array}{c c}{{I_{m_{1}}}}&{{\mu_{+ 1 2}}}\\{{0}}&{{I_{m_{2}}}}\\ \end{array}\right) ,
\begin{array}{c}{\sum_{i_{1}. . . i_{N}}d^{- N}\sum_{S}\prod_{j}{}\left( \varphi_{j}, \psi_{S ( j )}\right) \left( e_{i_{P ( j )}}, e_{i_{Q ( j )}}\right) =}\\{= \sum_{S}\prod_{j}\left( \varphi_{j}, \psi_{S ( j )}\right) \varphi^{\left( N \right)}( P S Q^{- 1})}\\ \end{array}
A_\pm(\xi) := \left[ \begin{matrix}-2\xi \pm 2c^{-2}(c\xi \mp \sqrt{-3c}\;\xi) \sqrt{-3c}\pm 2c^{-1}(\xi \mp \sqrt{-3c}\;\xi) \sqrt{-3c}\cdot \xi \pm \sqrt{-3c}\;\xi \cdot \pm c^{-1}\sqrt{-3c}\;\xi \xi\end{matrix}\right]
\sum_{k=1}^l \beta_k =\lambda_{n+1}\sum_{k=1}^l \frac{d(\gamma(t_{k-1}),\gamma(t_{k}))}{d(p_{n+1},q_{n+1})}\leq\frac{\lambda_{n+1}L(\gamma)}{d(p_{n+1},q_{n+1})}&<\lambda_{n+1}+\frac{\lambda_{n+1}\varepsilon\delta}{d(p_{n+1},q_{n+1})}\leq \lambda_{n+1}+\frac{\varepsilon}{2}.
P_{\textrm{isout}}^{\textrm{ISANC}}=\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NDN=0}\right\rbrace +\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{FDN=0}\right\rbrace.
(dg) g^{-1}= \check e_i \frac{dz}{(z-x)^2}, g u g^{-1}= u - \frac{1}{z-x}[\check e_i, u] = u + \frac{\langle u, \check \alpha_i \rangle}{z-x}\check e_i, g p_{-1}g^{-1}= p_{-1}- \frac{1}{z-x}[\check e_i, p_{-1}] + \frac{1}{2(z-x)^2}\big[ \check e_i, [\check e_i, p_{-1}] \big] = p_{-1}- \frac{\alpha_i}{z-x}- \frac{\check e_i}{(z-x)^2}.
P^\omega(D_t^c \mid S) = \frac{1}{P^\omega(S)}\left[P^\omega(D_t^c\cap S\cap C_t)+P^\omega(D_t^c\cap S\cap C_t^c)\right] &\leq c(\omega) \left[P^\omega(D_t^c\cap S_t\cap C_t)+P^\omega(D_t^c\cap S\cap C_t^c)\right] &\leq c(\omega) \left[P^\omega(D_t^c \mid C_t)+P^\omega(C_t^c \mid S)\right],
\begin{gathered}s_{1}(\theta) -t_{1}(\theta)=\arcsin(\frac{M\sin^2(\theta)-1}{\cos(\theta)}) -\arcsin(-\cos(\theta)) \leq \frac{1}{(1-\cos^2(\theta))^{\frac{1}{2}}}\frac{(M-1)\sin^{2}(\theta)}{\cos(\theta)}= \frac{(M-1)\sin(\theta)}{\cos(\theta)},\end{gathered}
Ric(T,T) =-m\frac{\bar{u}_{\bar{t}\bar{t}}}{\bar{u}}-(n-1)\frac{\bar{g}_{\bar{t}\bar{t}}}{\bar{g}}\ge-m\frac{C_3}{(1-C_3)^2\bar{t}^2}+2(n-1)\frac{C_3}{(1-C_3)^2\bar{t}^2}>0,
\int_{r_0-s}^{\infty}\frac{\d r}{\sqrt{{\sf K}\left(e^{-\frac{2\underline{\phi}_q(r)}{n-m}}r\right)}e^{-\frac{2\underline{\phi}_q(r)}{n-m}}}\geq\int_{r_0}^{\infty}\frac{\d r}{\sqrt{{\sf K}\left(e^{-\frac{2\underline{\phi}_p(r)}{n-m}}r\right)}e^{-\frac{2\underline{\phi}_p(r)}{n-m}}}=+\infty.
J:=\int_{z: \rho_L(z,0)>\varepsilon_L}\rho_L(z,0) d\mu(z)=\sum_{m=1}^\infty\frac 1{m^2}\int_{z: \rho_L(z,0)>\varepsilon_L}(1-|z|^2)^{mL}d\mu(z)\ .
