23,442 research outputs found
Generators, extremals and bases of max cones
Max cones are max-algebraic analogs of convex cones. In the present paper we
develop a theory of generating sets and extremals of max cones in . This theory is based on the observation that extremals are minimal
elements of max cones under suitable scalings of vectors. We give new proofs of
existing results suitably generalizing, restating and refining them. Of these,
it is important that any set of generators may be partitioned into the set of
extremals and the set of redundant elements. We include results on properties
of open and closed cones, on properties of totally dependent sets and on
computational bounds for the problem of finding the (essentially unique) basis
of a finitely generated cone.Comment: 15 pages, 1 figure; v2: new layout, several new references,
renumbering of result
An Efficient Linear Programming Algorithm to Generate the Densest Lattice Sphere Packings
Finding the densest sphere packing in -dimensional Euclidean space
is an outstanding fundamental problem with relevance in many
fields, including the ground states of molecular systems, colloidal crystal
structures, coding theory, discrete geometry, number theory, and biological
systems. Numerically generating the densest sphere packings becomes very
challenging in high dimensions due to an exponentially increasing number of
possible sphere contacts and sphere configurations, even for the restricted
problem of finding the densest lattice sphere packings. In this paper, we apply
the Torquato-Jiao packing algorithm, which is a method based on solving a
sequence of linear programs, to robustly reproduce the densest known lattice
sphere packings for dimensions 2 through 19. We show that the TJ algorithm is
appreciably more efficient at solving these problems than previously published
methods. Indeed, in some dimensions, the former procedure can be as much as
three orders of magnitude faster at finding the optimal solutions than earlier
ones. We also study the suboptimal local density-maxima solutions (inherent
structures or "extreme" lattices) to gain insight about the nature of the
topography of the "density" landscape.Comment: 23 pages, 3 figure
- …