We give geometric proofs of some of the basic structure theorems for compact Lie groups. The goal is to take a fresh look at these theorems, prove some that are difficult to find in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of p-compact groups.

Suppose that m is a positive integer, τ=(τ1, …, τm) ∈ℝm+ is a vector of strictly positive numbers, and Q is an infinite set of positive integers. Let WQ(m; τ) be the set

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In this paper we obtain the Hausdorff dimension of this set. We also consider a generalization of the set WQ(m; τ), where the error terms q−τi in the inequalities are replaced by ψi(q), for general functions ψi satisfying a certain condition at infinity.

In this paper we consider connections between three classes of optimal (in different senses) convex lattice polygons. A classical result is that if G is a strictly convex curve of length s, then the maximal number of integer points lying on G is essentially

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It is proved here that members of a class of digital convex polygons which have the maximal number of vertices with respect to their diameter are good approximations of these curves. We show that the number of vertices of these polygons is asymptotically

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where s is the perimeter of such a polygon. This result implies that the area of these polygons is asymptotically less than 0.0191612·n3, where n is the number of vertices of the observed polygon. This result is very close to the result given by Colbourn and Simpson, which is 15/784·n3≈0.0191326·n3. The previous upper bound for the minimal area of a convex lattice n-gon is improved to 1/54·n3≈0.0185185·n3 as n→∞.

Let A be a group of automorphisms of the finite group G such that ([mid ]A[mid ], [mid ]G[mid ])=1. Then [mid ]A[mid ]<[mid ]G[mid ]2, and the exponent 2 here is best possible. If, moreover, A is nilpotent of class at most 2, then [mid ]A[mid ]<[mid ]G[mid ]. If A is abelian, then A has a regular orbit on G.

In this paper we show that, given 0[les ]m, n[les ]∞, any monoid can be embedded into a monoid with left cohomological dimension m and right cohomological dimension n. The embedding can be chosen to preserve various monoid-theoretical properties.

Using techniques from geometric measure theory and descriptive set theory, we prove a general result concerning sets in the plane which meet each straight line in exactly two points. As an application, we show that no such ‘two-point’ set can be expressed as the union of countably many rectifiable sets together with a set with Hausdorff 1-measure zero. Also, as a corollary, we show that no analytic set can be a two-point set provided that every purely unrectifiable set meets some line in at least three points. Some generalizations are given to ‘n-point’ sets and some other geometric constructions.

Let D be an open set in Euclidean space IRm with boundary ∂D, and let ϕ[ratio ]∂D→[0, ∞) be a bounded, measurable function. Let u[ratio ]D∪∂D×[0, ∞)→[0, ∞) be the unique weak solution of the heat equation

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with initial condition

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and with inhomogeneous Dirichlet boundary condition

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Then u(x; t) represents the temperature at a point x∈D at time t if D has initial temperature 0, while the temperature at a point x∈∂D is kept fixed at ϕ(x) for all t>0. We define the total heat content (or energy) in D at time t by

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In this paper we wish to examine the effect of imposing additional cooling on some subset C on both u and ED.

We observe that the Bargmann transform arises from the polar decomposition of a simple restriction operator. By virtue of the Bargmann transform, the Wigner transform is imbedded in the geometric symmetry group of phase space. From this imbedding, we easily obtain that the spectrum of the Wigner transform consists of {eikπ/12[ratio ]k=0, 1, …, 23}.

The precise indeterminacy of Milnor's concordance μ invariants is determined. It is shown that the (nilpotent quotient) Artin representation of string links is surjective. Orr's computation of the number of linearly independent μ invariants of a fixed length is recovered. A structure theorem for the set of links up to concordance is proven. We define an action of 2l-component string links on l-component string links, which passes to concordance classes. It is shown that the set of links up to concordance is in bijection with the orbit space of the restriction of this action to the stabilizer of the identity. Via the Artin representation, the action passes to a unipotent action, defined purely algebraically and consequently algorithmically computable, on the corresponding automorphism groups.

Alan Breach Tayler, CBE, Director of the Oxford Centre for Industrial and Applied Mathematics, died on 28 January 1995, aged 63.

Alan went up to Oxford in 1951 to read Mathematics at Brasenose College. He obtained a first, and after a brief excursion to the Bristol Aircraft Company, he returned to work for a DPhil with George Temple. His thesis, completed in 1959 and entitled ‘Problems in compressible flow’, contained a mixture of analytic, approximate and numerical solutions which foreshadowed the new practical applied mathematics that he embraced later. He became University Lecturer and Tutorial Fellow of St Catherine's Society in 1959.

During the next twenty-five years, Alan Tayler brought a new ethos to applied mathematics. This change came about through his recognition that the status quo in the 1960s, which comprised a delicate balance between theory and practice in the area of applied mechanics, was capable of far-reaching generalisation; indeed, he saw that such a development was essential since the following decades were to be dominated by computers and an ever-increasing need for mathematical modelling. In 1967, with Leslie Fox, he initiated the mathematical Study Groups with Industry, wherein academic and industrial researchers interact in week-long workshops. These were an immediate success: (1) with industry, who found new insights into their problems and new recruiting possibilities; (2) with students, whose enthusiasm to use their theoretical knowledge soon led to the highly popular MSc in Mathematical Modelling and Numerical Analysis; and (3) with faculty, both pure and applied, who found an undreamed of source of fascinating new theoretical problems. For example, one intellectual consequence was the use of industrial case studies to uncover the new field of ‘free boundary problems’, on which several thousand learned articles have appeared since 1970.