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Advances and Applications of DSmT for Information Fusion (Collected Works) : Volume 2

By Smarandache, Florentin

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Book Id: WPLBN0002828226
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Reproduction Date: 7/17/2013

Title: Advances and Applications of DSmT for Information Fusion (Collected Works) : Volume 2  
Author: Smarandache, Florentin
Volume: Volume 2
Language: English
Subject: Non Fiction, Education, Dezert-Smarandache Theory (DSmT)
Collections: Mathematics, Probability Distribution, Applied Science, Critical Thinking, Algebra, Probability Theory, Logic, Mathematical Statistics, Authors Community, Classical Mechanics, Physics, Engineering, Math, Philosophy, Sociolinguistics, Statistics, Education, Chemistry, Most Popular Books in China, Fine Arts, Sociology, Literature, Language, Law
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Publication Date:
2013
Publisher: World Public Library
Member Page: Florentin Smarandache

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Smarandache, F., & Dezert, J. (2013). Advances and Applications of DSmT for Information Fusion (Collected Works) : Volume 2. Retrieved from http://www.self.gutenberg.org/


Description
This second book devoted on advances and applications of Dezert-Smarandache Theory (DSmT) for information fusion collects recent papers from different researchers working in engineering and mathematics. Part 1 of this book presents the current state-of-the-art on theoretical investigations while, Part 2 presents several applications of this new theory. Some ideas in this book are still under current development or improvements, but we think it is important to propose them in order to share ideas and motivate new debates with people interested in new reasoning methods and information fusion. So, we hope that this second volume on DSmT will continue to stir up some interests to researchers and engineers working in data fusion and in artificial intelligence.

Summary
Through this volume, the readers will discover a new family of Proportional Conflict Redistribution (PCR) rules for efficient combination of uncertain, imprecise and highly conflicting sources of information; new investigations on continuous belief functions; investigations on new fusion rules based on T-norms/T-conorms or N-norms/N-conorms (hence using fuzzy/neutrosophy logic in information fusion); an extension of DSmT for dealing with qualitative information expressed directly with linguistic labels; some proposals for new belief conditioning rules (BCR), and more. Also, applications of DSmT are showing up to multitarget tracking in clutter based on generalized data association, or target type tracking, to robot’s map reconstruction, sonar imagery and radar target classification.

Table of Contents
Preamble iii Prefaces v Part I Advances on DSmT 1 Chapter 1 Proportional Conflict Redistribution Rules for Information Fusion 3 by Florentin Smarandache and Jean Dezert 1.1 Introduction . . . . . . . . . . . 3 1.2 The principal rules of combination . . . . . . . 6 1.2.1 Notion of total and partial conflicting masses . . . . . 6 1.2.2 The conjunctive rule . . . . . . . . . 6 1.2.3 The disjunctive rule . . . . . . . . . 8 1.2.4 Dempster’s rule of combination . . . . . . . 8 1.2.5 Smets’ rule of combination . . . . . . . 9 1.2.6 Yager’s rule of combination . . . . . . . 9 1.2.7 Dubois & Prade’s rule of combination . . . . . . 9 1.2.8 The hybrid DSm rule . . . . . . . . 