harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\epsilon^2}{2}e^{-\alpha}\mu \sum_{k=0}^{\infty}k^2 (e^{-\alpha}\tilde{P})^k \tilde{P}^{(2)}(I-e^{-\alpha}\tilde{P})^{-1}r =\, \frac{\epsilon^2}{2}e^{-2\alpha}\mu \tilde P \left[ 2 e^{-\alpha}\tilde P(I - e^{-\alpha}\tilde P)^{-3}+ (I - e^{-\alpha}\tilde P)^{-2}\right] \tilde P^{(2)}(I - e^{-\alpha}\tilde P)^{-1}r.
||\psi_i||_H \leq \int_0^1\left(\max_{x\in M}H_i(x)-\min_{x\in M}H_i(x)\right)dt &= |\phi(x_i)|\left(\max_{|v|\leq\delta}H(v)-\min_{|v|\leq\delta}H(v)\right) &\leq 2\delta|\phi(x_i)|
F = \left( \begin{array}{c c c c c}{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{f}}&{{0}}&{{0}}\\{{0}}&{{- f}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
D^\sigma (g^{jk}-g^{jk0})=o(|x|^{-|\sigma |}),\tag*{$\textup{(\ref*{23-3-34})}_1$}&D^\sigma (V_j-V^0_j)=o(|x|^{m_1+1-|\sigma |})\forall \sigma :|\sigma |\le 1\tag*{$\textup{(\ref*{23-3-34})}_2$}
\rho_{\alpha\beta}(\omega)=\frac{1}{Z}\sum_{m,n}2\pi\delta(\xi_{\alpha}-\omega)\delta_{\xi_{\alpha},\xi_{\beta}}\langle m|\psi_{\alpha}|n\rangle\langle n|\psi_{\beta}^{\dagger}|m\rangle e^{-\beta E_{m}}(1-\zeta e^{-\beta\xi_{\alpha}}),
\left| (f^{\star}_0(x_\ell)-V^{\star}_{k_\ell}(x_\ell)+(f^{\star}_0(-x_\ell)-V^{\star}_{k_\ell}(-x_\ell))-2\, (f^{\star}_0(0)-V^{\star}_{k_\ell}(0)))\right|\leq &\leq 4C'_1\,\omega (\delta_{k_\ell}).
\begin{array}{r c l}{{\{\Gamma_{0}^{A}, \Gamma_{0}^{B}\}}}&{{=}}&{{0}}\\{{\{\Gamma_{0}^{A}, \Gamma_{1}^{B}\}}}&{{=}}&{{E^{A B}( \Gamma_{0})}}\\{{\{\Gamma_{1}^{A}, \Gamma_{1}^{B}\}}}&{{=}}&{{\Gamma_{1}^{C}\partial_{C}E^{A B}( \Gamma_{0}) \, .}}\\ \end{array}
\begin{aligned}\|\widehat{D}u\|^2\ = \ \|D(\phi s)\|^2 \ = \ \big(\,\|\phi Ds\|+\|[D,\phi]s\|\,\big)^2 \leq \ \big(\,\|Ds\|+a\,\|s\|\,\big)^2 \ \leq \ 2\,\|Ds\|^2\ + \ 2a^2\,\|s\|^2.\end{aligned}
\frac{d}{dr}f(rx) = \frac{d}{dr}f({\exp}_{\mathbb{G}}\left(r^{\nu_{1}}e_{1}(x)X_{1}+\ldots+r^{\nu_{n}}e_{n}(x)X_{n}\right)) & = \left[(\nu_{1}r^{\nu_{1}-1}e_{1}(x)X_{1}+\ldots+\nu_{n}r^{\nu_{n}-1}e_{n}(x)X_{n})f\right](rx) & =\frac{1}{r}\left(e(rx)\cdot A\nabla\right)f(rx).
\widehat{j}(0,a,b) =\frac{1}{\mathcal{H}^a_{q,r}(b,0)}\Big( \frac{W_q(b)}{W_q'(b+)}+\big[ \widehat{U}_1(b,a,b,0) + \widehat{U}_2(b,a,b)\big] \widehat{j}(0,a,b) \Big),
\sum_{j=1}^{p-2}\sum_{t=1}^{p-j-1}\sum_{s=1}^{t}x^{-2j+2t+2s}=\frac{x^{-2p+8}(1-x^{2p-4})(1-x^{2p-2})(1-x^{2p})}{(1-x^2)(1-x^4)(1-x^6)}
\frac{1}{i(i+1)}(x_1^2 + \cdots + x_n^2)\left(\frac{\partial^2h(x)}{\partial x_1^2}+ \cdots + \frac{\partial^2h(x)}{\partial x_n^2}\right) + \frac{(n-2)(i-1)}{i+1}h(x) = z(x)
z\rightarrow e^{\mp \partial_{x}\log z}=e^{\mp \partial_{x}\log \left(1-\beta u_{\beta}^{(1)}\right)}=1\pm \beta \partial_{x}u_{\beta}^{(1)}+\frac{\beta^{2}}{2}\left\{\left( \partial_{x}u_{\beta}^{(1)}\right)^{2}\pm \partial_{x}u_{\beta}^{(1)2}\right\}+O\left( \beta^{3}\right)
M \simeq \sum_{m=1}^{\infty}\frac 1{m^2}\sum_{j=1}^{n-1}\frac 1{(mL)^j}\frac{\log^{n+j}L}{L^{(n+1)m+n+j}}= \textrm{O}\left(\frac{\log^{n+j}L}{L^{2n+3}}\right)\ .
