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Handbook of Applicable Mathematics HANDBOOK OF APPLICABLE MATHEMATICS Chief Editor: Walter Ledermann Supplement Edited by Walter Ledermann University of Sussex Emlyn Lloyd University of Lancaster Steven Vajda University of Sussex and Carol Alexander University of Sussex A Wiley-Interscience Publication JOHN WILEY & SONS Chichester - New York - Brisbane - Toronto - Singapore Contents INTRODUCTION INTRODUCTION TO THE SUPPLEMENT 1. FURTHER NUMBER THEORY 1.1 Introduction 1.2. Base of notation .... 1.2.1. The representation theorem 1.3. Divisibility 1.3.1. The highest common factor (greatest common divisor) 1.3.2. Coprime integers 1.4. Modular arithmetic 1.4.1. Complete residue Systems 1.4.2. Reduced residue Systems 1.4.3. Quadratic residues 1.5. Finitefields 1.5.1. The fields GF(p) 1.5.2. The fields GF(p") 1.5.3. Vector Spaces over GF{q) 1.5.4. Polynomials over GF(q) FIBONACCI AND LUCAS NUMBERS 2.1. Definitions 2.2. Examples 2.3. Relationships 2.4. Diophantine equations 2.5. Convergents 2.6. Fibonacci series 2.7. Divisibility 2.8. Primefactors 2.9. Residue periods 3. CRYPTOGRAPHYAND CRYPTANALYSIS 3.1. Introduction 3.2. The Shannon approach vn Vlll Contents 3.2.1. Definition of a cipher System 3.2.2. Secrecy 3.2.3. Pure ciphers 3.2.4. Perfect secrecy 3.2.5. Cryptanalysis 3.2.6. Entropy and equivocation 3.2.7. Random ciphers 3.2.8. Comment 3.3. Stream ciphers 3.3.1. Stream ciphers and the one-time-päd 3.3.2. Shift registers 3.3.3. Pseudo-randomness 3.3.4. Linear complexity profiles 3.3.5. Cryptographically strong pseudo -random bit generators . 3.4. Conventional block ciphers 3.4.1. Feistel ciphers 3.4.2. Cryptanalysis 3.5. Public key Systems 3.5.1. One-way functions 3.5.2. Some public key Systems 3.5.3. The Diffie-Hellman key exchange . 3.5.4. Related mathematical problems .... 3.6. Personal identification and signatures 3.6.1. Signature schemes 3.6.2. Identification 3.7. Other mathematical themes 3.7.1. Extensions to the Shannon approach 3.7.2. Threshold schemes and key distribution patterns 3.7.3. Zero-knowledge proofs 4. CATALAN NUMBERS AND THEIR VARIOUS USES 4.0. Introduction 4.1. Catalan numbers and the ballot problem 4.2. Generalized Catalan numbers .... 4.3. Counting p-good paths 5. INTEGRAL EQUATIONS 5.1. Introduction 5.2. Fredholm integral equations 5.3. Hermitian kerneis, Hilbert-Schmidt theory 5.4. Positive kerneis 5.5. Linear Fredholm equations of the first kind 5.6. Volterra integral equations 5.7. Singular kerneis Contents ix 5.8. Difference kerneis 5.9. Non-linear equations 6. DYNAMICAL SYSTEMS 6.1. Introduction .... 6.1.1. Ordinary differential equations in the plane 6.2. Flows on manifolds 6.3. Basic concepts .... 6.3.1. Fixed points .... 6.3.2. Stable and unstable manifolds 6.3.3. Stability .... 6.3.4. Discrete and continuous dynamics 6.3.5. Liapunov functions 6.3.6. Periodic orbits 6.3.7. Asymptotic behaviour 6.3.8. Flows in the plane 6.3.9. Index theory in the plane 6.3.10. Special Systems 6.3.11. Structural stability 6.4. Attractors, bifurcations and chaos 6.4.1. Attractors 6.4.2. Bifurcations 6.4.3. Chaos 7. CONTROL THEORY 7.1. Introduction 7.2. Linear Systems 7.2.1. Equivalent Systems 7.2.2. Controllability 7.2.3. Observability 7.2.4. ControUable and unobservable subspaces 7.2.5. Module formulation 7.3. Non-linear Systems 7.3.1. Linear-analytic Systems 7.3.2. Accessibility 7.4. Optimal control IAA. Pontryagin's minimum principle 7.4.2. Singular control 7.4.3. Junction conditions 7.4.4. Dynamic programming THE FINITE ELEMENT METHOD 8.1. Introduction 8.2. First example 8.3. Steady-state (elliptic) problems 8.4. Finite element basis functions 8.5. The assembly ofthe finite element System 8.6. Accuracy of the finite element approximation 8.7. Time-dependent problems .... 9. COMPUTATIONALCOMPLEXITY 9.1. Introduction 9.1.1. Definition of an algorithm 9.1.2. Measures of complexity 9.2. Arithmetic complexity of computations 9.2.1. Matrix multiplication .... 