Alternating sign matrices and polytopes

This thesis deals with two types of mathematical objects: alternating sign matrices and polytopes. Alternating sign matrices were first defined in 1982 by Mills, Robbins and Rumsey. Since then, alternating sign matrices have led to some very captivating research (with multiple open problems still standing), an outline of which is presented in the opening chapter of this thesis. Convex polytopes are extremely relevant when considering enumerations of certain classes of integer valued matrices. An overview of the relevant properties of convex polytopes is presented, before a connection is made between polytopes and alternating sign matrices: the alternating sign matrix polytope. The vertex set of this new polytope is given, as well as a generalization of standard alternating sign matrices to give higher spin alternating sign matrices. From a result of Ehrhart a result concerning the enumeration of these matrices is obtained, namely, that for fixed size and variable line sum the enumeration is given by a particular polynomial. In Chapter 4, we give results concerning the symmetry classes of the alternating sign matrix polytope and in Chapter 3 we study symmetry classes of the Birkhoff polytope. For this classical polytope we give some new results. In the penultimate chapter, another polytope is defined that is a valid solution set of the transportation problem and for which a particular set of parameters gives the alternating sign matrix polytope. Importantly the transportation polytope is a subset of this new polytope

Higher spin alternating sign matrices

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r

Using singular spectrum analysis to obtain staffing level requirements in emergency units

Many operational research (OR) techniques use historical data to populate model input parameters. Although the majority of these models take into account stochastic variation of the inputs, they do not necessarily take into account seasonal variations and other stochastic effects that might arise. One of the major applications of OR lies within healthcare, where ever increasing pressure on healthcare systems is having major implications on those who plan the provision of such services. Coping with growing demand for healthcare, as well as the volatile nature of the number of arrivals at a healthcare facility makes modelling healthcare provision one of the most challenging fields of OR. This paper proposes the use of a relatively modern time series technique, Singular Spectrum Analysis (SSA), to improve existing algorithms that give required staffing levels. The methodology is demonstrated using data from a large teaching hospital's emergency unit. Using time dependent queueing theory, as well as SSA, staffing levels are obtained. The performance of our technique is analysed using a weighted mean square error measure, introduced in this paper

Modelling of psoriasis patient flows for the reconfiguration of secondary care services and treatments

This paper describes work in collaboration with a large dermatology directorate in South Wales to map out current patient flow and activity levels for psoriasis management. Psoriasis is a chronic skin disease which often has a high impact on patient quality of life. Clinical services for patients with moderate to severe psoriasis tend to be located in secondary care hospitals. The range of services that were studied, their geographical location in relation to the distribution of population, and the population demographics in this health board were not unique; similar profiles for these factors can be found throughout the NHS in England and Wales. The model was created to analyse patient flow through different therapies, with the aim of maximising throughput of patients, eliminating bottlenecks, improving patient access to services and improving patient safety. It was shown that reducing waiting times and improving access to phototherapy would lower overall service costs, as fewer patients would subsequently require systemic and biologic therapies. The model has been used to quantify how recent year-on-year increases in overall spend on psoriasis treatments might be slowed and eventually halted. This would require reallocation of notional cost-savings generated by reducing the rate of increase in the drug spend to fund the development of a more balanced and accessible network of more basic psoriasis services

Rostering staff at a mathematics support service using a finite-source queueing model

We study the problem of staffing university mathematics support services (MSSs) in which students drop in to the service (without appointment) for tutoring support. Our approach seeks to find the minimum sufficient number of tutors (with appropriate specialities) to present by hour and day to cover student demand with tolerable delays. We employ traditional operational research techniques to aid managers and administrators of MSSs to roster their services. The machine interference type queue is adopted to model the number of student queries within a mathematics support session. We define and solve an appropriate integer program to roster the number of tutors needed to run the service efficiently

Time-dependent stochastic methods for managing and scheduling Emergency Medical Services

