Mathematical modeling of random and deterministic phenomena

✍ Scribed by Manou-Abi, Solym Mawaki;Dabo-Niang, Sophie;Salone, Jean-Jacques

• Publisher: ISTE Ltd., Wiley
• Year: 2020
• Tongue: English
• Leaves: 314
• Category: Library

✦ Synopsis

This book highlights mathematical research interests that appear in real life, such as the study and modeling of random and deterministic phenomena. As such, it provides current research in mathematics, with applications in biological and environmental sciences, ecology, epidemiology and social perspectives.

The chapters can be read independently of each other, with dedicated references specific to each chapter. The book is organized in two main parts. The first is devoted to some advanced mathematical problems regarding epidemic models; predictions of biomass; space-time modeling of extreme rainfall; modeling with the piecewise deterministic Markov process; optimal control problems; evolution equations in a periodic environment; and the analysis of the heat equation. The second is devoted to a modelization with interdisciplinarity in ecological, socio-economic, epistemological, demographic and social problems.

Mathematical Modeling of Random and Deterministic Phenomena is aimed at expert readers, young researchers, plus graduate and advanced undergraduate students who are interested in probability, statistics, modeling and mathematical analysis.

✦ Table of Contents

Cover......Page 1
Half-Title Page......Page 3
Title Page......Page 5
Copyright Page......Page 6
Contents......Page 7
Preface......Page 13
Acknowledgments......Page 15
Introduction......Page 17
PART 1: Advances in Mathematical Modeling......Page 23
1.1. Introduction......Page 25
1.2.1. The SIS model......Page 27
1.2.2. The SIRS model......Page 28
1.2.3. The SIR model with demography......Page 29
1.3.1. The stochastic model......Page 30
1.3.2. Law of large numbers......Page 31
1.3.4. Large deviations and extinction of an epidemic......Page 32
1.4.1. CLT and extinction of an endemic disease......Page 34
1.4.2. Moderate deviations......Page 35
1.5. References......Page 51
2.1. Introduction......Page 53
2.2. Regression model and predictor......Page 56
2.3. Large sample properties......Page 58
2.4.1. Procedure of prediction......Page 61
2.4.2. Demersal coastal fish off Senegal data set......Page 62
2.4.3. Measuring prediction performance......Page 63
2.5. Conclusion......Page 70
2.6. References......Page 71
3.1.1. Rainfall-induced urban floods......Page 75
3.1.2. Sample hydraulic simulation of a rainfall-induced urban flood......Page 76
3.2.1. Spatial stochastic rainfall generator......Page 80
3.2.2. Modeling extreme events......Page 81
3.2.3. Stochastic rainfall generator geared towards extreme events......Page 85
3.3. Outlook......Page 86
3.4. References......Page 88
4.1. A quick introduction to stochastic control and change-point detection......Page 95
4.2. Model and problem setting......Page 98
4.2.1. Continuous-time PDMP model......Page 99
4.2.2. Optimal stopping problem under partial observations......Page 100
4.2.3. Fully observed optimal stopping problem......Page 102
4.3. Numerical approximation of the value functions......Page 104
4.3.1. Quantization......Page 105
4.3.2. Discretizations......Page 106
4.3.3. Construction of a stopping strategy......Page 109
4.4.1. Linear model......Page 111
4.4.2. Nonlinear model......Page 113
4.5. Conclusion......Page 114
4.6. References......Page 115
5.1. Introduction......Page 119
5.2. Statement of the problem......Page 121
5.2.1. Existence of a solution to the NTB uptake system......Page 122
5.3. Optimal control for the NTB problem with an unknown source......Page 124
5.3.1. Existence of a solution to the adjoint problem of NTB uptake system with an unknown source......Page 125
5.4. Characterization of the low-regret control for the NTB system......Page 129
5.5. Concluding remarks......Page 132
5.6. References......Page 133
