How many spheres of given diameter can be packed in a cube of given size? Or: What is the maximum diameter of k identical spheres if they can be packed in a cube of given size? These questions are obviously equivalent to the following problem:

Let d(Pi, Pj) denote the distance between the points Pi and Pj, and Γk the set of all configurations of k points Pi (1 ≤ i < j ≤ k) in a closed unit cube C.

The purpose of this paper is to locate five points Pi ( l ≤ i ≤ 5 ) in a closed unit cube C such that is as large as possible, where d(Pi, Pj) denotes the distance i j between Pi and Pj. We prove that this minimum distance cannot exceed (=m, say), and if it is equal to m, then the corresponding configuration is congruent to the set of points shown in fig. 1, namely P1 = A1 (0,0,0), P2 = A8 (1, 1, 1), P3 = B1 (0,1/2,1), P4 = B3 (1/2,1,0) and P5 = B5 (1, 0, 1/2).

This packing problem is obviously equivalent to the problem of locating six points Pi(l ≤ i ≤ 6) in a- closed unit cube C such that is as large as possible, where d(Pi, Pj) denotes the distance between Pi and Pj. We shall prove that this minimum distance cannot exceed (= m, say), and that 4 it attains this value only if the points form a configuration which is congruent to the one of the points Ri(l≤i≤6) shown in fig. 1. Note that , and so the six points are the vertices of a regular octahedron.

A Thomsen graph [2, p. 22] consists of six vertices partitioned into two classes of three each, with every vertex in one class connected to every vertex in the other; it is the graph of the “gas, water, and electricity” problem [1, p. 206]. (All graphs considered in this paper will be undirected, having neither loops nor multiple edges.)

Let A be the set of all functions from N, the natural numbers, to C the field of complex numbers. The Dirichlet product of elements f, g of A is given by

where the summation condition means sum over all positive integers d which divide n.

Beineke and Harary [l] recently showed that the maximum number of strong tournaments with k nodes that can be contained in a tournament with n nodes is

if 3 ≤ k ≤ n. The object of this note is to obtain some additional results of this type. We will use essentially the same terminology as was used in [ l ], so we will not repeat the standard definitions here.

Let Q(x) denote the number of squarefree integers ≤ x. Recently K. Rogers [ l ] has shown that Q(x) ≥ 53x / 88 for all x, with equality only at x = 176. Define R(x) to be Q(x) - 6/п2 X. (We observe that and ) Our objective will be to examine R(x). In particular, we show that for all x and observe that for x ≥ 8.

This is primarily an expository paper based on (and generalizing) some ideas of J. Pierpont [6], W.H. Young [8], R. L. Jeffery [4] and S. C. Fan [2]. Our aim is to give a simple and easily applicable theory of integration for arbitrary extended - real functions over arbitrary sets in a measure space. This will be achieved by using a generalized version of Pierpont1 s upper and lower integrals (with the upper integral playing the main role), and by appropriately defining the operations in the extended real number system, henceforth denoted by E*, so as to make it a commutative semigroup under addition and multiplication.

The aim of this paper is to extend the idea of a measure which takes on values in Euclidean n-space so as to allow it to assume infinite values while preserving its countable additivity over a given σ-ring. It is shown that in order to do this it is necessary to restrict the range of the measure to one infinite value.

It is well-known that Tychonoff's theorem on the product of compact spaces may be proved, for the special case of a countable number of metric spaces X1, X2…, Xn,…, in the following simple manner.

Kaplansky's Lemma [3] states: the number of k-combinations of 1, 2,…, n with no two consecutive integers in any selection is Using this, Abramson [l; lemma 3] solves the problem: find the, number of k-combinations so that no two integers i and i+2 appear in any selection. (This is generalized by Abramson in [2].) An interesting solution, also using Kaplansky's lemma, is obtained as follows.

If n = 2m, we choose s from among the m even integer s, no two consecutive, and k- s from among the m odd integers, no two consecutive.

Let X and Y be topological spaces. If Y is a uniform space then one of the most useful function space topologies for the class of continuous functions on X to Y (denoted by C) is the topology of uniform convergence. The reason for this usefulness is the fact that in this topology C is closed in YX (see Theorem 9, page 227 in [2]) and consequently, if Y is complete then C is complete. In this paper I shall show that a similar result is true for the function space of connectivity functions in the topology of uniform convergence and for the function space of semi-connectivity functions in the graph topology when X×Y is completely normal. In a subsequent paper the problem of connected functions will be discussed.

It is well-known that in a Hausdorff space, a sequence has at most one limit, but that the converse is not true. The condition that every sequence have at most one limit will be called the semi-Hausdorff condition. We will prove that the semi-Hausdorff condition is strictly stronger than the T1 -axiom and is thus between the T1 and T2 axioms. In this note, we investigate into some properties of the spaces satisfying the semi-Hausdorff condition.

By a semi-lattice we mean a system <L,.> where L is a set and. is a binary operation in L that is idempotent, commutative and associative. In a recent article [2] D.H. Potts considers the problem of reducing the number of axioms for a semi-lattice. His result was that the following two axioms viz. (1) xx=x, (2) (uv)((wx)(yz)) = ((uv)(xw))(zy) are sufficient to give a semi-lattice. But the second identity contains six elements instead of the original three. In the following we give a set of two simple identities with just three elements for a semi-lattice. This improves the above mentioned result.