harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-\Delta V -f'(U) V = & -\sum\limits_{a,b,i,j=1}^n\frac13 R_{iabj}(\xi)y_ay_b\partial^2_{ij}U-\sum\limits_{i, l,k=1}^n \partial_l\Gamma^k_{ii}(\xi)y_l\partial_kU -c(n)R_g(\xi) U+\nu(\xi)Z_0,
\frac{\partial \Gamma_{\Lambda}}{\partial \Lambda}= - N_{f}\mathrm{t r}\left( \frac{\partial \Delta_{I R}^{- 1}}{\partial \Lambda}_{\alpha}^{\phantom{\alpha}\beta}\left( \frac{\delta \Gamma_{\Lambda}}{\delta z}I + \Delta_{I R}^{- 1}\right)_{\phantom{- 1}\beta}^{- 1 \phantom{\beta}\alpha}\right) .
\|T_tx - T_ty\|^2 = \|tTx + (1-t)x -tTy - (1-t)y\|^2 = \|(x-y) + (-t)((x-Tx)-(y-Ty))\|^2 \leq \|x-y\|^2 -2tj(x-y)((x-Tx)-(y-Ty)) + dt^2\|(x-Tx)-(y-Ty)\|^2 \leq \|x-y\|^2 -t(1-k)\|(x-Tx)-(y-Ty)\|^2 + dt^2\|(x-Tx)-(y-Ty)\|^2 = \|x-y\|^2 +t(dt - (1-k))\|(x-Tx)-(y-Ty)\|^2 \leq \|x-y\|^2.
\textrm{I}_{n,2}=\{E\prod_{i=0}^{\tau}[\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda\rho(S_i)\rho(S_{i+1})}]\}^2 &\geq \{E\prod_{i=0}^{\tau}[\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda M^2}]\}^2 &=(E\rho)^4(\frac{\lambda}{1+\lambda M^2})^{2\tau+2}(E\rho^2)^{2\tau}.
\begin{array}{l}\lim\limits_{n\rightarrow \infty}\mathbb{E}^{m_0^n}[<\mu_{t-s}^{+s,\tilde e},\hat{\psi}_0>]=\lim\limits_{n\rightarrow \infty}\mathbb{E}^{\hat{\psi}_0}[<m_0^n,\psi_{t-s}^{t,\tilde e}>]= \mathbb{E}^{\hat{\psi}_0}[<m_0,\psi_{t-s}^{t,\tilde e}>]=\mathbb{E}^{m_0}[<\mu_{t-s}^{+s,\tilde e},\hat{\psi}_0>].\end{array}
\|P_{\gamma,\Gamma_{M_n}\cap(M_n,M_{n+1}]}\| =\|P_{\sigma_{M_n},\Gamma_{M_n}\cap(M_n,M_{n+1}]}\| &\geq \|P_{\sigma_{M_n},\Gamma_{M_n}}\|-\|P_{\sigma_{M_n},\Gamma_{M_n}\cap[1,M_{n}]}\| &=\|P_{\sigma_{M_n},\Gamma_{M_n}}\|-\|P_{\sigma,\Gamma_{M_n}\cap[1,M_{n}]}\|\geq M_n C^{-1}-DC.
\|h\|_2 =\sqrt{\|h_{S_0}\|_2^2+\|h_{S_0^c}\|_2^2}\leq\sqrt{\|h_{S_0}\|_2^2+(\|h_{S_0}\|_2+R)^2}\leq\sqrt{2\|h_{S_0}\|_2^2}+R\leq\sqrt{2}z+R \leq\frac{\sqrt{2(1+\delta)}}{1-\sqrt{t/(t-1)}\delta}\epsilon+\left( \frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1 \right) R.
\int{\frac{[ d \delta \phi ]}{\mathrm{V o l}( \mathrm{W e y l})}}\int_{0}^{\infty}d l J_{b}( l ;{\hat{g}}) e^{-{\frac{1 0 - d}{4 8 \pi}}S_{S L}[ \phi , \zeta , g ]}( \mathrm{d e t}^{\prime}\Delta )_{{\hat{g}}}^{- 5}\quad ,
G\left(\tau_2+\epsilon_2,\tau_1-\epsilon_1;\tau_2,\tau_1\right)=\frac 1 2|\tau_1-\tau_2-\epsilon_1-\epsilon_2|-\frac 1 2\left(\tau_1-\tau_2-\epsilon_1-\epsilon_2\right)-\frac{\epsilon_1\epsilon_2}{\tau_1-\tau_2}
\alpha_0\alpha_i\, =\sum_{a,b\in K}\gamma^{a+q^ib}=\sum_{a,b\in K}\gamma^{a(1+q^ib)}\cr =k|\{-q^{-i}\}\cap K|+\sum_{j=0}^{n-1}\alpha_j|(1+q^iK)\cap q^jK|.
Z_{s,\bar{s}}^\ell(t) := t^s \Vert (\sigma,u)^\ell(t)\Vert_{\dot{B}^{\bar{s}+s\alpha}_{2,1}}= t^s \sum_{j\leq j_0}2^{j(\bar{s}+s\alpha)}\|(\dot\Delta_j \sigma, \dot \Delta_j u)(t)\|_{L^2}= t^s \sum_{j\leq j_0}2^{j(\bar{s}+ s\alpha)}X_j(t),
\begin{aligned}\lim_{N\to\infty}\lim_{n\to\infty}&\sum_{j=1}^{n-N}\sum_{l=1}^k\frac{\binom{c_nkn}l(k)_l(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac k{k(n-j+1)-l}\frac{E_{n,k}(\tau(A^{(n)}_{c_nkn,j,l})|\mathcal{WA}^{(n,k)}_{M_n,j,l})}{nk}=0.\end{aligned}
g^i_t=g(t, x_i, S_t, \underline{\pi_t}), g^i_{t^-}=g(t, x_i, S_{t^-}, \underline{\pi_{t^-}}) \Delta g^i_t(z)=\Delta g(z;t, x_i, S_t, \underline{\pi_t})=g\left(t, x_i, S_{t^-}+z, (\pi^i_{t^-}+w^{f_i}(t,z))_{i=1,\dots,d}\right)-g(t, x_i, S_{t^-}, \underline{\pi_{t^-}})
\int_M |\Delta \nabla u|_g^2 d\mu_g=\int_M|\nabla^3 u|_g^2 d\mu_g+\int_M O\big(|{\rm Rm}||\nabla^2 u|_g+|\nabla{\rm Rm}||\nabla u|_g\big)|\nabla^2 u|_gd\mu_g.
