harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\lbrace\begin{tabular}{l l l l}$u(0,t) = h_0(t)$, $u(L,t) = h_1(t)$, $u_{x}(L,t) = h_2(t)$ in $(0,T)$, $v(0,t) =0$, $v(L,t) = 0$, $v_{x}(L,t) = g_2(t)$ in $(0,T)$.\end{tabular}\right.
a_\ell(x) := xf_\ell'(x), b_\ell(x) := x^2f_\ell''(x)+xf_\ell'(x) \quad c_\ell(x) := x^3f_\ell '''(x) + 3x^2f_\ell''(x) + xf_\ell'(x) \enspace .
a_{4}:=f\,(4\delta^{2}+\delta \,\acute{}\,^{2})+4\delta^{2}k,a_{3}:=4f\, \delta^{2}\, \delta \,\acute{}\, \lambda, a_{2}:=2f\, \delta^{2}[2\delta^{2}(\lambda^{2}+1)-\delta \,\acute{}\,^{2}]+4\delta^{4}(k-\lambda), a_{1}:=-4f\, \delta^{4}\delta \,\acute{}\, \lambda+4\delta^{4}\, \delta \,\acute{}\,,a_{0}:=f\, \delta^{4\,}\delta \,\acute{}\,^{2}
\lambda_k=\min \limits_{P\in P_k}\max \limits_{0\neq g\in P}\frac{\langle g,Mg \rangle}{\langle g,g \rangle}=\max \limits_{P\in P_{k-1}^{\perp}}\min \limits_{0\neq g\in P\in P_k}\frac{\langle g,Mg \rangle}{\langle g,g \rangle},
\epsilon L^{( 2 )}\theta = \tilde{h}_{1}^{( 2 )}v^{2}\phi^{j}( \epsilon \gamma^{i}\theta ) ( \theta \gamma^{i j}\theta ) + h_{1}^{( 2 )}v^{i}v^{j}\phi^{k}( \epsilon \gamma^{i}\theta ) ( \theta \gamma^{j k}\theta ) ,
J_p(v,B_1) =\; \int_{B_1}\Big( |\nabla v(x)|^p+\chi_{\{v>0\}}(x)\Big)\,dx \le\; \int_{B_{9/10}}|\nabla v(x)|^p\,dx +\int_{B_1\setminus B_{9/10}}|\nabla v(x)|^p\,dx+|B_1| \le\; \int_{B_1}\left|\nabla u\right|^p\,dx+\left|B_1\right|\le a^p+\left|B_1\right|.
Z_s^h(t) := t^s\Big(\Vert \sigma^h(t)\Vert_{\dot{B}^{\frac{N}{2}}_{2,1}}+ \Vert u^h(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}}\Big) = t^s \sum_{j > j_0}\Big(2^{j\frac{N}{2}}\|\dot \Delta_j \sigma\|_{L^2}+ 2^{j(\frac{N}{2}+1-\alpha )}\|\dot \Delta_j u\|_{L^2}\Big),
\sum_{k=0}^{n-1}\frac{(q^{d+1};q^d)_k^{d-2}(q;q^d)_k^2q^{dk}}{(q^d;q^d)_k^d}&\equiv \frac{(1-q)^2(q^d;q^d)_{n-1-(n+1)/d}}{(-1)^{(n+1)/d}(q^d;q^d)_{(n+1)/d}^{d-1}}q^{(d(d+n)(n+1)-(n+1)^2)/(2d)-2}\pmod{\Phi_n(q)^2}.
w_{( m )}^{a}= \frac{w_{( n )}^{a}}{w_{( n )}^{m}}, \quad \theta_{( m )}^{k}= \frac{\theta_{( n )}^{k}}{w_{( n )}^{m}}, \quad \mathrm{w h e r e}\quad w_{( n )}^{m}= ( w_{( n )}^{a}, w_{( n )}^{n}= 1 ) .
{\cal W}^{( 1 )}={\frac{1}{{\cal Z}^{( 1 )}N}}\sum_{R , S}\frac{d_{R}}{d_{S}}\exp \left[ -{\frac{g^{2}A_{1}}{2}}C_{2}( R ) -{\frac{g^{2}A_{2}}{2}}C_{2}( S ) \right] \int d U \mathrm{T r}[ U ] \chi_{R}( U ) \chi_{S}^{\dagger}( U ) ,
|V(P^*_{1,{\rm dir}}) \cap A_{i,h}| = \begin{cases}1 \textrm{for $i \in \{j, \dots, j'\}\setminus \{i_1, i_2, i_3 \}$,}m_k \textrm{for $i = i_k$ and $k \le 3$,}0 \textrm{otherwise.}\end{cases}
\frac{Z_3}{Z_\lambda \lambda}\, \langle \, \partial A^a_x \, \partial A^b_y \, \rangle \, = \, \widetilde Z_3 \, \Big( (1-\frac \alpha 2) \, \langle \, i (\partial D_r c)^a_x \, \bar c^b_y\, \rangle \, + \, \frac \alpha 2 \, \langle \, c^b_y\, i(\partial D_r \bar c)^a_x \, \rangle \Big) \;.
y= C_+ e^{-\gamma_+ (t-\tau)}+ C_- e^{-\gamma_- (t- \tau )}\,, \ \gamma_{\pm}=\frac{1\pm K}{2}\,, \K = \sqrt{1-4\varepsilon}\,, \C_{\pm}=\mp\frac{\tilde{w}+\gamma_{\mp}\tilde{u}}{K}\,.
