107,029 research outputs found
A branch-and-bound methodology within algebraic modelling systems
Through the use of application-specific branch-and-bound directives it is possible to find solutions to combinatorial models that would otherwise be difficult or impossible to find by just using generic branch-and-bound techniques within the framework of mathematical programming. {\sc Minto} is an example of a system which offers the possibility to incorporate user-provided directives (written in {\sc C}) to guide the branch-and-bound search. Its main focus, however, remains on mathematical programming models. The aim of this paper is to present a branch-and-bound methodology for particular combinatorial structures to be embedded inside an algebraic modelling language. One advantage is the increased scope of application. Another advantage is that directives are more easily implemented at the modelling level than at the programming level
Zero-sum subsets of decomposable sets in Abelian groups
A subset of an Abelian group is if . In the paper we give partial answer to an open problem asking whether
every finite decomposable subset of an Abelian group contains a non-empty
subset with . For every we present a
decomposable subset of cardinality in the cyclic group of order
such that , but for any proper non-empty subset
. On the other hand, we prove that every decomposable subset
of cardinality contains a non-empty subset
of cardinality with . For every
we present a subset of cardinality
such that for some subset of cardinality and
for any non-empty subset of cardinality
. Also we prove that every finite decomposable subset of
an Abelian group contains two non-empty subsets such that .Comment: 8 page
Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices
We prove that every infinite sequence of skew-symmetric or symmetric matrices
M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such
that M_i is isomorphic to a principal submatrix of the Schur complement of a
nonsingular principal submatrix in M_j, if those matrices have bounded
rank-width. This generalizes three theorems on well-quasi-ordering of graphs or
matroids admitting good tree-like decompositions; (1) Robertson and Seymour's
theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's
theorem for matroids representable over a fixed finite field having bounded
branch-width, and (3) Oum's theorem for graphs of bounded rank-width with
respect to pivot-minors.Comment: 43 page
Subexponential groups in 4-manifold topology
We present a new, more elementary proof of the Freedman-Teichner result that
the geometric classification techniques (surgery, s-cobordism, and
pseudoisotopy) hold for topological 4-manifolds with groups of subexponential
growth. In an appendix Freedman and Teichner give a correction to their
original proof, and reformulate the growth estimates in terms of coarse
geometry.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper14.abs.htm
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