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 k = 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 γδ(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
[ ] ................................................ 51
Appendix 2.В 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:
^'-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 Tainion, 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 S AV: A Decision-
Theoretic Analysis ....................................... 287
11.5 Voting for Political Parties ............................. 291
11.5.1 Bullet voting ..................................... 291
11.5.2 Formalization ..................................... 292
41.5.1 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
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