We show that all partial sums of 1 + σk−α cos kθ are non-negative for α > α0, where 0·308443 < α0 < 0·308444 and α0 is the unique root of the equation

The quasihyperbolic metric in a proper subdomain D of Rn is defined by

where the infimum is taken over all rectifiable arcs γ in D joining x and y. There always exists an arc, called a quasihyperbolic geodesic in D, for which the infimum above is attained. We refer to [3], [4], [16], and [17] for the motivation and basic properties of the quasihyperbolic metric.

(1·1) Introduction. This paper is concerned with the prime and primitive ideals of certain iterated skew polynomial rings in two variables. These rings include those constructed in [7] but an extra parameter, ρ, has been introduced to the construction. This leads to greater variety in the behaviour of the height one prime ideals.

It is well known that the so-called Beauville number B = 25.33.52.7.11.13.17.19 is a universal bound of the global indices of ℚ–Calabi–Yau 3-folds, but it has been unknown whether this number is best possible or not.

Let M be a smooth closed connected oriented 4-manifold; we shall say that M satisfies Winkelnkemper's inequality when its signature, σ(M), and Euler characteristic, X(M), are related by

This inequality is trivially true for manifolds M with first Betti number b1(M) ≤ 1.

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

Stable holomorphic vector bundles over complex projective space ℙn have been studied from both the differential-geometric and the algebraic-geometric points of view.

On the differential-geometric side, the stability of E -→ ℙn can be characterized by the existence of a unique hermitian–Einstein metric on E, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn (see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙn and the nullcorrelation bundle (see [9] and [10]).

We give a complete description of the additive Dyer-Lashof operations on the Hopf ring with coefficients in . We re-prove a result of Ravenel and Wilson [9] giving the operations on the element and give formulae for the operations on the other generators.

Let Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.

The main result of this paper is an extension to Poisson bundles [4] and Lie algebroids of the classical result that a linear map of Lie algebras is a morphism of Lie algebras if and only if its dual is a Poisson morphism. In formulating this extension we introduce a second class of structural maps for vector bundles, which we call comorphisms, alongside the standard morphisms, and we further show that this concept of comorphism, in conjunction with a corresponding concept for modules, allows one to extend to arbitrary base-changing morphisms of arbitrary vector bundles the familiar duality and section functors which are normally denned only in the base-preserving case.

In this paper, we consider limiting properties concerning linear operators of probabilistic type. Specifically, we show that gamma operators are limits of inverse Beta operators and that Bleimann–Butzer–Hahn, Szász and Baskakov operators are limits of generalized Bleimann–Butzer–Hahn operators. By duality, these results are closely related to the convergence in the total variation distance of the probability measures involved. In each case, rates of convergence are given.

In this paper we study the almost sure asymptotics of htXt where (Xt) satisfies a stochastic differential equation and ht is a deterministic differentiable function, ht ↓ 0 as t → ∞. We give necessary and sufficient criteria, in terms of deterministic integrals, for the process htXt to converge to zero, to oscillate boundedly and for the case of infinite oscillations. Necessary and sufficient criteria for convergence to zero, in terms of a related integral, was studied by Chan and Williams in [1], for a class of diffusions that arise in simulated annealing. In Chan [2] the results are extended to higher dimensions. For Brownian motion, Jeulin and Yor[3] give a general criterion for convergence to zero, when the function ht is merely continuous and non-negative. The results of [1] and [3] are both generalizations of the well known ‘Kolmogorov criterion’ for Brownian motion. Our approach is very close to the techniques developed by Chan and Williams in [1]. Indeed, our method is essentially a fine tuning of those techniques. We apply our results, in Section 3, to a class of diffusions and compute the critical rate ht for which bounded oscillations occur. For Brownian motion this is the law of iterated logarithm, modulo the size of the oscillation. We also consider two parametrized families of diffusions for which the critical rate is seen to depend smoothly on the parameter. These results appear to be new. Our method also gives estimates on the size of the oscillations (Section 6).

Every monic, degree n polynomial in Fq[x;] is the characteristic polynomial of at least one n × n matrix (with entries in the finite field Fq), but they do not appear with equal frequency. There is no a priori reason that the characteristic polynomial of a typical matrix should resemble a typical monic degree n polynomial. Nevertheless, we prove a precise version of the following heuristic statement: ‘Excepting its small factors, the characteristic polynomial of a random matrix is random.’

An interesting recent paper by Falk and Kaufmann[11] notes, with an element of surprise, that the percentile bootstrap applied to construct confidence intervals for quantiles produces two-sided intervals with coverage error of size n−½, where n denotes sample size. By way of contrast, the error would be O(n−1) for two-sided intervals in more classical problems, such as intervals for means or variances. In the present note we point out that the relatively poor performance in the case of quantiles is shared by a variety of related procedures. The coverage accuracy of two-sided bootstrap intervals may be improved to o(n−½) by smoothing the bootstrap. We show too that a normal approximation method, not involving the bootstrap but incorporating a density estimator as part of scale estimation, can have coverage error O(n−1+∈), for arbitrarily small ∈ > 0. Smoothed and unsmoothed versions of bootstrap percentile-t are also analysed.

In this article we study the convergence behaviour of least squares approximations of various types by radial basis functions, i.e. least squares approximations from spaces spanned by radially symmetric functions φ (‖· −xj‖). Here the xj are given ‘centres’ in ℝn which we assume to lie on a grid. The inner products with respect to which the least squares problem is considered are discrete and Sobolev, i.e. may involve derivative information. Favourable estimates for the least squares errors are found that are shown to decrease as powers of the gridspacing. The work on discrete least squares approximations also gives rise to the construction of prewavelets from radial function spaces with respect to a discrete Sobolev inner product, which are discussed in the paper as well.

We prove an equivariant version of Galin's theorem on versal deformations of infinitesimally symplectic matrices. Matrix families of codimension zero and one are classified, and the results are used to study the movement of eigenvalues in one parameter families.