Classical Recursion Theory

When do countries democratize? What facilitates the survival of authoritarian regimes? What determines the occurrence of revolutions, often leading to left-wing dictatorships, such as the Soviet regime? Although a large literature has developed since Aristotle through contemporary political science to answer these questions, we still lack a convincing understanding of the process of political development. Employing analytical tools borrowed from game theory, Carles Boix offers a complete theory of political transitions, in which political regimes ultimately hinge on the nature of economic assets, their distribution among individuals, and the balance of power among different social groups. Backed up by detailed historical work and extensive statistical analysis that goes back to the mid-nineteenth century, this 2003 book explains why democracy emerged in classical Athens. It also discusses the early triumph of democracy in both nineteenth-century agrarian Norway, Switzerland and northea

The focus of this book is on algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exh

Probabilistic risk analysis aims to quantify the risk caused by high technology installations. Increasingly, such analyses are being applied to a wider class of systems in which problems such as lack of data, complexity of the systems, uncertainty about consequences, make a classical statistical analysis difficult or impossible. The authors discuss the fundamental notion of uncertainty, its relationship with probability, and the limits to the quantification of uncertainty. Drawing on extensive experience in the theory and applications of risk analysis, the authors focus on the conceptual and mathematical foundations underlying the quantification, interpretation and management of risk. They cover standard topics as well as important new subjects such as the use of expert judgement and uncertainty propagation. The relationship of risk analysis with decision making is highlighted in chapters on influence diagrams and decision theory. Finally, the difficulties of choosing metrics to quanti

This book describes the research done on the problems of Bénard convection, as well as its modern offspring the Rayleigh–Bénard problem, and Taylor vortices. Bénard convection differs from Rayleigh–Bénard convection by the presence of surface tension, whilst Bénard convection is characterized by parallel rolls. Toroidal vortices characterize Taylor vortex flow. Convection and Taylor vortex flow deal with the consequences of the presence of infinitesimal disturbances in a fluid layer. Both problems are classical examples in the theory of hydrodynamic stability and share many features. Linear theory describing the onset of instability for both problems is practically completed; nonlinear problems have been at the forefront of research during the last 30 years. The impressive progress that has been made in the theoretical and experimental investigation of the nonlinear problems is described and the remaining basic problems are outlined.

These essays explore what happens when a skilful economist makes unconventional assumptions. Economic theory has traditionally relied upon a tacit and 'classical' set of assumptions that have gradually acquired a life of their own in defining how economists write and how they justify economic models. Similarly, these assumptions have acquired an autonomous character: they guide the way economists think about the world. In consequence, consideration of alternative assumptions has become taboo. These essays are substantively and stylistically novel because they break these taboos and bring new assumptions into economic theory. The papers apply this adventurous approach to a wide range of issues - from insurance markets and trade in underdeveloped countries to unemployment and discrimination. Some of the essays derive the implications for economic markets of costly asymmetric information. Others explore the findings of other social sciences such as anthropology, psychology and sociology.

Play is crucial to the learning and development of children in the early years. The third edition of Play in the Early Years is a comprehensive introduction to the theory and practice of play for children from birth to 8 years old. Drawing on classical and contemporary theories, this text examines social, cultural and institutional approaches to play, and explores a range of strategies for successfully integrating play into early years settings and primary classrooms. This edition features a new chapter on conceptual playworlds, which demonstrates what conceptual playworlds look like to infants and toddlers, pre-schoolers, and children transitioning into school. The text features classroom vignettes and photographs designed to help students connect theory to practice, and reflection questions and research activities encourage in-depth reflection and extend learning. Highly regarded by early childhood researchers and practitioners alike, Play in the Early Years remains an essential reso

Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.

Analytical mechanics is a set of mathematical tools used to describe a wide range of physical systems, both in classical mechanics and beyond. It offers a powerful and elegant alternative to Newtonian mechanics; however it can be challenging to learn due to its high degree of mathematical complexity. Designed to offer a more intuitive guide to this abstract topic, this guide explains the mathematical theory underlying analytical mechanics; helping students to formulate, solve and interpret complex problems using these analytical tools. Each chapter begins with an example of a physical system to illustrate the theoretical steps to be developed in that chapter, and ends with a set of exercises to further develop students' understanding. The book presents the fundamentals of the subject in depth before extending the theory to more elaborate systems, and includes a further reading section to ensure that this is an accessible companion to all standard textbooks.

