harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
K(\Phi) = \{K_\ell\}_{\ell \geq 0}, K_\ell = \left\{\widehat{h}\in E(G)_x \mid \widehat{h}\, \widehat{g}\, E(G)_x = \widehat{g}\, E(G)_x, \, \textrm{for all}\widehat{g}\in \widehat{U}_\ell \right\},
1= \theta(h)gh^{-1}g^{-1}= \theta(h)g_0 t^{\varepsilon_1}g_1 t^{\varepsilon_2}... g_{n-1}t^{\varepsilon_n}g_n h^{-1}g_n^{-1}t^{-\varepsilon_n}g^{-1}_{n-1}t^{-\varepsilon_{n-1}}... t^{-\varepsilon_1}g_0^{-1}h\in \Sigma_1.
\bigg( \sum_{k=-\infty}^{M-1}\bigg( \sum_{i=-\infty}^{k}2^{-i\frac{r}{p}}V_r(x_{i-1},x_i)^r a_i^r \bigg)^{\frac{q}{r}}\int_{x_k}^{x_{k+1}}u \bigg)^{\frac{1}{q}}\leq C' \bigg(\sum_{k=-\infty}^{M-1}a_k^p \bigg)^{\frac{1}{p}}
{\frac{d^{2}}{d \tau^{2}}}\varphi_{n}^{i}+ n^{2}\varphi_{n}^{i}- m_{i}^{2}( \tau ) \varphi_{n}^{i}- i n \sum_{j}\partial_{\tau}B_{i j}( \tau ) \varphi_{n}^{j}= 0 .
D\epsilon_{j}=\left( d+\frac{1}{2}[e^{a}{\bf \Gamma}_{a}+\frac{1}{2}\omega^{ab}{\bf \Gamma}_{ab}+\frac{1}{r!}b^{[r]}{\bf\Gamma}_{[r]}]\right) \epsilon_{j}-a_{j}^{i}\epsilon_{i}.
\frac{1}{t}\left(p_t\ast u_\lambda -u_\lambda\right)=\frac{\mathrm{e}^{\lambda t}-1}{t}\int_t^\infty \mathrm{e}^{-\lambda s}s^{-\frac{d}{\alpha}}p_1\left(\frac{x}{s^{1/\alpha}}\right)ds-\frac{1}{t}\int_0^t \mathrm{e}^{-\lambda s}s^{-\frac{d}{\alpha}}p_1\left(\frac{x}{s^{1/\alpha}}\right)ds.
p = 2-4r-4s,~~~q = \frac{1}{2}(r + s),~~~u = \frac{1}{2}(1- r - s), & v = \frac{1}{2}-r,~~w = \frac{1}{2}-s, ~~x = \frac{1}{2}(-1+3r+3s), ~~y=-1 + \frac{5}{2}(r + s).
\sum_{k=2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{t-1}}}^{2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{t}}-1}\tau(k) =\tau(2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{t-1}})+\sum_{m=1}^{2^{n_{t}}-1}\tau(2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{t-1}}+m) =t-1+\sum_{m=1}^{2^{n_{t}}-1}(\tau(m)+t-1)=(t-1)\, 2^{n_{t}}+n_{t}\, 2^{n_{t}-1}.
C_{\delta}\times C_{\delta}=(C_{\delta}^{2\varepsilon}\times C_{\delta}^{2\varepsilon})\cup(C_{\delta}^{\varepsilon}\times\hat{C}_{\delta}^{2\varepsilon})\cup(\hat{C}_{\delta}^{2\varepsilon}\times C_{\delta}^{\varepsilon})\cup(\hat{C}_{\delta}^{\varepsilon}\times\hat{C}_{\delta}^{\varepsilon})~.
\frac{1}{2}{\cal S}_{R N}= \beta_{R N}\Lambda \left[ \frac{r_{s}^{3}- r_{h}^{3}}{1 2}\right] \, + \, \frac{q^{2}}{4}\beta_{R N}\left( \frac{1}{r_{h}}- \frac{1}{r_{s}}\right) + 4 \pi \sigma r_{s}^{2}\beta_{R N}\sqrt{f ( r_{s})}
{}\displaystyle \sum_{n=1}^{\infty}S_n^{2-c}\frac{|A_{n}-A_{n-1}|^{2}}{n}\geq \frac{(c-1)^{2}}{4}\displaystyle \sum_{n=1}^{\infty}\frac{n}{S_n^{c}}|A_{n}|^{2},
Y_{\alpha}^{N}( \eta ) = a^{- \frac{n}{2}}e^{\pm i m_{n - 1}\theta_{1}}\prod_{k = 0}^{n - 2}( s i n{\theta_{n - k}})^{m_{k + 1}}C_{m_{k}- m_{k + 1}}^{m_{k + 1}+ \frac{1}{2}( n - k )}( c o s{\theta_{n - k}})
RHS= \int_{t_1-h}^{t_1}f(t)\chi_h'(t)dt+\int_{t_2}^{t_2+h}f(t)\chi_h'(t)dt = f(t_1-\theta_1(h)h)\int_{t_1-h}^{t_1}\chi_h'(t)dt+f(t_2+\theta_2(h)h)\int_{t_2}^{t_2+h}\chi_h'(t)dt = f(t_1-\theta_1(h)h)-f(t_2+\theta_2(h)h)\rightarrow f(t_1)-f(t_2),
\tilde f = \sqrt{\varphi (t,l_3)}e^{-s\varphi (t,l_3)}f + \left( \frac{1}{2}\varphi_t (t,l_3) \varphi^{-\frac{1}{2}}(t,l_3) -s\varphi_t (t,l_3) \sqrt{\varphi (t,l_3)}\right) e^{-s\varphi (t,l_3)}q,
h(x,t) =(\sqrt{L}a_{11})^{1/2}e^{-\pi i L a_{22}a_{21}t^2}&\sum_k \sum_{(q,m)\in\Gamma}b_{(q,m),k}\,e^{-\pi i a_{21}a_{11}(La_{22}t- (q-k))^2}& \hskip.25in H g(t- a_{11}(q-k)) \Phi_{(q,m)}(t , \, x -(q-k)a_{11}) \, e^{2\pi i (q-k)a_{21}t}\,.
