harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \begin{array}{c}{{a_{0}}}\\{{\rho_{+}}}\\{{\rho_{-}}}\\{{\eta_{1}}}\\ \end{array}\right) = \left( \begin{array}{c}{{\Delta^{3}a_{0 , 2}( z_{0}, z_{1})}}\\{{\Delta^{3}\rho_{+ , 2}( z_{0}, z_{1})}}\\{{\Delta \rho_{- , 0}( z_{1}) \cos \Omega z_{0}+ \Delta^{3}\rho_{- , 2}( z_{0}, z_{1})}}\\{{\Delta^{2}\eta_{1 , 1}( z_{0}, z_{1}) + \Delta^{3}\eta_{1 , 2}( z_{0}, z_{1})}}\\ \end{array}\right) .
\{g(1),\ldots,g(n+1)\}= \left\{\left(-\langle\mu,\varpi_1\rangle+\sum_{j=1}^{i-1}\left\langle\frac{\lambda+\mu}{2}+\rho,\alpha_j\right\rangle\right)^{}\ \middle\vert\ 1\leq i \leq n+1\right\}
\alpha(d,d, x_3, x_4, y_5) = \sum_{i \in [q], j \in [t]}\lambda_{ij}\tau_i(d, d, x_3) \tilde{\gamma}_j(x_4, y_5) + \sum_{i \in [q], j \in [t]}\lambda'_{ij}\tau_i(d, d, x_4) \tilde{\gamma}_j(x_3, y_5) + \sum_{i \in [s_3]}\beta''_{i}(d, y_5) \gamma''_i(x_3, x_4)
|u(x,t)-u(y,s)| \le |u(x,t)-u(x_0,t)|+|u(x_0,t)-u(y,s)| \le \tilde\rho(r)+\tilde\rho(|x_0-y|\vee |t-s|) \le \tilde\rho(2(|x-y|\vee|t-s|))+\tilde\rho((|x-y|+r)\vee |t-s|) \le \tilde\rho(2(|x-y|\vee|t-s|))+\tilde\rho(3(|x-y|\vee |t-s|)) \le 2\tilde\rho(3(|x-y|\vee |t-s|)).
q_{\infty}\left(\mathcal F_1^{c'},\mathcal F_2^{c'},\mathbf T^{'}\right)-q_{\infty}^{*}=\left(a_{1}-a_{n_2}\right)\left(f_{1,K_1^c+\min\{K_1^b,F_1^{b*}\},\infty}-f_{2,K_2^c,\infty}(T_{1}^*)\right).
f^{b c d}f^{d}{}_{[ e a}\Lambda_{a_{1}a_{2}. . . a_{2 m - 1}]}{}^{b}= 2 f^{c d}{}_{[ e}f^{d b}{}_{a}\Lambda_{a_{1}. . . a_{2 m - 1}]}{}^{b}= - 2 f^{c d}{}_{[ e}\Omega^{d}{}_{a a_{1}. . . a_{2 m - 1}]}= 0 \ .
(\begin{matrix}1&1&1&1&0&0&0&0\\ 1&1&0&0&1&1&0&0\\ 0&0&1&1&1&1&0&0\\ 1&0&1&0&0&0&1&1\\ 0&1&0&0&1&0&1&1\\ 0&0&0&1&0&1&1&1\\ \end{matrix})
W_t(\textbf{x}_t,f_t,q_t) =w_{t+K-1}\mathrm{E}[(q_t-A_{K-1,t}-D_{t+K-1})^+|f_t] &=w_{t+K-1}\sum\limits_{u=1}^{q_t}\mathrm{P}(A_{K-1,t}+D_{t+K-1}< u|f_t) &=w_{t+K-1}\sum\limits_{u=1}^{q_t}R_{K,t}(u).
\left. \frac{d^{2}}{d t^{2}}\int_{L}{\rm vol}_{\phi}[\iota_{t}] \right|_{t=0}= \int_{L}\left((2n+2) \eta (Y)^{2}- 2 g(Y, Y) -{\rm Ric}(Y, Y) \right. &\left. - g(\pi_{L}[Z, Y], H_{\phi}) + g(Y, H_{\phi})^{2}+ \left (\frac{{\rm div}(\rho_{\phi}[\iota] Y)}{\rho_{\phi}[\iota]}\right)^{2}\right){\rm vol}_{\phi}[\iota],
K \min_{k\in\{0,1,\ldots,K-1\}}\triangle \ell^k \leq \sum_{k=0}^{K-1}\triangle \ell^k \leq ({\epsilon}/{(D^p\rho)})^{-\frac{1}{p-1}}\left(\Phi(x^{0}) - \Phi(x^{K})\right)+ \frac{\epsilon}{2}K & \leq ({\epsilon}/{(D^p\rho)})^{-\frac{1}{p-1}}(\Phi(x^{0}) - \Phi^* )+ \frac{\epsilon}{2}K,
\Psi(a,\psi,\theta)\,=\,\psi-\frac{1}{2}\,\theta^{\alpha\beta}\,a_{\alpha}{\rm P}_{-}\partial_{\beta}\psi\,+\,\frac{i}{8}\,\theta^{\alpha\beta}\,[a_{\alpha},a_{\beta}]\,{\rm P}_{-}\psi\,+\,o(\theta^2).
