harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{P}_{n}\left( \mathcal{E}\right) =\frac{\left( -1\right)^{n}\left(\gamma +\frac{1}{2}-D)\right)_{n}}{\sqrt{n!\left( 2\gamma +1\right)_{n}}}_{3}\digamma_{2}\left(\begin{array}{ccc}-n 1-D+i\lambda 1-D-i\lambda . \gamma +\frac{1}{2}-D \gamma +\frac{1}{2}-D\end{array}\mid 1\right)
\begin{cases}\,\alpha_n(y)=\displaystyle y^nq^{n^2/2-n/2}begin{align*}2mm] \displaystyle \, \beta_n(y)=\frac{(-q,-y;q){}_n}{(yq;q){}_{2n}}\sum_{k=0}^{n}\frac{q^{2k}(y/q;q){}_{2k}}{\left(q^2,y^2;q^2\right){}_{k}}. \end{cases}
\sum^{\lfloor n/e\rfloor}_{\substack{r=1 (r,n)=1}}\frac{1}{r}\equiv -e\sum^{\lfloor n/e\rfloor}_{\substack{r=1 (r,n)=1}}\frac{1}{n-er}-\frac{n}{e}\sum^{\lfloor n/e\rfloor}_{\substack{r=1 (r,n)=1}}\frac{1}{r^2}\pmod{n^2}.
q_{\mu, \nu}^{\kappa,\tau}(t,\xi-\eta, \eta) = \frac{p_{\mu, \nu}^{\kappa,\tau}(\xi-\eta, \eta)\psi_{\geq 0}((1+t)^{5/4+3p_0}|\xi-\eta|)}{\mu|\xi|^{1/2}-\nu |\xi-\eta|^{1/2}-\kappa |\eta|^{1/2}}.
b_{\underline{n}\sigma}^\dagger= \frac{\pi/L_\perp}{\sqrt{8\pi^3 n}}b_{\underline{p}\sigma}^\dagger\,,\;\;a_{\underline{m}}^\dagger= \frac{\pi/L_\perp}{\sqrt{8\pi^3 m}}a_{\underline{q}}^\dagger\,,\;\;
\begin{aligned}\sup\limits_{t \in [0,T_0]}\|u_1(t,\cdot) - u_2(t,\cdot)\|_{C^0([0, m(t)])}+ \sup\limits_{t \in [0,T_0]}\|v_1(t,\cdot) - v_2(t,\cdot)\|_{C^0([0, n(t)])}+ \|h_1 - h_2\|_{C^1([0,T_0])}+ &+\|g_1 - g_2\|_{C^1([0,T_0])}\leq K(\|u_{01}-u_{02}\|_{C^2([0,m_0])}+\|v_{01}-v_{02}\|_{C^2([0,n_0])}+|h_{01}-h_{02}|+|g_{01}-g_{02}|).\end{aligned}
\tilde{sl}z=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{n+1}}(\sum_{l=0}^{n}2^{l}(\begin{matrix}2n+2\\ 2l+1\end{matrix})s_{l}t_{n-l})\frac{z^{2n+1}}{(2n+1)!},|z|<\frac{\varpi}{2}
A_{t}^{\psi}u(x) =\exp(-\psi(x))\sum_{i,j=1}^{n}\partial_{x_{i}}(a_{ij}(t,x)\partial_{x_{j}}[\exp(\psi(x))u(x)]) \quad-\exp(-\psi(x))\sum_{i=1}^{n}b_{i}(t,x)\partial_{x_{i}}[\exp(\psi(x))u(x)].
(\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\theta_A^*=\theta_A\theta_A^*(\lambda_U\otimes I_{\mathcal{H}_0}), ~ (\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi^*=\theta_A\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0}), (\lambda_U\otimes I_{\mathcal{H}_0})\theta_\Psi\theta_\Psi^*=\theta_\Psi\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0}).
\left( \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a_i 0 c_i a_i \end{pmatrix}p \right)^{\frac{1}{\sqrt{c_i}}}= \begin{pmatrix}1 \frac{1}{\sqrt{c_i}}0 1 \end{pmatrix}+ \begin{pmatrix}a'_i * \sqrt{c_i}a'_i \end{pmatrix}p
\left[C_{\gamma}(\mathcal{Z}_{m,n}^{\gamma}) \right](z) = \dfrac{- (\gamma+1)_{m+n}(-m-1)_{n}}{(\gamma +1)_{n}(m+1)}\overline{z}^{m+1-n}\left(1-|z|^2\right)^{\gamma+1}{_2F_1}\left( \begin{array}{c}1-n,\gamma+m+2 m-n+2\end{array}\bigg ||z|^2\right).
