Showing papers in "Colloquium Mathematicum in 1992"

A generalization of Davenport's constant and its arithmetical applications

Abstract: 1. For an additively written finite abelian group G, Davenport’s constant D(G) is defined as the maximal length d of a sequence (g1, . . . , gd) in G such that ∑d j=1 gj = 0, and ∑ j∈J gj 6= 0 for all ∅ 6= J {1, . . . , d}. It has the following arithmetical meaning: Let K be an algebraic number field, R its ring of integers and G the ideal class group of R. Then D(G) is the maximal number of prime ideals (counted with multiplicity) which can divide an irreducible element of R. This fact was first observed by H. Davenport (1966) and worked out by W. Narkiewicz [8] and A. Geroldinger [4]. For a subset Z ⊂ R and x > 1 we denote by Z(x) the number of principal ideals (α) of R with α ∈ Z and (R : (α)) ≤ x. If M denotes the set of irreducible integers of R, then it was proved by P. Rémond [12] that, as x →∞, M(x) ∼ Cx(log x)−1(log log x)D(G)−1 ,

The Mazur intersection property for families of closed bounded convex sets in Banach spaces

Abstract: Later, R. R. Phelps [11] provided a dual characterization of this property for finite-dimensional spaces. Nearly two decades later, Phelps’ results were extended by J. R. Giles, D. A. Gregory and B. Sims [9] to general normed linear spaces. They also showed that in dual Banach spaces the MIP implies reflexivity, and considered the weaker property that every weak* compact convex set in a dual space is the intersection of balls (Property weak*-I). Subsequently, there appeared several papers dealing with similar intersection properties for compact convex sets [14, 12], weakly compact convex sets [16] and compact convex sets with finite affine dimension [13]. In the present work, we give a unified treatment of the intersection properties for these diverse classes of sets by considering the MIP for the members of a general family of closed bounded convex sets in a Banach space, and show that all the known results follow as special cases of our result. We also introduce a new condition of separation of convex sets which turns out to be equivalent to the intersection property in all known cases. This strengthens the results of Zizler [15]. As another application of our result, we extend our previous work [1] on lifting the MIP from a Banach space X to the Lebesgue–Bochner space L(μ,X) (1 < p < ∞). We should point out that our proofs are usually modifications, refinements and adaptations to our very general set-up of arguments for particular cases to be found in [9], [12] and [14].

Some Borel measures associated with the generalized Collatz mapping

Abstract: 1. Abstract. This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T : Z → Z, which generalizes the famous 3x + 1 mapping of Collatz. We extended T to a mapping of the polyadic numbers Ẑ and construct finitely many ergodic Borel measures on Ẑ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.

On finite minimal non-p-supersoluble groups

Abstract: If F is a class of groups, then a minimal non-F-group (a dual minimal nonF-group resp.) is a group which is not in F but any of its proper subgroups (factor groups resp.) is in F. In many problems of classification of groups it is sometimes useful to know structure properties of classes of minimal non-F-groups and dual minimal non-F-groups. In fact, the literature on group theory contains many results directed to classify some of the most remarkable among the aforesaid classes. In particular, V. N. Semenchuk in [12] and [13] examined the structure of minimal non-F-groups for F a formation, proving, among other results, that if F is a saturated formation, then the structure of finite soluble, minimal non-F-groups can be determined provided that the structure of finite soluble, minimal non-F-groups with trivial Frattini subgroup is known. In this paper we use this result with regard to the formation of p-supersoluble groups (p prime), starting from the classification of finite soluble, minimal non-p-supersoluble groups with trivial Frattini subgroup given by N. P. Kontorovich and V. P. Nagrebetskĭı ([10]). The second part of this paper deals with non-soluble, minimal non-p-supersoluble finite groups. The problem is reduced to the case of simple groups. We classify the simple, minimal non-p-supersoluble groups, p being the smallest odd prime divisor of the group order, and provide a characterization of minimal simple groups. All the groups considered are finite.

Pseudocompactness - from compactifications to multiplication of borel sets

Abstract: 0. Introduction. All the spaces considered below are assumed to be completely regular and Hausdorff. For a space X, denote by K(X) the family of all compactifications of X; βX stands for the Čech–Stone compactification. If αX ∈ K(X), let Cα(X) stand for the set of those functions f ∈ C∗(X) which are continuously extendable over αX. For f ∈ Cα(X), let f be the continuous extension of f over αX and, for F ⊂ Cα(X), let F = {f : f ∈ F}. Suppose that F ⊂ C∗(X). Define ZF (X) as the family of all sets of the form ⋂∞ i=1 ⋃ni j=1 f −1 i,j ([ai,j ; bi,j ]) where fi,j ∈ F and ai,j ≤ bi,j (ai,j , bi,j ∈ R) for i ∈ N and j = 1, . . . , ni (ni ∈ N). Denote by BF (X) the smallest σ-algebra of subsets of X, containing ZF (X). Let SF (X) stand for the collection of all sets that are obtained from ZF (X) by the Souslin operation (cf. [11]). For αX ∈ K(X), put Zα(X) = ZF (X), Bα(X) = BF (X) and Sα(X) = SF (X) with F = Cα(X). Let E(X) be the family of all F ⊂ C∗(X) such that the diagonal mapping eF = ∆f∈F f is a homeomorphic embedding. If F ∈ E(X), then the closure of eF (X) in R|F | is a compactification of X called generated by F and denoted by eF X. By a slight modification of the proof of Theorem 6 of [13] we get

