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| communications math physics: Applied and Industrial Mathematics, Venice—2, 1998 Renato Spigler, 2012-12-06 In this volume, I have collected several papers which were presented at the international conference called Venice-2/Symposium on Applied and In dustrial Mathematics. Such a conference was held in Venice, Italy, between June 11 and 16,1998, and was intended as the follow-up of the very successful similar event (called Venice-1/Symposium on Applied and Industrial Math ematics), that was also organized in Venice in October 1989. The Venice-1 conference ended up with a Kluwer volume like this one. I am grateful to Kluwer for having accepted to publish the present volume, the aim of which is to update somehow the state-of-the-art in the field of Ap plied Mathematics as well as in that of the nowadays rather more developed area of Industrial Mathematics. The most of the invited (key-note) speakers contributed to this volume with a paper related to their talk. There are, in addition·, a few significant contributed papers, selected on the basis of their quality and relevance to the present-time research activities. The topics considered in the conference range from rather general sub jects in applied and numerical analysis, to more specialized subjects such as polymers and disordered media, granular flow, semiconductor mathematics, superconductors, elasticity, tomography and other inverse problems, financial modeling, photographic sciences, etc. The papers collected in this volume provide a selection of them. It is clear from the previous list that some attention has been paid to relatively new and emerging fields. |
| communications math physics: Number Theory in Science and Communication M.R. Schroeder, 2006-01-06 Number Theory in Science and Communication introductes non-mathematicians to the fascinating and diverse applications of number theory. This best-selling book stresses intuitive understanding rather than abstract theory. This revised fourth edition is augmented by recent advances in primes in progressions, twin primes, prime triplets, prime quadruplets and quintruplets, factoring with elliptic curves, quantum factoring, Golomb rulers and baroque integers. |
| communications math physics: The Mathematical Theory of Communication Claude Elwood Shannon, Warren Weaver, 1962 |
| communications math physics: The Mathematical Theory of Communication Claude Elwood Shannon, Warren Weaver, 1971 |
| communications math physics: Methods of Mathematical Physics Harold Jeffreys, Bertha Swirles Jeffreys, 1999-11-18 This book is a reissue of classic textbook of mathematical methods. |
| communications math physics: Nonlinear Optics D.L. Mills, 2012-12-06 Since the book was first published in 1991, the field of surface nonlinear optics has grown substantially to the point where an exposition of the principles of this field will prove useful to many. Thus, in this second edition, Chapter 8 addresses this area. Also, optical probes of magnetism of very thin films and multilayers are now widely used, and magneto-optic devices of increasing sophistication have appeared. Chapter 9 is thus devoted to magneto-optics, and associated nonlinear phenomena. The earlier chapter on Chaos appears as Chapter 10. The philosophy which underlies the first edition was also employed in the writing of the two new chapters. Irvine, CA D.L.Mills March 1998 Preface to the First Edition One intriguing aspect of physics is its dynamic and rapidly evolving nature; exciting new fields can become moribund within relatively few years, only to revive and grow again in a dramatic and explosive manner in response to new developments. |
| communications math physics: Algorithms for Computer Algebra Keith O. Geddes, Stephen R. Czapor, George Labahn, 1992-09-30 Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics. The book first develops the foundational material from modern algebra that is required for subsequent topics. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions. Numerous examples are integrated into the text as an aid to understanding the mathematical development. The algorithms developed for each topic are presented in a Pascal-like computer language. An extensive set of exercises is presented at the end of each chapter. Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields. |
| communications math physics: Quantum Information, Computation and Communication Jonathan A. Jones, Dieter Jaksch, 2012-07-19 Based on years of teaching experience, this textbook guides physics undergraduate students through the theory and experiment of the field. |
