harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J ={R^2\over 2 \pi\alpha'}e\omega\int_0^{2\pi}d\sigma \, \sin^2 \theta =4{R^2\over 2 \pi \alpha'}\omega \int_0^{\theta_0}d\theta \, \left(1 - \omega^2 \sin^2\theta \right )^{-1/2}\sin^2\theta\ .
\langle\mathcal{F}(x^*),\tilde{\eta}-x^*\rangle_{\mathcal{L}^2}= \int_{\Omega}\langle\mathcal{F}(x^*)(\omega),\tilde{\eta}(\omega)-x^*(\omega)\rangle P(d(\omega)) = \int_{A_{k,r}}\langle\mathcal{F}(x^*)(\omega),\eta(\omega)-x^*(\omega)\rangle P(d(\omega)) < -\frac{1}{k}P(A_{k,r}),
\left( \begin{array}{c}{{u_{n}^{1}( y )}}\\{{u_{n}^{2}( y )}}\\ \end{array}\right) \propto \left( \begin{array}{c}{{\cos \left( \mu_{n}y \right) + \frac{m_{-}}{\mu_{n}}\sin \left( \mu_{n}y \right)}}\\{{\frac{M_{n}- m_{+}}{\mu_{n}}\sin \left( \mu_{n}y \right)}}\\ \end{array}\right)
\overline{w}(x;\gamma, \delta,N) = \frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_x N!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_x x!}, \overline{h}_n (\gamma,\delta,N)= \left[\binom{\gamma+n}{n}\binom{N+\delta-n}{N-n}\right]^{-1}.
\mathfrak{S}_{n;p,q}^{z,w}:=\sum_{\substack{1\leq j\leq d(n,p,q)}}h_{p,q}^{j}(z,\bar{z})\overline{h_{p,q}^{j}(w,\bar{w})}&=\left( |z||w|\right)^{p+q}\sum_{\substack{1\leq j\leq d(n,p,q)}}h_{p,q}^{j}\left( \frac{z}{|z|},\frac{\bar{z}}{|z|}\right) \overline{h_{p,q}^{j}\left( \frac{w}{|w|},\frac{\bar{w}}{|w|}\right)}.
|W_2| \leq \biggl|W_{21}+\frac{1}{n}\biggr| + 2|W_{22}| + |W_{23}| = O\biggl(\frac{k^{1/2}}{n}\max\biggl\{\frac{k^{\beta/d}}{n^{\beta/d}}\, , \, \frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\biggr\}\biggr).
q\cdot N_d =q^2+q\phi(b)+\frac{\phi(-b)}{q-1}\sum_{m=0}^{q-2}G_{-m}G_{md}G_{-m(d-1)+\frac{q-1}{2}}G_{\frac{q-1}{2}}T^m\left(\frac{b^{d-1}}{a^d}\right).
\frac{M^{2}}{\cosh^{2}( M \omega )}[ \omega^{\prime \prime}- \ddot{\omega}- \frac 1 \tau \dot{\omega}] + 2 \frac{M^{3}\sinh ( M \omega )}{\cosh^{3}( M \omega )}[ \dot{\omega}^{2}- \omega^{\prime}{}^{\; 2}] = 2 \frac{M^{3}\sinh ( M \omega )}{\cosh^{3}( M \omega )}[ \dot{\omega}^{2}- \omega^{\prime}{}^{\; 2}] ,
E\left(\frac{a}{s^r}X_{ij}\right)E\left(s^dxY_{kl}\right)E\left(\frac{a}{s^r}X_{ij}\right)^{-1}=\left[E\left(\frac{a}{s^r}X_{ij}\right),E\left(s^dxY_{kl}\right)\right]E\left(s^dxY_{kl}\right) &= E\left(s^dxY_{kl}\right). &\textnormal{(by Lemma~\ref{l21}(i) and by Lemma~\ref{l21}~(iii)})
\left|{i_{1}, i_{2}, \ldots , i_{n}}\right\rangle ={\textnormal{T r}}( \phi_{i_{1}}\phi_{i_{2}}\ldots \phi_{i_{n}}) \left|{i_{1}}\right\rangle \left|{i_{2}}\right\rangle \ldots \left|{i_{n}}\right\rangle
\int_{t_{0}}^{t_{0}^{*}}d t \sqrt{\mu^{2}+ \phi^{2}( t )}= \int_{t_{0}}^{t_{0}^{*}}d t{\frac{d S_{0}}{d t}}\sqrt{\Omega + S_{0}^{2}}= \int_{i \sqrt{\Omega}}^{- i \sqrt{\Omega}}d S_{0}\sqrt{\Omega + S_{0}^{2}}= -{\frac{i \Omega \pi}{2}}.
\tilde{h}_1(t)=\psi(t)\tilde{f}(t)-\psi(t)e^{it\partial_x^2}\tilde{u}_0(x)-\psi(t)\mathcal{S}\big(\alpha \psi_T u\psi_Tv+\beta|\psi_Tu|^2\psi_Tu\big)(0,t), &\tilde{h}_2(t)=\psi(t)\tilde{g}(t)-\psi(t)e^{-t\partial_x^3}\tilde{v}_0(x)-\psi(t)\mathcal{K}\big(\gamma\partial_x(|\psi_T^2u|^2)-\tfrac{1}{2}\partial_x((\psi_T^2v)^2)\big)(x,t)+\big)(0,t).