S_{p,p-k}= \frac{(p+1+k)!}{p!(p-k)!}\sum_{j=0}^k(-1)^{j+1}\frac{1}{(k+j)!(k-j)!(p+1+j)}S(j+k,j) &= \frac{p+1}{(2k)!(p-k)!}\sum_{j=0}^k(-1)^{j+1}\binom{2k}{k+j}\frac{C(p,k)}{p+1+j}S(j+k,j)
\sup_{t\in (0,T)}\left( \|b^k(t)\|_1 + \|\ln b^k(t)\|_1\right)+ \int_0^T \|\nabla b^k\|_{2-\lambda}^{2-\lambda}+ \|b^k\|_{\frac{8-\lambda}{3}}^{\frac{8-\lambda}{3}}\; dt + \int_0^T \|\mu^k \nabla b^k\|_{\frac{8-\lambda}{7}}^{\frac{8-\lambda}{7}}\; dt\le C(\lambda^{-1})\textrm{for all}\lambda \in (0,1).
D_z^+\psi(F) =\psi'(F)D_z^+F +R_\psi^+(F,z)\bigl(D^+_zF\bigr)^2\quad D_z^-\psi(F) =\psi'(F)D_z^-F +R_\psi^-(F,z)\bigl(D^-_zF\bigr)^2\,.
: = \; \left\langle \prod_{b = 1}^{N}2 \left( \sum_{n_{b}= 0}^{\infty}\frac{1}{( 2 n_{b}) !}\Big ( \beta^{( b )}\Big )^{2 n_{b}}\; \left[ \chi_{+}^{( b )}[ Q^{W}] + \chi_{-}^{( b )}[ Q^{W}] \right]^{2 n_{b}}\right) \right\rangle_{Q^{\mu}}
\phi_{F_2}^{t}\left(v,w\right)=\left(\frac{v+w}{2}+(\cos t)\frac{v-w}{2}+(\sin t)\frac{w\times v}{\|v+w\|},\frac{v+w}{2}+(\cos t)\frac{w-v}{2}+(\sin t)\frac{v\times w}{\|v+w\|}\right).
y^{\delta}\tilde{\Gamma}^{\alpha}{}_{\beta \delta}\tilde{B}^{\beta}|_{1 + 3}= y^{\delta}d x^{\sigma}[ \Gamma^{\alpha}{}_{\beta \delta}B^{\beta}{}_{, \sigma}+ \Gamma^{\alpha}{}_{\beta \delta}\Gamma^{\beta}{}_{\sigma \tau}B^{\tau}+ \Gamma^{\alpha}{}_{\beta \delta , \sigma}B^{\beta}] .
\mathfrak{\widehat D}_{k}=-kX^{d}a^{k-2}+\frac{(k-3)k}{2!}X^{2d}a^{k-4}- \frac{(k-4)(k-5)k}{3!}X^{3d}a^{k-6}+ \cdots +(-1)^{\frac{k-3}{2}}\,\frac{(k-1)k(k+1)}{24}\,X^{\frac{d(k-3)}{2}}a^3+(-1)^{\frac{k-1}{2}}\,k\,X^{\frac{d(k-1)}{2}}a.
\begin{matrix}\begin{aligned}C_{0}=\mathrm{Cone}\left\{(0,1,0,0),(1,0,0,1),(0,0,1,0),(1,0,1,0),(0,1,0,1)\right\}, C_{NE}={\rm Cone}\left\{\begin{matrix}(1,0,0,0),(0,1,-1,1),(1,-1,1,0),(0,0,0,1) (0,0,1,-1),(-1,1,0,0)\end{matrix}\right\}.\end{aligned}\end{matrix}
f(S_{k+1}) =f(S_k\#_{\frac{1}{k+1}}Y_{k+1}) &\leq \frac{k}{k+1}f(S_k)+\frac{1}{k+1}f(Y_{k+1}) &\leq \frac{k}{k+1}Z_k+\frac{1}{k+1}f(Y_{k+1})=Z_{k+1}
\mathbf{B}(i,j)=\mathbf{A}^{\mathsf{T}}(i,j)\circ \mathbf{A}(i,j)=1, \and \mathbf{B}(j,i)=\mathbf{A}^{\mathsf{T}}(j,i)\circ \mathbf{A}(j,i)=1.
\int_0^j\frac{(j-s)^{-\alpha}q'(s)}{\Gamma(1-\alpha)}d s=\frac{B(m\alpha,1-\alpha)j^{(m-1)\alpha}}{\Gamma(1-\alpha)\Gamma(m\alpha)}=\frac{j^{(m-1)\alpha}}{\Gamma(1+(m-1)\alpha)},
\delta(y)+\delta(\overline{y})=1, \textrm{whenever}y, \overline{y}\in B; & \delta(y)+\delta(e_i+\overline{y})=1, \textrm{whenever}y, e_i+\overline{y}\in B; & \delta(y)+\delta(e_i+y)=0, \textrm{whenever}y, e_i+y \in B.
