Concrete Mathematics A Foundation for Computer. A Foundation for Computer Science 2nd Edition. Concrete Mathematics is the second most accessible book by. Concrete Mathematics A Foundation for Computer Science and a great selection of similar Used, New and Collectible Books available now at AbeBooks. Concrete Mathematics A Foundation for Computer Science. Concrete Mathematics is a blending of CONtinuous. This second edition includes important new material. This second edition includes important new material about. Concrete Mathematics A Foundation for Computer. A Foundation for Computer Science, 2nd Edition. Concrete Mathematics A Foundation For Computer Science Second Edition Pdf' title='Concrete Mathematics A Foundation For Computer Science Second Edition Pdf' />Concrete Mathematics A Foundation for Computer Science, 2nd Edition. Copyright 1. 99. 4Dimensions 7 38 x 9 18Pages 6. Edition 2nd. e. Book Watermarked ISBN 1. ISBN 1. 3 9. 78 0 1. 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. Concrete Mathematics A Foundation for Computer. The National Science Foundation and the Office of Naval Research have. This second edition features a new. C O N C R E T E MAT H E MAT I C S Second Edition. Concrete mathematics a foundation for computer science Ronald. Computer scienceMathematics. I. Online download concrete mathematics a foundation for computer science 2nd edition Concrete Mathematics A Foundation For Computer Science 2nd Edition. Concrete Mathematics, Second Edition. Based on the course Concrete Mathematics taught by Knuth at., not only in computer science but also in mathematics. Title Concrete Mathematics A Foundation for Computer Science 2nd Edition Authors Ronald L. Graham, Donald Ervin Knuth, Oren Patashnik Publisher AddisonWesley. It is an indispensable text and reference not only for computer scientists the authors themselves rely heavily on it Concrete Mathematics is a blending of CONtinuous and dis. CRETE 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 Knuths 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 5. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self study. Major topics include Sums. Thread Dimensions Program. 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. Ebook PDF version produced by Mathematical Sciences Publishers MSP, http msp. Table of ContentsMost chapters contain Exercises. Recurrent Problems. The Tower of Hanoi. Lines in the Plane. The Josephus Problem. Exercises. 2. Sums. Notation. Sums and Recurrences. Manipulation of Sums. Multiple Sums. General Methods. Finite and Infinite Calculus. Infinite Sums. Exercises. Integer Functions. Floors and Ceilings. FloorCeiling Applications. FloorCeiling Recurrences. The Binary Operation. FloorCeiling Sums. Exercises. 4. Number Theory. Divisibility. Factorial Factors. Relative Primality. The Congruence Relation. Independent Residues. Additional Applications. Phi and Mu. Exercises. Binomial Coefficients. Basic Identities. Basic Practice. Tricks of the Trade. Generating Functions. Hypergeometric Functions. Hypergeometric Transformations. Partial Hypergeometric Sums. Mechanical Summation. Exercises. 6. Special Numbers. Stirling Numbers. Eulerian Numbers. Harmonic Numbers. Harmonic Summation. Bernoulli Numbers. Fibonacci Numbers. Continuants. Exercises. Generating Functions. Domino Theory and Change. Basic Maneuvers. Solving Recurrences. Special Generating Functions. Convolutions. Exponential Generating Functions. Dirichlet Generating Functions. Exercises. 8. Discrete Probability. Definitions. Mean and Variance. Probability Generating Functions. Flipping Coins. Hashing. Exercises. 9. Asymptotics. A Hierarchy. O Notation. O Manipulation. Two Asymptotic Tricks. Eulers Summation Formula. Final Summations. Exercises. A. Answers to Exercises. B. Bibliography. C. Credits for Exercises. Index. List of Tables. T0. 40. 62. 00. 1. Concrete Mathematics A Foundation for Computer Science, 2nd Edition. Sample Pages. Download the sample pages includes Chapter 3 and IndexTable of ContentsMost chapters contain Exercises. Recurrent Problems. The Tower of Hanoi. Lines in the Plane. The Josephus Problem. Exercises. 2. Sums. Notation. Sums and Recurrences. Manipulation of Sums. Multiple Sums. General Methods. Finite and Infinite Calculus. Infinite Sums. Exercises. Integer Functions. Floors and Ceilings. FloorCeiling Applications. FloorCeiling Recurrences. The Binary Operation. FloorCeiling Sums. Exercises. 4. Number Theory. Divisibility. Factorial Factors. Relative Primality. The Congruence Relation. Independent Residues. Additional Applications. Phi and Mu. Exercises. Binomial Coefficients. Basic Identities. Basic Practice. Tricks of the Trade. Generating Functions. Hypergeometric Functions. Hypergeometric Transformations. Partial Hypergeometric Sums. Mechanical Summation. Exercises. 6. Special Numbers. Stirling Numbers. Eulerian Numbers. Harmonic Numbers. Harmonic Summation. Bernoulli Numbers. Fibonacci Numbers. Continuants. Exercises. Generating Functions. Domino Theory and Change. Basic Maneuvers. Solving Recurrences. Special Generating Functions. Convolutions. Exponential Generating Functions. Dirichlet Generating Functions. Exercises. 8. Discrete Probability. Definitions. Mean and Variance. Probability Generating Functions. Flipping Coins. Hashing. Exercises. 