\overline{a}=\alpha a;\overline{c}:=\alpha c;\overline{d}:=\alpha^{2}Rd;\alpha:=(\frac{4E^{*}}{3\pi\Delta\gamma R^{2}})^{\frac{1}{3}};\overline{A}:=\pi c^{2};\overline{F}=\frac{F}{\pi\Delta\gamma R}
\frac{G_{5}^{'}\left(r\right)}{G_{6}^{'}\left(r\right)}=\frac{\pi^{2}}{4}\frac{\left[\kappa\left(r\right)+\kappa_{a}\left(r\right)\right]\left[\kappa\left(r\right)-\kappa_{a}\left(r\right)\right]}{\left[r^{'2}\kappa^{2}\left(r\right)\kappa_{a}^{2}\left(r\right)\right]\left[\kappa\left(r\right)-\varepsilon\left(r\right)\right]}
\chi(\alpha)\,(=\chi(\Omega_\alpha))\,=\ell\,\,\,\mbox{if \,\,$\Omega_\alpha\bigcap X_{p_\ell}\ne \emptyset$\,\,and\,\,for any $\ell'>\ell$, \,$\Omega_\alpha\bigcap X_{p_{\ell'}}=\emptyset$},\,\,\, 1\le\ell, \ell'\le k.
\sum_{\ell=1}^\infty\frac{|B(x_0,r)|^{1/p}}{\Phi(r)^{1/p}}\cdot\frac{\Phi(2^{\ell+1}r)^{1/p}}{|B(x_0,2^{\ell+1}r)|^{1/p}}\leq C\sum_{\ell=1}^\infty\left(\frac{D(\Phi)}{2^{n}}\right)^{{(\ell+1)}/p}&\leq C.
\|\tilde{Z}_{i_k}\| = \| \tilde{Z}_{i_k}\tilde{Z}_{i_k}^* \|^{1/2}= \| (\sum Z_{i_k}) (\sum Z_{i_k}^* ) \|^{1/2}= \| 1 \otimes \cdots \otimes 1 \otimes (\sum A_{i_k}A_{i_k}^*) \|^{1/2}= \|\sum A_{i_k}A_{i_k}^* \|^{1/2}= \sqrt{n}
P_{k , m}^{\nu}= \frac{1}{( k^{2}- m^{2}+ i \epsilon )^{\nu}}, \overline{{k}}_{j}^{\mu}= \sum_{n = 1}^{L}\omega_{j n}k_{n}^{\mu}+ \sum_{m = 1}^{E}\eta_{j m}q_{m}^{\mu},
\|\theta(t_0)\|_2^2+\nu\int_0^{t_0}\|\Lambda^{\frac{\alpha}{2}}\theta(t)\|_2^2\, dt\lesssim \|\theta(0)\|_2^2+\frac 1\nu\int_0^{t_0}\|f\|_{H^{-\frac\alpha2}}^2\, dt \lesssim \|\theta(0)\|_2^2+\frac{t_0\lambda_0^{-\alpha}}\nu \|f\|_{2}^2 \lesssim \|\theta(0)\|_2^2+\frac{t_0\lambda_0^{\frac 4p-2-\alpha}}\nu \|f\|_{p}^2,
(f,g)_{2}=\int_{-1}^{1}\left( (1+t)^{\beta+2}f^{\prime\prime}(t)\overline{g}^{\prime\prime}(t)+(\beta+2)(1-t)^{-1}(1+t)^{\beta+1}f^{\prime}(t)\overline{g}^{\prime}(t)\right. & \left. +(1-t)^{-2}(1+t)^{\beta}f(t)\overline{g}(t)\right) dt.
{\cal H}^{( 1 )}( \xi ) \mid_{\Omega^{( 1 )}= 0}~ = ~ \frac{1}{2}( \pi_{i})^{2}+ \frac{1}{4}F_{i j}F^{i j}+ \frac{1}{2}m^{2}( A^{0})^{2}+ \frac{1}{2}m^{2}( A^{i})^{2}.
D = \left\{( 0 , ~ 0 , ~ \sigma , ~ \frac{\sigma ( q^{j}+ q^{- j}) - 2}{( q - q^{- 1})^{2}}\vert \sigma = \pm 1 , ~ j = 1 , ~ 2 , ~ \cdots , ~ \frac{p - 1}{2}\right\}~ .
