Dr. Robert A. Herrmann

Contents of Book 5.

If you would like a smaller font version that may not be as up-to-date as this version, please see the end of this page for the arxiv version.

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The following zip file (331KB) contains the title page, the contents pages and the first 8 chapters. (Latest revision 11:25 AM EDT, 7/31/03.) The contents of these chapters are as follows:

Chapter 1

Filters Ultrafilters, Cofinite Filter, Principle Ultrafilters, Free UltrafiltersChapter 2

A Simple Nonstandard Model for Analysis Equivalence Classes of Sequences, Totally Ordered Field of Equivalence Classes, The Hyper-extension of Sets and Relations, The Standard Object GeneratorChapter 3

Hyper-set Algebra And Infinite and Infinitesimal Numbers The Behavior of the Hyper and Standard Object Generators; Infinitesimals, The Infinite and Finite Numbers, *-Transform Process, Maximum IdealChapter 4

Basic Sequential Convergence Bounded Sequences, Convergent Sequences, Accumulation Point, Subsequences, Cauchy Criterion, Nonstandard Characteristics and ExamplesChapter 5

Advanced Sequential Convergence Double Sequences, Iterated Limits, Upper and Lower Limits, Nonstandard Characteristics and ExamplesChapter 6

Basic Infinite Series Hyperfinite Summation, Standard Results, Nonstandard Characteristics and ExamplesChapter 7

An Advance Infinite Series Concept Multiplying of Infinite Series, Nonstandard Characteristics and ExamplesChapter 8

Additional Real Number Properties Interior, Closure, Cluster, Accumulation, Isolated Points, Boundedness, Compactness, Nonstandard Characteristics

The zip file is ns1.zip The following zip file (333KB) contains chapters 9-15 and the Appendix, References, and Index. (Latest revision 1100 EDT, 8/1/03.) The contents of these chapters is as follows:

Chapter 9

Basic Continuous Function Concepts All Notions Generalized to Cluster Points, One-sided Limits and Continuity, Sum, Product and Composition of Continuous Functions, Extreme and Intermediate Value Theorems, Nonstandard CharacteristicsChapter 10

Slightly Advanced Continuous Function Concepts Nonstandard Analysis and Bolzano's Product Theorem, Inverse Images of Open Sets, Additive Functions, Uniform Continuity, Extensions of Continuous FunctionsChapter 11

Basic Derivative Concepts Nonstandard Characteristics for Finite and Infinite Derivatives at Cluster Points, The Infinitesimal Differential, The Fundamental Theorem of Differentials, Order Ideals, Basic Theorems, Generalized Mean Value Theorem, L'Hospital's RuleChapter 12

Some Advanced Derivative Concepts Nonstandard Analysis and the nth-Order Increments, nth-Order Ideals, Continuous Differentiability, Uniformly Differentiable, the Darboux Property, and Inverse Function TheoremsChapter 13

Riemann Integration The Simple Partition, Fine Partitions, Upper and Lower Sums, Upper and Lower Hyperfinite Sums, The Simple Integral, The Equivalence of The Simple Integral and The Riemann Integral, The Basic Integral Theorems and How The Generalized and Lebesgue Integral Relate to Fine Partitions.Chapter 14

What Does the Integral Measure? Additive Functions and The Rectangular PropertyChapter 15

Generalizations Metric and Normed Linear SpacesAppendix

Existence of Free Ultrafilters, Proof of *-Transform ProcessDate of original 7/30/03.

The zip file is ns2.zip

A smaller font version can be found at http://arxiv.org/abs/math.GM/031035

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