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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 Komornik–Loreti constant ： ウィキペディア英語版 Komornik–Loreti constant The Komornik–Loreti constant is a mathematical constant that represents the smallest number for which there still exists a unique ''q''-development. ==Definition== Given a real number ''q'' > 1, the series : $x\; =\; \backslash sum\_^\backslash infty\; a\_n\; q^$ is called the ''q''-expansion, or $\backslash beta$-expansion, of the positive real number ''x'' if, for all $n\; \backslash ge\; 0$, $0\; \backslash le\; a\_n\; \backslash le\; \backslash lfloor\; q\; \backslash rfloor$, where $\backslash lfloor\; q\; \backslash rfloor$ is the floor function and $a\_n$ need not be an integer. Any real number $x$ such that $0\; \backslash le\; x\; \backslash le\; q\; \backslash lfloor\; q\; \backslash rfloor\; /(q-1)$ has such an expansion, as can be found using the greedy algorithm. The special case of $x\; =\; 1$, $a\_0\; =\; 0$, and $a\_n\; =\; 0$ or 1 is sometimes called a $q$-development. $a\_n\; =\; 1$ gives the only 2-development. However, for almost all , there are an infinite number of different $q$-developments. Even more surprisingly though, there exist exceptional $q\; \backslash in\; (1,2)$ for which there exists only a single $q$-development. Furthermore, there is a smallest number known as the Komornik–Loreti constant for which there exists a unique $q$-development.〔Weissman, Eric W. "q-expansion" From (Wolfram MathWorld ). Retrieved on 2009-10-18.〕 The Komornik–Loreti constant is the value $q$ such that : $1\; =\; \backslash sum\_^\backslash infty\; \backslash frac$ where $t\_k$ is the Thue–Morse sequence, i.e., $t\_k$ is the parity of the number of 1's in the binary representation of $k$. It has approximate value : $q=1.787231650\backslash ldots.\; \backslash ,$ The constant $q$ is also the unique positive real root of : $\backslash prod\_^\backslash infty\; \backslash left\; (\; 1\; -\; \backslash frac\; \backslash right\; )^\; -\; 2.$ This constant is transcendental.〔Weissman, Eric W. "Komornik–Loreti Constant." From (Wolfram MathWorld ). Retrieved on 2010-12-27.〕 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「Komornik–Loreti constant」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.