Dominions in finitely generated nilpotent groups

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.     

On Rings with Locally Nilpotent Skew Derivations

In this article, we examine algebras with a locally nilpotent q-skew σ-derivation d when there is an element x such that d(x) = 1 and either q is not a root of 1 or q = 1 in characteristic zero. When characteristic p > 0, we also examine the situation where d is an ordinary derivation.     

Rigid Rings and Makar-Limanov Techniques

A ring is rigid if it admits no nonzero locally nilpotent derivation. Although a “generic” ring should be rigid, it is not trivial to show that a ring is rigid. We provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call “parametrization techniques.” and “filtration techniques.” We provide many tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple rings for which the question of rigidity remains unsettled.     

Hessian nilpotent polynomials and the Jacobian conjecture

Let and let be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture: for any homogeneous polynomial of degree , if for all , then when , or equivalently, when . It is also shown in this paper that the condition () above is equivalent to the condition that is Hessian nilpotent, i.e. the Hessian matrix is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.

     

Rank of the fundamental group of any component of a function space

We compute the rank of the fundamental group of any connected component of the space for and connected, nilpotent CW complexes of finite type with finite. For the component corresponding to a general homotopy class , we give a formula directly computable from the Sullivan model for . For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of and . When both and are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of explicitly in terms of the homomorphism induced by on rational cohomology.

     

Use of fluorescence spectroscopy and voltammetry for the analysis of metal-organic matter interactions in the New Caledonia lagoon

Fluorescence, polarographic and potentiometric analysis of sea water from the New Caledonia lagoon (located south of Noumea) allowed the determination of the specific properties of the dissolved and particulate phases of organic matter (OM)-metal complexes according to various regions of the lagoon. In particular, OM complexes with Ni, Zn, Pb, Cu, Cd were chosen in this study due to the sensitivity of these complexes to affect biocenosis of the nearby enclosed coral reef as well as their availability to enter the coast from erosion (terrigenous OM) or human activities from nickel extraction or pollution from waste sites (anthopogenic OM) that exist throughout New Caledonia. Combined with geochemical modelling, the OM-metal complexes analysis allowed the determination of their conditional stability constants which in turn helped in predicting the fate of the metal pollution in the lagoon. For the first time, fluorescence, polarographic and potentiometric techniques combined with geochemical models that employed discrete pKa distribution on OM enabled the determination of the origin of the OM, as either natural or anthopogenic.     

A note on blocks with dihedral defect groups

We prove that the 2-regular subspace of the center of a block with a dihedral defect group is multiplicatively closed. Received: 25 April 2008     

Allow problems concerning spectral properties of sign pattern matrices: A survey

An n×n sign pattern matrix has entries in {+,-,0}. This paper surveys the following problems concerning spectral properties of sign pattern matrices: sign patterns that allow all possible spectra (spectrally arbitrary sign patterns); sign patterns that allow all inertias (inertially arbitrary sign patterns); sign patterns that allow nilpotency (potentially nilpotent sign patterns); and sign patterns that allow stability (potentially stable sign patterns). Relationships between these four classes of sign patterns are given, and several open problems are identified.     

On potentially nilpotent double star sign patterns

A matrix whose entries come from the set {+, −, 0} is called a sign pattern matrix, or sign pattern. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by (m, 2), is introduced. We determine all potentially nilpotent sign patterns in (3, 2) and (5, 2), and prove that one sign pattern in (3, 2) is potentially stable. Supported by youth scientific funds of the Education Department of Jiangxi Province (No. GJJ09460), Natural Science Foundation of Jiangxi Normal University (No.2058), National Natural Science Foundation of China (No.10601001).