\liminf_{R\rightarrow+\infty}\left\{R^{-\frac{d-1}{2}}\left\Vert\mathcal{D}\left( R\Omega-x\right) \right\Vert_{L^{p}\left( \mathbb{T}^{d}\right)}\right\}=0\ \ \ \ \ \p<\frac{2d}{d-1}.
h_2(r) \geq \exp\left( \frac{\tilde \beta\, \tilde q^{1-m}(1)}{m}\int_1^r \rho^{\frac{n-2-nm}{m}-(1-m)\frac{\tilde\alpha}{\tilde\beta}-1}d\rho\right) &=\exp\left( \frac{\tilde \beta \, \tilde q^{1-m}(1)(r^{\frac{n-2-nm}{m}-(1-m)\frac{\tilde\alpha}{\tilde\beta}}-1) \,}{{n-2-nm}-m(1-m) \frac{\tilde\alpha}{\tilde\beta}}\right) \quad\forall r>1
0<n_2-n_1\theta\leqslant \frac{\det(\Lambda_d)\varepsilon B^{-\frac{1}{r}}}{(\lambda_2-\alpha\mu_2)^2}\times \frac{(\lambda_2-\alpha\mu_2)KB}{\det(\Lambda_d)}=\frac{K\varepsilon}{\lambda_2-\alpha\mu_2}B^{1-\frac{1}{r}}\leqslant \frac{1}{64}.
-{\textstyle\frac{1}{4}}\omega^{a}{}_{b}\gamma_{a}{}^{b}= -\vartheta\Gamma_{s}(M)-{\textstyle\frac{1}{8}}e^{c}f_{cb}{}^{a}\gamma_{a}{}^{b}\, ,\hspace{1cm}\Gamma_{s}(M) \equiv{\textstyle\frac{1}{2}}(\cos{\alpha}\gamma^{01}-\sin{\alpha}\gamma^{23})\, .
\begin{array}{c}t_0 = t, \theta_0 = 1 - \xi^{\sigma_1}, \theta_k = 1- \xi^{\sigma_{k+1}- \sigma_k}\, k \ge 1, begin{align*}0.3cm]t_{k+1}= (1-\theta_k)t_k = t \xi^{\sigma_{k+1}}.\end{array}
A_{n,s}= \sum_{k=\frac{n}{2}}^{\infty}\left( \frac{1}{(2k)^s}- \frac{1}{(2k+1)^s}\right) \frac{1}{2\left(n - \frac{1}{2}+ \epsilon\right)^s}= \sum_{k=\frac{n}{2}}^{\infty}\left( \frac{1}{2\left(2k-\frac{1}{2}+ \epsilon\right)^s}- \frac{1}{2\left(2k+\frac{3}{2}+ \epsilon\right)^s}\right),
\limsup_{n\to\infty}K_{\rm cl}\int_{\mathbb{R}^3}w_{n}^{(j)}(x)^{\frac{4}{3}}{\rm d}x \leq \limsup_{n\to\infty}\int_{\mathbb{R}^3}j_{m\epsilon_{n}}(w_{n}^{(j)}(x)){\rm d}x & \leq \limsup_{n\to\infty}\int_{\mathbb{R}^3}j_{m\epsilon_{n}}(w(x)){\rm d}x = K_{\rm cl}\int_{\mathbb{R}^3}w(x)^{\frac{4}{3}}{\rm d}x.
C_0 = 3\gamma^{-2}\|\xi\|^2_{\mathbb H^\mu}+6T ({L_f^*}^2+{K_f^*}^2\|\ell\|^2_{L^1}+\epsilon){\rho^*}^2, C_1 = 6({L_f^*}^2+{K_f^*}^2\|\ell\|^2_{L^1}+\epsilon).
\phantom{={}}\left( \int_0^1 |g(r)|^s \, (1-r)^{-u}\, r^{j-k}dr \right)^{1/s}\le C_2(s,n-k+1,n-k+1,j-k,u) \times \left[ \int_0^{1}\left[ M_p \left(\frac{d^k}{d\theta^k}(e^{-in\theta}f),r \right) \right]^s (1-r)^{-w}r^{n-k+1}\, dr\right]^{1/s}.
P_{1}=J_{Z_1}J_{Z_2}J_{Z_3}J_{Z_4},~P_{2}=J_{Z_1}J_{Z_2}J_{Z_5}J_{Z_6}, &P_{3}=J_{Z_1}J_{Z_3}J_{Z_5}J_{Z_7},~P_{4}=J_{Z_5}J_{Z_6}J_{Z_7},
= \; 2e \sum_{I=1}^N \left( \varepsilon_{\mu \nu}\partial_\mu A^{(I)}_\nu,K_{I I}\sum_{b=1}^N \sum_{j=1}^{n_b}U_{I b}\Big[ \delta_n(x_j^{(b)}) - \delta_n(y_j^{(b)}) \Big] \right) \; .
\mathcal{Q}_N:=\left\{q\in L^{\infty}(\Omega): \q \mbox{is piecewise constant}, \ \|q\|_{L^{\infty}(\Omega)}\leq \frac{\lambda_1}{2}\right\}, \d_{\mathcal{Q}_N}(q_1, q_2)= \|q_1-q_2\|_{L^{\infty}(\Omega)}.
\left(\frac{\omega(S_d)}{d+\alpha}\right)^{\lambda}\left(1+\lambda p\right)^{1/p}\leq\prod_{k=1}^m \left(\frac{\omega(S_d)}{d+\alpha_k}\right)^{\lambda_k}\left(1+\lambda_k p_k\right)^{1/p_k}
\cot(z)=\frac{\cot(z/2^n)}{2^n}-\frac{\tan(z/2^n)}{2^n}+\frac{1}{2^n}\sum_{j=1}^{2^{n-1}-1}\left(\cot\left(\frac{z+j\pi}{2^n}\right)+\cot\left(\frac{z-j\pi}{2^n}\right)\right).
{\frac{\imath}{4 \pi \alpha}}\int_{\Gamma}{\frac{\operatorname{c o t}{\frac{w}{2 \alpha}}~ d w}{\operatorname{s i n}^{2}{\frac{w}{2}}}}={\frac{1}{3}}({\frac{1}{\alpha^{2}}}- 1 ) \equiv 2 c_{2}( \alpha ) ~ ~ ,
\frac{\langle H\rangle^{d,\ell}_{min}}{E^{d, \ell}_{0}}=\frac{(\ell +\frac{d-1}{2})^2}{(\ell +\frac{d}{2})}\Big(\frac{\Gamma(\ell+\frac{d-1}{2})}{\Gamma(\ell+\frac{d}{2})}\Big)^2.