| communications math physics: Number Theory and Physics Jean-Marc Luck, Pierre Moussa, Michel Waldschmidt, 2012-12-06 7 Les Houches Number theory, or arithmetic, sometimes referred to as the queen of mathematics, is often considered as the purest branch of mathematics. It also has the false repu tation of being without any application to other areas of knowledge. Nevertheless, throughout their history, physical and natural sciences have experienced numerous unexpected relationships to number theory. The book entitled Number Theory in Science and Communication, by M.R. Schroeder (Springer Series in Information Sciences, Vol. 7, 1984) provides plenty of examples of cross-fertilization between number theory and a large variety of scientific topics. The most recent developments of theoretical physics have involved more and more questions related to number theory, and in an increasingly direct way. This new trend is especially visible in two broad families of physical problems. The first class, dynamical systems and quasiperiodicity, includes classical and quantum chaos, the stability of orbits in dynamical systems, K.A.M. theory, and problems with small denominators, as well as the study of incommensurate structures, aperiodic tilings, and quasicrystals. The second class, which includes the string theory of fundamental interactions, completely integrable models, and conformally invariant two-dimensional field theories, seems to involve modular forms and p adic numbers in a remarkable way. |
| communications math physics: Differential Forms in Mathematical Physics , 2009-06-17 Differential Forms in Mathematical Physics |
| communications math physics: Software Pioneers M. Broy, 2002-06-27 This book, coming with four DVDs, presents epochal works of 16 of the most influential software pioneers. Seminal historical papers, going back as far as to the 1950s, are complemented by new papers especially written by the software pioneers for inclusion in this book and by short biographical notes. The volume is based on a conference where the pioneers met and presented their assessment of the past, new ideas, and visions for the future. The volume editors coherently integrated the historical contributions with current aspects and future perspectives. The four DVDs included are an important supplement to the book providing more than 12 hours of video documentation. Besides a representative overview drawing together the highlights of the presentations, the video recording of each pioneer's talk together with the transparencies used is included. Together, the book and the four DVDs constitute a unique and major contribution to the history of software engineering. |
| communications math physics: Fundamental Math and Physics for Scientists and Engineers David Yevick, Hannah Yevick, 2014-12-31 Provides a concise overview of the core undergraduate physics and applied mathematics curriculum for students and practitioners of science and engineering Fundamental Math and Physics for Scientists and Engineers summarizes college and university level physics together with the mathematics frequently encountered in engineering and physics calculations. The presentation provides straightforward, coherent explanations of underlying concepts emphasizing essential formulas, derivations, examples, and computer programs. Content that should be thoroughly mastered and memorized is clearly identified while unnecessary technical details are omitted. Fundamental Math and Physics for Scientists and Engineers is an ideal resource for undergraduate science and engineering students and practitioners, students reviewing for the GRE and graduate-level comprehensive exams, and general readers seeking to improve their comprehension of undergraduate physics. Covers topics frequently encountered in undergraduate physics, in particular those appearing in the Physics GRE subject examination Reviews relevant areas of undergraduate applied mathematics, with an overview chapter on scientific programming Provides simple, concise explanations and illustrations of underlying concepts Succinct yet comprehensive, Fundamental Math and Physics for Scientists and Engineers constitutes a reference for science and engineering students, practitioners and non-practitioners alike. |
| communications math physics: Mathematics for Physics Michael Stone, Paul Goldbart, 2009-07-09 An engagingly-written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics – differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables. The authors' exposition avoids excess rigor whilst explaining subtle but important points often glossed over in more elementary texts. The topics are illustrated at every stage by carefully chosen examples, exercises and problems drawn from realistic physics settings. These make it useful both as a textbook in advanced courses and for self-study. Password-protected solutions to the exercises are available to instructors at www.cambridge.org/9780521854030. |
| communications math physics: Communications in mathematical physics [Anonymus AC01235699], 1965 |
| communications math physics: Mathematics for Physics Michael M. Woolfson, Malcolm S. Woolfson, 2007 Mathematics for Physics features both print and online support, with many in-text exercises and end-of-chapter problems, and web-based computer programs, to both stimulate learning and build understanding. |
