This paper is concerned with the convergence of sequences of subsets of a topological space, as defined by F. Hausdorff [6]. Such a sequence converges if and only if its limit inferior equals its limit superior, where its limit inferior (respectively, superior) is that set each of whose elements satisfies the condition that each of its neighborhoods has nonempty intersection with all but finitely (respectively, with infinitely) many terms of the sequence.

This is a generalization of the results in [3, § 5]. Some of the proofs presented here are actually the original proofs presented in [3]. Although we can find alternate proofs in the case p = 3, since [3] will not be published for a while yet, we feel that it is worthwhile to present the proof in [3] whenever it carries over in the case p = 3. The results in this paper will be used in the investigation of the quadratic pairs for the prime 3.

A directed partially ordered abelian group (G, ≦ ) is a tight Riesz group if for a1, a2, b1, b2 ∈ G with ai < bj, i, j = 1,2, there is an x ∈ G with ai < x < bj, i, j = 1, 2. The open interval topology on G is the topology having as a base the set of all open intervals (a, b) = {x ∈ G|a < x < b}. For any x ∈ G, a neighborhood base at x is the set of all open intervals (x — a, x + a) = x + ( — a, a) for a > 0.

A distance function d on a set X is a function X × X → [0, ∞ ) satisfying (1) d(x, y) = 0 if and only if x = y, and (2) d(x, y) = d(y, x). Such a function determines a topology T on X by agreeing that U is an open set if it contains an ∈-sphere N(p; ∈)( = {x: d(p, x) < ∈﹜} about each of its points. Equivalently, F is closed if and only if d(x, F) > 0 for each x ∈ X — F. A topological space is symmetrizable via a distance function d if its topology is determined by d as above, and semi-metrizahle via d if x ∈ Ā is equivalent to d(x, A) = 0.

The recent study of operational statistics (see [2; 4; 5; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16]) describes a generalized sample space which represents the set of all possible outcomes of a collection of coherently related operations (experiments). This approach to probability generalizes the classical notion of a sample space due to A. N. Kolmogorov [8], and it gives the concept somewhat wider applicability. For instance in [4] and [14], D. J. Foulis and C. H. Randall set out the start of a program wherein a generalized sample space (hereafter called a GSS) and its affiliated partially ordered set of generalized propositions could be a framework within which a genuinely operational interpretation of the so called “logic” of quantum mechanical systems may be found.

This paper will continue certain investigations into the geometric nature of the well-known traveling salesman problem: that of determining the extreme Hamiltonian circuits (H-circuits) of a graph.

Cartan subalgebras play a very important role in the classification of the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic zero. It is well-known [5, 273] that any two Cartan subalgebras of such an algebra are conjugate, i.e. images of one another under some automorphism of the algebra. On the other hand, there exist finitedimensional simple Lie algebras over fields of finite characteristic p possessing non-conjugate Cartan subalgebras [2; 3; 4]. The simple Lie algebras discovered by Zassenhaus [6] also possess non-conjugate Cartan subalgebras, and we shall give a complete classification of Cartan subalgebras of these algebras in this paper.

Let V be an n-dimensional vector space over the field F. Let ꕕm V be the rath tensor power of V. If ᓂ ∈ Sm, the symmetric group, there exists a linear operator P (ᓂ1) on ꕕm V such that

for all x1, … , xm ∈ V. (Here, x1 ꕕ … ꕕ xm denotes the decomposable tensor product of the indicated vectors.) If c is any function of Sm taking its values in F, we define

Let G be a group of finite order g, A a splitting field of G of characteristic p (which may be 0) and R = AG the group algebra of G over A. In [2], the author studied some of the properties of the Grothendieck ring K(R) of the category of all finitely generated R-modules, and derived a number of consequences. This paper continues the study carried out in [2]. The study is concerned with the structure and irreducible representations of K(R). The ring K(R) is proved to be semisimple and the primitive idempotents of K(R) are explicitly constructed. If the ring K(R) is identified with the ‘algebra of representations', then Robinson's idempotent [3; 4; 5] follow from our description as a special case.

