Mathematics for Digital Science 1

Gérard-Michel Cochard is Professor Emeritus at Université de Picardie Jules Verne, France, where he has held various senior positions. He has also served at the French Ministry of Education and the CNAM (Conservatoire National des Arts et Métiers). His research is conducted at the Eco-PRocédés, Optimisation et Aide à la Décision (EPROAD) laboratory, France.Mhand Hifi is Professor of Computer Science at Université de Picardie Jules Verne, France, where he heads the EPROAD UR 4669 laboratory and manages the ROD team. As an expert in operations research and NP-hard problem-solving, he actively contributes to numerous international conferences and journals in the field.

• Preface ixChapter 1. The Concept of Logic 11.1. Syllogisms 11.2. Elementary operations of propositional calculus 51.2.1. Negation 51.2.2. Conjunction 61.2.3. Disjunction 71.2.4. Conditional 81.2.5. Biconditional 101.2.6. Tautologies 101.3. Tools of deductive reasoning 141.4. Quantification 171.5. Prenex forms and pure forms 20Chapter 2. Sets and Relationships 252.1. Generalities 252.2. Algebra of sets 292.3. Parts and partitions 302.4. Cartesian product 312.5. Binary relationships 322.6. Properties of relationships 352.7. Applications 42Chapter 3. Counting and Combinatorial Analysis 473.1. Set cardinals 473.2. Permutations, arrangements, combinations 483.3. Properties of binomial coefficients 513.4. Stirling's formula 54Chapter 4. Boolean Algebra and Boolean Functions 614.1. Special elements of an ordered set 614.2. Lattice 664.3. Boolean algebra 684.4. Boolean functions 72Chapter 5. Logic Circuits 755.1. Canonical forms of a Boolean function 755.2. Reduction of a Boolean function 805.3. Karnaugh tables 845.4. Elementary logic circuits 875.5. A little on electronics 935.6. Construction of logical functions 975.7. A basic circuit: the binary adder 100Chapter 6. Arithmetic 1036.1. Reminder about integers 1036.2. Euclidean division 1046.3. Divisibility and prime numbers 1066.4. GCD and LCM 1126.5. Congruencies 1166.6. Elliptic curves 1216.7. Identity, theorem and Bézout algorithm 127Chapter 7. Error Protection 1317.1. General context 1317.2. Linear codes 1367.3. Polynomial codes 1407.4. Convolutional codes 142Chapter 8. Encryption Systems 1458.1. Substitution and transposition 1458.2. Substitution methods 1478.2.1. Replacement of a symbol by a symbol 1478.2.2. The code: replacement of a word by a word 1578.3. Transposition methods 1588.4. Asymmetric systems 1598.5. DES -- secret key system 1608.6. RSA -- public key system 1658.7. Cryptography with elliptic curves 1678.8. Quantum cryptography 1698.9. Appendix: the Crow and the Fox 176Chapter 9. Probabilities 1779.1. Chance 1779.2. Counting and probabilities 1789.3. Events and probabilities 1839.4. Statistics and probabilities 1899.5. Compound probabilities 1919.6. Graphs, states, transitions 1959.7. Markov chains 1999.7.1. Definition 1999.7.2. Transition matrix 2019.7.3. Evolution rules 2059.7.4. Ergodicity 206Chapter 10. Descriptive Statistics 21310.1. Statistical description 21310.1.1. A little vocabulary 21310.1.2. Tabular presentation and frequency 21510.1.3. Two-dimensional series 21910.2. Graphical representations 22010.2.1. Bar charts 22010.2.2. Histograms 22210.2.3. Cumulative diagrams 22310.2.4. Polar diagrams 22610.2.5. Pie charts 22710.2.6. Figurative diagrams 22710.2.7. Point clouds 22810.3. Position parameters 22910.3.1. Mode 22910.3.2. Median 23010.3.3. Averages 23510.3.4. Comparison of position parameters 23710.4. Dispersion parameters 23810.4.1. Range, interquartile range, mean range 23810.4.2. Variance and standard deviation 24010.4.3. Coefficient of variation and concentration index 24210.5. Linear adjustment 24310.5.1. Principle of adjustment 24310.5.2. Linear adjustment 24810.6. Chronological series 25410.6.1. Introduction 25410.6.2. Study of the general trend 25710.6.3. Study of seasonal variations 26010.7. Covariance and correlation 26510.7.1. Regression lines 26510.7.2. Linear correlation 268Chapter 11. Probability Laws and Simulation 27311.1. Random variables 27311.2. Mathematical expectation, variance and standard deviation 28311.3. Distribution function 28711.4. Usual probability laws 28911.4.1. Uniform law 28911.4.2. Binomial or Bernoulli's law 29011.4.3. Poisson's law 29311.4.4. Exponential law 29411.4.5. Normal law 29811.5. General information on simulation 30511.5.1. Weak law of large numbers 30511.5.2. Some simulations 30811.5.3. Generation of random numbers 31311.6. Generating programs 31611.6.1. Usual laws 31711.6.2. Any probability law 31811.7. M/M/1 waiting system 32011.7.1. Theoretical study 32211.7.2. Simulation 32611.8. Appendices 33311.8.1. Appendix 1: Table for Poisson's law 33311.8.2. Appendix 2: Table for normal law 33411.8.3. Appendix 3: Buffon's needle 33511.8.4. Appendix 4: Results for M/M/1 337References 339List of Authors 341Index 343