35,887 research outputs found
Intrinsic Volumes of Random Cubical Complexes
Intrinsic volumes, which generalize both Euler characteristic and Lebesgue
volume, are important properties of -dimensional sets. A random cubical
complex is a union of unit cubes, each with vertices on a regular cubic
lattice, constructed according to some probability model. We analyze and give
exact polynomial formulae, dependent on a probability, for the expected value
and variance of the intrinsic volumes of several models of random cubical
complexes. We then prove a central limit theorem for these intrinsic volumes.
For our primary model, we also prove an interleaving theorem for the zeros of
the expected-value polynomials. The intrinsic volumes of cubical complexes are
useful for understanding the shape of random -dimensional sets and for
characterizing noise in applications.Comment: 17 pages with 7 figures; this version includes a central limit
theore
Slices, slabs, and sections of the unit hypercube
Using combinatorial methods, we derive several formulas for the volume of
convex bodies obtained by intersecting a unit hypercube with a halfspace, or
with a hyperplane of codimension 1, or with a flat defined by two parallel
hyperplanes. We also describe some of the history of these problems, dating to
Polya's Ph.D. thesis, and we discuss several applications of these formulas.Comment: 11 pages; minor corrections to reference
Center vortex model for the infrared sector of Yang-Mills theory - Quenched Dirac spectrum and chiral condensate
The Dirac operator describing the coupling of continuum quark fields to SU(2)
center vortex world-surfaces composed of elementary squares on a hypercubic
lattice is constructed. It is used to evaluate the quenched Dirac spectral
density in the random vortex world-surface model, which previously has been
shown to quantitatively reproduce both the confinement properties and the
topological susceptibility of SU(2) Yang-Mills theory. Under certain conditions
on the modeling of the vortex gauge field, a behavior of the quenched chiral
condensate as a function of temperature is obtained which is consistent with
measurements in SU(2) lattice Yang-Mills theory.Comment: 36 LaTeX pages, 13 ps figures included via epsf; minor reformulations
and added cross-referencing for the purpose of clarit
Generation of initial molecular dynamics configurations in arbitrary geometries and in parallel
A computational pre-processing tool for generating initial configurations of molecules for molecular dynamics simulations in geometries described by a mesh of unstructured arbitrary polyhedra is described. The mesh is divided into separate zones and each can be filled with a single crystal lattice of atoms. Each zone is filled by creating an expanding cube of crystal unit cells, initiated from an anchor point for the lattice. Each unit cell places the appropriate atoms for the user-specified crystal structure and orientation. The cube expands until the entire zone is filled with the lattice; zones with concave and disconnected volumes may be filled. When the mesh is spatially decomposed into portions for distributed parallel processing, each portion may be filled independently, meaning that the entire molecular system never needs to fit onto a single processor, allowing very large systems to be created. The computational time required to fill a zone with molecules scales linearly with the number of cells in the zone for a fixed number of molecules, and better than linearly with the number of molecules for a fixed number of mesh cells. Our tool, molConfig, has been implemented in the open source C++ code OpenFOAM
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