On the beta distribution, the nonlinear Fourier transform and a combinatorial problem

Saksida, Pavle (author)

The paper describes some probabilistic and combinatorial aspects of the nonlinear Fourier transform associated with the AKNS-ZS problems. We show that the volumes of a family of polytopes that appear in a power expansion of the nonlinear Fourier transform are distributed according to the beta probability distribution. We establish this result by studying an Euler-type discretization of the nonlinear Fourier transform. This approach leads to the combinatorial problem of finding the number of alternating ordered partitions of an integer into a fixed number of distinct parts. We find the explicit formula for these numbers and show that they are essentially distributed according to a novel discretization of the beta distribution for a suitable choice of the shape parameters. We also find the generating functions of the numbers of alternating sums. These functions are expressed in terms of the our discrete nonlinear Fourier transform.

• Language: English
• Keywords: beta distribution, nonlinear Fourier transform, discretization
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2024
• Year: 2024
• Number of pages: 20 str.
• Numbering: Vol. 24, no. 1, article no. P1.08
• PID: 20.500.12556/RUL-154482
• UDC: 517.95:519.1
• ISSN on article: 1855-3966
• DOI: 10.26493/1855-3974.2976.f76
• COBISS.SI-ID: 165700867
• Publication date in RUL: 16.02.2024

Record is a part of a journal

• Title: Ars mathematica contemporanea
• Publisher: Društvo matematikov, fizikov in astronomov, Društvo matematikov, fizikov in astronomov, Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije
• ISSN: 1855-3966
• COBISS.SI-ID: 239049984

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: distribucija beta, nelinearna Fourierova transformacija, diskretizacija

Projects

• Funder: ARRS - Slovenian Research Agency
• Funding programme: Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije
• Project number: P1- 0291
• Name: Analiza in geometrija