Mathematics Books

Book SynopsisSince the emerging discipline of engineering enterprise systems extends traditional systems engineering to develop webs of systems and systems-of-systems, the engineering management and management science communities need new approaches for analyzing and managing risk in engineering enterprise systems. Advanced Risk Analysis in Engineering Enterprise Systems presents innovative methods to address these needs.With a focus on engineering management, the book explains how to represent, model, and measure risk in large-scale, complex systems that are engineered to function in enterprise-wide environments. Along with an analytical framework and computational model, the authors introduce new protocols: the risk co-relationship (RCR) index and the functional dependency network analysis (FDNA) approach. These protocols capture dependency risks and risk co-relationships that may exist in an enterprise. MovinTrade Review"The book develops several topics in risk analysis, including models and measurement of engineering risks, capability portfolio risk analysis and management, functional dependency network analysis, and extreme-event theory. Several chapters present some tools from applied probability and statistics … A reader with main interest in statistics may find in these chapters several ideas about the use of specific statistical tools in applied engineering. Questions and exercises are provided in each chapter to help the reader understand the main topics."—Fabrizio Durante, International Statistical Review, 2014"…excellent references … appropriate for risk engineers or quality professionals wishing to gain a comprehensive understanding of the engineering or mathematics behind advanced risk analysis. … I would certainly recommend this book to anyone working in high-risk, complex environments, such as nuclear, aerospace or explosives."—Paul Naysmith, Quality World"The book is a decidedly unique and rigorous treatment of selected topics in engineering systems risk analysis and management. The narrative is notably modern and clear. The mathematical formalism is comprehensive and advanced while remaining accessible for those involved in engineering complex systems. This is foremost a book of exciting and innovative ideas for the field, exceeding what might easily have been a rote assembly of worn methods or re-introduction of the works of others. It will be of long-standing appeal to practitioners engaged in the analysis of risk in engineering enterprise systems. The book will also appeal to scholars and researchers who will benefit from the advanced and fresh thinking it offers readers. The book will improve the systems engineering community’s ability to address enterprise design risk assessment and management across a system’s lifecycle."—Professor James Lambert, Associate Director, Center for Risk Management of Engineering Systems, University of VirginiaTable of ContentsEngineering Risk Management. Perspectives on Theories of Systems and Risk. Foundations of Risk and Decision Theory. A Risk Analysis Framework in Engineering Enterprise Systems. An Index to Measure Risk Co-Relationships. Functional Dependency Network Analysis. A Decision-Theoretic Algorithm for Ranking Risk Criticality. A Model for Measuring Risk in Engineering Enterprise Systems. Random Processes and Queuing Theory. Extreme Event Theory. Prioritization Systems in Highly Networked Environments. Risks of Extreme Events in Complex Queuing Systems. Appendix. References. Index.

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Book SynopsisParabola is a mathematics magazine published by UNSW, Sydney. Among other things, each issue of Parabola has contained a collection of puzzles/problems, on various mathematical topics and at a suitable level for younger (but mathematically sophisticated) readers.Parabolic Problems: 60 Years of Mathematical Puzzles in Parabola collects the very best of almost 1800 problems and puzzles into a single volume. Many of the problems have been re-mastered, and new illustrations have been added. Topics covered range across geometry, number theory, combinatorics, logic, and algebra. Solutions are provided to all problems, and a chapter has been included detailing some frequently useful problem-solving techniques, making this a fabulous resource for education and, most importantly, fun!Features Hundreds of diverting and mathematically interesting problems and puzzles. Accessible for anyone with a high school-level mathematics educati

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Book SynopsisLearning a computer language like R can be either frustrating, fun or boring. Having fun requires challenges that wake up the learner's curiosity but also provide an emotional reward for overcoming them. The book is designed so that it includes smaller and bigger challenges, in what I call playgrounds, in the hope that all readers will enjoy their path to R fluency. Fluency in the use of a language is a skill that is acquired through practice and exploration. For students and professionals in the biological sciences, humanities and many applied fields, recognizing the parallels between R and natural languages should help them feel at home with R. The approach I use is similar to that of a travel guide, encouraging exploration and describing the available alternatives and how to reach them. The intention is to guide the reader through the R landscape of 2024 and beyond.What is new in the second edition?Text expanded by more than 25% to include additional R features