{M}= \left( \begin{array}{c c}{C}&{\hat{0}}\\{\hat{0}}&{\bar{C}}\\ \end{array}\right) \left( \begin{array}{c c}{\sigma U}&{\bar{\sigma}V}\\{\bar{\sigma}V}&{\sigma U}\\ \end{array}\right) \left( \begin{array}{c c}{D^{T}}&{\hat{0}}\\{\hat{0}}&{\bar{D}^{T}}\\ \end{array}\! \right) .
\phi (\theta ,\varphi )=a\left\{\cos^{2}\frac{\theta}{2}|2\lambda_{\phi}>-\frac{\sqrt{2}}{2}\sin \theta e^{-i\varphi}|2\lambda_{\phi}-\alpha >+\sin^{2}\frac{\theta}{2}e^{-2i\varphi}|2\lambda_{\phi}-2\alpha >\right\}\, .
\overline{\mbox{\Large$v$}}_{3}(x)={2\pi^{3/2}\over g^{2}\Lambda}x+{1\over3}\left(4x-{{\cal E}^{2}\over x}\right)x^{1/2}{\mit\Gamma}(1/2,x)+{2\over3}(1-2x){\mathrm e}^{-x}-{2\over3}\ ,
S_{\vartheta}= \prod_{n > 0}\exp \left( - \cos 2 \vartheta \; a_{n}\bar{a}_{n}\right) \prod_{n > 0}( \sin 2 \vartheta )^{a_{n}a_{- n}+ \bar{a}_{n}\bar{a}_{- n}}\prod_{n > 0}\exp \left( \cos 2 \vartheta \; a_{- n}\bar{a}_{- n}\right)
\mathcal E_{0,j}:= 2\frac{a_{0,j,w_j}}{\phi^{2 \alpha_j}}- 2 \langle \nabla a_{0,j}, \nabla u_j\rangle\ \mathcal E_{1,j}:= (m+2-p)\,\frac{a_{1,j,w_j}}{\phi^{2 \alpha_j}}- 2 \langle \nabla a_{1,j}, \nabla u_j\rangle.
\mu=cp\sqrt{\pi}\frac{\Gamma(p)}{\Gamma(\frac{5}{2}+p)}\frac{\chi^{2p+2}}{2^{p+2}}\frac{M(p+1,\frac{5}{2}+p,-\frac{\chi^{2}}{2})}{\Gamma(p+1)-\Gamma(p+1,\frac{1}{2}\chi^{2})}
\displaystyle \frac{d^2\theta_i}{du^2}+\frac{2}{l}\frac{\sin(T/l)}{\cos(T/l)}\frac{dT}{d\lambda}\frac{d\theta_i}{d\lambda}-\sin\theta_i\cos\theta_i \prod_{j=i+1}^{j-1}\sin^2\theta_j\left(\frac{d\theta_j}{d\lambda}\right)^2+2\left(\sum_{k=1}^{i-1}\frac{\cos\theta_k}{\sin\theta_k}\frac{d\theta_k}{d\lambda}\right) \frac{d\theta_j}{d\lambda}= 0 \;.
{\mbox{SLER}}_n=\frac{{\vert{}{h}_{11,n}\vert{}}^2}{{\vert{}{h}_{21,n}\vert{}}^2+\beta{}\left(\max\left(\left(\frac{\bar{E}}{P}-{\vert{}{h}_{11,n}\vert{}}^2\right),0\right)\right)}.
{\cal U}_{\cal A}= \exp \left( - \frac{2 \pi{\mathrm i}}{l}\frac{\delta}{2 b}\left( \frac{l}{2 \pi}\hat{p}\right)^{2}\right) \exp \left( - \frac{2 \pi{\mathrm i}}{l}\frac{1}{2 b}\left[ ( 1 - a ) \frac{l}{2 \pi}\hat{p}+ b \frac{l}{2 \pi}\hat{q}\right]^{2}\right)
\begin{array}{lllllll}A^1_1 \oplus \langle a x_2+bh_2+cy_2\rangle \sim A^1_1 \oplus \langle h_2\rangle = (A_1 \oplus J)^1, ~ A^1_1 \oplus \langle a x_2+bh_2+cy_2\rangle \sim A^1_1 \oplus \langle x_2\rangle = (A_1 \oplus J)^2.\end{array}
\mathrm{c o e f f . \o f \}\varphi_{\varphi \pi}|_{\frac{1}{r}- p a r t}= - \frac{\lambda}{6 4 \pi}\cdot ( \mathrm{c o e f f . \o f \}\varphi_{\varphi \varphi}|_{c o n s t - p a r t}) \ .
