1 More than thirty years after its discovery by Abraham Robinson , the ideas and techniques of Nonstandard Analysis (NSA) are being applied across the whole mathematical spectrum,as well as constituting an im portant field of research in their own right. The current methods of NSA now greatly extend Robinson's original work with infinitesimals. However, while the range of applications is broad, certain fundamental themes re cur. The nonstandard framework allows many informal ideas (that could loosely be described as idealisation) to be made precise and tractable. For example, the real line can…mehr
1 More than thirty years after its discovery by Abraham Robinson , the ideas and techniques of Nonstandard Analysis (NSA) are being applied across the whole mathematical spectrum,as well as constituting an im portant field of research in their own right. The current methods of NSA now greatly extend Robinson's original work with infinitesimals. However, while the range of applications is broad, certain fundamental themes re cur. The nonstandard framework allows many informal ideas (that could loosely be described as idealisation) to be made precise and tractable. For example, the real line can (in this framework) be treated simultaneously as both a continuum and a discrete set of points; and a similar dual ap proach can be used to link the notions infinite and finite, rough and smooth. This has provided some powerful tools for the research mathematician - for example Loeb measure spaces in stochastic analysis and its applications, and nonstandard hulls in Banach spaces. The achievements of NSA can be summarised under the headings (i) explanation - giving fresh insight or new approaches to established theories; (ii) discovery - leading to new results in many fields; (iii) invention - providing new, rich structures that are useful in modelling and representation, as well as being of interest in their own right. The aim of the present volume is to make the power and range of appli cability of NSA more widely known and available to research mathemati cians.
Foundations of Nonstandard Analysis A Gentle Introduction to Nonstandard Extensions.- 1 Introduction.- 2 Nonstandard Extensions.- 3 Logical Formulas.- 4 Nonstandard Extensions of Multisets.- 5 Nonstandard Extensions of the Multiset (X, P(X)).- 6 Superstructures.- 7 Saturation.- Nonstandard Real Analysis.- 1 Introduction.- 2 Basic Properties of *?.- 3 Sequences and Series.- 4 Continuity.- 5 Differentiation.- 6 Riemann Integration.- 7 Topology on ?.- 8 Using Internal Subsets of *?.- 9 An Application: Differential Equations.- Nonstandard Analysis and Topology.- 1 Metric and Topological Spaces.- 2 Continuous Mappings.- 3 Convergence.- 4 More on Topologies.- 5 Compact Spaces.- 6 Product Spaces.- 7 Restricted or Relative Topologies.- 8 Uniform Continuity on Metric Spaces.- 9 Nonstandard Hulls.- 10 Compactifications.- 11 More Exercises.- Loeb Measure and Probability.- 1 Introduction.- 2 Finite Loeb Measure.- 3 Constructing Standard Measures.- 4 Representing Standard Measures.- 5 Measurable Functions.- 6 Integration Theory.- 7 Probability Theory.- 8 Advertisement.- 9 Exercises.- An Introduction to Nonstandard Functional Analysis.- 1 Elementary Nonstandard Analysis of Normed Linear Spaces.- 2 Advanced Theory of Banach Spaces.- 3 Elementary Theory of Linear Operators.- 4 Spectral Theory of Bounded Operators.- 5 Applications of Nonstandard Spectral Theory.- 6 Closed Operators.- Applications of Nonstandard Analysis in Ordinary Differential Equations.- 1 Introduction.- 2 Tools in NSA.- 3 Differential Equations and Recursive Sequences.- 4 Regular Perturbations.- 5 Example.- 6 Dynamical Systems: Notions of Stability.- 7 Singular Perturbations.- Better Nonstandard Universes with Applications.- 1 Introduction.- 2 The Isomorphism Property.- 3 The Special Model Axiom and FullSaturation.- 4 The ?-Bolzano-Weierstrass Property.- Internal Martingales and Stochastic Integration.- 1 Hyperfinite Probability Spaces.- 2 Poisson Processes.- 3 Brownian Motion.- 4 Internal Martingales.- 5 Doob's Inequality.- 6 Quadratic Variation.- 7 Standard Parts.- 8 S-continuity.- 9 Stochastic Integration.- 10 Itô's Formula.- 11 Lévy's Characterization of Brownian Motion.- 12 Connections to Standard Theory.- 13 Stochastic Integrals in Higher Dimensions.- 14 Stochastic Differential Equations.- 15 Brownian Local Time.- 16 The Infinite Dimensional Ornstein-Uhlenbeck Process.- Stochastic Differential Equations with Extra Properties.- 1 Introduction.- 2 Spaces of Stochastic Processes.- 3 Solutions of Stochastic Differential Equations.- 4 Solutions which are Markov Processes.- 5 A Fixed Point Theorem.- 6 Stochastic Differential Equations with Nondegenerate Coefficients.- Hyperfinite Mathematical Finance.- 1 Introduction.- 2 Finite Market Models.- 3 Pricing Options in a Hyperfinite CRR Model.- 4 Hyperfinite Trading Strategies.- 5 Convergence of Prices and Strategies.- 6 Further Developments.- Applications of Nsa to Mathematical Physics.- 1 A Kinetic Inequality.- 2 The Time Asymptotic Behaviour for Certain Rarefied Gases when the Incoming Fluxes at the Boundary are Given.- 3 On Semiclassical Limits for the Schrödinger Equation.- A Nonstandard Approach to Hydromechanics Navier-Stokes Equations.- 1 Introduction.- 2 Deterministic Navier-Stokes Equations.- 3 Statistical Solutions.- 4 Stochastic Equations.- 5 Some Open Problems.
