Let G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

It is observed that arrays which arise in the scheduling of tournaments exist if and only if there are Hamiltonian cycles in certain graphs. The graphs are generalizations of those which arise in the “Footballers of Croam” problem. It is proven that such Hamiltonian cycles exist in infinite classes of the graphs.

Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 35, 05 B 30.

We study the existence of zeros of a perturbed nonlinear operator near a zero of the unperturbed operator in the case where both operators are invariant under a symmetry group. To do this, we first correct some work of Rubinsztein on the G-homotopy groups of spheres.

The problem is the reconstruction of the shape of an object, whose shell is a surface star-shaped with respect to a point 0, from the knowledge of the volume of every “half-object” obtained by taking any plane through 0. Conditions for the existence and uniqueness of the solution are given. The main result consists in showing that any uniform a-priori bound on the mean curvature of the shell reestablishes continuous dependence on the data for bodies satisfying a certain symmetry condition.

Let ξ be a C*;-bundle over T with fibres {At}t∈A. Suppose that A is the C*-algebra of sections of ξ which vanish at infinity, and that (A, G, α) is a C*-dymanical system that, for each t ∈ T, the ideal It = {f ∈ A|f(t) =; 0} is G-invariant. If in addition, the stabiliser group of each P ∈ Prim(A) is amenable, then A ⋊αG is the section algebra of a C*-bundle with fibres {At ⋊αG}t∈T.

The above theorem may be used to prove a structure theorem for crossed products built from C*-dynamical systems (A, G, α) where the action of G on A is smooth. Assuming that the stabiliser groups are amenable, then A ⋊αG has a composition series such that each quotient is a section algebra of a C*-bundle where the fibres are of the form Aσ ⋊αG; moreover, the Aσ correspond to locally closed subsets of Prim(A), and G acts transitively on Prim(Aσ). In many cases, in particular when (G, A) is separable, the Aσ ⋊αG have been computed explicitly by other authors.

These results are actually proved for twisted C*dynamical systems.

By amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

Abstract: We show that a tree is stable if and only if its automorphism group contains a transposition. The method is constructive and so enables a stabilising sequence to be found.

In a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

The purpose of this paper is to show that higher order elliptic partial differential operators on smooth domains have an H∞ functional calculus and satisfy quadratic estimates in Lp spaces on these domains.

Recently M. Benedicks showed that if a function f ∈ L2(Rd) and its Fourier transform both have supports of finite measure, then f = 0 almot everywhere. In this paper we give a version of this result for all noncompact semisimple connected Lie groups with finite centres.

We prove that every Banach space X with characteristic of uniform convexity less than 2 has the fixed point property whenever X satisfies a certain orthogonality condition.

The quintuple product identity has appeared many times in the literature. Indeed, no fewer than 12 proofs have been given. We establish a more general identity from which the quintuple product identity follows in two ways.

A geometric property of convex sets which is equivalent to a minimax inequality of the Ky Fan type is formulated. This property is used directly to prove minimax inequalities of the von Neumann type, minimax inequalities of the Fan-Kneser type, and fixed point theorems for inward and outward maps.

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

We prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.

We consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann–Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi–Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^{n}$ and in minimally convex domains of $\mathbb{R}^{n}$, results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

We establish a duality between distributive bisemilattices and certain compact left normal bands. The main technique in the proof utilizes the idea of Plonka sums.

On analogy with functions if Lebesuge class Lα on the real line the author considers those multiplicative arthmetic functions which are bounded in mean α>1. Necessary and sufficient conditions are obtained in order that they should have a mean-value, zero or non-zero. An application is made to Ramanujan's τ-function.