Let ordq (a) be the exponent with which a prime q occurs in the factorization of a rational number a ≠ 0 and, if ordq (a) = 0, let Mq(a) be the multiplicative group generated by a modulo q. In the course of a group-theoretical investigation J. S. Wilson found he needed some results about integers a, b such that Mq(a) = Mq(b), indeed also for algebraic integers, and he proved some of what he needed. J. W. S. Cassels observed that Wilson's argument naturally proved the existence of infinitely many primes q with Mq(a) = Mq(b) for rational integers a, b with ab > 0, |a| > 1, |b| > 1. J. G. Thompson found a proof for the case of integers a, b with ab < 0, |a| > 1, |b| > 1. He also posed the problem for rational a, b (all this is unpublished). The aim of this paper is to prove that the answer to Thompson's question is affirmative. We also include the case ab > 0 settled by Thompson himself. We use the same technique devised by Wilson which has been elaborated by Cassels and Thompson. We thank Professor Cassels for the simplification of our original exposition and the referee for his suggestions.

The purpose of this paper is to study the structure of locally free modules over the ring of differential operators on projective space. Let be a non-singular, complex, algebraic variety. Denote by the sheaf of rings of differential operators over and by its ring of global sections. A -module M is called locally free if the associated sheaf ⊗ M is locally free as a sheaf of -modules. Locally free modules arise naturally in -module theory as inverse images of determined modules; see [1] for definitions and examples.

Suppose that K = {K0, K1} is a partition of some finite combinatorial power [S]r of a set S. Let X(K) be the compact subspace of the Tychonoff cube [0, 1]S consisting of those functions ƒ whose supports Sƒ = {ƒ > 0} are 0-homogeneous i.e., [Sƒ]r ⊆ K0. It can be said that almost every example of a space witnessing the distinction between various chain conditions is a variation of X(K) (see [2]). This comes from the fact that the subject is closely related to the subject of Partition Calculus. (See [11] for an explanation of this point.) To have examples which are even more topologically interesting one usually tries to make them small, i.e., as closely related to the unit interval as possible. The operation X(K) has a considerable drawback in that respect. For example, if we want X(K) to be ccc this becomes equivalent to the fact that the poset of all finite 0-homogeneous sets is ccc which amounts to the fact that K1 must be very small. Hence K0 is big in the sense that there exist large 0-homogeneous sets. This usually results in X(K) having large size and having points of large character. One attempt to solve this problem was given by van Douwen in [5] by going to the subspace Xm(K) of X(K) consisting of those ƒ for which Sƒ are maximal 0-homogeneous subsets of S. Unfortunately, while Xm(K) usually does have small character it is almost never compact. This might have been the reason for his question ([2], p. 207) whether the Continuum Hypothesis implies that the class of all first countable compacta distinguishes the standard chain conditions that lie between ‘ccc’ and ‘separable’. In this paper we solve this problem completely. Moreover, we shall not go beyond the usual axioms of set theory in constructing the examples. The sequence of examples will start with a compact space of small character whose chain condition is not productive and it will end with a compact space of size c and small character which is ccc in a strong sense but which fails to have calibre θ for some regular uncountable cardinal θ, i.e., it fails to have the property of Shanin. Note that one cannot go further and show that, for example, compact spaces of small character distinguish between ‘the property of Shanin’ and ‘separable’. This follows from an old result of Efimov [3] that, under CH, first-countable spaces of calibre ℵ1 are separable. The combinatorics behind our examples have been developed in a series of papers that deal with the subject of forcing axioms in general and Martin's axiom in particular ([10, 11, 12, 13, 14]). Martin's axiom was originally invented in connection with the Souslin Problem, i.e., to show that certain compact ccc spaces must be separable (see [8]). A result of the aforementioned study of MA showed that this axiom is nothing more than the statement that all compact ccc spaces of π-weight < c must be separable (see [12]). An analysis of the fact that MA implies SH, due to Hajnal and Juhasz[6] (see also [4], §43 for a definite result in that direction due to Shapirovskii), revealed that MA implies that every compact ccc space X with the property χ(X)+ < c must be separable. This result explains why the compact ccc non-separable spaces X that we construct in this paper have the property that χ(x, X) < c for all x m X rather than the stronger property χ(X) < c or even χ(X) = ℵ0. Note that our examples show, answering a question from [6], that the Hajnal–Juhasz result is sharp in the sense that the assumption χ(X)+ < c cannot be weakened to χ(X) < c. The first example to show this was constructed by Bell [1] using a consequence of MA rather than just ZFC for its construction. Another feature of our examples is that they all are remainders of certain compactifications of the integers. This is of independent interest in certain constructions of weak P-points in compact F-spaces. An explanation of this can be found in [1] and [9] where the first examples of ccc non-separable remainders were constructed and used.

An important problem in the topology of 3-manifolds is to classify manifolds obtained by Dehn surgeries on a knot in a closed 3-manifold, or equivalently, Dehn fillings of a 3-manifold M with boundary a torus.

All manifolds in this paper are piecewise linear (or smooth if one wishes).

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

In [2] we used the theory of harmonic sequences to obtain a congruence theorem for minimal immersions of surfaces into ℂPn. In this paper we show how these ideas may be used to give a simplified and unified treatment of some results of Kenmotsu [9, 10], Bryant [4] and Ohnita [13].