\psi^G_{1,1}= -\partial_y+\partial_z, \psi^G_{2,2}= \partial_z- \textstyle \frac 1 3 \overline \partial, &\psi^G_{i,j}= \psi^G_{i-1,j-1}+\alpha_{i,j}\psi^G_{i,j-1}\ \ \ \ \ \{\rm for}\ (i,j)\in \triangledown
{\rm Re}\ \check{W}_{\vec{k}, \vec{l}}(\vec{x}, \vec{y}) -{\rm Re}\ \check{W}_{\vec{k}, \vec{l}}(\vec{y}, \vec{x}) = 0,{\rm Im}\ \check{W}_{\vec{k}, \vec{l}}(\vec{x}, \vec{y}) +{\rm Im}\ \check{W}_{\vec{k}, \vec{l}}(\vec{y}, \vec{x}) = 0,
U_{1}( z ) = \sqrt{\frac{2 - \sqrt{2}}{2}}e^{\frac{1 1 \pi}{8}i}u_{1}+ u_{2}~ , \hspace{0 . 5 c m}U_{2}( z ) = - \sqrt{\frac{2 + \sqrt{2}}{2}}e^{\frac{\pi}{8}i}u_{1}+ u_{2}~ .
R_{v,j}^q={\rho_{v,j}^q B_{\rm VLC}\log_2 \left({1 +{\frac{\exp(1)}{2\pi}}\frac{p_{v,j}^{q}{\left({{R_{\rm{PD}}}{G_{v,j}^{q}}}\right)}^2}{\sum\limits_{j'\ne j}{\sum\limits_{v'\ne v}p_{v',j'}^{q}{\left({{R_{\rm{PD}}}{G_{v',j}^{q}}}\right)}^2}+ N_{\rm VLC}B_{\rm VLC}}}\right)},
\tilde{\Phi}:=\sup \left\{\begin{array}{c}\psi \colon \mathbb P^1\times \overline{\mathbb D}\to \mathbb R\cup \{-\infty\}: \psi \pi_{\mathbb P^1}^*\omega_{FS}+ dd^c\psi\ge 0 \psi(z,\tau)\le \phi(z) (z,\tau)\in \mathbb P^1\times \partial \mathbb D \nu_{(z=0,\tau=0)}(\psi)\ge 1\end{array}\right\}.
\frac{(-i)\pi R}{2}\frac{J_n(\omega R)}{J_n(\omega R_d)}\left(H_n^{(1)}(\omega R){J_n(\omega R_d)}- J_n(\omega R)H_n^{(1)}(\omega R_d) \right) = \begin{cases}R\ln\frac{R}{R_d}+ O(\omega\ln\omega) n = 0 -\frac{R}{2n}\left(1-\frac{R^{2n}}{R_d^{2n}}\right) + O(\omega) n\neq 0\end{cases}&= -\epsilon + O(\epsilon^2 + \omega\ln\omega),
v^*v = (\sigma_{(M\otimes R)}^\mathcal{U})^{-1}(q), vv^* = (1 \otimes 1 \otimes p)_\mathcal{U}, w^*w = (\sigma_{(M\otimes R)}^\mathcal{U})^{-1}(q^\perp), ww^* =(1 \otimes 1 \otimes p^\perp)_\mathcal{U}.
\begin{array}{l}{{\varphi_{1 E}\left( r , \theta \right) = \sum_{n = - \infty}^{\infty}f_{n}\left( r \right) e^{i \left( n + \frac{1}{2}\right) \theta}}}\\{{\chi_{1 E}\left( r , \theta \right) = \sum_{n = - \infty}^{\infty}g_{n}\left( r \right) e^{i \left( n + \frac{1}{2}\right) \theta}}}\\ \end{array}
\Gamma'^{+}_{H}(z) := \frac{1}{\sqrt{2}}\left(\frac{z+i}{2}\right)^{2H-2}\sum_{k=0}^{\infty}\sqrt{\frac{\Gamma(2-2H+k)}{\Gamma(2-2H)k!}}\left(\frac{z-i}{z+i}\right)^{k}(\xi^{(1)}_{k}+i\xi^{(2)}_{k})
\rho^{( S_{+}^{2})}( r ) = - \frac{1}{4 \pi}\int_{0}^{2 \pi}\int_{0}^{\pi / 2}\partial_{\theta}\sin \alpha d \theta d \varphi = - \left. \frac{1}{2}\sin \alpha \right|_{\theta = 0}^{\theta = \pi / 2}= \Theta ( r - z_{0}) \ .
e^{- \gamma \varphi ( \tau , \sigma )}= e^{- \gamma \phi ( \tau , \sigma )}\left( 1 + m^{2}\int \int d y d \bar{y}G ( z , \bar{z}; y , \bar{y}) e^{2 \gamma \phi ( y )}e^{2 \gamma \phi ( \bar{y})}\right) ,
{\bf D_3}(0,\sigma,0,\nu_1, \nu_2, \nu_3)= \frac{1}{{\rm i}\pi^{n/2}\Gamma\left(3-\frac{n}{2}\right)}\int \frac{{\rm d}^n p}{(p^2-m^2)^{\nu_3}}\left.J_{011}(\sigma,\nu_1,\nu_2;p^2,m)\right|_{m=1}\; .
Y\mathbf{d}=\sup_{y\in Y}y\mathbf{d}\leq\sup_{y\in Y}\liminf_\lambda(y\mathbf{d}x_\lambda+x_\lambda\underline{\mathbf{d}})\leq(x_\lambda)\underline{\mathbf{d}}. \mathbf{d}(x_\lambda)=\liminf_\lambda\mathbf{d}x_\lambda\leq\inf_{y\in Y}\liminf_\lambda(\overline{\mathbf{d}}y+y\mathbf{d}x_\lambda)=\overline{\mathbf{d}}Y.
\begin{pmatrix}-1 1 1 1\end{pmatrix}\begin{pmatrix}e^{i \pi \theta}0 0 1\end{pmatrix}{\begin{pmatrix}-1 1 1 1\end{pmatrix}}^{-1}~=~\frac{1}{2}\begin{pmatrix}1 + e^{i \pi \theta}1 - e^{i \pi \theta}1 - e^{i \pi \theta}1 + e^{i \pi \theta}\end{pmatrix}.