\prod_{i=1}^k \int_{\mathbb{R}^n}f_i(x_i) dx_i\le \Bigl( \int_{\mathbb{R}^{n}}\rho^{\frac{1}{k}}\Bigl( \frac{k(k-1)}{2}|u|^2 \Bigr) du\Bigr)^k,
\sum_{k=0}^{\infty}\frac{1}{(2k+1)(2k+2)}=\frac{1}{1\times2}+\frac{1}{3\times4}+\frac{1}{5\times6}+\frac{1}{7\times8}+\frac{1}{9\times10}+\cdot\cdot\cdot=ln2
\mathbb{E}[ C_1^] = \int_0^\infty\int_{x_1}^\infty \frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}\cdot\frac{1}{2}\log_2(1 + \xi x_1)dx_2 dx_1 = \frac{e^{\frac{2}{\beta\xi}}}{\ln(4)}E_1\left(\frac{2}{\beta\xi}\right),
\nu(\Gamma_{\mathbf{m}, k}(\underline{S}))= \max_{\substack{0\leq r < l_{\mathbf{m},k}0\leq s \leq r}}\nu \left(\frac{\Delta_{\mathbf{m}}(\underline{a}_0, \underline{a}_1, \ldots, \underline{a}_r)}{\Delta_{\mathbf{m}}(s;\underline{a}_0, \underline{a}_1, \ldots, \underline{a}_{r-1})}\right).
T_2^{*}:=\frac{1}{(cq)_N}\sum_{n=1}^N \frac{\left(cq\right)_{N-n}(cq)^n}{(q)_n (q)_{N-n}}\sum_{k=1}^n \left[\begin{matrix}n k\end{matrix}\right] \frac{\left(\frac{qc}{d}\right)_k \left(\frac{d}{c}\right)_{n-k}\left(\frac{d}{c}\right)^{k}}{1- q^k},
\begin{aligned}G\left(\left|m_u\left(B\left(z, r_1\right)\right)-m_u\left(B\left(z, r_2\right)\right)\right|\right) & \leq 4 b^2 2^{s+\alpha p_{0}}[u]_{W^{\alpha,G}_{s}(X,d,\mu)}^{p_{0}}\left(r_1^{\alpha p_{0}-s}+\frac{r_2^{\alpha p_{0}}}{r_1^s}\right)\end{aligned}
\frac{1}{( 4 \pi )^{2}}\int d^{8}z ( \frac{W^{2}\bar{W}^{2}}{\Phi^{2}\bar{\Phi}^{2}}+ \sum_{k = 1}^{\infty}( k + 1 ) \cdot \frac{W^{2}\bar{W}^{2}}{\Phi^{2 + k}\bar{\Phi}^{2 + k}}\cdot ( - ( \bar{Q}Q + \bar{\tilde{Q}}\tilde{Q}) )^{k}) .
A =\left(-i \int_{-\infty}^0x\bar{u}_{xx}u_xdx+i\alpha\int_{-\infty}^0x\bar{u}vu_xdx+i\beta \int_{-\infty}^0x|u|^2\bar{u}u_xdx \right) = - \int_{-\infty}^0x\bar{u}_{xx}u_xdx+\alpha\,\int_{-\infty}^0x\bar{u}vu_xdx+\beta\, \int_{-\infty}^0x|u|^2\bar{u}u_xdx :=A_1+A_2+A_3.
\omega_{\hat{\alpha}}^{\alpha \rho}= f^{\prime}, \quad \omega_{\hat{\theta}}^{\theta \rho}= g^{\prime}e^{g - f}, \quad \omega_{\hat{\varphi}}^{\varphi \rho}= g^{\prime}e^{g - f}\sin \theta , \quad \omega_{\hat{\varphi}}^{\varphi \theta}= \cos \theta ,
C_1(n,s):=2^{1+(n-s)/s}\,\Big(\frac{4\,|B|^{(n-s)/n}\,P(B)}{s\,(1-s)\,P_s(B)}\Big)^{1/s}\,,C_2(n,s):=\frac{2|B|^{(n-s)/n}}{(1-s)\,P_s(B)}\,.
\widetilde{\mathbf{y}}(t)\leq e^{(t-t_0)\widetilde{\mathbf{A}}_\nu}\widetilde{\mathbf{y}}(t_0)+\frac{1}{\nu}\int_{t_0}^te^{(t-u)\widetilde{\mathbf{A}}_\nu}\widetilde{\mathbf{M}}du.
\det\left( A-\lambda_{k}I\right)_{\operatorname{row}m=b}=\det\left(A_{G\backslash\left\{m\right\}}-\lambda_{k}I\right) \left(b_{m}-b_{\backslash m}^{T}\left( A_{G\backslash\left\{m\right\}}-\lambda_{k}I\right)^{-1}\left( a_{m}\right)_{\backslash m}\right)
\Psi[A_i]=\int D\phi\exp\left\{-\frac{1}{2}\int d^2xd^2y \left[A_i(x)-\frac{1}{g}\partial_i\phi(x)\right]G^{-1}(x-y)\left[A_i(y)-\frac{1}{g}\partial_i\phi_i(y)\right]\right\}
D(\rho_A^1(K_0(A)), \rho_A(K_0(A))) =\sup\{{\rm dist}(x,\rho_A^1(K_0(A)))\vert\, x\in\overline{\rho_A(K_0(A))}\}&=\sup\{{\rm dist}(x,\rho_A^1(K_0(A)))\vert\, x\in \rho_A(K_0(A))\}.
