In 1957 Hájek [1] proved that the extreme bilateral derivates of an arbitrary finite real valued function of a real variable, are Borel measurable of class ≦ 2. It was later shown by Staniszewska [3] that Hájek's result is the best possible (even among the class of functions satisfying a Lipschitz condition). Staniszewska exhibited a Eipschitz function whose extreme bilateral derivates are not in Borel class 1. Staniszewska's proof makes use of a result of Zahorski's [4] concerning kernel functions.

Nous nous proposons ici d'étendre certains résultats de la théorie des anneaux commutatifs à deux classes d'anneaux non- commutatifs. Il s'agit, d'une part, des anneaux inversifs à droite, définis par Thierrin [4], et des anneaux réversifs. Les premiers possèdent la propriété xyR = yxR, les seconds la propriété xRy = yRx, ce pour tout x, y ∈ R. Nous considérons aussi la réunion des parties d'un anneau qui possèdent respectivement l'une ou l'autre propriété.

The Bernstein polynomials

(1.1)

and the Bernstein power series

(1.2)

have been the subject of much research (e. g. [1; 2; 3; 6; 7; 8]). It is the purpose of this paper to demonstrate the relationship between these linear operators and certain classes of moment sequences defined below.

A family of sets {Aα} is said by Miller [3] to have property B if there exists a set S which meets all the sets Aα and contains none of them. Property B has been extensively studied in several recent papers (see the references in [2] and the last chapter of P. Erdös and A. Hajnal, On chromatic number of graphs and set systems, Acta. Math. Acad. Sci. Hung. 17 (1966) 61–99).

We prove in this paper that fifteen classes of rings coincide with the class of rings named in the title. One of them is the class of rings R such that X2 = X for each R-ideal X: we shall refer to rings with this property (and thus to the rings of the title) as fully idempotent rings. The simple rings and the (von Neumann) regular rings are fully idempotent. Indeed, every finitely generated right or left ideal of a regular ring is generated by an idempotent [l, p. 42], so that X2 = X holds for every one-sided ideal X.

Let X, Y be locally compact Hausdorff spaces and μ, ν be Radón outer measures on X and Y respectively. The classical product outer measure ϕ on X × Y generated by measurable rectangles, without direct reference to the topology, turns out to have some serious drawbacks. For example, one can only prove that closed sets (and hence Baire sets) are ϕ-measurable. It is unknown, even when X and Y are compact, whether closed sets are ϕ-measurable.

We suppose throughout that p is a non-negative integer, and use the following notations:

where log0x = x for x ≥ e0 = 1, and logn+1x = log(lognx) for x ≥ en+1 = een (n = 0, 1, 2,…);

All modules under consideration are finitely generated left modules. A G-lattice is a G-module which is R-torsion-free.

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1 (C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

We wish to study the problem of finding conditions under which a family of maps from one space into another, with a suitable topology, is compact. Some of the results obtained in this direction are in [1; 2; 3]. We propose to give conditions, to be called uniformly regular and regular (the terminology is motivated by [4]), under which "Ascoli" theorems can be proved. These notions turn out to be equivalent to even continuity of Kelley [1, page 235] under such conditions that all the theorems in the section on even continuity in it still hold when in their statements even continuity is replaced by either uniform regularity or regularity (see Theorem A below).

Let C1 denote the class of all groups with no non-trivial central factors. We prove the following theorem

There exist non-trivial locally solvable C1 groups; but there is no non-trivial locally k-step polynilpotent C1 group for any integer k.

It is well known that a minimal normal subgroup of a locally solvable group is abelian. Thus no non-trivial locally solvable group can be pluperfect - the class of all perfect groups in which every subnormal subgroup is also perfect.

This paper solves a question posed by P. Erdös:

If A1, A2, …, A are distinct subsets of n elements and if |Ai ∩ Ai| ≡ 0 mod 2 (i ≠ j), then

and for each n, there exists a collection of subsets which achieves this bound with equality.

The number of elements in the sum A + B of two complexes A and B of a group G which have multiple representations a + b = a '+ b' has been investigated by Scherk and Kemperman [1]. Kemperman [2] appealed to transfinite techniques (to order G) to prove:

Let A, B, denote two non-void finite complexes (= subsets) of the torsion free abelian group G,

Let d(A),… denote the maximum number of linearly independent elements of A,… and let n = n(A, B) denote the number of elements of A + B whose representation in the form a + b is unique. In the preceding paper, Tarwater and Entringer [1] proved that n ≥ d(A).

The purpose of this paper is two-fold. Sections 2 and 3 are motivated by an observation that certain theorems concerning "diminishing orbital diameters" (introduced in [1]) are true under weaker assumptions. Specifically, we investigate the relationship between that concept and alternate conditions such as "asymptotic regularity", and in the process we sharpen some metric space results established in [1;5]. Mention is made in these sections of examples which show that certain additional weakenings of our hypotheses cannot be made, but we include in detail only the one which seemed to us most intricate.

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.

A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.

Let A be an H-space and K a space. It is well known that [K, A] is a loop. Suppose A has a comultiplication as well, that is, cat A < 2. Then we shall prove that [K, A] is a Moufang loop. This generalises a result of C. W. Norman who proved this for the case where A is the circle, the 3-sphere or the 7-sphere. It also improves the known result that [K, A] is a diassociative loop if A has a comultiplication as well, since Moufang loops are diassociative.

The purpose of this note is to present a short proof for the following theorem.

Let A and B be two complex m x n matrices. If B*A = 0 and AB* = 0 then rank(A + B) = rank(A) + rank(B).

Let A† and B† be the generalized inverses of A and B, respectively, in the sense of Penrose [ 1]. Now,

Let ψ be the outer measure generated by a gauge g and a sequential covering class C of closed sets of a metric space X, and let D be the resulting strong upper density function. That is, for each A ⊆ X and each x ∈ X:

the infimum being taken over all countable subcollections {Iα} of C such that and, for all Iα in the subcollection, d(Iα) < ε where d(Iα) is the diameter of d(Iα).

We prove that the Frattini subgroups are trivial for finite groups whose orders are not divisible by squares of a prime.

If G is a group, we define its Frattini subgroup ϕ(G) as the intersection of all the maximal subgroups of G. An element x ∈ G is a nongenerator of G if whenever G = < X, x >, where X ⊆ G, then G = < X >.