Let Fn be the free group of rank n with basis x1, x2, …, xn, and let d(G) denote the minimal number of generators of the finitely generated group G. Suppose that n[ges ]d(G). There exists an exact sequence

formula here

and we may view the free abelian group R=R/R′ as a right ℤG-module by defining (rR′)g =rgϕ−1R′ for all g∈G, where gϕ−1 is any preimage of g under ϕ, and rgϕ−1 =(gϕ−1)−1r(gϕ−1), the conjugate of r by gϕ−1. We call R the relation module of G associated with the presentation (1), and say that R has ambient rank n. Furthermore, we call the group Fn/R′ the free abelianized extension of G associated with (1).

1. Introduction

The classical circle method of Hardy and Littlewood (see Baker [1]) ensures that the diophantine equation

a1xk1+[ctdot ]+ asxks =0 (1.1)

has a nontrivial solution in nonnegative integers x1, [ctdot ], xs, provided s[ges ]C(k) and the coefficients a1, [ctdot ], as are all integers and not all the same sign.

A C*-algebra A is said to be monotone (respectively monotone σ-) complete if every increasing net (respectively increasing sequence) of elements in the ordered space Ah of all hermitian elements of A has a supremum in Ah. It is straightforward to verify that every monotone complete C*-algebra is an AW*-algebra. For type I AW*-algebras, the converse is known to be true. However, for general AW*-algebras, this question is still open, although an impressive attack on the problem was made by E. Christensen and G. K. Pedersen, who showed that properly infinite AW*-algebras are monotone σ-complete [4].

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).

Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

The most natural candidates to the definition of equisingularity of germs of complex space curves are discussed, showing that they coincide in the case of Lagrangean curves (projectivizations of conic Lagrangean varieties of a symplectic manifold of dimension four).

Eric Primrose was appointed to a Lectureship in Pure Mathematics at the University College, Leicester, in 1947 and promoted to a Senior Lectureship in 1954. He came to Leicester direct from Oxford where he had spent the 1946–47 year completing his degree, which had been interrupted by war service. Eric was awarded his PhD in 1957 by the University of London.

Eric's secondary education was at Chigwell School and in 1939 he won an Open Scholarship in Mathematics and went up to St John's College, Oxford. He took ‘shortened finals’ in 1941 and then entered the RAF as a Technical Officer (Radar), reaching the rank of Flight Lieutenant, having served in various locations throughout Europe.

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?

It is shown that known solvability conditions and several new existence results for semilinear elliptic boundary value problems at resonance can be considered from a unified point of view. The proofs apply Leray–Schauder degree arguments.

Noel Baker was born on 10 August 1932 and died, of a heart attack, on 20 May 2001. He was the only child of a farming family living near the township of Virginia north of Adelaide in South Australia, and a fourth-generation Australian. From 1938 to 1944 he attended the local school, winning a scholarship to King's College, Adelaide, and from there another scholarship to Prince Alfred College, Adelaide. Here he was inspired by an enthusiastic mathematics teacher, Mr Williams, who expected pupils to practise mathematics assiduously, even on Saturday mornings. Prince Alfred College still aims to foster a love of mathematics today, and it has recently launched a centre for excellence in mathematics, named after Noel.

Let Γ be a discrete group acting on a compact manifold X, let V be a Γ-equivalent Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint Γ-equivalent differential operator acting on sections of V. This data is used to define Toeplitz operators with symbols in the transformation group C*-algebra C(X)[rtimes ]Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Γ is finite and acts freely on X, a geometric-topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol.

This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras.

Let $C$ be an elliptic curve defined over $\mathbb{Q}$. We can associate two formal groups with $C$: the formal group $\^{C}(X,Y)$ determined by the formal completion of the Néron model of $C$ over $\mathbb{Z}$ along the zero section, and the formal group $F_L(X,Y)$ of the L-series attached to $l$-adic representations on $C$ of the absolute Galois group of $\mathbb{Q}$. Honda shows that $F_L(X, Y)$ is defined over $\mathbb{Z}$, and it is strongly isomorphic over $\mathbb{Z}$ to $\^{C}(X,Y)$. In this paper we give a generalization of the result of Honda to building blocks over finite abelian extensions of $\mathbb{Q}$. The difficulty is to define new matrix L-series of building blocks. Our generalization contains the generalization of Deninger and Nart to abelian varieties of $\rm{GL}_2$-type. It also contains the generalization of our previous paper to $\mathbb{Q}$-curves over quadratic fields.

Oscillation criteria for the first-order linear delay differential equation

formula here

are established. Our results improve the known results in the literature.

For a semigroup $S$, the finitary power semigroup of $S$, denoted $P_f(S)$, consists of all finite subsets of $S$ under the usual multiplication. The main result of this paper asserts that $P_f(G)$ is not finitely generated for any infinite group $G$.

Let $G$ be a compact Lie group, and let $\pi$ be any prime or set of primes. A ‘$\pi$-perfection map’ is constructed: that is, a continuous function from the space of conjugacy classes of all closed subgroups of $G$ to the space of conjugacy classes of $\pi$-perfect subgroups with finite index in their normalizer. This is used to show that the idempotent elements of the Burnside ring of $G$ localized at $\pi$ are in bijective correspondence with the open and closed subsets of the space of conjugacy classes of $\pi$-perfect subgroups of $G$ with finite index in their normalizer.