SK1-like functors for division algebras

We investigate the group valued functor G(D) = D*/F*D' where D is a division algebra with center F and D' the commutator subgroup of D*. We show that G has the most important functorial properties of the reduced Whitehead group SK1. We then establish a fundamental connection between this group, its residue version, and relative value group when D is a Henselian division algebra. The structure of G(D) turns out to carry significant information about the arithmetic of D. Along these lines, we employ G(D) to compute the group SK1(D). As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method.

Dales, long lands, and the medieval division of land in eastern England

The long, parallel fields of the marshlands between the Fens and the Humber estuary in eastern England, which are recorded on nineteenth-century maps, were the result of the division of the wetlands that occurred particularly during the twelfth and early thirteenth centuries. Areas of common fen pasture were partitioned between tenants to provide land for grazing and arable. Similar division also took place on the coastal strip and in the peat fen for land for salt-making and cutting fuel. These long strips, known as dales, are compared to similar areas in open fields in parts of Yorkshire and Northamptonshire, which have been discussed elsewhere. It is argued that the field shape is the result of a type of division in eastern England in which considerable emphasis was placed on case of partitioning land equitably.

On maximal subgroups of the multiplicative group of a division algebra

This note studies the question whether a multiplicative group of a division ring has a maximal subgroup. It is published in J. Algebra. This is a reputable journal in the subject algebra. Most of submitted papers from 5* schools in RAE was in this journal.

A distributed contention-vector division multiple access (D-CVDMA) protocol for wireless networks

Traditional Time Division Multiple Access (TDMA) protocol provides deterministic periodic collision free data transmissions. However, TDMA lacks flexibility and exhibits low efficiency in dynamic environments such as wireless LANs. On the other hand contention-based MAC protocols such as the IEEE 802.11 DCF are adaptive to network dynamics but are generally inefficient in heavily loaded or large networks. To take advantage of the both types of protocols, a D-CVDMA protocol is proposed. It is based on the k-round elimination contention (k-EC) scheme, which provides fast contention resolution for Wireless LANs. D-CVDMA uses a contention mechanism to achieve TDMA-like collision-free data transmissions, which does not need to reserve time slots for forthcoming transmissions. These features make the D-CVDMA robust and adaptive to network dynamics such as node leaving and joining, changes in packet size and arrival rate, which in turn make it suitable for the delivery of hybrid traffic including multimedia and data content. Analyses and simulations demonstrate that D-CVDMA outperforms the IEEE 802.11 DCF and k-EC in terms of network throughput, delay, jitter, and fairness.

Promoting mental health in Northern Ireland: addressing division, inequality and stigma

This article provides an overview of the literature on the impact of ‘the Troubles’ on mental health in Nor thern Ireland. It identifies three main phases of professional and policy response from concerns about the effects of the violence in the early 1970s, through many years of collective denial and neglect, until acknowledgment, following the Good Friday Agreement in 1998 (Nor thern Ireland Office, 1998), of high levels of trauma and unmet need. The issues of inequality and stigma are also considered and it is argued that peace is necessary but insufficient for promoting mental health. The development of mental health services in Nor thern Ireland and the relatively recent focus on promoting mental health are also outlined and examined. It is suggested that attempts to address the needs arising as a result of ‘the Troubles’ and more general mental health promotion strategies have, to some extent, developed in parallel and that it may be impor tant to integrate these effor ts. The relative under-development of mental health services, the comprehensive Bamford Review of Mental Health and Learning Disability (2005; 2006) and the positive approach of the Public Health Agency mean that, even in the current economic climate, there are great oppor tunities for progress. Routine screening, in primary care and mental health services for trauma, including Troubles-related trauma, is recommended to identify and address these issues on an individual level. It is also argued, however, that more substantial political change is needed to effectively address societal division, inequality and stigma to the benefit of all.

Sexual division of antibacterial resource defence in breeding burying beetles, Nicrophorus vespilloides

A key component of parental care involves defending resources destined for offspring from a diverse array of potential interspecific competitors, such as social parasites, fungi and bacteria. 2. Just as with other aspects of parental care, such as offspring provisioning or brood defence, sexual conflict between parents may arise over how to share the costs of this form of care. There has been little previous work, however, to investigate how this particular burden might be shared. 3. Here, we describe a hitherto uncharacterized form of parental care in burying beetles Nicrophorus vespilloides, a species which prepares carrion for its young and faces competition from microbes for this resource. We found that parents defend the carcass with antibacterial anal exudates, and that the antibacterial activity of these exudates is only upregulated following the discovery of a corpse. At the same time, phenoloxidase activity in the anal exudates is downregulated, indicating parallels with the internal insect immune system. 4. In unmanipulated breeding pairs, females had higher antibacterial activity in their anal exudates than males, suggesting sex-specific roles in this aspect of parental care. 5. When we experimentally widowed males, we found that they increased levels of antibacterial activity in their anal exudates. Experimentally widowing females, however, led them to decrease levels of antibacterial activity in their anal exudates. Widowed beetles of each sex thus produced anal exudates of comparable antibacterial activity. We suggest that this flexible division of antibacterial activity may be coordinated by Juvenile Hormone. © 2009 British Ecological Society.

SK1 of graded division algebras

The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.