Our purpose is to deduce from a theorem of P. Hall the following observation.

Let G be a finitely generated group and F a free normal subgroup of G with G/F nilpotent. Then G is hopfian.

Here a group G is hopfian if every epiendomorphism G→G is an automorphism.

Recently Bruck initiated the study of the structure of the fixed-point set of a nonexpansive selfmap T of a Banach space, where T satisfies a conditional fixed point property. We generalize many of his results to a Hausdorff locally convex space X. Also, we generalize a result of Holmes and Narayanaswami and use it, along with a procedure of Kiang, to obtain a fixed point theorem for families of asymptotically nonexpansive mappings in X.

In this note, we prove three results on regular rings possessing a rank function: (a) the completion of a *-regular rank ring is a regular Baer *-ring; (b) (a) is used to construct regular Baer * factors of type IIf with centre any complex subfield closed under conjugation; (c) the units of a unit-regular rank ring form a dense topological subgroup of the units of the completion.

For two subsets Z and Y of a metric space (X, d) the set Z is said to be a bisector in Y iff Z ⊂ Y and there exist two distinct points y1, y2 ∊ Y such that Z = {z: d(z, y1) = d(z, y2) and z ∊ Y}. Considering chains of consecutive bisectors X ⊃ X1 ⊃ … ⊃ Xk we denote by b(X, d) the maximum of their length. The topological invariant b(X) is defined as the minimum of b(X, d) taken over the set of all metrizations of X. It is proved that if X is compact then dim(X) ≤ b(X) ≤ 2 dim(X) + 1, b(X) = 0 iff dim(X) = 0 and b(X) = n implies dim(X) = n for n = 1 and ∞. The sharp result b(En) = n for n = 1, 2, … is obtained for Euclidean space En.

We show that Krasnoselski's Theorem, which is usually derived from Helly's Theorem, is in fact equivalent to it.

Efforts to determine the orders of the sublattices of an arbitrary finite lattice date back at least to the early 1930's, and notably, in the work of Fritz Klein-Barmen [3], [4]. Nevertheless, very little that is new has appeared in the literature since that time.

In this paper, we define an integral logarithmic method of summability, extending the integral Abel-type methods defined by Jakimovski [6]. We examine the behaviour of the product of this method with integral Hausdorff methods. A full scale of strict inclusions for integral Abel-type methods is obtained and the integral logarithmic method is placed in this scale.

Local criteria are given which imply the lipschitzian character of mappings defined in complete and convex metric spaces.

In the study of quasi-analytic classes (see [3], pp. 372-379), a class C{Mn} is shown to have the following properties:

1. (a) If M0= 1 and (i.e. {Mn} is log convex), C{Mn} forms an algebra.

2. (b) C{Mn} is invariant under affine transformations.

3. (c) C{Mn} is quasi-analytic iff it contains non non-trivial function with compact support.

In [1] and [2], there was given a characterization for linear Jordan matrix algebras whose coordinatizing ring is *-prime Goldie or a Cayley-Dickson ring (C-D ring). If one considers the corresponding question in the more general setting of quadratic Jordan algebra as defined by McCrimmon in [11], then the result is similar. In this latter case the ample quadratic Jordan algebras, as studied by Montgomery in [12] and [13], are brought into play.

Initial and final value theorems for Hardy transformations and of a suitably chosen function f(x) under a certain set of conditions on v and p where

1

Jv(x) and Yv(x) being Bessel functions of the first and second kind, and

2

su, v(x) being Lommel's function, are proved.

A set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

All topological spaces are assumed to be completely regular. C(X) (resp. C*(X)) will denote the ring of all (resp. all bounded) continuous real-valued functions on X. βX is the Stone-Cech compactification of X. A real number t is said to be an accumulation point of a function f ∊ C(X) if and only if f-1[[t-ε, t + ε]] is not compact for every ε > 0.

I wish to present here some of the results of a research in the Theory of Fibrations initiated some time ago by Peter Booth, Philip Heath, and myself. The philosophy behind the work is to approach certain aspects of the Theory of Fibrations in a unified way through the systematic use of the sections of suitable fibrations; this yields general theorems, of which some well-known results are eventually particular cases.

The construction of images of morphisms in an elementary topos E has hithero required the use of colimits. For example, in [1], Freyd constructs the image of a morphism by taking the equalizer of the cokernel pair of the morphism. In particular, the construction of the direct image functor, or, as it is sometimes referred to, existential quantification along a morphism, has required the use of images, and hence colimits. However, Mikkelsen has defined existential quantification using only limits.

The purpose of this paper is to study a degenerate Sobolev type partial differential equation in the form of Mut + Lu = f, where M and L are second order partial differential operators defined in a domain (0, T]×Ω in Rn+1. The degenerate property of the equation is in the sense that both M and L are not necessarily strongly elliptic and their coefficients may vanish or be negative in some part of the domain (0, T]×Ω. Two types of boundary conditions are investigated.

The notion of entire function of Bounded Index is by now well established. It may be stated as follows.

An entire function f(z) is said to be of Bounded Index if for some fixed s

for all n and all z. (See [1], [2].)

This text is an elaboration of a lecture delivered at the First Winter Meeting of the Canadian Mathematical Congress, December 1975. Subsequently, this same lecture was presented at various occasions, (Colloquium at Université Pierre et Marie Curie, in Paris; Seventh Iranian Mathematical Congress in Tabriz; Université de Paris-Sud à Orsay, Illinois Institute of Technology in Chicago, Colloquium at the University of Toronto, and at the University of Waterloo).

In this note we continue the study of Abian's order for reduced rings initiated in papers such as [1], [5], [3], [4]. A simple proof is given of Abian's result that taking suprema commutes with ring multiplication. The properties of orthogonally complete rings and of rings satisfying chain conditions with respect to Abian's order are investigated.

Donné (n + 2) points dans l'intervalle [-1,1],

1.1

L. Fejér a montré ([3]) qu'il existe toujours un polynôme de degré inférieur ou égal à n + 1 tel que

1.2

et

1.3