harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\dot{\bar z}_i =A_i^0\bar z_i, \dot{\bar x}_{i}=S\bar x_i-\theta_idd^\top P\bar x_{vi}-d[\theta_i \mbox{sgn}[(d^\top P\bar x_{vi})]+v]+\bar g_i(\bar z_i,\,z_i,\,x_i) \dot{\theta}_i =||d^\top P\hat x_{vi}||_2^2+||d^\top P\hat x_{vi}||_1
0 = \frac{1}{2}x^3 w x^4 + \frac{1}{2}uwx^3w x^4 + \frac{1}{2}wx^3w x^4 + \frac{1}{4}x^4w x^4 + \frac{1}{4}uwx^4w x^4 & - \frac{1}{4}wx^4w x^4 & = \frac{1}{2}x^3 w x^4 + \frac{1}{2}(u+1)wx^3w x^4 + \frac{1}{4}x^4w x^4 + \frac{1}{4}(u-1)wx^4w x^4
g^E_{\mu\nu}dx^{\mu}dx^{\nu}=-{1\over{\left(H_1H_2H_3\right)^{2\over 3}}}fdt^2+\left(H_1H_2H_3\right)^{1\over 3}\left[f^{-1}dr^2+r^2d\Omega^2_3\right],
\eta_{S}^{mic,Q_{H}}(h\sigma) = \sum_{\xi \in \Xi({^\vee}H)^{Q_{H}}}e(\xi)(-1)^{\dim S_{\xi}- \dim S}\ \mathrm{tr}\left( \tau_{S}^{mic}(P(\xi))(h)\right) (\pi(\xi) \otimes 1).
P(\frac{\partial}{\partial y^{i}})=\stackrel{1}{P^l_i}\frac{\partial}{\partial y^l}+\stackrel{3}{P_{li}}\frac{\partial}{\partial p_l}\,,\,P(\frac{\partial}{\partial p_{j}})=\stackrel{2}{P^j_k}\frac{\partial}{\partial p_k}+\stackrel{4}{P^{kj}}\frac{\partial}{\partial y^k}.
F_{t}(A_{1},\bar{A}) =dA_{t}(A_{1},\bar{A})+\frac{1}{2}\left[ A_{t}(A_{1},\bar{A}),A_{t}(A_{1},\bar{A})\right] , & =td\omega+\frac{1}{2}\left[ t\omega,t\omega\right] , & =td\omega+\frac{t^{2}}{2}\left[ \omega,\omega\right] ,
H_2(n,{\textstyle \frac{1}{2}}) \ge \frac{\sqrt{2\pi}}{e^2}n^{1/2}\frac{(n-1)!}{2^{n+1}}(1-o(1)) =\frac{\sqrt{2\pi}}{e^2}n^{1/2}E(n,{\textstyle \frac{1}{2}})(1-o(1))\;.
S^{\mu}= R^{\mu}+ \frac{1}{3}\left( 1 - \frac{3T (G')}{2N_f T(R')}- \gamma_q \right) ( K^{\mu}_i + \widetilde{K}^{\mu}_i-2K^{\mu}_M ) - \frac{1}{3}(2\gamma_q + \gamma_M) K^{\mu}_M,
\alpha^{\natural}(\varphi,\tilde{\varphi};\chi)= \dfrac{L(1,\pi,{\rm Ad})\, L(1,\chi_{K/F})}{\zeta_{F}(2)\,\zeta_{F}(4)\, L(1/2,\pi\boxtimes\chi)}\,\alpha(\varphi,\tilde{\varphi};\chi)
F_{\mathrm{s i n g}}( N , g_{3}, g_{4}) \sim b \cdot [ \delta g_{3}- 4 . 4 7 2 2 \ldots \delta g_{4}]^{7 / 3}+ b^{\prime}\cdot [ \delta g_{3}- 6 . 2 0 8 7 \ldots \delta g_{4}]^{7 / 2}+ \cdots .
G_{{\cal D}}( \theta = 0 ) = [ 2 , 2 , a ] \frac{V^{\{\mu_{1}}\ldots V^{\mu_{s}\}}}{( x_{1 2}^{2})^{2 - \frac{d - s}{2}}( x_{1 3}^{2}x_{2 3}^{2})^{\frac{d - s}{2}}}\; .
\int_{B(0,\delta)}e^{\rho(\widetilde{\psi}-\widetilde{\psi}_B)}dx\int_{B(0,\delta)}e^{-\rho(\widetilde{\psi}-\widetilde{\psi}_B)}dx =\int_{B(0,\delta)}e^{\rho\widetilde{\psi}}dx\int_{B(0,\delta)}e^{-\rho\widetilde{\psi}}dx &\leq C\delta^{2n}
g'_j\left(\binom{n}{\frac{n+t}{2}-j}+\binom{n}{\frac{n+t}{2}+k+j-1}\right)\le g'_j\left(\binom{n}{\frac{n+t}{2}+j-1}+\binom{n}{\frac{n+t}{2}+k-j}\right).
[u]_{z',\delta/2,\delta; \,Q}:=[u]_{t,\delta/2; \,Q}+[u]_{x',\delta;\, Q},[u]_{z',(1+\delta)/2,1+\delta; \,Q}:=[u]_{t,(1+\delta)/2; \,Q}+[D_{x'}u]_{z',\delta/2,\delta; \,Q}.
{\cal H}_{C_{2}}^{( 1 )}={\frac{1}{2}}( p_{1}^{2}+ p_{2}^{2}) + m^{2}e^{q_{1}- q_{2}}+{\frac{m^{2}}{2}}\left( e^{2 q_{2}}+ e^{- 2 q_{1}}\right) .
