Hyperbolic differential operators with constant coefficients introduced and studied systematically by Gårding (4), were characterized by the existence of the fundamental solution with some cone condition, according to Hörmander (6). Recently Ehrenpreis, extending the notion of hyperbolicity due to Gårding, has defined hyperbolic operators for distributions with compact support in the convolution sense. Under the hypothesis that the operator is invertible as a distribution, he has established a theorem analogous to the theorem of Hörmander mentioned above (3).

A dissection of the disk will be a cell complex (1, p. 39) K with polyhedron the closed disk B2. It will further be required that:

• (a) every edge of K be incident with two distinct vertices (called its ends) ;

• (b) no two edges have the same ends ; and

• (b) no two edges have the same ends ; and

Define R to be a ring of characteristic not 2 or 3 satisfying the identity

(1)       (x,y,z) = (y,z,x)

for all x, y, z ∊ R, where by characteristic not p is meant x —> px is a one-toone mapping of R upon R. The associator (a, b, c) of R is defined by (a, b, c) = ab·c — a·bc. lf R also satisfies the identity

(2)      (x, x, x) = 0

In (2, 3, 4, and 5), the author has established a connection between orthomodular lattices and Baer *-semigroups. In brief, the connection is as follows. The lattice of closed projections of any Baer *-semigroup forms an orthomodular lattice. Conversely, if L is any orthomodular lattice, there exists a Baer *-semigroup S which co-ordinatizes L in the sense that L is isomorphic to the lattice of closed projections in S. In this note we shall assume that the reader is familiar with the results and the notation of the quoted papers.

The definition of the genus γ(G) of a graph G is very well known (König 2): it is the minimum genus among all orientable surfaces in which G can be drawn without intersections of its edges. But there are very few graphs whose genus is known. The purpose of this note is to answer this question for one family of graphs by determining the genus of the n-cube.

The graph Qn called the n-cube has 2n vertices each of which is a binary sequence a1a2. . . an of length n, where ai = 0 or 1.

A finite automaton may be thought of as a possible abstraction of a digital computer. Imagine a tape or sequence of letters from some alphabet being fed into a device with a finite number of internal states. When the device is in a particular state and receives an input letter, the system passes to another internal state and prints a letter on an output tape. This operation is continued until the entire input sequence has been processed.

Both Harary (6) and Ginsburg (4) have focused attention on the previously unsolved problem of counting the number of equivalence classes of finite automata. In the present paper, this problem is solved completely by proving a variety of theorems about the enumeration of functions.

Si un réseau G de Rn contient Zn et est permis pour l'«octaèdre généralisé» Ωn d'équation

l'indice de Zn dans G est borné supérieurement par n! d'après les théorèmes généraux de Minkowski.

In this paper I shall prove the following theorem:

Theorem. Let ϕ(r) be increasing and convex in log r with

(This condition is imposed to exclude certain trivial cases.) Then there is an integral function f(z) such that

A sequence {Mi} of non-trivial subspaces of a linear topological space X is a basis of subspaces for X if and only if corresponding to each x ∊ X there is a unique sequence {xi}, xi ∊ Mi, such that

Corresponding to a basis of subspaces {Mi} for X is a sequence of orthogonal projections {Ei} (Ei2 = Ei and EiEj = 0 if i ≠ j) defined by Ei(x) = xi if

Lyapunov's second method is a well-known and powerful tool for studying the behaviour of solutions of a system of differential equations. One approach to the theory is the comparison method developed by Corduneanu (4). This approach has the advantage that it also leads to other results on asymptotic behaviour which originally appeared to be unrelated to Lyapunov's method. Some of these results have been obtained by the author in (2). The purpose of this paper is to make use of the comparison method to obtain some refinements of Lyapunov's theory.

With this paper we begin a study of the structure of the group algebra RG of a finite group G over the ring of algebraic integers R in an algebraic number field k. The basic question is whether non-isomorphic groups can have isomorphic algebras over R. We shall show that this is impossible if G is

• (a) abelian,

• (b) Hamiltonian,

• (c) one of a special class of p-groups.

