Intractability is a growing concern across the cognitive sciences: while many models of cognition can describe and predict human behavior in the lab, it remains unclear how these models can scale to situations of real-world complexity. Cognition and Intractability is the first book to provide an accessible introduction to computational complexity analysis and its application to questions of intractability in cognitive science. Covering both classical and parameterized complexity analysis, it introduces the mathematical concepts and proof techniques that can be used to test one's intuition of (in)tractability. It also describes how these tools can be applied to cognitive modeling to deal with intractability, and its ramifications, in a systematic way. Aimed at students and researchers in philosophy, cognitive neuroscience, psychology, artificial intelligence, and linguistics who want to build a firm understanding of intractability and its implications in their modeling work, it is an ideal resource for teaching or self-study.

This book contains a rigorous coverage of those topics (and only those topics) that, in the author's judgement, are suitable for inclusion in a first course on Complex Functions. Roughly speaking, these can be summarized as being the things that can be done with Cauchy's integral formula and the residue theorem. On the theoretical side, this includes the basic core of the theory of differentiable complex functions, a theory which is unsurpassed in Mathematics for its cohesion, elegance and wealth of surprises. On the practical side, it includes the computational applications of the residue theorem. Some prominence is given to the latter, because for the more sceptical student they provide the justification for inventing the complex numbers. Analytic continuation and Riemann surfaces form an essentially different chapter of Complex Analysis. A proper treatment is far too sophisticated for a first course, and they are therefore excluded. The aim has been to produce the simplest possible rigorous treatment of the topics discussed. For the programme outlined above, it is quite sufficient to prove Cauchy'S integral theorem for paths in star-shaped open sets, so this is done. No form of the Jordan curve theorem is used anywhere in the book.

This book is a continuation of Volume I of the same title [Grund lehren der mathematischen Wissenschaften, Band 115 ]. We constantly 1 1. The textbook Real and cite definitions and results from Volume abstract analysis by E. HEWITT and K. R. STROMBERG [Berlin * Gottin gen *Heidelberg: Springer-Verlag 1965], which appeared between the publication of the two volumes of this work, contains many standard facts from analysis. We use this book as a convenient reference for such facts, and denote it in the text by RAAA. Most readers will have only occasional need actually to read in RAAA. Our goal in this volume is to present the most important parts of harmonic analysis on compact groups and on locally compact Abelian groups. We deal with general locally compact groups only where they are the natural setting for what we are considering, or where one or another group provides a useful counterexample. Readers who are interested only in compact groups may read as follows: 27, Appendix D, 28-30 [omitting subheads (30.6)-(30.60)ifdesired], (31.22)-(31.25), 32, 34-38, 44. Readers who are interested only in locally compact Abelian groups may read as follows: 31-33, 39-42, selected Mis cellaneous Theorems and Examples in 34-38. For all readers, 43 is interesting but optional. Obviously we have not been able to cover all of harmonic analysis.

This book is intended for someone learning functions of a complex variable and who enjoys using MATLAB. It will enhance the exprience of learning complex variable theory and will strengthen the knowledge of someone already trained in ths branch of advanced calculus. ABET, the accrediting board for engineering programs, makes it clear that engineering graduates must be skilled in the art of programming in a language such as MATLAB (R). Supplying students with a bridge between the functions of complex variable theory and MATLAB, this supplemental text enables instructors to easily add a MATLAB component to their complex variables courses. A MATLAB (R) Companion to Complex Variables provides readers with a clear understanding of the utility of MATLAB in complex variable calculus. An ideal adjunct to standard texts on the functions of complex variables, the book allows professors to quickly find and assign MATLAB programming problems that will strengthen students' knowledge of the language and concepts of complex variable theory. The book shows students how MATLAB can be a powerful learning aid in such staples of complex variable theory as conformal mapping, infinite series, contour integration, and Laplace and Fourier transforms. In addition to MATLAB programming problems, the text includes many examples in each chapter along with MATLAB code. Fractals, the most recent interesting topic involving complex variables, demands to be treated with a language such as MATLAB. This book concludes with a Coda, which is devoted entirely to this visually intriguing subject. MATLAB is not without constraints, limitations, irritations, and quirks, and there are subtleties involved in performing the calculus of complex variable theory with this language. Without knowledge of these subtleties, engineers or scientists attempting to use MATLAB for solutions of practical problems in complex variable theory suffer the risk of making major mistakes. This book serves as an early warning system about these pitfalls.

Spin glasses are disordered magnetic systems that have led to the development of mathematical tools with an array of real-world applications, from airline scheduling to neural networks. "Spin Glasses and Complexity" offers the most concise, engaging, and accessible introduction to the subject, fully explaining what spin glasses are, why they are important, and how they are opening up new ways of thinking about complexity.

