Mathematics

Sir David Cox is among the most important statisticians of the past half-century, making pioneering and highly influential contributions to a wide range of topics in statistics and applied probability. This book contains summaries of the invited talks at a meeting held at the University of Neuchâtel in July 2004 to celebrate David Cox’s 80th birthday. The chapters describe current developments across a wide range of topics, ranging from statistical theory and methods, through applied probability and modelling, to applications in areas including finance, epidemiology, hydrology, medicine, and social science. The book contains chapters by numerous well-known statisticians. It provides a summary of current thinking across a wide front by leading statistical thinkers.

Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and inspired generations of students and researchers in mathematics. This book summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles, presented as chapters, contain original research work, set in a broader context by the inclusion of review material.

The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. The book is about these two aspects of sets of natural numbers and about their interplay. For the first aspect, lowness and highness properties of sets are introduced. For the second aspect, firstly randomness of finite objects are studied, and then randomness of sets of natural numbers. A hierarchy of mathematical randomness notions is established. Each notion matches the intuition idea of randomness to some extent. The advantages and drawbacks of notions weaker and stronger than Martin-Löf randomness are discussed. The main topic is the interplay of the computability and randomness aspects. Research on this interplay has advanced rapidly in recent years. One chapter focuses on injury-free solutions to Post's problem. A core chapter contains a comprehensible treatment of lowness properties below the halting problem, and how they relate to K triviality. Each chapter exposes how the complexity properties are related to randomness. The book also contains analogs in the area of higher computability theory of results from the preceding chapters, reflecting very recent research.

How could we use living cells to perform computation? Would our definition of computation change as a consequence of this? Could such a cell-computer outperform digital computers? These are some of the questions that the study of Membrane Computing tries to answer and are at the base of what is treated by this monograph. Descriptional and computational complexity of models in Membrane Computing are the two lines of research on which is the focus here. In this context this book reports the results of only some of the models present in this framework. The models considered here represent a very relevant part of all the models introduced so far in the study of Membrane Computing. They are in between the most studied models in the field and they cover a broad range of features (using symbol objects or string objects, based only on communications, inspired by intra- and intercellular processes, having or not having a tree as underlying structure, etc.) that gives a grasp of the enormous flexibility of this framework. Links with biology and Petri nets are constant through this book. This book aims also to inspire research. This book gives suggestions for research of various levels of difficulty and this book clearly indicates their importance and the relevance of the possible outcomes. Readers new to this field of research will find the provided examples particularly useful in the understanding of the treated topics.

Credit scoring — the quantitative and statistical techniques which assess the credit risks when lending to consumers — has been one of the most successful if unsung applications of mathematics in business for the last fifty years. Now though, credit scoring is beginning to be used in relation to other decisions rather than the traditional one of assessing the default risk of a potential borrower. Lenders are changing their objectives from minimizing defaults to maximizing profits; using the internet and the telephone as application channels means lenders can price or customize their loans for individual consumers. The introduction of the Basel Capital Accord banking regulations and the credit crunch following the problems with securitizing sub prime mortgage mean one needs to be able to extend the default risk models from individual consumer loans to portfolios of such loans. Addressing these challenges requires new models that use credit scores as inputs. These in turn require extensions of what is meant by a credit score. This book reviews the current methodology for building scorecards, clarifies what a credit score really is, and the way that scoring systems are measured. It then looks at the models that can be used to address a number of these new challenges: how to obtain profitability based scoring systems; pricing new loans in a way that reflects their risk and also customise them to attract consumers; how the Basel Accord impacts on way credit scoring; and how credit scoring can be extended to assess the credit risk of portfolios of loans.

