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.

Let $G$ be a compact group and ${\cal C}(G)$ be the $C^{*}$ -algebra of continuous complex-valued functions on $G$ . The paper constructs an imbedding of the Fourier algebra ${\rm A}(G)$ of $G$ into the algebra ${\rm V}(G) = {\cal C}(G)\otimes^h {\cal C}(G)$ (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in $G \times G$ are sets of operator synthesis with respect to the Haar measure, using the definition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders defined by closed unital subsemigroups of $G$ .

Let $G$ be a finite group and suppose that $P$ is a soluble $\{2, 3\}^\prime$ -subgroup of $G$ . The reader will lose only a little by assuming that $P$ is a subgroup of prime order $p > 3$ . Define \[ \Sigma_G(P) = \{A \le G \mid A \hbox{ is soluble and } A = \langle P, P^a\rangle\hbox{ for some }a \in A\}. \] This set is partially ordered by inclusion and we let \[ \Sigma^*_G(P) \] denote the set of maximal members of $\Sigma_G(P)$ .

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.

The $n$ th quantum derivative ${\cal D}_nf(x)$ of the real-valued function $f$ is defined for each real non-zero $x$ as \[ \lim_{q\to 1}\frac{\displaystyle\sum\limits^n_{k=0}(-1)^k\left[\begin{array}{c}n\\k\end{array}\right]_qq^{(k-1)k/2}f(q^{n-k}x)}{q^{(n-1)n/2}(q-1)^nx^n}, \] where $\left[\begin{array}{c}n\\k\end{array}\right]_q$ the $q$ -binomial coefficient. If the $n$ th Peano derivative exists at $x$ , which is to say that if $f$ can be approximated by an $n$ th degree polynomial at the point $x$ , then it is not hard to see that ${\cal D}_nf(x)$ must also exist at that point. Consideration of the function $|1-x|$ at $x=1$ shows that the second quantum derivative is more general than the second Peano derivative. However, it can be shown that the existence of the $n$ th quantum derivative at each point of a set necessarily implies the existence of the $n$ th Peano derivative at almost every point of that set.

For the nonlinear recurrence relation \[ \alpha_2 = \alpha_1(1 - \alpha_1), \quad \alpha_{k+1} = k^2 \alpha_k + \sum\limits_{m=2}^{k-1} \alpha_m \alpha_{k+1-m}, \quad k\geqslant 2, \] it is proved that the limit \[ p_{\infty}(\alpha_1) = \lim_{k \rightarrow \infty} \alpha_k /[(k-1)!]^2 \] exists and defines an entire function of $\alpha_2 = \alpha_1(1-\alpha_1)$ .

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 $G$ .

Topologists say that a space is sequential if every sequentially closed set is closed. Directly from the definitions, metrizable ⇒ Fréchet–Urysohn ⇒ sequential ⇒ $k$ -space. Kąkol showed that for an (LM)-space (the inductive limit of a sequence of locally convex metrizable spaces), metrizable , Fréchet–Urysohn. The Cascales and Orihuela result that every (LM)-space is angelic proved that for an (LM)-space, sequential [hArr ] $k$ -space. Within the class of (LM)-spaces, then, the four notions become only two distinct ones bearing the relation metrizable ⇒ sequential. Webb proved that every infinite-dimensional Montel (DF)-space is sequential but not Fréchet–Urysohn, and equivalently, not metrizable, since Montel (DF)-spaces are (LB)-spaces and, a fortiori, (LM)-spaces. Does the converse hold in the (LB)-space, (DF)-space or (LM)-space settings? Yes, in all cases. If a (DF)-space or (LM)-space is sequential, then it is either metrizable or it is a Montel (DF)-space. Pfister's result that every (DF)-space is angelic is needed, and the paper provides elementary proofs for this and the similar theorem by Cascales and Orihuela. The strongest locally convex topology plays a vital role throughout.

