The distance between two vertices of a connected finite graph is the smallest number of edges forming a path that joins the two vertices, and the diameter of the graph is the largest integer which is realized as the distance between two vertices of the graph. We are concerned here with the diameters of two graphs associated with a d-dimensional convex polytope P (called henceforth a d-polytope). The graph Γ(P) of P is the 1- complex formed by the vertices and edges of P , and the polar graph II (P) of P is the 1-complex whose vertices correspond to the (d — 1)-faces of P, with two vertices joined by an edge in II (P) if and only if the corresponding (d — 1)- faces intersect in a (d — 2)-face of P. The diameters of Γ(P) and II (P) will be denoted respectively by δ(P) (called the diameter of P) and ϕ(Ps) (called the face-diameter of P ).

Given a set E of v elements and given positive integers k (k ≤ v) and λ, we understand by balanced incomplete block design (BIBD) B [k, λ, v] a system of blocks (subsets of E) having k elements each such that every pair of elements of E is contained in exactly λ blocks.

A necessary condition for the existence of a design B [k, λ, v] is known to be (4)

I t is straightforward, but tedious, to write down the integers whose representations in a given base do not have particular digits in certain positions. In the first section of this paper we give a computational scheme that enables us to carry out such operations in a rapid and simple fashion.

In the second section of the paper we derive a general identity involving the digits of integers in arbitrary Cantor systems of notation.

In the third section we apply this identity and deduce a number of results concerned with the splitting of integers into classes with equal power sums. The computational scheme of the first section leads us to an algorithm for the determination of such splittings.

In the article "Moulton Planes" (10), I studied F. R. Moulton's construction over any field containing a multiplicative subgroup of index 2. In "Collineations of Affine Moulton Planes" (11), I determined the collineations between two arbitrary affine Moulton planes.

The purpose now is to describe the collineations between two projective Moulton planes. Since the affine collineations are known from (11), we are concerned with collineations mapping ideal lines onto ordinary lines. Notations and conventions of (10) and (11) are retained.

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n > 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing. The densest lattice packings are known for spaces of up to eight dimensions (1, 2), but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

As is well known, the theory of linear inequalities is closely related to the study of convex polytopes. If the bounded subset P of euclidean d-space has a non-empty interior and is determined by i linear inequalities in d variables, then P is a d-dimensional convex polytope (here called a d-polytope) which may have as many as i faces of dimension d — 1, and the vertices of this polytope are exactly the basic solutions of the system of inequalities. Thus, to obtain an upper estimate of the size of the computation problem which must be faced in solving a system of linear inequalities, it suffices to find an upper bound for the number f0(P) of vertices of a d-polytope P which has a given number fd-1(P) of (d — l)-faces. A weak bound of this sort was found by Saaty (14), and several authors have posed the problem of finding a sharp estimate.

Let a be the Lebesgue measure on the unit circle |z| = 1 with

and let Lp be the space of complex-valued σ-measurable functions f such that

is finite. Hp is the closure in Lp of the algebra of analytic polynomials

A well-known consequence of the König theorem on maximum matchings and minimum covers in bipartite graphs (5) or of the P. Hall theorem on systems of distinct representatives for sets (4) asserts that an n by n (0, 1)-matrix A having precisely p ones in each row and column can be written as a sum of p permutation matrices:

Our main objective is a generalization of (1.1) along the following lines. Let A be an arbitrary m by n (0, 1)-matrix. Call an m by n (0, 1)-matrix P a permutation matrix if PPT = I, where PT is the transpose of P and I is the identity matrix of order m.

The purpose of this note is to point out some connexions between generalized Hadamard matrices (4, 5) and various tactical configurations such as group divisible designs (3), affine resolvable balanced incomplete block designs (1), and orthogonal arrays of strength two (2). Some constructions for these arrays are also indicated.

A balanced incomplete block design (BIBD) with parameters v, b, r, k, λ is an arrangement of v symbols called treatments into b subsets called blocks of k < v distinct treatments such that each treatment occurs in r blocks and any pair of treatments occurs in λ blocks.

