Nearly four hundred years ago, the cubic close-packing of equal spheres in R3 was discovered by Kepler [21], in which each sphere touches 12 others. In 1694, Gregory and Newton discussed the following thirteen spheres problem. Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? Gregory believed ‘yes’, while Newton thought ‘no’.

Well over two decades have now passed since the publication of the classic books Singular integrals and differentiability properties of functions by E. M. Stein [32] and An introduction to Fourier analysis on Euclidean spaces by E. M. Stein and G. Weiss [37]. These two texts would, I am sure, be universally regarded as defining the ‘common core’ of harmonic analysis in the Calderón–Zygmund tradition in the early 1970s. What has been going on in the subject since then?

The title alludes to a similar title of the paper [3] by Grunewald and Segal, in which it is shown how to solve a quadratic equation in integers. This latter procedure seems to be quite difficult, and the algorithm outlined in [3] is rather involved, although it is completely effective in the logical sense.

In 1956, A. Selberg introduced weakly symmetric spaces in the framework of his development of the trace formula, and proved that in a weakly symmetric space, the algebra of all invariant (with respect to the full isometry group) differential operators is commutative [22]. In this paper, Selberg asked whether the converse holds. In the present work, we shall answer this question by presenting examples of commutative spaces which are not weakly symmetric. These examples arise in the quaternionic analogues to the Heisenberg group, endowed with certain special metrics (see Theorem 5 and the explicit realization after it).

In this paper we are concerned with the asymptotic description of the sum of digits function of canonical number systems in number fields. We derive an asymptotic formula for its summatory function whose second term contains a periodic fluctuation. In imaginary quadratic fields this function is uniformly distributed with respect to the argument.

The theory of isometric group actions on R-trees is extended to actions by homeomorphisms with the following non-nesting property: no group element maps an arc properly into itself. A finitely presented group acting freely by homeomorphisms on an R-tree is free abelian or splits over a (possibly trivial) cyclic group.

Since the very beginning of the multidimensional theory of quasiregular mappings, it has been widely believed that the class of K-quasiregular mappings in ℝn is closed with respect to uniform convergence, where K stands for the linear dilatation. In this note we give a striking example which refutes this belief. The key element of our construction is that the linear dilatation function fails to be rank-one convex in dimensions higher than 2.

We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Möbius transformation by its critical points.

Let A1,…, An be Lipschitz functions on R such that A′1,…, A′n∈VMO. We show that on any bounded interval, the Calderón commutator associated with the kernel (A1(x)−A1(y)) …(An(x) −An(y))/(x−y) n+1 is a compact perturbation of −πiHMA′1… A′n, where H is the Hilbert transform.

In [8, 6] it was shown that for each k and n such that 2k>n, there exists a contractible k-dimensional complex Y and a continuous map ϕ[ratio ][ ]n→Y without the antipodal coincidence property, that is, ϕ(x)≠ϕ(−x) for all x∈[ ]n. In this paper it is shown that for each k and n such that 2k>n, and for each fixed-point free homeomorphism f of an n-dimensional paracompact Hausdorff space X onto itself, there is a contractible k-dimensional complex Y and a continuous map ϕ[ratio ]X→Y such that ϕ(x)≠ϕ(f(x)) for all x∈X. Various results along these lines are obtained.

The purpose of this note is to give a short proof of the Gross logarithmic Sobolev inequality using the asymptotics of the sharp L2 Sobolev constant and the product structure of Euclidean space. Let F∈Lr(IRn) for some positive r with ∥F∥r=1.

John Charles Burkill, born on 1 February 1900, was the only child of Hugh Roberson Burkill (1867–1951) and Bertha (née Bourne, 1866–1937). His father came from a family which had farmed in Lincolnshire for generations, whereas his mother came from a background of prosperous farming and building. On neither side was there a strong academic tradition, but Charles was soon to show evidence of intellectual distinction by winning a scholarship to St Paul's school at the age of 14. There he profited fully from the excellent teaching that the school offered and which was reflected not only by his mathematical prowess, which led to a scholarship to Trinity College, Cambridge, in 1918, but also in his ability in classical studies in which he maintained a lifelong interest. He was also a formidable chess player, and had a mischievous sense of humour which he retained, albeit in a more restrained mode, in later life. A striking example of his grasp of the essence of a practical joke is recorded in the story of how, as a boy on a visit to a house-proud aunt, he saw the comic potential of a trail of corn from the chicken run through the front door and upstairs to the bedrooms.

Held on Friday 23 May and Saturday 24 May 1997 at the University of Liverpool, Professor J. M. Ball, FRS, President, in the Chair. There were present about 30 members and visitors for all or part of the meeting.