\Sigma (\Omega r)(xy) = r(\Sigma (xy)) &= r\left((-1)^{|x|}(\pi_2^*x) \Sigma y \right) &= r_{[1]}\left(\pi_2^*((-1)^{|x|}x)\right) r_{[2]}(\Sigma y) &= \pi_2^*\left((\sigma^{|x|}r_{[1]})(x)\right) \Sigma (\Omega r_{[2]})(y) &= \Sigma \left( (-1)^{\deg_*(r_{[1]})}(\sigma^{|x|}r_{[1]})(x) (\Omega r_{[2]})(y) \right)
\int_{y}\Delta_{\varphi}( x_{1}, y ) \frac{\delta^{2}S_{B}}{\delta \varphi ( y ) \delta \varphi ( x_{2})}= \int_{y}\frac{\delta^{2}S_{B}}{\delta \varphi ( x_{1}) \delta \varphi ( y )}\Delta_{\varphi}( y , x_{2}) = \delta ( x_{1}- x_{2}) \; .
A^{+}=sym(e_{1}\mathfrak{a}e_{1}\oplus e_{2}\mathfrak{a}e_{2}),A^{-}=skew(e_{1}\mathfrak{a}e_{1}\oplus e_{2}\mathfrak{a}e_{2}),B=\mathcal{B}e_{2},B'=\mathcal{B}e_{1}, E =sym(e_{1}\mathfrak{a}e_{2}),C=skew(e_{1}\mathfrak{a}e_{2}),E'=sym(e_{2}\mathfrak{a}e_{1}),C'=skew(e_{2}\mathfrak{a}e_{1}),
\lambda_{\epsilon(1)}(T_1)E_{D_0}(\mu_2(T_2),\ldots, \mu_n(T_n)) = \left([L_{p_{\epsilon(1)}\left(T_1(1-p_{\epsilon(1)}) S_1 1_B\right)}(1-p_{\epsilon(k_{2})}) S_2 1_B] \otimes \cdots \otimes [(1-p_{\epsilon(k_{m})}) S_m1_B]\right) & \oplus \left([(1-p_{\epsilon(1)})T_1(1-p_{\epsilon(1)}) S_1 1_B] \otimes \cdots \otimes [(1-p_{\epsilon(k_{m})}) S_m1_B]\right).
\begin{aligned}\mathbb E_{\delta}\int_0^t \int_D S(t-s,x-y) |f(u(s,y))|^{2}dsdy <+\infty, & \mathbb E_{\delta}\int_0^t \int_D S(t-s,x-y) |u(s,y)|^{q}dsdy <+\infty, \end{aligned}
\begin{array}{lll}I_1^h = \displaystyle{\frac{1}{4}\int_{\partial B^h}\Big(\phi(z+h)n_1^{+}(z)+\phi(z-h)n_1^{-}(z)+i\phi(z+ih)n_2^{+}(z)}+i\phi(z-ih)n_2^{-}(z) \Big)f^h(z)dS^h(z)-\displaystyle{\int_{B^h}\phi\partial_{\bar z}^hf^hdV^h}.\end{array}
\frac{1}{2}H f(z_{0})=\lim_{\epsilon\rightarrow 0^{+}}\frac{1}{2\pi i}\int_{\gamma_{\epsilon}}\frac{f(w)}{w-z_{0}}dw=\lim_{\epsilon\rightarrow0^{+}}\lim_{z\rightarrow z_{0}}\frac{1}{2\pi i}\frac{f(w)}{w-z}dw.
\left\{e^{\imath\,\vartheta_{kA}(\varphi^{-1}_{k}(x),z_k)}\, e^{-\imath\,\vartheta_{kB}(\varphi^{-1}_{k'}(x),z_k)},...x\in\,M_4,\,z_k\in\,T_{\varphi^{-1}_k(x)}M^k_4,\,k=1,...,N\right\}
P = \big \lbrace \emptyset \neq P_i \subseteq \left \lbrace 1, \ldots, n \right \rbrace ~\forall i, P_i \cap P_j = \emptyset ~ \forall i\neq j, &P_1 \cup \ldots \cup P_k = \left \lbrace 1, \ldots, n \right \rbrace \big \rbrace.
