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Nonstandard Analysis

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ISBN-10: 0486432793

ISBN-13: 9780486432793

Edition: 2003

Authors: Alain M. Robert

List price: $14.95
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Description:

"This wonderful little book by Alain Robert should bring about a complete change in the learning of NSA. The author has accomplished a rare feat in the educational literature. He has succeeded in writing a book which is simple and brilliant, deep and witty, short and far-ranging. This is mathematics teaching at its best."—J.-M. Leacute;vy-Leblond, European Journal of Physics Brief and readable, this introduction to nonstandard analysis is based on the axiomatic IST (internal set theory) approach. The two-part treatment starts with a clear, rigorous exposition of theory, followed by self-contained chapters on applications. Exercises appear at the conclusion of each chapter, with hints in addition to full solutions. Theoretical topics include idealization, standardization and transfer, real numbers and numerical functions, continuity, differentiability, and integration. Chapters involving applications cover invariant means, approximation of functions, differential equations, perturbation of a Green function, and an invariant subspaces problem. 1988 edition.
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Book details

List price: $14.95
Copyright year: 2003
Publisher: Dover Publications, Incorporated
Publication date: 7/19/2011
Binding: Paperback
Pages: 176
Size: 6.00" wide x 9.00" long x 0.25" tall
Weight: 0.506
Language: English

Preface
Conventions, notation
Introduction
Idealization
Set theory, new predicate
Axiomatic: beginning
Comments
First applications of (I)
More comments suggested by the examples
Natural integers
Power sets
External sets
Exercises
Standardization and Transfer
Axiomatic: end
Consequences
An example
Standard finite sets
Functions and graphs
Relativization
Final comments on axiomatics
Exercises
Real Numbers and Numerical Functions
Basic concepts
Standard part
Generalization to topological spaces
Sequences
Exercises
Continuity
S-continuity
Examples showing the difference between continuity and S-continuity
Relations between continuity and S-continuity
Uniform continuity
Theorems on continuous functions
Exercises
Differentiability
Differentiable functions
Theorems for differentiable functions
Strictly differentiable functions
Higher derivatives
Finite differences and derivatives
Exercises
Integration
Method
Definite integral
Strict standard part of a function
Exercises
Invariant Means
Defining properties
Existence of invariant means
A result
More comments
Exercises
Approximation of Functions
Dirac functions
Periodic functions and trigonometric polynomials
Bernstein polynomial approximation
Approximation in quadratic mean
Exercises
Differential Equations
Review of some classical notions
Existence theorem
An example
Perturbation of a Green Function
Green function
Nonstandard perturbation
Origin of the problem
Invariant Subspaces Problem
Situation of the result
Preliminary results
Proof of the Bernstein-Robinson Theorem
Comments
Indications for Exercises
Solutions of Exercises
The Gibbs Phenomenon
Bibliography
Index
Basic Principles of NSA
IST Axioms for NSA