W = A ( \mathrm{d e t}\, T_{( 0 , 2 )}- M_{0}M_{1}M_{2}+ \Lambda_{1}^{4}M_{2}+ \Lambda_{2}^{4}M_{0}) + m_{0}M_{0}+ m_{1}M_{1}+ m_{2}M_{2}+ \mathrm{t r}\, ( c \, T_{( 0 , 2 )})
O_{\tilde{L}}( \{p , \zeta \}, \tilde{p}_{1}, \{t \}, \{q \}) = \exp \biggl [ \sum_{j = 1}^{2}\frac{( 2 \eta_{j}\zeta_{( j )}+ i p_{j}\vartheta_{j})}{8 \sqrt{z_{j}- u}}\hat{\Psi}\biggl ] ,
n(R_\ell + \delta_n) &\;\geq \sum_{i=1}^{n}\Big[I(X_{i};A_{i},V_{i},Y_i) \!-\!I(X_{i};Y_i|A_{i})\!+\!I(X_{i};Z_i|A_{i})\!-\!\epsilon_n &\;\qquad+ I(U_i;Y_i|A_{i})-I(U_i;Z_i|A_{i})\Big]
\left\langle w_{p}\right\rangle_{n,\omega\gamma,\overline{Q}}^{(m+\gamma,\frac{m+\gamma}{m})}\leq C\left\langle \left\langle w_{p}\right\rangle\right\rangle_{n,\omega\gamma,\overline{Q},y,\tau}^{(m+\gamma)(2s)}\leq2C=C.
z_{m+2}= sz_{m+1}+cx_{m+1}&= sz_{m+1}+c(sx_m+az_m) &= sz_{m+1}+acz_m+scx_m &= sz_{m+1}+acz_m+s(z_{m+1}-sz_m) &= 2sz_{m+1}+(ac-s^2)z_m &= 2sz_{m+1}-z_m.
f_\varepsilon(z)=\frac{1}{2\sqrt{\pi\varepsilon}}\int\limits_{-\infty}^{+\infty}e^{-\xi^2/4\varepsilon}\frac{1}{1-ze^{i\xi}}\,d\xi\,, \ \ \varepsilon>0\,,\,z\in\mathbb{D}.
\left(\frac{\partial T}{\partial t}\right)_{w,x^i}/\left(\frac{\partial r}{\partial t}\right)_{w,x^i}= - \left(\frac{\partial w}{\partial r}\right)_{T,x^i}/\left(\frac{\partial w}{\partial T}\right)_{r,x^i}= \mp\frac{\sqrt{f(r)+\dot{a}^2}}{f(r)\dot{a}(t)}.
\begin{aligned}\min \ \frac{1}{2}x^TAx+b^Tx \mbox{s.t.}\ \frac{1}{2}x^TB_ix+c_i^Tx+d_i\leq0,i=1,\dots,m, &\ -1\leq x_i \leq1, i=1,\dots,n,\end{aligned}
T^{\prime}(n)=\frac{(n+1)(\lfloor log_{2}(n)\rfloor+2)-2^{\lfloor log_{2}(n)\rfloor+1}}{(n+1)}=\lfloor log_{2}(n)\rfloor+2-2^{\lfloor log_{2}(n)\rfloor+1}/(n+1)
\lim_{N \to \infty}\frac{1}{|\Phi_N|}\sum_{g \in \Phi_N}\prod_{i=1}^k T_{[1,i]}^g f_i \prod_{j=1}^\ell S_{[1,j]}^g h_j=\left( \lim_{N \to \infty}\frac{1}{|\Phi_N|}\sum_{g \in \Phi_N}\prod_{i=1}^k T_{[1,i]}^g f_i\right)\!\!\!\left( \lim_{N \to \infty}\frac{1}{|\Phi_N|}\sum_{g \in \Phi_N}\prod_{j=1}^\ell S_{[1,j]}^g h_j \right)
\phi_{n}^{\lambda}(z) =\epsilon_nr^{n}\left[\frac{n+2\lambda}{2\lambda}P_{n}^{\lambda}(\cos\theta)+i\sin\theta P_{n-1}^{\lambda+1}(\cos\theta)\right], \bar{z}\overline{\phi_{n-1}^{\lambda}(z)}=\epsilon_{n-1}r^{n}\left[\frac{n}{2\lambda}P_{n}^{\lambda}(\cos\theta)- i\sin\theta P_{n-1}^{\lambda+1}(\cos\theta)\right],
f^{-n_{j+2}-m_{j+1}-n_{j+1}}(V_{j+2}) =f^{-n_{j+1}-n^*}\circ f^{-n_{j+2}-m_{j+1}+n^*}(V_{j+2}) &\subset f^{-n_{j+1}-n^*}(U_{j+2}) \subset f^{-n_{j+1}}(V_{j+1}).
C^{\mathrm{ul}}_{1}\left( N_{\mathrm{sc}}\right) = \sum_{k=1}^K \min \left\lbrace C^k_{\mathrm{u2s},1}\left( N_{\mathrm{sc}}\right), C^k_{\mathrm{s2b},1}\left( N_{\mathrm{sc}}\right) \right\rbrace.
[\begin{matrix}c_{3}c_{1}-c_{2}s_{3}s_{1}&s_{2}s_{3}&c_{3}s_{1}+s_{3}c_{2}c_{1}\\s_{1}s_{2}&c_{2}&-c_{1}s_{2}\\ -c_{2}c_{3}s_{1}-c_{1}s_{3}&c_{3}s_{2}&c_{3}c_{2}c_{1}-s_{3}s_{1}\end{matrix}]
T_2= \frac{d}{(c-d)(q)_N}\left(1-\frac{(dq)_N}{(cq)_N}\right) +\frac{1}{(cq)_N}\sum_{k=1}^N \frac{\left(\frac{cq}{d}\right)_k (dq)^k(dq)_{N-k}}{(q)_k (1-q^k)(q)_{N-k}}.