Optimization methods play a central role in financial modeling. This textbook is devoted to explaining how state-of-the-art optimization theory, algorithms, and software can be used to efficiently solve problems in computational finance. It discusses some classical mean–variance portfolio optimization models as well as more modern developments such as models for optimal trade execution and dynamic portfolio allocation with transaction costs and taxes. Chapters discussing the theory and efficient solution methods for the main classes of optimization problems alternate with chapters discussing their use in the modeling and solution of central problems in mathematical finance. This book will be interesting and useful for students, academics, and practitioners with a background in mathematics, operations research, or financial engineering. The second edition includes new examples and exercises as well as a more detailed discussion of mean–variance optimization, multi-period models, and add

Analytical mechanics is a set of mathematical tools used to describe a wide range of physical systems, both in classical mechanics and beyond. It offers a powerful and elegant alternative to Newtonian mechanics; however it can be challenging to learn due to its high degree of mathematical complexity. Designed to offer a more intuitive guide to this abstract topic, this guide explains the mathematical theory underlying analytical mechanics; helping students to formulate, solve and interpret complex problems using these analytical tools. Each chapter begins with an example of a physical system to illustrate the theoretical steps to be developed in that chapter, and ends with a set of exercises to further develop students' understanding. The book presents the fundamentals of the subject in depth before extending the theory to more elaborate systems, and includes a further reading section to ensure that this is an accessible companion to all standard textbooks.

With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York, Stony Brook, the book includes numerous exercises and worked examples, and is ideal for graduate courses on Lie groups and Lie algebras.

The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.

The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.

This book presents a detailed and contemporary account of the classical theory of convergence of semigroups and its more recent development treating the case where the limit semigroup, in contrast to the approximating semigroups, acts merely on a subspace of the original Banach space (this is the case, for example, with singular perturbations). The author demonstrates the far-reaching applications of this theory using real examples from various branches of pure and applied mathematics, with a particular emphasis on mathematical biology. The book may serve as a useful reference, containing a significant number of new results ranging from the analysis of fish populations to signaling pathways in living cells. It comprises many short chapters, which allows readers to pick and choose those topics most relevant to them, and it contains 160 end-of-chapter exercises so that readers can test their understanding of the material as they go along.

The Canonization of Islamic Law tells the story of the birth of classical Islamic law in the eighth and ninth centuries CE. It shows how an oral normative tradition embedded in communal practice was transformed into a systematic legal science defined by hermeneutic analysis of a clearly demarcated scriptural canon. This transformation was inaugurated by the innovative legal theory of Muhammad b. Idrīs al-Shāfi'ī (d. 820 CE), and it took place against the background of a crisis of identity and religious authority in ninth-century Egypt. By tracing the formulation, reception, interpretation and spread of al-Shāfi'ī's ideas, the author demonstrates how the canonization of scripture that lay at the heart of al-Shāfi'ī's theory formed the basis for the emergence of legal hermeneutics, the formation of the Sunni schools of law, and the creation of a shared methodological basis in Muslim thought.

The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in the first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. Volume 1 focuses on the analysis of real-valued functions of a real variable. Volume 2 goes on to consider metric and topological spaces. This third volume develops the classical theory of functions of a complex variable. It carefully establishes the properties of the complex plane, including a proof of the Jordan curve theorem. Lebesgue measure is introduced, and is used as a model for other measure spaces, where the theory of integration is developed. The Radon–Nikodym theorem is proved, and the differentiation of measures discussed.

The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in the first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. Volume 1 focuses on the analysis of real-valued functions of a real variable. Volume 2 goes on to consider metric and topological spaces. This third volume develops the classical theory of functions of a complex variable. It carefully establishes the properties of the complex plane, including a proof of the Jordan curve theorem. Lebesgue measure is introduced, and is used as a model for other measure spaces, where the theory of integration is developed. The Radon–Nikodym theorem is proved, and the differentiation of measures discussed.

The Canonization of Islamic Law tells the story of the birth of classical Islamic law in the eighth and ninth centuries CE. It shows how an oral normative tradition embedded in communal practice was transformed into a systematic legal science defined by hermeneutic analysis of a clearly demarcated scriptural canon. This transformation was inaugurated by the innovative legal theory of Muhammad b. Idrīs al-Shāfi'ī (d. 820 CE), and it took place against the background of a crisis of identity and religious authority in ninth-century Egypt. By tracing the formulation, reception, interpretation and spread of al-Shāfi'ī's ideas, the author demonstrates how the canonization of scripture that lay at the heart of al-Shāfi'ī's theory formed the basis for the emergence of legal hermeneutics, the formation of the Sunni schools of law, and the creation of a shared methodological basis in Muslim thought.

