The purpose of this paper is to present a brief proof of the well known result of Herstein which states that any Jordan derivation on a prime ring with characteristic not two is a derivation.

We say that a certain property in a Banach space E is stable by subspaces if every closed subspace of E enjoys the same property. It is well known that the Radon-Nikodym property is stable by subspaces while the Weak Radon-Nikodym property is not. In his recent memoir, Talagrand investigated the stability of the Weak*Radon-Nikodym property which is a generalization of the Weak Radon-Nikodym property and showed that under Axiom L, the Weak*Radon-Nikodym property is stable by subspaces. It is still an open problem whether or not this result holds without this extra set theoretical hypothesis. In this paper we show that in a dual Banach space, the Weak*Radon-Nikodym property is stable by subspaces without assuming Axiom L. Other related results are discussed.

We construct all Singleton arrays for the field GF(q) when q is odd. There exist φ(q − 1) arrays in this case.

We establish under nonuniform nonresonance conditions an existence and uniqueness theorem for a linear, and the solvability for a nonlinear, fourth-order boundary value problem which occurs frequently in plate deflection theory.

A linear complementarity problem, involving a given square matrix and vector, is generalised by including an element of the subdifferential of a convex function. The existence of a solution to this nonlinear complementarity problem is shown, under various conditions on the matrix. An application to convex nonlinear nondifferentiable programs is presented.

This paper characterises the integral domains R with the property that R/P is integrally closed for each prime ideal P of R. It is shown that Dedekind domains are the only Noetherian domains with this property. On the other hand, each integrally closed going-down domain has this property. Related properties and examples are also studied.

The concept of efficiency (Pareto optimum) is used to formulate duality for multiobjective fractional programming problems. We consider programs where the components of the objective function have non-negative and convex numerators while the denominators are concave and positive. For this case the Mond-Weir extension of Bector dual analogy is given. We also give the Schaible type vector dual. The case where functions are ρ-convex (weakly or strongly convex) is also considered.

It is shown that every minimal power-structure other than the singleton power-structure has at least two contenders; that is, any contender has at least one rival. The safe core of a power-structure is empty if and only if the power-structure is minimal.

This note presents a structure theorem for locally convex barrelled spaces. It is shown that, corresponding to any Hamel basis, there is a natural splitting of a barrelled space into a topological sum of two vector subspaces, one with its strongest locally convex topology. This yields a simple proof that a barrelled space has a dense infinite-codimensional vector subspace, provided that it does not have its strongest locally convex topology. Some further results and examples discuss the size of the codimension of a dense vector subspace.

We study the following Dirichlet problem for the degenerate elliptic Monge-Ampère equation: Given , f ≥ 0 and , find a solution , t ≥ 2, satisfying in Ω and u = g on ∂Ω. Since f is nonnegative, we cannot apply any standard elliptic methods. In this paper, we use an iteration scheme of Nash-Moser type and a priori estimates for degenerate elliptic operators, and solve the Dirichlet problem for a certain class of f and g.

The aim of this paper is to give an inversion theorem for set-valued maps involving both some known results for functions and set-valued maps. To do this we introduce a notion of strict differentiability for set-valued maps and we use a Newton like method assuming the derivative to be surjective. Moreover we prove the pseudo-Lipschitz regularity of the inverse.

A sequence of certain rational functions is determined where each member solves an ℓ2-minimisation problem on a “large” set of roots of unity. The sequence is compared with another sequence of L2-rational approximants. Our main result extends a result of Rivlin on Walsh type equiconvergence.

This paper gives some sufficient conditions under which upper, lower or both upper and lower quasi-continuity of multifunction in the process of semi-regularisation of a topological space are preserved. Analogous results for continuous maps are true.

In this paper we study the oscillatory behaviour of solutions of equations of the forms , where pi, and τi, for i = 1, 2,…, n, are positive constants and p is an arbitary constant. Three necessary and sufficient conditions for all solutions of (A) and (B) respectively to be oscillatory are established. These results extend and generalise previous results in a new approach.

The concept of uniformly strongly prime (usp) is introduced for Γ-ring, and a usp radical τ(M) is defined for a Γ-ring M. If M has left and right unities, then τ(L)+ = τ(M) = τ(R)*, where L and R denote, respectively, the left and right operator rings of M, and τ(·) denotes the usp radical of a ring. If m, n are positive integers, then τ(Mmn) = (τ(M))mn, where Mmn is the matrix Γnm-ring. τ is shown to be a special radical in the variety of Γ-rings. τ1 is the upper radical determined by the class of usp Γ-rings of bound 1. τ ⊆ τ1, but the reverse inclusion does not hold in general. The place of τ and τ1 in the hierarchy of radicals for Γ-rings is shown.

Maximum and minimum values are obtained for some sums invovling the divisor functions of number theory.