harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Y_{L}^{\alpha \beta}({\tilde{x}_{L}},{\tilde{\theta}}) ={{\tilde{x}_{L}}^{\alpha \beta}}+ \frac{1}{4}\left({{\tilde{\theta}}^{\alpha}}\psi^{\beta}({\tilde{x}_{L}},{\tilde{\theta}}) + \alpha \leftrightarrow \beta \right) ,
{\rm E}|G_t^{d,p}| ={\rm E}_A|G_t^{d,p}| &={\rm E}\sum_{x\in A}1_{\{\zeta_t^{d,p}(x)=1\}}=\sum_{x\in A}P(\zeta_t^{d,p}(x)=1) &=\sum_{x\in A}P(T_x>t)=|A|e^{-t}
M_{{\bf s},q}(j) = \left[\begin{array}{cccc}s_q s_{q+1}\cdots s_{q+j-1}s_{q+1}s_{q+2}\cdots s_{q+j}\vdots \vdots \ddots \vdots s_{q+j-1}s_{q+j}\cdots s_{q+2j-2}\end{array}\right] .
M_n^m = \sqrt{\frac{2n+1}{4 \pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}R^{n+1}\sin \theta_1 \times & \int_0^{2\pi}\left[ \sigma_1(\phi) P_n^{|m|}(\cos \theta_1) + \sigma_2(\phi) P_n^{|m|}(-\cos \theta_1) \right] e^{-im\phi}\,d\phi \, .
\sum_{x \in V}p(x, S_i) = \sum_{j = 1}^{\ln n}\sum_{x \in L_j}p(x, S_i) \leq \frac{n}{\ln n}\sum_{j=1}^{\ln n}z_j e^{-z_j + 1}\leq \frac{n}{\ln n}\sum_{j=1}^{\infty}z_j e^{-z_j + 1}.
H'(z^{(l)})= \begin{bmatrix}\nabla^2 F(x^{(l)}) + \sum_{i = 1}^m \lambda_i^{(l)}\nabla^2 g_i(x^{(l)}) 0 Dg(x^{(l)})^T A^T Dg(x^{(l)}) I 0 0 A 0 0 0 0 \Lambda^{(l)}S^{(l)}0\end{bmatrix},
\lim_{\varepsilon\rightarrow 0^+}\frac{|Du|(x+\varepsilon Q)}{\varepsilon^{N-1}}=0,\,\,\,\,\lim_{\varepsilon\rightarrow 0^+}\frac{|Du|(x+\varepsilon Q)}{\varepsilon^{N}}=+\infty, \,\, |D^cu|-\hbox{a.e.}\,\, x \in\Omega.
\int_{\Lambda^\infty}f^{m,v}\, dM = \sum_{\lambda \in D_v^J}c^{m,v}_\lambda M(Z(\lambda)) &= \sum_{\lambda \in D_v^J}c^{m,v}_\lambda \rho(\Lambda)^{(-j_1, \ldots, j_k)}x^\Lambda_{s(\lambda)}&= 0
\rho(\mathrm{d}x) =\sigma^{2}\delta_{0}(\mathrm{d}x)+\frac{x^{2}}{1+x^{2}}\nu(\mathrm{d}x), \eta =\gamma-\int_{\mathbb{R}}x\left( 1_{\left[ -1,1\right]}(x)-\frac{1}{1+x^{2}}\right) \nu(\mathrm{d}x),
{\rm Tr}e^{i F_9}-{\rm tr}e^{i F_9}\,{\rm tr}e^{i F_{\bar9}}+{\rm Tr}_S e^{i F_{\bar9}}={1\over2}\, ({\rm Str}e^{i F_9})^2 -{1\over 2}\,{\rm Str}e^{2i F_9}.
\Delta_{q}^{\alpha}\,a(x,\xi) := u_{\xi}^{-1}(x)\int_{\Omega}K_{A_{q^{\alpha}}}(x,y)u_{\xi}(y)dy{}& = u_{\xi}^{-1}(x)\int_{\Omega}q^{\alpha}(x,y)K_{A}(x,y)u_{\xi}(y)dy,
K_{[3,2]}:=K_{5,2}\cong \frac{\mathbb{C}\langle z_1,z_2 \rangle}{(z_1^2, z_2^3, z_1z_2,z_2^2z_1)}K_{[2,3]}:=K_{5,3}\cong \frac{\mathbb{C}\langle z_1,z_2 \rangle}{(z_1^3, z_2^2, z_1z_2,z_2z_1^2)}
\operatorname{col}\left\{u_j \right\}_{j=1}^N = \mathcal{B}_{N\times N}^{-1}\left[-\mathcal{G}_N+K \right]y^{N}(t), &K = \operatorname{diag}\left\{K_j \right\}_{j=1}^N \in \mathbb{R}^{N\times 3N},
\Delta \Gamma_{V}^{( 2 )}= \mathrm{R e}\int d^{2}\theta \, \frac{d^{4}p}{( 2 \pi )^{4}}W_{a}( p ) C^{a b}W_{b}( - p ) \Big ( \tilde{f}_{1}+ \tilde{f}_{2}+ \tilde{f}_{3}\Big ) ,
p^f_g(\bar{x};\gamma)=\arg\min_{x}[f(\bar{x})+ \langle \nabla f(\bar{x}), x-\bar{x}\rangle +\frac{\gamma}{2}\|x-\bar{x}\|^2+g(x)] &G^f_g(\bar{x};\gamma)=\gamma (\bar{x}-p^f_g(\bar{x};\gamma)).