Solving problems is an indispensable exercise for mastering the theory underlying the various branches of geophysics. This book is a collection of nearly 200 problems in geophysics, which are solved in detail showing each step of their solution, the equations used and the assumptions made. Simple figures are also included to help students understand how to reduce a problem to its key elements. The book introduces the equations most commonly used in solving geophysical problems, and presents a series of exercises for the main, classical areas of geophysics – gravity, geomagnetism, seismology, and heat flow and geochronology. Problems range from simple exercises for the most elementary courses to more complex problems suitable for graduate-level students. This handy book is the ideal adjunct to core course textbooks on geophysical theory. It is a convenient source of additional homework and exam questions for instructors, and provides students with a practice or revision aid.

This comprehensive text on entropy covers three major types of dynamics: measure preserving transformations; continuous maps on compact spaces; and operators on function spaces. Part I contains proofs of the Shannon–McMillan–Breiman Theorem, the Ornstein–Weiss Return Time Theorem, the Krieger Generator Theorem and, among the newest developments, the ergodic law of series. In Part II, after an expanded exposition of classical topological entropy, the book addresses symbolic extension entropy. It offers deep insight into the theory of entropy structure and explains the role of zero-dimensional dynamics as a bridge between measurable and topological dynamics. Part III explains how both measure-theoretic and topological entropy can be extended to operators on relevant function spaces. Intuitive explanations, examples, exercises and open problems make this an ideal text for a graduate course on entropy theory. More experienced researchers can also find inspiration for further research.

Frederic William Henry Myers (1843–1901) was a classical scholar who in mid-career turned to the investigation of psychic phenomena. After studying, and later teaching, Classics at Trinity College, Cambridge he resigned his lectureship in 1869, became an inspector of schools, and campaigned for women's higher education. With the encouragement of former colleagues he began a scientific investigation of spiritualism and related phenomena, and in 1882 he helped to found the Society for Psychical Research. This two-volume work, first published posthumously in 1903, contains the fullest statement of Myers' influential theory of the 'subliminal self', which he developed by combining his research into psychic phenomena with his in-depth reading about the latest advances in psychology and related fields. His deeply intellectual approach is evident throughout the book, which analyses a huge amount of interesting data. Volume 1 introduces his theory and discusses dreams and sensory automation.

This 2007 book presents a developed general conceptual and basic quantitative analysis as well as the theory of mechanical efficiency of heat engines that a level of ideality and generality compatible with the treatment given to thermal efficiency in classical thermodynamics. This yields broad bearing results concerning the overall cyclic conversion of heat into usable mechanical energy. The work reveals intrinsic limits on the overall performance of reciprocating heat engines. The theory describes the general effects of parameters such as compression ratio and external or buffer pressure on engine output. It also provides rational explanations of certain operational characteristics such as how engines generally behave when supercharged or pressurized. The results also identify optimum geometric configurations for engines operating in various regimes from isothermal to adiabatic and are extended to cover multi-workspace engines and heat pumps. Limited heat transfer due to finite-time e

This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between ell

Collective decision-making is a familiar feature of our social, political, and economic lives. It ranges from the relatively trivial (e.g. the choice of the next family car) to the globally significant (e.g. whether or not a country should go to war). Yet, whether trivial or globally significant, such decisions involve a number of challenging problems. These problems arise in the standard social choice setting, where individuals differ in their preferences. They also arise in the standard decision-making setting, where individuals share the same preferences, but differ in their decisional capabilities. The distinctive feature of Collective Preference and Choice is that it looks at classical aggregation problems that arise in three closely related areas: social choice theory, voting theory, and group decision-making under uncertainty. Using a series of exercises and examples, the book explains these problems with reference to a number of important contributions to the study of collecti

Collective decision-making is a familiar feature of our social, political, and economic lives. It ranges from the relatively trivial (e.g. the choice of the next family car) to the globally significant (e.g. whether or not a country should go to war). Yet, whether trivial or globally significant, such decisions involve a number of challenging problems. These problems arise in the standard social choice setting, where individuals differ in their preferences. They also arise in the standard decision-making setting, where individuals share the same preferences, but differ in their decisional capabilities. The distinctive feature of Collective Preference and Choice is that it looks at classical aggregation problems that arise in three closely related areas: social choice theory, voting theory, and group decision-making under uncertainty. Using a series of exercises and examples, the book explains these problems with reference to a number of important contributions to the study of collecti

