On the polynomial closure in a valued field

Let D be an integral domain with quotient field K. For any subset S of K, the D-polynomial closure of S is the largest subset T of K such that, for every polynomial f in K[X], if f maps S into D then f maps also T into D. When D is not local, the D-polynomial closure is not a topological closure. We prove here that, when D is any rank-one valuation domain, then there exists a topology on K such that the closed subsets for this topology are exactly the D-polynomially closed subsets of K.     

The relative approximation degree in valued function fields

We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author’s proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.     

Example of a constructive separable space with an inseparable completion

An example is constructed of a constructive uniform constructively-space with an enumerable fundamental system of encirclements, whose completion is not even a constructively-space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 207–208, 1976.     

Integer valued polynomials over a number field

A number field is called a Pólya field if the module of integer valued polynomials over that field is generated by (f_i) _i=0 over the ring of integers, with deg(f_i)=i, i=0, 1, 2,... In this paper bounds on the class numbers and on the number of ramified primes in Pólya fields are derived.     

The relative commutant of separable C-algebras of real rank zero

We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C^∗-algebra in its ultrapower depend on the choice of the ultrafilter?     

The relative equilibria of a tethered gyrostat in a central Newtonian field

The motion of an orbital tether system comprising a massive body and a gyrostat of small mass attached to it by a non-extensible weightless tether is examined. The body performs unperturbed motion in a Kepler orbit. There are several different equilibria of the system relative to a uniformly rotating system of coordinates. These equilibria are interpreted geometrically using Mohr circles. Despite being the simplest example of an orbital tether system with a gyrostat, it exhibits a wealth of dynamic properties. There are also more complex orbital tether systems which contain more than one gyrostat [1].     

On a relative uniform completion of an archimedean lattice ordered group

The notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups. Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) of G is dealt with. It is known that exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in . Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

Weak relative pseudo-complements of closure operators

We define the notion of weak relative pseudo-complement on meet semi-lattices, and we show that it is strictly weaker than relative pseudo-complementation, but stronger than pseudo-complementation. Our main result is that if a complete lattice is meet-continuous, then every closure operator on admits weak relative pseudo-complements with respect to continuous closure operators on .Presented by E. T. Schmidt.     

Extension closure of relative syzygy modules

In this paper we introduce the notion of relative syzygy modules.We then study the extensionclosure of the category of modules consisting of relative syzygy modules(resp.relative k-torsionfree modules).     

The support of a uniform semigroup valued measure

In this paper we introduce the notion of orthogonality for arbitrary families of elements of a partial abelian monoid, we study uniform semigroup valued measures on a partial abelian monoid, and we establish interesting results about the existence of the support of such measures. Received July 16, 2005; accepted in final form December 13, 2005.