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E-raamat: Nonstandard Analysis for the Working Mathematician

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  • Ilmumisaeg: 26-Aug-2015
  • Kirjastus: Springer
  • Keel: eng
  • ISBN-13: 9789401773270
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Nonstandard Analysis for the Working MathematicianSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 26-Aug-2015
  • Kirjastus: Springer
  • Keel: eng
  • ISBN-13: 9789401773270
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Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in almost all disciplines of mathematics; it is often used as a ‘secret weapon’ by those who know the technique.

This book illuminates the subject with some of the most striking applications in analysis, topology, functional analysis, probability and stochastic analysis, as well as applications in economics and combinatorial number theory. The first chapter is designed to facilitate the beginner in learning this technique by starting with calculus and basic real analysis. The second chapter provides the reader with the most important tools of nonstandard analysis: the transfer principle, Keisler’s internal definition principle, the spill-over principle, and saturation. The remaining chapters of the book study different fields for applications; each begins with a gentle introduction before then exploring solutions to open problems.

All chapters within this second edition have been reworked and updated, with several completely new chapters on compactifications and number theory. Nonstandard Analysis for the Working Mathematician will be accessible to both experts and non-experts, and will ultimately provide many new and helpful insights into the enterprise of mathematics.

Part I An Introduction to Nonstandard Analysis
1 Simple Nonstandard Analysis and Applications
3(34)
Peter A. Loeb
1.1 Introduction
3(4)
1.2 A Simple Construction of a Nonstandard Number System
7(4)
1.3 A Simple Language
11(1)
1.4 Interpretation of the Language L
12(3)
1.5 Transfer Principle for *R
15(3)
1.6 The Nonstandard Real Numbers
18(5)
1.7 Sequences
23(3)
1.8 Topology on the Reals
26(2)
1.9 Limits and Continuity
28(2)
1.10 Differentiation
30(2)
1.11 Riemann Integration
32(5)
References
35(2)
2 An Introduction to General Nonstandard Analysis
37(42)
Peter A. Loeb
2.1 Superstructures
37(1)
2.2 Language for Superstructures
38(1)
2.3 Interpretation of the Language for Superstructures
39(2)
2.4 Monomorphisms and the Transfer Principle
41(3)
2.5 Ultrapower Construction of Superstructures and Monomorphisms
44(6)
2.6 Special Index Sets Yielding Enlargements
50(2)
2.7 A Result in Infinite Graph Theory
52(1)
2.8 Internal and External Sets
53(5)
2.9 Saturation
58(21)
References
78(1)
3 Topology and Measure Theory
79(28)
Peter A. Loeb
3.1 Metric and Topological Spaces
79(4)
3.2 Continuous Mappings
83(1)
3.3 Convergence
83(1)
3.4 More on Topologies
84(1)
3.5 Compact Spaces
85(3)
3.6 Product Spaces
88(1)
3.7 Relative Topologies
88(1)
3.8 Uniform Continuity and Uniform Spaces
89(3)
3.9 Nonstandard Hulls
92(1)
3.10 Compactifications
93(1)
3.11 The Base and Antibase Operators
93(4)
3.12 Measure and Probability Theory
97(10)
3.12.1 The Martingale Convergence Theorem
99(2)
3.12.2 Representing Measures in Potential Theory
101(2)
References
103(4)
Part II Functional Analysis
4 Banach Spaces and Linear Operators
107(58)
Manfred P.H. Wolff
4.1 Introduction
107(1)
4.2 Basic Nonstandard Analysis of Normed Spaces
108(11)
4.2.1 Internal Normed Spaces and Their Nonstandard Hull
108(6)
4.2.2 Standard Continuous and Internal S--continuous Linear Operators
114(2)
