Embedding Dimensions of Matrices Whose Entries are Indefinite Distances in the Pseudo-Euclidean Space

IF 0.7 4区数学 Q2 MATHEMATICS

Bulletin of The Iranian Mathematical Society Pub Date : 2024-01-04 DOI:10.1007/s41980-023-00842-z

{"title":"Embedding Dimensions of Matrices Whose Entries are Indefinite Distances in the Pseudo-Euclidean Space","authors":"","doi":"10.1007/s41980-023-00842-z","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> A finite set of the Euclidean space is called an s-distance set provided that the number of Euclidean distances in the set is s. Determining the largest possible s-distance set for the Euclidean space of a given dimension is challenging. This problem was solved only when dealing with small values of s and dimensions. Lisoněk (J Combin Theory Ser A 77(2):318–338, 1997) achieved the classification of the largest 2-distance sets for dimensions up to 7, using computer assistance and graph representation theory. In this study, we consider a theory analogous to these results of Lisoněk for the pseudo-Euclidean space \$\\mathbb {R}^{p,q}\$ . We consider an s-indefinite-distance set in a pseudo-Euclidean space that uses the value $$\\begin{aligned} || \\varvec{x}-\\varvec{y}||&=(x_1-y_1)^2 +\\cdots +(x_p -y_p)^2 \\\\&\\quad -(x_{p+1}-y_{p+1})^2-\\cdots -(x_{p+q}-y_{p+q})^2 \\end{aligned}$$ instead of the Euclidean distance. We develop a representation theory for symmetric matrices in the context of s-indefinite-distance sets, which includes or improves the results of Euclidean s-distance sets with large s values. Moreover, we classify the largest possible 2-indefinite-distance sets for small dimensions.","PeriodicalId":9395,"journal":{"name":"Bulletin of The Iranian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of The Iranian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s41980-023-00842-z","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

A finite set of the Euclidean space is called an s-distance set provided that the number of Euclidean distances in the set is s. Determining the largest possible s-distance set for the Euclidean space of a given dimension is challenging. This problem was solved only when dealing with small values of s and dimensions. Lisoněk (J Combin Theory Ser A 77(2):318–338, 1997) achieved the classification of the largest 2-distance sets for dimensions up to 7, using computer assistance and graph representation theory. In this study, we consider a theory analogous to these results of Lisoněk for the pseudo-Euclidean space $\mathbb {R}^{p,q}$ . We consider an s-indefinite-distance set in a pseudo-Euclidean space that uses the value $$\begin{aligned} || \varvec{x}-\varvec{y}||&=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2 \\&\quad -(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \end{aligned}$$ instead of the Euclidean distance. We develop a representation theory for symmetric matrices in the context of s-indefinite-distance sets, which includes or improves the results of Euclidean s-distance sets with large s values. Moreover, we classify the largest possible 2-indefinite-distance sets for small dimensions.

查看原文本刊更多论文

伪欧几里得空间中条目为不定距离的矩阵的嵌入维数

摘要如果欧几里得空间的有限集合中的欧几里得距离数为 s，则该集合称为 s-距离集合。这个问题只有在处理较小的 s 值和维数时才能解决。Lisoněk (J Combin Theory Ser A 77(2):318-338, 1997)利用计算机辅助和图表示理论实现了维数不超过 7 的最大 2 距离集的分类。在本研究中，我们考虑了一个与 Lisoněk 在伪欧几里得空间 $\mathbb {R}^{p,q}$ 的这些结果类似的理论。我们考虑伪欧几里得空间中的一个 s-indefinite-distance 集，它使用的值是 $$\begin{aligned}.|| \varvec{x}-\varvec{y}||&=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2 \&\quad -(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \end{aligned}$$代替欧氏距离。我们在 s-indefinite-distance 集的背景下发展了对称矩阵的表示理论，其中包括或改进了具有大 s 值的欧氏 s-distance 集的结果。此外，我们还对小维度中可能存在的最大 2-indefinite-distance 集进行了分类。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Bulletin of The Iranian Mathematical Society Mathematics-General Mathematics

CiteScore

1.40

自引率

0.00%

发文量

期刊介绍： The Bulletin of the Iranian Mathematical Society (BIMS) publishes original research papers as well as survey articles on a variety of hot topics from distinguished mathematicians. Research papers presented comprise of innovative contributions while expository survey articles feature important results that appeal to a broad audience. Articles are expected to address active research topics and are required to cite existing (including recent) relevant literature appropriately. Papers are critically reviewed on the basis of quality in its exposition, brevity, potential applications, motivation, value and originality of the results. The BIMS takes a high standard policy against any type plagiarism. The editorial board is devoted to solicit expert referees for a fast and unbiased review process.