\bar{\lambda}_1 \leq \frac{\pi^2}{D^2}-\frac{K}{2}-\frac{(\pi^2-6)D^2K^2}{48\pi^2}-\frac{(120-20\pi^2+\pi^4)D^4K^3}{480\pi^4}-\frac{17(\pi^6-42\pi^4+840\pi^2-5040)D^6K^4}{80640\pi^6}.
\begin{gathered}\int \psi_x^\ell(\eta)^2 d\mu_\rho \leq \frac{c(\rho)}{\ell^2}, \int \Big( \psi_x^\ell(\eta) - \big(\eta^\ell(x) -\rho\big)^2 +\frac{\rho(1-\rho)}{\ell}\Big)^2 d\mu_\rho \leq \frac{c(\rho)}{\ell^3}.\end{gathered}
(1)bD\frac{d^{4}w_{x}}{dx^{4}}=q_{1}(x)-n_{1}(x)\frac{d^{2}w_{x}}{dx^{2}}-\frac{dn_{1}}{dx}\frac{dw_{x}}{dx}-\frac{1}{2}\frac{dn_{2}}{dx}\frac{d\theta_{x}}{dx}-\frac{n_{2}(x)}{2}\frac{d^{2}\theta_{x}}{dx^{2}}
\sin\left( \, \underline{r}\,\underline{\omega}^\intercal -(2\nu + 1)\frac{\pi}{4}\right) = - d_2 Q^\intercal C_{N+1}^{I}Q + d_1\begin{bmatrix}S_{N-1}^{I}\!\!\!\! \underline{0}\underline{0}^\intercal \!\!\!\! 0\end{bmatrix}.
k_{\lambda_1\lambda_2\cdots\lambda_r}^\rho=\frac{|DC_{\lambda_1}||DC_{\lambda_2}|\cdots|DC_{\lambda_r}|}{|G|}\sum_{\theta\in \mathcal{J}^{'}}\mathcal{X}^\theta(1)\omega^\theta_{\lambda_1}\omega^\theta_{\lambda_2}\cdots \omega^\theta_{\lambda_r}\overline{\omega^{\theta}_\rho}.
{k}_{\perp , \mathrm{m a x}}^{2}= \mathrm{M i n}\Bigl \{\Bigl ( \omega - q A_{0}( + \infty ) \Bigr )^{2}- m^{2}, \Bigl ( \omega - q A_{0}( - \infty ) \Bigr )^{2}- m^{2}\Bigr \}.
|Q^{-1/2}_t e^{tA}k|_H \le \frac{c}{t^{3/2}}|k|_K,\; \;\; k \in K = V \times U; |Q^{-1/2}_t e^{tA}Ga|_H \le \frac{c}{t^{1/2}}|a|_U,\; \;\; a \in U.
\sum_{n=0}^{\infty}\zeta(2n)x^{2n}=-\frac{\pi x}{2}cot(\pi x)=-\frac{1}{2}+\frac{\pi^{2}}{6}x^{2}+\frac{\pi^{4}}{90}x^{4}+\frac{\pi^{6}}{945}x^{6}+\cdot\cdot\cdot
h_0^{-\gamma_{i+1}\,p-p-1}\,\frac{i_0^{2\,p}}{s\,(1-s)}\,\|u_R\|^p_{L^p(B_1)}\le C\,i_0^{4\,p+1}\,\mathcal{A}_1,i=0,\dots,i_0-2,
\hat{S}= \int d^{11}\hat{x}\sqrt{|\hat{g}|}\left[ \hat{R}-{\textstyle\frac{1}{2 \cdot 4!}}\hat{G}{}^{2}-{\textstyle\frac{1}{(144)^{2}}}{\textstyle\frac{1}{\sqrt{|\hat{g}|}}}\, \hat{\epsilon}\hat{G}\hat{G}\hat{C}\right]\, .
0\to\nu(r)\to\Omega_{Y, d = 0}^{r}\overset{C - 1}{\longrightarrow}\Omega_{Y}^{r}\to0, 0\to\nu(r)\to\Omega_{Y}^{r}\overset{C^{-1}- 1}{\longrightarrow}\Omega_{Y}^{r}/ d \Omega_{Y}^{r - 1}\to0
\sum_{\Gamma\in{G}^{rt}_{2,n}}\frac{1}{|{\rm Aut}(\Gamma)|}\sum_{j=0}^{n-|E_2(\Gamma)|}{\xi_{\Gamma}}_*\left(\prod_{i=1}^n \left(1+3\omega_i\right) \prod_{(h,h')\in E_2(\Gamma)}\frac{1}{\psi_h-(1+3\omega_{h'})}\right)\cdot (-\lambda - \delta_1)^{j}.
