Gröbner basis techniques for certain problems in coding and systems theory

There is much common ground between the areas of coding theory and systems theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time modeling, multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as being generating sets for ideals or modules, Gröbner bases always contain a element which is minimal with respect tot the corresponding term order. Central to this thesis is a general algorithm, valid for any term order, that produces a Gröbner basis for the solution module (or ideal) of elements satisfying a sequence of generalized congruences. These congruences, based on shifts and homomorphisms, are applicable to a wide variety of problems, including key equations and interpolations. At the core of the algorithm is an incremental step. Iterating this step lends a recursive/iterative character to the algorithm. As a consequence, not all of the input to the algorithm need be available from the start and different "paths" can be taken to reach the final solution. The existence of a suitable chain of modules satisfying the criteria of the incremental step is a prerequisite for applying the algorithm.

Planning the deployment of fault-tolerant wireless sensor networks

Since Wireless Sensor Networks (WSNs) are subject to failures, fault-tolerance becomes an important requirement for many WSN applications. Fault-tolerance can be enabled in different areas of WSN design and operation, including the Medium Access Control (MAC) layer and the initial topology design. To be robust to failures, a MAC protocol must be able to adapt to traffic fluctuations and topology dynamics. We design ER-MAC that can switch from energy-efficient operation in normal monitoring to reliable and fast delivery for emergency monitoring, and vice versa. It also can prioritise high priority packets and guarantee fair packet deliveries from all sensor nodes. Topology design supports fault-tolerance by ensuring that there are alternative acceptable routes to data sinks when failures occur. We provide solutions for four topology planning problems: Additional Relay Placement (ARP), Additional Backup Placement (ABP), Multiple Sink Placement (MSP), and Multiple Sink and Relay Placement (MSRP). Our solutions use a local search technique based on Greedy Randomized Adaptive Search Procedures (GRASP). GRASP-ARP deploys relays for (k,l)-sink-connectivity, where each sensor node must have k vertex-disjoint paths of length ≤ l. To count how many disjoint paths a node has, we propose Counting-Paths. GRASP-ABP deploys fewer relays than GRASP-ARP by focusing only on the most important nodes – those whose failure has the worst effect. To identify such nodes, we define Length-constrained Connectivity and Rerouting Centrality (l-CRC). Greedy-MSP and GRASP-MSP place minimal cost sinks to ensure that each sensor node in the network is double-covered, i.e. has two length-bounded paths to two sinks. Greedy-MSRP and GRASP-MSRP deploy sinks and relays with minimal cost to make the network double-covered and non-critical, i.e. all sensor nodes must have length-bounded alternative paths to sinks when an arbitrary sensor node fails. We then evaluate the fault-tolerance of each topology in data gathering simulations using ER-MAC.

Irish neo-paganism: worldview, ritual and identity

Neo-paganism is a vibrant, dynamic global movement, which has had a significant cultural impact. Neo-paganism is an umbrella term for a wide range of spiritual practices, often described as nature- or earth-based spirituality. There are different “paths��� or spiritual traditions within this movement, of which Druidry, Wicca and other forms of Pagan Witchcraft are included in this research. The present work is an ethnographic study of the worldview and ritual practices of the Irish neo-pagan community. It is an enquiry into (a) what characterises the neo-pagan worldview and (b) how this worldview is expressed through ritual behaviour. In order to collect data, the methodology of participant observation and ethnographic interviewing was employed. The thesis comprises a collection of “insider” accounts of what it is like to be a neo-pagan in Ireland and analysis of these narratives, which gives insight into different aspects of neopagan culture. In the discussion, the use of mythology is examined in regard to how mythic narrative is connected to identity formation. Irish cultural symbols are observed as resources utilised in the construction of the movement’s overall character. The interconnectedness of the natural landscape, the numinous and mythology gives rise to creative expression through various forms of neo-pagan artworks, which are discussed herein. The identifying features and key issues of Irish neo-pagan culture are addressed. These key issues are expressed as prominent themes and symbols of their discourse. Neo-pagan dialogue often features discussion of the relationship that this cultural group has with the Irish landscape, history, and indigenous and popular Irish religion. Some of the specific aspects of neo-pagan culture examined are magical worldview, the notion of holism, different types of ritual practices (festivals, life cycle rituals, healing), and material culture. The thesis presents an in-depth analysis of neopagan cultural expressions and their significance as cultural processes