The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:

Lemma 1.

provided that R(m) > 0.

Suppose that {Xn} is a countable non-homogeneous Markov chain and

If converges for any i, l, m, j with , then

whenever lim , whereas if converges, then

where and . The behaviour of transition probabilities between various groups of states is studied and criteria for recurrence and transience are given.

Let F be a distribution function with support confined to ( − π, π), W a Wiener process on (0, 1), and W0 a Brownian bridge. We examine the order of the approximation to W(F) and W0(F) by their Fourier series. The results we obtain are of theoretical interest since the Wiener process possesses the unusual property of satisfying a Lipschitz condition of order ∝ for each 0 < ∝ < ½, but not one of order ∝. = ½. The classical results on the rate of convergence of Fourier series are not sufficiently specific to provide an exact rate in the case of the Wiener process. The work also has practical significance because many statistical procedures can be closely approximated by functions of a Wiener process, and several procedures rely heavily on Fourier series methods. Thus, some of our results may be applied to obtain limit theorems for quantities such as trigonometric series estimates of density and distribution functions.

1. It was shown by Barlotti(1) that the number, k, of points on a (k, n)-arc in a Galois plane S2, q, of order q, where n and q are coprime, satisfies

Regular arcs, in which all the points are of the same type have been studied by Basile and Brutti(2) and, for n = 3 by d'Orgeval(4). By means of an electronic computer Lunelli and Sce(5) have enumerated many arcs in Galois planes of low order. The object of this note is to show how the (11, 3)-arcs of S2, 5, none of which is regular, may be described using only geometrical properties.

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.

The possibility of constructing a quantum field theory by means of fields on Euclidean space is based on works by Schwinger and Symanzik (1). Probabilistic methods were used and Nelson has shown (2) that from so-called ‘Markoff fields’ one can construct Wightman fields. This idea turned out to be unusually fruitful as it made available the statistical mechanic's techniques for consideration of quantumfield theory problems. See for example (7). However as in the Minkowski space approach, the only two-dimensional space-time nontrivial models have been proved to fulfil all Wightman axioms. Since the problem for higher dimension theories is still open and extremely difficult, it is useful to have at least some criterion which allows us to eliminate certain procedures as leading to trivial theories. One such criterion existing in the Minkowski space approach is described by a notion of Borchers class (5).

The potential of an electric charge in a gravitational field has been discussed very recently by Professor Whittaker and by E. T. Copson. The latter gives the solution

for a point centre at “distance” a from a mass of gravitational radius α. The coordinates used are such that the metric is given by

To make a Bayes decision we choose the infimum of an expected loss function. Catastrophe theory classifies a wide class of functions locally in terms of their critical values. Firstly we will show how this local classification relates globally to some mixtures of symmetric expected loss functions. Secondly we shall indicate how such mixtures can arise and how the above classification can be usefully applied to the qualitative study of the behaviour of a Bayes decision-maker.

The fascinating chain of theorems due to Clifford and recently extended by Mr F. P. White originates in the elementary fact that the circumscribing circles of the triangles formed by four lines meet in a point. From a like simple germ, namely the fact that the centres of the above-mentioned circles lie on a circle, an infinite chain of theorems was first evolved by Pesci and may be enunciated as follows:

(i) The centres of the circumcircles of the triangles formed by four lines lie on a circle; thus from four lines we derive a point, namely the centre of the latter circle.

(ii) Five lines give five sets of four and the five derived points lie on a circle; thus from five lines we derive a point, namely the centre of the latter circle.

(iii) Six lines give six sets of five and the six derived points lie on a circle.

