The entities A, B, X, Y in the title are operators, by which we mean either linear transformations on a finite-dimensional vector space (matrices) or bounded (= continuous) linear transformations on a Banach space. (All scalars will be complex numbers.) The definitions and statements below are valid in both the finite-dimensional and the infinite-dimensional cases, unless the contrary is stated.

In [1] Almkvist and Meurman proved a result on the values of the Bernoulli polynomials (Theorem 5 below). Subsequently, Sury [5] and Bartz and Rutkowski [2] have given simpler proofs.

In this paper we show how this theorem can be obtained from classical results on the arithmetic of the Bernoulli numbers. The other ingredient is the remark that a polynomial with rational coefficients which is integer-valued on the integers is ℤ(p)-valued on ℤ(p). Here ℤ(p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. An application is given in Section 3 to the arithmetic of generalised Bernoulli numbers.

Let (R, [mfr ]) be a local Noetherian ring. We show that if R is complete, then an R-module M satisfies local duality if and only if the Bass numbers μi([mfr ], M) are finite for all i. The class of modules with finite Bass numbers includes all finitely generated, all Artinian, and all Matlis reflexive R-modules. If the ring R is not complete, we show by example that modules with finite Bass numbers need not satisfy local duality. We prove that Matlis reflexive modules satisfy local duality in general, where R is any local ring with a dualizing complex.

If G is a finitely presented group and [Kscr] is any (G,2)-complex (that is, a finite 2-complex with fundamental group G), then it is well known that χ([Kscr]) [ges ] ν(G), where ν(G) = 1−rk H1G+dH2G. We define χ(G) to be min{χ([Kscr]): [Kscr] a (G, 2)-complex}, and we say that G is efficient if χ(G) = ν(G). In this paper we give sufficient conditions for a Coxeter group to be efficient (Theorem 4.2). We also give examples of inefficient Coxeter groups (Theorem 5.1). In fact, we give an infinite family Gn(n = 2, 3, 4, . . . ) of Coxeter groups such that χ(Gn)−ν(Gn) [xrarr ] ∞ as n [xrarr ] ∞.

Given a field k and a finite group G acting on the rational function field k(X1, . . ., Xn) as a group of k-automorphisms, an important Noether's problem asks whether the invariant subfieldformula hereis purely transcendental over k.

A group is said to be (2,3)-generated if it can be generated by an element of order 2 and an element of order 3. The class of (2,3)-generated groups seems to be rather extensive. From results of Schupp [7] and Mason and Pride [3], it is known that there are 2ℵ0 isomorphism classes of (2,3)-generated groups, and moreover that every countable group can be embedded in a (2,3)-generated group. All finite non-abelian simple groups are (2,3)-generated, with the exception of some groups of low rank in characteristics 2 and 3 (see Wilson [13] or Sanchini and Tamburini [6]).

Suppose that K is a linear space of functions analytic in some domain D in thecomplex plane. A sequence Λ = (λk) of distinct points from D is said to be a set ofuniqueness for K if ƒ∈K and ƒ(λk) = 0 for all k imply ƒ≡0. Depending on thedispersion and the density of Λ on the one hand, and the growth of the functions inK on the other, one may often require only |ƒ(λk)| [les ]ak for some sequence of positivenumbers ak, and still conclude that ƒ≡0 for ƒ∈K. Of particular interest are sharpconditions on the decay of ak, which reflect the interplay between growth and decayof analytic functions.

We deal with weighted inequalities of the type

formula here

where:

T is either the Hardy–Littlewood maximal operator or a singular integral operator;G is any measurable subset of ℝn;ƒ is any measurable function, vanishing outside G, such that Tƒ is well-defined;v and w are weights, that is, nonnegative locallyintegrable functions on G;

p, q ∈(1, ∞).

We precisely evaluate the operator norm of the uncentred Hardy–Littlewood maximal function on Lp(ℝ1). Consequently, we compute the operator norm of the ‘strong’ maximal function on Lp(ℝn), and we observe that the operator norm of the uncentred Hardy–Littlewood maximal function over balls on Lp(ℝn) grows exponentially as n[xrarr ]∞.

Let [Ascr] be a complex algebra (with unit e), X a left [Ascr] module and a = (a1, . . ., am) a commuting tuple of elements in [Ascr]. Following Taylor [8], we have to consider the Koszul complex K·(a, X) in order to define the joint spectrum of the tuple a.

We generalise to Jordan–Banach algebras some results of Aupetit on perturbation by inessential elements. This enables us to show that the kh-socle is the largest inessential ideal, and to give some spectral characterisations of it.

This work is an investigation of the extent to which convex bodies are determined by the sizes of their projections. It is shown, on the one hand, that there are non-congruent bodies for which all corresponding projections have the same size. On the other hand, it is proved that in the usual Baire category sense, most convex bodies are determined, up to translation or reflection, by the combination of their widths and brightnesses in all directions.

In this paper the domain in which anytwo points are joined by finite numbers of arbitrarilysmall homeomorphisms is studied, and twoexamples are given.

We prove that it is consistent with the axioms of ZFC, relative to the consistency of ZFC itself, that the Baire number of the Stone representation space of the Boolean algebra [Pscr](ω)/fin equals the distributivity number of [Pscr](ω)/fin. This answers a question of Balcar, Pelant and Simon in [1].

The purpose of this note is to obtain a restriction on the fundamental groups of non-elliptic compact complex surfaces of class VII in Kodaira's classification [9]. We recall that these are the compact complex surfaces with first Betti number one and no non-constant meromorphic functions. This seems to be the class of compact complex surfaces whose structure is least understood. The first and simplest examples are the general Hopf surfaces [9, III]. Then there are various classes of examples found by Inoue [5, 6], and which have been studied in more detail in [11]. The only known topological restriction beyond the first Betti number is that intersection form in two-dimensional homology is negative definite. There seems to be little known as to how wide this class of surfaces is. We prove the following theorem.

An explanation of a duplication identity in law involving the variances of the Brownian bridge and Brownian motion is given, with the help of an elementary transformation in time and space of a two-dimensional Brownian motion.