Dinis B., van den Berg I. Neutrices and External Numbers: A Flexible Number System

New York: Chapman and Hall/CRC, 2019. — 361 p.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.FeaturesPresents 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 paradoxHalf Title
Series Page
Title Page
Copyright PageForewordIntroduction to Elementary Nonstandard Analysis
The axiomatic system ZFL and the Leibniz Rules
Internal and external sets, permanence
External Induction and the axiomatic system ENA
Orders of Magnitude
Nonstandard regularity properties of real internal functions
S-continuity
S-differentiability
S-integrability
Some models and calculations involving imprecisions
Validity of asymptotic approximation by a Taylor polynomial
Mass and tail of a random variable
The Mass Concentration Lemma
Application: Stirling’s formula
Jumps in singular perturbations
On linear equations
Neutrices and external numbers
External numbers and operations
Algebraic properties for addition and multiplication
External numbers and regular semigroups
Properties of neutral and inverse elements
Distributivity
Distributivity with neutrices
Distributivity with zeroless external numbers
Application: Binomial formulas
Advanced properties
Introduction to Internal Set Theory
Properties of
External sets
The nature of halflines, neutrices and external numbers
Generalized Dedekind completeness
Flexible sequences and functions
Flexible functions
Flexible sequences
Idempotent neutrices and ideals
Idempotent neutrices
Ideals and the product of neutrices
Sequences Convergence up to a neutrix
Notions of convergence for flexible sequences
Convergence for infinite sequences
Convergence with respect to an initial segment
Operations on flexible sequences
Boundedness and monotonicity
Operations
Cauchy flexible sequences
Functions of external numbers
Limits of flexible functions
Relation with convergence for sequences; strong convergence
Flexible continuity
Outer continuity
Inner continuity
M × N-derivation of flexible functions
Weak extrema and monotonicity
Integration of functions of external numbers
Integrals of internal functions on external intervals
Integrals of flexible functions
Elementary properties of integrals
Special integrals and applications
Mass and tail of probabilities and integrals
On local averaging
The concentration lemma and the Laplace method
Flexible systems of linear equations
Flexible systems
Determinants
On Gauss-Jordan elimination
Parameter method
Non-singular systems
Singular systems with strict rank equal to the number of equations
Singular systems with strict rank less than the number of equations
Applications in asymptotics
Nonstandard Borel-Ritt Theorem
Tools for solution of external equations
Matching principles
An external singular perturbation with canard solutions
External differentiable equations and their solutions
The external Riccati-Hermite equation
Solving the external Riccati-Hermite equation
Description of the canard behaviour
Influence of the singular point on the localization of canards
Applications in other fields
The Sorites paradox in philosophy
Forms of the paradox
Response proposals
External numbers as a model
External recurrence relations and near stability
On the size of fluctuations of the financial market
Further applications of external numbers
Near-optimization with uncertainties
On statistical estimation of uncertainties
External numbers as a complete arithmetical solid
The axioms
Algebraic axioms
Generalized Completeness axiom
Arithmetical axioms
A formal construction of the external numbers
The solid as a model for the axioms
On the axioms for the external numbers
Appendix A: Background on Nonstandard Analysis
On the foundations of external sets
Set theoretical Nonstandard Analysis
ZFC
Theories for internal sets: IST and BST
Theories for external sets: HST
HST axioms
Model theoretical nonstandard analysis
The superstructure approach
Appendix B: Solutions to selected exercises