TY - JOUR

T1 - Infinite bar-joint frameworks, crystals and operator theory

AU - Owen, J. C.

AU - Power, Stephen

PY - 2011

Y1 - 2011

N2 - A theory of fl exibility and rigidity is developed for general infinite bar-joint frameworks (G; p). 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 [35] fortetrahedral crystals.

AB - A theory of fl exibility and rigidity is developed for general infinite bar-joint frameworks (G; p). 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 [35] fortetrahedral crystals.

KW - Infinite bar-joint framework

KW - vanishing flexibility

KW - rigidity operator

M3 - Journal article

VL - 17

SP - 445

EP - 490

JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

SN - 1076-9803

ER -