Volume 19, Issue 2

Let α 1 , … , α n be non-zero algebraic numbers and K be a number field containing α 1 , … , α n . Denote by 𝔭 a prime ideal of the ring of integers in K . We present completely explicit upper bounds for , where with b 1 , … , b n being rational integers and Ξ ≠ 1. The cost n!/2 n −1 of the classical Kummer descent has been removed from the previous best upper bounds.

Let be a regular covering over a finite polyhedron with free abelian group of covering translations. Each nonzero cohomology class ξ ∈ H 1 ( X ; R ) with q *ξ = 0 determines a notion of “ infinity ” of the noncompact space . In this paper we characterize homology classes z in which can be realized in arbitrary small neighborhoods of infinity in . This problem was motivated by applications in the theory of critical points of closed 1-forms initiated in [Farber M.: Zeros of closed 1-forms, homoclinic orbits and Lusternik-Schnirelman theory. Topol. Methods Nonlinear Anal. 19 (2002), 123–152], [Farber M.: Lusternik-Schnirelman theory and dynamics. Lusternik-Schnirelmann Category and Related Topics. Contemporary Mathematics 316 (2002), 95–111].

Radical locally finite groups with min- p for all primes p in which every descendant subgroup is normal are studied in the paper. It turns out that these groups are precisely T -groups, that is, groups whose subnormal subgroups are normal.

Let R be an associative unital ring, T a left R -module and λ an infinite cardinal. We consider the class ⊥ T of all left R -modules M satisfying Ext R 1 ( M , T ) = 0 and search for λ-universal objects in suitable subclasses ℭ of ⊥ T . Here, an R -module U ∈ ℭ is λ-universal for T if | U | ≤ λ and every R -module M ∈ ℭ of cardinality less than or equal to λ embeds into U . We show that the existence of | T |-universal objects which are strong splitters implies the existence of λ-universal objects for sufficiently large λ if we assume ( V = L ). We then apply our results to module classes over small Dedekind domains to partially solve a generalized problem by Kulikov.

For multiparameter bilinear paraproduct operators B we prove the estimate Here, 1/ p + 1/ q = 1/ r and special attention is paid to the case of 0 < r < 1. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele [Mucalu Camil, Pipher Jill, Tao Terrance, and Thiele Christoph: Bi-parameter paraproducts. Acta Math. 193 (2004), 269–296, Mucalu Camil, Pipher Jill, Tao Terrance, and Thiele Christoph: Multi-parameter paraproducts. arxiv:math.CA/0411607]. We offer a different proof of these inequalities.

Given a pair of a partial abelian monoid M and a pointed space X , let C M (ℝ ∞ , X ) denote the configuration space of finite distinct points in ℝ ∞ parametrized by the partial monoid X ∧ M . In this note we will show that if M is embedded in a topological abelian group and if we put ± M = { a − b | a , b ∈ M } then the natural map C M (ℝ ∞ , X ) → C ± M (ℝ ∞ , X ) induced by the inclusion M ⊂ ± M is a group completion. This result can be applied to show that for any finite set M such that {0} ⊊ M ⊂ ℤ, C M (ℝ ∞ , X ) is weakly equivalent to the infinite loop space Ω ∞ Σ ∞ X if X is connected.

Let R be a ring, let F be a free group, and let X be a basis of F . Let ε : RF → R denote the usual augmentation map for the group ring RF , let X ∂ := { x − 1 | x ∈ X } ⊆ RF , let Σ denote the set of matrices over RF that are sent to invertible matrices by ε, and let ( RF )Σ −1 denote the universal localization of RF at Σ. A classic result of Magnus and Fox gives an embedding of RF in the power-series ring R 〈〈 X ∂〉〉. We show that if R is a commutative Bezout domain, then the division closure of the image of RF in R 〈〈 X ∂〉〉 is a universal localization of RF at Σ. We also show that if R is a von Neumann regular ring or a commutative Bezout domain, then ( RF )Σ −1 is stably flat as an RF -ring, in the sense of Neeman-Ranicki.