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Book SynopsisThe goal of this unique text is to provide an experience that would facilitate a better transition for mathematics majors to the advanced proof-based courses required for their major.If you feel like you love mathematics but hate proofs, this book is for you. The change from example-based courses such as Introductory Calculus to the proof-based courses in the major is often abrupt, and some students are left with the unpleasant feeling that a subject they loved has turned into material they find hard to understand.The book exposes students and readers to some fundamental content and essential methods of constructing mathematical proofs in the context of four main courses required for the mathematics major probability, linear algebra, real analysis, and abstract algebra.Following an optional foundational chapter on background material, four short chapters, each focusing on a particular course, provide a slow-paced but rigorous introduction. Students get

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Book SynopsisThis book was developed to address a need. Quantitative Literacy courses have been established in the mathematics curriculum for decades now. The students in these courses typically dislike and fear mathematics, and the result is often a class populated by many students who are unmotivated and uninterested in the material. This book is a text for such a course; however, it is focused on a single idea that most students seem to already have some intrinsic interest in and is written at an accessible level. It covers the basic ideas of discrete probability and shows how these ideas can be applied to familiar games (roulette, poker, blackjack, etc.). The gambling material is interweaved through the book and introduced as soon as the necessary mathematics has been developed. Throughout, mathematical formalism and symbolism have been avoided, and numerous examples are provided. The book starts with a simple definition of probability, goes through some basic

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Book SynopsisWhat sets Numerical Methods and Analysis with Mathematical Modelling apart are the modelling aspects utilizing numerical analysis (methods) to obtain solutions. The authors cover first the basic numerical analysis methods with simple examples to illustrate the techniques and discuss possible errors. The modelling prospective reveals the practical relevance of the numerical methods in context to real-world problems.At the core of this text are the real-world modelling projects. Chapters are introduced and techniques are discussed with common examples. A modelling scenario is introduced that will be solved with these techniques later in the chapter. Often, the modelling problems require more than one previously covered technique presented in the book.Fundamental exercises to practice the techniques are included. Multiple modelling scenarios per numerical methods illustrate the applications of the techniques introduced. Each chapter has several modelling

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Book SynopsisA Mathematical Tour introduces readers to a selection of mathematical topics chosen for their centrality, importance, historical significance, and intrinsic appeal and beauty.

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Book SynopsisLiterature and Ecotheology: From Chaos to Cosmos challenges us in a time of climate crisis to find more common ground between the dual projects of ecocriticism and ecotheology.This book argues that in our postsecular age, literature has become an important repository of theological wisdom that can, like formal work in ecotheology, provide the moral grounds for environmental care. However, for any cosmological understanding to be adequate to the challenges before us, it must be responsive to the often-painful contingencies and uncertainties that inhere in the cosmos, something that both ecocriticism and ecotheology have often neglected. After a treatment of the ecocritical and ecotheological questions that pertain to the religious/secular divide, the study then turns to four contemporary American writersAnnie Dillard, Cormac McCarthy, Marilynne Robinson, and David James Duncanas examples. Each uses the contingency of literary form and its promise of wholeness in order t

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Book SynopsisGaussian Integrals form an integral part of many subfields of applied mathematics and physics, especially in topics such as probability theory, statistics, statistical mechanics, quantum mechanics and so on. They are essential in computing quantities such as the statistical properties of normal random variables, solving partial differential equations involving diffusion processes, and gaining insight into the properties of particles. In Gaussian Integrals and their Applications, the author has condensed the material deemed essential for undergraduate and graduate students of physics and mathematics, such that for those who are very keen would know what to look for next if their appetite for knowledge remains unsatisfied by the time they finish reading this book. Features A concise and easily digestible treatment of the essentials of Gaussian Integrals Suitable for advanced undergraduates and graduate students in mathematics