10 1.3 The general weighted operator (WO) . . . . . . . 11 1.4 The weighted average operator (WAO) . . . . . . . 12 1.4.1 Definition . . . . . . . . . 12 1.4.2 Example for WAO . . . . . . . . . 13 1.4.3 Limitations of WAO . . . . . . . . . 13 1.5 Daniel’s minC rule of combination . . . . . . . 14 1.5.1 Principle of the minC rule . . . . . . . 14 1.5.2 Example for minC . . . . . . . . . 15 1.6 Principle of the PCR rules . . . . . . . . . 20 1.7 The PCR1 rule . . . . . . . . . 21 1.7.1 The PCR1 formula . . . . . . . . . 21 1.7.2 Example for PCR1 (degenerate case) . . . . . . 22 1.8 The PCR2 rule . . . . . . . . . 23 1.8.1 The PCR2 formula . . . . . . . . . 23 1.8.2 Example for PCR2 versus PCR1 . . . . . . . 24 1.8.3 Example of neutral impact of VBA for PCR2 . . . . . 25 1.9 The PCR3 rule . . . . . . . . . 25 1.9.1 Principle of PCR3 . . . . . . . . . 25 1.9.2 The PCR3 formula . . . . . . . . . 26 1.9.3 Example for PCR3 . . . . . . . . . 28 1.9.4 Example of neutral impact of VBA for PCR3 . . . . . 29 1.10 The PCR4 rule . . . . . . . . . 31 1.10.1 Principle of PCR4 . . . . . . . . . 31 1.10.2 The PCR4 formula . . . . . . . . . 31 1.10.3 Example for PCR4 versus minC . . . . . . . 32 1.10.4 Example of neutral impact of VBA for PCR4 . . . . . 33 1.10.5 A more complex example for PCR4 . . . . . . 34 1.11 The PCR5 rule . . . . . . . . . 36 1.11.1 Principle of PCR5 . . . . . . . . . 36 1.11.2 The PCR5 formula . . . . . . . . . 42 1.11.3 The PCR5 formula for Bayesian beliefs assignments . . . . 43 1.11.4 General procedure to apply the PCR5 . . . . . . 45 1.11.5 A 3-source example for PCR5 . . . . . . . 46 1.11.6 On the neutral impact of VBA for PCR5 . . . . . . 48 1.11.7 PCR6 as alternative to PCR5 when s > 2 . . . . . . 49 1.11.8 Imprecise PCR5 fusion rule (imp-PCR5) . . . . . . 49 1.11.9 Examples for imprecise PCR5 (imp-PCR5) . . . . . 50 1.12 More numerical examples and comparisons . . . . . . 53 1.12.1 Example 1 . . . . . . . . . 53 1.12.2 Example 2 . . . . . . . . . 55 1.12.3 Example 3 (Zadeh’s example) . . . . . . . 55 1.12.4 Example 4 (hybrid model) . . . . . . . 59 1.12.5 Example 5 (Target ID tracking) . . . . . . . 61 1.13 On Ad-Hoc-ity of fusion rules . . . . . . . . 63 1.14 On quasi-associativity and quasi-Markovian properties . . . . . 64 1.14.1 Quasi-associativity property . . . . . . . 64 1.14.2 Quasi-Markovian property . . . . . . . 64 1.14.3 Algorithm for Quasi-Associativity and Quasi-Markovian Requirement . . 64 1.15 Conclusion . . . . . . . . . . . 66 1.16 References . . . . . . . . . . . 66 Chapter 2 A new generalization of the proportional conflict redistribution rule stable in terms of decision 69 by Arnaud Martin and Christophe Osswald 2.1 Introduction . . . . . . . . . . . 69 2.2 Theory bases . . . . . . . . . . 70 2.2.1 Belief Function Models . . . . . . . . 70 2.2.2 Combination rules . . . . . . . . . 71 2.2.3 Decision rules . . . . . . . . . 72 2.3 The generalized PCR rules . . . . . . . . . 73 2.4 Discussion on the decision following the combination rules . . . . 75 2.4.1 Extending the PCR rule for more than two experts . . . . 76 2.4.2 Stability of decision process . . . . . . . 77 2.4.3 Calculi for two experts and two classes . . . . . . 78 2.5 Conclusion . . . . . . . . . . . 84 2.6 References . . . . . . . . . . . 84 2.7 Appendix: Algorithms . . . . . . . . . 86 Chapter 3 Classical Combination Rules Generalized to DSm Hyper-power Sets and their Comparison with the Hybrid DSm Rule 89 by Milan Daniel 3.1 Introduction . . . . . . . . . . . 89 3.2 Classic definitions . . . . . . . . . 