T^{(2)}_{\widetilde{m}^{\varepsilon}_{\mathbb{II},\mathbb{I}}}(f_{1},f_{2})(x):=\sum_{Q'\in\mathbb{Q}'_{\mathbb{II}}, \, Q''\in\mathbb{Q}''_{\mathbb{I}}}\int_{\mathbb{R}^{4}}\widetilde{m}^{\varepsilon}(\xi)\phi_{Q'}(\bar{\xi}_{1})\phi_{Q''}(\bar{\xi}_{2})\widehat{f_{1}}(\xi_{1})\widehat{f_{2}}(\xi_{2}) e^{2\pi ix\cdot(\xi_{1}+\xi_{2})}d\xi
F^{1 / 2}= \frac{2^{1 / 2}N^{1 / 2}g_{s}( M_{s}^{2}A_{T})}{5 \pi \sqrt{3}\, ( 2 \pi )^{3}\, ( 1 . 9 1 \times 1 0^{- 5}) ( 2 \pi R M_{s})^{6}}\, .
{n \choose k}2^{n-k}q(k) = \sum_{\sigma}\sum_{i=0}^{n-k}\frac{Q_{i,i+k}(\sigma)}{i! k! (n-i-k)!}&= \frac{1}{n!}{n \choose k}\sum_{\sigma}\sum_{i=0}^{n-k}{n-k \choose i}Q_{i,i+k}(\sigma).
f_{1}= 1 +{\frac{r_{1}^{2}}{r^{2}}}, f_{5}= 1 +{\frac{r_{5}^{2}}{r^{2}}}, f_{K}= 1 +{\frac{r_{K}^{2}}{r^{2}}}, d = 1 -{\frac{r_{0}^{2}}{r^{2}}},
\begin{aligned}\, 2 K_1^{\frac{p-1}{p-2+m}}\, \frac{C^{m-1}m (N-1) k \coth(k)}{a(m-1)}\, (T-t)^{-\alpha m+\beta}\leq \, C^{\frac{(p-1)(m-1)}{p-2+m}}\left(\frac{C^{m-1}m}{2 a^2(m-1)^2}\right)^{\frac{p-1}{p-2+m}}(T-t)^{\frac{-\alpha p(m-1)+ (2\beta-\alpha m)(p-1)}{p-2+m}}\forall t \in (0,T) \, .\end{aligned}
E(\sum_{j=1}^{n}X_{ij}^{2})+E(\sum_{1\le j,k\le n}\sum_{j\ne k}X_{ij}X_{ik})=n\cdot\frac{1}{k}+n(n-1)\cdot\frac{1}{k^{2}}=\frac{n^{2}+nk-n}{k^{2}}
L_{n}( \lambda , t ) = \left( \begin{array}{c c}{t^{Z_{n}- S_{n}^{3}}s h [ \gamma ( \lambda + i S_{n}^{3}) ]}&{i S_{n}^{-}s i n \gamma}\\{i S_{n}^{+}s i n \gamma}&{t^{- Z_{n}- S_{n}^{3}}s h [ \gamma ( \lambda - i S_{n}^{3}) ]}\\ \end{array}\right) ,
2{\partial A^i \over \partial z}=- \Omega_{-}F^i = \Omega_{-}K^{-1 i j^*}\frac{\partial{\cal W}^*}{\partial A^{*j}}, \quad \Omega_{-}\equiv i\frac{\langle -i Z_1^*-Z_2^* \rangle}{|\langle -i Z_1^*-Z_2^* \rangle|}.
Z_{N}= \prod_{j = 1}^{N}\int_{\mathrm{P B C}}{d \mu \left({\xi_{j}^{*}, \xi_{j}}\right)}\exp \left[ i J \left\{2 i \ln \left({\frac{1 + \xi_{j}^{*}\xi_{j}}{1 + \xi_{j}^{*}\xi_{j - 1}}}\right) + \epsilon h{\frac{1 - \xi_{j}^{*}\xi_{j - 1}}{1 + \xi_{j}^{*}\xi_{j - 1}}}\right\}\right] .
\mathcal{A}^{\varepsilon}(x^{\ast})=\bigcap\limits_{\eta>0}\bigcup\limits_{y^{\ast}\in Y^{\ast}}\Big\{x\in X\,:\,\left(x,0_{Y}\right) \in \left( M^{\varepsilon +\eta}F\right) (x^{\ast},y^{\ast})\Big\}.
\gamma_{\textrm{FHN}}^{\textrm{NOMA}}=\frac{P_\textrm{N}}{P_\textrm{F}+{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^2}/{|h_{\textrm{BN}}|^2}}+\frac{\beta_{\textrm{F}}\eta P_\textrm{B}|h_{\textrm{BF}}|^2 |h_{\textrm{NF}}|^2}{d_{\textrm{BF}}^{\alpha}d_{\textrm{NF}}^{\alpha}\sigma_{\textrm{N}}^2}.
