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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 Littlewood–Offord problem ： ウィキペディア英語版 Littlewood–Offord problem In mathematics, the Littlewood–Offord problem is the combinatorial question in geometry of bounding from above the number of subsums made out of vectors :$v\_1,\; v\_2,\; \backslash cdots,\; v\_n\; \backslash in\; \backslash mathbb^d$ that fall into a fixed convex set $A\; \backslash subset\; \backslash mathbb^d$. The first result (for ''d''=1, 2) was given in a paper from 1938 by John Edensor Littlewood and A. Cyril Offord, on random polynomials. This Littlewood–Offord lemma states that, for real or complex numbers ''v''''i'' of absolute value at least one and any disc of radius one, not more than $\backslash Big(\; c\; \backslash ,\; \backslash log\; n\; /\; \backslash sqrt\; \backslash Big)\; \backslash ,\; 2^n$ of the 2''n'' sums of ''v''''i'' fall into the disc. In 1945 Paul Erdős improved the upper bound for ''d''=1 to :$\backslash approx\; 2^n\; \backslash ,\; \backslash frac\{\backslash sqrt\{n\}\}$ using Sperner's theorem. This bound is sharp; equality is attained when all ''v''_i are equal. Then Kleitman in 1966 showed that the same bound held for complex numbers. He extended this (1970) to ''v''''i'' in a normed space. See for the proofs of these results. The semi-sum :''m'' = ½Σ ''v''''i'' can be subtracted from all the subsums. That is, by change of origin and then scaling by a factor of 2, we may as well consider sums :Σ ε''i''''v''''i'' in which ε''i'' takes the value 1 or −1. This makes the problem into a probabilistic one, in which the question is of the distribution of these ''random vectors'', and what can be said knowing nothing more about the ''v''''i''. ==References== 〔 Category:Probability theory Category:Lemmas Category:Mathematical problems 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「Littlewood–Offord problem」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.