Let X be a separable infinite dimensional Banach space. There exist a closed set A ⊂ X which contains a translate of any compact set in the unit ball of X, and a bi-Lipschitz homeomorphism f of X onto X so that every line in X intersects f(A) in a set of Lebesgue measure zero.

Nous donnons une majoration de la somme d'exponentielles SM = [sum ]Mm=1e(f(m)) lorsque la dérivée cinquième de f est d'un ordre de grandeur constant petit, noté λ, en fonction de M et de λ, améliorant un résultat ancien de Van Der Corput. La démonstration utilise un théorème de moyenne des puissances sixièmes de sommes d'exponentielles qui fait l'objet d'un article indépendant [6].

We give a bound for the exponential sum SM = [sum ]Mm=1e(f(m)) where f is a real-valued function whose fifth derivative is of a constant small size, say λ, by means of M and λ, improving an old result of Van Der Corput. The proof relies on a mean value theorem for sixth powers of exponential sums which is treated independently in [6].

Let V be a finite-dimensional vector space over a finite field. The general and special linear groups, GL(V) and SL(V), act on the exterior algebras $\Lambda^*V$ and $\Lambda^*V^*$ of V and its dual $V^*$, and on the symmetric algebra $\S^*V$. The subring of SL(V)-invariants of $\Lambda^*V\otimes\S^*V$ was determined by Dickson and Mui. This paper describes the equivalent, but simpler, calculation of the invariant subring of $\Lambda^*V^*\otimes\S^*V$ as a representation of GL(V)/SL(V).

Using the twistorial approach and some previous results, we prove the conjecture that the dimension of the moduli space of harmonic maps of area $4 \pi d$ from the 2-sphere to the $2n$-sphere is $2 d+n^2$ for the particular case $n=3$.

For any positive integer n we let $P(n)$ be the largest prime factor of n. We improve and generalize several results of P. Erdős and C. Stewart on $P(n!+1)$. In particular, we show that $\limsup_{n \to \infty}P(n!+1)/n \ge 2.5$, which improves their lower bound of $\limsup_{n \to \infty} P(n!+1)/n >2$.

Power series $\sum_{n=0}^\infty f(n)x^n$ with non-zero convergence radius $R(f)$ are considered, and the arithmetical nature (that is, irrationality, or even transcendence) of the corresponding multivariate series $\smash{\sum_{i_1,\ldots,i_m\!=0}^\infty f(i_1+\ldots +i_m)x_1^{i_1}\cdot\ldots \cdot x_m^{i_m}}$ is studied if $x_1,\ldots,x_m$ and the sequence $(f(n))$ satisfy appropriate arithmetical conditions. It follows that such arithmetical results can be written down easily if linear independence results on the function $F(x)$, defined in $|x|\,{<}\,R(f)$ by the original one-dimensional power series, and possibly its derivatives at the points $x_\mu$ are known. Some typical applications are explicitly stated.

In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.

The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture.

This paper proves that a subgroup of finite index in a positively finitely generated profinite group has maximal subgroup growth at most $n^{\log (n\!)}$. In particular such a subgroup cannot be free, answering a question by L. Pyber.

Richard Rado was born in Berlin; he was the second son of Leopold Rado, from Budapest. At one stage of his education he had to decide whether to become a concert pianist or a mathematician. He chose the latter in the belief that he could continue with music as a hobby, but that he could never treat mathematics in that way. He studied at the University of Berlin, but also spent some time in Göttingen. He took a DPh at Berlin with his thesis ‘Studien zur Kombinatorik’ [3] under Issai Schur in 1933. During this period he was also influenced by Erhard Schmidt.

On 16 March 1933, he married Luise Zadek, the elder daughter of Hermann Zadek, whom he had earlier come to know when he needed a partner to play piano duets. It was indeed a remarkable partnership.

As Hitler came to power in 1933, the Rados, being Jewish, made their way to England, Richard having obtained a scholarship of £300 p.a. from Sir Robert Mond through the recommendation of Professor Lindemann (later Lord Cherwell), who had interviewed him in Berlin, to enable him to study at Cambridge.

Let M and N be closed non-positively curved manifolds, and let f[ratio ]M→N be a smooth map. Results of Eells and Sampson [1] show that f is homotopic to a harmonic map ϕ, and Hartman [6] showed that this ϕ is unique when N is negatively curved and f∗(π1M) is not cyclic. Lawson and Yau conjectured that if f[ratio ]M→N is a homotopy equivalence, where M and N are negatively curved, then the unique harmonic map ϕ homotopic to f would be a diffeomorphism. Counterexamples to this conjecture appeared in [2], and later in [7] and [5].

There remains the question of whether a ‘topological’ Lawson–Yau conjecture holds.

Let $p$ be a monic complex polynomial of degree $n$, and let $K$ be a measurable subset of the complex plane. Then the area of $p(K)$, counted with multiplicity, is at least $\pi n ( \Area (K)/\pi )^n$, and the area of the pre-image of $K$ under $p$ is at most $\pi^{1-1/n} (\Area (K) )^{1/n}$. Both bounds are sharp. The special case of the pre-image result in which $K$ is a disc is a classical result, due to Pólya. The proof is based on Carleman's classical isoperimetric inequality for plane condensers.

The paper presents an improvement of the author's earlier estimate for the average number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin, provided that the field is close enough to a linear centre in a larger neighbourhood. The result follows from a new distributional inequality for the number of zeros of a family of univariate holomorphic functions depending holomorphically on a parameter.

Using basic harmonic analysis on the n-torus, it is shown that the non-commutative n-tori are the only primitive C*-algebras (up to isomorphism) on which the n-torus [ ]n acts freely and ergodically. An explicit construction recovers the generating unitaries in a natural way from the action.

If we consider the 14-sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, then we observe that we have a uniform, unifacial dessin whose automorphism group is transitive on the edges, but not on the directed edges of the dessin. We show that Klein's surface is the unique platonic surface with this property.

We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.

A new inequality concerning generalized characters of $p$-groups is obtained and applications to bounding the number of irreducible characters in blocks of finite groups are given.

Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω\{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.

We give a general method for expressing various characteristic numbers of the multiple-point manifolds of an immersion.

The preceding paper ‘Strong statements of analysis’ by A. R. D. Mathias defends a so-called full-blooded set theory without full detail [3]. He again objects to a weak set theory which he calls ‘Mac’, in which the usual Zermelo–Fraenkel separation scheme is required only for formulas with suitably ‘restricted’ quantifiers. I had proposed that such separation is adequate for all standard uses of set theory in mathematics. But Mathias has not produced any counter examples of actual mathematics which requires the use of a stronger separation.