\langle\eta,\partial_u (H^{(u)})^{-1}\eta\rangle = \langle T(u)^{-1}(H^{(u)})^{-1}\eta,e^{2\rho(u)}T(u)^{-1}(H^{(u)})^{-1}\eta\rangle &=\langle\widetilde\eta^{(u)},e^{2\rho(u)}\widetilde\eta^{(u)}\rangle = \sum_{i\in V}(\widetilde\eta_i^{(u)})^2 e^{2\rho_i(u)}.
\begin{cases}u_{tt}-\Delta u+u_t=|v|^{p}, x\in\mathbb{R}^n,\t>0, v_{tt}-\Delta v+v_t=|u|^{q}, x\in\mathbb{R}^n,\t>0, (u,u_t,v,v_t)(0,x)=(u_0,u_1,v_0,v_1)(x), x\in\mathbb{R}^n,\end{cases}
\varepsilon \rightarrow \varepsilon^{\prime}= W(0, u, v) \varepsilon W^T (0, u, v) = \varepsilon^{12}\left( \begin{array}{cccc}0 -v u 0 v 0 1 v -u -1 0 -u 0 -v u 0 \end{array}\right) = \varepsilon + \varepsilon^{12}\left( \begin{array}{cccc}0 -v u 0 v 0 0 v -u 0 0 -u 0 -v u 0\end{array}\right)
\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2s}\partial_{n+1}w=\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2s}\eta_{R}\partial_{n+1}\tilde{u}=c_{n,s}^{-1}q\eta_{R}u=c_{n,s}^{-1}qw\quad\mathbb{R}^{n}\times\{0\}.
\int_{0}^{\eta}\frac{\mathrm{d}\tau}{a^{2}({\tau})}= \frac{\eta_{0}}{2 \ell_{0}}\frac{1}{a ( \eta )}\biggl [ \frac{\eta}{\eta_{0}}+ \frac{\sqrt{2}}{\ell_{0}}a ( \eta ) \tan^{- 1}\biggl ( \frac{\eta}{\sqrt{2}\eta_{0}}\biggr ) \biggr ] .
\mathfrak{M}=\mathfrak{M}_{1}\times\mathfrak{M}_{2}=\{(\alpha_{1},\alpha_{2}):\alpha_{1}\in \mathfrak{M}_{1},\alpha_{2}\in \mathfrak{M}_{2}\},\ \mathfrak{m}=\left [\frac{1}{Q_{i}},1+\frac{1}{Q_{i}}\right ]^2\setminus\mathfrak{M}.
\Lambda f=\sum_{j,k=0}^n g^{jk}(x)\Big(\frac{\partial u}{\partial x_j}-iA_j(x)u\Big)\nu_k(x)\Big(\sum_{p,r=0}^n g^{pr}(x)\nu_r\nu_p\Big)^{-\frac{1}{2}}\Big|_S,
F_{\tilde{a}}=\{v\ |\v\in F_c{\rm\and}\ \max(v)\leq m\}\cup \{v\ |\v\cdot (v'+m)\in F_c{\rm\for\some}\v'\{\rm\and}\ \max(v)\leq m\}begin{align*}5pt]F_{\tilde{b}}=\{v'\ |\v'+m\in F_c \}\cup \{v'\ |\v\cdot(v'+m)\in F_c{\rm\for\some}\v\{\rm\such\that}\ \max(v)\leq m\}.
\dfrac{\hat{k}_2(\Delta)}{\hat{k}_2(\Psi)}= \dfrac{1}{1-x_{\sigma}R_{\sigma}}=\dfrac{x_{222}^*}{x_{111}}=\dfrac{(x_{1,1}+x_{1,2})(x_{2,1}+x_{2,2})(x_{3,1}+x_{3,2})}{x_{1,1}x_{2,1}x_{3,1}},
\sum_{k=0}^n \binom{p+k}{p}\binom{q+n-k}{q}\binom{k}{c}\binom{n-k}{d}H_{q+n-k}={}\binom{n+p+q+1}{n-c-d}\binom{p+c}{c}\binom{q+d}{d}(H_{n+p+q+1}-H_{p+q+c+d+1}+H_{q+d})
\hat{V}^{C}_{\sigma}(C_{1},C_{2}) = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{1}{N}\sum\limits_{i=1}^N \frac{1}{(4\pi^{2}\sigma^{4})}exp \left ( - \frac{1}{2\sigma^{2}}( a(u_1-b)^{2}+ a'(u_2-b')^{2}+ c \right ) \mathrm{d}u_1\mathrm{d}u_2
\delta \left( Z_{\; \; \beta_{2}}^{\alpha_{1}}\eta_{\alpha_{1}}^{*}+ \frac 1 2 C_{\beta_{2}}^{\alpha_{0}\beta_{0}}\Phi_{\alpha_{0}}^{*}\Phi_{\beta_{0}}^{*}\right) = D_{\; \; \beta_{2}}^{\alpha_{2}}\Phi_{\alpha_{2}}^{*},
\widehat{x}_p =\mathbf{q}_p^{\mathrm{H}}(\mathbf{y}+\widetilde{\mathbf{y}}^{(p)}) &=\mathbf{q}_p^{\mathrm{H}}\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{V}\mathbf{x}+\mathbf{q}_p^{\mathrm{H}}\mathbf{z}+\mathbf{q}_p^{\mathrm{H}}\widetilde{\mathbf{y}}^{(p)}.
exp\bigg( - \int_X \tilde{L}^{**}_{\omega +}\bigg) = \int{\cal D}R^+ exp\Bigg(- \int_X \epsilon^{\mu \nu \rho\sigma}\big(\tau^+ R^{+ab}_{\mu \nu}R^+_{ab\rho \sigma}+ i R_{\mu\nu}^{+ \ \ab}G^+_{\rho \sigma ab}\big) \Bigg).