T\gamma\left(\frac{\partial}{\partial t}\right)=\frac{\partial}{\partial t}+\sum_{j=1}^n\frac{\partial \gamma^j}{\partial t}\frac{\partial}{\partial p_j},T\gamma\left(\frac{\partial}{\partial q^i}\right)=\frac{\partial}{\partial q^i}+\sum_{j=1}^n\frac{\partial \gamma^j}{\partial q^i}\frac{\partial}{\partial p_j}
\psi_{l}( k , r ) = 2^{- 1 / 2}\, ( - 1 )^{m}\, i^{| l |}\, e^{i \delta_{l}}\, ( J_{\scriptscriptstyle{W}}( k r ) \, \cos \Delta_{\scriptscriptstyle{l}}- Y_{\scriptscriptstyle{W}}( k r ) \, \sin \Delta_{\scriptscriptstyle{l}}) \, ,
\mathbf{X}=[\mathbf{h}_1^{\perp}L_1^1(C_2,B_3)+\mathbf{h}_2^{\perp}L_2^1(A_3,C_1)+\mathbf{h}_3^{\perp}L_3^1(A_2,B_1), \mathbf{h}_1^{\perp}L_1^2(C_2,B_3)+\mathbf{h}_2^{\perp}L_2^2(A_3,C_1)+\mathbf{h}_3^{\perp}L_3^2(A_2,B_1) ].
(a_1 \star (a_2 \star q))(m) =(a_2 \star q) (ma_1)\cdot a_1^{-1}&=q(ma_1a_2)\cdot a_2^{-1}\cdot a_1^{-1}&= q(ma_1a_2)\cdot(a_1a_2)^{-1}&=((a_1a_2)\star q)(m).
v\in K_0,v= \begin{cases}u \mbox{a.e.~in}\Omega\setminus N, w \mbox{a.e.~in}N, \end{cases}\nabla v=\begin{cases}\nabla u \mbox{a.e.~in}\Omega\setminus N, \nabla w \mbox{a.e.~in}N. \end{cases}
w_1^K =\left(\rho_1e^{i\varphi_1}\frac{\|u_1^H\|_{L^2}^2}{\rho_1^2-\sigma_1^2}-\rho_2e^{i\varphi_2}\frac{\|u_2^H\|_{L^2}^2}{\sigma_1^2-\rho_2^2}\right)u_1^K &=\left(\rho_1e^{i\varphi_1}\frac{\rho_1^2-\sigma_2^2}{\rho_1^2-\rho_2^2}-\rho_2e^{i\varphi_2}\frac{\rho_2^2-\sigma_2^2}{\rho_1^2-\rho_2^2}\right) u_1^K.
\langle n\rangle=Tr\{\rho a^{\dagger}a\}=\frac{1}{Z}Tr\{D^{\dagger}(\alpha)a^{\dagger}D(\alpha)D^{\dagger}(\alpha)aD(\alpha)e^{-\beta\hbar\omega a^{\dagger}a}\}=\frac{1}{Z}Tr\{(a^{\dagger}+\alpha^{*})(a+\alpha)e^{-\beta\hbar\omega a^{\dagger}a}\}=
B(\xi)=\left[\begin{array}{c}F(i_2\mu_2+\cdots+i_n\mu_n) F(\mu_2) \vdots F(\mu_{n-1}) F(\mu_n)\end{array}\right]=\left[\begin{array}{c}i_2F(\mu_2)+\cdots+i_nF(\mu_n) F(\mu_2) \vdots F(\mu_{n-1}) F(\mu_n)\end{array}\right].
Q_{2\gamma}\equiv\Big(\frac{n-2\gamma}{2}\Big)^{-1}\cdot P_{2\gamma}(1)=\Big(\frac{n-2\gamma}{2}\Big)^{-1}e^{-\frac{n+2\gamma}{2}w}(-\triangle)^{\gamma}(e^{\frac{n-2\gamma}{2}w}).
\Phi^{+}_G =\left\{\widehat{x}_i-\widehat{x}_j\,,\; 1\leq i<j\leq n\,,\;\widehat{x}_i+\widehat{x}_j\,,\; 1\leq i\leq j\leq n\right\}\intertext{and}\Phi^{+}_K =\left\{\widehat{x}_i-\widehat{x}_j)\,,\; 1\leq i< j\leq n\right\}\,,
\sum_{i=1}^m \int_t^{\infty}\int\limits_{\{| \nabla_i P_s f | \leq M \|\nabla_i f \|_1 \}}|\nabla_i P_s f|^2 d\mu \, ds\leq \frac{1}{c}e^{-ct}M \sum_{i=1}^m \| \nabla_i f \|_1^2.
\langle \hat{E}_{0}| \hat{W}_{0}( f ) | \hat{E}_{0}\rangle = e^{- i \pi q ( \widetilde{f_{+}}( 0 ) - \widetilde{f_{-}}( 0 ) )}\langle \hat{V}_{0}| \hat{W}_{0}( f ) | \hat{V}_{0}\rangle .
\hat{\mathbb{E}}_{\tau+}[\varphi(X_{\tau+t_1}^x,\cdots,X_{\tau+t_m}^x)] =\mathbb{L}^1\lim_{n\rightarrow \infty}\hat{\mathbb{E}}_{\tau_n}[\varphi(X_{\tau_n+t_1}^x,\cdots,X_{\tau_n+t_m}^x)] &=\mathbb{L}^1\lim_{n\rightarrow \infty}\sum_{i=1}^{d_n}\hat{\mathbb{E}}_{t^n_i}[\varphi(X_{{t^n_i}+t_1}^x,\cdots,X_{{t^n_i}+t_m}^x)]I_{\{\tau_n={t^n_i}\}}.
{}\;\;\;\; \phi(\beta)'=\frac{(-3/2AB' + 9/4BA')}{(4A^3 + 27B^2)}\phi(\beta)^2+\frac{(2A^2A' + 9BB')}{(4A^3 + 27B^2)}\phi(\beta)+\frac{(-5/2A^2B' + 15/4ABA')}{(4A^3 + 27B^2)}.
