Softcover ISBN:  9780821894163 
Product Code:  MCL/16 
List Price:  $29.00 
Individual Price:  $21.75 
Electronic ISBN:  9781470425500 
Product Code:  MCL/16.E 
List Price:  $29.00 
Individual Price:  $21.75 

Book DetailsMSRI Mathematical Circles LibraryVolume: 16; 2015; 158 ppMSC: Primary 00; Secondary 34; 68; 20; 11;
One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the bestknown examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years.
This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments).
Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics.
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.ReadershipUndergraduate and graduate students and research mathematicians interested in mathematics.

Table of Contents

Cover

Title page

Preface to the English Translation

Introduction

Lecture 1. The Statistics of Topology and Algebra

1. Hilbert’s Sixteenth Problem

2. The Statistics of Smooth Functions

3. Statistics and the Topology of Periodic Functions and Trigonometric Polynomials

4. Algebraic Geometry of Trigonometric Polynomials

Editor’s notes

Lecture 2. Combinatorial Complexity and Randomness

1. Binary Sequences

2. Graph of the Operation of Taking Differences

3. Logarithmic Functions and Their Complexity

4. Complexity and Randomness of Tables of Galois Fields

Editor’s notes

Lecture 3. Random Permutations and Young Diagrams of Their Cycles

1. Statistics of Young Diagrams of Permutations of Small Numbers of Objects

2. Experimentation with Random Permutations of Larger Numbers of Elements

3. Random Permutations of 𝑝² Elements Generated by Galois Fields

4. Statistics of Cycles of Fibonacci Automorphisms

Editor’s notes

Lecture 4. The Geometry of Frobenius Numbers for Additive Semigroups

1. Sylvester’s Theorem and the Frobenius Numbers

2. Trees Blocked by Others in a Forest

3. The Geometry of Numbers

4. Upper Bound Estimate of the Frobenius Number

5. Average Values of the Frobenius Numbers

6. Proof of Sylvester’s Theorem

7. The Geometry of Continued Fractions of Frobenius Numbers

8. The Distribution of Points of an Additive Semigroup on the Segment Preceding the Frobenius Number

Editor’s notes

Bibliography

Back Cover


Additional Material

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One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the bestknown examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years.
This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments).
Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics.
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.
Undergraduate and graduate students and research mathematicians interested in mathematics.

Cover

Title page

Preface to the English Translation

Introduction

Lecture 1. The Statistics of Topology and Algebra

1. Hilbert’s Sixteenth Problem

2. The Statistics of Smooth Functions

3. Statistics and the Topology of Periodic Functions and Trigonometric Polynomials

4. Algebraic Geometry of Trigonometric Polynomials

Editor’s notes

Lecture 2. Combinatorial Complexity and Randomness

1. Binary Sequences

2. Graph of the Operation of Taking Differences

3. Logarithmic Functions and Their Complexity

4. Complexity and Randomness of Tables of Galois Fields

Editor’s notes

Lecture 3. Random Permutations and Young Diagrams of Their Cycles

1. Statistics of Young Diagrams of Permutations of Small Numbers of Objects

2. Experimentation with Random Permutations of Larger Numbers of Elements

3. Random Permutations of 𝑝² Elements Generated by Galois Fields

4. Statistics of Cycles of Fibonacci Automorphisms

Editor’s notes

Lecture 4. The Geometry of Frobenius Numbers for Additive Semigroups

1. Sylvester’s Theorem and the Frobenius Numbers

2. Trees Blocked by Others in a Forest

3. The Geometry of Numbers

4. Upper Bound Estimate of the Frobenius Number

5. Average Values of the Frobenius Numbers

6. Proof of Sylvester’s Theorem

7. The Geometry of Continued Fractions of Frobenius Numbers

8. The Distribution of Points of an Additive Semigroup on the Segment Preceding the Frobenius Number

Editor’s notes

Bibliography

Back Cover