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.

Using signed digit arithmetic for low-power multiplication

Hardware implementations of arithmetic operators using signed digit arithmetic have lost some of their earlier popularity. However, SD is revisited and used to realise an efficient radix-16 generic multiplier, which has particular potential for low-power implementation. The SD multiplier algorithm reduces the number of partial products to as much as 1/4, and in initial tests reduces the estimated power consumption to only about 50% of that of the Booth multiplier. It is different from other previous high-radix methods in that it employs a novel method to generate its partial products with zero arithmetic logic.

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.