Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field ${\bb Q}$ , are contained in the complex field ${\bb C}$ . Let $P$ be the set of all prime numbers. For any algebraic number field $F$ , let $C_F$ denote the ideal class group of $F$ and, writing $F^+$ for the maximal real subfield of $F$ , let $C^-_F$ denote the kernel of the norm map from $C_F$ to the ideal class group of $F^+$ ; for each $l \in P$ , let $C_F(l)$ denote the $l$ -class group of $F$ , that is, the $l$ -primary component of $C_F$ , and let $C^-_F(l)$ denote the $l$ -primary component of $C^-_F$ . Furthermore, for each $l \in P$ , we denote by ${\bb Z}_l$ the ring of $l$ -adic integers.

It is proved that all modules over an integral domain $R$ have strongly flat cover if and only if every flat $R$ -module is strongly flat. The domains satisfying this property are characterized by the property that all their proper quotients are perfect rings, and are called almost perfect. They are exactly the $h$ -local domains which are locally almost perfect. Various relevant classes of modules admitting or not admitting strongly flat cover are exhibited.

Let $F$ be an algebraically closed field of characteristic $0$ , and let $A$ be a $G$ -graded algebra over $F$ for some finite abelian group $G$ . Through $G$ being regarded as a group of automorphisms of $A$ , the duality between graded identities and $G$ -identities of $A$ is exploited. In this framework, the space of multilinear $G$ -polynomials is introduced, and the asymptotic behavior of the sequence of $G$ -codimensions of $A$ is studied.

Two characterizations are given of the ideal of $G$ -graded identities of such algebra in the case in which the sequence of $G$ -codimensions is polynomially bounded. While the first gives a list of $G$ -identities satisfied by $A$ , the second is expressed in the language of the representation theory of the wreath product $G \wr S_n$ , where $S_n$ is the symmetric group of degree $n$ .

As a consequence, it is proved that the sequence of $G$ -codimensions of an algebra satisfying a polynomial identity either is polynomially bounded or grows exponentially.

In 1998, Zalesskii proposed to classify all instances of irreducible characters of quasi- simple groups which are of prime power degree. In joint work, Malle and Zalesskii then dealt with all quasi-simple groups with the exception of the alternating groups and their double covers [5]. In an earlier article [1] we have classified all the irreducible characters of $S_n$ of prime power degree and have deduced from this also the corresponding classification for the alternating groups. In the present article we complete Zalesskii's programme by dealing with the final case left open in [5], the double covers of the alternating groups. We derive this result from a corresponding result on the double covers of the symmetric groups. If one is only interested in spin characters, one easily sees that only 2-powers can occur as prime power degrees. However from a combinatorial point of view it is natural to ask more generally: when is the number of shifted standard tableaux of a given shape a prime power? Since our method is independent of the prime, we will answer this question, showing that apart from a few accidental cases for small $n$ only the ‘obvious’ partitions satisfy the prime power condition. Thus in turn for the spin characters, apart from exceptions for small $n$ , the ‘obvious’ spin characters of 2-power degree are indeed the only ones (this confirms the conjecture stated in [5]).

A transitive finite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and affine group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups.

Part of the motivation for studying this class of groups was a conjecture due to Marušič, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.

A criterion is given for showing that certain one-relator groups are residually finite. This is applied to a one-relator group with torsion $G = \langle a_1, \ldots, a_r \mid W^n\rangle$ . It is shown that $G$ is residually finite provided that $W$ is outside the commutator subgroup and $n$ is sufficiently large. An important ingredient in the proof is a criterion which implies that a subgroup of a group is malnormal. A graded small-cancellation criterion is developed which detects whether a map $A \rightarrow B$ between graphs induces a $\pi_1$ -injection, and whether $\pi_1 A$ maps to a malnormal subgroup of $\pi_1 B$ .

