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Cambridge University Press Percolation

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Book SynopsisIn this book, first published in 2006, the authors' main aims are first to present classical results in a way that's accessible to non-specialists. Second, to describe results of Smirnov in conformal invariance. It is essential reading for all working in this exciting area.Trade Review'This book contains a complete account of most of the important results in the fascinating area of percolation. Elegant and straightforward proofs are given with minimal background in probability or graph theory. It is self-contained, accessible to a wide readership and widely illustrated with numerous examples. It will be of considerable interest for both beginners and advanced searchers alike.' Zentralblatt MATHTable of ContentsPreface; 1. Basic concepts; 2. Probabilistic tools; 3. Percolation on Z2 - the Harris-Kesten theorem; 4. Exponential decay and critical probabilities - theorems of Menshikov and Aizenman & Barsky; 5. Uniqueness of the infinite open cluster and critical probabilities; 6. Estimating critical probabilities; 7. Conformal invariance - Smirnov's theorem; 8. Continuum percolation; Bibliography; Index; List of notation.

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Cambridge University Press Partisan Families

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Book SynopsisA comprehensive introduction to the multidisciplinary applications of mathematical methods, revised and updated The second edition of Essentials of Mathematical Methods in Science and Engineering offers an introduction to the key mathematical concepts of advanced calculus, differential equations, complex analysis, and introductory mathematical physics for students in engineering and physics research. The book's approachable style is designed in a modular format with each chapter covering a subject thoroughly and thus can be read independently. This updated second edition includes two new and extensive chapters that cover practical linear algebra and applications of linear algebra as well as a computer file that includes Matlab codes. To enhance understanding of the material presented, the text contains a collection of exercises at the end of each chapter. The author offers a coherent treatment of the topics with a style that makes the essential mathematicTable of ContentsPreface xxiii Acknowledgments xxix 1 Functional Analysis 1 1.1 Concept of Function 1 1.2 Continuity and Limits 3 1.3 Partial Differentiation 6 1.4 Total Differential 8 1.5 Taylor Series 9 1.6 Maxima and Minima of Functions 13 1.7 Extrema of Functions with Conditions 17 1.8 Derivatives and Differentials of Composite Functions 21 1.9 Implicit Function Theorem 23 1.10 Inverse Functions 28 1.11 Integral Calculus and the Definite Integral 30 1.12 Riemann Integral 32 1.13 Improper Integrals 35 1.14 Cauchy Principal Value Integrals 38 1.15 Integrals Involving a Parameter 40 1.16 Limits of Integration Depending on a Parameter 44 1.17 Double Integrals 45 1.18 Properties of Double Integrals 47 1.19 Triple and Multiple Integrals 48 References 49 Problems 49 2 Vector Analysis 55 2.1 Vector Algebra: Geometric Method 55 2.1.1 Multiplication of Vectors 57 2.2 Vector Algebra: Coordinate Representation 60 2.3 Lines and Planes 65 2.4 Vector Differential Calculus 67 2.4.1 Scalar Fields and Vector Fields 67 2.4.2 Vector Differentiation 69 2.5 Gradient Operator 70 2.5.1 Meaning of the Gradient 71 2.5.2 Directional Derivative 72 2.6 Divergence and Curl Operators 73 2.6.1 Meaning of Divergence and the Divergence Theorem 75 2.7 Vector Integral Calculus in Two Dimensions 79 2.7.1 Arc Length and Line Integrals 79 2.7.2 Surface Area and Surface Integrals 83 2.7.3 An Alternate Way to Write Line Integrals 84 2.7.4 Green’s Theorem 86 2.7.5 Interpretations of Green’s Theorem 88 2.7.6 Extension to Multiply Connected Domains 89 2.8 Curl Operator and Stokes’s Theorem 92 2.8.1 On the Plane 92 2.8.2 In Space 96 2.8.3 Geometric Interpretation of Curl 99 2.9 Mixed Operations with the Del Operator 99 2.10 Potential Theory 102 2.10.1 Gravitational Field of a Star 105 2.10.2 Work Done by Gravitational Force 106 2.10.3 Path Independence and Exact Differentials 108 2.10.4 Gravity and Conservative Forces 109 2.10.5 Gravitational Potential 111 2.10.6 Gravitational Potential Energy of a System 113 2.10.7 Helmholtz Theorem 115 2.10.8 Applications of the Helmholtz Theorem 116 2.10.9 Examples from Physics 120 References 123 Problems 123 3 Generalized Coordinates and Tensors 133 3.1 Transformations between Cartesian Coordinates 134 3.1.1 Basis Vectors and Direction Cosines 134 3.1.2 Transformation Matrix and Orthogonality 136 3.1.3 Inverse Transformation Matrix 137 3.2 Cartesian Tensors 139 3.2.1 Algebraic Properties of Tensors 141 3.2.2 Kronecker Delta and the Permutation Symbol 145 3.3 Generalized Coordinates 148 3.3.1 Coordinate Curves and Surfaces 148 3.3.2 Why Upper and Lower Indices 152 3.4 General Tensors 