Let the manifold $X$ parametrise a family of compact complex submanifolds of the complex (or CR) manifold $Z$ . Under mild conditions the Penrose transform typically provides isomorphisms between a cohomology group of a holomorphic vector bundle $V\longrightarrow Z$ and the kernel of a differential operator between sections of vector bundles over $X$ . When the spaces in question are homogeneous for a group $G$ the Penrose transform provides an intertwining operator between representations.

The paper develops a Penrose transform for compactly supported cohomology on $Z$ . It provides a number of examples where a compactly supported cohomology group is shown to be isomorphic to the cokernel of a differential operator between compactly supported sections of vector bundles over $X$ . It considers also how the Serre duality pairing carries through the transform.

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.

Let $\omega \subseteq {\bb R}^d$ be an open domain. The sequentially complete DF-spaces $E$ are characterized such that for each (some) discrete sequence $(z_n) \subseteq \omega$ , a sequence of natural numbers $(k_n)$ and any family $(x_{n, \alpha})_{n \in {\bb N}, \vert \alpha\vert \leqslant k_n} \subseteq E$ the infinite system of equations \[ \left(\frac{\partial^{\vert \alpha\vert } f}{\partial z^{\alpha}}\right)(z_n) = x_{n,\alpha} \quad \hbox{for } n \in {\bb N}, \alpha \in {\bb N}^{d}, \vert \alpha\vert \leqslant k_n, \] has an $E$ -valued real analytic solution $f$ .

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.

Through the use of the so-called variational Lyapunov method, which is developed by combining the method of variation of parameters and the Lyapunov second method, stability and instability properties in terms of two measures for impulsive differential equations with variable moments of impulsive effects are discussed. Some stability and instability criteria are established. These theorems, together with an example, show that perturbation and impulsive effects may make a stable system uniformly asymptotically stable or unstable. These results significantly generalize the known ones.

The structure of the continuous strict completely positive linear maps between locally $C^{*}$ -algebras is described.

A systematic treatment is given of several classes of singular integrals. Their $L^p$ boundedness is proved when their kernels are given by functions $\Omega$ in $L\log L({\bf S}^{n-1})$ .

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.

The problem of domination for positive compact operators between Banach lattices was solved by Dodds and Fremlin in [6]: given a Banach lattice $E$ with order continuous dual norm, an order continuous Banach lattice $F$ and two positive operators $0 \leqslant S \leqslant T : E \longrightarrow F$ , the operator $S$ is compact if $T$ is. A similar problem has been considered in the class of weakly compact operators by Abramovich [1] and in a general form by Wickstead [26]. Precisely, Wickstead's result shows that the operator $S$ is weakly compact if $T$ is whenever one of the following two conditions holds: either $E^{\prime}$ is order continuous or $F$ is order continuous. When it comes to Dunford–Pettis operators, Kalton and Saab [15] have proved that the operator $S$ is Dunford–Pettis if $T$ is, provided that the Banach lattice $F$ is order continuous. On the other hand, Aliprantis and Burkinshaw settled the problem of domination for compact [2] and weakly compact [3] endomorphisms (that is, the case when $E = F$ ). For example, they proved that if either the norm on $E$ or the norm on $E^{\prime}$ is order continuous, then the compactness of $T$ is inherited by the power operator $S^2$ . Also, they showed that, for $E$ an arbitrary Banach lattice, $T$ being compact always implies that $S^3$ is compact. More recently, Wickstead studied converses for the Dodds–Fremlin and Kalton–Saab theorems in [27] and for the Aliprantis–Burkinshaw theorems in [28].

Maximal accretive realizations and bound-preserving self-adjoint extensions are two fundamental problems in applications of semi-bounded operator theory to differential equations. On the basis of using differential operator theory in direct sum spaces and Phillips theory for maximal accretive extensions of accretive operators, a complete characterization of the set of maximal accretive boundary conditions for Sturm–Liouville differential operators is presented. As an application, all possible forms of bound-preserving self-adjoint extensions of regular Sturm–Liouville operators are also characterized via various explicit boundary conditions. The methodology can also be applied to dealing with general classes of semi-bounded symmetric differential operators.

A family of Hecke $C^*$ -algebras can be realised as crossed products by semigroups of endomorphisms. It is shown by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.