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In this paper we discuss the extension of the Wishart probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions in the field of Herm(m,C), Herm(m,H), Herm(3,O) that denotes respectively the Jordan algebra of all Hermitian matrices of size m× m with complex entries, the skew field H of quaternions, and the algebra O of octonions. This density is characterised by the Vandermonde determinant structure and the exponential weight that is dependent on the trace of the given matrix.

The extreme points of Vandermonde determinants when optimized on surfaces like spheres and cubes have various applications in random matrix theory, electrostatics and financial mathematics. In this study, we apply the extreme points Vandermonde determinant when optimized on various surfaces to risk minimization in financial mathematics. We illustrate this by constructing the efficient frontiers represented by spheres, cubes and other general surfaces as applies to portfolio theory. The extreme points of Vandermonde determinant lying on such surfaces as efficient frontier would be used to determine the set of assets with minimum risk and maximum returns. This technique can also be applied in optimal portfolio selection and asset pricing.

In this paper we demonstrate the extreme points of the Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras. The extreme of points of theVandermonde4 determinant are defined to be a set of boundary points of the symmetric cones that occur in both the discrete and continuous part of the Gindikin set. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart ensembles in Herm(m,C), Herm(m,H), Herm(3,O) denoting respectively the Jordan algebra of all Hermitian matrices of size m × m with complex entries, the skew field H of quaternions, and the algebra O of octonions.

We will examine some properties of the extreme points of the probability densitydistribution of the Wishart matrix, using properties of the Vandermonde determinantand showing examples of the applications of these properties.

The problem of optimising the Vandermonde determinant on a few different surfaces defined by univariate polynomials is discussed. The coordinates of the extreme points are given as roots of polynomials. Applications in curve fitting and electrostatics are also briefly discussed.

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.

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 density distribution of a Wishart matrix using optimisation techniques for the Vandermonde determinant over certain surfaces implicitly defined by univariate polynomials.

Electric vehicles (EVs) poses new challenges for the Distribution System Operator (DSO). For example, EVs uses power electronic-based rectifiers for charging their batteries, an operation that could significantly impact Power Quality (PQ) in terms of harmonic distortion. The DSO responsibilities include ensuring grid code compliance confirmed by PQ metering. In general, 10 min rms values are sufficient. However, the large scale integration of non-linear loads, like EVs, could lead to new dynamic phenomena, possibly lost in the process of time aggregation.This paper presents an analysis of the impact on time aggregation (3 s-, 1min-and 10 min rms), when modelling current harmonics of aggregated EV loads, using power exponential functions. The results indicate that, while3 s rms and 1 min rms marginally affect the outcome, 10 min rms aggregation will lead to a significant deviation (>30%) in terms of maximum current harmonic magnitude. 

Electrostatic discharges (ESD) consequences are not only damages, but also very fast transients in electric circuits and devices, as well as in large power systems. This paper presents frequency spectrum analysis of the IEC 61000-4-2 standard current approximated by multi-peaked analytically extended function (MP-AEF) and of typical lightning currents according to the IEC 62305 standard. This analysis is important for the choice of protective devices and frequency range of measurement equipment used in experiments. A comparison of the measured and theoretical results is also given.

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.