harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bar{\mathcal{V}}_2 (\tau) = e^{\tau \Delta}(\bar{\mathcal{V}}_2 (\tau) ) + \int_0^\tau e^{(\tau - \tau')\Delta}\left( p \hat U_{K_0}^{p-1}\bar{\mathcal{V}}_2 - 2 (\mathcal{V}_2 \nabla \psi_*) + \mathcal{V}_2 \Delta \psi_* + p (U_1^{p-1}- \hat U_{K_0}^{p-1}) \psi_* V_2 + \psi_* \mathcal{G}_2 \right) (\tau') d\tau'.
I_4=C\left|\int^T_0 (v\cdot\nabla v_3|\Delta\psi)_{H^1}dt\right| \leq C\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}^2\|\nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))},
d^* \left( \frac{\beta_t}{\sqrt{t^{-2}+ |\beta_t|^2}}\right) = \frac{d^* \beta_t}{\sqrt{t^{-2}+ |\beta_t|^2}}- \frac{*(\frac{1}{2}\nabla|\beta_t|^2 \wedge *\beta_t)}{(t^{-2}+ |\beta_t|^2)^{3/2}}= 0
\mathbf{w}(0)=\mathbf{w}_0=\begin{bmatrix}g(x_1) \vdots g(x_{N^m-1})\end{bmatrix},\mathbf{w'}(0)=\mathbf w'_0= \begin{bmatrix}h(x_1) \vdots h(x_{N^m-1})\end{bmatrix}.
T^{\rho \sigma}={\frac{\rho_{_\mathrm{S}}}{{\mu_{_\mathrm{N}}}^{2}}}\mu^{\rho}\mu^{\sigma}+{\frac{\rho_{_\mathrm{N}}}{{s_{_\mathrm{S}}}^{2}}}s^{\rho}s^{\sigma}+ \Psi g^{\rho \sigma}\ ,
\begin{array}{l}{{\Psi ( z , \bar{z}) = S_{- 1 / 2}\phi ( z , \bar{z}) ;}}\\{{\bar{\Psi}( z , \bar{z}) = \bar{S}_{- 1 / 2}\phi ( z , \bar{z}) ;}}\\{{F ( z , \bar{z}) = S_{- 1 / 2}\bar{S}_{- 1 / 2}\phi ( z , \bar{z})}}\\ \end{array}
\begin{cases}e^{i\theta\infty}(w,\phi_i)_{L^2(\mathbb{R}^d)}+(\sigma_{i\infty}\nabla w_i,\nabla\phi_i)_{L^2(\mathbb{R}^d)}=0, e^{i\theta\infty}(w,\phi_e)_{L^2(\mathbb{R}^d)}-(\sigma_{e\infty}\nabla w_e,\nabla\phi_e)_{L^2(\mathbb{R}^d)}=0, w=w_i-w_e, \end{cases}
f(z_{1},...,z_{n})=\frac{1}{(2\pi i)^{n}}\int_{\partial D_{1}}\cdot\cdot\cdot\int_{\partial D_{n}}\frac{f(\zeta_{1},...,\zeta_{n})}{(\zeta_{1}-z_{1})\cdot\cdot\cdot(\zeta_{n}-z_{n})}d\zeta_{1}\cdot\cdot\cdot d\zeta_{n}
J_{1}:=\lim_{\delta\to 0}\int_{\Omega}\varepsilon_{n}\big (x-\xi,b(t,t-\delta)\big)(P\varphi)(\xi,t-\delta)d\xi &=\int_{\Omega}\delta(x-\xi)(P\varphi)(\xi,t)d\xi=(P\varphi)(x,t).
\begin{aligned}|\varepsilon_\delta(z_1,\bar{y},t)-\varepsilon_\delta(x_a,\bar{y}_a,t)| \le L_r \gamma |\epsilon_1(\eta_a,t)|+ |\bar{y}(t)-\bar{y}_a(t)| \le \gamma \delta L_r \left(2 + L_r\ ( 3 + L_r)\right) + |\bar{y}-z_2|.\end{aligned}
T=[\begin{matrix}a&*&*&*&*&*&*&*&*\\ 0&0&b&*&*&*&*&*&*\\ 0&0&0&c&*&*&*&*&*\\ 0&0&0&0&0&0&d&*&*\\ 0&0&0&0&0&0&0&0&e\\ 0&0&0&0&0&0&0&0&0\end{matrix}],
d^2(f_{n+1};\langle f_1,...,f_n\rangle)=\min_{t=(t_k)\in{\mathbb R}^n}\Vert f_{n+1}+\sum_{k=1}^nt_kf_k\Vert^2=\frac{\Gamma(f_1,f_2,...,f_{n+1})}{\Gamma(f_1,f_2,...,f_n)},
c_{2 r}^{\nu}( h^{2}) = c_{- 2 r}^{- \nu}( h^{2}) a n d \frac{c_{2 r}^{\nu}( h^{2})}{c_{0}^{\nu}( h^{2})}= \frac{c_{- 2 r}^{- \nu}( h^{2})}{c_{0}^{- \nu}( h^{2})}
|R(\eta)-\tau_0|= \frac{|P_1(\eta)+\tau_0P_2(\eta)|}{|P_2(\eta)|}\le \frac{c^{n'}+|P_1(\eta)-P_1(\l_0)|+|\tau_0||P_2(\eta)-P_2(\l_0)|}{|P_2(\eta)|}\le \frac{c^{n'}+(1+|\tau_0|)L(n')^2|\eta-\l_0|}{|P_2(\eta)|}.
\mathbb{E}V_{n,1}^2 =\sum_{0\le i<j\le M_n-1}\frac{\pi_n([0,i])^2}{\pi_n(i)p_{n,i}\pi_n(j)p_{n,j}}\mu_n^2 +\sum_{i=0}^{M_n-1}\frac{\pi_n([0,i])^2}{\pi_n(i)^2p_{n,i}^2}(\mu_n^2+\nu_n^2) \le (\mu_n^2+\nu_n^2)\alpha_{n,1}^2\le[(\mu_n+\nu_n)\alpha_{n,1}]^2.