\left[ \arraycolsep=3pt \begin{array}{rrrrrrrrrr}0 0 1 0 0 0 0 \hphantom{-}0 0 \hphantom{-}0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 0 0 0 0 0 0 0 2 0 0 -1 -1 -1 -1 0 -1 0 \end{array}\right]
\frac{\partial\overline{u}_{i}}{\partial t}+\overline{u}_{j}\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\overline{u_{j}^{\prime}\frac{\partial u_{i}^{\prime}}{\partial x_{j}}}=\overline{f}_{i}-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}}+\nu\frac{\partial^{2}\overline{u}_{i}}{\partial x_{j}\partial x_{j}}.
\prod_{q\mid \gamma(H)}\mathcal{F}_q(H,q^{k-2m}) =\prod_{q\mid \gamma(H)}\mathcal{F}_q(H,q^{-m+(p-1)\cdot t}) & \equiv \prod_{q\mid \gamma(H)}\mathcal{F}_q(H,q^{-m})=0 \pmod{p}.
\pi_{t}(\varphi ) \, :=\, \mathbb E [ \varphi ( \widetilde{X}_{t}) \vert \mathcal F^{X}_{t}] \, =\, \frac{\rho_{t}(\varphi)}{\rho_{t}({\bf 1})}\, , \rho_{t}(\varphi) \, :=\,{\mathbb E}_{0}[ Z_{t}^{-1}\varphi ( \widetilde{X}_{t}) \vert \mathcal F^{X}_{T}] \, ; 0 \le t \le T \,
P_h(x+\xi) = \exp -h\left((x+\xi)\frac{d^2}{dx^2}+ (\alpha + 1)\frac{d}{dx}\right)(x^kp(x)) &= \exp\left(-h\xi\frac{d^2}{dx^2}- hx\frac{d^2}{dx^2}- h (\alpha + 1)\frac{d}{dx}\right)(x^kp(x))
t^{\sigma}\|\Phi(u)(t)-\Phi(v)(t)\|_{L^a}\leq t^{\sigma}\int_0^t \left\|e^{-(t-\tau)H^{\beta}}(f(u(\tau))-f(v(\tau))) \, \right\|_{L^{a}}\, d\tau \leq C \sum_{k=0}^{\infty}\frac{\lambda^k}{k!}t^{\sigma}\int_0^t (t-\tau)^{-\frac d{2\beta}(\frac 1r-\frac 1a)}\|(u-v)(|u|^{pk+m-1}+|v|^{pk+m-1})\|_{L^r}\, d\tau,
\texttt{wptcoef[k]}:=\max\Bigl\{\texttt{mag}\Bigl(68\frac{(L +2) A(\sigma)}{\varepsilon_4}\frac{\Gamma(k+1/2)^{1/2}}{(B_1 a \sqrt{\pi})^k}\Bigr), \texttt{mag}(40(L+2))\Bigr\}.
e_{\hat{0}}^{\; \; \mu}e_{\alpha}^{\; \; \nu}\left. \frac{\partial \Gamma_{R}}{\partial b^{\mu \nu}}\right|_{g_{\mu \nu}}={\bar{e}}_{\alpha}^{\; \; i}{e}_{\hat{0}}^{\; \; 0}a \frac{\partial \Gamma_{R}}{\partial w^{i}},
C_P= \frac{1+b'\kappa_{r_0}}{\lambda}\leq \frac{8t}{n}\left(1+\frac{1}{4t}\exp\left(n+\frac{R^2}{8t}-1\right)\cdot Dr_0^2\exp\left(\frac{(r_0+R)^2}{2t}\right)\right) = \frac{8t}{n}+\frac{D}{e}\left(32t+\frac{4R^2}{n}\right)\exp\left(n+\frac{R^2}{8t}+\frac{(\sqrt{16nt+2R^2}+R)^2}{2t}\right).
\sum_{n=0}^{\infty}\frac{q^{(3n^{2}-n)/2+2mn}}{\left(q,q^{m+1/2},-q^{m+1/2};q\right)_{n}}=\frac{(-1)^{m}q^{-m(m-1)}}{\left(q^{2m+1};q^{2}\right)_{\infty}}\left\{\frac{a_{m}(q^{2})}{(q^{2},q^{8};q^{10})_{\infty}}-\frac{b_{m}(q^{2})}{(q^{4},q^{6};q^{10})_{\infty}}\right\}.
\theta^{(de)}:={f_{Y^{de}}}_*\phi^{(de)}_{(X,\Delta)/Y}\otimes\O_{X_{Y^{de}}}(R_{Y^{de}}):f^{(de)}_*\L^{(de)}_{(X,\Delta)/Y}(p^{de}R)\to{f_{Y^{de}}}_*\O_{X_{Y^{de}}}(R_{Y^{de}})
i\partial_{t}\phi_{t}^{(\mathrm{ref})}(x) = \left( -\frac{1}{2}\Delta +U*|\phi_{t}^{(\mathrm{ref})}|^{2}(x) -\left\Vert U\right\Vert_{1}\right) \phi_{t}^{(\mathrm{ref})}(x)\;,
\int_{{\mathbb R}^d}\varphi^{(j)}(K \cap M^x) \, \lambda(d x)= \varphi^{(j)}(K)\lambda (M) & + \sum_{m=j+1}^{d-1}\varphi_{m,d+j-m}^{(j)}(K,M) +\lambda (K)\varphi^{(j)}(M) .
\Big[B_i^\sigma,\big[B_i^\sigma, [B_i^\sigma,F_j]_{q^3}\big]_q \Big]_{q^{-1}}=\Big[B_i^\sigma,\big[F_i , [F_i ,F_j]_{q^3}\big]_q \Big]_{q^{-1}}+ \Big[B_i^\sigma,\big[K_i E_i, [F_i ,F_j]_{q^3}\big]_q \Big]_{q^{-1}}.
