It is proved that Vaught's conjecture is true for modules over an arbitrary countable Dedekind prime ring. It follows from the structural result that every module with few types over a countable Dedekind prime ring is ω-stable.

In [1], Hirano gives a method for constructing families of curves with a large number of singularities. The idea is to consider an abelian covering of ℙ2 ramified along three lines in general position, and to take the pull-back of a curve C intersecting the lines non-generically. Similar constructions are used by Shimada in [10] and Oka in [8]. We apply this method for the case where C is a conic, constructing a family of curves with the following asymptotic behaviour (see [9]):

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The goal of this paper is to calculate the fundamental group for the curves in this family as well as their Alexander polynomial.

The context of this note is as follows. One considers a connected reductive group G and a Frobenius endomorphism F[ratio ]G→G defining G over a finite field of order q. One denotes by GF the associated (finite) group of fixed points.

Let [lscr ] be a prime not dividing q. We are interested in the [lscr ]-blocks of the finite group GF. Such a block is called unipotent if there is a unipotent character (see, for instance, [6, Definition 12.1]) among its representations in characteristic zero. Roughly speaking, it is believed that the study of arbitrary blocks of GF might be reduced to unipotent blocks (see [2, Théorème 2.3], [5, Remark 3.6]). In view of certain conjectures about blocks (see, for instance, [9]), it would be interesting to further reduce the study of unipotent blocks to the study of principal blocks (blocks containing the trivial character). Our Theorem 7 is a step in that direction: we show that the local structure of any unipotent block of GF is very close to that of a principal block of a group of related type (notion of ‘control of fusion’, see [13, §49]).

A theorem of Maranda [1, Section 30] states that if F is a finite group, p is a prime and pe exactly divides [mid ]F[mid ], then a ℤpF-lattice M is determined up to isomorphism by its finite quotient M/pe+1M. If M is a free ℤp-module of rank d, this is equivalent to saying that representations of F in GLd(ℤp) are determined up to equivalence by their images modulo pe+1.

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

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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).

By a Kleinian group we mean a discrete subgroup of PSL(2, [Copf ]). We prove that abelian subgroups of finitely generated Kleinian groups are separable. In other words, if M=ℍ3/Γ is a hyperbolic 3-orbifold, with Γ finitely generated, then abelian subgroups of Γ are separable in Γ.

In this paper, we give some complete characterizations of the primitive of a Henstock integrable function in Euclidean space.

A deterministic self-similar Cantor set F defines a natural probability space, the elements of which are neighbourhoods of various sizes of all points in F. Considering random variables on this probability space, some porosity and density parameters are explicitly calculated.

Let Γ be a lattice in a connected Lie group. We show that, besides a few exceptional cases, the deficiency of Γ is nonpositive.

Oscillation criteria for the first-order linear delay differential equation

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

We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the Radon–Nikodym property contains l1. It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

We prove that the bound on the Lp norms of the Kakeya type maximal functions studied by Cordoba [2] and Bourgain [1] are sharp for p>2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r2 log (1/r) is the exact Minkowski dimension of the class of Kakeya sets in ℝ2, and prove that the exact Hausdorff dimension of these sets is between r2 log (1/r) and r2 log (1/r) [log log (1/r)]2+ε.

We construct a c0-valued random variable X such that (Sn/√n)n∈ℕ has a convergent subsequence, but X does not satisfy the central limit theorem.

Let K be a cone in ℝn. Then K is a uniqueness set for the spherical Radon transform if and only if it is not contained in the zero set of any (nontrivial) homogeneous harmonic polynomial. A local version of this result is also proved.

Based on a result of Hofer and Viterbo, we give an inequality for symplectic capacity as defined by Hofer and Zehnder, and, as an application, we give another proof of the symplectic packing inequality of Gromov.

Leslie Fox was born in Yorkshire in 1918, and spent most of his professional life in Oxford, as Director of the University Computing Laboratory and as the first Professor of Numerical Analysis in the University.

His mathematical education began at the Wheelwright Grammar School in Dewsbury, which produced a number of distinguished mathematicians and scientists at about the same time as Fox under the influence of an inspiring Headmaster and teacher (Leslie Sadler). Fox went to Oxford in 1936 as a scholar of Christ Church, and he gained a First Class Honours degree in Mathematics. He continued his studies for a DPhil under the direction of Professor Sir Richard Southwell, with a project in the area of computational and engineering mathematics which initiated some of the main interests of his career. His first appointment was at the Admiralty Computing Service in 1943; here he learnt the skills of table-making which he later used in a number of publications.

In 1945 Fox and several colleagues moved to the new Mathematics Division of the National Physical Laboratory. It was recognised at that time that the emerging technology of automatic computation would lead to requirements for effective mathematical methods which exploited the new machinery. The Mathematics Division embarked on an extensive programme of research in computational methods, in parallel with the construction of the Pilot ACE machine, which carried out its first computations in 1950. The Division remained a major source of ideas and methods in numerical mathematics for many years, with Fox taking a leading role until he was appointed to set up the Computing Laboratory in Oxford in 1957.