102,047 research outputs found
Counting configuration-free sets in groups
© 2017 Elsevier Ltd. We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.Postprint (updated version
When is a symmetric pin-jointed framework isostatic?
Maxwell's rule from 1864 gives a necessary condition for a framework to be
isostatic in 2D or in 3D. Given a framework with point group symmetry, group
representation theory is exploited to provide further necessary conditions.
This paper shows how, for an isostatic framework, these conditions imply very
simply stated restrictions on the numbers of those structural components that
are unshifted by the symmetry operations of the framework. In particular, it
turns out that an isostatic framework in 2D can belong to one of only six point
groups. Some conjectures and initial results are presented that would give
sufficient conditions (in both 2D and 3D) for a framework that is realized
generically for a given symmetry group to be an isostatic framework.Comment: 24 pages, 10 figures; added references, minor changes, revised last
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