1,791 research outputs found
On piecewise linear cell decompositions
In this note, we introduce a class of cell decompositions of PL manifolds and
polyhedra which are more general than triangulations yet not as general as CW
complexes; we propose calling them PLCW complexes. The main result is an analog
of Alexander's theorem: any two PLCW decompositions of the same polyhedron can
be obtained from each other by a sequence of certain "elementary" moves.
This definition is motivated by the needs of Topological Quantum Field
Theory, especially extended theories as defined by Lurie.Comment: LaTeX2e, 11 page
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Characterizing the Delaunay decompositions of compact hyperbolic surfaces
Given a Delaunay decomposition of a compact hyperbolic surface, one may
record the topological data of the decomposition, together with the
intersection angles between the `empty disks' circumscribing the regions of the
decomposition. The main result of this paper is a characterization of when a
given topological decomposition and angle assignment can be realized as the
data of an actual Delaunay decomposition of a hyperbolic surface.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper12.abs.htm
On examples of two-dimensional periodic continued fractions
This paper is a short survey of the recent results on examples of periodic
two-dimensional continued fractions (in Klein's model). In the last part of
this paper we formulate some questions, problems and conjectures on geometrical
properties concerning to this subject.Comment: 18 pages, 13 Postscript figure
Fidelity and Concurrence of conjugated states
We prove some new properties of fidelity (transition probability) and
concurrence, the latter defined by straightforward extension of Wootters
notation. Choose a conjugation and consider the dependence of fidelity or of
concurrence on conjugated pairs of density operators. These functions turn out
to be concave or convex roofs. Optimal decompositions are constructed. Some
applications to two- and tripartite systems illustrate the general theorem.Comment: 10 pages, RevTex, Correction: Enlarged, reorganized version. More
explanation
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