Mälardalen University Doctoral Dissertation 293 Karl Lundengård Lundengård Karl Extreme points of the Vandermonde determinant and phenomenological EXTREME POINTS OF THE VANDERMONDE DETERMINANT AND PHENOMENOLOGICAL MODELLING WITH POWER EXPONENTIAL FUNCTIONS modelling with power exponential functions Karl Lundengård Address: P.O. Box 883, SE-721 23 Västerås. Sweden ISBN 978-91-7485-431-2 Address: P.O. Box 325, SE-631 05 Eskilstuna. Sweden E-mail: [email protected] Web: www.mdh.se ISSN 1651-4238 2019 1 Mälardalen University Press Dissertations No. 293 EXTREME POINTS OF THE VANDERMONDE DETERMINANT AND PHENOMENOLOGICAL MODELLING WITH POWER EXPONENTIAL FUNCTIONS Karl Lundengård 2019 School of Education, Culture and Communication 2 Copyright © Karl Lundengård, 2019 ISBN 978-91-7485-431-2 ISSN 1651-4238 Printed by E-Print AB, Stockholm, Sweden 3 Mälardalen University Press Dissertations No. 293 EXTREME POINTS OF THE VANDERMONDE DETERMINANT AND PHENOMENOLOGICAL MODELLING WITH POWER EXPONENTIAL FUNCTIONS Karl Lundengård Akademisk avhandling som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras torsdagen den 26 september 2019, 13.15 i Delta, Mälardalens högskola, Västerås. Fakultetsopponent: Professor Palle Jorgensen, University of Iowa Akademin för utbildning, kultur och kommunikation 4 Abstract 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. ISBN 978-91-7485-431-2 ISSN 1651-4238 5 Acknowledgements Many thanks to all my coauthors and supervisors. My main supervisor, Pro- fessor Sergei Silvestrov, introduced me to the Vandermonde matrix and fre- quently suggested new problems and research directions throughout my time as a doctoral student. I have learned many lessons about mathematics and academia from him and my co-supervisor Professor Anatoliy Malyarenko. My other co-supervisor Dr. Milica Ranˇci´cplayed a crucial role and she is a role model with regards to conscientiousness, work ethic, communication and patience. I have learned invaluable lessons about interdisciplinary re- search, communication and time and resource management from her. I also want to thank Dr. Vesna Javor for her regular input that improved the research on electromagnetic compatibility considerably. Cooperating with other doctoral students was very valuable. Jonas Osterberg¨ and Asaph Keikara Muhumuza (with support from his super- visors Dr. John M. Mango and Dr. Godwin Kakuba) made important contributions to the research on the Vandermonde determinant and Samya Suleiman's understanding of mortality rate forecasting and other aspects of actuarial mathematics was necessary for the work to progress. I am also glad that I had the opportunity to take part in the supervision of talented master students Andromachi Boulogari and Belinda Strass and use the foundations they laid in their degree projects for further research. Many thanks to all my coworkers at M¨alardalenUniversity, especially to Dr. Christopher Engstr¨om,Dr. Johan Richter and Docent Linus Carls- son for managing the bachelor's and master's programmes in Engineering mathematics together with me. Perhaps most importantly, I thank my family for all the support, en- couragement and assistance you have given me. A special mention to my sister for help with translating from 18th century French, it is perfectly un- derstandable that you decided to move to the other side of the Earth after that. I will wonder my whole life how my father, whose entire mathematics career consisted of unsuccessfully solving a single problem on the blackboard in 9th grade, would have reacted to this dissertation if he were still with us. Fortunately my mother continues to be an endless source of support and encouragement. I am continually