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 κ=1/ √ 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
2.4.7 Functions γ δ (τ),δ >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