\frac{\# \mathbf{U}_\sigma(k_v)}{q_v^{\dim \mathbf{U}_\sigma}}= \frac{\# \mathbf{SU}_\sigma(k_v)}{q_v^{\dim \mathbf{SU}_\sigma}}\cdot \frac{\# \mathbf{U}_h(k_v)}{q_v^{\dim \mathbf{U}_h}}.
\displaystyle\frac{\partial J_\varepsilon}{\partial y^j_i}= B_j \langle\frac{\partial^2U_{y^j, \lambda_j}}{\partial \lambda_j\partial y^j_i}, v\rangle + \sum_{l = 1}^NC_{jl}\langle\frac{\partial^2 U_{y^j, \lambda_j}}{\partial y^j_l\partial y^j_i}, v\rangle,i = 1,\cdot\cdot\cdot,N, j = 1,2,
a^{\dagger}a\,e_{\lambda}^{a^{\dagger}a-\lambda}(t)=e_{\lambda}^{a^{\dagger}a-\lambda}(t)a^{\dagger}a=a^{\dagger}e_{\lambda}^{aa^{\dagger}-\lambda}(t)a=a^{\dagger}e_{\lambda}^{a^{\dagger}a+1-\lambda}(t)a.
\frac{F(X+bY,Y)}{{\Big(\prod\limits_{j=0}^{p-1}{\big(\delta_j(X + bY)-\gamma_j Y\big)}\Big)}^{t}}= \frac{F(X,Y)}{{\Big(\prod\limits_{j=0}^{p-1}{\big(\delta_jX-\gamma_jY\big)}\Big)}^{t}},
\frac{d\rho_i}{dt}=\sum_{j\in N(i)}\rho_j[ F_i(\rho)- F_j(\rho)+\beta(\log\rho_j-\log\rho_i)]_+ &-\sum_{j\in N(i)}\rho_{i}[ F_j(\rho)- F_i(\rho)+\beta(\log\rho_i-\log\rho_j)]_+ \ ,
\ P_\eta\Big(\mbox{there exists}j\in\{2,\ldots,n\}: \eta_{ij}=1, \{\rm and}\ \eta_{\ell j}=0 \{\rm if}\ \ell \not = i \Big) &=1-(1-(1-\delta)^{N-1}\delta)^{n-1}\geq 1-\frac{1}{4N}
\left(\int_x^\infty\!\!\phi_{N,d_j}(y)\phi_{N,d_l}(y)dy\right)_{1\le j,l\le M}=\left(\frac{\tilde{v}_{N;d_j,d_l}(x)}{u_N(x)}\cdot\frac{e^{-(k_{d_j}+k_{d_l})x}}{k_{d_j}+k_{d_l}}\right)_{1\le j,l\le M}&\propto \frac{\tilde{w}_{N,\mathcal{D}}(x)}{u_N(x)}e^{-2\sum_{j=1}^M k_{d_j}x}.
\lambda_{top}= \frac{1}{Z_\sigma}\int_{-\infty}^{\infty}(1 - x^2) \, e^{\frac{2}{\sigma^2}(\frac{1}{2}x^2 - \frac{1}{4}x^4)}\, dx &= \frac{2}{Z_\sigma}e^{\frac{1}{2 \sigma^2}}\int_0^\infty (1 - x^2) \, e^{-\frac{1}{2\sigma^2}\left(x^2-1\right)^2}\, dx.
\begin{aligned}\omega_T=\frac{d(x+\varphi y)}{x+\varphi y}\wedge\frac{d(x+\varphi' y)}{x + \varphi' y}=\frac{(dx+\varphi dy)\wedge(dx +\varphi' dy)}{x^2+xy- y^2}& = \frac{\varphi'-\varphi}{N_{E/F}(x+ \varphi y)}dx\wedge dy=\frac{-\sqrt{5}}{N_{E/F}(x+ \varphi y)}dx\wedge dy.\end{aligned}
\operatorname{a r c t a n}2 \lambda ( x ) = \pi x + \frac{\pi}{N}( \theta ( x - x_{1}) + \theta ( x - x_{2}) ) + \int_{- \frac{1}{4}}^{\frac{1}{4}}\operatorname{a r c t a n}( \lambda ( x ) - \lambda ( y ) ) d y \quad .
\gamma^-f(t,x,v)=g(t,x,v),(t,x,v)\in(0,T)\times\Sigma^-, & \rho(t,x)=\rho_B(x),(t,x)\in (0,T)\times\Gamma_{\rm{in}}, & u(t,x)=u_B(x),(t,x)\in (0,T)\times\partial\Omega,
S^{\delta_n^k-\alpha_k + 1}\left (\alpha \right )=\alpha, S^{\delta_n^k+2-\alpha_{k-1}}\left (\alpha \right )=\alpha, \dots, S^{\delta_n^k+1+u-\alpha_{k-u}}\left (\alpha \right )=\alpha, \dots, S^{\delta_n^k+k-\alpha_{1}}\left (\alpha \right )=\alpha
\frac{1}{T_{\mathrm{D R}}^{\mathrm{o n}}( E )}={\frac{1}{C_{\mathrm{D R}}}+ i \frac{2 \pi \mu}{\Gamma ( d / 2 ) ( 4 \pi )^{d / 2}}( 2 \mu E )^{d / 2 - 1}}.
I_1= \frac{1}{|Dw|^2\log|Dw|^2}\big[2\sum_{1\leq i,j,k\leq n}w_ka^{ij}w_{ijk}+2\sum_{1\leq i,j,k\leq n}a^{ij}w_{ki}w_{kj}\big] =: \frac{1}{|Dw|^2\log|Dw|^2}\big[I_{11}+I_{12}\big].
