Concrete Mathematics: A Foundation for Computer Science

(Most chapters contain Exercises.)

1. 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.

3. Integer Functions.

Floors and Ceilings.

Floor/Ceiling Applications.

Floor/Ceiling Recurrences.

'mod': The Binary Operation.

Floor/Ceiling Sums.

Exercises.

4. Number Theory.

Divisibility.

Factorial Factors.

Relative Primality.

'mod': The Congruence Relation.

Independent Residues.

Additional Applications.

Phi and Mu.

Exercises.

5. 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.

7. 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.

Euler's Summation Formula.

Final Summations.

Exercises.

A. Answers to Exercises.

B. Bibliography.

C. Credits for Exercises.

Index.

List of Tables. 0201558025T04062001
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Autorentext
Ronald L. Graham (19352020) was for many years the Chief Scientist at AT&T Labs Research. He was also a Professor of Computer and Information Science at the University of California, San Diego, and a former President of the American Mathematical Society and the Mathematical Association of America. He was the coauthor of seven other mathematics books.

Donald E. Knuth is Professor Emeritus of The Art of Computer Programming at Stanford University. His prolific writings include four volumes on The Art of Computer Programming, and five books related to his TEX and METAFONT typesetting systems.

Oren Patashnik is a member of the research staff at the Center for Communications Research, La Jolla, California. He is also the author of BibTEX, a widely used bibliography processor.

Klappentext

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

Zusammenfassung

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.

Inhalt

• 1. Recurrent Problems.
• 2. Sums.
• 3. Integer Functions.
• 4. Number Theory.
• 5. Binomial Coefficients.
• 6. Special Numbers.
• 7. Generating Functions.
• 8. Discrete Probability.
• 9. Asymptotics.
• A. Answers to Exercises.
• B. Bibliography.
• C. Credits for Exercises.
• Index.