6.1. Introduction......Page 135
6.2.1. Asymptotically periodic process and periodic limit processes......Page 137
6.2.2. Sectorial operators......Page 139
6.3. A stochastic integro-differential equation of fractional order......Page 140
6.4. An illustrative example......Page 159
6.5. References......Page 160
7.1. Introduction......Page 163
7.2. Preliminaries......Page 164
7.3. Main theorems......Page 166
7.4. The smoothness of the bounded solution......Page 173
7.5. Application to the Burgers equation......Page 178
7.6. References......Page 181
8.1. Introduction......Page 183
8.2. A physical invention is translated into mathematics thanks to the heat flow......Page 185
8.3. The proper story of proper modes......Page 186
8.3.1. Mathematical position of the lamina problem......Page 187
8.3.2. Simple modes are naturally involved......Page 188
8.3.3. A remarkable switch to proper modes......Page 189
8.4.1. A calculation that a priori imposes an extension to the function f at the base of the lamina......Page 191
8.4.2. A crazy calculation......Page 192
8.4.3. Fourier is happily confronted with the task of finding an explanation for the simplicity of the result about coefficients......Page 196
8.4.4. Criticisms of the modeling......Page 197
8.5. To invoke arbitrary functions leads to an interpretation of orthogonality relations......Page 199
8.5.1. Function is a leitmotiv in Fourier’s intellectual career......Page 202
8.6. The modeling of the temperature of the Earth and the greenhouse effect......Page 203
8.7.1. Another dictionary: the Fourier transform for tempered distributions......Page 206
8.7.2. Heisenberg inequalities may just be deduced from the existence of a scalar product......Page 207
8.8. Conclusion......Page 209
8.9. References......Page 211
PART 2: Some Topics on Mayotte and Its Region......Page 213
9.1.1. The ARESMA project......Page 215
9.1.2. Towards a methodology of interdisciplinary modeling......Page 216
9.2.1. Complex systems......Page 217
9.2.2. Methodology......Page 220
9.2.3. Results......Page 221
9.3.1. Hypergraphs and modeling of a complex system......Page 227
9.3.3. Results......Page 230
9.4. Discussion and perspectives......Page 234
9.5.2. Metric over an FHT......Page 236
9.6. References......Page 239
10.1. Introduction......Page 243
10.2.1. Exploration of post-forestry transitions in rainforests of Madagascar......Page 245
10.2.2. Applications to dry forests in southwestern Madagascar......Page 250
10.2.3. Viability......Page 251
10.3.1. Degradation, violation, sanction......Page 254
10.3.2. Local farmers’ maps and conceptual graphs......Page 256
10.4. Discussion and conclusion......Page 259
10.5. References......Page 262
11.1.1. Motivation......Page 267
11.1.2. Context......Page 268
11.1.3. About the literature on the birth curve in Mayotte......Page 269
11.3.1. Methodological approach......Page 270
11.3.3. Monthly trend......Page 271
11.3.4. Characterization of the explosion risk of the number of births......Page 272
11.3.5. Autocorrelation......Page 274
11.3.6. Modeling by an ARIMA process (p, d, q)......Page 275
11.3.7. Predictions for the year 2018......Page 278
11.4. Discussion......Page 279
11.6. References......Page 281
12.1. Introduction......Page 283
12.2. Justifying the mathematization of economics......Page 285
12.2.1. The ontological and linguistic arguments......Page 286
12.2.2. Towards a naturalization of modeling in economics......Page 287
12.2.3. A number of caveats......Page 289
12.3. For a reasonable mathematization of economics: the case of Mayotte......Page 290
12.3.2. From Mayotte’s formal economy to its informal one......Page 291
12.3.3. When the formal economy interacts with the informal one: some issues for the modelization of complex systems......Page 292
12.5. References......Page 295
List of Authors......Page 301
Index......Page 303
Other titles from iSTE in Mathematics and Statistics......Page 305
EULA......Page 312