\left[ \frac{\partial_{x}^{2}P_{n}}{\partial_{x}P_{n}}\right]_{x=x_{n,k}}-2x_{n,k}+\frac{1}{x_{n,k}-z}+\frac{1}{x_{n,k}+z}-\frac{1}{x_{n,k}-\zeta_{n}(z)}-\frac{1}{x_{n,k}+\zeta_{n}(z)}=0,
\sum_{\substack{p>2 p\notin A}}\sum_{k=1}^{K-1}f((A^k)_{2^k p}) \le (1-2^{1-K})\sum_{\substack{2<p\le 23 p\notin A}}b_pf(p) + \sum_{\substack{p> 23 p\notin A}}f(p)\Big((1-2^{1-J})b_p + 2^{-J}\frac{\log p}{\log(2p)}\Big) & \le (1-2^{1-K})\sum_{\substack{2<p\le 23 p\notin A}}b_pf(p) + \sum_{\substack{p> 23 p\notin A}}b_p f(p),
L^D_+=\left(\begin{array}{ll}(qt^{-1})^{S^3}t^Z \Omega S^- 0 (qt)^{-S^3}t^{-Z}\end{array}\right)\ \ \ ,\ \ \L^D_-=\left(\begin{array}{ll}(qt)^{-S^3}t^Z 0 -\Omega S^+ (qt^{-1})^{S^3}t^{-Z}\end{array}\right)\ ,
\tilde{D}(w_i) \leq \tilde{d}(w_i)-1 = \sharp\{j\in \{1,2,\dots,m-1\}\,|\,v_j \prec w_i\}&\leq \sharp\{j\in \{1,2,\dots,m-1\}\mid \tilde{D}(v_j)\leq i-2\}.
W(\Phi )=\sum_{i=0}^{N}M_{i}W(\imath e_{i})+\frac{1}{2}\alpha\sum_{l=0}^{N}s_{N-l}\sum_{m=0}^{l}{\rm Tr}(\Psi \Phi_{0}^{m}\Psi \Phi_{0}^{2l-m})+{\cal O}(\Psi^{3})\,.
\left( \ell_{1}, \dot{\ell}_{1}, h_{1}) ( \ell_{2}, \dot{\ell}_{2}, h_{2}) = ( \ell_{1}+ h_{1}\ell_{2}h_{1}^{- 1}, \dot{\ell}_{1}+ [ \ell_{1}, h_{1}\ell_{2}h_{1}^{- 1}] + h_{1}\dot{\ell}_{2}h_{1}^{- 1}, h_{1}h_{2}\right) \ .
\left[~\mbox{\begin{minipage}{15cm}All the Riemann curvatures of the metric $G$ vanish on $\omega\times \{0\}$:$$R_{ijkl}(x',0) = 0\mbox{for all}\; x'\in \omega\mbox{and all}\;i,j,k,l=1\ldots 3.$$\end{minipage}}\right.
P=\frac{1}{2\alpha'\dot{t}}\;\frac{\dot{R}^2-R^2}{(1+\frac{K}{4}R^2)^2}=\frac{1}{2\sqrt{b}\alpha'^2}[\frac{\dot{r}^2}{1-Kr^2}-r^2]=\frac{b\alpha'^2-2r^2}{2\sqrt{b}\alpha'^2}.
\int_{\omega_+}U_{\delta,e_n}^\frac{8n(1+\gamma)}{(n+2)(n-2)}=\begin{cases}O(\delta^\frac{n(n-2)(1+\gamma)}{n+2}) 4\leq n<6, O(\delta^\frac{4n(1+\gamma)}{n+2}) n\geq 6.\end{cases}
\frac{k \mu}{2 \pi}\widehat{L}\equiv - J^{-}( \theta ) ( L ( \theta ) - \partial_{\theta}J^{3}/ 2 ) , \; \frac{k \nu}{2 \pi}\widehat{\widetilde{L}}\equiv - \widetilde{J}^{+}( \theta ) ( \widetilde{L}( \theta ) + \partial_{\theta}\widetilde{J}^{3}/ 2 )
y(t)=\sum_{n=1}^{\infty}[lim_{r\rightarrow0}(\frac{x^{n}}{n!}\frac{d^{n-1}}{dr^{n-1}}[r^{n}(\frac{7}{2}(arcsin(\sqrt{r})-\sqrt{r-r^{2}}))^{-\frac{2}{3}n}])].
\mathbb{P}\left(|L_n(z)|\leq e^{-n\epsilon}\right) \leq C_\epsilon \left\{\sum_{i=1}^{l_n}\left(1-Q\left(a_{i_k},\left(d(z,K_z)+diam(K_z)\right)e^{-n\epsilon}\right)\right) \right\}^{-\frac{1}{2}}
\int\limits_{\psi_0}^{\pi/2}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi = (\cos\psi_0)^{n-4}\int\limits_0^\infty \int\limits_{S^{n-2}}\int\limits_{z\tan\psi_0}^{\infty}f_z(r\omega)(r^2-z^2\tan^2\psi_0)^{(n-4)/2}r dr d\omega dz.
(-\Delta)^m\xi(x) =\frac{R_0^{2m}}{|x|^{4m}}((-\Delta)^m\bar{w})\left(\frac{R_0x}{|x|^2}\right) & =\left(\frac{R_0}{|x|^2}\right)^{2m}K\left(\frac{R_0x}{|x|^2}\right)e^{2m\xi(x)}& =:f(x).
_{2}\phi_{4}\left(\sqrt{a},-\sqrt{a};i\sqrt{b}q^{3/4},-i\sqrt{b}q^{3/4},\sqrt{az},-\sqrt{az};\sqrt{q},-bzq\right)=\frac{\left(z;q\right)_{\infty}}{\left(az;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}S_{n}\left(b;q\right)z^{n}}{\left(-bq^{3/2};q\right)_{n}}.
\mathbf{x}_{1}^{a}=(_{c}\mathbf{L}^{-1})_{b}^{a}\mathbf{x}^{b}\longleftrightarrow(_{c}\mathbf{L}^{-1})_{B^{\prime}}^{A^{\prime}}(_{c}\mathbf{L}^{-1})_{B}^{A}\mathbf{x}^{BB^{\prime}}.
