1,690 research outputs found
A combinatorial non-positive curvature I: weak systolicity
We introduce the notion of weakly systolic complexes and groups, and initiate
regular studies of them. Those are simplicial complexes with
nonpositive-curvature-like properties and groups acting on them geometrically.
We characterize weakly systolic complexes as simply connected simplicial
complexes satisfying some local combinatorial conditions. We provide several
classes of examples --- in particular systolic groups and CAT(-1) cubical
groups are weakly systolic. We present applications of the theory, concerning
Gromov hyperbolic groups, Coxeter groups and systolic groups.Comment: 35 pages, 1 figur
On the Reconstruction of Geodesic Subspaces of
We consider the topological and geometric reconstruction of a geodesic
subspace of both from the \v{C}ech and Vietoris-Rips filtrations
on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique
leverages the intrinsic length metric induced by the geodesics on the subspace.
We consider the distortion and convexity radius as our sampling parameters for
a successful reconstruction. For a geodesic subspace with finite distortion and
positive convexity radius, we guarantee a correct computation of its homotopy
and homology groups from the sample. For geodesic subspaces of ,
we also devise an algorithm to output a homotopy equivalent geometric complex
that has a very small Hausdorff distance to the unknown shape of interest
Geometric Reasoning with polymake
The mathematical software system polymake provides a wide range of functions
for convex polytopes, simplicial complexes, and other objects. A large part of
this paper is dedicated to a tutorial which exemplifies the usage. Later
sections include a survey of research results obtained with the help of
polymake so far and a short description of the technical background
Linear colorings of simplicial complexes and collapsing
A vertex coloring of a simplicial complex is called a linear
coloring if it satisfies the property that for every pair of facets of , there exists no pair of vertices with the same
color such that and . We
show that every simplicial complex which is linearly colored with
colors includes a subcomplex with vertices such that is
a strong deformation retract of . We also prove that this deformation
is a nonevasive reduction, in particular, a collapsing.Comment: 18 page
Generalized Sums over Histories for Quantum Gravity II. Simplicial Conifolds
This paper examines the issues involved with concretely implementing a sum
over conifolds in the formulation of Euclidean sums over histories for gravity.
The first step in precisely formulating any sum over topological spaces is that
one must have an algorithmically implementable method of generating a list of
all spaces in the set to be summed over. This requirement causes well known
problems in the formulation of sums over manifolds in four or more dimensions;
there is no algorithmic method of determining whether or not a topological
space is an n-manifold in five or more dimensions and the issue of whether or
not such an algorithm exists is open in four. However, as this paper shows,
conifolds are algorithmically decidable in four dimensions. Thus the set of
4-conifolds provides a starting point for a concrete implementation of
Euclidean sums over histories in four dimensions. Explicit algorithms for
summing over various sets of 4-conifolds are presented in the context of Regge
calculus. Postscript figures available via anonymous ftp at
black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps.Comment: 82pp., plain TeX, To appear in Nucl. Phys. B,FF-92-
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