This one-of-a-kind guide to spin glasses begins by explaining the fundamentals of order and symmetry in condensed matter physics and how spin glasses fit into--and modify--this framework. It then explores how spin-glass concepts and ideas have found applications in areas as diverse as computational complexity, biological and artificial neural networks, protein folding, immune response maturation, combinatorial optimization, and social network modeling.

Providing an essential overview of the history, science, and growing significance of this exciting field, S"pin Glasses and Complexity" also features a forward-looking discussion of what spin glasses may teach us in the future about complex systems. This is a must-have book for students and practitioners in the natural and social sciences, with new material even for the experts.

Presents Real & Complex Analysis Together Using a Unified Approach
A two-semester course in analysis at the advanced undergraduate or first-year graduate level

Unlike other undergraduate-level texts, Real and Complex Analysis develops both the real and complex theory together. It takes a unified, elegant approach to the theory that is consistent with the recommendations of the MAA s 2004 Curriculum Guide.

By presenting real and complex analysis together, the authors illustrate the connections and differences between these two branches of analysis right from the beginning. This combined development also allows for a more streamlined approach to real and complex function theory. Enhanced by more than 1,000 exercises, the text covers all the essential topics usually found in separate treatments of real analysis and complex analysis. Ancillary materials are available on the book s website.

This book offers a unique, comprehensive presentation of both real and complex analysis. Consequently, students will no longer have to use two separate textbooks one for real function theory and one for complex function theory.

What's the point of calculating definite integrals since you can't possibly do them all? What makes doing the specific integrals in this book of value aren't the specific answers we'll obtain, but rather the methods we'll use in obtaining those answers; methods you can use for evaluating the integrals you will encounter in the future. This book, now in its second edition, is written in a light-hearted manner for students who have completed the first year of college or high school AP calculus and have just a bit of exposure to the concept of a differential equation. Every result is fully derived. If you are fascinated by definite integrals, then this is a book for you. New material in the second edition includes 25 new challenge problems and solutions, 25 new worked examples, simplified derivations, and additional historical discussion.

This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations - so-called phase portraits - which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions - even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits.

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based on the authors' extensive teaching experience, it covers topics of keen interest to these students, including ordinary differential equations, as well as Fourier and Laplace transform methods for solving partial differential equations arising in physical applications. Many worked examples, applications, and exercises are included. With this foundation, students can progress beyond the standard course and explore a range of additional topics, including generalized Cauchy theorem, Painleve equations, computational methods, and conformal mapping with circular arcs. Advanced topics are labeled with an asterisk and can be included in the syllabus or form the basis for challenging student projects.

Realism and Complexity in Social Science is an argument for a new approach to investigating the social world, that of complex realism. Complex realism brings together a number of strands of thought, in scientific realism, complexity science, probability theory and social research methodology. It proposes that the reality of the social world is that it is probabilistic, yet there exists enough invariance to make the discovery and explanation of social objects and causal mechanisms possible. This forms the basis for the development of a complex realist foundation for social research, that utilises a number of new and novel approaches to investigation, alongside the more traditional corpus of quantitative and qualitative methods. Research examples are drawn from research in sociology, epidemiology, criminology, social policy and human geography. The book assumes no prior knowledge of realism, probability or complexity and in the early chapters, the reader is introduced to these concepts and the arguments against them. Although the book is grounded in philosophical reasoning, this is in a direct and accessible style that will appeal both to social researchers with a methodological interest and philosophers with an interest in social investigation.

A world model: economies, trade, migration, security and development aid. This bookprovides the analytical capability to understand and explore the dynamics of globalisation. It is anchored in economic input-output models of over 200 countries and their relationships through trade, migration, security and development aid. The tools of complexity science are brought to bear and mathematical and computer models are developed both for the elements and for an integrated whole. Models are developed at a variety of scales ranging from the global and international trade through a European model of inter-sub-regional migration to piracy in the Gulf and the London riots of 2011. The models embrace the changing technology of international shipping, the impacts of migration on economic development along with changing patterns of military expenditure and development aid. A unique contribution is the level of spatial disaggregation which presents each of 200+ countries and their mutual interdependencies along with some finer scale analyses of cities and regions. This is the first global model which offers this depth of detail with fully work-out models, these provide tools for policy making at national, European and global scales. Global dynamics: * Presents in depth models of global dynamics. * Provides a world economic model of 200+ countries and their interactions through trade, migration, security and development aid. * Provides pointers to the deployment of analytical capability through modelling in policy development. * Features a variety of models that constitute a formidable toolkit for analysis and policy development. * Offers a demonstration of the practicalities of complexity science concepts. This book is for practitioners and policy analysts as well as those interested in mathematical model building and complexity science as well as advanced undergraduate and postgraduate level students.

