翻訳と辞書 Words near each other ・ Chebyshev center ・ Chebyshev distance ・ Chebyshev equation ・ Chebyshev filter ・ Chebyshev function ・ Chebyshev integral ・ Chebyshev iteration ・ Chebyshev linkage ・ Chebyshev nodes ・ Chebyshev polynomials ・ Chebyshev pseudospectral method ・ Chebyshev rational functions ・ Chebyshev's bias ・ Chebyshev's inequality ・ Chebyshev's Lambda Mechanism ・ Chebyshev's sum inequality ・ Chebyshev's theorem ・ Chebyshev–Gauss quadrature ・ Chebyshev–Markov–Stieltjes inequalities ・ Chebzie ・ Cheb–Hranice v Čechách railway ・ Cheb–Oberkotzau railway ・ CHEC ・ CHEC-FM ・ Checa ・ Checa (disambiguation) ・ Checa, Ecuador ・ Checacupe District ・ Checca ・ Checca District Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Chebyshev's sum inequality ： ウィキペディア英語版 Chebyshev's sum inequality In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if :$a\_1\; \backslash geq\; a\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; a\_n$ and :$b\_1\; \backslash geq\; b\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; b\_n,$ then :$\backslash sum\_^n\; a\_k\; \backslash cdot\; b\_k\; \backslash geq\; \backslash left(\backslash sum\_^n\; a\_k\backslash right)\backslash left(\backslash sum\_^n\; b\_k\backslash right).$ Similarly, if :$a\_1\; \backslash leq\; a\_2\; \backslash leq\; \backslash cdots\; \backslash leq\; a\_n$ and :$b\_1\; \backslash geq\; b\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; b\_n,$ then :$\backslash sum\_^n\; a\_kb\_k\; \backslash leq\; \backslash left(\backslash sum\_^n\; a\_k\backslash right)\backslash left(\backslash sum\_^n\; b\_k\backslash right).$ ==Proof== Consider the sum :$S\; =\; \backslash sum\_^n\; \backslash sum\_^n\; (a\_j\; -\; a\_k)\; (b\_j\; -\; b\_k).$ The two sequences are non-increasing, therefore and have the same sign for any . Hence . Opening the brackets, we deduce: :$0\; \backslash leq\; 2\; n\; \backslash sum\_^n\; a\_j\; b\_j\; -\; 2\; \backslash sum\_^n\; a\_j\; \backslash ,\; \backslash sum\_^n\; b\_k,$ whence :$\backslash frac\; \backslash sum\_^n\; a\_j\; b\_j\; \backslash geq\; \backslash left(\; \backslash frac\; \backslash sum\_^n\; a\_j\backslash right)\; \backslash ,\; \backslash left(\backslash frac\; \backslash sum\_^n\; b\_k\backslash right).$ An alternative proof is simply obtained with the rearrangement inequality. 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「Chebyshev's sum 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.