Best of
Mathematics

2008

The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine


Charles Petzold - 2008
    Turing Mathematician Alan Turing invented an imaginary computer known as the Turing Machine; in an age before computers, he explored the concept of what it meant to be "computable," creating the field of computability theory in the process, a foundation of present-day computer programming.The book expands Turing's original 36-page paper with additional background chapters and extensive annotations; the author elaborates on and clarifies many of Turing's statements, making the original difficult-to-read document accessible to present day programmers, computer science majors, math geeks, and others.Interwoven into the narrative are the highlights of Turing's own life: his years at Cambridge and Princeton, his secret work in cryptanalysis during World War II, his involvement in seminal computer projects, his speculations about artificial intelligence, his arrest and prosecution for the crime of "gross indecency," and his early death by apparent suicide at the age of 41.

Kiss My Math: Showing Pre-Algebra Who's Boss


Danica McKellar - 2008
    Kiss My Math: Showing Pre-Algebra Who's Boss

Euler's Gem: The Polyhedron Formula and the Birth of Topology


David S. Richeson - 2008
    Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea.From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map.Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.

What's Math Got to Do with It?: Helping Children Learn to Love Their Least Favorite Subject--and Why It's Important for America


Jo Boaler - 2008
    When the level of spending was taken into account, we sank to the very bottom of the list. According to Jo Boaler, who was a professor of mathematics education at Stanford University for nine years, statistics like these are becoming all too common—we have reached the point of crisis, and a new course of action is crucial. In this straightforward and inspiring book, Boaler outlines the nature of the problem by following the progress of students in middle and high schools over a number of years, to find out which teaching methods are exciting students and getting results. Based on her research, she presents concrete solutions that will help reverse the trend, including classroom approaches, essential strategies for students, advice for parents on how to help children enjoy mathematics, and ways to work with teachers in schools.The United States is continuing to fall rapidly behind the rest of the developed world when it comes to math education, and the future of our economy depends on the quality of teaching that our children receive today. In What’s Math Got to Do with It?, Jo Boaler offers us a new way forward, making this book in dispensable for all parents and educators, as well as anyone interested in the mathematical and scientific future of our society.

The Symmetries of Things


John H. Conway - 2008
    Repeat it in some way--translation, reflection over a line, rotation around a point--and you have created symmetry. Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments.This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

Problems From The Book


Titu Andreescu - 2008
    In this volume they present innumerable beautiful results, intriguing problems, and ingenious solutions. The problems range from elementary gems to deep truths. A trully delightful and highly instructive book, this will prepare the engaged reader not only for any mathematics competition they may enter but also for a life time of mathematical enjoyment. A must for the bookshelves of both aspiring and seasoned mathematicians.

Social and Economic Networks


Matthew O. Jackson - 2008
    The many aspects of our lives that are governed by social networks make it critical to understand how they impact behavior, which network structures are likely to emerge in a society, and why we organize ourselves as we do. In Social and Economic Networks, Matthew Jackson offers a comprehensive introduction to social and economic networks, drawing on the latest findings in economics, sociology, computer science, physics, and mathematics. He provides empirical background on networks and the regularities that they exhibit, and discusses random graph-based models and strategic models of network formation. He helps readers to understand behavior in networked societies, with a detailed analysis of learning and diffusion in networks, decision making by individuals who are influenced by their social neighbors, game theory and markets on networks, and a host of related subjects. Jackson also describes the varied statistical and modeling techniques used to analyze social networks. Each chapter includes exercises to aid students in their analysis of how networks function.This book is an indispensable resource for students and researchers in economics, mathematics, physics, sociology, and business.

Naive Lie Theory


John Stillwell - 2008
    In order to achieve this, the author focuses on the so-called "classical groups, '' viewed as matrix groups with real, complex, or quaternion entries. This allows them to be studied by elementary methods from calculus and linear algebra. Each chapter is enhanced with numerous exercises, discussion of further results, and historical comments.

Guaranteed to Mash Your Mind (Murderous Maths)


Kjartan Poskitt - 2008
    Maths with the laughs added in! \* How can maths save your life? \* Can you make a piece of one-sided paper? \* How can you be famous for ever? Find out in More Murderous Maths - all the tricks, tips and short-cuts they don't teach at school.

Logical Labyrinths


Raymond M. Smullyan - 2008
    It serves as a bridge from the author's puzzle books to his technical writing in the fascinating field of mathematical logic. Using the logic of lying and truth-telling, the author introduces the readers to informal reasoning preparing them for the formal study of symbolic logic, from propositional logic to first-order logic, a subject that has many important applications to philosophy, mathematics, and computer science. The book includes a journey through the amazing labyrinths of infinity, which have stirred the imagination of mankind as much, if not more, than any other subject.

