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.

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.

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.

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.

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.

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.

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.

Complicated groups of exponent four have been constructed from the ring of polynomials in associating non-commuting indeterminates with coefficients from the field of two elements. The justification of these constructions depends on a computational reduction result. In this note a further reduction is obtained. The expressions involved seem to have an interesting combinatorial structure.

A boolean space is a compact Hausdorff space which is zero-dimensional. In this paper, a boolean space X is characterized in terms of its ring of real-valued functions C(X). The result is sharpened for the case when X is an F-space (every finitely generated ideal of C(X) is principal).

It is shown that the correspondence which associates with each radical class τ of abelian groups the (radical) class of prime radical rings with additive groups in τ gives a complete classification of those radical classes of rings which are determined (as lower radicals) by zerorings.

The negative answer by Novikov and Adyan to the Burnside question (is every finitely generated group of finite exponent finite?) still leaves open some cases, in particular, all 2-power exponents. Some reduction results are known. In this note we present another kind of reduction result.

When a totally-ordered set Ω has no points fixed by all automorphisms ve equip Ω with the support topology which has fixed-sets of automorphisms as basic closed sets. There is a mapping from Ω into the set of prime dual ideals of the lattice Φ(Ω) of basic closed sets and this allows us to classify the points of Ω as excellent, isolated, static, or extraordinary. There is an action of the group of automorphisms of Ω on the lattice Φ(Ω) and this allows us to see that automorphisms of Ω preserve the prime dual ideal classification of points of Ω. When the empty set is a basic closed subset of Ω the dual spectrum of Φ(Ω) is a compact T0 space (Hausdorff when all points of Ω are excellent) containing Ω as a dense subspace and allowing an extension of each automorphism of Ω ťo a homeomorphism of the dual spectrum.

Certain new combinatorial numbers which arise in the counting of finite topologies are introduced and formulae obtained. These numbers are used to obtain a known formula for tn, the number of labelled topologies on n points in terms of the Stirling numbers S(n, p) and dn, the number of labelled T0-topologies on n points. The numbers dn are computed for n ≤ 5 with the the help of a method of Comtet (1966) (which seems to have been missed by later authors), reinterpreted for transitive digraphs.

A theoretical study is made of the free periods of oscillation of a global layer of inviscid, incompressible fluid under the action of gravitational and Coriolis forces. The leading approximations to the eigenvalues are found to be sensitive to variations in the Froude number and also to the shape of the globe. It is shown that for oceanic and atmospheric motions displaying essentially the same features as the model, it is not sufficient to consider the motion as horizontally non-divergent.

A constructive proof of a theorem of Worrell and Wicke, which states that every open covering of a θ-refinable space has a σ-distributively point-finite open refinement that covers the space minimally, is presented. Spaces for which every open covering has an open refinement which covers the space minimally are characterized by the use of discrete closed collections and some open questions relating to spaces of this type are included.

Let S be a compact semigroup and P(S) the set of probability measures on S. Suppose P(S) has zero θ and define a measure μ ε P(S) nilpotent if μn → θ. It is shown that any measure with support containing that of θ is nilpotent, and the set of nilpotent measures is convex and dense in P(S). A measure μ is called mean-nilpotent if (μ + μ2 + … + μn)/n → θ, and can be characterized in terms of its support.

In [1] I made a Remark which, when subsequently challenged by Anne Koehier, I was unable to justify. While writing up my PhD thesis, I tried to both prove and disprove the statement but failed on both counts. However, I did reduce it to a highly plausible statement which is presented here.