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Extreme points of the Vandermonde determinant on surfaces implicitly determined by a univariate polynomial
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, Busitema University, Kampala, Uganda.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.ORCID iD: 0000-0003-3204-617X
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.ORCID iD: 0000-0003-4554-6528
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper some results on optimising the Vandermonde determinanton a few different surfaces defined by univariate polynomials are discussed. The coordinates of the extreme points are given as roots of polynomials. Applications in curve-fitting and electrostatics are also briefly discussed.

Keywords [en]
Vandermonde Matrix, Vandermonde Determinant, Orthogonal Polynomials, n-Sphere, p-Norm, D-Optimal Design, Electrostatics
National Category
Mathematical Analysis
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-44569OAI: oai:DiVA.org:mdh-44569DiVA, id: diva2:1329174
Available from: 2019-06-24 Created: 2019-06-24 Last updated: 2019-06-26Bibliographically approved
In thesis
1. Electrostatic Discharge Currents Representation using the Multi-Peaked Analytically Extended Function by Interpolation on a D-Optimal Design
Open this publication in new window or tab >>Electrostatic Discharge Currents Representation using the Multi-Peaked Analytically Extended Function by Interpolation on a D-Optimal Design
2017 (English)Conference paper, Published paper (Refereed)
Abstract [en]

Multi-peaked analytically extended function (AEF), previously applied by the authors to modelling of lightning discharge currents, is used in this paper for representation of the electrostatic discharge (ESD) currents. The fitting to data is achieved by interpolation of certain data points. In order to minimize unstable behaviour, the exponents of the AEF are chosen from a certain arithmetic sequence and the interpolated points are chosen according to a D-optimal design. ESD currents’ modelling is illustrated through two examples: one corresponding to an approximation of the IEC Standard 61000-4-2 waveshape, and the other to representation of some measured ESD current. 

National Category
Computational Mathematics Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-36533 (URN)10.1109/ISEMC.2017.8077985 (DOI)000428753300006 ()2-s2.0-85039159873 (Scopus ID)978-1-5386-2231-5 (ISBN)978-1-5386-2229-2 (ISBN)
Conference
The 2017 IEEE International Symposium on Electromagnetic Compatibility, Signal and Power Integrity, Washington, USA, August 7-11, 2017
Available from: 2017-09-28 Created: 2019-06-24 Last updated: 2018-04-18Bibliographically approved
2. Extreme points of the Vandermonde determinant and phenomenological modelling with power exponential functions
Open this publication in new window or tab >>Extreme points of the Vandermonde determinant and phenomenological modelling with power exponential functions
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis discusses two topics, finding the extreme points of the Vandermonde determinant on various surfaces and phenomenological modelling using power-exponential functions. The relation between these two problems is that they are both related to methods for curve-fitting. Two applications of the mathematical models and methods are also discussed, modelling of electrostatic discharge currents for use in electromagnetic compatibility and modelling of mortality rates for humans. Both the construction and evaluation of models is discussed.

In the first chapter the basic theory for later chapters is introduced. First the Vandermonde matrix, a matrix whose rows (or columns) consists of monomials of sequential powers, its history and some of its properties are discussed. Next, some considerations and typical methods for a common class of curve fitting problems are presented, as well as how to analyse and evaluate the resulting fit. In preparation for the later parts of the thesis the topics of electromagnetic compatibility and mortality rate modelling are briefly introduced.

The second chapter discusses some techniques for finding the extreme points for the determinant of the Vandermonde matrix on various surfaces including spheres, ellipsoids and cylinders. The discussion focuses on low dimensions, but some results are given for arbitrary (finite) dimensions.

In the third chapter a particular model called the p-peaked Analytically Extended Function (AEF) is introduced and fitted to data taken either from a standard for electromagnetic compatibility or experimental measurements. The discussion here is entirely focused on currents originating from lightning or electrostatic discharges.

The fourth chapter consists of a comparison of several different methods for modelling mortality rates, including a model constructed in a similar way to the AEF found in the third chapter. The models are compared with respect to how well they can be fitted to estimated mortality rate for several countries and several years and the results when using the fitted models for mortality rate forecasting is also compared.

Place, publisher, year, edition, pages
Västerås: Mälardalen University, 2019
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 293
National Category
Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-44579 (URN)978-91-7485-431-2 (ISBN)
Public defence
2019-09-26, Delta, Mälardalens högskola, Västerås, 13:15 (English)
Opponent
Supervisors
Available from: 2019-06-24 Created: 2019-06-24 Last updated: 2019-08-09Bibliographically approved

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Muhumuza, Asaph KeikaraLundengård, KarlÖsterberg, JonasSilvestrov, Sergei

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Citation style
  • apa
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