\left( V , 1 , 1 \right) = \left( \begin{array}{c c c}{{1 + \frac{V^{2}}{2 a}}}&{{- \frac{V^{2}}{2 a}}}&{{- V^{T}}}\\{{\frac{V^{2}}{2 a}}}&{{1 - \frac{V^{2}}{2 a}}}&{{- V^{T}}}\\{{- V}}&{{V}}&{{{\bf 1}_{n - 2}}}\\ \end{array}\right) \quad ,
\omega \mid U_{\pm}= \int_{N}\frac{- 2 i S_{0}}{\left( 1 + z_{\pm}( \vec{x}) \overline{{{z}}}_{\pm}( \vec{x}) \right)^{2}}\delta \overline{{{z}}}_{\pm}( \vec{x}) \wedge \delta z_{\pm}( \vec{x}) d \mu ( \vec{x}) \, .
D_{\alpha a}D_{\beta b}= - ( \gamma^{\mu})_{[ \alpha \beta ]}\Omega_{[ a b ]}\partial_{\mu}+ \Omega_{[ a b ]}D_{( \alpha \beta )}+ ( \gamma^{i})_{[ a b ]}D_{i ( \alpha \beta )}+ ( \gamma^{i j})_{( a b )}D_{[ i j ] [ \alpha \beta ]},
\omega_h^{n+1}= \sum_{i = 0}^p\alpha_i\tau^i (\Pi L^i u(t^n)- L_h^i u_h^n) + \sum_{i = p+1}^s\alpha_i \tau^i \left(L_h^{i-(p+1)}\Pi L^{p+1}u(t^n) - L_h^{i}u_h^n\right),
\|v\|_{C^{(\alpha+k)/2+1, \alpha+k+2}}\le K_3 P_2(A_{\alpha+k})\big(\|f\|_{C^{(\alpha+k)/2,\alpha+k}}+\|v_0\|_{C^{\alpha+k+2}(\overline D)}&\; +\|\phi\|_{C^{(\alpha+k)/2+1,\alpha+k+2}(\overline{S_T})}\big),
i G^{[ s ]}( x , x^{\prime})_{m n}^{a b}= \int_{0}^{\infty}\, d T \int_{x ( 0 ) = x^{\prime}, x ( T ) = x}D x ( t ) e x p \left({\frac{- 1}{4}}\int_{0}^{T}d t \, \dot{x}^{2}\right) \Phi_{m n}^{[ s ]}[ C^{x x^{\prime}}] U [ C^{x x^{\prime}}]^{a b},
\frac{1}{n^{\gamma}}\, \bigg\| \frac{f}{w_t}\bigg\|_{\ell^{r}}< \frac{1}{n^{\frac{1}{r}}}\, \bigg\| \frac{f}{w_t}\bigg\|_{\ell^{p}}\iff n > \tilde{n}:= \Bigg(\frac{\|\frac{f}{w_{t}}\|_{\ell^r}}{\|\frac{f}{w_{t}}\|_{\ell^p}}\Bigg)^{\frac{pr}{p-r}}\,.
\mathbb{\tilde{E}}_{t_{n}}^{X^{n}}[\eta]=\sup \limits_{Q\in \Sigma}\mathbb{E}_{t_{n}}^{Q,X^{n}}\left[ \eta \right] =\sup_{Q\in \Sigma}\mathbb{E}_{P^{Q}}\left[ \left. \eta \right \vert X_{t_{n}}=X^{n}\right] ,
\frac{1}{N}\sum_{i=1}^{N}f(x_{i})-\int_{\overline{I}^{s}}f(u)du=\sum_{\emptyset\ne u\subseteq D}(-1)^{|u|}\int_{[0,1]^{|u|}}disc(x_{u},1)\frac{\partial^{|u|}}{\partial x_{u}}f(x_{u},1)dx_{u},
\sum_{(v_\nu)\in \mathcal{U}_\gamma}\prod_{\nu\in S}\min(1,e^{v_\nu}) \lesssim \sum_{M=0}^\infty \sum_{k=1}^{\#S-1}\begin{pmatrix}\#S-1 k\end{pmatrix}e^{-M}(M^{k+1}+M^{\#S}) \lesssim \sum_{M=0}^\infty e^{-M}M^{\#S}< \infty
\ell_R(J_n/\omega_{\beta n}\cap J_n)=\sum_{j=0}^{s-1}\ell_R(L_n^j/L_n^{j+1})=\sum_{j=0}^{s-1}\ell_{R_{j+1}}\left(L_n^jR_{j+1}/(L_n^jR_{j+1})\cap m_{R_{j+1}}^{\beta n}\right).
a^{2}\nabla_{\nu}b^{2}\geqslant a^{2}\sharp_{\nu}b^{2}+ \sum_{k=0}^{\infty}r_{k}\big[ a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}- a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big]^{2}
p_{y}(\textbf{z}) = P\{\delta = 1 \vert \textbf{Z}= \textbf{z}, Y=y\}=E(\delta \vert \textbf{Z}=\textbf{z},Y=y), begin{align*}5pt]q_{y}(\textbf{x}) = P\{\delta = 1 \vert \textbf{X}= \textbf{x}, Y=y\}=E(\delta \vert \textbf{X}=\textbf{x},Y=y).
\log \det ( I + KX^{-1}) \equiv \min_{Z \succ 0}\, \log \det (Z^{-1}) & \ \mbox{\rm s.t.}\ \left[ \begin{array}{@{}c@{\,\,}c@{}}I - Z K^{1/2}K^{1/2}X + K \end{array}\right] \succeq 0 \,.
