Nonstandard Analysis Theory and Applications

Nonstandard Analysis Theory and Applications / edited by Leif O. Arkeryd, Nigel J. Cutland, C. Ward Henson.

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 exten...

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Bibliographic Details
Corporate Author: SpringerLink (Online service)
Other Authors: Arkeryd, Leif O. (Editor), Cutland, Nigel J. (Editor), Henson, C. Ward (Editor)
Format: eBook
Language:English
Published: Dordrecht : Springer Netherlands : Imprint: Springer, 1997.
Edition:1st ed. 1997.
Series:Nato Science Series C:, Mathematical and Physical Sciences, 493
Springer eBook Collection.
Subjects:
Mathematical analysis.
Analysis (Mathematics).
Probabilities.
Functional analysis.
Applied mathematics.
Engineering mathematics.
Fluids.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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 Full Saturation
  • 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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