\delta I_{C S}= \frac{1}{8 \pi^{2}}\int_{T^{2}}d^{2}x \, \epsilon^{i j}\mathrm{T r}[ G_{m}\partial_{i}G_{m}^{- 1}({\cal U}_{s}^{- 1}\partial_{j}{\cal U}_{s}-{\cal U}_{s}\partial_{j}{\cal U}_{s}^{- 1}) ] .
\theta(x_{j,i})^{k_i}=\left(\theta(v_{j,i})^{x_i^{k_i-1}}\right) \cdots \left(\theta(v_{j,i})^{x_i^2}\right) \left(\theta(v_{j,i})^{x_i}\right) \left(\theta(v_{j,i})^1\right).
T_{c l}^{( m + n )}( z ) = \sum_{i = 1}^{p - 1}{\frac{1 - q_{i}^{2}}{2 h ( z - z_{i})^{2}}}-{\frac{1}{2 \pi h}}\sum_{i = 1}^{p - 3}\left({\frac{1}{z - z_{i}}}+{\frac{z_{i}- 1}{z}}-{\frac{z_{i}}{z - 1}}\right){\frac{\partial S_{c l}^{( 0 , m + n )}}{\partial z_{i}}}.
P_{nm}\left( \left| \alpha_{n}\right\rangle \otimes \left| \alpha_{m}\right\rangle \right) =\left| \alpha_{m}\right\rangle \otimes \left| \alpha_{n}\right\rangle ,\qquad \left| \alpha_{n}\right\rangle \otimes \left| \alpha_{m}\right\rangle \in{\cal H}_{n}\otimes{\cal H}_{m}.
\exp \left\{- \sum_{i , j = 1}^{4}c_{i j}\mathrm{\bf r}_{i}\cdot \mathrm{\bf r}_{j}- \sum_{i , j = 1}^{4}d_{i j}\hat{\mathrm{\bf k}}\cdot ( \mathrm{\bf r}_{i}\times \mathrm{\bf r}_{j}) \right\},
\left({\frac{\dot{R}}{R}}\right)^{2}={\frac{1}{3 6}}( \kappa_{5}^{4}\lambda_{0}^{2}- 3 6 \xi^{2}) R^{-{\frac{2}{3}}k^{2}}\equiv{\frac{8 \pi}{3}}G_{N}( R ) \rho_{Q}( R ) ,
t_{12}c_{23}+c_{12}t_{23}c^{-1}_{12}c_{23}=c_{23}t_{12}+c_{23}c_{12}t_{23}c^{-1}_{12}t_{23}c_{12}+c_{23}t_{12}c^{-1}_{23}c_{12}=c_{12}t_{23}+c_{12}c_{23}t_{12}c^{-1}_{23}
t_3(4n+3) =-3t_3(2n+1)-t_3(2n)=9t_3(n)+3t_3(n-1)-t_3(n)-3t_3(n-1)=8t_3(n), t_3(4n+2) =t_3(2n+1)+3t_3(2n)=-3t_3(n)-t_3(n-1)+3t_3(n)+9t_3(n-1)=8t_3(n-1).
\check{H}_{N}^{( 1 )}= \displaystyle \frac{1}{\check{q}_{N}^{( 0 )}}, \check{H}_{i}^{( 1 )}= \displaystyle \frac{\check{e}_{i}^{( 0 )}\check{H}_{i + 1}^{( 1 )}+ 1}{\check{q}_{i}^{( 0 )}}, i = 1, \dots , N - 1.
{\cal A}(r,v) = V_{26}\int_{0}^{\infty}{{dl}\over{l}}(8\pi^2\alpha'l)^{-p/2}e^{-r^2l/2\pi\alpha'}[l^{-1/2}\eta(i/l)]^{-22}[{{\eta(i/l)}\over{- \Theta_{11}(-iu/\pi,i/l)}}],
\Theta_{p,q}= \underset{n\rightarrow \infty}\limsup \frac{1}{n}\log d(\mathcal{F}_{i}^{n}(q),\mathcal{F}_{i}^{n}(p)) \Delta_{p,q}=\underset{n\rightarrow \infty}\limsup\frac{1}{n}\log d(\mathcal{F}_{i}^{-n}(q),\mathcal{F}_{i}^{-n}(p)).
\mathbf{S}=\mathbf{H}^*\mathbf{U}^*\mathbf{D}\mathbf{U}\mathbf{H}&=\left(\mathbf{U}\mathbf{H}\right)^*\mathbf{D}\left(\mathbf{U}\mathbf{H}\right) &=\mathbf{G}^* \mathbf{D}\mathbf{G},
{}_2F_1\bigg[\begin{matrix}\alpha \beta &\frac12+\frac12(\alpha+\beta)\end{matrix}\bigg|\,z\bigg]={}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12\beta &\frac12+\frac12(\alpha+\beta)\end{matrix}\bigg|\,4z(1-z)\bigg],
\rho(|P_2|) \geq \rho(|P_1|) + (1-\rho(|P_1|)) \Delta_{\{B_2\}}- \varepsilon &= \Delta_{\{B_2\}}+ (1 - \Delta_{\{B_2\}}) \rho(|P_1|) -\varepsilon &\geq \Delta_{\{B_1\}}+ \Delta_{\{B_2\}}- \Delta_{\{B_1\}}\Delta_{\{B_2\}}- 2\varepsilon &\geq 1 - (1 - \Delta_{\{B_1\}}+ \varepsilon)(1 - \Delta_{\{B_2\}}+ \varepsilon) \geq 1 - (1 - (s - \varepsilon))^2.
