A special Pisot number is a Pisot number α such that α/(α−1) is also a Pisot number. Lagarias, Porta and Stolarsky found 11 special Pisot numbers. Here it is proved that there are no more. Also, the one known Salem number τ with the property that τ/(τ−1) is a Pisot number is shown to be the only such number.

We prove that if A and B are semisimple Banach algebras, then the separating subspace of every Lie isomorphism from A onto B is contained in the centre of B.

We describe completely the possible Gorenstein Artin quotients of the coordinate ring of a set of points in projective space. In addition, we give a sufficient condition for an ideal of a Cohen–Macaulay graded k-algebra A to be isomorphic to the canonical module of A. This leads to a description of the canonical module in the case when A is the coordinate ring of points.

We give a structure theorem for pointed Hopf algebras of dimension 2n, having coradical kC2, where k is an algebraically closed field of characteristic zero.

We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to Fourier–Mukai transforms are discussed. In particular, we obtain a large number of such transforms for K3 surfaces.

A regular 2-complex has a group of cellular automorphisms which is transitive on 2-cells. Let p be an odd prime. By generalizing an example of Floyd and Parry [5], we obtain infinite families of compact, aspherical, regular 2-complexes with Euler characteristic p and infinite hyperbolic fundamental group.

The ‘isomorphism problem for integral group rings’ (IP) is the question whether for two finite groups G and H, the existence of an isomorphism of the integral group rings ℤG and ℤH implies that G and H are isomorphic. Though (IP) is not true in general [4], it is still an interesting question for which classes of finite groups (IP) has a positive solution. In this note, we want to show that (IP) holds for finite groups of Lie type associated to Chevalley groups of universal type.

Note that if U(ℤG) denotes the units of ℤG, and V(ℤG) stands for the units with augmentation 1, then U(ℤG)≅V(ℤG)×U(ℤ), and we can always assume that H[les ]V(ℤG).

Let ϕ[ratio ]G→G be an endomorphism of a finitely generated residually finite group. R. Hirshon asked if there exists n such that the restriction of ϕ to ϕn(G) is injective. We give an example to show that this is not always the case.

If two finitely generated, torsion-free, nilpotent groups of class two satisfy the two-arrow property (that is, they embed into each other with finite, relatively prime indices), then they necessarily belong to the same Mislin genus (that is, they have isomorphic localizations at every prime). Here we show that the other implication is false in general. We even provide counterexamples in the case where both groups have isomorphic localizations at every finite set of primes of bounded cardinality. The latter equivalence relation leads us to introduce the notion of n-genus for every positive integer n, which we show to be meaningful in various contexts. In particular, the two-arrow property is related to the n-genus in the context of topological spaces.

It is shown that every transcendental entire function f grows transcendentally in a sequence of cercles de remplissage. An example shows that if

formula here

then there may be no sequence of cercles de remplissage the union of which contains infinitely many zeros of f. It is also shown that every transcendental entire function f has a Hayman direction, that is, a direction θ such that, in every open sector containing θ, either f assumes all complex values infinitely often, or else every derivative of f assumes all complex values, except possibly zero, infinitely often.

We prove that if all the rank-one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank-one operator not attaining the numerical radius.

In this paper we characterise the kernel of a little Hankel operator on the Bergman space L2a(D) in terms of the inner divisors, and obtain a characterisation for finite rank little Hankel operators using the invariant subspace theory technique.

The question of which Riemannian manifolds admit simple closed geodesics is still a mystery. It is not known whether all closed Riemannian manifolds contain simple closed geodesics. For closed manifolds with nontrivial fundamental group, a simple closed geodesic can always be found by taking the shortest homotopically nontrivial closed geodesic. When the manifold is closed but simply connected, the question is open for dimensions three and above. In dimension two, it is known by the theorem of Lusternik and Schnirelmann [6] (see also [3] and [4]) that the 2-sphere equipped with any smooth Riemannian metric contains at least three distinct simple closed geodesics.

Non-compact manifolds do not necessarily contain closed geodesics, Euclidean space being an obvious example. Even if the manifold is not simply connected, it may not contain any simple closed geodesics, as with the hyperbolic thrice-punctured sphere. However, among the orientable, finite area, complete hyperbolic 2-manifolds, the thrice-punctured sphere is the only example that contains no simple closed geodesic. In this paper, we shall determine which orientable hyperbolic 3-manifolds do and do not contain simple closed geodesics. We shall prove that the Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite, or even that it has finitely generated fundamental group. The simple closed geodesic which we produce arises from an interesting class of elements of the fundamental group. It is the shortest closed geodesic corresponding to a screw motion induced by the action of the fundamental group on hyperbolic 3-space.

