Mathematical sciences have been playing an increasingly important role in modern society. They are in high demand for investigating complex problems in physical science, environmental and geophysical sciences, materials science, life science and chemical sciences.
This is a review volume on some timely and interesting topics in applied mathematical sciences. It surveys new developments and presents some future research directions in these topics. The chapters are written by experts in these fields, with a wide audience in mind and hence will be accessible to graduate students, junior researchers and other professionals who are interested in the subjects. The contributions of Professor Youzhong Guo, a leading expert in these areas, will be celebrated. His life and academic achievements are highlighted in the Preface and Postscript of the book. The underlying theme that binds the various chapters seamlessly is a set of dedicated ideas and techniques from partial differential equations and dynamical systems.
Author(s): Yisong Yang, Yisong Yang, Xinchu Fu, Jinqiao Duan
Series: Interdisciplinary Mathematical Sciences
Publisher: WS
Year: 2010
Language: English
Pages: 371
Contents......Page 16
Editorial Foreword......Page 8
Preface......Page 10
Selected Books Authored by Youzhong Guo......Page 14
1.1. Periodic Riemann Boundary Value Problems: Case of Closed Contours......Page 18
1.2. Periodic Riemann Boundary Value Problems: Case of Open Ares or Discontinuous Coefficients......Page 20
2. Periodic Problems for Isotropic Material in Elastic Theory......Page 21
2.1. Stress Functions......Page 22
2.2. Periodic Fundamental Problems of Elastic Half-plane......Page 24
2.3. Periodic Contact Problems......Page 26
3.1. The stress Functions......Page 29
3.2. Periodic Fundamental Problems of Anisotropic Half-plane......Page 30
3.3. Periodic Contact Problem for Anisotropic Medium......Page 32
4. Periodic Crack Problems in Plane Elasticity......Page 33
4.1. Fundamental Problems of Isotropic Plane with Periodic Collinear Cracks......Page 34
4.2. Fundamental Problems of Anisotropic Elastic Plane with Periodic Collinear Cracks......Page 37
5.1. Relaxing Conditions......Page 41
5.2. Generalized Plemelj Formulae and Singular Integral Equations......Page 42
5.3. Generalized Plemelj Formula and Automorphic Functions......Page 45
5.4. Singular Integral Equations and Boundary Value Problems......Page 46
5.5. Boundary Value Problems and Automorphic Functions......Page 47
6. Some Closed Formulae......Page 50
7. Some Remarks......Page 51
References......Page 52
1. Introduction......Page 54
2. Formulation of the problem......Page 55
3. Melnikov analysis......Page 56
4. Numerical simulations......Page 64
References......Page 68
3. Canonical Sample Spaces for Random Dynamical Systems Jinqiao Duan, Xingye Kan and Bjorn Schmalfuss......Page 70
1. Random dynamical systems......Page 71
2.1. Brownian Motion......Page 72
2.2. Wiener Measure......Page 75
2.3. Canonical sample space......Page 78
3.2. Canonical sample space......Page 79
4.1. L evy Motions......Page 80
4.2. Canonical sample space......Page 83
References......Page 84
4. Epidemic Propagation Dynamics on Complex Networks Xinchu Fu, Zengrong Liu, Michael Small and Chi Kong Tse......Page 88
1. Introduction......Page 89
2. The epidemic threshold for SIS model with piecewise linear infectivity......Page 90
2.1. Piecewise linear infectivity......Page 91
3. Model with different immunities and infectivities......Page 92
3.2. Multiple infected individuals......Page 93
3.3. Multiple-staged infected individuals......Page 94
4.1. Epidemic spreading without mobility of individuals......Page 95
4.2. Spreading of epidemic diseases among different cities......Page 96
4.3.2. The epidemic rate is β inside the same cities......Page 97
5. Multi-strain epidemic models......Page 98
5.2.1. Basic Reproduction Numbers (BRNs)......Page 100
5.2.3. Invasion Reproduction Numbers(IRNs)......Page 102
5.3. The case σ1 = σ2......Page 103
5.4.1. The model......Page 104
References......Page 105
1. Introduction......Page 110
2.1. Backward problem......Page 112
2.2. Inverse heat problem......Page 113
3. Abstract backward parabolic problems......Page 114
3.2. Hilbert space case for (3.1)......Page 116
3.2.2. Gajewski and Zachirias quasi-reversibility......Page 117
3.2.4. Modified quasi-reversibility method......Page 118
3.3.1. Lattes-Lions quasi-reversibility method......Page 119
3.3.3. Modified quasi-reversibility method......Page 121
3.4. Structural stability for (3.1)......Page 122
4. The linear inverse problem: recovering a source term......Page 125
