Random matrices arise from, and have important applications to, number theory, probability, combinatorics, representation theory, quantum mechanics, solid state physics, quantum field theory, quantum gravity, and many other areas of physics and mathematics. This 2001 volume of surveys and research results, based largely on lectures given at the Spring 1999 MSRI program of the same name, covers broad areas such as topologic and combinatorial aspects of random matrix theory; scaling limits, universalities and phase transitions in matrix models; universalities for random polynomials; and applications to integrable systems. Its stress on the interaction between physics and mathematics will make it a welcome addition to the shelves of graduate students and researchers in both fields, as will its expository emphasis.
Abner Shimony an eminent philosopher of science, whose work has exerted a profound influence in both the philosophy and physics communities. This two-volume collection of his essays written over a period of forty years explores the interrelations between science and philosophy. Shimony regards the knowing subject as an entity in nature whose faculties must be studied from the points of view of evolutionary biology and empirical psychology. He maintains that the twentieth century is one of the great ages of metaphysics, given the deep implications of quantum mechanics, relativity theory, and molecular biology. The first volume, Scientific Method and Epistemology, deals with the dialectic of subject and object, epistemic probability, induction and scientific theories, perception and conception, and fact and values. The focus of the second volume, Natural Sciences and Metaphysics, is on quantum mechanical measurement and non-locality, parts and wholes, time, and mind and matter.
Random matrices arise from, and have important applications to, number theory, probability, combinatorics, representation theory, quantum mechanics, solid state physics, quantum field theory, quantum gravity, and many other areas of physics and mathematics. This 2001 volume of surveys and research results, based largely on lectures given at the Spring 1999 MSRI program of the same name, covers broad areas such as topologic and combinatorial aspects of random matrix theory; scaling limits, universalities and phase transitions in matrix models; universalities for random polynomials; and applications to integrable systems. Its stress on the interaction between physics and mathematics will make it a welcome addition to the shelves of graduate students and researchers in both fields, as will its expository emphasis.
Now in a paperback edition for the first time, this second edition of Some Random Series of Functions covers random series in Banach and Hilbert spaces, random Taylor or Fourier series, Brownian motion and other Gaussian processes, plus certain types of random sets and measures. The subject matter of this book is important and has wide application in mathematics, statistics, engineering, and physics. Professor Kahane's presentation is suitable even for beginning graduate students in probability and analysis (exercises are provided throughout), as well as non-specialists in the other disciplines to which this subject has application.
This book develops and applies an analytical approach to deriving the probability laws of science in general. It is called 'extreme physical information' or EPI. EPI is an expression of the imperfection of observation: Owing to random interaction of a subject with its observer and other possible disturbances, its measurement contains less Fisher information than does the subject per se. Moreover, the information loss is an extreme value. An EPI output may alternatively be viewed as the payoff of a zero-sum game of information acquisition between the observer and a 'demon' in subject space. EPI derives, Escher-like, the very probability law that gave rise to the measurement. In applications, EPI is used to derive both existing and new analytical relations governing probability laws of physics, genetics, cancer growth, ecology and economics. This unified approach will be fascinating to students and those who seek a new mathematical tool of research.
This rigourous and self-contained book describes mathematical and, in particular, stochastic methods to assess the performance of networked systems. It consists of three parts. The first part is a review on probability theory. Part two covers the classical theory of stochastic processes (Poisson, renewal, Markov and queuing theory), which are considered to be the basic building blocks for performance evaluation studies. Part three focuses on the relatively new field of the physics of networks. This part deals with the recently obtained insights that many very different large complex networks - such as the Internet, World Wide Web, proteins, utility infrastructures, social networks - evolve and behave according to more general common scaling laws. This understanding is useful when assessing the end-to-end quality of communications services, for example, in Internet telephony, real-time video and interacting games. Containing problems and solutions, this book is ideal for graduate studen
The general area of geophysical fluid mechanics is truly interdisciplinary. Now ideas from statistical physics are being applied in novel ways to inhomogeneous complex systems such as atmospheres and oceans. In this book, the basic ideas of geophysics, probability theory, information theory, nonlinear dynamics and equilibrium statistical mechanics are introduced and applied to large time-selective decay, the effect of large scale forcing, nonlinear stability, fluid flow on a sphere and Jupiter's Great Red Spot. The book is the first to adopt this approach and it contains many recent ideas and results. Its audience ranges from graduate students and researchers in both applied mathematics and the geophysical sciences. It illustrates the richness of the interplay of mathematical analysis, qualitative models and numerical simulations which combine in the emerging area of computational science.