\alpha^{\prime \prime \prime}(s)=\left(\begin{array}{c}0,-\kappa^{\prime}\sinh \left( m\int \kappa (s)\,ds\right) -m\cosh \left(m\int \kappa (s)\,ds\right) \kappa^{2}(s), \cosh \left( m\int \kappa (s)\,ds\right) \kappa^{\prime}+m\sinh \left(m\int \kappa (s)\,ds\right) \kappa^{2}(s)\end{array}\right).
M^k_t - M^k_s = \frac{t - t^k_m}{\Delta t^k}(m^k_{t^k_{m+1}}- m^k_{t^k_{m}}) + \frac{t^k_{n+1}- s}{\Delta t^k}(m^k_{t^k_{n+1}}- m^k_{t^k_{n}}) + \sum_{l=n+1}^{m}(m^k_{t^k_{l+1}}- m^k_{t^k_{l}}).
F_{_{( 1 )}}= \left[ \begin{array}{c c c c}{0}&{- 1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{3 \alpha^{2}- k}\\{0}&{0}&{k - 3 \alpha^{2}}&{0}\\ \end{array}\right] .
{\cal J}= \frac{1}{4 \pi}\int \left( \frac{( | R_{z}|^{2}+ | R_{{\bar{z}}}|^{2}) ( 1 + | z |^{2})^{2}}{( 1 + | R |^{2})^{2}}\right) \frac{2 i ~ \mathrm{d}z \mathrm{d}{\bar{z}}}{( 1 + | z |^{2})^{2}}
{\rm Im}{\cal N}_{\Lambda\Sigma,\Gamma\Delta}f^{\Gamma\Delta}_{~~AB}=-{1\over 2\sqrt{2}}\left(e^{-\Phi_+}\tau^{\left(+\right)\Lambda\Sigma}\tau^{\left(+\right)AB}+e^{-\Phi_-}\tau^{\left(-\right)\Lambda\Sigma}\tau^{\left(-\right)AB}\right).
P_{\mu\nu}=\left(\begin{array}[c]{cc}0 F_{q}\medskip -F_{p}ie_{pqr}G_{r}\end{array}\right) ,Q^{\mu\nu}=\left(\begin{array}[c]{cc}0 -G_{q}\medskip G_{p}ie_{pqr}F_{r}\end{array}\right) ,
\Lambda(i)=\mathfrak{p}_F^{a_{n}+i}w_n\oplus\mathfrak{p}_F^{a_{(n-1)}+i}w_{n-1}\oplus\cdots\oplus \mathfrak{p}_F^{a_{(-n+1)}+i}w_{-n+1}\oplus \mathfrak{p}_F^{a_{(-n)}+i}w_{-n}.
\int_Q \partial_t u (v_h + \theta h \partial_t v_h ) \,dx dt + \nabla_x u \cdot \nabla_x (v_h + \theta h \partial_t v_h ) \,dx dt - \int_{\partial Q}n_x \cdot \nabla_x u (v_h + \theta h \partial_t v_h)\, ds =\int_Q f (v_h + \theta h \partial_t v_h ) \,dx dt.
\int_{l_+}\frac{d \zeta'}{\zeta'}\frac{\chi_{\gamma_e}(\zeta')}{1 - \chi_{\gamma_e}(\zeta')}= \sum_{n > 0}2e^{in\theta_{\gamma_e}}K_0(2\pi R |nc|) \int_{l_-}\frac{d \zeta'}{\zeta'}\frac{\chi_{\gamma_e}(\zeta')^{-1}}{1 - \chi_{\gamma_e}(\zeta')^{-1}}= \sum_{n < 0}2e^{in\theta_{\gamma_e}}K_0(2\pi R |nc|)
(\begin{matrix}\partial_{t}\tilde{u}_{k}(t)\\ \partial_{t}\tilde{v}_{k}(t)\end{matrix})=-k^{2}(\begin{matrix}D_{u}\tilde{u}_{k}(t)\\D_{v}\tilde{v}_{k}(t)\end{matrix})+R^{\prime}(\begin{matrix}\tilde{u}_{k}(t)\\ \tilde{v}_{k}(t)\end{matrix}).
\int_{x_1,\dots,x_n=-\infty}^\infty\, \prod_{1\leq j<k\leq n}|x_j-x_k|^{2c}\prod_{j=1}^n e^{-x_j^2/2}\,dx_j =(2\pi)^{n/2}\prod_{j=1}^n\frac{\Gamma(1+jc)}{\Gamma(1+c)}.
\sqrt{1-v^2}\sum_{n}\exp \Big[-\frac{\alpha^{\prime}(1-v^2)}{2R^2t}(n-\tilde qR\hat J_0)^2\Big]=\sqrt{\frac{2R^2t}{\alpha^{\prime}}}\sum_{w}\exp \Big[-\frac{2\pi R^2t}{\alpha^{\prime}(1-v^2)}w^2+2\pi i \tilde q R w \hat J_0\Big].
\{f,g\}.\{f',g'\}(\xi,\eta) =\{f,g\}(f'(\xi),g'(\xi)+Df'(\xi)\eta) * =(f\circ f'(\xi),g\circ f'(\xi)+Df(f'(\xi))(g'(\xi)+Df'(\xi)\eta)) * =(f\circ f'(\xi),h(\xi)+D(f\circ f')(\xi)\eta),
\left|\frac{1}{\zeta(2 + 2s)}\right| \ = \ \prod_p\left|1 - \frac{1}{p^{2 + s}}\right| \ \leq \ \prod_p\left(\frac{p^2 + 1}{p^2}\right) \ \leq \ \prod_{p}\left(\frac{p^2}{p^2 - 1}\right) \ = \ \zeta(2)
\aligned Y_{t}=X_{t_1}+\int_{t_1}^tb(s,Y_{s})ds+\int_{t_1}^th_{s}^\varepsilon ds+\int_{t_1}^t\sigma(s,Y_{s})dB_s &\quad+\int_{t_1}^{t+}\int_Uf_1(s,Y_{s-},u)\tilde{N}_k(ds,du)+\int_{t_1}^{t+}\int_Uf_2(s,Y_{s-},u)N_k(ds,du).\endaligned
\frac{A}{(r^*)^{\tilde p-2}}+ B(r^*)^p\,=\, A^{\frac{\tilde p}{2\tilde p-2}}B^{\frac{\tilde p-2}{2\tilde p-2}}\left[\left(\frac{\tilde p}{\tilde p-2}\right)^{\frac{\tilde p-2}{2\tilde p-2}}+ \left(\frac{\tilde p-2}{\tilde p}\right)^{\frac{\tilde p}{\tilde p-2}}\right]\,\ge\, A^{\frac{\tilde p}{2\tilde p-2}}B^{\frac{\tilde p-2}{2\tilde p-2}}.