The Alphabet That Changed the World: How Genesis Preserves a Science of Consciousness in Geometry and Gesture


Stan Tenen - 2008
    Author Stan Tenen demonstrates that each letter is also a hand gesture, and it is at this level that Hebrew forms a natural universal language. All people, including children before they speak and people without sight, make natural use of these gestures.   In The Alphabet That Changed the World, Tenen examines the Hebrew text of Genesis and its relationship to the alphabet. He shows how each letter is both concept and gesture, with the form of the gesture matching the function of the concept. There is thus an implicit relationship between the physical world of function and the conscious world of concept. Using over 200 color illustrations, Tenen demonstrates geometric metaphor as the best framework for understanding the deepest meaning of the text.   Such geometry models embryonic growth and self-organization and the core of many healing and meditative practices. Many subjects in contemporary science were derived from the methods and means available to the ancients; The Alphabet That Changed the World makes this authoritative recovery of the “science of consciousness” in Genesis accessible for the first time to the contemporary reading public.

Riemannian Geometry: Theory & Applications


Manfredo P. Do Carmo - 2008
    The author's treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed "superb" by teachers who have used the text.A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.

An Introduction to Mathematical Cryptography


Jeffrey Hoffstein - 2008
    It focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems.

Kiselev's Geometry / Book II. Stereometry


A.P. Kiselev - 2008
    It contains the chapters Lines and Planes, Polyhedra, Round Solids, which include the traditional material about dihedral and polyhedral angles, Platonic solids, symmetry and similarity of space figures, volumes and surface areas of prisms, pyramids, cylinders, cones and balls. The English edition also contains a new chapter Vectors and Foundations (written by A. Givental) about vectors, their applications, vector foundations of Euclidean geometry, and introduction to spherical and hyperbolic geometries. This volume completes the English adaptation of Kiselev's Geometry whose 1st part ( Book I. Planimetry ), dedicated to plane geometry, was published by Sumizdat in 2006 as ISBN 0977985202. Both volumes of Kiselev's Geometry are praised for precision, simplicity and clarity of exposition, and excellent collection of exercises. They dominated Russian math education for several decades, were reprinted in dozens of millions of copies, influenced geometry education in Eastern Europe and China, and are still active as textbooks for 7-11 grades. The books are adapted to the modern US curricula by a professor of mathematics from UC Berkeley.

Mathematics for Physical Chemistry: Opening Doors


Donald A. McQuarrie - 2008
    McQuarrie provides a quick review of the mathematical methods that are used throughout chemistry. The text is also ideal as a supplement with any traditional textbook on physical or quantum chemistry.

Feedback Systems: An Introduction for Scientists and Engineers


Karl Johan Åström - 2008
    It is an ideal textbook for undergraduate and graduate students, and is indispensable for researchers seeking a self-contained reference on control theory. Unlike most books on the subject, Feedback Systems develops transfer functions through the exponential response of a system, and is accessible across a range of disciplines that utilize feedback in physical, biological, information, and economic systems.Karl Åström and Richard Murray use techniques from physics, computer science, and operations research to introduce control-oriented modeling. They begin with state space tools for analysis and design, including stability of solutions, Lyapunov functions, reachability, state feedback observability, and estimators. The matrix exponential plays a central role in the analysis of linear control systems, allowing a concise development of many of the key concepts for this class of models. Åström and Murray then develop and explain tools in the frequency domain, including transfer functions, Nyquist analysis, PID control, frequency domain design, and robustness. They provide exercises at the end of every chapter, and an accompanying electronic solutions manual is available. Feedback Systems is a complete one-volume resource for students and researchers in mathematics, engineering, and the sciences. Covers the mathematics needed to model, analyze, and design feedback systems Serves as an introductory textbook for students and a self-contained resource for researchers Includes exercises at the end of every chapter Features an electronic solutions manual Offers techniques applicable across a range of disciplines

Analytic Combinatorics


Philippe Flajolet - 2008
    Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

General Relativity and the Einstein Equations


Yvonne Choquet-Bruhat - 2008
    It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. In this monograph, aimed at researchers in mathematics and physics, the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.

A Primer For The Mathematics Of Financial Engineering


Dan Stefanica - 2008
    The financial applications in the book range from basics such as the Put-Call parity, bond duration and convexity, and the Black-Scholes model, to more advanced topics, such as numerical estimation of the Greeks, implied volatility, and bootstrapping for finding interest rate curves. On the mathematical side, useful but sometimes overlooked topics are presented in detail: differentiating integrals with respect to nonconstant integral limits, numerical approximation of definite integrals, convergence of Taylor series expansions, finite difference approximations, Stirling's formula, Lagrange multipliers, polar coordinates, Newton's method for higher dimensional problems. Every chapter concludes with exercises that are a mix of mathematical and financial questions, with comments regarding their relevance to practice and to more advanced topics.