S_\lambda(\delta_\eta)(\nu) = \begin{cases}0, \nu \not\in R_\lambda \tau^{d(\lambda)}(\nu) \not\in G_i \delta_\eta(\tau^{d(\lambda)}(\nu))), \end{cases}& =\begin{cases}1, \lambda \eta = \nu x_j \cdots x_{i-1}0, \end{cases}
\frac{\partial F}{\partial r}= \frac{\partial\check{F}}{\partial r}+\frac{\partial u}{\partial r}\check{\nu}\frac{\partial F}{\partial \theta_i}= \frac{\partial u}{\partial \theta_i}\check{\nu}+ r\left(\sin \alpha - \frac{u}{r}\cos \alpha\right) \frac{\partial \Phi}{\partial \theta_i}.
\sigma_{a b s}^{S}= \frac{1}{8}A_{h}( \omega^{2}- m^{2})^{2}r_{e}\frac{( e^{\frac{\omega - m \phi_{K}}{T_{H}}}- 1 )}{( e^{\frac{( \omega - m )}{T_{H}}}- 1 ) ( e^{\frac{1}{2}\pi ( \omega + m ) r_{e}}- 1 )}.
h_{4}= - \frac{r_{g}^{2}}{1 6}\frac{\cosh^{2}u}{\left( 1 + \cosh u \right)^{4}}\left( \frac{\cosh u_{m}( f , \rho , \theta , t ) - \cosh u}{\cosh u_{m}( f , \rho , \theta , t ) + \cosh u}\right)^{2}, h_{5}= \frac{\sinh^{2}u \sin^{2}v}{\sinh^{2}u + \sin^{2}v},
\sup_{\zeta\in [0,t]}\|\Phi(u)(\cdot,\zeta)\| \le \|\xi\|_\infty + \int_0^t p(s)(1+ \|\xi\|_\infty+\sup_{\zeta\in [0,s]}\|u(\zeta)\| )ds &\le \|\xi\|_\infty + \int_0^t p(s)(1+ \|\xi\|_\infty+\psi(s))ds = \psi(t),
I^+_\lambda(\omega)= \frac{1}{2}\|\omega\|_X^2-\frac{\lambda}{2}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega^+(x,0)|^2\ln|\omega^+(x,0)|dx &+\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega^+(x,0)|^2dx-\frac{1}{2_\alpha^*}\displaystyle\int_{\mathbb{R}^N}b(x)|\omega^+(x,0)|^{2_\alpha^*}dx,
S_{( 5 )}= \frac{1}{2 \kappa_{( 5 )}^{2}}\int d^{4}x \int d y \sqrt{- g_{( 5 )}}\left[ R^{( 5 )}+ 1 2 k^{2}\right] - \int d^{4}x \left[ \sqrt{- g_{( + )}}\lambda_{( + )}+ \sqrt{- g_{( - )}}\lambda_{( - )}\right] ,
\sum_{n=0}^{\infty}Y_{n}^{\left( k\right)}\left( x;\lambda \right) \frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left( x\right)_{n}\lambda^{n}\frac{t^{n}}{n!}\sum_{n=0}^{\infty}Y_{n}^{\left( k\right)}\left( \lambda \right) \frac{t^{n}}{n!}.
\nabla_{i}\nabla_{j}b^{kl}=-\nabla_{i}(b^{kp}b^{lq}\nabla_{j}h_{pq}) &=-b^{kp}b^{ql}\nabla_{i}\nabla_{j}h_{pq}+b^{ks}b^{pt}b^{lq}\nabla_{i}h_{st}\nabla_{j}h_{pq}+b^{kp}b^{ls}b^{qt}\nabla_{i}h_{st}\nabla_{j}h_{pq}&=-b^{kp}b^{ql}\nabla_{i}\nabla_{j}h_{pq}+2b^{ks}b^{pt}b^{lq}\nabla_{i}h_{st}\nabla_{j}h_{pq}.
\omega^{(+)}_n \;=\; \frac{2 \pi}{\beta}n \;=\; \pi n M \;\;,\;\;\;\omega^{(-)}_n \;=\; \frac{2 \pi}{\beta}(n+\frac{1}{2}) \;=\; \pi(n+\frac{1}{2}) M \;,
C(p,q) = -2 \partial_p \left(\log \left| \frac{\theta_1(p) \theta_1(q)}{\theta_1(p-q)}\right|- \frac{2\pi}{\Im \tau}\left(\Im p\right) \left(\Im q\right)\right) dp = - \left( \frac{\theta_1'(p)}{\theta_1(p)}- \frac{\theta_1'(p-q)}{\theta_1(p-q)}+ \frac{2\pi i}{\Im \tau}\Im q\right) dp.
