We speak of rigidity, if partial information about the prime decomposition in an extension of number fields K¦k determines the decomposition law completely (and hence the zeta function ζK), or even fixes the field K itself. Several concepts of rigidity, depending on the degree of information we start from, are introduced and studied. The strongest concept (absolute rigidity) was only known to hold for the ground field and all quadratic extensions. Here a complete list of all Galois quartic extensions which are absolutely rigid is given. For the weaker concept of rigidity, all rigid situations among the fields of degree up to 8 are determined.

Defining a Radon-type integration process we extend the Alexandroff, Fichtengolts-KantorovichHildebrandt and Riesz integral representation theorems in partially ordered vector spaces.

We also identify some classes of operators with other classes of operator-valued set functions, the correspondence between operator and operator-valued set function being given by integration.

All these established results can be immediately applied in C* -algebras (especially in W* -algebras and AW* -algebras of type I), in Jordan algebras, in partially ordered involutory (O*-)algebras, in semifields, in quantum probability theory, as well as in the operator Feynman-Kac formula.

We use a theorem of Loxton and van der Poorten to prove the transcendence of certain real numbers defined by digit patterns. Among the results we prove are the following. If k is an integer at least 2, P is any nonzero pattern of digits base k, and counts the number of occurrences (mod r) of p in the base k representation of n, then is transcendental except when k = 3, P = 1 and r = 2. When (r, k − 1) = 1 the linear span of the numbers has infinite dimension over Q, where P ranges over all patterns base k without leading zeros.

If (G+) is a group and M is a nonempty set of endomorphisms of G operating on the left then G is said to be M-Goldie when (i) G has no infinite independent family of nonzero M-subgroups, and (ii) annihilators in M of subsets of G satisfy the a.c.c. (under set inclusion). Here we prove some results, analogous to those of a Noetherian module in some special cases, even when the set M of operators has no other algebraic structure than the existence of a zero element or in some cases M is at most a finite dimensional commutative near-ring. Precisely speaking, we prove that the collection of associated operating sets of G is finite and there exists a primary decomposition of 0 of such a Goldie M-group, and then if M is a finite dimensional commutative near-ring with unity, for any x belonging to each associated operating set of G, a power of it belongs to the annthilator of G.

We establish a duality between distributive bisemilattices and certain compact left normal bands. The main technique in the proof utilizes the idea of Plonka sums.

A slightly strengthened version of the union-closed sets conjecture is proposed. It is shown that this version holds for a minimum set size of one or two and an examination of a boundary function shows that it holds for collections containing up to 19 sets. Some related conjectures are outlined.

The main object of this paper is to provide the solution of an open problem raised by Professor Ron DeVore concerning constructing interpolating process Hn [f, x] satisfying the inequality (1.11). Results on simultaneous approximation are also obtained.

A topological space X is said to be S-closed if every cover of X by regular closed sets of X has a finite subcover. In this note some characterizations of S-closed Hausdorff spaces are obtained.

Garsia's discovery that functions in the periodic Besov space λ(p-1, p, 1), with 1 < p < ∞, have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in λ(p-1, p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to λ(α, p, q)

Let ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

In 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.

In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.