Softcover ISBN: 978-1-4704-6303-8
This book gives a lively development of the mathematics needed to answer the question, “How many times should a deck of cards be shuffled to mix it up?” The shuffles studied are the usual ones that real people use: riffle, overhand, and smooshing cards around on the table.
The mathematics ranges from probability (Markov chains) to combinatorics (symmetric function theory) to algebra (Hopf algebras). There are applications to magic tricks and gambling along with a careful comparison of the mathematics to the results of real people shuffling real cards. The book explores links between shuffling and higher mathematics?Lie theory, algebraic topology, the geometry of hyperplane arrangements, stochastic calculus, number theory, and more. It offers a useful springboard for seeing how probability theory is applied and leads to many corners of advanced mathematics.
The book can serve as a text for an upper division course in mathematics, statistics, or computer science departments and will be appreciated by graduate students and researchers in mathematics, statistics, and computer science, as well as magicians and people with a strong background in mathematics who are interested in games that use playing cards.
Cover
Title page
Contents
Preface
Acknowledgments
Chapter 1. Shuffling cards: An introduction
1.1. Riffle shuffling and total variation
1.2. The problem, its motivation, and some theorems
1.3. Outline of the book
1.4. Background for reading this book
Chapter 2. Practice and history of shuffling cards
2.1. Illustrations of some shuffles
2.2. Some early history
Chapter 3. Convergence rates for riffle shuffles
3.1. Basic properties of riffle shuffling
3.2. Total variation and relative entropy
3.3. Effect of cuts
3.4. Guessing strategies and games
3.5. Strong uniform times and separation distance
3.6. Coupling and riffle shuffles
3.7. Appendix: Distances between probability distributions
Chapter 4. Features
4.1. A single card
4.2. Shuffling with repeated cards
4.3. Different methods of dealing
4.4. Cycle structure, random polynomials, and Lie theory
4.5. Inversions
4.6. RSK shape, shuffling, and character theory of ??_{∞}
4.7. The sign function
4.8. Other techniques
Chapter 5. Eigenvectors and Hopf algebras
5.1. Overview
5.2. Combinatorics of words
5.3. The main theorem
5.4. Hopf algebras and Markov chains
5.5. Restriction/induction (with shuffling and Hopf algebras)
5.6. Shuffling and Hochschild homology
5.7. Appendix 1: Uses for eigenvectors
5.8. Appendix 2: Eigenvectors in this book
Chapter 6. Shuffling and carries
6.1. Introduction to carries
6.2. Connection with riffle shuffles
6.3. A bit of why
6.4. Convergence rates of the carries chain
6.5. Eigenvalues and eigenvectors of the carries chain
6.6. Balanced carries
6.7. Carries, number theory, and fractals
6.8. Other connections
6.9. Appendix: Some proofs
Chapter 7. Different models for riffle shuffling
7.1. Perfect shuffles
7.2. Thorp shuffle
7.3. Neat and clumpy shuffles: The Markov model
7.4. Dynamical systems and work of Lalley
7.5. Shuffling big decks
Chapter 8. Move-to-front shuffling and variations
8.1. Coupling and convergence rates for the move-to-front shuffle
8.2. Formula for move-to-front shuffle
8.3. Spectral aspects of the move-to-front shuffle
8.4. Statistics of features after iterations of top-to-random shuffle
8.5. Weighted move to front and the Plackett-Luce model
8.6. Move-to-front shuffle with Markov dependent requests
8.7. Move to root for binary search trees
8.8. Connection with Stein’s method
8.9. Random-to-random shuffle
Chapter 9. Shuffling and geometry
9.1. Hyperplane arrangements and random walks
9.2. Examples
9.3. General theory of hyperplane walks
9.4. An analog of riffle shuffling for general hyperplane arrangements
9.5. Stationary distribution, eigenfunctions, and lumped chains
9.6. Extensions
9.7. Connections and unification
9.8. Adjacent transpositions
9.9. Research problems
9.10. Appendix: Some proofs
Chapter 10. Shuffling and algebraic topology
10.1. A topology teaser