Channel coding lies at the heart of digital communication and data storage, and this detailed introduction describes the core theory as well as decoding algorithms, implementation details, and performance analyses. In this book, Professors Ryan and Lin provide clear information on modern channel codes, including turbo and low-density parity-check (LDPC) codes. They also present detailed coverage of BCH codes, Reed-Solomon codes, convolutional codes, finite geometry codes, and product codes, providing a one-stop resource for both classical and modern coding techniques. Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then extend to advanced topics such as code ensemble performance analyses and algebraic code design. 250 varied and stimulating end-of-chapter problems are also included to test and enhance learning, making this an essential resource for students and practitioners a

This final volume of papers by Lord Rayleigh covers the period from 1911 to his death in 1919. The first of the Solvay Conferences in 1911 played a key role in the foundation of quantum theory. Although invited, Rayleigh did not attend. His principal achievements lay in development and consolidation across classical physics, in which he continued to work. In a 1917 paper, he used electromagnetic theory to derive a formula for expressing the reflection properties from a regularly stratified medium. In 1919, he investigated the iridescent colours of birds and insects. Rayleigh continued his long-standing participation in the Society for Psychical Research, founded in 1882 for the study of 'debatable phenomena'. One of his last publications was his presidential address to that society, which considers several highly unorthodox views and practices. He concludes by asserting the importance to scientists of maintaining open minds in the pursuit of truth.

With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York, Stony Brook, the book includes numerous exercises and worked examples, and is ideal for graduate courses on Lie groups and Lie algebras.

Orthonormal Systems and Banach Space Geometry describes the interplay between orthonormal expansions and Banach space geometry. Using harmonic analysis as a starting platform, classical inequalities and special functions are used to study orthonormal systems leading to an understanding of the advantages of systems consisting of characters on compact Abelian groups. Probabilistic concepts such as random variables and martingales are employed and Ramsey's theorem is used to study the theory of super-reflexivity. The text yields a detailed insight into concepts including type and co-type of Banach spaces, B-convexity, super-reflexivity, the vector-valued Fourier transform, the vector-valued Hilbert transform and the unconditionality property for martingale differences (UMD). A long list of unsolved problems is included as a starting point for research. This book should be accessible to graduate students and researchers with some basic knowledge of Banach space theory, real analysis, proba

This 2007 book presents a developed general conceptual and basic quantitative analysis as well as the theory of mechanical efficiency of heat engines that a level of ideality and generality compatible with the treatment given to thermal efficiency in classical thermodynamics. This yields broad bearing results concerning the overall cyclic conversion of heat into usable mechanical energy. The work reveals intrinsic limits on the overall performance of reciprocating heat engines. The theory describes the general effects of parameters such as compression ratio and external or buffer pressure on engine output. It also provides rational explanations of certain operational characteristics such as how engines generally behave when supercharged or pressurized. The results also identify optimum geometric configurations for engines operating in various regimes from isothermal to adiabatic and are extended to cover multi-workspace engines and heat pumps. Limited heat transfer due to finite-time e

This book, first published in 2007, is an introductory textbook on quantum field theory in gravitational backgrounds intended for undergraduate and beginning graduate students in the fields of theoretical astrophysics, cosmology, particle physics, and string theory. The book covers the basic (but essential) material of quantization of fields in an expanding universe and quantum fluctuations in inflationary spacetime. It also contains a detailed explanation of the Casimir, Unruh, and Hawking effects, and introduces the method of effective action used for calculating the back-reaction of quantum systems on a classical external gravitational field. The broad scope of the material covered will provide the reader with a thorough perspective of the subject. Every major result is derived from first principles and thoroughly explained. The book is self-contained and assumes only a basic knowledge of general relativity. Exercises with detailed solutions are provided throughout the book.

According to Rousseau, the best relationship between unequals is one of 'benificence', giving, receiving and repaying benefits. This 1993 book addresses the problem implicit in his writings of whether it is indeed possible for a just and generous relationship to exist between non-equals. Judith Still draws together issues in Rousseau's work which are often treated in isolation: the state, just relations between individuals, sexual politics and the constructing of a feminine identity. She analyses his works, his classical sources, and the conceptual underpinnings of his ethics, crossing the boundary between study of Rousseau as a complex and sensitive writer of fiction and autobiography and consideration of his political and ethical theory. Using techniques of reading drawn from literary theory, particularly from the work of Derrida, de Man and Starobinski, she argues that for Rousseau it is sexual difference which disturbs the practice of benificence.