\mathcal D_4=\{\Phi_{(\psi,f)}|~ \psi\in D^{m+1,\alpha}_{\delta}(M)\psi|_{\partial M}=Id_{\partial M}; &~f \in C^{m+1,\alpha}_{\delta}(M) f|_{\partial M}=0; &\Phi_{(\psi,f)}:V^{(4)}\rightarrow V^{(4)}, &\Phi_{(\psi,f)}[t,p]=[t+f,\psi(p)],\quad\forall t\in\mathbb R,~p\in M.~\},
\gamma^{0}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) , \qquad \gamma^{1}= \left( \begin{array}{c c}{0}&{1}\\{- 1}&{0}\\ \end{array}\right) , \qquad \gamma^{2}= \left( \begin{array}{c c}{0}&{i}\\{i}&{0}\\ \end{array}\right)
\phi({\bf x},t) = \int\! \frac{d^{3}p}{(2\pi)^3}\,\frac{1}{\sqrt{2 E_{p}}}\left.\left[ a_{\bf p}e^{-ip \cdot x}+a^{\dag}_{\bf p}e^{ip \cdot x}\right]\right|_{p_{0}= E_{p}}
t-0=\frac{1}{g}[\frac{ln(1+\alpha v^{\prime})}{2\alpha}-\frac{ln(1-\alpha v^{\prime})}{2\alpha}+C]_{v^{\prime}=0}^{v^{\prime}=v}=\frac{1}{g}[\frac{ln\frac{1+\alpha v^{\prime}}{1-\alpha v^{\prime}}}{2\alpha}+C]_{v^{\prime}=0}^{v^{\prime}=v}
F(n,k)=\frac{(-1)^{n+k}\left(20 n^2-12 k n-2 k+8 n+1\right)\binom{2 n}{n}^3 \binom{2 k+2 n}{k+n}\binom{2 n-2 k}{n-k}\binom{k+n}{n-k}}{\binom{2 k}{k}4^{6 n-2 k}}
d_{\cal C}(\Upsilon_{\cal T}\gamma(0),\Upsilon_{\cal T}\gamma(\xi))/L-1 \leq d_{WP}(\gamma(0),\gamma(\xi)) \leq C\xi= Cd_{\cal T}(\gamma(0),\gamma(\xi)) \leq Cd_{\cal C}(\Upsilon_{\cal T}\gamma(0),\Upsilon_{\cal T}\gamma(\xi))/\theta.
\phi_{1}= \frac{\lambda}{m^{2}}{\bigg [}\left( \frac{1}{2 \sqrt{3}}+ \frac{3}{4 \pi}\right) \frac{1}{\cosh^{2}( m x / 2 )}- \frac{\sqrt{3}}{4}( m \partial_{m}+ 2 \lambda \partial_{\lambda}){\bigg ]}\phi_{\mathrm{k i n k}}.
\bigg[\partial_{n+1}x_{n+1}^{1-2s}\partial_{n+1}+x_{n+1}^{1-2s}\sum_{j,k=1}^{n}\partial_{j}a_{jk}\partial_{k}\bigg]u_{1}=0\quad\mathbb{R}_{+}^{n+1}, u_{1}=\zeta u\quad\mathbb{R}^{n}\times\{0\},
{P}_{2n}(q) = \frac{(-1)^n}{\sqrt{2}}{Q}_n\big(q^2-\alpha-1;\alpha,\beta,N\big), P_{2n+1}(q) = - \frac{(-1)^n}{\sqrt{2}}\frac{1}{(\alpha+1)}\sqrt{\frac{(n+\alpha+1)(n+\alpha+\beta+1)(2n+2+\alpha+\beta)}{(n+N+\alpha+\beta+2)(2n+\alpha+\beta+1)}}\hphantom{P_{2n+1}(q) =}{}\times q{Q}_n\big(q^2-\alpha-1;\alpha+1,\beta,N\big).
t_{k}=z^{k}\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+z^{2k}\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q}{2}+\frac{p^{3}}{27}}},k=0,1,2,
\frac{d}{dt}\sum_{\ell\le k\le N}\|\nabla^k f\|^2_{L^2_{x,p,\mathcal{I}}}+C \sum_{\ell\le k\le N}\|\nabla^k \{I-P\}f\|^2_{L^2_{x,p,\mathcal{I}}}\cr \le C\delta^2\Biggl(\sum_{\ell+1\le k\le N}\|\nabla^{k}f\|^2_{L^2_{x,p,\mathcal{I}}}+\sum_{\ell\le k\le N}\|\nabla^k \{I-P\}f\|^2_{L^2_{x,p,\mathcal{I}}}\Biggl).
