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On Hölder solutions to the spiral winding problem
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dc.contributor.author | Fraser, Jonathan | |
dc.date.accessioned | 2021-05-13T13:30:12Z | |
dc.date.available | 2021-05-13T13:30:12Z | |
dc.date.issued | 2021-05-07 | |
dc.identifier.citation | Fraser , J 2021 , ' On Hölder solutions to the spiral winding problem ' , Nonlinearity , vol. 34 , no. 5 , pp. 3251–3270 . https://doi.org/10.1088/1361-6544/abe75e | en |
dc.identifier.issn | 0951-7715 | |
dc.identifier.other | PURE: 273170241 | |
dc.identifier.other | PURE UUID: 2f01caf1-2637-4106-9346-de083c8c338c | |
dc.identifier.other | ORCID: /0000-0002-8066-9120/work/90112118 | |
dc.identifier.other | Scopus: 85107053675 | |
dc.identifier.other | WOS: 000649647600001 | |
dc.identifier.uri | https://hdl.handle.net/10023/23185 | |
dc.description | The author was supported by an EPSRC Standard Grant No. (EP/R015104/1) and a Leverhulme Trust Research Project Grant No. (RPG-2019-034). | en |
dc.description.abstract | The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret 'regularity' in terms of Hölder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp Hölder exponent of the forward map and its inverse satisfy a formula reminiscent of Sobolev conjugates. We also investigate the dimension theory of these spirals, in particular, the Assouad dimension, Assouad spectrum and box dimensions. The aim here is to compare the bounds on the Hölder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under Hölder image) with the sharp results. We find that the Assouad spectrum provides the best information, but that even this is not sharp. We also find that the Assouad spectrum is the only 'dimension' which distinguishes between spirals with different polynomial winding rates in the superlinear regime. | |
dc.format.extent | 20 | |
dc.language.iso | eng | |
dc.relation.ispartof | Nonlinearity | en |
dc.rights | Copyright © 2021 IOP Publishing Ltd & London Mathematical Society. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. | en |
dc.subject | Spiral | en |
dc.subject | Winding problem | en |
dc.subject | Holder exponents | en |
dc.subject | Assouad dimension | en |
dc.subject | Box dimension | en |
dc.subject | Assousad spectrum | en |
dc.subject | QA Mathematics | en |
dc.subject | T-NDAS | en |
dc.subject.lcc | QA | en |
dc.title | On Hölder solutions to the spiral winding problem | en |
dc.type | Journal article | en |
dc.contributor.sponsor | EPSRC | en |
dc.contributor.sponsor | The Leverhulme Trust | en |
dc.description.version | Publisher PDF | en |
dc.contributor.institution | University of St Andrews. Pure Mathematics | en |
dc.contributor.institution | University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra | en |
dc.identifier.doi | https://doi.org/10.1088/1361-6544/abe75e | |
dc.description.status | Peer reviewed | en |
dc.identifier.url | https://arxiv.org/abs/1905.07563 | en |
dc.identifier.grantnumber | EP/R015104/1 | en |
dc.identifier.grantnumber | RPG-2019-034 | en |
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