The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.

Let k ≧ 2 be an integer and each of ν1, ν2, …, νk and δ1, δ2, …, δk be 0 or 1. Then given any positive integer M and non-negative reals a1, a2, …, aM we put

where

and

If the elements of a symmetric matrix lie in the real field it is well known that the roots of its characteristic equation are real. This implies that the discriminant of that equation (i.e. the product of the squared differences of the roots) is a polynomial in the elements which is non-negative and the same must be true for the leading coefficients of all the other Sturm functions associated with the characteristic equation. One would expect that it should be possible to express them as a sum of squares. Conversely, such an expression would establish the reality of the roots.

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the condition

is both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

Normal right submodules and right ideals need not coincide in an arbitrary near-ring. Berman and Silverman (1) have shown that in a near-ring (N, +, ·) with a two-sided zero (i.e. x · 0 = 0 · x = 0, for all x ∈ N) a right ideal is also a right submodule. If (N, +, ·) is in fact a distributively generated near-ring, then all normal right submodules are also right ideals. (See (5).)

The inequality

which is valid for positive, non-null f, g in the spaces Lp(0, ∞), Lq(0, ∞), where p > 1, (1/p)+(1/q) = 1, is a well-known generalisation of the classical inequality of Hilbert (see for instance Chapter 9 of Hardy, Littlewood, and Polya (1)).

Varieties of topological groups have been investigated in several papers ((2) and (10)-(13)). In this note we investigate the varieties generated by classical Lie groups. In particular we show results of which the following is indicative: The variety generated by the unitary group U(n) contains U(m) if and only if m ≦ n. En route we introduce the notion of a variety of topological Lie algebras which provides a convenient setting in which to answer our questions.

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

Some axially symmetric boundary value problems of potential theory are formulated as integral equations of the first kind. In each case the kernel admits an expansion, for small values of a parameter of the problem, that leads to an approximate integral equation whose solution provides a direct asymptotic estimate for the physical quantity of primary interest. A manipulation of the original and modified integral equations provides an efficient formula for calculating higher order terms in the asymptotic expansion.

In a linear topological space E one often carries out various “ smoothing ” operations on a subset A, such as taking the convex hull co A and the closure A-. If E is also a (real) vector lattice, the solid hull

is also a natural “ smoothing out ” of A. If sol A = A then A is called solid, and if E has a base of solid neighbourhoods of 0 as do all the common topological vector lattices such as C(X), Lp, Köthe spaces and so on—then E is called a locally solid space.

Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping for x ∈ M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Rm(c) and 〈, 〉 the scalar product on Rm(c). If there exists a function λ on M such that 〈h(X, Y), H〉 = λ〈X, Y〉 for all tangent vectors X, Y, then M is called a pseudo-umbilical surface of Rm(c). Let D denote the covariant differentiation of Rm(c) and η be a normal vector field. If we denote by D*η the normal component of Dη, then D* defines a connection in the normal bundle. A normal vector field η is said to be parallel in the normal bundle if Dη = 0. The length of mean curvature vector is called the mean curvature.

A corollary of the main theorem presented in this note is a generalisation of the well-known result that a self-adjoint square root of a positive self-adjoint compact linear map in a Hilbert space is itself a compact linear map. The method used here exploits the techniques developed recently in the study of k-set contractions ((1), (2)).

Let G be a group and let K be a field. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be the vector space over K with basis elements ; let α: G ×G → K be a 2-cocycle and define a multiplication on Kt(G) by

extending this by linearity to Kt(G) yields an associative algebra. We are interested in information concerning the Jacobson radical of Kt(G), denoted by JKt(G).

In this paper two major questions concerning generalised quaternion groups and distributively generated (d.g.) near-rings are investigated. The d.g. near-rings generated, respectively, by the inner automorphisms, automorphisms, and endomorphisms of the group are described. It is also shown that these morphism near-rings are local near-rings and contain no non-trivial idempotents. Finally, it is demonstrated that exactly 16 d.g. near-rings can be defined on a given generalised quaternion group.

Let A be a commutative, semi-simple, convolution measure algebra in the sense of Taylor (6), and let S denote its structure semigroup. In (2) we initiated a study of some of the relationships between the topological structure of A^ (the spectrum of A), the algebraic properties of S, and the way that A lies in M(S). In particular, we asked when it is true that A is invariant in M(S) or an ideal of M(S) and also whether it is possible to characterise those measures on S which are elements of A. It appeared from (2) that if A is invariant in M(S) then S must be a union of groups and that A^ must be a space which is in some sense “ very disconnected ”. In (3) we showed that if A^ is discrete then A is “ approximately ” an ideal of M(S). (What is meant by “ approximately ” is explained in (3); it is the best one can expect since algebras which are approximately equal have identical structure semigroups and spectra.) In this paper we round off some of the results of (2) and (3). We show that if A is invariant in M(S) then A^ is totally disconnected, and that if A^ is totally disconnected then S is an inverse semigroup (union of groups). From these two crucial facts it is fairly straight-forward to obtain a complete characterisation of algebras A (and their structure semigroups) for which (i) A^ is totally disconnected, (ii) A is invariant in M(S), or (iii) A is an ideal of M(S).

G denotes a locally compact abelian group and M(G) the convolution algebra of regular bounded Borel measures on G. An ideal I of M(G) closed in the usual (total variation) norm topology is called an L-ideal if μ ∈ I, ν≪ μ (ν absolutely continuous with respect to μ) implies that ν ∈ I. Here we are concerned with the L-idealsL1(G), , and M0(G) where, as usual, L1(G) denotes the set of measures absolutely continuous with respect to Haar measure, denotes the radical of L1(G) in M(G) and M0(G) denotes the set of measures whose Fourier-Stieltjes transforms vanish at infinity.

It has been established by G. Lallement (3) that the set of idempotent-separating congruences on a regular semigroup S coincides with the set ∑() of congruences on S which are contained in Green's equivalence on S. In view of this and Lemma 10.3 of A. H. Clifford and G. B. Preston (1) it is obvious that the maximum idempotent-separating congruence on a regular semigroup S is given by

Relative uniform limits need not be unique in a non-archimedean partially ordered group, and order convergence need not imply metric convergence in a Banach lattice. We define a new type of convergence on partially ordered groups (R-convergence), which implies both the previous ones, and does not have these defects. Further R-convergence is equivalent to relative uniform convergence on divisible directed integrally closed partially ordered groups, and to order convergence on fully ordered groups.

In this paper it is shown that divisibility of a complete lattice ordered (abelian) group is closely related to the existence of a sufficient number of small elements in the positive cone.

We shall denote the set of all real numbers by R which symbol will be reserved for this purpose. All terms used are as defined in Birkhoff(1). For the reader's convenience we now define the two terms most used in the sequel.