Softcover ISBN:  9780821847565 
Product Code:  ULECT/49 
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Electronic ISBN:  9781470416447 
Product Code:  ULECT/49.E 
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Book DetailsUniversity Lecture SeriesVolume: 49; 2009; 250 ppMSC: Primary 60; 05; 91; Secondary 11;
Mathematics has been called the science of order. The subject is remarkably good for generalizing specific cases to create abstract theories. However, mathematics has little to say when faced with highly complex systems, where disorder reigns. This disorder can be found in pure mathematical arenas, such as the distribution of primes, the \(3n+1\) conjecture, and class field theory.
The purpose of this book is to provide examples—and rigorous proofs—of the complexity law:
(1) discrete systems are either simple or they exhibit advanced pseudorandomness;
(2) a priori probabilities often exist even when there is no intrinsic symmetry.
Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics.
This book considers randomness and complexity. The traditional approach to complexity—computational complexity theory—is to study very general complexity classes, such as P, NP and PSPACE. What Beck does is very different: he studies interesting concrete systems, which can give new insights into the mystery of complexity.
The book is divided into three parts. Part A is mostly an essay on the big picture. Part B is partly new results and partly a survey of real game theory. Part C contains new results about graph games, supporting the main conjecture. To make it accessible to a wide audience, the book is mostly selfcontained.ReadershipGraduate students and research mathematicians interested in discrete mathematics, combinatorics, and number theory.

Table of Contents

Part A. Reading the shadows on the wall and formulating a vague conjecture

Chapter 1. Complex systems

Chapter 2. Collecting data: Apparent randomness of digit sequences

Chapter 3. Collecting data: More randomness in number theory

Chapter 4. Laplace and the principle of insufficient reason

Chapter 5. Collecting proofs for the SLG conjecture

Part B. More evidence for the SLG conjecture: Exact solutions in real game theory

Chapter 6. Ramsey theory and games

Chapter 7. Practice session (I): More on Ramsey games and strategies

Chapter 8. Practice session (II): Connectivity games and more strategies

Chapter 9. What kind of games?

Chapter 10. Exact solutions of games: Understanding via the equiprobability postulate

Chapter 11. Equiprobability postulate with constraints (endgame policy)

Chapter 12. Constraints and threshold clustering

Chapter 13. Threshold clustering and a few bold conjectures

Part C. New evidence: Games and graphs, the surplus, and the square root law

Chapter 14. Yet another simplification: Sparse hypergraphs and the surplus

Chapter 15. Is surplus the right concept? (I)

Chapter 16. Is surplus the right concept? (II)

Chapter 17. Working with a gametheoretic partition function

Chapter 18. An attempt to save the variance

Chapter 19. Proof of theorem 1: Combining the variance with an exponential sum

Chapter 20. Proof of theoem 2: The upper bound

Chapter 21. Conclusion (I): More on theorem 1

Chapter 22. Conclusion (II): Beyond the SLG conjecture


Additional Material

Reviews

[T]he book gives a particular reading to a large portion of modern mathematics. ...[and] offers an interesting unifying approach and, in doing so, gives the reader a broad review of a number of combinatorial problems. These are accompanied by a very good choice of bibliographic references.
Michele Zito, Mathematical Reviews


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Mathematics has been called the science of order. The subject is remarkably good for generalizing specific cases to create abstract theories. However, mathematics has little to say when faced with highly complex systems, where disorder reigns. This disorder can be found in pure mathematical arenas, such as the distribution of primes, the \(3n+1\) conjecture, and class field theory.
The purpose of this book is to provide examples—and rigorous proofs—of the complexity law:
(1) discrete systems are either simple or they exhibit advanced pseudorandomness;
(2) a priori probabilities often exist even when there is no intrinsic symmetry.
Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics.
This book considers randomness and complexity. The traditional approach to complexity—computational complexity theory—is to study very general complexity classes, such as P, NP and PSPACE. What Beck does is very different: he studies interesting concrete systems, which can give new insights into the mystery of complexity.
The book is divided into three parts. Part A is mostly an essay on the big picture. Part B is partly new results and partly a survey of real game theory. Part C contains new results about graph games, supporting the main conjecture. To make it accessible to a wide audience, the book is mostly selfcontained.
Graduate students and research mathematicians interested in discrete mathematics, combinatorics, and number theory.

Part A. Reading the shadows on the wall and formulating a vague conjecture

Chapter 1. Complex systems

Chapter 2. Collecting data: Apparent randomness of digit sequences

Chapter 3. Collecting data: More randomness in number theory

Chapter 4. Laplace and the principle of insufficient reason

Chapter 5. Collecting proofs for the SLG conjecture

Part B. More evidence for the SLG conjecture: Exact solutions in real game theory

Chapter 6. Ramsey theory and games

Chapter 7. Practice session (I): More on Ramsey games and strategies

Chapter 8. Practice session (II): Connectivity games and more strategies

Chapter 9. What kind of games?

Chapter 10. Exact solutions of games: Understanding via the equiprobability postulate

Chapter 11. Equiprobability postulate with constraints (endgame policy)

Chapter 12. Constraints and threshold clustering

Chapter 13. Threshold clustering and a few bold conjectures

Part C. New evidence: Games and graphs, the surplus, and the square root law

Chapter 14. Yet another simplification: Sparse hypergraphs and the surplus

Chapter 15. Is surplus the right concept? (I)

Chapter 16. Is surplus the right concept? (II)

Chapter 17. Working with a gametheoretic partition function

Chapter 18. An attempt to save the variance

Chapter 19. Proof of theorem 1: Combining the variance with an exponential sum

Chapter 20. Proof of theoem 2: The upper bound

Chapter 21. Conclusion (I): More on theorem 1

Chapter 22. Conclusion (II): Beyond the SLG conjecture

[T]he book gives a particular reading to a large portion of modern mathematics. ...[and] offers an interesting unifying approach and, in doing so, gives the reader a broad review of a number of combinatorial problems. These are accompanied by a very good choice of bibliographic references.
Michele Zito, Mathematical Reviews