UID:
almahu_9949697919102882
Format:
1 online resource (343 p.)
ISBN:
1-281-76761-1
,
9786611767617
,
0-08-087388-X
Series Statement:
Pure and applied mathematics (Academic Press) ; 72
Content:
Introduction to the Theory of infiniteseimals
Note:
Description based upon print version of record.
,
Front Cover; INTRODUCTION TO THE THEORY OF INFINITESIMALS; Pure and Applied Mathematics; Copyright Page; CONTENTS; Preface; Acknowledgments; PART 1: CLASSICAL INFINITESIMALS; CHAPTER 1. INTRODUCTION: WHAT ARE INFINITESIMALS?; CHAPTER 2. A FIRST LOOK AT ULTRAPOWERS: A MODEL OF RATIONAL ANALYSIS; 2.1 A Free Ultrafilter u on a Countable Set J; 2.2 An Ultrapower of the Rational Numbers; 2.3 Some Calculus of Polynomials; 2.4 The Exponential Function; 2.5 Peano's Existence Theorem; 2.6 Summary; CHAPTER 3. SUPERSTRUCTURES AND THEIR NONSTANDARD MODELS; 3.1 Introduction
,
3.2 Definition of a Superstructure3.3 Superstructures Are Big Enough; 3.4 Nonstandard Models of Superstructures; 3.5 The Formal Language; 3.6 Interpretations of the Formal Language; 3.7 Models of a Superstructure; 3.8 Nonstandard Ultrapower Models; 3.9 Bounded Formal Sentences; 3.10 Embedding xj in Set Theory; 3.11 *-Transforms of Categories; 3.12 Postscript to Chapter; CHAPTER 4. SOME BASIC FACTS ABOUT HYPERREAL NUMBERS; 4.1 Addition, Multiplication, and Order in 'R; 4.2 Some Simplifications of the Notation; 4.3 'R Is Non-Archimedean; 4.4 Infinite, Infinitesimal, and Finite Numbers
,
4.5 Some External Entities4.6 Further Simplification of Notation and Classical Functions; 4.7 Hypercomplex Numbers; APPENDIX A. PRELIMINARY RESULTS ON ORDERED RINGS AND FIELDS; A.l Terminology; A.2 Ordered Rings and Fields; A.3 Archimedean Totally Ordered Fields; CHAPTER 5. FOUNDATIONS OF INFINITESIMAL CALCULUS; 5.1 Continuity and Limits; 5.2 Uniform Continuity; 5.3 Basic Definitions of Calculus; 5.4 The Mean Value Theorem; 5.5 The Fundamental Theorem of Calculus; 5.6 Landau's ""Oh-Calculus""; 5.7 Differential Vector Calculus; 5.8 Integral Vector Calculus; 5.9 Calculus on Manifolds
,
CHAPTER 6. TOPICS IN INFINITESIMAL CALCULUS6.1 Peano's Existence Theorem Revisited; 6.2 Interchanging Limits; 6.3 Euler's Product for the Sine Function; 6.4 Robinson's Lemma and Generalized Limits; 6.5 Dynamical Systems; 6.6 Geometry of the Unit Ball and Boundary Behavior; PART 2: INFINITESIMALS IN FUNCTIONAL ANALYSIS; CHAPTER 7. MORE TOOLS FROM MODEL THEORY; 7.1 Countable Ultrapowers; 7.2 Enlargements; 7.3 Comprehensive Models; 7.4 Saturated Models; 7.5 Ultralimits; 7.6 Properties of Polysaturated Models; 7.7 The Isomorphism Property of Ultralimits
,
CHAPTER 8. THE GENERAL THEORY OF MONADS AND INFINITESIMALS8.1 Monads with Respect to a Ring of Sets; 8.2 Chromatic Sets; 8.3 Topological Aspects of Monad Theory; 8.4 Uniform Infinitesimal Relations and Finite Points; 8.5 Topological lnfinitesimals at Remote Points; CHAPTER 9. COMPACTIFICATIONS; 9.1 Discrete Cech-Stone Compactification of N; 9.2 Measurable Infinitesimals; 9.3 The Samuel Compactification of the Hyperbolic Plane; 9.4 Normal Meromorphic Functions and Analytic Disks; 9.5 The Fatou-Lindelöf Boundary; 9.6 Bounded Holomorphic Functions and Gleason Parts
,
9.7 Fixed Points of Analytic Maps on A M
,
English
Additional Edition:
ISBN 0-12-674150-6
Language:
English