\theta\left(S(r+s)\right) =\int^{T_2}_{T_1}G\left(\frac{3T_1+T_2}{4},\tau\right)f_\downarrow(r(\tau)+s(\tau))\;d \tau &\le f_\downarrow(0)\int^{T_2}_{T_1}G\left(\frac{3T_1+T_2}{4},\tau\right)d\tau &<c.
\hat{Q}^{1}| \hat{Q}^{2}\phi > = \hat{Q}^{1}| \phi_{2}> = 0 , \; \; | \phi_{2}> \neq \hat{Q}^{1}| * > , \; \; | \hat{Q}^{2}\phi > \neq \hat{Q}^{1}| * > ,
\frac{1}{2}\sum_{\theta \in \{0,1\}^2}(-1)^{(m_1+\theta_1+1)(m_2+\theta_2+1) + j ( \theta_1 q_1 + \theta_2 q_2 )}= (-1)^{- j q_1(m_1+1) - jq_2(m_2+1) + j^2q_1q_2}.
\partial_{j}\partial_{k}= e^{-2t}\bigg(\omega_{j}\omega_{k}\partial_{t}^{2}+\omega_{j}\Omega_{k}\partial_{t}+\omega_{k}\Omega_{j}\partial_{t}+(\delta_{jk}-2\omega_{j}\omega_{k})\partial_{t}\qquad+\frac{1}{2}\Omega_{j}\Omega_{k}+\frac{1}{2}\Omega_{k}\Omega_{j}-\frac{1}{2}\omega_{j}\Omega_{k}-\frac{1}{2}\omega_{k}\Omega_{j}\bigg).
{\bf x}^+{\bf U}_{r+}^+ ({\bf W}_1 -{\bf W}_2 +{\bf M}){\bf U}_{r+}{\bf x}={\bf \tilde{x}}^+ ({\bf W}_1 -{\bf W}_2 +{\bf M}){\bf \tilde{x}}>{\bf 0}\ \forall{\bf x \neq 0}
\phi_{-}( \sigma ) = L \int_{0}^{\sigma}\frac{d x}{r_{-}^{2}( x )}= \pm \sqrt{\frac{L^{2}}{m^{2}+ L^{2}}}F [ \psi_{-}( r_{-}( \sigma ) ) , \sqrt{\frac{m^{2}}{m^{2}+ L^{2}}}] .
C_1 := \left\{(a^0,a^1,\dots,a^{m-1})\in (\mathbb{R}^n)^m: a^j\in\mathcal{A}^n,\,\forall j=0,1,\dots,m-1\right\}, C_2 := \left\{(a^0,a^1,\dots,a^{m-1})\in (\mathbb{R}^n)^m :\sum_{s=0}^{n-1}a_s^j=\alpha_j,\forall j=0,1,\dots,m-1\right\}, C_3 := \left\{(a^0,a^1,\dots,a^{m-1})\in (\mathbb{R}^n)^m:\sum_{j=0}^{m-1}a^j\star a^j=v\right\}.
\sum_{m=1}^{\infty}\frac{q^{m}}{1-q^{m}}\frac{(-1)_m}{(-q^N)_m}- \sum_{m=N}^{\infty}\frac{2q^m}{1-q^{2m}}= \sum_{m=1}^{N-1}\frac{q^m}{1-q^m}(-1)_m,
h(z)=\begin{cases}g_{\frac{2a}{\sigma^2d}-1,\frac{2a}{\sigma^2},\frac{c}{a-\frac{\sigma^2d}{2}}}(z), \ \rm{if}\ \Delta > 0, g_{\frac{\sigma^2d}{2a}-1,d,-\frac{c}{a-\frac{\sigma^2d}{2}}}(z), \ \ \ \ \rm{if}\ \Delta < 0,\end{cases}
\begin{array}{l}\bar\Omega_*:\bar\Omega_0=\mbox{Ker}_X^{K-1}\Omega_L\subset\bar\Omega_1=\mbox{Ker}_X^{K-2}\Omega_L \subset\cdots \subset\bar{\Omega}_{K-2}=\mbox{Ker}_X\Omega_L\subset\bar\Omega_{K-1}=\Omega_L\subset\bar\Omega_K=\Omega_{L+1}\subset\cdots\end{array}
\partial^2_{t_i t_j}(\Phi \circ \varphi) = \sum_{l,k}(\partial^2_{x_l x_k}\Phi) (\partial_{t_j}\varphi^l) (\partial_{t_i}\varphi^k) + \sum_l (\partial_{x_l}\Phi) (\partial^2_{t_i t_j}\varphi^l).
r_{s}=\frac{\sigma_{R}\sigma_{S}-\frac{1}{2n}\sum_{i=1}^{n}d_{i}^{2}}{\sigma_{R}\sigma_{S}}=1-\frac{\sum_{i=1}^{n}d_{i}^{2}}{2n\cdot\frac{n^{2}-1}{12}}=1-\frac{6\sum_{i=1}^{n}d_{i}^{2}}{n(n^{2}-1)}.
y_{u_0}^{(0)}= \mathbf{h}_{u_0}^{(0)}\mathbf{v}_{u_0}^{(0)}x_{u_0}^{(0)}+ \sum_{k=1}^{K}\mathbf{g}_{u_0}^{(k)}\mathbf{v}_{u_k}^{(k)}x_{u_k}^{(k)}+ n_{u_0}^{(0)},
A^{(D'\subset D, R)}&=A^{(D'\subset D)}\left[\frac{I_{\delta}}{\prod_{i\in I}t_{i}^{n_{i}}\otimes 1}, \frac{(1\otimes t_{1})-(t_{1}\otimes 1)}{(t_{1}\otimes 1)(\prod_{i\in I}t_{i}^{n_{i}}\otimes 1)}, \ldots, \frac{(1\otimes t_{r'})-(t_{r'}\otimes 1)}{(t_{r'}\otimes 1)(\prod_{i\in I}t_{i}^{n_{i}}\otimes 1)}\right].
