| eBook ISBN: | 978-0-88385-953-7 |
| Product Code: | NML/39.E |
| List Price: | $50.00 |
| MAA Member Price: | $37.50 |
| AMS Member Price: | $37.50 |
| eBook ISBN: | 978-0-88385-953-7 |
| Product Code: | NML/39.E |
| List Price: | $50.00 |
| MAA Member Price: | $37.50 |
| AMS Member Price: | $37.50 |
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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 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, 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 computer-aided geometric design.
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Table of Contents
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Chapters
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Chapter 1. Transformations and their Iteration
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Chapter 2. Arithmetic and Geometric Means
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Chapter 3. Isoperimetric Inequality for Triangles
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Chapter 4. Isoperimetric Quotient
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Chapter 5. Colored Marbles
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Chapter 6. Candy for School Children
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Chapter 7. Sugar Rather Than Candy
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Chapter 8. Checkers on a Circle
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Chapter 9. Decreasing Sets of Positive Integers
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Chapter 10. Matrix Manipulations
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Chapter 11. Nested Triangles
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Chapter 12. Morley’s Theorem and Napoleon’s Theorem
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Chapter 13. Complex Numbers in Geometry
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Chapter 14. Birth of an IMO Problem
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Chapter 15. Barycentric Coordinates
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Chapter 16. Douglas-Neumann Theorem
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Chapter 17. Lagrange Interpolation
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Chapter 18. The Isoperimetric Problem
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Chapter 19. Formulas for Iterates
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Chapter 20. Convergent Orbits
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Chapter 21. Finding Roots by Iteration
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Chapter 22. Chebyshev Polynomials
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Chapter 23. Sharkovskii’s Theorem
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Chapter 24. Variation Diminishing Matrices
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Chapter 25. Approximation by Bernstein Polynomials
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Chapter 26. Properties of Bernstein Polynomials
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Chapter 27. Bézier Curves
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Chapter 28. Cubic Interpolatory Splines
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Chapter 29. Moving Averages
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Chapter 30. Approximation of Surfaces
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Chapter 31. Properties of Triangular Patches
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Chapter 32. Convexity of Patches
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Appendix A. Approximation
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Appendix B. Limits and Continuity
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Appendix C. Convexity
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Reviews
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'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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RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Reviews
- Requests
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, 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 computer-aided 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. Douglas-Neumann 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
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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
