Convex Characteristics of Quaternionic Positive Definite Functions on Abelian Groups

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-06-25 DOI:10.1007/s00006-024-01336-9

Jingning Liu, Zeping Zhu

引用次数: 0

Abstract

This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group \({\mathbb {S}}\), i.e., the unit 3-sphere in the quaternion algebra. Secondly, we reveal a new phenomenon: The compact convex set of such functions is not a Bauer simplex except when G is of exponent \(\le 2\). In contrast, its complex counterpart is always a Bauer simplex, as is well known. We also present an integral representation for such functions as an application and some other minor results.

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阿贝尔群上四元正定函数的凸特性

本文关注任意无方群 G 上归一化四元值正定函数的拓扑空间，尤其是其凸特性。主要结果有两个。首先，我们证明了此类函数族中的极值元素正是从 G 到球面群 \({\mathbb {S}}\) 的同构，即四元数代数中的单位 3 球面。其次，我们揭示了一个新现象：除了当 G 的指数为 \(\le 2\) 时，这类函数的紧凑凸集不是鲍尔单纯形。相反，它的复数对应集总是鲍尔单纯形，这是众所周知的。作为应用，我们还提出了这类函数的积分表示法和其他一些次要结果。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.