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Book SynopsisAfter 30 years of research, the author of The History of Correlation organized his notes into a manuscript draft during the lockdown months of the COVID-19 pandemic. Getting it into shape for publication took another few years. It was a labor of love.Readers will enjoy learning in detail how correlation evolved from a completely non-mathematical concept to one today that is virtually always viewed mathematically. This book reports in detail on 19th- and 20th-century English-language publications; it discusses the good and bad of many dozens of 20th-century articles and statistics textbooks in regard to their presentation and explanation of correlation. The final chapter discusses 21st-century trends.Some topics included here have never been discussed in depth by any historian. For example: Was Francis Galton lying in the first sentence of his first paper about correlation? Why did he choose the word co-relation rather than correlation for his new coeff

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Book SynopsisIntroduction to Number Theory is a classroom-tested, student-friendly text that covers a diverse array of number theory topics, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments such as cryptography, the theory of elliptic curves, and the negative solution of Hilbertâs tenth problem. The authors illustrate the connections between number theory and other areas of mathematics, including algebra, analysis, and combinatorics. They also describe applications of number theory to real-world problems, such as congruences in the ISBN system, modular arithmetic and Eulerâs theorem in RSA encryption, and quadratic residues in the construction of tournaments. Ideal for a one- or two-semester undergraduate-level course, this Second Edition: Features a more flexible structure that offers a greater range of options for course design Adds new sections on the representations of integ

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Book SynopsisThis popular textbook provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space, with emphasis upon the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions. Measure Theory and Fine Properties of Functions, Second Edition includes many interesting items working mathematical analysts need to know, but are rarely taught. Topics covered include a review of abstract measure theory, including Besicovitchâs covering theorem, Rademacherâs theorem (on the differentiability a.e. of Lipschitz continuous functions), the area and coarea formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Aleksandrovâs theorem (on the twice differentiability a.e. of convex functions). The topics are carefully selected, and the proofs are succinct, but complete. This book provides ideal reading for mathematicians and gradua

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Book SynopsisThis book explores the critical intersection of human behaviour, cybersecurity, and the transformative potential of quantum technologies. It delves into the vulnerabilities and resilience of human intelligence in the face of cyber threats, examining how cognitive biases, social dynamics, and mental health can be exploited in the digital age.Cybersecurity 2050: Protecting Humanity in a Hyper-Connected World explores the cutting-edge applications of quantum computing in cybersecurity, discussing the efficiency of quantum security algorithms on Earth and over space communications such as those needed to inhabit Mars. The challenges and opportunities of human life on extraterrestrial worlds, such as Mars, will further shape the evolution of human intelligence. The isolated and confined environment of a Martian habitat, coupled with the reliance on advanced technologies for survival, will demand new forms of adaptability, resilience, and social cooperation. The author addresses the imminent revolution in cybersecurity regulations and transforms the attention of bright minds of businesses and policymakers for the challenges and opportunities of quantum advancements. This book attempts to bridge the gap between social intelligence and cybersecurity, offering a holistic and nuanced understanding of these interconnected domains. Through real-world case studies, the author provides practical insights and strategies for adapting to the evolving technological landscape and building a more secure digital future.This book is intended for futuristic minds, computer engineers, policymakers, or regulatory experts interested in the implications of the revolution of human intelligence on cybersecurity laws and regulations. It will be of interest to cybersecurity professionals and researchers looking for a historic and comprehensive understanding of the evolving landscape, including social intelligence, quantum computing and algorithm design.

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Book SynopsisHow does homebanking work? How are board games developed? How reliable can wind energy get? How do we discover forged paintings? Do smart girls stay single? How dangerous can a bioterrorist get? In all these questions (and many others), mathematics plays a crucial role in the search for an answer.?This book tells the story behind twenty of these questions. This is explicitly not a mathematics book, but a book about the crucial role that mathematics plays in devising the creative solutions the world needs. The questions are divided into three categories: mathematics in everyday life, mathematics for the workplace, and mathematics for tomorrow''s society. The themes illustrate not only the incredibly broad applicability of mathematics in the world around us, but also the great diversity of useful mathematical techniques.Table of ContentsForeword.- Preface.- Introduction.- 1. Worry-Free Online Shopping - 2. Reach Everyone, Everywhere in Just Six Steps.- 3. Are the Polls Right?.- 4. Cochlear Implants Help the Deaf Hear Again.- 5. Google PageRank, or the Needle in the Haystack.- 6. Fun and Game Theory.- 7. High Winds on High Mills.- 8. Reducing Factory Noise.- 9. Virtual Architecture and Engineering.- 10. The Art of Forgery.- 11. The Right Bike in the Right Place.- 12. 1. Can Computers Detect Fraud? (And Do We Want Them To?).- 13. How to Predict the Future.- 14. Do Smart Girls Stay Single Longer?.- 15. What if There's More Data than Storage?.- 16. Driving Without a Driver.- 17. How Vulnerable Are Our Banking Systems?.- 18. Can We Predict Climate Change?.- 19. War and Peace.- 20. Pandemics: From Ebola and HIV to Bioterrorism and the Coronavirus.- Bibliography.- Glossary.- Translator's Note.

