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
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Final published version
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 - Linear zero mode spectra for quasicrystals
AU - Power, Stephen
N1 - 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
PY - 2022/12/15
Y1 - 2022/12/15
N2 - 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 $\G$.
AB - 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 $\G$.
KW - Zero mode spectrum
KW - Parallelogram frameworks
KW - Multigrid quasicrystals
U2 - 10.1016/j.jmaa.2022.126534
DO - 10.1016/j.jmaa.2022.126534
M3 - Journal article
VL - 516
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 2
M1 - 126534
ER -