\tilde{L}(x,t,m,u,p,r,S)= \frac{1}{2}u^2-\frac{\varepsilon}{\rho}+p(x_m-\tau)+u(x_t-u)+rS_t = ux_t+p x_m+rS_t-\left(\frac{1}{2}u^2+w\right),
\overline{MH}=\sum_{i}\frac{\overline{p}_{i}\overline{q}_{i}}{\left(1-\alpha\right)\overline{p}_{i}+\alpha \overline{q}_{i}}\:\:;\:\overline{MA}=\sum_{i}\alpha \overline{p}_{i}+\left(1-\alpha\right)\overline{q}_{i}=1
S_{0}\! = \! \sum_{n = 0}^{\infty}\! \int \! d t d \xi_{2}d \xi_{3}\left\{| \partial_{t}\Phi_{\! n}( \xi_{2}, \xi_{3}, t ) |^{2}\! \! - \! | \partial_{\xi_{3}}\! \Phi_{\! n}( \xi_{2}, \xi_{3}, t ) |^{2}\! \! - \! M_{\! n}^{2}| \Phi_{\! n}( \xi_{2}, \xi_{3}, t ) |^{2}\! \right\},
{\cal L}^{(1)}= -\partial_\mu \overline \psi_1 \gamma_{\mu\nu}\partial_\nu \psi_1-\partial_\mu \overline \psi_2 \gamma_{\mu\nu}\partial_\nu \psi_2 - m^2 \overline \psi_1 \psi_1+ m^2 \overline \psi_2 \psi_2 .
A_{\alpha \beta}( \widetilde \phi^{2}) = - K^{3}\sum_{l = 1}^{\infty}{\frac{{\left({- \ln K}\right)^{l}}}{{l !}}}\left\{{\frac{1}{2}g_{\alpha \beta}\sum_{m = 1}^{2 l - 1}{\widetilde \phi_{m}^{2}}\phi_{2 l - m}+ \delta_{\alpha , 0}\delta_{\beta , 0}\sum_{m = 1}^{l}{\widetilde \phi_{2 m - 1}^{2}\phi_{2 l - 2 m + 1}}}\right\}.
{V}_{\mathrm{I}}^{\mathrm{g a u g e}}( r ) = - \, \frac{g_{\mathrm{Y M}}^{2}}{2}~ \left( \delta^{A E}\, \delta^{F D}- \delta^{A F}\, \delta^{D E}\right) ~ \frac{1}{7 \, \Omega_{8}\, r^{7}}~ ~ ,
M_m. f(s,t,v)=\lambda^mt f(s+m,t,v), &Y_m. f(s,t,v)=\lambda^m \Bigg[\bigg(m\sum\limits_{i=0}^{q}\tau_iv^i+v+am^2t+(1-\delta_{m,0})b\bigg)f(s+m,t,v) &+ mt\frac{\partial}{\partial v}f(s+m,t,v)\Bigg],
\mathcal{Z}(m, \mu )\times_{\mathcal{M}}\mathcal{M}_0 \simeq \bigsqcup_{\substack{m_1+m_2=m (\mu_1,\mu_2)\in ( \mu+L)/(L_0\oplus \Lambda)}}\mathcal{Z}_0(m_1,\mu_1) \times \Lambda_{m_2,\mu_2},
H_{n \times p}=\left(\begin{array}{ccccc}a_{n-1}a_{n-2}a_{n-3}\cdots a_{n-p}a_{n-2}a_{n-3}a_{n-4}\cdots a_{n-p-1}a_{n-3}a_{n-4}a_{n-5}\cdots a_{n-p-2}\vdots \vdots \vdots \cdots \vdots a_{0}a_{-1}a_{-2}\cdots a_{1-p}\end{array}\right).
\zeta_{g}=\frac{\partial v_{g}}{\partial x}-\frac{\partial u_{g}}{\partial y}=\frac{1}{f_{0}}(\frac{\partial^{2}\Phi}{\partial x^{2}}+\frac{\partial^{2}\Phi}{\partial y^{2}})=\frac{1}{f_{0}}\nabla^{2}\Phi
z C^{(1+\alpha)}_n(z) = \frac{n + 1}{2( n + \alpha + 1)}C^{(1+\alpha)}_{n+1}(z) + \frac{n + 2 \alpha + 1}{2( n + \alpha + 1)}C^{(1+\alpha)}_{n-1}(z), \ \ \n = 1,2,3,\ldots\ .
\big(\Sigma_3,D_3,\Phi_3,V_3\big)\ =\ \big(\widehat{\Sigma}_{N+},\widehat{D}_{N+},\zeta,\widehat{V}_{N+}\big)\oplus \big(\widehat{\Sigma}_{N-},\widehat{D}_{N-},-1,\widehat{V}_{N-}\big)\,.
0=|\Lambda_{3,1}^{(0)}|=|\Lambda_{3,2}^{(0)}|=|\Lambda_{3,3}^{(0)}|-1, 1=|\Lambda_{3,1}^{(1)}|=|\Lambda_{3,2}^{(1)}|=|\Lambda_{3,3}^{(1)}|, 2=|\Lambda_{3,1}^{(2)}|=|\Lambda_{3,2}^{(2)}|=|\Lambda_{3,3}^{(2)}|,
\mathsf{d}_{cc}(Z) =\mathsf{d}_{cc}(Z_1\cdot Z_2)=\mathsf{d}_{cc}(Z_1\cdot Z_2) \leq \mathsf{d}_{cc}(Z_1)+\mathsf{d}_{cc}(Z_2)\leq \|Z_1\|_1+2\mathsf{d}_{\mathrm{com}}(\{X_{ni}\}) \leq \frac{1}{2}+\frac{1}{2}=1.
