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John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry–Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, the book first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. The book achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. It then describes the whole theory and its subsequent developments and applications in their full generality.

This book introduces a new theoretical entity: Agent_Zero. This software individual, or “agent,” is endowed with distinct emotional/affective, cognitive/deliberative, and social modules. Grounded in contemporary neuroscience, these internal components interact to generate observed, often far-from-rational, individual behavior. When multiple agents of this new type move and interact spatially, they collectively generate an astonishing range of dynamics spanning the fields of social conflict, psychology, public health, law, network science, and economics. The book weaves a computational tapestry with threads from Plato, David Hume, Charles Darwin, Ivan Pavlov, Adam Smith, Leo Tolstoy, Karl Marx, William James, and Fyodor Dostoevsky, among others. This transformative synthesis of social philosophy, cognitive neuroscience, and agent-based modeling will fascinate scholars and students of every stripe. Computer programs are provided in the book or available online. This book is a signal departure in what it includes (e.g., a new synthesis of neurally grounded internal modules), what it eschews (e.g., standard behavioral imitation), the phenomena it generates (from genocide to financial panic), and the modeling arsenal it offers the scientific community. For generative social science, this book presents a ground-breaking vision and the tools to realize it.

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.

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.

This monograph presents a mathematical theory of concentration inequalities for functions of independent random variables. The basic phenomenon under investigation is that if a function of many independent random variables does not depend too much on any of them then it is concentrated around its expected value. This book offers a host of inequalities to quantify this statement. The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities, transportation arguments, to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are presented. The book offers a self-contained introduction to concentration inequalities, including a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed. Special attention is paid to applications to the supremum of empirical processes.

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 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.

Actuator and sensor delays are among the most common dynamic phenomena in engineering practice, and when disregarded, they render controlled systems unstable. Over the past sixty years, predictor feedback has been a key tool for compensating such delays, but conventional predictor feedback algorithms assume that the delays and other parameters of a given system are known. When incorrect parameter values are used in the predictor, the resulting controller may be as destabilizing as without the delay compensation. This book develops adaptive predictor feedback algorithms equipped with online estimators of unknown delays and other parameters. Such estimators are designed as nonlinear differential equations, which dynamically adjust the parameters of the predictor. The design and analysis of the adaptive predictors involves a Lyapunov stability study of systems whose dimension is infinite, because of the delays, and nonlinear, because of the parameter estimators. This book solves adaptive delay compensation problems for systems with single and multiple inputs/outputs, unknown and distinct delays in different input channels, unknown delay kernels, unknown plant parameters, unmeasurable finite-dimensional plant states, and unmeasurable infinite-dimensional actuator states. Presenting breakthroughs in adaptive control and control of delay systems, the book offers powerful new tools for the control engineer and the mathematician.

The book presents and develops the most recent ideas and concepts of the mathematical theory of viscous, compressible, and heat conducting fluids. Two main objectives are pursued: (i) global existence theory within the framework of variational solutions for the full Navier-Stokes-Fourier system supplemented with large data, and (ii) optimal existence results for barotropic flows with respect to the available a priori estimates.

This book is devoted to the mathematical modelling of electromagnetic materials. Electromagnetism in matter is developed with particular emphasis on material effects, which are ascribed to memory in time and nonlocality. Within the mathematical modelling, thermodynamics of continuous media plays a central role in that it places significant restrictions on the constitutive equations. Further, as shown in connection with uniqueness, existence and stability, variational settings, and wave propagation, a correct formulation of the pertinent problems is based on the knowledge of the thermodynamic restrictions for the material. The book is divided into four parts. Part I (chapters 1 to 4) reviews the basic concepts of electromagnetism, starting from the integral form of Maxwell’s equations and then addressing attention to the physical motivation for materials with memory. Part II (chapers 5 to 9) deals with thermodynamics of systems with memory and applications to evolution and initial/boundary-value problems. It contains developments and results which are unusual in textbooks on electromagnetism and arise from the research literature, mainly post-1960s. Part III (chapters 10 to 12) outlines some topics of materials modelling — nonlinearity, nonlocality, superconductivity, and magnetic hysteresis — which are of great interest both in mathematics and in applications.

