Willerton, S. (2020) The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. Discrete Analysis, 2020 (5). 12649. ISSN 2397-3129
Abstract
Magnitude is an invariant of metric spaces with origins in category theory. Using potential theoretic methods, Barceló and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schröder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 Simon Willerton. Licensed under a Creative Commons Attribution License (CC-BY) (http://creativecommons.org/licenses/by/3.0/) |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 26 Feb 2018 15:57 |
Last Modified: | 30 Jul 2020 15:59 |
Published Version: | https://discreteanalysisjournal.com/article/12649 |
Status: | Published |
Publisher: | Diamond Open Access Journals |
Refereed: | Yes |
Identification Number: | 10.19086/da.12649 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:127468 |