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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 Base flow (random dynamical systems) ： ウィキペディア英語版 Base flow (random dynamical systems) In mathematics, the base flow of a random dynamical system is the dynamical system defined on the "noise" probability space that describes how to "fast forward" or "rewind" the noise when one wishes to change the time at which one "starts" the random dynamical system. ==Definition== In the definition of a random dynamical system, one is given a family of maps $\backslash vartheta\_\; :\; \backslash Omega\; \backslash to\; \backslash Omega$ on a probability space $(\backslash Omega,\; \backslash mathcal,\; \backslash mathbb)$. The measure-preserving dynamical system $(\backslash Omega,\; \backslash mathcal,\; \backslash mathbb,\; \backslash vartheta)$ is known as the base flow of the random dynamical system. The maps $\backslash vartheta\_$ are often known as shift maps since they "shift" time. The base flow is often ergodic. The parameter $s$ may be chosen to run over * $\backslash mathbb$ (a two-sided continuous-time dynamical system); * $[0,\; +\; \backslash infty)\; \backslash subsetneq\; \backslash mathbb$ (a one-sided continuous-time dynamical system); * $\backslash mathbb$ (a two-sided discrete-time dynamical system); * $\backslash mathbb\; \backslash cup\; \backslash$ (a one-sided discrete-time dynamical system). Each map $\backslash vartheta\_$ is required * to be a $(\backslash mathcal,\; \backslash mathcal)$-measurable function: for all $E\; \backslash in\; \backslash mathcal$, $\backslash vartheta\_^\; (E)\; \backslash in\; \backslash mathcal$ * to preserve the measure $\backslash mathbb$: for all $E\; \backslash in\; \backslash mathcal$, $\backslash mathbb\; (\backslash vartheta\_^\; (E))\; =\; \backslash mathbb\; (E)$. Furthermore, as a family, the maps $\backslash vartheta\_$ satisfy the relations * $\backslash vartheta\_\; =\; \backslash mathrm\_\; :\; \backslash Omega\; \backslash to\; \backslash Omega$, the identity function on $\backslash Omega$; * $\backslash vartheta\_\; \backslash circ\; \backslash vartheta\_\; =\; \backslash vartheta\_$ for all $s$ and $t$ for which the three maps in this expression are defined. In particular, $\backslash vartheta\_^\; =\; \backslash vartheta\_$ if $-\; s$ exists. In other words, the maps $\backslash vartheta\_$ form a commutative monoid (in the cases $s\; \backslash in\; \backslash mathbb\; \backslash cup\; \backslash$ and $s\; \backslash in\; [0,\; +\; \backslash infty)$) or a commutative group (in the cases $s\; \backslash in\; \backslash mathbb$ and $s\; \backslash in\; \backslash mathbb$). 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「Base flow (random dynamical systems)」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.