Let G be a finite group, Sn be the symmetric group on n symbols and An be the corresponding alternating group. The conjugacy classes of the wreath product GSn (or monomial group as it is sometimes known) and the conjugacy classes of GAn have been described by Kerber (see [2] and [3]). The group Sn has a double cover n so that the faithful complex representations of this double cover may be regarded as protective representations of Sn. In Section 2, a particular double cover for GSn is constructed, the faithful complex representations of this group being the subject of a joint article with Peter Hoffman[1]. In the present paper, our task is to determine whether a conjugacy class of GSn corresponds to one or to two conjugacy classes in the double cover of GSn (and similarly for GAn). The main results, Theorems 1 and 2, are stated precisely in Section 2 and proved in Sections 3 and 4 respectively. The case when G = 1 provides classical results of Schur ([5], Satz IV). When G is a cyclic group, Read [4] has determined the conjugacy classes, not just for our particular double cover, but for all possible double covers of GSn.

It was first pointed out by Sommerfeld, around the turn of the century, that certain multi-valued solutions of the wave equation in ℝ3 can be used to deal with the problem of scattering by a wedge, or reflection in a corner. The older literature on this subject is extensive; see ((4), chapter 5) for references up to 1958. The object of this Note is to give an explicit and elementary construction of a forward fundamental solution of the wave equation, of this type, in ℝn+1, where n ≥ 2; for n = 2 this includes Sommer-feld's original result.

It is well known that every representation of the group Un of unitary matrices of order n × n is equivalent to a unitary representation (see e.g. Little-wood (6), ch. XI). Our object in the present paper is to discuss some properties of those representations, and to construct a specific unitary representation.

Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for which

These inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent to

for the case D < 0, or to

for the case D > 0.

The following paper is in two parts.

In Part 1 it is shown that Kronecker's theorem can be extended in the form

If is greater than ηfor all sets of integers l1, l2, … l3 less in absolute value than K/σ and not all zero, l also being an integer, then, for any x1, x2 … x3, an integer q less than l/(ησ8) can be found such that qv1–x1, qv2–x2…qv3–x4 all differ from integers by less than σ;. K, L depend only on s.

It is an immediate corollary that if

is greator than for all sets of integers l1, l2… l3, l less in absolute value than K/σ and not all zero while F(v1, v2… v3), l less in absolute value than K/σ and not all zero while F(v1, v2… v3, v) is periodic period 1, in v1, v2 … v3,v, then T can be found between 0 and such that

Where K, L depend on s, and N on s and the bounds of

We prove that in a separable infinite dimensional Hilbert space the Dynkin system generated by the family of all open balls (that is, the smallest collection containing the open balls and closed under complements and countable disjoint unions) does not contain all Borel sets.

We define and study a class of groups which produce simple, unique trace, C*-algebras. This class strictly contains the class of weak Powers groups, as shown by the fact that it is closed under extensions. We provide answers to some open problems involving Powers and weak Powers groups.

Let {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup asn. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

Spherically symmetric space-times have been studied in general relativity ever since the beginning when as the first exact solution of the Einstein equation the Schwarzschild space-time was given. Later on an elaborate local theory of spherically symmetric space-times was worked out with such fundamental results as Birkhoff's theorem and essential results concerning their global geometry have been achieved. Yet it seems that a general global theory is lacking even now. Some fundamental facts concerning the global geometry of spherically symmetric space-times are presented below. The concept of axis and of transverse submanifold are introduced and applied to study and classification.

Alexander (1, 2) has introduced certain topological invariants of a manifold which arise from the intersections of cycles of non-complementary dimensions, and he points out that they are not derivable from the Betti and torsion numbers, nor from the fundamental group. In the present paper we consider some topological invariants of this type on an algebraic surface, and, although we cannot define them completely, we show that they are intimately connected with the multiplications of the period matrix of the simple integrals of the first kind. We are then led to a concept which we call the “intersection group” of the surface, which is, by its definition, topologically invariant, and we show that it is also invariant under birational transformations. The proofs are based on Lefschetz's theory of cycles for an algebraic surface (4) and some simple properties of the period matrix of an algebraic curve. The results obtained here have a number of applications to the theory of ∞3 correspondences between algebraic surfaces, as we propose to show in a later paper.

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Plebański equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex structure in terms of systems of ordinary differential equations.