The algebraic trace form (as defined by O. Loos) of an element (x, y) of a (complex) Banach Jordan pair V, where x or y is in the socle, is equal to the sum of the products of all spectral values and their multiplicity. The trace form is calculated for two examples, the Banach Jordan pair of bounded linear operators between two Banach spaces, and the Banach Jordan pair of a quadratic form. Using analytic multifunctions, it is also shown that the complement of the socle of a Banach Jordan pair V is either dense or empty. In the last case, V has finite capacity.

This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below.

Contents

1 Historical background 452

2 Differential operators and integral Steenrod squares 460

3 Symmetric functions and differential operators 471

4 Bases, excess and conjugates 476

5 The stripping technique 483

6 Iteration theory and nilpotence 490

7 The hit problem and invariant theory 495

8 The dual of [Ascr ](n) and graph theory 504

9 The Steenrod group 507

10 Computing in the Steenrod algebra 508

References 511

In Section 1 the scene is set with a few remarks on the early history of the Steenrod algebra [Ascr ] at the prime 2 from a topologist's point of view, which puts into context some of the problems posed later. In Section 2 the subject is recast in an algebraic framework, by citing recent work on integral versions of the Steenrod algebra defined in terms of differential operators. In Section 3 there is an explanation of how the divided differential operator algebra [Dscr ] relates to the classical theory of symmetric functions. In Section 4 some comments are made on a few of the recently discovered bases for the Steenrod algebra. The stripping technique in Section 5 refers to a standard action of a Hopf algebra on its dual, which is particularly useful in the case of the Steenrod algebra for deriving relations from relations when implemented on suitable bases. In Section 6 a parallel is drawn between certain elementary aspects of the iteration theory of quadratic polynomials and problems about the nilpotence height of families of elements in the Steenrod algebra. The hit problem in Section 7 refers to the general question in algebra of finding necessary and sufficient conditions for an element in a graded module over a graded ring to be decomposable into elements of lower grading. Equivariant versions of this problem with respect to general linear groups over finite fields have attracted attention in the case of the Steenrod algebra acting on polynomials. Similar problems arise with respect to the symmetric groups and the algebra [Dscr ]. This subject relates to topics in classical invariant theory and modular representation theory. In Section 8 a number of statements about the dual Steenrod algebra are transcribed into the language of graph theory. In Section 9 a standard method is employed for passing from a nilpotent algebra over a finite field of characteristic p to a p-group, and questions are raised about the locally finite 2-groups that arise in this way from the Steenrod algebra. Finally, in Section 10 there are a few comments on the use of a computer in evaluating expressions and testing relations in the Steenrod algebra.

We give geometric proofs of some of the basic structure theorems for compact Lie groups. The goal is to take a fresh look at these theorems, prove some that are difficult to find in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of p-compact groups.

Investigations concerning the generating function associated with the kth powers,

formula here

originate with Hardy and Littlewood in their famous series of papers in the 1920s, ‘On some problems of “Partitio Numerorum”’ (see [7, Chapters 2 and 4]). Classical analyses of this and similar functions show that when P is large the function approaches P in size only for α in a subset of (0, 1) having small measure. Moreover, although it has never been proven, there is some expectation that for ‘most’ α, the generating function is about √P in magnitude. The main evidence in favour of this expectation comes from mean value estimates of the form

formula here

An asymptotic formula of the shape (1.2), with strong error term, is immediate from Parseval's identity when s=2, and follows easily when s=4 and k>2 from the work of Hooley [2, 3, 4], Greaves [1], Skinner and Wooley [5] and Wooley [9]. On the other hand, (1.2) is false when s>2k (see [7, Exercise 2.4]), and when s=4 and k=2. However, it is believed that when t<k, the total number of solutions of the diophantine equation

formula here

with 1[les ]xj, yj[les ]P (1[les ]j[les ]t), is dominated by the number of solutions in which the xi are merely a permutation of the yj, and the truth of such a belief would imply that (1.2) holds for even integers s with 0[les ]s<2k.

The purpose of this paper is to investigate the extent to which knowledge of the kind (1.2) for an initial segment of even integer exponents s can be used to establish information concerning the general distribution of fP(α), and the behaviour of the moments in (1.2) for general real s.

Algebraic topology is a young subject, and its foundations are not yet firmly in place. I shall give some history, examples and modern developments in that part of the subject called stable algebraic topology, or stable homotopy theory. This is by far the most calculationally accessible part of algebraic topology, although it is also the least intuitively grounded in visualizable geometric objects. It has a great many applications to other subjects such as algebraic geometry and geometric topology. Time will not allow me to say as much as I would like about that. Rather, I shall emphasize some foundational issues that have been central to this part of algebraic topology since the early 1960s, but that have been satisfactorily resolved only in the last few years.

Nearly four hundred years ago, the cubic close-packing of equal spheres in R3 was discovered by Kepler [21], in which each sphere touches 12 others. In 1694, Gregory and Newton discussed the following thirteen spheres problem. Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? Gregory believed ‘yes’, while Newton thought ‘no’.

We prove for a wide class of structures that if [Fscr ] is a family of substructures which are pairwise uniformly commensurable, then there is a commensurable substructure invariant under automorphisms stabilizing [Fscr ] setwise.

Well over two decades have now passed since the publication of the classic books Singular integrals and differentiability properties of functions by E. M. Stein [32] and An introduction to Fourier analysis on Euclidean spaces by E. M. Stein and G. Weiss [37]. These two texts would, I am sure, be universally regarded as defining the ‘common core’ of harmonic analysis in the Calderón–Zygmund tradition in the early 1970s. What has been going on in the subject since then?

