On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

Linear preservers of orthogonal decomposable Tensors

Let G be a subgroup of the symmetric group S_m and V be an n-dimensional unitary space where nm. Let V(G) be the symmetry class of tensors over V associated with G and the identity character. Let D(G) be the set of all decomposable elements of V(G) and O(G) be its subset consisting of all nonzero decomposable tensors x ₁ ^?…^? x_m such that {x ₁,…,x_m } is an orthogonal set. In this paper we study the structure of linear mappings on V(G) that preserve one of the following subsets: (i)O(G), (ii) D(G)\(O(G)?{0}).     

Dimension inequalities for the closure of the image of a multilinear function

In this note we give a simple proof and an extension of a dimension inequality of Howard concerning the range of a multilinear function with vector space range by using some results on algebraic varieties.     

Linear preservers of decomposable symmetric tensors

Let U be an n-dimensional vector space over an algebraically closed field F. Let U(^m) denote the mth symmetric power of U. For each positive integer k≤min{m,n}, let D_k denote the set of all nonzero decomposable elements x₁ …x_m in U(^m) such that dim(x₁ …x_m ) = k and E_k denote the set of all decomposable elements x₁ …x_m in U(^m) such that dim(x₁ …x_m ) ≤ k. In this paper we first show that E_k is an algebraic variety with D_k as a dense subset and determine the dimension of E_k . We next use these results to study the structure of linear mappings T on U^m such that T(D_k ) ? D_k or T(E_k ) ? E_k for some fixed k.     

Dimension inequalities for the closure of the image of a multilinear function

In this note we give a simple proof and an extension of a dimension inequality of Howard concerning the range of a multilinear function with vector space range by using some results on algebraic varieties.     

The group inverse of a companion matrix

A complete characterization is given for the group inverse of a companion matrix over an arbitrary ring to exist. Formulae are given for the actual group inverse and some consequences are drawn.     

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

Ideals associated with realcompactness in pointfree function rings

Abstract

Let RL denote the ring of continuous real-valued functions on a com- pletely regular frame L. The support of an α ∈ RL is the closed quotient ↑(coz α)?. We show that if supports are coz-quotients in L, then the set of functions with realcompact support is an ideal. If L satisfies the stronger condition that supports are C-quotients, then this ideal is the intersection of pure parts of the free maximal ideals of RL. The set of functions whose cozeroes are realcompact is always an ideal, which is free if and only if L is locally realcompact if and only if L is (isomorphic to) an open quotient of υL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only if υL → L is a one-point extension of L.     

The group inverse of a companion matrix

A complete characterization is given for the group inverse of a companion matrix over an arbitrary ring to exist. Formulae are given for the actual group inverse and some consequences are drawn.     

Orthogonality of cosets relative to irreducible characters of finite groups

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

On the apollonian metric of domains in

We study the apollonian metric considered for sets in ? _n by Beardon in 1995. This metric was first introduced for plane Jordan domains by Barbilian in 1934. For a special class of plane domains Beardon showed that conformal apollonian isometries are Möbius transformations. We give here a proof of Beardon's result without conformality assumption. We show that the apollonian metric of a domain D is either conformal at every point of D, at only one point of D or at no point of D. We also present a suprising relation between convex bodies of constant width and the apollonian metric.     

Subordinants of differential superordinations

Let Ω be any set in the complex plane ?, let p be analytic in the unit disk U and let ψ (r, s, t; z). In this article we consider the problem of determining properties of functions p that satisfy the differential superordination Ω ? {ψ(p(z), z ² p"(z);z)|z ∈ U}.     

Characteristic polynomials of symmetric matrices III: some counterexamples

When is a monic polynomial the characteristic polynomial of a symmetric matrix over an integral domain D? Known necessary conditions are shown to be insufficient when D is the field of 2-adic numbers and when D is the rational integers. The latter counterexamples lead to totally real cubic extensions of the rationals whose difierents are not narrowly equivalent to squares. Furthermorex³-4x+1 is the characteristic polynomial of a rational symmetric matrix and is the characteristic polynomial of an integral symmetric p-adic matrix for every prime p, but is not the characteristic polynomial of a rational integral symmetric matrix.     

On the sequences of the Weyl semi-radii associated with matricial Carathéodory and Schur sequences in both nondegenerate and degenerate cases

In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur.