The first nontrivial DCE (2-computably enumerable) Turing approximation to the class of computably enumerable degrees is obtained. This depends on the following extension of the splitting theorem for the DCE degrees. For any DCE degree ${\bf a}$ and any computably enumerable degree ${\bf b}$ , if ${\bf b} < {\bf a}$ , then there are DCE degrees ${\bf x_0}, {\bf x_1}$ such that ${\bf b} < {\bf x_0}, {\bf x_1} < {\bf a}$ and ${\bf a} = {\bf x_0} \lor {\bf x_1}$ . The construction is unusual in that it is incompatible with upper cone avoidance.

The object of the paper is a thorough analysis of the WP-Bailey tree, a recent extension of classical Bailey chains. The paper begins by observing how the WP-Bailey tree naturally requires a finite number of classical $q$ -hypergeometric transformation formulas. It then shows how to move beyond this closed set of results, and in the process, heretofore mysterious identities of Bressoud are explicated. Next, WP-Bailey pairs are used to provide a new proof of a recent formula of Kirillov. Finally, the relation between the approach in the paper and that of Burge is discussed.

The fact that smooth Schubert varieties in partial flag manifolds are iterated fiber bundles over Grassmannians is used to give a simple presentation for their integral cohomology ring, generalizing Borel's presentation for the cohomology of the partial flag manifold itself. More generally, such a presentation is shown to hold for a larger class of subvarieties of the partial flag manifolds (which are called subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.

Let $G$ be a finite nonabelian simple group and let $\Gamma$ be a connected undirected Cayley graph for $G$ . The possible structures for the full automorphism group ${\rm Aut}\Gamma$ are specified. Then, for certain finite simple groups $G$ , a sufficient condition is given under which $G$ is a normal subgroup of ${\rm Aut}\Gamma$ . Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups $G = J_1, J_4$ , Ly and BM, while others fall into two infinite families and involve the Ree simple groups and alternating groups. The two infinite families contain examples of half-transitive graphs of arbitrarily large valency.

Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, $h_1(p)$ , is further examined. Whereas Kummer conjectured that $h_1(p) \sim G(p) := 2p(p/4\pi^2)^{(p-1)/4}$ it is shown, under certain plausible assumptions, that there exist constants $a_\alpha, b_\alpha$ such that $h_1(p) \sim \alpha G(p)$ for $\sim a_\alpha x/\log^{b_\alpha} x$ primes $p \le x$ whenever $\log \alpha$ is rational. On the other hand, there are $\ll_A x/\log^A x$ such primes when $\log \alpha$ is irrational. Under a weak assumption it is shown that there are roughly the conjectured number of prime pairs $p, mp\pm 1$ if and only if there are $\gg_m x/\log^2 x$ primes $p \le x$ for which $h_1(p) \sim e^{\pm 1/2m} G(p)$ .

A Coxeter group $W$ is said to be rigid if, given any two Coxeter systems $(W, S)$ and $(W, S^\prime)$ , there is an automorphism $\rho: W \longrightarrow W$ such that $\rho(S) = S^\prime$ . The class of Coxeter systems $(W, S)$ for which the Coxeter graph $\Gamma_S$ is complete and has only odd edge labels is considered. (Such a system is said to be of type $K_n$ .) It is shown that if $W$ has a type $K_n$ system, then any other system for $W$ is also type $K_n$ . Moreover, the multiset of edge labels on $\Gamma_S$ and $\Gamma_{S^\prime}$ agree. In particular, if all but one of the edge labels of $\Gamma_S$ are identical, then $W$ is rigid.

A matrix $A$ with minimum polynomial $m_A$ and characteristic polynomial $c_A$ is said to be cyclic if $m_A = c_A$ , semisimple if $m_A$ has no repeated factors, and separable if it is both cyclic and semisimple. For any set $T$ of matrices, we write $C_T$ for the proportion of cyclic matrices in $T, SS_T$ for the proportion of semisimple matrices, and $S_T$ for the proportion of separable matrices. We will write $C_{{\rm GL}(\infty,q)}$ for $\lim_{d\rightarrow\infty}C_{{\rm GL}(d,q)}$ , and so on.

Results from the character theory of symmetric groups are used to obtain an asymptotic estimate for the subgroup growth of fundamental groups of closed 2-manifolds. The main result implies an affirmative answer, for the class of groups investigated, to a question of Lubotzky's concerning the relationship between the subgroup growth of a one-relator group and that of a free group of appropriately chosen rank. As byproducts, an interesting statistical property of commutators in symmetric groups and the fact that in a ‘large’ surface group almost all finite index subgroups are maximal are obtained, among other things. The approach requires an asymptotic estimate for the sum $\Sigma 1/(\chi_\lambda(1))^s$ taken over all partitions $\lambda$ of $n$ with fixed $s \ge 1$ , which is also established.

Let $f$ be a conformal map of the unit disk ${\bb D}$ onto the domain $G \subset \hat{\bb C} = {\bb C} \cup \{\infty\}$ . We shall always use the spherical metric in $\hat{\bb C}$ .

Carathéodory [3] introduced the concept of a prime end of $G$ in order to describe the boundary behaviour of $f$ in geometric terms; see for example [6, Chapter 9] or [12, Section 2.4]. There is a bijective map $\hat{f}$ of ${\bb T} = \partial {\bb D}$ onto the set of prime ends of $G$ .

Removable singularities for Hardy spaces $H^p(\Omega) = \{f \in \hbox{Hol}(\Omega):\vert f\vert^p \le u \hbox{ in } \Omega \hbox{ for some harmonic }u\}, 0 < p < \infty$ are studied. A set $E \subset \Omega$ is a weakly removable singularity for $H^p(\Omega\backslash E)$ if $H^p(\Omega\backslash E) \subset \hbox{Hol}(\Omega)$ , and a strongly removable singularity for $H^p(\Omega\backslash E)$ if $H^p(\Omega\backslash E) = H^p(\Omega)$ . The two types of singularities coincide for compact $E$ , and weak removability is independent of the domain $\Omega$ .

The paper looks at differences between weak and strong removability, the domain dependence of strong removability, and when removability is preserved under unions. In particular, a domain $\Omega$ and a set $E \subset \Omega$ that is weakly removable for all $H^p$ , but not strongly removable for any $H^p (\Omega\backslash E), 0 < p < \infty$ , are found.

It is easy to show that if $E$ is weakly removable for $H^p(\Omega\backslash E)$ and $q > p$ , then $E$ is also weakly removable for $H^q(\Omega\backslash E)$ . It is shown that the corresponding implication for strong removability holds if and only if $q/p$ is an integer.

Finally, the theory of Hardy space capacities is extended, and a comparison is made with the similar situation for weighted Bergman spaces.

One of the consequences of the uniformization theorem of Koebe and Poincaré is that any smooth complex algebraic curve $C$ of genus $g > 1$ is conformally equivalent to ${\bb H}/G$ , where $G \subset \hbox{PSL}_2({\bb R})$ is a Fuchsian group and is naturally endowed with a hyperbolic metric. Conversely, any compact hyperbolic surface is isomorphic to an algebraic curve. Hence any curve of genus $g > 1$ may be described in two ways, either by an equation or by a Fuchsian group.

This paper is devoted to the study of a family of oscillatory multipliers of the Laplace–Beltrami operator on symmetric spaces of the noncompact type. This family is related to the regularity (in space, for fixed time) of the solutions to the wave equation. Our method relies on estimates for the Poisson semigroup in complex time, which were obtained in our previous paper [6].

The question of when a $\phi$ -derivation on a Banach algebra has quasinilpotent values, and how this question is related to the noncommutative Singer–Wermer conjecture, is discussed.

A geometric property is introduced, and it is proved that Banach spaces with this property must carry weakly amenable algebras of approximable operators. These results are also applied to particular examples of Banach spaces, including the James spaces and the Tsirelson space.

The set of essentially Hankel operators is introduced and some of its properties are investigated. It is shown in particular that the set contains some operators not of the form ‘Hankel plus compact’, even when restricted to the class of essentially Hankel operators with trivial (Fredholm) index function.

In this paper, we are mainly concerned with $n$ -dimensional simplices in hyperbolic space ${\bb H}^n$ . We will also consider simplices with ideal vertices, and we suggest that the reader keeps the Poincaré unit ball model of hyperbolic space in mind, in which the sphere at infinity ${\bb H}^n(\infty)$ corresponds to the bounding sphere of radius 1. It is known that all hyperbolic simplices (even the ideal ones) have finite volume. However, explicit calculation of their volume is generally a very difficult problem (see, for example, [1] or [16]). Our first theorem states that, amongst all simplices in a closed geodesic ball, the simplex of maximal volume is regular. We call a simplex regular if every permutation of its vertices can be realized by an isometry of ${\bb H}^n$ . A corresponding result for simplices in the sphere has been proved by Böröczky [4].