Optimal Transport: Old and New


Cédric Villani - 2008
    This book presents a broad overview of this area. PhD students or researchers can read the entire book without any prior knowledge of the field.

From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory


Jean-Pierre Marquis - 2008
    The main thesis is that Klein's Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane's work in the early 1940's and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics.From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.

Sacred Mathematics: Japanese Temple Geometry


Hidetoshi Fukagawa - 2008
    During that time, a unique brand of homegrown mathematics flourished, one that was completely uninfluenced by developments in Western mathematics. People from all walks of life--samurai, farmers, and merchants--inscribed a wide variety of geometry problems on wooden tablets called sangaku and hung them in Buddhist temples and Shinto shrines throughout Japan. Sacred Mathematics is the first book published in the West to fully examine this tantalizing--and incredibly beautiful--mathematical tradition.Fukagawa Hidetoshi and Tony Rothman present for the first time in English excerpts from the travel diary of a nineteenth-century Japanese mathematician, Yamaguchi Kanzan, who journeyed on foot throughout Japan to collect temple geometry problems. The authors set this fascinating travel narrative--and almost everything else that is known about temple geometry--within the broader cultural and historical context of the period. They explain the sacred and devotional aspects of sangaku, and reveal how Japanese folk mathematicians discovered many well-known theorems independently of mathematicians in the West--and in some cases much earlier. The book is generously illustrated with photographs of the tablets and stunning artwork of the period. Then there are the geometry problems themselves, nearly two hundred of them, fully illustrated and ranging from the utterly simple to the virtually impossible. Solutions for most are provided.A unique book in every respect, Sacred Mathematics demonstrates how mathematical thinking can vary by culture yet transcend cultural and geographic boundaries.

The Lost Notebook and Other Unpublished Papers of Srinivasa Ramanujan


Srinivasa Ramanujan - 2008
    Ramanujan was brought to light in 1976 as part of the Watson bequest, by G.E. Andrews with whose introduction this collection of unpublished manuscripts opens. A major portion of the "Lost Notebook" - really just 90 unpaginated sheets of work on "q"-series and other topics - is reproduced here in facsimile. Letters from Ramanujan to Hardy as well as various other sheets of seemingly related notes are then included, on topics including coefficients in the 1/q3 and 1/q2 problems and the mock theta functions. The next 180 pages consist of unpublished manuscripts of Ramanujan, including 28 pages from the 'Loose Papers held in the Trinity College Library. Finally a number of interesting letters that were exchanged between Ramanujan, Littlewood, Hardy and Watson, with a bearing on Ramanujan's work are collected together here with other extracts and fragments.

Dynamical Processes on Complex Networks


Alain Barrat - 2008
    Until recently these systems were considered as haphazard sets of points and connections. Recent advances have generated a vigorous research effort in understanding the effect of complex connectivity patterns on dynamical phenomena. For example, a vast number of everyday systems, from the brain to ecosystems, power grids and the Internet, can be represented as large complex networks. This new and recent account presents a comprehensive explanation of these effects.

Structure and Randomness: Pages from Year One of a Mathematical Blog


Terence Tao - 2008
    This book is based on a selection of articles from the first year of that blog. These articles discuss a wide range of mathematics and its applications.

Discrete Mathematics: An Introduction to Proofs and Combinatorics


Kevin Ferland - 2008
    The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn. Throughout the text, engaging anecdotes and topics of interest inform as well as motivate learners. The text is ideal for one- or two-semester courses and for students who are typically mathematics, mathematics education, or computer science majors. Part I teaches student how to write proofs; Part II focuses on computation and problem solving. The second half of the book may also be suitable for introductory courses in combinatorics and graph theory.

Lectures on Elementary Mathematics


Joseph-Louis Lagrange - 2008
    He delivered these lectures on arithmetic, algebra, and geometry at the École Normale, a training school for teachers. An exemplar among elementary expositions, they feature both originality of thought and elegance of expression.

The Foundations of Mathematics


Thomas Q. Sibley - 2008
    The concepts are introduced in a pedagogically effective manner without compromising mathematical accuracy and completeness. Thus, in Part I students explore concepts before they use them in proofs. The exercises range from reading comprehension questions and many standard exercises to proving more challenging statements, formulating conjectures and critiquing a variety of false and questionable proofs. The discussion of metamathematics, including G�del's Theorems, and philosophy of mathematics provides an unusual and valuable addition compared to other similar texts

