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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 Beatty's theorem ： ウィキペディア英語版 Beatty sequence In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926. Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number. Beatty sequences can also be used to generate Sturmian words. ==Definition== A positive irrational number $r\backslash ,$ generates the Beatty sequence :$\backslash mathcal\_r\; =\; \backslash lfloor\; r\; \backslash rfloor,\; \backslash lfloor\; 2r\; \backslash rfloor,\; \backslash lfloor\; 3r\; \backslash rfloor,\backslash ldots\; =\; (\; \backslash lfloor\; nr\; \backslash rfloor)\_$ If $r\; >\; 1\; \backslash ,,$ then $s\; =\; r/(r-1)\backslash ,$ is also a positive irrational number. They naturally satisfy :$\backslash frac1r\; +\; \backslash frac1s\; =\; 1\; \backslash ,$ and the sequences :$\backslash mathcal\_r\; =\; (\; \backslash lfloor\; nr\; \backslash rfloor)\_$ and :$\backslash mathcal\_s\; =\; (\; \backslash lfloor\; ns\; \backslash rfloor)\_$ form a ''pair of complementary Beatty sequences''. A more general non-homogeneous Beatty sequence takes the form :$\backslash mathcal\_r\; =\; \backslash lfloor\; r+p\; \backslash rfloor,\; \backslash lfloor\; 2r+p\; \backslash rfloor,\; \backslash lfloor\; 3r+p\; \backslash rfloor,\backslash ldots\; =\; (\; \backslash lfloor\; nr+p\; \backslash rfloor)\_$ where $p\backslash ,$ is a real number. For $p=1\backslash ,$, the complementary non-homogeneous Beatty sequences can be found by making $t\; =\; 1/r\backslash ,$ so that :$\backslash mathcal\_r\; =\; (\; \backslash lfloor\; n(r+1)\; \backslash rfloor)\_$ and :$\backslash mathcal\_t\; =\; (\; \backslash lfloor\; n(t+1)\; \backslash rfloor)\_$ form a pair of complementary Beatty sequences. 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「Beatty sequence」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.