We suppose that 0 < sn ≤ 1 for every n, and denote by n(α, β) the number of S 0, S1, S 2, . . . , Sn which fall in the interval 0 ≤ α < x ≤ β ≤ 1. If there exists a function g(t), 0 ≤ t ≤ 1, such that

for every interval (α, β] with 0 ≤ β — α ≤ 1, the sequence (Sn ) is said to have a distribution function g(t), 0 ≤ t ≤ 1, in the interval [0, 1], (see 9, p. 87).

Let X = {x 1, x 2,...,x m} be a finite set of m points and = {A 1, A 2, . . . , An ] be a class of n subsets of X. Such a system of points and sets is called a complex (X, ). If every set of the class contains two points, the complex is a graph with m points x 1x 2, . . . , xm and n edges A 1, A 2, . . . , An . A complex (X, ) in which every set has the same number of points is called regular.

A well-known problem in topology is the so-called metrization problem. This consists of asking for the topological conditions that are necessary and sufficient in order to guarantee that a topological space be metrizable. The first solution of this problem was given in 1923 by P. Alexandroff and P. Urysohn (1).

An ordinary graph is a finite linear graph which contains no loops or multiple edges, and in which all edges are undirected. In such a graph G, let N, L, and T denote respectively the number of nodes, edges, and triangles. One problem, suggested by P. Erdös (1), is to determine the minimum number of triangles when the number of edges is specified, subject to suitable restrictions.

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.

A given semigroup is said to be embeddable in a group if there exists a group which contains a subsemigroup isomorphic to .

It can easily be proved that cancellation is a necessary condition for embeddability. It can also be shown that we can adjoin an identity to a semigroup without identity in such a way that the new semigroup is embeddable if and only if the original was embeddable. Therefore, we can, without loss of generality, restrict our attention to cancellation semigroups with identity, whenever this is convenient.

The following elementary facts about certain commutative diagrams, called "squares," are stated and proved in terms of abelian groups and their homomorphisms. However, they are valid for arbitrary abelian categories and can be proved also for them. This does not need to be shown, since every abelian category can be embedded into the category of abelian groups with preservation of exact sequences according to a result due to S. Lubkin (1). Proofs are often omitted or given only for one half of a theorem, the other half being dual to the first. Generalizations to a larger class of diagrams containing all finite commutative diagrams are possible.

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A 0, A 1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of A i+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.

Cet exposé fait suite à des travaux antérieurs et notamment à mon article (4). Il a fait l'objet d'une communication au Colloque de géométrie différentielle et de topologie algébrique de Zürich en juin 1960. Dans une première partie on aborde, par les méthodes cohomologiques, le problème de la restriction du groupe structural G d'un espace fibre à un sous-groupe N, ce qui, lorsque N n'est pas normal, conduit à une "cohomologie à coefficients dans un faisceau d'espaces homogènes." Dans le second on donne une interprétation cohomologique des espaces fibres en théorie simpliciale, une fonction tordante étant interprétée comme un cocycle.

Put

In a recent paper (4), Lohne showed that

A union curve on a surface in a euclidean 3-space, relative to a given congruence is characterized by the property that its osculating plane at each point contains the ray of the congruence through that point. Springer (2) and Pan (1) have studied union curves in a hypersurface Vn of a Riemannian V n+1. In the present paper we proceed to obtain the equations of union curves in a subspace Vn of a Riemannian Vm .

In what follows we prove that a finite graph of n nodes in which each node has degree ≥ 1 but ≤ possesses a set of at least n/(1 + ) pairwise disjoint edges. Our principal theorem states an analogue of this result for the case when each node has degree ≥2: we show that in this case the graph possesses a set of at least 2n/(2 + max(4, )) mutually disjoint edges.

Let Dn be the open unit ball in En , and let Sn be the n-sphere. The cluster set C(f, p) of a function f : Dn —> Sn at a point p on the boundary of Dn is the set of points y in Sn such that there exists a sequence of points xm —> p, xm in Dn , with f(xm ) —> y. Given an arc γ in , meeting the boundary bdy(Dn ) only in p, the arc cluster set C(f, γ) is the set of points y in Sn such that there exists a sequence xm —> p, xm in γ, with f(xm ) —> y.

Let λ, K, v be integers such that 0 < λ < k < v. Then the set of integers

is a difference set with parameters v, k, λ if each non-zero residue modulo v occurs precisely λ times and zero occurs precisely ktimes among the k2 numbers

In the mathematical justification of the formal calculations of axiomatic quantum field theory and the theory of dispersion relations, a strategic role is played by a theorem on analytic functions of several complex variables which has been given the euphonious name of the edge of the wedge theorem. The statement of the theorem seems to be due originally to N. Bogoliubov (cf. 3, Mathematical Appendix, pp. 654-673) but no complete proof which is fully satisfactory from the mathematical point of view has yet appeared in the literature.

In trying to extend the concept of torsion to rings more general than commutative integral domains the first thing that we notice is that if the definition is carried over word for word, integral domains are the only rings with torsion-free modules. Thus, if m is an element of any right module M over a ring containing a pair of non-zero elements x and y such that xy = 0, then either mx = 0 or (mx)y = 0. A second difficulty arises in the non-commutative case: Does the set of torsion elements of M form a submodule? The answer to this question will not even be "yes" for arbitrary non-commutative integral domains.

Valentine (1, Theorems 2 and 3) has defined a three-point property which he called P 3 and has shown that a closed subset of the euclidean plane possessing this property is expressible as the union of at most three convex sets. He also showed that if the number of isolated points of local non-convexity of such a set is one, finite and even, or infinite, the set is the union of two convex sets. In this paper we give properties which, together with Valentine's results, characterize those subsets of a plane which may be represented as a union of two closed, convex sets.

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and set

The Bergman-Whittaker operator defined by

Let k and t be positive integers; let q 1, q 2, . . . , q t be distinct prime numbers; and let ζ1, ζ 2, ... , ζ t be kth roots of unity, not necessarily primitive. Recent investigations on consecutive kth power residues have led to the following question: Under what conditions do there exist primes p that have a kth power character X such that

Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + 1 , . . . , r + m — 1, each of which is a kth power residue of p. Let r(k, m, p) denote the least such r.