M. Slater has shown that a prime alternative (not associative) ring R such that 3R≠0 is a Cayley-Dickson ring, [7], That is, if F is the field of quotients of the center, Z, of R then F ⊗Z R is a Cayley-Dickson algebra.

Let d=(d1,…,dn) be a sequence of positive integers. In this note we show that d is realizable by a graph whose square is hamiltonian if and only if (i) d is realizable by some graph, (ii) n≥3, and (iii) d1+…+dn≥2(n-1). In fact, we prove that if d is realizable by a connected graph, then d is realizable by a graph with a spanning caterpillar. From this it follows that if d is realizable by a connected graph, it is realizable by a graph whose square is pancyclic. We also prove that d is realizable by a graph with a spanning wreath if and only if d is realizable by some graph and d1+…+dn≥2n. (A wreath is a connected graph that has exactly one cycle and all vertices not in the cycle monovalent.)

The relationships between synchronous and asynchronous reversible Markov systems are investigated. It is shown that the invariant measure of such systems is a second order Markov random field. The conditions under which the invariant measure is a first order Markov random field are obtained.

A category space is a measure space which is also a topological space, the measure and the topology being related by ‘a set is measurable iff it has the Baire property’ and ‘a set is null iff it is nowhere dense’ [4]. We considered some category spaces in [3]; now we show that if a null set is deleted from the space, then the topology can be taken to be completely regular. The essential part of the construction consists of obtaining a suitable refinement of the original sequential covering class and using the consequent strong upper density function to define the required topology. Then the complete regularity follows much as in [1].

The origin and importance of multiparameter Sturm-Liouville problems in mathematical physics has recently been discussed by Atkinson [1, sects. 3 and 4], [2, introduction]. In spite of the importance of such problems, and in spite of the work done in this field by such early investigators as Klein and Hilbert, Atkinson points out that in recent years this field has been relatively neglected in contrast to the single parameter case. As an example, he states that as opposed to the one parameter case, the detailed behaviour of the eigenvalues and eigenfunctions in the multiparameter Sturm-Liouville case is still far from clear.

Throughout this paper all semigroups will have identity, and all S-systems (operands) will be right unitary S-systems. All homomorphisms will be S-homomorphisms unless specified.

Let M be a differentiable manifold of class C∞, with a given (1, 1) tensor field J of constant rank such that J2=λI (for some real constant λ). J defines a class of conjugate (G-structures on M. For λ>0, one particular representative structure is an almost product structure. Almost complex structure arises when λ<0. If the rank of J is maximum and λ=0, then we obtain an almost tangent structure. In the last two cases the dimension of the manifold is necessarily even. A Riemannian metric S on M is said to be related if one of the conjugate structures defined by S has a common subordinate structure with the G-structure defined by S. It is said to be J-metric if the orthogonal structure defined by S has a common subordinate structure.

We answer the following problem posed by Herrlich in the affirmative: “Can the Freudenthal compactification be regarded as a reflection in a sensible way?” This is accomplished by exploiting the one-to-one correspondence between proximities compatible with a given Tihonov space and compactifications of that space. We give topological characterizations of proximally continuous functions for the proximities associated with the Freudenthal and Fan-Gottesman compactifications.

Some results generalizing a fixed point theorem due to R. Kannan are presented.

0. Let be a family of self-mappings of a metric space (X, d).

Cette note a pour but d’expliciter une structure mathématique spéciale: . Cette structure sera aux fonctions finies (i.e. fonctions d’un ensemble fini dans un autre) ce que l’anneau ℤ des entiers est aux ensembles finis. Nous ne nous servirons des résultats de la théorie des catégories et des topos que comme outils de base pour les constructions considérées. Le lecteur ne doit donc pas s′attendre à des résultats catégoriques fondamentaux mais bien à un nouvel anneau pouvant servir en théorie des nombres, en combinatoire et peut-être même en physique.

The purpose of this note is to investigate boundary points of the numerical range of an operator in terms of inner and outer center points. Some applications on commutators are given.

Throughout this note, an operator will always mean a bounded linear operator on a Hilbert space X.