Suppose that m is a positive integer, τ=(τ1, …, τm) ∈ℝm+ is a vector of strictly positive numbers, and Q is an infinite set of positive integers. Let WQ(m; τ) be the set

formula here

In this paper we obtain the Hausdorff dimension of this set. We also consider a generalization of the set WQ(m; τ), where the error terms q−τi in the inequalities are replaced by ψi(q), for general functions ψi satisfying a certain condition at infinity.

The purpose of this note is to give a proof of a theorem of Serre, which states that if G is a p-group which is not elementary abelian, then there exist an integer m and non-zero elements x1, … xm∈H1 (G, Z/p) such that

formula here

with β the Bockstein homomorphism. Denote by mG the smallest integer m satisfying the above property. The theorem was originally proved by Serre [5], without any bound on mG. Later, in [2], Kroll showed that mG[les ]pk−1, with k=dimZ/pH1 (G, Z/p). Serre, in [6], also showed that mG[les ](pk−1)/ (p−1). In [3], using the Evens norm map, Okuyama and Sasaki gave a proof with a slight improvement on Serre's bound; it follows from their proof (see, for example, [1, Theorem 4.7.3]) that mG[les ](p+1) pk−2. However, mG can be sharpened further, as we see below.

For convenience, write H*ast;(G, Z/p)=H*(G). For every xi∈H1(G), set

formula here

Extending a theorem of S. Mori [6], we characterize two-dimensional factorial algebras which are graded by a finitely generated abelian group of rank one and which are affine over an algebraically closed field. The result is related to the classification of weighted projective lines [2, 5] and to the topic of canonical algebras [7].

Denote by f(n) the number of subgroups of the symmetric group Sym(n) of degree n, and by ftrans(n) the number of its transitive subgroups. It was conjectured by Pyber [9] that almost all subgroups of Sym(n) are not transitive, that is, ftrans(n)/f(n) tends to 0 when n tends to infinity. It is still an open question whether or not this conjecture is true. The difficulty comes from the fact that, from many points of view, transitivity is not a really strong restriction on permutation groups, and there are too many transitive groups [9, Sections 3 and 4]. In this paper we solve the problem in the particular case of permutation groups of prime power degree, proving the following result.

We classify all finite 2-groups G for which the maximum proportion of elements inverted by an automorphism of G is a half. These groups constitute 10 isoclinism families.

The title alludes to a similar title of the paper [3] by Grunewald and Segal, in which it is shown how to solve a quadratic equation in integers. This latter procedure seems to be quite difficult, and the algorithm outlined in [3] is rather involved, although it is completely effective in the logical sense.

In this paper we consider connections between three classes of optimal (in different senses) convex lattice polygons. A classical result is that if G is a strictly convex curve of length s, then the maximal number of integer points lying on G is essentially

formula here

It is proved here that members of a class of digital convex polygons which have the maximal number of vertices with respect to their diameter are good approximations of these curves. We show that the number of vertices of these polygons is asymptotically

formula here

where s is the perimeter of such a polygon. This result implies that the area of these polygons is asymptotically less than 0.0191612·n3, where n is the number of vertices of the observed polygon. This result is very close to the result given by Colbourn and Simpson, which is 15/784·n3≈0.0191326·n3. The previous upper bound for the minimal area of a convex lattice n-gon is improved to 1/54·n3≈0.0185185·n3 as n→∞.

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S, at least two distinct words of length n on these letters are equal in S. In particular, S is collapsing whenever it satisfies a law. Let [Uscr ](A) denote the group of units of a unitary associative algebra A over a field k of characteristic zero. If A is generated by its nilpotent elements, then the following conditions are equivalent: [Uscr ](A) is collapsing; [Uscr ](A) satisfies some semigroup law; [Uscr ](A) satisfies the Engel condition; [Uscr ](A) is nilpotent; A is nilpotent when considered as a Lie algebra.

The simplest example of the sort of representation formula that we shall study is the following familiar inequality for a smooth, real-valued function f(x) defined on a ball B in N-dimensional Euclidean space IRN:

formula here

where ∇f denotes the gradient of f, fB is the average [mid ]B[mid ]−1∫ Bf(y)dy, [mid ]B[mid ] is the Lebesgue measure of B, and C is a constant which is independent of f, x and B. This formula can be found, for example, in [4] and [12]; see also the closely related estimates in [20, pp. 228{231]. Indeed, such a formula holds in any bounded convex domain.

In this note we show that if an ideal I of a noetherian ring is principal, up to radical, then the local cohomology modules with support in V(I) are I-cofinite.

Let A be a group of automorphisms of the finite group G such that ([mid ]A[mid ], [mid ]G[mid ])=1. Then [mid ]A[mid ]<[mid ]G[mid ]2, and the exponent 2 here is best possible. If, moreover, A is nilpotent of class at most 2, then [mid ]A[mid ]<[mid ]G[mid ]. If A is abelian, then A has a regular orbit on G.

In 1956, A. Selberg introduced weakly symmetric spaces in the framework of his development of the trace formula, and proved that in a weakly symmetric space, the algebra of all invariant (with respect to the full isometry group) differential operators is commutative [22]. In this paper, Selberg asked whether the converse holds. In the present work, we shall answer this question by presenting examples of commutative spaces which are not weakly symmetric. These examples arise in the quaternionic analogues to the Heisenberg group, endowed with certain special metrics (see Theorem 5 and the explicit realization after it).