Number, shape, and symmetry : an introduction to number theory, geometry, and group theory

Pierre Antoine Grillet (Auth.)

"Abstract Algebra" is a clearly written, self-contained basic algebra text for graduate students, with a generous amount of additional material that suggests the scope of contemporary algebra. The first chapters blend standard contents with a careful introduction to proofs with arrows. The last chapters, on universal algebras and categories, including tripleability, give valuable general views of algebra. There are over 1400 exercises, at varying degrees of difficulty. For the new edition, the author has completely rewritten the entire text, streamlining the first chapters for rapid access to Galois theory, removing some material, and adding introductions to Groebner bases, Ext and Tor, and other topics. From a review of the First Edition: ...combines an exceptionally accessible discussion of the basic material with a just as thorough and well-organized treatment of the many additional (advanced) topics included.... represents an outstanding introduction to modern abstract algebra as a whole, with many unique features. It captivates the reader by its remarkable diversity, comprehensiveness, elegant succinctness, and coherence. - Werner Kleinert, Zentralblatt Leer más…

Michael D. Potter

Main subject categories: • Set theory • Philosophy of set theoryContent Headings:• I. Sets • Introduction to Part I • 1. Logic • 2. Collections • 3. The hierarchy • 4. The theory of sets • Conclusion to Part I• II. Numbers • Introduction to Part II • 5. Arithmetic • 6. Counting • 7. Lines • 8. Real numbers • Conclusion to Part II• III. Cardinals and Ordinals • Introduction to Part III • 9. Cardinals • 10. Basic cardinal arithmetic • 11. Ordinals • 12. Ordinal arithmetic • Conclusion to Part III• IV. Further Axioms • Introduction to Part IV • 13. Orders of infinity • 14. The axiom of choice • 15. Further cardinal arithmetic • Conclusion to Part IV• Appendices • A. Traditional axiomatizations • B. Classes • C. Sets and classes • References • List of symbols • Index of definitions • Index of namesMichael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. Leer más…

Miklos Bona, Miklós Bóna

There are 650 articles with the word permutation in the title whose primary classification is combinatorics, but, until now, there have been no books addressing the topic. The very first book to be published on the subject, Combinatorics of Permutations contains a comprehensive, up to date treatment of the subject. Covering both enumerative and external combinatorics, this book can be used as either a graduate text or as a reference for professional mathematicians. The book includes many applications from computer science, molecular biology, probabilistic methods, and pattern avoidance, and the numerous exercises show readers a fairly comprehensive list of recent results from the field. Leer más…

James E. Gentle

Matrix algebra is one of the most important areas of mathematics for data analysis and for statistical theory. The first part of this book presents the relevant aspects of the theory of matrix algebra for applications in statistics. This part begins with the fundamental concepts of vectors and vector spaces, next covers the basic algebraic properties of matrices, then describes the analytic properties of vectors and matrices in the multivariate calculus, and finally discusses operations on matrices in solutions of linear systems and in eigenanalysis. This part is essentially self-contained. The second part of the book begins with a consideration of various types of matrices encountered in statistics, such as projection matrices and positive definite matrices, and describes the special properties of those matrices. The second part also describes some of the many applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. The brief coverage in this part illustrates the matrix theory developed in the first part of the book. The first two parts of the book can be used as the text for a course in matrix algebra for statistics students, or as a supplementary text for various courses in linear models or multivariate statistics. The third part of this book covers numerical linear algebra. It begins with a discussion of the basics of numerical computations, and then describes accurate and efficient algorithms for factoring... Leer más…

H. S. M. Coxeter

This classic work is now available in an unabridged paperback edition. The Second Edition retains all the characterisitcs that made the first edition so popular: brilliant exposition, the flexibility permitted by relatively self-contained chapters, and broad coverage ranging from topics in the Euclidean plane, to affine geometry, projective geometry, differential geometry, and topology. The Second Edition incorporates improvements in the text and in some proofs, takes note of the solution of the 4-color map problem, and provides answers to most of the exercises. This unabridged paperback edition contains complete coverage, ranging from topics in the Euclidean plane to affine geometry, projective geometry, differential geometry and topology. Leer más…