| communications math physics: An Introduction to Quantum Communication Networks Mohsen Razavi, 2018-05-25 With the fast pace of developments in quantum technologies, it is more than ever necessary to make the new generation of students in science and engineering familiar with the key ideas behind such disruptive systems. This book intends to fill such a gap between experts and non-experts in the field by providing the reader with the basic tools needed to understand the latest developments in quantum communications and its future directions. This is not only to expand the audience knowledge but also to attract new talents to this flourishing field. To that end, the book as a whole does not delve into much detail and most often suffices to provide some insight into the problem in hand. The primary users of the book will then be students in science and engineering in their final year of undergraduate studies or early years of their post-graduate programmes. |
| communications math physics: Positive Linear Maps of Operator Algebras Erling Størmer, 2012-12-13 This volume, setting out the theory of positive maps as it stands today, reflects the rapid growth in this area of mathematics since it was recognized in the 1990s that these applications of C*-algebras are crucial to the study of entanglement in quantum theory. The author, a leading authority on the subject, sets out numerous results previously unpublished in book form. In addition to outlining the properties and structures of positive linear maps of operator algebras into the bounded operators on a Hilbert space, he guides readers through proofs of the Stinespring theorem and its applications to inequalities for positive maps. The text examines the maps’ positivity properties, as well as their associated linear functionals together with their density operators. It features special sections on extremal positive maps and Choi matrices. In sum, this is a vital publication that covers a full spectrum of matters relating to positive linear maps, of which a large proportion is relevant and applicable to today’s quantum information theory. The latter sections of the book present the material in finite dimensions, while the text as a whole appeals to a wider and more general readership by keeping the mathematics as elementary as possible throughout. |
| communications math physics: Number-Crunching Paul Nahin, 2011-08-08 More stimulating mathematics puzzles from bestselling author Paul Nahin How do technicians repair broken communications cables at the bottom of the ocean without actually seeing them? What's the likelihood of plucking a needle out of a haystack the size of the Earth? And is it possible to use computers to create a universal library of everything ever written or every photo ever taken? These are just some of the intriguing questions that best-selling popular math writer Paul Nahin tackles in Number-Crunching. Through brilliant math ideas and entertaining stories, Nahin demonstrates how odd and unusual math problems can be solved by bringing together basic physics ideas and today's powerful computers. Some of the outcomes discussed are so counterintuitive they will leave readers astonished. Nahin looks at how the art of number-crunching has changed since the advent of computers, and how high-speed technology helps to solve fascinating conundrums such as the three-body, Monte Carlo, leapfrog, and gambler's ruin problems. Along the way, Nahin traverses topics that include algebra, trigonometry, geometry, calculus, number theory, differential equations, Fourier series, electronics, and computers in science fiction. He gives historical background for the problems presented, offers many examples and numerous challenges, supplies MATLAB codes for all the theories discussed, and includes detailed and complete solutions. Exploring the intimate relationship between mathematics, physics, and the tremendous power of modern computers, Number-Crunching will appeal to anyone interested in understanding how these three important fields join forces to solve today's thorniest puzzles. |
| communications math physics: Mathematical Methods in Solid State and Superfluid Theory R.C. Clark, G.H. Derrick, 2013-12-17 |
| communications math physics: Quantum Fields in Curved Space N. D. Birrell, P. C. W. Davies, 1984-02-23 This book presents a comprehensive review of the subject of gravitational effects in quantum field theory. Although the treatment is general, special emphasis is given to the Hawking black hole evaporation effect, and to particle creation processes in the early universe. The last decade has witnessed a phenomenal growth in this subject. This is the first attempt to collect and unify the vast literature that has contributed to this development. All the major technical results are presented, and the theory is developed carefully from first principles. Here is everything that students or researchers will need to embark upon calculations involving quantum effects of gravity at the so-called one-loop approximation level. |
| communications math physics: Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization Houman Owhadi, Clint Scovel, 2019-10-24 Presents interplays between numerical approximation and statistical inference as a pathway to simple solutions to fundamental problems. |
