Every proper orthogonal matrix A can be written

where Q is a skew matrix [6], and conversely every such matrix A is orthogonal. It is also known that every proper orthogonal transformation in real Euclidean four-space may be characterized in term of quaternions [1, 3] by the equation

determines with the origin a vector having the coordinates (XQ, XI, x2, x3). The relationship between these two representations was clearly shown by Murnaghan [5].

1. Introduction. In his well-known book on graphs [1] König proposed the following problem: “When can a given abstract group be represented as the group of the automorphisms of a (finite) graph, and if possible how can the graph be constructed?”

To understand this problem well let us begin with the precise definition of a graph and its group (of automorphisms).

Introduction. The concept of a reducible integer was introduced recently [3] : if P(m) denotes the greatest prime factor of m then n is said to be reducible if P ( 1 + n2) < 2n. The reason for the term is that reducibility is a condition necessary and sufficient for the existence of a relation of the form where the ƒi are integers and the ni positive integers less than n. J. C. P. Miller pointed out to us the regularity of the distribution of the reducible integers (less than 600).

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.

Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.

Since the classical work of Minkowski and Jensen it is well known that many of the inequalities used in analysis may be considered as consequences of the convexity of certain functions. In several of these inequalities pairs of “conjugate” functions occur, for instance pairs of powers with exponents a and a related by 1/a + 1/a = 1. A more general example is the pair of positively homogeneous convex functions denned by Minkowski and known as the distance (or gauge) function and the function of support of a convex body. The purpose of the present paper is to explain the general (by the way rather elementary) idea underlying this correspondence.

The matrix solutions of an irreducible algebraic equation with integral coefficients were studied by Latimer and MacDuffee. They considered matrices with rational integers as elements. If A is such a matrix, then all matrices of the “class” S1 AS will again be solutions if S is a matrix of determinant ± 1. On the other hand, in general all solutions cannot be derived in this way from one solution only. It was in fact shown that the number of classes of matrix solutions coincides with the number of different classes of ideals in the ring generated by an algebraic root of the same equation.

A classical theorem of J. Liouville states that if z is a real algebraic number of degree n ≥ 2, then there exists a constant c > 0 such that for every pair of integers a, b with b ≠ 0.

Let A be a matrix of finite order n and finite degree d, whose characteristic roots are certain nth roots of unity a1, a2…, ad. We wish to prove a congruence (6) between the traces (tr) of certain powers of A, which is suggested by two somewhat simpler congruences (1) and (3). First, if tr (A) is a rational integer, it is easy to establish the familiar congruence

even though tr(Ap) may not itself be rational.

In this note, I shall establish necessary and sufficient conditions for the existence of critical lattices of an arbitrary point set, and I shall construct a non-trivial example of a point set without any critical lattice. In a previous paper, I proved that every star body of the finite type possesses at least one critical lattice.

Although the behaviour of Newtonian potentials inside n-dimensional distributions of mass or charge has been discussed in the sense of Lebesgue-Stieltjes integrals by various authors, the discussion of various important theorems seems to have been made only in the sense of Riemann integration, and assuming the Hälder conditions (or at least piecewise continuity) for the volume density p. We shall generalize these theorems below.

A projective plane geometry π is a mathematical system composed of undefined elements called points and undefined sets of points (at least two in number) called lines, subject to the following three postulates:

(P1) Two distinct points are contained in a unique line.

(P2) Two distinct lines contain a unique common point.

(P3) Each line contains at least three points.

During the last fifty years, the concept of the Euclidean space (an ndimensional coordinate space with a Pythagorean distance) has undergone various profound generalizations.

Hilbert introduced the infinitely-dimensional Euclidean space whose points are infinite sequences of coordinates having from the origin, and thus from each other, finite Pythagorean distances.

Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that for any group element other than the identity there is a representation sending it into a matrix other than the unit matrix. If G is commutative, the representations are necessarily one-dimensional. It is an immediate consequence of the Peter-Weyl theorem that the converse also holds: if every representation is one-dimensional, G is commutative. The main theorem in the present paper is a generalization of this result to the case where the representations have bounded degree.