Description
Neutrices and External Numbers
A Flexible Number System
Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series
Authors: Dinis Bruno, van den Berg Imme
Language: EnglishSubjects for Neutrices and External Numbers:
Keywords
External Numbers; Generalities; Regular Semigroups; Polynomials; Gauss Jordan Elimination; asymptotic approximations; Asymptotic equations; Convex Subgroup; singular perturbations; Strong Convergence Theorem; Cramer’s rule; Tikhonov’s Theorem; Axiomatics for external numbers; error propagation calculus; Flexible Function; Dedekind Completeness; real number system; Peano Arithmetic; algebraic laws; Slow Curve; scalar neutrices; Flexible Sequence; Maximal Solution; Maximal Ideal; Cauchy’s Principle; Initial Segment; External Sets; Asymptotic Expansion; Sorites Paradox; Vague Predicates; External Interval; Strict Rank; Bounded Formula; Taylor Polynomials
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Description
/li>Contents
/li>Biography
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Neutrices and External Numbers: A Flexible Number System introduces a new model of orders of magnitude and of error analysis, with particular emphasis on behaviour under algebraic operations. The model is formulated in terms of scalar neutrices and external numbers, in the form of an extension of the nonstandard set of real numbers. Many illustrative examples are given. The book starts with detailed presentation of the algebraic structure of external numbers, then deals with the generalized Dedekind completeness property, applications in analysis, domains of validity of approximations of solutions of differential equations, particularly singular perturbations. Finally, it describes the family of algebraic laws characterizing the practice of calculations with external numbers.
Features
- Presents scalar neutrices and external numbers, a mathematical model of order of magnitude within the real number system.
- Outlines complete algebraic rules for the neutrices and external numbers
- Conducts operational analysis of convergence and integration of functions known up to orders of magnitude
- Formalises a calculus of error propagation, covariant with algebraic operations
- Presents mathematical models of phenomena incorporating their necessary imprecisions, in particular related to the Sorites paradox
1 Introduction to Elementary Nonstandard Analysis
2 Some models and calculations involving imprecisions.
3 Neutrices and external numbers
4 Advanced properties
5 Sequences. Convergence up to a neutrix
6 Functions of external numbers
7 Integration of functions of external numbers
8 Flexible systems of linear equations
9 Applications in asymptotics
10 Applications in other fields
11 External numbers as a complete arithmetical solid
A Background on Nonstandard Analysis
B Solutions to selected exercises
Bruno Dinis is a postdoc at the Faculdade de Ciências, University of Lisbon, whose main area of interest is Mathematical Logic, Proof Theory, Nonstandard Analysis, and Philosophy of Mathematics.
Imme van den Berg is Associated Professor in Mathematics at the University of Évora, Portugal. His main area of interest lies in non-standard analysis. He is the author / co-author of 4 books and over 20 articles in the area of non-standard analysis.