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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 Ordered exponential ： ウィキペディア英語版 Ordered exponential The ordered exponential (also called the path-ordered exponential) is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras. == Definition == Let ''A'' be an algebra over a real or complex field ''K'', and ''a''(''t'') be a parameterized element of ''A'', : $a\; \backslash mathrel\; K\; \backslash to\; A.\; \backslash ,$ The parameter ''t'' in ''a''(''t'') is often referred to as the ''time parameter'' in this context. The ordered exponential of ''a'' is denoted : $\backslash operatorname()(t)\; \backslash equiv\; \backslash mathcal\; \backslash left\backslash \; \backslash equiv\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash int\_0^t\; \backslash cdots\; \backslash int\_0^\; \backslash mathcal\; \backslash left\backslash \; \backslash ,\; dt\text{'}\_1\; \backslash cdots\; dt\text{'}\_n$ where $\backslash mathcal$ is a higher-order operation that ensures the exponential is time-ordered: any product of ''a''(''t'') that occurs in the expansion of the exponential must be ordered such that the value of ''t'' is increasing from right to left of the product; a schematic example: : $\backslash mathcal\; \backslash left\backslash \; =\; a(9.5)\; a(4.1)\; a(1.2).$ This restriction is necessary as products in the algebra are not necessarily commutative. The operation maps a parameterized element onto another parameterized element, or symbolically, : $\backslash operatorname\; \backslash mathrel\; \backslash left(K\; \backslash to\; A\backslash right)\; \backslash to\; \backslash left(K\; \backslash to\; A\backslash right).$ There are various ways to define this integral more rigorously. 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「Ordered exponential」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.