_1\langle \varepsilon_1|\,_2\langle \varepsilon_2|\,_3\langle \varepsilon_3|\exp [\sum_{r=1}^{3}N^{rr}_{00}(\varepsilon_r^2-2\sqrt{2}\varepsilon_r)+\sum_{r,s=1}^{3}\sum_{n=1}^{\infty}N^{rs}_{0n}\varepsilon_r\phi^{(s)}_n],
3[\frac{\sqrt{\pi}Z_{\kappa}}{2\beta^{2/3}\sigma_{\kappa}^{4}}\frac{(2\kappa)^{-5/2}}{1+\frac{5}{2}\kappa}\frac{\Gamma(\frac{1}{2\kappa}-\frac{5}{4})}{\Gamma(\frac{1}{2\kappa}+\frac{5}{4})}-1]
r_{k}=\log_2\Biggl(1+\frac{P\mathbf{g}_{k}^{\dagger}(\mathbf{h}_{k}^{\mbox{\tiny{s}}})^{\dagger}\mathbf{h}_{k}^{\mbox{\tiny{s}}}\mathbf{g}_{k}}{\sigma_{k}^{2}+{\displaystyle \sum_{i=1, i\neq k}^{K}\mathbf{g}_{i}^{\dagger}\mathbf{T}^k_{i}\mathbf{g}_{i}}}\Biggr),
\mathbf{P}\Big( \{U^{*}_{n}(\theta_n,c^*_n)\ne \emptyset \}\cap \{\mathfrak W^{*,+\delta}(c_{\pi^*})=\emptyset \}\Big) \le \mathbf{P}\Big(\{U^{*}_{n}(\theta_n,c^*_n)\not\subseteq\mathfrak W^{*,+\delta}(c_{\pi^*}) \}\Big) &= \mathbf{P}(L_{n}^c)\le\mathbf{P}(A_{n})<\eta/2,~\forall n\ge N~,
\mathbb{V}_n(\kappa_n) = 4 \frac{\sum_{i = 1}^n c_{i,n}^2 \mathbb{V}_n(a_{i,n})}{\mathbb{V}^2_n(\sum_{i = 1}^n a_{i,n})}\leq 4 \frac{\max_{i = 1, \hdots, n}\mathbb{V}_n(a_{i,n})}{\mathbb{V}_n(\sum_{i = 1}^n a_{i,n})}~ d_n.
f^{[n]}(\lambda_0,\ldots,\lambda_n) &=\sum_{p=0}^n\sum_{0<j_1<\cdots<j_{p}\leq n}(-1)^{n-p}(fu^p)^{[p]}(\lambda_0,\lambda_{j_1},\ldots,\lambda_{j_p})u^{-1}(\lambda_1)\cdots u^{-1}(\lambda_n).
B^2(\delta',k,\eta) =\eta^{1/2}\int^\infty_{\sqrt{\eta}}|\frac{d}{dr}\psi^{\delta'}_k(r)|^2dr &\leq C_{\delta'}\eta^{1/2}k^{-\delta'}\int^\infty_{{\eta}/2}|\mathcal{L}^{\delta'+1}_{k-1}(r)|^2r^{{-\delta'-1/2}}dr &\;\;\;\;\;+C_{\delta'}\eta^{1/2}k^{-\delta'}\int^\infty_{{\eta}/2}|\mathcal{L}^{\delta'}_k(r)|^2r^{{-\delta'+1/2}}dr:=C_{\delta'}\cdot B.
U = \left(\begin{array}{cccc}0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 \end{array}\right), \;\; V = \left(\begin{array}{cccc}0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 \end{array}\right), \;\;W = \left(\begin{array}{cccc}1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 \end{array}\right),
H_{G / H}= \frac{1}{2}\left( \pi_{\alpha}- \frac{i \hbar}{2}\Gamma_{\alpha}\right) g^{\alpha \beta}\left( \pi_{\beta}+ \frac{i \hbar}{2}\Gamma_{\beta}\right) = \frac{1}{2}\pi_{\alpha}g^{\alpha \beta}\pi_{\beta}+ V_{G / H},
\left( \begin{array}{c c}{{( 1 - G^{- 1}U^{T}Z^{- 1}U )^{- 1}G^{- 1}}}&{{- ( 1 - G^{- 1}U^{T}Z^{- 1}U )^{- 1}G^{- 1}U^{T}Z^{- 1}}}\\{{- ( 1 - Z^{- 1}U G^{- 1}U^{T})^{- 1}Z^{- 1}U G^{- 1}}}&{{( 1 - Z^{- 1}U G^{- 1}U^{T})^{- 1}Z^{- 1}}}\\ \end{array}\right)
\sum_{1 \leq h_2,\dots,h_k \leq H}\left|\sum_{1 \leq n \leq X}1_{\mathcal S}g_1(a_1 n + b_1) \prod_{j=2}^k 1_{\mathcal S}g_j(a_j n + b_j + h_j)\right| \ll \frac{kA^2}{W^{1/20}}H^{k-1}X.
c_{\Omega_4}^{\Omega_4}= q^{-1/6}\varphi_1^{-4}(\chi_{3,4}^{1,2}\chi_{4,5}^{2,2}) = q^{-1/6}\varphi_1^{-4}(q^{1/24}V(q)^{-1})(q^{1/120}G_2V(q)^{-1}) = q^{-7/60}\varphi_1^{-4}(\varphi_1^{-2}\varphi_2^2)G_2 = q^{-7/60}\varphi_1^{-6}\varphi_2^2G_2.
\frac{1}{\beta}(\mu^{k+1}- \mu^k)^\top (\bar \mu - \mu^{k+1}) = \frac{1}{2\beta}(\|\mu^k - \bar \mu\|^2 - \|\mu^{k+1}- \bar \mu\|^2 - \|\mu^{k+1}- \mu^k\|^2).
