Let f(x) be a complex function of a real variable, defined over the whole real line, which possesses n derivatives (the nth at least almost everywhere) and is such that . Then, if k is any integer for which 0< k < n, Kolmogoroff's inequality may be written as

,

or, by putting ,

The constant K=K (k, n) known explicitly and is the best possible, i.e., there is a (real) function for which equality holds (see Bang [1]).

Let R be a ring and let S = Spec R. Let us consider the étale fini topology on S [5]. By a form of a given S-scheme T we mean any affine S-scheme W that is locally (in the étale fini topology) isomorphic to T. We shall consider forms of the R-schemes T = Spec R[X] and T = Spec R[X, Y].

A preliminary attempt is made to place the theory of completions of boolean algebras and of partially ordered sets in a wider context. The theory and construction of injective hulls in abelian categories is generalised and it is demonstrated that any variety with enough injectives admits injective hulls. Then the methods developed are applied to a non-algebraic bicategory, that of ordered sets.

In the definition of the spectrum of a linear operator, it is customary to assume that the underlying space is complete. However there are occasions for which it is neither desirable nor necessary to assume completeness in order to obtain a spectral theory for an operator; for example, completeness is not needed in the Riesz theory of a compact operator (see e.g. [1: XI. 3]). Several non-equivalent definitions for the spectrum of an operator on normed spaces have appeared in the literature. We shall discuss the relationship among these definitions and some of the difficulties that arise in applying these definitions to obtain a spectral theory.

The easiest way to construct automorphic functions is by means of the Poincaré series. If G is a Kleinian group with ∞ an ordinary point of G and if k ≧ 4, then

where Vz=(az+b)/(cz+d) and ad-bc=1. The convergence of this series is the crucial step in showing that the Poincaré series converges and is an automorphic form on G If ∞ is a limit point of ∞ the series in (1) may diverge and one can derive automorphic forms on ∞ from the Poincaré series of some conjugate group. These constructions are described in greater detail in /3, pp. 155–165].

In this note we construct certain Hilbert subspaces with Hilbert–Schmidt imbedding, for an arbitrary proper functional Hilbert space which consists of holomorphic functions. This work extends results of Chapter III in [1] and has applications in the regularity problem for generalised eigenfunctions (in particular to Theorem 2 in [2]). For an exposition of reproducing kernels and Bergman's kernel function we refer to [4].

A Baer semigroup is a semigroup 5 with 0 and 1 in which, for each x∈S, the left annihilator L(x) = {y∈S: yx = 0} of x is a principal left ideal generated by an idempotent and the right annihilator R(x) = {y∈S: xy = 0} of x is a principal right ideal generated by an idempotent. Baer semigroups are of interest because (see [5]) the left annihilators of the elements of a Baer semigroup S, ℒ(S) = {L(x): x∈S}, form a bounded lattice and (see [4]) every bounded lattice arises in this manner. In this note we look at a type of map φ on a Baer semigroup S which has the property that Sφ is a Baer semigroup. (The homomorphic image of a Baer semigroup need not be a Baer semigroup. For a case where it is, see [7].) When the Baer semigroup is specialized to a Boolean algebra, this type of map generalizes Halmos's notion of a quantifier.

This paper deals with the following problem: Can an arbitrary continuous function on [0, 1], which vanishes at the origin, be represented in some sense as a series of constant multiples of indefinite integrals of a complete orthonormal set of functions on [0,1]? Four contexts in which this problem arises naturally will be given in the introduction and the remainder of the paper will be devoted to giving a partial answer to the specific problem formulated in one of these contexts.

In a recent paper [6], this author has extended the method of the kernel function [1] to the boundary value problems of the generalized axially symmetric potentials

This method can also be applied to a more general class of singular differential equations, namely

or, equivalently,

We shall derive in the sequel explicit formulas for the Dirichlet problems of (1.1) in the first quadrant of the x-y plane in terms of sufficiently smooth boundary data, and obtain an error-bound for their approximate solutions. We shall also indicate how the Neumann problem can be solved.