9.2.2. Matrix product by Strassen's algorithm 9.2.3. Lower bounds on matrix multiplication 9.2.4. The fast Fourier transform (FFT) 9.2.5. Product of two integers 9.2.6. Worst-case and average-case analysis 9.3. Complexity of data processing problems 9.3.1. Search problem 9.3.2. A binary search algorithm 9.3.3. Comparison sort problem 9.4. Complexity of combinatorial problems 9.4.1. Polynomial and exponential time algorithms 9.4.2. Efficient algorithm design techniques 9.4.3. Graph searching 9.5. Theorem proving by machine 9.5.1. The first-order integer addition problem 9.5.2. Decidable and undecidable problems 9.5.3. Algorithms for decidable problems 9.6. Computational complexity and class NP 9.6.1. Non-deterministic algorithms 9.6.2. The JVP-space 9.6.3. Relationship between P and NP Spaces 9.6.4. NP-complete problems .... 9.6.5. Open problems 9.6.6. Beyond the NP-class .... 10. NON-COOPERATIVE FINITE GAMES 10.1. Introduction 10.2. Elementary theory of non-cooperative finite games 10.2.1. Finite games in extensive form 10.2.2. Normal form of two-person finite games 10.2.3. Subgame-perfect equilibria 10.3. Games with incomplete Information 10.3.1. Bayesian equilibria .... 10.3.2. Sequential equilibria .... Contents XI 10.3.3. Trembling hand perfect equilibria 10.4. Repeated games with complete Information 10.4.1. Finite repeated games .... 10.4.2. Infinitely repeated games without discounting 10.4.3. Infinitely repeated games with discounting 10.5. Repeated games with incomplete information 10.5.1. The two-person zero-sum case 10.5.2. The non-zero-sum case 10.6. Evolutionäry games 10.6.1. Symmetrie conflicts 10.6.2. Asymmetrie conflicts 10.6.3. The evolution of Cooperation and refinements of the ESS coneept .... 10.7. Bargaining theory 10.7.1. The axiomatic approach 10.7.2. The Strategie approach 10.7.3. Bargaining with incomplete information 10.8. Rational play in non-cooperative games 10.8.1. Rationalizable Strategie behaviour and Bayesian rationality 10.8.2. Refinements of the Nash equilibrium coneept 10.8.3. Bounded rationality .... 11. POPULATION STRUCTURES 11.1. Cohorts 11.2. Constant populations 11.3. The stationary population 11.4. Eigenvalues and eigenvectors 11.5. General transition matrices 11.6. The stable population 11.7. The semi-stationary population 11.8. Ageing and promotion 11.9. Continuous time 11.10. States in continuous time 12. QUEUEING AND RELATED THEORY 12.1. Introduction .... 12.2. Definitions 12.3. Exponential queues 12.3.1. M/M/l/oo .... (a) System State (b) System time (first come, first served) 12.3.2. M/M/2/oo .... 12.3.3. M/M/L/L .... 12.3.4. M/M/oo .... 12.4. Other demand and Service mechanisms Xll Contents 12.4.1. Generalized exponential Systems 12.4.2. M/G/l .... 12.4.3. G/M/l .... 12.5. Some special topics 12.5.1. Machine maintenance: a closed population 12.5.2. Batch queues 12.5.3. Networks 12.5.4. Heavy traffic analysis 12.5.5. Priority disciplines 12.6. Concluding remarks 13. BOOTSTRAP METHODS 13.1. Introduction 13.2. An example .... 13.3. Why the bootstrap works 13.3.1. Generalities 13.3.2. Estimation of F 13.4. More complex examples 13.4.1. Two-sample problem 13.4.2. Linear regression 13.4.3. Dosage-mortality relationship 13.4.4. Autoregressive time series 13.5. Efficient bootstrap methods 13.5.1. Von Mises expansions 13.5.2. Bias estimation 13.5.3. Variance estimation 13.5.4. An example 13.6. Bootstrap significance tests 13.7. Confidence intervals 13.8. Bibliographie notes 14. EXTREME VALUE THEORY 14.1. Introduction 14.2. The extreme value distributions 14.3. Derivation of the extreme value limit 14.4. Exceedances: the generalized Pareto distribution 14.5. Statistical methods: introduction 14.6. Estimation by maximum likelihood 14.7. Maximum likelihood in more complicated situations 14.8. The r-largest order statistics method 14.9. Threshold methods 14.10. Multivariate extremes 14.11. Miscellaneous Statistical applications INDEX 473 .
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