eBook ISBN:  9780883859537 
Product Code:  NML/39.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 
eBook ISBN:  9780883859537 
Product Code:  NML/39.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 

Book DetailsAnneli Lax New Mathematical LibraryVolume: 39; 1997; 309 pp
Suitable as supplemental reading in courses in differential and integral calculus, numerical analysis, approximation theory and computeraided 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, Bézier 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 computeraided geometric design.

Table of Contents

Chapters

Chapter 1. Transformations and their Iteration

Chapter 2. Arithmetic and Geometric Means

Chapter 3. Isoperimetric Inequality for Triangles

Chapter 4. Isoperimetric Quotient

Chapter 5. Colored Marbles

Chapter 6. Candy for School Children

Chapter 7. Sugar Rather Than Candy

Chapter 8. Checkers on a Circle

Chapter 9. Decreasing Sets of Positive Integers

Chapter 10. Matrix Manipulations

Chapter 11. Nested Triangles

Chapter 12. Morley’s Theorem and Napoleon’s Theorem

Chapter 13. Complex Numbers in Geometry

Chapter 14. Birth of an IMO Problem

Chapter 15. Barycentric Coordinates

Chapter 16. DouglasNeumann Theorem

Chapter 17. Lagrange Interpolation

Chapter 18. The Isoperimetric Problem

Chapter 19. Formulas for Iterates

Chapter 20. Convergent Orbits

Chapter 21. Finding Roots by Iteration

Chapter 22. Chebyshev Polynomials

Chapter 23. Sharkovskii’s Theorem

Chapter 24. Variation Diminishing Matrices

Chapter 25. Approximation by Bernstein Polynomials

Chapter 26. Properties of Bernstein Polynomials

Chapter 27. Bézier Curves

Chapter 28. Cubic Interpolatory Splines

Chapter 29. Moving Averages

Chapter 30. Approximation of Surfaces

Chapter 31. Properties of Triangular Patches

Chapter 32. Convexity of Patches

Appendix A. Approximation

Appendix B. Limits and Continuity

Appendix C. Convexity


Reviews

'Over and Over Again' explains the mathematics behind many of the problems that appear in the high school mathematics contests, particularly the International Math Olympiads (IMOs) and their national counterparts in China and the United States. Many, but not all, of the problems involve iteration, but the real focus of the book is contest problem solving and how it is related to the underlying mathematics. It starts to dispel the 'bag of tricks' air that clings to contest problem solving, and to replace it with a more systematic 'box of tools.'...This is a fine book for learning how people find and develop the problems they inflict on Math Olympiad participants. It also reveals contest problem solving as a learnable skill, distinct from other things called problem solving, and, perhaps also distinct from that thing we call 'mathematics.' It also contains some interesting and entertaining mathematics.
Ed Sandifer, MAA Reviews


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Suitable as supplemental reading in courses in differential and integral calculus, numerical analysis, approximation theory and computeraided 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, Bézier 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 computeraided geometric design.

Chapters

Chapter 1. Transformations and their Iteration

Chapter 2. Arithmetic and Geometric Means

Chapter 3. Isoperimetric Inequality for Triangles

Chapter 4. Isoperimetric Quotient

Chapter 5. Colored Marbles

Chapter 6. Candy for School Children

Chapter 7. Sugar Rather Than Candy

Chapter 8. Checkers on a Circle

Chapter 9. Decreasing Sets of Positive Integers

Chapter 10. Matrix Manipulations

Chapter 11. Nested Triangles

Chapter 12. Morley’s Theorem and Napoleon’s Theorem

Chapter 13. Complex Numbers in Geometry

Chapter 14. Birth of an IMO Problem

Chapter 15. Barycentric Coordinates

Chapter 16. DouglasNeumann Theorem

Chapter 17. Lagrange Interpolation

Chapter 18. The Isoperimetric Problem

Chapter 19. Formulas for Iterates

Chapter 20. Convergent Orbits

Chapter 21. Finding Roots by Iteration

Chapter 22. Chebyshev Polynomials

Chapter 23. Sharkovskii’s Theorem

Chapter 24. Variation Diminishing Matrices

Chapter 25. Approximation by Bernstein Polynomials

Chapter 26. Properties of Bernstein Polynomials

Chapter 27. Bézier Curves

Chapter 28. Cubic Interpolatory Splines

Chapter 29. Moving Averages

Chapter 30. Approximation of Surfaces

Chapter 31. Properties of Triangular Patches

Chapter 32. Convexity of Patches

Appendix A. Approximation

Appendix B. Limits and Continuity

Appendix C. Convexity

'Over and Over Again' explains the mathematics behind many of the problems that appear in the high school mathematics contests, particularly the International Math Olympiads (IMOs) and their national counterparts in China and the United States. Many, but not all, of the problems involve iteration, but the real focus of the book is contest problem solving and how it is related to the underlying mathematics. It starts to dispel the 'bag of tricks' air that clings to contest problem solving, and to replace it with a more systematic 'box of tools.'...This is a fine book for learning how people find and develop the problems they inflict on Math Olympiad participants. It also reveals contest problem solving as a learnable skill, distinct from other things called problem solving, and, perhaps also distinct from that thing we call 'mathematics.' It also contains some interesting and entertaining mathematics.
Ed Sandifer, MAA Reviews