Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.

Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.

In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

Soient K un corps local, sa clôture algébrique, R, L, M trois corps tels K ⊂ R ⊂ L ⊂ et K ⊂ R ⊂ M ⊂ . Supposons que L et M soient linéairement disjoints sur R. Quel est le complété du corps LM (compositum de L et M) pour une norme de R-algèbre qui coïncide avec la valeur absolue sur L ∪ M? On sait que LM est isomorphe en tant que R-algèbre au produit tensoriel L ⨂RM et il est facile de montrer que la valeur absolue est la plus petite norme du type considéré ci-dessus et que la norme tensorielle en est la plus grande. C'est pourquoi nous étudions d'abord le complété de L ⊗RM pour la norme tensorielle. Citons dans ce cas les résultats essentiels lorsque la caractéristique de K est nulle.

1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius

1

of a Hilbert space operator T. Among the many interesting developments, we may mention:

(a) C. Berger's proof of the “power inequality”

2

(b) R. Bouldin's result that

3

for any isometry V commuting with T;

(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;

(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

In this paper we study a question which, although somewhat special, has the virtue that its answer can be given in a very precise, definitive, and succinct way. It shows that the structure of a ring is very tightly determined by the imposition of a special behavior on one of its derivations.

The problem which we shall examine is: Suppose that R is a ring with unit element, 1, and that d ≠ 0 is a derivation of R such that for every x ∊ R, d(x) = 0 or d(x) is invertible in R; must R then have a very special structure?

As we shall see, the answer to this question is yes, in particular we show that except for a special case which occurs when 2R = 0, R must be a division ring D or the ring D2 of 2 × 2 matrices over a division ring.

The notion of a prime tangle is introduced by Kirby and Lickorish [7]. It is related deeply to the notion of a prime knot by the following result in [8]: summing together two prime tangles gives always a prime knot.

The purpose of this paper is to exploit this above mentioned result of Lickorish in creating or detecting prime knots which satisfy certain properties. First, we shall express certain knots (two-bridge knots and Terasaka slice knots [14]) as a sum of a prime tangle and an untangle (the existence of such a sum is proven to every knot in [7] and is not unique) in a natural way (natural means here depending on certain specific geometrical characters of the class of knots). Second, every Alexander polynomial (or Conway polynomial) is shown to be realized by a prime algebraic knot (algebraic in the sense of Conway [3], Bonahon-Siebenmann [2]) which can be expressed as the sum of two prime algebraic tangles.

In a paper published in 1975 [1, § 3], D. S. Mitrinovič and P. M. Vasič used the so-called “centroid method” to obtain two new inequalities which are complementary to (the discrete version of) Jensen's inequality for convex functions. In this paper we shall present a very general version of such inequalities using the same geometric ideas used in [1] but not using the centroid method itself. At the same time we shall extend the domain of the inequalities (even in the discrete case), and clarify the value of the constant (λ) appearing in the inequality. We give applications of the theorems to some general means and also to the classical means. Our first result is given as

THEOREM 1. Let v be a nonnegative measure on a σ-algebra of subsets of a set D and let q, f be real v-measurable functions on D such that q(x) > 0, –∞ < x1 ≦ f(x) ≦ x2 < ∞ for all x ∊ D and ∫D qdv = 1.

Our aim is to study the mixing sequences of a weak mixing transformation. An ergodic measure preserving transformation is weak mixing if and only if for each pair of sets there exists a sequence of density one on which the transformation mixes the sets [9]. An unpublished result of S. Kakutani implies there actually exists a single sequence of density one on which the transformation is mixing for all sets (see Section 3). This result motivated the general définition of a transformation being mixing on a sequence, as well as mixing of higher order on a sequence. Given a weak mixing transformation, there exist sequences along which it is mixing of all degrees. In particular, this is the case for an eventually independent sequence [7].

In Section 3 it will be shown that if T is weak mixing but not mixing, then a sequence on which T is two-mixing must have upper density zero.

M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.

The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.

Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.

We assume throughout this paper that A is a semi-simple, quasi-central, complex Banach algebra with a bounded approximate identity {eα}. The author [6] has shown that every central double centralizer T on A can be, under suitable conditions, represented as a bounded continuous complex-valued function ΦT on Prim A, the structure space of A with the hull-kernel topology, such that

Here x + P for P ∊ Prim A denotes the canonical image of x in A/P. This map Φ is called Dixmier's representation of Z(M(A)), the central double centralizer algebra of A. We denote by τ the canonical isomorphism of A into the Banach algebra D(A) with the restricted Arens product as defined in [6]. Also denote by μ Davenport's representation of Z(M(A)). In fact, this map μ is given by

for each T ∊ Z(M(A)).