D_{t}( \Phi (\mathrm m_{1}), \Phi (\mathrm m_{2})) \, \le \, C_{T}e^{C_{T}T}\int^{t}_{0}(u D_{v}(\mathrm m_{1}, \mathrm m_{2}) + (1-u) D_{v}( \mathrm m_{1}, \mathrm m_{2})){\mathrm d}v & \, =\, C_{T}e^{C_{T}T}\int^{t}_{0}D_{v}(\mathrm m_{1}, \mathrm m_{2}){\mathrm d}v
\frac{d(\mu \circ G_{\xi_{\Delta}}^{-1})}{d\mu}(\eta) = \mathbf{1}_{[\xi_{\Delta}]}(\eta)\sum_{\zeta_{\Delta}}\frac{\gamma_{\Delta}(\zeta_{\Delta}| \eta_{\Delta^c})}{\gamma_{\Delta}(\xi_{\Delta}| \eta_{\Delta^c})}, \eta \in \Omega.
D = \left( \begin{matrix}{{\gamma^{a}e_{a}^{\mu}\partial_{\mu}+ . . .}}&{{\gamma^{\mu}\gamma^{5}V_{\mu}}}\\{{\gamma^{\mu}\gamma^{5}V_{\mu}}}&{{\gamma^{a}e_{a}^{\mu}\partial_{\mu}+ . . .}}\\ \end{matrix}\right)
\sum_i A(v_1,...,Cv_i,...,v_k)=\sum_i \int_0^\infty B(e^{tC}v_1,...,e^{tC}Cv_i,...,e^{tC}v_k) dt = \int_0^\infty \frac{d}{dt}B(e^{tC}v_1,...,e^{tC}v_i,...,e^{tC}v_k) dt = -B(v_1,...,v_k).
|f'(r) | = \Big| \int_r^{\infty}f''\Big| \leq \frac{C}{r}\left|\int_r^{\infty}f'\right| +C \left|\int_r^{\infty}t^{-\beta -2}\right| \leq \frac{C}{r}|f(r) | + \frac{C}{\beta+2}r^{-\beta -1}\leq C r^{-\beta -1}.
(-1)^{m+n}\left[u_{1,0}-u_{0,1}-\frac{a(m)-b(n)}{u_{0,0}-u_{1,1}}-\left(u_{0,-1}-u_{-1,0}-\frac{a(m-1)-b(n-1)}{u_{-1,-1}-u_{0,0}}\right)\right]=0,
{\bf u}_m(\alpha)=\big[{\tilde u}_{1, 1}^\alpha, \ldots,{\tilde u}_{1, N_h}^\alpha,{\tilde u}_{2, 1}^\alpha, \ldots,{\tilde u}_{2, N_h}^\alpha, \ldots,{\tilde u}_{m, 1}^\alpha, \ldots,{\tilde u}_{m, N_h}^\alpha\big]^T\in \mathbb{R}^{mN_h}
+ q \exp ( - \sqrt{2 \kappa}\phi ) ( \overline{{{\psi}}}_{+}A \psi_{+}+ \overline{{{\psi}}}_{-}A \psi_{-}) + M \exp \! \Big ( - \sqrt{\frac{\kappa}{2}}\phi \Big ) ( \psi_{+}^{+}\psi_{-}+ \psi_{-}^{+}\psi_{+})
\overrightarrow{C}_{(m:n)}=\overrightarrow{C}_{(z:n)}\otimes \overrightarrow{C}_{(w:n)}&=\left(\bigoplus_i H_z(i,\phi(i))\right)\otimes\left(\bigoplus_j H_w(j,j)\right) &=\bigoplus_i \bigoplus_j H_z(i,\phi(i))\otimes H_w(j,j)
\Gamma_{3}= \frac{\lambda^{3}}{8 \left( 2 \pi \right)^{5}}\int^{\infty}\frac{\mathrm{d}|{\bf Q}|}{|{\bf Q}|}N \left( |{\bf Q}| \right) \int \mathrm{d}\Omega \left\{\frac{k_{1}^{2}\, k_{2}\cdot k_{3}}{\left( \hat{Q}\cdot k_{1}\right)^{2}\hat{Q}\cdot k_{2}\hat{Q}\cdot k_{3}}+ p e r m u t a t i o n s \right\},
P_1(x)\! =\! D_{0}x\! -\! D_1^{\,\prime}\, , \, P_n(x)\! = \! D_{n-1}x^{n}\! -\! D_n^{\,\prime}x^{n-1}\! +\! \sum\limits_{k=1}^{n-1}(-1)^{k-1}x^{n-1-k}\, D_{n}^{n-k, n+1}\, , \, n \geq 2 \, .
\overline{{\psi}}^{( 2 )}( x ) P_{+}\psi^{( 2 )}( x ) \longleftrightarrow \frac{1}{2 \pi}c^{( 2 )}: e^{- i a \varphi^{( 1 )}( x )}:_{M^{( 1 )}}: e^{+ i b \varphi^{( 2 )}( x )}:_{M^{( 2 )}}e^{i \frac{\theta}{2}},
-\sum_{\underline{a}\in E_k}\nu_k(\{\underline{a}\})\log\nu_k(\{\underline{a}\})+k\nu_k(\Phi) =\sum_{\underline{a}\in E_k}\nu_k(\{\underline{a}\})\left(-\log\nu_k(\{\underline{a}\})+S_{k}\Phi(\underline{a})\right) &=\log\sum_{\underline{a}\in E_k}\exp(S_{k}\Phi(\underline{a})),
2k_{1}= \rho\mu\frac{\left( 8 |a|^2 + 9b(a+\bar{a}) \sin^2 \alpha - 8 b^2 + 18 b^2\sin^2\alpha\right)}{(\bar{a}+b) \left(|a|^{2}+ \rho/2(-2 + 3\sin^{2}\alpha)\right)}\cot \alpha .