153 3.4.1 Einstein Summation Convention 156 3.4.2 Line Element 157 3.4.3 Metric Tensor 157 3.4.4 How to Raise and Lower Indices 158 3.4.5 Metric Tensor and the Basis Vectors 160 3.4.6 Displacement Vector 161 3.4.7 Line Integrals 162 3.4.8 Area Element in Generalized Coordinates 164 3.4.9 Area of a Surface 165 3.4.10 Volume Element in Generalized Coordinates 169 3.4.11 Invariance and Covariance 171 3.5 Differential Operators in Generalized Coordinates 171 3.5.1 Gradient 171 3.5.2 Divergence 172 3.5.3 Curl 174 3.5.4 Laplacian 178 3.6 Orthogonal Generalized Coordinates 178 3.6.1 Cylindrical Coordinates 179 3.6.2 Spherical Coordinates 184 References 189 Problems 189 4 Determinants and Matrices 197 4.1 Basic Definitions 197 4.2 Operations with Matrices 198 4.3 Submatrix and Partitioned Matrices 204 4.4 Systems of Linear Equations 207 4.5 Gauss’s Method of Elimination 208 4.6 Determinants 211 4.7 Properties of Determinants 214 4.8 Cramer’s Rule 216 4.9 Inverse of a Matrix 221 4.10 Homogeneous Linear Equations 224 References 225 Problems 225 5 Linear Algebra 233 5.1 Fields and Vector Spaces 233 5.2 Linear Combinations, Generators, and Bases 236 5.3 Components 238 5.4 Linear Transformations 241 5.5 Matrix Representation of Transformations 242 5.6 Algebra of Transformations 244 5.7 Change of Basis 246 5.8 Invariants under Similarity Transformations 247 5.9 Eigenvalues and Eigenvectors 248 5.10 Moment of Inertia Tensor 257 5.11 Inner Product Spaces 262 5.12 The Inner Product 262 5.13 Orthogonality and Completeness 265 5.14 Gram–Schmidt Orthogonalization 267 5.15 Eigenvalue Problem for Real Symmetric Matrices 268 5.16 Presence of Degenerate Eigenvalues 270 5.17 Quadratic Forms 276 5.18 Hermitian Matrices 279 5.19 Matrix Representation of Hermitian Operators 283 5.20 Functions of Matrices 284 5.21 Function Space and Hilbert Space 286 5.22 Dirac’s Bra and Ket Vectors 287 References 288 Problems 289 6 Practical Linear Algebra 293 6.1 Systems of Linear Equations 294 6.1.1 Matrices and Elementary Row Operations 295 6.1.2 Gauss-Jordan Method 295 6.1.3 Information From the Row-Echelon Form 300 6.1.4 Elementary Matrices 301 6.1.5 Inverse by Gauss-Jordan Row-Reduction 302 6.1.6 Row Space, Column Space, and Null Space 303 6.1.7 Bases for Row, Column, and Null Spaces 307 6.1.8 Vector Spaces Spanned by a Set of Vectors 310 6.1.9 Rank and Nullity 312 6.1.10 Linear Transformations 315 6.2 Numerical Methods of Linear Algebra 317 6.2.1 Gauss-Jordan Row-Reduction and Partial Pivoting 317 6.2.2 LU-Factorization 321 6.2.3 Solutions of Linear Systems by Iteration 325 6.2.4 Interpolation 328 6.2.5 Power Method for Eigenvalues 331 6.2.6 Solution of Equations 333 6.2.7 Numerical Integration 343 References 349 Problems 350 7 Applications of Linear Algebra 355 7.1 Chemistry and Chemical Engineering 355 7.1.1 Independent Reactions and Stoichiometric Matrix 356 7.1.2 Independent Reactions from a Set of Species 359 7.2 Linear Programming 362 7.2.1 The Geometric Method 363 7.2.2 The Simplex Method 367 7.3 Leontief Input–Output Model of Economy 375 7.3.1 Leontief Closed Model 375 7.3.2 Leontief Open Model 378 7.4 Applications to Geometry 381 7.4.1 Orbit Calculations 382 7.5 Elimination Theory 383 7.5.1 Quadratic Equations and the Resultant 384 7.6 Coding Theory 388 7.6.1 Fields and Vector Spaces 388 7.6.2 Hamming (7,4) Code 390 7.6.3 Hamming Algorithm for Error Correction 393 7.7 Cryptography 396 7.7.1 Single-Key Cryptography 396 7.8 Graph Theory 399 7.8.1 Basic Definition 399 7.8.2 Terminology 400 7.8.3 Walks, Trails, Paths and Circuits 402 7.8.4 Trees and Fundamental Circuits 404 7.8.5 Graph Operations 404 7.8.6 Cut Sets and Fundamental Cut Sets 405 7.8.7 Vector Space Associated with a Graph 407 7.8.8 Rank and Nullity 409 7.8.9 Subspaces in WG 410 7.8.10 Dot Product and Orthogonal vectors 411 7.8.11 Matrix Representation of Graphs 413 7.8.12 Dominance Directed Graphs 417 7.8.13 Gray Codes in Coding Theory 419 References 419 Problems 420 8 Sequences and Series 425 8.1 Sequences 426 8.2 Infinite Series 430 8.3 Absolute and Conditional Convergence 431 8.3.1 Comparison Test 431 8.3.2 Limit Comparison Test 431 8.3.3 Integral Test 431 8.3.4 Ratio Test 432 8.3.5 Root Test 432 8.4 Operations with Series 436 8.5 Sequences and Series of Functions 438 8.6 M-Test for Uniform Convergence 441 8.7 Properties of Uniformly Convergent Series 441 8.8 Power Series 443 8.9 Taylor Series and Maclaurin Series 446 8.10 Indeterminate Forms and Series 447 References 448 Problems 448 9 Complex Numbers and Functions 453 9.1 The Algebra of Complex Numbers 454 9.2 Roots of a Complex Number 458 9.3 Infinity and the Extended Complex Plane 460 9.4 Complex Functions 463 9.5 Limits and Continuity 465 9.6 Differentiation in the Complex Plane 467 9.7 Analytic Functions 470 9.8 Harmonic Functions 471 9.9 Basic Differentiation Formulas 474 9.10 Elementary Functions 475 9.10.1 Polynomials 475 9.10.2 Exponential Function 476 9.10.3 Trigonometric Functions 477 9.10.4 Hyperbolic Functions 478 9.10.5 Logarithmic Function 479 9.10.6 Powers of