{\bf 1 , 2 , 5 , 9 , 1 5 , 2 3 , 3 6}, 5 0 , 7 1 , 9 6 , 1 2 7 , 1 6 5 , 2 1 3 , 2 6 6 , 3 3 3 , 4 0 9 , 4 9 8 , 6 0 0 , 7 2 0 , 8 5 1 , 1 0 0 5 , \ldots
\Big{(}\mathcal{D}_tu_h^{n-\theta},u_h^{n-\theta}\Big{)}+\|\nabla u_h^{n-\theta}\|^2= -\gamma(\nabla \sigma_h^{n-\theta},\nabla u_h^{n-\theta})-(f^{n-\theta}(u_h),u_h^{n-\theta})+(g^{n-\theta},v_h) = -\gamma\|\sigma_h^{n-\theta}\|^2-(f^{n-\theta}(u_h),u_h^{n-\theta})+(g^{n-\theta},u_h^{n-\theta}).
\int_{\mathbb{R}^{N}}|u_k(x)|^{\alpha(x)}|v_k(x)|^{\beta(x)}\,dx= \int_{\mathbb{R}^{N}}\Phi^{q(x)}_{R}(x)|u_k(x)|^{\alpha(x)}|v_k(x)|^{\beta(x)}\,dx +\int_{\mathbb{R}^{N}}(1-\Phi^{q(x)}_{R}(x))|u_k(x)|^{\alpha(x)}|v_k(x)|^{\beta(x)}\,dx.
I+\sum_{0< i+j\le 2}B_{i,j}\alpha^i \beta^j = \begin{pmatrix}1 \alpha \, c \beta\, d \alpha \, c 1+\alpha^2 \, c^2 \alpha \beta\, cd \beta \, d \alpha\beta \, cd 1+\beta^2 \, d^2 \end{pmatrix},
\begin{aligned}\epsilon^{-1}\mathbb{E}[\int_0^T(Z^{u_1^\epsilon}(t)l_1^{u_1^\epsilon}(t)-Z(t)l_1(t))dt]\rightarrow \mathbb{E}\bigg[\int_0^TZ(t)\big(l_{1x}(t)x_1^1(t)+l_{1y}(t)y_1^1(t)+l_{1z}(t)z_1^1(t)+\sum_{j=1}^{2}l_{1z_j}(t)z_{j1}^1(t) + l_{1v_1}(t)v_1(t)\big)dt+\int_0^Tl_1(t)Z_1^1(t)dt\bigg].\end{aligned}
\frac{M_{i}}{k^{2}- M_{i}^{2}+ i 0}\bar{D}^{2}\delta^{4}( \theta_{1}- \theta_{2}) \quad \mathrm{a n d}\quad \frac{M_{i}}{k^{2}- M_{i}^{2}+ i 0}D^{2}\delta^{4}( \theta_{1}- \theta_{2})
\mathcal{F}(\Delta,\mathrm{Ind}_{H}^{\Delta}\rho) =\left\{\frac{\mathcal{F}(\Delta,\mathrm{Ind}_{H}^{\Delta}1_H)}{\mathcal{F}(H,1_H)}\right\}^{\mathrm{dim}\rho}\cdot\mathcal{F}(H,\rho) =\lambda_{H}^{\Delta}(\mathcal{F})^{\mathrm{dim}\rho}\mathcal{F}(H,\rho),
\vert \Xi_{0,p}[\alpha,\beta] \vert^2=\frac{\prod_{j=3}^{p-1}(-1)^{2m_{\beta_j}}q^{4im_{\beta_j}\rho_{\beta_j}}}{{\psi}_2{}_{(\alpha_1,\alpha_2,\beta_3)}\prod_{j=3}^{p-2}{\psi}_2{}_{(\beta_j,\alpha_j,\beta_{j+1})}{\psi}_2{}_{(\beta_{p-1},\alpha_{p-1},\alpha_{p})}}\Xi_{0,p}[\alpha,\beta]^2
\tilde{K_{1}}= \left[\begin{matrix}0 -3 0 -2 3 0 \end{matrix}\right],\ \tilde{K_{2}}= \left[\begin{matrix}0 5 0 0 0 2 \end{matrix}\right],\ \tilde{K_{3}}=\left[\begin{matrix}5 0 -3 0 0 3 \end{matrix}\right].
