A theorem of Macbeath asserts that μ(A+B)[ges ]min(1, μ(A)+μ(B)) for any subsets A and B of a finite-dimensional torus. We conjecture that, when the obvious exceptions are excluded, a stronger inequality

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holds, and we prove this conjecture under some technical restrictions.

The first person to consider the discrepancy of sequences of the type (αnσ)n[ges ]1 (where 0<σ<1) was H. Behnke. The subject was taken up again by one of the authors of this paper, who gave a detailed description of the discrepancy's behaviour if either 0<σ<½ or σ=½ and α2∉ℚ or σ=½ and α−2∈ℕ. In this paper, we study the case of sequences (α√n)n[ges ]1 where α>0 and α2∈ℚ. Both

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are expressed as maxima of finitely many numbers which involve class numbers of imaginary quadratic fields in many cases.

In an attempt to prove all the identities given in Ramanujan's first two letters to G. H. Hardy, G. N. Watson devoted a paper to the evaluation of G1353. At the end of his paper, Watson remarked that his proof was not rigorous as he had assumed certain identities (see (1.1) and (1.2)) which he found empirically. In this paper, we use class field theory, Galois theory and Kronecker's limit formula to justify Watson's assumptions. We shall then use our results to compute some new values of Gn.

For normalised generalized matrix functions f and g, we say that f dominates g if f(A)[ges ]g(A) for every M-matrix A. We first demonstrate a finite set of test matrices for any such inequality. Then, using results from group representation theory, all comparisons among immanants in certain classes are determined. This work parallels ongoing research into gmf inequalities on positive semidefinite matrices, for which no finite set of test matrices is available. However, the inequalities for the two classes are quite different, and the test matrices permit more rapid progress in the M-matrix case. Just as in the positive semidefinite case, the gmf inequalities we prove may be used to verify previously unknown determinantal inequalities for M-matrices, such as the symmetrized Fischer inequalities recently proved in the positive semidefinite case.

It is well known that the Horrocks–Mumford bundle [Fscr ] encodes a lot of very interesting geometric information. This is essentially the reason for the fact that much work has been done in order to find other rank-2 bundles on ℙ4. The only nonsplit vector bundles of rank 2 on ℙ4, known up to now, are twists of pullbacks of [Fscr ] by finite coverings f[ratio ]ℙ4→ℙ4. So it seems to be a natural question to consider, instead of ℙ4, other Fano 4-folds. It is the aim of this note to give an example of a rank-2 vector bundle on ℙ1×ℙ3 and to show that it also admits very interesting geometric properties.

The purpose of this note is to generalise, and give a more illuminating proof, of a theorem of [13] (Theorem 1.1 below). Before stating it, we provide some introductory information. Consider the following two sequences of pictures: in each we see a 1-parameter family Xℝ,t of real algebraic hypersurfaces, which undergoes a bifurcation when the parameter t is equal to 0. Note that in Figure 1, both (i) (a) and (i) (b), and in (ii) (b), the surface Xℝ,t has a purely 1-dimensional part, which we have indicated with a dotted line, and that in (i) (b) we have drawn a curve vertically along the middle of the surface to make clearer the way it passes through itself. The reader will observe that in (a) the surface Xℝ,t is homotopically a 2-sphere when t>0 and a 0-sphere when t<0, while in (b) Xℝ,t is a homotopy 1-sphere both for t<0 and t>0.

Such sequences are typical in singularity theory; each is in fact the family of algebraic closures of images of a versal deformation of a codimension 1 singularity of mapping.

Now suppose that the complexification X[Copf ],t is a homotopy n-sphere. In [13] the second author pointed out that it follows that Xℝ,t is a homotopy sphere for t≠0 (allowing the empty set as a −1-sphere). Indeed, in the local situation, or globally in the weighted homogeneous case, there are well-defined integers k+ and k− between −1 and n such that Xℝ,t≃Sk+ for t>0 and Xℝ,t≃Sk− for t<0.

We describe Xℝ,t for t∈ℝ−0 as ‘good’ if the homotopy dimension of Xℝ,t is equal to n. In this case the inclusion Xℝ,t[rarrhk ]Xt is a homotopy equivalence [13, 1.1].

A group G is conjugacy separable if whenever x and y are non-conjugate elements of G, there exists some finite quotient of G in which the images of x and y are non- conjugate. It is known that free products of conjugacy separable groups are again conjugacy separable [19, 12]. The property is not preserved in general by the formation of free products with amalgamation; but in [15] a method was introduced for showing that under certain circumstances, the free product of two conjugacy separable groups G1 and G2 amalgamating a cyclic subgroup is again conjugacy separable. The main result of [15] states that this is the case if G1 and G2 are free-by-finite or finitely generated and nilpotent-by-finite. We show here that the same conclusion holds for groups G1 and G2 in a considerably wider class, including, in particular, all polycyclic-by-finite groups. (This answers a question posed by C. Y. Tang, Problem 8.70 of the Kourovka Notebook [7], as well as two questions recently asked by Kim, MacCarron and Tang in G. Kim, J. MacCarron and C. Y. Tang, ‘On generalised free products of conjugacy separable groups’, J. Algebra 180 (1996) 121–135.)