Miles Reid, Balazs Szendroi

Geometry Provides A Whole Range Of Views On The Universe, Serving As The Inspiration, Technical Toolkit And Ultimate Goal For Many Branches Of Mathematics And Physics. This Book Introduces The Ideas Of Geometry, And Includes A Generous Supply Of Simple Explanations And Examples. The Treatment Emphasises Coordinate Systems And The Coordinate Changes That Generate Symmetries. The Discussion Moves From Euclidean To Non-euclidean Geometries, Including Spherical And Hyperbolic Geometry, And Then On To Affine And Projective Linear Geometries. Group Theory Is Introduced To Treat Geometric Symmetries, Leading To The Unification Of Geometry And Group Theory In The Erlangen Program. An Introduction To Basic Topology Follows, With The Möbius Strip, The Klein Bottle And The Surface With G Handles Exemplifying Quotient Topologies And The Homeomorphism Problem. Topology Combines With Group Theory To Yield The Geometry Of Transformation Groups,having Applications To Relativity Theory And Quantum Mechanics. A Final Chapter Features Historical Discussions And Indications For Further Reading. With Minimal Prerequisites, The Book Provides A First Glimpse Of Many Research Topics In Modern Algebra, Geometry And Theoretical Physics. The Book Is Based On Many Years' Teaching Experience, And Is Thoroughly Class-tested. There Are Copious Illustrations, And Each Chapter Ends With A Wide Supply Of Exercises. Further Teaching Material Is Available For Teachers Via The Web, Including Assignable... Leer más…

Allen Hatcher

In the TV series "Babylon 5" the Minbari had a saying: "Faith manages." If you are willing to take many small, some medium and a few very substantial details on faith, you will find Hatcher an agreeable fellow to hang out with in the pub and talk beer-coaster mathematics, you will be happy taking a picture as a proof, and you will have no qualms with tossing around words like "attach", "collapse", "twist", "embed", "identify", "glue" and so on as if making macaroni art. To be sure, the book bills itself as being "geometrically flavored", which over the years I have gathered is code in the mathematical community for there being a lot of cavalier hand-waving and prose that reads more like instructions for building a kite than the logical discourse of serious mathematics. Some folks really like that kind of stuff, I guess (judging from other reviews). Professors do, because they already know their stuff so the wand-waving doesn't bother them any more than it would bother the faculty at Hogwarts. When it comes to Hatcher some students do as well, I think because so often Hatcher's style of proof is similar to that of an undergrad: inconvenient details just "disappear" by the wayside if they're even brought up at all, and every other sentence features a leap in logic or an unremarked gap in reasoning that facilitates completion of an assignment by the due date. Some will say this is a book for mature math students, so any gaps should be filled in by the reader en route... Leer más…

K. Lee Lerner And Brenda Wilmoth Lerner, Editors

Cover Page......Page 1 Title Page - Volume 1......Page 3 Title Page - Volume 2......Page 4 ISBN 0787694223......Page 5 Volume 1: A–L......Page 6 Volume 2: M–Z......Page 7 Architectural Math......Page 8 Cartography......Page 9 Domain and Range......Page 10 Functions......Page 11 Inverse......Page 12 Multiplication......Page 13 Perspective......Page 14 Ratio......Page 15 Scientific Notation......Page 16 Topology......Page 17 Zero-Sum Games......Page 18 Introduction......Page 19 List of Advisors and Contributors......Page 21 ACKNOWLEDGMENTS......Page 22 A Brief History of Discovery and Development......Page 23 SPORTS AND FITNESS ADDITION......Page 25 FINANCIAL ADDITION......Page 26 POKER, PROBABILITY, AND OTHER USES OF ADDITION......Page 27 USING ADDITION TO PREDICT AND ENTERTAIN......Page 28 Potential Applications......Page 29 Where to Learn More......Page 30 Fundamental Mathematical Concepts and Terms......Page 31 A Brief History of Discovery and Development......Page 34 PERSONAL FINANCES......Page 35 COLLEGE FOOTBALL......Page 36 UPC BARCODES......Page 37 FLYING AN AIRPLANE......Page 38 SKYDIVING......Page 39 CRASH TESTS......Page 40 BUILDING SKYSCRAPERS......Page 41 BUYING LIGHT BULBS......Page 42 ART......Page 43 FINGERPRINT SCANNERS......Page 44 TELEPORTATION......Page 45 Where to Learn More......Page 46 A Brief History of Discovery and Development......Page 48 CREDIT CARD FRAUD DETECTION......Page 49 ENCRYPTION AND ENCRYPTION DEVICES......Page 50 INTERNET DATA... Leer más…

