24,802 research outputs found
A new structure for difference matrices over abelian -groups
A difference matrix over a group is a discrete structure that is intimately
related to many other combinatorial designs, including mutually orthogonal
Latin squares, orthogonal arrays, and transversal designs. Interest in
constructing difference matrices over -groups has been renewed by the recent
discovery that these matrices can be used to construct large linking systems of
difference sets, which in turn provide examples of systems of linked symmetric
designs and association schemes. We survey the main constructive and
nonexistence results for difference matrices, beginning with a classical
construction based on the properties of a finite field. We then introduce the
concept of a contracted difference matrix, which generates a much larger
difference matrix. We show that several of the main constructive results for
difference matrices over abelian -groups can be substantially simplified and
extended using contracted difference matrices. In particular, we obtain new
linking systems of difference sets of size in infinite families of abelian
-groups, whereas previously the largest known size was .Comment: 27 pages. Discussion of new reference [LT04
Topologically Stratified Energy Minimizers in a Product Abelian Field Theory
We study a recently developed product Abelian gauge field theory by Tong and
Wong hosting magnetic impurities. We first obtain a necessary and sufficient
condition for the existence of a unique solution realizing such impurities in
the form of multiple vortices. We next reformulate the theory into an extended
model that allows the coexistence of vortices and anti-vortices. The two
Abelian gauge fields in the model induce two species of magnetic vortex-lines
resulting from vortices and anti-vortices () realized as the
zeros and poles of two complex-valued Higgs fields, respectively. An existence
theorem is established for the governing equations over a compact Riemann
surface which states that a solution with prescribed vortices
and anti-vortices of two designated species exists if and only if the
inequalities hold simultaneously, which
give bounds for the `differences' of the vortex and anti-vortex numbers in
terms of the total surface area of . The minimum energy of these solutions
is shown to assume the explicit value given in
terms of several topological invariants, measuring the total tension of the
vortex-lines.Comment: 22 page
Independence in computable algebra
We give a sufficient condition for an algebraic structure to have a
computable presentation with a computable basis and a computable presentation
with no computable basis. We apply the condition to differentially closed, real
closed, and difference closed fields with the relevant notions of independence.
To cover these classes of structures we introduce a new technique of safe
extensions that was not necessary for the previously known results of this
kind. We will then apply our techniques to derive new corollaries on the number
of computable presentations of these structures. The condition also implies
classical and new results on vector spaces, algebraically closed fields,
torsion-free abelian groups and Archimedean ordered abelian groups.Comment: 24 page
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