surprised and delighted over how much of her work ethics, sense of quality and unhealthy work habits I seem to have inherited from her. Without the ideas, requests, remarks, questions, encouragements and patience of those around me this work would not have been completed. Karl Lundeng˚ard V¨aster˚as, September, 2019 3 6 Extreme points of the Vandermonde determinant and phenomenological modelling with power-exponential functions Popul¨arvetenskaplig sammanfattning Det finns m˚angaf¨oreteelser i v¨arldensom det ¨ar¨onskv¨artatt beskriva med en matematisk modell. I b¨astafall kan modellen h¨arledasifr˚anl¨amplig grundl¨aggandeteori men ibland ¨ardet inte m¨ojligtatt g¨oradet, antingen d¨arf¨oratt det inte finns n˚agonv¨alutvecklad teori eller f¨oratt den teori som finns kr¨aver information som inte ¨artillg¨anglig. I detta fall s˚abeh¨ovsen modell som, i n˚agonm˚an,st¨ammer ¨overens med teori och empiriska observa- tioner men som inte ¨arh¨arleddfr˚anden grundl¨aggandeteorin. S˚adana mod- eller kallas f¨orfenomenologiska modeller. I denna avhandling konstrueras fenomenologiska modeller av tv˚aolika fenomen, str¨ommeni elektrostatiska urladdningar och d¨odsrisk. Elektrostatiska urladdningar sker n¨arladdning snabbt fl¨odarfr˚anett objekt till ett annat. V¨albekanta exempel ¨arblixtnedslag eller sm˚ast¨otar orsakade av statisk elektricitet. F¨oringenj¨orer¨ardet viktigt att kunna beskriva denna typ av elektriska str¨ommarf¨oratt se till att elektroniska system inte ¨arf¨ork¨ansligaf¨orelektromagnetisk p˚averkan utifr˚anoch att de inte st¨orandra system d˚ade anv¨ands. D¨odsrisken beskriver sannolikheten f¨ord¨odvid en viss ˚alder.Den kan anv¨andasf¨oratt uppskatta livskvaliteten i ett land eller andra demografiska eller f¨ors¨akringsrelaterade¨andam˚al. En egenskap hos b˚adeelektrostatiska urladdningar och d¨odsrisksom kan vara utmanande att modellera ¨aromr˚adend¨aren brant ¨okningf¨oljs av en l˚angsams¨ankning.S˚adanam¨onsterf¨orekommer ofta i elektrostatiska urladdningar och i m˚angal¨ander¨okar d¨odsrisken kraftigt vid ¨overg˚angen fr˚anbarn till vuxen och f¨or¨andrassedan l˚angsamt fram till tidig medel˚alder. I denna avhandling anv¨andsen matematisk funktion som kallas poten- sexponentialfunktionen som en byggsten f¨oratt konstruera fenomenologiska modeller av str¨ommeni elektrostatiska urladdningar samt d¨odsriskutifr˚an empiriska data f¨orrespektive fenomen. F¨orelektrostatiska urladdningar f¨oresl˚asen metod som kan konstruera modeller med olika noggrannhet och komplexitet. F¨ord¨odsrisker f¨oresl˚asn˚agraenkla modeller som sedan j¨amf¨ors med tidigare f¨oreslagnamodeller. I avhandlingen diskuteras ocks˚aextrempunkterna hos Vandermonde de- terminanten. Detta ¨arett matematiskt problem som f¨orekommer inom flera olika omr˚adenmen f¨oravhandlingen ¨arden mest relevanta till¨ampningen att extrempunkterna kan hj¨alpatill att v¨aljal¨ampligadata att anv¨anda n¨ar man konstruerar modeller med hj¨alpav en teknik som kallas f¨oroptimal design. N˚agraallm¨annaresultat f¨orhur extrempunkterna kan hittas p˚adi- verse ytor, t.ex. sf¨areroch kuber, presenteras och det ges exempel p˚ahur resultaten kan till¨ampas. 4 7 Popular science summary There are many phenomena in the world that it is desirable to describe using a mathematical model. Ideally the mathematical model is derived from the appropriate fundamental theory but sometimes this is not feasible, either because the fundamental theory is not well understood or because the theory requires a lot of information to be applicable. In these cases it is necessary to create a model that, to some degree, matches the fundamental theory and the empirical observations, but is not derived from the fundamental theory.