24,544 research outputs found
Distance Geometry in Quasihypermetric Spaces. III
Let be a compact metric space and let denote the
space of all finite signed Borel measures on . Define by
and set , where ranges over the collection of signed
measures in of total mass 1. This paper, with two earlier
papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric
spaces. I and II], investigates the geometric constant and its
relationship to the metric properties of and the functional-analytic
properties of a certain subspace of when equipped with a
natural semi-inner product. Specifically, this paper explores links between the
properties of and metric embeddings of , and the properties of
when is a finite metric space.Comment: 20 pages. References [10] and [11] are arXiv:0809.0740v1 [math.MG]
and arXiv:0809.0744v1 [math.MG
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
Embedding Properties of sets with finite box-counting dimension
In this paper we study the regularity of embeddings of finite--dimensional
subsets of Banach spaces into Euclidean spaces. In 1999, Hunt and Kaloshin
[Nonlinearity 12 1263-1275] introduced the thickness exponent and proved an
embedding theorem for subsets of Hilbert spaces with finite box--counting
dimension. In 2009, Robinson [Nonlinearity 22 711-728] defined the dual
thickness and extended the result to subsets of Banach spaces. Here we prove a
similar result for subsets of Banach spaces, using the thickness rather than
the dual thickness. We also study the relation between the box-counting
dimension and these two thickness exponents for some particular subsets of
.Comment: Submitted, Referres comments addresse
Diversities and the Geometry of Hypergraphs
The embedding of finite metrics in has become a fundamental tool for
both combinatorial optimization and large-scale data analysis. One important
application is to network flow problems in which there is close relation
between max-flow min-cut theorems and the minimal distortion embeddings of
metrics into . Here we show that this theory can be generalized
considerably to encompass Steiner tree packing problems in both graphs and
hypergraphs. Instead of the theory of metrics and minimal distortion
embeddings, the parallel is the theory of diversities recently introduced by
Bryant and Tupper, and the corresponding theory of diversities and
embeddings which we develop here.Comment: 19 pages, no figures. This version: further small correction
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
Transactions on Information Theor
Green's J-order and the rank of tropical matrices
We study Green's J-order and J-equivalence for the semigroup of all n-by-n
matrices over the tropical semiring. We give an exact characterisation of the
J-order, in terms of morphisms between tropical convex sets. We establish
connections between the J-order, isometries of tropical convex sets, and
various notions of rank for tropical matrices. We also study the relationship
between the relations J and D; Izhakian and Margolis have observed that for the semigroup of all 3-by-3 matrices over the tropical semiring with
, but in contrast, we show that for all full matrix semigroups
over the finitary tropical semiring.Comment: 21 pages, exposition improve
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