Ebook: Nonstandard Analysis: Theory and Applications
- Tags: Analysis, Probability Theory and Stochastic Processes, Functional Analysis, Applications of Mathematics, Fluid- and Aerodynamics
- Series: NATO ASI Series 493
- Year: 1997
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xiv
Foundations of Nonstandard Analysis....Pages 1-49
Nonstandard Real Analysis....Pages 51-76
Nonstandard Analysis and Topology....Pages 77-89
Loeb Measure and Probability....Pages 91-120
An Introduction to Nonstandard Functional Analysis....Pages 121-151
Applications of Nonstandard Analysis in Ordinary Differential Equations....Pages 153-182
Better Nonstandard Universes with Applications....Pages 183-208
Internal Martingales and Stochastic Integration....Pages 209-258
Stochastic Differential Equations with Extra Properties....Pages 259-277
Hyperfinite Mathematical Finance....Pages 279-307
Applications of NSA to Mathematical Physics....Pages 309-339
A Nonstandard Approach to Hydromechanics....Pages 341-355
Back Matter....Pages 357-366
Content:
Front Matter....Pages i-xiv
Foundations of Nonstandard Analysis....Pages 1-49
Nonstandard Real Analysis....Pages 51-76
Nonstandard Analysis and Topology....Pages 77-89
Loeb Measure and Probability....Pages 91-120
An Introduction to Nonstandard Functional Analysis....Pages 121-151
Applications of Nonstandard Analysis in Ordinary Differential Equations....Pages 153-182
Better Nonstandard Universes with Applications....Pages 183-208
Internal Martingales and Stochastic Integration....Pages 209-258
Stochastic Differential Equations with Extra Properties....Pages 259-277
Hyperfinite Mathematical Finance....Pages 279-307
Applications of NSA to Mathematical Physics....Pages 309-339
A Nonstandard Approach to Hydromechanics....Pages 341-355
Back Matter....Pages 357-366
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