Foundations of Nonstandard Analysis A Gentle Introduction to Nonstandard Extensions.- 1 Introduction.- 2 Nonstandard Extensions.- 3 Logical Formulas.- 4 Nonstandard Extensions of Multisets.- 5 Nonstandard Extensions of the Multiset (X, P(X)).- 6 Superstructures.- 7 Saturation.- Nonstandard Real Analysis.- 1 Introduction.- 2 Basic Properties of *?.- 3 Sequences and Series.- 4 Continuity.- 5 Differentiation.- 6 Riemann Integration.- 7 Topology on ?.- 8 Using Internal Subsets of *?.- 9 An Application: Differential Equations.- Nonstandard Analysis and Topology.- 1 Metric and Topological Spaces.- 2 Continuous Mappings.- 3 Convergence.- 4 More on Topologies.- 5 Compact Spaces.- 6 Product Spaces.- 7 Restricted or Relative Topologies.- 8 Uniform Continuity on Metric Spaces.- 9 Nonstandard Hulls.- 10 Compactifications.- 11 More Exercises.- Loeb Measure and Probability.- 1 Introduction.- 2 Finite Loeb Measure.- 3 Constructing Standard Measures.- 4 Representing Standard Measures.- 5 Measurable Functions.- 6 Integration Theory.- 7 Probability Theory.- 8 Advertisement.- 9 Exercises.- An Introduction to Nonstandard Functional Analysis.- 1 Elementary Nonstandard Analysis of Normed Linear Spaces.- 2 Advanced Theory of Banach Spaces.- 3 Elementary Theory of Linear Operators.- 4 Spectral Theory of Bounded Operators.- 5 Applications of Nonstandard Spectral Theory.- 6 Closed Operators.- Applications of Nonstandard Analysis in Ordinary Differential Equations.- 1 Introduction.- 2 Tools in NSA.- 3 Differential Equations and Recursive Sequences.- 4 Regular Perturbations.- 5 Example.- 6 Dynamical Systems: Notions of Stability.- 7 Singular Perturbations.- Better Nonstandard Universes with Applications.- 1 Introduction.- 2 The Isomorphism Property.- 3 The Special Model Axiom and FullSaturation.- 4 The ?-Bolzano-Weierstrass Property.- Internal Martingales and Stochastic Integration.- 1 Hyperfinite Probability Spaces.- 2 Poisson Processes.- 3 Brownian Motion.- 4 Internal Martingales.- 5 Doob's Inequality.- 6 Quadratic Variation.- 7 Standard Parts.- 8 S-continuity.- 9 Stochastic Integration.- 10 Itô's Formula.- 11 Lévy's Characterization of Brownian Motion.- 12 Connections to Standard Theory.- 13 Stochastic Integrals in Higher Dimensions.- 14 Stochastic Differential Equations.- 15 Brownian Local Time.- 16 The Infinite Dimensional Ornstein-Uhlenbeck Process.- Stochastic Differential Equations with Extra Properties.- 1 Introduction.- 2 Spaces of Stochastic Processes.- 3 Solutions of Stochastic Differential Equations.- 4 Solutions which are Markov Processes.- 5 A Fixed Point Theorem.- 6 Stochastic Differential Equations with Nondegenerate Coefficients.- Hyperfinite Mathematical Finance.- 1 Introduction.- 2 Finite Market Models.- 3 Pricing Options in a Hyperfinite CRR Model.- 4 Hyperfinite Trading Strategies.- 5 Convergence of Prices and Strategies.- 6 Further Developments.- Applications of Nsa to Mathematical Physics.- 1 A Kinetic Inequality.- 2 The Time Asymptotic Behaviour for Certain Rarefied Gases when the Incoming Fluxes at the Boundary are Given.- 3 On Semiclassical Limits for the Schrödinger Equation.- A Nonstandard Approach to Hydromechanics Navier-Stokes Equations.- 1 Introduction.- 2 Deterministic Navier-Stokes Equations.- 3 Statistical Solutions.- 4 Stochastic Equations.- 5 Some Open Problems.
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