In proving our result, we use geometric methods to obtain results about isometries of hyperbolic 3-space. These results apply to elliptic, as well as to parabolic and loxodromic isometries, and we state them in full generality.

A related question is whether a hyperbolic 3-manifold always contains a non-simple closed geodesic. In [2], Alan Reid and Ted Chinburg utilized arithmetic hyperbolic 3-manifold theory to construct examples of closed hyperbolic 3-manifolds in which every closed geodesic is simple.

We shall be working with orientable 3-manifolds, so all of the isometries discussed will be orientation preserving.

Let f be an orientation-preserving univalent harmonic mapping of the unit disk U. Then f=h+[gmacron], where h and g are analytic in U. Furthermore, f satisfies the equation

formula here

in U, where a(z)=g′(z)/h′(z), and [mid ]a(z)[mid ]<1 in U. The function a(z) is the analytic dilatation of f.

In [2], Hengartner and Schober proved the following version of the Riemann mapping theorem for univalent harmonic mappings.

Throughout this paper, all 3-manifolds are closed and orientable. Our aim is to give some new examples of 3-manifolds which are virtual bundles, that is, have finite sheeted covers which are surface bundles over the circle. Thurston has raised the question as to whether all irreducible atoroidal 3-manifolds might have this property. The geometrisation conjecture would imply that all such 3-manifolds have hyperbolic metrics, and the virtual bundle question is then equivalent to showing the existence of geometrically infinite incompressible surfaces (compare [14]).

Not much progress has been made on this problem. Recently, Reid [9] has given an explicit example using an arithmetic hyperbolic 3-manifold, and Cooper, Long and Reid [3] have given methods of constructing immersed incompressible surface in surfaces bundles over the circle, which are sometimes geometrically infinite. In [10], examples are given of horizontal immersed incompressible surfaces in graph manifolds which do not lift to embeddings in any finite sheeted covering spaces. Similar examples are given in toroidal manifolds with cubings of non-positive curvatures in [9]. So the assumption that the manifold has the atoroidal property is essential in Thurston's question. Our aim is to give several simple constructions of large classes of examples using different polyhedral metrics of non-positive curvature on 3-manifolds (compare [2]).

We follow closely the construction of Thurston (compare [13]), where he observes that the right-angled regular hyperbolic dodecahedron can be viewed as a cube with arcs drawn on each face through midpoints of a pair of opposite sides so that no two such arcs share a common vertex. Then the foliation of the cube by planes orthogonal to a diagonal gives an induced foliation of any 3-manifold arising from a torsion-free finite index subgroup of the symmetry group of the tessellation of hyperbolic 3-space by the right-angled dodecahedron. Clearly, any leaf of this foliation is then an immersed geometrically infinite surface. One explicit example is then the 4-fold cyclic branched cover of the Borromean rings, since the Borromean rings has a flat 2-fold branched cover which is obtained by gluing two cubes together. This example is formed by a subgroup of index 4 in the above symmetry group.

Our examples come directly from this observation of Thurston, using different ways of dividing a 3-manifold into simple polyhedra so that there is an overall metric of non-positive curvature. We use some well-known tessellations of Euclidean 3-dimensional space, described in Coxeter [4], by truncated octahedra and tetrahedra, plus flying saucers (compare [2]), as well as the easy case of cubes.

In the final section we show that our classes of cubed examples and flying saucers admit geometric decompositions in the sense of Thurston, since they are virtual bundles. We would like to thank the referee for a number of helpful comments which improved the exposition.

Gordon Marshall Petersen, a member of the London Mathematical Society from 1958, was born on 25 November 1921 in San Francisco, and died on 9 November 1996 in Christchurch, New Zealand.

His father was Danish, born in 1883 in Esbønderup, a village on the island of Zealand. While still only 15, he suddenly announced to his family that, having secretly saved up his fare, he was emigrating to America the next day. He had little formal education, but was able to support his family comfortably. One day during the Depression, father and son, in their shabbiest clothes, visited a Cadillac showroom and were about to be ejected when the father produced a paper bag stuffed with enough banknotes to buy the car they had been inspecting. No wonder Gordon, an only child, adored his father!

Gordon's mother Catherine was born in 1883 in America to Richard and Ann Marshall, who had not long before emigrated from Nottingham in England. She had three older siblings and also a younger brother, Gordon, who operated several garages and built his own aeroplanes. He survived the First World War in the US Army Air Force, but was killed in an air crash in 1920, at the age of 32.