\|V^{(k,q)}_{I_{r;i}}\|\leq \|V^{(k,q-1)}_{I_{r;i}}\|+\sum_{j=1}^{k}\sum_{n=1}^{\infty}\frac{1}{n!}\,\| ad^{n}S_{I_{k;i+r-k}}(V^{(k,q-1)}_{I_{r-j;i}})\|\,.
\Delta_{2,5}:=-\int_0^{+\infty}\Big(<\nabla g_j(g^{-1}(v_-)),\int_\sigma^{+\infty}(F^s(z_-+x_-+y_-)-F^s(z_-+x_-))(\tau)d\tau>\Big)_{j=1\ldots n}d\sigma,
a(s')-a(s)=\frac{\beta}{m}\lambda^{-\frac{\rho_1}{\beta}}e^{-\frac{\rho_1}{\beta}s_1}(s'-s)\le\frac{\beta}{m}\lambda^{-\frac{\rho_1}{\beta}}e^{-\frac{\rho_1}{\beta}s}(s'-s).
|\Gamma(x,y)|\le c \frac{1}{d(x,y)^{q-2}}, |X_j \Gamma(x,y)|\le c\frac{1}{d(x,y)^{q-2+deg(X_j)}}, |X_i X_j \Gamma(x,y)|\le c\frac{1}{d(x,y)^{q-2+deg(X_j)+deg(X_i)}}
A(E,\nabla)=\left(\mu_2(E)-\frac{i}{2\pi}\frac{\left(\mu_1(E)^{+}+\left(\iota(\theta)\mu_2(E)\right)^{+}\right)^{2}}{\mu_{0}(E) +\iota(\theta)\mu_1(E) +\frac{1}{2}\iota(\theta)^2 \mu_2(E)}\right)[T],
\begin{array}{lll}C_1 = C(1/\theta,1/\theta,1/\theta) C_2 = C(1/\theta,1/\theta,1) C_3 = C(1/\theta,1/\theta,\theta) C_4 = C(1/\theta,1,1/\theta) C_5 = C(1/\theta,1,1) C_6 = C(1/\theta,1,\theta) \ldots \ldots \ldots\end{array}
C(\delta)= \bigcap_{\xi\in\mathrm{ext}(P)}\left\{y\in\mathbb{R}^{n}:\langle y, \xi\rangle=1+\frac{n|P^{\circ}|_n\|\xi\|}{|F_{\xi}|_{n-1}}\delta\right\}\cap\bigcap_{\xi,\xi'\in\mathrm{ext}(P),\xi\neq \xi'}\{y\in\mathbb{R}^{n}:\frac{1}{2}\langle\xi+\xi',y\rangle\leq 1\}.
\mathbf{\hat{d}}_{m_\textbf{s}}[n_{\textbf{s}}] = \mathbf{{d}}_{m_\textbf{s}}[n_{\textbf{s}}] + \sum\limits_{m_\textbf{i}\in \mathcal{M}_\textbf{i}}\mathbf{\eta}_{m_\textbf{i}\rightarrow m_\textbf{s}}[n_\textbf{s}] + \mathbf{w}_{\textbf{s}}[n_\textbf{s}],
2\sum_k(\alpha_{i,j}^{m+n,k}P_i^{k}+\beta_{i,j}^{m+n,k}P_j^{k}) =[\sum_k(\alpha_{t,i}^{m,k}P_t^{k}+\beta_{t,i}^{m,k}P_i^{k}), J_{t,j}^{n}]+[J_{t,i}^{m},\sum_k(\alpha_{t,j}^{n,k}P_t^{k}+\beta_{t,j}^{n,k}P_j^{k})] &=-\sum_k\alpha_{t,i}^{m,k}P_j^{n+k}+\sum_k\alpha_{t,j}^{n,k}P_i^{m+k}&=-\sum_k\alpha_{t,i}^{m,k-n}P_j^{k}+\sum_k\alpha_{t,j}^{n,k-m}P_i^{k}.
x\oplus_{Y}\partial_{Y}(\emptyset) = x x\oplus_{Y}(s_{Y}(z))= s_{Y}(x \oplus_{Y}z) & x\otimes_{Y}\partial_{Y}(\emptyset) =\partial_{Y}(\emptyset) x\otimes_{Y}(s_{Y}(z))= (x\otimes_{Y}z)\oplus_{Y}x
\epsilon_i = \ \alpha_i + \alpha_{i+1}+ \cdots + \alpha_n, 1 \leq i \leq n \epsilon_i + \epsilon_j = \ (\alpha_i + \alpha_{i+1}+ \cdots + \alpha_n) + (\alpha_j + \alpha_{j+1}+ \cdots + \alpha_n), 1 \leq i < j \leq n \epsilon_i - \epsilon_j = \ \alpha_i + \alpha_{i+1}+ \cdots + \alpha_{j-1}, 1 \leq i < j \leq n
\left. \begin{array}{l l}\frac{1}{\Lambda}\big( [x_1(y_2\!-\!y_3)+x_2(y_3\!-\!y_1)+x_3(y_1\!-\!y_2)]R_E^2 ~~ -[x_2(y_3\!-\!y_E)+x_3(y_E\!-\!y_2)+x_E(y_2\!-\!y_3)]R_1^2 ~~ -[x_3(y_1\!-\!y_E)+x_1(y_E\!-\!y_3)+x_E(y_3\!-\!y_1)]R_2^2 ~~ -[x_1(y_2\!-\!y_E)+x_2(y_E\!-\!y_1)+x_E(y_1\!-\!y_2)]R_3^2 \big) =0.\end{array}\right.
