Combinatorial Space Tiling

The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science

Algebraic Topology

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd

Computational Geometry Column 45

The algorithm of Edelsbrunner for surface reconstruction by ``wrapping'' a set of points in R^3 is described.Comment: 4 pages, to appear in SIGACT News and in IJCGA, 200

Energy-corrected FEM and explicit time-stepping for parabolic problems

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called "pollution effect". Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency

Witness (Delaunay) Graphs

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable generalizations. We introduce a generalization, witness graphs, that encompasses both the goal of more power and flexibility for graph drawing issues and a wider spectrum for neighborhood analysis. We study in detail two concrete examples, both related to Delaunay graphs, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200