91 3.3 Introduction to the DSm theory . . . . . . . 91 3.3.1 Dedekind lattice, basic DSm notions . . . . . . 91 3.3.2 DSm models . . . . . . . . . 92 3.3.3 The DSm rules of combination . . . . . . . 93 3.4 A generalization of Dempster’s rule . . . . . . . 94 3.4.1 The generalized non-normalized conjunctive rule . . . . . 95 3.4.2 The generalized Dempster’s rule . . . . . . . 95 3.5 A generalization of Yager’s rule . . . . . . . 96 3.6 A generalization of Dubois-Prade’s rule . . . . . . . 97 3.7 A comparison of the rules . . . . . . . . . 101 3.7.1 Examples . . . . . . . . . 101 3.7.2 A summary of the examples . . . . . . . 105 3.8 Open problems . . . . . . . . . 106 3.9 Conclusion . . . . . . . . . . . 107 3.10 References . . . . . . . . . . . 107 3.11 Appendix - proofs . . . . . . . . . 108 3.11.1 Generalized Dempster’s rule . . . . . . . 108 3.11.2 Generalized Yager’s rule . . . . . . . 109 3.11.3 Generalized Dubois-Prade rule . . . . . . . 110 3.11.4 Comparison statements . . . . . . . 112 Chapter 4 A Comparison of the Generalized minC Combination and the Hy- brid DSm Combination Rules 113 by Milan Daniel 4.1 Introduction . . . . . . . . . . . 113 4.2 MinC combination on classic frames of discernment . . . . . 114 4.2.1 Basic Definitions . . . . . . . . . 114 4.2.2 Ideas of the minC combination . . . . . . . 115 4.2.3 Formulas for the minC combination . . . . . . 116 4.3 Introduction to DSm theory . . . . . . . . . 117 4.3.1 Dedekind lattice and other basic DSm notions . . . . . 118 4.3.2 DSm models . . . . . . . . . 118 4.3.3 The DSm rule of combination . . . . . . . 119 4.4 MinC combination on hyper-power sets . . . . . . . 120 4.4.1 Generalized level of minC combination on hyper-power set . . . 120 4.4.2 MinC combination on the free DSm model Mf . . . . . 120 4.4.3 Static minC combination on hybrid DSm models . . . . . 121 4.4.4 Dynamic minC combination . . . . . . . 121 4.5 Examples of minC combination . . . . . . . 123 4.6 Comparison of the generalized minC combination and hybrid DSm combination rules . . . . . . . . . . . . 125 4.7 Related works. . . . . . . . . . 127 4.8 Conclusion . . . . . . . . . . . 128 4.9 References . . . . . . . . . . . 128 Chapter 5 Pre-Boolean algebra, ordered DSmT and DSm continuous models 131 by Fr´ed´eric Dambreville 5.1 Introduction . . . . . . . . . . . 131 5.2 A short introduction to the DSmT . . . . . . . 132 5.2.1 Boolean algebra . . . . . . . . . 133 5.2.2 Hyper-power sets . . . . . . . . . 134 5.2.3 Pre-Boolean algebra . . . . . . . . . 136 5.2.4 The free Dezert Smarandache Theory . . . . . . 140 5.2.5 Extensions to any insulated pre-Boolean algebra . . . . . 140 5.3 Ordered DSm model . . . . . . . . . 141 5.3.1 Ordered atomic propositions . . . . . . . 141 5.3.2 Associated pre-Boolean algebra and complexity. . . . . 142 5.3.3 General properties of the model . . . . . . . 144 5.4 Continuous DSm model . . . . . . . . . 145 5.4.1 Measurable increasing subsets . . . . . . . 145 5.4.2 Definition and manipulation of the belief . . . . . . 147 5.5 Implementation of the continuous model . . . . . . . 148 5.6 Conclusion . . . . . . . . . . . 149 5.7 References . . . . . . . . . . . 152 Chapter 6 Conflict Free Rule for Combining Evidences 155 by Fr´ed´eric Dambreville 6.1 Introduction . . . . . . . . . . . 155 6.2 Viewpoints in evidence theories . . . . . . . 156 6.2.1 Pre-Boolean algebra . . . . . . . . . 156 6.2.2 Belief . . . . . . . . . . 159 6.2.3 Fusion rules . . . . . . . . . 161 6.3 Entropic approach of the rule definition . . . . . . . 