Given a computably enumerable set W, there is a Turing degree which is the least jump of any set in which W is computably enumerable, namely 0′. Remarkably, this is not a phenomenon of computably enumerable sets. It is shown that for every subset A of ℕ, there is a Turing degree, c′μ(A), which is the least degree of the jumps of all sets X for which A is [sum ]01(X). In addition this result provides an isomorphism invariant method for assigning Turing degrees to certain torsion-free abelian groups.

In this paper we develop a structure theory for transitive permutation groups definable in o-minimal structures. We fix an o-minimal structure [Mscr ], a group G definable in [Mscr ], and a set Ω and a faithful transitive action of G on Ω definable in [Mscr ], and talk of the permutation group (G, Ω). Often, we are concerned with definably primitive permutation groups (G, Ω); this means that there is no proper non-trivial definable G-invariant equivalence relation on Ω, so definable primitivity is equivalent to a point stabiliser Gα being a maximal definable subgroup of G. Of course, since any group definable in an o-minimal structure has the descending chain condition on definable subgroups [23] we expect many questions on definable transitive permutation groups to reduce to questions on definably primitive ones.

Recall that a group G definable in an o-minimal structure is said to be connected if there is no proper definable subgroup of finite index. In some places, if G is a group definable in [Mscr ] we must distinguish between definability in the full ambient structure [Mscr ] and G-definability, which means definability in the pure group G:= (G, .); for example, G is G-definably connected means that G does not contain proper subgroups of finite index which are definable in the group structure. By definable, we always mean definability in [Mscr ]. In some situations, when there is a field R definable in [Mscr ], we say a set is R-semialgebraic, meaning that it is definable in (R, +, .). We call a permutation group (G, Ω) R-semialgebraic if G, Ω and the action of G on Ω can all be defined in the pure field structure of a real closed field R. If R is clear from the context, we also just write ‘semialgebraic’.

The paper gives a formula for the probability that a randomly chosen elliptic curve over a finite field has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen elliptic curve over a finite field has kq points where k is a small number and q is a prime.

The closure of the periodic points of rational maps over a non-archimedean field is studied. An analogue of Montel's theorem over non-archimedean fields is first proved. Then, it is shown that the (nonempty) Julia set of a rational map over a non-archimedean field is contained in the closure of the periodic points.

Let L1(x), L2(x), …, LN(x) be N real linear forms in N variables defined by

formula here

such that the associated N × N matrix of coefficients C = (cmn) is unimodular. In the classical theory of geometry of numbers, Minkowski's well-known linear forms theorem asserts that for any positive numbers ε1, ε2, …, εN satisfying ε1, ε2, …, εN [ges ] 1, there exists a lattice point p ∈ ℤN, p ≠ 0, such that

formula here

In 1937, Mordell [7] posed the following question which may be viewed, in some sense, as a converse to Minkowski's result. Does there exist a constant ΔN such that for each N × N unimodular real matrix C, there exist positive real numbers ε1, ε2, …, εN satisfying

formula here

such that the only integer solution to the inequalities of (1.1) is p = 0? Moreover, if ΔN does exist, what is the best possible value for ΔN, that is, what is the supremum of all such admissible values of ΔN? Clearly, by Minkowski's theorem, if ΔN exists, then ΔN < 1.