Digital Dice: Computational Solutions to Practical Probability Problems


Paul J. Nahin - 2008
    But even the hardest of these problems can often be solved with a computer and a Monte Carlo simulation, in which a random-number generator simulates a physical process, such as a million rolls of a pair of dice. This is what Digital Dice is all about: how to get numerical answers to difficult probability problems without having to solve complicated mathematical equations.Popular-math writer Paul Nahin challenges readers to solve twenty-one difficult but fun problems, from determining the odds of coin-flipping games to figuring out the behavior of elevators. Problems build from relatively easy (deciding whether a dishwasher who breaks most of the dishes at a restaurant during a given week is clumsy or just the victim of randomness) to the very difficult (tackling branching processes of the kind that had to be solved by Manhattan Project mathematician Stanislaw Ulam). In his characteristic style, Nahin brings the problems to life with interesting and odd historical anecdotes. Readers learn, for example, not just how to determine the optimal stopping point in any selection process but that astronomer Johannes Kepler selected his second wife by interviewing eleven women.The book shows readers how to write elementary computer codes using any common programming language, and provides solutions and line-by-line walk-throughs of a MATLAB code for each problem.Digital Dice will appeal to anyone who enjoys popular math or computer science.

How to Succeed in College Mathematics: A Guide for the College Mathematics Student


Richard Dahlke - 2008
    There is no book in existence today that addresses so comprehensively, authoritatively, and authentically the issues students face in college mathematics. The ideas in this guide, grounded in research and experience, could very well keep students from receiving an undesirable grade, dropping a course, changing an area of concentration, or dropping out of college. This guide will help students realize fully the mathematics potential that lies within them.There are many gems of wisdom for both the struggling and strong college mathematics student, and all levels in between. Readers will find out how to: improve as a problem solver; manage assignments; determine if they have course prerequisites; write mathematics; work with symbolic form; read their textbooks; get the most out of class; make the most of their learning style; work with classmates; select and work with their instructors; determine if they should drop a course; retake a course; decrease procrastination and anxiety; increase confidence and motivation; prepare for and take examinations; manage time, a job, and scheduling of classes.The primary organization of the book is this: A key topic is presented in a thorough way, and is often referenced again when related topics are presented in thorough ways. Hence, this spiraling effect reinforces knowledge of the topic. The book has an extensive Table of Contents and Index, clearly revealing the issues that are discussed and where they can be found. Thus, it is easy to find what is sought. Chapters may be related to each other, but they stand alone. Hence, the reader can begin to read at almost any place in the book, and move back and forth to chapters or sections of interest to them. Support for the relevance of the learning and study issues discussed includes: remarks from successful college mathematics students on what worked for them in their college mathematics courses, and from unsuccessful college mathematics students on what did not work for them; statements by college mathematic students which came from their free responses to questions on forms used to evaluate their instructors and courses; suggestions from experienced college mathematics instructors on issues they believe need to be addressed; the formal academic training, research activities, and extensive and varied college teaching experience of the author in mathematics and mathematics education.This resource is not only valuable for college mathematics students, but also for high students intending to take college mathematics; college and high school mathematics instructors; preservice teachers of high school mathematics; college and high school counselors and other academic support personnel.

Statistical Design


George Casella - 2008
    In this book the basic theoretical underpinnings are covered. It describes the principles that drive good designs and good statistics.Design played a key role in agricultural statistics and set down principles of good practice, principles that still apply today. Statistical design is all about understanding where the variance comes from, and making sure that is where the replication is. Indeed, it is probably correct to say that these principles are even more important today.

Finite Group Theory


I. Martin Isaacs - 2008
    It includes semidirect products, the Schur-Zassenhaus theorem, the theory of commutators, coprime actions on groups, transfer theory, Frobenius groups, primitive and multiply transitive permutation groups, the simplicity of the PSL groups, the generalized Fitting subgroup and also Thompson's J-subgroup and his normal $p$-complement theorem. Topics that seldom (or never) appear in books are also covered. These include subnormality theory, a group-theoretic proof of Burnside's theorem about groups with order divisible by just two primes, the Wielandt automorphism tower theorem, Yoshida's transfer theorem, the ``principal ideal theorem'' of transfer theory and many smaller results that are not very well known. Proofs often contain original ideas, and they are given in complete detail. In many cases they are simpler than can be found elsewhere. The book is largely based on the author's lectures, and consequently, the style is friendly and somewhat informal. Finally, the book includes a large collection of problems at disparate levels of difficulty. These should enable students to practice group theory and not just read about it. Martin Isaacs is professor of mathematics at the University of Wisconsin, Madison. Over the years, he has received many teaching awards and is well known for his inspiring teaching and lecturing. He received the University of Wisconsin Distinguished Teaching Award in 1985, the Benjamin Smith Reynolds Teaching Award in 1989, and the Wisconsin Section MAA Teaching Award in 1993, to name only a few. He was also honored by being the selected MAA Polya Lecturer in 2003-2005.