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Book SynopsisStop algebra from being a mathematical gatekeeper. With rich math tasks, all students can succeed.Every teacher strives to make instruction effective and interesting, yet traditional methods of teaching algebra are not working for many students! That's a problem. But the answer isn't to supplement the curriculum with random tasks.Classroom Ready-Rich Math Tasks for Grades 6-12equips you with a cohesive solution--50+ mathematical tasks that are rich, research-based, standards-aligned, and classroom-tested. The tasks: Are organized into learning progressions that help all students make the leap from arithmetic to algebra Offer students interesting mathematics problems to think about and solve so math is investigative, interactive, and engaging Provide opportunities for you to connect new content to prior knowledge or focus on an underdeveloped concept Engage students in conceptual understanding, procedural practice, and problem solving through critical thinking and application Come with downloadable planning tools, student resource pages, and extension questions Include additional support for students who may be struggling Every learner deserves opportunities to engage in meaningful, rigorous mathematics. And every teacher can develop mathematical thinking and reasoning abilities in students. Part of the bestselling series spanning elementary and middle school,Classroom-Ready Rich Algebra Tasks, Grades 6-12is a powerful add-on to any core mathematics program at your school.Trade ReviewIn this astounding book, Dougherty and Venenciano skillfully illustrate tasks that develop students’ reasoning, problem-solving, and communication skills. They thoughtfully include guiding and reflective questions along with suggestions for differentiation and extensions of each task. The wide range of topics makes this perfect for any algebra course! -- Laura Ashley YoungBring the love of learning mathematics back into your classroom by implementing effective mathematical practices, building numeracy skills in your students, and providing rich algebra tasks that are engaging. Dougherty and Venenciano have done some great work to support educators that will enhance students’ learning and grit. -- Katie MajeresDougherty and Venenciano provoke us to reframe our thinking from, "can my students get the correct answer for this task?" to "how can I use this task to help further my students’ understanding of the mathematics?" The authors provide a vision, tools, and practical advice for strategically using engaging tasks to enhance students’ understanding of mathematical ideas and processes. -- Dewey GottliebThis is a treasure trove of amazing tasks and supporting materials! Framing the tasks in an overview of research-grounded practices promotes fidelity and provides access for all students. Tasks presented in Part 2 spell out the mathematical topics, content and practice alignments, watch-fors, anticipated solutions, prompts, and post-task notes that allow them to be efficiently integrated. -- Jessica IvyIt’s important to implement rich tasks that support students’ math proficiency, drawing from their strengths and assets. It can be difficult locating or creating math tasks that are both rich and assetbased. I am excited to use tasks from this book in my work at the university with preservice teachers, in professional development with in-service teachers, and working with grades 6–12 students. -- Jonathan D. BostieClassroom-Ready Rich Algebra Tasks is a must-have resource for all algebra teachers. Dougherty and Venenciano outline how to implement these tasks in your classroom and give numerous examples that could be used today. After more than a decade of teaching algebra and supporting algebra teachers, this is a resource I wish I had when I started. -- Nick DaviesTable of ContentsForeword by Kevin Dykema Preface Acknowledgements About the Authors Part 1: GETTING STARTED WITH RICH ALGEBRA TASKS Chapter 1: RICH ALGEBRA TASKS: WHAT ARE THEY, WHY ARE THEY VALUABLE, AND HOW DO I PLAN FOR IMPLEMENTATION? Chapter 2: LAYING THE GROUNDWORK FOR TEACHING WITH RICH ALGEBRA TASKS Chapter 3: IMPLEMENTING RICH ALGEBRA TASK LESSONS Chapter 4: TASKS TO ESTABLISH MATHEMATICAL COMMUNITY Part 2: RICH ALGEBRA TASKS Chapter 5: RATIONAL NUMBER TASKS Chapter 6: EXPRESSIONS TASKS Chapter 7: EQUATIONS TASKS Chapter 8: LINEAR AND NON-LINEAR RELATIONSHIP TASKS Chapter 9: SYSTEMS OF EQUATIONS TASKS Chapter 10: POLYNOMIAL AND RATIONAL EXPRESSIONS AND EQUATIONS TASKS Chapter 11: Your Turn Appendix A References