c_{a_i,a_0}=\frac{u_\gamma (a_i)^2}{c_{\gamma}(a_0)^2}=\frac{u_\gamma(a_i)u_{\gamma}(s_i^{-1}a_i)}{u_\gamma (s_0^{-1}a_0)}=\frac{u_\gamma(s_0^{-1}\gamma_i)}{u_{\gamma}(s_0^{-1}a_0)}\frac{u_{\gamma}(a_i)}{u_{\gamma}(a_0)}
f^{i+j-1}=d^{i+j-2}\alpha_{jj}^i+\alpha_{jj}^{i+1}e_{\lbrace N-1 \rbrace}^{i+j-1}= d^{-N+j}_{\lbrace N-1\rbrace}\alpha_{(j+1)(j+1)}^{i-1}+\alpha_{(j+1)(j+1)}^{i}e^{j-1}
\left( \int_{\gamma_{\rho}^{+}}+ \sum_{k}\int_{C_{\rho}( i y_{k})}\right) \frac{H_{\mu}^{( 1 )}( \lambda z )}{H_{\nu}^{( 1 )}( z )}F ( z ) d z + \left( \int_{\gamma_{\rho}^{-}}+ \sum_{k}\int_{C_{\rho}( - i y_{k})}\right) \frac{H_{\mu}^{( 2 )}( \lambda z )}{H_{\nu}^{( 2 )}( z )}F ( z ) d z
\int_{A}P_{r_{kk_0}}(x,y) dy=C\int_{\tilde{A}}\left(\frac{1-|\tilde{x}|^2}{|\tilde{y}|^2-1}\right)^{s/2}\cdot\frac{1}{|\tilde{x}-\tilde{y}|^n}d\tilde{y}\geq \mu,
u^{\zeta}_{0,x_{i}}(0) = u_{0,x_{i}}(0) -\epsilon u^{\zeta}_{0,x_{i}x_{i}}(0) + H_{i}(0,0,u^{\zeta}_{0,x_{i}}(0)) = - \epsilon u^{\zeta}_{0,x_{1}x_{1}}(0) + H_{1}(0,0,u^{\zeta}_{0,x_{1}}(0))
\mathcal{G}^{1}(i,J,(k) ) = \left( (-p-x_1+i){\mathcal{A}}_{1,k}^{J}+(p-x_1-1-i)\tilde{\mathcal{D}}_{1,k}^{J}\right) \mathcal{G}^{0}(i+1,J\setminus k ,\emptyset) -(-p-x_1+1+i)(p-x_1-2-i)\mathcal{C}_{1,k}^{J}\,,
\delta\alpha_1\gamma(\mathbf{s}) =\{U_a(ab^m, ab^{m+1})^2, U_a(ab^m, ab^{m+1})^3, U_b(ab^m, ab^{m+1})^2, U_b(ab^m, ab^{m+1})^3\}&\subseteq\{ab^m, ab^{m+1}\}^{{+}}=:\mathbf{s}_2.
\nabla_{\dot\gamma(t)}\nu=\sum_{k=1}^n\left(\frac{d\nu_k(t)}{dt}+\sum_{i,j=1}^n\gamma_j'(t)\nu_i(t)\Gamma_{ji}^k(\gamma(t))\right) \left.\frac{\partial\}{\partial x_k}\right|_{\gamma(t)}\,\in T_{\gamma(t)}M.
(2 n_c +2 - n_f) \sum_{r=0}^{{n_f}/2-1}{}_{n_f}C_r \, +{2 n_c +2 - n_f \o 2}\,{}_{n_f}\!C_{n_f/2}= (2 n_c +2 - n_f) \, 2^{n_f-1}, \quad (n_f ={\hbox{\rm even}}),
\tilde{f}^{( 1 )}= \left( \begin{array}{c c c c}{0}&{- \delta_{i j}}&{2 a_{i}}&{0}\\{\delta_{i j}}&{0}&{0}&{0}\\{- 2 a_{i}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\ \end{array}\right) ,
G_1(x) =- \ln[x+ \beta(x)] + \ln\biggl(\frac1x-\frac1{2x^2}+\frac1{6x^3}-\frac1{30x^5}+\frac1{42x^7}\biggr) &+2\biggl( \ln x+\frac1{2x}-\frac1{12x^2}+\frac1{240x^4}\biggr)+\frac1{120x^4}.
0 =\ln\rho_i+\tilde{\xi}_i'\sigma_i^2+\ln(1+\tilde{\xi}_i'I_{\ell{i}}')+\sum_{k\ne{i},k\ne\ell}\ln(1+\tilde{\xi}_i'I_{ki}) &=\ln\rho_i+\tilde{\xi}_i\sigma_i^2+\ln(1+\tilde{\xi}_iI_{\ell{i}})+\sum_{k\ne{i},k\ne\ell}\ln(1+\tilde{\xi}_iI_{ki}) &<\ln\rho_i+\tilde{\xi}_i\sigma_i^2+\ln(1+\tilde{\xi}_iI_{\ell{i}}')+\sum_{k\ne{i},k\ne\ell}\ln(1+\tilde{\xi}_iI_{ki}),
A^{*2q}A^{2q}=A^{*q}A^{q*}A^qA^q=A^{*q}(A^{*}A)^qA^q &=A^{*q}A^q(A^{*}A)^q=(A^{*}A)^{2q}A^{*sq+1}A^{sq+1}=A^{*sq}A^{*}AA^{sq}=A^{*sq}A^{sq}A^{*}A=(A^{*}A)^{sq+1},
\mathcal{H}=\int d^{\,2}\,x\,\left[ \,\frac{1}{2}\,\left( \xi \,_{5}^{2}+\xi\,_{6}^{2}\right) +\frac{1}{2}\,\left( \partial_{1}\,\xi_{3}-\partial_{2}\,\xi_{2}\right)^{\,2}+\frac{1}{2}\,m^{2}\,\left( \xi \,_{2}^{2}+\xi\,_{3}^{2}\right) +m\left( \xi \,_{2}\,\xi \,_{6}-\xi \,_{3}\,\xi\,_{5}\right) \right] ;
\alpha \bigl( [ q_1 , q_0 ] \bigr) = [q_0] , \beta \bigl( [ q_1, q_0 ] \bigr) = [q_1] , m \bigl( [q_2 , q_1 ] , [ q_1 , q_0 ] \bigr) = [ q_2 , q_0 ] , \epsilon \bigl( [q] \bigr) = [ q, q ] , i \bigl( [ q_1 , q_0] \bigr) = [q_0, q_1].