Emergency Medical Services (EMS) are facing increasing pressures in many nations given that demands on the service are rising. This article focuses in particular on the operations of the Welsh Ambulance Service Trust (WAST), which is the only organisation that provides urgent paramedical care services on a day-to-day basis across the whole of Wales. In response to WAST’s aspiration to improve the quality of care it provides, this research investigates several interrelated advanced statistical and operational research (OR) methods, culminating in a suite of decision support tools to aid WAST with capacity planning issues. The developed techniques are integrated in a master workforce capacity planning tool that may be independently operated by WAST planners. By means of incorporating methods that seek to simultaneously better predict future demands, recommend minimum staffing requirements and generate low-cost rosters, the tool ultimately provides planners with an analytical base to effectively deploy resources. Whilst the tool is primarily developed for WAST, the generic nature of the methods considered means they could equally be applied to any service subject to demand that is of an urgent nature, cannot be backlogged, is heavily time-dependent and highly variabl

Differential Responses of Calcifying and Non-Calcifying Epibionts of a Brown Macroalga to Present-Day and Future Upwelling pCO2

Seaweeds are key species of the Baltic Sea benthic ecosystems. They are the substratum of numerous fouling epibionts like bryozoans and tubeworms. Several of these epibionts bear calcified structures and could be impacted by the high pCO2 events of the late summer upwellings in the Baltic nearshores. Those events are expected to increase in strength and duration with global change and ocean acidification. If calcifying epibionts are impacted by transient acidification as driven by upwelling events, their increasing prevalence could cause a shift of the fouling communities toward fleshy species. The aim of the present study was to test the sensitivity of selected seaweed macrofoulers to transient elevation of pCO2 in their natural microenvironment, i.e. the boundary layer covering the thallus surface of brown seaweeds. Fragments of the macroalga Fucus serratus bearing an epibiotic community composed of the calcifiers Spirorbis spirorbis (Annelida) and Electra pilosa (Bryozoa) and the non-calcifier Alcyonidium hirsutum (Bryozoa) were maintained for 30 days under three pCO2 conditions: natural 460±59 µatm, present-day upwelling1193±166 µatm and future upwelling 3150±446 µatm. Only the highest pCO2 caused a significant reduction of growth rates and settlement of S. spirorbis individuals. Additionally, S. spirorbis settled juveniles exhibited enhanced calcification of 40% during daylight hours compared to dark hours, possibly reflecting a day-night alternation of an acidification-modulating effect by algal photosynthesis as opposed to an acidification-enhancing effect of algal respiration. E. pilosa colonies showed significantly increased growth rates at intermediate pCO2 (1193 µatm) but no response to higher pCO2. No effect of acidification on A. hirsutum colonies growth rates was observed. The results suggest a remarkable resistance of the algal macro-epibionts to levels of acidification occurring at present day upwellings in the Baltic. Only extreme future upwelling conditions impacted the tubeworm S. spirorbis, but not the bryozoans

Natural resource exploitation in Western Sahara: new research directions

The authors wish to thank the Leverhulme Trust (through their Early Career scheme) and the Spanish Ministry of Economy, Industry and Competitivity, (CSO2017-86986-P, AEI/FEDER, UE) for funding this research. We would also like to thank Patricia Lalonde for her translations and editorial work, as well as all participants of the 'Analysis of the Management and Exploitation of Natural Resources in Situations of Conflict: The Case of Western Sahara' project (funded by the aforementioned ministry), for their constructive comments on earlier iterations of this paper.This review article provides an overview of research to date with an explicit focus on natural resource exploitation in Western Sahara. It integrates findings from various perspectives and disciplines, and synthesises the research done with a view to revealing gaps and, therefore, potential new research directions. As the issue of natural resource exploitation in Western Sahara has been conceptualised in very different ways and from the perspectives of a variety of disciplines, the authors have opted for a semi-systematic review of the work done encompassing academic, non-academic, and activist backgrounds.Leverhulme TrustSpanish Ministry of Economy, Industry and Competitivity (AEI/FEDER, UE) CSO2017-86986-

Function and regulation of phospholipid signalling in plants

As an important metabolic pathway, phosphatidylinositol metabolism generates both constitutive and signalling molecules that are crucial for plant growth and development. Recent studies using genetic and molecular approaches reveal the important roles of phospholipid molecules and signalling in multiple processes of higher plants, including root growth, pollen and vascular development, hormone effects and cell responses to environmental stimuli plants. The present review summarizes the current progress in our understanding of the functional mechanism of phospholipid signalling, with an emphasis on the regulation of Ins(1,4,5)P3–Ca2+ oscillation, the second messenger molecule phosphatidic acid and the cytoskeleton