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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 Shapiro inequality ： ウィキペディア英語版 Shapiro inequality In mathematics, the Shapiro inequality is an inequality proposed by H. Shapiro in 1954. ==Statement of the inequality== Suppose ''n'' is a natural number and $x\_1,\; x\_2,\; \backslash dots,\; x\_n$ are positive numbers and: * ''n'' is even and less than or equal to 12, or * ''n'' is odd and less than or equal to 23. Then the Shapiro inequality states that :$\backslash sum\_^\; \backslash frac\}\; \backslash geq\; \backslash frac$ where $x\_=x\_1,\; x\_=x\_2$. For greater values of ''n'' the inequality does not hold and the strict lower bound is $\backslash gamma\; \backslash frac$ with $\backslash gamma\; \backslash approx\; 0.9891\backslash dots$. The initial proofs of the inequality in the pivotal cases ''n'' = 12 (Godunova and Levin, 1976) and ''n'' = 23 (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for ''n'' = 12. The value of γ was determined in 1971 by Vladimir Drinfeld, who won a Fields Medal in 1990. Specifically, Drinfeld showed that the strict lower bound ''γ'' is given by $\backslash frac\; \backslash psi(0)$, where ''ψ'' is the function convex hull of ''f''(''x'') = ''e''−''x'' and $g(x)\; =\; \backslash frac\}\}$. (That is, the region above the graph of ''ψ'' is the convex hull of the union of the regions above the graphs of ''f'' and ''g''.) Interior local mimima of the left-hand side are always ≥ ''n''/2 (Nowosad, 1968). 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「Shapiro inequality」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.