\sum_{\imath=1}^{\infty}\frac{1}{\imath^\alpha}\bigg(\sum_{\jmath=\imath}^{\infty}\varrho(\jmath)\bigg)^p \leq \c{C}(\alpha,p)\sum_{\imath=1}^{\infty}\frac{1}{\imath^{\alpha-p}}\varrho^p(\imath);
s_0=2,\s_{2^{n-1}}=-2,\s_{2^{n-2}}=s_{3\cdot2^{n-2}}=0, s_{2^{n-3}}=s_{7\cdot2^{n-3}}=\sqrt{2},\s_{3\cdot2^{n-3}}=s_{5\cdot2^{n-3}}=-\sqrt{2}.
p_{(m,j)}\Big(e^{t \, d/dx}\phi\Big) = \sup_{|x|\leqslant j}\left| \dfrac{d^m}{dx^m}\phi(t+x) \right|\leqslant \sup_{|x|\leqslant j + \lceil |t| \rceil}\left| \dfrac{d^m}{dx^m}\phi(t+x) \right|= p_{(m, j + \lceil |t| \rceil)}(\phi),
\int_0^\infty \P(s<Z^+_1\leq s+m)ds= \sum_{k=0}^\infty \int_{km}^{(k+1)m}\P(s<Z^+_1\leq s+m)ds & \leq\sum_{k=0}^\infty \int_{km}^{(k+1)m}\Big(\P(km<Z^+_1\leq (k+1)m) + \P((k+1)m<Z^+_1\leq (k+2)m) \Big) ds & =\sum_{k=0}^\infty m \Big(\P(km<Z^+_1\leq (k+1)m) + \P((k+1)m<Z^+_1\leq (k+2)m) \Big) \leq 2 m < \infty.
g_tAg_t^{-1}, g_tBg_t^{-1}= \left[ \begin{smallmatrix}* t* 0 0 0 * 0 0 t* t^2* * t* 0 t* 0 * \end{smallmatrix}\right], \ \g_ta= \left[ \begin{smallmatrix}* 0 t* 0 \end{smallmatrix}\right], \ \bg_t^{-1}= \left[ \begin{smallmatrix}* t* 0 0 \end{smallmatrix}\right].
H^{\lambda,\theta,T}(y,y') = \frac{T}{n}\sum_{z^n=(-1)^{\theta_2}}z^{1+h-h'}\frac{e^{- (\lambda+\theta_1 \pi i +Tz)[t'-t]}}{1 - e^{- (\lambda+\theta_1 \pi i+Tz)}},
\sigma_{ij}(x^-,y^-;x^+) =\int\!\frac{dP^-}{4\pi}\frac{dp^+}{4\pi}\frac{dk^+}{4\pi}e^{-\frac{i}{2}P^- x^+}e^{-\frac{i}{2}(p^+x^--k^+y^-)}~\sigma_{ij}(p^+,k^+;P^-),
u^s (t,y) =\frac{1}{2\sqrt{\pi t}}\int^{+\infty}_{0}\Big(e^{-\frac{(y-\tilde{y})^2}{4 t}}-e^{-\frac{(y+\tilde{y})^2}{4 t}}\Big) u_0^s (\tilde{y}) d\tilde{y}&=\frac{1}{\sqrt{\pi}}\Big(\int^{+\infty}_{-{\frac{y}{2\sqrt t}}}e^{-\xi^2}u_0^s (2\sqrt t \xi +y) d\xi - \int^{+\infty}_{{\frac{y}{2\sqrt t}}}e^{-\xi^2}u_0^s (2\sqrt t \xi -y)d\xi \Big),
\nabla^N_{\partial_a}L =-l_a^c \partial_c-\eta_a L \nabla^N_{\partial_a}\partial_b =\gamma_{ab}^c\partial_c-l_{ab}\underline L-n_{ab}L \nabla^N_{\partial_a}\underline L =-n_a^c\partial_c+\eta_a \underline L,
\left(\frac{V_{k}(n,m)-kn/m}{\sqrt{n/m}}\right)_{1\leq k\leq m}\xRightarrow[n\to \infty]{}\left(\max_{\mathbf{t}\in I_{k,m}}\sum_{j=1}^k\sum_{l=j}^{m-k+j}\sqrt{(m-1)/m}\left(B_l(t_{j,l})-B_l(t_{j,l-1})\right)\right)_{1\leq k\leq m},
b_{\ell 1}=\sum\limits_{j \in \mathcal{J}}^{}u_j b_{j 1}+pc_{\ell 1}{b}_{\ell 2}=\sum\limits_{j \in \mathcal{J}}^{}v_j b_{j 1}+pc_{\ell 2}u_j, v_j \in \mathbb{Z}_p c_{\ell 1}, c_{\ell 2}\in \mathbb{Z}_{p^{a-1}}^{2n}
X(t) = X(0) + \sum_{k=1}^{K+1}\zeta_k \int_0^t 1_{\left[ \frac{q_{k-1}(s-)}{\overline \lambda_0}, \frac{q_{k}(s-)}{\overline \lambda_0}\right)}( \xi_{R_0(s-)})dR_0(s) R_0(s) = Y\big( \int_0^t \overline \lambda_0 ds\big),
S = - \int d^{4}\sigma \sqrt{- \mathrm{d e t}\, ( G_{i j}+ e^{- \frac{\phi}{2}}F_{i j}- b_{i j})}+ \int \left( C_{4}+ C_{2}\wedge ( e^{- \frac{\phi}{2}}F - b_{2}) + \frac{1}{2}C_{0}F \wedge F \right) \, .
