Let Ω be the weakly pseudoconvex domain

and let ∂Ω be its boundary. If ϕ ∈ L∞ (∂Ω), we denote by Tϕ, the Toephtz operator with symbol ϕ acting on the Hardy space H2(∂Ω), and by J(∂Ω) the C*-subalgebra of B(H2(∂Ω)) generated by the Toeplitz operators with continuous symbol. Our main theorem asserts that J(∂Ω) contains the ideal K of all compact operators on H2(∂Ω), and that the symbol map ϕ→Tϕ induces an isomorphism of C(∂Ω) onto the quotient C*-algebra ℑ(∂Ω)/K. Similar results have been established before for other domains, and in particular when Ω is strongly pseudoconvex. The main interest of our results lies in their proofs: ours are elementary, whereas those used in the strongly pseudoconvex case depend heavily on the theory of the tangential Cauchy-Riemann operator.

Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

By using pairwise relatively complete families of functions, which we define here, we obtain two characterizations of quasi-pseudometrizable bitopological spaces from which some known theorems can be derived as easy corollaries.

We supply proofs that are simple, and possibly partly new, for two theorems that appear in Rankin's book [6].

In this paper, we show some results involving classical geometric concepts. For example, we characterize rotundity and Efimov-Stechkin property by mean of faces of the unit ball. Also, we prove that every almost locally uniformly rotund Banach space is locally uniformly rotund if its norm is Fréchet differentiable. Finally, we also provide some theorems in which we characterize the (strongly) exposed points of the unit ball using renormings.

In his paper [8], N. Itô gives an elegant proof that the Sylow p-group of a finite solvable linear group of degree n over the field of complex numbers is necessarily normal if p > n+1. Moreover he shows that this bound on p is the best possible when p is a Fermat prime (i.e. a prime of the form 2sk + 1) but that the bound may be improved to p > n when p is not a Fermat prime.

A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.

In a previous paper ([14]) the author showed that a free inverse semigroup is determined by its lattice of inverse subsemigroups, in the sense that for any inverse semigroup T, implies . (In fact, the lattice isomorphism is induced by an isomorphism of upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and E-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

Let E be a Banach space whose dual E* has the approximation property, and let m be an index. We show that E* has the Radon-Nikodým property if and only if every m-homogeneous integral polynomial from E into any Banach space is nuclear. We also obtain factorization and composition results for nuclear polynomials.

Finite ρ-groups with all of their maximal subgroups isomorphic are studied by means of the coclass. All such groups of coclass I and 2 are determined, while those of coclass 3 are shown to have order at most ρ13. A general bound for the order is given as a function of ρ and the coclass only.

We prove some Banach–Stone type theorems for linear isometries of vector-valued continuous function spaces, by making use of the extreme point method.

In a series of papers ((1967), (1967a) and (1967b)) Magill has considered the semigroups J(X, Y; θ) (definition below), a natural, but extensive, generalization of the usual transformation semigroup J(X). They have also been studiedin Sullivan (to appear). Under the assumption that θ be onto Magill described their automorphisms and determined when one J(X, Y; θ) is isomorphic to another.

Schmidt has shown that if r and s are positive integers and there is no positive integer power of r which is also a positive integer power of s, then there exists an uncountable set of reals which are normal to base r but not even simply normal to base s. We give a structurally simple proof of this result

We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images.

We prove that for a large class of Banach function spaces continuity and holomorphy of superposition operators are equivalent and that bounded superposition operators are continuous. We also use techniques from infinite dimensional holomorphy to establish the boundedness of certain superposition operators. Finally, we apply our results to the study of superposition operators on weighted spaces of holomorphic functions and the $F(p, \alpha , \beta )$ spaces of Zhao. Some independent properties on these spaces are also obtained.

Our set-up will consist of the following: (i) a graph with vertex set V and edge set E; (ii) for each vertex ∈ V a non-trivial group Gv given by a presentation (xν; rν); (iii) for each edge e = {u, ν} ∈ E a group Ge given by a presentation (xu, xv; re) where re consists of the elements of ru ∪ rv, together with some further words on xu ∪ xv. We let G = (x; r) where x = ∪v∈v xv, r = ∪e∈E re. Ouraim is to try to describe the structure of G in terms of the groups Gv, (v ∈ V), Ge (e ∈ E). Under suitable conditions the natural homomorphisms Gv, → G (ν ∈ V), Ge → Ge (e ε E) are injective; and there is a short exact sequence (where, for any group H, IH is the augmentation ideal). Some (co)homological consequences of these resultsare derived.

The theory of alternating bilinear forms on finite dimensional vector spaces V is well understood; two forms on V are equivalent if and only if they have equal ranks. The situation for alternating trilinear forms is much harder. This is partly because the number of forms of a given dimension is not independent of the underlying field and so there is no useful canonical description of an alternating trilinear form.

Let A be a commutative Banach algebra with identity of norm 1, X a Banach A-module and G a locally compact abeian group with Haar measure. Then the multipliers from an A -valued function algebra into an X-valued function space is studied. We characterize the multiplier spaces as the following isometrically isomorphic relations under some appropriate conditions:

A sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

All known Banach spaces have an infinite-dimensional separable quotient and so do all nonnormable Fréchet spaces, although the general question for Banach spaces is still open. A properly separable topological vector space is defined, in such a way that separable and properly separable are equivalent for an infinite-dimensional complete metrisable space. The main result of this paper is that the strict inductive limit of a sequence of non-normable Fréchet spaces has a properly separable quotient.