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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 Dyson's transform ： ウィキペディア英語版 Dyson's transform Dyson's transform is a fundamental technique in additive number theory.〔Additive Number Theory: Inverse Problems and the Geometry of Sumsets By Melvyn Bernard Nathanson, Springer, Aug 22, 1996, ISBN 0-387-94655-1, http://books.google.com/books?id=PqlQjNhjkKUC&dq=%22e-transform%22&source=gbs_navlinks_s, p. 42〕 It was developed by Freeman Dyson as part of his proof of Mann's theorem〔 〕, is used to prove such fundamental results of Additive Number Theory as the Cauchy-Davenport theorem,〔 and was used by Olivier Ramaré in his work on the Goldbach conjecture that proved that every even integer is the sum of at most 6 primes.〔 〕 The term ''Dyson's transform'' for this technique is used by Ramaré.〔 Halberstam and Roth call it the τ-transformation.〔 This formulation of the transform is from Ramaré.〔 Let ''A'' be a sequence of natural numbers, and ''x'' be any real number. Write ''A''(''x'') for the number of elements of ''A'' which lie in (). Suppose $A=\; \backslash$ and $B=\; \backslash$ are two sequences of natural numbers. We write ''A'' + ''B'' for the sumset, that is, the set of all elements ''a'' + ''b'' where ''a'' is in ''A'' and ''b'' is in B; and similarly ''A'' − ''B'' for the set of differences ''a'' − ''b''. For any element ''e'' in ''A'', Dyson's transform consists in forming the sequences $A\text{'}=\; A\; \backslash cup\; \backslash$ and $\backslash ,B\text{'}=\; B\; \backslash cap\; \backslash$. The transformed sequences have the properties: * $A\text{'}\; +\; B\text{'}\; \backslash subset\; A\; +\; B$ * $\backslash \; +\; B\text{'}\; \backslash subset\; A\text{'}$ * $0\; \backslash in\; B\text{'}$ * $A\text{'}(m)+\; B\text{'}(m-e)\; =\; A(m)\; +\; B(m-e)\; \backslash$ ==References== 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「Dyson's transform」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.