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A number of models from mathematics, physics, probability theory and statistics can be described in terms of Wishart matrices and their eigenvalues. The most prominent example being the Laguerre ensembles of the spectrum of Wishart matrix. We aim to express extreme points of the joint eigenvalue probability densitydistribution of a Wishart matrix using optimisation techniques for the Vandermondedeterminant over certain surfaces implicitly defined by univariate polynomials.

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.

The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Grobner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.

The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed visually. The extreme points of the ordinary Vandermonde determinant on finite-dimensional unit spheres are given as the roots of rescaled Hermite polynomials and a recursion relation is provided for the polynomial coefficients. Analytical expressions for these roots are also given for dimension three to seven. A transformation of the optimization problem is provided and some relations between the ordinary and generalized Vandermonde matrices involving limits are discussed.

There are many models for the mortality rates of humans and other organisms. A phenomenon that complicates the modelling of human mortality rates is a rapid increase in mortality rate for young adults (in western Europe this is especially pronounced at the age of 25). We will examine models for mortality rates based on power-exponential functions, compare them to empirical data for mortality rates and other models.

In this paper the Analytically Extended Function (AEF) with p peaks is used for representation of the electrostatic discharge (ESD) currents and lightning discharge 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. The method is illustrated using several examples of currents taken from standards and measurements.

According to experimental results for lightning discharge currents, they are classified in the IEC 62305 Standard into waveshapes representing the first positive, first and subsequent negative strokes, and long-strokes. These waveshapes, especially shot-term pulses, are approximated with a few mathematical functions in literature, in order to be used in lightning discharge models for calculations of electromagnetic field and lightning induced effects. An analytically extended function (AEF) is presented in this paper and used for lightning currents modeling. The basic properties of this function with a finite number of peaks are examined. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks as well as for the given charge transfer and specific energy is described. This framework is used to find parameters for some common single-peak wave-forms and some advantages and disadvantages of the approach are also discussed.

An examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms, is presented in the paper. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks is presented. This framework is used to find parameters for some common waveforms with a single peak, such as Standard IEC 62305 lightning currents. Illustration of fitting a p-peak AEF to recorded lightning current data is also presented.

There are many models for mortality rates. A well-known problem that complicates modeling of human mortality rates is the “accident hump” occurring in early adulthood. Here, two models of mortality rate based on power-exponential functions are presented and compared to a few other models. The models will be fitted to known data of measured death rates from several different countries using numerical techniques for curve-fitting with the nonlinear least-squares method. The properties of the model with respect to forecasting with the Lee–Carter method will be discussed.