9. Asymptotics. A Hierarchy. O Notation. O Manipulation. Two Asymptotic Tricks. Eulers Summation Formula. Final Summations. Exercises. A. Answers to Exercises. B. Bibliography. C. Credits for Exercises. Index. List of Tables. T0. 40. 62. 00. 1Preface. This book is based on a course of the same name that has been taught annually at Stanford University since 1. About fifty students have taken it each year juniors and seniors, but mostly graduate students and alumni of these classes have begun to spawn similar courses elsewhere. Thus the time seems ripe to present the material to a wider audience including sophomores. It was dark and stormy decade when Concrete Mathematics was born. Long held values were constantly being questioned during those turbulent years college campuses were hotbeds of controversy. The college curriculum itself was challenged, and mathematics did not escape scrutiny. John Hammersley had just written a thought provoking article On the enfeeblement of mathematical skills by Modern Mathematics and by similar soft intellectual trash in schools and universities 1. Can mathematics be saved One of the present authors had embarked on a series of books called The Art of Computer Programming, and in writing the first volume he DEK had found that there were mathematical tools missing from his repertoire the mathematics he needed for a thorough, well grounded understanding of computer programs was quite different from what hed learned as a mathematics major in college. So he introduced a new course, teaching what he wished somebody had taught him. The course title Concrete Mathematics was originally intended as an antidote to Abstract Mathematics, since concrete classical results were rapidly being swept out of the modern mathematical curriculum by a new wave of abstract ideas popularly called the New Math. Abstract mathematics is a wonderful subject, and theres nothing wrong with it Its beautiful, general, and useful. But its adherents had become deluded that the rest of mathematics was inferior and no longer worthy of attention. The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality mathematical education needed a concrete counterweight in order to restore a healthy balance. When DEK taught Concrete Mathematics at Stanford for the first time he explained the somewhat strange title by saying that it was his attempt to teach a math course that was hard instead of soft. He announced that, contrary to the expectations of some of his colleagues, he was not going to teach the Theory of Aggregates, not Stones Embedding Theorem, nor even the Stone Cech compactification. Several students from the civil engineering department got up and quietly left the room. Although Concrete Mathematics began as a reaction against other trends, the main reasons for its existence were positive instead of negative. And as the course continued its popular place in the curriculum, its subject matter solidified and proved to be valuable in a variety of new applications. Meanwhile, independent confirmation for the appropriateness of the name came from another direction, when Z. A. Melzak published two volumes entitled Companion to Concrete Mathematics 2. The material of concrete mathematics may seem at first to be a disparate bag of tricks, but practice makes it into a disciplined set of tools. Indeed, the techniques have an underlying unity and a strong appeal for many people. When another one of the authors RLG first taught the course in 1. But what exactly is Concrete Mathematics It is a blend of continuous and discrete mathematics. More concretely, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. Once you, the reader, have learned the material in this book, all you will need is a cool head, a large sheet of paper, and fairly decent handwriting in order to evaluate horrendous looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. You will be so fluent in algebraic techniques that you will often find it easier to obtain exact results than to settle for approximate answers that are valid only in a limiting sense. The major topics treated in this book include sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods. The emphasis is on manipulative techniques rather than on existence theorems or combinatorial reasoning the goal is for each reader to become as familiar with discrete operation like the greatest integer function and finite summation as a student of calculus is familiar with continuous operations like the absolute value function and infinite integration Notice that this list of topics is quite different from what is usually taught nowadays in undergraduate course entitled Discrete Mathematics. Therefore the subject needs a distinctive name, and Concrete Mathematics has proved to be as suitable as another The original textbook for Stanfords course on concrete mathematics was the Mathematical Preliminaries section in The Art of Computer Programming 2. But the presentation in those 1. OP was inspired to draft a lengthy set of supplementary notes. The present book is an outgrowth of those notes it is an expansion of, and a more leisurely introduction to, the material if Mathematical Preliminaries. Some of the more advanced parts have been omitted on the other hand, several topics not found there have been included here so that the story will be complete The authors have enjoyed putting this book together because the subject began to jell and to take on a life of its own before our eyes this book almost seemed to write itself.