Support Vector Machines


Ingo Steinwart - 2008
    David Hilbert The goal of this book is to explain the principles that made support vector machines (SVMs) a successful modeling and prediction tool for a variety of applications. We try to achieve this by presenting the basic ideas of SVMs together with the latest developments and current research questions in a uni?ed style. In a nutshell, we identify at least three reasons for the success of SVMs: their ability to learn well with only a very small number of free parameters, their robustness against several types of model violations and outliers, and last but not least their computational e?ciency compared with several other methods. Although there are several roots and precursors of SVMs, these methods gained particular momentum during the last 15 years since Vapnik (1995, 1998) published his well-known textbooks on statistical learning theory with aspecialemphasisonsupportvectormachines. Sincethen, the?eldofmachine learninghaswitnessedintenseactivityinthestudyofSVMs, whichhasspread moreandmoretootherdisciplinessuchasstatisticsandmathematics. Thusit seems fair to say that several communities are currently working on support vector machines and on related kernel-based methods. Although there are many interactions between these communities, we think that there is still roomforadditionalfruitfulinteractionandwouldbegladifthistextbookwere found helpful in stimulating further research. Many of the results presented in this book have previously been scattered in the journal literature or are still under review. As a consequence, these results have been accessible only to a relativelysmallnumberofspecialists, sometimesprobablyonlytopeoplefrom one community but not the others.

The Humongous Book of Algebra Problems


W. Michael Kelley - 2008
    Students will learn how to interpret and solve problems as they are typically presented in algebra courses—and become prepared to solve those problems that were never discussed in class but always seem to find their way onto exams. • Annotations throughout the text clarify each problem and fill in missing steps needed to reach the solution, making this book like no other algebra workbook on the market

The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway


Jan Hendrik Bruinier - 2008
    The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture.Each part treats a number of beautiful applications.

Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach


Wassim M. Haddad - 2008
    Dynamical system theory lies at the heart of mathematical sciences and engineering. The application of dynamical systems has crossed interdisciplinary boundaries from chemistry to biochemistry to chemical kinetics, from medicine to biology to population genetics, from economics to sociology to psychology, and from physics to mechanics to engineering. The increasingly complex nature of engineering systems requiring feedback control to obtain a desired system behavior also gives rise to dynamical systems.Wassim Haddad and VijaySekhar Chellaboina provide an exhaustive treatment of nonlinear systems theory and control using the highest standards of exposition and rigor. This graduate-level textbook goes well beyond standard treatments by developing Lyapunov stability theory, partial stability, boundedness, input-to-state stability, input-output stability, finite-time stability, semistability, stability of sets and periodic orbits, and stability theorems via vector Lyapunov functions. A complete and thorough treatment of dissipativity theory, absolute stability theory, stability of feedback systems, optimal control, disturbance rejection control, and robust control for nonlinear dynamical systems is also given. This book is an indispensable resource for applied mathematicians, dynamical systems theorists, control theorists, and engineers.

Oxford Handbook of the History of Mathematics


Jacqueline A. Stedall - 2008
    It addresses questions of who creates mathematics, who uses it, and how. A broader understanding of mathematical practitioners naturally leads to a new appreciation of what counts as a historical source. Material and oral evidence is drawn upon as well as an unusual array of textual sources. Further, the ways in which people have chosen to express themselves are as historically meaningful as the contents of the mathematics they have produced. Mathematics is not a fixed and unchanging entity. New questions, contexts, and applications all influence what counts as productive ways of thinking. Because the history of mathematics should interact constructively with other ways of studying the past, the contributors to this book come from a diverse range of intellectual backgrounds in anthropology, archaeology, art history, philosophy, and literature, as well as history of mathematics more traditionally understood. The thirty-six self-contained, multifaceted chapters, each written by a specialist, are arranged under three main headings: 'Geographies and Cultures', 'Peoples and Practices', and 'Interactions and Interpretations'. Together they deal with the mathematics of 5000 years, but without privileging the past three centuries, and an impressive range of periods and places with many points of cross-reference between chapters. The key mathematical cultures of North America, Europe, the Middle East, India, and China are all represented here as well as areas which are not often treated in mainstream history of mathematics, such as Russia, the Balkans, Vietnam, and South America. This Handbook will be a vital reference for graduates and researchers in mathematics, historians of science, and general historians.

Computational Commutative Algebra 1


Martin Kreuzer - 2008
    (Douglas R. Hofstadter) Dear Reader, what you are holding in your hands now is for youabook. But for us, for our families and friends, it has been known as the book over the last three years. Three years of intense work just to ?ll three centimeters of your bookshelf! This amounts to about one centimeter per year, or roughly two-?fths of an inch per year if you are non-metric. Clearly we had ample opportunity to experience the full force of Hofstadter s Law. Writing a book about Computational Commutative Algebra is not - like computing a Gr] obner basis: you need unshakeable faith to believe that the project will ever end; likewise, you must trust in the Noetherianity of polynomial rings to believe that Buchberger s Algorithm will ever terminate. Naturally, we hope that the ?nal result proves our e?orts worthwhile. This is a book for learning, teaching, reading, and, most of all, enjoying the topic at hand."

