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.

We work in the smooth category.

An (oriented) (ordered) m-component n-(dimensional) link is a smooth oriented submanifold L = {K1, …, Km} of Sn+2 which is the ordered disjoint union of m manifolds, each PL-homeomorphic to the standard n-sphere. If m = 1, then L is called a knot.

We say that m-component n-dimensional links L0 and L1 are (link-)concordant or (link-)cobordant if there is a smooth oriented submanifold C˜ = {C1, …, Cm} of Sn+2 × [0, 1] which meets the boundary transversely in ∂C˜, is PL-homeomorphic to L0 × [0, 1], and meets Sn+2 × {l} in Ll (l = 0, 1). If m = 1, then we say that n-knots L0 and Ll are (knot-)concordant or (knot-)cobordant. Then we call C a concordance-cylinder of the two n-knots L0 and Ll.

If an n-link L is concordant to the trivial link, then we call L a slice link.

If an n-link L = {K1, …, Km} ⊂ Sn+2 = ∂Bn+3 ⊂ Bn+3 is slice, then there is a disjoint union of (n + 1)-discs Dn+11 [amalg ] … [amalg ] Dn+1m in Bn+3 such that Dn+1i ∩ Sn+2 = ∂Dn+1i = Ki. (Dn+11, …, Dn+1m) is called a set of slice discs for L. If m = 1, then Dn+11 is called a slice disc for the knot L.

Let K be a cone in ℝn. Then K is a uniqueness set for the spherical Radon transform if and only if it is not contained in the zero set of any (nontrivial) homogeneous harmonic polynomial. A local version of this result is also proved.

Let [Ascr ] be a unital von Neumann algebra of operators on a complex separable Hilbert space ([Hscr ]0, and let {Tt, t [ges ] 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on [Ascr ]. The infinitesimal generator [Lscr ] of {Tt} is a bounded linear operator on the Banach space [Ascr ]. For any Hilbert space [Kscr ], denote by [Bscr ]([Kscr ]) the von Neumann algebra of all bounded operators on [Kscr ]. Christensen and Evans [3] have shown that [Lscr ] has the form

formula here

where π is a representation of [Ascr ] in [Bscr ]([Kscr ]) for some Hilbert space [Kscr ], R: [Hscr ]0 → [Kscr ] is a bounded operator satisfying the ‘minimality’ condition that the set {(RX−π(X)R)u, u∈[Hscr ]0, X∈[Ascr ]} is total in [Kscr ], and K0 is a fixed element of [Ascr ]. The unitality of {Tt} implies that [Lscr ](1) = 0, and consequently K0 = iH−½R*R, where H is a hermitian element of [Ascr ]. Thus (1.1) can be expressed as

formula here

We say that the quadruple ([Kscr ], π, R, H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator [Lscr ] of the semigroup {Tt}. It is quite possible that another set ([Kscr ]′, π′, R′, H′) of CE parameters may determine the same generator [Lscr ]. In such a case, we say that these two sets of CE parameters are equivalent. In Section 2 we study this equivalence relation in some detail.

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.

Based on a result of Hofer and Viterbo, we give an inequality for symplectic capacity as defined by Hofer and Zehnder, and, as an application, we give another proof of the symplectic packing inequality of Gromov.

Basil Rennie was born in London on 24 December 1920. He came from a long line of engineers, a family tradition that surfaced in much of his later mathematical work. He attended the University College School in London, where he obtained a Mathematical Scholarship at Peterhouse, Cambridge. After graduating in 1941, he found employment first with the Rolls Royce Aero Engine division, then with Austin Motor Works. In 1943 he joined the Fleet Air Arm of the Royal Navy as a radio mechanic, and he served in the Pacific Fleet until the end of the war. This was his first contact with Australia, and he seems to have liked what he saw.

After his service with the Navy, Rennie resumed his studies at Peterhouse and received a PhD in 1949. Given his strong practical bent, it is perhaps surprising that he chose lattice theory as the subject of his thesis; apart from an article [1] in the Proceedings of the London Mathematical Society (he became a member in 1947) and a small booklet [2] published at his own expense, he never touched lattice theory again. It was at Peterhouse that he took up rowing, an activity which became a life-long interest.

In 1950 Rennie accepted an offer of a senior lectureship at the University of Adelaide in South Australia. This was a time of considerable post-war expansion at the University, and its forward-looking Vice-Chancellor A. P. Rowe recruited a number of young and promising staff from overseas, some to leading positions. For instance, he established a Mathematical Physics department (unique in Australia) with the 30-year-old H. S. Green as its head, which became one of the most active research departments in Australia.

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.

Leslie Fox was born in Yorkshire in 1918, and spent most of his professional life in Oxford, as Director of the University Computing Laboratory and as the first Professor of Numerical Analysis in the University.

His mathematical education began at the Wheelwright Grammar School in Dewsbury, which produced a number of distinguished mathematicians and scientists at about the same time as Fox under the influence of an inspiring Headmaster and teacher (Leslie Sadler). Fox went to Oxford in 1936 as a scholar of Christ Church, and he gained a First Class Honours degree in Mathematics. He continued his studies for a DPhil under the direction of Professor Sir Richard Southwell, with a project in the area of computational and engineering mathematics which initiated some of the main interests of his career. His first appointment was at the Admiralty Computing Service in 1943; here he learnt the skills of table-making which he later used in a number of publications.

In 1945 Fox and several colleagues moved to the new Mathematics Division of the National Physical Laboratory. It was recognised at that time that the emerging technology of automatic computation would lead to requirements for effective mathematical methods which exploited the new machinery. The Mathematics Division embarked on an extensive programme of research in computational methods, in parallel with the construction of the Pilot ACE machine, which carried out its first computations in 1950. The Division remained a major source of ideas and methods in numerical mathematics for many years, with Fox taking a leading role until he was appointed to set up the Computing Laboratory in Oxford in 1957.