ev^{\sigma (e) h-an}(\partial^{|n|}_{n(X)}P):= \left([\partial^{|n|}_{n(X)}P]\oplus \bigoplus_{n\subset m\in M(Y_1,...,Y_n)}\iota_{(n\subset m;P)(Y,Z;X)}([\partial^{|m|}_{m(X)}(P)])\right) &\in B_{\sigma (e) h \otimes I(n)}\langle Z_1,...,Z_n: I,S;X_1,...,X_n:I,T,C\rangle
\int(\delta u(x+he,y)-\delta u(x,y)) \frac{(2-\sigma)\langle A^h(x)y,y\rangle \rho(x+he,y)}{|y|^{n+\sigma+2}}dy &+\int\delta u(x,y)(\rho(x+he,y)-\rho(x,y))\frac{(2-\sigma)\langle A^h(x)y,y\rangle}{|y|^{n+\sigma+2}}dy &=f(x+he)-f(x)+k_h(x)
\begin{cases}\underline h^n(t):=(c^n-\delta) t+K, \ \ \t\geq 0, \underline U^n(t,x):=(1-\epsilon) [\Phi^n(x-\underline h^n(t))+\Phi^n(-x-\underline h^n(t))-\mathbf{u_n^*}], \ \ \t\geq 0,\x\in [-\underline h^n(t),\underline h^n(t)],\end{cases}
S_{1D}= - \int d^5 \sigma \left(\sqrt{-G}\sqrt{1+z_1+\frac{z_1^2}{2}-z_2}+\frac{\epsilon_{\mu\nu\lambda\sigma\tau}C^{\mu}\tilde{\cal H}^{\nu\lambda}\tilde{\cal H}^{\sigma\tau}}{8(1+C_1^2)}\right),
K_{p}( z ) \stackrel{( z \rightarrow 0 )}{=}\frac{1}{2}\left[ \Gamma ( p ) \left( \frac{z}{2}\right)^{- p}+ \Gamma ( - p ) \left( \frac{z}{2}\right)^{p}\right] \left[ 1 + O \left( z^{2}\right) \right] ,
\|A_f(v)\|_2^2 =\langle A_f (v),A_f (v)\rangle=\langle \Gamma_f (v),\Gamma_f (v)\rangle=\langle \Gamma^*_f\Gamma_f v,v\rangle=\Sigma_{j=0}^n|\lambda_j|^2s_j^2 &\geq\frac{1}{n+1}\left(\sum_{j=0}^n |\lambda_j|s_j\right)^2\geq\frac{M_n^2(f)}{n+1}\left(\sum_{j=0}^n|\lambda_j|\|v_j\|_\infty\right)^2\geq \frac{M_n^2(f)}{n+1},
\displaystyle{\prod_{i=1}^{e^{'}}}\big{[}Y_i\big{]}^{r}=\displaystyle{\prod_{i=1}^{e}}\big{[}Y_i\big{]}^{r}\times\displaystyle{\prod_{i=e+1}^{e^{'}}}\big{[}Y_i\big{]}^{r}\subset\displaystyle{\prod_{i=1}^{e}}\big{[}X_i\big{]}^{r}\times\displaystyle{\prod_{i=1}^{d^{'}}}\big{[}X_{e+\delta_{i}}\big{]}^{r}
{\mathbb E}\tilde x_i^T(T) y_i(T)-{\mathbb E}\tilde x_i^T(0) y_i(0) ={\mathbb E}\int_0^T\left\{\tilde x_i^T Q [x_i-(\Gamma m_i+\eta)] + y_i^T (B\tilde u_i +G \tilde m_i +\gamma \tilde p_i) \right\}dt .
\mathcal{L}_{\gamma_{SCHCN}}(s) =\sum_{i=1}^{N_c}\frac{\alpha_i}{(s+\lambda_{eq})^{i}}+\frac{\beta_0}{s+\lambda_{0}}+\sum_{j=1}^{N-N_c}\frac{\beta_j}{s+(1+\frac{j}{N_c})\lambda_{eq}},
\Big|\frac{1}{2 \varepsilon n^2}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \sum_{r=1}^{\varepsilon n}\int_{\Omega}\overleftarrow{\eta}^{\ell}(x) [\eta(x+1)-\eta(x+1+r) ] [f(\eta)-f(\eta^{x+1,x+1+r})] d\nu_{b}\Big|.
\sigma^{1}\, = \, \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) \; \; , \; \; \sigma^{2}\, = \, \left( \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) \; \; , \; \; \sigma^{3}\, = \, \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \; .
Z ( t , t^{*}) = c \, \sum_{\{h^{e}, h^{o}\}}{\frac{\prod_{i}( h_{i}^{e}- 1 ) ! ! h_{i}^{o}! !}{\prod_{i , j}( h_{i}^{e}- h_{j}^{o})}}~ \chi_{\{h \}}( A ) ~ \chi_{\{h \}}( B ) .
M^{r s}= \left( \begin{matrix}{{1 - x}}&{{- \sqrt{x ( 1 - x )}}}&{{- \sqrt{x}}}\\{{- \sqrt{x ( 1 - x )}}}&{{x}}&{{- \sqrt{1 - x}}}\\{{- \sqrt{x}}}&{{- \sqrt{1 - x}}}&{{0}}\\ \end{matrix}\right)
2\int_{0^-}^{\infty}e^{-s x}dG^n_{21}(x)=e^{-\Pi^n_2(0,\infty)}\frac{\mathbb E\left[e^{-s\overline{X}_{e(q_n)}}\right]}{\mathbb E\left[e^{-s\overline{X}_{e(p+q_n)}}\right]}+e^{\Pi^n_2(0,\infty)}\frac{\mathbb E\left[e^{-s\overline{X}_{e(p+q_n)}}\right]}{\mathbb E\left[e^{-s\overline{X}_{e(q_n)}}\right]},
\begin{aligned}\left((\rho^\gamma)_xr^2\right)_tu_t=\left[(\rho^\gamma r^2)_tu_t\right]_x+\frac{2\gamma}{(\gamma-1)^2}\partial_t(\rho^{\frac{\gamma-1}{2}})_t^2+\partial_t\left[3\rho^{\gamma-1}\frac{u^2}{r^2}+\frac{4}{\gamma-1}\rho^{\frac{\gamma-1}{2}}(\rho^{\frac{\gamma-1}{2}})_t\frac{u}{r}\right]+J.\end{aligned}
sl(\frac{\varpi}{\pi}x)=2e^{-\pi/4}sinx\prod_{n=1}^{\infty}\frac{1-2e^{-2n\pi}cos2x+e^{-4n\pi}}{1+2e^{-(2n-1)\pi}cos2x+e^{-(4n-2)\pi}},x\in\mathbb{C}
\|\partial_{t}^{n - 1}z(t, \cdot)\|_{H^{m + 2}(\Omega)}^{2}\leq\gamma_{1}\big( \|\partial_{t}^{n + 1}z(t, \cdot)\|_{H^{m}(\Omega)}^{2}+\|h^{n - 1}(t, \cdot)\|_{H^{m}(\Omega)}^{2}+ \|\partial_{t}^{n - 1}z(t, \cdot)\|_{H^{m}(\Omega)}^{2}\big) t \in [0, T].
