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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 Natural pseudodistance ： ウィキペディア英語版 Natural pseudodistance In size theory, the natural pseudodistance between two size pairs $(M,\backslash varphi:M\backslash to\; \backslash mathbb)\backslash$, $(N,\backslash psi:N\backslash to\; \backslash mathbb)\backslash$ is the value $\backslash inf\_h\; \backslash |\backslash varphi-\backslash psi\backslash circ\; h\backslash |\_\backslash infty\backslash$, where $h\backslash$ varies in the set of all homeomorphisms from the manifold $M\backslash$ to the manifold $N\backslash$ and $\backslash |\backslash cdot\backslash |\_\backslash infty\backslash$ is the supremum norm. If $M\backslash$ and $N\backslash$ are not homeomorphic, then the natural pseudodistance is defined to be $\backslash infty\backslash$. It is usually assumed that $M\backslash$, $N\backslash$ are $C^1\backslash$ closed manifolds and the measuring functions $\backslash varphi,\backslash psi\backslash$ are $C^1\backslash$. Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from $M\backslash$ to $N\backslash$. The concept of natural pseudodistance can be easily extended to size pairs where the measuring function $\backslash varphi\backslash$ takes values in $\backslash mathbb^m\backslash$ .〔Patrizio Frosini, Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society, 6:455-464, 1999.〕 ==Main properties== It can be proved 〔Pietro Donatini, Patrizio Frosini, ''Natural pseudodistances between closed manifolds'', Forum Mathematicum, 16(5):695-715, 2004.〕 that the natural pseudodistance always equals the Euclidean distance between two critical values of the measuring functions (possibly, of the ''same'' measuring function) divided by a suitable positive integer $k\backslash$. If $M\backslash$ and $N\backslash$ are surfaces, the number $k\backslash$ can be assumed to be $1\backslash$, $2\backslash$ or $3\backslash$.〔Pietro Donatini, Patrizio Frosini, ''Natural pseudodistances between closed surfaces'', Journal of the European Mathematical Society, 9(2):231–253, 2007.〕 If $M\backslash$ and $N\backslash$ are curves, the number $k\backslash$ can be assumed to be $1\backslash$ or $2\backslash$.〔Pietro Donatini, Patrizio Frosini, ''Natural pseudodistances between closed curves'', Forum Mathematicum, 21(6):981–999, 2009.〕 If an optimal homeomorphism $\backslash bar\; h\backslash$ exists (i.e., $\backslash |\backslash varphi-\backslash psi\backslash circ\; \backslash bar\; h\backslash |\_\backslash infty=\backslash inf\_h\; \backslash |\backslash varphi-\backslash psi\backslash circ\; h\backslash |\_\backslash infty\backslash$), then $k\backslash$ can be assumed to be $1\backslash$.〔 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「Natural pseudodistance」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.