I. Foundations ................................................ 1
1. The strength of nonstandard analysis ........................ 3
by H. Jerome Keisler
1.1. Introduction .......................................... 3
1.2. The theory РRAω ....................................... 4
1.3. The theory NPRAω ...................................... 6
1.4. The theory WNA ........................................ 7
1.5. Bounded minima and overspill .......................... 9
1.6. Standard parts ....................................... 11
1.7. Liftings of formulas ................................. 14
1.8. Choice principles in L(РRAω) ......................... 16
1.9. Saturation principles ................................ 17
1.10. Saturation and choice ................................ 20
1.11. Second order standard parts .......................... 21
1.12. Functional choice and ($2) ........................... 23
1.13. Conclusion ........................................... 25
2. The virtue of simplicity ................................... 27
by Edward Nelson
Part I. Technical ......................................... 27
Part II. General ........................................... 30
3. Analysis of various practices of referring in classical
or non standard mathematics ................................ 33
by Yves Peraire
3.1. Introduction ......................................... 33
3.2. Généralités sur la référentiation .................... 34
3.3. Le calcul de Dirac. L'égalité de Dirac ............... 37
3.4. Calcul de Heaviside sans transformée de Laplace.
L'égalité de Laplace ................................. 41
3.5. Exemples ............................................. 44
4. Stratified analysis? ....................................... 47
by Karel Hrbacek
4.1. The Robinsonian framework ............................ 48
4.2. Stratified analysis .................................. 53
4.3. An axiomatic system for stratified set theory ........ 58
5. ERNA at work ............................................... 64
by С. Impens and S. Sanders
5.1. Introduction ......................................... 64
5.2. The system ........................................... 65
5.2.1. The language .................................. 65
5.2.2. The axioms .................................... 67
6. The Sousa Pinto approach to nonstandard generalised
functions .................................................. 76
by R.F. Hoskins
6.1. Introduction ......................................... 76
6.1.1. Generalised functions and N.S.A ............... 77
6.2. Distributions, ultradistributions and
hyperfunctions ....................................... 77
6.2.1. Schwartz distributions ........................ 77
6.2.2. The Silva axioms .............................. 78
6.2.3. Fourier transforms and ultradistributions ..... 80
6.2.4. Sato hyperfunctions ........................... 81
6.2.5. Harmonic representation of hyperfunctions ..... 83
6.3. Prehyperfunctions and predistributions ............... 84
6.4. The differential algebra A(Ωε) ....................... 86
6.4.1. Predistributions of finite order .............. 86
6.4.2. Predistributions of local finite order ........ 88
6.4.3. Predistributions of infinite order ............ 88
6.5. Conclusion ........................................... 90
7. Neutrices in more dimensions ............................... 92
by Imme van den Berg
7.1. Introduction ......................................... 92
7.1.1. Motivation and objective ...................... 92
7.1.2. Setting ....................................... 94
7.1.3. Structure of this article ..................... 95
7.2. The decomposition theorem ............................ 95
7.3. Geometry of neutrices in K2 and proof of the
decomposition theorem ................................ 96
7.3.1. Thickness, width and length of neutrices ...... 97
7.3.2. On the division of neutrices ................. 103
7.3.3. Proof of the decomposition theorem ........... 111
II. Number theory ............................................ 117
8. Nonstandard methods for additive and combinatorial
number theory. A survey ................................... 119
by Renling Jin
8.1. The beginning ....................................... 119
8.2. Duality between null ideal and meager ideal ......... 120
8.3. Buy-one-get-one-free scheme ......................... 121
8.4. From Kneser to Banach ............................... 124
8.5. Inverse problem for upper asymptotic density ........ 125
8.6. Freiman's 3k — 3 + b conjecture ..................... 129
9. Nonstandard methods and the Erdõs-Turán conjecture ........ 133
by Steven С. Letii
9.1. Introduction 133
9.2. Near arithmetic progressions ........................ 134
9.3. The interval-measure property ....................... 140
III. Statistics, probability and measures ..................... 143
10. Nonstandard likelihood ratio test in exponential
families .................................................. 145
by Jacques Bosgiraud
10.1. Introduction ........................................ 145
10.1.1.A most powerful nonstandard test ............. 145
10.2. Some basic concepts of statistics ................... 146
10.2.1.Main definitions ............................. 146
10.2.2.Tests ........................................ 146
10.3. Exponential families ................................ 147
10.3.1.Basic concepts ............................... 147
10.3.2.Kullback-Leibler information number .......... 148
10.3.3.The nonstandard test ......................... 149