\mathcal{S}= \Sigma + \displaystyle \int d^{4}x \left( b^{a}\partial A^{a}+ h_{\nu}^{a}\partial_{\mu}B^{a \mu \nu}+ \omega^{a}\partial \xi^{a}+ h^{a \mu}\partial_{\mu}e^{a}+ \omega^{a}\lambda^{a}+ - \lambda^{a}\partial \overline{{{\xi}}}^{a}\right) \; ,
P_f[X_1+\cdots+X_k=n] = \sum_{\pi_1+\cdots+\pi_k=n}P[X_1=\pi_1]\cdots P[X_k=\pi_k] = \sum_{\pi_1+\cdots+\pi_k=n}f(\pi_1)\cdots f(\pi_k) =\binom{k}{n}_f,
V(f):=4\left\Vert f\right\Vert_{L^{1}\left( \mathbb{T}^{2}\right)}+2\left\Vert \frac{\partial f}{\partial t_{1}}\right\Vert_{L^{1}\left(\mathbb{T}^{2}\right)}+2\left\Vert \frac{\partial f}{\partial t_{2}}\right\Vert_{L^{1}\left( \mathbb{T}^{2}\right)}+\left\Vert \frac{\partial^{2}f}{\partial t_{1}\partial t_{2}}\right\Vert_{L^{1}\left(\mathbb{T}^{2}\right)}\;.
\left( \begin{matrix}{{{\hat{a}}}}\\{{{\hat{b}}^{\dagger}}}\\ \end{matrix}\right) \equiv \left( \begin{matrix}{{\alpha}}&{{- \beta^{*}}}\\{{\beta}}&{{\alpha^{*}}}\\ \end{matrix}\right) \left( \begin{matrix}{{a}}\\{{b^{\dagger}}}\\ \end{matrix}\right)
S:=\sum_{i_0=0}^\infty \sum_{i_1\geq i_0}^\infty \cdots \sum_{i_{d}\geq i_{d-1}}^\infty 2^{-(i_0+i_1+\cdots +i_{d})}~U(k, i_0, \ldots, i_d, \ell_0, \ell_1, \ldots, \ell_k)~q^{-k(d-k)}\lesssim 1.
\|r\|_{X}=\sup_{t\in (0,t_0]}\left(\frac{\|r\|_{L^\infty((0,t),L^2)}}{t^{3/4}}+\frac{\|\partial_\sigma r\|_{L^\infty((0,t),L^2)}}{t^{\gamma}}+8\frac{\|r\|_{L^\infty((0,t)\times I)}}{t^{1/2}}\right).
e^{-t}\sum_{i=N(t)}^{+\infty}\frac{t^i}{i!}\le e^{-t}\sum_{i=N(t)}^{+\infty}\left(\frac{et}{i}\right)^i \le e^{-t}\sum_{i=N(t)}^{+\infty}\left(\frac{et}{N(t)}\right)^i= e^{-t}\left(\frac{et}{N(t)}\right)^{N(t)}\frac{N(t)}{N(t)-et}.
\begin{array}{c c c c c c c}{{\nabla Z_{A B}}}&{{=}}&{{P_{A B C D}\overline{{{Z}}}^{C D}}}&{{\longrightarrow}}&{{\nabla \widehat{Z}_{A B}}}&{{=}}&{{{\cal P}_{A}^{A^{\prime}}\,{\cal P}_{B}^{B^{\prime}}P_{A^{\prime}B^{\prime}C D}\, \overline{{{Z}}}^{C D}\ ,}}\\ \end{array}
R_h: =\max_l\left(\max_{\Omega_h}\left( \hat{u}_h^l(z)-\hat{u}_l(z)+V_h(z)\right)\right) &=\max_l\left(\max_{\{{u}_h^l(z)\leq u_l(z)\}}\left( \hat{u}_h^l(z)-\hat{u}_l(z)+V_h(z)\right)\right).
(\forall \; x \; (R(x,y) \Rightarrow \varphi(x)))^{\Box}= \Box (\forall \; x \; \Box (R(x,y) \Rightarrow \varphi^{\Box}(x))) \equiv \; \Box \; \forall \; x \; (R(x,y) \Rightarrow \varphi^{\Box}(x))
\|\mathcal{B}f\|_0=\sup_{k\in\mathbb{N}}\sup_{\sigma>0}\int_{\mathbb{T}^\infty}\log(1+|\mathcal{B}_k f_\sigma|)dm_\infty=\sup_{\sigma>0}\sup_{k\in\mathbb{N}}\int_{\mathbb{T}^\infty}\log(1+|\mathcal{B}_k f_\sigma|)dm_\infty.
