翻訳と辞書 Words near each other ・ "O" Is for Outlaw ・ "O"-Jung.Ban.Hap. ・ "Ode-to-Napoleon" hexachord ・ "Oh Yeah!" Live ・ "Our Contemporary" regional art exhibition (Leningrad, 1975) ・ "P" Is for Peril ・ "Pimpernel" Smith ・ "Polish death camp" controversy ・ "Pro knigi" ("About books") ・ "Prosopa" Greek Television Awards ・ "Pussy Cats" Starring the Walkmen ・ "Q" Is for Quarry ・ "R" Is for Ricochet ・ "R" The King (2016 film) ・ "Rags" Ragland ・ ! (album) ・ ! (disambiguation) ・ !! ・ !!! ・ !!! (album) ・ !!Destroy-Oh-Boy!! ・ !Action Pact! ・ !Arriba! La Pachanga ・ !Hero ・ !Hero (album) ・ !Kung language ・ !Oka Tokat ・ !PAUS3 ・ !T.O.O.H.! ・ !Women Art Revolution 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 isomorphism ： ウィキペディア英語版 isomorphism In mathematics, an isomorphism (from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape") is a homomorphism (or more generally a morphism) that admits an inverse.〔For clarity, by ''inverse'' is meant ''inverse homomorphism'' or ''inverse morphism'' respectively, not ''inverse function''.〕 Two mathematical objects are isomorphic if an isomorphism exists between them. An ''automorphism'' is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective. In topology, where the morphisms are continuous functions, isomorphisms are also called ''homeomorphisms'' or ''bicontinuous functions''. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called ''diffeomorphisms''. A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space ''V'' to its second dual space is a canonical isomorphism; on the other hand, ''V'' is isomorphic to its dual space but not canonically in general. Isomorphisms are formalized using category theory. A morphism in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism in that category such that and , where 1''X'' and 1''Y'' are the identity morphisms of ''X'' and ''Y'', respectively. ==Examples== ===Logarithm and exponential=== Let $\backslash mathbb^+$ be the multiplicative group of positive real numbers, and let $\backslash mathbb$ be the additive group of real numbers. The logarithm function $\backslash log\; \backslash colon\; \backslash mathbb^+\; \backslash to\; \backslash mathbb$ satisfies $\backslash log(xy)\; =\; \backslash log\; x\; +\; \backslash log\; y$ for all $x,y\; \backslash in\; \backslash mathbb^+$, so it is a group homomorphism. The exponential function $\backslash exp\; \backslash colon\; \backslash mathbb\; \backslash to\; \backslash mathbb^+$ satisfies $\backslash exp(x+y)\; =\; (\backslash exp\; x)(\backslash exp\; y)$ for all $x,y\; \backslash in\; \backslash mathbb$, so it too is a homomorphism. The identities $\backslash log\; \backslash exp\; x\; =\; x$ and $\backslash exp\; \backslash log\; y\; =\; y$ show that $\backslash log$ and $\backslash exp$ are inverses of each other. Since $\backslash log$ is a homomorphism that has an inverse that is also a homomorphism, $\backslash log$ is an isomorphism of groups. Because $\backslash log$ is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale. 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「isomorphism」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.