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Lectures and Problems a Gift to Young Mathematicians 1st Edition by Arnold 9781470422592 147042259X

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Author(s): Arnold, V. I.; Mathematical Sciences Research Institute (Berkeley, Calif.) Staff;
ISBN(s): 9781470422592, 147042259X
Edition: 1
File Details: PDF, 1.57 MB
Year: 2015
Language: english
SKU: EB-34432948 Category: Tags: , , ,

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ISBN 10: 147042259X

ISBN 13: 9781470422592

Author: V. I. Arnold

Vladimir Arnold (1937-2010) was one of the great mathematical minds of the late 20th century. He did significant work in many areas of the field. On another level, he was keeping with a strong tradition in Russian mathematics to write for and to directly teach younger students interested in mathematics. This book contains some examples of Arnold’s contributions to the genre. “Continued Fractions” takes a common enrichment topic in high school math and pulls it in directions that only a master of mathematics could envision. “Euler Groups” treats a similar enrichment topic, but it is rarely treated with the depth and imagination lavished on it in Arnold’s text. He sets it in a mathematical context, bringing to bear numerous tools of the trade and expanding the topic way beyond its usual treatment. In “Complex Numbers” the context is physics, yet Arnold artfully extracts the mathematical aspects of the discussion in a way that students can understand long before they master the field of quantum mechanics. “Problems for Children 5 to 15 Years Old” must be read as a collection of the author’s favorite intellectual morsels. Many are not original, but all are worth thinking about, and each requires the solver to think out of his or her box. Dmitry Fuchs, a long-term friend and collaborator of Arnold, provided solutions to some of the problems. Readers are of course invited to select their own favorites and construct their own favorite solutions. In reading these essays, one has the sensation of walking along a path that is found to ascend a mountain peak and then being shown a vista whose existence one could never suspect from the ground. Arnold’s style of exposition is unforgiving. The reader–even a professional mathematician–will find paragraphs that require hours of thought to unscramble, and he or she must have patience with the ellipses of thought and the leaps of reason. These are all part of Arnold’s intent. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. A co-publication of the AMS and the Mathematical Sciences Research Institute.

Table of contents:

Part 1. Continued Fractions

  • Chapter 1: Continued Fractions

  • Chapter 2: What is a Continued Fraction?

  • Chapter 3: The Geometric Theory of Continued Fractions

  • Chapter 4: Kuzmin’s Theorem

  • Chapter 5: Multidimensional Continued Fractions

  • Chapter 6: A Generalization of Lagrange’s Theorem

  • Chapter 7: Editors’ Comments

Part 2. Geometry of Complex Numbers, Quaternions, and Spins

  • Chapter 8: Geometry of Complex Numbers, Quaternions, and Spins

  • Chapter 9: Complex Numbers

  • Chapter 10: Motions of the Plane

  • Chapter 11: A Digression Concerning Orientations

  • Chapter 12: The Generalization of Complex Numbers to the Concept of Quaternions

  • Chapter 13: Some Examples

  • Chapter 14: Newton’s Differential Equation

  • Chapter 15: From the Pythagorean Theorem to Riemann Surfaces

  • Chapter 16: Mathematical Trinities

  • Chapter 17: Spins and Braids

  • Chapter 18: Appendix

  • Chapter 19: Editors’ Comments

Part 3. Euler Groups and Arithmetic of Geometric Progressions

  • Chapter 20: Euler Groups and Arithmetic of Geometric Progressions

  • Chapter 21: Basic Definitions

  • Chapter 22: A Digression on the Euler Function

  • Chapter 23: Tables for Euler Groups

  • Chapter 24: Euler Groups of Products

  • Chapter 25: The Homomorphism Given by Reduction Modulo a, Γ(ab) → Γ(a)

  • Chapter 26: Proofs of the Theorems on Euler Groups

  • Chapter 27: Fermat-Euler Dynamical Systems

  • Chapter 28: Statistics of Geometric Progressions

  • Chapter 29: Measurement of the Degree of Randomness of a Subset

  • Chapter 30: The Average Value of the Parameter of Randomness

  • Chapter 31: Additional Remarks about Fermat-Euler Dynamics

  • Chapter 32: Primitive Roots of a Prime Modulus

  • Chapter 33: Patterns in Coordinates of Quadratic Residues

  • Chapter 34: Applications to Quadratic Congruences

Part 4. Problems for Children 5 to 15 Years Old

  • Chapter 35: Problems

  • Chapter 36: Solutions to Selected Problems

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