The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

I was recently challenged to find all the cases when the sum of three consecutive integral cubes is a square; that is to find all integral solutions x, y of

y2=(x−1)3+x3+(x+1)3

=3x(x2+2)

This is an example of a curve of genus 1. There is an effective procedure for finding all integral points on a given curve of genus 1 ([1, Theorem 4.2], [2]): that is, it can be guaranteed to find all the integral points and to show that no others exist with a finite amount of work. Unlike some effective procedures, which have only logical interest, this one can actually be carried out in practice, at least with the aid of a computer ([3], [5]). There are, however, older methods for dealing with problems of this kind which, while not effective, very often lead more easily to a complete set of solutions (and a proof that it is complete). I solve the problem here by a technique introduced in [4]. It requires only the elementary theory of algebraic number–fields. The motivation is p–adic, but it is simpler not to introduce p–adic theory overtly.

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

Let Tn denote the full transformation semigroup on the finite set = {1, 2, … n}, that is the set of all mappings from to , with function composition as the semigroup operation. In this paper algorithms are introduced to solve equations such as axmb = c and ax = xb (a, b, c ∊ Tn), which employ a representation of members of Tn as special directed graphs.

Let G be a simple, simply connected, compact Lie group, and let M be an orientable, smooth, connected, closed 4-manifold. In this paper, we calculate the homotopy type of the suspension of M and the homotopy types of the gauge groups of principal G-bundles over M when π1(M) is (1) ℤ*m, (2) ℤ/prℤ, or (3) ℤ*m*(*nj=1ℤ/prjjℤ), where p and the pj's are odd primes.

Semilocal and related classes of group rings have been investigated by many authors (cf. [10]). In particular, the following results have been obtained.

Theorem A[4,10]. Let K be a field and G a group.

(i) If ch K = 0, then K[G] is semilocal if and only if G is finite.

(ii) If ch K = p>0 and G is locally finite, then K[G] is semilocal if and only if G contains a p-subgroup of finite index.

In the case of semigroup rings some stronger conditions have been studied. Munn examined the semisimple artinian situation [6]. Zelmanov showed that if K[G]is artinian then G must be finite [11].

A semigroup endowed with a unary operation satisfying the identities

is a completely regular semigroup. In several recent papers devoted to the study of the lattice of subvarieties of the variety of completely regular semigroups, various results have been obtained which decompose special intervals in into either direct products or subdirect products. Petrich [14], Hall and Jones [6] and Rasin [20] have shown that certain intervals of the form , where is the trivial variety and are subdirect products of and Pastijn and Trotter [13] show that certain intervals of the form are direct products of the intervals and The main objective of this paper is to develop an appropriate lattice theoretic framework for these representations.

We prove that the conditions R·R=0 and R·S=0 are equivalent for hypersurfaces of a 5-dimensional semi-Riemannian space form N5(c). This solves a problem by P.J. Ryan in the case of hypersurfaces of dimension 4 in semi-Riemannian space forms.

Let $G$ be a locally soluble-by-finite group in which every non-subnormal subgroup has finite rank. It is proved that either $G$ has finite rank or $G$ is soluble and locally nilpotent (and even a Baer group). On the other hand, a group $G$ is constructed that has infinite rank and satisfies the given hypothesis, but does not have every subgroup subnormal.

We recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

G. B. Preston [10] proved that any semigroup can be embedded in a bisimple monoid. If S is a countable semigroup, his constructive proof yields a bisimple monoid which is also countable, but not necessarily finitely generated. The main result of this paper is that any countable semigroup can be embedded in a 2-generated bisimple monoid.

J. M. Howie [6] proved that any semigroup can be embedded in an idempotentgenerated semigroup. F. Pastijn [9] showed that any semigroup can be embedded in a bisimple idempotent-generated semigroup, and that any countable semigroup can be embedded in a semigroup which is generated by 3 idempotents. Easy proofs of these results using Rees matrix semigroups over a semigroup were given by the author [3]. In this paper, as a corollary to our main result, we deduce that any countable semigroup can be embedded in a bisimple semigroup which is generated by 3 idempotents.