Cryptography is a vital technology that underpins the security of information in computer networks. This book presents an introduction to the role that cryptography plays in providing information security for technologies such as the Internet, mobile phones, payment cards, and wireless local area networks. Focusing on the fundamental principles that ground modern cryptography as they arise in modern applications, it avoids both an over-reliance on transient current technologies and over-whelming theoretical research. A short appendix is included for those looking for a deeper appreciation of some of the concepts involved. By the end of this book, the reader will not only be able to understand the practical issues concerned with the deployment of cryptographic mechanisms, including the management of cryptographic keys, but will also be able to interpret future developments in this increasingly important area of technology.

This book is devoted to problems of shape identification in the context of (inverse) scattering problems and problems of impedance tomography. In contrast to traditional methods which are based on iterative schemes of solving sequences of corresponding direct problems, this book presents a completely different method. The Factorization Method avoids the need to solve the (time consuming) direct problems. Furthermore, no a-priori information about the type of scatterer (penetrable or impenetrable), type of boundary condition, or number of components is needed. The Factorization Method can be considered as an example of a Sampling Method. The book aims to construct a binary criterium on the known data to decide whether or not a given point z is inside or outside the unknown domain D. By choosing a grid of sampling points z in a region known to contain D, the characteristic function of D can be computed (in the case of finite data only approximately). The book also introduces some alternative Sampling Methods.

Graph theory goes back several centuries and revolves around the study of graphs—mathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematics—and some of its most famous problems. This book explores the questions and puzzles that have been studied, and often solved, through graph theory. It looks at graph theory's development and the vibrant individuals responsible for the field's growth. Introducing fundamental concepts, the book explores a diverse plethora of classic problems such as the Lights Out Puzzle, and each chapter contains math exercises for readers to savor. An eye-opening journey into the world of graphs, the book offers exciting problem-solving possibilities for mathematics and beyond.

This book is suitable for a first-year graduate course on Non-linear Elasticity Theory. It is aimed at graduate students, post-doctoral fellows and researchers working in Mechanics. Included is a modern treatment of elementary plasticity theory emphasizing the foundational role played by finite elasticity. The book covers fundamental and advanced material that should be mastered before embarking on research. Included are the concepts of frame invariance, material symmetry, kinematic constraints, a development of nonlinear membrane theory, energy minimizers as stable equilibria and various attendant convexity conditions.

The verification of control system software is critical to a host of technologies and industries, from aeronautics and medical technology to the cars we drive. The failure of controller software can cost people their lives. This book provides control engineers and computer scientists with an introduction to the formal techniques for analyzing and verifying this important class of software. Too often, control engineers are unaware of the issues surrounding the verification of software, while computer scientists tend to be unfamiliar with the specificities of controller software. The book provides a unified approach that is geared to graduate students in both fields, covering formal verification methods as well as the design and verification of controllers. It presents a wealth of new verification techniques for performing exhaustive analysis of controller software. These include new means to compute nonlinear invariants, the use of convex optimization tools, and methods for dealing with numerical imprecisions such as floating point computations occurring in the analyzed software. As the autonomy of critical systems continues to increase—as evidenced by autonomous cars, drones, and satellites and landers—the numerical functions in these systems are growing ever more advanced. The techniques presented here are essential to support the formal analysis of the controller software being used in these new and emerging technologies.

The main purpose of this book is to demonstrate, by way of example, the several advantages of using a formal game-theoretic framework to explain complex events, diplomatic history, and contentious interstate relationships, via causal mechanisms and rationality. Chapter 1 lays out the broad parameters and major concepts of the mathematical theory of games and its applications in the security studies literature. Chapter 2 explores a number of issues connected with the use of game-theoretic models to organize analytic narratives, both generally and specifically. Chapter 3 interprets the Moroccan crisis of 1905–6 in the context of an incomplete information game model. Chapter 4 surveys and evaluates several prominent attempts to use game theory to explain the strategic dynamic of the Cuban missile crisis of 1962. Chapter 5 offers a general explanation that answers all of the foundational questions associated with the Cuban crisis within the confines of a single, integrated, game-theoretic model with incomplete information. Chapter 6 uses the same game form to develop a logically consistent and empirically plausible explanation of the outbreak of war in Europe in early August 1914. Chapter 7 introduces perfect deterrence theory and contrasts it with the prevailing realist theory of interstate war prevention, and classical deterrence theory. Chapter 8 addresses the charge made by some behavioral economists (and many strategic analysts) that game theory is of limited utility for understanding interstate conflict behavior.