\begin{array}{c}s\left( \lambda ,\mu \right) =\frac t{\lambda -\mu}\left( \phi \left(\lambda \right)^{-1}-\phi \left( \mu \right)^{-1}\right) , a\left( \lambda ,\mu \right) =\frac t{\lambda -\mu}\left( \phi \left(\lambda \right)^{-1}-\phi \left( \mu \right)^{-1}\right) .\end{array}
g_\nu(\alpha)-g_\nu(1-\alpha)=-\frac{\pi}{2\sin(\nu\pi)}\left[ \frac{P_\nu(2\alpha-1)}{P_\nu(1-2\alpha)}-\frac{P_\nu(1-2\alpha)}{P_\nu(2\alpha-1)}\right]+\log\frac{\alpha}{1-\alpha}.
\pi_{ra}= k_{ra}- (\mbox{tr}_g k) g_{ra}&= \frac{1}{w}\left[ f_{ra}- \frac{f_a}{r}+ f_a \frac{m}{r^2}+ (f_{rr}+ r^{-2}\tilde{\Delta}f + 2r^{-1}f_r )f_rf_a\right] + o(r^{3p-3}).
k_j = N_j\frac{1-\frac{\prod_{k=1}^K\delta_k}{\delta_j}}{1-\prod_{k=1}^K\delta_k}+ \left(N_j\frac{1-\frac{\prod_{k=1}^K\delta_k}{\delta_j}}{1-\prod_{k=1}^K\delta_k}\right)^{3/4}, \mbox{and}n_1 = \max_{j}\frac{k_j+k_j^{3/4}}{(1-\delta_j)\frac{\prod_{k=1}^K\delta_k}{\delta_j}}.
\sum_{n=0}^\infty\lambda_F(p^n)t^n=\frac{P_{F,p}(t)}{Q_{F,p}(t)},\P_{F,p}(t)=1-p^{\mu-1}\chi(p)t^2, &Q_{F,p}(t)=1-\lambda_F(p)T+\{\lambda_F(p)^2-\lambda_F(p^2)-p^{\mu-1}\chi(p)\}t^2-\chi(p)p^{\mu}\lambda_F(p)t^3+\chi(p)^2p^{2\mu}t^4
{\cal Z}=1+\sum\limits_{N=1}^{\infty}\frac{\zeta^N}{N!}\left(\prod\limits_{a=1}^{N}\int d^3z_a\sum\limits_{\alpha_a=\pm 1,\pm 2,\pm 3}^{}\right)\exp\left[-\frac{g_m^2}{4\pi}\sum\limits_{a<b}^{}\frac{\vec q_{\alpha_a}\vec q_{\alpha_b}}{\left|\vec z_a-\vec z_b\right|}\right].
\left| 2^n \Delta f_n^{(k)}(\gamma) - f_n^{(k+1)}(\gamma) \right| \le \frac{\varepsilon_n}{2^{n(d-k)}}+ \sum_{\ell=2}^{d-k}\frac{|t_{k,k+\ell}|}{2^{n\ell}}\left| f_n^{(k+\ell)}(\gamma)\right|
I_{H}^{B H}={\frac{\beta}{r_{+}}}\left[ - r_{+}^{4}+ r_{+}^{2}( 1 + \sum_{i < j}^{4}q_{i}q_{j}) + r_{+}( \sum_{i = 1}^{4}q_{i}+ 2 \sum_{i < j < k}^{4}q_{i}q_{j}q_{k}) + 3 \prod_{i = 1}^{4}q_{i}\right] \ ,
N^{\cal P}~=~\left ( 2 \over{k+2}\right) ~\sum_{l=1}^{k+1}sin^2 \theta_l~\sum_{m_1 = -j_1}^{j_1}\cdots \sum_{m_p=-j_p}^{j_p}\exp \{2i(\sum_{n=1}^p m_n)~ \theta_l \}~,
\begin{array}{lcl}\Phi_{n+1,j}\big((\bar{r})\big) = \displaystyle(-1)^{n-k}\phi_{n}\big(\{\bar{r}',r_k\}\big)\wedge\frac{dr_j}{r_j}&= \displaystyle(-1)^{k+1}\log(1+i_k)\frac{d\bar{s}'}{\Pi_{\bar{s}'}}\wedge\frac{dr_j}{r_j}+(-1)^{n-k}\omega\wedge\frac{dr_k}{r_k}\wedge\frac{dr_j}{r_j},\end{array}
| \xi_{1}+ \xi_{i}|^{2}=2( |\xi_{1}|^{2}+ | \xi_{i}|^{2}) - | \xi_{1}- \xi_{i}|^{2}, =4 t^{2}- 4 ir^{2}, = 4 ( 2m-1-i) r^{2}, =4(i-1)r^{2}.
\mbox{Tr}_s \left(q^{L_0/\hbar}e^{i2\pi T_0^3/\hbar}\right) = \frac{q^{\hbar s(s+1)/k}\sum_{n=-\infty}^\infty (-1)^{2s+2 n k/\hbar}(2s+1 + 2 n k/\hbar) q^{{n^2 k/\hbar}+ (2s+1)n}}{\Pi_{m=1}^\infty (1-q^m)^3}.
E^{\mathbb{Q}}\big[ \exp\{L G_J(z) \}\big] = \smash{\prod_{j \in J}}E^{\mathbb{Q}}\big[ \exp\{\xi_j L p_n(x_j,z) \}\big]= \prod_{\smash{j \in J}}\Big(1 - L p_n(x_j,z) \Big)^{-1}.