Y_{\mathbf{d}, \mathcal{A}}(abc) = P_{\mathbf{d}, \mathcal{A}}(ab) \frac{P_{\mathbf{d}- \mathbf{e}_a - \mathbf{e}_b, \mathcal{A}}(bc)-Y_{\mathbf{d}- \mathbf{e}_a - \mathbf{e}_b, \mathcal{A}}(abc)}{1 - P_{\mathbf{d}- \mathbf{e}_a - \mathbf{e}_b, \mathcal{A}}(ab)}.
{\cal R}_0=m\sqrt{u^2}+eu^{\mu}A_{\mu}-\frac{u^{\mu}}{2}{\omega}_{\mu}^{ab}S_{ab}+\frac{eg}{4m}F^{ab}S_{ab}\sqrt{u^2}-\frac{\kappa}{8m}R^{abcd}S_{ab}S_{cd}\sqrt{u^2}\, .
1 = \mathbb{P}(||V^{\alpha}_{\omega}||_p < + \infty) = \mathbb{P}(\sup_{\varphi \in F_q}|V^{\alpha}_{\varphi}| < + \infty) = \mathbb{P}(\sup_{\varphi \in F_q}|U^{\alpha}_{\varphi}| < + \infty)
H_{L-1}+H_{k+1}+\sum_{a=1}^{k}\sum_{i=a}^{a+m-1}H_{i}\geq \left( \sum_{i=1}^{k+m-1}H_{i}+1 \right) + \left( \sum_{i=1}^{k+m-2}H_{i}+1 \right).
\sigma^{1}=\frac{\varphi+\bar{\varphi}}{1+\varphi\bar{\varphi}};~~~\sigma^{2}=\frac{i(\bar{\varphi}-\varphi)}{1+\varphi\bar{\varphi}};~~~\sigma^{3}=\frac{1-\varphi\bar{\varphi}}{1+\varphi\bar{\varphi}}
{\tilde{\delta}}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] = \int{\cal D}\lambda_{a}e x p i \int \lambda_{a}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] d t
|\hat{\phi}_X(u) -\phi_X(u) | \leq 2 |\phi_X(u)| +||\hat{\phi}_X(u)| - |\phi_X(u)|| = 2 |\phi_X(u)| +||\hat{\phi}(u)|^{\frac{1}{K}}- |\phi(u)|^{\frac{1}{K}}| \leq 2 |\phi_X(u)| +|\hat{\phi}(u) -\phi(u)|^{\frac{1}{K}},
T_{1}(\epsilon,v_1,v_2)=(-\Delta+\sigma)^{-1}[e^{v_2+h_{2\epsilon}}(1-e^{v_1+h_{1\epsilon}})+\sigma v_1-g_1], & T_{2}(\epsilon,v_1,v_2)=(-\Delta+\sigma)^{-1}[e^{v_1+h_{1\epsilon}}(1-e^{v_2+h_{2\epsilon}})+\sigma v_2-g_2]
\langle u, \mbox{div $\varphi$}\rangle=\left\langle\int_{\mathbb R^2}\Gamma_a(\cdot,y)f(y)\,dy, \mbox{div $\varphi$}\right\rangle=\int_{\mathbb R^2}\langle\Gamma_a(\cdot,y), \mbox{div $\varphi$}\rangle\,f(y)\,dy,
r_{p}(n_{p})=\sum_{k\geq 0}\left\lfloor \frac{1}{p^{k}}\left\lfloor \frac{n}{p-1}\right\rfloor \right\rfloor =\sum_{k\geq 0}\left\lfloor \frac{n}{p^{k}(p-1)}\right\rfloor .
F_{\mu\nu}= \left( \begin{array}{cccc}0 \check{E}_1 \check{E}_2 \check{E}_3 -\check{E}_1 0 -\check{B}_3 \check{B}_2 -\check{E}_2 \check{B}_3 0 -\check{B}_1 -\check{E}_3 -\check{B}_2 \check{B}_1 0 \end{array}\right) ,
CL_N \left(\left|g(L_N)\right|^2 + \left|h(L_N)\right|^2\right) \leq\|g\|^2 + \|h\|^2 & =2\|g\|^2 & =-\frac{2}{e^{2\varepsilon}-1}\Re\!\left(F_N\left(e^{i\theta+\varepsilon}\right)\right),
\lVert V\rVert_{\mathcal{S}(T)}=\max\left\{\sup_{0<t<T}\lVert V(t)\rVert_{L^\infty_H L^p_z},\sup_{0<t<T}t^{1/2}\lVert \nabla V(t)\rVert_{L^\infty_H L^p_z}\right\}
1=\frac{(k_{2}-k_{1})k_{1}}{k_{2}\cdot k_{1}}+\frac{(k_{3}-k_{2})k_{1}}{k_{3}\cdot k_{2}}+...+\frac{(k_{n}-k_{n-1})k_{1}}{k_{n}\cdot k_{n-1}}+\frac{1\cdot k_{1}}{k_{n}}
W_{\geq m,\ell}(\eta,x)=\frac{4\pi}{\ell^3}\sum_{\substack{|k|_\infty\geq m k\in \mathbb{Z}^3}}\frac{1}{\left|\frac{2\pi k}{\ell}-\eta\right|^2}\,e^{i\left(\frac{2\pi k}{\ell}-\eta\right)\cdot x}
\langle\mathcal{M}^N_{h,3}, \hat{\Lambda}^N_{F, G, H}\rangle_t= \int_0^t\mu_{s,2}^N\left(\psi(\cdot)h(\cdot)\left(e^{-G_s(\cdot)+H_s(\cdot)}-1\right)\right)ds &-\int_0^t\mu_{s,3}^N\left(\phi(\cdot)h(\cdot)\left(e^{-H_s(\cdot)}-1\right)\right)ds
\sum_{p=0}^{\infty}\sum_{k=0}^{n-1}(k-\lambda+1)_{p,\lambda}\frac{t^{p}}{p!}=\sum_{k=0}^{n-1}e_{\lambda}^{k+1-\lambda}(t) &=e_{\lambda}^{1-\lambda}(t)\frac{e_{\lambda}^{n}(t)-1}{e_{\lambda}(t)-1}=\sum_{p=0}^{\infty}\bigg(\frac{\beta_{p+1}(n+1-\lambda)-\beta_{p+1,\lambda}(1-\lambda)}{p+1}\bigg)\frac{t^{p}}{p!}.