T_{m,n,r,s}(j;k) =j^{m-1}(k-j)^{n-1}+(k-j)^{m-1}j^{n-1}-\alpha_{r,m,n,s}(j^{r-1}(k-j)^{s-1}+(k-j)^{r-1}j^{s-1}) =j^{r-1}(k-j)^{s-1}a(k/j-1;r-s, m-s, \alpha_{r,m,n,s}),
\begin{cases}K_{1}(z)\sim (\frac{\pi}{2})^{\frac{1}{2}}z^{-\frac{1}{2}}e^{-z}-2i(\frac{\pi}{2})^{\frac{1}{2}}z^{-\frac{1}{2}}e^{z}, \arg z\in(-\frac{3\pi}{2},\frac{3\pi}{2}); begin{align*}0.2cm]I_{1}(z)\sim -(\frac{1}{2\pi})^{\frac{1}{2}}z^{-\frac{1}{2}}e^{z}-i(\frac{1}{2\pi})^{\frac{1}{2}}z^{-\frac{1}{2}}e^{-z}, \arg z\in(\frac{\pi}{2},\frac{5\pi}{2}).\end{cases}
\Vert J^{\{u\}}_{-1}G^{\{v\}}_{-1/2}v_{\nu,\ell_0}\Vert^2= ((\theta_i|\xi+\nu)(\phi(u)|u)-\beta_k(\phi(u),u))\Vert G^{\{v\}}_{-1/2}v_{\nu,\ell_0}\Vert^2.
u^C_{\lambda,\nu}= \sum_{h+i+2j=k}\frac{2^{-2i-2h}\Gamma(\frac{2\nu+p'+2}{4})\Gamma(\frac{\lambda+\rho+\nu-\rho'}{2}+i)}{h!i!j!\Gamma(\frac{2\nu+p'+2}{4}-j)\Gamma(\frac{p''}{2}+i)\Gamma(\frac{\lambda+\rho+\nu-\rho'}{2})}\Delta_{\mathfrak{v}'}^h \Delta_{\mathfrak{v}''}^i \square^j\delta,
\widehat{D}_\ell(q)=\frac{1}{\vartheta_4^3(q)\vartheta_4^3(q^{\ell})}\,\operatorname{det}\!\begin{pmatrix}\vartheta_4(q) \vartheta_4(q^{\ell}) q\dfrac{d\vartheta_4(q)}{dq}q\dfrac{d\vartheta_4(q^{\ell})}{dq}\end{pmatrix}
\tilde t=\dfrac{at+b}{ct+d},\tilde x=\dfrac{e_2x+e_1t+e_0}{ct+d},\tilde u=\dfrac{e_2(ct+d)u-e_2cx-e_0c+e_1d}\Delta,\tilde \beta=\dfrac{e_2{}^5}{(ct+d)^3}\dfrac{\beta}\Delta,
u(x)=\sum_{k=0}^{m}p^+_k(x^{\prime \prime})\,\Psi_{m-k}(x\cdot e,x_{n+1})+\sum_{k=0}^{m}p^{-}_k(x^{\prime \prime})\,\Psi_{m-k}(-x\cdot e,x_{n+1}),
\partial_{\tau}\psi_1^{T}=\psi_2^{T}-\xi \cdot \nabla \psi_1^{T}-\tfrac{2}{p-1}\psi_1^{T}, \partial_{\tau}\psi_2^{T}=\Delta \psi_1^{T}-\xi \cdot \nabla \psi_2^{T}-\tfrac{p+1}{p-1}\psi_2^{T}+\psi_1^{T}|\psi_1^{T}|^{p-1}.
\frac{1}{\psi_h-(1+3\omega_{h'})}= -\frac{1}{1+3\omega_{h'}}\cdot \frac{1}{1-\dfrac{\psi_h}{1+3\omega_{h'}}}= -\frac{1}{1+3\omega_{h'}}\cdot \sum_{k=0}^\infty \left(\frac{\psi_h}{1+3\omega_{h'}}\right)^k.