\mathcal I_s^{\beta,q}(\mu)\leq C'\int_{\underline a\in\Sigma}\dfrac{\sum_{n=0}^{\infty}\beta^n\int_{\underline b' \in\Sigma:\underline b' \wedge \sigma^n\underline a = \phi}\left(\frac{\mu[a_1\cdots a_n]}{d(\underline a,a_1\cdots a_n\underline b')^{s'}}\right)^{q-1}d\mu(\underline b')}{\sum_{n=1}^{\infty}\beta^n}d\mu(\underline a)
\mathbf{x}(u,v)=\int^{u}\left( \begin{array}{c}-\emph{f}\left( 1+\emph{g}^{2}\right) \emph{f}\left( 1-\emph{g}^{2}\right) 2\emph{fg}\end{array}\right) du+\int^{v}\left( \begin{array}{c}\mathfrak{f}\left( 1+\mathfrak{g}^{2}\right) \mathfrak{f}\left( 1-\mathfrak{g}^{2}\right) 2\mathfrak{fg}\end{array}\right) dv.
I_{-\infty}^{\hat{Y}[y]}:= I_{-\infty}\cap \{1, \ldots, n_{{\textsc{g}}}\mkern-3.5mu-\mkern-2.5mu1\}, \;\;\;\;I_{+\infty}^{\hat{Y}[y]}:= I_{+\infty}\cap \{1, \ldots, n_{{\textsc{g}}}\mkern-3.5mu-\mkern-2.5mu1\}.
\sigma_{n}^M := T\wedge \inf\Big\{t\geq 0: \Vert X(t, x_n)\Vert_H> M \Big\}\wedge\inf\Big\{t\geq 0: \int_0^t\Vert X(s, x_n)\Vert_{V}^{\alpha}ds >M\Big\}\wedge \inf\Big\{t\geq 0: \Vert X(t, x)\Vert_H> M \Big\}\wedge\inf\Big\{t\geq 0: \int_0^t\Vert X(s, x)\Vert_{V}^{\alpha}ds >M\Big\}.
\left\{\begin{array}{c}{\left( \begin{array}{c l c}{\frac{\delta - \sqrt{\delta^{2}+ \omega^{\ast}\omega}}{\omega}}&{,}&{1}\\ \end{array}\right)}\\{\left( \begin{array}{c l c}{\frac{\delta + \sqrt{\delta^{2}+ \omega^{\ast}\omega}}{\omega}}&{,}&{1}\\ \end{array}\right)}\\ \end{array}\right\}.
u=\frac{\gamma}{2}x^{2}-\frac{\gamma^{2}}{2}(\frac{x^{5}}{5!})+\frac{11\gamma^{3}}{4}(\frac{x^{8}}{8!})-\frac{375\gamma^{4}}{8}(\frac{x^{11}}{11!})+\cdot\cdot\cdot
\omega_-^\psi = \left( \begin{array}{cc}\Sigma^{(1)}_- - \chi (\Sigma^{(1)}_- - \chi) \, \chi - \chi \,\Sigma^{(2)}_- + \partial_- \chi begin{align*}.7em]I_n \Sigma^{(2)}_- + \chi\end{array}\right).
\tilde{K}_i= -2e_2\big[f v^3+\frac{h'}{2(1-G_u)}\frac{Au_1}{w_1}-\frac{D}{1-G_u}\big]d_i &+2\frac{w_2^2}{w_1^4}\big[f v^3+\frac{h'}{2(1-G_u)}\frac{Au_1}{w_1}-\frac{D}{1-G_u}\big]w_i^2+K_id_2-K_2d_i.
L_{m}= P^{+}( m ) \alpha_{m}^{-}+ P^{-}( m ) \alpha_{m}^{+}+ \sum_{k \ne 0 , m}\alpha_{m - k}^{+}\alpha_{k}^{-}+ \sum_{s}( s + \frac{1}{2}) \psi_{m - s}^{+}\psi_{s}^{-}
\frac{\phi_1(z_0) - m U(z_0)}{[( \phi_1(z_0) - m U(z_0))^2 + 4 \lambda m_1 z_0^2]^{1/2}}= \frac{m U(z_0) - \phi_2(z_0)}{[( \phi_2(z_0) - m U(z_0))^2 + 4 \lambda m_2 z_0^2]^{1/2}},
\begin{array}{r l}{\}&{\Phi ( E , x ) = \}\\{\}&{E^{2}- \left( \omega^{2}+ \frac{\lambda^{2}}{4}+ \lambda E \right) \left( 2 x + 2 u - 1 \right)^{2}+ \frac{\lambda^{2}}{4}\left( 2 x + 2 u - 1 \right)^{4}.}\\ \end{array}
Tr_L \Phi^{\left[ 1 \right]}= D + \frac{{32}}{{T^2}}\bar \xi_4 \xi_4 \bar \xi_3 \xi_3 \bar \xi_2 \xi_2 \bar \xi_1 \xi_1 (\varepsilon^1 \cdot \varepsilon^4 \varepsilon^2 \cdot \varepsilon^3 - \varepsilon^1 \cdot \varepsilon^3 \varepsilon^2 \cdot \varepsilon^4 )\delta (u_4 - u_3 )\delta (u_2 - u_1 )+ \, f.t.
