The purpose for this short note is to describe the space of harmonic spinors on hyperbolic n-space Hn. This is a natural continuation of the study of harmonic functions on Hn in [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms on Hn were considered. The connection between invariant differential equations on a Riemannian semisimple symmetric space G/K and homological aspects of the representation theory of G, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

The authors consider the nonlinear neutral delay differential equation

and obtain results on the asymptotic behavior of solutions. Some of the results require that P(t) has arbitrarily large zeros or that P(t) oscillates about — 1

We give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.

We give a simple characterization of coefficient multipliers from the mixed norm space Hp,q,α, 2 ≤ p ≤ ∞, into Hu,v,β, 0 ≤ u ≤ 2, which includes the main results of Wojtaszczyk in [5]. We also calculate multipliers from the Hardy space Hp, 2 ≤ p ≤ ∞, into Hq, 0 < q ≤ 2.

An analog of the Hopf bifurcation theorem is proved for implicit neutral functional differential equations of the form F(xt, D′(xt, α), α) = 0. The proof is based on the method of S1-degree of convex-valued mappings. Examples illustrating the theorem are provided.

We derive a criterion for a generalized free product of groups to be cyclic subgroup separable. We see that most of the known results for cyclic subgroup separability are covered by this criterion, and we apply the criterion to polygonal products of groups. We show that a polygonal product of finitely generated abelian groups, amalgamating cyclic subgroups, is cyclic subgroup separable.

We study the asymptotic behavior of the n-widths of a class of entire functions in weighted approximation on subsets of the complex plane.

Compact right topological groups arise naturally as the enveloping semigroups of distal flows. Recently, John Pym and the author established the existence of Haar measure μ on such groups, which invites the consideration of the regular representations. We start here by characterizing the continuous representations of a compact right topological group G, and are led to the conclusion that the right regular representation r is not continuous (unless G is topological). The domain of the left regular representation l is generally taken to be the topological centre

or a tractable subgroup of it, furnished with a topology stronger than the relative topology from G (the goals being to have l both defined and continuous). An analysis of l and r on H = L2(G) for some non-topological compact right topological groups G shows, among other things, that:

1. (i) for the simplest (perhaps) G generated by ℤ, (l, H) decomposes into one copy of each irreducible representation of ℤ and c copies of the regular representation.

2. (ii) for the simplest (perhaps) G generated by the euclidean group of the plane , (l, H) decomposes into one copy of each of the continuous one-dimensional representations of and c copies of each continuous irreducible representation Ua,a > 0.

3. (iii) when Λ(G) is not dense in G, it can seem very reasonable to regard r as a continuous representation of a related compact topological group, and also, G can be almost completely "lost" in the measure space (G, μ).

We give characterizations of Toeplitz operators on generalised H2 spaces and derive some properties of the corresponding Toeplitz algebras. The proofs depend essentially on having a "sufficient" supply of inner functions.

A sequence of ideals Ik,n ⊆ BP* is introduced, with the property: Ik,n ⊆ Ann(γk,n), where γk,n is the toral class of the Brown-Peterson homology of the n-fold product BZ/pk × ··· × BZ/pk. These ideals seem to play an interesting and yet unclear role in understanding Ann(γk,n). They are defined by using the formal group law of the Brown-Peterson spectrum BP, and some of their elementary properties are established. By using classical theorems of Landweber and of Ravenel-Wilson, the author computes the radicals of Ik,n and Ann(γk,n), and discusses a few examples.

A commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

The non-separable Banach space of right continuous functions with left hand limits and the supremum norm is investigated to find the isomorphic types of complemented subspaces. It is shown that every isometric isomorph of c0 is complemented in this space which may be identified as a non-separable C(K) space. Sufficient conditions are given for other isomorphs of C0 to be complemented in the space and the complement of a C0 subspace is characterized isomorphically.

A known consequence of the theorem of the algebraic subgroup is a lower bound for the rank of matrices whose entries are linear combinations, with algebraic coefficients, of logarithms of algebraic numbers. We extend this kind of result to commutative algebraic groups.

Unlike for a smooth projective hypersurface, for an isolated hypersurface singularity, the pole order and Hodge filtrations do not in general coincide. This note studies the difference between the two.

The Hurwitz zeta function ζ(s, a) is defined by the series

for 0 < a ≤ 1 and σ = Re(s) > 1, and can be continued analytically to the whole complex plane except for a simple pole at s = 1 with residue 1. The integral functions C(s, a) and S(s, a) are defined in terms of the Hurwitz zeta function as follows:

Using integral representations of C(s, a) and S(s, a), we evaluate explicitly a class of improper integrals. For example if 0 < a < 1 we show that