Let ƒ(x) be a C-valued function which is integrable with respect to a positive measure m on the Borel σ-field of R = (– ∞, ∞). We shall give necessary and sufficient conditions under which the following Fourier-Laplace transform f(λ) of ƒ(x) decays exponentially as λ → ∞:

where θ is a constant with –π/2 ≤ θ ≤ π/2.

Singer [7] proved that in a Banach space all basic sequences are shrinking if and only if all of them are boundedly complete. Afterwards; Zippin [2] proved a similar result for Schauder bases, and it was extended [1] to Schauder decompositions.

We prove the result in the title. More generally we consider the Banach space υp, of all sequences (xn) of scalars such that

where the supremum runs over all increasing sequences n1 ≤ n2 ≤ …. We show that is of cotype 2 if p ≽ 2 and of cotype p′, where 1/p′ + 1/p = 1, if p ≤ 2. Similar results are obtained for the analogous function spaces.

The notion of packing measure, introduced in [12], [13] and [10], has been used by comparison with Hausdorif measure to study the regularity and rectifiability of sets in the plane [11]. Since a few technical mistakes can be found in [10], lemma 5·11 and [11], lemma 3·2, we wish in this paper to give the corresponding exact proofs, together with a natural development of this theory in higher dimensions.

An important class of systems of second order differential equations can be represented as sprays on a manifold M with tangent bundle TM↠ M; that is, as certain sections of the second tangent bundle TTM ↠ TM. We consider here quadratic sprays; they correspond to symmetric linear connections on TM ↠ M and hence to principal connections on the frame bundle LM ↠ M. Such connections over M constitute a system of connections, on which there is a universal connection and through which individual connections can be studied geometrically. Correspondingly, we obtain a universal spray-like field for the system of connections and each spray on M arises as a pullback of this ‘universal spray’. The Frölicher-Nijenhuis bracket determines for each spray (or connection) a Lie subalgebra of the Lie algebra of vector fields on M and this subalgebra consists precisely of those morphisms of TTM over TM which preserve the horizontal and vertical distributions; there is a universal version of this result. Each spray induces also a Riemannian structure on LM; it isometrically embeds this manifold as a section of the space of principal connections and gives a corresponding representation of TM as a section of the space of sprays. Such embeddings allow the formulation of global criteria for properties of sprays, in a natural context. For example, if LM is incomplete in a spray-metric then it is incomplete also in the spray-metric induced by a nearby spray, because that spray induces a nearby embedding. For Riemannian manifolds, completeness of LM is equivalent to completeness of M so in the above sense we can say that geodesic incompleteness is stable; it is known to be Whitney stable.

In a previous paper [6], the present authors established estimates for the continuous dependence of the solution on various data in the initial boundary value problem of linear elastodynamics on a bounded region of space. The main conclusion concerned continuous dependence on the body-force, but also it was shown how this result could be used to derive continuous dependence on the initial data, elasticities, boundary data and initial geometry. The method adopted was based upon logarithmic convexity arguments and hence led naturally to continuity in the sense of Hölder on compact sub-intervals of time. A special feature of the study entailed the lack of any sign-definiteness conditions on the elasticities which, of course, in the absence of any a priori constraint on the solution always gives rise to an ill-posed problem. (See, for instance, the comprehensive survey by Payne [10].)

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.

An element a of a Banach algebra A over ࠶ will be called quasi-central if, for some K ≥ 1,

The set Q(A) of all quasi-central elements of A will be called the quasi-centre of A and the set of elements a which satisfy (1) for a particular value of K will be denoted by Q(K, A). The motivation for these definitions is the result of Le Page ([1], proposition 3) that Q(1, A) coincides with the centre Z(A) of A. The reader is referred to [2] and [3] for a study of the properties of quasi-central elements.