In the theory of Kleinian groups, important examples of Kleinian groups are frequently constructed as algebraic limits of known sequences of Kleinian groups, for example quasi-conformal deformations of a Kleinian group. It is an important problem to determine the properties of the limit Kleinian group from information on the sequence converging to the limit. The topological properties of the domain of discontinuity and the limit set of a Kleinian group provide valuable pieces of information about the Kleinian group. It is reasonable to expect that in a good situation, the domain of discontinuity of the limit Kleinian group is the Carathéodory limit of the domains of discontinuity of the sequence, or, equivalently, the limit set of the limit Kleinian group is the Hausdorif limit of the limit sets of the sequence. Our main theorem (Theorem 5) shows that this is true when the sequence strongly converges to the limit, and the Kleinian groups of the sequence and the limit preserve the parabolicity in both directions and satisfy the condition (*) introduced by Bonahon.

A recent result of the first two authors has bridged the gap between the Timantype pointwise estimate and the norm estimate for polynomial approximation in C[– 1, 1]. Here we show that, when considering the second order modulus of smoothness, the aforementioned result holds with polynomials that preserve monotonicity, convexity and the values of the function at the end points of the interval [–1, 1]. This yields a generalization of results by Telyakovskii, Gopengauz, DeVore, Yu and the third author. Moreover, the construction here does not depend on the construction for the non-constrained approximation by polynomials.

A radial basis function approximation in n variables has the form

where ø:ℝn → ℝ denotes the n-variate, spherically symmetric function associated with a prescribed radial basis function ø+:ℝ+ → ℝ, i.e. ø = ø+(‖ · ‖), the norm being Euclidean. The are real coefficients (often, approximants s above are considered where only finitely many λjs are non-zero), and is a fixed set of points in ℝn (of course, only the xj with non-zero coefficient λj affect s). Thus s is a linear combination of translates of a radially symmetric function which can be of global support, the simplest choice being , where c is a positive parameter. The latter is referred to as the multiquadric function and is usefull in applications.

In this paper we study the question of characterizing those positive Borel measures μ on the unit disc Δ for which the differentiation operator D defined by Df = f′ maps the Hardy space Hp continuously into the Bergman space Bp(dμ) of all functions f analytic in Δ which belong to Lp (Δ, dμ).

Let G be a locally compact group with fixed left Haar measure dx. Recall that G is said to be amenable if there exists a left translation invariant mean on the space L∞(G), i.e. if there exists a positive, linear functional M on L∞(G) such that M(lG) = 1 and M(xø) = Mø for all ø∈L∞(G), x∈G, where xø denotes the left translate xø(y) = ø(xy). The class of amenable groups includes all soluble and all compact groups (concerning the theory of amenable groups we refer to [9]). It is easy to see that G is amenable if and only if ℂ1G, the space of the constant functions on G, has a closed left translationinvariant complement in L∞(G). This reformulation of amenability leads to the following more general question.

This paper deals with the problem of point realizations of isomorphisms of measure algebras.

The Hausdorff–Besicovitch dimension of a set A ⊆ ℝd, denoted dim (A), relates to the structure of A in the neighbourhood of its thickest point. If A is irregular then one may wish for an index which will describe the size of A in different places. The obvious definition of the local dimension of A near x is

and it is not hard to verify that

Let K be a standard singular integral kernel on ℝ satisfying the usual Hölder continuity condition of order δ, and define (where c is chosen so that the integral of w is 1), the mean of g with respect to the measure w(x) dx, and ‖·‖p the Lp norm with respect to w(x) dx. Although the inequality is not true in general, the centred norm inequality does hold for 1 < p < ∞ if α < δ.

Let X and Y denote normed spaces and T:D(T) ⊂ X → Y a linear transformation. It is shown that even in the case where both X and Y are incomplete, the quantity remains constant under both small and compact perturbation, provided that T is relatively open, R(T) is closed, and the perturbation is made in the right ‘direction’. If in addition and N(T) is topologically complemented, the topological complementation of the kernel is also preserved under small perturbations made in the right ‘direction’ and arbitrary compact perturbation. Various counter-examples are exhibited proving these results to be best possible.

In this paper, we present some results on the existence of periodic solutions to Volterra integro-differential equations of neutral type. The main idea is to show the convergence of an equibounded sequence of periodic solutions of certain limiting equations which are of finite delay. This makes it possible to apply the existing Liapunov–Razumikhin technique for neutral equations with finite delay to obtain existence of periodic solutions of Volterra neutral integro-differential equations (of infinite delay). Some comparisons between our results and the existing ideas are also provided.

Random sectioning of particles (compact sets in ℝ3 with interior points) is a familiar procedure in stereology where it is used to estimate particle quantities like volume or surface area from planar or linear sections (see, for example, the survey [23] or the book [20]). In the following, we study the problem whether the whole shape of a convex particle K can be estimated from random sections. If E is an IUR (isotropic, uniform, random) line or plane intersecting K then the intersection Xk = K ∩ E is a (k-dimensional, k = 1 or 2) random set. It is clear that the distribution of Xk determines K uniquely and that if E1,…, En are such flats, the most natural estimator for K would be the convex hull

Let {N(x), x ≥ 0} be a non-homogeneous, also called non-stationary, Poisson process with density function λ(x), x ≥ 0. We consider the problem of testing the null hypothesis of λ(x) having a constant value against the alternative that λ(x) is a function of x.