论文标题
矩阵的嵌入尺寸，其条目在伪欧国人空间中无限期距离
Embedding dimensions of matrices whose entries are indefinite distances in the pseudo-Euclidean space
论文作者
论文摘要
只要集合中的欧几里得距离数为$ S $，欧几里得空间的有限集称为$ s $ distance集。确定给定维度的欧几里得空间中最大的$ s $ distance集是具有挑战性的。仅在处理$ s $和尺寸的少量值时才解决此问题。 LisoněK（1997）使用计算机辅助和图表理论实现了最大尺寸的最大尺寸2距离集的分类。在这项研究中，我们考虑了一种类似于lisonk的伪欧克里德式空间$ \ mathbb {r}^{p，q} $的理论。我们考虑使用值\ [|| 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 \]，而不是euclidean demand。我们在$ s $ - 单次距离集的上下文中为对称矩阵开发了一种表示理论，其中包括或改善了具有较大$ s $值的Euclidean $ s $ distance set。此外，我们将最大的$ 2 $ - 单位距离设置分类为小尺寸。
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.
