Working out solutions to polynomial equations is a mathematical problem that dates from antiquity. Galois developed a theory in which the obstacle to solving a polynomial equation is an associated collection of symmetries. Obtaining a root requires "breaking" that symmetry. When the degree of an equation is at least five, Galois Theory established that there is no formula for the solutions like those found in lower degree cases. However, this negative result doesn't mean that the practice of equation-solving ends. In a recent breakthrough, Doyle and McMullen devised a solution to the fifth-degree equation that uses geometry, algebra, and dynamics to exploit icosahedral symmetry.
Polynomials, Dynamics, and Choice: The Price We Pay for Symmetry is organized in two parts, the first of which develops an account of polynomial symmetry that relies on considerations of algebra and geometry. The second explores beyond polynomials to spaces consisting of choices ranging from mundane decisions to evolutionary algorithms that search for optimal outcomes. The two algorithms in Part I provide frameworks that capture structural issues that can arise in deliberative settings. While decision-making has been approached in mathematical terms, the novelty here is in the use of equation-solving algorithms to illuminate such problems.
Features
- Treats the topic―familiar to many―of solving polynomial equations in a way that’s dramatically different from what they saw in school
- Accessible to a general audience with limited mathematical background
- Abundant diagrams and graphics.
Author(s): Scott Crass
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Pages: 189
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
List of Figures
List of Tables
Preface: Motivation
Acknowledgments
SECTION I: Polynomials: Symmetries and Solutions
CHAPTER 1: Solving Equations: A Fundamental Problem
1.1. POLYNOMIAL PRIMER
1.2. WHAT NUMBERS DO WE USE?
1.3. ROOTS AND COEFFICIENTS
CHAPTER 2: What Is Symmetry?
2.1. MIRRORS AND REFLECTIONS
2.2. MATHEMATICAL SYMMETRY
2.3. EXPLORING GEOMETRIC SYMMETRY
2.4. GROUPS IN THE ABSTRACT
2.5. POSING AND SOLVING PROBLEMS WITH SYMMETRY
2.6. STRUCTURE IN THE ABSTRACT
2.7. A LOOK AT HIGHER DIMENSIONS
2.8. WHAT IS GEOMETRY?
2.9. MOLECULAR SYMMETRY
2.10. CONSERVATION LAWS
2.11. THERMODYNAMIC SYSTEMS
CHAPTER 3: Geometry of Choice: Symmetry's Cost
3.1. SPACES WHERE THE ROOTS LIVE
3.2. SHUFFLING ROOTS AND SOLVING EQUATIONS
CHAPTER 4: Compute First, Then Choose
4.1. SIMPLIFYING A POLYNOMIAL
4.2. SOLUTIONS FROM A FORMULA AND A CHOICE
4.3. REDUCING A POLYNOMIAL'S SYMMETRY
4.4. WHAT GOES WRONG
CHAPTER 5: Choose First, Then Compute
5.1. A LINE THAT BECOMES A SPHERE
5.2. SYMMETRICAL STRUCTURES
5.3. FUNDAMENTALS OF DYNAMICS
5.4. DYNAMICAL GEOMETRY AND SYMMETRY
5.5. SOLVING EQUATIONS BY ITERATION
SECTION II: Beyond Equations
CHAPTER 6: Interlude: Modeling Choice
CHAPTER 7: Learning to Choose
7.1. MAKING RATIONAL DECISIONS
7.2. THE HEART HAS ITS REASONS
7.3. GIVE CHANCE A CHOICE
CHAPTER 8: Choosing to Learn
8.1. A CROWD DECIDES
8.2. WHEN IN DOUBT, SIMULATE
8.3. GIVE CHOICE A CHANCE
CHAPTER 9: Conclusion: The Price We Pay for Symmetry
9.1. SYMMETRY, MORE OR LESS
9.2. CHOOSING AS METAPHOR
9.3. RANDOM CHOICE IS UNAVOIDABLE
Bibliography
Index
