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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 1-parameter group ： ウィキペディア英語版 One-parameter group In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :$\backslash varphi\backslash colon\; \backslash mathbb\; \backslash rightarrow\; G$ from the real line $\backslash mathbb$ (as an additive group) to some other topological group $G$. That means that it is not in fact a group,〔("One-parameter group not a group? Why?" ), ''Stack Exchange'' Retrieved on 9 January 2015.〕 strictly speaking; if $\backslash varphi$ is injective then $\backslash varphi(\backslash mathbb)$, the image, will be a subgroup of $G$ that is isomorphic to $\backslash mathbb$ as additive group. One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations. According to Lie, an ''infinitesimal transformation'' is an infinitely small transformation of the one-parameter group that it generates.〔Sophus Lie (1893) (Vorlesungen über Continuierliche Gruppen ), English translation by D.H. Delphenich, §8, link from Neo-classical Physics〕 It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension. ==Discussion== That is, we start knowing only that :$\backslash varphi(s+t)\; =\; \backslash varphi(s)\backslash varphi(t)$ where $s$, $t$ are the 'parameters' of group elements in $G$. We may have :$\backslash varphi(s)\; =\; e$, the identity element in $G$, for some $s\; \backslash neq\; 0$. This happens for example if $G$ is the unit circle and :$\backslash varphi(s)\; =\; e^\_$; this may happen in cases where $\backslash varphi$ is injective. Think for example of the case where $G$ is a torus $T$, and $\backslash varphi$ is constructed by winding a straight line round $T$ at an irrational slope. Therefore a one-parameter group or one-parameter subgroup has to be distinguished from a group or subgroup itself, for the three reasons #it has a definite parametrization, #the group homomorphism may not be injective, and #the induced topology may not be the standard one of the real line. 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「One-parameter group」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.