This timely volume provides a broad survey of (2+1)-dimensional quantum gravity. It emphasises the 'quantum cosmology' of closed universes and the quantum mechanics of the (2+1)-dimensional black hole. It compares and contrasts a variety of approaches, and examines what they imply for a realistic theory of quantum gravity. General relativity in three spacetime dimensions has become a popular arena in which to explore the ramifications of quantum gravity. As a diffeomorphism-invariant theory of spacetime structure, this model shares many of the conceptual problems of realistic quantum gravity. But it is also simple enough that many programs of quantization can be carried out explicitly. After analysing the space of classical solutions, this book introduces some fifteen approaches to quantum gravity - from canonical quantization in York's 'extrinsic time' to Chern-Simons quantization, from the loop representation to covariant path integration to lattice methods. Relationships among quant

Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book

A systematic and comprehensive introduction to the study of nonlinear dynamical systems, in both discrete and continuous time, for nonmathematical students and researchers working in applied fields. An understanding of linear systems and the classical theory of stability are essential although basic reviews of the relevant material are provided. Further chapters are devoted to the stability of invariant sets, bifurcation theory, chaotic dynamics and the transition to chaos. In the final two chapters the authors approach the subject from a measure-theoretical point of view and compare results to those given for the geometrical or topological approach of the first eight chapters. Includes about one hundred exercises. A Windows-compatible software programme called DMC, provided free of charge through a website dedicated to the book, allows readers to perform numerical and graphical analysis of dynamical systems. Also available on the website are computer exercises and solutions to selecte

A systematic and comprehensive introduction to the study of nonlinear dynamical systems, in both discrete and continuous time, for nonmathematical students and researchers working in applied fields. An understanding of linear systems and the classical theory of stability are essential although basic reviews of the relevant material are provided. Further chapters are devoted to the stability of invariant sets, bifurcation theory, chaotic dynamics and the transition to chaos. In the final two chapters the authors approach the subject from a measure-theoretical point of view and compare results to those given for the geometrical or topological approach of the first eight chapters. Includes about one hundred exercises. A Windows-compatible software programme called DMC, provided free of charge through a website dedicated to the book, allows readers to perform numerical and graphical analysis of dynamical systems. Also available on the website are computer exercises and solutions to selecte

Spectral sequences are among the most elegant, most powerful, and most complicated methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first third of the book treats the algebraic foundations for this sort of homological algebra, starting from informal calculations, to give the novice a familiarity with the range of applications possible. The heart of the book is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Spectral sequences are among the most elegant, most powerful, and most complicated methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first third of the book treats the algebraic foundations for this sort of homological algebra, starting from informal calculations, to give the novice a familiarity with the range of applications possible. The heart of the book is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Most natural and industrial flows are turbulent. The atmosphere and oceans, automobile and aircraft engines, all provide examples of this ubiquitous phenomenon. In recent years, turbulence has become a very lively area of scientific research and application, attracting many newcomers who need a basic introduction to the subject. An Introduction to Turbulent Flow, first published in 2000, offers a solid grounding in the subject of turbulence, developing both physical insight and the mathematical framework needed to express the theory. It begins with a review of the physical nature of turbulence, statistical tools, and space and time scales of turbulence. Basic theory is presented next, illustrated by examples of simple turbulent flows and developed through classical models of jets, wakes, and boundary layers. A deeper understanding of turbulence dynamics is provided by spectral analysis and its applications. The final chapter introduces the numerical simulation of turbulent flows. This

This 1999 book is concerned with Diophantine approximation on smooth manifolds embedded in Euclidean space, and its aim is to develop a coherent body of theory comparable with that which already exists for classical Diophantine approximation. In particular, this book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. After setting out the necessary background material, the authors give a full discussion of Hausdorff dimension and its uses in Diophantine approximation. A wide range of techniques from the number theory arsenal are used to obtain the upper and lower bounds required, and this is an indication of the difficulty of some of the questions considered. The authors go on to consider briefly the p-adic case, and they conclude with a chapter on some applications of metric Diophantine approximation. All researchers with an interest in Diophantine approximation will welcome this book.