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dc.contributor.authorHolmes-Cerfon, Miranda
dc.contributor.authorTheran, Louis Simon
dc.contributor.authorGortler, Steven J.
dc.date.accessioned2021-12-19T00:41:13Z
dc.date.available2021-12-19T00:41:13Z
dc.date.issued2020-12-19
dc.identifier.citationHolmes-Cerfon , M , Theran , L S & Gortler , S J 2020 , ' Almost rigidity of frameworks ' , Communications on Pure and Applied Mathematics , vol. Early online , 21971 , pp. 1-63 . https://doi.org/10.1002/cpa.21971en
dc.identifier.issn0010-3640
dc.identifier.otherPURE: 270006843
dc.identifier.otherPURE UUID: 15890ade-8168-49ee-9289-e7e48ae156b6
dc.identifier.otherORCID: /0000-0001-5282-4800/work/85855377
dc.identifier.otherScopus: 85097747642
dc.identifier.otherWOS: 000600037300001
dc.identifier.urihttp://hdl.handle.net/10023/24537
dc.descriptionFunding: NSF-FRG Grants DMS-1564487and DMS-1564473. M. H.-C. acknowledges support from the US Department of Energy DE-SC0012296, and the Alfred P. Sloan Foundation.en
dc.description.abstractWe extend the mathematical theory of rigidity of frameworks (graphs embedded in d‐dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call “almost‐rigidity”; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost‐rigidity. We briefly discuss how the theory may be applied to tensegrities.
dc.format.extent63
dc.language.isoeng
dc.relation.ispartofCommunications on Pure and Applied Mathematicsen
dc.rightsCopyright © 2020 Wiley Periodicals LLC This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at ttps://doi.org/10.1002/cpa.21971en
dc.subjectQA Mathematicsen
dc.subjectT-NDASen
dc.subjectBDCen
dc.subjectR2Cen
dc.subject.lccQAen
dc.titleAlmost rigidity of frameworksen
dc.typeJournal articleen
dc.description.versionPostprinten
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.identifier.doihttps://doi.org/10.1002/cpa.21971
dc.description.statusPeer revieweden
dc.date.embargoedUntil2021-12-19
dc.identifier.urlhttps://arxiv.org/abs/1908.03802en


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