{r_n}= \sqrt{{p_u}}\alpha{\bf{g}}_n^H{{\bf{g}}_n}{x_n}+ \sqrt{{p_u}}\alpha \sum\limits_{i = 1,i \ne n}^N{{\bf{g}}_n^H{{\bf{g}}_i}{x_i}}+ \alpha{\bf{g}}_n^H{\bf{n}}\; +{\bf{g}}_n^H{{\bf{n}}_{\sf q}},
[RVR]_{\mu k}\overline{[(RV)^2R]}_{\nu k}= (\sum z^{1/2}\sigma_i^{1/2}X_{i \mu_1}^G R_{\mu a_1}R_{b_1 k})(\sum \sigma_i(X_{i \mu_1}^G)^2 \overline{zR_{\nu a_1}R_{b_1 a_2}R_{b_2 k}}),
t_{M,N}(h\cdot(m\otimes n))=t_{M,N}(h_{1}\cdot m\otimes h_{2}\cdot n)=t_{M,N}(l_{m}(h_{1})\otimes l_{n}(h_{2}))=t_{M,N}(l_{m}\otimes l_{n})(h_{1}\otimes h_{2}) &=(l_{m}\otimes l_{n})t_{H,H}(h\cdot(1_{H}\otimes 1_{H}))=h\cdot(l_{m}\otimes l_{n})t_{H,H}(1_{H}\otimes 1_{H})=h\cdot t_{M,N}(m\otimes n)
\mathrm{I m}Z ( \varphi ) = - \mathrm{I m}\log \frac{\vartheta ( \alpha , \tau )}{\vartheta ( 0 , \tau )}- \tau_{1}\pi \varphi_{1}^{2}+ \pi \varphi_{1}\varphi_{2}= \mathrm{I m}\Gamma ( \varphi ) .
\|r^q f^{q-2}_2\|^{1/q}_{L^\infty}= \left\| r^{q/(q-2)}f_2 \right\|^{(q-2)/q}_{L^\infty}\le \|f_2\|^{(q-2)/q}_{C^0_{w^2}},{q \over q - 2}\le{4 \over 3 + \varepsilon}.
F_{Q}({\bf x}) =\sum_{j\not\in Q}\big(-\tfrac{Z}{2}|x_j|+\tfrac{Z}{2}\sqrt{|x_j|^2+1}\big) \quad+\sum_{j,k\not\in Q,j<k}\big(\tfrac{1}{4}|x_j-x_k|-\tfrac{1}{4}\sqrt{|x_j-x_k|^2+1}\big) &\quad+\sum_{j,k\in Q, j<k}\big(\tfrac{1}{4}|x_j-x_k|- \tfrac{1}{4}\sqrt{|x_j-x_k|^2+1}\big)\,.
\frac{1}{M_0M_1}\sum_{m_0=1}^{M_0}\sum_{m_1=1}^{M_1}\sum_{s^n\in \mathcal{S}^n}p\left(s^n\right) p \left\{d_{y1}\left(y_1^n\right)\neq \left(m_0, m_1\right) ~or~ d_{y2}\left(y_2^n\right)\neq m_0 ~or~ d_{y3}\left(y_3^n\right)\neq m_0 \mid s^n,x^n \left( m_0,m_1,s^n\right)\right\}\leq \epsilon.
\varpi_j(\psi,\phi) = -\frac{1}{6}\sum_{m=1}^{\infty}\frac{(-1)^m\alpha^{jm}\Gamma(\frac{m}{6})}{\Gamma(m)\Gamma(1-\frac{m}{2})\Gamma^2(1-\frac{m}{6})}(12\psi)^mu_{-\frac{m}{6}}((-1)^j\phi)
e_{r}=\begin{pmatrix}\sin\theta\cos\phi \sin\theta\sin\phi \cos\theta\end{pmatrix},e_{\theta}=\begin{pmatrix}\cos\theta\cos\phi \cos\theta\sin\phi -\sin\theta\end{pmatrix},e_{\phi}=\begin{pmatrix}-\sin\phi \cos\phi 0\end{pmatrix}.
A_3 = -2^5 \cdot 3^3 \cdot 5^2 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 4 3 \cdot 47 \cdot 53 \cdot 59 \cdot 61 \cdot 113, A_1^3 - 27 A_3 = 197 \cdot 317 \cdot 3313949 \cdot 2831657657 \cdot 4 864617187.
S ( \theta ) = \left( \begin{array}{c c c c c c c}{{u}}&{{}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{t}}&{{r}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{r}}&{{t}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{u}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{S_{s b}}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{S_{b b}}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{}}&{{\ddots}}\\ \end{array}\right)
\Psi_{N \times N}\ni \left( \begin{matrix}{. \qquad . \qquad .}&{.}\\{. \qquad . \qquad .}&{.}\\{. \quad \Psi \quad .}&{| \theta \rangle}\\{. \qquad . \qquad .}&{.}\\{. \qquad . \qquad .}&{.}\\{. \quad \langle \theta | \quad .}&{\psi}\\ \end{matrix}\right)_{( N + 1 ) \times ( N + 1 )}.
S^{(d)}_z(x) =\int^{\infty}_{0}e^{-xt}\prod^{d}_{j=1}I_{z_j}(2t)dt (\Re(x)>2d), T^{(d)}_z(u) =S^{(d)}_z\Bigl(\frac{1+(2d-1)u^2}{u}\Bigr).
A_iX:=\left(\begin{array}{c}a_{1i}\partial_1X_1+a_{2i}\partial_2X_1+a_{3i}\partial_1X_2+a_{4i}\partial_2X_2 a_{5i}\partial_1X_1+a_{6i}\partial_2X_1+a_{7i}\partial_1X_2+a_{8i}\partial_2X_2\end{array}\right),\i=1,2,...,N.
(1) ~~\overline{C_{d}}=C_{d}, (2) ~~C_{d}=H_{d}+\sum\limits_{\substack{d'\in I_{k,n}\ell(d')<\ell(d)}}p_{d', d}H_{d'}, ~\mbox{where}p_{d', d}\in q^{-1}\mathbb{Z}[q^{-1}].
\alpha_{i,k}^2 = \min\left\{1,\frac{\epsilon_{i,k}\left( |\mathbf{v}_{i,k}^H\mathbf{h}_{i,ik}|^2 +\frac{1}{\rho}\right)}{|\mathbf{v}_{i,k}^H\mathbf{h}_{i,ik}|^2 (1+\epsilon_{i,k})}\right\}.
