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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 Bregman-Minc inequality ： ウィキペディア英語版 Bregman-Minc inequality The Bregman-Minc inequality or Bregman's theorem is a theorem in discrete mathematics, which allows to estimate the permanent of a binary matrix via its row or column sums. The inequality was conjectured in 1963 by Henryk Minc and first proved in 1973 by Lev M. Bregman.〔〔 Further entropy-based proofs have been given by Alexander Schrijver and Jaikumar Radhakrishnan.〔〔 The Bregman-Minc inequality is used, for example, in graph theory to obtain upper bounds for the number of perfect matchings in a bipartite graph. == Statement == The permanent of a square binary matrix $A\; =\; (a\_)$ of size $n$ with row sums $r\_i\; =\; a\_\; +\; \backslash ldots\; +\; a\_$ for $i=1,\; \backslash ldots\; ,\; n$ can be estimated by :$\backslash operatornameA\; \backslash leq\; \backslash prod\_^n\; (r\_i!)^.$ The permanent is therefore bounded by the product of the geometric means of the numbers from $1$ to $r\_i$ for $i=1,\; \backslash ldots\; ,\; n$. Equality holds if the matrix is a block diagonal matrix consisting of matrices of ones or results from row and/or column permutations of such a block diagonal matrix. Since the permanent is invariant under transposition, the inequality also holds for the column sums of the matrix accordingly.〔〔 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「Bregman-Minc 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.