We present a direct computation of the Rhodes expansions of the free objects in the varieties of bands, based in the manipulation of the invariants in introduced by Gerhard and Petrich [4] in the study of bands.

We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

Rings of invariants are identified for some automorphisms θ of certain iterated skew polynomial rings R, including the enveloping algebra of sl2(k), the Weyl algebra A1 and their quantizations. We investigate how finite-dimensional simple R-modules split over the ring of invariants Rθ and how finite-dimensional simple Rθ-modules extend to R.

We prove three results concerning the oscillation near a ray of solutions to (*)w″ + Aw = 0, where A is an entire function. The first result assumes that A is a polynomial and gives an upper bound on the number of its real zeros if (*) admits a solution with only real zeros and infinitely many. The second result proves that for A of finite order a solution w to (*) has “few” zeros “near” a ray if and only if the same is true for w′. The third result involves the density of the zeros of a solution to (*) “away” from a finite set of rays.

A Banach space X is called an HT space if the Hilbert transform is bounded from Lp(X) into Lp(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in Lp(X), 1 < p < ∞. Berkson, Gillespie and Muhly [5] showed that X ∈ HT ⇒ X ∈ ACF. In this note, we will show that X ∈ ACF ⇒ X ∈ UMD, thus providing a new proof of Bourgain's result X ∈ HT ⇒ X ∈ UMD.

We use Priestley's duality to characterize, via their dual space, the distributive double p-algebras on which all congruences are principal.

Existence results are established for the equation y″ + f(t, y) = 0, 0<t<1. Here f may be singular in y and f is allowed to change sign. Our boundary data include y(0) = y′(1) + ky(1) = 0, k> – 1 and y(0) = y′(1) + cy4(1) = 0, c>0.

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.

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<∞.

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}.

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.

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.

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.