The Experts below are selected from a list of 196224 Experts worldwide ranked by ideXlab platform
Hiroki Hashiguchi - One of the best experts on this subject based on the ideXlab platform.
-
heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta wishart matrix
2021Co-Authors: Koki Shimizu, Hiroki HashiguchiAbstract:Abstract This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the derivation of certain distributions for the Eigenvalues of singular beta-Wishart matrices. The joint density function of the Eigenvalues and the distribution of the largest eigenvalue can be expressed in terms of heterogeneous hypergeometric functions. Exact computation of the distribution of the largest eigenvalue is conducted for real and complex cases.
-
generalized heterogeneous hypergeometric functions and the distribution of the largest eigenvalue of an elliptical wishart matrix
2021Co-Authors: Aya Shinozaki, Koki Shimizu, Hiroki HashiguchiAbstract:In this study, we derive the exact distributions of Eigenvalues of a singular Wishart matrix under an elliptical model. We define generalized heterogeneous hypergeometric functions with two matrix arguments and provide convergence conditions for these functions. The joint density of Eigenvalues and the distribution function of the largest eigenvalue for a singular elliptical Wishart matrix are represented by these functions. Numerical computations for the distribution of the largest eigenvalue were conducted under the matrix-variate $t$ and Kotz-type models.
Jean Lagace - One of the best experts on this subject based on the ideXlab platform.
-
From Steklov to Neumann via homogenisation
2020Co-Authors: Alexandre Girouard, Antoine Henrot, Jean LagaceAbstract:We study a new link between the Steklov and Neumann Eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann Eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann Eigenvalues from known isoperimetric bounds for Steklov Eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.
-
large steklov Eigenvalues via homogenisation on manifolds
2020Co-Authors: Alexandre Girouard, Jean LagaceAbstract:Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov Eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace Eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov Eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev's upper bound of $8\pi$ for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two Eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than $2\pi$. This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus 0 in the unit ball with even larger area. The first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they verify a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois--El Soufi--Girouard is sharp and implies an upper bound for weighted Laplace Eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.
Hashiguchi Hiroki - One of the best experts on this subject based on the ideXlab platform.
-
Expressing the largest eigenvalue of a singular beta F-matrix with heterogeneous hypergeometric functions
2021Co-Authors: Shimizu Koki, Hashiguchi HirokiAbstract:In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of Eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of a nonsingular beta F -matrix to the singular case. We also give the joint density of Eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functions.Comment: The title is changed (the old title is "The exact distribution of the largest eigenvalue of a singular beta F-matrix for Roy's test"
-
Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix
2020Co-Authors: Shimizu Koki, Hashiguchi HirokiAbstract:This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the derivation of certain distributions for the Eigenvalues of singular beta-Wishart matrices. The joint density function of the Eigenvalues and the distribution of the largest eigenvalue can be expressed in terms of certain heterogeneous hypergeometric functions. Exact computation of the distribution of the largest eigenvalue is conducted here for a real case
David E Tyler - One of the best experts on this subject based on the ideXlab platform.
-
on wielandt s inequality and its application to the asymptotic distribution of the Eigenvalues of a random symmetric matrix
1991Co-Authors: Morris L Eaton, David E TylerAbstract:A relatively obscure eigenvalue inequality due to Wielandt is used to give a simple derivation of the asymptotic distribution of the Eigenvalues of a random symmetric matrix. The asymptotic distributions are obtained under a fairly general setting. An application of the general theory to the bootstrap distribution of the Eigenvalues of the sample covariance matrix is given. 1. Introduction and summary. The derivation of the asymptotic distribution of the Eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis. Although the main idea behind most of the derivations is quite basic, i.e., the expansion of the sample roots about the population roots, the derivations themselves are often quite involved. These complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem. One of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices. The method is based upon a relatively obscure eigenvalue inequality
Lei Zhang - One of the best experts on this subject based on the ideXlab platform.
-
multi input partial eigenvalue assignment for high order control systems with time delay
2016Co-Authors: Lei ZhangAbstract:Abstract In this paper, we consider the partial eigenvalue assignment problem for high order control systems with time delay. Ram et al. (2011) [1] have shown that a hybrid method can be used to solve partial quadratic eigenvalue assignment problem of single-input vibratory system. Based on this theory, a rather simple algorithm for solving multi-input partial eigenvalue assignment for high order control systems with time delay is proposed. Our method can assign the expected Eigenvalues and keep the no spillover property. The solution can be implemented with only partial information of the Eigenvalues and the corresponding eigenvectors of the matrix polynomial. Numerical examples are given to illustrate the efficiency of our approach.