\mathbf{P}(A_k \setminus \cup_{i=k_1}^{k-1}A_i) \geq \frac{c}{1+ \log \psi^{-1}(a(k))}\left(\mathbf{P}(C_k \cap G_k \cap B_k \setminus \cup_{i=k_1}^{k-1}A_i)-\sqrt{e^{-a(k)+a(k-1)+1}}\right)\vee 0.
\sum_{1\leq k<m\leq n}{\left|l_{k,n}(x)\cdot l_{m,n}(x)\right|}=\left[\left(\sum^{n}_{i=1}{\left|l_{i,n}(x)\right|}\right)^2-\left(\sum^{n}_{i=1}{l^2_{i,n}(x)}\right)\right]/2 &=\left[\Lambda^2_n(x)-\left(\sum^{n}_{i=1}{l^2_{i,n}(x)}\right)\right]/2.
\int_\Omega\phi(x,|\nabla u_\infty|)\,dx=\lim_{\lambda\to\infty}\int_\Omega \min\{\phi(x,|\nabla u_\infty|), \lambda |\nabla u_\infty|^p\}\,dx\le \limsup_{\lambda\to\infty}\int_\Omega \phi_\lambda(x,|\nabla u_\infty|)\,dx,
F_{\mu \nu \rho \sigma \tau}= - \frac{1}{5 !}\sqrt{- g}\epsilon_{\mu \nu \rho \sigma \tau \mu^{\prime}\nu^{\prime}\rho^{\prime}\sigma^{\prime}\tau^{\prime}}F^{\mu^{\prime}\nu^{\prime}\rho^{\prime}\sigma^{\prime}\tau^{\prime}}~ ,
\gamma_{\alpha \beta}^{0}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right) \, , \quad \gamma_{\alpha \beta}^{1}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) \, , \quad \gamma_{\alpha \beta}^{2}= \left( \begin{array}{c c}{{- 1}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right) \, .
\begin{array}{c}{\displaystyle E ( L ) = - \displaystyle \frac{\pi \tilde{c}( l )}{6 L}= - \displaystyle \frac{\pi}{6 L}\left( c - 1 2 ( \Delta + \bar{\Delta}) \right) + \ldots}\\{P ( L ) = \displaystyle \frac{2 \pi}{L}( \Delta - \bar{\Delta}) + \ldots}\\ \end{array}
\Bigg|\sum_{\substack{n\geq 1 P^{+}(n)\leq y}}\frac{\cos(2\pi n\alpha)}{n}\Bigg| \leq \Bigg|\sum_{\substack{n\geq 1 P^{+}(n)\leq y}}\frac{e(n\alpha)}{n}\Bigg|= \sum_{\substack{n\leq 1/||\alpha|| P^{+}(n)\leq y}}\frac1n+O(1),
R_{\ell}( r ) ={\frac{1}{\sqrt v}}\;{\frac{\Gamma ( \ell + 1 + i \eta )}{\Gamma ( 2 \ell + 1 )}}\; e^{- \pi \eta / 2}\; ( 2 i k r )^{\ell}\; e^{i k r}F ( \ell + 1 + i \eta \; , \; 2 \ell + 2 \; ; \; - 2 i k r )
w_{0}( x , \rho ) = \sum_{m_{1}, \cdots , m_{k}\geq 0}{\frac{\Gamma ( - \sum ( ( m_{k}+ \rho_{k}) l_{0}^{( k )}+ 1 )}{\prod_{1 \leq i \leq p}\Gamma ( \sum ( m_{k}+ \rho_{k}) l_{i}^{( k )}+ 1 )}}x^{m + \rho}.
H_N =\begin{pmatrix}\nu_1 \sigma^2 - \mu_1S(\gamma_1) - \mu_1 S(\gamma_1)^* \nu_2 \sigma^2 \ddots \ddots \ddots -\mu_{N-1}S(\gamma_{N-1}) -\mu_{N-1}S(\gamma_{N-1})^* \nu_N \sigma^2 \end{pmatrix}\,
\lim_{x_{n+1}\rightarrow0}\|x_{n+1}^{1-2s}\partial_{n+1}u_{1}\|_{H^{-2s}(\mathbb{R}^{n}\times\{0\})}=\|(-P)^{s}u_{1}\|_{H^{-2s}(\mathbb{R}^{n}\times\{0\})}\le C\|u_{1}\|_{L^{2}(\mathbb{R}^{n}\times\{0\})}\le C\|\tilde{u}\|_{L^{2}(B_{16}')},
\left( \begin{array}{cc}\boldsymbol{M_h}-\boldsymbol{K_h}-\boldsymbol{K_h}- \alpha^{-1}\boldsymbol{M_h}\end{array}\right) \left( \begin{array}{c}\underline{\boldsymbol{y}}_0^c \underline{\boldsymbol{p}}_0^c \end{array}\right) = \left( \begin{array}{c}{\underline{\boldsymbol{y}}_d^c}_0 0 \end{array}\right).
|B, p=0\rangle =\int Dx(\sigma)\exp\left({i\over 8\pi}F_{\mu\nu}{\int_0}^{2\pi}d\sigma ~x^{\mu}(\sigma)\cdot\partial_{\sigma}x^{\nu}-i{\int_0}^{2\pi}d\sigma P_\mu\cdot x^{\mu}\right)|B\rangle_{-1}
\sum_{l=2}^{k}x(l-1)\cdot d^l = xd\sum_{l=2}^{k}(ld^{l-1}-d^{l-1}) &= xd\left(f_{d}(k)-1-\frac{d^k-d}{d-1}\right) &\leq xd\left(\frac{kd^{k}}{(d-1)}-1-\frac{d^k-d}{d-1}\right) \leq \left(\frac{d+1}{d-1}\right)\cdot x(k-1)d^k.
