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.

J. H. E. Cohn solved the diophantine equations x2 + 74 = yn and x2 + 86 = yn, with the condition 5 ∤ n, by more or less elementary methods. We complete this work by solving these equations for 5 | n, by less elementary methods.

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.

In this paper we consider complex doubles of compact Klein surfaces that have large automorphism groups. It is known that a bordered Klein surface of algebraic genus g > 2 has at most 12(g − 1) automorphisms. Surfaces for which this bound is sharp are said to have maximal symmetry. The complex double of such a surface X is a compact Riemann surface X+ of genus g and it is easy to see that if G is the group of automorphisms of X then C2 × G is a group of automorphisms of X+. A natural question is whether X+ can have a group that strictly contains C2 × G. In [8] C. L. May claimed the following interesting result: there is a unique Klein surface X with maximal symmetry for which Aut X+ properly contains C2 × Aut X (where Aut X+ denotes the group of conformal and anticonformal automorphisms of X+).

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|).

A semigroup S with 0 and 1 is termed completely right injective provided every right unitary S-system is injective. A necessary condition for a semigroup to be com-pletely right injective is given in [2]; namely, every right ideal is generated by an idempotent. An example in section 3 of this paper shows the existence of semigroups with 0 and 1 satisfying this condition which are not completely right injective. In [3], it is shown that the condition that every right and left ideal is generated by an idempotent is necessary and sufficient in the case that S is both completely right and left injective (called completely injective). Such a semigroup is an inverse semigroup with 0 whose idempotents are dually well-ordered.

The object of this paper is to prove that the only non-trivial solution in positive integers of the equation of the title is X = 3, Y = 4.

Substituting x = 2X – 1, y = 2Y – 1 gives with a little manipulation

We shall show that there exists a chain, order isomorphic to the chain of real numbers, of semigroup varieties closed for the Bruck extension. The least semigroup variety closed for the Bruck extension will be obtained as the union of varieties in an infinite chain of semigroup varieties.

It is shown in [2] that a uqu space satisfies the following conditions.

(DC) There is no infinite, strictly decreasing sequence of open sets with open intersection.

(IC) There is no infinite, strictly increasing sequence of open sets.

In this note we show that for a transitive space these conditions are sufficient for the space to be uqu. This will follow as a consequence of the following result.

In this note it is shown that if S is a free inverse semigroup of rank at least two and if e, f are idempotents of S such that e > f then S can be embedded in the partial semigroup eSe/fSf. The proof makes use of Scheiblich's construction for free inverse semigroups [7, 8] and of Reilly's characterisation of a set of free generators in an inverse semigroup [4, 5].

The development of the theory of local rings has been greatly stimulated by the importance of the applications to algebraic geometry, but it is none the less true that this stimulus has produced a theory which, on aesthetic grounds, is somewhat unsatisfactory. In the first place, if a local ring Q arises in the ordinary way from a geometric problem, then Qwill have the same characteristic as its residue field. It is partly for this reason that our knowledge of equicharacteristic local rings is much more extensive than it is of those local rings which present the case of unequal characteristics. Again, in the geometric case, the integral closure of Q in its quotient field will be a finite Q-module. Here, once more, we have a special situation which it would be desirable to abandon from the point of view of a general abstract theory.

In his paper [11], Peter Neumann considered in detail the cycle structures of elements of Aut(ℚ), the group of all homeomorphisms of the “rational world” ℚ onto itself, and further analyses of Aut(ℚ) and its subgroups have been given by Mekler [9], Bruyns [1], and Truss [13]. My interest in Aut(ℚ) stems from its utility in proving an at first sight rather startling (to a general topologist) result concerning β ℚ, the so-called Stone-Čech compactification of ℚ, namely that βℚ\ℚ is separable, and in fact contains a homogeneous countable dense subspace. (A space X is “homogeneous” provided whenever x, y ∈ X, there is some g ∈ Aut(X) with g(x) = y.) This is in sharp contrast to the spaces βℕ\ℕ and βℝ\ℝ, which are both inseparable.

In Section 33 of [2], Bonsall and Duncan define an element t of a Banach algebra to act compactly on if the map a → tat is a compact operator on . In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elements b of a C*-algebra for which the maps a → ba, a → ab, a → ab + ba, a → bab are compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the C*-automorphisms of which are weakly compact perturbations of the identity.

A well-known product, referred to as the Dirichlet convolution product, is generalized to arithmetic functions defined on an order in a Cayley division algebra. Factorization results for orders, multiplicative functions and analogues of the Moebius inversion formula are discussed.

In this paper we shall derive for function fields in one variable over finite constant fields results analogous to [1], where algebraic number fields were considered. The ground field P will be the set of all rational functions in a given transcendent X, with coefficients in k = GF(q), q = pr, p a prime; thus P = k(X).

If E is a subset of the real line of positive measure, then the associated Hilbert transform H = HE,

where the integral is a Cauchy principal value, is a bounded self-adjoint operator on L2(E) (cf. Muskhelishvili [4]). In case E = (-∞, ∞) the transformation is also unitary with a spectrum consisting of 1 and -1, each of infinite multiplicity (Titchmarsh [10]). If E is a inite interval the spectral representation of H has been given by Koppelman and Pincus [3]; see also Putnam [6]. In particular the spectrum of H is in this case the closed interval [-1, 1]. Moreover, according to Widom [11], the spectrum of H is [-1, 1] whenever E ≠ (-∞, ∞), that is, whenever

Let G be a group and let ℓ(G) be the set of all conjugacy classes [H] of subgroups H of G, where a partial order ≤ is defined by [H1] ≤ [H2] if and only if H1, is contained in some conjugate of H2.

A number of papers (see for example [1] and the references mentioned there) deal with the question of characterizing groups G by the poset ℓ(G). For example, in [1] it was shown that if ℓ(G) and ℓ(H) are order-isomorphic and G is a noncyclic p-group then |G| = |H|. Moreover, if G is abelian, then G = H, and if G is metacyclic then H is metacyclic.

The modern algebraic treatment of geometry in projective spaces focuses attention on the properties of homogeneous ideals in polynomial and power-series rings. This inevitably raises questions concerning how far ordinary ideal theory needs to be modified if only homogeneous ideals are to be regarded as significant. In practice, one can usually answer any particular question of this type without undue difficulty when it arises but, it seems to the author, the topic has enough intrinsic interest to merit a connected discussion by itself.