17 research outputs found
On the Equivariance Properties of Self-adjoint Matrices
We investigate self-adjoint matrices with respect to
their equivariance properties. We show in particular that a matrix is
self-adjoint if and only if it is equivariant with respect to the action of a
group which is isomorphic to
. If the self-adjoint matrix possesses multiple
eigenvalues -- this may, for instance, be induced by symmetry properties of an
underlying dynamical system -- then is even equivariant with respect to the
action of a group where
are the multiplicities of the eigenvalues
of . We discuss implications of this result for
equivariant bifurcation problems, and we briefly address further applications
for the Procrustes problem, graph symmetries and Taylor expansions
A Hypergraph Framework for Optimal Model-Based Decomposition of Design Problems
Decomposition of large engineering system models is desirable sinceincreased model size reduces reliability and speed of numericalsolution algorithms. The article presents a methodology for optimalmodel-based decomposition (OMBD) of design problems, whether or notinitially cast as optimization problems. The overall model isrepresented by a hypergraph and is optimally partitioned into weaklyconnected subgraphs that satisfy decomposition constraints. Spectralgraph-partitioning methods together with iterative improvementtechniques are proposed for hypergraph partitioning. A known spectralK-partitioning formulation, which accounts for partition sizes andedge weights, is extended to graphs with also vertex weights. TheOMBD formulation is robust enough to account for computationaldemands and resources and strength of interdependencies between thecomputational modules contained in the model.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44780/1/10589_2004_Article_136837.pd