We begin with the following definitions.

Let f(z) be regular near z=0 and let f(0)=0, f'(0)≠0. Let α and λ be two real numbers such that |α|<π/2 and 0≤λ1. Then Rα,λ is the largest value of r such that the following conditions are satisfied for |z|<r

The purpose of this paper is to extend to almost periodic (a.p.) solutions a stability result on the periodic solutions of the autonomous equation

x′ = F(x),

(cf. Coppel [1], p. 82 or Coddington and Levinson [2], p. 323.)

In 1967, Powers [7] proved that there exists a one-parameter family of pairwise non-isomorphic hyperfinite factors of type III. Powers′ result on hyperfinite factors has been extended by Araki and Woods [1]. Connes [4], and Williams [11], with different proofs, showed that there exists a continuous family of mutually non-isomorphic non-hyperfinite factors of type III.

Let T, P denote two continuous operators from E into F, where E and F are locally convex spaces. It is proved by L. Schwartz [8] and G. KÖthe [6] that if E and F are Fréchet spaces, T is a ϕ_-operator and P a compact operator, then T+P is a ϕ_-operator.

This note answers affirmatively a question of the author [4, p. 456], by producing an example of an integrally closed quasi-local non valuation domain of global dimension 3, each of whose overrings is a goingdown ring. Although [4, Proposition 4.5] shows that such an example cannot be constructed by means of restrained power series, an approach via the more general D+M construction succeeds.

The purpose of this paper is to show that by a reorganization of the proofs of the main results concerning Radon-Nikodym derivatives in a von Neumann algebra of Pedersen and Takesaki in [12] and of Connes in paragraphs 1.1 and 1.2 of [5], considerable technical simplification can be achieved. Roughly speaking, the analytic vector techniques developed by these authors for the study of weights on a von Neumann algebra can be replaced, to a large extent, by the tensor product methods introduced by Connes, which are essentially algebraic in nature. In the exposition which follows, analytic vectors are not used at all (see, however, 4.4).

Some extensions of the open mapping and closed graph theorem are proved for certain classes of commutative topological semigroups, namely those embeddable as open subsets of topological groups. Preliminary results of independent interest include investigations of properties which “lift” from embeddable semigroups to the groups in which they are embedded, and from semigroup homomorphisms to homomorphisms of the groups.

Let X be a metrizable topological space and f:X→X a continuous selfmapping such that for every x ∈ X the sequence of iterates {fn(x)} converges. It is proved that under these conditions the following two statements are equivalent:

1. There is a metrization of X relative to which f is contractive in the sense of Edelstein.

2. For any nonempty f-invariant compact subset Y of X the intersection of all iterates fn(Y) is a one-point set. The relation between this type of contractivity and the Banach contraction principle is also discussed.

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

A friendship graph is a graph in which every two distinct vertices have exactly one common adjacent vertex (called a neighbour). Finite friendship graphs have been characterized by Erdós, Rényi and Sós [2]: Each finite friendship graph Fn which consists of n edge disjoint triangles such that all n>1 triangles have one vertex in common (F1 is a triangle i.e. the complete graph with three vertices). Thus Fn has 2n+1 vertices, 2n of them being of degree two and the remaining one (the common vertex of n triangles if n>1) being of degree 2n.

It is shown that the Lorentz structure of a compact prespace- time M can be so chosen such that M can not satisfy the strong energy condition. Thus, combining both the causal and the strong energy conditions, a stronger case against the compact space-times as proper arenas of physics can be made.

The purpose of the paper is to extend a tight set function on a lattice with values in a locally convex space of special type to a measure on the cr-ring generated by . This result generalizes the extension theorem of Thomas [12, p. 151], which in turn contains the extension theorems of Pauc [9, p. 710], Fox [4, p. 525] and J. J. Uhl, Jr. [14, Corollary 2].

The maximal number of pairwise edge disjoint 4-cycles in the complete graph Kn and the minimal number of 4-cycles whose union is Kn are determined.

Let (X, d) be a (nonempty) metric space. bc(X) will denote the family of all nonempty bounded closed subsets of X endowed with the Hausdorff metric D induced by d [2, pp. 205]. Let/be a map of X into bc(X). f is nonexpansive at a point x in X if for all y in X.

Let (Mn, F) be an n-dimensional Finsler space where Mn is the underlying n-dimensional manifold and is the Finsler fundamental function. F being a differentiable function of the point and element of support where T∞(Mn) is the tangent space of Mn at x and is positively homogeneous of degree one with respect to X.

All rings are associative but do not necessarily have identities. Definitions and basic results about radical classes can be found in [2]. A radical class is strong [3] if for every ring A, (A) contains all left and right -ideals of A.

J. Mycielski has conveyed to me the following problem by P. Erdös. Let {tn} be a sequence of natural numbers such that

(a) Does there exist an irrational number η such that the sequence {tnη (mod 1)} is not dense in [0,1)?

Inspired by a question of J. B. Miller, L. Fejes Tóth asked for a catalogue of those subsets of Euclidean n-space which are homothetic (similar and similarly situated) to their intersection with a suitably translated copy of themselves. For example, a triangle is homothetic to its intersection with an arbitrarily translated replica provided only that the intersection has non-void interior. In 3-space a cube is homothetic to its intersection with a replica translated part way along any body diagonal. With these two preliminary examples for motivation, let us make a definition.

Nous donnons, dans ce qui suit, la preuve de la proposition suivante.

Theorem. Letbe a function holomorphic in the disk, wherep is a natural number andIfthen then f(z) assumes every complex value infinitely often in every sector.

The purpose of this note is to prove the above result. To do this, we first observe that from the condition a<∞, we can easily show that the derivative f′(z) satisfying