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.

We solve a problem of Steinhaus (New Scottish Book, Problem 328; also Colloq. Math. 5 (1958), Problem 225); if p is a prime, does there exist a regular p-gon such that three of its diagonals cut in a point? The answer is in the negative, except in the case where the said diagonals are all parallel, which can clearly occur for any p ≥ 7.

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups.

The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorisation via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalisations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

1. We write L for the class of integrable functions in (− ∞, ∞), V for the class of functions of bounded variation, and define A, A to be the classes of functions F(x) which may be expressed in the forms

respectively.

Let A = (ank) be an infinite matrix of complex numbers; suppose I is the unit matrix and c is the space of convergent sequences. By (A) we denote the summability field of A, i.e. (A) = {x = (xk): Ax ∈ c}.

The inequality

for fεLp(− ∞, ∞)or Lp(0, ∞) (1≤p ≤ ∞), and its extension

for T an Hermitian or dissipative linear operator, in general unbounded, on a Banach space X, for xεX, have been considered by many authors. In particular, forms of inequality (1) have been given by Hadamard(7), Landau(15), and Hardy and Little-wood(8),(9). The second inequality has been discussed by Kallman and Rota(11), Bollobás (2) and Kato (12), and numerous further references may be found in the recent papers of Kwong and Zettl(i4) and Bollobás and Partington(3).

1. We use Eα to denote the Euler transformation obtained as the special case of the Hausdorff transformation (H, μn) in which μn = αn (see [5], §§64, 72; in the notation of Hardy's book [1], our Eα is (E, q) with q = (1 − α)/α). Eα is regular if and only if 0 < α < 1, and in this range Eα increases in strength as α decreases since EαEβ = Eαβ. Also, E1 = I, the identity transformation.

Processes related to the empirical distribution are shown to converge to a time changed Brownian motion. In particular, the time change of the Brownian motion differs from the time change obtained by using the distribution function: this gives rise to two different representations of a related Brownian bridge.

We give a local estimate of the shape of Schottky space at certain boundary points. This is done by studying the geometric limits of sequences of cyclic groups and applying the results to the generators of Schottky groups. We also obtain information on discrete, non-faithful representations close to the cusps.

Analysis of projections of a convex body is a familiar topic in tomography. However, instead of considering standard projection bodies, this work investigates a convex body introduced by Schneider [8] which is a Minkowski average of projections. The question addressed here is similar to that posed by Goodey and Weil [4] with respect to Minkowski averages of sections, as opposed to projections, that is, can the shape of a convex body be determined from random sections? Their main result shows that a body K is determined by the average of its two-dimensional sections, but not by the average of its one-dimensional sections. The goal of this study is to uncover the extent to which a convex body is determined by the average of its projections.

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.

It is well known that the trace X([0, ∞) of the 3-dimensional Brownian motion X has positive capacity and, therefore, the harmonic measure is well defined on this set. It is shown that a.s., this harmonic measure is singular with respect to the occupation measure Lebesgue ο X−1.

We show that every continuous simple curve with σ-finite length has a tangent at positively many points. We also apply this result to functions with finite lower scaled oscillation; and study the validity of the results in higher dimension.

A method is given for evaluating matrix elements of three particle operators in many nucleon states of [44…4] symmetry.

An identity of fundamental interest in General Relativity is the breakup or ‘decomposition’ of the scalar curvature density into an ordinary divergence in addition to a term constructed solely from the metric tensor and its first derivative. It is proven that this identity is the most elementary example of an identity satisfied by all members of a particularly important class of Riemannian invariants. For a subset of these invariants, the decomposition identity undergoes a simplification that connects this material with some problems of current research interest.

1. A series with partial sums {An} is said to be summable (e, c) (c > 0) if

exists, where it is to be understood that An+k = 0 when n+k < 0. The (e, c)-sum-mability method which is a regular method of summation was introduced by Hardy and Littlewood (3) (cf. also (5)) as an auxiliary method to prove the Tauberian theorem for Borel summability, viz, if Σan is summable (B) to A and an = 0(n−½), then it converges to A.

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

In (l) Khuri has proved the validity of a dispersion relation for non-relativistic potential scattering. More precisely, he has shown that if the potential V(r) is central and satisfies: then the scattering amplitude f(k, τ) = M(E, τ) (where E = k2 is the energy) satisfies the following dispersion relation for fixed momentum transfer τ ≤ 2α: In (2), Rj(τ) is the (real) residue of the scattering amplitude at the bound state Ej and is the Fourier transform of the potential .