In this paper we consider extensions of Livsic's well-known classical results on rigidity of measurable cocycles. We consider transformations which need not be hyperbolic, and cocycles which take values in certain non-abelian non-compact groups.

One important invariant of a closed Riemannian 3-manifold is the Chern–Simons invariant [1]. The concept was generalized to hyperbolic 3-manifolds with cusps in [11], and to geometric (spherical, euclidean or hyperbolic) 3-orbifolds, as particular cases of geometric cone-manifolds, in [7]. In this paper, we study the behaviour of this generalized invariant under change of orientation, and we give a method to compute it for hyperbolic 3-manifolds using virtually regular coverings [10]. We confine ourselves to virtually regular coverings because a covering of a geometric orbifold is a geometric manifold if and only if the covering is a virtually regular covering of the underlying space of the orbifold, branched over the singular locus. Therefore our work is the most general for the applications in mind; namely, computing volumes and Chern–Simons invariants of hyperbolic manifolds, using the computations for cone-manifolds for which a convenient Schläfli formula holds (see [7]). Among other results, we prove that every hyperbolic manifold obtained as a virtually regular covering of a figure-eight knot hyperbolic orbifold has rational Chern–Simons invariant. We give explicit examples with computations of volumes and Chern–Simons invariants for some hyperbolic 3-manifolds, to show the efficiency of our method. We also give examples of different hyperbolic manifolds with the same volume, whose Chern–Simons invariants (mod ½) differ by a rational number, as well as pairs of different hyperbolic manifolds with the same volume and the same Chern–Simons invariant (mod ½). (Examples of this type were also obtained in [12] and [9], but using mutation and surgery techniques, respectively, instead of coverings as we do here.)

We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the Radon–Nikodym property contains l1. It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

In 1980, Robertson, Tweddle and Yeomans solved the metrizable BCE problem in certain special cases, for example, in the normable case under assumption of the Continuum Hypothesis (CH). A complete positive solution was recently given by the author and L. M. Sánchez Ruiz under an assumption, weaker than CH, that scales exist. A better partial solution for the general BCE problem obviates scales or other extra-ZFC assumptions, and entirely lays to rest the metrizable BCE problem. The proof stimulates thought about how the least infinite-dimensionality of normed barrelled spaces impinges on the separable quotient problem for Banach spaces.

Let P be an n-dimensional polytope admitting a finite reflection group G as its symmetry group. Consider the set [Hscr ]P(k) of all continuous functions on Rn satisfying the mean value property with respect to the k-skeleton P(k) of P, as well as the set [Hscr ]G of all G-harmonic functions. Then a necessary and sufficient condition for the equality [Hscr ]P(k)=[Hscr ]G is given in terms of a distinguished invariant basis, called the canonical invariant basis, of G.

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.

We shall construct a map Ω0(G/H)→ΠαS2nα−1× ΠβΩS2nβ−1 which is a rational equivalence with applications to phantom maps, spaces of the same n-type and genus sets. This answers in the affirmative a question of C. McGibbon.

In the model of the OK Corral formulated by Williams and McIlroy [2]: ‘Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side.’ They are interested in the number S of survivors when the shooting ceases, and the surprising result, for which they give both numerical and heuristic evidence, is that when m=n,

formula here

as m→∞ (where E denotes expectation).

It is the occurrence of this curious power of m, rather than any application to real gunfights, which makes the Williams–McIlroy process of interest. The purpose of this paper is to give an essentially elementary proof of the fact that if m and n are not too different, then S/m3/4 has a simple asymptotic distribution that leads at once to (1.1).

We prove that the bound on the Lp norms of the Kakeya type maximal functions studied by Cordoba [2] and Bourgain [1] are sharp for p>2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r2 log (1/r) is the exact Minkowski dimension of the class of Kakeya sets in ℝ2, and prove that the exact Hausdorff dimension of these sets is between r2 log (1/r) and r2 log (1/r) [log log (1/r)]2+ε.

It is well known that there exist infinite closed subsets E of [ ] such that A(E) = C(E) (see, for example, [3]). Such sets are called Helson sets. Let E be a closed subset of [ ], let 0 < α < 1, and let Aα(E) be the restriction of the Beurling algebra Aα([ ]). Then Aα(E) ⊂ lipαE. We shall show that Aα(E) = lipαE if and only if E is finite. This answers a question raised by Pedersen [5], where partial results were obtained.

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.

A theorem on gambling systems for Bernoulli sequences is extended to arbitrary sequences of random variables by using the notion of the likelihood ratio, and the martingale technique together with the tool of moment generating functions for the study of a.e. convergence is presented.

Let a be an element in a complex Banach algebra with unit, and let r be a nonnegative number. The Gelfand spectral radius formula implies that the spectrum of a is included in the disk {z∈C:[mid ]z[mid ][les ]r} if and only if

formula here

Kunihiko Kodaira, who died on 26 July 1997, was the outstanding Japanese mathematician of the post-war period, his fame established by the award of the Fields Medal at the Amsterdam Congress in 1954.

He was born on 16 March 1915, the son of an agricultural scientist who at one time was Vice Minister of Agriculture in the Japanese Government and had also played an active role in agricultural developments in South America. Kodaira studied at Tokyo University, taking degrees in both mathematics and physics. From 1944 to 1951 he was an associate professor of physics at the University. His PhD thesis was published in the Annals of Mathematics [18], and it immediately attracted international attention. Essentially this filled a significant lacuna in the basic theorem of W. V. D. Hodge on harmonic integrals. Kodaira had worked on this for many years but, because of the war, his research was carried out in isolation from the international community and did not become known until much later.

Hermann Weyl, who had been a keen supporter of Hodge's work, realised the importance of Kodaira's thesis, and arranged for him to come to the Institute for Advanced Study in Princeton in 1949. This was the start of Kodaira's 18-year residence in the United States, a fruitful period which saw the full blossoming of his research, much of it in collaboration with Donald Spencer. Kodaira spent many years at Princeton, divided between the Institute and the University, but the years 1961–67 were more unsettled, seeing him successively at Harvard, Johns Hopkins and finally Stanford. In 1967 he returned to a professorship at the University of Tokyo, where he remained until the normal retiring age. From 1975 to 1985 he worked at Gakushuin University, where retirement restrictions did not apply.

We construct a c0-valued random variable X such that (Sn/√n)n∈ℕ has a convergent subsequence, but X does not satisfy the central limit theorem.

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.