# The public defense of Karl Lundengård's doctoral thesis in Mathematics/Applied Mathematics

Doctoral thesis and Licentiate seminars

The public defense of Karl Lundengård´s doctoral thesis in Mathematics/Applied Mathematics will take place at Mälardalen University, room Delta, at 13.15 on September 26, 2019.

**Title:** Extreme points of the Vandermonde determinant and phenomenological modelling with power exponential functions

**Serial number:** 293

The faculty examiner is Professor Thomas Curtright, University of Miami, and the examining committee consists of Professor Gradimir Milovanovic, Mathematical Institute of SASA, Professor Mikhail Kotchetov, Memorial University, Professor Blas Torrecillas, University of Almeria.

Reserve; Professor Christos Skiadas, Technical University of Crete

**Abstract:**

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. Such models are called phenomenological models. In this thesis phenomenological models are constructed for two phenomena, electrostatic discharge currents and mortality rates.

Electrostatic discharge currents are rapid flows of electric charge from one object to another. Well known examples are lightning strikes or small electric chocks caused by static electricity. Describing such currents is important when engineers want to ensure that electronic systems are not disturbed too much by external electromagnetic disturbances or disturbs other systems when used.

Mortality rate describes the probability of a dying at a certain age. It can be used to assess the quality of life in a country or for other demographical or actuarial purposes.

For electrostatic discharge currents and mortality rates an important feature that can be challenging to model is a steep increase followed by a slower decrease. This pattern is often observed in electrostatic discharge currents and in many countries the mortality rate increases rapidly in the transition from childhood to adulthood and then changes slowly until the beginning of middle age.

In this thesis a mathematical function called the power-exponential function is used as a building block to construct phenomenological models of electrostatic discharge currents and mortality rates based on empirical data for the respective phenomena. For electrostatic discharge currents a methodology for constructing models with different accuracy and complexity is proposed. For the mortality rates a few simple models are suggested and compared to previously suggested models.

The thesis also discusses the extreme points of the Vandermonde determinant. This is a mathematical problem that appears in many areas but for this thesis the most relevant application is that it helps choosing the appropriate data to use when constructing a model using a technique called optimal design. Some general results for finding the extreme points of the Vandermonde determinant on various surfaces, e.g. spheres or cubes, and applications of these results are discussed.