Recountings: Conversations with Mit Mathematicians


Joel Segel - 2008
    The process reveals much about the motivation, path, and impact of research mathematicians in a society that owes so much to this little understood and often mystifying section of its intellectual fabric. At a time when the mathematical experience touches and attracts more laypeople than ever, such a book contributes to our understanding and entertains through its personal approach.

Applied Calculus of Variations for Engineers


Louis Komzsik - 2008
    Applied Calculus of Variations for Engineers addresses this very important mathematical area applicable to many engineering disciplines. Its unique, application-oriented approach sets it apart from the theoretical treatises of most texts. It is aimed at enhancing the engineer's understanding of the topic as well as aiding in the application of the concepts in a variety of engineering disciplines.The first part of the book presents the fundamental variational problem and its solution via the Euler-Lagrange equation. It also discusses variational problems subject to constraints, the inverse problem of variational calculus, and the direct solution techniques of variational problems, such as the Ritz, Galerkin, and Kantorovich methods. With an emphasis on applications, the second part details the geodesic concept of differential geometry and its extensions to higher order spaces. It covers the variational origin of natural splines and the variational formulation of B-splines under various constraints. This section also focuses on analytic and computational mechanics, explaining classical mechanical problems and Lagrange's equations of motion.

Integrated Algebra


Lawrence S. Leff - 2008
    These ever popular guides contain study tips, test-taking strategies, score analysis charts, and other valuable features. They are an ideal source of practice and test preparation. The detailed answer explanations make each exam a practical learning experience. The book reviews all pertinent math topics, including sets, algebraic language, linear equations and formulas, ratios, rates and proportions, polynomials and factoring, rational expressions and factoring, radicals and right triangles, area and volume, quadratic and exponential functions, and much more.

Markov Chains and Mixing Times


David Asher Levin - 2008
    The main goals of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space.

Mathematics Emerging: A Sourcebook 1540 - 1900


Jacqueline A. Stedall - 2008
    Mathematics has an amazingly long and rich history, it has been practised in every society and culture, with written records reaching back in some cases as far as four thousand years. This book will focus on just a small part of the story, in a sense the most recent chapter of it: the mathematics of western Europe from the sixteenth to the nineteenth centuries. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. Almost every source is given in its original form, not just in the language in which it was first written, but as far as practicable in the layout and typeface in which it was read by contemporaries. This book is designed to provide mathematics undergraduates with some historical background to the material that is now taught universally to students in their final years at school and the first years at college or university: the core subjects of calculus, analysis, and abstract algebra, along with others such as mechanics, probability, and number theory. All of these evolved into their present form in a relatively limited area of western Europe from the mid sixteenth century onwards, and it is there that we find the major writings that relate in a recognizable way to contemporary mathematics.

Geometric Integration Theory


Steven G. Krantz - 2008
    In more modern times, the Plateau problem is considered to be the wellspring of questions in geometric measure theory. Named in honor of the nineteenth century Belgian physicist Joseph Plateau, who studied surface tension phenomena in general, andsoap?lmsandsoapbubblesinparticular, thequestion(initsoriginalformulation) was to show that a ?xed, simple, closed curve in three-space will bound a surface of the type of a disk and having minimal area. Further, one wishes to study uniqueness for this minimal surface, and also to determine its other properties. Jesse Douglas solved the original Plateau problem by considering the minimal surfacetobeaharmonicmapping(whichoneseesbystudyingtheDirichletintegral). For this work he was awarded the Fields Medal in 1936. Unfortunately, Douglas's methods do not adapt well to higher dimensions, so it is desirable to ?nd other techniques with broader applicability. Enter the theory of currents. Currents are continuous linear functionals on spaces of differential forms.

Chaos and Coarse Graining in Statistical Mechanics


Patrizia Castiglione - 2008
    This book presents the basic aspects of chaotic systems, with emphasis on systems composed by huge numbers of particles. Firstly, the basic concepts of chaotic dynamics are introduced, moving on to explore the role of ergodicity and chaos for the validity of statistical laws, and ending with problems characterized by the presence of more than one significant scale. Also discussed is the relevance of many degrees of freedom, coarse graining procedure, and instability mechanisms in justifying a statistical description of macroscopic bodies. Introducing the tools to characterize the non asymptotic behaviors of chaotic systems, this text will interest researchers and graduate students in statistical mechanics and chaos.