162 6.3.1 Independent sources and entropy . . . . . . . 163 6.3.2 Definition of a new rule for the DSmT . . . . . . 163 6.3.3 Feasibility of the rule . . . . . . . . . 164 6.3.4 Generalizations . . . . . . . . . 164 6.4 Implementation and properties . . . . . . . 165 6.4.1 Properties . . . . . . . . . 166 6.4.2 Algorithm . . . . . . . . . 168 6.4.3 Examples . . . . . . . . . 169 6.5 Logical ground of the rule . . . . . . . . . 171 6.5.1 The belief as a probability of a modal proposition . . . . 171 6.5.2 Definition of the logic . . . . . . . . 173 6.5.3 Fusion rule . . . . . . . . . 175 6.6 Conclusion . . . . . . . . . . . 181 6.7 References . . . . . . . . . . . 181 Chapter 7 DSm models and Non-Archimedean Reasoning 183 by Andrew Schumann 7.1 Introduction . . . . . . . . . . . 183 7.2 Standard many-valued logics . . . . . . . . 185 7.3 Many-valued logics on DSm models . . . . . . . 189 7.4 Hyper-valued Reasoning . . . . . . . . . 192 7.4.1 Hyper-valued matrix logics . . . . . . . 192 7.4.2 Hyper-valued probability theory and hyper-valued fuzzy logic . . . 195 7.5 p-Adic Valued Reasoning . . . . . . . . . 197 7.5.1 p-Adic valued matrix logic . . . . . . . 197 7.5.2 p-Adic probability theory . . . . . . . 199 7.5.3 p-Adic fuzzy logic . . . . . . . . . 202 7.6 Conclusion . . . . . . . . . . . 203 7.7 References . . . . . . . . . . . 203 Chapter 8 An In-Depth Look at Quantitative Information Fusion Rules 205 by Florentin Smarandache 8.1 Introduction . . . . . . . . . . . 206 8.2 Conjunctive Rule . . . . . . . . . 206 8.3 Disjunctive Rule . . . . . . . . . 207 8.4 Exclusive Disjunctive Rule . . . . . . . . . 207 8.5 Mixed Conjunctive-Disjunctive Rule . . . . . . . 207 8.6 Conditioning Rule . . . . . . . . . 207 8.7 Dempster’s Rule . . . . . . . . . 208 8.8 Modified Dempster-Shafer rule (MDS) . . . . . . . 208 8.9 Murphy’s Statistical Average Rule . . . . . . . 208 8.10 Dezert-Smarandache Classic Rule (DSmC) . . . . . . 209 8.11 Dezert-Smarandache Hybrid Rule (DSmH) . . . . . . 209 8.12 Smets’ TBM Rule . . . . . . . . . 210 8.13 Yager’s Rule . . . . . . . . . . . 210 8.14 Dubois-Prade’s Rule . . . . . . . . . 210 8.15 Weighted Operator (Unification of the Rules) . . . . . . 210 8.16 Inagaki’s Unified Parameterized Combination Rule . . . . . 211 8.17 The Adaptive Combination Rule (ACR) . . . . . . . 211 8.18 The Weighted Average Operator (WAO) . . . . . . . 212 8.19 The Ordered Weighted Average operator (OWA) . . . . . . 212 8.20 The Power Average Operator (PAO) . . . . . . . 212 8.21 Proportional Conflict Redistribution Rules (PCR) . . . . . 213 8.21.1 PCR1 Fusion rule . . . . . . . . . 213 8.21.2 PCR2-PCR4 Fusion rules . . . . . . . 214 8.21.3 PCR5 Fusion Rule . . . . . . . . . 216 8.21.4 PCR6 Fusion Rule . . . . . . . . . 218 8.22 The minC Rule . . . . . . . . . 219 8.23 The Consensus Operator . . . . . . . . . 220 8.24 Zhang’s Center Combination Rule . . . . . . . 221 8.25 The Convolutive x-Averaging . . . . . . . . 222 8.26 The _-junctions Rules . . . . . . . . . 222 8.27 The Cautious Rule . . . . . . . . . 222 8.28 Other fusion rules . . . . . . . . . 223 8.29 Fusion rules based on T-norm and T-conorm . . . . . . 223 8.30 Improvements of fusion rules . . . . . . . . 224 8.31 Extension of bba on neutrosophic sets . . . . . . . 227 8.32 Unification of Fusion Rules (UFR) . . . . . . . 230 8.33 Unification of Fusion Theories (UFT) . . . . . . . 231 8.34 References . . . . . . . . . . . 233 Chapter 9 Belief Conditioning Rules 237 by Florentin Smarandache and Jean Dezert 9.1 Introduction . . . . . . . . . . . 