In this note we consider some problems related to the following question: What is the smallest integer m(n) for which there exists a family Fn of sets A1, A2,…, Am(n) with the following properties, (i) each member of Fn has n elements and (ii) if S is a set which meets each member of Fn, then S contains at least one member of Fn?

Erdős and Hajnal [1] observed that

and that m(l) = 1, m(2) = 3, m(3) = 7.

In the first part of this paper [l] there was introduced a hypothetical computing device, the Q-machine. It was derived by abstracting from the process of calculating carried out by a man on his fingers, assuming an adequate supply of hands and the ability to grow fingers at will. The Q-machine was shown to be equal in computing power to a universal Turing machine. That is, the Q-machine could compute any number regarded as computable by any theory of computability developed so far. It may be recalled here that Turing machines were obtained by Turing [2] by abstracting from the process of calculating carried out by a man on some concrete 'symbol space' (tape, piece of paper, blackboard) by means of fixed but arbitrary symbols. Hence the contrast between the Q-machine and the Turing machines is that between arithmetical manipulation of counters and logical manipulation of symbols. In particular, one might say, loosely, that in a Turing machine, as in arithmetic, numbers are represented by signs whereas in the Q-machine, as on a counting frame, numbers represent themselves.

In the present note we shall study some properties of the Mathieu groups.

We shall give an invariant characterisation of the 2-Sylow subgroups. The 2-Sylow subgroup of M24 is the holomorph of the elementary abelian group of type (1, 1, 1, 1), and for the 2-Sylow subgroups of the other Mathieu groups there are similar characterisations.

It is known [4] that the finite two- dimensional unitary group U(2, p2n) is generated by two reflections if p ≠ 2. The present note completes that result by giving two generating reflections for U(2, 22n), n > 1. As in [4] this implies that the points of the "unit circle " in the unitary plane over GF(22n), n > 1, are the vertices of a "regular unitary polygon" whose group of auto- 2n morphisms is U(2, 22n).

The final section gives abstract definitions for the particular groups U(2,24) and U(2,52) in terms of their generating reflections.

We obtain in this article some results concerning partitions of the natural numbers, the most important of which is a generalization of that quoted immediately below. Some intuitive material is included.

In 1954, J. Lambek and L. Moser [l] showed that "Two non-decreasing sequences f and g (of non-negative integers) are inverses if and only if the corresponding sets F and G of positive integers, defined by F(m) = the mth element of F = f(m) + m and G(n) = g(n) + n are complementary."

The purpose of this expository note is to establish the fact mentioned in the title. While this is not difficult and requires no new ideas, it seems worth doing, as such a simple characterization does not appear explicitly in a recent treatise on the subject of flat modules [Bourbaki XXVII, Chapter 1].

In Theorem 4 of [5] Stone stated that the theory of Boolean rings was "mathematically equivalent" to the theory of Boolean spaces without, however, properly defining the phrase "mathematically equivalent". It is the main purpose of this note to establish a precise reformulation of Theorem 4 in [5]. This is accomplished by introducing special classes of maps between Boolean lattices, Boolean rings and Boolean spaces respectively, and showing the categories arising in conjunction with these maps to be equivalent in the sense of Grothendieck [2]. Thus the notion of equivalence of categories will replace the phrase "mathematically equivalent" in [5]. In addition the well-known axiomatic characterization of meet and complementation of Boolean lattices with unit is discussed in analogous terms.

The permanent of an n-square matrix A = (aij) is defined by

where the summation extends over all permutations σ of the symmetric group Sn. A matrix is said to be a (0, 1)-matrix if each of its entries is either 0 or 1. A (0, 1)-matrix of n-1 the form , where θj = 0 or 1, j = 1,…, n, and Pn is the n-square permutation matrix with ones in the (1, 2), (2, 3),…, (n-1, n), (n, 1) positions, is called a (0, 1)-circulant. Denote the (0, 1)-circulant . It has been conjectured that

1

It is well-known in elementary calculus that if a differentiable function has a monotone increasing curvature, then its curvature is continuous and the circles of curvature at distinct points have no points in common. In particular, two one-sided osculating circles at distinct points of an arc A3 of cyclic order three have no points in common; cf. [l], [2], [3]. The conformai proof given here that any two general osculating circles at distinct points of A3 are disjoint (Theorem 1), may be of interest. We also prove that all but a countable number of points of A3 are strongly conformally differentiable (Theorem 2).

The subject of alternants and alternating functions was widely studied during the last century (cf. Muir [6]). One of the best-known alternants is actually a double alternant (rows and columns) defined by Cauchy [2] in 1841. Cauchy's result may be stated as follows:If D = [dpq] p, q = l, …, n, where dpq = (xp+yq)-1, then

1.

In this paper we are concerned with functional equations of the type

1.

in which x, y do not appear explicitly. J. Aczel [1] has given a method for finding real solutions of some of these equations. We prove a theorem which can sometimes be used to solve problems concerning the uniqueness of solutions of such equations.

A. P. Guinand [1] has shown that for appropriate infinite sequences an, αn,(an, αn real and positive for all n) the relation

1.

implies that

2.

In this note we do not intend to establish new results but only to suggest a very simple proof of Lusin's theorem, direct for σ-finite regular measures, a proof that bypasses the usual procedure of first establishing this theorem for sets of finite measure only. The proposed proof utilizes the notion of subuniform convergence, a method which seems not yet to have been used, despite its simplicity and adaptability. Simultaneously, a useful supplement to Egoroff's theorem will be obtained.

In this note, we present two matrix lemmas (one without proof) which have interesting applications in statistical estimation theory.

LEMMA 1. Let A be a k X k positive definite matrix. Then for any k X 1 vector c, we have that

1.

The identity of Abel [1] [2] we deal with here can be stated in the following form: If n is a positive integer,

.

(In order that all terms be defined we require a ≠ 0, b ≠ n.)

This identity and deductions from it have been very useful in many problems, for instance in mathematical statistics [3]. Usually this identity is established by means of the Lagrange - Bűrman theorem [4]. Here we will derive it very simply.