— In this paper, we consider the problem of deriving new eigenvalue distributions of real-valued Wishart matrices that arises in many scientific and engineering applications. The distributions are derived using the tools from the theory of skew symmetric matrices. In particular, we relate the multiple integrals of a determinant, which arises while finding the eigenvalue distributions, in terms ...

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric different...

Let P = f" + (I ,,) , the direct sum of the p x p identity matrix and the negative of the q x q ident ity matrix. The fo llowing theorem is proved. TH EOHEM: If X = cZ where Z is a 4 x 4 P-orthogonal , P-skew-symmetric matrix and Ie I .;;; 2, there exist matrices A an.d B, both of which are P-orthogollal and P-skew-symmetric, sach that X = AB BA. Methods for o btaining certain matrices which sa...

Given an undamped gyroscopic system GðλÞ 1⁄4 Mλ þ CλþK with M , K symmetric and C skew-symmetric, this paper presents a real-valued spectral decomposition of GðλÞ by a real standard pair ðX;TÞ and a skew-symmetric parameter matrix S . When T is assumed to be a block diagonal matrix, the parameter matrix S has a special structure. This spectral decomposition is applied to solve the quadratic inv...

We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute b...

We consider the question: can the isotropy representation of an irreducible pseudoRiemannian symmetric space be realized as a conformal holonomy group? Using recent results of Čap, Gover and Hammerl, we study the representations of SO(2, 1), PSU(2, 1) and PSp(2, 1) as isotropy groups of irreducible symmetric spaces of signature (3, 2), (4, 4) and (6, 8), respectively, describing the geometry in...

We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition i...