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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク A-equivalence ： ウィキペディア英語版 A-equivalence In mathematics, $\backslash mathcal$-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. Let $M$ and $N$ be two manifolds, and let $f,\; g\; :\; (M,x)\; \backslash to\; (N,y)$ be two smooth map germs. We say that $f$ and $g$ are $\backslash mathcal$-equivalent if there exist diffeomorphism germs $\backslash phi\; :\; (M,x)\; \backslash to\; (M,x)$ and $\backslash psi\; :\; (N,y)\; \backslash to\; (N,y)$ such that $\backslash psi\; \backslash circ\; f\; =\; g\; \backslash circ\; \backslash phi.$ In other words, two map germs are $\backslash mathcal$-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. $M$) and the target (i.e. $N$). Let $\backslash Omega(M\_x,N\_y)$ denote the space of smooth map germs $(M,x)\; \backslash to\; (N,y).$ Let $\backslash mbox(M\_x)$ be the group of diffeomorphism germs $(M,x)\; \backslash to\; (M,x)$ and $\backslash mbox(N\_y)$ be the group of diffeomorphism germs $(N,y)\; \backslash to\; (N,y).$ The group $G\; :=\; \backslash mbox(M\_x)\; \backslash times\; \backslash mbox(N\_y)$ acts on $\backslash Omega(M\_x,N\_y)$ in the natural way: $(\backslash phi,\backslash psi)\; \backslash cdot\; f\; =\; \backslash psi^\; \backslash circ\; f\; \backslash circ\; \backslash phi.$ Under this action we see that the map germs $f,\; g\; :\; (M,x)\; \backslash to\; (N,y)$ are $\backslash mathcal$-equivalent if, and only if, $g$ lies in the orbit of $f$, i.e. $g\; \backslash in\; \backslash mbox\_G(f)$ (or vice versa). A map germ is called stable if its orbit under the action of $G\; :=\; \backslash mbox(M\_x)\; \backslash times\; \backslash mbox(N\_y)$ is open relative to the Whitney topology. Since $\backslash Omega(M\_x,N\_y)$ is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking $k$-jets for every $k$ and taking open neighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions of these base sets. Consider the orbit of some map germ $orb\_G(f).$ The map germ $f$ is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs $(\backslash mathbb^n,0)\; \backslash to\; (\backslash mathbb,0)$ for $1\; \backslash le\; n\; \backslash le\; 3$ are the infinite sequence $A\_k$ ($k\; \backslash in\; \backslash mathbb$), the infinite sequence $D\_$ ($k\; \backslash in\; \backslash mathbb$), $E\_6,$ $E\_7,$ and $E\_8.$ ==See also== * K-equivalence (contact equivalence) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「A-equivalence」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.