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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 Multiplicative sequence ： ウィキペディア英語版 Multiplicative sequence In mathematics, a multiplicative sequence or ''m''-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology. ==Definition== Let ''K''''n'' be polynomials over a ring ''A'' in indeterminates ''p''1,... weighted so that ''p''''i'' has weight ''i'' (with ''p''0 = 1) and all the terms in ''K''''n'' have weight ''n'' (so that ''K''''n'' is a polynomial in ''p''1, ..., ''p''''n''). The sequence ''K''''n'' is ''multiplicative'' if an identity :$\backslash sum\_i\; p\_i\; z^i\; =\; \backslash sum\_i\; p\text{'}\_i\; z^i\; \backslash cdot\; \backslash sum\_i\; p\text{'}\text{'}\_i\; z^i$ implies :$\backslash sum\_i\; K\_i(p\_1,\backslash ldots,p\_i)\; z^i\; =\; \backslash sum\_j\; K\_j(p\text{'}\_1,\backslash ldots,p\text{'}\_j)\; z^j\; \backslash cdot\; \backslash sum\_k\; K\_k(p\text{'}\text{'}\_1,\backslash ldots,p\text{'}\text{'}\_k)\; z^k$ The power series :$\backslash sum\; K\_n(1,0,\backslash ldots,0)\; z^n$ is the ''characteristic power series'' of the ''K''''n''. A multiplicative sequence is determined by its characteristic power series ''Q''(''z''), and every power series with constant term 1 gives rise to a multiplicative sequence. To recover a multiplicative sequence from a characteristic power series ''Q''(''z'') we consider the coefficient of ''z''''j'' in the product :$\backslash prod\_^m\; Q(\backslash beta\_i\; z)\; \backslash$ for any ''m'' > ''j''. This is symmetric in the ''β''''i'' and homogeneous of weight ''j'': so can be expressed as a polynomial ''K''''j''(''p''1, ..., ''p''''j'') in the elementary symmetric functions ''p'' of the ''β''. Then ''K''''j'' defines a multiplicative sequence. 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「Multiplicative 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.