Some well-known results about the 2-density topology on ${\mathcal R}$ (in particular in the context of the Lusin–Menchoff property) are extended to τbm , i.e. the m-density topology on ${\mathcal R}$ n with m ∈ (n,+∞). Every set of finite perimeter in ${\mathcal R}$ n is equivalent (in measure) to a set in τb m 0 , where m 0=n+1+ ${1\over n-1}$ . There exists a set of finite perimeter in ${\mathcal R}$ n which is not equivalent (in measure) to any member in the a.e.-modification of τbm , whatever m ∈ [n,+∞).

The investigation of general F-abundant semigroups is initiated. After obtaining some properties of such semigroups, the structure of a class of F-abundant semigroups is established. In addition, a problem raised in [2] is positively answered.

Given a normed linear space X, let S(X), X′, B(X) denote respectively the unit sphere {x: ∥x∥ = 1} of X, the dual space of X, and the algebra of all bounded linear mappings of X into X. For each x ∊ S(X) and T ∊ B(X), let Dx(x) = {f e X′:∥f∥ = f(x)= 1}, and V(T; x) = {f(Tx):f∊Dx(x)}. The numerical range V(T) is then defined by

A (bounded, linear) operator H on a Banach space is said to be hermitian if ∥exp(itH)∥ = 1 for all real t. An operator N on is said to be normal if N = H + iK, where H and K are commuting hermitian operators. These definitions generalize those familiar concepts of operators on Hilbert spaces. Also, the normal derivations defined in [1] are normal operators. For more details about hermitian operators and normal operators on general Banach spaces, see [4]. The main result concerning normal operators in the present paper is the following theorem.

We investigate the radial behavior of holomorphic functions in the unit ball $B$ of $\mtc^n$. In particular, we prove the existence of universal holomorphic functions $f$ in the following sense : given any measurable function $\vphi$ on $\partial B$, there is a sequence $(r_n)_{n\geq 1}$, $0<r_n<1$, that converges to 1, such that $f(r_n\xi)$ converges to $\vphi(\xi)$ for almost every $\xi\,{\in}\,\partial B$.

It is known that every complete Boolean algebra of projections on a Banach space X is strongly closed and bounded and that, although the converse of this result fails in general, it is valid if X is weakly sequentially complete [1, XVII. 3, pp. 2194–2201]. In the present note it is shown that this converse is in fact valid precisely when X contains no subspace isomorphics to the sequence space c0. More explicitly, the following two results are proved. In both, X may be a real or complex space, but c0 will consist of the null sequences in the underlying scalar field.

A ring $R$ is called left morphic if, for every $a\in R$, $R/Ra\cong {\bf l}(a)$ where ${\bf l}(a)$ denotes the left annihilator of $a$ in $R$. Right morphic rings are defined analogously. In this paper, we investigate when the trivial extension $R\propto M$ of a ring $R$ and a bimodule $M$ over $R$ is (left) morphic. Several new families of (left) morphic rings are identified through the construction of trivial extensions. For example, it is shown here that if $R$ is strongly regular or semisimple, then $R\propto R$ is morphic; for an integer $n>1$, ${\mathbb Z}_n\propto {\mathbb Z}_n$ is morphic if and only if $n$ is a product of distinct prime numbers; if $R$ is a principal ideal domain with classical quotient ring $Q$, then the trivial extension $R\propto {Q}/{R}$ is morphic; for a bimodule $M$ over $\mathbb Z$, ${\mathbb Z}\propto M$ is morphic if and only if $M\cong{\mathbb Q}/{\mathbb Z}$. Thus, ${\mathbb Z}\propto {\mathbb Q}/{\mathbb Z}$ is a morphic ring which is not clean. This example settled two questions both in the negative raised by Nicholson and Sánchez Campos, and by Nicholson, respectively.

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.

Suppose throughout that λ > 0, κ > - l. γ is real and that

The series is said to be

(i) summable (C, k) to s if

(ii) strongly summable (C, k + 1) with index λ, or summable |C, k + 1|λ, to s if

(iii) absolutely summable (C, k) with indices γ, λ, or summable |C, k + 1|λ, if