Evolution PDEs and augmented eigenfunctions. half line.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

Dying young: a palaeopathological analysis of child health in Roman Britain

Children represent the most vulnerable members of society, and as such provide valuable insight into past lifeways. Adverse environmental conditions translate more readily into the osteological record of children, making them primary evidence for the investigation of ill-health in the past. To date, most information on growing up in Roman Britain has been based on the Classical literature, or discussed in palaeopathological studies with a regional focus, e.g. Dorset or Durnovaria. Thus, the lifestyles and everyday realities of children throughout Britannia remained largely unknown. This study sets out to fill this gap by providing the first large scale analysis of Romano-British children from town and country. The palaeopathological analysis of 1643 non-adult (0-17 years) skeletons, compiled from the literature (N=690) and primary osteological analysis (N=953), from 27 urban and rural settlements has highlighted diverse patterns in non-adult mortality and morbidity. The distribution of ages-at-death suggest that older children and adolescents migrated from country to town, possibly for commencing their working lives. True prevalence rates suggest that caries (1.8%) and enamel hypoplasia (11.4%) were more common in children from major urban towns, whereas children in the countryside displayed higher frequencies of scurvy (6.9%), cribra orbitalia (27.7%), porotic hyperostosis (6.2%) and endocranial lesions (10.9%). Social inequality in late Roman Britain may have been the driving force behind these urban-rural dichotomies. The results may point to exploitation of the peasantry on the one hand, and higher status of the urban population as a more ‘Romanised’ group on the other. Comparison with Iron Age and post-medieval non-adults also demonstrated a decline in health in the Roman period, with some levels of ill-health, particularly in the rural children, similar to those from post-medieval London. This research provides the most comprehensive study of non-adult morbidity and mortality in Roman Britain to date. It has provided new insights into Romano-British lifeways and presents suggestions for further work.

A comparison between classical techniques and eigenstructure assignment in the design of a longitudinal instrument landing system

Although modern control techniques such as eigenstructure assignment have been given extensive coverage in control literature there is a reluctance to use them in practice as they are often not believed to be as `visible' or as simple as classical methods. A simple aircraft example is used, and it is shown that eigenstructure assignment can be used easily to produce a more viable controller than with simple classical techniques.