arXiv subjects
Publications and source records attributed to Byungchang So.
In coordinate geometry, geometric objects such as lines, disks, and cubes are not perfectly evenly divided because the points on the border should belong to any one of the parts. To mathematically formalize such division, which is physically straightforward, this article proposes a novel representation of geometric objects, called ``rasped representation.''
Magnitude, obtained as a special case of Euler characteristic of enriched category, represents a sense of the size of metric spaces and is related to classical notions such as cardinality, dimension, and volume. While the studies have explained the meaning of magnitude from various perspectives, continuity also gives a valuable view of magnitude. Based on established results about continuity of magnitude and maximum diversity, this article focuses on continuity of weighting, a distribution whose totality is magnitude, and its variation corresponding to maximum diversity. Meanwhile, recent studies also illuminated the connection between magnitude and data analysis by applying magnitude theory to point clouds representing the data or the set of model parameters. This article will also provide an application for time series analysis by introducing a new kind of invariants of periodic time series, where the invariance follows directly from the continuity results. As a use-case, a simple machine learning experiment is conducted with real-world data, in which the suggested invariants improved the performance.
The magnitude of metric spaces does not appear to possess a simple, convenient continuity property, and previous studies have presented affirmative results under additional constraints or weaker notions, as well as counterexamples. In this vein, we discuss the continuity of magnitude of finite positive definite metric spaces with respect to the Gromov-Hausdorff distance, but with a restriction of the domain based on a canonical partition of a sufficiently small neighborhood of a finite metric space. As a result, the main theorem of this article explains a condition on the cardinality of metric spaces that determines the continuity of magnitude. This study takes advantage of the geometric interpretation of magnitude as the circumradius of the corresponding finite Euclidean subset. Such a transformation is especially useful for constructing counterexamples, as we can depend on Euclidean geometric intuition.