\begin{aligned}_xI_b^{1-\alpha}u'(x_{n-1}) =\sum_{j=n}^{P}\int_{I_j}\omega_{1-\alpha}(s-x_{n-1})u'(s)\,ds &= \int_{I_P}\omega_{1-\alpha}(s-x_{n-1})u'(s)\,ds +\sum_{j=n+1}^{P}\int_{I_j}\omega_{1-\alpha}(q-x_{n})u'(q-h)\,dq,\end{aligned}
\displaystyle \lim\limits_{s\to 0^+}s\widetilde{D}_s(\varphi)=\frac{\prod\limits_{i=1}^N h_\ell}{\sum_{\ell=1}^Nh_\ell}(2\pi)^{N-1}\psi_0(0)=\frac{D}{n}(2\pi)^{N-1}2^{1-c}\psi_0(0)
\begin{aligned}\gamma_{k}= \frac{|\tilde{\mathbf{h}}_{k}^H \mathbf{w}_{k}|^2}{\sum_{p \in \mathcal{K}, p\ne k}|\tilde{\mathbf{h}}_{k}^H \mathbf{w}_{p}|^2 + \sigma_{k}^2}, \forall i \in \{\mathrm{t,r}\}, \forall k \in \mathcal{K}_i. \end{aligned}
R_1 \le I(X_1;Y_k) + I(X_2;Y_k|X_1) + I(X_3;Y_k|X_1,X_2), R_2 \le I(W_1;Y_k|X_1,X_2,X_3) + I(W_2;Y_k|X_1,X_2,X_3,W_1).
\int_0^ta(s) \frac{d}{ds}{\Vert u_n(s;x,f_1) - u_n(s;x,f_2)\Vert}^2ds \le \ 2\int_0^n ds \int_s^nb\Vert f_1-f_2\Vert \cdot \Vert u_n(\tau;x,f_1) - u_n(\tau; x,f_2)\Vert \, d\tau = \ 2\int_0^n sb\Vert f_1-f_2\Vert\cdot \Vert u_n(s;x,f_1) - u_n(s; x,f_2) \Vert \, ds.
S_{l , l^{\prime}}^{( 2 )}( E ) = \delta_{l 0}\, \delta_{l^{\prime}0}\; \frac{\operatorname{l n}\left( E / | E_{_\mathrm{( g s )}}| \right) + i \pi}{\operatorname{l n}\left( E / | E_{_\mathrm{( g s )}}| \right) - i \pi}\; .
\mathbb{P}(E) =\frac{t}{\pi}\int_0^\pi\mathbb{P}\bigl( \lbrace\alpha\in B\rbrace \lbrace x=z\rbrace\bigr)\mathbb{P}\bigl( \bigl\lbrace\eta'+z[\pi]\leq t\bigr\rbrace \bigr)\,dz, \mathbb{P}(E) =\frac{t}{\pi}\mathbb{P}\bigl( \lbrace\alpha\in B\rbrace\bigr).
\begin{array}{c}{{B^{-}B^{+}- Q B^{+}B^{-}~ = ~ Q^{- N_{B}}}}\\{{B^{-}B^{+}- Q^{- 1}B^{+}B^{-}~ = ~ Q^{N_{B}}}}\\{{Q^{N_{B}}B^{+}Q^{- N_{B}}~ = ~ Q B^{+}}}\\{{Q^{N_{B}}B^{-}Q^{- N_{B}}~ = ~ Q^{- 1}B^{-}}}\\{{Q^{N_{B}}Q^{- N_{B}}~ = ~ Q^{- N_{B}}Q^{N_{B}}~ = ~ 1}}\\ \end{array}
\frac{dx^a}{d\tau}\rightarrow \frac{dx'^a}{d\tau}= \frac{d}{d\tau}(\delta^a_{\;\;b}+ \xi^i(\Gamma_i)^a_{\;\;b})x^b &= G^a_{\;\;b}\dot{x}^b + \dot{\xi}^i(\Gamma_i)^a_{\;\;b}x^b.
(f)_{\mu}^{\sim}({\bf l}) =\int_{0}^{2\pi R}d\theta_{1}\cdots \int_{0}^{2\pi R}d\theta_{d}\exp \bigl[ iR^{- 1}\sum_{k = 1}^{d}l_{k}\theta_{k}\bigr] (f)_{\mu}(\mbox{\boldmath{$\theta$}}).
\delta_{[k = \frac{1}{2}]}\delta_{[h = a^2]}\bigg(R'_h X^{\frac{1}{2}}\log X +R'_hX^{\frac{1}{2}}\frac{\pi}{Y}\bigg)\frac{\exp(\frac{\pi}{4Y})}{Y}+ R_k^k X^k\frac{\exp(\frac{\pi k^2}{Y})}{Y}+ O(\frac{X^{k-\frac{1}{2}}}{Y}) + O(X^{\frac{k}{2}+ \epsilon}Y^{3k + \frac{17}{2}+ \epsilon}).
S_{^{1}T^{\ast}T^{\ast}G}^{\prime}=\left\{\left( g,\mu,ad_{\frac{\delta\bar{H}}{\delta \mu}}^{\ast}\mu -T_{e}^{\ast}R_{g}\frac{\delta \bar{H}}{\delta g},-\frac{\delta \bar{H}}{\delta \mu}\right) \in \text{}^{1}T^{\ast}T^{\ast}G:\left( g,\mu \right) \in G\circledS\mathfrak{g}^{\ast}\right\}
r_\ell(L) \le L\sum\limits_{k=1}^{K_\ell(L)}\frac{N(\ell^k)}{\ell^k}+dK_\ell(L) &\le L\sum\limits_{k=1}^{d}\frac{d}{\ell}+L\sum\limits_{k=d+1}^{\infty}2\ell^{-k/d}+dK_\ell(L) &=d^2L\ell^{-1}+\frac{2L\ell^{-1}}{\ell^{1/d}-1}+dK_\ell(L).