\vartheta(\hbar) [\mathfrak{X}^+(u, \lambda_1) , \mathfrak{X}^{-}(v, \lambda_2)]=\frac{\vartheta(u-v+\lambda_{1})}{\vartheta(u-v) \vartheta(\lambda_{1})}\Phi(v)+\frac{\vartheta(u-v-\lambda_{2})}{\vartheta(u-v) \vartheta(\lambda_{2})}\Phi(u).
\mbox{LHS}=\frac{1}{2}\sum_{t=1}^{k}z_{t}\eta_{t}\left\Vert \nabla f(x_{t})\right\Vert^{2}+z_{k}\Delta_{k+1}-z_{1}\Delta_{1}+\sum_{t=2}^{k}\left(z_{k-1}-z_{k}\right)\Delta_{k}\ge\frac{1}{2}\sum_{t=1}^{k}z_{t}\eta_{t}\left\Vert \nabla f(x_{t})\right\Vert^{2}+z_{k}\Delta_{k+1}-z_{1}\Delta_{1}
\int_{\mathbb{R}^n}|y|^{\frac{n\theta}{p}}|\nabla_r v(y)|^{{\theta}}\frac{dy}{|y|^{{n}}}= \int_{{B_{R}}}|x|^{\frac{n\theta}{p}}|\nabla_r u(x)|^{\theta}\frac{dx}{|x|^{{n}}}.
\|(- A)^\gamma E(t)\|_{\mathcal{L}(H)}\leq C t^{-\gamma}, t >0, \gamma \geq 0, \|(- A)^{-\rho}(I-E(t))\|_{\mathcal{L}(H)}\leq C t^{\rho}, t >0, \rho \in [0,1].
A_{33,\ell}= \int_{C/\ell}^{\pi -C/\ell}\frac{a^2_{3,\ell}(\phi)}{(1-4 \alpha_2^2/\lambda_{\ell}^2)^{1/2}(1-4 \alpha_1^2/\lambda_{\ell}^2)^{1/2}}\sin \phi d \phi
f(x)=\sum_{i=1}^k\sum_{j=1}^{d_i}f(\alpha_{ij})\frac{x(x-1)\prod_{(k,l)\neq (i,j)}(x-\alpha_{kl})}{\alpha_{ij}(\alpha_{ij}-1)\prod_{(k,l)\neq (i,j)}(\alpha_{ij}-\alpha_{kl})}+f(0)\frac{(x-1)G(x)}{S'(0)}+f(1)\frac{xG(x)}{S'(1)}.
0 \longrightarrow R(n-1)^n \xrightarrow{C^t_n}R(m+n-1)^{n+1}\xrightarrow{E_n := \left[-\delta_1 \; \delta_2 \;\cdots \; (-1)^{n+1}\delta_{n+1}\right]}R(mn+m+n-1).
\mathcal{R}_0 = \mathcal{R}\setminus ( \mathcal{R}_i \sqcup \mathcal{R}_j ) \mathcal{R}_1 = \mathcal{R}\cap \mathcal{G}_{l_1}\mathcal{R}_2 = \mathcal{R}\cap \mathcal{G}_{l_2}
\sigma_{e_\beta}(z_k)=\sum_{j_1\in J_2^\prime}c_{\alpha_1,\beta}^{\gamma_1,j_1}z_1^{j_1}\sigma_{e_{\beta_{(j_1)}}}^{(2)}(z_k)+\begin{cases}z_1\sigma_{h_{\alpha_1}}^{(2)}(z_k), \textrm{if}\beta=\alpha_1\textrm{and}\gamma_1=e, 0, \textrm{otherwise.}\end{cases}
1+ 2^{-\beta_{\ref{boundA}}(n-2)^2}+2^{N^2-\beta (n-2)}\leq 2+2^{N^2-\beta (n-2)}\leq \begin{cases}4 \textnormal{if}N^2-\beta(n-2) \leq 1, 2(2^{N^2-\beta (n-2)}) \textnormal{if}N^2-\beta(n-2) > 1. \end{cases}