Maths for Social Sciences

Preface......Page 3
Contents......Page 4
1.1.1 Introductory Examples......Page 7
1.2.1 Vector Representation in mathbbR2 and in mathbbR3......Page 10
1.2.2 Vector Addition......Page 11
1.2.3 Scalar Multiplication......Page 13
1.2.4 Linear Combination of Vectors......Page 14
1.2.5 Linear Dependence/Independence of Vectors......Page 16
1.3.1 Types of Matrices......Page 22
1.3.2 Operations on Matrices: An Inventory......Page 24
1.3.3 Transposition......Page 25
1.3.4 Matrix Addition......Page 26
1.3.5 Multiplication of a Scalar by a Matrix......Page 27
1.3.6 Multiplication of Matrices......Page 28
1.4.1 Notion and Computation......Page 47
1.4.2 How to Invert a Matrix Using Determinants......Page 56
1.4.3 Determinants and ``Viability'' of a Leontief System......Page 57
1.5 Rank of a Matrix......Page 58
1.6 Statistical Applications of Linear Algebra......Page 64
1.7 Linear Applications......Page 73
1.8 Linear Algebraic Systems......Page 77
1.8.1 A Special Case: Cramer's Systems......Page 78
1.8.2 The General Case......Page 80
1.9 Applications to Networks......Page 92
1.10 Some Complements on Square Matrices......Page 93
1.11 Exercises......Page 99
2.1 What's a Function......Page 104
2.1.1 Intervals......Page 105
2.1.3 Elementary Functions......Page 106
2.1.4 Continuous Functions......Page 115
2.1.5 An Annoying Detail and a Tribute to L.D. Landau......Page 117
2.1.6 The Small o Algebra......Page 119
2.1.7 Some Rankings......Page 120
2.2.1 Local Behavior, Derivative and Differential......Page 123
2.2.2 Notation for Derivatives......Page 130
2.2.3 Derivative and Differential: What's the Most Important Notion......Page 131
2.2.4 The Computation of Derivatives, also of Order >1......Page 133
2.3 What's a Function of a Vector......Page 157
2.3.1 Graphic Representation of a Function of n2 Variables......Page 160
2.3.2 How Big is a Vector?......Page 162
2.3.3 Derivatives of Functions of a Vector......Page 163
2.3.4 Unconstrained Extrema for Functions f:mathbbRnrightarrowmathbbR......Page 168
2.3.5 Constrained Extrema......Page 175
2.3.6 The General Case......Page 183
2.4 Exercises......Page 197
3.1 Integrals and Areas......Page 201
3.2 Fundamental Theorem of Integral Calculus......Page 202
3.3 Antiderivative Calculus......Page 205
3.3.1 Integration by Parts......Page 206
3.3.2 Integration by Substitution......Page 207
3.4 An Immediate Application: Mean and Expected Values......Page 208
3.4.1 Expectation and the Law of Large Numbers......Page 210
3.4.2 Density Function and Distribution Function......Page 211
3.4.3 Discrete Distributions......Page 213
3.5 Frequency/Probability Density Functions: Some Cases......Page 214
3.5.1 A Special Distribution......Page 215
3.6 People Survival......Page 218
3.7 Exercises......Page 221
4.1 Introduction......Page 224
4.2.1 Discrete Time......Page 227
4.2.2 Continuous Time......Page 229
4.2.3 Motion Law of a DS......Page 231
4.2.4 Autonomous Systems......Page 232
4.3 Extracting Info from a Motion Law......Page 235
4.4 Classic Approach......Page 236
4.4.1 Linear Discrete Systems......Page 237
4.4.2 About Some Special Discrete Systems......Page 249
4.4.3 Continuous-Time Systems......Page 252
4.4.4 Continuous Systems: Separable Equations......Page 253
4.4.5 Continuous Systems: Linear Differential Equations of the First Order......Page 256
4.4.6 An Interesting Socio-demographic Model/1......Page 257
4.4.7 Linear Continuous Systems......Page 263
4.5.1 Discrete Systems......Page 285
4.5.2 Continuous Systems......Page 286
4.6.1 Equilibria: Notion and General Systems......Page 288
4.6.2 How to Find Equilibria for Autonomous Systems?......Page 290
4.6.3 Nature of an Equilibrium Point......Page 292
4.7.1 Notion......Page 293
4.7.2 Equilibria in a Phase Diagram......Page 296
4.7.3 Behaviors Revealed by a Phase Diagram......Page 298
4.7.4 Continuous Systems......Page 310
4.8 Some Politically Relevant Applications......Page 312
4.8.1 Consensus Diffusion......Page 313
4.8.2 Arab Springs......Page 318
4.8.3 Growth and Demographic Trap......Page 321
4.9.1 War......Page 324
4.9.2 Guerrilla......Page 325
4.10 USA Against USSR: An Old Story......Page 327
4.10.1 An Electoral Application......Page 332
4.11.1 The B Model......Page 335
4.11.2 The R Model......Page 336
4.11.3 The RI Model......Page 337
4.11.4 The RIV Model......Page 338
4.12 Exercises......Page 340
Refs......Page 344
Index......Page 345