\{\tilde f^+_j, \tilde b^+_k\}^2 = \frac 12 \{\{\tilde f^+_j, \tilde b^+_k\}, \{\tilde f^+_j, \tilde b^+_k\}\}& = \frac12 \{\{\{\tilde f^+_j, \tilde b^+_k\}, \tilde f^+_j \}, \tilde b^+_k\}- \frac12 [ \tilde f^+_j, [ \{\tilde f^+_j, \tilde b^+_k\}, \tilde b^+_k] ] = 0
\sqrt{3}\hat{T}_{R}={\frac{-{\frac{f_{1}+ f_{2}}{2}}+ f_{3}}{\sum_{k = 1}^{3}f_{k}}}={\frac{-{\frac{f_{1}f_{2}( f_{1}+ f_{2})}{2}}+ 1}{f_{1}f_{2}( f_{1}+ f_{2}) + 1}}
\aligned\frac{4(n-1)}{n-2}\Delta \psi +R\psi= -\frac{n-1}{n}\tau^{2a_0/t}\psi^{N-1}&+\bigg[|\sigma+LW|^2+\bigg(2\max\big\{\|\varphi_0\|_{L^\infty},\,2\big\}-\|\varphi\|_{L^\infty}\bigg)_+\bigg(\|\sigma+LW_0\|^2_{L^\infty}+k_0^2\bigg)\bigg]\psi^{-N-1}, -\frac{1}{2}L^*LW= \frac{n-1}{n}\psi^Nd\tau^{a_0/t}.\endaligned
[P,R]_{\scriptscriptstyle SN}\; \hat{\longrightarrow}\; \sum_{i=1}^p(-1)^{i+1}\widehat{X}_{(1)}\cdots{\widehat{\widehat X}}_{(i)}\cdots\widehat{X}_{(p)}\{-\{\widehat{X}_{(i)}, Q\},\widehat{R}\}
\prod_{v\in S \cup \{v_0\}}\gamma( s, \Pi_v \otimes \mathcal{X}_v , R, \Psi_v) = \frac{L^{S \cup \{v_0\}}( s, \Pi \otimes \mathcal{X}_v, R)}{L^{S \cup \{v_0 \}}( 1-s ,\Pi \otimes \mathcal{X}_v, R^{\vee})},
\Phi^{\Sigma}_1(x) = \frac{1}{8}a_{(1,1,-1)}(x)\langle d\varphi \rangle^{-3}\left[3 P_0(1, d\varphi \langle d\varphi\rangle, (d\varphi)^2; \, \langle d\varphi\rangle, d\varphi) \right. & \left. + P_2(1, d\varphi \langle d\varphi\rangle, (d\varphi)^2; \, \langle d\varphi\rangle, d\varphi) \right].
|M^{\pm}_{[\alpha, \psi_{\alpha}]i}| \leq A^\pm_{[\alpha, \psi_{\alpha}][\alpha, \psi_{\alpha}]i}\;\;\;\;\;{\rm and}\;\;\;\;\; M^\pm_{[\alpha, \psi_{\alpha}]i}= A^\pm_{[\alpha, \psi_{\alpha}][\alpha, \psi_{\alpha}]i}\;\;\;\;{\rm mod}\;\; 2 \;\;\;\; ,
\sigma^{2}q^{\frac{2}{p}}((1+3\lambda^{2})\frac{B(\frac{3}{p},q-\frac{2}{p})}{B(\frac{1}{p},q)}-4\lambda^{2}\frac{B(\frac{2}{p},q-\frac{1}{p})^{2}}{B(\frac{1}{p},q)^{2}})
\varphi_{\alpha}(t) = \mathbb{E}[e^{itX}]= \int_{\mathbb{R}}e^{itx}f_X(x)dx &= \frac{\alpha\Lambda}{2\Gamma(1/\alpha)}e^{it\mu}\int_{\mathbb{R}}e^{itx}\exp\left(-\Lambda^{\alpha}|x|^{\alpha}\right)dx.
V_{\cal N}^-(q) = \frac{C_1^2}{2}q^6 + C_1 C_2 q^4 + \frac12\{C_2^2-(4{\cal N}-1-2 C_3) C_1\}q^2+\frac{C_3(C_3 + 1)}{2}\frac{1}{q^2}.
\left( \sum_{i=0}^{2n+1-m}a_i t^i \right) \equiv \left(\sum_{i=0}^{n+1-m}(-1)^i \beta_i t^{2i}+ \sum_{i=n+1}^{2n+1-m}(-1)^i \beta_i t^{2i}\right)\left(\sum_{i=0}^m \delta_i t^i + \sum_{i \geq 2n+3-m}\delta_i t^i \right).
\begin{aligned}\int_\Omega \theta^2dx+ \iint_{Q_t}\kappa \theta_x^2dxds \leq C +C\int_0^t\left[\|\theta\|_{L^\infty(\Omega)}+\|\theta\|_{L^\infty(\Omega)}^{2}\right]\int_{\Omega}u_x^2 dxds +C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2}. \end{aligned}
\frac{\partial L_0}{\partial \varphi_{\overline{\mu}}}= \frac{\partial^2 F^\lambda}{\partial \varphi_{\overline{\mu}}\partial x^\lambda}+ \sum_{0\le |{\overline{\nu}}|\le l-1}\frac{\partial^2 F^\lambda}{\partial \varphi_{\overline{\mu}}\partial \varphi_{\overline{\nu}}}\varphi_{\overline{\nu}\lambda}+ p^{({\overline{\mu}})}\ ,
\frac{\partial}{\partial x_0}E_1(x,\zeta) = -i(\zeta_1 e_1+\zeta_2 e_2) E_1(x,\zeta) = -\left(e_1 \frac{\partial}{\partial x_1}+e_2 \frac{\partial}{\partial x_2}\right) E_1(x,\zeta) &= -\left( \frac{\partial}{\partial x_1}E_1 e_1+ \frac{\partial}{\partial x_2}E_1 e_2\right).
