519,576 research outputs found
Subsets of finite groups exhibiting additive regularity
In this article we aim to develop from first principles a theory of sum sets
and partial sum sets, which are defined analogously to difference sets and
partial difference sets. We obtain non-existence results and characterisations.
In particular, we show that any sum set must exhibit higher-order regularity
and that an abelian sum set is necessarily a reversible difference set. We next
develop several general construction techniques under the hypothesis that the
over-riding group contains a normal subgroup of order 2. Finally, by exploiting
properties of dihedral groups and Frobenius groups, several infinite classes of
sum sets and partial sum sets are introduced
Numerical studies of entangled PPT states in composite quantum systems
We report here on the results of numerical searches for PPT states with
specified ranks for density matrices and their partial transpose. The study
includes several bipartite quantum systems of low dimensions. For a series of
ranks extremal PPT states are found. The results are listed in tables and
charted in diagrams. Comparison of the results for systems of different
dimensions reveal several regularities. We discuss lower and upper bounds on
the ranks of extremal PPT states.Comment: 18 pages, 4 figure
Semifields, relative difference sets, and bent functions
Recently, the interest in semifields has increased due to the discovery of
several new families and progress in the classification problem. Commutative
semifields play an important role since they are equivalent to certain planar
functions (in the case of odd characteristic) and to modified planar functions
in even characteristic. Similarly, commutative semifields are equivalent to
relative difference sets. The goal of this survey is to describe the connection
between these concepts. Moreover, we shall discuss power mappings that are
planar and consider component functions of planar mappings, which may be also
viewed as projections of relative difference sets. It turns out that the
component functions in the even characteristic case are related to negabent
functions as well as to -valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite
fields", 09-13 December 2013, Linz, Austria. This article will appear in the
proceedings volume for this workshop, published as part of the "Radon Series
on Computational and Applied Mathematics" by DeGruyte
Dihedral symmetries of multiple logarithms
This paper finds relationships between multiple logarithms with a dihedral
group action on the arguments. I generalize the combinatorics developed in
Gangl, Goncharov and Levin's R-deco polygon representation of multiple
logarithms to find these relations. By writing multiple logarithms as iterated
integrals, my arguments are valid for iterated integrals as over an arbitrary
field
- …