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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 Bhargava cube ： ウィキペディア英語版 Bhargava cube In mathematics, in number theory, Bhargava cube (also called Bhargava's cube) is a configuration consisting of eight integers placed at the eight corners of a cube. This configuration was extensively used by Manjul Bhargava, an Indian-American Fields Medal winning mathematician, to study the composition laws of binary quadratic forms and other such forms. To each pair of opposite faces of a Bhargava cube one can associate an integer binary quadratic form thus getting three binary quadratic forms corresponding to the three pairs of opposite faces of the Bhargava cube. These three quadratic forms all have the same discriminant and Manjul Bhargava proved that their composition in the sense of Gauss is the identity element in the associated group of equivalence classes of primitive binary quadratic forms. Using this property as the starting point for a theory of composition of binary quadratic forms Manjul Bhargava went on to define fourteen different composition laws using a cube. ==Integer binary quadratic forms== An expression of the form $Q(x,y)=ax^2+bxy+cy^2$, where ''a'', ''b'' and ''c'' are fixed integers and ''x'' and ''y'' are variable integers, is called an integer binary quadratic form. The discriminant of the form is defined as :$D\; =\; b^2\; -4ac.$ The form is said to be primitive if the coefficients ''a'', ''b'', ''c'' are relatively prime. Two forms :$Q(x,y)\; =\; ax^2+bxy+cy^2,\; \backslash quad\; Q^\backslash prime(x,y)=a^\backslash prime\; x^2+b^\backslash prime\; xy\; +\; c^\backslash prime\; y^2$ are said to be equivalent if there exists a transformation :$x\backslash mapsto\; \backslash alpha\; x\; +\; \backslash beta\; y,\backslash quad\; y\backslash mapsto\; \backslash gamma\; x\; +\; \backslash delta\; y$ with integer coefficients satisfying $\backslash alpha\backslash delta\; -\; \backslash beta\backslash gamma\; =1$ which transforms $Q(x,y)$ to $Q^\backslash prime(x,y)$. This relation is indeed an equivalence relation in the set of integer binary quadratic forms and it preserves discriminants and primitivity. 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「Bhargava cube」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.