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.

Cordially dedicated to Professor Alexander Weinstein, on the occasion of his Seventyseventh birthday, January 21, 1974.

This note contains the proof of an extension of Alexander Weinstein's mean value theorem for Generalized Axially Symmetric Potential Theory.

A metrizable topological space has a metric taking values in a closed subset of the real numbers having Čech dimension zero if and only if the space itself has Čech dimension zero. We call a development D = {Dn} for a topological space (X, T) a sieve for X if the sets in each Dn are pairwise disjoint. Then a Hausdorff topological space (X, T) has a compatible metric taking values in a closed subset of the real numbers having Čech dimension zero if and only if there exists a sieve for X.

A homological characterization is given of when a twisted group ring relative to an automorphism of an arbitrary field has all of its simple right modules injective (= a right V-ring). This answers a question raised by Osofsky. A “Hilbert Theorem 90” type theorem determines the cardinality of the isomorphism classes of one-dimensional simple modules.

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.

An α-distributive (respectively α-implicative) semilattice S is a lower semilattice (with greatest lower bound denoted by juxtaposition) in which the annihilator 〈x, a〉, that is {y ∈ S: xy ≤ α}, is an ideal (respectively a principal ideal) for the fixed element α and any x of S. These semilattices appear as natural links between general and distributive semi-lattices on the one hand, and between pseudo-complemented and implicative semilattices on the other hand. Prime and dense elements, as well as maximal and prime filters, are essential. Mandelker's result, a lattice L is distributive if and only if 〈x, y〉 is an ideal for any x, y ∈ L is extended to semi-lattices.

This paper studies the connection between the best possible value of a constant in the compact abelian case of a known inequality due to Helgason and the Λ2-constants of sets of characters. Various estimates of and expressions for the best possible value are given.

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.

The structure of inverse semigroup extensions of one inverse semigroup R by any other is analyzed in the case where R is a semilattice. Both a representation and method of construction are given. A brief preliminary examination is made of a certain class of congruences, on inverse semigroups, which are intimately related to such extensions.

A class of near rings which generalizes the class of integral domains is defined. The definition of near domain is derived from the desirability of embedding near domains in near fields. The near domains presented here are shown to contain the D-rings of Berman and Siverman.

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

Let H be a fixed Hilbert space and B(H, H) be the Banach space of bounded linear operators from H to H with the uniform operator topology. Oscillation criteria are obtained for the operator differential equation

where the coefficients A, C are linear operators from B(H, H) to B(H, H), for each t ≤ 0. A solution Y: R+ → B(H, H) is said to be oscillatory if there exists a sequence of points ti ∈ R+, so that ti → ∞ as i → ∞, and Y(ti) fails to have a bounded inverse. The main theorem states that a solution Y is oscillatory if an associated scalar differential equation is oscillatory.

The object of this present paper is to study a nonlinear perturbed stochastic integral equation of the form

where ω ∈ Ω, the supporting set of the complete probability measure space (Ω A, μ). We are concerned with the existence and uniqueness of a random solution to the above equation. A random solution, x(t; ω), of the above equation is defined to be a vector random variable which satisfies the equation μ almost everywhere.

We prove some new theorems and reprove some old ones about finitely generated soluble groups and Lie algebras by a uniform method. Among the applications are Gruenberg's Theorem on Engel groups, for which we obtain a very short proof; and the Milnor and Wolf polynomial growth theorem. It is shown that a finitely generated soluble group with all 2-generator subgroups polycyclic is itself polycyclic, and that a finitely generated soluble Lie algebra, all of whose inner derivations are algebraic, is finite-dimensional. This last result enables us to give a partial answer to a question of Jacobson.

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

Let R be a metrization with distance xy of an open convex set D in the 2-dimensional real affine plane such that xy + yz = xz whenever x, y, z lie on an affine line with y between x and z and such that all the balls px ≤ ρ are compact. The study of such metrics, called open plane projective metrics falls under the topic of Hilbert's Problem IV of his famous mathematical problems. In this paper it is proved that if in R the sets of points equidistant from lines lie again on lines then D must be the entire affine plane and the distance must in fact be a norm. The paper contributes to and gives extensions of similar results proved earlier. The novel features of the present result are that in the space collinearity of points x, y, z is taken only as a sufficient condition for the equality xy + yz = xz. Consequently the solution encompasses all normed linear planes, that is, norms whose unit circles are not necessarily strictly convex are also admitted.

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.

It is shown that a saturated formation F has the property that in each group G each F-projector of G is complemented if and only if F is the formation of finite soluble π-groups for some set π of primes.

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.

The proliferation of classifying spaces in recent years owes much to the theorem of Edgar H. Brown, Jr on the representability of homotopy functors. Since the theorem only gives a representation for functors defined on the category of spaces having the homotopy type of a CW complex, there is some interest in finding conditions under which the domain category may be enlarged. It appears that a version of the theorem holds for any small full subcategory of Htp, the category of topological spaces and homotopy classes of continuous maps, but that the resulting classifying space is generally intractable.