The first international workshop on the role and impact of mathematics in medicine: a collective account

The First International Workshop on The Role and Impact of Mathematics in Medicine (RIMM) convened in Paris in June 2010. A broad range of researchers discussed the difficulties, challenges and opportunities faced by those wishing to see mathematical methods contribute to improved medical outcomes. Finding mechanisms for inter- disciplinary meetings, developing a common language, staying focused on the medical problem at hand, deriving realistic mathematical solutions, obtaining

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Adaptation-induced collective dynamics of a single-cell protozoan

We investigate the behavior of a single-cell protozoan in a narrow tubular ring. This environment forces them to swim under a one-dimensional periodic boundary condition. Above a critical density, single-cell protozoa aggregate spontaneously without external stimulation. The high-density zone of swimming cells exhibits a characteristic collective dynamics including translation and boundary fluctuation. We analyzed the velocity distribution and turn rate of swimming cells and found that the regulation of the turing rate leads to a stable aggregation and that acceleration of velocity triggers instability of aggregation. These two opposing effects may help to explain the spontaneous dynamics of collective behavior. We also propose a stochastic model for the mechanism underlying the collective behavior of swimming cells.

Can valuation factors be prescribed? The case of collective enfranchisement and leasehold extension of flats in England and Wales

Recent concerns over the valuation process in collective leasehold enfranchisement and lease extension cases have culminated in new legislation. To underpin this, the Government (Department of Environment Transport and the Regions (DETR)) commissioned new research, which examined whether the valuation of the freehold in such cases could be simplified through the prescription of either yield or marriage value/relativity. This paper, which is based on that research, examines whether it is possible or desirable to prescribe such factors in the valuation process. Market, settlement and Local Valuation Tribunal (LVT) decisions are analysed, and the basis of 'relativity charts' used in practice is critically examined. Ultimately the imperfect nature of the market in freehold investment sales and leasehold vacant possession sales means that recommendations must rest on an analysis of LVT data. New relativity curves are developed from this data and used in conjunction with an alternative approach to valuation yields (based on other investment assets). However, the paper concludes that although the prescription of yields and relativity is possible, it is not fully defensible because of problems in determining risk premia; that the evidential basis for relativity consists of LVT decisions; and that a formula approach would tend to 'lead' the market as a whole.