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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 Lattice (group) ： ウィキペディア英語版 Lattice (group) In mathematics, especially in geometry and group theory, a lattice in $\backslash mathbb^n$ is a subgroup of $\backslash mathbb^n$ which is isomorphic to $\backslash mathbb^n$, and which spans the real vector space $\backslash mathbb^n$. In other words, for any basis of $\backslash mathbb^n$, the subgroup of all linear combinations with integer coefficients forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for the "frame work" of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics. ==Symmetry considerations and examples== A lattice is the symmetry group of discrete translational symmetry in ''n'' directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. As a group (dropping its geometric structure) a lattice is a finitely-generated free abelian group, and thus isomorphic to $\backslash mathbb^n$. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translate of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense. A simple example of a lattice in $\backslash mathbb^n$ is the subgroup $\backslash mathbb^n$. More complicated examples include the E8 lattice, which is a lattice in $\backslash mathbb^$, and the Leech lattice in $\backslash mathbb^$. The period lattice in $\backslash mathbb^2$ is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras; for example, the E8 lattice is related to a Lie algebra that goes by the same name. 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「Lattice (group)」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.