Applied Mathematical Methods In Theoretical Physics

Contents
Preface IX
Introduction 1
1 Function Spaces, Linear Operators and Green’s Functions 5
1.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Orthonormal System of Functions . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 The Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Green’s Functions for Differential Equations . . . . . . . . . . . . . . . . . 16
1.8 Review of Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9 Review of Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Integral Equations and Green’s Functions 33
2.1 Introduction to Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Relationship of Integral Equations with Differential Equations and Green’s
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Sturm–Liouville System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Green’s Function for Time-Dependent Scattering Problem . . . . . . . . . . 48
2.5 Lippmann–Schwinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6 Problems for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Integral Equations of Volterra Type 63
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind . . . . . 63
3.2 Solvable cases of Volterra Integral Equation . . . . . . . . . . . . . . . . . 66
3.3 Problems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Integral Equations of the Fredholm Type 75
4.1 Iterative Solution to the Fredholm Integral Equation of the Second Kind . . 75
4.2 Resolvent Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Pincherle–Goursat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Fredholm Theory for a Bounded Kernel . . . . . . . . . . . . . . . . . . . . 86
4.5 Solvable Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Fredholm Integral Equation with a Translation Kernel . . . . . . . . . . . . 95
4.7 System of Fredholm Integral Equations of the Second Kind . . . . . . . . . 100
4.8 Problems for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 Hilbert–Schmidt Theory of Symmetric Kernel 109
5.1 Real and Symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Real and Symmetric Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 Bounds on the Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5 Completeness of Sturm–Liouville Eigenfunctions . . . . . . . . . . . . . . 129
5.6 Generalization of Hilbert–Schmidt Theory . . . . . . . . . . . . . . . . . . 131
5.7 Generalization of Sturm–Liouville System . . . . . . . . . . . . . . . . . . 138
5.8 Problems for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6 Singular Integral Equations of Cauchy Type 149
6.1 Hilbert Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 Cauchy Integral Equation of the First Kind . . . . . . . . . . . . . . . . . . 153
6.3 Cauchy Integral Equation of the Second Kind . . . . . . . . . . . . . . . . 157
6.4 Carleman Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.6 Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7 Wiener–Hopf Method and Wiener–Hopf Integral Equation 177
7.1 The Wiener–Hopf Method for Partial Differential Equations . . . . . . . . . 177
7.2 Homogeneous Wiener–Hopf Integral Equation of the Second Kind . . . . . 191
7.3 General Decomposition Problem . . . . . . . . . . . . . . . . . . . . . . . 207
7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind . . . . 216
7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation . . . . . . . . . . . . . . . 227
7.6 Wiener–Hopf Integral Equation of the First Kind and Dual Integral Equations 235
7.7 Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8 Nonlinear Integral Equations 249
8.1 Nonlinear Integral Equation of Volterra type . . . . . . . . . . . . . . . . . 249
8.2 Nonlinear Integral Equation of Fredholm Type . . . . . . . . . . . . . . . . 253
8.3 Nonlinear Integral Equation of Hammerstein type . . . . . . . . . . . . . . 257
8.4 Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9 Calculus of Variations: Fundamentals 263
9.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.3 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.4 Generalization of the Basic Problems . . . . . . . . . . . . . . . . . . . . . 272
9.5 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
9.6 Differential Equations, Integral Equations, and Extremization of Integrals . . 278
9.7 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
9.8 Weierstrass–Erdmann Corner Relation . . . . . . . . . . . . . . . . . . . . 297
9.9 Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10 Calculus of Variations: Applications 303
10.1 Feynman’s Action Principle in Quantum Mechanics . . . . . . . . . . . . . 303
10.2 Feynman’s Variational Principle in Quantum Statistical Mechanics . . . . . 308
10.3 Schwinger–Dyson Equation in Quantum Field Theory . . . . . . . . . . . . 312
10.4 Schwinger–Dyson Equation in Quantum Statistical Mechanics . . . . . . . 329
10.5 Weyl’s Gauge Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
10.6 Problems for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Bibliography 365
Index 373

