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\displaystyle x|D(R(x)|D(y))\circ R(x)|D(y)\circ R(x)|D(y)\circ R(R(x)|D(y))|y\mbox{ by (TC4)}
\displaystyle 2\pi\int^{\tilde{Q}}_{Q}d\lambda\sigma^{\rm imp}(\lambda).
\displaystyle x|D(R(x)|D(y))\circ R(x)|D(y)\circ R(R(x)|D(y))|y\mbox{ by (SMU2)}
\displaystyle x\otimes y\mbox{ by Proposition \ref{lrequal},}
\displaystyle(s\otimes D(t))\otimes(R(s)\otimes t)
\displaystyle(s|D(t))\otimes(R(s)|t)
\displaystyle(s|D(t))|D(R(s)|t)\circ R(s|D(t))|(R(s)|t)
\displaystyle(s|D(t))|R(s|D(t))\circ D(R(s)|t)|(R(s)|t)\mbox{ by (SMU1)}
\displaystyle s|D(t)\circ R(s)|t
\displaystyle s\otimes t\mbox{ by Proposition \ref{strongood},}
u_{\nu}=\frac{1}{\mu}(\epsilon_{,\nu}+\zeta\beta_{,\nu}+\theta S_{,\nu})
u^{\nu}u_{\nu}=-1.
\displaystyle p^{\rm imp}_{q}(q)
ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)g_{ij}dx^{i}dx^{j},
{\cal H}=-\frac{p_{a}^{2}}{12a}+\Lambda a^{3}-3ka+\frac{p_{T}}{a^{3\alpha}}\,\,,
\displaystyle-ka+\frac{2}{3}\Lambda a^{3}+\frac{1-3\alpha}{6}a^{-3\alpha}p_{T},
\displaystyle-3\dot{a}^{2}+\Lambda a^{4}-3ka^{2}+a^{1-3\alpha}p_{T}.
(\Phi,\Psi)=\int_{0}^{\infty}a^{1-3\alpha}\Phi^{*}\Psi da,
\psi(a,t)=e^{iEt}\psi(a),
\Psi(a,t)=\sum_{n=0}^{m}C_{n}(E_{n})\psi_{n}(a)e^{iE_{n}t},
-\frac{d^{2}\psi(x)}{dx^{2}}+\hat{f}[x]\psi(x)=E\,\hat{g}[x]\,\,\psi(x),
\displaystyle\hat{f}\,\,\psi(x)
\displaystyle\hat{g}\,\,\psi(x)
\displaystyle p_{1\lambda}^{\rm imp}(\lambda)
\displaystyle\left(\frac{n\pi}{L}\right)^{2}A_{n}+B_{n}=E\,B^{\prime}_{n}.
\displaystyle B_{n}=\sum_{m}C_{m,n}\,\,A_{m},
\displaystyle B^{\prime}_{n}=\sum_{m}C^{\prime}_{m,n}\,\,A_{m},
\displaystyle D\,A=E\,D^{\prime}\,A,
\displaystyle D^{\prime-1}D\,A=E\,A,
-\frac{d^{2}\psi(a)}{da^{2}}+(36ka^{2}-12\Lambda a^{4})\psi(a)=12E\psi(a),
-\frac{d^{2}\psi(a)}{da^{2}}+(36ka^{2}-12\Lambda a^{4})\psi(a)=12Ea\psi(a),
-\frac{d^{2}\psi(a)}{da^{2}}+36ka^{2}\psi(a)=12Ea\psi(a),
-\frac{d^{2}\xi}{dx^{2}}+\left[-2\lambda+w^{2}x^{2}\right]\xi(a)=0,
E_{n}=\pm\sqrt{6(2n+1)}\,\,,\mbox{       }n=0,1,2,...\quad.
\displaystyle\theta_{n}(\lambda)
{\Psi}_{n}(a,t)=e^{-iE_{n}t}{\varphi}_{n}\left(12a-E_{n}\right),
{\varphi}_{n}(x)=H_{n}\bigg{(}\frac{x}{\sqrt{12}}\bigg{)}e^{-x^{2}/24}\,\,,
V_{1D}(r)=-\frac{C_{3}}{(r-r_{0})^{3}}
r-r_{0}\to r(x,y)=x-A\sin{(\frac{2\pi}{L}y+\phi)}\,,
V_{2D}(x,y)=-\frac{C_{3}}{(x-A\sin{(\frac{2\pi}{L}y+\phi))^{3}}}\,,
\displaystyle 2\tan^{-1}(\frac{2}{n}\lambda)-\pi;
\displaystyle\Sigma_{nm}(\lambda)
\mathsf{W}
\mathsf{0}
\mathrm{d}
\mathsf{P}
f\left(x\right)=\cos\left(5x\right)
\displaystyle\theta_{|n-m|}(\lambda)
\frac{\,\mathrm{d}f\left(x\right)}{\,\mathrm{d}x}=-5\sin\left(5x\right).
\mathsf{P}_{c}^{\textrm{T}}\mathsf{W}_{c}\mathsf{P}_{c}=\mathsf{I}.
\mathsf{U}^{\textrm{T}}\mathsf{U}=\mathsf{I},
\mathsf{U}=\mathsf{W}_{c}^{\frac{1}{2}}\mathsf{P}_{c}.
\eta_{m}=-\log_{10}\left(\epsilon_{m}\right).
