Let K/Kobe a finitely generated field extension of transcendence degree 1. Let u0 be a valuation of Koand v a valuation of Kextending v0such that the residue field of vis a transcendental extension ofthe residue field k0of vo/such a prolongation vwill be called a residually transcendental prolongation of v0. Byan element with the uniqueness propertyfor (K, v)/(K0, v0) (or more briefly for v/v0)we mean an element / of Khaving u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of vonto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over ko; that is vcoincides with the Gaussian valuation on the subfield K0(t) defined by (ii) vis the only valuation of K (up to equivalence) extending the valuation .

We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relation

with polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

In the paper [3] the following lemma was proved.

Lemma. Let a, b and c be positive integers such that a and be are relatively prime. Then there are infinitely many primes p in the arithmetic progression ax + b (x = 0,1,2,…) such that

A group G is called Dedekindian if every subgroup ofG is normal in G.

The structure of the finite Dedekindian groups is well-known [3, Satz 7.12]. They are either abelian or direct products of the form Q × A × B, where Q is the quaternion group of order 8, Ais abelian of odd order and exp(B) ≤ 2.

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

The introduction of curvature considerations in the past decade into Combinatorial Group Theory has had a profound effect on the study of infinite discrete groups. In particular, the theory of negatively curved groups has enjoyed significant and extensive development since Cannon's seminal study of cocompact hyperbolic groups in the early eighties [7]. Unarguably the greatest influence on the direction of this development has been Gromov's tour de force, his foundational essay in [12] entitled Hyperbolic Groups. Therein Gromov hints at the prospect of developing a corresponding theory of “non-positively curved groups” in his non-definition (Gromov's terminology) of a semihyperbolic group as a group that “looks as if it admits a discrete co-compact isometric action on a space of nonpositive curvature”; [12, p. 81]. Such a development is now occurring and is closely related to the other notable outgrowth of the theory of negatively curved groups, that of automatic groups [10]; we mention here the works [3] and [6] as developments of a theory of nonpositively curved groups along with Chapter 6 of Gromov's more recent treatise [13]. A natural question that serves both to guide and organize the developing theory is: to what extent is the well-developed theory of negatively curved groups reflected in and subsumed under the developing theory of nonpositively curved groups? Our overall interest is in one aspect of this question—namely, as the question relates to the boundaries of groups and spaces: can one define the boundary of a nonpositively curved group intrinsically in a way that generalizes that of negatively curved groups and retains some of their essential features?

An N-tuple ℐ= (T1…, TN) of commuting contractions on a Hilbert space H is said to be a joint isometry if for all x in H, or, equivalently, if Athavale in [1] characterized the joint isometries as subnormal N-tuples whose minimal normal extensions have joint spectra in the unit sphere S2N−X a geometric perspective of this is given in [4]. Subsequently, V. Müller and F.-H. Vasilescu proved that commuting N-tuples which are joint contractions, i.e. , can be represented as restrictions of certain weighted shifts direct sum a joint isometry. In this paper we adapt the canonical models of [3], and also construct a new canonical model, which completes the previous descriptions by showing joint isometries are indeed restrictions of specific multivariable weighted shifts [2].

We show how the solution to certain diophantine equations involving the discriminant of complex quadratic fields leads to the divisibility of the class numbers of the underlying fields. This not only generalizes certain results in the literature such as [2], [4]–[6] but also shows why certain hypotheses made in these results are actually unnecessary since, as our criteria demonstrate, these hypotheses are forced by the solution of the diophantine equations involved. Our methods are based only on the most elementary properties of a principal ideal in a complex quadratic field.

The axiom of comonotonic independence for a preference ordering was introduced by Schmeidler [9]. It leads to the comonotonic additivity for the functional representing the preference ordering, which is necessarily a Choquet integral.

The aim of this paper is to illuminate the concepts of comonotonicity, comonotonic independence and comonotonic additivity. For example the seemingly weaker condition of weak comonotonic independence used by Chateauneuf in [2] is seen to be equivalent to comonotonic independence. Comonotonic additivity is characterized as additivity on chains of sets. From this the characterization of Choquet integrals in [4], [1], [8] follows easily.