Complex Numbers 481 9.10.7 Inverse Trigonometric Functions 483 References 483 Problems 484 10 Complex Analysis 491 10.1 Contour Integrals 492 10.2 Types of Contours 494 10.3 The Cauchy–Goursat Theorem 497 10.4 Indefinite Integrals 500 10.5 Simply and Multiply Connected Domains 502 10.6 The Cauchy Integral Formula 503 10.7 Derivatives of Analytic Functions 505 10.8 Complex Power Series 506 10.8.1 Taylor Series with the Remainder 506 10.8.2 Laurent Series with the Remainder 510 10.9 Convergence of Power Series 514 10.10 Classification of Singular Points 514 10.11 Residue Theorem 517 References 522 Problems 522 11 Ordinary Differential Equations 527 11.1 Basic Definitions for Ordinary Differential Equations 528 11.2 First-Order Differential Equations 530 11.2.1 Uniqueness of Solution 530 11.2.2 Methods of Solution 532 11.2.3 Dependent Variable is Missing 532 11.2.4 Independent Variable is Missing 532 11.2.5 The Case of Separable f(x, y) 532 11.2.6 Homogeneous f(x, y) of Zeroth Degree 533 11.2.7 Solution When f(x, y) is a Rational Function 533 11.2.8 Linear Equations of First-order 535 11.2.9 Exact Equations 537 11.2.10 Integrating Factors 539 11.2.11 Bernoulli Equation 542 11.2.12 Riccati Equation 543 11.2.13 Equations that Cannot Be Solved for y' 546 11.3 Second-Order Differential Equations 548 11.3.1 The General Case 549 11.3.2 Linear Homogeneous Equations with Constant Coefficients 551 11.3.3 Operator Approach 556 11.3.4 Linear Homogeneous Equations with Variable Coefficients 557 11.3.5 Cauchy–Euler Equation 560 11.3.6 Exact Equations and Integrating Factors 561 11.3.7 Linear Nonhomogeneous Equations 564 11.3.8 Variation of Parameters 564 11.3.9 Method of Undetermined Coefficients 566 11.4 Linear Differential Equations of Higher Order 569 11.4.1 With Constant Coefficients 569 11.4.2 With Variable Coefficients 570 11.4.3 Nonhomogeneous Equations 570 11.5 Initial Value Problem and Uniqueness of the Solution 571 11.6 Series Solutions: Frobenius Method 571 11.6.1 Frobenius Method and First-order Equations 581 References 582 Problems 582 12 Second-Order Differential Equations and Special Functions 589 12.1 Legendre Equation 590 12.1.1 Series Solution 590 12.1.2 Effect of Boundary Conditions 593 12.1.3 Legendre Polynomials 594 12.1.4 Rodriguez Formula 596 12.1.5 Generating Function 597 12.1.6 Special Values 599 12.1.7 Recursion Relations 600 12.1.8 Orthogonality 601 12.1.9 Legendre Series 603 12.2 Hermite Equation 606 12.2.1 Series Solution 606 12.2.2 Hermite Polynomials 610 12.2.3 Contour Integral Representation 611 12.2.4 Rodriguez Formula 612 12.2.5 Generating Function 613 12.2.6 Special Values 614 12.2.7 Recursion Relations 614 12.2.8 Orthogonality 616 12.2.9 Series Expansions in Hermite Polynomials 618 12.3 Laguerre Equation 619 12.3.1 Series Solution 620 12.3.2 Laguerre Polynomials 621 12.3.3 Contour Integral Representation 622 12.3.4 Rodriguez Formula 623 12.3.5 Generating Function 623 12.3.6 Special Values and Recursion Relations 624 12.3.7 Orthogonality 624 12.3.8 Series Expansions in Laguerre Polynomials 625 References 626 Problems 626 13 Bessel’s Equation and Bessel Functions 629 13.1 Bessel’s Equation and Its Series Solution 630 13.1.1 Bessel Functions J±m(x), Nm(x), and H(1,2)m (x) 634 13.1.2 Recursion Relations 639 13.1.3 Generating Function 639 13.1.4 Integral Definitions 641 13.1.5 Linear Independence of Bessel Functions 642 13.1.6 Modified Bessel Functions Im(x) and Km(x) 644 13.1.7 Spherical Bessel Functions jl(x), nl(x), and h(1,2)l (x) 645 13.2 Orthogonality and the Roots of Bessel Functions 648 13.2.1 Expansion Theorem 652 13.2.2 Boundary Conditions for the Bessel Functions 652 References 656 Problems 656 14 Partial Differential Equations and Separation of Variables 661 14.1 Separation of Variables in Cartesian Coordinates 662 14.1.1 Wave Equation 665 14.1.2 Laplace Equation 666 14.1.3 Diffusion and Heat Flow Equations 671 14.2 Separation of Variables in Spherical Coordinates 673 14.2.1 Laplace Equation 677 14.2.2 Boundary Conditions for a Spherical Boundary 678 14.2.3 Helmholtz Equation 682 14.2.4 Wave Equation 683 14.2.5 Diffusion and Heat Flow Equations 684 14.2.6 Time-Independent Schrödinger Equation 685 14.2.7 Time-Dependent Schrödinger Equation 685 14.3 Separation of Variables in Cylindrical Coordinates 686 14.3.1 Laplace Equation 688 14.3.2 Helmholtz Equation 689 14.3.3 Wave Equation 690 14.3.4 Diffusion and Heat Flow Equations 691 References 701 Problems 701 15 Fourier Series 705 15.1 Orthogonal Systems of Functions 705 15.2 Fourier Series 711 15.3 Exponential Form of the Fourier Series 712 15.4 Convergence of Fourier Series 713 15.5 Sufficient Conditions for Convergence 715 15.6 The Fundamental Theorem 716 15.7 Uniqueness of Fourier Series 717 15.8 Examples of Fourier Series 717 15.8.1 Square Wave 717 15.8.2 Triangular Wave 719 15.8.3 Periodic Extension 720 15.9 Fourier Sine and Cosine Series 721 15.10 Change of Interval 722 15.11 Integration and Differentiation of Fourier Series 723 References 724 Problems 724 16 Fourier and Laplace Transforms 