Over and Over Again

Gengzhe Chang; Thomas W. Sederberg

doi:10.5948/UPO9780883859537

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. Among topics treated are classical and not so classical inequalities, Sharkovskii's theorem, interpolation, Bernstein polynomials, Bzier curves and surfaces, and splines. Most of the book requires only high school mathematics; the last part requires elementary calculus. This book would be an excellent supplement to courses in calculus, differential equations,, numerical analysis, approximation theory and computer-aided geometric design.

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.

Source: 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.

Source: The American Mathematical Monthly

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