534 research outputs found
Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8
We show that the Leech lattice gives a sphere covering which is locally least
dense among lattice coverings. We show that a similar result is false for the
root lattice E8. For this we construct a less dense covering lattice whose
Delone subdivision has a common refinement with the Delone subdivision of E8.
The new lattice yields a sphere covering which is more than 12% less dense than
the formerly best known given by the lattice A8*. Currently, the Leech lattice
is the first and only known example of a locally optimal lattice covering
having a non-simplicial Delone subdivision. We hereby in particular answer a
question of Dickson posed in 1968. By showing that the Leech lattice is rigid
our answer is even strongest possible in a sense.Comment: 13 pages; (v2) major revision: proof of rigidity corrected, full
discussion of E8-case included, src of (v3) contains MAGMA program, (v4) some
correction
Inhomogeneous extreme forms
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two
reduction theories for lattices, well-suited for sphere packing and covering
problems. In his first memoir a characterization of locally most economic
packings is given, but a corresponding result for coverings has been missing.
In this paper we bridge the two classical memoirs.
By looking at the covering problem from a different perspective, we discover
the missing analogue. Instead of trying to find lattices giving economical
coverings we consider lattices giving, at least locally, very uneconomical
ones. We classify local covering maxima up to dimension 6 and prove their
existence in all dimensions beyond.
New phenomena arise: Many highly symmetric lattices turn out to give
uneconomical coverings; the covering density function is not a topological
Morse function. Both phenomena are in sharp contrast to the packing problem.Comment: 22 pages, revision based on suggestions by referee, accepted in
Annales de l'Institut Fourie
Three ways to look at mutually unbiased bases
This is a review of the problem of Mutually Unbiased Bases in finite
dimensional Hilbert spaces, real and complex. Also a geometric measure of
"mubness" is introduced, and applied to some recent calculations in six
dimensions (partly done by Bjorck and by Grassl). Although this does not yet
solve any problem, some appealing structures emerge.Comment: 18 pages. Talk at the Vaxjo Conference on Foundations of Probability
and Physics, June 200
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
50 Years of the Golomb--Welch Conjecture
Since 1968, when the Golomb--Welch conjecture was raised, it has become the
main motive power behind the progress in the area of the perfect Lee codes.
Although there is a vast literature on the topic and it is widely believed to
be true, this conjecture is far from being solved. In this paper, we provide a
survey of papers on the Golomb--Welch conjecture. Further, new results on
Golomb--Welch conjecture dealing with perfect Lee codes of large radii are
presented. Algebraic ways of tackling the conjecture in the future are
discussed as well. Finally, a brief survey of research inspired by the
conjecture is given.Comment: 28 pages, 2 figure
On contact numbers of totally separable unit sphere packings
Contact numbers are natural extensions of kissing numbers. In this paper we
give estimates for the number of contacts in a totally separable packing of n
unit balls in Euclidean d-space for all n>1 and d>1.Comment: 11 page
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