!!! Aplied Combinatorial Mathematics
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Transcript of !!! Aplied Combinatorial Mathematics
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APPLIED COMBINATORIAL
MATHEMATICS
The Authors
GEORGE POLYA
DERRICK H. L E H ME R
MONTGOMERY PHISTER, Jr.
JOHN RIORDAN
ELLIOTT W. MONTROLL
N. G. D E BRUIJN
FRANK HARARY
RICHARD BELLMAN
ROBERT KALABA
EDWIN L. PETERSON
LEO BREIMAN
ALBERTW.
TUCKER
EDWIN F. BECKENBACH
MARSHALL HALL, Jr.
JACOB WOLFOWITZ
CHARLES B. TOMPKINS
KENNETH N. TRUEBLOOD
GEORGE GAMOW
HERMANN WE Y L
Editor
EDWIN F. BECKENBACHProfessor of MathematicsUniversity of CaliforniaLos Angeles
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CONTENTS
INTRODUCTION 1
PART 1A COMPUTATION AND EVALUATION
chapter 1 THE MACHINE TOOLS OF COMBINATORICS 5DERRICK H. LEHMER
1.1 Introduction 5
1.2 Representation and Processing of Digital Information 6
1.3 Application of Representations 11
1.4 Signatures and Their Orderly Generation 17
1.5 Orderly Listing of Permutations 19
1.6 Orderly Listing of Combinations 24
1.7 Orderly Listing of Compositions 24
1.8 Orderly Listing of Partitions 25
1.9 The Back-Track Procedure 26
1.10 Ranking of Combinations 27
chapter 2 TECHNIQUES FOR SIMPLIFYING LOGICAL
NETWORKS 32MONTGOMERY PHISTER, JR.
2.1 Introduction 32
Simplifying a Given Sequential Circuit 37
2.2 Simplest AND-OR Gates 37
2.3 Multiple-Output Function Simplifications 40
2.4 Elimination of Memory Elements 42
2.5 Alternative Methods of Assigning Flip-Flop States 49
C/ioosing Decision and Memory Elements 53
2.6 Memory Elements Other Than Delay Flip-Flops 53
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xvi Contents
2.7 Decision Elements Other Than AND and OR Gates
2.8 Conclusion
57
61
PART IB COUNTING AND ENUMERATION
chapter 3 GENERATING FUNCTIONS 67JOHN RIORDAN
3.1 Introduction 67
3.2 Generating Functions for Combinations 69
3.3 Generating Functions for Permutations 71
3.4 Elementary Relations: Ordinary Generating Function 72
3.5 Elementary Relations: Exponential Generating Function 79
3.6 Generating Functions in Probability and Statistics 84
3.7 Polya s Enumeration Theorem 90
chapter 4 LATTICE STATISTICS 96ELLIOTT W. MONTROLL
4.1 Introduction 96
4.2 Random Walks on Lattices and the Polya Problem 97
4.3 More General Random Walks on Lattices 103
4.4 The PfafRan and the Dimer Problem 105
4.5 Cyclic Matrices 113
4.6 Evaluation of Dimer Pfaffian 116
4.7 The Ising Problem 121
4.8 Some Remarks on Periodic Boundary Conditions 132
4.9 Lattice Statistics of Slightly Defective Lattices 134
chapter 5 POLYA S THEORY OF COUNTING 144
N. G. DE BRUIJN
5.1 Introduction 144
5.2 Cycle Index of a Permutation Group 145
5.3 The Main Lemma 150
5.4 Functions and Patterns 151
5.5 Weight of a Function; Weight of a Pattern 153
5.6 Store and Inventory 154
5.7 Inventory of a Function 155
5.8 The Pattern Inventory; Polya s Theorem 157
5.9 Generalization of Polya s Theorem 161
5.10 Patterns of One-to-One Mappings 164
5.11 Labeling and Delabeling 168
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Contents xvii
5.12 The Total Number of Patterns 171
5.13 The Kranz Group 176
5.14 Epilogue 180
chapter 6 COMBINATORIAL PROBLEMS IN GRAPHICALENUMERATION 185
FRANK HARARY
6.1 Introduction 185