\begin{cases}dx_i =( Ax_i +B\check u_i +G m_i+\gamma p_i) dt +D dW_i, \dot{m_i}= (A + G)m_i +B\bar u +\gamma p_i, \dot{p_i}= -(A+G)^T p_i -(I-\Gamma)^T Q [\mathbb{E}x_i- (\Gamma m_i+\eta)], dy_i=\left\{-A^T y_i +Q[x_i-(\Gamma m_i+\eta)]\right\}dt +\zeta_i dW_i, \end{cases}
Y_\alpha = (Y_\alpha(1),\dots,Y_\alpha(l(Y_\alpha))) = ( \underbrace{F_\alpha(1), \dots, F_\alpha(1)}_{G_\alpha(1)}, \dots, \underbrace{F_\alpha(m_\alpha), \dots, F_\alpha(m_\alpha)}_{G_\alpha(m_\alpha)})
(\begin{matrix}1&1&0\\ 1&0&1\\ 0&1&1\end{matrix}),(\begin{matrix}1&1&0&0\\ 0&1&1&0\\ 0&0&1&1\\ 1&0&0&1\\ \end{matrix}),(\begin{matrix}1&1&0&0&1&0\\ 0&1&1&0&0&1\\ 0&0&1&1&1&0\\ 1&0&0&1&0&1\\ \end{matrix})
J_q(n,k,t) = S_k^{k-t}=\frac{1}{q^{(k-t)^2}{k \choose t}_q}\sum\limits_{i=0}^{k-t}(-1)^{k-t-i}q^{{k-t-i \choose 2}}{k-t \choose i}_q{k+i \choose i}_q N_k^{i}
\sum_{n=1}^N\mathbb{E}\left[f\left(2^{n_3-n_1\vee 0}x_1,2^{|n-n_3|-|n_1-n|\vee 0}x_2\right)f\left(2^{n_1-n_3\vee 0}x_1,2^{|n_1-n|-|n-n_3|\vee 0}x_2\right)\right]\neq 0
\mathbf{v}_f = (v_{f(1)}\wedge\cdots\wedge v_{f(n+1)})\otimes (v_{f(n+2)}\wedge\cdots\wedge v_{f(n+d)})\otimes \cdots \otimes (v_{f(n+2+(n-1)(d-1))}\wedge\cdots\wedge v_{f(nd+1)})
\tilde{s}^{( 2 )}( p^{2}; m^{2}) ={\frac{1}{2 \pi}}\,{\frac{1}{p^{2}\sqrt{- 1 - 4 m^{2}/ p^{2}}}}\, \operatorname{a r c t a n}{\frac{1}{\sqrt{- 1 - 4 m^{2}/ p^{2}}}}\, \raise 2 p t \mathrm{,}\quad \mathrm{f o r}\ 0 < - p^{2}< 4 m^{2}.
2\exp\Big(2\pi \frac{1}{\sqrt{3}}\frac{_2F_1(\frac{1}{3},\frac{2}{3},1,1-\alpha_0)}{_2F_1(\frac{1}{3},\frac{2}{3},1,\alpha_0)}\Big)=27\frac{(1+8\alpha_0)^3}{\alpha_0(1-\alpha_0)^3}.
\int_{x}^{a}\left(1-\frac{k}{n}\right)P(y)f_{n,k}(y;x)dy+\theta_{p}^{n,k}(x) =\left(1-F(a;x)\right)F_{n-1,k}(a;x)+\left(1-F(a;x)\right)\left(1-F_{n-1,k}(a;x)\right) &=1-F(a;x)= P(x)
E(r) = \frac1r E'(r)\left[ 1+(r-1)s_{r-1}-\frac1{\delta-2r+2}p_r \right] E'(r) E'(r+1) = E(r)\left[ 1 - r\left(\frac{\delta-2r+2}{\delta-2r+1}\right)h_r \right] E(r)
\mathrm{MDTE}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{P})=\left(\begin{array}{ccccccccccc}3.4459 4.6047\end{array}\right),\mathrm{MDTCov}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{P})=\left(\begin{array}{ccccccccccc}0.6008 0.2436 0.2436 0.2760\end{array}\right).
\begin{array}{lll}S_m(x) \approx -i x^{-{1\over2}}\left[1-\left(m+{1\over2}\right)(i x)^{-1}+\cdots \right] ~, \quad |x|>\!\!>1 S_m(x) \approx \left(m-{1\over2}\right)^{-1}x^{1\over2}~~, \quad |x|\approx0\hbox{~{\rm{and}}~}m>0 S_0(x) \approx\pi^{1\over 2}e^{-i\pi/4}~~, \quad|x|\approx 0 ~~.\end{array}
\alpha = k_4+k_5-\frac{1}{2}+\frac{1}{2k_3}\left[k_2-\sqrt{(k_2-k_3)^2-4A}\right], \beta = k_4+k_5-\frac{1}{2}+\frac{1}{2k_3}\left[k_2+\sqrt{(k_2-k_3)^2-4A}\right], \gamma = 2k_4+k_1.
{\cal L}_{\mathrm{e f f}}= \frac{1}{8 \pi}\mathrm{I m}\! \int \! \! \mathrm{d}^{2}\theta \, \tau_{D}( A_{D}) \, W_{D}^{2}- \frac{1}{4 \pi}\mathrm{I m}\! \int \! \! \mathrm{d}^{2}\theta \mathrm{d}^{2}\bar{\theta}\, \bar{A}_{D}\, A ( A_{D}) \, .
L_{1}^{\ast}=\frac{\xi_{p}^{2}\phi(\xi_{p})(1+\sqrt{2\pi}\phi(\xi_{q}))-\xi_{q}^{2}\phi(\xi_{q})(1+\sqrt{2\pi}\phi(\xi_{p}))}{\Psi_{2}^{\ast}(-1,\frac{1}{2},1)F_{Y}(\xi_{p},\xi_{q})(1+\sqrt{2\pi}\phi(\xi_{p}))(1+\sqrt{2\pi}\phi(\xi_{q}))},
p_1-r_1\leq(1-c_0)p_1+c_0<\frac{n[(1-c^0)p_0+c^0]}{n-2p_0}\leq\frac{n(p_0-r_0)}{n-2p_0}=\frac{n\frac{n(p_0-r_0)}{n-2r_0}}{n-\frac{2n(p_0-r_0)}{n-2r_0}},
f(\kappa,{\textstyle \frac{3}{2}})\simeq - \int \frac{d^3 p}{(2 \pi)^3}\log[18(3-2 \sqrt{2}) \{s^2{\bf p}^2 + \frac{s^4}{2}+ \frac{1}{4}( p_2^2 p_3^2 + p_3^2 p_1^2 + p_1^2 p_2^2) + O(p^6) \}].