L_{B}^{r}= q^{1 / 2}p_{\mu}\dot{x}^{\mu}- \; \frac{e}{1 + q^{2}}( p^{2}- m^{2}) +{\cal B}\; \dot{e}+ \frac{{\cal B}^{2}}{1 + q^{2}}+ \dot{\bar{\lambda}}\frac{d}{d \tau}( \lambda \; e ) ,
\Phi_F^{S_d}(t_1(H_{L,M})_f) = e^{-2\pi\,{\rm Tr}(S_d)}(LM)^{-k}a(\phi_{L,M}(c_1)), \Phi_F^{S_d}(t_2(H_{L,M})_f) = e^{-2\pi\,{\rm Tr}(S_d)}(LM)^{-k}a(\phi_{L,M}(c_2)).
\begin{array}{l}{v_{k + 1}= W \left[ \left( 2 F ( u_{k}, v_{k}) - F ( u_{k}, v_{k - 1}) \right) , u_{k}\right] ,}\\{u_{k + 1}= \bar{W}\left[ \left( 2 \bar{F}( u_{k}, v_{k}) - \bar{F}( u_{k - 1}, v_{k}) \right) , v_{k}\right] .}\\ \end{array}
\Theta_{t}^{-1}d\Theta_{t}=\frac{\tau}{2}\sum_{r=1}^{3}(X_{r}\otimes \rho_{t}(\zeta)^{-1})^{2}dt-\sum_{r=1}^{3}X_{r}\otimes \rho_{t}(\zeta)^{-1}dB_{t}^{(r)}.
sep(R_{X_1}\Lambda_1R_{X_1}^{-1},R_{V_2}\Lambda_2R_{V_2}^{-1})\geq \frac{sep(\Lambda_1,\Lambda_2)}{\kappa_2(R_{X_1})\kappa_2(R_{V_2})}=\frac{\delta_\lambda}{\kappa_2(X_1)},
S_5=\left(\frac{\phi(\mathfrak{m})}{|\mathfrak{m}|}\right)^{k-1}c_K^{k-1}(\log R)^{k-1}I_{2k}^{(m)}(F) +O\left(F_{\max}^2 \left(\frac{\phi(\mathfrak{m})}{|\mathfrak{m}|}\right)^{k-1}\frac{1}{D_{0}}(\log R)^{k-1}\right)
P\left(\frac{|S_k^\mathbf{I}(\mathbf{Z})\cap[0,m-1]|}{m}\geq \frac{(1+p)z+(1-p)w}{2}\right) = P\left(\frac{1}{m}\sum_{j=0}^{m-1}\mathbf{1}_{S_k^\mathbf{I}(\mathbf{Z})}(j)\geq\frac{(1+p)z+(1-p)w}{2}\right) \geq \frac{(1-p)(z-w)}{2}.
h'_{g_{u}}(1)= -2\beta A(u)+\left((1-\frac{3}{2}\beta)p+3\beta-2\right)C(u)+(2-\beta)B(u) &+\frac{\beta}{4}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{-|x-y|}u^{2}(x)u^{2}(y)dxdy.
f:\widehat{A^{(j+1)}}[x_j^{\pm 1};\sigma'_j]{}~ \longrightarrow ~ \widehat{A^{(j+1)}}[x_j;\sigma'_j,\delta'_j]S_j^{-1}\widehat{A^{(j+1)}}~ \ni ~ a{}~ \longmapsto ~ f(a)=\sum_{n=0}^{\infty}q_j^{\frac{n(n+1)}{2}}(q_j-1)^{-n}d'_{j,n}\circ (\sigma'_j)^{-n}(a)x_j^{-n}x_j{}~ \longmapsto ~ x_j.
I(\sigma):=\int_{T^{\ast}M}\overline{\psi^{\prime}\circ\pi}e^{i\sigma E/\hbar}\psi\circ\pi\circ\Phi_{\sigma}\left( \sqrt{d\mathrm{vol}_{g}},\sqrt{\Phi_{\sigma}^{\ast}d\mathrm{vol}_{g}}\right)_{}\frac{\omega^{n}}{(2\pi\hbar)^{n}n!}.
[\cdot,\cdot]_\tau:\mathfrak{h}_{6}^{(\tau)}\times\mathfrak{h}_{6}^{(\tau)}\longrightarrow\mathfrak{h}_{6}^{(\tau)}\quad{}[\mathcal{A}, \mathcal{B}]_\tau=\mathcal{A}\mathcal{B}-\mathcal{B}\mathcal{A},\mbox{~~for all~~}\mathcal{A}, \mathcal{B}\in\mathfrak{h}_{6}^{(\tau)}.
\Gamma^{\mu \nu}= ( p^{2}+ A ) \left( \eta^{\mu \nu}-{\frac{p^{\mu}p^{\nu}}{p^{2}}}\right) + B{\frac{\bar{\theta}^{\mu}\bar{\theta}^{\nu}}{\bar{\theta}^{2}}}+{\frac{p^{\mu}p^{\nu}}{\xi}}
\underset{[0,T]}{\sup}\int u(x,t) \varphi_{\delta}(x)\,dx \le \int u_0(x) \varphi_{\delta}(x)\,dx\; e^{\int_0^T N(t)\,dt}&\le \int u_0(x) \varphi_0(x) \,dx\; e^{\int_0^T N(t)\,dt}
\lambda_{112}\lambda_{qrr}[a_{(1\,r)},a_{(2\,q)}]_c=-3\xi^2 \lambda_{112}\lambda_{122}\lambda_{qrr}\chi_2(g_{(3\,r)})a_{(3\,r)}a_{(3\,q)}&+3\lambda_{112}\lambda_{qrr}\xi^2\lambda_{qqr}\chi_{(2\,p)}(g_{(1\,r)})a_{(2\,p)}a_{(1\,p)}g_{qqr}.