\chi^m_1 \to g^{m \bar{n}}\frac{\partial}{\partial (d\bar{z}^{\bar{n}})}, \quad \chi^m_2 \to d z^m, \quad \bar{\chi}^{\bar{m},1}\to d\bar{z}^{\bar{n}}, \quad \bar{\chi}^{\bar{m},2}\to g^{n \bar{m}}\frac{\partial}{\partial (dz^n)}
{\varphi_0}^{\pm}={\varphi_{01}}^{\pm}+i{\varphi_{02}}^{\pm}, begin{align*}1mm]{\varphi_j^1}^{\pm}={\varphi_{j1}^{1}}^{\pm}\cos{j\theta}+i{\varphi_{j2}^{1}}^{\pm}\sin{j\theta}, begin{align*}1mm]{\varphi_j^2}^{\pm}={\varphi_{j1}^2}^{\pm}\sin{j\theta}+i{\varphi_{j2}^2}^{\pm}\cos{j\theta},
(1+a)\big(Q_{\Sigma}(v, v)+ C''\|v\|^2_{L^2(\Sigma)}\big) \geq\ a\int_{\Sigma}|\nabla v|^2 + \Big(Q_{\Sigma}(v, v) - \vartheta\int_{\Sigma}v^2/\rho^2 + C''\|v\|^2_{L^2(\Sigma)}\Big) \geq a\int_{\Sigma}|\nabla v|^2
\begin{array}{l}{\Phi_{j}\bar{\Phi}_{i}= \delta_{i j}+ q R_{k i j l}\bar{\Phi}_{l}\Phi_{k},}\\{\Phi_{l}\Phi_{k}= q^{- 1}R_{j i k l}\Phi_{j}\Phi_{i}, \qquad i , j = 1 , 2 , . . . , N ,}\\ \end{array}
\big\langle \big[A^{\frac{1}{2}}_{p'}\big]^* \Phi^{-1}A^{-\frac{1}{2}}_{p}f , v \big\rangle_{W^{-1 , p}_{\sigma}, W^{1 , p^{\prime}}_{0 , \sigma}}= \big\langle A_2^{- \frac{1}{2}}f , A_2^{\frac{1}{2}}\big\rangle_{L^2_{\sigma}, L^2_{\sigma}}= \big\langle \Phi^{-1}f , v \big\rangle_{W^{-1 , p}_{\sigma}, W^{1 , p^{\prime}}_{0 , \sigma}}.
\mathcal{V}^{\pm}_1 \left(\frac{n_{1}^2 n_{2}}{(p_1q)^3r}\right) = \frac{1}{2\pi^{3/2}}\bigg\{G_{0}\left(\frac{n_{1}^2 n_{2}}{(p_1q)^3r}\right) \mp i G_{1}\left(\frac{n_{1}^2 n_{2}}{(p_1q)^3r}\right)\bigg\},
\sum_{r \geq 0}\nu^{r}\sum_{u + v = r \atop u , v \geq 0}\rho_{u}( \rho_{v}( F G ) H ) = \sum_{r \geq 0}\nu^{r}\sum_{u + v = r \atop u , v \geq 0}\rho_{u}( F \rho_{v}( G H ) ) , \quad \forall F , G , H \in N [ \nu ] .
I_f(u) =\int_{M\setminus\{p\}}\eta r^{2-n}\Delta_g(\eta r^{2-n})\,dv^g+\int_{M\setminus\{p\}}u\Delta_g(\eta r^{2-n})\,dv^g &{}+\int_{M\setminus\{p\}}\eta r^{2-n}\Delta_gu\,dv^g+\int_{M}uP_fu\,dv^g.
< x_{f}^{\mu}| U_{\mathrm{p h y s i c a l}}( \tau_{f}, \tau_{i}) | x_{i}^{\mu}> = \int_{( \infty )}\frac{d^{D}p^{\mu}}{( 2 \pi )^{D}}e^{i \Delta x \cdot p}\int_{- \infty}^{\infty}\frac{d \gamma}{2 \pi}e^{- \frac{1}{2}i \gamma ( p^{2}+ m^{2})}\ .
\begin{aligned}\sum_{n\geq 1}\int_{I_n}\left(\frac{du|_{I_n}}{d\ss_n}\right)^2d\ss_n =\sum_{n\geq 1}\int_{I_n}\left(\frac{du|_{I_n}}{d\tilde{\ss}_n}\right)^2d\tilde{\ss}_n &=\sum_{n\geq 1}\int_{I_n}\left(\varphi'\circ\ss\right)^2d\tilde{\ss}_n &\leq \int_I (\varphi'\circ \ss)^2d\ss &<\infty.\end{aligned}
H \alpha^{\prime}P = \frac{[ H \dot{r}\cos H t + H^{2}\sqrt{b}\alpha^{\prime}r \sin H t ]^{2}- H^{2}r^{2}( 1 + H^{2}r^{2}) [ ( 1 + H^{2}r^{2}) \cos^{2}H t - H^{2}r^{2}]}{2 [ H^{2}r \dot{r}\sin H t + H \sqrt{b}\alpha^{\prime}\cos H t ] \sqrt{1 + H^{2}r^{2}}\sqrt{( 1 + H^{2}r^{2}) \cos^{2}H t - H^{2}r^{2}}}.
\ln \left[F(\bar N)\right] =\ln h_{\bar N}+ 2 \cdot 3^{\bar N}\ln |x_0| +\sum_{j=0}^{\bar N-1}\ln h_{j}^{2 \cdot 3^{\bar N-1-j}}&= \ln h_{\bar N}+ 2 \cdot 3^{\bar N}\ln |x_0| + \frac{2}{3}\cdot \sum_{j=0}^{\bar N -1}3^{\bar N-j}\ln h_{j}.
V_{\tilde{A}}: = V_{\tilde{A}s}= V_{\tilde{A}t}=V_{\tilde{C}s}=V_{\tilde{D}t}, V_{\tilde{B}}: = V_{\tilde{B}s}= V_{\tilde{B}t}=V_{\tilde{D}s}=V_{\tilde{C}t}.
