Let G1 and G2 be two locally compact abelian groups and let 1 ≤ p ∞. We prove that G1 and G2 are isomorphic as topological groups provid∈d there exists a bipositive or isometric algebra isomorphism of M(Ap (G1)) onto M(Ap (G2)). As a consequence of this, we prove that G1 and G2 are isomorphic as topological groups provided there exists a bipositive or isometric algebra isomorphism of Ap (G1) onto Ap (G2). Similar results about the algebras L1 ∩ Lp and L1 ∩ C0 are also established.

The object of this paper is to investigate under very general conditions the existence and mean-square stability of a random solution of a class of stochastic integral equations in the form

for t ≥ 0, where a random solution is a second order stochastic process {x(t; w) t ≥ 0} which satisfies the equation almost certainly. A random solution x(t; w) is defined to be stable in mean-square if E[|x(t; w)|2] ≤ p for all t ≥ 0 and some p > 0 or exponentially stable in mean-square if E[|x(t; w)|2] ≤ pe-at, t ≥ 0, for some constants ρ > 0 and α > 0.

Let κ and λ be cardinal numbers. Take any family A = {Aν; ν ∈ N} where each Aν is a product Aν = Bν x Cν with |Bν = |Cν| = Nα, such that if B x CAμ x Aν (for μ ≠ ν) then |B|, |c| < λ. We investigate under what conditions on α, κ, λ and |N| there will be a set T with 1 ≤ |T∩Aν| < κ for each ν.

Operators are defined that yield basic Galois closure operations for almost every category. These give rise to a new and more general approach for characterization of epireflective subcategories, and construction of epireflective hulls. As a by-product, satisfactory characterizations of classes of perfect morphisms and ω-extendable epimorphisms are obtained. Detailed proofs and examples will appear elsewhere.

Certain topological spaces X may bear various uniform structures compatible with the topology of X; to each uniform structure there corresponds a completion of X, that is, a complete space Z containing X as a dense subspace. For compact completions, there has been extensive study of the relationship between X and the possible remainders Z\X. This paper begins a study of the more general, and apparently easier, problem of the relationship between X and its not necessarily compact remainders. We find that for spaces X admitting a complete metric, every space Y which satisfies certain conditions obviously necessary for Y to be the remainder of a completion of X in fact occurs as such a completion.

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

It is shown that any sheaf of R˜-modules, all of whose stalks are injective, is necessarily a flabby sheaf. This generalizes the result or Grothendieck that the sheaf M˜ determined by an injective module M over a commutative noetherian ring with 1 is flabby.

The following conditions for a group G are investigated:

(i) maximal class n subgroups are normal,

(ii) normal closures of elements have nilpotency class n at most,

(iii) normal closures are n–Engel groups,

(iv) G is an (n+1 )-Engel group.

Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1. Here the question is settled for n = 2 as follows: conditions (ii), (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.

Using the characterization of the free inverse semigroup F on a set X, given by Scheiblich, a necessary and sufficient condition is found for a subset K of an inverse semigroup S to be a set of free generators for the inverse sub semigroup of S generated by K. It is then shown that any non-idempotent element of F generates the free inverse semigroup on one generator and that if |X| > 2 then F contains the free inverse semigroup on a countable number of generators. In addition, it is shown that if |X| = 1 then F does not contain the free inverse semigroup on two generators.

We give the representation formula of a function in a disc in terms of its arithmetic means on the boundary. We also give applications of this formula to the solutions of some elementary differential equations with prescribed means of boundary or initial data.

Let (X, T) be a topological space (we assume T1. throughout) where every point is a limit point. The purpose of this note is to present an internal construction of a maximal perfect topology on (X, T). The existence of a maximal connected Hausdorff space has not been demonstrated. However, this construction of a maximal perfect topology is useful in constructing connected Hausdorff spaces which cannot be embedded in a maximal connected Hausdorff space.

It is shown that, in a variety which does not contain all metabelian groups and is contained in a product of (finitely many) varieties each of which is soluble or locally finite, every group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent.

The question has been raised [R.J. Loy, Bull. Austral. Math. Soc. 5 (1970), 253–260] as to whether the existence of a bounded (left) approximate identity in the tensor product A ⊗αB of Banach algebras A and B (for a a crossnorm on A ⊗ B ) implies the existence of a bounded (left) approximate identity in A and B. This is known [David A. Robbins, Bull. Austral. Math. Soc. 6 (1972), 443–445] to be the case for α equal to the greatest crossnorm. This paper answers the general question affirmatively.