u\left(t,x\right)=\frac{e^{i\frac{x^{2}}{4t}}}{\sqrt{-2it}}(\mathcal{F}f) \left(t,-\frac{x}{2t}\right) + \mathcal{O}\big({\big\| \partial_{k}(\mathcal{F}f) (t) \big\|}_{L_k^2}\, t^{-\frac{3}{4}}\big), t\geq 1,
{\bf 70}\, \stackrel{Usp(4) \, \times \, SU(4) \, \times \, U(1)}{\longrightarrow}\, \left({\bf 1}+{\bar{\bf 1}},{\bf 1},{\bf 1}\right) \, \oplus\, \left({\bf 1},{\bf 5},{\bf 6}\right) \, \oplus \,\left({\bf 1},{\bf 1},{\bf 6}\right)
d s^{2}={\frac{r^{2}}{L^{2}}}\left[ - \left( 1 -{\frac{r_{0}^{4}}{r^{4}}}\right) d t^{2}+ d \vec{x}^{2}\right] +{\frac{L^{2}}{r^{2}}}\left( 1 -{\frac{r_{0}^{4}}{r^{4}}}\right)^{- 1}d r^{2}+ L^{2}d \Omega_{5}^{2}\ ,
L =\frac{\partial}{\partial x^0}+\tilde{X}^i \frac{\partial}{\partial x^i}\underline L =\frac{1}{2}(\frac{\partial}{\partial x^0}-\tilde{X}^i\frac{\partial}{\partial x^i}) e_a =(\tilde{\nabla}_a \tilde{X}^i) \frac{\partial}{\partial x^i}.
( Y_{d})^{a b}= e^{\hat{K}/ 2}\sum_{l = 1}^{3}( t_{3 3}^{1 / 2})^{l a}( t_{3 7}^{1 / 2})^{l b}( t_{3 7}^{1 / 2})^{l c}\langle h_{d}\rangle_{c}.
\tilde{m}_{\mu, \nu}(\xi -\eta, \eta) = \Big(- \frac{c_{\mu}c_{\nu}}{4 |\xi|}\frac{\xi \cdot (\xi -\eta)}{|\xi |}\frac{(\xi - \eta) \cdot \eta}{|\xi - \eta| |\eta|}- \frac{\xi \cdot (\xi - \eta)}{4 |\xi|^{2}}-\frac{i c_{\mu}\,\xi\cdot (\xi-\eta)}{4|\xi| |\xi-\eta|}\Big) \Big( (\xi - \eta) \times \eta \Big),
J_{\mathrm{Lie}}^{s}\overline{\eta}_{\mathrm{L}}=\overline{\eta}_{\mathrm{BrL}}^{s}J_{\mathrm{Lie}}^{s}J_{\mathrm{Bialg}}^{s}\overline{\epsilon}_{\mathrm{L}}=\overline{\epsilon}_{\mathrm{BrL}}^{s}J_{\mathrm{Bialg}}^{s},
f(x)=2^{2^{-1}}\pi^{-2^{-1}}M^{3^{1}2^{-1}}R^{-3^{1}2^{-1}}T^{-3^{1}2^{-1}}x^{2}e^{-2^{-1}\pi^{-1}M^{1}R^{-1}T^{-1}x^{2}}
S_{\rm min}= \int_{t_{i}}^{t_{f}}dt \int_0^1 d\lambda \Big [ -({\partial u^\mu \over \partial t}{\partial u_\mu \over \partial t})({\partial u^\nu \over \partial \lambda}{\partial u_\nu \over \partial \lambda})+ ({\partial u^\mu \over \partial t}{\partial u_\mu \over \partial \lambda})^2 \Big ]^{1\over 2}\, ,
c_{i,j}=Lcoef(e_{j})\cdot\frac{\partial e_{i}}{\partial x_{i_{k}}}-Lcoef(e_{i})\cdot\frac{\partial^{p_{1}+\cdot\cdot\cdot+p_{l}}e_{j}}{\partial x_{j_{1}}^{p_{1}}\cdot\cdot\cdot\partial x_{j_{l}}^{p_{l}}}
\left( \Delta_{1}\right)_{\mu \nu}^{a b}= - \left( \bar{D}^{2}\right)^{a b}\delta_{\mu \nu}+ 2 f^{a c b}\bar{F}_{\mu \nu}^{c}\, , \quad \left( \Delta_{0}\right)^{a b}= - \left( \bar{D}^{2}\right)^{a b}\, , \quad \Delta_{1 / 2}= i \bar{\, / \hspace{- 0 . 2 9 c m}D}\, ,
\partial_z\left(\theta^{n}A_n\right)+\partial_{\bar{z}}\left(\theta^n\bar{A}_n\right) = -\theta^n\sum_{p=2}^{n-2}\left(\!\!\begin{array}{c}n p\end{array}\!\!\right)\!\bar{A}_{p+1}A_{n-p+1}-\theta^n\left.\frac{d^n}{d\varphi^n}V\left(\varphi\right)\right|_{\varphi=\varphi_0}
\frac{\partial h}{\partial \dot{x}^{1}}=-\frac{(1-b^2)^{\frac{3}{2}}}{2}x^{2}+\lambda\left( \frac{\dot{x}^{1}}{\sqrt{(\dot{x}^{1})^{2}+(\dot{x}^{2})^{2}}}+b\cos (t+c)\right)
\begin{array}{c}\int_{M}\langle J\nabla_{b}\varphi ,\nabla_{b}\varphi_{0}\rangle d\mu+n\int_{M}\varphi_{0}^{2}d\mu =\frac{1}{2}C_{n}\int_{\Sigma}\varphi_{0}\varphi_{e_{n}}d\Sigma_{p}.\end{array}
\frac{d}{dt}\begin{pmatrix}E_d^{\lambda,\gamma,\delta}\zeta_t(O) E_d^{\lambda,\gamma,\delta}\theta_t(O)\end{pmatrix}=\begin{pmatrix}-1 \gamma & 2d\lambda -(1+\gamma+\delta)\end{pmatrix}\begin{pmatrix}E_d^{\lambda,\gamma,\delta}\zeta_t(O) E_d^{\lambda,\gamma,\delta}\theta_t(O)\end{pmatrix}.