10.2. First steps: Homology
10.3. Triangulating a product of simplices (and shuffling)
10.4. The Kunneth formula
10.5. Translating shuffling theorems into homology
10.6. A different application of shuffling to topology
10.7. Final remarks
Chapter 11. Type B shuffles and shelf-shuffling machines
11.1. Models of Type B shuffles
11.2. Cycle structure and RSK shape
11.3. Shelf shufflers
11.4. Other types
11.5. Research problems
11.6. Appendix: Proof of Proposition 11.3.2
Chapter 12. Descent algebras, ??-partitions, and quasisymmetric functions
12.1. Descent theory
12.2. ??-partitions and shuffling
12.3. Quasisymmetric functions and shuffles with biased cuts
12.4. Algebras of shuffles
12.5. A strange inequality
12.6. And then …
Chapter 13. Overhand shuffling
13.1. Introduction to the overhand shuffle
13.2. Mathematical models
13.3. Some theorems
13.4. Entertaining (and cheating) with overhand shuffles
13.5. The over-under shuffle
13.6. Coupling and the coin tossing model
13.7. Braid arrangement and overhand shuffles
13.8. Interval exchange maps and overhand shuffles
Chapter 14. “Smoosh” shuffle
14.1. Introduction
14.2. Tests of mixing for smoosh shuffles
14.3. A mathematical model for spatial mixing
14.4. Some analysis
14.5. Bells, whistles, and computer implementation
14.6. A different model for spatial mixing
14.7. Combining shuffles and some practical tests
Chapter 15. How to shuffle perfectly (randomly)
Chapter 16. Applications to magic tricks, traffic merging, and statistics
16.1. Rising sequences
16.2. The Gilbreath principle
16.3. Tops and bottoms are special
16.4. A performable magic trick
16.5. A homework problem
16.6. Riffle stacking
16.7. Close stays close
16.8. Reds and blacks
16.9. Overhand shuffle
16.10. Smooshing
16.11. An application of shuffling to cars merging in traffic
16.12. Statistics of permutations
Chapter 17. Shuffling and multiple zeta values
17.1. Multiple zeta values
17.2. A bit of proof
17.3. Chen’s integrals
17.4. Periods
Bibliography
Index
Back Cover
Readership
Graduate students and researchers interested in applications of mathematics to card shuffling.
Paperback
9780262544849
Published: January 3, 2023
An accessible introduction to philosophical logic, suitable for undergraduate courses and above.
Rigorous yet accessible, Logical Methods introduces logical tools used in philosophy?including proofs, models, modal logics, meta-theory, two-dimensional logics, and quantification?for philosophy students at the undergraduate level and above. The approach developed by Greg Restall and Shawn Standefer is distinct from other texts because it presents proof construction on equal footing with model building and emphasizes connections to other areas of philosophy as the tools are developed.
Throughout, the material draws on a broad range of examples to show readers how to develop and master tools of proofs and models for propositional, modal, and predicate logic; to construct and analyze arguments and to find their structure; to build counterexamples; to understand the broad sweep of formal logic's development in the twentieth and twenty-first centuries; and to grasp key concepts used again and again in philosophy.
This text is essential to philosophy curricula, regardless of specialization, and will also find wide use in mathematics and computer science programs.
? An accessible introduction to proof theory for readers with no background in logic
? Covers proofs, models, modal logics, meta-theory, two-dimensional logics, quantification, and many other topics
? Provides tools and techniques of particular interest to philosophers and philosophical logicians
? Features short summaries of key concepts and skills at the end of each chapter
? Offers chapter-by-chapter exercises in two categories: basic, designed to reinforce important ideas; and challenge, designed to push students' understanding and developing skills in new directions
Series:Annals of Mathematics Studies
Hardcover
ISBN:
9780691249551
Published:
Jul 11, 2023
Pages:
256
Size:
0.24 x 0.36 in.
Illus:
11 b/w illus.