Let A be a commutative ring. A graded A-algebra U = [oplus ]n[ges ]0 is a standard A-algebra if U0 = A and U = A[U1] is generated as an A-algebra by the elements of U1. A graded U-module F = [oplus ]n[ges ]0Fn is a standard U-module if F is generated as a U-module by the elements of F0, that is, Fn = UnF0 for all n [ges ] 0. In particular, Fn = U1Fn−1 for all n [ges ] 1. Given I, J, two ideals of A, we consider the following standard algebras: the Rees algebra of I, [Rscr ](I) = [oplus ]n[ges ]0Intn = A[It] ⊂ A[t], and the multi-Rees algebra of I and J, [Rscr ](I, J) = [oplus ]n[ges ]0([oplus ]p+q=nIpJqupvq) = A[Iu, Jv] ⊂ A[u, v]. Consider the associated graded ring of I, [Gscr ](I) = [Rscr ](I) [otimes ] A/I = [oplus ]n[ges ]0In/In+1, and the multi-associated graded ring of I and J, [Gscr ](I, J) = [Rscr ](I, J) [otimes ] A/(I+J) = [oplus ]n[ges ]0([oplus ]p+q= nIpJq/(I+J)IpJq). We can always consider the tensor product of two standard A-algebras U = [oplus ]p[ges ]0Up and V = [oplus ]q[ges ]0Vq as a standard A-algebra with the natural grading U [otimes ] V = [oplus ]n[ges ]0([oplus ]p+q=nUp [otimes ] Vq). If M is an A-module, we have the standard modules: the Rees module of I with respect to M, [Rscr ](I; M) = [oplus ]n[ges ]0InMtn = M[It] ⊂ M[t] (a standard [Rscr ](I)-module), and the multi-Rees module of I and J with respect to M, [Rscr ](I, J; M) = [oplus ]n[ges ]0([oplus ]p+q=nIpJqMupvq) = M[Iu, Jv] ⊂ M[u, v] (a standard [Rscr ](I, J)-module). Consider the associated graded module of M with respect to I, [Gscr ](I; M) = [Rscr ](I; M) [otimes ] A/I = [oplus ]n[ges ]0InM/In+1M (a standard [Gscr ](I)-module), and the multi-associated graded module of M with respect to I and J, [Gscr ](I, J; M) = [Rscr ](I, J; M) [otimes ] A/(I+J) = [oplus ]n[ges ]0([oplus ]p+q= nIpJqM/ (I+J)IpJqM) (a standard [Gscr ](I, J)-module). If U, V are two standard A-algebras, F is a standard U-module and G is a standard V-module, then F [otimes ] G = [oplus ]n[ges ]0([oplus ]p+q= nFp [otimes ] Gq) is a standard U [otimes ] V-module.

Denote by π[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I, J; M) and σ[ratio ][Rscr ](I, J; M) → [Rscr ](I+J; M) the natural surjective graded morphisms of standard [Rscr ](I) [otimes ] [Rscr ](J)-modules. Let ϕ[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I+J; M) be σ∘π. Denote by &πmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I, J; M) and &σmacr;[ratio ][Gscr ](I, J; M) → [Gscr ](I+J; M) the tensor product of π and σ by A/(I+J); these are two natural surjective graded morphisms of standard [Gscr ](I) [otimes ] [Gscr ](J)-modules. Let &ϕmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I+J; M) be &σmacr;∘&πmacr;. The first purpose of this paper is to prove the following theorem.

The paper is concerned with rings of polynomial invariants of finite groups. In particular, it will be shown that these rings are isomorphic as modules over the Steenrod algebra [Pscr ]* if and only if the group representations are pointwise conjugate. An application to cohomology is the construction of classifying spaces of finite groups which are not homotopy equivalent, but where the cohomology rings are isomorphic as unstable modules over the (topological) Steenrod algebra.

It is proved that an excellent ring which is the homomorphic image of a regular ring of finite Krull dimension and having a resolution of singularities satisfies the uniform Artin–Rees property. Extensions of some results of Lipman, Sathaye and Teissier are also obtained.

Let k[x](x) be the polynomial ring k[x] localized in the maximal ideal (x) ⊂ k[x]. The paper studies the Hilbert functor parametrizing ideals of colength n in this ring having support at the origin. The main result is that this functor is not representable. A complete description of the functor as a limit of representable functors is also given.

Let M be a smooth, compact, oriented, odd-dimensional Riemannian manifold and let Γ → Mˆ → M be a normal covering of M. It is proved that the relative von Neumann eta-invariant ρ(2)(Mˆ) of Cheeger and Gromov is a homotopy invariant when Γ is torsion-free, discrete and the Baum–Connes assembly map μmax[ratio ]K0(BΓ) → K0(C*Γ) is an isomorphism.

The paper gives an explicit classification of the finite groups which are homogeneous in the sense of Fraïssé.

A point λ in the complex plane is said to be free if the group generated by the matrices

formula here

is free. The paper gives an infinite family of polynomials whose roots are the non-free points. The main idea in the paper is to employ a symmetry relation.

The aim of this paper is to determine which (finite) 3-dimensional classical groups are completions of the Goldschmidt G3-amalgam. We recall, first, that an amalgam (of rank 2) consists of three groups P1, P2, B and two group monomorphisms ϕ1, ϕ2 such that

formula here

Usually, when ϕ1 and ϕ2 are understood, this amalgam is denoted by [Ascr ](P1, P2, B). Now a group G is a completion of [Ascr ](P1, P2, B) if there exist group homomorphisms ψi[ratio ]Pi → G (i = 1, 2) satisfying G = 〈im ψ1, im ψ2〉 and ψ1ϕ1 = ψ2ϕ2[ratio ]B→G. The Goldschmidt G3-amalgam, which appears in [6], is defined as follows: P1 ≅ Sym(4) ≅ P2, B ≅ Dih(8) (Dih(n) denotes the dihedral group of order n) with ϕ−11(O2(P1)) ≠ ϕ−12(O2(P2)).

