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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 Quasi-isometry ： ウィキペディア英語版 Quasi-isometry In mathematics, quasi-isometry is an equivalence relation on metric spaces that ignores their small-scale details in favor of their coarse structure. The concept is especially important in Gromov's geometric group theory.〔 〕 ==Definition== Suppose that $f$ is a (not necessarily continuous) function from one metric space $(M\_1,d\_1)$ to a second metric space $(M\_2,d\_2)$. Then $f$ is called a ''quasi-isometry'' from $(M\_1,d\_1)$ to $(M\_2,d\_2)$ if there exist constants $A\backslash ge\; 1$, $B\backslash ge\; 0$, and $C\backslash ge\; 0$ such that the following two properties both hold:〔P. de la Harpe, ''Topics in geometric group theory''. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6〕 #For every two points $x$ and $y$ in $M\_1$, the distance between their images is (up to the additive constant $B$) within a factor of $A$ of their original distance. More formally: #:$\backslash forall\; x,y\backslash in\; M\_1:\; \backslash frac\backslash ;\; d\_1(x,y)-B\backslash leq\; d\_2(f(x),f(y))\backslash leq\; A\backslash ;\; d\_1(x,y)+B.$ #Every point of $M\_2$ is within the constant distance $C$ of an image point. More formally: #:$\backslash forall\; z\backslash in\; M\_2:\backslash exists\; x\backslash in\; M\_1:\; d\_2(z,f(x))\backslash le\; C.$ The two metric spaces $(M\_1,d\_1)$ and $(M\_2,d\_2)$ are called quasi-isometric if there exists a quasi-isometry $f$ from $(M\_1,d\_1)$ to $(M\_2,d\_2)$. 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「Quasi-isometry」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.