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 nelements 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 hypotheticalcomputing device, the Q-machine. It was derived by abstracting from theprocess of calculating carried out by a man on his fingers, assuming anadequate supply of hands and the ability to grow fingers at will. TheQ-machine was shown to be equal in computing power to a universal Turingmachine. That is, the Q-machine could compute any number regarded ascomputable by any theory of computability developed so far. It may berecalled here that Turing machines were obtained by Turing [2] byabstracting from the process of calculating carried out by a man on someconcrete 'symbol space' (tape, piece of paper, blackboard) by means of fixedbut arbitrary symbols. Hence the contrast between the Q-machine and theTuring machines is that between arithmetical manipulation of counters andlogical manipulation of symbols. In particular, one might say, loosely, thatin a Turing machine, as in arithmetic, numbers are represented by signswhereas in the Q-machine, as on a counting frame, numbers representthemselves.

In the present note we shall study some properties of the Mathieugroups.

We shall give an invariant characterisation of the 2-Sylow subgroups. The2-Sylow subgroup of M24 is the holomorph of the elementaryabelian group of type (1, 1, 1, 1), and for the 2-Sylow subgroups of theother 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 presentnote 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 theunitary 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 generatingreflections.

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

In 1954, J. Lambek and L. Moser [l] showed that "Two non-decreasingsequences f and g (of non-negative integers) are inverses if and only if thecorresponding sets F and G of positive integers, defined by F(m) = the mthelement 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 inthe title. While this is not difficult and requires no new ideas, it seemsworth doing, as such a simple characterization does not appear explicitly ina recent treatise on the subject of flat modules [Bourbaki XXVII, Chapter1].

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 themain purpose of this note to establish a precise reformulation of Theorem 4in [5]. This is accomplished by introducing special classes of maps betweenBoolean lattices, Boolean rings and Boolean spaces respectively, and showingthe categories arising in conjunction with these maps to be equivalent inthe sense of Grothendieck [2]. Thus the notion of equivalence of categorieswill replace the phrase "mathematically equivalent" in [5]. In addition thewell-known axiomatic characterization of meet and complementation of Booleanlattices 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 itsentries 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. Denotethe (0, 1)-circulant . It hasbeen conjectured that

1

It is well-known in elementary calculus that if a differentiable functionhas a monotone increasing curvature, then its curvature is continuous andthe circles of curvature at distinct points have no points in common. Inparticular, 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 osculatingcircles at distinct points of A3 are disjoint (Theorem 1), may beof 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 studiedduring the last century (cf. Muir [6]). One of the best-known alternants isactually a double alternant (rows and columns) defined by Cauchy [2] in1841. 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 forfinding real solutions of some of these equations. We prove a theorem whichcan sometimes be used to solve problems concerning the uniqueness ofsolutions of such equations.

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

1 .

implies that

2 .

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

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

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

1 .

The identity of Abel [1] [2] we deal with here can be stated in thefollowing 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 isestablished by means of the Lagrange - Bűrman theorem [4]. Here we willderive it very simply.