The space βℕ is the Stone-Čech compactification of the discrete space of positive integers. The set of elements of βℕ which are in the kernel of every continuous homomorpnism from βℕ to a topological group is a compact semigroup containing the idempotents. At first glance it would seem a good candidate for the smallest such semigroup. We produce an infinite nested sequence of smaller such semigroups all defined naturally in terms of addition on ℕ.

We obtain some new results about the maximal operator space structure which can be put on a normed space. These results are used to prove some dilation results for contractive linear maps from a normed space into B(H). Finally, we prove CB(MIN(X), MAX(y)) = Γ2(X, Y) and apply this result to prove some new Grothendieck-type inequalities and some new estimates on spans of “free” unitaries.

For distributive double p-algebras, a close connection is established between being congruence coherent and congruence regular. Every congruence regular distributive double p-algebra is congruence coherent. Even though every congruence coherent distributive double p-algebra that has either a non-empty core or finite range is congruence regular, an example is given that is congruence coherent but not congruence regular.

In this paper we are dealing with oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the form

Some sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) is considered also.

We consider the finite-dimensional Banach spaces ℓp(n), where p>l. On these spaces there is a unique homogeneous semi-inner-product [.,.] consistent with the norm. If p≠2 this semi-inner product is not symmetric. We define a pair of vectors x and y to be biorthogonal if [x, y] = [y, x] = 0. For a given non-zero x, let τ(X) be the number of elements in a maximal linearly independent set of vectors biorthogonal to x. If p = 2 it is well-known that this number is n–1. The aim of this paper is to find τ(X) when p≠2. Our investigation shows that the situation differs from the Euclidean case in that the value of τ(X) can be either n–l or n –2. The ‘exceptional’ vectors x for which τ(x) = n –2 are characterised.

The author has proved that an outer function in the Hardy space H1 can be factored into a product in which one factor is strongly outer and the other is the sum of two inner functions. In an endeavor to understand better the latter factor, we introduce a class of functions containing sums of inner functions as a special case. Using it, we describe the solutions of extremal problems in the Hardy spaces Hp for 1≦p<∞.

The initial-boundary value problem for the nonlinear heat equation u1 = Δu + λf(u) might possibly have global classical unbounded solutions, for some “critical” initial data . The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x;λ) for some values of λ We find, for radial symmetric solutions, that u*(r, t)→w(r) for any 0<r≤l but supu*(·, t) = u*(0, t)→∞, as t→∞. Furthermore, if , where is some such critical initial data, then û = u(x, t; û0) blows up in finite time provided that f grows sufficiently fast.

In this paper we consider the quartic diophantine equation 3(y2 – 1) = 2x2(x2 – 1) in integers x and y. We show that this equation does not have any other solutions (x, y) with x≧0 than those given by x = 0,1,2,3,6,91. Two approaches are emphasized, one based on diophantine approximation techniques, the other depends on the structure of certain quartic number fields.

In this paper we generalize recent results of Kreer and Penrose by showing that solutions to the discrete Smoluchowski equations

with general exponentially decreasing initial data, with density p, have the following asymptotic behaviour

where J = {j: cj(t)>0, t>0} and q = gcd{j: cj(0)>0}.

We show that every continuous nest of bounded multiplicity is unitarily equivalent to itself in a non-trivial way. Along the way, it is shown that no finite (measurable) partition of the unit interval can separate absolutely continuous homeomorphisms.

Groups called amalgamated sums that arise as inductive limits of systems of groups and injective homomorphisms are studied. The problem is to find conditions under which the groups in the system do not collapse in the limit. Such a condition is given by J. Tits when certain subsystems are associated to buildings. This condition can be phrased to apply to certain systems of abstract groups and injective homomorphisms. It is shown to imply that no collapse occurs in the limit in a strong sense; namely the natural homomorphism of the amalgamated sum of any subsystem into the amalgamated sum of the full system is injective. This answers a question of S. J. Pride.

If G is a hyperfinite locally soluble group and A an artinian ZG-module then Zaĭcev proved that A has an f-decomposition. For G being a hyper-(cyclic or finite) locally soluble group, Z. Y. Duan has shown that any periodic artinian ZG-module A has an f-decomposition. Here we prove that: if G is a hyper-(cyclic or finite) group, then any artinian ZG-module A has an f-decomposition.

The structure of completely prime ideals in any structural matrix near-rings is determined. Partial descriptions are obtained for prime, nil, nilpotent, and locally nilpotent ideals of structural matrix near-rings. Their associated radicals are also studied in this paper.

Introducing the technique of subharmonic functions, we prove that the local spectrum Spu(λ)(T) is almost constant if u is an analytic family of vectors and if the spectrum of T is thin, a result which is similar to the finite-dimensional situation. We apply this result to improve a former result of C. Foiaş [7] on generalized scalar operators and results of C. Foiaş and F. -H. Vasilescu [8] on generalized commutators.

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

Let be a polynomial with distinct real zeros whose separation is defined by δ(f) = min i≠j(ai-aj ). We establish upper estimates for δ(f′-kf) in terms of n, k, and δ(f). The results give sufficient conditions for the inverse operator (D – kl)−1 to preserve the reality of the zeros of a polynomial.

In this paper we present an existence result for a class of nonlinear evolutions inclusions. A result on the compactness of the solution set for a differential inclusion is also established.

The conjecture in question is that the proportion of the first n positive integers which are quadratic residues of an arbitrary prime p is bounded below by a positive. δ. This is established here as a corollary of a more general result concerning multiplicative functions; the problem of the sharp δ is left open.

A semigroup is said to be completely regular if and only if each of its elements lies in a subgroup. It is shown that the algebra of a completely regular monoid (semigroup with identity) over a field of characteristic zero is directly finite.

Let G be a Moore group. Then, for each f∈L1(G), the convolution operator Lf: L1(G)→L1(G) is decomposable. On the other hand, there is a discrete probability measure µ on a compact group G such that Lµ: Ll(G)→Ll(G) fails to be decomposable.