727 16.1 Types of Signals 727 16.2 Spectral Analysis and Fourier Transforms 730 16.3 Correlation with Cosines and Sines 731 16.4 Correlation Functions and Fourier Transforms 735 16.5 Inverse Fourier Transform 736 16.6 Frequency Spectrums 736 16.7 Dirac-Delta Function 738 16.8 A Case with Two Cosines 739 16.9 General Fourier Transforms and Their Properties 740 16.10 Basic Definition of Laplace Transform 743 16.11 Differential Equations and Laplace Transforms 746 16.12 Transfer Functions and Signal Processors 748 16.13 Connection of Signal Processors 750 References 753 Problems 753 17 Calculus of Variations 757 17.1 A Simple Case 758 17.2 Variational Analysis 759 17.2.1 Case I: The Desired Function is Prescribed at the End Points 761 17.2.2 Case II: Natural Boundary Conditions 762 17.3 Alternate Form of Euler Equation 763 17.4 Variational Notation 765 17.5 A More General Case 767 17.6 Hamilton’s Principle 772 17.7 Lagrange’s Equations of Motion 773 17.8 Definition of Lagrangian 777 17.9 Presence of Constraints in Dynamical Systems 779 17.10 Conservation Laws 783 References 784 Problems 784 18 Probability Theory and Distributions 789 18.1 Introduction to Probability Theory 790 18.1.1 Fundamental Concepts 790 18.1.2 Basic Axioms of Probability 791 18.1.3 Basic Theorems of Probability 791 18.1.4 Statistical Definition of Probability 794 18.1.5 Conditional Probability and Multiplication Theorem 795 18.1.6 Bayes’ Theorem 796 18.1.7 Geometric Probability and Buffon’s Needle Problem 798 18.2 Permutations and Combinations 800 18.2.1 The Case of Distinguishable Balls with Replacement 800 18.2.2 The Case of Distinguishable Balls without Replacement 801 18.2.3 The Case of Indistinguishable Balls 802 18.2.4 Binomial and Multinomial Coefficients 803 18.3 Applications to Statistical Mechanics 804 18.3.1 Boltzmann Distribution for Solids 805 18.3.2 Boltzmann Distribution for Gases 807 18.3.3 Bose–Einstein Distribution for Perfect Gases 808 18.3.4 Fermi–Dirac Distribution 810 18.4 Statistical Mechanics and Thermodynamics 811 18.4.1 Probability and Entropy 811 18.4.2 Derivation of β 812 18.5 Random Variables and Distributions 814 18.6 Distribution Functions and Probability 817 18.7 Examples of Continuous Distributions 819 18.7.1 Uniform Distribution 819 18.7.2 Gaussian or Normal Distribution 820 18.7.3 Gamma Distribution 821 18.8 Discrete Probability Distributions 821 18.8.1 Uniform Distribution 822 18.8.2 Binomial Distribution 822 18.8.3 Poisson Distribution 824 18.9 Fundamental Theorem of Averages 825 18.10 Moments of Distribution Functions 826 18.10.1 Moments of the Gaussian Distribution 827 18.10.2 Moments of the Binomial Distribution 827 18.10.3 Moments of the Poisson Distribution 829 18.11 Chebyshev’s Theorem 831 18.12 Law of Large Numbers 832 References 833 Problems 834 19 Information Theory 841 19.1 Elements of Information Processing Mechanisms 844 19.2 Classical Information Theory 846 19.2.1 Prior Uncertainty and Entropy of Information 848 19.2.2 Joint and Conditional Entropies of Information 851 19.2.3 Decision Theory 854 19.2.4 Decision Theory and Game Theory 856 19.2.5 Traveler’s Dilemma and Nash Equilibrium 862 19.2.6 Classical Bit or Cbit 866 19.2.7 Operations on Cbits 869 19.3 Quantum Information Theory 871 19.3.1 Basic Quantum Theory 872 19.3.2 Single-Particle Systems and Quantum Information 878 19.3.3 Mach–Zehnder Interferometer 880 19.3.4 Mathematics of the Mach–Zehnder Interferometer 882 19.3.5 Quantum Bit or Qbit 886 19.3.6 The No-Cloning Theorem 889 19.3.7 Entanglement and Bell States 890 19.3.8 Quantum Dense Coding 895 19.3.9 Quantum Teleportation 896 References 900 Problems 901 Further Reading 907 Index 915

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John Wiley & Sons Inc Case Studies in Bayesian Statistical Modelling

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Book SynopsisProvides an accessible foundation to Bayesian analysis using real world models This book aims to present an introduction to Bayesian modelling and computation, by considering real case studies drawn from diverse fields spanning ecology, health, genetics and finance. Each chapter comprises a description of the problem, the corresponding model, the computational method, results and inferences as well as the issues that arise in the implementation of these approaches. Case Studies in Bayesian Statistical Modelling and Analysis: Illustrates how to do Bayesian analysis in a clear and concise manner using real-world problems. Each chapter focuses on a real-world problem and describes the way in which the problem may be analysed using Bayesian methods. Features approaches that can be used in a wide area of application, such as, health, the environment, genetics, information science, medicine, biology, industry and remote sensing.Trade Review“As such, this book can serve as a handy reference for proficient statisticians and programmers.” (The Quarterly Review of Biology, 1 October 2015) Table of ContentsPreface xvii List of contributors xix 1 Introduction 1Clair L. Alston, Margaret