Titu Andreescu, Oleg Mushkarov, Luchezar Stoyanov, Titu Andreescu

Questions Of Maxima And Minima Have Great Practical Significance, With Applications To Physics, Engineering, And Economics; They Have Also Given Rise To Theoretical Advances, Notably In Calculus And Optimization. Indeed, While Most Texts View The Study Of Extrema Within The Context Of Calculus, This Carefully Constructed Problem Book Takes A Uniquely Intuitive Approach To The Subject: It Presents Hundreds Of Extreme-value Problems, Examples, And Solutions Primarily Through Euclidean Geometry. Key Features And Topics: * Comprehensive Selection Of Problems, Including Greek Geometry And Optics, Newtonian Mechanics, Isoperimetric Problems, And Recently Solved Problems Such As Malfatti’s Problem * Unified Approach To The Subject, With Emphasis On Geometric, Algebraic, Analytic, And Combinatorial Reasoning * Presentation And Application Of Classical Inequalities, Including Cauchy--schwarz And Minkowski’s Inequality; Basic Results In Calculus, Such As The Intermediate Value Theorem; And Emphasis On Simple But Useful Geometric Concepts, Including Transformations, Convexity, And Symmetry * Clear Solutions To The Problems, Often Accompanied By Figures * Hundreds Of Exercises Of Varying Difficulty, From Straightforward To Olympiad-caliber Written By A Team Of Established Mathematicians And Professors, This Work Draws On The Authors’ Experience In The Classroom And As Olympiad Coaches. By Exposing Readers To A Wealth Of Creative Problem-solving Approaches, The Text Communicates Not... Leer más…

Benjamin Fine, Gerhard Rosenberger, Fine, Benjamin

This book provides an introduction and overview of number theory based on the distribution and properties of primes. This unique approach provides both a firm background in the standard material as well as an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. Key Topics and Features: * Solid introduction to analytic number theory, including full proofs of Dirichlet’s Theorem and the Prime Number Theorem * Solid treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals * First treatment in book form of the AKS algorithm that shows that primality testing is of polynomial time * Many interesting side topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers The book’s user-friendly style, historical context, and wide range of exercises from simple to quite difficult (with solutions and hints provided for select ones) make it ideal for self study as well as classroom use. Intended for upper level undergraduates and beginning graduate students, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced in the book. Leer más…

David Wells, David G. Wells

A fascinating journey into the mind-bending world of prime numbers Cicadas of the genus Magicicada appear once every 7, 13, or 17 years. Is it just a coincidence that these are all prime numbers? How do twin primes differ from cousin primes, and what on earth (or in the mind of a mathematician) could be sexy about prime numbers? What did Albert Wilansky find so fascinating about his brother-in-law's phone number? Mathematicians have been asking questions about prime numbers for more than twenty-five centuries, and every answer seems to generate a new rash of questions. In Prime Numbers: The Most Mysterious Figures in Math, you'll meet the world's most gifted mathematicians, from Pythagoras and Euclid to Fermat, Gauss, and Erd?o?s, and you'll discover a host of unique insights and inventive conjectures that have both enlarged our understanding and deepened the mystique of prime numbers. This comprehensive, A-to-Z guide covers everything you ever wanted to know--and much more that you never suspected--about prime numbers, including: * The unproven Riemann hypothesis and the power of the zeta function * The "Primes is in P" algorithm * The sieve of Eratosthenes of Cyrene * Fermat and Fibonacci numbers * The Great Internet Mersenne Prime Search * And much, much more Leer más…

Robert Lawlor

An introduction to the geometry which, as modern science now confirms, underlies the structure of the universe.The thinkers of ancient Egypt, Greece and India recognized that numbers governed much of what they saw in their world and hence provided an approach to its divine creator. Robert Lawlor sets out the system that determines the dimension and the form of both man-made and natural structures, from Gothic cathedrals to flowers, from music to the human body. By also involving the reader in practical experiments, he leads with ease from simple principles to a grasp of the logarithmic spiral, the Golden Proportion, the squaring of the circle and other ubiquitous ratios and proportions.About the Art & Imagination Series: Explore a range of interests, philosophies, religions, and cultures: from Kabbalah to Freemasonry, Buddhism to Hinduism, myth to magic. The distinguished authors bring a wealth of knowledge, visionary thinking, and accessible writing to each intriguing subject in these lavishly illustrated, large-format paperback books. Leer más…