| communications math physics: Deaf Education : Costs and Student Characteristics at Federally Assisted Schools , 1986 |
| communications math physics: The Mathematical World of Walter Noll Yurie A. Ignatieff, 2013-03-09 The book stresses particularly Noll's method of axiomatization of physical theories, his axiomatics of continuum mechanics, thermodynamics of materials, special relativity theory, his discovery of the neo-classical space-time of mechanics, his theories of inhomogeneities in simple bodies, fit regions, contact interactions, annihilators of linear differential operators, and finite-dimensional spaces. |
| communications math physics: Exactly Solved Models in Statistical Mechanics Rodney J. Baxter, 2007-01-01 Exploration of two-dimensional lattice models examines basic statistical mechanics, Ising models, spherical models, ice-type models, corner transfer matrices, and elliptic functions. 1982 edition, with author's 2007 update on subsequent developments. |
| communications math physics: Electronic Information and Communication in Mathematics Fengshan Bai, Bernd Wegner, 2003-09-03 This book constitutes the thoroughly refereed post-proceedings of the ICM 2002 International Satellite Conference on Electronic Information and Communication in Mathematics, held in Beijing, China, in August 2002. The 18 revised and reviewed papers assess the state of the art of the production and dissemination of electronic information in mathematics. Among the topics addressed are models and standards for information and metainformation representation; data search, discovery, retrieval, and analysis; access to distributed and heterogeneous digital collections; intelligent user interfaces to digital libraries; information agents, and cooperative work on mathematical data; digital collection generation; business models; and data security and protection. |
| communications math physics: Trends in Interactive Visualization Elena Zudilova-Seinstra, Tony Adriaansen, Robert van Liere, 2008-12-17 II Challenges in Data Mapping Part II deals with one of the most challenging tasks in Interactive Visualization, mapping and teasing out information from large complex datasets and generating visual representations. This section consists of four chapters. Binh Pham, Alex Streit, and Ross Brown provide a comprehensive requirement analysis of information uncertainty visualizations. They examine the sources of uncertainty, review aspects of its complexity, introduce typical models of uncertainty, and analyze major issues in visualization of uncertainty, from various user and task perspectives. Alfred Inselberg examines challenges in the multivariate data analysis. He explains how relations among multiple variables can be mapped uniquely into ?-space subsets having geometrical properties and introduces Parallel Coordinates meth- ology for the unambiguous visualization and exploration of a multidimensional geometry and multivariate relations. Christiaan Gribble describes two alternative approaches to interactive particle visualization: one targeting desktop systems equipped with programmable graphics hardware and the other targeting moderately sized multicore systems using pack- based ray tracing. Finally, Christof Rezk Salama reviews state-of-the-art strategies for the assignment of visual parameters in scientific visualization systems. He explains the process of mapping abstract data values into visual based on transfer functions, clarifies the terms of pre- and postclassification, and introduces the state-of-the-art user int- faces for the design of transfer functions. |
| communications math physics: The Factorization Method for Inverse Problems Andreas Kirsch, Natalia Grinberg, 2008 The 'factorization method', discovered by Professor Kirsch, is a relatively new method for solving certain types of inverse scattering problems and problems in tomography. The text introduces the reader to this promising approach and discusses the wide applicability of this method by choosing typical examples. |
| communications math physics: Completely Bounded Maps and Operator Algebras Vern Paulsen, 2002 In this book, first published in 2003, the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will also want this book for their library since the author illustrates the power of methods he has developed with new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature. An indispensable introduction to the theory of operator spaces for all who want to know more. |
| communications math physics: Quantum Field Theory for Mathematicians Robin Ticciati, 1999-06-13 This should be a useful reference for anybody with an interest in quantum theory. |
| communications math physics: The Math Book Clifford A. Pickover, 2009 This book covers 250 milestones in mathematical history, beginning millions of years ago with ancient ant odometers and moving through time to our modern-day quest for new dimensions. |
| communications math physics: Fiber Optic Communications Palais, 2005 |