\left[ \varphi (x),\varphi_0^{*}(x^{\prime})\right]_{t=t^{\prime}}=i\delta (\mathbf{x}-\mathbf{x}^{\prime})\hspace{0.3in}\left[ \widetilde{\varphi}(x),\widetilde{\varphi}_0^{*}(x^{\prime})\right]_{t=t^{\prime}}=-i\delta(\mathbf{x}-\mathbf{x}^{\prime})
\lim_{s\rightarrow 1}\int_{\partial \Delta'_{s}}\dot{u}_t^*(-u^{*,i,j}_{t,j}+I_{H,i}(a))\nu_iP_{DH}'d\sigma= \sum_Y \int_{\Delta'_Y}2n_Y \frac{\nu}{\mu_Y}\dot{u}_t^*P_{DH}'d\sigma
\lambda_{1}v_{1}(y_{1})+\sum_{i=2}^{n}\lambda_{i}v_{i}(w_{i})\sim<\lambda_{1}v_{1}(x_{1})+\sum_{i=2}^{n}\lambda_{i}v_{i}(w_{i}):\lambda_{1}v_{1}(z_{1})+\sum_{i=2}^{n}\lambda_{i}v_{i}(w_{i})>
X = x + h(\lambda y + \mu z) + g\mu y^2 - h \lambda\mu (cx^2 + dxy + ey^2) + h^2 \lambda^2\mu y (cx + dy) - ch^3 \lambda^3\mu y^2 Z = z + \lambda (cx^2 + dxy + ey^2) - h \lambda^2 y(cx + dy) + ch^2 \lambda^3 y^2
A^{ijkl,pq}(x) =\int_{0}^{1}\frac{\partial a^{ijkl}}{\partial u_{pq}}\left( (1-t)D^{2}u(x)+tD^{2}u(x+he_{m})\right) dt & =\frac{\partial a^{ijkl}}{\partial u_{pq}}\left( M^{\ast}(x)\right)
\frac{1}{i\partial_{-}- eA_{-}}= \frac{1}{i\partial_{-}}+\left(\frac{1}{i\partial_{-}}eA_{-}\right) \ast \frac{1}{i\partial_{-}- eA_{-}}= \frac{1}{i\partial_{-}}+\frac{1}{i\partial_{-}- e A_{-}}\ast \left( eA_{-}\frac{1}{i\partial_{-}}\right).
\mathrm{SINR}_{l}^{LR}(\mathbf{r}_{l})=\frac{\mathrm{SNR}_{l}^{LR}}{1+\sum_{k:\textrm{\ensuremath{kl\in\mathcal{E}}}}[(1-r_{kl})\mathrm{INR}_{kl}^{LR}+r_{kl}\mathrm{INR}_{kl}^{RR}]}
S_{(isentrpic)}[J^{\mu},\varphi^{'},{\tilde\alpha}^r,\beta_r,{}^4g^{\mu\nu}]=\int d^4x \Big[ -\sqrt{{}^4g}\rho ({{|J|}\over{\sqrt{{}^4g}}})+J^{\mu}(\partial_{\mu}\varphi^{'}+\beta_r\partial_{\mu}{\tilde\alpha}^r)\Big]
g_{\mathrm{w}}( x ) \; = \;{\frac{{\frac{\sqrt{x}}{8 \beta}}- 1}{\sqrt{( x - b^{2}) ( x - a^{2})}}}-{\frac{1}{8 \pi \beta}}\int_{0}^{\infty}d y \,{\frac{\sqrt{y}}{x + y}}{\frac{1}{\sqrt{( y + b^{2}) ( y + a^{2})}}}\; .
n_1R_1-\epsilon_n \leq I(W_1;Y^{n_1}), n_2R_2-\epsilon_n \leq I(W_2;Y^{n_1}) + I(X_{2,n_1+1}^{n_2};Y_{n_1+1}^{n_2}|X_{1,n_1+1}^{n_2}= \psi_{1,n_1+1}^{n_2}), n_1R_1+n_2R_2-\epsilon_n \leq I(W_1,W_2;Y^{n_1}) + I(X_{2,n_1+1}^{n_2};Y_{n_1+1}^{n_2}|X_{1,n_1+1}^{n_2}= \psi_{1,n_1+1}^{n_2}),
\sum_{j=1}^{k -1}x_{v_{j},v_{j+1}}+ x_{v_k,v_1}+ 2 x([\{v_1\}: \{v_3, \ldots, v_k\}]) + \sum_{j = 4}^{k}x([\{v_j\}: \{v_3, \ldots, v_{j - 1}\}]) \leq k - 1.
\mathcal{T}^\nu_{1,\mp}= \left(\frac{\nu}{\pi}\right)^{\frac{1}{2}}\mathcal{C}_{\psi_1}\circ \mathcal{B}^{1,\nu}(\mathcal{B}^{1,\nu})^{-1}\circ \Gamma_{{\mp}i}= \left(\frac{\nu}{\pi}\right)^{\frac{1}{2}}\Gamma_{{\mp}i}\circ \mathcal{C}_{\psi_1}= \left(\frac{\nu}{\pi}\right)^{\frac{1}{2}}\mathcal{C}_{{\mp}i\psi_1}
\frac{n_{e,A}\pi^{2}u_{ph}^{2}\dot{\gamma}_{ph,e,sp}}{\omega_{e,g}^{2}\int_{\omega}d\omega}=\frac{n_{e,A}\pi\omega_{e,g}|\mu_{e}|^{2}}{3\epsilon_{o}\hbar u_{ph}\int_{\omega}d\omega}
i) \hbox{}\bigcup_{p+q=k}\Sigma_{\delta ,p}(\rho_1)\times\Sigma_{\delta ,q}(\rho_2)\subseteq\Sigma_{\delta ,k}(\rho)\subseteq Sp_{\delta ,k}(\rho)\subseteq &\bigcup_{p+q=k}Sp_{\delta ,p}(\rho_1)\times Sp_{\delta,q}(\rho_2),
n_{i}^{\prime\prime}(\phi\alpha n_{i}^{\prime})=\frac{\phi}{2}(n_{i}^{\prime}+\frac{1}{\phi}+\frac{1}{\alpha-1}-\sqrt{(n_{i}^{\prime}+\frac{1}{\phi}+\frac{1}{\alpha-1})^{2}-\frac{4\alpha n_{i}^{\prime}}{(\alpha-1)\phi}})
\widetilde{\mathfrak{L}}_\alpha f = \widetilde{Q}_{\alpha}^{-1}\delta_x^{\alpha -1}(\delta_x^{\alpha-1})^* \widetilde{Q}_{\alpha}f &= \widetilde{Q}_{\alpha}^{-1}(\delta_x^{\alpha})^*\delta_x^{\alpha}\widetilde{Q}_{\alpha}f +\frac12 f = \widetilde{Q}_{\alpha}^{-1}Q{\mathfrak{L}}_\alpha Q^{-1}\widetilde{Q}_{\alpha}f +\frac12 f
\sum_{n\in \mathbb{Z}^2}h(x+2^qn) =2^{-2q}\sum_{k\in \mathbb{Z}^2}\phi(2^{-q}k)e^{i\frac{2\pi 2^{-q}k \cdot x}{L}}&=2^{-2q}h_q(2^{-q}x).
