2023/01/12 by Olga Anosova, Anosova, Olga, Vitaliy Kurlin +1 · 1 citation
Business, Management and Accounting · Chemistry · #51F20 #51F30 #51K05 #51N20 #68U05 #Chemistry and Stereochemistry Studies #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #History and advancements in chemistry #Metric Geometry (math.MG) #Optics and Image Analysis
paper · pdf · doi:10.48550/arxiv.2301.05137
openalex publication_date 2023/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations. The motivations come from periodic crystals whose structures are determined in a rigid form but any minimal cells can discontinuously change due to small noise in measurements. For any integer k>=0, the density function of a periodic set S was previously defined as the fractional volume of all k-fold intersections (within a minimal cell) of balls that have a variable radius t and centers at all points of S. This paper introduces the density functions for periodic sets of points with different initial radii motivated by atomic radii of chemical elements and by continuous events occupying disjoint intervals in time series. The contributions are explicit descriptions of the densities for periodic sequences of intervals. The new densities are strictly stronger and distinguish periodic sequences that have identical densities in the case of zero radii.