237 9.2 Shafer’s conditioning rule (SCR) . . . . . . . 238 9.3 Belief Conditioning Rules (BCR) . . . . . . . 238 9.3.1 Belief Conditioning Rule no. 1 (BCR1) . . . . . . 240 9.3.2 Belief Conditioning Rule no. 2 (BCR2) . . . . . . 241 9.3.3 Belief Conditioning Rule no. 3 (BCR3) . . . . . . 242 9.3.4 Belief Conditioning Rule no. 4 (BCR4) . . . . . . 242 9.3.5 Belief Conditioning Rule no. 5 (BCR5) . . . . . . 243 9.3.6 Belief Conditioning Rule no. 6 (BCR6) . . . . . . 243 9.3.7 Belief Conditioning Rule no. 7 (BCR7) . . . . . . 243 9.3.8 Belief Conditioning Rule no. 8 (BCR8) . . . . . . 244 9.3.9 Belief Conditioning Rule no. 9 (BCR9) . . . . . . 245 9.3.10 Belief Conditioning Rule no. 10 (BCR10) . . . . . . 245 9.3.11 Belief Conditioning Rule no. 11 (BCR11) . . . . . . 245 9.3.12 More Belief Conditioning Rules (BCR12-BCR21) . . . . . 245 9.4 Examples . . . . . . . . . . . 248 9.4.1 Example no. 1 (free DSm model with non-Bayesian bba) . . . 248 9.4.2 Example no. 2 (Shafer’s model with non-Bayesian bba) . . . 256 9.4.3 Example no. 3 (Shafer’s model with Bayesian bba) . . . . 258 9.5 Classification of the BCRs . . . . . . . . . 259 9.6 Properties for all BCRs . . . . . . . . . 261 9.7 Open question on conditioning versus fusion . . . . . . 262 9.7.1 Examples of non commutation of BCR with fusion . . . . 263 9.8 Conclusion . . . . . . . . . . . 267 9.9 References . . . . . . . . . . . 267 Chapter 10 Fusion of qualitative beliefs 269 by Florentin Smarandache and Jean Dezert 10.1 A brief historic of previous works . . . . . . . 269 10.2 Qualitative Operators . . . . . . . . . 271 10.2.1 Qualitative Belief Assignment . . . . . . . 272 10.2.2 Qualitative Conjunctive Rule (qCR) . . . . . . 273 10.2.3 Qualitative DSm Classic rule (q-DSmC) . . . . . . 273 10.2.4 Qualitative DSm Hybrid rule (q-DSmH) . . . . . . 273 10.3 Qualitative Average Operator . . . . . . . . 274 10.4 Qualitative PCR5 rule (q-PCR5) . . . . . . . 275 10.5 A simple example . . . . . . . . . 275 10.5.1 Qualitative Fusion of qba’s . . . . . . . 276 10.5.2 Fusion with a crisp mapping . . . . . . . 279 10.5.3 Fusion with an interval mapping . . . . . . . 280 10.6 Conclusion . . . . . . . . . . . 281 10.7 References . . . . . . . . . . . 282 Part II Applications of DSmT 287 Chapter 11 Generalized proportional conflict redistribution rule applied to Sonar imagery and Radar targets classification 289 by Arnaud Martin and Christophe Osswald 11.1 Introduction . . . . . . . . . . . 289 11.2 Backgrounds on combination rules . . . . . . . 290 11.3 Experts fusion in Sonar imagery . . . . . . . 293 11.3.1 Models . . . . . . . . . 294 11.3.2 Experimentation . . . . . . . . . 298 11.4 Classifiers fusion in Radar target recognition . . . . . . 300 11.4.1 Classifiers . . . . . . . . . 300 11.4.2 Database . . . . . . . . . 300 11.4.3 Model . . . . . . . . . . 301 11.4.4 Results . . . . . . . . . 301 11.5 Conclusion . . . . . . . . . . . 303 11.6 References . . . . . . . . . . . 303 Chapter 12 Multitarget Tracking in Clutter based on Generalized Data Association: Performance Evaluation of Fusion Rules 305 by Jean Dezert, Albena Tchamova, Tzvetan Semerdjiev and Pavlina Konstantinova 12.1 Introduction . . . . . . . . . . . 306 12.2 General principle of GDA-MTT . . . . . . . 307 12.2.1 Kinematics Likelihood Ratios for GDA . . . . . . 308 12.2.2 Attribute Likelihood Ratios for GDA . . . . . . 308 12.3 Fusion rules proposed for GDA-MTT . . . . . . . 309 12.4 Simulation scenario and results . . . . . . . 312 12.4.1 Simulation scenario . . . . . . . . . 312 12.4.2 Simulation results . . . . . . . . . 