\big( (\cdot \otimes g)^{\prime}(\lambda \otimes \omega) \big) (\phi) = (\lambda \otimes \omega) \big( (\cdot \otimes g) (\phi) \big)= (\lambda \otimes \omega) (\phi \otimes g) &=\lambda (\phi) \omega (g) =\big( \omega (g) \lambda \big) (\phi),
F_0^\delta\mathcal O_{hol}(\tilde C)=\mathbb C,F_{n+1}^\delta\mathcal O_{hol}(\tilde C):=\{f\in \mathcal O_{hol}(\tilde C)\,|\,df\in F_n^\delta\mathcal O_{hol}(\tilde C)\cdot p^*\Omega(C)\}n\geq 0,
\begin{aligned}v_{ijk}= (1 - \phi_u d) u_{ijk}- (\phi_{uu}d\, u_k + \phi_u d_k + \phi_{ku}d)u_{ij}- \phi_{uuu}d\,u_i u_j u_k & - (\phi_{uu}d\, u_j + \phi_u d_j + \phi_{ju}d) u_{ik}- (\phi_{uu}d\, u_i + \phi_u d_i + \phi_{iu}d) u_{jk}+ Y_{ijk},\end{aligned}
p^0(x) = 0, u^0(x) = (U(x_2), 0), U(x_2) = \frac{p-1}{p}\left(\sqrt{2}^{-\frac{p}{(p-1)}}- \sqrt{2}^{\frac{p}{(p-1)}}\left|x_2 - \frac{1}{2}\right|^{\frac{p}{p-1}}\right).
\begin{aligned}h_q+G_\beta(a_q, \eta_q, Y_q, K\cap \{v(\cdot, t_q)\leq v(y_q, t_q)\})\geq 0, &k_q+G_\beta(b_q, \zeta_q, Z_q, K\cap \{v(\cdot, t_q)\leq v(z_q, t_q)\})\geq 0,\end{aligned}
\|\tilde{f}\|_{\textup{L}^p(\mathbb{R})}\|\tilde{g}\|_{\textup{L}^q(\mathbb{R})}\|\tilde{h}\|_{\textup{L}^r(\mathbb{R})}= |x_2|^{-1/p}\|f(\cdot,x_2)\|_{\textup{L}^p(\mathbb{R})}\, |y_2|^{-1/q}\|g(\cdot,y_2)\|_{\textup{L}^q(\mathbb{R})}\, |z_2|^{-1/r}\|h(\cdot,z_2)\|_{\textup{L}^r(\mathbb{R})}
Z_{2}( 0 ) = \frac{1}{4}\, \Gamma \left( \frac{3}{2}\right) \Gamma \left( \frac{1}{2}\right) \frac{1}{8}\left( \lim_{s \to 0}^{}\zeta ( 1 + s ) + \frac{1}{2}\right) = \frac{\pi}{6 4}\left( \left. \frac{1}{s}\, \right|_{s \to 0}+ \gamma \right) + \frac{\pi}{1 2 8}\,{.}
\left| \beta_{k}\right| = \frac{\left| 3 m + i \right|}{\sqrt{\pi}}\frac{2^{n_{s}- 3}\left| \Gamma \left( n_{s}\right) \right|}{| n_{1}( 1 - n_{1}) |}z_{S}^{m - 3 / 2}\left( \frac{g_{S}}{g_{1}}\right)^{- l}\left( \frac{\omega}{\omega_{s}}\right)^{2 - n_{s}}\left( \frac{M}{M_{s}}\right)_{.}^{- n_{1}/ 2}
\frac{1}{|b_i|}\int_{b_i}u(P)ds(P)=u(X)+\mathcal{R}_i(X), ~~ i\in\mathcal{I},~~~~~ \textrm{where}~~ \mathcal{R}_i(X)=\frac{1}{|b_i|}\int_{b_i}R_i(P,X)ds(P).
{\cal L}= \frac{\theta}{4}\epsilon^{\mu\nu\alpha}F_{\mu\nu}A_{\alpha}+\bar{\psi}(i \not \! \partial-m) \psi+\bar{\varphi}(i \not \! \partial+m)\varphi +e \bar \psi\gamma^\mu \psi A_\mu+e \bar \varphi\gamma^\mu \varphi A_\mu,
\hat{u}(k,t):=i\alpha(k,t)\frac{k^{\perp}}{|k|}&\hat{g}(k):=i\gamma(k)\frac{k^{\perp}}{|k|}&\hat{B}(u,u)(k):=i\beta(u,u)(k)\frac{k^{\perp}}{|k|}=[Q_k(\hat{u},\hat{u})-\frac{Q_k(\hat{u},\hat{u})\cdot k}{|k|^2}k],
\nabla_{X\wedge Y}Z =X\wedge \nabla_Y Z+(-1)^{kl}Y\wedge \nabla_X Z &=X\wedge \widetilde{\nabla}_Y Z+X\wedge F^{l,m}(Y,Z) &+(-1)^{kl}Y\wedge \widetilde{\nabla}_X Z+(-1)^{kl}Y\wedge F^{k,m}(X,Z).
