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]).

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.

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].

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.

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].

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].

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.

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.

An ancipital form is a form [a, b, c] in which b= 0 or b= a; these fall into pairs of associates: [a, 0, c] and [c, 0, a] (type 1), and [a, a, c] and [4c–a, 4c–a, c] (type 2). The set of discriminantal divisors of discriminant d is formed by choosing, from each pair of primitive associate ancipital forms of discriminant d, exactly one of the two leading coefficients. In this article we study representations of discriminantal divisors of a given discriminant by binary quadratic forms of that discriminant, previously studied by the author and by G. Pall. We are concerned here with discriminants d= 4kpq, where k ≥ 1, p = 1, q = 3 (mod 4) are primes, and d = 4kp, where k ≥ 1 and p is an odd prime. This investigation arose in connection with the search for integral solutions of x2 –Dy2 = – 1.

We give a simple proof of the lemma of Ford [1] on the existence of self-adjoint square roots in Banach*-algebras in which continuity of the involution is not assumed.

Bellman [1], [2, p. 116] proved that, if all solutions of the equation

are in L2, ∞) and b(t) is bounded, then all solutions of

are also in L2(a, ∞). The purpose of this paper is to present conditions on the function f that guarantee that all solutions of

be in the class L2(a, ∞) whenever all solutions of the equation

have this property. It is assumed that r(t) >0, r and qare continuous on a half line (a, ∞) and f is continuous. Actually the continuity assumptions may be weakened to local integrability and L2 (a, ∞) may be replaced by Lp(a, ∞) for any p > 1.

Let Rbe an associative ring, Jan infinite index set, and Rj the ring of all J × J row-finite matrices over R. The Jacobson radical of Rwill be denoted by Γ(R).

Let T = (tmn) be a regular matrix, and CTbe its bounded convergence field. Necessary and sufficient conditions for CT to contain the space of almost convergent sequences are well known. (See, e.g., [7, p.62]). G. M. Petersen has suggested as a problem for research the discovery of necessary and sufficient conditions for the reverse inclusion: When is CT contained in the space of almost convergent sequences? [7, p. 137, research problem 9]. In this paper we deal with this question in a more general context. First we need some notation.