| communications math physics: Concise Optics Ajawad I. Haija, M. Z. Numan, W. Larry Freeman, 2018-02-21 This introductory text is a reader friendly treatment of geometrical and physical optics emphasizing problems and solved examples with detailed analysis and helpful commentary. The authors are seasoned educators with decades of experience teaching optics. Their approach is to gradually present mathematics explaining the physical concepts. It covers ray tracing to the wave nature of light, and introduces Maxwell’s equations in an organic fashion. The text then moves on to explains how to analyze simple optical systems such as spectacles for improving vision, microscopes, and telescopes, while also being exposed to contemporary research topics. Ajawad I. Haija is a professor of physics at Indiana University of Pennsylvania. M. Z. Numan is professor and chair of the department of physics at Indiana University of Pennsylvania. W. Larry Freeman is Emeritus Professor of Physics at Indiana University of Pennsylvania. |
| communications math physics: Tau Functions and their Applications John Harnad, Ferenc Balogh, 2021-02-04 Tau functions are a central tool in the modern theory of integrable systems. This volume provides a thorough introduction, starting from the basics and extending to recent research results. It covers a wide range of applications, including generating functions for solutions of integrable hierarchies, correlation functions in the spectral theory of random matrices and combinatorial generating functions for enumerative geometrical and topological invariants. A self-contained summary of more advanced topics needed to understand the material is provided, as are solutions and hints for the various exercises and problems that are included throughout the text to enrich the subject matter and engage the reader. Building on knowledge of standard topics in undergraduate mathematics and basic concepts and methods of classical and quantum mechanics, this monograph is ideal for graduate students and researchers who wish to become acquainted with the full range of applications of the theory of tau functions. |
| communications math physics: Holomorphic Dynamics and Renormalization Mikhail Lyubich, Michael Yampolsky, Schwarzian derivatives and cylinder maps by A. Bonifant and J. Milnor Holomorphic dynamics: Symbolic dynamics and self-similar groups by V. Nekrashevych Are there critical points on the boundaries of mother hedgehogs? by D. K. Childers Finiteness for degenerate polynomials by L. DeMarco Cantor webs in the parameter and dynamical planes of rational maps by R. L. Devaney Simple proofs of uniformization theorems by A. A. Glutsyuk The Yoccoz combinatorial analytic invariant by C. L. Petersen and P. Roesch Bifurcation loci of exponential maps and quadratic polynomials: Local connectivity, triviality of fibers, and density of hyperbolicity by L. Rempe and D. Schleicher Rational and transcendental Newton maps by J. Ruckert Newton's method as a dynamical system: Efficient root finding of polynomials and the Riemann $\zeta$ function by D. Schleicher The external boundary of $M_2$ by V. Timorin Renormalization: Renormalization of vector fields by H. Koch Renormalization of arbitrary weak noises for one-dimensional critical dynamical systems: Summary of results and numerical explorations by O. Diaz-Espinosa and R. de la Llave KAM for the nonlinear Schrodinger equation--A short presentation by H. L. Eliasson and S. B. Kuksin Siegel disks and renormalization fixed points by M. Yampolsky |
| communications math physics: Surveys in Applied Mathematics Joseph B. Keller, David W. McLaughlin, George C. Papanicolaou, 2013-12-21 Partial differential equations play a central role in many branches of science and engineering. Therefore it is important to solve problems involving them. One aspect of solving a partial differential equation problem is to show that it is well-posed, i. e. , that it has one and only one solution, and that the solution depends continuously on the data of the problem. Another aspect is to obtain detailed quantitative information about the solution. The traditional method for doing this was to find a representation of the solution as a series or integral of known special functions, and then to evaluate the series or integral by numerical or by asymptotic methods. The shortcoming of this method is that there are relatively few problems for which such representations can be found. Consequently, the traditional method has been replaced by methods for direct solution of problems either numerically or asymptotically. This article is devoted to a particular method, called the ray method, for the asymptotic solution of problems for linear partial differential equations governing wave propagation. These equations involve a parameter, such as the wavelength. . \, which is small compared to all other lengths in the problem. The ray method is used to construct an asymptotic expansion of the solution which is valid near . . \ = 0, or equivalently for k = 21r I A near infinity. |
