Stan Wagon – författare

This book is an example-based introduction to techniques, from elementary to advanced, of using Mathematica, a revolutionary tool for mathematical computation and exploration. By integrating the basic functions of mathematics with a powerful and easy-to-use programming language, Mathematica allows us to carry out projects that would be extremely laborious in traditional programming environments. And the new developments that began with version 6 — allowing the user to dyna- cally manipulate output using sliders or other controls — add amazing power to the interface. Animations have always been part of Mathematica, but the new design allows the manipulation of any number of variables, an important enhancement. Mathematica in Action illustrates this power by using demonstrations and ani- tions, three-dimensional graphics, high-precision number theory computations, and sophisticated geometric and symbolic programming to attack a diverse collection of problems.. It is my hope that this book will serve a mathematical purpose as well, and I have interspersed several unusual or complicated examples among others that will be more familiar. Thus the reader may have to deal simultaneously with new mat- matics and new Mathematica techniques. Rarely is more than undergraduate mathematics required, however. An underlying theme of the book is that a computational way of looking at a mathematical problem or result yields many benefits. For example: Well-chosen computations can shed light on familiar relations and reveal new patterns. One is forced to think very precisely; gaps in understanding must be eliminated if a program is to work.

This book is an example-based introduction to techniques, from elementary to advanced, of using Mathematica, a revolutionary tool for mathematical computation and exploration. By integrating the basic functions of mathematics with a powerful and easy-to-use programming language, Mathematica allows us to carry out projects that would be extremely laborious in traditional programming environments. And the new developments that began with version 6 — allowing the user to dyna- cally manipulate output using sliders or other controls — add amazing power to the interface. Animations have always been part of Mathematica, but the new design allows the manipulation of any number of variables, an important enhancement. Mathematica in Action illustrates this power by using demonstrations and ani- tions, three-dimensional graphics, high-precision number theory computations, and sophisticated geometric and symbolic programming to attack a diverse collection of problems.. It is my hope that this book will serve a mathematical purpose as well, and I have interspersed several unusual or complicated examples among others that will be more familiar. Thus the reader may have to deal simultaneously with new mat- matics and new Mathematica techniques. Rarely is more than undergraduate mathematics required, however. An underlying theme of the book is that a computational way of looking at a mathematical problem or result yields many benefits. For example: Well-chosen computations can shed light on familiar relations and reveal new patterns. One is forced to think very precisely; gaps in understanding must be eliminated if a program is to work.

Calculus and change. The two words go together. Calculus is about change, and approaches to teaching calculus are changing dramatically. Thus it is both timely and appropriate to apply techniques of animation to the varied and important graphical aspects of calculus. AB a computer algebra system, Mathematica is an excellent tool for numerical and symbolic computation. It also has the power to generate striking and colorful graphical images and to animate them dynamically. The combination of these capabilities makes Mathematica a natural resource for exploring the changing world of calculus and approaches to mastering it. In addition, Mathematica notebooks are easy to edit, allowing flexible input for commands to Mathematica and stylish text for explanation to the reader. Much has been written about the use and importance of technology in the teaching and learning of calculus. We will not repeat the arguments or feign objectivity. We are enthusiastic believers in the value of a significant laboratory experience as part oflearning calculus, and we think Mathematica notebooks are a most appropriate and exciting way to provide that experience. The notebooks that follow represent our choice of laboratory topics for a course in one-variable calculus. They offer a balance between what we think belongs in a first-year calculus course and what lends itself well to exploration in a Mathematica laboratory setting.

A Course in Computational Number Theory uses the computer as a tool for motivation and explanation. The book is designed for the reader to quickly access a computer and begin doing personal experiments with the patterns of the integers. It presents and explains many of the fastest algorithms for working with integers. Traditional topics are covered, but the text also explores factoring algorithms, primality testing, the RSA public-key cryptosystem, and unusual applications such as check digit schemes and a computation of the energy that holds a salt crystal together. Advanced topics include continued fractions, Pell’s equation, and the Gaussian primes.

This book takes readers on a thrilling tour of some of the most important and powerful areas of contemporary numerical mathematics. The tour is organized along the 10 problems of the SIAM 100-Digit Challenge, a contest posed by Nick Trefethen of Oxford University in the January/February 2002 issue of SIAM News. The complete story of the contest as well as a lively interview with Nick Trefethen are also included.The authors, members of teams that solved all 10 problems, show in detail multiple approaches for solving each problem, ranging from elementary to sophisticated, from brute-force to schemes that can be scaled to provide thousands of digits of accuracy and that can solve even larger related problems. The authors touch on virtually every major technique of modern numerical analysis: matrix computation, iterative linear methods, limit extrapolation and convergence acceleration, numerical quadrature, contour integration, discretization of PDEs, global optimization, Monte Carlo and evolutionary algorithms, error control, interval and high-precision arithmetic, and many more.The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing gives concrete examples of how to justify the validity of every single digit of a numerical answer. Methods range from carefully designed computer experiments to a posteriori error estimates and computer-assisted proofs based on interval arithmetic.This book will aid readers in developing problem-solving skills for making judicious method selections. The chapters may be read independently. Appendices A and B include basic methods of convergence acceleration and details of computing the solutions to very high accuracy. Full code for all the methods, examples, tables, and figures is given (partly in Appendix C, completely on the accompanying web page. The code is written in a variety of languages, including Mathematica, MATLAB, Maple, C, Octave, and PARI/GP. Appendix D offers a sample of additional challenging problems for those who master some of the techniques discussed here.

