Let denote a compact convex subset of the Euclidean plane containing the origin as an inner point and assume that the boundary ∂ of is a C∞-image of the 1-torus with finite nonvanishing curvature throughout. As a generalization of the classical circle problem in analytic number theory we consider, for a large parameter t, the number A(t) of lattice points (of the standard lattice ℤ2) in the ‘blown up’ domain and define the ‘lattice rest’ by (where is the area of ).

The aim of this paper is to establish a result on a family of congruences arising from the Fermat quotient: this result has an interesting application to the Fermat problem. For over 150 years the Fermat problem has been divided into two cases; the First Case being the assertion that for each odd prime p,

has no solution in non-zero integers x, y, z. Kummer's work on ideal numbers led, in 1847, to the complete solution of the Fermat problem for regular primes.

Let p be an odd prime. We write g(p) for the least positive integer z such that

In ([7], p. 181, problem 7) Maury and Raynaud pose the following question: ‘Let J be a ring and G a group such that the group ring J[G] is an order in a quotient ring Q; when is J[G] a maximal order in Q?’ The question is interesting for two reasons. In the first place, the analogous question for universal enveloping algebras of finite-dimensional Lie algebras has been settled very satisfactorily by Chamarie([2], corollaire 2·3·2). Secondly, it has been pointed out by several authors that maximal orders have some very desirable properties – see for example [3], [4], [6], [7] and [12].

In the following (Q, m, k) will denote a quasi-unmixed local ring of Krull dimension d with maximal ideal m and residue field k. We recall that Q is quasi-unmixed if dim Q̅/p = d for every minimal prime ideal p of the completion of Q.

Let X be an algebraic variety over a field k, z a point of X and Let t = {t1..., ts} be part of a system of parameters of R. Then these parameters are algebraically independent over k, and therefore they define (near z) a projection f: X → As(k) to an s-dimensional affine space over k, sending z to the origin 0∊s(k). In general, the Hilbert function (resp. multiplicity) of will be worse than that of , and we will call the system t transversal, if the Hilbert function (resp. multiplicity) of R and R/tR agree. This gives two notions of transversality: one for Hilbert functions (H-transversal), and a weaker one for multiplicities (e-transversal). For s = dim R we recover the notion of a transversal system of parameters introduced by Zariski for studying equisingularity problems (see e.g. [17]). In the above set-up the numerical characters are defined with respect to the maximal ideal of R, but we will consider this problem for arbitrary ideals I using generalized Hilbert functions of type H[x, I, R] (see [7] and [10]).

We make use of the universal connection on the bundle of principal connections; the bundle structure is governed by the action of the group on the first jet bundle. Each section determines a connection in the principal bundle, which in the case of the frame bundle allows a metric completion projecting onto the corresponding b-completion. It is shown that b-incompleteness of the base manifold is stable under perturbations of the chosen section. Hence, for instance, for (pseudo) Riemannian manifolds including spacetimes, b-incompleteness is a stable condition under conformal deformations of the metric. This reinforces the belief that general relativistic singularities cannot be removed by quantization.

An almost tangent structure on a manifold N is a type (1,1) tensor field S onN with the property that at each point y ∊N the kernel of Sy (regarded as a linear endo-morphism of TyN) coincides with its image. An almost tangent structure is said to be integrable if its Nijenhuis tensor vanishes.

The axiom of uniform visibility on simply connected manifolds was introduced in [8] by Eberlein and O'Neill. Using the axiom, they were able to extend to a class of simply connected manifolds of non-positive sectional curvature many results which were known to be true for the case of strictly negative curvature. The axiom proved very useful in obtaining results about the geodesic flow on manifolds of non-positive sectional curvature [3], [4].

In the theory of locales (or ‘pointless topology'), several authors have tried to find a suitable form of a T1-separation axiom. Their proposals are unsatisfactory in the sense that they do not coincide for topological spaces with the T1-axiom. Our main result is that sublocales of sober T1-spaces are exactly locales in which primes are dual atoms. Hence, these locales should be called T1, despite the fact that they include any locale without points. Our T1-locales are also closed with respect to products; in fact they form the smallest epireflective subcategory of locales containing all sober T1-spaces. Besides T1-locales we will also consider T2-locales and epireflective sub-categories of locales in general.

