We establish the existence of self-homeomorphisms of Rn, n [ges ] 2, which are chaotic in the sense of Devaney, preserve volume and are spatially periodic. Moreover, we show that in the space of volume-preserving homeomorphisms of the n-torus with mean rotation zero, those with chaotic lifts to Rn are dense, with respect to the uniform topology. An application is given for fixed points of 2-dimensional torus homeomorphisms (Conley–Zehnder–Franks Theorem).

This paper proves that a difference field (E, σ) admits quantifier elimination if and only if E is an algebraically closed field, and σ is an integer power of the Frobenius automorphism.

In this paper the domain in which any two points are joined by finite numbers of arbitrarily small homeomorphisms is studied, and two examples are given.

The distribution on the torus ℝ/ℤ of a set of fractions of the form

(formula here)

is investigated, where q is a large integer, m is the inverse of m modulo q, R(x) is a rational function defined modulo q, and [Uscr ], [Mscr ], [Nscr ] are subsets of {1,…,q}. Under some natural assumptions, it is shown that the set [Rscr ] is uniformly distributed on ℝ/ℤ.

Sufficient conditions are given for the global existence of solutions to the Rayleigh equation.

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in ${\rm SL}(2,{\mathbb Z})$. It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example of a holomorphic almost modular form is the logarithm of $\prod_{n=1}^\infty (1-\exp(2\pi\i n^2 z))$. The paper is motivated by the author's previous studies [Int. Math. Res. Not. 39 (2003) 2131–2151] on the connection between almost modular functions and the distribution of the sequence $n^2x$ modulo one.

In a previous paper, the author proved a recent Wiener–Hopf formula, established by Alili and Doney, by means of a path transform. We derive here two other results using this path transform.

It is proved that each mod 2 homology class of a compact Nash manifold can be represented by a semialgebraic arc-symmetric subset.

It is shown that the mod $3$ cohomology of a $1$-connected, homotopy associative mod $3$$H$-space that is rationally equivalent to the Lie group $E_6$ is isomorphic to that of $E_6$ as an algebra. Moreover, it is shown that the mod $3$ cohomology of a nilpotent, homotopy-associative mod $3$$H$-space that is rationally equivalent to $E_6$, and whose fundamental group localized at $3$ is non-trivial, is isomorphic to that of the Lie group $\Ad E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra. It is also shown that the mod $3$ cohomology of the universal cover of such an $H$-space is isomorphic to that of $E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra.

A criterion of isomorphism for symmetry groups of set ideals is provided in terms of ideal quotients and cardinal invariants. Furthermore, set ideals with complete symmetry group are characterized. They form a wide class, comprising, for example, all uniform dense ideals and the ideals of ‘thin’ sets in separable metric spaces. If the symmetry group of a set ideal is not complete, then its outer automorphism group is shown to be cyclic of order 2.

It is known [10, 11] that if T is an integral operator with an extended totally positive kernel, then T has a countably infinite family of simple, positive eigenvalues. We prove a similar result for a rather larger class of kernels and, writing the eigenvalues of T in decreasing order as (λn)n∈IN, we use results obtained in [4] and [5] to give a formula for the ratio λn+1/ lambda;n analogous to that given in [3] for the case of a strictly totally positive matrix, and to the spectral radius formula

formula here

This may be regarded as a generalisation of inequalities due to Hopf [8, 9].

A theorem of Beurling and Nevanlinna gives a lower bound for the harmonic measure of a compact set in the unit disk in terms of the harmonic measure of a projection of the set. In this paper, a generalization of that result is proved, involving the sum of harmonic measures evaluated at two points.

Let $s:\Q\,{\longrightarrow}\,\Q$ be the Dedekind sum, given by $s(h/k)=\sum_{\nu=1}^{k-1}({\nu/k}\,{-}\,{1/2})(\{{h\nu/k}\}\,{-}\,{1/2})$ when $\gcd(h,k)\,{=}\,1$. Then for every rational $\alpha\,{\ne}\,1/12$ there are infinitely many rational $x$ such that $s(x)\,{=}\,\alpha x$. Also, the fixed points of $s$ are dense in the real line.

