For any semi-group S and any ring Λ with unit 1 (always taken to be distinct from 0, the neutral element of Λ under addition) there is known to exist a ring Λ[S] ⊇S which is a A-bimodule such that (i) S is a sub semi-group of the multiplicative semi-group of Λ[S], (ii) λs = sλ, (iii) λ(st) = (λs)t = s(λt) ( s, t ∊ S and λ∊Λ) and (iv) Sis a Λ -basis of Λ[S]. This ring is uniquely determined by these conditions and is usually called the semi-group ring of S over Λ. It may be described explicitly as consisting of the functions f: S → Λ which vanish at all but finitely many places, with functional addition (f+g) (s) = f(s) + g(s) and convolution (fg) (s) = Σf(u) g(v) (uv = s) as the ring operations, the functional A-bimodule operations (λf) (s) = λf(s) and (fλ) (s) - f(s)λ, and each s ∊ S identified with the characteristic function of { s} with values in Λ.

In [5, p. 105] attention has been called to a set of propositions, due to H. Cox [3, p. 67], which are related to another set, due to Clifford [2, p. 145; 4, p. 447], concerning points and circles in the plane or on the sphere. One may state Cox's chain of theorems as follows:

In a projective 3-space, S3, let (1), (2), (3), (4) be four points lying in a plane α such that no three of them are collinear. Every two determine a line; let one plane such as [12], pass through each line. There are six such planes. The planes [12], [23], [13] determine a point (123); there are four such points. The first theorem of the chain states that they all lie in one plane [1234], It is not difficult to see that this is, in fact, a rewording of Möbius's theorem on mutually inscribed pairs of tetrahedra [4, p. 444].

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.

The Tarry-Escott problem in Diophantine analysis is the following: consider the system of equations

1

for the unknowns a1, …, ak, b1, …, bk, what is the smallest integer K = K(n) in the set of all k! s for which the system (1) possesses a non - trivial solution in integers ? By an integer we mean a rational integer and a solution is called trivial if the sets { a1, …, ak} and {b1, …, bk} are permutations of each other. By the Tarry-Escott problem one may also mean the problem of finding an actual solution of (1) or finding all such solutions but we shall be concerned with the bound K(n) only. However, the method given here is constructive, that is to say, with each estimate K for K it leads to finding actual non - trivial solutions of (1) with k = K1.

Write

1

2

where the maximum is over all real θ, and the lower bound is over all sets of positive integers a1 ≤ a2 ≤ … ≤ an. The problem of the order of magnitude of f(n) was posed by Erdös and Szekeres [1], side by side with a number of other interesting questions. Writing g(n) = log f(n), it is obvious that g(n) is sub-additive, in the sense that g(m+n) ≤ g(m) + g(n), and also that g(1) = log 2, so that g(n) ≤ n log 2.

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.

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then

1

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.

On an orientable surface of genus p, a set of 2p fundamental circuits can be selected which all pass through a single point A. After cutting along the 2p circuits, the surface can be unfolded into a flat region bounded by a 4p-gon so that: the set of vertices corresponds to the one point A; and the 2p pairs of edges to the 2p fundamental circuits; and the interior of the polygon to the remainder of the surface. If the edges of the polygon are directed, the 2 edges which correspond to one fundamental circuit will be directed in opposite sense, since the surface is orientable [1]. The sequence and direction of the edges is the same as the sequence of the fundamental circuits.

In 1954 A.J. Hoffman and O. Taussky [1] showed that if A is an n-square complex matrix with eigenvalues λ = (λ1, …, λn ) and P is a permutation matrix for which αA + βA* has eigenvalues for some αβ ≠ 0 then A is normal. Here is the conjugate vector of λ. As a companion result they also proved that if the eigenvalues of AA* are , i = 1, …, n then A is normal.

The purpose here is to study a type of perturbation problem, arising from a differential equation, which is not included in the realm of analytic or asymptotic perturbation theory [4], [6]. Such a problem arises when the domain of the differential operator has been subjected to a variation (rather than the formal operator). We propose to outline one simple problem of this type, concerned with a second order ordinary differential operator. Our purpose is to obtain asymptotic estimates for the characteristic values of a regular Sturm-Liouville problem on a closed interval [a, b] when b is near a singular point of the differential operator. Similar results have been obtained in [3], [8], and [9] by other methods.

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.

A fundamental problem in combinatorial analysis is the classification of the permutations of 1, 2, …, n which satisfy a system of constraints. Thus one may ask such questions as how many permutations are there which have exactly r k-cycles; how many have at least s cycles regardless of cycle length. Again, one may ask how many permutations are there in which k ascending sequences appear; or how many permutations are there in which specified numbers may not appear in specified places or at specified distances from other numbers. The literature on these problems is quite extensive. References [1,2,5,7,10,14,17] give an indication of the present, status of these problems.

The purpose of the present note is to help popularize a section of Artin' s “Geometric Algebra” (chapter 1,4; Interscience, New York (1957) ) by elaborating on its contents. The author will have succeeded when the reader discovers that his results are either presented more simply in Artin' s book or that they are trivial corollaries of its theorems, in particular of theorem 1.11.

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.

By a composition of a positive integer n is meant a representation of n as a sum of one or more positive integers where the order of the summands is taken into account. Thus for example 4 has the eight compositions 4 = 3 + 1 = 1 +3 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1. Now n can be written in the form 1 + 1 + … + 1 with n - 1 plus signs. Deletion of any subset of these plus signs breaks n into parts which form a composition of n. Conversely, any composition of n corresponds to a subset of plus signs, so that the number of compositions of n is the number of subsets of a set with n - 1 elements, namely 2n - 1. In this note we obtain a number of generalizations of this rather obvious remark by making use of the notion of a weighted composition and the method of generating series.

Let E be a finite dimensional vector space over an arbitrary field. In E a bilinear form is given. It associates with every sub s pa ce V its right orthogonal sub space V* and its left orthogonal subspace *V. In general we cannot expect that dim V* = dim *V. However this relation will hold in some interesting special cases.

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.

In his Geometric Algebra (New York, 1957) E. Artin poses the problem of co-ordinatizing an affine plane in the following terms.

How little do we have to assume, from a geometric point of view, about an affine plane, in order to be able to describe its points by pairs of elements of a field, and its lines by linear equations?