Q^{\mu_1\mu_2;\nu_1}= \frac{1}{2}\big(\omega^{\mu_1\nu_1}S^{\mu_2}+ \omega^{\mu_2\nu_1}S^{\mu_1}+ \omega^{\nu_1\mu_1}S^{\mu_2}+ \omega^{\nu_1\mu_2}S^{\mu_2}\big)
C_n =C_n(a,b,c,d;q) &=\frac{2\pi(abcdq^{2n-1},abcdq^{2n};q)_\infty}{4^n(q^{n+1},abq^n,acq^n,adq^n,bcq^n,bdq^n,cdq^n,abcdq^{n-1};q)_\infty}.
df \big( \sum_{\sigma \in S_n}sgn(\sigma)^{m - 1}[x_{\sigma(1)}| \dots | x_{\sigma(n)}] \big) & = \sum_{\sigma \in S_n}sgn(\sigma)^{m - 1}( (-1)^{\delta_n'}+ 1) f([x_{\sigma(1)}| \dots | x_{\sigma(n-1)}] ) x_{\sigma(n)}
\| \sum_j \partial^{\alpha}f_{>j-2}D^{s,\alpha}g_j \|_p = \| \sum_j \partial^{\alpha}f_j D^{s,\alpha}g_{<j+2}\|_p \lesssim \| D^{s_1}f\|_{p_1}\| D^{s_2}g\|_{p_2}; \| \sum_j \partial^{\beta}g_{>j-2}D^{s,\beta}f_j \|_p = \| \sum_j \partial^{\beta}g_j D^{s,\beta}f_{<j+2}\| \lesssim \| D^{s_1}f \|_{p_1}\cdot\| D^{s_2}g \|_{p_2}.
\cdot\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}\sum_{m_{3}=-j_{3}}^{j_{3}}C_{m_{1},m_{2},m_{12}}^{j_{1},j_{2},j_{12}}C_{m_{12},m_{3},m}^{j_{12},j_{3},j}C_{m_{2},m_{3},m_{23}}^{j_{2},j_{3},j_{23}}C_{m_{23},m_{1},m}^{j_{23},j_{1},j}
M^{\bf b}_f :=\texttt{v}^{b_1}_{f(1)}\otimes\texttt{v}^{b_2}_{f(2)}\otimes\cdots\otimes\texttt{v}^{b_{m+n}}_{f(m+n)}, \qquad \texttt{v}^{b_i}:=\begin{cases}v,b_i={0}, w,b_i={1}.\end{cases}
\begin{aligned}\mathbf{S}\mathbf{\Xi}\,\, +\,\, \mathbf{I}_\omega^T\mathbf{\Phi}\quad =\mathbf{S}\mathbf{P}_{\omega,K}, \mathbf{I}_\omega \mathbf{\Xi}=\mathbf{0}_{N_\omega \times 2^d},\end{aligned}
f_{\,\kappa}\,(x)= \sqrt{\frac{2\beta\kappa}{\pi}}\left(1+\frac{1}{2}\kappa\right)\frac{\Gamma\!\left(\frac{1}{2\kappa}+\frac{1}{4}\right)}{\Gamma\!\left(\frac{1}{2\kappa}-\frac{1}{4}\right)}\,\,\, \exp_{\kappa}(-\beta \,x^2) \ \ .
P_{\rm{tot}}=\sum_{n=1}^{N}\frac{p_{n}}{\xi}+ \sum_{n=1}^{N}\sum_{k=1}^{K}x_k\frac{p_{k,n}}{\xi}+\sum_{k=1}^{K}x_k\frac{q_{k}}{\xi}+P_\mathrm{c}.
\left\langle{^0 t}, \left( w - w_{\Lambda}\right) x^{\alpha_{1}}x^{\alpha_{2}}\right\rangle = - \left({D ( \Lambda )^{\alpha_{1}}}_{\beta_{1}}{D ( \Lambda )^{\alpha_{2}}}_{\beta_{2}}- \delta_{\beta_{1}}^{\alpha_{1}}\delta_{\beta_{2}}^{\alpha_{2}}\right) c^{\beta_{1}\beta_{2}}.
{\bf \Omega}\wedge{\bf \Omega}= \sum_{m , n = 2}^{\infty}\Big [{\frac{\partial{B}_{n}}{\partial t_{m}}}-{\frac{\partial{B}_{m}}{\partial t_{n}}}+ \{{B}_{n},{B}_{m}\}\Big ] d \lambda \wedge d x \wedge d t_{m}\wedge d t_{n}
I = \int d^{4}x d^{4}y J^{\mu a}( x ) \check{G}_{\mu}^{a b}( x - y ) \eta^{b}( y ) + \frac 1 2 \int d^{4}x d^{4}y A^{\mu a}( x ) \check{\Gamma}_{\mu \nu}^{a b}( x - y ) A^{\nu b}( y ) + \cdots
f(x):= 8 \pi^3\left(\frac xe\right)^{6x}\Gamma^{-6}(x+1)=\frac{\exp(E_0(x))}{\Phi_0(x)}, f(x):= \frac{\exp(E_k(x))}{\Phi_0(x)+\mathrm{MC}_k(x)},k\ge 1,
\partial_t\gamma_T^{s\to t}\circ\alpha_{V(t)}^{(t-s)T}(A) =\gamma_T^{s\to t}(-\i[\widetilde{H}_t,\alpha_{V(t)}^{(t-s)T}(A)])+\gamma_T^{s\to t}(T\alpha_{V(t)}^{(t-s)T}\circ\delta_{V(t)}(A)+\i[\widetilde{H}_t,\alpha_{V(t)}^{(t-s)T}(A)]) &=T\gamma_T^{s\to t}\circ\alpha_{V(t)}^{(t-s)T}\circ\delta_{V(t)}(A).
\|g_k\|_{p, \alpha}^p = \left(\frac{\alpha p}{2\pi}\right)\int_{\mathbb{C}}|z|^{kp}e^{-(\alpha p/2) |z|^2}\,\lambda(dz)= \left(\frac{\alpha p}{2\pi}\right)\int_0^\infty r\int_{S^{1}}r^{kp}e^{-(\alpha p/2) r^2}dSdr &= (\alpha p) \int_0^\infty (r^2)^{kp/2}re^{-(\alpha p/2) r^2}dr.
