Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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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 -