J. C. Owen and S. C. Power
Infinite bar-joint frameworks, crystals and operator theory
||August 11, 2011
||Infinite bar-joint framework, vanishing flexibility, rigidity operator
A theory of flexibility and rigidity is developed for general infinite bar-joint frameworks
Determinations of nondeformability through vanishing flexibility are obtained as well
as sufficient conditions for deformability.
Forms of infinitesimal flexibility are defined in terms of the operator theory of the
associated infinite rigidity matrix R(G,p). The matricial symbol function of an abstract crystal framework is introduced, being the multi-variable matrix-valued function on the d-torus representing R(G,p) as a Hilbert space operator.
The symbol function is related to infinitesimal flexibility, deformability and isostaticity.
Various generic abstract crystal frameworks which
are in Maxwellian equilibrium, such as certain 4-regular planar frameworks, are proven to be square-summably infinitesimally rigid as well as smoothly deformable in infinitely many ways.
The symbol function of a three-dimensional crystal framework
determines the infinitesimal wave flexes in models for the
low energy vibrational modes (RUMs) in material crystals.
For crystal frameworks
with inversion symmetry it is shown that the RUMS
generally appear in surfaces, generalising a result of F. Wegner  for
J. C. Owen:
D-Cubed, Siemens PLM Software, Park House, Castle Park, Cambridge UK
S. C. Power:
Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom LA1 4YF