Mathematical and Computational Modeling

List of Contributors xiii

Preface xv

Section 1 Introduction 1

1 Universality of Mathematical Models in Understanding Nature Society and Man-Made World 3 Roderick Melnik

1.1 Human Knowledge Models and Algorithms 3

1.2 Looking into the Future from a Modeling Perspective 7

1.3 What This Book Is About 10

1.4 Concluding Remarks 15

References 16

Section 2 Advanced Mathematical and Computational Models in Physics and Chemistry 17

2 Magnetic Vortices Abrikosov Lattices and Automorphic Functions 19 Israel Michael Sigal

2.1 Introduction 19

2.2 The Ginzburg-Landau Equations 20

2.2.1 Ginzburg-Landau energy 21

2.2.2 Symmetries of the equations 21

2.2.3 Quantization of flux 22

2.2.4 Homogeneous solutions 22

2.2.5 Type I and Type II superconductors 23

2.2.6 Self-dual case ¿=1/ v 2 24

2.2.7 Critical magnetic fields 24

2.2.8 Time-dependent equations 25

2.3 Vortices 25

2.3.1 n-vortex solutions 25

2.3.2 Stability 26

2.4 Vortex Lattices 30

2.4.1 Abrikosov lattices 31

2.4.2 Existence of Abrikosov lattices 31

2.4.3 Abrikosov lattices as gauge-equivariant states 34

2.4.4 Abrikosov function 34

2.4.5 Comments on the proofs of existence results 35

2.4.6 Stability of Abrikosov lattices 40

>0 42

2.4.8 Key ideas of approach to stability 45

2.5 Multi-Vortex Dynamics 48

2.6 Conclusions 51

Appendix 2.A Parameterization of the equivalence classes [L] 51

Appendix 2.B Automorphy factors 52

References 54

3 Numerical Challenges in a Cholesky-Decomposed Local Correlation Quantum Chemistry Framework 59 David B. Krisiloff, Johannes M. Dieterich, Florian Libisch and Emily A. Carter

3.1 Introduction 59

3.2 Local MRSDCI 61

3.2.1 Mrsdci 61

3.2.2 Symmetric group graphical approach 62

3.2.3 Local electron correlation approximation 64

3.2.4 Algorithm summary 66

3.3 Numerical Importance of Individual Steps 67

3.4 Cholesky Decomposition 68

3.5 Transformation of the Cholesky Vectors 71

3.6 Two-Electron Integral Reassembly 72

3.7 Integral and Execution Buffer 76

3.8 Symmetric Group Graphical Approach 77

3.9 Summary and Outlook 87

References 87

4 Generalized Variational Theorem in Quantum Mechanics 92 Mel Levy and Antonios Gonis

4.1 Introduction 92

4.2 First Proof 93

4.3 Second Proof 95

4.4 Conclusions 96

References 97

Section 3 Mathematical and Statistical Models in Life And Climate Science Applications 99

5 A Model for the Spread of Tuberculosis with Drug-Sensitive and Emerging Multidrug-Resistant and Extensively Drug-Resistant Strains 101 Julien Arino and Iman A. Soliman

5.1 Introduction 101

5.1.1 Model formulation 102

5.1.2 Mathematical Analysis 107

5.1.2.1 Basic properties of solutions 107

5.1.2.2 Nature of the disease-free equilibrium 108

5.1.2.3 Local asymptotic stability of the DFE 108

5.1.2.4 Existence of subthreshold endemic equilibria 110

5.1.2.5 Global stability of the DFE when the bifurcation is "forward" 113

5.1.2.6 Strain-specific global stability in "forward" bifurcation cases 115

5.2 Discussion 117

References 119

6 The Need for More Integrated Epidemic Modeling with Emphasis on Antibiotic Resistance 121 Eili Y. Klein, Julia Chelen, Michael D. Makowsky and Paul E. Smaldino

6.1 Introduction 121

6.2 Mathematical Modeling of Infectious Diseases 122

6.3 Antibiotic Resistance Behavior and Mathematical Modeling 125

6.3.1 Why an integrated approach? 125

6.3.2 The role of symptomology 127

6.4 Conclusion 128

References 129

Section 4 Mathematical Models and Analysis for Science and Engineering 135

7 Data-Driven Methods for Dynamical Systems: Quantifying Predictability and Extracting Spatiotemporal Patterns 137 Dimitrios Giannakis and Andrew J. Majda

