翻訳と辞書 Words near each other ・ Aritao ・ Aritar, Sikkim ・ Aritatsu Ogi ・ Arith ・ ARITH Symposium on Computer Arithmetic ・ ARITH-MATIC ・ Aritha Van Herk ・ Arithmancy ・ Arithmaurel ・ Arithmetic ・ Arithmetic (song) ・ Arithmetic and geometric Frobenius ・ Arithmetic circuit complexity ・ Arithmetic coding ・ Arithmetic combinatorics ・ Arithmetic derivative ・ Arithmetic dynamics ・ Arithmetic for Parents ・ Arithmetic function ・ Arithmetic genus ・ Arithmetic group ・ Arithmetic hyperbolic 3-manifold ・ Arithmetic IF ・ Arithmetic logic unit ・ Arithmetic mean ・ Arithmetic number ・ Arithmetic of abelian varieties ・ Arithmetic overflow ・ Arithmetic progression ・ Arithmetic rope Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Arithmetic derivative ： ウィキペディア英語版 Arithmetic derivative In number theory, the Lagarias arithmetic derivative, or number derivative, is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. Remark: There are many versions of "Arithmetic Derivatives", there are the ones as in this article (Lagarias Arithmetic Derivative), Ihara's Arithmetic Derivative, and Buium's Arithmetic Derivatives. ==Definition== For natural numbers the arithmetic derivative is defined as follows: * $p\text{'}\; \backslash ;=\backslash ;\; 1\; \backslash !$ for any prime $p\; \backslash !$. * $(ab)\text{'}\backslash ;=\backslash ;a\text{'}b\backslash ,+\backslash ,ab\text{'}\; \backslash !$ for any $a\; \backslash textrm\backslash ,\; b\; \backslash ;\backslash in\backslash ;\; \backslash mathbb$ (Leibniz rule). To coincide with the Leibniz rule $1\text{'}$ is defined to be $0$, as is $0\text{'}$. Explicitly, assume that :$x\; =\; p\_1^\backslash cdots\; p\_k^\backslash textrm$ where $p\_1,\backslash ,\; \backslash dots,\backslash ,\; p\_k$ are distinct primes and $e\_1,\backslash ,\; \backslash dots,\backslash ,\; e\_k$ are positive integers. Then :$x\text{'}\; =\; \backslash sum\_^k\; e\_ip\_1^\backslash cdots\; p\_^p\_^\; =\; \backslash sum\_^k\; e\_i\backslash frac.$ The arithmetic derivative also preserves the power rule (for primes): :$(p^a)\text{'}\; =\; ap^\backslash textrm\backslash !$ where $p$ is prime and $a$ is a positive integer. For example, : $$ \begin 81' = (3^4)' & = (9\cdot 9)' = 9'\cdot 9 + 9\cdot 9' = 2(3)' ) \\ & = 2(3 + 3\cdot 3') ) = 2(6 ) = 108 = 4\cdot 3^3. \end The sequence of number derivatives for ''k'' = 0, 1, 2, ... begins : :0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, .... E. J. Barbeau was most likely the first person to formalize this definition. He also extended it to all integers by proving that $(-x)\text{'}\; \backslash ;=\backslash ;\; -(x\text{'})$ uniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers,showing that the familiar quotient rule gives a well-defined derivative on Q: :$\backslash left(\backslash frac\backslash right)\text{'}\; =\; \backslash frac\; \backslash \; .$ Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents $e\_i$ are allowed to be arbitrary rational numbers. The ''logarithmic derivative'' $\backslash frac$ is a totally additive function. 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「Arithmetic derivative」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.