\begin{cases}-\Delta \psi = \lambda m \psi + \lambda a (p-1) w^{p-2}\psi + \gamma \psi \mbox{in $\Omega$}, \partial_{\mathbf{n}}\psi = \lambda b (q-1) w^{q-2}\psi + \gamma \psi \mbox{on $\partial \Omega$}.\end{cases}
S_{e f f}^{m}= \int d^{4}x \sqrt{- \bar{g}}[ \frac{I_{1}}{2 k}{\bar{g}}^{n m}D_{n}H^{\dagger}D_{m}H - \frac{I_{2}}{2 k}\lambda ( | H |^{2}- \upsilon_{0}^{2})^{2}] .
p_i(\mathfrak d_i)q(p_i(\mathfrak d_i)) E_{ii}= \mathfrak h\mathfrak v_iq(\mathfrak v_i)\mathfrak h_i^{-1}E_{ii}= \mathfrak h\mathfrak b_i\mathfrak c_iq(\mathfrak v_i)\mathfrak h_i^{-1}E_{ii}= \mathfrak h\mathfrak b_i\mathfrak c_i \mathfrak t_iE_{ii}= \mathfrak h\mathfrak UW(x)\mathfrak U^*\mathfrak c_i \mathfrak t_iE_{ii}.
\int_0^{\frac1{b(\kappa)\kappa}}\,g\big(b(\kappa)\kappa\,r\big)\,dr =\frac1{b(\kappa)\kappa}\int_0^1g(t)\,dt &=\frac1{2b(\kappa)\kappa}\left(L^{4/3}E(L)+\varepsilon_1(L)\right)\,,
\begin{array}{c c l}{\left( \begin{array}{c}{1}\\{r}\\{\tilde{r}}\\{\tilde{r}}\\ \end{array}\right)}&{=}&{\mathcal{S}[ a , b ; E ; V , | W | ] \left( \begin{array}{c}{t}\\{t}\\{\tilde{t}}\\{\tilde{t}}\\ \end{array}\right) ,}\\ \end{array}
T_{0}(z^{\prime})T(z,\theta)=\frac{\frac{c(\theta)}{2}I(\theta)}{(z^{\prime}-z)^{4}}+\frac{2+\theta}{(z^{\prime}-z)^{2}}T(z,\theta)+\frac{\partial_{z}T(z,\theta)}{z^{\prime}-z}+\cdots\:\:\:,
\kappa = \kappa ( d , \eta ; C ) = 2 v_{d}\int_{0}^{\infty}d w \left[ ( d - 2 ) \frac{w^{d / 2 - 2}C ( w )}{w + C ( w )}- \eta \frac{w^{d / 2 - 1}C ( w )}{( w + C ( w ) )^{2}}\right] .
\||x|^{\gamma_1}\hat{u}\|_{L^s(K)}\le C\left(\||x|^{\gamma_2}\nabla u\|_{L^p(K)}^{a'}\||x|^{\gamma_3}u\|_{L^q(K)}^{1-a'}+\||x|^{\gamma_2}\nabla u\|_{L^p(K)}^{a''}\||x|^{\gamma_3}u\|_{L^q(K)}^{1-a''}\right) =2C\||x|^{\gamma_2}\nabla \hat{u}\|_{L^p(K)}^a\||x|^{\gamma_3}\hat{u}\|_{L^{q}(K)}^{1-a}.
e^{ic_+(x)}\partial_{x}\left(\bar{u}_x(x) e^{ic_+(x)}\right) =-\frac{1}{\pi}\int_{\mathbb{R}}r_2(z) e^{2 iz x}\left[M_{-,22}(x;z)+\bar{r}_1(z)e^{-2izx}M_{-,21}(x ;z)\right] \mathrm{d}z &=-\frac{1}{\pi}\int_{\mathbb{R}}r_2(z) e^{2isx}M_{+,22}(x;z)\mathrm{d}z,
\sigma_Y^*P \simeq \{(v,y)\in P\times Y\,\mid\, h(v) = \sigma_Y(y)\}& = \{(v,y)\,\mid\,y=(\sigma_Y\circ h)(v)\}& =\{(v,\left(\sigma_Y\circ h)(v)\right)\,\mid\,v\in P\}
c_j:= (2N+1)b_j^{(N-M-1)}(-\frac n2-M-1)+a_{j-1}^{(N-M-1)}(-\frac n2-M-1), d_j:= (2N+1)b_j^{(N-M-1)}(-\frac n2-M-1)-a_{j}^{(N-M-1)}(-\frac n2-M-1),
T\otimes I_{(j)}=I_{L^2(X_1,\nu_1)}\otimes \cdots \otimes I_{L^2(X_{j-1},\nu_{j-1})}\otimes T\otimes I_{L^2(X_{j+1},\nu_{j+1})}\otimes \cdots \otimes I_{L^2(X_d,\nu_d)}.
\gamma_{g , 9}\gamma_{S_{1}, 9}\gamma_{g , 9}^{- 1}= \gamma_{S_{2}, 9}, \gamma_{g , 9}\gamma_{S_{2}, 9}\gamma_{g , 9}^{- 1}= \gamma_{S_{1}, 9}, \gamma_{g , 9}\gamma_{S_{3}, 9}\gamma_{g , 9}^{- 1}= - \gamma_{S_{3}, 9}.
P\left[ C_{H}^{\circ}\left(a,v,\varphi\right) \right] = P\left[\right. \bigcup_{\varepsilon>0}C_{H}\left(a,v,\varphi-\varepsilon \right) \left.\right] = \bigcup_{\varepsilon>0}P\left[ C_{H}\left(a,v,\varphi-\varepsilon \right) \right]\mbox{.}
v(\delta)-v(\delta_0) +\sigma_-(\delta_0) v'(\delta_0)\int_\delta^{\delta_0}\sigma_-^{-1}\:ds = \int_\delta^{\delta_0}\sigma_- (v^ps^{\beta(p-1)}+\beta\frac{N-1}{(R-s)s}v)\:ds\int_\delta^s\sigma_-^{-1}\:d\xi.
