54 research outputs found
Bounds on generalized Frobenius numbers
Let and let be relatively prime integers.
The Frobenius number of this -tuple is defined to be the largest positive
integer that has no representation as where
are non-negative integers. More generally, the -Frobenius
number is defined to be the largest positive integer that has precisely
distinct representations like this. We use techniques from the Geometry of
Numbers to give upper and lower bounds on the -Frobenius number for any
nonnegative integer .Comment: We include an appendix with an erratum and addendum to the published
version of this paper: two inaccuracies in the statement of Theorem 2.2 are
corrected and additional bounds on s-Frobenius numbers are derive
Energy minimization, periodic sets and spherical designs
We study energy minimization for pair potentials among periodic sets in
Euclidean spaces. We derive some sufficient conditions under which a point
lattice locally minimizes the energy associated to a large class of potential
functions. This allows in particular to prove a local version of Cohn and
Kumar's conjecture that , , and the
Leech lattice are globally universally optimal, regarding energy minimization,
and among periodic sets of fixed point density.Comment: 16 pages; incorporated referee comment
On Lattice-Free Orbit Polytopes
Given a permutation group acting on coordinates of , we
consider lattice-free polytopes that are the convex hull of an orbit of one
integral vector. The vertices of such polytopes are called \emph{core points}
and they play a key role in a recent approach to exploit symmetry in integer
convex optimization problems. Here, naturally the question arises, for which
groups the number of core points is finite up to translations by vectors fixed
by the group. In this paper we consider transitive permutation groups and prove
this type of finiteness for the -homogeneous ones. We provide tools for
practical computations of core points and obtain a complete list of
representatives for all -homogeneous groups up to degree twelve. For
transitive groups that are not -homogeneous we conjecture that there exist
infinitely many core points up to translations by the all-ones-vector. We prove
our conjecture for two large classes of groups: For imprimitive groups and
groups that have an irrational invariant subspace.Comment: 27 pages, 2 figures; with minor adaptions according to referee
comments; to appear in Discrete and Computational Geometr
Local Energy Optimality of Periodic Sets
We study the local optimality of periodic point sets in for
energy minimization in the Gaussian core model, that is, for radial pair
potential functions with . By considering suitable
parameter spaces for -periodic sets, we can locally rigorously analyze the
energy of point sets, within the family of periodic sets having the same point
density. We derive a characterization of periodic point sets being
-critical for all in terms of weighted spherical -designs contained
in the set. Especially for -periodic sets like the family
we obtain expressions for the hessian of the energy function, allowing to
certify -optimality in certain cases. For odd integers we can
hereby in particular show that is locally -optimal among
periodic sets for all sufficiently large~.Comment: 27 pages, 2 figure
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