V\tilde\pi_1(S_2)V^* e_{\sigma_2^k(m_{j''})}= V\tilde\pi_1(S_2)e_{\sigma_2^k(m_{j'})}= V e_{\sigma_2^{k+1}(m_{j'})}= e_{\sigma_2^{k+1}(m_{j''})}= \tilde\pi_2(S_2) e_{\sigma_2^k(m_{j''})}
\frac{2\binom{n-k}{k}}{|H|}\sum_{h\in H}\chi^{\left[2,1^{n-2}\right]}(xh) = \frac{2\binom{n-k}{k}}{|H|}\sum_{h\in H}\chi^{[1^n]}(xh) \chi^{[n-1,1]}(xh) = -2\binom{n-k-1}{k-1}.
V ( \overline{{{\rho}}}) = \frac{3 e^{4}}{6 4 \pi^{2}}\overline{{{\rho}}}^{4}\left( \ln \frac{\overline{{{\rho}}}^{2}}{\sigma^{2}}- \frac{1}{2}\right) + \frac{e^{2}}{8}T^{2}\overline{{{\rho}}}^{2}- \frac{e^{3}}{4 \pi}T | \overline{{{\rho}}}|^{3}
X_{\eta_X}(\pi_F^a\varepsilon_1,\pi_F^b\varepsilon) =\eta_X(\varepsilon_1)^b\cdot\eta_X(\varepsilon_2)^{-a}=X(\varepsilon_1,\pi_F)^b\cdot X(\varepsilon_2,\pi_F)^{-a}=X(\varepsilon_1,\pi_F)^b\cdot X(\pi_F,\varepsilon_2)^{a}=X(\varepsilon_1,\pi_F^b)\cdot X(\pi_F^a,\varepsilon_2) =X(\pi_F^a\varepsilon_1,\pi_F^b\varepsilon).
( f^{( 2 )})^{- 1}= \left( \begin{array}{c c c c}{{0}}&{{\delta_{i j}- a_{i}a_{j}}}&{{a_{i}}}&{{0}}\\{{- \delta_{i j}+ a_{i}a_{j}}}&{{a_{j}\pi_{i}- a_{i}\pi_{j}}}&{{- \pi_{i}}}&{{a_{i}}}\\{{- 2 a_{i}}}&{{- 2 \pi_{i}}}&{{0}}&{{- \delta_{i j}}}\\{{0}}&{{2 a_{i}}}&{{\delta{i j}}}&{{0}}\\ \end{array}\right)
\partial \beta = \partial \sum_{i=1}^k c_i\tilde x_i = \sum_{i=1}^k \bigg(\sum_{j=1}^{m+1}(-1)^jc_id_j\tilde x_i + c_id_0\tilde x_i\bigg) = \sum_{i=1}^k c_i\sum_{j=1}^{m+1}(-1)^jd_j\tilde x_i+ \epsilon = \epsilon.
\gcd\left(d(v_2^2+v_3^2),d(v_1^2+v_3^2),d(v_1^2+v_2^2)\right) =d\gcd\left(v_2^2+v_3^2,v_1^2+v_3^2,v_1^2+v_2^2\right){}=d\gcd\left(v_2^2+v_3^2,v_1^2+v_3^2,v_1^2+v_2^2,v_2^2-v_3^2,v_1^2-v_3^2,v_1^2-v_2^2\right){}=d\gcd\left(2v_1^2,2v_2^2,2v_3^2,v_2^2+v_3^2,v_1^2+v_3^2,v_1^2+v_2^2\right).
SS^{\star}=\frac{(1-\frac{1}{R^2})(1-\frac{1}{{R^{\star}}^2})}{(e^{i\pi\nu}-\frac{e^{-i\pi\nu}}{R^2})(e^{-i\pi\nu^{\star}}-\frac{e^{i\pi\nu^{\star}}}{{R^{\star}}^2})}
F_{\acute{\gamma}(n)}(x) = F_{\hat{\gamma}(n)}\left(\frac{x+1}{\left( \frac{2^B}{d_n}\right)^{\frac{1}{n_T-1}}}\right) = 1 - \frac{2^B}{(1+x)^{n_T-1}}+ O\left(\frac{1}{x}\right),
\Bigg(\sum_{j=1}^\infty \mu(D(z_j,r))^{\frac{p}{p-1}}\Bigg)^{{\frac{p-1}{p}}}\Bigg( \sum_{j=1}^\infty \Bigg(\int_{D(z_j,2r)}|f(z)| e^{-\frac{\alpha}{2}|z|^2}dV\bigg)^p\Bigg)^{\frac{1}{p}}
\| \partial_\nu v\|_{L^2(\Sigma)}=\| (\Lambda_{q_1,\lambda}^0-\Lambda_{q_2,\lambda}^0)(v_2{_{|\partial M}})\|_{L^2(\Sigma)}&\le \| \Lambda_{q_1,\lambda}^0-\Lambda_{q_2,\lambda}^0\| \|v_2\|_{H^{3/2}(\Sigma)}&\le \lambda^2\mathbf{e}_\lambda e^{c/\epsilon}\| \Lambda_{q_1,\lambda}^0-\Lambda_{q_2,\lambda}^0\| \|u_2\|_{L^2(\mathrm{M}_0)},
u\in D(A_{i,e}):=\left\{u\in W^{2,p}_{av}(\Omega)\mid\sigma_{i,e}\nabla u\cdot n=0{\rm~a.e.~in~}\partial\Omega\right\}\subset L^p_{av}(\Omega), &A_{i,e}u:=-\nabla\cdot(\sigma_{i,e}\nabla u).
\begin{aligned}\mathbb F^f(a-3\epsilon, b+3\epsilon) = \mathbb F^f(a-\epsilon, b+\epsilon) , \mathbb F^f(a+\epsilon, b-\epsilon) = \mathbb F^f(a+3\epsilon, b-3\epsilon), \mathbb F^f(a+\epsilon, b+3\epsilon) = \mathbb F^f(a+3\epsilon, b+\epsilon), \mathbb F^f(a-3\epsilon, b-\epsilon) = \mathbb F^f(a-\epsilon, b-3\epsilon).\end{aligned}
(1,-2) = 0\cdot(-1,0) + 3\cdot(0,-1) + 1\cdot(1,1) (2,3) = 1 \cdot (-1,0) + 0 \cdot(0,-1)+ 3 \cdot(1,1) (-4,1) = 5 \cdot(-1,0) + 0 \cdot(0,-1) + 1 \cdot(1,1).