\frac{h}{r^3}\frac{d}{dr}(h r^{3}\frac{dR}{dr}) + [\frac{(\omega r_{1}r_{K}r_{5})^{2}}{r^{6}}+ \frac{\omega^{2}( r_{1}^{2}r_{5}^{2}+ r_{1}^{2}r_{K}^{2}+ r_{5}^{2}r_{K}^{2})}{r^4}]R = 0.
\begin{array}{rcl}\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}A_{p}^{\top}\tilde{\mu}_{p}= \left(A_{p}\hat{v}_{p}-A_{p}\tilde{v}_{p}+a_{p}-a_{p}\right)^{\top}\tilde{\mu}_{p}= \left(A_{p}\hat{v}_{p}-a_{p}\right)^{\top}\tilde{\mu}_{p}-\left(A_{p}\tilde{v}_{p}-a_{p}\right)^{\top}\tilde{\mu}_{p}= 0\end{array}
\lim_{\lambda \to \infty}\left[ \mathrm{R e}\Gamma ( \varphi , \lambda ) \right]_{\lambda - d e p}= \varphi_{1}^{2}\lim_{\lambda \to \infty}\frac{\pi}{2}\sum_{n_{1}}\frac{\tau_{2}^{2}\lambda^{2}}{( \lambda^{2}+ \tau_{2}^{2}n_{1}^{2})^{\frac{3}{2}}}\,
\vec{\phi}_{T K 2^{*}}[ x ; a_{1}, a_{2}] = \frac{( - 1 )^{\alpha}}{2 \sqrt{2}}\tanh ( x + a_{1}) \vec{\varepsilon}_{1}+ \frac{( - 1 )^{\beta}}{2 \sqrt{2}}\tanh ( x + a_{2}) \vec{\varepsilon}_{2}\quad ,
\mathfrak{R}_{2k}(\lambda) = -\frac{1}{2(2k-1)}\big[z^{-2k}\big] \left\{\exp \left((2k-1) \sum_{j=1}^\infty \frac{2p_{2j-1}(\lambda)}{2j-1}z^{-(2j-1)}\right) \right. \left.\hphantom{\mathfrak{R}_{2k}(\lambda) =}{}\times \exp \left(-(2k-1) \sum_{j=1}^\infty \frac{2p_{2j-1}(\lambda)}{2j-1}(z-1)^{-(2j-1)}\right) \right\}.
\begin{cases}\rho(x) u_{tt}- \operatorname{div}(K(x) \nabla u) + v^2u =0 ~\hbox{in}~ \Omega \times (0,T), \rho(x) v_{tt}- \operatorname{div}(K(x) \nabla v) + u^2v=0 ~\hbox{in}~ \Omega\times (0,T), u_t=v_t=0 ~\hbox{in}~ \omega\times(0,T)\end{cases}
I_3 \leq C \sum_{n=1}^\infty \int^\infty_{n+1}\frac{|f(\mu )|^2\mu^{2\alpha}}{( |\mu -n| + 1)^{2-2\beta}}d\mu \leq C \int^\infty_{0}{|f(\mu )|^2 \mu^{2\alpha}}\sum_{n=1}^\infty \frac{1}{( |\mu -n| + 1)^{2-2\beta}}d\mu &\leq C\int^\infty_{0}{|f(\mu )|^2 \mu^{2\alpha}}d\mu.
b_{uv}=(-\tau n)_{v}=(\lambda\tau^{2})t+(-\tau_{v})n+(-\tau\gamma)b, b_{vu}=(-\lambda_{u}t+\gamma n)_{u}=(-\lambda_{uu}-\kappa\gamma)t+(\lambda_{u}\kappa+\gamma_{u})n+(-\tau\gamma)b.
(g_{21}y_1+g_{22}y_2+g_{23}y_3)\{(g_{11}+g_{21})y_1+g_{22}y_2+g_{23}y_3\}^{b-1}=\alpha y_1^a+\beta y_2(y_1+y_2)^{a-1}+g_{23}^b y_3(y_1+y_2+y_3)^{b-1}
\left(-zq^{1/2},-z;q\right)_{\infty}=\left(-z;q^{1/2}\right)_{\infty}=\sum_{n=0}^{\infty}\frac{q^{n(n-1)/4}z^{n}}{\left(q^{1/2};q^{1/2}\right)_{n}}=\sum_{k=0}^{\infty}z^{k}q^{k^{2}/2}S_{k}\left(-q^{-k-1/2};q\right),
I_{Q_2}^{Q_1}(e_{Q_2}(\sigma_{\ell_{Q_1}- \ell_P}))_{\ell - \ell_{Q_1}}= I_{Q_2}^{Q_1}(e_{Q_2}(\sigma_{\ell_{Q_2}- \ell_P})_{\ell_{Q_1}- \ell_{Q_2}})_{\ell - \ell_{Q_1}}& = I_{Q_2}^{Q_1}(e_{Q_2}(\sigma_{\ell_{Q_2}- \ell_P})_{\ell - \ell_{Q_2}}) & = I_{Q_2}^{Q_1}(e_{Q_2}(\sigma_{\ell - \ell_P})).
(\Psi^T_\lambda(t)J\Psi_\lambda(t))' =(\Psi'_\lambda(t))^TJ\Psi_\lambda(t)+\Psi^T_\lambda(t)J\Psi'_\lambda(t) &=(JS_\lambda(t)\Psi_\lambda(t))^TJ\Psi_\lambda(t)+\Psi^T_\lambda(t)J^2S_\lambda(t)\Psi_\lambda(t)=0,
\lim_{t\to\infty}\frac{f(y(t-K(t)))}{y(t-K(t))}K(t) =\lim_{t\to\infty}\frac{f(y(\kappa(t)))}{y(\kappa(t))}K(t)=\lim_{z\to\infty}\frac{f(y(z))}{y(z)}K(\kappa^{-1}(z)) &=\lim_{z\to\infty}\frac{f(y(z))}{y(z)}K(z) \cdot \frac{K(\kappa^{-1}(z))}{K(z)}&=\lim_{z\to\infty}\frac{f(F^{-1}(Mz))}{F^{-1}(Mz)}K(z) \cdot \frac{K(\kappa^{-1}(z))}{K(z)}&=+\infty
H ( \{q_{N_{s}}\}) = ( u_{1}- u_{2}) ( v_{1}- u_{2}) \left[ 1 - \frac{\mu_{1}\mu_{2}}{2 ( u_{1}- u_{2})}- \frac{\nu_{1}\mu_{2}}{2 ( v_{1}- u_{2})}\right] \prod_{s = 1}^{n}( u_{s}- v_{s}- \mu_{s}\nu_{s})^{- 1}
\sup_{v \in L^{\widetilde A}(\Omega )}\frac{\int_\Omega uv\,dx}{\|v\|_{L^{\widetilde A}(\Omega )}}= \sup_{v \in L^\infty(\Omega)}\frac{\int_\Omega uv\, dx}{\|v\|_{L^{\widetilde A}(\Omega )}},
\mathcal{A}_{\textrm{approx,1}}:= \alpha \left(\begin{array}{cc}G_{0}\otimes{A}_{11}^{0}G_{0}\otimes{A}_{21}^{0}G_{0}\otimes{A}_{12}^{0}G_{0}\otimes{A}_{22}^{0}\end{array}\right), \mathcal{A}_{\textrm{approx},2}:= \frac{1}{\alpha \beta}\left(G_{0}\otimes D_{0}\right).
