A few results on uniform and uniform ultimate boundedness properties of solutions of impulsively perturbed systems are presented in this paper, assuming the corresponding properties of solutions of the unperturbed nonlinear system.

A general process for the desingularization of smooth actions of compact Lie groups is described. If G is a compact Lie group, it is shown that there is naturally associated to any compact G manifold M a compact G × (Z/2)p manifold on which G acts principally. Here Z/2 denotes the cyclic group of order two and p + 1 is the number of orbit types of the G action on M.

In this paper we present further properties of the kth variation of a function, and obtain an integral representation for a function having bounded kth variation and an absolutely continuous (k-l)th derivative. The absolute continuity-requirement replaces a previous stronger condition that required the kth derivative of a function to be continuous except on a set of Lebesgue measure zero.

The idea of duality is now a widely accepted and useful idea in the analysis of optimization problems posed in real finite dimensional vector spaces. Although similar ideas have filtered over to the analysis of optimization problems in complex space, these have mainly been concerned with problems of the linear and quadratic programming variety. In this paper we present a general duality theory for convex mathematical programs in finite dimensional complex space, and, by means of an example, show that this formulation captures all previous results in the area.

P.S. Kenderov has shown that every monotone operator on an Asplund Banach space is continuous on a dense Gδ subset of the interior of its domain. We prove a general result which yields as special cases both Kenderov's Theorem and a theorem of Collier on the Fréchet differentiability of weak* lower semicontinuous convex functions.

A classical theorem of Camille Jordan concerning finite linear groups is used to give two structural characterisations of profinite groups which have only finitely many pairwise inequivalent, continuous, irreducible, unitary representations of each degree. Our characterisations of such groups involve properties of their open normal subgroups.

It is shown that the set of chaotic self-maps of the unit interval contains an open dense subset of the space of all continuous self-maps of the unit interval. Other aspects of chaotic behaviour are also considered together with some illustrative examples.

Consider the class of stochastic processes with stationary independent increments and finite variances; notable members are brownian motion, and the Poisson process. Now for Xt any member of this class of processes, we wish to find the optimum sampling points of Xt, for predicting . This design question is shown to be directly related to finding sampling points of Yt for estimating β in the regression equation, Yt = β + Xt. Since processes with stationary independent increments have linear drift, the regression equation for Yt is the first type of departure we might look for; namely quadratic drift, and unchanged covariance structure.

We prove in this paper several results on lattice related measures and images of such measures under mappings. If we apply these to various areas of point set topology we obtain as corollaries many known and new results on sequential compactness, repleteness, and measure repleteness – areas of recent considerable interest to mathematicians.

This paper characterizes topological spaces which have an Abian structure - a uniform-like structure of the covering type.

Let be a Lie algebra, a complemented ideal of , and W an -module. Hochschild and Mostow have described the construction of a -module “induced” from W, which is finite-dimensional provided W is finite-dimensional and satisfies a nilpotent action condition. This note describes a modification of their construction which is functorial and a weak adjoint to the restriction functor from –modules to -modules.

A study of centered composition operators on l2 is made in this paper. Also the spectrum of surjective composition operators is computed. A necessary and sufficient condition is obtained for the closed unit disc to be the spectrum of a surjective composition operator.

Suppose n competitors each compete in r races and a ranking function F assigns a score F(j) to the competitor finishing in the jth position in each race. The sum of the scores over r races gives each competitor a final ranking. If n is fixed, the ranking function F bifurcates as r increases. The complete bifurcation behaviour is determined for n = 3 and some information obtained for n > 3.

In 1929 Rolf Nevanlinna posed a problem attributed to Bloch which has since been known as the Bloch-Nevanlinna conjecture. It can be stated as follows: Is the derivative of a function of bounded characteristic of bounded characteristic? A variety of different counterexamples have provided negative answers to this question. The purpose of the paper is to survey these counterexamples and then give a truly elementary proof of the following theorem.

Michael A.B. Deakin (Bull. Austral. Math. Soc. 17 (1977), 161–175) states (on p. 173) a conjecture, which, if true, would lead to simpler proofs of a number of known theorems. This conjecture is, however, shown to be false by means of a counter-example.

We prove a Frobenius reciprocity type theorem for coinduced comodules.

Two nonlinear ordinary differential systems, one of which contains impulses are considered in this note, and assuming the existence of bounded solutions, some results on asymptotic equivalence type correspondence between them are obtained. At the end an open question is posed.

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

We show that there exists a conjugate Banach space E = B* with a basis, such that the distance from E to the collection of all conjugate Banach spaces can be made arbitrarily large, by suitable renorming of E. This solves a problem raised by B.V. Godun, Dokl. Akad. Nauk SSSR. 236 (1977), 18–20.