\frac{\partial^2 l}{\partial \theta_i^2}= -\frac{6\phi}{\theta_i^4}- \frac{1}{\phi};\frac{\partial^2 l}{\partial \theta_i \partial \phi}= \frac{2}{\theta_i^3}+ \frac{\theta_i}{\phi^2};\frac{\partial^2 l}{\partial \phi^2}= \frac{1}{2\phi^2}- \frac{\theta_i^2}{\phi^3};\frac{\partial^3 l}{\partial \theta_i^3}= \frac{24\phi}{\theta_i^5}.
\begin{aligned}cap_p(X_\varepsilon) \leq \int_X|\nabla^{g^+}u|_{g^+}^pdV_{g^+}=(\int_0^\varepsilon T^{\frac{1}{1-p}}(x)dx)^{1-p}&=(\int_0^\varepsilon x^{\frac{p-n-1}{1-p}}\cdot [Vol(\Sigma_x)]^{\frac{1}{1-p}}dx)^{1-p}\end{aligned}
\left| \frac{\d p_{k,n}}{\d v'_h}\right| = \left| \frac{k_h h^{-s}}{2 \sqrt{1+\lambda_h/c^2}}\right| \gtrsim \; \frac{1}{2 h^s \sqrt{1+h^{max(s,2)}}}\geq \frac{1}{2 N^s \sqrt{1+N^{max(s,2)}}}\; > \; 0,
\mathbf{n}\left(e^{-(q+r) \tau_0^-}\Big(Z^{(q)}(X(\tau_0^-)-a,\theta ) - Z^{(q)}(-a,\theta ) \frac{W^{(q)}(X(\tau_0^-)-a)}{W^{(q)}(-a)}\Big) ; \tau_0^-< \tau_b^+ \right) =-\frac{\hat{J}_a^{(q,r)}(b, \theta)}{W^{(q+r)}(b)}.
\eqref{va1}= T(x)\psi(x)\sqrt{\tfrac{it}{2\pi}}\int_+ e^{-\frac{it}{2}(\xi-x)^2}\,d\xi & + \sqrt{\tfrac{it}{2\pi}}\int_+ e^{-\frac{it}{2}(\xi-x)^2}[T(\xi)\psi(\xi)-T(x)\psi(x)]\,d\xi.
\bigcup_{p=1}^n \{(k_1, \cdots, k_n)\in\mathbb{Z}^{\oplus n}|\ 0\leq k_r\leq l-1,\k_1+\cdots+k_n=lk-p\}\times \{(k_1', \cdots, k_n')\in\mathbb{Z}^{\oplus n}|\ 0\leq k_r\leq m-1,\k_1'+\cdots+k_n'=mp-q\}\longrightarrow \{(k_1'', \cdots, k_n'')\in\mathbb{Z}^{\oplus n}|\ 0\leq k_r''\leq ml-1,\k_1''+\cdots+k_n''=mlk-q\}
\widehat{m_1(\lambda+)}= ~ -\frac{i\epsilon}{\xi-(\lambda+i\epsilon)}F(m_1(\lambda+)-1)-\frac{\lambda+i\epsilon}{\xi-(\lambda+i\epsilon)}\widehat 1 &+\frac{1}{\xi-(\lambda+i\epsilon)}\chi_{\mathbb{R}^+}F(um_1(\lambda+)).
\mathcal{L}^{0,\hat{v}}\psi_k^{0,\hat{v}}(x) = \bigl \langle \bigtriangledown_x \psi_k^{0,\hat{v}}(x), f_k(x, \hat{v}_k(x)) \bigr\rangle + \sum\nolimits_{j=1}^n \gamma_{kj}(x) \bigl[\psi_j^{0,\hat{v}}(x) - \psi_k^{0,\hat{v}}(x)\bigr] & = 0, x \in D, k=1,2, \ldots, n,
\left. \begin{array}{l l}\mu_E = y_1(z_2\!-\!z_3)+y_2(z_3\!-\!z_1)+y_3(z_1\!-\!z_2) \mu_1=y_2(z_3\!-\!z_E)+y_3(z_E\!-\!z_2)+y_E(z_2\!-\!z_3) \mu_2 = y_3(z_1\!-\!z_E)+y_1(z_E\!-\!z_3)+y_E(z_3\!-\!z_1) \mu_3 = y_1(z_2\!-\!z_E)+y_2(z_E\!-\!z_1)+y_E(z_1\!-\!z_2)\end{array}\right.
\log{h(z;\boldsymbol{\xi})}=\log{z^R}+\log{h_{-R}}+\log{(1+\sum_{r=1}^{\infty}\frac{h_{-R+r}}{h_{-R}}z^{-r})}&=\log{z^R}+\sum_{r=0}^{\infty}\eta_r z^{-r}
A_{\mu}^{( I )}: = \sum_{b = 1}^{N}U_{I b}a_{\mu}^{( b )}\stackrel{U^{T}= U^{- 1}}{\Longleftarrow \! \! = \! \! = \! \! = \! \! = \! \! = \! \! \! \Longrightarrow}a_{\mu}^{( b )}= \sum_{I = 1}^{N}U_{I b}A_{\mu}^{( I )}.