B ( x , y , z ) = \sum_{r > 0}{\frac{1}{r}}\left({\frac{y}{x}}\right)^{r}d + d \sum_{r , r^{\prime}> 0}B_{r r^{\prime}}\left({\frac{z}{x}}\right)^{r}\left({\frac{y}{z}}\right)^{r^{\prime}}\, .
I(s,x) := \int_0^\infty \mathcal{Q}(x,k)\phi(k)e^{ik^{2}s}h(k)\,dk = \int_0^{\infty}\mathcal{Q}(x,\sqrt{\lambda})\phi(\sqrt{\lambda})e^{is\lambda}h(\sqrt{\lambda})\frac{1}{2\sqrt{\lambda}}\,d\lambda.
R(e_1,e_2) = \frac{3}{4m_{12}}(-e_2\wedge e_5 + e_3\wedge e_5) + \frac{4m_{12}-3}{4m_{12}}\, e_4\wedge e_5 + \frac{1}{2}\,e_4\wedge e_6, R(e_1,e_3) = e_4\wedge e_5 + \frac{1}{2}\, e_4\wedge e_6 ,\enskip R(e_1,e_6) = -\frac{1}{4m_{12}}\, e_5\wedge e_6, R(e_2, e_6) = R(e_3, e_6) = -\frac{1}{2m_{12}}\,e_4\wedge e_5.
\left[V_{v}\right]^{\varepsilon_v}=\sum_{\underline{i}=(i_{1},\ldots, i_{q_{0}})}\sum_{l=1}^{\dim V_{\mu_{v}}^{\vee}}X_{\underline{i}}^{\ast}\otimes \varphi^{\varepsilon_v}_{v,\underline{i},l}\otimes b_{v,l}^{\vee}.
A(G) = \left({\begin{array}{*{20}c}{A(F)}{\left({\frac{{I_k \otimes \textbf{1}_{m - 1}^T}}{0}}\right)}{(I_k \otimes \textbf{1}_{m - 1}|0)}{I_k \otimes A(H_1 )}\end{array}}\right).
\int_{\Theta}v^{2}(w)\|\chi_{w}\pi_{F(w)}(I_{U}+L)^{\ast}\pi_{(L+G)F(w)}f\|^{2}d\mu(w) =\int_{\Theta}v^{2}(w)\|\chi_{w}\pi_{F(w)}(I_{U}+L)^{\ast}f\|^{2}d\mu(w) &\leq B\|(I_{U}+L)^{\ast}f\|^{2}&\leq B\|I_{U}+L\|^{2}\|f\|^{2}
dA_{t}^{\nu^{+}}\le \mathbf{1}_{\{u=h_{1}\}}(X_{t})(-f_{h_{1}}(X_{t})\,dt-dA_{t}^{\mu}+dA_{t}^{\alpha})^{+}&=\mathbf{1}_{\{u=h_{1}\}}(X_{t})(f_{h_{1}}(X_{t})\,dt+dA_{t}^{\mu}-dA_{t}^{\alpha})^{-},
\varphi_\varepsilon^*(u) =e^{-A}\left(1+\sum_{n=1}^{\infty}\left(1-e^{-A}\right)^nH^{*n}(u)\right) &=\varphi_\varepsilon^*(0)\left(1+\sum_{n=1}^{\infty}\left(\psi_\varepsilon^*(0)\right)^nH^{*n}(u)\right),\,u\geqslant0,
740t^4+2073t^3+92 < 740t^4+2073t^3+92\biggr\rvert_{t=0.6263716633 + 10^{-10}}& = 715.3525597141428299499534408273089356632640 & < 780.82181422656832973952 = 1698t^2+183t\biggr\rvert_{t=0.6263716633 - 10^{-10}}< 1698t^2+183t ,
\delta_i ={1 \over 2}\hat{\rho}_i \hat{\gamma}_i \tilde{b}_i ( 1 + \hat{\lambda}_i b_i + p_{i,i}) &+\sum_{j=i+1}^{i+N-1}\hat{\rho}_j\big({1 \over 2}\hat{\gamma}_j \tilde{b}_i(\hat{\lambda}_i \tilde{b}_j + p_{j,i})+\sum_{k=i}^{j-1}\hat{\gamma}_k \tilde{b}_i ( \hat{\lambda}_i b_k + p_{k,i})\big).
\|D_{z}F_R(t)\|_4 =\left\|\int_{B_R}D_{z}u(t,x) dx \right\|_4 \leq \int_{B_R}\|D_{z}u(t,x) \|_4 dx \leq C \int_{B_R}\int_0^t G_{t-s}(x-z) dsdx.
