Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω\{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.

In this note we give some sharp estimates for norms of polynomials via the products of norms of their linear terms. Different convex norms on the unit disc are considered.

For a very general class of ordinary quasi-differential expressions M with matrix-valued coefficients, the maximal operator TM and the minimal operator T0M are defined as operators of a subspace of Lp into Lq for arbitrary p, q∈[1, ∞], where the weak topologies induced by the dualities 〈Lp, Lp′〉 and 〈Lq, Lq′〉 with 1/p+1/p′=1 and 1/q+1/q′=1 are used instead of the norm topologies. It is shown that these operators and their adjoints possess the usual properties which are known for scalar differential expressions and the two cases p, q∈[1, ∞) and p, q∈(1, ∞].

Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R+: u′(t)=Au(t)+ϕ(t) (t[ges ]0). Suppose that u has uniformly convergent means, σ(A)∩iR is countable, and ϕ is asymptotically almost periodic. Then u is asymptotically almost periodic. Related results have been obtained by Ruess and Vũ, and by Basit, using different methods. A direct proof is given of a Tauberian theorem of Batty, van Neerven and Räbiger, and applications to Volterra equations are discussed.

As a special case of a well-known conjecture of Artin, it is expected that a system of R additive forms of degree k, say

formula here

with integer coefficients aij, has a non-trivial solution in ℚp for all primes p whenever

formula here

Here we adopt the convention that a solution of (1) is non-trivial if not all the xi are 0. To date, this has been verified only when R=1, by Davenport and Lewis [4], and for odd k when R=2, by Davenport and Lewis [7]. For larger values of R, and in particular when k is even, more severe conditions on N are required to assure the existence of p-adic solutions of (1) for all primes p. In another important contribution, Davenport and Lewis [6] showed that the conditions

formula here

are sufficient. There have been a number of refinements of these results. Schmidt [13] obtained N[Gt ]R2k3 log k, and Low, Pitman and Wolff [10] improved the work of Davenport and Lewis by showing the weaker constraints

formula here

to be sufficient for p-adic solubility of (1).

A noticeable feature of these results is that for even k, one always encounters a factor k3 log k, in spite of the expected k2 in (2). In this paper we show that one can reach the expected order of magnitude k2.

In the early 1930s, Wiener proved that if f(x) is a strictly positive periodic function whose Fourier series is absolutely convergent, then the Fourier series of g(x)=1/f(x) is also absolutely convergent [8, pp. 10–14]. This phenomenon can be easily understood nowadays using Banach algebra techniques (see, for example, [4, pp. 202–203]). In fact, these techniques allow us to study the absolute convergence of g(x)=F(f(x)), where F is holomorphic in an open subset of [Copf ] that contains the range of f(x) (for x∈ℝ). In this context, Wiener's original problem corresponds to the choice F(z)=1/z.

In this work we want to analyse the constraints on the simultaneous rate of vanishing of the Fourier coefficients fˆ(n) and ĝ(n) as n→∞. We shall focus on g=1/f, but we shall also study the general case g=F(f). In either case, there are obviously no constraints when f is a constant function.

Although this problem does not seem to be directly related to uncertainty inequalities for the Fourier Transform, we observe that there are some analogies, both in the nature of the results and in the proof techniques. The general fact with which we are dealing is that fˆ(n) and ĝ(n) cannot vanish too quickly at the same time as n→∞, unless f(x) is constant. The general fact that underlies uncertainty inequalities is that a non-periodic function ϕ(x) and its Fourier Transform ϕcirc;(u) cannot vanish too quickly at the same time as x→∞ and u→∞, unless ϕ(x) is zero (almost everywhere). For a simple introduction to some aspects of uncertainty inequalities, see [5]; for a thorough and recent introduction to this vast subject, see [3].

Throughout this paper, X will denote a Banach space, S=S(X) and B=B(X) will be the unit sphere and the closed unit ball of X, respectively, and [Gscr ]=[Gscr ](X) will stand for the group of all surjective linear isometries on X. Unless explicitly stated otherwise, all Banach spaces will be assumed to be real. Nevertheless, by passing to real structures, the results remain true for complex spaces.

Let T=(T1, …, Tn) be a commuting n-tuple of bounded linear operators acting on some complex Banach space [Xscr ]. We show that if T has the single-valued extension property, then the local spectrum σT(x) coincides with the spectrum σ(T), for all vectors x∈[Xscr ], except on a set of the first Baire category.

We prove that a large and natural class of subhomogeneous C*-algebras with one-dimensional spectrum have the fragility property, that is, every morphism out of such a C*-algebra and into a C*-algebra of real rank zero can be approximated by a morphism with finite-dimensional range, provided that a K-theoretical obstruction vanishes.

Let Ω be an open subset of IRd, and let Tp for p∈[1, ∞) be consistent C0-semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting their generators by Ap, we show that the spectrum σ(Ap) is independent of p∈[1, ∞). We also treat the case of weighted Lp-spaces for weights that satisfy a subexponential growth condition. An example shows that independence of the spectrum may fail for an exponential weight.

One important invariant of a closed Riemannian 3-manifold is the Chern–Simons invariant [1]. The concept was generalized to hyperbolic 3-manifolds with cusps in [11], and to geometric (spherical, euclidean or hyperbolic) 3-orbifolds, as particular cases of geometric cone-manifolds, in [7]. In this paper, we study the behaviour of this generalized invariant under change of orientation, and we give a method to compute it for hyperbolic 3-manifolds using virtually regular coverings [10]. We confine ourselves to virtually regular coverings because a covering of a geometric orbifold is a geometric manifold if and only if the covering is a virtually regular covering of the underlying space of the orbifold, branched over the singular locus. Therefore our work is the most general for the applications in mind; namely, computing volumes and Chern–Simons invariants of hyperbolic manifolds, using the computations for cone-manifolds for which a convenient Schläfli formula holds (see [7]). Among other results, we prove that every hyperbolic manifold obtained as a virtually regular covering of a figure-eight knot hyperbolic orbifold has rational Chern–Simons invariant. We give explicit examples with computations of volumes and Chern–Simons invariants for some hyperbolic 3-manifolds, to show the efficiency of our method. We also give examples of different hyperbolic manifolds with the same volume, whose Chern–Simons invariants (mod ½) differ by a rational number, as well as pairs of different hyperbolic manifolds with the same volume and the same Chern–Simons invariant (mod ½). (Examples of this type were also obtained in [12] and [9], but using mutation and surgery techniques, respectively, instead of coverings as we do here.)