Let X be a complex normed space, with dual space X′, and T a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x∊X, f∊ X′, ∥x∥ = ∥f∥ = f(x) = 1}. Let ⃒V(T)⃒ denote sup {⃒λ⃒: λ∊ V(T)}. Our purpose is to prove the following theorem.

Given any finite group G, we wish to determine a relationship between the highest power of a prime p dividing the order of G, denoted by |G|p, and |A(G)|p, where A(G) is the automorphism group of G. It was shown by Herstein and Adney [8] that |A(G)|p ≧ p whenever |G|p= ≧P2. Later Scott [16] showed that A(G)p≧P2. For the special case where G is abelian, Hilton [9] proved that Adney [1] showed that this result holds if a Sylow p-subgroup of G is abelian, and gave an example where |G|p= p4 and |A(G)|P =p2. We are able to show in Theorem 4.5 that, if |G|p = ≧ p5, then |A(G)| = ≧ p3.

An n-tournament is a set of n labelled points, each pair A, B of which is joined either by the oriented line AB or by the oriented line BA. There are N = n(n –1)/2 such pairs and so Fn different n-tournaments, where Fn = 2N. A tournament is reducible if the points can be separated into two non-empty subsets and ℬ, such that every line joining a point in to a point in ℬ is directed towards the point in ℬ. Rado [3] showed that an irreducible tournamentis strongly connected; i.e. for every ordered pair of points A, B, there is a sequence of correctly oriented lines AC1, C1C2, …, ChB in the tournament, and conversely that a strongly connected tournament is irreducible.

Let Pn be the class of normalised polynomials of the form

of degree n which are univalent in U = {|z| < 1}. In this note we discuss the coefficients of polynomials in Pn and in some of its subclasses.

In [1, p. 130] the following partition theorem was deduced from a general theorem concerning the limit of a recurrent sequence.

The purpose of this paper is to generalize a result of K. Iseki [1]. In his note, Iseki proves that, in a normal space S, for every countable discrete collection ℋ = {H1, H2, …} of sets from S, there exists a countable collection = {U1, U2, …} of mutually disjoint open sets from S such that Hi ⊂ Ui for every i.

In [1, p. 97], Bruck and Bose ask the question ”Has every (right) Veblen-Wedderburn system finite dimension over its left operator skew-field?” It is the purpose of this note to show that, in general, this question has a negative answer.

This paper is concerned with three basic transforms

The first of these has been studied by Widder [1], who points out that f(t) can be interpreted as the temperature u(0, t) on the time axis, where u(x, t) is the solution of the heat equation withsymmetric initial temperature u(x, 0) = g(|x|). The second has also been studied by Widder [2], where it is pointed out that f(t) can be interpreted as the value of the harmonic function u(x, t) on the t-axis arising from the boundary data u(x, 0) = g(|x|).

In two papers [3] and [4], the author has extended the inequality of Schur (Theorem 319 of [2]) to cases involving kernels which satisfy identities of the form

The purpose of this paper is to prove a general inequality, which includes the above and also the inequality of Young (Theorem 281 of [2]) as special cases. We shall give the results a general setting by considering functions defined on abstract measure spaces. From this we shall deduce an extension to n dimensions of the results given in [3], which also generalises a similar extension of the Schur inequality given by Stein and Weiss. In fact some cases of the other results given in [5] will follow directly from our theorem.

Since all the semigroups considered in this paper are commutative, we shall use the terms “semigroup” and “group” where we actually mean “commutative semigroup” and “commutative group”. Some basic results from the theory of semigroups are required and will be used without explicit mention; these results may be found in [1, § 4.3]. We shall denote the additive semigroups of integers, positive integers, negative integers, positive rationals by Z, Z+, Z-, Q+ respectively.

Let E be a locally compact space which can be expressed as the union of an increasing sequence of compact subsets Kn (n =1, 2, …) and let μ be a positive Radon measure on E. Ω is the space of equivalence classes of locally integrable functions on E. We denote the equivalence class of a function f by and if is an equivalence class then f denotes any function belonging to f. Provided with the topology defined by the sequence of seminorms

Ω is a Fréchet space. The dual of Ω is the space φ of equivalence classes of measurable, p.p. bounded functions vanishing outside a compact subset of E. For a subset Γ of Ω, the collection Λ of all ∊Ω, such that for each g∊Γ the product fg is integrable, is called a Köthe space and Γ is said to be the denning set of Λ. The Köthe space Λx which has Λ as a denning set is called the associated Kothe space of Λ. Λ and Λx are put into duality by the bilinear form

This paper may be regarded as a continuation of the investigations begun in [2]; certain intrinsic lattice topologies are studied, especially the order and ideal topologies in Boolean algebras, bicompactly generated lattices, and other more general structures. The results of [1], [2], and [3] are shown to be closely related. It is proved that the ideal topology on any Boolean algebra has a closed subbase consisting of all sublattices, whereas the order topology on an atomic Boolean algebra has a closed subbase consisting of all sub-complete lattices. It is also shown that the order topology on an atomic Boolean algebra is autouniformizable (in the sense defined by Rema [3]) and, if the ground set is infinite, strictly coarser than the ideal topology. The conditions Cl and C3 on a lattice, introduced by Kent [1], are shown to be slightly stronger than the condition “ bicompactly generated ”, and in complete lattices, where these conditions are satisfied, the order topology is shown to be coarser than the ideal topology.

This paper is a continuation of [1]. We begin with the notations for the sequence spaces considered in this paper. Let Γ be the space of sequences x = {xp} of complex numbers such that |xp|1/p⃗0 as p⃗∞. Γ can also be regarded as the space of integral functions f(z) = . The sequence space Γ is a vector space over the complex numbers with seminorms

In this paper we prove the following:

Let S be a regular semigroup anda, bany elements of S such that Jb = ≦ Ja. Then, for each idempotent e∈Ja, there exists an idempotent f∈ Jb such that f = ≦e.