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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 Mahler's compactness theorem ： ウィキペディア英語版 Mahler's compactness theorem In mathematics, Mahler's compactness theorem, proved by , is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could degenerate (''go off to infinity'') in a sequence of lattices. In intuitive terms it says that this is possible in just two ways: becoming ''coarse-grained'' with a fundamental domain that has ever larger volume; or containing shorter and shorter vectors. It is also called his selection theorem, following an older convention used in naming compactness theorems, because they were formulated in terms of sequential compactness (the possibility of selecting a convergent subsequence). Let ''X'' be the space :$\backslash mathrm\_n(\backslash mathbb)/\backslash mathrm\_n(\backslash mathbb)$ that parametrises lattices in $\backslash mathbb^n$, with its quotient topology. There is a well-defined function Δ on ''X'', which is the absolute value of the determinant of a matrix – this is constant on the cosets, since an invertible integer matrix has determinant 1 or −1. Mahler's compactness theorem states that a subset ''Y'' of ''X'' is relatively compact if and only if Δ is bounded on ''Y'', and there is a neighbourhood ''N'' of in $\backslash mathbb^n$ such that for all ''Λ'' in ''Y'', the only lattice point of Λ in ''N'' is 0 itself. The assertion of Mahler's theorem is equivalent to the compactness of the space of unit-covolume lattices in $\backslash mathbb^n$ whose systole is larger or equal than any fixed $\backslash epsilon>0$. Mahler's compactness theorem was generalized to semisimple Lie groups by Mumford; see Mumford's compactness theorem. ==References== *William Andrew Coppel (2006), ''Number theory'', p. 418. * 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「Mahler's compactness theorem」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.