\widetilde{G}_n(x, n +z) = \frac{n^{nd}e^{-nd}}{(n!)^d}\det(e^{(n-1) L_M (1 + z_j/n - x_i/n)-(z_j - x_i)}))_{i, j= 1}^d + O(n^{-\alpha}) \sim (2\pi n)^{-d/2}\det\left(\sum_{p=0}^M \frac{1}{p!}\left[(n-1)\sum_{k=2}^M \frac{(x_i - z_j)^k}{k n^k}+ \frac{x_i - z_j}{n^2}\right]^p\right)_{i, j= 1}^d + O(n^{-\alpha}).
\lim_{n\to\infty}\mathbb E_x\left[h\circ X_n \mid \tau>n\right] = \lim_{n\to\infty}\frac{\mathcal P^n h(x)}{\mathcal P^n(x,M)}= \int_M h(y) \mu(\d y), \mbox{for $\mu$-almost every $x\in M$}.
\operatorname{diam}(G_{X}) \leq\lbrack1+f(a_{1})]+[f(a_{1})+f(a_{2})]+\cdots+[f(a_{t-1})+f(a_{t})]+[f(a_{t})+1] & =2{\textstyle\sum_{i=1}^{t}}f(a_{i})+2.
\Big\|\Omega_{\mu}(\pi(c)) |D_{\mu}|^{2-p}e^{-s^2D_0^2}-\Omega_{\mu}(\pi(c))|D_1|^{2-p}e^{-s^2D_0^2}\Big\|_1 \leq\Big\|\Omega_{\mu}(\pi(c))|D_1|^{-p}e^{-s^2D_0^2}\Big\|_1\Big\|(|D_{\mu}|^{2-p}-|D_1|^{2-p})\cdot |D_1|^{p-2}\Big\|_{\infty}=O(s^{-1}).
n_1^p(t) = \sqrt{{{\rm{2}}\mathord{\left/{\vphantom{{\rm{2}}T}}\right. \kern-\nulldelimiterspace}T}}w_1^i\left( t \right)\cos \left({2\pi f't +{{\varphi '}_n}}\right) - \sqrt{{{\rm{2}}\mathord{\left/{\vphantom{{\rm{2}}T}}\right.\kern-\nulldelimiterspace}T}}w_1^q\left( t \right)\sin \left({2\pi f't +{{\varphi '}_n}}\right)
b_1 = 1 b_2 = -2a_1 b_3 = -6a_2 + 12a_1^2 b_4 = -24a_3 + 120a_1a_2 - 120 a_1^3 b_5 = -120a_4 + 720a_1a_3 + 360a_2^2 -2520a_1^2a_2 + 1680a_1^4
u_t -u_{xxt}+(b+1)u^ku_x+(2c-3k)u^{k-1}u_xu_{xx}-u^ku_{xxx}+(3k-9a-b-2c)u^{k-2}u_x^3+6au^{k-2}u_xu_{xx}^2+3au^{k-2}u_x^2u_{xxx}=0.
< 0 | T_{\mu \nu}( x ) | 0 > = \mathop{{\sum}^{\prime}}_{\gamma}{\frac{\Delta_{\gamma}( x , x )^{1 / 2}}{\pi^{2}s_{\gamma}( x , x )^{4}}}\; t_{\mu \nu}( x ; \gamma ) + O ( s_{\gamma}( x , x )^{- 3}) .
\|\nabla u\|_{L^\infty(\Omega_\delta)}=\left[1+o(1)\right]\begin{cases}\displaystyle \frac{\mathcal{R}_o}{\mathcal{C}_{12}}\frac{1}{\delta^{1/2}}, d=2 begin{align*}8pt]\displaystyle \frac{\mathcal{R}_o}{\mathcal{C}_{12}}\frac{1}{\delta |\ln \delta|}, d=3\end{cases}\mbox{for}~\delta\ll 1,
F_4 :=2E C_R \int_{t_0}^{u \wedge \tau_{nR}}|x_s^n-x_{\kappa(n,s)}^n|I_{B(R)}ds \leq f(R) K \sup_{t_0 \leq t \leq t_1}E|x_t^n-x_{\kappa(n,t)}^n|
f \mathcal R_{11}g = \sum_j f_{<j-2}\mathcal R_{11}g_j+ \sum_j f_j \mathcal R_{11}g_{<j-2}+ \sum_j f_j \mathcal R_{11}\tilde g_j, & (\mathcal R_{11}f) g = \sum_j (\mathcal R_{11}f)_{<j-2}g_j+\sum_j (\mathcal R_{11}f)_j g_{<j-2}+ \sum_j (\mathcal R_{11}f)_j \tilde g_j.
\lim_{\lambda \to \infty}\left[{\rm Re}\, \Gamma (\varphi , \lambda)\right]_{\lambda - dep}\;=\; \varphi_1^2 \lim_{\lambda \to \infty}\frac{\pi}{2}\, \sum_{n_1}\,\frac{\tau_2^2 \lambda^2}{(\lambda^2 + \tau_2^2n_1^2)^{\frac{3}{2}}}\,
G_0 =G_{0,0}-G_{0,0}v\Bigl(M_0^\dagger +Q-m_1^\dagger m_3m_1^\dagger &-m_1^\dagger M_2(M_0^\dagger +Q)-(M_0^\dagger +Q) M_2 m_1^\dagger \Bigr)v^*G_{0,0}&-G_{0,0}vm_1^\dagger v^*G_{0,2}-G_{0,2}vm_1^\dagger v^*G_{0,0}.