\begin{array}{rcl}{\cal R}^{(3)}= 3\partial a^{(2)\prime}+{\textstyle\frac{1}{2\pi\alpha^{\prime}}}C^{(3)}-3{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})b\partial b-3{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})B b & & -\left(C^{(1)}-{\textstyle\frac{m}{2}b}\right)\left({\cal F}+{\cal L}^{(2)}\right)\, , \end{array}
P^\omega(E_t, S_t)\leq P^\omega\left(E_{1,t}, S_{1,t}, F_{1,t}\right) \prod_{j=2}^{t/h(t)}P^\omega\left(E_{j,t}, S_{j,t}, F_{j,t}\:\bigg\vert\: S_{j-1,t}\, ,\:\bigcap_{k=1}^{j-1}(E_{k,t}, F_{k,t})\right) + \sum_{j=1}^{t/h(t)}P^\omega(F_{j,t}^c)
\begin{array}{ll}F^0\ast^g_{sp}\Gamma^v_{\nu}(t,x,v):=\int_{{\mathbb R}^{2d}}F^0(y)\Gamma^v_{\nu}(t,x,v;0,y)dy Q^S(F^{\nu},F^{\nu})\ast^g \Gamma^v_{\nu}(t,x,v):=\int_0^t\int_{{\mathbb R}^{2d}}Q^S(F^{\nu},F^{\nu})(s,y) \Gamma^v_{\nu}(t,x,v;s,y)dyds.\end{array}
ds_{}^{2}=ds_{}^{2}+ds_{}^{2}=\frac{4}{\mu_{A}^{\prime2}}d\mu_{A}^{\prime2}+\frac{1}{\sigma_{B}^{\prime2}}d\mu_{B}^{\prime2}+\frac{2}{\sigma_{B}^{\prime2}}d\sigma_{B}^{\prime2}.
E\{(U_{1}^{\top}\Gamma_{1}^{\top}\Gamma_{1}U_{1}-(\Sigma_{1}))^{4}\}=O(\{(\Gamma_{1}^{\top}\Gamma_{1}\Gamma_{1}^{\top}\Gamma_{1})\}^{2})=O(^{2}(\Sigma_{1}^{2}))=O(p^{2}).
\int d^{3}p \sum_{\lambda}\left[ U_{\vec{p}, \lambda}( t , \vec{x}) U_{\vec{p}, \lambda}^{+}( t , \vec{x}^{\, \prime}) + V_{\vec{p}, \lambda}( t , \vec{x}) V_{\vec{p}, \lambda}^{+}( t , \vec{x}^{\, \prime}) \right] = e^{- 3 \omega t}\delta^{3}( \vec{x}- \vec{x}^{\, \prime}) \, .
Z_{2}( \{b_{k}\}, \{a_{k}\}) = \prod_{k = 1}^{\infty}\exp \left( - 2 k \left( \frac{1 + q^{2 k}}{1 - q^{2 k}}( a_{k}a_{- k}+ b_{k}b_{- k}) - \frac{2 q^{k}}{1 - q^{2 k}}( a_{k}b_{- k}+ b_{k}a_{- k}) \right) \right) .
{\bf v}_{0}= \frac{\sqrt 2}l \left[ \begin{array}{c}{{\, 1}}\\{{\, \, a^{2}\,}}\\ \end{array}\right] , \, \, \,{\bf v}_{i}= \frac{{\sqrt 2}a}l \left[ \begin{array}{c}{{1 / J_{2}( j_{i})}}\\{{- 1 \,}}\\ \end{array}\right] \ .
I(g,A)= \frac{2k}{\pi}\int d^2 z (1-uv) [A- \frac{u \partial v - v\partial u}{4(1-uv)}][ \overline{A}+ \frac{u \overline{\partial}v - v \overline{\partial}u}{4 (1-uv)}]-\frac{k}{4 \pi}\int d^2 z \frac{\partial u \overline{\partial}v -\partial v \overline{\partial}u}{1-uv}
\delta \left( \zeta^{+}( 0 )^{\prime}- \zeta^{-}( 0 )^{\prime}\right) = - 2 \zeta^{+}( \delta \phi , 0 ) - 2 \zeta^{-}( \delta \phi , 0 ) + 2 \zeta^{\tilde{F}+ F}( \delta \phi , 0 ) \, .
|{\bf 2k-1}+\Phi(x)+\overline{\Phi(y)}| &=|{\bf 2k-1}+\Phi(x)-{\Phi(\overline{y})}|=\frac{|{\bf 2k-1}+\Phi(x)|}{|y|}|\Phi({\bf 2k-1}+\Phi(x))-\overline{y}|.
l i m \,_{Q \rightarrow q}\frac 1 k Q^{- N_{B}}[ Q^{- N_{B}}B^{-}, [ Q^{- N_{B}}B^{-}, [ . . . . [ Q^{- N_{B}}B^{-}, ( B^{+})^{k}]_{Q^{2 k}}. . . ]_{Q^{4}}]_{Q^{2}}]
\left. \begin{array}{l}{\gamma = x ( \partial \Sigma )}\\{={\gamma_{\mu}}^{\prime}\circ{\gamma_{\mu - 1}}^{\prime}\circ \cdots \circ{\gamma_{3}}^{\prime}\circ{\gamma_{2}}^{\prime}\circ \gamma_{g + 1}\circ \gamma_{g}\circ \cdots \circ \gamma_{2}\circ \gamma_{1}}\\ \end{array}\right.
k_{e_{n}}=\limsup_{z\rightarrow 0}\frac{\ln\|(z-T)^{-1}e_{n}\|_{p}}{\ln\|(z-T)^{-1}\|}&=\lim_{z\rightarrow 0}\frac{\ln\|(z-T)^{-1}e_{n}\|_{p}}{\ln\|(z-T)^{-1}e_{0}\|_p}\limsup_{z\rightarrow 0}\frac{\ln\|(z-T)^{-1}e_{0}\|_{p}}{\ln\|(z-T)^{-1}\|}&=k_{e_{0}}.
\left<p_i(t_0,s_0), r^0_iD^j_i\right>_{u^0_i}+ \left<p_i(t_1,s_1), r^1_iD^j_i\right>_{u^1_i}= \left<r^0_jD^i_j, p_j(t_0,s_0)\right>_{u^0_j}+ \left<r^1_jD^i_j, p_j(t_1,s_1)\right>_{u^1_j}.
y_1=x_2, \, y_2=\frac{1}{\sqrt{\mu}}x_3, \, y_3=x_1; &[y_1,y_2]= -\frac{1-w}{\sqrt{\mu}}y_1, \,[y_2,y_3]= \frac{1+w}{\sqrt{\mu}}y_3, \,[y_3,y_1]= 0.
