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

DOI DOI PDF Web Site 参考文献22件

書誌事項

公開日
2024-01-04
資源種別
journal article
権利情報
  • https://www.springernature.com/gp/researchers/text-and-data-mining
  • https://www.springernature.com/gp/researchers/text-and-data-mining
DOI
  • 10.1007/s41980-023-00842-z
  • 10.48550/arxiv.2210.11749
公開者
Springer Science and Business Media LLC

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説明

A finite set of the Euclidean space is called an $s$-distance set provided 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 (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 \[ || x-y ||=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2-(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \] 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.

25 pages, 34 figures

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