The spaces P(ω), 1 ⩽ p ⩽ ∞, ω real are defined to consist of those analytic functions ƒ(s), regular for Re s > ω and for which μP(ƒ;x) is bounded for x > ω where

(1.1), 1 ⩽ p ⩽ ∞and

(1.2)

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

If the differential expression l(y) = — y” + g(x)y generates a closed operator L on L2(— ∞, ∞), with domain D consisting of those functions y ∈ L2 with absolutely continuous derivatives and such that l(y) ∈ L2. The case where g(x) is real-valued has been extensively investigated and yields an expansion of any ƒ ∈ L2 in terms of the characteristic functions of L. We shall investigate the case where g is complex-valued.

Let a, β, μ, ν, z be complex numbers such that 2μ and 2ν are not negative integers. Using the notation of (4) for generalized hypergeometric series, we set

(1) and define an = an(α,β, μ, ν)

(2)

The classical problème des ménages asks for the number of ways of seating at a circular table n married couples, husbands and wives alternating, so that no husband is next to his own wife.

An outline of the history of the problem to 1946 was given by Kaplansky and Riordan (11). They also presented a bibliography, which is augmented and brought up to date in the bibliography of the present paper.

The 8-cage (3) may be defined as the simplest cubic graph having no circuit of fewer than eight edges. To construct it we first observe that it must contain a tree T whose vertices are of degrees 1 and 3 and in which each vertex of degree 1 is separated by an arc of just three edges from a central edge AB. These properties fix the structure of T uniquely.

In the preceding paper, Tutte described the forty-five chords of an “ellipsoid” in the finite space PG(3, 3), showing that they may be regarded as the edges of a remarkable graph whose group is the group of automorphisms of the symmetric group . The object of this sequel is to relate Tutte's idea to Sylvester's combinatorial investigation of the fifteen “duads” and fifteen “synthemes” formed by six symbols, and to Richmond's discussion of the figure of six points in a projective 4-space.

“The remarkable analogies which exist between the complete quadrilateral and the desmic system of points suggest that it may be possible to extend the properties considered above to spaces of higher dimensions”, remarks Prof. N. A. Court at the end of his paper (2). Here is an attempt in that direction in Euclidean as well as in elliptic 4-space, suggesting extensions in higher spaces. The corresponding figure, called an S-configuration, is discussed.

In non-Euclidean geometry the three medians of a triangle A1A2A3 (each joining a vertex Ai with the internal midpoint Gi of the opposite side) are concurrent; their common point is the centroid G. But the Euclidean theorem ' which depends on similarity, does not hold.

The purpose of this paper is to integrate the theorems on enumerating subgraphs and supergraphs in (2) and (3) respectively by generalizing to a result which includes both of these as special cases. In this process we again utilize the powerful enumeration method of Pólya (4).

For the purpose of analysing bipartite graphs (hereinafter called simply graphs) the concept of an exterior covering is introduced. In terms of this concept it is possible in a natural way to decompose any graph into two parts, an inadmissible part and a core. It is also possible to decompose the core into irreducible parts and thus obtain a canonical reduction of the graph. The concept of irreducibility is very easily and naturally expressed in terms of exterior coverings. The role of the inadmissible edges of a graph is to obstruct certain natural coverings of the graph.

The simplest examples of figures which possess inner orientation are a sensed line, and a plane in which a sense of rotation is specified. Suppose two sensed lines, which intersect in a finite point, are given in a definite order. Then there is only one way in which the first can be rotated to coincide with the second without passing through the second. In this way, two ordered, sensed lines determine a sense of rotation in the plane which contains them.

In many cases Lattice Theory has proven itself to be useful in the study of the totality of mathematical systems of a given type. In this paper we shall continue one of such studies by investigating further the lattice of all topologies on a given set S. A considerable amount of research has been done in this field (1; 2; 3; 5; 6). This research, besides satisfying the intrinsic interest in the lattice theoretic properties of this lattice, has aided the study of interconnections of different properties of point set topologies.

La notion d'ensemble projectif de classe finie Pn peut être généralisée pour les classes transfinies Pα(α < Ω). Nous allons ici construire effectivement un ensemble qui n'est pas projectif, même dans ce sens généralisé.

Let ƒ and λ be positive integers and p a positive odd prime. Suppose further that P is an ideal of norm pf in a finite extension F of the rational field. In (2), which will also be referred to as I in the present paper, we obtained the number of solutions Ns(m) of the quadratic congruence,

In this note we discuss some properties of well distributed sequences. We take 0 ⩽ a < b ⩽ 1 and let I(x) denote the characteristic function of the interval [a, b], so that .

If n is a non-negative integer, define pr(n) by otherwise define pr(n) as 0. (Here and in what follows all sums will be extended from 0 to ∞ and all products from 1 to ∞ unless otherwise stated.) pr(n) is thus generated by the powers of ,

It is known (Perron (10); Frobenius (5, 6)) that if A = (aik) is a finite matrix with elements aik ⩾ 0, then A has a real, nonnegative eigenvalue μ, satisfying μ =max|λ| where λ is in the spectrum of A, with a corresponding eigenvector x = (x1, … , xn) for which xi≥ 0. Moreover if aik > 0, then μ is a simple point of the spectrum with an eigenvector x (unique, except for constant multiples) with components xi ≥0. Much has been written on this and related issues; cf., for example, the recent papers (4, 12) wherein are given several references.

In Part I of this paper we present a theory of Padé-approximants for Laurent series, and discuss their relation to orthogonal polynomials. For earlier results in this direction we may refer to (1 ; 7; 8). It is also indicated how this theory can be extended, for example, to matrix polynomials.