The strength of nonstandard analysis

Contents......Page 13
I. Foundations......Page 21
1.1 Introduction......Page 22
1.2 The theory PRA[sup(ω)]......Page 23
1.3 The theory NPRA[sup(ω)]......Page 25
1.4 The theory WNA......Page 26
1.5 Bounded minima and overspill......Page 28
1.6 Standard parts......Page 30
1.7 Liftings of formulas......Page 33
1.8 Choice principles in L(PRA[sup(ω)])......Page 35
1.9 Saturation principles......Page 36
1.10 Saturation and choice......Page 39
1.11 Second order standard parts......Page 40
1.12 Functional choice and (∃[sup(2)])......Page 42
1.13 Conclusion......Page 44
Part I. Technical......Page 46
Part II. General......Page 49
3.1 Introduction......Page 52
3.2 Généralités sur la référentiation......Page 53
3.3 Le calcul de Dirac. L'égalité de Dirac......Page 56
3.4 Calcul de Heaviside sans transformée de Laplace. L'égalité de Laplace......Page 60
3.5 Exemples......Page 63
4. Stratified analysis?......Page 66
4.1 The Robinsonian framework......Page 67
4.2 Stratified analysis......Page 72
4.3 An axiomatic system for stratified set theory......Page 77
5.1 Introduction......Page 83
5.2 The system......Page 84
6.1 Introduction......Page 95
6.2 Distributions, ultradistributions and hyperfunctions......Page 96
6.3 Prehyperfunctions and predistributions......Page 103
6.4 The differential algebra A(Ω[sub(ε)])......Page 105
6.5 Conclusion......Page 109
7.1 Introduction......Page 111
7.2 The decomposition theorem......Page 114
7.3 Geometry of neutrices in ℝ[sup(2)] and proof of the decomposition theorem......Page 115
II. Number theory......Page 136
8.1 The beginning......Page 137
8.2 Duality between null ideal and meager ideal......Page 138
8.3 Buy-one-get-one-free scheme......Page 139
8.4 From Kneser to Banach......Page 142
8.5 Inverse problem for upper asymptotic density......Page 143
8.6 Freiman's 3k — 3+b conjecture......Page 147
9.1 Introduction......Page 151
9.2 Near arithmetic progressions......Page 152
9.3 The interval-measure property......Page 158
III. Statistics, probability and measures......Page 161
10.1 Introduction......Page 162
10.2 Some basic concepts of statistics......Page 163
10.3 Exponential families......Page 164
10.4 The nonstandard likelihood ratio test......Page 176
10.5 Comparison with nonstandard tests based on [ ]......Page 182
11.1 Introduction......Page 187
11.2 Construction of the least upper bound of sums in IST......Page 189
11.3 The global part of the infinitesimal generator......Page 191
11.4 Remarks......Page 193
12.1 Introduction......Page 194
12.2 A fair price for a multiply traded asset......Page 195
12.3 Fairness-enhancing effects of a currency transaction tax......Page 197
12.4 How to minimize "unfairness"......Page 198
13.1 Introduction......Page 206
13.2 Abstract quantum probability spaces......Page 208
13.3 Quantum Bernoulli experiments......Page 209
13.4 The internal quantum processes......Page 211
13.5 From the internal to the standard world......Page 213
13.6 The symmetric Fock space and its embedding into L......Page 219
14.1 Introduction......Page 223
14.2 Recent work of Yeneng Sun......Page 224
14.3 Purification of measure-valued maps......Page 227
15.2 Loeb measures and S-measures......Page 234
15.3 Egoroff's Theorem......Page 238
15.4 A Theorem of Riesz......Page 239
16.1 Introduction and notation......Page 244
16.2 The existing literature......Page 245
16.3 The nonstandard approach......Page 247
16.4 A nonstandard vector-vector integral......Page 249
16.5 Uniform convexity......Page 250
16.6 Vector-vector derivatives without uniform convexity......Page 251
16.7 Remarks......Page 253
17.1 Introduction......Page 255
17.2 S-differentiability of internal measures......Page 257
17.3 Differentiability of Loeb measures......Page 260
IV. Differential systems and equations......Page 267
18.1 Preliminaries......Page 268
18.2 Smoothness......Page 272
18.3 Smoothness and finite points......Page 276
18.4 Smoothness and the nonstandard hull......Page 278
19.1 Preliminaries......Page 286
19.2 The Palais-Smale condition......Page 288
19.3 Nonstandard Palais-Smale conditions......Page 289
19.4 Palais-Smale conditions per level......Page 295
19.5 Nonstandard variants of Palais-Smale conditions per level......Page 296
19.6 Mountain Pass Theorems......Page 297
20.1 Introduction......Page 301
20.2 Deformations and perturbations......Page 302
20.3 Averaging in ordinary differential equations......Page 305
20.4 Functional differential equations......Page 312
21.1 Heuristic arguments and definitions......Page 321
21.2 Bounds for the ∗-measure and the ∗-Green function......Page 324
21.3 Solution to the Fokker-Planck equation......Page 325
22.1 Introduction......Page 332
22.2 Preliminaries......Page 334
22.3 Optimal control for d = 2......Page 340
22.4 Optimal control for d = 3......Page 352
23.1 Introduction......Page 364
23.2 A discretization for the diffusion-advection equations in the torus......Page 366
23.3 Some standard estimates for the solution of the discrete problem......Page 368
23.4 Main estimates on the hyperfinite discrete problem......Page 371
23.5 Existence and uniqueness of solution......Page 376
V. Infinitesimals and education......Page 382
24.1 Intuitive proofs with "small" quantities......Page 383
24.2 Keisler's axioms......Page 396
25.1 Introduction......Page 409
25.2 Standard part......Page 410
25.3 Stratified analysis......Page 411
25.4 Derivative......Page 413
25.5 Transfer and closure......Page 414