Extreme points of the vandermonde determinant in numerical approximation, random matrix theory and financial mathematics / Asaph Keikara Muhumuza.

Contents;

1. Introduction
1.1 Historical background
1.2 Vandermonde determinant and symmetric polynomials
1.3 Orthogonal polynomials
1.4 Applications and occurences of the vandermonde matrix and its determinants
1.5 Random matrix theory
etc.

2. The generalized Vandermonde interpolation polynomial based on divided differences
2.1 Generalized divided differences and Vandermonde determinant
2.2 Weighted Fekete points and Fekete polynomials
2.3 Weighted Lebegue constant and Lebegue function
2.4 The optimizatiom of Gaussian ensembles as weighted Fekete points
2.5 Fitting interpolating polynomial to experimental data

3. Extreme points of the Vandermonde determinant on surfaces implicitly determined by a univariate polynomial
3.1 Extreme points of the Vandermonde determinant on surfaces defined by low degree univariate polynomial
3.2 Critical points on the sphere defined by a p-norm
3.3 Some results for cubes and intersections of planes

4. Symmetric group properties of extreme points of vandermonde determinant and schur polynomials
4.1 Symmetric group properties of Vandermonde matrix and its determinants
4.2 Derivatives, extreme points of Vandermode determinant and schur polynomials
4.3 The extreme points of schur polynomials on certain surfaces
4.4 The extreme points of Vandermonde determinant, schur polynomial and maximum szego limit theorem
4.5 Interpolation with extreme points of schur polynomial

5. Optimiization of the Wishart joint eigenvalue probability density distribution based on the Vandermonde determinant
5.1 The vandermonde determinant and joint eigenvalue probability densities for random matrices
5.2 Optimising the joint eigenvalue probability density function

6. Properties of the extreme points of the joint eigenvalue probability density function of the Wishart type matrix
6.1 Polynomial factorization of the vandemande matrix and Wishart matrix
6.2 Matrix norm of the Vandemonde and Wishart matrices
6.3 Condition number of the Vandemonde and Wishart matrix

7. Connections between the extreme points of Vandemonde determinants and minimising risk measure in financial mathematics
7.1 Pricing with points Vandemonde determinant
7.2 Optimum value of value of generalized variance with extreme points of Vandemonde determinant

8. The Wishart distribution on symmetric cones
8.1 The Wishart ensembles on symmetric cones
8.2 Lassalle measure on symmetric cones and probability distribution
8.3 Degenerate Wishart ensembles on symmetric cones

9. Extreme points of the vandemonde determinant and Wishart ensembles on symmetric cones
9.1 The Gindikin set and Wishart joint eigenvalue distribution
9.2 A quick jump into Wishart distribution on symmetric cones
9.3 Extreme points of the degenerate Wishart distribution and vandemonde determinant

References : p. 300-333 . _ Index : p. 334