This paperback, which comprises the first part of Introduction to Measure and Probability by J. F. C. Kingman and S. J. Taylor, gives a self-contained treatment of the theory of finite measures in general spaces at the undergraduate level. It sets the material out in a form which not only provides an introduction for intending specialists in measure theory but also meets the needs of students of probability. The theory of measure and integration is presented for general spaces, with Lebesgue measure and the Lebesgue integral considered as important examples whose special properties are obtained. The introduction to functional analysis which follows covers the material to probability theory and also the basic theory of L2-spaces, important in modern physics. A large number of examples is included; these form an essential part of the development.
The common cause principle says that every correlation is either due to a direct causal effect linking the correlated entities or is brought about by a third factor, a so-called common cause. The principle is of central importance in the philosophy of science, especially in causal explanation, causal modeling and in the foundations of quantum physics. Written for philosophers of science, physicists and statisticians, this book contributes to the debate over the validity of the common cause principle, by proving results that bring to the surface the nature of explanation by common causes. It provides a technical and mathematically rigorous examination of the notion of common cause, providing an analysis not only in terms of classical probability measure spaces, which is typical in the available literature, but in quantum probability theory as well. The authors provide numerous open problems to further the debate and encourage future research in this field.
Hidden Markov processes (HMPs) are important objects of study in many areas of pure and applied mathematics, including information theory, probability theory, dynamical systems and statistical physics, with applications in electrical engineering, computer science and molecular biology. This collection of research and survey papers presents important new results and open problems, serving as a unifying gateway for researchers in these areas. Based on talks given at the Banff International Research Station Workshop, 2007, this volume addresses a central problem of the subject: computation of the Shannon entropy rate of an HMP. This is a key quantity in statistical physics and information theory, characterising the fundamental limit on compression and closely related to channel capacity, the limit on reliable communication. Also discussed, from a symbolic dynamics and thermodynamical viewpoint, is the problem of characterizing the mappings between dynamical systems which map Markov measur
Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers inter
Time asymmetric phenomena are successfully predicted by statistical mechanics. Yet the foundations of this theory are surprisingly shaky. Its explanation for the ease of mixing milk with coffee is incomplete, and even implies that un-mixing them should be just as easy. In this book the authors develop a new conceptual foundation for statistical mechanics that addresses this difficulty. Explaining the notions of macrostates, probability, measurement, memory, and the arrow of time in statistical mechanics, they reach the startling conclusion that Maxwell's Demon, the famous perpetuum mobile, is consistent with the fundamental physical laws. Mathematical treatments are avoided where possible, and instead the authors use novel diagrams to illustrate the text. This is a fascinating book for graduate students and researchers interested in the foundations and philosophy of physics.
Combining physics and philosophy, this is a uniquely interdisciplinary examination of quantum information science which provides an up-to-date examination of developments in this field. The authors provide coherent definitions and theories of information, taking clearly defined approaches to considering information in connection with quantum mechanics, probability, and correlations. Concepts addressed include entanglement of quantum states, the relation of quantum correlations to quantum information, and the meaning of the informational approach for the foundations of quantum mechanics. Furthermore, the mathematical concept of information in the communicational context, and the notion of pragmatic information are considered. Suitable as both a discussion of the conceptual and philosophical problems of this field and a comprehensive stand-alone introduction, this book will benefit both experienced and new researchers in quantum information and the philosophy of physics.
