Symbol functions for symmetric frameworks

Eleftherios Kastis; Derek Kitson; John E. McCarthy

doi:10.1016/j.jmaa.2020.124895

Symbol functions for symmetric frameworks

Eleftherios Kastis, Derek Kitson, John E. McCarthy

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ²(Z) are unitarily equivalent to multiplication operators on L²(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

Original language	English
Article number	124895
Journal	Journal of Mathematical Analysis and Applications
Volume	497
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2020.124895
State	Published - May 15 2021

Keywords

Bar-joint framework
Infinitesimal flex
Multiplication operator
Rigid unit modes
Rigidity matrix

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10.1016/j.jmaa.2020.124895

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title = "Symbol functions for symmetric frameworks",

abstract = "We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.",

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author = "Eleftherios Kastis and Derek Kitson and McCarthy, {John E.}",

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AU - Kitson, Derek

AU - McCarthy, John E.

N1 - Funding Information: E.K. and D.K. supported by the Engineering and Physical Sciences Research Council [grant number EP/S00940X/1 ]. J.E.M. partially supported by National Science Foundation Grant DMS 156243 . Publisher Copyright: © 2020 The Authors

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N2 - We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

AB - We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

KW - Bar-joint framework

KW - Infinitesimal flex

KW - Multiplication operator

KW - Rigid unit modes

KW - Rigidity matrix

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VL - 497

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 124895

ER -

Symbol functions for symmetric frameworks

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Keywords

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