Paperback
ISBN:
9780691249551
To gain insight into the nature of turbulent fluids, mathematicians start from experimental facts, translate them into mathematical properties for solutions of the fundamental fluids PDEs, and construct solutions to these PDEs that exhibit turbulent properties. This book belongs to such a program, one that has brought convex integration techniques into hydrodynamics. Convex integration techniques have been used to produce solutions with precise regularity, which are necessary for the resolution of the Onsager conjecture for the 3D Euler equations, or solutions with intermittency, which are necessary for the construction of dissipative weak solutions for the Navier-Stokes equations. In this book, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions enjoy a spatial regularity index in L^2 that can be taken as close as desired to 1/2, thus lying at the threshold of all known convex integration methods. This property matches the measured intermittent nature of turbulent flows. The construction of such solutions requires technology specifically adapted to the inhomogeneities inherent in intermittent solutions. The main technical contribution of this book is to develop convex integration techniques at the local rather than global level. This localization procedure functions as an ad hoc wavelet decomposition of the solution, carrying information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates.
Format: Paperback / softback, 225 pages, height x width: 235x155 mm, weight: 379 g, XVIII, 225 p., 1 Paperback / softback
Pub. Date: 29-Jan-2023
ISBN-13: 9789811681646
This book brings together a variety of non-Gaussian autoregressive-type models to analyze time-series data. This book collects and collates most of the available models in the field and provide their probabilistic and inferential properties. This book classifies the stationary time-series models into different groups such as linear stationary models with non-Gaussian innovations, linear stationary models with non-Gaussian marginal distributions, product autoregressive models and minification models. Even though several non-Gaussian time-series models are available in the literature, most of them are focusing on the model structure and the probabilistic properties.
1. Basics of Time Series.-
2. Statistical Inference for Stationary Time
Series.-
3. AR Models with Stationary Non-Gaussian Positive Marginals.-
4. AR
Models with Stationary Non-Gaussian Real-Valued Marginals.-
5. Some Nonlinear
AR-type Models for Non-Gaussian Time series.-
6. Linear Time Series Models
with Non-Gaussian Innovations.-
7. Autoregressive-type Time Series of Counts.
Format: Paperback / softback, 251 pages, height x width: 235x155 mm, weight: 415 g, XIII, 251 p., 1 Paperback / softback
Series: Developments in Mathematics 71
Pub. Date: 27-Jan-2023
ISBN-13: 9783030922061
This book examines in detail approximate fixed point theory in different classes of topological spaces for general classes of maps. It offers a comprehensive treatment of the subject that is up-to-date, self-contained, and rich in methods, for a wide variety of topologies and maps. Content includes known and recent results in topology (with proofs), as well as recent results in approximate fixed point theory. This work starts with a set of basic notions in topological spaces. Special attention is given to topological vector spaces, locally convex spaces, Banach spaces, and ultrametric spaces. Sequences and function spaces-and fundamental properties of their topologies-are also covered. The reader will find discussions on fundamental principles, namely the Hahn-Banach theorem on extensions of linear (bounded) functionals; the Banach open mapping theorem; the Banach-Steinhaus uniform boundedness principle; and Baire categories, including some applications. Also included are weak topologies and their properties, in particular the theorems of Eberlein-Smulian, Goldstine, Kakutani, James and Grothendieck, reflexive Banach spaces, l_{1}- sequences, Rosenthal's theorem, sequential properties of the weak topology in a Banach space and weak* topology of its dual, and the Frechet-Urysohn property. The subsequent chapters cover various almost fixed point results, discussing how to reach or approximate the unique fixed point of a strictly contractive mapping of a spherically complete ultrametric space. They also introduce synthetic approaches to fixed point problems involving regular-global-inf functions. The book finishes with a study of problems involving approximate fixed point property on an ambient space with different topologies. By providing appropriate background and up-to-date research results, this book can greatly benefit graduate students and mathematicians seeking to advance in topology and fixed point theory.
Preface.- Basic Concepts.- Almost Fixed Points.- Approximate Fixed
Points in Ultrametric Spaces.- Synthetic Approaches to Problems of Fixed
Points.- Approximate Fixed Theory in Topological Vector Spaces.- Bibliography.