Based around a series of real-life scenarios, this engaging introduction to statistical reasoning will teach you how to apply powerful statistical, qualitative and probabilistic tools in a technical context. From analysis of electricity bills, baseball statistics, and stock market fluctuations, through to profound questions about physics of fermions and bosons, decaying nuclei, and climate change, each chapter introduces relevant physical, statistical and mathematical principles step-by-step in an engaging narrative style, helping to develop practical proficiency in the use of probability and statistical reasoning. With numerous illustrations making it easy to focus on the most important information, this insightful book is perfect for students and researchers of any discipline interested in the interwoven tapestry of probability, statistics, and physics.
Representation theory of big groups is an important and quickly developing part of modern mathematics, giving rise to a variety of important applications in probability and mathematical physics. This book provides the first concise and self-contained introduction to the theory on the simplest yet very nontrivial example of the infinite symmetric group, focusing on its deep connections to probability, mathematical physics, and algebraic combinatorics. Following a discussion of the classical Thoma's theorem which describes the characters of the infinite symmetric group, the authors describe explicit constructions of an important class of representations, including both the irreducible and generalized ones. Complete with detailed proofs, as well as numerous examples and exercises which help to summarize recent developments in the field, this book will enable graduates to enhance their understanding of the topic, while also aiding lecturers and researchers in related areas.
This self-contained book is a graduate-level introduction for mathematicians and for physicists interested in the mathematical foundations of the field, and can be used as a textbook for a two-semester course on mathematical statistical mechanics. It assumes only basic knowledge of classical physics and, on the mathematics side, a good working knowledge of graduate-level probability theory. The book starts with a concise introduction to statistical mechanics, proceeds to disordered lattice spin systems, and concludes with a presentation of the latest developments in the mathematical understanding of mean-field spin glass models. In particular, progress towards a rigorous understanding of the replica symmetry-breaking solutions of the Sherrington-Kirkpatrick spin glass models, due to Guerra, Aizenman-Sims-Starr and Talagrand, is reviewed in some detail.
This original text develops a deep, conceptual understanding of thermal physics, highlighting the important links between thermodynamics and statistical physics, and examining how thermal physics fits within physics as a whole, from an empirical perspective. The first part of the book is devoted to elementary, mesoscopic topics such as Brownian motion, which leads to intuitive uses of large deviation theory, one of the pillars of modern probability theory. The book then introduces the key concepts behind statistical thermodynamics, and the final part describes more advanced and applied topics from thermal physics such as phase transitions and critical phenomena. This important subject is presented from a fresh perspective and in a highly pedagogical manner, with numerous worked examples and relevant cultural side notes throughout, making it ideal as either a textbook for advanced thermal physics courses or for self-study by undergraduate and graduate students in physics and engineering
Starting from physical motivations and leading to practical applications, this book provides an interdisciplinary perspective on the cutting edge of ultrametric pseudodifferential equations. It shows the ways in which these equations link different fields including mathematics, engineering, and geophysics. In particular, the authors provide a detailed explanation of the geophysical applications of p-adic diffusion equations, useful when modeling the flows of liquids through porous rock. p-adic wavelets theory and p-adic pseudodifferential equations are also presented, along with their connections to mathematical physics, representation theory, the physics of disordered systems, probability, number theory, and p-adic dynamical systems. Material that was previously spread across many articles in journals of many different fields is brought together here, including recent work on the van der Put series technique. This book provides an excellent snapshot of the fascinating field of ultrame
Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probabi
This Element analyzes the various forms that design arguments for the existence of God can take, but the main focus is on two such arguments. The first concerns the complex adaptive features that organisms have. Creationists who advance this argument contend that evolution by natural selection cannot be the right explanation. The second design argument - the argument from fine-tuning - begins with the fact that life could not exist in our universe if the constants found in the laws of physics had values that differed more than a little from their actual values. Since probability is the main analytical tool used, the Element provides a primer on probability theory.
