In this paper we seek to establish some “complex inversion formulae” for the generalized Stieltjes transform for all s in the cut plane, supposing that p is any complex number except zero and the negative integers. The “cut plane” means all complex numbers except those which are negative real or zero.

In this paper it is shown that if c is a point of the region of convergence of an analytic function f(z) = Σ∞ncnZn then in every neighborhood of c there exists a point e such that the value f(c) of the function f(z) is attained by some truncation Σkn=0cnZn off(z) at z = e, i.e., Σkn=0cnen = Σ∞n=0cncn. Also it is shown that the above does not hold in the case of real-valued functions of a real variable.

A number of completions have been applied to p.o.-groups — the Dede kind-Macneille completion of archimedean l.o. groups; the lateral completion of l.o. groups (Conrad [2]); and the orthocompletion of l.o. groups (Bernau [1]). Fuchs in [3] has considered a completion of p.o. groups having a non-trivial open interval topology — the only l.o. groups of this form being fully ordered. He applies an ordering, which arises from the original partial order, to the group of round Cauchy filters over this topology; Kowaisky in [6] has shown that group, imbued with a suitable topology, is in fact the topological completion of the original group under its open interval topology. In this paper a slightly different ordering, also arising from the original order, is proposed for the group of round Cauchy filters; Fuchs' ordering can be obtained from this one as the associated order.

Birkhoff and Pierce [2] introduced the concept of an ƒ-ring and showed that an l-ring is an f-ring if and only if it is a subdirect product of totallyordered rings. An l-ideal of an f-ring R is an algebraic ideal which is at the same time a lattice ideal of R. Structure spaces (i.e. sets of prime ideals endowed with the so-called hull-kernel or Stone topology) for ordinary rings have been studied by many authors. In this paper we consider certain analogues for ƒ-rings, and give characterisations of ƒ-rings for which these structure spaces are discrete.

The present paper investigates the thermal instability of a non-homogeneous fluid rotating between two co-axial cylinders when the inner cylinder is being heated uniformly. The conditions are established under which the oscillatory and non-oscillatory modes exist and further it has been shown that the oscillatory modes are amplified due to the adverse temperature gradient. In the case of non-oscillatory modes, sufficient conditions for stability and instability are obtained.

Kent [4] showed that each uniform space (S, u) could be embedded in a complete, uniform lattice, called the scale of (S, u). The scale was first introduced by Bushaw [3] for studying stability in topological dynamics. In [5], the notions of connectedness and local connectedness were studied. This note is a follow-up of [5]; the purpose being to characterise the uniform subspaces of the reals, R, which have connected (locally connected) scales. The reader is asked to refer to [5] for definitions and notation not given here.

This note is a continuation of the author's work [6], describing the structure of a finite group given some information about the distribution of the subnormal subgroups in the lattice of all subgoups.

DEFINITION. An upper chain of length n in the finite group G is a sequence of subgroups of G; G = Go > G1 > … > Gn, such that for each i, Gi is a maximal subgroup of Gi-1. Let h(G) = n if every upper chain in G of length n contains a proper ( ≠ G) subnormal entry, and there is at least one upper chain in G of length (n – 1) which contains no proper subnormal entry.

Terminology and notation may be found in Dekker [1] and [2]. Briefly, we fix a recursively enumerable (r.e.) field F with recursive structure, and let Ū be the vector space over F consisting of ultimately vanishing countable sequences of elements of F with the usual definitions of vector addition and multiplication by a scalar. A subspace V of Ū is called an α-space if V has a basis B which is contained in some r.e. linearly independent set S.

In a recent issue of this Journal, Pu [3] has given an interesting construction of a nonmeasurable subset A of R such that for all intervals I in R. [Throughout this note, the symbol λ denotes Lebesgue outer measure on R or Haar outer measure on a general locally compact group.] This solves a problem stated in [2], p. 295, Exercise (18.30).

Let R be a ring with identity. We will use J(R) and P(R) to denote the Jacobson and prime radicals of R, respectively. If G is a group, the group ring of G over R will be denoted by RG.

A near ring is a triple (R, +, · ) such that (R, + ) is a group, (R, ·) is a semigroup, and · is left distributive over +; i.e. w(x + z) = wx + wz for each w, x, z in R. A normal subgroup K of a near ring R is an ideal if (i) (m + k)n − mn is in K for all m, n in R and k in K, and (ii) RK ⊆ K. In particular, kernels of near ring homomorphisms are ideals. For various other definitions and elementary facts about near rings, see [5,8]. For each x in a near ring R, let A(x) = {y ∈ R: xy = 0}. A survey on several recent papers on near rings [2,3,6,7,8] shows that the concept of A(x) being an ideal was the main technique. The purpose of this note is to initiate a study of near rings having the property that each A(x) is an ideal.

Let R be the set of real numbers, and let S1 denote the class of all real valued functions f on R which are smooth to the first order (i.e. the derivative ƒ(1) exists and is continuous) and have compact support. The first order variation off on an open set U is given by and in the case where U = R we have the total first order variation of f, usually denoted by I1(f).

We consider subspaces of a vector space UF, which is countably infinite dimensional over a recursively enumerable field F with recursive operations, where the operations in UF are also recursive, and where, of course, F and UF are sets of natural numbers. It is the object of this paper to investigate recursive equivalence types of such vector spaces and the ways in which their properties are analogous to and depend on properties of recursive equivalence types of sets.

Chen and Grätzer [3], [4] have had great success in describing the properties of Stone lattices by way of representing them as triples. Their triple representation has recently been generalized to distributive pseudo-complemented lattices by Katriňák [6]. By varying the approach slightly Katriňák [5] has been able to obtain a triple representation for distributive pseudo-complemented semilattices that enabled him to characterize semilattices from distributive pseudo-complemented semilattices through to Stone lattices and Brouwer lattices in an elegant unified manner.

The purpose of this note is to introduce some “canonical” characteristic, subgroups and to observe some of their more obvious properties.

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

Abstract: Under certain suitable conditions, the Stepanov-bounded solution of an abstract differential equation corresponding to a Stepanov almost periodic function is strongly (weakly) almost periodic.