This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.

Markov Chains and Mixing Times

Preface. List of Symbols. Leonardo Fibonacci. The Rabbit Problem. Fibonacci Numbers in Nature. Fibonacci Numbers: Additional Occurrances. Fibonacci and Lucas Identities. Geometric Paradoxes. Generalized Fibonacci Numbers. Additional Fibonacci and Lucas Formulas. The Euclidean Algorithm. Solving Recurrence Relations. Completeness Theorems. Pascal's Triangle. Pascal-Like Triangles. Additional Pascal-Like Triangles. Hosoya's Triangle. Divisibility Properties. Generalized Fibonacci Numbers Revisited. Generating Functions. Generating Functions Revisited. The Golden Ratio. The Golden Ratio Revisited. Golden Triangles. Golden Rectangles. Fibonacci Geometry. Regular Pentagons. The Golden Ellipse and Hyperbola. Continued Fractions. Weighted Fibonacci and Lucas Sums. Weighted Fibonacci and Lucas Sums Revisited. The Knapsack Problem. Fibonacci Magic Squares. Fibonacci Matrices. Fibonacci Determinants. Fibonacci and Lucas Congruences. Fibonacci and Lucas Periodicity. Fibonacci and Lucas Series. Fibonacci Polynomials. Lucas Polynomials. Jacobsthal Polynomials. Zeros of Fibonacci and Lucas Polynomials. Morgan-Voyce Polynomials. Fibonometry. Fibonacci and Lucas Subscripts. Gaussian Fibonacci and Lucas Numbers. Analytic Extensions. Tribonacci Numbers. Tribonacci Polynomials. Appendix 1: Fundamentals. Appendix 2: The First 100 Fibonacci and Lucas Numbers. Appendix 3: The First 100 Fibonacci Numbers and Their Prime Factorizations. Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations. References. Solutions to Odd-Numbered Exercises. Index.

Fibonacci and Lucas Numbers with Applications

Unlike most books reviewed in the Intelligencer this is definitely a textbook. It assumes knowledge one might acquire in the first two years of an undergraduate mathematics program – basic mathematical probability, plus linear algebra, a little graph theory and the infamous concept of “mathematical maturity”. It has the theorem-proof style of pure mathematics, but with friendly explanations of intuition and motivation.

Markov Chains and Mixing Times: Second Edition

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

In this paper we survey most of the games that, throughout history, can be classified as mathematical games. This paper is based on a talk given at the conference ‘Numeracy: historical, philosophical, and educational perspectives’ at St Anne's College, Oxford, December 2009.

On mathematical games

(2003). The Fibonacci Numbers—Exposed More Discretely. Mathematics Magazine: Vol. 76, No. 3, pp. 182-192.

/pdf/the-fibonacci-numbers-exposed-more-discretely-4aza7fh0zt.pdf

The Fibonacci Numbers -- Exposed More Discretely

(1999). Recounting Fibonacci and Lucas Identities. The College Mathematics Journal: Vol. 30, No. 5, pp. 359-366.

/pdf/recounting-fibonacci-and-lucas-identities-1bypjf6m7e.pdf

Recounting Fibonacci and Lucas Identities

(2002). A Stirling Encounter with Harmonic Numbers. Mathematics Magazine: Vol. 75, No. 2, pp. 95-103.

/pdf/a-stirling-encounter-with-harmonic-numbers-2ow3n8hobd.pdf

A Stirling Encounter with Harmonic Numbers

The fascinating world of graph theory goes back several centuries and revolves around the study of graphsmathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematicsand some of its most famous problems. For example, what is the shortest route for a traveling salesman seeking to visit a number of cities in one trip? What is the least number of colors needed to fill in any map so that neighboring regions are always colored differently? Requiring readers to have a math background only up to high school algebra, this book explores the questions and puzzles that have been studied, and often solved, through graph theory. In doing so, the book looks at graph theorys development and the vibrant individuals responsible for the fields growth. Introducing graph theorys fundamental concepts, the authors explore a diverse plethora of classic problems such as the Lights Out Puzzle, the Minimum Spanning Tree Problem, the Knigsberg Bridge Problem, the Chinese Postman Problem, a Knights Tour, and the Road Coloring Problem. They present every type of graph imaginable, such as bipartite graphs, Eulerian graphs, the Petersen graph, and trees. Each chapter contains math exercises and problems for readers to savor. An eye-opening journey into the world of graphs, this book offers exciting problem-solving possibilities for mathematics and beyond.

The Fascinating World of Graph Theory

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 5 k 5 1 5 m, the subset graph S,(k, 1) is a bipartite graph whose vertices are the k- and 1-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that sq(Sm(k, 1)) = (&) and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. @ 1997 John Wiley & Sons, Inc.

Strong chromatic index of subset graphs

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