Mathematics for Digital Science 1 - Gérard-Michel Cochard

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Résumé :

Preface ix

Chapter 1. The Concept of Logic 1

1.1. Syllogisms 1

1.2. Elementary operations of propositional calculus 5

1.2.1. Negation 5

1.2.2. Conjunction 6

1.2.3. Disjunction 7

1.2.4. Conditional 8

1.2.5. Biconditional 10

1.2.6. Tautologies 10

1.3. Tools of deductive reasoning 14

1.4. Quantification 17

1.5. Prenex forms and pure forms 20

Chapter 2. Sets and Relationships 25

2.1. Generalities 25

2.2. Algebra of sets 29

2.3. Parts and partitions 30

2.4. Cartesian product 31

2.5. Binary relationships 32

2.6. Properties of relationships 35

2.7. Applications 42

Chapter 3. Counting and Combinatorial Analysis 47

3.1. Set cardinals 47

3.2. Permutations, arrangements, combinations 48

3.3. Properties of binomial coefficients 51

3.4. Stirling's formula 54

Chapter 4. Boolean Algebra and Boolean Functions 61

4.1. Special elements of an ordered set 61

4.2. Lattice 66

4.3. Boolean algebra 68

4.4. Boolean functions 72

Chapter 5. Logic Circuits 75

5.1. Canonical forms of a Boolean function 75

5.2. Reduction of a Boolean function 80

5.3. Karnaugh tables 84

5.4. Elementary logic circuits 87

5.5. A little on electronics 93

5.6. Construction of logical functions 97

5.7. A basic circuit: the binary adder 100

Chapter 6. Arithmetic 103

6.1. Reminder about integers 103

6.2. Euclidean division 104

6.3. Divisibility and prime numbers 106

6.4. GCD and LCM 112

6.5. Congruencies 116

6.6. Elliptic curves 121

6.7. Identity, theorem and B?zout algorithm 127

Chapter 7. Error Protection 131

7.1. General context 131

7.2. Linear codes 136

7.3. Polynomial codes 140

7.4. Convolutional codes 142

Chapter 8. Encryption Systems 145

8.1. Substitution and transposition 145

8.2. Substitution methods 147

8.2.1. Replacement of a symbol by a symbol 147

8.2.2. The code: replacement of a word by a word 157

8.3. Transposition methods 158

8.4. Asymmetric systems 159

8.5. DES -- secret key system 160

8.6. RSA -- public key system 165

8.7. Cryptography with elliptic curves 167

8.8. Quantum cryptography 169

8.9. Appendix: the Crow and the Fox 176

Chapter 9. Probabilities 177

9.1. Chance 177

9.2. Counting and probabilities 178

9.3. Events and probabilities 183

9.4. Statistics and probabilities 189

9.5. Compound probabilities 191

9.6. Graphs, states, transitions 195

9.7. Markov chains 199

9.7.1. Definition 199

9.7.2. Transition matrix 201

9.7.3. Evolution rules 205

9.7.4. Ergodicity 206

Chapter 10. Descriptive Statistics 213

10.1. Statistical description 213

10.1.1. A little vocabulary 213

10.1.2. Tabular presentation and frequency 215

10.1.3. Two-dimensional series 219

10.2. Graphical representations 220

10.2.1. Bar charts 220

10.2.2. Histograms 222

10.2.3. Cumulative diagrams 223

10.2.4. Polar diagrams 226

10.2.5. Pie charts 227

10.2.6. Figurative diagrams 227

10.2.7. Point clouds 228

10.3. Position parameters 229

10.3.1. Mode 229

10.3.2. Median 230

10.3.3. Averages 235

10.3.4. Comparison of position parameters 237

10.4. Dispersion parameters 238

10.4.1. Range, interquartile range, mean range 238

10.4.2. Variance and standard deviation 240

10.4.3....