\partial_{x_j}(W_{\phi_\lambda}u(x,\lambda\xi)) =\lambda^{\frac{1}{2}}W_{(\partial_{x_j}\phi)_\lambda}u(x,\lambda\xi), \partial_{\xi_j}(W_{\phi_\lambda}u(x,\lambda\xi)) =-i\lambda W_{\phi_\lambda}[yu](x,\lambda\xi)
x_{1}=T^{*3(j-l-2)}x-\sum_{i=0}^{3l+3}\alpha_{i}e_{i}\in M_{3l+6}\setminus M_{3l+3}, y_{1}=T^{*3(j-r-l-2)}y - \sum_{i=0}^{3l+3}\beta_{i}e_{i}\in M_{3l+6}\setminus M_{3l+3}, z_{1}= T^{*3(n-2j+r-l-3)}z-\sum_{i=0}^{3l+3}\gamma_{i}e_{i}\in M_{3l+6}\setminus M_{3l+3}.
U=\begin{cases}{U}_2^{(0)}=({u}_2^{(0)},0,{p}_2^{(0)},{\rho}_2^{(0)},0),{y}^{(0)}<y<0,{U}_1^{(0)}=({u}_1^{(0)},0,{p}_1^{(0)},{\rho}_1^{(0)},0), y<{y}^{(0)},\end{cases}
\mathcal{L}_k = 4\nu_k \left({i \over 2}\left( \langle\vec{n}_k,\vec{n}_{k,t}\rangle-\langle\vec{n}_{k,t},\vec{n}_k\rangle \right) +2\mu_k{d\xi_k \over dt}-{d\delta_k \over dt}-2\mu_k^2 +{2\nu_k^2 \over 3}\right) -\int_{x=-\infty}^{\infty}V(x) \langle\vec{u}_k\,^\dag, \vec{u}_k\rangle.
\lim_{J_2\ni m\to\infty}\int_{\varphi_n(\mathsf{U}_n)}\Bigl|\sum_{1\leq k\leq d}u_{ik}^{(n)}(y) v_{kj}^{(m,n)}\left(\phi_{m,n}^{-1}(y)\right) \, \Bigr|^p J_{\phi_{m,n}^{-1}}(y)\,d\mu(y) &=\int_{\varphi_n(\mathsf{U}_n)}\left|u_{ij}(y)\right|^p\,d\mu(y) \ .
\big( \Lambda (v_T), v_T\big)_{L^2_{(L-x)dx}}= \langle u(T,.), v_T\rangle_{L^2_{\frac{1}{L-x}dx}, L^2_{(L-x)dx}}=\int_0^T ||\rho (x)v_x||^2_{L^2}dt \ge C ||v_T||^2_{L^2_{(L-x) dx}},
p_{\mathrm{S}_0}(m|n) = \int_{0}^\infty p_{\mathrm{S}_0}(t_\mathrm{p}|n)\frac{(\lambda_\mathrm{B}t_\mathrm{p})^{m}}{m!}e^{-\lambda_\mathrm{B}t_\mathrm{p}}dt_\mathrm{p}
\frac{1}{(\begin{matrix}z\\n\end{matrix})}=\sum_{i=0}^{n-1}(-1)^{n-1-i}(\begin{matrix}n\\i\end{matrix})\frac{n-i}{z-i},\frac{1}{(\begin{matrix}z+n\\n\end{matrix})}=\sum_{i=1}^{n}(-1)^{i-1}(\begin{matrix}n\\i\end{matrix})\frac{i}{z+i}.
G_{k n}\stackrel{k \rightarrow \infty}{\longrightarrow}\, - \frac{a_{n}}{\pi}\left( - 1 \right)^{k}\left( \frac{1}{A}\right)^{k}k ! \, k^{p ( n ) - 1}\left[ 1 +{\cal O}\left( 1 / k \right) \right] .
\left<L, V\right>=L\overline{V}^{t},\quad\mbox{for $L \in \mathcal{M}_{m,n}\left(\mathbb{C}\right)$ and $V \in \mathcal{M}_{n,p}\left(\mathbb{C}\right)$, for $m,n,p\in\mathbb{N}^{\star}$,}
{\cal B}_{W Z W C}^{( \varphi )}+{\cal B}_{W Z W}^{( \varphi )}= \frac \ell{1 6 \pi G}\int_{\cal V}\partial_{\varphi}\left[ 2 \, Q_{2}^{- 1}\partial_{r}Q_{2}Q^{- 1}\partial_{t}Q - Q^{- 1}\partial_{r}Q Q^{- 1}\partial_{t}Q \right] \, d r \, d \varphi \, d t \, .
W^{2,\sigma}(U)\hookrightarrow W^{1,\frac{2\sigma}{2-\sigma}}(U) =\begin{cases}W^{1,\frac{pt}{p+t}}(U), \textnormal{if}\frac{6p}{2+p}\le t; W^{1,\frac{2t}{8-t}}(U), \textnormal{if}\frac{6p}{2+p}\ge t. \end{cases}
w_{\mu}(\bar{x}t^{\delta(n)};q,t) = q^{-|\mu|}(x;q^{-1}, t^{-1})_{\mu}\! \prod_{1\leq i < j\leq n}\dfrac{(t^{j-i+1})_{\mu_i-\mu_j}}{(t^{j-i})_{\mu_i -\mu_j}}