翻訳と辞書 Words near each other ・ Noviforum ・ Noviglio ・ Novigrad ・ Novigrad Castle ・ Novigrad na Dobri ・ Novigrad Podravski ・ Novigrad, Istria County ・ Novigrad, Zadar County ・ Novik ・ Novik, Iran ・ Novik-class frigate ・ Novika ・ Noviken VLF Transmitter ・ Novikov ・ Novikov conjecture ・ Novikov ring ・ Novikov self-consistency principle ・ Novikov's compact leaf theorem ・ Novikov's condition ・ Novikovo ・ Novikov–Shubin invariant ・ Novikov–Veselov equation ・ Novillard ・ Novillars ・ Novillas ・ Noville ・ Noville Peninsula ・ Noville, Switzerland ・ Novillero ・ Novillers 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 Novikov ring ： ウィキペディア英語版 Novikov ring :''For a concept in quantum cohomology, see the linked article.'' In mathematics, given an additive subgroup $\backslash Gamma\; \backslash subset\; \backslash mathbb$, the Novikov ring $\backslash operatorname(\backslash Gamma)$ of $\backslash Gamma$ is the subring of $\backslash mathbb$〔Here, $\backslash mathbb$ is the ring consisting of the formal sums $\backslash sum\_\; n\_\backslash gamma\; t^\backslash gamma$, $n\_\backslash gamma$ integers and ''t'' a formal variable, such that the multiplication is an extension of a multiplication in the integral group ring $\backslash mathbb()$.〕 consisting of formal sums $\backslash sum\; n\_\; t^$ such that $\backslash gamma\_1\; >\; \backslash gamma\_2\; >\; \backslash cdots$ and $\backslash gamma\_i\; \backslash to\; -\backslash infty$. The notion was introduced by S. P. Novikov in the papers that initiated the generalization of Morse theory using a closed one-form instead of a function. The Novikov ring $\backslash operatorname(\backslash Gamma)$ is a principal ideal domain. Let ''S'' be the subset of $\backslash mathbb()$ consisting of those with leading term 1. Since the elements of ''S'' are unit elements of $\backslash operatorname(\backslash Gamma)$, the localization $\backslash operatorname(\backslash Gamma)()$ of $\backslash operatorname(\backslash Gamma)$ with respect to ''S'' is a subring of $\backslash operatorname(\backslash Gamma)$ called the "rational part" of $\backslash operatorname(\backslash Gamma)$; it is also a principal ideal domain. == Novikov numbers == Given a smooth function ''f'' on a smooth manifold ''M'' with nondegenerate critical points, the usual Morse theory constructs a free chain complex $C\_$ *(f) such that the (integral) rank of $C\_p$ is the number of critical points of ''f'' of index ''p'' (called the Morse number). It computes the homology of ''M'': $H^$ *(C_ *(f)) \approx H^ *(M, \mathbf) (cf. Morse homology.) In an analogy with this, one can define "Novikov numbers". Let ''X'' be a connected polyhedron with a base point. Each cohomology class $\backslash xi\; \backslash in\; H^1(X,\; \backslash mathbb)$ may be viewed as a linear functional on the first homology group $H\_1(X,\; \backslash mathbb)$ and, composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism $\backslash xi:\; \backslash pi=\backslash pi\_1(X)\; \backslash to\; \backslash mathbb$. By the universal property, this map in turns gives a ring homomorphism $\backslash phi\_\backslash xi:\; \backslash mathbb()\; \backslash to\; \backslash operatorname\; =\; \backslash operatorname(\backslash mathbb)$, making $\backslash operatorname$ a module over $\backslash mathbb()$. Since ''X'' is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a $\backslash mathbb()$-module. Let $L\_\backslash xi$ be a local coefficient system corresponding to $\backslash operatorname$ with module structure given by $\backslash phi\_\backslash xi$. The homology group $H\_p(X,\; L\_\backslash xi)$ is a finitely generated module over $\backslash operatorname,$ which is, by the structure theorem, a direct sum of the free part and the torsion part. The rank of the free part is called the Novikov Betti number and is denoted by $b\_p(\backslash xi)$. The number of cyclic modules in the torsion part is denoted by $q\_p(\backslash xi)$. If $\backslash xi\; =\; 0$, $L\_\backslash xi$ is trivial and $b\_p(0)$ is the usual Betti number of ''X''. The analog of Morse inequalities holds for Novikov numbers as well (cf. the reference for now.) 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「Novikov ring」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.