We prove here that the (saturated) formation generated by a finite soluble group has only finitely many (saturated) subformations. This answers a question asked by Professor W. Gaschütz. Some partial results are also given in the case of a formation generated by an arbitrary finite group.

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

This paper presents an algorithm, motivated by Morse Theory, for the topological configuration of the components of a real algebraic curve {f(x, y) = 0}. The running time of the algorithm is O(n12 (d + log n)2 log n), where n, d are the degree and maximum coefficient size of f(x, y).

In one of their most recent works, George Andrews and Peter Paule continue their study of partition functions via MacMahon's Partition Analysis by considering partition functions associated with directed graphs which consist of chains of hexagons. In the process, they prove a congruence related to one of these partition functions and conjecture a number of similar congruence results. Our first goal in this note is to reprove this congruence by explicitly finding the generating function in question. We then prove one of the conjectures posed by Andrews and Paule as well as a number of congruences not mentioned by them. All of our results follow from straightforward generating function manipulations.

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math.13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,t∈(0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

The equivalence of a fixed point theorem and the Fan-Knaster-Kuratowski-Mazurkiewicz theorem in H-space has been established. The fixed point theorem is then applied to obtain a theorem on sets with H-convex sections, and also results on minimax inequalities.

The present paper deals with some oscillating and asymptotic properties of the functional differential equations of the form

where λ is an arbitrary positive constant and τ > 0 is a constant delay.

In a uniformly convex Banach space the convergence of Ishikawa iterates to a fixed point is discussed for nonexpansive and generalised nonexpansive mappings.

Some characterisations of generalised convex functions are established by means of Clarke's subdifferential and directional derivatives.

We establish a link between regular subgroups of the affine group and radical circle algebras on the underlying vector space.

A ring R with unity is called a (quasi-) Baer ring if the left annihilator of every (left ideal) nonempty subset of R is generated (as a left ideal) by an idempotent. Armendariz has shown that if R is a reduced PI-ring whose centre is Baer, then R is Baer. We generalise his result by considering the broader question: when does the (quasi-) Baer condition extend to a ring from a subring? Also it is well known that a regular ring is Baer if and only if its lattice of principal right ideals is complete. Analogously, we prove that a biregular ring is quasi-Baer if and only if its lattice of principal ideals is complete.

A refinement of the Sobolev imbedding theorem, due to Trudinger, is shown to be optimal in a natural sense.

In various papers on some of the theoretical aspects of computing, the reader is confronted with a mathematical entity called a “multiset”, and is told that a multiset is “an unordered collection of elements that may have repeated occurrences of identical elements” (see, for example, Nachum Dershowitz and Zohar Manna, “Proving termination with multiset orderings”, Lecture Notes in Computer Science, 71). These entities are then manipulated in accordance with the classical laws of set algebra. The purpose of this note is to present a formal foundation for this concept of multiset, and to show by means of examples that if this foundation is adopted then, although certain sections of classical set algebra can be applied to multisets, it certainly cannot be applied in total.

For r-partite and for r-uniform hypergraphs bounds are given for the minimum degree which ensures d independent edges.

It is shown that if a retarded delay differential equation has a global attractor in the space C ([—τ0, ], ℝd) for a given nonzero constant delay τ0, then the equation has an attractor Aτ in the space C ([—τ, 0], ℝd) for nearby constant delays τ. Moreover the attractors Aτ converge upper semi continuously to in C ([—τ0, 0], ℝd) in the sense that they are identified through corresponding segments of entire trajectories in ℝd with nonempty compact subsets of C ([—τ0, 0], ℝd) which converge upper semi continuously to in C ([—τ0, 0], ℝd).

Let Sa (h) denote the class of analytic functions f on the unit disc E with f (0) =0 = f′ (0) −1 satisfying , where (a real), denotes the Hadamard product of Ka with f, and h is a convex univalent function on E, with Re h > 0. Let . It is proved that F ε Sa (h) whenever f ε Sa (h) and also that for a ≥ 1. Three more such classes are introduced and studied here. The method of differential subordination due to Eenigenburg et al. is used.

Until recently most authors have devoted their research to the theory of perturbed systems under continuous perturbations. In this paper, Liapunov's second method is employed to investigate sufficient conditions for integral and integral asymptotic stability of ordinary differential systems with respect to impulsive perturbations.

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.