Browse

Categories for Quantum Theory: An Introduction

Chris Heunen and Jamie Vicary

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 ... More

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.Less

Computing with Cells: Advances in Membrane Computing

Pierluigi Frisco

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 ... More

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.Less

Concentration Inequalities: A Nonasymptotic Theory of Independence

Stéphane Boucheron, Gábor Lugosi, and Pascal Massart

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 ... More

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.Less

Consumer Credit Models: Pricing, Profit and Portfolios

Lyn C. Thomas

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 ... More

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.Less

Credit Risk Management: Basic Concepts: Financial Risk Components, Rating Analysis, Models, Economic and Regulatory Capital

Tony Van Gestel and Bart Baesens

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 ... More

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.Less

Dynamics of Viscous Compressible Fluids

Eduard Feireisl

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 ... More

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.Less

Electromagnetism of Continuous Media: Mathematical Modelling and Applications

Mauro Fabrizio and Angelo Morro

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 ... More

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.Less

Everyday Cryptography: Fundamental Principles and Applications

Keith M. Martin

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 ... More

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.Less

The Factorization Method for Inverse Problems

Andreas Kirsch and Natalia Grinberg

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 ... More

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.Less

Finite Elasticity Theory

David J. Steigmann

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 ... More

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.Less

Game Theory, Diplomatic History and Security Studies

Frank C. Zagare

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 ... More

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.Less

Gamma-Convergence for Beginners

Andrea Braides

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 ... More

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.Less

Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics

Daniel Canarutto

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 ... More

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.Less

General Relativity and the Einstein Equations

Yvonne Choquet-Bruhat

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 ... More

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.Less

How to Free Your Inner Mathematician: Notes on Mathematics and Life

Susan D'Agostino

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. ... More

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.Less

Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

Moody Chu and Gene Golub

The basic goal of an inverse eigenvalue problem is to reconstruct the physical parameters of a certain system from the knowledge or desire of its dynamical behavior. Depending on the application, ... More

The basic goal of an inverse eigenvalue problem is to reconstruct the physical parameters of a certain system from the knowledge or desire of its dynamical behavior. Depending on the application, inverse eigenvalue problems appear in many different forms. This book discusses the fundamental questions, some known results, many applications, mathematical properties, a variety of numerical techniques, as well as several open problems.Less

The Joy of Statistics: A Treasury of Elementary Statistical Tools and their Applications

Steve Selvin

The Joy of Statistics consists of a series of 42 “short stories,” each illustrating how elementary statistical methods are applied to data to produce insight and solutions to the questions data are ... More

The Joy of Statistics consists of a series of 42 “short stories,” each illustrating how elementary statistical methods are applied to data to produce insight and solutions to the questions data are collected to answer. The text contains brief histories of the evolution of statistical methods and a number of brief biographies of the most famous statisticians of the 20th century. Also throughout are a few statistical jokes, puzzles, and traditional stories. The level of the Joy of Statistics is elementary and explores a variety of statistical applications using graphs and plots, along with detailed and intuitive descriptions and occasionally using a bit of 10th grade mathematics. Examples of a few of the topics are gambling games such as roulette, blackjack, and lotteries as well as more serious subjects such as comparison of black/white infant mortality rates, coronary heart disease risk, and ethnic differences in Hodgkin’s disease. The statistical description of these methods and topics are accompanied by easy to understand explanations labeled “how it works.”Less

Krylov Subspace Methods: Principles and Analysis

Jörg Liesen and Zdenek Strakos

This book offers a detailed treatment of the mathematical theory of Krylov subspace methods with focus on solving systems of linear algebraic equations. Starting from the idea of projections, Krylov ... More