\chi(x):= C^{-1}_1 \int_{r_0}^{\Lambda_{0}}\phi(x;r)\,f(r)\,r^2\,\mathrm{d}r= C^{-1}_1 \int_{(x-\varepsilon_2)\vee r_0}^{(x+\varepsilon_{2})\wedge\Lambda_0}\phi(x;r)\,f(r)\,r^2\,\mathrm{d}r,x\in[r_0-\varepsilon_2,\Lambda_0+\varepsilon_2].
\dim \,C_k^r(\Delta) \leq \binom{k+3}{3}+f_2^0\binom{k+2-r)}{3}&-\sum_{i=1}^{f_1^0}\biggl[s_i\binom{k+2-r)}{3}- b_i\binom{k+3- \Omega_i}{3}- a_i\binom{k+2-\Omega_i}{3}\biggr] &+f_0^0\binom{k+3}{3}-\sum_{i=1}^{f_0^0}\biggl(\sum_{j=0}^k F(t_i,r+1,3)_j\biggr),
2\rho^{1/4}\binom{|Y|}{\ell}\binom{|Y|-\ell}{2k-2\ell-1}+ 2\sqrt{\rho}\binom{|Y|}{k-1}\binom{|Y|-k+1}{k- \ell}\le 3\binom{2k-\ell-1}{\ell}\rho^{1/4}\binom{|Y|}{2k-\ell-1},
f_2 \ge \frac{h-1}{2}\ge 1+\frac{h-1}{8}\ge 1+h_1=1+e_2\quad f_3 \ge \frac{3h-3}{4}\ge 1+\frac{h-1}{8}+\frac{h-7}{2}\ge 1+h_1+\max\{h_1,\dots,h_4\}\ge 1+e_2+e_3.
g(z_{t-1},u_t) = I(Y_t;X_t,X_{t-1}|z_{t-1},u_t) &= H(Y_t|z_{t-1},u_t) - H(Y_t|X_t,X_{t-1},z_{t-1},u_t) &\stackrel{(a)}= H_{ter}((1-\delta_t)\bar{\epsilon}, \epsilon , \delta_t\bar{\epsilon}) - H_b(\epsilon) &\stackrel{(b)}= H_{b}(\epsilon)+ \bar{\epsilon}H_b(\delta_t) - H_b(\epsilon) &= \bar{\epsilon}H_b(\delta_t),
|u^o_{n,k+1}- x^o_{n,k+1}| \leq \frac{1}{2}|u^o_{n,k}-x^o_{n,k}| + \frac{D}{n^2}&\leq \frac{1}{2}\left(\frac{\left(\frac{1}{2}\right)^kC}{k_n^2}+ \sum_{j=1}^{k}\frac{\left(\frac{1}{2}\right)^{j-1}D}{n^2}\right) + \frac{D}{n^2}&=\frac{\left(\frac{1}{2}\right)^{k+1}C}{k_n^2}+ \sum_{j=1}^{k+1}\frac{\left(\frac{1}{2}\right)^{j-1}D}{n^2},
{X_{\varepsilon}}:\left(\begin{array}{c}\dot x \dot y \dot \varphi\end{array}\right)=\left(\begin{array}{ccc}0 -1 0 1 0 0 0 0 E\end{array}\right)\left(\begin{array}{c}x y \varphi\end{array}\right)+ \underline G (x,y,\varphi , \varepsilon)
\widehat{\mathcal{I}}(\delta) := \inf_{\lambda \in \mathbb{R}_{+}^2}\left[ \langle \lambda, \delta \rangle + \sup_{\varpi \in \Pi(\widehat{\mu}_{13}, \widehat{\mu}_{23})}\int_{\mathcal{V}}f_{\theta, \lambda}\, d \varpi \right] .
-i\overline{p}\Psi_p^{(1/2)}=\overline{\varepsilon}m\Psi_p^{(1/2)},\hspace{0.5in}\overline{\sigma}_p^{(1/2)}\Psi_p^{(1/2)}=\overline{s}_p\Psi_p^{(1/2)},
I_{\mathrm{GMI}}= \sup_{a\in\mathbb{C},\theta<0}\mathbb{E}_{\mathbf{h}}\Bigg(\theta\mathbb{E}_{x,\mathbf{z}}[|f(x,\mathbf{h},\mathbf{z})-ax|^2|\mathbf{h}]- &\frac{\theta\mathbb{E}_{x,\mathbf{z}}[|f(x,\mathbf{h},\mathbf{z})|^2|\mathbf{h}]}{1-\theta|a|^2\mathcal{E}_\mathrm{s}}+\log(1-\theta|a|^2\mathcal{E}_\mathrm{s})\Bigg).
\|u\|_{\dot B^s_{p,r}}^\ell := \big\|\{2^{js}\|\dot \Delta_j u\|_{L^p}\}_{j\leq j_0}\big\|_{\ell^r}, \textrm{and}\|u\|_{\dot B^s_{p,r}}^h := \big\|\{2^{js}\|\dot \Delta_j u\|_{L^p}\}_{j\geq j_0}\big\|_{\ell^r}.
