For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G.

G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n).

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .

Fuchs, in [3], problem 14, proposes the study of pure-high subgroups of an abelian group. In this paper we show that in abelian torsion groups, pure-high subgroups are also high. A natural problem arises, that of characterizing the pure-absolute summands. We show that this concept is the same as absolute summands in torsion groups, but that it is more general in mixed abelian groups. There is a definite connection between the existence of pure TV-high subgroups and the splitting of mixed groups. The notation is that of [3].

We consider finite dimensional complex vector bundles over a compact connected Hausdorff space X, as defined, for example, in [1]. It is well known that if ξ is such a bundle, then there is a bundle η such that ξ⊕η is trivial.

Several separation axioms, defined in terms of continuous functions, were examined by van Est and Freudenthal [3], in 1951. Since that time, a number of new topological properties which were called separation axioms were defined by Aull and Thron [1], and later by Robinson and Wu [2], This paper gives a general definition of separation axiom, defined in terms of continuous functions, and shows that the standard separation axioms, and all but one of these new topological properties, fit this definition.

In this note all Banach space are assumed to be real and separable and their norms will be denoted by || ||. The canonical bilinear form between a Banach space B and its topological dual B′ will be denoted by 〈x, y〉, x ∊ B, y ∊ B′.

In this paper we consider the equation

(1.1) (r(t)y′(t))′+p(t)f(y(t)) = 0

under the conditions

((H 0): the real valued functions r, r′ and p are continuous on a non-trivial interval J of reals, and r(t)>0 for t∈J;

and

(H1):f:R→R is continuously differentiable and odd with f'(y)>0 for all real y. We also consider the equation

(1.2) y″(t)+m(t)y′(t)+n(t)f(y(t)) = 0

under the conditions (H 1) and

(H 2): the real valued functions m and n are continuous on a non-trivial interval J of reals.

We name a functional equation with restricted argument one in which at least one of the variables is restricted to a certain discrete subset of the domain of the other variable(s). In particular, the subset may consist of a single element.

The purpose of this paper is to present a functional equation satisfied only by cosine functions.

The purpose of this note is to offer a equalization of the concept of Goldie dimension and to prove a structure theorem (Theorem 3.1) for modules satisfying the conditions of this dualization. In this paper, all rings considered are associative with unit and all modules are unital.

Let {X a }a∈A be a non-empty family of compact spaces and let F be a subset of the m-product P{X a :a∈A}. The Tychonoff problem is the determination of the sets F which are pointwise compact. It is known that P{X a : a ∈ A} is pointwise compact [5, p. 400]. Also the set F of all point-closed multifunctions of P{X a :a∈ A) is pointwise compact.

Let S be a non-empty family of real valued continuous functions on [a, b]. Diaz and McLaughlin [1], [2], and Dunham [3] have considered the problem of simultaneously approximating two continuous functions f1 and f2 by elements of S. If || • || denotes the supremum norm, then the problem is to find an element * ∈ S if it exists, for which

Let Un(f) denote the class of all n × n (0, 1)-matrices with precisely r-ones, r≥3, in each row and column. Then

T. Nakayama showed in [2, Theorem 13] that symmetric algebras have the property that the left and right annihilators of their two-sided ideals are equal. He also gave examples [2, p. 630] to show that QF algebras with this property are not necessarily symmetric, and that weakly symmetric algebras need not have this property.

In [1], Zhelobenko introduced the concept of a Gauss decomposition ZtDZ of a topological group and gave characterizations of irreducible representations of the classical groups. In this setting, vectors of representation spaces are polynomial solutions of a system of differential equations and the problem of obtaining branching theorem with respect to a subgroup G0 is to find all polynomial solutions that are invariant under Z ∩ G0 and have dominant weight with respect to D ∩ G0

Let A = (aij) be an m×n (0, 1)-matrix, m≤n. The permanent of A is defined by

where the summation is over all one-one functions σ:{1, 2, . . . , m}→{1, 2, . . . , n}. If A is regarded as the incidence matrix of a configuration of m subsets of an n-set, then Per (A) is the number of systems of distinct representatives in the configuration.

We shall work in the piecewise-linear category, so that all manifolds and subsets thereof, as well as all maps are assumed to be piecewise-linear. If M is a manifold, denote by #k  M the k-fold connected sum of copies of M and by 2M the double of M, that is the manifold obtained by sewing two copies of M together by the identity map on their boundaries.

In recent papers Brewer and Mott have studied integral domains which have finite character globally. This paper concentrates on domains which have finite character locally. Examples include global finite character domains plus Prufer, almost Dedekind, and almost Krull domains. General properties are given, including a valuation-theoretic characterization. The effect of requiring essential and/or rank one valuations is also studied.

Given a complete Serre class τ this determines a torsion theory with T the class of torsion modules. It also determines the torsion free modules. For the classical torsion in the category of abelian groups the torsion free modules are flat and visa-versa. Which rings are characterized by this property? More precisely: Which rings admit a torsion theory for which the concepts of torsion free and flat are equivalent? We also dispose of the cases when R admits a toision theory for which torsion free is equivalent to injective and when projective is equivalent to torsion free.

Let A be a complex Banach algebra without order. Following Kellogg [4] and Ching and Wong [2], a mapping T of A into itself is called a right (left) multiplier on A if T(ab)=(Ta)b(T(ab)=a(Tb)) for all a, b in A. T is said to be a multiplier on A if it is both a right and left multiplier on A. Let M(A)(RM(A), LM(A)) be the set of all (right, left) multipliers on A.

It has been shown by E. Granirer that for certain infinite amenable discrete groups G there exists a nested family of left almost convergent subsets of G on which every left invariant mean on m(G) attains as its range the entire [0,1] interval. This paper examines the range of left invariant means on L ∞(G) for infinite locally compact abelian groups G and demonstrates the existence in every such group of a nested family of left almost convergent Borel subsets on which every left invariant mean on L ∞ (G) attains as its range the interval [0,1],