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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 Size functor ： ウィキペディア英語版 Size functor Given a size pair $(M,f)\backslash$ where $M\backslash$ is a manifold of dimension $n\backslash$ and $f\backslash$ is an arbitrary real continuous function defined on it, the $i\backslash$-th '' size functor'',〔Francesca Cagliari, Massimo Ferri, Paola Pozzi, ''Size functions from a categorical viewpoint'', Acta Applicandae Mathematicae, 67(3):225-235, 2001.〕 with $i=0,\backslash ldots,n\backslash$, denoted by $F\_i\backslash$, is the functor in $Fun(\backslash mathrm,\backslash mathrm)\backslash$, where $\backslash mathrm\backslash$ is the category of ordered real numbers, and $\backslash mathrm\backslash$ is the category of Abelian groups, defined in the following way. For $x\backslash le\; y\backslash$, setting $M\_x=\backslash \backslash$, $M\_y=\backslash \backslash$, $j\_\backslash$ equal to the inclusion from $M\_x\backslash$ into $M\_y\backslash$, and $k\_\backslash$ equal to the morphism in $\backslash mathrm\backslash$ from $x\backslash$ to $y\backslash$, * for each $x\backslash in\backslash R\backslash$, $F\_i(x)=H\_i(M\_x);\backslash$ * $F\_i(k\_)=H\_i(j\_).\backslash$ In other words, the size functor studies the process of the birth and death of homology classes as the lower level set changes. When $M\backslash$ is smooth and compact and $f\backslash$ is a Morse function, the functor $F\_0\backslash$ can be described by oriented trees, called $H\_0\backslash$ − trees. The concept of size functor was introduced as an extension to homology theory and category theory of the idea of size function. The main motivation for introducing the size functor originated by the observation that the size function $\backslash ell\_(x,\; y)\backslash$ can be seen as the rank of the image of $H\_0(j\_)\; :\; H\_0(M\_x)\; \backslash rightarrow\; H\_0(M\_y)$. The concept of size functor is strictly related to the concept of persistent homology group ,〔Herbert Edelsbrunner, David Letscher, Afra Zomorodian, ''Topological Persistence and Simplification'', Discrete and Computational Geometry, 28(4):511-533, 2002.〕 studied in persistent homology. It is worth to point out that the $i\backslash$-th persistent homology group coincides with the image of the homomorphism $F\_i(k\_)=H\_i(j\_):\; H\_i(M\_x)\; \backslash rightarrow\; H\_i(M\_y)$. ==References== 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「Size functor」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.