翻訳と辞書 Words near each other ・ Simplexvirus ・ Simpli ・ Simplic ・ Simplicala ・ Simplice Guedet Manzela ・ Simplicia ・ Simplicia (moth) ・ Simplicia (plant) ・ Simplicia armatalis ・ Simplicia cornicalis ・ Simplicia erebina ・ Simplicia extinctalis ・ Simplicia inareolalis ・ Simplicia inflexalis ・ Simplicia mistacalis ・ Simplicial approximation theorem ・ Simplicial category ・ Simplicial commutative ring ・ Simplicial complex ・ Simplicial group ・ Simplicial homology ・ Simplicial homotopy ・ Simplicial Lie algebra ・ Simplicial localization ・ Simplicial manifold ・ Simplicial map ・ Simplicial polytope ・ Simplicial presheaf ・ Simplicial set ・ Simplicial space Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Simplicial approximation theorem ： ウィキペディア英語版 Simplicial approximation theorem In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices — that is, finite simplicial complexes. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (''affine''-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of the simplices of the domain, and (ii) replacement of the actual mapping by a homotopic one. This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness). It served to put the homology theory of the time — the first decade of the twentieth century — on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mappings could in a given case be expressed in a finitary way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological, in some other areas. This initiated, one could say, the era of combinatorial topology. There is a further simplicial approximation theorem for homotopies, stating that a homotopy between continuous mappings can likewise be approximated by a combinatorial version. ==Formal statement of the theorem== Let $K$ and $L$ be two simplicial complexes. A simplicial mapping $f\; :\; K\; \backslash to\; L$ is called a simplicial approximation of a continuous function $F\; :\; |K|\; \backslash to\; |L|$ if for every point $x\; \backslash in\; |K|$, $|f|(x)$ belongs to the minimal closed simplex of $L$ containing the point $F(x)$. If $f$ is a simplicial approximation to a continuous map $F$, then the geometric realization of $f$, $|f|$ is necessarily homotopic to $F$. The simplicial approximation theorem states that given any continuous map $F\; :\; |K|\; \backslash to\; |L|$ there exists a natural number $n\_0$ such that for all $n\; \backslash ge\; n\_0$ there exists a simplicial approximation $f\; :\; \backslash mathrm^n\; K\; \backslash to\; L$ to $F$ (where $\backslash mathrm\backslash ;\; K$ denotes the barycentric subdivision of $K$, and $\backslash mathrm^n\; K$ denotes the result of applying barycentric subdivision $n$ times.) 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「Simplicial approximation theorem」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.