\tau^2 =\sigma_W^2+\sum_{p=1}^{P}\delta_p^{-1}\mathbb{E}\left[\left(\eta(X+\tau Z)-X\right)^2\right] &\overset{(a)}{=}\sigma_W^2+\delta^{-1}\mathbb{E}\left[\left(\eta(X+\tau Z)-X\right)^2\right],
\int_{t_0}^{t_1}\int_{K_{R}}\textbf{A}(x,t,u,Du)\cdot D\varphi \,dx \,dt \geq C_0 \int_{t_0}^{t_1}\int_{K_{R}}G(|Du|)h(\Psi^2)[4g_0 - 2 + \Psi] \left(\Psi'\right)^2 \zeta^{q}\,dx \,dt - 2q C_1 \int_{t_0}^{t_1}\int_{K_{R}}g(|Du|) h(\Psi^2)\Psi \left|\Psi'\right| \zeta^{q-1}|D\zeta| \,dx \,dt.
h^{i}\equiv f_{+}^{- i}+ f_{-}^{+ i}= 2 f_{+}^{+ i}={\frac{1}{2}}u^{\underline{{m}}i}u^{\underline{{n}}-}u^{\underline{{k}}+}H_{\underline{{k}}\underline{{n}}\underline{{m}}}( X ( \xi ) )
\int_{\mathbb{R}^n \times G_k (\mathbb{R}^n)}\phi (x,S) \, dV(x,S):= \int_{\mathbb{R}^n}\phi (x,T_x \mu ) \, d\mu (x) \ \phi \in C_c(\mathbb{R}^n\times G_k (\mathbb{R}^n)),
\begin{array}{rl}\bar{\phi}_{ab}(z):= (\bar{\phi}_{ab}^{(0)}(z), \cdots , \bar{\phi}_{ab}^{(n-1)}(z)), \bar{\phi}_{aa+\bar{\epsilon}_j}^{(i)}(z):= (\tilde{\Phi}(z))_{j}^{i}/Ch(z-(n-1)/2), \end{array}
\begin{cases}\nabla\times E-ik\mu H=0 \mbox{in}\Omega\backslash\bar{D}, \nabla\times H+i\epsilon E=0 \mbox{in}\Omega\backslash\bar{D}, \nu\times E=f \mbox{on}\partial\Omega, \nu\times H=0 \mbox{on}\partial D,\end{cases}
\sum_{i=1}^3\Phi^{ij}(p,q)\left(\frac{q_i}{q^0}-\frac{p_i}{p^0}\right)=\sum_{j=1}^3\Phi^{ij}(p,q)\left(\frac{q_j}{q^0}-\frac{p_j}{p^0}\right)=0,
\Psi_{i,j,k}^{\alpha,\beta,\gamma}=-[c_+\ \Theta_{i,j,k}^{\alpha,\beta,\gamma}\times\Upsilon_{i,k}^{\alpha,\beta,\gamma}\times x^{-\alpha}+c_-\ \overline{\Theta}_{j,k}^{\alpha,\beta,\gamma}\times\overline{\Upsilon}_{i,k}^{\beta,\gamma}\times (L-x)^{-\alpha}].
\langle A^{-1}_{\varepsilon, \delta}\overline{\sigma_{l}}, \overline{\sigma_s}\rangle_{\omega_{\varepsilon}}= \langle \biggl( \sum_{1 \leq m \leq k}\frac{1 + \delta \lambda_{m}}{\varepsilon + \lambda_{m}}+ \frac{n-k-1}{\varepsilon}\biggl)^{-1}\overline{\sigma_{l}}, \overline{\sigma_{s}}\rangle_{\omega_{\varepsilon}}\leq \frac{\varepsilon}{n-k-1}\delta_{l s}
\left( 1 - \frac{w N}{6}e^{2 \phi}\right) \left( \dot{\phi}^{\prime}- \dot{\rho}\phi^{\prime}- \rho^{\prime}\dot{\phi}\right) - \dot{\phi}\phi^{\prime}= \frac{N}{1 2}e^{2 \phi}\left[ \dot{Z}Z^{\prime}+ 2 \dot{Z}^{\prime}- 2 \left( \dot{\rho}Z^{\prime}+ \rho^{\prime}\dot{Z}\right) \right] .
\begin{array}{c}\displaystyle\frac{\Gamma(\frac{1}{\gamma})\Gamma(\frac{5}{\gamma})}{\Gamma(\frac{3}{\gamma})^2}=\frac{1}{(1+\delta)^2}\left(\delta^2\frac{\Gamma(\frac{1}{\alpha})\Gamma(\frac{5}{\alpha})}{\Gamma(\frac{3}{\alpha})^2}+\frac{\Gamma(\frac{1}{\beta})\Gamma(\frac{5}{\beta})}{\Gamma(\frac{3}{\beta})^2}+6\delta\right),\end{array}
\limsup_{R\to \infty}\,\sup_{t\in \mathbb{R}}\,\frac{1}{R}\,\int_{[t, t+R]^s}\,\big|\sum_{i=1}^m\,c_i\,e^{-2\pi i (\sum_{j=1}^s\,n_{ij}\,a_j\,\xi_j)}\big|^2\,d\nu_{\mu}(\xi_1,\dots,\xi_s) &=\limsup_{R\to \infty}\,\sup_{t\in \mathbb{R}}\,\frac{1}{R}\int_{[t, t+R]}\,\big|\sum_{i=1}^m\,c_i\,e^{-2\pi i x_i\,\lambda}\big|^2\,d\mu(\lambda):=L.