This is the second book of a six volume edition of the complete correspondence of one of the leading figures in the scientific revolution of the 17th century, the Oxford mathematician and theologian John Wallis (1616-1703). It covers the period 1660 to September 1668 and thus some of the most decisive years of political and scientific reorganization in England during that century. The volume begins shortly before the restoration of the monarchy in 1660 and witnesses the emergence of the Royal Society from scientific circles, which had existed earlier in London and Oxford. Wallis's involvement in the Royal Society stretches back to its beginnings. After its official establishment, he became one of its most active members, corresponding regularly with its secretary Henry Oldenburg and attending meetings whenever he was in London. Wallis contributed extensively to contemporary scientific debate both in England and on the continent, and many of his letters to Oldenburg on mathematical and physical topics were edited and published in the journal Philosophical Transactions to this purpose. The correspondence contained in the volume, much of which is previously unpublished, throws new light on the background to the scientific revolution and on university politics during this time. As Keeper of the Archives, Wallis was often called upon to prepare papers aimed at defending the University of Oxford's ancient rights and privileges, and was also required to spend a considerable amount of his time in London. To this extent, at least his university commitments and scientific interests were able to go hand-in-hand.

This book is the first of a six volume edition of the complete correspondence of John Wallis (1616-1703). It begins with his earliest known letters written shortly before the outbreak of the first Civil War while he was serving as a private chaplain, and ends on the eve of the restoration of the monarchy in 1660, by which time he was already an established figure within the Republic of Letters. The period covered is thus a momentous one in Wallis's life. It witnesses his election to Savilian professor of geometry at the University of Oxford in 1649 and his subsequent rise to become one of the leading mathematicians of his day, particularly through his introduction of new arithmetical approaches to Cavalieri's method of quadratures. The correspondence reflects the full breadth of his professional activities in theology and mathematics, and provides insights not only into religious debates taking place during the revolutionary years but also into the various questions with which the mathematically-orientated scientific community was concerned. Many of the previously unpublished letters also throw light on University affairs. After his controversial election to the post of Keeper of the Archives in 1658, Wallis fought vigorously to uphold the rights of the University of Oxford whenever necessary, and to prevent as far as possible outside interference from political and religious quarters.

This book is the first book of a series of three that provides an overview of all aspects, steps, and issues that should be considered when undertaking credit risk management, including the Basel II Capital Accord, which all major banks must comply with in 2008. The introduction of the recently suggested Basel II Capital Accord has raised many issues and concerns about how to appropriately manage credit risk. Managing credit risk is one of the next big challenges facing financial institutions. The importance and relevance of efficiently managing credit risk is evident from the huge investments that many financial institutions are making in this area, the booming credit industry in emerging economies (e.g. Brazil, China, India), the many events (courses, seminars, workshops) that are being organised on this topic, and the emergence of new academic journals and magazines in the field (e.g., Journal of Credit Risk,Journal of Risk Model Validation, Journal of Risk Management in Financial Institutions). Financial risk management, an area of increasing importance with the recent Basel II developments, is discussed in terms of practical business impact and the increasing profitability competition, laying the foundation for the other two books in the series.

This book provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property. The finiteness conditions and homological properties are closely interrelated in the sense that either hypothesis induces the other in some form. The main objective behind writing this volume is the absence of a book that contains most of the relevant material on the subject. Since before the last half century, numerous authors including Armendariz, Beidar, Camillo, Chatters, Clark, Cohen, Cozzens, Faith, Farkas, Fisher, Goodearl, Gómez Pardo, Guil Asensio, Hajarnavis, Huynh, Jain, Kohler, Levy, López-Permouth, Mohamed, Ornstein, Osofsky, Singh, Skornyakov, Smith, Tuganbaev, and Wisbauer have investigated rings whose factor rings or factor modules have a finiteness condition or a homological property. They made important contributions leading to new directions and questions that have been listed at the end of each chapter for the benefit of future researchers. The bibliography has more than 200 references and is not claimed to be exhaustive.

During the early part of the last century, F. G. Frobenius raised, in his lectures, the following problem (called the Diophantine Frobenius Problem FP): given relatively prime positive integers a1, . . . , an, find the largest natural number (called the Frobenius number and denoted by g(a1, . . . , an)) that is not representable as a nonnegative integer combination of a1, . . . , an. It turned out that the knowledge of g(a1, . . . , an) has been extremely useful to investigate many different problems. A number of methods, from several areas of mathematics, have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. The main intention of this book is to highlight such ‘methods, ideas, viewpoints, and applications’ for as wide an audience as possible. This book aims to provide a comprehensive exposition of what is known today on FP.