1. Introduction. The ternary orthogonal group of projectivities over a finite field leaves a non-singular conic ✗ invariant, but the geometry consequent thereupon does not appear to have been investigated. The group is isomorphic to a binary group of fractional substitutions over the same field and this fact may, since these binary groups and their subgroups are so well known, have inhibited projects to embark on a detailed description of the geometry of the ternary group.

The convex hull of an arbitrary set M in real Euclidean n-space is known to consist of all the points within the r-simplexes with r + 1 vertices from M, r ≤ n. This note shows that if M is specialized to be a curve An of real order n, then its convex hull consists of all the points within the r-simplexes with r + 1 vertices on An, n = 2r + 1 or n = 2r.

Introduction. In hyperbolic geometry we have three compasses, namely an ordinary compass for drawing ordinary circles with a given centre and a given radius, a hypercompass for drawing hypercycles with a given axis and a given radius, and a horocompass for drawing horocycles with a given diameter and passing through a given point.

1. Introduction. An extension E* of a topological space E (that is, a space containing E as a dense subspace) determines a family of filters (u) on E, given by the traces U ∩ E of the neighbourhoods U ⊆ E* of each u ∈ E* − E. Many topological properties of an extension E* of a given space E can be related to properties of these trace filters (as we shall call them) belonging to E*.

Introduction. The problem discussed in this paper was formulated by T. Harris as follows:

“Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.”

1. Introduction. The number of connected linear graphs having V vertices labelled 1, … , V and λ (unlabelled) lines is found below. Similar formulas are found for graphs in which slings, lines “in parallel,” or both are allowed and for directed graphs with or without slings or parallel lines. Some of these graphs are also counted when the lines are labelled and the vertices are unlabelled.

1. Introduction. The theorem of von Neumann, that every finite, zero-sum two-person game has a value, has been extended in various ways to infinite games. In particular Wald (6) has shown that any bounded game in which one player has finitely many pure strategies, has a value. Our interest was aroused by the infinite analogue of the game of “hide and seek” as described by von Neumann (5), which does not appear to fit any of the known cases, unless the matrix is bounded.

1. Introduction. In a recent paper (2) Halperin and the author considered separable Banach spaces Lλ of real valued functions on general measure spaces and proved the existence of 1-regular (§2) Haar or σ-Haar bases when λ was the classical p-norm or any levelling length function (3) and, more generally, of K-regular Haar or σ-Haar bases when λ was a continuous length function satisfying certain additional conditions (2, Theorem 3.2).

1. The theorem of Landau in question may be stated in the form that if the function F(Z) is regular for |Z| < 1 and does not take the values 0 and 1, while

F(Z) = a0 + a1Z + …

is its Taylor expansion about Z = 0, then |a1| has a bound depending only on a0. In fact |a1| has a bound depending only on |a0| and Hayman (1) gave the explicit bound

We shall discuss solutions of linear partial differential equations of the form

1

where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;

2

j = 1,2, ߪ,s − 1,

whenever the function P is annihilated by a finite iteration of the operator Φ.

Introduction. The purpose of this paper is to discuss implications of Hadamard's condition for elastic stability (2, §269) with respect to uniqueness of solutions of boundary value problems in the theory of small deformations superimposed on large. We show that a slightly refined form of his condition implies a uniqueness theorem for displacement boundary value problems. We construct a counter-example showing that his condition does not imply uniqueness of solutions for one type of stress boundary value problem.

1. Introduction. The averaging operator has an extensive literature, the most detailed account being that of Nörlund

(4). In discussing solutions of the functional relation

1.1

he defines a “principal solution” (4, p. 41) by means of a summability process, and later, working in terms of complex numbers, he obtains (4, p. 70) a principal solution of (1.1) by means of a contour integral. He distinguishes his principal solution from other solutions, by showing that it is continuous at h = 0.

If F(z) is an analytic function for z ∉ [− ∞, −l],g(t) of bounded variation and real valued for 0 ≤ t ≤ 1 and

then the Stieltjes type of inversion between F(z) and g(t) (cf. 1, p. 339, Theorem 7a) is

where 0 ≤ v < u ≤ 1, ImF(z) is the imaginary part of F(z) and z = − t−l+iy.

1. Introduction. As is well known, the spectral theorem plays an important part in mathematics because of its many applications. Unfortunately, the theorem is valid for normal operators only. In view of this, attempts have been made by several mathematicians to obtain a theorem about a more general class of operators, which will reduce to the ordinary spectral theorem if the operator is normal. Brown (1) has developed a unitary equivalence theory for a certain class of operators.

1. Introduction. Semi-groups of bounded positive operators on certain function spaces enter the theory of stochastic processes of the diffusion type in an essential way. It is a matter of experience that these semi-groups cannot be imbedded in groups of positive operators, or, in more special terms, that the solution of a diffusion equation does not define a one-parameter group of positive operators on the natural function space.

Introduction. Certain aspects of the theory of subspaces of a Finsler space had been treated by the present author in earlier papers (7). These developments were based on an approach essentially different from the classical theory of Cartan (2) and subsequent writers, whose use of the element of support enables one to introduce the so-called “euclidean connection,” which effects the vanishing of the covariant derivative of the metric tensor.

Introduction. In §1 of this paper we consider the complex differential equation

1, |z| < 1,

where q(z) is a regular function in the open unit circle. We shall give a lower bound for the non-Euclidean distance of any pair of zeros of any non-trivial (i.e., not identically zero) solution u(z) of (1).

Introduction. This paper concerns the problem of extending a given measure defined on a Boolean ring to a measure on the generated σ-ring. Two general methods are familiar to the literature, that of Lebesgue (outer measure) and a method proposed by Borel using transfinite induction (4, 49-134; 2, 228-238).

In (1) Fan showed that if A is a Hermitian matrix with eigenvalues λ1 ≤ … ≤ λn then, for k ≤ n,

,

,

where X1, … , xk run over all sets of k orthonormal (o.n.) vectors in unitary n-space V.

1. Introduction and summary. In her papers (12) and (13) R. Moufang discusses projective plane geometries which satisfy the axiom of the uniqueness of the fourth harmonic point. Her main result is that in such geometries non-homogeneous co-ordinates may be assigned to the points of the plane (except for the “line at infinity”) in such a way that straight lines have equations of the forms aαx + y + β = 0, or x + γ − 0.