If a linear operator A in a Banach space satisfies certain conditions, then the spectrum of any almost periodic solution of the differential equation u′ = Au + f is shown to be identical with the spectrum of f, where f is a Stepanov almost periodic function.

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

Spitzbart [1] has considered a generalization of Hermite's interpolation formula in one variable and has obtained a polynomial p(x) of degree n + Σnj=0 = rj in x which interpolates to the values of a function and its derivatives up to order rj at xj, j = 0, 1,···n. Ahlin [2] has considered a bivariate generalization of Hermite's interpolation formula. He has developed a bivariate osculatory interpolation polynomial which agrees with f(x, y) and its partial and mixed partial derivatives up to a specified order at each of the nodes of a Cartesian grid. However, the above interpolation problem considered by Ahlin assumes that the values of partial and mixed partial derivatives of the same fixed order k – 1 are available at every point of the rectangular grid. It may also be observed that Ahlin's formula is essentially a Cartesian product of a special case of Spitzbart's formula in one variable.

Magill in [4] first proved that two Boolean rings are isomorphic if and only if their respective endomorphism semigroups are isomorphic. His proof, however, relied on topological techniques. More recently Maxson has published a proof of the above using purely algebraic techniques [5]. In this paper, structure theorems are given which allow us to extend the above result to the pk-rings of Foster [1]. As a special case, the result is shown to apply also to p-rings. An example is given to show that a further extension to J-rings is impossible.

Suppose that A = [αnk], (n, k = 0, 1, 2, …), is an indinite matrix with complex entries. A transforms a complex sequence a = {an to a complex sequence {bn = b = Aa where assuming that the series in (1) converges. Each sequence a = {an is uniquely associated with a power series In this way the matrix A transforms a power series into a power series. Specifically, the power series (2) is mapped to the power series where the bn's are given by (1). We are only interested in matrices having the property that each power series analytic in Δ = {z: |z| < 1} maps to a power series analytic in Δ.

In a previous paper [2], we gave an explicit description of the structure of all semirings with a completely simple additive semigroup. The next step is then clearly to consider semirings with a completely 0-simple additive semigroup. We are able to classify these semirings according to the multiplicative nature of their additive zero. Let R be a semiring whose additive semigroup is completely 0-simple with zero ∞. First, if ∞ ∞ ≠ ∞, then the multiplication of R is trivial. Besides these trivial semirings, another class of semirings with a completely 0-simple additive semigroup can be easily obtained by adjoining an element ∞ which is together an additive zero and a multiplicative zero.

Dickson's construction [1] of radical and semi-simple classes for certain abelian categories is a rather straightforward procedure in comparison with the methods traditionally used in more general situations. In §2 of the present paper we use a well-known characterization of the lower radical class to obtain, via consideration of maps with accessible images, a similar “homomorphic orthogonality” characterization of radical and semi-simple classes of associative rings. By substituting certain other subring properties for accessibility, we are then able to obtain simple constructions of various types of radical classes, including those which are strict in the sense first used by Kurosh [3] for groups.

Several classes of topological spaces which are either the infimum or supremum of a family of metrizable topologies in the lattice of topologies were given by Anderson [1]. In this paper we investigate the completion of the set of metrizable topologies in four distinct convergence structure lattices on an infinite set. We also observe the influence on these completions of the question regarding the existence of nonnormal ultrafilters which Čech [3] asserted could be neither provednor disproved using present mathematical axiom systems.

In this paper, we consider factor covered graphs, which are defined basically as connected graphs in which every edge belongs to a 1-factor. The main theorem is that for any two edges e and e′ of a factor covered graph, there is a cycle C passing through e and e′ such that the edge set of C is the symmetric difference of two 1-factors.

Investigations of closure operators on an involution poset T lead to a certain type of closure operators (so called c-closure operators) that are closely related to projections on T.

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.

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.

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.

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.

The purpose of this paper is to generalize some results in a recent paper by Tomiuk and the author ([11]).

For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].

The qualitative behaviour of control systems based on ordinary differential equations has been investigated with clarity and elegance using axiomatically defined General Control Systems. Here an attainablity set function, evolving in semigroup fashion, is the main entity of interest [1], [2], [3], [4].

In this paper our graphs will be finite, undirected, and without loops or multiple edges. We will denote the set of vertices of a graph G by V(G). If G is a graph and u, v∈V(G), then we will write u ∼ v to denote that u and v are adjacent and u ≁ v otherwise. If A ⊆ V(G), then we let N(A) = {u∈ V(G)|u ∼ a for each a ∈A}. However we write N(v) instead of N({v}). When there is no chance of confusion, we will not distinguish between a subset A ⊆ V(G) of vertices of G and the subgraph that it induces. We will denote the cardinality of a set A by |A|. The degree of a vertex v is δ(v) = |N(v)|. Any undefined terminology in this paper will generally conform with Behzad and Chartrand [1].

In [1] Atherton has asked whether the ideal topology (see [1]) is Hausdorff on every distributive lattice. He also gave some sufficient conditions (Kent's conditions cl and c3 [1]) under which the ideal topology on a complete distributive lattice is Hausdorff. In this paper we determine some more classes of lattices in which the ideal topology is Hausdorff and give examples to show that the classes determined are independent even in a complete distributive lattice. We also show by an example that Atherton's conditions are not necessary even in the case of a complete distributive lattice.