The present work is concerned with an integration by parts formula for the Pk-integral of De Sarkar and Das, and of the equivalent Pk-integral of Bullen. The process involves a simpler and updated version of that for the Zk−1-integral of Bergin. If f is Pk – (Zk−1)-integrable and G is of bounded kth variation, then fG is Pk – (Zk−1)-integrable.

Let R be a ring with prime radical P. The main theorems of this paper are (1) The following conditions are equivalent.: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x3 = x. (2) If R is a PI-ring in which every element is the sum of two idempotents, then R/P satisfies the identity x3 = x. (3) Let R be a semi-perfect ring in which every element is the sum of two idempotents. If RRR is quasi-projective, then R is a finite direct sum of copies of GF(2) and/or GF(3).

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 obtain some explicit formulae for series of the type

where f is an entire function of exponential type τ, bounded on the real exis (and satisfying in the first case). These series are expressed in terms of the derivatives of f and Bernoulli numbers. We examine the case where f is a trigonometric polynomial which lead us, in particular, to a new representation of the associated Fejér mean.

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.

In this paper we give another version of the generalised dominated convergence theorem, which is better than other convergence theorems for Perron integrals in the sense that it can be applied more easily.

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

We find conditions for which a manifold endowed with an integrable almost transversal structure is the total space of a vector bundle isomorphic to the transversal bundle of a foliation.

In this note a general solution of the problem of the characterisation of quadratic functionals posed by Vukman is given.

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.

In this paper we deal with nonlinear second order boundary value problems for ordinary differential equations including the case of jumping nonlinearities. The set of generalised eigenvalues in the case of nonconstant coefficients is described. It is proved that these generalised eigenvalues are simultaneously bifurcation points of the problem with coefficients also depending on the solution u = u(x).

We give an extension, to Orlicz-Sobolev spaces, of a theorem of Marcus and Mizel on the demicontinuity of Nemitsky operators on Sobolev spaces.

We prove the analogue of the Baer Criterion for injectivity in the category AbShℒ of abelian groups in a topos of sheaves on a locale, that is, we show A is injective in AbShℒ if and only if it is injective relative to all S ↣ Zℒ where Zℒ is the group of integers in Shℒ. for a well-ordered locale we describe the injective hulls in AbShℒ in terms of injective hulls in Ab. Further we show that the global functor A → AE preserves injective hulls if and only if ℒ is a finite boolean locale. Finally We characterise injectives in AbShℒ for some special locales.

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.

In this paper, we prove that if h is a generator of a Z2n, action on S1 × S1 × S1, and Fix(hn) consists of two disjoint tori, one torus, four simple closed curves, or two simple closed curves, then h is equivalent to the obvious actions.

We show that, if two finitely generated nilpotent groups are elementarily equivalent, and more generally if they satisfy the same sentences with one alternation of quantifiers, then, they are quite similar, though not necessarily isomorphic. For instance, each of them is isomorphic to a subgroup of finite index of the other and they have the same finite images, the same Mislin genus and the same Pickel genus.

Suppose the edges of the complete graph on 17 vertices are coloured in three colours. It is shown that at least five monochromatic triangles must aris.

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.

In the investigation of factorised groups one often encounters groups G = AB = AK = BK which have a triple factorisation as a product of two subgroups A and B and a normal subgroup K of G. It is of particular interest to know whether G satisfies some nilpotency requirement whenever the three subgroups A, B and K satisfy this same nilpotency requirement. A positive answer to this problem for the classes of nilpotent, hypercentral and locally nilpotent groups is given under the hypothesis that K is a minimax group or G has finite abelian section rank. The results become false if K has only finite Prüfer rank. Some applications of the main theorems are pointed out.

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.