[i][c]
Maurer, Astephen B. & Ralston, Anthony
Discrete Algorithmic Mathematics
ISBN: 9780201155853 #informatica #matematica
ig01#informatica ig01#matematica
ig02#informatica ig02#matematica

[i][c] INDICE:
 List of Algorithms 0.13 Contents 0.17 Symbols, Notation and Conventions 0.17 A. Algorithms 0.17 B. Notation related to problems 0.17 C. Numbering conventions 0.18 D. General mathematical notation E. Specific mathematical symbols 1 Prologue What Is Discrete algorithmic Mathematics? 8 Chapter 0. Mathematical Preliminares 8 0.1 Sets 18 0.2 Relations 21 0.3 General Properties of Functions 26 0.4 Some Important Functions 39 0.5 Summation and Product Notation 49 0.6 Matrix Algebra 56 0.7 Proof and Logic Concepts 66 Supplementary Problems 68 Chapter 1. Algorithms 68 1.1 Introduction 71 1.2 The Notion of an Algorithm 88 1.3 Algorithmic Language 100 1.4 Recursive Algorithms 110 1.5 Algorithmic Language - Procedures and Functions 122 1.6 The Analysis of Algorithms 134 Supplementary Problems 137 Chapter 2. Mathematical Induction 137 2.1 Introduction 139 2.2 Examples of Induction 155 2.3 Strong Induction and Other Variants 162 2.4 How to Guess What to Prove 170 2.5 Faulty Inductions 178 2.6 Induction and Algorrithms 184 2.7 Inductive Definitions 190 Supplementary Problems 194 Chapter 3. Graphs and Trees 194 3.1 Introduction and Examples 201 3.2 Terminology and Notation 214 3.3 Paths and Cycles - The Adjacency Matrix 227 3.4 Eulerian and Hamiltonian Paths and Cycles 240 3.5 A Shortest Path Algorithm 249 3.6 Breadth First Search and Depth First Search 256 3.7 Coloring Problems 267 3.8 Trees 281 Supplementary Problems 288 Chapter 4. Fundamental Counting Methods 288 4.1 Introduction 289 4.2 First Examples: The Sum and Product Rules 295 4.3 Subtler Examples and the Division Rule 303 4.4 Permutations and Combinations 310 4.5 Combinatorial Identities aand Combinatorial Arguments 315 4.6 The Binomial Theorem 325 4.7 Four Common Problems with Balls and Urns 335 4.8 Inclusion-Exclusion 343 4.9 Combinatorial Algorithms 357 4.10 Algorithmic Pigeonholes 364 Summary 364 Supplementary Problems 366 Chapter 5. Difference Equations 366 5.1 Introduction 368 5.2 Modeling with Different Equations 379 5.3 Getting Information from Difference Equations 387 5.4 Solving Difference Equations: Preliminaries 390 5.5 Secord-Order, Constant Coefficient, Homogeneous Difference Equations 403 5.6 Difference Equations of Arbitrary Order 408 5.7 Nonhomogeneous Difference Equations 415 5.8 The General First-Order Linear Difference Equation 420 5.9 Applications to Algorithms 433 Summary 434 Supplementary Problems 438 Chapter 6. Probability 438 6.1 Introduction 441 6.2 Probability Space 451 6.3 Conditional Probability, Independence, and Bayes' Theorem 465 6.4 Random Variables and Probability Distributions 480 6.5 Expected Value and Variance 493 6.6 Applications to Algorithms: Proofs of Prior Claims 508 6.7 Recursive Methods in Probability 518 Supplementary Problems 522 Chapter 7. An Introduction to Mathematical Logic 522 7.1 Introduction, Terminology, and Notation 528 7.2 The Propositional Calculus 551 7.3 Natural Deduction 559 7.4 Algorithm Verification 564 7.5 Boolean Algebra 581 7.6 The Predicate Calculus 594 7.7 Algorithm Verification Using the Predicate Calculus 600 7.8 Wffs and Algorithms 606 Supplementary Problems 608 Chapter 8. Algorithmic Linear Algebra 608 8.1 Introduction 610 8.2 Gaussian Elimination: Square Systems 625 8.3 Gaussian Elimination: General Case 640 8.4 Gaussian Elimination: A Closer Look 646 8.5 Algorithm Lu-Gauss 655 8.6 Gaussian Elimination and Matrix Algebra 666 8.7 Vector Spaces: Definition and Examples 675 8.8 Vector Spaces: Basic Theory 693 8.9 Eigenvalues 707 8.10 Markov Chains 718 Supplementary Problems 722 Chapter 9. Infinite Processes in Discrete Mathematics 722 9.1 Introduction 725 9.2 Sequences and Their Limits 738 9.3 Growth Rates and Order Notation 746 9.4 Finite Differences and Factorial Functions 757 9.5 Infinite Series and Their Summation 773 9.6 Generating Function 784 9.7 Approximation Algorithms 795 Supplementary Problems 798 Epilogue. Sorting Things Out with Sorting 798 E.1 Comparison of Previous Methods 802 E.2 The Complexity of SOrting by Comparisons 815 E.3 Quicksort 829 Final Problems References Appendixes 1. Summary of Algorithmic Language 2. Abbreviations Hints and Answers Index

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