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Book SynopsisProbabilistic numerical computation formalises the connection between machine learning and applied mathematics. Numerical algorithms approximate intractable quantities from computable ones. They estimate integrals from evaluations of the integrand, or the path of a dynamical system described by differential equations from evaluations of the vector field. In other words, they infer a latent quantity from data. This book shows that it is thus formally possible to think of computational routines as learning machines, and to use the notion of Bayesian inference to build more flexible, efficient, or customised algorithms for computation. The text caters for Masters'' and PhD students, as well as postgraduate researchers in artificial intelligence, computer science, statistics, and applied mathematics. Extensive background material is provided along with a wealth of figures, worked examples, and exercises (with solutions) to develop intuition.Trade Review'Computational methods for solving numerical problems lie at the heart of many of the technological advances in science and engineering over the last five decades, and underpin fields as diverse as artificial intelligence, climate modelling, and epidemiology. This impressive text rethinks numerical problems through the lens of probabilistic inference and decision making. This fresh perspective opens up a new chapter in this field, and suggests new and highly efficient methods. A landmark achievement!' Zoubin Ghahramani, University of Cambridge'This beautiful book is both timely and important with deep roots in powerful early exposition in numerical analysis. In this stunning and comprehensive new book, early developments from Kac and Larkin have been comprehensively built upon, formalised and extended by including modern day machine learning, numerical analysis and the formal Bayesian statistical methodology. Probabilistic Numerical methodology is of enormous importance for this age of data-centric science and Hennig, Osborne and Kersting are to be congratulated in providing us with this definitive volume.' Mark Girolami, University of Cambridge and The Alan Turing Institute'Numerical analysis is at the very heart of digital computing: every result of a computation on a digital computer is a only finite-precision representation of the true mathematical quantity where the precision is the tradeoff between computation time and accuracy. This book presents an in-depth overview of both the past and present of the newly emerging area of probabilistic numerics, where recent advances in probabilistic machine learning are used to develop principled improvements which are both faster and more accurate than classical numerical analysis algorithms. A must-read for every algorithm developer and practitioner in optimization!' Ralf Herbrich, Hasso Plattner Institute'Probabilistic Numerics spans from the intellectual fireworks of the dawn of a new field to its practical algorithmic consequences. It is precise but accessible and rich in wide-ranging, principled examples. This convergence of ideas from diverse fields in lucid style is the very fabric of good science.' Carl Edward Rasmussen, University of Cambridge'An important read for anyone who has thought about uncertainty in numerical methods; an essential read for anyone who hasn't …' John Cunningham, Columbia University'This is a rare example of a textbook that essentially founds a new field, re-casting numerics on stronger, more general foundations. A tour de force.' David Duvenaud, University of Toronto'The idea of applying probabilistic inference to the problem of numerical analysis must appear bold, possibly outrageous, even to an entrenched Bayesian statistician. Many in machine learning are now familiar with the application of Bayesian methods to problems that involve randomness, say, the estimation of quantities from noisy data. But to apply the 'calculus of uncertainty' to unknown mathematical facts, where the uncertainty arises only from our lack of knowledge, opens up a universe of new possibilities. This elegant idea is at the core of Probabilistic Numerics, and the authors succeed in demonstrating its potential to transform the way we think about computation itself. And that's not even considering what would happen if we were to apply probabilistic numerics to the numerical problems that arise from probabilistic numerics itself!' Thore Graepel, Senior Vice President, Altos Labs'… the machine learning background of the authors comes through clearly in the book … I thoroughly recommend it.' Chris J. Oates, SIAM ReviewTable of ContentsIntroduction; 1. Mathematical background; 2. Integration; 3. Linear algebra; 4. Local optimisation; 5. Global optimisation; 6. Solving ordinary differential equations; 7. The frontier; Solutions to exercises; References; Index.