10.3.4.Large deviations for  ...................... 150
10.3.5.n-regular sets ............................... 151
10.3.6.n-regular sets defined by Kullback-Leibler
information .................................. 156
10.4. The nonstandard likelihood ratio test ............... 159
10.4.1. infinitesimal ............................ 162
10.4.2. .................................. 163
10.4.3. .................................... 164
10.5. Comparison with nonstandard tests based on ....... 165
10.5.1.Regular nonstandard tests .................... 165
10.5.2.Case when  is convex ....................... 167
11. A finitary approach for the representation of the
infinitesimal generator of a markovian semigroup .......... 170
by Schérazade Benhabib
11.1. Introduction ........................................ 170
11.2. Construction of the least upper bound of sums in
IST ................................................. 172
11.3. The global part of the infinitesimal generator ...... 174
11.4. Remarks ............................................. 176
12. On two recent applications of nonstandard analysis to
the theory of financial markets ........................... 177
by Frederik S. Herzberg
12.1. Introduction ........................................ 177
12.2. A fair price for a multiply traded asset ............ 178
12.3. Fairness-enhancing effects of a currency
transaction tax ..................................... 180
12.4. How to minimize "unfairness" ........................ 181
13. Quantum Bernoulli experiments and quantum stochastic
processes ................................................. 189
by Manfred Wolff
13.1. Introduction ........................................ 189
13.2. Abstract quantum probability spaces ................. 191
13.3. Quantum Bernoulli experiments ....................... 192
13.4. The internal quantum processes ...................... 194
13.5. From the internal to the standard world ............. 196
13.5.1.Brownian motion .............................. 197
13.5.2.The nonstandard hulls of the basic internal
processes .................................... 198
13.6. The symmetric Fock space and its embedding into L ... 202
14. Applications of rich measure spaces formed from
nonstandard models ........................................ 206
by Peter Loeb
14.1. Introduction ........................................ 206
14.2. Recent work of Yeneng Sun ........................... 207
14.3. Purification of measure-valued maps ................. 210
15. More on S-measures ........................................ 217
by David A. Ross
15.1. Introduction ........................................ 217
15.2. Loeb measures and S-measures ........................ 217
15.3. Egoroff's Theorem ................................... 221
15.4. A Theorem of Riesz .................................. 222
15.4.1.Conditional expectation ...................... 225
16. A Radon-Nikodým theorem for a vector-valued reference
measure ................................................... 227
by G. Beate Zimmer
16.1. Introduction and notation ........................... 227
16.2. The existing literature ............................. 228
16.3. The nonstandard approach ............................ 230
16.4. A nonstandard vector-vector integral ................ 232
16.5. Uniform convexity ................................... 233
16.6. Vector-vector derivatives without uniform
convexity ........................................... 234
16.7. Remarks ............................................. 236
17. Differentiability of Loeb measures ........................ 238
by Eva Aigner
17.1. Introduction ........................................ 238
17.2. S-differentiability of internal measures ............ 240
17.3. Differentiability of Loeb measures .................. 243
IV. Differential systems and equations ........................ 251
18. The power of Gâteaux differentiability .................... 253
by Vítor Neves
18.1. Preliminaries ....................................... 253
18.2. Smoothness .......................................... 257
18.3. Smoothness and finite points ........................ 261
18.4. Smoothness and the nonstandard hull ................. 263
18.4.1.Strong uniform differentiability ............. 264
18.4.2.The non-standard hull ........................ 266
19. Nonstandard Palais-Smale conditions ....................... 271
by Natália Martins and Vítor Neves
19.1. Preliminaries ....................................... 271
19.2. The Palais-Smale condition .......................... 273
19.3. Nonstandard Palais-Smale conditions ................. 274
19.4. Palais-Smale conditions per level ................... 280
19.5. Nonstandard variants of Palais-Smale conditions
per level ........................................... 281
19.6. Mountain Pass Theorems .............................. 282
20. Averaging for ordinary differential equations and
functional differential equations ......................... 286
by Tewfik Sari
20.1. Introduction ........................................ 286
20.2. Deformations and perturbations ...................... 287
20.2.1.Deformations ................................. 287
20.2.2.Perturbations ................................ 288
20.3. Averaging in ordinary differential equations ........ 290
20.3.1.KBM vector fields ............................ 291
20.3.2.Almost solutions ............................. 292
20.3.3.The stroboscopic method for ODEs ............. 294
20.3.4.Proof of Theorem 2 for almost periodic
