Let ${\mathcal A}$ be a line arrangement in the complex projective plane $\mathds{C}\mathds{P}^2$. We define and describe the inclusion map of the boundary manifold, the boundary of a closed regular neighbourhood of ${\mathcal A}$, in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computations provide a new minimal presentation of the fundamental group of the complement.

We derive a two-term Edgeworth expansion for the distribution of Stein's double sampling statistic in the case of a non-normal parent population. The first term describes the main effect of skewness, while the second describes the main effect of kurtosis and the secondary effect of skewness. The remainder is shown to be uniformly negligible in comparison with these effects. Explicit conditions are given for the expansion to be valid. These conditions are considerably weaker than those which were imposed in an earlier derivation of the ‘exact’ distribution of Stein's statistic.

Let m be the set of all real sequences x = (xn) with norm . A linear functional L on m is said to be a Banach limit (see Banach(1), p. 32) if it has the following properties:

(i) L(x) ≥ 0, if x ≥ 0 (i.e. xn ≥ 0, for all n ∈ N)

(ii) L{e) = 1, where e = (1, 1, 1,…),

(iii) L(Sx) = L(x), where (Sx)n = xn+1.

1. In the course of a systematic analysis of certain problems in quantum mechanics it has been observed that their exact solutions can be expressed in terms of the Appell function F2 defined by means of (see e.g. (8), p. 211)

where, as usual,

and for convergence of the double series,

A non-enumerable set of real numbers, Z, is concentrated provided that there exists an enumerable set K such that if G is open and G ⊂ K then Z − G is at most enumerable. Z is said to be universally concentrated if we can take for K any enumerable set H for which

In this paper we shall describe that part of the canonical stratification of a versal unfolding of a function germ which deals with simple singularities. These results are doubtless known to many workers in this field; see especially the papers of E. J. N. Looijenga (e.g. (12)). We fill in the details however since we have in mind some geometrical applications (given in § 2) concerning natural partitions of the spaces of binary forms, cubic surfaces and quartic curves. The first two stratifications were considered in the author's Liverpool Ph.D. thesis; the quartic curves were considered in the Liverpool thesis of C. M. Hui, where the results were obtained by specific calculation. See (4) for an illustration of the techniques employed in these theses.

Introduction: Let X1, X2, …, Xn be n (n ≤ 2) independent observations on a random variable X with distribution function F. Also let L = L (X1, X2, …, Xn) be a linear statistic and Q = Q (X1, X2, …, Xn) be a homogeneous quadratic statistic. In this paper, we consider the problem of characterizing a class of probability distributions by the linear regression of the statistic Q on the other statistic L. In section 2, we obtain a characterization of a class of probability distributions, which includes the normal and the Poisson distributions. In section 3, a class of distributions including the gamma, the binomial and the negative binomial distributions is characterized.

We use the theory of the integral closure of an ideal to study the equivalence of map-germs under C0 coordinate changes in the target. We also derive a formula for the number of double points of a map germ from Cn → C2n in terms of the Segre number of dimension 0 of an ideal associated to the double point locus of f, and the number of Whitney umbrellas of the composition of f with a generic projection to C2n−1.

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined by

where V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured that

where ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

In this paper, I prove some new results which are identities connecting groups of general infinite products, and I give two tables, calculated using EDSAC, of the function 1/II(1 − aqn−1) over the range a = − 0·90(0·05) + 0·95, q = 0·0(0·05) + 1·00, and of the function 1/n(1 − qn) over the range q = 0(0·005)+ 0·890.

In this paper the authors consider a problem, frequently encountered in diffraction theory, of expanding a given function f(r) in terms of a set of orthogonal functions which is not complete. The expansion in question can be written as

where r < a and

The upper portion of the β-ray spectrum of thorium C″ has been investigated by measuring the tracks produced in a Wilson chamber. No trace of a “tail“ is found, no tracks of Hρ greater than 10,800 being observed in some 600 to 800 disintegrations. The end-point is placed at 9400 Hρ, somewhat higher than the value obtained by other workers.

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equation

where g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

This paper is concerned with the properties of precompact and compact linear operators from a locally convex Hausdorff space into itself, the field of scalars being the complex number field.

The Riesz–Schauder theory for the equations

where T is a compact linear operator from the Bacach space into itself, is well known (see, for example, Banach(1), Chapter 10, §2) and has been extended to the more general setting of locally convx Hausdorff spaces by Leray (4). In particular, the ‘Fredholm alternative theorem’ remains valid.

It is a familiar fact that the Picard surface (or hyperelliptic surface of rank 1) admits a completely transitive permutable continuous group of ∞2 automorphisms. There are, however, other non-scrollar surfaces which possess continuous groups of automorphisms, namely, the elhptic surfaces. Every elliptic surface V2 contains a pencil of birationally equivalent elhptic curves, which are the trajectories of the group in question; it also contains a second, elliptic, pencil of birationally equivalent curves; the intersection number of the two pencils is an important character, known as the determinant of V2. Just as any Picard surface can be mapped on a multiple Picard surface of divisor unity, so V2 can be mapped on a multiple elliptic surface of determinant unity, the branch curve (if any) corresponding to a certain number of trajectories.

Combining Freiman's theorem with Balog–Szemerédi–Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ${\mathbb{R}^d$ and ${\mathbb{Z}^d$.

1. Mazur(1) has shown that any normed algebra A over the real field in which the norm is multiplicative in the sense that

is equivalent (i.e. algebraically isomorphic and isometric under one and the same mapping) to one of the following algebras: (i) the real numbers, (ii) the complex numbers, (iii) the real quaternions, each of these sets being regarded as normed algebras over the real field. Completeness of A is not assumed by Mazur. A relevant discussion is given also in Lorch (2).

We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

Improving on estimates of Erdős, Halász and Ruzsa, we provide new upper and lower bounds for the concentration function of the limit law of certain additive arithmetic functions under hypotheses involving only their average behaviour on the primes. In particular we partially confirm a conjecture of Erdős and Kátai. The upper bound is derived via a reappraisal of the method of Diamond and Rhoads, resting upon the theory of functions with bounded mean oscillation.

1. Introduction. In [2], a short synthetic proof of the Riesz representation theorem was given; this used the Hahn-Banach theorem, the Stone-Čech compactification of a discrete space and the Caratheodory extension procedure for measures. In this note, we show how the theorem can be proved using ultrapowers in place of the Stone-Čech compactification. We also describe how the proof can be expressed in a non-standard way (a rather different non-standard proof has been given by Loeb [4]).