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Book SynopsisWritten for senior undergraduates in all disciplines of physical science and engineering, the plain language style of this concise guide to numerical methods concentrates on developing computational skills and avoids potentially intimidating formal mathematical proofs. Including numerous worked examples and exercises, this textbook explains the practical realities of numerical techniques.Table of ContentsPreface; 1. Fitting functions to data; 2. Ordinary differential equations; 3. Two-point boundary conditions; 4. Partial differential equations; 5. Diffusion: parabolic PDEs; 6. Elliptic problems and iterative matrix solution; 7. Fluid dynamics and hyperbolic equations; 8. Boltzmann's equation and its solution; 9. Energy-resolved diffusive transport; 10. Atomistic and particle-in-cell simulation; 11. Monte Carlo techniques; 12. Monte Carlo radiation transport; 13. Next steps; Appendix A. Summary of matrix algebra; Index.

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Book SynopsisThis book introduces prime numbers and explains the celebrated, unsolved Riemann hypothesis in a direct manner. Suitable for both scholars and those with a minimal mathematical background.Trade Review'This is an extraordinary book, really one of a kind. Written by two supreme experts, but aimed at the level of an undergraduate or curious amateur, it emphasizes the really powerful ideas, with the bare minimum of math notation and the maximum number of elegant and suggestive visuals. The authors explain why this legendary problem is so beautiful, why it is difficult, and why you should care.' Will Hearst, Hearst Corporation'This book is a soaring ride, starting from the simplest ideas and ending with one of the deepest unsolved problems of mathematics. Unlike in many popular math books puffed up with anecdotal material, the authors here treat the reader as seriously interested in prime numbers and build up the real math in four stages with compelling graphical demonstrations revealing in deeper and deeper ways the hidden music of the primes. If you have ever wondered why so many mathematicians are obsessed with primes, here's the real deal.' David Mumford, Brown University, Rhode Island'This is a delightful little book, not quite like anything else that I am aware of … a splendid piece of work, informative and valuable. Undergraduate mathematics majors, and the faculty who teach them, should derive considerable benefit from looking at it.' Mark Hunacek, MAA Reviews'This book is divided into four parts, and succeeds beautifully in giving both an overview for the general audience and a sense of the details needed to understand how quickly the number of primes grows. This is accomplished through a very clear exposition and numerous illuminating pictures.' Steven Joel Miller, MathSciNet'Where popularizers of mathematics usually succumb either to a journalist's penchant for 'man bites dog' irony and spectacle or a schoolteacher's iron will to simplify away the terror, one might call the distinctive approach here 'take a lay reader to work'. Computers now provide mathematicians a laboratory, and the authors exploit this modern power to exhibit graphics, making the key equivalence a luminous phenomenon of experimental mathematics … for its clarity and the importance of its topic, this book deserves the same classic status as A Brief History of Time (CH, Jul'88). Summing Up: Essential. All readers.' D. V. Feldman, CHOICE'Prime Numbers and the Riemann Hypothesis is an agile, unusual book written over a decade, one week per year; it can be considered a sort of collaborative work, in that each version was put online with the purpose of getting feedback.' Massimo Nespolo, Acta Crystallographica Section A: Foundations and Advances'… a great gift for a curious student. Using the graphical methods found in calculus reform texts, this beautiful little book allows a patient reader with a good grasp of first-year calculus to explore the most famous unsolved problem in mathematics, the so-called Riemann Hypothesis, and to understand why it points to as yet undiscovered regularities in the distribution of prime numbers.' Donal O'Shea, The Herald Tribune'The book under review succeeds handsomely in making the case for the Riemann Hypothesis to a wide audience … Beginning with the definition of prime numbers, the authors weave their way through concrete and picturesque presentations of elementary techniques and descriptions of unsolved problems connected with the primes. They provide many insightful footnotes, concrete and illuminating figures, pointers to arXiv pages for added information, and a rich set of endnotes that contain further descriptions and details with varying levels of sophistication. After 23 short sections (a few pages each) they have arrived at a formulation of the Riemann Hypothesis in terms of counting primes up to a given size. By this point in their masterful and compelling presentation, the Hypothesis appears to be completely natural and inevitable … I have no doubt that many newcomers to the subject who have read to the end of the book will be eager to learn more and will be drawn into this fertile playground.' Peter Sarnak, Bulletin of the AMS'I really recommend this book if you want to get a feeling for the Riemann hypothesis without sinking into technicalities.' John Baez, The n-Category Café (http://golem.ph.utexas.edu/category)Table of Contents1. Thoughts about numbers; 2. What are prime numbers?; 3. 'Named' prime numbers; 4. Sieves; 5. Questions about primes; 6. Further questions about primes; 7. How many primes are there?; 8. Prime numbers viewed from a distance; 9. Pure and applied mathematics; 10. A probabilistic 'first' guess; 11. What is a 'good approximation'?; 12. Square root error and random walks; 13. What is Riemann's hypothesis?; 14. The mystery moves to the error term; 15. Césaro smoothing; 16. A view of Li(X) - π(X); 17. The prime number theorem; 18. The staircase of primes; 19. Tinkering with the staircase of primes; 20. Computer music files and prime numbers; 21. The word 'spectrum'; 22. Spectra and trigonometric sums; 23. The spectrum and the staircase of primes; 24. To our readers of part I; 25. Slopes and graphs that have no slopes; 26. Distributions; 27. Fourier transforms: second visit; 28. Fourier transform of delta; 29. Trigonometric series; 30. A sneak preview; 31. On losing no information; 32. Going from the primes to the Riemann spectrum; 33. How many θi's are there?; 34. Further questions about the Riemann spectrum; 35. Going from the Riemann spectrum to the primes; 36. Building π(X) knowing the spectrum; 37. As Riemann envisioned it; 38. Companions to the zeta function.