Scientific Computing with Case Studies


Dianne P. O'Leary - 2008
    The book provides a practical guide to the numerical solution of linear an nonlinear equations, differential equations, optimization problems, and eigenvalue problems. It treats, standard problems and introduces important variants such as space systems, differential-algebraic equations, constrained optimization, Monte Carlo simulations, and parametric studies. Stability and error analysis is emphasized, and the MATLAB algorithms are grounded in sound principles of software design and in the understanding of machine arithmetic and memory management.Nineteen case studies allow readers to become familiar with mathematical modeling and algorithm design, motivated by problems in physics, engineering, epidemiology, chemistry, and biology. A Web site provides solutions to the challenges that are offered throughout the book and also supplies relevant MATLAB codes derivations, and supplementary notes and slides.The book is intended as a primary text for courses in numerical analysis, scientific computing, and computational science for advanced undergraduate and early graduate students. Physicist, chemists, biologists, earth scientists, astronomers, and engineers whose work involves numerical computing also will find the book useful as a reference and tool for self-study.--back cover

Foundation Maths


Anthony Croft - 2008
    It is ideally suited to those studying marketing, business studies, management, science, engineering, social science, geography, combined studies and design. It will be useful for those who lack confidence and who need careful, steady guidance in mathematical methods. For those whose mathematical expertise is already established, the book will be a helpful revision and reference guide. The style of the book also makes it suitable for self-study and distance learning.

Information Geometry: Near Randomness and Near Independence


Khadiga Arwini - 2008
    Our approach uses information geometry to provide a c- mon context but we need only rather elementary material from di?erential geometry, information theory and mathematical statistics. Introductory s- tions serve together to help those interested from the applications side in making use of our methods and results. We have available Mathematica no- books to perform many of the computations for those who wish to pursue their own calculations or developments. Some 44 years ago, the second author ?rst encountered, at about the same time, di?erential geometry via relativity from Weyl's book [209] during - dergraduate studies and information theory from Tribus [200, 201] via spatial statistical processes while working on research projects at Wiggins Teape - searchandDevelopmentLtd-cf. theForewordin[196]and[170,47,58]. H- ing started work there as a student laboratory assistant in 1959, this research environment engendered a recognition of the importance of international c- laboration, and a lifelong research interest in randomness and near-Poisson statistical geometric processes, persisting at various rates through a career mainly involved with global di?erential geometry. From correspondence in the 1960s with Gabriel Kron [4, 124, 125] on his Diakoptics, and with Kazuo Kondo who in?uenced the post-war Japanese schools of di?erential geometry and supervised Shun-ichi Amari's doctorate [6], it was clear that both had a much wider remit than traditionally pursued elsewhere.

Group Theory: Application to the Physics of Condensed Matter


Mildred S. Dresselhaus - 2008
    The approach centers on the conviction that teaching group theory along with applications helps students to learn, understand and use it as a springboard to more sophisticated concepts."

Probability and Statistics by Example: Volume 2, Markov Chains: A Primer in Random Processes and Their Applications


Yuri Suhov - 2008
    Because of this, students can find it very difficult to make a successful transition from lectures to examinations to practice, since the problems involved can vary so much in nature. Since the subject is critical in many modern applications such as mathematical finance, quantitative management, telecommunications, signal processing, bioinformatics, as well as traditional ones such as insurance, social science and engineering, the authors have rectified deficiencies in traditional lecture-based methods by collecting together a wealth of exercises with complete solutions, adapted to needs and skills of students. Following on from the success of Probability and Statistics by Example: Basic Probability and Statistics, the authors here concentrate on random processes, particularly Markov processes, emphasizing models rather than general constructions. Basic mathematical facts are supplied as and when they are needed and historical information is sprinkled throughout.

Group Theory: Birdtracks, Lie's, and Exceptional Groups


Predrag Cvitanovic - 2008
    This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional.The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, "birdtracks" notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as "negative dimensional" relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.

A Combinatorial Approach to Matrix Theory and Its Applications


Richard A. Brualdi - 2008
    After reviewing the basics of graph theory, elementary counting formulas, fields, and vector spaces, the book explains the algebra of matrices and uses the K�nig digraph to carry out simple matrix operations. It then discusses matrix powers, provides a graph-theoretical definition of the determinant using the Coates digraph of a matrix, and presents a graph-theoretical interpretation of matrix inverses. The authors develop the elementary theory of solutions of systems of linear equations and show how to use the Coates digraph to solve a linear system. They also explore the eigenvalues, eigenvectors, and characteristic polynomial of a matrix; examine the important properties of nonnegative matrices that are part of the Perron-Frobenius theory; and study eigenvalue inclusion regions and sign-nonsingular matrices. The final chapter presents applications to electrical engineering, physics, and chemistry.Using combinatorial and graph-theoretical tools, this book enables a solid understanding of the fundamentals of matrix theory and its application to scientific areas.