Some possible covers of measure zero sets

Abstract: n ⋃ k≥n Ik. A partial ordering was introduced in [8] to identify the measure zero sets with similar covering properties, and it was shown that at least four essential differences exist. In [10], it was shown that this number cannot be improved using the usual axioms of set theory (ZFC); more precisely, we showed that there are exactly four classes of measure zero sets assuming u < g, a forcing axiom known to be relatively consistent with ZFC (see [2]); we describe here explicitly (what should have been done in [10]) the covering properties of those four classes. In [4], Borel defined regular measure zero sets in order to extended Weierstrass’ theory of analytic functions. They are equipped with special covers and Borel needed a regular measure zero set with a fast enough rate of convergence for the series of lengths of one of its covers to pursue his theory. In an attempt to understand which measure zero sets could satisfy Borel’s condition, we investigate the rate of convergence of these covers compared with the previous classification; under u < g, both types of covers have the same properties.

A subfield of a complex Banach algebra is not necessarily topologically isomorphic to a subfield of ℂ

Abstract: The classical Mazur–Gelfand theorem ([1]–[5]) implies that any subfield of a complex Banach algebra A is topologically isomorphic to C, provided it is a linear subspace of A. Here we present a somewhat surprising observation that if F is a subfield of A which is just a subring, and not a subalgebra, it need not be topologically isomorphic to a subfield of C. Let A be a complex Banach algebra and let F be a subfield of A. Denote by A0 the smallest closed subalgebra of A containing F . This is a commutative algebra with unit element equal to the unity of F . Thus A0 has a non-zero multiplicative-linear functional mapping isomorphically F into C. Therefore any subfield of A is isomorphic to a subfield of C under a continuous isomorphism. We shall show that in certain cases such an isomorphism cannot be a homeomorphic map. Denote by Q the set of all rational complex numbers, i.e. numbers of the form % = r1 + ir2 with rational r1 and r2. Denote by W the field of all rational functions in a variable t, with coefficients in Q; it contains the subfield of all constant functions, i.e. quotients of elements in Q. This subfield is clearly a dense subset of the complex plane C. Fixing a transcendental number c we obtain an isomorphic imbedding of W into C given by w → w(c), w ∈ W (a function w is uniquely determined by its value w(c) and this value is a well defined complex number, since c is transcendental). One can easily see that each isomorphism h of W into C is of the form w → w̃(d), where d is a transcendental number given by d = h(t), and w̃ is either w or w, depending on whether h(i) = i or h(i) = −i. Here w is an element of W obtained by replacing in w all coefficients by their complex conjugates. Take a complex Banach space X, dim X > 1, and take as A the algebra L(X) of all continuous endomorphisms of X. One can easily see that A contains a non-zero operator T satisfying

Common extension of a family of group-valued, finitely additive measures

Abstract: We deal with the problem of finding common extensions of finitely additive measures (“charges”) taking values in a group G. All groups will be assumed Abelian, and the usual additive notation for Abelian groups will be employed. Let X be a non-empty set and let A be a field of subsets of X. A function μ : A → G is a (G-valued) charge if μ(∅) = 0 and μ(A1 ∪A2) = μ(A1) + μ(A2) whenever A1 and A2 are disjoint sets in A. Now suppose that A and B are fields of subsets of X and that μ : A → G and ν : B → G are G-valued charges. We say that μ and ν are consistent if μ(C) = ν(C) whenever C ∈ A ∩ B. It is natural to ask when two such consistent charges have a common extension, i.e. a charge % such that %(A) = μ(A) if A ∈ A and %(B) = ν(B) if B ∈ B. The charge % is to be defined on A ∨ B, the field generated by A ∪ B. Say that a group G has the 2-extension property if every pair of consistent G-valued charges has a common extension. The following result is to be found in [1] and [3].

Compact 3-manifolds with a flat Carnot-Carathéodory metric

Abstract: § 0. Introduction. A nonintegrable subbundle ∆ of a tangent bundle TM yields an interesting geometry on a manifold M . A Riemannian metric on ∆ enables one to measure the length of differentiable curves tangent to ∆. The metric obtained by minimizing the length of such curves is called the Carnot–Carathéodory metric. The standard example of such a metric arises if we take for ∆ the invariant 2-dimensional subbundle in the tangent bundle of the Heisenberg group generated by two noncentral vectors of its Lie algebra, and then equip ∆ with the invariant Riemannian metric. This metric space, described more precisely in §1, will be denoted by Hc. The aim of this paper is to classify, up to isometry, all 3-dimensional compact manifolds with a Carnot–Carathéodory metric satisfying the property of being locally differentiably isometric to Hc. This property of a Carnot– Carathéodory metric will be called flatness. The above problem is in many aspects similar to the classification problem for compact flat Riemannian manifolds. All classified manifolds have the same universal covering space, isometric to Hc, thus they are quotients of it by a free discrete action of some group of isometries (cf. §2). The Heisenberg group translations of Hc behave like ordinary translations of R, in particular, an analogue of the Bieberbach theorem holds: the holonomy group of a compact manifold with flat Carnot–Carathéodory metric is finite, thus the manifold is finitely covered by a nilmanifold (Theorem 5.3). The Heisenberg group nilmanifolds (an analogue of tori in the Riemannian case) are dealt with in §4, where they are classified up to isometry. In §5 all possible holonomy groups (in the above sense) are determined, and the classification is completed in §6 by considering each holonomy group separately.