8,717 research outputs found
On the outer automorphism groups of triangular alternation limit algebras
Let denote the alternation limit algebra, studied by Hopenwasser and
Power, and by Poon, which is the closed direct limit of upper triangular matrix
algebras determined by refinement embeddings of multiplicity and standard
embeddings of multiplicity . It is shown that the quotient of the
isometric automorphism group by the approximately inner automorphisms is the
abelian group \ZZ ^d where is the number of primes that are divisors of
infinitely many terms of each of the sequences and . This group
is also the group of automorphisms of the fundamental relation of .Comment: 12 pages, Late
Frameworks, Symmetry and Rigidity
Symmetry equations are obtained for the rigidity matrix of a bar-joint
framework in R^d. These form the basis for a short proof of the Fowler-Guest
symmetry group generalisation of the Calladine-Maxwell counting rules. Similar
symmetry equations are obtained for the Jacobian of diverse framework systems,
including constrained point-line systems that appear in CAD, body-pin
frameworks, hybrid systems of distance constrained objects and infinite
bar-joint frameworks. This leads to generalised forms of the Fowler-Guest
character formula together with counting rules in terms of counts of
symmetry-fixed elements. Necessary conditions for isostaticity are obtained for
asymmetric frameworks, both when symmetries are present in subframeworks and
when symmetries occur in partition-derived frameworks.Comment: 5 Figures. Replaces Dec. 2008 version. To appear in IJCG
The rigidity of infinite graphs
A rigidity theory is developed for the Euclidean and non-Euclidean placements
of countably infinite simple graphs in R^d with respect to the classical l^p
norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and
Henneberg combinatorial characterisations of generic infinitesimal rigidity for
finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation
of the rigidity of generic finite body-bar frameworks in d-dimensional
Euclidean space is generalised to the non-Euclidean l^p norms and to countably
infinite graphs. For all dimensions and norms it is shown that a generically
rigid countable simple graph is the direct limit of an inclusion tower of
finite graphs for which the inclusions satisfy a relative rigidity property.
For d>2 a countable graph which is rigid for generic placements in R^d may fail
the stronger property of sequential rigidity, while for d=2 the equivalence
with sequential rigidity is obtained from the generalised Laman
characterisations. Applications are given to the flexibility of non-Euclidean
convex polyhedra and to the infinitesimal and continuous rigidity of compact
infinitely-faceted simplicial polytopes.Comment: 51 page
Rigidity of Frameworks Supported on Surfaces
A theorem of Laman gives a combinatorial characterisation of the graphs that
admit a realisation as a minimally rigid generic bar-joint framework in
\bR^2. A more general theory is developed for frameworks in \bR^3 whose
vertices are constrained to move on a two-dimensional smooth submanifold \M.
Furthermore, when \M is a union of concentric spheres, or a union of parallel
planes or a union of concentric cylinders, necessary and sufficient
combinatorial conditions are obtained for the minimal rigidity of generic
frameworks.Comment: Final version, 28 pages, with new figure
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