d_{j}^+(\mu)= \frac{\kappa(\mu)}{\kappa(\mu+\epsilon_j)}c_{j}^+(\mu)=\frac{c_{j}^-(\mu+\epsilon_j)}{d_{j}^-(\mu+\epsilon_j)}c_{j}^+(\mu)=\frac{c_{j}^+(\mu)^2}{d_{j}^-(\mu+\epsilon_j)}
(A_kX_kB_k)_{ij}=\sum_{\ell=1}^{n}(A_k)_{i\ell}\sum_{t=1}^n(X_k)_{\ell t}(B_k)_{t j}=\sum_{\ell=i}^n\sum_{t=j}^{n}(A_k)_{i\ell}(X_k)_{\ell t}(B_k)_{tj},
\begin{array}{r c l c r c l}{{\mathrm{T r}[ T_{a}, \cdots ]}}&{{=}}&{{0 ,}}&{{{\qquad}}}&{{\mathrm{T r}[ \Sigma_{\alpha}, \cdots ]}}&{{=}}&{{0 ,}}\\{{\mathrm{S t r}[ T_{a}, \cdots ]}}&{{=}}&{{0 ,}}&{{{\qquad}}}&{{\mathrm{S t r}\{\Sigma_{\alpha}, \cdots \}}}&{{=}}&{{0 ,}}\\ \end{array}
{}^{( i )}\langle \zeta_{1}, \ldots , \zeta_{N}| ={}^{( i )}\langle W^{*}| \Phi^{* ( i , 1 - i )}( \zeta_{N}) \otimes \cdots \otimes \Phi^{* ( 1 - i , i )}( \zeta_{1}) | V^{*}\rangle^{( i )},
r_{A}=(\frac{sin^{3}(\theta_{A})}{2-3cos(\theta_{A})+cos^{3}(\theta_{A})})^{\frac{1}{3}};r_{R}=(\frac{sin^{3}(\theta_{R})}{2-3cos(\theta_{R})+cos^{3}(\theta_{R})})^{\frac{1}{3}}
\left[ e^{i\gamma \varphi}\right]_{n}(\theta_{1,}\theta_{2,}\dots,\theta_{n})=\left[ e^{i\gamma \varphi}\right]_{1}(\theta_{1})\,\left[e^{i\gamma \varphi}\right]_{n-1}(\theta_{2,}\dots ,\theta_{n})+O(e^{-\theta_{1}})\,.
z_{\sf Aff}({\cal U}) \leq z_{\sf Aff}({\cal V}) &= O \left( \frac{\log n}{\log \log n}\right) \cdot z_{\sf AR}({\cal V}) & \leq O\left( \frac{\log n}{\log \log n}\right) \cdot L \cdot z_{\sf AR}({\cal U}),
w_{1}({y},{\bar{y}}\mid x ) = \sum_{n , m = 0}^{\infty}\frac{\lambda^{1 - | \frac{n - m}{2}|}}{2 i \, n ! m !}w_{1}^{\alpha_{1}\ldots \alpha_{n}}{}_{,}{}^{\dot{\beta}_{1}\ldots \dot{\beta}_{m}}( x ){y}_{\alpha_{1}}\ldots{y}_{\alpha_{n}}{\bar{y}}_{\dot{\beta}_{1}}\ldots{\bar{y}}_{\dot{\beta}_{m}}\,
\mathcal B^*_u \Phi_{\lambda^p_k}= \xi_{\lambda^p_k}(1) + \eta^\prime_{\lambda^p_k}(1), \forall k \geq k_0, &\mathcal B^*_u \Phi_{\lambda^h_k}= \xi_{\lambda^h_k}(1) + \eta^\prime_{\lambda^h_k}(1), \forall |k|\geq k_0.
\begin{array}{@{}l@{}}{\displaystyle W = \big( w_{ij}\big) \in \mathbb{R}^{N\times N}, \quad\vec{r}= (r_1,\cdots, r_N)^{T}\in \mathbb{R}^{N},}begin{align*}2mm]{\displaystyle S = \big(s_{ij}\big)\in \mathbb{R}^{N\times N}, Q = \big(q_{ij}\big)\in \mathbb{R}^{N\times N},}\end{array}
\frac{d}{dt}\sum_{i=1}^N (\frac{1}{2}+ \frac{\kappa_i}{2}) |\nabla u_{i\varepsilon}|_{L^{2}(\Omega)}^2 + \frac{1}{2}|u_{\varepsilon t}|_{L^2(\Omega)}^2+ \frac{\mu}{4}\sum_{i=1}^N |\Delta u_{i\varepsilon}|_{L^{2}(\Omega)}^2 \leq C_2( |\nabla \theta_{\varepsilon}|_{L^4(\Omega)}^4 +1) \mbox{a.e. on}[0,T].
\frac{\delta}{\delta U}\Big|_{(U,\rho,S)}= \frac{\delta}{\delta U}\Big|_{(U,\rho,p)}, & \frac{\delta}{\delta \rho}\Big|_{(U,\rho,S)}= \frac{\delta}{\delta \rho}\Big|_{(U,\rho,p)}+a^2\frac{\delta}{\delta p}\Big|_{(U,\rho,p)}, & \frac{\delta}{\delta S}\Big|_{(U,\rho,S)}= (\partial p/\partial S)|_\rho \frac{\delta}{\delta p}\Big|_{(U,\rho,p)}.
{\tilde{H}}_{c o l l}^{\beta , \pm}= \sum_{n > 0}( 1 - \beta ) n \alpha_{n}^{\dagger}\alpha_{n}\mp \sqrt{\beta}\sum_{n , n^{\prime}\geq 0}( \alpha_{n}^{\dagger}\alpha_{n^{\prime}}^{\dagger}\alpha_{n + n^{\prime}}+ \alpha_{n + n^{\prime}}^{\dagger}\alpha_{n}\alpha_{n^{\prime}}) .
\delta_\epsilon V^{(1)\mu_1\ldots \mu_{p-2k-3}}=\partial^{\left[ \mu_1\right.}\bar \epsilon^{\left. \mu_2\ldots \mu_{p-2k-3}\right]}+M\epsilon^{\mu_1\ldots \mu_{p-2k-3}}+\partial_\mu\bar \epsilon^{\mu \mu_1\ldots \mu_{p-2k-3}},
u^{(k)}_{B_e}\in W^{(k)}:=\{\check{N}_{i,p}^{(l)}\,|\, \,\{\check{N}_{i,p}^{(l)}\}\cap\Gamma^{(k)}_e \neq \emptyset l\in{\mathcal{I}}_{\mathcal{F}}^{(k)}\cup\{k\}\}.