S_{\mathrm{Y M}}= - \, \frac{1}{4 g^{2}}\int d^{4}x \, F_{\alpha \beta}^{a}F_{a}^{\alpha \beta}+ \frac{\theta_{\mathrm{Y M}}}{3 2 \pi^{2}}\int d^{4}x \, F_{\alpha \beta}^{a}{\widetilde{F}}_{a}^{\alpha \beta}
Y(t) = Y(0)+ \int_0^t\int_{\mathbb{R}^d_0}{^{p}}E[D_{s,y}Y_s| \mathcal{F}_s] (J-\mu)(ds\dy), &=Y(0)+ \int_0^t\int_{\mathbb{R}^d_0}{^{p}}E[\tilde{D}_{s,y}Y_s| \mathcal{F}_s] (J-\mu)(ds\dy), &=Y(0)+ \int_0^t\int_{\mathbb{R}^d_0}{^{p}}E[\hat{D}_{s,y}Y_s| \mathcal{F}_s] (J-\mu)(ds\dy),
\frac{1}{\sqrt{-\overline{g}}}\partial_{\mu}[\pm\sqrt{-\overline{g}}\enspace\overline{g}^{\mu\gamma}\overline{g}^{\nu\delta}\tilde{F}_{\gamma\delta}]-\frac{\tilde{e}^{2}}{8\omega}\varphi^{2}\overline{g}^{\alpha\nu}\tilde{A}_{\alpha}=0
(H,\rho, u)|_{t=0}=(H_0(x), \rho_0(x), u_0(x)), x\in \Omega, u|_{\partial{\Omega}}=0, (H(t,x),\rho(t,x), u(t,x),P(t,x)) \rightarrow (0,\overline{\rho},0,\overline{P}) |x|\rightarrow \infty,t>0,
\rho (\eta ) \simeq\frac{1}{16\pi^2\beta (\beta -1)a^4}\int_{\cal K}{\rm d}k\,k^2\left(\frac{\eta_2}{\eta_1}\right)^{2\beta-1}\left|\gamma_2\eta_2\right|^2\left|1+\frac{\gamma_1\eta_1}{1-2\beta}\right|^2\omega(k,\eta),
h_n(X) = \hskip-1ex \sum_{k_1 \le k_2 \le \cdots \le k_n}\hskip-3ex x_{k_1}x_{k_2}\cdots x_{k_n}\, \quad\mbox{and}e_n(X) = \hskip-1ex \sum_{k_1 < k_2 < \cdots < k_n}\hskip-3ex x_{k_1}x_{k_2}\cdots x_{k_n}\,.
\| T_{\gamma_*}g \|_{L^{q,\infty}(\mathbb{R}^n)}= \big\| \widehat{gd\sigma}\langle \cdot \rangle^{-\gamma_*}\big\|_{L^{q,\infty}(\mathbb{R}^n)}\lesssim \| g \|_{L^{p,1}(\mathbb{S}^{n-1})}
\begin{aligned}2\varphi_{1}(z)+\varphi_{2}(z) = \omega z^{1/3}f_{1}(z) + \omega^2 z^{2/3}f_{2}(z), \varphi_{2}(z)-\varphi_{1}(z) = \omega^2 z^{1/3}f_{1}(z) + \omega z^{2/3}f_{2}(z), -\varphi_{1}(z)-2\varphi_{2}(z) = z^{1/3}f_{1}(z) + z^{2/3}f_{2}(z),\end{aligned}
\begin{aligned}\ \ d(\sigma_{k+1}^\phi(\Bar{\sigma}_k),\sigma_{k+1}^\phi(\hat{\sigma}_k)) \leq \phi^{-1}(y) \cdot \left( \int_{\mathbb{R}^n}|\Bar{\sigma}_k(\xi)-\hat{\sigma}_k(\xi)|^pd\xi \right)^{1/p}\cdot \left( \int_{\mathbb{R}^n}|\Tilde{T}(u,y,\xi)|^qd\xi \right)^{1/q}, \end{aligned}
K_{n \ell}= ( n + \ell ) ! \sum_{{\mathrm{\scriptsize ~ p a r t i t i o n s ~ o f ~ n ~}}\atop{\mathrm{\scriptsize ~ o f ~ l e n g t h ~ \ell ~}}}\; \prod_{j = 1}^{\infty}\frac{1}{{n_{j}}!}\left( \frac{1}{( j + 1 ) !}\right)^{n_{j}}
G_p(r) :=b^{\frac{1}{p}}\cdot \nabla (\mu -\Delta )^{-\frac{1}{2}-\frac{1}{2r}}, Q_p(q) \upharpoonright{\mathcal E}:=(\mu -\Delta )^{-\frac{1}{2}+\frac{1}{2q}}|b|^{1-\frac{1}{p}}, T_p \upharpoonright{\mathcal E}:=b^{\frac{1}{p}}\cdot \nabla(\mu - \Delta)^{-1}|b|^{1-\frac{1}{p}}.