References......Page 127
1. Introduction and main results......Page 132
2. Preliminary Results......Page 137
3. The Proof of Theorem 1.1......Page 140
References......Page 150
1. Introduction......Page 152
2. The bifurcations of phase portraits of system (5)......Page 154
3. The Melnikov analysis for the perturbed system (6) and numerical examples......Page 159
References......Page 161
1. Introduction......Page 162
2.1. Chaos control......Page 163
2.2. Chaos synchronization......Page 165
2.3. Complex networks......Page 166
3.1. Chaos control......Page 167
3.2. Chaos synchronization......Page 171
3.3. Complex networks......Page 175
3.4. The analysis of the problems related to emergence......Page 187
4. Some perspectives......Page 188
References......Page 189
1. Introduction......Page 198
1.1. Physical background......Page 199
2. The cold plasma model......Page 200
2.1. Equations of motion......Page 201
2.2. The dielectric tensor......Page 202
2.3. The plasma dispersion relation......Page 204
2.4. Electrostatic waves......Page 206
2.4.1. Plane-layered media......Page 207
2.4.2. A two-dimensional inhomogeneity......Page 208
2.4.3. The type of the governing equation......Page 210
2.4.4. Geometry of the resonance curve ( after Piliya and Fedorov)......Page 211
2.5. Analytic difficulties in the electromagnetic case (after H. Weitzner)......Page 212
2.5.1. Choices of potential and gauge......Page 213
2.5.2. Variational interpretation......Page 216
2.6. A conjecture about the singular set......Page 218
3. Analysis of the model equation......Page 219
3.1. The closed Dirichlet problem for distribution solutions......Page 220
3.2. Mixed boundary value problems with closed boundary data......Page 223
References......Page 226
1. Introduction......Page 228
2. Layout......Page 231
2.1. Gravitational systems in all dimensions: Einstein hierarchy......Page 232
2.2. The Yang–Mills hierarchy......Page 233
2.3. Higgs models on RD......Page 235
2.4. Static spherically symmetric fields......Page 237
3.1.1. Einstein–Yang-Mills solutions in four dimensions......Page 240
3.1.2. Einstein–Yang-Mills solutions in higher dimensions......Page 242
3.2. EYM solutions with Euclidean signature......Page 248
4. Summary and outlook......Page 250
References......Page 252
1. Extended knot families......Page 258
2.1. Family {ψi}22......Page 261
2.2. Family {ψi}43......Page 262
3.2. Contribution of the rank five term C1µ 1···µ5 Rµ1···µ5......Page 263
3.3. Contribution of the rank six term (C2µ 1···µ6 + C3µ 1···µ6 )Rµ1···µ6......Page 265
3.4. Contribution of the rank six term (C1µ 1···µ6 + C4µ 1···µ6 + C5µ 1···µ6 )Rµ1···µ6......Page 266
3.5. Action of H0 over the family {ψi}64......Page 267
4. The Mandelstam identities......Page 268
References......Page 269
1. Introduction......Page 270
2.1. LB model for DE......Page 271
2.2. Version of LB Model for Complex DE......Page 272
3. Simulation Results......Page 273
References......Page 277
Appendix: Derivation of Macroscopic Equation......Page 280
1. Introduction......Page 282
2. Global existence for continuous initial data......Page 284
3. Decay estimate of solution......Page 287
Acknowledgments......Page 290
References......Page 291
1. Introduction......Page 292
2. Linear Stability of the Trivial Solution......Page 296
3. Local Behavior near L1 and L2......Page 297
4. Global Bounds for Solutions of Eq. (1.5)......Page 299
5. Stationary Solutions in the Limit as a 0+......Page 300
6. Stationary Solutions in the Limit as a a*+ 1......Page 302
6.1. Stationary solutions......Page 303
6.2. Stability......Page 304
7. Another Structure in the Bifurcation Diagrams......Page 305
References......Page 307
15. A New GL Method for Mathematical and Physical Problems Ganquan Xie and Jianhua Li......Page 310
1. Introduction......Page 311
2.1. Wave Equation......Page 313
2.3. Integral Equation......Page 314
4.1. Fundamental Theorem......Page 315
5. GL Method for Solving Parabolic Type Partial Differential Equation......Page 316
7.1. Discretization of GL Method......Page 317
7.3. Computational Simulation of GL Method......Page 318
8. Memories and Discussion......Page 320
References......Page 322
1. Introduction......Page 324
2. Harmonic Maps, Hodge Theory, and Instantons......Page 325
3. Quantum Tunneling, Imaginary Time, Instantons, and Liouville- Type Equations......Page 336
4. Atiyah–Singer Index Theorem and Calculation of Dimension of Moduli Space......Page 344
5. Topological Classes and Instantons in All 4m Dimensions and Nonlinear Elliptic Equations......Page 351
References......Page 364
Postscript......Page 370
