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Sergei Silvestrov - One of the best experts on this subject based on the ideXlab platform.
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Extreme Points of the Vandermonde Determinant on the Sphere and Some Limits Involving the Generalized Vandermonde Determinant
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Karl Lundengård, Jonas Österberg, Sergei SilvestrovAbstract: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.
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Extreme points of the Vandermonde Determinant on surfaces implicitly determined by a univariate polynomial
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract: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.
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Optimization of the Wishart Joint Eigenvalue Probability Density Distribution Based on the Vandermonde Determinant.
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract: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.
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properties of the extreme points of the joint eigenvalue probability density function of the wishart matrix
ASMDA2019 18th Applied Stochastic Models and Data Analysis International Conference, 2019Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract:We will examine some properties of the extreme points of the probability density distribution of the Wishart matrix using properties of the Vandermonde Determinant and show examples of applications ...
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Optimization of the Determinant of the Vandermonde Matrix and Related Matrices
Methodology and Computing in Applied Probability, 2017Co-Authors: Karl Lundengård, Jonas Österberg, Sergei SilvestrovAbstract: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.
Karl Lundengård - One of the best experts on this subject based on the ideXlab platform.
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Extreme Points of the Vandermonde Determinant on the Sphere and Some Limits Involving the Generalized Vandermonde Determinant
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Karl Lundengård, Jonas Österberg, Sergei SilvestrovAbstract: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.
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Extreme points of the Vandermonde Determinant on surfaces implicitly determined by a univariate polynomial
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract: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.
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Optimization of the Wishart Joint Eigenvalue Probability Density Distribution Based on the Vandermonde Determinant.
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract: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.
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Extreme points of the Vandermonde Determinant and phenomenological modelling with power exponential functions
2019Co-Authors: Karl LundengårdAbstract:This thesis discusses two topics, finding the extreme points of the Vandermonde Determinant on various surfaces and phenomenological modelling using power-exponential functions. The relation betwee ...
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properties of the extreme points of the joint eigenvalue probability density function of the wishart matrix
ASMDA2019 18th Applied Stochastic Models and Data Analysis International Conference, 2019Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract:We will examine some properties of the extreme points of the probability density distribution of the Wishart matrix using properties of the Vandermonde Determinant and show examples of applications ...
Cristina Ballantine - One of the best experts on this subject based on the ideXlab platform.
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Powers of the Vandermonde Determinant, Schur functions, and the dimension game
2013Co-Authors: Cristina BallantineAbstract:Abstract. Since every even power of the Vandermonde Determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function sµ in the decomposition of an even power of the Vandermonde Determinant in n + 1 variables in terms of the coefficient of the Schur function sλ in the decomposition of the same even power of the Vandermonde Determinant in n variables if the Young diagram of µ is obtained from the Young diagram of λ by adding a tetris type shape to the top or to the left. Résumé. Comme toute puissance paire du déterminant de Vandermonde est un polynôme symétrique, nous voulons comprendre sa décomposition dans la base des fonctions de Schur. Nous allons étudier plusieurs propriétés combinatoires des coefficients de la décomposition. En particulier, nous allons donner une approche récursive pour le calcul du coefficient de la fonction de Schur sµ dans la décomposition d’une puissance paire du déterminant de Vandermonde en n + 1 variables, en fonction du coefficient de la fonction de Schur sλ dans la décomposition de la même puissance paire du déterminant de Vandermonde en n variables, lorsque le diagramme de Young de µ est obtenu à partir du diagramme de Young de λ par l’addition d’une forme de type tetris vers le haut ou vers la gauche
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Powers of the Vandermonde Determinant, Schur functions and recursive formulas
Journal of Physics A: Mathematical and Theoretical, 2012Co-Authors: Cristina BallantineAbstract:The decomposition of an even power of the Vandermonde Determinant in terms of the basis of Schur functions matches the decomposition of the Laughlin wavefunction as a linear combination of Slater wavefunctions and thus contributes to the understanding of the quantum Hall effect. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function sμ in the decomposition of an even power of the Vandermonde Determinant in n + 1 variables in terms of the coefficient of the Schur function sλ in the decomposition of the same even power of the Vandermonde Determinant in n variables if the Young diagram of μ is obtained from the Young diagram of λ by adding a tetris type shape to the top or to the left.
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Powers of the Vandermonde Determinant, Schur Functions, and recursive formulas
Journal of Physics A: Mathematical and Theoretical, 2012Co-Authors: Cristina BallantineAbstract:Since every even power of the Vandermonde Determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function $s_{\m}$ in the decomposition of an even power of the Vandermonde Determinant in $n + 1$ variables in terms of the coefficient of the Schur function $s_{\l}$ in the decomposition of the same even power of the Vandermonde Determinant in $n$ variables if the Young diagram of $\m$ is obtained from the Young diagram of $\l$ by adding a tetris type shape to the top or to the left. An extended abstract containing the statement of the results presented here appeared in the Proceedings of FPSAC11
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Powers of the Vandermonde Determinant, Schur functions, and the dimension game
Discrete Mathematics & Theoretical Computer Science, 2011Co-Authors: Cristina BallantineAbstract:Since every even power of the Vandermonde Determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde Determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde Determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left.
Jonas Österberg - One of the best experts on this subject based on the ideXlab platform.
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Extreme Points of the Vandermonde Determinant on the Sphere and Some Limits Involving the Generalized Vandermonde Determinant
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Karl Lundengård, Jonas Österberg, Sergei SilvestrovAbstract: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.
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Extreme points of the Vandermonde Determinant on surfaces implicitly determined by a univariate polynomial
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract: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.
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Optimization of the Wishart Joint Eigenvalue Probability Density Distribution Based on the Vandermonde Determinant.
Springer Proceedings in Mathematics & Statistics, 2020Co-Authors: Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg, Sergei Silvestrov, John Magero Mango, Godwin KakubaAbstract: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.
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Optimization of the Determinant of the Vandermonde Matrix and Related Matrices
Methodology and Computing in Applied Probability, 2017Co-Authors: Karl Lundengård, Jonas Österberg, Sergei SilvestrovAbstract: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.
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Optimization of the Determinant of the Vandermonde matrix on the sphere and related surfaces
2015Co-Authors: Karl Lundengård, Jonas Österberg, Sergei SilvestrovAbstract: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.
Itai Ben Yaacov - One of the best experts on this subject based on the ideXlab platform.
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2016Co-Authors: Itai Ben Yaacov, Xd X, Xdd XAbstract:Abstract. We generalise the (projective) Vandermonde Determinant identity to projective dimension higher than one. The Vandermonde Determinant identity asserts that in any commutative ring A, det 1 x0...
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A multivariate version of the Vandermonde Determinant identity
arXiv: Commutative Algebra, 2014Co-Authors: Itai Ben YaacovAbstract:We give a multivariate version of the Vandermonde Determinant identity, measuring whether a family of points in projective space are in as general a position as possible.
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a higher dimensional version of the Vandermonde Determinant identity
2014Co-Authors: Itai Ben YaacovAbstract:We generalise the (projective) Vandermonde Determinant identity to projective dimension higher than one.