Donald, Kerrie L. Mengersen and Anthony N. Pettitt 1.1 Introduction 1 1.2 Overview 1 1.3 Further reading 8 1.3.1 Bayesian theory and methodology 8 1.3.2 Bayesian methodology 10 1.3.3 Bayesian computation 10 1.3.4 Bayesian software 11 1.3.5 Applications 13 References 13 2 Introduction to MCMC 17Anthony N. Pettitt and Candice M. Hincksman 2.1 Introduction 17 2.2 Gibbs sampling 18 2.2.1 Example: Bivariate normal 18 2.2.2 Example: Change-point model 19 2.3 Metropolis–Hastings algorithms 19 2.3.1 Example: Component-wise MH or MH within Gibbs 20 2.3.2 Extensions to basic MCMC 21 2.3.3 Adaptive MCMC 22 2.3.4 Doubly intractable problems 22 2.4 Approximate Bayesian computation 24 2.5 Reversible jump MCMC 25 2.6 MCMC for some further applications 26 References 27 3 Priors: Silent or active partners of Bayesian inference? 30Samantha Low Choy 3.1 Priors in the very beginning 30 3.1.1 Priors as a basis for learning 32 3.1.2 Priors and philosophy 32 3.1.3 Prior chronology 33 3.1.4 Pooling prior information 34 3.2 Methodology I: Priors defined by mathematical criteria 35 3.2.1 Conjugate priors 35 3.2.2 Impropriety and hierarchical priors 37 3.2.3 Zellner’s g-prior for regression models 37 3.2.4 Objective priors 38 3.3 Methodology II: Modelling informative priors 40 3.3.1 Informative modelling approaches 40 3.3.2 Elicitation of distributions 42 3.4 Case studies 44 3.4.1 Normal likelihood: Time to submit research dissertations 44 3.4.2 Binomial likelihood: Surveillance for exotic plant pests 47 3.4.3 Mixture model likelihood: Bioregionalization 50 3.4.4 Logistic regression likelihood: Mapping species distribution via habitat models 53 3.5 Discussion 57 3.5.1 Limitations 57 3.5.2 Finding out about the problem 58 3.5.3 Prior formulation 59 3.5.4 Communication 60 3.5.5 Conclusion 61 Acknowledgements 61 References 61 4 Bayesian analysis of the normal linear regression model 66Christopher M. Strickland and Clair L. Alston 4.1 Introduction 66 4.2 Case studies 67 4.2.1 Case study 1: Boston housing data set 67 4.2.2 Case study 2: Production of cars and station wagons 67 4.3 Matrix notation and the likelihood 67 4.4 Posterior inference 68 4.4.1 Natural conjugate prior 69 4.4.2 Alternative prior specifications 73 4.4.3 Generalizations of the normal linear model 74 4.4.4 Variable selection 78 4.5 Analysis 81 4.5.1 Case study 1: Boston housing data set 81 4.5.2 Case study 2: Car production data set 85 References 88 5 Adapting ICU mortality models for local data: A Bayesian approach 90Petra L. Graham, Kerrie L. Mengersen and David A. Cook 5.1 Introduction 90 5.2 Case study: Updating a known risk-adjustment model for local use 91 5.3 Models and methods 92 5.4 Data analysis and results 96 5.4.1 Updating using the training data 96 5.4.2 Updating the model yearly 98 5.5 Discussion 100 References 101 6 A Bayesian regression model with variable selection for genome-wide association studies 103Carla Chen, Kerrie L. Mengersen, Katja Ickstadt and Jonathan M. Keith 6.1 Introduction 103 6.2 Case study: Case–control of Type 1 diabetes 104 6.3 Case study: GENICA 105 6.4 Models and methods 105 6.4.1 Main effect models 105 6.4.2 Main effects and interactions 108 6.5 Data analysis and results 109 6.5.1 WTCCC TID 109 6.5.2 GENICA 110 6.6 Discussion 112 Acknowledgements 115 References 115 6.A Appendix: SNP IDs 117 7 Bayesian meta-analysis 118Jegar O. Pitchforth and Kerrie L. Mengersen 7.1 Introduction 118 7.2 Case study 1: Association between red meat consumption and breast cancer 119 7.2.1 Background 119 7.2.2 Meta-analysis models 121 7.2.3 Computation 125 7.2.4 Results 125 7.2.5 Discussion 129 7.3 Case study 2: Trends in fish growth rate and size 130 7.3.1 Background 130 7.3.2 Meta-analysis models 131 7.3.3 Computation 134 7.3.4 Results 134 7.3.5 Discussion 135 Acknowledgements 137 References 138 8 Bayesian mixed effects models 141Clair L. Alston, Christopher M. Strickland, Kerrie L. Mengersen and Graham E. Gardner 8.1 Introduction 141 8.2 Case studies 142 8.2.1 Case study 1: Hot carcase weight of sheep carcases 142 8.2.2 Case study 2: Growth of primary school girls 142 8.3 Models and methods 146 8.3.1 Model for Case study 1 147 8.3.2 Model for Case study 2 148 8.3.3 MCMC estimation 149 8.4 Data analysis and results 150 8.5 Discussion 158 References 158 9 Ordering of hierarchies in hierarchical models: Bone mineral density estimation 159Cathal D. Walsh and Kerrie L. Mengersen 9.1 Introduction 159 9.2 Case study 160 9.2.1 Measurement of bone mineral density 160 9.3 Models 161 9.3.1 Hierarchical model 162 9.3.2 Model H1 163 9.3.3 Model H2 163 9.4 Data analysis and results 164 9.4.1 Model H1 164 9.4.2 Model H2 165 9.4.3 Implication of ordering 166 9.4.4 Simulation study 166 9.4.5 Study design 166 9.4.6 Simulation study results 167 9.5 Discussion 168 References 168 9.A Appendix: Likelihoods 170 10 Bayesian Weibull survival model for gene expression data 171Sri Astuti Thamrin, James M. McGree and Kerrie L. Mengersen 10.1 Introduction 171 10.2 Survival analysis 172 10.3 Bayesian inference for the Weibull