We construct a T1-space that is not hereditarily compact, although each of its open sets is the intersection of two compact open sets. The search for such a space was motivated by a problem in the theory of quasi-proximities.

Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of X

where EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extension

where G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order

Let G: Rn → Rn be a continuous mapping such that the origin 0 ∈ Rn is isolated in G-1(0). Then deg0G will denote the local topological degree of G at the origin, i.e. the topological degree of the mapping

where Sr denotes a sphere in Rn centered at the origin with small radius r > 0.

Let be a Banach algebra of bounded linear operators such that contains every operator with finite dimensional range. Then contains every nuclear operator.

An interesting concept of semigroups (and also rings) of (left) quotients, based on the notion of group inverse in a semigroup, was developed by J. B. Fountain, V. Gould and M. Petrich, in a series of papers (see [5]-[12]). Among the most interesting are semigroups having a semigroup of (left) quotients that is a union of groups. Such semigroups have been widely studied. Recall from [3] that a semigroup has a group of left quotients if and only if it is right reversible and cancellative. A more general result was obtained by V. Gould [10]. She proved that a semigroup has a semilattice of groups as its semigroup of left quotients if and only if it is a semilattice of right reversible, cancellative semigroups. This result has been since generalized by A. El-Qallali [4]. He proved that a semigroup has a left regular band of groups as its semigroup of left quotients if and only if it is a left regular band of right reversible, cancellative semigroups. Moreover, he proved that such semigroups can be also characterised as punched spined products of a left regular band and a semilattice of right reversible, cancellative semigroups. If we consider the proofs of their theorems, we will observe that the principal problem treated there can be formulated in the following way: Given a semigroup S that is a band B of right reversible, cancellative semigroups Si, i ε B, to each Si, we can associate its group of left quotients Gi. When is it possible to define a multiplication of such that Q becomes a semigroup having S as its left order, and especially, that Q becomes a band B of groups Gi, i E B?Applying the methods developed in [1] (see also [2]), in the present paper we show how this problem can be solved for Qto become a strong band of groups (that is in fact a band of groups whose idempotents form a subsemigroup, by [16, Theorem 2]. Moreover, we show how Gould's and El-Quallali's constructions of semigroups of left quotients of a semilattice and a left regular band of right reversible, cancellative semigroups, can be simplified.

A Banach space E is said to be regular if every bounded linear operator from E into E′ is weakly compact. This property was studied in [7, 9] under the name Property (w). In [7], using James type spaces as constructed in [4], examples were given of regular Banach spaces which fail to have weakly sequentially complete duals, answering a question raised in [9]. In this paper, we present some more results concerning the regularity of James type spaces.

If Г is a discrete Möbius group acting on the upper half-plane ℋ of the complex plane, the quotient space ℋ/Г is a Riemann surface ℛ and the automorphic functions on Г correspond to meromorphic functions on ℛ. If Г is a nondiscrete Möbius group acting on ℋ, then ℋ/Г is no longer a Riemann surface, and it is obvious that in this case there are no nonconstant automorphic functions on Г. The situation for automorphic forms is quite different. Automorphic forms of integral dimension for a discrete group Г correspond to meromorphic differentials on ℛ, but even if Г is nondiscrete it may still support nontrivial automorphic forms. The problem of classifying those nondiscrete Möbius groups which act on ℋ and which support nonconstant automorphic forms of arbitrary real dimension was raised and solved (rather indirectly) in [2] where, roughly speaking, function-theoretic methods are used to analyse all possible polynomial automorphic forms of integral dimension, and the results obtained then used to analyse the more general situation.

Consider the following hyperlogistic equation

where r, K, τj ∈ (0, ∝), and αj = pj/qj are rational numbers with qj odd, pj and qj, are co-prime, 1 ≤ j ≤ m, and .