We are concerned with the following problem. Let F be a Fréchet Montel space and let E be a Fréchet space with a certain property (P). When does it follow that the complete projective tensor product has the property (P)? (We consider the following properties: being Montel, reflexive, satisfying the density condition.) In this paper we provide a positive answer if F is a Montel generalized Dubinsky sequence space with decreasing steps.

The kernel–trace approach to congruences on a regular semigroup S can be refined by introducing the left and right traces. This induces eight operators on the lattice of congruences on S: t1, k, tr,; Tt, K, Tr; t, T. We describe the lattice of congruences on S generated by six 3-element subsets of the set {ωt1, ωk, ωtr, εTt, εK, εTr} where ω and ε denote the universal and the equality relations. This is effected by means of a diagram and in terms of generators and relations on a free distributive lattice, or a homomorphic image thereof. We perform the same analysis for the lattice of congruences on S generated by the set {εK, ωk, εT, ωt}.

Implicit operations (new operations commuting with all old homomorphisms) on pseudovarieties have been shown to play an important role in the study of these classes. They may be used to axiomatize sub-pseudovariaties and to describe recognizable subsets of (relatively) free objects. This paper presents a case study for the pseudovariety CS consisting of all finite simple semigroups. Based on a result of profinite group theory, a structural description of semigroups of implicit operations on finite simple semigroups is used to deduce that CS is join-irreducible.

Let R be a ring and G a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and G, when the group ring R[G] is a prime Krull order and when it is a price v-HC order. The key ingredient in obtaining both characterizations is the first author's earlier study of height one prime ideals in the ring R[G[.

An isomorphism (as groups) is established between an arbitrary connected module over a structural matrix near-ring and a direct sum of appropriate modules over the base near-ring. This isomorphism leads to a characterization of the 2-primitive ideals of a structural matrix near-ring.

A ring R is prime essential if R is semiprime and for each prime ideal P of R, P ∩ I ≠0 whenever I is a nonzero two-sided ideal of R. Examples of prime essential rings include rings of continuous functions and infinite products modulo infinite sums. We show that the class of prime essential rings is closed under many familiar operations; in particular, we consider polynomial rings, matix rings, fixed rings and skew group rings. Also, we explore the relationship between prime essential rings and special radical classes, and we demonstrate how prime essential rings can be used to construct radical classes which are not special.

Using a classical result of Nagata, Achilles, Huneke and Vogel gave a criterion for the Stückrad-Vogel multiplicity to take the value one. We use Huneke's extension of Nagata's theorem to give a necessary condition for the Stückrad-Vogel multiplicity to have an arbitrary preassigned bound, under certain conditions. A usable criterion of multiplicity n results (given mild hypotheses). We also revisit some basic results in the Stückrad-Vogel theory in the light of the behaviour of tensor products of affine primary rings, and also revisit some arguments of Achilles, Huneke and Vogel from the point of view of fibre rings.

A semiband is defined as a semigroup generated by idempotents. It is known that every finite semigroup is embeddable in a finite semiband. For a class C of semigroups and an integer n≧2, the number σC (n) is defined as the smallest k with the property that every semigroup of order n in the class C is embeddable in a semiband of order not exceeding k. It is shown that for the class Gp of groups σGp(n) = nq(ρGp(n)), where

and

Estimates are known (and are quoted) for the function q. Estimates are considered for the function pC for various C

It is shown also that if C0S, CS denote respectively the classes of completely 0-simple and completely simple semigroups, then

Although it is well-known that tempered distributions on ℝn can be expanded into series of Herrnite functions, it does not seem to be known, however, that expansions of this type are accessible through the elementary concept of orthonorma! expansions in Hilbert space. This approach is developed here complementing previous work on a Hilbert space approach to distributions. The basis of the development is the observation that the Hermite functions are a complete orthogonal set in each space of a certain scale of Sobolev type Hilbert spaces associated with the family of differential operators defined by

Here Ф denotes a smooth function with compact support. The setting is first developed in the one-dimensional case. By use of the usual multi-index notation this can be extended to the higher-dimensional case. As applications various imbedding results are derived. The paper concludes with a characterization of tempered distributions by convergent Hermite expansions.

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

An action of a group G on a tree, and an associated Lyndon length function l, give rise to a hyperbolic length function L and a normal subgroup K having bounded action. The Theorem in Section 1 shows that for two Lyndon length functions l, l′ to arise from the same action of G on some tree, L = L′ and K = K′. Moreover for L non-abelian L = L′ implies K = K′. That this is not so for abelian L is shown in Section 2 where two examples of Lyndon length functions l, l′ on an H.N.N. group are given, with their associated actions on trees, for which L = L′ is abelian but K≠K′.

Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:

Let R have its natural involution * in which each element of P is Hermitian. We show that the Banach *-algebra ℓ1(R) has a separating family of finite dimensional *-representations and consequently is *-semisimple. This generalizes a result of B. A. Barnes and J. Duncan (J. Funct. Anal.18 (1975), 96–113.) dealing with the case where P has two elements.

We calculate K0 of the rational group algebra of a certain crystallographic group, showing that it contains an element of order 2. We show that this element is the Euler class, and use our calculation to produce a whole family of groups with Euler class of order 2.