(x_{1}+x_{2}+\cdot\cdot\cdot+x_{m})^{n}=\sum_{k_{1}+k_{2}+\cdot\cdot\cdot+k_{m}=n;k_{1},k_{2},\cdot\cdot\cdot,k_{m}\ge0}(\begin{matrix}n\\k_{1},k_{2},...,k_{m}\end{matrix})\prod_{t=1}^{m}x_{t}^{k_{t}},
\gamma = \mathrm{res}_v\left(\left(L^\mathrm{AH}(\varepsilon(v))\frac{d L^{\mathrm{AH}}(\eta(v))}{dv}- L^{\mathrm{AH}}(\varepsilon(v))d_{\mathrm{log}}(B(v)) + L^{\mathrm{AH}}(\eta(v)) d_{\mathrm{log}}(A(v))\right)\left(\frac{1}{z(v)^p-1}\right)\right),
\Sigma_{a_1,\ldots,a_p;\; b_1,\ldots,b_q;c}^{\; i_1,\ldots,i_p; \;j_1,\ldots,j_q}(u)\equiv\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right)}\frac{u^j}{j^c}[S_{a_1}(j\!-\!1)]^{i_1}\ldots [S_{a_p}(j\!-\!1)]^{i_p}\;[S_{b_1}(2j\!-\!1)]^{j_1}\ldots [S_{b_q}(2j\!-\!1)]^{j_q},
\frac{1}{N}\mathrm{tr\,}A^{k}~ =~\,\frac{1}{N^{1+k/2}}\sum_{r=1}^{k}\sum_{\underline{i}:\,\rho (\underline{i})=r}\,X_{N}(\underline{i}) & =~ \frac{1}{N^{1+k/2}}\sum_{r=1}^{k}\sum_{s=0}^k \sum_{\substack{\rho(\underline{i})= r \; \sigma(\underline{i}) = s}}\,X_{N}(\underline{i}) .
\begin{array}{lll}{\mathcal{F}}_{A_\pm, \lambda_\pm,f_\pm}(v) := \displaystyle \int\limits_{\Omega \cap\{v>0\}}\left \{\dfrac{1}{2}|\nabla v(x)|^2_{A_+}+ \lambda_{+}(x) + f_{+}v \right \}\,dx &+ \displaystyle \int\limits_{\Omega \cap\{v\le0\}}\left \{\dfrac{1}{2}|\nabla v(x)|^2_{A_-}+ \lambda_{-}(x) + f_{-}v \right \}\,dx,\end{array}
L=\big(\dot{\bar\psi}\big)^i m_{ij}\big(\dot{\psi}\big)^j + \big(\dot{\bar\psi}\big)^i c_{ij}\big({\psi}\big)^j- \big({\bar\psi}\big)^i \tilde c_{ij}\big({\dot\psi}\big)^j - \big({\bar\psi}\big)^i \tilde r_{ij}\big({\psi}\big)^j,
a(u-u_{h},u-u_{h}) -\l b(u-u_{h},u-u_{h})=a(u_{h},u_{h})-\l b(u_{h},u_{h}) & =a(u_{h},u_{h})-a_{h}(u_{h},u_{h})+\l_{h}^{(j)}b_{h}(u_{h},u_{h})-\l b(u_{h},u_{h}) & =a(u_{h},u_{h})-a_{h}(u_{h},u_{h})+(\l_{h}^{(j)}-\l)b_{h}(u_{h},u_{h})+\l[b_h(u_{h},u_{h})-b(u_{h},u_{h})],
\lambda_{1}=m_{12}+m_{13}+m_{14}+n_{14}+n_{13}+n_{12}, \lambda_{2}=m_{23}+m_{14}+m_{13}+m_{24}+n_{24}+n_{23}+n_{13}+2n_{12}+n_{14}, \lambda_{3}=m_{34}+m_{14}+m_{24}+n_{23}+n_{13}+n_{12}, \lambda_{4}=n_{34}+n_{14}+n_{24}+n_{23}+n_{13}+n_{12},
\begin{array}{@{}l@{}}{\displaystyle \big|\tau\sum_{k=1}^{m}\mathrm{Re}\big[ V_{2}^{k}(\partial_{\tau}{\theta_{\Psi}^{k}})\big]\big|\leq C\big(h^{2r}+\tau^{4}\big)+C \|\theta_{\Psi}^{m}\|_{\mathcal{L}^2}^{2}+ C\tau \sum_{k=1}^{m-1}{\|\theta_{\Psi}^{k}\|_{\mathcal{L}^2}^{2}}.}\end{array}
\frac{1}{2}\partial^{2}J( \alpha^{i}\varphi_{i})h_{1}h_{2}= k_{\alpha^{i}\varphi_{i}}^{\frac{2-n}{n}}[\int L_{g_{0}}h_{1}h_{2}-c_{n}n(n+2)\int \sum_{i}\tilde\varphi_{i}^{\frac{4}{n-2}}h_{1}h_{2}] & +o_{\varepsilon}(\Vert h_{1}\Vert \Vert h_{2}\Vert),
\begin{gathered}PUP=Q_{LC}U_{LC}Q_L + Q_RU_{CR}Q_{CR}+ Q_{LC}U_{LC}U_{CR}Q_{CR}, (P_L-Q_L)UP = (P_L-Q_L)U_{LC}Q_L + (P_L-Q_L)U_{LC}U_{CR}Q_{CR}, PU(P_R-Q_R) = Q_RU_{CR}(P_R-Q_R) + Q_{LC}U_{LC}U_{CR}(P_R-Q_R), PU(P_L-Q_L) = Q_{LC}U_{LC}(P_L-Q_L), (P_R-Q_R)UP = (P_R-Q_R)U_{CR}Q_{CR}.\end{gathered}
\frac{\partial}{\partial t}k^2 = (k^2)''-2\langle( \nabla_SH)^\perp,(\nabla_SH)^\perp\rangle+2k^4 &-2{\rm Ric}_g(H,H)+4k^2{\rm Ric}_g(S,S) +2\widehat{Rm}(\widehat H,S,H,S) & +2{\rm Ric}_g(S,S){\rm Ric}_g(H,H) -2(\nabla^g_S({\rm Ric}_g))(S,H).
