Carleson and Hunt proved that the space of functions with almost everywhere convergent Fourier series contains Lp (p > l) as a subspace. We shall give two kinds of subspaces of the spaces of functions with everywhere convergent or uniformly convergent Fourier series.

The first theorem shows that the subspaces of the space of functions with everywhere convergent Fourier series, defined in our previous paper, is a good subspace. The second theorem shows that convergence criterion in the previous paper is the proper generalization of Lebesgue's Convergence Criterion.

Let F be the upper radical determined by all fields. The supernilpotent radical classes which are not special have thus far always contained F properly. The purpose of this note is to construct a countably infinite number of supernilpotent radical classes which are not special and each of which is properly contained in F. The construction involves a ring due to Leavitt which is interesting in its own right and is not generally known. All rings considered are associative.

It is shown that a finite loop with no proper nontrivial subloops has a finite basis for its laws.

Completely semi-stable trees are characterised by the complete absence of three types of subtrees.

It is shown that a condition of Kurzwell concerning fixed-points of certain operators on a finite group G is sufficient to ensure that G is soluble. The result generalizes those of Martineau on elementary abelian fixed-point-free operator groups.

In a recent paper Martineau has shown that the study of groups with a fixed-point-free automorphism group can be applied to help in the investigation of signalizer functors. We prove further results about signalizer functors by this method.

Berman has raised the question in his work of whether Hermite-Fejér interpolation based on the so-called “practical” Chebyshev points, , 0(1)n, is uniformly convergent for all continuous functions on the interval [−1, 1]. In spite of similar negative results by Berman and Szegö, this paper shows this result is true, which is in accord with the great similarities of Lagrangian interpolation based on these points versus the points , 1(1)n.

Some problems in the behavioural and physical sciences arise in the context of an incomplete knowledge of the fine detail of underlying practical situations. This paper presents a general mathematical framework for the discussion of such problems. This framework provides an algebraic language for the discussion of ecological analysis in the social sciences, aggregation in economics and macroscopic descriptions in statistical physics. Here, however, only the mathematical framework is presented; detailed applications will be presented elsewhere.

In deriving macroscopic descriptions of microscopic phenomena one often uses a vector valued summary function which is defined on a cartesian product in terms of component summary functions. We show that any character of such a summary function is the product of characters of its component summary functions.

An initial value investigation is made of the propagation of capillary-gravity waves generated by an oscillating pressure distribution acting at the free surface of a running stream of finite, infinite, and shallow depth. The solution for the free surface elevation is obtained explicitly by using the generalized Fourier transform and its asymptotic expansion. It is found that the solution consists of both the steady state and the transient components. The latter decays asymptotically as t → ∞ and the ultimate steady state is attained. It is shown that the steady state consists of two or four progressive capillary-gravity waves travelling both upstream and downstream according as the basic stream velocity is less or greater than the critical speed. Special attention is given to the existence of the critical values associated with the running stream of finite, infinite, and shallow depth. A comparison is made between the unsteady wave motions in an inviscid fluid with or without surface tension.

Power and commutator structure tables are obtained for a number of factor groups of Burnside groups.

The following statement is proved: Let X be a set having at most continuously many elements and f: X → X a mapping such that each iteration fn (n = 1, 2, …) has a unique fixed point. Then for every number c ∈ (0, 1) there exists a metric p on X such that the metric space (X, p) is separable and the mapping f is a.contraction with the Lipschitz constant c.

According to the Sees Theorem, every completely 0-simple semigroup can be represented by a Rees matrix semigroup over a group with zero. A characterization of all subsemigroups of the latter is given in terms of the structure group, structure sets, and two mappings. Next all congruences on such subsemigroups are described, along with conditions for comparability. Finally, an algorithm for computing the number of nonisomorphic inverse subsemigroups is constructed.