Phase Transitions for the Minimizers of the [math]-Frame Potentials in [math]

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-07-26 DOI:10.1137/22m1539915

Radel Ben-Av, Xuemei Chen, Assaf Goldberger, Shujie Kang, Kasso A. Okoudjou

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2243-2259, September 2024.
Abstract. Given [math] points [math] on the unit circle in [math] and a number [math], we investigate the minimizers of the functional [math]. While it is known that each of these minimizers is a spanning set for [math], less is known about their number as a function of [math] and [math] especially for relatively small [math]. In this paper we show that there is unique minimum for this functional for all [math] and all odd [math]. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for [math] odd, there exists a sequence of points [math] so that a unique (up to some isometries) minimizer exists on each of the subintervals [math].

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[数学]中[数学]框架势最小值的相变

SIAM 离散数学杂志》，第 38 卷第 3 期，第 2243-2259 页，2024 年 9 月。摘要。给定[math]中单位圆上的[math]点[math]和一个数[math]，我们研究函数[math]的最小值。虽然我们知道这些最小化子中的每一个都是[math]的跨集，但对于它们的数量与[math]和[math]的函数关系却知之甚少，尤其是对于相对较小的[math]。在本文中，我们证明了对于所有[math]和所有奇数[math]，这个函数都有唯一的最小值。此外，我们还提出了一些数值结果，表明这些最小值出现了相变现象。更具体地说，对于奇数[math]，存在一个点序列[math]，因此在每个子区间[math]上都存在一个唯一的（直到某些等距）最小值。

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.