\phi_{1 2}^{+}\, \widetilde T_{1 2 , 2 3}^{3 D}\, \phi_{2 3}\, = \, ( - 2 i \pi )^{2}\, \left[ \overline{{\Phi}}_{1 2}\, T_{1 2 , 2 3}\, \Phi_{2 3}\right]_{3 1}^{\delta \delta}\, \beta_{1}
det \left|\begin{array}{ccc}Q(\lambda-3i/2) Q(\lambda \pm i/2) Q(\lambda+3i/2) P(\lambda-3i/2) P(\lambda \pm i/2) P(\lambda+3i/2) R(\lambda-3i/2) R(\lambda \pm i/2) R(\lambda+3i/2) \end{array}\right| = T^{\pm}(\lambda).
\dfrac{d}{dt}\|U\|^2_{I_2 \otimes P_x}+ \vec U^T (I_2 \otimes B){\bf A}\vec U=({\bf D_x}\vec U)^T (I_2 \otimes P_x){\bf A}\vec U-({\bf A}\vec U)^T (I_2 \otimes P_x){\bf D_x}\vec U,
g^{-1}= (g^{ij})= \frac{1}{\left[ a(y^1)^2+\tilde{b}^(y^2)^2\right]^3}\begin{pmatrix}(y^1)^4\left\lbrace a (y^1)^2+3\tilde{b}(y^2)^2\right\rbrace 2\tilde{b}(y^1)^3 (y^2)^3 2\tilde{b}(y^1)^3 (y^2)^3 \frac{(y^1)^2}{2\tilde{b}}\left\lbrace a^2(y^1)^4+3\tilde{b}^2(y^2)^4\right\rbrace \end{pmatrix}.
\sup\{\Re\lambda:\lambda\in\operatorname{Sp}(\mathcal{L}_{\alpha}) \}=\sup \{\Re\lambda:\lambda\in \operatorname{Sp}(\hat{\mathcal{L}}) \}&=\max(\alpha^2-c\alpha,\epsilon\alpha^2-c\alpha-\kappa e^{-\kappa}) &=\alpha^2-c\alpha.
u(x+h)+u(x-h)-2u(x) \leqslant g(y^+)+\Phi^{\Omega}_L(y^+,x+h)+g(y^-)+\Phi^{\Omega}_L(y^-,x-h)-2g(\bar{x})-2A^{\Omega}_{T(x)}(\bar{x},x) \leqslant \{g(y^+)+g(y^-)-2g(\bar{x})\}+\{\Phi^{\Omega}_L(y^+,x^+)+\Phi^{\Omega}_L(y^-,x^-)\}+c_1|h|^2.
\partial{}_{\mu}\overline{F}^{\prime a\mu \nu}-\overline{f}^{a}{}_{cb}\overline{F}^{\prime c\mu \nu}\overline{A}^{\prime b}{}_{\mu}{}+\overline{A}^{\prime b}{}_{\mu}\overline{R}^{\prime a}{}_{b}{}^{\mu \nu}-\overline{C}^{a}{}_{d\mu}\overline{F}^{\prime d\mu \nu}=0,
e^{ih_2'}=(w^{(2)}_1,x^{(2)}_1)(w^{(2)}_2,x^{(2)}_2)e^{ih_2''}, &=(w^{(2)}_1,x^{(2)}_1)(w^{(2)}_2,x^{(2)}_2)(y^{(2)},z^{(2)})e^{ih_3},
\begin{pmatrix}12\lambda_i(\lambda_i-4)^2 -12\lambda_i(\lambda_i-4) 0 -12\lambda_i(\lambda_i-4) \frac{29}{2}\lambda_i 5\lambda_i 0 5\lambda_i 20\lambda_i+40\end{pmatrix},\begin{pmatrix}3\lambda_i(\lambda_i-4) 0 0 0 2\lambda_i(\lambda_i-1) 15\lambda_i 0 15\lambda_i 205 \end{pmatrix}.
\sum_{k=0}^m\beta_k\left[(f+g)(x^k)-(f+g)(x_*)\right]\le \frac{1}{2}(\|x^0-x_*\|^2-\|x^{m+1}-x_*\|^2)+(f+g)(x^0)-(f+g)(x^{m+1}) \le \frac{1}{2}\|x^0-x_*\|^2+(f+g)(x^0)-(f+g)(x_*).
\pi\circ F_{r_n(x)}^{n_j+nq}(\tilde{a}_j\circ r_n(x)) = f_{r_n(x)}^{n_j+nq}\circ\pi (\tilde{a}_j\circ r_n(x)) = f_{r_n(x)}^{n_j+nq}(a_j\circ r_n(x)) & = \pi\circ\sigma_j\left(a^{n,j}(x)\right).
\partial_{i}x^{j}= \tilde{J_{i}^{\alpha}}v_{\alpha}x^{j}= \partial_{i}(x^{j}) +\underbrace{\tilde{J_{i}^{\alpha}}O_{\alpha}{}^{\beta}(x^{j})J_{\beta}{}^{k}}_{L_{i}{}^{k}(x^{j})}\partial_{k}.
\partial^2_y \left(\frac{1}{u^s_y +{u}_y}\right) = - \partial_y\left(\frac{u_{yy}^s +{u}_{yy}}{(u^s_y +{u}_y)^2}\right)= -\frac{u_{yyy}^s +{u}_{yyy}}{(u^s_y +{u}_y)^2}+ 2 \left(\frac{u_{yy}^s +{u}_{yy}}{(u^s_y +{u}_y)}\right)^2 \frac{1}{u^s_y +{u}_y},
\begin{array}{rl}\partial_t\theta h+\partial_b(\theta h v^b)= \mathfrak{M}, \partial_t(\theta hv_a)+\partial_b(hv_av^b)+\theta h\partial_aw= -{\cal K}(h,\theta)|\boldsymbol{v}|v_a, a=1,2. \end{array}
R_r =-{\rm tr}(R_{NN}T_{r-1})N=\sum_i (R(N,T_{r-1}e_i)e_i)^{\bot}, H_r =\sigma_{r-1}N, S_r ={\rm tr}(AT_r)N=(r+1)\sigma_r N, W_r ={\rm div}({\rm div}T_{r-1})N.