M(\theta, \varphi(t)) = \frac{v_{J(\varphi(t))}(\theta)}{v_{{\rm sign}(x)}(\theta)}{\rm e}^{\theta(\xi(\varphi(t)) - \log |x|)}\mathbf{1}_{(\varphi(t)<\infty)}= \frac{v_{{\rm sign}(X_t)}(\theta)}{v_{{\rm sign}(x)}(\theta)}\frac{|X_t|^\theta}{|x|^\theta}\mathbf{1}_{(t<\tau^{\{0\}})}, t\geq 0.
q_{ij}^2 q_{jk}^2 +q_{ij}^2 q_{ik}^2 + h^4q_{ik}^2 q_{jk}^2 -2q_{ij}q_{ik}q_{jk}(q_{ij}+q_{jk}+q_{ik}){}-2 h^2 q_{ik}(q_{ij}q_{jk}+q_{ij}q_{ik}+q_{jk}q_{ik}) = 8 h q_{ij}q_{ik}q_{jk}p_{ik},
g :=x_1^{-1}(f^{lt+1}+r^t)=x_1^{-1}(f^{lt}(x_1x_3-x_2^t)+(f^lx_2+x_1^m)^t) =f^{lt}x_3+g^*+x_1^{mt-1},g^*:=x_1^{-1}((f^lx_2+x_1^m)^t-f^{lt}x_2^t-x_1^{mt}).
\langle \phi(x)^2\rangle_{\rm reg}^{(0)}= (4\pi)^{-2}\left[-(\gamma-1)\bar{\tau}_1^{(0)}+ \bar{\tau}_2^{(0)}(\bar{\tau}_1^{(0)})^{-1}+\bar{\tau}_3^{(0)}(\bar{\tau}_1^{(0)})^{-2}+...\right]
\langle{\cal L}( 0 ) \tilde{q}^{k}( 1 ) q^{k}( 2 ) q^{k}( 3 ) \tilde{q}^{k}( 4 ) \rangle = \Theta^{2 2 2 2}\F^{( k - 2 )}( x , u ) + O ( \theta \bar{\theta}) .
e^{-M}\left\Vert u\left( .,0\right) \right\Vert_{X}\leq \left\Vert u\left(.,t\right) \right\Vert_{X}=\left\Vert \tilde{u}\left( .,s\right)\right\Vert_{X}\leq e^{M}\left\Vert u\left( .,0\right) \right\Vert_{X}t,s\in \left[ 0,1\right] ,
< \pi(F)\xi,\xi> = \int F(x) d\mu_{\xi,\xi}(x) = \int f(g)1_{Xg}(x)dg d \mu_{\xi,\xi}(x) = \int f(g) \Big (\int 1_{Xg}(x)d\mu_{\xi,\xi}(x) \Big) dg = \int f(g) < \pi(1_{Xg\cap X})\xi,\xi> dg = \int f(g)<E_{g}\xi,\xi> dg.
s(p,\mathcal C_n(\hat{c}_{n,1-\alpha}))\equiv\sup_{\theta\in\Theta}~p'\theta \sqrt{n}\frac{\bar{m}_{n,j}(\theta )}{\hat{\sigma}_{n,j}(\theta )}\leq \hat{c}_{n,1-\alpha}(\theta),~j=1,\dots ,J
\lim_{A \to \infty}\lim_{x \to \infty}\frac{H_2(x)}{\overline{\mu}(x)}= \lim_{A \to \infty}e^{\gamma A}\overline{\mu}(A) \le \lim_{A \to \infty}\int_{A+}^{\infty}e^{\gamma x}\mu(dx)=0.
\tilde{l}_1(\tilde{t})=\tilde{t}(2\delta_1-\tilde{t}), &\tilde{\rho}_1(x,\tilde{t})=\frac{e^{\lambda\psi(x)}}{\tilde{l}_1(\tilde{t})}, &\tilde{\theta}_1(x,\tilde{t})=\frac{e^{\lambda\psi(x)}-e^{2\lambda\|\psi(x)\|_{C(\overline{\Omega})}}}{\tilde{l}_1(\tilde{t})},
E_{c i r}^{D}= \frac{1}{2}\zeta_{c i r}^{D}( - 1 ) = \frac{1}{a}\left( 0 . 0 0 2 3 5 9 5 - \left. \frac{1}{1 2 8}\frac{1}{s}\right|_{s \to 0}\right) .
h(\mathcal{I})\cdot h(\mathcal{I}_{\alpha\beta}^{ij})=h(\mathcal{I}_{\alpha}^{i})\cdot h(\mathcal{I}_{\beta}^{j})+Y_{\alpha\beta}^{ij}\cdot h(\mathcal{I}_{\beta}^{i})\cdot h(\mathcal{I}_{\alpha}^{j})+\frac{h(\mathcal{I}_{\alpha}^{i})\cdot h(\mathcal{I}_{\beta}^{j})}{Y_{\alpha\beta}^{ij}}+h(\mathcal{I}_{\beta}^{i})\cdot h(\mathcal{I}_{\alpha}^{j}).
=\frac{1}{\left( f^{2}+4g\right)(gh^{2}-k^{2}+fhk)}\left|\left[\begin{array}[c]{cc}R_{n}\left( f^{2}h+2gh-fk\right) \ \gR_{n}\left( 2k-fh\right) R_{n}\left( 2k-fh\right) \ \ \R_{n}\left( 2gh+fk\right)\end{array}\right] \right| =R_{n}^{2}.