We shall say that the sequence x = {sk} is summable- (Z, p) to the value s if

where p is a positive integer and

Ignoring the values of n, 1 — p ≤ n < 0, which are clearly irrelevant, the transformation (Z, p) coincides with the regular Nörlund transformation defined by the sequence ( 1 , 1 , . . . , 1, 0, 0 , . . .) containing p initial 1's. This class of methods was first studied systematically by Silverman and Szász (8).

We consider a combinatorial enumeration problem involving certain collections of labelled rooted trees having coloured nodes and edges. Notations and definitions are introduced in §§2 and 3, and the problem is described in §4. We give recursion formulas for its solution in §5. Then, by using a modification of a method of Priifer, we obtain a direct solution in terms of multinomial coefficients and power products in §§6 and 7. These results are combined in §8. Working in a formal power series algebra we find a formal solution of the equation

which expresses the unknown X as a multiple power series in the Ai and the Bi.

F. B. Jones (6) has shown that, if , then every separable normal Moore space is metrizable. It is not known whether this assumption is necessary, though perhaps some progress is made in (5). However, it is easily seen from R. H. Bing's Example E in (3) that a certain condition (see (1) below) implied by is necessary. Also in (3), Bing showed that every screenable normal Moore space is metrizable. In this paper we establish that: (1) every separable normal Moore space is metrizable if and only if every uncountable subspace M of E1 contains a subset which is not an Fσ (in M); (2) if every pointwise paracompact normal Moore space is metrizable, then so is every separable normal Moore space; (3) every screenable Moore space is pointwise paracompact but not conversely; (4) a T3-space is a pointwise paracompact Moore space if and only if it has a uniform base (in the sense of (1, p. 40), not a uniformity).

In general, given the fact that every element in a ring is nilpotent, one cannot conclude that the ring itself is nilpotent. However, there are theorems which do assert that, in the presence of certain side conditions, nil implies nilpotent. We shall prove some theorems of this nature here; among them they contain or subsume many of the earlier known theorems of this sort.

The purpose of this paper is to prove the following theorem:

Theorem. Given a convex curve C, of perimeter length I and three points M, N, and P which divide the perimeter of C into three parts of equal length, the perimeter length of the triangle MNP is never less than ½l. Equality holds only in the case where C is an equilateral triangle and M, N, and P are the mid-points of the three sides.

In (1) Bruck introduced the notion of a difference set in a finite group. Let G be a finite group of v elements and let D = {di}, i = 1, . . . , k be a k-subset of G such that in the set of differences {di-1dj} each element ≠ 1 in G appears exactly λ times, where 0 < λ < k <v— 1. When this occurs we say that (G, D) is a v, k, λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that a v, k, λ group difference set is equivalent to a transitive v, k, λ configuration, i.e., a v, k, λ configuration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parameters v, k and λ, then, we have the relation shown by Ryser (5)

Dans (6), l'auteur a défini, pour chaque entier n ≥ 0, un groupe fini Gn d'ordre q3(q —1)(q3 + 1), où q = 32n+1. Les groupes G1, G2, . . . sont simples, tandis que est simple.

En réalisant G = Gn comme un groupe de permutations, opérant à droite, de l'ensemble de q3 + 1 classes à droite modulo le normalisateur N(P) d'un 3-sous-groupe de Sylow de G, on voit aisément que G satisfait aux conditions (0.1)-(0.4) suivantes :

• (0.1) G est un groupe de permutations doublement transitif d'un ensemble E de m + 1 lettres, où m ≥ 3;

• (0.2) si a et b sont deux lettres distinctes quelconques de E, le sous-groupe formé de la totalité des permutations dans G qui laissent invariantes a et b contient une, et une seule, permutation ≠ 1 qui laisse au moins 3 lettres invariantes;

• (0.3) toute involution dans G laisse au moins 3 lettres invariantes (une involution dans un group est par définition un élément d'ordre 2);

• (0.4) m est impair.

It is well known that for finite connected graphs the following are equivalent:

1. (i) X is Euler (i.e., every vertex of X has positive even degree);

2. (ii) X is traceable (i.e., the edges of X can be arranged in a sequence e1, . . . ,en such that ei ≠ ej if i ≠ j, and ei, ei+1 are adjacent, i = 1, . . . , n, subscripts considered mod n) ;

3. (iii) X is cyclically coverable (i.e., X contains a family of non-overlapping circuits whose union is X).