The aim of this paper is to continue our investigation of the Lebesgue function of weighted Lagrange interpolation by considering Erdős weights on ℝ and weights on [−1, 1]. The main results give lower bounds for the Lebesgue function on large subsets of the relevant domains.

Lagrangian necessary and sufficient conditions for a nonsmooth vector-valued minimax in terms of Clarke's generalized Jacobians are established under suitable invexity hypotheses.

We consider the space L1 (ν, X) of all real functions that are integrable with respect to a measure v with values in a real Fréchet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of l∞ in the space of all linear continuous operators from a complete DF-space Y to a Fréchet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure ν and the space X that imply that L1 (ν, X) has the Dunford-Pettis property. Applications of these results to Fréchet AL-spaces and Köthe sequence spaces are also given.

The Plancherel formula for various semisimple homogeneous spaces with non-reductive stability group is derived within the framework of the Bonnet Plancherel formula for the direct integral decomposition of a quasi-regular representation. These formulas represent a continuation of the author's program to establish a new paradigm for concrete Plancherel analysis on homogeneous spaces wherein the distinction between finite and infinite multiplicity is de-emphasized. One interesting feature of the paper is the computation of the Bonnet nuclear operators corresponding to certain exponential representations (roughly those induced from infinite-dimensional representations of a subgroup). Another feature is a natural realization of the direct integral decomposition over a canonical set of concrete irreducible representations, rather than over the unitary dual.

A nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset X of R we have xXy ≠ 0 whenever 0 ≠ x, y ∈ R. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n = 1. This paper is devoted primarily to an investigation of uniform bounds of primeness in matrix rings over fields. It is shown that the existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the n × n matrix ring over F.

Groups, all proper factor-groups of which are hypercentral of finite torsion-free rank, are studied in this article.

This is an expanded version of two lectures given at the conference held at Sydney University in December 1997 on the 50th anniversary of the death of G. H. Hardy.

Let A ∈ ℒ(Cn) and A1, A2 be the unique Hermitian operators such that A = A1 + i A2. The paper is concerned with the differential structure of the numerical range map nA: x ↦ ((A1x, x), (A1x, x)) and its connection with certain natural subsets of the numerical range W(A) of A. We completely characterize the various sets of critical and regular points of the map nA as well as their respective images within W(A). In particular, we show that the plane algebraic curves introduced by R. Kippenhahn appear naturally in this context. They basically coincide with the image of the critical points of nA.

The algebra consisting of those linear transformations of a complex inner product space that have a formal adjoint is shown to possess a special involution. Two earlier results concerning special involutions are then generalized.