\limsup_{M\rightarrow \infty}\int_{\frac{1}{6}Q_0}\min\left\lbrace\mathcal{S}^M ,( \mathcal{S}^M)^{\frac{1}{2}}\right\rbrace \zeta \leq c\,\min \Big\{2^{\frac{-k}{p}},2^{\frac{-k}{2}}\Big \}.
\mathcal{O}_i^+ := \left\{\left((A^+,\underline{A}^+),(B^+,\underline{B}^+)\right),\left((A^+,\underline{A}^+),(B^-,\underline{B}^-)\right) : A,B\in \binom{[n]}{k}\mbox{and}|A\cap B| = k-i \right\},
B_R^{+}= \left\{(x,y) \in \mathbb{R}_{+}^{N+1}\mid \sqrt{x^2+|y|^2}< R \right\}\Omega_R^{+}= \left\{(x,y) \in \mathbb{R}_{+}^{N+1}\mid \sqrt{x^2+|y|^2}> R \right\}\Gamma_R = \left\{(0,y) \in \partial \mathbb{R}_{+}^{N+1}\mid |y| \geq R \right\}.
\begin{array}{rcl}\varphi^{out}= (\varphi_1(0),\dots,\varphi_m(0),\varphi_{m+1}(1),\dots,\varphi_n(1)), [1mm] \varphi^{in}= (\varphi_1(1),\dots,\varphi_m(1),\varphi_{m+1}(0),\dots,\varphi_n(0)). \end{array}
P'''_\ell(\cos \phi)= \frac{\ell+1}{\sin^6 \phi}\sum_{u=0}^{2}\ell^u \sum_{n=0}^{1}\sum_{v=0}^{3}{_3}\omega_{u,v}(\cos \phi) p_{n,\ell+v}(\phi) + \phi^{-4}O(\ell).
{\bf \tau}( E ) = \frac{1}{\det{\bf \tau}^{- 1}}\left( \begin{array}{c c}{{- \frac{C}{C_{2}^{2}}- I_{5}- 2 \mu E I_{3}- 4 \mu^{2}E^{2}I ( E )}}&{{I_{3}+ 2 \mu E \, I ( E ) - \frac{1}{C_{2}}}}\\{{I_{3}+ 2 \mu E \, I ( E ) - \frac{1}{C_{2}}}}&{{- I ( E )}}\\ \end{array}\right) ,
\begin{aligned}x_i - x^L_i \geq 0 x^U_i - x_i \geq 0, i = 1, \ldots, d , f_{i}(x) - a_{i0}(x) \geq 0 a_{in}(x) - f_{i}(x) \geq 0, i = 1, \ldots, d , f_{i}(x) - u_{ij}(x) \geq 0 a_{ij}(x) - u_{ij}(x) \geq 0, i = 1, \ldots, d, \; j = 0, \ldots, n. \end{aligned}
{\bf E}\bigl[(M^{\star})^{q/\tau}_{(1/\tau, \tau\lambda_1, \tau\lambda_2)}\bigr] =\frac{\Gamma(1-q)}{\Gamma(1-q/\tau)}\,{\bf E}\bigl[(M^{\star})^q_{(\tau, \lambda_1, \lambda_2)}\bigr].
A_{\pi_{2}}^{}=d\pi_{2}=\left(\theta^{8}-\theta^{7}\right)\otimes e_{1}+\left(\theta^{9}+\theta^{6}\right)\otimes e_{2}+\left(\theta^{10}-\theta^{5}\right)\otimes e_{3}=A_{}^{}-\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right).
\int_{\Omega_1}F^p\left(\frac{\partial v}{\partial x_1}, k \nabla_{x'}v\right)\dx=k^{n-1}\int_{\Omega_k}F^p(\nabla u )\dx, \int_{\Omega_1}|v|^p=k^{n-1}\int_{\Omega_k}|u|^p\dx, \int_{\Omega_1}|v|^{p-2}v\dx=k^{n-1}\int_{\Omega_k}|u|^{p-2}u\dx=0.
\int_{I_R(x_0)}(e^{\alpha{{\tilde{w}(x,0)}}^2}-1)dx \ge \frac{1}{\| w\|_{\mathcal{C}}}\int_{I_R(x_0)}\sum_{k=1}^{\infty}\frac{\alpha^k |w|^{2k}}{k!}dx &= \frac{1}{\|w\|_{\mathcal{C}}}\int_{I_R(x_0)}(e^{\alpha (w(x,0))^2}-1) dx.
\mathbb{E}\left\{\mathbf{h}_l^{(g_k)}\left(\mathbf{h}_{l'}^{(g'_{k'})}\right)^H\right\}=\rho_l^{(g)}\mathbf{R}_l^{(g)}\delta_{gg'}\delta_{kk'}\delta_{ll'},\textrm{where}\sum_{l=0}^{L_g-1}\rho_l^{(g)}=1,\;\operatorname{Tr}\left\{\mathbf{R}_l^{(g)}\right\}=1
\left[(-1)^{\ell(\sigma_J)}\mathbf{z}^{\sigma_J(\mu_-)}\prod_{1 \leq j \leq n}\frac{(1+tz_jx)}{(1-z_j x)}\right] \Big\vert_{\mathbf{z}^\lambda}= \left[(-1)^{\ell(\sigma_J)}\prod_{1 \leq j \leq n}\frac{(1+tz_jx)}{(1-z_jx)}\right]\Big\vert_{\mathbf{z}^{\lambda-\sigma_J(\mu_-)}}&= (-1)^{|J|}(1+t)^{s(\lambda;\mu)+1-|J|}x^{|\lambda|-|\mu|},
\frac{\sum_{i \in Q}\exp(w_{f_i}+ \sum_{j \in Q \setminus \{i\}}\nu_{z_j})}{\exp(\nu_z + \sum_{j \in Q}\nu_{z_j}) + \sum_{i \in Q}\exp(w_{f_i}+ \sum_{j \in Q \setminus \{i\}}\nu_{z_j})}= \frac{\sum_{i \in Q}\exp(w_{f_i}- \nu_{z_i}- \nu_z)}{1 + \sum_{i \in Q}\exp(w_{f_i}- \nu_{z_i}- \nu_z)}.