L\Bigl( \frac{1}{2}, f \times u_j\Bigr ) = \sum_{m \geq 1}\sum_{n \geq 1}\frac{\lambda_j(n) A(n, m)}{(m^2 n)^{\frac{1}{2}}}V_-(m^2 n, t_j) +{}&\qquad{}+ \sum_{m \geq 1}\sum_{n \geq 1}\frac{\lambda_j(n) A(m, n)}{(m^2 n)^{\frac{1}{2}}}V_+(m^2 n, t_j)
\lim_{n \to \infty}n^{3-d(r,s)}E(S_{\lfloor nt \rfloor}(r)\overline{S_{\lfloor nu \rfloor}(s)}) = \frac{\sigma(r,s)}{(3-d(r,s))(2-d(r,s))}\left[c(r,s)u^{3-d(r,s)}+\overline{c(s,r)}(t^{3-d(r,s)}-(t-u)^{3-d(r,s)}) \right]
{\mathbb E}\left[{{{\left\|{{\bf{\bar q}}\left[i \right]}\right\|}^2}}\right] = \frac{1}{K}\sum\limits_{k = 0}^{K - 1}{{{\Big\|{\sum\limits_{{l}= 1}^{L'}{{{\bf{\bar H}}_{l}^\bot{\bf{\bar X}}_{l}}{{\bf{u}}_k}{e^{- j\frac{{2\pi}}{{K}}k{\kappa_{l}}}}}}\Big\|^2}}}.
a_n = \frac{1}{(\alpha_1 - \alpha_2) (\alpha_2 - \alpha_3) (\alpha_3 - \alpha_1)}\left[ \frac{\alpha_3 - \alpha_2}{\alpha_1^{n + 1}}+ \frac{\alpha_1 - \alpha_3}{\alpha_2^{n + 1}}+ \frac{\alpha_2 - \alpha_1}{\alpha_3^{n + 1}}\right] .
\sum_{n=0}^{\infty}\frac{\left(b;q\right)_{n}S_{n}\left(x;q\right)z^{n}}{\left(c;q\right)_{n}}=\frac{\left(c/b,bz;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(b;q\right)_{n}\left(\frac{c}{b}\right)^{n}}{\left(bz;q\right)_{n}}S_{n}\left(\frac{xbz}{c}\right),
A_{u v}( x , y ) \; = \; \int \frac{d^{4}q}{( 2 \pi )^{4}}\: \hat{V}( q ) \int \frac{d^{4}p}{( 2 \pi )^{4}}\: A_{u v}( p + \frac{q}{2}, p - \frac{q}{2}) \; e^{- i p ( x - y )}\: e^{- i \: \frac{q}{2}\: ( x + y )}\; ,
p(\mathbf u , \mathbf y | \mathbf u \in \Lambda_{c,n}) = p(\mathbf y | \mathbf u )p(\mathbf u| \mathbf u \in \Lambda_{c,n}) = \prod_{k=1}^n p(y_k | u_k)\prod_{k=0}^n p(u_k) = \prod_{k=1}^n p(u_k , y_k).
(\ddot{\theta},v_h )+a(\theta,v_h)-\int_0^t K(t-s) a(\theta(s),v_h)ds = (f,v_h)-(R_h\ddot{u},v_h)-a(u,v_h)+\int_0^t K(t-s) a(u(s),v_h)ds = (\ddot{u},v_h)-(R_h\ddot{u},v_h) =-(\ddot{\omega},v_h).
m_{j_{1},j_{2},j_{3}}(\xi,\tau) =e((\Phi(\xi)-\tau)\cdot (x_{4},x_{5},x_{6}))\prod_{i=1}^{3}M_{j_{i}}(\frac{N(\tau_{i}-\Phi_{i}(\xi))}{2})\prod_{i=1}^{3}H(N^{\frac{1}{2}}(\xi_{i}-c_{i})),
h^{\mu \nu}= \frac{e}{4 f}\left( \epsilon^{\alpha \beta \sigma \nu}\partial_{\sigma}T_{( \alpha \beta )}{}^{\mu}+ \epsilon^{\alpha \beta \sigma \mu}\partial_{\sigma}T_{( \alpha \beta )}{}^{\nu}\right) - g^{\mu \nu}\frac{k e}{2 \left( f + 4 k \right) f}\epsilon^{\alpha \beta \rho \kappa}\partial_{\rho}T_{( \alpha \beta ) \kappa}.
|\int_{0}^{1}\frac{u(x)-u(-x)}{x}dx|\le\int_{0}^{1}\frac{|u(x)-u(-x)|}{x}dx\le\int_{0}^{1}\frac{2x}{x}sup_{x\in\mathbb{R}}|u^{\prime}(x)|dx\le2sup_{x\in\mathbb{R}}|u^{\prime}(x)|.
