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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.22457 |
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| _version_ | 1866909556100038656 |
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| author | Kastis, Eleftherios Kitson, Derek |
| author_facet | Kastis, Eleftherios Kitson, Derek |
| contents | We establish several fundamental properties of the Rigid Unit Mode (RUM) spectrum for symmetric frameworks with a discrete abelian symmetry group and arbitrary linear constraints. In particular, we identify a nonempty subset of the RUM spectrum which derives from the joint eigenvalues of generators for the linear part of the symmetry group. These joint eigenvalues give rise to $χ$-symmetric flexes which span the space of translations for the framework. We show that the RUM spectrum is a union of Bohr-Fourier spectra arising from twisted almost-periodic flexes of the framework. We also characterise frameworks for which every almost periodic flex is a translation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Rigid Unit Mode spectrum for symmetric frameworks Kastis, Eleftherios Kitson, Derek Functional Analysis Metric Geometry 52C25, 47B91, 47A56, 43A60 We establish several fundamental properties of the Rigid Unit Mode (RUM) spectrum for symmetric frameworks with a discrete abelian symmetry group and arbitrary linear constraints. In particular, we identify a nonempty subset of the RUM spectrum which derives from the joint eigenvalues of generators for the linear part of the symmetry group. These joint eigenvalues give rise to $χ$-symmetric flexes which span the space of translations for the framework. We show that the RUM spectrum is a union of Bohr-Fourier spectra arising from twisted almost-periodic flexes of the framework. We also characterise frameworks for which every almost periodic flex is a translation. |
| title | The Rigid Unit Mode spectrum for symmetric frameworks |
| topic | Functional Analysis Metric Geometry 52C25, 47B91, 47A56, 43A60 |
| url | https://arxiv.org/abs/2503.22457 |