Throughout this note, let be a probability space with an increasing sequence of sub σ-fields of whose union generates . Let be a sequence of random variables adapted to (see [3], p. 65) and henceforth be referred to as a game. As in [1], the game will be said to become fairer with time if, for every ε > ε,

as n, m → ∞ with n ≧ m

In this paper we presuppose a familiarity with the terminology and the basic results of radical theory, all of which can be found in Divinsky [2].

In a recent paper Cooke [1] obtained a solution of the integral equation

by using the identity

and the technique, first used by Copson, of interchanging the orders of integration and hence reducing the problem to that of the successive solution of two Abel integral equations. It is also shown in [1] that the above identity can also be used to solve the dual series equations

The kernel in equation (1) is a particular member of a general class of kernels which the author [6] has shown to be such that the resulting integral equation is directly soluble by using Copson's technique. The particular example of equation (1) is given in [6] and the identity of equation (2) was used by the author [7] to obtain the solution of equation (3).

Suppose that G is a p-primary abelian group. The subgroup G[p] = {x∈G:px=0} is called the socle of G and any subgroup S of G[p] is called a subsocle of G. If each subsocle of G supports a pure subgroup, then G is said to be pure-complete [1]. It is well known that, if G a direct sum of cyclic groups, then G is necessarily pure-complete. Further results about pure-complete groups are contained in [1] and [3].

In [2], K. D. Magill, Jr. has proved that every automorphism of the semigroup (with respect to composition) of all real-valued differentiable functions of a real variable is inner. The purpose of this paper is to generalize this fact to arbitrary finite-dimensional real Banach spaces.

The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra 𝒜 has “enough” hermitian elements, then 𝒜 can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.

We show that the commutative V*-algebras with relatively weakly compact unit spheres are those that are representable by means of hermitian spectral measures. This provides a more unified approach to the results of [15], and allows us to generalise some of them.

In this paper, a problem of B. H. Neumann and Hanna Neumann [7] about the finite embeddability of an embeddable finite amalgam is discussed. After proving a “reduction theorem” for a finite amalgam to have a finite embedding, we examine some known embeddable amalgams (cf. [3]) as regards their embeddability in a finite group. Since the existence of the generalised free product and the embeddability of an amalgam are synonymous terms, Theorem 3.1 generalises a result in [4]. A sufficient condition for an amalgam of type S to have a finite embedding is also given.

Ogasawara and Yoshinaga [9] have shown that a B*-algebra is weakly completely continuous (w.c.c.) if and only if it is *-isomorphic to the B*(∞)-sum of algebras LC(HX), where each LC(HX)is the algebra of all compact linear operators on the Hilbert space Hx. As Kaplansky [5] has shown that a B*-algebra is B*-isomorphic to the B*(∞)-sum of algebras LC(HX) if and only if it is dual, it follows that a 5*-algebra A is w.c.c. if and only if it is dual. We have observed that, if only certain key elements of a B*-algebra A are w.c.c, then A is already dual. This observation constitutes our main theorem which goes as follows. A B*-algebra A is dual if and only if for every maximal modular left ideal M there exists a right identity modulo M that is w.c.c.

Recently, several papers have investigated conditions under which the range of a vectorvalued measure is a compact convex set (see e.g. [1], [2], [3]). It therefore seems of interest to characterise the extremal points of the range in such cases.

Certain questions about the range of finite embeddings of a finite amalgam were discussed in [3]. Another pertinent question is the following.

If a finite reduced amalgam has both an infinite and a finite embedding, does it have a maximal finite embedding such that all other finite embeddings are its homomorphic images? We give a counter example to answer this question in the negative. The finite amalgam considered will involve a group from the family of groups of the type (l, m; n, k) discussed by Coxeter [2] having the following presentation

These groups may be regarded as factor groups of

which is known to be finite if

and infinite otherwise.