The class of subexponential distributions S is characterized by F(0) = 0, 1 − F(2)(x) ~ 2(1 − F(x)) as x → ∞. In this paper we consider a subclass of S for which the relation 1 − F(2)(x) − 2(1 − F(x)) + (1 − F(x))2 = o(a(x)) as x → ∞ holds, where α is a positive function satisfying α(X) = 0(1 − F(x)) (x → ∞).

A topological space X is said to be S-closed if every cover of X by regular closed sets of X has a finite subcover. In this note some characterizations of S-closed Hausdorff spaces are obtained.

Some properties of v-semiprime (v = 0, 1, 2) near-rings are pointed out. In particular v semiprime near-rings which contain nil non-nilpotent ideals are studied.

We provide an existence theorem for stable sets which is equivalent to Zorn's lemma and study the connections between the unique stable set for majorization and the stable sets for the dominance relation.

There are two types of quartic normal extensions of the rational field, depending on the Galois group of the generating equation. All such extensions are described here in a uniquely parametrized form.

Garsia's discovery that functions in the periodic Besov space λ(p-1, p, 1), with 1 < p < ∞, have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in λ(p-1, p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to λ(α, p, q)

Let be the class of normalized univalent functions in the unit disk. For f ∈ let Sf be the set of all star center points of f. Let 0 = where is the interior of Sf. The influence that the size of the set has on the Taylor coefficients of a function f ∈ 0 is examined, and estimates of these coefficients depending only on , as well as other results, are obtained.

Let ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

Let E and F constitute a Banach pairing. We prove that the algebra of F-nuclear operators on E, Nf (E), is amenable if and only if E is finite dimensional and is weakly amenable if and only if dim KF ≦ 1, and the trace on E⊗F is injective on KF. Here KF is the kernel of the canonical map E⊗^F →NF(E). On the route we find the corresponding statements for the associated tensor algebra, E⊗^F.

A locally convex space E is said to be ordered suprabarrelled if given any increasing sequence of subspaces of E covering E there is one of them which is suprabarrelled. In this paper we show that the space m0(X, Σ), where X is any set and Σ is a σ-algebra on X, is ordered suprabarrelled, given an affirmative answer to a previously raised question. We also include two applications of this result to the theory of vector measures.

A complete classification is given of harmonic morphisms to a surface and conformal foliations by geodesics, with or without isolated singularities, of a simply-connected space form. The method is to associate to any such a holomorphic map from a Riemann surface into the space of geodesics of the space form. Properties such as nonintersecting fibres (or leaves) are translated into conditions on the holomorphic mapping which show it must have a simple form corresponding to a standard example.

Let S, T1,… Tk be finite semigroups and Ψ: S → Ti, be embeddings. When C[S] is semisimple, we find necessary and sufficient conditions for the semigroup amalgam (T1,…, Tk; S) to be embeddable in a finite semigroup. As a consequence we show that if S is a finite semigroup with C[S] semisimple, then S is an amalgamation base for the class of finite semigroups if and only if the principal ideals of S are linearly ordered. Our proof uses both the theory of representations by transformations and the theory of matrix representations as developed by Clifford, Munn and Ponizovskii

In 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.

In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.

This paper is the first part of a long delayed revision of the manuscript ‘The growth conditions recurrence sequences’ (circulated in 1982) in which the authors outlined a proof of the now well known theorem on the finiteness of the number of solutions of S-unit equations. The argument lifting the result from number fields to arbitrary fields of characteristic zero has original features.

The purpose of this paper is to provide examples and explore properties of a wide variety of zero square (left) near rings. Among the main results are complete classifications of (i) finite Abelian groups which are the additive group of a zero square near-ring and (ii) finite non-Abelian groups which support 3-nilpotent distributive zero square near-rings.