\int_\sigma^rdt \int_{\varrho \leq t}(\Delta \varrho)^{n-k}\wedge \Delta u_1\wedge\dots\wedge \Delta u_k =- \int_{\varrho \leq r}u_k(\Delta \varrho)^{n-k+1}\wedge \Delta u_1\wedge\dots\wedge \Delta u_{k-1}.
f_{\cal N}( z ) ={\frac{P_{\cal N}( z )}{{\cal A}{\cal B}\sum_{i = 1}^{\cal N}\left({\frac{1}{z - a_{i}}}-{\frac{1}{z - b_{i}}}\right)}}\ , \quad \chi_{\cal N}( z ) ={\cal B}f_{\cal N}( z ) \ ,
\Delta_{\omega}\ge0 \Longleftrightarrow\Big(\kappa m-{\frac{\sigma^2}{2}}\Big)^2\ge{\frac{2\epsilon_3\mu}{2\epsilon_3\mu-1}}\left[{\frac{2-2\epsilon_3\mu}{2\epsilon_3\mu-1}}-{\frac{2\beta}{\mu\sigma^2}}\right]\sigma^4=:g(2\epsilon_3\mu)\sigma^4,
gl(3n) \downarrow gl(3)+gl(n) \quad{\{}\overline{1^{K}}{\}}\cdot{\{}1^{L}{\}}\downarrow \sum_{\rho \vdash K, \sigma \vdash L}{\{}\overline{\rho'}{\}}\cdot{\{}\sigma'{\}}\times{\{}\overline{\rho}{\}}\cdot{\{}\sigma{\}},
\sigma_{0}^{B = 0}={\frac{\pi^{4}( \tilde{\omega}R_{0})^{8}}{8 \omega^{5}}}\left\{1 -{\frac{1}{6}}\left( \tilde{\omega}R_{0}\right)^{4}\log ( \tilde{\omega}\bar{R}_{0}) +{\frac{7}{7 2}}( \tilde{\omega}R_{0})^{4}+ \cdots \right\}
H_{\mathrm{e f f}}= \frac{1}{2}p_{r}^{2}+ \frac{1}{2 r^{2}}p_{\theta}^{2}+ \frac{1}{2}p_{z}^{2}+ \frac{1}{2}\omega^{2}r^{2}+ \xi \Big [ p_{z}+ g p_{\theta}\Big ] + F ( \xi ) p_{\xi}- F^{\prime}( \xi ){\cal P}_{1}\eta^{1}-{\cal P}_{2}\eta^{1}\ \ \ ,
\|B^{\dagger}EA^{\dagger}\|_{F}^{2}=\|\widetilde{V}_{1}^{\ast}V_{1}\Sigma_{1}^{-1}-\widetilde{\Sigma}_{1}^{-1}\widetilde{U}_{1}^{\ast}U_{1}\|_{F}^{2}, &\|A^{\dagger}EB^{\dagger}\|_{F}^{2}=\|\Sigma_{1}^{-1}U_{1}^{\ast}\widetilde{U}_{1}-V_{1}^{\ast}\widetilde{V}_{1}\widetilde{\Sigma}_{1}^{-1}\|_{F}^{2}.
{\rm there \exist \constants\}\kappa > \kappa' > 1{\rm \which \satisfy}&\begin{cases}(\kappa ' +\frac{21}{40}+ \frac{\gamma}{10})\vee 2\kappa ' < \kappa < \gamma -\frac{C_f}{\mu}, 1< \kappa ' <\frac{1}{20}+ \frac{\gamma}{5}.\end{cases}
{\cal H}= \frac{i}{2}\sum_{k = 1}^{r}p_{k}p_{k}- \hbar^{2}t r ( H^{2}H^{2}) \hspace{0 . 7 c m}, \hspace{0 . 7 c m}{\mathrm{w}h e r e}\hspace{0 . 5 c m}p_{k}= \frac{1}{2}( p_{k}^{l}+ p_{k}^{r})
\langle u,u_t\rangle_{\o_T^+}+\sum_{j=1}^{d-1}\langle D_j^m u, D_j^m u\rangle_{\o_T^+}+\langle D_d^mu, A^{\hat{\alpha}\hat{\alpha}}D^m_d u\rangle+\lambda \langle u,u\rangle_{\o_T^+}=\sum_{|\alpha|\le m}(-1)^{|\alpha|}\langle D^\alpha u,f_\alpha \rangle_{\o_T^+}.
E_{q\bar{q}}=\lim_{{\tilde \Lambda}\rightarrow\infty}\frac{1}{2\pi}\int^{{\tilde \Lambda}}_{U_0}dx \sqrt{{U^{\prime}}^2+\frac{U^4-U^4_T}{R^4}}= \frac{U_T}{\pi}\lim_{\Lambda \rightarrow \infty}\int_a^{\Lambda}d y \sqrt{\frac{y^4 - 1}{y^4 - a^4}}
\frac{\partial{log\,p_{DI}(\textbf{y}_k| \textbf{x}, \textbf{W}_k)}}{\partial{x_p}}\!=\!-\frac{1}{\lvert\boldsymbol{\Sigma}_{\textbf{y},k}\rvert}\frac{\partial\lvert\boldsymbol{\Sigma}_{\textbf{y},k}\rvert}{\partial{x}_p}\!-\!\textbf{y}_k^\frac{\partial \boldsymbol{\Sigma}_{\textbf{y},k}^{-1}}{\partial{x}_p}\textbf{y}_k,
{\tilde D}=1+{k_{11}^2 T_6 \over 4\pi^2}\left({1\over [(x-x_0)^2+(x^{11}-x^{11}_0)^2]^{3/2}}+{1\over [(x-x_0)^2+(x^{11}+x^{11}_0)^2]^{3/2}}\right) \ .