Statistical Models and Methods for Financial Markets


Tze Leung Lai - 2008
    S. program in ?nancial mathematics at Stanford, which is an interdisciplinary program that aims to provide a master's-level education in applied mathematics, statistics, computing, ?nance, and economics. Students in the programhad di?erent backgroundsin statistics. Some had only taken a basic course in statistical inference, while others had taken a broad spectrum of M. S. - and Ph. D. -level statistics courses. On the other hand, all of them had already taken required core courses in investment theory and derivative pricing, and STATS 240 was supposed to link the theory and pricing formulas to real-world data and pricing or investment strategies. Besides students in theprogram, thecoursealso attractedmanystudentsfromother departments in the university, further increasing the heterogeneity of students, as many of them had a strong background in mathematical and statistical modeling from the mathematical, physical, and engineering sciences but no previous experience in ?nance. To address the diversity in background but common strong interest in the subject and in a potential career as a "quant" in the nancialindustry, thecoursematerialwascarefullychosennotonlytopresent basic statistical methods of importance to quantitative ?nance but also to summarize domain knowledge in ?nance and show how it can be combined with statistical modeling in ?nancial analysis and decision making. The course material evolved over the years, especially after the second author helped as the head TA during the years 2004 and 2005.

The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators


Alexander Soifer - 2008
    van der Waerden, and Henry Baudet.

Basic Quadratic Forms


Larry J. Gerstein - 2008
    This book is a self-contained introduction to quadratic forms that is based on graduate courses the author has taught many times. It leads the reader from foundation material up to topics of current research interest--with special attention to the theory over the integers and over polynomial rings in one variable over a field--and requires only a basic background in linear and abstract algebra as a prerequisite. Whenever possible, concrete constructions are chosen over more abstract arguments. The book includes many exercises and explicit examples, and it is appropriate as a textbook for graduate courses or for independent study. To facilitate further study, a guide to the extensive literature on quadratic forms is provided.

Discrete Differential Geometry


Alexander I. Bobenko - 2008
    It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. The carefully edited collection of essays gives a lively, multi-facetted introduction to this emerging field.

Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing


Pierre Henry-Labordère - 2008
    It even obtains new results when only approximate and partial solutions were previously available.Through the problem of option pricing, the author introduces powerful tools and methods, including differential geometry, spectral decomposition, and supersymmetry, and applies these methods to practical problems in finance. He mainly focuses on the calibration and dynamics of implied volatility, which is commonly called smile. The book covers the Black-Scholes, local volatility, and stochastic volatility models, along with the Kolmogorov, Schr�dinger, and Bellman-Hamilton-Jacobi equations.Providing both theoretical and numerical results throughout, this book offers new ways of solving financial problems using techniques found in physics and mathematics.

Elementary Calculus of Financial Mathematics (Monographs on Mathematical Modeling & Computation) (Monographs on Mathematical Modeling and Computation)


A.J. Roberts - 2008
    This book introduces the fascinating area of financial mathematics and its calculus in an accessible manner for undergraduate students. Using little high-level mathematics, the author presents the basic methods for evaluating financial options and building financial simulations. By emphasising relevant applications and illustrating concepts with colour graphics, Elementary Calculus of Financial Mathematics presents the crucial concepts needed to understand financial options among these fluctuations. Among the topics covered are the binomial lattice model for evaluating financial options, the Black-Scholes and Fokker-Planck equations, and the interpretation of Ito's formula in financial applications. Each chapter includes exercises for student practice and the appendices offer MATLAB(R) and SCILAB code as well as alternate proofs of the Fokker-Planck equation and Kolmogorov backward equation.

Patterns of Change: Linguistic Innovations in the Development of Classical Mathematics


Ladislav Kvasz - 2008
    This approach is for mathematics what the history and philosophy of science is for science. Yet the historical approach to the philosophy of science appeared much earlier than the historical approach to the philosophy of mathematics. The first significant work in the history and philosophy of science is perhaps William Whewell's Philosophy of the Inductive Sciences, founded upon their History. This was originally published in 1840, a second, enlarged edition appeared in 1847, and the third edition appeared as three separate works published between 1858 and 1860. Ernst Mach's The Science of Mechanics: A Critical and Historical Account of Its Development is certainly a work of history and philosophy of science. It first appeared in 1883, and had six further editions in Mach's lifetime (1888, 1897, 1901, 1904, 1908, and 1912). Duhem's Aim and Structure of Physical Theory appeared in 1906 and had a second enlarged edition in 1914. So we can say that history and philosophy of science was a well-established field by the end of the 19th and the beginning of the 20th century. By contrast the first significant work in the history and philosophy of mathematics is Lakatos's Proofs and Refutations, which was published as a series of papers in the years 1963 and 1964."