vector fields ................................ 294
20.3.5.Proof of Theorem 2 for KBM vector fields ..... 295
20.4. Functional differential equations ................... 297
20.4.1.Averaging for FDEs in the form
z'(ε) = τf(τ,zτ) ............................. 298
20.4.2.The stroboscopic method for ODEs revisited ... 299
20.4.3.Averaging for FDEs in the form
x(t) = f(t/ε,xi) ............................. 300
20.4.4.The stroboscopic method for FDEs ............. 301
21. Path-space measure for stochastic differential equation
with a coefficient of polynomial growth ................... 306
by Toru Nakamura
21.1. Heuristic arguments and definitions ................. 306
21.2. Bounds for the *-measure and the *-Grccn function ... 309
21.3. Solution to the Fokker-Planck equation .............. 310
22. Optimal control for Navier-Stokes equations ............... 317
by Nigel J. Cutland and Katarzyna Grzesiak
22.1. Introduction ........................................ 317
22.2. Preliminaries ....................................... 319
22.2.1.Nonstandard analysis ......................... 319
22.2.2.The stochastic Navier-Stokes equations ....... 319
22.2.3.Controls ..................................... 323
22.3. Optimal control for d = 2 .......................... 325
22.3.1.Controls with no feedback .................... 325
22.3.2.Costs ........................................ 326
22.3.3.Solutions for internal controls .............. 327
22.3.4.Optimal controls ............................. 328
22.3.5.Holder continuous feedback controls
(d = 2) ...................................... 329
22.3.6.Controls based on digital observations
(d = 2) ...................................... 331
22.3.7.The space H .................................. 331
22.3.8.The observations ............................. 332
22.3.9.Ordinary and relaxed feedback controls
for digital observations ..................... 332
22.3.10.Costs for digitally observed controls ....... 334
22.3.11.Solution of the equations ................... 334
22.3.12.Optimal control ............................. 336
22.4. Optimal control for d = 3 ........................... 337
22.4.1.Existence of solutions for any control ....... 338
22.4.2.The control problem for 3D stochastic
Navier-Stokes equations ...................... 338
22.4.3.The space Ω .................................. 339
22.4.4.Approximate solutions ........................ 340
22.4.5.Optimal control .............................. 342
22.4.6.Hölder continuous feedback controls
(d = 3) ...................................... 343
22.4.7.Approximate solutions for Hölder continuous
controls ..................................... 344
Appendix: Nonstandard representations of the spaces Hr .... 345
23. Local-in-time existence of strong solutions of the
n-dimensional Burgers equation via discretizations ........ 349
by João Paulo Teixeira
23.1. Introduction ........................................ 349
23.2. A discretization for the diffusion-advection
equations in the torus .............................. 351
23.3. Some standard estimates for the solution of the
discrete problem .................................... 353
23.4. Main estimates on the hyperfmite discrete problem ... 356
23.5. Existence and uniqueness of solution ................ 361
V. Infinitesimals and education .............................. 367
24. Calculus with infinitesimals .............................. 369
by Keith D. Stroyan
24.1. Intuitive proofs with "small" quantities ............ 369
24.1.1.Continuity and extreme values ................ 369
24.1.2.Microscopic tangency in one variable ......... 370
24.1.3.The Fundamental Theorem of Integral
Calculus ..................................... 372
24.1.4.Telescoping sums and derivatives ............. 372
24.1.5.Continuity of the derivative ................. 373
24.1.6.Trig, polar coordinates, and Holditch's
formula ..................................... 375
24.1.7.The polar area differential .................. 376
24.1.8.Leibniz's formula for radius of curvature .... 379
24.1.9.Changes ...................................... 379
24.1.10.Small changes ............................... 380
24.1.11.The natural exponential ..................... 381
24.1.12.Concerning the history of the calculus ...... 382
24.2. Keisler's axioms .................................... 382
24.2.1.Small, medium, and large hyperreal
numbers ...................................... 382
24.2.2.Keisler's algebra axiom ...................... 383
24.2.3.The uniform derivative of x3 ................. 385
24.2.4.Keisler's function extension axiom ........... 385
24.2.5.Logical real expressions ..................... 386
24.2.6.Logical real formulas ........................ 386
24.2.7.Logical real statements ...................... 387
24.2.8.Continuity and extreme values ................ 388
24.2.9.Microscopic tangency in one variable ......... 389
24.2.10.The Fundamental Theorem of Integral
Calculus .................................... 390
24.2.11.The Local Inverse Function Theorem .......... 391
24.2.12.Second differences and higher order
smoothness .................................. 392
25. Pre-University Analysis ................................... 395
by Richard O'Donovan
25.1. Introduction ........................................ 395
25.2. Standard part ....................................... 396
25.3. Stratified analysis ................................. 397
25.4. Derivative .......................................... 399
25.5. Transfer and closure ................................ 400
|