\sum_{k=0}^{l}\frac{1}{r_k}= \frac{1}{2},\frac{1}{2}\leq \frac{1}{r_0}+ \frac{L-|\alpha|}{d}{\rm and}\frac{1}{2}\leq \frac{1}{r_k}+ \frac{L-|\beta_k|-1}{d}.
N^{\beta}_{2}f(x) = \chi_{(0,1)}(x)\int_{0}^{\infty}\bigg[ (xy)^{-2\beta}\sup_{1/4 < s < 1}(1-s)^{-2\beta}K_s^{-\beta}(x,y)\bigg] f(y)\, d\nu_{\beta}(y), f \ge 0, N^{\gamma}_{3}f(x) = \chi_{(0,1)}(x)\int_{0}^{\infty}\bigg[\sup_{1/4 < s < 1}K_s^{\gamma}(x,y)\bigg] f(y)\, d\nu_{\gamma}(y), f \ge 0,
\left( \frac{1}{|B_{kr}|}\int_{B_{kr}}|\overline{L}u(x)|^{p}dx\right)^{1/p}\leq\left( \frac{1}{|B_{kr}|}\int_{B_{kr}}|Lu(x)|^{p}dx\right)^{1/p}\ & +c(a_{R}^{\sharp})^{1/\beta p}\sum_{i,j=1}^{q}\left( \frac{1}{|B_{kr}|}\int_{B_{kr}}|X_{i}X_{j}u(x)|^{p\alpha}dx\right)^{1/\alpha p}.
\phi_{\alpha,b}(t) \equiv E[ e^{itZ_{\alpha, b}}] = \begin{cases}\exp\big[ -b|t|^{\alpha}\big(1 - i \tan(\frac{\pi \alpha}{2}) \mathrm{sgn}(t) \big) \big], ~~\alpha \not= 1, \exp\big[ -b|t| \big(1 + \frac{2i}{\pi}\log|t| \mathrm{sgn}(t) \big) \big], ~~~~~\alpha = 1.\end{cases}
\begin{cases}\partial_t u+u\cdot \nabla u+\nabla P=-\nabla\cdot(\nabla d\odot\nabla d)+\nabla\cdot(\sigma^L(u,d)), \nabla\cdot u = 0, \partial_t d+u\cdot\nabla d-\omega d+\frac{\lambda_2}{\lambda_1}Ad=\frac{1}{|\lambda_1|}\left(\Delta d+|\nabla d|^2d\right)+\frac{\lambda_2}{\lambda_1}\left(d^TAd\right) d.\end{cases}
(\Delta(z)) =p\mathfrak{a}_{\mathfrak{0}}+\mathfrak{a}_\infty, (\Delta_p(z)) =\mathfrak{a}_{\mathfrak{0}}+p\mathfrak{a}_\infty, (\eta(z)^{12}\eta(pz)^{12}) =\frac{p+1}{2}\mathfrak{a}_{\mathfrak{0}}+\frac{p+1}{2}\mathfrak{a}_\infty ,
\tilde{Q}= F(0)+\sum_{n=1}^{\infty}\left.\frac{1}{n!}\frac{d^nF}{d^nt}t^n\right|_{t=1}= Q+\sum_{n=1}^{\infty}\frac{(-1)^n \epsilon^n}{n!}[K,[K,\dots,[K,Q]]] \ .
\mathcal{M}_2\left(s\right)=L\sum_{j=r+1}^{m}\sum_{i=1}^{m}\mathcal{D}\left(i,j\right)\Gamma\left(s+j-1\right)\Biggl( \theta_{n-m+i}^{n-m+s+j-2}& -\sum_{l=1}^{n-m}\sum_{k=1}^{n-m}\left[\mathbf{\Psi}^{-1}\right]_{k,l}\theta_{l}^{n-m+s+j-2}\theta_{n-m+i}^{k-1}\Biggr).
+\sum_{i=1,2}\frac{1}{8}(\Phi^{\alpha}+ \overline{\Phi^\alpha})Q_{2}\big(\Gamma^\alpha((-1)^{i}2R_{3-i}\phi_0 + c_{+}R_i\Phi + c_{-}R_i \overline{\Phi}),(-1)^{i}2R_{3-i}\phi_0 + c_{+}R_i\Phi + c_{-}R_i\overline{\Phi}) \Big).
\begin{array}{l l}{Y_{1}= X_{2}+ \sqrt{- 1}\X_{1}, \quad}&{Y_{2}= - X_{2}+ \sqrt{- 1}\X_{1},}\\{Y_{3}= X_{4}+ \sqrt{- 1}\X_{3}, \quad}&{Y_{4}= - X_{4}+ \sqrt{- 1}\X_{3}.}\\ \end{array}
S=\frac{1}{2}\int dt \, \left[{\cal G}_{\alpha\beta}\left({\dot z}^{\alpha}{\dot z}^{\beta}+i{\overline \psi}^{\alpha}\gamma^{0}D_{t}\psi^{\beta}\right) + \frac{1}{6}R_{\alpha\beta\gamma\delta}{\overline \psi}^{\alpha}\psi^{\gamma}{\overline \psi}^{\beta}\psi^{\delta}\right] -\frac{4\pi v}{e^{2}}k \, ,
\bar{G}_{\dot{\alpha}\dot{\beta}}\equiv - \frac 1 2 G_{\mu \nu}( \bar{\sigma}^{\mu})_{\dot{\alpha}\gamma}( \sigma^{\nu})_{\dot{\beta}}^{\gamma}\equiv ({E}^{a}+ i{B}^{a}) ({\bar{\sigma}}^{a})_{\dot{\alpha}\dot{\beta}}= 0 .