J^{\mu \nu}= E g^{\mu \nu}, \ \ \j_{1}^{\mu}={\frac{\nabla^{\mu}\phi}{( \nabla \phi )^{2}}}, \ \ \j_{2}^{\mu}= j_{\cal R}^{\mu}+ V{\frac{\nabla^{\mu}\phi}{( \nabla \phi )^{2}}}, \ \ \j_{3}^{\mu}= E j_{1}^{\mu},
u_{i}^{\u}= \frac{1}{\sqrt{1 + t \bar{t}}}\left( \begin{array}{c c}{t}&{1}\\{- 1}&{\bar{t}}\\ \end{array}\right) , \u_{u}^{\i}= \frac{1}{\sqrt{1 + t \bar{t}}}\left( \begin{array}{c c}{\bar{t}}&{- 1}\\{1}&{t}\\ \end{array}\right) ,
= \; 2^{N}\prod_{a = 1}^{N}\sum_{n_{a}= 0}^{\infty}\frac{( \beta^{( a )})^{2 n_{a}}}{( 2 n_{a}) !}\prod_{b = 1}^{N}\sum_{q_{b}= 0}^{2 n_{b}}{\binom{2 n_{b}}{q_{b}}}E^{\mu}( \{q , n \}) \; ,
\lim_{n \to \infty}\int_\Omega \sum_{l=1}^d w_{ln}\, \overline{\partial_l \varphi}\, (\pi_i - \varepsilon_n \chi^{(n)}_i)= \int_\Omega \sum_{l=1}^d w_l \, \overline{\partial_l \varphi}\, \pi_i.
\mathrm{\frac{1}{2}}\epsilon_{a b}\mathbf{s}_{m}^{b}\mathbf{s}_{m}^{a}A^{i}= R_{\alpha_{0}}^{i}B^{\alpha_{0}}+ \mathrm{\frac{1}{2}}\epsilon_{a b}R_{\alpha_{0}, j}^{i}R_{\beta_{0}}^{j}C^{\beta_{0}b}C^{\alpha_{0}a},
\frac{\log L_{n}}{\log q}=n-k + \sum_{i=1}^{k}\left(a_{2i}+1\right) = n+\sum_{i=1}^{k}a_{2i}= n-\sum_{i=1}^{k}a_{2i-1}+\sum_{i=1}^{k}a_{2i-1}+ \sum_{i=1}^{k}a_{2i}= n-n_{k}+s_{2k},
\hat \imath(c_k\langle \alpha_k \rangle, L_{g_0}) \leq (2+\rho_k q)c_kl_{g_0}(\alpha_k) \leq (2+\rho_k q)c_kl_{g_1}(\alpha_k) & \leq \hat \imath(c_k\langle \alpha_k \rangle, L_{g_1})+\rho_kqc_kl_{g_1}(\alpha_k)
\star Q_\Omega u:=(-1)^{\ell-1}R_\Omega(\star\,u),\ \star L_\Omega u:=K_\Omega(\star\,u)\mbox{for an $\ell$-form}u; &T_\Omega u:=Q_\Omega^*u; &\star S_\Omega u:=(-1)^{\ell-1}T_\Omega(\star\,u),\ \quad\mbox{for an $\ell$-form}u.
\begin{aligned}S_1 = \{(2,0,0), (1,1,0), (0,2,0), (1,0,1) \}S_2 = \{(2,0,0), (1,1,0), (0,2,0), (0,1,1) \}S_3 = \{(1,0,0), (0,1,0), (0,0,1), (0,0,0) \}\end{aligned}
\Phi_{c}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=\underset{V_{c}\in S_{c}}{}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}V_{c}-\frac{\lambda}{2}V_{c}^{\top}Q_{c}V_{c}.
\begin{aligned}\rho-1\in{\widetilde{L}^\infty (\mathbb{R}^+;\widetilde{B}^{s,s+\alpha-1})}, u\in{\widetilde{L}^\infty (\mathbb{R}^+;\dot{B}^s_{2,1})}\cap{L^1(\mathbb{R}^+;\dot{B}^{s+\alpha}_{2,1}}).\end{aligned}
\begin{array}{lll}h_1(d) = d-R_1+ x^* = d-R_1 + \frac{R_0^2 + R_1^2 - d^2}{2 R_1}\mbox{and}h_2(d) = R_0 - x^* = R_0 - \frac{R_0^2 + R_1^2 - d^2}{2 R_1}.\end{array}
\delta_{\epsilon , \epsilon^{*}}\langle X \rangle = - \frac{1}{2 \pi i}\oint_{C}d z \epsilon ( z ) \langle T ( z ) X^{\prime}\rangle - \frac{1}{2 \pi i}\oint_{\bar{C}}d z \epsilon ( z ) \langle T ( z ) X^{\prime}\rangle ,
\stackrel{\circ}{T}{^a}= \stackrel{\circ}{D}e^{a}\equiv d e^{a}+ \stackrel{\circ}{\omega}{_{\cdot b}^{a}}\wedge e^{b}, \stackrel{\circ}{R}{^{a b}}= d \stackrel{\circ}{\omega}{^{a b}}+ \stackrel{\circ}{\omega}{^a_{\cdot c}}\wedge \stackrel{\circ}{\omega}{^{c b}},
\tilde{\varepsilon}_1(x) = 2Q(x) + \{\tau(\xi(x)) + Q(x)\}\circ \nabla_{\partial \Omega}(\tau - \nu)(\xi(x)) \circ (\tilde{x}- a), &\tilde{\varepsilon}_2(x) = -2Q(x) \circ \left(\dfrac{\tilde{x}-a}{\tilde{r}}\otimes i_x\left(\dfrac{\tilde{x}-a}{\tilde{r}}\right) \right).
