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.

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.

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.

• Plenty of examples and case studies utilize Mathematica 7's newest tools, such as dynamic manipulations and adaptive three-dimensional plotting.
• Emphasizes the breadth of Mathematica and the impressive results of combining techniques from different areas.
• Whenever possible, the book shows how Mathematica can be used to discover new things. Striking examples include the design of a road on which a square wheel bike can ride, the design of a drill that can drill square holes, and new and surprising formulas for p.
• Visualization is emphasized throughout, with finely crafted graphics in each chapter.
• All Mathematica code is included on a CD, saving the reader hours of typing.

Dieses Buch fuhrt seine Leser auf einen packenden Streifzug durch die wichtigsten und leistungsfahigsten Bereiche zeitgenossischer numerischer Mathematik. Die Route orientiert sich an den 10 Wettbewerbsaufgaben der "SIAM 10 x 10-Digit Challenge," mit der Nick Trefethen aus Oxford im Fruhjahr 2002 seine Kollegen und ihre Doktoranden weltweit herausforderte: jede der in wenigen Satzen formulierten, hochgradig nichttrivialen Aufgaben verlangt dabei nach der Berechnung der ersten 10 Ziffern einer reellen Zahl.

Fur jede Aufgabe wird eine Vielfalt von Losungsmethoden diskutiert und dabei so gut wie jede bedeutendere Technik moderner numerischer Mathematik gestreift. Viele aus der Praxis stammende Hinweise helfen dem Leser, Kompetenzen im Losen numerischer Probleme zu entwickeln und zu lernen, unter konkurrierenden Methoden eine wohluberlegte Wahl zu treffen. Der vollstandige lauffahige Programmtext aller Verfahren, Beispiele, Tabellen und Figuren steht auf der begleitenden Webseite www.numerikstreifzug.de zur Verfugung.

"