Neutrices and External Numbers: A Flexible Number System

Description

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 Cover Half Title Series Page Title Page Copyright Page Contents Foreword Preface 1. Introduction to Elementary Nonstandard Analysis 1.1 The axiomatic system ZFL and the Leibniz Rules 1.2 Internal and external sets, permanence 1.3 External Induction and the axiomatic system ENA 1.4 Orders of Magnitude 1.5 Nonstandard regularity properties of real internal functions 1.5.1 S-continuity 1.5.2 S-differentiability 1.5.3 S-integrability 2. Some models and calculations involving imprecisions 2.1 Validity of asymptotic approximation by a Taylor polynomial 2.2 Mass and tail of a random variable 2.2.1 The Mass Concentration Lemma 2.2.2 Application: Stirling’s formula 2.3 Jumps in singular perturbations 2.4 On linear equations 3. Neutrices and external numbers 3.1 External numbers and operations 3.2 Algebraic properties for addition and multiplication 3.2.1 External numbers and regular semigroups 3.2.2 Properties of neutral and inverse elements 3.3 Distributivity 3.3.1 Distributivity with neutrices 3.3.2 Distributivity with zeroless external numbers 3.3.3 Application: Binomial formulas 4. Advanced properties 4.1 Introduction to Internal Set Theory 4.1.1 Properties of 4.1.2 External sets 4.2 The nature of halflines, neutrices and external numbers 4.3 Generalized Dedekind completeness 4.4 Flexible sequences and functions 4.4.1 Flexible functions 4.4.2 Flexible sequences 4.5 Idempotent neutrices and ideals 4.5.1 Idempotent neutrices 4.5.2 Ideals and the product of neutrices 5. Sequences. Convergence up to a neutrix 5.1 Notions of convergence for flexible sequences 5.1.1 Convergence for infinite sequences 5.1.2 Convergence with respect to an initial segment 5.2 Operations on flexible sequences 5.2.1 Boundedness and monotonicity 5.2.2 Operations 5.3 Cauchy flexible sequences 6. Functions of external numbers 6.1 Limits of flexible functions 6.1.1 Relation with convergence for sequences; strong convergence 6.2 Flexible continuity 6.2.1 Outer continuity 6.2.2 Inner continuity 6.3 M × N-derivation of flexible functions 6.4 Weak extrema and monotonicity 7. Integration of functions of external numbers 7.1 Integrals of internal functions on external intervals 7.2 Integrals of flexible functions 7.3 Elementary properties of integrals 7.4 Special integrals and applications 7.4.1 Mass and tail of probabilities and integrals 7.4.2 On local averaging 7.4.3 The concentration lemma and the Laplace method 8. Flexible systems of linear equations 8.1 Flexible systems 8.2 Determinants 8.3 On Gauss-Jordan elimination 8.4 Parameter method 8.4.1 Non-singular systems 8.4.2 Singular systems with strict rank equal to the number of equations 8.4.3 Singular systems with strict rank less than the number of equations 9. Applications in asymptotics 9.1 Nonstandard Borel-Ritt Theorem 9.2 Tools for solution of external equations 9.3 Matching principles 9.4 An external singular perturbation with canard solutions 9.4.1 External differentiable equations and their solutions 9.4.2 The external Riccati-Hermite equation 9.4.3 Solving the external Riccati-Hermite equation 9.4.4 Description of the canard behaviour 9.4.5 Influence of the singular point on the localization of canards 10. Applications in other fields 10.1 The Sorites paradox in philosophy 10.1.1 Forms of the paradox 10.1.2 Response proposals 10.1.3 External numbers as a model 10.2 External recurrence relations and near stability 10.3 On the size of fluctuations of the financial market 10.4 Further applications of external numbers 10.4.1 Near-optimization with uncertainties 10.4.2 On statistical estimation of uncertainties 11. External numbers as a complete arithmetical solid 11.1 The axioms 11.1.1 Algebraic axioms 11.1.2 Generalized Completeness axiom 11.1.3 Arithmetical axioms 11.2 A formal construction of the external numbers 11.3 The solid as a model for the axioms 11.4 On the axioms for the external numbers Appendix A: Background on Nonstandard Analysis A.1 On the foundations of external sets A.2 Set theoretical Nonstandard Analysis A.2.1 ZFC A.2.2 Theories for internal sets: IST and BST A.2.3 Theories for external sets: HST A.2.3.1 HST axioms A.3 Model theoretical nonstandard analysis A.3.1 The superstructure approach Appendix B: Solutions to selected exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Bibliography Index