\bigl((\rho' \circ \rho) \otimes_0 (\varphi' \circ \varphi)\bigr)_{\sigma}^{\sigma''}= \sum_{\sigma' \in \Sigma'}(\rho' \otimes_0 \varphi')_{\sigma'}^{\sigma''}\circ (\rho \otimes_0 \varphi)_{\sigma}^{\sigma'}.
\widetilde{(Q^{n,\pm})}_l = \frac{i\varepsilon^2}{\alpha \beta_l}\int_0^\tau e^{-i \beta_l^\pm w}f_l^n(w)dw \mp (p_l^\pm \widehat{(f(\psi(t_n)))}_l + q_l^\pm \widehat{(g(\psi(t_n)))}_l),
\begin{array}[c]{l}Y_{1}\left( t\right) =p\left( t\right) X_{1}\left( t\right) , Z_{1}(t)=(1-p(t)\sigma_{z}(t))^{-1}p(t)(\delta\sigma(t,\Delta)-\sigma_{z}(t)\Delta(t))I_{E_{\epsilon}}(t)+K_{1}(t)X_{1}\left( t\right) .\end{array}
\tilde{K}_k^{(1)}(Y)= \sum_{j\in I_p}\sum_{l \in I_p}\tilde K_{kjl}^{(1)}Y_{j}Y_{l}, \tilde{K}_k^{(3)}(Z)=\sum_{j \in J_p}\sum_{l \in J_p}\tilde K_{kjl}^{(3)}Z_{j}Z_{l},
{\cal E}(r){\cal T}=- 2\kappa_{(10)}^2 \int d^{10}x \int d^{10}{\tilde x}\ \Bigl[ 4 j_{\Phi}\Delta{\tilde j_{\Phi}}- j_{C}\Delta{\tilde j_{C}}+T_{\mu\nu}\Delta^{\mu\nu,\rho\tau}{\tilde T}_{\rho\tau}\Bigr]
\frac{N'}{\mu}(2 -\frac 2{x_i^2})+2\sum_{j\neq i}\left(\frac 1{x_i-x_j}-\frac 1{x_i+x_j}\right) + \sum_{a=0,1}\frac2{x_i-\lambda_a}-\frac2{x_i+\lambda_a}
\sigma_a(b)=(\lambda_a(1))^{-}\circ b=(-1)^a+(-1)^{{(-1)}^a}b=\begin{cases}1-b, \quad{\rm if}\; a\;{\rm is\; even,}-1-b=(2^m-1)(1+b), \quad{\rm if}\; a\;{\rm is\; odd}\end{cases}
| v |^2_{k-1,w}= \sum\limits_{D\in\mathcal{T}_h}\int_{D}\nabla_{w,k-1,D}v\cdot\nabla_{w,k-1,D}v\{\rm d}x+ h_D^{-1}\langle v_0-v_b, v_0-v_b\rangle_{\partial D},
\mathfrak{X}_m^l([0,T]):=\bigcap_{k=0}^m C^k([0,T];H^{l+m-k}(\Omega)), \mathfrak{Y}_m^l([0,T]):=\bigcap_{k=0}^{[m/2]}C^k([0,T];H^{l+m-2k}(\Omega)),\quad\m,l=0,1,2,\cdots, \mathfrak{Z}([0,T]):=C^2([0,T];H^2(\Omega)).
{\cal{G}}_{k_{\perp}}^{\pm ( \eta ) , s c}( t , t^{\prime}) = -{\frac{\Gamma ( - \nu )}{4 \sqrt{\pi}}}e^{-{\frac{i \pi}{4}}}{\frac{1}{k_{\perp}}}{\frac{\partial \Omega}{\partial k_{\perp}}}S_{0}^{\prime}D_{\nu}\left({\frac{1 + i}{\sqrt{\hbar}}}S_{0}( t ) \right) D_{\nu}\left( -{\frac{1 + i}{\sqrt{\hbar}}}S_{0}( t^{\prime}) \right) .
T_{*}= \frac{4 \pi m_{W}^{2}}{g^{2}\left[ \left( 5 + 8 \frac{m_{W}^{2}}{m_{H}^{2}}\right) \ln \frac{m_{W}}{g^{2}}+ O \left( \ln \ln \frac{m_{W}}{g^{2}}\right) \right]},
\int_{| k | < \frac{\Lambda}{s}}\phi_{s}( 4 ) \phi_{s}( 3 ) \phi_{s}( 2 ) \phi_{s}( 1 ) \biggl < \int_{\frac{\Lambda}{s}< | k | < \Lambda}\phi_{f}( 8 ) \phi_{f}( 7 ) \phi_{f}( 6 ) \phi_{f}( 5 ){\cal P}( k_{8}, k_{7}, \cdots , k_{1}) \biggr >_{0 f},
\frac{1}{\mu_f A_f}\prod_{g \in \Gamma(f)}(1+\mu_g) = \frac{d}{Z}\sum_{S \subseteq \Gamma(f)}\prod_{g \in S}\frac{Z}{d A_g}= \frac{1-\delta}{\mathrm{e}}\sum_{i = 0}^{d}\left( \frac{1}{d}\right)^{i}\binom{d}{i}= \frac{1-\delta}{\mathrm{e}}\left( 1 + \frac{1}{d}\right)^d \le 1 -\delta \enspace.
\widetilde{C}^c{}_{ab}=(K^{-1})^c{}_{d}(K^e{}_{a}K^f{}_{b}C^d{}_{ef}+K^e{}_{a}(X^{-1}_*\rho)_e( K^d{}_{b})-K^e{}_{b}(X^{-1}_*\rho)_e(K^d{}_{a})),