This book introduces the main concepts of the theory of De Giorgi's Gamma-convergence and gives a description of its main applications to the study of asymptotic variational problems. The content is based on results obtained during thirty years of research. The book is divided into sixteen short chapters, an Introduction, and an Appendix. After explaining how a notion of variational convergence arises naturally from the study of the asymptotic behaviour of variational problems, the Introduction presents a number of examples that show how diversified the applications of this notion may be. The first chapter covers the abstract theory of Gamma-convergence, including its links with lower semicontinuity and relaxation, and the fundamental results on the convergence of minimum problems. The following ten chapters are all set in a one-dimensional framework to illustrate the main issues of convergence without the burden of high-dimensional technicalities. These include variational problems in Sobolev spaces, in particular homogenization theory, limits of discrete systems, segmentation and phase-transition problems, free-discontinuity problems and their approximation, etc. Chapters 12-15 are devoted to problems in a higher-dimensional setting, showing how some one-dimensional reasoning may be extended, if properly formulated, to a more general setting, and how some concepts already introduced can be integrated with vectorial issues. The final chapter introduces the more general and abstract localization methods of Gamma-convergence. All chapters are complemented by a guide to the literature, and a short description of extensions and developments.

This monograph addresses the need to clarify basic mathematical concepts at the crossroad between gravitation and quantum physics. Selected mathematical and theoretical topics are exposed within a not-too-short, integrated approach that exploits standard and non-standard notions in natural geometric language. The role of structure groups can be regarded as secondary even in the treatment of the gauge fields themselves. Two-spinors yield a partly original ‘minimal geometric data’ approach to Einstein-Cartan-Maxwell-Dirac fields. The gravitational field is jointly represented by a spinor connection and by a soldering form (a ‘tetrad’) valued in a vector bundle naturally constructed from the assumed 2-spinor bundle. We give a presentation of electroweak theory that dispenses with group-related notions, and we introduce a non-standard, natural extension of it. Also within the 2-spinor approach we present: a non-standard view of gauge freedom; a first-order Lagrangian theory of fields with arbitrary spin; an original treatment of Lie derivatives of spinors and spinor connections. Furthermore we introduce an original formulation of Lagrangian field theories based on covariant differentials, which works in the classical and quantum field theories alike and simplifies calculations. We offer a precise mathematical approach to quantum bundles and quantum fields, including ghosts, BRST symmetry and anti-fields, treating the geometry of quantum bundles and their jet prolongations in terms Frölicher's notion of smoothness. We propose an approach to quantum particle physics based on the notion of detector, and illustrate the basic scattering computations in that context.

General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. This book overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.

How to Free Your Inner Mathematician: Notes on Mathematics and Life offers readers guidance in managing the fear, freedom, frustration, and joy that often accompany calls to think mathematically. With practical insight and years of award-winning mathematics teaching experience, DAgostino offers more than 300 hand-drawn sketches alongside accessible descriptions of fractals, symmetry, fuzzy logic, knot theory, Penrose patterns, infinity, the Twin Prime Conjecture, Arrows Impossibility Theorem, Fermats Last Theorem, and other intriguing mathematical topics. Readers are encouraged to embrace change, proceed at their own pace, mix up their routines, resist comparison, have faith, fail more often, look for beauty, exercise their imaginations, and define success for themselves. Mathematics students and enthusiasts will learn advice for fostering courage on their journey regardless of age or mathematical background. How to Free Your Inner Mathematician delivers not only engaging mathematical content but provides reassurance that mathematical success has more to do with curiosity and drive than innate aptitude.