H=\begin{pmatrix}2\cos \frac{2\pi}{n}-1 0 \cdots 0 -1 -1 2\cos \frac{2\pi}{n}-1 0 \cdots 0 0 -1 2\cos \frac{2\pi}{n}-1 \ddots \vdots \vdots 0 \ddots \ddots \ddots 0 0 \vdots \ddots -1 2\cos \frac{2\pi}{n}-1 -1 0 \cdots 0 -1 2\cos \frac{2\pi}{n}\end{pmatrix}_{n \times n}.
M_G\left( 1+m(1+x)(1+y)(1+z)\right) = 8m+1, & M_G\left( 2+m(1+x)(1+y)(1+z)\right) = 2^8(4m+1), &M_G\left( 3+z+k(1+x)(1+y)(1+z)\right) = 2^{12}(2k+1), & M_G\left( x+y+z-3+(1-x)(1-y)(1-z) +k(1+x)(1+y)(1+z)\right) = 2^{12}(2k),
|x_j-x_k| \ge |x_k|-|x_j|\ge \frac{1}{4}\frac{4}{4^J}|x_1|-\frac34\frac{1}{4^J}|x_1| &=\frac{1}{4}\frac{1}{4^J}|x_1|\ge \frac{1}{4}\frac{1}{4^N}|x_1|=c_N|x_1|\,,
\sum_{j\geq0}t^{j}T_{2j+1}(x)T_{2j+1}(y)= \frac{(1-t)xy(1+6t+t^{2}-4t(x^{2}+y^{2}))}{(1-t)^{4}+8t(1-t)^{2}(x^{2}+y^{2})-16t(1+t^{2})x^{2}y^{2}+16t^{2}(x^{4}+y^{4})}.
{\cal W}(\theta)=\left({\cal P}\int d\theta'\rho(\theta')\cot\frac{\theta-\theta'}{2}\right)^2-2{\cal P}\int d\theta'\rho(\theta')d\theta''\rho(\theta'')\cot\frac{\theta-\theta'}{2}\cot\frac{\theta'-\theta''}{2}\; .
\rho = \lim_{r \to \infty}\dfrac{s_{m,k,n,mn}(r)r^{mn}}{|B_r^{n(k+m)}|}= \lim_{r \to \infty}\dfrac{s_{m,k,n,mn}(r)r^{mn}}{(2r)^{n(k+m)}}= \lim\limits_{r \to \infty}\dfrac{F_{m,k,n}(r)}{(2r)^{nk}}.
g_\pm := \left[ \begin{matrix}g_{11}\mp c^{-1}g_{11}g_{23}(cg_{33}\mp g_{23})(cg_{33}^2-g_{23}^2)^{-1}g_{11}g_{23}(g_{23}\mp g_{33})(cg_{33}^2-g_{23}^2)^{-1}\cdot \pm g_{33}g_{23}\cdot \pm c^{-1}g_{23}g_{33}\end{matrix}\right].
\textstyle \sqrt[3]{a - c \cdot t_1}\ + \ \sqrt[3]{a - c \cdot t_2}\ +\ \sqrt[3]{a - c \cdot t_3}\ = \ \sqrt[3]{\left(\frac{3+B}{2}\right) \ - \ 6 \ + \ 3\sqrt[3]{\frac{27 + B^2}{4}}},
L'(t)= -I(u(x,t))=\frac{q-p}{p}\|\nabla u\|_p^p+\frac{1}{q}\|u\|_q^q-qJ(u(x,t)) &\ge \frac{q-p}{p}\Big(\frac{2}{\widetilde{C}}L(t)-1\Big)-qJ(u(x,t)) &=\frac{q-p}{p}\frac{2}{\widetilde{C}}\Big(L(t)-C_1-C_2J(u(x,t)\Big),
\mathbf{III}= -\int_{\partial B_R(0) \cap \Omega}\left[ \left( \varphi -{m \cdot x\over \gamma_n |x|^n}\right) (c \cdot N) - (c \cdot x) N \cdot ( \nabla \varphi + V) \right] \, dS + O\left(\frac{1}{R^n}\right).
d\varphi_{s,t}(x) =-X(\circ dt,\varphi_{s,t}(x)) =-\sum_{i=1}^{\infty}X^{(i)}(t,\varphi_{s,t})\,\circ dB_{i}(t)-X^{(0)}(t,\varphi_{s,t})\, dt\,, \varphi_{s,s}(x) =x\,.
\lim_{\epsilon \to 0}R^{\epsilon}_0(x,0)(z) = \frac{1}{( 2 \pi)^d}\sum_{k=0}^{\frac{d-3}{2}}\frac{c_k (-i)^{s_k}}{|x|^{d-1-s_k}}\partial_s^{s_k}\Big\{e^{izs}E_i(- izs) + e^{-izs}E_i( izs) + 2 \pi i e^{iz s}\Big\}\Big|_{(s=|x|)}.
\begin{array}{rcccl}\partial_{p^2}\Gamma^T_{ZZ}(p^2=M_Z^2) = 1 \Rightarrow z_{ZZ}\partial_{p^2}\Gamma^T_{AA}(p^2=0) = 1 \Rightarrow z_{AA}\Gamma^T_{AZ}(p^2=0) = 0 \Rightarrow \theta_V \Gamma^T_{AZ}(p^2=M_Z^2) = 0 \Rightarrow z_{AZ}\Gamma^T_{ZZ}(p^2=M_Z^2) = 0 \end{array}
\big(u+vR_0(\lambda-\kappa^2)\;\!v\big)^{-1}=2\kappa\;\!\big(I_0(\kappa)+S_0\big)^{-1}+\big(I_0(\kappa)+S_0\big)^{-1}S_0I_1(\kappa)^{-1}S_0\big(I_0(\kappa)+S_0\big)^{-1},
\omega_{2}~ = ~ - \left[ 4 ( \mu - 1 ) ( \mu - 2 ) \hat{\Psi}( \mu ) ~ + ~ \frac{( 1 6 \mu^{4}- 9 6 \mu^{3}+ 1 8 8 \mu^{2}- 1 3 6 \mu + 2 5 )}{2 ( 2 \mu - 3 ) ( \mu - 1 )}\right] \eta_{1}^{2}~ + ~ \Pi_{2}
Y_{\nu}dC_{\nu}\cdot C_{\nu}^{-1}Y_{\nu}^{-1}=-\bar{\omega}_{u}-dY_{\nu}\cdot Y_{\nu}^{-1}\sim-\bar{\omega}_{u}+zdU-[dU,G_{1}]-dH_{0}\cdot H_{0}^{-1}+O(z^{-1}).
