It is well known that for a Noetherian ring R, an ideal I of R, and M a finitely generated R-module, the local cohomology modules HiI(M) are not always finitely generated. On the other hand, if R is local and m is its maximal ideal, then Him(M) are Artinian modules, which is equivalent to the following two properties:

(i) SuppR(Him(M)) ⊆{m};

(ii) the vector space HomR(k, Him(M)) has finite dimension over k, where k = R/m.

Taking these facts into account, Grothendieck [9] made the following conjecture.

Let [Mscr ]n denote the space of complex n×n matrices, and let Ωn denote the spectral unit ball of [Mscr ]n, namely the set of matrices in [Mscr ]n whose eigenvalues lie within the open unit disc. As a step towards the eventual classification of the holomorphic automorphisms of Ωn, we prove that every such automorphism F satisfying F(0) = 0 and F′(0) = I has the property that F(x) is conjugate to x for each x∈Ωn. This result is obtained by combining earlier work of Ransford and White with a general theorem about spectrum-preserving maps proved in this paper.

An upper bound on operator norms of compound matrices is presented, and special cases that involve the l1, l2 and l∞ norms are investigated. The results are then used to obtain bounds on products of the largest or smallest eigenvalues of a matrix.

It is well known that the second homology group of any Kac–Moody Lie algebra and the Virasoro algebra is trivial. This is equivalent to saying that any Kac–Moody Lie algebra (or the Virasoro algebra) is its own universal covering algebra; see, for example, [1] and [4]. Recently, Loday and Pirashvili in [12] proved that the second Leibniz homology group of the Virasoro algebra is also trivial. The Leibniz homology is a noncommutative analogue of Lie algebra homology developed by J-L. Loday (see [11] and [12]). It can be extended for a larger class of algebras called Leibniz algebras.

In this paper, we study the second Leibniz homology group HL2(g(A)) of a Kac–Moody Lie algebra g(A). We shall show that HL2(g(A)) = 0 for any non-affine Kac–Moody algebra g(A), but HL2(g(A)) ≠ 0 for any affine Kac–Moody algebra g(A). In the latter case, we use Wilson's idea [15] to carry out HL2(g(A)) explicitly in terms of certain subspaces of Kähler differentials over the Laurent polynomial ring. More precisely, we have the following result.

We define a Henstock-type integral on measure spaces with metric topologies, and give an example of a function which is integrable but whose absolute value is not.

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <[mid ]X[mid ], the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory.

We prove uniform and tangential approximation theorems for superharmonic functions in abstract harmonic spaces. Our tangential approximation theorem differs from traditional ones by the absence of the so-called long islands condition. The result is new also in the case of the classical potential theory.

It is proved that the infinitesimal generator A of a strongly continuous semigroup of linear operators on a Hilbert space also generates a strongly continuous group if and only if the resolvent of −A, (λ+A)−1, is also a bounded function on some right-hand-side half plane of complex numbers, and converges strongly to zero as the real part of λ tends to infinity. An application to a partial differential equation is given.

A potential for the one-dimensional Dirac operator is constructed such that its essential spectrum does not cover the whole real line, whereas the potential q(x) tends to ∞ as [mid ]x[mid ] → ∞. Furthermore, a criterion by Hartman and Wintner for points of the essential spectrum of Sturm–Liouville operators is generalised to a purely operator-theoretical setting, and a simplified proof is given.

Let [Mscr ]S be the universal maximal operator over unit vectors of arbitrary directions. This operator is not bounded in L2(R2). We consider a sequence of operators over sets of finite equidistributed directions converging to [Mscr ]S. We provide a new proof of N. Katz's bound for such operators. As a corollary, we deduce that [Mscr ]S is bounded from some subsets of L2 to L2. These subsets are composed of positive functions whose Fourier transforms have a logarithmic decay or which are supported on a disc.

Let G be the group given by the following presentation:

formula here

The subgroup generated by ab is infinite-cyclic and normal, with quotient the dihedral group of order 6, so G is cyclic-by-finite. The subgroups H = 〈a, c〉 and K = 〈b, c〉 are both dihedral of order 6, and G is isomorphic to the free product of H and K amalgamating L = H∩K. We study K0(kG), the Grothendieck group of isomorphism classes of finitely generated projective kG-modules, and, in particular, the dependence of K0(kG) on the choice of field k. As usual, let ℚ, ℝ and [Copf ] stand for the rationals, reals and complex numbers, respectively. We prove the following.

Let G be either a free product with amalgamation A[midast ]CB or an HNN group A[midast ]C, where all normal subgroups of C are finitely generated. Suppose that both A and B have no non-trivial finitely generated normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which intersects A or B non-trivially but is not contained in C, then the index of N in G is finite.

Let Γ be a group having finite virtual cohomological dimension. Let BΓ be its classifying space, and let C(Γ) be its Quillen category, whose objects are elementary abelian 2-subgroups V of Γ and whose morphisms are restrictions of conjugacy of Γ. The group Γ is called a Quillen group if the natural map

formula here

is an isomorphism. The symmetric groups (see [8]) and the Coxeter groups of finite order (see [7]) are examples of Quillen groups.

J. H. Gunawardena, J. Lannes and S. Zarati obtained the characterization of Quillen groups in terms of unstable modules over the mod 2 Steenrod algebra [8] (see Proposition 2 below). By making use of this result, M. Errokh and F. Grazzini obtained the classification of Coxeter groups on three generators which are Quillen groups [8]. The purpose of the present paper is to obtain a necessary and sufficient condition for aspherical Coxeter groups to be Quillen groups (Theorem 1). Our result is very simple, and the previously mentioned result by Errokh and Grazzini follows easily from ours (see Example 1).

The paper is organized as follows. In Section 2, after giving relevant definitions concerning Coxeter groups, we shall state our main result. The next two sections are devoted to preliminaries. In Section 3, we shall review necessary definitions and facts concerning unstable modules over the mod 2 Steenrod algebra. In Section 4, we shall discuss the Leray spectral sequences converging to the cohomology of Coxeter groups. The proof of Theorem 1 will be given in Section 5. In Section 6, we shall give examples illustrating Theorem 1.

Notation. The cardinality of a finite set X is denoted by [mid ]X[mid ].

Eric Charles Milner was born on 17 May 1928 and brought up in London. His father was an engineer, but times were hard and work was often difficult to obtain. So his mother had to help out by working as a seamstress, and Eric was often looked after by his grandmother. At the age of 11, he won a scholarship to the Haberdashers' Aske's Boys' School, but never attended it in its permanent London buildings because the outbreak of the Second World War caused all London schools and their pupils to be evacuated to safer parts of the country. As a result, Eric, an only child and knowing none of his new schoolfellows, was billeted at a home near Reading where he was extremely unhappy. In despair, he ran away and returned to London, where, after unsuccessful attempts to find him another billet, he roamed the streets and missed school. After some time, he was eventually found another billet where he received kindness and was much happier. Despite these disruptions and the other inevitable shortcomings of a war-time education, Eric's intelligence more than sufficed to surmount such hurdles, and in later life he could speak and write better than most of us.

From 1946 to 1951, Eric attended King's College, London. He graduated with First Class Honours in 1949, when he was awarded the Drew Gold Medal as the most distinguished Mathematics student in that year, and a Research Studentship. He then studied for an MSc degree, taking ‘Modern algebra’ and ‘Quantum mechanics (Wave mechanics)’ as his selected subjects, his supervisors being Richard Rado (then a Reader at King's College) and Professor Charles Coulson. He received the MSc degree, with distinction, in 1950. This was followed by a year's research in quantum mechanics under the supervision of Professor Coulson.