Let ƒ be a C1 diffeomorphism of a compact C∞ boundary–less manifold, and let ƒ# be the operator on the bounded or continuous sections of the tangent bundle (with supremum norm) defined by ƒ#η = Tƒ о η о ƒ−1. The main result of this paper is that ƒ is quasi-Anosov if and only if 1 – f# is injective and has closed range.

A class of replacement systems is studied which satisfies a “subcommutativity” condition. Examples of systems satisfying this condition are many of the systems of graph-like expressions which have recently been studied in connection with the efficient evaluation of recursive definitions. An optimality theory of subcommutative systems is developed and is used to give conditions which are sufficient to ensure that an evaluation (reduction) procedure is optimal. The optimality theory is also applied to develop conditions under which a given subcommutative system speeds up, in a natural sense, another replacement system.

Consider a monopolistic firm selling each business period's production at the end of that period at a price determined by the buyers, and wishing to determine the production for which business period profit is maximal. In this paper we announce the results of our investigations into what happens when the firm employs a certain algorithm (based on linear approximations to its average contribution profit function) in an attempt to determine this optimal production. Our results are as follows: firstly, for many apparently feasible average contribution profit functions the algorithm generates production sequences globally convergent to the optimal production, the convergence being linear with convergence ratio dependent on the average contribution profit function; secondly, in certain cases a lower bound for the initial rate of convergence of the algorithm can be obtained. Proofs are for the most part given only in outline, and will be published in full later.

Roughly speaking, a system of linear differential equations has an exponential dichotomy if it has a subspace of solutions shrinking exponentially and a complementary subspace of solutions growing exponentially. In the case of constant coefficients, this happens if and only if the eigenvalues of the coefficient matrix have nonzero real parts. In the general case, Lazer has shown that if the coefficient matrix function is bounded and satisfies a diagonal dominance condition (which, in the constant case, is a sufficient but not necessary condition that the eigenvalues have nonzero real parts) then the system has an exponential dichotomy. In this paper we prove the same result with a weaker diagonal dominance condition, thus generalizing a theorem of Nakajima.

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

Groups of nilpotent length four containing a subgroup which covers and avoids the same chief factors as an -injector for some Fitting class but which is not itself an -injector have been constructed by F.P. Lockett (in his PhD thesis) and T.R. Berger and John Cossey (in preparation). Graham A. Chambers (J. Algebra 16 (1970), 442–455) has shown that such a subgroup cannot exist in a group of p–length one for all primes p. The main result of this paper closes the small gap remaining: it includes Chambers' result and establishes also that such a subgroup cannot exist in a group of nilpotent length three.

This paper deals with the study of uniform asymptotic stability of the measure differential system Dx = F(t, x) + G(t, x)Du, where the symbol D stands for the derivative in the sense of distributions. The system is viewed as a perturbed system of the ordinary differential system x' = F(t, x), where the perturbation term G(t, x)Du is impulsive and the state of the system changes suddenly at the points of discontinuity of u. It is shown, under certain conditions, that the uniform asymptotic stability property of the unperturbed system is shared by the perturbed system. To do this, the well-known Gronwall integral inequality is generalized so as to be applicable to Lebesgue-Stieltjes integrals.

It has “been open for some time whether, given an algebraic theory (triple, monad) Π in a cocomplete category K, also the category KΠ of Π-algebras must be cocomplete. We solve this in the negative by exhibiting a free algebraic theory Π in the category Gra of graphs such that GraΠ is not cocomplete. Further, we improve somewhat the well-known colimit theorem of Barr and Linton by showing that the base category need not be complete.

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) − X(u) = c or X(u) = 0. It is shown that c is an even positive integer and all such groups with c ≤ 8 are described.

In recent years some fixed point theorems have been proved for nonexpansive mappings in Hilbert space, which have non-convex domains. The purpose of this paper is to present a simple but very useful new result of that kind and to indicate some of its consequences.

In this note, a positive solution is given to the Hahn-Banach problem for an important class of convergence vector spaces. As well, a topological vector space characterization is obtained for fully complete and Br-complete spaces.

A simple argument yields the following generalization of Theorem 6 of [1] (whose notation is retained without further explanation).