We study the distributions Fθ,p of the random sums [sum ]∞1εnθn, where ε1, ε2, … are i.i.d. Bernoulli-p and θ is the inverse of a Pisot number (an algebraic integer β whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p=.5, Fθ,p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain β of small degree, simulation gives the Hausdorff dimension to several decimal places.

The paper provides necessary and sufficient conditions for the existence of weak solutions for the Dirichlet problem within Bishop's constructive mathematics. It also considers the continuity of those weak solutions relative to the boundary data and the domain.

This paper shows that a subharmonic function in the half-space which does not grow too rapidly near the boundary and which does not have asymptotic value +∞ at too many points must have finite minimal fine limits at a boundary set of positive measure. For harmonic functions, the conclusion may be expressed in terms of non-tangential limits. A related Phragmén–Lindelöf theorem is also established.

Asymptotic results are given for the heat content of planar regions with cusps.

Let G be a locally compact group not necessarily unimodular. Let μ be a regular and bounded measure on G. We study, in this paper, the following integral equation,

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This equation generalizes the functional equation for spherical functions on a Gel'fand pair. We seek solutions ϕ in the space of continuous and bounded functions on G. If π is a continuous unitary representation of G such that π(μ) is of rank one, then tr(π(μ)π(x)) is a solution of [Escr ](μ). (Here, tr means trace). We give some conditions under which all solutions are of that form. We show that [Escr ](μ) has (bounded and) integrable solutions if and only if G admits integrable, irreducible and continuous unitary representations. We solve completely the problem when G is compact. This paper contains also a list of results dealing with general aspects of [Escr ](μ) and properties of its solutions. We treat examples and give some applications.

Weakly almost periodic compactifications have been seriously studied for over 30 years. In the pioneering papers of de Leeuw and Glicksberg [4] and [5], the approach adopted was operator-theoretic. The current definition is more likely to be created from the perspective of universal algebra (see [1, Chapter 3]). For a discrete group or semigroup S, the weakly almost periodic compactification wS is the largest compact semigroup which (i) contains S as a dense subsemigroup, and (ii) has multiplication continuous in each variable separately (where largest means that any other compact semigroup with the properties (i) and (ii) is a quotient of wS). A third viewpoint is to envisage wS as the Gelfand space of the C*-algebra of bounded weakly almost periodic functions on S (for the definition of such functions, see below).

In this paper, we are concerned only with the simplest semigroup (ℕ, +). The three approaches described above give three methods of obtaining information about wℕ. An early striking result about wℕ, that it contains more than one idempotent, was obtained by T. T. West using operator theory [13]. He considered the weak operator closure of the semigroup {T, T2, T3, …} of iterates of a single operator T on the Hilbert space L2(μ) for a particular measure μ on [0, 1]. Brown and Moran, in a series of papers culminating in [2], used sophisticated techniques from harmonic analysis to produce measures μ that permitted the detection of further structure in wℕ; in particular, they found 2[cfr ] distinct idempotents. However, for many years, no other way of showing the existence of more than one idempotent in wℕ was found.

The breakthrough came in 1991, and it was made by Ruppert [11]. In his paper, he created a direct construction of a family of weakly almost periodic functions which could detect 2[cfr ] different idempotents in wℕ. His method was very ingenious (he used a unique variant of the p-adic expansion of integers) and rather complicated. Our main aim in this paper is to construct weakly almost periodic functions which are easy to describe and so appear more ‘natural’ than Ruppert's. We also show that there are enough functions of our type to distinguish 2[cfr ] idempotents in wℕ.

We investigate classes of uniform pseudodifferential operators on Riemannian manifolds with bounded geometry. We prove that operators belonging to the classes are bounded in function spaces of Hardy–Sobolev–Besov type defined on the manifold. The classes contain the powers of the Laplace–Beltrami operators so we get the fractional lift property for the function spaces.

Given two self-adjoint Hilbert space operators A, B, and a continuous function f, we prove several inequalities of the form

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involving the Lp-norm of the derivative f′ and the Besov Br∞,1-norm of f.

We establish a Weitzenböck formula for harmonic morphisms between Riemannian manifolds and show that under suitable curvature conditions, such a map is totally geodesic. As an application of the Weitzenböck formula we obtain some non-existence results of a global nature for harmonic morphisms and totally geodesic horizontally conformal maps between compact Riemannian manifolds. In particular, it is shown that the only harmonic morphisms from a Riemannian symmetric space of compact type to a compact Riemann surface of genus at least 1 are the constant maps.

This paper is mainly devoted to the study of the index of a map at a zero, and the index of a polynomial map over ℝn. For semi-quasi-homogeneous maps we prove that the index at a zero coincides with the index at this zero of its quasi-homogeneous part. For a class of polynomial maps with finite zero set we provide a method which makes easier the computation of its index over ℝn. Finally we relate the index and the multiplicity.