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

Doctoral thesis and Licentiate seminars

The public defense of Karl Lundengård's licentiate thesis in Mathematics/Applied Mathematics will take place at Mälardalen University on March 23, 2017, at 13.15 PM in room Kappa, Västerås.

**The title of the thesis is** “Generalized Vandermonde matrices and determinants in electromagnetic compatibility”.

The examining committee consists of Professor Dietrich von Rosen, Swedish University of Agricultural Sciences; Professor Dragan Poljak; University of Split; Professor Anders Logg, Chalmers University of Technology. Among the members of the examining committee, Professor Dietrich von Rosen has been appointed the faculty examiner.

Reserve: Docent Nenad Cvetković, University of Niš.

The Licentiate thesis has serial number 253

## Abstract

This licentiate thesis discusses to different topics, optimisation of the Vandermonde determinant over different volumes in various dimensions and how certain class of functions can be used to approximate the current in electrostatic discharges used in ensuring electromagnetic compatibility. An example of how the two topics can be connected is given in the final part of the thesis.

A Vandermonde matrix is a matrix with rows (or column) are given by increasing powers and such matrices appear in many different circumstances, both in abstract mathematics and various applications. In the thesis a brief history of the Vandermonde matrix is given as well as a discussion of some applications of the Vandermonde matrix and some related matrices. The main topics will be interpolation and regression which can be described as methods for fitting a mathematical description to collected data from for example experimental measurements.

The determinant of a matrix is a number calculated from the elements of the matrix in a particular way and it can describe properties of the matrix or the system it describes in a compact way. In this thesis it is discussed how to choose the elements of the Vandermonde matrix to maximise the determinant under the constraint that the elements that define the Vandermonde determinant are interpreted as points in a certain volume (that can have a dimension higher than three). Examined volumes include spheres, cubes, ellipsoids and tori.

One way to motivate the usefulness of knowing how to maximise the Vandermonde determinant is that it can be used in \emph{optimal experiment design}, that is determining how to choose the data points in an experiment to construct the best possible mathematical model. An example of how to do this can be found in the final section of the thesis.

One area where it is useful to construct mathematical model from experimental data is electromagnetic compatibility. This is the study if how to ensure that a system that contains electronics is not disturbed too much by external electromagnetic disturbances or disturbs other systems when it is used. An important aspect of this field is examining how systems respond to different external disturbances described in different construction standards. In this thesis we discuss how a specific class of functions can be used to construct mathematical models based on specifications in standards or experimental data. Well-known phenomenon such as lightning strikes and electrostatic discharges from a human being to a metal object are discussed.