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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 Iverson bracket ： ウィキペディア英語版 Iverson bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly, : where is a statement that can be true or false. This notation was introduced by Kenneth E. Iverson in his programming language APL,〔Kenneth E. Iverson, ''A Programming Language'', New York: Wiley, p. 11, 1962.〕〔Ronald Graham, Donald Knuth, and Oren Patashnik. ''Concrete Mathematics'', Section 2.2: Sums and Recurrences.〕 while the specific restriction to square brackets was advocated by Donald Knuth to avoid ambiguity in parenthesized logical expressions.〔Donald Knuth, "Two Notes on Notation", ''American Mathematical Monthly'', Volume 99, Number 5, May 1992, pp. 403–422. ((TeX ), ).〕 ==Uses== The Iverson bracket converts a Boolean value to an integer value through the natural map $\backslash textbf\backslash mapsto\; 0;\; \backslash textbf\backslash mapsto1$, which allows counting to be represented as summation. For instance, the Euler phi function that counts the number of positive integers up to ''n'' which are coprime to ''n'' can be expressed by : $\backslash phi(n)=\backslash sum\_^(),\backslash qquad\backslash textn\backslash in\backslash mathbb\; N^+.$ More generally the notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. For example, : $\backslash sum\_\; i^2\; =\; \backslash sum\_\; i^2(\backslash le\; i\; \backslash le\; 10).$ In the first sum, the index $i$ is limited to be in the range 1 to 10. The second sum is allowed to range over all integers, but where ''i'' is strictly less than 1 or strictly greater than 10, the summand is 0, contributing nothing to the sum. Such use of the Iverson bracket can permit easier manipulation of these expressions. Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula :$\backslash sum\_\backslash !\backslash !k\; =\; \backslash fracn\backslash varphi(n)$ is valid for but is off by for . To get an identity valid for all positive integers ''n'' (i.e., all values for which $\backslash phi(n)$ is defined), a correction term involving the Iverson bracket may be added: :$\backslash sum\_\backslash !\backslash !k\; =\; \backslash fracn(\backslash varphi(n)+())$ 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「Iverson bracket」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.