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**Over and Over Again** describes a potpourri of mathematical problems and results on the theme of repetition, broadly construed. Chang and Sederberg write at varying levels, for a mixed audience of lay readers, advanced high school students, and college students and up...A highly useful compilation. Recommended for all libraries. — *Choice*

Beautiful book of problems featuring the basic notions of transformations and iteration. Many problems in the first half come from mathematical Olympiads...and require only high school mathematics. The second half is more advanced... but appendices provide prerequisites for high school and beginning college students, and laypersons. — *The American Mathematical Monthly*

**Suitable as supplemental reading in courses in differential and integral calculus, numerical analysis, approximation theory and computer-aided geometric design **

Relations of mathematical objects to each other are expressed by transformations. The repeated application of a transformation over and over again, i.e., its *iteration* leads to solution of equations, as in Newton's method for finding roots, or Picard's method for solving differential equations. This book studies a treasure trove of iterations, in number theory, analysis and geometry, and applied them to various problems, many of them taken from international and national Mathematical Olympiad competitions.

The eminent mathematician and expositor Philip Davis has said about this book: "Selecting functional iterations as their specific theme, the authors, both experienced mathematicians as well as skillful expositors, introduce us to topics that will delight, instruct, and challenge."

Among topics treated are classical and not so classical inequalities, Sharkovskii's theorem, interpolation, Bernstein polynomials, Bézier curves and surfaces, and splines. Most of the book requires only high school mathematics; the last part requires elementary calculus.

**Contents:** Transformations and their Iteration; Arithmetic and Geometric Means; Isoperimetric Inequality for Triangles; Isoperimetric Quotient; Colored Marbles; Candy for School Children; Sugar Rather than Candy; Checkers on a Circle; Decreasing Sets of Positive Integers; Matrix Manipulations; Nested Triangles; Morley's Theorem and Napoleon's Theorem; Complex Numbers in Geometry; Birth of an IMO Problem; Barycentric Coordinates; Douglas-Neumann Theorem; Lagrange Interpolation; The Isoperimetric Problem; Formulas for Iterates; Convergent Orbits; Finding Roots by Iteration; Chebyshev Polynomials; Sharkovskii's Theorem; Variation Diminishing Matrices; Approximation by Bernstein Polynomials; Properties of Bernstein Polynomials Bézier Curves; Cubic Interpolatory Splines; Moving Averages; Approximation of Surfaces; Properties of Triangular Patches; Convexity of Patches; Appendix A: Approximations; Appendix B: Limits and Continuity; Appendix C: Convexity; Hints and Solutions; Bibliography; Index.

Print ISBN 9780883856413

Electronic ISBN 9780883859537

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Over and Over Again |
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