If T is a linear transformation on ℝn with singular values α1 ≥ α2 ≥ … ≥ αn, the singular value function øs is defined by where m is the smallest integer greater than or equal to s. Let T1, …, Tk be contractive linear transformations on ℝn. Let where the sum is over all finite sequences (i1, …, ir) with 1 ≤ ij ≤ k. Then for almost all (a1, …, ak) ∈ ℝnk, the unique non-empty compact set F satisfying has Hausdorff dimension min{d, n}. Moreover the ‘box counting’ dimension of F is almost surely equal to this number.

§1. A real-valued arithmetic function f is additive if it satisfies the relation f(ab) = f(a) + f(b) for all mutually prime positive integers a, b. In the present paper I establish three theorems concerning the value distribution of such functions on intervals.

We consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support. It is first shown that the set of far field patterns of the electric fields corresponding to incident plane waves propagating in arbitrary directions is complete in the space of square-integrable tangential vector fields defined on the unit sphere. We then show that under certain conditions the electric far field patterns satisfy an integral identity involving the unique solution of a new class of boundary value problems for Maxwell's equations called the interior transmission problem for electromagnetic waves. Finally, it is indicated how this integral identity can be used to formulate an optimization scheme yielding an optimal solution of the inverse scattering problem for electromagnetic waves.

In 1950, Dvoretzky, Erdös and Kakutani [2] showed that in ℝ3 almost all paths of Brownian motion X have double points, or self-intersections of order 2 (there are no triple points [4]); later the same authors proved that almost all sample paths of Brownian motion in the plane have points of arbitrarily high multiplicity (a point x in ℝ2 is a k-tuple point for the path ω, or a self-intersection of order k, if there are times tl < t2 < … < tk such that x = X(t1, ω) = X(t2, ω) = … X(tk, ω)).

Let (Xt: t ≥ 0) be the Markov branching process (MBP) with a density independent catastrophe component. It is denned to be the Feller process on the non-negative integers having the generator

Here {pj} is the offspring distribution which satisfies p1 = 0 and p0 < 1, ρ is the per capita birth rate, κ is the rate of occurrence of catastrophe events, {δj: j ≥ 0} is the decrement distribution and . Thus Xt can be interpreted as the size of a population in which individuals reproduce according to the rules of a MBP – see Athreya and Ney[1], chap, III – and where there is an external and independent Poisson process of catastrophe events, κ per unit time, and if j < i each such event reduces the population size by j with probability δj. Usually we assume that δ0 = 0 on the basis that a catastrophe always reduces the population size. Let f(s) = Σpjsj and assume that . This ensures that the MBP obtained by setting κ = 0 is regular ([1], p. 105) and hence (Xt) is the unique Markov process corresponding to the above generator when κ > 0.

A stochastic process X = {X(t): t ∈ [0, 1]} on a probability space (Ω, , ℙ) is said to have finite expectation if the function defined on the measureable rectangles in Ω × [0, 1] by

for A ∈ and (s, t) ⊂ [0, 1] gives rise to a complex measure in each of its two coordinates (see [1], definition 1·1). Equivalently, X has finite expectation if

is finite. The function defined by (1), effectively a generalization of the Doléans measure (see e.g. [4] pp. 33–35), is extendible to a bona fide complex measure on Ω × [0, 1] if and only if its ‘total variation’

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ3 equivariant with respect to a particular finite subgroup G ⊂ O(3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

Categories of rigid/plastic response are investigated along arbitrary paths of homogeneous deformation. Successive yield surfaces are presumed to be convex and self-similar relative to certain canonical measures of stress and finite strain. After a general analysis attention is focussed on path-sensitive response in configurations where the canonical rate of hardening vanishes. The analysis is subsequently extended to other types of eigenstate, especially those associated with actual or prospective loading devices. The results are illustrated in the context of formability tests on sheet metal.

Volume 102 (1987), 17–23

‘The maximal subgroups of Fi22’

We reported on some computer calculations which we used to complete the enumeration of the maximal subgroups of the sporadic simple group Fi22 and of its automorphism group Fi22:2. Unfortunately there was an error in these calculations. We have therefore repeated all the calculations, incorporating much more thorough checking routines.