\frac{1}{\gamma}\frac{d}{dt}\int_{\Omega}u^{\gamma}(t)+\frac{4d_1(\gamma-1)}{\gamma^2}\int_{\Omega}|\nabla u^{\frac{\gamma}{2}}(t)|^2 =(\gamma-1)\chi_1\int_{\Omega}u^{\gamma-1}(t)\nabla u (t)\cdot \nabla w(t) +\int_{\Omega}u^{\gamma}(t)\Big[a_0-a_1u(t)-a_2v-a_3\int_{\Omega}u(t)-a_4\int_{\Omega}v(t) \Big].
\frac{\tau_k(p^{\alpha}) \tau_{\ell}(p^{\alpha})}{p^{\alpha}}\Big(1 - \frac{2}{p}\Big) + \sum_{j=1}^{\infty}\frac{\tau_k(p^{\alpha}) \tau_{\ell}(p^{\alpha+j})}{p^{\alpha}}\Big( \frac{1}{p^{j}}- \frac{1}{p^{j+1}}\Big).
\int_{\Omega_r}D (u-f)\cdot A(x)D (u-f)\,dx = \int_{\Omega_r}D u\cdot A(x)D u\,dx- \int_{\Omega_r}D f\cdot A(x)D f\,dx &\le(1+C_gr^\alpha) \left(\int_{\Omega_r}|D u|^2\,dx-\frac{1-C_gr^\alpha}{1+C_gr^\alpha}\int_{\Omega_r}|D f|^2\,dx\right) &\le(1+C_gr^\alpha) \left(|\Omega_r|+C_gr^\alpha\int_{\Omega_r}|D f|^2\,dx\right).
T_{( \mu \nu ) \beta}= - \frac{1}{2 e}\epsilon_{\, \, \, \, \, \, \, \mu \nu}^{\alpha \sigma}\partial_{\alpha}h_{\sigma \beta}- \frac{1}{2 e M^{2}}\epsilon_{\, \, \, \, \mu \nu \beta}^{\sigma}\partial_{\eta}\Theta_{\sigma}^{\eta}\, ,
| 0 \rangle \rightarrow | 0 \rangle_{\beta}= ( 1 + \mathrm{e}^{- \beta \epsilon})^{- 1 / 2}\{| 0 \rangle_{a}\otimes | 0 \rangle_{b}+ \mathrm{e}^{- \beta \epsilon / 2}a^{\dagger}{\tilde{a}}| 0 \rangle_{a}\otimes b^{\dagger}{\tilde{b}}| 0 \rangle_{b}\}.
\mathcal{F}_{1}\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_{k-1}\circ\mathcal{F}_k(P)=\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}S^{[k]}_{i_k},...,S^{[2]}_{i_2}S^{[1]}_{i_1}P.
b = \sum_{i \in \mathbb{F}_{p}}(x+ia)^{-1}= \sum_{i \in \mathbb{F}_{p}}(ia)^{-1}= a^{-1}\sum_{i=1}^{p-1}i^{-1}= a^{-1}\sum_{i = 1}^{p-1}i = 0.
\left\langle \bar{T}( r ) \bar{T}( 0 ) \right\rangle_{T , m = 0}^{( 1 )}= - \frac{\pi \sqrt{3}}{2 \nu^{2}}T^{4}\left[ \Phi ( - 1 , 4 , r T ) + \Phi ( - 1 , 4 , 1 - r T ) \right] .
X =2\binom m3+\binom m2+3\binom M3+3\binom N3+\binom M2+(M+1)\binom N2 &\quad+\left(\binom M2+\binom N2+m\left(M+\frac m4-\frac 12\right)\right)n+\frac 12\left(\binom{M+N+n}3-\binom n3\right) &\quad+\frac m2\left(\left[\frac{(M+N+n-1)^2}4\right]-\left[\frac{(n-1)^2}4\right]\right)
\mathrm{II}^{(2)}(T;j)(y_1, y_2) = y_2^{-1}y_1^{j-1}D(T)(y_2, y_1^{-1}y_2^{-1}) - (y_1 y_2)^{j-1}D(T)(y_1, y_1^{-1}y_2^{-1}).
{\cal L}= \int d^4 x e^{-2\alpha}\left( \partial_\mu \bar{\phi}\partial^\mu \phi + \bar{\psi}{\not\!\partial}\psi - \bar{F}F\right) + \int d^4 x e^{-2\alpha}m \left( \frac{1}{2}\psi \psi + \phi F \right) + \mbox{h.c.}
{\cal S}_{5 D}= \int d^{4}x d y \, \sqrt{G}\, [ 2 M_{5}^{3}R - \Lambda ( y ) ] - \sum_{i}\int d^{4}x \sqrt{\hat{g}}V_{i}+ \int d^{4}x d y \sqrt{\hat{g}}{\cal L}_{m}( \Phi_{i}, \hat{g})
\begin{array}{c c}{{\left( \bar{C}_{0}\, \eta_{0}\,{\cal A}. . . \right) \; \, \left( \bar{C}_{1}\, \eta_{0}\, B . . . \right)}}\\{{\left( \bar{C}_{1}\, \eta_{1}\,{\cal A}. . . \right) \; \, \left( \bar{C}_{1}\, \eta_{0}\,{\cal B}. . . \right)}}\\ \end{array}
\allowdisplaybreaks-u_{n-1}+ z'_j = -2u_{n-1}+ ((u_n - 1) - \lambda) \vee (-1)) & = -(u_n - 1) + ((u_n - 1) - \lambda) \vee (-1)) & \leq (-\lambda) \vee (-u_n) = z'_n \vee (-u_n) \, .
\delta \bar{\psi}_{\rho}= \bar{\epsilon}\left( \overleftarrow{\partial}_{\rho}- \frac{1}{2}\omega_{\rho m n}\sigma^{m n}+ \frac{1}{2}\alpha^{a}A_{\rho}^{a}\right) - \frac{1}{2 \sqrt{2}}e^{\phi}\bar{\epsilon}