The aim of this paper is two-fold: first, to extend the results of (4) to the exceptional finite Lie groups recently discovered by Chevalley (1), and, secondly, to give a construction which works simultaneously for the groups An, Bn, Cn, Dn, En, F4 and G2 (in the usual Lie group notation), and which depends only on intrinsic structural properties of these groups.

The purpose of this paper is to establish some results, of a somewhat miscellaneous nature, concerning certain classes of ideals in polynomial rings. Although the results will be formulated for a ring R[x] of polynomials in one indeterminate over a given ring R, they can be easily extended to polynomial rings in any finite number of indeterminates.

The number of binomial coefficients , which are congruent to j , 0 ≤ j ≤ p − 1, modulo the prime number p is denoted by θj(n). In this paper we give systems of simultaneous linear difference equations with constant coefficients whose solutions would yield the quantities θj(n) explicitly.

This paper is concerned with a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri (i = 1, … , m) and let the sum of column i of A be denoted by si (i = 1, … , n).

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p2n for every positive integer n and every odd prime p.

A perspectivity in a projective plane is a collineation which leaves some line pointwise fixed. Baer (4) has considered the coordinatization of planes which admit certain groups of perspectivities. André (1;2;3) has made an extensive study of the Veblen-Wedderburn plane in terms of its perspectivities. The author has shown (7) that finite doubly transitive planes are Desarguesian if the number of points on a line is n + 1 where n is an odd non-square.

McShane (5, 6) has introduced the concept of “Dedekind completeness” for partially ordered sets, which seems to be a natural generalization of the usual concept of completeness for lattices. It is the purpose of this paper to discuss some of the properties of Dedekind completeness, particularly with respect to a rather natural class of partially ordered sets which we call “uniform.”

The classical separation theorems regarding the relative position of the zeros of two consecutive orthogonal polynomials, or of the positive zeros of two consecutive Bessel Functions, are well known. Less well known is a theorem of Bôcher and Porter (1, p. 237), which states that the positive zeros of three consecutive Bessel Functions occur in cyclic order.

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm

(f ∊ ).

In 1943 Gelfand and Neumark (3) characterized uniformly closed self-adjoint algebras of bounded operators on a Hilbert space as Banach algebras with an involution (a conjugate linear anti-isomorphism of period two) satisfying several additional conditions. The main purpose of this paper is to point out that if we consider algebras of bounded operators on complex Banach spaces more general than Hilbert space, then we can represent a larger class of algebras by essentially the same methods.

A. H. Stone (9), E. Michael (3, 4), J. L. Kelley and J. S. Griff en (2) have established many necessary and sufficient conditions that a regular Hausdorff space be paracompact. It is the purpose of this note to show that if the word “countable” is inserted in the appropriate places in the above-mentioned conditions they become, in general, necessary and sufficient conditions that a normal space be countably paracompact.

Von Neumann's fundamental theorem of the theory of games has been extended by various authors and recently in two different directions by Kneser (8) and Nikaidô (9). We present here a form of the theorem, which is more general than that of both these authors. We develop some consequences of this theorem, which make it easier to decide whether certain classes of games have a value and we give several illustrative examples

The inversion theory of the Gauss transformation has been the subject of recent work by several authors. If the transformation is defined by

1.1,

then operational methods indicate that

,

under a suitable definition of the differential operator.

If μ is a measure defined on a space and fn(x) is a sequence of μ-integrable functions converging to f(x), then under suitable conditions of boundedness, one has the theorem of Lebesgue that

.

Let C be the field of complex numbers and E a locally compact topological space. The authors' theory of C-bimeasures Λ and their Λ-integrals in (1; 2) leads to integral representation of bounded operators from A to B' where A and B are MT-spaces as defined in (3). These MT-spaces include the -spaces and Orlicz spaces as special cases.

1. Notation. Let s(x) be a function integrable in every finite interval of x ≥ 0. Then the Riemann-Liouville integral of s(x), of order a > 0, is defined for x ≥ 0 by

(1).

The object of this note is to prove a Tauberian theorem for sα(x) in the case in which α is a positive integer p, employing certain difference formulae due to Karamata (4, Lemma 2) and Bosanquet (1, Theorem 1) used already for a broadly similar purpose in an earlier paper (12) where a is any positive number.

1. Introduction. As is well known, the functions

(1.1)

have various applications in theoretical physics and engineering. It is thus worthwhile to study their behaviour for real and complex values of the argument.

Sometimes it is possible to characterize topological properties of a metrizable space M by claiming that a certain (topologypreserving) metric ρ can be introduced in M.

n 1938 Frucht (2) proved the following theorem:

(1.1). Theorem. Given any finite group G there exist infinitely many nonisomorphic connected graphs X whose automorphism group is isomorphic to G.