The Resource Nonstandard Analysis : Theory and Applications, edited by Leif O. Arkeryd, Nigel J. Cutland, C. Ward Henson, (electronic resource)

Nonstandard Analysis : Theory and Applications, edited by Leif O. Arkeryd, Nigel J. Cutland, C. Ward Henson, (electronic resource)

Label
Nonstandard Analysis : Theory and Applications
Title
Nonstandard Analysis
Title remainder
Theory and Applications
Statement of responsibility
edited by Leif O. Arkeryd, Nigel J. Cutland, C. Ward Henson
Title variation
Proceedings of the NATO Advanced Study Institute on Nonstandard Analysis and its Applications, International Centre for Mathematical Studies, Endinburgh, Scotland, 30 June-13 July 1996
Contributor
Editor
Editor
Subject
Language
  • eng
  • eng
Summary
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
Member of
Dewey number
515
http://bibfra.me/vocab/relation/httpidlocgovvocabularyrelatorsedt
  • 4Mt_7_A4BaQ
  • SDIfIFxPKPw
  • Q3LU3EqPazk
Image bit depth
0
Language note
English
LC call number
QA299.6-433
Literary form
non fiction
Nature of contents
dictionaries
http://library.link/vocab/relatedWorkOrContributorName
  • Arkeryd, Leif O.
  • Cutland, Nigel J.
  • Henson, C. Ward.
Series statement
Nato Science Series C:, Mathematical and Physical Sciences,
Series volume
493
http://library.link/vocab/subjectName
  • Global analysis (Mathematics)
  • Distribution (Probability theory
  • Functional analysis
  • Mathematics
  • Analysis
  • Probability Theory and Stochastic Processes
  • Functional Analysis
  • Applications of Mathematics
  • Fluid- and Aerodynamics
Label
Nonstandard Analysis : Theory and Applications, edited by Leif O. Arkeryd, Nigel J. Cutland, C. Ward Henson, (electronic resource)
Instantiates
Publication
Note
"Proceedings of the NATO Advanced Study Institute on Nonstandard Analysis and its Applications, held at the International Centre for Mathematical Science, Edinburgh, Scotland, June 30-July 13, 1996"--Title page verso
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Color
not applicable
Content category
text
Content type code
  • txt
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
Dimensions
unknown
Edition
1st ed. 1997.
Extent
1 online resource (XIV, 366 p.)
File format
multiple file formats
Form of item
online
Isbn
9789401155441
Level of compression
uncompressed
Media category
computer
Media type code
  • c
Other control number
10.1007/978-94-011-5544-1
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
  • (CKB)3400000000123585
  • (SSID)ssj0001245426
  • (PQKBManifestationID)11705918
  • (PQKBTitleCode)TC0001245426
  • (PQKBWorkID)11323069
  • (PQKB)10863685
  • (DE-He213)978-94-011-5544-1
  • (MiAaPQ)EBC3566973
  • (EXLCZ)993400000000123585
Label
Nonstandard Analysis : Theory and Applications, edited by Leif O. Arkeryd, Nigel J. Cutland, C. Ward Henson, (electronic resource)
Publication
Note
"Proceedings of the NATO Advanced Study Institute on Nonstandard Analysis and its Applications, held at the International Centre for Mathematical Science, Edinburgh, Scotland, June 30-July 13, 1996"--Title page verso
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Color
not applicable
Content category
text
Content type code
  • txt
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
Dimensions
unknown
Edition
1st ed. 1997.
Extent
1 online resource (XIV, 366 p.)
File format
multiple file formats
Form of item
online
Isbn
9789401155441
Level of compression
uncompressed
Media category
computer
Media type code
  • c
Other control number
10.1007/978-94-011-5544-1
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
  • (CKB)3400000000123585
  • (SSID)ssj0001245426
  • (PQKBManifestationID)11705918
  • (PQKBTitleCode)TC0001245426
  • (PQKBWorkID)11323069
  • (PQKB)10863685
  • (DE-He213)978-94-011-5544-1
  • (MiAaPQ)EBC3566973
  • (EXLCZ)993400000000123585

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