\alpha_q(\eta) \leq \limsup_{t\rightarrow\infty}\frac{\hat{m}(N_t - N_{z(t)})}{D_t}= \limsup_{t\rightarrow\infty}\frac{N_t - N_{z(t)}}{N_t}& = 1 -\liminf_{t\rightarrow\infty}\frac{N_{z(t)}}{N_t}\leq 1 - \eta\Big(1-\frac{1}{n^*q^{\hat{m}}}\Big)^{-1},
\begin{aligned}\mathbf{R}_{\mathrm{c},{\pi(k)}}=\mathbf{I}+\sum_{j=1}^K\mathbf{H}_{\pi(k)}^H\mathbf{P}_{\pi(j)}\mathbf{P}_{\pi(j)}^H\mathbf{H}_{\pi(k)}, &\mathbf{R}_{\pi(k)}=\mathbf{I}+\sum_{j>k}^K\mathbf{H}_{\pi(k)}^H\mathbf{P}_{\pi(j)}\mathbf{P}_{\pi(j)}^H\mathbf{H}_{\pi(k)}.\end{aligned}
E_{n}=-(\frac{Z^{2}\mu e^{4}}{32\pi^{2}\epsilon_{0}^{2}\hbar^{2}})\frac{1}{n^{2}}=-(\frac{Z^{2}\hbar^{2}}{2\mu a_{\mu}^{2}})\frac{1}{n^{2}}=-\frac{\mu c^{2}Z^{2}\alpha^{2}}{2n^{2}}.
exp(it)=(1-\frac{t^{2}}{2!}+\frac{t^{4}}{4!}-\frac{t^{6}}{6!}+\cdot\cdot\cdot)+i(t-\frac{t^{3}}{3!}+\frac{t^{5}}{5!}-\frac{t^{7}}{7!}+\cdot\cdot\cdot).
|f(Z)|_k =e_1|f_1(z_1)|+e_2|f_2(z_2)| &\leq' e_1\left(\frac{\|f_1\|_{H^2(\mathbb{U})}}{\sqrt{1-|z_1|^2}}\right)+e_2\left(\frac{\|f_2\|_{H^2(\mathbb{U})}}{\sqrt{1-|z_2|^2}}\right) &=\frac{\|f\|_{\mathbb{D},{H^2(\mathbb{U}_{\mathbb{B}\mathbb{C}})}}}{\sqrt{1-|Z|^2_k}}\;.
\sum_{n=0}^\infty\frac{CC_{3,n}}{\Pi(n)}x^n =\dfrac{-\dfrac{x^{27}}{L_3}}{\sum_{i=3}^\infty(-1)^i\dfrac{x^{3^i}}{L_i}}&=1+\frac{L_3}{L_4}x^{54}+\frac{L_3^2}{L_4^2}x^{108}+\frac{L_3^3}{L_4^3}x^{162}&\quad+\left(\frac{L_3^5}{L_4^5}-\frac{2 L_3^2}{L_4 L_5}\right)x^{270}+\left(\frac{L_3^6}{L_4^6}-\frac{3 L_3^3}{L_4^2 L_5}\right)x^{324}+\cdots\,.
\textstyle |\epsilon| \int_{\Omega_{\le R}}|v|^{2}\le \textstyle \int_{\Omega_{\le2R}}|f \overline{v}| +\frac{N}{R}\int_{\Omega_{R\le|x|\le 2R}}|v||\nabla^{b}v| \le 2R\|f\|_{\dot Y^{*}}\|v\|_{\dot X}+ 3NR\|v\|_{\dot X}\|\nabla^{b}v\|_{\dot Y}.
s_p(h s) = \begin{cases}(c,-d)_p cd \neq 0, \mathrm{ord}_p(d) , 1 \end{cases}x(h s) = \begin{cases}-d d \neq 0, c d = 0, \end{cases}x(h\beta_m) = \begin{cases}-dp^m d \neq 0, cp^{-m}d = 0. \end{cases}
{\cal H}=\frac{i}{2}\sum_{k=1}^rp_k p_k-\hbar^2 tr (H^2H^2) \hskip 0.7 cm ,\hskip 0.7 cm{\mbox where}\hskip 0.5 cm p_k=\frac{1}{2}(p_k^l+p_k^r)
I_\varepsilon(v) = \frac 12 \int_{\mathcal{C}_\Omega}y^{1-2\alpha}|\nabla v|^2\,dxdy -\frac 1{2^*_\alpha-\varepsilon}\int_{\Omega\times\{0\}}|v|^{2^*_\alpha-\varepsilon}\,dx -\frac \lambda 2\int_{\Omega\times\{0\}}|v|^{2}\,dx
\|v_n\|_{L^{\infty}(-2,0; \dot B^{s_p}_{p,p})}= \|u\|_{L^{\infty}(t_n,T_*; \dot B^{s_p}_{p,p})}\leq \widetilde{M}, \|v_{0,n}\|_{\dot B^{s_p}_{p,p}}= \|u(t_n)\|_{\dot B^{s_p}_{p,p}}\leq \widetilde{M}.
B = \frac{\pi}{2}+ \gamma + \log{8\pi}- \log \operatorname{rd}(K) - \int_0^\infty \frac{1-F(x)}{2}\left( \frac{1}{\sinh{\frac{x}{2}}}+ \frac{1}{\cosh{\frac{x}{2}}}\right) \, dx & + 2 \sum_{p \in S}\sum_{m=1}^\infty \frac{\log p}{p^{m/2}}F(m \log p) > 0.1796.
\delta_{V}^{1}( x_{1}^{j}( \xi , \eta ) ) = \sum_{k}{\cal D}( A_{2}+ \xi_{1}C_{2}, ( \tilde{D}_{2}- \tilde{C}_{2}\xi_{1})^{- 1})_{k}^{i}\Delta ( x_{1}^{k}( \xi , \eta ) ) .
\pi_0 = 1\ 10\ 11\ 12 / 2\ 4\ 5\ 8/ 3\ 6\ 7\ 9 \pi_1 = 1\ \textbf{2}\ 11\ 12 / \textbf{3\ 5\ 6\ 9}/ \textbf{4\ 7\ 8\ 10}\pi_2 = 1\ 2\ 11\ 12 / 3\ \textbf{4}\ 6\ 9/ \textbf{5}\ 7\ 8\ 10 \pi_3 = 1\ 2\ 11\ 12 / 3\ 4\ \textbf{7}\ 9/ 5\ \textbf{6}\ 8\ 10