Brillouin Zones of Integer Lattices and Their Perturbations

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-06-07 DOI:10.1137/22m1489071

Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian, Mathijs Wintraecken

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1784-1807, June 2024.
Abstract. For a locally finite set, [math], the [math]th Brillouin zone of [math] is the region of points [math] for which [math] is the [math]th smallest among the Euclidean distances between [math] and the points in [math]. If [math] is a lattice, the [math]th Brillouin zones of the points in [math] are translates of each other, and together they tile space. Depending on the value of [math], they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in [math], and the convergence of the maximum volume of a chamber to zero for the integer lattice.

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整数网格的布里渊区及其扰动

SIAM 离散数学杂志》，第 38 卷，第 2 期，第 1784-1807 页，2024 年 6 月。摘要。对于局部有限集[math]，[math]的[math]th 布里渊区是[math]与[math]中各点的欧氏距离中[math]th 最小的点[math]区域。如果[math]是一个晶格，那么[math]中各点的[math]th布里渊区就是彼此的平移，它们一起平铺空间。根据[math]值的不同，它们表达了集合中的中程或远程秩序。我们研究了布里渊区的基本几何和组合性质，重点是整数网格及其扰动。我们的研究结果包括布里渊区在扰动下的稳定性、[math]中点阵的布里渊区腔室数量的线性上限，以及整数点阵的腔室最大体积趋近于零。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.