Let D(n) denote a function of an integral variable n ≥ 0 such that

1. (1) D(1) = D(0) = 0

2. (2) D(p) = 1 for every prime p

3. (3) D(n1n2) = n1D(n2) + n2D(n1) for every pair of non-negative integers n1, n2.

Orthonormal sequences, o.n. s., {ϕn} defined on [0,1] and satisfying

1

have been studied in [3] and [1]. One of the objects of this paper is to indicate that the methods used to study such o. n. s. can be used for a much wider class, and that, although there seems to be no super theorem to cover all cases, a knowledge of the results and methods of proof in some fairly broad special cases enables one to state and prove theorems for other classes of o. n. s.

D. Hawkins [1] propose un modèle stochastique simple de la distribution des nombres premiers. II construit un crible aléatoire analogue à celui d' Eratosthène. La règle pour obtenir une suite de nombres premiers aléatoires est la suivante: 2 est un nombre premier aléatoire, que nous désignerons par P1; chaque entier plus grand que P1 est ensuite criblé avec probabilité 1/P1 d' être éliminé par le crible; si P2 est le premier entier plus grand que P1 qui n' a pas été éliminé par le crible, on crible chaque nombre plus grand que p2, qui n' a pas été éliminé par le crible à la première opération, avec probabilité 1/P2 d'être élimine; à la niéme étape, si Pn est le premier nombre plus grand que Pn, qui n' a pas été éliminé par les n - i premières opérations du crible, on crible chaque nombre plus grand que Pn, qui n' a pas été éliminé par les n - 1 premières opérations du crible, avec probabilité l / Pn d' être éliminé. On obtient ainsi une suite { Pn} n = 1, 2, 3, … de nombres premiers aléatoires.

There is a well-known elementary problem:

(S3) Given a triangle T with the vertices a1, a2, a3, to find in the plane of T the point p which minimize s the sum of the distances |pa1| + |pa2| + |pa3|.

p, called the Steiner point of T, is unique: if an angle of T is ≥ 2π/3 then p is its vertex, otherwise p lies inside T and the sides of T subtend at p the angle 2π/3. In the latter case p is called the S-point of T, and it can be found by the following simple construction: let a12 be the third vertex of the equilateral triangle whose other two vertices are a1 and a2, and whose interior does not overlap that of T, let C be the circle through a1, a2 a12; then p is the intersection of C and the straight segment a12a3. It is easily proved that any one of the three ellipses through p with two of the vertices of T as foci is tangent at p to the circle through p about the third vertex of T.

In his book [1] Combinatorial Analysis, J. Riordan (p. 32) refers to the continual rediscovery of the Stirling numbers. The author of this note has been surprised on many occasions by the number of different environments in which these numbers make a natural appearance and, in fact, this article is concerned with just such an occurrence. The connection is made in a study of the exponential generating function of nr.

The maximum density of packings of a given type into the whole of a Euclidean space is defined to be the limit of the maximum density of such packings into a cube as the edge of the cube goes to infinity.

For E2 in particular, a number of well known results such as those due to A. Thue [1], L. Fejes-Toth [2], and C. A. Rogers [3] yield precise information about packings into the whole space. They are however of limited applicability to problems of finite packing in so-far as each requires some restriction upon the boundary of the configuration.

A graph G is defined by a set V(G) of vertices, a set E(G) of edges, and a relation of incidence which associates with each edge two distinct vertices called its ends. We consider only the case in which V(G) and E(G) are both finite.

An n-colouring of G is usually defined as a mapping f of V(G) into the set of integers { 1, 2,…, n} which maps the two ends of any edge onto distinct integers. The integers 1 to n are the n "colours". Much work has been done on n-colourings in recent years, especially by G. A. Dirac.

Given n functions of n variables, in the real domain, by the equations

1

we have in various contexts to consider whether the equations are soluble for the xr when the yr are given. Such questions receive fairly complete answers in complex variable theory; a complex variable relation w = f(z) is of course brought under the heading of the real equations (1) by setting w = y1 + iy2, z = x1 + ix2. For example, if f(z) is a polynomial the fundamental theorem of algebra asserts that the equations are soluble, though not in general uniquely. Again, a basic theorem on conformal mapping gives conditions under which the equations are uniquely soluble, to the effect that a (1,1) mapping of the boundaries of domain and range implies a (1,1) mapping of the interiors.

Let V be a vector space over an arbitrary field F. In V a bilinear form

is given. If f is symmetric [(x, y) ≡ (y, x)] or skew-symmetric [(x, y) + (y, x) ≡ 0], then

1

Thus right and left orthogonality coincide. It is well known that (1) implies conversely that f is either symmetric or skew-symmetric in V. We wish to give a simple proof of this result.

The energy of a system of indistinguishable particles in two-body interaction may be expressed in terms of the one and two-particle density matrices. In order to work directly with the density matrices we must know what restrictions are imposed on them by the statistics of the system. This report summarizes some results giving a partial answer to the question "What functions can arise as density matrices of a system of n fermions?".

We consider the following problem. Let A = (aij) be a symmetric n x n matrix of non-negative numbers with aii = 0 for all i, and let n points x1, x2, …, xn be chosen at random from the interval [0, L]. What is the probability P = P(n, A, L) that for all i and j, |xi - xj| ≥ aij?