7.1 Quantifying Long-Range Predictability and Model Error through Data Clustering and Information Theory 138

7.1.1 Background 138

7.1.2 Information theory predictability and model error 140

7.1.2.1 Predictability in a perfect-model environment 140

7.1.2.2 Quantifying the error of imperfect models 143

7.1.3 Coarse-graining phase space to reveal long-range predictability 144

7.1.3.1 Perfect-model scenario 144

7.1.3.2 Quantifying the model error in long-range forecasts 147

7.1.4 K-means clustering with persistence 149

7.1.5 Demonstration in a double-gyre ocean model 152

7.1.5.1 Predictability bounds for coarse-grained observables 154

7.1.5.2 The physical properties of the regimes 157

7.1.5.3 Markov models of regime behavior in the 1.5-layer ocean model 159

7.1.5.4 The model error in long-range predictions with coarse-grained Markov models 162

7.2 NLSA Algorithms for Decomposition of Spatiotemporal Data 163

7.2.1 Background 163

7.2.2 Mathematical framework 165

7.2.2.1 Time-lagged embedding 166

7.2.2.2 Overview of singular spectrum analysis 167

7.2.2.3 Spaces of temporal patterns 167

7.2.2.4 Discrete formulation 169

7.2.2.5 Dynamics-adapted kernels 171

7.2.2.6 Singular value decomposition 173

7.2.2.7 Setting the truncation level 174

7.2.2.8 Projection to data space 175

7.2.3 Analysis of infrared brightness temperature satellite data for tropical dynamics 175

7.2.3.1 Dataset description 176

7.2.3.2 Modes recovered by NLSA 176

7.2.3.3 Reconstruction of the TOGA COARE MJOs 183

7.3 Conclusions 184

References 185

8 On Smoothness Concepts in Regularization for Nonlinear Inverse Problems in Banach Spaces 192 Bernd Hofmann

8.1 Introduction 192

8.2 Model Assumptions Existence and Stability 195

8.3 Convergence of Regularized Solutions 197

8.4 A Powerful Tool for Obtaining Convergence Rates 200

8.5 How to Obtain Variational Inequalities? 206

8.5.1 Bregman distance as error measure: the benchmark case 206

8.5.2 Bregman distance as error measure: violating the benchmark 210

8.5.3 Norm distance as error measure: l 1 -regularization 213

8.6 Summary 215

References 215

9 Initial and Initial-Boundary Value Problems for First-Order Symmetric Hyperbolic Systems with Constraints 222 Nicolae Tarfulea

9.1 Introduction 222

9.2 FOSH Initial Value Problems with Constraints 223

9.2.1 FOSH initial value problems 224

9.2.2 Abstract formulation 225

9.2.3 FOSH initial value problems with constraints 228

9.3 FOSH Initial-Boundary Value Problems with Constraints 230

9.3.1 FOSH initial-boundary value problems 232

9.3.2 FOSH initial-boundary value problems with constraints 234

9.4 Applications 236

9.4.1 System of wave equations with constraints 237

9.4.2 Applications to Einstein's equations 240

9.4.2.1 Einstein-Christoffel formulation 243

9.4.2.2 Alekseenko-Arnold formulation 246

References 250

10 Information Integration Organization and Numerical Harmonic Analysis 254 Ronald R. Coifman, Ronen Talmon, Matan Gavish and Ali Haddad

10.1 Introduction 254

10.2 Empirical Intrinsic Geometry 257

10.2.1 Manifold formulation 259

10.2.2 Mahalanobis distance 261

10.3 Organization and Harmonic Analysis of Databases/Matrices 263

10.3.1 Haar bases 264

10.3.2 Coupled partition trees 265

10.4 Summary 269

References 270

Section 5 Mathematical Methods in Social Sciences And Arts 273

11 Satisfaction Approval Voting 275 Steven J. Brams and D. Marc Kilgour

11.1 Introduction 275

11.2 Satisfaction Approval Voting for Individual Candidates 277

11.3 The Game Theory Society Election 285

11.4 Voting for Multiple Candidates under SAV: A Decision-Theoretic Analysis 287

11.5 Voting for Political Parties 291

11.5.1 Bullet voting 291

11.5.2 Formalization 292

11.5.3 Multiple-party voting 294

11.6 Conclusions 295

11.7 Summary 296

References 297

12 Modeling Musical Rhythm Mutations with Geometric Quantization 299 Godfried T. Toussaint

12.1 Introduction 299

12.2 Rhythm Mutations 301

12.2.1 Musicological rhythm mutations 301

12.2.2 Geometric rhythm mutations 302

12.3 Similarity-Based Rhythm Mutations 303

12.3.1 Global rhythm similarity measures 304

12.4 Conclusion 306

References 307

Index 309