6.2 Graphical Preliminaries 185
6.3 Some Unsolved Problems 189
6.4 Problems Involving Directed Graphs 189
6.5 Problems Involving Partitions 190
6.6 Topological Problems 191
6.7 Problems Involving Connectivity 193
6.8 Problems Involving Groups 195
6.9 Electrical Problems 197
6.10 Physical Problems 199
6.11 Graph-Counting Methods 202
6.12 Tree-Counting Methods 205
6.13 Comparison of Solved and Unsolved Problems 208
6.14 Some Applications of Enumeration Problems to Other Fields 212
PART 2 CONTROL AND EXTREMIZATION
chapter 7 DYNAMIC PROGRAMMING AND MARKOVIANDECISION PROCESSES, WITH PARTICULAR
APPLICATION TO BASEBALL AND CHESS 221RICHARD BELLMAN
7.1 Introduction 221
7.2 Baseball as a Multistage Decision Process; State Variables 221
7.3 Decisions 222
7.4 Criterion Function 223
7.5 Policies, Optimum Policies/ and Combinatorics 224
7.6 Principle of Optimality 226
7.7 Functional Equations 226
7.8 Existence and Uniqueness 227
7.9 Analytic Aspects 227
7.10 Direct Computational Approach 229
7.11 Stratification 229
7.12 Approximation in Policy Space 230
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xviii Contents
7.13 Utilization of Experience an d In tuition 231
7 14 Two-Team Version of Baseball 232
7.15 Probability of Tying or Winning 232
7.16 Inventory and Replacement Processes and Asymptotic Behavior 2327.17 Chess as a Multistage Decision Process 233
7.18 Computational Aspects 234
chapter 8 GRAPH THEORY AND AUTOMATIC CONTROL 237ROBERT KALABA
8.1 Introduction 237
8.2 Time-Optimal Control and the Nature of Feedback 238
8.3 Formulation 239
8.4 Uniqueness 240
8.5 Successive Approximations 241
8.6 Observations on the Approximation Scheme 242
8.7 Other Approaches 242
8.8 Arbitrary Terminal States 243
8.9 Preferred Suboptima l Trajectories 243
8.10 A Stochastic Time-Optimal Control Process 244
8.11 Minimax Control Processes 246
8.12 Use of Functional Equations 246
8.13 A Special Case 246
8.14 Minimal Spanning Trees 247
8.15 Comments and Interconnections 248
8.16 Multiple Stresses 249
8.17 Discussion 250
chapter 9 OPTIMUM MULT1VARIABLE CONTROL 253EDWIN L. PETERSON
9.1 Problem Structure 253
9.2 Alternative Approaches 257
9.3 Polynomial Approximation 264
9.4 Linearization and Successive Approximation 272
9.5 Physical Implementation 279
chapter 10 STOPPING-RULE PR O B LEMS 284LEO BREIMAN
10.1 How to Recognize a Stopping-Rule Problem 284
10.2 Examples 285
10.3 Formulation 287
10.4 What Is a Stopping Rule? 290
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Contents xix
10.5 What Is a Solution? 291
10.6 Stop When You Are Ahead; the Stability Problem 292
10.7 The Functional Equation 294
10.8 Elimination of Forced Continuation 296
10.9 Entrance-Fee Problems and Reduction to Them 298
10.10 The Linear-Programming Solution 302
Binary Decision Renewal Problems 305
10.11 Introduction and Examples 305
10.12 Formulation 306
10.13 What Is a Solution? 307
10.14 Reduction to a Stopping-Rule Problem by Cycle Analysis 308
10.15 A Direct Approach 312
10.16 The Linear-Programming Solution 314
10.17 A Duality Relation 317
chapter COMBINATORIAL A L G E B R A OF MATRIX GAMESAND LINEAR PROGRAMS 320ALBERT W. TUCKER
11.1 Introduction 320
11.2 Game Example 320
11.3 Key Equation and Basic Solutions 322
11.4 Schematic Representation 324
