Concrete Mathematics: A Foundation for Computer Science (2nd Edition)

This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them. 0201558025B04062001 This book, updated and improved, introduces the mathematics that support advanced computer programming and the analysis of algorithms. The book's primary aim is to provide a solid and relevant base of mathematical skills. It is an indispensable text and reference for computer scientists and serious programmers in virtually every discipline. 1 Recurrent Problems 1 1.1 The Tower of Hanoi 1 1.2 Lines in the Plane 4 1.3 The Josephus Problem 8 Exercises 17 2 Sums 21 2.1 Notation 21 2.2 Sums and Recurrences 25 2.3 Manipulation of Sums 30 2.4 Multiple Sums 34 2.5 General Methods 41 2.6 Finite and In nite Calculus 47 2.7 In nite Sums 56 Exercises 62 3 Integer Functions 67 3.1 Floors and Ceilings 67 3.2 Floor/Ceiling Applications 70 3.3 Floor/Ceiling Recurrences 78 3.4 `mod': The Binary Operation 81 3.5 Floor/Ceiling Sums 86 Exercises 95 4 Number Theory 102 4.1 Divisibility 102 4.2 Primes 105 4.3 Prime Examples 107 4.4 Factorial Factors 111 4.5 Relative Primality 115 4.6 `mod': The Congruence Relation 123 4.7 Independent Residues 126 4.8 Additional Applications 129 4.9 Phi and Mu 133 Exercises 144 5 Binomial Coefficients 153 5.1 Basic Identities 153 5.2 Basic Practice 172 5.3 Tricks of the Trade 186 xii CONTENTS xiii 5.4 Generating Functions 196 5.5 Hypergeometric Functions 204 5.6 Hypergeometric Transformations 216 5.7 Partial Hypergeometric Sums 223 5.8 Mechanical Summation 229 Exercises 242 6 Special Numbers 257 6.1 Stirling Numbers 257 6.2 Eulerian Numbers 267 6.3 Harmonic Numbers 272 6.4 Harmonic Summation 279 6.5 Bernoulli Numbers 283 6.6 Fibonacci Numbers 290 6.7 Continuants 301 Exercises 309 7 Generating Functions 320 7.1 Domino Theory and Change 320 7.2 Basic Maneuvers 331 7.3 Solving Recurrences 337 7.4 Special Generating Functions 350 7.5 Convolutions 353 7.6 Exponential Generating Functions 364 7.7 Dirichlet Generating Functions 370 Exercises 371 8 Discrete Probability 381 8.1 De nitions 381 8.2 Mean and Variance 387 8.3 Probability Generating Functions 394 8.4 Flipping Coins 401 8.5 Hashing 411 Exercises 427 9 Asymptotics 439 9.1 A Hierarchy 440 9.2 O Notation 443 9.3 O Manipulation 450 9.4 Two Asymptotic Tricks 463 9.5 Euler's Summation Formula 469 9.6 Final Summations 476 Exercises 489 A Answers to Exercises 497 B Bibliography 604 C Credits for Exercises 632 Index 637 List of Tables 657 Es un libro de texto integral que proporciona una sólida base de principios y técnicas matemáticas necesarias para la programación informática avanzada y el análisis de algoritmos. Escrito por Ronald L. Graham, Donald E. Knuth y Oren Patashnik, este libro abarca una mezcla de matemáticas continuas y discretas para presentar una visión clara de los aspectos relevantes de las matemáticas contemporáneas tal como se aplican a la informática. Los temas incluyen, pero no se limitan a, combinatoria, teoría de números y probabilidad. Es un excelente recurso para cualquier persona que busque profundizar su comprensión de la informática desde una perspectiva matemática. "This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline."--from Amazon.com.