Periodic point sets model all solid crystalline materials whose structures are determined in a rigid form and should be studied up to rigid motion or isometry preserving inter-point distances. In 2021 H.Edelsbrunner et al. introduced an infinite sequence of density functions that are continuous isometry invariants of periodic point sets. These density functions turned out to be highly non-trivial even in dimension 1 for periodic sequences of points in the line. This paper fully describes the density functions of any periodic sequence and their symmetry properties. The explicit description theoretically confirms coincidences of density functions that were previously computed only through finite samples.
翻译:周期点设置所有固晶质材料的模型,其结构以僵硬的形式确定,应研究到固态运动或保持点间距离的等离子测量。2021年,H.Edelsbrunner等人引入了无穷的密度函数序列,这些密度函数是周期点数连续数组的连续异差。这些密度函数即使在第1维层面,对于线内各点的周期序列来说也是高度非三角的。本文充分描述了任何周期序列的密度函数及其对称特性。明确的描述理论上证实了先前仅通过有限样本计算的密度函数的巧合。