survival model 174 10.3.1 Weibull model without covariates 174 10.3.2 Weibull model with covariates 175 10.3.3 Model evaluation and comparison 176 10.4 Case study 178 10.4.1 Weibull model without covariates 178 10.4.2 Weibull survival model with covariates 180 10.4.3 Model evaluation and comparison 182 10.5 Discussion 182 References 183 11 Bayesian change point detection in monitoring clinical outcomes 186Hassan Assareh, Ian Smith and Kerrie L. Mengersen 11.1 Introduction 186 11.2 Case study: Monitoring intensive care unit outcomes 187 11.3 Risk-adjusted control charts 187 11.4 Change point model 188 11.5 Evaluation 189 11.6 Performance analysis 190 11.7 Comparison of Bayesian estimator with other methods 194 11.8 Conclusion 194 References 195 12 Bayesian splines 197Samuel Clifford and Samantha Low Choy 12.1 Introduction 197 12.2 Models and methods 197 12.2.1 Splines and linear models 197 12.2.2 Link functions 198 12.2.3 Bayesian splines 198 12.2.4 Markov chain Monte Carlo 204 12.2.5 Model choice 206 12.2.6 Posterior diagnostics 207 12.3 Case studies 207 12.3.1 Data 207 12.3.2 Analysis 208 12.4 Conclusion 216 12.4.1 Discussion 216 12.4.2 Extensions 217 12.4.3 Summary 218 References 218 13 Disease mapping using Bayesian hierarchical models 221Arul Earnest, Susanna M. Cramb and Nicole M. White 13.1 Introduction 221 13.2 Case studies 224 13.2.1 Case study 1: Spatio-temporal model examining the incidence of birth defects 224 13.2.2 Case study 2: Relative survival model examining survival from breast cancer 225 13.3 Models and methods 225 13.3.1 Case study 1 225 13.3.2 Case study 2 229 13.4 Data analysis and results 230 13.4.1 Case study 1 230 13.4.2 Case study 2 231 13.5 Discussion 234 References 237 14 Moisture, crops and salination: An analysis of a three-dimensional agricultural data set 240Margaret Donald, Clair L. Alston, Rick Young and Kerrie L. Mengersen 14.1 Introduction 240 14.2 Case study 241 14.2.1 Data 242 14.2.2 Aim of the analysis 242 14.3 Review 243 14.3.1 General methodology 243 14.3.2 Computations 243 14.4 Case study modelling 243 14.4.1 Modelling framework 243 14.5 Model implementation: Coding considerations 246 14.5.1 Neighbourhood matrices and CAR models 246 14.5.2 Design matrices vs indexing 246 14.6 Case study results 247 14.7 Conclusions 249 References 250 15 A Bayesian approach to multivariate state space modelling: A study of a Fama–French asset-pricing model with time-varying regressors 252Christopher M. Strickland and Philip Gharghori 15.1 Introduction 252 15.2 Case study: Asset pricing in financial markets 253 15.2.1 Data 254 15.3 Time-varying Fama–French model 254 15.3.1 Specific models under consideration 255 15.4 Bayesian estimation 256 15.4.1 Gibbs sampler 256 15.4.2 Sampling Σε 257 15.4.3 Sampling β 1:n 257 15.4.4 Sampling Σ α 259 15.4.5 Likelihood calculation 260 15.5 Analysis 261 15.5.1 Prior elicitation 261 15.5.2 Estimation output 261 15.6 Conclusion 264 References 265 16 Bayesian mixture models: When the thing you need to know is the thing you cannot measure 267Clair L. Alston, Kerrie L. Mengersen and Graham E. Gardner 16.1 Introduction 267 16.2 Case study: CT scan images of sheep 268 16.3 Models and methods 270 16.3.1 Bayesian mixture models 270 16.3.2 Parameter estimation using the Gibbs sampler 273 16.3.3 Extending the model to incorporate spatial information 274 16.4 Data analysis and results 276 16.4.1 Normal Bayesian mixture model 276 16.4.2 Spatial mixture model 278 16.4.3 Carcase volume calculation 281 16.5 Discussion 284 References 284 17 Latent class models in medicine 287Margaret Rolfe, Nicole M. White and Carla Chen 17.1 Introduction 287 17.2 Case studies 288 17.2.1 Case study 1: Parkinson’s disease 288 17.2.2 Case study 2: Cognition in breast cancer 288 17.3 Models and methods 289 17.3.1 Finite mixture models 290 17.3.2 Trajectory mixture models 292 17.3.3 Goodness of fit 296 17.3.4 Label switching 297 17.3.5 Model computation 298 17.4 Data analysis and results 300 17.4.1 Case study 1: Phenotype identification in PD 300 17.4.2 Case study 2: Trajectory groups for verbal memory 302 17.5 Discussion 306 References 307 18 Hidden Markov models for complex stochastic processes: A case study in electrophysiology 310Nicole M. White, Helen Johnson, Peter Silburn, Judith Rousseau and Kerrie L. Mengersen 18.1 Introduction 310 18.2 Case study: Spike identification and sorting of extracellular recordings 311 18.3 Models and methods 312 18.3.1 What is an HMM? 312 18.3.2 Modelling a single AP: Application of a simple HMM 313 18.3.3 Multiple neurons: An application of a factorial HMM 315 18.3.4 Model estimation and inference 317 18.4 Data analysis and results 320 18.4.1 Simulation study 320 18.4.2 Case study: Extracellular recordings collected during deep brain stimulation 323 18.5 Discussion 326 References 327 19 Bayesian classification and regression trees 330Rebecca A. O’Leary, Samantha Low Choy, Wenbiao Hu and Kerrie L. Mengersen 19.1 Introduction 330 19.2 Case studies 332 19.2.1 Case study 1: Kyphosis 332 19.2.2 Case study 2: Cryptosporidium 332 19.3 Models and methods 334 19.3.1 CARTs 334 19.3.2 Bayesian CARTs 335 19.4 Computation 337 19.4.1 Building the BCART model – stochastic search 337 19.4.2 Model diagnostics and identifying good trees 339 19.5 Case studies – results 341 19.5.1 Case study 1: Kyphosis 341 19.5.2 Case study 2: Cryptosporidium 343 19.6 Discussion 345 References 346 20 Tangled webs: Using Bayesian networks in the fight against infection 348Mary Waterhouse and Sandra Johnson 20.1 Introduction to Bayesian network modelling 348 20.1.1 Building a BN 349 20.2 Introduction to case study 351 20.3 Model 352 20.4 Methods 354 20.5 Results 355 20.6 Discussion 357 References 359 21 Implementing adaptive dose finding studies using sequential Monte Carlo 361James M. McGree, Christopher C. Drovandi and Anthony N. Pettitt 21.1 Introduction 361 21.2 Model and priors 363 21.3 SMC for dose finding studies 364 21.3.1 Importance sampling 364 21.3.2 SMC 365 21.3.3 Dose selection procedure 367 21.4 Example 369 21.5 Discussion 371 References 372 21.A Appendix: Extra example 373 22 Likelihood-free inference for transmission rates of nosocomial pathogens 374Christopher C. Drovandi and Anthony N. Pettitt 22.1 Introduction 374 22.2 Case study: Estimating transmission rates of nosocomial pathogens 375 22.2.1 Background 375 22.2.2 Data 376 22.2.3 Objective 376 22.3 Models and methods 376 22.3.1 Models 376 22.3.2 Computing the likelihood 379 22.3.3 Model simulation 380 22.3.4 ABC 381 22.3.5 ABC algorithms 382 22.4 Data analysis and results 384 22.5 Discussion 385 References 386 23 Variational Bayesian inference for mixture models 388Clare A. McGrory 23.1 Introduction 388 23.2 Case study: Computed tomography (CT) scanning of a loin portion of a pork carcase 390 23.3 Models and methods 392 23.4 Data analysis and results 397 23.5 Discussion 399 References 399 23.A Appendix: Form of the variational posterior for a mixture of multivariate normal densities 401 24 Issues in designing hybrid algorithms 403Jeong E. Lee, Kerrie L. Mengersen and Christian P. Robert 24.1 Introduction 403 24.2 Algorithms and hybrid approaches 406 24.2.1 Particle system in the MCMC context 407 24.2.2 MALA 407 24.2.3 DRA 408 24.2.4 PS 409 24.2.5 Population Monte Carlo (PMC) algorithm 410 24.3 Illustration of hybrid algorithms 412 24.3.1 Simulated data set 412 24.3.2 Application: Aerosol particle size 415 24.4 Discussion 417 References 418 25 A Python package for Bayesian estimation using Markov chain Monte Carlo 421Christopher M. Strickland, Robert J. Denham, Clair L. Alston and Kerrie L. Mengersen 25.1 Introduction 421 25.2 Bayesian analysis 423 25.2.1 MCMC methods and implementation 424 25.2.2 Normal linear Bayesian regression model 433 25.3 Empirical illustrations 437 25.3.1 Example 1: Linear regression model – variable selection and estimation 438 25.3.2 Example 2: Loglinear model 441 25.3.3 Example 3: First-order autoregressive regression 446 25.4 Using PyMCMC efficiently 451 25.4.1 Compiling code in Windows 455 25.5 PyMCMC interacting with R 457 25.6 Conclusions 458 25.7 Obtaining PyMCMC 459 References 459 Index 461

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Book SynopsisLinear systems have all the necessary elements (modeling, identification, analysis and control), from an educational point of view, to help us understand the discipline of automation and apply it efficiently. This book is progressive and organized in such a way that different levels of readership are possible. It is addressed both to beginners and those with a good understanding of automation wishing to enhance their knowledge on the subject. The theory is rigorously developed and illustrated by numerous examples which can be reproduced with the help of appropriate computation software. 60 exercises and their solutions are included to enable the readers to test and enhance their knowledge.Table of ContentsPreface xi Introduction xvii Chapter 1. Reminders and General Points 1 1.1. Lines 1 1.2. Adaptation and stationary waves 4 1.3. Smith chart 6 1.4. Power in a line 6 1.5. Line sections 7 1.6. Lines with losses 9 Chapter 2. Measurements in HF 11 2.1. Material 11 2.2. The power bench11 2.3. Measurements on the network analyzer 14 Chapter 3. Resonant Cavities 25 3.1. Resonance 25 3.2. Coaxial cavities 28 3.3. Quarter-wave cavities 29 Chapter 4. Fabrication and Tuning of Cavities 47 4.1. Standard structures 47 4.2. Materials 53 4.3. Assembly 56 4.4. Temperature stability 58 4.5. Cavity tuning 70 Chapter 5. The Band-pass Filter 83 5.1. The band-pass function 83 5.2. Calculation of a Tchebycheff band-pass 85 5.3. Technologies 87 Chapter 6. The Combline Filter 97 6.1. Architecture 97 6.2. Dimension calculations. Dishal’s Method 105 6.3. Tuning of filters 114 Chapter 7. Channel Multiplexing 129 7.1. Definitions 129 7.2. The duplexer 129 7.3. The combiner 140 Chapter 8. Auxiliary Devices 157 8.1. Introduction 157 8.2. Circulators 157 8.3. The antenna alarms 165 8.4. Loads and attenuators 170 8.5. Reception amplifiers 177 8.6. The impedance adaptor 186 8.7. The 