This note is a continuation of our attempts (see [3]) to give a satisfactory representation-theoretic justification of the following formula of D. E. Littlewood:

where sI is the Schur symmetric function corresponding to a partition I, |I| is the weight of I, r(I) is the rank of I, and the summation ranges over all self-conjugate partitions (i.e. partitions I such that I = I where I is the partition conjugate to I).

In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequality

for almost all α (in the sense of Lebesgue measure on Iℝ), where , and both m and n are restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):

Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sum

diverges, or converges, respectively.

In Theorem 1·2 of [3] the following extension of the classical theorem of Shult is proved.

Theorem. Let G be a p-solvable group and let V be a faithful KG-module, where char. Assume that G contains an element x of order pn acting fixed-pointfreely on V. If p is a Fermat prime suppose further that the Sylow 2-subgroups of G are abelian. If p = 2 assume that the Sylow q-subgroups of G for each Mersenne prime q less than 2n are abelian. Then

Our object is to describe all of the-primitive permutation groups of degree less than 1000 together with some of their significant properties. We think that such a list is of interest in illustrating in concrete form the kinds of primitive groups which arise, in suggesting conjectures about primitive groups, and in settling small exceptional cases which often occur in proofs of theorems about permutation groups. The range that we consider is large enough to allow examples of most of the types of primitive group to appear. Earlier lists (of varying completeness and accuracy) of primitive groups of degree d have been published by: C. Jordan (1872) [21] for d ≤ 17, by W. Burnside (1897) [5] for d ≤ 8, by Manning (1929) [34–38] for d ≤ 15, by C. C. Sims (1970) [45] for d ≤ 20, and by B. A. Pogorelev (1980) [42] for d ≤ 50. Unpublished lists have also been prepared by C. C. Sims for d ≤ 50 and by Mizutani[41] for d ≤ 48. Using the classification of finite simple groups which was completed in 1981 we have been able to extend the list much further. Our task has been greatly simplified by the detailed information about many finite simple groups which is available in the Atlas of Finite Groups which we will refer to as the Atlas [8].

For any category A and group G there is a category AG of A-objects with G-action. An object of AG is an A-object A together with a group morphism, θA:G → Isom (A, A), of G into the group of invertible A-morphisms A → A. A morphism of AG is an A-morphism f: A → B which commutes with the G-action; that is, for each g∈G, the following diagram commutes.

In this study of complete, or integrally closed, ideals in a two-dimensional regular local ring (R, m), Zariski established a one-to-one correspondence between prime divisors of R, i.e. rank 1 discrete valuations v birationally dominating R with residue field of transcendence degree 1 over R/m, and m-primary simple complete ideals Iv in R; cf. [17] and [18]. In this correspondence, the blow-up of such an ideal has unique exceptional prime and the localization at this prime is the valuation ring of a prime divisor of R. In this paper, we will study such ideals in a more general setting, so we begin by recalling some definitions and background results.

In the theory of operator algebras the rings of finite matrices over such algebras play a very important role (see [10]). For commutative operator algebras, the Gelfand-Naimark representation allows one to concentrate on matrices over rings of continuous complex functions on compact Hausdorif spaces.

A non-zero n-by-n matrix A is said to be completely positive if there exist non-negative vectors b1,…, bk, such that

The smallest such integer k is called the factorization index of (completely positive) A, and is denoted by ø(A). Completely positive matrices are important in the study of block designs [4], statistics and modelling of energy demand [3].

Several authors have considered relationships between individuals specified by the genes which they have in common, where two genes are regarded as the same if and only if they are identical by descent from some common ancestor. In particular Thompson [6] considered n pairs of genes, one pair from each of n individuals, and studied the number Nn of gene identity states, the number Mn, k of such states in which there are exactly k distinct genes and the number Dn of genetically distinct states (see Definitions 2 and 4 below). In this type of work the nature of the genes themselves is of no interest, but only which genes are identical to which, and precise definitions are best given in terms of orbits under group actions. Thompson did use some group theory in her work, but not the full machinery of Redfield-Pólya-de Bruijn enumeration.

Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q + 1, q + 1)/S[U(q + 1, q) × U(1)], and SU(q + 1, q + 1)/SL(n, ℂ) ⋊ H(n) (p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

A simple proof of Denjoy's theorem on topological conjugacy of circle maps with irrational rotation number to uniform rotation is presented.

Exact sequences which can be used for calculating the group (X) of homotopy classes of homotopy self-equivalences of a space X are known in the case of cellular decompositions (some references are given in § 1 of [8]), and in the case of Postnikov decompositions (for example 3·1 of [5]). Here we obtain exact sequences which can be used to calculate (X) using a simply-connected homology decomposition.

In a 1928 paper [1], Hardy and Littlewood published the following inequality which is valid for all non-negative measurable functions ƒ and g on ( − ∞, ∞):

where 1 < p, q < ∞, 0 < λ < 1, l/p + 1/q + λ = 2 and Cλp, q depends only on λ, p, q; ‖ƒ‖p and ‖g‖q are the Lp and Lq norms of ƒ and g respectively. See also p. 288 of the monograph [2] by Hardy, Littlewood and Pólya for a discussion of this inequality and related matters.

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.

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.

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.

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’

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.