Alfred S. Posamentier, Ingmar Lehmann

Acknowledgments --, Introduction --, history and introduction to the Fibonacci numbers --, Fibonacci numbers in nature --, Fibonacci numbers and the Pascal triangle --, Fibonacci numbers and the golden ratio --, Fibonacci numbers and continued fractions --, potpourri of Fibonacci number applications --, Fibonacci numbers found in art and architecture --, Fibonacci numbers and musical form --, famous Binet formula for finding a particular Fibonacci number --, Fibonacci numbers and fractals --, Epilogue --, Afterword /, Appendix A: List of the first 500 Fibonacci numbers, with the first 200 Fibonacci numbers factored --, Appendix B: Proofs of Fibonacci relationships --, References --, Index.; Herbert A. Hauptman Leer más…

The Math Forum, Drexel University; Cartoons By Jessica Wolk-Stanley

you, Too, Can Understand Geometry - Just Ask Dr. Math ! Have You Started Studying Geometry In Math Class? Do You Get Totally Lost Trying To Find The Perimeter Of A Rectangle Or The Circumference Of A Circle? Don't Worry. Grasping The Basics Of Geometry Doesn't Have To Be As Scary As It Sounds. Dr. Math-the Popular Online Math Resource-is Here To Help! Students Just Like You Have Been Turning To Dr. Math For Years Asking Questions About Math Problems, And The Math Doctors At The Math Forum Have Helped Them Find The Answers With Lots Of Clear Explanations And Helpful Hints. Now, With Dr. Math Introduces Geometry, You'll Learn Just What It Takes To Succeed In This Subject. You'll Find The Answers To Dozens Of Real Questions From Students Who Needed Help Understanding The Basic Concepts Of Geometry, From Lines, Rays, And Angles To Measuring Three-dimensional Objects And Applying Geometry In The Real World. Pretty Soon, Everything From Recognizing Types Of Quadrilaterals To Finding Surface Area To Counting Lines Of Symmetry Will Make Sense. Plus, You'll Get Plenty Of Tips For Working With Tricky Problems Submitted By Other Kids Who Are Just As Confused As You Are. You Won't Find A Better Introduction To The World And Language Of Geometry Anywhere! sally Niezgoda - Children's Literature this Book Covers Topics Of Geometry Including The Undefined Terms (point, Line, And Plane), Two-dimensional And Three-dimensional Figures, And Symmetry. There Are Some Real World Correlations,... Leer más…

Clifford Alan Pickover

Math’s infinite mysteries and beauty unfold in this follow-up to the best-selling The Science Book. Beginning millions of years ago with ancient “ant odometers” and moving through time to our modern-day quest for new dimensions, it covers 250 milestones in mathematical history. Among the numerous delights readers will learn about as they dip into this inviting anthology: cicada-generated prime numbers, magic squares from centuries ago, the discovery of pi and calculus, and the butterfly effect. Each topic gets a lavishly illustrated spread with stunning color art, along with formulas and concepts, fascinating facts about scientists’ lives, and real-world applications of the theorems. Leer más…

Glendinning, Paul

Paul Glendinning is Professor of Applied Mathematics at the University of Manchester. He was founding Head of School for Mathematics at the combined University of Manchester and has published over fifty academic articles and an undergraduate textbook on chaos theory. Both simple and accessible, Math in Minutes is a visually led introduction to 200 key mathematical concepts. Each concept is described by means of an easy-to-understand illustration and a compact, 200-word explanation. Concepts span all of the key areas of mathematics, including Fundamentals of Mathematics, Sets and Numbers, Geometry, Equations, Limits, Functions and Calculus, Vectors and Algebra, Complex Numbers, Combinatorics, Number Theory, and more. From the Trade Paperback edition. MATHEMATICS Leer más…

Paul Renteln

Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at (http://www.cambridge.org/9781107042193) www.cambridge.org/9781107042193 . Leer más…

Ian Stewart; David Tall; Oxford University Press

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows... Leer más…