\phi^{-1}(b\star b^{\prime})=\phi^{-1}(\phi(a)\star\phi(a^{\prime}))=\phi^{-1}(\phi(a\ast a^{\prime}))=a\ast a^{\prime}=\phi^{-1}(b)\ast\phi^{-1}(b^{\prime}).
\displaystyle\partial_{q}p^{\rm imp}_{q}(q,T)=2\pi\rho^{\rm imp}(q,T);
\{([n],\star):([n],\star)\cong([n],\ast)\}=\{([n],\ast_{\sigma}):\sigma\in S_{n}\}.
\displaystyle a\ \ast_{\sigma\circ\tau}\ a^{\prime}
\displaystyle=\sigma\circ\tau((\sigma\circ\tau)^{-1}(a)\ast(\sigma\circ\tau)^{-1}(a^{\prime}))
\displaystyle=\sigma(\tau(\tau^{-1}(\sigma^{-1}(a))\ast\tau^{-1}(\sigma^{-1}(a^{\prime}))))
\displaystyle=\sigma(\sigma^{-1}(a)\ast_{\tau}\sigma^{-1}(a^{\prime}))
\displaystyle\delta_{el\uparrow}(T)
\displaystyle\iff\ast_{\sigma}=\ast
\displaystyle\iff\forall b,b^{\prime}\in[n]\colon b\ast_{\sigma}b^{\prime}=b\ast b^{\prime}
\displaystyle\iff\forall b,b^{\prime}\in[n]\colon\sigma(\sigma^{-1}(b)\ast\sigma^{-1}(b^{\prime}))=b\ast b^{\prime}
\displaystyle\iff\forall a,a^{\prime}\in[n]\colon\sigma(a\ast a^{\prime})=\sigma(a)\ast\sigma(a^{\prime})
\sigma^{k}(x_{p,q})=\sigma^{k}(\sigma^{p}(a)\ast\sigma^{q}(b))=\sigma^{k+p}(a)\ast\sigma^{k+q}(b)=x_{(k+p)\bmod r,\,(k+q)\bmod s}
\displaystyle 2\pi\int^{q}_{-D}dq\rho^{\rm imp}(q,T)
\frac{n!}{1^{j_{1}}\cdots n^{j_{n}}j_{1}!\cdots j_{n}!}=\frac{n!}{\prod_{i=1}^{n}i^{j_{i}}j_{i}!}
\ast_{\sigma}=\sigma\circ\ast\circ(\sigma^{-1}\times\sigma^{-1}),
\phi\circ\ast=\star\circ\phi^{\times k}.
\displaystyle\ast_{\sigma\circ\tau}
\displaystyle=\sigma\circ\tau\circ\ast\circ((\sigma\circ\tau)^{-1})^{\times k}
\displaystyle=\sigma\circ\tau\circ\ast\circ(\tau^{-1})^{\times k}\circ(\sigma^{-1})^{\times k}
\displaystyle\epsilon_{el}=\epsilon_{q}(q)-\epsilon(\lambda).
\displaystyle=\sigma\circ\ast_{\tau}\circ(\sigma^{-1})^{\times k}
\displaystyle\iff\sigma\circ\ast\circ(\sigma^{-1})^{\times k}=\ast
\displaystyle\iff\sigma\circ\ast=\ast\circ\sigma^{\times k}
L[\hat{a}]\hat{u}=\hat{f}.
\delta_{el}(\epsilon,T)=\delta_{el}(-\epsilon,T).
\frac{(n-1)!}{(n-i)!}t_{i}(n-i)!Z_{n-i}=(n-1)!t_{i}Z_{n-i},
\sum_{i=1}^{n}(n-1)!t_{i}Z_{n-i}=n!Z_{n},
d_{E}(\mathbf{x},\mathbf{y})=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}
d_{max}(\mathbf{x},\mathbf{y})=max\{|x_{1}-x_{2}|,|y_{1}-y_{2}|\}
d_{1}(\mathbf{x},\mathbf{y})=\{|x_{1}-x_{2}|+|y_{1}-y_{2}|\}
\lim_{p_{i}\rightarrow p}{K(p_{i})}<\infty
\displaystyle\delta_{el\uparrow}(\epsilon_{el},T)
K_{l.c.t}(p)=\frac{1}{d(p,q_{2})}
S\bigcap B(k,r)=\{p\}
\displaystyle x_{1}(t_{1})
\displaystyle x_{2}(t_{1})
\displaystyle x_{1}(t_{2})
\displaystyle x_{2}(t_{2})
\displaystyle(x_{1}(t_{1})-x_{10})^{2}+\left(x_{2}(t_{1})-x_{20}\right)^{2}
\displaystyle(x_{1}(t_{2})-x_{10})^{2}+\left(x_{2}(t_{2})-x_{20}\right)^{2}
\rho(r)\propto{1\over(r/r_{v})^{\alpha}(1/c+r/r_{v})^{2}},
\displaystyle\epsilon_{el}
\lambda={JE^{1/2}\over GM^{5/2}},
{\cal J}(r)\propto\left({r\over r_{v}}\right)^{\beta},
{r_{\rm es}}={1\over\sqrt{2}}{j_{d}\over m_{d}}\lambda r_{v},
{m_{\rm tot}}M(<r_{i})\geq 3M(<r_{f}).