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• zeromodes_jmaa

Rights statement: This is the author’s version of a work that was accepted for publication in Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Mathematical Analysis and Applications, 512, 2, 2022 DOI: 10.1016/j.jmaa.2022.126534

Accepted author manuscript, 273 KB, PDF document

Embargo ends: 26/07/23

## Linear zero mode spectra for quasicrystals

Research output: Contribution to Journal/MagazineJournal articlepeer-review

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of {linear zero mode spectra} for an aperiodic bar-joint framework $\G$ that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the {limit spectrum} ${\bf L}_{\rm lim}(\G,\ul{a})$ is defined in this way, as a subset of the reciprocal space for a reference basis $\ul{a}$ of the ambient space. A smaller spectrum, the \emph{slippage spectrum} ${\bf L}_{\rm slip}(\G,\ul{a})$, is also defined. For quasicrystal parallelogram frameworks associated with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra coincide and are determined in terms of the geometry of $\G$.