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Ant Colony Optimization

Contents
Preface ix
Acknowledgments xiii
1 From Real to Artificial Ants 1
1.1 Ants’ Foraging Behavior and Optimization 1
1.2 Toward Artificial Ants 7
1.3 Artificial Ants and Minimum Cost Paths 9
1.4 Bibliographical Remarks 21
1.5 Things to Remember 22
1.6 Thought and Computer Exercises 23
2 The Ant Colony Optimization Metaheuristic 25
2.1 Combinatorial Optimization 25
2.2 The ACO Metaheuristic 33
2.3 How Do I Apply ACO? 38
2.4 Other Metaheuristics 46
2.5 Bibliographical Remarks 60
2.6 Things to Remember 61
2.7 Thought and Computer Exercises 63
3 Ant Colony Optimization Algorithms for the Traveling Salesman
Problem 65
3.1 The Traveling Salesman Problem 65
3.2 ACO Algorithms for the TSP 67
3.3 Ant System and Its Direct Successors 69
3.4 Extensions of Ant System 76
3.5 Parallel Implementations 82
3.6 Experimental Evaluation 84
3.7 ACO Plus Local Search 92
3.8 Implementing ACO Algorithms 99
3.9 Bibliographical Remarks 114
3.10 Things to Remember 117
3.11 Computer Exercises 117
4 Ant Colony Optimization Theory 121
4.1 Theoretical Considerations on ACO 121
4.2 The Problem and the Algorithm 123
4.3 Convergence Proofs 127
4.4 ACO and Model-Based Search 138
4.5 Bibliographical Remarks 149
4.6 Things to Remember 150
4.7 Thought and Computer Exercises 151
5 Ant Colony Optimization for NP -Hard Problems 153
5.1 Routing Problems 153
5.2 Assignment Problems 159
5.3 Scheduling Problems 167
5.4 Subset Problems 181
5.5 Application of ACO to Other NP -Hard Problems 190
5.6 Machine Learning Problems 204
5.7 Application Principles of ACO 211
5.8 Bibliographical Remarks 219
5.9 Things to Remember 220
5.10 Computer Exercises 221
6 AntNet: An ACO Algorithm for Data Network Routing 223
6.1 The Routing Problem 223
6.2 The AntNet Algorithm 228
6.3 The Experimental Settings 238
6.4 Results 243
6.5 AntNet and Stigmergy 252
6.6 AntNet, Monte Carlo Simulation, and Reinforcement Learning 254
6.7 Bibliographical Remarks 257
6.8 Things to Remember 258
6.9 Computer Exercises 259
7 Conclusions and Prospects for the Future 261
7.1 What Do We Know about ACO? 261
7.2 Current Trends in ACO 263
7.3 Ant Algorithms 271
Appendix: Sources of Information about the ACO Field 275
References 277
Index 301

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An Introduction to the Theory of Formal Languages and Automata

Table of contents
Preface ix
chapter 1
Grammars as formal systems 1
1.1 Grammars, automata, and inference 1
1.2 The definition of ‘grammar’ 3
1.3 Examples 5
chapter 2
The hierarchy of grammars 9
2.1 Classes of grammars 9
2.2 Regular grammars 12
2.3 Context-free grammars 16
2.3.1 The Chomsky normal-form 16
2.3.2 The Greibach normal-form 18
2.3.3 Self-embedding 20
2.3.4 Ambiguity 23
2.3.5 Linear grammars 25
2.4 Context-sensitive grammars 26
2.4.1 Context-sensitive productions 26
2.4.2 The Kuroda normal-form 29
chapter 3
Probabilistic grammars 33
3.1 Definitions and concepts 33
3.2 Classification 35
3.3 Regular probabilistic grammars 36
3.4 Context-free probabilistic grammars 41
3.4.1 Normal-forms 41
3.4.2 Consistency conditions for context-free
probabilistic grammars 46
chapter 4
Finite automata 49
4.1 Definitions and concepts 50
4.2 Nondeterministic finite automata 55
4.3 Finite automata and regular grammars 58
4.4 Probabilistic finite automata 62
chapter 5
Push-down automata 69
5.1 Definitions and concepts 70
5.2 Nondeterministic push-down automata
and context-free languages 79
chapter 6
Linear-bounded automata 85
6.1 Definitions and concepts 85
6.2 Linear-bounded automata
and context-sensitive grammars 89
chapter 7
Turing machines 95
7.1 Definitions and concepts 96
7.2 A few elementary procedures 97
7.3 Turing machines and type-0 languages 100
7.4 Mechanical procedures, recursive enumerability,
and recursiveness 103
chapter 8
Grammatical inference 109
8.1 Hypotheses, observations, and evaluation 109
8.2 The classical estimation of parameters
for proba bilistic grammars 112
8.3 The ‘learnability’ of nonprobabilistic languages 114
8.4 Inference by means of Bayes’ theorem 118
Historical and bibliographical remarks 125
Appendix. Some references to new developments 129
Bibliography 131
Index of authors 135
Index of subjects 137


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