I wish I could write four papers in three weeks. The title just means that I submitted four papers to the arXiv in the last three weeks—somehow, after the stress of doing my taxes ended, four of my papers converged to their final state very fast. Here are the papers with their abstracts:
- On k-visibility graphs (with Matthew Babbitt and Jesse Geneson). We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we improve the upper bound on the maximum thickness of bar k-visibility graphs from 2k(9k−1) to 6k, and prove that the maximum thickness must be at least k+1. We also show that the maximum thickness of semi-bar k-visibility graphs is between the ceiling of 2(k+1)/3 and 2k. Moreover we bound the maximum thickness of rectangle k-visibility graphs. We also bound the maximum number of edges and the chromatic number of arc and circle k-visibility graphs. Furthermore we give a method for finding the number of edges in semi-bar k-visibility graphs based on skyscraper puzzles.
- Skyscraper Numbers (with Joel Brewster Lewis). We introduce numbers depending on three parameters which we call skyscraper numbers. We discuss properties of these numbers and their relationship with Stirling numbers of the first kind, and we also introduce a skyscraper sequence.
- Connected Components of Underlying Graphs of Halving Lines (with Dai Yang). In this paper we discuss the connected components of underlying graphs of halving lines’ configurations. We show how to create a configuration whose underlying graph is the union of two given underlying graphs. We also prove that every connected component of the underlying graph is itself an underlying graph.
- Efficient Calculation of Determinants of Symbolic Matrices with Many Variables (with Ziv Scully). Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant algorithms for numerical matrices are often not the fastest for symbolic matrices with many variables. We compare the performance of two algorithms, fraction-free Gaussian elimination and minor expansion, on symbolic matrices with many variables. We show that, under a simplified theoretical model, minor expansion is faster in most situations. We then propose optimizations for minor expansion and demonstrate their effectiveness with empirical data.