2^{-ia}=\sup_{r\in(0,\infty)}\left(\frac{r}{2^ir}\right)^a\leq\sup_{x_0\in\mathbb{R}^n,\,r\in(0,\infty)}\left(\frac{|x_0|+r}{|x_0|+2^ir}\right)^a &\leq\sup_{x_0\in\mathbb{R}^n,\,r\in(0,\infty)}\left(\frac{|x_0|+r}{2^i|x_0|+2^ir}\right)^a=2^{-ia},
2 T^0_0 = w \Phi'^2 + \frac{{F}}{r^2}e^{-2\Phi}+\frac{4 \beta}{r^2}\left\{2w(w -1)(\Phi'' - 2\Phi'^2) +\Phi' w'(3w - 1)\right\}e^{-2\Phi}\; .
S(e^{i\pi}\theta_\pm z) = S(\omega^{-1/2}\theta_\pm z) =\sum_k(-i)^k c_k\left( \frac{3f}{2z^{3/2}\theta_\pm^{3/2}}\right)^k = 1 - i c_1\frac{3f}{2z^{3/2}}+f^2 z^{-5/2}O\left( 1 + |z|^{-1/2}\right).
W^{(1)}= \frac{a + b_1}{2}y^5+\left(- 9 a + 16 b_1 + \frac{b_2+ b}{2}\right)y^9 &+\left(\frac{153a- 553 b_1-c}{2}- 7 b_2 - 8 b\right)y^{13}&+(152192 - 408 a - 18592 b_1 - 339 b_2 + 60 b + 7 c) y^{17}+ \cdots.
-g_{11}\kappa_g=\left(-\frac{\partial \Gamma_{11}^2}{\partial s_2}-\frac{e^{-4s_2}\left(1-e^{-2s_2}\right)}{\left(1+e^{-2s_2}\right)^3}-\frac{e^{-4s_2}}{\left(1+e^{-2s_2}\right)^2}\right).
a_{n}^{\alpha}= \int_{0}^{\pi}\frac{d \sigma_{+}}{\pi}\exp \left[ 4 i n \frac{e_{\mu}X_{L}^{\mu}( \sigma_{+})}{e_{\mu}P_{L}^{\mu}}\right] \xi_{i}^{\alpha}\partial_{+}X_{L}^{i}( \sigma_{+})
I=\int_{0}^{2\pi}(\mathcal{J}_{\varepsilon}u^{\varepsilon})\partial_{x}^{5}(\mathcal{J}_{\varepsilon}u^{\varepsilon})-2(\mathcal{J}_{\varepsilon}u^{\varepsilon})^{2}(\mathcal{J}_{\varepsilon}u^{\varepsilon}_{x})+\omega(\mathcal{J}u^{\varepsilon})(\mathcal{J}_{\varepsilon}u^{\varepsilon}_{x})\dx.
L_0 = \frac{1}{2}\Big( \frac{1}{2}:\chi_{-\frac{1}{2}}\chi_{\frac{1}{2}}: -\frac{3}{2}:\chi_{-\frac{3}{2}}\chi_{\frac{3}{2}}: +\frac{5}{2}:\chi_{-\frac{5}{2}}\chi_{\frac{5}{2}}: -\dots \Big).
A_i(u) P_{\lambda}= g_i(u) \displaystyle \prod_{a=1}^{rN}\dfrac{u + t m(\lambda)_a + \hbar'(a - \dfrac{3}{2}+ \delta(j(\lambda)_a \leq i))}{u + t m(\lambda)_a+ \hbar'(a - \dfrac{3}{2})}P_{\lambda}
t r \ \gamma^{5}\gamma^{\lambda}\gamma^{\mu}\gamma^{\rho}\gamma^{\sigma}= - 4 \epsilon^{\lambda \mu \rho \sigma}\ \ \ ( \gamma^{5}= \gamma^{4}\gamma^{1}\gamma^{2}\gamma^{3}\ , \epsilon^{1 2 3 4}= + 1 ) ,
E \sim \int_0 dr ~r^2~ \left( \kappa_1 (f^\prime)^2 +\kappa_2 \frac{1}{r^2}(\sin f)^2 + \kappa_3 \frac{1}{r^2}(f^\prime)^2 (\sin f)^2 +\kappa_4 \frac{1}{r^4}(\sin f)^4 \right).
\|f\|_{\alpha, p}^p =c(\alpha)\int_{\mathbb{B}}|f(x)|^p\Phi^\alpha_n(|x|) (1-|x|^2)^{-n}\frac{dV(x)}{{\omega_n}}&={c(\alpha)}\int_0^1 \Psi(r) dr\int_{\mathbb{S}}|f(r\zeta)|^p d\sigma(\zeta) &\le \|f\|^p_{p},
h_{p,s}:\mathbb C^\times\rightarrow\mathbb C,h_{p,s}=Q^{m_1}p^{m_2}s^{m_3},m_1\neq0, &H_{p,s}:\mathbb C\rightarrow\mathbb C,H_{p,s}(z):=G_{\boldsymbol\epsilon}(z,p^{a_1}s^{b_1},\dots,p^{a_n}s^{b_n}).