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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 Curvature of a measure ： ウィキペディア英語版 Curvature of a measure In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov; accordingly, it may be referred to as the Melnikov curvature or Menger-Melnikov curvature. Melnikov and Verdera (1995) established a powerful connection between the curvature of measures and the Cauchy kernel. ==Definition== Let ''μ'' be a Borel measure on the Euclidean plane R2. Given three (distinct) points ''x'', ''y'' and ''z'' in R2, let ''R''(''x'', ''y'', ''z'') be the radius of the Euclidean circle that joins all three of them, or +∞ if they are collinear. The Menger curvature ''c''(''x'', ''y'', ''z'') is defined to be :$c(x,\; y,\; z)\; =\; \backslash frac,$ with the natural convention that ''c''(''x'', ''y'', ''z'') = 0 if ''x'', ''y'' and ''z'' are collinear. It is also conventional to extend this definition by setting ''c''(''x'', ''y'', ''z'') = 0 if any of the points ''x'', ''y'' and ''z'' coincide. The Menger-Melnikov curvature ''c''2(''μ'') of ''μ'' is defined to be :$c^\; (\backslash mu)\; =\; \backslash iiint\_\}\; c(x,\; y,\; z)^\; \backslash ,\; \backslash mathrm\; \backslash mu\; (x)\; \backslash mathrm\; \backslash mu\; (y)\; \backslash mathrm\; \backslash mu\; (z).$ More generally, for ''α'' ≥ 0, define ''c''2''α''(''μ'') by :$c^\; (\backslash mu)\; =\; \backslash iiint\_\}\; c(x,\; y,\; z)^\; \backslash ,\; \backslash mathrm\; \backslash mu\; (x)\; \backslash mathrm\; \backslash mu\; (y)\; \backslash mathrm\; \backslash mu\; (z).$ One may also refer to the curvature of ''μ'' at a given point ''x'': :$c^\; (\backslash mu;\; x)\; =\; \backslash iint\_\}\; c(x,\; y,\; z)^\; \backslash ,\; \backslash mathrm\; \backslash mu\; (y)\; \backslash mathrm\; \backslash mu\; (z),$ in which case :$c^\; (\backslash mu)\; =\; \backslash int\_\}\; c^\; (\backslash mu;\; x)\; \backslash ,\; \backslash mathrm\; \backslash mu\; (x).$ 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「Curvature of a measure」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.