Consider a round robin tournament in which each of n players is required to play precisely one game with each other player, and assume that each game ends in a win or a loss. The results of such a tournament can be conveniently recorded in a square (0, 1)-matrix T = (tij) of order n by setting tij = 1 if player i defeats player j , tij = 0 if player i loses to player j , and tii = 0. Thus T has 0's along the main diagonal, and in the off-diagonal positions T satisfies the "skew-symmetry" condition that tij = 1 if and only if tji = 0. We call such a (0, 1)-matrix T a tournament matrix.

Let G be a finite group with commutator subgroup G′. In an earlier paper (4) it was shown that each element of G′ is a product of n commutators, if 4n ≥ |G′|. The object of this paper is to improve this result in two directions:

Theorem 1a. If (n + 2)!n! > 2|G′| — 2, then each element of G′ is a product of n commutators.

Theorem 1b. If G is a p-group, with |G′| = pa, and if n(n + 1) > a, then each element of G′ is a product of n commutators.

In this paper we consider undirected graphs, with no edges joining a vertex to itself, but with possibly several edges joining pairs of vertices. The first part of the paper deals with the question of characterizing those sets of non-negative integers d1d2 . . . , dn and {cij}, 1 ≤ i < j ≤ n, such that there exists a graph G with n vertices whose valences (degrees) are the numbers di, and with the additional property that the number of edges joining i and j is at most cij. This problem has been studied extensively, in the general case (1, 2, 9, 11), in the case where the graph is bipartite (3, 5, 7, 10), and in the case where the Cij are all 1 (6).

We shall say that a plane set D has the Kakeya property if a unit segment can be turned continuously in D through 360° back to its original position. The famous solution of this problem by A. S. Besicovitch (1; 2; 4; 5; 6), to the effect that there are sets of arbitrarily small area having the Kayeka property, leaves open the problem obtained by adding the new condition that the set D be also simply connected. Since we do not know whether there is an attainable minimum, we define the Kakeya constant K to be the greatest lower bound of areas of simply connected sets having the Kakeya property. We shall refer to such sets as Kakeya sets.

Maximum of a square-free quadratic form on a simplex. The following question was suggested by a problem of J. E. MacDonald Jr. (1):

Given a graph G with vertices 1, 2, . . . , n. Let S be the simplex in En given by xi ≥ 0, Σxi = 1. What is

We say that the restricted cancellation law for ideals (RCL) holds in the commutative ring R if from the equation AB = AC, where A, B, and C are ideals of R and AB ≠ (0), it follows that B = C. RCL is a weakened form of the cancellation law for ideals (CL) : If A, B, and C are ideals of R such that AB = AC and A ≠ (0), then B = C. A ring in which CL holds is an integral domain and in an integral domain, RCL is equivalent to CL.

A category is made up of its class of objects , of a function M (or ) which assigns to each (A,B) ∊ × the class M(A, B) of all morphisms from A to B, of an operation μ which is a class of maps

A, B, C ∊ , and of a family of maps

The class of developable topological spaces, which includes the metrizable spaces, has been fundamentally involved in investigations in point set topology. One example is the remarkable edifice of theorems relating to these spaces constructed by R. L. Moore (13). Another is the role played by the developable property in several metrization theorems, including Alexandroff and Urysohn's original solution of the general metrization problem (1).

This paper presents an anslysis of the concept of developable space in terms of certain more extensive classes of spaces satisfying the first axiom of countability : spaces with a base of countable order and those having what is here called a θ-base. The analysis is given in the characterizations of Theorems 3 and 4 below.

In this paper we consider finite graphs without loops and multiple edges. A graph is considered to be an ordered pair 〈G, *〉 where G is a finite set the elements of which are called the vertices of while * is a subset of [G]2 (where [G]2 is the set of all subsets of two elements of G). The elements of * are called the edges of . If {P, Q} ∊ *, we say that Q is adjacent to P. The degree of a vertex is the number of vertices adjacent to it. Let k be an integer. We say that is the complete k-graph if G has k elements and * = [G]2.