Systems biology came about as growing numbers of engineers and scientists from other fields created algorithms which supported the analysis of biological data in incredible quantities. Whereas biologists of the past had been forced to study one item or aspect at a time, due to technical and biological limitations, it suddenly became possible to study biological phenomena within their natural contexts. This interdisciplinary field offers a holistic approach to interpreting these processes, and has been responsible for some of the most important developments in the science of human health and environmental sustainability. This Very Short Introduction outlines the exciting processes and possibilities in the new field of systems biology. Eberhard O. Voit describes how it enabled us to learn how intricately the expression of every gene is controlled, how signaling systems keep organisms running smoothly, and how complicated even the simplest cells are. He explores what this field is about, why it is needed, and how it will affect our understanding of life, particularly in the areas of personalized medicine, drug development, food and energy production, and sustainable stewardship of our environments. Throughout he considers how new tools are being provided from the fields of mathematics, computer science, engineering, physics, and chemistry to grasp the complexity of the countless interacting processes in cells which would overwhelm the cognitive and analytical capabilities of the human mind. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

The central theme of this reference book is the metric geometry of complex analysis in several variables. Bridging a gap in the current literature, the text focuses on the fine behavior of the Kobayashi metric of complex manifolds and its relationships to dynamical systems, hyperbolicity in the sense of Gromov and operator theory, all very active areas of research. The modern points of view expressed in these notes, collected here for the first time, will be of interest to academics working in the fields of several complex variables and metric geometry. The different topics are treated coherently and include expository presentations of the relevant tools, techniques and objects, which will be particularly useful for graduate and PhD students specializing in the area.

Aimed at graduate students, this textbook provides an accessible and comprehensive introduction to operator theory. Rather than discuss the subject in the abstract, this textbook covers the subject through twenty examples of a wide variety of operators, discussing the norm, spectrum, commutant, invariant subspaces, and interesting properties of each operator. The text is supplemented by over 600 end-of-chapter exercises, designed to help the reader master the topics covered in the chapter, as well as providing an opportunity to further explore the vast operator theory literature. Each chapter also contains well-researched historical facts which place each chapter within the broader context of the development of the field as a whole.

The focus of this book is on open conformal dynamical systems corresponding to the escape of a point through an open Euclidean ball. The ultimate goal is to understand the asymptotic behavior of the escape rate as the radius of the ball tends to zero. In the case of hyperbolic conformal systems this has been addressed by various authors. The conformal maps considered in this book are far more general, and the analysis correspondingly more involved. The asymptotic existence of escape rates is proved and they are calculated in the context of (finite or infinite) countable alphabets, uniformly contracting conformal graph-directed Markov systems, and in particular, conformal countable alphabet iterated function systems. These results have direct applications to interval maps, rational functions and meromorphic maps. Towards this goal the authors develop, on a purely symbolic level, a theory of singular perturbations of Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hoelder continuous summable potentials. This leads to a fairly full account of the structure of the corresponding open dynamical systems and their associated surviving sets.

John J. Benedetto has had a profound influence not only on the direction of harmonic analysis and its applications, but also on the entire community of people involved in the field. The chapters in this volume - compiled on the occasion of his 80th birthday - are written by leading researchers in the field and pay tribute to John's many significant and lasting achievements. Covering a wide range of topics in harmonic analysis and related areas, these chapters are organized into four main parts: harmonic analysis, wavelets and frames, sampling and signal processing, and compressed sensing and optimization. An introductory chapter also provides a brief overview of John's life and mathematical career. This volume will be an excellent reference for graduate students, researchers, and professionals in pure and applied mathematics, engineering, and physics.

This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kahler-Ricci flow and its current state-of-the-art. While several excellent books on Kahler-Einstein geometry are available, there have been no such works on the Kahler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research.
The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman's celebrated proof of the Poincare conjecture. When specialized for Kahler manifolds, it becomes the Kahler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampere equation).
As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kahler-Ricci flow on Kahler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman's ideas: the Kahler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman's surgeries."

Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wealth of connections between rectifiability and other areas of analysis and geometry, including deep links with the calculus of variations and complex and harmonic analysis. This short book provides an easily digestible overview of this wide and active field, including discussions of historical background, the basic theory in Euclidean and non-Euclidean settings, and the appearance of rectifiability in analysis and geometry. The author avoids complicated technical arguments and long proofs, instead giving the reader a flavour of each of the topics in turn while providing full references to the wider literature in an extensive bibliography. It is a perfect introduction to the area for researchers and graduate students, who will find much inspiration for their own research inside.

This second edition presents a collection of exercises on the theory of analytic functions, including completed and detailed solutions. It introduces students to various applications and aspects of the theory of analytic functions not always touched on in a first course, while also addressing topics of interest to electrical engineering students (e.g., the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). It provides examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space), and also includes a section reviewing essential aspects of topology, functional analysis and Lebesgue integration. Benefits of the 2nd edition Rational functions are now covered in a separate chapter. Further, the section on conformal mappings has been expanded.