We extend results of McMullen about the dynamics of entire functions for which the orbits of the critical values stay away from the Julia set. In particular we show that such functions are expanding on their Julia sets which have self-similarity properties. Under suitable further conditions the Julia sets have plane measure zero.

Weak precompactness in spaces of vector measures and in the space of Bochner integrable functions is studied. Uniform countable additivity and uniform integrability are characterized in terms of weak precompactness. Through this, a connection between strongly bounded operators and operators having weakly precompact adjoints on abstract continuous function spaces is established. These operators are compared with weakly completely continuous operators.

The Rikitake two-disc system [10] has been proposed as a model of the Earth's magnetic dipole. Numerical integration of the system by Rikitake[10] and Allan [1] showed the phenomenon of polarity reversal which is seen in terrestrial paleomagnetic data [6]. Reversals in the simulations occurred at irregular and apparently unpredictable time intervals, in accord with the capricious character of the real data, though the agreement was qualitative rather than quantitative.

We consider invariant positive sesquilinear forms on a (partial) *-algebra A without unit. First we investigate the relationship between extendability and representability for such a form ø; in particular we discuss under which conditions the two concepts are equivalent. Then we introduce the notions of weak representability and strict unrepresentability, and we show that every fully invariant positive sesquilinear form on A × A is uniquely decomposed into a weakly representable part and a strictly unrepresentable part.

The data analytical problem of testing whether an empirical set of n given points in the plane could be considered to contain too many straight line configurations in a situation where the generating mechanism of the n points is unknown, was recently reintroduced to the literature by D. G. Kendall and W. S. Kendall[7]. Taking the Land's end data problem (cf. also Broadbent [1] and further references given there) as the anchor point and motivation for their discussion, D. G. and W. S. Kendall developed a new testing device, called the pontogram. The pontogram is a one- parameter stochastic process defined pointwise by

where N denotes a Poisson process in [0, 1] with N(0) = 0 and unknown intensity parameter μ > 0 and where R = N(1) gives the number of Poisson events in the entire [0, 1] section.

In this paper we give a description of the qualitative behaviour of the orbits near a non-trivial compact minimal set of a continuous flow on a 2-manifold. The first results in this direction were obtained in [1] and the present paper can be regarded as a continuation of that work. The main result can be stated as follows:

Theorem 1·1. Let (ℝ, M, f) be a continuous flow on a 2-manifold M and A ⊂ M a non-trivial compact minimal set.

Throughout the paper, A will denote a unital Banach algebra with centre Z over the complex field ℂ. An element a of A is called quasi-central if, for some K ≥ 1,

The set of elements a satisfying (1) for a particular value of K will be denoted by Q(K) and Q = UK≥1Q(K) will be called the quasi-centre of A.

When two purely imaginary eigenvalues of opposite Krein signature coalesce, in a Hamiltonian system, a small perturbation can drive them off of the imaginary axis resulting in a linear instability. The most celebrated example of this instability occurs in the restricted 3-body problem at Routh's critical mass ratio. In this paper the collision of eigenvalues is treated as a singularity. A variational form of the Lyapunov–Schmidt method and distinguished parameter ࡃ2-equivariant singularity theory, with the frequency as distinguished parameter, are used to determine the effect of the degeneracy on the branches of periodic solutions in a neighbourhood. Previous results of Meyer and Schmidt[13], Sokol'skij [16] and van der Meer [12] are recovered in the formulation as a co-dimension 1 singularity. The results are extended to include the effect of an additional degeneracy (a co-dimension 2 singularity). The theory is applied to a spinning double pendulum.

Let T be an operator on a complex Banach space such that sup {‖Tn‖: n ≥ 0} < ∞; such operators are said to be power-bounded. For such an operator one defines the peripheral spectrum to be the set

where σ(T) denotes the spectrum of T, and T is the unit circle in ℂ. Esterle proved in [3] that limnn→∞ ‖Tn−Tn+1‖ = 0 provided that ‖T‖ ≤ 1 and σ{T} = {1}. Later Katznelson and Tzafriri [6] proved the same result for power-bounded operators such that σΓ(T) = {1}, and in fact they proved a more general result. More precisely, let be an absolutely convergent Taylor series, i.e. , and assume that f is of spectral synthesis for σΓ(T) (in particular f|σΓ(T) ≡ 0). Then limn→∞ ‖f(T) Tn‖ = 0. Katznelson and Tzafriri prove an analogous result, in which T is replaced by a commuting k-tuple (T1, T2, …, Tk) of contractions. In this case they consider functions of the form f(T1, T2, …, Tk), where f is either an absolutely convergent Taylor series if k is odd, or an element of a smaller algebra if k is even. The result of Katznelson and Tzafriri was further extended in the case of a single contraction T by Esterle, Strouse and Zouakia[4], who consider functions f(T) where f is allowed to be in a more general function algebra. For instance, if T acts on a Hilbert space, they allow f to be an element of the disk algebra.

The purpose of this paper is to investigate the asymptotic behaviour as r → 1− of the integrals

and f is an analytic function on the unit disk Δ which has non-tangential limits at almost every point on ∂Δ. The paper is divided into three parts. In the first part we consider the case where λ ≠ 1/k, in the second the somewhat more delicate case when λ = 1/k and in the third part we concentrate on some problems related to the case λ = k = 1.

Let O denote the origin of ℝn (where n ≥ 2), let B(X, r) denote the open ball in ℝn of centre X and radius r, and define φn: (0, + ∞) → ℝ by φn(t) = t2-n if n ≥ 3 and φ2(t) = log (1/t). A sequence (Xk) in B(O, 1) is said to converge regularly to O if Xk → O and there is a constant k > 1 such that φn(|Xk+1|) < Kφn(|Xk|) for each k ∈ ℕ. If n ≥ 3, this is clearly equivalent to saying that there is a constant k′ ε (0, 1) such that |Xk+1| > k′|Xk| for each k (cf. [18], p. 149).

Of late the link between operator algebras and certain tensor products has been reiterated [5]. We prove here that the projective and Haagerup tensor products of two infinite-dimensional C*-algebras is not even topologically isomorphic to an algebra of operators on a Hilbert space. Estimates are given for the distance of the tensor product from such an algebra. Nonetheless with respect to a natural multiplication the Haagerup tensor product of two algebras of Hilbert space operators is completely isometrically isomorphic to an algebra of operators on some B(ℋ).

Artstein has given a characterization of the distributions induced by selections of a random compact subset of a complete separable metric space. In this paper, Artstein's results are extended to spaces which may be neither metric nor separable.

Conditions are given on the real coefficients p, q and r and the weight w, for the fourth order formally symmetric differential expression

to have the properties of being strong limit-2 and Dirichlet at ∞, when considered in the weighted Hilbert space, . These extend existing results due to both W. N. Everitt and V. Krishna Kumar and cover an expression which is important in the study of certain orthogonal polynomials.