h(\lambda_k) - h(\lambda^\star) = L(x_k, \lambda_k) - L(x^\star,\lambda^\star) & \le L(x^\star, \lambda_k) - L(x^\star, \lambda^\star) & = (\lambda_k - \lambda^\star)^T g(x^\star) & \le \| \lambda_k - \lambda^\star \|_2 \| g(x^\star) \|_2
=\frac{e^{4}}{(k-k^{\prime})^{4}}Tr((\sum_{r^{\prime}}v_{k^{\prime}}\overline{v}_{k^{\prime}})\gamma^{\mu}(\sum_{r}v_{k}\overline{v}_{k})\gamma^{\nu})Tr((\sum_{s}u_{p}\overline{u}_{p})\gamma_{\mu}(\sum_{s^{\prime}}u_{p^{\prime}}\overline{u}_{p^{\prime}})\gamma_{\nu})
\big|A_{k_\epsilon,\varrho}(t)\big| =\big|A_{k_\epsilon,(1-\sigma)\varrho}(t)\cup (A_{k_\epsilon,\varrho}(t)\setminus A_{k_\epsilon,(1-\sigma)\varrho}(t))\big| &\leq \big|A_{k_\epsilon,(1-\sigma)\varrho}(t)\big|+|K_\varrho\setminus K_{(1-\sigma)\varrho}| &\leq\big |A_{k_\epsilon,(1-\sigma)\varrho}(t)\big|+N\sigma |K_\varrho|.
\sum_{\xi_{\Theta}}\int_{\Omega}h_{\Delta}(\eta)(\hat{c}_{\Theta}(\eta, \xi_{\Theta})-c_{\Theta}(\eta, \xi_{\Theta}))\mu(d\eta) = \sum_{\xi_{\Theta}}\int_{\Omega}c_{\Theta}(\eta, \xi_{\Theta})\left(h_{\Delta}(\xi_{\Theta}\eta_{\Theta^c})- h_{\Delta}(\eta)\right)\mu(d\eta).
\|\tilde{\ell_{2}}- \bar{\ell_{2}}\|\leq \frac{2\varepsilon}{\lambda_{2}^2}(\lambda_{2}+ \|x_{2}\|), \ \ \ \bar{a}_{2}-\tilde{a_{2}}=\frac{\varepsilon}{\lambda_{2}^2}\ \ \tilde{c_{2}}-\bar{c}_{2}\leq \frac{\varepsilon}{\lambda_{2}^2}\|x_{2}\|^2.
\bar{H}_{\mathrm{D s}}( q^{r}, p^{r}) = H_{\mathrm{D}}( q^{r}, p^{r}) + \lambda_{a}^{( 1 ) s_{a}}\pi_{a}^{s_{a}}+ \lambda_{a}^{( 2 ) s_{a}}\psi_{a}^{s_{a}}+ \lambda_{A}\phi_{A},
\Gamma\left( r\right) \sum_{k=1}^{n}\frac{h_{k}^{\left( r\right)}}{k^{\overline{r}}}=\frac{\left( H_{n}\right)^{2}+H_{n}^{\left(2\right)}}{2}-H_{r-1}\left( H_{n+r-1}-H_{n}\right) -\left( \psi\left(r\right) +\gamma\right) H_{n}& +\sum_{j=1}^{r-1}\frac{H_{j}}{j}+\sum_{j=1}^{r-1}\frac{H_{j-1}}{j+n}.
Q(g_2) = Q\left(g-\frac{Q(g)}{Q'(g)}\right) &= Q(g)+\left[\frac{-Q(g)}{Q'(g)}\right]Q'(g)+\left[\frac{Q(g)}{Q'(g)}\right]^2 \hat{Q}(\cdots) \ \ \ \ \ \hat{Q}&= (f^{n-m}h_1h_2^{-1})^2 \ \hat{Q}(\cdots) &\equiv 0 \pmod{f^{n+1}}\ \ \ \ \n > 2m.
\kappa_\theta = \frac{V_1^2+V_2^2+2V_1V_2\cos{\theta}}{2\sigma^2 \mu}= \frac{\left( V_{1}^{2}+V_{2}^{2}\right)\left( 1+\Delta \cos \theta \right)}{2{{\sigma}^{2}}\mu}= K\left( 1+\Delta \cos \theta \right).
\hat{V}^{C}_{\sigma}(C_{1},C_{2}) = \left ( \frac{1}{2\pi4\theta^2}\right ) \frac{1}{N}\sum\limits_{i=1}^N exp \left( - \frac{(x_{i}-y_{i})^2 + (z_{i}-s_{i})^2}{8\theta^2}\right )
I_2 = \sum_{r,s \geq 1}A_k^{(2)}(m,r) \overline{A_k^{(2)}(m,s)}\int_{-1/2}^{1/2}e^{\pi i x (r-s)}\int_{\sqrt{1-x^2}}^\infty \frac{y^{k-2}}{(x^2+y^2)^k}e^{-\pi y(r+s)}dy dx.
\Vert\nabla(u_{\varepsilon}-u_{\varepsilon,h})\Vert_{A_{\varepsilon}}^{2}\, \leq\,\mathcal{M}_{\Omega}^{2}(u_{\varepsilon,h},y,\gamma) & :=(1+\gamma)\Vert A_{\varepsilon}\nabla u_{\varepsilon,h}-y\Vert_{A_{\varepsilon}^{-1}}^{2}+\left( 1+\frac{1}{\gamma}\right) C_{\Omega}^{2}\Vert\operatorname{div}y+f\Vert_{2,\Omega}^{2}.
