The inequalities related to Mill's Ratio were conjectured true by Birnbaum (2) and were proved by Murty (4) for sufficiently large x > 0. In the present paper, these are proved for all finite x and have been used to calculate an upper limit for Mill's Ratio over the range x > – 1.

The hysteresis losses and demagnetizing coefficients of rectangular Fe-Si strips of various dimensions have been investigated. The hysteresis loss is approximately independent of the value of N, in agreement with a simple theoretical argument. The value of the demagnetizing coefficient is in fair agreement with the value calculated for the inscribed ellipsoid.

Let f be an entire transcendental function of finite order and Δ be a forward invariant bounded Siegel disk for f with rotation number in Herman's class $\mathcal{H}$. We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

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

In this paper we present two versions in several variables of the following result:

Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.

In this paper, it will be shown how it is possible, using Bernoulli polynomials of the first kind, to derive correction terms for any quadrature approximation in the form of a weighted sum over a finite interval. The terms derived will involve either the difference or the sum of derivatives taken at the ends of the range of integration, and they will be described as being of either Type A or Type B respectively. There will also be some discussion of the uses of such terms.

For a compact irreducible 3-manifold containing 2-sided projective planes and admitting a ℤ2-action, we consider the problem of when there exists a ℤ2-equivariant hierarchy. We show that a ℤ2-equivariant hierarchy always exists if either every irreducible summand of the orientable double cover has infinite first homology or every 2-sided projective plane is boundary parallel.

Chinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

One of the central problems in higher-dimensional knot theory is the classification of links up to concordance. In 14, Le Dimet constructed a universal model for (disk) link complements, which allowed him to formulate this problem in the framework of surgery theory by applying the Cappell-Shaneson program for studying codimension two embeddings of manifolds 1. The concordance classification was reduced to questions in L-theory (-groups 1) and homotopy theory (of Vogel local spaces 14). While recent results of Cochran and Orr2 (see also 18) provide rich information on the -theoretic part of the problem (in particular, they settle the question of the existence of links not concordant to boundary links), little is known about Le Dimet's homotopy invariant of links; for example, it is not known whether it may ever be non-trivial, or phrasing it more geometrically (according to 19), whether there are links that are not concordant to sublinks of homology boundary links. This motivated us to look at simpler classes of links, for which a more direct geometric approach to the problem is also possible, in an attempt to get some insight on the geometry carried by the homotopy invariants.

In Theorem 1·2 of [3] the following extension of the classical theorem of Shult is proved.

Theorem. Let G be a p-solvable group and let V be a faithful KG-module, where char. Assume that G contains an element x of order pn acting fixed-pointfreely on V. If p is a Fermat prime suppose further that the Sylow 2-subgroups of G are abelian. If p = 2 assume that the Sylow q-subgroups of G for each Mersenne prime q less than 2n are abelian. Then

Abel's series(1)

may be regarded as a generalization of the Taylor expansion

This note generalizes the two-point series of Lidstone and Whittaker (see (9)) in a similar way. Alternatively, the series discussed might be regarded as generalizations of Abel's series.

Mackey's imprimitivity theorem characterizes the unitary representations of a locally compact group G which have been induced from representations of a closed subgroup K; Rieffel's influential reformulation says that the group C*-algebra C*(K) is Morita equivalent to the crossed product C0(G/K)×G [14]. There have since been many important generalizations of this theorem, especially by Rieffel [15, 16] and by Green [3, 4]. These are all special cases of the symmetric imprimitivity theorem of [11], which gives a Morita equivalence between two crossed products of induced C*-algebras.

Lemma 4·1, which appears on p. 289 of [2], is false. Nevertheless, Theorem 4·2, the only result dependent upon it, is true. The following supplies the necessary modification to the proof.

We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.

We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly.

Let ℝd* = ℝd ∪ {[midast ]} be the one-point compactification of Euclidean space ℝd and d [ges ] 2. Given a permutation f of the set ℕ of positive integers, let [Cscr ]f(ℝd*) denote the set of all sets C ⊆ ℝd* for which there is a series [sum ]an in ℝd with zero sum such that C is the cluster set in ℝd* of the sequence of partial sums of [sum ]af(n). Every C ∈ [Cscr ]f(ℝd*) is non-empty, connected and closed in ℝd*. We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of ℝd* belong to [Cscr ]f(ℝd*). For every permutation f of ℕ, we determine all C ∈ [Cscr ]f(ℝd*) which contain [midast ].

Let (A, M) be a local Noetherian integral domain of dimension two and X = Spec A. For an ideal IA the graded ring RA(I) = noIn denotes the Rees algebra of A with respect to I. The projective scheme Y = Proj RA(I) is called the blowing-up of X (resp. A) along I. Then there exists a proper mapping : YX. The preimage Z = 1(V(I)) is called the exceptional fibre. Note that induces an isomorphism

A strong approximation for randomly indexed U-statistics is derived. The special case when the indices are stopping times based upon the jackknife estimator of the variance is discussed.

Some recent work by the authors (1) on the distribution of the residues of a cubic polynomial modulo an odd prime p led to the conjecture that, in general, two cubic polynomials with integer coefficients possessing the same residues modulo p (not necessarily occurring to the same multiplicity) are equivalent, that is are related by a linear transformation modulo p. The purpose of the present paper is to prove this conjecture. We establish the following theorem.

Introduction. Infinite systems

are discussed in which k(θ) and b(n) are differentiable. The aim is to determine the behaviour of a(n) for large n and to reduce (1) to a finite system. This involves solving

for the high order a(n), n ≥ N, in terms of the preceding a(n), with a large enough N to facilitate the iteration procedure employed. The desired N − 1, N − 1 system follows on substitution for the high order a(n) in the first N − 1 equations (1).