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Book SynopsisImre Lakatos's influential and enduring work on the nature of mathematic discovery and development continues to be relevant to philosophers of mathematics. Including a specially commissioned preface written by Paolo Mancosu, and presented in a fresh twenty-first-century series livery, it is now available for a new generation of readers.Trade Review'For anyone interested in mathematics who has not encountered the work of the late Imre Lakatos before, this book is a treasure; and those who know well the famous dialogue, first published in 1963–4 in the British Journal for the Philosophy of Science, that forms the greater part of this book, will be eager to read the supplementary material … the book, as it stands, is rich and stimulating, and, unlike most writings on the philosophy of mathematics, succeeds in making excellent use of detailed observations about mathematics as it is actually practised.' Michael Dummett, Nature'The whole book, as well as being a delightful read, is of immense value to anyone concerned with mathematical education at any level.' C. W. Kilmister, The Times Higher Education Supplement'In this book the late Imre Lakatos explores 'the logic of discovery' and 'the logic of justification' as applied to mathematics … The arguments presented are deep … but the author's lucid literary style greatly facilitates their comprehension … The book is destined to become a classic. It should be read by all those who would understand more about the nature of mathematics, of how it is created and how it might best be taught.' Education'How is mathematics really done, and - once done - how should it be presented? Imre Lakatos had some very strong opinions about this. The current book, based on his PhD work under George Polya, is a classic book on the subject. It is often characterized as a work in the philosophy of mathematics, and it is that - and more. The argument, presented in several forms, is that mathematical philosophy should address the way that mathematics is done, not just the way it is often packaged for delivery.' William J. Satzer, MAA ReviewsTable of ContentsPreface to this edition Paolo Mancosu; Editors' preface; Acknowledgments; Author's introduction; Part I: 1. A problem and a conjecture; 2. A proof; 3. Criticism of the proof by counterexamples which are local but not global; 4. Criticism of the conjecture by global counterexamples; 5. Criticism of the proof-analysis by counterexamples which are global but not local. The problem of rigour; 6. Return to criticism of the proof by counterexamples which are local but not global. The problem of content; 7. The problem of content revisited; 8. Concept-formation; 9. How criticism may turn mathematical truth into logical truth; Part II: Editors' introduction; Appendix 1. Another case-study in the method of proofs and refutations; Appendix 2. The deductivist versus the heuristic approach; Bibliography; Index of names; Index of subjects.

