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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 Complete set of invariants ： ウィキペディア英語版 Complete set of invariants In mathematics, a complete set of invariants for a classification problem is a collection of maps :$f\_i\; :\; X\; \backslash to\; Y\_i\; \backslash ,$ (where ''X'' is the collection of objects being classified, up to some equivalence relation, and the $Y\_i$ are some sets), such that $x\; \backslash sim\; x\text{'}$ if and only if $f\_i(x)\; =\; f\_i(x\text{'})$ for all ''i''. In words, such that two objects are equivalent if and only if all invariants are equal.〔. See in particular (p. 97 ).〕 Symbolically, a complete set of invariants is a collection of maps such that :$\backslash prod\; f\_i\; :\; (X/\backslash sim)\; \backslash to\; \backslash prod\; Y\_i$ is injective. As invariants are, by definition, equal on equivalent objects, equality of invariants is a ''necessary'' condition for equivalence; a ''complete'' set of invariants is a set such that equality of these is ''sufficient'' for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants). ==Examples== * In the classification of two-dimensional closed manifolds, Euler characteristic (or genus) and orientability are a complete set of invariants. * Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not. 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「Complete set of invariants」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.