For a principal fibre bundle (p, q, B, G) with structure group G, the group of G-equivariant self equivalences of the total space P is investigated by using bundle map theory. Computations are given for well-known principal fibre bundles.

Let V be a smooth, 1-connected 4-manifold. It has long been known that there may be 2-dimensional homology classes in V which are not represented by smoothly embedded 2-spheres, owing to the failure of the Whitney lemma in this dimension. One of the simplest examples of this failure may be described as follows. Let K be a smooth knot in S3 and r an integer. By V(K, r) we mean the 4-manifold obtained by attaching a 2-handle to B4 along an r-framed tubular neighbourhood of K. Evidently each V(K, r) is homotopy equivalent to S2 and we say that K is an r-shake-slice knot if there is a 2-sphere, smoothly embedded in V(K, r), which represents a generator of the second homology group. For example, if K is a slice knot then it is r-shake-slice for each integer r. On the other hand, it is known that arf (K) = 0 whenever K is r-shake-slice and so in particular a trefoil is never such a knot.

A 2 component link with Alexander polynomial 1 is TOP concordant to the Hopf link. Our argument is modelled closely on Freedman's analysis of the problem of slicing Alexander polynomial 1 knots, and uses his theory of 4-dimensional surgery over groups with polynomial growth. A similar argument shows that certain F-isotopies may be realized by TOP concordances.

An n-knot will be a smooth, oriented submanifold Kn ⊂ Sn+2 such that Kn is homeomorphic to Sn. Given two knots and , we define their connected sum as in [13], p. 39. The cancellation problem for n-knots is the following.

Gleason's theorem characterizes the totally additive measures on the closed sub-spaces of a separable real or complex Hilbert space of dimension greater than two. This paper presents an elementary proof of Gleason's theorem which is accessible to undergraduates having completed a first course in real analysis.

There are many results which discuss ergodicity in terms of approximate product properties. Here we work throughout with stochastically independent σ-algebras (ℰ, ℱ are independent of μ if μ (E ∩ F) =μ (E)μ(F) for all E ∊ ℰ,F ∊ ℱ), to obtain an exact product property characteristic of (a large class of) ergodic measures. The ideas are based on work of A. V. Skorokhod on admissible translates of probabilities on Hilbert space.

1. Preliminaries. Let G be a non-elementary finitely generated Fuchsian group of the second kind acting on the unit disc Δ, and let Λ(G) be the limit set of G. It is well known that Λ(G) is a perfect non-dense set on the boundary of A. The Hausdorff dimension of the limit set Λ(G) of G is defined as the non-negative number

where mt(Λ(G)) denotes the t-dimensional measure of Λ(G) ([2], [3]). We say G has the type (g; m) if S = Δ/G is obtained from a compact surface of genus g by removing m(≥ 0) conformal discs.

For a random walk on a finite group, the distribution after n steps will converge, as n→∞, to the uniform distribution (under mild conditions). The asymptotic behaviour of such walks can be studied easily using standard Markov chain theory. But there are many natural problems about the non-asymptotic behaviour which have no simple solution in general. For example, what is

the time until a specified element is first visited?

the time until all element have been visited?

the time until the distribution approaches the uniform distribution?

1. Consider the Cauchy problem for the classical wave equation in ℝn:

with initial conditions u(0, x) = uo(x) and ∂tu(0, x)= u1(x), where uj (j = 0,1) are compactly supported C∞ functions on ℝn. In this paper we shall establish the following result.

This paper proves the existence of solutions for the discrete coagulation–fragmentation equation over all times, even when source and efflux terms are present. The hypotheses required cover most physical applications. Roughly speaking, the hypotheses ensure a finite flux of mass through the system. The techniques used, which extend those in I of this series, may apply to other infinite systems of differential equations.