The Banach-Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher's famous 'Angel and Devils' woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.

The Banach-Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher's famous 'Angel and Devils' woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.

The Banach–Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher''s famous ''Angel and Devils'' woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.

The Banach–Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher''s famous ''Angel and Devils'' woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.

This title presents new ideas on the visualization of differential equations with user-configurable tools. The authors use the widely-used computer algebra system, Mathematica, to provide an integrated environment for programming, visualizing graphics, and running commentary for learning and working with differential equations.

Bicycle or Unicycle? is a collection of 105 mathematical puzzles whose defining characteristic is the surprise encountered in their solutions. Solvers will be surprised, even occasionally shocked, at those solutions. The problems unfold into levels of depth and generality very unusual in the types of problems seen in contests. In contrast to contest problems, these are problems meant to be savored; many solutions, all beautifully explained, lead to unanswered research questions. At the same time, the mathematics necessary to understand the problems and their solutions is all at the undergraduate level. The puzzles will, nonetheless, appeal to professionals as well as to students and, in fact, to anyone who finds delight in an unexpected discovery.These problems were selected from the Macalester College Problem of the Week archive. The Macalester tradition of a weekly problem was started by Joseph Konhauser in 1968. In 1993 Stan Wagon assumed problem-generating duties. A previous book written by Wagon, Konhauser, and Dan Velleman, Which Way Did the Bicycle Go?, gathered problems from the first twenty-five years of the archive. The title problem in that collection was inspired by an error in logic made by Sherlock Holmes, who attempted to determine the direction of a bicycle from the tracks of its wheels. Here the title problem asks whether a bicycle track can always be distinguished from a unicycle track. You'll be surprised by the answer.

Victor Klee and Stan Wagon discuss some of the unsolved problems in number theory and geometry, many of which can be understood by readers with a very modest mathematical background. The presentation is organized around 24 central problems, many of which are accompanied by other, related problems. The authors place each problem in its historical and mathematical context, and the discussion is at the level of undergraduate mathematics. Each problem section is presented in two parts. The first gives an elementary overview discussing the history and both the solved and unsolved variants of the problem. The second part contains more details, including a few proofs of related results, a wider and deeper survey of what is known about the problem and its relatives, and a large collection of references. Both parts contain exercises, with solutions. The book is aimed at both teachers and students of mathematics who want to know more about famous unsolved problems.

This collection will give students (high school or beyond), teachers, and university professors a chance to experience the pleasure of wrestling with some beautiful problems of elementary mathematics. Readers can compare their sleuthing talents with those of Sherlock Holmes, who made a bad mistake regarding the first problem in the collection: Determine the direction of travel of a bicycle that has left its tracks in a patch of mud.Which Way did the Bicycle Go? contains a variety of other unusual and interesting problems in geometry, algebra, combinatorics, and number theory. For example, if a pizza is sliced into eight 45-degree wedges meeting at a point other than the center of the pizza, and two people eat alternate wedges, will they get equal amounts of pizza? Or: What is the rightmost nonzero digit of the product $1\cdot 2\cdot 3\cdots 1,000,000$? Or: Is a manufacturer's claim that a certain unusual combination lock allows thousands of combinations justified? Complete solutions to the 191 problems are included along with problem variations and topics for investigation.

Dieses Buch führt seine Leser auf einen packenden Streifzug durch die wichtigsten und leistungsfähigsten Bereiche zeitgenössischer numerischer Mathematik. Die Route orientiert sich an den 10 Wettbewerbsaufgaben der "SIAM 10 x 10-Digit Challenge", mit der Nick Trefethen aus Oxford im Frühjahr 2002 seine Kollegen und ihre Doktoranden weltweit herausforderte: jede der in wenigen Sätzen formulierten, hochgradig nichttrivialen Aufgaben verlangt dabei nach der Berechnung der ersten 10 Ziffern einer reellen Zahl.Für jede Aufgabe wird eine Vielfalt von Lösungsmethoden diskutiert und dabei so gut wie jede bedeutendere Technik moderner numerischer Mathematik gestreift. Viele aus der Praxis stammende Hinweise helfen dem Leser, Kompetenzen im Lösen numerischer Probleme zu entwickeln und zu lernen, unter konkurrierenden Methoden eine wohlüberlegte Wahl zu treffen. Der vollständige lauffähige Programmtext aller Verfahren, Beispiele, Tabellen und Figuren steht auf der begleitenden Webseite www.numerikstreifzug.de zur Verfügung.