We study the behaviour of non-negative solutions to the equation

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This equation is an ‘explicitly’ soluble special case of a class of non-local evolution equations, of which the behaviour of solutions has been studied by Uchiyama [9]. In this paper, certain fine properties of solutions such as a local relaxation within their supports are obtained for the present special case. The asymptotic forms (for large time) of the solutions whose initial measures have compact supports are identified within the error of the order O(t−3/2). It is also shown that the comparison principle does not hold. With a simple change of variables, the equation ([midast ]) is transformed to the equation arising in Dyson's approach to the Wigner semi-circle law of eigenvalues of random matrices, and the present results have immediate consequences on the latter.

In this paper, the behaviour of the positive eigenfunction $\phi$ of $\Lp u=\la |u|^{p-2}u$ in $\Om$, $u_{|\p \Om} =0$, $p>1$, is studied near its critical points. Under some convexity and symmetry assumptions on $\Om$, $\phi$ is seen to have a unique critical point at $x=0$; also, the behaviour of both $\phi$ and $\nabla\phi$ is determined nearby. Positive solutions $u$ to some general problems $\Lp u=f(u)$ in $\Om$, $u_{|\p \Om} =0$, are also considered, with some convexity restrictions on $u$.

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).

We study the conditions for truncated symmetric products of manifolds to be manifolds. In particular, we show that suitably defined spaces of systems of real roots of real polynomials are homeomorphic to real projective spaces.

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.

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

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

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

In a beautiful result, Herrero (D. A. Herrero, ‘Normal limits of nilpotent operators’, Indiana Univ. Math. J. 23 (1973/74) 1097–1108) showed that a normal operator on $l^2$ lies in the closure of the set of nilpotent operators if and only if its spectrum is connected and contains zero. In the quest for an automatic continuity result for algebra homomorphisms between ${\rm C}^*$-algebras, Dales showed that, if a discontinuous algebra homomorphism $\theta :{\cal A} \rightarrow {\cal U}$ exists between ${\rm C}^*$-algebras ${\cal A}$ and ${\cal U}$, and if $\theta({\cal A})$ is dense in ${\cal U}$, then there is a ${\rm C}^*$-algebra ${\cal U}_2$ with a dense subalgebra ${\cal N} \subset {\cal U}_2$ such that every $x \in {\cal N}$ is quasinilpotent (see p. 685 of H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs 24, Oxford University Press, 2001). (A discontinuous homomorphism $\theta_2 :{\cal A}_2 \rightarrow {\cal U}_2$ can be defined with the same basic properties as $\theta$, but the revised target space ${\cal U}_2$ has a dense subalgebra consisting of quasinilpotent elements.) As remarked by Dales, no such ${\rm C}^*$-algebra was then known; but here we present one. Indeed, using the full power of Herrero's result, one may arrange that every $x \in {\cal N}$ is nilpotent. The ${\rm C}^*$-algebra is constructed in a ‘neat’ way; it is most naturally constructed as a non-separable, concrete ${\rm C}^*$-algebra of operators on a separable Hilbert space ${\cal K}$; but one can arrange that the algebra ${\cal U}$ itself be separable if desired.

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

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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.

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\backslash G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion that involves root systems.

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.

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.

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?

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).

Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

The most natural candidates to the definition of equisingularity of germs of complex space curves are discussed, showing that they coincide in the case of Lagrangean curves (projectivizations of conic Lagrangean varieties of a symplectic manifold of dimension four).

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

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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).

The nonexistence of a submersion from the 23-sphere to the Cayley projective plane is proved.