This book offers a detailed treatment of the mathematical theory of Krylov subspace methods with focus on solving systems of linear algebraic equations. Starting from the idea of projections, Krylov subspace methods are characterised by their orthogonality and minimisation properties. Projections onto highly nonlinear Krylov subspaces can be linked with the underlying problem of moments, and therefore Krylov subspace methods can be viewed as matching moments model reduction. This allows enlightening reformulations of questions from matrix computations into the language of orthogonal polynomials, Gauss–Christoffel quadrature, continued fractions, and, more generally, of Vorobyev method of moments. Using the concept of cyclic invariant subspaces conditions are studied that allow generation of orthogonal Krylov subspace bases via short recurrences. The results motivate the practically important distinction between Hermitian and non-Hermitian problems. Finally, the book thoroughly addresses the computational cost while using Krylov subspace methods. The investigation includes effects of finite precision arithmetic and focuses on the method of conjugate gradients (CG) and generalised minimal residuals (GMRES) as major examples. The book emphasises that algebraic computations must always be considered in the context of solving real-world problems, where the mathematical modelling, discretisation, and computation cannot be separated from each other. Moreover, the book underlines the importance of the historical context and it demonstrates that knowledge of early developments can play an important role in understanding and resolving very recent computational problems. Many extensive historical notes are therefore included as an inherent part of the text. The book ends with formulating some omitted issues and challenges which need to be addressed in future work. The book is intended as a research monograph which can be used in a wide scope of graduate courses on related subjects. It can be beneficial also for readers interested in the history of mathematics.Less

Matheron's Theory of Regionalised Variables

Georges Matheron

Vera Pawlowsky-Glahn and Jean Serra (eds)

This book has never been published before, although its contents have provided the basis for hundreds of papers, theses, and books on geostatistics. The chapters are based on the lectures of a summer ... More

This book has never been published before, although its contents have provided the basis for hundreds of papers, theses, and books on geostatistics. The chapters are based on the lectures of a summer course given by Georges Matheron in 1970; initially written in French, they were translated into English by Charles Huijbregts. They do not contain mathematical technicalities or practical case studies; instead, they present major topics like estimation variances, kriging systems, mining estimation, and intrinsic theory, all of which are established by simple proofs. The reader is invited to wonder about the physical meaning of the notions Matheron deals with. When Matheron wrote these lectures, he considered the theory of linear geostatistics complete; however, what was an ending for Matheron has been the starting point for most geostatisticians. Many discovered the book’s content indirectly, via the many borrowings one can find in several books; in such a situation, it is always instructive to come back to the original document, where the author's motivations, his physical intuitions, and his thoughts on the meaning of what he does are detailed. The decision to publish this book was motivated by the desire to introduce Matheron’s work to a larger audience. The book has remained faithful to the original notes while introducing a common structure for the chapters and sections, numbering equations sequentially within each chapter, numbering the figures (most of which were redrawn) sequentially, and adding captions. In addition, Matheron’s comments on the exercises, or suggestions for solutions, have been added.Less

Modern Risk Quantification in Complex Projects: Non-linear Monte Carlo and System Dynamics Methodologies

Yuri G. Raydugin

There are multiple complaints that existing project risk quantification methods—both parametric and Monte Carlo—fail to produce accurate project duration and cost-risk contingencies in a majority of ... More

There are multiple complaints that existing project risk quantification methods—both parametric and Monte Carlo—fail to produce accurate project duration and cost-risk contingencies in a majority of cases. It is shown that major components of project risk exposure—non-linear risk interactions—pertaining to complex projects are not taken into account. It is argued that a project system consists of two interacting subsystems: a project structure subsystem (PSS) and a project delivery subsystem (PDS). Any misalignments or imbalances between these two subsystems (PSS–PDS mismatches) are associated with the non-linear risk interactions. Principles of risk quantification are developed to take into account three types of non-linear risk interactions in complex projects: internal risk amplifications due to existing ‘chronic’ project system issues, knock-on interactions, and risk compounding. Modified bowtie diagrams for the three types of risk interactions are developed to identify and address interacting risks. A framework to visualize dynamic risk patterns in affinities of interacting risks is proposed. Required mathematical expressions and templates to factor relevant risk interactions to Monte Carlo models are developed. Business cases are discussed to demonstrate the power of the newly-developed non-linear Monte Carlo methodology (non-linear integrated schedule and cost risk analysis (N-SCRA)). A project system dynamics methodology based on rework cycles is adopted as a supporting risk quantification tool. Comparison of results yielded by the non-linear Monte Carlo and system dynamics models demonstrates a good alignment of the two methodologies. All developed Monte Carlo and system dynamics models are available on the book’s companion website.Less