We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

We shall solve the equation

where 0 <a<b, and f(x) is a continuous function on the interval (a, b).

A block design is a finite set p of elements called points, where |p| = v, together with certain distinguished subsets of p called blocks, such that

(i) each block contains k points,

(ii) each point is contained in r blocks, and

(iii) two distinct points are contained in precisely λ blocks.

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

We exhibit a large class K* of real 2 × 2 matrices of determinant ±1 such that, for nearly all A and B in K*, the group generated by A and B1 (the transpose of B) is the free product of the cyclic groups It is shown that K* contains all matrices of determinant ±1 with integer entries satisfying |b| > |a|, |c|, |d|. This gives a generalization of a theorem of Goldberg and Newman [2]. We also prove related results concerning the dominance of b and the discreteness of the free products .

In [1], we introduced the notion of multiplicative forms on associative algebras of finite rank over integral domains D, and obtained a complete classification when D ⊆, the complex field. We propose here to remove the hypothesis of associativity, using a refinement of the technique of Schafer [2]. In [l], it was noted that multiplicative forms extend uniquely under the adjunction of an identity when is associative but not unitary; this appears difficult to verify in the general case, so that some mild restriction on is required. We shall assume that is biregular, that is that contains elements eL, eR such that the linear maps x eL x and x xeR, are bijective on We can then (§1) reduce the biregular case to the unitary case, which is handled in §2.

In their paper “ The enumeration of tree-like polyhexes”, Harary and Read [6] consider structures obtained by assembling hexagons subject to certain restrictions. Their problem is introduced as a simplified hexagonal cell-growth problem.

In his recent study of free inverse semigroups, Munn [2] introduced and used extensively the concept of a word-tree. In this note the number of such trees is found.

A permutation group is quasiregular if it acts regularly on each of its orbits (i.e. the stabiliser of an element fixes every other element in its orbit). So, in particular, any permutation representation of an abelian or hamiltonian group must be quasiregular.

In a recent paper, Boudjelkha and Diaz [1] have considered the solutions of the Dirichlet problems for Laplace's equation

and for Helmholtz's equation

in¼–ℝn. The purpose of this brief note is to show that their formulae may be derived easily by the use of the theory of multiple Fourier transforms.

If R is a local (Noetherian) ring, it is well known that R is regular if and only if its completion is regular. It is the purpose of this note to show that a similar result is true for Noether lattices.

Let γ and γ' be non-negative integers. We say that the graph G is (γ, γ') bi-embeddable if G can be embedded in a surface of genus γ and the complement Ḡ of G can be embedded in a surface of genus γ'. Let N(γ, γ') be the least integer such that every graph with at least N(γ, γ') points is not (γ, γ') bi-embeddable. It has been shown in [1] and [5] that N(0, 0) = 9; this result was also obtained by John R. Ball of the Carnegie Institute of Technology. Our object here is to obtain upper and lower bounds for N(γ, γ').

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

Let be a C*-algebra acting on the Hilbert space H and let be the self-adjoint elements of . The following characterization of commutativity is due to I. Kaplansky (see Dixmier [3, p. 58]).

Throughout the following note R will denote an associative ring with unit element 1. We shall denote by R-mod [resp. mod-R] the category of all unitary left [resp. right] R-modules. Morphisms in these categories will be written as acting on the side opposite scalar multiplication. All other functions will be written as acting on the left. If is a category, we shall abuse notation and write “A∈ when we mean “A is an object of ”.

A (v, k, λ)-configuration, also called a symmetric balanced incomplete block design, is an arrangement of v distinct objects called points or varieties into v subsets called lines or blocks such that each line contains exactly k points and each pair of distinct lines contains exactly λ points in common. To avoid certain trivial configurations, one assumes that 0<λ<k<v–1.

1. Equation (4) should read

2. On p. 83 below equation (7) replace

3. On p. 83 two lines above equation (9) replace