\mathbf{v}:=D_{k}\mathbf{\lambda}+\left[\begin{array}[c]{c}D_{U}\mathbf{x}_{0}0\end{array}\right] ;\mathbf{y}:=N_{k}\mathbf{\lambda}+\left[\begin{array}[c]{c}N_{U}\mathbf{x}_{0}0\end{array}\right]
\begin{aligned}R_k(\phi^\circ_k)= \{\Tilde{u}_k\in U_k: V(\sigma^\circ_k,\phi^\circ_k,\Tilde{u}_k,k) = \min_{u_k\in U_k}V(\sigma^\circ_k,\phi^\circ_k,u_k,k) \}. \end{aligned}
\frac{1}{2}\frac{d}{dt}\|\nabla \theta(t)\|_{L^{2}}^{2}+ \|\Lambda_{x_{1}}^{\alpha}\nabla\theta\|_{L^{2}}^{2}+ \|\Lambda_{x_{2}}^{\beta}\nabla\theta\|_{L^{2}}^{2}= \int_{\mathbb{R}^{2}}{(u \cdot \nabla)\theta\Delta\theta\,dx}= \mathcal{H}_{1}+\mathcal{H}_{2}+\mathcal{H}_{3}+\mathcal{H}_{4},
\|T_m^{(n)}\|_{l,a}\le 2^{-2m}D_{et}^{-\frac{m}{2}-n+1}\prod_{i=1}^n\Bigg(\sum_{q_i=0}^1\Bigg)\prod_{j=2}^n\Bigg(\sum_{r_j=0}^1\Bigg)1_{\sum_{i=1}^nq_i+\sum_{j=2}^nr_j=a}&\cdot\prod_{k=2}^n\|\widetilde{C_o}\|_{l,r_k}\prod_{p=1}^n\Bigg(\sum_{m_p=2}^N2^{3m_p}D_{et}^{\frac{m_p}{2}}\|J_{m_p}\|_{l,q_p}\Bigg)1_{\sum_{j=1}^nm_j-2n+2\ge m}.
\begin{aligned}2 n \left(A + 1\right) S_n (X) \geq \frac{A \delta_0 t}{4}\sum_l \left(S_{n - 1;l}(X) - \frac{c}{C^m_n}S_{m - 1;l}(X)\right) + \frac{(n - m) A t}{2}S_n (X) \geq \frac{A (n - m)\delta_0 t}{4 n}S_{n - 1}(X) + \frac{(n - m) A t}{2}S_n (X) ,\end{aligned}
B^{ij}= - (g^{ij}(\mathbf{b}_x)\partial_x^{}+ c^{ij}_k(\mathbf{b}_x)b^k_{xx}+ c^\alpha w^i_{\alpha k}(\mathbf{b}_x)b^k_{xx}\partial_x^{-1}w^j_{\alpha h}(\mathbf{b}_x)b^h_{xx})
\begin{array}{r c l}{{G}}&{{=}}&{{( \varphi ( x ) \varphi ( y ) ) \star i \Delta ( z , m^{2}) + i \Delta ( z , m^{2}) \star ( \varphi ( y ) \varphi ( x ) )}}\\{{}}&{{}}&{{+ \varphi ( x ) \star_{x}i \Delta ( z , m^{2}) \star_{y}\varphi ( y ) + \varphi ( y ) \star_{y}i \Delta ( z , m^{2}) \star_{x}\varphi ( x ) ,}}\\ \end{array}
v_1 \wedge v_0 = \frac{q^{-2}}{[2]_{q^2}}\pi_+(V_1), v_1 \wedge v_{-1}= \frac{1}{[2]_{q^2}}\pi_+(V_0), v_0 \wedge v_{-1}= \frac{q^{-2}}{[2]_{q^2}}\pi_+(V_{-1}), w_{-1}\wedge w_0 = \frac{q^{-2}}{[2]_{q^2}}\pi_-(W_{-1}), w_{-1}\wedge w_1 = \frac{1}{[2]_{q^2}}\pi_-(W_0), w_0 \wedge w_1 = \frac{q^{-2}}{[2]_{q^2}}\pi_-(W_1).
\sum_{i=1}^\ell\left(c_i(d_{\ell,i}(\mathcal{F}_2(f)))+\sum_{\xi \in X^{\circ}_i(F)}I_i(d_{\ell,i}(\mathcal{F}_2(f)))(\xi)\right)+\kappa d_{\ell,0}(\mathcal{F}_2(f))(0_{V_0},0,0).
J^{-1}=(\begin{matrix}\frac{x}{r}&\frac{y}{r}&\frac{z}{r}\\ \\ \frac{xz}{r^{2}\sqrt{x^{2}+y^{2}}}&\frac{yz}{r^{2}\sqrt{x^{2}+y^{2}}}&\frac{-(x^{2}+y^{2})}{r^{2}\sqrt{x^{2}+y^{2}}}\\ \\ \frac{-y}{x^{2}+y^{2}}&\frac{x}{x^{2}+y^{2}}&0\end{matrix}).
A_{\mu}^{a}= \left( \begin{matrix}{V^{M}{}_{\mu}}\\{B_{\mu M}}\\{A^{I}{}_{\mu}}\\ \end{matrix}\right){}F_{\mu \nu}^{a}= \left( \begin{matrix}{V^{M}{}_{\mu \nu}}\\{H_{\mu \nu M}}\\{F^{I}{}_{\mu \nu}}\\ \end{matrix}\right)