A general definition for a measure of nonprecompactness for bounded subsets of a locally convex linear topological space is given. Fixed point theorems for condensing multivalued mappings have been proved. These fixed point theorems are further generalizations of Kakutani's fixed point theorems.

We show that if the normal, closure of an element a, of an orderable group, G, is abelian, then G can be embedded in an orderable group, G#, which contains an n-th root of a for every positive integer, n. Furthermore, every order of G extends to an order of G#.

It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.

In this paper the right global divisible dimension and the right global codivisible dimension of a ring R are studied relative to a torsion theory of modR. The main result shows that if (A, B) is a central splitting torsion theory on modR. then the right global divisible dimension of R with respect to (B, A) is equal to the right global codivisible dimension of R with respect to (A, B).

The object of this paper is to give a general theorem which implies Izumi's Theorem and Kanno's Theorem on the absolute Nörlund summability factors of Fourier series and deduce to several known and new results from the theorem.

This paper is concerned with the stability, boundedness, and asymptotic behavior of solutions of integrodifferential systems of the form

We shall also investigate the behavioral relationships between the solutions of two integrodifferential systems related to this system.

Giles and Joseph (Bull. Austral. Math. Soc. 11 (1974), 31–36), proved that the numerical range of an unbounded operator on a Banach space has a certain density property. They showed, in particular, that the numerical range of an unbounded operator on certain Banach spaces is dense in the scalar field. We prove that the numerical range of an unbounded operator on a Banach space is always dense in the scalar field.

We display the relevance of Martin's Axiom to suitable forms of some long-known results in classical measure and category theory (for example, outer-measure-preserving partitions of sets of positive outer measure). While our theorems are at best fragmentarily new, the proofs are very simple, and, we think, instructive as regards the force of the axiom.

Infinite families of amicable orthogonal designs are constructed with

(i) both of type (1, q) in order q + 1 when q ≡ 3 (mod 4) is a prime power,

(ii) both of type (1, q) in order 2(q + 1) where q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix,

(iii) both of type (2, 2q) in order 2(q + 1) when q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix.

A question of L.G. Kovács, Joachim Neubüser, B.H. Neumann (J. Austral. Math. Soc. 12 (1971), 287–300) on the existence of ‘secretive’ prime-power groups of large rank is settled affirmatively by proving the following result: given a prime p and integer d ≥ 2, there exists a finite p–group P with cyclic centre and minimal number of generators d and having the property that every element not in its Frattini subgroup has a non-trivial power in its centre.

The theory of F-finite linear operators developed by Robert T. Moore is used to construct a differential calculus in locally convex spaces. This note contains the fundamental theory up to the implicit function theorem.

Let G be a graph with vertex degree sequence d1 ≤ d2 ≤ … ≤ dp It is shown that if di + dp–i+1 ≥ p for some i, then G is uniquely reconstructable from its collection of maximal (edge deleted) subgraphs. This generalizes considerably a result of Lovász. As a corollary, it is shown that Chvátal's existence condition for hamiltonian cycles implies edge reconstructability as well.

Analogues of Artin-Wedderburn and Goldie structure theorems are obtained for a class of halfrings which includes the semisubtractive ones. In the semisubtractive case, precise results are obtained which show that non-ring examples of these structures are relatively limited.

For a random normed space of mappings into a separable normed linear space, convergence of identically distributed elements in i the random norm (norm distribution) is shown to be equivalent to convergence in measure in the weak linear topology. Convergence in measure in each coordinate of a Schauder basis is also shown to be a necessary and sufficient condition for convergence in the random norm topology. These results have laws of large numbers for random elements in separable normed linear spaces as almost immediate corollaries and illustrate some of the recently obtained laws of large numbers for random elements. Similar results are also given for elements which need not have the same norm distributions, and the results are extended to linear metric spaces. Finally, applications of the results to stochastic processes are considered.

The maximal left ideal, the maximal right ideal and the maximal two-sided ideal are all characterized for quite general semigroups of endomorphisms. These results are then applied to such semigroups as the semigroup of all continuous selfmaps of a topological space, the semigroup of all closed selfmaps of a topological space, and the semigroup of all linear endomorphisms of a vector space.

An elementary proof is given of a result obtained by concerning the sum of an infinite series involving a 4F3.

The ring C(X) is the ring of all continuous real-valued functions on a completely regular Hausdorff space X, and βX is the Stone-Čech compactification of X.

The author proves a theorem which leads to a characterization of those zero-sets in X whose closures (in βX) are zero-sets in βX, and relates this characterization to the ideals in the ring C(X).

For a V-category B, where V is a symmetric monoidal closed category, various limit-like notions have been recognized: ordinary limits (in the underlying category B0 ) preserved by the V-valued representable functors; cotensor products; ends; pointwise Kan extensions. It has further been recognized that, to be called complete, B should admit all of these; for which it suffices to demand the first two. Hitherto, however, there has been no single limit-notion of which all these are special cases, and particular instances of which may exist even when B is not complete or even cotensored. In consequence it has not been possible even to state, say, the representability criterion for a V-functor T: B → V, or even to define, say, pointwise Kan extensions into B, except for cotensored B. (It is somewhat as if, for ordinary categories, we had the notions of product and equalizer, but lacked that of general limit, and could not discuss pullbacks in the absence of products.) In this paper we provide such a general limit-notion for V-categories.

A general method is described for constructing examples of soluble groups whose normal subgroups form a well-ordered chain under the ordering of inclusion. This method is a variant of one introduced in a recent paper by Heineken and Wilson. Each of the resulting groups is obtained by an embedding procedure from a pair of iterated wreath products A1 wr A2 wr … wr An, B1 wr B2 wr … wr Bn, where the constituent groups Ai., Bi are each either cyclic of prime power order or quasicyclic. Here n may be chosen arbitrarily, and the choice of constituent groups is subject only to a condition on the sequences of prime numbers that may occur as orders of elements in the groups

respectively. The construction is applied to give certain examples which illustrate the limitations of results on particular classes of soluble groups satisfying the minimal condition for normal subgroups obtained in recent papers by Hartley, McDougall, and the present author.

In this note we obtain a characterization of the class of convergence spaces for which Richardson's compactification is the largest Hausdorff compactification and a characterization of the class of convergence spaces which possess largest Hausdorff compactifications.