A variation on the unique product property of groups is described which seems natural from a geometric point of view. It is stronger than the unique product property, and hence implies, for example, that the group rings have no zero divisors. Some of its closure properties are described. It is shown that most surface groups satisfy this condition, and various other examples are given. It is explained how these ideas can give a more geometric interpretation of Promislow's example of a non-unique-product group.

In the study of discrete groups it is important to find conditions for a group to be discrete. Given a discrete subgroup of Möbius transformations containing a parabolic element with fixed point ∞, a classical result, called Shimizu's lemma, gives a uniform bound on the radii of isometric circles of those elements of the group not fixing ∞. Recently Parker [8] has shown that if a discrete subgroup G of PU(1, n; C) contains a Heisenberg translation g, then any element of G not sharing a fixed point with g has an isometric sphere whose radius is bounded above by a function of the translation length of g at its centres. Parker's theorem is considered as a generalization of Shimizu's lemma. In [1] Basmajian and Miner have independently obtained qualitatively similar results for discrete subgroups of PU(1, 2; C) by using their stable basin theorem.

The purpose of this paper is twofold. First we improve the stable basin theorem, and second we show that under some conditions Parker's theorem yields the discreteness conditions of Basmajian and Miner for groups with a Heisenberg translation. The latter answers a question posed in [8].

Let v be a measure on a separable metric space. For t, q ∈ R, the centred Hausdorff measures μh with the gauge function h(x, r) = rt(vB(x, r))q is studied. The dimension defined by these measures plays an important role in the study of multifractals. It is shown that if v is a doubling measure, then μh is equivalent to the usual spherical measure, and thus they define the same dimension. Moreover, it is shown that this is true even without the doubling condition, if q [ges ] 1 and t [ges ] 0 or if q [les ] 0. An example in R2 is also given to show the surprising fact that the above assertion is not necessarily true if 0 < q < 1. Another interesting question, which has been asked several times about the centred Hausdorff measure, is whether it is Borel regular. A positive answer is given, using the above equivalence for all gauge functions mentioned above.

Let us consider the boundary value problem

formula here

where Ω ⊂ ℝN is a bounded domain with smooth boundary (for example, such that certain Sobolev imbedding theorems hold). Let

formula here

Then, if ϕ(s) = [mid ]s[mid ]p−1s, p > 1, problem (1) is fairly well understood and a great variety of existence results are available. These results are usually obtained using variational methods, monotone operator methods or fixed point and degree theory arguments in the Sobolev space W1,0p(Ω). If, on the other hand, we assume that ϕ is an odd nondecreasing function such that

formula here

and

formula here

then a Sobolev space setting for the problem is not appropriate and very general results are rather sparse. The first general existence results using the theory of monotone operators in Orlicz–Sobolev spaces were obtained in [5] and in [9, 10]. Other recent work that puts the problem into this framework is contained in [2] and [8].

It is in the spirit of these latter papers that we pursue the study of problem (1) and we assume that F[ratio ]Ω×ℝ→ℝ is a Carathéodory function that satisfies certain growth conditions to be specified later.

We note here that the problems to be studied, when formulated as operator equations, lead to the use of the topological degree for multivalued maps (cf. [4, 16]).

We shall see that a natural way of formulating the boundary value problem will be a variational inequality formulation of the problem in a suitable Orlicz–Sobolev space. In order to do this we shall have need of some facts about Orlicz–Sobolev spaces which we shall give now.

The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*-algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of s-resolving, finite-to-one factor maps, then the construction of the stable C*-algebra is contravariant, while that of the unstable C*-algebra is covariant. The paper also discusses the constructions of these C*-algebras for Smale spaces which are not mixing.

The paper studies the heat kernel of the Schrödinger operator with magnetic fields and of uniformly elliptic operators with non-negative electric potentials in the reverse Hölder class which includes non-negative polynomials as typical examples. The main aim of the paper is to give a pointwise estimate of the heat kernel of the operators above which is affected by magnetic fields and non-negative degenerate electric potentials. A weighted smoothing estimate for the semigroup generated by the operators above is also given.

The paper considers finite subsets Λ ⊂ Zd which possess the extension property, namely that every collection {ck}k∈Λ−Λ of complex numbers which is positive definite with respect to Λ is the restriction of the Fourier coefficients of some positive measure on Td. All finite subsets of Z2 which possess the extension property are described.