H_2(x)-H_2(x+1) = \sum_{k=0}^{\infty}\frac{2\left(p(k+x)+q\right)}{\left(p(k+x)+q\right)^4+s\left(p(k+x)+q\right)^2+r}-\sum_{k=0}^{\infty}\frac{2\left(p(k+1+x)+q\right)}{\left(p(k+1+x)+q\right)^4+s\left(p(k+1+x)+q\right)^2+r}= \frac{2\left(px+q\right)}{\left(px+q\right)^4+s\left(px+q\right)^2+r}.
\Psi^{(0)}(\lambda,x) =\tilde{\Psi}^{(0)}(\lambda,x)\lambda^{\frac{1}{2}\Theta\sigma_3}{\rm as}~\lambda\rightarrow0, \Psi^{(1)}(\lambda,x) =\tilde{\Psi}^{(1)}(\lambda,x)(\lambda-1)^{\frac{1}{2}\Theta\sigma_3}{\rm as}~\lambda\rightarrow1,
R_1^{n-\theta}= \mathcal{D}_t(u(t_{n-\theta})-u^{n-\theta})+\gamma\bigtriangleup (\sigma(t_{n-\theta})-\sigma^{n-\theta})-\bigtriangleup (u(t_{n-\theta})-u^{n-\theta})+(f(u(t_{n-\theta}))-f^{n-\theta}(u)) = O(\Delta t^2),
\begin{array}{ll}\hat{\bar{p}}= \left( \frac{1}{|\Omega|}\int\limits_{\Omega}\hat{p}_0^{\frac{1}{\gamma}}\,\mathrm{d}x \right)^{\gamma},\hat{\bar{S}}= \frac{1}{\Omega}\int\limits_{\Omega}\hat{S}_0 \,\mathrm{d}x, \hat{\bar{\varrho}}= \frac{1}{\Omega}\int\limits_{\Omega}\hat{\varrho}_0 \,\mathrm{d}x,\end{array}
\begin{aligned}|h^{\epsilon}(t)|\leq B\left(\rho^{2}+|Y^{\epsilon}(t-)|^{2}\right)^{\frac{1}{2}}, &|\gamma^{\epsilon}(t)|\leq M\left(\rho^{2}+|Y^{\epsilon}(t-)|^{2}\right)^{\frac{1}{2}}.\end{aligned}
\mathrm{P}_{m}^i = 1 - \mathrm{P}(_{m}^l >R_l,\forall l \in\{0, \cdots, i\}) = \mathrm{P}\left( L(||x_m||)>\min\left\{\frac{\rho \zeta_{l}}{{\epsilon}_l}, \forall l \in\{0, \cdots, i\}\right\}\right).
\varepsilon_{N,l-1}-\varepsilon_{N,l}=(1-b_N\varepsilon_N)\sqrt{N-1}\left(\frac{1}{\sqrt{(N-l)-1}}-\frac{1}{\sqrt{N-l}}\right)> 0, &\varepsilon_{N}-\varepsilon_{N,l}=(1-b_N\varepsilon_N) \left(\sqrt{\frac{N-1}{(N-l)-1}}-1\right)+\varepsilon_N(1-b_N)> 0.
\widetilde{\mathcal E}=z(\mathop{\mathrm{Ker}}S M_0)\oplus\bigl(\mathbb C\mathbf n \cap \mathop{\mathrm{Ker}}v^*\bigr), \mathcal E =z(\mathop{\mathrm{Ker}}M_0), \mathsf E =z(\mathop{\mathrm{Ker}}M_0\cap\mathop{\mathrm{Ker}}M_1)=z(\mathop{\mathrm{Ker}}M_0\cap\mathop{\mathrm{Ker}}m_0).
\frac{1}{y}=\{0,\frac{1+(y_{R}-y)(\frac{1}{y})_{L}}{y_{R}},\frac{1+(y_{L}-y)(\frac{1}{y})_{R}}{y_{L}}|\frac{1+(y_{L}-y)(\frac{1}{y})_{L}}{y_{L}},\frac{1+(y_{R}-y)(\frac{1}{y})_{R}}{y_{R}}\}
\left. \partial^2_t (\Delta_g \eta^3) \right|_{t=0}= \,\delta^{ij}\partial^2_{ij}\partial_t v^3(0) + \partial_t g^{ij}(0) \partial^2_{ij}v^3(0)- \delta^{ij}\partial^k v^3(0) \partial^2_{ij}v_k (0) = \, \delta^{ij}\partial^2_{ij}\partial_t v^3(0) + F_0,
{\cal L}= a^3 \left [6 F(\varphi) \left ( \frac{\dot{a}}{a}\right )^2+ 6 F'(\varphi) \dot{\varphi}\left ( \frac{\dot{a}}{a}\right )+ \frac{\gamma}{2}\dot{\varphi}^2 - V(\varphi) \right ] \ .
E[F|\mathcal{F}_{m\Delta}]=\sum_{j_{m+1}=0}^{\infty}\ldots\sum_{j_{M=0}}^{\infty}\Big(\prod\limits_{k=m+1}^{M}\gamma(k-1,j_{k})\Big)D_{(m+1)\Delta}^{j_{m+1}}\ldots D_{M\Delta}^{j_{M}}F.
\eta^{\mu \nu}k_{\mu}k_{\nu}\int_{w_{-}}^{w_{+}}d w e^{A}F_{( T )}^{2}= - \int_{w_{-}}^{w_{+}}d w e^{- 3 A}\left[ ( e^{2 A}F_{( T )})^{\cdot}\right]^{2}\leq 0 .