We consider the following strengthening of Hadwiger's Conjecture.

Let G be any graph of chromatic number k, S any subset of V(G) which takes all k colours in each proper k-colouring of G. Then there are k pairwise adjacent connected subgraphs of G, each of whose vertex sets has non-trivial intersection with S.

We show that the truth of this conjecture for all graphs of chromatic number [les ]k implies the truth of Hadwiger's Conjecture for all graphs of chromatic number [les ]k + 1. We also show that its truth implies the following statement (which is at first sight even stronger).

For any graph G of chromatic number k and any subset S of V(G), define χ(S; G) to be the least number of colours that can appear on S in any proper k-colouring of G, and h(S; G) to be the largest number of pairwise adjacent connected subgraphs of G each having non-trivial intersection with S. Then χ(S; G) [les ]h(S; G).

We define the number w(S; G) to be the largest cardinality of a subset T of S such that, however T is partitioned into pairs (possibly with one spare element), there are vertex-disjoint paths linking the elements in each pair, none passing through the spare element if it exists. We show that χ(S; G) [les ]([mid ]S[mid ]+w(S; G))/2 for any graph G and subset S of V(G).

Finally, we show that for any graph G, χ(S; G) [les ]h(S; G) whenever χ(S; G) [les ]3.

Philip Holgate was born in Chesterfield on 8 December 1934. His family moved from Derbyshire to Devon in 1945, and he was educated at Newton Abbot Grammar School from 1945 to 1952, and the University College of the South-West (now Exeter University, but which then awarded London degrees) from 1952 to 1955. He qualified as both a teacher (King's College, London, 1955–56) and a statistician (University College, London, 1956–57).

After teaching mathematics and physics at Burgess Hill School, in Hampstead and then in Borehamwood, Philip joined the Statistics Section at Rothamsted Experimental Station (1961–62). He then spent five years in the Biometrics Section of the Nature Conservancy. His first publications date from this period, and the interests he acquired then were to develop into what became his most enduring, and distinctive, scientific interests.

Philip joined the Department of Statistics, Birkbeck College, University of London, in 1967, and remained at Birkbeck for the rest of his career, until his death from a heart attack on 13 April 1993.

Noel Baker was born on 10 August 1932 and died, of a heart attack, on 20 May 2001. He was the only child of a farming family living near the township of Virginia north of Adelaide in South Australia, and a fourth-generation Australian. From 1938 to 1944 he attended the local school, winning a scholarship to King's College, Adelaide, and from there another scholarship to Prince Alfred College, Adelaide. Here he was inspired by an enthusiastic mathematics teacher, Mr Williams, who expected pupils to practise mathematics assiduously, even on Saturday mornings. Prince Alfred College still aims to foster a love of mathematics today, and it has recently launched a centre for excellence in mathematics, named after Noel.

Let Fn be the free group of rank n with basis x1, x2, …, xn, and let d(G) denote the minimal number of generators of the finitely generated group G. Suppose that n[ges ]d(G). There exists an exact sequence

formula here

and we may view the free abelian group R=R/R′ as a right ℤG-module by defining (rR′)g =rgϕ−1R′ for all g∈G, where gϕ−1 is any preimage of g under ϕ, and rgϕ−1 =(gϕ−1)−1r(gϕ−1), the conjugate of r by gϕ−1. We call R the relation module of G associated with the presentation (1), and say that R has ambient rank n. Furthermore, we call the group Fn/R′ the free abelianized extension of G associated with (1).

Based on a result of Hofer and Viterbo, we give an inequality for symplectic capacity as defined by Hofer and Zehnder, and, as an application, we give another proof of the symplectic packing inequality of Gromov.