314 12.5 Conclusions . . . . . . . . . . . 318 12.6 References . . . . . . . . . . . 319 Chapter 13 Target Type Tracking with Different Fusion Rules: A Compara- tive Analysis 323 by Jean Dezert, Albena Tchamova, Florentin Smarandache and Pavlina Konstantinova 13.1 Introduction . . . . . . . . . . . 324 13.2 Fusion Rules proposed for Target Type Tracking . . . . . . 325 13.2.1 Basics on DST and DSmT . . . . . . . 325 13.2.2 Fusion rules . . . . . . . . . 326 13.3 The Target Type Tracking Problem . . . . . . . 327 13.3.1 Formulation of the problem . . . . . . . 327 13.3.2 Proposed issues . . . . . . . . . 327 13.4 Simulation results . . . . . . . . . 328 13.4.1 Results for classifier 1 . . . . . . . . 329 13.4.2 Results for classifier 2 . . . . . . . . 336 13.5 Conclusions . . . . . . . . . . . 341 13.6 References . . . . . . . . . . . 341 Chapter 14 A DSmT-based Fusion Machine for Robot’s Map Reconstruction 343 by Xinhan Huang, Xinde Li and Min Wang 14.1 Introduction . . . . . . . . . . . 344 14.2 The fusion machine . . . . . . . . . 344 14.2.1 General principle . . . . . . . . . 344 14.2.2 Basis of DSmT . . . . . . . . . 345 14.2.3 The PCR5 fusion rule . . . . . . . . 347 14.3 Modeling of Sonar Grid Map Building Based on DSmT . . . . . 348 14.4 Sonar Grid Map Building Based on Other Methods . . . . . 351 14.5 Simulation Experiment . . . . . . . . . 356 14.6 Conclusion . . . . . . . . . . . 361 14.7 References . . . . . . . . . . . 361 Chapter 15 Reducing DSmT hybrid rule complexity through optimization of the calculation algorithm 365 by Pascal Djiknavorian and Dominic Grenier 15.1 Introduction . . . . . . . . . . . 365 15.2 Theories . . . . . . . . . . . 366 15.2.1 Definitions . . . . . . . . . 366 15.2.2 Dempster-Shafer Theory . . . . . . . 366 15.2.3 Dezert-Smarandache Theory . . . . . . . 367 15.3 How to avoid the complexity . . . . . . . . 369 15.3.1 Simpler way to view the DSmT hybrid rule of combination . . . 369 15.3.2 Notation system used . . . . . . . . 371 15.3.3 How simple can it be . . . . . . . . . 373 15.3.4 Optimization in the calculation algorithm . . . . . . 375 15.3.5 Performances analysis . . . . . . . . 381 15.4 Conclusion . . . . . . . . . . . 387 15.5 Acknowledgements . . . . . . . . . 387 15.6 References . . . . . . . . . . . 387 15.7 Appendix: MatlabTMcode listings . . . . . . . 388 15.7.1 File : aff ensemble.m . . . . . . . . . 388 15.7.2 File : aff matrice.m . . . . . . . . . 388 15.7.3 File : bon ordre.m . . . . . . . . . 389 15.7.4 File : calcul DSm hybrid auto.m . . . . . . . 390 15.7.5 File : calcul DSm hybride.m . . . . . . . 391 15.7.6 File : croyance.m . . . . . . . . . 394 15.7.7 File : dedouble.m . . . . . . . . . 395 15.7.8 File : depart.m . . . . . . . . . 397 15.7.9 File : DSmH auto.m . . . . . . . . . 399 15.7.10 File : enlever contrainte.m . . . . . . . 404 15.7.11 File : ensemble.m . . . . . . . . . 406 15.7.12 File : faire contraire.m . . . . . . . . 408 15.7.13 File : hybride.m . . . . . . . . . 409 15.7.14 File : intersection matrice.m . . . . . . . 413 15.7.15 File : ordre grandeur.m . . . . . . . 414 15.7.16 File : plausibilite.m . . . . . . . . . 415 15.7.17 File : produit somme complet.m . . . . . . . 417 15.7.18 File : separation.m . . . . . . . . . 420 15.7.19 File : separation unique.m . . . . . . . 423 15.7.20 File : somme produit complet.m . . . . . . . 424 15.7.21 File : tri.m . . . . . . . . . 427 15.7.22 File : union matrice.m . . . . . . . . 429 Biographies of contributors 431

 

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