Ruth R. Struik investigated the nilpotent group , where G is a free product of a finite number of cyclic groups, not all of which are of infinite order, and Gm is the mth subgroup of the lower central series of G. Making use of the “collection process” first given by Philip Hall in [8], she determined completely for 1 ≦ n ≦ p + 1, where p is the smallest prime with the property that it divides the order of at least one of the free factors of G. However, she was unable to proceed beyond n = p + 1.

If (G ; U, V) is a triad with G a group we define

where [g, u] = gug-1u-1 is the commutator. CG(U, V) will be called the (left) center of U in G modulo V or in brief a (left) C-space. If G is a topological group it will be understood that the topology on CG(U, V) is the relative topology of G.

Un anneau commutatif unitaire est dit réflexif si tous ses idéaux sont divisoriels [3; 6]. On dira qu'un anneau commutatif unitaire et intègre est un anneau de Mori, si ses idéaux divisoriels entiers vérifient la condition de chaîne ascendante [8]. Enfin, on appellera provisoirement, M-anneau, un anneau de Mori, tel que tout idéal engendré par deux éléments est divisoriel. Un anneau noethérien réflexif est un M -anneau; dans la première partie de cette note, nous établissons la réciproque. La troisième partie étudie la clôture intégrale de certains anneaux locaux réflexifs non noethériens.

The study of metrization has led to the development of a number of new topological spaces, called generalized metric spaces, within the past fifteen years. For a survey of results in metrization theory involving many of these spaces, the reader is referred to [13]. Quite a few of these generalized metric spaces have been studied extensively, somewhat independently of their role in metrization theorems. Specifically, we refer here to characterizations of these spaces by various workers as images of metric spaces. Results in this area have been obtained by Alexander [2], Arhangel'skii [3], Burke [5], Heath [10], Michael [15], Nagata [16], and the author [1], to mention a few. Later we will recall specifically some of these results.

Oscillation criteria will be obtained for the linear elliptic partial differential equation

in an unbounded domain G of general type in n-dimensional Euclidean space En. The differential operator D is defined as usual by

where each α (i), i = 1, … , n, is a non-negative integer.

In a lecture in 1966 (see [6]), the second author considered briefly those first order properties which hold for a lattice if and only if they hold for the lattice of all ideals of . The best known examples of such properties are those given by identities. The well-known connection between the modular identity ϵ and the five-element nonmodular lattice transforms the above result for ϵ into the following statement: is a sublattice of a lattice if and only if is a sublattice of .

In 1950, R. C. James [7] exhibited a nonreflexive Banach space with a basis that is of finite codimension in its second dual. This space is the first example of a quasi-reflexive space. General results on quasi-reflexive spaces have been obtained by P. Civin and B. Yood [3], and quasi-reflexive spaces with bases have been studied by D. Dean, B. L. Lin, and I. Singer [4 ; 12].

A well-known theorem in the classical kinetic theory for a gas states that the entropy is an increasing function of time. However, in order to obtain this theorem for a confined gas, some assumption about molecular response to the container wall is required. For example, it is enough to suppose that the wall reflects the molecules specularly [4].

Let ffl and be Hilbert spaces, the set of all densely defined linear operators from to , and its subset of bounded ones. Let and σ(T) denote the adjoint, range, domain, closure and spectrum of T respectively. Ri(z) will denote the resolvent (z — Ti)-1.

Un exemple de Fatou-Bieberbach [1, p. 45] montre qu'il existe des applications holomorphes

h:Cn → Cn

à Jacobien J(h) partout non nul, mais dont le complémentaire de l'image Cn\h(Cn) est un fermé à intérieur non vide. Néanmoins on a montré récemment [3; 5] qu'une application holomorphe générique de Cn dans Cn est surjective. Nous voulons montrer qu'il en est de même pour les applications prennant leurs valeurs dans l'espace projectif Pn.

In any group G, the commutator subgroup G' is contained in G2, the subgroup of G generated by the squares in G. Thus any product of commutators can be written as a product of squares in G. For instance, the commutator [x, y] ( = xyx-1y-1) can be expressed as the product of three squares: [x, y] = x2(x-1y)2(y-1)2. Roger Lyndon and Morris Newman have made the interesting observation [4, Theorem 1] that, in this case, the number three is minimal in the sense that there are groups which contain commutators not expressible as the product of fewer than three squares.

In this paper we introduce a new class of mappings and apply it to study some local properties of continua. A solution is obtained to a problem raised in [14] by the first author (see 4.4 below). By a mapping we always mean a continuous function.

Of recent years the author has been interested in developing a representation theory of the algebra of representations [5; 6] of a finite group G, and dually of its classes [7]. In this paper Frobenius’ Reciprocity Theorem provides a starting point for the introduction of the inverses R-1 and I-1 of the restricting and inducing operators R and I. The condition under which such inverse operations are available is that the classes of G do not splitin the subgroup Ĝ. When this condition is satisfied the application of these operations to inner products is of interest.

In [3] Remeslennikov proves that a finitely generated metabelian group G has a faithful representation of finite degree over some field F of characteristic zero (respectively, p > 0) if its derived group G’ is torsion-free (respectively, of exponent p). By the Lie-Kolchin-Mal'cev theorem any metabelian subgroup of GL(n, F) has a subgroup of finite index whose derived group is torsion-free if char F = 0 and is a p-group of finite exponent if char F = p > 0. Moreover every finite extension of a group with a faithful representation (of finite degree) has a faithful representation over the same field. Thus Remeslennikov's results have a gap which we propose here to fill.

In recent papers Horst Herrlich [4; 5] has demonstrated the usefulness of topological categories for applications to a large variety of special structures. A particularly striking result is his characterization of cartesian closedness for topological categories (see [5]). Spaces satisfying a separation axiom usually cannot form a topological category in Herrlich's sense however and some interesting special cases, e.g. Hausdorff C-spaces, remain excluded from his theory despite having many analogous properties. It therefore seems worthwhile to undertake a similar study in a wider setting. To this end we relax one of the axioms for a topological category and show that in the resulting initially structured categories a significant selection of results can still be proved, including the characterization of cartesian closedness.

Let S be a subset of Rd. A point x in 5 is a point of local convexity of S if and only if there is some neighborhood N of x such that, if y, z ∈ N ᑎ 5, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S then q is called a point of local nonconvexity (lnc point) of S.

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of line segments determined by these points lies in S. A point x in S is called a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z ∈ N⋂ S, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S.

Let X be a Hausdorff space. Let 2X denote the set of all non-empty closed subsets of X. For a subset A of X, we set 2A = {F 6 2X : F ⊆ A}. Recall that the finite topology on 2X is that topology having as a sub-basis the family {2G : G is open in X} U }2X — 2F : F is closed in X). When endowed with this topology, 2X is referred to as the hyper space of X. For the fundamental properties of hyperspaces, we refer the reader to [6; 7]. Following [6], we adopt the following notation: If A0, A1, … , An are subsets of X, we set and for all .

1. Let G be a compact group, let I b e a subset of its dual object 𝚪, which, without loss of generality, will be assumed to be a countable subset. Let Di, i ∈ I , be irreducible representations of G of degree di. The Fourier series of a function F in L1(G) is denned by

where