4.2.3 Special Banach Spaces and Their Nonstandard Hulls
116(3)
4.2.4 Notes
119(1)
4.3 Advanced Theory of Banach Spaces
119(14)
4.3.1 A Brief Excursion to Locally Convex Vector Spaces
119(5)
4.3.2 General Banach Spaces
124(6)
4.3.3 Banach Lattices
130(3)
4.3.4 Notes
133(1)
4.4 Elementary Theory of Linear Operators
133(4)
4.4.1 Compact Operators
133(2)
4.4.2 Fredholm Operators
135(2)
4.4.3 Notes
137(1)
4.5 Spectral Theory of Operators
137(8)
4.5.1 Basic Definitions and Facts
137(2)
4.5.2 The Spectrum of an S--bounded Internal Operator
139(2)
4.5.3 The Spectrum of Compact Operators and the Essential Spectrum
141(1)
4.5.4 Closed Operators and Pseudoresolvents
142(2)
4.5.5 Notes
144(1)
4.6 Selected Applications
145(20)
4.6.1 Strongly Continuous Semigroups
145(2)
4.6.2 Approximation of Operators and of Their Spectra
147(5)
4.6.3 Super Properties
152(3)
4.6.4 The Fixed Point Property
155(3)
4.6.5 References to Further Applications of Nonstandard Analysis To operator Theory
158(1)
4.6.6 Notes
158(1)
References
159(6)
Part III Compactifications
5 General and End Compactifications
165(14)
Matt Insall
Peter A. Loeb
Malgorzata Aneta Marciniak
5.1 Introduction
165(2)
5.2 General Compactifications
167(3)
5.3 End Compactifications
170(4)
5.4 Product Spaces
174(5)
References
176(3)
Part IV Measure and Probability Theory
6 Measure Theory and Integration
179(54)
Horst Osswald
6.1 Introduction
179(3)
6.2 Loeb Measures
182(15)
6.2.1 Loeb Measure Spaces
182(3)
6.2.2 Loeb Measures over Gaußian Measures
185(1)
6.2.3 Loeb Measurable Functions
186(2)
6.2.4 Loeb Spaces over the Product of Internal Spaces
188(1)
6.2.5 The Hyperfinite Time Line T
189(1)
6.2.6 Lebesgue Measure as a Counting Measure
190(5)
6.2.7 Adapted Loeb Spaces
195(2)
6.3 Standard Integrability for Internal Measures
197(25)
6.3.1 The Definition of S-integrability and Equivalent Conditions
197(3)
6.3.2 μL-integrability and Sμ-integrability
200(5)
6.3.3 Integrable Functions defined on Nn × Λ × [ 0, ∞[ m
205(5)
6.3.4 Standard Part of the Conditional Expectation
210(1)
6.3.5 Characterization of S-integrability
211(2)
6.3.6 Keisler's Fubini Theorem
213(4)
6.3.7 Hyperfinite Representation of the Tensor Product
217(3)
6.3.8 On Symmetric Functions
220(2)
6.4 Internal and Standard Martingales
222(11)
6.4.1 Stopping Times and Doob's Upcrossing Result
223(1)
6.4.2 The Maximum Inequality
224(1)
6.4.3 Doob's Inequality
224(1)
6.4.4 The Burkholder Davis Gundy Inequalities
225(1)
6.4.5 S-integrability of Internal Martingales
225(1)
6.4.6 S-continuity of Internal Martingales
226(1)
6.4.7 The Standard Part of Internal Martingales
226(4)
References
230(3)
7 Stochastic Analysis
233(88)
Horst Osswald
7.1 Introduction
233(4)
7.2 The Ito Integral for the Brownian Motion
237(15)
7.2.1 The S-Continuity of the Internal Integral
238(5)
7.2.2 The S-Square-Integrability of the Internal Ito Integral
243(2)
7.2.3 Adaptedness and Predictability
245(2)
7.2.4 The Standard Ito Integral
247(1)
7.2.5 Integrability of the Ito Integral
248(2)
7.2.6 The Wiener Measure
250(2)
7.3 The Iterated Integral
252(12)
7.3.1 The Definition of the Iterated Integral
252(4)
7.3.2 On Products of Iterated Integrals
256(3)
7.3.3 The Continuity of the Standard Iterated Integral Process
259(1)
7.3.4 The WCH-Measurability of the Iterated Ito Integral
260(2)
7.3.5 Imn(f) is a Continuous Version of the Standard Part of Imn(F)
262(1)
7.3.6 Continuous Versions of Iterated Integral Processes
263(1)
7.4 Beginning of Malliavin Calculus
264(24)
7.4.1 Chaos Decomposition
265(5)
7.4.2 A Lifting Theorem for Functionals in L2W(ΓL)
270(1)
7.4.3 Computation of the Kernels
271(2)
7.4.4 The Kernels of the Product of Wiener Functionals