2 \kappa^{2}S_{\Phi}={\frac{\kappa^{2}N}{2 \pi g_{s}}}{\frac{1}{\sqrt{G_{\Omega \Omega}}}}\left({\frac{3}{4}}{e^{3 \Phi ( r ) / 4}}\mu_{6}+{\frac{1}{4}}{\frac{e^{- \Phi ( r ) / 4}}{V_{\mathrm{E}}( r )}}\mu_{2}\right) .
\beta_{i}(P,f;C)=\sum\limits_{C_{k}\in C^{i}}\sum\limits_{R \subseteq C\backslash C^{i}}\sum\limits_{T \subseteq C_{k}\backslash \{i\}}\frac{1}{2^{c+c_{k}-2}}\Bigl(f\bigl(T \cup S_R \cup \{i\}\bigr)- f(T \cup S_R)\Bigr),
\Delta = \Omega + C^{A}V_{A}^{B}\bar{\cal P}_{B}( - 1 )^{\varepsilon_{B}}+ \frac{1}{4}C^{B}C^{A}V_{A B}^{C D}\bar{\cal P}_{D}\bar{\cal P}_{C}( - 1 )^{\varepsilon_{B}+ \varepsilon_{D}}+ \ldots .
\begin{aligned}\star_7 \beta = (-1)^k 2^{(-2+k)}|F|^{(\frac{4}{3}- \frac{2}{3}k)}G dr \wedge \star_6 \beta, \star_7 (dr \wedge \beta) = 2^{(-4+k)}|F|^{(\frac{8}{3}- \frac{2}{3}k)}G^{-1}\star_6 \beta.\end{aligned}
Q_{r,s,g,n}\coloneqq Q_{r(p-1)}Q_{s(p-1)}-q^{\tfrac{p-1}{2}}(-1)^{\tfrac{s(p-1)}{2}}\sum_i (-1)^{\tfrac{r-i}{2}}\binom{(i-s)\tfrac{p-1}{2}-1}{\tfrac{r-i}{2}-1}Q_{(r+ps-pi)(p-1)}Q_{i(p-1)}
\frac{\gamma_\beta(T)}{h_\beta(T)}=\frac{\beta^{2T}\Gamma(y(T)+1)}{\left(\beta^2\right)^{1-\frac{1}{\beta^2}}\sqrt{2\pi y(T)}\left(\frac{\beta^2y(T)}{e}\right)^{T-1+\frac{1}{\beta^2}}}=\frac{\Gamma\left(y(T)+1\right)}{\sqrt{2\pi y(T)}\left(\frac{y(T)}{e}\right)^{y(T)}},
\frac{b_V-a_V}{2 N \rho_V(x)}K_{N,V}\left( \lambda_V\left( x+ \frac{s}{N \rho_V(x)}\right),\lambda_V\left( x+ \frac{t}{N \rho_V(x)}\right) \right) & = \frac{\sin(\pi (s-t))}{\pi (s-t)}+ \mathcal{O}\left( \frac{1+|s|+|t|}{N}\right).
de_{t}(\zeta) =\overline{e}_{t}(\zeta)dt +\sum_{r=1}^{3}e_{t}^{r}(\zeta)dB_{t}^{(r)}, dh_{t}(\zeta) =\overline{h}_{t}(\zeta)dt +\sum_{r=1}^{3}h_{t}^{r}(\zeta)dB_{t}^{(r)}, df_{t}(\zeta) =\overline{f}_{t}(\zeta)dt +\sum_{r=1}^{3}f_{t}^{r}(\zeta)dB_{t}^{(r)}.
{\rm II}_{x}=\sum_{s\in G}\mu(s)\frac{ds\nu}{d\nu}(\pi(x))\sum_{\xi_{n}^{x}\in L_{x}\backslash G}\log\left(P_{\mu,x}^{n-1}(L_{x}s,\xi_{n}^{x})\right)P_{\mu,x}^{n-1}(L_{x}s,\xi_{n}^{x})
\Big\|\Big\{\sum_{k\in \mathbb Z}{\mathfrak R}^{-k\beta}|D_k(a)|\Big\}\Big\|_{L^{p'}_{\omega}}&\le \Big\|\Big\{\sum_{k=k_0+1}^\infty{\mathfrak R}^{-k\beta}|D_k(a)|\Big\}\Big\|_{L^{p'}_{\omega}}+ \Big\|\Big\{\sum_{k=-\infty}^{k_0}{\mathfrak R}^{-k\beta}|D_k(a)|\Big\}\Big\|_{L^{p'}_{\omega}}=:I_1+I_2.
F^{i}:=\left (\begin{array}{cccccc}I K^{i, 1}\frac{1}{2}K^{i, 1}K^{i, 2}\frac{1}{6}K^{i, 1}K^{i, 2}K^{i, 3}\cdots \frac{1}{N!}K^{i, 1}K^{i, 2}\cdots K^{i, N}0 I K^{i, 2}\frac{1}{2}K^{i, 2}K^{i, 3}\cdots \frac{1}{(N-1)!}K^{i, 2}K^{i, 3}\cdots K^{i, N}& \cdots \cdots 0 0 0 0 \cdots I \end{array}\right ).
|\mathcal{A}|\leq e^{-(C_0/5)(|k|+|\xi-\xi_1-\cdots-\xi_p|)}(1+|\xi|)\prod_{i=1}^r\frac{|\mu_{j_{i-1}}-kt|}{(\mu_{j_i}-kt)^2+k^2}\cdot(1+|\mu_{j_{i-1}}-kt|+|\mu_{j_{i-1}}-\mu_{j_i}|),
\frac{\sqrt{y}}{|\zeta(1+2it)|}(t\Delta)^{-1/4}\sum_{n = \frac{t}{2 \pi y}+ O(\frac{\Delta}{y})}\tau_0(n) \ll \frac{\sqrt{y}\log t}{(t\Delta)^{1/4}}\Big(\frac{\Delta}{y}\log t + t^{\varepsilon}\Big),
f_k(r) =c_ke^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}] &=c_k\frac{\Gamma(2l_k+\frac{m}{2})}{\Gamma(l_k-\lambda)}e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big) &=C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big).