The paper examines blow-up phenomena for the inequality \renewcommand{\theequation}{*} \begin{equation} u_t - Lu - \vert u \vert^{q-1} u \leqslant v_t - Lv - \vert v \vert^{q-1}v \end{equation} in the half-space ${\bb R}^1_+ \times {\bb R}^n,\; n \geqslant 1$ , where $L$ is a linear second-order partial differential operator in divergence form.

The paper studies weak solutions of (*) that belong only locally to the corresponding Sobolev spaces in the half-space ${\bb R}^1_+ \times {\bb R}^n$ . It also requires no conditions for the behavior of solutions of (*) on the hyperplane $t = 0$ .

The existence of critical blow-up exponents is obtained for solutions of (*) as a special case of a comparison principle for the corresponding solutions of (*). For example, the well-known Fujita result is a consequence of the comparison principle.

The approach developed in the paper is directly applicable to the study of analogous problems involving nonlinear differential operators. Its elliptic analogue has been recently developed by the authors.

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 $G$ .

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 $\alpha_2 = \alpha_1(1-\alpha_1)$ .

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 $\omega \subseteq {\bb R}^d$ 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 $f$ .

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

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].

Guided by the Hopf fibration, a family (indexed by a positive constant $K$ ) of right invariant Riemannian metrics on the Lie group $S^3$ is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each $K > 1$ . Each such Riemannian metric couples with the corresponding Killing field to produce a $y$ -global and explicit Randers metric on $S^3$ . Employing the machinery of spray curvature and Berwald's formula, it is proved directly that the said Randers metric has constant positive flag curvature $K$ , as predicted by Yasuda–Shimada. It is explained why this family of Finslerian space forms is not projectively flat.

The paper shows that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (that is, is tangent to a smooth variation through harmonic maps). This provides one of the few known answers to the problem of integrability, which was raised in different contexts of geometry and analysis. It implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from $S^2$ to ${\bb C} P^2$ thus giving the nullity of any such harmonic map; it also has a bearing on the behaviour of weakly harmonic $E$ -minimizing maps from a 3-manifold to ${\bb C} P^2$ near a singularity and the structure of the singular set of such maps from any manifold to ${\bb C} P^2$ .

Parthasarathy and Sunder have proved that the set of coherent vectors associated with the indicator functions of Borel sets is total in the boson Fock space $\Gamma(L^2({\bb R}^+;{\bb C}))$ . The paper studies the space generated by coherent vectors associated with the union of $n$ intervals. A complete characterization is given of their orthogonal space in terms of their chaos expansion. Parthasarathy and Sunder's result is recovered in a very simple way. In the cases of the Brownian motion or Poisson process interpretation of the Fock space, the result characterizes those random variables that are orthogonal to the exponential of any sum of $n$ increments of the Brownian motion or Poisson process.

Let $D$ be an open set in euclidean space ${\bb R}^m$ with non-empty boundary $\partial D$ , and let $p_D : D \times D \times; [0,\infty)) \longrightarrow {\bb R}$ be the Dirichlet heat kernel for the parabolic operator ${-}\Delta + \partial/\partial t$ , where ${-}\Delta$ is the Dirichlet laplacian on $L^2(D)$ . Since the Dirichlet heat kernel is non-negative, we may define the (open) set function \renewcommand{\theequation}{1.1} \begin{equation} P_D = \int\nolimits^{\infty}_0 \int\nolimits_D \int\nolimits_D p_D (x,y;t)\,dx\,dy\,dt. \end{equation} We say that $D$ has finite torsional rigidity if $P_D < \infty$ . It is well known that if $D$ has finite volume, then $D$ has finite torsional rigidity [11]. As we shall see, the converse is not true. The main purpose of this paper is to obtain necessary and sufficient conditions on the geometry of $D$ to guarantee finite torsional rigidity and to gain some understanding of the behaviour of the expected lifetime of brownian motion in a certain natural class of domains that do not have finite torsional rigidity.