Let A be a Banach algebra and A** its second conjugate space. Arens has denned two natural extensions of the product on A to A**. Under either Arens product, A** becomes a Banach algebra. Let A be a semisimple Banach algebra which is a dense two-sided ideal of a B*-algebra B and R** the radical of (A**, o). We show that A** = Q ⊕ R**, where Q is a closed two-sided ideal of A**, o). This was inspired by Alexander's recent result for simple dual A*-algebras (see [1, p. 573, Theorem 5]). We also obtain that if A is commutative, then A is Arens regular.

Transferable and sharply transferable lattices were defined and characterized in [1]. Finite sublattices of free lattices were studied in [2] and a characterization of them given in [3]. In this paper, we will show that the class of finite sharply transferable lattices coincides with the class of finite sublattices of free lattices.

Evaluation subgroups of the homotopy groups have been objects of extensive study recently by Gottlieb, Haslam, Jerrold Siegel, G. E. Lang (Jr), etc. In [8] one of the authors has introduced the notions of ‘cyclic' and ‘cocyclic’ maps and studied generalizations of evaluation subgroups and their duals in the set up of Eckmann-Hilton duality. This paper continues the study of these generalized Gottlieb and dual Gottlieb subsets. All the spaces, except the function spaces, will be arc connected locally compact CW-complexes with base point at a vertex. For any X, Y the set of base point preserving homotopy classes of maps of X to Y is denoted by [X, Y].

In this paper all rings R are associative rings with identity and all modules are members of R-mod, the category of unital left R-modules, unless the contrary is specifically stated.

Let be a measure space and the usual Banach spaces. A linear operator T : Lp → Lpis called a positive contraction if it transforms non-negative functions into non-negative functions and if its norm is not more than one. The purpose of this note is to show that if 1 < p < ∞ and if T : Lp → Lp is a positive contraction then

In this paper we are concerned with partition functions pϒ(n) that have generating functions of the form

where γ(n) ≧ 0. We shall obtain an asymptotic relation for pϒ(n) under suitable restrictions on ϒ (see Theorem 1.1). These restrictions are weaker than those of Brigham [2] who considered this problem previously.

The purpose of this paper is to extend the class of knot groups whose commutator subgroups are known to be residually a finite pgroup (i.e., residually of order a power of the prime p). Such a knot group is known to be residually finite (see, e.g., [10]), and although this class is quite restricted we will show that it includes all the groups of knots in the classical knot table [15].

The main theorem of this paper characterizes the class of essentially T1 spaces having bases of countable order as those spaces of the class of essentially T1 spaces having primitive bases in which closed sets are sets of interior condensation. In addition we deduce some corollaries of this theorem, derive some other characterizations, and prove a lemma concerning primitive sequences which is a key to the proof of the main theorem and has other applications.

Let E be a separable Fréchet space, and let E* be its topological dual space. We recall that a Fréchet space is, by definition, a complete metrizable locally convex topological vector space. A sequence {Ln} of continuous linear functional is said to be interpolating if for every sequence {An} of complex numbers, there exists an ƒ ∈ E such that Ln(ƒ) = An for n = 1, 2, 3, … . In this paper, we give necessary and sufficient conditions that {Ln} be an interpolating sequence. They are different from the conditions in [2] and don't seem to be easily interderivable with them.

Let R be a ring with involution *, and let Z denote the center of R. In R let S = {x ∈ R|x* = x} be the set of symmetric elements of R. We shall study rings which are conditioned in the following way: given s ∈ S, then for some integer and some polynomial p(t), with integer coefficients which depend on . What can one hope to say about such rings? Certainly all rings in which every symmetric element is nilpotent fall into this class.

By an RE (recursively enumerable) sequence we mean a sequence V(0), V(1), … of uniformly RE sets, denotes the class of all RE sets. If is an RE class if either for some RE sequence V(0), V(1) , … .