2nd harmonic rejecter 186 Chapter 9. Directive Couplers 189 9.1. Introduction 189 9.2. Technologies 191 9.3. The hybrid transformer 194 9.4. The 180° hybrid ring 196 9.5. The wireline 198 9.6. The “groundless” coupler 199 9.7. The “catnose” coupler 201 9.8. Discrete-elements coupler 202 9.9. Numerical data 204 9.10. Applications 210 Chapter 10. Helical Resonators 213 10.1. Introduction 213 10.2. Functioning 214 10.3. Structures 215 10.4. Tapping and coupling 219 10.5. Quality coefficient 220 10.6. Set-up rules 223 10.7. Applications 224 Chapter 11. Multicouplers 225 11.1. Transmitter multicouplers (TX) 225 11.2. Receiver multicouplers (RX) 235 11.3. TX/RX multicouplers 236 11.4. TMA 240 11.5. Power and intermodulations 243 11.6. Multiband coupling 254 Chapter 12. Utilities 257 12.1. BASIC programs 257 12.2. Varia 264 Chapter 13. Various Questions and Exploratory Ways 271 13.1. The coupler without intrinsic loss 271 13.2. Infinite rejection band-pass 275 13.3. Helix TX multicoupler 276 13.4. Conclusion 278 Bibliography 281 Index 283

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Modeling, Dynamics, Optimization and Bioeconomics IV: DGS VI JOLATE, Madrid, Spain, May 2018, and ICABR, Berkeley, USA, May–June 2017—Selected Contributions

Springer Nature Switzerland AG Modeling, Dynamics, Optimization and Bioeconomics IV: DGS VI JOLATE, Madrid, Spain, May 2018, and ICABR, Berkeley, USA, May–June 2017—Selected Contributions

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Book SynopsisThis book, following the three published volumes of the book, provides the main purpose to collect research papers and review papers to provide an overview of the main issues, results, and open questions in the cutting-edge research on the fields of modeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of this book is to collect papers of the world experts in mathematics, economics, and other applied sciences that is seminal to the future research developments. The majority of the papers presented in this book is authored by the participants in The Joint Meeting 6th International Conference on Dynamics, Games, and Science – DGSVI – JOLATE and in the 21st ICABR Conference. The scientific scope of the conferences is focused on the fields of modeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology, and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of the conference is to bring together some of the world experts in mathematics, economics, and other applied sciences that reinforce ongoing projects and establish future works and collaborations.Table of ContentsA. Afsar, F. Martins, Bruno M. P. M. Oliveira, and A. A. Pinto, Immune response model fitting to CD4+ T cell data in lymphocytic choriomeningitis virus LCMV infection.- U. Agyüz, V. Purutçuoglu, E. Purutçuoglu and Y. Ürün, Construction of a New Model to Investigate Breast Cancer Data.- I. Baltas, M. Szczepanski, L. Dopierala, K. Kolodziejczyk, G.-W. Weber and A. N. Yannacopoulos, Optimal Pension Fund Management Under Risk and Uncertainty: The Case Study of Poland.- M. Bujidos-Casado, J. Navío-Marco and B. Rodrigo-Moya, Collaborative Innovation of Spanish SMEs in the European context: A compared study.- G. G. de Castro, A. O. Lopes and G. Mantovani, Haar systems, KMS states on von Neumann algebras and C*-algebras on dynamically defined groupoids and Noncommutative Integration.- C. Çıtak, T. Aksu, Ö. Harputlu and Gerhard-Wilhelm Weber, Mixed Compression Air-Intake Design for High-Speed Transportation.- D. Czerkawski, J. Małecka, G. Wilhelm Weber and B. Kjamili, Social Entrepreneurship Business Models for Handicapped People - Polish & Turkish case study of sharing public goods by doing business.- H. H. Ferreira, A. O. Lopes and E. R. Oliveira, An iterative process for approximating subactions.- A. D. Garcia and M. A. Szybisz, "Beat the gun": The phenomenon of liquidity.- E. Gómez-Escalonilla and Laura Parte, Board Knowledge and Bank Risk-Taking. An International Analysis.- F. Jiménez-Delgado, M. Dolores Reina-Paz, Israel J ThuissardVasallo and David Sanz-Rosa, The shopping experience in virtual sales: A study of the influence of website atmosphere on purchase intention.- Kyung B. Kim and José M. Labeaga, European Mobile Phone Industry: Demand Estimation Using Discrete Random Coefficients Models.- A. O. Lopes and M. Sebastiani, On Bertelson-Gromov Dynamical Morse Entropy, Rogério Martins, Synchronisation of weakly coupled oscillators.- Z. Kamisli Ozturk, Y. Cetin, Y. Isik and Z. I. Erzurum Cicek, Demand Forecasting with Clustering and Artificial Neural Networks Methods: an Application for Stock Keeping Units.- O. Palanci, S.Z. Alparslan Gok and Gerhard-Wilhelm Weber, On the Grey Obligation Rules.- Juan Diego Paredes-Gázquez, Eva Pardo and José Miguel Rodríguez-Fernández, Robustness checks in composite indicators: A responsible approach.- Elena V. Ravve, Zeev Volkovich, Gerhard-Wilhelm Weber, A Logic-Based Approach to Incremental Reasoning on Multi-Agent Systems.

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