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Pramy, F. A.; Mestel, B. D. and Gilbert, A. D.
(2023).
DOI: https://doi.org/10.1080/14689367.2022.2139224
Abstract
The Stretch-Fold-Shear (SFS) operator Sα is a functional linear operator acting on complex-valued functions of a real variable x on some domain containing [−1,1] in R. It arises from a stylized model in kinematic dynamo theory where magnetic field growth corresponds to an eigenvalue of modulus greater than 1. When the shear parameter α is zero, the spectrum of Sα can be determined exactly, and the eigenfunctions corresponding to non-zero eigenvalues are related to the Bernoulli polynomials. The spectrum for α>0 has not been rigorously determined although the spectrum has been approximated numerically. In this paper, a computer-assisted proof is presented to provide rigorous bounds on the leading eigenvalue for α∈[0,5], showing inter alia that Sα has an eigenvalue of modulus greater than 1 for all α satisfying π/2<α≤5, thereby partially confirming an outstanding conjecture on the SFS operator.
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About
- Item ORO ID
- 85622
- Item Type
- Journal Item
- ISSN
- 1468-9375
- Project Funding Details
-
Funded Project Name Project ID Funding Body Analysis of waves, instabilities and mean flows in MHD systems EP/T023139/1 EPSRC Studentship Not Set The Open University (OU) - Keywords
- kinematic dynamo; stretch-fold-shear map; operator spectrum; computer-assisted proof
- Academic Unit or School
-
Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM) - Copyright Holders
- © 2022 The Authors
- Depositing User
- Benjamin Mestel