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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 Pythagorean means ： ウィキペディア英語版 Pythagorean means In mathematics, the three classical Pythagorean means are the arithmetic mean (''A''), the geometric mean (''G''), and the harmonic mean (''H''). They are defined by: * $AM(x\_1,\; \backslash ldots,\; x\_n)\; =\; \backslash frac(x\_1\; +\; \backslash cdots\; +\; x\_n)$ * $GM(x\_1,\; \backslash ldots,\; x\_n)\; =\; \backslash sqrt()$ * $HM(x\_1,\; \backslash ldots,\; x\_n)\; =\; \backslash frac\; +\; \backslash cdots\; +\; \backslash frac\}$ Each mean has the following properties: * Value preservation: $M(x,x,\; \backslash ldots,x)\; =\; x$ * First order homogeneity: $M(bx\_1,\; \backslash ldots,\; bx\_n)\; =\; b\; M(x\_1,\; \backslash ldots,\; x\_n)$ * Invariance under exchange: $M(\backslash ldots,\; x\_i,\; \backslash ldots,\; x\_j,\; \backslash ldots\; )\; =\; M(\backslash ldots,\; x\_j,\; \backslash ldots,\; x\_i,\; \backslash ldots)$ for any $i$ and $j$. * Averaging: $\backslash min(x\_1,\backslash ldots,x\_n)\; \backslash leq\; M(x\_1,\backslash ldots,x\_n)\; \backslash leq\; \backslash max(x\_1,\backslash ldots,x\_n)$ These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. The harmonic and arithmetic means are reciprocal duals of each other for positive arguments ($HM(1/x\_1\; \backslash ldots\; 1/x\_n\; )\; =\; 1/AM(x\_1\; \backslash ldots\; x\_n)$) while the geometric mean is its own reciprocal dual. == Inequalities among means == There is an ordering to these means (if all of the $x\_i$ are positive) : $\backslash min\; \backslash leq\; HM\; \backslash leq\; GM\; \backslash leq\; AM\; \backslash leq\; \backslash max$ with equality holding if and only if the $x\_i$ are all equal. This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the arithmetic-geometric mean inequality, $AM\; \backslash leq\; \backslash max$, and reciprocal duality ($\backslash min$ and $\backslash max$ are also reciprocal dual to each other). The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments, so both concave and convex. 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「Pythagorean means」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.