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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."
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."
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