273(3)
7.4.5 The Malliavin Derivative
276(1)
7.4.6 A Commutation Rule for Derivative and Limit
277(1)
7.4.7 The Clark-Ocone Formula
278(2)
7.4.8 A Lifting Theorem for the Derivative
280(1)
7.4.9 The Skorokhod Integral
281(3)
7.4.10 Product and Chain Rules for the Malliavin Derivative
284(4)
7.5 Stochastic Integration for Symmetric Poisson Processes
288(14)
7.5.1 Orthogonal Increments
288(2)
7.5.2 From Internal Random Walks to the Standard Poisson Integral
290(3)
7.5.3 Iterated Integrals
293(4)
7.5.4 Multiple Integrals
297(1)
7.5.5 The σ-Algebra D generated by the Wiener-Levy Integrals
298(4)
7.6 Malliavin Calculus for Poisson Processes
302(19)
7.6.1 Chaos
302(3)
7.6.2 Malliavin Derivative
305(1)
7.6.3 Exchange of Derivative and Limit
306(1)
7.6.4 The Clark-Ocone Formula
307(2)
7.6.5 The Skorokhod Integral
309(1)
7.6.6 Smooth Representations
310(1)
7.6.7 The Product Rule
311(4)
7.6.8 The Chain Rule
315(2)
References
317(4)
8 New Understanding of Stochastic Independence
321(28)
Yeneng Sun
8.1 The General Context
321(1)
8.2 The Specific Problems
322(2)
8.3 Difficulties in the Classical Framework
324(2)
8.4 The Resolution
326(1)
8.5 Exact Law of Large Numbers
327(3)
8.6 Converse Law of Large Numbers
330(2)
8.7 Almost Equivalence of Pairwise and Mutual Independence
332(4)
8.8 Duality of Independence and Exchangeability
336(2)
8.9 Grand Unification of Multiplicative Properties
338(2)
8.10 Discrete Interpretations
340(3)
8.11 Notes
343(6)
References
344(5)
Part V Economics and Nonstandard Analysis
9 Nonstandard Analysis in Mathematical Economics
349(54)
Yeneng Sun
9.1 Introduction
349(7)
9.2 Distribution and Integration of Correspondences
356(7)
9.2.1 Distribution of Correspondences
356(5)
9.2.2 Integration of Correspondences
361(2)
9.3 Nash Equilibria in Games with Many Players
363(5)
9.3.1 General Existence of Nash Equilibria in the Loeb Setting
364(1)
9.3.2 Nonexistence of Nash Equilibria in the Lebesgue Setting
365(3)
9.4 Nash Equilibria in Finite Games with Incomplete Information
368(7)
9.4.1 Nonexistence of Nash Equilibria for Games with Information
368(2)
9.4.2 Approximate Nash Equilibria for Large Finite Games and Idealizations
370(3)
9.4.3 General Existence of Nash Equilibria for Games with Information
373(2)
9.5 Exact Law of Large Numbers and Independent Set-Valued Processes
375(5)
9.6 Competitive Equilibria in Random Economies
380(3)
9.7 General Risk Analysis and Asset Pricing
383(6)
9.7.1 General Risk Analysis for Large Markets
383(5)
9.7.2 The Equivalence of Exact No Arbitrage and APT Pricing
388(1)
9.8 Independent Universal Random Matching
389(3)
9.9 Notes
392(11)
References
396(7)
Part VI Combinatorial Number Theory
10 Density Problems and Freiman's Inverse Problems
403(40)
Renling Jin
10.1 Introduction
403(2)
10.2 Applications to Density Problems
405(12)
10.2.1 Sumset Phenomenon
407(4)
10.2.2 Plunnecke Type of Inequalities for Densities
411(6)
10.3 Applications to Freiman's Inverse Problems
417(26)
10.3.1 Freiman's Inverse Problem for Cuts
419(13)
10.3.2 Freiman's 3|A| -- 3 + b Conjecture
432(7)
10.3.3 Freiman's Inverse Problem for Upper Asymptotic Density
439(1)
References
440(3)
11 Hypernatural Numbers as Ultrafilters
443(32)
Mauro Di Nasso
11.1 Introduction
443(2)
11.2 The u-equivalence
445(5)
11.3 Hausdorff S-topologies and Hausdorff Ultrafilters
450(4)
11.4 Regular and Good Ultrafilters
454(4)
11.5 Ultrafilters Generated by Pairs
458(4)
11.6 Hyper-Shifts
462(4)
11.7 Nonstandard Characterizations in the Space (βN)
466(2)
11.8 Idempotent Ultrafilters
468(3)
11.9 Final Remarks and Open Questions
471(4)
References
473(2)
Index 475
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