This paper encompasses a motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network. The linking threads are the discrete Laplacian on a graph and the solution of the associated Dirichlet problem. Thirty years ago, this subject was dismissed by many as a trivial specialisation of cohomology theory, but it has now been shown to have hidden depths. Plumbing these depths leads to new theoretical advances, many of which throw light on the diverse applications of the theory.

We prove that each interval of the lattice [Escr ] of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [11]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in [7] by giving an example of a non-definable subclass of [Escr ]* which has an arithmetical index set and is invariant under automorphisms.

Let Ω be the set {1, 2, [ctdot ], n}, and let [emptyv ] be the empty set. Let [Gscr ] be the family of all non-empty sets of subsets of Ω. For [Ascr ]∈[Gscr ] and X⊆Ω, put

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An important discovery is the following.

We determine all primitive groups which do not have a regular orbit on the power set of the permutation domain. As a corollary, we also determine all families of orbit equivalent primitive permutation groups.

We estimate the growth order in n of the Taylor coefficients of bounded univalent functions in the unit disk. Our estimate depends on the norm of the Grunsky operator. In particular, we improve the known results for coefficients of quasiconformally extendible univalent functions.

The present paper deals with real infinite-dimensional normed spaces; some of the main concepts here make sense, and have been treated in the literature, in the general context of topological Hausdorff linear spaces over reals.

A subset of a normed space X is a body if it is different from X itself and is the closure of its non-empty interior. A covering of X by bodies is called a tiling ofX whenever any two different members of it have disjoint interiors. The elements of such a covering are called tiles. A tiling is bounded (respectively convex) whenever each tile is bounded (respectively convex).

Let Ω be a bounded connected open set in ℝN, N[ges ]2, and let −ΔΩ[ges ]0 be the Dirichlet Laplacian defined in L2(Ω). Let λΩ>0 be the smallest eigenvalue of −ΔΩ, and let ϕΩ>0 be its corresponding eigenfunction, normalized by ∥ϕΩ∥2=1. For sufficiently small ε>0 we let R(ε) be a connected open subset of Ω satisfying

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Let −Δε[ges ]0 be the Dirichlet Laplacian on R(ε), and let λε>0 and ϕε>0 be its ground state eigenvalue and ground state eigenfunction, respectively, normalized by ∥ϕε∥2=1. For functions f defined on Ω, we let Sεf denote the restriction of f to R(ε). For functions g defined on R(ε), we let Tεg be the extension of g to Ω satisfying

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We give a formula for the Betti numbers of 3-Sasakian manifolds or orbifolds which can be obtained as 3-Sasakian quotients of a sphere by a torus. This answers a question of Galicki and Salamon about the topology of 3-Sasakian manifolds.

We show that the zero curvature limit of the space of hyperbolic monopoles gives the Euclidean monopoles, settling a conjecture of Atiyah. We also study the infinite curvature limit of the space of hyperbolic monopoles, and show that the associated rational maps appear explicitly here.

With the death of Brian Kuttner on 2 January 1992, near Birmingham, England, summability theory has lost one of its leading exponents. During the period 1934–1991 he published over 100 research papers, including work on Fourier series, strong summability, Riesz means, Nörlund methods and Tauberian theory. Since his death, further papers with joint authorship have continued to appear; his influence persists.