We consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

The main result is that d(Ssm) = n+2 for every finite non-abelian two-generator simple group S of order s and every integer n > 0. This is applied to give a very close estimate on d(Gn) for any finite group G whose simple images are two-generator. The article is based on the author's previous papers with similar titles.

Let δ be a root system and let V be the Hahn group of real-valued functions on δ Then δ can be order-embedded into P(δ), the root system of prime l-ideals of V. In this note we identify P(δ) in terms of δ without explicit reference to V, up to the convex subgroup structure of the additive groups of real closed η1-fields. In particular, we characterize the minimal prime 1-ideals of V in terms of δ by an ultrafilter construction which generalizes the well-known method when δ is trivially ordered.

If L is any semilattice, let TL denote the Munn semigroup of L, and Aut (L) the automorphism group of L.

We show that every semilattice L can be isomorphically embedded as a convex subsemilattice in a semilattice L' which has a transitive automorphism group in such a way that (i) every partial isomorphism α of L can be extended to an automorphism of L', (ii) every partial isomorphism: α: eL → fL of L can be extended to a partial isomorphism αL′: eL′ → fL′ of L′ such that TL → TL′, α → αL′ embeds TL' isomorphically in TL′, (iii) every automorphism γ of L can be extended to an automorphism γL′ of L′ such that Aut (L) → Aut (L′), γ → γL embeds Aut (L) isomorphically in Aut (L′).

A group G is termed conjugacy separable (c.s.) if any pair of distinct conjugacy classes may be mapped to distinct conjugacy classes in some finite epimorph of G. The free product of A and B with cyclic amalgamated subgroup H is shown to be c.s. if A and B are both free, or are both finitely generated nilpotent groups. Further, one-relator groups with nontrivial center and HNN extensions with c.s. base group and finite associated subgroups are also c.s.

In this paper, we define the function space H1P(G) on a LCA group G with the algebraically ordered dual, and construct a multiplier on H1P(G) similar to the one given by Gaudry (1968).

In this paper we investigate near-rings of polynomials and polynomial functions. After some results which belong to universal algebra we turn our attention to the familiar case of polynomials and polynomial functions over a commutative ring with identity. We study the relation between ring- and near-ring homomorphisms, and the behaviour of polynomial near-rings when the ring splits into a direct sum. A discussion of the structure of these polynomial near-rings (radical, semisimplicity) finishes this paper. These investigations are motivated by Clay and Doi (1973).

This paper gives characterizations of functionally compact spaces in terms of filterbases and nets. Also, a topological property that is weaker than countable compactness but stronger than first countable minimality is investigated.

In a manuscript discovered in 1976 by George E. Andrews, Ramanujan states a formula for a certain continued fraction, without proof. In this paper we establish formulae for the convergents to the continued fraction, from which Ramanujan's result is easily deduced.

We consider, on an Archimedean Riesz space, the spaces of all linear operators lying between two multiples of the identity (for the order), those leaving all ideals invariant and the order bounded orthomorphisms. We find, if E is uniformly complete, necessary and sufficient conditions for all such operators defined on sublattices of E to extend to the whole of E. Examples are given to show the role of uniform completeness. For the space of all orthomorphisms we give a sufficient condition on E for such an extension to exist.

Gelfand-type duality results can be obtained for locally convex algebras using a quasitopological structure on the spectrum of an algebra (as opposed to the topologies traditionally considered). In this way, the duality between (commutative, with identity) C*-algebras and compact spaces can be extended to pro-C*-algebras and separated quasitopologies. The extension is provided by a functional representation of an algebra A as the algebra of all continuous numerical functions on a quasitopological space. The first half of the paper deals with uniform spaces and quasitopologies, and has independent interest.