Boundary value problems for ordinary differential equations have long been the subject of extensive research activity. In particular, questions concerning the existence and uniqueness of solutions for these problems have received much attention, and algebraic fixed-point theorems have served as important tools in such investigations. For example Picard [8] based his pioneering work in this area on the use of successive approximation techniques, and recently his classical methods have been refined and extended to more general nonlinear problems (see [1] and [4]). The standard procedure for applying these techniques requires that the boundary value problem under consideration first be converted into an equivalent integral equation through the choice of a suitable Green’s function. The resulting theory is consequently limited to problems for which such a formulation is possible.

It is well known that if T is a compact self-adjoint operator on a Hilbert space whose distinct non-zero eigenvalues {λn} are arranged so that |λn|≥|λn+1| for n = 1, 2…. and if En in the spectral projection corresponding to λn, then with convergence in the uniform operator topology. With the generalisation of self-adjoint operators on Hilbert spaces to Hermitian operators on Banach spaces by Vidav and Lumer, Bonsall gave a partial analogue of this result for Banach spaces when he proved the following theorem.

This article presents some new algebraic and module theoretic characterizations of strongly regular rings. The latter uses Lambek’s notion of symmetry. Strongly regular rings are shown to admit an involution and form an equational category. An example due to Paré shows that the category of regular rings and ring homomorphisms between them is not equational. Remarks on quasiinverses and the generalized inverse of a matrix are included. The author acknowledges support from the NRC (A7752) and improvements from W. Blair received after announcement of the results.

One of the fundamental tools in the theory of totally ordered groups is Hahn’s Theorem (a detailed discussion may be found in Fuchs [3]), which asserts, roughly, that every abelian totally ordered group can be embedded in a lexicographically ordered (unrestricted) direct sum of copies of the ordered group of real numbers. Almost any general question regarding the structure of abelian totally ordered groups can be answered by reference to Hahn’s theorem. For the class of nonabelian totally ordered groups, a theorem which parallels Hahn’s Theorem is given in [5], and states that each totally ordered group can be o-embedded in an ordered wreath product of subgroups of the real numbers. In order to extend this theorem to include an “if and only if” statement, one must consider lattice ordered groups, as an ordered wreath product of subgroups of the real numbers is, in general, not totally-ordered, but lattice ordered.

All algebras are assumed to be finite dimensional and not necessarily associative. An involution of an algebra is an algebra automorphism of order two. A quasi-division algebra is any algebra in which the non-zero elements form a quasi-group under multiplication. The purpose of this short paper is to determine the structure of all involutions of quasi-division algebras and to give an application of this result.

R. Arens and I. Kaplansky ([1]) call a ring A biregular if every two sided principal ideal of A is generated by a central idempotent and a ring A strongly regular if for any a in A there exists b in A such that a=a2b. In ([1], Sections 2 and 3), a lot of interesting properties of a biregular ring and a strongly regular ring are given. Some more properties can also be found in [3], [5], [8], [9] and [13]. For example, J. Dauns and K. Hofmann ([3]) show that a biregular ring A is isomorphic with the global sections of the sheaf of simple rings A/K where K are maximal ideals of A. The converse is also proved by R. Pierce ([9], Th. 11–1). Moreover, J. Lambek ([5], Th. 1) extends the above representation of a biregular ring to a symmetric module.

The Bell numbers Bn can be defined by Bn= where S(n, k) is t n e Stirling number of the second kind. In this note we employ a technique developed by Rota (which formalizes the umbral calculus) to derive a veriety of facts concerning the related numbers and polynomials .

Borsuk has asked whether there exists for each compact metric absolute neighbourhood retract X an integer l (depending only on X) with the property that if X is homotopy equivalent to a cartesian product of more than l spaces than at least one of these spaces is contractible. The answer to this question is still not known. The following theorem has been proved by Ganea and Hilton [Theorem 1.3 of [1]].

Let and let Gm be the group generated by A and the transpose of A. The problem of determining complex numbers m such that Gm is a free group had been studied by several authors [1, 2, 3]. In this note we characterize those rational values of m for which Gm is torsion free.