\frac{(m-1)k_{\mu}k^{\lambda}l^{\mu \nu}_{\,\,\,\,\nu \lambda}}{m^2 (k^2)^{2-1/m}((k^2)^{1/m}- \lambda)^3}-\frac{(m-1) k^{\mu}k^{\lambda}l_{\nu}^{\,\,\nu \mu \lambda}}{m^2(k^2)^{2-1/m}((k^2)^{1/m}- \lambda)^3}
l_{ab}+r^{-a}\sigma_{ab}= -\frac{1}{3}r^3\alpha_{ab}+O(r^4) n_{cd}-\frac{1}{2}r^{-1}\sigma_{cd}= r^3( \frac{1}{2}\rho \tilde \sigma_{ab}- \frac{1}{6}\alpha_{ab})+O(r^4).
(a_t\delta^h \delta^hu_t,u_t)_{l_2(\mathbb{G}_h)}= -( \delta^hu_t,\delta^h(a_tu_t))_{l_2(\mathbb{G}_h)}= - (\delta^hu_t,(\delta^ha_t)T^hu_t)_{l_2(\mathbb{G}_h)}-( \delta^hu_t,(T^ha_t)\delta^hu_t)_{l_2(\mathbb{G}_h)}.
\begin{aligned}H_j[\lambda,\xi,\dot{\lambda},\dot{\xi},\phi]:= \Bigg\{\mu_{0j}^{\frac{n+2s}{2}}S_{\mu, \xi, j}(\xi_j + \mu_{0j}y, t)+ B_j[\phi_j] + B_j^0[\phi_j] &+ p\mu_{0j}^{\frac{n-2s}{2}}\frac{\mu_{0j}^{2s}}{\mu_j^{2s}}U^{p-1}\left(\frac{\mu_{0j}}{\mu_j}y\right)\psi(\xi_j + \mu_{0j}y, t)\Bigg\}\chi_{B_{2R}(0)}(y)\end{aligned}
B^\infty_{(\psi,\alpha)}(|g|)(z) = \frac{|g'(z)|}{1+|z|}e^{\frac{\alpha}{2}\big(|\psi(z)|^2-|z|^2\big)},\ \M^\infty_{(\psi,\alpha)}(|g|)(z) = \frac{|g'(\psi(z))|e^{\frac{\alpha}{2}\big(|\psi(z)|^2-|z|^2\big)}}{(1+|z|)(|\psi'(z)|^{-1})}
\Pi_{h_{\prime}\; \rho \,}^{\rho \; , \, \mu \nu}= - \kappa \frac{a + 4 c + 2 d}{a \left( a + 4 b \right)}\; \Gamma_{h_{\prime}}^{\mu \nu}\equiv - \tilde{\Gamma}_{h_{\prime}}^{\mu \nu},
\int_0^{m-x}m \frac{2}{\sqrt{T-t}}\varphi\Big(\frac{z}{\sqrt{T-t}}\Big) dz \ = \ 2m \int_0^{\frac{m-x}{\sqrt{T-t}}}\varphi(z) dz \ = \ 2m \Big(\Phi\Big(\frac{m-x}{\sqrt{T-t}}\Big) - \frac{1}{2}\Big),
(X,{\delta_X}\dot{Df}\wedge \delta_X Dg) = (X,{\delta_X}{D^2f}(\dot X, \cdot) \wedge \delta_X Dg) =-(X, \delta_X Dg\wedge{\delta_X}{D^2f}(\dot X, \cdot) ) &=-(d_X \rho_X \delta_X dDg,{D^2f}(\dot X, \cdot) )=-(\nu_{X}Dg,{D^2f}(\dot X, \cdot) ) & =-(\dot X,{D^2f}( \nu_{X}Dg, \cdot) )
\int_\Omega \theta\cdot\tau +\int_\Omega u{\rm div}( \tau) =0{\rm for\,\,all}\,\,\tau\in H(div,\Omega), -\varepsilon\int_\Omega{\rm div}(\theta) v+\int_\Omega cuv =(f,v){\rm for\,\,all}\,\,v\in L^2(\Omega).
\Delta I(\frac{x_{2}}{\lambda f},\frac{y_{2}}{\lambda f})=\frac{2r_{o}}{\lambda f}S(\frac{x_{2}}{\lambda f},\frac{y_{2}}{\lambda f})cos[2\pi\alpha y_{2}+\psi(\frac{x_{2}}{\lambda f},\frac{y_{2}}{\lambda f})].
\widetilde{M}_k(\alpha,\theta,E,n)=\begin{pmatrix}\widetilde{P}_k(\alpha,\theta,E,n) -\widetilde{P}_{k-1}(\alpha,\theta,E,n+1) \widetilde{P}_{k-1}(\alpha,\theta,E,n) -\widetilde{P}_{k-2}(\alpha,\theta,E,n+1) \end{pmatrix}.
P_1(A^p,B^p) =P_{th}(A^p,B^p)=P_{h(1/t)}(B^p,A^p) &\le P_{h(1/t)}(g(A)^p,A^p)=A^p h\left(\left({A\over g(A)}\right)^p\right) &=A^p h\left(\left({1\over{f(A)}}\right)^p\right) &\le \left( A h\left({1\over{f(A)}}\right)\right)^p=f(A)^{2(n-1)p}.