A 2-complex K is called almost-acyclic if H2(K) = 0 and H1(K) is torsion-free. This class of complexes was introduced in a previous paper (2), and applied to a problem of J. H. C. Whitehead concerning aspherical 2-complexes. In this note, the methods developed in (2) are used to study the finitely-generated subgroups of the fundamental group of an almost-acyclic 2-complex.

Let {Ai; i ∈ Ω} be a family of C*-algebras acting on a Hubert space H and let A be the C*-algebra that they generate. We shall assume throughout that C*-algebras always contain the identity operator. Let M(A) denote the space of characters, that is, multiplicative linear functionals, acting on A, with the weak star topology. We obtain here a natural characterisation of M(A) as a subset of the product space determined by {M(Ai); i ∈ Ω}. In the case of singly generated C*-algebras this characterisation is related to the joint normal spectrum (5) of a family of operators.

One of the aims of this paper is to examine the following conjecture, attributed to Mahowald on p. 255 of (2), Part 2. Let M be a closed connected smooth manifold of odd dimension q (q≠l,3,7) and with tangent bundle τ. Let the inclusion of a compactified fibre in the Thorn complex of τ be written μ: Sq → Tτ.

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X; when E is the real or complex field this space will be denoted by C(X). The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.

Consider the equation

where is the generalised derivative of x defined as follows: for every t ≧ T

.

In this note we are concerned with the permutability of congruence relations on semilattices and lattices with pseudocomplementation. There are some results in the literature along these lines. For example, in (8) H. P. Sankappanavar characterises those pseudocomplemented semilattices whose congruence lattice is modular and employs the result in conjunction with the well-known fact that algebras with permuting congruences are congruence-modular to characterise those pseudocomplemented semilattices with permuting congruences. Our first result is a direct, short proof of his result. In (2), J. Berman shows that for all congruences on a distributive lattice L with pseudocomplementation to permute it is necessary and sufficient that D(L), the dense filter of L, be relatively complemented. Our second result is a generalisation of that result to an important equational class of lattices with pseudocomplementation which properly contains the modular lattices with pseudocomplementation.

A formally self-adjoint differential operator L is said to be of limit circle type at infinity if its highest order coefficient is zero-free and all solutions x of L(x) = 0 are square-integrable on [a, ∞). (We will drop reference to “at infinity” in what follows.)

For the second-order case

Dunford and Schwartz (3) p. 1409 prove that given

then L is of limit circle type if and only if

Let R ⊊ T be an extension of commutative rings having the same identity. A. Wadsworth (10) studies the situation when R and T are integral domains, and all rings between R and T are Noetherian. In this case (R, T) is called a Noetherian pair. In a similar vein, E. Davis (4) studies normal pairs and I. Papick (8) shows when coherent pairs are Noetherian pairs.

In this paper we prove that if the strong dual of an echelon space fulfils the Mackey convergence condition the echelon space is quasi-normable. Also we give a characterisation of the quasi-normable echelon spaces and we deduce that every non-quasi-normable echelon space is the strong dual of a non-complete (LB)-space.

A partially ordered set, is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.