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Book SynopsisTheodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. It can be used as a course text or for self study.Trade ReviewReview of previous edition: '… highly readable and enjoyable … The book will make an excellent course text or self-study manual for this interesting subject.' Physics TodayReview of previous edition: 'This book provides a highly detailed account of the intricacies involved in considering geometrical concepts.' Contemporary Physics'If you're looking for a well-written and well-motivated introduction to differential geometry, this one looks hard to beat.' Fernando Q. Gouvêa, MAA Online'… a first rate introductory textbook … the style is lively and exposition is clear which make the text easy to read … This book will be beneficial to students and scientists wishing to learn the foundations of differential geometry and algebraic topology as well as geometric formulations of modern physical theories.' Pure and Applied Geophysics'… this book should not be missing in any physics or mathematics library.' European Mathematical Society'This book is a great read and has a lot to offer to graduate students in both mathematics and physics. I wish I had had it on my desk when I began studying geometry.' AMS ReviewReview of previous edition: 'The layout, the typography and the illustrations of this advanced textbook on modern mathematical methods are all very impressive and so are the topics covered in the text.' Zentralblatt für Mathematik und ihre Grenzgebiete'… contains a wealth of interesting material for both the beginning and the advanced levels. The writing may feel informal but it is precise - a masterful exposition. Users of this 'introduction' will be well prepared for further study of differential geometry and its use in physics and engineering … As did earlier editions, this third edition will continue to promote the language with which mathematicians and scientists can communicate.' Jay P. Fillmore, SIAM ReviewTable of ContentsPreface to the Third Edition; Preface to the Second Edition; Preface to the revised printing; Preface to the First Edition; Overview; Part I. Manifolds, Tensors, and Exterior Forms: 1. Manifolds and vector fields; 2. Tensors and exterior forms; 3. Integration of differential forms; 4. The Lie derivative; 5. The Poincaré Lemma and potentials; 6. Holonomic and nonholonomic constraints; Part II. Geometry and Topology: 7. R3 and Minkowski space; 8. The geometry of surfaces in R3; 9. Covariant differentiation and curvature; 10. Geodesics; 11. Relativity, tensors, and curvature; 12. Curvature and topology: Synge's theorem; 13. Betti numbers and De Rham's theorem; 14. Harmonic forms; Part III. Lie Groups, Bundles, and Chern Forms: 15. Lie groups; 16. Vector bundles in geometry and physics; 17. Fiber bundles, Gauss–Bonnet, and topological quantization; 18. Connections and associated bundles; 19. The Dirac equation; 20. Yang–Mills fields; 21. Betti numbers and covering spaces; 22. Chern forms and homotopy groups; Appendix A. Forms in continuum mechanics; Appendix B. Harmonic chains and Kirchhoff's circuit laws; Appendix C. Symmetries, quarks, and Meson masses; Appendix D. Representations and hyperelastic bodies; Appendix E. Orbits and Morse–Bott theory in compact Lie groups.

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Book SynopsisWritten for non-science majors taking introductory astronomy courses, this book explains how to use mathematics to solve problems in astronomy. It features plain-language explanations, dozens of fully worked examples, over 150 exercises and homework problems, and a rich set of online supporting materials.Trade Review'For the budding student of astronomy with a phobia of numbers and equations comes this book to the rescue … a great introduction to the maths of astronomy.' Astronomy Now'A strong feature of the book is the excellent selection of instructive problems at the end of each chapter. Could be useful if you are desperately trying to think of some questions for your mid-term exams.' The ObservatoryTable of Contents1. Fundamentals; 2. Gravity; 3. Light; 4. Parallax, angular size, and angular resolution; 5. Stars; 6. Black holes and cosmology; References; Index.

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