11.5 Equivalence by Pivot Steps; Simplex Method 327
11.6 Linear Programming Examples 331
11.7 Key Equation and Basic Solutions for Linear Programs 333
11.8 Canonical Forms 336
11.9 Inverse Basis Procedure 340
11.10 General Formulations 343
chapter 12 NETWORK FLOW PROBLEMS 348
EDWIN F. BECKENBACH
12.1 Introduction 348
12.2 Networks 348
12.3 Cuts 350
12.4 Flows 351
12.5 Flows and Cuts 352
12.6 Bounds on Capacities of Cuts and on Values of Flows 355
12.7 The Max-Flow—Min-Cut Theorem 357
12.8 The Ford-Fulkerson Algorithm 358
12.9 Example 359
12.10 Extensions and Applications 361
12.11 Graph Theory and Combinatorial Problems 362
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Contents
PART 3 CONSTRUCTION AND EXISTENCE
chapter 13 BLOCK DESIGNS 369
MARSHALL HALL, JR.13.1 Introduction 369
13.2 Block Designs and Latin Squares; Examples and Definitions 370
13.3 Applications of Block Designs 373
13.4 General Theory of Block Designs 377
13.5 Construction of Block Designs and Orthogonal Latin Squares 387
chapter 14 INTRODUCTION TO INFORMATION THEORY 406
JACOB WOLFOWITZ
14.1 Introduction 406
14.2 Channels 406
14.3 Codes 407
14.4 Entropy; Generated Sequences 408
14.5 The Discrete Memoryless Channel 410
14.6 Compound Channels 412
14.7 Other Channels 414
chapter 15 SPERNER S LEMMA AND SOME EXTENSIONS 416CHARLES B. TOMPKINS
15.1 Introduction 416
15.2 Sperner s Lemma 418
15.3 Brouwer s Fixed-Point Theorem 424
15.4 Subdivisions of a Simplex 427
15.5 Review of Position—Second Introduction 430
15.6 Oriented Simplexes, Their Boundaries/ and Oriented Simplicial
Mappings 434
15.7 Homology—fc-Chains, Cycles/ Bounding Cycles 437
15.8 The Fundamental Index Theorem 441
15.9 Some Applications 445
15.10 Toward Greater Rigor 453
chapter 16 CRYSTALLOGRAPHY 456KENNETH N. TRUEBLOOD
16.1 Introduction 456
Nature of the Crystalline Sfate 457
16.2 Lattices 457
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Contents xxi
16.3 Point Symmetry 461
16.4 Space Symmetry 465
16.5 Antisymmetry 467
Crystal Structure Analysis by X Ray Diffraction 468
16.6 Diffraction468
16.7 Fourier Transforms of Atoms and Groups of Atoms 474
16.8 The Phase Problem 478
Approaches to Solving the Phase Problem 482
16.9 The Patterson Function 482
16.10 Heavy-Atom Methods 485
16.11 Isomorphous Replacement 488
16.12 Direct Methods 491
16.13 Refinement 495
Some Aspects of Atomic Arrangements 500
16.14 Topology and Shapes of Molecules and Assemblies of Molecules 500
16.15 Packing of Spheres, Molecules, and Polyhedra 504
16.16 Disordered Structures and Order-Disorder Phenomena 507
chapter 17 COMBINATORIAL PRINCIPLES IN GENETICS 515
GEORGE GAMOW
17.1 Introduction 515
17.2 Amino-Acid Sequences in Proteins 515
17.3 Double-Stranded Sequences in DNA 519
17.4 Combinatorial Principles 522
17.5 The Overlapping-Code Hypothesis 522
17.6 Statistical Investigations 523
17.7 Geometric Implications 525
17.8 Statistical Assignments 526
17.9 Monte Carlo Methods 528
17.10 Experimental Results 532
chapter 18 APPENDICES 536HERMANN W E Y L
18.1 Ars Combinatoria 536
18.2 Quantum Physics and Causality 551
18.3 Chemical Valence and the Hierarchy of Structures 563
18.4 Physics and Biology 572
ANSWERS TO MULTIPLE-CHOICE REVIEW P ROBLEMS 583
AUTHOR INDEX 585
SUBJECT INDEX 591