The strength of nonstandard analysis (Wien, 2007) - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаThe strength of nonstandard analysis / ed. by Berg I. van den, Neves V. - Wien; New York: Springer, 2007. - 401 c. - ISBN 978-3-211-49904-7
 

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Оглавление / Contents
 
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 fig.1 ...................... 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.fig.2 infinitesimal ............................ 162
          10.4.2.fig.3 .................................. 163
          10.4.3.fig.4 .................................... 164
    10.5. Comparison with nonstandard tests based on fig.1 ....... 165
          10.5.1.Regular nonstandard tests .................... 165
          10.5.2.Case when fig.5 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


 
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