2HOT: An Improved Parallel Hashed Oct-Tree N-Body Algorithm for Cosmological Simulation

We report on improvements made over the past two decades to our adaptive treecode N-body method (HOT). A mathematical and computational approach to the cosmological N-body problem is described, with performance and scalability measured up to 256k ($2^{18}$) processors. We present error analysis and scientific application results from a series of more than ten 69 billion ($4096^3$) particle cosmological simulations, accounting for $4 \times 10^{20}$ floating point operations. These results include the first simulations using the new constraints on the standard model of cosmology from the Planck satellite. Our simulations set a new standard for accuracy and scientific throughput, while meeting or exceeding the computational efficiency of the latest generation of hybrid TreePM N-body methods.Comment: 12 pages, 8 figures, 77 references; To appear in Proceedings of SC '1

Homotopy theoretic models of identity types

This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.Comment: 11 page

Graph-theoretical conditions for inscribability and Delaunay realizability

We present new graph-theoretical conditions for inscribable polyhedra and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is inscribable. These results have several consequences:All 4-connected polyhedra are inscribable.All simplical polyhedra in which all vertex degrees are between 4 and 6, inclusive, are inscribable.All triangulations without chords or nonfacial triangles are realizable as Delaunay triangulations.We also strengthen some earlier results about matchings in inscribable polyhedra. Specifically, we show that any nonbipartite inscribable polyhedron has a perfect matching containing any specified edge, and that any bipartite inscribable polyhedron has a perfect matching containing any two specified disjoint edges. We give examples showing that these results are best possible