| communications math physics: Quantum Communications Gianfranco Cariolaro, 2015-04-08 This book demonstrates that a quantum communication system using the coherent light of a laser can achieve performance orders of magnitude superior to classical optical communications Quantum Communications provides the Masters and PhD signals or communications student with a complete basics-to-applications course in using the principles of quantum mechanics to provide cutting-edge telecommunications. Assuming only knowledge of elementary probability, complex analysis and optics, the book guides its reader through the fundamentals of vector and Hilbert spaces and the necessary quantum-mechanical ideas, simply formulated in four postulates. A turn to practical matters begins with and is then developed by: development of the concept of quantum decision, emphasizing the optimization of measurements to extract useful information from a quantum system; general formulation of a transmitter–receiver system particular treatment of the most popular quantum communications systems—OOK, PPM, PSK and QAM; more realistic performance evaluation introducing thermal noise and system description with density operators; consideration of scarce existing implementations of quantum communications systems and their difficulties with suggestions for future improvement; and separate treatment of quantum information with discrete and continuous states. Quantum Communications develops the engineering student’s exposure to quantum mechanics and shows physics students that its theories can have practically beneficial application in communications systems. The use of example and exercise questions (together with a downloadable solutions manual for instructors, available from http://extras.springer.com/) will help to make the material presented really sink in for students and invigorate subsequent research. |
| communications math physics: Chance and Chaos David Ruelle, 2020-06-16 How do scientists look at chance, or randomness, and chaos in physical systems? In answering this question for a general audience, Ruelle writes in the best French tradition: he has produced an authoritative and elegant book--a model of clarity, succinctness, and a humor bordering at times on the sardonic. |
| communications math physics: Instantons and Four-Manifolds Daniel S. Freed, Karen K. Uhlenbeck, 2012-12-06 From the reviews of the first edition: This book exposes the beautiful confluence of deep techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional manifolds, compact spaces locally modeled on the world in which we live and operate... The book is filled with insightful remarks, proofs, and contributions that have never before appeared in print. For anyone attempting to understand the work of Donaldson and the applications of gauge theories to four-dimensional topology, the book is a must. #Science#1 I would strongly advise the graduate student or working mathematician who wishes to learn the analytic aspects of this subject to begin with Freed and Uhlenbeck's book. #Bulletin of the American Mathematical Society#2 |
| communications math physics: Statistical Mechanics Giovanni Gallavotti, 2013-11-11 This clear book presents a critical and modern analysis of the conceptual foundations of statistical mechanics as laid down in Boltzmann's works. The author emphasises the relation between microscopic reversibility and macroscopic irreversibility, explaining fundamental concepts in detail. |
| communications math physics: Information Luciano Floridi, 2010-02-25 We live an information-soaked existence - information pours into our lives through television, radio, books, and of course, the Internet. Some say we suffer from 'infoglut'. But what is information? The concept of 'information' is a profound one, rooted in mathematics, central to whole branches of science, yet with implications on every aspect of our everyday lives: DNA provides the information to create us; we learn through the information fed to us; we relate to each other through information transfer - gossip, lectures, reading. Information is not only a mathematically powerful concept, but its critical role in society raises wider ethical issues: who owns information? Who controls its dissemination? Who has access to information? Luciano Floridi, a philosopher of information, cuts across many subjects, from a brief look at the mathematical roots of information - its definition and measurement in 'bits'- to its role in genetics (we are information), and its social meaning and value. He ends by considering the ethics of information, including issues of ownership, privacy, and accessibility; copyright and open source. For those unfamiliar with its precise meaning and wide applicability as a philosophical concept, 'information' may seem a bland or mundane topic. Those who have studied some science or philosophy or sociology will already be aware of its centrality and richness. But for all readers, whether from the humanities or sciences, Floridi gives a fascinating and inspirational introduction to this most fundamental of ideas. 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. |
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