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 ).

Let V be a smooth complex algebraic threefold with k(V) = 0. Consider the space of holomorphic 2-forms on V, and let s denote the rank of the subsheaf of generated by these global forms. It was conjectured by Uenoin[8] that ≤ 3-here as always, hi denotes the dimension of the corresponding cohomology group Hi.

Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(ε2), where ε runs through the group of units and tr denotes the absolute trace. Hidden in Cusick's proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusick's for some families of functions related to tr (ε2) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].

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.

The quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.

In problems of additive number theory one frequently needs to obtain a non-trivial estimate for the absolute value of a trigonometric sum of the type

where f(X) ε ℤ [X] and 1 ≤ m ε ℤ. The general procedure is first to reduce the estimation to the case where m is a prime power, by means of the Chinese Remainder Theorem. The case m = pr (p prime) can often be reduced to that of a lower power of p, by a substitution of the type x = u + vps (where 0 ≤ u < ps and 0 ≤ v < pr-s), followed by the use of a p-adic Taylor expansion f(u+psv) = f(u) +psvf′(u) +…. Frequently this gives T(f, pr) = 0 when r ≥ 2, or at least allows one to reduce to the case m = p. In the latter case an appeal to Weil's estimate

usually gives a good estimate for (0.1), at least if deg f = o(√p).

In this paper we prove that, for any graph G, there exists a graph F such that G is a subgraph of F and

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

Let C be a set with at least two, and at most ℵ0, members, and for any set X let [X]2 denote the set of its 2-element subsets. If Γ is a countable set, and Fc is a function from [Γ]2 into C, then the structure Γc = (Γ, Fc) is called the countable universal C-coloured graph if the following condition is satisfied:

Whenever α is a map from a finite subset of Γ into C there is xεΓ–dom α such that (∀yεdom α) Fc {x, y} = α(y).

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].

We may define a mobility problem to be one of the type: ‘Given a (fully or partlys pecified) collection of bodies in ℝn, assumed not to interpenetrate, can they be moved in some (fully or partly specified) fashion?’ Early examples of mobility problems nclude ‘Chinese puzzles’ and Sam Loyd's ‘15 puzzle’ (to which could be added the more recent Rubik's Cube!) Less combinatorial examples include the ‘piano mover’ problem (see, e.g. [2, 3, 9]) and the several recent papers on collision avoidance, such as [1, 5, 6, 7, 8].

In this paper I shall define a deformation of hyperbolic structures on n-manifolds, called ‘bending’, by constructing a family of quasiconformal homeomorphisms of hyperbolic (n+ l)-space. The deformation can be applied to manifolds which contain totally geodesic submanifolds of codimension 1.

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.

Exact octagons, i.e. circular eight-term exact sequences, have cropped up recently in a few places in the literature. Papers of the author [11], and implicitly [10], the book of M. Warshauer[14], and the notes [5], all contain exact octagons. The first three references involve octagons of Witt groups of quadratic and other kinds of forms, the last reference extending the octagons to the setting of L-groups, i.e. surgery obstruction groups.

In an earlier paper [3], we introduced a new notion of completely 0-simple semigroup of quotients. The definition is motivated by a connection between Artinian simple rings and completely 0-simple semigroups. Given an Artinian simple ring Q, one obtains a completely 0-simple multiplicative sub-semigroup by taking the union of the subsets eiQej(i, j = l,…, n), where 1 = e1 + … + en and the ei are primitive orthogonal idempotents. If Q is the classical ring of left quotients of R, then is a semigroup of left quotients of in the sense that every element q in can be written as q = a-1b for some elements a, b in with a2 ≠ 0.

Chatters has asked whether a unique factorization domain (UFD) R, equipped with an automorphism σ transitive on the height 1 prime ideals of R, is necessarily Dedekind. If R contains an uncountable field, then Chatters observed that the answer is affirm ative. In this note was show:

Theorem. Let R be a local UFD equipped with an automorphism σ which is transitive on the height 1 prime ideals of R. Suppose that σ induces an automorphism of finite order on the residue field κ of R (for example, if κ is a global or finite field), Then R is Dedekind.

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 prove that the Poincaré conjecture implies that the 4-dimensional version of the realization theorem for Whitehead torsion is false. We show that infinitely many examples may be constructed to demonstrate this.

Let A be a (commutative Noetherian) local ring (with identity) having maximal ideal m and positive dimension. This note is concerned with, among other things, a complex of A -modules which was studied in [8] and which involves modules of generalized fractions derived from A and subsets of systems of parameters for A; in ([8], 3·5), the complex was shown to have connections with local cohomology. The complex is described as follows.

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.