From the viewpoint of Birman and Menasco, a particular type of singular foliation on a surface with boundary induces an embedding of the surface in 3-space, such that the boundary of the surface is braided relative to the z-axis; hence the foliation determines a conjugacy class in the braid group. Birman and Hirsch describe an explicit algorithm to find a braid word representing this conjugacy class, given the foliation. A braid word β ∈ Bn is said to be irreducible if it is not conjugate to a braid of the form bσ±1n−1, with b ∈ Bn−1. We exhibit a foliation of a disk and show that the corresponding braid word is an irreducible element of B4. We give an explicit geometric description of the embedding induced by the foliation and describe a particularly nice form of symmetry possessed by this example.

It is a classical result of Postnikov [15] that homotopy types X with at most two non-trivial homotopy groups πmX = A and πnX = B, 2 [les ] m < n, are classified by the k-invariant

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Here the cohomology group of the Eilenberg–MacLane space K(A, m) was computed by Eilenberg–MacLane [11] and Cartan [5]. Let p be a prime and let ℤp ⊂ ℚ be the smallest subring of ℚ containing 1/q for all primes q with q ≠ p. We consider finitely generated ℤp-modules A and B and the stable range n < 2m − 1. Hence X is a p-local space with at most two non-trivial homotopy groups in a stable range. Then the homotopy type of X admits a product decomposition

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where all Xi with 1 [les ] i [les ] j are indecomposable and this decomposition is unique up to permutation. We classify in this paper the indecomposable factors in (2) by the following result.

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. In this paper we study Lagrangian immersions in complex space forms which are ideal. We prove that all Lagrangian ideal immersions in a complex space form are minimal. We also determine ideal Lagrangian submanifolds in complex space forms.

Let M be a complete hyperbolic n-manifold of finite volume. By a systole of M we mean a shortest closed geodesic in M. By the systole length of M we mean the length of a systole. We denote this by sl (M). In the case when M is closed, the systole length is simply twice the injectivity radius of M. In the presence of cusps, injectivity radius becomes arbitrarily small and it is for this reason we use the language of ‘systole length’.

In the context of hyperbolic surfaces of finite volume, much work has been done on systoles; we refer the reader to [2, 10–12] for some results. In dimension 3, little seems known about systoles. The main result in this paper is the following (see below for definitions):

We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.

The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.

A family of sets {Fd}d is said to be ‘represented by the measure μ’ if, for each d, the set Fd comprises those points at which the local dimension of μ takes some specific value (depending on d). Finding the Hausdorff dimension of these sets may then be thought of as finding the dimension spectrum, or multifractal spectrum, of μ. This situation pertains surprisingly often, with many familiar families of sets representable by measures which have simple dimension spectra. Examples are given from Diophantine approximation, Kleinian groups and hyperbolic dynamical systems.

An orthonormal basis of analytic functions in the Bergman space is a doubly orthogonal system if it simultaneously is an orthogonal system in a larger domain. The case when one of the basis elements is a monomial is characterized as belonging to a pair of domains of concentric discs.

Suppose [Cscr ] and [Cscr ]′ are two sets of simple closed curves on a hyperbolic surface F. We give necessary and sufficient conditions for the existence of a pseudo-Anosov map g such that g([Cscr ]) ≅ [Cscr ]′.

In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension.

We prove that in a separable infinite dimensional Hilbert space the Dynkin system generated by the family of all open balls (that is, the smallest collection containing the open balls and closed under complements and countable disjoint unions) does not contain all Borel sets.

Consider an (L, α)-superdiffusion X, 1 < α [les ] 2, in a smooth cylinder Q = ℝ+ × D. Where L is a uniformly elliptic operator on ℝ+ × ℝd and D is a bounded smooth domain in ℝd. Criteria for determining which (internal) subsets of Q are not hit by the graph [Gscr ] of X were established by Dynkin [5] in terms of Bessel capacity and according to Sheu [14] in terms of restricted Hausdorff dimension (partial results were also obtained by Barlow, Evans and Perkins [3]). While using Poisson capacity on the lateral boundary ∂Q of Q, Kuznetsov [10] recently characterized complete subsets of ∂Q which have no intersection with [Gscr ]. In this work, we examine the relations between Poisson capacity and restricted Hausdorff measure. According to our results, the critical restricted Hausdorff dimension for the lateral [Gscr ]-polarity is d − (3 − α)/(α − 1). (A similar result also holds for the case d = (3 − α)/(α − 1)). This investigation provides a different proof for the critical dimension of the boundary polarity for the range of X (as established earlier by Le Gall [12] for L = Δ, α = 2 and by Dynkin and Kuznetsov [7] for the general case).

The symmetric imprimitivity theorem provides a Morita equivalence between two crossed products of induced C*-algebras and includes as special cases many other important Morita equivalences such as Green's imprimitivity theorem. We show that the symmetric imprimitivity theorem is compatible with various inflated actions and coactions on the crossed products.

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.

The Pisier algebra [Pscr ] consists of those continuous functions f on the unit circle [ ] for which

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is continuous on [ ] almost surely, that is, for almost every choice of a sequence of signs (±1, ±1, …). In this paper, we prove that spectral synthesis holds in [Pscr ]. Moreover, we show that certain closed ideals in [Pscr ] with infinite hull have bounded approximate identities and give an example of a closed ideal in [Pscr ] which does not have a bounded approximate identity.

Sanov's Theorem states that the sequence of empirical measures associated with a sequence of i.d.d. random variables satisfies the large deviation principle (LDP) in the weak topology with rate function given by a relative entropy. We present a derivative which allows one to establish LDPs for symmetric functions of many i.d.d. random variables under the condition that (i) a law of large numbers holds whatever the underlying distribution and (ii) the functions are uniformly Lipschitz. The heuristic (of the title) is that the LDP follows from (i) provided the functions are ‘sufficiently smooth’. As an application, we obtain large deviations results for the stochastic bin-packing problem.

Mourre's commutator method is used together with Besov spaces, in this paper, to settle the question of uniqueness of solutions to the scattering problem for the acoustic wave propagation in perturbed stratified fluids in Rn with n [ges ] 2: The uniqueness of solutions is established by introducing a radiation condition in the framework of Besov spaces, and is then used to prove resolvent estimates in the Besov space setting for the acoustic propagator in perturbed stratified fluids. These estimates are ‘sharp’ in a sense made precise in the text and are important in establishing existence of solutions to the scattering problem.

Let f: C → C denote a transcendental entire function and let m(r, f) equal min {[mid ]f(z)[mid ][ratio ][mid ]z[mid ] = r}. In this note, we mainly prove that if

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then the set of normality N(f) of f has no unbounded preperiodic and periodic components. In particular, f has no Baker domains. We also discuss non-existence of unbounded components of normality under other conditions.

Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov–Poisson system, which is the non-relativistic case, or of the Vlasov–Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function Φ(E, L) of the particle energy and angular momentum. We give a new condition on Φ which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour of Φ for E → E0 needs to be prescribed, where E0 is a cut-off energy above which no particles exist.

(Volume 125 (1999), 433–440)

On page 438 of this article Figure 1 was incorrectly printed. The correct version appears below.

Fig. 1.

The Editors and Publishers apologise for any confusion this oversight may have caused.