Q/Q(2) \downarrow L' = (1,0,000) + (0,1,100), Q(2)/Q(3) \downarrow L' = (1,1,100), Q(3)/Q(4) \downarrow L' = (1,0,010), Q(4)/Q(5) \downarrow L' = (0,0,010), Q(5)/Q(6) \downarrow L' = (0,1,001), Q(6)/Q(7) \downarrow L' = (0,0,000), Q(7) \downarrow L' = (1,0,000).
[ a_{n}^{( r ) \mu}, a_{m}^{( s ) \nu \dagger}] = \eta^{\mu \nu}\delta_{n m}\delta^{r s}, \quad \{c_{n}^{( r )}, b_{m}^{( s ) \dagger}\}= \{b_{n}^{( r )}, c_{m}^{( s ) \dagger}\}= \delta_{n m}\delta^{r s}, \quad ( n , m \ge 1 ) ,
\|x_0\|_1+\rho \geq\|\hat{x}\|_1 =\|x_0+h\|_1 =\|x_{0,T_0}+h_{T_0}+x_{0,T_0^c}+h_{T_0^c}\|_1 \geq \|x_{0,T_0}\|_1-\|h_{T_0}\|_1-\|x_{0,T_0^c}\|_1+\|h_{T_0^c}\|_1.
\begin{array}{lll}\|\mathcal L_{\mu_2}(f,g)^t\|^2 =\alpha \int_{\mathbb R}| |\xi| \coth(\sqrt{\mu_2}|\xi|)-|\xi||^2|\widehat{g}(\xi)|^2d\xi &\leqq \frac{\alpha}{\mu_2}\|g\|^2\to 0\end{array}
S ( 0 , a_{1}, \ldots , a_{n}) \ = \ ( - 1 )^{n}( n - 1 ) \, a_{1}\,{\mathcal R}_{Q , Q^{\prime}}+ ( - 1 )^{n}n \, a_{1}\,{\mathcal R}_{Q , Q^{\prime}}\ = \ ( - 1 )^{n}( 2 n - 1 ) \, a_{1}\,{\mathcal R}_{Q , Q^{\prime}}
\Theta\left[\begin{array}{c}r/k 0\end{array}\right]\Big(x\Big|-\frac{k}{\tau}\Big)=\frac{1}{k}\,\left(-ik\tau\right)^{1/2}\,e^{i\pi\tau r^2/k}\,\sum_{r'=0}^{k-1}\,e^{2i\pi rr'/k}\,\Theta\left[\begin{array}{c}r'/k 0\end{array}\right]\Big(-\tau x\Big|k\tau\Big)\ \ \ .
F_{N, n}(z,s) := \sum_{\gamma\in \Gamma_0(N)_{\infty}\backslash\Gamma_0(N)}\pi \left| n\mathrm{Im}(\gamma z)\right|^{1/2}I_{s-\frac12}(|2\pi n\mathrm{Im}(\gamma z)|)e(-n\mathrm{Re}(\gamma z)),
\widetilde{\varphi}(p) ={w_{{}_1}}^+(p) \, f_+(p) \, + \,{w_{{}_1}}^-(p) \, f_-(p) \,+ \, w_{{}_{r^{-2}}}(p)\, f_{0+}(p)\, + \, w_{{}_{r^2}}(p) \, f_{0-}(p)
\left( \begin{matrix}{{t^{{\vec{\Lambda}}_{1}}= p^{{\vec{\Lambda}}_{1}}+ \mathrm{i}\, q_{{\vec{\Lambda}}_{1}}\,}}\\{{{\bar{t}}_{{\vec{\Lambda}}_{1}}= p^{{\vec{\Lambda}}_{1}}- \mathrm{i}\, q_{{\vec{\Lambda}}_{1}}\,}}\\ \end{matrix}\right)
{M_{c}= \left( \begin{matrix}{0}&{1}&{0}&{0}&{\cdots}&{0}\\{0}&{0}&{1}&{0}&{\cdots}&{0}\\{0}&{0}&{0}&{1}&{\cdots}&{0}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\{1}&{0}&{0}&{0}&{\cdots}&{0}\\ \end{matrix}\right) ,}
\frac{1}{4\pi}L_{CM}= \frac{vg}{2k}\left(\sum_{j=1}^k\vec{V}_j\right)^2-\frac{v}{2gk}\left(\sum_{j=1}^kq_j\right)^2+\frac{b}{k}\left(\sum_{j=1}^k q_j\right)\left(\sum_{l=1}^k V_{l\chi}\right),
\|u(t_n)-u_H^{n}\|+\gamma^{\frac12}\Big{(}\Delta t\sum_{k=1}^n\|\sigma(t_{k-\theta})-\sigma_H^{k-\theta}\|^2\Big{)}^{\frac12}+\Big{(}\Delta t\sum_{k=1}^n\|\nabla (u(t_{k-\theta})-u_{H}^{k-\theta})\|^2\Big{)}^{\frac12}\leq C[\Delta t^2+H^{m+1}],